
// Make newform 6762.2.a.q in Magma, downloaded from the LMFDB on 24 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_q();".
// 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_q();".
// 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], [-6], [2], [0], [0], [-1], [6], [-8], [0], [-10], [-12], [8], [2], [12], [-4], [-12], [0], [10], [-6], [-14], [0], [6], [6], [-14], [14], [-16], [-8], [12], [8], [-12], [-12], [-18], [-12], [-16], [-12], [-8], [18], [-16], [-8], [0], [14], [-6], [-2], [20], [-12], [10], [24], [10], [-16], [6], [6], [6], [4], [-18], [-28], [26], [16], [-24], [6], [20], [-16], [26], [22], [4], [-10], [-8], [-26], [2], [24], [14], [-32], [0], [-4], [6], [-6], [-32], [10], [26], [4], [-12], [2], [16], [36], [2], [26], [30], [-40], [-18], [-24], [8], [-32], [-20], [12], [10], [-4], [4], [26], [28], [6], [-14], [16], [4], [-2], [-16], [6], [-32], [-10], [4], [-4], [-32], [0], [-2], [16], [-44], [40], [-22], [42], [4], [-46], [-6], [-24], [4], [2], [4], [48], [18], [-36], [-20], [4], [-38], [-32], [-30], [26], [30], [28], [18], [-2], [-28], [-6], [-24], [18], [-42], [12], [-6], [-2], [4], [32], [10], [32], [36], [16], [16], [-20], [10], [54], [-2], [-54], [8], [-24], [-8], [10], [12], [-52], [-16], [-10], [-22], [-34], [-6], [62], [-44], [14], [20], [-36], [20], [6], [34], [-16], [0], [54], [-54], [10], [20], [50], [-10], [-28], [-10], [24], [-14], [-24], [-12], [34], [30], [4], [58], [44], [-28], [24], [22], [44], [46], [26], [30], [-38], [-14], [-20], [2], [-20], [-50], [-54], [10], [-42], [0], [-14], [-28], [-38], [8], [-66], [-66], [-34], [6], [14], [20], [54], [8], [24], [-10], [-12], [34], [0], [50], [54], [64], [-28], [-50], [14], [0], [-40], [6], [-28], [-24], [78], [6], [56], [28], [-22], [-48], [48], [48], [-60], [-68], [74], [-46], [36], [40], [-12], [-26], [42], [46], [62], [78], [-44], [-34], [-16], [30], [-20], [64], [58], [-42], [-76], [34], [-38], [38], [-40], [-60], [6], [34], [30], [24], [46], [40], [60], [-4], [48], [-22], [-66], [0], [66], [22], [26], [-66], [18], [-14], [50], [58], [54], [36], [12], [-6], [-66], [-42], [-12], [32], [70], [-4], [48], [60], [-58], [24], [10], [60], [68], [-60], [26], [-84], [32], [2], [26], [-20], [46], [18], [52], [-24], [22], [-20], [-28], [72], [-26], [-22], [-22], [56], [72], [16], [-48], [0], [-24], [58], [-22], [-58], [72], [10], [62], [-6], [-36], [-46], [28], [-68], [-34], [68], [-22], [82], [-58], [-36], [-6], [-84], [-90], [22], [-64], [-44], [42], [48], [-50], [52], [-74], [2], [-54], [-82], [-72], [20], [-24], [-70], [-46], [-74], [-32], [-52], [-50], [-20], [2], [18], [60], [20], [18], [64], [-32], [-100], [-42], [4], [12], [46], [-42], [44], [-20], [68], [66], [-42], [-52], [-28], [54], [-46], [-66], [50], [16], [10], [44], [0], [22], [64], [104], [-18], [54], [30], [-28], [46], [74], [-16], [-104], [28], [48], [38], [104], [-24], [92], [14], [-18], [-26], [-54], [-60], [-48]];
    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_q();".
// 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_q(: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_q();".
// 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_q( : 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![6, 1]>,<13,R![-2, 1]>,<17,R![0, 1]>],Snew);
    return Vf;
end function;