Properties

Label 9.6.c.a
Level 9
Weight 6
Character orbit 9.c
Analytic conductor 1.443
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 9.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.44345437832\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \(x^{8} + 40 x^{6} + 568 x^{4} + 3363 x^{2} + 7056\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} + \beta_{4} + \beta_{5} ) q^{2} + ( -3 + 3 \beta_{1} - \beta_{3} + \beta_{5} + \beta_{6} ) q^{3} + ( -13 \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{6} - \beta_{7} ) q^{4} + ( 20 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{5} + ( 24 - 3 \beta_{1} + 2 \beta_{3} - 12 \beta_{4} - 4 \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{6} + ( 4 - 4 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - 11 \beta_{4} - 10 \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{7} + ( -95 + 4 \beta_{2} - 6 \beta_{3} + 5 \beta_{4} - 7 \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{8} + ( -72 + 27 \beta_{1} - 3 \beta_{2} + 12 \beta_{4} - 24 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{1} + \beta_{4} + \beta_{5} ) q^{2} + ( -3 + 3 \beta_{1} - \beta_{3} + \beta_{5} + \beta_{6} ) q^{3} + ( -13 \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{6} - \beta_{7} ) q^{4} + ( 20 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{5} + ( 24 - 3 \beta_{1} + 2 \beta_{3} - 12 \beta_{4} - 4 \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{6} + ( 4 - 4 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - 11 \beta_{4} - 10 \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{7} + ( -95 + 4 \beta_{2} - 6 \beta_{3} + 5 \beta_{4} - 7 \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{8} + ( -72 + 27 \beta_{1} - 3 \beta_{2} + 12 \beta_{4} - 24 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} ) q^{9} + ( 18 - 6 \beta_{2} - 8 \beta_{3} + \beta_{4} + 62 \beta_{5} + 13 \beta_{6} - 7 \beta_{7} ) q^{10} + ( 109 - 109 \beta_{1} + 7 \beta_{2} - 9 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + 12 \beta_{6} + 2 \beta_{7} ) q^{11} + ( 327 + 54 \beta_{1} + 10 \beta_{3} + 45 \beta_{4} + 63 \beta_{5} - 9 \beta_{6} + 9 \beta_{7} ) q^{12} + ( -62 \beta_{1} + 2 \beta_{2} + 5 \beta_{3} + 69 \beta_{4} - 3 \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{13} + ( 344 \beta_{1} - 5 \beta_{2} - 3 \beta_{3} + 18 \beta_{4} - 2 \beta_{5} - 9 \beta_{6} - 7 \beta_{7} ) q^{14} + ( 12 - 546 \beta_{1} - 3 \beta_{2} - 20 \beta_{3} - 93 \beta_{4} - 41 \beta_{5} + 13 \beta_{6} - 18 \beta_{7} ) q^{15} + ( -97 + 97 \beta_{1} + 3 \beta_{2} - 15 \beta_{3} - 108 \beta_{4} - 117 \beta_{5} - 6 \beta_{6} + 12 \beta_{7} ) q^{16} + ( -531 - 19 \beta_{2} + 51 \beta_{3} - 35 \beta_{4} + 41 \beta_{5} + 3 \beta_{6} + 16 \beta_{7} ) q^{17} + ( -1521 + 990 \beta_{1} + 27 \beta_{2} - 3 \beta_{3} - 18 \beta_{4} - 39 \beta_{5} + 33 \beta_{6} - 9 \beta_{7} ) q^{18} + ( 119 + 39 \beta_{2} + 45 \beta_{3} - 3 \beta_{4} - 123 \beta_{5} - 81 \beta_{6} + 42 \beta_{7} ) q^{19} + ( 1676 - 1676 \beta_{1} - 38 \beta_{2} + 54 \beta_{3} - 2 \beta_{4} - 24 \beta_{5} - 60 \beta_{6} - 16 \beta_{7} ) q^{20} + ( 864 + 426 \beta_{1} - 9 \beta_{2} - 36 \beta_{3} + 34 \beta_{5} + 25 \beta_{6} - 63 \beta_{7} ) q^{21} + ( 261 \beta_{1} + 8 \beta_{2} - 16 \beta_{3} - 57 \beta_{4} + 24 \beta_{5} + 56 \beta_{6} + 32 \beta_{7} ) q^{22} + ( 2216 \beta_{1} - 35 \beta_{2} - 84 \beta_{3} - 69 \beta_{4} + 49 \beta_{5} + 63 \beta_{6} + 14 \beta_{7} ) q^{23} + ( 1551 - 2913 \beta_{1} + 