
// Make newform 3328.2.a.d in Magma, downloaded from the LMFDB on 14 January 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_3328_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_3328_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_3328_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_3328_a();"
function MakeCharacter_3328_a()
    N := 3328;
    order := 1;
    char_gens := [1535, 261, 769];
    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_3328_a_Hecke(Kf)
    return MakeCharacter_3328_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]];
    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], [-1], [-1], [-2], [1], [1], [8], [6], [6], [-8], [-1], [-4], [5], [11], [-6], [-8], [4], [-2], [-3], [-8], [-10], [-6], [-16], [-2], [-8], [-6], [-4], [-1], [-6], [22], [-3], [-14], [13], [6], [5], [-4], [6], [16], [-14], [13], [22], [-18], [6], [-23], [-2], [-15], [-3], [-24], [21], [-7], [11], [-20], [-12], [-15], [-16], [-4], [3], [24], [-8], [-32], [-9], [8], [24], [11], [6], [-26], [-25], [-15], [-25], [-18], [-32], [-22], [24], [34], [-15], [-20], [-22], [10], [6], [15], [15], [15], [31], [-20], [35], [-6], [32], [-37], [-16], [-12], [35], [-16], [13], [-24], [0], [14], [11], [4], [1], [-1], [-15], [-31], [-1], [7], [-8], [-18], [34], [-16], [45], [32], [-38], [-20], [0], [-47], [42], [14], [4], [36], [12], [42], [1], [-18], [18], [16], [22], [-42], [-42], [-26], [1], [-20], [3], [20], [22], [12], [-32], [-33], [-8], [-36], [3], [4], [-45], [8], [-52], [46], [24], [33], [-18], [-36], [39], [-7], [-19], [31], [-40], [17], [-18], [36], [48], [46], [-27], [-42], [-57], [-59], [27], [-10], [23], [-40], [50], [0], [-18], [-30], [-31], [-56], [-8], [-26], [-17], [28], [54], [36], [34], [56], [36], [-34], [-8], [24], [-4], [-20], [28], [-20], [36], [21], [4], [-29], [-42], [-29], [10], [24], [-54], [-22], [-40], [-69], [-30], [-27], [-23], [0], [68], [5], [-24], [-18], [-53], [-66], [26], [2], [40], [13], [-2], [40], [-63], [48], [9], [36], [-35], [-48], [17], [-57], [32], [-33], [34], [60], [54], [64], [64], [24], [27], [63], [-53], [0], [9], [76], [22], [52], [61], [-56], [-39], [-54], [-50], [-15], [40], [46], [-16], [-53], [-42], [1], [55], [-6], [22], [22], [-28], [66], [-36], [4], [1], [41], [-18], [-48], [5], [-21], [42], [-24], [-28], [6], [-40], [-46], [-62], [-38], [-44], [-68], [78], [64], [63], [42], [-75], [78], [-26], [22], [44], [33], [-7], [-13], [-18], [60], [81], [-20], [-26], [34], [24], [50], [0], [79], [-2], [-33], [3], [-70], [-20], [67], [-46], [-71], [-4], [44], [-8], [38], [-75], [3], [28], [-27], [-23], [-36], [32], [8], [-74], [0], [-48], [30], [25], [-72], [74], [86], [24], [-42], [54], [50], [0], [82], [-66], [-32], [-48], [-14], [-4], [22], [10], [-73], [-53], [75], [-16], [9], [-15], [52], [6], [-35], [-46], [28], [-58], [-85], [56], [-70], [22], [21], [62], [-49], [-23], [-22], [-10], [-38], [-38], [20], [-51], [-19], [40], [29], [-61], [-49], [-5], [-44], [80], [2], [42], [24], [86], [26], [10], [30], [-26], [-4], [2], [50], [-9], [-45], [34], [-72], [-58], [-72], [-55], [0], [13], [-5], [-71], [-19], [21], [84], [-65], [7], [98], [-59], [42], [89], [40], [69], [-12], [-18], [44], [29], [48], [15], [-55], [1], [-2], [81], [36], [6], [68], [-30], [86], [72], [35], [34], [-50], [101], [24], [-28], [-54]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3328_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3328_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_3328_2_a_d(:prec:=1)
    chi := MakeCharacter_3328_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_3328_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_3328_2_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3328_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 1]>,<5,R![1, 1]>,<7,R![1, 1]>],Snew);
    return Vf;
end function;