Properties

Label 91.8.a.c
Level $91$
Weight $8$
Character orbit 91.a
Self dual yes
Analytic conductor $28.427$
Analytic rank $1$
Dimension $10$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 91.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.4270373191\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 2 x^{9} - 957 x^{8} + 1224 x^{7} + 310102 x^{6} - 241884 x^{5} - 40367312 x^{4} + 11067840 x^{3} + 1840757376 x^{2} + 541859072 x - 4516262912\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 + \beta_{1} ) q^{2} + ( -8 - \beta_{1} + \beta_{3} ) q^{3} + ( 68 - 3 \beta_{1} + \beta_{2} ) q^{4} + ( -93 + 4 \beta_{1} - \beta_{6} ) q^{5} + ( -140 - 10 \beta_{1} - 4 \beta_{3} + \beta_{4} - \beta_{8} ) q^{6} + 343 q^{7} + ( -503 + 59 \beta_{1} - 5 \beta_{2} + 9 \beta_{3} - \beta_{4} + \beta_{5} + 4 \beta_{6} + 4 \beta_{8} ) q^{8} + ( 365 - 6 \beta_{1} - 5 \beta_{2} - 15 \beta_{3} - 5 \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{8} + 2 \beta_{9} ) q^{9} +O(q^{10})\) \( q + ( -2 + \beta_{1} ) q^{2} + ( -8 - \beta_{1} + \beta_{3} ) q^{3} + ( 68 - 3 \beta_{1} + \beta_{2} ) q^{4} + ( -93 + 4 \beta_{1} - \beta_{6} ) q^{5} + ( -140 - 10 \beta_{1} - 4 \beta_{3} + \beta_{4} - \beta_{8} ) q^{6} + 343 q^{7} + ( -503 + 59 \beta_{1} - 5 \beta_{2} + 9 \beta_{3} - \beta_{4} + \beta_{5} + 4 \beta_{6} + 4 \beta_{8} ) q^{8} + ( 365 - 6 \beta_{1} - 5 \beta_{2} - 15 \beta_{3} - 5 \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{8} + 2 \beta_{9} ) q^{9} + ( 964 - 134 \beta_{1} - 7 \beta_{2} - 26 \beta_{3} - 5 \beta_{5} + 6 \beta_{6} + \beta_{7} - 6 \beta_{8} - 3 \beta_{9} ) q^{10} + ( 78 + 45 \beta_{1} - 19 \beta_{2} - 32 \beta_{3} - \beta_{4} + 7 \beta_{5} + 7 \beta_{6} - \beta_{7} + 3 \beta_{8} - 5 \beta_{9} ) q^{11} + ( -883 - 12 \beta_{1} - 32 \beta_{2} + 21 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} - 6 \beta_{6} - 3 \beta_{7} + 7 \beta_{8} + 5 \beta_{9} ) q^{12} -2197 q^{13} + ( -686 + 343 \beta_{1} ) q^{14} + ( -475 - 134 \beta_{1} + 21 \beta_{2} - 179 \beta_{3} + 19 \beta_{4} + 7 \beta_{5} + 14 \beta_{6} + \beta_{8} + 9 \beta_{9} ) q^{15} + ( 4314 - 566 \beta_{1} + 67 \beta_{2} - 274 \beta_{3} + 9 \beta_{4} + 7 \beta_{5} + 12 \beta_{6} + 18 \beta_{7} - 10 \beta_{8} + 5 \beta_{9} ) q^{16} + ( 712 - 611 \beta_{1} - 34 \beta_{2} + 27 \beta_{3} + 10 \beta_{4} - 52 \beta_{5} - 17 \beta_{6} + 15 \beta_{7} - 10 \beta_{8} + 15 \beta_{9} ) q^{17} + ( -1494 - 112 \beta_{1} + 10 \beta_{2} - 260 \beta_{3} + 8 \beta_{4} + 34 \beta_{5} - 48 \beta_{6} - 15 \beta_{7} - 11 \beta_{8} - 28 \beta_{9} ) q^{18} + ( -9648 - 481 \beta_{1} + 14 \beta_{2} + 23 \beta_{3} + 24 \beta_{4} + 8 \beta_{5} - 33 \beta_{6} - 16 \beta_{7} - 42 \beta_{8} - 39 \beta_{9} ) q^{19} + ( -16728 + 197 \beta_{1} - 152 \beta_{2} + 316 \beta_{3} - 9 \beta_{4} + 33 \beta_{5} - 96 \beta_{6} - 52 \beta_{7} + 56 \beta_{8} + 37 \beta_{9} ) q^{20} + ( -2744 - 343 \beta_{1} + 343 \beta_{3} ) q^{21} + ( 7999 - 1949 \beta_{1} + 192 \beta_{2} - 269 \beta_{3} - 99 \beta_{4} + 22 \beta_{5} - 36 \beta_{6} + 46 \beta_{7} + 3 \beta_{8} + 51 \beta_{9} ) q^{22} + ( -1301 - 1287 \beta_{1} - 37 \beta_{2} - 352 \beta_{3} + 81 \beta_{4} - 27 \beta_{5} + 86 \beta_{6} - 17 \beta_{7} - 5 \beta_{8} - 61 \beta_{9} ) q^{23} + ( 20399 - 3414 \beta_{1} + 112 \beta_{2} - 573 \beta_{3} - 12 \beta_{4} - 102 \beta_{5} - 38 \beta_{6} + 21 \beta_{7} - 125 \beta_{8} - 73 \beta_{9} ) q^{24} + ( 17049 - 3362 \beta_{1} + 120 \beta_{2} - 504 \beta_{3} - 42 \beta_{4} + 4 \beta_{5} + 24 \beta_{6} + 11 \beta_{7} + 16 \beta_{8} + 10 \beta_{9} ) q^{25} + ( 