Properties

Label 29.18.a.a
Level $29$
Weight $18$
Character orbit 29.a
Self dual yes
Analytic conductor $53.134$
Analytic rank $1$
Dimension $18$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 29.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.1344053299\)
Analytic rank: \(1\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Defining polynomial: \(x^{18} - 1610997 x^{16} - 28978880 x^{15} + 1054878119348 x^{14} + 33471007935200 x^{13} - 361118629330503872 x^{12} - 14867597249727884800 x^{11} + 69075461759211999549440 x^{10} + 3136429534292709271797760 x^{9} - 7293684107327839501793427456 x^{8} - 310705174164268019482139033600 x^{7} + 392089226833876108917621043232768 x^{6} + 11987811702756552456984769970831360 x^{5} - 8875073388959894229615422940880306176 x^{4} - 110123126416280653106554268008725872640 x^{3} + 38261887665727248243207924189998746697728 x^{2} - 570869901591701973911550639846099485982720 x - 1366002559879573583573099965753694433050624\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{42}\cdot 3^{14}\cdot 17 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( -856 - 2 \beta_{1} + \beta_{2} ) q^{3} + ( 47927 + 27 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{4} + ( -31340 - 405 \beta_{1} - 14 \beta_{2} - \beta_{3} - \beta_{4} ) q^{5} + ( -370307 + 800 \beta_{1} - 78 \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{5} ) q^{6} + ( -2440320 - 3152 \beta_{1} + 84 \beta_{2} - 8 \beta_{3} + 3 \beta_{4} - \beta_{5} + \beta_{7} ) q^{7} + ( 4829561 + 39291 \beta_{1} + 599 \beta_{2} + 57 \beta_{3} + 3 \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} ) q^{8} + ( 27141574 - 67214 \beta_{1} - 2028 \beta_{2} - 107 \beta_{3} + 3 \beta_{4} - 5 \beta_{5} - 4 \beta_{7} - \beta_{8} + \beta_{16} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( -856 - 2 \beta_{1} + \beta_{2} ) q^{3} + ( 47927 + 27 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{4} + ( -31340 - 405 \beta_{1} - 14 \beta_{2} - \beta_{3} - \beta_{4} ) q^{5} + ( -370307 + 800 \beta_{1} - 78 \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{5} ) q^{6} + ( -2440320 - 3152 \beta_{1} + 84 \beta_{2} - 8 \beta_{3} + 3 \beta_{4} - \beta_{5} + \beta_{7} ) q^{7} + ( 4829561 + 39291 \beta_{1} + 599 \beta_{2} + 57 \beta_{3} + 3 \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} ) q^{8} + ( 27141574 - 67214 \beta_{1} - 2028 \beta_{2} - 107 \beta_{3} + 3 \beta_{4} - 5 \beta_{5} - 4 \beta_{7} - \beta_{8} + \beta_{16} ) q^{9} + ( -72318678 - 129252 \beta_{1} + 3354 \beta_{2} - 625 \beta_{3} + 112 \beta_{4} - 38 \beta_{5} - 5 \beta_{7} - 6 \beta_{8} + \beta_{13} - \beta_{14} - 3 \beta_{16} ) q^{10} + ( 23015518 + 73941 \beta_{1} + 4714 \beta_{2} - 194 \beta_{3} + 221 \beta_{4} - 57 \beta_{5} - 2 \beta_{6} - 8 \beta_{7} - 7 \beta_{8} + 3 \beta_{9} + \beta_{10} - 2 \beta_{11} - \beta_{12} - 2 \beta_{13} + 4 \beta_{14} + \beta_{15} - 4 \beta_{16} - 2 \beta_{17} ) q^{11} + ( 256309529 + 144098 \beta_{1} + 29410 \beta_{2} - 492 \beta_{3} + 835 \beta_{4} - 98 \beta_{5} + 9 \beta_{6} - 34 \beta_{7} - 7 \beta_{8} - 15 \beta_{9} - 5 \beta_{10} + 9 \beta_{11} + 5 \beta_{12} - 8 \beta_{13} + 4 \beta_{14} - 7 \beta_{15} + 9 \beta_{16} + 13 \beta_{17} ) q^{12} + ( -94903492 + 1433491 \beta_{1} - 34670 \beta_{2} - 2648 \beta_{3} + 910 \beta_{4} - 73 \beta_{5} + 9 \beta_{6} - 36 \beta_{7} + 17 \beta_{8} + 2 \beta_{9} + \beta_{10} + 8 \beta_{11} - 17 \beta_{13} - 27 \beta_{14} + 12 \beta_{15} - 4 \beta_{16} - 24 \beta_{17} ) q^{13} + ( -565421783 - 3426010 \beta_{1} - 142050 \beta_{2} - 13946 \beta_{3} + 329 \beta_{4} - 52 \beta_{5} - 76 \beta_{6} + 118 \beta_{7} + 109 \beta_{8} + 55 \beta_{9} + 14 \beta_{10} - 66 \beta_{11} - 24 \beta_{12} + 115 \beta_{13} + 17 \beta_{14} + 17 \beta_{15} + 31 \beta_{16} + 14 \beta_{17} ) q^{14} + ( -1996523238 + 3519385 \beta_{1} + 18698 \beta_{2} - 31890 \beta_{3} + 2436 \beta_{4} - 278 \beta_{5} + 36 \beta_{6} + 249 \beta_{7} + 153 \beta_{8} + 27 \beta_{9} + 31 \beta_{10} - 2 \beta_{11} + 7 \beta_{12} + 32 \beta_{13} + 10 \beta_{14} - 67 \beta_{15} + 14 \beta_{16} - 34 \beta_{17} ) q^{15} + ( 744937528 + 9890579 \beta_{1} - 68574 \beta_{2} + 19084 \beta_{3} + 8971 \beta_{4} + 807 \beta_{5} + 85 \beta_{6} - 47 \beta_{7} - 45 \beta_{8} - 70 \beta_{9} - 73 \beta_{10} + 154 \beta_{11} + 5 \beta_{12} - 227 \beta_{13} - 22 \beta_{14} + 27 \beta_{15} - 15 \beta_{16} + 22 \beta_{17} ) q^{16} + ( -1729612064 - 9071592 \beta_{1} - 522961 \beta_{2} - 47746 \beta_{3} + 4475 \beta_{4} + 1380 \beta_{5} + 51 \beta_{6} + 184 \beta_{7} - 544 \beta_{8} - 283 \beta_{9} - 69 \beta_{10} - 15 \beta_{11} + 126 \beta_{12} - 271 \beta_{13} + \beta_{14} + 42 \beta_{15} - 76 \beta_{16} + 184 \beta_{17} ) q^{17} + ( -12007639567 + 9158445 \beta_{1} - 1000169 \beta_{2} - 151736 \beta_{3} + 447 \beta_{4} - 214 \beta_{5} + 54 \beta_{6} + 876 \beta_{7} - 566 \beta_{8} - 183 \beta_{9} + 17 \beta_{10} + 58 \beta_{11} - 10 \beta_{12} - 48 \beta_{13} + 138 \beta_{14} + 119 \beta_{15} - 208 \beta_{16} - 241 \beta_{17} ) q^{18} + ( -13126607322 - 30161199 \beta_{1} - 2015240 \beta_{2} - 108747 \beta_{3} - 4735 \beta_{4} + 3274 \beta_{5} - 300 \beta_{6} - 100 \beta_{7} + 374 \beta_{8} + 534 \beta_{9} + 80 \beta_{10} + 39 \beta_{11} - 284 \beta_{12} + 796 \beta_{13} + 215 \beta_{14} + 23 \beta_{15} + 189 \beta_{16} + 394 \beta_{17} ) q^{19} + ( -19080606381 - 95490363 \beta_{1} - 5299718 \beta_{2} - 440769 \beta_{3} - 59958 \beta_{4} + 3888 \beta_{5} + 270 \beta_{6} + 392 \beta_{7} + 574 \beta_{8} + 1274 \beta_{9} + 626 \beta_{10} - 878 \beta_{11} + 22 \beta_{12} + 36 \beta_{13} - 752 \beta_{14} - 1010 \beta_{15} - 386 \beta_{16} - 894 \beta_{17} ) q^{20} + ( 16263418312 - 113400100 \beta_{1} - 7469816 \beta_{2} - 67227 \beta_{3} - 48475 \beta_{4} + 4317 \beta_{5} - 1681 \beta_{6} - 3458 \beta_{7} + 489 \beta_{8} + 42 \beta_{9} - 996 \beta_{10} - 410 \beta_{11} - 291 \beta_{12} + 1703 \beta_{13} + 586 \beta_{14} + 340 \beta_{15} + 221 \beta_{16} + 252 \beta_{17} ) q^{21} + ( 13175032063 + 780742 \beta_{1} - 9871213 \beta_{2} - 153178 \beta_{3} - 113391 \beta_{4} + 5419 \beta_{5} + 668 \beta_{6} - 6609 \beta_{7} - 1409 \beta_{8} - 1020 \beta_{9} + 957 \beta_{10} + 1122 \beta_{11} + 1196 \beta_{12} - 1894 \beta_{13} - 1040 \beta_{14} + 1656 \beta_{15} + 348 \beta_{16} - 865 \beta_{17} ) q^{22} + ( 24928353951 - 68573105 \beta_{1} - 6178960 \beta_{2} + 45253 \beta_{3} + 20568 \beta_{4} - 10433 \beta_{5} + 1767 \beta_{6} - 1197 \beta_{7} - 634 \beta_{8} - 3120 \beta_{9} - 396 \beta_{10} + 1088 \beta_{11} - 1485 \beta_{12} + 913 \beta_{13} + 216 \beta_{14} + 852 \beta_{15} + 2094 \beta_{16} + 2792 \beta_{17} ) q^{23} + ( 73953733564 + 103889853 \beta_{1} - 11278332 \beta_{2} + 448576 \beta_{3} - 97081 \beta_{4} - 14997 \beta_{5} + 3751 \beta_{6} - 3395 \beta_{7} + 1651 \beta_{8} - 438 \beta_{9} - 1299 \beta_{10} - 50 \beta_{11} + 47 \beta_{12} - 5157 \beta_{13} + 490 \beta_{14} - 971 \beta_{15} + 1471 \beta_{16} - 294 \beta_{17} ) q^{24} + ( 167576770921 - 419306303 \beta_{1} - 7590430 \beta_{2} + 1577761 \beta_{3} - 49637 \beta_{4} - 11830 \beta_{5} - 2654 \beta_{6} - 7814 \beta_{7} + 4912 \beta_{8} - 764 \beta_{9} - 943 \beta_{10} + 150 \beta_{11} + 2999 \beta_{12} - 2992 \beta_{13} + 143 \beta_{14} - 4972 \beta_{15} - 1997 \beta_{16} + 1292 \beta_{17} ) q^{25} + ( 256952676660 - 477943744 \beta_{1} - 9987343 \beta_{2} + 4062394 \beta_{3} + 241018 \beta_{4} - 50174 \beta_{5} - 6396 \beta_{6} + 5632 \beta_{7} + 4063 \beta_{8} + 3122 \beta_{9} - 1251 \beta_{10} + 4924 \beta_{11} + 684 \beta_{12} - \beta_{13} + 2833 \beta_{14} + 2692 \beta_{15} - 4161 \beta_{16} - 5587 \beta_{17} ) q^{26} + ( -201851377694 - 557054627 \beta_{1} + 5304792 \beta_{2} + 1173340 \beta_{3} + 190761 \beta_{4} - 45215 \beta_{5} + 9276 \beta_{6} + 20852 \beta_{7} + 3791 \beta_{8} + 4857 \beta_{9} + 6957 \beta_{10} - 1548 \beta_{11} + 4953 \beta_{12} + 1620 \beta_{13} - 3864 \beta_{14} - 759 \beta_{15} - 6092 \beta_{16} + 1422 \beta_{17} ) q^{27} + ( -291935830970 - 2142737762 \beta_{1} - 26414088 \beta_{2} - 2455474 \beta_{3} + 335844 \beta_{4} - 183728 \beta_{5} - 9832 \beta_{6} + 112736 \beta_{7} - 812 \beta_{8} + 10628 \beta_{9} + 1032 \beta_{10} - 22020 \beta_{11} - 22040 \beta_{12} + 30396 \beta_{13} + 11088 \beta_{14} - 2872 \beta_{15} + 6432 \beta_{16} + 1468 \beta_{17} ) q^{28} -500246412961 q^{29} + ( 629031390686 - 5755304102 \beta_{1} - 50182023 \beta_{2} + 9640510 \beta_{3} + 943004 \beta_{4} + 136967 \beta_{5} - 25820 \beta_{6} + 1903 \beta_{7} - 44294 \beta_{8} - 2259 \beta_{9} + 7739 \beta_{10} + 8388 \beta_{11} - 1592 \beta_{12} - 191 \beta_{13} - 18063 \beta_{14} + 21995 \beta_{15} + 3431 \beta_{16} + 8089 \beta_{17} ) q^{30} + ( 238131830253 - 3163022589 \beta_{1} + 39659958 \beta_{2} + 5134914 \beta_{3} + 1503304 \beta_{4} + 186496 \beta_{5} + 39913 \beta_{6} - 40237 \beta_{7} - 53295 \beta_{8} - 36839 \beta_{9} - 14765 \beta_{10} + 6169 \beta_{11} + 32376 \beta_{12} - 25677 \beta_{13} - 17619 \beta_{14} + 8672 \beta_{15} - 36403 \beta_{16} - 30000 \beta_{17} ) q^{31} + ( 1138290130417 - 2544711019 \beta_{1} + 50691817 \beta_{2} + 14200711 \beta_{3} + 598767 \beta_{4} + 320051 \beta_{5} + 18322 \beta_{6} - 198631 \beta_{7} - 