
// Make newform 361.2.e.g in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_361_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_361_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_361_2_e_g();".
// 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_361_2_e_g();".
// 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, -1, 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_361_e();"
function MakeCharacter_361_e()
    N := 361;
    order := 9;
    char_gens := [2];
    v := [4];
    // 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_361_e_Hecke();"
function MakeCharacter_361_e_Hecke(Kf)
    N := 361;
    order := 9;
    char_gens := [2];
    char_values := [[0, 0, 0, 0, 0, -1]];
    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, -1, -1, 1, 1, 0], [1, 1, 1, 0, 0, 0], [0, 0, 0, 1, -1, 1], [0, -1, 1, 0, 0, 0], [0, -1, 2, 0, 2, -1], [-1, -1, 0, -2, 3, 0], [-1, 2, 0, 0, -2, 1], [0, 0, 0, 0, 0, 0], [-2, 2, 0, 0, 0, 2], [0, -1, -5, -1, 0, 0], [3, 3, -1, -3, -2, -2], [0, 3, 3, 0, -2, -1], [-1, 4, -4, -3, 0, 3], [-5, 0, 5, 3, 0, -2], [-1, -2, -2, -2, -1, 0], [2, -3, 0, 1, 1, 2], [0, -7, 2, -2, 7, 0], [0, 4, 0, -4, -4, -3], [6, -4, 6, -4, 6, 0], [2, 0, -2, 0, -10, 2], [-4, 0, 0, 4, 0, -4], [-1, 3, 6, 7, 0, -7], [0, 3, 6, 0, -9, -9], [3, 3, 5, -1, -2, -5], [2, 5, 4, -4, -5, -2], [3, 3, 5, -4, 1, -5], [0, -4, 3, 6, 3, -4], [0, -3, -3, 0, 6, 6], [2, 0, -2, -4, 5, -2], [-3, -8, -8, 0, 1, 7], [-8, -8, 1, 7, 1, -1], [6, -7, -1, 1, 7, -6], [-6, 1, 0, 5, 5, -10], [0, 0, -6, 5, -5, 6], [-7, -8, -4, 3, 0, -3], [-10, 3, 3, 0, 0, -3], [8, 0, -8, -5, -3, 3], [0, -3, -2, 0, -2, -3], [8, 0, 0, -8, -8, 11], [-14, 2, -2, 2, -2, 14], [0, 9, -6, 9, -6, 9], [-8, 10, 2, 10, -8, 0], [9, 7, 7, 0, -11, 4], [3, -9, 5, 2, 0, -2], [6, -11, 1, -6, 10, 10], [-8, 0, 12, 0, -8, 0], [1, 3, -6, 6, -3, -1], [-4, 0, 4, 1, -9, -3], [-6, -6, -6, 0, 9, -3], [3, -14, -14, 0, 8, 6], [5, 0, -5, 9, -1, 14], [0, 10, -11, 3, -11, 10], [-2, -2, -9, -3, 5, 9], [7, -10, 0, 3, 3, 2], [2, -5, 7, -5, 2, 0], [4, 1, -9, -13, 0, 13], [-10, 4, -13, -3, 0, 3], [-13, 0, 13, -1, -7, -14], [0, -12, 3, -5, 3, -12], [-3, 13, 0, -10, -10, -4], [8, -7, 7, -7, 7, -8], [0, -1, 8, -3, 8, -1], [8, -11, 13, 5, 0, -5], [-9, 2, -7, 9, 5, 5], [10, -17, -10, -17, 10, 0], [-1, -1, -28, 1, 0, 28], [0, 4, -10, -15, -10, 4], [-8, 0, 8, 3, 15, -5], [-3, 0, 3, -2, -1, -5], [0, 11, -17, 0, 6, 6], [0, 3, 3, 15, 3, 3], [-5, 14, 1, -1, -14, 5], [-8, -8, 9, -6, 14, -9], [-14, 6, -21, 14, 15, 15], [15, -8, -8, 0, 5, 3], [-6, 16, -11, -5, 0, 5], [-1, -8, -8, -8, -1, 0], [-10, -12, -13, 13, 12, 10], [-4, 5, 3, -3, -5, 4], [-2, 16, -7, 16, -2, 0], [3, 11, 11, 0, 5, -16], [2, -2, -2, -4, 0, 4], [13, 13, 