
// Make newform 9280.2.a.bc in Magma, downloaded from the LMFDB on 18 January 2026.
// 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_bc();".
// 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_bc();".
// 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, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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, 1], [1, 0], [-3, 1], [-2, 2], [-4, -1], [3, -3], [4, -2], [-4, 1], [-1, 0], [1, 3], [4, -6], [-4, -2], [1, 1], [0, 0], [-3, 3], [6, -3], [10, -3], [0, -4], [0, 0], [-9, 5], [-2, 3], [-4, -2], [-10, -2], [8, -3], [6, 3], [0, -8], [4, 2], [12, -2], [8, 1], [-20, 0], [-6, 12], [-12, -3], [9, -1], [-6, -6], [0, -2], [-10, 8], [-12, -4], [5, 1], [-12, -3], [-6, 3], [8, 2], [-9, -3], [-4, -9], [-3, 3], [10, -6], [14, 6], [-4, 5], [10, -4], [1, -3], [14, 4], [-6, 0], [-2, 1], [-4, -2], [-16, 10], [4, -4], [1, 5], [-4, -4], [-14, 0], [-1, 7], [-8, 10], [-12, -6], [-8, 4], [-18, -3], [-18, 8], [2, 10], [18, 2], [-3, -1], [0, 6], [-8, 0], [8, 4], [22, 5], [20, 2], [25, 3], [-26, -2], [-7, 1], [18, 0], [-5, -3], [26, 1], [0, -4], [7, -7], [-2, 12], [20, -14], [-6, -4], [10, 12], [-16, 19], [-20, 2], [-10, 0], [9, -15], [-4, -16], [-3, -15], [-9, 9], [-4, -13], [24, -12], [2, -9], [12, 0], [12, -6], [7, -1], [-28, 6], [17, -7], [-8, -14], [12, -15], [24, -3], [12, 12], [10, 7], [27, 5], [-10, 4], [6, 6], [-27, 3], [-42, 2], [-8, 12], [17, 11], [-3, 3], [6, -4], [-18, 16], [-12, 6], [-14, -2], [0, -12], [18, 6], [-30, 6], [-2, 6], [8, 18], [-28, 16], [0, 18], [-31, 3], [14, -2], [2, -10], [-4, -14], [-30, -2], [-24, 16], [4, 4], [-32, -4], [-4, -16], [-20, 18], [-15, 21], [22, 10], [22, 8], [4, 10], [-24, 24], [20, -20], [-5, 13], [22, 14], [10, 12], [-25, 7], [-7, -19], [20, 4], [-4, 14], [2, -14], [24, 2], [-12, 3], [-36, 1], [-8, -10], [16, -2], [-24, -12], [39, -1], [-14, -19], [-8, -12], [-19, 1], [-28, 6], [-36, 12], [4, 17], [-10, -8], [2, -22], [-12, -6], [-18, -6], [10, 20], [-30, -2], [-10, -22], [19, -5], [6, -18], [0, 12], [4, -6], [-6, 18], [-32, 7], [0, 4], [-37, -5], [-9, 5], [30, -15], [20, -4], [26, -4], [38, -10], [-6, -6], [-25, -7], [-28, 4], [-29, 23], [-28, 16], [-12, 10], [12, -13], [22, -26], [36, -12], [28, -8], [13, -7], [56, -6], [-9, 3], [-40, 7], [-12, 30], [-38, 22], [-16, 11], [26, -8], [-32, 5], [-18, -9], [32, 8], [50, 2], [-40, -6], [-32, -4], [-7, 7], [-50, 9], [30, -30], [2, 13], [50, -6], [42, -4], [-22, -8], [-14, -6], [20, 4], [46, -13], [-26, 13], [-32, -3], [4, 2], [12, 0], [38, -26], [-27, 1], [54, -2], [-4, 4], [-6, -8], [-2, 2], [-26, -3], [18, -24], [-26, 23], [34, 0], [-9, -15], [28, -18], [58, -5], [-36, 16], [34, -16], [8, 18], [-48, -6], [-52, 12], [-26, 14], [15, -27], [4, -4], [-69, 3], [-6, -19], [11, 23], [-6, -5], [-40, 4], [48, -18], [49, 9], [-33, 21], [59, -3], [10, -16], [14, -16], [6, -24], [20, -20], [-2, -8], [52, 2], [-15, -21], [-10, 5], [-24, -1], [-32, -13], [-8, 16], [6, 28], [-4, 7], [64, -6], [-40, 8], [-26, 8], [-18, 2], [-35, 5], [-24, 30], [44, -3], [-24, 21], [-56, 18], [30, -16], [20, -12], [35, 5], [24, 5], [-18, 22], [21, -15], [-18, 22], [22, -14], [12, 0], [-32, -13], [4, 0], [-32, 29], [-40, 26], [12, -7], [15, 9], [28, 25], [-18, -24], [-4, 8], [-6, 12], [-2, 20], [-18, -18], [50, -14], [-56, -10], [-24, -8], [2, -8], [52, -12], [3, -3], [-10, 31], [-29, 7], [7, -23], [-57, 9], [-1, -19], [44, -11], [16, -14], [32, 6], [-20, 8], [-4, -22], [-72, -6], [2, 8], [15, 3], [-31, 1], [-43, -11], [-26, 40], [-64, 13], [-21, 47], [2, 16], [0, 36], [-24, 20], [-22, 4], [20, -12], [38, -12], [30, 12], [22, -12], [32, 4], [14, 6], [1, -17], [38, -24], [-60, 24], [38, -32], [-26, 24], [48, -21], [10, -12], [13, -7], [-54, -10], [30, -30], [52, -12], [-46, 16], [-32, -20], [20, 17], [-46, 8], [-34, -8], [13, -7], [26, 20], [18, 12], [48, -6], [50, 5], [52, -5], [22, -49], [-36, 24], [18, 17], [-32, 4], [66, -15], [61, -3], [-26, 12], [48, -15], [-43, 31], [-38, 5], [-48, 3], [-21, 3], [-22, 32], [66, 9], [12, 30], [-13, 37], [-76, -3], [-80, 7], [-20, 32], [-32, 21], [28, 16], [-36, 0], [-10, 9], [-37, -5], [18, 21], [52, 6], [4, -9], [-18, 12], [-14, -7], [-36, -7], [-18, 21], [-14, -2], [-50, 35], [-34, 4], [2, -10], [-54, 12], [-64, 24], [51, 3], [-1, 17], [11, 5], [16, -17], [-32, 8], [-66, 8], [6, -12], [30, -30], [-20, 37], [3, -27], [-22, 24], [28, -6], [28, -40], [48, -28], [-55, 25], [50, 2], [-45, -15], [-6, -14], [-4, 28], [-2, 26], [16, 18], [-12, -17], [-20, 8], [-7, -3], [-2, -13], [-6, 38], [0, 39], [0, 18], [43, -11], [44, -15], [-70, -8], [-42, 45], [-8, -12], [3, 45], [-58, 22], [-48, 18], [4, 24], [36, -30], [22, -4], [44, -24], [12, -15], [-12, -30], [-48, 12], [-4, 24], [-19, 19]];
    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_bc();".
// 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_bc(:prec:=2)
    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_bc();".
// 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_bc( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9280_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-3, -1, 1]>,<7,R![3, 5, 1]>,<11,R![-12, 2, 1]>,<13,R![17, 9, 1]>,<19,R![-4, -6, 1]>],Snew);
    return Vf;
end function;