24863 \beta_{8} - 25338 \beta_{9} - 33766 \beta_{10} + 50570 \beta_{11} + 39858 \beta_{12} - 55452 \beta_{13} + 13460 \beta_{14} + 14142 \beta_{15} + 26222 \beta_{16} + 6502 \beta_{17} ) q^{32} + ( 681803224241 - 8601503520 \beta_{1} + 44461577 \beta_{2} + 7837459 \beta_{3} + 381170 \beta_{4} + 898527 \beta_{5} - 44025 \beta_{6} - 155522 \beta_{7} + 75757 \beta_{8} + 34323 \beta_{9} + 31164 \beta_{10} + 5237 \beta_{11} - 54221 \beta_{12} - 19585 \beta_{13} - 48276 \beta_{14} - 59490 \beta_{15} + 18860 \beta_{16} + 16604 \beta_{17} ) q^{33} + ( -1618740442615 - 8847129760 \beta_{1} + 157476753 \beta_{2} - 37478511 \beta_{3} - 2889607 \beta_{4} - 151244 \beta_{5} + 62150 \beta_{6} + 138157 \beta_{7} + 72178 \beta_{8} + 23451 \beta_{9} - 7591 \beta_{10} - 90726 \beta_{11} - 16306 \beta_{12} + 39109 \beta_{13} + 75793 \beta_{14} - 94247 \beta_{15} + 91333 \beta_{16} + 56475 \beta_{17} ) q^{34} + ( -1907714844965 + 2824010989 \beta_{1} + 410631157 \beta_{2} - 65258 \beta_{3} + 2043280 \beta_{4} + 394608 \beta_{5} + 70011 \beta_{6} - 161377 \beta_{7} + 126701 \beta_{8} - 5155 \beta_{9} - 5437 \beta_{10} + 70287 \beta_{11} - 36754 \beta_{12} + 33549 \beta_{13} + 48831 \beta_{14} + 42938 \beta_{15} - 7053 \beta_{16} - 40732 \beta_{17} ) q^{35} + ( -1909174220486 - 22595990326 \beta_{1} + 231028336 \beta_{2} - 21739458 \beta_{3} - 3375276 \beta_{4} - 202892 \beta_{5} - 62520 \beta_{6} + 23044 \beta_{7} - 49108 \beta_{8} + 8184 \beta_{9} + 74840 \beta_{10} - 1880 \beta_{11} + 48056 \beta_{12} + 41928 \beta_{13} - 16968 \beta_{14} + 86120 \beta_{15} - 19328 \beta_{16} + 32408 \beta_{17} ) q^{36} + ( -1874807773043 - 16638683887 \beta_{1} + 178393448 \beta_{2} - 23032963 \beta_{3} - 6483107 \beta_{4} + 552936 \beta_{5} - 224759 \beta_{6} - 177934 \beta_{7} + 110940 \beta_{8} + 36632 \beta_{9} - 3018 \beta_{10} - 113596 \beta_{11} + 1297 \beta_{12} + 119557 \beta_{13} + 85688 \beta_{14} + 67180 \beta_{15} - 94362 \beta_{16} - 62980 \beta_{17} ) q^{37} + ( -5376547748492 - 30092289084 \beta_{1} + 512554867 \beta_{2} - 68746351 \beta_{3} - 3194292 \beta_{4} - 2412392 \beta_{5} + 5036 \beta_{6} + 425003 \beta_{7} - 100355 \beta_{8} - 48144 \beta_{9} - 5353 \beta_{10} + 40718 \beta_{11} + 2284 \beta_{12} - 30548 \beta_{13} - 77718 \beta_{14} - 54004 \beta_{15} - 75590 \beta_{16} - 47371 \beta_{17} ) q^{38} + ( -5807693548035 - 8922196163 \beta_{1} + 238015296 \beta_{2} - 3094082 \beta_{3} - 4883092 \beta_{4} + 611428 \beta_{5} + 78587 \beta_{6} - 368759 \beta_{7} + 18643 \beta_{8} + 116355 \beta_{9} - 128105 \beta_{10} - 102503 \beta_{11} + 43138 \beta_{12} - 103071 \beta_{13} - 34497 \beta_{14} + 213574 \beta_{15} - 168505 \beta_{16} - 59844 \beta_{17} ) q^{39} + ( -7556940737709 - 62759089133 \beta_{1} + 439889045 \beta_{2} - 122722797 \beta_{3} - 7610985 \beta_{4} - 4158959 \beta_{5} - 174426 \beta_{6} + 1030991 \beta_{7} - 264007 \beta_{8} - 81812 \beta_{9} + 125554 \beta_{10} - 18588 \beta_{11} + 2038 \beta_{12} + 54422 \beta_{13} - 141884 \beta_{14} - 152718 \beta_{15} - 34426 \beta_{16} + 37484 \beta_{17} ) q^{40} + ( -3494334815043 - 20281411861 \beta_{1} + 418704633 \beta_{2} - 78897 \beta_{3} + 3063330 \beta_{4} + 457302 \beta_{5} + 100290 \beta_{6} - 295130 \beta_{7} + 71270 \beta_{8} + 10439 \beta_{9} - 37771 \beta_{10} + 201711 \beta_{11} - 163079 \beta_{12} - 254404 \beta_{13} - 174783 \beta_{14} - 208702 \beta_{15} + 160178 \beta_{16} + 79684 \beta_{17} ) q^{41} + ( -20206980531011 - 5028691614 \beta_{1} + 809944567 \beta_{2} - 155661629 \beta_{3} - 422803 \beta_{4} - 7050724 \beta_{5} + 769630 \beta_{6} + 1610243 \beta_{7} - 164848 \beta_{8} + 28407 \beta_{9} - 324873 \beta_{10} - 108346 \beta_{11} - 84890 \beta_{12} - 198335 \beta_{13} + 2329 \beta_{14} - 191219 \beta_{15} + 31653 \beta_{16} - 17939 \beta_{17} ) q^{42} + ( 2419811329146 - 37121693137 \beta_{1} + 274947683 \beta_{2} + 80707169 \beta_{3} + 8456153 \beta_{4} + 3191658 \beta_{5} - 58790 \beta_{6} - 350740 \beta_{7} + 7706 \beta_{8} - 56810 \beta_{9} + 417958 \beta_{10} + 244495 \beta_{11} + 191340 \beta_{12} + 282982 \beta_{13} + 191285 \beta_{14} + 98051 \beta_{15} + 428003 \beta_{16} + 319434 \beta_{17} ) q^{43} + ( -2756404352609 - 25783867558 \beta_{1} + 945814758 \beta_{2} + 95691948 \beta_{3} + 30652665 \beta_{4} - 5870002 \beta_{5} + 228363 \beta_{6} + 525566 \beta_{7} - 955493 \beta_{8} - 740281 \beta_{9} + 108313 \beta_{10} + 1088663 \beta_{11} + 313663 \beta_{12} - 223668 \beta_{13} - 301812 \beta_{14} + 339851 \beta_{15} - 233149 \beta_{16} - 363189 \beta_{17} ) q^{44} + ( 7435147919263 + 17444692541 \beta_{1} - 2359904927 \beta_{2} + 223932171 \beta_{3} - 6712068 \beta_{4} + 8941258 \beta_{5} - 683628 \beta_{6} - 1808792 \beta_{7} - 127756 \beta_{8} + 72339 \beta_{9} - 365218 \beta_{10} - 613739 \beta_{11} + 293324 \beta_{12} + 140496 \beta_{13} + 617748 \beta_{14} + 377966 \beta_{15} + 115087 \beta_{16} - 12928 \beta_{17} ) q^{45} + ( -12196048874063 + 17111118294 \beta_{1} - 1499654516 \beta_{2} - 40241242 \beta_{3} - 23261843 \beta_{4} - 393150 \beta_{5} - 726520 \beta_{6} + 976724 \beta_{7} + 1192201 \beta_{8} + 565765 \beta_{9} + 17568 \beta_{10} - 833638 \beta_{11} - 556180 \beta_{12} + 685937 \beta_{13} + 768863 \beta_{14} - 640181 \beta_{15} + 646133 \beta_{16} + 419732 \beta_{17} ) q^{46} + ( -7865314546936 + 25578235891 \beta_{1} - 2704848627 \beta_{2} + 106569281 \beta_{3} + 21985514 \beta_{4} + 12813967 \beta_{5} + 566096 \beta_{6} + 88067 \beta_{7} + 437142 \beta_{8} + 286962 \beta_{9} + 429454 \beta_{10} - 842215 \beta_{11} - 438686 \beta_{12} + 121976 \beta_{13} - 185881 \beta_{14} - 1183059 \beta_{15} + 723065 \beta_{16} + 372070 \beta_{17} ) q^{47} + ( -14845608068431 + 97363134427 \beta_{1} - 4196666063 \beta_{2} + 520936339 \beta_{3} - 975903 \beta_{4} + 5439377 \beta_{5} - 2072300 \beta_{6} - 1897693 \beta_{7} + 347595 \beta_{8} + 719670 \beta_{9} - 258152 \beta_{10} - 327614 \beta_{11} - 247884 \beta_{12} + 483778 \beta_{13} - 159960 \beta_{14} + 1692652 \beta_{15} - 1633488 \beta_{16} - 1054626 \beta_{17} ) q^{48} + ( 25171806543758 + 138779933287 \beta_{1} - 4313918735 \beta_{2} + 787166200 \beta_{3} + 37859447 \beta_{4} + 24932539 \beta_{5} - 159381 \beta_{6} - 5490256 \beta_{7} + 165683 \beta_{8} - 249283 \beta_{9} - 155583 \beta_{10} - 226611 \beta_{11} + 939384 \beta_{12} - 501735 \beta_{13} + 320471 \beta_{14} + 159026 \beta_{15} - 26861 \beta_{16} + 165232 \beta_{17} ) q^{49} + ( -74925170705789 + 332935361471 \beta_{1} - 766560690 \beta_{2} + 101654593 \beta_{3} + 20587293 \beta_{4} + 636682 \beta_{5} + 4175614 \beta_{6} - 2143041 \beta_{7} + 538827 \beta_{8} + 106419 \beta_{9} - 511442 \beta_{10} + 425070 \beta_{11} + 266766 \beta_{12} - 2604212 \beta_{13} - 1597654 \beta_{14} - 505793 \beta_{15} - 801576 \beta_{16} - 170232 \beta_{17} ) q^{50} + ( -75896398197016 + 162347313090 \beta_{1} - 3372798311 \beta_{2} + 441259397 \beta_{3} + 139964441 \beta_{4} + 2871786 \beta_{5} - 1455416 \beta_{6} + 6549551 \beta_{7} + 710187 \beta_{8} + 668541 \beta_{9} + 1855975 \beta_{10} + 689843 \beta_{11} - 3149125 \beta_{12} + 2786372 \beta_{13} - 1226241 \beta_{14} - 40682 \beta_{15} - 354567 \beta_{16} - 307580 \beta_{17} ) q^{51} + ( -72903549498451 + 509188226047 \beta_{1} - 948914702 \beta_{2} + 429773589 \beta_{3} - 62711666 \beta_{4} + 1590260 \beta_{5} + 638486 \beta_{6} - 8822092 \beta_{7} + 2110490 \beta_{8} - 204078 \beta_{9} + 28570 \beta_{10} + 2058194 \beta_{11} + 1906574 \beta_{12} - 572796 \beta_{13} + 1285800 \beta_{14} + 2554318 \beta_{15} + 141358 \beta_{16} + 699602 \beta_{17} ) q^{52} + ( -26995408769343 + 299445981912 \beta_{1} - 3892984015 \beta_{2} + 646967124 \beta_{3} + 120247071 \beta_{4} - 5592588 \beta_{5} + 2714564 \beta_{6} - 5058936 \beta_{7} - 5836890 \beta_{8} - 4433407 \beta_{9} - 2707828 \beta_{10} + 3075699 \beta_{11} + 3660390 \beta_{12} - 4041348 \beta_{13} - 1563814 \beta_{14} - 653050 \beta_{15} - 1827763 \beta_{16} - 1785432 \beta_{17} ) q^{53} + ( -99752709022375 - 77450247648 \beta_{1} - 6375426065 \beta_{2} - 386403644 \beta_{3} - 124752729 \beta_{4} - 20730169 \beta_{5} - 5956320 \beta_{6} + 3679521 \beta_{7} + 1736041 \beta_{8} + 1641726 \beta_{9} + 1805873 \beta_{10} - 464882 \beta_{11} - 875056 \beta_{12} + 1080114 \beta_{13} - 1651788 \beta_{14} - 1449550 \beta_{15} - 2478892 \beta_{16} - 1022617 \beta_{17} ) q^{54} + ( -190377967534571 + 657833392559 \beta_{1} - 682952869 \beta_{2} - 947787917 \beta_{3} - 7756142 \beta_{4} - 32372961 \beta_{5} + 3425819 \beta_{6} + 10507791 \beta_{7} + 3644914 \beta_{8} + 648448 \beta_{9} - 1883726 \beta_{10} - 1412188 \beta_{11} - 2291027 \beta_{12} + 2591841 \beta_{13} + 5301942 \beta_{14} - 3600188 \beta_{15} + 6301672 \beta_{16} + 3566344 \beta_{17} ) q^{55} + ( -309152682373130 - 241162843054 \beta_{1} - 10818571878 \beta_{2} - 3834288370 \beta_{3} - 350602014 \beta_{4} - 55432658 \beta_{5} - 2291648 \beta_{6} + 19949890 \beta_{7} - 2549002 \beta_{8} - 373224 \beta_{9} + 1757392 \beta_{10} - 6556712 \beta_{11} - 3192288 \beta_{12} + 5564952 \beta_{13} + 1286160 \beta_{14} - 3111408 \beta_{15} + 4932992 \beta_{16} + 4230408 \beta_{17} ) q^{56} + ( -289187128534143 + 530950658707 \beta_{1} - 4542090956 \beta_{2} - 