
// Make newform 1140.2.f.a in Magma, downloaded from the LMFDB on 15 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1140_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1140_f_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_1140_2_f_a();".
// 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_1140_2_f_a();".
// 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_1140_f();"
function MakeCharacter_1140_f()
    N := 1140;
    order := 2;
    char_gens := [571, 761, 457, 781];
    v := [2, 2, 1, 2];
    // 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_1140_f_Hecke();"
function MakeCharacter_1140_f_Hecke( : Kf := false)
    N := 1140;
    order := 2;
    char_gens := [571, 761, 457, 781];
    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, 0, 0, -1, 0, 0], [0, -1, 0, 0, 1, 0], [0, 0, 0, 0, 0, -1], [-1, 0, 1, 0, 0, 0], [0, 0, 0, 2, -2, -1], [0, 0, 0, -3, 1, -1], [-1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 1, -3], [-1, 0, 1, 0, 0, 0], [-3, -1, 3, 0, 0, 0], [0, 0, 0, -2, -4, 1], [3, -2, 5, 0, 0, 0], [0, 0, 0, 2, 0, -1], [0, 0, 0, 1, 1, -3], [0, 0, 0, 3, -3, 3], [2, 4, 4, 0, 0, 0], [-3, -3, 1, 0, 0, 0], [0, 0, 0, -2, 4, -4], [-4, 2, -2, 0, 0, 0], [0, 0, 0, 4, 2, -2], [6, -4, 6, 0, 0, 0], [0, 0, 0, 7, -1, -3], [-5, -4, 3, 0, 0, 0], [0, 0, 0, -8, -4, 3], [0, 4, -4, 0, 0, 0], [0, 0, 0, 2, -2, -2], [0, 0, 0, -2, 6, -4], [0, -6, -2, 0, 0, 0], [0, 0, 0, 7, 5, 1], [0, 0, 0, 4, -6, -2], [-1, 2, -5, 0, 0, 0], [0, 0, 0, -6, 4, -4], [10, 2, -2, 0, 0, 0], [-12, 6, -10, 0, 0, 0], [7, -5, 9, 0, 0, 0], [0, 0, 0, 0, -2, 4], [0, 0, 0, 12, -4, 5], [0, 0, 0, 5, -5, 7], [0, 0, 0, 15, 1, 1], [-10, -4, -4, 0, 0, 0], [2, -4, -8, 0, 0, 0], [11, 2, 5, 0, 0, 0], [0, 0, 0, 8, -12, 7], [0, 0, 0, -1, -5, -3], [2, 2, 6, 0, 0, 0], [-2, -8, 2, 0, 0, 0], [0, 0, 0, 2, 2, 2], [0, 0, 0, -7, -5, 7], [-5, -11, -1, 0, 0, 0], [0, 0, 0, 2, 4, 6], [5, -4, 9, 0, 0, 0], [10, 0, 4, 0, 0, 0], [7, -8, -1, 0, 0, 0], [0, 0, 0, -5, 5, 7], [0, 0, 0, 3, -5, 3], [-3, 2, -3, 0, 0, 0], [4, 12, -4, 0, 0, 0], [0, 0, 0, -8, 4, 4], [1, -2, -11, 0, 0, 0], [0, 0, 0, 8, 0, 5], [0, 0, 0, -1, 9, 7], [0, 0, 0, 16, -12, 2], [-1, 2, 11, 0, 0, 0], [0, 0, 0, 12, 6, -16], [0, 0, 0, -19, 3, -1], [-9, -1, -15, 0, 0, 0], [0, 0, 0, -8, 2, 9], [0, 0, 0, 3, 11, 1], [10, -2, 16, 0, 0, 0], [0, 0, 0, -8, 14, 4], [-3, -2, -9, 0, 0, 0], [0, 0, 0, -2, 8, -3], [0, 0, 0, 18, -2, -1], [-3, 5, -5, 0, 0, 0], [0, 0, 0, 2, 2, -10], [-10, 8, 6, 0, 0, 0], [0, 0, 0, -14, -6, -6], [-3, 6, -5, 0, 0, 0], [8, 8, 2, 0, 0, 0], [-25, 2, 5, 0, 0, 0], [10, -10, 0, 0, 0, 0], [10, -12, 18, 0, 0, 0], [0, 0, 0, 14, -6, 13], [8, 0, 12, 0, 0, 0], [0, 0, 0, 5, 5, -1], [-1, -4, 17, 0, 0, 0], [0, 0, 0, -14, 0, -6], [6, 0, 0, 0, 0, 0], [0, 0, 0, -16, 