Properties

Label 43.8.a.b
Level 43
Weight 8
Character orbit 43.a
Self dual yes
Analytic conductor 13.433
Analytic rank 0
Dimension 13
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 43.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.4325560958\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{1} ) q^{2} + ( 7 + \beta_{3} ) q^{3} + ( 71 + 2 \beta_{1} - \beta_{3} + \beta_{4} ) q^{4} + ( 76 + \beta_{1} + \beta_{3} - \beta_{8} ) q^{5} + ( 14 - 3 \beta_{1} + 5 \beta_{3} + \beta_{4} - \beta_{10} ) q^{6} + ( 100 + 16 \beta_{1} + 3 \beta_{3} - \beta_{9} + \beta_{10} ) q^{7} + ( 278 + 90 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{8} + \beta_{9} - \beta_{11} ) q^{8} + ( 755 + 58 \beta_{1} + 2 \beta_{2} + 11 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - \beta_{6} - \beta_{7} + 4 \beta_{8} + 2 \beta_{9} + 4 \beta_{10} + 2 \beta_{11} - \beta_{12} ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{1} ) q^{2} + ( 7 + \beta_{3} ) q^{3} + ( 71 + 2 \beta_{1} - \beta_{3} + \beta_{4} ) q^{4} + ( 76 + \beta_{1} + \beta_{3} - \beta_{8} ) q^{5} + ( 14 - 3 \beta_{1} + 5 \beta_{3} + \beta_{4} - \beta_{10} ) q^{6} + ( 100 + 16 \beta_{1} + 3 \beta_{3} - \beta_{9} + \beta_{10} ) q^{7} + ( 278 + 90 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{8} + \beta_{9} - \beta_{11} ) q^{8} + ( 755 + 58 \beta_{1} + 2 \beta_{2} + 11 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - \beta_{6} - \beta_{7} + 4 \beta_{8} + 2 \beta_{9} + 4 \beta_{10} + 2 \beta_{11} - \beta_{12} ) q^{9} + ( 332 + 113 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} - \beta_{4} + \beta_{5} + 3 \beta_{6} + 4 \beta_{7} + \beta_{8} + 4 \beta_{9} - \beta_{10} + 2 \beta_{11} + 4 \beta_{12} ) q^{10} + ( 98 + 76 \beta_{1} - 5 \beta_{2} + 13 \beta_{3} - 10 \beta_{4} + \beta_{5} + \beta_{6} - 4 \beta_{7} - 2 \beta_{8} - 5 \beta_{9} + \beta_{10} - \beta_{11} - 3 \beta_{12} ) q^{11} + ( -1540 + 79 \beta_{1} - 4 \beta_{2} + 38 \beta_{3} - \beta_{4} + 5 \beta_{5} - 6 \beta_{6} + \beta_{7} + \beta_{8} - 10 \beta_{9} - 11 \beta_{10} - 6 \beta_{11} - 5 \beta_{12} ) q^{12} + ( 1019 + 91 \beta_{1} - 2 \beta_{2} - 13 \beta_{3} - 24 \beta_{4} - 3 \beta_{5} - 6 \beta_{6} - 6 \beta_{7} - 5 \beta_{8} - 6 \beta_{9} - \beta_{10} - 8 \beta_{11} + 5 \beta_{12} ) q^{13} + ( 3391 + 133 \beta_{1} + 5 \beta_{2} - 93 \beta_{3} + 2 \beta_{4} - 7 \beta_{5} + 6 \beta_{6} + 5 \beta_{7} + \beta_{8} + 8 \beta_{9} + 4 \beta_{11} + 5 \beta_{12} ) q^{14} + ( 2382 + 25 \beta_{1} + 19 \beta_{2} + 39 \beta_{3} - 9 \beta_{4} + 5 \beta_{5} + 14 \beta_{6} + 9 \beta_{7} - 25 \beta_{8} + 12 \beta_{9} + \beta_{10} + 19 \beta_{11} - 11 \beta_{12} ) q^{15} + ( 8746 + 226 \beta_{1} - 23 \beta_{2} - 109 \beta_{3} + 36 \beta_{4} + 7 \beta_{5} + \beta_{6} - 16 \beta_{7} - 7 \beta_{8} + 8 \beta_{9} - 2 \beta_{10} + 6 \beta_{11} ) q^{16} + ( 8577 - 103 \beta_{1} - \beta_{2} - 66 \beta_{3} - \beta_{4} - 2 \beta_{5} - 10 \beta_{6} + 31 \beta_{7} - 13 \beta_{8} + 7 \beta_{9} + 9 \beta_{10} + 3 \beta_{11} + 34 \beta_{12} ) q^{17} + ( 12309 + 224 \beta_{1} + 35 \beta_{2} - 307 \beta_{3} + 43 \beta_{4} - 34 \beta_{5} - 4 \beta_{6} - 21 \beta_{7} + 49 \beta_{8} + 13 \beta_{9} + 21 \beta_{10} - 7 \beta_{11} - 23 \beta_{12} ) q^{18} + ( 8053 + 220 \beta_{1} - 15 \beta_{2} - 124 \beta_{3} - 2 \beta_{4} + 10 \beta_{5} + 7 \beta_{6} + 2 \beta_{7} - 8 \beta_{8} - 12 \beta_{9} + 13 \beta_{10} - 25 \beta_{11} - 32 \beta_{12} ) q^{19} + ( 12970 - 203 \beta_{1} + \beta_{2} - 157 \beta_{3} + 107 \beta_{4} + 30 \beta_{5} - 17 \beta_{6} - 51 \beta_{7} + 14 \beta_{8} - 36 \beta_{9} - 11 \beta_{10} - 18 \beta_{11} + 19 \beta_{12} ) q^{20} + ( 10250 - 1417 \beta_{1} - 34 \beta_{2} + 359 \beta_{3} - 15 \beta_{4} + 5 \beta_{5} - 33 \beta_{6} + 19 \beta_{7} + 19 \beta_{8} - 21 \beta_{9} - 21 \beta_{10} - 14 \beta_{11} + 21 \beta_{12} ) q^{21} + ( 15760 - 828 \beta_{1} - 60 \beta_{2} - 151 \beta_{3} + 76 \beta_{4} - 9 \beta_{5} + 18 \beta_{6} + 72 \beta_{7} + 24 \beta_{8} - 13 \beta_{9} + 16 \beta_{10} + 21 \beta_{11} - 24 \beta_{12} ) q^{22} + ( 12566 - 1632 \beta_{1} + 47 \beta_{2} + 340 \beta_{3} - 31 \beta_{4} + 26 \beta_{5} + 56 \beta_{6} - 39 \beta_{7} - 16 \beta_{8} - \beta_{9} - 9 \beta_{10} + 11 \beta_{11} - 6 \beta_{12} ) q^{23} + ( 12781 - 1992 \beta_{1} - 31 \beta_{2} + 466 \beta_{3} + 43 \beta_{4} + 7 \beta_{5} - 26 \beta_{6} - 3 \beta_{7} + 37 \beta_{8} - 32 \beta_{9} - 41 \beta_{10} + 18 \beta_{11} + \beta_{12} ) q^{24} + ( 20911 - 1754 \beta_{1} + 65 \beta_{2} + 775 \beta_{3} - 183 \beta_{4} + 13 \beta_{5} + 34 \beta_{6} - 61 \beta_{7} - 72 \beta_{8} + 87 \beta_{9} - 34 \beta_{10} + 19 \beta_{11} + 29 \beta_{12} ) q^{25} + ( 21306 - 1066 \beta_{1} + 164 \beta_{2} - 331 \beta_{3} - 12 \beta_{4} - 89 \beta_{5} + 50 \beta_{6} + 168 \beta_{7} + 44 \beta_{8} + 67 \beta_{9} + 76 \beta_{10} + 73 \beta_{11} + 40 \beta_{12} ) q^{26} + ( 19995 - 4223 \beta_{1} - 92 \beta_{2} + 1504 \beta_{3} - 216 \beta_{4} - 26 \beta_{5} - 8 \beta_{6} - 48 \beta_{7} + 23 \beta_{8} + 38 \beta_{9} - 64 \beta_{10} + 40 \beta_{11} + 8 \beta_{12} ) q^{27} + ( 15447 + 2713 \beta_{1} + 21 \beta_{2} - 384 \beta_{3} - 138 \beta_{4} - 4 \beta_{5} - 64 \beta_{6} - 80 \beta_{7} - 40 \beta_{8} - 12 \beta_{9} + 32 \beta_{10} - 110 \beta_{11} - 82 \beta_{12} ) q^{28} + ( 22390 - 1721 \beta_{1} - 27 \beta_{2} + 219 \beta_{3} - 154 \beta_{4} - 94 \beta_{5} - 113 \beta_{6} - 14 \beta_{7} + 115 \beta_{8} - 49 \beta_{9} + 38 \beta_{10} - 115 \beta_{11} - 30 \beta_{12} ) q^{29} + ( 7989 + 239 \beta_{1} + 40 \beta_{2} + 473 \beta_{3} - 180 \beta_{4} + 53 \beta_{5} - 14 \beta_{6} - 88 \beta_{7} - 154 \beta_{8} + 23 \beta_{9} - 94 \beta_{10} + 15 \beta_{11} + 142 \beta_{12} ) q^{30} + ( -7972 + 336 \beta_{1} - 150 \beta_{2} + 660 \beta_{3} - 153 \beta_{4} + 98 \beta_{5} - 59 \beta_{6} + 63 \beta_{7} - 106 \beta_{8} - 122 \beta_{9} - 27 \beta_{10} - 98 \beta_{11} - 106 \beta_{12} ) q^{31} + ( 14536 + 4400 \beta_{1} - 323 \beta_{2} - 961 \beta_{3} + 440 \beta_{4} + 215 \beta_{5} + 61 \beta_{6} + 166 \beta_{7} - 233 \beta_{8} - 118 \beta_{9} + 82 \beta_{10} + 112 \beta_{11} - 82 \beta_{12} ) q^{32} + ( 41160 - 948 \beta_{1} - 20 \beta_{2} + 85 \beta_{3} - 4 \beta_{4} - 154 \beta_{5} + 60 \beta_{6} + 4 \beta_{7} + 188 \beta_{8} + 163 \beta_{9} + 103 \beta_{10} + 104 \beta_{11} - 62 \beta_{12} ) q^{33} + ( -8304 + 9009 \beta_{1} + 178 \beta_{2} - 1551 \beta_{3} - 175 \beta_{4} + 210 \beta_{5} + 113 \beta_{6} - 361 \beta_{7} - 470 \beta_{8} + 32 \beta_{9} - 79 \beta_{10} - 44 \beta_{11} + 209 \beta_{12} ) q^{34} + ( -14002 - 237 \beta_{1} + 200 \beta_{2} - 1155 \beta_{3} + 637 \beta_{4} + 109 \beta_{5} + 201 \beta_{6} + 163 \beta_{7} + 