
// Make newform 1240.2.a.k in Magma, downloaded from the LMFDB on 17 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_1240_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_1240_2_a_k();".
// 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_1240_2_a_k();".
// 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, 13, -9, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-5, 1, 1, 0], [5, -8, 0, 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_1240_a();"
function MakeCharacter_1240_a()
    N := 1240;
    order := 1;
    char_gens := [311, 621, 497, 561];
    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_1240_a_Hecke( : Kf := Kf)
    return MakeCharacter_1240_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, -1, 0, 0], [-1, 0, 0, 0], [1, -1, 0, 1], [-1, -1, 0, 1], [-1, 1, 0, -1], [1, 0, 0, 1], [-1, -1, -1, 0], [3, -1, 0, -1], [1, 1, -2, -1], [-1, 0, 0, 0], [-1, 2, 0, 1], [4, 0, 1, 1], [-1, -2, 2, 1], [4, 2, 0, 0], [2, 1, 0, 0], [0, 0, 1, -1], [-4, 2, 2, 2], [1, 3, -2, -1], [5, -1, -1, -2], [6, 1, -2, 0], [-1, -3, 0, 3], [-1, 0, 0, 3], [11, -1, 0, 1], [8, 0, 2, -4], [-3, 1, -1, 4], [-4, 2, -2, 2], [4, 2, 0, 0], [-3, -1, 3, 0], [3, 5, 0, 1], [-1, 5, -2, -3], [-3, -1, 1, 0], [4, 1, 0, -4], [-1, 3, 2, -1], [4, -2, 3, -1], [-11, 1, 2, 3], [-9, 5, 0, -3], [0, -2, -2, -2], [11, 0, 0, 1], [1, -5, 0, -1], [5, 3, 2, -3], [4, 2, 0, -4], [-10, 4, 2, 0], [4, -6, 0, 2], [-5, -3, 0, 3], [-7, -3, 4, 1], [-8, 2, -2, -2], [6, 1, 4, -4], [9, 1, 0, 5], [1, -3, -2, -1], [-3, 1, -2, 1], [0, 8, -2, -2], [-5, -1, -4, 1], [0, 2, 0, -2], [-1, 1, -2, 3], [10, 1, 0, -4], [9, -5, 0, -3], [-1, 7, 2, -3], [-14, -1, -2, 6], [-7, 7, -1, -4], [-8, -2, 4, 0], [-13, 5, 2, -1], [-9, 5, 0, -1], [2, 0, 1, -3], [22, -5, 2, 4], [-16, -8, 0, 2], [6, -10, 0, 4], [5, 2, 2, 3], [13, -1, 2, -3], [12, -4, -1, -1], [11, -5, -2, -5], [-6, 6, 0, -6], [-3, 0, -2, 1], [3, 1, 2, -1], [-12, 6, 3, -3], [8, 5, 4, -6], [6, 4, 0, 2], [8, -6, 0, -2], [-1, 1, -2, -1], [13, 3, -4, -1], [10, 8, 1, -3], [0, 2, 5, 1], [-5, -9, -1, 0], [9, -3, 6, 5], [10, -4, -7, -3], [8, -2, 0, 0], [8, 10, 0, 2], [11, 3, -2, -1], [5, 5, -4, 1], [-17, -5, 4, -1], [12, 8, 0, 4], [16, -2, 2, -6], [14, -5, 0, -4], [10, 0, -2, 6], [-7, 9, -2, -3], [3, -3, -2, -3], [-2, -2, 0, -6], [-13, 5, 3, 2], [-5, 10, -8, -5], [-8, -6, -4, 6], [11, 1, 0, 3], [-19, 5, -2, -5], [27, -5, 0, 3], [4, 0, 0, 2], [-19, -7, 2, 1], [-5, -5, 0, -3], [15, -7, 6, 7], [-2, -4, 2, -2], [4, 2, -2, -6], [7, -5, 0, 5], [9, -11, -4, 1], [-9, 2, -2, -9], [19, -9, -2, 3], [-20, 4, -4, 2], [14, 2, 4, -2], [0, -6, -2, -4], [15, 12, -2, -3], [-3, 2, -2, -7], [-18, -12, 2, 2], [17, -1, 9, -4], [3, 3, -1, -10], [23, -2, 2, 5], [5, -6, 0, 3], [7, 3, -2, -3], [6, -2, 1, -7], [6, -4, 4, 12], [6, 16, 4, -6], [-20, -6, -6, 6], [6, -4, 6, 4], [28, -10, -2, 4], [-6, 4, 4, 0], [3, -2, 6, 3], [-14, 