Properties

Label 8.16.b.a
Level 8
Weight 16
Character orbit 8.b
Analytic conductor 11.415
Analytic rank 0
Dimension 14
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 16 \)
Character orbit: \([\chi]\) = 8.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(11.415480408\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{91}\cdot 3^{6}\cdot 5^{4}\cdot 31^{2} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q\) \( + ( -6 - \beta_{1} ) q^{2} \) \( + \beta_{2} q^{3} \) \( + ( 3672 + 5 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{4} \) \( + ( -54 + 126 \beta_{1} - 3 \beta_{2} - \beta_{5} ) q^{5} \) \( + ( -13531 - 7 \beta_{2} + \beta_{4} ) q^{6} \) \( + ( -118147 + 1156 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} - \beta_{8} ) q^{7} \) \( + ( 136595 - 3769 \beta_{1} - 153 \beta_{2} - 10 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{8} \) \( + ( -4099555 - 348 \beta_{1} - 18 \beta_{2} + 38 \beta_{3} + \beta_{4} - \beta_{9} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -6 - \beta_{1} ) q^{2} \) \( + \beta_{2} q^{3} \) \( + ( 3672 + 5 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{4} \) \( + ( -54 + 126 \beta_{1} - 3 \beta_{2} - \beta_{5} ) q^{5} \) \( + ( -13531 - 7 \beta_{2} + \beta_{4} ) q^{6} \) \( + ( -118147 + 1156 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} - \beta_{8} ) q^{7} \) \( + ( 136595 - 3769 \beta_{1} - 153 \beta_{2} - 10 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{8} \) \( + ( -4099555 - 348 \beta_{1} - 18 \beta_{2} + 38 \beta_{3} + \beta_{4} - \beta_{9} ) q^{9} \) \( + ( 4176589 - 606 \beta_{1} + 547 \beta_{2} - 134 \beta_{3} - 2 \beta_{4} + 6 \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{13} ) q^{10} \) \( + ( -26735 + 62124 \beta_{1} + 501 \beta_{2} + 228 \beta_{3} + 12 \beta_{4} - 57 \beta_{5} + 4 \beta_{6} + 15 \beta_{7} - \beta_{8} + \beta_{12} + \beta_{13} ) q^{11} \) \( + ( 28521939 + 8519 \beta_{1} + 1549 \beta_{2} - 83 \beta_{3} - 17 \beta_{4} - 119 \beta_{5} - 3 \beta_{6} + 17 \beta_{7} + 20 \beta_{8} + 6 \beta_{9} - \beta_{11} + 2 \beta_{12} ) q^{12} \) \( + ( -18196 + 42087 \beta_{1} + 8546 \beta_{2} + 410 \beta_{3} - 89 \beta_{4} - 26 \beta_{5} + 18 \beta_{6} + 37 \beta_{7} + 3 \beta_{8} - \beta_{10} - 2 \beta_{11} + 2 \beta_{12} - 6 \beta_{13} ) q^{13} \) \( + ( -37123620 + 129153 \beta_{1} + 9978 \beta_{2} - 818 \beta_{3} - 20 \beta_{4} + 249 \beta_{5} - 4 \beta_{6} + 41 \beta_{7} + 82 \beta_{8} - 14 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} + 4 \beta_{12} - 6 \beta_{13} ) q^{14} \) \( + ( 50756093 + 262756 \beta_{1} + 1876 \beta_{2} - 3886 \beta_{3} - 498 \beta_{4} - 156 \beta_{5} - 84 \beta_{6} + 16 \beta_{7} + 9 \beta_{8} + 8 \beta_{9} - 2 \beta_{10} + 4 \beta_{11} + 8 \beta_{12} - 8 \beta_{13} ) q^{15} \) \( + ( -102523960 - 104402 \beta_{1} + 22912 \beta_{2} + 5158 \beta_{3} - 76 \beta_{4} + 1404 \beta_{5} + 20 \beta_{6} + 122 \beta_{7} - 268 \beta_{8} - 20 \beta_{9} + 4 \beta_{10} - 14 \beta_{11} + 12 \beta_{12} - 8 \beta_{13} ) q^{16} \) \( + ( 52548484 - 1183020 \beta_{1} + 1534 \beta_{2} - 5182 \beta_{3} + 899 \beta_{4} - 136 \beta_{6} + 8 \beta_{7} + 84 \beta_{8} + 25 \beta_{9} - 20 \beta_{10} - 24 \beta_{11} + 16 \beta_{12} - 16 \beta_{13} ) q^{17} \) \( + ( 35881878 + 4085747 \beta_{1} - 22436 \beta_{2} - 1292 \beta_{3} + 96 \beta_{4} - 6958 \beta_{5} - 32 \beta_{6} + 226 \beta_{7} - 764 \beta_{8} + 4 \beta_{9} + 36 \beta_{10} + 28 \beta_{11} + 24 \beta_{12} + 4 \beta_{13} ) q^{18} \) \( + ( -1538251 + 3571956 \beta_{1} - 74535 \beta_{2} + 13764 \beta_{3} + 3348 \beta_{4} - 469 \beta_{5} + 36 \beta_{6} + 275 \beta_{7} - 61 \beta_{8} - 40 \beta_{10} + 48 \beta_{11} + 5 \beta_{12} - 59 \beta_{13} ) q^{19} \) \( + ( -244638714 - 4056790 \beta_{1} + 197490 \beta_{2} + 5114 \beta_{3} - 10 \beta_{4} - 14418 \beta_{5} + 90 \beta_{6} - 210 \beta_{7} + 1152 \beta_{8} - 116 \beta_{9} + 72 \beta_{10} - 74 \beta_{11} - 12 \beta_{12} - 48 \beta_{13} ) q^{20} \) \( + ( 2279458 - 5427521 \beta_{1} - 289011 \beta_{2} + 116270 \beta_{3} - 7611 \beta_{4} - 449 \beta_{5} + 22 \beta_{6} - 33 \beta_{7} + 377 \beta_{8} - 179 \beta_{10} - 102 \beta_{11} + 6 \beta_{12} + 110 \beta_{13} ) q^{21} \) \( + ( 2026330537 - 158848 \beta_{1} + 556381 \beta_{2} - 52384 \beta_{3} + 1309 \beta_{4} + 38592 \beta_{5} - 280 \beta_{6} - 952 \beta_{7} + 336 \beta_{8} + 328 \beta_{9} + 288 \beta_{10} + 144 \beta_{11} - 32 \beta_{12} + 88 \beta_{13} ) q^{22} \) \( + ( -2542420033 + 7723084 \beta_{1} + 97696 \beta_{2} - 204146 \beta_{3} - 9170 \beta_{4} - 2204 \beta_{5} + 1004 \beta_{6} + 976 \beta_{7} - 129 \beta_{8} - 248 \beta_{9} - 354 \beta_{10} + 196 \beta_{11} - 120 \beta_{12} + 120 \beta_{13} ) q^{23} \) \( + ( 2880795874 - 29305378 \beta_{1} - 509190 \beta_{2} - 22048 \beta_{3} + 2450 \beta_{4} + 85740 \beta_{5} + 130 \beta_{6} - 2282 \beta_{7} + 888 \beta_{8} + 472 \beta_{9} + 568 \beta_{10} - 140 \beta_{11} - 200 \beta_{12} + 80 \beta_{13} ) q^{24} \) \( + ( -5420644031 - 1478912 \beta_{1} + 164504 \beta_{2} - 356884 \beta_{3} + 11946 \beta_{4} + 2944 \beta_{5} + 2824 \beta_{6} - 520 \beta_{7} + 364 \beta_{8} - 262 \beta_{9} - 940 \beta_{10} - 104 \beta_{11} - 272 \beta_{12} + 272 \beta_{13} ) q^{25} \) \( + ( 1262487647 - 1307530 \beta_{1} - 2438671 \beta_{2} - 128450 \beta_{3} + 6730 \beta_{4} - 104766 \beta_{5} - 187 \beta_{6} - 8507 \beta_{7} + 4618 \beta_{8} - 421 \beta_{9} + 1344 \beta_{10} + 224 \beta_{11} - 448 \beta_{12} + 101 \beta_{13} ) q^{26} \) \( + ( -2800627 + 5517620 \beta_{1} - 4650854 \beta_{2} + 1023972 \beta_{3} - 5036 \beta_{4} - 5773 \beta_{5} - 2492 \beta_{6} + 283 \beta_{7} + 1867 \beta_{8} - 1800 \beta_{10} + 112 \beta_{11} - 291 \beta_{12} + 925 \beta_{13} ) q^{27} \) \( + ( 5687996200 + 38650472 \beta_{1} + 6553592 \beta_{2} + 6768 \beta_{3} + 12696 \beta_{4} - 173216 \beta_{5} + 344 \beta_{6} + 1248 \beta_{7} - 17488 \beta_{8} + 784 \beta_{9} + 2544 \beta_{10} + 248 \beta_{11} - 176 \beta_{12} + 992 \beta_{13} ) q^{28} \) \( + ( 7903558 - 19983382 \beta_{1} - 6485189 \beta_{2} + 1536272 \beta_{3} + 21848 \beta_{4} + 28073 \beta_{5} - 2608 \beta_{6} - 12536 \beta_{7} + 2648 \beta_{8} - 3176 \beta_{10} + 560 \beta_{11} - 592 \beta_{12} - 912 \beta_{13} ) q^{29} \) \( + ( -8902015556 - 42026023 \beta_{1} + 17010090 \beta_{2} + 266206 \beta_{3} + 15308 \beta_{4} + 208785 \beta_{5} + 4444 \beta_{6} - 18239 \beta_{7} - 24702 \beta_{8} - 3294 \beta_{9} + 4018 \beta_{10} - 782 \beta_{11} - 284 \beta_{12} - 406 \beta_{13} ) q^{30} \) \( + ( -7480990916 - 168375280 \beta_{1} + 968834 \beta_{2} - 2416868 \beta_{3} + 54498 \beta_{4} - 28420 \beta_{5} - 4172 \beta_{6} + 25392 \beta_{7} + 8922 \beta_{8} + 