
// Make newform 6864.2.a.bl in Magma, downloaded from the LMFDB on 24 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_bl();".
// 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_bl();".
// 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 := [-8, -1, 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_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], [1, 0], [0, 1], [0, 1], [1, 0], [-1, 0], [-4, 0], [0, 2], [0, 1], [0, -1], [-2, -2], [-2, 0], [2, 1], [6, -1], [4, 0], [-10, 0], [8, 1], [-2, 3], [-2, -3], [8, 2], [2, -3], [10, 0], [4, -2], [8, 0], [-10, 0], [-8, -2], [4, 3], [-4, 1], [6, 2], [-6, -1], [-2, -4], [8, -1], [8, -2], [6, -6], [10, 2], [-12, 4], [-6, 4], [-6, -1], [8, -3], [0, 5], [4, 4], [10, 0], [-16, -1], [2, -4], [-6, -2], [-12, 5], [2, 2], [-6, 2], [4, -8], [2, -7], [12, -2], [-8, -5], [14, 0], [12, -6], [6, -3], [4, 4], [-6, -6], [4, -4], [-14, -1], [26, -1], [26, -1], [-14, -4], [8, -10], [-8, 0], [-18, -3], [0, -5], [-30, 1], [6, 8], [16, 4], [-6, 2], [-4, 2], [8, -3], [-8, -8], [-10, 3], [14, -6], [0, -10], [10, -8], [22, 1], [20, -4], [-14, 7], [20, 4], [-22, -3], [0, 8], [-6, 3], [22, -2], [-4, 0], [-16, 8], [-2, -1], [22, -6], [-22, -4], [-20, 8], [0, 9], [-10, 12], [32, -1], [-18, -5], [4, 4], [20, 2], [26, -3], [-14, 2], [18, 0], [10, 3], [-6, 0], [-16, 8], [16, -6], [2, 3], [-14, -6], [8, -1], [-10, 8], [24, -3], [-6, -2], [14, -8], [-10, 10], [-8, -2], [-2, -3], [-30, 0], [-14, 13], [18, 2], [-8, -8], [-6, 4], [4, -4], [22, 0], [10, -6], [12, 6], [16, -15], [2, 14], [16, -13], [26, 1], [24, 7], [-4, 0], [-18, -8], [-4, 12], [-8, 13], [-32, 1], [10, -8], [-34, 3], [-26, 1], [24, -6], [-28, 8], [-6, 10], [-32, -2], [0, 0], [22, -8], [-20, -7], [4, 12], [30, -8], [20, -2], [-14, -6], [24, 8], [8, 0], [-16, -4], [18, 10], [-14, -5], [-8, 14], [-4, 8], [-4, 12], [16, 8], [14, -18], [0, -12], [10, -14], [-26, -2], [-12, 0], [-16, 8], [40, -1], [12, 2], [-12, 0], [12, 2], [-36, -1], [-10, -7], [10, -13], [-30, 10], [20, -10], [-10, 3], [-20, 4], [-34, 5], [36, -5], [-34, 9], [4, -6], [8, -7], [-30, 10], [6, 7], [14, 0], [-8, -8], [26, -12], [10, -9], [4, 2], [-10, -14], [-46, -4], [6, -17], [14, 5], [40, -5], [14, -12], [-28, 2], [-16, 0], [-12, 3], [16, -8], [14, -16], [-26, 12], [-2, 12], [-14, -5], [-12, -6], [-2, 8], [-42, -6], [-14, -3], [-58, 0], [28, 7], [-30, 12], [10, -6], [0, 11], [34, 4], [24, 6], [18, 0], [6, -6], [-8, -3], [60, 0], [32, 0], [34, 2], [-26, -6], [16, -12], [-40, -6], [0, -14], [10, -19], [-8, -13], [-36, 12], [-22, 6], [12, 3], [-2, -20], [38, 8], [-8, -11], [2, -8], [44, 8], [-10, 0], [-42, 9], [-4, 7], [-4, 22], [16, 0], [-16, 8], [-34, -4], [50, 4], [-12, -2], [24, -16], [0, 9], [42, -1], [-24, 17], [14, -16], [-10, 9], [-4, -6], [22, 0], [-44, 7], [44, -12], [12, -16], [34, 4], [-30, 8], [-16, -6], [30, 5], [-12, 5], [44, -2], [42, 10], [56, 6], [34, -14], [30, -7], [-54, 2], [4, -7], [-6, -14], [22, -15], [16, -20], [-12, 18], [-52, -1], [-40, 8], [8, -6], [0, 10], [30, -14], [-22, -14], [-50, 0], [-18, 14], [-42, 14], [26, 12], [-16, -1], [-18, -6], [50, -13], [28, -15], [16, -6], [-66, 2], [-12, -2], [22, -6], [28, 12], [64, 1], [-6, -12], [22, 14], [-14, 2], [28, -20], [-30, 8], [12, 22], [54, -7], [44, 10], [30, 11], [32, 9], [-28, -13], [54, -2], [32, -1], [32, -8], [-62, 4], [-10, 14], [34, -6], [-36, -2], [-40, -6], [-18, 24], [52, 2], [-26, -4], [28, -12], [38, 9], [-60, -4], [-16, 19], [12, -12], [20, 14], [24, -4], [22, 6], [28, -21], [-40, -7], [22, 18], [44, -4], [2, 10], [-30, -2], [-48, -9], [-38, -4], [-18, 11], [22, -9], [6, 7], [-2, -10], [-12, -18], [16, -18], [14, -18], [62, -10], [26, -24], [28, 4], [28, -6], [-8, 12], [-2, 5], [26, 3], [-42, 8], [-22, 18], [-26, 9], [-18, 19], [-14, -18], [52, 12], [-28, 21], [-28, -5], [30, -12], [-6, -19], [-16, 11], [-26, -6], [-12, 18], [-42, -8], [-48, -6], [-18, 22], [14, 5], [16, -10], [-60, -4], [-12, -6], [0, 12], [12, -30], [-70, -2], [8, 2], [0, 15], [20, -8], [28, -10], [38, 13], [-26, -4], [12, 9], [2, -2], [44, 2], [48, -4], [8, 15], [-44, 10], [30, -16], [22, -6], [-36, 4], [28, -4], [-46, -8], [44, -12], [54, 6], [-44, 7], [-4, 2], [80, -1], [26, 11], [24, 6], [-56, -10], [-6, 6], [38, -12], [12, -4], [16, -28], [-10, -14], [96, -3], [96, 0], [12, -10], [16, 22], [-2, 18], [52, -13], [-38, 13], [-80, 2], [-6, 20], [-62, 14], [38, -13], [-8, -9], [-24, -2], [70, 2], [34, -22], [42, -21], [-2, -25], [34, -7], [-8, -17], [2, -6], [8, 22], [-28, 8], [6, -3], [-34, 2], [-54, 0], [-24, -17], [-12, 12], [56, -6], [0, -16], [34, 8], [-14, 22], [-24, 18], [4, 7], [10, 16], [16, 7], [20, -22], [-50, 8], [66, -5], [-40, 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_bl();".
// 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_bl(:prec:=2)
    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_bl();".
// 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_bl( : 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, -1, 1]>,<7,R![-8, -1, 1]>,<17,R![4, 1]>,<19,R![-32, -2, 1]>],Snew);
    return Vf;
end function;