Properties

Label 10.17.c.a
Level 10
Weight 17
Character orbit 10.c
Analytic conductor 16.232
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 10 = 2 \cdot 5 \)
Weight: \( k \) = \( 17 \)
Character orbit: \([\chi]\) = 10.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(16.2324543857\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{8}\cdot 5^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -128 + 128 \beta_{1} ) q^{2} + ( -673 - 673 \beta_{1} - \beta_{3} ) q^{3} -32768 \beta_{1} q^{4} + ( 23107 + 152225 \beta_{1} + 10 \beta_{2} + 4 \beta_{3} - \beta_{6} ) q^{5} + ( 172288 + 128 \beta_{2} + 128 \beta_{3} ) q^{6} + ( -198187 + 198191 \beta_{1} + 574 \beta_{2} + 2 \beta_{3} + 10 \beta_{4} - 3 \beta_{5} + 11 \beta_{6} + 9 \beta_{7} ) q^{7} + ( 4194304 + 4194304 \beta_{1} ) q^{8} + ( -37 + 18865264 \beta_{1} + 588 \beta_{2} - 625 \beta_{3} - 119 \beta_{4} + 16 \beta_{5} - 74 \beta_{6} + 66 \beta_{7} ) q^{9} +O(q^{10})\) \( q + ( -128 + 128 \beta_{1} ) q^{2} + ( -673 - 673 \beta_{1} - \beta_{3} ) q^{3} -32768 \beta_{1} q^{4} + ( 23107 + 152225 \beta_{1} + 10 \beta_{2} + 4 \beta_{3} - \beta_{6} ) q^{5} + ( 172288 + 128 \beta_{2} + 128 \beta_{3} ) q^{6} + ( -198187 + 198191 \beta_{1} + 574 \beta_{2} + 2 \beta_{3} + 10 \beta_{4} - 3 \beta_{5} + 11 \beta_{6} + 9 \beta_{7} ) q^{7} + ( 4194304 + 4194304 \beta_{1} ) q^{8} + ( -37 + 18865264 \beta_{1} + 588 \beta_{2} - 625 \beta_{3} - 119 \beta_{4} + 16 \beta_{5} - 74 \beta_{6} + 66 \beta_{7} ) q^{9} + ( -22442496 - 16527104 \beta_{1} - 1792 \beta_{2} + 768 \beta_{3} - 128 \beta_{5} + 128 \beta_{6} ) q^{10} + ( 38490934 + 60 \beta_{1} - 7794 \beta_{2} - 7734 \beta_{3} + 17 \beta_{4} + 120 \beta_{5} + 206 \beta_{6} + 283 \beta_{7} ) q^{11} + ( -22052864 + 22052864 \beta_{1} - 32768 \beta_{2} ) q^{12} + ( 25253629 + 25253171 \beta_{1} + 229 \beta_{2} - 32222 \beta_{3} + 418 \beta_{4} - 647 \beta_{5} + 269 \beta_{6} - 1145 \beta_{7} ) q^{13} + ( -512 - 50736384 \beta_{1} - 73728 \beta_{2} + 73216 \beta_{3} - 2432 \beta_{4} + 1792 \beta_{5} - 1024 \beta_{6} + 128 \beta_{7} ) q^{14} + ( -517442003 - 343593983 \beta_{1} + 367761 \beta_{2} - 27781 \beta_{3} + 250 \beta_{4} + 669 \beta_{5} + 1679 \beta_{6} - 1375 \beta_{7} ) q^{15} -1073741824 q^{16} + ( -1352984979 + 1352981993 \beta_{1} + 616613 \beta_{2} - 1493 \beta_{3} - 7465 \beta_{4} + 10034 \beta_{5} - 16006 \beta_{6} - 14513 \beta_{7} ) q^{17} + ( -2414749056 - 2414758528 \beta_{1} + 4736 \beta_{2} + 155264 \beta_{3} + 6784 \beta_{4} - 11520 \beta_{5} + 7424 \beta_{6} - 23680 \beta_{7} ) q^{18} + ( 8144 - 10226203204 \beta_{1} - 536673 \beta_{2} + 544817 \beta_{3} + 2151 \beta_{4} + 44562 \beta_{5} + 16288 \beta_{6} - 38569 \beta_{7} ) q^{19} + ( 4988108800 - 757170176 \beta_{1} + 131072 \beta_{2} - 327680 \beta_{3} + 32768 \beta_{5} ) q^{20} + ( -34915808031 - 29231 \beta_{1} - 2088888 \beta_{2} - 2118119 \beta_{3} - 44133 \beta_{4} - 58462 \beta_{5} - 28658 \beta_{6} - 102022 \beta_{7} ) q^{21} + ( -4926847232 + 4926831872 \beta_{1} + 1987584 \beta_{2} - 7680 \beta_{3} - 38400 \beta_{4} + 11008 \beta_{5} - 41728 \beta_{6} - 34048 \beta_{7} ) q^{22} + ( -7270757671 - 7270737707 \beta_{1} - 9982 \beta_{2} + 154370 \beta_{3} - 153353 \beta_{4} + 163335 \beta_{5} + 123407 \beta_{6} + 49910 \beta_{7} ) q^{23} + ( -5645533184 \beta_{1} + 4194304 \beta_{2} - 4194304 \beta_{3} ) q^{24} + ( 6096802280 + 31808853085 \beta_{1} - 2733220 \beta_{2} + 7615835 \beta_{3} - 180375 \beta_{4} - 67780 \beta_{5} - 61540 \beta_{6} - 328250 \beta_{7} ) q^{25} + ( -6464870400 + 58624 \beta_{1} + 4095104 \beta_{2} + 4153728 \beta_{3} + 93056 \beta_{4} + 117248 \beta_{5} + 48384 \beta_{6} + 200064 \beta_{7} ) q^{26} + ( -54214591176 + 54214691328 \beta_{1} + 32470920 \beta_{2} + 50076 \beta_{3} + 250380 \beta_{4} + 82152 \beta_{5} + 118152 \beta_{6} + 68076 \beta_{7} ) q^{27} + ( 6494322688 + 6494191616 \beta_{1} + 65536 \beta_{2} - 18808832 \beta_{3} + 294912 \beta_{4} - 360448 \beta_{5} - 98304 \beta_{6} - 327680 \beta_{7} ) q^{28} + ( -75228 - 225689977216 \beta_{1} - 10405029 \beta_{2} + 10329801 \beta_{3} - 397139 \beta_{4} + 342910 \beta_{5} - 150456 \beta_{6} - 20999 \beta_{7} ) q^{29} + ( 110212606208 - 22252546560 \beta_{1} - 43517440 \beta_{2} + 50629376 \beta_{3} + 144000 \beta_{4} + 129280 \beta_{5} - 300544 \beta_{6} + 208000 \beta_{7} ) q^{30} + ( -187995969398 - 142312 \beta_{1} - 51835706 \beta_{2} - 51978018 \beta_{3} - 204653 \beta_{4} - 284624 \beta_{5} - 159942 \beta_{6} - 506907 \beta_{7} ) q^{31} + ( 137438953472 - 137438953472 \beta_{1} ) q^{32} + ( 445358145604 + 445358480450 \beta_{1} - 167423 \beta_{2} - 149405137 \beta_{3} - 4909 \beta_{4} + 172332 \beta_{5} - 497360 \beta_{6} + 837115 \beta_{7} ) q^{33} + ( 382208 - 346363772416 \beta_{1} - 78735360 \beta_{2} + 79117568 \beta_{3} + 2813184 \beta_{4} - 3333120 \beta_{5} + 764416 \beta_{6} + 902144 \beta_{7} ) q^{34} + ( 170212105087 + 778965521933 \beta_{1} + 37897984 \beta_{2} + 225231179 \beta_{3} + 2653625 \beta_{4} - 479794 \beta_{5} + 2997784 \beta_{6} + 5006625 \beta_{7} ) q^{35} + ( 618176970752 + 1212416 \beta_{1} - 20480000 \beta_{2} - 19267584 \beta_{3} + 2162688 \beta_{4} + 2424832 \beta_{5} + 524288 \beta_{6} + 3899392 \beta_{7} ) q^{36} + ( 715046694239 - 715048293903 \beta_{1} + 413692025 \beta_{2} - 799832 \beta_{3} - 3999160 \beta_{4} - 4358723 \beta_{5} + 1159395 \beta_{6} + 1959227 \beta_{7} ) q^{37} + ( 1308952967680 + 1308955052544 \beta_{1} - 1042432 \beta_{2} - 138430720 \beta_{3} + 4661504 \beta_{4} - 3619072 \beta_{5} - 7788800 \beta_{6} + 5212160 \beta_{7} ) q^{38} + ( -2092100 + 1924020227798 \beta_{1} - 276437652 \beta_{2} + 274345552 \beta_{3} - 6826375 \beta_{4} + 1100150 \beta_{5} - 4184200 \beta_{6} + 3634125 \beta_{7} ) q^{39} + ( -541560143872 + 735395708928 \beta_{1} + 25165824 \beta_{2} + 58720256 \beta_{3} - 4194304 \beta_{5} - 4194304 \beta_{6} ) q^{40} + ( -2703517260571 - 515767 \beta_{1} - 871273295 \beta_{2} - 871789062 \beta_{3} - 5203436 \beta_{4} - 1031534 \beta_{5} + 8343804 \beta_{6} + 2624601 \beta_{7} ) q^{41} + ( 4469227169536 - 4469219686400 \beta_{1} + 538496896 \beta_{2} + 3741568 \beta_{3} + 18707840 \beta_{4} + 3814912 \beta_{5} + 11151360 \beta_{6} + 7409792 \beta_{7} ) q^{42} + ( 86864371715 + 86859030307 \beta_{1} + 2670704 \beta_{2} + 59638541 \beta_{3} - 14009338 \beta_{4} + 11338634 \beta_{5} + 22021450 \beta_{6} - 13353520 \beta_{7} ) q^{43} + ( 1966080 - 1261270925312 \beta_{1} - 253427712 \beta_{2} + 255393792 \beta_{3} + 9273344 \beta_{4} - 6750208 \beta_{5} + 3932160 \beta_{6} - 557056 \beta_{7} ) q^{44} + ( -15733425742240 + 1232181382442 \beta_{1} - 324117449 \beta_{2} + 698892480 \beta_{3} - 31180500 \beta_{4} + 12853969 \beta_{5} - 10028980 \beta_{6} - 40366625 \beta_{7} ) q^{45} + ( 1861311408384 - 2555392 \beta_{1} - 18481664 \beta_{2} - 21037056 \beta_{3} + 13240704 \beta_{4} - 5110784 \beta_{5} - 36702976 \beta_{6} - 26017664 \beta_{7} ) q^{46} + ( 2181927826129 - 2181936817461 \beta_{1} + 715099204 \beta_{2} - 4495666 \beta_{3} - 22478330 \beta_{4} + 23374499 \beta_{5} - 41357163 \beta_{6} - 36861497 \beta_{7} ) q^{47} + ( 722628247552 + 722628247552 \beta_{1} + 1073741824 \beta_{3} ) q^{48} + ( 11287721 - 17966180965492 \beta_{1} + 3201532350 \beta_{2} - 3190244629 \beta_{3} - 722283 \beta_{4} + 69170892 \beta_{5} + 22575442 \beta_{6} - 57160888 \beta_{7} ) q^{49} + ( -4851923886720 - 3291142503040 \beta_{1} - 624974720 \beta_{2} - 1324679040 \beta_{3} + 65104000 \beta_{4} + 798720 \beta_{5} + 16552960 \beta_{6} + 18928000 \beta_{7} ) q^{50} + ( -35333998359726 - 16653044 \beta_{1} + 4490076375 \beta_{2} + 4473423331 \beta_{3} - 46109868 \beta_{4} - 33306088 \beta_{5} + 25607560 \beta_{6} - 37155352 \beta_{7} ) q^{51} + ( 827495907328 - 827510915072 \beta_{1} - 1055850496 \beta_{2} - 7503872 \beta_{3} - 37519360 \beta_{4} - 8814592 \beta_{5} - 21200896 \beta_{6} - 13697024 \beta_{7} ) q^{52} + ( 39491025987847 + 39491020684355 \beta_{1} + 2651746 \beta_{2} - 2752616843 \beta_{3} - 4249819 \beta_{4} + 1598073 \beta_{5} + 12205057 \beta_{6} - 13258730 \beta_{7} ) q^{53} + ( -12819456 - 13878948160512 \beta_{1} - 4162687488 \beta_{2} + 4149868032 \beta_{3} - 40762368 \beta_{4} + 4608000 \beta_{5} - 25638912 \beta_{6} + 23334912 \beta_{7} ) q^{54} + ( -21297283073526 + 27248777801920 \beta_{1} + 3425663605 \beta_{2} - 2259949297 \beta_{3} - 54136750 \beta_{4} + 20975290 \beta_{5} + 24038318 \beta_{6} + 87236500 \beta_{7} ) q^{55} + ( -1662529830912 + 16777216 \beta_{1} + 2399141888 \beta_{2} + 2415919104 \beta_{3} + 4194304 \beta_{4} + 33554432 \beta_{5} + 58720256 \beta_{6} + 79691776 \beta_{7} ) q^{56} + ( 27358650508012 - 27358641213728 \beta_{1} - 23285558168 \beta_{2} + 4647142 \beta_{3} + 23235710 \beta_{4} - 57836210 \beta_{5} + 76424778 \beta_{6} + 71777636 \beta_{7} ) q^{57} + ( 28888326712832 + 28888307454464 \beta_{1} + 9629184 \beta_{2} - 2654058240 \beta_{3} + 53521664 \beta_{4} - 63150848 \beta_{5} - 24634112 \beta_{6} - 48145920 \beta_{7} ) q^{58} + ( -28685544 - 68307507496916 \beta_{1} + 5906944305 \beta_{2} - 5935629849 \beta_{3} + 27121077 \beta_{4} - 226355418 \beta_{5} - 57371088 \beta_{6} + 170548797 \beta_{7} ) q^{59} + ( -11258887634944 + 16955539554304 \beta_{1} - 910327808 \beta_{2} - 12050792448 \beta_{3} - 45056000 \beta_{4} - 55017472 \beta_{5} + 21921792 \beta_{6} - 8192000 \beta_{7} ) q^{60} + ( -32587267392295 + 83494405 \beta_{1} - 6892871635 \beta_{2} - 6809377230 \beta_{3} + 184867264 \beta_{4} + 166988810 \beta_{5} - 35756908 \beta_{6} + 232604761 \beta_{7} ) q^{61} + ( 24063502298880 - 24063465867008 \beta_{1} + 13288156672 \beta_{2} + 18215936 \beta_{3} + 91079680 \beta_{4} + 15959296 \beta_{5} + 56904448 \beta_{6} + 38688512 \beta_{7} ) q^{62} + ( 159961571528073 + 159961669980877 \beta_{1} - 49226402 \beta_{2} + 48505602586 \beta_{3} + 175046087 \beta_{4} - 125819685 \beta_{5} - 322725293 \beta_{6} + 246132010 \beta_{7} ) q^{63} + 35184372088832 \beta_{1} q^{64} + ( 55307196583912 - 79129560665848 \beta_{1} + 13309649991 \beta_{2} - 4895487226 \beta_{3} + 467113000 \beta_{4} - 291724636 \beta_{5} - 115901916 \beta_{6} - 31074625 \beta_{7} ) q^{65} + ( -114011728134912 - 42860288 \beta_{1} + 19145287680 \beta_{2} + 19102427392 \beta_{3} - 106522368 \beta_{4} - 85720576 \beta_{5} + 41603584 \beta_{6} - 107779072 \beta_{7} ) q^{66} + ( 79483846042131 - 79483672092923 \beta_{1} - 8577317423 \beta_{2} + 86974604 \beta_{3} + 434873020 \beta_{4} - 55494226 \beta_{5} + 403392642 \beta_{6} + 316418038 \beta_{7} ) q^{67} + ( 44334513946624 + 44334611791872 \beta_{1} - 48922624 \beta_{2} - 20205174784 \beta_{3} - 475561984 \beta_{4} + 524484608 \beta_{5} + 328794112 \beta_{6} + 244613120 \beta_{7} ) q^{68} + ( 48315603 - 5168740786315 \beta_{1} + 8729505791 \beta_{2} - 8681190188 \beta_{3} + 340405998 \beta_{4} - 390918378 \beta_{5} + 96631206 \beta_{6} + 98827983 \beta_{7} ) q^{69} + ( -121494736258560 - 77920437356288 \beta_{1} - 33680532864 \beta_{2} - 23978648960 \beta_{3} - 980512000 \beta_{4} + 445129984 \beta_{5} - 322302720 \beta_{6} - 301184000 \beta_{7} ) q^{70} + ( -143769401166894 - 84372432 \beta_{1} - 40783237458 \beta_{2} - 40867609890 \beta_{3} + 185045987 \beta_{4} - 168744864 \beta_{5} - 707581702 \beta_{6} - 606908147 \beta_{7} ) q^{71} + ( -79126807445504 + 79126497067008 \beta_{1} + 5087690752 \beta_{2} - 155189248 \beta_{3} - 775946240 \beta_{4} - 243269632 \beta_{5} - 377487360 \beta_{6} - 222298112 \beta_{7} ) q^{72} + ( 32082210874745 + 32082153985749 \beta_{1} + 28444498 \beta_{2} - 45684902852 \beta_{3} - 86200304 \beta_{4} + 57755806 \beta_{5} + 171533798 \beta_{6} - 142222490 \beta_{7} ) q^{73} + ( 204756992 + 183052158482176 \beta_{1} - 52850200704 \beta_{2} + 53054957696 \beta_{3} + 261111424 \beta_{4} + 706319104 \beta_{5} + 409513984 \beta_{6} - 762673536 \beta_{7} ) q^{74} + ( 187200598419495 - 496256800293785 \beta_{1} + 18473756870 \beta_{2} + 123341892215 \beta_{3} + 835016000 \beta_{4} + 374551630 \beta_{5} + 515742090 \beta_{6} + 764630750 \beta_{7} ) q^{75} + ( -335092226588672 - 266862592 \beta_{1} + 17852563456 \beta_{2} + 17585700864 \beta_{3} - 1263828992 \beta_{4} - 533725184 \beta_{5} + 1460207616 \beta_{6} - 70483968 \beta_{7} ) q^{76} + ( 179539908655566 - 179540257376256 \beta_{1} + 65034389889 \beta_{2} - 174360345 \beta_{3} - 871801725 \beta_{4} + 679659418 \beta_{5} - 1377100798 \beta_{6} - 1202740453 \beta_{7} ) q^{77} + ( -246274321369344 - 246274856946944 \beta_{1} + 267788800 \beta_{2} - 70500250112 \beta_{3} + 408608000 \beta_{4} - 676396800 \beta_{5} + 394758400 \beta_{6} - 1338944000 \beta_{7} ) q^{78} + ( -140361068 + 1324097527935220 \beta_{1} + 75362807988 \beta_{2} - 75503169056 \beta_{3} - 1430327468 \beta_{4} + 2018488528 \beta_{5} - 280722136 \beta_{6} - 728522128 \beta_{7} ) q^{79} + ( -24810952327168 - 163450349158400 \beta_{1} - 10737418240 \beta_{2} - 4294967296 \beta_{3} + 1073741824 \beta_{6} ) q^{80} + ( -1093160310643692 + 195107553 \beta_{1} - 4982922513 \beta_{2} - 4787814960 \beta_{3} + 338195466 \beta_{4} + 390215106 \beta_{5} + 104039280 \beta_{6} + 637342299 \beta_{7} ) q^{81} + ( 346050275371264 - 346050143334912 \beta_{1} + 223111981696 \beta_{2} + 66018176 \beta_{3} + 330090880 \beta_{4} + 1200043264 \beta_{5} - 935970560 \beta_{6} - 1001988736 \beta_{7} ) q^{82} + ( -514417306788553 - 514418245649721 \beta_{1} + 469430584 \beta_{2} - 87033485477 \beta_{3} + 951839736 \beta_{4} - 1421270320 \beta_{5} + 456452016 \beta_{6} - 2347152920 \beta_{7} ) q^{83} + ( -957841408 + 1144121197559808 \beta_{1} - 69406523392 \beta_{2} + 68448681984 \beta_{3} - 3343056896 \beta_{4} + 939065344 \beta_{5} - 1915682816 \beta_{6} + 1446150144 \beta_{7} ) q^{84} + ( 706401693430042 - 2091530603737662 \beta_{1} - 