620647654 \beta_{3} - 32684628 \beta_{4} - 48968043 \beta_{5} + 2123846 \beta_{6} + 11479562 \beta_{7} - 7865245 \beta_{8} - 1195950 \beta_{9} - 1823635 \beta_{10} - 1081448 \beta_{11} + 3862995 \beta_{12} - 2309876 \beta_{13} + 274375 \beta_{14} + 3930372 \beta_{15} - 3530286 \beta_{16} - 3031892 \beta_{17} ) q^{57} -500246412961 \beta_{1} q^{58} + ( -252629484528699 + 170479329711 \beta_{1} - 2430978215 \beta_{2} + 1405994506 \beta_{3} + 30531320 \beta_{4} + 13378820 \beta_{5} - 799075 \beta_{6} - 13209403 \beta_{7} - 8869585 \beta_{8} + 1038371 \beta_{9} + 2901341 \beta_{10} + 8027881 \beta_{11} + 7541034 \beta_{12} - 6430293 \beta_{13} - 5562191 \beta_{14} + 5895698 \beta_{15} - 911107 \beta_{16} + 145924 \beta_{17} ) q^{59} + ( -767729748379551 + 1008228692544 \beta_{1} + 10836834902 \beta_{2} - 5512852758 \beta_{3} - 555012623 \beta_{4} - 60513226 \beta_{5} + 349839 \beta_{6} + 17405606 \beta_{7} + 7322379 \beta_{8} + 4878903 \beta_{9} + 948389 \beta_{10} - 7457233 \beta_{11} - 3976157 \beta_{12} + 7328056 \beta_{13} + 6894252 \beta_{14} + 3193487 \beta_{15} + 170127 \beta_{16} - 3253581 \beta_{17} ) q^{60} + ( -340056901421989 + 221917075653 \beta_{1} - 1617421194 \beta_{2} + 656065198 \beta_{3} + 236311884 \beta_{4} + 31685341 \beta_{5} - 7219680 \beta_{6} + 13839088 \beta_{7} + 5462441 \beta_{8} + 4465568 \beta_{9} + 1788808 \beta_{10} - 1715160 \beta_{11} - 17584388 \beta_{12} + 606932 \beta_{13} - 4454268 \beta_{14} - 3369456 \beta_{15} + 7040367 \beta_{16} + 3976320 \beta_{17} ) q^{61} + ( -566625540169766 + 730613128594 \beta_{1} + 13906354126 \beta_{2} - 3257770017 \beta_{3} - 705751084 \beta_{4} + 98977257 \beta_{5} - 2289636 \beta_{6} - 54915252 \beta_{7} + 6582555 \beta_{8} + 317823 \beta_{9} - 3484208 \beta_{10} + 3620342 \beta_{11} + 4558032 \beta_{12} - 10861345 \beta_{13} + 761885 \beta_{14} - 6584723 \beta_{15} + 7058203 \beta_{16} + 3377944 \beta_{17} ) q^{62} + ( -813447478459415 + 1218692036123 \beta_{1} + 48951813976 \beta_{2} + 2336350971 \beta_{3} + 532204365 \beta_{4} - 38216618 \beta_{5} - 1355859 \beta_{6} + 17091874 \beta_{7} + 11398436 \beta_{8} + 8041938 \beta_{9} + 3264746 \beta_{10} + 5188732 \beta_{11} + 1394375 \beta_{12} + 2043915 \beta_{13} - 3613488 \beta_{14} + 1757942 \beta_{15} - 19349774 \beta_{16} - 14211820 \beta_{17} ) q^{63} + ( -553510541629636 + 1481721809747 \beta_{1} + 45737328194 \beta_{2} - 541548384 \beta_{3} - 164053673 \beta_{4} + 151187355 \beta_{5} + 7888465 \beta_{6} - 80785859 \beta_{7} - 539873 \beta_{8} - 15310070 \beta_{9} - 7881941 \beta_{10} + 11111634 \beta_{11} + 14490865 \beta_{12} - 8165967 \beta_{13} + 5429178 \beta_{14} + 10601751 \beta_{15} - 6281051 \beta_{16} - 11676402 \beta_{17} ) q^{64} + ( -745600753740359 - 1052671192736 \beta_{1} + 21597051692 \beta_{2} + 2821985475 \beta_{3} + 949992301 \beta_{4} + 14673431 \beta_{5} - 3665000 \beta_{6} + 21483000 \beta_{7} - 36932733 \beta_{8} - 18286820 \beta_{9} - 3253190 \beta_{10} - 2682160 \beta_{11} + 3865770 \beta_{12} + 7922068 \beta_{13} - 8201258 \beta_{14} + 15040860 \beta_{15} - 12530589 \beta_{16} + 1668600 \beta_{17} ) q^{65} + ( -1540156599611631 + 1390614304088 \beta_{1} + 119306893636 \beta_{2} - 5903540471 \beta_{3} + 364099287 \beta_{4} + 52793854 \beta_{5} + 43795254 \beta_{6} - 2110829 \beta_{7} + 14648123 \beta_{8} - 5432127 \beta_{9} + 3826472 \beta_{10} + 5373734 \beta_{11} + 3031198 \beta_{12} - 12850424 \beta_{13} - 4696134 \beta_{14} - 20805487 \beta_{15} - 355616 \beta_{16} + 5701974 \beta_{17} ) q^{66} + ( -984223347682581 + 522721847505 \beta_{1} + 35597114162 \beta_{2} + 1606341287 \beta_{3} - 226473501 \beta_{4} + 124489564 \beta_{5} - 1175581 \beta_{6} - 3749038 \beta_{7} + 52790114 \beta_{8} + 24519204 \beta_{9} + 4993960 \beta_{10} - 24269298 \beta_{11} - 10157009 \beta_{12} + 9610721 \beta_{13} + 9041718 \beta_{14} - 29494602 \beta_{15} + 13608272 \beta_{16} + 11715532 \beta_{17} ) q^{67} + ( -1359550121720956 - 4660776116526 \beta_{1} + 7814147428 \beta_{2} - 5082862178 \beta_{3} + 433362558 \beta_{4} + 104435220 \beta_{5} - 38905714 \beta_{6} + 53633356 \beta_{7} - 3050790 \beta_{8} + 8760130 \beta_{9} - 2541838 \beta_{10} - 20450966 \beta_{11} - 32057306 \beta_{12} + 58105044 \beta_{13} + 7835272 \beta_{14} - 8687010 \beta_{15} + 18580806 \beta_{16} + 16769962 \beta_{17} ) q^{68} + ( -953341297826955 - 1271679812253 \beta_{1} + 55599674218 \beta_{2} + 9247790550 \beta_{3} + 575695216 \beta_{4} - 27536445 \beta_{5} + 13607226 \beta_{6} + 11108864 \beta_{7} - 17853829 \beta_{8} - 10921428 \beta_{9} + 13339866 \beta_{10} - 4314532 \beta_{11} + 26563132 \beta_{12} - 25897582 \beta_{13} - 14950422 \beta_{14} + 2134932 \beta_{15} - 14316077 \beta_{16} + 12907208 \beta_{17} ) q^{69} + ( 500358400500541 - 1267884001226 \beta_{1} + 23588727418 \beta_{2} + 22109278066 \beta_{3} + 649367277 \beta_{4} + 601654792 \beta_{5} - 23421652 \beta_{6} - 271323930 \beta_{7} - 38812267 \beta_{8} - 36410873 \beta_{9} - 8714098 \beta_{10} + 67649722 \beta_{11} + 27159344 \beta_{12} - 66987213 \beta_{13} - 17150395 \beta_{14} + 37536989 \beta_{15} + 8640627 \beta_{16} + 4171842 \beta_{17} ) q^{70} + ( -1048768275849676 - 2673427924164 \beta_{1} - 10447942318 \beta_{2} + 12194519460 \beta_{3} + 1344591040 \beta_{4} - 354904900 \beta_{5} - 29170912 \beta_{6} + 134026480 \beta_{7} - 5463840 \beta_{8} + 14338872 \beta_{9} - 16341696 \beta_{10} - 18614852 \beta_{11} - 8052760 \beta_{12} + 59303688 \beta_{13} + 46710052 \beta_{14} + 20690088 \beta_{15} + 2212704 \beta_{16} - 6091600 \beta_{17} ) q^{71} + ( -2473996992919326 - 5698701133286 \beta_{1} + 29794582486 \beta_{2} - 4623653566 \beta_{3} - 225342390 \beta_{4} - 121683066 \beta_{5} - 11556564 \beta_{6} + 112770074 \beta_{7} + 26894982 \beta_{8} + 46114344 \beta_{9} + 26472068 \beta_{10} - 15667160 \beta_{11} - 14252692 \beta_{12} + 25159740 \beta_{13} - 13747944 \beta_{14} - 16222540 \beta_{15} + 1155612 \beta_{16} - 3210856 \beta_{17} ) q^{72} + ( -646704359422903 + 643475337449 \beta_{1} - 122237444208 \beta_{2} + 11811148515 \beta_{3} - 914441669 \beta_{4} - 208146734 \beta_{5} + 11533223 \beta_{6} - 23544106 \beta_{7} + 29703386 \beta_{8} + 5246588 \beta_{9} - 24517282 \beta_{10} - 20523320 \beta_{11} - 13996561 \beta_{12} - 20814985 \beta_{13} + 16501032 \beta_{14} + 5999524 \beta_{15} + 18567492 \beta_{16} - 29452188 \beta_{17} ) q^{73} + ( -2980895911053190 - 4396740954086 \beta_{1} + 22640107470 \beta_{2} - 5667070400 \beta_{3} + 1368478414 \beta_{4} - 191233992 \beta_{5} + 84718320 \beta_{6} + 146465024 \beta_{7} - 98469964 \beta_{8} - 9992570 \beta_{9} + 456462 \beta_{10} + 15231980 \beta_{11} - 1823480 \beta_{12} - 40612380 \beta_{13} - 13417420 \beta_{14} + 15244934 \beta_{15} + 15936700 \beta_{16} + 2624286 \beta_{17} ) q^{74} + ( -1168501707396308 - 993244532470 \beta_{1} + 20977952773 \beta_{2} + 16407469675 \beta_{3} + 1439150526 \beta_{4} - 1066409943 \beta_{5} - 15637554 \beta_{6} + 208786394 \beta_{7} + 32167993 \beta_{8} - 19007853 \beta_{9} - 15393349 \beta_{10} + 20141893 \beta_{11} - 32781563 \beta_{12} + 18282882 \beta_{13} + 51796855 \beta_{14} - 9423402 \beta_{15} - 602351 \beta_{16} - 3913484 \beta_{17} ) q^{75} + ( -3673370303613124 - 9197597594628 \beta_{1} - 204380474992 \beta_{2} - 27250855536 \beta_{3} - 3830021376 \beta_{4} + 25475052 \beta_{5} + 21423312 \beta_{6} + 39610812 \beta_{7} - 53460784 \beta_{8} + 18449180 \beta_{9} + 19264424 \beta_{10} - 69542332 \beta_{11} + 23561968 \beta_{12} - 29278308 \beta_{13} - 58675552 \beta_{14} - 10196208 \beta_{15} - 49842504 \beta_{16} - 76230068 \beta_{17} ) q^{76} + ( -491138023560043 - 10948859557555 \beta_{1} - 289303663468 \beta_{2} + 22415825762 \beta_{3} - 1286051406 \beta_{4} - 83096403 \beta_{5} - 52359454 \beta_{6} - 79592084 \beta_{7} + 14764277 \beta_{8} + 37230674 \beta_{9} + 56604866 \beta_{10} + 48667678 \beta_{11} + 18483012 \beta_{12} - 15497490 \beta_{13} - 60315998 \beta_{14} - 7054540 \beta_{15} + 8333783 \beta_{16} + 9592 \beta_{17} ) q^{77} + ( -1600005349755034 - 5786717379852 \beta_{1} + 60565958864 \beta_{2} + 11175890603 \beta_{3} + 919442964 \beta_{4} - 440108019 \beta_{5} - 59088408 \beta_{6} - 36873960 \beta_{7} + 23836957 \beta_{8} - 36611553 \beta_{9} - 82160454 \beta_{10} + 44354874 \beta_{11} + 17447252 \beta_{12} - 35964615 \beta_{13} + 31593115 \beta_{14} - 24416135 \beta_{15} - 41401175 \beta_{16} + 4360058 \beta_{17} ) q^{78} + ( -2455322120523302 - 3051786765359 \beta_{1} - 255384448288 \beta_{2} - 11861431552 \beta_{3} - 4952104140 \beta_{4} - 19274776 \beta_{5} + 70145646 \beta_{6} - 229587795 \beta_{7} + 26978019 \beta_{8} - 42924727 \beta_{9} + 79664111 \beta_{10} + 26870856 \beta_{11} + 53760721 \beta_{12} - 54712302 \beta_{13} - 53105882 \beta_{14} - 40588951 \beta_{15} - 8045854 \beta_{16} - 4071738 \beta_{17} ) q^{79} + ( -8740251770550214 - 9860189745553 \beta_{1} - 98792847992 \beta_{2} - 51168269606 \beta_{3} + 242263411 \beta_{4} - 453040217 \beta_{5} - 22357705 \beta_{6} + 515537641 \beta_{7} - 87063905 \beta_{8} + 32289402 \beta_{9} + 31793093 \beta_{10} - 87709934 \beta_{11} - 79705529 \beta_{12} + 220454315 \beta_{13} + 67734542 \beta_{14} + 85697145 \beta_{15} + 20147171 \beta_{16} + 65967998 \beta_{17} ) q^{80} + ( -2307247235704109 + 3760140113662 \beta_{1} - 256096365915 \beta_{2} - 2422056596 \beta_{3} - 2496513099 \beta_{4} - 519213674 \beta_{5} + 113092971 \beta_{6} - 240668506 \beta_{7} + 46261292 \beta_{8} - 115829877 \beta_{9} - 64467552 \beta_{10} + 32776509 \beta_{11} + 9248571 \beta_{12} - 36329373 \beta_{13} + 10933008 \beta_{14} + 5899998 \beta_{15} + 80513665 \beta_{16} + 135460140 \beta_{17} ) q^{81} + ( -3635612562734131 - 3543447789036 \beta_{1} + 9537496719 \beta_{2} - 17307832963 \beta_{3} - 529032551 \beta_{4} + 1101714692 \beta_{5} + 47151514 \beta_{6} - 327748307 \beta_{7} + 108873408 \beta_{8} - 1485225 \beta_{9} - 24199045 \beta_{10} + 12147946 \beta_{11} + 29180362 \beta_{12} - 3658521 \beta_{13} - 6700381 \beta_{14} - 33001635 \beta_{15} - 33771033 \beta_{16} + 2002193 \beta_{17} ) q^{82} + ( -530157518133908 - 4797610319532 \beta_{1} - 569167020199 \beta_{2} + 8066407435 \beta_{3} - 3723862797 \beta_{4} + 778629538 \beta_{5} - 81787984 \beta_{6} - 223295497 \beta_{7} + 167935867 \beta_{8} + 139053285 \beta_{9} - 84008253 \beta_{10} - 75014147 \beta_{11} + 18705647 \beta_{12} + 21874508 \beta_{13} + 85161241 \beta_{14} - 148904052 \beta_{15} - 51525489 \beta_{16} - 44881352 \beta_{17} ) q^{83} + ( -3044202459372342 - 22164099155448 \beta_{1} - 141372473432 \beta_{2} - 23329018112 \beta_{3} - 1978143946 \beta_{4} + 673023204 \beta_{5} - 226280058 \beta_{6} + 41494892 \beta_{7} + 54221042 \beta_{8} + 62830698 \beta_{9} + 27465514 \beta_{10} - 114801470 \beta_{11} + 8547550 \beta_{12} + 41391332 \beta_{13} - 53300264 \beta_{14} - 65751994 \beta_{15} - 58530930 \beta_{16} + 75707666 \beta_{17} ) q^{84} + ( -1895402692028460 + 18353142157546 \beta_{1} - 788283641792 \beta_{2} + 11952558541 \beta_{3} - 2234952107 \beta_{4} + 1983825371 \beta_{5} + 42946621 \beta_{6} - 960471318 \beta_{7} - 76676617 \beta_{8} - 137979262 \beta_{9} - 49059450 \beta_{10} + 157401766 \beta_{11} - 90490795 \beta_{12} + 45268677 \beta_{13} - 19708284 \beta_{14} + 312716908 \beta_{15} + 121693117 \beta_{16} + 52205940 \beta_{17} ) q^{85} + ( -6646993875110231 + 10822373278614 \beta_{1} + 355177194501 \beta_{2} - 36881329608 \beta_{3} + 7383162643 \beta_{4} + 709606807 \beta_{5} + 12600184 \beta_{6} + 410441349 \beta_{7} - 218042813 \beta_{8} - 114726794 \beta_{9} + 127117899 \beta_{10} + 130847594 \beta_{11} - 9574120 \beta_{12} - 110505808 \beta_{13} + 15511170 \beta_{14} + 103158922 \beta_{15} - 17348346 \beta_{16} - 97912339 \beta_{17} ) q^{86} + ( 428210929494616 + 1000492825922 \beta_{1} - 500246412961 \beta_{2} ) q^{87} + ( -6351937511489984 + 7925817571567 \beta_{1} + 253714110328 \beta_{2} - 23921639164 \beta_{3} + 291041981 \beta_{4} + 1887021641 \beta_{5} - 129473039 \beta_{6} - 264059009 \beta_{7} + 420281257 \beta_{8} + 390138910 \beta_{9} + 78095915 \beta_{10} - 43686294 \beta_{11} - 54732375 \beta_{12} + 192388853 \beta_{13} + 15074726 \beta_{14} - 13820013 \beta_{15} - 66895479 \beta_{16} - 185235698 \beta_{17} ) q^{88} + ( -2696002008134211 + 18876978382831 \beta_{1} - 811455340393 \beta_{2} - 73574374289 \beta_{3} - 5412060196 \beta_{4} + 1342383446 \beta_{5} + 255119916 \beta_{6} + 80714682 \beta_{7} - 123783278 \beta_{8} + 102517593 \beta_{9} - 23778163 \beta_{10} - 106614483 \beta_{11} + 20212671 \beta_{12} - 417848966 \beta_{13} - 166082843 \beta_{14} - 346105070 \beta_{15} - 24686744 \beta_{16} - 297463076 \beta_{17} ) q^{89} + ( 3155466983545708 + 30301412586082 \beta_{1} + 1533812455533 \beta_{2} + 25968449863 \beta_{3} + 20463125856 \beta_{4} - 2478086732 \beta_{5} + 353958300 \beta_{6} + 612992765 \beta_{7} - 132011959 \beta_{8} - 354486300 \beta_{9} - 210782625 \beta_{10} + 140203260 \beta_{11} + 25225260 \beta_{12} - 263893806 \beta_{13} + 185896746 \beta_{14} + 57897090 \beta_{15} + 177575938 \beta_{16} + 122023155 \beta_{17} ) q^{90} + ( -5820560871980214 + 7766158643254 \beta_{1} - 569621999355 \beta_{2} - 108105792613 \beta_{3} - 10308296501 \beta_{4} + 749334834 \beta_{5} - 23678474 \beta_{6} + 45625125 \beta_{7} - 408111047 \beta_{8} - 66943153 \beta_{9} + 183270217 \beta_{10} - 138665729 \beta_{11} + 150917547 \beta_{12} + 146744310 \beta_{13} - 43893013 \beta_{14} + 59147248 \beta_{15} - 141470887 \beta_{16} - 125894992 \beta_{17} ) q^{91} + ( -186161833074688 - 8633427608134 \beta_{1} + 979240642556 \beta_{2} + 6964641550 \beta_{3} + 13657233306 \beta_{4} - 213437428 \beta_{5} - 73561110 \beta_{6} + 1152483564 \beta_{7} - 148147962 \beta_{8} + 244637654 \beta_{9} + 14153966 \beta_{10} - 313727122 \beta_{11} + 71479154 \beta_{12} + 25503868 \beta_{13} - 13268408 \beta_{14} - 55513910 \beta_{15} - 129779038 \beta_{16} - 113236506 \beta_{17} ) q^{92} + ( 7119539541832971 + 13231516174614 \beta_{1} - 277507667403 \beta_{2} + 52652609049 \beta_{3} + 2427163678 \beta_{4} - 1602524819 \beta_{5} - 270835673 \beta_{6} - 177439964 \beta_{7} - 104112787 \beta_{8} - 85991481 \beta_{9} + 114239937 \beta_{10} + 307737823 \beta_{11} - 49868936 \beta_{12} + 135545301 \beta_{13} - 184000969 \beta_{14} + 233583746 \beta_{15} + 229526939 \beta_{16} + 145845464 \beta_{17} ) q^{93} + ( 4611907244908174 - 983534817752 \beta_{1} + 2214520108285 \beta_{2} - 50819918954 \beta_{3} + 7691589900 \beta_{4} - 1882155875 \beta_{5} - 74152484 \beta_{6} + 634290937 \beta_{7} + 37547624 \beta_{8} - 258069789 \beta_{9} + 24710647 \beta_{10} - 230234264 \beta_{11} - 135936232 \beta_{12} + 187225969 \beta_{13} + 162697617 \beta_{14} - 384966259 \beta_{15} + 264059967 \beta_{16} + 462800141 \beta_{17} ) q^{94} + ( 11112199677091421 + 31086590816188 \beta_{1} + 336466109350 \beta_{2} + 199110606082 \beta_{3} + 15882729556 \beta_{4} + 2969159912 \beta_{5} - 256264215 \beta_{6} - 57797012 \beta_{7} - 450251544 \beta_{8} + 114991966 \beta_{9} - 3022026 \beta_{10} + 205531603 \beta_{11} + 30202903 \beta_{12} - 282273901 \beta_{13} + 11327017 \beta_{14} + 198605515 \beta_{15} - 24560679 \beta_{16} + 452981634 \beta_{17} ) q^{95} + ( 7796558342174546 + 28486212950825 \beta_{1} + 3002477849712 \beta_{2} + 126830089994 \beta_{3} + 17818733537 \beta_{4} - 7730868371 \beta_{5} + 77745205 \beta_{6} + 624170451 \beta_{7} + 588755269 \beta_{8} + 210816126 \beta_{9} + 96520527 \beta_{10} + 278259374 \beta_{11} + 126111669 \beta_{12} + 367500769 \beta_{13} - 92953102 \beta_{14} + 213535139 \beta_{15} - 132888159 \beta_{16} - 247436046 \beta_{17} ) q^{96} + ( 2963753341191358 + 22774664445284 \beta_{1} - 504396735811 \beta_{2} - 24542948820 \beta_{3} - 10353657129 \beta_{4} + 102712826 \beta_{5} - 370731087 \beta_{6} - 248045972 \beta_{7} + 770363866 \beta_{8} + 534858799 \beta_{9} - 334150297 \beta_{10} - 366265881 \beta_{11} - 426596700 \beta_{12} + 1179279003 \beta_{13} + 158642033 \beta_{14} - 102442094 \beta_{15} + 102696632 \beta_{16} + 216691488 \beta_{17} ) q^{97} + ( 24907321901842566 + 111241216079065 \beta_{1} + 4413170189926 \beta_{2} + 253793473982 \beta_{3} + 31691497974 \beta_{4} - 2435772936 \beta_{5} + 772662784 \beta_{6} - 548325670 \beta_{7} + 79913044 \beta_{8} - 726547094 \beta_{9} - 307405970 \beta_{10} + 619433440 \beta_{11} + 193979608 \beta_{12} - 1051117738 \beta_{13} + 152404510 \beta_{14} - 117065098 \beta_{15} + 207649482 \beta_{16} + 157974234 \beta_{17} ) q^{98} + ( 6553432951568750 + 98330514629635 \beta_{1} + 502077105467 \beta_{2} - 73759036872 \beta_{3} - 12496143351 \beta_{4} - 8656284619 \beta_{5} + 442891140 \beta_{6} + 1596412532 \beta_{7} - 1402463861 \beta_{8} - 870738303 \beta_{9} - 422124575 \beta_{10} - 263608408 \beta_{11} + 212072593 \beta_{12} - 553276680 \beta_{13} + 314555556 \beta_{14} + 367773985 \beta_{15} - 453017188 \beta_{16} - 137479418 \beta_{17} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q - 15400q^{3} + 862698q^{4} - 564228q^{5} - 6666170q^{6} - 43925040q^{7} + 86936640q^{8} + 488532554q^{9} + O(q^{10}) \) \( 18q - 15400q^{3} + 862698q^{4} - 564228q^{5} - 6666170q^{6} - 43925040q^{7} + 86936640q^{8} + 488532554q^{9} - 1301706588q^{10} + 414318256q^{11} + 4613809340q^{12} - 1708529620q^{13} - 10178671680q^{14} - 35937136948q^{15} + 13408243234q^{16} - 31137019060q^{17} - 216144895280q^{18} - 236294644572q^{19} - 343491571178q^{20} + 292681980344q^{21} + 237072099770q^{22} + 448660830360q^{23} + 1331075294514q^{24} + 3016314845934q^{25} + 4625052436620q^{26} - 3633286593580q^{27} - 5255043772340q^{28} - 9004435433298q^{29} + 11322123726866q^{30} + 4286667897456q^{31} + 20489566928480q^{32} + 12272773628920q^{33} - 29135914295852q^{34} - 34335586657384q^{35} - 34363200450796q^{36} - 33745027570060q^{37} - 96773461186360q^{38} - 104536576294796q^{39} - 136020881729180q^{40} - 62894681812676q^{41} - 363718470035260q^{42} + 43558449431040q^{43} - 49608048285572q^{44} + 133812803620916q^{45} - 219540697042836q^{46} - 141597817069240q^{47} - 267256681151460q^{48} + 453054608269810q^{49} - 1348659632272640q^{50} - 1366163894523096q^{51} - 1312273323620630q^{52} - 485951074130100q^{53} - 1795598060978246q^{54} - 3426804650590972q^{55} - 5564818824556288q^{56} - 5205401672025040q^{57} - 4547356113106800q^{59} - 13819026055113000q^{60} - 6121039899052148q^{61} - 10199136673985490q^{62} - 14641672030091800q^{63} - 9962823646631806q^{64} - 