
// Make newform 6864.2.a.bn in Magma, downloaded from the LMFDB on 18 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_bn();".
// 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_bn();".
// 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, -5, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-3, -1, 1]];
    Rf_basisdens := [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], [-1, 0, 0], [-1, 0, -1], [2, 1, -1], [-1, 0, 0], [1, 0, 0], [-1, 3, 0], [2, -1, 1], [0, 3, -3], [-1, 1, 0], [5, -2, 1], [-4, 2, 2], [0, -3, -1], [-3, 3, 0], [4, 0, -2], [-4, -2, 0], [-2, 4, -2], [2, 2, -4], [9, -4, -1], [2, -2, 2], [2, -1, -1], [7, -1, 2], [-6, 4, -2], [-1, 4, -3], [-8, 2, -2], [-7, 3, 2], [4, 4, 0], [2, 0, -6], [-2, -3, -3], [4, -4, 2], [3, 5, -2], [0, -6, 6], [-13, 4, 1], [3, -1, 0], [-4, -7, 3], [14, -1, 5], [0, -1, -1], [11, 4, -5], [4, 2, 6], [-3, 9, 0], [6, -5, -1], [0, 1, -3], [6, -4, -2], [2, 9, -3], [-6, -1, -1], [-4, 4, -4], [3, 5, 0], [7, -2, 7], [-10, -5, 5], [-14, 0, -2], [17, 5, -2], [-2, -11, 3], [-4, -6, 8], [-14, -4, 6], [10, -10, 2], [-2, -8, 10], [8, 2, 0], [12, 5, -3], [2, -6, 6], [10, -7, -3], [1, 11, -10], [14, -1, 3], [10, -3, -1], [4, -3, 11], [14, -7, -5], [11, -8, 1], [-13, 10, -3], [16, 4, 2], [2, 12, -4], [12, -2, 4], [-3, 0, 5], [-10, 7, -11], [6, -2, 4], [-12, 4, 6], [21, -8, 1], [14, -6, 0], [4, 8, 2], [-4, 2, 2], [5, -2, -5], [2, -5, 3], [-6, -1, 3], [22, 2, -2], [20, 6, -6], [2, 0, -8], [-3, -13, 0], [-4, 4, 4], [-13, -2, 5], [26, 2, -2], [0, 9, -9], [3, 6, 5], [12, -1, 9], [22, -1, 5], [5, 0, -11], [-10, -2, -2], [-19, 6, -11], [-18, 16, 6], [-3, -4, 3], [-2, 0, -12], [-1, 1, 4], [-26, 4, 4], [-15, 17, 4], [8, 1, 7], [-4, -2, 2], [21, -3, 0], [19, 7, -10], [-6, 6, -6], [18, 4, -6], [-2, 11, -9], [2, 8, -2], [2, -4, 12], [19, -11, 6], [10, -5, -9], [-13, -2, -7], [-15, -10, 7], [7, -2, 1], [10, -6, 2], [27, -4, -3], [-14, 20, 2], [-6, 6, -2], [-12, 2, 4], [20, -18, 0], [12, 0, -4], [17, -9, -8], [-14, -6, -2], [-17, 18, 5], [9, -1, 14], [-2, 10, -8], [-16, -12, 4], [10, -14, 8], [22, 11, -9], [6, -1, 17], [2, -6, 4], [-26, 10, -4], [0, -9, 3], [10, -11, 5], [-10, 20, 0], [-13, -8, 9], [10, -5, -7], [-10, 4, 12], [-3, -11, 12], [-10, 5, -13], [-14, 1, 9], [-10, -6, 8], [-18, -1, -11], [18, 12, -8], [-8, 2, 6], [10, -6, 14], [11, -19, 0], [-14, -18, 16], [14, -6, -4], [0, 14, -4], [-14, 20, 4], [0, -4, -4], [12, 18, -16], [-14, -6, 16], [-14, 0, -14], [-29, -7, 2], [-13, -6, 11], [-24, -10, 6], [12, 21, -9], [-12, -6, -4], [31, -17, -2], [8, 15, -17], [8, 13, 3], [-39, 0, -9], [-8, 10, -2], [14, -8, -6], [-24, -4, 10], [-18, -5, 7], [-18, -14, 2], [-22, 10, 8], [16, 6, -10], [-40, 4, -6], [-12, -2, 0], [32, -12, 8], [2, 16, -8], [34, -11, 3], [-17, 22, -5], [3, -11, 0], [-4, 4, 6], [-7, -10, 5], [8, 24, -16], [-24, 7, -9], [4, -11, 19], [0, -26, 16], [0, 14, -4], [48, 4, -4], [5, -16, 9], [-48, 6, 4], [10, -12, -6], [-12, -1, 11], [-36, -2, 10], [4, 12, -20], [17, -2, 17], [-4, 6, 6], [6, 4, 16], [-14, 0, 12], [28, -2, 0], [10, 7, -1], [0, -6, -4], [-18, 1, 1], [3, 1, 22], [-6, -4, 8], [22, 0, 10], [0, 18, -8], [-32, 4, -18], [7, -4, -17], [4, -6, 18], [-22, 17, 13], [20, 0, 8], [-24, -30, 18], [38, 8, 0], [18, 14, -12], [-28, 22, 10], [22, -1, 9], [-14, -4, -6], [9, -1, 4], [-9, 11, -10], [14, -10, 2], [-1, -10, 5], [6, 10, 6], [16, 9, 1], [17, -18, -9], [-33, 2, -5], [-16, -24, 14], [-40, 10, 12], [-4, 