
// Make newform 3040.2.a.q in Magma, downloaded from the LMFDB on 02 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_3040_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_3040_2_a_q();".
// 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_3040_2_a_q();".
// 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 := [18, 8, -10, -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, 0, 1, 0], [-5, -7, 1, 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_3040_a();"
function MakeCharacter_3040_a()
    N := 3040;
    order := 1;
    char_gens := [191, 2661, 1217, 1921];
    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_3040_a_Hecke( : Kf := Kf)
    return MakeCharacter_3040_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, 0, 1, 0], [1, 1, 0, 1], [1, 0, 0, 1], [-2, 1, 1, 1], [-1, 0, 0, 0], [5, 0, 1, 0], [0, -1, -1, 1], [0, 2, 0, 2], [3, -1, -1, 0], [0, -2, 2, -2], [-1, 1, -2, -1], [7, -1, -2, -1], [-1, -2, 0, 1], [2, 1, -1, 3], [-1, -1, 0, -1], [-6, 1, 2, 2], [0, 0, 2, 0], [0, 1, 1, 3], [0, -2, -2, 0], [3, 1, -2, -1], [6, 0, -2, -2], [-1, 3, -3, -2], [1, 1, -2, -1], [8, -2, 1, -1], [6, -1, -2, -2], [-4, -3, -1, -1], [-1, -3, -3, 0], [8, 4, -3, 1], [4, -4, 2, -2], [4, 3, 1, -1], [7, -1, -4, -1], [-5, -5, 4, -1], [-2, 0, -4, -2], [2, 8, 0, 0], [-3, 3, -2, 1], [12, 0, -1, 3], [-1, 3, 3, -2], [8, -4, 0, -2], [-2, 4, -4, 0], [4, -3, 1, 1], [-1, -5, 1, -2], [8, 2, 6, 0], [-2, -1, 5, -1], [0, -5, 1, 1], [-2, -2, -1, -7], [8, 3, -4, 2], [-5, -3, 4, -1], [-8, 2, -2, 0], [0, 1, -1, -5], [10, -6, 0, -2], [-6, 6, 0, -2], [-9, 3, 3, 0], [-3, -3, -2, 1], [8, 2, -4, 0], [3, 0, -3, 0], [-14, 2, -2, -2], [-8, -2, -2, -2], [-11, -5, -2, 1], [9, 6, -6, 1], [-8, -6, -5, 1], [-9, 8, 1, 0], [14, -5, 1, -3], [-5, 8, 2, -3], [-26, -1, -1, 1], [-15, 1, -5, 4], [-5, 5, 4, 1], [8, 2, 6, 0], [6, -1, -1, 7], [9, -4, 3, 0], [13, -7, 0, -5], [1, 2, -2, 1], [-6, -3, 7, -3], [12, -2, -1, 5], [8, -4, -4, 2], [-8, 4, -4, -2], [-4, -2, 0, 6], [8, -6, -4, -4], [6, -8, 4, -6], [-4, 3, -5, -1], [-4, 8, -2, -2], [-9, -11, 5, -4], [2, -4, 2, 6], [-5, 3, -2, 7], [-10, 0, 0, -2], [-4, -7, 3, -7], [6, 8, -4, 8], [-9, 7, 2, -1], [7, -9, 2, -9], [11, 3, 0, 3], [-2, -2, -1, 3], [-10, 12, 2, 0], [9, 3, 2, -1], [1, 0, 1, -8], [-12, 6, -6, 4], [0, 4, -10, 0], [-4, 9, 4, 4], [3, -1, 2, -7], [-14, 8, 3, 1], [22, -2, 2, -2], [18, 4, 1, 3], [-20, 4, 2, 6], [-11, 3, -2, -1], [-6, 5, 7, 3], [5, -9, 6, -3], [-10, -8, -4, 0], [-6, 0, 6, 2], [-10, 8, 10, 4], [20, 8, -5, 1], [6, -2, -4, -8], [6, -4, -6, -2], [-13, -1, -12, -1], [-11, -3, 8, 5], [-2, -2, -12, -6], [-7, -1, 4, -1], [9, 4, -13, -2], [18, -2, 0, -4], [22, 7, -3, 1], [-4, -3, 9, 1], [15, -7, 3, -2], [-11, 6, -6, 9], [-2, 0, -3, -1], [-11, 3, 2, 3], [1, 1, 8, -3], [0, -6, -2, 4], [3, -8, 1, -4], [3, 2, -1, 0], [10, 14, -8, 4], [-4, -2, -10, 6], [-12, 0, -3, 5], [0, -6, -4, 