10, -8, -6], [-10, -3, -2, -6], [18, 2, 8, -8], [16, -2, 1, -7], [9, -7, 6, 3], [1, 11, 2, -3], [11, -5, 2, 5], [25, -11, -6, 1], [-6, 6, -1, -5], [9, 3, 6, -5], [8, -9, 2, 0], [10, 1, -6, 0], [-2, -2, 2, 8], [21, -7, 1, 8], [-10, 0, -6, 2], [5, 15, 4, -5], [-13, 7, -4, -7], [11, 0, 6, 7], [-3, -5, 8, -1], [-19, -1, -4, 11], [22, -3, 6, -4], [-11, -3, -4, 5], [-6, -4, 2, 12], [-21, -9, -4, 3], [-31, 1, -5, 2], [-14, 4, -2, -2], [-7, -17, 0, 3], [14, -12, 4, 14], [12, 1, 2, -10], [-8, 15, -6, -6], [7, 9, -10, -3], [12, 4, 7, -7], [-29, -5, 2, -1], [21, -11, -4, 1], [3, -7, 6, -3], [24, 10, -2, -4], [9, 5, 0, -5], [-13, -3, 0, -1], [-2, -4, 4, 4], [6, 4, -2, 0], [38, -10, 1, 5], [-3, -1, 4, 7], [-5, -15, 1, -6], [12, -8, -2, 8], [-8, -4, -7, 3], [4, 4, -2, -12], [-23, 5, -4, 1], [3, 3, 0, 3], [-15, 15, 4, -3], [-3, -21, 0, 3], [-1, -13, 12, 5], [22, 9, -4, 2], [-35, 3, 8, 5], [7, 9, -6, -7], [13, -3, -4, 7], [-21, 5, 6, 9], [-16, -6, 2, 8], [-22, 0, 5, 9], [-9, -7, 0, -7], [2, -4, -4, -2], [6, -8, -4, 12], [22, 2, -8, -2], [-22, 10, 4, -2], [-5, 21, 0, -11], [21, -5, 2, 13], [14, -12, 2, -6], [6, 2, -2, -14], [-18, 6, -2, -16], [-38, -2, -5, 1], [-18, 6, 6, 8], [18, -4, 4, 12], [-6, 24, 0, -12], [-12, 0, 0, 12], [-1, 4, -10, -5], [0, -4, -5, -11], [2, -9, 4, -10], [-15, -1, 11, 4], [-19, -13, 11, 2], [6, -21, -2, 8], [-5, 7, 0, -5], [17, -1, 2, 7], [19, 21, -6, 1], [-2, -4, 6, -4], [37, -7, -1, 0], [-14, 0, 6, 0], [-8, 10, 8, -6], [18, 1, 10, 2], [-20, 8, 2, -4], [9, 21, -2, -5], [-30, -4, 2, -4], [-3, -9, 7, 4], [-4, -18, 6, -6], [6, 0, -8, 2], [-8, -6, -2, -10], [-19, 15, -2, 1], [15, 9, 4, -9], [14, 9, 6, -10], [-18, 8, -2, -8], [20, 7, -2, 4], [-26, 4, 2, 4], [4, -2, 3, 13], [-6, 10, 0, 4], [11, 9, 2, -13], [1, -7, 0, -19], [21, 21, -1, -4], [-19, -11, -8, 7], [2, 10, 0, 8], [9, -7, -6, 3], [-8, 4, -4, 0], [4, -6, 6, 8], [13, 10, -2, 13], [-33, -5, 0, -1], [-12, 13, 0, 0], [28, 4, 4, 0], [7, -2, 0, 1], [28, 4, -6, 0], [-55, -1, -4, 3], [-27, 11, -6, -5], [-7, 13, 4, 7], [17, 7, -5, 4], [-9, -5, 8, -5], [-2, 12, -5, -3], [-9, 11, 2, 15], [-8, 22, 1, -1], [-18, 0, -4, 4], [-2, 5, -10, 4], [21, -9, -8, -5], [23, 1, 0, 9], [-2, -8, -6, 8], [-17, 13, -10, -13], [21, 7, 4, -7], [-24, 16, -2, -4], [3, -5, 10, -5], [30, -2, 4, 2], [14, 10, -6, -2], [-36, 6, -4, -10], [45, -5, 4, 13], [-16, 0, -6, 8], [21, 9, -1, 4], [23, -7, -10, -5], [3, 16, 0, 3], [16, 14, 4, 0], [-15, -21, 6, 1], [-2, 6, 2, 12], [-4, -10, 10, 6], [-11, 3, -12, -3], [3, 29, 2, -13], [-16, 4, 0, -10], [16, -2, 6, -2], [36, -10, -1, 1], [-43, 13, 2, -1], [27, 5, 3, 6], [19, -3, -2, -11], [-35, -1, -12, 5], [37, 1, 2, 5], [20, -27, 2, 16], [0, 4, 0, 8], [19, -21, -2, 3], [4, -2, 3, 3], [-21, 5, 6, -7], [9, -4, 4, -7], [-18, 4, 7, 7], [13, 1, -6, -5], [20, 20, 0, 4], [0, -6, 4, 4], [-5, -13, 6, 3], [19, 1, -2, -1], [34, 5, -12, -6], [55, 3, -2, 3], [6, -9, 