16, -22], [10, -8, 6], [-41, 5, 4], [-2, 4, -10], [14, -12, 0], [-23, 1, -10], [6, 11, -11], [17, 7, -12], [8, 4, 4], [24, 4, 4], [-24, 8, -22], [-18, 9, 1], [6, -1, 15], [20, -5, 1], [6, 2, -4], [29, -5, 10], [44, 10, -6], [-8, 16, 14], [-14, 11, 11], [-8, -10, 16], [1, 0, 5], [8, -6, -8], [-18, 7, -15], [-24, 6, -4], [-32, -12, 14], [-8, 1, -25], [26, -2, 22], [-6, 5, -29], [5, 9, 10], [-22, 13, 7], [-2, -13, -3], [-22, -19, 7], [4, -22, 4], [-4, 12, -22], [9, 1, -16], [24, -12, -6], [4, 0, -10], [36, 4, 6], [11, -18, -5], [24, 16, -20], [-19, 28, -3], [35, -18, 1], [6, -23, 11], [-25, -3, 12], [-4, -26, 20], [-49, 6, 11], [-32, -20, 18], [3, 9, -18], [4, -2, -8], [27, 6, -19], [46, -10, -2], [-10, 3, 19], [-10, 9, 13], [32, -16, 2], [-26, 16, 4], [37, -32, -11], [24, -24, 6], [10, -15, 13], [-16, -23, 13], [48, 0, 16], [-2, 12, -12], [-2, -3, 13], [67, -2, -3], [13, -15, 10], [30, -2, -2], [56, -11, -5], [-12, 27, -15], [22, -3, -13], [-54, 8, -6], [-27, -3, 20], [22, -14, -20], [-20, -2, 24], [-22, 22, -18], [-30, 15, 9], [-18, 2, -14], [12, 11, -3], [-63, -1, -2], [8, 7, -7], [16, 26, -18], [-24, 24, -4], [30, -11, -11], [28, -10, 12], [24, -10, 10], [20, -10, -8], [2, 18, 0], [-37, -14, 1], [-33, 9, -2], [-26, -8, -2], [-50, -16, 6], [-8, 16, 10], [8, -22, 14], [-46, -5, 5], [-12, 26, -20], [-7, 7, -8], [-27, 11, 6], [-26, -8, 24], [19, -13, 14], [-9, 18, -19], [-12, -3, 23], [-8, 27, -9], [-43, 2, -15], [13, 26, -21], [-34, 8, 14], [-21, -21, -2], [10, 23, -5], [-6, -14, -2], [25, -7, 20], [46, -1, 17], [0, -17, 11], [2, -8, -2], [12, -14, 18], [-6, 3, -17], [-44, -6, -8], [-19, -15, 12], [-8, 6, 22], [24, 0, -2], [16, 9, -17], [-14, 36, -2], [-43, -12, 11], [-10, -34, 20], [26, 0, -16], [-45, 2, 13], [-22, 3, -15], [14, 18, -26], [6, -19, -9], [44, 2, 12], [-1, -12, -9], [5, -19, 14], [-22, -8, 6], [-7, 13, 12], [12, 18, -28], [2, -31, 7], [-7, 21, 16], [-4, 8, -10], [22, -6, 6], [-49, 9, -16], [-6, 8, 22], [-14, -3, -29], [60, 0, -4], [-44, -7, 1], [2, -15, 1], [4, 29, -11], [4, 18, -2], [18, -28, -10], [-36, -4, 16], [-34, -17, -7], [19, -5, -12], [17, -1, 16], [-14, 28, -28], [24, 8, -2], [-14, 20, 18], [10, -19, 17], [5, -25, -4], [6, -10, -4], [61, -4, 9], [-15, -34, 19], [6, -11, 19], [28, 3, -15], [-12, 29, -11], [38, 4, 0], [51, 0, 21], [0, -14, 28], [-10, -21, 29], [54, -15, 13], [-12, 2, 30], [-49, -8, 5], [-6, 4, -24], [-20, -20, 24], [26, -16, 26], [14, 26, 4], [18, 37, -13], [22, 12, 0], [64, 12, -12], [-32, 9, -25], [-2, -18, -18], [-30, -42, 30], [-34, -13, 21], [43, 10, 7], [-12, 15, 1], [5, -27, -12], [26, -6, -16], [48, 17, -1], [-13, -24, 17], [30, -16, -4], [22, 24, -8], [17, 19, -18], [14, 0, 14], [5, 25, -30], [-8, 10, -8], [58, 2, 10], [-10, 1, -5], [-26, -2, 0], [-45, -12, 11], [-10, 4, -16], [2, 24, 0], [-20, 2, 24], [-46, -30, 18], [-50, 32, 2], [22, -30, 2], [-17, 8, 3], [44, 16, 4], [96, 5, -1], [35, 14, 13], [22, 8, 20]];
    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_bn();".
// 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_bn(:prec:=3)
    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_bn();".
// 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_bn( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6864_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-14, -2, 4, 1]>,<7,R![12, 4, -6, 1]>,<17,R![-74, -48, 0, 1]>,<19,R![4, 4, -6, 1]>],Snew);
    return Vf;
end function;