Properties

Label 91.8.a.d
Level $91$
Weight $8$
Character orbit 91.a
Self dual yes
Analytic conductor $28.427$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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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: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 3 x^{9} - 816 x^{8} + 2298 x^{7} + 213848 x^{6} - 507132 x^{5} - 19919976 x^{4} + 24331248 x^{3} + 727257184 x^{2} - 56397312 x - 7335224320\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( -10 - \beta_{1} - \beta_{3} ) q^{3} + ( 36 + \beta_{2} ) q^{4} + ( 23 - 2 \beta_{1} - \beta_{3} - \beta_{6} ) q^{5} + ( 104 + 16 \beta_{1} + \beta_{2} - 6 \beta_{3} - \beta_{6} + \beta_{7} + \beta_{8} ) q^{6} -343 q^{7} + ( 39 - 35 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{8} + ( 1253 - 97 \beta_{1} + 13 \beta_{2} + 6 \beta_{3} - 2 \beta_{6} + 2 \beta_{8} + \beta_{9} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( -10 - \beta_{1} - \beta_{3} ) q^{3} + ( 36 + \beta_{2} ) q^{4} + ( 23 - 2 \beta_{1} - \beta_{3} - \beta_{6} ) q^{5} + ( 104 + 16 \beta_{1} + \beta_{2} - 6 \beta_{3} - \beta_{6} + \beta_{7} + \beta_{8} ) q^{6} -343 q^{7} + ( 39 - 35 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{8} + ( 1253 - 97 \beta_{1} + 13 \beta_{2} + 6 \beta_{3} - 2 \beta_{6} + 2 \beta_{8} + \beta_{9} ) q^{9} + ( 241 + 19 \beta_{1} + 6 \beta_{2} - 36 \beta_{3} + 2 \beta_{4} + \beta_{6} + \beta_{8} ) q^{10} + ( 37 + 28 \beta_{1} - 2 \beta_{2} - 16 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} - 5 \beta_{8} ) q^{11} + ( -1618 - 85 \beta_{1} - 9 \beta_{2} - 101 \beta_{3} + 3 \beta_{4} - \beta_{5} - 4 \beta_{6} + 12 \beta_{7} - 4 \beta_{9} ) q^{12} -2197 q^{13} + 343 \beta_{1} q^{14} + ( 2903 - 674 \beta_{1} + 38 \beta_{2} - 53 \beta_{3} - 4 \beta_{4} + 6 \beta_{5} + 8 \beta_{6} - 3 \beta_{7} + 8 \beta_{8} - 3 \beta_{9} ) q^{15} + ( 1227 - 132 \beta_{1} - \beta_{2} - 33 \beta_{3} + 2 \beta_{4} + \beta_{5} + 14 \beta_{6} - 8 \beta_{7} - 7 \beta_{8} + 3 \beta_{9} ) q^{16} + ( -659 - 659 \beta_{1} - 15 \beta_{2} - 8 \beta_{3} + 14 \beta_{4} - \beta_{5} + 26 \beta_{6} - 16 \beta_{7} - 11 \beta_{8} + 13 \beta_{9} ) q^{17} + ( 16716 - 2837 \beta_{1} + 142 \beta_{2} - 307 \beta_{3} - 8 \beta_{4} - 11 \beta_{5} - 22 \beta_{6} - 33 \beta_{7} - 35 \beta_{8} - 11 \beta_{9} ) q^{18} + ( 1362 - 1184 \beta_{1} + 3 \beta_{2} - 65 \beta_{3} - 8 \beta_{4} - 4 \beta_{5} + 17 \beta_{6} - \beta_{7} - 12 \beta_{8} + 12 \beta_{9} ) q^{19} + ( -8062 - 599 \beta_{1} - 72 \beta_{2} - 159 \beta_{3} - 3 \beta_{4} - 11 \beta_{5} + 36 \beta_{6} + 48 \beta_{7} + 32 \beta_{8} ) q^{20} + ( 3430 + 343 \beta_{1} + 343 \beta_{3} ) q^{21} + ( -5871 + 562 \beta_{1} - 96 \beta_{2} + 500 \beta_{3} - 10 \beta_{4} + 8 \beta_{5} + 47 \beta_{6} - 52 \beta_{7} + 49 \beta_{8} + 6 \beta_{9} ) q^{22} + ( -4462 - 2444 \beta_{1} - 72 \beta_{2} + 137 \beta_{3} - 22 \beta_{4} + 17 \beta_{5} + 39 \beta_{6} - 124 \beta_{7} + \beta_{8} - 20 \beta_{9} ) q^{23} + ( -5139 + 556 \beta_{1} - 153 \beta_{2} - 235 \beta_{3} + 42 \beta_{4} - 13 \beta_{5} + 46 \beta_{6} + 140 \beta_{7} + 67 \beta_{8} + 9 \beta_{9} ) q^{24} + ( 4886 - 367 \beta_{1} - 222 \beta_{2} - \beta_{3} - 26 \beta_{4} + 11 \beta_{5} - 93 \beta_{6} + 65 \beta_{7} - 55 \beta_{8} - 3 \beta_{9} ) q^{25} + 2197 \beta_{1} q^{26} + ( -14428 - 4420 \beta_{1} - 384 \beta_{2} - 1299 \beta_{3} + 10 \beta_{4} - 11 \beta_{5} - 154 \beta_{6} + 169 \beta_{7} + 