Properties

Label 1680.3.bd.a
Level $1680$
Weight $3$
Character orbit 1680.bd
Analytic conductor $45.777$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1680.bd (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(45.7766844125\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \( x^{16} - 8 x^{15} + 96 x^{14} - 532 x^{13} + 3236 x^{12} - 12864 x^{11} + 49526 x^{10} - 141436 x^{9} + 362298 x^{8} - 722060 x^{7} + 1208164 x^{6} - 1570812 x^{5} + \cdots + 33750 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 210)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{6} q^{5} + (\beta_{9} + \beta_{2} - \beta_1) q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{6} q^{5} + (\beta_{9} + \beta_{2} - \beta_1) q^{7} + 3 q^{9} + ( - \beta_{12} + \beta_{11} + \beta_{3} - 6) q^{11} + (\beta_{9} + \beta_{8} - \beta_{6} - \beta_{5} - \beta_{2} - 3 \beta_1) q^{13} + (\beta_{11} + \beta_{3} + 2) q^{15} + (\beta_{6} + \beta_{5} - 2 \beta_{2} - \beta_1) q^{17} + (\beta_{9} - \beta_{8} - 2 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_1) q^{19} + ( - \beta_{14} + \beta_{13} + \beta_{12} + \beta_{3} + 2) q^{21} + ( - 2 \beta_{15} + 2 \beta_{13} + 3 \beta_{12} + 3 \beta_{11} - 4 \beta_{10} + 3) q^{23} + (\beta_{15} + \beta_{14} - \beta_{13} + \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} - 3 \beta_{3} + \beta_1 + 2) q^{25} - 3 \beta_1 q^{27} + (2 \beta_{15} + 4 \beta_{14} - 2 \beta_{13} - 4 \beta_{11} + 2) q^{29} + (2 \beta_{15} + 2 \beta_{13} + \beta_{6} - \beta_{5} + 5 \beta_{4} - \beta_1) q^{31} + ( - \beta_{9} - \beta_{8} - 2 \beta_{6} - 2 \beta_{5} - \beta_{2} + 6 \beta_1) q^{33} + ( - 3 \beta_{15} - 2 \beta_{14} + \beta_{13} + 5 \beta_{12} + 2 \beta_{11} - 2 \beta_{10} + \cdots + 4) q^{35}+ \cdots + ( - 3 \beta_{12} + 3 \beta_{11} + 3 \beta_{3} - 18) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 48 q^{9} - 96 q^{11} + 24 q^{15} + 24 q^{21} + 24 q^{25} + 64 q^{29} + 8 q^{35} + 144 q^{39} + 224 q^{49} + 48 q^{51} + 368 q^{65} + 384 q^{71} + 608 q^{79} + 144 q^{81} - 440 q^{85} - 224 q^{91} + 560 q^{95} - 288 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 96 x^{14} - 532 x^{13} + 3236 x^{12} - 12864 x^{11} + 49526 x^{10} - 141436 x^{9} + 362298 x^{8} - 722060 x^{7} + 1208164 x^{6} - 1570812 x^{5} + \cdots + 33750 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1304 \nu^{14} - 9128 \nu^{13} + 115446 \nu^{12} - 574012 \nu^{11} + 3591842 \nu^{10} - 12914984 \nu^{9} + 49995515 \nu^{8} - 128433344 \nu^{7} + 320844898 \nu^{6} + \cdots + 99070875 ) / 560625 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 19306 \nu^{14} + 135142 \nu^{13} - 1709079 \nu^{12} + 8497628 \nu^{11} - 53165468 \nu^{10} + 191153301 \nu^{9} - 739735340 \nu^{8} + \cdots - 1406219625 ) / 3027375 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 181 \nu^{14} + 1267 \nu^{13} - 16034 \nu^{12} + 79733 \nu^{11} - 499338 \nu^{10} + 1796001 \nu^{9} - 6961130 \nu^{8} + 17893811 \nu^{7} - 44790412 \nu^{6} + \cdots - 14271525 ) / 26325 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1984 \nu^{15} - 14880 \nu^{14} + 171958 \nu^{13} - 892047 \nu^{12} + 4997626 \nu^{11} - 18170922 \nu^{10} + 58997141 \nu^{9} - 142860075 \nu^{8} + \cdots + 575687250 ) / 85899825 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 447494447 \nu^{15} - 3678104460 \nu^{14} + 43437004320 \nu^{13} - 245298593960 \nu^{12} + 1472808594649 \nu^{11} + \cdots - 41918527948500 ) / 83752329375 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 447494447 \nu^{15} + 3229118109 \nu^{14} - 40294099863 \nu^{13} + 205531523111 \nu^{12} - 1275063927496 \nu^{11} + \cdots + 