The Labour Party has spent the last month celebrating unexpected success. And rightly so. The party started so far back that it could never have won the 2017 election. It did as well as anyone could have reasonably hoped – and is in an excellent position to win the next election. It was a huge achievement.

Now however – one month on – Labour needs to put the euphoria behind it and look to the future. The party can win the next election and transform the country once in power. But to do so it must find answers to some very tough questions.

On Saturday the Fabian summer conference will start that process. The Path to Power conference and will bring together politicians, commentators and activitists from all strands of left opinion.

The day will seek to celebrate and understand what Labour has just achieved. But more importantly, the conference will look to the future and begin the search for answers. Here are the five questions that will be debated at the Fabian conference and which Labour must address to win.

1. How does Labour attract left-leaning “conservatives” as well as “liberals”?

The contours of Labour’s success in the election are now well established. The party did very well with the liberal half of Britain, especially the young and the highly educated. But it only did “okay” with older, working-class “left conservatives” – people who are socially conservative but lean left on economic issues. It is true that, in the wake of the UKIP collapse, Labour won extra votes in strongly working-class constituencies; but the Tories made still more progress and won six seats from Labour as a consequence.

Labour will not win the marginal constituencies needed for victory at the next election by convincing yet more liberal-minded voters. In the bell-weather seats of the new towns, market towns and outer suburbs, and in the former Labour heartlands we’ve recently lost, the party will only move forward by building coalitions that bring together social liberals and social conservatives; young and old; professional and working-class voters.

To succeed with social conservatives the party will need to look and feel like it is patriotic and serious with security; respectful of tradition and mindful of the pace of change; concerned about responsibility and contribution; and trustworthy as stewards of the national finances. Importantly, there is nothing on this list that is incompatible with left-wing economics. But Labour will need a more communitarian version of Corbynism.

2. How does Labour build on success in Scotland?

The Fabian conference may be in London, but the line-up includes a panel of Scots who will debate how Scottish Labour can now move forward to build on its surprise success. The prospects look good, not only because of the six seats we gained but also the many others we turned into tight marginals. While before the election, most SNP majorities looked impregnable, now a path has formed for Labour’s comeback in Scotland.

But Scottish Labour still needs to do more to turn the political conversation to domestic questions so that it can expose the failing record of the SNP in Holyrood and the Tories in Westminster. That is the way to lure back traditional supporters who are starting to tire of the SNP and to win over people who voted Conservative this time, as a pro-union choice.

In most of our Scottish marginals Labour is the obvious anti-SNP choice and will be able to squeeze the Tories and Lib Dems. However, it will be more complicated in the handful of seats which are three-way marginals, where the de-toxified Scottish Tories are a real threat. Overall Labour has good reason to be optimistic about its prospects in Scotland, but we have only just started down the road to recovery.

3. How does Labour “campaign in poetry” but also prove it can “govern in prose”?

Labour ran an inspiring, high-energy campaign, with flagship policies that were popular and memorable. From now until the next election we need more of the same. But Labour also needs to look credible and make itself ready to govern. In the 2017 election the party’s policies did not receive much scrutiny, which was to Labour’s advantage. While many of the individual ideas were genuinely popular, the whole package was vulnerable to attack. Had the media thought Labour could win, or had Theresa May not run such a terrible campaign, the election could have ended very differently.

Three examples illustrate the point. First, the IFS exposed that our fiscal maths did not add up, though their verdict came late in the campaign and no one really noticed. Second, Labour had no answer when it was asked why it simultaneously planned to cut benefits for the poor and expand public services for the rich. Third, for some reason the Tories never presented our proposals for nationalisations as a single package. There was a good case for each in isolation, but taken together they could have easily been made to look impractical and extreme. Next time, Labour’s policies must be developed and road-tested in much more detail. They must be able to withstand a tougher campaign and be implementation-ready for when we retake power.

Looking credible will also depend on how the party now conducts itself in opposition. Can it look like a heavyweight alternative to Theresa May’s chaotic, disintegrating minority government? And will all Labour’s talent have the chance to shine? The party needs to give everyone of its leading lights a role to showcase the strength of the whole Labour team: on the frontbench, in select committees and on party policy commissions

4. How does Labour square the circle on Brexit?

Just weeks after the election, Labour’s position on Brexit is already causing problems. Fortunately the public barely noticed last week’s Queen’s Speech rebellion as, for now, Labour is a sideshow beside the Tory Brexit civil war. But the incident illustrated the pressures Labour faces.

The party leadership is trying to set the government tests for Brexit that it knows cannot be met. The goal is to pin blame for failure on the Tories but to avoid specifying our own position with too much definition. This approach is frustrating for the most pro-EU MPs, who see defence of single market membership as their patriotic duty. But Jeremy Corbyn and Keir Starmer know they cannot get ahead of public opinion.

Right now, if Labour calls for anything that looks too close to free movement the party will be told it is defying the referendum result. Labour’s fragile coalition of leave and remain voters will shatter and power will become impossible. But, more importantly, these tactics will be futile, as hard Brexit will still happen whatever Labour says. It is not the way to serve the national economic interest.

Instead Labour must try both to follow and to lead public opinion, by slowly opening people’s eyes to the reality of hard Brexit. Only when the public understands that hard Brexit will be a national calamity will Labour be in a position to stop it happening. In particular, the party must do all it can to shift the public mood on EU immigration, because views here must soften before the UK can negotiate membership of the single market or anything remotely similar.

5. How does Labour stay civil, inclusive and united?

Labour will lose if it returns to civil war. But, just a month after a campaign that saw a united Labour party in action, too many people, on all sides, seem to have forgotten that lesson. Most of Jeremy Corbyn’s critics are remaining silent in public, but in their private remarks some of them sound as if they think there are “right” and “wrong” sorts of Labour voters. They need to remain more open-minded about the leaders’ capacity to attract new supporters and perform day-in-day-out. After all he has surprised pretty much everyone so far.

But Corbyn’s most ardent supporters need to reflect too. Corbyn’s personal appeal was only one of many factors that explained Labour’s success in this election. Left liberals may have flocked to the party because of its anti-establishment leader and the clarity of his manifesto. But Labour also succeeded because it was the standard-bearer for soft Brexit; because people backed their local candidates; and because the Tories, the Lib Dems, Ukip and the SNP all blunderred in their campaigns. Corbynites do not have a monopoly of wisdom on where to go from here.

Attempts to change party rules and deselect MPs look vindictive and short-sighted. People considering such moves must ask themselves, is Corbyn more likely to win as the leader of a broad-church party that can connect with many different sorts of people? Or at the head of narrow faction that seems more interested in driving people away than drawing them in?

It will take patience and generosity – from all sides – for Labour to come together and take the fight to the Tories. But it will be worth it, when the prize is power. Respect, comradeship and tolerance of difference are now the essential ingredients, if Labour is to move forward again.

Andrew Harrop is general secretary of the Fabian Society. The Fabian summer conference Path to Power takes place on Saturday in London. Tickets are available here.