Properties

Label 98.13.b.c
Level $98$
Weight $13$
Character orbit 98.b
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.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(89.5713940931\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 6613776 x^{14} + 17532494948988 x^{12} + 23874431083308410544 x^{10} + 17814546644991454004799078 x^{8} + 7194684838492777650788022411888 x^{6} + 1439579890626389563511626657632573372 x^{4} + 111400901558110496276819595183126508110288 x^{2} + 166146008225686888975158615912586831592255841\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{56}\cdot 3^{8}\cdot 7^{24} \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_{1} q^{3} + 2048 q^{4} + ( 2 \beta_{1} - 2 \beta_{9} + \beta_{10} ) q^{5} + ( -\beta_{1} + 14 \beta_{9} - \beta_{10} - \beta_{12} ) q^{6} + 2048 \beta_{2} q^{8} + ( -295281 + 280 \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{2} q^{2} + \beta_{1} q^{3} + 2048 q^{4} + ( 2 \beta_{1} - 2 \beta_{9} + \beta_{10} ) q^{5} + ( -\beta_{1} + 14 \beta_{9} - \beta_{10} - \beta_{12} ) q^{6} + 2048 \beta_{2} q^{8} + ( -295281 + 280 \beta_{2} + \beta_{3} ) q^{9} + ( -330 \beta_{1} - 275 \beta_{9} + 22 \beta_{10} + 5 \beta_{11} - 8 \beta_{12} - \beta_{14} ) q^{10} + ( -259011 + 1915 \beta_{2} - 2 \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{11} + 2048 \beta_{1} q^{12} + ( 1557 \beta_{1} + 500 \beta_{9} + 90 \beta_{10} + 101 \beta_{11} - 50 \beta_{12} - 5 \beta_{13} - 5 \beta_{15} ) q^{13} + ( -1697733 + 131346 \beta_{2} - 7 \beta_{3} + 12 \beta_{4} - 7 \beta_{5} + 4 \beta_{6} - 12 \beta_{7} + 16 \beta_{8} ) q^{15} + 4194304 q^{16} + ( -3517 \beta_{1} - 12399 \beta_{9} - 248 \beta_{10} - 189 \beta_{11} - 86 \beta_{12} + 29 \beta_{13} + 24 \beta_{14} - 3 \beta_{15} ) q^{17} + ( 574080 - 295278 \beta_{2} - 5 \beta_{3} - 34 \beta_{4} + 13 \beta_{5} + 7 \beta_{6} + 28 \beta_{7} - 18 \beta_{8} ) q^{18} + ( 24936 \beta_{1} - 21288 \beta_{9} - 310 \beta_{10} - 239 \beta_{11} + 187 \beta_{12} - 53 \beta_{13} - 65 \beta_{14} + 6 \beta_{15} ) q^{19} + ( 4096 \beta_{1} - 4096 \beta_{9} + 2048 \beta_{10} ) q^{20} + ( 3923520 - 258981 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + 62 \beta_{5} + 21 \beta_{6} + 84 \beta_{7} + 107 \beta_{8} ) q^{22} + ( -3203820 - 439717 \beta_{2} + 187 \beta_{3} - 91 \beta_{4} - 19 \beta_{5} + 23 \beta_{6} + 137 \beta_{7} + 106 \beta_{8} ) q^{23} + ( -2048 \beta_{1} + 28672 \beta_{9} - 2048 \beta_{10} - 2048 \beta_{12} ) q^{24} + ( -65150726 - 3028790 \beta_{2} + 126 \beta_{3} - 164 \beta_{4} + 86 \beta_{5} + 86 \beta_{6} - 122 \beta_{7} - 140 \beta_{8} ) q^{25} + ( 58564 \beta_{1} + 174437 \beta_{9} + 1954 \beta_{10} + 560 \beta_{11} - 1756 \beta_{12} + 320 \beta_{13} - 65 \beta_{14} + 160 \beta_{15} ) q^{26} + ( -250962 \beta_{1} - 169592 \beta_{9} + 1922 \beta_{10} - 3333 \beta_{11} - 6155 \beta_{12} + 57 \beta_{13} - 671 \beta_{14} + 214 \beta_{15} ) q^{27} + ( 33272559 - 1809050 \beta_{2} + 619 \beta_{3} + 260 \beta_{4} + 10 \beta_{5} + 1210 \beta_{6} + 570 \beta_{7} - 540 \beta_{8} ) q^{29} + ( 268989696 - 1697385 \beta_{2} - 1046 \beta_{3} + 214 \beta_{4} - 256 \beta_{5} + 559 \beta_{6} - 228 \beta_{7} - 5 \beta_{8} ) q^{30} + ( -72255 \beta_{1} + 554119 \beta_{9} - 1399 \beta_{10} - 15450 \beta_{11} - 14493 \beta_{12} - 614 \beta_{13} - 989 \beta_{14} - 173 \beta_{15} ) q^{31} + 4194304 \beta_{2} q^{32} + ( -468792 \beta_{1} - 739819 \beta_{9} - 11448 \beta_{10} + 19836 \beta_{11} - 11156 \beta_{12} - 556 \beta_{13} + 1644 \beta_{14} - 1008 \beta_{15} ) q^{33} + ( 181770 \beta_{1} - 345651 \beta_{9} - 10090 \beta_{10} - 16621 \beta_{11} + 5592 \beta_{12} + 192 \beta_{13} + 319 \beta_{14} - 928 \beta_{15} ) q^{34} + ( -604735488 + 573440 \beta_{2} + 2048 \beta_{3} ) q^{36} + ( -720565505 - 299713 \beta_{2} - 119 \beta_{3} + 1594 \beta_{4} + 279 \beta_{5} - 4367 \beta_{6} - 