
// Make newform 5610.2.a.bj in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5610_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_5610_2_a_bj();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_5610_2_a_bj();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_5610_a();"
function MakeCharacter_5610_a()
    N := 5610;
    order := 1;
    char_gens := [1871, 3367, 1531, 3301];
    v := [1, 1, 1, 1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_5610_a_Hecke(Kf)
    return MakeCharacter_5610_a();
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..(prec+1)]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 2;
    raw_aps := [[1], [1], [-1], [5], [1], [-4], [-1], [-4], [-3], [9], [5], [2], [0], [11], [0], [12], [-6], [2], [-4], [-6], [2], [8], [12], [0], [-7], [-6], [5], [-15], [-16], [18], [2], [-12], [-3], [17], [-12], [20], [-10], [11], [-6], [-18], [18], [5], [-21], [2], [-24], [-16], [-1], [-19], [-15], [-4], [-21], [6], [-1], [-24], [3], [9], [18], [17], [-10], [9], [14], [-9], [20], [-24], [23], [3], [5], [14], [-12], [-10], [-3], [6], [2], [-22], [8], [-12], [6], [-34], [-15], [32], [-21], [32], [15], [-40], [-34], [-15], [-21], [-10], [18], [32], [-12], [33], [38], [0], [8], [36], [30], [-30], [11], [20], [-34], [9], [-24], [-18], [-28], [-10], [-9], [-18], [-3], [-22], [8], [-34], [42], [-10], [14], [-9], [29], [-42], [27], [-45], [-40], [-4], [6], [12], [-28], [6], [-10], [-18], [53], [20], [-22], [-36], [-37], [-7], [15], [-22], [-48], [8], [0], [-6], [-40], [3], [-34], [-9], [-46], [54], [53], [-27], [-13], [-24], [5], [57], [20], [0], [5], [36], [-25], [-21], [14], [-18], [18], [30], [50], [0], [18], [21], [-25], [-25], [41], [51], [-27], [-28], [-60], [29], [-4], [30], [-4], [-54], [20], [5], [-19], [-12], [-22], [-6], [-36], [18], [5], [-64], [59], [0], [53], [36], [47], [-57], [-3], [24], [50], [26], [-45], [39], [54], [26], [-19], [8], [66], [0], [62], [-24], [-36], [65], [-52], [-63], [8], [21], [-24], [32], [26], [39], [0], [42], [-43], [-4], [72], [-46], [-12], [35], [18], [-54], [38], [-54], [-7], [-25], [-19], [9], [56], [-69], [-40], [36], [-3], [0], [48], [-10], [41], [38], [-3], [33], [38], [57], [-46], [-39], [-52], [-72], [24], [-46], [21], [-51], [-10], [-70], [63], [-34], [44], [-78], [-22], [-61], [0], [-19], [-9], [24], [-22], [21], [23], [38], [-70], [-55], [5], [20], [12], [11], [-22], [-36], [-39], [32], [24], [83], [-22], [72], [2], [-54], [26], [-51], [78], [0], [-24], [66], [17], [-18], [62], [-60], [-30], [5], [41], [-57], [-10], [-24], [-52], [2], [-15], [-46], [87], [62], [66], [18], [24], [-4], [-39], [-52], [-60], [24], [26], [15], [-91], [23], [-36], [71], [6], [-25], [-52], [35], [-60], [12], [26], [3], [26], [36], [-52], [12], [-16], [57], [32], [74], [-10], [18], [3], [-82], [-24], [-18], [2], [-28], [33], [15], [-76], [62], [-9], [44], [14], [-66], [-81], [24], [63], [-54], [-55], [-42], [72], [-18], [-52], [80], [-30], [-34], [-22], [18], [20], [66], [-54], [-13], [80], [60], [81], [-31], [54], [38], [72], [51], [8], [-81], [-1], [-87], [-46], [53], [74], [0], [20], [93], [30], [62], [0], [-28], [92], [-42], [-94], [-84], [44], [-30], [-40], [60], [18], [-82], [2], [90], [14], [-33], [26], [54], [36], [-55], [17], [-78], [0], [92], [84], [-48], [6], [2], [24], [36], [86], [78], [-60], [-18], [-67], [-81]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5610_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5610_2_a_bj();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_5610_2_a_bj(:prec:=1)
    chi := MakeCharacter_5610_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_5610_2_a_bj();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_5610_2_a_bj( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5610_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-5, 1]>,<13,R![4, 1]>,<19,R![4, 1]>,<23,R![3, 1]>],Snew);
    return Vf;
end function;