Properties

Label 10.15.c.b
Level 10
Weight 15
Character orbit 10.c
Analytic conductor 12.433
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 10 = 2 \cdot 5 \)
Weight: \( k \) = \( 15 \)
Character orbit: \([\chi]\) = 10.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.4328968152\)
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: \( 2^{14}\cdot 3^{4}\cdot 5^{9} \)
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.

\(f(q)\) \(=\) \( q + ( 64 - 64 \beta_{1} ) q^{2} + ( 175 + 176 \beta_{1} - \beta_{3} ) q^{3} -8192 \beta_{1} q^{4} + ( -12603 - 5042 \beta_{1} + 10 \beta_{2} - \beta_{3} + \beta_{4} ) q^{5} + ( 22464 + 64 \beta_{2} - 64 \beta_{3} ) q^{6} + ( 166656 - 166656 \beta_{1} - 6 \beta_{2} - \beta_{3} + 6 \beta_{4} - \beta_{7} ) q^{7} + ( -524288 - 524288 \beta_{1} ) q^{8} + ( -301 + 4060945 \beta_{1} - 283 \beta_{2} - 319 \beta_{3} - 62 \beta_{4} + 28 \beta_{5} - 12 \beta_{6} + 2 \beta_{7} ) q^{9} +O(q^{10})\) \( q + ( 64 - 64 \beta_{1} ) q^{2} + ( 175 + 176 \beta_{1} - \beta_{3} ) q^{3} -8192 \beta_{1} q^{4} + ( -12603 - 5042 \beta_{1} + 10 \beta_{2} - \beta_{3} + \beta_{4} ) q^{5} + ( 22464 + 64 \beta_{2} - 64 \beta_{3} ) q^{6} + ( 166656 - 166656 \beta_{1} - 6 \beta_{2} - \beta_{3} + 6 \beta_{4} - \beta_{7} ) q^{7} + ( -524288 - 524288 \beta_{1} ) q^{8} + ( -301 + 4060945 \beta_{1} - 283 \beta_{2} - 319 \beta_{3} - 62 \beta_{4} + 28 \beta_{5} - 12 \beta_{6} + 2 \beta_{7} ) q^{9} + ( -1128640 + 483840 \beta_{1} + 704 \beta_{2} + 576 \beta_{3} + 64 \beta_{4} + 64 \beta_{5} ) q^{10} + ( 1022661 - 4 \beta_{1} - 1909 \beta_{2} + 1917 \beta_{3} + 56 \beta_{4} - 146 \beta_{5} - 16 \beta_{6} - 26 \beta_{7} ) q^{11} + ( 1441792 - 1441792 \beta_{1} + 8192 \beta_{2} ) q^{12} + ( -1323509 - 1308775 \beta_{1} + 178 \beta_{2} - 14912 \beta_{3} - 225 \beta_{4} + 443 \beta_{5} + 53 \beta_{6} - 25 \beta_{7} ) q^{13} + ( -384 - 21332032 \beta_{1} - 320 \beta_{2} - 448 \beta_{3} + 384 \beta_{4} + 384 \beta_{5} + 64 \beta_{6} - 64 \beta_{7} ) q^{14} + ( 90236862 + 4350243 \beta_{1} + 42857 \beta_{2} + 25612 \beta_{3} - 342 \beta_{4} - 598 \beta_{5} - 125 \beta_{6} - 250 \beta_{7} ) q^{15} -67108864 q^{16} + ( -208743109 + 208743109 \beta_{1} - 20504 \beta_{2} - 1858 \beta_{3} + 2648 \beta_{4} + 3060 \beta_{5} - 340 \beta_{6} - 158 \beta_{7} ) q^{17} + ( 259863104 + 259899328 \beta_{1} + 2304 \beta_{2} - 38528 \beta_{3} - 5760 \beta_{4} - 2176 \beta_{5} - 896 \beta_{6} - 640 \beta_{7} ) q^{18} + ( 87874 - 196617044 \beta_{1} + 90842 \beta_{2} + 84906 \beta_{3} - 2422 \beta_{4} + 8288 \beta_{5} - 602 \beta_{6} - 588 \beta_{7} ) q^{19} + ( -41222144 + 103235584 \beta_{1} + 8192 \beta_{2} + 81920 \beta_{3} + 8192 \beta_{5} ) q^{20} + ( 24064386 + 711 \beta_{1} + 455403 \beta_{2} - 456825 \beta_{3} + 5306 \beta_{4} - 6476 \beta_{5} - 971 \beta_{6} - 1101 \beta_{7} ) q^{21} + ( 65327872 - 65327872 \beta_{1} - 244864 \beta_{2} + 512 \beta_{3} + 12928 \beta_{4} - 5760 \beta_{5} + 640 \beta_{6} - 2688 \beta_{7} ) q^{22} + ( -597642404 - 597087105 \beta_{1} - 8763 \beta_{2} - 546536 \beta_{3} + 18270 \beta_{4} - 1828 \beta_{5} + 1387 \beta_{6} + 2030 \beta_{7} ) q^{23} + ( 524288 - 184549376 \beta_{1} + 524288 \beta_{2} + 524288 \beta_{3} ) q^{24} + ( 1931621490 - 540067390 \beta_{1} - 330525 \beta_{2} + 1542305 \beta_{3} - 1430 \beta_{4} - 19350 \beta_{5} + 500 \beta_{6} + 6000 \beta_{7} ) q^{25} + ( -168454784 - 11392 \beta_{1} + 965760 \beta_{2} - 942976 \beta_{3} - 42752 \beta_{4} + 13952 \beta_{5} + 4992 \beta_{6} + 1792 \beta_{7} ) q^{26} + ( -2608632060 + 2608632060 \beta_{1} - 6666906 \beta_{2} + 27486 \beta_{3} - 104166 \beta_{4} - 21870 \beta_{5} + 2430 \beta_{6} + 15336 \beta_{7} ) q^{27} + ( -1365295104 - 1365254144 \beta_{1} + 8192 \beta_{2} - 49152 \beta_{3} + 49152 \beta_{5} + 8192 \beta_{6} ) q^{28} + ( 961828 - 1644983748 \beta_{1} + 881736 \beta_{2} + 1041920 \beta_{3} + 40838 \beta_{4} - 235192 \beta_{5} + 11918 \beta_{6} + 18752 \beta_{7} ) q^{29} + ( 6056317568 - 5495104448 \beta_{1} + 1103680 \beta_{2} + 4382016 \beta_{3} + 