3512 \beta_{9} - 5646 \beta_{10} - 1508 \beta_{11} + 568 \beta_{12} - 568 \beta_{13} ) q^{31} \) \( + ( -2730269080 + 95716572 \beta_{1} - 17988616 \beta_{2} - 407148 \beta_{3} + 12528 \beta_{4} - 34776 \beta_{5} - 816 \beta_{6} + 35844 \beta_{7} + 10456 \beta_{8} - 4792 \beta_{9} + 6904 \beta_{10} + 1708 \beta_{11} + 1224 \beta_{12} + 400 \beta_{13} ) q^{32} \) \( + ( -10871529726 + 295209236 \beta_{1} + 2219142 \beta_{2} - 4399722 \beta_{3} - 199111 \beta_{4} - 26496 \beta_{5} - 19024 \beta_{6} - 28464 \beta_{7} + 17352 \beta_{8} + 1375 \beta_{9} - 6984 \beta_{10} + 2064 \beta_{11} + 1696 \beta_{12} - 1696 \beta_{13} ) q^{33} \) \( + ( 38421323492 - 71381124 \beta_{1} - 28726844 \beta_{2} + 358508 \beta_{3} - 1248 \beta_{4} + 264542 \beta_{5} + 7072 \beta_{6} - 31794 \beta_{7} - 420 \beta_{8} + 5468 \beta_{9} + 7740 \beta_{10} - 3836 \beta_{11} + 3368 \beta_{12} - 932 \beta_{13} ) q^{34} \) \( + ( -127109778 + 290944768 \beta_{1} - 4633352 \beta_{2} + 5749576 \beta_{3} - 260480 \beta_{4} - 87414 \beta_{5} + 24008 \beta_{6} + 139002 \beta_{7} + 16410 \beta_{8} - 10408 \beta_{10} - 4304 \beta_{11} + 2606 \beta_{12} - 6674 \beta_{13} ) q^{35} \) \( + ( 24195142520 - 6832597 \beta_{1} + 86096818 \beta_{2} - 1991745 \beta_{3} - 23888 \beta_{4} + 737600 \beta_{5} - 10448 \beta_{6} + 120256 \beta_{7} + 30304 \beta_{8} - 480 \beta_{9} + 10464 \beta_{10} + 2800 \beta_{11} + 2208 \beta_{12} - 8768 \beta_{13} ) q^{36} \) \( + ( -2406892 - 448667 \beta_{1} - 13534002 \beta_{2} + 5899342 \beta_{3} + 201013 \beta_{4} - 249622 \beta_{5} + 25014 \beta_{6} - 45265 \beta_{7} - 2103 \beta_{8} - 8643 \beta_{10} + 2170 \beta_{11} + 6406 \beta_{12} + 4846 \beta_{13} ) q^{37} \) \( + ( 117828621317 + 22601344 \beta_{1} + 112292969 \beta_{2} - 1198112 \beta_{3} - 54935 \beta_{4} - 1504064 \beta_{5} - 35320 \beta_{6} - 209176 \beta_{7} + 115088 \beta_{8} + 17640 \beta_{9} + 7584 \beta_{10} - 3888 \beta_{11} + 4960 \beta_{12} - 200 \beta_{13} ) q^{38} \) \( + ( -160242123241 - 1358171444 \beta_{1} - 475990 \beta_{2} - 1576270 \beta_{3} + 69424 \beta_{4} - 294816 \beta_{5} - 2080 \beta_{6} + 289472 \beta_{7} + 8685 \beta_{8} - 29760 \beta_{9} - 6352 \beta_{10} - 1120 \beta_{11} + 320 \beta_{12} - 320 \beta_{13} ) q^{39} \) \( + ( 89295028940 + 159605204 \beta_{1} - 165473252 \beta_{2} - 1129664 \beta_{3} + 118444 \beta_{4} - 2731224 \beta_{5} - 12276 \beta_{6} + 332772 \beta_{7} - 178032 \beta_{8} + 26256 \beta_{9} + 1936 \beta_{10} - 968 \beta_{11} - 2224 \beta_{12} - 10656 \beta_{13} ) q^{40} \) \( + ( -4759684410 + 2231280192 \beta_{1} + 445224 \beta_{2} + 3379524 \beta_{3} + 262398 \beta_{4} + 565504 \beta_{5} + 32600 \beta_{6} - 498392 \beta_{7} - 146268 \beta_{8} - 2546 \beta_{9} + 2204 \beta_{10} - 2680 \beta_{11} - 2992 \beta_{12} + 2992 \beta_{13} ) q^{41} \) \( + ( -174044492572 - 64196600 \beta_{1} - 242737444 \beta_{2} + 442344 \beta_{3} + 12056 \beta_{4} + 4390296 \beta_{5} - 63764 \beta_{6} - 796308 \beta_{7} - 212904 \beta_{8} - 33292 \beta_{9} - 8256 \beta_{10} + 9888 \beta_{11} - 11584 \beta_{12} + 1612 \beta_{13} ) q^{42} \) \( + ( -1319124358 + 3089489192 \beta_{1} - 11756291 \beta_{2} - 12927288 \beta_{3} + 817768 \beta_{4} - 1339578 \beta_{5} - 81528 \beta_{6} + 636822 \beta_{7} - 42762 \beta_{8} + 23344 \beta_{10} + 14176 \beta_{11} - 8934 \beta_{12} + 22682 \beta_{13} ) q^{43} \) \( + ( -243258505641 - 1879817829 \beta_{1} + 240843209 \beta_{2} + 7802409 \beta_{3} + 364035 \beta_{4} + 2287541 \beta_{5} + 53049 \beta_{6} + 1026077 \beta_{7} + 196420 \beta_{8} - 25202 \beta_{9} - 34752 \beta_{10} - 12077 \beta_{11} - 7846 \beta_{12} + 41600 \beta_{13} ) q^{44} \) \( + ( 3344157560 - 7772333755 \beta_{1} + 75937168 \beta_{2} - 27546522 \beta_{3} - 1254887 \beta_{4} + 3304064 \beta_{5} - 121874 \beta_{6} - 1859429 \beta_{7} + 19165 \beta_{8} + 41025 \beta_{10} - 14462 \beta_{11} - 31266 \beta_{12} - 19994 \beta_{13} ) q^{45} \) \( + ( -237127726892 + 2709267395 \beta_{1} + 290699822 \beta_{2} + 1538090 \beta_{3} + 293252 \beta_{4} - 729813 \beta_{5} + 208308 \beta_{6} - 1587141 \beta_{7} - 65930 \beta_{8} - 46506 \beta_{9} - 55930 \beta_{10} + 31430 \beta_{11} - 25972 \beta_{12} + 7406 \beta_{13} ) q^{46} \) \( + ( 898110040022 - 7618928744 \beta_{1} - 19593614 \beta_{2} + 27806992 \beta_{3} - 1383770 \beta_{4} - 1933516 \beta_{5} + 86044 \beta_{6} + 1949264 \beta_{7} - 153660 \beta_{8} + 165032 \beta_{9} + 88406 \beta_{10} + 30036 \beta_{11} - 11608 \beta_{12} + 11608 \beta_{13} ) q^{47} \) \( + ( -458837985568 - 2987485300 \beta_{1} - 497886384 \beta_{2} + 17365036 \beta_{3} - 1263432 \beta_{4} - 2562376 \beta_{5} + 106424 \beta_{6} + 2594916 \beta_{7} + 501160 \beta_{8} - 75048 \beta_{9} - 107704 \beta_{10} - 26076 \beta_{11} - 9576 \beta_{12} + 73840 \beta_{13} ) q^{48} \) \( + ( 597171298361 + 11103207072 \beta_{1} - 17303312 \beta_{2} + 61401712 \beta_{3} + 2643816 \beta_{4} + 3979136 \beta_{5} + 190848 \beta_{6} - 3259648 \beta_{7} + 939072 \beta_{8} - 10920 \beta_{9} + 115136 \beta_{10} - 29568 \beta_{11} - 16128 \beta_{12} + 16128 \beta_{13} ) q^{49} \) \( + ( 82026478962 + 5242404751 \beta_{1} - 404817048 \beta_{2} - 9088136 \beta_{3} + 28096 \beta_{4} - 3596020 \beta_{5} + 374464 \beta_{6} - 3047060 \beta_{7} + 829016 \beta_{8} + 95832 \beta_{9} - 136424 \beta_{10} + 28904 \beta_{11} + 6416 \beta_{12} + 12120 \beta_{13} ) q^{50} \) \( + ( -6920256261 + 16236415044 \beta_{1} + 280788362 \beta_{2} - 95706868 \beta_{3} + 2287204 \beta_{4} + 1399533 \beta_{5} - 27028 \beta_{6} + 4276741 \beta_{7} - 221355 \beta_{8} + 169696 \beta_{10} + 26560 \beta_{11} - 1461 \beta_{12} - 6837 \beta_{13} ) q^{51} \) \( + ( -173491630718 - 1191324722 \beta_{1} + 110690854 \beta_{2} + 374910 \beta_{3} - 3407022 \beta_{4} - 16580358 \beta_{5} - 84322 \beta_{6} + 5377658 \beta_{7} - 894976 \beta_{8} + 169348 \beta_{9} - 188008 \beta_{10} - 24430 \beta_{11} - 6660 \beta_{12} - 102672 \beta_{13} ) q^{52} \) \( + ( 9023429688 - 20960494441 \beta_{1} + 403558968 \beta_{2} - 89000062 \beta_{3} + 175043 \beta_{4} + 3499736 \beta_{5} + 279514 \beta_{6} - 5628487 \beta_{7} - 220689 \beta_{8} + 191227 \beta_{10} - 17674 \beta_{11} + 64810 \beta_{12} + 62210 \beta_{13} ) q^{53} \) \( + ( 239876527562 - 90005888 \beta_{1} - 132603646 \beta_{2} - 4270880 \beta_{3} - 1774262 \beta_{4} + 27644608 \beta_{5} - 914488 \beta_{6} - 7703128 \beta_{7} - 945904 \beta_{8} - 4184 \beta_{9} - 200544 \beta_{10} - 19888 \beta_{11} + 43104 \beta_{12} - 15496 \beta_{13} ) q^{54} \) \( + ( -2165029137599 - 41365885932 \beta_{1} - 90166018 \beta_{2} + 118259022 \beta_{3} + 2141640 \beta_{4} - 8159184 \beta_{5} - 246704 \beta_{6} + 8500480 \beta_{7} - 105661 \beta_{8} - 