7 \beta_{8} - 85 \beta_{9} ) q^{27} + ( -12348 - 343 \beta_{2} ) q^{28} + ( 52426 - 800 \beta_{1} + 90 \beta_{2} + 714 \beta_{3} + 28 \beta_{4} - 42 \beta_{5} - 115 \beta_{6} + 139 \beta_{7} - 24 \beta_{8} + 83 \beta_{9} ) q^{29} + ( 108992 - 7268 \beta_{1} + 641 \beta_{2} - 755 \beta_{3} - 72 \beta_{4} - 9 \beta_{5} - 69 \beta_{6} + 4 \beta_{7} - 30 \beta_{8} - 5 \beta_{9} ) q^{30} + ( 70503 - 4449 \beta_{1} + 37 \beta_{2} - 615 \beta_{3} + 80 \beta_{4} + 16 \beta_{5} - 103 \beta_{6} - 182 \beta_{7} - 82 \beta_{8} - 9 \beta_{9} ) q^{31} + ( 14249 + 3470 \beta_{1} - 291 \beta_{2} + 1171 \beta_{3} + 84 \beta_{4} + 125 \beta_{5} + 136 \beta_{6} + 66 \beta_{7} + 49 \beta_{8} + 9 \beta_{9} ) q^{32} + ( 41181 + 8905 \beta_{1} - 314 \beta_{2} + 1100 \beta_{3} + 96 \beta_{4} + 160 \beta_{5} + 16 \beta_{6} - 253 \beta_{7} + 114 \beta_{8} + 15 \beta_{9} ) q^{33} + ( 105629 + 2272 \beta_{1} + 128 \beta_{2} + 2233 \beta_{3} - 46 \beta_{4} - 23 \beta_{5} - 381 \beta_{6} - 191 \beta_{7} - 20 \beta_{8} - 21 \beta_{9} ) q^{34} + ( -7889 + 686 \beta_{1} + 343 \beta_{3} + 343 \beta_{6} ) q^{35} + ( 289839 - 17736 \beta_{1} + 2228 \beta_{2} + 2847 \beta_{3} - 138 \beta_{4} - 131 \beta_{5} - 204 \beta_{6} + 114 \beta_{7} + 355 \beta_{8} + 99 \beta_{9} ) q^{36} + ( -3180 + 6092 \beta_{1} - 595 \beta_{2} + 3062 \beta_{3} - 82 \beta_{4} - 197 \beta_{5} + 70 \beta_{6} - 107 \beta_{7} - 177 \beta_{8} + 120 \beta_{9} ) q^{37} + ( 191079 - 1903 \beta_{1} + 1514 \beta_{2} + 994 \beta_{3} - 66 \beta_{4} + 6 \beta_{5} + 39 \beta_{6} - 198 \beta_{7} + 53 \beta_{8} - 58 \beta_{9} ) q^{38} + ( 21970 + 2197 \beta_{1} + 2197 \beta_{3} ) q^{39} + ( 59932 + 11667 \beta_{1} + 83 \beta_{2} - 1358 \beta_{3} - 149 \beta_{4} + 54 \beta_{5} - 60 \beta_{6} + 681 \beta_{7} - 95 \beta_{8} - 141 \beta_{9} ) q^{40} + ( 34091 + 3496 \beta_{1} - 351 \beta_{2} - 1881 \beta_{3} + 174 \beta_{4} - 281 \beta_{5} - 432 \beta_{6} + 54 \beta_{7} - 151 \beta_{8} - 33 \beta_{9} ) q^{41} + ( -35672 - 5488 \beta_{1} - 343 \beta_{2} + 2058 \beta_{3} + 343 \beta_{6} - 343 \beta_{7} - 343 \beta_{8} ) q^{42} + ( 127324 - 6441 \beta_{1} - 511 \beta_{2} - 2121 \beta_{3} - 210 \beta_{4} + 75 \beta_{5} - 303 \beta_{6} - 396 \beta_{7} - 303 \beta_{8} - 111 \beta_{9} ) q^{43} + ( -72480 + 14437 \beta_{1} - 241 \beta_{2} + 1987 \beta_{3} + 205 \beta_{4} - 149 \beta_{5} + 1016 \beta_{6} - 464 \beta_{7} - 650 \beta_{8} - 62 \beta_{9} ) q^{44} + ( 126251 - 6399 \beta_{1} - 145 \beta_{2} - 7846 \beta_{3} + 254 \beta_{4} + 127 \beta_{5} + 379 \beta_{6} + 607 \beta_{7} + 415 \beta_{8} - 120 \beta_{9} ) q^{45} + ( 399776 + 20607 \beta_{1} + 2620 \beta_{2} + 7540 \beta_{3} - 204 \beta_{4} + 280 \beta_{5} + 236 \beta_{6} - 590 \beta_{7} - 546 \beta_{8} + 330 \beta_{9} ) q^{46} + ( 120945 + 10832 \beta_{1} - 102 \beta_{2} - 8736 \beta_{3} + 34 \beta_{4} + 13 \beta_{5} + 407 \beta_{6} + 367 \beta_{7} + 299 \beta_{8} - 121 \beta_{9} ) q^{47} + ( 105025 + 25472 \beta_{1} - 937 \beta_{2} - 1269 \beta_{3} - 2 \beta_{4} + 69 \beta_{5} + 20 \beta_{6} + 48 \beta_{7} + 183 \beta_{8} + 103 \beta_{9} ) q^{48} + 117649 q^{49} + ( 54332 + 21848 \beta_{1} + 142 \beta_{2} + 1347 \beta_{3} + 336 \beta_{4} + 191 \beta_{5} + 1558 \beta_{6} + 63 \beta_{7} + 749 \beta_{8} - 47 \beta_{9} ) q^{50} + ( 124537 + 52611 \beta_{1} - 2068 \beta_{2} - 4467 \beta_{3} - 534 \beta_{4} + 57 \beta_{5} + 234 \beta_{6} + 256 \beta_{7} + 417 \beta_{8} + 76 \beta_{9} ) q^{51} + ( -79092 - 2197 \beta_{2} ) q^{52} + ( 95463 - 11103 \beta_{1} + 1140 \beta_{2} - 3859 \beta_{3} + 130 \beta_{4} - 3 \beta_{5} - 161 \beta_{6} - 1235 \beta_{7} - 741 \beta_{8} - 421 \beta_{9} ) q^{53} + ( 640523 + 64672 \beta_{1} + 3685 \beta_{2} - 16385 \beta_{3} + 1006 \beta_{4} + 181 \beta_{5} + 376 \beta_{6} + 4094 \beta_{7} + 2711 \beta_{8} + 275 \beta_{9} ) q^{54} + ( 333949 + 27917 \beta_{1} - 3609 \beta_{2} - 2610 \beta_{3} - 1134 \beta_{4} + 157 \beta_{5} - 3278 \beta_{6} - 860 \beta_{7} - 509 \beta_{8} + 159 \beta_{9} ) q^{55} + ( -13377 + 12005 \beta_{1} - 686 \beta_{2} - 343 \beta_{3} + 343 \beta_{4} + 343 \beta_{5} + 686 \beta_{6} + 343 \beta_{7} ) q^{56} + ( 287941 + 38250 \beta_{1} - 2442 \beta_{2} - 8965 \beta_{3} + 464 \beta_{4} - 260 \beta_{5} - 368 \beta_{6} + 1197 \beta_{7} + 1306 \beta_{8} + 197 \beta_{9} ) q^{57} + ( 182251 - 74827 \beta_{1} + 1729 \beta_{2} - 5525 \beta_{3} + 354 \beta_{4} - 379 \beta_{5} + 516 \beta_{6} - 1216 \beta_{7} - 539 \beta_{8} - 699 \beta_{9} ) q^{58} + ( 307394 + 24633 \beta_{1} - 2311 \beta_{2} - 4231 \beta_{3} - 338 \beta_{4} + 651 \beta_{5} + 48 \beta_{6} - 575 \beta_{7} - 1201 \beta_{8} + 616 \beta_{9} ) q^{59} + ( 802527 - 96131 \beta_{1} + 6554 \beta_{2} + 3902 \beta_{3} - 129 \beta_{4} - 1236 \beta_{5} - 1424 \beta_{6} + 26 \beta_{7} - 115 \beta_{8} + 417 \beta_{9} ) q^{60} + ( 641259 - 40244 \beta_{1} + 3855 \beta_{2} - 9981 \beta_{3} - 478 \beta_{4} - 1107 \beta_{5} - 1640 \beta_{6} - 1272 \beta_{7} + 331 \beta_{8} + 195 \beta_{9} ) q^{61} + ( 673829 - 58303 \beta_{1} + 1916 \beta_{2} + 11915 \beta_{3} + 26 \beta_{4} - 161 \beta_{5} - 2581 \beta_{6} - 103 \beta_{7} + 1012 \beta_{8} + 639 \beta_{9} ) q^{62} + ( -429779 + 33271 \beta_{1} - 4459 \beta_{2} - 2058 \beta_{3} + 686 \beta_{6} - 686 \beta_{8} - 343 \beta_{9} ) q^{63} + ( -666675 + 21696 \beta_{1} - 12625 \beta_{2} + 8057 \beta_{3} - 338 \beta_{4} + 19 \beta_{5} - 1468 \beta_{6} + 58 \beta_{7} - 455 \beta_{8} - 843 \beta_{9} ) q^{64} + ( -50531 + 4394 \beta_{1} + 2197 \beta_{3} + 2197 \beta_{6} ) q^{65} + ( -1432690 + 8576 \beta_{1} - 18048 \beta_{2} + 4317 \beta_{3} - 160 \beta_{4} + 553 \beta_{5} - 1014 \beta_{6} - 2731 \beta_{7} - 3259 \beta_{8} - 51 \beta_{9} ) q^{66} + ( 400087 - 26404 \beta_{1} + 5607 \beta_{2} - 6031 \beta_{3} + 1330 \beta_{4} - 79 \beta_{5} - 930 \beta_{6} + 188 \beta_{7} + 767 \beta_{8} - 849 \beta_{9} ) q^{67} + ( -169453 - 23017 \beta_{1} + 4578 \beta_{2} + 13368 \beta_{3} - 1371 \beta_{4} + 286 \beta_{5} - 420 \beta_{6} - 1446 \beta_{7} - 1315 \beta_{8} - 1095 \beta_{9} ) q^{68} + ( -75690 + 210493 \beta_{1} - 12418 \beta_{2} + 4097 \beta_{3} + 820 \beta_{4} + 2022 \beta_{5} - 4132 \beta_{6} + 1602 \beta_{7} + 2202 \beta_{8} + 916 \beta_{9} ) q^{69} + ( -82663 - 6517 \beta_{1} - 2058 \beta_{2} + 12348 \beta_{3} - 686 \beta_{4} - 343 \beta_{6} - 343 \beta_{8} ) q^{70} + ( 591213 + 36312 \beta_{1} + 5870 \beta_{2} - 7719 \beta_{3} - 704 \beta_{4} + 180 \beta_{5} + 4612 \beta_{6} + 6709 \beta_{7} - 658 \beta_{8} + 37 \beta_{9} ) q^{71} + ( 1040274 - 218353 \beta_{1} + 16437 \beta_{2} - 3572 \beta_{3} - 309 \beta_{4} - 296 \beta_{5} + 3530 \beta_{6} - 989 \beta_{7} - 2635 \beta_{8} - 99 \beta_{9} ) q^{72} + ( 183372 - 201307 \beta_{1} - 3454 \beta_{2} - 10899 \beta_{3} - 1376 \beta_{4} - 1884 \beta_{5} - 2923 \beta_{6} - 1839 \beta_{7} - 1570 \beta_{8} + 837 \beta_{9} ) q^{73} + ( -837201 + 64556 \beta_{1} + 3278 \beta_{2} + 37750 \beta_{3} + 