Properties

Label 19.3.f.a
Level $19$
Weight $3$
Character orbit 19.f
Analytic conductor $0.518$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [19,3,Mod(2,19)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(19, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("19.2");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 19.f (of order \(18\), degree \(6\), 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: \(0.517712502285\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 24x^{10} + 216x^{8} + 905x^{6} + 1770x^{4} + 1395x^{2} + 361 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{10} q^{2} + ( - \beta_{11} - \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_1) q^{3} + (\beta_{11} - \beta_{10} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_1) q^{4} + (\beta_{11} - \beta_{10} + \beta_{9} + 2 \beta_{8} + \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{5} + (2 \beta_{9} + \beta_{8} - 4 \beta_{6} + \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1 - 4) q^{6} + ( - \beta_{11} - 2 \beta_{10} - \beta_{8} - 2 \beta_{7} - 4 \beta_{6} - 2 \beta_{5} - 2 \beta_{3} + \cdots - 1) q^{7}+ \cdots + (\beta_{10} - 2 \beta_{9} - \beta_{8} - 2 \beta_{7} + 4 \beta_{6} - 3 \beta_{5} + 4 \beta_{3} + \cdots + 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{10} q^{2} + ( - \beta_{11} - \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_1) q^{3} + (\beta_{11} - \beta_{10} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_1) q^{4} + (\beta_{11} - \beta_{10} + \beta_{9} + 2 \beta_{8} + \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{5} + (2 \beta_{9} + \beta_{8} - 4 \beta_{6} + \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1 - 4) q^{6} + ( - \beta_{11} - 2 \beta_{10} - \beta_{8} - 2 \beta_{7} - 4 \beta_{6} - 2 \beta_{5} - 2 \beta_{3} + \cdots - 1) q^{7}+ \cdots + (7 \beta_{11} - 12 \beta_{10} + 3 \beta_{9} - 19 \beta_{8} - 7 \beta_{7} + 4 \beta_{6} + \cdots + 40) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{2} - 6 q^{5} - 36 q^{6} + 6 q^{7} - 9 q^{8} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{2} - 6 q^{5} - 36 q^{6} + 6 q^{7} - 9 q^{8} - 24 q^{9} + 51 q^{10} - 18 q^{11} + 63 q^{12} + 21 q^{13} + 9 q^{14} + 63 q^{15} - 12 q^{16} - 3 q^{17} - 24 q^{19} - 90 q^{20} + 30 q^{21} - 78 q^{22} - 102 q^{23} - 12 q^{24} - 156 q^{25} + 21 q^{26} - 27 q^{27} + 12 q^{28} + 147 q^{29} + 24 q^{30} + 99 q^{31} + 165 q^{32} + 84 q^{33} + 132 q^{34} + 96 q^{35} + 63 q^{36} + 72 q^{38} - 108 q^{39} - 138 q^{40} - 144 q^{41} - 237 q^{42} - 27 q^{43} - 123 q^{44} - 3 q^{45} - 54 q^{46} - 99 q^{47} - 51 q^{48} - 24 q^{49} + 72 q^{50} - 42 q^{51} + 93 q^{52} + 111 q^{53} + 21 q^{54} + 162 q^{55} - 168 q^{57} - 132 q^{58} + 3 q^{59} - 30 q^{60} + 150 q^{61} + 108 q^{62} + 234 q^{63} + 27 q^{64} + 126 q^{65} + 168 q^{66} + 135 q^{67} - 30 q^{68} + 72 q^{69} + 225 q^{70} - 168 q^{71} - 102 q^{72} - 90 q^{73} - 231 q^{74} + 42 q^{76} + 246 q^{77} - 189 q^{78} - 75 q^{79} + 21 q^{80} - 159 q^{81} - 117 q^{82} - 156 q^{83} + 99 q^{84} - 300 q^{85} - 144 q^{86} + 69 q^{87} - 405 q^{88} - 558 q^{89} - 66 q^{90} - 453 q^{91} + 48 q^{92} - 57 q^{93} - 69 q^{95} + 558 q^{96} + 465 q^{97} + 777 q^{98} + 462 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 24x^{10} + 216x^{8} + 905x^{6} + 1770x^{4} + 1395x^{2} + 361 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - \nu^{11} + 