Properties

Label 38.8.c.b
Level 38
Weight 8
Character orbit 38.c
Analytic conductor 11.871
Analytic rank 0
Dimension 14
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 38.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.8706309684\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \(x^{14} - x^{13} + 9740 x^{12} + 37173 x^{11} + 71393485 x^{10} + 196352446 x^{9} + 218671355941 x^{8} - 33116965187 x^{7} + 501909199155445 x^{6} + 488917878843726 x^{5} + 115037439673768301 x^{4} + 1132246613431747493 x^{3} + 24536677825193566084 x^{2} + 126800218755690130647 x + 666410398757822143881\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 8 \beta_{3} q^{2} + ( \beta_{1} + \beta_{2} + 8 \beta_{3} ) q^{3} + ( -64 + 64 \beta_{3} ) q^{4} + ( -18 \beta_{3} + \beta_{6} - \beta_{8} ) q^{5} + ( -64 + 8 \beta_{1} + 64 \beta_{3} ) q^{6} + ( 137 - 2 \beta_{2} + \beta_{7} ) q^{7} -512 q^{8} + ( -659 + 13 \beta_{1} + 659 \beta_{3} - \beta_{12} ) q^{9} +O(q^{10})\) \( q + 8 \beta_{3} q^{2} + ( \beta_{1} + \beta_{2} + 8 \beta_{3} ) q^{3} + ( -64 + 64 \beta_{3} ) q^{4} + ( -18 \beta_{3} + \beta_{6} - \beta_{8} ) q^{5} + ( -64 + 8 \beta_{1} + 64 \beta_{3} ) q^{6} + ( 137 - 2 \beta_{2} + \beta_{7} ) q^{7} -512 q^{8} + ( -659 + 13 \beta_{1} + 659 \beta_{3} - \beta_{12} ) q^{9} + ( 144 - 144 \beta_{3} - 8 \beta_{8} ) q^{10} + ( 49 + 19 \beta_{2} - \beta_{4} + \beta_{5} ) q^{11} + ( -512 - 64 \beta_{2} ) q^{12} + ( -181 - 40 \beta_{1} - \beta_{2} + 181 \beta_{3} - \beta_{4} - \beta_{6} + 7 \beta_{8} - 2 \beta_{9} + \beta_{10} + 2 \beta_{12} - \beta_{13} ) q^{13} + ( -16 \beta_{1} - 16 \beta_{2} + 1096 \beta_{3} + 8 \beta_{7} + 8 \beta_{11} ) q^{14} + ( 1253 - 11 \beta_{1} - 1253 \beta_{3} - 28 \beta_{8} + 7 \beta_{11} - \beta_{13} ) q^{15} -4096 \beta_{3} q^{16} + ( -175 \beta_{1} - 174 \beta_{2} + 1480 \beta_{3} + 6 \beta_{4} + \beta_{5} - 14 \beta_{6} - 8 \beta_{7} + 15 \beta_{8} - \beta_{9} + 2 \beta_{10} - 8 \beta_{11} - 5 \beta_{12} + \beta_{13} ) q^{17} + ( -5272 - 104 \beta_{2} - 8 \beta_{4} ) q^{18} + ( -6479 + 34 \beta_{1} - 144 \beta_{2} + 418 \beta_{3} - 7 \beta_{4} + 3 \beta_{5} - 46 \beta_{6} - 2 \beta_{7} + 26 \beta_{8} - 2 \beta_{10} - 12 \beta_{11} - \beta_{13} ) q^{19} + ( 1152 - 64 \beta_{6} ) q^{20} + ( 318 \beta_{1} + 315 \beta_{2} - 4341 \beta_{3} - 7 \beta_{4} + 3 \beta_{5} + 105 \beta_{6} + 48 \beta_{7} - 108 \beta_{8} + 3 \beta_{9} - 6 \beta_{10} + 48 \beta_{11} + 4 \beta_{12} + 3 \beta_{13} ) q^{21} + ( 152 \beta_{1} + 152 \beta_{2} + 392 \beta_{3} - 8 \beta_{4} + 8 \beta_{5} + 8 \beta_{12} + 8 \beta_{13} ) q^{22} + ( -16289 + 211 \beta_{1} + 16289 \beta_{3} - 44 \beta_{8} - 13 \beta_{11} - 22 \beta_{12} + 5 \beta_{13} ) q^{23} + ( -512 \beta_{1} - 512 \beta_{2} - 4096 \beta_{3} ) q^{24} + ( -7247 + 591 \beta_{1} - 5 \beta_{2} + 7247 \beta_{3} - 5 \beta_{4} - 5 \beta_{6} + 21 \beta_{8} - 10 \beta_{9} + 5 \beta_{10} + 8 \beta_{11} + 25 \beta_{12} + \beta_{13} ) q^{25} + ( -1448 + 304 \beta_{2} + 8 \beta_{5} + 40 \beta_{6} - 8 \beta_{9} - 8 \beta_{10} ) q^{26} + ( -22951 - 682 \beta_{2} - 7 \beta_{4} - 8 \beta_{5} - 34 \beta_{6} + 8 \beta_{7} + 6 \beta_{9} + 6 \beta_{10} ) q^{27} + ( -8768 - 128 \beta_{1} + 8768 \beta_{3} + 64 \beta_{11} ) q^{28} + ( 24024 - 516 \beta_{1} + 2 \beta_{2} - 24024 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} - 175 \beta_{8} + 4 \beta_{9} - 2 \beta_{10} + 12 \beta_{12} - 18 \beta_{13} ) q^{29} + ( 10024 + 88 \beta_{2} + 8 \beta_{5} - 224 \beta_{6} - 56 \beta_{7} ) q^{30} + ( 14387 + 60 \beta_{2} - 14 \beta_{4} + 6 \beta_{5} + 220 \beta_{6} + 77 \beta_{7} + 8 \beta_{9} + 8 \beta_{10} ) q^{31} + ( 32768 - 32768 \beta_{3} ) q^{32} + ( -1528 \beta_{1} - 1513 \beta_{2} + 53692 \beta_{3} + 43 \beta_{4} - 45 \beta_{5} - 336 \beta_{6} + 351 \beta_{8} - 15 \beta_{9} + 30 \beta_{10} - 28 \beta_{12} - 45 \beta_{13} ) q^{33} + ( -11840 - 1400 \beta_{1} - 8 \beta_{2} + 11840 \beta_{3} - 8 \beta_{4} - 8 \beta_{6} + 120 \beta_{8} - 16 \beta_{9} + 8 \beta_{10} - 64 \beta_{11} - 40 \beta_{12} + 8 \beta_{13} ) q^{34} + ( 3171 \beta_{1} + 3161 \beta_{2} - 4231 \beta_{3} + 105 \beta_{4} - 4 \beta_{5} + 120 \beta_{6} - 182 \beta_{7} - 130 \beta_{8} + 10 \beta_{9} - 20 \beta_{10} - 182 \beta_{11} - 115 \beta_{12} - 4 \beta_{13} ) q^{35} + ( -832 \beta_{1} - 832 \beta_{2} - 42176 \beta_{3} - 64 \beta_{4} + 64 \beta_{12} ) q^{36} + ( 27297 + 584 \beta_{2} + 42 \beta_{4} + 7 \beta_{5} + 934 \beta_{6} - 48 \beta_{7} + 5 \beta_{9} + 5 \beta_{10} ) q^{37} + ( -3344 - 1136 \beta_{1} - 1408 \beta_{2} - 48488 \beta_{3} - 40 \beta_{4} + 32 \beta_{5} - 144 \beta_{6} + 80 \beta_{7} + 352 \beta_{8} + 16 \beta_{9} - 16 \beta_{10} - 16 \beta_{11} + 40 \beta_{12} + 24 \beta_{13} ) q^{38} + ( 98752 + 2593 \beta_{2} + 154 \beta_{4} - 55 \beta_{5} + 208 \beta_{6} + 286 \beta_{7} ) q^{39} + ( 9216 \beta_{3} - 512 \beta_{6} + 512 \beta_{8} ) q^{40} + ( 617 \beta_{1} + 617 \beta_{2} + 9740 \beta_{3} - 47 \beta_{4} - 16 \beta_{5} - 98 \beta_{6} - 384 \beta_{7} + 98 \beta_{8} - 384 \beta_{11} + 47 \beta_{12} - 16 \beta_{13} ) q^{41} + ( 34728 + 2544 \beta_{1} + 24 \beta_{2} - 34728 \beta_{3} + 24 \beta_{4} + 24 \beta_{6} - 864 \beta_{8} + 48 \beta_{9} - 24 \beta_{10} + 384 \beta_{11} + 32 \beta_{12} + 24 \beta_{13} ) q^{42} + ( 2659 \beta_{1} + 2645 \beta_{2} + 104192 \beta_{3} - 84 \beta_{4} - 41 \beta_{5} + 600 \beta_{6} + 48 \beta_{7} - 614 \beta_{8} + 14 \beta_{9} - 28 \beta_{10} + 48 \beta_{11} + 70 \beta_{12} - 41 \beta_{13} ) q^{43} + ( -3136 + 1216 \beta_{1} + 3136 \beta_{3} + 64 \beta_{12} + 64 \beta_{13} ) q^{44} + ( 30874 + 320 \beta_{2} + 100 \beta_{4} - 30 \beta_{5} + 402 \beta_{6} - 540 \beta_{7} + 30 \beta_{9} + 30 \beta_{10} ) q^{45} + ( -130312 - 1688 \beta_{2} - 176 \beta_{4} - 40 \beta_{5} - 352 \beta_{6} + 104 \beta_{7} ) q^{46} + ( -338993 - 815 \beta_{1} + 20 \beta_{2} + 338993 \beta_{3} + 20 \beta_{4} + 20 \beta_{6} + 1924 \beta_{8} + 40 \beta_{9} - 20 \beta_{10} - 