163 \beta_{7} + 1702 \beta_{8} ) q^{37} + ( -444025 \beta_{1} - 23701 \beta_{9} - 2151 \beta_{10} - 14982 \beta_{11} - 31061 \beta_{12} - 384 \beta_{13} + 823 \beta_{14} + 1696 \beta_{15} ) q^{38} + ( -1315760409 - 23988467 \beta_{2} + 4624 \beta_{3} + 3011 \beta_{4} + 3470 \beta_{5} + 6379 \beta_{6} + 693 \beta_{7} - 38 \beta_{8} ) q^{39} + ( -675840 \beta_{1} - 563200 \beta_{9} + 45056 \beta_{10} + 10240 \beta_{11} - 16384 \beta_{12} - 2048 \beta_{14} ) q^{40} + ( -40145 \beta_{1} - 1052326 \beta_{9} - 73656 \beta_{10} + 8631 \beta_{11} - 16154 \beta_{12} - 743 \beta_{13} + 5060 \beta_{14} + 1541 \beta_{15} ) q^{41} + ( -4183089500 + 2329502 \beta_{2} - 1902 \beta_{3} - 4186 \beta_{4} + 5314 \beta_{5} + 3862 \beta_{6} - 3142 \beta_{7} - 4156 \beta_{8} ) q^{43} + ( -530454528 + 3921920 \beta_{2} - 4096 \beta_{5} + 2048 \beta_{6} - 2048 \beta_{7} - 4096 \beta_{8} ) q^{44} + ( 62901 \beta_{1} + 6808984 \beta_{9} - 546724 \beta_{10} - 18435 \beta_{11} - 216182 \beta_{12} + 8835 \beta_{13} + 7036 \beta_{14} - 2825 \beta_{15} ) q^{45} + ( -900401664 - 3205609 \beta_{2} + 3531 \beta_{3} - 2788 \beta_{4} - 653 \beta_{5} + 9202 \beta_{6} + 8552 \beta_{7} - 7009 \beta_{8} ) q^{46} + ( 670265 \beta_{1} - 2309975 \beta_{9} - 515021 \beta_{10} - 206484 \beta_{11} - 29313 \beta_{12} + 2700 \beta_{13} - 6417 \beta_{14} + 7677 \beta_{15} ) q^{47} + 4194304 \beta_{1} q^{48} + ( -6203005056 - 65147058 \beta_{2} + 196 \beta_{3} - 13696 \beta_{4} + 7316 \beta_{5} + 368 \beta_{6} - 1024 \beta_{7} - 1028 \beta_{8} ) q^{50} + ( 2912407047 - 71337573 \beta_{2} - 7666 \beta_{3} - 5076 \beta_{4} + 6572 \beta_{5} - 19835 \beta_{6} - 8037 \beta_{7} - 3770 \beta_{8} ) q^{51} + ( 3188736 \beta_{1} + 1024000 \beta_{9} + 184320 \beta_{10} + 206848 \beta_{11} - 102400 \beta_{12} - 10240 \beta_{13} - 10240 \beta_{15} ) q^{52} + ( -4891770225 + 85922973 \beta_{2} + 9607 \beta_{3} - 21010 \beta_{4} + 8865 \beta_{5} + 7419 \beta_{6} + 751 \beta_{7} - 12958 \beta_{8} ) q^{53} + ( 8010207 \beta_{1} - 12557119 \beta_{9} + 1016469 \beta_{10} - 83286 \beta_{11} + 66403 \beta_{12} - 13696 \beta_{13} + 4761 \beta_{14} - 1824 \beta_{15} ) q^{54} + ( -6706806 \beta_{1} - 35308839 \beta_{9} - 278591 \beta_{10} + 227715 \beta_{11} + 45472 \beta_{12} - 4175 \beta_{13} - 8836 \beta_{14} - 6565 \beta_{15} ) q^{55} + ( -20515247505 + 175326732 \beta_{2} + 19256 \beta_{3} + 44384 \beta_{4} - 20612 \beta_{5} + 17444 \beta_{6} + 16116 \beta_{7} - 3352 \beta_{8} ) q^{57} + ( -3704674560 + 33267446 \beta_{2} + 6335 \beta_{3} - 41346 \beta_{4} - 8383 \beta_{5} - 377 \beta_{6} + 21212 \beta_{7} + 21018 \beta_{8} ) q^{58} + ( -4878875 \beta_{1} + 24776146 \beta_{9} + 1181782 \beta_{10} + 214488 \beta_{11} + 103458 \beta_{12} - 22184 \beta_{13} - 27830 \beta_{14} + 29878 \beta_{15} ) q^{59} + ( -3476957184 + 268996608 \beta_{2} - 14336 \beta_{3} + 24576 \beta_{4} - 14336 \beta_{5} + 8192 \beta_{6} - 24576 \beta_{7} + 32768 \beta_{8} ) q^{60} + ( -223383 \beta_{1} - 41110226 \beta_{9} - 752329 \beta_{10} + 189451 \beta_{11} - 284002 \beta_{12} + 15957 \beta_{13} + 23020 \beta_{14} - 34119 \beta_{15} ) q^{61} + ( 26710977 \beta_{1} - 35838507 \beta_{9} + 1021535 \beta_{10} + 575918 \beta_{11} - 255091 \beta_{12} + 11072 \beta_{13} + 7705 \beta_{14} + 19648 \beta_{15} ) q^{62} + 8589934592 q^{64} + ( -19790337987 - 314378080 \beta_{2} - 2623 \beta_{3} + 106792 \beta_{4} - 5188 \beta_{5} + 111512 \beta_{6} - 11184 \beta_{7} + 19200 \beta_{8} ) q^{65} + ( 20214090 \beta_{1} + 32626964 \beta_{9} - 902300 \beta_{10} - 1198131 \beta_{11} + 852092 \beta_{12} + 64512 \beta_{13} - 22156 \beta_{14} + 17792 \beta_{15} ) q^{66} + ( -13470889990 + 32961426 \beta_{2} - 48700 \beta_{3} - 20267 \beta_{4} + 904 \beta_{5} - 158934 \beta_{6} - 22266 \beta_{7} + 62476 \beta_{8} ) q^{67} + ( -7202816 \beta_{1} - 25393152 \beta_{9} - 507904 \beta_{10} - 387072 \beta_{11} - 176128 \beta_{12} + 59392 \beta_{13} + 49152 \beta_{14} - 6144 \beta_{15} ) q^{68} + ( -76878054 \beta_{1} - 90402408 \beta_{9} + 171575 \beta_{10} + 100248 \beta_{11} - 91792 \beta_{12} + 38648 \beta_{13} - 60728 \beta_{14} - 33128 \beta_{15} ) q^{69} + ( -71891189334 - 587785672 \beta_{2} - 7370 \beta_{3} + 83816 \beta_{4} - 14962 \beta_{5} - 73660 \beta_{6} + 42452 \beta_{7} + 116200 \beta_{8} ) q^{71} + ( 1175715840 - 604729344 \beta_{2} - 10240 \beta_{3} - 69632 \beta_{4} + 26624 \beta_{5} + 14336 \beta_{6} + 57344 \beta_{7} - 36864 \beta_{8} ) q^{72} + ( -70748244 \beta_{1} - 88847435 \beta_{9} + 2208978 \beta_{10} - 1127012 \beta_{11} + 2566024 \beta_{12} - 109980 \beta_{13} + 58416 \beta_{14} - 100876 \beta_{15} ) q^{73} + ( -612847296 - 720595697 \beta_{2} - 19636 \beta_{3} + 121336 \beta_{4} - 17004 \beta_{5} - 98484 \beta_{6} - 21424 \beta_{7} - 79832 \beta_{8} ) q^{74} + ( -145567004 \beta_{1} - 185226106 \beta_{9} + 5427434 \beta_{10} - 3316530 \beta_{11} + 2386776 \beta_{12} - 125750 \beta_{13} - 165968 \beta_{14} + 120790 \beta_{15} ) q^{75} + ( 51068928 \beta_{1} - 43597824 \beta_{9} - 634880 \beta_{10} - 489472 \beta_{11} + 382976 \beta_{12} - 108544 \beta_{13} - 133120 \beta_{14} + 12288 \beta_{15} ) q^{76} + ( -49130555520 - 1315731732 \beta_{2} - 173245 \beta_{3} - 405662 \beta_{4} - 80567 \beta_{5} - 52463 \beta_{6} - 75452 \beta_{7} - 38748 \beta_{8} ) q^{78} + ( -28454008096 - 1643167319 \beta_{2} + 277421 \beta_{3} + 239837 \beta_{4} + 63627 \beta_{5} - 49363 \beta_{6} + 54307 \beta_{7} - 124114 \beta_{8} ) q^{79} + ( 8388608 \beta_{1} - 8388608 \beta_{9} + 4194304 \beta_{10} ) q^{80} + ( 48998022300 - 3092237880 \beta_{2} - 251505 \beta_{3} - 26580 \beta_{4} - 277176 \beta_{5} + 64368 \beta_{6} + 22512 \beta_{7} - 78096 \beta_{8} ) q^{81} + ( 46232976 \beta_{1} + 35506969 \beta_{9} - 7200626 \beta_{10} - 2686964 \beta_{11} + 1397384 \beta_{12} - 98624 \beta_{13} - 176797 \beta_{14} + 23776 \beta_{15} ) q^{82} + ( -121966688 \beta_{1} - 63312514 \beta_{9} + 6503462 \beta_{10} + 1277688 \beta_{11} + 2663874 \beta_{12} - 11736 \beta_{13} + 370314 \beta_{14} + 126342 \beta_{15} ) q^{83} + ( 44388106005 + 522697147 \beta_{2} - 464565 \beta_{3} + 330238 \beta_{4} + 9395 \beta_{5} - 539203 \beta_{6} - 372767 \beta_{7} + 16702 \beta_{8} ) q^{85} + ( 4763149056 - 4183008910 \beta_{2} + 3470 \beta_{3} - 322880 \beta_{4} + 88678 \beta_{5} - 122984 \beta_{6} - 323936 \beta_{7} + 41858 \beta_{8} ) q^{86} + ( -50566749 \beta_{1} + 238802602 \beta_{9} + 872768 \beta_{10} - 16816107 \beta_{11} + 5016805 \beta_{12} + 588303 \beta_{13} + 136321 \beta_{14} + 157096 \beta_{15} ) q^{87} + ( 8035368960 - 530393088 \beta_{2} + 8192 \beta_{3} + 4096 \beta_{4} + 126976 \beta_{5} + 43008 \beta_{6} + 172032 \beta_{7} + 219136 \beta_{8} ) q^{88} + ( 285902214 \beta_{1} + 302805863 \beta_{9} + 4980850 \beta_{10} - 10492278 \beta_{11} - 686428 \beta_{12} + 351222 \beta_{13} + 450248 \beta_{14} - 197842 \beta_{15} ) q^{89} + ( 581272404 \beta_{1} + 74848991 \beta_{9} - 9891822 \beta_{10} + 1008450 \beta_{11} + 2175496 \beta_{12} + 180800 \beta_{13} + 464997 \beta_{14} - 282720 \beta_{15} ) q^{90} + ( -6561423360 - 900540416 \beta_{2} + 382976 \beta_{3} - 186368 \beta_{4} - 38912 \beta_{5} + 47104 \beta_{6} + 280576 \beta_{7} + 217088 \beta_{8} ) q^{92} + ( 57333596085 - 10875562691 \beta_{2} - 690477 \beta_{3} - 414482 \beta_{4} - 283755 \beta_{5} + 40131 \beta_{6} + 195159 \beta_{7} - 6222 \beta_{8} ) q^{93} + ( 183997659 \beta_{1} - 247458443 \beta_{9} - 9069743 \beta_{10} - 5034922 \beta_{11} + 30199 \beta_{12} - 491328 \beta_{13} + 433661 \beta_{14} - 86400 \beta_{15} ) q^{94} + ( 23312982870 - 929193755 \beta_{2} + 668915 \beta_{3} - 447995 \beta_{4} + 239785 \beta_{5} + 223645 \beta_{6} + 38635 \beta_{7} + 480350 \beta_{8} ) q^{95} + ( -4194304 \beta_{1} + 58720256 \beta_{9} - 4194304 \beta_{10} - 4194304 \beta_{12} ) q^{96} + ( -267544437 \beta_{1} + 518458302 \beta_{9} - 18368288 \beta_{10} + 11153255 \beta_{11} - 5002278 \beta_{12} - 269927 \beta_{13} - 497296 \beta_{14} + 14913 \beta_{15} ) q^{97} + ( 245073772161 - 6936328953 \beta_{2} + 991344 \beta_{3} + 1139139 \beta_{4} + 217710 \beta_{5} + 1209081 \beta_{6} - 6393 \beta_{7} - 1208082 \beta_{8} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 32768q^{4} - 4724496q^{9} + O(q^{10}) \) \( 16q + 32768q^{4} - 4724496q^{9} - 4144176q^{11} - 27163728q^{15} + 67108864q^{16} + 9185280q^{18} + 62776320q^{22} - 51261120q^{23} - 1042411616q^{25} + 532360944q^{29} + 4303835136q^{30} - 9675767808q^{36} - 11529048080q^{37} - 21052166544q^{39} - 66929432000q^{43} - 8487272448q^{44} - 14406426624q^{46} - 99248080896q^{50} + 46598512752q^{51} - 78268323600q^{53} - 328243960080q^{57} - 59274792960q^{58} - 55631314944q^{60} + 137438953472q^{64} - 316645407792q^{65} - 215534239840q^{67} - 1150259029344q^{71} + 18811453440q^{72} - 9805556736q^{74} - 786088888320q^{78} - 455264129536q^{79} + 783968356800q^{81} + 710209696080q^{85} + 76210384896q^{86} + 128565903360q^{88} - 104982773760q^{92} + 917337537360q^{93} + 373007725920q^{95} + 3921180354576q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 6613776 x^{14} + 17532494948988 x^{12} + 23874431083308410544 x^{10} + 17814546644991454004799078 x^{8} + 7194684838492777650788022411888 x^{6} + 1439579890626389563511626657632573372 x^{4} + 111400901558110496276819595183126508110288 x^{2} + 166146008225686888975158615912586831592255841\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-1833263318430833872861825909044965 \nu^{14} - 11784858787618912111176900733973516985933 \nu^{12} - 29809609268437465030909277551816879994376366897 \nu^{10} - 37525326005789436062898698051661429541328483615973513 \nu^{8} - 24429281106164464899341491642554383610492607546568328581103 \nu^{6} - 7658589423615956590988846040707488803578535072590917657396909191 \nu^{4} - 883709693404455358986878618578209726606770576285371768478432418275835 \nu^{2} - 5260324009289780797756128473448618259075543856752051320965679098628568227\)\()/ \)\(86\!\cdots\!00\)\( \)
\(\beta_{3}\)\(=\)\((\)\(12832843229015837110032781363314755 \nu^{14} + 82494011513332384778238305137814618901531 \nu^{12} + 208667264879062255216364942862718159960634568279 \nu^{10} + 262677282040526052440290886361630006789299385311814591 \nu^{8} + 171004967743151254295390441497880685273448252825978300067721 \nu^{6} + 53610125965311696136921922284952421625049745508136423601778364337 \nu^{4} + 8359078200169312701367875122617901252384784113112262388734682461349085 \nu^{2} + 1833380399810375997638093499277553975788964207983094311526087667588994186869\)\()/ \)\(21\!\cdots\!40\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(75\!\cdots\!49\)\( \nu^{14} - \)\(42\!\cdots\!45\)\( \nu^{12} - \)\(91\!\cdots\!45\)\( \nu^{10} - \)\(88\!\cdots\!73\)\( \nu^{8} - \)\(37\!\cdots\!27\)\( \nu^{6} - \)\(48\!\cdots\!83\)\( \nu^{4} + \)\(17\!\cdots\!69\)\( \nu^{2} + \)\(71\!\cdots\!69\)\(\)\()/ \)\(77\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(47\!\cdots\!67\)\( \nu^{14} + \)\(30\!\cdots\!85\)\( \nu^{12} + \)\(77\!\cdots\!55\)\( \nu^{10} + \)\(96\!\cdots\!29\)\( \nu^{8} + \)\(62\!\cdots\!61\)\( \nu^{6} + \)\(19\!\cdots\!79\)\( \nu^{4} + \)\(20\!\cdots\!13\)\( \nu^{2} - \)\(36\!\cdots\!37\)\(\)\()/ \)\(37\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(25\!\cdots\!17\)\( \nu^{14} + \)\(14\!\cdots\!58\)\( \nu^{12} + \)\(32\!\cdots\!87\)\( \nu^{10} + \)\(32\!\cdots\!32\)\( \nu^{8} + \)\(15\!\cdots\!99\)\( \nu^{6} + \)\(27\!\cdots\!30\)\( \nu^{4} + \)\(14\!\cdots\!93\)\( \nu^{2} + \)\(14\!\cdots\!60\)\(\)\()/ \)\(13\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(12\!