13420652091655824q^{65} - 27721841054867288q^{66} - 17715742779810920q^{67} - 24471819430698220q^{68} - 17159735572757960q^{69} + 9006543780480744q^{70} - 18877957007949512q^{71} - 44531687124196680q^{72} - 11641701802444700q^{73} - 53655919274849056q^{74} - 21032918050227748q^{75} - 66122191206313032q^{76} - 8842889451812280q^{77} - 28799653307053650q^{78} - 44197796400782136q^{79} - 157325108946937442q^{80} - 41532487375317502q^{81} - 65440892426610260q^{82} - 9547427423528400q^{83} - 54796688181105048q^{84} - 34123627920307712q^{85} - 119642903919749046q^{86} + 7703794759599400q^{87} - 114332767598176050q^{88} - 48534239108944036q^{89} + 56810599078278752q^{90} - 104774225723277192q^{91} - 3343093822255980q^{92} + 128149288210966200q^{93} + 83032272917645478q^{94} + 200021462885519192q^{95} + 140361624410708610q^{96} + 53343624485398380q^{97} + 448366099002416160q^{98} + 117966192294311356q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{18} - 1610997 x^{16} - 28978880 x^{15} + 1054878119348 x^{14} + 33471007935200 x^{13} - 361118629330503872 x^{12} - 14867597249727884800 x^{11} + 69075461759211999549440 x^{10} + 3136429534292709271797760 x^{9} - 7293684107327839501793427456 x^{8} - 310705174164268019482139033600 x^{7} + 392089226833876108917621043232768 x^{6} + 11987811702756552456984769970831360 x^{5} - 8875073388959894229615422940880306176 x^{4} - 110123126416280653106554268008725872640 x^{3} + 38261887665727248243207924189998746697728 x^{2} - 570869901591701973911550639846099485982720 x - 1366002559879573583573099965753694433050624\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(16\!\cdots\!43\)\( \nu^{17} + \)\(46\!\cdots\!60\)\( \nu^{16} + \)\(29\!\cdots\!67\)\( \nu^{15} - \)\(67\!\cdots\!20\)\( \nu^{14} - \)\(20\!\cdots\!48\)\( \nu^{13} + \)\(39\!\cdots\!00\)\( \nu^{12} + \)\(79\!\cdots\!52\)\( \nu^{11} - \)\(11\!\cdots\!40\)\( \nu^{10} - \)\(17\!\cdots\!24\)\( \nu^{9} + \)\(16\!\cdots\!80\)\( \nu^{8} + \)\(21\!\cdots\!96\)\( \nu^{7} - \)\(99\!\cdots\!80\)\( \nu^{6} - \)\(14\!\cdots\!16\)\( \nu^{5} + \)\(32\!\cdots\!00\)\( \nu^{4} + \)\(37\!\cdots\!20\)\( \nu^{3} + \)\(96\!\cdots\!20\)\( \nu^{2} - \)\(17\!\cdots\!04\)\( \nu - \)\(84\!\cdots\!20\)\(\)\()/ \)\(19\!\cdots\!80\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(16\!\cdots\!43\)\( \nu^{17} - \)\(46\!\cdots\!60\)\( \nu^{16} - \)\(29\!\cdots\!67\)\( \nu^{15} + \)\(67\!\cdots\!20\)\( \nu^{14} + \)\(20\!\cdots\!48\)\( \nu^{13} - \)\(39\!\cdots\!00\)\( \nu^{12} - \)\(79\!\cdots\!52\)\( \nu^{11} + \)\(11\!\cdots\!40\)\( \nu^{10} + \)\(17\!\cdots\!24\)\( \nu^{9} - \)\(16\!\cdots\!80\)\( \nu^{8} - \)\(21\!\cdots\!96\)\( \nu^{7} + \)\(99\!\cdots\!80\)\( \nu^{6} + \)\(14\!\cdots\!16\)\( \nu^{5} - \)\(32\!\cdots\!00\)\( \nu^{4} - \)\(37\!\cdots\!20\)\( \nu^{3} - \)\(86\!\cdots\!80\)\( \nu^{2} + \)\(16\!\cdots\!24\)\( \nu - \)\(16\!\cdots\!40\)\(\)\()/ \)\(98\!\cdots\!40\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(15\!\cdots\!77\)\( \nu^{17} - \)\(44\!\cdots\!60\)\( \nu^{16} - \)\(24\!\cdots\!13\)\( \nu^{15} + \)\(19\!\cdots\!80\)\( \nu^{14} + \)\(15\!\cdots\!92\)\( \nu^{13} + \)\(14\!\cdots\!20\)\( \nu^{12} - \)\(53\!\cdots\!88\)\( \nu^{11} - \)\(11\!\cdots\!20\)\( \nu^{10} + \)\(10\!\cdots\!76\)\( \nu^{9} + \)\(30\!\cdots\!80\)\( \nu^{8} - \)\(10\!\cdots\!24\)\( \nu^{7} - \)\(32\!\cdots\!60\)\( \nu^{6} + \)\(56\!\cdots\!44\)\( \nu^{5} + \)\(11\!\cdots\!40\)\( \nu^{4} - \)\(12\!\cdots\!00\)\( \nu^{3} - \)\(81\!\cdots\!00\)\( \nu^{2} + \)\(60\!\cdots\!76\)\( \nu - \)\(29\!\cdots\!20\)\(\)\()/ \)\(65\!\cdots\!60\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(77\!\cdots\!79\)\( \nu^{17} + \)\(20\!\cdots\!56\)\( \nu^{16} - \)\(12\!\cdots\!71\)\( \nu^{15} - \)\(33\!\cdots\!24\)\( \nu^{14} + \)\(74\!\cdots\!44\)\( \nu^{13} + \)\(21\!\cdots\!16\)\( \nu^{12} - \)\(23\!\cdots\!36\)\( \nu^{11} - \)\(69\!\cdots\!24\)\( \nu^{10} + \)\(37\!\cdots\!72\)\( \nu^{9} + \)\(11\!\cdots\!48\)\( \nu^{8} - \)\(28\!\cdots\!88\)\( \nu^{7} - \)\(93\!\cdots\!92\)\( \nu^{6} + \)\(73\!\cdots\!68\)\( \nu^{5} + \)\(26\!\cdots\!72\)\( \nu^{4} + \)\(15\!\cdots\!80\)\( \nu^{3} - \)\(50\!\cdots\!20\)\( \nu^{2} + \)\(14\!\cdots\!12\)\( \nu + \)\(16\!\cdots\!48\)\(\)\()/ \)\(19\!\cdots\!80\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(13\!\cdots\!55\)\( \nu^{17} - \)\(54\!\cdots\!28\)\( \nu^{16} - \)\(19\!\cdots\!55\)\( \nu^{15} + \)\(74\!\cdots\!52\)\( \nu^{14} + \)\(12\!\cdots\!20\)\( \nu^{13} - \)\(40\!\cdots\!88\)\( \nu^{12} - \)\(39\!\cdots\!20\)\( \nu^{11} + \)\(10\!\cdots\!72\)\( \nu^{10} + \)\(73\!\cdots\!60\)\( \nu^{9} - \)\(12\!\cdots\!84\)\( \nu^{8} - \)\(81\!\cdots\!20\)\( \nu^{7} + \)\(46\!\cdots\!16\)\( \nu^{6} + \)\(49\!\cdots\!80\)\( \nu^{5} + \)\(32\!\cdots\!04\)\( \nu^{4} - \)\(11\!\cdots\!00\)\( \nu^{3} - \)\(18\!\cdots\!80\)\( \nu^{2} - \)\(20\!\cdots\!00\)\( \nu + \)\(22\!\cdots\!96\)\(\)\()/ \)\(10\!\cdots\!60\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(63\!\cdots\!75\)\( \nu^{17} + \)\(50\!\cdots\!36\)\( \nu^{16} + \)\(10\!\cdots\!15\)\( \nu^{15} - \)\(59\!\cdots\!64\)\( \nu^{14} - \)\(68\!\cdots\!80\)\( \nu^{13} + \)\(26\!\cdots\!56\)\( \nu^{12} + \)\(23\!\cdots\!80\)\( \nu^{11} - \)\(50\!\cdots\!04\)\( \nu^{10} - \)\(47\!\cdots\!60\)\( \nu^{9} + \)\(27\!\cdots\!48\)\( \nu^{8} + \)\(52\!\cdots\!40\)\( \nu^{7} + \)\(41\!\cdots\!48\)\( \nu^{6} - \)\(29\!\cdots\!20\)\( \nu^{5} - \)\(56\!\cdots\!48\)\( \nu^{4} + \)\(70\!\cdots\!00\)\( \nu^{3} + \)\(17\!\cdots\!60\)\( \nu^{2} - \)\(22\!\cdots\!80\)\( \nu + \)\(15\!\cdots\!28\)\(\)\()/ \)\(19\!\cdots\!80\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(31\!\cdots\!08\)\( \nu^{17} + \)\(46\!\cdots\!05\)\( \nu^{16} + \)\(51\!\cdots\!92\)\( \nu^{15} + \)\(22\!\cdots\!35\)\( \nu^{14} - \)\(33\!\cdots\!28\)\( \nu^{13} - \)\(64\!\cdots\!60\)\( \nu^{12} + \)\(11\!\cdots\!12\)\( \nu^{11} + \)\(34\!\cdots\!00\)\( \nu^{10} - \)\(21\!\cdots\!24\)\( \nu^{9} - \)\(81\!\cdots\!80\)\( \nu^{8} + \)\(22\!\cdots\!16\)\( \nu^{7} + \)\(88\!\cdots\!20\)\( \nu^{6} - \)\(12\!\cdots\!76\)\( \nu^{5} - \)\(39\!\cdots\!20\)\( \nu^{4} + \)\(26\!\cdots\!00\)\( \nu^{3} + \)\(56\!\cdots\!80\)\( \nu^{2} - \)\(10\!\cdots\!84\)\( \nu + \)\(13\!\cdots\!80\)\(\)\()/ \)\(82\!\cdots\!20\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(27\!\cdots\!67\)\( \nu^{17} + \)\(12\!\cdots\!28\)\( \nu^{16} + \)\(44\!\cdots\!63\)\( \nu^{15} - \)\(12\!\cdots\!32\)\( \nu^{14} - \)\(30\!\cdots\!32\)\( \nu^{13} + \)\(28\!\cdots\!08\)\( \nu^{12} + \)\(10\!\cdots\!68\)\( \nu^{11} + \)\(53\!\cdots\!68\)\( \nu^{10} - \)\(21\!\cdots\!96\)\( \nu^{9} - \)\(35\!\cdots\!76\)\( \nu^{8} + \)\(24\!\cdots\!44\)\( \nu^{7} + \)\(59\!\cdots\!04\)\( \nu^{6} - \)\(14\!\cdots\!64\)\( \nu^{5} - \)\(40\!\cdots\!24\)\( \nu^{4} + \)\(34\!\cdots\!60\)\( \nu^{3} + \)\(85\!\cdots\!40\)\( \nu^{2} - \)\(13\!\cdots\!76\)\( \nu + \)\(16\!\cdots\!04\)\(\)\()/ \)\(19\!\cdots\!80\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(33\!\cdots\!95\)\( \nu^{17} - \)\(20\!\cdots\!28\)\( \nu^{16} + \)\(54\!\cdots\!55\)\( \nu^{15} + \)\(40\!\cdots\!72\)\( \nu^{14} - \)\(34\!\cdots\!00\)\( \nu^{13} - \)\(29\!\cdots\!48\)\( \nu^{12} + \)\(11\!\cdots\!20\)\( \nu^{11} + \)\(10\!\cdots\!72\)\( \nu^{10} - \)\(21\!\cdots\!60\)\( \nu^{9} - \)\(19\!\cdots\!84\)\( \nu^{8} + \)\(22\!\cdots\!80\)\( \nu^{7} + \)\(18\!\cdots\!76\)\( \nu^{6} - \)\(11\!\cdots\!40\)\( \nu^{5} - \)\(69\!\cdots\!56\)\( \nu^{4} + \)\(23\!\cdots\!40\)\( \nu^{3} + \)\(73\!\cdots\!00\)\( \nu^{2} - \)\(89\!\cdots\!80\)\( \nu + \)\(52\!\cdots\!56\)\(\)\()/ \)\(19\!\cdots\!80\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(53\!\cdots\!81\)\( \nu^{17} + \)\(71\!\cdots\!14\)\( \nu^{16} - \)\(85\!\cdots\!69\)\( \nu^{15} - \)\(12\!\cdots\!86\)\( \nu^{14} + \)\(54\!\cdots\!56\)\( \nu^{13} + \)\(87\!\cdots\!44\)\( \nu^{12} - \)\(18\!\cdots\!64\)\( \nu^{11} - \)\(30\!\cdots\!36\)\( \nu^{10} + \)\(32\!\cdots\!88\)\( \nu^{9} + \)\(55\!\cdots\!12\)\( \nu^{8} - \)\(31\!\cdots\!12\)\( \nu^{7} - \)\(50\!\cdots\!88\)\( \nu^{6} + \)\(14\!\cdots\!12\)\( \nu^{5} + \)\(19\!\cdots\!08\)\( \nu^{4} - \)\(26\!\cdots\!60\)\( \nu^{3} - \)\(24\!\cdots\!20\)\( \nu^{2} + \)\(89\!\cdots\!68\)\( \nu + \)\(33\!\cdots\!92\)\(\)\()/ \)\(24\!\cdots\!60\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(56\!\cdots\!91\)\( \nu^{17} + \)\(10\!\cdots\!68\)\( \nu^{16} + \)\(91\!\cdots\!79\)\( \nu^{15} + \)\(15\!\cdots\!68\)\( \nu^{14} - \)\(60\!\cdots\!36\)\( \nu^{13} - \)\(19\!\cdots\!92\)\( \nu^{12} + \)\(20\!\cdots\!84\)\( \nu^{11} + \)\(89\!\cdots\!28\)\( \nu^{10} - \)\(40\!\cdots\!48\)\( \nu^{9} - \)\(20\!\cdots\!96\)\( \nu^{8} + \)\(43\!\cdots\!12\)\( \nu^{7} + \)\(22\!\cdots\!04\)\( \nu^{6} - \)\(23\!\cdots\!92\)\( \nu^{5} - \)\(11\!\cdots\!24\)\( \nu^{4} + \)\(52\!\cdots\!80\)\( \nu^{3} + \)\(20\!\cdots\!20\)\( \nu^{2} - \)\(13\!\cdots\!68\)\( \nu - \)\(10\!\cdots\!36\)\(\)\()/ \)\(19\!\cdots\!80\)\( \)
