Properties

Label 98.13.d.a
Level $98$
Weight $13$
Character orbit 98.d
Analytic conductor $89.571$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 98.d (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(89.5713940931\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 8 x^{15} - 2204556 x^{14} - 87623088 x^{13} + 1948666431190 x^{12} + 195028079162640 x^{11} - 880028445307457796 x^{10} - 151849761581608663584 x^{9} + 211028190361269390536623 x^{8} + 54528516937299786222168960 x^{7} - 23832744423106572444860463196 x^{6} - 9141197551096679767843476458896 x^{5} + 523508549551073444838291611810734 x^{4} + 566732587460008152706924066069468392 x^{3} + 73514373627956541910802883627148139944 x^{2} + 2445664577552004486793879639585121170640 x + 24120197826518417347025654428599335187025\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{48}\cdot 3^{8}\cdot 7^{8} \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + ( -2 \beta_{2} + 5 \beta_{3} - \beta_{5} ) q^{3} + 2048 \beta_{1} q^{4} + ( -756 + 756 \beta_{1} - 93 \beta_{2} + 48 \beta_{3} + 3 \beta_{4} - \beta_{7} ) q^{5} + ( 4928 + 9856 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} - \beta_{11} ) q^{6} + ( 2048 \beta_{2} - 2048 \beta_{3} ) q^{8} + ( 295281 + 295281 \beta_{1} - 70 \beta_{2} + 219 \beta_{3} - 73 \beta_{4} + 148 \beta_{5} + 18 \beta_{6} + 9 \beta_{7} - 2 \beta_{8} - 5 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - 5 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{9} +O(q^{10})\) \( q + \beta_{3} q^{2} + ( -2 \beta_{2} + 5 \beta_{3} - \beta_{5} ) q^{3} + 2048 \beta_{1} q^{4} + ( -756 + 756 \beta_{1} - 93 \beta_{2} + 48 \beta_{3} + 3 \beta_{4} - \beta_{7} ) q^{5} + ( 4928 + 9856 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} - \beta_{11} ) q^{6} + ( 2048 \beta_{2} - 2048 \beta_{3} ) q^{8} + ( 295281 + 295281 \beta_{1} - 70 \beta_{2} + 219 \beta_{3} - 73 \beta_{4} + 148 \beta_{5} + 18 \beta_{6} + 9 \beta_{7} - 2 \beta_{8} - 5 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - 5 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{9} + ( 190848 + 95424 \beta_{1} + 624 \beta_{2} - 1585 \beta_{3} + \beta_{4} + 329 \beta_{5} + 23 \beta_{6} + 23 \beta_{7} + 2 \beta_{9} - 3 \beta_{10} + 6 \beta_{11} + \beta_{12} + 3 \beta_{13} - \beta_{14} - \beta_{15} ) q^{10} + ( -259011 \beta_{1} - 1668 \beta_{2} + 331 \beta_{3} + 662 \beta_{4} - 335 \beta_{5} + 37 \beta_{6} - 37 \beta_{7} + 2 \beta_{8} + 5 \beta_{9} - 3 \beta_{10} - 7 \beta_{11} + 8 \beta_{12} - \beta_{13} + 4 \beta_{14} + 8 \beta_{15} ) q^{11} + ( 10240 \beta_{2} - 4096 \beta_{3} + 2048 \beta_{4} ) q^{12} + ( 168245 + 336490 \beta_{1} - 29816 \beta_{2} - 29781 \beta_{3} - 1111 \beta_{4} + 1101 \beta_{5} - 95 \beta_{6} - 50 \beta_{8} + 40 \beta_{9} + 15 \beta_{10} - 70 \beta_{11} + 10 \beta_{12} + 15 \beta_{13} + 15 \beta_{14} + 20 \beta_{15} ) q^{13} + ( -1697733 + 130908 \beta_{2} - 131016 \beta_{3} + 2727 \beta_{4} + 2703 \beta_{5} + 211 \beta_{6} + 422 \beta_{7} + 45 \beta_{8} - 67 \beta_{9} - 89 \beta_{10} - 45 \beta_{11} + 24 \beta_{12} + 132 \beta_{13} - 132 \beta_{14} ) q^{15} + ( -4194304 - 4194304 \beta_{1} ) q^{16} + ( 8463420 + 4231710 \beta_{1} + 56494 \beta_{2} - 115778 \beta_{3} - 50 \beta_{4} + 2653 \beta_{5} - 269 \beta_{6} - 269 \beta_{7} + 103 \beta_{9} - 53 \beta_{10} + 76 \beta_{11} - 50 \beta_{12} + 32 \beta_{13} + 71 \beta_{14} + 50 \beta_{15} ) q^{17} + ( 574080 \beta_{1} + 300877 \beta_{2} + 6671 \beta_{3} + 14021 \beta_{4} - 6993 \beta_{5} - 45 \beta_{6} + 45 \beta_{7} + 210 \beta_{8} - 252 \beta_{9} - 217 \beta_{10} + 140 \beta_{11} - 35 \beta_{12} + 65 \beta_{13} - 187 \beta_{14} - 35 \beta_{15} ) q^{18} + ( -7288827 + 7288827 \beta_{1} - 1479 \beta_{2} + 13915 \beta_{3} + 25979 \beta_{4} + 112 \beta_{5} + 251 \beta_{7} - 323 \beta_{8} + 154 \beta_{9} + 231 \beta_{10} + 112 \beta_{11} - 224 \beta_{12} - 154 \beta_{13} + 7 \beta_{14} - 112 \beta_{15} ) q^{19} + ( -1548288 - 3096576 \beta_{1} + 92160 \beta_{2} + 92160 \beta_{3} - 6144 \beta_{4} + 6144 \beta_{5} - 2048 \beta_{6} ) q^{20} + ( 3923520 - 262006 \beta_{2} + 262565 \beta_{3} + 16120 \beta_{4} + 15685 \beta_{5} - 214 \beta_{6} - 428 \beta_{7} - 146 \beta_{8} + 49 \beta_{9} - 360 \beta_{10} + 146 \beta_{11} + 435 \beta_{12} - 124 \beta_{13} + 124 \beta_{14} ) q^{22} + ( 3203820 + 3203820 \beta_{1} - 35770 \beta_{2} - 467585 \beta_{3} - 36201 \beta_{4} + 72442 \beta_{5} - 1136 \beta_{6} - 568 \beta_{7} - 68 \beta_{8} - 1055 \beta_{9} + 652 \beta_{10} - 96 \beta_{11} - 1055 \beta_{13} - 652 \beta_{14} - 40 \beta_{15} ) q^{23} + ( -20185088 - 10092544 \beta_{1} - 2048 \beta_{2} + 2048 \beta_{5} - 2048 \beta_{6} - 2048 \beta_{7} + 2048 \beta_{11} ) q^{24} + ( -65150726 \beta_{1} + 3078696 \beta_{2} + 60306 \beta_{3} + 123524 \beta_{4} - 61960 \beta_{5} - 4340 \beta_{6} + 4340 \beta_{7} + 3200 \beta_{8} + 342 \beta_{9} - 54 \beta_{10} + 1204 \beta_{11} + 396 \beta_{12} - 232 \beta_{13} + 110 \beta_{14} + 396 \beta_{15} ) q^{25} + ( 59778432 - 59778432 \beta_{1} + 346950 \beta_{2} - 143421 \beta_{3} + 58274 \beta_{4} - 415 \beta_{5} - 1859 \beta_{7} + 1401 \beta_{8} + 705 \beta_{9} + 1285 \beta_{10} - 415 \beta_{11} + 830 \beta_{12} - 705 \beta_{13} + 2115 \beta_{14} + 415 \beta_{15} ) q^{26} + ( -57968505 - 115937010 \beta_{1} + 2278873 \beta_{2} + 2272678 \beta_{3} + 222601 \beta_{4} - 223001 \beta_{5} - 2379 \beta_{6} - 4299 \beta_{8} - 3097 \beta_{9} - 2298 \beta_{10} - 5099 \beta_{11} + 400 \beta_{12} - 1199 \beta_{13} - 1199 \beta_{14} + 800 \beta_{15} ) q^{27} + ( 33272559 - 1834738 \beta_{2} + 1834983 \beta_{3} + 116725 \beta_{4} + 117243 \beta_{5} + 4419 \beta_{6} + 8838 \beta_{7} - 9058 \beta_{8} + 1058 \beta_{9} + 223 \beta_{10} + 9058 \beta_{11} - 518 \beta_{12} - 763 \beta_{13} + 763 \beta_{14} ) q^{29} + ( -268989696 - 268989696 \beta_{1} + 108878 \beta_{2} - 1603924 \beta_{3} + 106889 \beta_{4} - 215679 \beta_{5} - 33680 \beta_{6} - 16840 \beta_{7} - 220 \beta_{8} + 3765 \beta_{9} - 3633 \beta_{10} - 2341 \beta_{11} + 3765 \beta_{13} + 3633 \beta_{14} + 1901 \beta_{15} ) q^{30} + ( -381984862 - 190992431 \beta_{1} + 6041598 \beta_{2} - 12057228 \beta_{3} + 1776 \beta_{4} - 36543 \beta_{5} - 237 \beta_{6} - 237 \beta_{7} - 746 \beta_{9} - 1030 \beta_{10} + 13743 \beta_{11} + 1776 \beta_{12} - 181 \beta_{13} - 565 \beta_{14} - 1776 \beta_{15} ) q^{31} -4194304 \beta_{2} q^{32} + ( -252289905 + 252289905 \beta_{1} + 10838452 \beta_{2} - 5729660 \beta_{3} - 586988 \beta_{4} - 80 \beta_{5} + 12084 \beta_{7} + 15636 \beta_{8} - 7136 \beta_{9} - 11348 \beta_{10} - 80 \beta_{11} + 160 \beta_{12} + 7136 \beta_{13} - 11188 \beta_{14} + 80 \beta_{15} ) q^{33} + ( -118164480 - 236328960 \beta_{1} + 4063266 \beta_{2} + 4071995 \beta_{3} - 265847 \beta_{4} + 265430 \beta_{5} + 9481 \beta_{6} + 5755 \beta_{8} + 3901 \beta_{9} + 5662 \beta_{10} + 4921 \beta_{11} + 