
// Make newform 3330.2.a.d in Magma, downloaded from the LMFDB on 19 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_d();".
// 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_d();".
// 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], [2], [0], [2], [-6], [2], [0], [-6], [-10], [1], [6], [-4], [6], [-6], [6], [-10], [2], [0], [2], [-10], [6], [6], [2], [-18], [-4], [-6], [14], [-6], [2], [6], [6], [-4], [-18], [20], [-10], [-16], [-12], [18], [-6], [-22], [6], [2], [18], [-22], [20], [14], [-24], [-10], [-18], [-6], [-22], [-18], [-6], [6], [6], [-16], [26], [-18], [-16], [-6], [26], [18], [-22], [18], [-10], [2], [12], [26], [30], [-36], [2], [-34], [-16], [0], [-6], [14], [6], [2], [-24], [-10], [-30], [2], [-22], [-6], [30], [26], [-6], [-40], [0], [18], [20], [-36], [14], [0], [-30], [-30], [-4], [14], [44], [30], [-36], [6], [-28], [38], [36], [6], [36], [38], [32], [-34], [30], [-4], [14], [-30], [32], [-48], [18], [36], [14], [26], [42], [-36], [8], [-30], [38], [36], [32], [14], [20], [-18], [-4], [-10], [6], [26], [18], [-10], [30], [6], [20], [42], [-34], [-24], [-34], [0], [-10], [-42], [50], [-54], [-10], [42], [56], [42], [8], [30], [-46], [30], [26], [30], [-48], [-42], [-4], [36], [-18], [-18], [-34], [-10], [-10], [42], [-6], [38], [-30], [26], [-40], [30], [-22], [30], [2], [62], [8], [54], [50], [6], [-42], [42], [38], [20], [-22], [-36], [-34], [30], [50], [-30], [54], [6], [-22], [26], [-42], [-36], [-6], [8], [-58], [62], [-24], [-42], [-64], [-6], [6], [-52], [14], [-30], [-16], [18], [-42], [-10], [-64], [-66], [-12], [18], [-22], [-64], [-30], [56], [-42], [-10], [54], [-12], [-10], [18], [-34], [-16], [20], [6], [-10], [-42], [50], [6], [6], [-54], [-12], [50], [-34], [62], [-18], [42], [-28], [66], [-4], [12], [-22], [-30], [-42], [2], [-78], [-12], [-10], [26], [-30], [74], [-16], [-36], [-58], [62], [18], [8], [6], [-42], [-58], [-66], [-10], [-16], [62], [-22], [2], [-34], [-42], [14], [74], [42], [-54], [-58], [-42], [-10], [80], [60], [14], [66], [26], [-42], [-6], [36], [54], [36], [38], [66], [-4], [-30], [-6], [62], [-22], [42], [20], [-36], [-58], [-58], [-60], [14], [0], [2], [6], [30], [18], [14], [24], [2], [84], [-6], [2], [78], [-82], [50], [42], [38], [6], [-70], [-40], [-16], [-24], [-6], [14], [-30], [38], [84], [50], [-6], [-82], [6], [-70], [-34], [-82], [6], [18], [50], [-30], [-18], [38], [-58], [18], [-6], [56], [2], [-30], [-76], [26], [54], [-66], [-54], [78], [-36], [-22], [-42], [-12], [-18], [74], [74], [54], [20], [2], [78], [26], [54], [-66], [-22], [-82], [-72], [48], [50], [-66], [-70], [-78], [-54], [-88], [-42], [14], [54], [32], [-58], [20], [60], [50], [-6], [-66], [-64], [0], [74], [50], [-42], [8], [42], [50], [-18], [8], [102], [66], [56], [-34], [30], [-34], [0], [-46], [18], [-24], [-22], [-22], [30], [96], [74], [30], [24], [90], [38], [18], [24], [74], [66], [66], [-42], [8], [6]];
    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_d();".
// 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_d(: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_d();".
// 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_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3330_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-2, 1]>,<11,R![0, 1]>,<13,R![-2, 1]>,<17,R![6, 1]>],Snew);
    return Vf;
end function;