16384 \beta_{4} - 60160 \beta_{5} + 8000 \beta_{6} - 24000 \beta_{7} ) q^{30} + ( 4473477636 + 65491 \beta_{1} + 813993 \beta_{2} - 944975 \beta_{3} + 132146 \beta_{4} + 161254 \beta_{5} - 291 \beta_{6} + 32309 \beta_{7} ) q^{31} + ( -4294967296 + 4294967296 \beta_{1} ) q^{32} + ( -16284236388 - 16268130330 \beta_{1} - 124074 \beta_{2} - 15981984 \beta_{3} + 197190 \beta_{4} - 196694 \beta_{5} - 14524 \beta_{6} + 21910 \beta_{7} ) q^{33} + ( -1312256 + 26718999040 \beta_{1} - 1193344 \beta_{2} - 1431168 \beta_{3} - 26368 \beta_{4} + 365312 \beta_{5} - 11648 \beta_{6} - 31872 \beta_{7} ) q^{34} + ( -3751111200 - 29366067650 \beta_{1} + 7645997 \beta_{2} + 7568208 \beta_{3} + 54312 \beta_{4} + 118502 \beta_{5} - 20250 \beta_{6} + 63250 \beta_{7} ) q^{35} + ( 33264943104 - 147456 \beta_{1} + 2613248 \beta_{2} - 2318336 \beta_{3} - 229376 \beta_{4} - 507904 \beta_{5} - 16384 \beta_{6} - 98304 \beta_{7} ) q^{36} + ( -3936303599 + 3936303599 \beta_{1} - 8832694 \beta_{2} - 111918 \beta_{3} + 393883 \beta_{4} + 99945 \beta_{5} - 11105 \beta_{6} - 56393 \beta_{7} ) q^{37} + ( -12572052992 - 12583680768 \beta_{1} + 379904 \beta_{2} + 11247872 \beta_{3} - 685440 \beta_{4} + 375424 \beta_{5} - 896 \beta_{6} - 76160 \beta_{7} ) q^{38} + ( 6698746 + 129325326437 \beta_{1} + 6924421 \beta_{2} + 6473071 \beta_{3} - 434350 \beta_{4} + 512450 \beta_{5} - 89925 \beta_{6} - 15275 \beta_{7} ) q^{39} + ( 3969384448 + 9250537472 \beta_{1} - 4718592 \beta_{2} + 5767168 \beta_{3} - 524288 \beta_{4} + 524288 \beta_{5} ) q^{40} + ( 11774846935 - 7178 \beta_{1} - 5944401 \beta_{2} + 5958757 \beta_{3} - 21388 \beta_{4} - 3002 \beta_{5} + 1758 \beta_{6} - 952 \beta_{7} ) q^{41} + ( 1569312000 - 1569312000 \beta_{1} + 58382592 \beta_{2} - 91008 \beta_{3} + 754048 \beta_{4} - 74880 \beta_{5} + 8320 \beta_{6} - 132608 \beta_{7} ) q^{42} + ( -117782662255 - 117833119874 \beta_{1} + 21842 \beta_{2} + 50435777 \beta_{3} - 211320 \beta_{4} - 455948 \beta_{5} - 95558 \beta_{6} - 23480 \beta_{7} ) q^{43} + ( -15671296 - 8361934848 \beta_{1} - 15704064 \beta_{2} - 15638528 \beta_{3} + 1196032 \beta_{4} + 458752 \beta_{5} + 212992 \beta_{6} - 131072 \beta_{7} ) q^{44} + ( 365868393138 - 146006152143 \beta_{1} - 23983774 \beta_{2} - 172837475 \beta_{3} + 829360 \beta_{4} - 949249 \beta_{5} + 23375 \beta_{6} - 508875 \beta_{7} ) q^{45} + ( -76463249408 + 560832 \beta_{1} + 34417472 \beta_{2} - 35539136 \beta_{3} + 1286272 \beta_{4} + 1052288 \beta_{5} - 41152 \beta_{6} + 218688 \beta_{7} ) q^{46} + ( -224690933092 + 224690933092 \beta_{1} + 72501334 \beta_{2} - 574977 \beta_{3} - 1431388 \beta_{4} + 1757250 \beta_{5} - 195250 \beta_{6} + 401273 \beta_{7} ) q^{47} + ( -11744051200 - 11811160064 \beta_{1} + 67108864 \beta_{3} ) q^{48} + ( 42768271 + 153866922685 \beta_{1} + 42197687 \beta_{2} + 43338855 \beta_{3} - 473494 \beta_{4} - 2035264 \beta_{5} - 49994 \beta_{6} + 223524 \beta_{7} ) q^{49} + ( 89038308800 - 158089380800 \beta_{1} - 119861120 \beta_{2} + 77553920 \beta_{3} + 1146880 \beta_{4} - 1329920 \beta_{5} - 352000 \beta_{6} + 416000 \beta_{7} ) q^{50} + ( -247785332079 - 551574 \beta_{1} - 275427171 \beta_{2} + 276530319 \beta_{3} - 1026724 \beta_{4} - 1541336 \beta_{5} - 19106 \beta_{6} - 304446 \beta_{7} ) q^{51} + ( -10720026624 + 10720026624 \beta_{1} + 122159104 \beta_{2} + 1458176 \beta_{3} - 3629056 \beta_{4} - 1843200 \beta_{5} + 204800 \beta_{6} + 434176 \beta_{7} ) q^{52} + ( 67544280905 + 67565283141 \beta_{1} - 603444 \beta_{2} - 20398792 \beta_{3} + 2947005 \beta_{4} + 4565461 \beta_{5} + 1033781 \beta_{6} + 327445 \beta_{7} ) q^{53} + ( -426681984 + 333906662784 \beta_{1} - 428441088 \beta_{2} - 424922880 \beta_{3} - 5266944 \beta_{4} - 8066304 \beta_{5} - 825984 \beta_{6} + 1137024 \beta_{7} ) q^{54} + ( 98587231634 - 611827442949 \beta_{1} + 279536265 \beta_{2} - 163954967 \beta_{3} - 4133458 \beta_{4} + 3730220 \beta_{5} + 844625 \beta_{6} + 1991125 \beta_{7} ) q^{55} + ( -174754627584 - 524288 \beta_{1} + 3670016 \beta_{2} - 2621440 \beta_{3} - 3145728 \beta_{4} + 