7 \beta_{11} - 40 \beta_{12} + 83 \beta_{13} ) q^{47} + ( 32768 - 4096 \beta_{1} - 32768 \beta_{3} ) q^{48} + ( 423394 + 15287 \beta_{2} + 151 \beta_{4} + 10 \beta_{5} - 1740 \beta_{6} + 200 \beta_{7} + 10 \beta_{9} + 10 \beta_{10} ) q^{49} + ( -57976 - 4808 \beta_{2} + 120 \beta_{4} - 8 \beta_{5} + 88 \beta_{6} - 64 \beta_{7} - 40 \beta_{9} - 40 \beta_{10} ) q^{50} + ( 464109 + 12516 \beta_{1} - 464109 \beta_{3} + 1584 \beta_{8} - 798 \beta_{11} + 201 \beta_{12} + 21 \beta_{13} ) q^{51} + ( 2560 \beta_{1} + 2496 \beta_{2} - 11584 \beta_{3} + 64 \beta_{4} + 64 \beta_{5} + 384 \beta_{6} - 448 \beta_{8} + 64 \beta_{9} - 128 \beta_{10} - 128 \beta_{12} + 64 \beta_{13} ) q^{52} + ( 24445 - 8086 \beta_{1} + 35 \beta_{2} - 24445 \beta_{3} + 35 \beta_{4} + 35 \beta_{6} + 611 \beta_{8} + 70 \beta_{9} - 35 \beta_{10} + 300 \beta_{11} - 280 \beta_{12} - 13 \beta_{13} ) q^{53} + ( -5552 \beta_{1} - 5504 \beta_{2} - 183608 \beta_{3} - 104 \beta_{4} - 64 \beta_{5} - 320 \beta_{6} + 64 \beta_{7} + 368 \beta_{8} - 48 \beta_{9} + 96 \beta_{10} + 64 \beta_{11} + 152 \beta_{12} - 64 \beta_{13} ) q^{54} + ( -9018 \beta_{1} - 9038 \beta_{2} - 8741 \beta_{3} + 120 \beta_{4} - 8 \beta_{5} + 2648 \beta_{6} - 469 \beta_{7} - 2668 \beta_{8} + 20 \beta_{9} - 40 \beta_{10} - 469 \beta_{11} - 140 \beta_{12} - 8 \beta_{13} ) q^{55} + ( -70144 + 1024 \beta_{2} - 512 \beta_{7} ) q^{56} + ( -125267 - 19823 \beta_{1} - 15927 \beta_{2} - 392730 \beta_{3} - 99 \beta_{4} - 158 \beta_{5} - 715 \beta_{6} + 536 \beta_{7} + 2293 \beta_{8} - 96 \beta_{9} + 105 \beta_{10} - 40 \beta_{11} + 7 \beta_{12} - 149 \beta_{13} ) q^{57} + ( 192192 + 4160 \beta_{2} + 128 \beta_{4} + 144 \beta_{5} - 1368 \beta_{6} + 16 \beta_{9} + 16 \beta_{10} ) q^{58} + ( 4640 \beta_{1} + 4690 \beta_{2} - 263029 \beta_{3} - 339 \beta_{4} + 50 \beta_{5} + 832 \beta_{6} + 168 \beta_{7} - 782 \beta_{8} - 50 \beta_{9} + 100 \beta_{10} + 168 \beta_{11} + 389 \beta_{12} + 50 \beta_{13} ) q^{59} + ( 704 \beta_{1} + 704 \beta_{2} + 80192 \beta_{3} + 64 \beta_{5} - 1792 \beta_{6} - 448 \beta_{7} + 1792 \beta_{8} - 448 \beta_{11} + 64 \beta_{13} ) q^{60} + ( -229446 + 8940 \beta_{1} - 112 \beta_{2} + 229446 \beta_{3} - 112 \beta_{4} - 112 \beta_{6} + 1849 \beta_{8} - 224 \beta_{9} + 112 \beta_{10} + 812 \beta_{11} - 140 \beta_{12} - 152 \beta_{13} ) q^{61} + ( 352 \beta_{1} + 416 \beta_{2} + 115096 \beta_{3} - 176 \beta_{4} + 48 \beta_{5} + 1696 \beta_{6} + 616 \beta_{7} - 1632 \beta_{8} - 64 \beta_{9} + 128 \beta_{10} + 616 \beta_{11} + 240 \beta_{12} + 48 \beta_{13} ) q^{62} + ( -436612 - 2448 \beta_{1} + 84 \beta_{2} + 436612 \beta_{3} + 84 \beta_{4} + 84 \beta_{6} - 7784 \beta_{8} + 168 \beta_{9} - 84 \beta_{10} + 1664 \beta_{11} - 504 \beta_{12} - 188 \beta_{13} ) q^{63} + 262144 q^{64} + ( 466079 - 408 \beta_{2} + 160 \beta_{4} + 317 \beta_{5} - 7387 \beta_{6} + 1456 \beta_{7} - 135 \beta_{9} - 135 \beta_{10} ) q^{65} + ( -429536 - 12224 \beta_{1} - 120 \beta_{2} + 429536 \beta_{3} - 120 \beta_{4} - 120 \beta_{6} + 2808 \beta_{8} - 240 \beta_{9} + 120 \beta_{10} - 224 \beta_{12} - 360 \beta_{13} ) q^{66} + ( -1277289 + 30848 \beta_{1} - 54 \beta_{2} + 1277289 \beta_{3} - 54 \beta_{4} - 54 \beta_{6} - 1738 \beta_{8} - 108 \beta_{9} + 54 \beta_{10} + 526 \beta_{11} + 193 \beta_{12} + 238 \beta_{13} ) q^{67} + ( -94720 + 11072 \beta_{2} - 448 \beta_{4} - 64 \beta_{5} + 832 \beta_{6} + 512 \beta_{7} - 64 \beta_{9} - 64 \beta_{10} ) q^{68} + ( -642320 - 64952 \beta_{2} - 348 \beta_{4} + 92 \beta_{5} + 781 \beta_{6} + 532 \beta_{7} + 48 \beta_{9} + 48 \beta_{10} ) q^{69} + ( 33848 + 25368 \beta_{1} + 80 \beta_{2} - 33848 \beta_{3} + 80 \beta_{4} + 80 \beta_{6} - 1040 \beta_{8} + 160 \beta_{9} - 80 \beta_{10} - 1456 \beta_{11} - 920 \beta_{12} - 32 \beta_{13} ) q^{70} + ( 10957 \beta_{1} + 10957 \beta_{2} + 74912 \beta_{3} - 554 \beta_{4} + 365 \beta_{5} + 3292 \beta_{6} + 1490 \beta_{7} - 3292 \beta_{8} + 1490 \beta_{11} + 554 \beta_{12} + 365 \beta_{13} ) q^{71} + ( 337408 - 6656 \beta_{1} - 337408 \beta_{3} + 512 \beta_{12} ) q^{72} + ( -505 \beta_{1} - 555 \beta_{2} - 1365070 \beta_{3} - 15 \beta_{4} + 358 \beta_{5} + 8118 \beta_{6} - 1064 \beta_{7} - 8168 \beta_{8} + 50 \beta_{9} - 100 \beta_{10} - 1064 \beta_{11} - 35 \beta_{12} + 358 \beta_{13} ) q^{73} + ( 4592 \beta_{1} + 4632 \beta_{2} + 218376 \beta_{3} + 296 \beta_{4} + 56 \beta_{5} + 7432 \beta_{6} - 384 \beta_{7} - 7392 \beta_{8} - 40 \beta_{9} + 80 \beta_{10} - 384 \beta_{11} - 256 \beta_{12} + 56 \beta_{13} ) q^{74} + ( -1754599 + 30247 \beta_{2} - 245 \beta_{4} + 57 \beta_{5} + 2064 \beta_{6} + 876 \beta_{7} - 120 \beta_{9} - 120 \beta_{10} ) q^{75} + ( 387904 - 11264 \beta_{1} - 2048 \beta_{2} - 414656 \beta_{3} + 128 \beta_{4} + 64 \beta_{5} + 1792 \beta_{6} + 768 \beta_{7} + 1152 \beta_{8} + 128 \beta_{9} + 640 \beta_{11} + 320 \beta_{12} + 256 \beta_{13} ) q^{76} + ( -654777 + 10486 \beta_{2} - 1704 \beta_{4} + 129 \beta_{5} - 5590 \beta_{6} + 1144 \beta_{7} - 81 \beta_{9} - 81 \beta_{10} ) q^{77} + ( 20744 \beta_{1} + 20744 \beta_{2} + 790016 \beta_{3} + 1232 \beta_{4} - 440 \beta_{5} + 1664 \beta_{6} + 2288 \beta_{7} - 1664 \beta_{8} + 2288 \beta_{11} - 1232 \beta_{12} - 440 \beta_{13} ) q^{78} + ( 13493 \beta_{1} + 13481 \beta_{2} + 1139844 \beta_{3} + 334 \beta_{4} - 373 \beta_{5} - 14728 \beta_{6} - 714 \beta_{7} + 14716 \beta_{8} + 12 \beta_{9} - 24 \beta_{10} - 714 \beta_{11} - 346 \beta_{12} - 373 \beta_{13} ) q^{79} + ( -73728 + 73728 \beta_{3} + 4096 \beta_{8} ) q^{80} + ( -28003 \beta_{1} - 28003 \beta_{2} - 593116 \beta_{3} + 721 \beta_{4} + 376 \beta_{5} - 208 \beta_{6} - 1288 \beta_{7} + 208 \beta_{8} - 1288 \beta_{11} - 721 \beta_{12} + 376 \beta_{13} ) q^{81} + ( -77920 + 4936 \beta_{1} + 77920 \beta_{3} + 784 \beta_{8} - 3072 \beta_{11} + 376 \beta_{12} - 128 \beta_{13} ) q^{82} + ( 2178406 - 65642 \beta_{2} - 1178 \beta_{4} + 85 \beta_{5} + 542 \beta_{6} - 1190 \beta_{7} - 