4394 - 2197 \beta_{1} ) q^{26} + ( -17901 - 1843 \beta_{1} + 316 \beta_{2} + 729 \beta_{3} - 46 \beta_{4} - 72 \beta_{5} + 127 \beta_{6} + 111 \beta_{7} + 68 \beta_{8} + 50 \beta_{9} ) q^{27} + ( 23324 - 1029 \beta_{1} + 343 \beta_{2} ) q^{28} + ( -34042 + 932 \beta_{1} + 213 \beta_{2} - 663 \beta_{3} + 45 \beta_{4} + 99 \beta_{5} + 171 \beta_{6} - 102 \beta_{7} + 69 \beta_{8} + 45 \beta_{9} ) q^{29} + ( -33734 + 2817 \beta_{1} - 250 \beta_{2} + 3214 \beta_{3} - 173 \beta_{4} + 6 \beta_{5} + 296 \beta_{6} + 39 \beta_{7} + 364 \beta_{8} + 70 \beta_{9} ) q^{30} + ( -13957 - 5234 \beta_{1} + 91 \beta_{2} + 1261 \beta_{3} - 237 \beta_{4} + 31 \beta_{5} + 111 \beta_{6} + 4 \beta_{7} + 21 \beta_{8} + 234 \beta_{9} ) q^{31} + ( -65750 + 5588 \beta_{1} - 477 \beta_{2} + 3002 \beta_{3} - 289 \beta_{4} + 229 \beta_{5} - 136 \beta_{6} - 62 \beta_{7} + 338 \beta_{8} + 59 \beta_{9} ) q^{32} + ( -61961 - 3467 \beta_{1} + 367 \beta_{2} + 382 \beta_{3} + 347 \beta_{4} + 141 \beta_{5} + 160 \beta_{6} - 12 \beta_{7} - 193 \beta_{8} - 223 \beta_{9} ) q^{33} + ( -116287 - 1386 \beta_{1} - 758 \beta_{2} + 545 \beta_{3} + 537 \beta_{4} - 204 \beta_{5} - 612 \beta_{6} - 281 \beta_{7} - 152 \beta_{8} - 143 \beta_{9} ) q^{34} + ( -31899 + 1372 \beta_{1} - 343 \beta_{6} ) q^{35} + ( -77819 - 1665 \beta_{1} - 415 \beta_{2} + 1857 \beta_{3} - 56 \beta_{4} - 154 \beta_{5} + 150 \beta_{6} + 231 \beta_{7} - 151 \beta_{8} - 155 \beta_{9} ) q^{36} + ( -61859 - 2740 \beta_{1} - 259 \beta_{2} - 1771 \beta_{3} + 351 \beta_{4} + 305 \beta_{5} + 735 \beta_{6} - 151 \beta_{7} - 183 \beta_{8} - 414 \beta_{9} ) q^{37} + ( -79210 - 9316 \beta_{1} - 1353 \beta_{2} + 1076 \beta_{3} - 298 \beta_{4} - 323 \beta_{5} - 402 \beta_{6} + 13 \beta_{7} - 142 \beta_{8} + 281 \beta_{9} ) q^{38} + ( 17576 + 2197 \beta_{1} - 2197 \beta_{3} ) q^{39} + ( -30469 - 22523 \beta_{1} + 1350 \beta_{2} - 4413 \beta_{3} + 666 \beta_{4} - 542 \beta_{5} + 264 \beta_{6} + 392 \beta_{7} - 1508 \beta_{8} - 185 \beta_{9} ) q^{40} + ( -206384 + 12462 \beta_{1} + 84 \beta_{2} - 1096 \beta_{3} - 136 \beta_{4} + 598 \beta_{5} + 313 \beta_{6} + 499 \beta_{7} + 620 \beta_{8} + 365 \beta_{9} ) q^{41} + ( -48020 - 3430 \beta_{1} - 1372 \beta_{3} + 343 \beta_{4} - 343 \beta_{8} ) q^{42} + ( -184197 + 17593 \beta_{1} - 1350 \beta_{2} + 899 \beta_{3} - 438 \beta_{4} - 704 \beta_{5} - 102 \beta_{6} - 339 \beta_{7} + 50 \beta_{8} + 683 \beta_{9} ) q^{43} + ( -400613 + 22386 \beta_{1} - 1268 \beta_{2} + 3239 \beta_{3} - 386 \beta_{4} + 512 \beta_{5} + 522 \beta_{6} - 125 \beta_{7} + 657 \beta_{8} - 151 \beta_{9} ) q^{44} + ( -195286 + 27909 \beta_{1} + 961 \beta_{2} - 4786 \beta_{3} + 353 \beta_{4} + 309 \beta_{5} - 320 \beta_{6} - 357 \beta_{7} - 871 \beta_{8} - 1276 \beta_{9} ) q^{45} + ( -271517 + 3601 \beta_{1} + 83 \beta_{2} + 4937 \beta_{3} - 399 \beta_{4} - 593 \beta_{5} - 1206 \beta_{6} + 99 \beta_{7} + 839 \beta_{8} + 926 \beta_{9} ) q^{46} + ( -161822 + 2169 \beta_{1} + 1524 \beta_{2} - 4303 \beta_{3} + 524 \beta_{4} - 32 \beta_{5} - 518 \beta_{6} + 103 \beta_{7} - 862 \beta_{8} + 380 \beta_{9} ) q^{47} + ( -614561 + 41238 \beta_{1} - 1984 \beta_{2} + 715 \beta_{3} - 68 \beta_{4} + 462 \beta_{5} - 2014 \beta_{6} - 531 \beta_{7} + 367 \beta_{8} - 341 \beta_{9} ) q^{48} + 117649 q^{49} + ( -697295 + 36303 \beta_{1} - 3717 \beta_{2} + 669 \beta_{3} - 645 \beta_{4} + 615 \beta_{5} + 478 \beta_{6} - 54 \beta_{7} + 1134 \beta_{8} - 168 \beta_{9} ) q^{50} + ( 178886 + 14450 \beta_{1} + 1115 \beta_{2} + 5135 \beta_{3} - 2043 \beta_{4} - 1019 \beta_{5} + 