
// Make newform 1480.2.a.j in Magma, downloaded from the LMFDB on 13 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1480_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_1480_2_a_j();".
// 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_1480_2_a_j();".
// 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 := [-2, 3, 10, -6, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0], [0, 3, -5, -1, 1], [0, -13, 5, 3, -1], [4, 7, -7, -1, 1], [6, 3, -7, -1, 1]];
    Rf_basisdens := [1, 2, 2, 2, 2];
    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_1480_a();"
function MakeCharacter_1480_a()
    N := 1480;
    order := 1;
    char_gens := [1111, 741, 297, 1001];
    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_1480_a_Hecke( : Kf := Kf)
    return MakeCharacter_1480_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, 0, 0, 1], [-1, 0, 0, 0, 0], [0, 0, -1, 0, 0], [2, -1, 0, 0, 0], [-1, 0, 0, -1, 1], [-1, 0, 1, -1, 2], [2, -1, 1, 0, 0], [0, 2, 0, 0, 0], [2, 2, -1, 0, -1], [-2, 1, 2, 0, 1], [-1, 0, 0, 0, 0], [4, -3, 0, 0, 2], [6, 0, 1, -2, -1], [2, -2, 0, -2, -1], [2, -1, 0, -2, 0], [6, -1, -1, 0, -2], [-4, 0, 1, 2, 3], [4, 1, -3, 2, -4], [0, 1, 1, -2, 1], [2, -1, -1, 2, -1], [0, -1, 3, -2, 0], [4, 2, 0, 0, 3], [4, 2, -2, 2, -4], [-4, 0, 1, -2, -1], [3, -1, -1, 1, 0], [4, -4, 0, 0, 2], [4, -1, -3, 2, -2], [-2, 0, 1, 0, -3], [3, 2, 1, -1, -2], [-6, 0, 2, -2, 1], [-2, 3, 3, -4, 2], [-2, 0, -2, 4, -2], [6, -2, 1, -2, -1], [7, -1, -1, -3, -4], [-4, 2, -2, 0, 0], [-4, -3, -2, 0, -4], [6, 2, -1, 2, -1], [0, -2, 2, 0, 0], [-2, 3, 0, 2, 0], [-2, 3, -1, 4, 0], [-7, -3, 1, -1, 0], [6, -1, -2, 4, -5], [-2, 0, -4, 0, -4], [0, -1, 5, -4, -1], [0, -1, 3, 2, -2], [10, -3, 0, 4, -2], [0, -2, -1, 0, 6], [-2, 0, 7, -2, -1], [6, 3, -3, 2, 1], [-2, 0, 2, 4, -2], [6, -1, 2, 0, -3], [-6, 8, -2, 0, -2], [-2, -3, 3, 0, 10], [-3, -6, 0, 1, 3], [12, 0, -7, 0, -2], [7, 0, 2, 3, -1], [8, -3, 1, -2, 5], [-13, 2, -2, -1, -5], [-2, -2, -4, 0, 0], [0, 6, 2, -4, -4], [4, -2, -1, 6, -3], [-8, 8, 0, -4, -5], [-6, -7, -4, 4, -1], [-14, -4, 4, -4, 4], [0, -2, 1, 2, -1], [10, 3, -3, 0, -2], [-2, 3, -5, 6, 3], [0, -2, -2, 4, -2], [3, 4, -2, -1, -1], [4, 0, 7, -2, -7], [0, -5, 5, -6, -1], [-12, 1, -6, -2, -1], [-14, 1, -1, -2, -1], [0, 1, 3, 2, 9], [0, 4, 0, 4, 4], [2, 0, 7, 4, 7], [8, -9, 1, 4, 3], [14, -4, 0, 4, 0], [0, 2, 8, -2, 2], [12, 1, -1, -2, 1], [-8, 2, -2, 6, 0], [-6, -1, 6, 0, -3], [-8, 1, 7, 0, -3], [-22, -1, 2, -4, 3], [4, 8, 0, 0, -7], [8, -2, -2, -2, 4], [-12, -2, -5, 2, 9], [8, 2, 9, -6, -5], [12, 