14756076901 \beta_{2} + 228942244839 \beta_{3} - 3428852375 \beta_{4} - 405750709 \beta_{5} - 1419265531 \beta_{6} - 4720366625 \beta_{7} ) q^{85} + ( -22236595458816 + 683700224 \beta_{1} - 7975583360 \beta_{2} - 7291883136 \beta_{3} + 3502445824 \beta_{4} + 1367400448 \beta_{5} - 4270090752 \beta_{6} - 83944704 \beta_{7} ) q^{86} + ( 485421634186340 - 485420564358628 \beta_{1} - 137270258530 \beta_{2} + 534913856 \beta_{3} + 2674569280 \beta_{4} + 430782638 \beta_{5} + 1708872786 \beta_{6} + 1173958930 \beta_{7} ) q^{87} + ( 161442426781696 + 161442930098176 \beta_{1} - 251658240 \beta_{2} - 65129152512 \beta_{3} - 1115684864 \beta_{4} + 1367343104 \beta_{5} + 360710144 \beta_{6} + 1258291200 \beta_{7} ) q^{88} + ( 1124824448 + 2413649044661064 \beta_{1} - 119350595598 \beta_{2} + 120475420046 \beta_{3} + 5043459430 \beta_{4} - 3337972172 \beta_{5} + 2249648896 \beta_{6} - 580662810 \beta_{7} ) q^{89} + ( 1856159278054144 - 2171597711959296 \beta_{1} - 47971203968 \beta_{2} - 130945270912 \beta_{3} + 9158032000 \beta_{4} - 2929017472 \beta_{5} - 361598592 \beta_{6} + 1175824000 \beta_{7} ) q^{90} + ( -3594309180767302 - 1005684740 \beta_{1} - 288978303367 \beta_{2} - 289983988107 \beta_{3} - 4306015694 \beta_{4} - 2011369480 \beta_{5} + 4589292428 \beta_{6} - 722408006 \beta_{7} ) q^{91} + ( -238247533182976 + 238248187363328 \beta_{1} + 5058396160 \beta_{2} + 327090176 \beta_{3} + 1635450880 \beta_{4} - 4043800576 \beta_{5} + 5352161280 \beta_{6} + 5025071104 \beta_{7} ) q^{92} + ( 3291677906944736 + 3291682308348682 \beta_{1} - 2200701973 \beta_{2} + 268023033193 \beta_{3} - 3269922287 \beta_{4} + 5470624260 \beta_{5} - 3332183632 \beta_{6} + 11003509865 \beta_{7} ) q^{93} + ( 1150890496 + 558574674379520 \beta_{1} - 90957252864 \beta_{2} + 92108143360 \beta_{3} + 7595497856 \beta_{4} - 8285652736 \beta_{5} + 2301780992 \beta_{6} + 1841045376 \beta_{7} ) q^{94} + ( 5085416120527360 - 4840594940458840 \beta_{1} - 17408480095 \beta_{2} - 407420529905 \beta_{3} - 6534877375 \beta_{4} + 3984429870 \beta_{5} - 1291077980 \beta_{6} + 14403989625 \beta_{7} ) q^{95} + ( -184992831373312 - 137438953472 \beta_{2} - 137438953472 \beta_{3} ) q^{96} + ( -2232339196823191 + 2232335739353083 \beta_{1} + 207738152694 \beta_{2} - 1728735054 \beta_{3} - 8643675270 \beta_{4} - 9072527720 \beta_{5} + 2157587504 \beta_{6} + 3886322558 \beta_{7} ) q^{97} + ( 2299669718754688 + 2299672608411264 \beta_{1} - 1444828288 \beta_{2} + 818147453312 \beta_{3} + 7409045888 \beta_{4} - 5964217600 \beta_{5} - 11743530752 \beta_{6} + 7224141440 \beta_{7} ) q^{98} + ( -2294793152 + 6876989832475346 \beta_{1} + 548275708194 \beta_{2} - 550570501346 \beta_{3} - 3835937527 \beta_{4} - 6096883858 \beta_{5} - 4589586304 \beta_{6} + 7638028233 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 1024q^{2} - 5382q^{3} + 184830q^{5} + 1377792q^{6} - 1586702q^{7} + 33554432q^{8} + O(q^{10}) \) \( 8q - 1024q^{2} - 5382q^{3} + 184830q^{5} + 1377792q^{6} - 1586702q^{7} + 33554432q^{8} - 179537920q^{10} + 307957276q^{11} - 176357376q^{12} + 202095228q^{13} - 4140218430q^{15} - 8589934592q^{16} - 10825054172q^{17} - 19318270464q^{18} + 39905198080q^{20} - 279317583684q^{21} - 39418531328q^{22} - 58166716742q^{23} + 48765928900q^{25} - 51736378368q^{26} - 433782808920q^{27} + 51993051136q^{28} + 881686264320q^{30} - 1503757815484q^{31} + 1099511627776q^{32} + 3563163295596q^{33} + 1361150225890q^{35} + 4945477238784q^{36} + 5719558248048q^{37} + 10471905756160q^{38} - 4332632145920q^{40} - 21624661426724q^{41} + 35752650711552q^{42} + 694778360778q^{43} - 125868018043710q^{45} + 14890679485952q^{46} + 17454156046938q^{47} + 5778878496768q^{48} - 38811694553600q^{50} - 282689731949724q^{51} + 6622256431104q^{52} + 315933715243808q^{53} - 170380752243540q^{55} - 13310221090816q^{56} + 218915538682320q^{57} + 231112067379200q^{58} - 90045006151680q^{60} - 260671832048484q^{61} + 192481000381952q^{62} + 1279595714205978q^{63} + 442440687522120q^{65} - 912169803672576q^{66} + 635885550643738q^{67} + 354715375108096q^{68} - 971840253967360q^{70} - 1149989311263004q^{71} - 633021086564352q^{72} + 256748998181048q^{73} + 1497316176176850q^{75} - 2680807873576960q^{76} + 1436195093152556q^{77} - 1970051682083328q^{78} - 198459701329920q^{80} - 8745265883258892q^{81} + 2767956662620672q^{82} - 4115160605935662q^{83} + 5650805123574940q^{85} - 177863260359168q^{86} + 3883634567812800q^{87} + 1291666434555904q^{88} + 14849617970903040q^{90} - 28753310620553964q^{91} - 1906006974201856q^{92} + 26332871866839036q^{93} + 40684157497310600q^{95} - 1479392895172608q^{96} - 17859102248834952q^{97} + 18395730493909504q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 7213550 x^{5} + 3043721913 x^{4} - 386278388950 x^{3} + 26017651801250 x^{2} - 635609377818800 x + 7763945882921536\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(126726460968045093775 \nu^{7} + 3382480904508191810786 \nu^{6} + 227825870965985051417350 \nu^{5} - 898802527111956815636760000 \nu^{4} + 362353977543769146009419135025 \nu^{3} - 39776119992323254944808878758098 \nu^{2} + 2608563326792295289471985515784250 \nu - 38922153660323998955711811411575000\)\()/ \)\(24\!\cdots\!16\)\( \)
