Properties

Label 531.8.a.b
Level $531$
Weight $8$
Character orbit 531.a
Self dual yes
Analytic conductor $165.876$
Analytic rank $1$
Dimension $16$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 531.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(165.876448532\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 6 x^{15} - 1493 x^{14} + 8791 x^{13} + 890490 x^{12} - 5107725 x^{11} - 269092298 x^{10} + 1488374176 x^{9} + 42885295136 x^{8} - 226132003872 x^{7} - 3353576629440 x^{6} + 16796366777600 x^{5} + 99470801612800 x^{4} - 494039551757568 x^{3} - 493048066650624 x^{2} + 3193975642099712 x - 2385018853548032\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 177)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 61 + \beta_{2} ) q^{4} + ( 5 - \beta_{1} - \beta_{4} ) q^{5} + ( -148 + 5 \beta_{1} + \beta_{2} - \beta_{9} ) q^{7} + ( -67 + 42 \beta_{1} + \beta_{3} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 61 + \beta_{2} ) q^{4} + ( 5 - \beta_{1} - \beta_{4} ) q^{5} + ( -148 + 5 \beta_{1} + \beta_{2} - \beta_{9} ) q^{7} + ( -67 + 42 \beta_{1} + \beta_{3} ) q^{8} + ( -204 - 41 \beta_{1} - 3 \beta_{2} + \beta_{3} - \beta_{7} + 3 \beta_{9} - \beta_{10} - \beta_{13} + \beta_{14} ) q^{10} + ( -73 + 37 \beta_{1} - 3 \beta_{2} + 3 \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{12} + 2 \beta_{14} ) q^{11} + ( -527 + 42 \beta_{1} - 15 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{8} + 2 \beta_{10} - \beta_{11} - 3 \beta_{12} + \beta_{13} - \beta_{15} ) q^{13} + ( 852 - 67 \beta_{1} + 10 \beta_{2} - 7 \beta_{3} + 30 \beta_{4} + 3 \beta_{5} - 5 \beta_{6} - 4 \beta_{7} + 2 \beta_{8} - \beta_{9} - 4 \beta_{10} - 3 \beta_{11} + 6 \beta_{12} - 6 \beta_{13} - 3 \beta_{14} - 2 \beta_{15} ) q^{14} + ( 218 - 19 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 19 \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} + 3 \beta_{8} + 3 \beta_{9} + 2 \beta_{10} - 3 \beta_{11} + 3 \beta_{12} + 4 \beta_{13} - 4 \beta_{14} - \beta_{15} ) q^{16} + ( 2857 - 110 \beta_{1} + 44 \beta_{2} - \beta_{3} + 9 \beta_{4} + \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 4 \beta_{11} - 3 \beta_{12} + 7 \beta_{13} + \beta_{15} ) q^{17} + ( -2481 - 118 \beta_{1} - 109 \beta_{2} + 2 \beta_{3} - 14 \beta_{4} - 6 \beta_{5} + 4 \beta_{6} + 5 \beta_{7} - 2 \beta_{8} + \beta_{9} - 4 \beta_{10} + 7 \beta_{11} + 2 \beta_{12} + 6 \beta_{13} + 3 \beta_{14} + 8 \beta_{15} ) q^{19} + ( -8074 - 175 \beta_{1} + 89 \beta_{2} + 7 \beta_{3} - 64 \beta_{4} - 16 \beta_{5} + 25 \beta_{6} + 9 \beta_{7} - 8 \beta_{8} - 18 \beta_{9} + 10 \beta_{10} - 2 \beta_{11} - 6 \beta_{12} + 17 \beta_{13} + 3 \beta_{14} + 6 \beta_{15} ) q^{20} + ( 6881 - 147 \beta_{1} - 48 \beta_{2} - 22 \beta_{3} - 15 \beta_{4} + 3 \beta_{6} + 16 \beta_{7} + 4 \beta_{8} - 7 \beta_{9} - 9 \beta_{10} + \beta_{11} + 12 \beta_{12} - 22 \beta_{13} - 5 \beta_{14} - 7 \beta_{15} ) q^{22} + ( 90 + 277 \beta_{1} + 70 \beta_{2} + 8 \beta_{3} - 6 \beta_{4} - 9 \beta_{5} - 8 \beta_{6} + 24 \beta_{7} - 5 \beta_{8} + 4 \beta_{9} + 23 \beta_{10} + 4 \beta_{11} + 12 \beta_{12} - 15 \beta_{14} + 6 \beta_{15} ) q^{23} + ( 17915 - 124 \beta_{1} - 41 \beta_{2} - 51 \beta_{3} - 6 \beta_{4} + \beta_{5} - 32 \beta_{6} + 21 \beta_{7} + 19 \beta_{8} + 23 \beta_{9} + 15 \beta_{10} + 21 \beta_{11} - \beta_{12} - \beta_{13} - 3 \beta_{14} + 5 \beta_{15} ) q^{25} + ( 8835 - 2022 \beta_{1} + 87 \beta_{2} + 6 \beta_{3} + 48 \beta_{4} + 11 \beta_{5} - 16 \beta_{6} + 29 \beta_{7} + 51 \beta_{8} + 71 \beta_{9} + 14 \beta_{10} + 44 \beta_{11} - 9 \beta_{12} - 21 \beta_{13} - 12 \beta_{14} - 10 \beta_{15} ) q^{26} + ( 6175 + 2451 \beta_{1} - 155 \beta_{2} - 4 \beta_{3} - 96 \beta_{4} - 21 \beta_{5} + 32 \beta_{6} + 6 \beta_{7} + 9 \beta_{8} - 108 \beta_{9} + 37 \beta_{10} + 28 \beta_{11} + 13 \beta_{12} - 10 \beta_{13} - 39 \beta_{14} + 28 \beta_{15} ) q^{28} + ( -8188 - 2323 \beta_{1} - 91 \beta_{2} - 44 \beta_{3} + 17 \beta_{4} + 21 \beta_{5} - 7 \beta_{6} + 18 \beta_{7} - 29 \beta_{8} - 19 \beta_{9} - 38 \beta_{10} - 6 \beta_{11} + 10 \beta_{12} - 14 \beta_{13} - 19 \beta_{14} + 20 \beta_{15} ) q^{29} + ( -10334 + 3825 \beta_{1} - 214 \beta_{2} - 4 \beta_{3} - 28 \beta_{4} - 53 \beta_{5} + 42 \beta_{6} + 2 \beta_{7} - 6 \beta_{8} - 6 \beta_{9} - 14 \beta_{10} - 29 \beta_{11} + 22 \beta_{12} + 14 \beta_{13} + 10 \beta_{14} - 10 \beta_{15} ) q^{31} + ( 4541 - 6197 \beta_{1} - 106 \beta_{2} + 6 \beta_{3} + 35 \beta_{4} - 58 \beta_{5} + 10 \beta_{6} - 19 \beta_{7} - 16 \beta_{8} + 40 \beta_{9} - 6 \beta_{10} + 20 \beta_{11} + 16 \beta_{12} + 20 \beta_{13} + 2 \beta_{14} + 8 \beta_{15} ) q^{32} + ( -24213 + 7668 \beta_{1} - 227 \beta_{2} + 40 \beta_{3} + 117 \beta_{4} + 58 \beta_{5} - 41 \beta_{6} + 26 \beta_{7} - 53 \beta_{8} + 49 \beta_{9} + 25 \beta_{10} - 4 \beta_{11} - 45 \beta_{12} + 142 \beta_{13} - 5 \beta_{14} - 9 \beta_{15} ) q^{34} + ( 9846 - 12971 \beta_{1} + 367 \beta_{2} - 41 \beta_{3} + 286 \beta_{4} + 14 \beta_{5} + 8 \beta_{6} - 35 \beta_{7} + 76 \beta_{8} + 41 \beta_{9} + 30 \beta_{10} - 73 \beta_{11} + 71 \beta_{12} + 31 \beta_{13} - 26 \beta_{14} - 47 \beta_{15} ) q^{35} + ( -20829 + 5539 \beta_{1} - 21 \beta_{2} - 14 \beta_{3} + 397 \beta_{4} + 68 \beta_{5} + 9 \beta_{6} - 9 \beta_{7} + 15 \beta_{8} + 146 \beta_{9} - 2 \beta_{10} - 65 \beta_{11} - 96 \beta_{12} + 72 \beta_{13} - 7 \beta_{14} - 30 \beta_{15} ) q^{37} + ( -14758 - 14974 \beta_{1} + 218 \beta_{2} - 221 \beta_{3} - 200 \beta_{4} + 63 \beta_{5} - 7 \beta_{6} - 77 \beta_{7} - 84 \beta_{8} + 99 \beta_{9} - 24 \beta_{10} - 134 \beta_{11} + 12 \beta_{12} + 184 \beta_{13} + 62 \beta_{14} + 29 \beta_{15} ) q^{38} + ( -14374 + 3317 \beta_{1} - 700 \beta_{2} - 42 \beta_{3} + 970 \beta_{4} + 32 \beta_{5} - 59 \beta_{6} - 25 \beta_{7} - 20 \beta_{8} + 138 \beta_{9} - 74 \beta_{10} - 154 \beta_{11} + 78 \beta_{12} - 3 \beta_{13} + 55 \beta_{14} - 42 \beta_{15} ) q^{40} + ( -38117 - 8031 \beta_{1} + 186 \beta_{2} - 118 \beta_{3} - 166 \beta_{4} - 123 \beta_{5} + 9 \beta_{6} - 84 \beta_{7} - 38 \beta_{8} - 379 \beta_{9} - 123 \beta_{10} - 54 \beta_{11} - 22 \beta_{12} - 96 \beta_{13} - 29 \beta_{14} - 52 \beta_{15} ) q^{41} + ( -93992 + 11917 \beta_{1} - 807 \beta_{2} - 6 \beta_{3} - 194 \beta_{4} - 13 \beta_{5} + 67 \beta_{6} - 173 \beta_{7} - 70 \beta_{8} + 30 \beta_{9} - 73 \beta_{10} + 47 \beta_{11} + 180 \beta_{12} - 142 \beta_{13} - 72 \beta_{14} + 10 \beta_{15} ) q^{43} + ( -14557 - 4839 \beta_{1} - 1604 \beta_{2} - 120 \beta_{3} + 329 \beta_{4} + 156 \beta_{5} - 38 \beta_{6} - 28 \beta_{7} - 20 \beta_{8} + 44 \beta_{9} - 60 \beta_{10} - 41 \beta_{11} - 92 \beta_{12} - 174 \beta_{13} + 85 \beta_{14} + 53 \beta_{15} ) q^{44} + ( 51875 + 6443 \beta_{1} + 469 \beta_{2} + 139 \beta_{3} + 1062 \beta_{4} - 65 \beta_{5} - 212 \beta_{6} - 3 \beta_{7} + 218 \beta_{8} + 290 \beta_{9} + 77 \beta_{10} + 98 \beta_{11} - 46 \beta_{12} - 90 \beta_{13} - 88 \beta_{14} - 3 \beta_{15} ) q^{46} + ( 105509 - 20309 \beta_{1} - 2207 \beta_{2} + 149 \beta_{3} + 1235 \beta_{4} + 199 \beta_{5} - 42 \beta_{6} - 71 \beta_{7} + 215 \beta_{8} + 254 \beta_{9} + 223 \beta_{10} + 222 \beta_{11} - \beta_{12} + 43 \beta_{13} - 36 \beta_{14} - 133 \beta_{15} ) q^{47} + ( 118775 - 980 \beta_{1} + 731 \beta_{2} + 11 \beta_{3} - 73 \beta_{4} + 156 \beta_{5} + 81 \beta_{6} - 187 \beta_{7} + 94 \beta_{8} + 630 \beta_{9} + 145 \beta_{10} + 20 \beta_{11} + 5 \beta_{12} - 161 \beta_{13} + 47 \beta_{14} + 85 \beta_{15} ) q^{49} + ( -23630 + 9196 \beta_{1} - 4419 \beta_{2} + 107 \beta_{3} + 1342 \beta_{4} - 45 \beta_{5} - 191 \beta_{6} + 255 \beta_{7} - 299 \beta_{8} - 205 \beta_{9} - 24 \beta_{10} + 209 \beta_{11} - 251 \beta_{12} - 88 \beta_{13} - 72 \beta_{14} - 57 \beta_{15} ) q^{50} + ( -317278 + 15682 \beta_{1} - 1827 \beta_{2} - 88 \beta_{3} + 303 \beta_{4} - 115 \beta_{5} - 56 \beta_{6} + 168 \beta_{7} - 445 \beta_{8} - 343 \beta_{9} - 82 \beta_{10} + 40 \beta_{11} - 387 \beta_{12} + 123 \beta_{13} + 84 \beta_{14} - 32 \beta_{15} ) q^{52} + ( -30712 - 18819 \beta_{1} - 2750 \beta_{2} + 440 \beta_{3} + 1836 \beta_{4} + 50 \beta_{5} + 139 \beta_{6} + 65 \beta_{7} + 252 \beta_{8} + 207 \beta_{9} + 145 \beta_{10} + 182 \beta_{11} - 2 \beta_{12} - 30 \beta_{13} - 132 \beta_{14} + 118 \beta_{15} ) q^{53} + ( -294432 + 14476 \beta_{1} + 257 \beta_{2} + 315 \beta_{3} - 574 \beta_{4} - 53 \beta_{5} + 212 \beta_{6} - 51 \beta_{7} + 83 \beta_{8} + 187 \beta_{9} - 93 \beta_{10} + 73 \beta_{11} - 11 \beta_{12} - 351 \beta_{13} - 19 \beta_{14} - 315 \beta_{15} ) q^{55} + ( 364649 - 9102 \beta_{1} - 2139 \beta_{2} - 202 \beta_{3} + 1762 \beta_{4} + 62 \beta_{5} - 292 \beta_{6} - 277 \beta_{7} - 16 \beta_{8} + 232 \beta_{9} - 242 \beta_{10} - 201 \beta_{11} + 232 \beta_{12} - 782 \beta_{13} + 115 \beta_{14} - 11 \beta_{15} ) q^{56} + ( -430877 - 17438 \beta_{1} - 3686 \beta_{2} - 378 \beta_{3} + 284 \beta_{4} + 589 \beta_{5} - 6 \beta_{6} - 60 \beta_{7} + 280 \beta_{8} + 171 \beta_{9} + 216 \beta_{10} + 264 \beta_{11} - 288 \beta_{12} + 133 \beta_{13} + 373 \beta_{14} + 295 \beta_{15} ) q^{58} -205379 q^{59} + ( -386219 + 9867 \beta_{1} + 56 \beta_{2} - 500 \beta_{3} - 667 \beta_{4} - 512 \beta_{5} - 202 \beta_{6} + 100 \beta_{7} - 67 \beta_{8} - 221 \beta_{9} - 291 \beta_{10} - 372 \beta_{11} + 440 \beta_{12} - 848 \beta_{13} + 180 \beta_{14} - 198 \beta_{15} ) q^{61} + ( 737550 - 37632 \beta_{1} + 3802 \beta_{2} + 220 \beta_{3} - 901 \beta_{4} - 485 \beta_{5} + 509 \beta_{6} - 237 \beta_{7} - 266 \beta_{8} + 206 \beta_{9} - 85 \beta_{10} - 348 \beta_{11} - 90 \beta_{12} + 269 \beta_{13} + 547 \beta_{14} + 139 \beta_{15} ) q^{62} + ( -1191377 - 3508 \beta_{1} - 5760 \beta_{2} + 373 \beta_{3} - 138 \beta_{4} - 266 \beta_{5} + 242 \beta_{6} + 432 \beta_{7} - 222 \beta_{8} - 266 \beta_{9} + 108 \beta_{10} + 202 \beta_{11} - 182 \beta_{12} + 192 \beta_{13} + 504 \beta_{14} + 318 \beta_{15} ) q^{64} + ( 346608 - 39465 \beta_{1} - 1883 \beta_{2} - 551 \beta_{3} + 2002 \beta_{4} - 303 \beta_{5} + 195 \beta_{6} - 84 \beta_{7} - 62 \beta_{8} + 432 \beta_{9} - 729 \beta_{10} - 842 \beta_{11} + 733 \beta_{12} - 479 \beta_{13} + 750 \beta_{14} - 131 \beta_{15} ) q^{65} + ( -1018619 - 34597 \beta_{1} + 118 \beta_{2} - 655 \beta_{3} - 766 \beta_{4} + 721 \beta_{5} - 163 \beta_{6} - 47 \beta_{7} + 207 \beta_{8} + 318 \beta_{9} - 298 \beta_{10} - 371 \beta_{11} - 43 \beta_{12} + 577 \beta_{13} - 47 \beta_{14} + 137 \beta_{15} ) q^{67} + ( 1093752 - 30501 \beta_{1} + 5203 \beta_{2} + 587 \beta_{3} - 934 \beta_{4} + 710 \beta_{5} - 149 \beta_{6} + 433 \beta_{7} + 466 \beta_{8} + 