478 \beta_{4} + 574 \beta_{5} + 4847 \beta_{6} - 4424 \beta_{7} - 2849 \beta_{8} + 40 \beta_{9} ) q^{74} + ( 146796 + 39248 \beta_{1} + 9110 \beta_{2} + 16839 \beta_{3} + 578 \beta_{4} + 397 \beta_{5} + 2082 \beta_{6} - 2667 \beta_{7} + 2723 \beta_{8} + 613 \beta_{9} ) q^{75} + ( 248376 - 222829 \beta_{1} + 7310 \beta_{2} + 20205 \beta_{3} - 805 \beta_{4} - 547 \beta_{5} - 3732 \beta_{6} - 2644 \beta_{7} - 462 \beta_{8} - 918 \beta_{9} ) q^{76} + ( -12691 - 9604 \beta_{1} + 686 \beta_{2} + 5488 \beta_{3} + 686 \beta_{4} - 1029 \beta_{5} + 1372 \beta_{6} + 686 \beta_{7} + 1715 \beta_{8} ) q^{77} + ( -228488 - 35152 \beta_{1} - 2197 \beta_{2} + 13182 \beta_{3} + 2197 \beta_{6} - 2197 \beta_{7} - 2197 \beta_{8} ) q^{78} + ( -1521103 + 97999 \beta_{1} - 10510 \beta_{2} + 160 \beta_{3} + 2280 \beta_{4} + 2040 \beta_{5} + 7837 \beta_{6} + 2880 \beta_{7} - 856 \beta_{8} + 714 \beta_{9} ) q^{79} + ( -906044 - 29020 \beta_{1} - 2790 \beta_{2} + 1542 \beta_{3} + 688 \beta_{4} + 1050 \beta_{5} + 326 \beta_{6} - 800 \beta_{7} + 1704 \beta_{8} - 442 \beta_{9} ) q^{80} + ( 2571980 - 8113 \beta_{1} + 41842 \beta_{2} + 24009 \beta_{3} - 1638 \beta_{4} - 903 \beta_{5} - 1310 \beta_{6} + 2466 \beta_{7} + 4949 \beta_{8} + 160 \beta_{9} ) q^{81} + ( -710181 + 27266 \beta_{1} + 2867 \beta_{2} - 7835 \beta_{3} + 2342 \beta_{4} - 657 \beta_{5} - 4054 \beta_{6} + 5958 \beta_{7} + 4707 \beta_{8} + 1161 \beta_{9} ) q^{82} + ( -1427676 + 45631 \beta_{1} - 2200 \beta_{2} + 9884 \beta_{3} + 1834 \beta_{4} - 127 \beta_{5} - 885 \beta_{6} - 500 \beta_{7} - 4059 \beta_{8} - 1724 \beta_{9} ) q^{83} + ( 554974 + 29155 \beta_{1} + 3087 \beta_{2} + 34643 \beta_{3} - 1029 \beta_{4} + 343 \beta_{5} + 1372 \beta_{6} - 4116 \beta_{7} + 1372 \beta_{9} ) q^{84} + ( -1143417 - 105984 \beta_{1} - 9144 \beta_{2} + 17725 \beta_{3} + 5240 \beta_{4} + 3104 \beta_{5} + 9748 \beta_{6} - 385 \beta_{7} + 4142 \beta_{8} - 1611 \beta_{9} ) q^{85} + ( 883170 - 10845 \beta_{1} + 10294 \beta_{2} + 35033 \beta_{3} - 116 \beta_{4} + 1309 \beta_{5} + 3176 \beta_{6} - 707 \beta_{7} + 3861 \beta_{8} + 1815 \beta_{9} ) q^{86} + ( -2919491 - 161553 \beta_{1} - 4513 \beta_{2} - 91206 \beta_{3} - 1394 \beta_{4} - 2769 \beta_{5} + 7408 \beta_{6} + 1892 \beta_{7} - 83 \beta_{8} - 2353 \beta_{9} ) q^{87} + ( -1520459 + 24 \beta_{1} - 8509 \beta_{2} + 80565 \beta_{3} - 706 \beta_{4} - 1733 \beta_{5} - 10558 \beta_{6} + 5548 \beta_{7} - 2293 \beta_{8} + 2397 \beta_{9} ) q^{88} + ( 703071 - 114093 \beta_{1} + 38931 \beta_{2} - 3824 \beta_{3} - 3162 \beta_{4} - 2153 \beta_{5} + 5077 \beta_{6} + 6585 \beta_{7} - 241 \beta_{8} + 1386 \beta_{9} ) q^{89} + ( 611994 - 116357 \beta_{1} - 9723 \beta_{2} - 102196 \beta_{3} + 656 \beta_{4} - 682 \beta_{5} - 11175 \beta_{6} + 15445 \beta_{7} + 7839 \beta_{8} - 1204 \beta_{9} ) q^{90} + 753571 q^{91} + ( -2292114 - 436570 \beta_{1} - 12443 \beta_{2} + 114728 \beta_{3} - 2990 \beta_{4} - 3860 \beta_{5} - 172 \beta_{6} - 8528 \beta_{7} - 8430 \beta_{8} + 1758 \beta_{9} ) q^{92} + ( 1643123 + 226482 \beta_{1} + 12386 \beta_{2} - 83242 \beta_{3} - 3254 \beta_{4} + 3997 \beta_{5} - 1230 \beta_{6} + 4680 \beta_{7} + 7389 \beta_{8} + 410 \beta_{9} ) q^{93} + ( -2267138 - 91841 \beta_{1} - 14439 \beta_{2} - 85981 \beta_{3} + 104 \beta_{4} - 55 \beta_{5} - 8875 \beta_{6} + 14276 \beta_{7} + 8392 \beta_{8} - 477 \beta_{9} ) q^{94} + ( -979106 + 119553 \beta_{1} + 11879 \beta_{2} - 41509 \beta_{3} - 590 \beta_{4} + 