# Properties

 Label 13.7.d.a Level 13 Weight 7 Character orbit 13.d Analytic conductor 2.991 Analytic rank 0 Dimension 12 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$13$$ Weight: $$k$$ = $$7$$ Character orbit: $$[\chi]$$ = 13.d (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.99070308706$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2\cdot 5^{2}\cdot 13^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} + ( \beta_{1} + \beta_{2} - \beta_{3} - \beta_{7} ) q^{3} + ( -1 + \beta_{1} - \beta_{2} - 26 \beta_{3} - \beta_{4} - \beta_{9} ) q^{4} + ( 7 - 4 \beta_{2} - 7 \beta_{3} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{5} + ( -49 + 7 \beta_{2} + 49 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{6} + ( 36 - 4 \beta_{1} + 36 \beta_{3} - \beta_{7} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{7} + ( -83 + 17 \beta_{1} - 83 \beta_{3} - 2 \beta_{4} - 3 \beta_{7} - 2 \beta_{8} - 3 \beta_{9} - \beta_{10} ) q^{8} + ( 159 - 13 \beta_{1} - 13 \beta_{2} + 13 \beta_{3} + \beta_{5} + \beta_{6} + 3 \beta_{7} + 4 \beta_{8} + \beta_{10} - \beta_{11} ) q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{2} + ( \beta_{1} + \beta_{2} - \beta_{3} - \beta_{7} ) q^{3} + ( -1 + \beta_{1} - \beta_{2} - 26 \beta_{3} - \beta_{4} - \beta_{9} ) q^{4} + ( 7 - 4 \beta_{2} - 7 \beta_{3} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{5} + ( -49 + 7 \beta_{2} + 49 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{6} + ( 36 - 4 \beta_{1} + 36 \beta_{3} - \beta_{7} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{7} + ( -83 + 17 \beta_{1} - 83 \beta_{3} - 2 \beta_{4} - 3 \beta_{7} - 2 \beta_{8} - 3 \beta_{9} - \beta_{10} ) q^{8} + ( 159 - 13 \beta_{1} - 13 \beta_{2} + 13 \beta_{3} + \beta_{5} + \beta_{6} + 3 \beta_{7} + 4 \beta_{8} + \beta_{10} - \beta_{11} ) q^{9} + ( -22 + 22 \beta_{1} - 22 \beta_{2} - 307 \beta_{3} - 5 \beta_{4} + 3 \beta_{5} - 3 \beta_{9} - 3 \beta_{10} ) q^{10} + ( 145 - 26 \beta_{1} + 145 \beta_{3} + 8 \beta_{4} + 9 \beta_{7} + 8 \beta_{8} + 9 \beta_{9} - \beta_{10} - 8 \beta_{11} ) q^{11} + ( 74 - 74 \beta_{1} + 74 \beta_{2} + 561 \beta_{3} + 15 \beta_{4} - 2 \beta_{5} + 8 \beta_{6} + 43 \beta_{9} + 2 \beta_{10} + 8 \beta_{11} ) q^{12} + ( -336 + 61 \beta_{1} + 33 \beta_{2} - 110 \beta_{3} - 8 \beta_{4} + 3 \beta_{5} + 8 \beta_{6} - 10 \beta_{7} - 12 \beta_{8} - 37 \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{13} + ( 428 - 57 \beta_{1} - 57 \beta_{2} + 57 \beta_{3} + \beta_{5} - 8 \beta_{6} + 93 \beta_{7} + 11 \beta_{8} + \beta_{10} + 8 \beta_{11} ) q^{14} + ( -1256 - 4 \beta_{2} + 1256 \beta_{3} + 40 \beta_{4} - 6 \beta_{5} - 7 \beta_{6} - 11 \beta_{7} - 40 \beta_{8} + 11 \beta_{9} ) q^{15} + ( 178 + 155 \beta_{1} + 155 \beta_{2} - 155 \beta_{3} - 3 \beta_{5} + 8 \beta_{6} - 57 \beta_{7} + 37 \beta_{8} - 3 \beta_{10} - 8 \beta_{11} ) q^{16} + ( 297 - 297 \beta_{1} + 297 \beta_{2} - 1386 \beta_{3} - 52 \beta_{4} - 7 \beta_{6} + 59 \beta_{9} - 7 \beta_{11} ) q^{17} + ( 1022 - 416 \beta_{2} - 1022 \beta_{3} - 22 \beta_{4} - \beta_{5} - 16 \beta_{6} + 117 \beta_{7} + 22 \beta_{8} - 117 \beta_{9} ) q^{18} + ( 50 - 284 \beta_{2} - 50 \beta_{3} + 9 \beta_{5} - 8 \beta_{6} - 122 \beta_{7} + 122 \beta_{9} ) q^{19} + ( -1716 + 463 \beta_{1} - 1716 \beta_{3} - 45 \beta_{4} - 248 \beta_{7} - 45 \beta_{8} - 248 \beta_{9} - 3 \beta_{10} + 16 \beta_{11} ) q^{20} + ( -135 + 918 \beta_{1} - 135 \beta_{3} - 36 \beta_{4} - 76 \beta_{7} - 36 \beta_{8} - 76 \beta_{9} + 4 \beta_{10} + \beta_{11} ) q^{21} + ( 2389 - 521 \beta_{1} - 521 \beta_{2} + 521 \beta_{3} - 7 \beta_{5} + 8 \beta_{6} - 10 \beta_{7} + 28 \beta_{8} - 7 \beta_{10} - 8 \beta_{11} ) q^{22} + ( 8 - 8 \beta_{1} + 8 \beta_{2} + 1652 \beta_{3} + 56 \beta_{4} - 28 \beta_{5} + 7 \beta_{6} + 190 \beta_{9} + 28 \beta_{10} + 7 \beta_{11} ) q^{23} + ( 3242 - 1333 \beta_{1} + 3242 \beta_{3} + 45 \beta_{4} + 168 \beta_{7} + 45 \beta_{8} + 168 \beta_{9} + 11 \beta_{10} + 32 \beta_{11} ) q^{24} + ( 151 - 151 \beta_{1} + 151 \beta_{2} - 5041 \beta_{3} + 20 \beta_{4} + 23 \beta_{5} - 33 \beta_{6} + 141 \beta_{9} - 23 \beta_{10} - 33 \beta_{11} ) q^{25} + ( -4716 + 670 \beta_{1} + 940 \beta_{2} + 3709 \beta_{3} + 27 \beta_{4} - 28 \beta_{5} - 40 \beta_{6} + 63 \beta_{7} - 70 \beta_{8} - 296 \beta_{9} - 23 \beta_{10} - 8 \beta_{11} ) q^{26} + ( -1860 + 629 \beta_{1} + 629 \beta_{2} - 629 \beta_{3} - 5 \beta_{5} + 40 \beta_{6} + 387 \beta_{7} - 8 \beta_{8} - 5 \beta_{10} - 40 \beta_{11} ) q^{27} + ( -510 - 1439 \beta_{2} + 510 \beta_{3} - 82 \beta_{4} + 61 \beta_{5} + 48 \beta_{6} + 295 \beta_{7} + 82 \beta_{8} - 295 \beta_{9} ) q^{28} + ( -7444 + 956 \beta_{1} + 956 \beta_{2} - 956 \beta_{3} + 27 \beta_{5} - 41 \beta_{6} - 110 \beta_{7} + 24 \beta_{8} + 27 \beta_{10} + 41 \beta_{11} ) q^{29} + ( 3335 - 3335 \beta_{1} + 3335 \beta_{2} + 2952 \beta_{3} + 105 \beta_{4} + 3 \beta_{5} + 48 \beta_{6} + 683 \beta_{9} - 3 \beta_{10} + 48 \beta_{11} ) q^{30} + ( 4668 - 784 \beta_{2} - 4668 \beta_{3} + 72 \beta_{4} + 5 \beta_{5} + 156 \beta_{6} - 8 \beta_{7} - 72 \beta_{8} + 8 \beta_{9} ) q^{31} + ( -6512 - 2333 \beta_{2} + 6512 \beta_{3} + 16 \beta_{4} - 25 \beta_{5} + 48 \beta_{6} - 241 \beta_{7} - 16 \beta_{8} + 241 \beta_{9} ) q^{32} + ( -4063 + 686 \beta_{1} - 4063 \beta_{3} + 140 \beta_{4} - 275 \beta_{7} + 140 \beta_{8} - 275 \beta_{9} - 36 \beta_{10} - 164 \beta_{11} ) q^{33} + ( 19825 + 3301 \beta_{1} + 19825 \beta_{3} + 238 \beta_{4} + 325 \beta_{7} + 238 \beta_{8} + 325 \beta_{9} + 31 \beta_{10} ) q^{34} + ( 13396 - 2747 \beta_{1} - 2747 \beta_{2} + 2747 \beta_{3} + 2 \beta_{5} - 116 \beta_{6} - 21 \beta_{7} - 240 \beta_{8} + 2 \beta_{10} + 116 \beta_{11} ) q^{35} + ( -2422 + 2422 \beta_{1} - 2422 \beta_{2} - 21708 \beta_{3} - 214 \beta_{4} + 85 \beta_{5} - 56 \beta_{6} - 360 \beta_{9} - 85 \beta_{10} - 56 \beta_{11} ) q^{36} + ( -1533 - 3292 \beta_{1} - 1533 \beta_{3} - 312 \beta_{4} - 159 \beta_{7} - 312 \beta_{8} - 159 \beta_{9} - 40 \beta_{10} + 65 \beta_{11} ) q^{37} + ( 160 - 160 \beta_{1} + 160 \beta_{2} - 34282 \beta_{3} - 346 \beta_{4} - 106 \beta_{5} - 72 \beta_{6} - 862 \beta_{9} + 106 \beta_{10} - 72 \beta_{11} ) q^{38} + ( 12596 + 80 \beta_{1} + 2772 \beta_{2} + 24560 \beta_{3} - 48 \beta_{4} + 83 \beta_{5} - 17 \beta_{6} + 395 \beta_{7} + 552 \beta_{8} + 285 \beta_{9} + 103 \beta_{10} + 124 \beta_{11} ) q^{39} + ( -8233 + 3942 \beta_{1} + 3942 \beta_{2} - 3942 \beta_{3} - 24 \beta_{5} + 24 \beta_{6} - 1187 \beta_{7} - 235 \beta_{8} - 24 \beta_{10} - 24 \beta_{11} ) q^{40} + ( -15910 - 74 \beta_{2} + 15910 \beta_{3} - 356 \beta_{4} - 220 \beta_{5} - 100 \beta_{6} - 351 \beta_{7} + 356 \beta_{8} + 351 \beta_{9} ) q^{41} + ( -79359 + 2226 \beta_{1} + 2226 \beta_{2} - 2226 \beta_{3} - 74 \beta_{5} - 32 \beta_{6} - 769 \beta_{7} - 965 \beta_{8} - 74 \beta_{10} + 32 \beta_{11} ) q^{42} + ( 2553 - 2553 \beta_{1} + 2553 \beta_{2} + 4400 \beta_{3} + 600 \beta_{4} - 33 \beta_{5} - 92 \beta_{6} - 1447 \beta_{9} + 33 \beta_{10} - 92 \beta_{11} ) q^{43} + ( 40075 - 2142 \beta_{2} - 40075 \beta_{3} + 13 \beta_{4} + 22 \beta_{5} - 400 \beta_{6} - 671 \beta_{7} - 13 \beta_{8} + 671 \beta_{9} ) q^{44} + ( -17308 - 8342 \beta_{2} + 17308 \beta_{3} - 220 \beta_{4} + 2 \beta_{5} - 16 \beta_{6} + 1347 \beta_{7} + 220 \beta_{8} - 1347 \beta_{9} ) q^{45} + ( -1155 - 6790 \beta_{1} - 1155 \beta_{3} + 239 \beta_{4} + 2865 \beta_{7} + 239 \beta_{8} + 2865 \beta_{9} + 218 \beta_{10} + 448 \beta_{11} ) q^{46} + ( 43968 + 6564 \beta_{1} + 43968 \beta_{3} - 496 \beta_{4} - 831 \beta_{7} - 496 \beta_{8} - 831 \beta_{9} - 239 \beta_{10} - 69 \beta_{11} ) q^{47} + ( 77109 - 2076 \beta_{1} - 2076 \beta_{2} + 2076 \beta_{3} + 104 \beta_{5} + 424 \beta_{6} - 55 \beta_{7} + 561 \beta_{8} + 104 \beta_{10} - 424 \beta_{11} ) q^{48} + ( 829 - 829 \beta_{1} + 829 \beta_{2} - 23077 \beta_{3} + 356 \beta_{4} - 26 \beta_{5} + 141 \beta_{6} - 201 \beta_{9} + 26 \beta_{10} + 141 \beta_{11} ) q^{49} + ( 5118 + 4426 \beta_{1} + 5118 \beta_{3} - 1751 \beta_{7} - 1751 \beta_{9} + 9 \beta_{10} - 368 \beta_{11} ) q^{50} + ( 591 - 591 \beta_{1} + 591 \beta_{2} - 90732 \beta_{3} + 336 \beta_{4} + 231 \beta_{5} + 492 \beta_{6} + 1167 \beta_{9} - 231 \beta_{10} + 492 \beta_{11} ) q^{51} + ( -44348 - 1927 \beta_{1} + 5696 \beta_{2} + 81564 \beta_{3} + 397 \beta_{4} - \beta_{5} + 344 \beta_{6} - 716 \beta_{7} - 347 \beta_{8} + 2790 \beta_{9} - 200 \beta_{10} - 472 \beta_{11} ) q^{52} + ( 42806 - 1980 \beta_{1} - 1980 \beta_{2} + 1980 \beta_{3} + 238 \beta_{5} - 399 \beta_{6} - 842 \beta_{7} + 744 \beta_{8} + 238 \beta_{10} + 399 \beta_{11} ) q^{53} + ( -71587 + 1171 \beta_{2} + 71587 \beta_{3} + 707 \beta_{4} + 227 \beta_{5} + 80 \beta_{6} - 600 \beta_{7} - 707 \beta_{8} + 600 \beta_{9} ) q^{54} + ( -108028 - 2630 \beta_{1} - 2630 \beta_{2} + 2630 \beta_{3} - 13 \beta_{5} + 517 \beta_{6} + 5228 \beta_{7} + 1800 \beta_{8} - 13 \beta_{10} - 517 \beta_{11} ) q^{55} + ( -4089 + 4089 \beta_{1} - 4089 \beta_{2} - 82658 \beta_{3} - 1217 \beta_{4} + 135 \beta_{5} + 24 \beta_{6} - 2119 \beta_{9} - 135 \beta_{10} + 24 \beta_{11} ) q^{56} + ( 82636 + 8000 \beta_{2} - 82636 \beta_{3} + 312 \beta_{4} - 210 \beta_{5} + 222 \beta_{6} - 946 \beta_{7} - 312 \beta_{8} + 946 \beta_{9} ) q^{57} + ( -81905 + 6310 \beta_{2} + 81905 \beta_{3} + 729 \beta_{4} + 54 \beta_{5} - 432 \beta_{6} + 1373 \beta_{7} - 729 \beta_{8} - 1373 \beta_{9} ) q^{58} + ( -82546 - 492 \beta_{1} - 82546 \beta_{3} - 704 \beta_{4} + 76 \beta_{7} - 704 \beta_{8} + 76 \beta_{9} - 287 \beta_{10} - 246 \beta_{11} ) q^{59} + ( 207579 - 16029 \beta_{1} + 207579 \beta_{3} + 910 \beta_{4} + 2993 \beta_{7} + 910 \beta_{8} + 2993 \beta_{9} + 491 \beta_{10} + 400 \beta_{11} ) q^{60} + ( 85006 + 7284 \beta_{1} + 7284 \beta_{2} - 7284 \beta_{3} - 290 \beta_{5} - 622 \beta_{6} + 456 \beta_{7} - 1096 \beta_{8} - 290 \beta_{10} + 622 \beta_{11} ) q^{61} + ( -1278 + 1278 \beta_{1} - 1278 \beta_{2} - 67472 \beta_{3} - 514 \beta_{4} - 320 \beta_{5} - 40 \beta_{6} - 1562 \beta_{9} + 320 \beta_{10} - 40 \beta_{11} ) q^{62} + ( 71926 - 1988 \beta_{1} + 71926 \beta_{3} + 1432 \beta_{4} + 2130 \beta_{7} + 1432 \beta_{8} + 2130 \beta_{9} + 299 \beta_{10} - 236 \beta_{11} ) q^{63} + ( -2737 + 2737 \beta_{1} - 2737 \beta_{2} - 209216 \beta_{3} + 265 \beta_{4} - 145 \beta_{5} - 312 \beta_{6} - 3261 \beta_{9} + 145 \beta_{10} - 312 \beta_{11} ) q^{64} + ( -99113 + 6359 \beta_{1} - 8473 \beta_{2} + 138284 \beta_{3} - 1400 \beta_{4} - 450 \beta_{5} - 147 \beta_{6} - 4753 \beta_{7} - 2100 \beta_{8} - 872 \beta_{9} + 38 \beta_{10} + 631 \beta_{11} ) q^{65} + ( -33711 + 605 \beta_{1} + 605 \beta_{2} - 605 \beta_{3} - 603 \beta_{5} + 288 \beta_{6} - 3848 \beta_{7} - 786 \beta_{8} - 603 \beta_{10} - 288 \beta_{11} ) q^{66} + ( -115019 + 7578 \beta_{2} + 115019 \beta_{3} + 512 \beta_{4} + 479 \beta_{5} - 406 \beta_{6} - 3015 \beta_{7} - 512 \beta_{8} + 3015 \beta_{9} ) q^{67} + ( -191406 - 14873 \beta_{1} - 14873 \beta_{2} + 14873 \beta_{3} + 325 \beta_{5} - 696 \beta_{6} - 3037 \beta_{7} + 689 \beta_{8} + 325 \beta_{10} + 696 \beta_{11} ) q^{68} + ( -29770 + 29770 \beta_{1} - 29770 \beta_{2} - 134280 \beta_{3} - 1488 \beta_{4} - 202 \beta_{5} - 335 \beta_{6} - 4644 \beta_{9} + 202 \beta_{10} - 335 \beta_{11} ) q^{69} + ( 242139 - 5745 \beta_{2} - 242139 \beta_{3} - 2635 \beta_{4} + 443 \beta_{5} - 32 \beta_{6} + 2326 \beta_{7} + 2635 \beta_{8} - 2326 \beta_{9} ) q^{70} + ( -45872 + 17388 \beta_{2} + 45872 \beta_{3} - 1144 \beta_{4} + 229 \beta_{5} + 721 \beta_{6} - 2019 \beta_{7} + 1144 \beta_{8} + 2019 \beta_{9} ) q^{71} + ( -147840 + 14238 \beta_{1} - 147840 \beta_{3} - 1794 \beta_{4} - 3330 \beta_{7} - 1794 \beta_{8} - 3330 \beta_{9} - 648 \beta_{10} - 336 \beta_{11} ) q^{72} + ( 229338 - 22074 \beta_{1} + 229338 \beta_{3} - 2524 \beta_{4} - 6153 \beta_{7} - 2524 \beta_{8} - 6153 \beta_{9} + 182 \beta_{10} - 922 \beta_{11} ) q^{73} + ( 280627 + 18018 \beta_{1} + 18018 \beta_{2} - 18018 \beta_{3} - 29 \beta_{5} + 320 \beta_{6} - 1605 \beta_{7} + 2493 \beta_{8} - 29 \beta_{10} - 320 \beta_{11} ) q^{74} + ( 21168 - 21168 \beta_{1} + 21168 \beta_{2} - 103044 \beta_{3} + 1440 \beta_{4} + 335 \beta_{5} - 158 \beta_{6} + 5884 \beta_{9} - 335 \beta_{10} - 158 \beta_{11} ) q^{75} + ( 3764 + 33782 \beta_{1} + 3764 \beta_{3} + 378 \beta_{4} + 1324 \beta_{7} + 378 \beta_{8} + 1324 \beta_{9} - 574 \beta_{10} + 2208 \beta_{11} ) q^{76} + ( -496 + 496 \beta_{1} - 496 \beta_{2} - 177068 \beta_{3} + 3448 \beta_{4} - 455 \beta_{5} - 1113 \beta_{6} + 8722 \beta_{9} + 455 \beta_{10} - 1113 \beta_{11} ) q^{77} + ( -4545 - 27713 \beta_{1} - 43487 \beta_{2} + 274610 \beta_{3} + 1657 \beta_{4} + 677 \beta_{5} - 1488 \beta_{6} + 10414 \beta_{7} + 1166 \beta_{8} - 5689 \beta_{9} + 499 \beta_{10} + 160 \beta_{11} ) q^{78} + ( -10124 - 26970 \beta_{1} - 26970 \beta_{2} + 26970 \beta_{3} + 115 \beta_{5} + 1371 \beta_{6} + 9428 \beta_{7} - 512 \beta_{8} + 115 \beta_{10} - 1371 \beta_{11} ) q^{79} + ( -225601 + 2241 \beta_{2} + 225601 \beta_{3} + 1465 \beta_{4} - 1091 \beta_{5} + 1408 \beta_{6} + 8324 \beta_{7} - 1465 \beta_{8} - 8324 \beta_{9} ) q^{80} + ( -334743 + 4172 \beta_{1} + 4172 \beta_{2} - 4172 \beta_{3} - 62 \beta_{5} - 998 \beta_{6} + 1752 \beta_{7} + 376 \beta_{8} - 62 \beta_{10} + 998 \beta_{11} ) q^{81} + ( 5194 - 5194 \beta_{1} + 5194 \beta_{2} - 58756 \beta_{3} + 434 \beta_{4} - 151 \beta_{5} + 1760 \beta_{6} + 20896 \beta_{9} + 151 \beta_{10} + 1760 \beta_{11} ) q^{82} + ( 91459 + 32038 \beta_{2} - 91459 \beta_{3} + 4208 \beta_{4} + 335 \beta_{5} + 1280 \beta_{6} + 517 \beta_{7} - 4208 \beta_{8} - 517 \beta_{9} ) q^{83} + ( -228107 + 76703 \beta_{2} + 228107 \beta_{3} + 1299 \beta_{4} - 897 \beta_{5} + 1248 \beta_{6} - 6148 \beta_{7} - 1299 \beta_{8} + 6148 \beta_{9} ) q^{84} + ( 4811 + 12014 \beta_{1} + 4811 \beta_{3} + 4060 \beta_{4} - 9938 \beta_{7} + 4060 \beta_{8} - 9938 \beta_{9} + 1784 \beta_{10} - 665 \beta_{11} ) q^{85} + ( 319630 - 27603 \beta_{1} + 319630 \beta_{3} + 3259 \beta_{4} + 5084 \beta_{7} + 3259 \beta_{8} + 5084 \beta_{9} - 1815 \beta_{10} + 528 \beta_{11} ) q^{86} + ( 173604 + 8730 \beta_{1} + 8730 \beta_{2} - 8730 \beta_{3} + 1120 \beta_{5} + 175 \beta_{6} + 8144 \beta_{7} - 920 \beta_{8} + 1120 \beta_{10} - 175 \beta_{11} ) q^{87} + ( -6707 + 6707 \beta_{1} - 6707 \beta_{2} - 133003 \beta_{3} - 856 \beta_{4} + 577 \beta_{5} - 688 \beta_{6} - 2282 \beta_{9} - 577 \beta_{10} - 688 \beta_{11} ) q^{88} + ( -74302 - 16574 \beta_{1} - 74302 \beta_{3} - 4660 \beta_{4} + 6811 \beta_{7} - 4660 \beta_{8} + 6811 \beta_{9} - 688 \beta_{10} - 928 \beta_{11} ) q^{89} + ( -9800 + 9800 \beta_{1} - 9800 \beta_{2} - 651954 \beta_{3} - 8810 \beta_{4} + 1379 \beta_{5} - 16 \beta_{6} - 11616 \beta_{9} - 1379 \beta_{10} - 16 \beta_{11} ) q^{90} + ( 59098 + 36699 \beta_{1} - 22841 \beta_{2} + 137267 \beta_{3} - 1040 \beta_{4} + 221 \beta_{5} + 1586 \beta_{6} - 2171 \beta_{7} + 4888 \beta_{8} - 9464 \beta_{9} - 806 \beta_{10} - 988 \beta_{11} ) q^{91} + ( 299951 - 34031 \beta_{1} - 34031 \beta_{2} + 34031 \beta_{3} + 1969 \beta_{5} - 1296 \beta_{6} + 19986 \beta_{7} + 5440 \beta_{8} + 1969 \beta_{10} + 1296 \beta_{11} ) q^{92} + ( -5898 + 23532 \beta_{2} + 5898 \beta_{3} - 5328 \beta_{4} - 278 \beta_{5} - 490 \beta_{6} - 6052 \beta_{7} + 5328 \beta_{8} + 6052 \beta_{9} ) q^{93} + ( -507944 - 9571 \beta_{1} - 9571 \beta_{2} + 9571 \beta_{3} - 969 \beta_{5} + 1912 \beta_{6} - 31477 \beta_{7} - 9059 \beta_{8} - 969 \beta_{10} - 1912 \beta_{11} ) q^{94} + ( -34422 + 34422 \beta_{1} - 34422 \beta_{2} + 41040 \beta_{3} + 8720 \beta_{4} + 751 \beta_{5} - 1452 \beta_{6} + 1770 \beta_{9} - 751 \beta_{10} - 1452 \beta_{11} ) q^{95} + ( 62813 - 20807 \beta_{2} - 62813 \beta_{3} + 43 \beta_{4} - 2455 \beta_{5} + 384 \beta_{6} + 2878 \beta_{7} - 43 \beta_{8} - 2878 \beta_{9} ) q^{96} + ( 66213 - 38064 \beta_{2} - 66213 \beta_{3} - 376 \beta_{4} + 48 \beta_{5} - 3766 \beta_{6} - 6474 \beta_{7} + 376 \beta_{8} + 6474 \beta_{9} ) q^{97} + ( 83406 + 3812 \beta_{1} + 83406 \beta_{3} + 1482 \beta_{4} + 3005 \beta_{7} + 1482 \beta_{8} + 3005 \beta_{9} + 363 \beta_{10} + 416 \beta_{11} ) q^{98} + ( 247870 - 9212 \beta_{1} + 247870 \beta_{3} - 560 \beta_{4} - 2988 \beta_{7} - 560 \beta_{8} - 2988 \beta_{9} + 1103 \beta_{10} + 2014 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 6q^{2} - 4q^{3} + 108q^{5} - 640q^{6} + 398q^{7} - 912q^{8} + 1940q^{9} + O(q^{10})$$ $$12q + 6q^{2} - 4q^{3} + 108q^{5} - 640q^{6} + 398q^{7} - 912q^{8} + 1940q^{9} + 1686q^{11} - 3926q^{13} + 5484q^{14} - 15268q^{15} + 2132q^{16} + 15254q^{18} + 1766q^{19} - 19044q^{20} + 3428q^{21} + 28832q^{22} + 31608q^{24} - 58266q^{26} - 20464q^{27} + 4092q^{28} - 90108q^{29} + 61014q^{31} - 64932q^{32} - 44452q^{33} + 259896q^{34} + 158772q^{35} - 40212q^{37} + 137852q^{39} - 104196q^{40} - 190416q^{41} - 959204q^{42} + 489372q^{44} - 151444q^{45} - 44412q^{46} + 562446q^{47} + 930308q^{48} + 82422q^{50} - 578500q^{52} + 509136q^{53} - 871432q^{54} - 1264036q^{55} + 939908q^{57} - 1019980q^{58} - 994458q^{59} + 2407804q^{60} + 1013696q^{61} + 865778q^{63} - 1130064q^{65} - 418352q^{66} - 1442386q^{67} - 2313132q^{68} + 2958968q^{70} - 655866q^{71} - 1706508q^{72} + 2588228q^{73} + 3373752q^{74} + 246984q^{76} + 77480q^{78} - 75316q^{79} - 2685408q^{80} - 4016140q^{81} + 894966q^{83} - 3220504q^{84} + 105396q^{85} + 3704832q^{86} + 2109064q^{87} - 977376q^{89} + 1088750q^{91} + 3682872q^{92} - 216268q^{93} - 6238300q^{94} + 896384q^{96} + 983388q^{97} + 1039302q^{98} + 2894714q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 6 x^{11} + 18 x^{10} + 488 x^{9} + 36205 x^{8} - 155430 x^{7} + 399962 x^{6} + 9502784 x^{5} + 275595012 x^{4} - 541321656 x^{3} + 523196552 x^{2} - 242221824 x + 56070144$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$357114067457877400091 \nu^{11} - 2128261731755613609450 \nu^{10} + 6345490399526750340438 \nu^{9} + 174860636719079419952536 \nu^{8} + 12925952551033055488927871 \nu^{7} - 54942822772347566670635754 \nu^{6} + 140832422401403167276894318 \nu^{5} + 3408416628369323032624656352 \nu^{4} + 98035211725237854012633477804 \nu^{3} - 188190316198958158128876499272 \nu^{2} + 181983402678115217418930098200 \nu - 84271945543930352398314518784$$$$)/$$$$41\!\cdots\!00$$ $$\beta_{3}$$ $$=$$ $$($$$$3241403613457336365199 \nu^{11} - 17992588404576068826450 \nu^{10} + 50993985296434053794382 \nu^{9} + 1601071496665980448309304 \nu^{8} + 118081786138293446450576419 \nu^{7} - 450359340256545622236943506 \nu^{6} + 1120793045650815378413957702 \nu^{5} + 31218989329148360235860700128 \nu^{4} + 907491573934919722127792205756 \nu^{3} - 1337326456492378754493918632808 \nu^{2} + 1299516318703631221526793457400 \nu - 395539390258771826127888936576$$$$)/$$$$20\!\cdots\!00$$ $$\beta_{4}$$ $$=$$ $$($$$$-22912445860669066253413 \nu^{11} + 101527098802980281143875 \nu^{10} - 256306667489467113090084 \nu^{9} - 11528098898159150707367048 \nu^{8} - 847855458750438196463051953 \nu^{7} + 2230473399748094133745196847 \nu^{6} - 5762775984044183221026812924 \nu^{5} - 224937926601773064178639285136 \nu^{4} - 6672498646502570675933302792572 \nu^{3} + 1943965170319355491379614921596 \nu^{2} - 9544084396769930498809171568000 \nu + 2904568872219137156875798874112$$$$)/$$$$55\!\cdots\!00$$ $$\beta_{5}$$ $$=$$ $$($$$$-92073754120669108051559 \nu^{11} + 585294363815167122063150 \nu^{10} - 1823874315754389341950512 \nu^{9} - 44447331103509094213158064 \nu^{8} - 3322262130728207350541254379 \nu^{7} + 15527333843268327969982514046 \nu^{6} - 40805763389300746844112120832 \nu^{5} - 865648917115961028856086597448 \nu^{4} - 25169942396198623513457361024396 \nu^{3} + 59300502400341256538607672406728 \nu^{2} - 57157125302550536238255245617600 \nu + 26428359035169628420344790358016$$$$)/$$$$16\!\cdots\!00$$ $$\beta_{6}$$ $$=$$ $$($$$$1280371007734425507571841 \nu^{11} - 15841998837522733266745050 \nu^{10} + 64140261566921047002107088 \nu^{9} + 520679752698828203132183536 \nu^{8} + 42255335220777824890377513821 \nu^{7} - 498435274698423750267666347754 \nu^{6} + 1493613446546286664243328987968 \nu^{5} + 9974969093535582133925928569752 \nu^{4} + 272725350192007604776556851298004 \nu^{3} - 3020283937772783448085994684638872 \nu^{2} + 2880417746017754970252043077313600 \nu - 1325333929538630898153683679504384$$$$)/$$$$16\!\cdots\!00$$ $$\beta_{7}$$ $$=$$ $$($$$$-88005002947366457599642 \nu^{11} + 442999245001071970552725 \nu^{10} - 1114382541663357479277956 \nu^{9} - 44209257227011221123982232 \nu^{8} - 3228526341860911283090556402 \nu^{7} + 10580426150111151354121985873 \nu^{6} - 23433632862012524401290141516 \nu^{5} - 863418945334969351238539131624 \nu^{4} - 25079013543394492511478818033248 \nu^{3} + 23822003024067271348959931592564 \nu^{2} - 10940434783497474177671455075200 \nu - 519024354207783574933614920192$$$$)/$$$$92\!