
// Make newform 6762.2.a.h in Magma, downloaded from the LMFDB on 02 November 2025.
// 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_h();".
// 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_h();".
// 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], [6], [-2], [6], [-2], [1], [-6], [-8], [8], [-6], [2], [0], [-12], [0], [-8], [-10], [0], [-14], [8], [-6], [-6], [10], [6], [4], [-6], [-4], [6], [8], [0], [-6], [16], [12], [8], [4], [-16], [-24], [18], [-24], [-8], [-12], [-22], [-6], [16], [-4], [-8], [-18], [-8], [-6], [-24], [10], [-6], [18], [-24], [18], [16], [26], [6], [10], [-24], [4], [24], [34], [18], [20], [-22], [-24], [10], [-18], [36], [16], [-16], [-10], [0], [24], [-26], [-30], [22], [18], [8], [12], [-38], [-8], [0], [6], [-34], [-30], [-16], [-30], [-24], [-40], [0], [-16], [-12], [42], [-42], [-26], [26], [-28], [0], [-18], [18], [-10], [10], [0], [-30], [24], [-14], [40], [20], [6], [-38], [20], [-30], [34], [24], [6], [18], [-8], [14], [-24], [-36], [-8], [0], [8], [0], [-8], [-20], [8], [-48], [20], [-52], [6], [10], [36], [-14], [-12], [-6], [16], [-42], [-40], [-30], [-38], [0], [22], [54], [4], [24], [50], [-54], [-52], [-24], [2], [12], [32], [-30], [46], [48], [0], [42], [-40], [-18], [-54], [-60], [8], [-14], [38], [-18], [-6], [58], [48], [50], [-8], [18], [-64], [30], [-20], [-44], [56], [-54], [50], [6], [-12], [36], [38], [-14], [-10], [24], [-14], [36], [2], [-18], [-54], [-30], [2], [8], [66], [0], [24], [-8], [-14], [46], [-42], [42], [16], [12], [6], [4], [14], [54], [44], [6], [-24], [58], [32], [30], [24], [6], [14], [-44], [-30], [20], [-36], [-10], [42], [-24], [-20], [-24], [-22], [-38], [-52], [54], [34], [12], [34], [24], [-36], [72], [-18], [-20], [16], [38], [54], [0], [-8], [-54], [50], [24], [68], [-6], [-48], [46], [-42], [0], [-52], [22], [18], [-74], [-52], [-78], [10], [16], [30], [34], [-24], [-30], [50], [-30], [10], [2], [70], [-52], [-50], [-56], [-60], [14], [26], [-66], [-24], [68], [-84], [-56], [76], [-24], [14], [0], [16], [-30], [36], [-6], [6], [-42], [-22], [-48], [40], [-42], [-12], [-80], [58], [24], [-16], [12], [62], [2], [-24], [-56], [24], [-82], [48], [-60], [18], [68], [36], [-26], [36], [24], [-26], [78], [10], [-10], [54], [-64], [42], [-86], [-46], [16], [0], [48], [-46], [30], [64], [0], [-22], [12], [-16], [-30], [22], [28], [86], [42], [-18], [8], [24], [48], [22], [80], [-60], [-24], [-68], [38], [-6], [-80], [44], [6], [24], [-30], [66], [96], [44], [42], [-72], [30], [-80], [-10], [-42], [-16], [-34], [-72], [76], [-48], [-48], [52], [-34], [-72], [36], [70], [78], [-2], [-84], [-6], [56], [78], [22], [-48], [56], [58], [14], [60], [74], [-54], [12], [64], [12], [-58], [52], [-90], [-22], [30], [-50], [78], [-40], [6], [-90], [64], [44], [-66], [-86], [36], [-62], [102], [66], [-10], [26], [-54], [0], [-32], [-42], [60], [0], [28], [-72], [0], [-74], [-42], [-18], [-54], [-56], [48]];
    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_h();".
// 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_h(: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_h();".
// 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_h( : 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![-6, 1]>,<13,R![2, 1]>,<17,R![-6, 1]>],Snew);
    return Vf;
end function;