\(\beta_{13}\)\(=\)\((\)\(\)\(28\!\cdots\!01\)\( \nu^{17} + \)\(17\!\cdots\!00\)\( \nu^{16} - \)\(45\!\cdots\!49\)\( \nu^{15} - \)\(36\!\cdots\!60\)\( \nu^{14} + \)\(29\!\cdots\!76\)\( \nu^{13} + \)\(27\!\cdots\!60\)\( \nu^{12} - \)\(98\!\cdots\!44\)\( \nu^{11} - \)\(10\!\cdots\!80\)\( \nu^{10} + \)\(18\!\cdots\!68\)\( \nu^{9} + \)\(19\!\cdots\!00\)\( \nu^{8} - \)\(18\!\cdots\!52\)\( \nu^{7} - \)\(18\!\cdots\!40\)\( \nu^{6} + \)\(91\!\cdots\!32\)\( \nu^{5} + \)\(72\!\cdots\!00\)\( \nu^{4} - \)\(18\!\cdots\!40\)\( \nu^{3} - \)\(88\!\cdots\!20\)\( \nu^{2} + \)\(74\!\cdots\!88\)\( \nu - \)\(21\!\cdots\!20\)\(\)\()/ \)\(98\!\cdots\!40\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(60\!\cdots\!87\)\( \nu^{17} - \)\(42\!\cdots\!12\)\( \nu^{16} + \)\(96\!\cdots\!83\)\( \nu^{15} + \)\(82\!\cdots\!68\)\( \nu^{14} - \)\(61\!\cdots\!52\)\( \nu^{13} - \)\(60\!\cdots\!52\)\( \nu^{12} + \)\(20\!\cdots\!28\)\( \nu^{11} + \)\(21\!\cdots\!48\)\( \nu^{10} - \)\(38\!\cdots\!76\)\( \nu^{9} - \)\(39\!\cdots\!56\)\( \nu^{8} + \)\(38\!\cdots\!04\)\( \nu^{7} + \)\(35\!\cdots\!04\)\( \nu^{6} - \)\(18\!\cdots\!64\)\( \nu^{5} - \)\(12\!\cdots\!24\)\( \nu^{4} + \)\(39\!\cdots\!40\)\( \nu^{3} + \)\(10\!\cdots\!80\)\( \nu^{2} - \)\(16\!\cdots\!56\)\( \nu + \)\(10\!\cdots\!84\)\(\)\()/ \)\(19\!\cdots\!80\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-\)\(93\!\cdots\!13\)\( \nu^{17} - \)\(32\!\cdots\!44\)\( \nu^{16} + \)\(15\!\cdots\!97\)\( \nu^{15} + \)\(75\!\cdots\!56\)\( \nu^{14} - \)\(98\!\cdots\!48\)\( \nu^{13} - \)\(60\!\cdots\!24\)\( \nu^{12} + \)\(33\!\cdots\!52\)\( \nu^{11} + \)\(22\!\cdots\!56\)\( \nu^{10} - \)\(63\!\cdots\!24\)\( \nu^{9} - \)\(43\!\cdots\!32\)\( \nu^{8} + \)\(66\!\cdots\!76\)\( \nu^{7} + \)\(41\!\cdots\!48\)\( \nu^{6} - \)\(35\!\cdots\!76\)\( \nu^{5} - \)\(16\!\cdots\!48\)\( \nu^{4} + \)\(79\!\cdots\!60\)\( \nu^{3} + \)\(20\!\cdots\!40\)\( \nu^{2} - \)\(34\!\cdots\!04\)\( \nu + \)\(14\!\cdots\!88\)\(\)\()/ \)\(28\!\cdots\!40\)\( \)
\(\beta_{16}\)\(=\)\((\)\(\)\(30\!\cdots\!69\)\( \nu^{17} + \)\(34\!\cdots\!24\)\( \nu^{16} - \)\(49\!\cdots\!41\)\( \nu^{15} - \)\(14\!\cdots\!76\)\( \nu^{14} + \)\(32\!\cdots\!84\)\( \nu^{13} + \)\(14\!\cdots\!24\)\( \nu^{12} - \)\(11\!\cdots\!36\)\( \nu^{11} - \)\(61\!\cdots\!76\)\( \nu^{10} + \)\(21\!\cdots\!92\)\( \nu^{9} + \)\(12\!\cdots\!52\)\( \nu^{8} - \)\(22\!\cdots\!68\)\( \nu^{7} - \)\(12\!\cdots\!48\)\( \nu^{6} + \)\(11\!\cdots\!08\)\( \nu^{5} + \)\(52\!\cdots\!08\)\( \nu^{4} - \)\(26\!\cdots\!60\)\( \nu^{3} - \)\(63\!\cdots\!00\)\( \nu^{2} + \)\(10\!\cdots\!32\)\( \nu - \)\(73\!\cdots\!68\)\(\)\()/ \)\(89\!\cdots\!40\)\( \)
\(\beta_{17}\)\(=\)\((\)\(-\)\(18\!\cdots\!97\)\( \nu^{17} - \)\(23\!\cdots\!56\)\( \nu^{16} + \)\(29\!\cdots\!25\)\( \nu^{15} + \)\(90\!\cdots\!04\)\( \nu^{14} - \)\(19\!\cdots\!52\)\( \nu^{13} - \)\(85\!\cdots\!24\)\( \nu^{12} + \)\(65\!\cdots\!88\)\( \nu^{11} + \)\(35\!\cdots\!00\)\( \nu^{10} - \)\(12\!\cdots\!12\)\( \nu^{9} - \)\(72\!\cdots\!72\)\( \nu^{8} + \)\(13\!\cdots\!24\)\( \nu^{7} + \)\(72\!\cdots\!08\)\( \nu^{6} - \)\(70\!\cdots\!84\)\( \nu^{5} - \)\(29\!\cdots\!84\)\( \nu^{4} + \)\(15\!\cdots\!40\)\( \nu^{3} + \)\(33\!\cdots\!96\)\( \nu^{2} - \)\(62\!\cdots\!16\)\( \nu + \)\(66\!\cdots\!28\)\(\)\()/ \)\(39\!\cdots\!36\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 2 \beta_{2} + 27 \beta_{1} + 178999\)
\(\nu^{3}\)\(=\)\(\beta_{8} - \beta_{7} + \beta_{5} + 3 \beta_{4} + 57 \beta_{3} + 599 \beta_{2} + 301435 \beta_{1} + 4829561\)
\(\nu^{4}\)\(=\)\(22 \beta_{17} - 15 \beta_{16} + 27 \beta_{15} - 22 \beta_{14} - 227 \beta_{13} + 5 \beta_{12} + 154 \beta_{11} - 73 \beta_{10} - 70 \beta_{9} - 45 \beta_{8} - 47 \beta_{7} + 85 \beta_{6} + 807 \beta_{5} + 8971 \beta_{4} + 412300 \beta_{3} + 717858 \beta_{2} + 20507411 \beta_{1} + 53950339128\)
\(\nu^{5}\)\(=\)\(6502 \beta_{17} + 26222 \beta_{16} + 14142 \beta_{15} + 13460 \beta_{14} - 55452 \beta_{13} + 39858 \beta_{12} + 50570 \beta_{11} - 33766 \beta_{10} - 25338 \beta_{9} + 499425 \beta_{8} - 722919 \beta_{7} + 18322 \beta_{6} + 844339 \beta_{5} + 2171631 \beta_{4} + 44085127 \beta_{3} + 364740329 \beta_{2} + 103954434709 \beta_{1} + 3670371007985\)
\(\nu^{6}\)\(=\)\(2741518 \beta_{17} - 16111451 \beta_{16} + 28296471 \beta_{15} - 8988742 \beta_{14} - 156932687 \beta_{13} + 17767665 \beta_{12} + 112037074 \beta_{11} - 55723221 \beta_{10} - 61185270 \beta_{9} - 30031073 \beta_{8} - 111587779 \beta_{7} + 63594065 \beta_{6} + 680062875 \beta_{5} + 5715180887 \beta_{4} + 166584164512 \beta_{3} + 310034316866 \beta_{2} + 12138319874899 \beta_{1} + 18604107098580796\)
\(\nu^{7}\)\(=\)\(3257270462 \beta_{17} + 16924258150 \beta_{16} + 14194420086 \beta_{15} + 7254310276 \beta_{14} - 53692095596 \beta_{13} + 31466629210 \beta_{12} + 46183694034 \beta_{11} - 26054319230 \beta_{10} - 24524212642 \beta_{9} + 208172355169 \beta_{8} - 408090734911 \beta_{7} + 18201583994 \beta_{6} + 542466300635 \beta_{5} + 1348304462847 \beta_{4} + 27509823957215 \beta_{3} + 195367996045745 \beta_{2} + 38807499771006717 \beta_{1} + 2172227124583392913\)
\(\nu^{8}\)\(=\)\(-2096037017234 \beta_{17} - 9462019876619 \beta_{16} + 17586320843399 \beta_{15} - 3038311388294 \beta_{14} - 86485179236095 \beta_{13} + 15080037777569 \beta_{12} + 63690935079474 \beta_{11} - 30692957591301 \beta_{10} - 36746540084662 \beta_{9} - 12425725487057 \beta_{8} - 98921927398131 \beta_{7} + 36428566334273 \beta_{6} + 413178791712587 \beta_{5} + 2894102870378279 \beta_{4} + 68904376593057744 \beta_{3} + 145774796633839458 \beta_{2} + 6696189851272215155 \beta_{1} + 6944619437812280315436\)
\(\nu^{9}\)\(=\)\(746776349354846 \beta_{17} + 7659830827259430 \beta_{16} + 9623865302697462 \beta_{15} + 2739396199784964 \beta_{14} - 36138050389379660 \beta_{13} + 18731815396921178 \beta_{12} + 30290238626231986 \beta_{11} - 15031060004191422 \beta_{10} - 16585381597249922 \beta_{9} + 83005393348839169 \beta_{8} - 209985046870137279 \beta_{7} + 12664320461118330 \beta_{6} + 304824939669408667 \beta_{5} + 787875768050192863 \beta_{4} + 15718917613087902143 \beta_{3} + 98669697704802665105 \beta_{2} + 15338552190038103591485 \beta_{1} + 1198151719066378043095825\)
\(\nu^{10}\)\(=\)\(-2212971802671486930 \beta_{17} - 4719030764488811915 \beta_{16} + 9484255662945015431 \beta_{15} - 1131133540093787142 \beta_{14} - 44247642368680973887 \beta_{13} + 9662574598989003553 \beta_{12} + 33242797871664444530 \beta_{11} - 15239381488655870213 \beta_{10} - 19424992409154635638 \beta_{9} - 3825797393893466033 \beta_{8} - 66511274712988728531 \beta_{7} + 18940618412073057985 \beta_{6} + 222435409249996934571 \beta_{5} + 1363988378429430215815 \beta_{4} + 29266814889405022426544 \beta_{3} + 70116022388975456821442 \beta_{2} + 3575658695557017009020499 \beta_{1} + 2744663719631879086257191884\)
\(\nu^{11}\)\(=\)\(-208911805051130180578 \beta_{17} + 2937017998582930048134 \beta_{16} + 5601282649634391268630 \beta_{15} + 844505930226437874884 \beta_{14} - 21037106463189113341932 \beta_{13} + 10045634741259859025210 \beta_{12} + 17405251413742847570930 \beta_{11} - 7871605786891235373854 \beta_{10} - 9652840559921071616578 \beta_{9} + 32851870890250217717153 \beta_{8} - 103499757965604622474079 \beta_{7} + 7571236280838749059418 \beta_{6} + 159288334099962167796539 \beta_{5} + 437207602160137977828991 \beta_{4} + 8539807563512651132351807 \beta_{3} + 48418959290068145283683793 \beta_{2} + 6333040383152855749407635677 \beta_{1} + 639734873250637953743618967953\)
\(\nu^{12}\)\(=\)\(-1476605295500288676938898 \beta_{17} - 2221486478841218160723019 \beta_{16} + 4850095354674449317547975 \beta_{15} - 502916698990535845673606 \beta_{14} - 21898212930870175447391679 \beta_{13} + 5503549568755159875027425 \beta_{12} + 16683909148617436462175026 \beta_{11} - 7278800430876011609983301 \beta_{10} - 9719306208624068131244598 \beta_{9} - 661095462108893218824017 \beta_{8} - 39367581031674168400652403 \beta_{7} + 9411671930036350102340993 \beta_{6} + 113281843831287225172989899 \beta_{5} + 626879763626475218485042727 \beta_{4} + 12727338017768114288160535888 \beta_{3} + 33763259031292994550869183522 \beta_{2} + 1868420717308717794338232504883 \beta_{1} + 1133169659392693447264636184355884\)
\(\nu^{13}\)\(=\)\(-\)\(38\!\cdots\!06\)\( \beta_{17} + \)\(99\!\cdots\!78\)\( \beta_{16} + \)\(30\!\cdots\!70\)\( \beta_{15} + \)\(19\!\cdots\!32\)\( \beta_{14} - \)\(11\!\cdots\!84\)\( \beta_{13} + \)\(51\!\cdots\!98\)\( \beta_{12} + \)\(93\!\cdots\!10\)\( \beta_{11} - \)\(39\!\cdots\!62\)\( \beta_{10} - \)\(52\!\cdots\!74\)\( \beta_{9} + \)\(13\!\cdots\!73\)\( \beta_{8} - \)\(49\!\cdots\!39\)\( \beta_{7} + \)\(41\!\cdots\!50\)\( \beta_{6} + \)\(79\!\cdots\!11\)\( \beta_{5} + \)\(23\!\cdots\!79\)\( \beta_{4} + \)\(44\!\cdots\!95\)\( \beta_{3} + \)\(23\!\cdots\!93\)\( \beta_{2} + \)\(27\!\cdots\!45\)\( \beta_{1} + \)\(33\!\cdots\!65\)\(\)
\(\nu^{14}\)\(=\)\(-\)\(84\!\cdots\!06\)\( \beta_{17} - \)\(10\!\cdots\!79\)\( \beta_{16} + \)\(24\!\cdots\!07\)\( \beta_{15} - \)\(25\!\cdots\!90\)\( \beta_{14} - \)\(10\!\cdots\!91\)\( \beta_{13} + \)\(29\!\cdots\!05\)\( \beta_{12} + \)\(82\!\cdots\!38\)\( \beta_{11} - \)\(34\!\cdots\!93\)\( \beta_{10} - \)\(47\!\cdots\!78\)\( \beta_{9} + \)\(22\!\cdots\!63\)\( \beta_{8} - \)\(21\!\cdots\!91\)\( \beta_{7} + \)\(45\!\cdots\!33\)\( \beta_{6} + \)\(56\!\cdots\!35\)\( \beta_{5} + \)\(28\!\cdots\!03\)\( \beta_{4} + \)\(56\!\cdots\!52\)\( \beta_{3} + \)\(16\!\cdots\!06\)\( \beta_{2} + \)\(96\!\cdots\!87\)\( \beta_{1} + \)\(48\!\cdots\!08\)\(\)