1883 \beta_{9} + 991 \beta_{10} + 577 \beta_{11} - 482 \beta_{12} + 1796 \beta_{13} - 145 \beta_{14} + 51 \beta_{15} ) q^{68} + ( -2472995 + 56644 \beta_{1} - 13976 \beta_{2} + 1606 \beta_{3} + 690 \beta_{4} - 970 \beta_{5} + 885 \beta_{6} - 230 \beta_{7} - 1189 \beta_{8} - 2179 \beta_{9} - 167 \beta_{10} - 125 \beta_{11} - 177 \beta_{12} + 650 \beta_{13} - 48 \beta_{14} + 92 \beta_{15} ) q^{70} + ( 680947 - 12116 \beta_{1} - 5457 \beta_{2} + 1339 \beta_{3} + 4414 \beta_{4} - 1343 \beta_{5} + 1399 \beta_{6} - 281 \beta_{7} - 753 \beta_{8} - 1766 \beta_{9} - 226 \beta_{10} - 226 \beta_{11} - 753 \beta_{12} - 585 \beta_{13} - 124 \beta_{14} + 167 \beta_{15} ) q^{71} + ( -1328867 + 25474 \beta_{1} - 6451 \beta_{2} + 557 \beta_{3} + 1803 \beta_{4} + 158 \beta_{5} - 712 \beta_{6} + 181 \beta_{7} + 429 \beta_{8} + 257 \beta_{9} - 135 \beta_{10} + 358 \beta_{11} + 271 \beta_{12} + 119 \beta_{13} - 739 \beta_{14} - 565 \beta_{15} ) q^{73} + ( 1046161 - 2540 \beta_{1} + 1917 \beta_{2} + 2188 \beta_{3} - 2667 \beta_{4} - 235 \beta_{5} + 832 \beta_{6} + 1748 \beta_{7} - 440 \beta_{8} + 31 \beta_{9} + 1144 \beta_{10} + 1455 \beta_{11} - 1464 \beta_{12} + 1389 \beta_{13} + 198 \beta_{14} + 24 \beta_{15} ) q^{74} + ( -2558134 + 7170 \beta_{1} - 12403 \beta_{2} + 2258 \beta_{3} - 1677 \beta_{4} + 167 \beta_{5} + 1226 \beta_{6} + 160 \beta_{7} - 331 \beta_{8} + 540 \beta_{9} + 257 \beta_{10} + 683 \beta_{11} - 555 \beta_{12} + 1520 \beta_{13} + 686 \beta_{14} + 29 \beta_{15} ) q^{76} + ( 216064 - 9652 \beta_{1} - 6385 \beta_{2} + 1350 \beta_{3} + 1142 \beta_{4} + 92 \beta_{5} + 71 \beta_{6} - 633 \beta_{7} - 511 \beta_{8} - 856 \beta_{9} + 386 \beta_{10} + 995 \beta_{11} - 364 \beta_{12} + 1644 \beta_{13} - 379 \beta_{14} + 670 \beta_{15} ) q^{77} + ( -209482 - 66750 \beta_{1} + 3869 \beta_{2} - 1154 \beta_{3} - 2484 \beta_{4} - 1050 \beta_{5} - 303 \beta_{6} - 119 \beta_{7} + 952 \beta_{8} - 163 \beta_{9} - 487 \beta_{10} - 491 \beta_{11} + 1176 \beta_{12} + 188 \beta_{13} - 763 \beta_{14} + 904 \beta_{15} ) q^{79} + ( 1727817 - 23662 \beta_{1} + 2617 \beta_{2} - 383 \beta_{3} - 1623 \beta_{4} + 213 \beta_{5} - 802 \beta_{6} + 309 \beta_{7} + 557 \beta_{8} - 1649 \beta_{9} + 368 \beta_{10} + 1185 \beta_{11} - 249 \beta_{12} + 841 \beta_{13} - 737 \beta_{14} + 79 \beta_{15} ) q^{80} + ( -1550026 - 41754 \beta_{1} - 13695 \beta_{2} - 887 \beta_{3} + 5713 \beta_{4} - 209 \beta_{5} - 1139 \beta_{6} - 459 \beta_{7} + 2042 \beta_{8} - 19 \beta_{9} - 454 \beta_{10} - 123 \beta_{11} + 2270 \beta_{12} - 3872 \beta_{13} - 359 \beta_{14} - 388 \beta_{15} ) q^{82} + ( 108681 + 30161 \beta_{1} - 2422 \beta_{2} - 3368 \beta_{3} + 3210 \beta_{4} + 41 \beta_{5} - 1807 \beta_{6} + 234 \beta_{7} - 854 \beta_{8} - 613 \beta_{9} - 2327 \beta_{10} - 2878 \beta_{11} + 2292 \beta_{12} - 2366 \beta_{13} + 341 \beta_{14} - 502 \beta_{15} ) q^{83} + ( -1426791 - 57033 \beta_{1} + 6025 \beta_{2} + 704 \beta_{3} - 8550 \beta_{4} - 1054 \beta_{5} + 1779 \beta_{6} - 484 \beta_{7} - 269 \beta_{8} - 1232 \beta_{9} + 440 \beta_{10} - 214 \beta_{11} + 88 \beta_{12} + 1452 \beta_{13} - 438 \beta_{14} + 582 \beta_{15} ) q^{85} + ( 2315364 - 174324 \beta_{1} + 14369 \beta_{2} - 2184 \beta_{3} - 7287 \beta_{4} + 764 \beta_{5} + 828 \beta_{6} - 1842 \beta_{7} + 1302 \beta_{8} - 1728 \beta_{9} + 12 \beta_{10} - 927 \beta_{11} + 1102 \beta_{12} - 522 \beta_{13} + 855 \beta_{14} + 1269 \beta_{15} ) q^{86} + ( -1688029 - 140269 \beta_{1} - 5813 \beta_{2} + 39 \beta_{3} - 1629 \beta_{4} + 106 \beta_{5} + 598 \beta_{6} - 632 \beta_{7} - 870 \beta_{8} - 2556 \beta_{9} + 826 \beta_{10} + 1029 \beta_{11} - 1650 \beta_{12} + 26 \beta_{13} + 241 \beta_{14} + 855 \beta_{15} ) q^{88} + ( -642950 - 31348 \beta_{1} + 1116 \beta_{2} - 1367 \beta_{3} + 1029 \beta_{4} + 1090 \beta_{5} - 830 \beta_{6} + 424 \beta_{7} + 3038 \beta_{8} - 239 \beta_{9} - 924 \beta_{10} - 501 \beta_{11} + 3135 \beta_{12} - 2365 \beta_{13} - 1418 \beta_{14} - 785 \beta_{15} ) q^{89} + ( 106518 - 172430 \beta_{1} - 882 \beta_{2} - 818 \beta_{3} + 7029 \beta_{4} + 967 \beta_{5} - 3731 \beta_{6} + 988 \beta_{7} + 1551 \beta_{8} + 3065 \beta_{9} - 360 \beta_{10} + 769 \beta_{11} + 308 \beta_{12} + 54 \beta_{13} + 714 \beta_{14} - 32 \beta_{15} ) q^{91} + ( 1216284 + 115470 \beta_{1} + 12313 \beta_{2} + 1304 \beta_{3} - 6549 \beta_{4} - 2107 \beta_{5} - 84 \beta_{6} + 36 \beta_{7} + 283 \beta_{8} - 5428 \beta_{9} - 209 \beta_{10} + 97 \beta_{11} - 733 \beta_{12} + 410 \beta_{13} - 2500 \beta_{14} - 1893 \beta_{15} ) q^{92} + ( -3686652 - 65779 \beta_{1} - 23447 \beta_{2} - 3608 \beta_{3} + 4694 \beta_{4} + 2020 \beta_{5} - 896 \beta_{6} + 536 \beta_{7} - 2393 \beta_{8} - 5140 \beta_{9} + 516 \beta_{10} - 683 \beta_{11} - 1857 \beta_{12} + 2872 \beta_{13} - 1866 \beta_{14} - 938 \beta_{15} ) q^{94} + ( 1789684 + 105868 \beta_{1} + 422 \beta_{2} - 2008 \beta_{3} + 3584 \beta_{4} + 4053 \beta_{5} - 6239 \beta_{6} - 501 \beta_{7} + 2186 \beta_{8} + 5845 \beta_{9} + 459 \beta_{10} + 2035 \beta_{11} + 60 \beta_{12} - 1960 \beta_{13} - 2648 \beta_{14} + 320 \beta_{15} ) q^{95} + ( -1504208 - 198778 \beta_{1} - 18212 \beta_{2} + 1005 \beta_{3} + 11283 \beta_{4} + 438 \beta_{5} + 264 \beta_{6} - 1150 \beta_{7} - 1885 \beta_{8} - 438 \beta_{9} + 427 \beta_{10} - 