606112 \beta_{9} + 189064 \beta_{10} - 29456 \beta_{11} + 27616 \beta_{12} - 27616 \beta_{13} ) q^{55} \) \( + ( 541643997768 - 5641178392 \beta_{1} + 424525736 \beta_{2} - 32886384 \beta_{3} + 4739304 \beta_{4} + 17863536 \beta_{5} - 259992 \beta_{6} + 10739880 \beta_{7} + 1278080 \beta_{8} + 46592 \beta_{9} - 171648 \beta_{10} + 36224 \beta_{11} + 50944 \beta_{12} - 243968 \beta_{13} ) q^{56} \) \( + ( 1278498207226 + 62991835940 \beta_{1} + 20087326 \beta_{2} + 70932574 \beta_{3} - 6166363 \beta_{4} + 10397952 \beta_{5} - 951344 \beta_{6} - 11767760 \beta_{7} - 4887432 \beta_{8} + 74883 \beta_{9} + 170248 \beta_{10} + 64624 \beta_{11} + 88672 \beta_{12} - 88672 \beta_{13} ) q^{57} \) \( + ( -572674346445 + 386821470 \beta_{1} + 660438237 \beta_{2} + 44609414 \beta_{3} - 2989054 \beta_{4} - 18120454 \beta_{5} - 1630207 \beta_{6} - 14192639 \beta_{7} + 777218 \beta_{8} + 8959 \beta_{9} - 139776 \beta_{10} - 128768 \beta_{11} + 85504 \beta_{12} - 55551 \beta_{13} ) q^{58} \) \( + ( -24350963488 + 56810582360 \beta_{1} + 422287309 \beta_{2} + 201680 \beta_{3} - 9613672 \beta_{4} - 46722552 \beta_{5} + 921744 \beta_{6} + 16942984 \beta_{7} + 242824 \beta_{8} + 9112 \beta_{10} - 177232 \beta_{11} + 102528 \beta_{12} - 231872 \beta_{13} ) q^{59} \) \( + ( -3830556970776 + 8836232360 \beta_{1} - 1989465032 \beta_{2} - 10454672 \beta_{3} + 12463576 \beta_{4} - 56168352 \beta_{5} - 65640 \beta_{6} + 18747872 \beta_{7} + 933936 \beta_{8} - 468848 \beta_{9} + 74864 \beta_{10} + 158264 \beta_{11} + 121552 \beta_{12} + 55520 \beta_{13} ) q^{60} \) \( + ( 24044557124 - 56129736993 \beta_{1} + 50390398 \beta_{2} + 31578682 \beta_{3} + 12482135 \beta_{4} + 14844762 \beta_{5} + 199218 \beta_{6} - 18062667 \beta_{7} - 326381 \beta_{8} - 78673 \beta_{10} + 173150 \beta_{11} + 58754 \beta_{12} - 78598 \beta_{13} ) q^{61} \) \( + ( 5560561098736 + 5683275732 \beta_{1} - 1977223096 \beta_{2} + 117154904 \beta_{3} + 1427120 \beta_{4} + 37392692 \beta_{5} + 2651120 \beta_{6} - 18098316 \beta_{7} + 2855144 \beta_{8} + 431208 \beta_{9} + 180904 \beta_{10} - 251032 \beta_{11} + 141776 \beta_{12} - 45432 \beta_{13} ) q^{62} \) \( + ( 2633919202139 - 121932017796 \beta_{1} - 17697468 \beta_{2} - 188775922 \beta_{3} + 10816666 \beta_{4} - 23387700 \beta_{5} - 441756 \beta_{6} + 24221232 \beta_{7} + 938295 \beta_{8} + 1347416 \beta_{9} - 442134 \beta_{10} - 255444 \beta_{11} + 69720 \beta_{12} - 69720 \beta_{13} ) q^{63} \) \( + ( 5433116590320 + 3310231064 \beta_{1} + 4127628304 \beta_{2} - 22109336 \beta_{3} - 20510464 \beta_{4} - 28592944 \beta_{5} - 128192 \beta_{6} + 25465320 \beta_{7} - 6200272 \beta_{8} + 374672 \beta_{9} + 579440 \beta_{10} + 209496 \beta_{11} - 28272 \beta_{12} + 255520 \beta_{13} ) q^{64} \) \( + ( 335900299420 + 122902295008 \beta_{1} + 313250616 \beta_{2} - 447255964 \beta_{3} - 12147442 \beta_{4} + 25089536 \beta_{5} + 443768 \beta_{6} - 27610104 \beta_{7} + 15319380 \beta_{8} - 114290 \beta_{9} - 641300 \beta_{10} + 201192 \beta_{11} - 64496 \beta_{12} + 64496 \beta_{13} ) q^{65} \) \( + ( -9586400539912 + 15525681326 \beta_{1} + 6904183132 \beta_{2} - 62325260 \beta_{3} + 8339808 \beta_{4} + 69558482 \beta_{5} + 4724960 \beta_{6} - 30181662 \beta_{7} - 7966460 \beta_{8} - 1005564 \beta_{9} + 840740 \beta_{10} - 101604 \beta_{11} - 253416 \beta_{12} - 16892 \beta_{13} ) q^{66} \) \( + ( -39652021515 + 91890256556 \beta_{1} - 2699955655 \beta_{2} + 610611028 \beta_{3} - 7802868 \beta_{4} + 94310771 \beta_{5} - 2014860 \beta_{6} + 29708507 \beta_{7} + 1256331 \beta_{8} - 1049776 \beta_{10} + 19872 \beta_{11} - 246299 \beta_{12} + 763493 \beta_{13} ) q^{67} \) \( + ( -4966722974624 - 39668508134 \beta_{1} - 5322064052 \beta_{2} - 68719870 \beta_{3} - 36415024 \beta_{4} + 48354496 \beta_{5} + 196944 \beta_{6} + 31302720 \beta_{7} + 4266656 \beta_{8} + 126176 \beta_{9} + 1238048 \beta_{10} + 65424 \beta_{11} - 237984 \beta_{12} + 400448 \beta_{13} ) q^{68} \) \( + ( 70242467358 - 164503737999 \beta_{1} - 3953942029 \beta_{2} + 660817394 \beta_{3} - 9077909 \beta_{4} + 65304897 \beta_{5} - 2806390 \beta_{6} - 44741327 \beta_{7} + 1952823 \beta_{8} - 1430813 \beta_{10} + 51910 \beta_{11} - 625830 \beta_{12} - 374670 \beta_{13} ) q^{69} \) \( + ( 9570335515200 - 4805288448 \beta_{1} - 6922441824 \beta_{2} - 528633280 \beta_{3} + 5960048 \beta_{4} - 50867840 \beta_{5} - 4347600 \beta_{6} - 52578704 \beta_{7} + 3012448 \beta_{8} - 1205904 \beta_{9} + 1641408 \beta_{10} + 428000 \beta_{11} - 782784 \beta_{12} + 191824 \beta_{13} ) q^{70} \) \( + ( -12306495148851 - 159051620828 \beta_{1} + 222050600 \beta_{2} - 803401254 \beta_{3} - 19333230 \beta_{4} - 44529252 \beta_{5} + 3442580 \beta_{6} + 41580464 \beta_{7} + 404037 \beta_{8} - 888584 \beta_{9} - 1582558 \beta_{10} + 444860 \beta_{11} - 388744 \beta_{12} + 388744 \beta_{13} ) q^{71} \) \( + ( 11680732766621 - 23592860759 \beta_{1} + 7184617481 \beta_{2} + 138700138 \beta_{3} + 78022897 \beta_{4} - 199230818 \beta_{5} + 1562609 \beta_{6} + 35639153 \beta_{7} - 2313472 \beta_{8} - 968192 \beta_{9} + 1719552 \beta_{10} - 317440 \beta_{11} - 291840 \beta_{12} + 989696 \beta_{13} ) q^{72} \) \( + ( -13084974998664 + 189099269444 \beta_{1} + 377788894 \beta_{2} - 460758634 \beta_{3} + 42928617 \beta_{4} + 46772608 \beta_{5} + 6380928 \beta_{6} - 38624000 \beta_{7} - 32128448 \beta_{8} - 391977 \beta_{9} - 1979456 \beta_{10} - 505728 \beta_{11} - 587520 \beta_{12} + 587520 \beta_{13} ) q^{73} \) \( + ( 146750413061 + 2851471730 \beta_{1} + 6656384363 \beta_{2} + 238079370 \beta_{3} + 1525710 \beta_{4} + 238784886 \beta_{5} - 6867385 \beta_{6} - 46480185 \beta_{7} + 6194958 \beta_{8} + 2876057 \beta_{9} + 1994688 \beta_{10} + 746144 \beta_{11} + 47808 \beta_{12} + 547239 \beta_{13} ) q^{74} \) \( + ( -99133906162 + 230475140192 \beta_{1} - 5247944347 \beta_{2} + 754337800 \beta_{3} + 48071136 \beta_{4} - 446651638 \beta_{5} - 1409656 \beta_{6} + 37746810 \beta_{7} + 109210 \beta_{8} - 1567432 \beta_{10} + 814320 \beta_{11} - 170418 \beta_{12} - 256754 \beta_{13} ) q^{75} \) \( + ( -17700515428677 - 116129871121 \beta_{1} - 4194252059 \beta_{2} - 44604475 \beta_{3} + 111281335 \beta_{4} + 227012385 \beta_{5} - 74603 \beta_{6} + 25531369 \beta_{7} - 12225228 \beta_{8} + 3068630 \beta_{9} + 1305920 \beta_{10} - 982969 \beta_{11} - 491918 \beta_{12} - 1149824 \beta_{13} ) q^{76} \) \( + ( 43694754722 - 103153453393 \beta_{1} + 2135773933 \beta_{2} + 1281440846 \beta_{3} - 74352139 \beta_{4} - 33844129 \beta_{5} + 5064630 \beta_{6} - 13231889 \beta_{7} + 2794921 \beta_{8} - 1546883 \beta_{10} - 1212934 \beta_{11} + 1177830 \beta_{12} + 2037454 \beta_{13} ) q^{77} \) \( + ( 45399256908820 + 