
// Make newform 1350.2.e.n in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1350_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1350_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_1350_2_e_n();".
// 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_1350_2_e_n();".
// 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 := [4, 0, -2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 0, 1, 0], [0, 2, 0, 1], [0, 4, 0, -1]];
    Rf_basisdens := [1, 2, 2, 2];
    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_1350_e();"
function MakeCharacter_1350_e()
    N := 1350;
    order := 3;
    char_gens := [1001, 1027];
    v := [2, 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_1350_e_Hecke();"
function MakeCharacter_1350_e_Hecke(Kf)
    N := 1350;
    order := 3;
    char_gens := [1001, 1027];
    char_values := [[0, -1, 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 := 2;
    raw_aps := [[1, -1, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [2, -2, 1, -1], [0, 0, 2, -2], [0, 2, -1, 0], [0, 0, 0, 2], [5, 0, 0, 1], [0, 0, 1, 0], [0, 0, 1, -1], [0, -2, -1, 0], [4, 0, 0, 3], [0, 9, 0, 0], [5, -5, 1, -1], [6, -6, -2, 2], [-6, 0, 0, 1], [0, -3, -1, 0], [-8, 8, 0, 0], [0, -7, 3, 0], [-6, 0, 0, -3], [1, 0, 0, 0], [-2, 2, 6, -6], [3, -3, -1, 1], [9, 0, 0, 0], [-1, 1, -4, 4], [6, -6, -1, 1], [0, 2, 6, 0], [-3, 0, 0, 5], [8, 0, 0, -1], [0, 9, -2, 0], [4, 0, 0, -3], [0, 6, -4, 0], [-3, 3, 0, 0], [0, 4, 0, 0], [0, -6, 0, 0], [-20, 20, 0, 0], [0, -10, -4, 0], [-5, 0, 0, -1], [0, 0, -8, 0], [0, 0, 4, -4], [3, 0, 0, -5], [8, 0, 0, 4], [-6, 6, -2, 2], [0, 20, 0, 0], [12, 0, 0, -5], [-4, 0, 0, -4], [0, 1, 1, 0], [8, -8, -4, 4], [-21, 21, -3, 3], [0, 4, -7, 0], [-9, 0, 0, -6], [0, 12, 1, 0], [13, -13, 4, -4], [3, 0, 0, -1], [0, 3, 4, 0], [-18, 18, 1, -1], [-12, 0, 0, -1], [8, 0, 0, 3], [-4, 4, -1, 1], [24, -24, 2, -2], [0, -1, -5, 0], [0, 6, -9, 0], [-8, 0, 0, -6], [0, -6, 2, 0], [-1, 1, 2, -2], [0, 0, 7, -7], [1, -1, 3, -3], [0, -16, -2, 0], [0, -12, 0, 0], [-14, 14, 0, 0], [9, -9, 0, 0], [24, 0, 0, -4], [2, -2, 6, -6], [0, -4, -8, 0], [-4, 0, 0, -2], [0, -24, 1, 0], [6, -6, -8, 8], [-2, 0, 0, 8], [0, 24, 6, 0], [0, -5, 2, 0], [0, -9, -7, 0], [4, -4, -9, 9], [-18, 0, 0, -3], [10, 0, 0, -8], [4, -4, 3, -3], [-18, 18, -4, 4], [9, 0, 0, 8], [-1, 1, -6, 6], [12, -12, 3, -3], [0, 2, -3, 0], [-3, 0, 0, -3], [-12, 12, -5, 5], [22, 0, 0, 1], [0, -15, -5, 0], [0, 1, 3, 0], [24, 0, 0, -1], [0, 0, 0, 0], [0, 0, 0, 12], [13, 0, 0, 3], [-28, 0, 0, -4], [23, -23, 3, -3], [12, 0, 0, 12], [0, -3, 11, 0], [24, -24, -4, 4], [0, 1, -11, 0], [-11, 0, 0, -12], [12, -12, -6, 6], [-3, 0, 0, -2], [0, -30, 3, 0], [-8, 8, -10, 10], [0, -4, 3, 0], [10, 0, 0, -9], [0, -15, -12, 0], [-5, 5, -15, 15], [-16, 0, 0, 4], [15, -15, 12, -12], [0, -25, 3, 0], [12, 0, 0, 5], [0, -6, -5, 0], [27, -27, 7, -7], [0, 22, -12, 0], [8, -8, -6, 6], [-6, 6, -13, 13], [15, 0, 0, 11], [-17, 17, 9, -9], [-24, 0, 0, -4], [-2, 2, 1, -1], [-30, 0, 0, 9], [-16, 16, 0, 0], [0, 26, 0, 0], [-19, 0, 0, 9], [0, -42, -3, 0], [0, 28, -9, 0], [16, 0, 0, 9], [0, 9, 2, 0], [0, -11, 16, 0], [-30, 0, 0, 8], [0, 20, 6, 0], [0, 18, 5, 0], [21, 0, 0, -6], [-31, 0, 0, 1], [-12, 12, -16, 16], [0, -22, 2, 0], [-9, 0, 0, -11], [-10, 0, 0, -15], [30, -30, -3, 3], [-46, 