Properties

Label 29.10.a.b
Level $29$
Weight $10$
Character orbit 29.a
Self dual yes
Analytic conductor $14.936$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 29.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.9360392488\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 4 x^{11} - 4803 x^{10} + 14952 x^{9} + 8248476 x^{8} - 14809944 x^{7} - 6122244486 x^{6} - 1026481216 x^{5} + 1899055440249 x^{4} + 3992697832188 x^{3} - 174144148490503 x^{2} - 282853247198664 x + 4071614738753338\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{1} ) q^{2} + ( 20 + \beta_{1} - \beta_{3} ) q^{3} + ( 291 + 3 \beta_{1} + \beta_{2} ) q^{4} + ( 149 - 4 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{7} ) q^{5} + ( 451 + 15 \beta_{1} + 2 \beta_{2} - 10 \beta_{3} + \beta_{5} ) q^{6} + ( 984 + 73 \beta_{1} - 8 \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} ) q^{7} + ( 1996 + 364 \beta_{1} - \beta_{2} - 23 \beta_{3} - \beta_{4} - 2 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} + 3 \beta_{8} + \beta_{9} + 3 \beta_{10} + \beta_{11} ) q^{8} + ( 3503 + 261 \beta_{1} - 9 \beta_{2} - 35 \beta_{3} + 5 \beta_{4} - 5 \beta_{5} + 2 \beta_{7} - 5 \beta_{9} - \beta_{10} - 4 \beta_{11} ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{1} ) q^{2} + ( 20 + \beta_{1} - \beta_{3} ) q^{3} + ( 291 + 3 \beta_{1} + \beta_{2} ) q^{4} + ( 149 - 4 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{7} ) q^{5} + ( 451 + 15 \beta_{1} + 2 \beta_{2} - 10 \beta_{3} + \beta_{5} ) q^{6} + ( 984 + 73 \beta_{1} - 8 \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} ) q^{7} + ( 1996 + 364 \beta_{1} - \beta_{2} - 23 \beta_{3} - \beta_{4} - 2 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} + 3 \beta_{8} + \beta_{9} + 3 \beta_{10} + \beta_{11} ) q^{8} + ( 3503 + 261 \beta_{1} - 9 \beta_{2} - 35 \beta_{3} + 5 \beta_{4} - 5 \beta_{5} + 2 \beta_{7} - 5 \beta_{9} - \beta_{10} - 4 \beta_{11} ) q^{9} + ( -4099 + 594 \beta_{1} - 16 \beta_{2} - 26 \beta_{3} - 7 \beta_{4} + 7 \beta_{5} - 3 \beta_{6} + 6 \beta_{7} - 5 \beta_{8} + 8 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{10} + ( 1939 + 272 \beta_{1} - 13 \beta_{2} + 15 \beta_{3} - \beta_{4} + 6 \beta_{5} - 9 \beta_{6} - 9 \beta_{7} + 5 \beta_{8} - 6 \beta_{9} - 13 \beta_{10} + 4 \beta_{11} ) q^{11} + ( -1550 + 1150 \beta_{1} - 2 \beta_{2} - 99 \beta_{3} + 8 \beta_{4} + 15 \beta_{5} + 18 \beta_{6} + 2 \beta_{7} - 11 \beta_{8} + 7 \beta_{9} + 7 \beta_{10} + 12 \beta_{11} ) q^{12} + ( 8726 + 642 \beta_{1} - 18 \beta_{2} + 151 \beta_{3} - 6 \beta_{4} - 5 \beta_{5} - 2 \beta_{6} - 18 \beta_{7} - 9 \beta_{8} + 5 \beta_{9} - 4 \beta_{10} - 21 \beta_{11} ) q^{13} + ( 56104 + 815 \beta_{1} + 17 \beta_{2} + 593 \beta_{3} - 2 \beta_{4} - 18 \beta_{5} - 36 \beta_{6} - 29 \beta_{7} + 48 \beta_{8} - 27 \beta_{9} + 11 \beta_{10} - 18 \beta_{11} ) q^{14} + ( 41753 + 993 \beta_{1} - 37 \beta_{2} + 187 \beta_{3} + 23 \beta_{4} - 33 \beta_{5} + 18 \beta_{6} - 48 \beta_{7} - 24 \beta_{8} - 4 \beta_{9} + 34 \beta_{10} + 20 \beta_{11} ) q^{15} + ( 137707 + 1446 \beta_{1} + 284 \beta_{2} + 415 \beta_{3} - 26 \beta_{4} + 12 \beta_{5} + 80 \beta_{6} + 59 \beta_{7} + 10 \beta_{8} + 7 \beta_{9} - 35 \beta_{10} - 18 \beta_{11} ) q^{16} + ( 81389 + 1419 \beta_{1} + 10 \beta_{2} + 83 \beta_{3} - \beta_{4} - 25 \beta_{5} - 2 \beta_{6} + 39 \beta_{7} + 2 \beta_{8} + 59 \beta_{9} + 9 \beta_{10} + 64 \beta_{11} ) q^{17} + ( 197626 - 1835 \beta_{1} + 145 \beta_{2} + 304 \beta_{3} + 74 \beta_{4} + 8 \beta_{5} - 76 \beta_{6} + 151 \beta_{7} - 27 \beta_{8} - 20 \beta_{9} - 20 \beta_{10} - 5 \beta_{11} ) q^{18} + ( 176037 - 7367 \beta_{1} - 113 \beta_{2} - 174 \beta_{3} - 119 \beta_{4} + 46 \beta_{5} - 67 \beta_{6} - 29 \beta_{7} - 59 \beta_{8} + 38 \beta_{9} - 27 \beta_{10} - 10 \beta_{11} ) q^{19} + ( 394049 - 9755 \beta_{1} + 239 \beta_{2} + 418 \beta_{3} - 68 \beta_{4} + 64 \beta_{5} + 4 \beta_{6} - 96 \beta_{7} + 66 \beta_{8} - 134 \beta_{9} - 50 \beta_{10} + 14 \beta_{11} ) q^{20} + ( 247757 - 14663 \beta_{1} - 380 \beta_{2} - 2919 \beta_{3} + 27 \beta_{4} + 83 \beta_{5} + 32 \beta_{6} - 61 \beta_{7} + 18 \beta_{8} - 59 \beta_{9} + 3 \beta_{10} - 86 \beta_{11} ) q^{21} + ( 228499 - 4716 \beta_{1} - 410 \beta_{2} - 771 \beta_{3} + 209 \beta_{4} - 84 \beta_{5} - 303 \beta_{6} + 402 \beta_{7} - 210 \beta_{8} + 173 \beta_{9} + 85 \beta_{10} - 63 \beta_{11} ) q^{22} + ( 253124 - 5929 \beta_{1} - 562 \beta_{2} - 5222 \beta_{3} + 114 \beta_{4} - 115 \beta_{5} + 341 \beta_{6} - 5 \beta_{7} + 239 \beta_{8} + 124 \beta_{9} + 167 \beta_{10} - 110 \beta_{11} ) q^{23} + ( 662383 - 10942 \beta_{1} + 520 \beta_{2} - 2533 \beta_{3} + 50 \beta_{4} + 72 \beta_{5} + 596 \beta_{6} - 209 \beta_{7} + 22 \beta_{8} - 181 \beta_{9} - 71 \beta_{10} + 10 \beta_{11} ) q^{24} + ( 537862 - 29581 \beta_{1} - 836 \beta_{2} - 1650 \beta_{3} - 347 \beta_{4} + 144 \beta_{5} - 130 \beta_{6} - 849 \beta_{7} + 79 \beta_{8} - 192 \beta_{9} - 53 \beta_{10} + 567 \beta_{11} ) q^{25} + ( 571957 - 2829 \beta_{1} - 64 \beta_{2} + 5073 \beta_{3} - 350 \beta_{4} - 118 \beta_{5} - 98 \beta_{6} - 