
// Make newform 6864.2.a.cb in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6864_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_6864_2_a_cb();".
// 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_6864_2_a_cb();".
// 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 := [2, -3, -11, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [2, -9, -2, 1], [-3, -10, -1, 1], [2, 12, 1, -1]];
    Rf_basisdens := [1, 2, 1, 1];
    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_6864_a();"
function MakeCharacter_6864_a()
    N := 6864;
    order := 1;
    char_gens := [2575, 1717, 4577, 4369, 2641];
    v := [1, 1, 1, 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_6864_a_Hecke(Kf)
    return MakeCharacter_6864_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+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, 0, 0, 0], [0, -1, 0, 0], [-1, 0, -1, 0], [-1, 0, 0, 0], [1, 0, 0, 0], [0, -1, 0, 1], [-1, 0, -1, 0], [2, 2, 0, 1], [3, 1, -1, 0], [-2, -1, 0, 0], [-1, -2, 1, 1], [1, 0, -1, -2], [0, -1, 2, 1], [-3, -4, 1, -1], [-2, -2, 0, 0], [0, -2, 0, 0], [5, 0, -1, -1], [-2, -1, 0, -2], [-5, -2, -1, -1], [1, -2, 1, 0], [-4, -3, 2, 1], [2, 2, 2, 2], [11, 1, -1, -1], [5, -2, 3, -1], [6, 1, -2, -1], [-2, -4, 2, 0], [-4, -2, 0, 0], [3, -2, -1, 0], [4, 2, 2, 0], [-5, 3, 1, 0], [8, 0, 0, -2], [3, 1, 3, -1], [1, 3, -1, 0], [7, 0, 1, 2], [1, 4, -3, -2], [-2, 0, 2, -1], [-5, -5, 1, -1], [-10, 0, -2, 0], [2, 5, -2, 1], [8, 4, 2, 3], [6, 0, -2, 1], [16, -2, 0, 0], [1, -6, 1, 2], [-7, 2, -1, 0], [-4, 0, 4, 2], [-4, 1, 2, 3], [10, 1, 4, 0], [15, 4, 1, 4], [-3, 0, -1, -3], [9, 3, 1, 2], [-3, 0, -1, -4], [-10, -6, 4, 2], [8, -2, 4, 0], [-8, 0, 2, 2], [2, -6, 2, 2], [0, 6, 2, 0], [-15, -6, 1, -2], [0, 4, -2, 0], [-3, 2, -1, 0], [-7, 1, -1, 0], [-19, 2, -1, 0], [1, 0, -3, 0], [0, -2, 2, 1], [2, 2, 2, 1], [-19, -3, 1, 1], [3, 3, -3, 1], [-6, 2, 0, 0], [21, -2, 1, -1], [20, 6, -2, 0], [6, 1, 2, 2], [-1, -8, 1, 0], [2, 2, -2, 2], [-16, -2, 2, 2], [9, 5, 3, 1], [0, -2, 0, 2], [-10, 2, 0, -4], [3, 2, 5, 1], [0, -3, -4, -2], [17, 10, 1, 4], [-6, 4, -4, -3], [-2, 0, 4, 2], [-2, 8, -2, 0], [2, 8, -4, 0], [-24, -1, -2, 1], [16, 0, 4, 0], [0, -5, -4, -2], [-14, 4, 0, 2], [1, 0, 3, 6], [-15, -1, -1, -1], [2, 2, -4, -7], [-13, 4, 1, 0], [15, 5, 1, -1], [-5, -2, -5, -5], [4, -5, -6, -2], [-4, -6, 4, 4], [1, 3, -3, 1], [2, -4, -4, 0], [-14, -7, 4, 1], [-14, 4, -4, 4], [-9, 1, 1, 4], [-1, 8, -7, -2], [-12, 4, -4, -2], [12, -3, -4, -1], [-4, 5, 2, 5], [-8, -4, -6, -4], [10, -6, 2, 2], [7, 2, 5, 2], [-4, 2, -4, 4], [-4, -4, -6, 2], [3, -5, 1, -4], [-23, 2, -3, 4], [14, -9, -2, 0], [5, 1, 3, 3], [-7, -1, 3, 3], [-8, -8, 2, -4], [10, 1, 4, -2], [20, 6, -4, 0], [8, 4, 6, 2], [7, -12, 7, -1], [24, 6, 2, 0], [-9, -10, 1, -1], [3, -1, 7, 2], [7, -6, -1, -1], [17, -9, 3, 3], [22, -3, 2, -1], [7, -12, 5, -3], [4, 16, -4, 0], [-18, -10, 6, 0], [-21, 