50 \beta_{9} - 50 \beta_{10} ) q^{83} + ( 277824 - 19968 \beta_{2} + 640 \beta_{4} - 192 \beta_{5} - 6528 \beta_{6} - 3072 \beta_{7} + 192 \beta_{9} + 192 \beta_{10} ) q^{84} + ( 1003197 - 46968 \beta_{1} + 135 \beta_{2} - 1003197 \beta_{3} + 135 \beta_{4} + 135 \beta_{6} + 1395 \beta_{8} + 270 \beta_{9} - 135 \beta_{10} + 1716 \beta_{11} + 1230 \beta_{12} + 267 \beta_{13} ) q^{85} + ( -833536 + 21272 \beta_{1} + 112 \beta_{2} + 833536 \beta_{3} + 112 \beta_{4} + 112 \beta_{6} - 4912 \beta_{8} + 224 \beta_{9} - 112 \beta_{10} + 384 \beta_{11} + 560 \beta_{12} - 328 \beta_{13} ) q^{86} + ( 1801577 + 64709 \beta_{2} + 852 \beta_{4} - 341 \beta_{5} - 10120 \beta_{6} - 1987 \beta_{7} + 84 \beta_{9} + 84 \beta_{10} ) q^{87} + ( -25088 - 9728 \beta_{2} + 512 \beta_{4} - 512 \beta_{5} ) q^{88} + ( 1693230 - 6767 \beta_{1} - 135 \beta_{2} - 1693230 \beta_{3} - 135 \beta_{4} - 135 \beta_{6} + 4489 \beta_{8} - 270 \beta_{9} + 135 \beta_{10} + 5824 \beta_{11} + 975 \beta_{12} + 75 \beta_{13} ) q^{89} + ( 2080 \beta_{1} + 2320 \beta_{2} + 246992 \beta_{3} + 560 \beta_{4} - 240 \beta_{5} + 2976 \beta_{6} - 4320 \beta_{7} - 2736 \beta_{8} - 240 \beta_{9} + 480 \beta_{10} - 4320 \beta_{11} - 320 \beta_{12} - 240 \beta_{13} ) q^{90} + ( 596158 - 160028 \beta_{1} - 168 \beta_{2} - 596158 \beta_{3} - 168 \beta_{4} - 168 \beta_{6} + 26088 \beta_{8} - 336 \beta_{9} + 168 \beta_{10} - 558 \beta_{11} + 1792 \beta_{12} - 8 \beta_{13} ) q^{91} + ( -13504 \beta_{1} - 13504 \beta_{2} - 1042496 \beta_{3} - 1408 \beta_{4} - 320 \beta_{5} - 2816 \beta_{6} + 832 \beta_{7} + 2816 \beta_{8} + 832 \beta_{11} + 1408 \beta_{12} - 320 \beta_{13} ) q^{92} + ( -12866 \beta_{1} - 12887 \beta_{2} + 53771 \beta_{3} - 917 \beta_{4} - 219 \beta_{5} + 8997 \beta_{6} + 3228 \beta_{7} - 9018 \beta_{8} + 21 \beta_{9} - 42 \beta_{10} + 3228 \beta_{11} + 896 \beta_{12} - 219 \beta_{13} ) q^{93} + ( -2711944 + 6840 \beta_{2} - 664 \beta_{5} + 15712 \beta_{6} + 56 \beta_{7} + 160 \beta_{9} + 160 \beta_{10} ) q^{94} + ( 2275687 - 49833 \beta_{1} - 9061 \beta_{2} - 3535520 \beta_{3} + 2310 \beta_{4} + 239 \beta_{5} - 3792 \beta_{6} - 5663 \beta_{7} + 7824 \beta_{8} + 280 \beta_{9} - 560 \beta_{10} - 3214 \beta_{11} - 2150 \beta_{12} + 337 \beta_{13} ) q^{95} + ( 262144 + 32768 \beta_{2} ) q^{96} + ( 80916 \beta_{1} + 80552 \beta_{2} - 1133665 \beta_{3} - 824 \beta_{4} + 544 \beta_{5} + 2114 \beta_{6} - 4864 \beta_{7} - 2478 \beta_{8} + 364 \beta_{9} - 728 \beta_{10} - 4864 \beta_{11} + 460 \beta_{12} + 544 \beta_{13} ) q^{97} + ( 122136 \beta_{1} + 122216 \beta_{2} + 3387152 \beta_{3} + 1128 \beta_{4} + 80 \beta_{5} - 14000 \beta_{6} + 1600 \beta_{7} + 14080 \beta_{8} - 80 \beta_{9} + 160 \beta_{10} + 1600 \beta_{11} - 1048 \beta_{12} + 80 \beta_{13} ) q^{98} + ( 3530517 + 74723 \beta_{1} + 618 \beta_{2} - 3530517 \beta_{3} + 618 \beta_{4} + 618 \beta_{6} - 1810 \beta_{8} + 1236 \beta_{9} - 618 \beta_{10} - 6236 \beta_{11} + 1153 \beta_{12} + 458 \beta_{13} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q + 56q^{2} + 55q^{3} - 448q^{4} - 126q^{5} - 440q^{6} + 1928q^{7} - 7168q^{8} - 4602q^{9} + O(q^{10}) \) \( 14q + 56q^{2} + 55q^{3} - 448q^{4} - 126q^{5} - 440q^{6} + 1928q^{7} - 7168q^{8} - 4602q^{9} + 1008q^{10} + 646q^{11} - 7040q^{12} - 1308q^{13} + 7712q^{14} + 8740q^{15} - 28672q^{16} + 10528q^{17} - 73632q^{18} - 87463q^{19} + 16128q^{20} - 30584q^{21} + 2584q^{22} - 113822q^{23} - 28160q^{24} - 50143q^{25} - 20928q^{26} - 319898q^{27} - 61696q^{28} + 167706q^{29} + 139840q^{30} + 201780q^{31} + 229376q^{32} + 377473q^{33} - 84224q^{34} - 33168q^{35} - 294528q^{36} + 380924q^{37} - 384064q^{38} + 1379564q^{39} + 64512q^{40} + 66301q^{41} + 244672q^{42} + 726564q^{43} - 20672q^{44} + 428936q^{45} - 1821152q^{46} - 2373788q^{47} + 225280q^{48} + 5898846q^{49} - 802288q^{50} + 3264054q^{51} - 83712q^{52} + 161792q^{53} - 1279592q^{54} - 53424q^{55} - 987136q^{56} - 4487994q^{57} + 2683296q^{58} - 1845767q^{59} + 559360q^{60} - 1600418q^{61} + 807120q^{62} - 3064040q^{63} + 3670016q^{64} + 6534852q^{65} - 3019784q^{66} - 8911929q^{67} - 1347584q^{68} - 8860208q^{69} + 265344q^{70} + 517154q^{71} + 2356224q^{72} - 9558049q^{73} + 1523696q^{74} - 24621450q^{75} + 2525120q^{76} - 9188192q^{77} + 5518256q^{78} + 7963520q^{79} - 516096q^{80} - 4125855q^{81} - 530408q^{82} + 30616886q^{83} + 3914752q^{84} + 6973266q^{85} - 5812512q^{86} + 25084136q^{87} - 330752q^{88} + 11829436q^{89} + 1715744q^{90} + 4017336q^{91} - 7284608q^{92} + 396810q^{93} - 37980608q^{94} + 7058892q^{95} + 3604480q^{96} - 8033723q^{97} + 23595384q^{98} + 24812606q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14} - x^{13} + 9740 x^{12} + 37173 x^{11} + 71393485 x^{10} + 196352446 x^{9} + 218671355941 x^{8} - 33116965187 x^{7} + 501909199155445 x^{6} + 488917878843726 x^{5} + 115037439673768301 x^{4} + 1132246613431747493 x^{3} + 24536677825193566084 x^{2} + 126800218755690130647 x + 666410398757822143881\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(29\!\cdots\!90\)\( \nu^{13} + \)\(10\!\cdots\!12\)\( \nu^{12} - \)\(21\!\cdots\!64\)\( \nu^{11} - \)\(46\!\cdots\!52\)\( \nu^{10} - \)\(14\!\cdots\!95\)\( \nu^{9} + \)\(19\!\cdots\!76\)\( \nu^{8} - \)\(15\!\cdots\!64\)\( \nu^{7} + \)\(29\!\cdots\!48\)\( \nu^{6} - \)\(33\!\cdots\!30\)\( \nu^{5} + \)\(14\!\cdots\!16\)\( \nu^{4} + \)\(31\!\cdots\!16\)\( \nu^{3} + \)\(36\!\cdots\!88\)\( \nu^{2} + \)\(19\!\cdots\!60\)\( \nu + \)\(38\!\cdots\!24\)\(\)\()/ \)\(72\!\cdots\!95\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(16\!\cdots\!96\)\( \nu^{13} - \)\(79\!\cdots\!86\)\( \nu^{12} + \)\(15\!\cdots\!52\)\( \nu^{11} + \)\(67\!\cdots\!44\)\( \nu^{10} + \)\(11\!\cdots\!08\)\( \nu^{9} + \)\(36\!\cdots\!21\)\( \nu^{8} + \)\(35\!\cdots\!12\)\( \nu^{7} - \)\(10\!\cdots\!16\)\( \nu^{6} + \)\(82\!\cdots\!68\)\( \nu^{5} + \)\(80\!\cdots\!66\)\( \nu^{4} + \)\(18\!\cdots\!12\)\( \nu^{3} + \)\(95\!\cdots\!44\)\( \nu^{2} + \)\(40\!