\cdots\!85\)\( \nu^{14} - \)\(79\!\cdots\!41\)\( \nu^{12} - \)\(19\!\cdots\!09\)\( \nu^{10} - \)\(22\!\cdots\!01\)\( \nu^{8} - \)\(13\!\cdots\!11\)\( \nu^{6} - \)\(40\!\cdots\!47\)\( \nu^{4} - \)\(45\!\cdots\!95\)\( \nu^{2} - \)\(32\!\cdots\!79\)\(\)\()/ \)\(30\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(19\!\cdots\!45\)\( \nu^{14} + \)\(12\!\cdots\!83\)\( \nu^{12} + \)\(29\!\cdots\!77\)\( \nu^{10} + \)\(35\!\cdots\!63\)\( \nu^{8} + \)\(21\!\cdots\!03\)\( \nu^{6} + \)\(62\!\cdots\!81\)\( \nu^{4} + \)\(64\!\cdots\!75\)\( \nu^{2} - \)\(65\!\cdots\!03\)\(\)\()/ \)\(45\!\cdots\!00\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(63\!\cdots\!44\)\( \nu^{15} - \)\(39\!\cdots\!57\)\( \nu^{13} - \)\(95\!\cdots\!88\)\( \nu^{11} - \)\(11\!\cdots\!85\)\( \nu^{9} - \)\(72\!\cdots\!84\)\( \nu^{7} - \)\(24\!\cdots\!47\)\( \nu^{5} - \)\(40\!\cdots\!56\)\( \nu^{3} - \)\(28\!\cdots\!19\)\( \nu\)\()/ \)\(18\!\cdots\!00\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(64\!\cdots\!51\)\( \nu^{15} - \)\(40\!\cdots\!03\)\( \nu^{13} - \)\(98\!\cdots\!47\)\( \nu^{11} - \)\(11\!\cdots\!15\)\( \nu^{9} - \)\(74\!\cdots\!21\)\( \nu^{7} - \)\(22\!\cdots\!13\)\( \nu^{5} - \)\(23\!\cdots\!69\)\( \nu^{3} + \)\(86\!\cdots\!39\)\( \nu\)\()/ \)\(29\!\cdots\!00\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(31\!\cdots\!85\)\( \nu^{15} + \)\(21\!\cdots\!13\)\( \nu^{13} + \)\(55\!\cdots\!89\)\( \nu^{11} + \)\(73\!\cdots\!25\)\( \nu^{9} + \)\(51\!\cdots\!03\)\( \nu^{7} + \)\(18\!\cdots\!91\)\( \nu^{5} + \)\(27\!\cdots\!43\)\( \nu^{3} + \)\(10\!\cdots\!15\)\( \nu\)\()/ \)\(14\!\cdots\!04\)\( \)
\(\beta_{12}\)\(=\)\((\)\(\)\(22\!\cdots\!03\)\( \nu^{15} + \)\(14\!\cdots\!75\)\( \nu^{13} + \)\(35\!\cdots\!15\)\( \nu^{11} + \)\(44\!\cdots\!11\)\( \nu^{9} + \)\(28\!\cdots\!69\)\( \nu^{7} + \)\(85\!\cdots\!61\)\( \nu^{5} + \)\(89\!\cdots\!37\)\( \nu^{3} - \)\(11\!\cdots\!43\)\( \nu\)\()/ \)\(59\!\cdots\!00\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(22\!\cdots\!27\)\( \nu^{15} - \)\(12\!\cdots\!69\)\( \nu^{13} - \)\(24\!\cdots\!71\)\( \nu^{11} - \)\(19\!\cdots\!33\)\( \nu^{9} - \)\(66\!\cdots\!05\)\( \nu^{7} + \)\(66\!\cdots\!73\)\( \nu^{5} + \)\(32\!\cdots\!87\)\( \nu^{3} + \)\(47\!\cdots\!01\)\( \nu\)\()/ \)\(29\!\cdots\!00\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(13\!\cdots\!31\)\( \nu^{15} - \)\(89\!\cdots\!34\)\( \nu^{13} - \)\(22\!\cdots\!01\)\( \nu^{11} - \)\(27\!\cdots\!16\)\( \nu^{9} - \)\(15\!\cdots\!97\)\( \nu^{7} - \)\(35\!\cdots\!70\)\( \nu^{5} + \)\(20\!\cdots\!01\)\( \nu^{3} + \)\(98\!\cdots\!60\)\( \nu\)\()/ \)\(74\!\cdots\!00\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(52\!\cdots\!67\)\( \nu^{15} + \)\(33\!\cdots\!75\)\( \nu^{13} + \)\(86\!\cdots\!95\)\( \nu^{11} + \)\(11\!\cdots\!59\)\( \nu^{9} + \)\(77\!\cdots\!61\)\( \nu^{7} + \)\(27\!\cdots\!89\)\( \nu^{5} + \)\(45\!\cdots\!33\)\( \nu^{3} + \)\(24\!\cdots\!73\)\( \nu\)\()/ \)\(65\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 280 \beta_{2} - 826722\)
\(\nu^{3}\)\(=\)\(214 \beta_{15} - 671 \beta_{14} + 57 \beta_{13} - 6155 \beta_{12} - 3333 \beta_{11} + 1922 \beta_{10} - 169592 \beta_{9} - 1313844 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-78096 \beta_{8} + 22512 \beta_{7} + 64368 \beta_{6} - 277176 \beta_{5} - 26580 \beta_{4} - 1845828 \beta_{3} - 3538648320 \beta_{2} + 1084630385025\)
\(\nu^{5}\)\(=\)\(-605724156 \beta_{15} + 1492588260 \beta_{14} - 118372224 \beta_{13} + 13531854972 \beta_{12} + 10551563712 \beta_{11} - 2792819880 \beta_{10} + 169479804084 \beta_{9} + 1977623710533 \beta_{1}\)
\(\nu^{6}\)\(=\)\(117215835768 \beta_{8} - 96982337772 \beta_{7} - 16916257764 \beta_{6} + 715128585984 \beta_{5} + 266623110072 \beta_{4} + 3203978505993 \beta_{3} + 8145523823811516 \beta_{2} - 1631914776372501018\)