2, -5, -2], [-17, -4, -5, -6], [-2, 2, -6, -4], [-4, 2, -4, -4], [16, -4, 4, -3], [1, -6, -1, 2], [2, 8, 0, -4], [25, 1, -7, -3], [11, 4, -1, -2], [16, 4, 6, 4], [26, 13, 2, 5], [9, 0, -3, -4], [-27, 6, -1, -4], [4, -6, 8, 2], [5, -4, 7, 6], [10, 4, 0, 4], [-6, 4, -2, -8], [0, -4, 2, -4], [-1, -3, 7, 0], [14, 6, -2, 0], [2, -10, 2, 4], [-6, 10, -4, -6], [22, 0, 0, 4], [-4, 12, 0, -2], [-1, 4, 11, 3], [-10, -2, -2, 6], [-10, -6, 6, 2], [19, -1, 5, 4], [7, 1, -5, 1], [-17, 2, -3, -3], [-7, -8, 7, -2], [11, -4, -1, -1], [5, 1, 1, 0], [-17, 6, -1, 0], [4, 2, -6, -3], [-13, 7, -1, 3], [2, 8, -2, -8], [0, 14, -4, 0], [10, 2, -4, 4], [13, -6, -3, 0], [18, 12, -8, 0], [30, -6, 6, -4], [-1, -10, 9, 5], [-17, 8, -9, 5], [-24, 2, -2, 0], [-12, 8, 0, 2], [-20, 4, -10, 0], [1, 4, -7, -2], [-13, -3, 7, 1], [-32, 1, 2, -1], [-4, -14, 6, -2], [2, -13, 4, -4], [10, 8, 6, -2], [6, 8, -6, -1], [17, 0, -5, -2], [-21, -8, 3, 3], [-14, -2, -4, 4], [-5, -6, -7, 3], [-7, -19, 3, -3], [-12, 10, 2, 0], [2, 6, -2, -2], [12, 8, -4, -7], [4, -4, -8, -2], [10, -8, -2, 0], [19, 11, 7, 1], [4, -16, 4, -2], [-32, -12, -2, -4], [-24, -4, -2, -2], [-4, -6, -6, -4], [-11, 6, 7, 2], [-27, 8, 1, -3], [-5, -10, 1, -2], [-15, 3, -9, 2], [-14, 4, 4, 0], [-15, -8, 3, 1], [7, 0, 3, -5], [-4, 2, -2, -10], [15, 3, 9, 7], [-10, 0, -2, 4], [33, 10, -1, 4], [-10, 0, 2, -4], [-15, 6, -9, -1], [30, 4, 0, -4], [-28, -10, -4, -4], [-8, 0, -8, 2], [-7, 4, 3, 6], [-9, -4, -11, -1], [10, -5, 0, -1], [11, -3, 3, 2], [11, 10, -1, -1], [-34, -5, -2, 0], [4, 12, -6, 0], [17, 6, 5, 4], [4, -7, 8, 2], [10, 17, -4, 0], [1, 0, -3, -5], [26, -6, 0, 2], [-4, -2, 2, 2], [36, 6, -4, 0], [24, -7, -2, -1], [-8, 2, 8, 0], [16, 8, -10, -6], [0, -1, 10, 11], [-17, 0, -5, -4], [-15, 7, 1, 4], [28, 8, 8, 2], [20, 4, 4, 0], [-4, -6, 6, 6], [-7, -2, 15, 6], [-6, -4, -12, -3], [-18, 2, -4, 1], [18, -2, -2, -8], [-3, 13, 3, 0], [-8, 0, 4, 6], [-26, 2, -4, -4], [31, -6, -3, -2], [-33, 0, 3, 3], [-27, 9, -1, -7], [-3, 4, -7, -3], [55, 4, 3, 2], [-41, -4, -5, -5], [-10, -14, 0, -4], [27, -4, 9, -8], [27, 6, 7, -1], [-2, 18, -2, 5], [19, -1, 7, 0], [-17, 0, -3, 10], [-22, -10, 2, -1], [-15, -12, -3, 0], [-8, -10, -10, -8], [-36, 2, 2, 6], [1, -7, -1, 2], [37, 12, 9, 7], [2, 10, -4, -8], [-16, 10, -14, 2], [-39, 5, 1, -5], [24, 0, -4, 0], [0, -11, 8, -4], [-42, -9, -10, -4], [-25, -8, 3, 2], [-9, 9, -5, -4], [18, 6, 4, -2], [16, 13, -2, 6], [4, 10, -2, -6], [9, 15, -5, -6], [16, -14, 2, 4], [-19, 3, -1, -1], [23, -10, -1, -5], [9, -6, -3, 0], [-13, 6, 3, 4], [-7, 4, -3, -5], [-33, 6, -9, 5], [27, 13, 5, 11], [-13, -8, -5, -3], [-9, 2, 7, 8], [-11, 6, 9, 0], [24, -20, 8, 0], [-24, 4, -6, -2], [-17, -2, 1, 2], [-47, 3, -5, -1], [7, 15, -1, -4], [-22, 0, 4, 4], [12, 2, 10, 1], [-7, 12, -5, 6], [-9, 0, -15, -2], [37, 20, 3, 9], [-2, -15, -2, 1], [-10, -2, 6, 0], [-22, -2, -4, -12], [21, 2, -11, -3], [15, -12, 