3145728 \beta_{5} + 524288 \beta_{6} + 524288 \beta_{7} ) q^{56} + ( 810646481280 - 810646481280 \beta_{1} - 172319586 \beta_{2} + 6727626 \beta_{3} + 1710244 \beta_{4} - 15147360 \beta_{5} + 1683040 \beta_{6} - 1687574 \beta_{7} ) q^{57} + ( -105160971776 - 105273833984 \beta_{1} - 10251776 \beta_{2} + 123113984 \beta_{3} + 17665920 \beta_{4} - 12438656 \beta_{5} - 437376 \beta_{6} + 1962880 \beta_{7} ) q^{58} + ( 471018456 + 227459723254 \beta_{1} + 472060662 \beta_{2} + 469976250 \beta_{3} + 13694646 \beta_{4} + 9755076 \beta_{5} + 2355396 \beta_{6} - 1917666 \beta_{7} ) q^{59} + ( 35988275200 - 739010560000 \beta_{1} - 209813504 \beta_{2} + 351084544 \beta_{3} + 4898816 \beta_{4} - 2801664 \beta_{5} + 2048000 \beta_{6} - 1024000 \beta_{7} ) q^{60} + ( -2489248065730 - 1338893 \beta_{1} + 111756131 \beta_{2} - 109078345 \beta_{3} - 5778798 \beta_{4} + 3242418 \beta_{5} + 775253 \beta_{6} + 493433 \beta_{7} ) q^{61} + ( 286358855680 - 286358855680 \beta_{1} + 112573952 \beta_{2} - 8382848 \beta_{3} - 1862912 \beta_{4} + 18777600 \beta_{5} - 2086400 \beta_{6} + 2049152 \beta_{7} ) q^{62} + ( 3230707860544 + 3231499112867 \beta_{1} + 19565217 \beta_{2} - 810817540 \beta_{3} - 42029730 \beta_{4} + 642052 \beta_{5} - 3784633 \beta_{6} - 4669970 \beta_{7} ) q^{63} + 549755813888 \beta_{1} q^{64} + ( -878886662348 - 909878305447 \beta_{1} + 266529427 \beta_{2} + 500333347 \beta_{3} - 9448252 \beta_{4} - 4159928 \beta_{5} - 5845250 \beta_{6} - 6215500 \beta_{7} ) q^{65} + ( -2083359410688 + 7940736 \beta_{1} + 1014906240 \beta_{2} - 1030787712 \beta_{3} + 25208576 \beta_{4} + 31744 \beta_{5} - 2331776 \beta_{6} + 472704 \beta_{7} ) q^{66} + ( 1693098815512 - 1693098815512 \beta_{1} - 2183440493 \beta_{2} - 24958722 \beta_{3} + 29599832 \beta_{4} + 43254900 \beta_{5} - 4806100 \beta_{6} - 928222 \beta_{7} ) q^{67} + ( 1709855580160 + 1710008328192 \beta_{1} + 15220736 \beta_{2} - 167968768 \beta_{3} - 25067520 \beta_{4} + 21692416 \beta_{5} + 1294336 \beta_{6} - 2785280 \beta_{7} ) q^{68} + ( 1591199614 + 4578977582819 \beta_{1} + 1607939263 \beta_{2} + 1574459965 \beta_{3} - 34719756 \beta_{4} + 36834294 \beta_{5} - 7111701 \beta_{6} - 838749 \beta_{7} ) q^{69} + ( -2119010102592 - 1638872847488 \beta_{1} + 4978496 \beta_{2} + 973709120 \beta_{3} - 4108160 \beta_{4} + 11060096 \beta_{5} - 5344000 \beta_{6} + 2752000 \beta_{7} ) q^{70} + ( -6442686819004 - 3030829 \beta_{1} + 1725089177 \beta_{2} - 1719027519 \beta_{3} + 14674226 \beta_{4} - 51640826 \beta_{5} - 5183971 \beta_{6} - 9291371 \beta_{7} ) q^{71} + ( 2129114169344 - 2129114169344 \beta_{1} + 315621376 \beta_{2} + 18874368 \beta_{3} + 17825792 \beta_{4} - 47185920 \beta_{5} + 5242880 \beta_{6} - 7340032 \beta_{7} ) q^{72} + ( 5547659887835 + 5548274849151 \beta_{1} - 39000854 \beta_{2} - 575960462 \beta_{3} + 88510500 \beta_{4} + 11857376 \beta_{5} + 10171646 \beta_{6} + 9834500 \beta_{7} ) q^{73} + ( -565292416 + 503839697920 \beta_{1} - 558129664 \beta_{2} - 572455168 \beta_{3} + 18812032 \beta_{4} + 31604992 \beta_{5} + 2898432 \beta_{6} - 4319872 \beta_{7} ) q^{74} + ( -2425423252035 - 13219057622490 \beta_{1} + 644302940 \beta_{2} - 109212485 \beta_{3} + 76981510 \beta_{4} + 20956590 \beta_{5} + 19417000 \beta_{6} + 4961500 \beta_{7} ) q^{75} + ( -1609942646784 - 24313856 \beta_{1} - 695549952 \beta_{2} + 744177664 \beta_{3} - 67895296 \beta_{4} - 19841024 \beta_{5} + 4816896 \beta_{6} - 4931584 \beta_{7} ) q^{76} + ( 9360887809668 - 9360887809668 \beta_{1} + 107024426 \beta_{2} + 45527252 \beta_{3} - 102601262 \beta_{4} - 61402410 \beta_{5} + 6822490 \beta_{6} + 11414802 \beta_{7} ) q^{77} + ( 8277692774656 + 8276806448768 \beta_{1} + 28886400 \beta_{2} + 857439488 \beta_{3} - 60595200 \beta_{4} + 4998400 \beta_{5} - 4777600 \beta_{6} - 6732800 \beta_{7} ) q^{78} + ( -3306913896 + 2696263062456 \beta_{1} - 3368645120 \beta_{2} - 3245182672 \beta_{3} - 9337944 \beta_{4} - 200481744 \beta_{5} + 1983376 \beta_{6} + 19254824 \beta_{7} ) q^{79} + ( 845773012992 + 338362892288 \beta_{1} - 