\(\beta_{2}\)\(=\)\((\)\(23441122427338353891800017 \nu^{7} + 3828610288599785859695939678 \nu^{6} + 257874766543996949412777509050 \nu^{5} - 173811402343219981023543173647236 \nu^{4} + 39843134333648860601890920221462523 \nu^{3} + 148254406984259536997252094003124382 \nu^{2} - 89833901091961231076152113379636288350 \nu + 2749737163092590157124836368641219934416\)\()/ \)\(15\!\cdots\!00\)\( \)
\(\beta_{3}\)\(=\)\((\)\(34379026952574818969991275 \nu^{7} + 65846550551950569529392022 \nu^{6} - 287460465673639609877376899554 \nu^{5} - 289443395857179577828906515396036 \nu^{4} + 101373455987899451872499662694318625 \nu^{3} - 10865729250217415807434989736000556282 \nu^{2} + 360044257068674696359027685599832370174 \nu - 3962856938229778084195416240761648133184\)\()/ \)\(15\!\cdots\!00\)\( \)
\(\beta_{4}\)\(=\)\((\)\(107118076555598905528220129 \nu^{7} - 192253069698529644221139772082 \nu^{6} - 9835588856228570250894739971574 \nu^{5} - 1268404750935502889998842225498648 \nu^{4} + 1603979818262371205954679055691027751 \nu^{3} - 566114133047694755974774947922768404758 \nu^{2} + 47850620768345599535170354511013088361694 \nu - 1896356924074939166025997257110458393958512\)\()/ \)\(30\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(2194668622323851927851920107 \nu^{7} - 40397449558278905332171810142 \nu^{6} - 3207449278450938338481403848790 \nu^{5} - 15968284824029094973013568745226016 \nu^{4} + 6955320850004179553342637718603379733 \nu^{3} - 953483837537593246739849055233987805698 \nu^{2} + 65374573131624543237029603334646958531190 \nu - 1399491919519183822670303084525388874064404\)\()/ \)\(15\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(6672351345721779440482261715 \nu^{7} + 156961583237255151954791712586 \nu^{6} - 1492920588496609224151611278482 \nu^{5} - 48310317818496685430883729750625608 \nu^{4} + 19102631905017924480426451672283409285 \nu^{3} - 2099611638366292358897120168753113198466 \nu^{2} + 108470031112158655514267201873766045386442 \nu - 360102126957387087369125175079467286612552\)\()/ \)\(30\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-8053453434909124533967268477 \nu^{7} - 157000812978056583373127759854 \nu^{6} + 517293155476480381042264227502 \nu^{5} + 58278476245529954745777489050528184 \nu^{4} - 23403317701534594996559675359930233963 \nu^{3} + 2636045948277316575716056644958204260374 \nu^{2} - 143979623495478580036820697793123564550262 \nu + 1634515992863349066059787017589195476608096\)\()/ \)\(30\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(5 \beta_{7} + 2 \beta_{6} + 6 \beta_{5} - 5 \beta_{4} + 17 \beta_{3} - \beta_{2} + 8 \beta_{1} + 6\)\()/18000\)
\(\nu^{2}\)\(=\)\((\)\(9 \beta_{7} + 4 \beta_{6} - 26 \beta_{5} + 19 \beta_{4} + 233 \beta_{3} - 231 \beta_{2} + 9638926 \beta_{1} + 2\)\()/360\)
\(\nu^{3}\)\(=\)\((\)\(-312535 \beta_{7} - 361422 \beta_{6} + 165874 \beta_{5} - 244435 \beta_{4} - 48887 \beta_{3} - 696979 \beta_{2} - 48691185678 \beta_{1} + 48691087904\)\()/18000\)
\(\nu^{4}\)\(=\)\((\)\(1738786 \beta_{7} + 1969358 \beta_{6} + 502738 \beta_{5} - 481941 \beta_{4} - 11071380 \beta_{3} - 11322749 \beta_{2} + 251369 \beta_{1} - 547874610637\)\()/360\)
\(\nu^{5}\)\(=\)\((\)\(-5111067865 \beta_{7} - 4342021946 \beta_{6} - 8430876238 \beta_{5} + 7408662665 \beta_{4} + 14952242059 \beta_{3} + 1022213573 \beta_{2} + 1448543288923416 \beta_{1} + 1448545333350562\)\()/6000\)
\(\nu^{6}\)\(=\)\((\)\(-31512905463 \beta_{7} - 41379522940 \beta_{6} + 145784856806 \beta_{5} - 134961712813 \beta_{4} - 580101834383 \beta_{3} + 559412072913 \beta_{2} - 37212173675178130 \beta_{1} - 20689761470\)\()/360\)
\(\nu^{7}\)\(=\)\((\)\(539400346293155 \beta_{7} + 610685248912006 \beta_{6} - 325545638436602 \beta_{5} + 356424513094255 \beta_{4} + 71284902618851 \beta_{3} - 2126382990732433 \beta_{2} + 111932622948647137894 \beta_{1} - 111932480378841900192\)\()/6000\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/10\mathbb{Z}\right)^\times\).