724 \beta_{7} + 57 \beta_{8} + 73 \beta_{9} + 75 \beta_{10} + 235 \beta_{11} ) q^{26} + ( 662503 - 16424 \beta_{1} - 603 \beta_{2} - 3423 \beta_{3} + 257 \beta_{4} - 144 \beta_{5} - 315 \beta_{6} + 417 \beta_{7} + 379 \beta_{8} - 120 \beta_{9} - 167 \beta_{10} - 712 \beta_{11} ) q^{27} + ( 410314 + 43338 \beta_{1} + 36 \beta_{2} + 9650 \beta_{3} + 242 \beta_{4} - 664 \beta_{5} - 838 \beta_{6} + 1122 \beta_{7} - 754 \beta_{8} + 810 \beta_{9} + 174 \beta_{10} + 134 \beta_{11} ) q^{28} + 707281 q^{29} + ( 905645 + 8139 \beta_{1} + 1639 \beta_{2} + 12888 \beta_{3} - 663 \beta_{4} + 750 \beta_{5} - 497 \beta_{6} - 355 \beta_{7} - 178 \beta_{8} - 348 \beta_{9} - 606 \beta_{10} - 419 \beta_{11} ) q^{30} + ( 1491772 - 1906 \beta_{1} + 2778 \beta_{2} - 533 \beta_{3} + 400 \beta_{4} + 375 \beta_{5} + 635 \beta_{6} + 767 \beta_{7} - 613 \beta_{8} - 606 \beta_{9} - 459 \beta_{10} - 24 \beta_{11} ) q^{31} + ( 385038 + 131788 \beta_{1} + 3719 \beta_{2} + 6915 \beta_{3} + 115 \beta_{4} - 118 \beta_{5} + 2795 \beta_{6} - 127 \beta_{7} + 123 \beta_{8} + 435 \beta_{9} + 585 \beta_{10} + 317 \beta_{11} ) q^{32} + ( -83539 + 35477 \beta_{1} - 1634 \beta_{2} - 8880 \beta_{3} + 1139 \beta_{4} + 602 \beta_{5} - 984 \beta_{6} + 1837 \beta_{7} - 361 \beta_{8} - 174 \beta_{9} + 589 \beta_{10} + 745 \beta_{11} ) q^{33} + ( 1272534 + 90040 \beta_{1} + 3817 \beta_{2} + 27824 \beta_{3} - 235 \beta_{4} - 543 \beta_{5} - 893 \beta_{6} - 119 \beta_{7} + 2384 \beta_{8} - 1470 \beta_{9} - 796 \beta_{10} - 1549 \beta_{11} ) q^{34} + ( -1880334 + 75539 \beta_{1} - 3152 \beta_{2} - 6762 \beta_{3} + 566 \beta_{4} - 547 \beta_{5} + 319 \beta_{6} - 313 \beta_{7} - 773 \beta_{8} + 1222 \beta_{9} + 2173 \beta_{10} + 264 \beta_{11} ) q^{35} + ( -2987622 + 152648 \beta_{1} - 3772 \beta_{2} + 4446 \beta_{3} - 188 \beta_{4} + 674 \beta_{5} - 1708 \beta_{6} - 342 \beta_{7} + 1856 \beta_{8} + 338 \beta_{9} + 298 \beta_{10} + 1538 \beta_{11} ) q^{36} + ( 48918 - 20606 \beta_{1} + 5862 \beta_{2} - 4472 \beta_{3} - 2092 \beta_{4} + 1050 \beta_{5} + 812 \beta_{6} + 1538 \beta_{7} - 38 \beta_{8} + 392 \beta_{9} - 2622 \beta_{10} + 426 \beta_{11} ) q^{37} + ( -5759238 + 89393 \beta_{1} - 8464 \beta_{2} + 10557 \beta_{3} - 1749 \beta_{4} + 23 \beta_{5} - 773 \beta_{6} - 6514 \beta_{7} - 234 \beta_{8} + 251 \beta_{9} + 955 \beta_{10} + 679 \beta_{11} ) q^{38} + ( -2421914 - 112466 \beta_{1} - 3882 \beta_{2} - 24671 \beta_{3} - 2356 \beta_{4} - 445 \beta_{5} + 413 \beta_{6} - 655 \beta_{7} - 531 \beta_{8} + 96 \beta_{9} - 185 \beta_{10} - 3312 \beta_{11} ) q^{39} + ( -5203524 + 226184 \beta_{1} - 2679 \beta_{2} - 25985 \beta_{3} + 3893 \beta_{4} - 4002 \beta_{5} + 4049 \beta_{6} + 1181 \beta_{7} - 2287 \beta_{8} + 559 \beta_{9} + 2021 \beta_{10} + 735 \beta_{11} ) q^{40} + ( -5687327 - 185201 \beta_{1} - 1194 \beta_{2} - 24721 \beta_{3} + 577 \beta_{4} + 157 \beta_{5} - 2282 \beta_{6} + 1973 \beta_{7} - 2166 \beta_{8} + 1751 \beta_{9} - 3851 \beta_{10} - 576 \beta_{11} ) q^{41} + ( -12539628 - 14274 \beta_{1} - 13369 \beta_{2} - 83804 \beta_{3} + 1383 \beta_{4} + 5051 \beta_{5} + 1097 \beta_{6} + 111 \beta_{7} - 4888 \beta_{8} + 1706 \beta_{9} - 600 \beta_{10} + 2073 \beta_{11} ) q^{42} + ( 569109 - 361180 \beta_{1} + 4533 \beta_{2} + 15113 \beta_{3} + 2523 \beta_{4} - 4752 \beta_{5} - 2189 \beta_{6} - 3383 \beta_{7} + 723 \beta_{8} - 4248 \beta_{9} + 1719 \beta_{10} + 162 \beta_{11} ) q^{43} + ( -4808908 - 183074 \beta_{1} - 12272 \beta_{2} + 32717 \beta_{3} + 3962 \beta_{4} - 5205 \beta_{5} - 7928 \beta_{6} - 7274 \beta_{7} + 6607 \beta_{8} - 9641 \beta_{9} - 249 \beta_{10} - 5650 \beta_{11} ) q^{44} + ( -5225823 - 229780 \beta_{1} - 461 \beta_{2} - 37645 \beta_{3} - 3916 \beta_{4} + 1349 \beta_{5} - 396 \beta_{6} + 4849 \beta_{7} + 3219 \beta_{8} + 3471 \beta_{9} + 1794 \beta_{10} + 4877 \beta_{11} ) q^{45} + ( -6357614 - 74199 \beta_{1} + 22337 \beta_{2} - 37795 \beta_{3} - 190 \beta_{4} + 7760 \beta_{5} + 5864 \beta_{6} + 8439 \beta_{7} - 312 \beta_{8} + 677 \beta_{9} - 5245 \beta_{10} + 2 \beta_{11} ) q^{46} + ( -3931463 - 470437 \beta_{1} + 18193 \beta_{2} + 37337 \beta_{3} - 1977 \beta_{4} + 3109 \beta_{5} + 3084 \beta_{6} - 5272 \beta_{7} + 3644 \beta_{8} + 972 \beta_{9} - 516 \beta_{10} + 3002 \beta_{11} ) q^{47} + ( -8347014 + 311076 \beta_{1} + 5703 \beta_{2} - 68965 \beta_{3} - 5781 \beta_{4} + 1246 \beta_{5} + 12163 \beta_{6} + 11957 \beta_{7} + 1295 \beta_{8} + 5187 \beta_{9} + 3921 \beta_{10} - 1883 \beta_{11} ) q^{48} + ( 6222165 - 431934 \beta_{1} + 23732 \beta_{2} + 138466 \beta_{3} - 1714 \beta_{4} - 6146 \beta_{5} - 6998 \beta_{6} + 8700 \beta_{7} + 7548 \beta_{8} + 598 \beta_{9} + 5442 \beta_{10} - 6018 \beta_{11} ) q^{49} + ( -23448214 - 80284 \beta_{1} - 19065 \beta_{2} + 30403 \beta_{3} - 4597 \beta_{4} + 9171 \beta_{5} - 8295 \beta_{6} + 1847 \beta_{7} - 17227 \beta_{8} + 10397 \beta_{9} + 989 \beta_{10} - 4002 \beta_{11} ) q^{50} + ( 396986 - 762589 \beta_{1} + 12342 \beta_{2} - 10484 \beta_{3} + 6580 \beta_{4} + 3125 \beta_{5} + 6947 \beta_{6} - 8625 \beta_{7} - 595 \beta_{8} - 5718 \beta_{9} + 