
// Make newform 2394.2.a.x in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2394_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_2394_2_a_x();".
// 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_2394_2_a_x();".
// 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
        poly := [-3, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [0, 2]];
    Rf_basisdens := [1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_2394_a();"
function MakeCharacter_2394_a()
    N := 2394;
    order := 1;
    char_gens := [533, 1711, 1009];
    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_2394_a_Hecke(Kf)
    return MakeCharacter_2394_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], [0, 0], [0, 1], [-1, 0], [2, 1], [0, -1], [-4, -1], [1, 0], [2, 1], [6, 0], [0, 2], [10, 0], [2, 0], [0, -2], [-4, -2], [-2, 0], [-4, 0], [2, -2], [6, 1], [4, -2], [10, 0], [-2, -3], [-8, 0], [2, 0], [8, -1], [-8, 1], [0, -2], [4, 0], [2, 0], [2, -4], [-14, -1], [12, -2], [-6, 2], [-4, 4], [10, -2], [-2, 1], [6, 4], [8, -2], [12, 2], [-6, 2], [-4, 4], [-12, 1], [2, 1], [-22, 0], [2, -2], [-4, -6], [-2, 1], [-12, 0], [-8, -2], [10, -2], [-6, -6], [-6, -7], [4, 5], [-12, 2], [-18, -2], [6, -5], [10, 2], [-16, -4], [22, 0], [26, 0], [-20, 0], [-14, 2], [0, 6], [-16, 4], [-6, 0], [6, -4], [18, -1], [-10, 6], [-10, -5], [22, 4], [12, -5], [-18, -5], [8, 0], [-10, 6], [2, -1], [-8, 0], [-2, -8], [6, 4], [-6, -4], [0, -9], [-12, -2], [-2, 6], [28, 2], [24, -5], [16, -6], [-6, 5], [-6, 4], [26, -4], [0, -11], [24, 0], [-8, -8], [0, -8], [-22, -5], [30, 3], [-16, -2], [8, 4], [10, 10], [2, -8], [-8, 10], [-34, 0], [14, -7], [2, -2], [-12, -4], [18, 8], [-4, 12], [10, -4], [8, 0], [-16, 1], [-20, 2], [24, -5], [-4, -4], [-10, -4], [-6, -2], [-8, 10], [-4, 6], [2, 4], [-8, -2], [0, 0], [6, 8], [-28, 4], [28, 5], [-10, -2], [6, 4], [-24, -2], [-20, 0], [14, -8], [14, 0], [-16, 0], [8, 0], [-2, 0], [4, -4], [0, -8], [10, 3], [-10, 4], [-8, 1], [-6, -4], [18, -6], [-32, 6], [10, 10], [-18, 0], [40, -2], [42, 2], [-8, -4], [12, 0], [-16, 3], [12, -2], [42, -2], [-6, -12], [-8, -6], [-32, 4], [2, 0], [-12, -9], [0, 14], [0, 8], [-38, -1], [12, -10], [-8, -12], [-36, 3], [14, 2], [26, -6], [-10, 11], [18, 4], [-12, -2], [-24, 10], [-6, 4], [12, 6], [46, -3], [-34, -4], [10, 4], [48, 1], [-24, 6], [4, 1], [30, -5], [34, 4], [4, 8], [-24, 1], [-12, 0], [22, -12], [12, 12], [10, 2], [4, 6], [8, 10], [10, -8], [-2, 2], [-2, 7], [24, -3], [14, 14], [52, 0], [22, 6], [52, -2], [12, 13], [-18, 11], [-2, -7], [30, -4], [-18, 3], [18, 8], [10, -12], [-2, 4], [28, -1], [-24, 8], [6, 8], [4, 0], [12, 1], [4, 1], [48, 0], [-24, -3], [8, 4], [-48, -2], [-42, 4], [32, -10], [2, -12], [16, 1], [24, 4], [-8, -6], [-4, -18], [12, 