0], [20, -10, -2, -4], [-21, -8, 7, -6], [-9, 6, 5, -4], [3, -4, -2, 1], [4, -9, 8, 4], [-17, 2, -8, -3], [6, -3, -1, -5], [-6, -5, 7, -1], [-8, 16, 0, 2], [1, -10, 1, 0], [12, -1, -2, 2], [0, 9, 3, 7], [6, -6, 4, -4], [4, -2, 8, 10], [-21, -3, -3, 2], [-18, -2, 2, -12], [-14, 0, -5, 9], [9, 2, -2, -3], [-23, -1, -14, 7], [13, 11, 4, -1], [14, -4, -1, -5], [-22, -1, 2, 2], [-4, 8, 2, 2], [-20, 19, 5, 1], [0, 15, -3, 3], [12, 9, -7, 11], [14, 1, -3, 1], [13, 9, -2, -3], [9, 3, -3, 4], [13, -11, 0, -9], [2, -10, -6, 6], [23, 3, 5, 0], [30, 2, 9, 1], [-10, -8, 4, 2], [-20, -6, 4, 2], [-22, -6, 4, 0], [8, 6, -10, -4], [22, 3, 1, 3], [2, 5, 9, 5], [-13, 3, 16, 7], [-18, -7, -3, -7], [2, 4, 14, 4], [5, 3, -4, -3], [0, -2, 4, -8], [-23, 7, 0, -3], [-6, 2, 5, 11], [17, -1, 10, -1], [25, -10, 1, -12], [32, -3, -1, 5], [5, -7, -3, 2], [-19, -9, 11, -4], [15, -6, 5, 6], [6, -4, 0, -4], [7, -12, 0, -9], [16, 2, 9, -3], [-2, 10, -10, -4], [36, -3, -7, -7], [-1, 17, -3, -6], [3, 3, -4, 1], [-24, 4, -6, -4], [2, -5, 1, 7], [3, 7, -2, 11], [9, -15, 3, 0], [4, -1, 1, 7], [-8, -12, 4, 2], [-14, 13, 1, 1], [-7, 14, 11, -2], [-16, 2, -10, 0], [-4, -8, -6, -2], [-25, 2, -12, 3], [6, 0, 14, 6], [13, 7, 6, -1], [-16, -12, 0, -2], [-3, -6, 1, 2], [24, 3, -12, -6], [10, 1, 7, 7], [26, 9, -7, -3], [48, -3, -1, -3], [3, -1, 4, -9], [27, 11, -6, -1], [14, 3, 12, -10], [-4, 10, -10, 2], [-14, 6, 8, 10], [31, -2, 1, 6], [-4, -14, 0, -2], [-24, 4, 13, 5], [6, -8, 10, 6], [26, 8, -6, 0], [-6, -8, -2, 0], [26, -2, -4, -4], [15, -6, -5, 4], [22, -1, -2, -2], [-26, 4, -4, -8], [-3, 1, 9, 2], [10, 10, -8, 2], [34, -4, -5, -1], [30, -20, -2, -16], [2, -10, -10, 6], [14, -3, -9, -3], [2, 6, 6, -8], [2, -4, 2, -4], [-7, -5, 6, 5], [17, -6, -3, 0], [-45, 1, -2, 1], [20, -2, -6, 4], [13, -5, -6, -1], [-22, -12, 0, -2], [20, -14, -3, -5], [-12, -4, 6, -6], [25, -10, -5, 2], [32, -9, -5, -11], [-1, -9, 5, -18], [28, 4, 16, -8], [-15, -3, -16, -5], [24, -4, 6, -2], [34, 2, 8, -2], [-9, -10, 3, 4], [-18, -2, 0, 4], [0, -3, -9, 9], [-9, 6, 9, -6], [34, -18, -6, -4], [22, -4, 0, -8], [36, -4, -4, -8], [28, 6, 6, 0], [-8, -2, -15, -9], [-9, 3, 9, -6], [18, -4, -2, 6], [2, 10, -1, -9], [10, -12, 5, -5], [17, -1, 0, 5], [-17, 6, 4, 9], [-10, -4, -6, 6], [-24, 7, -5, 1], [4, -11, -7, 3], [17, -2, -5, -12], [26, 3, -2, 10], [-24, -12, 2, 4], [-8, 7, -3, -5], [-10, 3, 8, -6], [34, -9, -3, 5], [-13, 6, 11, 2], [-49, 9, 3, 0], [23, -4, 9, 18], [-31, 1, 2, 1], [32, -9, 11, -15], [0, 12, -16, -4], [39, 15, -8, 3], [30, -2, -7, -11], [25, 10, -15, -6], [-21, 6, -19, 10], [42, 1, 1, -3], [-11, -11, 10, -15], [19, -9, -8, -5], [-6, -5, 5, 5], [-3, 1, 5, 10], [-16, -1, -3, 5], [14, 12, 4, 2], [-8, -4, 8, -14], [-9, -5, -12, 9], [5, -1, -9, 6], [46, -2, 6, 0], [-5, -1, -9, -2], [-35, 23, 10, 11], [10, 16, 8, -4], [14, -16, 8, -12], [0, -15, 5, -7], [-31, -7, 12, -1], [0, -2, 12, 6], [-17, 4, 