1835 \beta_{11} + 2793 \beta_{12} - 2651 \beta_{13} - 206 \beta_{14} - 1015 \beta_{15} ) q^{97} + ( -221055 + 237718 \beta_{1} - 11120 \beta_{2} + 3418 \beta_{3} - 16786 \beta_{4} - 2408 \beta_{5} + 3387 \beta_{6} + 1335 \beta_{7} - 2639 \beta_{8} - 4935 \beta_{9} - 725 \beta_{10} + 614 \beta_{11} - 343 \beta_{12} - 2032 \beta_{13} + 31 \beta_{14} + 487 \beta_{15} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 6q^{2} + 974q^{4} + 68q^{5} - 2343q^{7} - 819q^{8} + O(q^{10}) \) \( 16q + 6q^{2} + 974q^{4} + 68q^{5} - 2343q^{7} - 819q^{8} - 3479q^{10} - 898q^{11} - 8172q^{13} + 13315q^{14} + 3138q^{16} + 44985q^{17} - 40137q^{19} - 130657q^{20} + 109394q^{22} + 2833q^{23} + 285746q^{25} + 129420q^{26} + 112890q^{28} - 144375q^{29} - 141759q^{31} + 36224q^{32} - 341332q^{34} + 78859q^{35} - 297971q^{37} - 329075q^{38} - 203048q^{40} - 659077q^{41} - 1431608q^{43} - 254916q^{44} + 873113q^{46} + 1574073q^{47} + 1893545q^{49} - 302533q^{50} - 4972548q^{52} - 587736q^{53} - 4624036q^{55} + 5798506q^{56} - 6991380q^{58} - 3286064q^{59} - 6117131q^{61} + 11570258q^{62} - 19063011q^{64} + 5335514q^{65} - 16518710q^{67} + 17284669q^{68} - 39189486q^{70} + 10882582q^{71} - 21097441q^{73} + 16717030q^{74} - 40864952q^{76} + 3404601q^{77} - 3784458q^{79} + 27466195q^{80} - 24990117q^{82} + 1951425q^{83} - 23238675q^{85} + 35910572q^{86} - 27843055q^{88} - 10499443q^{89} + 699217q^{91} + 20062766q^{92} - 59358988q^{94} + 29236333q^{95} - 25158976q^{97} - 2120460q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 6 x^{15} - 1493 x^{14} + 8791 x^{13} + 890490 x^{12} - 5107725 x^{11} - 269092298 x^{10} + 1488374176 x^{9} + 42885295136 x^{8} - 226132003872 x^{7} - 3353576629440 x^{6} + 16796366777600 x^{5} + 99470801612800 x^{4} - 494039551757568 x^{3} - 493048066650624 x^{2} + 3193975642099712 x - 2385018853548032\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 189 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 298 \nu + 67 \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(72\!\cdots\!81\)\( \nu^{15} - \)\(79\!\cdots\!72\)\( \nu^{14} + \)\(49\!\cdots\!17\)\( \nu^{13} + \)\(10\!\cdots\!15\)\( \nu^{12} + \)\(14\!\cdots\!20\)\( \nu^{11} - \)\(57\!\cdots\!15\)\( \nu^{10} - \)\(20\!\cdots\!52\)\( \nu^{9} + \)\(15\!\cdots\!68\)\( \nu^{8} + \)\(67\!\cdots\!68\)\( \nu^{7} - \)\(19\!\cdots\!64\)\( \nu^{6} - \)\(86\!\cdots\!32\)\( \nu^{5} + \)\(11\!\cdots\!64\)\( \nu^{4} + \)\(39\!\cdots\!32\)\( \nu^{3} - \)\(27\!\cdots\!36\)\( \nu^{2} - \)\(45\!\cdots\!44\)\( \nu + \)\(98\!\cdots\!76\)\(\)\()/ \)\(16\!\cdots\!20\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(14\!\cdots\!49\)\( \nu^{15} - \)\(34\!\cdots\!24\)\( \nu^{14} + \)\(17\!\cdots\!69\)\( \nu^{13} + \)\(45\!\cdots\!99\)\( \nu^{12} - \)\(76\!\cdots\!24\)\( \nu^{11} - \)\(23\!\cdots\!23\)\( \nu^{10} + \)\(13\!\cdots\!60\)\( \nu^{9} + \)\(55\!\cdots\!00\)\( \nu^{8} - \)\(52\!\cdots\!20\)\( \nu^{7} - \)\(63\!\cdots\!16\)\( \nu^{6} - \)\(90\!\cdots\!80\)\( \nu^{5} + \)\(30\!\cdots\!12\)\( \nu^{4} + \)\(53\!\cdots\!96\)\( \nu^{3} - \)\(52\!\cdots\!12\)\( \nu^{2} - \)\(77\!\cdots\!96\)\( \nu + \)\(18\!\cdots\!64\)\(\)\()/ \)\(80\!\cdots\!16\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(36\!\cdots\!57\)\( \nu^{15} - \)\(43\!\cdots\!56\)\( \nu^{14} - \)\(55\!\cdots\!09\)\( \nu^{13} + \)\(60\!\cdots\!45\)\( \nu^{12} + \)\(34\!\cdots\!00\)\( \nu^{11} - \)\(33\!\cdots\!65\)\( \nu^{10} - \)\(10\!\cdots\!56\)\( \nu^{9} + \)\(91\!\cdots\!84\)\( \nu^{8} + \)\(17\!\cdots\!04\)\( \nu^{7} - \)\(12\!\cdots\!92\)\( \nu^{6} - \)\(15\!\cdots\!16\)\( \nu^{5} + \)\(87\!\cdots\!92\)\( \nu^{4} + \)\(51\!\cdots\!56\)\( \nu^{3} - \)\(23\!\cdots\!08\)\( \nu^{2} - \)\(47\!\cdots\!12\)\( \nu + \)\(12\!\cdots\!48\)\(\)\()/ \)\(16\!\cdots\!20\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(50\!\cdots\!83\)\( \nu^{15} + \)\(18\!\cdots\!96\)\( \nu^{14} - \)\(69\!\cdots\!11\)\( \nu^{13} - \)\(21\!\cdots\!25\)\( \nu^{12} + \)\(38\!\cdots\!20\)\( \nu^{11} + \)\(95\!\cdots\!05\)\( \nu^{10} - \)\(10\!\cdots\!04\)\( \nu^{9} - \)\(17\!\cdots\!84\)\( \nu^{8} + \)\(14\!\cdots\!36\)\( \nu^{7} + \)\(10\!\cdots\!72\)\( \nu^{6} - \)\(98\!\cdots\!84\)\( \nu^{5} + \)\(48\!\cdots\!88\)\( \nu^{4} + \)\(29\!\cdots\!84\)\( \nu^{3} - \)\(29\!\cdots\!32\)\( \nu^{2} - \)\(31\!\cdots\!68\)\( \nu + \)\(10\!\cdots\!52\)\(\)\()/ \)\(16\!\cdots\!20\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(50\!\cdots\!91\)\( \nu^{15} + \)\(13\!\cdots\!68\)\( \nu^{14} + \)\(77\!\cdots\!47\)\( \nu^{13} - \)\(21\!\cdots\!15\)\( \nu^{12} - \)\(46\!\cdots\!20\)\( \nu^{11} + \)\(13\!\cdots\!35\)\( \nu^{10} + \)\(14\!\cdots\!88\)\( \nu^{9} - \)\(40\!\cdots\!92\)\( \nu^{8} - \)\(22\!\cdots\!52\)\( \nu^{7} + \)\(63\!\cdots\!76\)\( \nu^{6} + \)\(17\!\cdots\!88\)\( \nu^{5} - \)\(46\!\cdots\!56\)\( \nu^{4} - \)\(53\!\cdots\!28\)\( \nu^{3} + \)\(12\!\cdots\!04\)\( \nu^{2} + \)\(30\!\cdots\!16\)\( \nu - \)\(48\!\cdots\!64\)\(\)\()/ \)\(16\!\cdots\!20\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(11\!\cdots\!16\)\( \nu^{15} + \)\(14\!\cdots\!67\)\( \nu^{14} - \)\(15\!\cdots\!77\)\( \nu^{13} - \)\(14\!\cdots\!45\)\( \nu^{12} + \)\(90\!\cdots\!00\)\( \nu^{11} + \)\(43\!\cdots\!85\)\( \nu^{10} - \)\(25\!\cdots\!53\)\( \nu^{9} + \)\(16\!\cdots\!57\)\( \nu^{8} + \)\(38\!