
// Make newform 3332.2.a.b in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3332_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_3332_2_a_b();".
// 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_3332_2_a_b();".
// 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_3332_a();"
function MakeCharacter_3332_a()
    N := 3332;
    order := 1;
    char_gens := [1667, 885, 785];
    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_3332_a_Hecke(Kf)
    return MakeCharacter_3332_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 := [[0], [-1], [0], [0], [-5], [-5], [-1], [-6], [4], [4], [0], [8], [4], [-6], [-6], [11], [10], [0], [10], [9], [4], [9], [4], [-15], [-18], [2], [-16], [-17], [16], [0], [-4], [-4], [-15], [-7], [21], [-2], [17], [4], [-21], [18], [-24], [6], [-12], [10], [2], [-1], [20], [18], [-9], [22], [-12], [18], [-20], [-4], [3], [-8], [6], [24], [24], [5], [9], [9], [16], [-3], [6], [18], [10], [8], [-12], [-34], [11], [-26], [-15], [-1], [-25], [18], [21], [14], [-30], [35], [-5], [6], [-17], [14], [-23], [2], [14], [14], [-25], [20], [16], [24], [-19], [-22], [-15], [-13], [-26], [-28], [6], [10], [-43], [-17], [-26], [1], [-4], [-19], [34], [45], [30], [2], [7], [17], [16], [17], [18], [22], [41], [28], [42], [-30], [-22], [-24], [12], [13], [24], [23], [-10], [35], [26], [-5], [-18], [7], [21], [33], [13], [-7], [1], [-44], [-29], [30], [7], [-8], [3], [-17], [-7], [8], [2], [26], [-40], [-52], [14], [38], [34], [27], [28], [28], [8], [40], [-14], [4], [-53], [-3], [-18], [-22], [-46], [49], [-9], [58], [-14], [36], [4], [35], [13], [53], [46], [-42], [-37], [-52], [16], [-35], [4], [8], [40], [33], [-30], [19], [36], [-34], [52], [-59], [0], [57], [54], [-55], [-33], [-30], [58], [-14], [-26], [-6], [-48], [-21], [-7], [-26], [8], [-42], [18], [16], [-10], [60], [68], [-18], [8], [-12], [-45], [8], [-26], [59], [15], [-9], [-35], [5], [-14], [-53], [16], [-42], [14], [-40], [16], [59], [39], [-7], [22], [-57], [-26], [68], [-22], [6], [36], [-4], [24], [50], [20], [49], [-56], [-15], [1], [60], [-16], [52], [41], [-62], [0], [64], [41], [20], [58], [-52], [-8], [-38], [16], [-18], [4], [-16], [-68], [40], [3], [-33], [7], [-6], [-32], [-2], [57], [-14], [-15], [-6], [-12], [-65], [-7], [-28], [-46], [32], [-9], [-1], [56], [78], [56], [72], [-10], [-26], [60], [-43], [-73], [-84], [-14], [4], [-26], [11], [60], [62], [2], [-19], [-77], [43], [53], [2], [72], [-2], [24], [1], [60], [-86], [36], [56], [-72], [69], [38], [57], [-8], [-90], [28], [-52], [66], [-6], [-30], [90], [-68], [49], [-32], [86], [-40], [30], [-56], [-22], [-3], [15], [35], [64], [14], [-88], [3], [-59], [-24], [52], [-69], [-35], [-94], [38], [-39], [-46], [50], [-16], [-23], [8], [41], [-11], [90], [3], [-52], [54], [-50], [18], [38], [4], [-6], [65], [-54], [24], [-40], [70], [-44], [-9], [27], [62], [-38], [-39], [-21], [-93], [-55], [79], [-46], [-35], [48], [-36], [20], [28], [-26], [94], [59], [-24], [-82], [50], [2], [-26], [-12], [2], [-10], [85], [-21], [19], [-38], [21], [12], [-16], [33], [17], [94], [-75], [-36], [-84], [46], [17], [-38], [-45], [47], [64], [-55], [17], [72], [32], [-69], [-26], [63], [54], [-94], [-59], [36], [-66], [-62], [53]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3332_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3332_2_a_b();".
// 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_3332_2_a_b(:prec:=1)
    chi := MakeCharacter_3332_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_3332_2_a_b();".
// 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_3332_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3332_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 1]>,<5,R![0, 1]>],Snew);
    return Vf;
end function;