21 \beta_{2} + 67 \beta_{3} + 306 \beta_{4} + 163 \beta_{5} - 68 \beta_{6} + 30 \beta_{7} ) q^{24} + ( -331 + 331 \beta_{1} + 55 \beta_{2} - 70 \beta_{3} + 370 \beta_{4} + 410 \beta_{5} + 95 \beta_{6} + 15 \beta_{7} ) q^{25} + ( -3164 - 2 \beta_{2} - 168 \beta_{3} + 83 \beta_{4} - 40 \beta_{5} + 87 \beta_{6} - 85 \beta_{7} ) q^{26} + ( -1647 + 918 \beta_{1} - 90 \beta_{2} + 36 \beta_{3} + 171 \beta_{4} + 342 \beta_{5} - 135 \beta_{6} - 45 \beta_{7} ) q^{27} + ( -250 - 72 \beta_{2} - 24 \beta_{3} - 24 \beta_{4} - 30 \beta_{5} + 120 \beta_{6} - 48 \beta_{7} ) q^{28} + ( 2998 - 2998 \beta_{1} + 21 \beta_{2} - 72 \beta_{3} - 56 \beta_{4} - 86 \beta_{5} - 9 \beta_{6} + 51 \beta_{7} ) q^{29} + ( 1314 + 2628 \beta_{1} + 72 \beta_{2} + 42 \beta_{3} - 387 \beta_{4} - 834 \beta_{5} + 3 \beta_{6} + 135 \beta_{7} ) q^{30} + ( 478 \beta_{1} + 29 \beta_{2} + 98 \beta_{3} - 711 \beta_{4} - 69 \beta_{5} - 109 \beta_{6} - 40 \beta_{7} ) q^{31} + ( 2223 \beta_{1} + 145 \beta_{2} + 285 \beta_{3} - 49 \beta_{4} - 140 \beta_{5} - 135 \beta_{6} + 5 \beta_{7} ) q^{32} + ( -246 - 1893 \beta_{1} - 45 \beta_{2} - 25 \beta_{3} + 141 \beta_{4} - 103 \beta_{5} + 161 \beta_{6} - 6 \beta_{7} ) q^{33} + ( 2583 - 2583 \beta_{1} - 219 \beta_{2} + 321 \beta_{3} + 90 \beta_{4} - 27 \beta_{5} - 336 \beta_{6} - 102 \beta_{7} ) q^{34} + ( -1966 + 111 \beta_{2} + 243 \beta_{3} - 66 \beta_{4} - 217 \beta_{5} - 288 \beta_{6} + 177 \beta_{7} ) q^{35} + ( -3213 + 2961 \beta_{1} + 105 \beta_{2} - 165 \beta_{3} - 1032 \beta_{4} - 117 \beta_{5} + 192 \beta_{6} + 210 \beta_{7} ) q^{36} + ( -2140 - 6 \beta_{2} - 306 \beta_{3} + 150 \beta_{4} - 654 \beta_{5} + 162 \beta_{6} - 156 \beta_{7} ) q^{37} + ( -2743 + 2743 \beta_{1} + 189 \beta_{2} - 99 \beta_{3} + 452 \beta_{4} + 731 \beta_{5} + 468 \beta_{6} - 90 \beta_{7} ) q^{38} + ( 2634 - 2862 \beta_{1} - 189 \beta_{2} + 50 \beta_{3} - 279 \beta_{4} + 549 \beta_{5} - 117 \beta_{6} - 18 \beta_{7} ) q^{39} + ( -2448 \beta_{1} - 280 \beta_{2} - 352 \beta_{3} + 984 \beta_{4} + 72 \beta_{5} - 136 \beta_{6} - 208 \beta_{7} ) q^{40} + ( 119 \beta_{1} - 72 \beta_{2} - 126 \beta_{3} + 718 \beta_{4} + 54 \beta_{5} + 36 \beta_{6} - 18 \beta_{7} ) q^{41} + ( 1578 - 438 \beta_{1} + 15 \beta_{2} - 371 \beta_{3} + 240 \beta_{4} + 436 \beta_{5} - 41 \beta_{6} + 45 \beta_{7} ) q^{42} + ( -1517 + 1517 \beta_{1} + 181 \beta_{2} - 181 \beta_{3} + 205 \beta_{4} + 386 \beta_{5} + 362 \beta_{6} ) q^{43} + ( 6353 - 16 \beta_{2} - 30 \beta_{3} + 7 \beta_{4} + 709 \beta_{5} + 39 \beta_{6} - 23 \beta_{7} ) q^{44} + ( 5760 - 1260 \beta_{1} + 90 \beta_{2} + 291 \beta_{3} + 1359 \beta_{4} - 33 \beta_{5} + 264 \beta_{6} - 135 \beta_{7} ) q^{45} + ( 6372 + 42 \beta_{2} + 628 \beta_{3} - 293 \beta_{4} + 1796 \beta_{5} - 377 \beta_{6} + 335 \beta_{7} ) q^{46} + ( -188 + 188 \beta_{1} - 190 \beta_{2} + 81 \beta_{3} - 991 \beta_{4} - 1290 \beta_{5} - 489 \beta_{6} + 109 \beta_{7} ) q^{47} + ( -7614 + 735 \beta_{1} + 81 \beta_{2} - 135 \beta_{3} + 1809 \beta_{4} + 758 \beta_{5} + 83 \beta_{6} - 189 \beta_{7} ) q^{48} + ( 2355 \beta_{1} + 416 \beta_{2} + 407 \beta_{3} - 339 \beta_{4} + 9 \beta_{5} + 434 \beta_{6} + 425 \beta_{7} ) q^{49} + ( -14099 \beta_{1} - 335 \beta_{2} - 525 \beta_{3} - 851 \beta_{4} + 190 \beta_{5} + 45 \beta_{6} - 145 \beta_{7} ) q^{50} + ( -8289 + 14067 \beta_{1} - 9 \beta_{2} + 792 \beta_{3} - 2232 \beta_{4} - 2052 \beta_{5} - 693 \beta_{6} - 153 \beta_{7} ) q^{51} + ( -8870 + 8870 \beta_{1} + 336 \beta_{2} - 576 \beta_{3} - 2874 \beta_{4} - 2778 \beta_{5} + 432 \beta_{6} + 240 \beta_{7} ) q^{52} + ( 2052 - 390 \beta_{2} - 450 \beta_{3} + 30 \beta_{4} - 1038 \beta_{5} + 810 \beta_{6} - 420 \beta_{7} ) q^{53} + ( 9369 - 17199 \beta_{1} - 270 \beta_{2} + 234 \beta_{3} + 297 \beta_{4} - 1791 \beta_{5} - 1224 \beta_{6} - 324 \beta_{7} ) q^{54} + ( -4482 + 195 \beta_{2} + 43 \beta_{3} + 76 \beta_{4} - 301 \beta_{5} - 314 \beta_{6} + 119 \beta_{7} ) q^{55} + ( -15218 + 15218 \beta_{1} - 434 \beta_{2} + 450 \beta_{3} - 120 \beta_{4} - 538 \beta_{5} - 852 \beta_{6} - 16 \beta_{7} ) q^{56} + ( -6351 - 12279 \beta_{1} + 351 \beta_{2} + 16 \beta_{3} + 216 \beta_{4} + 2360 \beta_{5} + 227 \beta_{6} - 27 \beta_{7} ) q^{57} + ( 1440 \beta_{1} + 341 \beta_{2} + 359 \beta_{3} + 2586 \beta_{4} - 18 \beta_{5} + 305 \beta_{6} + 323 \beta_{7} ) q^{58} + ( 1019 \beta_{1} + 377 \beta_{2} + 363 \beta_{3} - 2014 \beta_{4} + 14 \beta_{5} + 405 \beta_{6} + 391 \beta_{7} ) q^{59} + ( 3180 + 16572 \beta_{1} + 198 \beta_{2} + 58 \beta_{3} - 762 \beta_{4} + 2728 \beta_{5} + 1456 \beta_{6} - 84 \beta_{7} ) q^{60} + ( 12532 - 12532 \beta_{1} - 545 \beta_{2} + 284 \beta_{3} + 1180 \beta_{4} + 374 \beta_{5} - 1351 \beta_{6} + 261 \beta_{7} ) q^{61} + ( 30550 + 22 \beta_{2} + 732 \beta_{3} - 355 \beta_{4} + 542 \beta_{5} - 399 \beta_{6} + 377 \beta_{7} ) q^{62} + ( 20016 - 7110 \beta_{1} - 42 \beta_{2} - 1749 \beta_{3} + 1311 \beta_{4} + 876 \beta_{5} + 1107 \beta_{6} - 21 \beta_{7} ) q^{63} + ( -2735 + 84 \beta_{2} - 1110 \beta_{3} + 597 \beta_{4} + 465 \beta_{5} + 429 \beta_{6} - 513 \beta_{7} ) q^{64} + ( -2384 + 2384 \beta_{1} + 63 \beta_{2} + 306 \beta_{3} + 4042 \beta_{4} + 4474 \beta_{5} + 495 \beta_{6} - 369 \beta_{7} ) q^{65} + ( -17073 + 7992 \beta_{1} - 189 \beta_{2} - 93 \beta_{3} - 549 \beta_{4} - 1935 \beta_{5} + 18 \beta_{6} - 72 \beta_{7} ) q^{66} + ( -1655 \beta_{1} - 955 \beta_{2} - 931 \beta_{3} - 504 \beta_{4} - 24 \beta_{5} - 1003 \beta_{6} - 979 \beta_{7} ) q^{67} + ( -32085 \beta_{1} + 29 \beta_{2} + 93 \beta_{3} + 5275 \beta_{4} - 64 \beta_{5} - 99 \beta_{6} - 35 \beta_{7} ) q^{68} + ( -27042 + 21648 \beta_{1} - 66 \beta_{2} - 2027 \beta_{3} + 1509 \beta_{4} - 1463 \beta_{5} - 212 \beta_{6} + 315 \beta_{7} ) q^{69} + ( -234 + 234 \beta_{1} + 175 \beta_{2} + 311 \beta_{3} + 379 \beta_{4} + 1040 \beta_{5} + 836 \beta_{6} - 486 \beta_{7} ) q^{70} + ( -7956 + 716 \beta_{2} - 516 \beta_{3} + 616 \beta_{4} + 3188 \beta_{5} - 816 \beta_{6} + 100 \beta_{7} ) q^{71} + ( 7056 - 3501 \beta_{1} + 45 \beta_{2} + 2001 \beta_{3} - 4077 \beta_{4} + 2100 \beta_{5} + 957 \beta_{6} + 585 \beta_{7} ) q^{72} + ( 14699 - 873 \beta_{2} + 1161 \beta_{3} - 1017 \beta_{4} - 1845 \beta_{5} + 729 \beta_{6} + 144 \beta_{7} ) q^{73} + ( -41776 + 41776 \beta_{1} + 1242 \beta_{2} - 1854 \beta_{3} - 6154 \beta_{4} - 5524 \beta_{5} + 1872 \beta_{6} + 612 \beta_{7} ) q^{74} + ( 7830 - 20883 \beta_{1} - 360 \beta_{2} + 585 \beta_{3} - 3645 \beta_{4} - 3991 \beta_{5} - 706 \beta_{6} + 1125 \beta_{7} ) q^{75} + ( -10943 \beta_{1} + 163 \beta_{2} + 475 \beta_{3} - 5487 \beta_{4} - 312 \beta_{5} - 461 \beta_{6} - 149 \beta_{7} ) q^{76} + ( 10658 \beta_{1} + 174 \beta_{2} + 1143 \beta_{3} - 2129 \beta_{4} - 969 \beta_{5} - 1764 \beta_{6} - 795 \beta_{7} ) q^{77} + ( 38352 - 32046 \beta_{1} - 654 \beta_{2} + 2236 \beta_{3} + 5625 \beta_{4} - 1232 \beta_{5} - 2921 \beta_{6} + 111 \beta_{7} ) q^{78} + ( -14054 + 14054 \beta_{1} - 392 \beta_{2} + 1685 \beta_{3} + 4201 \beta_{4} + 5102 \beta_{5} + 509 \beta_{6} - 1293 \beta_{7} ) q^{79} + ( 13648 + 464 \beta_{2} + 672 \beta_{3} - 104 \beta_{4} - 9264 \beta_{5} - 1032 \beta_{6} + 568 \beta_{7} ) q^{80} + ( 14823 + 20817 \beta_{1} + 1350 \beta_{2} + 1593 \beta_{3} - 2889 \beta_{4} - 1701 \beta_{5} - 2484 \beta_{6} + 1485 \beta_{7} ) q^{81} + ( -29637 + 180 \beta_{2} - 896 \beta_{3} + 538 \beta_{4} - 1213 \beta_{5} + 178 \beta_{6} - 358 \beta_{7} ) q^{82} + ( 28918 - 28918 \beta_{1} + 846 \beta_{2} - 765 \beta_{3} - 1097 \beta_{4} - 170 \beta_{5} + 1773 \beta_{6} - 81 \beta_{7} ) q^{83} + ( -6 + 25926 \beta_{1} - 1242 \beta_{2} - 218 \beta_{3} + 54 \beta_{4} - 1834 \beta_{5} - 538 \beta_{6} - 810 \beta_{7} ) q^{84} + ( 7920 \beta_{1} - 648 \beta_{2} - 1590 \beta_{3} - 3786 \beta_{4} + 942 \beta_{5} + 1236 \beta_{6} + 294 \beta_{7} ) q^{85} + ( -2797 \beta_{1} - 386 \beta_{2} - 1110 \beta_{3} - 5451 \beta_{4} + 724 \beta_{5} + 1062 \beta_{6} + 338 \beta_{7} ) q^{86} + ( -21846 + 8016 \beta_{1} + 270 \beta_{2} + 191 \beta_{3} + 2877 \beta_{4} + 4262 \beta_{5} + 3023 \beta_{6} + 381 \beta_{7} ) q^{87} + ( 27747 - 27747 \beta_{1} + 311 \beta_{2} - 1139 \beta_{3} + 5948 \beta_{4} + 5431 \beta_{5} - 206 \beta_{6} + 828 \beta_{7} ) q^{88} + ( -56034 - 716 \beta_{2} - 1788 \beta_{3} + 536 \beta_{4} + 4948 \beta_{5} + 1968 \beta_{6} - 1252 \beta_{7} ) q^{89} + ( -66798 - 1296 \beta_{1} - 1278 \beta_{2} - 4752 \beta_{3} + 5841 \beta_{4} + 6894 \beta_{5} + 1989 \beta_{6} - 2367 \beta_{7} ) q^{90} + ( 13702 + 195 \beta_{2} - 97 \beta_{3} + 146 \beta_{4} - 1181 \beta_{5} - 244 \beta_{6} + 49 \beta_{7} ) q^{91} + ( 37514 - 37514 \beta_{1} - 2436 \beta_{2} + 2124 \beta_{3} + 13274 \beta_{4} + 10526 \beta_{5} - 5184 \beta_{6} + 312 \beta_{7} ) q^{92} + ( 17484 - 8532 \beta_{1} + 2700 \beta_{2} - 805 \beta_{3} + 1395 \beta_{4} - 2745 \beta_{5} + 1854 \beta_{6} - 261 \beta_{7} ) q^{93} + ( 41904 \beta_{1} + 1835 \beta_{2} + 2813 \beta_{3} + 2826 \beta_{4} - 978 \beta_{5} - 121 \beta_{6} + 857 \beta_{7} ) q^{94} + ( 70528 \beta_{1} + 112 \beta_{2} - 300 \beta_{3} + 12364 \beta_{4} + 412 \beta_{5} + 936 \beta_{6} + 524 \beta_{7} ) q^{95} + ( 36639 - 61614 \beta_{1} + 522 \beta_{2} - 2628 \beta_{3} - 6003 \beta_{4} + 567 \beta_{5} + 1233 \beta_{6} - 1305 \beta_{7} ) q^{96} + ( 6691 - 6691 \beta_{1} + 154 \beta_{2} - 2296 \beta_{3} - 9200 \beta_{4} - 11188 \beta_{5} - 1834 \beta_{6} + 2142 \beta_{7} ) q^{97} + ( 9873 - 1682 \beta_{2} + 768 \beta_{3} - 1225 \beta_{4} + 13261 \beta_{5} + 2139 \beta_{6} - 457 \beta_{7} ) q^{98} + ( -6381 + 20430 \beta_{1} - 2241 \beta_{2} + 1698 \beta_{3} - 666 \beta_{4} - 1011 \beta_{5} - 858 \beta_{6} - 1143 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 3q^{2} - 12q^{3} - 49q^{4} + 78q^{5} + 171q^{6} + 28q^{7} - 750q^{8} - 414q^{9} + O(q^{10}) \) \( 8q + 3q^{2} - 12q^{3} - 49q^{4} + 78q^{5} + 171q^{6} + 28q^{7} - 750q^{8} - 414q^{9} + 60q^{10} + 444q^{11} + 2724q^{12} - 182q^{13} + 1392q^{14} - 2052q^{15} - 289q^{16} - 4356q^{17} - 8100q^{18} + 952q^{19} + 6684q^{20} + 8670q^{21} + 1011q^{22} + 8844q^{23} + 549q^{24} - 1654q^{25} - 24888q^{26} - 10152q^{27} - 1604q^{28} + 12018q^{29} + 22104q^{30} + 1132q^{31} + 8703q^{32} - 8820q^{33} + 10125q^{34} - 16224q^{35} - 14589q^{36} - 15176q^{37} - 11145q^{38} + 8220q^{39} - 8736q^{40} + 1248q^{41} + 10098q^{42} - 6092q^{43} + 49530q^{44} + 43038q^{45} + 45960q^{46} - 60q^{47} - 57405q^{48} + 9090q^{49} - 57057q^{50} - 9396q^{51} - 32510q^{52} + 20952q^{53} + 8181q^{54} - 