
// Make newform 135.4.a.c in Magma, downloaded from the LMFDB on 01 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_135_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_135_4_a_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_135_4_a_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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_135_a();"
function MakeCharacter_135_a()
    N := 135;
    order := 1;
    char_gens := [56, 82];
    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;

function MakeCharacter_135_a_Hecke( : Kf := Kf)
    return MakeCharacter_135_a();
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 := 4;
    raw_aps := [[1], [0], [5], [-6], [-47], [-5], [-131], [-56], [3], [-157], [225], [-70], [140], [397], [-347], [4], [748], [-338], [492], [32], [970], [-1257], [-102], [-1488], [974], [-1335], [686], [-1098], [-700], [-1055], [-1646], [-1833], [1098], [-1042], [-2941], [511], [-571], [713], [1596], [4134], [1828], [-520], [4826], [1670], [-1380], [4357], [-4162], [-5956], [4940], [4344], [-5202], [1546], [-3659], [1221], [-6255], [836], [-2231], [-4832], [6450], [-1050], [-1584], [6594], [-4343], [-2124], [-7516], [-6880], [-4986], [904], [8860], [-4454], [-8781], [2928], [9102], [-8183], [6136], [5643], [8991], [-12449], [8076], [-2833], [-4777], [-6464], [10680], [11566], [-1448], [-2376], [-14894], [16204], [-5082], [-10326], [4184], [-15576], [10220], [-2692], [5764], [-2437], [-5849], [-17032], [4147], [-3942], [-13751], [7944], [-6702], [-2760], [8930], [6944], [-4206], [-6571], [-9490], [11861], [-518], [15163], [19011], [-7906], [-3416], [4830], [12549], [8164], [23768], [21472], [12982], [-6006], [1164], [26496], [17110], [30251], [18820], [31890], [-11452], [7094], [-3200], [-20831], [-15605], [21349], [3702], [-1393], [-6906], [30493], [33488], [-15304], [-40122], [25098], [-43492], [11206], [-23964], [-34606], [-18477], [-41342], [21898], [-18487], [7593], [-3038], [-16732], [-8031], [-38487], [-5120], [28075], [-12856], [-1374], [8543], [13506], [21775], [3854], [12933], [17521], [-12573], [8945], [-58179]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_135_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_135_4_a_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_135_4_a_c(:prec:=1)
    chi := MakeCharacter_135_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_135_4_a_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_135_4_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_135_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![-1, 1]>],Snew);
    return Vf;
end function;