150391444579 \beta_{1} - 4295799378 \beta_{2} + 1151394986 \beta_{3} + 5144644 \beta_{4} - 192282933 \beta_{5} + 2014836 \beta_{6} - 13850021 \beta_{7} - 22289226 \beta_{8} + 472982 \beta_{9} + 1244614 \beta_{10} + 730054 \beta_{11} + 876428 \beta_{12} + 101038 \beta_{13} ) q^{78} \) \( + ( -21015699778350 - 24697889752 \beta_{1} + 108039428 \beta_{2} - 326741812 \beta_{3} - 67077288 \beta_{4} - 33010928 \beta_{5} - 1916176 \beta_{6} + 20241024 \beta_{7} - 1126066 \beta_{8} - 4940768 \beta_{9} - 269160 \beta_{10} + 1128656 \beta_{11} + 78752 \beta_{12} - 78752 \beta_{13} ) q^{79} \) \( + ( 54773386701504 - 99071907128 \beta_{1} + 3082184544 \beta_{2} - 328600056 \beta_{3} - 176971824 \beta_{4} - 62577328 \beta_{5} - 1885744 \beta_{6} + 11794392 \beta_{7} + 35446256 \beta_{8} - 404208 \beta_{9} - 364624 \beta_{10} - 1232168 \beta_{11} + 492432 \beta_{12} - 4162912 \beta_{13} ) q^{80} \) \( + ( 18394153181899 - 102364675500 \beta_{1} - 21856122 \beta_{2} - 112884506 \beta_{3} + 13467857 \beta_{4} - 7401216 \beta_{5} - 14082864 \beta_{6} - 149712 \beta_{7} + 55130616 \beta_{8} + 1802455 \beta_{9} + 402312 \beta_{10} - 970896 \beta_{11} + 1505376 \beta_{12} - 1505376 \beta_{13} ) q^{81} \) \( + ( -72971352418108 + 16432397610 \beta_{1} - 3439497032 \beta_{2} - 2236331544 \beta_{3} - 10829504 \beta_{4} - 53572124 \beta_{5} - 3347392 \beta_{6} + 20184324 \beta_{7} + 22887560 \beta_{8} - 328568 \beta_{9} - 1231160 \beta_{10} + 491832 \beta_{11} + 822320 \beta_{12} - 712824 \beta_{13} ) q^{82} \) \( + ( 57876546120 - 133669325240 \beta_{1} + 13305551553 \beta_{2} - 1414950320 \beta_{3} + 32329480 \beta_{4} + 1061216720 \beta_{5} + 11890832 \beta_{6} - 405104 \beta_{7} - 3549936 \beta_{8} + 2303528 \beta_{10} - 146480 \beta_{11} + 1539368 \beta_{12} - 4047512 \beta_{13} ) q^{83} \) \( + ( -121635377996840 + 203922290088 \beta_{1} + 14086171272 \beta_{2} + 841664040 \beta_{3} - 239365352 \beta_{4} - 295913416 \beta_{5} + 2783656 \beta_{6} - 8935624 \beta_{7} - 11432576 \beta_{8} - 8499792 \beta_{9} - 3001184 \beta_{10} - 60904 \beta_{11} + 2458448 \beta_{12} + 1360448 \beta_{13} ) q^{84} \) \( + ( -95975912222 + 226384280961 \beta_{1} - 1944531771 \beta_{2} - 2534538006 \beta_{3} + 31308831 \beta_{4} - 31377401 \beta_{5} + 3199266 \beta_{6} + 60410893 \beta_{7} - 4533605 \beta_{8} + 4164151 \beta_{10} + 131694 \beta_{11} + 106066 \beta_{12} - 2800630 \beta_{13} ) q^{85} \) \( + ( 101332717607777 - 7346612992 \beta_{1} + 31002026309 \beta_{2} - 2814835264 \beta_{3} - 27008227 \beta_{4} + 239330688 \beta_{5} + 8270480 \beta_{6} + 122878544 \beta_{7} - 11925472 \beta_{8} + 4413008 \beta_{9} - 5608128 \beta_{10} - 1434976 \beta_{11} + 2606272 \beta_{12} - 1168400 \beta_{13} ) q^{86} \) \( + ( 106513435205955 + 363811552604 \beta_{1} - 759726756 \beta_{2} + 2425860366 \beta_{3} + 81804386 \beta_{4} + 120095868 \beta_{5} - 21873484 \beta_{6} - 112509264 \beta_{7} + 649863 \beta_{8} + 18461368 \beta_{9} + 5609586 \beta_{10} - 2569956 \beta_{11} + 2444344 \beta_{12} - 2444344 \beta_{13} ) q^{87} \) \( + ( 139113270720890 + 199935594822 \beta_{1} - 42250506574 \beta_{2} + 484073056 \beta_{3} + 275992682 \beta_{4} + 590483196 \beta_{5} - 300902 \beta_{6} - 110126306 \beta_{7} - 21031144 \beta_{8} + 3758072 \beta_{9} - 6845736 \beta_{10} + 1608420 \beta_{11} + 866776 \beta_{12} + 5156880 \beta_{13} ) q^{88} \) \( + ( 22686463616360 - 569874440284 \beta_{1} - 2261053362 \beta_{2} + 3771882550 \beta_{3} - 118709255 \beta_{4} - 163377024 \beta_{5} - 11190304 \beta_{6} + 150946208 \beta_{7} - 44945328 \beta_{8} - 867977 \beta_{9} + 9092784 \beta_{10} + 1953184 \beta_{11} + 923712 \beta_{12} - 923712 \beta_{13} ) q^{89} \) \( + ( -255349582457101 + 26384971006 \beta_{1} - 58409409763 \beta_{2} + 7368744742 \beta_{3} - 2207102 \beta_{4} - 1043677478 \beta_{5} + 34055201 \beta_{6} + 222738593 \beta_{7} - 38157118 \beta_{8} - 15386561 \beta_{9} - 10086720 \beta_{10} - 3306208 \beta_{11} - 670272 \beta_{12} - 2632447 \beta_{13} ) q^{90} \) \( + ( 349093826218 - 809669070752 \beta_{1} + 4451728072 \beta_{2} - 4446823528 \beta_{3} - 169199904 \beta_{4} - 2075432866 \beta_{5} - 13687272 \beta_{6} - 268484434 \beta_{7} - 3766514 \beta_{8} + 9605864 \beta_{10} - 2242224 \beta_{11} - 1354902 \beta_{12} + 9154986 \beta_{13} ) q^{91} \) \( + ( -217560310752392 + 294851985336 \beta_{1} + 39017399528 \beta_{2} - 940250160 \beta_{3} + 320516808 \beta_{4} - 762792416 \beta_{5} - 9049592 \beta_{6} - 258521440 \beta_{7} + 39137040 \beta_{8} + 3720496 \beta_{9} - 10348080 \beta_{10} + 3456232 \beta_{11} - 992784 \beta_{12} - 1563744 \beta_{13} ) q^{92} \) \( + ( -475674603984 + 1117461221400 \beta_{1} + 49473889496 \beta_{2} - 8133010704 \beta_{3} + 358005480 \beta_{4} - 109582008 \beta_{5} - 16330704 \beta_{6} + 237635640 \beta_{7} - 19489464 \beta_{8} + 12314664 \beta_{10} + 5286480 \beta_{11} - 3398160 \beta_{12} - 5946576 \beta_{13} ) q^{93} \) \( + ( 243775688291688 - 931585189690 \beta_{1} + 26743294236 \beta_{2} + 8019666068 \beta_{3} - 39034296 \beta_{4} + 959799190 \beta_{5} - 33038680 \beta_{6} + 221204022 \beta_{7} + 123560556 \beta_{8} - 9457364 \beta_{9} - 12222324 \beta_{10} - 1550644 \beta_{11} - 8481128 \beta_{12} - 17956 \beta_{13} ) q^{94} \) \( + ( -61476410844611 + 1553749298948 \beta_{1} - 2121005770 \beta_{2} + 7782341750 \beta_{3} + 312150296 \beta_{4} + 483331536 \beta_{5} + 35971440 \beta_{6} - 395882688 \beta_{7} - 15149049 \beta_{8} - 27255648 \beta_{9} + 11283416 \beta_{10} - 2467760 \beta_{11} - 3350368 \beta_{12} + 3350368 \beta_{13} ) q^{95} \) \( + ( 335978562227152 + 363817483736 \beta_{1} - 75995617424 \beta_{2} + 127568712 \beta_{3} - 366744800 \beta_{4} + 1327541904 \beta_{5} - 2002720 \beta_{6} - 416987544 \beta_{7} - 115740816 \beta_{8} + 3832528 \beta_{9} - 9535824 \beta_{10} + 5663928 \beta_{11} - 1857584 \beta_{12} + 4436384 \beta_{13} ) q^{96} \) \( + ( -47275838104924 - 1791770129452 \beta_{1} - 4278805794 \beta_{2} + 5894114706 \beta_{3} - 6029157 \beta_{4} - 370772352 \beta_{5} + 77967320 \beta_{6} + 443835176 \beta_{7} - 108783772 \beta_{8} - 9262415 \beta_{9} + 11205596 \beta_{10} + 4876040 \beta_{11} - 8284336 \beta_{12} + 8284336 \beta_{13} ) q^{97} \) \( + ( -366922861207222 - 562860791993 \beta_{1} - 73284955040 \beta_{2} - 12961556448 \beta_{3} - 93888768 \beta_{4} - 1161577008 \beta_{5} - 68457728 \beta_{6} + 434415696 \beta_{7} - 2772320 \beta_{8} + 29231776 \beta_{9} - 10338400 \beta_{10} - 1996192 \beta_{11} - 901184 \beta_{12} + 6949536 \beta_{13} ) q^{98} \) \( + ( 1266125749320 - 2949915587832 \beta_{1} + 25848648843 \beta_{2} - 3522354480 \beta_{3} - 249945912 \beta_{4} + 5077017360 \beta_{5} - 12497136 \beta_{6} - 606198576 \beta_{7} + 3192144 \beta_{8} + 6337512 \beta_{10} - 3306672 \beta_{11} - 2916312 \beta_{12} + 1222248 \beta_{13} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(14q \) \(\mathstrut -\mathstrut 90q^{2} \) \(\mathstrut +\mathstrut 51444q^{4} \) \(\mathstrut -\mathstrut 189428q^{6} \) \(\mathstrut -\mathstrut 1647088q^{7} \) \(\mathstrut +\mathstrut 1889640q^{8} \) \(\mathstrut -\mathstrut 57395630q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(14q \) \(\mathstrut -\mathstrut 90q^{2} \) \(\mathstrut +\mathstrut 51444q^{4} \) \(\mathstrut -\mathstrut 189428q^{6} \) \(\mathstrut -\mathstrut 1647088q^{7} \) \(\mathstrut +\mathstrut 1889640q^{8} \) \(\mathstrut -\mathstrut 57395630q^{9} \) \(\mathstrut +\mathstrut 58467784q^{10} \) \(\mathstrut +\mathstrut 399357832q^{12} \) \(\mathstrut -\mathstrut 518960496q^{14} \) \(\mathstrut +\mathstrut 712135312q^{15} \) \(\mathstrut -\mathstrut 1435931120q^{16} \) \(\mathstrut +\mathstrut 728554812q^{17} \) \(\mathstrut +\mathstrut 526853306q^{18} \) \(\mathstrut -\mathstrut 3449250768q^{20} \) \(\mathstrut +\mathstrut 28367364252q^{22} \) \(\mathstrut -\mathstrut 35548816080q^{23} \) \(\mathstrut +\mathstrut 40155187088q^{24} \) \(\mathstrut -\mathstrut 75899954794q^{25} \) \(\mathstrut +\mathstrut 17666210712q^{26} \) \(\mathstrut +\mathstrut 79863955680q^{28} \) \(\mathstrut -\mathstrut 124878825712q^{30} \) \(\mathstrut -\mathstrut 105758138816q^{31} \) \(\mathstrut -\mathstrut 37651613280q^{32} \) \(\mathstrut -\mathstrut 150458001384q^{33} \) \(\mathstrut +\mathstrut 537472307308q^{34} \) \(\mathstrut +\mathstrut 338679650892q^{36} \) \(\mathstrut +\mathstrut 1649727781164q^{38} \) \(\mathstrut -\mathstrut 2251546247120q^{39} \) \(\mathstrut +\mathstrut 1251083710304q^{40} \) \(\mathstrut -\mathstrut 53229185940q^{41} \) \(\mathstrut -\mathstrut 2437011096800q^{42} \) \(\mathstrut -\mathstrut 3416842360344q^{44} \) \(\mathstrut -\mathstrut 3303531082064q^{46} \) \(\mathstrut +\mathstrut 12527998446432q^{47} \) \(\mathstrut -\mathstrut 6441543679584q^{48} \) \(\mathstrut +\mathstrut 8427385380990q^{49} \) \(\mathstrut +\mathstrut 1179755527374q^{50} \) \(\mathstrut -\mathstrut 2436018627056q^{52} \) \(\mathstrut +\mathstrut 3357642572216q^{54} \) \(\mathstrut -\mathstrut 30557833792176q^{55} \) \(\mathstrut +\mathstrut 7549064859072q^{56} \) \(\mathstrut +\mathstrut 18277230892472q^{57} \) \(\mathstrut -\mathstrut 8014960165320q^{58} \) \(\mathstrut -\mathstrut 53574657402912q^{60} \) \(\mathstrut +\mathstrut 77882578979904q^{62} \) \(\mathstrut +\mathstrut 36142362113776q^{63} \) \(\mathstrut +\mathstrut 76083381630528q^{64} \) \(\mathstrut +\mathstrut 5437123965600q^{65} \) \(\mathstrut -\mathstrut 134116957601160q^{66} \) \(\mathstrut -\mathstrut 69772560247896q^{68} \) \(\mathstrut +\mathstrut 133952399750848q^{70} \) \(\mathstrut -\mathstrut 173249927708016q^{71} \) \(\mathstrut +\mathstrut 163390222317848q^{72} \) \(\mathstrut -\mathstrut 182057837882196q^{73} \) \(\mathstrut +\mathstrut 2072780135688q^{74} \) \(\mathstrut -\mathstrut 248503439494072q^{76} \) \(\mathstrut +\mathstrut 636498768647600q^{78} \) \(\mathstrut -\mathstrut 294370273271392q^{79} \) \(\mathstrut +\mathstrut 766230078246336q^{80} \) \(\mathstrut +\mathstrut 256903428263798q^{81} \) \(\mathstrut -\mathstrut 1021513680215332q^{82} \) \(\mathstrut -\mathstrut 1701668269684544q^{84} \) \(\mathstrut +\mathstrut 1418597672590812q^{86} \) \(\mathstrut +\mathstrut 1493385390675312q^{87} \) \(\mathstrut +\mathstrut 1948789255860816q^{88} \) \(\mathstrut +\mathstrut 314213951649228q^{89} \) \(\mathstrut -\mathstrut 3574690367103304q^{90} \) \(\mathstrut -\mathstrut 3044080509008736q^{92} \) \(\mathstrut +\mathstrut 3407319354561120q^{94} \) \(\mathstrut -\mathstrut 851301047679984q^{95} \) \(\mathstrut +\mathstrut 4705878559349312q^{96} \) \(\mathstrut -\mathstrut 672574291859236q^{97} \) \(\mathstrut -\mathstrut 5140373067292458q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14}\mathstrut -\mathstrut \) \(3\) \(x^{13}\mathstrut -\mathstrut \) \(6354\) \(x^{12}\mathstrut +\mathstrut \) \(136110\) \(x^{11}\mathstrut +\mathstrut \) \(41390651\) \(x^{10}\mathstrut -\mathstrut \) \(1368564777\) \(x^{9}\mathstrut -\mathstrut \) \(361745089708\) \(x^{8}\mathstrut +\mathstrut \) \(52039630950804\) \(x^{7}\mathstrut -\mathstrut \) \(1350212102075609\) \(x^{6}\mathstrut -\mathstrut \) \(202833046637545317\) \(x^{5}\mathstrut +\mathstrut \) \(15458794901317392766\) \(x^{4}\mathstrut +\mathstrut \) \(1792289418481984019550\) \(x^{3}\mathstrut -\mathstrut \) \(184645940533558790286339\) \(x^{2}\mathstrut -\mathstrut \) \(14754933471339430268597775\) \(x\mathstrut +\mathstrut \) \(2433324183085485312818967200\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(57787459611\) \(\nu^{13}\mathstrut +\mathstrut \) \(1737913630882\) \(\nu^{12}\mathstrut +\mathstrut \) \(1457668688612416\) \(\nu^{11}\mathstrut +\mathstrut \) \(49465007556401126\) \(\nu^{10}\mathstrut -\mathstrut \) \(4840039639061597163\) \(\nu^{9}\mathstrut +\mathstrut \) \(584100205207734152212\) \(\nu^{8}\mathstrut -\mathstrut \) \(26977907232684731704856\) \(\nu^{7}\mathstrut +\mathstrut \) \(1982309909141547091425196\) \(\nu^{6}\mathstrut +\mathstrut \) \(560369599997266260543656927\) \(\nu^{5}\mathstrut +\mathstrut \) \(41331461862700986965580629002\) \(\nu^{4}\mathstrut -\mathstrut \) \(984700974818799065991650348136\) \(\nu^{3}\mathstrut -\mathstrut \) \(315822772842349713922518368913586\) \(\nu^{2}\mathstrut +\mathstrut \) \(35747336042143281846256439940666983\) \(\nu\mathstrut -\mathstrut \) \(832667899000943747969648174128105376\)\()/\)\(14\!\cdots\!24\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(57787459611\) \(\nu^{13}\mathstrut +\mathstrut \) \(1737913630882\) \(\nu^{12}\mathstrut +\mathstrut \) \(1457668688612416\) \(\nu^{11}\mathstrut +\mathstrut \) \(49465007556401126\) \(\nu^{10}\mathstrut -\mathstrut \) \(4840039639061597163\) \(\nu^{9}\mathstrut +\mathstrut \) \(584100205207734152212\) \(\nu^{8}\mathstrut -\mathstrut \) \(26977907232684731704856\) \(\nu^{7}\mathstrut +\mathstrut \) \(1982309909141547091425196\) \(\nu^{6}\mathstrut +\mathstrut \) \(560369599997266260543656927\) \(\nu^{5}\mathstrut +\mathstrut \) \(41331461862700986965580629002\) \(\nu^{4}\mathstrut -\mathstrut \) \(984700974818799065991650348136\) \(\nu^{3}\mathstrut +\mathstrut \) \(2657860129713967767312665562838862\) \(\nu^{2}\mathstrut +\mathstrut \) \(46155226201090393030579583701800551\) \(\nu\mathstrut -\mathstrut \) \(3535745657424636338412430368091080608\)\()/\)\(74\!