36120q^{55} - 61170q^{56} - 104298q^{57} + 8328q^{58} + 2076q^{59} + 88308q^{60} + 48142q^{61} + 241764q^{62} + 133524q^{63} - 20926q^{64} - 13146q^{65} - 100998q^{66} - 7148q^{67} - 123129q^{68} - 125982q^{69} - 654q^{70} - 71856q^{71} + 35451q^{72} + 122452q^{73} - 160320q^{74} - 18732q^{75} - 49571q^{76} + 39534q^{77} + 181422q^{78} - 59516q^{79} + 124512q^{80} + 194562q^{81} - 233598q^{82} + 117696q^{83} + 108354q^{84} + 28836q^{85} - 15915q^{86} - 142956q^{87} + 104523q^{88} - 451728q^{89} - 539892q^{90} + 111392q^{91} + 134034q^{92} + 113898q^{93} + 169464q^{94} + 294888q^{95} + 42768q^{96} + 33976q^{97} + 57654q^{98} + 33696q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 40 x^{6} + 568 x^{4} + 3363 x^{2} + 7056\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{7} + 40 \nu^{5} + 484 \nu^{3} + 1683 \nu + 84 \)\()/168\)
\(\beta_{2}\)\(=\)\((\)\( -3 \nu^{7} - 4 \nu^{6} - 96 \nu^{5} - 88 \nu^{4} - 984 \nu^{3} - 592 \nu^{2} - 3273 \nu - 1644 \)\()/24\)
\(\beta_{3}\)\(=\)\((\)\( -5 \nu^{7} - 158 \nu^{5} - 42 \nu^{4} - 1580 \nu^{3} - 798 \nu^{2} - 4929 \nu - 3108 \)\()/28\)
\(\beta_{4}\)\(=\)\((\)\( -9 \nu^{7} + 28 \nu^{6} - 276 \nu^{5} + 868 \nu^{4} - 2592 \nu^{3} + 8428 \nu^{2} - 7167 \nu + 25116 \)\()/168\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{6} - 31 \nu^{4} - 301 \nu^{2} - 897 \)\()/3\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{6} + 31 \nu^{4} + 310 \nu^{2} - 9 \nu + 987 \)\()/3\)
\(\beta_{7}\)\(=\)\((\)\( 51 \nu^{7} - 28 \nu^{6} + 1620 \nu^{5} - 1372 \nu^{4} + 16368 \nu^{3} - 18508 \nu^{2} + 51477 \nu - 67452 \)\()/168\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{7} - 3 \beta_{6} - 2 \beta_{5} - \beta_{4} - 2 \beta_{2}\)\()/18\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{7} + 3 \beta_{6} + 4 \beta_{5} - \beta_{4} - 2 \beta_{2} - 180\)\()/18\)
\(\nu^{3}\)\(=\)\((\)\(7 \beta_{7} + 18 \beta_{6} + 20 \beta_{5} + 22 \beta_{4} - 3 \beta_{3} + 11 \beta_{2} - 18 \beta_{1} + 9\)\()/9\)
\(\nu^{4}\)\(=\)\((\)\(17 \beta_{7} - 57 \beta_{6} - 74 \beta_{5} + 23 \beta_{4} - 6 \beta_{3} + 40 \beta_{2} + 2088\)\()/18\)
\(\nu^{5}\)\(=\)\((\)\(-199 \beta_{7} - 471 \beta_{6} - 632 \beta_{5} - 793 \beta_{4} + 126 \beta_{3} - 272 \beta_{2} + 1080 \beta_{1} - 540\)\()/18\)
\(\nu^{6}\)\(=\)\((\)\(-113 \beta_{7} + 432 \beta_{6} + 518 \beta_{5} - 206 \beta_{4} + 93 \beta_{3} - 319 \beta_{2} - 13347\)\()/9\)
\(\nu^{7}\)\(=\)\((\)\(2867 \beta_{7} + 6465 \beta_{6} + 9286 \beta_{5} + 12107 \beta_{4} - 2136 \beta_{3} + 3598 \beta_{2} - 22752 \beta_{1} + 11376\)\()/18\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/9\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
3.62198i
2.34949i
3.84183i
2.56934i
3.62198i
2.34949i
3.84183i
2.56934i
−3.77467 6.53793i −15.5883 + 0.0716805i −12.4963 + 21.6443i 43.7155 75.7175i 59.3094 + 101.645i 20.6033 + 35.6859i −52.9007 242.990 2.23475i −660.048
4.2 −1.47673 2.55778i 11.3240 10.7129i 11.6385 20.1585i −32.4255 + 56.1625i −44.1239 13.1441i 80.0952 + 138.729i −163.259 13.4657 242.627i 191.535
4.3 1.78978 + 3.09998i 1.34189 + 15.5306i 9.59340 16.6162i 4.05388 7.02152i −45.7429 + 31.9561i −87.7139 151.925i 183.226 −239.399 + 41.6808i 29.0221
4.4 4.96163 + 8.59380i −3.07760 15.2816i −33.2356 + 57.5657i 23.6560 40.9735i 116.057 102.270i 1.01546 + 1.75882i −342.066 −224.057 + 94.0614i 469.490
7.1 −3.77467 + 6.53793i −15.5883 0.0716805i −12.4963 21.6443i 43.7155 + 75.7175i 59.3094 101.645i 20.6033 35.6859i −52.9007 242.990 + 2.23475i −660.048