8, -5], [-5, -16, 9, -12], [-2, -11, 0, 10], [8, 18, -8, 4], [13, 13, 3, 0], [22, -7, 6, -12], [6, -9, 5, -7], [32, -16, -12, -8], [-20, -12, 12, -6], [-45, 3, 2, -9], [13, -3, -2, -9], [-46, 10, 6, 2], [-14, 4, 1, -9], [9, 1, 0, -9], [8, -6, -6, 16], [5, -1, -10, -9], [10, -2, 10, 8], [0, 3, -1, -7], [18, -12, 4, 12], [28, -15, 3, -11], [8, 11, -5, 5], [14, 13, -1, 5], [-6, 8, -5, 3], [24, -17, 5, -17], [32, -4, 0, 12], [-40, 15, 3, 9], [4, -7, -4, 8], [10, -2, -3, -5], [16, -2, 14, 0], [19, 3, 4, -1], [-19, -2, 8, -7], [13, -6, -3, 2], [27, -15, 4, 5], [-3, -9, 6, 11], [-23, 3, -6, 5], [-48, 5, 7, 7], [-65, 9, 5, 2], [-40, -3, -5, 1], [-11, -6, -3, 12], [-3, -10, -12, -3], [-16, 11, 9, -5], [6, -19, -11, 7], [14, -12, 12, -8], [9, -8, -4, 11], [-14, -2, 8, -12], [-30, 4, 4, -4], [4, -11, 12, 8], [22, -2, 18, -10], [26, 10, 12, 4], [-12, -8, -10, -4], [1, 7, 7, 8], [51, -9, -4, -5], [42, -12, -5, -13], [-32, -2, -10, 0], [-59, -5, -1, -8], [12, -15, -5, -9], [24, -8, 18, -6], [-6, 21, 5, -7], [24, 16, -3, -3], [-16, -14, 4, 6], [37, -17, 0, 5], [12, 2, 13, 15], [8, 11, -9, -1], [-51, 17, 8, 7], [-15, 21, -1, 4], [36, -22, -10, -12], [-20, -6, 3, -3], [8, 10, 6, -10], [-3, 3, 6, -9], [-32, -3, 9, 7], [-5, -30, -3, 2], [28, 7, 5, 13], [50, 2, -4, 10], [8, 4, 18, 0], [24, -7, 17, 1], [-2, -10, 8, -2], [8, -6, 2, -4], [-18, -11, -9, 3], [-9, 11, -15, -14], [-30, 8, -16, 4], [12, -10, -4, -10], [-11, 14, -3, 8], [-8, 10, -6, -8], [-46, 21, 11, 19], [16, 4, -17, -3], [-15, -4, -5, 16], [2, -8, -16, -4], [-3, -13, 8, -5], [-28, 10, -7, -9], [-4, -10, -8, -4], [-33, 20, 4, 19], [22, 0, 20, 2], [-33, 11, 0, -7], [-30, 20, 8, 10], [-7, 1, -19, 2], [-24, 14, -14, 4], [-20, 20, 0, 2], [48, -20, -10, -16], [26, -4, -24, -8], [0, -18, 8, 0], [-8, 2, -24, -10], [2, -6, -21, -11], [43, 3, -19, -4], [-6, -20, 4, -2], [-14, 19, 3, -3], [-24, -4, 16, -6], [42, -22, -6, -6], [40, 9, 12, -4], [-4, -22, -2, 2], [9, 9, 7, -6], [36, -10, 2, 16], [-48, -1, 4, 2], [-39, 21, 4, -3], [8, -3, 19, -11], [-4, 2, 30, 12], [-2, -6, -8, -8], [30, 6, 5, -5], [32, -5, 11, -11], [-22, 28, -1, 11], [-20, 10, -12, 20], [27, -16, 10, -17], [-21, 16, 3, -8], [-52, -7, 13, 5], [19, -3, -15, 0], [5, -2, 0, -7], [-8, 3, -6, 6], [59, 0, -11, -4], [-20, -12, 6, -2], [5, 6, -19, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3040_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3040_2_a_q();".
// 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_3040_2_a_q(:prec:=4)
    chi := MakeCharacter_3040_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_3040_2_a_q();".
// 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_3040_2_a_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3040_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![18, -8, -10, 1, 1]>,<7,R![-28, 40, -4, -5, 1]>,<11,R![32, 28, -8, -6, 1]>],Snew);
    return Vf;
end function;