
// Make newform 117.3.d.a in Magma, downloaded from the LMFDB on 26 December 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_117_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_117_d_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_117_3_d_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_117_3_d_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 := [81, 0, 18, 0, -3, 0, 2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 9, 0, 3, 0, -2, 0, -1], [-9, 0, 12, 0, -2, 0, -1, 0], [-27, 0, 6, 0, 4, 0, -1, 0], [0, 81, 0, -21, 0, 1, 0, 2], [0, 108, 0, 24, 0, 10, 0, 2], [0, -63, 0, -3, 0, 5, 0, -2], [162, 0, 24, 0, 4, 0, 8, 0]];
    Rf_basisdens := [1, 27, 9, 9, 27, 27, 27, 9];
    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_117_d();"
function MakeCharacter_117_d()
    N := 117;
    order := 2;
    char_gens := [92, 28];
    v := [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_117_d_Hecke();"
function MakeCharacter_117_d_Hecke(Kf)
    N := 117;
    order := 2;
    char_gens := [92, 28];
    char_values := [[-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]];
    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 := 3;
    raw_aps := [[0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, -1, 0, 0, 0], [0, 2, 0, 0, 0, 0, 3, 0], [-3, 0, -1, 0, -1, -1, 0, 0], [0, 0, 0, -1, 0, 0, 0, -1], [0, 0, 0, 0, 3, -2, 0, 0], [0, 0, 0, -2, 0, 0, 0, -1], [0, 0, 0, -3, 0, 0, 0, 2], [0, 0, 0, 0, -5, 1, 0, 0], [0, 0, 0, 0, 6, -1, 0, 0], [0, 4, 0, 0, 0, 0, -1, 0], [2, 0, 6, 0, 0, 0, 0, 0], [0, -26, 0, 0, 0, 0, -1, 0], [0, 0, 0, -11, 0, 0, 0, 1], [0, -12, 0, 0, 0, 0, 7, 0], [-30, 0, 4, 0, 0, 0, 0, 0], [0, 0, 0, 0, -5, -7, 0, 0], [0, 40, 0, 0, 0, 0, 23, 0], [0, 0, 0, 0, -12, 9, 0, 0], [72, 0, -2, 0, 0, 0, 0, 0], [0, -16, 0, 0, 0, 0, 21, 0], [0, -20, 0, 0, 0, 0, -19, 0], [0, 0, 0, 0, 20, -9, 0, 0], [0, 0, 0, -3, 0, 0, 0, 5], [-34, 0, 30, 0, 0, 0, 0, 0], [0, 0, 0, -34, 0, 0, 0, -1], [0, 0, 0, 0, 12, 14, 0, 0], [0, 0, 0, 33, 0, 0, 0, 1], [-24, 0, -32, 0, 0, 0, 0, 0], [0, 0, 0, -28, 0, 0, 0, 3], [0, 48, 0, 0, 0, 0, -29, 0], [-118, 0, 0, 0, 0, 0, 0, 0], [0, 4, 0, 0, 0, 0, -55, 0], [0, 0, 0, 0, 17, -5, 0, 0], [44, 0, -36, 0, 0, 0, 0, 0], [0, 0, 0, 0, 15, -14, 0, 0], [0, -34, 0, 0, 0, 0, 9, 0], [0, 0, 0, -7, 0, 0, 0, 1], [0, 0, 0, 4, 0, 0, 0, 1], [-46, 0, -12, 0, 0, 0, 0, 0], [0, 0, 0, 24, 0, 0, 0, -7], [0, 0, 0, 0, -26, 26, 0, 0], [0, 120, 0, 0, 0, 0, 37, 0], [108, 0, 28, 0, 0, 0, 0, 0], [120, 0, -2, 0, 0, 0, 0, 0], [0, 0, 0, 0, -19, -21, 0, 0], [0, 22, 0, 0, 0, 0, 65, 0], [0, 0, 0, 0, 46, -6, 0, 0], [0, 0, 0, -25, 0, 0, 0, 2], [0, -12, 0, 0, 0, 0, -11, 0], [0, 0, 0, 0, -10, -25, 0, 0], [0, 0, 0, -50, 0, 0, 0, -14], [0, 0, 0, 19, 0, 0, 0, -4], [0, 0, 0, 10, 0, 0, 0, 12], [0, 0, 0, 49, 0, 0, 0, -2], [0, 0, 0, 0, -27, 30, 0, 0], [-4, 0, 54, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, -21, 0], [-36, 0, 94, 0, 0, 0, 0, 0], [0, 116, 0, 0, 0, 0, 51, 0], [0, 0, 0, 0, -51, -21, 0, 0], [0, 0, 0, -80, 0, 0, 0, 5], [252, 0, 34, 0, 0, 0, 0, 0], [0, -116, 0, 0, 0, 0, -37, 0], [0, 0, 0, 0, 11, -37, 0, 0], [234, 0, 40, 0, 0, 0, 0, 0], [0, 0, 0, -8, 0, 0, 0, 2], [0, 0, 0, 0, -64, -7, 0, 0], [0, -80, 0, 0, 0, 0, 71, 0], [0, 134, 0, 0, 0, 0, -33, 0], [-226, 0, 66, 0, 0, 0, 0, 0], [-324, 0, -50, 0, 0, 0, 0, 0], [0, 0, 