
// Make newform 6762.2.a.bc in Magma, downloaded from the LMFDB on 25 April 2024.
// 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_bc();".
// 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_bc();".
// 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)
    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+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], [2], [0], [4], [-2], [-6], [0], [1], [-2], [-4], [6], [6], [12], [12], [6], [4], [10], [4], [-16], [-2], [8], [16], [-6], [2], [14], [0], [-12], [-2], [-6], [-8], [-4], [10], [-4], [6], [0], [18], [-4], [-20], [6], [4], [2], [8], [18], [-18], [0], [28], [4], [24], [18], [10], [8], [10], [-24], [6], [-24], [6], [-20], [-10], [-22], [-24], [34], [20], [-12], [-14], [-10], [28], [-22], [-12], [6], [14], [-16], [24], [22], [20], [0], [38], [-34], [18], [-26], [-16], [-26], [16], [2], [-20], [20], [-30], [-6], [22], [-32], [-8], [-40], [8], [-12], [-36], [-40], [6], [26], [8], [-18], [-28], [14], [0], [42], [-20], [-10], [12], [6], [40], [-2], [-20], [-10], [-6], [-32], [16], [-30], [-24], [-36], [-34], [-12], [18], [34], [34], [36], [20], [-34], [6], [-28], [-16], [-14], [-4], [0], [0], [38], [-18], [10], [26], [40], [34], [-6], [-28], [30], [32], [20], [-18], [24], [38], [22], [12], [-8], [-26], [-6], [-36], [44], [-28], [-24], [-16], [46], [-30], [-38], [-36], [34], [24], [24], [18], [24], [-56], [6], [18], [-26], [-28], [22], [-32], [18], [-60], [-38], [-36], [6], [-24], [34], [8], [-48], [-50], [22], [-16], [10], [46], [-24], [-6], [-4], [-2], [36], [28], [-18], [-44], [-6], [-22], [62], [-38], [44], [46], [-20], [6], [-38], [0], [-18], [-48], [28], [42], [-4], [-62], [-2], [56], [48], [-44], [2], [-32], [-42], [64], [-18], [46], [-40], [2], [24], [60], [-34], [42], [24], [-16], [28], [-26], [-40], [-40], [42], [56], [8], [-22], [-10], [44], [-60], [20], [-44], [-44], [-2], [54], [20], [-20], [-32], [44], [72], [-34], [-62], [-40], [-30], [-10], [-76], [38], [-64], [22], [-58], [16], [20], [-10], [-22], [-2], [56], [46], [-54], [-20], [-26], [38], [12], [-22], [-64], [-58], [52], [60], [-82], [-22], [40], [28], [-72], [-24], [-14], [44], [-56], [34], [-42], [-52], [14], [-34], [8], [-22], [-56], [46], [-38], [64], [-26], [-4], [28], [26], [46], [80], [-52], [-52], [-78], [-68], [26], [-24], [-26], [-12], [54], [82], [-68], [-24], [-86], [68], [24], [-14], [-30], [-64], [-62], [-10], [64], [66], [26], [-12], [76], [-8], [6], [-18], [-18], [-20], [44], [28], [-52], [-66], [66], [-58], [0], [70], [-70], [54], [56], [-34], [84], [70], [60], [-80], [-78], [44], [-54], [22], [-64], [22], [-58], [48], [-80], [66], [-72], [-26], [22], [80], [-36], [28], [74], [-34], [-40], [-6], [-36], [28], [-80], [54], [-16], [-2], [-28], [80], [74], [-62], [78], [18], [82], [16], [66], [-56], [20], [88], [-26], [-4], [16], [-54], [54], [-4], [-76], [48], [66], [-16], [74], [-76], [6], [-74], [-14], [-72], [-86], [-26], [-20], [-82], [66], [48], [-60], [-82], [-58], [92], [28], [34], [-42], [80], [-20], [50], [-56], [30], [10], [48], [36], [-10], [-58], [-52], [42], [-84], [84]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6762_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6762_2_a_bc();".
// 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_bc(: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_bc();".
// 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_bc( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6762_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, 1]>,<11,R![-4, 1]>,<13,R![2, 1]>,<17,R![6, 1]>],Snew);
    return Vf;
end function;