
// Make newform 136.2.n.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_136_n();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_136_n_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_136_2_n_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_136_2_n_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, 0, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_136_n();"
function MakeCharacter_136_n()
    N := 136;
    order := 8;
    char_gens := [103, 69, 105];
    v := [8, 8, 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_136_n_Hecke();"
function MakeCharacter_136_n_Hecke(Kf)
    N := 136;
    order := 8;
    char_gens := [103, 69, 105];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [0, 1, 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 := 2;
    raw_aps := [[0, 0, 0, 0], [-1, -1, 0, 0], [1, -1, 0, 0], [-1, -2, -2, -1], [1, 0, 0, -1], [0, 0, 0, 0], [1, 0, -4, 0], [-3, 0, 3, 2], [1, 4, -4, -1], [-3, 3, 0, 0], [3, 3, 0, 0], [1, 1, 0, 0], [-3, -4, -4, -3], [3, -6, 3, 0], [0, 2, -10, 2], [5, 0, -5, -4], [7, 2, 7, 0], [-7, -4, -4, -7], [4, -4, 0, 4], [3, 3, -4, 4], [1, -1, 8, 8], [9, -4, 4, -9], [-3, 0, 3, -6], [0, 4, 4, 4], [-7, 7, -4, -4], [10, 0, 0, 0], [-8, -4, 0, 4], [-7, 7, -2, -2], [-3, 8, 8, -3], [1, -4, 4, -1], [3, 6, 3, 0], [-7, 7, -2, -2], [14, 0, 0, 0], [-9, -9, -4, 4], [0, 4, 12, 4], [-7, 0, 7, -10], [0, 8, -4, 8], [-5, -6, -6, -5], [-1, -1, 0, 0], [-15, -15, 4, -4], [3, -14, 3, 0], [1, -4, 4, -1], [0, -2, 6, -2], [1, 0, 0, -1], [5, -4, -4, 5], [13, -13, -2, -2], [15, 2, 2, 15], [-11, 0, 11, -2], [5, 4, -4, -5], [-7, 0, -7, 0], [-3, 3, 8, 8], [0, -8, 0, 8], [1, 1, 8, -8], [0, 2, 2, 2], [-3, 0, 3, -4], [7, 6, 7, 0], [1, 1, 8, -8], [0, 8, 0, -8], [5, -5, 0, 0], [9, 16, 9, 0], [-15, 8, -8, 15], [0, -8, -4, -8], [-4, 16, 0, -16], [-3, 3, 2, 2], [1, 8, 8, 1], [-11, 4, -4, 11], [3, 18, 3, 0], [1, 1, -8, 8], [-1, -10, -10, -1], [-7, 12, -7, 0], [0, -12, 8, -12], [-3, 0, 3, -18], [3, 10, 10, 3], [2, -24, 0, 24], [9, -9, -2, -2], [-9, -2, -9, 0], [9, 0, -9, 16], [9, -12, 12, -9], [13, -4, -4, 13], [18, 0, 0, 0], [1, 4, -4, -1], [0, -8, 0, -8], [-3, 0, 0, 3], [1, -4, 1, 0], [7, 7, 8, -8], [-12, 20, 0, -20], [13, 8, 8, 13], [5, 0, -5, -12], [-11, 0, 11, 12], [0, 10, -10, 10], [11, 2, 11, 0], [11, 11, -16, 16], [-23, 0, 0, 23], [13, 0, -13, 2], [-7, -12, 12, 7], [-1, -2, -2, -1], [-10, 20, 0, -20], [-3, 20, -20, 3], [0, 10, 10, 10], [5, 5, -12, 12], [-1, -1, 24, -24], [0, 0, 16, 0], [17, 0, -17, 2], [-27, -8, -27, 0], [15, -6, -6, 15], [14, 0, 0, 0], [19, 18, 19, 0], [21, 0, -21, -8], [0, 2, -42, 2], [1, -16, 16, -1], [17, -17, 10, 10], [-18, 8, 0, -8], [-15, 15, 4, 4], [3, 10, 10, 3], [1, 0, -1, -26], [13, -13, -16, -16], [3, 3, -20, 20], [0, -32, 0, 32], [17, 8, 8, 17], [0, -26, -14, -26], [1, 0, -1, -28], [5, -8, -8, 5], [-23, -23, -4, 4], [-25, -25, 8, -8], [-23, 8, -8, 23], [0, -8, -40, -8], [25, -25, -12, -12], [13, -13, -6, -6], [0, -14, -2, -14], [9, 0, -9, 12], [-5, 2, -5, 0], [-15, 15, -6, -6], [7, 7, 16, -16], [-7, -24, -7, 0], [0, 12, 20, 12], [0, 28, -4, 28], [5, 40, 5, 0], [-3, 3, 22, 22], [1, 0, -1, 12], [-7, 4, 4, -7], [-19, 19, -10, -10], [-31, 31, 12, 12], [-1, 10, 10, -1], [1, -4, 4, -1], [0, 24, -20, 24], [-7, 16, -16, 7], [-11, -11, 4, -4], [1, -8, -8, 1], [-17, 18, -17, 0], [0, 10, 30, 10], [1, 12, 12, 1], [-7, -7, -16, 16], [-4, -32, 0, 32], [-9, -9, -12, 12], [21, -4, 4, -21], [-5, 26, 26, -5], [16, -20, 0, 20], [17, -8, 8, -17], [-11, 0, 11, 16], [-7, -24, 24, 7], [-27, 27, -2, -2], [-50, 0, 0, 0], [-23, 0, 23, -2], [9, 0, -9, 42], [13, -16, 13, 0], [7, 7, 24, -24], [25, -25, 6, 6], [21, 4, -4, -21]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_136_n_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_136_2_n_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_136_2_n_a(:prec:=4)
    chi := MakeCharacter_136_n();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    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_136_2_n_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_136_2_n_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_136_n();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, 4, 6, 4, 1]>],Snew);
    return Vf;
end function;