417 \beta_{12} - 2178 \beta_{13} - 2178 \beta_{14} + 834 \beta_{15} ) q^{34} + ( -604735488 + 589824 \beta_{2} - 600064 \beta_{3} - 153600 \beta_{4} - 149504 \beta_{5} - 18432 \beta_{6} - 36864 \beta_{7} - 4096 \beta_{8} + 4096 \beta_{9} - 6144 \beta_{10} + 4096 \beta_{11} - 4096 \beta_{12} + 6144 \beta_{13} - 6144 \beta_{14} ) q^{36} + ( 720565505 + 720565505 \beta_{1} + 459305 \beta_{2} + 46709 \beta_{3} + 490226 \beta_{4} - 980122 \beta_{5} + 52972 \beta_{6} + 26486 \beta_{7} + 28286 \beta_{8} - 3721 \beta_{9} + 6026 \beta_{10} + 56902 \beta_{11} - 3721 \beta_{13} - 6026 \beta_{14} - 330 \beta_{15} ) q^{37} + ( 12458880 + 6229440 \beta_{1} + 7134904 \beta_{2} - 14646659 \beta_{3} - 2135 \beta_{4} + 326590 \beta_{5} - 4670 \beta_{6} - 4670 \beta_{7} - 6958 \beta_{9} + 9093 \beta_{10} + 33475 \beta_{11} - 2135 \beta_{12} + 2779 \beta_{13} - 9737 \beta_{14} + 2135 \beta_{15} ) q^{38} + ( -1315760409 \beta_{1} + 22910404 \beta_{2} - 1396972 \beta_{3} - 2695328 \beta_{4} + 1350008 \beta_{5} - 80537 \beta_{6} + 80537 \beta_{7} + 128510 \beta_{8} + 12603 \beta_{9} + 17291 \beta_{10} + 68943 \beta_{11} - 4688 \beta_{12} - 4863 \beta_{13} + 7740 \beta_{14} - 4688 \beta_{15} ) q^{39} + ( -195428352 + 195428352 \beta_{1} - 3252224 \beta_{2} + 1286144 \beta_{3} - 675840 \beta_{4} + 2048 \beta_{5} - 47104 \beta_{7} + 10240 \beta_{8} + 2048 \beta_{9} + 10240 \beta_{10} + 2048 \beta_{11} - 4096 \beta_{12} - 2048 \beta_{13} + 6144 \beta_{14} - 2048 \beta_{15} ) q^{40} + ( -354831435 - 709662870 \beta_{1} - 6080474 \beta_{2} - 6055841 \beta_{3} + 153805 \beta_{4} - 147947 \beta_{5} + 80257 \beta_{6} - 33618 \beta_{8} + 11988 \beta_{9} + 929 \beta_{10} - 21902 \beta_{11} - 5858 \beta_{12} + 16917 \beta_{13} + 16917 \beta_{14} - 11716 \beta_{15} ) q^{41} + ( -4183089500 + 2410812 \beta_{2} - 2389134 \beta_{3} - 735050 \beta_{4} - 741890 \beta_{5} - 132148 \beta_{6} - 264296 \beta_{7} - 57392 \beta_{8} - 22112 \beta_{9} + 30110 \beta_{10} + 57392 \beta_{11} + 6840 \beta_{12} - 14838 \beta_{13} + 14838 \beta_{14} ) q^{43} + ( 530454528 + 530454528 \beta_{1} - 673792 \beta_{2} + 3438592 \beta_{3} - 669696 \beta_{4} + 1355776 \beta_{5} - 151552 \beta_{6} - 75776 \beta_{7} - 2048 \beta_{8} - 8192 \beta_{9} - 2048 \beta_{10} + 12288 \beta_{11} - 8192 \beta_{13} + 2048 \beta_{14} - 16384 \beta_{15} ) q^{44} + ( -4771112682 - 2385556341 \beta_{1} - 13083950 \beta_{2} + 25771366 \beta_{3} - 13046 \beta_{4} + 187747 \beta_{5} - 550935 \beta_{6} - 550935 \beta_{7} + 35753 \beta_{9} - 22707 \beta_{10} + 212584 \beta_{11} - 13046 \beta_{12} + 2932 \beta_{13} + 32821 \beta_{14} + 13046 \beta_{15} ) q^{45} + ( -900401664 \beta_{1} + 3464877 \beta_{2} + 294402 \beta_{3} + 802066 \beta_{4} - 403349 \beta_{5} + 88037 \beta_{6} - 88037 \beta_{7} + 128950 \beta_{8} - 39840 \beta_{9} - 44472 \beta_{10} + 59843 \beta_{11} + 4632 \beta_{12} + 21976 \beta_{13} - 17864 \beta_{14} + 4632 \beta_{15} ) q^{46} + ( -754840485 + 754840485 \beta_{1} - 81811608 \beta_{2} + 41664564 \beta_{3} + 1418061 \beta_{4} - 3960 \beta_{5} + 516281 \beta_{7} + 15039 \beta_{8} + 49779 \beta_{9} + 52839 \beta_{10} - 3960 \beta_{11} + 7920 \beta_{12} - 49779 \beta_{13} + 60759 \beta_{14} + 3960 \beta_{15} ) q^{47} + ( -12582912 \beta_{2} - 12582912 \beta_{3} - 4194304 \beta_{4} + 4194304 \beta_{5} ) q^{48} + ( -6203005056 - 64420834 \beta_{2} + 64451978 \beta_{3} - 3778396 \beta_{4} - 3789924 \beta_{5} - 188468 \beta_{6} - 376936 \beta_{7} - 42260 \beta_{8} - 9704 \beta_{9} + 17792 \beta_{10} + 42260 \beta_{11} + 11528 \beta_{12} - 19616 \beta_{13} + 19616 \beta_{14} ) q^{50} + ( -2912407047 - 2912407047 \beta_{1} - 1458597 \beta_{2} - 72596049 \beta_{3} - 1308477 \beta_{4} + 2623890 \beta_{5} + 590314 \beta_{6} + 295157 \beta_{7} + 85905 \beta_{8} + 59854 \beta_{9} - 2575 \beta_{10} + 178746 \beta_{11} + 59854 \beta_{13} + 2575 \beta_{14} - 6936 \beta_{15} ) q^{51} + ( -689131520 - 344565760 \beta_{1} - 58798080 \beta_{2} + 119922688 \beta_{3} + 20480 \beta_{4} - 2275328 \beta_{5} + 194560 \beta_{6} + 194560 \beta_{7} - 30720 \beta_{9} + 10240 \beta_{10} + 122880 \beta_{11} + 20480 \beta_{12} - 81920 \beta_{13} + 51200 \beta_{14} - 20480 \beta_{15} ) q^{52} + ( -4891770225 \beta_{1} - 80737560 \beta_{2} + 6275985 \beta_{3} + 12914682 \beta_{4} - 6466956 \beta_{5} - 299782 \beta_{6} + 299782 \beta_{7} + 266436 \beta_{8} - 38523 \beta_{9} - 57753 \beta_{10} + 113988 \beta_{11} + 19230 \beta_{12} + 10722 \beta_{13} - 27801 \beta_{14} + 19230 \beta_{15} ) q^{53} + ( -4381341888 + 4381341888 \beta_{1} - 114541802 \beta_{2} + 61990150 \beta_{3} + 9487381 \beta_{4} + 10759 \beta_{5} - 1013532 \beta_{7} - 40248 \beta_{8} - 2009 \beta_{9} - 54845 \beta_{10} + 10759 \beta_{11} - 21518 \beta_{12} + 2009 \beta_{13} - 76363 \beta_{14} - 10759 \beta_{15} ) q^{54} + ( -12093817386 - 24187634772 \beta_{1} - 68937386 \beta_{2} - 68953283 \beta_{3} + 8369233 \beta_{4} - 8388809 \beta_{5} + 263190 \beta_{6} + 74370 \beta_{8} + 16452 \beta_{9} + 6803 \beta_{10} + 35218 \beta_{11} + 19576 \beta_{12} - 9927 \beta_{13} - 9927 \beta_{14} + 39152 \beta_{15} ) q^{55} + ( -20515247505 + 172802852 \beta_{2} - 172916236 \beta_{3} + 12628268 \beta_{4} + 12681148 \beta_{5} + 557684 \beta_{6} + 1115368 \beta_{7} - 30704 \beta_{8} + 75936 \beta_{9} - 83560 \beta_{10} + 30704 \beta_{11} - 52880 \beta_{12} + 60504 \beta_{13} - 60504 \beta_{14} ) q^{57} + ( 3704674560 + 3704674560 \beta_{1} - 14175980 \beta_{2} + 22400053 \beta_{3} - 14136072 \beta_{4} + 28238709 \beta_{5} - 833790 \beta_{6} - 416895 \beta_{7} + 94755 \beta_{8} - 85837 \beta_{9} + 64425 \beta_{10} + 156075 \beta_{11} - 85837 \beta_{13} - 64425 \beta_{14} + 33435 \beta_{15} ) q^{58} + ( -16820454924 - 8410227462 \beta_{1} - 2123946 \beta_{2} - 963609 \beta_{3} + 20136 \beta_{4} + 5022975 \beta_{5} + 1179734 \beta_{6} + 1179734 \beta_{7} - 115016 \beta_{9} + 94880 \beta_{10} - 114750 \beta_{11} + 20136 \beta_{12} + 114266 \beta_{13} - 229282 \beta_{14} - 20136 \beta_{15} ) q^{59} + ( -3476957184 \beta_{1} - 273862656 \beta_{2} - 5173248 \beta_{3} - 11120640 \beta_{4} + 5584896 \beta_{5} + 432128 \beta_{6} - 432128 \beta_{7} - 184320 \beta_{8} + 270336 \beta_{9} + 319488 \beta_{10} - 43008 \beta_{11} - 49152 \beta_{12} - 137216 \beta_{13} + 133120 \beta_{14} - 49152 \beta_{15} ) q^{60} + ( -14032510597 + 14032510597 \beta_{1} + 176273281 \beta_{2} - 89408569 \beta_{3} - 1860330 \beta_{4} - 4858 \beta_{5} + 741230 \beta_{7} + 302986 \beta_{8} - 243691 \beta_{9} - 151424 \beta_{10} - 4858 \beta_{11} + 9716 \beta_{12} + 243691 \beta_{13} - 141708 \beta_{14} + 4858 \beta_{15} ) q^{61} + ( -12350378880 - 24700757760 \beta_{1} - 205140870 \beta_{2} - 205224471 \beta_{3} - 24471526 \beta_{4} + 24509951 \beta_{5} - 994182 \beta_{6} - 286782 \beta_{8} - 99765 \beta_{9} - 60686 \beta_{10} - 209932 \beta_{11} - 38425 \beta_{12} - 