
// Make newform 6864.2.a.ca in Magma, downloaded from the LMFDB on 23 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_ca();".
// 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_ca();".
// 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, 10, -10, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [2, -10, 0, 1], [-7, 10, 1, -1]];
    Rf_basisdens := [1, 1, 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, -1], [-1, 0, 0, 0], [-1, 0, 0, 0], [-1, 1, 1, -1], [-1, 2, 0, 1], [1, 2, -1, 1], [1, -1, 2, -1], [2, 1, 1, 0], [0, -2, 0, 0], [3, 0, 1, -1], [1, 1, -2, -1], [4, 0, -2, 2], [2, -2, -2, 0], [2, -4, 1, -2], [-4, 2, -3, 0], [2, 3, -2, 0], [2, -2, -2, 0], [1, -2, -1, -3], [3, 1, 1, -3], [6, -2, 0, -2], [2, -1, -1, 0], [0, 2, 0, 4], [3, -1, 1, 3], [8, -2, 3, 0], [6, -4, 3, -2], [-7, 0, -2, 1], [-2, 0, 1, 0], [3, -3, -1, -1], [4, 2, 3, 0], [2, 3, 1, -4], [-11, 1, -3, 1], [3, 2, 0, -1], [-1, -2, -2, 1], [-1, -2, -4, 1], [2, 1, 0, 2], [16, -2, -1, 0], [-1, 1, 0, -1], [1, 2, -4, 5], [-3, 2, 0, -1], [8, -4, 5, -4], [-3, 0, 4, 5], [-13, 4, -6, 3], [12, -2, 5, 0], [-3, 1, 3, 3], [10, -1, 3, 0], [-5, 2, -2, 1], [0, 2, -3, 4], [-13, -1, -1, 1], [3, -4, 3, -3], [2, 2, 2, -4], [10, 2, 0, -2], [-12, 2, -3, 2], [8, 2, -2, 0], [0, 2, 4, 2], [-1, 0, -4, 1], [-6, 2, -5, 0], [1, -6, 1, -3], [-13, -3, -4, 3], [3, 0, 4, -5], [-3, -6, 0, 3], [5, 0, -2, -3], [-7, 2, 3, 3], [-4, 1, -4, 6], [-4, 9, 2, 0], [10, -2, -2, 4], [-2, -6, 4, -4], [6, -2, -4, 0], [-10, -3, -7, 6], [1, 4, -3, 7], [12, 6, 2, -4], [-4, 4, 1, -2], [-8, 5, 3, 4], [2, -2, 4, 2], [6, 6, -2, 4], [10, -4, 3, -2], [8, -7, -1, -4], [17, -10, 1, -3], [5, 2, -4, -3], [2, -6, -5, 4], [26, 0, 0, -2], [-16, -2, -1, -6], [-1, -1, -7, 9], [4, 4, 0, 0], [20, -1, 5, 0], [2, 6, -3, 0], [1, -6, 0, 5], [14, -5, -1, -6], [7, 4, -2, -1], [-15, 0, -11, 7], [-28, -3, 1, 0], [-6, -4, 7, -8], [4, -1, -4, 2], [-4, 2, 6, 0], [2, -3, -1, 8], [-12, 6, 1, 6], [-3, -3, -1, -5], [-2, 4, 8, 4], [-3, -11, 4, -3], [15, 6, -2, -5], [22, 0, -4, 6], [1, 3, -7, 1], [-31, -5, 0, 3], [-8, -8, 6, 2], [2, 8, -1, 2], [-9, 0, 0, -1], [8, 0, 3, 4], [10, -4, 2, -4], [-5, -3, 7, -1], [3, 0, 6, -9], [16, -9, 7, -4], [-12, -9, 4, 0], [2, -13, 3, -6], [-10, -2, -7, -4], [-20, -1, 3, -10], [6, -6, -6, -2], [-14, 4, 8, -8], [-20, 0, -10, 2], [8, 6, -2, -2], [-8, 6, -6, 0], [5, -1, 3, 7], [-4, 4, -5, 6], [2, 3, -7, 6], [17, 9, 6, -7], [-14, 10, -3, 2], [2, -2, -9, 6], [-2, 14, 2, -2], [-25, -4, 0, -1], [-19, 2, -14, 3], [-2, 8, 1, 6], [6, -8, -5, 6], [11, -6, 8, -7], [15, -2, -3, 7], [12, 2, -1, 10], [26, 7, -3, -4], [9, -10, 2, -9], [-12, 4, 6, -6], [5, 11, -3, 9], [-19, 6, -10, 3], [-21, 4, 0, -1], [-24, 8, 3, 0], [7, -14, -2, 1], [-2, -12, 0, 0], [6, -4, -10, -2], [-28, -4, -2, 