
// Make newform 3360.2.a.n in Magma, downloaded from the LMFDB on 19 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_n();".
// 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_n();".
// 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], [0], [-4], [2], [8], [10], [2], [4], [4], [10], [4], [-2], [4], [0], [-6], [8], [-4], [-6], [10], [18], [-16], [-4], [-2], [6], [8], [-12], [-2], [8], [18], [8], [18], [-20], [-4], [18], [-12], [6], [0], [26], [-6], [-8], [8], [0], [-4], [-2], [6], [-24], [2], [20], [-10], [-4], [-6], [-16], [-22], [18], [28], [26], [28], [0], [10], [2], [0], [-6], [-12], [14], [-18], [-16], [8], [18], [16], [-4], [-30], [26], [10], [-38], [12], [30], [8], [-14], [-24], [-36], [-6], [10], [34], [-24], [-12], [24], [0], [-28], [-8], [44], [-22], [-22], [28], [-18], [28], [-30], [-44], [-30], [32], [10], [36], [6], [48], [26], [0], [26], [22], [8], [-16], [2], [-4], [-12], [34], [12], [-42], [-6], [-6], [-12], [8], [18], [-34], [-48], [16], [26], [-8], [4], [0], [-38], [-6], [-30], [42], [20], [2], [-38], [16], [-38], [56], [-12], [-18], [-16], [-30], [54], [-8], [-36], [10], [34], [-52], [-20], [44], [-8], [-40], [-30], [2], [2], [-28], [-34], [-56], [-12], [14], [60], [-40], [-22], [-14], [-22], [36], [-42], [24], [-30], [-32], [-22], [8], [-30], [-8], [-26], [64], [-12], [58], [-2], [36], [-6], [10], [60], [10], [0], [2], [12], [-8], [-14], [-52], [30], [66], [26], [6], [28], [-54], [56], [-38], [50], [36], [34], [16], [-36], [18], [-56], [10], [-22], [-16], [12], [-8], [58], [40], [18], [60], [-14], [-34], [-8], [18], [-24], [-28], [-50], [-34], [16], [0], [-52], [42], [-72], [-40], [2], [44], [28], [66], [-54], [-44], [64], [60], [24], [-24], [-34], [14], [-40], [-8], [12], [-72], [12], [34], [58], [68], [-22], [34], [76], [22], [20], [10], [42], [0], [20], [46], [10], [70], [32], [26], [66], [12], [-54], [6], [-20], [-38], [8], [-14], [-64], [-12], [-2], [10], [68], [68], [-8], [52], [-2], [-4], [48], [-46], [-70], [8], [42], [10], [36], [-42], [44], [-14], [34], [-40], [42], [20], [28], [10], [66], [16], [4], [32], [-62], [-68], [22], [40], [-30], [36], [42], [-38], [-76], [-4], [-22], [52], [-48], [-46], [82], [-72], [18], [-70], [-40], [-90], [34], [-80], [4], [-52], [26], [70], [-78], [0], [28], [8], [52], [-58], [-66], [-86], [-56], [2], [-42], [50], [-24], [2], [84], [-2], [12], [0], [-22], [16], [74], [18], [0], [-58], [54], [-48], [-36], [-90], [-36], [34], [-78], [12], [-12], [-92], [10], [2], [16], [10], [76], [-56], [52], [-22], [0], [34], [20], [88], [-38], [34], [66], [-46], [-34], [88], [-2], [-56], [36], [-64], [-46], [-92], [-68], [-46], [58], [-12], [52], [24], [-86], [56], [-30], [64], [2], [-90], [-90], [72], [-34], [34], [-88], [98], [-6], [-4], [44], [-6], [-22], [-76], [80], [-46], [-54], [96], [72], [6], [84], [-70], [-22], [12], [12], [-54], [50], [-92], [10], [24], [0]];
    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_n();".
// 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_n(: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_n();".
// 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_n( : 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![0, 1]>,<23,R![4, 1]>],Snew);
    return Vf;
end function;