654 \beta_{13} - 654 \beta_{14} - 76850 \beta_{15} ) q^{62} + 8589934592 q^{64} + ( 19790337987 + 19790337987 \beta_{1} + 38348114 \beta_{2} - 284346377 \beta_{3} + 37240407 \beta_{4} - 74614412 \beta_{5} - 1586670 \beta_{6} - 793335 \beta_{7} - 610670 \beta_{8} - 87605 \beta_{9} - 275834 \beta_{10} - 1354938 \beta_{11} - 87605 \beta_{13} + 275834 \beta_{14} + 133598 \beta_{15} ) q^{65} + ( -22407006720 - 11203503360 \beta_{1} + 262653269 \beta_{2} - 499402174 \beta_{3} - 60148 \beta_{4} - 25459312 \beta_{5} - 897936 \beta_{6} - 897936 \beta_{7} + 331288 \beta_{9} - 271140 \beta_{10} - 1025428 \beta_{11} - 60148 \beta_{12} + 395684 \beta_{13} - 64396 \beta_{14} + 60148 \beta_{15} ) q^{66} + ( -13470889990 \beta_{1} - 25251944 \beta_{2} + 9412723 \beta_{3} + 16695038 \beta_{4} - 8326039 \beta_{5} - 487490 \beta_{6} + 487490 \beta_{7} - 2298340 \beta_{8} - 105446 \beta_{9} - 62486 \beta_{10} - 1106210 \beta_{11} - 42960 \beta_{12} + 90290 \beta_{13} - 15156 \beta_{14} - 42960 \beta_{15} ) q^{67} + ( -8666542080 + 8666542080 \beta_{1} - 236806144 \beta_{2} + 115619840 \beta_{3} - 5330944 \beta_{4} - 102400 \beta_{5} + 550912 \beta_{7} + 258048 \beta_{8} - 145408 \beta_{9} - 139264 \beta_{10} - 102400 \beta_{11} + 204800 \beta_{12} + 145408 \beta_{13} + 65536 \beta_{14} + 102400 \beta_{15} ) q^{68} + ( -31016712258 - 62033424516 \beta_{1} + 214531223 \beta_{2} + 214338023 \beta_{3} + 77238263 \beta_{4} - 77293471 \beta_{5} - 265431 \beta_{6} + 162168 \beta_{8} - 204264 \beta_{9} + 121480 \beta_{10} + 51752 \beta_{11} + 55208 \beta_{12} - 380952 \beta_{13} - 380952 \beta_{14} + 110416 \beta_{15} ) q^{69} + ( -71891189334 - 593198540 \beta_{2} + 593256552 \beta_{3} + 29054798 \beta_{4} + 28868790 \beta_{5} + 830326 \beta_{6} + 1660652 \beta_{7} + 806362 \beta_{8} + 158698 \beta_{9} - 472702 \beta_{10} - 806362 \beta_{11} + 186008 \beta_{12} + 127996 \beta_{13} - 127996 \beta_{14} ) q^{71} + ( -1175715840 - 1175715840 \beta_{1} - 14622720 \beta_{2} - 615823360 \beta_{3} - 14393344 \beta_{4} + 28715008 \beta_{5} + 184320 \beta_{6} + 92160 \beta_{7} - 215040 \beta_{8} + 382976 \beta_{9} + 133120 \beta_{10} - 501760 \beta_{11} + 382976 \beta_{13} - 133120 \beta_{14} + 71680 \beta_{15} ) q^{72} + ( 61534523610 + 30767261805 \beta_{1} + 513854384 \beta_{2} - 1087573110 \beta_{3} + 152440 \beta_{4} + 63970818 \beta_{5} + 2251438 \beta_{6} + 2251438 \beta_{7} - 464980 \beta_{9} + 312540 \beta_{10} - 2229232 \beta_{11} + 152440 \beta_{12} - 1018672 \beta_{13} + 553692 \beta_{14} - 152440 \beta_{15} ) q^{73} + ( -612847296 \beta_{1} + 693078653 \beta_{2} - 37313460 \beta_{3} - 74586684 \beta_{4} + 37156188 \beta_{5} - 518228 \beta_{6} + 518228 \beta_{7} - 1068216 \beta_{8} - 417072 \beta_{9} - 691380 \beta_{10} - 808416 \beta_{11} + 274308 \beta_{12} + 190260 \beta_{13} - 226812 \beta_{14} + 274308 \beta_{15} ) q^{74} + ( -64042860000 + 64042860000 \beta_{1} - 3267134968 \beta_{2} + 1574360518 \beta_{3} - 122375666 \beta_{4} + 170928 \beta_{5} - 5472612 \beta_{7} - 2638140 \beta_{8} + 1016458 \beta_{9} + 819920 \beta_{10} + 170928 \beta_{11} - 341856 \beta_{12} - 1016458 \beta_{13} + 478064 \beta_{14} - 170928 \beta_{15} ) q^{75} + ( -14927517696 - 29855035392 \beta_{1} - 25139200 \beta_{2} - 26157056 \beta_{3} - 53434368 \beta_{4} + 53204992 \beta_{5} + 514048 \beta_{6} + 661504 \beta_{8} - 14336 \beta_{9} - 544768 \beta_{10} + 202752 \beta_{11} + 229376 \beta_{12} + 301056 \beta_{13} + 301056 \beta_{14} + 458752 \beta_{15} ) q^{76} + ( -49130555520 - 1282659281 \beta_{2} + 1282799918 \beta_{3} - 144501217 \beta_{4} - 144481162 \beta_{5} + 2343975 \beta_{6} + 4687950 \beta_{7} + 911155 \beta_{8} - 348605 \beta_{9} + 529352 \beta_{10} - 911155 \beta_{11} - 20055 \beta_{12} - 160692 \beta_{13} + 160692 \beta_{14} ) q^{78} + ( 28454008096 + 28454008096 \beta_{1} + 74551876 \beta_{2} - 1586781803 \beta_{3} + 74677363 \beta_{4} - 148683854 \beta_{5} + 8655416 \beta_{6} + 4327708 \beta_{7} - 770720 \beta_{8} - 651641 \beta_{9} + 876976 \beta_{10} - 870568 \beta_{11} - 651641 \beta_{13} - 876976 \beta_{14} - 670872 \beta_{15} ) q^{79} + ( 6341787648 + 3170893824 \beta_{1} + 201326592 \beta_{2} - 390070272 \beta_{3} - 12582912 \beta_{5} + 4194304 \beta_{6} + 4194304 \beta_{7} ) q^{80} + ( 48998022300 \beta_{1} + 3108176106 \beta_{2} + 26221458 \beta_{3} + 50263956 \beta_{4} - 25192305 \beta_{5} + 9192201 \beta_{6} - 9192201 \beta_{7} - 2738004 \beta_{8} - 219195 \beta_{9} - 339849 \beta_{10} - 1489656 \beta_{11} + 120654 \beta_{12} - 208938 \beta_{13} - 428133 \beta_{14} + 120654 \beta_{15} ) q^{81} + ( 12607418880 - 12607418880 \beta_{1} - 697601546 \beta_{2} + 376879175 \beta_{3} + 53953998 \beta_{4} + 251645 \beta_{5} + 7047605 \beta_{7} - 1805375 \beta_{8} + 418077 \beta_{9} + 734289 \beta_{10} + 251645 \beta_{11} - 503290 \beta_{12} - 418077 \beta_{13} + 230999 \beta_{14} - 251645 \beta_{15} ) q^{82} + ( -22318410870 - 44636821740 \beta_{1} + 116024442 \beta_{2} + 117732246 \beta_{3} + 126254676 \beta_{4} - 125769756 \beta_{5} - 6006806 \beta_{6} + 1414854 \beta_{8} + 652518 \beta_{9} + 85446 \beta_{10} + 2384694 \beta_{11} - 484920 \beta_{12} + 1051992 \beta_{13} + 1051992 \beta_{14} - 969840 \beta_{15} ) q^{83} + ( 44388106005 + 498558039 \beta_{2} - 497919042 \beta_{3} + 107653394 \beta_{4} + 107584040 \beta_{5} - 2567708 \beta_{6} - 5135416 \beta_{7} + 3778010 \beta_{8} - 995122 \beta_{9} + 1495411 \beta_{10} - 3778010 \beta_{11} + 69354 \beta_{12} - 569643 \beta_{13} + 569643 \beta_{14} ) q^{85} + ( -4763149056 - 4763149056 \beta_{1} - 126962444 \beta_{2} - 4286414586 \beta_{3} - 128007822 \beta_{4} + 256208432 \beta_{5} + 11290796 \beta_{6} + 5645398 \beta_{7} + 47354 \beta_{8} - 436084 \beta_{9} - 849436 \beta_{10} + 287496 \beta_{11} - 436084 \beta_{13} + 849436 \beta_{14} - 192788 \beta_{15} ) q^{86} + ( -163084806990 - 81542403495 \beta_{1} + 5394982804 \beta_{2} - 10744471334 \beta_{3} - 881720 \beta_{4} - 44127530 \beta_{5} + 579351 \beta_{6} + 579351 \beta_{7} + 2354681 \beta_{9} - 1472961 \beta_{10} - 5920769 \beta_{11} - 881720 \beta_{12} + 2572633 \beta_{13} - 217952 \beta_{14} + 881720 \beta_{15} ) q^{87} + ( 8035368960 \beta_{1} + 503529472 \beta_{2} - 32675840 \beta_{3} - 65136640 \beta_{4} + 33013760 \beta_{5} - 438272 \beta_{6} + 438272 \beta_{7} + 598016 \beta_{8} - 253952 \beta_{9} + 636928 \beta_{10} + 1189888 \beta_{11} - 890880 \beta_{12} + 100352 \beta_{13} - 153600 \beta_{14} - 890880 \beta_{15} ) q^{88} + ( 103541976783 - 103541976783 \beta_{1} - 5915212426 \beta_{2} + 3132711458 \beta_{3} + 353820716 \beta_{4} - 603628 \beta_{5} - 4728444 \beta_{7} + 1488108 \beta_{8} - 1988522 \beta_{9} - 1944480 \beta_{10} - 603628 \beta_{11} + 1207256 \beta_{12} + 1988522 \beta_{13} - 737224 \beta_{14} + 603628 \beta_{15} ) q^{89} + ( 26241444096 + 52482888192 \beta_{1} - 2725844993 \beta_{2} - 2720610974 \beta_{3} - 562429709 \beta_{4} + 562792786 \beta_{5} + 10074099 \beta_{6} + 1244025 \beta_{8} + 1802671 \beta_{9} + 2705194 \beta_{10} + 1970179 \beta_{11} - 363077 \beta_{12} - 539446 \beta_{13} - 539446 \beta_{14} - 726154 \beta_{15} ) q^{90} + ( -6561423360 - 883695616 \beta_{2} + 882788352 \beta_{3} - 74221568 \beta_{4} - 74139648 \beta_{5} + 1163264 \beta_{6} + 2326528 \beta_{7} - 139264 \beta_{8} + 1335296 \beta_{9} - 2078720 \beta_{10} + 139264 \beta_{11} - 81920 \beta_{12} + 825344 \beta_{13} - 825344 \beta_{14} ) q^{92} + ( -57333596085 - 57333596085 \beta_{1} - 117408725 \beta_{2} - 10957647577 \beta_{3} - 113562770 \beta_{4} + 226089250 \beta_{5} - 29993484 \beta_{6} - 14996742 \beta_{7} + 2741658 \beta_{8} + 2726349 \beta_{9} - 585762 \beta_{10} + 4447026 \beta_{11} + 2726349 \beta_{13} + 585762 \beta_{14} + 1036290 \beta_{15} ) q^{93} + ( 168441557760 + 84220778880 \beta_{1} + 837684606 \beta_{2} - 1466594257 \beta_{3} + 144067 \beta_{4} - 206233972 \beta_{5} - 9417004 \beta_{6} - 9417004 \beta_{7} - 3151162 \beta_{9} + 3007095 \beta_{10} + 1819779 \beta_{11} + 144067 \beta_{12} - 3611895 \beta_{13} + 460733 \beta_{14} - 144067 \beta_{15} ) q^{94} + ( 23312982870 \beta_{1} + 1055759790 \beta_{2} + 164970010 \beta_{3} + 328408070 \beta_{4} - 164021675 \beta_{5} - 8164090 \beta_{6} + 8164090 \beta_{7} + 5758580 \beta_{8} + 3462905 \beta_{9} + 3827625 \beta_{10} + 3244010 \beta_{11} - 364720 \beta_{12} + 286310 \beta_{13} + 3749215 \beta_{14} - 364720 \beta_{15} ) q^{95} + ( 20669530112 - 20669530112 \beta_{1} - 4194304 \beta_{3} - 4194304 \beta_{4} + 4194304 \beta_{7} + 4194304 \beta_{8} ) q^{96} + ( 179404856785 + 358809713570 \beta_{1} - 3448988094 \beta_{2} - 3452573557 \beta_{3} + 346537813 \beta_{4} - 347290123 \beta_{5} + 17885905 \beta_{6} - 3795230 \beta_{8} - 471832 \beta_{9} - 1609011 \beta_{10} - 5299850 \beta_{11} + 752310 \beta_{12} + 384869 \beta_{13} + 384869 \beta_{14} + 1504620 \beta_{15} ) q^{97} + ( 245073772161 - 7031364696 \beta_{2} + 7028665251 \beta_{3} + 372485022 \beta_{4} + 375245766 \beta_{5} - 11782515 \beta_{6} - 23565030 \beta_{7} - 12748281 \beta_{8} + 535719 \beta_{9} + 2286324 \beta_{10} + 12748281 \beta_{11} - 2760744 \beta_{12} - 61299 \beta_{13} + 61299 \beta_{14} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 16384q^{4} - 18144q^{5} + 2362248q^{9} + O(q^{10}) \) \( 16q - 16384q^{4} - 18144q^{5} + 2362248q^{9} + 2290176q^{10} + 2072088q^{11} - 27163728q^{15} - 33554432q^{16} + 101561040q^{17} - 4592640q^{18} - 174931848q^{19} + 62776320q^{22} + 25630560q^{23} - 242221056q^{24} + 521205808q^{25} + 1434682368q^{26} + 532360944q^{29} - 2151917568q^{30} - 4583818344q^{31} - 6054957720q^{33} - 9675767808q^{36} + 5764524040q^{37} + 149506560q^{38} + 10526083272q^{39} - 4690280448q^{40} - 66929432000q^{43} + 4243636224q^{44} - 57253352184q^{45} + 7203213312q^{46} - 18116171640q^{47} - 99248080896q^{50} - 23299256376q^{51} - 8269578240q^{52} + 39134161800q^{53} - 105152205312q^{54} - 328243960080q^{57} + 29637396480q^{58} - 201845459088q^{59} + 27815657472q^{60} - 336780254328q^{61} + 137438953472q^{64} + 158322703896q^{65} - 268884080640q^{66} + 107767119920q^{67} - 207997009920q^{68} - 1150259029344q^{71} - 9405726720q^{72} + 738414283320q^{73} + 4902778368q^{74} - 1537028640000q^{75} - 786088888320q^{78} + 227632064768q^{79} + 76101451776q^{80} - 391984178400q^{81} + 302578053120q^{82} + 710209696080q^{85} - 38105192448q^{86} - 1957017683880q^{87} - 64282951680q^{88} + 2485007442792q^{89} - 104982773760q^{92} - 458668768680q^{93} + 2021298693120q^{94} - 186503862960q^{95} + 496068722688q^{96} + 3921180354576q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 8 x^{15} - 2204556 x^{14} - 87623088 x^{13} + 1948666431190 x^{12} + 195028079162640 x^{11} - 880028445307457796 x^{10} - 151849761581608663584 x^{9} + 211028190361269390536623 x^{8} + 54528516937299786222168960 x^{7} - 23832744423106572444860463196 x^{6} - 9141197551096679767843476458896 x^{5} + 523508549551073444838291611810734 x^{4} + 566732587460008152706924066069468392 x^{3} + 73514373627956541910802883627148139944 x^{2} + 2445664577552004486793879639585121170640 x + 24120197826518417347025654428599335187025\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-\)\(33\!\cdots\!20\)\( \nu^{15} + \)\(96\!\cdots\!52\)\( \nu^{14} + \)\(70\!\cdots\!60\)\( \nu^{13} - \)\(16\!\cdots\!70\)\( \nu^{12} - \)\(59\!\cdots\!08\)\( \nu^{11} + \)\(10\!\cdots\!52\)\( \nu^{10} + \)\(26\!\cdots\!16\)\( \nu^{9} - \)\(23\!\cdots\!25\)\( \nu^{8} - \)\(62\!\cdots\!16\)\( \nu^{7} - \)\(21\!\cdots\!16\)\( \nu^{6} + \)\(79\!\cdots\!00\)\( \nu^{5} + \)\(77\!\cdots\!90\)\( \nu^{4} - \)\(39\!\cdots\!28\)\( \nu^{3} - \)\(76\!\cdots\!36\)\( \nu^{2} - \)\(27\!\cdots\!16\)\( \nu - \)\(28\!\cdots\!00\)\(\)\()/ \)\(33\!\cdots\!65\)\( \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(60\!\cdots\!71\)\( \nu^{15} + \)\(36\!\cdots\!98\)\( \nu^{14} + \)\(11\!\cdots\!77\)\( \nu^{13} - \)\(67\!\cdots\!43\)\( \nu^{12} - \)\(87\!\cdots\!04\)\( \nu^{11} + \)\(46\!\cdots\!21\)\( \nu^{10} + \)\(33\!\cdots\!63\)\( \nu^{9} - \)\(15\!\cdots\!40\)\( \nu^{8} - \)\(68\!\cdots\!24\)\( \nu^{7} + \)\(22\!\cdots\!25\)\( \nu^{6} + \)\(76\!\cdots\!55\)\( \nu^{5} - \)\(11\!\cdots\!92\)\( \nu^{4} - \)\(36\!\cdots\!81\)\( \nu^{3} - \)\(11\!\cdots\!33\)\( \nu^{2} - \)\(18\!\cdots\!66\)\( \nu - \)\(39\!\cdots\!45\)\(\)\()/ \)\(40\!\cdots\!20\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(91\!\cdots\!71\)\( \nu^{15} - \)\(83\!\cdots\!93\)\( \nu^{14} - \)\(20\!\cdots\!62\)\( \nu^{13} + \)\(11\!\cdots\!63\)\( \nu^{12} + \)\(18\!\cdots\!89\)\( \nu^{11} - \)\(15\!\cdots\!96\)\( \nu^{10} - \)\(86\!\cdots\!93\)\( \nu^{9} - \)\(39\!\cdots\!15\)\( \nu^{8} + \)\(22\!\cdots\!89\)\( \nu^{7} + \)\(22\!\cdots\!60\)\( \nu^{6} - \)\(28\!\cdots\!25\)\( \nu^{5} - \)\(46\!\cdots\!63\)\( \nu^{4} + \)\(14\!\cdots\!86\)\( \nu^{3} + \)\(33\!\cdots\!13\)\( \nu^{2} + \)\(10\!\cdots\!41\)\( \nu + \)\(91\!\cdots\!90\)\(\)\()/ \)\(40\!\cdots\!20\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(10\!\cdots\!39\)\( \nu^{15} + \)\(30\!\cdots\!96\)\( \nu^{14} + \)\(22\!\cdots\!91\)\( \nu^{13} - \)\(54\!\cdots\!95\)\( \nu^{12} - \)\(19\!\cdots\!66\)\( \nu^{11} + \)\(33\!\cdots\!87\)\( \nu^{10} + \)\(84\!\cdots\!75\)\( \nu^{9} - \)\(75\!\cdots\!42\)\( \nu^{8} - \)\(20\!\cdots\!70\)\( \nu^{7} - \)\(99\!\cdots\!81\)\( \nu^{6} + \)\(25\!\cdots\!15\)\( \nu^{5} + \)\(25\!\cdots\!78\)\( \nu^{4} - \)\(12\!\cdots\!03\)\( \nu^{3} - \)\(24\!\cdots\!37\)\( \nu^{2} - \)\(90\!\cdots\!04\)\( \nu - \)\(91\!\cdots\!75\)\(\)\()/ \)\(40\!\cdots\!20\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(11\!\cdots\!53\)\( \nu^{15} - \)\(33\!\cdots\!45\)\( \nu^{14} - \)\(24\!\cdots\!24\)\( \nu^{13} + \)\(59\!\cdots\!57\)\( \nu^{12} + \)\(21\!\cdots\!65\)\( \nu^{11} - \)\(36\!