\cdots\!00$$ $$\beta_{8}$$ $$=$$ $$($$$$88037053331792348246522 \nu^{11} - 443182717922660954688725 \nu^{10} + 1115691367884769143073796 \nu^{9} + 44201785535279230498722712 \nu^{8} + 3229778379044534211147241682 \nu^{7} - 10584869781771561117273768593 \nu^{6} + 23466607933025157737019703756 \nu^{5} + 862566403154558956660818090984 \nu^{4} + 25090398570457238614441836191968 \nu^{3} - 23832797348704030556843055121524 \nu^{2} + 10945387839612531410160577891200 \nu - 7735872675020737540437472892928$$$$)/$$$$92\!\cdots\!00$$ $$\beta_{9}$$ $$=$$ $$($$$$-6528610652683580397349 \nu^{11} + 39294880871740546305399 \nu^{10} - 115090599091069741302540 \nu^{9} - 3199736340743872899054344 \nu^{8} - 236264736620067253323865393 \nu^{7} + 1020587240325504485325671955 \nu^{6} - 2513817665804821796544677540 \nu^{5} - 62372791415195418092622157136 \nu^{4} - 1797190261476222261974023861020 \nu^{3} + 3594756045422440548635250133644 \nu^{2} - 2574841647580113523311974255072 \nu + 783763789746483243998432451072$$$$)/$$$$66\!\cdots\!32$$ $$\beta_{10}$$ $$=$$ $$($$$$1206107845578338721073207 \nu^{11} - 6674359625525494178037600 \nu^{10} + 18300524098582539654248376 \nu^{9} + 598492614910151512229725472 \nu^{8} + 43942846036790063072591724667 \nu^{7} - 167134087416288733042825165608 \nu^{6} + 396565556869553713880142802136 \nu^{5} + 11673245849259161340248963845304 \nu^{4} + 337771952224323265068968287674108 \nu^{3} - 497807711938793955881411855894544 \nu^{2} + 378237620862801487477085843595200 \nu - 69974418310187098813340655034368$$$$)/$$$$83\!\cdots\!00$$ $$\beta_{11}$$ $$=$$ $$($$$$18818270548927347405555317 \nu^{11} - 104120432833520459028858600 \nu^{10} + 286035933925434023882990856 \nu^{9} + 9338203466932758402662828032 \nu^{8} + 685623199938251875643472304577 \nu^{7} - 2606835122282705077212062708448 \nu^{6} + 6160360793754360550118500469416 \nu^{5} + 182211381749628968703296147441224 \nu^{4} + 5270192220960805727664242174976948 \nu^{3} - 7767270297607681075250087377547664 \nu^{2} + 5060507631507659690918549648507200 \nu - 1091811988682700188423951481643008$$$$)/$$$$16\!\cdots\!00$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{9} + \beta_{4} + 90 \beta_{3} - \beta_{2} + \beta_{1} - 1$$ $$\nu^{3}$$ $$=$$ $$-\beta_{8} + \beta_{5} + \beta_{4} + 203 \beta_{3} - 145 \beta_{2} - 203$$ $$\nu^{4}$$ $$=$$ $$-8 \beta_{11} + \beta_{10} - 151 \beta_{8} - 237 \beta_{7} + 8 \beta_{6} + \beta_{5} + 229 \beta_{3} - 229 \beta_{2} - 229 \beta_{1} - 12974$$ $$\nu^{5}$$ $$=$$ $$-32 \beta_{11} + 271 \beta_{10} - 196 \beta_{9} - 227 \beta_{8} - 196 \beta_{7} - 227 \beta_{4} - 18442 \beta_{3} - 21909 \beta_{1} - 18442$$ $$\nu^{6}$$ $$=$$ $$-2200 \beta_{11} + 531 \beta_{10} - 47025 \beta_{9} - 2200 \beta_{6} - 531 \beta_{5} - 23503 \beta_{4} - 1971602 \beta_{3} + 46621 \beta_{2} - 46621 \beta_{1} + 46621$$ $$\nu^{7}$$ $$=$$ $$-79632 \beta_{9} + 47607 \beta_{8} + 79632 \beta_{7} - 12896 \beta_{6} - 56887 \beta_{5} - 47607 \beta_{4} - 7589847 \beta_{3} + 3427153 \beta_{2} + 7589847$$ $$\nu^{8}$$ $$=$$ $$467992 \beta_{11} - 162311 \beta_{10} + 3755579 \beta_{8} + 8718005 \beta_{7} - 467992 \beta_{6} - 162311 \beta_{5} - 9165741 \beta_{3} + 9165741 \beta_{2} + 9165741 \beta_{1} + 311456790$$ $$\nu^{9}$$ $$=$$ $$3532960 \beta_{11} - 10914595 \beta_{10} + 22071372 \beta_{9} + 9671615 \beta_{8} + 22071372 \beta_{7} + 9671615 \beta_{4} + 854289306 \beta_{3} + 550736589 \beta_{1} + 854289306$$ $$\nu^{10}$$ $$=$$ $$90849720 \beta_{11} - 40051887 \beta_{10} + 1567964169 \beta_{9} + 90849720 \beta_{6} + 40051887 \beta_{5} + 612691199 \beta_{4} + 50643771034 \beta_{3} - 1779235357 \beta_{2} + 1779235357 \beta_{1} - 1779235357$$ $$\nu^{11}$$ $$=$$ $$5203970696 \beta_{9} - 1928696959 \beta_{8} - 5203970696 \beta_{7} + 822529632 \beta_{6} + 2011466823 \beta_{5} + 1928696959 \beta_{4} + 259296282423 \beta_{3} - 90386183129 \beta_{2} - 259296282423$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/13\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-\beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 9.50218 + 9.50218i 7.85712 + 7.85712i 0.561862 + 0.561862i 0.388469 + 0.388469i −6.57888 − 6.57888i −8.73075 − 8.73075i 9.50218 − 9.50218i 7.85712 − 7.85712i 0.561862 − 0.561862i 0.388469 − 0.388469i −6.57888 + 6.57888i −8.73075 + 8.73075i
−8.50218 + 8.50218i −17.1101 80.5742i 73.6481 73.6481i 145.473 145.473i −214.232 214.232i 140.917 + 140.917i −436.244 1252.34i
5.2 −6.85712 + 6.85712i 38.7457 30.0401i −127.232 + 127.232i −265.684 + 265.684i 215.792 + 215.792i −232.867 232.867i 772.229 1744.89i
5.3 0.438138 0.438138i −27.8396 63.6161i −78.9164 + 78.9164i −12.1976 + 12.1976i −88.1833 88.1833i 55.9134 + 55.9134i 46.0426 69.1525i
5.4 0.611531 0.611531i 18.7379 63.2521i 134.607 134.607i 11.4588 11.4588i 241.226 + 241.226i 77.8186 + 77.8186i −377.891 164.633i
5.5 7.57888 7.57888i 26.7789 50.8787i −59.0322 + 59.0322i 202.954 202.954i −226.950 226.950i 99.4446 + 99.4446i −11.8881 894.795i