2 \nu^{10} - 27 \nu^{9} + 28 \nu^{8} - 277 \nu^{7} + 38 \nu^{6} - 1304 \nu^{5} - 774 \nu^{4} - 2566 \nu^{3} - 2742 \nu^{2} - 1177 \nu - 1520 ) / 304 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{11} - 5 \nu^{10} - \nu^{9} - 89 \nu^{8} + 239 \nu^{7} - 437 \nu^{6} + 2078 \nu^{5} - 22 \nu^{4} + 5308 \nu^{3} + 2960 \nu^{2} + 2697 \nu + 1881 ) / 304 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{11} + 5 \nu^{10} - \nu^{9} + 89 \nu^{8} + 239 \nu^{7} + 437 \nu^{6} + 2078 \nu^{5} + 22 \nu^{4} + 5308 \nu^{3} - 2960 \nu^{2} + 2697 \nu - 1881 ) / 304 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{11} - 2 \nu^{10} - 27 \nu^{9} - 28 \nu^{8} - 277 \nu^{7} - 38 \nu^{6} - 1304 \nu^{5} + 774 \nu^{4} - 2566 \nu^{3} + 2742 \nu^{2} - 1177 \nu + 1520 ) / 304 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3 \nu^{11} + 17 \nu^{10} + 55 \nu^{9} + 333 \nu^{8} + 315 \nu^{7} + 2185 \nu^{6} + 530 \nu^{5} + 5334 \nu^{4} - 24 \nu^{3} + 3464 \nu^{2} + 417 \nu + 171 ) / 608 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3 \nu^{11} - 17 \nu^{10} + 55 \nu^{9} - 333 \nu^{8} + 315 \nu^{7} - 2185 \nu^{6} + 530 \nu^{5} - 5334 \nu^{4} - 24 \nu^{3} - 3464 \nu^{2} + 417 \nu - 171 ) / 608 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{10} + 21\nu^{8} + 153\nu^{6} + 454\nu^{4} + 504\nu^{2} + 8\nu + 171 ) / 16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{10} - 21\nu^{8} - 153\nu^{6} - 454\nu^{4} - 504\nu^{2} + 8\nu - 171 ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3 \nu^{11} + \nu^{10} - 56 \nu^{9} + 14 \nu^{8} - 329 \nu^{7} + 19 \nu^{6} - 549 \nu^{5} - 387 \nu^{4} + 411 \nu^{3} - 1295 \nu^{2} + 650 \nu - 380 ) / 152 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 17 \nu^{11} + 17 \nu^{10} + 333 \nu^{9} + 333 \nu^{8} + 2185 \nu^{7} + 2185 \nu^{6} + 5334 \nu^{5} + 5334 \nu^{4} + 3464 \nu^{3} + 3768 \nu^{2} + 171 \nu + 1083 ) / 608 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 9\nu^{11} + 197\nu^{9} + 1545\nu^{7} + 5238\nu^{5} + 7304\nu^{3} + 2979\nu - 152 ) / 304 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{8} + \beta_{7} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{9} - \beta_{5} + \beta_{4} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -5\beta_{8} - 5\beta_{7} + 2\beta_{6} + 2\beta_{5} + 2\beta_{4} + \beta_{3} + \beta_{2} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 7 \beta_{10} - 7 \beta_{9} - \beta_{8} + \beta_{7} + 2 \beta_{6} + 5 \beta_{5} - 11 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 4 \beta _1 + 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{11} - 3 \beta_{10} + 3 \beta_{9} + 32 \beta_{8} + 32 \beta_{7} - 18 \beta_{6} - 15 \beta_{5} - 23 \beta_{4} - 13 \beta_{3} - 13 \beta_{2} - 26 \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 50 \beta_{10} + 50 \beta_{9} + 14 \beta_{8} - 14 \beta_{7} - 30 \beta_{6} - 20 \beta_{5} + 102 \beta_{4} + 23 \beta_{3} - 23 \beta_{2} - 52 \beta _1 - 153 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 38 \beta_{11} + 44 \beta_{10} - 44 \beta_{9} - 226 \beta_{8} - 226 \beta_{7} + 158 \beta_{6} + 114 \beta_{5} + 200 \beta_{4} + 130 \beta_{3} + 130 \beta_{2} + 244 \beta _1 + 25 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 384 \beta_{10} - 384 \beta_{9} - 149 \beta_{8} + 149 \beta_{7} + 334 \beta_{6} + 50 \beta_{5} - 888 \beta_{4} - 200 \beta_{3} + 200 \beta_{2} + 