
// Make newform 1040.2.k.b in Magma, downloaded from the LMFDB on 02 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1040_k();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1040_k_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_1040_2_k_b();".
// 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_1040_2_k_b();".
// 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, -4, 4, 2, 2, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [38, 2, -1, -4, 8, -1], [29, 10, -5, -20, 17, -5], [-13, 32, 30, 5, -10, 7], [27, -70, -11, -21, 19, -11], [26, -64, -37, -10, 20, -14]];
    Rf_basisdens := [1, 23, 23, 23, 23, 23];
    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_1040_k();"
function MakeCharacter_1040_k()
    N := 1040;
    order := 2;
    char_gens := [911, 261, 417, 561];
    v := [2, 2, 2, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1040_k_Hecke();"
function MakeCharacter_1040_k_Hecke( : Kf := false)
    N := 1040;
    order := 2;
    char_gens := [911, 261, 417, 561];
    char_values := [[1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0], [0, 0, 0, -1, 1, -1], [0, 0, 0, 1, 1, 2], [2, -1, 2, 0, 1, 0], [2, -2, 2, 0, 0, 0], [0, 0, 0, -1, 1, -2], [-4, 1, -2, 0, 0, 0], [-1, 3, 3, 0, 0, 0], [0, 0, 0, -3, 5, 0], [0, 0, 0, -3, 1, 1], [0, 0, 0, 0, -2, -4], [6, -3, 0, 0, 0, 0], [0, 0, 0, -3, 3, -3], [0, 4, 2, 0, 0, 0], [0, 0, 0, 1, -1, 0], [-7, -1, -3, 0, 0, 0], [0, 0, 0, -3, -1, -1], [0, 0, 0, -1, 5, -6], [0, 0, 0, -1, 3, -5], [-2, -2, 6, 0, 0, 0], [0, 0, 0, 7, -7, 1], [0, 0, 0, -2, -2, 2], [0, 0, 0, 0, 2, -6], [6, 2, -4, 0, 0, 0], [2, 5, 2, 0, 0, 0], [8, -3, 2, 0, 0, 0], [0, 0, 0, 0, 2, -2], [-8, -4, 4, 0, 0, 0], [-6, -5, 6, 0, 0, 0], [-6, -2, -2, 0, 0, 0], [0, 0, 0, 2, -8, 6], [6, 6, -2, 0, 0, 0], [0, 0, 0, 12, -2, -2], [0, 0, 0, 3, 5, -6], [4, 2, 0, 0, 0, 0], [0, 0, 0, -3, -5, -3], [0, 0, 0, -5, 1, 5], [2, 2, 4, 0, 0, 0], [-8, -2, -8, 0, 0, 0], [-7, -3, -7, 0, 0, 0], [0, 8, -8, 0, 0, 0], [0, 0, 0, -2, 4, -2], [0, 0, 0, -4, -10, -4], [8, -6, 8, 0, 0, 0], [0, -2, -8, 0, 0, 0], [0, 0, 0, 9, 3, -7], [0, 0, 0, 7, -3, -7], [0, 0, 0, 4, -4, 6], [16, -6, 2, 0, 0, 0], [0, 0, 0, 17, 1, 2], [0, 0, 0, -2, 4, 6], [-2, 8, -6, 0, 0, 0], [14, -4, 4, 0, 0, 0], [8, 3, -8, 0, 0, 0], [-10, -2, 0, 0, 0, 0], [0, 0, 0, -15, 7, 4], [0, -4, 6, 0, 0, 0], [0, 0, 0, 6, -4, 6], [-14, 5, 4, 0, 0, 0], [0, 0, 0, -5, -13, -5], [0, 0, 0, 7, -11, -3], [-4, -14, -4, 0, 0, 0], [8, -4, -4, 0, 0, 0], [0, 0, 0, 9, -3, -11], [0, 0, 0, 21, 1, -2], [2, 2, 14, 0, 0, 0], [10, -3, 2, 0, 0, 0], [0, 0, 0, -10, -6, 12], [0, 0, 0, -17, -1, 7], [0, 0, 0, 13, -5, -2], [16, 5, 8, 0, 0, 0], [-12, 2, 0, 0, 0, 0], [0, 0, 0, -13, -1, 6], [0, 0, 0, 5, 7, -5], [14, -4, -4, 0, 0, 0], [0, 0, 0, -23, 17, -7], [0, 0, 0, -4, 12, -16], [0, 0, 0, 24, 6, 0], [2, 8, -10, 0, 0, 0], [0, 0, 0, 16, 