671088640 \beta_{2} + 67108864 \beta_{3} - 67108864 \beta_{4} ) q^{80} + ( -40340458451862 + 31924098 \beta_{1} - 5694943377 \beta_{2} + 5631095181 \beta_{3} - 8802852 \beta_{4} + 234193722 \beta_{5} + 18162762 \beta_{6} + 43206192 \beta_{7} ) q^{81} + ( 753209302784 - 753209302784 \beta_{1} - 761802112 \beta_{2} + 918784 \beta_{3} - 1176704 \beta_{4} - 1560960 \beta_{5} + 173440 \beta_{6} + 51584 \beta_{7} ) q^{82} + ( 16784623378201 + 16783626471890 \beta_{1} - 78570214 \beta_{2} + 1075476525 \beta_{3} + 104660910 \beta_{4} - 180696534 \beta_{5} - 20425264 \beta_{6} + 11628990 \beta_{7} ) q^{83} + ( 3736485888 - 200877760512 \beta_{1} + 3742310400 \beta_{2} + 3730661376 \beta_{3} + 53051392 \beta_{4} + 43466752 \beta_{5} + 9019392 \beta_{6} - 7954432 \beta_{7} ) q^{84} + ( 18164177984635 - 12814880610960 \beta_{1} - 3887794213 \beta_{2} - 1694237737 \beta_{3} - 144382593 \beta_{4} - 163624533 \beta_{5} - 28403500 \beta_{6} + 21533000 \beta_{7} ) q^{85} + ( -15079408658368 - 1397888 \beta_{1} - 3226491840 \beta_{2} + 3229287616 \beta_{3} + 15656192 \beta_{4} - 42705152 \beta_{5} - 4612992 \beta_{6} - 7618432 \beta_{7} ) q^{86} + ( 8557836894744 - 8557836894744 \beta_{1} + 9872289618 \beta_{2} - 52895250 \beta_{3} + 22508000 \beta_{4} + 106150860 \beta_{5} - 11794540 \beta_{6} + 6077450 \beta_{7} ) q^{87} + ( -537171853312 - 535161733120 \beta_{1} - 4194304 \beta_{2} - 2005925888 \beta_{3} + 47185920 \beta_{4} + 105906176 \beta_{5} + 22020096 \beta_{6} + 5242880 \beta_{7} ) q^{88} + ( -10417195450 + 19696247117970 \beta_{1} - 10446868780 \beta_{2} - 10387522120 \beta_{3} + 168636940 \beta_{4} - 14897900 \beta_{5} + 31504950 \beta_{6} - 11112190 \beta_{7} ) q^{89} + ( 14069648462144 - 32771032496384 \beta_{1} + 9526636864 \beta_{2} - 12596559936 \beta_{3} + 113830976 \beta_{4} - 7672896 \beta_{5} + 34064000 \beta_{6} - 31072000 \beta_{7} ) q^{90} + ( -29361477460411 + 11393338 \beta_{1} + 4120706753 \beta_{2} - 4143493429 \beta_{3} + 146387868 \beta_{4} - 234169188 \beta_{5} - 30900298 \beta_{6} - 40653778 \beta_{7} ) q^{91} + ( -4891409350656 + 4891409350656 \beta_{1} + 4477222912 \beta_{2} - 71786496 \beta_{3} + 14974976 \beta_{4} + 149667840 \beta_{5} - 16629760 \beta_{6} + 11362304 \beta_{7} ) q^{92} + ( 8351640291252 + 8336296817550 \beta_{1} + 148377078 \beta_{2} + 15195096624 \beta_{3} - 250409970 \beta_{4} + 194679218 \beta_{5} + 9260428 \beta_{6} - 27823330 \beta_{7} ) q^{93} + ( 4640085376 + 28760402637248 \beta_{1} + 4676883904 \beta_{2} + 4603286848 \beta_{3} - 204072832 \beta_{4} + 20855168 \beta_{5} - 38177472 \beta_{6} + 13185472 \beta_{7} ) q^{94} + ( 3338311351890 - 46164174505040 \beta_{1} - 11239015260 \beta_{2} - 2189489060 \beta_{3} + 166602960 \beta_{4} + 421986390 \beta_{5} - 17946000 \beta_{6} - 8957000 \beta_{7} ) q^{95} + ( -1507533520896 - 4294967296 \beta_{2} + 4294967296 \beta_{3} ) q^{96} + ( 20940455568641 - 20940455568641 \beta_{1} + 17924648510 \beta_{2} + 161274218 \beta_{3} + 106483942 \beta_{4} - 386686530 \beta_{5} + 42965170 \beta_{6} - 53551632 \beta_{7} ) q^{97} + ( 9852920873152 + 9847519569216 \beta_{1} - 73034752 \beta_{2} + 5474338688 \beta_{3} + 99953280 \beta_{4} - 160560512 \beta_{5} - 17505152 \beta_{6} + 11105920 \beta_{7} ) q^{98} + ( 8028098965 + 130178319585068 \beta_{1} + 8353351909 \beta_{2} + 7702846021 \beta_{3} - 503551786 \beta_{4} + 796190864 \beta_{5} - 107994606 \beta_{6} - 36421244 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 512q^{2} + 1404q^{3} - 100860q^{5} + 179712q^{6} + 1333276q^{7} - 4194304q^{8} + O(q^{10}) \) \( 8q + 512q^{2} + 1404q^{3} - 100860q^{5} + 179712q^{6} + 1333276q^{7} - 4194304q^{8} - 9034240q^{10} + 8181256q^{11} + 11501568q^{12} - 10529136q^{13} + 721621020q^{15} - 536870912q^{16} - 1669855424q^{17} + 2079049728q^{18} - 330137600q^{20} + 192520776q^{21} + 523600384q^{22} - 4778918036q^{23} + 15448124800q^{25} - 1347729408q^{26} - 20842498800q^{27} - 10922196992q^{28} + 48428597760q^{30} + 35788345016q^{31} - 34359738368q^{32} - 130209466872q^{33} - 