504 \beta _1 + 1104 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 494 \beta_{11} - 483 \beta_{10} + 483 \beta_{9} + 1688 \beta_{8} + 1688 \beta_{7} - 1400 \beta_{6} - 917 \beta_{5} - 1589 \beta_{4} - 1186 \beta_{3} - 1186 \beta_{2} - 2072 \beta _1 - 236 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3088 \beta_{10} + 3088 \beta_{9} + 1433 \beta_{8} - 1433 \beta_{7} - 3332 \beta_{6} + 244 \beta_{5} + 7532 \beta_{4} + 1589 \beta_{3} - 1589 \beta_{2} - 4444 \beta _1 - 8372 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 5420 \beta_{11} + 4765 \beta_{10} - 4765 \beta_{9} - 13049 \beta_{8} - 13049 \beta_{7} + 12374 \beta_{6} + 7609 \beta_{5} + 12211 \beta_{4} + 10398 \beta_{3} + 10398 \beta_{2} + 16976 \beta _1 + 2055 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/19\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{1} - \beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
2.88811i
0.918492i
2.01431i
0.728740i
2.88811i
0.918492i
2.01431i
0.728740i
2.57727i
1.89323i
2.57727i
1.89323i
−2.84423 0.501515i 3.28392 3.91363i 4.07936 + 1.48477i −3.00117 + 1.09234i −11.3030 + 9.48432i 3.87208 + 6.70664i −0.853313 0.492661i −2.96949 16.8408i 9.08386 1.60173i
2.2 0.904538 + 0.159494i −0.464845 + 0.553981i −2.96602 1.07954i 0.295437 0.107530i −0.508827 + 0.426957i −0.328846 0.569578i −5.69245 3.28654i 1.47202 + 8.34824i 0.284385 0.0501447i
3.1 −1.29478 + 1.54305i 0.621128 + 1.70654i −0.00997859 0.0565914i 1.24445 7.05761i −3.43750 1.25115i 0.422527 + 0.731838i −6.87755 3.97075i 4.36793 3.66513i 9.27900 + 11.0583i
3.2 0.468425 0.558247i −1.14207 3.13782i 0.602375 + 3.41624i −1.13111 + 6.41483i −2.28665 0.832274i −5.47943 9.49065i 4.71370 + 2.72146i −1.64718 + 1.38215i 3.05122 + 3.63630i
10.1 −2.84423 + 0.501515i 3.28392 + 3.91363i 4.07936 1.48477i −3.00117 1.09234i −11.3030 9.48432i 3.87208 6.70664i −0.853313 + 0.492661i −2.96949 + 16.8408i 9.08386 + 1.60173i
10.2 0.904538 0.159494i −0.464845 0.553981i −2.96602 + 1.07954i 0.295437 + 0.107530i −0.508827 0.426957i −0.328846 + 0.569578i −5.69245 + 3.28654i 1.47202 8.34824i 0.284385 + 0.0501447i
13.1 −1.29478 1.54305i 0.621128 1.70654i −0.00997859 + 0.0565914i 1.24445 + 7.05761i −3.43750 + 1.25115i 0.422527 0.731838i −6.87755 + 3.97075i 4.36793 + 3.66513i 9.27900 11.0583i
13.2 0.468425 + 0.558247i −1.14207 + 3.13782i 0.602375 3.41624i −1.13111 6.41483i −2.28665 + 0.832274i −5.47943 + 9.49065i 4.71370 2.72146i −1.64718 1.38215i 3.05122 3.63630i
14.1 −0.881480 2.42185i −0.384565 0.0678091i −2.02415 + 1.69847i 1.73199 + 1.45331i 0.174763 + 0.991129i 5.72163 + 9.91015i −3.03027 1.74952i −8.31394 3.02603i 1.99298 5.47566i
14.2 0.647524 + 1.77906i −1.91357 0.337414i 0.318417 0.267183i −2.13959 1.79533i −0.638803 3.62283i −1.20796 2.09224i 7.23987 + 4.17994i −4.90934 1.78685i 1.80856 4.96897i
15.1 −0.881480 + 2.42185i −0.384565 + 0.0678091i −2.02415 1.69847i 1.73199 1.45331i 0.174763 0.991129i 5.72163 9.91015i −3.03027 + 1.74952i −8.31394 + 3.02603i 1.99298 + 5.47566i
15.2 0.647524 1.77906i −1.91357 + 0.337414i 0.318417 + 0.267183i −2.13959 + 1.79533i −0.638803 + 3.62283i −1.20796 + 2.09224i 7.23987 4.17994i −4.90934 + 1.78685i 1.80856 + 4.96897i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.f odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 19.3.f.a 12
3.b odd 2 1 171.3.ba.b 12
4.b odd 2 1 304.3.z.a 12
19.b odd 2 1 361.3.f.g 12