\cdots\!42\)\( \nu^{7} - \)\(27\!\cdots\!16\)\( \nu^{6} - \)\(28\!\cdots\!08\)\( \nu^{5} + \)\(41\!\cdots\!36\)\( \nu^{4} + \)\(85\!\cdots\!08\)\( \nu^{3} - \)\(16\!\cdots\!04\)\( \nu^{2} - \)\(61\!\cdots\!56\)\( \nu + \)\(10\!\cdots\!64\)\(\)\()/ \)\(25\!\cdots\!80\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(72\!\cdots\!59\)\( \nu^{15} + \)\(61\!\cdots\!88\)\( \nu^{14} - \)\(10\!\cdots\!23\)\( \nu^{13} - \)\(31\!\cdots\!25\)\( \nu^{12} + \)\(59\!\cdots\!80\)\( \nu^{11} - \)\(20\!\cdots\!95\)\( \nu^{10} - \)\(17\!\cdots\!12\)\( \nu^{9} + \)\(19\!\cdots\!88\)\( \nu^{8} + \)\(25\!\cdots\!28\)\( \nu^{7} - \)\(51\!\cdots\!64\)\( \nu^{6} - \)\(18\!\cdots\!32\)\( \nu^{5} + \)\(53\!\cdots\!84\)\( \nu^{4} + \)\(52\!\cdots\!52\)\( \nu^{3} - \)\(17\!\cdots\!16\)\( \nu^{2} - \)\(27\!\cdots\!84\)\( \nu + \)\(79\!\cdots\!16\)\(\)\()/ \)\(16\!\cdots\!20\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(77\!\cdots\!87\)\( \nu^{15} - \)\(25\!\cdots\!84\)\( \nu^{14} + \)\(11\!\cdots\!39\)\( \nu^{13} + \)\(29\!\cdots\!65\)\( \nu^{12} - \)\(62\!\cdots\!60\)\( \nu^{11} - \)\(12\!\cdots\!65\)\( \nu^{10} + \)\(17\!\cdots\!76\)\( \nu^{9} + \)\(19\!\cdots\!96\)\( \nu^{8} - \)\(26\!\cdots\!24\)\( \nu^{7} - \)\(70\!\cdots\!68\)\( \nu^{6} + \)\(19\!\cdots\!76\)\( \nu^{5} - \)\(22\!\cdots\!92\)\( \nu^{4} - \)\(62\!\cdots\!16\)\( \nu^{3} + \)\(11\!\cdots\!88\)\( \nu^{2} + \)\(63\!\cdots\!52\)\( \nu - \)\(87\!\cdots\!08\)\(\)\()/ \)\(16\!\cdots\!20\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(42\!\cdots\!39\)\( \nu^{15} - \)\(40\!\cdots\!68\)\( \nu^{14} + \)\(58\!\cdots\!43\)\( \nu^{13} + \)\(52\!\cdots\!05\)\( \nu^{12} - \)\(31\!\cdots\!40\)\( \nu^{11} - \)\(25\!\cdots\!45\)\( \nu^{10} + \)\(84\!\cdots\!52\)\( \nu^{9} + \)\(58\!\cdots\!92\)\( \nu^{8} - \)\(11\!\cdots\!68\)\( \nu^{7} - \)\(61\!\cdots\!56\)\( \nu^{6} + \)\(77\!\cdots\!12\)\( \nu^{5} + \)\(21\!\cdots\!76\)\( \nu^{4} - \)\(22\!\cdots\!12\)\( \nu^{3} - \)\(87\!\cdots\!84\)\( \nu^{2} + \)\(20\!\cdots\!64\)\( \nu - \)\(20\!\cdots\!96\)\(\)\()/ \)\(80\!\cdots\!60\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(63\!\cdots\!26\)\( \nu^{15} - \)\(21\!\cdots\!27\)\( \nu^{14} + \)\(90\!\cdots\!92\)\( \nu^{13} + \)\(25\!\cdots\!85\)\( \nu^{12} - \)\(50\!\cdots\!85\)\( \nu^{11} - \)\(11\!\cdots\!80\)\( \nu^{10} + \)\(14\!\cdots\!83\)\( \nu^{9} + \)\(21\!\cdots\!18\)\( \nu^{8} - \)\(20\!\cdots\!32\)\( \nu^{7} - \)\(10\!\cdots\!84\)\( \nu^{6} + \)\(14\!\cdots\!88\)\( \nu^{5} - \)\(95\!\cdots\!36\)\( \nu^{4} - \)\(43\!\cdots\!48\)\( \nu^{3} + \)\(63\!\cdots\!84\)\( \nu^{2} + \)\(31\!\cdots\!16\)\( \nu - \)\(41\!\cdots\!04\)\(\)\()/ \)\(10\!\cdots\!20\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(13\!\cdots\!11\)\( \nu^{15} + \)\(26\!\cdots\!38\)\( \nu^{14} + \)\(20\!\cdots\!67\)\( \nu^{13} - \)\(42\!\cdots\!45\)\( \nu^{12} - \)\(11\!\cdots\!90\)\( \nu^{11} + \)\(26\!\cdots\!75\)\( \nu^{10} + \)\(35\!\cdots\!58\)\( \nu^{9} - \)\(82\!\cdots\!92\)\( \nu^{8} - \)\(57\!\cdots\!52\)\( \nu^{7} + \)\(13\!\cdots\!76\)\( \nu^{6} + \)\(46\!\cdots\!08\)\( \nu^{5} - \)\(10\!\cdots\!56\)\( \nu^{4} - \)\(15\!\cdots\!68\)\( \nu^{3} + \)\(32\!\cdots\!24\)\( \nu^{2} + \)\(12\!\cdots\!36\)\( \nu - \)\(22\!\cdots\!24\)\(\)\()/ \)\(20\!\cdots\!40\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(69\!\cdots\!33\)\( \nu^{15} - \)\(51\!\cdots\!59\)\( \nu^{14} - \)\(10\!\cdots\!11\)\( \nu^{13} + \)\(70\!\cdots\!20\)\( \nu^{12} + \)\(67\!\cdots\!15\)\( \nu^{11} - \)\(37\!\cdots\!35\)\( \nu^{10} - \)\(21\!\cdots\!09\)\( \nu^{9} + \)\(99\!\cdots\!06\)\( \nu^{8} + \)\(36\!\cdots\!36\)\( \nu^{7} - \)\(13\!\cdots\!08\)\( \nu^{6} - \)\(31\!\cdots\!64\)\( \nu^{5} + \)\(80\!\cdots\!68\)\( \nu^{4} + \)\(10\!\cdots\!84\)\( \nu^{3} - \)\(20\!\cdots\!32\)\( \nu^{2} - \)\(89\!\cdots\!08\)\( \nu + \)\(11\!\cdots\!92\)\(\)\()/ \)\(10\!\cdots\!20\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 189\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 298 \beta_{1} - 67\)
\(\nu^{4}\)\(=\)\(-\beta_{15} - 4 \beta_{14} + 4 \beta_{13} + 3 \beta_{12} - 3 \beta_{11} + 2 \beta_{10} + 3 \beta_{9} + 3 \beta_{8} - 2 \beta_{7} - \beta_{6} - \beta_{5} - 19 \beta_{4} - 2 \beta_{3} + 387 \beta_{2} - 19 \beta_{1} + 56410\)
\(\nu^{5}\)\(=\)\(8 \beta_{15} + 2 \beta_{14} + 20 \beta_{13} + 16 \beta_{12} + 20 \beta_{11} - 6 \beta_{10} + 40 \beta_{9} - 16 \beta_{8} - 19 \beta_{7} + 10 \beta_{6} - 58 \beta_{5} + 35 \beta_{4} + 518 \beta_{3} - 106 \beta_{2} + 97227 \beta_{1} - 29763\)
\(\nu^{6}\)\(=\)\(-322 \beta_{15} - 2056 \beta_{14} + 2752 \beta_{13} + 1738 \beta_{12} - 1718 \beta_{11} + 1388 \beta_{10} + 1654 \beta_{9} + 1698 \beta_{8} - 848 \beta_{7} - 398 \beta_{6} - 906 \beta_{5} - 12298 \beta_{4} - 907 \beta_{3} + 143616 \beta_{2} - 15668 \beta_{1} + 18428719\)
\(\nu^{7}\)\(=\)\(3715 \beta_{15} + 1540 \beta_{14} + 14628 \beta_{13} + 11923 \beta_{12} + 7041 \beta_{11} - 7546 \beta_{10} + 30515 \beta_{9} - 12781 \beta_{8} - 13624 \beta_{7} + 5243 \beta_{6} - 36341 \beta_{5} + 25687 \beta_{4} + 218800 \beta_{3} - 49769 \beta_{2} + 33214587 \beta_{1} - 12727306\)
\(\nu^{8}\)\(=\)\(-54326 \beta_{15} - 837682 \beta_{14} + 1420036 \beta_{13} + 734014 \beta_{12} - 710438 \beta_{11} + 722166 \beta_{10} + 738550 \beta_{9} + 771358 \beta_{8} - 286527 \beta_{7} - 66320 \beta_{6} - 512220 \beta_{5} - 6123319 \beta_{4} - 241240 \beta_{3} + 53275788 \beta_{2} - 6922547 \beta_{1} + 6300524291\)