30069746420q^{35} + 266118365184q^{36} - 31454650344q^{37} - 100622935040q^{38} + 31750881280q^{40} + 94198718056q^{41} + 12321329664q^{42} - 942463128516q^{43} + 2927734430100q^{45} - 611701508608q^{46} - 1797815170164q^{47} - 94220845056q^{48} + 712475699200q^{50} - 1982287069224q^{51} - 86254682112q^{52} + 540438256184q^{53} + 788235527880q^{55} - 1398041214976q^{56} + 6485834218080q^{57} - 841739223040q^{58} + 287341117440q^{60} - 19913995236984q^{61} + 2290454081024q^{62} + 25848827893644q^{63} - 7034160749880q^{65} - 16666811759616q^{66} + 13553624120956q^{67} + 13679455633408q^{68} - 16955995571200q^{70} - 51541518798664q^{71} + 17031575371776q^{72} + 44383738947944q^{73} - 19405526378100q^{75} - 12879735685120q^{76} + 74886492270632q^{77} + 66217996893696q^{78} + 6768600023040q^{80} - 322723412222112q^{81} + 6028717955584q^{82} + 134272999400364q^{83} + 145335752004880q^{85} - 120635280450048q^{86} + 68423417580480q^{87} - 4289334345728q^{88} + 112569467389440q^{90} - 234891728536584q^{91} - 39148896550912q^{92} + 66751748435208q^{93} + 26760204832400q^{95} - 12060268167168q^{96} + 167451300858216q^{97} + 78801761769472q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 513401 x^{6} + 81983771116 x^{4} + 4511941511282436 x^{2} + 65023716741123799296\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 9772303 \nu^{7} + 5013078268271 \nu^{5} + 721334070988048924 \nu^{3} + 23831357983732936482252 \nu \)\()/ \)\(15\!\cdots\!00\)\( \)
\(\beta_{2}\)\(=\)\((\)\(1104024388163 \nu^{7} - 71452148119836 \nu^{6} + 505763475967392151 \nu^{5} - 33079912668389459532 \nu^{4} + 67321304473302929593784 \nu^{3} - 5542229826789380287596528 \nu^{2} + 2489788103336045869015673652 \nu - 291047676618223143718254787104\)\()/ \)\(63\!\cdots\!00\)\( \)
\(\beta_{3}\)\(=\)\((\)\(1104024388163 \nu^{7} + 71452148119836 \nu^{6} + 505763475967392151 \nu^{5} + 33079912668389459532 \nu^{4} + 67321304473302929593784 \nu^{3} + 5542229826789380287596528 \nu^{2} + 2489788103336045869015673652 \nu + 290984598999955115428401409104\)\()/ \)\(63\!\cdots\!00\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-4407175440013 \nu^{7} - 7796484773643456 \nu^{6} - 2018476963410637181 \nu^{5} - 3307504975632395167872 \nu^{4} - 268626639886399629707524 \nu^{3} - 359267028216863298330583488 \nu^{2} - 14983603844947298493324638532 \nu - 7768590150425121839960258909184\)\()/ \)\(42\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-1104160093943 \nu^{7} - 18699648858099 \nu^{6} - 442546531908405541 \nu^{5} - 24835985815928743863 \nu^{4} - 40261764624245990397014 \nu^{3} - 5589653035127711509629252 \nu^{2} - 117698470449010572267601752 \nu - 204527215831877369931592779786\)\()/ \)\(26\!\cdots\!50\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-30820479348545 \nu^{7} + 37595107279646048 \nu^{6} - 15126662941945185265 \nu^{5} + 14859808685493497673376 \nu^{4} - 2311303512041380161037460 \nu^{3} + 1423818067298037940936387904 \nu^{2} - 157910037332920349419299424980 \nu + 24977165904727886511983815044672\)\()/ \)\(42\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(1328110165567 \nu^{7} - 135051672968520 \nu^{6} + 546659214860534879 \nu^{5} - 90409540937250634440 \nu^{4} + 54572332856855579969116 \nu^{3} - 15558297965176276353922560 \nu^{2} + 926687918164199945320282188 \nu - 508012307740998262772053854480\)\()/ \)\(51\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} - 3 \beta_{6} + 2 \beta_{5} - 16 \beta_{4} - 164 \beta_{3} - 158 \beta_{2} + 1661 \beta_{1} - 161\)\()/6000\)
\(\nu^{2}\)\(=\)\((\)\(27 \beta_{7} - 11 \beta_{6} + 124 \beta_{5} + 218 \beta_{4} + 25622 \beta_{3} - 25796 \beta_{2} + 87 \beta_{1} - 154020387\)\()/1200\)
\(\nu^{3}\)\(=\)\((\)\(-253279 \beta_{7} + 502737 \beta_{6} + 521842 \beta_{5} + 2766964 \beta_{4} + 60221756 \beta_{3} + 59730482 \beta_{2} - 88010704619 \beta_{1} + 59976119\)\()/6000\)
\(\nu^{4}\)\(=\)\((\)\(-18604191 \beta_{7} - 1084537 \beta_{6} - 94105492 \beta_{5} - 63571394 \beta_{4} - 7198611326 \beta_{3} + 7266520868 \beta_{2} - 33954771 \beta_{1} + 29883947325471\)\()/1200\)