19 \beta_{6} + 375 \beta_{7} + 1483 \beta_{8} + 379 \beta_{9} ) q^{51} + ( -149396 + 6591 \beta_{1} - 2197 \beta_{2} ) q^{52} + ( -20180 + 23468 \beta_{1} + 5444 \beta_{2} - 11980 \beta_{3} - 108 \beta_{4} - 600 \beta_{5} - 426 \beta_{6} - 245 \beta_{7} - 560 \beta_{8} - 364 \beta_{9} ) q^{53} + ( -294033 + 26754 \beta_{1} + 768 \beta_{2} - 261 \beta_{3} + 1028 \beta_{4} + 1894 \beta_{5} + 1100 \beta_{6} - 489 \beta_{7} + 2247 \beta_{8} - 243 \beta_{9} ) q^{54} + ( 75992 - 3009 \beta_{1} + 2642 \beta_{2} - 3461 \beta_{3} - 1236 \beta_{4} - 1728 \beta_{5} - 1161 \beta_{6} + 1337 \beta_{7} - 1366 \beta_{8} + 593 \beta_{9} ) q^{55} + ( -172529 + 20237 \beta_{1} - 1715 \beta_{2} + 3087 \beta_{3} - 343 \beta_{4} + 343 \beta_{5} + 1372 \beta_{6} + 1372 \beta_{8} ) q^{56} + ( 211197 + 32536 \beta_{1} - 2551 \beta_{2} - 17123 \beta_{3} + 1261 \beta_{4} + 1651 \beta_{5} + 2272 \beta_{6} - 564 \beta_{7} + 1233 \beta_{8} + 571 \beta_{9} ) q^{57} + ( 206726 - 4845 \beta_{1} + 2441 \beta_{2} + 10080 \beta_{3} - 933 \beta_{4} - 129 \beta_{5} + 2058 \beta_{6} + 774 \beta_{7} + 1302 \beta_{8} + 453 \beta_{9} ) q^{58} + ( -818997 + 45293 \beta_{1} + 1023 \beta_{2} + 2758 \beta_{3} + 1189 \beta_{4} - 445 \beta_{5} - 1325 \beta_{6} + 35 \beta_{7} + 427 \beta_{8} - 446 \beta_{9} ) q^{59} + ( 823710 - 39186 \beta_{1} + 13404 \beta_{2} - 14322 \beta_{3} + 1548 \beta_{4} + 324 \beta_{5} - 360 \beta_{6} + 984 \beta_{7} - 3012 \beta_{8} - 1734 \beta_{9} ) q^{60} + ( 232652 - 20960 \beta_{1} + 942 \beta_{2} - 196 \beta_{3} + 384 \beta_{4} - 488 \beta_{5} - 2085 \beta_{6} - 1191 \beta_{7} - 448 \beta_{8} + 2387 \beta_{9} ) q^{61} + ( -891504 - 7173 \beta_{1} - 5019 \beta_{2} - 5582 \beta_{3} + 2332 \beta_{4} + 2867 \beta_{5} + 858 \beta_{6} - 832 \beta_{7} - 1349 \beta_{8} - 2009 \beta_{9} ) q^{62} + ( 125195 - 2058 \beta_{1} - 1715 \beta_{2} - 5145 \beta_{3} - 1715 \beta_{4} - 343 \beta_{5} + 686 \beta_{6} + 343 \beta_{8} + 686 \beta_{9} ) q^{63} + ( 825320 - 72962 \beta_{1} + 4099 \beta_{2} - 19448 \beta_{3} + 1143 \beta_{4} - 1139 \beta_{5} + 3000 \beta_{6} - 134 \beta_{7} - 6054 \beta_{8} - 2667 \beta_{9} ) q^{64} + ( 204321 - 8788 \beta_{1} + 2197 \beta_{6} ) q^{65} + ( -602864 - 15127 \beta_{1} - 4134 \beta_{2} + 32852 \beta_{3} - 2484 \beta_{4} - 170 \beta_{5} + 392 \beta_{6} + 603 \beta_{7} + 3599 \beta_{8} + 3540 \beta_{9} ) q^{66} + ( 193770 + 50518 \beta_{1} + 12726 \beta_{2} - 5790 \beta_{3} - 528 \beta_{4} + 448 \beta_{5} - 237 \beta_{6} - 1033 \beta_{7} - 572 \beta_{8} - 3185 \beta_{9} ) q^{67} + ( -154864 - 133110 \beta_{1} - 1226 \beta_{2} - 340 \beta_{3} + 1882 \beta_{4} - 4558 \beta_{5} - 1472 \beta_{6} - 962 \beta_{7} - 7890 \beta_{8} - 802 \beta_{9} ) q^{68} + ( -469830 + 58851 \beta_{1} - 1974 \beta_{2} - 17985 \beta_{3} + 248 \beta_{4} - 170 \beta_{5} - 1594 \beta_{6} - 1074 \beta_{7} + 1212 \beta_{8} - 3060 \beta_{9} ) q^{69} + ( 330652 - 45962 \beta_{1} - 2401 \beta_{2} - 8918 \beta_{3} - 1715 \beta_{5} + 2058 \beta_{6} + 343 \beta_{7} - 2058 \beta_{8} - 1029 \beta_{9} ) q^{70} + ( -712485 - 16908 \beta_{1} - 5693 \beta_{2} - 25623 \beta_{3} + 3449 \beta_{4} - 2195 \beta_{5} + 632 \beta_{6} + 2562 \beta_{7} - 1661 \beta_{8} + 1329 \beta_{9} ) q^{71} + ( 114818 - 104927 \beta_{1} + 39 \beta_{2} + 25110 \beta_{3} + 495 \beta_{4} - 1409 \beta_{5} - 574 \beta_{6} + 135 \beta_{7} + 1913 \beta_{8} + 4737 \beta_{9} ) q^{72} + ( 141492 - 26425 \beta_{1} + 5173 \beta_{2} + 32472 \beta_{3} + 267 \beta_{4} + 3815 \beta_{5} + 633 \beta_{6} + 2686 \beta_{7} - 499 \beta_{8} + 1121 \beta_{9} ) q^{73} + ( -535971 - 63129 \beta_{1} + 3072 \beta_{2} + 39881 \beta_{3} - 6229 \beta_{4} + 910 \beta_{5} - 5484 \beta_{6} + 1402 \beta_{7} + 5765 \beta_{8} + 6581 \beta_{9} ) q^{74} + ( -891163 + 3521 \beta_{1} - 1918 \beta_{2} + 34907 \beta_{3} - 2312 \beta_{4} - 486 \beta_{5} - 733 \beta_{6} + 195 \beta_{7} + 4402 \beta_{8} + 508 \beta_{9} ) q^{75} + ( -240184 - 182267 \beta_{1} - 15830 \beta_{2} - 33276 \beta_{3} + 3429 \beta_{4} - 2501 \beta_{5} - 4464 \beta_{6} - 736 \beta_{7} - 4988 \beta_{8} + 727 \beta_{9} ) q^{76} + ( 26754 + 15435 \beta_{1} - 6517 \beta_{2} - 10976 \beta_{3} - 343 \beta_{4} + 2401 \beta_{5} + 2401 \beta_{6} - 343 \beta_{7} + 1029 \beta_{8} - 1715 \beta_{9} ) q^{77} + ( 307580 + 21970 \beta_{1} + 8788 \beta_{3} - 2197 \beta_{4} + 2197 \beta_{8} ) q^{78} + ( -900103 - 12843 \beta_{1} - 478 \beta_{2} + 6951 \beta_{3} + 2016 \beta_{4} + 6198 \beta_{5} + 4767 \beta_{6} - 4878 \beta_{7} + 210 \beta_{8} - 1260 \beta_{9} ) q^{79} + ( -2439360 + 134164 \beta_{1} - 32306 \beta_{2} + 124516 \beta_{3} - 3588 \beta_{4} + 2556 \beta_{5} - 4712 \beta_{6} - 2398 \beta_{7} + 11786 \beta_{8} + 796 \beta_{9} ) q^{80} + ( 1158987 + 10340 \beta_{1} - 8747 \beta_{2} + 4181 \beta_{3} - 975 \beta_{4} - 4409 \beta_{5} - 7611 \beta_{6} - 309 \beta_{7} + 2445 \beta_{8} - 55 \beta_{9} ) q^{81} + ( 2842761 - 216290 \beta_{1} + 20360 \beta_{2} + 11253 \beta_{3} - 4422 \beta_{4} + 8710 \beta_{5} + 19368 \beta_{6} + 3179 \beta_{7} + 7909 \beta_{8} - 1817 \beta_{9} ) q^{82} + ( -1517087 + 58860 \beta_{1} - 26477 \beta_{2} - 19693 \beta_{3} - 1733 \beta_{4} - 283 \beta_{5} - 3918 \beta_{6} + 397 \beta_{7} - 1569 \beta_{8} + 2165 \beta_{9} ) q^{83} + ( -302869 - 4116 \beta_{1} - 10976 \beta_{2} + 7203 \beta_{3} - 1372 \beta_{4} - 686 \beta_{5} - 2058 \beta_{6} - 1029 \beta_{7} + 2401 \beta_{8} + 1715 \beta_{9} ) q^{84} + ( -1005355 + 276434 \beta_{1} - 6671 \beta_{2} + 12647 \beta_{3} + 7005 \beta_{4} + 7955 \beta_{5} + 6264 \beta_{6} + 368 \beta_{7} + 2877 \beta_{8} - 5289 \beta_{9} ) q^{85} + ( 3914435 - 348548 \beta_{1} + 18589 \beta_{2} - 38311 \beta_{3} + 13291 \beta_{4} - 4927 \beta_{5} - 13242 \beta_{6} - 4028 \beta_{7} - 14658 \beta_{8} - 6910 \beta_{9} ) q^{86} + ( -1592140 + 108949 \beta_{1} + 6540 \beta_{2} - 40361 \beta_{3} + 4190 \beta_{4} + 690 \beta_{5} - 13077 \beta_{6} - 1017 \beta_{7} - 7736 \beta_{8} - 8007 \beta_{9} ) q^{87} + ( 4287759 - 314908 \beta_{1} + 20468 \beta_{2} - 44821 \beta_{3} + 12672 \beta_{4} - 1174 \beta_{5} + 7182 \beta_{6} - 1297 \beta_{7} - 7527 \beta_{8} - 5745 \beta_{9} ) q^{88} + ( -1170316 + 76027 \beta_{1} + 1979 \beta_{2} + 39100 \beta_{3} - 5335 \beta_{4} - 5597 \beta_{5} + 398 \beta_{6} + 6005 \beta_{7} + 519 \beta_{8} - 3392 \beta_{9} ) q^{89} + ( 5380205 - 116183 \beta_{1} + 9987 \beta_{2} + 24955 \beta_{3} - 16598 \beta_{4} - 3689 \beta_{5} - 10594 \beta_{6} + 1173 \beta_{7} + 8140 \beta_{8} + 9288 \beta_{9} ) q^{90} -753571 q^{91} + ( 1702341 - 130785 \beta_{1} + 11282 \beta_{2} - 44363 \beta_{3} + 5855 \beta_{4} - 3917 \beta_{5} + 3934 \beta_{6} + 2357 \beta_{7} - 15021 \beta_{8} - 4394 \beta_{9} ) q^{92} + ( 3621206 + 16923 \beta_{1} + 5359 \beta_{2} + 100288 \beta_{3} - 17137 \beta_{4} - 3559 \beta_{5} - 2479 \beta_{6} + 4371 \beta_{7} + 2727 \beta_{8} + 4577 \beta_{9} ) q^{93} + ( 484377 - 24458 \beta_{1} - 30055 \beta_{2} + 131259 \beta_{3} - 3558 \beta_{4} + 1309 \beta_{5} + 5214 \beta_{6} - 3838 \beta_{7} + 9265 \beta_{8} - 908 \beta_{9} ) q^{94} + ( 2947503 + 136001 \beta_{1} - 1280 \beta_{2} - 19517 \beta_{3} - 13386 \beta_{4} - 