\(\nu^{7}\)\(=\)\(1287550407931110 \beta_{15} - 2772314522998047 \beta_{14} + 338889714452937 \beta_{13} - 24546279175435179 \beta_{12} - 24691444358197797 \beta_{11} + 1883387439288954 \beta_{10} - 25244998660296612 \beta_{9} - 3159316753064754192 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-189303187777447392 \beta_{8} + 183481608689796672 \beta_{7} - 214692944851620864 \beta_{6} - 1459842934816194480 \beta_{5} - 876798324156649896 \beta_{4} - 5556319299123900984 \beta_{3} - 15676868291448215952288 \beta_{2} + 2607086126607260898076497\)
\(\nu^{9}\)\(=\)\(-2485557134307793337112 \beta_{15} + 4920001646498064755352 \beta_{14} - 889955574587495256240 \beta_{13} + 42697576646463285470088 \beta_{12} + 51314117185876384301040 \beta_{11} + 3327815687056018788816 \beta_{10} - 381113303582981266811640 \beta_{9} + 5203998978756305261859225 \beta_{1}\)
\(\nu^{10}\)\(=\)\(364756869872693654826096 \beta_{8} - 286374363727543742493144 \beta_{7} + 779681245238900864596152 \beta_{6} + 2752278787657343622158208 \beta_{5} + 2209711481275758388963344 \beta_{4} + 9669980691397570147658145 \beta_{3} + 28987672207248444634441660512 \beta_{2} - 4295079850917298263317394301602\)
\(\nu^{11}\)\(=\)\(4606494186470752586582577846 \beta_{15} - 8606515330743450439330570383 \beta_{14} + 2080054674518779752671032137 \beta_{13} - 74113466521789494882330934203 \beta_{12} - 100416406068906674397708420981 \beta_{11} - 18490789830667835218939180974 \beta_{10} + 1315083482001283771108978712208 \beta_{9} - 8728278215155011099785916553788 \beta_{1}\)
\(\nu^{12}\)\(=\)\(-769514990560565485737114317808 \beta_{8} + 432985626236494244802940594992 \beta_{7} - 1966702106833271227016082922896 \beta_{6} - 5004080829866022255063621402984 \beta_{5} - 4950469954731768092198230657212 \beta_{4} - 16875434157102803542481603653788 \beta_{3} - 53391943472006905098294039361262496 \beta_{2} + 7205129286575577291890439659008702065\)
\(\nu^{13}\)\(=\)\(-8369214730095433074316839508819860 \beta_{15} + 14988328870505178445377035764929372 \beta_{14} - 4489786345993458851422325557945872 \beta_{13} + 129877335132980754795358377382377444 \beta_{12} + 189917474439862646654670492580130832 \beta_{11} + 55455777667709280857427644488778856 \beta_{10} - 3295460494061928567024690351425241348 \beta_{9} + 14814874644367477767143189213211111549 \beta_{1}\)
\(\nu^{14}\)\(=\)\(1654773829907649203352372339862513512 \beta_{8} - 684230747394542034054652572866908932 \beta_{7} + 4315812099200432689344434107174438164 \beta_{6} + 8924965930252591960893276483036518400 \beta_{5} + 10397988472931888280121209442735908936 \beta_{4} + 29510650515666157133463395993422622601 \beta_{3} + 98819587944728387000805864088285741655748 \beta_{2} - 12231438723371678259161011903097219133418746\)
\(\nu^{15}\)\(=\)\(15043944501442948466671360712031459414150 \beta_{15} - 26082257765544331494873174321553660129839 \beta_{14} + 9207141483159832357306410971077928412729 \beta_{13} - 230187264693648411333736048386821350811643 \beta_{12} - 351845920239721374070977790938272558993781 \beta_{11} - 138247986995930896541169715809467965937142 \beta_{10} + 7311119315190429459234856184979653794199076 \beta_{9} - 25362689524659027957122194895748750426180408 \beta_{1}\)

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
1340.81i
1138.15i
512.782i
39.0012i
39.0012i
512.782i
1138.15i
1340.81i
1277.57i
861.382i
811.301i
473.046i
473.046i
811.301i
861.382i
1277.57i
−45.2548 1340.81i 2048.00 16424.1i 60678.2i 0 −92681.9 −1.26633e6 743268.i
97.2 −45.2548 1138.15i 2048.00 14010.2i 51506.9i 0 −92681.9 −763948. 634031.i
97.3 −45.2548 512.782i 2048.00 14650.7i 23205.9i 0 −92681.9 268495. 663015.i
97.4 −45.2548 39.0012i 2048.00 2865.82i 1764.99i 0 −92681.9 529920. 129692.i
97.5 −45.2548 39.0012i 2048.00 2865.82i 1764.99i 0 −92681.9 529920. 129692.i
97.6 −45.2548 512.782i 2048.00 14650.7i 23205.9i 0 −92681.9 268495. 663015.i