31 \beta_{8} - 137 \beta_{9} + 53 \beta_{10} + 124 \beta_{11} - 87 \beta_{12} ) q^{35} + ( -48602 + 12952 \beta_{1} + 318 \beta_{2} - 3816 \beta_{3} - 112 \beta_{4} - 326 \beta_{5} - 41 \beta_{6} + 153 \beta_{7} + 462 \beta_{8} - 52 \beta_{9} + 213 \beta_{10} - 344 \beta_{11} - 155 \beta_{12} ) q^{36} + ( 14148 - 1591 \beta_{1} + 89 \beta_{2} - 512 \beta_{3} + 240 \beta_{4} - 94 \beta_{5} - 111 \beta_{6} + 8 \beta_{7} - 35 \beta_{8} + 64 \beta_{9} - 45 \beta_{10} + 141 \beta_{11} + 210 \beta_{12} ) q^{37} + ( 49167 + 9476 \beta_{1} - 204 \beta_{2} - 3909 \beta_{3} + 591 \beta_{4} - 292 \beta_{5} + 113 \beta_{6} - 139 \beta_{7} + 334 \beta_{8} + 230 \beta_{9} + 261 \beta_{10} + 200 \beta_{11} + 17 \beta_{12} ) q^{38} + ( -59960 - 5634 \beta_{1} - 90 \beta_{2} + 2027 \beta_{3} + 392 \beta_{4} + 50 \beta_{5} - 106 \beta_{6} + 156 \beta_{7} + 450 \beta_{8} - 91 \beta_{9} - 313 \beta_{10} + 42 \beta_{11} + 406 \beta_{12} ) q^{39} + ( -71867 + 14982 \beta_{1} - 158 \beta_{2} - 1315 \beta_{3} + 69 \beta_{4} - 48 \beta_{5} - 115 \beta_{6} + 641 \beta_{7} - 140 \beta_{8} + 280 \beta_{9} + 87 \beta_{10} + 184 \beta_{11} + 41 \beta_{12} ) q^{40} + ( -30743 + 194 \beta_{1} + 292 \beta_{2} - 1289 \beta_{3} + 1224 \beta_{4} + 3 \beta_{5} - 396 \beta_{6} - 462 \beta_{7} + 496 \beta_{8} - 237 \beta_{9} - 106 \beta_{10} - 214 \beta_{11} - 235 \beta_{12} ) q^{41} + ( -264631 + 2411 \beta_{1} - 219 \beta_{2} + 4802 \beta_{3} - 624 \beta_{4} + 318 \beta_{5} - 27 \beta_{6} - 342 \beta_{7} - 219 \beta_{8} - 283 \beta_{9} - 780 \beta_{10} - 109 \beta_{11} - 116 \beta_{12} ) q^{42} -79507 q^{43} + ( -170995 + 15674 \beta_{1} - 889 \beta_{2} - 118 \beta_{3} + 483 \beta_{4} + 397 \beta_{5} - 437 \beta_{6} - 1106 \beta_{7} + 241 \beta_{8} - 590 \beta_{9} - 302 \beta_{10} - 724 \beta_{11} - 550 \beta_{12} ) q^{44} + ( -82375 + 2287 \beta_{1} + 1106 \beta_{2} + 1309 \beta_{3} - 162 \beta_{4} + 112 \beta_{5} + 1236 \beta_{6} + 278 \beta_{7} - 509 \beta_{8} + 1143 \beta_{9} + 273 \beta_{10} + 704 \beta_{11} - 752 \beta_{12} ) q^{45} + ( -308074 + 3625 \beta_{1} + 231 \beta_{2} + 3910 \beta_{3} - 2113 \beta_{4} - 267 \beta_{5} + 1077 \beta_{7} - 543 \beta_{8} + 216 \beta_{9} + 33 \beta_{10} + 342 \beta_{11} + 435 \beta_{12} ) q^{46} + ( -29082 - 9150 \beta_{1} - 647 \beta_{2} - 4263 \beta_{3} + 681 \beta_{4} - 175 \beta_{5} - 166 \beta_{6} - 337 \beta_{7} + 582 \beta_{8} - 643 \beta_{9} + 428 \beta_{10} - 611 \beta_{11} - 59 \beta_{12} ) q^{47} + ( -183749 + 310 \beta_{1} - 133 \beta_{2} + 6128 \beta_{3} - 1401 \beta_{4} - 43 \beta_{5} + 496 \beta_{6} - 199 \beta_{7} - 149 \beta_{8} + 616 \beta_{9} + 281 \beta_{10} + 386 \beta_{11} + 213 \beta_{12} ) q^{48} + ( -117087 - 12507 \beta_{1} - 730 \beta_{2} - 2161 \beta_{3} - 1921 \beta_{4} - 353 \beta_{5} - 279 \beta_{6} + 793 \beta_{7} - 963 \beta_{8} - 271 \beta_{9} + 305 \beta_{10} - 470 \beta_{11} + 467 \beta_{12} ) q^{49} + ( -311433 - 12040 \beta_{1} + 944 \beta_{2} + 5682 \beta_{3} - 875 \beta_{4} + 21 \beta_{5} + 117 \beta_{6} + 1367 \beta_{7} - 852 \beta_{8} + 561 \beta_{9} - 187 \beta_{10} + 647 \beta_{11} + 757 \beta_{12} ) q^{50} + ( -178949 - 26436 \beta_{1} - 531 \beta_{2} + 8451 \beta_{3} - 778 \beta_{4} + 950 \beta_{5} - 669 \beta_{6} - 38 \beta_{7} - 336 \beta_{8} - 947 \beta_{9} - 116 \beta_{10} - 541 \beta_{11} + 1704 \beta_{12} ) q^{51} + ( -322793 + 2622 \beta_{1} + 2123 \beta_{2} + 984 \beta_{3} - 2159 \beta_{4} - 663 \beta_{5} + 95 \beta_{6} - 1858 \beta_{7} - 91 \beta_{8} - 158 \beta_{9} + 770 \beta_{10} - 76 \beta_{11} - 758 \beta_{12} ) q^{52} + ( 313769 - 21398 \beta_{1} - 370 \beta_{2} - 5216 \beta_{3} + 1777 \beta_{4} - 98 \beta_{5} - 409 \beta_{6} - 847 \beta_{7} - 334 \beta_{8} + 853 \beta_{9} + 392 \beta_{10} + 104 \beta_{11} + 442 \beta_{12} ) q^{53} + ( -802796 - 25388 \beta_{1} - 1577 \beta_{2} + 20483 \beta_{3} - 1964 \beta_{4} + 1111 \beta_{5} - 527 \beta_{6} + 150 \beta_{7} - 163 \beta_{8} - 1858 \beta_{9} - 1406 \beta_{10} - 716 \beta_{11} - 1486 \beta_{12} ) q^{54} + ( 37311 - 26065 \beta_{1} - 743 \beta_{2} - 3619 \beta_{3} - 258 \beta_{4} - 292 \beta_{5} + 939 \beta_{6} - 142 \beta_{7} - 567 \beta_{8} - 48 \beta_{9} + 719 \beta_{10} + 815 \beta_{11} - 872 \beta_{12} ) q^{55} + ( 125470 - 10932 \beta_{1} + 1204 \beta_{2} - 4232 \beta_{3} + 3514 \beta_{4} - 938 \beta_{5} - 68 \beta_{6} + 426 \beta_{7} + 2042 \beta_{8} + 1324 \beta_{9} + 1152 \beta_{10} + 1024 \beta_{11} + 198 \beta_{12} ) q^{56} + ( -230306 - 48978 \beta_{1} - 409 \beta_{2} + 6004 \beta_{3} + 1410 \beta_{4} + 798 \beta_{5} + 269 \beta_{6} + 842 \beta_{7} - 1130 \beta_{8} - 1043 \beta_{9} - 686 \beta_{10} + 445 \beta_{11} - 2148 \beta_{12} ) q^{57} + ( -310022 - 911 \beta_{1} + 1582 \beta_{2} - 2393 \beta_{3} - 1463 \beta_{4} - 1974 \beta_{5} - 50 \beta_{6} - 572 \beta_{7} + 2614 \beta_{8} + 332 \beta_{9} + 511 \beta_{10} + 100 \beta_{11} - 904 \beta_{12} ) q^{58} + ( 181420 - 11483 \beta_{1} + 896 \beta_{2} - 16331 \beta_{3} + 2891 \beta_{4} - 441 \beta_{5} + 227 \beta_{6} + 161 \beta_{7} + 1809 \beta_{8} + 393 \beta_{9} - 609 \beta_{10} - 1184 \beta_{11} - 529 \beta_{12} ) q^{59} + ( -218684 - 16892 \beta_{1} - 939 \beta_{2} + 2639 \beta_{3} + 1372 \beta_{4} + 435 \beta_{5} - 1157 \beta_{6} + 1924 \beta_{7} + 903 \beta_{8} - 546 \beta_{9} - 836 \beta_{10} - 1368 \beta_{11} + 3244 \beta_{12} ) q^{60} + ( 478863 + 1669 \beta_{1} + 695 \beta_{2} - 9909 \beta_{3} + 2386 \beta_{4} + 748 \beta_{5} + 729 \beta_{6} - 902 \beta_{7} - 1677 \beta_{8} + 1544 \beta_{9} - 295 \beta_{10} + 1813 \beta_{11} + 500 \beta_{12} ) q^{61} + ( 77534 - 34379 \beta_{1} - 1550 \beta_{2} - 3575 \beta_{3} + 3959 \beta_{4} + 260 \beta_{5} + 681 \beta_{6} - 1105 \beta_{7} + 202 \beta_{8} + 482 \beta_{9} - 1877 \beta_{10} + 962 \beta_{11} + 465 \beta_{12} ) q^{62} + ( 895685 + 18259 \beta_{1} + 245 \beta_{2} - 11073 \beta_{3} + 1506 \beta_{4} - 1644 \beta_{5} - 1485 \beta_{6} - 762 \beta_{7} + 4405 \beta_{8} + 1864 \beta_{9} + 1671 \beta_{10} - 1037 \beta_{11} + 1372 \beta_{12} ) q^{63} + ( -252170 + 63098 \beta_{1} - 4275 \beta_{2} - 7099 \beta_{3} + 3916 \beta_{4} + 3805 \beta_{5} + 1235 \beta_{6} - 714 \beta_{7} - 1691 \beta_{8} - 1498 \beta_{9} - 1650 \beta_{10} - 1060 \beta_{11} + 418 \beta_{12} ) q^{64} + ( 443139 - 26771 \beta_{1} - 155 \beta_{2} - 23513 \beta_{3} + 918 \beta_{4} + 56 \beta_{5} - 821 \beta_{6} + 462 \beta_{7} - 2453 \beta_{8} - 748 \beta_{9} - 2893 \beta_{10} - 2089 \beta_{11} + 1532 \beta_{12} ) q^{65} + ( -175693 + 36369 \beta_{1} - 1745 \beta_{2} + 515 \beta_{3} - 886 \beta_{4} - 203 \beta_{5} - 1236 \beta_{6} - 2431 \beta_{7} + 1907 \beta_{8} - 2494 \beta_{9} + 1612 \beta_{10} - 2090 \beta_{11} - 3263 \beta_{12} ) q^{66} + ( -159168 + 20354 \beta_{1} + 2685 \beta_{2} - 325 \beta_{3} - 1082 \beta_{4} + 751 \beta_{5} + 287 \beta_{6} + 852 \beta_{7} - 308 \beta_{8} - 2143 \beta_{9} - 1089 \beta_{10} + 629 \beta_{11} - 1829 \beta_{12} ) q^{67} + ( 715832 + 29894 \beta_{1} + 4382 \beta_{2} - 18806 \beta_{3} + 7132 \beta_{4} + 608 \beta_{5} + 4155 \beta_{6} + 6495 \beta_{7} - 2396 \beta_{8} + 5324 \beta_{9} + 1955 \beta_{10} + 4392 \beta_{11} + 2051 \beta_{12} ) q^{68} + ( 1020097 + 20005 \beta_{1} + 1710 \beta_{2} + 4383 \beta_{3} - 5566 \beta_{4} - 940 \beta_{5} + 972 \beta_{6} + 658 \beta_{7} - 1827 \beta_{8} + 2885 \beta_{9} + 2167 \beta_{10} + 2148 \beta_{11} - 40 \beta_{12} ) q^{69} + ( -103361 + 68673 \beta_{1} - 683 \beta_{2} + 1030 \beta_{3} - 6584 \beta_{4} - 310 \beta_{5} - 581 \beta_{6} - 1260 \beta_{7} - 1827 \beta_{8} - 263 \beta_{9} - 292 \beta_{10} - 757 \beta_{11} + 1270 \beta_{12} ) q^{70} + ( 376973 + 9083 \beta_{1} - 1259 \beta_{2} + 10660 \beta_{3} - 3652 \beta_{4} + 560 \beta_{5} - 1363 \beta_{6} - 552 \beta_{7} + 177 \beta_{8} - 2995 \beta_{9} - 776 \beta_{10} - 981 \beta_{11} - 3992 \beta_{12} ) q^{71} + ( 891169 - 56064 \beta_{1} + 3191 \beta_{2} - 10928 \beta_{3} - 2471 \beta_{4} - 5555 \beta_{5} - 1031 \beta_{6} - 1425 \beta_{7} + 880 \beta_{8} - 905 \beta_{9} + 5299 \beta_{10} + 381 \beta_{11} + 931 \beta_{12} ) q^{72} + ( 622856 + 32614 \beta_{1} - 5360 \beta_{2} + 7045 \beta_{3} - 4292 \beta_{4} + 24 \beta_{5} - 4816 \beta_{6} - 2272 \beta_{7} - 1046 \beta_{8} - 5131 \beta_{9} - 2305 \beta_{10} - 2300 \beta_{11} - 928 \beta_{12} ) q^{73} + ( -295909 + 53438 \beta_{1} + 2127 \beta_{2} + 16052 \beta_{3} - 5135 \beta_{4} + 1851 \beta_{5} - 180 \beta_{6} + 713 \beta_{7} - 813 \beta_{8} - 264 \beta_{9} - 731 \beta_{10} - 1208 \beta_{11} + 369 \beta_{12} ) q^{74} + ( 2065262 + 36632 \beta_{1} + 5385 \beta_{2} + 27565 \beta_{3} - 9610 \beta_{4} - 1984 \beta_{5} + 4047 \beta_{6} + 94 \beta_{7} - 3672 \beta_{8} + 7596 \beta_{9} + 5287 \beta_{10} + 6715 \beta_{11} + 1474 \beta_{12} ) q^{75} + ( 791127 + 164160 \beta_{1} - 3404 \beta_{2} - 16553 \beta_{3} + 6529 \beta_{4} + 206 \beta_{5} - 2219 \beta_{6} - 1737 \beta_{7} + 5034 \beta_{8} - 2480 \beta_{9} + 5879 \beta_{10} - 1156 \beta_{11} - 2321 \beta_{12} ) q^{76} + ( 1339373 - 49817 \beta_{1} - 2849 \beta_{2} - 10819 \beta_{3} - 5524 \beta_{4} - 1210 \beta_{5} + 1947 \beta_{6} + 3704 \beta_{7} - 3615 \beta_{8} - 1834 \beta_{9} - 845 \beta_{10} + 1097 \beta_{11} + 2442 \beta_{12} ) q^{77} + ( -1175961 - 55445 \beta_{1} - 2273 \beta_{2} + 58385 \beta_{3} - 4156 \beta_{4} + 4353 \beta_{5} - 2244 \beta_{6} - 1827 \beta_{7} - 5905 \beta_{8} - 5414 \beta_{9} - 5990 \beta_{10} - 3218 \beta_{11} - 1563 \beta_{12} ) q^{78} + ( 489690 + 167934 \beta_{1} - 1334 \beta_{2} + 22003 \beta_{3} - 8801 \beta_{4} + 1657 \beta_{5} + 1201 \beta_{6} + 497 \beta_{7} + 3560 \beta_{8} + 650 \beta_{9} - 4646 \beta_{10} - 3066 \beta_{11} + 4425 \beta_{12} ) q^{79} + ( 1224795 - 30856 \beta_{1} - 3898 \beta_{2} - 9863 \beta_{3} + 5609 \beta_{4} + 712 \beta_{5} + 1655 \beta_{6} - 6539 \beta_{7} - 4682 \beta_{8} + 1210 \beta_{9} - 1203 \beta_{10} - 906 \beta_{11} - 3779 \beta_{12} ) q^{80} + ( 2673607 + 131165 \beta_{1} - 27 \beta_{2} - 19285 \beta_{3} - 4629 \beta_{4} - 2321 \beta_{5} + 3624 \beta_{6} - 4015 \beta_{7} + 2325 \beta_{8} + 4090 \beta_{9} + 6159 \beta_{10} + 87 \beta_{11} + 1825 \beta_{12} ) q^{81} + ( -62215 + 125432 \beta_{1} + 6754 \beta_{2} + 5624 \beta_{3} + 405 \beta_{4} - 4627 \beta_{5} - 39 \beta_{6} + 6185 \beta_{7} + 10380 \beta_{8} + 7965 \beta_{9} - 769 \beta_{10} + 2717 \beta_{11} + 2005 \beta_{12} ) q^{82} + ( 1752808 + 24511 \beta_{1} + 295 \beta_{2} + 47500 \beta_{3} + 8051 \beta_{4} + 1650 \beta_{5} - 3422 \beta_{6} - 861 \beta_{7} + 7413 \beta_{8} + 1560 \beta_{9} - 3080 \beta_{10} + 819 \beta_{11} - 2426 \beta_{12} ) q^{83} + ( -1040802 - 209390 \beta_{1} + 282 \beta_{2} + 50640 \beta_{3} + 10832 \beta_{4} + 3044 \beta_{5} + 2542 \beta_{6} + 5510 \beta_{7} - 4884 \beta_{8} - 1340 \beta_{9} - 6150 \beta_{10} + 2996 \beta_{11} - 2990 \beta_{12} ) q^{84} + ( -88156 + 226146 \beta_{1} + 2241 \beta_{2} - 24149 \beta_{3} + 3320 \beta_{4} + 4698 \beta_{5} - 1959 \beta_{6} + 5424 \beta_{7} - 3924 \beta_{8} - 3242 \beta_{9} - 8515 \beta_{10} - 8427 \beta_{11} + 1554 \beta_{12} ) q^{85} + ( -79507 - 79507 \beta_{1} ) q^{86} + ( 707090 - 62124 \beta_{1} - 3618 \beta_{2} + 52869 \beta_{3} + 6223 \beta_{4} - 643 \beta_{5} - 4495 \beta_{6} - 339 \beta_{7} + 11058 \beta_{8} - 6106 \beta_{9} - 762 \beta_{10} - 1842 \beta_{11} - 2195 \beta_{12} ) q^{87} + ( 866556 + 48260 \beta_{1} - 488 \beta_{2} - 12122 \beta_{3} + 28906 \beta_{4} + 654 \beta_{5} + 1017 \beta_{6} + 5418 \beta_{7} + 10165 \beta_{8} + 9817 \beta_{9} - 2054 \beta_{10} + 4453 \beta_{11} + 1910 \beta_{12} ) q^{88} + ( 723159 - 71443 \beta_{1} + 3113 \beta_{2} + 4977 \beta_{3} - 3768 \beta_{4} + 5584 \beta_{5} + 5105 \beta_{6} - 3840 \beta_{7} - 3797 \beta_{8} - 3656 \beta_{9} - 3539 \beta_{10} + 55 \beta_{11} + 4480 \beta_{12} ) q^{89} + ( 260723 - 168987 \beta_{1} - 910 \beta_{2} - 18364 \beta_{3} - 1362 \beta_{4} - 3654 \beta_{5} - 3451 \beta_{6} - 6793 \beta_{7} - 4412 \beta_{8} - 458 \beta_{9} + 5242 \beta_{10} - 344 \beta_{11} + 2127 \beta_{12} ) q^{90} + ( 1959909 + 57519 \beta_{1} + 6015 \beta_{2} - 14901 \beta_{3} + 11580 \beta_{4} - 3270 \beta_{5} + 8079 \beta_{6} + 3732 \beta_{7} + 5697 \beta_{8} + 2724 \beta_{9} + 7281 \beta_{10} + 10173 \beta_{11} + 3750 \beta_{12} ) q^{91} + ( -1006594 - 427738 \beta_{1} - 651 \beta_{2} - 15495 \beta_{3} - 2778 \beta_{4} - 1595 \beta_{5} - 8644 \beta_{6} - 9549 \beta_{7} - 10179 \beta_{8} - 9014 \beta_{9} - 5181 \beta_{10} - 5592 \beta_{11} + 1119 \beta_{12} ) q^{92} + ( 2020233 - 31384 \beta_{1} + 3435 \beta_{2} - 56101 \beta_{3} + 17982 \beta_{4} - 2068 \beta_{5} + 1749 \beta_{6} + 3466 \beta_{7} + 240 \beta_{8} + 3 \beta_{9} + 7830 \beta_{10} + 901 \beta_{11} - 5108 \beta_{12} ) q^{93} + ( -1912846 + 140603 \beta_{1} - 3466 \beta_{2} - 60738 \beta_{3} - 10259 \beta_{4} - 5767 \beta_{5} + 2223 \beta_{6} + 3827 \beta_{7} + 10102 \beta_{8} + 1815 \beta_{9} + 9625 \beta_{10} + 381 \beta_{11} - 4875 \beta_{12} ) q^{94} + ( 2393634 - 155241 \beta_{1} - 11535 \beta_{2} + 21012 \beta_{3} - 16086 \beta_{4} - 5714 \beta_{5} - 6697 \beta_{6} + 2598 \beta_{7} - 3261 \beta_{8} - 2006 \beta_{9} + 10767 \beta_{10} + 1233 \beta_{11} - 1962 \beta_{12} ) q^{95} + ( -1680477 - 155168 \beta_{1} - 1949 \beta_{2} - 68576 \beta_{3} + 4133 \beta_{4} + 2903 \beta_{5} + 3810 \beta_{6} + 2191 \beta_{7} - 12973 \beta_{8} + 182 \beta_{9} + 6319 \beta_{10} - 2664 \beta_{11} - 2337 \beta_{12} ) q^{96} + ( 730205 + 229431 \beta_{1} - 4412 \beta_{2} - 60498 \beta_{3} + 8561 \beta_{4} + 1406 \beta_{5} - 3067 \beta_{6} - 11279 \beta_{7} - 2883 \beta_{8} - 2454 \beta_{9} - 479 \beta_{10} - 4558 \beta_{11} - 10304 \beta_{12} ) q^{97} + ( -2418480 - 288956 \beta_{1} + 1803 \beta_{2} - 44272 \beta_{3} - 18054 \beta_{4} - 750 \beta_{5} + 4079 \beta_{6} - 9146 \beta_{7} - 3177 \beta_{8} - 4281 \beta_{9} + 4626 \beta_{10} - 1055 \beta_{11} - 1172 \beta_{12} ) q^{98} + ( 405264 - 130819 \beta_{1} + 6859 \beta_{2} + 84350 \beta_{3} - 217 \beta_{4} - 4902 \beta_{5} - 2214 \beta_{6} - 729 \beta_{7} + 10079 \beta_{8} + 11780 \beta_{9} - 836 \beta_{10} + 4991 \beta_{11} - 1486 \beta_{12} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q + 16q^{2} + 94q^{3} + 922q^{4} + 998q^{5} + 183q^{6} + 1360q^{7} + 3870q^{8} + 10011q^{9} + O(q^{10}) \) \( 13q + 16q^{2} + 94q^{3} + 922q^{4} + 998q^{5} + 183q^{6} + 1360q^{7} + 3870q^{8} + 10011q^{9} + 4667q^{10} + 1620q^{11} - 19681q^{12} + 13550q^{13} + 44160q^{14} + 31412q^{15} + 114026q^{16} + 110880q^{17} + 159267q^{18} + 105058q^{19} + 167251q^{20} + 129840q^{21} + 201504q^{22} + 160184q^{23} + 161289q^{24} + 270149q^{25} + 272104q^{26} + 252544q^{27} + 208172q^{28} + 285546q^{29} + 107580q^{30} - 99616q^{31} + 200126q^{32} + 531468q^{33} - 80941q^{34} - 187104q^{35} - 608975q^{36} + 176038q^{37} + 652165q^{38} - 794680q^{39} - 895387q^{40} - 410260q^{41} - 3413218q^{42} - 1033591q^{43} - 2177076q^{44} - 1051178q^{45} - 3975765q^{46} - 424556q^{47} - 2360477q^{48} - 1561359q^{49} - 4063801q^{50} - 2375738q^{51} - 4172312q^{52} + 3992458q^{53} - 10438626q^{54} + 406960q^{55} + 1559556q^{56} - 3116152q^{57} - 4052005q^{58} + 2248836q^{59} - 2911436q^{60} + 6210394q^{61} + 885317q^{62} + 11622368q^{63} - 3096318q^{64} + 5600420q^{65} - 2174604q^{66} - 1993648q^{67} + 9327135q^{68} + 13366240q^{69} - 1105098q^{70} + 4978064q^{71} + 11370663q^{72} + 8224814q^{73} - 3613563q^{74} + 27115592q^{75} + 10687121q^{76} + 17261892q^{77} - 15226630q^{78} + 6945708q^{79} + 15822799q^{80} + 35113185q^{81} - 508449q^{82} + 22937328q^{83} - 14010106q^{84} - 575532q^{85} - 1272112q^{86} + 9081380q^{87} + 11202656q^{88} + 9291302q^{89} + 2841402q^{90} + 25581108q^{91} - 14388137q^{92} + 25930480q^{93} - 24645805q^{94} + 30750464q^{95} - 22461255q^{96} + 10001852q^{97} - 32304856q^{98} + 5055452q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{13} - 3 x^{12} - 1279 x^{11} + 3765 x^{10} + 598742 x^{9} - 1518614 x^{8} - 124677082 x^{7} + 193428526 x^{6} + 11160446785 x^{5} + 1754605765 x^{4} - 349352939351 x^{3} - 481872751923 x^{2} + 2098464001560 x + 1551032970660\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-37067748216477627507880951097 \nu^{12} - 8824957003163581128923882930 \nu^{11} + 43862779143599504444638103397325 \nu^{10} - 19958809290909131950834736227596 \nu^{9} - 18274537871897921730748731778193890 \nu^{8} + 17369960127436577248810344724568316 \nu^{7} + 3123782290750141165224584116401763702 \nu^{6} - 2888829201925713307885514699909321072 \nu^{5} - 188523210247301180573718184132559420329 \nu^{4} - 82714028618438574902769612076620476522 \nu^{3} + 2848608966659910630806249671235576164301 \nu^{2} + 11700211150804525769117415217256616592284 \nu - 8577673914048078368597032100995151576652\)\()/ \)\(10\!\cdots\!92\)\( \)
\(\beta_{3}\)\(=\)\((\)\(191590755258224372380382007533 \nu^{12} - 1337230315893986401319183245558 \nu^{11} - 237686733830652456821240679341953 \nu^{10} + 1676956685288974787039426199380508 \nu^{9} + 105769272392877638573688904761177322 \nu^{8} - 719804417434332761155324012203188844 \nu^{7} - 20139079551474772420695951582015782446 \nu^{6} + 119056858190946237429490759019074536080 \nu^{5} + 1524010461526012671138758047002558328573 \nu^{4} - 5835448902401378857536433463918115495550 \nu^{3} - 35715095975552186480810775470506398124577 \nu^{2} + 48064799726629005291385214900703202426164 \nu + 80415072838101207822336497542706077690620\)\()/ \)\(43\!\cdots\!68\)\( \)
\(\beta_{4}\)\(=\)\((\)\(191590755258224372380382007533 \nu^{12} - 1337230315893986401319183245558 \nu^{11} - 237686733830652456821240679341953 \nu^{10} + 1676956685288974787039426199380508 \nu^{9} + 105769272392877638573688904761177322 \nu^{8} - 719804417434332761155324012203188844 \nu^{7} - 20139079551474772420695951582015782446 \nu^{6} + 119056858190946237429490759019074536080 \nu^{5} + 1524010461526012671138758047002558328573 \nu^{4} - 5835448902401378857536433463918115495550 \nu^{3} - 35280985104471072651878678728131051915809 \nu^{2} + 48064799726629005291385214900703202426164 \nu - 5538879635959330306218657447612471645444\)\()/ \)\(43\!\cdots\!68\)\( \)
\(\beta_{5}\)\(=\)\((\)\(487982563054357232794705117663 \nu^{12} - 6105015878674681774245408152210 \nu^{11} - 614886357847863779573600534809499 \nu^{10} + 7413029775881980608295684111654452 \nu^{9} + 277836110491847036427194357390083502 \nu^{8} - 3125848793975508793208298901260816996 \nu^{7} - 53673595867605654919738431125128120346 \nu^{6} + 525415430235896052832349158646271144304 \nu^{5} + 4123850390830985586541205525831848844495 \nu^{4} - 29233627053939810350021442147531234691754 \nu^{3} - 98641097955135611784114276249442995306043 \nu^{2} + 371369155643928932156431777415265593264124 \nu + 280577990332649437337365093780561726502292\)\()/ \)\(43\!\cdots\!68\)\( \)
\(\beta_{6}\)\(=\)\((\)\(209018642508274696653159244129 \nu^{12} - 982194329650686472924511282190 \nu^{11} - 261593884750772328773104776022021 \nu^{10} + 1253389232387199468546987196961292 \nu^{9} + 118132636737884963094046844952162962 \nu^{8} - 539713669242519784827660954509953820 \nu^{7} - 23059233636897520955657676902682410086 \nu^{6} + 86699180940427292434726263638860361296 \nu^{5} + 1814387939100513901841289970090838183473 \nu^{4} - 3626535373787601827318726701576660597814 \nu^{3} - 42764132636474151422575640218576506604709 \nu^{2} + 12339691446440321979920453723345456950020 \nu + 64448009490660149528748523636375452960492\)\()/ \)\(10\!\cdots\!92\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-24992578471150783154244183649 \nu^{12} + 195703268281163002221991229150 \nu^{11} + 31070124206596915822610458074965 \nu^{10} - 243287489856443495876973730936620 \nu^{9} - 13844982965588441780961310089336242 \nu^{8} + 103988109155544631123672934807115612 \nu^{7} + 2634879383615760241491488121944024678 \nu^{6} - 17290688295734251421795589104348871472 \nu^{5} - 198432771700395471132447922417840288337 \nu^{4} + 878734437763276248056317769729733435302 \nu^{3} + 4609190572474045175612001192429089050837 \nu^{2} - 8529847978020689305124023975420311036804 \nu - 10546049398806805833155508099995978561580\)\()/ \)\(90\!\cdots\!16\)\( \)
