
// Make newform 560.4.e.c in Magma, downloaded from the LMFDB on 20 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_560_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_560_e_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_560_4_e_c();".
// 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_560_4_e_c();".
// 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 := [16, 8, -16, 18, 23, -9, -4, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [2624, -120, 40, -95, -205, -70, -3, 25], [-1072, 768, 1036, -1517, 173, 74, 111, -41], [16, -128, 60, 95, -15, -14, -5, 3], [557, -192, 64, 1072, -328, -112, -66, 40], [68, -52, -66, 97, -15, -4, -7, 3], [64, -24, 8, 125, -41, -14, -7, 5], [-172, 2556, -1226, -1691, 533, 284, 69, -65]];
    Rf_basisdens := [1, 1224, 204, 12, 153, 6, 8, 102];
    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_560_e();"
function MakeCharacter_560_e()
    N := 560;
    order := 2;
    char_gens := [351, 421, 337, 241];
    v := [1, 2, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_560_e_Hecke();"
function MakeCharacter_560_e_Hecke(Kf)
    N := 560;
    order := 2;
    char_gens := [351, 421, 337, 241];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 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 := 4;
    raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, -1, 0, 0], [0, 5, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 7, 0, 0, 0], [0, 0, -18, 0, 0, 0, 1, 0], [0, -9, 0, 8, 0, 0, 0, 0], [0, -23, 0, 9, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-27, 0, 0, 0, 0, 0, 0, -13], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 117, -20, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 146, 0], [0, -234, 0, 0, 0, 0, 0, 0], [0, 0, 59, 0, 0, 0, -117, 0], [0, 0, 0, 0, 18, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 369, 0, 91, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -151, 216, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [1133, 0, 0, 0, 0, 0, 0, 9], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-2466, 0, 0, 0, 0, 0, 0, 0], [0, 0, 697, 0, 0, 0, 207, 0], [0, 1746, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 657, -286, 0, 0], [0, -985, 0, -108, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -794, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1053, 0, 0, 0, -269, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -598, 0, 0, 0, 477, 0], [0, 0, 0, 0, -979, 468, 0, 0], [0, 0, 0, 0, 927, 533, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1431, 0, 0, 0, 395, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 1502, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [4059, 0, 0, 0, 0, 0, 0, -202], [0, 0, 0, 0, -421, -1035, 0, 0], [0, 1507, 0, 702, 0, 0, 0, 0], [0, 0, 0, 0, 1703, 657, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -2403, 0, -253, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1602, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -1391, 0, 1107, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1874, 0], [0, 0, 0, 0, 2653, -90, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -738, 0], [0, 0, 0, 0, 5382, 0, 0, 0], [-6903, 0, 0, 0, 0, 0, 0, -283], [0, -3537, 0, 26, 0, 0, 0, 0], [6399, 0, 0, 0, 0, 0, 0, -410], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [1961, 0, 0, 0, 0, 0, 0, 711], [0, 0, -963, 0, 0, 0, -1477, 0], [0, -5778, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-5697, 0, 0, 0, 0, 0, 0, -644], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 2007, -1919, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 3168, 0, 0, 0, -1495, 0], [0, 0, -1886, 0, 0, 0, 1773, 0], [0, 0, 0, 0, -4239, -272, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -2758, 0, 0, 0], [-4859, 0, 0, 0, 0, 0, 0, 981], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -8622, 0, 0, 0], [10026, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2718, 0], [0, 4545, 0, 1781, 0, 0, 0, 0], [0, 0, 0, 0, -9126, 0, 0, 0], [0, -247, 0, 2727, 0, 0, 0, 0], [0, 0, -2691, 0, 0, 0, -2305, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1717, -3186, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 3713, 0, 0, 0, 2367, 0], [-28782, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -5681, 1413, 0, 0], [0, 0, 0, 0, 5886, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 7956, 0, 0, 0, -971, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -7061, 0, -1476, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-16731, 0, 0, 0, 0, 0, 0, 679], [-1063, 0, 0, 0, 0, 0, 0, 1593], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -4306, 0, 0, 0], [0, -8073, 0, 1558, 0, 0, 0, 0], [0, 0, -5764, 0, 0, 0, 2781, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 6773, 0, 0, 0, 2619, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -1157, 0, -4032, 0, 0, 0, 0], [0, 0, 0, 0, -2873, 4635, 0, 0], [0, 9175, 0, -1746, 0, 0, 0, 0], [17667, 0, 0, 0, 0, 0, 0, 1246], [0, 0, 0, 0, 0, 0, 0, 0], [22401, 0, 0, 0, 0, 0, 0, -607], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -16902, 0, 0, 0, 0, 0, 0], [0, -3322, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -7074, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -7670, 0], [0, 0, 13891, 0, 0, 0, 351, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 4149, 0, 5129, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1449, -6838, 0, 0], [0, 0, 0, 0, 0, 0, -10278, 0], [0, -8829, 0, -4634, 0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_560_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_560_4_e_c();".
// 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_560_4_e_c(:prec:=8)
    chi := MakeCharacter_560_e();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_560_4_e_c();".
// 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_560_4_e_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_560_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![2809, 0, 134, 0, 1]>],Snew);
    return Vf;
end function;