
// Make newform 2240.2.b.f in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2240_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2240_b_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_2240_2_b_f();".
// 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_2240_2_b_f();".
// 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 := [40, -12, 1, 16, -3, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [-660, -236, 115, 53, -30, -11], [-2510, 2293, 80, -679, 250, -27], [315, 768, -510, 1, 75, -17], [-6, -7, -92, 11, 24, -9], [100, -358, -395, 54, 45, -28]];
    Rf_basisdens := [1, 445, 890, 445, 178, 445];
    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_2240_b();"
function MakeCharacter_2240_b()
    N := 2240;
    order := 2;
    char_gens := [1471, 1541, 897, 1921];
    v := [2, 1, 2, 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_2240_b_Hecke();"
function MakeCharacter_2240_b_Hecke(Kf)
    N := 2240;
    order := 2;
    char_gens := [1471, 1541, 897, 1921];
    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 := true, 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+1)]];
    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, 0, 0, -1], [0, 0, 0, 0, -1, 0], [1, 0, 0, 0, 0, 0], [0, 0, -1, 0, -1, -1], [0, 0, -1, 0, -1, 1], [0, -1, 0, 2, 0, 0], [0, 0, 0, 0, 0, -2], [-2, -2, 0, -2, 0, 0], [0, 0, 2, 0, -2, 1], [-2, 0, 0, 2, 0, 0], [0, 0, 2, 0, -2, -2], [-6, -2, 0, 0, 0, 0], [0, 0, 0, 0, -2, 0], [3, 1, 0, -1, 0, 0], [0, 0, -2, 0, 6, 0], [0, 0, 2, 0, -6, 0], [0, 0, -2, 0, 8, -2], [0, 0, 0, 0, -6, -2], [-8, -2, 0, 0, 0, 0], [-8, -2, 0, -2, 0, 0], [2, -5, 0, -2, 0, 0], [0, 0, 0, 0, 0, -2], [-8, 2, 0, 2, 0, 0], [4, 1, 0, -2, 0, 0], [0, 0, 0, 0, 2, 4], [9, -1, 0, 1, 0, 0], [0, 0, -4, 0, -2, 0], [0, 0, 2, 0, 6, 5], [0, 4, 0, 2, 0, 0], [2, 0, 0, 2, 0, 0], [0, 0, -2, 0, -6, 2], [-4, 2, 0, 2, 0, 0], [0, 0, -2, 0, 2, 2], [0, 0, -4, 0, -4, 0], [2, 3, 0, -2, 0, 0], [0, 0, 0, 0, -14, -4], [0, 0, -2, 0, 0, -4], [-1, 9, 0, -1, 0, 0], [0, 0, 5, 0, 1, -1], [0, 0, 2, 0, 4, -2], [0, 0, 4, 0, 2, 6], [-2, -5, 0, 2, 0, 0], [8, -6, 0, 2, 0, 0], [0, 0, -2, 0, -10, -8], [-6, -4, 0, -6, 0, 0], [0, 0, -1, 0, 11, -5], [-1, 9, 0, -1, 0, 0], [0, 0, 4, 0, -16, 1], [0, 0, -4, 0, 2, 2], [-4, 2, 0, -6, 0, 0], [10, -1, 0, -2, 0, 0], [14, 4, 0, 0, 0, 0], [0, 0, 2, 0, -2, -2], [4, -6, 0, -2, 0, 0], [16, -2, 0, 0, 0, 0], [0, 0, -2, 0, -4, -4], [-8, -4, 0, 0, 0, 0], [0, 0, -2, 0, 6, -2], [-11, -5, 0, -1, 0, 0], [0, 0, -8, 0, 16, -3], [0, 0, 1, 0, -3, 3], [0, 0, -4, 0, 8, -1], [-22, 2, 0, -2, 0, 0], [4, -7, 0, 2, 0, 0], [0, 0, -4, 0, 0, -10], [0, 0, 6, 0, 12, -6], [12, -4, 0, -2, 0, 0], [0, 0, -6, 0, 4, -8], [0, 0, -4, 0, -2, 8], [4, -11, 0, -6, 0, 0], [4, -6, 0, 4, 0, 0], [-7, -5, 0, 1, 0, 0], [0, 0, 0, 0, 12, -4], [0, 0, 2, 0, 8, -6], [-8, 4, 0, -8, 0, 0], [0, 0, 2, 0, -2, -1], [0, 