5890964889375 ) / 83752329375 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 472182246 \nu^{15} - 3541366845 \nu^{14} + 43501742065 \nu^{13} - 229050592940 \nu^{12} + 1408375647077 \nu^{11} + \cdots - 20360648194875 ) / 83752329375 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 755997166 \nu^{15} + 5904909925 \nu^{14} - 71246353625 \nu^{13} + 387215467260 \nu^{12} - 2354325589857 \nu^{11} + \cdots + 47103825674250 ) / 83752329375 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 755997166 \nu^{15} - 5240241701 \nu^{14} + 66593676057 \nu^{13} - 328372443834 \nu^{12} + 2061752257685 \nu^{11} + \cdots + 2190542081625 ) / 83752329375 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 5163308 \nu^{15} - 38724810 \nu^{14} + 475369020 \nu^{13} - 2502572345 \nu^{12} + 15373400596 \nu^{11} - 58317525310 \nu^{10} + \cdots - 200352316500 ) / 404600625 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 49652491 \nu^{15} - 376911377 \nu^{14} + 4602946389 \nu^{13} - 24466123483 \nu^{12} + 149826974288 \nu^{11} - 573281443581 \nu^{10} + \cdots - 2269018608750 ) / 3641405625 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 49652491 \nu^{15} - 367875988 \nu^{14} + 4539698666 \nu^{13} - 23665127672 \nu^{12} + 145843219821 \nu^{11} - 548311933059 \nu^{10} + \cdots - 1551811091625 ) / 3641405625 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1309724492 \nu^{15} + 9822933690 \nu^{14} - 120588431330 \nu^{13} + 634843642680 \nu^{12} - 3900144384104 \nu^{11} + \cdots + 51303336994125 ) / 83752329375 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 65666397 \nu^{15} - 477141136 \nu^{14} + 5938288402 \nu^{13} - 30467772959 \nu^{12} + 188760325727 \nu^{11} - 699352435848 \nu^{10} + \cdots - 1356819889500 ) / 3641405625 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 2046364168 \nu^{15} + 15347731260 \nu^{14} - 188413764520 \nu^{13} + 991915545270 \nu^{12} - 6093859902766 \nu^{11} + \cdots + 80293733362125 ) / 83752329375 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( - \beta_{15} - \beta_{13} - 2 \beta_{10} + \beta_{9} - \beta_{8} - 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - 3 \beta _1 + 6 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 7 \beta_{15} - 12 \beta_{14} + 5 \beta_{13} + 6 \beta_{12} + 6 \beta_{11} - 2 \beta_{10} + 7 \beta_{9} + 5 \beta_{8} - 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - 6 \beta_{3} + 6 \beta_{2} - 3 \beta _1 - 90 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 11 \beta_{15} - 18 \beta_{14} + 11 \beta_{13} - 9 \beta_{12} - 9 \beta_{11} + 61 \beta_{10} - 14 \beta_{9} + 32 \beta_{8} + 28 \beta_{7} - 46 \beta_{6} + 46 \beta_{5} - 79 \beta_{4} - 9 \beta_{3} + 9 \beta_{2} + 69 \beta _1 - 156 ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 143 \beta_{15} + 204 \beta_{14} - 97 \beta_{13} - 108 \beta_{12} - 168 \beta_{11} + 124 \beta_{10} - 227 \beta_{9} - 133 \beta_{8} + 58 \beta_{7} - 106 \beta_{6} + 82 \beta_{5} - 160 \beta_{4} + 132 \beta_{3} - 144 \beta_{2} + \cdots + 1386 ) / 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 88 \beta_{15} + 540 \beta_{14} - 328 \beta_{13} + 255 \beta_{12} + 105 \beta_{11} - 1493 \beta_{10} + 37 \beta_{9} - 967 \beta_{8} - 404 \beta_{7} + 878 \beta_{6} - 938 \beta_{5} + 1937 \beta_{4} + 345 \beta_{3} + \cdots + 4236 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 1496 \beta_{15} - 1815 \beta_{14} + 814 \beta_{13} + 1095 \beta_{12} + 2160 \beta_{11} - 2395 \beta_{10} + 2960 \beta_{9} + 1336 \beta_{8} - 679 \beta_{7} + 1432 \beta_{6} - 1528 \beta_{5} + 3106 \beta_{4} + \cdots - 11916 