\(\nu^{5}\)\(=\)\((\)\(55897578691 \beta_{7} - 107558290773 \beta_{6} - 128742623818 \beta_{5} - 593689032556 \beta_{4} - 16660560017324 \beta_{3} - 16565712326378 \beta_{2} + 37584534458738351 \beta_{1} - 16613136171851\)\()/6000\)
\(\nu^{6}\)\(=\)\((\)\(6518835619071 \beta_{7} + 1355325485497 \beta_{6} + 33949503580852 \beta_{5} + 12522087621314 \beta_{4} + 1875003003575006 \beta_{3} - 1892946393138308 \beta_{2} + 8971694781651 \beta_{1} - 6776022317862069951\)\()/1200\)
\(\nu^{7}\)\(=\)\((\)\(-12417904234321339 \beta_{7} + 25383052198788717 \beta_{6} + 22646833548058522 \beta_{5} + 139333165228264924 \beta_{4} + 4501398457648611596 \beta_{3} + 4474373674259384762 \beta_{2} - 11854833123495648736679 \beta_{1} + 4487886065953998179\)\()/6000\)

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
148.763i
267.359i
394.425i
514.022i
148.763i
267.359i
394.425i
514.022i
64.0000 64.0000i −2738.23 2738.23i 8192.00i −76838.8 14117.8i −350493. 390848. 390848.i −524288. 524288.i 1.02128e7i −5.82123e6 + 4.01414e6i
3.2 64.0000 64.0000i −423.414 423.414i 8192.00i 72594.1 + 28872.4i −54197.0 346999. 346999.i −524288. 524288.i 4.42441e6i 6.49385e6 2.79819e6i
3.3 64.0000 64.0000i 803.510 + 803.510i 8192.00i −66844.3 + 40439.5i 102849. −657264. + 657264.i −524288. 524288.i 3.49171e6i −1.68991e6 + 6.86616e6i
3.4 64.0000 64.0000i 3060.13 + 3060.13i 8192.00i 20659.0 75344.0i 391697. 586054. 586054.i −524288. 524288.i 1.39459e7i −3.49984e6 6.14419e6i
7.1 64.0000 + 64.0000i −2738.23 + 2738.23i 8192.00i −76838.8 + 14117.8i −350493. 390848. + 390848.i −524288. + 524288.i 1.02128e7i −5.82123e6 4.01414e6i
7.2 64.0000 + 64.0000i −423.414 + 423.414i 8192.00i 72594.1 28872.4i −54197.0 346999. + 346999.i −524288. + 524288.i 4.42441e6i 6.49385e6 + 2.79819e6i
7.3 64.0000 + 64.0000i 803.510 803.510i 8192.00i −66844.3 40439.5i 102849. −657264. 657264.i −524288. + 524288.i 3.49171e6i −1.68991e6 6.86616e6i
7.4 64.0000 + 64.0000i 3060.13 3060.13i 8192.00i 20659.0 + 75344.0i 391697. 586054. + 586054.i −524288. + 524288.i 1.39459e7i −3.49984e6 + 6.14419e6i
\(n\): 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 10.15.c.b 8
3.b odd 2 1 90.15.g.b 8
4.b odd 2 1 80.15.p.b 8
5.b even 2 1 50.15.c.e 8
5.c odd 4 1 inner 10.15.c.b 8
5.c odd 4 1 50.15.c.e 8
15.e even 4 1 90.15.g.b 8
20.e even 4 1 80.15.p.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.15.c.b 8 1.a even 1 1 trivial
10.15.c.b 8 5.c odd 4 1 inner
50.15.c.e 8 5.b even 2 1
50.15.c.e 8 5.c odd 4 1
80.15.p.b 8 4.b odd 2 1
80.15.p.b 8 20.e even 4 1
90.15.g.b 8 3.b odd 2 1
90.15.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 \(S_{15}^{\mathrm{new}}(10, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 128 T + 8192 T^{2} )^{4} \)
$3$ \( 1 - 1404 T + 985608 T^{2} + 4247805564 T^{3} + 35387353734768 T^{4} - 2379282410867004 T^{5} - 22515620380185266280 T^{6} + \)\(83\!\cdots\!00\)\( T^{7} + \)\(11\!\cdots\!66\)\( T^{8} + \)\(39\!\cdots\!00\)\( T^{9} - \)\(51\!\cdots\!80\)\( T^{10} - \)\(26\!\cdots\!36\)\( T^{11} + \)\(18\!\cdots\!28\)\( T^{12} + \)\(10\!\cdots\!36\)\( T^{13} + \)\(11\!\cdots\!48\)\( T^{14} - \)\(80\!\cdots\!56\)\( T^{15} + \)\(27\!\cdots\!41\)\( T^{16} \)
$5$ \( 1 + 100860 T - 2637692600 T^{2} - 261079989187500 T^{3} + 16542779034667968750 T^{4} - \)\(15\!\cdots\!00\)\( T^{5} - \)\(98\!\cdots\!00\)\( T^{6} + \)\(22\!\cdots\!00\)\( T^{7} + \)\(13\!\cdots\!25\)\( T^{8} \)
$7$ \( 1 - 1333276 T + 888812446088 T^{2} - 1274150359547644964 T^{3} + \)\(98\!\cdots\!88\)\( T^{4} + \)\(10\!\cdots\!64\)\( T^{5} - \)\(20\!\cdots\!60\)\( T^{6} + \)\(41\!\cdots\!60\)\( T^{7} - \)\(66\!\cdots\!34\)\( T^{8} + \)\(28\!\cdots\!40\)\( T^{9} - \)\(95\!\cdots\!60\)\( T^{10} + \)\(34\!\cdots\!36\)\( T^{11} + \)\(20\!\cdots\!88\)\( T^{12} - \)\(18\!\cdots\!36\)\( T^{13} + \)\(86\!\cdots\!88\)\( T^{14} - \)\(88\!\cdots\!24\)\( T^{15} + \)\(44\!\cdots\!01\)\( T^{16} \)