2603 \beta_{10} + 9598 \beta_{11} ) q^{51} + ( -4230029 + 175597 \beta_{1} + 13465 \beta_{2} + 117018 \beta_{3} - 7330 \beta_{4} + 44 \beta_{5} - 6054 \beta_{6} + 4268 \beta_{7} - 548 \beta_{8} + 3826 \beta_{9} + 2086 \beta_{10} + 8400 \beta_{11} ) q^{52} + ( 305952 - 328328 \beta_{1} + 8644 \beta_{2} + 52725 \beta_{3} + 10400 \beta_{4} - 4931 \beta_{5} + 13948 \beta_{6} - 3020 \beta_{7} - 8257 \beta_{8} - 3637 \beta_{9} - 3974 \beta_{10} + 1385 \beta_{11} ) q^{53} + ( -13969541 + 394860 \beta_{1} - 32598 \beta_{2} - 73255 \beta_{3} + 10941 \beta_{4} - 7704 \beta_{5} - 10539 \beta_{6} + 12758 \beta_{7} - 534 \beta_{8} - 879 \beta_{9} - 4471 \beta_{10} + 3953 \beta_{11} ) q^{54} + ( 7170576 - 828524 \beta_{1} + 22584 \beta_{2} + 158579 \beta_{3} + 7896 \beta_{4} - 2901 \beta_{5} + 3311 \beta_{6} - 20555 \beta_{7} + 2753 \beta_{8} - 2034 \beta_{9} + 5713 \beta_{10} - 14306 \beta_{11} ) q^{55} + ( 9899690 + 34520 \beta_{1} - 110 \beta_{2} + 252808 \beta_{3} - 2470 \beta_{4} - 15340 \beta_{5} - 17466 \beta_{6} - 27788 \beta_{7} + 21930 \beta_{8} - 25848 \beta_{9} - 19768 \beta_{10} - 8482 \beta_{11} ) q^{56} + ( 5822261 - 298912 \beta_{1} - 49361 \beta_{2} + 10825 \beta_{3} - 15014 \beta_{4} - 6295 \beta_{5} - 1438 \beta_{6} - 7775 \beta_{7} - 4871 \beta_{8} + 2053 \beta_{9} - 3490 \beta_{10} - 9627 \beta_{11} ) q^{57} + ( 707281 + 707281 \beta_{1} ) q^{58} + ( 4246030 + 93739 \beta_{1} - 15884 \beta_{2} + 86576 \beta_{3} + 3674 \beta_{4} + 14359 \beta_{5} - 2855 \beta_{6} - 13651 \beta_{7} - 5375 \beta_{8} + 582 \beta_{9} - 14845 \beta_{10} + 8652 \beta_{11} ) q^{59} + ( -9466586 + 1487596 \beta_{1} - 36244 \beta_{2} - 254403 \beta_{3} - 13232 \beta_{4} - 1403 \beta_{5} - 390 \beta_{6} + 9380 \beta_{7} - 2393 \beta_{8} + 20379 \beta_{9} + 5039 \beta_{10} + 6872 \beta_{11} ) q^{60} + ( 20817767 + 257485 \beta_{1} - 20324 \beta_{2} - 8469 \beta_{3} + 2829 \beta_{4} + 17295 \beta_{5} - 7244 \beta_{6} + 14385 \beta_{7} - 25236 \beta_{8} + 12303 \beta_{9} + 13503 \beta_{10} + 5828 \beta_{11} ) q^{61} + ( -749047 + 3013952 \beta_{1} - 34897 \beta_{2} - 353369 \beta_{3} + 9018 \beta_{4} + 2817 \beta_{5} + 39816 \beta_{6} + 12011 \beta_{7} + 6944 \beta_{8} + 2297 \beta_{9} + 24415 \beta_{10} + 9410 \beta_{11} ) q^{62} + ( 47488714 + 852198 \beta_{1} - 45424 \beta_{2} - 137712 \beta_{3} + 8564 \beta_{4} - 20466 \beta_{5} - 244 \beta_{6} + 5728 \beta_{7} + 24522 \beta_{8} - 12276 \beta_{9} + 10602 \beta_{10} - 26398 \beta_{11} ) q^{63} + ( 37744647 + 2080738 \beta_{1} + 95358 \beta_{2} - 271679 \beta_{3} - 8428 \beta_{4} + 8872 \beta_{5} + 53558 \beta_{6} + 2481 \beta_{7} + 12864 \beta_{8} + 11321 \beta_{9} + 34275 \beta_{10} + 17568 \beta_{11} ) q^{64} + ( 24589544 + 1364019 \beta_{1} - 53485 \beta_{2} + 44497 \beta_{3} - 42029 \beta_{4} + 52351 \beta_{5} - 29684 \beta_{6} + 17592 \beta_{7} - 9674 \beta_{8} + 20859 \beta_{9} - 16539 \beta_{10} + 22338 \beta_{11} ) q^{65} + ( 25815545 - 922427 \beta_{1} - 1081 \beta_{2} - 492553 \beta_{3} + 37397 \beta_{4} + 11353 \beta_{5} - 34965 \beta_{6} - 6461 \beta_{7} - 3589 \beta_{8} - 37371 \beta_{9} - 27671 \beta_{10} - 13766 \beta_{11} ) q^{66} + ( 61397714 + 1021382 \beta_{1} + 60724 \beta_{2} + 195800 \beta_{3} - 36496 \beta_{4} + 25310 \beta_{5} - 5328 \beta_{6} - 2052 \beta_{7} + 33586 \beta_{8} + 31848 \beta_{9} + 3610 \beta_{10} + 29846 \beta_{11} ) q^{67} + ( 40712824 + 3727054 \beta_{1} + 30698 \beta_{2} + 41300 \beta_{3} + 21740 \beta_{4} - 62162 \beta_{5} - 8972 \beta_{6} + 87770 \beta_{7} - 34270 \beta_{8} + 20504 \beta_{9} + 15796 \beta_{10} - 13196 \beta_{11} ) q^{68} + ( 123271301 + 659313 \beta_{1} - 24456 \beta_{2} - 353291 \beta_{3} + 46793 \beta_{4} - 83559 \beta_{5} + 27780 \beta_{6} + 28729 \beta_{7} + 8630 \beta_{8} - 56835 \beta_{9} - 5369 \beta_{10} - 38246 \beta_{11} ) q^{69} + ( 56877944 - 3844821 \beta_{1} + 147435 \beta_{2} + 156681 \beta_{3} - 22312 \beta_{4} + 38336 \beta_{5} - 21982 \beta_{6} - 69763 \beta_{7} + 6256 \beta_{8} - 38495 \beta_{9} - 49061 \beta_{10} - 10268 \beta_{11} ) q^{70} + ( 40860968 - 621414 \beta_{1} - 8980 \beta_{2} + 245334 \beta_{3} + 8656 \beta_{4} + 40180 \beta_{5} - 27600 \beta_{6} - 37220 \beta_{7} - 41712 \beta_{8} + 20 \beta_{9} - 26600 \beta_{10} - 6888 \beta_{11} ) q^{71} + ( 21482318 - 3446544 \beta_{1} + 99422 \beta_{2} - 143828 \beta_{3} - 7594 \beta_{4} - 17228 \beta_{5} - 37638 \beta_{6} - 30676 \beta_{7} - 20398 \beta_{8} + 9612 \beta_{9} - 31612 \beta_{10} - 25954 \beta_{11} ) q^{72} + ( 119991664 - 1690980 \beta_{1} + 29526 \beta_{2} - 38750 \beta_{3} + 49814 \beta_{4} - 25860 \beta_{5} + 2868 \beta_{6} + 64820 \beta_{7} - 7154 \beta_{8} - 48714 \beta_{9} - 49164 \beta_{10} - 24250 \beta_{11} ) q^{73} + ( -18529312 + 3625368 \beta_{1} - 5808 \beta_{2} + 35078 \beta_{3} - 9384 \beta_{4} - 32820 \beta_{5} + 86128 \beta_{6} + 7384 \beta_{7} + 54418 \beta_{8} + 28806 \beta_{9} + 77882 \beta_{10} + 8882 \beta_{11} ) q^{74} + ( 38900406 - 1193066 \beta_{1} - 63972 \beta_{2} + 211936 \beta_{3} - 35250 \beta_{4} - 24494 \beta_{5} - 13176 \beta_{6} - 100946 \beta_{7} + 830 \beta_{8} + 37456 \beta_{9} + 32772 \beta_{10} + 75614 \beta_{11} ) q^{75} + ( -18785662 - 7276868 \beta_{1} + 172970 \beta_{2} + 744606 \beta_{3} - 1314 \beta_{4} + 26422 \beta_{5} - 39546 \beta_{6} - 65384 \beta_{7} - 63548 \beta_{8} + 22890 \beta_{9} - 16406 \beta_{10} + 4782 \beta_{11} ) q^{76} + ( 31723039 + 5921907 \beta_{1} - 210412 \beta_{2} + 176993 \beta_{3} - 5533 \beta_{4} - 12171 \beta_{5} + 864 \beta_{6} + 103363 \beta_{7} + 63024 \beta_{8} + 31617 \beta_{9} + 109333 \beta_{10} + 31788 \beta_{11} ) q^{77} + ( -102075103 - 5003016 \beta_{1} - 44847 \beta_{2} + 72055 \beta_{3} - 55032 \beta_{4} + 19667 \beta_{5} + 41778 \beta_{6} - 120403 \beta_{7} - 23180 \beta_{8} + 36605 \beta_{9} + 29487 \beta_{10} + 53944 \beta_{11} ) q^{78} + ( 34778877 - 4304051 \beta_{1} + 70805 \beta_{2} + 454493 \beta_{3} - 25175 \beta_{4} + 67513 \beta_{5} + 30808 \beta_{6} - 22366 \beta_{7} - 36214 \beta_{8} + 36642 \beta_{9} - 15856 \beta_{10} + 2388 \beta_{11} ) q^{79} + ( -35829435 - 2559734 \beta_{1} + 256516 \beta_{2} + 757609 \beta_{3} + 12570 \beta_{4} + 43316 \beta_{5} + 34416 \beta_{6} + 67045 \beta_{7} + 66318 \beta_{8} - 18399 \beta_{9} - 21221 \beta_{10} - 57734 \beta_{11} ) q^{80} + ( 18058272 - 2203922 \beta_{1} - 140707 \beta_{2} - 892743 \beta_{3} + 1832 \beta_{4} + 33061 \beta_{5} - 17468 \beta_{6} + 127515 \beta_{7} - 34057 \beta_{8} + 10457 \beta_{9} - 26176 \beta_{10} - 38831 \beta_{11} ) q^{81} + ( -163906880 - 7396196 \beta_{1} - 415477 \beta_{2} + 579440 \beta_{3} + 28949 \beta_{4} - 27831 \beta_{5} - 74393 \beta_{6} - 58169 \beta_{7} + 135206 \beta_{8} - 73286 \beta_{9} + 18500 \beta_{10} - 33303 \beta_{11} ) q^{82} + ( 125769340 + 3708485 \beta_{1} + 128168 \beta_{2} + 1288264 \beta_{3} - 36130 \beta_{4} - 60795 \beta_{5} + 55769 \beta_{6} + 27869 \beta_{7} + 25551 \beta_{8} - 69614 \beta_{9} - 64355 \beta_{10} - 32282 \beta_{11} ) q^{83} + ( -177740522 - 14351944 \beta_{1} + 113888 \beta_{2} - 776300 \beta_{3} + 22588 \beta_{4} + 162938 \beta_{5} + 14676 \beta_{6} - 161082 \beta_{7} - 23922 \beta_{8} - 47456 \beta_{9} - 30076 \beta_{10} + 42964 \beta_{11} ) q^{84} + ( -32548507 + 1157741 \beta_{1} - 247528 \beta_{2} - 761809 \beta_{3} + 128319 \beta_{4} - 149639 \beta_{5} + 123620 \beta_{6} - 77473 \beta_{7} - 25120 \beta_{8} + 10147 \beta_{9} + 74583 \beta_{10} + 5520 \beta_{11} ) q^{85} + ( -286563777 + 1893802 \beta_{1} - 461814 \beta_{2} + 1045011 \beta_{3} - 3281 \beta_{4} - 29958 \beta_{5} - 139641 \beta_{6} + 138666 \beta_{7} - 67230 \beta_{8} + 21471 \beta_{9} - 22945 \beta_{10} - 28881 \beta_{11} ) q^{86} + ( 14145620 + 707281 \beta_{1} - 707281 \beta_{3} ) q^{87} + ( -259582679 - 9159238 \beta_{1} - 354218 \beta_{2} + 731027 \beta_{3} + 10464 \beta_{4} - 59720 \beta_{5} - 215386 \beta_{6} + 147359 \beta_{7} - 209288 \beta_{8} + 45147 \beta_{9} - 124799 \beta_{10} + 51852 \beta_{11} ) q^{88} + ( 54141099 + 10842611 \beta_{1} + 75792 \beta_{2} - 1027391 \beta_{3} - 43955 \beta_{4} + 89307 \beta_{5} - 9330 \beta_{6} + 91919 \beta_{7} + 64848 \beta_{8} - 32563 \beta_{9} - 6895 \beta_{10} + 28234 \beta_{11} ) q^{89} + ( -199396108 - 4660453 \beta_{1} + 194480 \beta_{2} + 199299 \beta_{3} - 7025 \beta_{4} - 7851 \beta_{5} + 14607 \beta_{6} - 21462 \beta_{7} + 44476 \beta_{8} - 22939 \beta_{9} - 36271 \beta_{10} - 59951 \beta_{11} ) q^{90} + ( 130013964 + 2310421 \beta_{1} + 97320 \beta_{2} + 253694 \beta_{3} - 9258 \beta_{4} - 109617 \beta_{5} - 39961 \beta_{6} - 17681 \beta_{7} + 7345 \beta_{8} - 29342 \beta_{9} + 5927 \beta_{10} - 123446 \beta_{11} ) q^{91} + ( -210764546 + 10483902 \beta_{1} + 171816 \beta_{2} - 1302792 \beta_{3} + 40450 \beta_{4} + 68050 \beta_{5} + 228150 \beta_{6} + 135190 \beta_{7} - 10340 \beta_{8} - 20212 \beta_{9} + 99072 \beta_{10} + 142046 \beta_{11} ) q^{92} + ( 18848923 + 12354092 \beta_{1} + 304803 \beta_{2} - 1960236 \beta_{3} - 74720 \beta_{4} + 50034 \beta_{5} - 49346 \beta_{6} + 122667 \beta_{7} + 90814 \beta_{8} + 125234 \beta_{9} + 25352 \beta_{10} + 7096 \beta_{11} ) q^{93} + ( -368574605 + 6439387 \beta_{1} - 368299 \beta_{2} - 862654 \beta_{3} - 28751 \beta_{4} - 16696 \beta_{5} + 150003 \beta_{6} + 207075 \beta_{7} - 34874 \beta_{8} + 127810 \beta_{9} + 66332 \beta_{10} + 20053 \beta_{11} ) q^{94} + ( 16674559 + 9103965 \beta_{1} + 451837 \beta_{2} + 405592 \beta_{3} + 5839 \beta_{4} + 77616 \beta_{5} - 124675 \beta_{6} - 415117 \beta_{7} - 118221 \beta_{8} - 10976 \beta_{9} + 15335 \beta_{10} - 71100 \beta_{11} ) q^{95} + ( -121141981 + 2949258 \beta_{1} + 998498 \beta_{2} + 107705 \beta_{3} - 141984 \beta_{4} + 38272 \beta_{5} + 240506 \beta_{6} - 60607 \beta_{7} + 119100 \beta_{8} + 115441 \beta_{9} + 118251 \beta_{10} + 91492 \beta_{11} ) q^{96} + ( 206108595 + 6414549 \beta_{1} - 30002 \beta_{2} - 2094737 \beta_{3} - 52463 \beta_{4} - 35797 \beta_{5} - 21290 \beta_{6} - 34043 \beta_{7} + 83636 \beta_{8} - 139785 \beta_{9} - 104681 \beta_{10} + 68310 \beta_{11} ) q^{97} + ( -295044895 + 23045789 \beta_{1} - 411024 \beta_{2} + 2408628 \beta_{3} - 5172 \beta_{4} - 403236 \beta_{5} - 151444 \beta_{6} + 85176 \beta_{7} + 78632 \beta_{8} - 16020 \beta_{9} - 59204 \beta_{10} + 28044 \beta_{11} ) q^{98} + ( 174542219 + 9039199 \beta_{1} + 819725 \beta_{2} - 1575918 \beta_{3} + 204517 \beta_{4} + 58668 \beta_{5} - 109447 \beta_{6} + 228985 \beta_{7} - 57845 \beta_{8} + 60756 \beta_{9} + 145381 \beta_{10} + 41236 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 16q^{2} + 242q^{3} + 3498q^{4} + 1762q^{5} + 5446q^{6} + 12080q^{7} + 25350q^{8} + 43026q^{9} + O(q^{10}) \) \( 12q + 16q^{2} + 242q^{3} + 3498q^{4} + 1762q^{5} + 5446q^{6} + 12080q^{7} + 25350q^{8} + 43026q^{9} - 46678q^{10} + 24474q^{11} - 14210q^{12} + 107722q^{13} + 677768q^{14} + 505426q^{15} + 1656882q^{16} + 982120q^{17} + 2364102q^{18} + 2084360q^{19} + 4689410q^{20} + 2911344q^{21} + 2725230q^{22} + 3004004q^{23} + 7893170q^{24} + 6339542q^{25} + 6863698q^{26} + 7881014q^{27} + 5116944q^{28} + 8487372q^{29} + 10924626q^{30} + 17872478q^{31} + 5122946q^{32} - 860442q^{33} + 15662848q^{34} - 22252312q^{35} - 35199980q^{36} + 452980q^{37} - 68665276q^{38} - 29528222q^{39} - 61623214q^{40} - 69039804q^{41} - 150603216q^{42} + 5379186q^{43} - 58283762q^{44} - 63687756q^{45} - 76817844q^{46} - 49104062q^{47} - 99120062q^{48} + 73113148q^{49} - 281373726q^{50} + 1578252q^{51} - 49849646q^{52} + 2253998q^{53} - 166064634q^{54} + 82907066q^{55} + 119369464q^{56} + 69024164q^{57} + 11316496q^{58} + 51587572q^{59} - 107912622q^{60} + 251179296q^{61} + 2421010q^{62} + 573206808q^{63} + 460030950q^{64} + 301434554q^{65} + 305189958q^{66} + 741046264q^{67} + 503103116q^{68} + 1480618500q^{69} + 666826600q^{70} + 488700124q^{71} + 243154096q^{72} + 1432375020q^{73} - 208138340q^{74} + 462882236q^{75} - 253709644q^{76} + 406327616q^{77} - 1244370462q^{78} + 400834638q^{79} - 440320610q^{80} + 207205984q^{81} - 1992598260q^{82} + 1525085236q^{83} - 2191854376q^{84} - 387675996q^{85} - 3425646378q^{86} + 171162002q^{87} - 3147673814q^{88} + 691159332q^{89} - 2412410836q^{90} + 1569278264q^{91} - 2491626380q^{92} + 270455138q^{93} - 4397366402q^{94} + 236293724q^{95} - 1448270346q^{96} + 2494422276q^{97} - 3443098784q^{98} + 2123567852q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 4 x^{11} - 4803 x^{10} + 14952 x^{9} + 8248476 x^{8} - 14809944 x^{7} - 6122244486 x^{6} - 1026481216 x^{5} + 1899055440249 x^{4} + 3992697832188 x^{3} - 174144148490503 x^{2} - 282853247198664 x + 4071614738753338\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 802 \)
\(\beta_{3}\)\(=\)\((\)\(-406357569792591614600449 \nu^{11} + 792740960952341270243465 \nu^{10} + 1946897167526761992899986518 \nu^{9} - 6712088880929415894051437974 \nu^{8} - 3405661853467477668694262223278 \nu^{7} + 18712934197692512271820503963454 \nu^{6} + 2666653672613879857062358916979792 \nu^{5} - 19931301320056918022766233205488080 \nu^{4} - 915506622997210906928284554033584873 \nu^{3} + 6878481599835692742819625326592184977 \nu^{2} + 111061179397074402684043349968380403954 \nu - 335753949777084521700124495172805331698\)\()/ \)\(22\!\cdots\!00\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-20954337749949728684837 \nu^{11} + 3783746072333098824069005 \nu^{10} + 277882506078844403377843054 \nu^{9} - 14559639005950895887699553582 \nu^{8} - 905663837449204843910826340134 \nu^{7} + 15641676469532502184483619126902 \nu^{6} + 1120141619623571963688885231422096 \nu^{5} - 1879636186354497852038107476868880 \nu^{4} - 514834838172270175650361100691613869 \nu^{3} - 2438578228075615918386563489167833419 \nu^{2} + 51680932772173174400226425080873946362 \nu + 103084432816170144881442629967066154886\)\()/ \)\(49\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-1412264404957649671622855 \nu^{11} + 5986454417813301729080607 \nu^{10} + 9079152502565722004322701594 \nu^{9} - 3288804156701992592626942106 \nu^{8} - 21672879013147718931483315954050 \nu^{7} - 5208436913796535239269198967246 \nu^{6} + 22174151392976670522336967850986096 \nu^{5} - 17785321242142583780166478371939824 \nu^{4} - 8370252097407327678785892902482950623 \nu^{3} + 9662822115078003556951910706381809943 \nu^{2} + 734259276227140551938028127627564490910 \nu - 998414362451671007380476026894758073822\)\()/ \)\(11\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(130954898376300260473341 \nu^{11} + 419209129123254015271671 \nu^{10} - 651235829127750591316038470 \nu^{9} - 1547451555798535275394080962 \nu^{8} + 1163954419353459558136766395102 \nu^{7} + 2111142863256113232023794952506 \nu^{6} - 881382412534043984905157401957080 \nu^{5} - 1322770239062952047803547610352072 \nu^{4} + 243154178493258598992494518600531773 \nu^{3} + 85711892576171581449652167009675031 \nu^{2} - 8256840519634810793834994507646715386 \nu + 112213961840027926487184770998504175426\)\()/ \)\(63\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(2463650464212841101251053 \nu^{11} + 33052619622405672307992691 \nu^{10} - 10355805603675583640826858894 \nu^{9} - 181227596508382094339246717090 \nu^{8} + 14724988984580422388855221219526 \nu^{7} + 341148426187180356571617569119514 \nu^{6} - 8263781435348814219207256637819136 \nu^{5} - 