\(\beta_{8}\)\(=\)\((\)\(624073470526447525104225874691 \nu^{12} - 4895759513885580038512061652186 \nu^{11} - 771163209292372336855015668373247 \nu^{10} + 6139070126618706157432036665451908 \nu^{9} + 340391203173506000944472364388743574 \nu^{8} - 2651539138199561122789526845346392084 \nu^{7} - 63709353561034188524773938134625738098 \nu^{6} + 448239028964783083313631226323771963536 \nu^{5} + 4634925905507830736664114689492853546899 \nu^{4} - 23811079312520454994041307407807759345586 \nu^{3} - 99625850885489863855892767308783034044223 \nu^{2} + 265086251682926946732831603687950680178316 \nu + 136934476445703302559538119252708706152324\)\()/ \)\(21\!\cdots\!84\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(16\!\cdots\!69\)\( \nu^{12} + \)\(12\!\cdots\!78\)\( \nu^{11} + \)\(20\!\cdots\!85\)\( \nu^{10} - \)\(16\!\cdots\!48\)\( \nu^{9} - \)\(89\!\cdots\!94\)\( \nu^{8} + \)\(69\!\cdots\!40\)\( \nu^{7} + \)\(16\!\cdots\!66\)\( \nu^{6} - \)\(11\!\cdots\!20\)\( \nu^{5} - \)\(12\!\cdots\!21\)\( \nu^{4} + \)\(61\!\cdots\!30\)\( \nu^{3} + \)\(25\!\cdots\!57\)\( \nu^{2} - \)\(73\!\cdots\!16\)\( \nu - \)\(20\!\cdots\!96\)\(\)\()/ \)\(43\!\cdots\!68\)\( \)
\(\beta_{10}\)\(=\)\((\)\(107525739150652196629996703789 \nu^{12} - 877749607755402966241489032534 \nu^{11} - 134003197555938894320843208608033 \nu^{10} + 1083058938649188283080181886878044 \nu^{9} + 59856148512062987936606693376334762 \nu^{8} - 459182427466778914671755701469788780 \nu^{7} - 11418321160218438112884882118445272622 \nu^{6} + 75594516124130659846873307571020623952 \nu^{5} + 861979215858014172542021796390661397885 \nu^{4} - 3774683877399590707189064176810747092766 \nu^{3} - 19908033325787211541501949226173867067265 \nu^{2} + 34850882501581800494432786219136357256500 \nu + 38520235380823128842225623160585171910012\)\()/ \)\(27\!\cdots\!48\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(20\!\cdots\!45\)\( \nu^{12} - \)\(16\!\cdots\!30\)\( \nu^{11} - \)\(25\!\cdots\!61\)\( \nu^{10} + \)\(21\!\cdots\!44\)\( \nu^{9} + \)\(11\!\cdots\!30\)\( \nu^{8} - \)\(90\!\cdots\!88\)\( \nu^{7} - \)\(21\!\cdots\!18\)\( \nu^{6} + \)\(15\!\cdots\!56\)\( \nu^{5} + \)\(15\!\cdots\!05\)\( \nu^{4} - \)\(80\!\cdots\!98\)\( \nu^{3} - \)\(36\!\cdots\!81\)\( \nu^{2} + \)\(94\!\cdots\!52\)\( \nu + \)\(96\!\cdots\!76\)\(\)\()/ \)\(43\!\cdots\!68\)\( \)
\(\beta_{12}\)\(=\)\((\)\(1098499500962555551979565862501 \nu^{12} - 9478472343580770759804427738502 \nu^{11} - 1364368378293181126693041976305113 \nu^{10} + 11724277216827980027875735228649916 \nu^{9} + 605824970572077132243164161510062938 \nu^{8} - 4991315231080282667493816805386009996 \nu^{7} - 114204735008813809511399329448925036542 \nu^{6} + 828417689196912330589781218182889582096 \nu^{5} + 8379316476116621200738993002222781213685 \nu^{4} - 42331188268516850028549827028791815594958 \nu^{3} - 178977666795837493145636361662018168478073 \nu^{2} + 429924660555079285717592772216335707799316 \nu + 163492579846930852703297358371618877587868\)\()/ \)\(21\!\cdots\!84\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} - \beta_{3} + 198\)
\(\nu^{3}\)\(=\)\(-\beta_{11} + \beta_{9} + 2 \beta_{8} + \beta_{5} - \beta_{4} + 4 \beta_{3} - \beta_{2} + 343 \beta_{1} - 61\)
\(\nu^{4}\)\(=\)\(10 \beta_{11} - 2 \beta_{10} + 4 \beta_{9} - 15 \beta_{8} - 16 \beta_{7} + \beta_{6} + 3 \beta_{5} + 418 \beta_{4} - 503 \beta_{3} - 19 \beta_{2} - 382 \beta_{1} + 67833\)
\(\nu^{5}\)\(=\)\(-82 \beta_{12} - 440 \beta_{11} + 92 \beta_{10} + 364 \beta_{9} + 846 \beta_{8} + 246 \beta_{7} + 56 \beta_{6} + 702 \beta_{5} - 626 \beta_{4} + 2036 \beta_{3} - 730 \beta_{2} + 130875 \beta_{1} - 101744\)
\(\nu^{6}\)\(=\)\(910 \beta_{12} + 5290 \beta_{11} - 3452 \beta_{10} + 1358 \beta_{9} - 11062 \beta_{8} - 12190 \beta_{7} + 1524 \beta_{6} + 4008 \beta_{5} + 171903 \beta_{4} - 229051 \beta_{3} - 14310 \beta_{2} - 283736 \beta_{1} + 25891624\)
\(\nu^{7}\)\(=\)\(-43958 \beta_{12} - 171917 \beta_{11} + 60312 \beta_{10} + 135381 \beta_{9} + 320802 \beta_{8} + 210482 \beta_{7} + 42038 \beta_{6} + 376669 \beta_{5} - 313791 \beta_{4} + 1063282 \beta_{3} - 403673 \beta_{2} + 52540593 \beta_{1} - 67032993\)
\(\nu^{8}\)\(=\)\(648870 \beta_{12} + 2282548 \beta_{11} - 2602726 \beta_{10} + 147986 \beta_{9} - 6399797 \beta_{8} - 7148222 \beta_{7} + 949613 \beta_{6} + 2874435 \beta_{5} + 72279140 \beta_{4} - 100477853 \beta_{3} - 8029893 \beta_{2} - 156362010 \beta_{1} + 10400440171\)
\(\nu^{9}\)\(=\)\(-16786808 \beta_{12} - 64621264 \beta_{11} + 28254332 \beta_{10} + 57213756 \beta_{9} + 125357310 \beta_{8} + 131196368 \beta_{7} + 25185846 \beta_{6} + 184518630 \beta_{5} - 142801076 \beta_{4} + 538991130 \beta_{3} - 199812850 \beta_{2} + 21809332945 \beta_{1} - 35512340756\)
\(\nu^{10}\)\(=\)\(326459580 \beta_{12} + 898705500 \beta_{11} - 1543929920 \beta_{10} - 189972212 \beta_{9} - 3373872160 \beta_{8} - 3809775548 \beta_{7} + 448580004 \beta_{6} + 1679026308 \beta_{5} + 30953163861 \beta_{4} - 43426864713 \beta_{3} - 4076880528 \beta_{2} - 77278294568 \beta_{1} + 4318834662810\)
\(\nu^{11}\)\(=\)\(-5161725412 \beta_{12} - 23579751209 \beta_{11} + 11736596944 \beta_{10} + 26700665177 \beta_{9} + 51550129066 \beta_{8} + 72532507884 \beta_{7} + 13933907740 \beta_{6} + 86528972593 \beta_{5} - 61899138149 \beta_{4} + 263120727360 \beta_{3} - 93906894985 \beta_{2} + 9264534511443 \beta_{1} - 17229301256645\)
\(\nu^{12}\)\(=\)\(139155983092 \beta_{12} + 329457520662 \beta_{11} - 821451174890 \beta_{10} - 207979112680 \beta_{9} - 1692994078659 \beta_{8} - 1938413731572 \beta_{7} + 184449462297 \beta_{6} + 891826642051 \beta_{5} + 13426613909262 \beta_{4} - 18695134385227 \beta_{3} - 1977787024591 \beta_{2} - 36303164798374 \beta_{1} + 1834895374668485\)

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
−21.3781
−20.1662
−16.1540
−8.80627
−4.99525
−4.71679
−0.684341
2.37903
7.08185
15.0703
16.3440
17.8435
21.1822
−20.3781 −17.6937 287.267 405.085 360.564 52.5965 −3245.57 −1873.93 −8254.88
1.2 −19.1662 −36.5899 239.342 −174.722 701.287 −1126.43 −2133.99 −848.182 3348.75
1.3 −15.1540 48.8090 101.645 210.950 −739.653 1100.87 399.390 195.321 −3196.74
1.4 −7.80627 −2.71381 −67.0621 −402.005 21.1847 −356.766 1522.71 −2179.64 3138.16
1.5 −3.99525 84.4924 −112.038 −383.451 −337.568 1003.83 959.011 4951.96 1531.98
1.6 −3.71679 −24.7485 −114.185 385.350 91.9851 −1546.77 900.153 −1574.51 −1432.27
1.7 0.315659 82.9732 −127.900 531.642 26.1913 −349.841 −80.7773 4697.55 167.818
1.8 3.37903 −66.5036 −116.582 −241.175 −224.718 −173.086 −826.450 2235.73 −814.939
1.9 8.08185 −1.19690 −62.6836 164.718 −9.67319 1314.61 −1541.08 −2185.57 1331.23