\(\nu^{9}\)\(=\)\(1081644 \beta_{15} + 1020800 \beta_{14} + 7884360 \beta_{13} + 6213940 \beta_{12} + 1058156 \beta_{11} - 5009772 \beta_{10} + 17333224 \beta_{9} - 7295620 \beta_{8} - 7331266 \beta_{7} + 1834072 \beta_{6} - 16583580 \beta_{5} + 13533134 \beta_{4} + 86669521 \beta_{3} - 16175238 \beta_{2} + 11689751028 \beta_{1} - 4947659831\)
\(\nu^{10}\)\(=\)\(9178535 \beta_{15} - 317171524 \beta_{14} + 657958388 \beta_{13} + 275901219 \beta_{12} - 259385355 \beta_{11} + 336047130 \beta_{10} + 308264147 \beta_{9} + 325066915 \beta_{8} - 83949594 \beta_{7} + 19910823 \beta_{6} - 243727129 \beta_{5} - 2768075571 \beta_{4} - 29133994 \beta_{3} + 19841524915 \beta_{2} - 2072757707 \beta_{1} + 2218326673922\)
\(\nu^{11}\)\(=\)\(192171720 \beta_{15} + 611574026 \beta_{14} + 3800835556 \beta_{13} + 2776451832 \beta_{12} - 369416716 \beta_{11} - 2662505174 \beta_{10} + 8697745536 \beta_{9} - 3626096648 \beta_{8} - 3496352747 \beta_{7} + 487732514 \beta_{6} - 6729436450 \beta_{5} + 6493743835 \beta_{4} + 33456096830 \beta_{3} - 4025081962 \beta_{2} + 4198394323443 \beta_{1} - 1752912931667\)
\(\nu^{12}\)\(=\)\(14484661278 \beta_{15} - 116815310624 \beta_{14} + 289230246224 \beta_{13} + 97765146802 \beta_{12} - 88736673590 \beta_{11} + 147941094428 \beta_{10} + 124943198350 \beta_{9} + 132108919402 \beta_{8} - 20221237448 \beta_{7} + 25572353802 \beta_{6} - 106658166258 \beta_{5} - 1191879523154 \beta_{4} + 14768573773 \beta_{3} + 7420591546960 \beta_{2} - 311700004700 \beta_{1} + 796832833506959\)
\(\nu^{13}\)\(=\)\(-21120806365 \beta_{15} + 332609278476 \beta_{14} + 1734027120372 \beta_{13} + 1141712395035 \beta_{12} - 429566174527 \beta_{11} - 1267333743370 \beta_{10} + 4078194850923 \beta_{9} - 1669184390405 \beta_{8} - 1559118630512 \beta_{7} + 75884758643 \beta_{6} - 2580600494013 \beta_{5} + 3001940238351 \beta_{4} + 12776997190808 \beta_{3} - 574272081961 \beta_{2} + 1529257432641459 \beta_{1} - 570094464406250\)
\(\nu^{14}\)\(=\)\(9321682989434 \beta_{15} - 42615689960586 \beta_{14} + 123413715766356 \beta_{13} + 33342192214390 \beta_{12} - 29070197630774 \beta_{11} + 63088856352646 \beta_{10} + 49897316740638 \beta_{9} + 52566842529846 \beta_{8} - 2844933629239 \beta_{7} + 16261067736600 \beta_{6} - 44553716705652 \beta_{5} - 498779313344495 \beta_{4} + 15193934723680 \beta_{3} + 2785776883737756 \beta_{2} + 124691688780213 \beta_{1} + 290243769849439443\)
\(\nu^{15}\)\(=\)\(-43827714804276 \beta_{15} + 167484351779752 \beta_{14} + 765851434829912 \beta_{13} + 446629910493564 \beta_{12} - 262328813562644 \beta_{11} - 565726399182620 \beta_{10} + 1833434032720608 \beta_{9} - 731877362581724 \beta_{8} - 666505217257930 \beta_{7} - 18705578790416 \beta_{6} - 959151595388804 \beta_{5} + 1355259616083414 \beta_{4} + 4860395355063145 \beta_{3} + 152225579361114 \beta_{2} + 562659361261144860 \beta_{1} - 169125649025696791\)

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
−19.6562
−18.7189
−17.9442
−14.0604
−13.0039
−7.02227
−3.09726
1.05882
1.97136
4.55626
7.00808
14.7989
15.0467
16.2952
19.0314
19.7363
−19.6562 0 258.365 436.761 0 956.841 −2562.48 0 −8585.06
1.2 −18.7189 0 222.398 −443.832 0 −1695.79 −1767.03 0 8308.06
1.3 −17.9442 0 193.993 −1.22320 0 −719.259 −1184.20 0 21.9494
1.4 −14.0604 0 69.6949 −153.219 0 −215.221 819.793 0 2154.32
1.5 −13.0039 0 41.1009 −167.303 0 887.373 1130.03 0 2175.59
1.6 −7.02227 0 −78.6878 266.773 0 665.758 1451.42 0 −1873.35
1.7 −3.09726 0 −118.407 156.435 0 11.3597 763.187 0 −484.520
1.8 1.05882 0 −126.879 −151.597 0 −1574.54 −269.870 0 −160.513
1.9 1.97136 0 −124.114 339.775 0 364.700 −497.007 0 669.820
1.10 4.55626 0 −107.241 540.445 0 −1238.10 −1071.82 0 2462.40
1.11 7.00808 0 −78.8868 −449.079 0 −271.717 −1449.88 0 −3147.18
1.12 14.7989 0 91.0084 296.536 0 −1410.76 −547.437 0 4388.42
1.13 15.0467 0 98.4026 −159.890 0 980.332 −445.343 0 −2405.81
1.14 16.2952 0 137.534 −495.569 0 565.301 155.363 0 −8075.41
1.15 19.0314 0 234.195 −37.6075 0 1158.54 2021.05 0 −715.724
1.16 19.7363 0 261.522 90.5952 0 −807.818 2635.23 0 1788.01
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(59\) \(1\)

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 531.8.a.b 16
3.b odd 2 1 177.8.a.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.8.a.a 16 3.b odd 2 1
531.8.a.b 16 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(33\!\cdots\!40\)\( T_{2}^{6} + \)\(16\!\cdots\!00\)\( T_{2}^{5} + \)\(99\!\cdots\!00\)\( T_{2}^{4} - \)\(49\!\cdots\!68\)\( T_{2}^{3} - \)\(49\!\cdots\!24\)\( T_{2}^{2} + \)\(31\!\cdots\!12\)\( T_{2} - \)\(23\!\cdots\!32\)\( \)">\(T_{2}^{16} - \cdots\) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(531))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2385018853548032 + 3193975642099712 T - 493048066650624 T^{2} - 494039551757568 T^{3} + 99470801612800 T^{4} + 16796366777600 T^{5} - 3353576629440 T^{6} - 226132003872 T^{7} + 42885295136 T^{8} + 1488374176 T^{9} - 269092298 T^{10} - 5107725 T^{11} + 890490 T^{12} + 8791 T^{13} - 1493 T^{14} - 6 T^{15} + T^{16} \)
$3$ \( T^{16} \)
