
// Make newform 9280.2.a.cf in Magma, downloaded from the LMFDB on 20 October 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_9280_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_9280_2_a_cf();".
// 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_9280_2_a_cf();".
// 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, 4, -4, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [-3, 0, 1, 0], [-2, -2, 1, 0], [2, -3, -2, 1]];
    Rf_basisdens := [1, 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_9280_a();"
function MakeCharacter_9280_a()
    N := 9280;
    order := 1;
    char_gens := [4351, 581, 1857, 321];
    v := [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_9280_a_Hecke( : Kf := Kf)
    return MakeCharacter_9280_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, 0], [0, 0, 0, -1], [1, 0, 0, 0], [0, 0, 0, -1], [-1, -1, 1, -1], [-1, -1, -1, 0], [-2, -1, 0, -2], [1, 1, 1, -1], [-4, 0, 2, -1], [-1, 0, 0, 0], [-3, 1, 1, -1], [4, 1, -2, 0], [-1, 1, -1, -2], [4, 0, 0, 1], [2, 0, 2, 1], [-1, 1, -1, 2], [-4, 0, 2, -2], [-5, -1, 3, -2], [2, -2, 4, 1], [2, 0, 0, 2], [1, 0, 1, -4], [-1, -3, 1, -1], [6, 2, -2, 1], [0, 0, 2, -2], [-1, 4, -1, 2], [-2, 2, 0, 0], [-4, 0, -2, 3], [6, -4, 2, -1], [-9, 3, -1, 0], [7, 2, -5, 0], [-10, -2, 2, -1], [-1, 5, -5, -3], [10, -1, 4, 2], [4, 2, 4, 2], [2, 0, -4, 2], [-6, -2, 0, 2], [1, 4, 1, 0], [0, 2, 4, -3], [-14, 0, 0, -3], [7, 3, -1, 2], [8, 2, 0, 0], [1, -3, 1, -6], [-5, 1, -3, -5], [-10, -3, 8, -4], [8, 0, 6, 2], [-14, 0, -4, -2], [11, -5, 1, 1], [-18, 0, 4, -1], [12, 0, -2, 1], [6, 0, 4, 4], [0, -2, 2, 2], [2, 2, 0, -2], [-4, -2, -6, 0], [3, 1, 5, 3], [5, 5, -7, 2], [-2, 4, 6, 1], [-13, -3, 3, 2], [13, -3, -1, 1], [1, -1, 1, -10], [-7, 1, -3, 4], [8, 0, -6, 3], [1, 0, -3, -2], [10, -2, 2, -3], [9, 5, 5, 3], [7, -5, 7, 4], [2, 3, -4, -8], [-5, 3, -1, 5], [1, -2, 1, 10], [-10, 4, -2, -1], [6, -4, 0, -8], [11, 5, -1, 6], [23, 3, -5, 3], [-4, -6, 0, -7], [4, -4, -2, -4], [5, -9, -1, -5], [-12, -2, -8, -3], [7, 1, -5, 2], [-4, 0, 6, -10], [3, -1, 3, 10], [3, 7, -5, 6], [-8, -4, 6, -4], [-1, 3, -5, -4], [-6, -6, -2, -4], [-8, 1, 2, 4], [-4, 4, 6, -4], [-16, 2, -4, 3], [-7, 1, -7, -2], [1, -1, -7, -2], [8, 4, 6, 0], [-14, 0, 0, 3], [-4, 6, -4, -7], [-9, -1, -7, -7], [8, 2, 0, 11], [-3, -1, 1, 1], [-6, 2, 2, -4], [4, -10, 4, 5], [17, -7, 1, 0], [15, -1, 3, 0], [-6, 4, -4, 7], [28, 4, -6, 6], [4, 4, 4, 3], [6, 4, -8, -2], [12, -8, 0, 1], [3, -5, -5, -4], [2, -10, -2, 4], [12, 3, 6, 4], [6, 8, 0, 9], [-11, 3, -3, -8], [1, -3, 7, -5], [25, -1, -3, 2], [-14, 0, 2, -7], [-24, 2, 2, -4], [25, -2, 5, 2], [-13, -7, -3, 1], [2, 4, -2, 10], [11, 5, -9, -2], [0, 4, 2, 3], [18, 4, -10, 7], [-13, 0, 7, -4], [7, -5, -5, -5], [-7, 1, -11, 4], [-4, 0, 2, -10], [-3, 0, 5, -6], [-16, 2, -6, -1], [12, 6, 2, 12], [21, 1, 1, 4], [-5, 7, 3, 4], [10, -4, 8, 2], [-10, 10, 6, 1], [8, 7, -6, 14], [1, -3, 5, -13], [-10, -2, -2, -7], [5, 3, 