259559031410422278864759857903124032 \nu^{4} + 1807110720705679365633694625870260037 \nu^{3} + 68207724373387725292875777051816876155 \nu^{2} - 165664326664658475041813437402628236138 \nu - 2964422719762596678047459119034204591910\)\()/ \)\(11\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-10222000904814770809959925 \nu^{11} + 96988551901512996413492437 \nu^{10} + 49239747452383981534233380414 \nu^{9} - 322277832360503484927597381646 \nu^{8} - 86346571974912802459392811447350 \nu^{7} + 261893584633867321870774576752854 \nu^{6} + 66326105212356590631845014265608576 \nu^{5} + 66774171191694656059401444501004736 \nu^{4} - 20209488917566614089920194490627925133 \nu^{3} - 95297243125648102506424504607421313267 \nu^{2} + 993821782776948391861400069927066836730 \nu + 3970822051972805212405347627599721910678\)\()/ \)\(11\!\cdots\!00\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-2310504861598103090572611 \nu^{11} - 29193370261959189610054565 \nu^{10} + 11726421680038231025469210242 \nu^{9} + 134022370959145201909231852734 \nu^{8} - 21202491681449770653364965597322 \nu^{7} - 234361107104378346583240595164614 \nu^{6} + 16354401753224731673099521650260208 \nu^{5} + 191239328542227920033869732226978960 \nu^{4} - 4943449522796186496947520204967208187 \nu^{3} - 61883912019547235856885258300172662157 \nu^{2} + 261962715382221412557312531485811526486 \nu + 2811754981431937824191718488729159140778\)\()/ \)\(25\!\cdots\!00\)\( \)
\(\beta_{10}\)\(=\)\((\)\(13946693724274117635951743 \nu^{11} - 194561160171337553376325055 \nu^{10} - 64812427118726102503306875706 \nu^{9} + 786816114045207463817635904618 \nu^{8} + 105531633522735766009429963765026 \nu^{7} - 965389262027007416238428061204418 \nu^{6} - 71611032839362422917863660099728704 \nu^{5} + 292410470617485851126583828063605120 \nu^{4} + 19081061053534120324147988346089857671 \nu^{3} + 48302762314739268179776152084489726041 \nu^{2} - 941986456291964868272837702413730401198 \nu - 5454973756011664143043738208002003261634\)\()/ \)\(11\!\cdots\!00\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-3869818852820710101769067 \nu^{11} + 68401404783109515234107211 \nu^{10} + 17853702182380372391806888706 \nu^{9} - 298018111402160171792762256050 \nu^{8} - 28752269213046024746469969009674 \nu^{7} + 432034241536145004895654355727594 \nu^{6} + 19237341141124356122890630544212864 \nu^{5} - 232672742779665681658953263103637952 \nu^{4} - 5148121853428593174358067601638544083 \nu^{3} + 42853153679153933558724644966219265555 \nu^{2} + 326647540505681018060851685994014755782 \nu - 2202991263788485598089411893467823867350\)\()/ \)\(19\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 802\)
\(\nu^{3}\)\(=\)\(\beta_{11} + 3 \beta_{10} + \beta_{9} + 3 \beta_{8} + 3 \beta_{7} + 3 \beta_{6} - 2 \beta_{5} - \beta_{4} - 23 \beta_{3} - 4 \beta_{2} + 1382 \beta_{1} + 613\)
\(\nu^{4}\)\(=\)\(-22 \beta_{11} - 47 \beta_{10} + 3 \beta_{9} - 2 \beta_{8} + 47 \beta_{7} + 68 \beta_{6} + 20 \beta_{5} - 22 \beta_{4} + 507 \beta_{3} + 1830 \beta_{2} + 516 \beta_{1} + 1101706\)
\(\nu^{5}\)\(=\)\(2465 \beta_{11} + 6934 \beta_{10} + 2458 \beta_{9} + 6247 \beta_{8} + 5752 \beta_{7} + 8569 \beta_{6} - 4294 \beta_{5} - 1813 \beta_{4} - 42494 \beta_{3} - 7449 \beta_{2} + 2171565 \beta_{1} + 260885\)
\(\nu^{6}\)\(=\)\(-42992 \beta_{11} - 96284 \beta_{10} + 14428 \beta_{9} + 952 \beta_{8} + 118244 \beta_{7} + 205864 \beta_{6} + 65096 \beta_{5} - 63760 \beta_{4} + 1038540 \beta_{3} + 3199003 \beta_{2} - 204405 \beta_{1} + 1729803152\)
\(\nu^{7}\)\(=\)\(4998295 \beta_{11} + 13174201 \beta_{10} + 5060579 \beta_{9} + 11052189 \beta_{8} + 10182857 \beta_{7} + 18594493 \beta_{6} - 7192182 \beta_{5} - 3396151 \beta_{4} - 84646517 \beta_{3} - 10600360 \beta_{2} + 3608124680 \beta_{1} - 355340293\)
\(\nu^{8}\)\(=\)\(-64582010 \beta_{11} - 156075269 \beta_{10} + 41345617 \beta_{9} + 11520058 \beta_{8} + 223440781 \beta_{7} + 480197076 \beta_{6} + 160759060 \beta_{5} - 139020858 \beta_{4} + 1464287049 \beta_{3} + 5592787848 \beta_{2} + 29471898 \beta_{1} + 2869577136274\)
\(\nu^{9}\)\(=\)\(9607745489 \beta_{11} + 23867759212 \beta_{10} + 9927622412 \beta_{9} + 18795167111 \beta_{8} + 18000883146 \beta_{7} + 37146186245 \beta_{6} - 10976806246 \beta_{5} - 6426610781 \beta_{4} - 172466792956 \beta_{3} - 12084356323 \beta_{2} + 6160418187643 \beta_{1} - 407253696883\)
\(\nu^{10}\)\(=\)\(-84569969644 \beta_{11} - 233629946502 \beta_{10} + 97038898062 \beta_{9} + 38774614140 \beta_{8} + 392820002278 \beta_{7} + 1022302513016 \beta_{6} + 354912971944 \beta_{5} - 276281172556 \beta_{4} + 1522879388478 \beta_{3} + 9818597125181 \beta_{2} + 3687943127769 \beta_{1} + 4889414255473516\)
\(\nu^{11}\)\(=\)\(18065382227179 \beta_{11} + 42678308589903 \beta_{10} + 19052616072789 \beta_{9} + 31684894303913 \beta_{8} + 32105084155011 \beta_{7} + 71812545228733 \beta_{6} - 15932604987070 \beta_{5} - 12071860768819 \beta_{4} - 347807829477043 \beta_{3} - 7669074098702 \beta_{2} + 10674799597445358 \beta_{1} + 1867815484159671\)

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
−41.5077
−41.0148