$5$ \( -\)\(25\!\cdots\!00\)\( - \)\(21\!\cdots\!00\)\( T - \)\(55\!\cdots\!00\)\( T^{2} + \)\(39\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!50\)\( T^{4} - \)\(11\!\cdots\!50\)\( T^{5} - \)\(71\!\cdots\!25\)\( T^{6} - \)\(30\!\cdots\!20\)\( T^{7} + \)\(20\!\cdots\!29\)\( T^{8} + 1276883506994508700 T^{9} - 30163601555694730 T^{10} - 12282106911840 T^{11} + 220666998180 T^{12} + 47355522 T^{13} - 765561 T^{14} - 68 T^{15} + T^{16} \)
$7$ \( \)\(23\!\cdots\!80\)\( - \)\(20\!\cdots\!76\)\( T - \)\(81\!\cdots\!16\)\( T^{2} + \)\(45\!\cdots\!48\)\( T^{3} + \)\(12\!\cdots\!53\)\( T^{4} - \)\(40\!\cdots\!35\)\( T^{5} - \)\(52\!\cdots\!76\)\( T^{6} + \)\(15\!\cdots\!17\)\( T^{7} + \)\(10\!\cdots\!28\)\( T^{8} - \)\(27\!\cdots\!71\)\( T^{9} - 13028951743237034522 T^{10} + 25724670378028207 T^{11} + 10148751808072 T^{12} - 12238111473 T^{13} - 4790292 T^{14} + 2343 T^{15} + T^{16} \)
$11$ \( -\)\(51\!\cdots\!72\)\( - \)\(57\!\cdots\!12\)\( T + \)\(72\!\cdots\!08\)\( T^{2} + \)\(11\!\cdots\!00\)\( T^{3} + \)\(63\!\cdots\!83\)\( T^{4} - \)\(40\!\cdots\!30\)\( T^{5} - \)\(99\!\cdots\!68\)\( T^{6} + \)\(62\!\cdots\!82\)\( T^{7} + \)\(20\!\cdots\!35\)\( T^{8} - \)\(46\!\cdots\!60\)\( T^{9} - \)\(18\!\cdots\!28\)\( T^{10} + 16194688134834128320 T^{11} + 7779298719542601 T^{12} - 220257477422 T^{13} - 148053196 T^{14} + 898 T^{15} + T^{16} \)
$13$ \( \)\(25\!\cdots\!00\)\( - \)\(31\!\cdots\!80\)\( T - \)\(70\!\cdots\!90\)\( T^{2} + \)\(97\!\cdots\!14\)\( T^{3} + \)\(15\!\cdots\!63\)\( T^{4} - \)\(83\!\cdots\!66\)\( T^{5} - \)\(13\!\cdots\!18\)\( T^{6} + \)\(31\!\cdots\!20\)\( T^{7} + \)\(49\!\cdots\!63\)\( T^{8} - \)\(63\!\cdots\!28\)\( T^{9} - \)\(93\!\cdots\!98\)\( T^{10} + \)\(67\!\cdots\!06\)\( T^{11} + 95656696630982041 T^{12} - 3704745067846 T^{13} - 492040454 T^{14} + 8172 T^{15} + T^{16} \)
$17$ \( -\)\(41\!\cdots\!40\)\( - \)\(11\!\cdots\!12\)\( T + \)\(74\!\cdots\!92\)\( T^{2} + \)\(10\!\cdots\!88\)\( T^{3} - \)\(52\!\cdots\!27\)\( T^{4} - \)\(97\!\cdots\!33\)\( T^{5} + \)\(14\!\cdots\!94\)\( T^{6} - \)\(74\!\cdots\!67\)\( T^{7} - \)\(11\!\cdots\!92\)\( T^{8} + \)\(92\!\cdots\!87\)\( T^{9} + \)\(28\!\cdots\!08\)\( T^{10} - \)\(39\!\cdots\!49\)\( T^{11} + 45590257642588608 T^{12} + 71095321312497 T^{13} - 1070737138 T^{14} - 44985 T^{15} + T^{16} \)
$19$ \( -\)\(23\!\cdots\!20\)\( - \)\(64\!\cdots\!60\)\( T + \)\(36\!\cdots\!76\)\( T^{2} + \)\(16\!\cdots\!20\)\( T^{3} + \)\(79\!\cdots\!00\)\( T^{4} - \)\(15\!\cdots\!20\)\( T^{5} - \)\(18\!\cdots\!13\)\( T^{6} + \)\(57\!\cdots\!89\)\( T^{7} + \)\(10\!\cdots\!35\)\( T^{8} - \)\(10\!\cdots\!70\)\( T^{9} - \)\(21\!\cdots\!09\)\( T^{10} + \)\(86\!\cdots\!59\)\( T^{11} + 19674435440125746613 T^{12} - 312748284754822 T^{13} - 7492545475 T^{14} + 40137 T^{15} + T^{16} \)
$23$ \( \)\(16\!\cdots\!80\)\( - \)\(29\!\cdots\!00\)\( T + \)\(17\!\cdots\!36\)\( T^{2} - \)\(10\!\cdots\!36\)\( T^{3} - \)\(25\!\cdots\!32\)\( T^{4} + \)\(82\!\cdots\!52\)\( T^{5} + \)\(56\!\cdots\!03\)\( T^{6} - \)\(61\!\cdots\!17\)\( T^{7} + \)\(50\!\cdots\!91\)\( T^{8} + \)\(16\!\cdots\!70\)\( T^{9} - \)\(24\!\cdots\!69\)\( T^{10} - \)\(18\!\cdots\!95\)\( T^{11} + \)\(43\!\cdots\!05\)\( T^{12} + 792812076231546 T^{13} - 33662419967 T^{14} - 2833 T^{15} + T^{16} \)
$29$ \( -\)\(67\!\cdots\!20\)\( - \)\(15\!\cdots\!92\)\( T - \)\(14\!\cdots\!08\)\( T^{2} - \)\(62\!\cdots\!16\)\( T^{3} - \)\(13\!\cdots\!94\)\( T^{4} - \)\(83\!\cdots\!96\)\( T^{5} + \)\(18\!\cdots\!41\)\( T^{6} + \)\(31\!\cdots\!31\)\( T^{7} + \)\(39\!\cdots\!69\)\( T^{8} - \)\(27\!\cdots\!30\)\( T^{9} - \)\(10\!\cdots\!63\)\( T^{10} + \)\(11\!\cdots\!85\)\( T^{11} + \)\(60\!\cdots\!51\)\( T^{12} - 20469385938299166 T^{13} - 131675926081 T^{14} + 144375 T^{15} + T^{16} \)
$31$ \( \)\(10\!\cdots\!60\)\( - \)\(10\!\cdots\!72\)\( T - \)\(21\!\cdots\!48\)\( T^{2} + \)\(14\!\cdots\!76\)\( T^{3} + \)\(15\!\cdots\!06\)\( T^{4} - \)\(65\!\cdots\!48\)\( T^{5} - \)\(54\!\cdots\!95\)\( T^{6} + \)\(13\!\cdots\!51\)\( T^{7} + \)\(97\!\cdots\!65\)\( T^{8} - \)\(14\!\cdots\!02\)\( T^{9} - \)\(95\!\cdots\!55\)\( T^{10} + \)\(81\!\cdots\!57\)\( T^{11} + \)\(51\!\cdots\!43\)\( T^{12} - 20197683250020434 T^{13} - 128454996345 T^{14} + 141759 T^{15} + T^{16} \)
$37$ \( -\)\(50\!\cdots\!84\)\( - \)\(10\!\cdots\!20\)\( T + \)\(65\!\cdots\!34\)\( T^{2} + \)\(16\!\cdots\!16\)\( T^{3} + \)\(31\!\cdots\!59\)\( T^{4} - \)\(77\!\cdots\!87\)\( T^{5} - \)\(24\!\cdots\!54\)\( T^{6} + \)\(15\!\cdots\!89\)\( T^{7} + \)\(53\!\cdots\!86\)\( T^{8} - \)\(15\!\cdots\!95\)\( T^{9} - \)\(56\!\cdots\!12\)\( T^{10} + \)\(85\!\cdots\!39\)\( T^{11} + \)\(31\!\cdots\!66\)\( T^{12} - 250272440202312489 T^{13} - 878197839972 T^{14} + 297971 T^{15} + T^{16} \)
$41$ \( -\)\(45\!\cdots\!00\)\( + \)\(15\!\cdots\!20\)\( T + \)\(37\!\cdots\!44\)\( T^{2} + \)\(13\!\cdots\!04\)\( T^{3} - \)\(13\!\cdots\!15\)\( T^{4} - \)\(86\!\cdots\!71\)\( T^{5} + \)\(89\!\cdots\!62\)\( T^{6} + \)\(14\!\cdots\!87\)\( T^{7} + \)\(80\!\cdots\!24\)\( T^{8} - \)\(11\!\cdots\!83\)\( T^{9} - \)\(12\!\cdots\!92\)\( T^{10} + \)\(44\!\cdots\!25\)\( T^{11} + \)\(61\!\cdots\!32\)\( T^{12} - 858512390358027885 T^{13} - 1278569786638 T^{14} + 659077 T^{15} + T^{16} \)