\cdots\!12\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(3186889963709\) \(\nu^{13}\mathstrut -\mathstrut \) \(2179236426857362\) \(\nu^{12}\mathstrut -\mathstrut \) \(113203248679496256\) \(\nu^{11}\mathstrut +\mathstrut \) \(4945823139808601930\) \(\nu^{10}\mathstrut -\mathstrut \) \(1014868686502680458093\) \(\nu^{9}\mathstrut +\mathstrut \) \(96348574192534441129100\) \(\nu^{8}\mathstrut -\mathstrut \) \(10006073868997642270509736\) \(\nu^{7}\mathstrut -\mathstrut \) \(962706235455435645284432460\) \(\nu^{6}\mathstrut -\mathstrut \) \(58660141144718248945978217703\) \(\nu^{5}\mathstrut +\mathstrut \) \(224084439511075988634784177222\) \(\nu^{4}\mathstrut +\mathstrut \) \(423517139946379971857258968688936\) \(\nu^{3}\mathstrut -\mathstrut \) \(50470055195292640638903730075139294\) \(\nu^{2}\mathstrut +\mathstrut \) \(3406383378120069483222242397030808785\) \(\nu\mathstrut -\mathstrut \) \(261945462122800281578392546533324436832\)\()/\)\(14\!\cdots\!24\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(1396821254639\) \(\nu^{13}\mathstrut +\mathstrut \) \(221715839114138\) \(\nu^{12}\mathstrut -\mathstrut \) \(6204346467628224\) \(\nu^{11}\mathstrut +\mathstrut \) \(161625821725085902\) \(\nu^{10}\mathstrut -\mathstrut \) \(158409430689092235903\) \(\nu^{9}\mathstrut -\mathstrut \) \(5342340200281390501180\) \(\nu^{8}\mathstrut -\mathstrut \) \(924254945977297458679736\) \(\nu^{7}\mathstrut -\mathstrut \) \(71601889971988258348862980\) \(\nu^{6}\mathstrut +\mathstrut \) \(7937750028734907547243968931\) \(\nu^{5}\mathstrut -\mathstrut \) \(215422818181177857191181750430\) \(\nu^{4}\mathstrut +\mathstrut \) \(19610514967222836208816721577784\) \(\nu^{3}\mathstrut -\mathstrut \) \(4630770081594388005573109059055466\) \(\nu^{2}\mathstrut +\mathstrut \) \(527136782729922022717226992073189387\) \(\nu\mathstrut -\mathstrut \) \(30959075543892027641123782211196018976\)\()/\)\(49\!\cdots\!08\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(1002455217403\) \(\nu^{13}\mathstrut +\mathstrut \) \(89631572299234\) \(\nu^{12}\mathstrut +\mathstrut \) \(2965466763883584\) \(\nu^{11}\mathstrut -\mathstrut \) \(627988705015857242\) \(\nu^{10}\mathstrut -\mathstrut \) \(8161278705140497099\) \(\nu^{9}\mathstrut -\mathstrut \) \(5021255893899396151148\) \(\nu^{8}\mathstrut +\mathstrut \) \(851830668333072593988328\) \(\nu^{7}\mathstrut -\mathstrut \) \(56238976113303032527178964\) \(\nu^{6}\mathstrut +\mathstrut \) \(1423357404687413740024597311\) \(\nu^{5}\mathstrut -\mathstrut \) \(6263610553396653477012008374\) \(\nu^{4}\mathstrut +\mathstrut \) \(356688997044746924960925504688792\) \(\nu^{3}\mathstrut +\mathstrut \) \(2250559042434423436009156213088270\) \(\nu^{2}\mathstrut -\mathstrut \) \(417863375083402744168050409462922553\) \(\nu\mathstrut +\mathstrut \) \(35701572321905024816442027798078927968\)\()/\)\(47\!\cdots\!04\)
\(\beta_{7}\)\(=\)\((\)\(35879304688071\) \(\nu^{13}\mathstrut +\mathstrut \) \(430293662123350\) \(\nu^{12}\mathstrut -\mathstrut \) \(268128982936612160\) \(\nu^{11}\mathstrut +\mathstrut \) \(6934135338776093186\) \(\nu^{10}\mathstrut +\mathstrut \) \(1154738599785138761943\) \(\nu^{9}\mathstrut -\mathstrut \) \(73793018184279511783268\) \(\nu^{8}\mathstrut -\mathstrut \) \(17279028573936934310266760\) \(\nu^{7}\mathstrut +\mathstrut \) \(1933563540854412456717243556\) \(\nu^{6}\mathstrut -\mathstrut \) \(34781379027709194785373133723\) \(\nu^{5}\mathstrut -\mathstrut \) \(8472792323690117619039951237554\) \(\nu^{4}\mathstrut +\mathstrut \) \(701871920338724534964241952662536\) \(\nu^{3}\mathstrut +\mathstrut \) \(55134370532886766297539853845159674\) \(\nu^{2}\mathstrut -\mathstrut \) \(4568898098883460559600153110344844867\) \(\nu\mathstrut -\mathstrut \) \(629028365633154357907814856027384627680\)\()/\)\(14\!\cdots\!24\)
\(\beta_{8}\)\(=\)\((\)\(26368160926857\) \(\nu^{13}\mathstrut +\mathstrut \) \(2699441110893770\) \(\nu^{12}\mathstrut +\mathstrut \) \(149482884168560960\) \(\nu^{11}\mathstrut +\mathstrut \) \(28584304439284669534\) \(\nu^{10}\mathstrut +\mathstrut \) \(458989418570724675705\) \(\nu^{9}\mathstrut -\mathstrut \) \(4437043847267081742172\) \(\nu^{8}\mathstrut -\mathstrut \) \(5160162564726114976406008\) \(\nu^{7}\mathstrut -\mathstrut \) \(402177458933808062609971684\) \(\nu^{6}\mathstrut -\mathstrut \) \(53933105212730426191921358453\) \(\nu^{5}\mathstrut -\mathstrut \) \(17595381831248209599335632407982\) \(\nu^{4}\mathstrut +\mathstrut \) \(520648265367461512819597859640696\) \(\nu^{3}\mathstrut +\mathstrut \) \(6347282716246585135237527206499814\) \(\nu^{2}\mathstrut +\mathstrut \) \(1067525175886733496022867466549832883\) \(\nu\mathstrut -\mathstrut \) \(216845590222232551945837178605091403808\)\()/\)\(74\!\cdots\!12\)
\(\beta_{9}\)\(=\)\((\)\(127928962719909\) \(\nu^{13}\mathstrut -\mathstrut \) \(9166442428904414\) \(\nu^{12}\mathstrut -\mathstrut \) \(2592296782770668480\) \(\nu^{11}\mathstrut +\mathstrut \) \(1793898105508952678\) \(\nu^{10}\mathstrut -\mathstrut \) \(37919996680691749023531\) \(\nu^{9}\mathstrut -\mathstrut \) \(502311222603325476802796\) \(\nu^{8}\mathstrut +\mathstrut \) \(219748058325195527146860520\) \(\nu^{7}\mathstrut +\mathstrut \) \(15081773856874497889364640940\) \(\nu^{6}\mathstrut +\mathstrut \) \(1103942642514218866267934622239\) \(\nu^{5}\mathstrut -\mathstrut \) \(152006762628222995925763382393462\) \(\nu^{4}\mathstrut +\mathstrut \) \(11258387394866317144820898216902040\) \(\nu^{3}\mathstrut -\mathstrut \) \(672596291021470568240171320562020402\) \(\nu^{2}\mathstrut +\mathstrut \) \(14195432511010232676933216697632699047\) \(\nu\mathstrut -\mathstrut \) \(390200531306600670128542775739202887584\)\()/\)\(14\!\cdots\!24\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(72522157664001\) \(\nu^{13}\mathstrut -\mathstrut \) \(17695585989469690\) \(\nu^{12}\mathstrut +\mathstrut \) \(726634622654179520\) \(\nu^{11}\mathstrut +\mathstrut \) \(115852470061927641106\) \(\nu^{10}\mathstrut -\mathstrut \) \(5601796753602554298225\) \(\nu^{9}\mathstrut -\mathstrut \) \(856636438342448628805252\) \(\nu^{8}\mathstrut +\mathstrut \) \(80510422374545712263463992\) \(\nu^{7}\mathstrut +\mathstrut \) \(2187112578109345791651763652\) \(\nu^{6}\mathstrut -\mathstrut \) \(1062917020205453602991110782323\) \(\nu^{5}\mathstrut +\mathstrut \) \(38284931683616891926303860765374\) \(\nu^{4}\mathstrut +\mathstrut \) \(2464099958781728632683619648404552\) \(\nu^{3}\mathstrut -\mathstrut \) \(359727997528966357870266899837936438\) \(\nu^{2}\mathstrut -\mathstrut \) \(22963242858368037147541011742498870619\) \(\nu\mathstrut +\mathstrut \) \(5002497498271045345722827203245851741984\)\()/\)\(37\!\cdots\!56\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(338295464497805\) \(\nu^{13}\mathstrut +\mathstrut \) \(111549167043911502\) \(\nu^{12}\mathstrut +\mathstrut \) \(2062148203907543488\) \(\nu^{11}\mathstrut -\mathstrut \) \(1573299921739270285334\) \(\nu^{10}\mathstrut -\mathstrut \) \(61664435424729433593917\) \(\nu^{9}\mathstrut +\mathstrut \) \(6450834079721164421137740\) \(\nu^{8}\mathstrut -\mathstrut \) \(561029742438178540521235752\) \(\nu^{7}\mathstrut +\mathstrut \) \(4709843935751430077666482420\) \(\nu^{6}\mathstrut +\mathstrut \) \(198178514431840651607050785449\) \(\nu^{5}\mathstrut -\mathstrut \) \(590537111979197338202015329614682\) \(\nu^{4}\mathstrut -\mathstrut \) \(39284687376945897965656935017026648\) \(\nu^{3}\mathstrut +\mathstrut \) \(1931577335247117918042811680335298882\) \(\nu^{2}\mathstrut +\mathstrut \) \(290681419651251996652635703293167849441\) \(\nu\mathstrut -\mathstrut \) \(40178435603509287227480828443513012407136\)\()/\)\(14\!\cdots\!24\)