1], [8, 4], [42, -8], [20, 2], [-2, 12], [-46, 4], [4, -4], [-22, 4], [8, 0], [-8, 10], [14, 12], [30, -6], [-32, 4], [0, -6], [6, -5], [-34, 8], [16, -14], [2, 3], [18, -4], [12, 12], [-28, -2], [-22, -4], [18, 2], [2, -3], [8, 4], [-16, -6], [4, 8], [4, 2], [-66, 2], [38, 10], [-8, 12], [8, -8], [-48, 6], [-46, 3], [6, -17], [-2, -16], [-24, -3], [18, 13], [28, -7], [-20, -1], [-30, 13], [-18, -16], [44, 0], [30, 8], [-34, -6], [-14, 11], [0, 6], [30, 12], [-32, -9], [-16, -3], [44, 4], [-26, -8], [24, 9], [30, 1], [-18, 0], [20, 17], [18, -5], [-38, -8], [28, 6], [-16, 7], [-4, -6], [2, -15], [-10, 6], [2, -4], [8, -12], [-12, -14], [-16, 20], [48, 0], [24, -9], [-68, 0], [-18, 3], [2, 4], [-26, -12], [-44, 10], [-50, 6], [-30, 6], [-8, 8], [10, -16], [12, -12], [-2, 2], [-12, -1], [16, -6], [-8, 1], [-76, 0], [-16, 6], [-26, 6], [22, 16], [36, 2], [8, 6], [16, 2], [-2, -2], [32, 14], [-40, -1], [66, 5], [-26, -12], [-20, 6], [-2, 4], [18, 12], [-2, 5], [-30, -11], [16, -13], [-56, 4], [12, -6], [-2, 6], [10, 2], [20, -4], [-14, 16], [30, 12], [30, 1], [-10, 8], [28, 1], [-62, -5], [-32, -6], [-36, -2], [-18, 8], [46, -8], [62, 0], [-24, 16], [-12, 14], [24, -2], [0, 8], [-6, -4], [-70, 0], [54, -10], [40, -4], [-26, -10], [10, 2], [6, 4], [-6, 7], [-50, -8], [60, 8], [22, 0], [14, 17], [-60, 2], [8, 1], [4, -4], [10, 16], [-54, -2], [12, -8], [-50, -6], [42, 4], [-36, -6], [12, 14], [-10, 20], [-60, 6], [70, 0], [-72, -3], [12, 2], [-8, 14], [20, -8], [30, 10], [12, 15], [-46, 3], [42, 4], [-46, 13], [-4, -12], [38, -1], [38, 0], [24, 20], [30, 12], [-64, -6], [-22, 9], [-22, 20], [-16, -3], [-60, 1], [30, 4], [-82, 0], [-60, 2], [-26, 12], [-4, -20], [0, -4], [44, 6], [-30, 10], [-8, -18], [60, -2], [-18, -14], [-42, 12], [30, -13], [-16, -2], [-24, 8], [-50, 2], [-16, -6], [-82, 2], [-46, -5], [30, 0], [32, 7], [-50, -4], [-26, -19], [-74, -2], [26, 14], [32, 12], [30, -8], [66, 4], [32, 2], [8, -12], [56, -9], [-34, 16], [36, 4], [8, 18], [90, -4], [-28, -5], [-48, 0], [44, 8], [28, -9], [40, 0], [30, 0], [-68, -3], [62, -1], [20, -20], [-56, -1], [-54, 10], [36, 4], [34, -10], [40, -10], [12, -18]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2394_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2394_2_a_x();".
// 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_2394_2_a_x(:prec:=2)
    chi := MakeCharacter_2394_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_2394_2_a_x();".
// 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_2394_2_a_x( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2394_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-12, 0, 1]>,<11,R![-8, -4, 1]>,<13,R![-12, 0, 1]>,<17,R![4, 8, 1]>],Snew);
    return Vf;
end function;