5.6 9.73075 9.73075i −41.3128 125.375i 110.925 110.925i −402.005 + 402.005i 271.347 + 271.347i −597.226 597.226i 977.751 2158.77i
8.1 −8.50218 8.50218i −17.1101 80.5742i 73.6481 + 73.6481i 145.473 + 145.473i −214.232 + 214.232i 140.917 140.917i −436.244 1252.34i
8.2 −6.85712 6.85712i 38.7457 30.0401i −127.232 127.232i −265.684 265.684i 215.792 215.792i −232.867 + 232.867i 772.229 1744.89i
8.3 0.438138 + 0.438138i −27.8396 63.6161i −78.9164 78.9164i −12.1976 12.1976i −88.1833 + 88.1833i 55.9134 55.9134i 46.0426 69.1525i
8.4 0.611531 + 0.611531i 18.7379 63.2521i 134.607 + 134.607i 11.4588 + 11.4588i 241.226 241.226i 77.8186 77.8186i −377.891 164.633i
8.5 7.57888 + 7.57888i 26.7789 50.8787i −59.0322 59.0322i 202.954 + 202.954i −226.950 + 226.950i 99.4446 99.4446i −11.8881 894.795i
8.6 9.73075 + 9.73075i −41.3128 125.375i 110.925 + 110.925i −402.005 402.005i 271.347 271.347i −597.226 + 597.226i 977.751 2158.77i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 8.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.d odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.7.d.a 12
3.b odd 2 1 117.7.j.b 12
4.b odd 2 1 208.7.t.c 12
13.d odd 4 1 inner 13.7.d.a 12
39.f even 4 1 117.7.j.b 12
52.f even 4 1 208.7.t.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.7.d.a 12 1.a even 1 1 trivial
13.7.d.a 12 13.d odd 4 1 inner
117.7.j.b 12 3.b odd 2 1
117.7.j.b 12 39.f even 4 1
208.7.t.c 12 4.b odd 2 1
208.7.t.c 12 52.f even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{7}^{\mathrm{new}}(13, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 6 T + 18 T^{2} + 140 T^{3} - 6655 T^{4} + 38334 T^{5} - 100414 T^{6} + 726352 T^{7} - 18453952 T^{8} + 65125376 T^{9} + 109600768 T^{10} - 13164609536 T^{11} + 244439318528 T^{12} - 842535010304 T^{13} + 448924745728 T^{14} + 17072226566144 T^{15} - 309605938757632 T^{16} + 779914521346048 T^{17} - 6900397536968704 T^{18} + 168594714956660736 T^{19} - 1873215970009415680 T^{20} + 2522015791327477760 T^{21} + 20752587082923245568 T^{22} -$$$$44\!\cdots\!84$$$$T^{23} +$$$$47\!\cdots\!96$$$$T^{24}$$
$3$ $$( 1 + 2 T + 1704 T^{2} + 6330 T^{3} + 2136456 T^{4} + 7744698 T^{5} + 1696048146 T^{6} + 5645884842 T^{7} + 1135400313096 T^{8} + 2452371695370 T^{9} + 481259930163624 T^{10} + 411782264189298 T^{11} + 150094635296999121 T^{12} )^{2}$$
$5$ $$1 - 108 T + 5832 T^{2} + 31196 T^{3} - 770587876 T^{4} + 59922609780 T^{5} - 1977086768200 T^{6} - 361180635738500 T^{7} + 326413718853155000 T^{8} - 16465642976655137500 T^{9} +$$$$13\!\cdots\!00$$$$T^{10} +$$$$21\!\cdots\!00$$$$T^{11} -$$$$96\!\cdots\!50$$$$T^{12} +$$$$33\!\cdots\!00$$$$T^{13} +$$$$33\!\cdots\!00$$$$T^{14} -$$$$62\!\cdots\!00$$$$T^{15} +$$$$19\!\cdots\!00$$$$T^{16} -$$$$33\!\cdots\!00$$$$T^{17} -$$$$28\!\cdots\!00$$$$T^{18} +$$$$13\!\cdots\!00$$$$T^{19} -$$$$27\!\cdots\!00$$$$T^{20} +$$$$17\!\cdots\!00$$$$T^{21} +$$$$50\!\cdots\!00$$$$T^{22} -$$$$14\!\cdots\!00$$$$T^{23} +$$$$21\!\cdots\!25$$$$T^{24}$$
$7$ $$1 - 398 T + 79202 T^{2} - 28188670 T^{3} - 10722345216 T^{4} + 3939021777462 T^{5} - 321198923447794 T^{6} - 176763204490308034 T^{7} +$$$$55\!\cdots\!08$$$$T^{8} -$$$$12\!\cdots\!14$$$$T^{9} +$$$$16\!\cdots\!54$$$$T^{10} -$$$$90\!\cdots\!14$$$$T^{11} -$$$$37\!\cdots\!10$$$$T^{12} -$$$$10\!\cdots\!86$$$$T^{13} +$$$$23\!\cdots\!54$$$$T^{14} -$$$$20\!\cdots\!86$$$$T^{15} +$$$$10\!\cdots\!08$$$$T^{16} -$$$$39\!\cdots\!66$$$$T^{17} -$$$$85\!\cdots\!94$$$$T^{18} +$$$$12\!\cdots\!38$$$$T^{19} -$$$$39\!\cdots\!16$$$$T^{20} -$$$$12\!\cdots\!30$$$$T^{21} +$$$$40\!\cdots\!02$$$$T^{22} -$$$$23\!\cdots\!02$$$$T^{23} +$$$$70\!\cdots\!01$$$$T^{24}$$
$11$ $$1 - 1686 T + 1421298 T^{2} - 9449711854 T^{3} + 21784578331358 T^{4} - 16083795355398150 T^{5} + 40803428417812876874 T^{6} -$$$$11\!\cdots\!94$$$$T^{7} +$$$$99\!\cdots\!75$$$$T^{8} -$$$$10\!\cdots\!04$$$$T^{9} +$$$$34\!\cdots\!28$$$$T^{10} -$$$$39\!\cdots\!32$$$$T^{11} +$$$$22\!\cdots\!72$$$$T^{12} -$$$$69\!\cdots\!52$$$$T^{13} +$$$$10\!\cdots\!88$$$$T^{14} -$$$$57\!\cdots\!24$$$$T^{15} +$$$$97\!\cdots\!75$$$$T^{16} -$$$$19\!\cdots\!94$$$$T^{17} +$$$$12\!\cdots\!14$$$$T^{18} -$$$$88\!\cdots\!50$$$$T^{19} +$$$$21\!\cdots\!98$$$$T^{20} -$$$$16\!\cdots\!14$$$$T^{21} +$$$$43\!\cdots\!98$$$$T^{22} -$$$$90\!\cdots\!46$$$$T^{23} +$$$$95\!\cdots\!21$$$$T^{24}$$
$13$ $$1 + 3926 T - 3214718 T^{2} - 41376896314 T^{3} - 59058634955853 T^{4} + 107869141290743820 T^{5} +$$$$51\!\cdots\!12$$$$T^{6} +$$$$52\!\cdots\!80$$$$T^{7} -$$$$13\!\cdots\!93$$$$T^{8} -$$$$46\!\cdots\!06$$$$T^{9} -$$$$17\!\cdots\!98$$$$T^{10} +$$$$10\!\cdots\!74$$$$T^{11} +$$$$12\!\cdots\!41$$$$T^{12}$$
$17$ $$1 - 95568036 T^{2} + 4520630311261512 T^{4} -$$$$18\!\cdots\!40$$$$T^{6} +$$$$66\!\cdots\!80$$$$T^{8} -$$$$19\!\cdots\!68$$$$T^{10} +$$$$48\!\cdots\!06$$$$T^{12} -$$$$11\!\cdots\!48$$$$T^{14} +$$$$22\!\cdots\!80$$$$T^{16} -$$$$35\!\cdots\!40$$$$T^{18} +$$$$52\!\cdots\!92$$$$T^{20} -$$$$64\!\cdots\!36$$$$T^{22} +$$$$39\!\cdots\!61$$$$T^{24}$$