\cdots\!52\)\( \nu + \)\(20\!\cdots\!72\)\(\)\()/ \)\(20\!\cdots\!05\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(16\!\cdots\!92\)\( \nu^{13} - \)\(70\!\cdots\!00\)\( \nu^{12} - \)\(12\!\cdots\!10\)\( \nu^{11} - \)\(67\!\cdots\!11\)\( \nu^{10} - \)\(12\!\cdots\!01\)\( \nu^{9} - \)\(49\!\cdots\!98\)\( \nu^{8} - \)\(33\!\cdots\!10\)\( \nu^{7} - \)\(14\!\cdots\!76\)\( \nu^{6} - \)\(29\!\cdots\!46\)\( \nu^{5} - \)\(34\!\cdots\!58\)\( \nu^{4} + \)\(56\!\cdots\!90\)\( \nu^{3} - \)\(79\!\cdots\!51\)\( \nu^{2} - \)\(41\!\cdots\!74\)\( \nu - \)\(20\!\cdots\!08\)\(\)\()/ \)\(72\!\cdots\!95\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(20\!\cdots\!74\)\( \nu^{13} + \)\(30\!\cdots\!33\)\( \nu^{12} + \)\(19\!\cdots\!54\)\( \nu^{11} + \)\(30\!\cdots\!59\)\( \nu^{10} + \)\(28\!\cdots\!12\)\( \nu^{9} + \)\(22\!\cdots\!90\)\( \nu^{8} + \)\(12\!\cdots\!54\)\( \nu^{7} + \)\(67\!\cdots\!79\)\( \nu^{6} + \)\(12\!\cdots\!72\)\( \nu^{5} + \)\(15\!\cdots\!70\)\( \nu^{4} + \)\(20\!\cdots\!54\)\( \nu^{3} + \)\(35\!\cdots\!99\)\( \nu^{2} + \)\(18\!\cdots\!38\)\( \nu + \)\(31\!\cdots\!97\)\(\)\()/ \)\(79\!\cdots\!80\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(26\!\cdots\!07\)\( \nu^{13} - \)\(53\!\cdots\!37\)\( \nu^{12} - \)\(20\!\cdots\!74\)\( \nu^{11} - \)\(53\!\cdots\!13\)\( \nu^{10} - \)\(16\!\cdots\!19\)\( \nu^{9} - \)\(38\!\cdots\!22\)\( \nu^{8} - \)\(33\!\cdots\!74\)\( \nu^{7} - \)\(11\!\cdots\!17\)\( \nu^{6} - \)\(24\!\cdots\!51\)\( \nu^{5} - \)\(27\!\cdots\!42\)\( \nu^{4} + \)\(18\!\cdots\!94\)\( \nu^{3} - \)\(62\!\cdots\!13\)\( \nu^{2} - \)\(32\!\cdots\!64\)\( \nu - \)\(63\!\cdots\!37\)\(\)\()/ \)\(79\!\cdots\!28\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(46\!\cdots\!82\)\( \nu^{13} + \)\(12\!\cdots\!33\)\( \nu^{12} + \)\(35\!\cdots\!94\)\( \nu^{11} + \)\(12\!\cdots\!73\)\( \nu^{10} + \)\(30\!\cdots\!36\)\( \nu^{9} + \)\(92\!\cdots\!42\)\( \nu^{8} + \)\(70\!\cdots\!94\)\( \nu^{7} + \)\(27\!\cdots\!93\)\( \nu^{6} + \)\(57\!\cdots\!76\)\( \nu^{5} + \)\(64\!\cdots\!62\)\( \nu^{4} - \)\(27\!\cdots\!66\)\( \nu^{3} + \)\(14\!\cdots\!73\)\( \nu^{2} + \)\(78\!\cdots\!14\)\( \nu + \)\(17\!\cdots\!49\)\(\)\()/ \)\(79\!\cdots\!80\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(98\!\cdots\!84\)\( \nu^{13} - \)\(90\!\cdots\!54\)\( \nu^{12} + \)\(96\!\cdots\!03\)\( \nu^{11} - \)\(40\!\cdots\!94\)\( \nu^{10} + \)\(70\!\cdots\!22\)\( \nu^{9} - \)\(37\!\cdots\!26\)\( \nu^{8} + \)\(21\!\cdots\!83\)\( \nu^{7} - \)\(17\!\cdots\!14\)\( \nu^{6} + \)\(49\!\cdots\!62\)\( \nu^{5} - \)\(34\!\cdots\!26\)\( \nu^{4} + \)\(10\!\cdots\!43\)\( \nu^{3} + \)\(49\!\cdots\!26\)\( \nu^{2} + \)\(12\!\cdots\!58\)\( \nu - \)\(29\!\cdots\!32\)\(\)\()/ \)\(11\!\cdots\!60\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(15\!\cdots\!70\)\( \nu^{13} - \)\(82\!\cdots\!16\)\( \nu^{12} + \)\(15\!\cdots\!37\)\( \nu^{11} - \)\(71\!\cdots\!84\)\( \nu^{10} + \)\(11\!\cdots\!20\)\( \nu^{9} - \)\(15\!\cdots\!38\)\( \nu^{8} + \)\(37\!\cdots\!77\)\( \nu^{7} - \)\(14\!\cdots\!64\)\( \nu^{6} + \)\(88\!\cdots\!40\)\( \nu^{5} - \)\(27\!\cdots\!78\)\( \nu^{4} + \)\(39\!\cdots\!57\)\( \nu^{3} + \)\(10\!\cdots\!36\)\( \nu^{2} - \)\(14\!\cdots\!10\)\( \nu + \)\(45\!\cdots\!18\)\(\)\()/ \)\(11\!\cdots\!60\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(22\!\cdots\!01\)\( \nu^{13} + \)\(36\!\cdots\!07\)\( \nu^{12} - \)\(20\!\cdots\!59\)\( \nu^{11} - \)\(68\!\cdots\!85\)\( \nu^{10} - \)\(14\!\cdots\!03\)\( \nu^{9} - \)\(28\!\cdots\!68\)\( \nu^{8} - \)\(41\!\cdots\!99\)\( \nu^{7} + \)\(62\!\cdots\!15\)\( \nu^{6} - \)\(89\!\cdots\!23\)\( \nu^{5} - \)\(32\!\cdots\!88\)\( \nu^{4} + \)\(22\!\cdots\!01\)\( \nu^{3} - \)\(17\!\cdots\!05\)\( \nu^{2} + \)\(10\!\cdots\!98\)\( \nu + \)\(51\!\cdots\!45\)\(\)\()/ \)\(11\!\cdots\!60\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(46\!\cdots\!97\)\( \nu^{13} - \)\(43\!\cdots\!82\)\( \nu^{12} + \)\(45\!\cdots\!99\)\( \nu^{11} - \)\(19\!\cdots\!62\)\( \nu^{10} + \)\(33\!\cdots\!46\)\( \nu^{9} - \)\(17\!\cdots\!98\)\( \nu^{8} + \)\(10\!\cdots\!79\)\( \nu^{7} - \)\(83\!\cdots\!42\)\( \nu^{6} + \)\(23\!\cdots\!66\)\( \nu^{5} - \)\(16\!\cdots\!98\)\( \nu^{4} + \)\(52\!\cdots\!19\)\( \nu^{3} + \)\(23\!\cdots\!78\)\( \nu^{2} + \)\(69\!\cdots\!09\)\( \nu - \)\(13\!\cdots\!96\)\(\)\()/ \)\(22\!\cdots\!20\)\( \)
\(\beta_{12}\)\(=\)\((\)\(\)\(45\!\cdots\!94\)\( \nu^{13} - \)\(42\!\cdots\!16\)\( \nu^{12} + \)\(44\!\cdots\!82\)\( \nu^{11} + \)\(16\!\cdots\!63\)\( \nu^{10} + \)\(32\!\cdots\!72\)\( \nu^{9} + \)\(86\!\cdots\!48\)\( \nu^{8} + \)\(99\!\cdots\!02\)\( \nu^{7} - \)\(44\!\cdots\!92\)\( \nu^{6} + \)\(22\!\cdots\!32\)\( \nu^{5} + \)\(12\!\cdots\!18\)\( \nu^{4} + \)\(52\!\cdots\!42\)\( \nu^{3} + \)\(26\!\cdots\!28\)\( \nu^{2} + \)\(10\!\cdots\!03\)\( \nu - \)\(30\!\cdots\!16\)\(\)\()/ \)\(20\!\cdots\!05\)\( \)
\(\beta_{13}\)\(=\)\((\)\(\)\(67\!\cdots\!03\)\( \nu^{13} - \)\(10\!\cdots\!38\)\( \nu^{12} + \)\(65\!\cdots\!41\)\( \nu^{11} - \)\(65\!\cdots\!78\)\( \nu^{10} + \)\(47\!\cdots\!94\)\( \nu^{9} - \)\(52\!\cdots\!92\)\( \nu^{8} + \)\(14\!\cdots\!01\)\( \nu^{7} - \)\(19\!\cdots\!98\)\( \nu^{6} + \)\(34\!\cdots\!74\)\( \nu^{5} - \)\(41\!\cdots\!32\)\( \nu^{4} + \)\(73\!\cdots\!21\)\( \nu^{3} + \)\(29\!\cdots\!82\)\( \nu^{2} + \)\(11\!\cdots\!91\)\( \nu - \)\(30\!\cdots\!74\)\(\)\()/ \)\(22\!\cdots\!20\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{12} - \beta_{4} - 2782 \beta_{3} + 3 \beta_{2} + 3 \beta_{1}\)
\(\nu^{3}\)\(=\)\(-6 \beta_{10} - 6 \beta_{9} - 8 \beta_{7} + 34 \beta_{6} + 8 \beta_{5} - 17 \beta_{4} + 4936 \beta_{2} - 9337\)
\(\nu^{4}\)\(=\)\(-632 \beta_{13} - 5616 \beta_{12} + 1544 \beta_{11} + 192 \beta_{10} - 384 \beta_{9} + 1264 \beta_{8} - 192 \beta_{6} - 192 \beta_{4} + 13709153 \beta_{3} - 192 \beta_{2} - 45936 \beta_{1} - 13709153\)