\(\nu^{15}\)\(=\)\(-\)\(30\!\cdots\!66\)\( \beta_{17} + \)\(28\!\cdots\!14\)\( \beta_{16} + \)\(15\!\cdots\!70\)\( \beta_{15} + \)\(10\!\cdots\!60\)\( \beta_{14} - \)\(59\!\cdots\!36\)\( \beta_{13} + \)\(25\!\cdots\!98\)\( \beta_{12} + \)\(48\!\cdots\!02\)\( \beta_{11} - \)\(19\!\cdots\!58\)\( \beta_{10} - \)\(26\!\cdots\!38\)\( \beta_{9} + \)\(52\!\cdots\!73\)\( \beta_{8} - \)\(23\!\cdots\!63\)\( \beta_{7} + \)\(22\!\cdots\!98\)\( \beta_{6} + \)\(39\!\cdots\!75\)\( \beta_{5} + \)\(12\!\cdots\!55\)\( \beta_{4} + \)\(23\!\cdots\!03\)\( \beta_{3} + \)\(11\!\cdots\!57\)\( \beta_{2} + \)\(11\!\cdots\!93\)\( \beta_{1} + \)\(17\!\cdots\!81\)\(\)
\(\nu^{16}\)\(=\)\(-\)\(45\!\cdots\!02\)\( \beta_{17} - \)\(46\!\cdots\!67\)\( \beta_{16} + \)\(11\!\cdots\!23\)\( \beta_{15} - \)\(13\!\cdots\!30\)\( \beta_{14} - \)\(51\!\cdots\!23\)\( \beta_{13} + \)\(15\!\cdots\!25\)\( \beta_{12} + \)\(39\!\cdots\!14\)\( \beta_{11} - \)\(16\!\cdots\!21\)\( \beta_{10} - \)\(22\!\cdots\!90\)\( \beta_{9} + \)\(34\!\cdots\!47\)\( \beta_{8} - \)\(11\!\cdots\!51\)\( \beta_{7} + \)\(21\!\cdots\!45\)\( \beta_{6} + \)\(27\!\cdots\!75\)\( \beta_{5} + \)\(13\!\cdots\!15\)\( \beta_{4} + \)\(25\!\cdots\!88\)\( \beta_{3} + \)\(77\!\cdots\!62\)\( \beta_{2} + \)\(48\!\cdots\!87\)\( \beta_{1} + \)\(21\!\cdots\!44\)\(\)
\(\nu^{17}\)\(=\)\(-\)\(19\!\cdots\!38\)\( \beta_{17} + \)\(61\!\cdots\!70\)\( \beta_{16} + \)\(78\!\cdots\!82\)\( \beta_{15} - \)\(26\!\cdots\!96\)\( \beta_{14} - \)\(29\!\cdots\!56\)\( \beta_{13} + \)\(12\!\cdots\!94\)\( \beta_{12} + \)\(24\!\cdots\!18\)\( \beta_{11} - \)\(95\!\cdots\!58\)\( \beta_{10} - \)\(13\!\cdots\!82\)\( \beta_{9} + \)\(21\!\cdots\!37\)\( \beta_{8} - \)\(11\!\cdots\!51\)\( \beta_{7} + \)\(11\!\cdots\!38\)\( \beta_{6} + \)\(19\!\cdots\!51\)\( \beta_{5} + \)\(61\!\cdots\!23\)\( \beta_{4} + \)\(11\!\cdots\!67\)\( \beta_{3} + \)\(54\!\cdots\!29\)\( \beta_{2} + \)\(53\!\cdots\!13\)\( \beta_{1} + \)\(87\!\cdots\!57\)\(\)

Embeddings

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

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

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−615.069
−608.916
−509.707
−483.296
−415.129
−315.569
−235.315
−87.1238
−2.09672
19.4887
54.5616
229.733
235.511
421.353
516.149
525.865
577.016
692.544
−615.069 −9589.87 247238. 540648. 5.89843e6 1.54300e7 −7.14501e7 −3.71746e7 −3.32536e8
1.2 −608.916 12508.7 239707. −1.45732e6 −7.61677e6 1.09811e7 −6.61493e7 2.73284e7 8.87386e8
1.3 −509.707 −18148.6 128729. 1.42408e6 9.25045e6 −2.65975e7 1.19411e6 2.00230e8 −7.25865e8
1.4 −483.296 19868.0 102503. −300011. −9.60214e6 −1.05230e7 1.38073e7 2.65598e8 1.44994e8
1.5 −415.129 11060.0 41259.9 1.55169e6 −4.59133e6 5.04285e6 3.72836e7 −6.81605e6 −6.44152e8
1.6 −315.569 −7907.47 −31488.2 −995824. 2.49535e6 −2.13865e7 5.12990e7 −6.66120e7 3.14251e8
1.7 −235.315 964.844 −75699.1 864499. −227042. 6.42601e6 4.86562e7 −1.28209e8 −2.03429e8
1.8 −87.1238 11891.8 −123481. −509857. −1.03606e6 −8.46695e6 2.21777e7 1.22759e7 4.44206e7
1.9 −2.09672 −3304.37 −131068. 820444. 6928.31 6.94935e6 549632. −1.18221e8 −1.72024e6
1.10 19.4887 −18855.2 −130692. −785915. −367464. −8.64265e6 −5.10144e6 2.26380e8 −1.53164e7
1.11 54.5616 −10196.6 −128095. −1.33117e6 −556341. 2.01292e7 −1.41406e7 −2.51701e7 −7.26309e7
1.12 229.733 12724.1 −78294.6 93032.9 2.92314e6 1.03405e7 −4.80985e7 3.27614e7 2.13728e7
1.13 235.511 −19946.7 −75606.8 778691. −4.69767e6 −1.72672e7 −4.86750e7 2.68732e8 1.83390e8
1.14 421.353 14930.9 46466.2 −57183.5 6.29116e6 −1.89653e7 −3.56489e7 9.37907e7 −2.40944e7
1.15 516.149 −14037.1 135338. −336022. −7.24525e6 1.38849e7 2.20183e6 6.79008e7 −1.73438e8
1.16 525.865 −5290.76 145463. 1.28880e6 −2.78223e6 −1.26851e7 7.56747e6 −1.01148e8 6.77737e8
1.17 577.016 2597.06 201875. −942114. 1.49854e6 2.00475e7 4.08544e7 −1.22395e8 −5.43614e8
1.18 692.544 5331.24 348545. −1.21070e6 3.69212e6 −2.86223e7 1.50609e8 −1.00718e8 −8.38462e8
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(29\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 29.18.a.a 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.18.a.a 18 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(10\!\cdots\!48\)\( T_{2}^{14} + \)\(33\!\cdots\!00\)\( T_{2}^{13} - \)\(36\!\cdots\!72\)\( T_{2}^{12} - \)\(14\!\cdots\!00\)\( T_{2}^{11} + \)\(69\!\cdots\!40\)\( T_{2}^{10} + \)\(31\!\cdots\!60\)\( T_{2}^{9} - \)\(72\!\cdots\!56\)\( T_{2}^{8} - \)\(31\!\cdots\!00\)\( T_{2}^{7} + \)\(39\!\cdots\!68\)\( T_{2}^{6} + \)\(11\!\cdots\!60\)\( T_{2}^{5} - \)\(88\!\cdots\!76\)\( T_{2}^{4} - \)\(11\!\cdots\!40\)\( T_{2}^{3} + \)\(38\!\cdots\!28\)\( T_{2}^{2} - \)\(57\!\cdots\!20\)\( T_{2} - \)\(13\!\cdots\!24\)\( \)">\(T_{2}^{18} - \cdots\) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(29))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -\)\(13\!\cdots\!24\)\( - \)\(57\!\cdots\!20\)\( T + \)\(38\!\cdots\!28\)\( T^{2} - \)\(11\!\cdots\!40\)\( T^{3} - \)\(88\!\cdots\!76\)\( T^{4} + \)\(11\!\cdots\!60\)\( T^{5} + \)\(39\!\cdots\!68\)\( T^{6} - \)\(31\!\cdots\!00\)\( T^{7} - \)\(72\!\cdots\!56\)\( T^{8} + \)\(31\!\cdots\!60\)\( T^{9} + \)\(69\!\cdots\!40\)\( T^{10} - 14867597249727884800 T^{11} - 361118629330503872 T^{12} + 33471007935200 T^{13} + 1054878119348 T^{14} - 28978880 T^{15} - 1610997 T^{16} + T^{18} \)
$3$ \( -\)\(10\!\cdots\!00\)\( + \)\(10\!\cdots\!60\)\( T + \)\(24\!\cdots\!41\)\( T^{2} - \)\(20\!\cdots\!00\)\( T^{3} - \)\(27\!\cdots\!24\)\( T^{4} + \)\(11\!\cdots\!20\)\( T^{5} + \)\(12\!\cdots\!52\)\( T^{6} - \)\(25\!\cdots\!80\)\( T^{7} - \)\(26\!\cdots\!52\)\( T^{8} + \)\(31\!\cdots\!80\)\( T^{9} + \)\(29\!\cdots\!66\)\( T^{10} - \)\(22\!\cdots\!60\)\( T^{11} - \)\(18\!\cdots\!96\)\( T^{12} + \)\(89\!\cdots\!20\)\( T^{13} + 676758021695164296 T^{14} - 18514882052940 T^{15} - 1287947744 T^{16} + 15400 T^{17} + T^{18} \)
$5$ \( \)\(40\!\cdots\!00\)\( + \)\(52\!\cdots\!00\)\( T - \)\(58\!\cdots\!75\)\( T^{2} - \)\(49\!\cdots\!00\)\( T^{3} - \)\(19\!\cdots\!00\)\( T^{4} + \)\(44\!\cdots\!00\)\( T^{5} + \)\(48\!\cdots\!00\)\( T^{6} - \)\(15\!\cdots\!00\)\( T^{7} - \)\(21\!\cdots\!00\)\( T^{8} + \)\(29\!\cdots\!00\)\( T^{9} + \)\(42\!\cdots\!50\)\( T^{10} - \)\(30\!\cdots\!00\)\( T^{11} - \)\(45\!\cdots\!00\)\( T^{12} + \)\(17\!\cdots\!00\)\( T^{13} + \)\(27\!\cdots\!84\)\( T^{14} - 4981536691589292688 T^{15} - 8215435883100 T^{16} + 564228 T^{17} + T^{18} \)
$7$ \( -\)\(11\!\cdots\!24\)\( + \)\(32\!\cdots\!40\)\( T + \)\(51\!\cdots\!88\)\( T^{2} - \)\(18\!\cdots\!60\)\( T^{3} - \)\(87\!\cdots\!88\)\( T^{4} + \)\(44\!\cdots\!40\)\( T^{5} + \)\(76\!\cdots\!04\)\( T^{6} - \)\(56\!\cdots\!80\)\( T^{7} - \)\(45\!\cdots\!40\)\( T^{8} + \)\(42\!\cdots\!60\)\( T^{9} + \)\(30\!\cdots\!72\)\( T^{10} - \)\(19\!\cdots\!20\)\( T^{11} - \)\(18\!\cdots\!84\)\( T^{12} + \)\(51\!\cdots\!80\)\( T^{13} + \)\(70\!\cdots\!56\)\( T^{14} - \)\(73\!\cdots\!80\)\( T^{15} - 1355497360518968 T^{16} + 43925040 T^{17} + T^{18} \)
$11$ \( -\)\(41\!\cdots\!16\)\( + \)\(14\!\cdots\!36\)\( T - \)\(13\!\cdots\!99\)\( T^{2} + \)\(27\!\cdots\!72\)\( T^{3} + \)\(24\!\cdots\!72\)\( T^{4} - \)\(11\!\cdots\!76\)\( T^{5} - \)\(50\!\cdots\!76\)\( T^{6} + \)\(10\!\cdots\!80\)\( T^{7} - \)\(94\!\cdots\!72\)\( T^{8} - \)\(37\!\cdots\!76\)\( T^{9} + \)\(52\!\cdots\!70\)\( T^{10} + \)\(64\!\cdots\!32\)\( T^{11} - \)\(10\!\cdots\!28\)\( T^{12} - \)\(56\!\cdots\!92\)\( T^{13} + \)\(10\!\cdots\!12\)\( T^{14} + \)\(24\!\cdots\!28\)\( T^{15} - 5293777741847066216 T^{16} - 414318256 T^{17} + T^{18} \)
$13$ \( \)\(51\!\cdots\!84\)\( + \)\(16\!\cdots\!60\)\( T - \)\(36\!\cdots\!55\)\( T^{2} - \)\(52\!\cdots\!20\)\( T^{3} - \)\(21\!\cdots\!56\)\( T^{4} + \)\(52\!\cdots\!60\)\( T^{5} + \)\(32\!\cdots\!04\)\( T^{6} - \)\(20\!\cdots\!00\)\( T^{7} - \)\(14\!\cdots\!64\)\( T^{8} + \)\(39\!\cdots\!60\)\( T^{9} + \)\(31\!\cdots\!06\)\( T^{10} - \)\(44\!\cdots\!40\)\( T^{11} - \)\(35\!\cdots\!04\)\( T^{12} + \)\(30\!\cdots\!80\)\( T^{13} + \)\(22\!\cdots\!24\)\( T^{14} - \)\(11\!\cdots\!20\)\( T^{15} - 74541276657492054580 T^{16} + 1708529620 T^{17} + T^{18} \)
$17$ \( -\)\(15\!\cdots\!16\)\( + \)\(11\!\cdots\!20\)\( T + \)\(24\!\cdots\!52\)\( T^{2} - \)\(16\!\cdots\!40\)\( T^{3} - \)\(33\!\cdots\!84\)\( T^{4} + \)\(13\!\cdots\!20\)\( T^{5} + \)\(17\!\cdots\!64\)\( T^{6} - \)\(52\!\cdots\!40\)\( T^{7} - \)\(44\!\cdots\!40\)\( T^{8} + \)\(11\!\cdots\!60\)\( T^{9} + \)\(67\!\cdots\!32\)\( T^{10} - \)\(12\!\cdots\!00\)\( T^{11} - \)\(60\!\cdots\!24\)\( T^{12} + \)\(81\!\cdots\!40\)\( T^{13} + \)\(31\!\cdots\!76\)\( T^{14} - \)\(25\!\cdots\!20\)\( T^{15} - \)\(88\!\cdots\!36\)\( T^{16} + 31137019060 T^{17} + T^{18} \)
$19$ \( -\)\(17\!\cdots\!00\)\( - \)\(25\!\cdots\!60\)\( T + \)\(19\!\cdots\!04\)\( T^{2} + \)\(17\!\cdots\!36\)\( T^{3} - \)\(32\!\cdots\!20\)\( T^{4} - \)\(45\!\cdots\!48\)\( T^{5} - \)\(46\!\cdots\!08\)\( T^{6} + \)\(35\!\cdots\!40\)\( T^{7} + \)\(85\!\cdots\!76\)\( T^{8} - \)\(64\!\cdots\!00\)\( T^{9} - \)\(37\!\cdots\!16\)\( T^{10} - \)\(17\!\cdots\!72\)\( T^{11} + \)\(64\!\cdots\!32\)\( T^{12} + \)\(70\!\cdots\!72\)\( T^{13} - \)\(31\!\cdots\!16\)\( T^{14} - \)\(75\!\cdots\!92\)\( T^{15} - \)\(14\!\cdots\!80\)\( T^{16} + 236294644572 T^{17} + T^{18} \)