\cdots\!18\)\( \nu^{10} - \)\(92\!\cdots\!63\)\( \nu^{9} + \)\(82\!\cdots\!17\)\( \nu^{8} + \)\(22\!\cdots\!57\)\( \nu^{7} + \)\(10\!\cdots\!22\)\( \nu^{6} - \)\(28\!\cdots\!75\)\( \nu^{5} - \)\(27\!\cdots\!15\)\( \nu^{4} + \)\(13\!\cdots\!40\)\( \nu^{3} + \)\(27\!\cdots\!75\)\( \nu^{2} + \)\(97\!\cdots\!09\)\( \nu + \)\(99\!\cdots\!20\)\(\)\()/ \)\(40\!\cdots\!20\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(28\!\cdots\!52\)\( \nu^{15} - \)\(39\!\cdots\!92\)\( \nu^{14} + \)\(66\!\cdots\!10\)\( \nu^{13} + \)\(23\!\cdots\!32\)\( \nu^{12} - \)\(61\!\cdots\!86\)\( \nu^{11} - \)\(39\!\cdots\!25\)\( \nu^{10} + \)\(29\!\cdots\!69\)\( \nu^{9} + \)\(29\!\cdots\!64\)\( \nu^{8} - \)\(76\!\cdots\!00\)\( \nu^{7} - \)\(11\!\cdots\!09\)\( \nu^{6} + \)\(10\!\cdots\!89\)\( \nu^{5} + \)\(19\!\cdots\!02\)\( \nu^{4} - \)\(50\!\cdots\!46\)\( \nu^{3} - \)\(13\!\cdots\!11\)\( \nu^{2} - \)\(53\!\cdots\!55\)\( \nu - \)\(59\!\cdots\!10\)\(\)\()/ \)\(48\!\cdots\!20\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(34\!\cdots\!24\)\( \nu^{15} + \)\(14\!\cdots\!11\)\( \nu^{14} + \)\(69\!\cdots\!65\)\( \nu^{13} - \)\(26\!\cdots\!14\)\( \nu^{12} - \)\(56\!\cdots\!78\)\( \nu^{11} + \)\(17\!\cdots\!97\)\( \nu^{10} + \)\(23\!\cdots\!33\)\( \nu^{9} - \)\(50\!\cdots\!98\)\( \nu^{8} - \)\(53\!\cdots\!16\)\( \nu^{7} + \)\(53\!\cdots\!83\)\( \nu^{6} + \)\(65\!\cdots\!59\)\( \nu^{5} + \)\(14\!\cdots\!66\)\( \nu^{4} - \)\(32\!\cdots\!72\)\( \nu^{3} - \)\(44\!\cdots\!18\)\( \nu^{2} - \)\(13\!\cdots\!84\)\( \nu - \)\(10\!\cdots\!90\)\(\)\()/ \)\(48\!\cdots\!20\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(28\!\cdots\!11\)\( \nu^{15} + \)\(55\!\cdots\!62\)\( \nu^{14} + \)\(61\!\cdots\!29\)\( \nu^{13} - \)\(91\!\cdots\!55\)\( \nu^{12} - \)\(53\!\cdots\!72\)\( \nu^{11} + \)\(48\!\cdots\!97\)\( \nu^{10} + \)\(24\!\cdots\!99\)\( \nu^{9} - \)\(43\!\cdots\!08\)\( \nu^{8} - \)\(60\!\cdots\!36\)\( \nu^{7} - \)\(35\!\cdots\!19\)\( \nu^{6} + \)\(77\!\cdots\!43\)\( \nu^{5} + \)\(10\!\cdots\!92\)\( \nu^{4} - \)\(38\!\cdots\!85\)\( \nu^{3} - \)\(85\!\cdots\!29\)\( \nu^{2} - \)\(32\!\cdots\!82\)\( \nu - \)\(34\!\cdots\!25\)\(\)\()/ \)\(11\!\cdots\!80\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(38\!\cdots\!63\)\( \nu^{15} - \)\(22\!\cdots\!86\)\( \nu^{14} - \)\(86\!\cdots\!45\)\( \nu^{13} + \)\(22\!\cdots\!31\)\( \nu^{12} + \)\(78\!\cdots\!48\)\( \nu^{11} + \)\(10\!\cdots\!15\)\( \nu^{10} - \)\(36\!\cdots\!83\)\( \nu^{9} - \)\(22\!\cdots\!48\)\( \nu^{8} + \)\(93\!\cdots\!00\)\( \nu^{7} + \)\(10\!\cdots\!75\)\( \nu^{6} - \)\(12\!\cdots\!39\)\( \nu^{5} - \)\(20\!\cdots\!24\)\( \nu^{4} + \)\(60\!\cdots\!65\)\( \nu^{3} + \)\(14\!\cdots\!29\)\( \nu^{2} + \)\(44\!\cdots\!54\)\( \nu + \)\(36\!\cdots\!05\)\(\)\()/ \)\(11\!\cdots\!80\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(80\!\cdots\!27\)\( \nu^{15} - \)\(14\!\cdots\!92\)\( \nu^{14} - \)\(17\!\cdots\!11\)\( \nu^{13} + \)\(24\!\cdots\!03\)\( \nu^{12} + \)\(15\!\cdots\!10\)\( \nu^{11} - \)\(12\!\cdots\!35\)\( \nu^{10} - \)\(69\!\cdots\!43\)\( \nu^{9} + \)\(85\!\cdots\!38\)\( \nu^{8} + \)\(17\!\cdots\!18\)\( \nu^{7} + \)\(10\!\cdots\!73\)\( \nu^{6} - \)\(22\!\cdots\!23\)\( \nu^{5} - \)\(30\!\cdots\!90\)\( \nu^{4} + \)\(11\!\cdots\!15\)\( \nu^{3} + \)\(23\!\cdots\!21\)\( \nu^{2} + \)\(79\!\cdots\!24\)\( \nu + \)\(72\!\cdots\!15\)\(\)\()/ \)\(11\!\cdots\!80\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(89\!\cdots\!77\)\( \nu^{15} + \)\(23\!\cdots\!59\)\( \nu^{14} + \)\(19\!\cdots\!30\)\( \nu^{13} - \)\(40\!\cdots\!01\)\( \nu^{12} - \)\(16\!\cdots\!35\)\( \nu^{11} + \)\(24\!\cdots\!64\)\( \nu^{10} + \)\(73\!\cdots\!47\)\( \nu^{9} - \)\(49\!\cdots\!47\)\( \nu^{8} - \)\(17\!\cdots\!27\)\( \nu^{7} - \)\(38\!\cdots\!56\)\( \nu^{6} + \)\(22\!\cdots\!23\)\( \nu^{5} + \)\(24\!\cdots\!09\)\( \nu^{4} - \)\(11\!\cdots\!94\)\( \nu^{3} - \)\(22\!\cdots\!51\)\( \nu^{2} - \)\(83\!\cdots\!39\)\( \nu - \)\(86\!\cdots\!50\)\(\)\()/ \)\(11\!\cdots\!80\)\( \)
\(\beta_{12}\)\(=\)\((\)\(\)\(19\!\cdots\!59\)\( \nu^{15} - \)\(47\!\cdots\!13\)\( \nu^{14} - \)\(41\!\cdots\!14\)\( \nu^{13} + \)\(83\!\cdots\!43\)\( \nu^{12} + \)\(36\!\cdots\!93\)\( \nu^{11} - \)\(48\!\cdots\!52\)\( \nu^{10} - \)\(16\!\cdots\!97\)\( \nu^{9} + \)\(91\!\cdots\!09\)\( \nu^{8} + \)\(39\!\cdots\!33\)\( \nu^{7} + \)\(11\!\cdots\!08\)\( \nu^{6} - \)\(49\!\cdots\!97\)\( \nu^{5} - \)\(56\!\cdots\!39\)\( \nu^{4} + \)\(24\!\cdots\!54\)\( \nu^{3} + \)\(50\!\cdots\!17\)\( \nu^{2} + \)\(17\!\cdots\!41\)\( \nu + \)\(16\!\cdots\!30\)\(\)\()/ \)\(11\!\cdots\!80\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(13\!\cdots\!81\)\( \nu^{15} + \)\(40\!\cdots\!29\)\( \nu^{14} + \)\(29\!\cdots\!92\)\( \nu^{13} - \)\(70\!\cdots\!25\)\( \nu^{12} - \)\(24\!\cdots\!17\)\( \nu^{11} + \)\(43\!\cdots\!70\)\( \nu^{10} + \)\(10\!\cdots\!51\)\( \nu^{9} - \)\(99\!\cdots\!89\)\( \nu^{8} - \)\(26\!\cdots\!81\)\( \nu^{7} - \)\(49\!\cdots\!94\)\( \nu^{6} + \)\(32\!\cdots\!39\)\( \nu^{5} + \)\(31\!\cdots\!31\)\( \nu^{4} - \)\(16\!\cdots\!40\)\( \nu^{3} - \)\(31\!\cdots\!95\)\( \nu^{2} - \)\(11\!\cdots\!85\)\( \nu - \)\(11\!\cdots\!60\)\(\)\()/ \)\(11\!\cdots\!80\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(15\!\cdots\!93\)\( \nu^{15} + \)\(43\!\cdots\!47\)\( \nu^{14} + \)\(32\!\cdots\!90\)\( \nu^{13} - \)\(77\!\cdots\!37\)\( \nu^{12} - \)\(27\!\cdots\!43\)\( \nu^{11} + \)\(47\!\cdots\!68\)\( \nu^{10} + \)\(12\!\cdots\!91\)\( \nu^{9} - \)\(10\!\cdots\!39\)\( \nu^{8} - \)\(29\!\cdots\!43\)\( \nu^{7} - \)\(15\!\cdots\!12\)\( \nu^{6} + \)\(36\!\cdots\!75\)\( \nu^{5} + \)\(36\!\cdots\!53\)\( \nu^{4} - \)\(18\!\cdots\!78\)\( \nu^{3} - \)\(35\!\cdots\!43\)\( \nu^{2} - \)\(12\!\cdots\!91\)\( \nu - \)\(13\!\cdots\!90\)\(\)\()/ \)\(11\!\cdots\!80\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(19\!\cdots\!96\)\( \nu^{15} - \)\(59\!\cdots\!95\)\( \nu^{14} - \)\(41\!\cdots\!05\)\( \nu^{13} + \)\(10\!\cdots\!76\)\( \nu^{12} + \)\(35\!\cdots\!05\)\( \nu^{11} - \)\(65\!\cdots\!27\)\( \nu^{10} - \)\(15\!\cdots\!10\)\( \nu^{9} + \)\(15\!\cdots\!37\)\( \nu^{8} + \)\(37\!\cdots\!85\)\( \nu^{7} - \)\(17\!\cdots\!83\)\( \nu^{6} - \)\(46\!\cdots\!10\)\( \nu^{5} - \)\(43\!\cdots\!19\)\( \nu^{4} + \)\(23\!\cdots\!01\)\( \nu^{3} + \)\(43\!\cdots\!84\)\( \nu^{2} + \)\(15\!\cdots\!55\)\( \nu + \)\(15\!\cdots\!45\)\(\)\()/ \)\(11\!\cdots\!80\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} + \beta_{4} - 7 \beta_{3} + 7 \beta_{2} - 3 \beta_{1}\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(3 \beta_{14} - 3 \beta_{13} + 2 \beta_{12} - 2 \beta_{11} + 3 \beta_{10} - 2 \beta_{9} + 2 \beta_{8} + 18 \beta_{7} + 9 \beta_{6} + 71 \beta_{5} + 79 \beta_{4} + 295 \beta_{3} - 272 \beta_{2} - 3 \beta_{1} + 826719\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(6 \beta_{15} - 1650 \beta_{14} + 1671 \beta_{13} + 406 \beta_{12} + 1433 \beta_{11} - 1829 \beta_{10} - 224 \beta_{9} - 1421 \beta_{8} + 1613 \beta_{7} + 766 \beta_{6} + 428130 \beta_{5} + 429208 \beta_{4} - 4604820 \beta_{3} + 4603150 \beta_{2} - 2480166 \beta_{1} + 57968502\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(1612 \beta_{15} + 1661672 \beta_{14} - 1666426 \beta_{13} + 1024264 \beta_{12} - 1526536 \beta_{11} + 1768980 \beta_{10} - 1139158 \beta_{9} + 1507764 \beta_{8} + 15697190 \beta_{7} + 7843756 \beta_{6} + 28722040 \beta_{5} + 34889562 \beta_{4} + 1184588158 \beta_{3} - 1160043844 \beta_{2} - 236834349 \beta_{1} + 361538501343\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(5117320 \beta_{15} - 1361854928 \beta_{14} + 1381505158 \beta_{13} + 261277260 \beta_{12} + 1062610356 \beta_{11} - 1335135992 \beta_{10} - 285340876 \beta_{9} - 1044982426 \beta_{8} + 856934622 \beta_{7} + 369602646 \beta_{6} + 210577393088 \beta_{5} + 211313208123 \beta_{4} - 2993670075216 \beta_{3} + 2986998399353 \beta_{2} - 1808296993422 \beta_{1} + 19195727425833\)\()/3\)