\(\nu^{5}\)\(=\)\(-84448 \beta_{13} - 86440 \beta_{12} + 37072 \beta_{11} + 92256 \beta_{10} - 46128 \beta_{9} + 549368 \beta_{8} + 37072 \beta_{7} - 503240 \beta_{6} - 84448 \beta_{5} + 132568 \beta_{4} + 131263664 \beta_{3} - 26330689 \beta_{2} - 26376817 \beta_{1}\)
\(\nu^{6}\)\(=\)\(1897296 \beta_{10} + 1897296 \beta_{9} + 14418040 \beta_{7} - 8575904 \beta_{6} - 6397720 \beta_{5} + 35698289 \beta_{4} - 460223571 \beta_{2} + 72769391118\)
\(\nu^{7}\)\(=\)\(649978584 \beta_{13} + 817692189 \beta_{12} - 209113944 \beta_{11} - 313126038 \beta_{10} + 626252076 \beta_{9} - 4301721462 \beta_{8} + 313126038 \beta_{6} + 313126038 \beta_{4} - 1301961499113 \beta_{3} + 313126038 \beta_{2} + 146856529108 \beta_{1} + 1301961499113\)
\(\nu^{8}\)\(=\)\(49174040112 \beta_{13} + 185901785440 \beta_{12} - 104815097232 \beta_{11} - 29655377280 \beta_{10} + 14827688640 \beta_{9} - 101953826976 \beta_{8} - 104815097232 \beta_{7} + 87126138336 \beta_{6} + 49174040112 \beta_{5} - 200729474080 \beta_{4} - 403965075057025 \beta_{3} + 3898103159712 \beta_{2} + 3912930848352 \beta_{1}\)
\(\nu^{9}\)\(=\)\(-2035526940384 \beta_{10} - 2035526940384 \beta_{9} - 1484921597216 \beta_{7} + 25423234501168 \beta_{6} + 4516792800512 \beta_{5} - 10593038007824 \beta_{4} + 841797726210913 \beta_{2} - 10904972866728928\)
\(\nu^{10}\)\(=\)\(-344339829603440 \beta_{13} - 1106725327361313 \beta_{12} + 694377216837488 \beta_{11} + 106628601102816 \beta_{10} - 213257202205632 \beta_{9} + 790125277274944 \beta_{8} - 106628601102816 \beta_{6} - 106628601102816 \beta_{4} + 2322692838203932670 \beta_{3} - 106628601102816 \beta_{2} - 30025802729643363 \beta_{1} - 2322692838203932670\)
\(\nu^{11}\)\(=\)\(-30101324108608744 \beta_{13} - 48069793558037557 \beta_{12} + 11684937137323048 \beta_{11} + 25990913386484556 \beta_{10} - 12995456693242278 \beta_{9} + 191131811712485822 \beta_{8} + 11684937137323048 \beta_{7} - 178136355019243544 \beta_{6} - 30101324108608744 \beta_{5} + 61065250251279835 \beta_{4} + 83391260507327521241 \beta_{3} - 4989545113843143406 \beta_{2} - 5002540570536385684 \beta_{1}\)
\(\nu^{12}\)\(=\)\(737335513363338240 \beta_{10} + 737335513363338240 \beta_{9} + 4406087149026143896 \beta_{7} - 4432802550670378256 \beta_{6} - 2316181887193690408 \beta_{5} + 8178845652594992720 \beta_{4} - 216457182535896013008 \beta_{2} + 13724515810546138757217\)
\(\nu^{13}\)\(=\)\(196895375815948217184 \beta_{13} + 339043295082927392760 \beta_{12} - 92657339977961914608 \beta_{11} - 82399636834028263440 \beta_{10} + 164799273668056526880 \beta_{9} - 1206634958424149463528 \beta_{8} + 82399636834028263440 \beta_{6} + 82399636834028263440 \beta_{4} - 603968162830619249681808 \beta_{3} + 82399636834028263440 \beta_{2} + 30192779111753470366225 \beta_{1} + 603968162830619249681808\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/38\mathbb{Z}\right)^\times\).

\(n\) \(21\)
\(\chi(n)\) \(-1 + \beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
40.1624 69.5634i
28.5226 49.4026i
8.67249 15.0212i
−3.15555 + 5.46558i
−5.55281 + 9.61775i
−32.4887 + 56.2721i
−35.6605 + 61.7657i
40.1624 + 69.5634i
28.5226 + 49.4026i
8.67249 + 15.0212i
−3.15555 5.46558i
−5.55281 9.61775i
−32.4887 56.2721i
−35.6605 61.7657i
4.00000 + 6.92820i −36.1624 62.6352i −32.0000 + 55.4256i −36.8530 63.8313i 289.300 501.082i 866.283 −512.000 −1521.94 + 2636.09i 294.824 510.650i
7.2 4.00000 + 6.92820i −24.5226 42.4744i −32.0000 + 55.4256i 81.5095 + 141.179i 196.181 339.795i −230.715 −512.000 −109.217 + 189.169i −652.076 + 1129.43i
7.3 4.00000 + 6.92820i −4.67249 8.09299i −32.0000 + 55.4256i −40.1604 69.5598i 37.3799 64.7439i −541.879 −512.000 1049.84 1818.37i 321.283 556.478i
7.4 4.00000 + 6.92820i 7.15555 + 12.3938i −32.0000 + 55.4256i −222.650 385.641i −57.2444 + 99.1503i 1436.09 −512.000 991.096 1716.63i 1781.20 3085.13i
7.5 4.00000 + 6.92820i 9.55281 + 16.5459i −32.0000 + 55.4256i 226.101 + 391.619i −76.4225 + 132.368i −488.812 −512.000 910.988 1577.88i −1808.81 + 3132.95i
7.6 4.00000 + 6.92820i 36.4887 + 63.2003i −32.0000 + 55.4256i −170.433 295.199i −291.910 + 505.603i −1668.31 −512.000 −1569.35 + 2718.20i 1363.47 2361.59i
7.7 4.00000 + 6.92820i 39.6605 + 68.6939i −32.0000 + 55.4256i 99.4859 + 172.315i −317.284 + 549.551i 1591.34 −512.000 −2052.40 + 3554.87i −795.887 + 1378.52i
11.1 4.00000 6.92820i −36.1624 + 62.6352i −32.0000 55.4256i −36.8530 + 63.8313i 289.300 + 501.082i 866.283 −512.000 −1521.94 2636.09i 294.824 + 510.650i
11.2 4.00000 6.92820i −24.5226 + 42.4744i −32.0000 55.4256i 81.5095 141.179i 196.181 + 339.795i −230.715 −512.000 −109.217 189.169i −652.076 1129.43i
11.3 4.00000 6.92820i −4.67249 + 8.09299i −32.0000 55.4256i −40.1604 + 69.5598i 37.3799 + 64.7439i −541.879 −512.000 1049.84 + 1818.37i 321.283 + 556.478i
11.4 4.00000 6.92820i 7.15555 12.3938i −32.0000 55.4256i −222.650 + 385.641i −57.2444 99.1503i 1436.09 −512.000 991.096 + 1716.63i 1781.20 + 3085.13i
11.5 4.00000 6.92820i 9.55281 16.5459i −32.0000 55.4256i 226.101 391.619i −76.4225 132.368i −488.812 −512.000 910.988 + 1577.88i −1808.81 3132.95i
11.6 4.00000 6.92820i 36.4887 63.2003i −32.0000 55.4256i −170.433 + 295.199i −291.910 505.603i −1668.31 −512.000 −1569.35 2718.20i 1363.47 + 2361.59i
11.7 4.00000 6.92820i 39.6605 68.6939i −32.0000 55.4256i 99.4859 172.315i −317.284 549.551i 1591.34 −512.000 −2052.40 3554.87i −795.887 1378.52i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 38.8.c.b 14
19.c even 3 1 inner 38.8.c.b 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.8.c.b 14 1.a even 1 1 trivial
38.8.c.b 14 19.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{14} - \cdots\) acting on \(S_{8}^{\mathrm{new}}(38, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 8 T + 64 T^{2} )^{7} \)