−24.9566
−21.0654
−11.2767
−7.09515
4.88142
9.20723
26.3150
29.2132
38.0522
43.2474
−40.5077 −111.357 1128.88 2140.75 4510.81 −10261.8 −24988.3 −7282.70 −86716.9
1.2 −40.0148 −3.22123 1089.19 −146.392 128.897 8248.48 −23096.1 −19672.6 5857.84
1.3 −23.9566 161.985 61.9199 2644.17 −3880.61 2059.91 10782.4 6556.14 −63345.5
1.4 −20.0654 −72.4764 −109.378 −1017.86 1454.27 −11164.8 12468.2 −14430.2 20423.8
1.5 −10.2767 52.0955 −406.390 −2693.22 −535.369 4178.53 9437.99 −16969.1 27677.4
1.6 −6.09515 250.876 −474.849 −546.084 −1529.13 6563.37 6015.00 43256.0 3328.46
1.7 5.88142 −88.8951 −477.409 959.601 −522.829 −2615.28 −5819.13 −11780.7 5643.82
1.8 10.2072 −225.321 −407.813 −1619.94 −2299.90 −263.065 −9388.73 31086.4 −16535.1
1.9 27.3150 101.195 234.111 2383.64 2764.14 3032.73 −7590.56 −9442.56 65109.1
1.10 30.2132 241.333 400.835 −389.202 7291.44 3096.10 −3358.64 38558.7 −11759.0
1.11 39.0522 −174.323 1013.08 −303.923 −6807.72 12556.4 19568.2 10705.6 −11868.9
1.12 44.2474 110.108 1445.83 350.462 4872.00 −3350.56 41319.6 −7559.17 15507.0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(29\) \(-1\)

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 29.10.a.b 12
3.b odd 2 1 261.10.a.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.10.a.b 12 1.a even 1 1 trivial
261.10.a.e 12 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(18\!\cdots\!32\)\( T_{2}^{4} - \)\(34\!\cdots\!96\)\( T_{2}^{3} - \)\(17\!\cdots\!72\)\( T_{2}^{2} + \)\(69\!\cdots\!72\)\( T_{2} + \)\(41\!\cdots\!60\)\( \)">\(T_{2}^{12} - \cdots\) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(29))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4178225122344960 + 69848353103872 T - 174808936988672 T^{2} - 3492322282496 T^{3} + 1813447029632 T^{4} + 34937153888 T^{5} - 5789879376 T^{6} - 79685232 T^{7} + 7898928 T^{8} + 62542 T^{9} - 4693 T^{10} - 16 T^{11} + T^{12} \)
$3$ \( \)\(51\!\cdots\!72\)\( + \)\(15\!\cdots\!24\)\( T - \)\(15\!\cdots\!05\)\( T^{2} - 78821686171672221618 T^{3} + 517095853414573005 T^{4} + 13065167411073072 T^{5} - 68623887122442 T^{6} - 903557330188 T^{7} + 4193159218 T^{8} + 25621252 T^{9} - 110329 T^{10} - 242 T^{11} + T^{12} \)
$5$ \( -\)\(19\!\cdots\!00\)\( - \)\(21\!\cdots\!00\)\( T - \)\(47\!\cdots\!75\)\( T^{2} + \)\(16\!\cdots\!50\)\( T^{3} + \)\(63\!\cdots\!45\)\( T^{4} + \)\(17\!\cdots\!16\)\( T^{5} - \)\(10\!\cdots\!02\)\( T^{6} - 50764604678212284 T^{7} + 57573410101830 T^{8} + 19430541380 T^{9} - 13336199 T^{10} - 1762 T^{11} + T^{12} \)
$7$ \( -\)\(14\!\cdots\!08\)\( - \)\(42\!\cdots\!16\)\( T + \)\(48\!\cdots\!68\)\( T^{2} - \)\(12\!\cdots\!24\)\( T^{3} - \)\(10\!\cdots\!32\)\( T^{4} + \)\(23\!\cdots\!08\)\( T^{5} + \)\(51\!\cdots\!68\)\( T^{6} - \)\(19\!\cdots\!52\)\( T^{7} + 6476071267346240 T^{8} + 2927626876160 T^{9} - 205715016 T^{10} - 12080 T^{11} + T^{12} \)
$11$ \( -\)\(13\!\cdots\!72\)\( - \)\(48\!\cdots\!04\)\( T - \)\(15\!\cdots\!33\)\( T^{2} - \)\(33\!\cdots\!30\)\( T^{3} + \)\(44\!\cdots\!89\)\( T^{4} + \)\(55\!\cdots\!80\)\( T^{5} - \)\(36\!\cdots\!90\)\( T^{6} - \)\(24\!\cdots\!40\)\( T^{7} + \)\(12\!\cdots\!42\)\( T^{8} + 419974474031316 T^{9} - 18523408385 T^{10} - 24474 T^{11} + T^{12} \)
$13$ \( \)\(20\!\cdots\!56\)\( + \)\(58\!\cdots\!44\)\( T + \)\(66\!\cdots\!29\)\( T^{2} - \)\(41\!\cdots\!14\)\( T^{3} - \)\(10\!\cdots\!35\)\( T^{4} + \)\(51\!\cdots\!68\)\( T^{5} - \)\(44\!\cdots\!06\)\( T^{6} - \)\(87\!\cdots\!32\)\( T^{7} + \)\(85\!\cdots\!54\)\( T^{8} + 5258002796803796 T^{9} - 51812555079 T^{10} - 107722 T^{11} + T^{12} \)
$17$ \( \)\(31\!\cdots\!32\)\( - \)\(19\!\cdots\!68\)\( T + \)\(31\!\cdots\!24\)\( T^{2} - \)\(68\!\cdots\!44\)\( T^{3} - \)\(14\!\cdots\!76\)\( T^{4} + \)\(56\!\cdots\!52\)\( T^{5} + \)\(17\!\cdots\!88\)\( T^{6} - \)\(95\!\cdots\!24\)\( T^{7} - \)\(35\!\cdots\!32\)\( T^{8} + 556670127460868480 T^{9} - 335564738256 T^{10} - 982120 T^{11} + T^{12} \)
$19$ \( \)\(78\!\cdots\!60\)\( + \)\(25\!\cdots\!64\)\( T + \)\(19\!\cdots\!24\)\( T^{2} - \)\(87\!\cdots\!04\)\( T^{3} - \)\(11\!\cdots\!48\)\( T^{4} + \)\(12\!\cdots\!20\)\( T^{5} + \)\(18\!\cdots\!36\)\( T^{6} - \)\(21\!\cdots\!20\)\( T^{7} - \)\(93\!\cdots\!96\)\( T^{8} + 1666478040478268048 T^{9} + 382240753700 T^{10} - 2084360 T^{11} + T^{12} \)
$23$ \( \)\(47\!\cdots\!76\)\( + \)\(73\!\cdots\!36\)\( T - \)\(18\!\cdots\!64\)\( T^{2} - \)\(28\!\cdots\!60\)\( T^{3} + \)\(21\!\cdots\!28\)\( T^{4} + \)\(32\!\cdots\!36\)\( T^{5} - \)\(14\!\cdots\!00\)\( T^{6} - \)\(16\!\cdots\!96\)\( T^{7} + \)\(55\!\cdots\!92\)\( T^{8} + 36105137753568161696 T^{9} - 11843059506540 T^{10} - 3004004 T^{11} + T^{12} \)
$29$ \( ( -707281 + T )^{12} \)
$31$ \( -\)\(22\!\cdots\!04\)\( + \)\(13\!\cdots\!40\)\( T + \)\(18\!\cdots\!07\)\( T^{2} - \)\(12\!\cdots\!38\)\( T^{3} - \)\(72\!\cdots\!55\)\( T^{4} + \)\(26\!\cdots\!92\)\( T^{5} + \)\(84\!\cdots\!94\)\( T^{6} - \)\(24\!\cdots\!32\)\( T^{7} - \)\(34\!\cdots\!14\)\( T^{8} + \)\(10\!\cdots\!20\)\( T^{9} + 20655209997527 T^{10} - 17872478 T^{11} + T^{12} \)
