
// Make newform 3330.2.a.c in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3330_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_3330_2_a_c();".
// 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_3330_2_a_c();".
// 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_3330_a();"
function MakeCharacter_3330_a()
    N := 3330;
    order := 1;
    char_gens := [371, 667, 631];
    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_3330_a_Hecke(Kf)
    return MakeCharacter_3330_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], [-4], [-2], [-2], [4], [8], [2], [8], [1], [-10], [12], [0], [-6], [-4], [-10], [-4], [-8], [-6], [-8], [4], [-10], [10], [18], [-16], [-20], [-10], [14], [-8], [-12], [-2], [12], [18], [-8], [-2], [-12], [-8], [2], [-12], [-10], [-8], [-22], [26], [16], [-28], [-24], [28], [-10], [14], [-8], [-14], [12], [-2], [-8], [-22], [32], [22], [-10], [12], [-6], [12], [0], [2], [18], [-36], [2], [-12], [14], [-2], [8], [8], [22], [-20], [-16], [10], [14], [-34], [10], [4], [-2], [-24], [2], [0], [12], [-18], [18], [-30], [-24], [12], [-24], [0], [-4], [4], [-24], [26], [6], [-20], [-42], [20], [18], [12], [22], [12], [-38], [-12], [-10], [-24], [26], [-8], [-26], [30], [-4], [-16], [-2], [-28], [-40], [18], [36], [-2], [34], [-6], [20], [52], [-14], [-18], [-16], [48], [-34], [-12], [-8], [-16], [22], [-10], [2], [-54], [-4], [34], [-26], [44], [-30], [16], [36], [38], [24], [54], [6], [-36], [0], [-18], [30], [20], [24], [-36], [40], [0], [14], [26], [42], [-4], [-50], [32], [28], [-2], [-24], [40], [-42], [-14], [-54], [44], [-42], [48], [26], [-32], [-10], [12], [-30], [-48], [-34], [-24], [-60], [-26], [22], [-48], [18], [-2], [-44], [-38], [16], [58], [-4], [36], [42], [-60], [14], [66], [14], [38], [-24], [50], [-48], [-10], [2], [12], [-30], [-32], [68], [6], [-20], [-22], [-6], [-16], [-68], [0], [42], [8], [30], [-24], [-30], [-10], [-8], [-66], [24], [68], [46], [-34], [-64], [48], [-36], [14], [-44], [-48], [-26], [4], [-32], [-46], [-54], [60], [16], [44], [28], [-16], [22], [-42], [64], [56], [-60], [-68], [-48], [-34], [30], [-24], [10], [-14], [-12], [-10], [36], [42], [34], [72], [12], [70], [-34], [-18], [4], [-6], [-42], [-44], [26], [6], [52], [34], [-24], [18], [64], [12], [-26], [10], [52], [-48], [48], [-40], [-42], [-52], [-16], [-38], [42], [48], [46], [18], [-52], [46], [60], [14], [74], [-80], [-54], [-36], [28], [-62], [-78], [-16], [-4], [76], [-38], [4], [-10], [8], [86], [16], [-30], [-18], [44], [88], [58], [68], [-72], [34], [30], [-28], [-6], [-78], [40], [-50], [50], [4], [28], [0], [-22], [46], [-30], [40], [12], [-68], [-52], [-50], [22], [90], [-72], [-10], [-18], [-6], [16], [-30], [36], [-58], [-60], [24], [42], [20], [-38], [50], [-24], [22], [78], [-24], [28], [38], [56], [22], [-42], [-16], [-4], [60], [18], [-78], [-96], [-6], [-12], [28], [48], [66], [-16], [-2], [-44], [0], [50], [-66], [-6], [-94], [6], [-64], [-26], [68], [-24], [-96], [-10], [-20], [32], [66], [-6], [60], [84], [-72], [26], [-88], [54], [-4], [34], [78], [14], [40], [6], [42], [8], [-18], [30], [-20], [-12], [-46], [90], [-60], [4], [-62], [90], [80], [0], [54], [-24], [-30], [-10], [-96], [60], [74], [82], [52], [6], [12], [-48]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3330_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3330_2_a_c();".
// 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_3330_2_a_c(:prec:=1)
    chi := MakeCharacter_3330_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_3330_2_a_c();".
// 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_3330_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3330_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![0, 1]>,<11,R![4, 1]>,<13,R![2, 1]>,<17,R![2, 1]>],Snew);
    return Vf;
end function;