97.7 −45.2548 1138.15i 2048.00 14010.2i 51506.9i 0 −92681.9 −763948. 634031.i
97.8 −45.2548 1340.81i 2048.00 16424.1i 60678.2i 0 −92681.9 −1.26633e6 743268.i
97.9 45.2548 1277.57i 2048.00 28642.8i 57816.2i 0 92681.9 −1.10074e6 1.29623e6i
97.10 45.2548 861.382i 2048.00 8697.53i 38981.7i 0 92681.9 −210538. 393605.i
97.11 45.2548 811.301i 2048.00 21137.0i 36715.3i 0 92681.9 −126768. 956553.i
97.12 45.2548 473.046i 2048.00 21038.2i 21407.6i 0 92681.9 307668. 952080.i
97.13 45.2548 473.046i 2048.00 21038.2i 21407.6i 0 92681.9 307668. 952080.i
97.14 45.2548 811.301i 2048.00 21137.0i 36715.3i 0 92681.9 −126768. 956553.i
97.15 45.2548 861.382i 2048.00 8697.53i 38981.7i 0 92681.9 −210538. 393605.i
97.16 45.2548 1277.57i 2048.00 28642.8i 57816.2i 0 92681.9 −1.10074e6 1.29623e6i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.13.b.c 16
7.b odd 2 1 inner 98.13.b.c 16
7.c even 3 1 14.13.d.a 16
7.c even 3 1 98.13.d.a 16
7.d odd 6 1 14.13.d.a 16
7.d odd 6 1 98.13.d.a 16
21.g even 6 1 126.13.n.a 16
21.h odd 6 1 126.13.n.a 16
28.f even 6 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.c even 3 1
14.13.d.a 16 7.d odd 6 1
98.13.b.c 16 1.a even 1 1 trivial
98.13.b.c 16 7.b odd 2 1 inner
98.13.d.a 16 7.c even 3 1
98.13.d.a 16 7.d odd 6 1
112.13.s.c 16 28.f even 6 1
112.13.s.c 16 28.g odd 6 1
126.13.n.a 16 21.g even 6 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 \(17\!\cdots\!88\)\( T_{3}^{12} + \)\(23\!\cdots\!44\)\( T_{3}^{10} + \)\(17\!\cdots\!78\)\( T_{3}^{8} + \)\(71\!\cdots\!88\)\( T_{3}^{6} + \)\(14\!\cdots\!72\)\( T_{3}^{4} + \)\(11\!\cdots\!88\)\( T_{3}^{2} + \)\(16\!\cdots\!41\)\( \)">\(T_{3}^{16} + \cdots\) acting on \(S_{13}^{\mathrm{new}}(98, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -2048 + T^{2} )^{8} \)
$3$ \( \)\(16\!\cdots\!41\)\( + \)\(11\!\cdots\!88\)\( T^{2} + \)\(14\!\cdots\!72\)\( T^{4} + \)\(71\!\cdots\!88\)\( T^{6} + \)\(17\!\cdots\!78\)\( T^{8} + 23874431083308410544 T^{10} + 17532494948988 T^{12} + 6613776 T^{14} + T^{16} \)
$5$ \( \)\(11\!\cdots\!25\)\( + \)\(17\!\cdots\!00\)\( T^{2} + \)\(49\!\cdots\!00\)\( T^{4} + \)\(58\!\cdots\!00\)\( T^{6} + \)\(36\!\cdots\!50\)\( T^{8} + \)\(12\!\cdots\!00\)\( T^{10} + 2445287494934154204 T^{12} + 2474330808 T^{14} + T^{16} \)
$7$ \( T^{16} \)
$11$ \( ( \)\(54\!\cdots\!89\)\( - \)\(85\!\cdots\!44\)\( T - \)\(27\!\cdots\!32\)\( T^{2} + \)\(39\!\cdots\!28\)\( T^{3} + \)\(28\!\cdots\!98\)\( T^{4} - 28271466183284641152 T^{5} - 13637290138200 T^{6} + 2072088 T^{7} + T^{8} )^{2} \)
$13$ \( \)\(20\!\cdots\!56\)\( + \)\(19\!\cdots\!48\)\( T^{2} + \)\(53\!\cdots\!12\)\( T^{4} + \)\(68\!\cdots\!48\)\( T^{6} + \)\(44\!\cdots\!48\)\( T^{8} + \)\(15\!\cdots\!84\)\( T^{10} + \)\(28\!\cdots\!68\)\( T^{12} + 265309230378576 T^{14} + T^{16} \)
$17$ \( \)\(73\!\cdots\!41\)\( + \)\(10\!\cdots\!16\)\( T^{2} + \)\(46\!\cdots\!32\)\( T^{4} + \)\(84\!\cdots\!00\)\( T^{6} + \)\(68\!\cdots\!66\)\( T^{8} + \)\(26\!\cdots\!40\)\( T^{10} + \)\(45\!\cdots\!32\)\( T^{12} + 3500425248369816 T^{14} + T^{16} \)
$19$ \( \)\(72\!\cdots\!25\)\( + \)\(78\!\cdots\!00\)\( T^{2} + \)\(32\!\cdots\!00\)\( T^{4} + \)\(66\!\cdots\!00\)\( T^{6} + \)\(72\!\cdots\!06\)\( T^{8} + \)\(41\!\cdots\!64\)\( T^{10} + \)\(12\!\cdots\!72\)\( T^{12} + 18058964580932496 T^{14} + T^{16} \)
$23$ \( ( -\)\(21\!\cdots\!71\)\( - \)\(18\!\cdots\!60\)\( T + \)\(16\!\cdots\!92\)\( T^{2} + \)\(33\!\cdots\!80\)\( T^{3} + \)\(17\!\cdots\!58\)\( T^{4} - \)\(89\!\cdots\!00\)\( T^{5} - 85516528711990104 T^{6} + 25630560 T^{7} + T^{8} )^{2} \)
$29$ \( ( -\)\(24\!\cdots\!84\)\( - \)\(33\!\cdots\!76\)\( T - \)\(10\!\cdots\!24\)\( T^{2} + \)\(17\!\cdots\!56\)\( T^{3} + \)\(86\!\cdots\!68\)\( T^{4} + \)\(37\!\cdots\!28\)\( T^{5} - 1674687874740006456 T^{6} - 266180472 T^{7} + T^{8} )^{2} \)