$3$ \( 1 - 55 T - 3841 T^{2} + 293156 T^{3} + 3675062 T^{4} - 640984438 T^{5} + 26323137955 T^{6} - 81322224003 T^{7} - 106222969848024 T^{8} + 4965424185332169 T^{9} + 113356987360233165 T^{10} - 14398516630706326110 T^{11} + \)\(38\!\cdots\!57\)\( T^{12} + \)\(15\!\cdots\!15\)\( T^{13} - \)\(16\!\cdots\!22\)\( T^{14} + \)\(34\!\cdots\!05\)\( T^{15} + \)\(18\!\cdots\!33\)\( T^{16} - \)\(15\!\cdots\!30\)\( T^{17} + \)\(25\!\cdots\!65\)\( T^{18} + \)\(24\!\cdots\!83\)\( T^{19} - \)\(11\!\cdots\!16\)\( T^{20} - \)\(19\!\cdots\!49\)\( T^{21} + \)\(13\!\cdots\!55\)\( T^{22} - \)\(73\!\cdots\!26\)\( T^{23} + \)\(91\!\cdots\!38\)\( T^{24} + \)\(16\!\cdots\!28\)\( T^{25} - \)\(45\!\cdots\!21\)\( T^{26} - \)\(14\!\cdots\!85\)\( T^{27} + \)\(57\!\cdots\!89\)\( T^{28} \)
$5$ \( 1 + 126 T - 240428 T^{2} - 27369376 T^{3} + 28299463426 T^{4} + 2248681694810 T^{5} - 1790358868898250 T^{6} + 47858855429768750 T^{7} + 37651821549849909375 T^{8} - \)\(34\!\cdots\!50\)\( T^{9} + \)\(46\!\cdots\!00\)\( T^{10} + \)\(39\!\cdots\!50\)\( T^{11} - \)\(62\!\cdots\!25\)\( T^{12} - \)\(15\!\cdots\!50\)\( T^{13} + \)\(54\!\cdots\!50\)\( T^{14} - \)\(11\!\cdots\!50\)\( T^{15} - \)\(38\!\cdots\!25\)\( T^{16} + \)\(18\!\cdots\!50\)\( T^{17} + \)\(17\!\cdots\!00\)\( T^{18} - \)\(99\!\cdots\!50\)\( T^{19} + \)\(85\!\cdots\!75\)\( T^{20} + \)\(85\!\cdots\!50\)\( T^{21} - \)\(24\!\cdots\!50\)\( T^{22} + \)\(24\!\cdots\!50\)\( T^{23} + \)\(23\!\cdots\!50\)\( T^{24} - \)\(18\!\cdots\!00\)\( T^{25} - \)\(12\!\cdots\!00\)\( T^{26} + \)\(50\!\cdots\!50\)\( T^{27} + \)\(31\!\cdots\!25\)\( T^{28} \)
$7$ \( ( 1 - 964 T + 1872337 T^{2} - 1615811624 T^{3} + 1808662214969 T^{4} - 581011503608780 T^{5} + 748209055127077481 T^{6} - 43927976075692954640 T^{7} + \)\(61\!\cdots\!83\)\( T^{8} - \)\(39\!\cdots\!20\)\( T^{9} + \)\(10\!\cdots\!83\)\( T^{10} - \)\(74\!\cdots\!24\)\( T^{11} + \)\(70\!\cdots\!91\)\( T^{12} - \)\(30\!\cdots\!36\)\( T^{13} + \)\(25\!\cdots\!07\)\( T^{14} )^{2} \)
$11$ \( ( 1 - 323 T + 35272703 T^{2} + 59049498580 T^{3} + 620645521891300 T^{4} + 1484973402846849056 T^{5} + \)\(14\!\cdots\!10\)\( T^{6} + \)\(19\!\cdots\!26\)\( T^{7} + \)\(28\!\cdots\!10\)\( T^{8} + \)\(56\!\cdots\!96\)\( T^{9} + \)\(45\!\cdots\!00\)\( T^{10} + \)\(85\!\cdots\!80\)\( T^{11} + \)\(99\!\cdots\!53\)\( T^{12} - \)\(17\!\cdots\!83\)\( T^{13} + \)\(10\!\cdots\!91\)\( T^{14} )^{2} \)
$13$ \( 1 + 1308 T + 14219084 T^{2} + 411594681004 T^{3} - 2795323299098726 T^{4} + 11851587983821375892 T^{5} - \)\(26\!\cdots\!90\)\( T^{6} - \)\(96\!\cdots\!64\)\( T^{7} + \)\(14\!\cdots\!47\)\( T^{8} - \)\(96\!\cdots\!04\)\( T^{9} - \)\(50\!\cdots\!60\)\( T^{10} - \)\(22\!\cdots\!20\)\( T^{11} - \)\(16\!\cdots\!65\)\( T^{12} + \)\(16\!\cdots\!80\)\( T^{13} - \)\(23\!\cdots\!38\)\( T^{14} + \)\(10\!\cdots\!60\)\( T^{15} - \)\(66\!\cdots\!85\)\( T^{16} - \)\(54\!\cdots\!60\)\( T^{17} - \)\(78\!\cdots\!60\)\( T^{18} - \)\(93\!\cdots\!28\)\( T^{19} + \)\(90\!\cdots\!43\)\( T^{20} - \)\(37\!\cdots\!72\)\( T^{21} - \)\(64\!\cdots\!90\)\( T^{22} + \)\(17\!\cdots\!24\)\( T^{23} - \)\(26\!\cdots\!74\)\( T^{24} + \)\(24\!\cdots\!32\)\( T^{25} + \)\(52\!\cdots\!24\)\( T^{26} + \)\(30\!\cdots\!96\)\( T^{27} + \)\(14\!\cdots\!29\)\( T^{28} \)
$17$ \( 1 - 10528 T - 765279628 T^{2} + 13866949754092 T^{3} + 93590884203881794 T^{4} - \)\(41\!\cdots\!08\)\( T^{5} + \)\(82\!\cdots\!98\)\( T^{6} - \)\(18\!\cdots\!56\)\( T^{7} - \)\(17\!\cdots\!25\)\( T^{8} + \)\(87\!\cdots\!48\)\( T^{9} - \)\(80\!\cdots\!16\)\( T^{10} + \)\(24\!\cdots\!16\)\( T^{11} - \)\(58\!\cdots\!53\)\( T^{12} - \)\(11\!\cdots\!24\)\( T^{13} + \)\(64\!\cdots\!18\)\( T^{14} - \)\(46\!\cdots\!52\)\( T^{15} - \)\(98\!\cdots\!37\)\( T^{16} + \)\(17\!\cdots\!72\)\( T^{17} - \)\(22\!\cdots\!56\)\( T^{18} + \)\(10\!\cdots\!64\)\( T^{19} - \)\(81\!\cdots\!25\)\( T^{20} - \)\(37\!\cdots\!32\)\( T^{21} + \)\(66\!\cdots\!38\)\( T^{22} - \)\(13\!\cdots\!04\)\( T^{23} + \)\(12\!\cdots\!06\)\( T^{24} + \)\(77\!\cdots\!84\)\( T^{25} - \)\(17\!\cdots\!88\)\( T^{26} - \)\(98\!\cdots\!24\)\( T^{27} + \)\(38\!\cdots\!09\)\( T^{28} \)
$19$ \( 1 + 87463 T + 1218560863 T^{2} - 132318486715668 T^{3} - 5980277100285867204 T^{4} - \)\(38\!\cdots\!20\)\( T^{5} + \)\(38\!\cdots\!86\)\( T^{6} + \)\(16\!\cdots\!78\)\( T^{7} + \)\(34\!\cdots\!54\)\( T^{8} - \)\(30\!\cdots\!20\)\( T^{9} - \)\(42\!\cdots\!76\)\( T^{10} - \)\(84\!\cdots\!88\)\( T^{11} + \)\(69\!\cdots\!37\)\( T^{12} + \)\(44\!\cdots\!43\)\( T^{13} + \)\(45\!\cdots\!79\)\( T^{14} \)
$23$ \( 1 + 113822 T - 2000886916 T^{2} - 373831550598132 T^{3} + 5359450061941137686 T^{4} - \)\(11\!\cdots\!34\)\( T^{5} - \)\(14\!\cdots\!54\)\( T^{6} + \)\(57\!\cdots\!94\)\( T^{7} + \)\(42\!\cdots\!43\)\( T^{8} + \)\(25\!\cdots\!94\)\( T^{9} + \)\(54\!\cdots\!76\)\( T^{10} + \)\(12\!\cdots\!50\)\( T^{11} + \)\(33\!\cdots\!79\)\( T^{12} - \)\(16\!\cdots\!82\)\( T^{13} - \)\(36\!\cdots\!58\)\( T^{14} - \)\(56\!\cdots\!54\)\( T^{15} + \)\(38\!\cdots\!11\)\( T^{16} + \)\(48\!\cdots\!50\)\( T^{17} + \)\(72\!\cdots\!56\)\( T^{18} + \)\(11\!\cdots\!58\)\( T^{19} + \)\(66\!\cdots\!47\)\( T^{20} + \)\(30\!\cdots\!22\)\( T^{21} - \)\(26\!\cdots\!94\)\( T^{22} - \)\(70\!\cdots\!78\)\( T^{23} + \)\(11\!\cdots\!14\)\( T^{24} - \)\(26\!\cdots\!96\)\( T^{25} - \)\(48\!\cdots\!56\)\( T^{26} + \)\(94\!\cdots\!94\)\( T^{27} + \)\(28\!\cdots\!69\)\( T^{28} \)
$29$ \( 1 - 167706 T - 49993022276 T^{2} + 4730414346909376 T^{3} + \)\(20\!\cdots\!50\)\( T^{4} - \)\(43\!\cdots\!38\)\( T^{5} - \)\(49\!\cdots\!38\)\( T^{6} - \)\(24\!\cdots\!18\)\( T^{7} + \)\(85\!\cdots\!67\)\( T^{8} + \)\(88\!\cdots\!22\)\( T^{9} - \)\(61\!\cdots\!52\)\( T^{10} - \)\(16\!\cdots\!02\)\( T^{11} - \)\(62\!\cdots\!09\)\( T^{12} + \)\(11\!\cdots\!82\)\( T^{13} + \)\(27\!\cdots\!50\)\( T^{14} + \)\(19\!\cdots\!38\)\( T^{15} - \)\(18\!\cdots\!29\)\( T^{16} - \)\(85\!\cdots\!58\)\( T^{17} - \)\(54\!\cdots\!72\)\( T^{18} + \)\(13\!\cdots\!78\)\( T^{19} + \)\(22\!\cdots\!47\)\( T^{20} - \)\(11\!\cdots\!42\)\( T^{21} - \)\(38\!\cdots\!98\)\( T^{22} - \)\(58\!\cdots\!82\)\( T^{23} + \)\(47\!\cdots\!50\)\( T^{24} + \)\(19\!\cdots\!84\)\( T^{25} - \)\(34\!\cdots\!56\)\( T^{26} - \)\(20\!\cdots\!74\)\( T^{27} + \)\(20\!\cdots\!61\)\( T^{28} \)