0, 0, 3, -34, 0, 0], [0, 70, 0, 0, 0, 0, -25, 0], [0, 0, 0, -139, 0, 0, 0, 4], [0, 0, 0, 0, 0, -7, 0, 0], [0, 116, 0, 0, 0, 0, 51, 0], [0, 0, 0, 0, 40, -29, 0, 0], [0, 0, 0, 38, 0, 0, 0, -27], [0, 0, 0, 0, 2, -24, 0, 0], [0, 98, 0, 0, 0, 0, -141, 0], [306, 0, -26, 0, 0, 0, 0, 0], [-382, 0, -54, 0, 0, 0, 0, 0], [0, 0, 0, 50, 0, 0, 0, -23], [0, -280, 0, 0, 0, 0, -51, 0], [0, 0, 0, 0, 8, -19, 0, 0], [0, 272, 0, 0, 0, 0, 39, 0], [0, 0, 0, 0, -11, 60, 0, 0], [0, 0, 0, -118, 0, 0, 0, 2], [0, 144, 0, 0, 0, 0, -95, 0], [0, 0, 0, 0, 75, -1, 0, 0], [0, 0, 0, 134, 0, 0, 0, 18], [0, 0, 0, 0, -41, 29, 0, 0], [0, 0, 0, 16, 0, 0, 0, 40], [0, 120, 0, 0, 0, 0, 31, 0], [0, 0, 0, 15, 0, 0, 0, 3], [-166, 0, -42, 0, 0, 0, 0, 0], [0, 0, 0, 0, 102, 44, 0, 0], [494, 0, -6, 0, 0, 0, 0, 0], [0, -248, 0, 0, 0, 0, -103, 0], [0, 0, 0, -70, 0, 0, 0, -3], [0, 0, 0, -23, 0, 0, 0, -6], [180, 0, -50, 0, 0, 0, 0, 0], [0, 0, 0, 0, -132, 24, 0, 0], [0, -108, 0, 0, 0, 0, 85, 0], [0, 68, 0, 0, 0, 0, 63, 0], [0, 0, 0, 122, 0, 0, 0, 6], [-504, 0, 106, 0, 0, 0, 0, 0], [-190, 0, 72, 0, 0, 0, 0, 0], [0, 0, 0, 0, -30, 39, 0, 0], [0, -164, 0, 0, 0, 0, 47, 0], [0, 0, 0, 0, -61, -17, 0, 0], [0, 0, 0, 0, -75, 65, 0, 0], [0, 0, 0, -71, 0, 0, 0, -33], [0, 0, 0, 0, -83, 17, 0, 0], [0, 0, 0, 182, 0, 0, 0, -10], [0, 0, 0, 89, 0, 0, 0, 36], [0, 0, 0, -56, 0, 0, 0, -24], [0, 0, 0, 0, -122, -49, 0, 0], [248, 0, 162, 0, 0, 0, 0, 0], [0, 0, 0, -135, 0, 0, 0, 5], [0, -258, 0, 0, 0, 0, -17, 0], [0, 0, 0, 0, 59, 35, 0, 0], [0, 0, 0, -61, 0, 0, 0, 8], [0, 0, 0, 0, -34, -46, 0, 0], [0, 0, 0, -26, 0, 0, 0, -11], [132, 0, -86, 0, 0, 0, 0, 0], [0, 0, 0, 0, 88, -42, 0, 0], [0, 0, 0, 0, -63, -23, 0, 0], [0, -8, 0, 0, 0, 0, -211, 0], [-478, 0, -180, 0, 0, 0, 0, 0], [-264, 0, -38, 0, 0, 0, 0, 0], [0, -80, 0, 0, 0, 0, -193, 0], [0, 0, 0, 0, -110, -50, 0, 0], [0, -256, 0, 0, 0, 0, 45, 0], [0, 0, 0, 0, 109, -91, 0, 0], [0, 0, 0, -99, 0, 0, 0, -26], [0, 0, 0, -17, 0, 0, 0, -5], [0, 0, 0, 0, 231, -98, 0, 0], [0, -644, 0, 0, 0, 0, -223, 0], [566, 0, -36, 0, 0, 0, 0, 0], [0, 238, 0, 0, 0, 0, 113, 0], [720, 0, -26, 0, 0, 0, 0, 0], [0, 268, 0, 0, 0, 0, 113, 0], [0, 0, 0, 0, 62, -17, 0, 0], [0, 0, 0, 57, 0, 0, 0, -34], [-432, 0, 220, 0, 0, 0, 0, 0], [0, -102, 0, 0, 0, 0, -113, 0], [0, 0, 0, 0, -290, 103, 0, 0], [0, 0, 0, -135, 0, 0, 0, 48], [-252, 0, 298, 0, 0, 0, 0, 0], [0, 0, 0, -222, 0, 0, 0, -17], [-828, 0, -20, 0, 0, 0, 0, 0], [0, 0, 0, 18, 0, 0, 0, 17], [-84, 0, 88, 0, 0, 0, 0, 0], [0, -324, 0, 0, 0, 0, 283, 0], [486, 0, -206, 0, 0, 0, 0, 0], [0, 360, 0, 0, 0, 0, -143, 0], [0, 144, 0, 0, 0, 0, 139, 0], [0, 0, 0, 103, 0, 0, 0, 58], [0, 0, 0, 0, -11, 5, 0, 0], [0, 0, 0, -98, 0, 0, 0, -50], [0, 152, 0, 0, 0, 0, 51, 0], [0, 60, 0, 0, 0, 0, 91, 0], [698, 0, 102, 0, 0, 0, 0, 0], [62, 0, 12, 0, 0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_117_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_117_3_d_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_117_3_d_a(:prec:=8)
    chi := MakeCharacter_117_d();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    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_117_3_d_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_117_3_d_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_117_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;