
// Make newform 2394.2.o.l in Magma, downloaded from the LMFDB on 30 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_2394_o();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2394_o_Hecke();"
// 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_o_l();".
// 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_o_l();".
// 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 := [16, 4, 5, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [16, -5, 5, -1], [4, 0, 0, 1]];
    Rf_basisdens := [1, 1, 20, 5];
    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_o();"
function MakeCharacter_2394_o()
    N := 2394;
    order := 3;
    char_gens := [533, 1711, 1009];
    v := [3, 3, 2];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2394_o_Hecke();"
function MakeCharacter_2394_o_Hecke( : Kf := false)
    N := 2394;
    order := 3;
    char_gens := [533, 1711, 1009];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [-1, 0, 1, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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]];
    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, 0, -1, 0], [0, 0, 0, 0], [0, -1, 0, 0], [-1, 0, 0, 0], [5, 0, 0, 1], [2, -1, -2, -1], [0, 0, 2, 0], [3, 2, -1, 1], [2, 1, -2, 1], [-4, -4, 4, -4], [-2, 0, 0, 0], [2, 0, 0, -2], [0, 3, -3, 0], [0, -2, 4, 0], [-4, -2, 4, -2], [6, 2, -6, 2], [0, 0, 11, 0], [12, -1, -12, -1], [-1, 3, 1, 3], [0, -5, -2, 0], [0, 1, -3, 0], [0, -4, -4, 0], [1, 0, 0, 0], [6, 6, -6, 6], [0, -1, -5, 0], [-4, -6, 4, -6], [-6, 0, 0, 2], [0, 0, 0, 2], [0, -4, 0, 0], [17, 0, 0, 0], [-6, 3, 6, 3], [0, -4, 5, 0], [1, -6, -1, -6], [13, 1, -13, 1], [0, -6, 6, 0], [-2, 0, 0, -1], [0, 3, 12, 0], [-11, 0, 0, 3], [16, -2, -16, -2], [0, 3, 10, 0], [-9, 0, 0, -7], [-10, -5, 10, -5], [-8, 0, 0, 5], [0, -3, -6, 0], [10, 0, 0, 10], [10, 0, -10, 0], [0, -8, 4, 0], [0, 8, -10, 0], [-7, 0, 0, -12], [4, 0, 0, -5], [0, 2, -21, 0], [-16, 0, 0, -5], [7, -7, -7, -7], [9, 6, -9, 6], [-1, -1, 1, -1], [0, 9, 4, 0], [0, -2, 12, 0], [0, 6, 12, 0], [4, 0, 0, 2], [-9, -3, 9, -3], [0, -2, -1, 0], [-8, 0, 0, 7], [0, 0, 3, 0], [10, 0, 0, 8], [15, 11, -15, 11], [-2, -8, 2, -8], [-3, 0, 0, 5], [0, 10, 5, 0], [0, 3, 7, 0], [-4, 0, 0, 6], [-7, 0, 0, -3], [0, -6, 6, 0], [-14, -6, 14, -6], [-28, 0, 0, 2], [20, 0, 0, 6], [0, 0, -32, 0], [-4, -12, 4, -12], [0, 12, 6, 0], [0, -6, -3, 0], [-1, 3, 1, 3], [-20, 0, 0, -8], [0, 6, 14, 0], [-22, -6, 22, -6], [14, 10, -14, 10], [0, 12, -10, 0], [-7, -9, 7, -9], [-9, 0, 0, -12], [21, 0, 0, 4], [0, -5, 26, 0], [18, 0, 0, -1], [-5, 0, 0, -5], [-20, 2, 20, 2], [-18, 0, 0, -14], [0, 12, 8, 0], [0, -11, 1, 0], [-14, 4, 14, 4], [12, 3, -12, 3], [3, 0, 0, -7], [-12, -4, 12, -4], [20, -4, -20, -4], [-12, -18, 12, -18], [40, 2, -40, 2], [-31, 0, 0, 0], [26, 0, 0, 1], [9, 0, 0, -13], [-25, 0, 0, -9], [0, 0, 12, 0], [7, 7, -7, 7], [10, -3, -10, -3], [-17, 0, 0, 9], [0, 0, 0, -12], [0, -8, -10, 0], [-29, -6, 29, -6], [8, 0, 0, -5], [24, 8, -24, 8], [0, 2, -31, 0], [0, 4, 7, 0], [-18, 0, 0, -12], [20, 0, 0, -6], [12, -2, -12, -2], [-24, -5, 24, -5], [2, 0, 0, 3], [-14, 0, 0, 8], [-12, 0, 0, 8], [0, 0, 0, -9], [0, -2, -28, 0], [6, 6, -6, 6], [0, -4, 6, 0], [0, -16, 24, 0], [-14, 0, 0, -9], [0, 15, 5, 0], [0, 15, 14, 0], [26, 10, -26, 10], [0, 20, 2, 0], [27, 0, 0, -1], [-2, -8, 2, -8], [16, 21, -16, 21], [-27, 0, 0, -6], [-38, 0, 