1.10 16.0703 41.5098 130.256 431.884 667.077 218.970 36.2566 −463.938 6940.53
1.11 17.3440 73.8355 172.814 −122.945 1280.60 247.011 777.246 3264.69 −2132.35
1.12 18.8435 −90.3374 227.079 70.1157 −1702.28 1065.82 1867.00 5973.84 1321.23
1.13 22.1822 2.16379 364.050 122.553 47.9976 −90.8173 5236.10 −2182.32 2718.48
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
Significant digits:
Format:

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 43.8.a.b 13
3.b odd 2 1 387.8.a.d 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.8.a.b 13 1.a even 1 1 trivial
387.8.a.d 13 3.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(43\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{13} - \cdots\) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(43))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 16 T + 499 T^{2} - 7226 T^{3} + 129754 T^{4} - 1623108 T^{5} + 25061980 T^{6} - 301751896 T^{7} + 4320739680 T^{8} - 53577914560 T^{9} + 686733102400 T^{10} - 8227010823296 T^{11} + 97673737613312 T^{12} - 1101231869857792 T^{13} + 12502238414503936 T^{14} - 134791345328881664 T^{15} + 1440183699164364800 T^{16} - 14382211926442639360 T^{17} + \)\(14\!\cdots\!40\)\( T^{18} - \)\(13\!\cdots\!84\)\( T^{19} + \)\(14\!\cdots\!60\)\( T^{20} - \)\(11\!\cdots\!88\)\( T^{21} + \)\(11\!\cdots\!32\)\( T^{22} - \)\(85\!\cdots\!24\)\( T^{23} + \)\(75\!\cdots\!28\)\( T^{24} - \)\(30\!\cdots\!56\)\( T^{25} + \)\(24\!\cdots\!48\)\( T^{26} \)
$3$ \( 1 - 94 T + 13628 T^{2} - 1019826 T^{3} + 84521770 T^{4} - 4911562460 T^{5} + 301148239904 T^{6} - 13554949900530 T^{7} + 666960819834852 T^{8} - 22345833199134114 T^{9} + 943680642806331405 T^{10} - 20201450216794018380 T^{11} + \)\(96\!\cdots\!90\)\( T^{12} - \)\(14\!\cdots\!92\)\( T^{13} + \)\(21\!\cdots\!30\)\( T^{14} - \)\(96\!\cdots\!20\)\( T^{15} + \)\(98\!\cdots\!15\)\( T^{16} - \)\(51\!\cdots\!54\)\( T^{17} + \)\(33\!\cdots\!64\)\( T^{18} - \)\(14\!\cdots\!70\)\( T^{19} + \)\(72\!\cdots\!32\)\( T^{20} - \)\(25\!\cdots\!60\)\( T^{21} + \)\(96\!\cdots\!90\)\( T^{22} - \)\(25\!\cdots\!74\)\( T^{23} + \)\(74\!\cdots\!64\)\( T^{24} - \)\(11\!\cdots\!14\)\( T^{25} + \)\(26\!\cdots\!47\)\( T^{26} \)
$5$ \( 1 - 998 T + 870740 T^{2} - 535689006 T^{3} + 300509982542 T^{4} - 143467127408470 T^{5} + 64011052241216400 T^{6} - 25802469413530458250 T^{7} + \)\(98\!\cdots\!00\)\( T^{8} - \)\(34\!\cdots\!50\)\( T^{9} + \)\(11\!\cdots\!75\)\( T^{10} - \)\(37\!\cdots\!00\)\( T^{11} + \)\(11\!\cdots\!50\)\( T^{12} - \)\(32\!\cdots\!00\)\( T^{13} + \)\(88\!\cdots\!50\)\( T^{14} - \)\(22\!\cdots\!00\)\( T^{15} + \)\(56\!\cdots\!75\)\( T^{16} - \)\(12\!\cdots\!50\)\( T^{17} + \)\(28\!\cdots\!00\)\( T^{18} - \)\(58\!\cdots\!50\)\( T^{19} + \)\(11\!\cdots\!00\)\( T^{20} - \)\(19\!\cdots\!50\)\( T^{21} + \)\(32\!\cdots\!50\)\( T^{22} - \)\(45\!\cdots\!50\)\( T^{23} + \)\(57\!\cdots\!00\)\( T^{24} - \)\(51\!\cdots\!50\)\( T^{25} + \)\(40\!\cdots\!25\)\( T^{26} \)
$7$ \( 1 - 1360 T + 7058509 T^{2} - 7851071624 T^{3} + 22977827763868 T^{4} - 21067270496738864 T^{5} + 46647610898487061092 T^{6} - \)\(35\!\cdots\!92\)\( T^{7} + \)\(68\!\cdots\!45\)\( T^{8} - \)\(44\!\cdots\!60\)\( T^{9} + \)\(78\!\cdots\!53\)\( T^{10} - \)\(44\!\cdots\!96\)\( T^{11} + \)\(74\!\cdots\!48\)\( T^{12} - \)\(39\!\cdots\!32\)\( T^{13} + \)\(61\!\cdots\!64\)\( T^{14} - \)\(30\!\cdots\!04\)\( T^{15} + \)\(43\!\cdots\!71\)\( T^{16} - \)\(20\!\cdots\!60\)\( T^{17} + \)\(25\!\cdots\!35\)\( T^{18} - \)\(11\!\cdots\!08\)\( T^{19} + \)\(11\!\cdots\!44\)\( T^{20} - \)\(44\!\cdots\!64\)\( T^{21} + \)\(40\!\cdots\!24\)\( T^{22} - \)\(11\!\cdots\!76\)\( T^{23} + \)\(83\!\cdots\!63\)\( T^{24} - \)\(13\!\cdots\!60\)\( T^{25} + \)\(80\!\cdots\!43\)\( T^{26} \)
$11$ \( 1 - 1620 T + 132588019 T^{2} - 123981604500 T^{3} + 7703760283965880 T^{4} - 3713200717818796164 T^{5} + \)\(25\!\cdots\!00\)\( T^{6} - \)\(12\!\cdots\!20\)\( T^{7} + \)\(55\!\cdots\!44\)\( T^{8} - \)\(83\!\cdots\!32\)\( T^{9} + \)\(85\!\cdots\!00\)\( T^{10} - \)\(37\!\cdots\!84\)\( T^{11} + \)\(11\!\cdots\!86\)\( T^{12} - \)\(96\!\cdots\!20\)\( T^{13} + \)\(22\!\cdots\!06\)\( T^{14} - \)\(14\!\cdots\!44\)\( T^{15} + \)\(62\!\cdots\!00\)\( T^{16} - \)\(11\!\cdots\!92\)\( T^{17} + \)\(15\!\cdots\!44\)\( T^{18} - \)\(68\!\cdots\!20\)\( T^{19} + \)\(27\!\cdots\!00\)\( T^{20} - \)\(77\!\cdots\!04\)\( T^{21} + \)\(31\!\cdots\!80\)\( T^{22} - \)\(97\!\cdots\!00\)\( T^{23} + \)\(20\!\cdots\!49\)\( T^{24} - \)\(48\!\cdots\!20\)\( T^{25} + \)\(58\!\cdots\!11\)\( T^{26} \)
$13$ \( 1 - 13550 T + 451004117 T^{2} - 4457271716692 T^{3} + 93112309019999992 T^{4} - \)\(74\!\cdots\!76\)\( T^{5} + \)\(12\!\cdots\!04\)\( T^{6} - \)\(87\!\cdots\!48\)\( T^{7} + \)\(13\!\cdots\!96\)\( T^{8} - \)\(81\!\cdots\!96\)\( T^{9} + \)\(11\!\cdots\!04\)\( T^{10} - \)\(63\!\cdots\!80\)\( T^{11} + \)\(82\!\cdots\!86\)\( T^{12} - \)\(42\!\cdots\!16\)\( T^{13} + \)\(51\!\cdots\!62\)\( T^{14} - \)\(24\!\cdots\!20\)\( T^{15} + \)\(28\!\cdots\!52\)\( T^{16} - \)\(12\!\cdots\!16\)\( T^{17} + \)\(12\!\cdots\!72\)\( T^{18} - \)\(53\!\cdots\!12\)\( T^{19} + \)\(48\!\cdots\!92\)\( T^{20} - \)\(17\!\cdots\!16\)\( T^{21} + \)\(14\!\cdots\!24\)\( T^{22} - \)\(42\!\cdots\!08\)\( T^{23} + \)\(26\!\cdots\!61\)\( T^{24} - \)\(50\!\cdots\!50\)\( T^{25} + \)\(23\!\cdots\!37\)\( T^{26} \)
$17$ \( 1 - 110880 T + 7269241380 T^{2} - 350308303802712 T^{3} + 13915691873194028506 T^{4} - \)\(48\!\cdots\!14\)\( T^{5} + \)\(15\!\cdots\!04\)\( T^{6} - \)\(43\!\cdots\!00\)\( T^{7} + \)\(11\!\cdots\!91\)\( T^{8} - \)\(30\!\cdots\!04\)\( T^{9} + \)\(72\!\cdots\!29\)\( T^{10} - \)\(16\!\cdots\!04\)\( T^{11} + \)\(35\!\cdots\!06\)\( T^{12} - \)\(74\!\cdots\!34\)\( T^{13} + \)\(14\!\cdots\!38\)\( T^{14} - \)\(27\!\cdots\!16\)\( T^{15} + \)\(49\!\cdots\!93\)\( T^{16} - \)\(85\!\cdots\!64\)\( T^{17} + \)\(13\!\cdots\!63\)\( T^{18} - \)\(20\!\cdots\!00\)\( T^{19} + \)\(29\!\cdots\!88\)\( T^{20} - \)\(38\!\cdots\!34\)\( T^{21} + \)\(45\!\cdots\!78\)\( T^{22} - \)\(47\!\cdots\!88\)\( T^{23} + \)\(40\!\cdots\!60\)\( T^{24} - \)\(25\!\cdots\!80\)\( T^{25} + \)\(93\!\cdots\!33\)\( T^{26} \)
$19$ \( 1 - 105058 T + 10894929034 T^{2} - 729074101697894 T^{3} + 46770979546573980358 T^{4} - \)\(24\!\cdots\!36\)\( T^{5} + \)\(12\!\cdots\!42\)\( T^{6} - \)\(53\!\cdots\!86\)\( T^{7} + \)\(22\!\cdots\!20\)\( T^{8} - \)\(85\!\cdots\!82\)\( T^{9} + \)\(31\!\cdots\!91\)\( T^{10} - \)\(10\!\cdots\!04\)\( T^{11} + \)\(35\!\cdots\!64\)\( T^{12} - \)\(10\!\cdots\!20\)\( T^{13} + \)\(31\!\cdots\!96\)\( T^{14} - \)\(85\!\cdots\!84\)\( T^{15} + \)\(22\!\cdots\!29\)\( T^{16} - \)\(54\!\cdots\!62\)\( T^{17} + \)\(12\!\cdots\!80\)\( T^{18} - \)\(27\!\cdots\!46\)\( T^{19} + \)\(55\!\cdots\!18\)\( T^{20} - \)\(99\!\cdots\!16\)\( T^{21} + \)\(17\!\cdots\!22\)\( T^{22} - \)\(23\!\cdots\!94\)\( T^{23} + \)\(31\!\cdots\!26\)\( T^{24} - \)\(27\!\cdots\!18\)\( T^{25} + \)\(23\!\cdots\!19\)\( T^{26} \)
