
// Make newform 3330.2.a.x in Magma, downloaded from the LMFDB on 24 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_x();".
// 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_x();".
// 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], [1], [1], [2], [7], [-2], [0], [-9], [7], [-1], [11], [-11], [-8], [11], [10], [-1], [-8], [0], [4], [12], [6], [-6], [-19], [20], [-12], [6], [7], [13], [-16], [6], [6], [21], [-6], [-6], [23], [13], [-12], [3], [-24], [-26], [-19], [14], [18], [-4], [-5], [3], [23], [-16], [-24], [-27], [6], [16], [-14], [-21], [0], [8], [8], [-6], [12], [9], [10], [9], [-18], [-5], [-8], [22], [-28], [-32], [31], [-14], [17], [-6], [-16], [-24], [21], [-14], [6], [-18], [-28], [-2], [1], [20], [1], [30], [-8], [-11], [13], [2], [-3], [-24], [18], [12], [-28], [12], [-24], [-15], [-24], [-10], [-11], [-2], [-13], [-30], [11], [2], [-43], [-42], [-4], [-23], [-10], [33], [2], [-19], [-25], [27], [-7], [-32], [-4], [24], [41], [4], [6], [15], [-33], [6], [35], [-26], [14], [-15], [-31], [-21], [8], [-10], [43], [-2], [9], [-12], [48], [24], [-44], [-46], [-40], [-21], [19], [26], [-50], [-31], [18], [-27], [23], [-21], [19], [47], [-20], [-24], [16], [21], [32], [22], [47], [-36], [28], [-49], [-5], [-51], [37], [-18], [-19], [18], [50], [7], [-23], [36], [18], [58], [18], [54], [-51], [-12], [-8], [16], [32], [33], [9], [-46], [53], [13], [-10], [-12], [34], [56], [22], [-40], [18], [42], [-68], [16], [-14], [-20], [-26], [14], [35], [5], [63], [-30], [-70], [-42], [-12], [-69], [29], [-1], [-30], [46], [-45], [7], [26], [-32], [-42], [25], [68], [58], [-45], [16], [52], [18], [72], [68], [-67], [18], [19], [60], [22], [46], [46], [51], [-50], [-22], [-2], [-48], [-3], [-36], [64], [41], [-54], [21], [-48], [64], [-25], [-78], [-16], [34], [45], [-30], [22], [-1], [-58], [-22], [47], [51], [-34], [0], [12], [-5], [78], [-9], [-40], [12], [8], [-2], [46], [-5], [22], [19], [54], [-13], [10], [-30], [41], [-38], [72], [-35], [-21], [20], [36], [-42], [-67], [35], [-64], [60], [5], [-36], [-50], [33], [-33], [38], [-54], [-2], [-44], [48], [13], [18], [36], [64], [-60], [-17], [-37], [-46], [50], [60], [24], [44], [-1], [-26], [26], [-16], [51], [-76], [62], [48], [64], [42], [-63], [-13], [52], [-28], [75], [-1], [2], [37], [-10], [-50], [-61], [60], [34], [78], [-22], [-38], [-10], [0], [-26], [54], [46], [-65], [14], [2], [-28], [14], [9], [-37], [35], [-16], [-89], [4], [50], [14], [35], [42], [18], [-12], [78], [-14], [94], [42], [-86], [47], [-30], [-50], [18], [-36], [98], [-49], [6], [-11], [-33], [-64], [48], [-16], [-72], [-8], [-21], [70], [52], [-48], [-2], [7], [-10], [-50], [-57], [92], [-21], [44], [-10], [67], [-12], [4], [-8], [-55], [95], [40], [-58], [-11], [-2], [-42], [-54], [-16], [28], [85], [-40], [21], [64], [-83], [-79], [-10], [23], [-84], [-18], [-77], [4], [-54], [-54], [0], [-84], [-59], [76], [-81], [24], [1], [-68], [-23]];
    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_x();".
// 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_x(: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_x();".
// 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_x( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3330_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-1, 1]>,<11,R![-1, 1]>,<13,R![-2, 1]>,<17,R![-7, 1]>],Snew);
    return Vf;
end function;