
// Make newform 3360.2.a.g in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3360_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_3360_2_a_g();".
// 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_3360_2_a_g();".
// 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_3360_a();"
function MakeCharacter_3360_a()
    N := 3360;
    order := 1;
    char_gens := [1471, 421, 1121, 2017, 1921];
    v := [1, 1, 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_3360_a_Hecke(Kf)
    return MakeCharacter_3360_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], [1], [-1], [4], [-2], [-2], [8], [0], [-2], [0], [-6], [2], [4], [8], [-6], [0], [-2], [4], [4], [-2], [4], [-12], [18], [6], [6], [16], [-12], [6], [-6], [0], [8], [18], [0], [-2], [-20], [14], [4], [24], [6], [4], [-10], [-4], [10], [2], [24], [12], [-16], [28], [-26], [10], [20], [10], [0], [22], [16], [-2], [-8], [-14], [10], [-4], [22], [20], [-16], [-26], [-14], [-20], [26], [12], [-2], [6], [20], [8], [-22], [20], [-8], [-2], [-18], [2], [-22], [-16], [-26], [-12], [14], [24], [-4], [18], [-14], [30], [-40], [-28], [0], [-8], [-12], [28], [-24], [-2], [42], [-4], [-26], [4], [26], [-28], [-22], [20], [22], [-36], [-34], [20], [-30], [8], [-38], [-38], [16], [-4], [34], [-44], [-24], [34], [-12], [-42], [-46], [-18], [36], [-32], [-50], [-26], [48], [-8], [6], [12], [-24], [-28], [26], [-38], [26], [6], [4], [6], [26], [8], [-42], [-40], [-36], [46], [8], [54], [14], [-16], [0], [-38], [42], [-20], [48], [-52], [20], [-52], [50], [54], [14], [-4], [-22], [32], [-16], [-6], [-8], [-52], [46], [-46], [-50], [28], [30], [-12], [-46], [8], [18], [36], [-34], [48], [-34], [8], [-8], [58], [-42], [-32], [-10], [2], [44], [-22], [-32], [14], [-20], [4], [-2], [-12], [6], [34], [26], [-42], [16], [-2], [-8], [-18], [10], [40], [-18], [0], [-4], [-70], [-56], [66], [-10], [64], [12], [48], [34], [-40], [34], [24], [34], [-26], [56], [-30], [16], [-52], [-66], [-50], [60], [-8], [60], [26], [64], [44], [-6], [12], [-32], [-62], [42], [-76], [48], [44], [-24], [-32], [-2], [14], [0], [-56], [-32], [44], [48], [10], [-14], [32], [26], [30], [60], [14], [60], [54], [22], [16], [60], [-58], [6], [6], [-24], [62], [-70], [-44], [10], [14], [20], [-42], [4], [62], [8], [68], [-66], [-70], [0], [16], [12], [0], [-74], [20], [12], [26], [-62], [-64], [-14], [30], [12], [-6], [-56], [26], [14], [-16], [-18], [-24], [-52], [-66], [-30], [4], [4], [60], [2], [60], [-18], [-12], [74], [-24], [-26], [-62], [-76], [-48], [34], [-64], [84], [-34], [34], [-24], [-38], [-34], [8], [-54], [42], [44], [-20], [24], [-78], [-2], [-62], [-16], [36], [52], [68], [38], [-2], [50], [16], [-22], [2], [-10], [68], [26], [-44], [38], [-28], [-56], [-26], [32], [18], [-50], [-24], [-26], [30], [88], [-72], [34], [-48], [26], [58], [32], [12], [28], [-14], [-58], [-40], [90], [60], [-96], [-48], [-66], [88], [26], [-88], [68], [-74], [-94], [-42], [46], [-6], [-16], [-6], [-32], [-40], [-76], [-34], [4], [0], [-46], [-74], [-44], [-60], [-92], [-70], [-40], [34], [-36], [22], [14], [66], [8], [6], [-42], [16], [10], [-14], [28], [-8], [-34], [-30], [36], [-4], [18], [62], [-36], [80], [-58], [56], [-26], [62], [-80], [-96], [70], [-66], [52], [90], [-56], [-56]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3360_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3360_2_a_g();".
// 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_3360_2_a_g(:prec:=1)
    chi := MakeCharacter_3360_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_3360_2_a_g();".
// 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_3360_2_a_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3360_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![-4, 1]>,<13,R![2, 1]>,<17,R![2, 1]>,<19,R![-8, 1]>,<23,R![0, 1]>],Snew);
    return Vf;
end function;