1609 \beta_{5} - 7365 \beta_{6} + 1930 \beta_{7} + 1675 \beta_{8} - 2399 \beta_{9} ) q^{95} + ( -3630763 - 50856 \beta_{1} - 9917 \beta_{2} - 6615 \beta_{3} - 4562 \beta_{4} + 2811 \beta_{5} - 5456 \beta_{6} - 17394 \beta_{7} - 9947 \beta_{8} - 2475 \beta_{9} ) q^{96} + ( -1258854 + 124069 \beta_{1} - 6500 \beta_{2} + 61795 \beta_{3} + 2764 \beta_{4} - 1006 \beta_{5} + 1757 \beta_{6} - 9031 \beta_{7} - 5844 \beta_{8} + 15 \beta_{9} ) q^{97} -117649 \beta_{1} q^{98} + ( -4252001 - 192714 \beta_{1} - 45588 \beta_{2} + 20683 \beta_{3} - 2192 \beta_{4} + 2112 \beta_{5} + 6914 \beta_{6} - 23047 \beta_{7} - 7434 \beta_{8} - 2481 \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 3 q^{2} - 101 q^{3} + 361 q^{4} + 226 q^{5} + 1105 q^{6} - 3430 q^{7} + 291 q^{8} + 12247 q^{9} + O(q^{10}) \) \( 10 q - 3 q^{2} - 101 q^{3} + 361 q^{4} + 226 q^{5} + 1105 q^{6} - 3430 q^{7} + 291 q^{8} + 12247 q^{9} + 2548 q^{10} + 451 q^{11} - 16241 q^{12} - 21970 q^{13} + 1029 q^{14} + 27184 q^{15} + 11897 q^{16} - 8654 q^{17} + 159348 q^{18} + 10130 q^{19} - 82012 q^{20} + 34643 q^{21} - 57863 q^{22} - 52155 q^{23} - 49227 q^{24} + 47190 q^{25} + 6591 q^{26} - 155171 q^{27} - 123823 q^{28} + 520154 q^{29} + 1070236 q^{30} + 692605 q^{31} + 149835 q^{32} + 436053 q^{33} + 1059060 q^{34} - 77518 q^{35} + 2843742 q^{36} - 20511 q^{37} + 1905286 q^{38} + 221897 q^{39} + 636320 q^{40} + 355049 q^{41} - 379015 q^{42} + 1256772 q^{43} - 687913 q^{44} + 1259926 q^{45} + 4043075 q^{46} + 1260721 q^{47} + 1128551 q^{48} + 1176490 q^{49} + 609035 q^{50} + 1411976 q^{51} - 793117 q^{52} + 928854 q^{53} + 6642607 q^{54} + 3423196 q^{55} - 99813 q^{56} + 3014966 q^{57} + 1612588 q^{58} + 3144446 q^{59} + 7738848 q^{60} + 6322923 q^{61} + 6545331 q^{62} - 4200721 q^{63} - 6629943 q^{64} - 496522 q^{65} - 14343317 q^{66} + 3944507 q^{67} - 1787356 q^{68} - 148281 q^{69} - 873964 q^{70} + 6032248 q^{71} + 9760866 q^{72} + 1248533 q^{73} - 8263279 q^{74} + 1573413 q^{75} + 1788254 q^{76} - 154693 q^{77} - 2427685 q^{78} - 14947605 q^{79} - 9147616 q^{80} + 25716334 q^{81} - 6987095 q^{82} - 14177784 q^{83} + 5570663 q^{84} - 11788444 q^{85} + 8748840 q^{86} - 29484448 q^{87} - 15390723 q^{88} + 6734836 q^{89} + 5994972 q^{90} + 7535710 q^{91} - 24493215 q^{92} + 17307847 q^{93} - 22760149 q^{94} - 9329708 q^{95} - 36488483 q^{96} - 12365397 q^{97} - 352947 q^{98} - 43198042 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 3 x^{9} - 816 x^{8} + 2298 x^{7} + 213848 x^{6} - 507132 x^{5} - 19919976 x^{4} + 24331248 x^{3} + 727257184 x^{2} - 56397312 x - 7335224320\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 164 \)
\(\beta_{3}\)\(=\)\((\)\(2710730315 \nu^{9} - 10591077029 \nu^{8} - 2080232699020 \nu^{7} + 8256593138470 \nu^{6} + 475250169039696 \nu^{5} - 1887482839380516 \nu^{4} - 29505722844621864 \nu^{3} + 108035987258967312 \nu^{2} + 459226885383275744 \nu - 2026471443331588352\)\()/ 6604053619701888 \)
\(\beta_{4}\)\(=\)\((\)\(-3194242487 \nu^{9} - 1907850873301 \nu^{8} + 3542382315514 \nu^{7} + 1446371109921230 \nu^{6} - 1352801131155108 \nu^{5} - 324395969902423524 \nu^{4} + 161974354745208864 \nu^{3} + 19185289940564666448 \nu^{2} + 7458647501074047040 \nu - 239545848385365834496\)\()/ 6604053619701888 \)
\(\beta_{5}\)\(=\)\((\)\(8702011532 \nu^{9} + 219416775281 \nu^{8} - 6637241612701 \nu^{7} - 166817327091610 \nu^{6} + 1504914496875870 \nu^{5} + 37830391473107604 \nu^{4} - 87075056020249188 \nu^{3} - 2345441329966057200 \nu^{2} - 656398435870468144 \nu + 29275385637077899904\)\()/ 1100675603283648 \)