19.c even 3 1 361.3.f.e 12
19.c even 3 1 361.3.f.f 12
19.d odd 6 1 361.3.f.b 12
19.d odd 6 1 361.3.f.c 12
19.e even 9 1 361.3.b.c 12
19.e even 9 1 361.3.d.d 12
19.e even 9 1 361.3.d.f 12
19.e even 9 1 361.3.f.b 12
19.e even 9 1 361.3.f.c 12
19.e even 9 1 361.3.f.g 12
19.f odd 18 1 inner 19.3.f.a 12
19.f odd 18 1 361.3.b.c 12
19.f odd 18 1 361.3.d.d 12
19.f odd 18 1 361.3.d.f 12
19.f odd 18 1 361.3.f.e 12
19.f odd 18 1 361.3.f.f 12
57.j even 18 1 171.3.ba.b 12
76.k even 18 1 304.3.z.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.3.f.a 12 1.a even 1 1 trivial
19.3.f.a 12 19.f odd 18 1 inner
171.3.ba.b 12 3.b odd 2 1
171.3.ba.b 12 57.j even 18 1
304.3.z.a 12 4.b odd 2 1
304.3.z.a 12 76.k even 18 1
361.3.b.c 12 19.e even 9 1
361.3.b.c 12 19.f odd 18 1
361.3.d.d 12 19.e even 9 1
361.3.d.d 12 19.f odd 18 1
361.3.d.f 12 19.e even 9 1
361.3.d.f 12 19.f odd 18 1
361.3.f.b 12 19.d odd 6 1
361.3.f.b 12 19.e even 9 1
361.3.f.c 12 19.d odd 6 1
361.3.f.c 12 19.e even 9 1
361.3.f.e 12 19.c even 3 1
361.3.f.e 12 19.f odd 18 1
361.3.f.f 12 19.c even 3 1
361.3.f.f 12 19.f odd 18 1
361.3.f.g 12 19.b odd 2 1
361.3.f.g 12 19.e even 9 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(19, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 6 T^{11} + 18 T^{10} + 39 T^{9} + \cdots + 361 \) Copy content Toggle raw display
$3$ \( T^{12} + 12 T^{10} + 63 T^{9} + \cdots + 289 \) Copy content Toggle raw display
$5$ \( T^{12} + 6 T^{11} + 96 T^{10} + \cdots + 87616 \) Copy content Toggle raw display
$7$ \( T^{12} - 6 T^{11} + 177 T^{10} + \cdots + 1700416 \) Copy content Toggle raw display
$11$ \( T^{12} + 18 T^{11} + \cdots + 131774082049 \) Copy content Toggle raw display
$13$ \( T^{12} - 21 T^{11} + \cdots + 3568865052736 \) Copy content Toggle raw display
$17$ \( T^{12} + 3 T^{11} + \cdots + 21556993329 \) Copy content Toggle raw display
$19$ \( T^{12} + 24 T^{11} + \cdots + 22\!\cdots\!61 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 114585206984704 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 484594358358016 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 125789503910464 \) Copy content Toggle raw display
$37$ \( T^{12} + 7554 T^{10} + \cdots + 58527273697344 \) Copy content Toggle raw display
$41$ \( T^{12} + 144 T^{11} + \cdots + 17\!\cdots\!69 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 126559282520449 \) Copy content Toggle raw display
$47$ \( T^{12} + 99 T^{11} + \cdots + 51\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{12} - 111 T^{11} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{12} - 3 T^{11} + \cdots + 48\!\cdots\!41 \) Copy content Toggle raw display
$61$ \( T^{12} - 150 T^{11} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{12} - 135 T^{11} + \cdots + 22\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{12} + 168 T^{11} + \cdots + 42\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{12} + 90 T^{11} + \cdots + 17\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{12} + 75 T^{11} + \cdots + 30\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( T^{12} + 156 T^{11} + \cdots + 15\!\cdots\!89 \) Copy content Toggle raw display
$89$ \( T^{12} + 558 T^{11} + \cdots + 14\!\cdots\!89 \) Copy content Toggle raw display
$97$ \( T^{12} - 465 T^{11} + \cdots + 11\!\cdots\!01 \) Copy content Toggle raw display
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