
// Make newform 5031.2.a.b in Magma, downloaded from the LMFDB on 05 March 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_5031_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_5031_2_a_b();".
// 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_5031_2_a_b();".
// 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_5031_a();"
function MakeCharacter_5031_a()
    N := 5031;
    order := 1;
    char_gens := [560, 1549, 820];
    v := [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_5031_a_Hecke( : Kf := Kf)
    return MakeCharacter_5031_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]];
    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], [0], [-2], [0], [-2], [-1], [-4], [8], [-6], [-6], [8], [-10], [0], [1], [-10], [4], [14], [6], [4], [-4], [-2], [0], [10], [-6], [18], [-4], [16], [18], [-2], [2], [-8], [20], [10], [12], [-6], [24], [-18], [-4], [-6], [0], [16], [-6], [-12], [-18], [8], [20], [-20], [-20], [28], [14], [14], [2], [14], [2], [-22], [-24], [12], [32], [-14], [-28], [-4], [-24], [20], [-6], [-6], [-20], [8], [2], [-24], [14], [0], [-6], [-8], [-2], [-4], [12], [-10], [34], [24], [6], [28], [2], [22], [26], [0], [-30], [30], [-2], [-20], [4], [8], [38], [-32], [12], [8], [-4], [-36], [-30], [16], [-30], [-36], [36], [-18], [0], [-12], [42], [-16], [-18], [30], [2], [20], [2], [-28], [-44], [8], [-34], [44], [32], [38], [10], [-22], [26], [18], [-30], [-32], [12], [10], [10], [48], [-14], [-44], [-36], [28], [-38], [-30], [-34], [14], [4], [-16], [52], [12], [12], [-48], [50], [50], [0], [-10], [-24], [44], [32], [-50], [-28], [4], [48], [-12], [24], [40], [-14], [58], [-12], [-26], [34], [-16], [18], [12], [-4], [-20], [-18], [-18], [-12], [-12], [-34], [28], [6], [-20], [40], [16], [-54], [-24], [30], [16], [-10], [-14], [42], [16], [14], [10], [44], [38], [-12], [34], [16], [44], [50], [-48], [-34], [-26], [14], [56], [12], [0], [-40], [-62], [14], [60], [10], [16], [50], [-54], [44], [2], [-44], [8], [22], [28], [22], [-8], [-54], [48], [24], [-46], [0], [42], [32], [-8], [-10], [-20], [24], [-28], [-24], [34], [-52], [16], [2], [28], [-38], [-74], [60], [-48], [62], [0], [-20], [-48], [14], [6], [18], [-44], [30], [36], [-46], [-10], [0], [62], [62], [6], [-36], [34], [-52], [42], [-22], [-40], [44], [-26], [-14], [-26], [40], [-30], [-36], [56], [72], [46], [-48], [46], [36], [-34], [-56], [74], [6], [10], [-40], [-26], [16], [-22], [-34], [-68], [24], [14], [54], [-20], [80], [-12], [22], [16], [84], [-22], [-24], [56], [-36], [-66], [44], [42], [-50], [-24], [30], [24], [-34], [58], [50], [0], [46], [-16], [-54], [-40], [4], [-58], [70], [-6], [66], [-82], [18], [-12], [-86], [34], [56], [-6], [-82], [84], [68], [6], [14], [18], [-52], [36], [-76], [12], [-86], [-34], [-50], [14], [44], [74], [78], [74], [24], [84], [-78], [58], [-44], [-48], [-48], [-12], [-82], [-80], [-92], [86], [66], [52], [-72], [0], [74], [-26], [-58], [60], [-72], [-28], [26], [66], [72], [34], [-24], [20], [-34], [42], [80], [10], [96], [10], [34], [-30], [46], [24], [0], [-88], [-22], [-76], [-62], [24], [70], [-20], [42], [-82], [42], [-100], [44], [16], [54], [96], [46], [-64], [-38], [94], [56], [-16], [8], [-18], [16], [-102], [28], [-60], [78], [102], [-38], [68], [44], [70], [-8], [38], [40], [72], [32], [18], [38], [48], [90], [58], [-62], [-24], [-34], [92], [-28]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5031_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5031_2_a_b();".
// 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_5031_2_a_b(:prec:=1)
    chi := MakeCharacter_5031_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_5031_2_a_b();".
// 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_5031_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5031_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 1]>,<7,R![0, 1]>,<11,R![2, 1]>],Snew);
    return Vf;
end function;