\(\beta_{6}\)\(=\)\((\)\(-77448321017 \nu^{9} - 118851619051 \nu^{8} + 58643694893350 \nu^{7} + 91889448816290 \nu^{6} - 13112355777656796 \nu^{5} - 22934572829074620 \nu^{4} + 763336811734808160 \nu^{3} + 2040479023768700784 \nu^{2} - 8015617409167559744 \nu - 21333548537638031104\)\()/ 6604053619701888 \)
\(\beta_{7}\)\(=\)\((\)\(27147386411 \nu^{9} + 204615595672 \nu^{8} - 20771638781257 \nu^{7} - 155247362966420 \nu^{6} + 4755818968563294 \nu^{5} + 35348820970636656 \nu^{4} - 297274827829562508 \nu^{3} - 2268088978451838432 \nu^{2} + 2762106303633476912 \nu + 25656397174687265824\)\()/ 1651013404925472 \)
\(\beta_{8}\)\(=\)\((\)\(-19136930095 \nu^{9} - 96570802877 \nu^{8} + 14586243188762 \nu^{7} + 73109374683374 \nu^{6} - 3310768376771748 \nu^{5} - 16795865975028196 \nu^{4} + 201942480187656800 \nu^{3} + 1138764405671422352 \nu^{2} - 2024655463680966592 \nu - 12870878638627250048\)\()/ 733783735522432 \)
\(\beta_{9}\)\(=\)\((\)\(131832796597 \nu^{9} + 712892783321 \nu^{8} - 100446640439600 \nu^{7} - 544641329504038 \nu^{6} + 22756691833994232 \nu^{5} + 126705574187831268 \nu^{4} - 1375386805042063368 \nu^{3} - 8777582157342320688 \nu^{2} + 13016678190206610976 \nu + 107398239353653408256\)\()/ 3302026809850944 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 164\)
\(\nu^{3}\)\(=\)\(\beta_{7} + 2 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_{2} + 291 \beta_{1} - 39\)
\(\nu^{4}\)\(=\)\(3 \beta_{9} - 7 \beta_{8} - 8 \beta_{7} + 14 \beta_{6} + \beta_{5} + 2 \beta_{4} - 33 \beta_{3} + 383 \beta_{2} - 132 \beta_{1} + 47819\)
\(\nu^{5}\)\(=\)\(-9 \beta_{9} - 49 \beta_{8} + 446 \beta_{7} + 888 \beta_{6} + 387 \beta_{5} + 428 \beta_{4} - 1683 \beta_{3} - 733 \beta_{2} + 96370 \beta_{1} - 34217\)
\(\nu^{6}\)\(=\)\(1077 \beta_{9} - 4935 \beta_{8} - 5062 \beta_{7} + 7492 \beta_{6} + 659 \beta_{5} + 942 \beta_{4} - 13063 \beta_{3} + 134191 \beta_{2} - 62784 \beta_{1} + 15912781\)
\(\nu^{7}\)\(=\)\(-12079 \beta_{9} - 34639 \beta_{8} + 172474 \beta_{7} + 325824 \beta_{6} + 136231 \beta_{5} + 160342 \beta_{4} - 1061051 \beta_{3} - 238425 \beta_{2} + 32895336 \beta_{1} - 14490419\)
\(\nu^{8}\)\(=\)\(303237 \beta_{9} - 2550843 \beta_{8} - 2497490 \beta_{7} + 3250664 \beta_{6} + 313483 \beta_{5} + 351318 \beta_{4} - 4270631 \beta_{3} + 46699871 \beta_{2} - 22807272 \beta_{1} + 5459291269\)
\(\nu^{9}\)\(=\)\(-7698359 \beta_{9} - 17800439 \beta_{8} + 65138942 \beta_{7} + 115752360 \beta_{6} + 47493695 \beta_{5} + 58790582 \beta_{4} - 527514763 \beta_{3} - 75670077 \beta_{2} + 11356612544 \beta_{1} - 5176466335\)

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
19.2053
17.3557
9.01767
8.95987
3.66228
−4.83483
−6.50047
−6.95530
−17.8475
−19.0627
−19.2053 20.2953 240.842 −234.137 −389.776 −343.000 −2167.15 −1775.10 4496.66
1.2 −17.3557 −89.1399 173.221 −9.67118 1547.09 −343.000 −784.843 5758.93 167.850
1.3 −9.01767 −31.1400 −46.6816 514.920 280.810 −343.000 1575.22 −1217.30 −4643.38
1.4 −8.95987 −21.0756 −47.7208 −55.0053 188.835 −343.000 1574.43 −1742.82 492.840
1.5 −3.66228 42.3449 −114.588 −251.753 −155.079 −343.000 888.424 −393.907 921.992
1.6 4.83483 62.4820 −104.624 314.530 302.090 −343.000 −1124.70 1717.00 1520.70
1.7 6.50047 −52.7234 −85.7438 295.835 −342.727 −343.000 −1389.44 592.761 1923.06
1.8 6.95530 −32.1557 −79.6238 −381.581 −223.652 −343.000 −1444.09 −1153.01 −2654.01