$31$ \( ( 1 - 100890 T + 119900944537 T^{2} - 9963144467980740 T^{3} + \)\(69\!\cdots\!29\)\( T^{4} - \)\(51\!\cdots\!70\)\( T^{5} + \)\(26\!\cdots\!85\)\( T^{6} - \)\(17\!\cdots\!56\)\( T^{7} + \)\(74\!\cdots\!35\)\( T^{8} - \)\(38\!\cdots\!70\)\( T^{9} + \)\(14\!\cdots\!99\)\( T^{10} - \)\(57\!\cdots\!40\)\( T^{11} + \)\(18\!\cdots\!87\)\( T^{12} - \)\(43\!\cdots\!90\)\( T^{13} + \)\(11\!\cdots\!71\)\( T^{14} )^{2} \)
$37$ \( ( 1 - 190462 T + 364636942663 T^{2} - 49366607659083012 T^{3} + \)\(63\!\cdots\!01\)\( T^{4} - \)\(66\!\cdots\!90\)\( T^{5} + \)\(76\!\cdots\!87\)\( T^{6} - \)\(69\!\cdots\!40\)\( T^{7} + \)\(72\!\cdots\!71\)\( T^{8} - \)\(59\!\cdots\!10\)\( T^{9} + \)\(54\!\cdots\!37\)\( T^{10} - \)\(40\!\cdots\!52\)\( T^{11} + \)\(28\!\cdots\!59\)\( T^{12} - \)\(13\!\cdots\!78\)\( T^{13} + \)\(69\!\cdots\!77\)\( T^{14} )^{2} \)
$41$ \( 1 - 66301 T - 603724117711 T^{2} + 16645519220623550 T^{3} + \)\(14\!\cdots\!56\)\( T^{4} + \)\(92\!\cdots\!40\)\( T^{5} - \)\(22\!\cdots\!99\)\( T^{6} - \)\(66\!\cdots\!23\)\( T^{7} + \)\(42\!\cdots\!60\)\( T^{8} + \)\(19\!\cdots\!91\)\( T^{9} - \)\(80\!\cdots\!45\)\( T^{10} - \)\(31\!\cdots\!36\)\( T^{11} + \)\(58\!\cdots\!59\)\( T^{12} + \)\(20\!\cdots\!59\)\( T^{13} + \)\(30\!\cdots\!42\)\( T^{14} + \)\(40\!\cdots\!79\)\( T^{15} + \)\(22\!\cdots\!99\)\( T^{16} - \)\(23\!\cdots\!76\)\( T^{17} - \)\(11\!\cdots\!45\)\( T^{18} + \)\(55\!\cdots\!91\)\( T^{19} + \)\(23\!\cdots\!60\)\( T^{20} - \)\(70\!\cdots\!03\)\( T^{21} - \)\(46\!\cdots\!59\)\( T^{22} + \)\(37\!\cdots\!40\)\( T^{23} + \)\(11\!\cdots\!56\)\( T^{24} + \)\(25\!\cdots\!50\)\( T^{25} - \)\(17\!\cdots\!71\)\( T^{26} - \)\(38\!\cdots\!41\)\( T^{27} + \)\(11\!\cdots\!21\)\( T^{28} \)
$43$ \( 1 - 726564 T - 923581956088 T^{2} + 855538102544616312 T^{3} + \)\(38\!\cdots\!10\)\( T^{4} - \)\(47\!\cdots\!92\)\( T^{5} - \)\(12\!\cdots\!98\)\( T^{6} + \)\(17\!\cdots\!44\)\( T^{7} + \)\(45\!\cdots\!83\)\( T^{8} - \)\(50\!\cdots\!88\)\( T^{9} - \)\(15\!\cdots\!96\)\( T^{10} + \)\(11\!\cdots\!60\)\( T^{11} + \)\(41\!\cdots\!63\)\( T^{12} - \)\(13\!\cdots\!56\)\( T^{13} - \)\(10\!\cdots\!26\)\( T^{14} - \)\(35\!\cdots\!92\)\( T^{15} + \)\(30\!\cdots\!87\)\( T^{16} + \)\(23\!\cdots\!80\)\( T^{17} - \)\(85\!\cdots\!96\)\( T^{18} - \)\(74\!\cdots\!16\)\( T^{19} + \)\(18\!\cdots\!67\)\( T^{20} + \)\(18\!\cdots\!92\)\( T^{21} - \)\(37\!\cdots\!98\)\( T^{22} - \)\(38\!\cdots\!44\)\( T^{23} + \)\(85\!\cdots\!90\)\( T^{24} + \)\(51\!\cdots\!16\)\( T^{25} - \)\(15\!\cdots\!88\)\( T^{26} - \)\(32\!\cdots\!48\)\( T^{27} + \)\(12\!\cdots\!49\)\( T^{28} \)
$47$ \( 1 + 2373788 T + 1370507061080 T^{2} - 261963874590034752 T^{3} + \)\(41\!\cdots\!14\)\( T^{4} + \)\(90\!\cdots\!96\)\( T^{5} + \)\(16\!\cdots\!42\)\( T^{6} + \)\(33\!\cdots\!16\)\( T^{7} + \)\(62\!\cdots\!79\)\( T^{8} + \)\(12\!\cdots\!72\)\( T^{9} - \)\(23\!\cdots\!88\)\( T^{10} + \)\(11\!\cdots\!84\)\( T^{11} + \)\(44\!\cdots\!87\)\( T^{12} + \)\(42\!\cdots\!24\)\( T^{13} + \)\(72\!\cdots\!06\)\( T^{14} + \)\(21\!\cdots\!12\)\( T^{15} + \)\(11\!\cdots\!03\)\( T^{16} + \)\(14\!\cdots\!48\)\( T^{17} - \)\(15\!\cdots\!68\)\( T^{18} + \)\(41\!\cdots\!96\)\( T^{19} + \)\(10\!\cdots\!11\)\( T^{20} + \)\(29\!\cdots\!72\)\( T^{21} + \)\(73\!\cdots\!82\)\( T^{22} + \)\(19\!\cdots\!08\)\( T^{23} + \)\(46\!\cdots\!86\)\( T^{24} - \)\(14\!\cdots\!24\)\( T^{25} + \)\(39\!\cdots\!80\)\( T^{26} + \)\(34\!\cdots\!64\)\( T^{27} + \)\(73\!\cdots\!89\)\( T^{28} \)
$53$ \( 1 - 161792 T - 5012387856388 T^{2} + 1826276832278361028 T^{3} + \)\(13\!\cdots\!86\)\( T^{4} - \)\(68\!\cdots\!76\)\( T^{5} - \)\(22\!\cdots\!50\)\( T^{6} + \)\(15\!\cdots\!68\)\( T^{7} + \)\(24\!\cdots\!23\)\( T^{8} - \)\(24\!\cdots\!88\)\( T^{9} - \)\(12\!\cdots\!32\)\( T^{10} + \)\(24\!\cdots\!04\)\( T^{11} - \)\(77\!\cdots\!25\)\( T^{12} - \)\(11\!\cdots\!36\)\( T^{13} + \)\(21\!\cdots\!34\)\( T^{14} - \)\(13\!\cdots\!32\)\( T^{15} - \)\(10\!\cdots\!25\)\( T^{16} + \)\(40\!\cdots\!12\)\( T^{17} - \)\(23\!\cdots\!52\)\( T^{18} - \)\(53\!\cdots\!16\)\( T^{19} + \)\(63\!\cdots\!07\)\( T^{20} + \)\(48\!\cdots\!44\)\( T^{21} - \)\(80\!\cdots\!50\)\( T^{22} - \)\(29\!\cdots\!52\)\( T^{23} + \)\(66\!\cdots\!14\)\( T^{24} + \)\(10\!\cdots\!64\)\( T^{25} - \)\(34\!\cdots\!28\)\( T^{26} - \)\(13\!\cdots\!24\)\( T^{27} + \)\(95\!\cdots\!89\)\( T^{28} \)
$59$ \( 1 + 1845767 T - 11341077687337 T^{2} - 14324535590263921004 T^{3} + \)\(83\!\cdots\!30\)\( T^{4} + \)\(61\!\cdots\!86\)\( T^{5} - \)\(44\!\cdots\!13\)\( T^{6} - \)\(19\!\cdots\!61\)\( T^{7} + \)\(17\!\cdots\!00\)\( T^{8} + \)\(43\!\cdots\!99\)\( T^{9} - \)\(61\!\cdots\!59\)\( T^{10} - \)\(79\!\cdots\!54\)\( T^{11} + \)\(18\!\cdots\!81\)\( T^{12} + \)\(80\!\cdots\!53\)\( T^{13} - \)\(48\!\cdots\!98\)\( T^{14} + \)\(19\!\cdots\!07\)\( T^{15} + \)\(11\!\cdots\!41\)\( T^{16} - \)\(12\!\cdots\!86\)\( T^{17} - \)\(23\!\cdots\!39\)\( T^{18} + \)\(41\!\cdots\!01\)\( T^{19} + \)\(42\!\cdots\!00\)\( T^{20} - \)\(11\!\cdots\!79\)\( T^{21} - \)\(65\!\cdots\!33\)\( T^{22} + \)\(22\!\cdots\!94\)\( T^{23} + \)\(76\!\cdots\!30\)\( T^{24} - \)\(32\!\cdots\!76\)\( T^{25} - \)\(64\!\cdots\!57\)\( T^{26} + \)\(25\!\cdots\!53\)\( T^{27} + \)\(34\!\cdots\!21\)\( T^{28} \)
$61$ \( 1 + 1600418 T - 9134366904084 T^{2} + 765879370215665640 T^{3} + \)\(59\!\cdots\!86\)\( T^{4} - \)\(64\!\cdots\!54\)\( T^{5} - \)\(16\!\cdots\!38\)\( T^{6} + \)\(37\!\cdots\!26\)\( T^{7} + \)\(67\!\cdots\!59\)\( T^{8} - \)\(59\!\cdots\!18\)\( T^{9} + \)\(24\!\cdots\!28\)\( T^{10} - \)\(15\!\cdots\!70\)\( T^{11} + \)\(44\!\cdots\!79\)\( T^{12} + \)\(45\!\cdots\!78\)\( T^{13} - \)\(26\!\cdots\!66\)\( T^{14} + \)\(14\!\cdots\!38\)\( T^{15} + \)\(44\!\cdots\!39\)\( T^{16} - \)\(48\!\cdots\!70\)\( T^{17} + \)\(24\!\cdots\!68\)\( T^{18} - \)\(18\!\cdots\!18\)\( T^{19} + \)\(65\!\cdots\!39\)\( T^{20} + \)\(11\!\cdots\!66\)\( T^{21} - \)\(15\!\cdots\!18\)\( T^{22} - \)\(19\!\cdots\!74\)\( T^{23} + \)\(55\!\cdots\!86\)\( T^{24} + \)\(22\!\cdots\!40\)\( T^{25} - \)\(84\!\cdots\!44\)\( T^{26} + \)\(46\!\cdots\!98\)\( T^{27} + \)\(91\!\cdots\!81\)\( T^{28} \)