6, -6, 4, -2], [-2, 2, -1, 2, -9], [-8, -3, -7, 6, 6], [4, 2, 0, 4, -10], [12, 0, 0, 0, 0], [-6, 5, -5, 8, 0], [0, 0, -4, -4, -8], [22, 1, 1, -2, 1], [5, 1, 0, 3, -5], [8, 6, -2, 4, -2], [-8, 2, 2, 6, 8], [2, -10, 3, -2, 5], [-1, 8, -4, -1, 5], [2, -4, -1, 2, -3], [-6, -2, -6, -4, -2], [14, -7, 0, 0, 2], [5, 6, 0, 1, -13], [-14, -2, -1, -2, 1], [12, 3, -1, 0, 1], [12, -5, 1, -2, 7], [-5, 2, -5, 3, -4], [0, -4, -2, -4, 2], [12, -3, 4, -8, 6], [-2, 1, 5, -6, 13], [2, -3, -2, 4, 4], [10, 9, 2, 0, 9], [19, -3, -2, 5, -1], [2, 6, -5, 6, -13], [-4, 0, 8, 0, 0], [2, 6, -6, 4, 2], [-8, 1, 3, -6, 11], [8, -8, -1, -2, -3], [-6, 3, 5, -6, 1], [-2, -3, -1, -6, -1], [18, -4, -5, 10, -7], [6, 0, -3, -6, 5], [-4, 6, -6, 2, 0], [8, 8, -3, -2, -5], [-12, 9, 1, 2, 3], [12, 4, -8, 8, -12], [4, -7, -4, 0, 10], [18, 1, -6, 4, -12], [-4, 2, -1, 0, -4], [-8, -1, -5, 2, 5], [10, 4, -8, 8, -8], [8, 3, 2, 0, 4], [-10, 0, 6, 4, 6], [-10, -3, 4, -10, 8], [-4, 6, -2, -4, -5], [19, -2, 4, 7, 1], [-6, -4, -8, 8, -8], [4, -1, 1, -6, 13], [10, 1, 5, 2, 9], [-10, 1, 9, -4, 10], [26, -2, -7, -2, 1], [16, -4, -3, -2, -1], [-12, 0, 0, 4, 0], [-6, 0, -8, 0, 4], [4, 8, 1, -2, -9], [2, -5, -5, 4, -6], [24, -1, 4, -2, 7], [-16, 6, 1, -6, 3], [-3, 6, -3, 1, -8], [-2, 4, 3, -2, 11], [4, 4, 5, 4, -4], [20, -6, -2, -8, 4], [24, -9, -9, 6, -2], [-8, -9, -9, 6, 2], [-8, 2, -7, 2, -1], [-12, 5, 15, -4, 1], [21, -4, -10, -3, 1], [-2, -8, -5, 6, 3], [2, 9, 1, -2, 1], [28, -4, 0, 8, -2], [-2, 2, 3, -10, 13], [4, 4, 11, -2, -7], [12, 6, -7, 4, -6], [2, -1, -6, 4, -3], [-18, 8, 10, 4, -2], [5, -1, 10, -1, -11], [-30, 2, -6, 4, -2], [6, -1, -7, 8, -6], [-14, -10, 7, 0, 15], [-22, 5, 0, 8, -9], [-36, -1, 3, 0, -7], [8, 1, -11, 6, -15], [18, 6, -6, 4, -10], [22, 11, 1, 0, 8], [7, -8, -14, 11, -1], [-20, -8, -5, 4, 0], [-10, -4, 2, 0, -2], [-4, 4, 2, -8, -6], [10, -7, -3, 12, -8], [-11, -2, 4, 5, 3], [-4, -8, 1, -2, 15], [8, 9, 2, -2, 11], [2, -3, 3, -6, -5], [22, 7, -6, 2, -6], [14, -4, 9, -2, -9], [-6, 0, -2, 0, -6], [-4, 1, -1, 2, -3], [-17, 4, -2, -5, 3], [-4, 1, -3, 10, 0], [34, 1, 3, -8, -2], [-26, 1, 9, -2, 9], [-8, 8, 12, -4, -1], [-8, 13, -5, 8, 1], [-12, 10, -6, 10, 0], [-35, -2, 2, -3, 9], [-20, 3, -3, 8, 7], [-8, 0, 2, 4, 4], [12, -10, 6, 2, 4], [-4, -7, -5, 10, -1], [12, -2, 1, 6, 3], [-5, -3, -8, 5, 1], [2, 0, 13, 6, 13], [25, -6, -6, 9, -3], [-12, -8, 2, 4, 2], [-12, -1, -1, 2, -4], [-8, -2, -6, -10, -4], [-2, 5, 8, 0, 2], [5, 2, 1, -3, 8], [4, -2, -13, 6, -3], [-12, 4, 8, 8, 0], [24, -1, 1, 2, 14], [2, -1, 0, 4, -7], [-1, -7, 