
// Make newform 4368.2.a.bn in Magma, downloaded from the LMFDB on 27 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_4368_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_4368_2_a_bn();".
// 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_4368_2_a_bn();".
// 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 := [-5, -8, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 2, 0], [-5, -1, 1]];
    Rf_basisdens := [1, 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_4368_a();"
function MakeCharacter_4368_a()
    N := 4368;
    order := 1;
    char_gens := [3823, 1093, 1457, 1249, 2017];
    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_4368_a_Hecke( : Kf := Kf)
    return MakeCharacter_4368_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]];
    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, 0], [-1, 0, 0], [0, 0, 1], [1, 0, 0], [0, -1, 0], [1, 0, 0], [4, 0, 0], [-2, 1, 1], [0, 1, -1], [4, -1, -1], [-2, -1, 1], [2, 0, 2], [-2, 1, 2], [-6, -1, 1], [0, 0, 3], [-4, 1, -1], [-2, 1, 0], [2, 0, 0], [0, 0, 2], [8, 1, -2], [4, 1, -3], [2, 1, -3], [8, 0, -1], [0, 2, -1], [0, -1, 1], [8, 0, -2], [-4, 0, 0], [2, -2, 2], [-6, -2, 0], [-4, 1, -3], [4, -2, -4], [8, 2, 0], [8, -1, -2], [-4, 2, 2], [-8, 1, 0], [0, 0, 4], [2, 2, -2], [-4, 2, 0], [4, 2, 1], [4, -2, -4], [8, -1, 3], [2, 0, -4], [10, 0, 2], [2, -2, 4], [-4, -1, 0], [0, 0, -4], [-6, 5, 1], [14, -1, 3], [6, -3, -4], [-6, -2, -4], [-4, -1, -1], [12, 1, -2], [20, 1, 1], [8, 0, 2], [16, 0, 2], [4, 3, 1], [8, 2, 0], [4, -2, 0], [0, 3, 1], [-12, 1, -4], [-4, 0, -4], [8, 2, 5], [-2, 1, 3], [-4, 0, -2], [-2, 0, 8], [16, -3, -2], [0, -2, -2], [-8, 5, 1], [2, 0, -6], [-12, 5, 1], [2, -3, 2], [24, 1, -2], [0, 2, 2], [14, 2, 4], [24, 2, -2], [6, 1, 4], [-10, 0, 0], [8, -1, -7], [4, 1, 2], [0, -5, -3], [-8, -4, 2], [6, -6, -2], [16, 1, 4], [-6, -2, -2], [-8, -2, -10], [12, 1, 7], [0, -3, -6], [18, 0, -6], [10, 3, -6], [12, -4, 2], [-16, 0, -6], [-8, -4, 1], [28, 2, 2], [-2, 6, -2], [-4, -4, -4], [-16, 0, 2], [0, 4, 1], [0, 2, -4], [-4, 2, -6], [-10, -2, 6], [-14, -3, 3], [28, -5, -2], [-32, 0, 0], [12, 3, -1], [-10, -3, -7], [-18, 0, 0], [12, 4, -5], [0, 0, 5], [28, 1, 3], [6, 0, 8], [-16, -2, -6], [18, -2, -6], [4, -7, -4], [-8, -2, -8], [-20, 0, -2], [-12, 3, 5], [-4, 0, -4], [-16, 2, 6], [10, 2, 4], [-24, -3, -3], [-28, -3, 5], [20, 3, 1], [-4, -2, 6], [-16, -3, 6], [-14, -5, -1], [-12, -1, 5], [10, 6, -4], [0, 0, 10], [-4, 2, -6], [-32, 1, -3], [8, -8, -2], [8, -7, 2], [-2, 3, 1], [-28, 5, 1], [8, -2, -3], [-12, -1, 3], [22, -3, -2], [14, -3, -7], [12, 2, -10], [-4, -3, 1], [-28, 4, 0], [8, -3, -4], [-20, 4, -4], [12, 5, -2], [-18, -4, -4], [-6, 5, -4], [28, 1, -5], [8, 8, -6], [16, 6, -2], [0, -5, -2], [-30, 2, -2], [-12, 4, 2], [-4, 6, -4], [-4, -4, 4], [2, 5, -3], [12, -3, -3], [-24, -6, 4], [-16, 6, 1], [-14, 6, 2], [8, 0, 3], [-4, -3, -12], [-28, -3, 9], [20, 8, -6], [-32, -2, -6], [24, 3, 0], [20, 4, 9], [32, 0, 4], [10, 6, -6], [-10, 0, -4], [-20, -2, -4], [-28, 1, 0], [10, 4, 0], [24, 3, -3], [-6, 6, 6], [20, 0, -8], [-36, 0, 0], [-20, -6, 8], [-12, 1, 4], [16, -6, -6], [-38, 0, 4], [-16, 2, 4], [-8, 2, -6], [10, -2, 8], [24, -4, 1], [-8, -5, -12], [-4, -2, 14], [16, -3, -1], [-14, -5, -7], [14, -4, -4], [-8, 6, 9], [-14, 6, 6], [-24, -5, -4], [18, -5, 1], [0, -6, -3], [-8, 1, 1], [8, -2, -4], [10, -2, 8], [-8, -7, 1], [-16, -2, -17], [24, -6, -6], [-16, -3, -2], [-4, 0, -4], [32, -3, 1], [26, 2, -2], [20, 6, -1], [32, 4, 4], [18, 5, -11], [0, 1, 7], [-16, -7, 