558 \beta_{5} + 8274 \beta_{6} + 593 \beta_{7} + 13268 \beta_{8} + 12655 \beta_{9} ) q^{95} + ( 6641631 - 526702 \beta_{1} + 17584 \beta_{2} - 49525 \beta_{3} + 44 \beta_{4} - 4570 \beta_{5} + 13562 \beta_{6} + 4617 \beta_{7} - 10125 \beta_{8} + 5171 \beta_{9} ) q^{96} + ( 346262 - 130777 \beta_{1} - 21495 \beta_{2} - 12030 \beta_{3} + 1581 \beta_{4} - 1609 \beta_{5} - 6069 \beta_{6} + 4390 \beta_{7} + 7493 \beta_{8} + 7601 \beta_{9} ) q^{97} + ( -235298 + 117649 \beta_{1} ) q^{98} + ( 1894757 + 390104 \beta_{1} + 7517 \beta_{2} - 129169 \beta_{3} + 2899 \beta_{4} - 6753 \beta_{5} - 23386 \beta_{6} - 6618 \beta_{7} - 7463 \beta_{8} + 5845 \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 18 q^{2} - 80 q^{3} + 670 q^{4} - 927 q^{5} - 1419 q^{6} + 3430 q^{7} - 4878 q^{8} + 3612 q^{9} + O(q^{10}) \) \( 10 q - 18 q^{2} - 80 q^{3} + 670 q^{4} - 927 q^{5} - 1419 q^{6} + 3430 q^{7} - 4878 q^{8} + 3612 q^{9} + 9420 q^{10} + 876 q^{11} - 8765 q^{12} - 21970 q^{13} - 6174 q^{14} - 5320 q^{15} + 41370 q^{16} + 6294 q^{17} - 16027 q^{18} - 97401 q^{19} - 166650 q^{20} - 27440 q^{21} + 74171 q^{22} - 15255 q^{23} + 196187 q^{24} + 162145 q^{25} + 39546 q^{26} - 181820 q^{27} + 229810 q^{28} - 340533 q^{29} - 325020 q^{30} - 148675 q^{31} - 642762 q^{32} - 624400 q^{33} - 1161518 q^{34} - 317961 q^{35} - 773917 q^{36} - 621782 q^{37} - 805092 q^{38} + 175760 q^{39} - 350478 q^{40} - 2043336 q^{41} - 486717 q^{42} - 1801391 q^{43} - 3953667 q^{44} - 1908807 q^{45} - 2707731 q^{46} - 1624701 q^{47} - 6068625 q^{48} + 1176490 q^{49} - 6891516 q^{50} + 1811700 q^{51} - 1471990 q^{52} - 199965 q^{53} - 2895913 q^{54} + 739086 q^{55} - 1673154 q^{56} + 2159088 q^{57} + 2071092 q^{58} - 8098908 q^{59} + 8096436 q^{60} + 2271618 q^{61} - 8910225 q^{62} + 1238916 q^{63} + 8099930 q^{64} + 2036619 q^{65} - 5999191 q^{66} + 1970272 q^{67} - 1766238 q^{68} - 4622962 q^{69} + 3231060 q^{70} - 7145820 q^{71} + 984975 q^{72} + 1409431 q^{73} - 5498643 q^{74} - 8857892 q^{75} - 2749534 q^{76} + 300468 q^{77} + 3117543 q^{78} - 9011055 q^{79} - 23850522 q^{80} + 11613490 q^{81} + 27962597 q^{82} - 15006567 q^{83} - 3006395 q^{84} - 9416628 q^{85} + 38357850 q^{86} - 15828996 q^{87} + 42205269 q^{88} - 11472777 q^{89} + 53425712 q^{90} - 7535710 q^{91} + 16755837 q^{92} + 36339848 q^{93} + 5133371 q^{94} + 29637939 q^{95} + 65329611 q^{96} + 3228571 q^{97} - 2117682 q^{98} + 19367194 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 2 x^{9} - 957 x^{8} + 1224 x^{7} + 310102 x^{6} - 241884 x^{5} - 40367312 x^{4} + 11067840 x^{3} + 1840757376 x^{2} + 541859072 x - 4516262912\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 192 \)
\(\beta_{3}\)\(=\)\((\)\(-1176111762173 \nu^{9} - 12574063170642 \nu^{8} + 876995328306269 \nu^{7} + 11515664425143836 \nu^{6} - 151357859667407914 \nu^{5} - 3131934393841946428 \nu^{4} - 4950798924727042264 \nu^{3} + 274403785845235565168 \nu^{2} + 1435362147548441825536 \nu - 2502527564156239212160\)\()/ 59680876045268916864 \)
\(\beta_{4}\)\(=\)\((\)\(-2790653758009 \nu^{9} - 499024731176506 \nu^{8} + 1963439477763629 \nu^{7} + 399663409553784820 \nu^{6} - 24670125491794246 \nu^{5} - 91336185656434071500 \nu^{4} - 104130260202995462752 \nu^{3} + 5429871451046257657600 \nu^{2} + 9955435843020709111680 \nu + 30794608272687706674176\)\()/ \)\(11\!\cdots\!28\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-6214679484235 \nu^{9} - 367159236899342 \nu^{8} + 6731203112384271 \nu^{7} + 311027096288686444 \nu^{6} - 2069343979725860250 \nu^{5} - 78925698274966324580 \nu^{4} + 156327729152503243920 \nu^{3} + 5763216561320182971808 \nu^{2} + 6199683768286110471040 \nu - 1397135855455063529472\)\()/ \)\(11\!\cdots\!28\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-711607147759 \nu^{9} - 687274083168 \nu^{8} + 619904788708757 \nu^{7} + 1206981571162370 \nu^{6} - 172814347326744056 \nu^{5} - 418194659513348736 \nu^{4} + 17949255712535764684 \nu^{3} + 38916257941304712952 \nu^{2} - 616057452027620213440 \nu - 335992622475665049024\)\()/ 3315604224737162048 \)