$31$ \( \)\(11\!\cdots\!21\)\( + \)\(11\!\cdots\!00\)\( T^{2} + \)\(36\!\cdots\!64\)\( T^{4} + \)\(38\!\cdots\!00\)\( T^{6} + \)\(14\!\cdots\!66\)\( T^{8} + \)\(24\!\cdots\!00\)\( T^{10} + \)\(19\!\cdots\!24\)\( T^{12} + 7325103328316912400 T^{14} + T^{16} \)
$37$ \( ( \)\(11\!\cdots\!81\)\( + \)\(42\!\cdots\!40\)\( T + \)\(46\!\cdots\!68\)\( T^{2} + \)\(77\!\cdots\!80\)\( T^{3} - \)\(16\!\cdots\!86\)\( T^{4} - \)\(93\!\cdots\!20\)\( T^{5} - 6094649194767914372 T^{6} + 5764524040 T^{7} + T^{8} )^{2} \)
$41$ \( \)\(33\!\cdots\!76\)\( + \)\(30\!\cdots\!64\)\( T^{2} + \)\(11\!\cdots\!16\)\( T^{4} + \)\(21\!\cdots\!64\)\( T^{6} + \)\(24\!\cdots\!04\)\( T^{8} + \)\(15\!\cdots\!56\)\( T^{10} + \)\(59\!\cdots\!08\)\( T^{12} + \)\(12\!\cdots\!80\)\( T^{14} + T^{16} \)
$43$ \( ( \)\(41\!\cdots\!04\)\( + \)\(21\!\cdots\!40\)\( T + \)\(24\!\cdots\!68\)\( T^{2} - \)\(16\!\cdots\!80\)\( T^{3} - \)\(32\!\cdots\!32\)\( T^{4} - \)\(95\!\cdots\!40\)\( T^{5} + \)\(31\!\cdots\!56\)\( T^{6} + 33464716000 T^{7} + T^{8} )^{2} \)
$47$ \( \)\(90\!\cdots\!01\)\( + \)\(15\!\cdots\!72\)\( T^{2} + \)\(10\!\cdots\!40\)\( T^{4} + \)\(36\!\cdots\!12\)\( T^{6} + \)\(69\!\cdots\!94\)\( T^{8} + \)\(73\!\cdots\!88\)\( T^{10} + \)\(40\!\cdots\!40\)\( T^{12} + \)\(10\!\cdots\!28\)\( T^{14} + T^{16} \)
$53$ \( ( -\)\(73\!\cdots\!31\)\( + \)\(30\!\cdots\!20\)\( T + \)\(13\!\cdots\!72\)\( T^{2} + \)\(14\!\cdots\!40\)\( T^{3} - \)\(11\!\cdots\!22\)\( T^{4} - \)\(22\!\cdots\!20\)\( T^{5} - \)\(43\!\cdots\!36\)\( T^{6} + 39134161800 T^{7} + T^{8} )^{2} \)
$59$ \( \)\(19\!\cdots\!61\)\( + \)\(54\!\cdots\!80\)\( T^{2} + \)\(25\!\cdots\!28\)\( T^{4} + \)\(29\!\cdots\!24\)\( T^{6} + \)\(95\!\cdots\!34\)\( T^{8} + \)\(11\!\cdots\!16\)\( T^{10} + \)\(54\!\cdots\!76\)\( T^{12} + \)\(12\!\cdots\!16\)\( T^{14} + T^{16} \)
$61$ \( \)\(56\!\cdots\!41\)\( + \)\(32\!\cdots\!80\)\( T^{2} + \)\(45\!\cdots\!68\)\( T^{4} + \)\(17\!\cdots\!00\)\( T^{6} + \)\(28\!\cdots\!58\)\( T^{8} + \)\(22\!\cdots\!00\)\( T^{10} + \)\(84\!\cdots\!08\)\( T^{12} + \)\(15\!\cdots\!80\)\( T^{14} + T^{16} \)
$67$ \( ( -\)\(31\!\cdots\!51\)\( - \)\(24\!\cdots\!60\)\( T - \)\(49\!\cdots\!20\)\( T^{2} + \)\(13\!\cdots\!00\)\( T^{3} + \)\(93\!\cdots\!94\)\( T^{4} - \)\(22\!\cdots\!20\)\( T^{5} - \)\(19\!\cdots\!36\)\( T^{6} + 107767119920 T^{7} + T^{8} )^{2} \)
$71$ \( ( \)\(25\!\cdots\!56\)\( + \)\(61\!\cdots\!92\)\( T - \)\(17\!\cdots\!64\)\( T^{2} - \)\(64\!\cdots\!48\)\( T^{3} - \)\(39\!\cdots\!36\)\( T^{4} + \)\(67\!\cdots\!12\)\( T^{5} + \)\(11\!\cdots\!96\)\( T^{6} + 575129514672 T^{7} + T^{8} )^{2} \)
$73$ \( \)\(80\!\cdots\!01\)\( + \)\(18\!\cdots\!76\)\( T^{2} + \)\(93\!\cdots\!96\)\( T^{4} + \)\(18\!\cdots\!16\)\( T^{6} + \)\(17\!\cdots\!94\)\( T^{8} + \)\(82\!\cdots\!64\)\( T^{10} + \)\(19\!\cdots\!48\)\( T^{12} + \)\(22\!\cdots\!60\)\( T^{14} + T^{16} \)
$79$ \( ( \)\(82\!\cdots\!25\)\( - \)\(10\!\cdots\!00\)\( T - \)\(38\!\cdots\!00\)\( T^{2} + \)\(14\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!14\)\( T^{4} - \)\(34\!\cdots\!44\)\( T^{5} - \)\(20\!\cdots\!60\)\( T^{6} + 227632064768 T^{7} + T^{8} )^{2} \)
$83$ \( \)\(52\!\cdots\!96\)\( + \)\(12\!\cdots\!00\)\( T^{2} + \)\(26\!\cdots\!88\)\( T^{4} + \)\(13\!\cdots\!24\)\( T^{6} + \)\(32\!\cdots\!44\)\( T^{8} + \)\(39\!\cdots\!36\)\( T^{10} + \)\(24\!\cdots\!56\)\( T^{12} + \)\(79\!\cdots\!16\)\( T^{14} + T^{16} \)
$89$ \( \)\(78\!\cdots\!01\)\( + \)\(21\!\cdots\!80\)\( T^{2} + \)\(83\!\cdots\!12\)\( T^{4} + \)\(67\!\cdots\!60\)\( T^{6} + \)\(12\!\cdots\!38\)\( T^{8} + \)\(74\!\cdots\!60\)\( T^{10} + \)\(19\!\cdots\!12\)\( T^{12} + \)\(22\!\cdots\!80\)\( T^{14} + T^{16} \)
$97$ \( \)\(16\!\cdots\!36\)\( + \)\(51\!\cdots\!56\)\( T^{2} + \)\(67\!\cdots\!00\)\( T^{4} + \)\(46\!\cdots\!04\)\( T^{6} + \)\(18\!\cdots\!24\)\( T^{8} + \)\(42\!\cdots\!44\)\( T^{10} + \)\(54\!\cdots\!00\)\( T^{12} + \)\(36\!\cdots\!36\)\( T^{14} + T^{16} \)
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