0, -9], [11, 0, 0, 16], [-8, -12, 8, -12], [26, 2, -26, 2], [-2, 7, 2, 7], [-7, -3, 7, -3], [-40, 0, 0, 1], [0, 2, 8, 0], [0, 20, -2, 0], [0, 5, -23, 0], [43, -5, -43, -5], [18, 0, 0, -2], [0, -12, 0, 0], [-27, 0, 0, -1], [-37, 5, 37, 5], [16, -6, -16, -6], [0, -13, -25, 0], [-22, 0, 0, 3], [-6, 0, 0, 7], [0, -1, -47, 0], [9, -13, -9, -13], [26, 11, -26, 11], [0, -2, 4, 0], [0, 1, -43, 0], [0, 3, -10, 0], [0, 8, 17, 0], [15, 0, 0, 10], [0, 6, 32, 0], [0, -13, -4, 0], [-28, 7, 28, 7], [0, -4, 18, 0], [-30, 8, 30, 8], [-1, 0, 0, 19], [0, 4, -42, 0], [0, -10, 2, 0], [-13, 0, 0, -3], [22, -2, -22, -2], [-29, 1, 29, 1], [9, 17, -9, 17], [0, 6, 28, 0], [-20, 0, 0, 0], [0, -13, -10, 0], [2, 3, -2, 3], [32, 0, 0, 3], [32, 2, -32, 2], [0, 3, -21, 0], [-12, 0, 0, -25], [-20, 0, 0, -2], [28, 14, -28, 14], [0, 1, 4, 0], [17, 0, 0, 2], [-26, 0, 0, -18], [-15, 15, 15, 15], [-4, 10, 4, 10], [51, 0, 0, -1], [0, -11, 4, 0], [1, 5, -1, 5], [34, -2, -34, -2], [-35, 6, 35, 6], [0, -16, -20, 0], [1, 0, 0, 1], [-18, 0, 0, -12], [-38, -2, 38, -2], [-32, 6, 32, 6], [0, 3, -38, 0], [0, -17, 9, 0], [0, -5, 45, 0], [42, 15, -42, 15], [-26, 6, 26, 6], [3, -15, -3, -15], [0, -1, 47, 0], [-15, 0, 0, -7], [0, -2, 15, 0], [22, 23, -22, 23], [20, 0, 0, 12], [-27, -10, 27, -10], [-6, 0, 0, -8], [6, -12, -6, -12], [0, 2, 22, 0], [-6, 0, 0, -18], [-22, 0, 0, -5], [0, 2, -22, 0], [18, 0, -18, 0], [-2, 0, 0, -16], [0, 12, -3, 0], [0, 5, -48, 0], [0, -14, 19, 0], [62, -2, -62, -2], [5, 0, 0, -15], [0, -17, -12, 0], [0, -16, 12, 0], [20, 0, 0, 20], [4, 2, -4, 2], [23, 0, -23, 0], [14, 0, 0, 3], [5, 0, 0, -12], [-1, 0, 0, 7], [0, 14, 18, 0], [21, 0, 0, -17], [-30, 0, 0, 12], [0, 3, -43, 0], [22, 14, -22, 14], [0, 7, 3, 0], [-19, 0, 0, -10], [-12, -2, 12, -2], [-26, -2, 26, -2], [0, 2, -54, 0], [-2, 0, 0, -28], [20, 28, -20, 28], [-27, 5, 27, 5], [0, 13, 27, 0], [-40, 5, 40, 5], [8, 0, 0, 2], [0, 8, 6, 0], [-30, 0, 0, -22], [-7, -1, 7, -1], [0, -24, 38, 0], [12, -4, -12, -4], [34, 14, -34, 14], [23, 0, 0, 10], [0, 24, -6, 0], [30, 14, -30, 14], [-6, -15, 6, -15], [0, 25, -25, 0], [0, -3, 60, 0], [0, 14, -8, 0], [-23, 5, 23, 5], [-33, 0, 0, 7], [-16, 0, 0, 4], [41, 0, 0, -13], [-28, -12, 28, -12], [-30, -2, 30, -2], [-22, 0, 0, 4], [-44, 0, 0, 0], [0, 5, -3, 0], [-44, 0, 0, -4], [-6, 10, 6, 10], [0, 2, 26, 0], [11, 0, 0, -9], [0, 6, -24, 0], [21, 0, -21, 0], [-40, 1, 40, 1], [30, 0, 0, 28], [-10, 0, 0, 1], [-36, -30, 36, -30], [-18, 0, 0, -36], [0, 11, 24, 0], [0, 11, 0, 11], [-27, 0, 0, 0], [8, 22, -8, 22], [0, -10, 8, 0], [-30, 0, 0, -22], [2, 0, 0, -18], [16, 0, 0, -17], [-13, -6, 13, -6], [8, 0, 0, 5], [0, 2, -8, 0], [-10, 0, 0, -13], [6, -14, -6, -14], [0, -7, 32, 0], [0, -7, -64, 0], [-19, 0, 0, -39], [0, -17, -1, 0], [0, -28, 38, 0], [-12, -23, 12, -23]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2394_o_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2394_2_o_l();".
// 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_o_l(:prec:=4)
    chi := MakeCharacter_2394_o();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_o_l();".
// 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_o_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2394_o();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![16, -4, 5, 1, 1]>,<11,R![16, -9, 1]>,<13,R![4, -10, 23, -5, 1]>,<17,R![4, -2, 1]>],Snew);
    return Vf;
end function;