5, -3], [29, -2, 1, -2], [22, -4, 4, 8], [-10, -2, 12, -8], [-22, -1, 0, 2], [-24, 0, -4, -5], [-11, -6, 1, -2], [-4, 8, -6, -10], [4, -8, -16, 2], [0, -10, 10, -6], [-16, 0, 0, 5], [22, 2, 6, 11], [3, -7, -5, -6], [19, 5, -7, 1], [-24, 3, -2, 6], [5, -11, -11, -2], [33, 3, -1, -7], [-4, -2, 8, 1], [15, -3, -5, 2], [-25, 5, -5, 4], [16, -4, 4, 1], [16, 6, -4, 7], [22, 6, -6, 7], [11, 1, -1, 13], [-10, -2, -8, -14], [-4, -2, 2, 6], [33, 3, 1, 4], [-14, -8, 0, 0], [-6, 0, 2, -7], [8, 10, -6, 20], [-16, 4, 8, 3], [-25, -5, -1, 15], [-3, 5, 13, 2], [4, -8, -4, -3], [-8, 0, 10, -14], [-13, 8, 3, 0], [-16, 10, 6, 4], [16, 3, 10, 4], [-24, -4, -2, 0], [-2, -4, -12, 4], [-24, 4, 2, 8], [-14, -5, 0, 0], [-34, 0, -8, -2], [7, 7, -1, 12], [-6, 12, 0, 8], [22, -6, 0, -2], [18, -6, 2, 1], [-4, 2, -10, 0], [-6, -8, -10, -5], [-9, 7, -9, -3], [-6, 2, -16, 0], [13, 1, -7, 0], [6, 0, 8, 11], [-22, 0, -12, -4], [14, 7, 16, 4], [20, -8, 8, -7], [-35, 1, -7, 0], [2, -4, 10, 0], [14, 6, -4, -14], [-10, 0, 6, -7], [7, -9, 3, 3], [19, 7, 7, -4], [30, 0, -6, -7], [-24, 8, 2, 2], [-37, -3, 3, -8], [36, 4, -14, 8], [24, -4, -6, 6], [-8, 16, 0, -3], [8, 10, -6, -2], [-6, -2, -2, -10], [-6, 5, -12, 10], [20, 8, -10, -4], [-33, -5, 7, -3], [12, -12, 10, 8], [19, -1, 11, 3], [-20, -6, -10, -9], [-33, -1, -1, -8], [13, -5, -1, 13], [4, 1, 6, 6], [18, 4, -4, -10], [8, 10, 12, -5], [8, -4, -4, -5], [21, 7, 11, -1], [-2, 12, -4, 10], [-18, 0, 2, -5], [-14, 8, 0, 10], [28, -6, 0, -1], [11, 4, 7, -10], [5, -11, 1, 8], [8, -2, -12, -2], [8, -4, -10, 12], [0, 6, -4, 5], [-2, -6, 18, -5], [-21, -11, 7, 2], [-14, 7, 4, -12], [-21, 3, 1, -11], [-16, 2, -8, -19], [22, 4, 0, 12], [-8, 5, 6, 4], [20, 10, -14, 4], [-13, 9, 1, -5], [-1, 1, -1, -12], [20, 10, -4, 17], [12, -10, -20, 7], [-25, -9, -5, -2], [-23, -4, 5, 8], [-18, 0, 2, -18], [-5, 3, 19, -1], [0, -4, -8, -9], [-22, 8, -4, -4], [-24, -6, -8, 5], [-20, -6, 2, 8], [4, 10, 2, 10], [34, 4, -2, -4], [14, 10, -14, 3], [-10, -4, -20, -6], [-12, 8, -14, -8], [-14, 0, 14, -3], [-20, 7, 2, 2], [6, 4, -8, -16], [8, -10, 12, -9], [-19, -3, -11, 16], [6, -11, 0, -4], [14, 6, -10, -2], [-18, 6, -12, -14], [24, 4, 4, 7], [26, 4, -12, 18], [10, 2, 0, 6], [-2, 4, -10, 11], [-4, 12, -8, 15], [-38, 0, 0, -4], [34, 7, 12, -6], [-3, 6, -7, -10], [5, -7, 3, 11], [-26, -4, 20, -18], [10, 4, 8, -10], [-12, -8, -12, -1], [6, 4, -16, 14], [-4, 2, -22, 0], [14, 8, 8, -11], [-20, 16, -6, 0], [-3, 15, -21, -13], [-18, 9, 4, -4], [26, -4, -14, 1], [26, -4, 6, 13], [23, -13, 15, 6], [-5, -15, 11, -2], [-32, 18, 6, 2], [16, -6, 8, -11], [-10, -8, -10, 6], [-2, 2, -6, -21], [-25, 7, 7, -4], [-6, 2, -2, -19], [-47, -1, 1, 9], [15, 0, -17, -6], [14, -15, 12, 10], [10, 0, 6, -2], [16, -6, -10, 8], [-28, 2, -18, -8], [10, 8, 6, -17], [-52, 8, -2, -2], [-3, -17, 19, 7], [-3, 12, -7, 18], [-3, 13, -7, -4], [-21, 7, 7, 9], [5, 7, -19, 2], [-12, 2, 6, 2], [-4, 12, 8, -9], [9, 6, -15, 2], [-13, -3, 3, -4], [-49, -5, 