1.9 17.8475 86.1795 190.532 250.298 1538.09 −343.000 1116.04 5239.91 4467.19
1.10 19.0627 −86.0671 235.388 −217.435 −1640.67 −343.000 2047.10 5220.54 −4144.91
\(n\): 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.d 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.8.a.d 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

$p$ $F_p(T)$
$2$ \( -7335224320 + 56397312 T + 727257184 T^{2} - 24331248 T^{3} - 19919976 T^{4} + 507132 T^{5} + 213848 T^{6} - 2298 T^{7} - 816 T^{8} + 3 T^{9} + T^{10} \)
$3$ \( 39502613236151808 + 2052856949954400 T - 109539306717408 T^{2} - 6995852893656 T^{3} + 22654161816 T^{4} + 5376507204 T^{5} + 32490260 T^{6} - 1352210 T^{7} - 11958 T^{8} + 101 T^{9} + T^{10} \)
$5$ \( \)\(31\!\cdots\!00\)\( + \)\(38\!\cdots\!50\)\( T + 68426832004052315775 T^{2} - 8753316903155360 T^{3} - 3166169499742990 T^{4} - 3318493729288 T^{5} + 53404092472 T^{6} + 53550872 T^{7} - 388682 T^{8} - 226 T^{9} + T^{10} \)
$7$ \( ( 343 + T )^{10} \)
$11$ \( -\)\(10\!\cdots\!20\)\( + \)\(89\!\cdots\!28\)\( T + \)\(43\!\cdots\!12\)\( T^{2} - \)\(28\!\cdots\!12\)\( T^{3} - \)\(10\!\cdots\!40\)\( T^{4} + 13920095146632385708 T^{5} + 6461146520665052 T^{6} - 144647825502 T^{7} - 138374754 T^{8} - 451 T^{9} + T^{10} \)
$13$ \( ( 2197 + T )^{10} \)
$17$ \( -\)\(90\!\cdots\!44\)\( + \)\(17\!\cdots\!84\)\( T + \)\(49\!\cdots\!84\)\( T^{2} - \)\(19\!\cdots\!04\)\( T^{3} - \)\(53\!\cdots\!44\)\( T^{4} + \)\(72\!\cdots\!00\)\( T^{5} + 1934207046484823300 T^{6} - 14283851697540 T^{7} - 2446710978 T^{8} + 8654 T^{9} + T^{10} \)
$19$ \( \)\(12\!\cdots\!44\)\( - \)\(53\!\cdots\!98\)\( T + \)\(55\!\cdots\!11\)\( T^{2} + \)\(41\!\cdots\!08\)\( T^{3} - \)\(66\!\cdots\!14\)\( T^{4} - \)\(13\!\cdots\!92\)\( T^{5} + 2304389011404901120 T^{6} + 23931800477192 T^{7} - 2779703538 T^{8} - 10130 T^{9} + T^{10} \)
$23$ \( \)\(40\!\cdots\!12\)\( + \)\(98\!\cdots\!63\)\( T - \)\(16\!\cdots\!79\)\( T^{2} - \)\(70\!\cdots\!20\)\( T^{3} - \)\(36\!\cdots\!92\)\( T^{4} + \)\(19\!\cdots\!26\)\( T^{5} + \)\(21\!\cdots\!94\)\( T^{6} - 1802551003101324 T^{7} - 26598140036 T^{8} + 52155 T^{9} + T^{10} \)
$29$ \( \)\(97\!\cdots\!12\)\( + \)\(67\!\cdots\!42\)\( T - \)\(79\!\cdots\!83\)\( T^{2} - \)\(34\!\cdots\!92\)\( T^{3} + \)\(54\!\cdots\!44\)\( T^{4} + \)\(57\!\cdots\!88\)\( T^{5} - \)\(50\!\cdots\!66\)\( T^{6} + 25653743750267328 T^{7} + 32572021804 T^{8} - 520154 T^{9} + T^{10} \)
$31$ \( -\)\(12\!\cdots\!20\)\( + \)\(11\!\cdots\!49\)\( T - \)\(96\!\cdots\!13\)\( T^{2} - \)\(38\!\cdots\!86\)\( T^{3} + \)\(51\!\cdots\!38\)\( T^{4} + \)\(22\!\cdots\!20\)\( T^{5} - \)\(57\!\cdots\!68\)\( T^{6} + 15636436463998842 T^{7} + 115972438466 T^{8} - 692605 T^{9} + T^{10} \)
$37$ \( \)\(31\!\cdots\!80\)\( - \)\(16\!\cdots\!00\)\( T + \)\(21\!\cdots\!56\)\( T^{2} + \)\(15\!\cdots\!36\)\( T^{3} - \)\(33\!\cdots\!12\)\( T^{4} - \)\(48\!\cdots\!12\)\( T^{5} + \)\(77\!\cdots\!32\)\( T^{6} + 13052517093030974 T^{7} - 507451621388 T^{8} + 20511 T^{9} + T^{10} \)
$41$ \( \)\(73\!\cdots\!40\)\( - \)\(70\!\cdots\!20\)\( T + \)\(43\!\cdots\!88\)\( T^{2} + \)\(13\!\cdots\!60\)\( T^{3} - \)\(26\!\cdots\!84\)\( T^{4} - \)\(92\!\cdots\!28\)\( T^{5} + \)\(23\!\cdots\!36\)\( T^{6} + 291300210818398600 T^{7} - 820326494378 T^{8} - 355049 T^{9} + T^{10} \)
$43$ \( -\)\(50\!\cdots\!12\)\( + \)\(47\!\cdots\!92\)\( T + \)\(16\!\cdots\!21\)\( T^{2} - \)\(26\!\cdots\!68\)\( T^{3} + \)\(10\!\cdots\!68\)\( T^{4} + \)\(23\!\cdots\!48\)\( T^{5} - \)\(26\!\cdots\!18\)\( T^{6} + 574610072326096284 T^{7} + 21156175684 T^{8} - 1256772 T^{9} + T^{10} \)