\(\beta_{7}\)\(=\)\((\)\(25960034868425 \nu^{9} + 56774085353906 \nu^{8} - 27582279809630605 \nu^{7} - 51782708096290220 \nu^{6} + 10231704826006173638 \nu^{5} + 15099261549325353052 \nu^{4} - 1576484629234070407552 \nu^{3} - 1961207436399028598144 \nu^{2} + 83910884288767060434048 \nu + 80342767409024923304960\)\()/ \)\(11\!\cdots\!28\)\( \)
\(\beta_{8}\)\(=\)\((\)\(31766366680659 \nu^{9} + 48358777692646 \nu^{8} - 27454992279548623 \nu^{7} - 73112748158717988 \nu^{6} + 7413595335824638130 \nu^{5} + 26046090669402376692 \nu^{4} - 659168669806256056672 \nu^{3} - 2748979037781645496512 \nu^{2} + 7467021716738575932032 \nu + 32807545066926530316800\)\()/ \)\(11\!\cdots\!28\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-13936442615075 \nu^{9} - 2356424471438 \nu^{8} + 12331210262698483 \nu^{7} + 7198965679261796 \nu^{6} - 3453853452498122630 \nu^{5} - 1769948172874794244 \nu^{4} + 340196819020665701752 \nu^{3} + 39818918619808626512 \nu^{2} - 8697702288188436017280 \nu + 6101129501351127337984\)\()/ 19893625348422972288 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 192\)
\(\nu^{3}\)\(=\)\(4 \beta_{8} + 4 \beta_{6} + \beta_{5} - \beta_{4} + 9 \beta_{3} + \beta_{2} + 309 \beta_{1} + 145\)
\(\nu^{4}\)\(=\)\(5 \beta_{9} + 22 \beta_{8} + 18 \beta_{7} + 44 \beta_{6} + 15 \beta_{5} + \beta_{4} - 202 \beta_{3} + 435 \beta_{2} + 762 \beta_{1} + 59730\)
\(\nu^{5}\)\(=\)\(109 \beta_{9} + 2446 \beta_{8} + 118 \beta_{7} + 2192 \beta_{6} + 851 \beta_{5} - 751 \beta_{4} + 5230 \beta_{3} + 1353 \beta_{2} + 112976 \beta_{1} + 119766\)
\(\nu^{6}\)\(=\)\(1541 \beta_{9} + 16218 \beta_{8} + 11722 \beta_{7} + 34984 \beta_{6} + 12813 \beta_{5} - 2329 \beta_{4} - 118488 \beta_{3} + 184491 \beta_{2} + 481814 \beta_{1} + 21929496\)
\(\nu^{7}\)\(=\)\(86067 \beta_{9} + 1226770 \beta_{8} + 89618 \beta_{7} + 1040712 \beta_{6} + 516339 \beta_{5} - 420631 \beta_{4} + 2297336 \beta_{3} + 979737 \beta_{2} + 45245650 \beta_{1} + 80561872\)
\(\nu^{8}\)\(=\)\(384913 \beta_{9} + 9584946 \beta_{8} + 6145098 \beta_{7} + 20408544 \beta_{6} + 7766169 \beta_{5} - 2514621 \beta_{4} - 56223408 \beta_{3} + 79325107 \beta_{2} + 278600722 \beta_{1} + 8791834440\)
\(\nu^{9}\)\(=\)\(47808719 \beta_{9} + 580883410 \beta_{8} + 52781794 \beta_{7} + 484274136 \beta_{6} + 273774531 \beta_{5} - 211377687 \beta_{4} + 930228476 \beta_{3} + 585485729 \beta_{2} + 19062064646 \beta_{1} + 48382466052\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−20.0501
−16.8829
−11.8703
−9.51136
−1.80018
1.45069
10.4944
12.2698
16.1785
21.7215
−22.0501 −47.1151 358.208 −467.623 1038.89 343.000 −5076.12 32.8291 10311.1
1.2 −18.8829 14.3398 228.563 382.689 −270.777 343.000 −1898.92 −1981.37 −7226.26
1.3 −13.8703 79.8279 64.3848 −355.678 −1107.24 343.000 882.360 4185.49 4933.36
1.4 −11.5114 23.9058 4.51150 −301.396 −275.188 343.000 1421.52 −1615.51 3469.48
1.5 −3.80018 −76.5407 −113.559 −339.464 290.868 343.000 917.967 3671.47 1290.03
1.6 −0.549314 −7.31646 −127.698 243.344 4.01903 343.000 140.459 −2133.47 −133.672
1.7 8.49443 45.4613 −55.8447 −130.405 386.168 343.000 −1561.66 −120.267 −1107.72