$19$ $$1 - 1766 T + 1559378 T^{2} - 696664382926 T^{3} + 1154247310387398 T^{4} + 28961508664554671754 T^{5} +$$$$18\!\cdots\!30$$$$T^{6} -$$$$58\!\cdots\!02$$$$T^{7} -$$$$11\!\cdots\!37$$$$T^{8} -$$$$42\!\cdots\!84$$$$T^{9} +$$$$33\!\cdots\!00$$$$T^{10} -$$$$27\!\cdots\!60$$$$T^{11} +$$$$24\!\cdots\!32$$$$T^{12} -$$$$13\!\cdots\!60$$$$T^{13} +$$$$73\!\cdots\!00$$$$T^{14} -$$$$43\!\cdots\!44$$$$T^{15} -$$$$55\!\cdots\!77$$$$T^{16} -$$$$13\!\cdots\!02$$$$T^{17} +$$$$20\!\cdots\!30$$$$T^{18} +$$$$14\!\cdots\!94$$$$T^{19} +$$$$27\!\cdots\!18$$$$T^{20} -$$$$78\!\cdots\!46$$$$T^{21} +$$$$82\!\cdots\!78$$$$T^{22} -$$$$44\!\cdots\!46$$$$T^{23} +$$$$11\!\cdots\!61$$$$T^{24}$$
$23$ $$1 - 814212660 T^{2} + 317916327712063914 T^{4} -$$$$86\!\cdots\!44$$$$T^{6} +$$$$19\!\cdots\!11$$$$T^{8} -$$$$36\!\cdots\!04$$$$T^{10} +$$$$58\!\cdots\!32$$$$T^{12} -$$$$79\!\cdots\!84$$$$T^{14} +$$$$92\!\cdots\!51$$$$T^{16} -$$$$90\!\cdots\!84$$$$T^{18} +$$$$73\!\cdots\!34$$$$T^{20} -$$$$41\!\cdots\!60$$$$T^{22} +$$$$11\!\cdots\!21$$$$T^{24}$$
$29$ $$( 1 + 45054 T + 3480794474 T^{2} + 115344513309902 T^{3} + 5022872858956651955 T^{4} +$$$$12\!\cdots\!56$$$$T^{5} +$$$$39\!\cdots\!64$$$$T^{6} +$$$$75\!\cdots\!76$$$$T^{7} +$$$$17\!\cdots\!55$$$$T^{8} +$$$$24\!\cdots\!22$$$$T^{9} +$$$$43\!\cdots\!94$$$$T^{10} +$$$$33\!\cdots\!54$$$$T^{11} +$$$$44\!\cdots\!21$$$$T^{12} )^{2}$$
$31$ $$1 - 61014 T + 1861354098 T^{2} - 53209300169710 T^{3} + 2357813398795510374 T^{4} -$$$$94\!\cdots\!82$$$$T^{5} +$$$$27\!\cdots\!46$$$$T^{6} -$$$$72\!\cdots\!38$$$$T^{7} +$$$$25\!\cdots\!35$$$$T^{8} -$$$$10\!\cdots\!80$$$$T^{9} +$$$$34\!\cdots\!28$$$$T^{10} -$$$$96\!\cdots\!40$$$$T^{11} +$$$$27\!\cdots\!24$$$$T^{12} -$$$$85\!\cdots\!40$$$$T^{13} +$$$$26\!\cdots\!08$$$$T^{14} -$$$$73\!\cdots\!80$$$$T^{15} +$$$$15\!\cdots\!35$$$$T^{16} -$$$$39\!\cdots\!38$$$$T^{17} +$$$$13\!\cdots\!26$$$$T^{18} -$$$$41\!\cdots\!02$$$$T^{19} +$$$$90\!\cdots\!34$$$$T^{20} -$$$$18\!\cdots\!10$$$$T^{21} +$$$$56\!\cdots\!98$$$$T^{22} -$$$$16\!\cdots\!34$$$$T^{23} +$$$$23\!\cdots\!61$$$$T^{24}$$
$37$ $$1 + 40212 T + 808502472 T^{2} - 94108369185124 T^{3} - 2212353752837064900 T^{4} +$$$$60\!\cdots\!48$$$$T^{5} +$$$$30\!\cdots\!64$$$$T^{6} +$$$$15\!\cdots\!64$$$$T^{7} -$$$$30\!\cdots\!60$$$$T^{8} -$$$$71\!\cdots\!72$$$$T^{9} +$$$$73\!\cdots\!72$$$$T^{10} +$$$$75\!\cdots\!96$$$$T^{11} +$$$$81\!\cdots\!62$$$$T^{12} +$$$$19\!\cdots\!64$$$$T^{13} +$$$$48\!\cdots\!32$$$$T^{14} -$$$$12\!\cdots\!88$$$$T^{15} -$$$$13\!\cdots\!60$$$$T^{16} +$$$$17\!\cdots\!36$$$$T^{17} +$$$$86\!\cdots\!24$$$$T^{18} +$$$$44\!\cdots\!12$$$$T^{19} -$$$$41\!\cdots\!00$$$$T^{20} -$$$$45\!\cdots\!36$$$$T^{21} +$$$$99\!\cdots\!72$$$$T^{22} +$$$$12\!\cdots\!08$$$$T^{23} +$$$$81\!\cdots\!81$$$$T^{24}$$
$41$ $$1 + 190416 T + 18129126528 T^{2} + 1488204135776624 T^{3} + 64138724847656792258 T^{4} -$$$$31\!\cdots\!60$$$$T^{5} -$$$$64\!\cdots\!96$$$$T^{6} -$$$$70\!\cdots\!64$$$$T^{7} -$$$$50\!\cdots\!05$$$$T^{8} -$$$$13\!\cdots\!76$$$$T^{9} +$$$$57\!\cdots\!56$$$$T^{10} +$$$$14\!\cdots\!24$$$$T^{11} +$$$$14\!\cdots\!88$$$$T^{12} +$$$$68\!\cdots\!84$$$$T^{13} +$$$$12\!\cdots\!36$$$$T^{14} -$$$$14\!\cdots\!96$$$$T^{15} -$$$$25\!\cdots\!05$$$$T^{16} -$$$$17\!\cdots\!64$$$$T^{17} -$$$$74\!\cdots\!36$$$$T^{18} -$$$$16\!\cdots\!60$$$$T^{19} +$$$$16\!\cdots\!18$$$$T^{20} +$$$$18\!\cdots\!64$$$$T^{21} +$$$$10\!\cdots\!28$$$$T^{22} +$$$$52\!\cdots\!56$$$$T^{23} +$$$$13\!\cdots\!81$$$$T^{24}$$
$43$ $$1 - 36018404640 T^{2} +$$$$71\!\cdots\!12$$$$T^{4} -$$$$98\!\cdots\!24$$$$T^{6} +$$$$10\!\cdots\!56$$$$T^{8} -$$$$87\!\cdots\!36$$$$T^{10} +$$$$60\!\cdots\!62$$$$T^{12} -$$$$34\!\cdots\!36$$$$T^{14} +$$$$16\!\cdots\!56$$$$T^{16} -$$$$62\!\cdots\!24$$$$T^{18} +$$$$18\!\cdots\!12$$$$T^{20} -$$$$36\!\cdots\!40$$$$T^{22} +$$$$40\!\cdots\!01$$$$T^{24}$$
$47$ $$1 - 562446 T + 158172751458 T^{2} - 33342810078438670 T^{3} +$$$$58\!\cdots\!68$$$$T^{4} -$$$$81\!\cdots\!86$$$$T^{5} +$$$$91\!\cdots\!62$$$$T^{6} -$$$$79\!\cdots\!34$$$$T^{7} +$$$$33\!\cdots\!20$$$$T^{8} +$$$$42\!\cdots\!90$$$$T^{9} -$$$$12\!\cdots\!38$$$$T^{10} +$$$$19\!\cdots\!94$$$$T^{11} -$$$$23\!\cdots\!42$$$$T^{12} +$$$$21\!\cdots\!26$$$$T^{13} -$$$$14\!\cdots\!58$$$$T^{14} +$$$$52\!\cdots\!10$$$$T^{15} +$$$$45\!\cdots\!20$$$$T^{16} -$$$$11\!\cdots\!66$$$$T^{17} +$$$$14\!\cdots\!02$$$$T^{18} -$$$$13\!\cdots\!74$$$$T^{19} +$$$$10\!\cdots\!48$$$$T^{20} -$$$$65\!\cdots\!30$$$$T^{21} +$$$$33\!\cdots\!58$$$$T^{22} -$$$$12\!\cdots\!34$$$$T^{23} +$$$$24\!\cdots\!41$$$$T^{24}$$
$53$ $$( 1 - 254568 T + 108652497230 T^{2} - 20683934542944136 T^{3} +$$$$52\!\cdots\!83$$$$T^{4} -$$$$79\!\cdots\!72$$$$T^{5} +$$$$14\!\cdots\!32$$$$T^{6} -$$$$17\!\cdots\!88$$$$T^{7} +$$$$26\!\cdots\!03$$$$T^{8} -$$$$22\!\cdots\!04$$$$T^{9} +$$$$26\!\cdots\!30$$$$T^{10} -$$$$13\!\cdots\!32$$$$T^{11} +$$$$11\!\cdots\!21$$$$T^{12} )^{2}$$
$59$ $$1 + 994458 T + 494473356882 T^{2} + 189903786626655986 T^{3} +$$$$68\!\cdots\!90$$$$T^{4} +$$$$22\!\cdots\!94$$$$T^{5} +$$$$67\!\cdots\!70$$$$T^{6} +$$$$18\!\cdots\!14$$$$T^{7} +$$$$48\!\cdots\!75$$$$T^{8} +$$$$11\!\cdots\!88$$$$T^{9} +$$$$27\!\cdots\!76$$$$T^{10} +$$$$61\!\cdots\!96$$$$T^{11} +$$$$13\!\cdots\!24$$$$T^{12} +$$$$25\!\cdots\!36$$$$T^{13} +$$$$48\!\cdots\!56$$$$T^{14} +$$$$89\!\cdots\!48$$$$T^{15} +$$$$15\!\cdots\!75$$$$T^{16} +$$$$24\!\cdots\!14$$$$T^{17} +$$$$37\!\cdots\!70$$$$T^{18} +$$$$54\!\cdots\!14$$$$T^{19} +$$$$68\!\cdots\!90$$$$T^{20} +$$$$80\!\cdots\!46$$$$T^{21} +$$$$88\!\cdots\!82$$$$T^{22} +$$$$74\!\cdots\!78$$$$T^{23} +$$$$31\!\cdots\!81$$$$T^{24}$$
