
// Make newform 6762.2.a.bw 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_bw();".
// 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_bw();".
// 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
        poly := [-4, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
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, 0], [1, 0], [0, -1], [0, 0], [2, -2], [2, -1], [-2, -2], [2, -2], [1, 0], [6, -1], [0, 2], [0, 1], [-2, 1], [2, -3], [-4, -1], [8, -4], [-8, -2], [-4, -2], [2, 2], [0, 6], [-6, 2], [8, -4], [6, -2], [-14, 0], [14, 1], [-10, 4], [-8, 7], [-2, 2], [12, 1], [2, 3], [-12, 3], [-8, -2], [-2, -1], [4, 5], [0, 4], [12, 3], [-8, 4], [-8, 0], [8, -4], [-6, 8], [4, 5], [12, -2], [-20, 0], [-2, -3], [14, -7], [4, -11], [-4, 8], [0, 2], [14, 5], [4, 4], [-6, 8], [-8, -8], [6, -9], [2, 1], [2, 0], [12, -7], [-6, 4], [-8, 4], [10, -2], [-6, -7], [-2, -8], [-12, 10], [8, 7], [0, 12], [2, 4], [-10, 5], [-12, 8], [18, -6], [4, -7], [10, -4], [26, -5], [-8, -1], [20, 1], [12, -2], [6, -9], [0, -4], [-20, 6], [14, 4], [10, -4], [-18, 10], [-2, 14], [16, -11], [-8, -3], [-10, 3], [8, -10], [20, 7], [6, 4], [-2, -6], [-22, 12], [-12, 1], [2, 3], [-16, 10], [-4, 1], [-8, 12], [32, -6], [4, -18], [-6, -6], [6, 4], [-14, 0], [-6, 6], [12, 10], [-4, 16], [-18, 1], [-18, 17], [34, -6], [26, 4], [0, -16], [-10, 13], [8, 12], [-6, -6], [32, -10], [-4, -5], [38, 4], [-2, 4], [-4, -8], [-18, -3], [38, 0], [16, 8], [10, -17], [-10, -10], [-36, -2], [18, -3], [-12, -6], [-20, -8], [36, -9], [28, -4], [20, 6], [-12, -1], [8, 4], [-32, -4], [8, -10], [0, 16], [12, -8], [-16, 6], [22, 12], [30, -11], [12, -13], [14, 2], [-4, 15], [2, -14], [20, -3], [-18, 24], [28, 3], [30, -12], [-6, 12], [8, 10], [2, 1], [-6, -3], [16, -11], [-8, -12], [-14, 14], [2, 0], [-12, -2], [-24, 0], [26, -17], [-16, -9], [-32, 12], [38, 1], [10, -11], [-16, -9], [-44, 7], [50, 0], [-16, -4], [-14, -10], [-34, -5], [-12, 14], [0, 4], [-14, 24], [-34, 4], [6, -26], [22, 0], [22, 7], [4, 5], [-46, 8], [-16, 8], [26, 0], [-16, 14], [6, 4], [8, -27], [-32, -6], [-16, 4], [-38, -9], [50, -2], [-14, 11], [-8, -5], [-4, -9], [-58, 0], [-34, -10], [-6, -7], [20, -1], [18, 12], [0, -1], [22, -2], [14, 12], [22, -8], [18, -10], [2, -12], [52, 2], [-54, 8], [-20, 1], [28, -2], [-32, -12], [-2, 15], [-10, 20], [-2, 7], [-22, 6], [4, 9], [28, -8], [54, 4], [-36, 0], [-62, 1], [-42, -4], [20, 0], [-6, 22], [60, -7], [10, -4], [-24, 4], [2, 21], [-4, 21], [-54, -5], [6, -14], [-4, 20], [34, -16], [28, 8], [4, 24], [-42, 16], [-6, 22], [-32, 8], [16, -9], [16, 8], [-46, 0], [-18, -16], [20, -28], [-18, -10], [-18, 2], [4, 4], [-34, 3], [12, -18], [-24, -1], [-56, -4], [18, 16], [20, -8], [-16, -4], [6, 28], [-46, 11], [12, -19], [40, -14], [34, -3], [-14, 29], [28, -27], [-52, -1], [2, -20], [52, -7], [2, -9], [-50, 6], [-52, 13], [-36, -15], [2, 12], [42, -8], [46, -14], [68, 4], [62, 0], [-10, -17], [-12, 10], [46, -12], [-26, 24], [-20, 18], [34, -2], [2, -11], [-14, 0], [6, -9], [-26, -2], [-14, 37], [-20, 22], [46, 4], [-40, 10], [8, 9], [14, 4], [-22, -24], [14, -31], [-68, 7], [68, -6], [-20, 18], [-12, -24], [40, -7], [-32, 2], [-66, 2], [20, -6], [32, -14], [-42, 9], [-8, -12], [10, 3], [-18, 2], [-18, -5], [2, 12], [-20, 30], [12, 13], [14, -30], [-28, -10], [20, -7], [6, -17], [4, 4], [-24, 2], [-8, 11], [6, 1], [-18, -17], [-12, 13], [-44, 8], [-28, 19], [-2, -20], [28, 17], [32, -6], [-70, 0], [-12, -16], [-52, -8], [14, -8], [-4, 20], [-16, 20], [-46, 17], [6, -10], [14, 20], [-34, 20], [6, -8], [-28, -5], [-50, 7], [-6, -28], [38, -30], [52, -11], [60, 3], [-20, 24], [-14, 2], [50, -13], [-16, 18], [48, 8], [-22, 29], [4, 16], [-40, 13], [42, 0], [22, 16], [-16, 19], [-26, -24], [-10, -11], [30, -16], [72, -4], [20, 28], [-4, -31], [-38, -9], [-48, 14], [28, 4], [-32, -15], [36, 16], [34, -9], [-2, -31], [16, -8], [-20, -19], [-30, 31], [0, -18], [14, 30], [26, 6], [-48, -2], [48, 2], [-34, 25], [16, -8], [-22, 10], [20, 25], [42, 9], [-34, 26], [32, 8], [-66, 2], [16, -20], [0, 27], [64, -14], [-20, 12], [-52, 20], [-10, 2], [32, -14], [-8, 35], [2, -7], [-18, 0], [-42, -22], [0, -10], [-30, 12], [8, 24], [46, -8], [-22, -14], [28, -15], [28, 13], [-58, 19], [46, -5], [60, -22], [10, -28], [82, 4], [32, -15], [52, -27], [4, 16], [-2, 24], [40, -25], [-50, -18], [-6, 33], [-14, 35], [-14, 35], [46, 0], [32, -18], [46, 18], [-10, 12], [-88, 10], [36, 15], [22, -10], [6, 2], [-4, 20], [-14, 24], [42, -33], [-90, 10], [98, 2], [74, 4], [-6, 2], [-16, 10], [-8, -6], [-42, -8], [28, -22], [20, -6], [-32, 20], [-56, -4], [80, -8], [-50, 2], [-34, -18], [26, 10], [-30, -10], [-28, 23], [-52, 7]];
    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_bw();".
// 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_bw(:prec:=2)
    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_bw();".
// 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_bw( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6762_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-4, 1, 1]>,<11,R![-16, -2, 1]>,<13,R![-2, -3, 1]>,<17,R![-8, 6, 1]>],Snew);
    return Vf;
end function;