46, 0, 0], [45, -45, -2, 2], [0, 13, 3, 0], [-12, 0, 0, -13], [0, -22, -4, 0], [-24, 0, 0, -6], [-38, 0, 0, -4], [0, -30, 10, 0], [29, -29, -3, 3], [0, 0, 3, -3], [-4, 0, 0, 3], [-12, 12, 16, -16], [-17, 0, 0, 4], [0, 12, 17, 0], [-9, 9, 7, -7], [12, 0, 0, -4], [0, 32, 0, 0], [9, 0, 0, 13], [0, 9, 0, 0], [18, -18, 2, -2], [-16, 0, 0, 0], [14, -14, 9, -9], [-13, 0, 0, 0], [0, 24, -11, 0], [-33, 33, -5, 5], [0, -8, 12, 0], [0, 0, -11, 0], [-1, 1, 10, -10], [0, -8, -5, 0], [0, 51, -4, 0], [22, -22, 4, -4], [18, 0, 0, -8], [10, 0, 0, -2], [16, -16, -4, 4], [-10, 10, -13, 13], [-21, 21, 7, -7], [0, -10, -12, 0], [-9, 0, 0, 8], [0, 30, 1, 0], [-12, 12, -8, 8], [-26, 0, 0, 0], [-25, 25, -1, 1], [0, 34, 8, 0], [6, 0, 0, 11], [40, 0, 0, -4], [-15, 15, 5, -5], [2, 0, 0, 8], [-12, 12, -12, 12], [36, 0, 0, -6], [0, -3, 14, 0], [0, 19, 6, 0], [8, -8, -9, 9], [-24, 24, -10, 10], [42, 0, 0, -2], [0, -30, 5, 0], [-14, 14, -18, 18], [0, -46, 0, 0], [-8, 8, -12, 12], [-3, 0, 0, -27], [-48, 0, 0, -4], [-10, 0, 0, -11], [0, 9, -19, 0], [-39, 39, 6, -6], [0, 1, -7, 0], [-17, 0, 0, -10], [0, 12, 7, 0], [-16, 16, -21, 21], [39, -39, -3, 3], [0, 36, -8, 0], [28, -28, 12, -12], [0, 2, -10, 0], [-15, 15, -14, 14], [0, 0, 0, -4], [0, 30, 12, 0], [0, -14, -13, 0], [0, -14, 6, 0], [0, 0, 22, 0], [-26, 0, 0, 6], [0, -3, -9, 0], [-62, 62, -1, 1], [21, -21, 18, -18], [-6, 0, 0, 6], [2, -2, -13, 13], [-33, 33, 7, -7], [0, -28, -7, 0], [5, 0, 0, -3], [0, -50, -2, 0], [0, 3, 2, 0], [-4, 4, 0, 0], [-12, 12, -1, 1], [0, -23, 12, 0], [-6, 0, 0, -9], [0, -12, 0, 0], [-24, 0, 0, -11], [-9, 9, -11, 11], [5, 0, 0, 15], [0, 50, 6, 0], [-4, 0, 0, -27], [0, 30, 8, 0], [-36, 36, 3, -3], [-26, 0, 0, 2], [0, -39, 1, 0], [0, -23, 5, 0], [-6, 0, 0, 10], [0, 32, 0, 0], [39, 0, 0, -4], [0, -30, -7, 0], [-5, 5, 24, -24], [42, -42, 11, -11], [-33, 0, 0, 13], [2, 0, 0, 3], [-7, 7, 15, -15], [-6, 0, 0, -1], [-8, 0, 0, 0], [2, -2, 26, -26], [27, -27, 3, -3], [0, -38, 3, 0], [-62, 0, 0, 4], [0, -15, -2, 0], [-20, 20, 0, 0], [18, 0, 0, 16], [-51, 51, 6, -6], [0, 26, 12, 0], [0, -12, -12, 0], [0, -8, -11, 0], [1, 0, 0, 3], [32, -32, 18, -18], [0, 4, 24, 0], [0, -37, -14, 0], [-32, 0, 0, -12], [0, 51, -5, 0], [-20, 20, -20, 20], [-43, 0, 0, -8], [18, -18, 20, -20], [0, -24, -11, 0], [0, -26, 21, 0], [-12, 12, -8, 8], [-38, 38, 12, -12], [0, -28, 18, 0], [18, 0, 0, 10], [22, 0, 0, 16], [0, -18, -1, 0], [4, -4, 9, -9], [-39, 0, 0, 14], [54, -54, 12, -12], [-3, 0, 0, 11], [0, 21, -10, 0], [0, -72, 2, 0], [-16, 16, -15, 15], [0, -12, -13, 0], [-2, 2, 7, -7], [6, -6, -27, 27], [36, 0, 0, 0], [-13, 13, 15, -15], [0, 56, -6, 0], [-12, 0, 0, -5], [-22, 0, 0, 14]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1350_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1350_2_e_n();".
// 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_1350_2_e_n(:prec:=4)
    chi := MakeCharacter_1350_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(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_1350_2_e_n();".
// 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_1350_2_e_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1350_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![4, 8, 18, -4, 1]>,<11,R![576, 0, 24, 0, 1]>,<17,R![-24, 0, 1]>],Snew);
    return Vf;
end function;