$61$ $$( 1 - 506848 T + 282902598538 T^{2} - 105182501059034848 T^{3} +$$$$35\!\cdots\!39$$$$T^{4} -$$$$97\!\cdots\!56$$$$T^{5} +$$$$24\!\cdots\!88$$$$T^{6} -$$$$49\!\cdots\!16$$$$T^{7} +$$$$94\!\cdots\!19$$$$T^{8} -$$$$14\!\cdots\!88$$$$T^{9} +$$$$19\!\cdots\!58$$$$T^{10} -$$$$18\!\cdots\!48$$$$T^{11} +$$$$18\!\cdots\!61$$$$T^{12} )^{2}$$
$67$ $$1 + 1442386 T + 1040238686498 T^{2} + 574663178423383274 T^{3} +$$$$28\!\cdots\!54$$$$T^{4} +$$$$12\!\cdots\!82$$$$T^{5} +$$$$49\!\cdots\!98$$$$T^{6} +$$$$18\!\cdots\!90$$$$T^{7} +$$$$66\!\cdots\!55$$$$T^{8} +$$$$23\!\cdots\!36$$$$T^{9} +$$$$77\!\cdots\!36$$$$T^{10} +$$$$25\!\cdots\!32$$$$T^{11} +$$$$78\!\cdots\!36$$$$T^{12} +$$$$22\!\cdots\!08$$$$T^{13} +$$$$63\!\cdots\!96$$$$T^{14} +$$$$17\!\cdots\!24$$$$T^{15} +$$$$44\!\cdots\!55$$$$T^{16} +$$$$11\!\cdots\!10$$$$T^{17} +$$$$26\!\cdots\!38$$$$T^{18} +$$$$61\!\cdots\!98$$$$T^{19} +$$$$12\!\cdots\!14$$$$T^{20} +$$$$23\!\cdots\!46$$$$T^{21} +$$$$38\!\cdots\!98$$$$T^{22} +$$$$47\!\cdots\!34$$$$T^{23} +$$$$30\!\cdots\!61$$$$T^{24}$$
$71$ $$1 + 655866 T + 215080104978 T^{2} + 141159966904204586 T^{3} +$$$$93\!\cdots\!48$$$$T^{4} +$$$$25\!\cdots\!74$$$$T^{5} +$$$$62\!\cdots\!38$$$$T^{6} +$$$$33\!\cdots\!78$$$$T^{7} +$$$$73\!\cdots\!60$$$$T^{8} -$$$$16\!\cdots\!30$$$$T^{9} -$$$$63\!\cdots\!38$$$$T^{10} -$$$$36\!\cdots\!50$$$$T^{11} -$$$$20\!\cdots\!42$$$$T^{12} -$$$$46\!\cdots\!50$$$$T^{13} -$$$$10\!\cdots\!58$$$$T^{14} -$$$$33\!\cdots\!30$$$$T^{15} +$$$$19\!\cdots\!60$$$$T^{16} +$$$$11\!\cdots\!78$$$$T^{17} +$$$$27\!\cdots\!98$$$$T^{18} +$$$$14\!\cdots\!34$$$$T^{19} +$$$$68\!\cdots\!28$$$$T^{20} +$$$$13\!\cdots\!66$$$$T^{21} +$$$$25\!\cdots\!78$$$$T^{22} +$$$$99\!\cdots\!86$$$$T^{23} +$$$$19\!\cdots\!41$$$$T^{24}$$
$73$ $$1 - 2588228 T + 3349462089992 T^{2} - 2980435039024420492 T^{3} +$$$$20\!\cdots\!26$$$$T^{4} -$$$$11\!\cdots\!48$$$$T^{5} +$$$$49\!\cdots\!84$$$$T^{6} -$$$$16\!\cdots\!44$$$$T^{7} +$$$$27\!\cdots\!95$$$$T^{8} +$$$$10\!\cdots\!08$$$$T^{9} -$$$$12\!\cdots\!48$$$$T^{10} +$$$$77\!\cdots\!00$$$$T^{11} -$$$$34\!\cdots\!20$$$$T^{12} +$$$$11\!\cdots\!00$$$$T^{13} -$$$$29\!\cdots\!08$$$$T^{14} +$$$$36\!\cdots\!52$$$$T^{15} +$$$$14\!\cdots\!95$$$$T^{16} -$$$$12\!\cdots\!56$$$$T^{17} +$$$$59\!\cdots\!24$$$$T^{18} -$$$$20\!\cdots\!92$$$$T^{19} +$$$$56\!\cdots\!06$$$$T^{20} -$$$$12\!\cdots\!28$$$$T^{21} +$$$$21\!\cdots\!92$$$$T^{22} -$$$$24\!\cdots\!92$$$$T^{23} +$$$$14\!\cdots\!21$$$$T^{24}$$
$79$ $$( 1 + 37658 T + 781884604234 T^{2} - 44764114305384646 T^{3} +$$$$35\!\cdots\!55$$$$T^{4} -$$$$18\!\cdots\!44$$$$T^{5} +$$$$10\!\cdots\!76$$$$T^{6} -$$$$44\!\cdots\!24$$$$T^{7} +$$$$20\!\cdots\!55$$$$T^{8} -$$$$64\!\cdots\!06$$$$T^{9} +$$$$27\!\cdots\!54$$$$T^{10} +$$$$31\!\cdots\!58$$$$T^{11} +$$$$20\!\cdots\!21$$$$T^{12} )^{2}$$
$83$ $$1 - 894966 T + 400482070578 T^{2} - 28674110295070414 T^{3} -$$$$31\!\cdots\!42$$$$T^{4} +$$$$47\!\cdots\!86$$$$T^{5} -$$$$29\!\cdots\!02$$$$T^{6} +$$$$46\!\cdots\!02$$$$T^{7} -$$$$19\!\cdots\!85$$$$T^{8} -$$$$15\!\cdots\!48$$$$T^{9} +$$$$81\!\cdots\!84$$$$T^{10} -$$$$39\!\cdots\!56$$$$T^{11} +$$$$26\!\cdots\!72$$$$T^{12} -$$$$12\!\cdots\!64$$$$T^{13} +$$$$87\!\cdots\!24$$$$T^{14} -$$$$54\!\cdots\!32$$$$T^{15} -$$$$22\!\cdots\!85$$$$T^{16} +$$$$17\!\cdots\!98$$$$T^{17} -$$$$35\!\cdots\!62$$$$T^{18} +$$$$18\!\cdots\!54$$$$T^{19} -$$$$41\!\cdots\!22$$$$T^{20} -$$$$12\!\cdots\!06$$$$T^{21} +$$$$55\!\cdots\!78$$$$T^{22} -$$$$40\!\cdots\!54$$$$T^{23} +$$$$14\!\cdots\!61$$$$T^{24}$$
$89$ $$1 + 977376 T + 477631922688 T^{2} + 751481099068671296 T^{3} +$$$$91\!\cdots\!82$$$$T^{4} +$$$$32\!\cdots\!88$$$$T^{5} +$$$$15\!\cdots\!80$$$$T^{6} +$$$$21\!\cdots\!88$$$$T^{7} -$$$$48\!\cdots\!97$$$$T^{8} -$$$$14\!\cdots\!96$$$$T^{9} -$$$$50\!\cdots\!00$$$$T^{10} -$$$$76\!\cdots\!96$$$$T^{11} -$$$$10\!\cdots\!76$$$$T^{12} -$$$$38\!\cdots\!56$$$$T^{13} -$$$$12\!\cdots\!00$$$$T^{14} -$$$$18\!\cdots\!76$$$$T^{15} -$$$$29\!\cdots\!77$$$$T^{16} +$$$$66\!\cdots\!88$$$$T^{17} +$$$$23\!\cdots\!80$$$$T^{18} +$$$$24\!\cdots\!48$$$$T^{19} +$$$$34\!\cdots\!42$$$$T^{20} +$$$$13\!\cdots\!36$$$$T^{21} +$$$$43\!\cdots\!88$$$$T^{22} +$$$$44\!\cdots\!36$$$$T^{23} +$$$$22\!\cdots\!21$$$$T^{24}$$
$97$ $$1 - 983388 T + 483525979272 T^{2} - 532915345861301044 T^{3} +$$$$79\!\cdots\!30$$$$T^{4} -$$$$71\!\cdots\!48$$$$T^{5} +$$$$46\!\cdots\!32$$$$T^{6} -$$$$68\!\cdots\!88$$$$T^{7} +$$$$13\!\cdots\!95$$$$T^{8} -$$$$89\!\cdots\!64$$$$T^{9} +$$$$46\!\cdots\!56$$$$T^{10} -$$$$52\!\cdots\!88$$$$T^{11} +$$$$52\!\cdots\!08$$$$T^{12} -$$$$43\!\cdots\!52$$$$T^{13} +$$$$32\!\cdots\!96$$$$T^{14} -$$$$51\!\cdots\!96$$$$T^{15} +$$$$66\!\cdots\!95$$$$T^{16} -$$$$27\!\cdots\!12$$$$T^{17} +$$$$15\!\cdots\!72$$$$T^{18} -$$$$19\!\cdots\!32$$$$T^{19} +$$$$18\!\cdots\!30$$$$T^{20} -$$$$10\!\cdots\!36$$$$T^{21} +$$$$77\!\cdots\!72$$$$T^{22} -$$$$13\!\cdots\!52$$$$T^{23} +$$$$11\!\cdots\!41$$$$T^{24}$$