1.8 10.2698 −81.6519 −22.5320 313.060 −838.546 343.000 −1545.93 4480.04 3215.05
1.9 14.1785 6.78261 73.0290 −4.23520 96.1671 343.000 −779.405 −2141.00 −60.0487
1.10 19.7215 −37.6933 260.937 −267.290 −743.369 343.000 2621.72 −766.212 −5271.36
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(13\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.8.a.c 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.8.a.c 10 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{10} + \cdots\) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(91))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 3385168384 + 6785706432 T + 992532576 T^{2} - 272954592 T^{3} - 24903776 T^{4} + 3157452 T^{5} + 222070 T^{6} - 13416 T^{7} - 813 T^{8} + 18 T^{9} + T^{10} \)
$3$ \( -685212736352256 + 57408056536320 T + 14433182782560 T^{2} - 1105049368368 T^{3} - 23705666388 T^{4} + 1746933648 T^{5} + 23538940 T^{6} - 756700 T^{7} - 9541 T^{8} + 80 T^{9} + T^{10} \)
$5$ \( -\)\(73\!\cdots\!00\)\( - \)\(18\!\cdots\!00\)\( T - \)\(22\!\cdots\!50\)\( T^{2} - 167250796972891875 T^{3} + 6679173581703625 T^{4} + 20167356683025 T^{5} - 50597070975 T^{6} - 259855317 T^{7} - 42033 T^{8} + 927 T^{9} + T^{10} \)
$7$ \( ( -343 + T )^{10} \)
$11$ \( \)\(14\!\cdots\!04\)\( - \)\(32\!\cdots\!96\)\( T + \)\(15\!\cdots\!28\)\( T^{2} + \)\(64\!\cdots\!60\)\( T^{3} - \)\(43\!\cdots\!88\)\( T^{4} - 4075277046322126824 T^{5} + 3567917018390176 T^{6} + 97789908594 T^{7} - 109979147 T^{8} - 876 T^{9} + T^{10} \)
$13$ \( ( 2197 + T )^{10} \)
$17$ \( -\)\(16\!\cdots\!56\)\( + \)\(24\!\cdots\!12\)\( T + \)\(30\!\cdots\!04\)\( T^{2} + \)\(13\!\cdots\!32\)\( T^{3} - \)\(15\!\cdots\!48\)\( T^{4} - \)\(97\!\cdots\!84\)\( T^{5} + 3201728060124255268 T^{6} + 14548037205024 T^{7} - 2972676240 T^{8} - 6294 T^{9} + T^{10} \)
$19$ \( -\)\(12\!\cdots\!68\)\( + \)\(45\!\cdots\!60\)\( T - \)\(13\!\cdots\!40\)\( T^{2} - \)\(88\!\cdots\!49\)\( T^{3} + \)\(47\!\cdots\!03\)\( T^{4} + \)\(48\!\cdots\!71\)\( T^{5} - 1920907669129564949 T^{6} - 115061146634631 T^{7} + 1120143449 T^{8} + 97401 T^{9} + T^{10} \)
$23$ \( \)\(10\!\cdots\!88\)\( + \)\(32\!\cdots\!39\)\( T + \)\(10\!\cdots\!49\)\( T^{2} - \)\(13\!\cdots\!84\)\( T^{3} - \)\(46\!\cdots\!16\)\( T^{4} + \)\(14\!\cdots\!82\)\( T^{5} + 51326383986442740862 T^{6} - 319408338086004 T^{7} - 14352019620 T^{8} + 15255 T^{9} + T^{10} \)
$29$ \( \)\(15\!\cdots\!32\)\( - \)\(77\!\cdots\!60\)\( T - \)\(10\!\cdots\!10\)\( T^{2} - \)\(11\!\cdots\!25\)\( T^{3} + \)\(33\!\cdots\!57\)\( T^{4} + \)\(76\!\cdots\!27\)\( T^{5} - \)\(33\!\cdots\!87\)\( T^{6} - 10572056838525327 T^{7} - 10206038397 T^{8} + 340533 T^{9} + T^{10} \)
$31$ \( -\)\(13\!\cdots\!68\)\( + \)\(40\!\cdots\!47\)\( T + \)\(23\!\cdots\!05\)\( T^{2} - \)\(11\!\cdots\!28\)\( T^{3} - \)\(41\!\cdots\!04\)\( T^{4} + \)\(98\!\cdots\!34\)\( T^{5} + \)\(45\!\cdots\!14\)\( T^{6} - 25177392335497688 T^{7} - 136464806680 T^{8} + 148675 T^{9} + T^{10} \)
$37$ \( \)\(91\!\cdots\!28\)\( + \)\(15\!\cdots\!12\)\( T - \)\(18\!\cdots\!08\)\( T^{2} + \)\(84\!\cdots\!76\)\( T^{3} + \)\(32\!\cdots\!80\)\( T^{4} + \)\(13\!\cdots\!16\)\( T^{5} - \)\(29\!\cdots\!80\)\( T^{6} - 244755872413253386 T^{7} - 251304362917 T^{8} + 621782 T^{9} + T^{10} \)
$41$ \( \)\(51\!\cdots\!44\)\( - \)\(55\!\cdots\!52\)\( T + \)\(35\!\cdots\!80\)\( T^{2} + \)\(84\!\cdots\!76\)\( T^{3} + \)\(29\!\cdots\!64\)\( T^{4} + \)\(87\!\cdots\!92\)\( T^{5} - \)\(11\!\cdots\!68\)\( T^{6} - 1574563820198118936 T^{7} + 479608989959 T^{8} + 2043336 T^{9} + T^{10} \)