\(\beta_{12}\)\(=\)\((\)\(269874431012467\) \(\nu^{13}\mathstrut +\mathstrut \) \(98653412059961678\) \(\nu^{12}\mathstrut +\mathstrut \) \(274779618236101056\) \(\nu^{11}\mathstrut -\mathstrut \) \(281574718738932742678\) \(\nu^{10}\mathstrut -\mathstrut \) \(27755708375724205500221\) \(\nu^{9}\mathstrut +\mathstrut \) \(7611627045343157771569484\) \(\nu^{8}\mathstrut -\mathstrut \) \(10104829775872883611753768\) \(\nu^{7}\mathstrut -\mathstrut \) \(3481342454604731235965430540\) \(\nu^{6}\mathstrut -\mathstrut \) \(212062494625215119965187160663\) \(\nu^{5}\mathstrut +\mathstrut \) \(23015668941751840137583898101414\) \(\nu^{4}\mathstrut -\mathstrut \) \(1723905676751408076610846915894360\) \(\nu^{3}\mathstrut -\mathstrut \) \(397679725322260965405627352222369982\) \(\nu^{2}\mathstrut +\mathstrut \) \(247577910381109059146421652374916901601\) \(\nu\mathstrut -\mathstrut \) \(21060829563161330284155095251414109383520\)\()/\)\(74\!\cdots\!12\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(652797590290385\) \(\nu^{13}\mathstrut +\mathstrut \) \(32787279165828838\) \(\nu^{12}\mathstrut -\mathstrut \) \(2430818254685092672\) \(\nu^{11}\mathstrut +\mathstrut \) \(260631263864064691890\) \(\nu^{10}\mathstrut +\mathstrut \) \(301295593295319452863\) \(\nu^{9}\mathstrut +\mathstrut \) \(4284612193536687622022716\) \(\nu^{8}\mathstrut +\mathstrut \) \(103882788381347719698624952\) \(\nu^{7}\mathstrut -\mathstrut \) \(10130706132669283945605356284\) \(\nu^{6}\mathstrut +\mathstrut \) \(2194356775854112314576280149277\) \(\nu^{5}\mathstrut -\mathstrut \) \(154646399495612325411439023760098\) \(\nu^{4}\mathstrut +\mathstrut \) \(5174800278739398966512482944186056\) \(\nu^{3}\mathstrut -\mathstrut \) \(1308867415111680118620198361094237974\) \(\nu^{2}\mathstrut +\mathstrut \) \(173493915485816469131217857982977611957\) \(\nu\mathstrut -\mathstrut \) \(6998656279622972366993761669445037827808\)\()/\)\(74\!\cdots\!12\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut -\mathstrut \) \(7\) \(\beta_{1}\mathstrut +\mathstrut \) \(3636\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(8\) \(\beta_{3}\mathstrut +\mathstrut \) \(189\) \(\beta_{2}\mathstrut +\mathstrut \) \(3787\) \(\beta_{1}\mathstrut -\mathstrut \) \(202259\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(4\) \(\beta_{13}\mathstrut +\mathstrut \) \(6\) \(\beta_{12}\mathstrut -\mathstrut \) \(7\) \(\beta_{11}\mathstrut +\mathstrut \) \(2\) \(\beta_{10}\mathstrut -\mathstrut \) \(10\) \(\beta_{9}\mathstrut -\mathstrut \) \(134\) \(\beta_{8}\mathstrut +\mathstrut \) \(49\) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(726\) \(\beta_{5}\mathstrut -\mathstrut \) \(50\) \(\beta_{4}\mathstrut +\mathstrut \) \(2567\) \(\beta_{3}\mathstrut +\mathstrut \) \(9404\) \(\beta_{2}\mathstrut -\mathstrut \) \(97321\) \(\beta_{1}\mathstrut -\mathstrut \) \(49228208\)\()/8\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(20\) \(\beta_{13}\mathstrut -\mathstrut \) \(198\) \(\beta_{12}\mathstrut -\mathstrut \) \(161\) \(\beta_{11}\mathstrut -\mathstrut \) \(878\) \(\beta_{10}\mathstrut +\mathstrut \) \(674\) \(\beta_{9}\mathstrut -\mathstrut \) \(302\) \(\beta_{8}\mathstrut -\mathstrut \) \(4893\) \(\beta_{7}\mathstrut +\mathstrut \) \(72\) \(\beta_{6}\mathstrut -\mathstrut \) \(1008\) \(\beta_{5}\mathstrut -\mathstrut \) \(1236\) \(\beta_{4}\mathstrut +\mathstrut \) \(31731\) \(\beta_{3}\mathstrut +\mathstrut \) \(2170082\) \(\beta_{2}\mathstrut -\mathstrut \) \(11403999\) \(\beta_{1}\mathstrut +\mathstrut \) \(718614158\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(16600\) \(\beta_{13}\mathstrut +\mathstrut \) \(1392\) \(\beta_{12}\mathstrut +\mathstrut \) \(16464\) \(\beta_{11}\mathstrut +\mathstrut \) \(51884\) \(\beta_{10}\mathstrut +\mathstrut \) \(11960\) \(\beta_{9}\mathstrut -\mathstrut \) \(373036\) \(\beta_{8}\mathstrut +\mathstrut \) \(1676079\) \(\beta_{7}\mathstrut -\mathstrut \) \(9443\) \(\beta_{6}\mathstrut -\mathstrut \) \(1817380\) \(\beta_{5}\mathstrut -\mathstrut \) \(1256551\) \(\beta_{4}\mathstrut -\mathstrut \) \(2125319\) \(\beta_{3}\mathstrut +\mathstrut \) \(218231923\) \(\beta_{2}\mathstrut +\mathstrut \) \(417713690\) \(\beta_{1}\mathstrut +\mathstrut \) \(330007825365\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(1129800\) \(\beta_{13}\mathstrut +\mathstrut \) \(2007260\) \(\beta_{12}\mathstrut -\mathstrut \) \(2573638\) \(\beta_{11}\mathstrut +\mathstrut \) \(671308\) \(\beta_{10}\mathstrut +\mathstrut \) \(11379852\) \(\beta_{9}\mathstrut +\mathstrut \) \(5480972\) \(\beta_{8}\mathstrut -\mathstrut \) \(191285825\) \(\beta_{7}\mathstrut -\mathstrut \) \(2574003\) \(\beta_{6}\mathstrut -\mathstrut \) \(528054122\) \(\beta_{5}\mathstrut -\mathstrut \) \(200571387\) \(\beta_{4}\mathstrut +\mathstrut \) \(159305226\) \(\beta_{3}\mathstrut -\mathstrut \) \(33233999467\) \(\beta_{2}\mathstrut +\mathstrut \) \(348019358329\) \(\beta_{1}\mathstrut -\mathstrut \) \(171680455469587\)\()/8\)
\(\nu^{8}\)\(=\)\((\)\(324650588\) \(\beta_{13}\mathstrut +\mathstrut \) \(287521870\) \(\beta_{12}\mathstrut +\mathstrut \) \(36361941\) \(\beta_{11}\mathstrut -\mathstrut \) \(153436126\) \(\beta_{10}\mathstrut +\mathstrut \) \(41973870\) \(\beta_{9}\mathstrut -\mathstrut \) \(404692758\) \(\beta_{8}\mathstrut +\mathstrut \) \(1395872279\) \(\beta_{7}\mathstrut -\mathstrut \) \(62539884\) \(\beta_{6}\mathstrut -\mathstrut \) \(86192968618\) \(\beta_{5}\mathstrut +\mathstrut \) \(16833277164\) \(\beta_{4}\mathstrut +\mathstrut \) \(165779806243\) \(\beta_{3}\mathstrut +\mathstrut \) \(1522357392122\) \(\beta_{2}\mathstrut -\mathstrut \) \(88981624484743\) \(\beta_{1}\mathstrut +\mathstrut \) \(14001792998445782\)\()/8\)