\(\nu^{6}\)\(=\)\((\)\(1552323690 \beta_{15} + 925356336552 \beta_{14} - 930028528653 \beta_{13} + 541222171732 \beta_{12} - 1090606966378 \beta_{11} + 973282435748 \beta_{10} - 599157369987 \beta_{9} + 1070040123070 \beta_{8} + 10336639354375 \beta_{7} + 5164463435096 \beta_{6} + 4724626408948 \beta_{5} + 9062149438181 \beta_{4} + 905262505837433 \beta_{3} - 883308148411900 \beta_{2} - 120601031792895 \beta_{1} + 181318439730093780\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(3783175801132 \beta_{15} - 914572651260432 \beta_{14} + 929352138757983 \beta_{13} + 131097138410774 \beta_{12} + 639973610477214 \beta_{11} - 846309496732384 \beta_{10} - 197009135100562 \beta_{9} - 621086052412573 \beta_{8} + 285653636372583 \beta_{7} + 88558843496415 \beta_{6} + 110431514411398040 \beta_{5} + 110705494267522130 \beta_{4} - 1882330624082285033 \beta_{3} + 1875359159424227883 \beta_{2} - 1269670168866404280 \beta_{1} + 828354495587831562\)\()/3\)
\(\nu^{8}\)\(=\)\((\)\(1033666136647176 \beta_{15} + 527333005894102628 \beta_{14} - 531419579140670056 \beta_{13} + 289282800491136632 \beta_{12} - 728466060547301864 \beta_{11} + 550598716762005980 \beta_{10} - 321568946308807516 \beta_{9} + 712254014142728648 \beta_{8} + 6271154591322810420 \beta_{7} + 3133863319855678812 \beta_{6} - 3920292562596444524 \beta_{5} - 977806632785486352 \beta_{4} + 579351468011801109488 \beta_{3} - 561723033600914467968 \beta_{2} - 11707205059586891529 \beta_{1} + 96553667286715001099448\)\()/3\)
\(\nu^{9}\)\(=\)\((\)\(2598961804163172792 \beta_{15} - 575150319943675817592 \beta_{14} + 585585250567641551064 \beta_{13} + 59797626958421147888 \beta_{12} + 368464027380389888932 \beta_{11} - 518966808271724206700 \beta_{10} - 114051848656670582088 \beta_{9} - 351626814595013031352 \beta_{8} + 6152449533548896976 \beta_{7} - 39255045550424026148 \beta_{6} + 59960408274146211068217 \beta_{5} + 59952791664266852262890 \beta_{4} - 1154468614751813106837381 \beta_{3} + 1148754706455474500127272 \beta_{2} - 869058542134596506464341 \beta_{1} - 4935476825172528152325321\)\()/3\)
\(\nu^{10}\)\(=\)\((\)\(585004718149266786470 \beta_{15} + 306527339936388895506987 \beta_{14} - 309926430558027788118150 \beta_{13} + 156282921081819894523374 \beta_{12} - 465108153265679973361160 \beta_{11} + 319453065361716096997567 \beta_{10} - 176214866909675672263231 \beta_{9} + 453682459976083269790116 \beta_{8} + 3691718259864798721162131 \beta_{7} + 1845812947996199171104761 \beta_{6} - 5738089746183091532805089 \beta_{5} - 3783322127232248817978858 \beta_{4} + 356448870045111155712174938 \beta_{3} - 343261769768446888721331088 \beta_{2} + 45009212121340561310282238 \beta_{1} + 53021331928826632372979130207\)\()/3\)
\(\nu^{11}\)\(=\)\((\)\(1715966077774348283842090 \beta_{15} - 351614579408727713026542766 \beta_{14} + 358767630759249361574862956 \beta_{13} + 25416062083300100910062184 \beta_{12} + 212451692863004053156502059 \beta_{11} - 314041444582020378363122441 \beta_{10} - 61479782441954612905229230 \beta_{9} - 199016731903093149071430824 \beta_{8} - 98960180764962463176345898 \beta_{7} - 79937930141672090972934341 \beta_{6} + 33260053031816578861923477948 \beta_{5} + 33127510823097107970603301042 \beta_{4} - 696522640120464426568903695293 \beta_{3} + 692228570510999789836891226215 \beta_{2} - 583010999786032168202577633960 \beta_{1} - 5633553006533583977384107440858\)\()/3\)
\(\nu^{12}\)\(=\)\((\)\(294716253842154390396179112 \beta_{15} + 180829728133412196234434665998 \beta_{14} - 183512669196587627303578610334 \beta_{13} + 85345991328826970966533250332 \beta_{12} - 288609611202565969760347575092 \beta_{11} + 188283149180288434516567353774 \beta_{10} - 97851669027538797856908248392 \beta_{9} + 280877218168557268930890951152 \beta_{8} + 2147273236215529593386567167292 \beta_{7} + 1074527258210391036557853338902 \beta_{6} - 5111932855430032968967258734682 \beta_{5} - 3831683269103293545974650910382 \beta_{4} + 218555753451020862182431223993630 \beta_{3} - 209153822151039014353422212097244 \beta_{2} + 64106055381948227369238995317620 \beta_{1} + 29647239195940237364029659546354009\)\()/3\)
\(\nu^{13}\)\(=\)\((\)\(\)\(11\!\cdots\!12\)\( \beta_{15} - \)\(21\!\cdots\!28\)\( \beta_{14} + \)\(21\!\cdots\!10\)\( \beta_{13} + \)\(98\!\cdots\!16\)\( \beta_{12} + \)\(12\!\cdots\!90\)\( \beta_{11} - \)\(18\!\cdots\!38\)\( \beta_{10} - \)\(31\!\cdots\!76\)\( \beta_{9} - \)\(11\!\cdots\!06\)\( \beta_{8} - \)\(12\!\cdots\!34\)\( \beta_{7} - \)\(80\!\cdots\!84\)\( \beta_{6} + \)\(18\!\cdots\!75\)\( \beta_{5} + \)\(18\!\cdots\!77\)\( \beta_{4} - \)\(41\!\cdots\!05\)\( \beta_{3} + \)\(41\!\cdots\!79\)\( \beta_{2} - \)\(38\!\cdots\!17\)\( \beta_{1} - \)\(46\!\cdots\!34\)\(\)\()/3\)
\(\nu^{14}\)\(=\)\((\)\(\)\(13\!\cdots\!64\)\( \beta_{15} + \)\(10\!\cdots\!49\)\( \beta_{14} - \)\(10\!\cdots\!63\)\( \beta_{13} + \)\(47\!\cdots\!86\)\( \beta_{12} - \)\(17\!\cdots\!26\)\( \beta_{11} + \)\(11\!\cdots\!69\)\( \beta_{10} - \)\(54\!\cdots\!44\)\( \beta_{9} + \)\(17\!\cdots\!22\)\( \beta_{8} + \)\(12\!\cdots\!04\)\( \beta_{7} + \)\(62\!\cdots\!63\)\( \beta_{6} - \)\(38\!\cdots\!95\)\( \beta_{5} - \)\(30\!\cdots\!89\)\( \beta_{4} + \)\(13\!\cdots\!87\)\( \beta_{3} - \)\(12\!\cdots\!24\)\( \beta_{2} + \)\(63\!\cdots\!83\)\( \beta_{1} + \)\(16\!\cdots\!15\)\(\)\()/3\)
\(\nu^{15}\)\(=\)\((\)\(\)\(70\!\cdots\!78\)\( \beta_{15} - \)\(12\!\cdots\!78\)\( \beta_{14} + \)\(12\!\cdots\!39\)\( \beta_{13} + \)\(32\!\cdots\!50\)\( \beta_{12} + \)\(74\!\cdots\!27\)\( \beta_{11} - \)\(11\!\cdots\!99\)\( \beta_{10} - \)\(16\!\cdots\!84\)\( \beta_{9} - \)\(66\!\cdots\!97\)\( \beta_{8} - \)\(10\!\cdots\!35\)\( \beta_{7} - \)\(67\!\cdots\!84\)\( \beta_{6} + \)\(10\!\cdots\!82\)\( \beta_{5} + \)\(10\!\cdots\!02\)\( \beta_{4} - \)\(24\!\cdots\!74\)\( \beta_{3} + \)\(24\!\cdots\!26\)\( \beta_{2} - \)\(25\!\cdots\!54\)\( \beta_{1} - \)\(34\!\cdots\!12\)\(\)\()/3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/98\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