2, 2], [-28, 6, -2, 2], [19, -3, -4, -5], [-4, -6, 4, -2], [-19, 21, 0, -17], [-9, 4, 6, -3], [-12, -14, 0, -2], [-8, 0, -10, -10], [5, 15, 10, -3], [25, -10, 12, 7], [0, -12, 18, 2], [-4, 8, -3, -15], [39, 4, -2, 1], [-6, -6, -10, -4], [-11, -19, -6, 9], [14, 16, 2, 10], [-4, 0, 2, 22], [9, 3, -14, -3], [0, 14, -2, -16], [7, -5, -14, -1], [9, -17, -4, -13], [-8, -16, -6, 8], [20, 4, 6, -10], [-48, 4, 6, 0], [-13, -7, 2, -9], [-24, 2, 2, 10], [10, -20, 0, -2], [11, -14, 4, 13], [-11, 8, 2, -7], [23, 1, 1, -4], [8, 22, -6, -8], [4, 20, 8, -12], [-3, 9, -4, -13], [11, 23, -3, -6], [-28, 14, 12, 8], [-15, 9, 4, 1], [-22, -2, -6, 10], [-9, -1, 1, 4], [2, 1, 8, 0], [4, -1, 2, -8], [3, 19, -2, 7], [-28, 2, -2, 22], [-41, -1, 2, 9], [26, -12, -8, 4], [13, 1, -3, 0], [-11, -5, 1, 2], [1, -17, 2, -3], [-19, -9, 0, 1], [9, 9, -4, -13], [55, -3, -4, -3], [-33, -6, 6, -7], [-15, -9, 13, 2], [-1, 7, 8, -5], [-52, -8, 6, 4], [-49, 1, -4, -3], [27, 1, -4, 11], [-46, -2, 8, 0], [-7, -14, -2, 7], [14, 8, -2, 12], [31, 15, -2, -9], [-50, -2, 0, 8], [-15, 14, -14, -3], [12, -6, 2, 14], [-9, -7, 2, 21], [-1, -13, 10, 11], [-31, 5, 0, 3], [8, -9, 10, -4], [-17, -27, -7, 16], [8, 4, -12, -16], [-29, -3, 1, -2], [-46, 14, -2, -12], [26, 2, 9, -13], [-32, -4, 4, -4], [13, 22, -2, -11], [15, 5, -12, 1], [-48, -10, -2, 6], [-15, 15, -8, -5], [-16, 12, -9, -13], [-16, 27, 8, -4], [8, -4, 2, 10], [14, -13, -4, -6], [5, 2, 4, 11], [-6, -9, -2, 0], [36, -16, 0, -6], [-44, -12, 4, 8], [3, -15, 3, -16], [-45, -16, 6, 3], [7, -3, -6, -1], [-3, -2, 4, 23], [2, 0, -6, -18], [24, -11, 0, 2], [-12, -16, 4, -12], [-5, -3, 8, -13], [-18, -8, 3, -1], [34, 0, 12, -8], [5, 21, -8, -11], [-7, 9, 13, -6], [37, 15, -2, 1], [-23, -5, 0, -3], [26, -26, -2, 8], [-15, -11, -4, -5], [19, 3, 16, -9], [2, 2, -4, -4], [8, 8, 2, -2], [-21, 13, -11, -10], [-26, 4, -4, 16], [-16, 12, -14, -4], [6, -15, -2, 0], [3, -1, -10, 7], [80, -7, 0, 4], [5, -1, -2, -3], [-29, 22, -10, -15], [-4, 0, 0, 14], [51, 3, 0, 9], [6, -30, -1, 13], [17, -11, -6, -5], [-4, 16, -14, -14], [38, -6, -6, -14], [-41, -1, 2, -7], [29, -13, 2, -5], [-36, 5, -2, 12], [-17, -2, 0, -3], [19, -33, -9, 4], [46, -2, 10, -2], [47, 12, 0, -3], [-29, 1, -8, 11], [-9, 9, -8, -7], [-7, -9, 12, 15], [-18, -16, -12, 10]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1240_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1240_2_a_k();".
// 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_1240_2_a_k(:prec:=4)
    chi := MakeCharacter_1240_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_1240_2_a_k();".
// 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_1240_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1240_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-2, -13, -9, 1, 1]>,<7,R![-32, 52, -12, -4, 1]>,<11,R![8, -12, -12, 4, 1]>],Snew);
    return Vf;
end function;