2, -7, -1], [0, -6, -10, 8, -2], [12, -8, 2, -2, 0], [20, 4, 4, 0, 2], [-4, -6, 3, -2, 21], [9, -13, -13, 3, -6], [16, -9, -7, 10, 5], [-5, 5, 6, -3, -9], [-12, 6, 6, -12, 6], [-4, 10, -4, 0, -25], [-50, -2, 2, 0, -2], [-8, 8, -6, -6, 12], [-8, 5, -7, -2, -9], [8, -2, 9, 4, -4], [-26, 3, 5, -4, -2], [-8, 9, -2, -4, 4], [8, -3, -9, -6, -3], [12, 5, -13, 2, 5], [-22, 9, 7, -2, 7], [-4, -8, -8, 8, 1], [0, 14, 3, -8, 0], [-10, -12, 8, -12, 8], [-1, -4, 8, -1, 9], [30, 1, 1, 0, -4], [-20, 1, -7, 6, 14], [14, 6, -5, 2, 1], [6, -9, 7, 8, 18], [6, 12, -2, 2, -21], [-26, 10, 1, 12, 1], [18, -4, -16, 0, 0], [-26, 7, -10, -4, 1], [8, -4, -6, 4, -4], [-10, -3, 1, -4, -6], [-22, 5, 0, -4, 2], [0, -6, -6, 0, 16], [21, 2, 8, -11, -9], [-15, -4, 6, -11, 17], [-4, 9, 18, -18, 7], [-2, -18, -12, 8, -8], [2, 0, 4, 0, 4], [-18, -3, -8, 0, -2], [-6, 5, -3, 2, -11], [20, -7, -5, 10, -6], [-2, -1, -4, -14, -12], [0, 2, -7, -2, 7], [-4, 8, -2, 8, -10], [16, -20, 4, -8, 8], [-18, 0, -12, 8, -4], [-2, 6, -3, 12, -19], [7, 6, 4, -1, -15], [18, -3, -4, 4, 0], [15, 1, 15, 1, 4], [4, 10, 0, 2, -18], [16, -6, 8, 4, -15], [53, 2, 0, -3, -9], [38, -14, -2, -8, 2], [10, -4, -5, 18, -9], [-15, -12, 2, 1, -7], [-6, 9, 2, -4, 3], [10, -4, 0, -8, -4], [4, 3, 10, 6, -3], [-24, 1, 3, -2, -10], [34, -10, -6, 4, 6], [8, -11, 6, 4, 12], [-6, -5, 7, -12, 0], [-2, 0, -2, 6, -25], [-6, 3, 6, -4, -15], [8, 5, 12, -10, 17], [-22, 1, -3, 10, -23], [0, 4, 4, -8, -2], [-8, 1, 5, 6, -21], [21, 2, 1, -3, -16], [16, -2, -7, 2, -17], [-28, -13, 11, -6, 14], [-4, 12, -8, 6, -18], [-38, 0, 3, -4, -1], [8, -6, 2, 4, -6], [-4, -7, 13, -12, -1], [-26, -5, 0, -4, 22], [-10, -13, 12, -6, 4], [22, -5, 1, 2, -15], [-8, 7, -5, -6, -23], [-12, -8, 4, 6, -10], [-18, -13, 9, -8, 2], [-28, 10, 10, -8, 7], [-16, -6, -1, -2, -11], [-2, 17, 7, -6, -5], [16, 9, -1, 6, 9], [-12, 8, -4, -4, 0], [22, 13, -1, 4, -8], [43, 6, -21, 7, -8], [-62, 8, -1, 2, -3], [-14, 0, 0, -8, 4], [16, 11, 3, 2, 11], [-11, -12, -4, 1, 11], [22, 0, -6, 6, -11], [-25, 9, 5, 9, -14], [-25, 3, 0, 5, -11], [-24, -6, 4, 8, 25], [12, -12, 0, 8, 4], [30, 2, 0, -8, 8], [10, -9, 2, 0, 4], [-24, 15, -1, -6, -16], [-22, -7, 13, -10, 21], [-26, 8, -10, 4, 6], [6, -3, -15, 12, -14], [44, -17, -3, 0, 11], [10, -10, 13, -16, -7], [16, -5, -6, 10, -15], [-2, -3, -3, 10, -11], [30, 2, -6, 0, -2], [10, -12, -7, -2, 17], [-46, -8, -5, 2, -13], [52, -2, -14, 4, -6], [-6, 14, -1, 0, 7], [18, -3, 3, 2, -13], [18, -10, -2, 4, 18], [34, -5, 0, 4, 7], [-8, 12, 8, 0, -12], [22, 3, -11, 12, -4], [-28, 3, 3, -6, -4], [10, 13, 