
// Make newform 192.8.c.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_192_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_192_c_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_192_8_c_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_192_8_c_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 := [1, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [-1, 2]];
    Rf_basisdens := [1, 1];
    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_192_c();"
function MakeCharacter_192_c()
    N := 192;
    order := 2;
    char_gens := [127, 133, 65];
    v := [1, 2, 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_192_c_Hecke();"
function MakeCharacter_192_c_Hecke(Kf)
    N := 192;
    order := 2;
    char_gens := [127, 133, 65];
    char_values := [[-1, 0], [1, 0], [-1, 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 := 8;
    raw_aps := [[0, 0], [0, -27], [0, 0], [0, -1006], [0, 0], [-14614, 0], [0, 0], [0, 9578], [0, 0], [0, 0], [0, -161286], [-279710, 0], [0, 0], [0, -72078], [0, 0], [0, 0], [0, 0], [3535546, 0], [0, 2833994], [0, 0], [6274810, 0], [0, 90730], [0, 0], [0, 0], [-12245198, 0], [0, 0], [0, 11937746], [0, 0], [-16824022, 0], [0, 0], [0, 5632698], [0, 0], [0, 0], [0, -17788670], [0, 0], [0, 20205346], [-82011398, 0], [0, -39139142], [0, 0], [0, 0], [0, 0], [-39803374, 0], [0, 0], [-96518014, 0], [0, 0], [0, 97420178], [0, -92778854], [0, -116307078], [0, 0], [-201186238, 0], [0, 0], [0, 0], [-57587054, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, -254046230], [5850962, 0], [0, 0], [0, -356450478], [0, 0], [0, 576206586], [0, 0], [-1084883206, 0], [0, 0], [0, -247733630], [-1275706622, 0], [0, 0], [695221114, 0], [0, 0], [0, 0], [0, 813260426], [-608468654, 0], [0, 149194562], [0, 0], [0, 0], [-1908239222, 0], [0, 0], [2127270058, 0], [0, 0], [-2237031742, 0], [0, 0], [-1115656414, 0], [0, -815841918], [0, 0], [0, 0], [-4067093510, 0], [0, 0], [0, -2394320566], [0, 0], [0, 0], [0, 1538307794], [0, 0], [0, -2463203718], [0, 0], [0, 0], [0, 0], [0, -1260205102], [-3687820966, 0], [0, 2756052922], [0, 0], [0, 0], [0, 0], [0, 3751839026], [4239333458, 0], [0, 0], [0, 0], [0, 0], [7416536426, 0], [0, 6214755482], [-11282581790, 0], [0, 0], [0, 1202000162], [0, 2731826146], [0, 0], [0, -7760471526], [0, 0], [0, 0], [0, 0], [-6618596078, 0], [-4999175950, 0], [0, 0], [0, 0], [0, 4814286090], [0, 0], [3033183362, 0], [0, 0], [0, -11031530814], [-19773242950, 0], [0, -6296403990], [0, 0], [0, 10422790570], [7127874962, 0], [0, 0], [-6300532222, 0], [0, 0], [0, 6617682794], [0, 0], [0, 0], [0, -16826188686], [0, 0], [0, -1607952542], [0, 0], [6330170906, 0], [0, 0], [24407273074, 0], [0, 0], [0, 18519482530], [0, 0], [36275909578, 0], [0, 0], [0, -13834141238], [0, 0], [0, -9420118318], [0, 0], [0, -16204675518], [0, 0], [-3585136438, 0], [0, 0], [0, 0], [0, 0], [0, 21433006258], [0, 0], [0, 0], [0, 0], [0, -22903171174], [-56143662110, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_192_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_192_8_c_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_192_8_c_a(:prec:=2)
    chi := MakeCharacter_192_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 8));
    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_192_8_c_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_192_8_c_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_192_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,8,sign)));
    Vf := Kernel([<5,R![0, 1]>],Snew);
    return Vf;
end function;