2, -14], [0, 0, 0, 1, 5, 8], [-2, -4, -8, 0, 0, 0], [2, -4, 14, 0, 0, 0], [6, 11, 10, 0, 0, 0], [0, 0, 0, -14, 12, 2], [0, 0, 0, -12, 14, -2], [0, 0, 0, -4, 8, 2], [0, 0, 0, -3, 19, 5], [-6, 1, 14, 0, 0, 0], [0, 0, 0, 21, -13, 10], [0, 0, 0, 11, 1, 3], [8, 4, 0, 0, 0, 0], [0, 0, 0, -1, 13, -18], [-16, 5, 6, 0, 0, 0], [0, 0, 0, 12, 12, 10], [-9, -5, -1, 0, 0, 0], [8, -3, 20, 0, 0, 0], [0, 0, 0, -18, 14, -4], [0, 5, -2, 0, 0, 0], [0, 0, 0, -3, 9, 5], [22, -11, 20, 0, 0, 0], [-3, -9, -11, 0, 0, 0], [-18, -2, -2, 0, 0, 0], [0, 0, 0, 1, 13, 5], [0, 0, 0, -25, -3, -1], [0, 0, 0, 16, 2, -14], [20, 2, 0, 0, 0, 0], [14, -8, 12, 0, 0, 0], [12, -1, -10, 0, 0, 0], [0, 0, 0, -10, -8, 24], [0, 0, 0, 2, -8, -2], [0, 0, 0, -7, 11, -4], [0, 0, 0, -13, 3, -6], [-6, 12, 8, 0, 0, 0], [0, 0, 0, 31, 9, 3], [2, -13, -2, 0, 0, 0], [-6, -8, 2, 0, 0, 0], [-10, -4, -14, 0, 0, 0], [0, 0, 0, -6, 4, 4], [0, 4, 12, 0, 0, 0], [-10, -6, -4, 0, 0, 0], [0, 0, 0, 7, 5, 1], [0, 0, 0, -9, -1, 12], [-10, 6, 12, 0, 0, 0], [0, 0, 0, 2, -6, -12], [-18, -2, -2, 0, 0, 0], [-4, 3, -18, 0, 0, 0], [0, 0, 0, 28, -6, 4], [0, 0, 0, 27, 11, -6], [0, 0, 0, 17, 3, -1], [10, 0, -10, 0, 0, 0], [6, 10, 0, 0, 0, 0], [0, 0, 0, -30, -2, 2], [0, 0, 0, 22, -18, 2], [0, 0, 0, -17, -5, 3], [0, 0, 0, 11, -3, 5], [-6, -12, -14, 0, 0, 0], [-31, 13, -15, 0, 0, 0], [0, 0, 0, -13, -9, 8], [0, 0, 0, 0, -18, 26], [-12, 11, -24, 0, 0, 0], [0, 0, 0, -19, 19, -9], [-1, -3, -9, 0, 0, 0], [0, 0, 0, -25, -13, 4], [0, 0, 0, 23, -9, 3], [-4, -2, -26, 0, 0, 0], [-24, 8, -16, 0, 0, 0], [0, 0, 0, 17, -17, 1], [0, 0, 0, 6, 16, 8], [3, 5, 3, 0, 0, 0], [12, 23, -6, 0, 0, 0], [-2, 7, -10, 0, 0, 0], [2, 5, -18, 0, 0, 0], [24, -2, 4, 0, 0, 0], [8, -2, 28, 0, 0, 0], [0, 0, 0, 36, 10, 8], [-24, 14, -10, 0, 0, 0], [0, 0, 0, -8, 20, -22], [0, 0, 0, 13, 7, 11], [24, 2, 6, 0, 0, 0], [0, 0, 0, 15, 5, 17], [-8, 14, 20, 0, 0, 0], [0, 0, 0, 5, -7, 21], [0, 0, 0, 29, -25, 5], [-14, -20, -2, 0, 0, 0], [22, -14, 0, 0, 0, 0], [0, 0, 0, 2, 10, 6], [44, 0, 6, 0, 0, 0], [0, 0, 0, 37, 5, 8], [0, 0, 0, 8, -32, -2], [8, -24, 20, 0, 0, 0], [0, 0, 0, -9, -1, 15], [14, 4, 10, 0, 0, 0], [2, 16, -24, 0, 0, 0], [0, 0, 0, -7, -9, -2], [0, 0, 0, 30, -14, 4], [-10, 15, -4, 0, 0, 0], [18, 24, -8, 0, 0, 0], [0, 0, 0, 5, -9, -15], [-16, -14, 8, 0, 0, 0], [14, -18, -4, 0, 0, 0], [0, 0, 0, 22, 8, 2], [0, 0, 0, -25, -7, 5], [-17, -1, -17, 0, 0, 0], [-38, 16, -22, 0, 0, 0], [0, 0, 0, -17, -15, -13], [0, 0, 0, -30, -8, -6], [0, 0, 0, -17, -19, 10], [-14, -10, -2, 0, 0, 0], [0, 0, 0, 23, -15, 1], [12, -14, 32, 0, 0, 0], [0, 0, 0, 34, -20, 4], [-14, -1, 20, 0, 0, 0], [-34, 4, -4, 0, 0, 0], [0, 0, 0, 10, 0, 22], [30, 4, 2, 0, 0, 0], [0, 0, 0, 5, -7, -11], [34, -5, 0, 0, 0, 0], [0, 0, 0, -10, 26, -24], [-2, 6, -2, 0, 0, 0], [0, 0, 0, -42, 12, -8], [-19, -3, -15, 0, 0, 0], [0, 0, 0, 39, 3, -18], [-10, 