7.2 −1.47673 + 2.55778i 11.3240 + 10.7129i 11.6385 + 20.1585i −32.4255 56.1625i −44.1239 + 13.1441i 80.0952 138.729i −163.259 13.4657 + 242.627i 191.535
7.3 1.78978 3.09998i 1.34189 15.5306i 9.59340 + 16.6162i 4.05388 + 7.02152i −45.7429 31.9561i −87.7139 + 151.925i 183.226 −239.399 41.6808i 29.0221
7.4 4.96163 8.59380i −3.07760 + 15.2816i −33.2356 57.5657i 23.6560 + 40.9735i 116.057 + 102.270i 1.01546 1.75882i −342.066 −224.057 94.0614i 469.490
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9.6.c.a 8
3.b odd 2 1 27.6.c.a 8
4.b odd 2 1 144.6.i.c 8
9.c even 3 1 inner 9.6.c.a 8
9.c even 3 1 81.6.a.c 4
9.d odd 6 1 27.6.c.a 8
9.d odd 6 1 81.6.a.d 4
12.b even 2 1 432.6.i.c 8
36.f odd 6 1 144.6.i.c 8
36.h even 6 1 432.6.i.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.6.c.a 8 1.a even 1 1 trivial
9.6.c.a 8 9.c even 3 1 inner
27.6.c.a 8 3.b odd 2 1
27.6.c.a 8 9.d odd 6 1
81.6.a.c 4 9.c even 3 1
81.6.a.d 4 9.d odd 6 1
144.6.i.c 8 4.b odd 2 1
144.6.i.c 8 36.f odd 6 1
432.6.i.c 8 12.b even 2 1
432.6.i.c 8 36.h even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(9, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 3 T - 35 T^{2} + 300 T^{3} - 272 T^{4} - 7056 T^{5} + 47392 T^{6} + 37824 T^{7} - 1500608 T^{8} + 1210368 T^{9} + 48529408 T^{10} - 231211008 T^{11} - 285212672 T^{12} + 10066329600 T^{13} - 37580963840 T^{14} - 103079215104 T^{15} + 1099511627776 T^{16} \)
$3$ \( 1 + 12 T + 279 T^{2} + 6156 T^{3} + 29160 T^{4} + 1495908 T^{5} + 16474671 T^{6} + 172186884 T^{7} + 3486784401 T^{8} \)
$5$ \( 1 - 78 T - 2381 T^{2} + 191094 T^{3} + 7253005 T^{4} - 66450636 T^{5} - 21330994286 T^{6} + 411629383632 T^{7} - 16622006693174 T^{8} + 1286341823850000 T^{9} - 208310491074218750 T^{10} - 2027912475585937500 T^{11} + \)\(69\!\cdots\!25\)\( T^{12} + \)\(56\!\cdots\!50\)\( T^{13} - \)\(22\!\cdots\!25\)\( T^{14} - \)\(22\!\cdots\!50\)\( T^{15} + \)\(90\!\cdots\!25\)\( T^{16} \)
$7$ \( 1 - 28 T - 37767 T^{2} - 688508 T^{3} + 752155661 T^{4} + 29978584680 T^{5} - 6360743348222 T^{6} - 326958516789136 T^{7} + 37987499418745374 T^{8} - 5495191791675008752 T^{9} - \)\(17\!\cdots\!78\)\( T^{10} + \)\(14\!\cdots\!40\)\( T^{11} + \)\(60\!\cdots\!61\)\( T^{12} - \)\(92\!\cdots\!56\)\( T^{13} - \)\(85\!\cdots\!83\)\( T^{14} - \)\(10\!\cdots\!04\)\( T^{15} + \)\(63\!\cdots\!01\)\( T^{16} \)
$11$ \( 1 - 444 T - 421874 T^{2} + 122372328 T^{3} + 147603288625 T^{4} - 24043789512072 T^{5} - 33881306185300610 T^{6} + 1343287267601416260 T^{7} + \)\(64\!\cdots\!56\)\( T^{8} + \)\(21\!\cdots\!60\)\( T^{9} - \)\(87\!\cdots\!10\)\( T^{10} - \)\(10\!\cdots\!72\)\( T^{11} + \)\(99\!\cdots\!25\)\( T^{12} + \)\(13\!\cdots\!28\)\( T^{13} - \)\(73\!\cdots\!74\)\( T^{14} - \)\(12\!\cdots\!44\)\( T^{15} + \)\(45\!\cdots\!01\)\( T^{16} \)
$13$ \( 1 + 182 T - 1009641 T^{2} - 162494414 T^{3} + 553084541681 T^{4} + 66263508549516 T^{5} - 217064082764017418 T^{6} - 12452829446314121656 T^{7} + \)\(75\!\cdots\!98\)\( T^{8} - \)\(46\!\cdots\!08\)\( T^{9} - \)\(29\!\cdots\!82\)\( T^{10} + \)\(33\!\cdots\!12\)\( T^{11} + \)\(10\!\cdots\!81\)\( T^{12} - \)\(11\!\cdots\!02\)\( T^{13} - \)\(26\!\cdots\!09\)\( T^{14} + \)\(17\!\cdots\!74\)\( T^{15} + \)\(36\!\cdots\!01\)\( T^{16} \)
$17$ \( ( 1 + 2178 T + 4502417 T^{2} + 6473305458 T^{3} + 7835841277908 T^{4} + 9191168067679506 T^{5} + 9076845209277885233 T^{6} + \)\(62\!\cdots\!54\)\( T^{7} + \)\(40\!\cdots\!01\)\( T^{8} )^{2} \)