\(n\) \(7\)
\(\chi(n)\) \(-\beta_{1}\)

Embeddings

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

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

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
110.950 + 110.950i
15.9615 + 15.9615i
−191.774 191.774i
64.8621 + 64.8621i
110.950 110.950i
15.9615 15.9615i
−191.774 + 191.774i
64.8621 64.8621i
−128.000 + 128.000i −7760.96 7760.96i 32768.0i 147024. + 361900.i 1.98681e6 7.60621e6 7.60621e6i 4.19430e6 + 4.19430e6i 7.74184e7i −6.51423e7 2.75042e7i
3.2 −128.000 + 128.000i −1354.68 1354.68i 32768.0i −371534. 120625.i 346799. 232276. 232276.i 4.19430e6 + 4.19430e6i 3.93764e7i 6.29964e7 3.21163e7i
3.3 −128.000 + 128.000i −1321.92 1321.92i 32768.0i 390295. 16042.9i 338411. −6.21755e6 + 6.21755e6i 4.19430e6 + 4.19430e6i 3.95518e7i −4.79043e7 + 5.20113e7i
3.4 −128.000 + 128.000i 7746.56 + 7746.56i 32768.0i −73370.4 + 383673.i −1.98312e6 −2.41428e6 + 2.41428e6i 4.19430e6 + 4.19430e6i 7.69718e7i −3.97187e7 5.85015e7i
7.1 −128.000 128.000i −7760.96 + 7760.96i 32768.0i 147024. 361900.i 1.98681e6 7.60621e6 + 7.60621e6i 4.19430e6 4.19430e6i 7.74184e7i −6.51423e7 + 2.75042e7i
7.2 −128.000 128.000i −1354.68 + 1354.68i 32768.0i −371534. + 120625.i 346799. 232276. + 232276.i 4.19430e6 4.19430e6i 3.93764e7i 6.29964e7 + 3.21163e7i
7.3 −128.000 128.000i −1321.92 + 1321.92i 32768.0i 390295. + 16042.9i 338411. −6.21755e6 6.21755e6i 4.19430e6 4.19430e6i 3.95518e7i −4.79043e7 5.20113e7i
7.4 −128.000 128.000i 7746.56 7746.56i 32768.0i −73370.4 383673.i −1.98312e6 −2.41428e6 2.41428e6i 4.19430e6 4.19430e6i 7.69718e7i −3.97187e7 + 5.85015e7i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 10.17.c.a 8
3.b odd 2 1 90.17.g.b 8
4.b odd 2 1 80.17.p.a 8
5.b even 2 1 50.17.c.d 8
5.c odd 4 1 inner 10.17.c.a 8
5.c odd 4 1 50.17.c.d 8
15.e even 4 1 90.17.g.b 8
20.e even 4 1 80.17.p.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.17.c.a 8 1.a even 1 1 trivial
10.17.c.a 8 5.c odd 4 1 inner
50.17.c.d 8 5.b even 2 1
50.17.c.d 8 5.c odd 4 1
80.17.p.a 8 4.b odd 2 1
80.17.p.a 8 20.e even 4 1
90.17.g.b 8 3.b odd 2 1
90.17.g.b 8 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{8} + \cdots\) acting on \(S_{17}^{\mathrm{new}}(10, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 256 T + 32768 T^{2} )^{4} \)
$3$ \( 1 + 5382 T + 14482962 T^{2} + 247802520942 T^{3} + 1790915843467008 T^{4} - 2854503825980313438 T^{5} - \)\(10\!\cdots\!30\)\( T^{6} - \)\(21\!\cdots\!90\)\( T^{7} - \)\(41\!\cdots\!14\)\( T^{8} - \)\(92\!\cdots\!90\)\( T^{9} - \)\(19\!\cdots\!30\)\( T^{10} - \)\(22\!\cdots\!18\)\( T^{11} + \)\(61\!\cdots\!48\)\( T^{12} + \)\(36\!\cdots\!42\)\( T^{13} + \)\(92\!\cdots\!02\)\( T^{14} + \)\(14\!\cdots\!62\)\( T^{15} + \)\(11\!\cdots\!61\)\( T^{16} \)
$5$ \( 1 - 184830 T - 7301900000 T^{2} + 2453457761718750 T^{3} - \)\(23\!\cdots\!50\)\( T^{4} + \)\(37\!\cdots\!50\)\( T^{5} - \)\(17\!\cdots\!00\)\( T^{6} - \)\(65\!\cdots\!50\)\( T^{7} + \)\(54\!\cdots\!25\)\( T^{8} \)
$7$ \( 1 + 1586702 T + 1258811618402 T^{2} + \)\(30\!\cdots\!22\)\( T^{3} + \)\(10\!\cdots\!88\)\( T^{4} + \)\(56\!\cdots\!42\)\( T^{5} + \)\(44\!\cdots\!50\)\( T^{6} + \)\(27\!\cdots\!30\)\( T^{7} + \)\(29\!\cdots\!26\)\( T^{8} + \)\(90\!\cdots\!30\)\( T^{9} + \)\(49\!\cdots\!50\)\( T^{10} + \)\(20\!\cdots\!42\)\( T^{11} + \)\(12\!\cdots\!88\)\( T^{12} + \)\(12\!\cdots\!22\)\( T^{13} + \)\(16\!\cdots\!02\)\( T^{14} + \)\(71\!\cdots\!02\)\( T^{15} + \)\(14\!\cdots\!01\)\( T^{16} \)
$11$ \( ( 1 - 153978638 T + 150796147434775648 T^{2} - \)\(18\!\cdots\!46\)\( T^{3} + \)\(95\!\cdots\!70\)\( T^{4} - \)\(85\!\cdots\!06\)\( T^{5} + \)\(31\!\cdots\!08\)\( T^{6} - \)\(14\!\cdots\!78\)\( T^{7} + \)\(44\!\cdots\!41\)\( T^{8} )^{2} \)