$23$ \( 1 - 160184 T + 36979701828 T^{2} - 4753092322433576 T^{3} + \)\(65\!\cdots\!70\)\( T^{4} - \)\(68\!\cdots\!64\)\( T^{5} + \)\(71\!\cdots\!92\)\( T^{6} - \)\(64\!\cdots\!20\)\( T^{7} + \)\(55\!\cdots\!51\)\( T^{8} - \)\(42\!\cdots\!96\)\( T^{9} + \)\(31\!\cdots\!15\)\( T^{10} - \)\(21\!\cdots\!56\)\( T^{11} + \)\(13\!\cdots\!02\)\( T^{12} - \)\(83\!\cdots\!40\)\( T^{13} + \)\(47\!\cdots\!94\)\( T^{14} - \)\(24\!\cdots\!04\)\( T^{15} + \)\(12\!\cdots\!45\)\( T^{16} - \)\(57\!\cdots\!76\)\( T^{17} + \)\(25\!\cdots\!57\)\( T^{18} - \)\(10\!\cdots\!80\)\( T^{19} + \)\(38\!\cdots\!96\)\( T^{20} - \)\(12\!\cdots\!04\)\( T^{21} + \)\(40\!\cdots\!90\)\( T^{22} - \)\(99\!\cdots\!24\)\( T^{23} + \)\(26\!\cdots\!84\)\( T^{24} - \)\(38\!\cdots\!44\)\( T^{25} + \)\(82\!\cdots\!27\)\( T^{26} \)
$29$ \( 1 - 285546 T + 180604853400 T^{2} - 41013344500675266 T^{3} + \)\(14\!\cdots\!66\)\( T^{4} - \)\(27\!\cdots\!34\)\( T^{5} + \)\(71\!\cdots\!40\)\( T^{6} - \)\(11\!\cdots\!42\)\( T^{7} + \)\(24\!\cdots\!24\)\( T^{8} - \)\(32\!\cdots\!66\)\( T^{9} + \)\(61\!\cdots\!47\)\( T^{10} - \)\(73\!\cdots\!48\)\( T^{11} + \)\(12\!\cdots\!62\)\( T^{12} - \)\(13\!\cdots\!56\)\( T^{13} + \)\(21\!\cdots\!58\)\( T^{14} - \)\(21\!\cdots\!88\)\( T^{15} + \)\(31\!\cdots\!63\)\( T^{16} - \)\(29\!\cdots\!26\)\( T^{17} + \)\(36\!\cdots\!76\)\( T^{18} - \)\(29\!\cdots\!22\)\( T^{19} + \)\(32\!\cdots\!60\)\( T^{20} - \)\(21\!\cdots\!14\)\( T^{21} + \)\(19\!\cdots\!74\)\( T^{22} - \)\(95\!\cdots\!66\)\( T^{23} + \)\(72\!\cdots\!00\)\( T^{24} - \)\(19\!\cdots\!26\)\( T^{25} + \)\(11\!\cdots\!29\)\( T^{26} \)
$31$ \( 1 + 99616 T + 241614413464 T^{2} + 18338344889599376 T^{3} + \)\(27\!\cdots\!10\)\( T^{4} + \)\(14\!\cdots\!12\)\( T^{5} + \)\(20\!\cdots\!92\)\( T^{6} + \)\(65\!\cdots\!12\)\( T^{7} + \)\(10\!\cdots\!63\)\( T^{8} + \)\(16\!\cdots\!72\)\( T^{9} + \)\(42\!\cdots\!31\)\( T^{10} + \)\(15\!\cdots\!08\)\( T^{11} + \)\(14\!\cdots\!42\)\( T^{12} - \)\(52\!\cdots\!48\)\( T^{13} + \)\(38\!\cdots\!62\)\( T^{14} + \)\(11\!\cdots\!68\)\( T^{15} + \)\(89\!\cdots\!61\)\( T^{16} + \)\(92\!\cdots\!52\)\( T^{17} + \)\(16\!\cdots\!13\)\( T^{18} + \)\(28\!\cdots\!32\)\( T^{19} + \)\(24\!\cdots\!32\)\( T^{20} + \)\(48\!\cdots\!72\)\( T^{21} + \)\(25\!\cdots\!10\)\( T^{22} + \)\(45\!\cdots\!76\)\( T^{23} + \)\(16\!\cdots\!04\)\( T^{24} + \)\(18\!\cdots\!36\)\( T^{25} + \)\(51\!\cdots\!31\)\( T^{26} \)
$37$ \( 1 - 176038 T + 1002592896582 T^{2} - 172621415570488610 T^{3} + \)\(48\!\cdots\!62\)\( T^{4} - \)\(79\!\cdots\!90\)\( T^{5} + \)\(14\!\cdots\!62\)\( T^{6} - \)\(22\!\cdots\!06\)\( T^{7} + \)\(31\!\cdots\!64\)\( T^{8} - \)\(45\!\cdots\!18\)\( T^{9} + \)\(50\!\cdots\!45\)\( T^{10} - \)\(65\!\cdots\!96\)\( T^{11} + \)\(61\!\cdots\!44\)\( T^{12} - \)\(71\!\cdots\!12\)\( T^{13} + \)\(58\!\cdots\!52\)\( T^{14} - \)\(59\!\cdots\!44\)\( T^{15} + \)\(43\!\cdots\!65\)\( T^{16} - \)\(36\!\cdots\!78\)\( T^{17} + \)\(24\!\cdots\!52\)\( T^{18} - \)\(16\!\cdots\!14\)\( T^{19} + \)\(10\!\cdots\!74\)\( T^{20} - \)\(52\!\cdots\!90\)\( T^{21} + \)\(30\!\cdots\!86\)\( T^{22} - \)\(10\!\cdots\!90\)\( T^{23} + \)\(56\!\cdots\!94\)\( T^{24} - \)\(94\!\cdots\!18\)\( T^{25} + \)\(50\!\cdots\!13\)\( T^{26} \)
$41$ \( 1 + 410260 T + 1275774355044 T^{2} + 520566650539856152 T^{3} + \)\(84\!\cdots\!34\)\( T^{4} + \)\(33\!\cdots\!10\)\( T^{5} + \)\(37\!\cdots\!04\)\( T^{6} + \)\(14\!\cdots\!96\)\( T^{7} + \)\(12\!\cdots\!59\)\( T^{8} + \)\(44\!\cdots\!08\)\( T^{9} + \)\(32\!\cdots\!41\)\( T^{10} + \)\(11\!\cdots\!44\)\( T^{11} + \)\(73\!\cdots\!58\)\( T^{12} + \)\(23\!\cdots\!38\)\( T^{13} + \)\(14\!\cdots\!98\)\( T^{14} + \)\(42\!\cdots\!84\)\( T^{15} + \)\(24\!\cdots\!81\)\( T^{16} + \)\(63\!\cdots\!68\)\( T^{17} + \)\(34\!\cdots\!59\)\( T^{18} + \)\(76\!\cdots\!76\)\( T^{19} + \)\(39\!\cdots\!44\)\( T^{20} + \)\(68\!\cdots\!10\)\( T^{21} + \)\(33\!\cdots\!14\)\( T^{22} + \)\(40\!\cdots\!52\)\( T^{23} + \)\(19\!\cdots\!64\)\( T^{24} + \)\(12\!\cdots\!60\)\( T^{25} + \)\(57\!\cdots\!41\)\( T^{26} \)
$43$ \( ( 1 + 79507 T )^{13} \)
$47$ \( 1 + 424556 T + 3688252683440 T^{2} + 2420585272729575804 T^{3} + \)\(67\!\cdots\!76\)\( T^{4} + \)\(56\!\cdots\!60\)\( T^{5} + \)\(83\!\cdots\!12\)\( T^{6} + \)\(77\!\cdots\!80\)\( T^{7} + \)\(81\!\cdots\!18\)\( T^{8} + \)\(74\!\cdots\!56\)\( T^{9} + \)\(64\!\cdots\!19\)\( T^{10} + \)\(53\!\cdots\!64\)\( T^{11} + \)\(40\!\cdots\!52\)\( T^{12} + \)\(30\!\cdots\!32\)\( T^{13} + \)\(20\!\cdots\!76\)\( T^{14} + \)\(13\!\cdots\!16\)\( T^{15} + \)\(83\!\cdots\!93\)\( T^{16} + \)\(48\!\cdots\!16\)\( T^{17} + \)\(27\!\cdots\!74\)\( T^{18} + \)\(13\!\cdots\!20\)\( T^{19} + \)\(71\!\cdots\!04\)\( T^{20} + \)\(24\!\cdots\!60\)\( T^{21} + \)\(14\!\cdots\!48\)\( T^{22} + \)\(26\!\cdots\!96\)\( T^{23} + \)\(20\!\cdots\!80\)\( T^{24} + \)\(12\!\cdots\!36\)\( T^{25} + \)\(14\!\cdots\!03\)\( T^{26} \)
$53$ \( 1 - 3992458 T + 15589250135817 T^{2} - 39082970394967823704 T^{3} + \)\(92\!\cdots\!16\)\( T^{4} - \)\(17\!\cdots\!84\)\( T^{5} + \)\(31\!\cdots\!44\)\( T^{6} - \)\(49\!\cdots\!88\)\( T^{7} + \)\(73\!\cdots\!96\)\( T^{8} - \)\(10\!\cdots\!68\)\( T^{9} + \)\(13\!\cdots\!44\)\( T^{10} - \)\(16\!\cdots\!28\)\( T^{11} + \)\(19\!\cdots\!82\)\( T^{12} - \)\(21\!\cdots\!60\)\( T^{13} + \)\(22\!\cdots\!34\)\( T^{14} - \)\(22\!\cdots\!32\)\( T^{15} + \)\(21\!\cdots\!32\)\( T^{16} - \)\(19\!\cdots\!48\)\( T^{17} + \)\(16\!\cdots\!72\)\( T^{18} - \)\(12\!\cdots\!92\)\( T^{19} + \)\(96\!\cdots\!52\)\( T^{20} - \)\(63\!\cdots\!64\)\( T^{21} + \)\(39\!\cdots\!32\)\( T^{22} - \)\(19\!\cdots\!96\)\( T^{23} + \)\(91\!\cdots\!21\)\( T^{24} - \)\(27\!\cdots\!98\)\( T^{25} + \)\(81\!\cdots\!97\)\( T^{26} \)
$59$ \( 1 - 2248836 T + 18986189218171 T^{2} - 37576890307888851744 T^{3} + \)\(17\!\cdots\!18\)\( T^{4} - \)\(31\!\cdots\!68\)\( T^{5} + \)\(11\!\cdots\!54\)\( T^{6} - \)\(17\!\cdots\!32\)\( T^{7} + \)\(51\!\cdots\!59\)\( T^{8} - \)\(76\!\cdots\!40\)\( T^{9} + \)\(18\!\cdots\!17\)\( T^{10} - \)\(25\!\cdots\!88\)\( T^{11} + \)\(56\!\cdots\!40\)\( T^{12} - \)\(70\!\cdots\!64\)\( T^{13} + \)\(14\!\cdots\!60\)\( T^{14} - \)\(15\!\cdots\!68\)\( T^{15} + \)\(29\!\cdots\!03\)\( T^{16} - \)\(29\!\cdots\!40\)\( T^{17} + \)\(49\!\cdots\!41\)\( T^{18} - \)\(42\!\cdots\!92\)\( T^{19} + \)\(65\!\cdots\!06\)\( T^{20} - \)\(46\!\cdots\!88\)\( T^{21} + \)\(65\!\cdots\!22\)\( T^{22} - \)\(34\!\cdots\!44\)\( T^{23} + \)\(43\!\cdots\!49\)\( T^{24} - \)\(12\!\cdots\!96\)\( T^{25} + \)\(14\!\cdots\!59\)\( T^{26} \)
