
// Make newform 3366.2.a.c in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3366_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_3366_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_3366_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_3366_a();"
function MakeCharacter_3366_a()
    N := 3366;
    order := 1;
    char_gens := [749, 1531, 2179];
    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_3366_a_Hecke(Kf)
    return MakeCharacter_3366_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], [-2], [-2], [1], [4], [-1], [2], [-2], [-2], [-4], [6], [-6], [2], [0], [12], [14], [6], [-4], [2], [-8], [-2], [12], [-6], [-6], [-2], [-8], [-4], [-6], [-12], [-8], [-20], [-18], [0], [-6], [-4], [-14], [-12], [-12], [-6], [10], [-10], [-12], [-4], [6], [-20], [8], [0], [12], [-22], [18], [0], [-4], [2], [22], [-8], [-10], [-8], [-14], [-22], [8], [-6], [22], [-30], [34], [-30], [-20], [8], [12], [-4], [6], [32], [4], [20], [4], [-4], [-24], [-14], [-28], [30], [20], [-34], [16], [14], [-10], [-34], [-32], [22], [18], [-16], [6], [-12], [28], [8], [4], [-24], [-24], [28], [-2], [-10], [28], [42], [-36], [6], [-8], [46], [-2], [-30], [4], [-8], [-42], [12], [-16], [36], [24], [-28], [36], [-8], [6], [24], [18], [44], [-26], [28], [-4], [-22], [-6], [6], [-32], [-36], [10], [-16], [-16], [38], [-30], [-22], [0], [-4], [20], [-18], [44], [10], [-32], [20], [-34], [2], [26], [-42], [52], [44], [-22], [0], [-16], [36], [-4], [-46], [4], [24], [14], [-30], [-24], [2], [32], [-10], [30], [-18], [-24], [46], [-32], [-54], [48], [10], [36], [-18], [-24], [-20], [28], [26], [32], [8], [-56], [-20], [-14], [42], [40], [0], [-2], [-20], [-40], [44], [-22], [60], [16], [-8], [-20], [20], [0], [-14], [22], [-32], [18], [-18], [-62], [-30], [-14], [12], [16], [-40], [-18], [40], [-68], [-12], [52], [10], [48], [-46], [-28], [-42], [-42], [40], [-44], [-54], [6], [56], [48], [44], [-36], [18], [36], [12], [-26], [16], [28], [-30], [16], [-32], [-30], [-54], [-20], [52], [-28], [58], [56], [-62], [26], [32], [-72], [-52], [10], [-48], [12], [-30], [-48], [-18], [-70], [-32], [-14], [68], [2], [38], [34], [20], [30], [-54], [48], [-32], [72], [6], [64], [54], [-26], [68], [-38], [-52], [62], [-40], [14], [-40], [-74], [-20], [-56], [44], [-8], [44], [68], [48], [66], [-38], [24], [30], [-10], [-28], [54], [-20], [6], [30], [-8], [-12], [44], [86], [58], [-30], [-70], [-72], [-20], [-42], [-78], [-18], [60], [-16], [80], [-18], [-62], [-28], [24], [-2], [30], [-4], [26], [-6], [0], [86], [74], [0], [42], [30], [-52], [44], [16], [54], [-70], [50], [-42], [-60], [68], [24], [-6], [-6], [14], [80], [-14], [22], [26], [0], [-76], [68], [26], [-4], [60], [90], [-14], [-50], [0], [70], [76], [-26], [-72], [-12], [38], [28], [-42], [-30], [-60], [60], [-48], [-58], [30], [-92], [-8], [-90], [36], [-20], [-50], [-56], [-14], [24], [-76], [90], [-18], [-26], [62], [54], [-22], [78], [-56], [30], [-56], [78], [-76], [-52], [-22], [-78], [26], [76], [36], [-76], [-12], [6], [52], [46], [94], [-90], [-8], [-28], [42], [-42], [-58], [62], [76], [-32], [-104], [-62], [-2], [80], [66], [-78], [20], [92], [-36], [-96], [64], [106], [98], [56], [2], [32], [-52], [-78], [80], [24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3366_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3366_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_3366_2_a_c(:prec:=1)
    chi := MakeCharacter_3366_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_3366_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_3366_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3366_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2, 1]>,<7,R![2, 1]>,<13,R![-4, 1]>,<19,R![-2, 1]>,<23,R![2, 1]>],Snew);
    return Vf;
end function;