
// Make newform 6762.2.a.bj in Magma, downloaded from the LMFDB on 25 August 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_6762_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_6762_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_6762_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_6762_a();"
function MakeCharacter_6762_a()
    N := 6762;
    order := 1;
    char_gens := [2255, 3727, 3823];
    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_6762_a_Hecke( : Kf := Kf)
    return MakeCharacter_6762_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], [1], [0], [0], [-4], [-2], [2], [-6], [1], [-2], [2], [-2], [-8], [4], [-6], [6], [-4], [0], [-4], [0], [-4], [-12], [-6], [-10], [6], [-14], [4], [12], [2], [-2], [16], [-12], [-6], [8], [10], [-16], [4], [-8], [-10], [-6], [12], [-16], [20], [6], [18], [-4], [-8], [14], [6], [-16], [10], [8], [-22], [26], [-28], [8], [-6], [-30], [30], [18], [10], [16], [8], [18], [-22], [-22], [-20], [14], [-16], [10], [0], [-4], [20], [2], [4], [24], [-30], [-2], [10], [-24], [34], [6], [8], [6], [-10], [8], [14], [-22], [-6], [-8], [-18], [-4], [16], [24], [-8], [0], [-22], [-10], [-14], [-10], [20], [18], [-6], [26], [-28], [-28], [0], [12], [-40], [-28], [-14], [42], [2], [-26], [-12], [2], [-34], [-10], [6], [-4], [16], [2], [40], [8], [48], [30], [-50], [-2], [12], [-20], [16], [-52], [-40], [26], [-36], [50], [36], [46], [28], [-26], [-12], [10], [24], [12], [-18], [-32], [46], [40], [48], [16], [-46], [18], [-24], [22], [-4], [-36], [-20], [-24], [2], [0], [24], [6], [40], [50], [-38], [48], [-40], [22], [18], [-2], [-36], [-14], [-12], [14], [-34], [30], [-12], [-50], [-32], [44], [8], [26], [-30], [-56], [-24], [-4], [-34], [-34], [2], [-14], [-16], [24], [52], [22], [12], [6], [-2], [-62], [18], [62], [6], [-54], [14], [-6], [-26], [-18], [16], [56], [-54], [48], [14], [-30], [16], [30], [42], [-22], [0], [-20], [-12], [30], [-54], [-32], [-30], [28], [-52], [-30], [22], [72], [-52], [4], [-30], [46], [-4], [70], [-62], [12], [-22], [-18], [60], [10], [-28], [52], [2], [62], [4], [34], [2], [66], [-52], [-12], [-18], [38], [-24], [-54], [-6], [-20], [-6], [62], [-18], [-8], [-32], [12], [-26], [16], [-48], [22], [-14], [-74], [-12], [46], [-62], [12], [46], [40], [4], [2], [-12], [22], [-22], [-66], [42], [0], [-16], [-40], [80], [-48], [34], [10], [-2], [16], [-30], [46], [-2], [2], [62], [-16], [16], [14], [40], [-36], [-42], [10], [-56], [40], [-52], [-18], [-44], [-64], [-8], [-26], [-90], [-6], [50], [8], [-8], [-26], [-24], [40], [-66], [78], [-54], [22], [-14], [-16], [86], [50], [-28], [40], [-36], [-26], [-18], [-18], [-78], [-52], [-4], [88], [-58], [-2], [-40], [32], [-58], [50], [-42], [72], [42], [-60], [-58], [-68], [-88], [0], [-4], [58], [10], [32], [58], [-88], [56], [-42], [30], [-48], [38], [40], [48], [-36], [-52], [26], [22], [-72], [58], [-24], [56], [48], [42], [-28], [-18], [32], [-80], [-62], [-50], [52], [-60], [14], [96], [18], [94], [-46], [40], [-82], [44], [-56], [22], [-42], [-4], [-24], [-12], [-74], [16], [86], [84], [-18], [-22], [62], [76], [6], [-74], [-82], [-62], [-50], [50], [80], [-64], [38], [-84], [-44], [-50], [-62], [8], [-2], [-18], [-48], [30], [28], [-48], [-28], [-24], [58], [-68], [-58], [-24], [-102]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6762_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6762_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_6762_2_a_bj(:prec:=1)
    chi := MakeCharacter_6762_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_6762_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_6762_2_a_bj( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6762_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<11,R![4, 1]>,<13,R![2, 1]>,<17,R![-2, 1]>],Snew);
    return Vf;
end function;