10, -11], [0, 0, 0, -7, -15, -1], [-9, -4, 3, 0, 0, 0], [0, 0, 0, 8, 2, -16], [31, 6, -3, 0, 0, 0], [14, 2, 2, 0, 0, 0], [0, 0, 0, -5, -17, -5], [-9, 2, -3, 0, 0, 0], [9, -14, 17, 0, 0, 0], [0, 0, 0, 4, 6, 16], [10, -8, 20, 0, 0, 0], [0, 0, 0, -10, 18, 4], [0, 0, 0, 16, -18, 6], [0, 0, 0, -1, 9, -11], [-37, 0, -7, 0, 0, 0], [-8, 6, 12, 0, 0, 0], [0, 0, 0, 10, 12, -18], [0, 0, 0, 1, -7, 19], [0, 0, 0, 0, -14, -2], [-2, 6, -4, 0, 0, 0], [6, 12, 12, 0, 0, 0], [0, 0, 0, -6, 16, -10], [0, 0, 0, -28, 10, -4], [0, 0, 0, -3, -3, -17], [4, 10, -4, 0, 0, 0], [-8, 6, 8, 0, 0, 0], [-1, 10, -5, 0, 0, 0], [0, 0, 0, 0, 6, 11], [0, 0, 0, 9, 1, -15], [0, 0, 0, -5, -1, -11], [-4, -16, 10, 0, 0, 0], [6, 2, -8, 0, 0, 0], [0, 0, 0, -2, -8, 1], [0, 0, 0, -29, 9, 1], [0, 0, 0, 18, -18, 0], [0, 14, 16, 0, 0, 0], [14, 4, 8, 0, 0, 0], [7, -9, -5, 0, 0, 0], [-25, -18, 7, 0, 0, 0], [0, 0, 0, 42, 6, 3], [0, 0, 0, 20, 6, -2], [36, -2, -4, 0, 0, 0], [0, 0, 0, -6, 14, -2], [-7, 5, -13, 0, 0, 0], [0, 0, 0, 14, 2, -4], [-4, 18, 12, 0, 0, 0], [6, -4, 24, 0, 0, 0], [0, 0, 0, 5, 3, -27], [0, 0, 0, 8, -2, 16], [0, 0, 0, -33, 17, -5], [-16, -6, -18, 0, 0, 0], [-8, 16, -8, 0, 0, 0], [6, -6, 2, 0, 0, 0], [0, 0, 0, -24, 10, -3], [0, 0, 0, -5, 9, 21], [18, -2, 24, 0, 0, 0], [0, 6, 6, 0, 0, 0], [0, 0, 0, -10, -10, 12], [0, 0, 0, -17, 13, -9], [18, 4, -2, 0, 0, 0], [0, 0, 0, -34, -2, 12], [0, 0, 0, 6, 8, -9], [-14, -26, 4, 0, 0, 0], [0, 0, 0, -2, -2, 15], [0, 0, 0, -8, 12, -4], [0, 0, 0, 26, 4, 14], [10, -14, 2, 0, 0, 0], [-2, 24, -6, 0, 0, 0], [0, -4, 4, 0, 0, 0], [0, 0, 0, -24, 8, 2], [-35, -8, 1, 0, 0, 0], [0, 0, 0, 27, -21, 13], [0, 0, 0, -25, 9, -3], [0, 0, 0, 2, -18, 27], [-8, -4, 6, 0, 0, 0], [0, 0, 0, 19, -7, 5], [0, 0, 0, -3, -9, 23], [-40, -8, 0, 0, 0, 0], [0, 0, 0, -34, -8, -6], [16, -18, -10, 0, 0, 0], [0, 0, 0, -8, -6, 22], [-18, -4, -2, 0, 0, 0], [10, 2, -8, 0, 0, 0], [3, 14, -11, 0, 0, 0], [0, 0, 0, 2, 10, -10], [6, 8, 14, 0, 0, 0], [2, -18, 26, 0, 0, 0], [-4, 0, -16, 0, 0, 0], [-20, -4, -12, 0, 0, 0], [0, 0, 0, -32, 22, -4], [-22, 14, -24, 0, 0, 0], [0, 0, 0, 2, 8, 15], [30, 2, -16, 0, 0, 0], [0, 0, 0, -6, 0, -11], [0, 0, 0, -7, -21, 9], [0, 0, 0, -41, 15, -9], [12, -2, 18, 0, 0, 0], [0, 0, 0, 10, -8, 25], [0, 0, 0, 6, -16, -10], [-10, -4, -16, 0, 0, 0], [31, -12, 13, 0, 0, 0], [0, 0, 0, -2, 12, 7], [0, 0, 0, -13, -1, 5], [-10, -12, 2, 0, 0, 0], [-9, 18, 3, 0, 0, 0], [0, 0, 0, 9, -27, -1], [0, 0, 0, -13, 17, -7], [-11, 15, 9, 0, 0, 0], [0, 0, 0, -14, -4, -4], [0, 0, 0, -1, 11, -3], [0, 0, 0, -29, -13, -7], [-19, 18, -7, 0, 0, 0], [18, 12, -6, 0, 0, 0], [0, 0, 0, -12, -18, -11], [22, 16, 4, 0, 0, 0], [-11, -30, 5, 0, 0, 0], [0, 0, 0, -13, 3, -7], [0, -12, 12, 0, 0, 0], [0, 0, 0, 9, 11, 19], [20, 20, -2, 0, 0, 0], [-24, 