$37$ \( -\)\(18\!\cdots\!12\)\( + \)\(15\!\cdots\!60\)\( T - \)\(27\!\cdots\!56\)\( T^{2} - \)\(57\!\cdots\!68\)\( T^{3} + \)\(15\!\cdots\!96\)\( T^{4} + \)\(77\!\cdots\!68\)\( T^{5} - \)\(28\!\cdots\!20\)\( T^{6} - \)\(48\!\cdots\!80\)\( T^{7} + \)\(22\!\cdots\!68\)\( T^{8} + \)\(12\!\cdots\!44\)\( T^{9} - 808707982845196 T^{10} - 452980 T^{11} + T^{12} \)
$41$ \( \)\(35\!\cdots\!52\)\( + \)\(38\!\cdots\!40\)\( T + \)\(97\!\cdots\!28\)\( T^{2} - \)\(11\!\cdots\!76\)\( T^{3} - \)\(50\!\cdots\!16\)\( T^{4} + \)\(51\!\cdots\!44\)\( T^{5} + \)\(58\!\cdots\!04\)\( T^{6} + \)\(19\!\cdots\!44\)\( T^{7} - \)\(15\!\cdots\!52\)\( T^{8} - \)\(85\!\cdots\!08\)\( T^{9} + 102546229658156 T^{10} + 69039804 T^{11} + T^{12} \)
$43$ \( -\)\(16\!\cdots\!40\)\( + \)\(48\!\cdots\!32\)\( T - \)\(29\!\cdots\!17\)\( T^{2} - \)\(74\!\cdots\!74\)\( T^{3} + \)\(19\!\cdots\!37\)\( T^{4} + \)\(27\!\cdots\!56\)\( T^{5} - \)\(32\!\cdots\!18\)\( T^{6} - \)\(24\!\cdots\!88\)\( T^{7} + \)\(22\!\cdots\!94\)\( T^{8} + \)\(23\!\cdots\!28\)\( T^{9} - 2851424060241689 T^{10} - 5379186 T^{11} + T^{12} \)
$47$ \( \)\(45\!\cdots\!92\)\( + \)\(19\!\cdots\!16\)\( T - \)\(20\!\cdots\!09\)\( T^{2} + \)\(53\!\cdots\!38\)\( T^{3} + \)\(25\!\cdots\!21\)\( T^{4} - \)\(46\!\cdots\!52\)\( T^{5} + \)\(11\!\cdots\!50\)\( T^{6} + \)\(15\!\cdots\!52\)\( T^{7} + \)\(95\!\cdots\!98\)\( T^{8} - \)\(19\!\cdots\!64\)\( T^{9} - 3329893663530425 T^{10} + 49104062 T^{11} + T^{12} \)
$53$ \( -\)\(21\!\cdots\!08\)\( - \)\(32\!\cdots\!80\)\( T - \)\(61\!\cdots\!19\)\( T^{2} + \)\(87\!\cdots\!70\)\( T^{3} + \)\(28\!\cdots\!61\)\( T^{4} - \)\(70\!\cdots\!44\)\( T^{5} - \)\(28\!\cdots\!22\)\( T^{6} + \)\(21\!\cdots\!60\)\( T^{7} + \)\(10\!\cdots\!34\)\( T^{8} - \)\(21\!\cdots\!60\)\( T^{9} - 17908324054244359 T^{10} - 2253998 T^{11} + T^{12} \)
$59$ \( -\)\(25\!\cdots\!00\)\( + \)\(16\!\cdots\!00\)\( T + \)\(35\!\cdots\!00\)\( T^{2} - \)\(15\!\cdots\!80\)\( T^{3} + \)\(63\!\cdots\!96\)\( T^{4} + \)\(14\!\cdots\!64\)\( T^{5} - \)\(72\!\cdots\!76\)\( T^{6} - \)\(49\!\cdots\!12\)\( T^{7} + \)\(25\!\cdots\!56\)\( T^{8} + \)\(88\!\cdots\!04\)\( T^{9} - 29150254796429084 T^{10} - 51587572 T^{11} + T^{12} \)
$61$ \( \)\(22\!\cdots\!20\)\( - \)\(12\!\cdots\!92\)\( T - \)\(19\!\cdots\!96\)\( T^{2} - \)\(48\!\cdots\!28\)\( T^{3} + \)\(49\!\cdots\!68\)\( T^{4} + \)\(17\!\cdots\!72\)\( T^{5} - \)\(62\!\cdots\!92\)\( T^{6} - \)\(22\!\cdots\!64\)\( T^{7} + \)\(68\!\cdots\!16\)\( T^{8} + \)\(12\!\cdots\!88\)\( T^{9} - 43402681623264880 T^{10} - 251179296 T^{11} + T^{12} \)
$67$ \( \)\(96\!\cdots\!20\)\( + \)\(75\!\cdots\!56\)\( T - \)\(59\!\cdots\!00\)\( T^{2} + \)\(68\!\cdots\!88\)\( T^{3} + \)\(53\!\cdots\!24\)\( T^{4} - \)\(13\!\cdots\!84\)\( T^{5} + \)\(96\!\cdots\!76\)\( T^{6} - \)\(59\!\cdots\!76\)\( T^{7} - \)\(23\!\cdots\!40\)\( T^{8} + \)\(89\!\cdots\!60\)\( T^{9} + 47172886248785136 T^{10} - 741046264 T^{11} + T^{12} \)
$71$ \( \)\(31\!\cdots\!76\)\( + \)\(80\!\cdots\!24\)\( T - \)\(15\!\cdots\!04\)\( T^{2} - \)\(14\!\cdots\!80\)\( T^{3} - \)\(10\!\cdots\!80\)\( T^{4} + \)\(10\!\cdots\!72\)\( T^{5} + \)\(34\!\cdots\!80\)\( T^{6} - \)\(38\!\cdots\!48\)\( T^{7} + \)\(21\!\cdots\!08\)\( T^{8} + \)\(69\!\cdots\!80\)\( T^{9} - 96515803642596740 T^{10} - 488700124 T^{11} + T^{12} \)
$73$ \( -\)\(53\!\cdots\!16\)\( + \)\(16\!\cdots\!96\)\( T + \)\(28\!\cdots\!92\)\( T^{2} - \)\(17\!\cdots\!64\)\( T^{3} + \)\(24\!\cdots\!08\)\( T^{4} - \)\(13\!\cdots\!60\)\( T^{5} + \)\(12\!\cdots\!56\)\( T^{6} + \)\(31\!\cdots\!68\)\( T^{7} - \)\(12\!\cdots\!44\)\( T^{8} + \)\(72\!\cdots\!68\)\( T^{9} + 595859034265004916 T^{10} - 1432375020 T^{11} + T^{12} \)
$79$ \( -\)\(64\!\cdots\!00\)\( - \)\(24\!\cdots\!80\)\( T + \)\(96\!\cdots\!79\)\( T^{2} - \)\(49\!\cdots\!58\)\( T^{3} - \)\(82\!\cdots\!07\)\( T^{4} + \)\(80\!\cdots\!76\)\( T^{5} + \)\(23\!\cdots\!10\)\( T^{6} - \)\(20\!\cdots\!44\)\( T^{7} + \)\(31\!\cdots\!70\)\( T^{8} + \)\(16\!\cdots\!56\)\( T^{9} - 362372748915616473 T^{10} - 400834638 T^{11} + T^{12} \)
$83$ \( \)\(21\!\cdots\!76\)\( + \)\(29\!\cdots\!00\)\( T + \)\(45\!\cdots\!88\)\( T^{2} - \)\(78\!\cdots\!72\)\( T^{3} - \)\(17\!\cdots\!76\)\( T^{4} + \)\(94\!\cdots\!28\)\( T^{5} + \)\(13\!\cdots\!12\)\( T^{6} - \)\(61\!\cdots\!72\)\( T^{7} - \)\(18\!\cdots\!56\)\( T^{8} + \)\(16\!\cdots\!76\)\( T^{9} - 548906193657479196 T^{10} - 1525085236 T^{11} + T^{12} \)
$89$ \( -\)\(32\!\cdots\!00\)\( - \)\(17\!\cdots\!20\)\( T - \)\(20\!\cdots\!64\)\( T^{2} + \)\(12\!\cdots\!88\)\( T^{3} + \)\(10\!\cdots\!72\)\( T^{4} + \)\(17\!\cdots\!56\)\( T^{5} - \)\(16\!\cdots\!48\)\( T^{6} - \)\(27\!\cdots\!08\)\( T^{7} + \)\(80\!\cdots\!44\)\( T^{8} + \)\(84\!\cdots\!64\)\( T^{9} - 1523577836114488436 T^{10} - 691159332 T^{11} + T^{12} \)
$97$ \( \)\(95\!\cdots\!04\)\( - \)\(71\!\cdots\!56\)\( T - \)\(75\!\cdots\!44\)\( T^{2} + \)\(43\!\cdots\!04\)\( T^{3} - \)\(75\!\cdots\!36\)\( T^{4} - \)\(71\!\cdots\!96\)\( T^{5} + \)\(38\!\cdots\!00\)\( T^{6} - \)\(29\!\cdots\!16\)\( T^{7} - \)\(35\!\cdots\!84\)\( T^{8} + \)\(58\!\cdots\!44\)\( T^{9} - 562672787603623956 T^{10} - 2494422276 T^{11} + T^{12} \)
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