
// Make newform 3360.2.a.s 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_s();".
// 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_s();".
// 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], [6], [-2], [-4], [0], [-6], [8], [6], [6], [-8], [4], [-2], [0], [10], [8], [0], [-10], [-4], [-4], [-10], [-18], [-6], [8], [20], [6], [-10], [16], [0], [6], [12], [18], [4], [14], [24], [-4], [10], [-4], [-14], [-8], [-14], [-10], [8], [20], [8], [12], [10], [6], [-24], [-6], [8], [-18], [24], [-14], [0], [22], [-6], [-4], [-30], [-12], [24], [-34], [-10], [-12], [34], [-12], [-14], [-18], [-24], [-24], [22], [-4], [36], [18], [-26], [-30], [10], [-16], [38], [-16], [-26], [-16], [-12], [-6], [18], [-30], [-32], [28], [-24], [-8], [-36], [-28], [-36], [-30], [14], [36], [-2], [8], [-18], [-28], [26], [-12], [30], [-20], [-2], [0], [-30], [-16], [6], [6], [-12], [12], [-38], [28], [-28], [22], [-12], [-14], [10], [18], [36], [-28], [42], [-18], [24], [32], [14], [4], [40], [20], [-10], [54], [-46], [-30], [52], [-38], [-6], [-12], [-30], [-16], [-44], [-30], [-24], [46], [6], [-44], [24], [46], [6], [-32], [-28], [16], [0], [12], [30], [14], [-46], [12], [-10], [-16], [-40], [-42], [-28], [-52], [-10], [-30], [34], [60], [-38], [-24], [10], [0], [14], [20], [-30], [40], [34], [-32], [-40], [6], [-42], [56], [42], [-34], [52], [-38], [-24], [-58], [44], [28], [-62], [52], [6], [18], [46], [-2], [60], [18], [-56], [46], [18], [0], [-54], [-24], [36], [34], [60], [-38], [-54], [16], [-28], [0], [10], [-8], [70], [16], [54], [-50], [56], [50], [8], [-4], [22], [54], [-56], [-8], [36], [-66], [-44], [-4], [66], [60], [-28], [-30], [62], [60], [-16], [12], [-36], [-32], [38], [30], [-24], [-16], [-40], [4], [-24], [-34], [22], [56], [50], [-6], [4], [30], [-4], [-30], [46], [-56], [-44], [-46], [-74], [-34], [44], [66], [-50], [-64], [30], [-54], [32], [-50], [-20], [-42], [56], [36], [-26], [-22], [48], [28], [68], [52], [-38], [-20], [80], [-70], [54], [-40], [-18], [-14], [-52], [86], [-32], [-2], [50], [40], [10], [-16], [-52], [-10], [-42], [-28], [-12], [12], [34], [60], [-22], [-80], [6], [4], [-30], [-62], [-24], [-8], [-70], [0], [24], [22], [-38], [-4], [-46], [82], [40], [-18], [-54], [60], [-44], [-80], [6], [-50], [-74], [16], [28], [-12], [28], [14], [-34], [10], [64], [-10], [-18], [-6], [-60], [94], [-52], [26], [-32], [-88], [18], [-52], [82], [10], [8], [14], [38], [48], [88], [-2], [-40], [-26], [86], [8], [76], [-52], [-14], [-62], [-16], [10], [-68], [-76], [40], [-70], [-48], [-34], [32], [-24], [-2], [22], [46], [50], [-26], [-16], [-2], [-20], [4], [-20], [38], [-24], [20], [-14], [-70], [84], [-44], [96], [74], [24], [78], [-76], [74], [98], [14], [-48], [38], [-38], [48], [78], [-54], [-4], [24], [-42], [-90], [-52], [-68], [-38], [82], [64], [80], [30], [-4], [90], [-10], [16], [-56], [-34], [42], [44], [50], [-28], [-72]];
    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_s();".
// 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_s(: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_s();".
// 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_s( : 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![-6, 1]>,<17,R![2, 1]>,<19,R![4, 1]>,<23,R![0, 1]>],Snew);
    return Vf;
end function;