0, 7, 0, -9, 5], [13, 3, 0, 3, 0, 0], [4, 4, 0, 2, 0, 0], [0, 0, 2, 0, 18, 4], [0, 0, -2, 0, -2, 1], [-6, -9, 0, 6, 0, 0], [24, 6, 0, -2, 0, 0], [20, 2, 0, 0, 0, 0], [0, 0, 0, 0, 2, -12], [5, -9, 0, -1, 0, 0], [-14, -2, 0, 4, 0, 0], [0, 0, -10, 0, 4, 2], [-4, 8, 0, 4, 0, 0], [0, 0, 0, 0, 12, 3], [0, 6, 0, 4, 0, 0], [22, -6, 0, -2, 0, 0], [0, 0, -5, 0, -1, -1], [0, 0, 1, 0, 9, 13], [-21, -3, 0, -1, 0, 0], [0, 0, 2, 0, 0, 4], [0, -4, 0, 6, 0, 0], [0, 0, 0, 0, 4, -2], [0, 0, 2, 0, 18, -5], [0, 0, -2, 0, 24, -2], [0, 0, 4, 0, 8, 8], [0, 0, 0, 0, 20, 6], [6, -12, 0, 0, 0, 0], [0, 0, -2, 0, 12, 2], [28, -3, 0, 2, 0, 0], [0, 0, 4, 0, -36, 2], [0, -1, 0, -2, 0, 0], [38, 1, 0, 2, 0, 0], [4, 8, 0, -10, 0, 0], [11, -7, 0, -5, 0, 0], [0, 0, 8, 0, 16, -6], [-4, 12, 0, -2, 0, 0], [0, 0, 10, 0, 10, -10], [-34, 1, 0, 2, 0, 0], [-18, -8, 0, 0, 0, 0], [0, 0, -4, 0, 4, 9], [12, -4, 0, 8, 0, 0], [0, 0, -4, 0, -4, -10], [0, 0, -9, 0, -1, 11], [0, 0, -6, 0, 8, 2], [28, 6, 0, 2, 0, 0], [0, 0, -3, 0, 5, 7], [0, 0, -2, 0, 4, 2], [0, 0, 4, 0, 4, 8], [0, 0, 6, 0, 2, 13], [0, 0, 10, 0, -10, 9], [12, 10, 0, 0, 0, 0], [-4, -12, 0, -8, 0, 0], [0, 0, -7, 0, 9, -5], [0, 0, -13, 0, 19, -1], [-2, 12, 0, 6, 0, 0], [6, -11, 0, -6, 0, 0], [0, 0, 0, 0, -16, 12], [18, 0, 0, 8, 0, 0], [-12, -12, 0, 6, 0, 0], [0, 0, -3, 0, 25, -5], [0, 0, -4, 0, 8, 7], [0, 0, -1, 0, -25, 5], [-25, 1, 0, 5, 0, 0], [0, 0, -6, 0, 2, -10], [0, 0, -2, 0, 6, 11], [14, -4, 0, -2, 0, 0], [0, 0, 6, 0, 4, -8], [0, 0, 4, 0, 2, -14], [-12, 2, 0, -12, 0, 0], [0, 0, 0, 0, -10, -16], [12, 10, 0, -2, 0, 0], [0, 0, -16, 0, 16, -8], [-2, 8, 0, -10, 0, 0], [0, 0, 0, 0, -8, 10], [-16, 10, 0, -2, 0, 0], [0, 0, 10, 0, -16, 2], [12, -4, 0, 8, 0, 0], [0, 0, -2, 0, -12, 0], [-4, 10, 0, -4, 0, 0], [-18, 9, 0, -6, 0, 0], [-10, 10, 0, -8, 0, 0], [28, -7, 0, -2, 0, 0], [0, 0, 2, 0, 4, -14], [0, 0, 12, 0, -6, 6], [-18, -6, 0, -4, 0, 0], [-24, 6, 0, 0, 0, 0], [0, 0, 6, 0, -26, 4], [-6, -8, 0, -8, 0, 0], [-7, -1, 0, 1, 0, 0], [12, -10, 0, 4, 0, 0], [0, 0, 1, 0, -27, 3], [29, -9, 0, 3, 0, 0], [0, 0, 0, 0, 6, 8], [0, 0, 10, 0, -4, -6], [0, 0, 16, 0, -22, 4], [-22, -3, 0, 6, 0, 0], [4, 0, 0, 10, 0, 0], [-38, -2, 0, 6, 0, 0], [-24, -2, 0, 2, 0, 0], [0, 0, -9, 0, 3, -9], [0, 0, 10, 0, 22, -15], [8, -4, 0, 8, 0, 0], [0, 0, 2, 0, 44, -2], [20, 4, 0, 12, 0, 0], [0, 0, 0, 0, -20, -4], [0, 0, 8, 0, -32, 2], [-8, 2, 0, 6, 0, 0], [-4, 0, 0, -4, 0, 0], [0, 0, -6, 0, -20, 0], [0, 0, -18, 0, 22, -8], [0, 0, -8, 0, 20, 5], [19, -11, 0, 5, 0, 0], [8, -8, 0, -4, 0, 0], [16, -2, 0, 10, 0, 0], [0, 0, -12, 0, 6, -4], [0, 0, -15, 0, 17, -3], [0, 0, 0, 0, 2, 10], [0, 0, 6, 0, -28, 6], [24, -5, 0, 10, 0, 0], [19, 1, 0, 17, 0, 0], [0, 0, 6, 0, -14, 0], [8, 3, 0, 6, 0, 0], [-13, 13, 0, -1, 0, 0], [0, 0, 4, 0, -4, 0], [30, -2, 0, 2, 0, 0], [0, 0, 4, 0, 14, 20], [0, 12, 0, 6, 0, 0], [0, 0, -4, 0, 24, 