738.105 + 0.866025i
468.905 + 0.866025i
−272.613 + 0.866025i
−496.819 + 0.866025i
657.612 + 0.866025i
−22.0174 + 0.866025i
−295.555 + 0.866025i
−773.618 + 0.866025i
738.105 0.866025i
468.905 0.866025i
−272.613 0.866025i
−496.819 0.866025i
657.612 0.866025i
−22.0174 0.866025i
−295.555 0.866025i
−773.618 0.866025i
−22.6274 + 39.1918i −1106.41 + 638.785i −1024.00 1773.62i −24805.4 14321.4i 57816.2i 0 92681.9 550372. 953273.i 1.12257e6 648114.i
19.2 −22.6274 + 39.1918i −702.607 + 405.650i −1024.00 1773.62i 18305.2 + 10568.5i 36715.3i 0 92681.9 63384.0 109784.i −828399. + 478276.i
19.3 −22.6274 + 39.1918i 409.670 236.523i −1024.00 1773.62i −18219.6 10519.1i 21407.6i 0 92681.9 −153834. + 266449.i 824525. 476040.i
19.4 −22.6274 + 39.1918i 745.979 430.691i −1024.00 1773.62i 7532.28 + 4348.76i 38981.7i 0 92681.9 105269. 182331.i −340872. + 196803.i
19.5 22.6274 39.1918i −985.668 + 569.076i −1024.00 1773.62i 12133.2 + 7005.11i 51506.9i 0 −92681.9 381974. 661598.i 549087. 317015.i
19.6 22.6274 39.1918i 33.7761 19.5006i −1024.00 1773.62i −2481.87 1432.91i 1764.99i 0 −92681.9 −264960. + 458924.i −112317. + 64846.2i
19.7 22.6274 39.1918i 444.082 256.391i −1024.00 1773.62i 12687.9 + 7325.35i 23205.9i 0 −92681.9 −134248. + 232524.i 574187. 331507.i
19.8 22.6274 39.1918i 1161.18 670.406i −1024.00 1773.62i −14223.7 8212.03i 60678.2i 0 −92681.9 633167. 1.09668e6i −643689. + 371634.i
31.1 −22.6274 39.1918i −1106.41 638.785i −1024.00 + 1773.62i −24805.4 + 14321.4i 57816.2i 0 92681.9 550372. + 953273.i 1.12257e6 + 648114.i
31.2 −22.6274 39.1918i −702.607 405.650i −1024.00 + 1773.62i 18305.2 10568.5i 36715.3i 0 92681.9 63384.0 + 109784.i −828399. 478276.i
31.3 −22.6274 39.1918i 409.670 + 236.523i −1024.00 + 1773.62i −18219.6 + 10519.1i 21407.6i 0 92681.9 −153834. 266449.i 824525. + 476040.i
31.4 −22.6274 39.1918i 745.979 + 430.691i −1024.00 + 1773.62i 7532.28 4348.76i 38981.7i 0 92681.9 105269. + 182331.i −340872. 196803.i
31.5 22.6274 + 39.1918i −985.668 569.076i −1024.00 + 1773.62i 12133.2 7005.11i 51506.9i 0 −92681.9 381974. + 661598.i 549087. + 317015.i
31.6 22.6274 + 39.1918i 33.7761 + 19.5006i −1024.00 + 1773.62i −2481.87 + 1432.91i 1764.99i 0 −92681.9 −264960. 458924.i −112317. 64846.2i
31.7 22.6274 + 39.1918i 444.082 + 256.391i −1024.00 + 1773.62i 12687.9 7325.35i 23205.9i 0 −92681.9 −134248. 232524.i 574187. + 331507.i
31.8 22.6274 + 39.1918i 1161.18 + 670.406i −1024.00 + 1773.62i −14223.7 + 8212.03i 60678.2i 0 −92681.9 633167. + 1.09668e6i −643689. 371634.i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.13.d.a 16
7.b odd 2 1 14.13.d.a 16
7.c even 3 1 14.13.d.a 16
7.c even 3 1 98.13.b.c 16
7.d odd 6 1 98.13.b.c 16
7.d odd 6 1 inner 98.13.d.a 16
21.c even 2 1 126.13.n.a 16
21.h odd 6 1 126.13.n.a 16
28.d even 2 1 112.13.s.c 16
28.g odd 6 1 112.13.s.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.13.d.a 16 7.b odd 2 1
14.13.d.a 16 7.c even 3 1
98.13.b.c 16 7.c even 3 1
98.13.b.c 16 7.d odd 6 1
98.13.d.a 16 1.a even 1 1 trivial
98.13.d.a 16 7.d odd 6 1 inner
112.13.s.c 16 28.d even 2 1
112.13.s.c 16 28.g odd 6 1
126.13.n.a 16 21.c even 2 1
126.13.n.a 16 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(76\!\cdots\!22\)\( T_{3}^{12} - \)\(27\!\cdots\!40\)\( T_{3}^{11} - \)\(88\!\cdots\!64\)\( T_{3}^{10} + \)\(88\!\cdots\!20\)\( T_{3}^{9} + \)\(74\!\cdots\!87\)\( T_{3}^{8} - \)\(20\!\cdots\!80\)\( T_{3}^{7} - \)\(29\!\cdots\!88\)\( T_{3}^{6} + \)\(12\!\cdots\!20\)\( T_{3}^{5} + \)\(76\!\cdots\!58\)\( T_{3}^{4} - \)\(63\!\cdots\!60\)\( T_{3}^{3} + \)\(14\!\cdots\!56\)\( T_{3}^{2} - \)\(82\!\cdots\!60\)\( T_{3} + \)\(16\!\cdots\!41\)\( \)">\(T_{3}^{16} - \cdots\) acting on \(S_{13}^{\mathrm{new}}(98, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4194304 + 2048 T^{2} + T^{4} )^{4} \)
$3$ \( \)\(16\!\cdots\!41\)\( - \)\(82\!\cdots\!60\)\( T + \)\(14\!\cdots\!56\)\( T^{2} - \)\(63\!\cdots\!60\)\( T^{3} + \)\(76\!\cdots\!58\)\( T^{4} + \)\(12\!\cdots\!20\)\( T^{5} - \)\(29\!\cdots\!88\)\( T^{6} - \)\(20\!\cdots\!80\)\( T^{7} + \)\(74\!\cdots\!87\)\( T^{8} + \)\(88\!\cdots\!20\)\( T^{9} - 8848813170319704864 T^{10} - 276426941444640 T^{11} + 7637014892322 T^{12} - 3306888 T^{14} + T^{16} \)
$5$ \( \)\(11\!\cdots\!25\)\( + \)\(37\!\cdots\!00\)\( T - \)\(26\!\cdots\!00\)\( T^{2} - \)\(22\!\cdots\!00\)\( T^{3} + \)\(34\!\cdots\!50\)\( T^{4} + \)\(11\!\cdots\!00\)\( T^{5} - \)\(25\!\cdots\!00\)\( T^{6} + \)\(39\!\cdots\!00\)\( T^{7} + \)\(13\!\cdots\!75\)\( T^{8} - \)\(37\!\cdots\!00\)\( T^{9} - \)\(39\!\cdots\!00\)\( T^{10} + \)\(11\!\cdots\!00\)\( T^{11} + 874098909300438954 T^{12} - 21451613968512 T^{13} - 1072563036 T^{14} + 18144 T^{15} + T^{16} \)
$7$ \( T^{16} \)
$11$ \( \)\(29\!\cdots\!21\)\( + \)\(46\!\cdots\!16\)\( T + \)\(22\!\cdots\!84\)\( T^{2} + \)\(19\!\cdots\!76\)\( T^{3} + \)\(91\!\cdots\!34\)\( T^{4} + \)\(73\!\cdots\!00\)\( T^{5} + \)\(19\!\cdots\!32\)\( T^{6} + \)\(30\!\cdots\!52\)\( T^{7} + \)\(14\!\cdots\!27\)\( T^{8} - \)\(20\!\cdots\!44\)\( T^{9} + \)\(10\!\cdots\!76\)\( T^{10} - \)\(30\!\cdots\!80\)\( T^{11} + \)\(21\!\cdots\!78\)\( T^{12} - 28285267118686720704 T^{13} + 17930838817944 T^{14} - 2072088 T^{15} + T^{16} \)
$13$ \( \)\(20\!\cdots\!56\)\( + \)\(19\!\cdots\!48\)\( T^{2} + \)\(53\!\cdots\!12\)\( T^{4} + \)\(68\!\cdots\!48\)\( T^{6} + \)\(44\!\cdots\!48\)\( T^{8} + \)\(15\!\cdots\!84\)\( T^{10} + \)\(28\!\cdots\!68\)\( T^{12} + 265309230378576 T^{14} + T^{16} \)
$17$ \( \)\(73\!\cdots\!41\)\( + \)\(72\!\cdots\!00\)\( T - \)\(50\!\cdots\!08\)\( T^{2} - \)\(51\!\cdots\!00\)\( T^{3} + \)\(33\!\cdots\!90\)\( T^{4} + \)\(88\!\cdots\!40\)\( T^{5} + \)\(75\!\cdots\!04\)\( T^{6} - \)\(19\!\cdots\!20\)\( T^{7} - \)\(33\!\cdots\!49\)\( T^{8} - \)\(51\!\cdots\!00\)\( T^{9} + \)\(10\!\cdots\!24\)\( T^{10} + \)\(13\!\cdots\!00\)\( T^{11} - \)\(22\!\cdots\!70\)\( T^{12} + \)\(31\!\cdots\!20\)\( T^{13} + 3407109798755892 T^{14} - 101561040 T^{15} + T^{16} \)
$19$ \( \)\(72\!\cdots\!25\)\( - \)\(90\!\cdots\!00\)\( T + \)\(16\!\cdots\!00\)\( T^{2} + \)\(25\!\cdots\!00\)\( T^{3} - \)\(75\!\cdots\!50\)\( T^{4} - \)\(57\!\cdots\!00\)\( T^{5} + \)\(25\!\cdots\!00\)\( T^{6} + \)\(44\!\cdots\!00\)\( T^{7} - \)\(24\!\cdots\!69\)\( T^{8} - \)\(32\!\cdots\!96\)\( T^{9} + \)\(16\!\cdots\!36\)\( T^{10} + \)\(22\!\cdots\!16\)\( T^{11} - \)\(38\!\cdots\!42\)\( T^{12} - \)\(68\!\cdots\!72\)\( T^{13} + 6271093431881304 T^{14} + 174931848 T^{15} + T^{16} \)
$23$ \( \)\(45\!\cdots\!41\)\( - \)\(39\!\cdots\!60\)\( T + \)\(37\!\cdots\!32\)\( T^{2} - \)\(11\!\cdots\!40\)\( T^{3} + \)\(35\!\cdots\!82\)\( T^{4} - \)\(62\!\cdots\!20\)\( T^{5} + \)\(89\!\cdots\!32\)\( T^{6} - \)\(88\!\cdots\!20\)\( T^{7} + \)\(73\!\cdots\!03\)\( T^{8} - \)\(43\!\cdots\!00\)\( T^{9} + \)\(24\!\cdots\!16\)\( T^{10} - \)\(10\!\cdots\!20\)\( T^{11} + \)\(58\!\cdots\!58\)\( T^{12} - \)\(15\!\cdots\!60\)\( T^{13} + 86173454317903704 T^{14} - 25630560 T^{15} + T^{16} \)