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1450 \beta_{15} - 14616 \beta_{14} + 9932 \beta_{13} - 6321 \beta_{12} + 1659 \beta_{11} + 31705 \beta_{10} + 6625 \beta_{9} + 26723 \beta_{8} + 5944 \beta_{7} - 18136 \beta_{6} + 17674 \beta_{5} + \cdots - 111504 ) / 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 61883 \beta_{15} + 65724 \beta_{14} - 21481 \beta_{13} - 50412 \beta_{12} - 106032 \beta_{11} + 149464 \beta_{10} - 138761 \beta_{9} - 42991 \beta_{8} + 30250 \beta_{7} - 76510 \beta_{6} + \cdots + 413190 ) / 12 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 103439 \beta_{15} + 385758 \beta_{14} - 270907 \beta_{13} + 143883 \beta_{12} - 154917 \beta_{11} - 611849 \beta_{10} - 321620 \beta_{9} - 700294 \beta_{8} - 85022 \beta_{7} + 385532 \beta_{6} + \cdots + 2878206 ) / 12 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 1245547 \beta_{15} - 1144488 \beta_{14} + 95411 \beta_{13} + 1245744 \beta_{12} + 2473284 \beta_{11} - 4214378 \beta_{10} + 3045589 \beta_{9} + 410735 \beta_{8} - 661784 \beta_{7} + \cdots - 6713352 ) / 12 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 3766714 \beta_{15} - 9997614 \beta_{14} + 6779426 \beta_{13} - 3036957 \beta_{12} + 6542943 \beta_{11} + 10534555 \beta_{10} + 11151253 \beta_{9} + 17603231 \beta_{8} + \cdots - 73078230 ) / 12 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 12045634 \beta_{15} + 8786955 \beta_{14} + 3666955 \beta_{13} - 15943449 \beta_{12} - 27259074 \beta_{11} + 56069540 \beta_{10} - 31399507 \beta_{9} + 3274735 \beta_{8} + \cdots + 44325837 ) / 6 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 114248852 \beta_{15} + 254014020 \beta_{14} - 158151874 \beta_{13} + 57387759 \beta_{12} - 221707941 \beta_{11} - 144608033 \beta_{10} - 338639183 \beta_{9} + \cdots + 1821064524 ) / 12 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 436459717 \beta_{15} - 180750738 \beta_{14} - 388881655 \beta_{13} + 824613468 \beta_{12} + 1120617078 \beta_{11} - 2864827628 \beta_{10} + 1192678201 \beta_{9} + \cdots - 387024930 ) / 12 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 3176368585 \beta_{15} - 6304996872 \beta_{14} + 3451153529 \beta_{13} - 864267417 \beta_{12} + 6713172693 \beta_{11} + 715237729 \beta_{10} + 9540516082 \beta_{9} + \cdots - 44352101334 ) / 12 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(421\) \(1121\) \(1471\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
769.1
0.500000 + 2.68650i
0.500000 2.68650i
0.500000 0.971291i
0.500000 + 0.971291i
0.500000 + 4.96598i
0.500000 4.96598i
0.500000 + 0.422343i
0.500000 0.422343i
0.500000 + 1.83656i
0.500000 1.83656i
0.500000 + 3.55177i
0.500000 3.55177i
0.500000 + 0.442923i
0.500000 0.442923i
0.500000 + 4.10071i
0.500000 4.10071i
0 −1.73205 0 −4.59769 1.96501i 0 6.81963 + 1.57881i 0 3.00000 0
769.2 0 −1.73205 0 −4.59769 + 1.96501i 0 6.81963 1.57881i 0 3.00000 0
769.3 0 −1.73205 0 −2.40341 4.38447i 0 −6.94781 0.853218i 0 3.00000 0
769.4 0 −1.73205 0 −2.40341 + 4.38447i 0 −6.94781 + 0.853218i 0 3.00000 0
769.5 0 −1.73205 0 −1.38028 4.80571i 0 −5.24961 + 4.63050i 0 3.00000 0
769.6 0 −1.73205 0 −1.38028 + 4.80571i 0 −5.24961 4.63050i 0 3.00000 0
769.7 0 −1.73205 0 4.91728 0.905717i 0 1.91369 + 6.73333i 0 3.00000 0
769.8 0 −1.73205 0 4.91728 + 0.905717i 0 1.91369 6.73333i 0 3.00000 0
769.9 0 1.73205 0 −4.91728 0.905717i 0 −1.91369 6.73333i 0 3.00000 0
769.10 0 1.73205 0 −4.91728 + 0.905717i 0 −1.91369 + 6.73333i 0 3.00000 0
769.11 0 1.73205 0 1.38028 4.80571i 0 5.24961 4.63050i 0 3.00000 0
769.12 0 1.73205 0 1.38028 + 4.80571i 0 5.24961 + 4.63050i 0 3.00000 0
769.13 0 1.73205 0 2.40341 4.38447i 0 6.94781 + 0.853218i 0 3.00000 0