6], [-13, -3, -9, 5], [-2, -10, 2, 2], [30, -6, -2, -6], [6, -6, -4, 4], [-8, 10, -1, 10], [-10, -12, 6, 2], [-8, 12, 2, -2], [20, -2, 10, 0], [2, -6, -2, -2], [-13, 9, -7, 11], [-4, -5, -5, -6], [-2, -14, 6, -10], [-11, -6, 0, 1], [14, 8, -6, 16], [-7, 9, -1, 11], [1, -2, 1, -9], [-35, -4, 0, 1], [-10, 9, -3, 4], [6, 12, -10, 6], [10, -4, 5, -6], [-8, -4, -9, 6], [13, -10, -11, 1], [6, -4, 10, -8], [-4, -14, -4, 4], [18, 0, -7, 6], [2, 16, -6, 0], [-8, 0, 9, -14], [16, 10, -5, -4], [10, -14, -3, 0], [-17, 18, 0, 9], [2, 9, -6, 0], [29, 1, 11, -5], [-8, -4, -3, 2], [18, 9, 7, -8], [16, -4, 8, 4], [-49, 6, 4, 1], [3, 4, -5, -9], [10, 8, 8, -4], [-2, -6, 0, -4], [8, 2, -8, -8], [14, -5, -6, 6], [24, 0, 1, 2], [20, -12, 15, -4], [-9, 10, 0, 1], [-26, -4, 6, -6], [14, -8, -4, -2], [2, 7, 10, 0], [-18, 8, -4, 6], [-24, 4, 0, 2], [-26, -8, 4, 4], [-30, -2, 2, 0], [27, -10, 5, -1], [24, 4, 0, 2], [23, 12, 0, -1], [-17, 17, -11, 15], [-8, -2, 11, -6], [-12, 4, -2, 4], [-4, -2, 7, -14], [14, 14, -6, 0], [32, -17, -3, -4], [-8, -2, -1, -8], [-5, 0, -8, 3], [-6, -8, 2, 2], [-6, -2, -4, 6], [-6, 0, -10, 4], [10, 8, -9, 10], [-10, 8, -4, 2], [-21, 2, -2, 9], [-22, 8, 12, -2], [-25, 3, 5, 5], [17, 1, 11, 3], [10, 6, 18, -8], [4, -9, 7, -4], [6, 2, -3, -4], [-1, 2, -7, 9], [16, -11, 15, -12], [22, 7, -1, 12], [12, -14, 13, -14], [4, -6, 14, -18], [6, 14, -2, 12], [4, 4, 15, -8], [29, -7, -7, -9], [-2, 2, 6, -14], [18, 8, -4, 0], [-9, -7, -16, 13], [-7, -12, 3, 3], [-23, 7, -17, 15], [10, -16, -8, 6], [-8, 8, -16, 16], [6, 6, 10, -4], [29, -12, 12, -11], [-7, 14, 2, 1], [23, -8, -2, -5], [36, 12, 1, -4], [19, 9, 16, -13], [-16, 6, -11, 4], [2, 18, -6, 12], [-31, 10, -9, 13], [4, -4, 0, -6], [48, 1, 5, 0], [0, -14, 7, -14], [-7, 6, 14, -9], [8, -8, -2, -2], [4, -14, -6, -6], [9, 2, 2, 5], [2, 2, -2, 0], [37, -6, 3, -9], [-15, -7, -10, 7], [-3, 6, 0, 11], [-35, -16, 4, -1], [-15, -2, 12, -17], [-8, -2, 12, 2], [22, 12, -3, 0], [7, -21, 7, -1], [-24, 14, 11, -2], [-40, 6, 0, 6], [-28, -16, -1, 6], [32, -1, -1, -10], [16, -12, 6, -4], [-12, -5, 0, 4], [12, -5, -11, 8], [-47, 6, -8, 3], [-29, -11, 1, -3], [2, 6, 12, -12], [22, 7, -9, 16], [-38, 2, 2, -8], [-17, -7, 5, 3], [28, -10, 8, -6], [18, -1, 11, -6], [-20, -4, -7, 14], [-17, 16, -18, 19], [3, -10, -3, 11], [0, -2, -5, -2], [6, 14, -10, 8], [12, -15, -5, -8], [-26, -4, 4, 4], [9, 12, 14, -7], [-23, -4, -12, 11], [46, -6, 1, 2], [-10, -4, 4, 0], [-17, 12, -2, -1], [34, -5, 1, 2], [-1, -17, -15, 9], [22, -4, -4, 8], [-55, -4, 4, -3], [-25, 4, 5, -1], [-19, 14, 0, 11], [-50, 10, -15, 10], [-9, 1, -8, 11], [24, 4, -7, 4], [-10, -18, -8, 4], [-18, 8, -9, 8], [-5, 10, -10, 5], [-20, 4, -8, 14], [55, 2, 0, 7], [5, 11, -9, 1], [-1, -8, -8, 3], [-20, 8, 10, 