17, -9, -4, -17], [9, -17, 0, 8, 8, 9], [8, -18, -1, 1, 18, -8], [20, 20, -7, -2, -18, 7], [-15, 5, 11, 15, -16, -16], [0, -13, -13, 0, 3, 10], [-1, 0, 1, -6, -25, -7], [15, 10, 6, -15, -16, -16], [0, -3, 0, -21, 0, -3], [19, -1, 0, -18, -18, -4], [1, -12, 9, -1, 3, 3], [15, 7, 16, 1, 0, -1], [12, 0, -12, 2, -3, 14], [0, 17, 1, 17, 0, 0], [-21, 1, 0, 20, 20, -1], [0, -8, -8, 18, -8, -8], [-21, 2, 0, 2, -21, 0], [-1, 4, 9, 4, -1, 0], [-2, -4, 0, 6, 6, -6], [1, 1, -13, -6, 5, 13], [9, 5, -19, -9, 14, 14], [-12, -3, -3, 0, 12, -9], [0, 6, 6, 0, 4, -10], [7, -3, 6, -7, -3, -3], [6, -28, -1, 1, 28, -6], [-2, 2, 0, 0, 0, 8], [13, 9, -17, 9, 13, 0], [27, -7, 13, -27, -6, -6], [-6, 24, 24, 0, -11, -13], [2, 0, -2, 2, 14, 4], [-23, 12, -23, 12, -23, 0], [0, 2, 6, -18, 6, 2], [-10, 0, 0, 10, 10, -18], [1, 0, -1, 8, 14, 9], [9, -33, 5, -4, 0, 4], [-3, 5, 5, 0, 14, -19], [-27, 0, 0, 27, 0, 0], [-19, -19, -1, 25, -6, 1], [13, -2, 0, -11, -11, 10], [0, 1, 8, 0, 8, 1], [0, 4, 1, 0, -5, -5], [12, -1, -1, 0, -4, 5], [-8, -24, 12, 8, 12, 12], [19, 11, -2, 2, -11, -19], [12, 12, 21, -7, -5, -21], [-8, 13, 21, 29, 0, -29], [-21, 0, 21, 17, 27, -4], [0, 6, -6, 25, -6, 6], [13, 3, 15, -15, -3, -13], [-8, 17, 4, -4, -17, 8], [13, -6, -3, -6, 13, 0], [-11, -18, 4, 15, 0, -15], [-3, 4, 4, 0, 1, -5], [1, 0, 21, 0, 1, 0], [13, 13, 2, 0, -13, -2], [0, -9, 22, -12, 22, -9], [9, 11, 11, 0, 11, -22], [0, 19, -23, -27, -23, 19], [-8, -8, -2, 1, 7, 2], [-10, -3, 0, 13, 13, 8], [-11, -11, -26, 1, 10, 26], [0, 2, 37, 2, 0, 0], [21, 2, 7, -21, -9, -9], [20, 19, 8, -12, 0, 12], [-11, -15, 18, -18, 15, 11], [-4, -16, 3, -3, 16, 4], [-8, -1, 0, 9, 9, 3], [0, -28, 17, 6, 17, -28], [-7, 15, -9, -2, 0, 2], [-33, -8, 1, 33, 7, 7], [17, 1, -15, 1, 17, 0], [4, 4, 23, -21, 17, -23], [23, 0, -23, -25, -1, -2], [-36, 11, 11, 0, -10, -1], [-3, -9, -13, 3, 22, 22], [12, 11, -7, 7, -11, -12], [-6, -6, 9, -1, 7, -9], [20, -14, -23, -14, 20, 0], [4, 4, -9, -13, 0, 13], [-17, -14, -9, 8, 0, -8], [-8, 24, 0, 0, -24, 8], [20, -22, 12, -12, 22, -20], [0, 18, -24, 3, -24, 18], [-27, 0, 27, -11, -7, -38], [21, 0, -7, -28, 0, 28], [14, -11, 12, -11, 14, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_361_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_361_2_e_g();".
// 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_361_2_e_g(:prec:=6)
    chi := MakeCharacter_361_e();
    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_361_2_e_g();".
// 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_361_2_e_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_361_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![9, 0, 9, 3, 0, -3, 1]>,<3,R![1, -3, 6, -8, 12, -6, 1]>],Snew);
    return Vf;
end function;