$61$ \( 1 - 6210394 T + 38426666503003 T^{2} - \)\(15\!\cdots\!76\)\( T^{3} + \)\(60\!\cdots\!88\)\( T^{4} - \)\(18\!\cdots\!92\)\( T^{5} + \)\(56\!\cdots\!64\)\( T^{6} - \)\(14\!\cdots\!80\)\( T^{7} + \)\(36\!\cdots\!17\)\( T^{8} - \)\(83\!\cdots\!78\)\( T^{9} + \)\(18\!\cdots\!55\)\( T^{10} - \)\(36\!\cdots\!04\)\( T^{11} + \)\(70\!\cdots\!72\)\( T^{12} - \)\(12\!\cdots\!32\)\( T^{13} + \)\(22\!\cdots\!12\)\( T^{14} - \)\(35\!\cdots\!64\)\( T^{15} + \)\(56\!\cdots\!55\)\( T^{16} - \)\(81\!\cdots\!18\)\( T^{17} + \)\(11\!\cdots\!17\)\( T^{18} - \)\(14\!\cdots\!80\)\( T^{19} + \)\(17\!\cdots\!24\)\( T^{20} - \)\(18\!\cdots\!12\)\( T^{21} + \)\(18\!\cdots\!28\)\( T^{22} - \)\(14\!\cdots\!76\)\( T^{23} + \)\(11\!\cdots\!63\)\( T^{24} - \)\(57\!\cdots\!54\)\( T^{25} + \)\(29\!\cdots\!61\)\( T^{26} \)
$67$ \( 1 + 1993648 T + 36332499483043 T^{2} + 80369020063411411764 T^{3} + \)\(68\!\cdots\!56\)\( T^{4} + \)\(16\!\cdots\!32\)\( T^{5} + \)\(89\!\cdots\!48\)\( T^{6} + \)\(22\!\cdots\!60\)\( T^{7} + \)\(90\!\cdots\!36\)\( T^{8} + \)\(22\!\cdots\!72\)\( T^{9} + \)\(75\!\cdots\!16\)\( T^{10} + \)\(18\!\cdots\!92\)\( T^{11} + \)\(53\!\cdots\!18\)\( T^{12} + \)\(12\!\cdots\!16\)\( T^{13} + \)\(32\!\cdots\!14\)\( T^{14} + \)\(69\!\cdots\!68\)\( T^{15} + \)\(16\!\cdots\!72\)\( T^{16} + \)\(30\!\cdots\!52\)\( T^{17} + \)\(73\!\cdots\!48\)\( T^{18} + \)\(11\!\cdots\!40\)\( T^{19} + \)\(26\!\cdots\!56\)\( T^{20} + \)\(29\!\cdots\!92\)\( T^{21} + \)\(75\!\cdots\!28\)\( T^{22} + \)\(53\!\cdots\!36\)\( T^{23} + \)\(14\!\cdots\!61\)\( T^{24} + \)\(48\!\cdots\!08\)\( T^{25} + \)\(14\!\cdots\!83\)\( T^{26} \)
$71$ \( 1 - 4978064 T + 87202612193659 T^{2} - \)\(39\!\cdots\!40\)\( T^{3} + \)\(37\!\cdots\!18\)\( T^{4} - \)\(15\!\cdots\!52\)\( T^{5} + \)\(10\!\cdots\!70\)\( T^{6} - \)\(37\!\cdots\!88\)\( T^{7} + \)\(20\!\cdots\!19\)\( T^{8} - \)\(66\!\cdots\!92\)\( T^{9} + \)\(29\!\cdots\!13\)\( T^{10} - \)\(88\!\cdots\!48\)\( T^{11} + \)\(34\!\cdots\!08\)\( T^{12} - \)\(91\!\cdots\!08\)\( T^{13} + \)\(31\!\cdots\!28\)\( T^{14} - \)\(73\!\cdots\!88\)\( T^{15} + \)\(22\!\cdots\!23\)\( T^{16} - \)\(45\!\cdots\!12\)\( T^{17} + \)\(12\!\cdots\!69\)\( T^{18} - \)\(21\!\cdots\!08\)\( T^{19} + \)\(52\!\cdots\!70\)\( T^{20} - \)\(70\!\cdots\!92\)\( T^{21} + \)\(15\!\cdots\!98\)\( T^{22} - \)\(15\!\cdots\!40\)\( T^{23} + \)\(30\!\cdots\!69\)\( T^{24} - \)\(15\!\cdots\!84\)\( T^{25} + \)\(29\!\cdots\!71\)\( T^{26} \)
$73$ \( 1 - 8224814 T + 92481350221839 T^{2} - \)\(53\!\cdots\!16\)\( T^{3} + \)\(38\!\cdots\!04\)\( T^{4} - \)\(18\!\cdots\!96\)\( T^{5} + \)\(10\!\cdots\!88\)\( T^{6} - \)\(45\!\cdots\!64\)\( T^{7} + \)\(22\!\cdots\!37\)\( T^{8} - \)\(85\!\cdots\!50\)\( T^{9} + \)\(36\!\cdots\!23\)\( T^{10} - \)\(12\!\cdots\!68\)\( T^{11} + \)\(49\!\cdots\!68\)\( T^{12} - \)\(15\!\cdots\!60\)\( T^{13} + \)\(54\!\cdots\!96\)\( T^{14} - \)\(15\!\cdots\!12\)\( T^{15} + \)\(49\!\cdots\!79\)\( T^{16} - \)\(12\!\cdots\!50\)\( T^{17} + \)\(36\!\cdots\!09\)\( T^{18} - \)\(83\!\cdots\!56\)\( T^{19} + \)\(21\!\cdots\!44\)\( T^{20} - \)\(41\!\cdots\!56\)\( T^{21} + \)\(94\!\cdots\!68\)\( T^{22} - \)\(14\!\cdots\!84\)\( T^{23} + \)\(27\!\cdots\!67\)\( T^{24} - \)\(27\!\cdots\!74\)\( T^{25} + \)\(36\!\cdots\!77\)\( T^{26} \)
$79$ \( 1 - 6945708 T + 117121136837052 T^{2} - \)\(56\!\cdots\!28\)\( T^{3} + \)\(65\!\cdots\!68\)\( T^{4} - \)\(24\!\cdots\!60\)\( T^{5} + \)\(24\!\cdots\!68\)\( T^{6} - \)\(74\!\cdots\!48\)\( T^{7} + \)\(72\!\cdots\!74\)\( T^{8} - \)\(19\!\cdots\!48\)\( T^{9} + \)\(18\!\cdots\!35\)\( T^{10} - \)\(46\!\cdots\!04\)\( T^{11} + \)\(42\!\cdots\!92\)\( T^{12} - \)\(97\!\cdots\!28\)\( T^{13} + \)\(80\!\cdots\!28\)\( T^{14} - \)\(17\!\cdots\!24\)\( T^{15} + \)\(13\!\cdots\!65\)\( T^{16} - \)\(26\!\cdots\!28\)\( T^{17} + \)\(18\!\cdots\!26\)\( T^{18} - \)\(37\!\cdots\!68\)\( T^{19} + \)\(23\!\cdots\!92\)\( T^{20} - \)\(45\!\cdots\!60\)\( T^{21} + \)\(23\!\cdots\!52\)\( T^{22} - \)\(38\!\cdots\!28\)\( T^{23} + \)\(15\!\cdots\!68\)\( T^{24} - \)\(17\!\cdots\!48\)\( T^{25} + \)\(48\!\cdots\!79\)\( T^{26} \)
$83$ \( 1 - 22937328 T + 403848212345231 T^{2} - \)\(46\!\cdots\!40\)\( T^{3} + \)\(46\!\cdots\!36\)\( T^{4} - \)\(35\!\cdots\!72\)\( T^{5} + \)\(24\!\cdots\!76\)\( T^{6} - \)\(14\!\cdots\!68\)\( T^{7} + \)\(83\!\cdots\!60\)\( T^{8} - \)\(44\!\cdots\!24\)\( T^{9} + \)\(27\!\cdots\!48\)\( T^{10} - \)\(15\!\cdots\!32\)\( T^{11} + \)\(97\!\cdots\!38\)\( T^{12} - \)\(51\!\cdots\!32\)\( T^{13} + \)\(26\!\cdots\!26\)\( T^{14} - \)\(11\!\cdots\!28\)\( T^{15} + \)\(54\!\cdots\!84\)\( T^{16} - \)\(24\!\cdots\!84\)\( T^{17} + \)\(12\!\cdots\!20\)\( T^{18} - \)\(57\!\cdots\!52\)\( T^{19} + \)\(26\!\cdots\!28\)\( T^{20} - \)\(10\!\cdots\!32\)\( T^{21} + \)\(37\!\cdots\!32\)\( T^{22} - \)\(10\!\cdots\!60\)\( T^{23} + \)\(23\!\cdots\!13\)\( T^{24} - \)\(36\!\cdots\!88\)\( T^{25} + \)\(43\!\cdots\!67\)\( T^{26} \)
$89$ \( 1 - 9291302 T + 323406546094099 T^{2} - \)\(31\!\cdots\!04\)\( T^{3} + \)\(56\!\cdots\!92\)\( T^{4} - \)\(51\!\cdots\!56\)\( T^{5} + \)\(68\!\cdots\!68\)\( T^{6} - \)\(56\!\cdots\!24\)\( T^{7} + \)\(61\!\cdots\!61\)\( T^{8} - \)\(45\!\cdots\!46\)\( T^{9} + \)\(42\!\cdots\!63\)\( T^{10} - \)\(28\!\cdots\!48\)\( T^{11} + \)\(23\!\cdots\!60\)\( T^{12} - \)\(14\!\cdots\!00\)\( T^{13} + \)\(10\!\cdots\!40\)\( T^{14} - \)\(55\!\cdots\!68\)\( T^{15} + \)\(36\!\cdots\!07\)\( T^{16} - \)\(17\!\cdots\!26\)\( T^{17} + \)\(10\!\cdots\!89\)\( T^{18} - \)\(42\!\cdots\!04\)\( T^{19} + \)\(22\!\cdots\!12\)\( T^{20} - \)\(75\!\cdots\!16\)\( T^{21} + \)\(36\!\cdots\!48\)\( T^{22} - \)\(90\!\cdots\!04\)\( T^{23} + \)\(40\!\cdots\!71\)\( T^{24} - \)\(52\!\cdots\!82\)\( T^{25} + \)\(24\!\cdots\!89\)\( T^{26} \)
$97$ \( 1 - 10001852 T + 551581550822996 T^{2} - \)\(38\!\cdots\!08\)\( T^{3} + \)\(14\!\cdots\!86\)\( T^{4} - \)\(69\!\cdots\!90\)\( T^{5} + \)\(23\!\cdots\!80\)\( T^{6} - \)\(80\!\cdots\!08\)\( T^{7} + \)\(31\!\cdots\!27\)\( T^{8} - \)\(74\!\cdots\!80\)\( T^{9} + \)\(34\!\cdots\!81\)\( T^{10} - \)\(66\!\cdots\!32\)\( T^{11} + \)\(32\!\cdots\!38\)\( T^{12} - \)\(56\!\cdots\!82\)\( T^{13} + \)\(26\!\cdots\!94\)\( T^{14} - \)\(43\!\cdots\!08\)\( T^{15} + \)\(18\!\cdots\!57\)\( T^{16} - \)\(31\!\cdots\!80\)\( T^{17} + \)\(10\!\cdots\!11\)\( T^{18} - \)\(22\!\cdots\!72\)\( T^{19} + \)\(53\!\cdots\!60\)\( T^{20} - \)\(12\!\cdots\!90\)\( T^{21} + \)\(20\!\cdots\!78\)\( T^{22} - \)\(46\!\cdots\!92\)\( T^{23} + \)\(52\!\cdots\!52\)\( T^{24} - \)\(77\!\cdots\!12\)\( T^{25} + \)\(62\!\cdots\!53\)\( T^{26} \)
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