7, 0], [20, -16, 6, -4], [11, -12, 9, 2], [-37, 5, 9, 2], [24, 10, -6, -1], [-4, 8, 4, -8], [29, -2, 15, -3], [4, -28, 10, 1], [34, 6, 8, 2], [-34, -8, 2, 0], [20, 10, 10, 4], [-10, -10, -14, -4], [32, 19, -4, 2], [38, 3, -10, -3], [-30, -6, -6, 0], [0, -18, 4, -4], [24, 10, 6, -2], [26, 4, 2, 0], [-13, 0, -3, -4], [-20, -10, -6, -4], [3, -17, 11, -2], [-37, -23, 5, -2], [-12, -8, -6, 2], [37, -15, 1, 0], [9, 15, 3, 3], [-7, -4, -1, -4], [-18, 20, -10, 3], [-8, 11, -2, -6], [11, 1, -3, -7], [38, 10, 6, 10], [9, 5, 9, 0], [17, -6, 1, -4], [-12, 4, 14, 4], [-5, -3, -11, 0], [43, 12, 5, 12], [11, 6, -13, -8], [12, -14, -4, 0], [41, 22, 3, 11], [3, 2, 1, 0], [4, -10, 2, 4], [-31, 9, -5, -2], [57, 10, 3, 7], [28, 22, 6, 6], [-1, 8, 1, -6], [-14, 6, 10, 2], [-27, -13, 1, 1], [2, 2, -6, 8], [-8, -16, -6, -6], [-17, 11, 1, 9], [-3, 4, -9, 2], [-2, -12, -12, -2], [9, 16, 5, 6], [24, -6, 6, 0], [14, 11, 2, -6], [-8, 11, -2, -9], [51, -8, 9, -5], [12, -7, -4, -1], [-3, -20, 5, 1], [-31, -12, -1, 0], [18, -23, 2, 5], [-37, -4, -1, -11], [34, 8, 4, 10], [-33, -11, 3, 4], [52, -6, 4, 0], [-7, -4, -9, 0], [-2, 4, -6, 2], [8, -12, 12, -5], [1, 14, 7, 4], [13, 18, -7, 2], [-41, -10, -3, -7], [28, 10, 0, 4], [-12, 4, -4, 2], [5, 4, -9, -6], [24, 3, -4, -7], [12, -21, 2, 5], [28, 8, -18, -14], [7, 16, -17, -3], [32, -2, -8, 0], [-43, -2, -3, -6], [59, -5, 3, 2], [27, 12, 1, -3], [10, -31, 6, 2], [-2, 21, -10, 8], [-23, -12, 1, 2], [43, 12, 1, 6], [-5, 18, 7, 4], [49, -2, 9, -1], [16, 1, 10, -4], [22, 2, -8, -2], [3, 28, -1, 2], [8, 4, 2, 5], [57, 22, 7, 11], [-15, 5, -7, 1], [-17, 6, -5, -3], [-6, 16, -2, -2], [0, 10, -4, -4], [13, 0, 11, -1], [-47, -8, 9, 4], [-14, 4, 8, 0], [36, 20, -4, 6], [43, -20, 1, 2], [8, 24, -6, 4], [34, -12, 0, 8], [21, -12, -7, 0], [-43, -3, 13, 7], [15, 0, -3, 10], [-21, -15, 5, 0], [-17, 0, -3, -7], [-11, -4, -5, -16], [-46, -11, 8, -2], [23, 10, 11, 9], [24, -16, 6, -6], [-28, -3, 8, 1], [16, -6, -4, 8], [0, 7, -2, -9], [-26, 6, -8, -6], [-24, 8, 2, 10], [-1, -8, -7, -10], [14, 26, -14, 0], [-26, -5, 22, 10], [-11, -2, -3, 7], [-26, 0, 0, 6], [60, 6, 6, 8], [19, -2, -17, -5], [-14, -14, 6, 10], [28, 4, -10, -12], [-26, 3, -6, -6], [2, 16, 6, 2], [-5, 12, -7, 2], [5, -13, -9, -9], [11, 14, -5, -11]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6864_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6864_2_a_cb();".
// 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_6864_2_a_cb(:prec:=4)
    chi := MakeCharacter_6864_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_6864_2_a_cb();".
// 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_6864_2_a_cb( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6864_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-8, 22, -10, -2, 1]>,<7,R![72, -60, -28, 2, 1]>,<17,R![300, 186, -64, -2, 1]>,<19,R![72, -60, -28, 2, 1]>],Snew);
    return Vf;
end function;