6], [0, -2, -10], [4, -1, -1], [4, -8, -8], [-16, -6, 12], [14, -5, -8], [4, -2, -5], [-16, -3, 7], [22, -3, 1], [-20, 0, -4], [28, -3, 2], [-16, 3, -10], [12, -5, -7], [-20, -2, -8], [-40, -3, -2], [-4, 2, -2], [20, 0, -2], [12, 5, 19], [24, -4, -4], [20, 3, -3], [-20, -2, -14], [56, -3, 0], [54, 4, -4], [8, 8, 8], [-32, 0, 0], [-14, -8, 4], [-44, -2, 2], [12, 0, -9], [-26, -2, 8], [-4, 7, -6], [4, 1, 1], [-28, 6, 2], [-40, -3, 2], [-16, 2, -10], [40, 0, 0], [-34, 10, 4], [16, -10, 3], [16, 2, 4], [-42, 7, 3], [-20, 0, 5], [-36, 6, 4], [4, 3, -3], [22, -4, 6], [-30, 1, 10], [0, 5, 0], [10, 0, -8], [-16, -4, 10], [28, 5, -6], [-12, -1, -1], [24, 2, -8], [24, 8, 4], [-8, -11, 9], [-14, -11, 5], [-36, -3, -3], [-20, 11, 5], [54, -4, 0], [-2, 7, 2], [0, 2, 10], [-36, 7, 4], [0, 0, 15], [0, 0, -6], [2, -2, 4], [-6, 4, 0], [-20, -8, 4], [-10, 6, -4], [-30, -3, 3], [-10, -6, -6], [12, -2, 0], [-16, 7, 2], [34, 8, -6], [-2, -4, -2], [28, 2, 0], [-4, -10, 2], [-20, -8, 2], [-8, -4, -20], [40, -1, -7], [-8, -10, 0], [8, 3, 15], [-4, 3, -9], [4, -5, -6], [24, 4, 0], [-36, 2, -4], [-12, 5, 3], [36, 4, 4], [48, -5, 4], [-12, -4, 1], [4, -5, -7], [-56, 0, -8], [-32, -2, -6], [20, -4, -12], [8, 4, -20], [6, 1, 21], [-22, 12, 8], [-28, -7, 6], [-6, -13, 5], [-4, 7, -1], [50, 5, -5], [22, -2, -12], [-6, 8, 6], [14, 2, -10], [-24, 5, -4], [28, -13, -1], [28, 2, 4], [24, 3, -12], [-30, 10, 8], [14, 3, 3], [-28, -7, 6], [-10, 6, -14], [-16, -4, 1], [-28, -7, 8], [-12, 1, 3], [-36, 3, 9], [-8, 12, 0], [-10, -4, 2], [-4, 2, 18], [0, 4, -12], [-24, -7, -6], [-58, 2, -2], [12, -6, -8], [-10, 9, -9], [-6, 0, 6], [50, 2, 0], [-38, 2, 4], [-20, 15, 11], [4, 2, 6], [-52, 0, 3], [16, 4, 2], [8, 10, -3], [-30, -2, 14], [-34, 5, 10], [42, 0, -4], [-24, -2, 10], [26, 6, 14], [-22, 6, 0], [18, -7, -2], [48, 0, -8], [4, 1, 6], [14, -4, 6], [38, -8, -4], [52, 0, -4], [0, -2, -15], [-28, -6, 2], [-38, -1, 5], [-2, 4, -20], [-4, 7, -10], [-44, -6, 6], [-2, 10, 12], [28, 8, -10], [68, 4, -1], [-28, 2, 15], [6, 6, 16], [0, 5, 8], [6, 0, -20], [44, 2, -6], [52, 5, -1], [24, -5, -2], [52, 2, -10], [48, -3, 3], [48, 0, 5], [44, 4, 18], [16, 1, 19], [46, -10, -2], [-44, 10, 2], [-24, -3, 18], [4, -3, -5], [0, -8, 12], [-4, 5, 15], [6, -9, 8], [-42, 12, 6], [22, 12, -12], [12, 0, 0], [-34, -8, -8], [0, -6, -15], [-56, 1, -12], [8, 0, 0], [16, -7, -2], [14, 5, 11], [8, -6, -1], [-8, -8, 20], [50, -2, -2], [-12, 10, -12], [16, 2, 0], [-10, 12, -4], [16, 0, 7], [20, -15, -10], [28, 0, 8], [8, -7, 6], [0, 5, 9], [-16, 4, 0], [-32, 10, -2], [2, -13, 11], [4, 1, 6], [36, 1, 11], [20, 1, -1], [-36, 4, 2], [56, 2, 15], [32, 4, -19], [-40, 16, 8], [-18, -6, 6], [16, -3, 10], [26, -5, -5], [-52, 0, -1], [-58, 6, 2], [36, 7, -7], [36, 5, -7], [-20, 12, 16], [-8, 1, -5], [-20, -4, 16], [-44, -7, 18], [-8, -4, -16], [14, -5, -10], [16, -6, 0], [28, 7, 3], [6, -2, 24], [-24, 1, -2], [56, 2, -12], [26, -10, -16], [40, -2, 9], [4, -7, 7], [12, 1, -2], [-18, -3, -13], [24, -2, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4368_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4368_2_a_bn();".
// 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_4368_2_a_bn(:prec:=3)
    chi := MakeCharacter_4368_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_4368_2_a_bn();".
// 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_4368_2_a_bn( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4368_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![20, -14, -1, 1]>,<11,R![40, -32, 0, 1]>,<17,R![-4, 1]>],Snew);
    return Vf;
end function;