$11$ \( ( 1 - 4090628 T + 618343617046408 T^{2} + \)\(41\!\cdots\!84\)\( T^{3} + \)\(18\!\cdots\!70\)\( T^{4} + \)\(15\!\cdots\!44\)\( T^{5} + \)\(89\!\cdots\!48\)\( T^{6} - \)\(22\!\cdots\!88\)\( T^{7} + \)\(20\!\cdots\!61\)\( T^{8} )^{2} \)
$13$ \( 1 + 10529136 T + 55431352453248 T^{2} + \)\(11\!\cdots\!44\)\( T^{3} - \)\(22\!\cdots\!52\)\( T^{4} - \)\(54\!\cdots\!04\)\( T^{5} + \)\(20\!\cdots\!20\)\( T^{6} - \)\(95\!\cdots\!00\)\( T^{7} + \)\(30\!\cdots\!86\)\( T^{8} - \)\(37\!\cdots\!00\)\( T^{9} + \)\(32\!\cdots\!20\)\( T^{10} - \)\(33\!\cdots\!76\)\( T^{11} - \)\(54\!\cdots\!32\)\( T^{12} + \)\(10\!\cdots\!56\)\( T^{13} + \)\(20\!\cdots\!28\)\( T^{14} + \)\(15\!\cdots\!44\)\( T^{15} + \)\(57\!\cdots\!81\)\( T^{16} \)
$17$ \( 1 + 1669855424 T + 1394208568531109888 T^{2} + \)\(86\!\cdots\!56\)\( T^{3} + \)\(44\!\cdots\!08\)\( T^{4} + \)\(17\!\cdots\!84\)\( T^{5} + \)\(60\!\cdots\!80\)\( T^{6} + \)\(18\!\cdots\!80\)\( T^{7} + \)\(61\!\cdots\!86\)\( T^{8} + \)\(31\!\cdots\!20\)\( T^{9} + \)\(17\!\cdots\!80\)\( T^{10} + \)\(84\!\cdots\!76\)\( T^{11} + \)\(35\!\cdots\!48\)\( T^{12} + \)\(11\!\cdots\!44\)\( T^{13} + \)\(31\!\cdots\!48\)\( T^{14} + \)\(64\!\cdots\!16\)\( T^{15} + \)\(64\!\cdots\!61\)\( T^{16} \)
$19$ \( 1 - 3514167312421177768 T^{2} + \)\(61\!\cdots\!48\)\( T^{4} - \)\(74\!\cdots\!16\)\( T^{6} + \)\(67\!\cdots\!70\)\( T^{8} - \)\(47\!\cdots\!56\)\( T^{10} + \)\(25\!\cdots\!88\)\( T^{12} - \)\(91\!\cdots\!28\)\( T^{14} + \)\(16\!\cdots\!61\)\( T^{16} \)
$23$ \( 1 + 4778918036 T + 11419028797403048648 T^{2} + \)\(17\!\cdots\!64\)\( T^{3} + \)\(24\!\cdots\!28\)\( T^{4} + \)\(15\!\cdots\!16\)\( T^{5} + \)\(47\!\cdots\!80\)\( T^{6} + \)\(13\!\cdots\!20\)\( T^{7} + \)\(30\!\cdots\!66\)\( T^{8} + \)\(15\!\cdots\!80\)\( T^{9} + \)\(64\!\cdots\!80\)\( T^{10} + \)\(24\!\cdots\!64\)\( T^{11} + \)\(44\!\cdots\!08\)\( T^{12} + \)\(37\!\cdots\!36\)\( T^{13} + \)\(27\!\cdots\!68\)\( T^{14} + \)\(13\!\cdots\!84\)\( T^{15} + \)\(32\!\cdots\!21\)\( T^{16} \)
$29$ \( 1 - \)\(69\!\cdots\!48\)\( T^{2} + \)\(19\!\cdots\!08\)\( T^{4} - \)\(46\!\cdots\!96\)\( T^{6} - \)\(72\!\cdots\!30\)\( T^{8} - \)\(41\!\cdots\!56\)\( T^{10} + \)\(15\!\cdots\!68\)\( T^{12} - \)\(48\!\cdots\!88\)\( T^{14} + \)\(61\!\cdots\!41\)\( T^{16} \)
$31$ \( ( 1 - 17894172508 T + \)\(21\!\cdots\!08\)\( T^{2} - \)\(31\!\cdots\!36\)\( T^{3} + \)\(22\!\cdots\!70\)\( T^{4} - \)\(23\!\cdots\!56\)\( T^{5} + \)\(12\!\cdots\!28\)\( T^{6} - \)\(77\!\cdots\!88\)\( T^{7} + \)\(32\!\cdots\!81\)\( T^{8} )^{2} \)
$37$ \( 1 + 31454650344 T + \)\(49\!\cdots\!68\)\( T^{2} + \)\(50\!\cdots\!76\)\( T^{3} + \)\(17\!\cdots\!48\)\( T^{4} + \)\(82\!\cdots\!64\)\( T^{5} + \)\(17\!\cdots\!40\)\( T^{6} + \)\(56\!\cdots\!60\)\( T^{7} + \)\(16\!\cdots\!06\)\( T^{8} + \)\(50\!\cdots\!40\)\( T^{9} + \)\(14\!\cdots\!40\)\( T^{10} + \)\(60\!\cdots\!16\)\( T^{11} + \)\(11\!\cdots\!68\)\( T^{12} + \)\(29\!\cdots\!24\)\( T^{13} + \)\(26\!\cdots\!48\)\( T^{14} + \)\(15\!\cdots\!76\)\( T^{15} + \)\(43\!\cdots\!81\)\( T^{16} \)
$41$ \( ( 1 - 47099359028 T + \)\(15\!\cdots\!88\)\( T^{2} - \)\(53\!\cdots\!96\)\( T^{3} + \)\(85\!\cdots\!70\)\( T^{4} - \)\(20\!\cdots\!56\)\( T^{5} + \)\(21\!\cdots\!48\)\( T^{6} - \)\(25\!\cdots\!68\)\( T^{7} + \)\(20\!\cdots\!41\)\( T^{8} )^{2} \)
$43$ \( 1 + 942463128516 T + \)\(44\!\cdots\!28\)\( T^{2} + \)\(18\!\cdots\!24\)\( T^{3} + \)\(70\!\cdots\!88\)\( T^{4} + \)\(23\!\cdots\!16\)\( T^{5} + \)\(67\!\cdots\!80\)\( T^{6} + \)\(20\!\cdots\!20\)\( T^{7} + \)\(58\!\cdots\!06\)\( T^{8} + \)\(14\!\cdots\!80\)\( T^{9} + \)\(37\!\cdots\!80\)\( T^{10} + \)\(93\!\cdots\!84\)\( T^{11} + \)\(21\!\cdots\!88\)\( T^{12} + \)\(39\!\cdots\!76\)\( T^{13} + \)\(72\!\cdots\!28\)\( T^{14} + \)\(11\!\cdots\!84\)\( T^{15} + \)\(88\!\cdots\!01\)\( T^{16} \)
$47$ \( 1 + 1797815170164 T + \)\(16\!\cdots\!48\)\( T^{2} + \)\(10\!\cdots\!76\)\( T^{3} + \)\(75\!\cdots\!68\)\( T^{4} + \)\(52\!\cdots\!24\)\( T^{5} + \)\(31\!\cdots\!60\)\( T^{6} + \)\(16\!\cdots\!20\)\( T^{7} + \)\(82\!\cdots\!06\)\( T^{8} + \)\(42\!\cdots\!80\)\( T^{9} + \)\(20\!\cdots\!60\)\( T^{10} + \)\(87\!\cdots\!16\)\( T^{11} + \)\(32\!\cdots\!28\)\( T^{12} + \)\(12\!\cdots\!24\)\( T^{13} + \)\(46\!\cdots\!88\)\( T^{14} + \)\(13\!\cdots\!96\)\( T^{15} + \)\(18\!\cdots\!41\)\( T^{16} \)
