
// Make newform 1386.2.k.s in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1386_k();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1386_k_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_1386_2_k_s();".
// 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_1386_2_k_s();".
// 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 := [49, 0, 7, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
    Rf_basisdens := [1, 1, 7, 7];
    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_1386_k();"
function MakeCharacter_1386_k()
    N := 1386;
    order := 3;
    char_gens := [155, 199, 1135];
    v := [3, 1, 3];
    // 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_1386_k_Hecke();"
function MakeCharacter_1386_k_Hecke(Kf)
    N := 1386;
    order := 3;
    char_gens := [155, 199, 1135];
    char_values := [[1, 0, 0, 0], [-1, 0, -1, 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 := 2;
    raw_aps := [[-1, 0, -1, 0], [0, 0, 0, 0], [1, 1, 1, 0], [0, 1, 0, 0], [0, 0, -1, 0], [5, 0, 0, 0], [0, 0, -6, 0], [3, -1, 3, 0], [-1, -1, -1, 0], [-1, 0, 0, -2], [0, 0, -4, 0], [-1, 1, -1, 0], [-3, 0, 0, -3], [-4, 0, 0, 0], [8, 2, 8, 0], [0, 1, 1, 1], [0, 1, -2, 1], [-9, 2, -9, 0], [0, 3, -4, 3], [-7, 0, 0, 1], [0, 1, 3, 1], [0, -1, 0, 0], [-8, 0, 0, 2], [-4, -4, -4, 0], [-11, 0, 0, -2], [0, 0, -3, 0], [-5, -3, -5, 0], [-3, 3, -3, 0], [0, 4, 0, 4], [13, 0, 0, 2], [8, 0, 0, 3], [13, 1, 13, 0], [0, -6, -3, -6], [-4, 0, 0, 0], [-10, 2, -10, 0], [0, 1, -6, 1], [0, -8, 0, -8], [2, 1, 2, 0], [-2, 0, 0, 5], [5, 2, 5, 0], [0, -3, 12, -3], [-10, 0, 0, 0], [-8, -2, -8, 0], [0, -7, 7, -7], [-3, 0, 0, 6], [0, -5, -9, -5], [-7, 0, 0, 3], [-15, 0, 0, -1], [0, 2, 8, 2], [16, 0, 16, 0], [9, 3, 9, 0], [4, 0, 0, 5], [0, 9, -1, 9], [-2, 0, 0, 2], [-11, 4, -11, 0], [0, 3, -12, 3], [0, -4, -16, -4], [10, 3, 10, 0], [0, 2, -17, 2], [15, 0, 0, 3], [0, -1, -5, -1], [-12, 0, 0, 0], [9, 0, 0, 5], [0, -3, -9, -3], [-13, 4, -13, 0], [12, 0, 12, 0], [18, -1, 18, 0], [17, 0, 0, -3], [0, -9, -3, -9], [14, 0, 0, -6], [0, 8, 14, 8], [-18, 3, -18, 0], [0, 5, -23, 5], [-5, 6, -5, 0], [14, 0, 0, -3], [-5, -11, -5, 0], [0, 9, -3, 9], [-10, -8, -10, 0], [11, -4, 11, 0], [0, 3, 11, 3], [-6, 0, 0, 6], [-12, 0, 0, 8], [0, 5, -16, 5], [-16, 0, 0, 0], [4, 3, 4, 0], [8, -10, 8, 0], [6, 0, 0, 6], [18, 8, 18, 0], [11, 0, 0, -8], [4, 0, 0, 10], [16, -2, 16, 0], [0, 3, 18, 3], [0, -4, -20, -4], [-16, 0, 0, -2], [-2, -6, -2, 0], [12, 0, 0, -3], [10, 4, 10, 0], [0, -6, 18, -6], [3, -7, 3, 0], [3, 2, 3, 0], [9, 0, 0, -7], [0, -12, 0, -12], [0, 3, -21, 3], [-20, -8, -20, 0], [0, -3, -37, -3], [0, -10, 1, -10], [0, 0, 0, -3], [15, 3, 15, 0], [0, 12, -12, 12], [-14, 0, 0, 4], [-16, -2, -16, 0], [0, 0, 2, 0], [11, 0, 0, -2], [0, 4, 24, 4], [11, 0, 0, -9], [0, 6, 3, 6], [-12, 0, 0, -7], [0, -9, -15, -9], [5, -1, 5, 0], [-4, 0, 0, 4], [0, 4, 18, 4], [-7, 0, 0, 3], [-18, 6, -18, 0], [0, 3, 6, 3], [32, -5, 32, 0], [-9, 0, 0, -6], [-17, 9, -17, 0], [12, -6, 12, 0], [-12, 0, 0, 2], [-15, -10, -15, 0], [0, -6, 8, -6], [40, 0, 0, 2], [16, 0, 16, 0], [-2, 0, 0, -8], [-6, 0, -6, 0], [6, 0, 0, 8], [0, -4, 8, -4], [0, -9, 23, -9], [14, 0, 0, -8], [0, -4, -40, -4], [-4, 0, 0, -12], [-13, 2, -13, 0], [0, -6, -28, -6], [1, 0, 0, -13], [0, 13, 9, 