Properties

Label 588.2.o.a
Level $588$
Weight $2$
Character orbit 588.o
Analytic conductor $4.695$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [588,2,Mod(19,588)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(588, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("588.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 588.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.69520363885\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.432972864.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + x^{6} + 4x^{5} - 6x^{4} + 8x^{3} + 4x^{2} - 8x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{6} - \beta_1) q^{2} + \beta_{2} q^{3} + \beta_{5} q^{4} + ( - \beta_{7} + \beta_{6} - \beta_{4} + \beta_1) q^{5} + \beta_1 q^{6} + ( - \beta_{7} - 2) q^{8} + ( - \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} - \beta_1) q^{2} + \beta_{2} q^{3} + \beta_{5} q^{4} + ( - \beta_{7} + \beta_{6} - \beta_{4} + \beta_1) q^{5} + \beta_1 q^{6} + ( - \beta_{7} - 2) q^{8} + ( - \beta_{2} - 1) q^{9} + ( - \beta_{5} - \beta_{4} + 2 \beta_{2}) q^{10} + ( - \beta_{6} - \beta_{5}) q^{11} + \beta_{3} q^{12} + ( - 2 \beta_{5} - 2 \beta_{3} + 2 \beta_1) q^{13} + (\beta_{7} - \beta_1) q^{15} + ( - \beta_{7} + 2 \beta_{6} - \beta_{4} + 2 \beta_{2} + 2 \beta_1 + 2) q^{16} + ( - \beta_{6} - 2 \beta_{5} - \beta_{4}) q^{17} + \beta_{6} q^{18} + ( - \beta_{7} - 2 \beta_{6} - \beta_{4} - \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{19} + (2 \beta_{7} + 2 \beta_1) q^{20} + (\beta_{7} + \beta_{5} + \beta_{3} + 2) q^{22} + ( - \beta_{6} + \beta_{3} - \beta_1) q^{23} + ( - \beta_{4} - 2 \beta_{2}) q^{24} + (2 \beta_{6} - \beta_{5} + \beta_{4} - \beta_{2}) q^{25} + (2 \beta_{7} + 2 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + 4) q^{26} + q^{27} - 2 q^{29} + (\beta_{7} + \beta_{4} - \beta_{3} - 2 \beta_{2} - 2) q^{30} + ( - 2 \beta_{6} - 2 \beta_{5} - \beta_{4} + 2 \beta_{2}) q^{32} + (\beta_{6} - \beta_{3} + \beta_1) q^{33} + (3 \beta_{7} + \beta_{5} + \beta_{3} + 2) q^{34} + ( - \beta_{5} - \beta_{3}) q^{36} + (\beta_{7} + 2 \beta_{6} + \beta_{4} + \beta_{3} + 4 \beta_{2} + 2 \beta_1 + 4) q^{37} + ( - 2 \beta_{6} + 2 \beta_{5} + 4 \beta_{2}) q^{38} + (2 \beta_{6} + 2 \beta_{5}) q^{39} + (2 \beta_{7} + 2 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} - 4) q^{40} + ( - \beta_{7} - 2 \beta_{5} - 2 \beta_{3} + 3 \beta_1) q^{41} + ( - 3 \beta_{7} - \beta_{5} - \beta_{3} + 4 \beta_1) q^{43} + ( - 2 \beta_{6} - 4 \beta_{2} - 2 \beta_1 - 4) q^{44} + ( - \beta_{6} + \beta_{4}) q^{45} + (\beta_{5} - \beta_{4} - 2 \beta_{2}) q^{46} + ( - 2 \beta_{7} - 4 \beta_{6} - 2 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - 4 \beta_1 - 4) q^{47} + (\beta_{7} - 2 \beta_1 - 2) q^{48} + ( - 2 \beta_{5} - 2 \beta_{3} - \beta_1 + 4) q^{50} + (\beta_{7} + \beta_{6} + \beta_{4} - 2 \beta_{3} + \beta_1) q^{51} + ( - 4 \beta_{6} - 8 \beta_{2}) q^{52} + ( - 4 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{2}) q^{53} + ( - \beta_{6} - \beta_1) q^{54} + ( - \beta_{7} - \beta_{5} - \beta_{3} - 2 \beta_1 - 2) q^{55} + (\beta_{7} + \beta_{5} + \beta_{3} + 2 \beta_1 - 2) q^{57} + (2 \beta_{6} + 2 \beta_1) q^{58} - 4 \beta_{2} q^{59} + (2 \beta_{6} + 2 \beta_{4}) q^{60} + ( - 2 \beta_{7} + 4 \beta_{6} - 2 \beta_{4} - 2 \beta_{3} + 4 \beta_1) q^{61} + (3 \beta_{7} + 2 \beta_{5} + 2 \beta_{3} + 2 \beta_1 + 2) q^{64} + ( - 2 \beta_{7} - 4 \beta_{6} - 2 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - 4 \beta_1 - 4) q^{65} + ( - \beta_{5} + \beta_{4} + 2 \beta_{2}) q^{66} + ( - 2 \beta_{6} + \beta_{5} + 3 \beta_{4}) q^{67} + (2 \beta_{7} - 2 \beta_{6} + 2 \beta_{4} - 8 \beta_{2} - 2 \beta_1 - 8) q^{68} + ( - \beta_{5} - \beta_{3} + \beta_1) q^{69} + (2 \beta_{7} + 3 \beta_{5} + 3 \beta_{3} - 5 \beta_1) q^{71} + (\beta_{7} + \beta_{4} + 2 \beta_{2} + 2) q^{72} + ( - 2 \beta_{6} + 2 \beta_{4}) q^{73} + ( - 4 \beta_{6} - 2 \beta_{5} - 4 \beta_{2}) q^{74} + ( - \beta_{7} - 2 \beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{75} + ( - 2 \beta_{7} + 2 \beta_{5} + 2 \beta_{3} + 4 \beta_1 - 4) q^{76} + ( - 2 \beta_{7} - 2 \beta_{5} - 2 \beta_{3} - 4) q^{78} + ( - \beta_{7} + 2 \beta_{6} - \beta_{4} - \beta_{3} + 2 \beta_1) q^{79} + (4 \beta_{6} - 8 \beta_{2}) q^{80} + \beta_{2} q^{81} + (\beta_{7} + \beta_{4} + 3 \beta_{3} + 6 \beta_{2} + 6) q^{82} + ( - 2 \beta_{7} - 2 \beta_{5} - 2 \beta_{3} - 4 \beta_1) q^{83} + ( - 3 \beta_{7} - 3 \beta_{5} - 3 \beta_{3} - 6 \beta_1 + 2) q^{85} + ( - 2 \beta_{7} - 2 \beta_{4} + 4 \beta_{3} + 8 \beta_{2} + 8) q^{86} - 2 \beta_{2} q^{87} + (4 \beta_{6} + 2 \beta_{5}) q^{88} + ( - \beta_{7} + 3 \beta_{6} - \beta_{4} - 2 \beta_{3} + 3 \beta_1) q^{89} + ( - \beta_{7} + \beta_{5} + \beta_{3} + 2) q^{90} + ( - 2 \beta_1 - 4) q^{92} + (4 \beta_{6} + 4 \beta_{5} + 8 \beta_{2}) q^{94} + (2 \beta_{6} - 4 \beta_{5} - 6 \beta_{4}) q^{95} + (\beta_{7} + 2 \beta_{6} + \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{96} + ( - 2 \beta_{7} + 4 \beta_{5} + 4 \beta_{3} - 2 \beta_1) q^{97} + (\beta_{5} + \beta_{3} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} - 4 q^{3} - q^{4} - 2 q^{6} - 14 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{2} - 4 q^{3} - q^{4} - 2 q^{6} - 14 q^{8} - 4 q^{9} - 8 q^{10} - q^{12} + 7 q^{16} + q^{18} + 12 q^{19} - 8 q^{20} + 12 q^{22} + 7 q^{24} + 8 q^{25} + 12 q^{26} + 8 q^{27} - 16 q^{29} - 8 q^{30} - 9 q^{32} + 8 q^{34} + 2 q^{36} + 12 q^{37} - 20 q^{38} - 20 q^{40} - 14 q^{44} + 6 q^{46} - 8 q^{47} - 14 q^{48} + 38 q^{50} + 28 q^{52} - 16 q^{53} + q^{54} - 8 q^{55} - 24 q^{57} - 2 q^{58} + 16 q^{59} + 4 q^{60} + 2 q^{64} - 8 q^{65} - 6 q^{66} - 32 q^{68} + 7 q^{72} + 14 q^{74} + 8 q^{75} - 40 q^{76} - 24 q^{78} + 36 q^{80} - 4 q^{81} + 20 q^{82} + 16 q^{83} + 40 q^{85} + 30 q^{86} + 8 q^{87} + 2 q^{88} + 16 q^{90} - 28 q^{92} - 32 q^{94} - 9 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + x^{6} + 4x^{5} - 6x^{4} + 8x^{3} + 4x^{2} - 8x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} - \nu^{5} + \nu^{4} - 2\nu^{2} + 4\nu - 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - \nu^{6} + \nu^{5} - 2\nu^{3} + 4\nu^{2} - 4\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - \nu^{6} + 5\nu^{5} - 2\nu^{3} + 8\nu^{2} - 4\nu + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - \nu^{6} + \nu^{5} - 6\nu^{4} - 2\nu^{3} + 4\nu^{2} - 4\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} - \nu^{6} + \nu^{5} - 6\nu^{4} - 2\nu^{3} + 4\nu^{2} - 20\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 3\nu^{6} + 3\nu^{5} + 2\nu^{4} - 6\nu^{3} + 12\nu^{2} - 4\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{6} + \nu^{5} - \nu^{4} + 6\nu^{3} + 2\nu^{2} - 4\nu + 12 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + 2\beta_{6} + \beta_{4} - \beta_{3} + 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} - \beta_{5} - \beta_{3} - 2\beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{6} + \beta_{5} - 3\beta_{4} - 4\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{7} - 2\beta_{6} - \beta_{4} + 5\beta_{3} - 4\beta_{2} - 2\beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( \beta_{7} + 3\beta_{5} + 3\beta_{3} + 10\beta _1 + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -6\beta_{6} - 3\beta_{5} + \beta_{4} + 20\beta_{2} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(1\) \(-1\) \(1 + \beta_{2}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−1.41156 0.0865986i
0.121053 + 1.40902i
0.630783 1.26575i
1.15972 + 0.809347i
−1.41156 + 0.0865986i
0.121053 1.40902i
0.630783 + 1.26575i
1.15972 0.809347i
−1.41156 + 0.0865986i −0.500000 0.866025i 1.98500 0.244478i 1.46890 + 0.848071i 0.780776 + 1.17915i 0 −2.78078 + 0.516994i −0.500000 + 0.866025i −2.14688 1.06990i
19.2 0.121053 1.40902i −0.500000 0.866025i −1.97069 0.341134i 2.88831 + 1.66757i −1.28078 + 0.599676i 0 −0.719224 + 2.73546i −0.500000 + 0.866025i 2.69928 3.86783i
19.3 0.630783 + 1.26575i −0.500000 0.866025i −1.20422 + 1.59682i −1.46890 0.848071i 0.780776 1.17915i 0 −2.78078 0.516994i −0.500000 + 0.866025i 0.146883 2.39420i
19.4 1.15972 0.809347i −0.500000 0.866025i 0.689916 1.87724i −2.88831 1.66757i −1.28078 0.599676i 0 −0.719224 2.73546i −0.500000 + 0.866025i −4.69928 + 0.403728i
31.1 −1.41156 0.0865986i −0.500000 + 0.866025i 1.98500 + 0.244478i 1.46890 0.848071i 0.780776 1.17915i 0 −2.78078 0.516994i −0.500000 0.866025i −2.14688 + 1.06990i
31.2 0.121053 + 1.40902i −0.500000 + 0.866025i −1.97069 + 0.341134i 2.88831 1.66757i −1.28078 0.599676i 0 −0.719224 2.73546i −0.500000 0.866025i 2.69928 + 3.86783i
31.3 0.630783 1.26575i −0.500000 + 0.866025i −1.20422 1.59682i −1.46890 + 0.848071i 0.780776 + 1.17915i 0 −2.78078 + 0.516994i −0.500000 0.866025i 0.146883 + 2.39420i
31.4 1.15972 + 0.809347i −0.500000 + 0.866025i 0.689916 + 1.87724i −2.88831 + 1.66757i −1.28078 + 0.599676i 0 −0.719224 + 2.73546i −0.500000 0.866025i −4.69928 0.403728i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
28.d even 2 1 inner
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.2.o.a 8
4.b odd 2 1 588.2.o.c 8
7.b odd 2 1 588.2.o.c 8
7.c even 3 1 84.2.b.b yes 4
7.c even 3 1 inner 588.2.o.a 8
7.d odd 6 1 84.2.b.a 4
7.d odd 6 1 588.2.o.c 8
21.g even 6 1 252.2.b.e 4
21.h odd 6 1 252.2.b.d 4
28.d even 2 1 inner 588.2.o.a 8
28.f even 6 1 84.2.b.b yes 4
28.f even 6 1 inner 588.2.o.a 8
28.g odd 6 1 84.2.b.a 4
28.g odd 6 1 588.2.o.c 8
56.j odd 6 1 1344.2.b.f 4
