
// Make newform 192.3.e.f in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_192_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_192_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_192_3_e_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_192_3_e_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 := [9, 0, 8, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, -10, 0, -2], [0, 22, 0, 2], [12, 5, 3, 1]];
    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_192_e();"
function MakeCharacter_192_e()
    N := 192;
    order := 2;
    char_gens := [127, 133, 65];
    v := [2, 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_e_Hecke();"
function MakeCharacter_192_e_Hecke(Kf)
    N := 192;
    order := 2;
    char_gens := [127, 133, 65];
    char_values := [[1, 0, 0, 0], [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 := 3;
    raw_aps := [[0, 0, 0, 0], [0, 1, 0, 1], [0, 0, 1, 0], [0, 1, 0, 2], [0, -5, 0, 0], [-10, 0, 0, 0], [0, 0, 0, 0], [0, 5, 0, 10], [0, 6, 0, 0], [0, 0, -5, 0], [0, 5, 0, 10], [-10, 0, 0, 0], [0, 0, 2, 0], [0, -11, 0, -22], [0, 4, 0, 0], [0, 0, 5, 0], [0, 35, 0, 0], [-90, 0, 0, 0], [0, 1, 0, 2], [0, 10, 0, 0], [-30, 0, 0, 0], [0, 5, 0, 10], [0, 9, 0, 0], [0, 0, 10, 0], [10, 0, 0, 0], [0, 0, -15, 0], [0, 9, 0, 18], [0, -1, 0, 0], [70, 0, 0, 0], [0, 0, 20, 0], [0, 29, 0, 58], [0, 25, 0, 0], [0, 0, -30, 0], [0, 25, 0, 50], [0, 0, -3, 0], [0, -15, 0, -30], [-170, 0, 0, 0], [0, -19, 0, -38], [0, -86, 0, 0], [0, 0, 15, 0], [0, -75, 0, 0], [262, 0, 0, 0], [0, 120, 0, 0], [210, 0, 0, 0], [0, 0, 25, 0], [0, 65, 0, 130], [0, -55, 0, -110], [0, -59, 0, -118], [0, 21, 0, 0], [358, 0, 0, 0], [0, 0, 30, 0], [0, -60, 0, 0], [-110, 0, 0, 0], [0, -105, 0, 0], [0, 0, -40, 0], [0, -54, 0, 0], [0, 0, -57, 0], [0, -75, 0, -150], [-490, 0, 0, 0], [0, 0, 18, 0], [0, 1, 0, 2], [0, 0, -15, 0], [0, 9, 0, 18], [0, -90, 0, 0], [-230, 0, 0, 0], [0, 0, 75, 0], [0, 25, 0, 50], [-70, 0, 0, 0], [0, 191, 0, 0], [-298, 0, 0, 0], [0, 0, 40, 0], [0, -190, 0, 0], [0, 21, 0, 42], [-410, 0, 0, 0], [0, 125, 0, 250], [0, -56, 0, 0], [0, 0, 33, 0], [230, 0, 0, 0], [0, 0, 40, 0], [-590, 0, 0, 0], [0, -35, 0, 0], [70, 0, 0, 0], [0, 180, 0, 0], [10, 0, 0, 0], [0, -95, 0, -190], [0, 111, 0, 0], [0, 0, 28, 0], [-430, 0, 0, 0], [0, 0, 15, 0], [0, -19, 0, -38], [0, 49, 0, 0], [0, -120, 0, 0], [0, -71, 0, -142], [0, -185, 0, 0], [0, 105, 0, 210], [0, 294, 0, 0], [0, 0, 75, 0], [0, 0, -30, 0], [0, -171, 0, -342], [518, 0, 0, 0], [0, 189, 0, 378], [0, 0, 55, 0], [0, -231, 0, 0], [0, 0, -82, 0], [0, 65, 0, 130], [-390, 0, 0, 0], [0, 231, 0, 0], [0, 0, -60, 0], [0, -170, 0, 0], [130, 0, 0, 0], [0, 29, 0, 58], [630, 0, 0, 0], [0, 0, 150, 0], [0, 105, 0, 210], [0, 185, 0, 370], [0, 0, 52, 0], [0, -51, 0, -102], [0, -414, 0, 0], [0, 0, 95, 0], [0, 45, 0, 0], [-170, 0, 0, 0], [-310, 0, 0, 0], [0, 0, -175, 0], [0, 411, 0, 0], [0, 5, 0, 10], [0, 0, 3, 0], [182, 0, 0, 0], [0, -140, 0, 0], [0, 9, 0, 18], [-650, 0, 0, 0], [0, 65, 0, 130], [0, 234, 0, 0], [0, -115, 0, -230], [-1050, 0, 0, 0], [0, 0, -90, 0], [1138, 0, 0, 0], [0, 0, 105, 0], [0, -171, 0, -342], [0, 0, -165, 0], [0, 0, -110, 0], [0, 5, 0, 10], [0, 0, 13, 0], [0, 169, 0, 338], [0, -101, 0, 0], [-170, 0, 0, 0], [0, 250, 0, 0], [-570, 0, 0, 0], [0, 0, 70, 0], [0, -275, 0, -550], [0, 64, 0, 0], [-490, 0, 0, 0], [0, 0, 132, 0], [0, 181, 0, 362], [0, 486, 0, 0], [0, 101, 0, 202], [0, 60, 0, 0], [0, 105, 0, 210], [0, 0, 12, 0], [-1270, 0, 0, 0], [0, 0, -25, 0], [0, -99, 0, 0], [0, 0, 30, 0], [0, -39, 0, -78], [0, -605, 0, 0], [0, 0, 80, 0], [0, -546, 0, 0], [0, -115, 0, -230], [-810, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_192_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_192_3_e_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_192_3_e_f(:prec:=4)
    chi := MakeCharacter_192_e();
    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_192_3_e_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_192_3_e_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_192_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![56, 0, 1]>,<7,R![-28, 0, 1]>],Snew);
    return Vf;
end function;