769.14 0 1.73205 0 2.40341 + 4.38447i 0 6.94781 0.853218i 0 3.00000 0
769.15 0 1.73205 0 4.59769 1.96501i 0 −6.81963 1.57881i 0 3.00000 0
769.16 0 1.73205 0 4.59769 + 1.96501i 0 −6.81963 + 1.57881i 0 3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 769.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.b odd 2 1 inner
35.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1680.3.bd.a 16
4.b odd 2 1 210.3.h.a 16
5.b even 2 1 inner 1680.3.bd.a 16
7.b odd 2 1 inner 1680.3.bd.a 16
12.b even 2 1 630.3.h.e 16
20.d odd 2 1 210.3.h.a 16
20.e even 4 2 1050.3.f.e 16
28.d even 2 1 210.3.h.a 16
35.c odd 2 1 inner 1680.3.bd.a 16
60.h even 2 1 630.3.h.e 16
84.h odd 2 1 630.3.h.e 16
140.c even 2 1 210.3.h.a 16
140.j odd 4 2 1050.3.f.e 16
420.o odd 2 1 630.3.h.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.h.a 16 4.b odd 2 1
210.3.h.a 16 20.d odd 2 1
210.3.h.a 16 28.d even 2 1
210.3.h.a 16 140.c even 2 1
630.3.h.e 16 12.b even 2 1
630.3.h.e 16 60.h even 2 1
630.3.h.e 16 84.h odd 2 1
630.3.h.e 16 420.o odd 2 1
1050.3.f.e 16 20.e even 4 2
1050.3.f.e 16 140.j odd 4 2
1680.3.bd.a 16 1.a even 1 1 trivial
1680.3.bd.a 16 5.b even 2 1 inner
1680.3.bd.a 16 7.b odd 2 1 inner
1680.3.bd.a 16 35.c odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{4} + 24T_{11}^{3} + 28T_{11}^{2} - 1776T_{11} - 4844 \) acting on \(S_{3}^{\mathrm{new}}(1680, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} - 12 T^{14} + \cdots + 152587890625 \) Copy content Toggle raw display
$7$ \( T^{16} - 112 T^{14} + \cdots + 33232930569601 \) Copy content Toggle raw display
$11$ \( (T^{4} + 24 T^{3} + 28 T^{2} - 1776 T - 4844)^{4} \) Copy content Toggle raw display
$13$ \( (T^{8} - 1044 T^{6} + 313732 T^{4} + \cdots + 586802176)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 996 T^{6} + 270436 T^{4} + \cdots + 338560000)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 1796 T^{6} + \cdots + 2149991424)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 2932 T^{6} + \cdots + 11186869824)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 16 T^{3} - 2088 T^{2} + \cdots + 19600)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + 3740 T^{6} + 1609684 T^{4} + \cdots + 747256896)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 8536 T^{6} + \cdots + 3617786594304)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 932 T^{6} + 280612 T^{4} + \cdots + 1141899264)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 12880 T^{6} + \cdots + 9151544623104)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 10904 T^{6} + \cdots + 14545741676544)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 15604 T^{6} + \cdots + 38389325511744)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 16112 T^{6} + \cdots + 203235065856)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 13824 T^{6} + \cdots + 72666906624)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 17224 T^{6} + \cdots + 404129746944)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 96 T^{3} - 6116 T^{2} + \cdots - 18715436)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} - 24020 T^{6} + \cdots + 105841627140096)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 152 T^{3} + 3952 T^{2} + \cdots - 12612096)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} - 15480 T^{6} + \cdots + 13129665286144)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 24860 T^{6} + \cdots + 135150669186624)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 20468 T^{6} + \cdots + 5898136817664)^{2} \) Copy content Toggle raw display
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