
// Make newform 128.5.d.c in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_128_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_128_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_128_5_d_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_128_5_d_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 := [9, 0, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 0, 8, 0], [0, 12, 0, -4], [0, 96, 0, 32]];
    Rf_basisdens := [1, 3, 3, 3];
    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_128_d();"
function MakeCharacter_128_d()
    N := 128;
    order := 2;
    char_gens := [127, 5];
    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_128_d_Hecke();"
function MakeCharacter_128_d_Hecke(Kf)
    N := 128;
    order := 2;
    char_gens := [127, 5];
    char_values := [[-1, 0, 0, 0], [-1, 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 := 5;
    raw_aps := [[0, 0, 0, 0], [0, 0, -1, 0], [0, -1, 0, 0], [0, 0, 0, 1], [0, 0, 11, 0], [0, -27, 0, 0], [-162, 0, 0, 0], [0, 0, -45, 0], [0, 0, 0, -9], [0, -163, 0, 0], [0, 0, 0, -8], [0, -189, 0, 0], [1890, 0, 0, 0], [0, 0, -297, 0], [0, 0, 0, 18], [0, 247, 0, 0], [0, 0, 231, 0], [0, 297, 0, 0], [0, 0, 171, 0], [0, 0, 0, -99], [2750, 0, 0, 0], [0, 0, 0, -102], [0, 0, -953, 0], [2430, 0, 0, 0], [7454, 0, 0, 0], [0, 187, 0, 0], [0, 0, 0, 33], [0, 0, -109, 0], [0, -1863, 0, 0], [-702, 0, 0, 0], [0, 0, 0, -44], [0, 0, 979, 0], [-32670, 0, 0, 0], [0, 0, 135, 0], [0, -109, 0, 0], [0, 0, 0, -121], [0, -3591, 0, 0], [0, 0, -3177, 0], [0, 0, 0, 693], [0, 3157, 0, 0], [0, 0, -913, 0], [0, -4509, 0, 0], [0, 0, 0, 396], [41374, 0, 0, 0], [0, 8455, 0, 0], [0, 0, 0, -319], [0, 0, -2673, 0], [0, 0, 0, -264], [0, 0, -7953, 0], [0, -1809, 0, 0], [-2754, 0, 0, 0], [0, 0, 0, -1170], [-97570, 0, 0, 0], [0, 0, -6921, 0], [80514, 0, 0, 0], [0, 0, 0, 189], [0, 5609, 0, 0], [0, 0, 0, 1166], [0, -11313, 0, 0], [23166, 0, 0, 0], [0, 0, 7227, 0], [0, -4721, 0, 0], [0, 0, -3933, 0], [0, 0, 0, -1089], [93602, 0, 0, 0], [0, 6149, 0, 0], [0, 0, 1683, 0], [-93758, 0, 0, 0], [0, 0, 22495, 0], [0, -6075, 0, 0], [108162, 0, 0, 0], [0, 0, 0, -639], [0, 0, 0, -782], [0, 6831, 0, 0], [0, 0, -10593, 0], [0, 0, 0, 1584], [0, 16139, 0, 0], [0, 29025, 0, 0], [151902, 0, 0, 0], [28450, 0, 0, 0], [0, 0, 29739, 0], [0, -14985, 0, 0], [0, 0, 0, -2178], [215518, 0, 0, 0], [0, 0, 0, 959], [0, 0, -18909, 0], [177822, 0, 0, 0], [-203294, 0, 0, 0], [0, -35831, 0, 0], [0, 0, 0, -4246], [0, 0, 5995, 0], [0, 0, 0, -2376], [0, 0, 0, -4635], [0, 0, 10975, 0], [0, 0, 39303, 0], [0, 0, 0, 2691], [0, -21439, 0, 0], [-184734, 0, 0, 0], [0, 0, 42867, 0], [0, -28863, 0, 0], [0, 0, 4059, 0], [0, 2893, 0, 0], [0, 0, -50125, 0], [-127710, 0, 0, 0], [0, 0, 3987, 0], [-270718, 0, 0, 0], [0, 0, 27899, 0], [-62910, 0, 0, 0], [0, 0, 0, 3339], [203902, 0, 0, 0], [0, 0, 0, -4716], [0, 27783, 0, 0], [363582, 0, 0, 0], [0, 0, 32427, 0], [0, 0, 0, -4445], [67230, 0, 0, 0], [0, 0, -7605, 0], [0, 0, 0, -4851], [0, 11161, 0, 0], [0, 0, 22775, 0], [0, 70011, 0, 0], [-810722, 0, 0, 0], [0, 81971, 0, 0], [0, 0, 19311, 0], [0, 0, -58113, 0], [0, -64903, 0, 0], [0, 114939, 0, 0], [0, 0, 0, 8370], [0, 0, 0, 6235], [0, 88317, 0, 0], [0, 0, 17631, 0], [0, 0, 0, -3195], [0, 0, 0, -6842], [0, 58779, 0, 0], [-691038, 0, 0, 0], [304030, 0, 0, 0], [0, 66715, 0, 0], [0, 0, -119889, 0], [0, -69047, 0, 0], [803682, 0, 0, 0], [0, 0, -7029, 0], [0, -108449, 0, 0], [0, 0, 0, -385], [0, 0, 88251, 0], [0, 56241, 0, 0], [0, 0, 0, -10215], [0, -15201, 0, 0], [-1264734, 0, 0, 0], [0, 0, 98703, 0], [0, 0, 0, 5472], [0, 64881, 0, 0], [434754, 0, 0, 0], [0, 0, 48411, 0], [0, 0, 0, 3807], [0, 0, -70785, 0], [0, 0, 0, 14454], [0, 0, 0, -6721], [-371682, 0, 0, 0], [-532418, 0, 0, 0], [0, 76825, 0, 0], [0, 0, 59603, 0], [604962, 0, 0, 0], [0, 0, 0, 17785], [0, 0, 67583, 0], [-815778, 0, 0, 0], [0, 0, 0, -15741], [0, 0, 0, 5324], [0, 225315, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_128_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_128_5_d_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_128_5_d_c(:prec:=4)
    chi := MakeCharacter_128_d();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 5));
    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_128_5_d_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_128_5_d_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_128_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,5,sign)));
    Vf := Kernel([<3,R![-96, 0, 1]>],Snew);
    return Vf;
end function;