0], [30, -2, 0, 10], [-13, -6, -6, 7], [-10, 10, -10, 0], [-22, 8, 0, 2], [-22, 0, 3, 12], [-2, 6, 2, -2], [26, -3, -2, 14], [-23, 1, -3, -3], [22, -2, -4, 10], [-10, -6, -2, 14], [26, 2, 16, 0], [8, -2, 1, -4], [-3, 12, -7, 3], [8, -26, 16, -10], [45, -5, 9, -1], [-3, -9, 11, -7], [38, 8, 10, -8], [-1, -21, 6, 1], [-18, -13, -1, -10], [33, 0, -1, 1], [13, -12, -9, 7], [28, -9, 8, -14], [-2, 13, -17, 14], [16, -10, 0, 0], [5, -7, 13, -9], [-13, 8, 10, -9], [16, 12, -6, 14], [-9, 15, 5, 7], [-29, -6, 6, -7], [-9, 20, -14, 21], [0, 2, -14, 8], [30, -8, -9, -10], [-9, 14, 1, -1], [-32, -2, -18, 6], [-9, -1, -1, 3], [-42, -8, 3, -10], [12, 4, 9, -10], [7, 6, 0, -5], [44, 2, 6, 4], [32, -17, -4, 2], [-20, 4, -15, 16], [-18, 8, 12, -4], [44, -7, 6, 0], [17, 4, 7, 19], [-12, -12, 6, 14], [11, 10, 24, -19], [28, 2, 22, -14], [26, 3, 11, -18], [-11, 7, -11, 7], [26, 6, -1, 10], [13, -1, -15, -11], [-14, 0, -10, -8], [41, -14, 4, -5], [-55, -5, -1, -3], [-32, 0, 8, -6], [-4, 8, 18, -14], [-5, -11, 9, -15], [-4, -12, -19, 16], [9, -10, -10, -13], [40, 0, 2, 8], [-13, -4, -15, -7], [47, 0, 0, 11], [13, -10, -9, -5], [34, -2, -6, -6], [-14, -14, 8, -2], [10, -12, 8, -10], [21, -4, 11, -1], [11, -7, 7, 3], [7, -25, 3, -7], [-2, -20, 2, 4], [-12, 16, 2, -2], [12, -18, 18, 0], [-71, 12, -12, 9], [-27, 7, -7, 7], [10, 0, -12, 6], [24, -1, -12, 0], [-8, -3, -3, 12], [5, -8, -13, -9], [57, 8, 11, -7], [-13, 12, -6, -3], [20, -18, 18, -18], [-20, 27, -1, 6], [-10, -10, -18, 8], [21, -6, 22, -5], [-13, 10, -4, 3], [44, -2, 10, -16], [-48, 5, -18, 6], [-56, 10, -4, 6], [0, -4, 6, 8], [28, -18, 16, -20], [50, -8, 0, 2], [15, -4, 15, -3], [0, -22, -3, 10], [10, 20, -2, 6], [11, -10, -14, 7], [32, -16, 22, -14], [-16, -10, 15, -22], [-17, 4, -13, 1], [60, 3, 5, 2], [19, 2, -12, 27], [-17, -1, 3, -9], [46, 6, 19, -10], [13, -8, -13, 21], [18, 5, 8, -4], [-2, 12, 13, 0], [-34, 24, -2, 16], [-57, -1, -7, 7], [2, -14, -2, 14], [-21, 13, -4, 5], [4, -6, 4, -18], [18, -8, -12, -8], [11, 4, -13, -15], [26, -2, -10, 18], [10, -9, -13, -2], [2, -6, 0, 0], [24, 0, 8, -6], [-30, 6, 0, -12], [-2, -14, 10, 4], [18, 4, 17, -18], [40, -16, -8, -6], [-12, 9, 6, -12], [-12, -4, 10, -20], [47, -14, 14, -17], [-4, -3, 4, 4], [38, 2, 0, 8]];
    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_ca();".
// 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_ca(: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_ca();".
// 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_ca( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6864_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, 10, -10, -1, 1]>,<7,R![28, 8, -18, -1, 1]>,<17,R![232, -90, -22, 6, 1]>,<19,R![72, -108, -48, 0, 1]>],Snew);
    return Vf;
end function;