\(\nu^{9}\)\(=\)\((\)\(6962854628\) \(\beta_{13}\mathstrut -\mathstrut \) \(4578219874\) \(\beta_{12}\mathstrut -\mathstrut \) \(1344782523\) \(\beta_{11}\mathstrut -\mathstrut \) \(10220322330\) \(\beta_{10}\mathstrut -\mathstrut \) \(29158122858\) \(\beta_{9}\mathstrut +\mathstrut \) \(4767519910\) \(\beta_{8}\mathstrut +\mathstrut \) \(88912265071\) \(\beta_{7}\mathstrut +\mathstrut \) \(35864638038\) \(\beta_{6}\mathstrut -\mathstrut \) \(1622718116172\) \(\beta_{5}\mathstrut -\mathstrut \) \(265617335198\) \(\beta_{4}\mathstrut -\mathstrut \) \(11925566735999\) \(\beta_{3}\mathstrut -\mathstrut \) \(9006528046212\) \(\beta_{2}\mathstrut +\mathstrut \) \(1839934136290504\) \(\beta_{1}\mathstrut -\mathstrut \) \(168209035884989856\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(1241250818976\) \(\beta_{13}\mathstrut +\mathstrut \) \(187901689480\) \(\beta_{12}\mathstrut -\mathstrut \) \(1587647469956\) \(\beta_{11}\mathstrut +\mathstrut \) \(1670154050368\) \(\beta_{10}\mathstrut -\mathstrut \) \(129478674440\) \(\beta_{9}\mathstrut +\mathstrut \) \(22662117398992\) \(\beta_{8}\mathstrut +\mathstrut \) \(24402738406838\) \(\beta_{7}\mathstrut -\mathstrut \) \(12440385246154\) \(\beta_{6}\mathstrut +\mathstrut \) \(139474786997232\) \(\beta_{5}\mathstrut +\mathstrut \) \(11917058380574\) \(\beta_{4}\mathstrut +\mathstrut \) \(2098002712752701\) \(\beta_{3}\mathstrut +\mathstrut \) \(1890301422697084\) \(\beta_{2}\mathstrut -\mathstrut \) \(177367945118838453\) \(\beta_{1}\mathstrut -\mathstrut \) \(361952283115171750\)\()/4\)
\(\nu^{11}\)\(=\)\((\)\(-\)\(201599671400976\) \(\beta_{13}\mathstrut -\mathstrut \) \(146754756519032\) \(\beta_{12}\mathstrut +\mathstrut \) \(214591658187884\) \(\beta_{11}\mathstrut +\mathstrut \) \(20108268299048\) \(\beta_{10}\mathstrut -\mathstrut \) \(134425550777816\) \(\beta_{9}\mathstrut +\mathstrut \) \(5364014873761960\) \(\beta_{8}\mathstrut -\mathstrut \) \(8173531077998579\) \(\beta_{7}\mathstrut +\mathstrut \) \(1696150213141329\) \(\beta_{6}\mathstrut -\mathstrut \) \(38894821621137986\) \(\beta_{5}\mathstrut -\mathstrut \) \(3728384822665759\) \(\beta_{4}\mathstrut -\mathstrut \) \(82536283768154380\) \(\beta_{3}\mathstrut +\mathstrut \) \(3628604344989811733\) \(\beta_{2}\mathstrut +\mathstrut \) \(3122926213245486695\) \(\beta_{1}\mathstrut -\mathstrut \) \(5165209415755795781771\)\()/8\)
\(\nu^{12}\)\(=\)\((\)\(4884583077582236\) \(\beta_{13}\mathstrut +\mathstrut \) \(28321743975795430\) \(\beta_{12}\mathstrut -\mathstrut \) \(5441763314168087\) \(\beta_{11}\mathstrut +\mathstrut \) \(26572527972474002\) \(\beta_{10}\mathstrut -\mathstrut \) \(47199371670001162\) \(\beta_{9}\mathstrut +\mathstrut \) \(45934236246981674\) \(\beta_{8}\mathstrut +\mathstrut \) \(353172173557645037\) \(\beta_{7}\mathstrut +\mathstrut \) \(7188214586389666\) \(\beta_{6}\mathstrut +\mathstrut \) \(5978946628242347782\) \(\beta_{5}\mathstrut -\mathstrut \) \(1776273527434170654\) \(\beta_{4}\mathstrut -\mathstrut \) \(2677897477058527657\) \(\beta_{3}\mathstrut -\mathstrut \) \(124767040394415481760\) \(\beta_{2}\mathstrut -\mathstrut \) \(2728780741371335341317\) \(\beta_{1}\mathstrut +\mathstrut \) \(466674573231494952386724\)\()/8\)
\(\nu^{13}\)\(=\)\((\)\(-\)\(2251013119321286700\) \(\beta_{13}\mathstrut +\mathstrut \) \(760685633371961510\) \(\beta_{12}\mathstrut -\mathstrut \) \(1224313363811073423\) \(\beta_{11}\mathstrut -\mathstrut \) \(3156315759548824946\) \(\beta_{10}\mathstrut +\mathstrut \) \(3294079883966536510\) \(\beta_{9}\mathstrut +\mathstrut \) \(19973006507600689358\) \(\beta_{8}\mathstrut -\mathstrut \) \(21026270941801720895\) \(\beta_{7}\mathstrut -\mathstrut \) \(3732228024195140948\) \(\beta_{6}\mathstrut -\mathstrut \) \(117231024034999487768\) \(\beta_{5}\mathstrut +\mathstrut \) \(31361915938232456520\) \(\beta_{4}\mathstrut -\mathstrut \) \(731368821007291070019\) \(\beta_{3}\mathstrut +\mathstrut \) \(762030484578352506978\) \(\beta_{2}\mathstrut +\mathstrut \) \(119514734434502765845119\) \(\beta_{1}\mathstrut +\mathstrut \) \(5475845055933646890430710\)\()/4\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
85.4255 + 19.3112i
85.4255 19.3112i
73.6274 + 48.1688i
73.6274 48.1688i
48.9348 + 74.1268i
48.9348 74.1268i
11.1806 + 89.3919i
11.1806 89.3919i
−53.8860 + 74.8506i
−53.8860 74.8506i
−70.4661 + 60.3351i
−70.4661 60.3351i
−93.3164 + 5.91227i
−93.3164 5.91227i
−176.851 38.6225i 4900.64i 29784.6 + 13660.8i 174283.i 189275. 866683.i 3.39868e6 −4.73983e6 3.56629e6i −9.66732e6 6.73124e6 3.08221e7i
5.2 −176.851 + 38.6225i 4900.64i 29784.6 13660.8i 174283.i 189275. + 866683.i 3.39868e6 −4.73983e6 + 3.56629e6i −9.66732e6 6.73124e6 + 3.08221e7i
5.3 −153.255 96.3377i 27.2769i 14206.1 + 29528.4i 212327.i 2627.80 4180.32i −3.74922e6 667546. 5.89396e6i 1.43482e7 −2.04551e7 + 3.25401e7i
5.4 −153.255 + 96.3377i 27.2769i 14206.1 29528.4i 212327.i 2627.80 + 4180.32i −3.74922e6 667546. + 5.89396e6i 1.43482e7 −2.04551e7 3.25401e7i
5.5 −103.870 148.254i 6357.96i −11190.2 + 30798.1i 267219.i −942590. + 660399.i 120583. 5.72825e6 1.53999e6i −2.60748e7 3.96161e7 2.77559e7i
5.6 −103.870 + 148.254i 6357.96i −11190.2 30798.1i 267219.i −942590. 660399.i 120583. 5.72825e6 + 1.53999e6i −2.60748e7 3.96161e7 + 2.77559e7i
5.7 −28.3613 178.784i 2380.16i −31159.3 + 10141.1i 9943.39i 425534. 67504.4i 1.24365e6 2.69677e6 + 5.28316e6i 8.68373e6 −1.77772e6 + 282007.i
5.8 −28.3613 + 178.784i 2380.16i −31159.3 10141.1i 9943.39i 425534. + 67504.4i 1.24365e6 2.69677e6 5.28316e6i 8.68373e6 −1.77772e6 282007.i
5.9 101.772 149.701i 3583.46i −12052.9 30470.8i 82632.2i −536448. 364695.i −153730. −5.78817e6 1.29673e6i 1.50775e6 −1.23701e7 8.40964e6i
5.10 101.772 + 149.701i 3583.46i −12052.9 + 30470.8i 82632.2i −536448. + 364695.i −153730. −5.78817e6 + 1.29673e6i 1.50775e6 −1.23701e7 + 8.40964e6i
5.11 134.932 120.670i 6537.39i 3645.36 32564.6i 116500.i 788869. + 882104.i −2.94378e6 −3.43770e6 4.83390e6i −2.83885e7 1.40581e7 + 1.57196e7i
5.12 134.932 + 120.670i 6537.39i 3645.36 + 32564.6i 116500.i 788869. 882104.i −2.94378e6 −3.43770e6 + 4.83390e6i −2.83885e7 1.40581e7 1.57196e7i
5.13 180.633 11.8245i 1858.96i 32488.4 4271.80i 290191.i −21981.3 335789.i 1.26027e6 5.81795e6 1.15579e6i 1.08932e7 3.43137e6 + 5.24180e7i
5.14 180.633 + 11.8245i 1858.96i 32488.4 + 4271.80i 290191.i −21981.3 + 335789.i 1.26027e6 5.81795e6 + 1.15579e6i 1.08932e7 3.43137e6 5.24180e7i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.14
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
8.b Even 1 yes

Hecke kernels

There are no other newforms in \(S_{16}^{\mathrm{new}}(8, [\chi])\).