$43$ \( -\)\(10\!\cdots\!08\)\( - \)\(85\!\cdots\!12\)\( T + \)\(37\!\cdots\!48\)\( T^{2} + \)\(33\!\cdots\!68\)\( T^{3} - \)\(48\!\cdots\!83\)\( T^{4} - \)\(47\!\cdots\!16\)\( T^{5} + \)\(25\!\cdots\!86\)\( T^{6} + \)\(33\!\cdots\!04\)\( T^{7} - \)\(20\!\cdots\!01\)\( T^{8} - \)\(13\!\cdots\!76\)\( T^{9} - \)\(30\!\cdots\!32\)\( T^{10} + \)\(28\!\cdots\!16\)\( T^{11} + \)\(12\!\cdots\!87\)\( T^{12} - 3138661091232654520 T^{13} - 1825919654306 T^{14} + 1431608 T^{15} + T^{16} \)
$47$ \( -\)\(82\!\cdots\!24\)\( + \)\(18\!\cdots\!28\)\( T + \)\(48\!\cdots\!56\)\( T^{2} - \)\(26\!\cdots\!20\)\( T^{3} - \)\(18\!\cdots\!68\)\( T^{4} + \)\(22\!\cdots\!24\)\( T^{5} - \)\(35\!\cdots\!39\)\( T^{6} - \)\(77\!\cdots\!97\)\( T^{7} + \)\(31\!\cdots\!67\)\( T^{8} + \)\(14\!\cdots\!06\)\( T^{9} - \)\(73\!\cdots\!79\)\( T^{10} - \)\(14\!\cdots\!27\)\( T^{11} + \)\(82\!\cdots\!37\)\( T^{12} + 7529304337576894050 T^{13} - 4570866681065 T^{14} - 1574073 T^{15} + T^{16} \)
$53$ \( \)\(21\!\cdots\!48\)\( + \)\(10\!\cdots\!16\)\( T + \)\(12\!\cdots\!16\)\( T^{2} + \)\(33\!\cdots\!28\)\( T^{3} - \)\(13\!\cdots\!62\)\( T^{4} - \)\(61\!\cdots\!10\)\( T^{5} - \)\(51\!\cdots\!17\)\( T^{6} + \)\(22\!\cdots\!32\)\( T^{7} + \)\(14\!\cdots\!53\)\( T^{8} - \)\(32\!\cdots\!20\)\( T^{9} - \)\(29\!\cdots\!22\)\( T^{10} + \)\(21\!\cdots\!44\)\( T^{11} + \)\(23\!\cdots\!36\)\( T^{12} - 6276448053670807446 T^{13} - 8103424856601 T^{14} + 587736 T^{15} + T^{16} \)
$59$ \( ( 205379 + T )^{16} \)
$61$ \( -\)\(45\!\cdots\!36\)\( + \)\(15\!\cdots\!72\)\( T + \)\(11\!\cdots\!72\)\( T^{2} - \)\(11\!\cdots\!24\)\( T^{3} - \)\(65\!\cdots\!86\)\( T^{4} + \)\(39\!\cdots\!92\)\( T^{5} + \)\(16\!\cdots\!41\)\( T^{6} - \)\(32\!\cdots\!85\)\( T^{7} - \)\(16\!\cdots\!11\)\( T^{8} - \)\(49\!\cdots\!50\)\( T^{9} + \)\(69\!\cdots\!45\)\( T^{10} + \)\(35\!\cdots\!65\)\( T^{11} - \)\(98\!\cdots\!93\)\( T^{12} - 82817544807912341990 T^{13} - 2914515679197 T^{14} + 6117131 T^{15} + T^{16} \)
$67$ \( \)\(35\!\cdots\!68\)\( + \)\(71\!\cdots\!84\)\( T + \)\(20\!\cdots\!68\)\( T^{2} + \)\(63\!\cdots\!12\)\( T^{3} - \)\(49\!\cdots\!56\)\( T^{4} - \)\(75\!\cdots\!68\)\( T^{5} - \)\(24\!\cdots\!95\)\( T^{6} + \)\(32\!\cdots\!94\)\( T^{7} + \)\(33\!\cdots\!05\)\( T^{8} + \)\(72\!\cdots\!28\)\( T^{9} - \)\(56\!\cdots\!86\)\( T^{10} - \)\(38\!\cdots\!12\)\( T^{11} - \)\(64\!\cdots\!50\)\( T^{12} + \)\(18\!\cdots\!84\)\( T^{13} + 99098156876413 T^{14} + 16518710 T^{15} + T^{16} \)
$71$ \( -\)\(74\!\cdots\!48\)\( + \)\(11\!\cdots\!72\)\( T + \)\(61\!\cdots\!22\)\( T^{2} - \)\(10\!\cdots\!74\)\( T^{3} - \)\(20\!\cdots\!63\)\( T^{4} + \)\(35\!\cdots\!88\)\( T^{5} + \)\(33\!\cdots\!28\)\( T^{6} - \)\(60\!\cdots\!70\)\( T^{7} - \)\(24\!\cdots\!45\)\( T^{8} + \)\(57\!\cdots\!56\)\( T^{9} + \)\(26\!\cdots\!26\)\( T^{10} - \)\(30\!\cdots\!98\)\( T^{11} + \)\(10\!\cdots\!07\)\( T^{12} + \)\(89\!\cdots\!20\)\( T^{13} - 58727757650844 T^{14} - 10882582 T^{15} + T^{16} \)
$73$ \( -\)\(69\!\cdots\!00\)\( - \)\(88\!\cdots\!60\)\( T + \)\(29\!\cdots\!56\)\( T^{2} + \)\(58\!\cdots\!88\)\( T^{3} + \)\(25\!\cdots\!92\)\( T^{4} - \)\(62\!\cdots\!12\)\( T^{5} - \)\(43\!\cdots\!11\)\( T^{6} + \)\(45\!\cdots\!17\)\( T^{7} + \)\(16\!\cdots\!27\)\( T^{8} + \)\(49\!\cdots\!54\)\( T^{9} - \)\(12\!\cdots\!99\)\( T^{10} - \)\(10\!\cdots\!77\)\( T^{11} - \)\(14\!\cdots\!75\)\( T^{12} + \)\(38\!\cdots\!74\)\( T^{13} + 160731807982939 T^{14} + 21097441 T^{15} + T^{16} \)
$79$ \( -\)\(49\!\cdots\!16\)\( - \)\(85\!\cdots\!68\)\( T - \)\(25\!\cdots\!88\)\( T^{2} + \)\(58\!\cdots\!84\)\( T^{3} + \)\(16\!\cdots\!67\)\( T^{4} - \)\(13\!\cdots\!58\)\( T^{5} - \)\(48\!\cdots\!88\)\( T^{6} + \)\(15\!\cdots\!02\)\( T^{7} + \)\(57\!\cdots\!55\)\( T^{8} - \)\(89\!\cdots\!04\)\( T^{9} - \)\(34\!\cdots\!12\)\( T^{10} + \)\(29\!\cdots\!08\)\( T^{11} + \)\(10\!\cdots\!81\)\( T^{12} - \)\(52\!\cdots\!18\)\( T^{13} - 164311559623024 T^{14} + 3784458 T^{15} + T^{16} \)
$83$ \( -\)\(91\!\cdots\!72\)\( + \)\(17\!\cdots\!56\)\( T - \)\(66\!\cdots\!44\)\( T^{2} - \)\(81\!\cdots\!48\)\( T^{3} + \)\(14\!\cdots\!07\)\( T^{4} + \)\(13\!\cdots\!51\)\( T^{5} - \)\(30\!\cdots\!24\)\( T^{6} - \)\(11\!\cdots\!87\)\( T^{7} + \)\(27\!\cdots\!30\)\( T^{8} + \)\(43\!\cdots\!03\)\( T^{9} - \)\(11\!\cdots\!98\)\( T^{10} - \)\(80\!\cdots\!81\)\( T^{11} + \)\(25\!\cdots\!82\)\( T^{12} + \)\(67\!\cdots\!61\)\( T^{13} - 256440516265952 T^{14} - 1951425 T^{15} + T^{16} \)
$89$ \( \)\(38\!\cdots\!60\)\( - \)\(54\!\cdots\!28\)\( T - \)\(45\!\cdots\!48\)\( T^{2} + \)\(25\!\cdots\!56\)\( T^{3} + \)\(10\!\cdots\!24\)\( T^{4} - \)\(39\!\cdots\!16\)\( T^{5} - \)\(11\!\cdots\!05\)\( T^{6} + \)\(29\!\cdots\!39\)\( T^{7} + \)\(61\!\cdots\!75\)\( T^{8} - \)\(11\!\cdots\!66\)\( T^{9} - \)\(18\!\cdots\!89\)\( T^{10} + \)\(25\!\cdots\!29\)\( T^{11} + \)\(31\!\cdots\!09\)\( T^{12} - \)\(26\!\cdots\!90\)\( T^{13} - 281604197712695 T^{14} + 10499443 T^{15} + T^{16} \)
$97$ \( \)\(16\!\cdots\!80\)\( + \)\(71\!\cdots\!20\)\( T - \)\(40\!\cdots\!24\)\( T^{2} - \)\(65\!\cdots\!60\)\( T^{3} - \)\(24\!\cdots\!92\)\( T^{4} + \)\(39\!\cdots\!36\)\( T^{5} + \)\(34\!\cdots\!31\)\( T^{6} + \)\(23\!\cdots\!80\)\( T^{7} - \)\(11\!\cdots\!91\)\( T^{8} - \)\(16\!\cdots\!40\)\( T^{9} + \)\(10\!\cdots\!38\)\( T^{10} + \)\(24\!\cdots\!16\)\( T^{11} + \)\(67\!\cdots\!02\)\( T^{12} - \)\(13\!\cdots\!12\)\( T^{13} - 372261609039309 T^{14} + 25158976 T^{15} + T^{16} \)
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