$47$ \( \)\(68\!\cdots\!72\)\( + \)\(12\!\cdots\!05\)\( T - \)\(11\!\cdots\!61\)\( T^{2} - \)\(11\!\cdots\!46\)\( T^{3} + \)\(23\!\cdots\!90\)\( T^{4} - \)\(39\!\cdots\!12\)\( T^{5} - \)\(66\!\cdots\!32\)\( T^{6} + 2196811713209202814 T^{7} - 1082581165810 T^{8} - 1260721 T^{9} + T^{10} \)
$53$ \( -\)\(10\!\cdots\!00\)\( - \)\(27\!\cdots\!90\)\( T + \)\(17\!\cdots\!81\)\( T^{2} + \)\(48\!\cdots\!40\)\( T^{3} - \)\(29\!\cdots\!36\)\( T^{4} - \)\(25\!\cdots\!08\)\( T^{5} + \)\(17\!\cdots\!14\)\( T^{6} + 4425635112957776172 T^{7} - 3809172816676 T^{8} - 928854 T^{9} + T^{10} \)
$59$ \( \)\(58\!\cdots\!00\)\( + \)\(16\!\cdots\!08\)\( T - \)\(13\!\cdots\!04\)\( T^{2} + \)\(18\!\cdots\!00\)\( T^{3} + \)\(33\!\cdots\!40\)\( T^{4} - \)\(59\!\cdots\!60\)\( T^{5} + \)\(26\!\cdots\!36\)\( T^{6} + 26572368044049123144 T^{7} - 6748019155212 T^{8} - 3144446 T^{9} + T^{10} \)
$61$ \( \)\(74\!\cdots\!00\)\( - \)\(99\!\cdots\!00\)\( T - \)\(98\!\cdots\!60\)\( T^{2} + \)\(19\!\cdots\!72\)\( T^{3} + \)\(50\!\cdots\!00\)\( T^{4} - \)\(86\!\cdots\!76\)\( T^{5} - \)\(53\!\cdots\!28\)\( T^{6} + 26618871183221541200 T^{7} + 6727194303282 T^{8} - 6322923 T^{9} + T^{10} \)
$67$ \( \)\(65\!\cdots\!80\)\( + \)\(94\!\cdots\!80\)\( T + \)\(31\!\cdots\!56\)\( T^{2} + \)\(25\!\cdots\!48\)\( T^{3} - \)\(19\!\cdots\!00\)\( T^{4} - \)\(22\!\cdots\!24\)\( T^{5} + \)\(71\!\cdots\!00\)\( T^{6} + 54188162940512110872 T^{7} - 13812128658620 T^{8} - 3944507 T^{9} + T^{10} \)
$71$ \( \)\(18\!\cdots\!24\)\( - \)\(16\!\cdots\!52\)\( T - \)\(45\!\cdots\!72\)\( T^{2} + \)\(41\!\cdots\!68\)\( T^{3} + \)\(32\!\cdots\!32\)\( T^{4} - \)\(65\!\cdots\!96\)\( T^{5} + \)\(75\!\cdots\!40\)\( T^{6} + \)\(34\!\cdots\!60\)\( T^{7} - 52251307367118 T^{8} - 6032248 T^{9} + T^{10} \)
$73$ \( -\)\(50\!\cdots\!06\)\( - \)\(19\!\cdots\!23\)\( T + \)\(81\!\cdots\!19\)\( T^{2} + \)\(11\!\cdots\!34\)\( T^{3} - \)\(46\!\cdots\!42\)\( T^{4} - \)\(34\!\cdots\!28\)\( T^{5} + \)\(12\!\cdots\!52\)\( T^{6} + \)\(14\!\cdots\!42\)\( T^{7} - 67238801026160 T^{8} - 1248533 T^{9} + T^{10} \)
$79$ \( \)\(31\!\cdots\!48\)\( + \)\(25\!\cdots\!49\)\( T - \)\(54\!\cdots\!43\)\( T^{2} - \)\(25\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!12\)\( T^{4} + \)\(52\!\cdots\!54\)\( T^{5} - \)\(13\!\cdots\!50\)\( T^{6} - \)\(46\!\cdots\!08\)\( T^{7} + 26226167934992 T^{8} + 14947605 T^{9} + T^{10} \)
$83$ \( \)\(14\!\cdots\!56\)\( - \)\(37\!\cdots\!16\)\( T - \)\(31\!\cdots\!21\)\( T^{2} + \)\(48\!\cdots\!16\)\( T^{3} + \)\(46\!\cdots\!02\)\( T^{4} + \)\(77\!\cdots\!92\)\( T^{5} - \)\(24\!\cdots\!56\)\( T^{6} - \)\(73\!\cdots\!36\)\( T^{7} + 923666600758 T^{8} + 14177784 T^{9} + T^{10} \)
$89$ \( -\)\(24\!\cdots\!00\)\( + \)\(47\!\cdots\!20\)\( T - \)\(32\!\cdots\!85\)\( T^{2} + \)\(89\!\cdots\!76\)\( T^{3} - \)\(27\!\cdots\!50\)\( T^{4} - \)\(27\!\cdots\!24\)\( T^{5} + \)\(29\!\cdots\!56\)\( T^{6} + \)\(24\!\cdots\!76\)\( T^{7} - 326948334079878 T^{8} - 6734836 T^{9} + T^{10} \)
$97$ \( \)\(60\!\cdots\!54\)\( + \)\(83\!\cdots\!67\)\( T + \)\(17\!\cdots\!67\)\( T^{2} - \)\(14\!\cdots\!06\)\( T^{3} - \)\(12\!\cdots\!22\)\( T^{4} + \)\(82\!\cdots\!60\)\( T^{5} + \)\(81\!\cdots\!44\)\( T^{6} - \)\(17\!\cdots\!90\)\( T^{7} - 175877533217180 T^{8} + 12365397 T^{9} + T^{10} \)
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