$23$ \( \)\(49\!\cdots\!76\)\( + \)\(10\!\cdots\!40\)\( T - \)\(30\!\cdots\!28\)\( T^{2} + \)\(26\!\cdots\!00\)\( T^{3} + \)\(64\!\cdots\!40\)\( T^{4} - \)\(67\!\cdots\!40\)\( T^{5} - \)\(38\!\cdots\!48\)\( T^{6} + \)\(51\!\cdots\!20\)\( T^{7} + \)\(54\!\cdots\!88\)\( T^{8} - \)\(16\!\cdots\!00\)\( T^{9} + \)\(13\!\cdots\!44\)\( T^{10} + \)\(22\!\cdots\!40\)\( T^{11} - \)\(36\!\cdots\!56\)\( T^{12} - \)\(14\!\cdots\!80\)\( T^{13} + \)\(28\!\cdots\!56\)\( T^{14} + \)\(41\!\cdots\!20\)\( T^{15} - \)\(90\!\cdots\!68\)\( T^{16} - 448660830360 T^{17} + T^{18} \)
$29$ \( ( 500246412961 + T )^{18} \)
$31$ \( \)\(75\!\cdots\!64\)\( - \)\(17\!\cdots\!36\)\( T - \)\(15\!\cdots\!39\)\( T^{2} + \)\(37\!\cdots\!52\)\( T^{3} - \)\(45\!\cdots\!20\)\( T^{4} - \)\(15\!\cdots\!76\)\( T^{5} + \)\(40\!\cdots\!24\)\( T^{6} + \)\(28\!\cdots\!64\)\( T^{7} - \)\(87\!\cdots\!20\)\( T^{8} - \)\(28\!\cdots\!80\)\( T^{9} + \)\(89\!\cdots\!82\)\( T^{10} + \)\(16\!\cdots\!36\)\( T^{11} - \)\(48\!\cdots\!00\)\( T^{12} - \)\(50\!\cdots\!88\)\( T^{13} + \)\(13\!\cdots\!76\)\( T^{14} + \)\(77\!\cdots\!08\)\( T^{15} - \)\(19\!\cdots\!08\)\( T^{16} - 4286667897456 T^{17} + T^{18} \)
$37$ \( -\)\(10\!\cdots\!00\)\( - \)\(94\!\cdots\!00\)\( T - \)\(19\!\cdots\!96\)\( T^{2} + \)\(50\!\cdots\!60\)\( T^{3} + \)\(21\!\cdots\!64\)\( T^{4} + \)\(12\!\cdots\!60\)\( T^{5} - \)\(29\!\cdots\!88\)\( T^{6} - \)\(34\!\cdots\!00\)\( T^{7} + \)\(12\!\cdots\!92\)\( T^{8} + \)\(26\!\cdots\!80\)\( T^{9} + \)\(93\!\cdots\!52\)\( T^{10} - \)\(90\!\cdots\!80\)\( T^{11} - \)\(16\!\cdots\!56\)\( T^{12} + \)\(15\!\cdots\!20\)\( T^{13} + \)\(37\!\cdots\!72\)\( T^{14} - \)\(11\!\cdots\!60\)\( T^{15} - \)\(33\!\cdots\!84\)\( T^{16} + 33745027570060 T^{17} + T^{18} \)
$41$ \( \)\(91\!\cdots\!00\)\( - \)\(17\!\cdots\!00\)\( T + \)\(82\!\cdots\!00\)\( T^{2} - \)\(94\!\cdots\!00\)\( T^{3} - \)\(22\!\cdots\!00\)\( T^{4} + \)\(49\!\cdots\!20\)\( T^{5} + \)\(11\!\cdots\!84\)\( T^{6} - \)\(66\!\cdots\!12\)\( T^{7} + \)\(56\!\cdots\!36\)\( T^{8} + \)\(39\!\cdots\!68\)\( T^{9} + \)\(14\!\cdots\!00\)\( T^{10} - \)\(11\!\cdots\!48\)\( T^{11} - \)\(76\!\cdots\!00\)\( T^{12} + \)\(13\!\cdots\!28\)\( T^{13} + \)\(14\!\cdots\!00\)\( T^{14} - \)\(61\!\cdots\!48\)\( T^{15} - \)\(86\!\cdots\!76\)\( T^{16} + 62894681812676 T^{17} + T^{18} \)
$43$ \( -\)\(37\!\cdots\!36\)\( - \)\(14\!\cdots\!00\)\( T + \)\(97\!\cdots\!21\)\( T^{2} - \)\(23\!\cdots\!40\)\( T^{3} + \)\(23\!\cdots\!96\)\( T^{4} - \)\(87\!\cdots\!80\)\( T^{5} - \)\(22\!\cdots\!76\)\( T^{6} + \)\(22\!\cdots\!40\)\( T^{7} - \)\(20\!\cdots\!56\)\( T^{8} - \)\(13\!\cdots\!60\)\( T^{9} + \)\(20\!\cdots\!42\)\( T^{10} + \)\(32\!\cdots\!80\)\( T^{11} - \)\(60\!\cdots\!64\)\( T^{12} - \)\(37\!\cdots\!40\)\( T^{13} + \)\(77\!\cdots\!72\)\( T^{14} + \)\(20\!\cdots\!40\)\( T^{15} - \)\(45\!\cdots\!00\)\( T^{16} - 43558449431040 T^{17} + T^{18} \)
$47$ \( \)\(13\!\cdots\!84\)\( - \)\(13\!\cdots\!40\)\( T - \)\(20\!\cdots\!11\)\( T^{2} - \)\(24\!\cdots\!80\)\( T^{3} + \)\(17\!\cdots\!20\)\( T^{4} + \)\(29\!\cdots\!40\)\( T^{5} - \)\(37\!\cdots\!56\)\( T^{6} - \)\(88\!\cdots\!00\)\( T^{7} + \)\(17\!\cdots\!84\)\( T^{8} + \)\(10\!\cdots\!60\)\( T^{9} + \)\(19\!\cdots\!18\)\( T^{10} - \)\(63\!\cdots\!80\)\( T^{11} - \)\(23\!\cdots\!20\)\( T^{12} + \)\(18\!\cdots\!40\)\( T^{13} + \)\(92\!\cdots\!12\)\( T^{14} - \)\(26\!\cdots\!00\)\( T^{15} - \)\(15\!\cdots\!04\)\( T^{16} + 141597817069240 T^{17} + T^{18} \)
$53$ \( \)\(16\!\cdots\!76\)\( + \)\(23\!\cdots\!60\)\( T - \)\(36\!\cdots\!63\)\( T^{2} - \)\(17\!\cdots\!20\)\( T^{3} - \)\(30\!\cdots\!12\)\( T^{4} + \)\(45\!\cdots\!80\)\( T^{5} + \)\(12\!\cdots\!08\)\( T^{6} - \)\(62\!\cdots\!20\)\( T^{7} - \)\(18\!\cdots\!12\)\( T^{8} + \)\(49\!\cdots\!40\)\( T^{9} + \)\(14\!\cdots\!78\)\( T^{10} - \)\(22\!\cdots\!80\)\( T^{11} - \)\(63\!\cdots\!80\)\( T^{12} + \)\(61\!\cdots\!80\)\( T^{13} + \)\(15\!\cdots\!00\)\( T^{14} - \)\(85\!\cdots\!40\)\( T^{15} - \)\(19\!\cdots\!96\)\( T^{16} + 485951074130100 T^{17} + T^{18} \)
$59$ \( -\)\(19\!\cdots\!00\)\( - \)\(23\!\cdots\!00\)\( T - \)\(62\!\cdots\!00\)\( T^{2} + \)\(14\!\cdots\!00\)\( T^{3} + \)\(63\!\cdots\!00\)\( T^{4} - \)\(15\!\cdots\!00\)\( T^{5} - \)\(21\!\cdots\!00\)\( T^{6} - \)\(56\!\cdots\!60\)\( T^{7} + \)\(35\!\cdots\!36\)\( T^{8} + \)\(18\!\cdots\!28\)\( T^{9} - \)\(31\!\cdots\!68\)\( T^{10} - \)\(21\!\cdots\!68\)\( T^{11} + \)\(14\!\cdots\!56\)\( T^{12} + \)\(13\!\cdots\!76\)\( T^{13} - \)\(22\!\cdots\!76\)\( T^{14} - \)\(39\!\cdots\!44\)\( T^{15} - \)\(30\!\cdots\!92\)\( T^{16} + 4547356113106800 T^{17} + T^{18} \)
$61$ \( -\)\(15\!\cdots\!64\)\( - \)\(12\!\cdots\!84\)\( T - \)\(30\!\cdots\!64\)\( T^{2} - \)\(28\!\cdots\!80\)\( T^{3} - \)\(73\!\cdots\!72\)\( T^{4} + \)\(26\!\cdots\!68\)\( T^{5} + \)\(17\!\cdots\!20\)\( T^{6} + \)\(32\!\cdots\!64\)\( T^{7} + \)\(10\!\cdots\!48\)\( T^{8} - \)\(30\!\cdots\!40\)\( T^{9} - \)\(32\!\cdots\!04\)\( T^{10} - \)\(48\!\cdots\!00\)\( T^{11} + \)\(13\!\cdots\!56\)\( T^{12} + \)\(47\!\cdots\!00\)\( T^{13} - \)\(14\!\cdots\!84\)\( T^{14} - \)\(10\!\cdots\!76\)\( T^{15} - \)\(51\!\cdots\!12\)\( T^{16} + 6121039899052148 T^{17} + T^{18} \)
$67$ \( \)\(11\!\cdots\!16\)\( - \)\(13\!\cdots\!60\)\( T - \)\(11\!\cdots\!44\)\( T^{2} + \)\(73\!\cdots\!80\)\( T^{3} + \)\(11\!\cdots\!08\)\( T^{4} - \)\(45\!\cdots\!40\)\( T^{5} + \)\(21\!\cdots\!24\)\( T^{6} + \)\(10\!\cdots\!00\)\( T^{7} - \)\(35\!\cdots\!60\)\( T^{8} - \)\(12\!\cdots\!40\)\( T^{9} - \)\(99\!\cdots\!32\)\( T^{10} + \)\(60\!\cdots\!20\)\( T^{11} + \)\(19\!\cdots\!24\)\( T^{12} - \)\(46\!\cdots\!20\)\( T^{13} - \)\(29\!\cdots\!16\)\( T^{14} - \)\(24\!\cdots\!40\)\( T^{15} + \)\(82\!\cdots\!20\)\( T^{16} + 17715742779810920 T^{17} + T^{18} \)
$71$ \( \)\(10\!\cdots\!56\)\( + \)\(22\!\cdots\!96\)\( T - \)\(62\!\cdots\!72\)\( T^{2} - \)\(10\!\cdots\!92\)\( T^{3} + \)\(14\!\cdots\!72\)\( T^{4} + \)\(22\!\cdots\!44\)\( T^{5} - \)\(17\!\cdots\!44\)\( T^{6} - \)\(27\!\cdots\!28\)\( T^{7} + \)\(11\!\cdots\!36\)\( T^{8} + \)\(20\!\cdots\!60\)\( T^{9} - \)\(44\!\cdots\!44\)\( T^{10} - \)\(91\!\cdots\!04\)\( T^{11} + \)\(89\!\cdots\!28\)\( T^{12} + \)\(24\!\cdots\!68\)\( T^{13} - \)\(54\!\cdots\!88\)\( T^{14} - \)\(33\!\cdots\!96\)\( T^{15} - \)\(79\!\cdots\!64\)\( T^{16} + 18877957007949512 T^{17} + T^{18} \)
$73$ \( -\)\(75\!\cdots\!00\)\( - \)\(40\!\cdots\!00\)\( T - \)\(61\!\cdots\!00\)\( T^{2} - \)\(63\!\cdots\!60\)\( T^{3} + \)\(44\!\cdots\!16\)\( T^{4} + \)\(16\!\cdots\!00\)\( T^{5} - \)\(10\!\cdots\!92\)\( T^{6} - \)\(39\!\cdots\!00\)\( T^{7} + \)\(86\!\cdots\!12\)\( T^{8} + \)\(38\!\cdots\!20\)\( T^{9} - \)\(24\!\cdots\!24\)\( T^{10} - \)\(17\!\cdots\!80\)\( T^{11} - \)\(18\!\cdots\!24\)\( T^{12} + \)\(38\!\cdots\!80\)\( T^{13} + \)\(18\!\cdots\!92\)\( T^{14} - \)\(37\!\cdots\!60\)\( T^{15} - \)\(26\!\cdots\!56\)\( T^{16} + 11641701802444700 T^{17} + T^{18} \)
$79$ \( \)\(11\!\cdots\!00\)\( - \)\(14\!\cdots\!00\)\( T + \)\(13\!\cdots\!25\)\( T^{2} + \)\(39\!\cdots\!20\)\( T^{3} - \)\(15\!\cdots\!44\)\( T^{4} + \)\(37\!\cdots\!36\)\( T^{5} + \)\(70\!\cdots\!64\)\( T^{6} - \)\(82\!\cdots\!40\)\( T^{7} - \)\(11\!\cdots\!76\)\( T^{8} + \)\(20\!\cdots\!84\)\( T^{9} + \)\(81\!\cdots\!62\)\( T^{10} - \)\(22\!\cdots\!20\)\( T^{11} - \)\(26\!\cdots\!40\)\( T^{12} + \)\(12\!\cdots\!04\)\( T^{13} + \)\(90\!\cdots\!08\)\( T^{14} - \)\(36\!\cdots\!00\)\( T^{15} - \)\(46\!\cdots\!64\)\( T^{16} + 44197796400782136 T^{17} + T^{18} \)
$83$ \( \)\(83\!\cdots\!64\)\( + \)\(26\!\cdots\!80\)\( T - \)\(15\!\cdots\!80\)\( T^{2} + \)\(62\!\cdots\!40\)\( T^{3} + \)\(10\!\cdots\!84\)\( T^{4} - \)\(10\!\cdots\!40\)\( T^{5} - \)\(24\!\cdots\!48\)\( T^{6} + \)\(36\!\cdots\!40\)\( T^{7} + \)\(98\!\cdots\!64\)\( T^{8} - \)\(39\!\cdots\!80\)\( T^{9} + \)\(14\!\cdots\!36\)\( T^{10} + \)\(79\!\cdots\!40\)\( T^{11} - \)\(53\!\cdots\!32\)\( T^{12} - \)\(47\!\cdots\!20\)\( T^{13} + \)\(70\!\cdots\!60\)\( T^{14} - \)\(74\!\cdots\!60\)\( T^{15} - \)\(42\!\cdots\!24\)\( T^{16} + 9547427423528400 T^{17} + T^{18} \)
$89$ \( -\)\(53\!\cdots\!00\)\( + \)\(12\!\cdots\!80\)\( T + \)\(10\!\cdots\!84\)\( T^{2} - \)\(12\!\cdots\!32\)\( T^{3} - \)\(71\!\cdots\!48\)\( T^{4} + \)\(75\!\cdots\!48\)\( T^{5} + \)\(23\!\cdots\!20\)\( T^{6} - \)\(25\!\cdots\!20\)\( T^{7} - \)\(40\!\cdots\!56\)\( T^{8} + \)\(46\!\cdots\!16\)\( T^{9} + \)\(40\!\cdots\!04\)\( T^{10} - \)\(43\!\cdots\!28\)\( T^{11} - \)\(22\!\cdots\!80\)\( T^{12} + \)\(21\!\cdots\!08\)\( T^{13} + \)\(74\!\cdots\!92\)\( T^{14} - \)\(51\!\cdots\!08\)\( T^{15} - \)\(13\!\cdots\!08\)\( T^{16} + 48534239108944036 T^{17} + T^{18} \)
$97$ \( -\)\(58\!\cdots\!24\)\( + \)\(36\!\cdots\!20\)\( T + \)\(52\!\cdots\!60\)\( T^{2} - \)\(23\!\cdots\!60\)\( T^{3} - \)\(17\!\cdots\!08\)\( T^{4} + \)\(51\!\cdots\!20\)\( T^{5} + \)\(28\!\cdots\!84\)\( T^{6} - \)\(41\!\cdots\!60\)\( T^{7} - \)\(23\!\cdots\!76\)\( T^{8} + \)\(85\!\cdots\!60\)\( T^{9} + \)\(81\!\cdots\!92\)\( T^{10} + \)\(17\!\cdots\!00\)\( T^{11} - \)\(13\!\cdots\!12\)\( T^{12} - \)\(20\!\cdots\!60\)\( T^{13} + \)\(12\!\cdots\!32\)\( T^{14} + \)\(19\!\cdots\!80\)\( T^{15} - \)\(56\!\cdots\!36\)\( T^{16} - 53343624485398380 T^{17} + T^{18} \)
show more
show less