1, -6, 1], [0, 23, 11, -12, -7], [-16, 6, -14, -2, 24], [38, 5, -11, 4, -14], [26, 5, 11, -2, -13], [6, 3, 1, 6, 1], [-4, -12, -4, 6, 6], [-10, -13, 4, -8, 13], [-2, 0, 6, -4, -6], [-34, 1, 5, -4, 4], [54, 9, -1, -2, 7], [-4, -12, -16, 8, -7], [-2, -7, 12, 0, 15], [-4, -7, -10, 0, 16], [30, 4, -7, 10, 1], [-16, 11, 3, -4, 9], [40, 4, -23, 14, -13], [-4, 0, 12, -24, 18], [-23, -4, -20, 9, 11], [-16, -4, -2, -6, 0], [10, -21, 8, -8, -3], [2, -3, -5, -12, -12], [-8, 7, 21, -4, -1], [4, 4, 8, -8, -6], [-9, 16, 16, -1, 9], [2, 9, 11, -10, 23], [8, -4, 10, -12, 16], [18, 5, -7, 12, 4], [0, 4, -8, 12, -5], [-32, 10, 1, 6, -13], [-7, -16, -8, -3, -13], [4, 4, 12, 0, -2], [-4, -8, 8, 6, 22], [-2, 9, -3, 0, -24], [-18, 1, -9, 4, -2], [4, -15, -20, 6, -3], [19, 4, -8, 15, -15], [22, -15, 4, 8, 13], [-8, -13, 16, -16, 6], [0, 18, 2, 8, 4], [-32, 19, 5, -6, -3], [-6, 24, 2, -8, -14], [-28, 4, -8, -10, -6], [-10, 5, 9, 2, -3], [18, 24, 5, -4, 9], [39, -10, 4, -9, 9], [32, 4, -10, -4, -6], [-8, -4, -4, -14, -14], [-30, 1, -9, -8, -12], [20, -6, -9, 4, -12], [-8, 3, 1, -14, 9], [30, 4, 2, -4, 2], [38, -4, 7, -10, -11], [28, -16, -8, 0, 2], [4, -3, -4, 8, 18], [-26, 8, -10, 12, -2], [-6, -3, -13, 4, 14], [-6, -2, 1, 6, -7], [-8, -3, -7, 22, 1], [-18, -4, -12, 4, -4], [-32, 17, 1, -14, -8], [-12, -1, -4, 4, 6], [-2, 11, 6, 4, -10], [-18, -8, -2, -8, -14], [-23, 21, 3, -1, -6], [-24, -2, 11, -10, 17], [-8, 0, 2, 12, -10], [-10, -16, 0, 0, -16], [-26, -8, 0, -12, 8], [-16, 11, -3, 10, -7], [6, -8, 6, 4, 26], [36, -18, 5, -2, 7], [-16, 3, 7, -10, 20], [-2, -5, 8, -12, -8], [46, -1, -9, 14, -1], [-58, -3, -12, 6, 4], [-38, 30, 7, -2, -13], [26, -1, -7, 0, -20], [0, -8, -9, 10, -7], [-3, 12, 6, -11, -15], [12, 4, 4, -4, -20], [-36, 2, -1, -12, -12], [4, 4, 0, -2, 6], [-4, 16, 2, 0, 10], [-6, 2, -2, -12, 2], [34, -16, 4, -8, 12], [22, -11, 13, -12, -6], [6, -19, 6, 0, -10], [6, -4, 8, -12, 0], [30, -15, -6, -2, -6], [4, -9, -1, 14, -12], [20, -5, -24, 12, -2], [8, 6, 2, -8, 8], [6, 13, 4, -4, 1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1480_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1480_2_a_j();".
// 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_1480_2_a_j(:prec:=5)
    chi := MakeCharacter_1480_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_1480_2_a_j();".
// 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_1480_2_a_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1480_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-8, 12, 8, -10, -1, 1]>],Snew);
    return Vf;
end function;