$13$ \( 1 - 202095228 T + 20421240590185992 T^{2} - \)\(32\!\cdots\!28\)\( T^{3} + \)\(47\!\cdots\!28\)\( T^{4} + \)\(24\!\cdots\!52\)\( T^{5} - \)\(78\!\cdots\!40\)\( T^{6} + \)\(81\!\cdots\!20\)\( T^{7} + \)\(31\!\cdots\!26\)\( T^{8} + \)\(54\!\cdots\!20\)\( T^{9} - \)\(34\!\cdots\!40\)\( T^{10} + \)\(72\!\cdots\!92\)\( T^{11} + \)\(93\!\cdots\!08\)\( T^{12} - \)\(42\!\cdots\!28\)\( T^{13} + \)\(17\!\cdots\!72\)\( T^{14} - \)\(11\!\cdots\!68\)\( T^{15} + \)\(38\!\cdots\!21\)\( T^{16} \)
$17$ \( 1 + 10825054172 T + 58590898913367302792 T^{2} + \)\(42\!\cdots\!52\)\( T^{3} + \)\(93\!\cdots\!68\)\( T^{4} - \)\(12\!\cdots\!28\)\( T^{5} - \)\(10\!\cdots\!20\)\( T^{6} - \)\(75\!\cdots\!00\)\( T^{7} - \)\(52\!\cdots\!34\)\( T^{8} - \)\(36\!\cdots\!00\)\( T^{9} - \)\(24\!\cdots\!20\)\( T^{10} - \)\(14\!\cdots\!48\)\( T^{11} + \)\(52\!\cdots\!28\)\( T^{12} + \)\(11\!\cdots\!52\)\( T^{13} + \)\(77\!\cdots\!52\)\( T^{14} + \)\(69\!\cdots\!92\)\( T^{15} + \)\(31\!\cdots\!41\)\( T^{16} \)
$19$ \( 1 + \)\(12\!\cdots\!52\)\( T^{2} + \)\(22\!\cdots\!08\)\( T^{4} + \)\(19\!\cdots\!04\)\( T^{6} + \)\(25\!\cdots\!70\)\( T^{8} + \)\(16\!\cdots\!44\)\( T^{10} + \)\(15\!\cdots\!68\)\( T^{12} + \)\(71\!\cdots\!12\)\( T^{14} + \)\(47\!\cdots\!41\)\( T^{16} \)
$23$ \( 1 + 58166716742 T + \)\(16\!\cdots\!82\)\( T^{2} - \)\(66\!\cdots\!18\)\( T^{3} + \)\(39\!\cdots\!48\)\( T^{4} + \)\(54\!\cdots\!02\)\( T^{5} + \)\(53\!\cdots\!10\)\( T^{6} - \)\(79\!\cdots\!30\)\( T^{7} - \)\(29\!\cdots\!94\)\( T^{8} - \)\(48\!\cdots\!30\)\( T^{9} + \)\(20\!\cdots\!10\)\( T^{10} + \)\(12\!\cdots\!62\)\( T^{11} + \)\(55\!\cdots\!68\)\( T^{12} - \)\(58\!\cdots\!18\)\( T^{13} + \)\(89\!\cdots\!02\)\( T^{14} + \)\(18\!\cdots\!82\)\( T^{15} + \)\(20\!\cdots\!81\)\( T^{16} \)
$29$ \( 1 - \)\(16\!\cdots\!68\)\( T^{2} + \)\(12\!\cdots\!48\)\( T^{4} - \)\(56\!\cdots\!16\)\( T^{6} + \)\(17\!\cdots\!70\)\( T^{8} - \)\(35\!\cdots\!56\)\( T^{10} + \)\(48\!\cdots\!88\)\( T^{12} - \)\(40\!\cdots\!28\)\( T^{14} + \)\(15\!\cdots\!61\)\( T^{16} \)
$31$ \( ( 1 + 751878907742 T + \)\(23\!\cdots\!48\)\( T^{2} + \)\(15\!\cdots\!74\)\( T^{3} + \)\(23\!\cdots\!70\)\( T^{4} + \)\(11\!\cdots\!94\)\( T^{5} + \)\(12\!\cdots\!28\)\( T^{6} + \)\(28\!\cdots\!22\)\( T^{7} + \)\(27\!\cdots\!21\)\( T^{8} )^{2} \)
$37$ \( 1 - 5719558248048 T + \)\(16\!\cdots\!52\)\( T^{2} - \)\(84\!\cdots\!48\)\( T^{3} + \)\(15\!\cdots\!28\)\( T^{4} + \)\(57\!\cdots\!12\)\( T^{5} - \)\(22\!\cdots\!80\)\( T^{6} + \)\(13\!\cdots\!60\)\( T^{7} - \)\(79\!\cdots\!34\)\( T^{8} + \)\(17\!\cdots\!60\)\( T^{9} - \)\(34\!\cdots\!80\)\( T^{10} + \)\(10\!\cdots\!52\)\( T^{11} + \)\(35\!\cdots\!08\)\( T^{12} - \)\(24\!\cdots\!48\)\( T^{13} + \)\(57\!\cdots\!32\)\( T^{14} - \)\(24\!\cdots\!88\)\( T^{15} + \)\(53\!\cdots\!21\)\( T^{16} \)
$41$ \( ( 1 + 10812330713362 T + \)\(93\!\cdots\!68\)\( T^{2} + \)\(88\!\cdots\!34\)\( T^{3} + \)\(56\!\cdots\!70\)\( T^{4} + \)\(56\!\cdots\!94\)\( T^{5} + \)\(38\!\cdots\!08\)\( T^{6} + \)\(28\!\cdots\!02\)\( T^{7} + \)\(16\!\cdots\!61\)\( T^{8} )^{2} \)
$43$ \( 1 - 694778360778 T + \)\(24\!\cdots\!42\)\( T^{2} + \)\(27\!\cdots\!42\)\( T^{3} + \)\(98\!\cdots\!88\)\( T^{4} - \)\(48\!\cdots\!18\)\( T^{5} + \)\(41\!\cdots\!90\)\( T^{6} + \)\(56\!\cdots\!90\)\( T^{7} - \)\(10\!\cdots\!14\)\( T^{8} + \)\(76\!\cdots\!90\)\( T^{9} + \)\(77\!\cdots\!90\)\( T^{10} - \)\(12\!\cdots\!18\)\( T^{11} + \)\(34\!\cdots\!88\)\( T^{12} + \)\(13\!\cdots\!42\)\( T^{13} + \)\(15\!\cdots\!42\)\( T^{14} - \)\(61\!\cdots\!78\)\( T^{15} + \)\(12\!\cdots\!01\)\( T^{16} \)
$47$ \( 1 - 17454156046938 T + \)\(15\!\cdots\!22\)\( T^{2} + \)\(10\!\cdots\!22\)\( T^{3} + \)\(18\!\cdots\!08\)\( T^{4} - \)\(17\!\cdots\!78\)\( T^{5} + \)\(33\!\cdots\!30\)\( T^{6} - \)\(29\!\cdots\!50\)\( T^{7} - \)\(11\!\cdots\!74\)\( T^{8} - \)\(16\!\cdots\!50\)\( T^{9} + \)\(10\!\cdots\!30\)\( T^{10} - \)\(32\!\cdots\!58\)\( T^{11} + \)\(18\!\cdots\!48\)\( T^{12} + \)\(63\!\cdots\!22\)\( T^{13} + \)\(50\!\cdots\!62\)\( T^{14} - \)\(32\!\cdots\!58\)\( T^{15} + \)\(10\!\cdots\!61\)\( T^{16} \)
