
// Make newform 6864.2.a.bd 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_bd();".
// 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_bd();".
// 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 := [-10, -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], [-2, 1], [1, 0], [-1, 0], [4, 0], [-6, 0], [4, -1], [-4, 1], [2, -2], [0, 2], [-4, 1], [2, -3], [-4, 0], [0, 2], [10, 1], [8, -1], [-6, 3], [4, 2], [2, -1], [-4, 2], [-8, -2], [-6, 2], [-6, 4], [6, 0], [14, -1], [-8, -1], [6, -2], [8, 3], [-4, 2], [-8, -3], [2, -4], [-16, 0], [-6, 6], [0, 0], [2, -4], [-6, -3], [-2, 1], [-4, -5], [-4, 4], [-22, 0], [-8, 5], [-16, 2], [-14, 2], [14, -3], [-8, 4], [-14, -2], [-8, -4], [-2, -1], [12, 2], [2, -1], [-20, -2], [0, 2], [0, 1], [0, 0], [16, 4], [-12, -4], [4, 3], [-16, -1], [2, 1], [14, 0], [-10, 0], [18, 2], [-4, -7], [12, -7], [-6, -5], [-18, 0], [-6, -2], [-14, 6], [14, 0], [-18, 1], [12, -4], [0, -9], [0, 0], [-16, 2], [0, -2], [-14, 7], [-10, 6], [-14, 5], [-4, 4], [-10, -5], [0, -8], [16, 3], [0, -4], [-16, -4], [6, -10], [10, -7], [-18, 6], [-6, -4], [-8, 12], [-26, 5], [10, 8], [8, -3], [14, -7], [-32, 0], [14, 8], [-8, -3], [-4, 8], [-30, 0], [-2, 1], [18, 0], [-18, -2], [-12, 2], [18, 5], [-14, 6], [-34, -1], [10, -4], [0, 7], [18, 2], [-4, 6], [14, 6], [-10, -8], [6, -1], [-34, -4], [-4, 1], [24, -4], [14, -10], [-16, 10], [-26, -2], [8, 2], [14, 2], [-30, 0], [-6, 1], [-20, 4], [-16, 9], [6, -1], [20, -7], [-12, -8], [26, 4], [2, 2], [14, 9], [22, 5], [26, -8], [-32, -1], [6, 3], [10, -8], [-2, 2], [20, 0], [-12, -10], [22, -2], [2, -4], [-18, 9], [-8, 16], [-26, -8], [-8, -2], [-2, -14], [-16, 8], [-20, -4], [8, 4], [-22, -2], [0, -9], [-32, 2], [36, 0], [-20, 4], [-18, -2], [8, 4], [30, 2], [38, 2], [-18, -6], [28, -8], [8, -8], [26, -9], [20, -2], [-6, -10], [20, -10], [-6, -9], [4, -11], [-14, 1], [-28, 8], [-8, 6], [10, -3], [40, 0], [-6, 3], [30, 7], [4, 9], [34, 4], [-4, -13], [-32, -4], [-4, 7], [18, -4], [-10, 14], [-6, 4], [36, 3], [16, -6], [8, 8], [46, 0], [-6, 9], [-6, -5], [50, -1], [-10, 20], [20, 6], [-12, 4], [8, -7], [42, -6], [-14, 4], [8, -10], [18, -8], [20, -17], [24, -14], [-14, 4], [-32, -8], [6, 11], [-46, -4], [-2, -13], [-12, -10], [34, -10], [-4, 5], [-6, -12], [-64, 2], [-10, -4], [-32, -8], [34, -11], [12, 0], [-48, 0], [10, -10], [32, -8], [8, -12], [28, 2], [34, 0], [16, 9], [18, -5], [-14, 10], [-58, -2], [-60, 1], [14, 4], [14, 0], [40, -9], [-12, 10], [-24, 4], [34, 4], [2, 3], [26, -13], [44, 2], [12, 4], [8, -16], [-6, 8], [-26, 0], [-4, 10], [-30, 2], [2, -15], [14, -11], [38, -3], [0, 2], [-32, 1], [-4, 6], [30, 8], [-34, 3], [-6, -6], [12, 16], [-18, -8], [6, -4], [48, -10], [-20, -7], [4, -9], [-8, -2], [18, 6], [42, 4], [48, -4], [14, -5], [16, 4], [8, 7], [6, -14], [-16, 17], [-18, 10], [24, 10], [-4, -7], [-10, -18], [6, -12], [50, -8], [30, -2], [-32, 12], [-26, 0], [8, 4], [70, 2], [-10, 16], [22, -9], [30, 14], [-26, 5], [40, 7], [-20, -6], [-10, 22], [-12, 18], [2, -14], [42, -6], [24, -13], [-66, 0], [34, -18], [-2, 10], [-40, 0], [-48, -10], [-4, 18], [-56, 1], [-4, -18], [24, 7], [-32, 19], [6, 15], [-12, 8], [20, -15], [10, -14], [-50, -8], [38, 2], [-64, 4], [-8, 14], [32, 6], [-2, 8], [0, 18], [34, 0], [0, -16], [36, -11], [-44, 4], [4, 17], [12, 4], [-16, -10], [-8, 20], [-50, -6], [-10, 7], [38, 5], [2, -6], [80, 0], [-44, -8], [-2, 14], [-8, 13], [58, 4], [4, -13], [-16, 11], [-10, 13], [8, -4], [56, 6], [18, -12], [-54, 14], [-32, -4], [-44, 2], [-64, 0], [-22, 4], [-44, 0], [10, 11], [48, 11], [-14, 12], [-56, -4], [26, 15], [8, -13], [-54, 10], [-16, -16], [-16, 7], [-8, 5], [-58, 4], [30, 3], [66, -9], [28, -12], [30, 0], [-34, 0], [26, 16], [8, -4], [40, -11], [8, 18], [8, 0], [-60, -2], [-32, -12], [28, -2], [14, -10], [-16, 6], [60, -7], [-16, -12], [-16, 6], [-4, 9], [50, -8], [2, 17], [54, -10], [-4, 14], [-52, -8], [-62, 3], [54, -8], [-44, -10], [70, -10], [-36, 12], [-64, 0], [4, -22], [4, 4], [50, -18], [-24, -3], [-6, 12], [-22, -1], [20, 15], [-38, -4], [-20, -10], [-70, -6], [22, 4], [-54, 2], [-38, -2], [2, -14], [-36, -9], [-28, 20], [32, -2], [44, 6], [-6, -14], [54, -13], [52, 9], [0, -18], [-30, 4], [30, 6], [28, -1], [-38, -17], [18, -4], [-14, -14], [8, -12], [-42, -3], [44, -9], [34, 3], [6, -1], [18, -18], [-18, 16], [-16, -12], [-2, -5], [-10, 6], [-12, 18], [18, 3], [40, 8], [34, -8], [4, 12], [32, -10], [-18, -10], [-32, -18], [36, -19], [-50, -4], [-16, 9], [20, -10], [-30, 12], [-58, -3], [8, -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_bd();".
// 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_bd(: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_bd();".
// 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_bd( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6864_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-10, 1, 1]>,<7,R![-8, 3, 1]>,<17,R![-4, 1]>,<19,R![6, 1]>],Snew);
    return Vf;
end function;