56.k odd 6 1 1344.2.b.f 4
56.m even 6 1 1344.2.b.e 4
56.p even 6 1 1344.2.b.e 4
84.j odd 6 1 252.2.b.d 4
84.n even 6 1 252.2.b.e 4
168.s odd 6 1 4032.2.b.j 4
168.v even 6 1 4032.2.b.n 4
168.ba even 6 1 4032.2.b.n 4
168.be odd 6 1 4032.2.b.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.b.a 4 7.d odd 6 1
84.2.b.a 4 28.g odd 6 1
84.2.b.b yes 4 7.c even 3 1
84.2.b.b yes 4 28.f even 6 1
252.2.b.d 4 21.h odd 6 1
252.2.b.d 4 84.j odd 6 1
252.2.b.e 4 21.g even 6 1
252.2.b.e 4 84.n even 6 1
588.2.o.a 8 1.a even 1 1 trivial
588.2.o.a 8 7.c even 3 1 inner
588.2.o.a 8 28.d even 2 1 inner
588.2.o.a 8 28.f even 6 1 inner
588.2.o.c 8 4.b odd 2 1
588.2.o.c 8 7.b odd 2 1
588.2.o.c 8 7.d odd 6 1
588.2.o.c 8 28.g odd 6 1
1344.2.b.e 4 56.m even 6 1
1344.2.b.e 4 56.p even 6 1
1344.2.b.f 4 56.j odd 6 1
1344.2.b.f 4 56.k odd 6 1
4032.2.b.j 4 168.s odd 6 1
4032.2.b.j 4 168.be odd 6 1
4032.2.b.n 4 168.v even 6 1
4032.2.b.n 4 168.ba even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(588, [\chi])\):

\( T_{5}^{8} - 14T_{5}^{6} + 164T_{5}^{4} - 448T_{5}^{2} + 1024 \) Copy content Toggle raw display
\( T_{11}^{8} - 10T_{11}^{6} + 92T_{11}^{4} - 80T_{11}^{2} + 64 \) Copy content Toggle raw display
\( T_{19}^{4} - 6T_{19}^{3} + 44T_{19}^{2} + 48T_{19} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{7} + T^{6} + 4 T^{5} - 6 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} - 14 T^{6} + 164 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 10 T^{6} + 92 T^{4} - 80 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$13$ \( (T^{4} + 40 T^{2} + 128)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 46 T^{6} + 1604 T^{4} + \cdots + 262144 \) Copy content Toggle raw display
$19$ \( (T^{4} - 6 T^{3} + 44 T^{2} + 48 T + 64)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 10 T^{6} + 92 T^{4} - 80 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$29$ \( (T + 2)^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} - 6 T^{3} + 44 T^{2} + 48 T + 64)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 62 T^{2} + 128)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 148 T^{2} + 5408)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 4 T^{3} + 80 T^{2} - 256 T + 4096)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 8 T^{3} + 116 T^{2} - 416 T + 2704)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 4 T + 16)^{4} \) Copy content Toggle raw display
$61$ \( T^{8} - 112 T^{6} + 10496 T^{4} + \cdots + 4194304 \) Copy content Toggle raw display
$67$ \( T^{8} - 124 T^{6} + 14864 T^{4} + \cdots + 262144 \) Copy content Toggle raw display
$71$ \( (T^{4} + 170 T^{2} + 2312)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 56 T^{6} + 2624 T^{4} + \cdots + 262144 \) Copy content Toggle raw display
$79$ \( T^{8} - 28 T^{6} + 656 T^{4} + \cdots + 16384 \) Copy content Toggle raw display
$83$ \( (T^{2} - 4 T - 64)^{4} \) Copy content Toggle raw display
$89$ \( T^{8} - 62 T^{6} + 3716 T^{4} + \cdots + 16384 \) Copy content Toggle raw display
$97$ \( (T^{4} + 184 T^{2} + 8192)^{2} \) Copy content Toggle raw display
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