$53$ \( 1 - 315933715243808 T + \)\(49\!\cdots\!32\)\( T^{2} - \)\(59\!\cdots\!48\)\( T^{3} + \)\(63\!\cdots\!08\)\( T^{4} - \)\(57\!\cdots\!68\)\( T^{5} + \)\(45\!\cdots\!40\)\( T^{6} - \)\(33\!\cdots\!60\)\( T^{7} + \)\(21\!\cdots\!66\)\( T^{8} - \)\(12\!\cdots\!60\)\( T^{9} + \)\(68\!\cdots\!40\)\( T^{10} - \)\(33\!\cdots\!48\)\( T^{11} + \)\(14\!\cdots\!48\)\( T^{12} - \)\(52\!\cdots\!48\)\( T^{13} + \)\(16\!\cdots\!72\)\( T^{14} - \)\(41\!\cdots\!28\)\( T^{15} + \)\(50\!\cdots\!61\)\( T^{16} \)
$59$ \( 1 - \)\(97\!\cdots\!28\)\( T^{2} + \)\(43\!\cdots\!68\)\( T^{4} - \)\(12\!\cdots\!76\)\( T^{6} + \)\(27\!\cdots\!70\)\( T^{8} - \)\(56\!\cdots\!56\)\( T^{10} + \)\(93\!\cdots\!48\)\( T^{12} - \)\(98\!\cdots\!48\)\( T^{14} + \)\(46\!\cdots\!21\)\( T^{16} \)
$61$ \( ( 1 + 130335916024242 T + \)\(10\!\cdots\!68\)\( T^{2} + \)\(98\!\cdots\!54\)\( T^{3} + \)\(52\!\cdots\!70\)\( T^{4} + \)\(36\!\cdots\!94\)\( T^{5} + \)\(14\!\cdots\!28\)\( T^{6} + \)\(64\!\cdots\!02\)\( T^{7} + \)\(18\!\cdots\!41\)\( T^{8} )^{2} \)
$67$ \( 1 - 635885550643738 T + \)\(20\!\cdots\!22\)\( T^{2} - \)\(12\!\cdots\!58\)\( T^{3} + \)\(92\!\cdots\!68\)\( T^{4} - \)\(33\!\cdots\!98\)\( T^{5} + \)\(97\!\cdots\!10\)\( T^{6} - \)\(54\!\cdots\!30\)\( T^{7} + \)\(30\!\cdots\!86\)\( T^{8} - \)\(90\!\cdots\!30\)\( T^{9} + \)\(26\!\cdots\!10\)\( T^{10} - \)\(14\!\cdots\!18\)\( T^{11} + \)\(68\!\cdots\!28\)\( T^{12} - \)\(14\!\cdots\!58\)\( T^{13} + \)\(40\!\cdots\!82\)\( T^{14} - \)\(21\!\cdots\!18\)\( T^{15} + \)\(54\!\cdots\!41\)\( T^{16} \)
$71$ \( ( 1 + 574994655631502 T + \)\(12\!\cdots\!48\)\( T^{2} + \)\(61\!\cdots\!14\)\( T^{3} + \)\(71\!\cdots\!70\)\( T^{4} + \)\(25\!\cdots\!94\)\( T^{5} + \)\(21\!\cdots\!68\)\( T^{6} + \)\(41\!\cdots\!22\)\( T^{7} + \)\(30\!\cdots\!81\)\( T^{8} )^{2} \)
$73$ \( 1 - 256748998181048 T + \)\(32\!\cdots\!52\)\( T^{2} - \)\(14\!\cdots\!08\)\( T^{3} + \)\(10\!\cdots\!48\)\( T^{4} - \)\(20\!\cdots\!28\)\( T^{5} + \)\(27\!\cdots\!80\)\( T^{6} - \)\(12\!\cdots\!00\)\( T^{7} + \)\(56\!\cdots\!86\)\( T^{8} - \)\(81\!\cdots\!00\)\( T^{9} + \)\(11\!\cdots\!80\)\( T^{10} - \)\(56\!\cdots\!68\)\( T^{11} + \)\(19\!\cdots\!68\)\( T^{12} - \)\(17\!\cdots\!08\)\( T^{13} + \)\(24\!\cdots\!72\)\( T^{14} - \)\(12\!\cdots\!08\)\( T^{15} + \)\(32\!\cdots\!81\)\( T^{16} \)
$79$ \( 1 - \)\(63\!\cdots\!68\)\( T^{2} + \)\(16\!\cdots\!48\)\( T^{4} - \)\(33\!\cdots\!16\)\( T^{6} + \)\(79\!\cdots\!70\)\( T^{8} - \)\(17\!\cdots\!56\)\( T^{10} + \)\(45\!\cdots\!88\)\( T^{12} - \)\(94\!\cdots\!28\)\( T^{14} + \)\(78\!\cdots\!61\)\( T^{16} \)
$83$ \( 1 + 4115160605935662 T + \)\(84\!\cdots\!22\)\( T^{2} + \)\(21\!\cdots\!42\)\( T^{3} + \)\(10\!\cdots\!68\)\( T^{4} + \)\(30\!\cdots\!02\)\( T^{5} + \)\(61\!\cdots\!10\)\( T^{6} + \)\(15\!\cdots\!70\)\( T^{7} + \)\(40\!\cdots\!86\)\( T^{8} + \)\(79\!\cdots\!70\)\( T^{9} + \)\(15\!\cdots\!10\)\( T^{10} + \)\(39\!\cdots\!82\)\( T^{11} + \)\(68\!\cdots\!28\)\( T^{12} + \)\(73\!\cdots\!42\)\( T^{13} + \)\(14\!\cdots\!82\)\( T^{14} + \)\(35\!\cdots\!82\)\( T^{15} + \)\(43\!\cdots\!41\)\( T^{16} \)
$89$ \( 1 - \)\(76\!\cdots\!88\)\( T^{2} + \)\(28\!\cdots\!88\)\( T^{4} - \)\(66\!\cdots\!36\)\( T^{6} + \)\(11\!\cdots\!70\)\( T^{8} - \)\(16\!\cdots\!56\)\( T^{10} + \)\(16\!\cdots\!08\)\( T^{12} - \)\(10\!\cdots\!68\)\( T^{14} + \)\(33\!\cdots\!81\)\( T^{16} \)
$97$ \( 1 + 17859102248834952 T + \)\(15\!\cdots\!52\)\( T^{2} + \)\(17\!\cdots\!12\)\( T^{3} + \)\(18\!\cdots\!08\)\( T^{4} + \)\(12\!\cdots\!52\)\( T^{5} + \)\(78\!\cdots\!60\)\( T^{6} + \)\(60\!\cdots\!20\)\( T^{7} + \)\(47\!\cdots\!46\)\( T^{8} + \)\(37\!\cdots\!20\)\( T^{9} + \)\(29\!\cdots\!60\)\( T^{10} + \)\(29\!\cdots\!72\)\( T^{11} + \)\(25\!\cdots\!48\)\( T^{12} + \)\(14\!\cdots\!12\)\( T^{13} + \)\(85\!\cdots\!92\)\( T^{14} + \)\(58\!\cdots\!32\)\( T^{15} + \)\(20\!\cdots\!61\)\( T^{16} \)
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