$53$ \( 1 - 540438256184 T + \)\(14\!\cdots\!28\)\( T^{2} - \)\(19\!\cdots\!56\)\( T^{3} + \)\(31\!\cdots\!68\)\( T^{4} - \)\(25\!\cdots\!64\)\( T^{5} + \)\(91\!\cdots\!40\)\( T^{6} - \)\(27\!\cdots\!60\)\( T^{7} + \)\(74\!\cdots\!86\)\( T^{8} - \)\(37\!\cdots\!40\)\( T^{9} + \)\(17\!\cdots\!40\)\( T^{10} - \)\(66\!\cdots\!76\)\( T^{11} + \)\(11\!\cdots\!28\)\( T^{12} - \)\(95\!\cdots\!44\)\( T^{13} + \)\(10\!\cdots\!68\)\( T^{14} - \)\(51\!\cdots\!76\)\( T^{15} + \)\(13\!\cdots\!41\)\( T^{16} \)
$59$ \( 1 - \)\(17\!\cdots\!88\)\( T^{2} + \)\(15\!\cdots\!88\)\( T^{4} - \)\(10\!\cdots\!36\)\( T^{6} + \)\(60\!\cdots\!70\)\( T^{8} - \)\(38\!\cdots\!56\)\( T^{10} + \)\(22\!\cdots\!08\)\( T^{12} - \)\(96\!\cdots\!68\)\( T^{14} + \)\(21\!\cdots\!81\)\( T^{16} \)
$61$ \( ( 1 + 9956997618492 T + \)\(75\!\cdots\!88\)\( T^{2} + \)\(34\!\cdots\!84\)\( T^{3} + \)\(13\!\cdots\!70\)\( T^{4} + \)\(34\!\cdots\!44\)\( T^{5} + \)\(73\!\cdots\!28\)\( T^{6} + \)\(95\!\cdots\!32\)\( T^{7} + \)\(95\!\cdots\!61\)\( T^{8} )^{2} \)
$67$ \( 1 - 13553624120956 T + \)\(91\!\cdots\!68\)\( T^{2} - \)\(94\!\cdots\!64\)\( T^{3} - \)\(40\!\cdots\!92\)\( T^{4} + \)\(35\!\cdots\!24\)\( T^{5} - \)\(10\!\cdots\!40\)\( T^{6} - \)\(73\!\cdots\!80\)\( T^{7} + \)\(85\!\cdots\!66\)\( T^{8} - \)\(26\!\cdots\!20\)\( T^{9} - \)\(13\!\cdots\!40\)\( T^{10} + \)\(17\!\cdots\!36\)\( T^{11} - \)\(74\!\cdots\!52\)\( T^{12} - \)\(62\!\cdots\!36\)\( T^{13} + \)\(22\!\cdots\!28\)\( T^{14} - \)\(12\!\cdots\!04\)\( T^{15} + \)\(33\!\cdots\!61\)\( T^{16} \)
$71$ \( ( 1 + 25770759399332 T + \)\(37\!\cdots\!08\)\( T^{2} + \)\(34\!\cdots\!24\)\( T^{3} + \)\(30\!\cdots\!70\)\( T^{4} + \)\(28\!\cdots\!44\)\( T^{5} + \)\(26\!\cdots\!88\)\( T^{6} + \)\(14\!\cdots\!12\)\( T^{7} + \)\(46\!\cdots\!21\)\( T^{8} )^{2} \)
$73$ \( 1 - 44383738947944 T + \)\(98\!\cdots\!68\)\( T^{2} - \)\(14\!\cdots\!56\)\( T^{3} + \)\(17\!\cdots\!28\)\( T^{4} - \)\(20\!\cdots\!44\)\( T^{5} + \)\(27\!\cdots\!00\)\( T^{6} - \)\(36\!\cdots\!40\)\( T^{7} + \)\(44\!\cdots\!86\)\( T^{8} - \)\(44\!\cdots\!60\)\( T^{9} + \)\(40\!\cdots\!00\)\( T^{10} - \)\(38\!\cdots\!76\)\( T^{11} + \)\(39\!\cdots\!08\)\( T^{12} - \)\(40\!\cdots\!44\)\( T^{13} + \)\(32\!\cdots\!88\)\( T^{14} - \)\(17\!\cdots\!36\)\( T^{15} + \)\(49\!\cdots\!21\)\( T^{16} \)
$79$ \( 1 - \)\(95\!\cdots\!48\)\( T^{2} + \)\(83\!\cdots\!08\)\( T^{4} - \)\(42\!\cdots\!96\)\( T^{6} + \)\(19\!\cdots\!70\)\( T^{8} - \)\(57\!\cdots\!56\)\( T^{10} + \)\(15\!\cdots\!68\)\( T^{12} - \)\(24\!\cdots\!88\)\( T^{14} + \)\(34\!\cdots\!41\)\( T^{16} \)
$83$ \( 1 - 134272999400364 T + \)\(90\!\cdots\!48\)\( T^{2} - \)\(43\!\cdots\!16\)\( T^{3} + \)\(17\!\cdots\!08\)\( T^{4} - \)\(59\!\cdots\!64\)\( T^{5} + \)\(18\!\cdots\!40\)\( T^{6} - \)\(52\!\cdots\!60\)\( T^{7} + \)\(14\!\cdots\!46\)\( T^{8} - \)\(38\!\cdots\!40\)\( T^{9} + \)\(99\!\cdots\!40\)\( T^{10} - \)\(23\!\cdots\!96\)\( T^{11} + \)\(50\!\cdots\!48\)\( T^{12} - \)\(93\!\cdots\!84\)\( T^{13} + \)\(14\!\cdots\!08\)\( T^{14} - \)\(15\!\cdots\!76\)\( T^{15} + \)\(86\!\cdots\!61\)\( T^{16} \)
$89$ \( 1 - \)\(46\!\cdots\!28\)\( T^{2} + \)\(19\!\cdots\!68\)\( T^{4} - \)\(55\!\cdots\!76\)\( T^{6} + \)\(12\!\cdots\!70\)\( T^{8} - \)\(21\!\cdots\!56\)\( T^{10} + \)\(29\!\cdots\!48\)\( T^{12} - \)\(26\!\cdots\!48\)\( T^{14} + \)\(21\!\cdots\!21\)\( T^{16} \)
$97$ \( 1 - 167451300858216 T + \)\(14\!\cdots\!28\)\( T^{2} - \)\(36\!\cdots\!44\)\( T^{3} + \)\(70\!\cdots\!68\)\( T^{4} - \)\(15\!\cdots\!36\)\( T^{5} + \)\(16\!\cdots\!40\)\( T^{6} - \)\(70\!\cdots\!40\)\( T^{7} + \)\(23\!\cdots\!86\)\( T^{8} - \)\(45\!\cdots\!60\)\( T^{9} + \)\(71\!\cdots\!40\)\( T^{10} - \)\(43\!\cdots\!24\)\( T^{11} + \)\(12\!\cdots\!28\)\( T^{12} - \)\(43\!\cdots\!56\)\( T^{13} + \)\(10\!\cdots\!68\)\( T^{14} - \)\(84\!\cdots\!24\)\( T^{15} + \)\(32\!\cdots\!41\)\( T^{16} \)
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