7, -11], [-6, -4, 18, 1], [-21, -3, 23, -11], [11, -9, -13, 12], [-24, -4, 16, 3], [-42, -4, 12, -16], [-12, -2, 2, 4], [20, -8, -2, 14], [10, -2, -12, 15], [-20, 6, 2, 2], [-7, 5, 1, -6], [4, 12, 8, 9], [-24, 0, 4, -13], [-15, 9, -3, 10], [1, 11, -21, 7], [34, 4, 2, -4], [27, -9, -5, 8], [25, -7, -3, -14], [-31, -7, -1, 15], [39, 7, 7, 0], [45, 1, 9, -8], [10, -18, 10, -13], [-24, 6, 2, -6], [-35, 19, 1, -2], [-20, 4, 8, 4], [-28, 0, -4, 1], [-16, 16, 4, -9], [-14, -8, 0, -8], [21, 13, -3, 10], [27, 3, 11, -14], [-38, 4, 6, -12], [34, 8, 6, 19], [29, 13, 9, -3], [-18, -6, -14, -9], [28, 10, -22, 24], [18, -7, 4, 16], [-31, 3, -11, 8], [30, 8, 10, 11], [-29, -6, 7, -2], [42, 6, -4, 0], [-29, -15, 19, -20], [14, -2, -2, 12], [-27, -3, 1, 6], [11, -15, 9, 3], [5, 1, 1, 4], [-18, 4, 22, -17], [21, -7, -13, -1], [-21, 14, 15, 4], [-36, 16, 2, 6], [-31, 19, -11, 4], [-21, 3, -13, 2], [24, -8, -10, -9], [-52, -6, -6, 2], [-21, -2, 3, -12], [17, 7, -19, -1], [2, 10, -10, 14], [0, 3, -10, -6], [-16, 10, -12, 17], [0, 8, -10, -10], [46, 4, 20, 12], [-14, -8, -14, -7], [10, 2, -10, 22], [0, -2, 0, -17], [-8, 1, 10, -14], [-6, -5, 4, 12], [-16, 14, 10, -1], [-15, 15, -7, 8], [57, 9, -15, 9], [48, -6, 8, 6], [-62, -8, -8, -5], [-1, -19, -21, 4], [12, -2, 10, 18], [10, -12, 0, 12], [-41, 5, 3, -11], [29, -15, -5, 5], [-3, 0, -11, 2], [-13, 23, 3, 12], [-27, 9, 9, -8], [-26, -10, 4, 18], [19, -9, -13, 4], [0, 2, 0, -5], [-10, -7, -12, -16], [42, -4, -12, 2], [-16, -12, -2, 7], [-13, 21, -3, 3], [33, -11, 17, 12], [-2, -8, 10, 9], [6, 0, 14, 11], [-17, -5, 15, 2], [47, -11, 3, -8], [7, 1, -5, 15], [6, -14, 2, 9], [9, 15, -5, 7], [1, 1, -3, 6], [-2, -10, 14, -20], [26, 10, 0, 22], [-18, -10, 16, -6], [43, -13, 11, 8], [24, -6, 22, 2], [78, 3, -8, 4], [-44, 12, -8, 15], [56, -4, 6, 4], [-11, 13, 9, 8], [-62, -8, -18, -16], [-4, -6, -10, -18], [6, -16, -8, 12], [32, 8, -4, -3], [-34, -4, -6, 20], [-5, -21, 15, -16], [13, 7, -7, -6], [22, -4, 26, 13], [30, -6, 12, 2], [32, 7, 10, 2], [-18, -6, 12, -30], [-15, -13, 7, 13], [8, -10, -14, 9], [27, -16, -13, 0], [-32, -10, 24, -17], [23, 15, 11, 8], [23, 16, -1, 28], [44, -4, 4, 1], [19, 3, -3, 21], [16, -22, 6, -24], [10, 0, -12, -20], [44, 10, -20, 25], [-9, -13, 7, -26], [54, 10, 12, 22], [25, 5, 5, 25]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9280_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9280_2_a_cf();".
// 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_9280_2_a_cf(:prec:=4)
    chi := MakeCharacter_9280_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_9280_2_a_cf();".
// 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_9280_2_a_cf( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9280_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-4, 16, -8, -2, 1]>,<7,R![-4, 16, -8, -2, 1]>,<11,R![20, -32, -24, 2, 1]>,<13,R![160, -48, -24, 4, 1]>,<19,R![36, 160, -28, -6, 1]>],Snew);
    return Vf;
end function;