$43$ \( \)\(46\!\cdots\!56\)\( + \)\(12\!\cdots\!16\)\( T - \)\(35\!\cdots\!36\)\( T^{2} - \)\(19\!\cdots\!03\)\( T^{3} + \)\(23\!\cdots\!43\)\( T^{4} + \)\(10\!\cdots\!21\)\( T^{5} - \)\(29\!\cdots\!01\)\( T^{6} - 2369740544383264837 T^{7} - 585794365903 T^{8} + 1801391 T^{9} + T^{10} \)
$47$ \( \)\(85\!\cdots\!08\)\( + \)\(16\!\cdots\!57\)\( T - \)\(18\!\cdots\!59\)\( T^{2} - \)\(41\!\cdots\!56\)\( T^{3} + \)\(77\!\cdots\!72\)\( T^{4} + \)\(27\!\cdots\!34\)\( T^{5} - \)\(33\!\cdots\!94\)\( T^{6} - 1746831467377868892 T^{7} - 583366591300 T^{8} + 1624701 T^{9} + T^{10} \)
$53$ \( \)\(63\!\cdots\!84\)\( + \)\(43\!\cdots\!48\)\( T + \)\(38\!\cdots\!98\)\( T^{2} - \)\(12\!\cdots\!21\)\( T^{3} - \)\(16\!\cdots\!83\)\( T^{4} + \)\(97\!\cdots\!55\)\( T^{5} + \)\(17\!\cdots\!21\)\( T^{6} - 2706218322209792415 T^{7} - 7009145534457 T^{8} + 199965 T^{9} + T^{10} \)
$59$ \( -\)\(58\!\cdots\!84\)\( + \)\(31\!\cdots\!80\)\( T + \)\(78\!\cdots\!16\)\( T^{2} + \)\(51\!\cdots\!88\)\( T^{3} - \)\(26\!\cdots\!76\)\( T^{4} - \)\(43\!\cdots\!84\)\( T^{5} - \)\(93\!\cdots\!56\)\( T^{6} + 26338165212666267264 T^{7} + 23936536520524 T^{8} + 8098908 T^{9} + T^{10} \)
$61$ \( -\)\(13\!\cdots\!52\)\( - \)\(75\!\cdots\!40\)\( T + \)\(52\!\cdots\!68\)\( T^{2} + \)\(52\!\cdots\!80\)\( T^{3} - \)\(84\!\cdots\!76\)\( T^{4} - \)\(56\!\cdots\!28\)\( T^{5} + \)\(46\!\cdots\!84\)\( T^{6} + 20203753832086818996 T^{7} - 11011769081803 T^{8} - 2271618 T^{9} + T^{10} \)
$67$ \( \)\(57\!\cdots\!04\)\( - \)\(20\!\cdots\!92\)\( T + \)\(86\!\cdots\!00\)\( T^{2} + \)\(23\!\cdots\!56\)\( T^{3} - \)\(12\!\cdots\!16\)\( T^{4} - \)\(76\!\cdots\!76\)\( T^{5} + \)\(39\!\cdots\!24\)\( T^{6} + 75240325653085451174 T^{7} - 37247716092411 T^{8} - 1970272 T^{9} + T^{10} \)
$71$ \( \)\(80\!\cdots\!44\)\( - \)\(10\!\cdots\!00\)\( T - \)\(24\!\cdots\!40\)\( T^{2} - \)\(73\!\cdots\!00\)\( T^{3} + \)\(53\!\cdots\!88\)\( T^{4} + \)\(27\!\cdots\!08\)\( T^{5} - \)\(17\!\cdots\!88\)\( T^{6} - \)\(26\!\cdots\!52\)\( T^{7} - 22156634466320 T^{8} + 7145820 T^{9} + T^{10} \)
$73$ \( \)\(39\!\cdots\!54\)\( - \)\(13\!\cdots\!43\)\( T + \)\(75\!\cdots\!13\)\( T^{2} + \)\(65\!\cdots\!80\)\( T^{3} - \)\(49\!\cdots\!96\)\( T^{4} - \)\(94\!\cdots\!42\)\( T^{5} + \)\(87\!\cdots\!14\)\( T^{6} + 59658142683960502796 T^{7} - 54597235417738 T^{8} - 1409431 T^{9} + T^{10} \)
$79$ \( \)\(40\!\cdots\!12\)\( + \)\(55\!\cdots\!11\)\( T - \)\(26\!\cdots\!59\)\( T^{2} - \)\(33\!\cdots\!88\)\( T^{3} + \)\(35\!\cdots\!80\)\( T^{4} + \)\(63\!\cdots\!26\)\( T^{5} + \)\(21\!\cdots\!18\)\( T^{6} - \)\(42\!\cdots\!76\)\( T^{7} - 37576721727988 T^{8} + 9011055 T^{9} + T^{10} \)
$83$ \( \)\(15\!\cdots\!36\)\( + \)\(62\!\cdots\!56\)\( T - \)\(47\!\cdots\!16\)\( T^{2} - \)\(18\!\cdots\!43\)\( T^{3} + \)\(47\!\cdots\!91\)\( T^{4} + \)\(20\!\cdots\!53\)\( T^{5} - \)\(15\!\cdots\!09\)\( T^{6} - \)\(93\!\cdots\!01\)\( T^{7} - 11387842873887 T^{8} + 15006567 T^{9} + T^{10} \)
$89$ \( \)\(13\!\cdots\!76\)\( + \)\(14\!\cdots\!68\)\( T - \)\(96\!\cdots\!62\)\( T^{2} - \)\(47\!\cdots\!05\)\( T^{3} - \)\(51\!\cdots\!27\)\( T^{4} + \)\(45\!\cdots\!15\)\( T^{5} + \)\(51\!\cdots\!81\)\( T^{6} - \)\(13\!\cdots\!63\)\( T^{7} - 132984502384749 T^{8} + 11472777 T^{9} + T^{10} \)
$97$ \( \)\(52\!\cdots\!18\)\( + \)\(10\!\cdots\!45\)\( T - \)\(52\!\cdots\!59\)\( T^{2} - \)\(30\!\cdots\!36\)\( T^{3} - \)\(37\!\cdots\!04\)\( T^{4} + \)\(84\!\cdots\!90\)\( T^{5} + \)\(20\!\cdots\!90\)\( T^{6} - \)\(25\!\cdots\!08\)\( T^{7} - 249839278163730 T^{8} - 3228571 T^{9} + T^{10} \)
show more
show less