12, -8, 0, 0, 0], [0, 0, 0, 2, 4, 18], [0, 12, 26, 0, 0, 0], [0, 0, 0, -10, -8, -5], [0, 0, 0, 37, 3, 3], [-24, 14, 16, 0, 0, 0], [38, -6, 8, 0, 0, 0], [0, 0, 0, 48, 2, -1], [-29, 0, 15, 0, 0, 0], [0, 0, 0, -15, -25, 3], [0, 0, 0, -13, -17, 25], [-12, 10, -18, 0, 0, 0], [15, -5, -7, 0, 0, 0], [15, 16, -17, 0, 0, 0], [0, 0, 0, 6, -20, 13], [0, 0, 0, -3, 3, -1], [22, 6, 20, 0, 0, 0], [0, 0, 0, 7, 21, -13], [0, -10, -16, 0, 0, 0], [0, 0, 0, 10, -12, -16], [37, 12, 5, 0, 0, 0], [0, 0, 0, -6, -26, -20], [-29, 11, -15, 0, 0, 0], [-16, -40, 8, 0, 0, 0], [17, 0, -7, 0, 0, 0], [0, 0, 0, 14, -10, -11], [0, 0, 0, 35, -9, -7], [19, 23, 7, 0, 0, 0], [0, 0, 0, -2, -16, 16], [-11, 26, -15, 0, 0, 0], [-20, 2, -8, 0, 0, 0], [0, 0, 0, 18, -14, -22], [28, -20, 40, 0, 0, 0], [0, 0, 0, -20, 16, 9], [-14, -26, 0, 0, 0, 0], [0, 0, 0, -13, 5, -3], [-21, 14, 21, 0, 0, 0], [0, 0, 0, -10, 24, -17], [6, 14, 22, 0, 0, 0], [-25, -25, 5, 0, 0, 0], [0, 0, 0, -3, -3, -5], [0, 0, 0, 12, -28, -6], [-28, -10, -10, 0, 0, 0], [0, 0, 0, 41, -3, -9], [-6, 16, 12, 0, 0, 0], [0, 0, 0, -21, 27, -27], [11, 14, -17, 0, 0, 0], [35, -17, -5, 0, 0, 0], [0, 0, 0, -34, -22, -6], [0, 0, 0, 23, 29, 15], [0, 0, 0, -6, 28, -32], [0, 0, 0, -18, -18, 18], [0, 0, 0, -24, 8, 8], [-6, -2, 16, 0, 0, 0], [0, 0, 0, 0, -10, -5], [0, 0, 0, -37, -13, -15], [1, -25, 3, 0, 0, 0], [3, -24, 23, 0, 0, 0], [-26, -2, -14, 0, 0, 0], [0, 0, 0, 18, 10, -2], [0, 0, 0, 19, 15, -27], [-26, -2, 0, 0, 0, 0], [0, 0, 0, -58, -4, -4], [0, 0, 0, 14, -14, -8], [6, -8, 10, 0, 0, 0], [0, 0, 0, 38, 8, 3], [0, 0, 0, -22, -14, -19], [0, 0, 0, 13, 25, 3], [-24, -22, -14, 0, 0, 0], [32, 10, -2, 0, 0, 0], [-1, -18, 25, 0, 0, 0], [0, 0, 0, 34, -10, 0], [-14, 14, 30, 0, 0, 0], [0, 0, 0, -23, -3, 3], [14, 6, -8, 0, 0, 0], [0, 0, 0, 12, -36, -3], [5, -24, -3, 0, 0, 0], [0, 0, 0, -8, 28, 8], [0, 0, 0, -5, 9, 17], [20, -28, 8, 0, 0, 0], [-33, 0, -23, 0, 0, 0], [32, 4, 4, 0, 0, 0], [0, 0, 0, 37, -7, -9], [0, 0, 0, 45, -41, 23], [28, 18, 8, 0, 0, 0], [0, 0, 0, -12, 12, -11], [50, -18, 28, 0, 0, 0], [8, 0, 16, 0, 0, 0], [0, 0, 0, 5, -19, 7], [2, 2, 38, 0, 0, 0], [0, 0, 0, 12, 16, -11], [0, 0, 0, 16, -6, -10], [0, 0, 0, -7, 3, 21], [-14, 26, -14, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1140_f_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1140_2_f_a();".
// 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_1140_2_f_a(:prec:=6)
    chi := MakeCharacter_1140_f();
    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_1140_2_f_a();".
// 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_1140_2_f_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1140_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![4, 0, 16, 0, 8, 0, 1]>],Snew);
    return Vf;
end function;