6, -20, 0, 0, 0], [0, 0, 0, 1, 1, -26], [-42, -1, 6, 0, 0, 0], [0, 0, 0, 8, 16, -16], [-8, 20, 4, 0, 0, 0], [-18, 4, -8, 0, 0, 0], [-39, 3, 1, 0, 0, 0], [-26, -1, 0, 0, 0, 0], [0, 0, 0, 19, 5, -21], [0, 0, 0, -45, -3, 8], [0, 0, 0, 10, -24, 18], [-40, 13, -20, 0, 0, 0], [30, -14, -20, 0, 0, 0], [0, 0, 0, 13, 7, -11], [0, 0, 0, -28, 2, -8], [13, -5, -11, 0, 0, 0], [0, 0, 0, -9, 19, 0], [0, 0, 0, -32, 12, 0], [0, 0, 0, 27, 5, -23], [50, 5, -2, 0, 0, 0], [-17, -19, 11, 0, 0, 0], [-8, 18, 18, 0, 0, 0], [-38, 0, -10, 0, 0, 0], [-2, -23, 12, 0, 0, 0], [0, 0, 0, -21, 1, 4], [2, -36, 22, 0, 0, 0], [-12, -8, -4, 0, 0, 0], [0, 0, 0, 9, 27, -4], [22, -14, 16, 0, 0, 0], [16, -7, -28, 0, 0, 0], [0, 0, 0, -21, 13, -5], [0, 0, 0, 20, -22, -4], [0, 0, 0, 23, 15, 15], [-36, 6, -24, 0, 0, 0], [6, -16, 6, 0, 0, 0], [0, 0, 0, 9, -25, 11], [40, -14, 16, 0, 0, 0], [-52, -7, -6, 0, 0, 0], [0, 0, 0, 62, -2, 4], [0, 0, 0, -34, 0, 6], [-34, -4, -10, 0, 0, 0], [0, 0, 0, -3, -9, 13], [0, 0, 0, -13, 7, -24], [0, 0, 0, -3, -11, 12], [-6, 1, -2, 0, 0, 0], [0, 0, 0, -11, -19, 1], [0, 0, 0, -58, 22, -10], [0, 0, 0, 19, 13, 7], [-6, -20, 4, 0, 0, 0], [-14, -16, 18, 0, 0, 0], [0, 0, 0, -17, 37, -2], [5, 29, -11, 0, 0, 0], [0, 0, 0, 29, -9, 1], [28, 6, 20, 0, 0, 0], [0, 0, 0, 1, 17, -35], [30, -13, 4, 0, 0, 0], [6, 17, 22, 0, 0, 0], [0, 0, 0, -36, -16, 14], [2, 2, -32, 0, 0, 0], [0, 0, 0, -19, 5, -3], [12, -2, -12, 0, 0, 0], [0, 0, 0, -24, 36, -18], [0, 0, 0, 24, -40, 20], [0, 0, 0, 57, -1, -13], [28, 0, -14, 0, 0, 0], [-10, -14, 24, 0, 0, 0], [0, 0, 0, 11, -27, 23], [0, 0, 0, 14, 0, -2], [20, -26, 20, 0, 0, 0], [14, 6, -10, 0, 0, 0], [0, 0, 0, 37, 19, 3], [0, 0, 0, -1, -7, 25], [0, 0, 0, 28, -32, 18], [0, 0, 0, -22, 36, -26], [-34, -14, -14, 0, 0, 0], [-2, 19, 12, 0, 0, 0], [0, 0, 0, -57, 3, -6], [22, 1, 4, 0, 0, 0], [0, 0, 0, 28, 4, 18], [0, 0, 0, -9, -27, 25], [-30, -12, 22, 0, 0, 0], [12, 18, -14, 0, 0, 0], [0, 0, 0, -10, -4, -8], [0, 0, 0, 7, -19, 8], [-3, -5, -7, 0, 0, 0], [-11, -1, 17, 0, 0, 0], [-24, -27, 6, 0, 0, 0], [0, 0, 0, 0, -14, -22], [0, 0, 0, 55, -13, 4], [-42, 4, -18, 0, 0, 0], [-22, 12, -24, 0, 0, 0], [12, -6, -4, 0, 0, 0], [-30, -4, 6, 0, 0, 0], [52, 2, -4, 0, 0, 0], [0, 0, 0, 37, -33, 3], [44, -6, 10, 0, 0, 0], [0, 0, 0, -30, 28, -28], [36, -2, 16, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1040_k_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1040_2_k_b();".
// 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_1040_2_k_b(:prec:=6)
    chi := MakeCharacter_1040_k();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_1040_2_k_b();".
// 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_1040_2_k_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1040_k();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, -4, 0, 1]>],Snew);
    return Vf;
end function;