13], [-12, 0, 0, 6], [18, 0, 0, 8], [0, 0, -36, 0], [-14, -9, -14, 0], [-31, -7, -31, 0], [7, -6, 7, 0], [9, 0, 0, 6], [14, 0, 0, 3], [16, -11, 16, 0], [0, -10, 16, -10], [-4, 0, 0, -2], [30, 2, 30, 0], [25, 4, 25, 0], [-7, 0, 0, 15], [0, 18, -3, 18], [-18, -6, -18, 0], [7, 0, 0, -7], [16, 0, 0, 16], [-18, -3, -18, 0], [0, -10, -10, -10], [0, 2, 32, 2], [0, 15, 17, 15], [0, -4, -32, -4], [-52, 0, 0, 0], [6, -12, 6, 0], [0, 10, 16, 10], [17, 0, 0, -3], [48, 3, 48, 0], [0, 8, 34, 8], [0, 1, 45, 1], [-7, 0, 0, -2], [-11, 0, 0, 1], [0, 16, -2, 16], [-18, 0, 0, 14], [10, 0, 10, 0], [3, 5, 3, 0], [32, 0, 0, 10], [24, 0, 0, 8], [-12, 0, -12, 0], [0, 15, 21, 15], [0, 11, 5, 11], [0, -2, -51, -2], [0, -3, 50, -3], [3, 11, 3, 0], [0, -5, -26, -5], [-14, -6, -14, 0], [-27, 0, 0, 9], [6, 11, 6, 0], [-15, -15, -15, 0], [0, 0, -24, 0], [0, 0, -15, 0], [0, 4, 24, 4], [53, -5, 53, 0], [29, 0, 0, 1], [10, -5, 10, 0], [0, -10, -13, -10], [-26, 0, 0, 7], [-35, -3, -35, 0], [0, 9, -37, 9], [-16, 0, 0, -11], [0, -16, -4, -16], [16, 0, 16, 0], [13, 13, 13, 0], [-15, 0, 0, 15], [0, -7, -14, -7], [23, -11, 23, 0], [18, 0, 0, -12], [32, 0, 0, 12], [6, -18, 6, 0], [0, -3, 24, -3], [3, -4, 3, 0], [0, 7, 24, 7], [0, -9, 15, -9], [28, 4, 28, 0], [30, 0, 0, 12], [21, 8, 21, 0], [-42, 0, 0, -7], [3, 12, 3, 0], [-5, -21, -5, 0], [0, 0, 0, -6], [9, 0, 0, -16], [-45, 0, -45, 0], [0, -15, 27, -15], [4, 15, 4, 0], [24, -6, 24, 0], [0, 4, -12, 4], [0, -2, -24, -2], [-40, 0, 0, -9], [0, 4, 10, 4], [12, 0, 0, -13], [0, 3, 54, 3], [-30, 14, -30, 0], [-23, 10, -23, 0], [46, 0, 0, 5], [31, 0, 0, 11], [0, -2, 10, -2], [-4, 16, -4, 0], [0, -8, -36, -8], [-17, -15, -17, 0], [-11, 0, 0, 17], [0, -6, 0, 0], [-37, 0, 0, -15], [0, 11, -10, 11], [0, -13, -29, -13], [-28, 0, 0, -5], [33, 0, 0, -4], [-28, -16, -28, 0], [0, 7, -17, 7], [0, 0, 0, 8], [0, 4, 13, 4], [-16, 2, -16, 0], [0, -8, 6, -8], [0, 0, 32, 0], [-19, 0, 0, -11], [45, 5, 45, 0], [0, -4, 4, -4], [12, 0, 0, -24], [0, -8, 60, -8], [19, 0, 0, -8], [0, -2, 7, -2], [56, 1, 56, 0], [20, 0, 0, 16], [-21, 0, 0, 18], [3, 0, 0, 5], [0, 20, -16, 20], [-22, -8, -22, 0], [0, -12, 8, -12], [0, -8, -3, -8], [-14, 18, -14, 0], [15, 0, 0, -7], [11, 1, 11, 0], [6, 15, 6, 0], [0, 6, 29, 6], [19, 21, 19, 0], [-5, -17, -5, 0], [0, 15, -18, 15], [0, 22, -12, 22], [18, 0, 0, -3], [63, 0, 0, 2], [-28, 10, -28, 0], [24, -6, 24, 0], [0, -2, 33, -2], [-22, 0, 0, 16], [0, 1, 9, 1], [48, 0, 0, 0], [0, 8, 50, 8], [0, -19, -28, -19], [39, -9, 39, 0], [0, 0, 0, -6], [27, 0, 0, 8], [0, -4, -37, -4], [15, -13, 15, 0], [-12, 0, 0, 0], [28, 22, 28, 0], [-13, 0, 0, -27], [-21, -13, -21, 0], [57, 0, 57, 0], [0, 15, -10, 15]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1386_k_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1386_2_k_s();".
// 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_1386_2_k_s(:prec:=4)
    chi := MakeCharacter_1386_k();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_1386_2_k_s();".
// 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_1386_2_k_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1386_k();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![36, 12, 10, -2, 1]>,<13,R![-5, 1]>,<17,R![36, -6, 1]>,<23,R![36, -12, 10, 2, 1]>],Snew);
    return Vf;
end function;