6], [0, 0, -1, 0, -13, -11], [-4, 18, 0, 0, 0, 0], [0, 0, -2, 0, -40, -10], [-2, -8, 0, -8, 0, 0], [0, 0, 0, 0, 32, 4], [-2, 10, 0, 0, 0, 0], [0, 0, -14, 0, -8, 0], [-2, 8, 0, -10, 0, 0], [0, 0, 0, 0, -24, 13], [14, -14, 0, -6, 0, 0], [-12, 2, 0, 2, 0, 0], [2, -10, 0, -14, 0, 0], [-26, 6, 0, 0, 0, 0], [-40, 10, 0, 0, 0, 0], [0, 0, 10, 0, 14, -2], [0, 0, -12, 0, 24, 4], [10, -6, 0, 6, 0, 0], [25, -5, 0, 3, 0, 0], [44, 6, 0, 4, 0, 0], [0, 0, -12, 0, 0, 1], [0, 0, -4, 0, -24, 4], [4, -5, 0, -10, 0, 0], [-6, -7, 0, -2, 0, 0], [13, 19, 0, -15, 0, 0], [0, 0, -6, 0, -8, -6], [0, 0, 0, 0, -32, -8], [0, 0, 0, 0, 4, -2], [0, -10, 0, -16, 0, 0], [1, -1, 0, -13, 0, 0], [0, 0, -4, 0, 20, 5], [-29, -7, 0, 3, 0, 0], [10, 10, 0, -12, 0, 0], [0, 0, 4, 0, 16, -4], [0, 0, 7, 0, -5, 15], [-24, -14, 0, -4, 0, 0], [0, 0, -14, 0, -28, 14], [0, 0, 0, 0, 28, 12], [31, 1, 0, 3, 0, 0], [0, 0, -4, 0, 52, -8], [-20, 5, 0, 2, 0, 0], [18, -22, 0, 2, 0, 0], [-16, -4, 0, -4, 0, 0], [0, 0, -12, 0, -4, -6], [0, 0, -7, 0, -19, 5], [-16, 6, 0, -16, 0, 0], [0, 0, 8, 0, -36, 2], [-20, -16, 0, -14, 0, 0], [-18, -8, 0, -2, 0, 0], [-12, -10, 0, 6, 0, 0], [0, 0, 12, 0, 10, -8], [0, 0, -13, 0, -1, -1], [0, 0, 0, 0, -12, 4], [0, 0, 12, 0, 12, 14], [0, 0, 5, 0, 25, -13], [-32, 5, 0, 6, 0, 0], [-4, 6, 0, -4, 0, 0], [0, 0, -10, 0, 0, 4], [0, 0, -8, 0, -22, 14], [0, 0, -9, 0, -5, 1], [16, -10, 0, -2, 0, 0], [0, 0, 10, 0, -14, 10], [0, 0, 2, 0, 18, 21], [32, 12, 0, -6, 0, 0], [0, 0, -8, 0, 34, 2], [0, 0, -8, 0, 24, -2], [0, 0, 4, 0, -34, 6], [0, 0, 8, 0, 6, -2], [16, 19, 0, 18, 0, 0], [28, -6, 0, 12, 0, 0], [40, 10, 0, -6, 0, 0], [-11, -9, 0, -15, 0, 0], [0, 0, -2, 0, 64, -2], [0, 0, 8, 0, 16, 12], [9, 3, 0, 7, 0, 0], [0, 0, -8, 0, 0, 10], [-45, 1, 0, 3, 0, 0], [10, -19, 0, -2, 0, 0], [0, 20, 0, -4, 0, 0], [0, 0, 6, 0, 12, 4], [0, 0, 12, 0, 28, -13], [-2, -15, 0, 10, 0, 0], [2, -28, 0, 0, 0, 0], [0, 0, 0, 0, 60, -2], [-10, -4, 0, -2, 0, 0], [-22, 22, 0, 0, 0, 0], [0, 0, -2, 0, -2, 3], [0, 0, 0, 0, -4, -11], [58, -2, 0, 4, 0, 0], [0, 0, 6, 0, 22, -8], [0, 0, -4, 0, 0, -14], [0, 0, 4, 0, 6, 14], [56, -10, 0, 0, 0, 0], [0, 0, 5, 0, 53, -1], [0, 0, -4, 0, -4, -20], [0, 0, -4, 0, -60, -9], [16, -13, 0, -2, 0, 0], [0, 0, 6, 0, -22, -6], [-22, -5, 0, 14, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2240_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2240_2_b_f();".
// 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_2240_2_b_f(:prec:=6)
    chi := MakeCharacter_2240_b();
    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_2240_2_b_f();".
// 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_2240_2_b_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2240_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![16, 0, 49, 0, 14, 0, 1]>,<23,R![-160, -60, 4, 1]>,<31,R![-32, -24, 8, 1]>],Snew);
    return Vf;
end function;