$67$ \( 1 + 8911929 T + 21649850201615 T^{2} + 2818391344931485652 T^{3} + \)\(57\!\cdots\!90\)\( T^{4} + \)\(22\!\cdots\!54\)\( T^{5} - \)\(15\!\cdots\!33\)\( T^{6} - \)\(69\!\cdots\!31\)\( T^{7} - \)\(71\!\cdots\!72\)\( T^{8} - \)\(11\!\cdots\!67\)\( T^{9} - \)\(81\!\cdots\!31\)\( T^{10} - \)\(45\!\cdots\!74\)\( T^{11} + \)\(70\!\cdots\!69\)\( T^{12} + \)\(14\!\cdots\!83\)\( T^{13} + \)\(11\!\cdots\!06\)\( T^{14} + \)\(90\!\cdots\!09\)\( T^{15} + \)\(26\!\cdots\!01\)\( T^{16} - \)\(10\!\cdots\!58\)\( T^{17} - \)\(10\!\cdots\!71\)\( T^{18} - \)\(95\!\cdots\!81\)\( T^{19} - \)\(35\!\cdots\!08\)\( T^{20} - \)\(21\!\cdots\!57\)\( T^{21} - \)\(28\!\cdots\!73\)\( T^{22} + \)\(25\!\cdots\!02\)\( T^{23} + \)\(38\!\cdots\!10\)\( T^{24} + \)\(11\!\cdots\!04\)\( T^{25} + \)\(53\!\cdots\!15\)\( T^{26} + \)\(13\!\cdots\!07\)\( T^{27} + \)\(90\!\cdots\!09\)\( T^{28} \)
$71$ \( 1 - 517154 T - 37070234195980 T^{2} - 7109081551216262324 T^{3} + \)\(61\!\cdots\!10\)\( T^{4} + \)\(46\!\cdots\!38\)\( T^{5} - \)\(84\!\cdots\!70\)\( T^{6} - \)\(29\!\cdots\!26\)\( T^{7} + \)\(12\!\cdots\!07\)\( T^{8} - \)\(10\!\cdots\!46\)\( T^{9} - \)\(15\!\cdots\!68\)\( T^{10} - \)\(35\!\cdots\!14\)\( T^{11} + \)\(15\!\cdots\!47\)\( T^{12} + \)\(38\!\cdots\!50\)\( T^{13} - \)\(14\!\cdots\!86\)\( T^{14} + \)\(34\!\cdots\!50\)\( T^{15} + \)\(12\!\cdots\!07\)\( T^{16} - \)\(26\!\cdots\!94\)\( T^{17} - \)\(10\!\cdots\!48\)\( T^{18} - \)\(65\!\cdots\!46\)\( T^{19} + \)\(69\!\cdots\!87\)\( T^{20} - \)\(15\!\cdots\!06\)\( T^{21} - \)\(39\!\cdots\!70\)\( T^{22} + \)\(19\!\cdots\!18\)\( T^{23} + \)\(23\!\cdots\!10\)\( T^{24} - \)\(25\!\cdots\!84\)\( T^{25} - \)\(11\!\cdots\!80\)\( T^{26} - \)\(15\!\cdots\!34\)\( T^{27} + \)\(26\!\cdots\!61\)\( T^{28} \)
$73$ \( 1 + 9558049 T + 9940048244901 T^{2} - \)\(12\!\cdots\!58\)\( T^{3} - \)\(19\!\cdots\!36\)\( T^{4} + \)\(18\!\cdots\!56\)\( T^{5} - \)\(20\!\cdots\!11\)\( T^{6} - \)\(27\!\cdots\!85\)\( T^{7} - \)\(59\!\cdots\!28\)\( T^{8} + \)\(14\!\cdots\!17\)\( T^{9} - \)\(12\!\cdots\!89\)\( T^{10} - \)\(10\!\cdots\!12\)\( T^{11} + \)\(15\!\cdots\!71\)\( T^{12} + \)\(14\!\cdots\!53\)\( T^{13} + \)\(51\!\cdots\!34\)\( T^{14} + \)\(16\!\cdots\!41\)\( T^{15} + \)\(18\!\cdots\!39\)\( T^{16} - \)\(13\!\cdots\!76\)\( T^{17} - \)\(19\!\cdots\!09\)\( T^{18} + \)\(23\!\cdots\!69\)\( T^{19} - \)\(10\!\cdots\!12\)\( T^{20} - \)\(55\!\cdots\!05\)\( T^{21} - \)\(44\!\cdots\!71\)\( T^{22} + \)\(44\!\cdots\!52\)\( T^{23} - \)\(52\!\cdots\!64\)\( T^{24} - \)\(37\!\cdots\!74\)\( T^{25} + \)\(32\!\cdots\!41\)\( T^{26} + \)\(34\!\cdots\!73\)\( T^{27} + \)\(40\!\cdots\!69\)\( T^{28} \)
$79$ \( 1 - 7963520 T - 15171701371008 T^{2} + \)\(21\!\cdots\!40\)\( T^{3} - \)\(11\!\cdots\!62\)\( T^{4} - \)\(71\!\cdots\!56\)\( T^{5} - \)\(55\!\cdots\!58\)\( T^{6} + \)\(27\!\cdots\!48\)\( T^{7} + \)\(57\!\cdots\!51\)\( T^{8} - \)\(18\!\cdots\!48\)\( T^{9} + \)\(64\!\cdots\!72\)\( T^{10} + \)\(19\!\cdots\!68\)\( T^{11} - \)\(74\!\cdots\!61\)\( T^{12} + \)\(18\!\cdots\!64\)\( T^{13} - \)\(60\!\cdots\!74\)\( T^{14} + \)\(36\!\cdots\!76\)\( T^{15} - \)\(27\!\cdots\!41\)\( T^{16} + \)\(13\!\cdots\!72\)\( T^{17} + \)\(87\!\cdots\!92\)\( T^{18} - \)\(49\!\cdots\!52\)\( T^{19} + \)\(28\!\cdots\!91\)\( T^{20} + \)\(26\!\cdots\!12\)\( T^{21} - \)\(10\!\cdots\!18\)\( T^{22} - \)\(25\!\cdots\!84\)\( T^{23} - \)\(75\!\cdots\!62\)\( T^{24} + \)\(28\!\cdots\!60\)\( T^{25} - \)\(38\!\cdots\!48\)\( T^{26} - \)\(38\!\cdots\!80\)\( T^{27} + \)\(92\!\cdots\!61\)\( T^{28} \)
$83$ \( ( 1 - 15308443 T + 212048914964135 T^{2} - \)\(18\!\cdots\!36\)\( T^{3} + \)\(15\!\cdots\!32\)\( T^{4} - \)\(95\!\cdots\!92\)\( T^{5} + \)\(60\!\cdots\!90\)\( T^{6} - \)\(31\!\cdots\!50\)\( T^{7} + \)\(16\!\cdots\!30\)\( T^{8} - \)\(70\!\cdots\!68\)\( T^{9} + \)\(30\!\cdots\!56\)\( T^{10} - \)\(99\!\cdots\!76\)\( T^{11} + \)\(31\!\cdots\!45\)\( T^{12} - \)\(61\!\cdots\!27\)\( T^{13} + \)\(10\!\cdots\!03\)\( T^{14} )^{2} \)
$89$ \( 1 - 11829436 T - 73165976926900 T^{2} + \)\(60\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!38\)\( T^{4} - \)\(33\!\cdots\!24\)\( T^{5} - \)\(55\!\cdots\!26\)\( T^{6} - \)\(20\!\cdots\!16\)\( T^{7} + \)\(27\!\cdots\!87\)\( T^{8} + \)\(16\!\cdots\!68\)\( T^{9} + \)\(39\!\cdots\!64\)\( T^{10} - \)\(12\!\cdots\!68\)\( T^{11} - \)\(63\!\cdots\!05\)\( T^{12} + \)\(14\!\cdots\!20\)\( T^{13} + \)\(50\!\cdots\!78\)\( T^{14} + \)\(62\!\cdots\!80\)\( T^{15} - \)\(12\!\cdots\!05\)\( T^{16} - \)\(10\!\cdots\!52\)\( T^{17} + \)\(15\!\cdots\!84\)\( T^{18} + \)\(28\!\cdots\!32\)\( T^{19} + \)\(20\!\cdots\!27\)\( T^{20} - \)\(67\!\cdots\!44\)\( T^{21} - \)\(81\!\cdots\!86\)\( T^{22} - \)\(21\!\cdots\!56\)\( T^{23} + \)\(31\!\cdots\!38\)\( T^{24} + \)\(76\!\cdots\!00\)\( T^{25} - \)\(41\!\cdots\!00\)\( T^{26} - \)\(29\!\cdots\!04\)\( T^{27} + \)\(10\!\cdots\!81\)\( T^{28} \)
$97$ \( 1 + 8033723 T - 287993815511247 T^{2} - \)\(43\!\cdots\!06\)\( T^{3} + \)\(35\!\cdots\!92\)\( T^{4} + \)\(10\!\cdots\!52\)\( T^{5} - \)\(33\!\cdots\!23\)\( T^{6} - \)\(14\!\cdots\!31\)\( T^{7} - \)\(62\!\cdots\!48\)\( T^{8} + \)\(14\!\cdots\!79\)\( T^{9} + \)\(12\!\cdots\!19\)\( T^{10} - \)\(95\!\cdots\!96\)\( T^{11} - \)\(15\!\cdots\!05\)\( T^{12} + \)\(29\!\cdots\!75\)\( T^{13} + \)\(14\!\cdots\!94\)\( T^{14} + \)\(24\!\cdots\!75\)\( T^{15} - \)\(10\!\cdots\!45\)\( T^{16} - \)\(50\!\cdots\!12\)\( T^{17} + \)\(53\!\cdots\!59\)\( T^{18} + \)\(49\!\cdots\!47\)\( T^{19} - \)\(17\!\cdots\!32\)\( T^{20} - \)\(32\!\cdots\!27\)\( T^{21} - \)\(61\!\cdots\!83\)\( T^{22} + \)\(14\!\cdots\!96\)\( T^{23} + \)\(42\!\cdots\!08\)\( T^{24} - \)\(41\!\cdots\!22\)\( T^{25} - \)\(22\!\cdots\!07\)\( T^{26} + \)\(50\!\cdots\!19\)\( T^{27} + \)\(50\!\cdots\!89\)\( T^{28} \)
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