$29$ \( ( -\)\(24\!\cdots\!84\)\( - \)\(33\!\cdots\!76\)\( T - \)\(10\!\cdots\!24\)\( T^{2} + \)\(17\!\cdots\!56\)\( T^{3} + \)\(86\!\cdots\!68\)\( T^{4} + \)\(37\!\cdots\!28\)\( T^{5} - 1674687874740006456 T^{6} - 266180472 T^{7} + T^{8} )^{2} \)
$31$ \( \)\(11\!\cdots\!21\)\( - \)\(20\!\cdots\!36\)\( T + \)\(12\!\cdots\!88\)\( T^{2} - \)\(17\!\cdots\!36\)\( T^{3} - \)\(18\!\cdots\!38\)\( T^{4} + \)\(10\!\cdots\!00\)\( T^{5} + \)\(25\!\cdots\!52\)\( T^{6} - \)\(17\!\cdots\!48\)\( T^{7} - \)\(58\!\cdots\!65\)\( T^{8} + \)\(38\!\cdots\!72\)\( T^{9} + \)\(10\!\cdots\!92\)\( T^{10} - \)\(19\!\cdots\!00\)\( T^{11} - \)\(97\!\cdots\!58\)\( T^{12} - \)\(73\!\cdots\!36\)\( T^{13} + 6843143641236994968 T^{14} + 4583818344 T^{15} + T^{16} \)
$37$ \( \)\(14\!\cdots\!61\)\( - \)\(50\!\cdots\!40\)\( T + \)\(12\!\cdots\!92\)\( T^{2} - \)\(17\!\cdots\!60\)\( T^{3} + \)\(19\!\cdots\!90\)\( T^{4} - \)\(16\!\cdots\!00\)\( T^{5} + \)\(11\!\cdots\!84\)\( T^{6} - \)\(71\!\cdots\!00\)\( T^{7} + \)\(41\!\cdots\!11\)\( T^{8} - \)\(19\!\cdots\!00\)\( T^{9} + \)\(82\!\cdots\!44\)\( T^{10} - \)\(25\!\cdots\!00\)\( T^{11} + \)\(74\!\cdots\!70\)\( T^{12} - \)\(15\!\cdots\!60\)\( T^{13} + 39324386602505835972 T^{14} - 5764524040 T^{15} + T^{16} \)
$41$ \( \)\(33\!\cdots\!76\)\( + \)\(30\!\cdots\!64\)\( T^{2} + \)\(11\!\cdots\!16\)\( T^{4} + \)\(21\!\cdots\!64\)\( T^{6} + \)\(24\!\cdots\!04\)\( T^{8} + \)\(15\!\cdots\!56\)\( T^{10} + \)\(59\!\cdots\!08\)\( T^{12} + \)\(12\!\cdots\!80\)\( T^{14} + T^{16} \)
$43$ \( ( \)\(41\!\cdots\!04\)\( + \)\(21\!\cdots\!40\)\( T + \)\(24\!\cdots\!68\)\( T^{2} - \)\(16\!\cdots\!80\)\( T^{3} - \)\(32\!\cdots\!32\)\( T^{4} - \)\(95\!\cdots\!40\)\( T^{5} + \)\(31\!\cdots\!56\)\( T^{6} + 33464716000 T^{7} + T^{8} )^{2} \)
$47$ \( \)\(90\!\cdots\!01\)\( - \)\(15\!\cdots\!00\)\( T - \)\(64\!\cdots\!36\)\( T^{2} + \)\(12\!\cdots\!00\)\( T^{3} + \)\(35\!\cdots\!74\)\( T^{4} - \)\(78\!\cdots\!80\)\( T^{5} - \)\(58\!\cdots\!64\)\( T^{6} + \)\(18\!\cdots\!80\)\( T^{7} + \)\(94\!\cdots\!79\)\( T^{8} - \)\(28\!\cdots\!20\)\( T^{9} - \)\(10\!\cdots\!36\)\( T^{10} + \)\(25\!\cdots\!60\)\( T^{11} + \)\(13\!\cdots\!74\)\( T^{12} - \)\(85\!\cdots\!60\)\( T^{13} - \)\(36\!\cdots\!64\)\( T^{14} + 18116171640 T^{15} + T^{16} \)
$53$ \( \)\(53\!\cdots\!61\)\( + \)\(22\!\cdots\!20\)\( T + \)\(10\!\cdots\!32\)\( T^{2} - \)\(44\!\cdots\!20\)\( T^{3} + \)\(15\!\cdots\!02\)\( T^{4} - \)\(27\!\cdots\!80\)\( T^{5} + \)\(43\!\cdots\!16\)\( T^{6} - \)\(45\!\cdots\!80\)\( T^{7} + \)\(54\!\cdots\!07\)\( T^{8} - \)\(45\!\cdots\!40\)\( T^{9} + \)\(44\!\cdots\!52\)\( T^{10} - \)\(20\!\cdots\!60\)\( T^{11} + \)\(12\!\cdots\!18\)\( T^{12} - \)\(28\!\cdots\!40\)\( T^{13} + \)\(19\!\cdots\!36\)\( T^{14} - 39134161800 T^{15} + T^{16} \)
$59$ \( \)\(19\!\cdots\!61\)\( - \)\(15\!\cdots\!64\)\( T + \)\(35\!\cdots\!28\)\( T^{2} + \)\(50\!\cdots\!16\)\( T^{3} - \)\(94\!\cdots\!70\)\( T^{4} - \)\(12\!\cdots\!36\)\( T^{5} + \)\(17\!\cdots\!76\)\( T^{6} + \)\(35\!\cdots\!04\)\( T^{7} + \)\(25\!\cdots\!35\)\( T^{8} + \)\(85\!\cdots\!80\)\( T^{9} + \)\(13\!\cdots\!48\)\( T^{10} - \)\(17\!\cdots\!96\)\( T^{11} - \)\(26\!\cdots\!54\)\( T^{12} + \)\(14\!\cdots\!08\)\( T^{13} + \)\(14\!\cdots\!64\)\( T^{14} + 201845459088 T^{15} + T^{16} \)
$61$ \( \)\(56\!\cdots\!41\)\( + \)\(17\!\cdots\!08\)\( T + \)\(12\!\cdots\!12\)\( T^{2} - \)\(21\!\cdots\!28\)\( T^{3} - \)\(11\!\cdots\!90\)\( T^{4} + \)\(25\!\cdots\!28\)\( T^{5} + \)\(33\!\cdots\!36\)\( T^{6} + \)\(16\!\cdots\!52\)\( T^{7} + \)\(34\!\cdots\!03\)\( T^{8} + \)\(15\!\cdots\!92\)\( T^{9} - \)\(10\!\cdots\!24\)\( T^{10} + \)\(13\!\cdots\!08\)\( T^{11} + \)\(15\!\cdots\!10\)\( T^{12} + \)\(38\!\cdots\!72\)\( T^{13} + \)\(49\!\cdots\!52\)\( T^{14} + 336780254328 T^{15} + T^{16} \)
$67$ \( \)\(97\!\cdots\!01\)\( - \)\(75\!\cdots\!60\)\( T + \)\(57\!\cdots\!80\)\( T^{2} - \)\(94\!\cdots\!00\)\( T^{3} + \)\(35\!\cdots\!94\)\( T^{4} - \)\(45\!\cdots\!00\)\( T^{5} + \)\(13\!\cdots\!52\)\( T^{6} - \)\(14\!\cdots\!40\)\( T^{7} + \)\(32\!\cdots\!67\)\( T^{8} - \)\(27\!\cdots\!60\)\( T^{9} + \)\(52\!\cdots\!44\)\( T^{10} - \)\(36\!\cdots\!60\)\( T^{11} + \)\(51\!\cdots\!02\)\( T^{12} - \)\(23\!\cdots\!20\)\( T^{13} + \)\(30\!\cdots\!36\)\( T^{14} - 107767119920 T^{15} + T^{16} \)
$71$ \( ( \)\(25\!\cdots\!56\)\( + \)\(61\!\cdots\!92\)\( T - \)\(17\!\cdots\!64\)\( T^{2} - \)\(64\!\cdots\!48\)\( T^{3} - \)\(39\!\cdots\!36\)\( T^{4} + \)\(67\!\cdots\!12\)\( T^{5} + \)\(11\!\cdots\!96\)\( T^{6} + 575129514672 T^{7} + T^{8} )^{2} \)
$73$ \( \)\(80\!\cdots\!01\)\( + \)\(65\!\cdots\!40\)\( T + \)\(16\!\cdots\!12\)\( T^{2} - \)\(47\!\cdots\!20\)\( T^{3} - \)\(45\!\cdots\!82\)\( T^{4} + \)\(15\!\cdots\!60\)\( T^{5} + \)\(98\!\cdots\!52\)\( T^{6} + \)\(25\!\cdots\!40\)\( T^{7} - \)\(70\!\cdots\!33\)\( T^{8} - \)\(15\!\cdots\!40\)\( T^{9} + \)\(38\!\cdots\!04\)\( T^{10} - \)\(22\!\cdots\!40\)\( T^{11} - \)\(90\!\cdots\!74\)\( T^{12} + \)\(15\!\cdots\!00\)\( T^{13} + \)\(16\!\cdots\!20\)\( T^{14} - 738414283320 T^{15} + T^{16} \)
$79$ \( \)\(68\!\cdots\!25\)\( + \)\(88\!\cdots\!00\)\( T + \)\(14\!\cdots\!00\)\( T^{2} - \)\(38\!\cdots\!00\)\( T^{3} + \)\(16\!\cdots\!50\)\( T^{4} + \)\(25\!\cdots\!00\)\( T^{5} + \)\(75\!\cdots\!00\)\( T^{6} + \)\(21\!\cdots\!00\)\( T^{7} + \)\(19\!\cdots\!71\)\( T^{8} + \)\(40\!\cdots\!16\)\( T^{9} + \)\(32\!\cdots\!76\)\( T^{10} - \)\(18\!\cdots\!36\)\( T^{11} + \)\(35\!\cdots\!78\)\( T^{12} - \)\(22\!\cdots\!08\)\( T^{13} + \)\(25\!\cdots\!84\)\( T^{14} - 227632064768 T^{15} + T^{16} \)
$83$ \( \)\(52\!\cdots\!96\)\( + \)\(12\!\cdots\!00\)\( T^{2} + \)\(26\!\cdots\!88\)\( T^{4} + \)\(13\!\cdots\!24\)\( T^{6} + \)\(32\!\cdots\!44\)\( T^{8} + \)\(39\!\cdots\!36\)\( T^{10} + \)\(24\!\cdots\!56\)\( T^{12} + \)\(79\!\cdots\!16\)\( T^{14} + T^{16} \)
$89$ \( \)\(78\!\cdots\!01\)\( + \)\(42\!\cdots\!64\)\( T + \)\(94\!\cdots\!08\)\( T^{2} - \)\(38\!\cdots\!36\)\( T^{3} + \)\(50\!\cdots\!58\)\( T^{4} + \)\(37\!\cdots\!04\)\( T^{5} - \)\(23\!\cdots\!80\)\( T^{6} - \)\(19\!\cdots\!28\)\( T^{7} + \)\(20\!\cdots\!75\)\( T^{8} - \)\(68\!\cdots\!44\)\( T^{9} + \)\(10\!\cdots\!40\)\( T^{10} - \)\(23\!\cdots\!12\)\( T^{11} - \)\(95\!\cdots\!82\)\( T^{12} + \)\(28\!\cdots\!32\)\( T^{13} + \)\(19\!\cdots\!92\)\( T^{14} - 2485007442792 T^{15} + T^{16} \)
$97$ \( \)\(16\!\cdots\!36\)\( + \)\(51\!\cdots\!56\)\( T^{2} + \)\(67\!\cdots\!00\)\( T^{4} + \)\(46\!\cdots\!04\)\( T^{6} + \)\(18\!\cdots\!24\)\( T^{8} + \)\(42\!\cdots\!44\)\( T^{10} + \)\(54\!\cdots\!00\)\( T^{12} + \)\(36\!\cdots\!36\)\( T^{14} + T^{16} \)
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