$19$ \( ( 1 - 476 T + 5379895 T^{2} - 1785507140 T^{3} + 16120413258280 T^{4} - 4421092443846860 T^{5} + 32984492705012310895 T^{6} - \)\(72\!\cdots\!24\)\( T^{7} + \)\(37\!\cdots\!01\)\( T^{8} )^{2} \)
$23$ \( 1 - 8844 T + 29643313 T^{2} - 58247545548 T^{3} + 172348926961837 T^{4} - 574153269020383320 T^{5} + \)\(14\!\cdots\!02\)\( T^{6} - \)\(52\!\cdots\!88\)\( T^{7} + \)\(17\!\cdots\!06\)\( T^{8} - \)\(33\!\cdots\!84\)\( T^{9} + \)\(60\!\cdots\!98\)\( T^{10} - \)\(15\!\cdots\!40\)\( T^{11} + \)\(29\!\cdots\!37\)\( T^{12} - \)\(64\!\cdots\!64\)\( T^{13} + \)\(21\!\cdots\!37\)\( T^{14} - \)\(40\!\cdots\!08\)\( T^{15} + \)\(29\!\cdots\!01\)\( T^{16} \)
$29$ \( 1 - 12018 T + 15291415 T^{2} + 58796364810 T^{3} + 2512265879646529 T^{4} - 12711396421291774644 T^{5} - \)\(20\!\cdots\!74\)\( T^{6} - \)\(10\!\cdots\!36\)\( T^{7} + \)\(21\!\cdots\!82\)\( T^{8} - \)\(21\!\cdots\!64\)\( T^{9} - \)\(86\!\cdots\!74\)\( T^{10} - \)\(10\!\cdots\!56\)\( T^{11} + \)\(44\!\cdots\!29\)\( T^{12} + \)\(21\!\cdots\!90\)\( T^{13} + \)\(11\!\cdots\!15\)\( T^{14} - \)\(18\!\cdots\!82\)\( T^{15} + \)\(31\!\cdots\!01\)\( T^{16} \)
$31$ \( 1 - 1132 T - 60658635 T^{2} + 62431167700 T^{3} + 1166392900733297 T^{4} - 126919079777751576 T^{5} - \)\(56\!\cdots\!38\)\( T^{6} - \)\(20\!\cdots\!72\)\( T^{7} + \)\(28\!\cdots\!06\)\( T^{8} - \)\(60\!\cdots\!72\)\( T^{9} - \)\(46\!\cdots\!38\)\( T^{10} - \)\(29\!\cdots\!76\)\( T^{11} + \)\(78\!\cdots\!97\)\( T^{12} + \)\(12\!\cdots\!00\)\( T^{13} - \)\(33\!\cdots\!35\)\( T^{14} - \)\(17\!\cdots\!32\)\( T^{15} + \)\(45\!\cdots\!01\)\( T^{16} \)
$37$ \( ( 1 + 7588 T + 181007992 T^{2} + 1702118220604 T^{3} + 15462039197985406 T^{4} + \)\(11\!\cdots\!28\)\( T^{5} + \)\(87\!\cdots\!08\)\( T^{6} + \)\(25\!\cdots\!84\)\( T^{7} + \)\(23\!\cdots\!01\)\( T^{8} )^{2} \)
$41$ \( 1 - 1248 T - 412882754 T^{2} + 256860139200 T^{3} + 101750473497083809 T^{4} - 31706402029277721984 T^{5} - \)\(17\!\cdots\!98\)\( T^{6} + \)\(13\!\cdots\!52\)\( T^{7} + \)\(23\!\cdots\!00\)\( T^{8} + \)\(15\!\cdots\!52\)\( T^{9} - \)\(23\!\cdots\!98\)\( T^{10} - \)\(49\!\cdots\!84\)\( T^{11} + \)\(18\!\cdots\!09\)\( T^{12} + \)\(53\!\cdots\!00\)\( T^{13} - \)\(99\!\cdots\!54\)\( T^{14} - \)\(34\!\cdots\!48\)\( T^{15} + \)\(32\!\cdots\!01\)\( T^{16} \)
$43$ \( 1 + 6092 T - 493633434 T^{2} - 1717030281896 T^{3} + 152938627936979465 T^{4} + \)\(29\!\cdots\!88\)\( T^{5} - \)\(33\!\cdots\!22\)\( T^{6} - \)\(15\!\cdots\!68\)\( T^{7} + \)\(57\!\cdots\!20\)\( T^{8} - \)\(23\!\cdots\!24\)\( T^{9} - \)\(72\!\cdots\!78\)\( T^{10} + \)\(92\!\cdots\!16\)\( T^{11} + \)\(71\!\cdots\!65\)\( T^{12} - \)\(11\!\cdots\!28\)\( T^{13} - \)\(49\!\cdots\!66\)\( T^{14} + \)\(90\!\cdots\!44\)\( T^{15} + \)\(21\!\cdots\!01\)\( T^{16} \)
$47$ \( 1 + 60 T - 695011823 T^{2} + 3293074754652 T^{3} + 272693815319492413 T^{4} - \)\(15\!\cdots\!00\)\( T^{5} - \)\(70\!\cdots\!78\)\( T^{6} + \)\(17\!\cdots\!56\)\( T^{7} + \)\(15\!\cdots\!82\)\( T^{8} + \)\(41\!\cdots\!92\)\( T^{9} - \)\(37\!\cdots\!22\)\( T^{10} - \)\(18\!\cdots\!00\)\( T^{11} + \)\(75\!\cdots\!13\)\( T^{12} + \)\(20\!\cdots\!64\)\( T^{13} - \)\(10\!\cdots\!27\)\( T^{14} + \)\(20\!\cdots\!80\)\( T^{15} + \)\(76\!\cdots\!01\)\( T^{16} \)
$53$ \( ( 1 - 10476 T + 1064413976 T^{2} - 16431260960628 T^{3} + 549720542476264830 T^{4} - \)\(68\!\cdots\!04\)\( T^{5} + \)\(18\!\cdots\!24\)\( T^{6} - \)\(76\!\cdots\!32\)\( T^{7} + \)\(30\!\cdots\!01\)\( T^{8} )^{2} \)
$59$ \( 1 - 2076 T - 2248193858 T^{2} + 15695201332392 T^{3} + 2863253985413117089 T^{4} - \)\(21\!\cdots\!96\)\( T^{5} - \)\(25\!\cdots\!86\)\( T^{6} + \)\(73\!\cdots\!56\)\( T^{7} + \)\(18\!\cdots\!76\)\( T^{8} + \)\(52\!\cdots\!44\)\( T^{9} - \)\(12\!\cdots\!86\)\( T^{10} - \)\(77\!\cdots\!04\)\( T^{11} + \)\(74\!\cdots\!89\)\( T^{12} + \)\(29\!\cdots\!08\)\( T^{13} - \)\(30\!\cdots\!58\)\( T^{14} - \)\(19\!\cdots\!24\)\( T^{15} + \)\(68\!\cdots\!01\)\( T^{16} \)
$61$ \( 1 - 48142 T - 755431725 T^{2} + 43590251951830 T^{3} + 1116615498545065565 T^{4} - \)\(17\!\cdots\!08\)\( T^{5} - \)\(17\!\cdots\!86\)\( T^{6} + \)\(66\!\cdots\!20\)\( T^{7} + \)\(16\!\cdots\!34\)\( T^{8} + \)\(56\!\cdots\!20\)\( T^{9} - \)\(12\!\cdots\!86\)\( T^{10} - \)\(10\!\cdots\!08\)\( T^{11} + \)\(56\!\cdots\!65\)\( T^{12} + \)\(18\!\cdots\!30\)\( T^{13} - \)\(27\!\cdots\!25\)\( T^{14} - \)\(14\!\cdots\!42\)\( T^{15} + \)\(25\!\cdots\!01\)\( T^{16} \)
$67$ \( 1 + 7148 T - 3268771122 T^{2} - 16753922588168 T^{3} + 4825996342110423185 T^{4} + \)\(80\!\cdots\!92\)\( T^{5} - \)\(77\!\cdots\!58\)\( T^{6} + \)\(18\!\cdots\!60\)\( T^{7} + \)\(12\!\cdots\!20\)\( T^{8} + \)\(25\!\cdots\!20\)\( T^{9} - \)\(14\!\cdots\!42\)\( T^{10} + \)\(19\!\cdots\!56\)\( T^{11} + \)\(16\!\cdots\!85\)\( T^{12} - \)\(75\!\cdots\!76\)\( T^{13} - \)\(19\!\cdots\!78\)\( T^{14} + \)\(58\!\cdots\!64\)\( T^{15} + \)\(11\!\cdots\!01\)\( T^{16} \)
$71$ \( ( 1 + 35928 T + 5285725772 T^{2} + 127659705990840 T^{3} + 12554595869440882758 T^{4} + \)\(23\!\cdots\!40\)\( T^{5} + \)\(17\!\cdots\!72\)\( T^{6} + \)\(21\!\cdots\!28\)\( T^{7} + \)\(10\!\cdots\!01\)\( T^{8} )^{2} \)
$73$ \( ( 1 - 61226 T + 6621012721 T^{2} - 212085627051026 T^{3} + 16124371439127547300 T^{4} - \)\(43\!\cdots\!18\)\( T^{5} + \)\(28\!\cdots\!29\)\( T^{6} - \)\(54\!\cdots\!82\)\( T^{7} + \)\(18\!\cdots\!01\)\( T^{8} )^{2} \)
$79$ \( 1 + 59516 T - 3717013179 T^{2} - 374196053042180 T^{3} - 3765094998545504239 T^{4} + \)\(30\!\cdots\!72\)\( T^{5} - \)\(95\!\cdots\!50\)\( T^{6} + \)\(90\!\cdots\!24\)\( T^{7} + \)\(17\!\cdots\!98\)\( T^{8} + \)\(27\!\cdots\!76\)\( T^{9} - \)\(90\!\cdots\!50\)\( T^{10} + \)\(87\!\cdots\!28\)\( T^{11} - \)\(33\!\cdots\!39\)\( T^{12} - \)\(10\!\cdots\!20\)\( T^{13} - \)\(31\!\cdots\!79\)\( T^{14} + \)\(15\!\cdots\!84\)\( T^{15} + \)\(80\!\cdots\!01\)\( T^{16} \)
$83$ \( 1 - 117696 T - 5262215903 T^{2} + 507938387214576 T^{3} + 83906673955765181161 T^{4} - \)\(37\!\cdots\!00\)\( T^{5} - \)\(42\!\cdots\!38\)\( T^{6} + \)\(12\!\cdots\!52\)\( T^{7} + \)\(24\!\cdots\!34\)\( T^{8} + \)\(47\!\cdots\!36\)\( T^{9} - \)\(65\!\cdots\!62\)\( T^{10} - \)\(22\!\cdots\!00\)\( T^{11} + \)\(20\!\cdots\!61\)\( T^{12} + \)\(48\!\cdots\!68\)\( T^{13} - \)\(19\!\cdots\!47\)\( T^{14} - \)\(17\!\cdots\!72\)\( T^{15} + \)\(57\!\cdots\!01\)\( T^{16} \)
$89$ \( ( 1 + 225864 T + 38022042668 T^{2} + 4189883418729720 T^{3} + \)\(36\!\cdots\!22\)\( T^{4} + \)\(23\!\cdots\!80\)\( T^{5} + \)\(11\!\cdots\!68\)\( T^{6} + \)\(39\!\cdots\!36\)\( T^{7} + \)\(97\!\cdots\!01\)\( T^{8} )^{2} \)
$97$ \( 1 - 33976 T - 11979920730 T^{2} + 1870850391619312 T^{3} + 3904959771682438313 T^{4} - \)\(16\!\cdots\!64\)\( T^{5} + \)\(12\!\cdots\!22\)\( T^{6} + \)\(57\!\cdots\!68\)\( T^{7} - \)\(11\!\cdots\!00\)\( T^{8} + \)\(49\!\cdots\!76\)\( T^{9} + \)\(91\!\cdots\!78\)\( T^{10} - \)\(10\!\cdots\!52\)\( T^{11} + \)\(21\!\cdots\!13\)\( T^{12} + \)\(87\!\cdots\!84\)\( T^{13} - \)\(48\!\cdots\!70\)\( T^{14} - \)\(11\!\cdots\!68\)\( T^{15} + \)\(29\!\cdots\!01\)\( T^{16} \)
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