# Properties

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

# Related objects

## 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: $$2^{13}\cdot 3^{2}\cdot 5^{11}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 128 + 128 \beta_{1} ) q^{2} + ( -1305 + 1305 \beta_{1} + \beta_{3} ) q^{3} + 32768 \beta_{1} q^{4} + ( -27045 + 45491 \beta_{1} + 17 \beta_{2} - 27 \beta_{3} + \beta_{6} ) q^{5} + ( -334080 - 128 \beta_{2} + 128 \beta_{3} ) q^{6} + ( 390884 + 390906 \beta_{1} - 72 \beta_{2} + 11 \beta_{3} - 20 \beta_{4} - 9 \beta_{5} - 2 \beta_{6} + 31 \beta_{7} ) q^{7} + ( -4194304 + 4194304 \beta_{1} ) q^{8} + ( 18 + 4142703 \beta_{1} - 2459 \beta_{2} - 2477 \beta_{3} - 9 \beta_{4} + 9 \beta_{5} + 12 \beta_{6} - 24 \beta_{7} ) q^{9} +O(q^{10})$$ $$q + ( 128 + 128 \beta_{1} ) q^{2} + ( -1305 + 1305 \beta_{1} + \beta_{3} ) q^{3} + 32768 \beta_{1} q^{4} + ( -27045 + 45491 \beta_{1} + 17 \beta_{2} - 27 \beta_{3} + \beta_{6} ) q^{5} + ( -334080 - 128 \beta_{2} + 128 \beta_{3} ) q^{6} + ( 390884 + 390906 \beta_{1} - 72 \beta_{2} + 11 \beta_{3} - 20 \beta_{4} - 9 \beta_{5} - 2 \beta_{6} + 31 \beta_{7} ) q^{7} + ( -4194304 + 4194304 \beta_{1} ) q^{8} + ( 18 + 4142703 \beta_{1} - 2459 \beta_{2} - 2477 \beta_{3} - 9 \beta_{4} + 9 \beta_{5} + 12 \beta_{6} - 24 \beta_{7} ) q^{9} + ( -9284608 + 2361088 \beta_{1} + 5632 \beta_{2} - 1280 \beta_{3} + 128 \beta_{5} + 128 \beta_{6} ) q^{10} + ( 62986790 + 6 \beta_{1} - 16699 \beta_{2} + 16705 \beta_{3} + 557 \beta_{4} + 569 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} ) q^{11} + ( -42762240 - 42762240 \beta_{1} - 32768 \beta_{2} ) q^{12} + ( 152206794 - 152203688 \beta_{1} + 1553 \beta_{2} - 67445 \beta_{3} - 3288 \beta_{4} + 1371 \beta_{5} - 182 \beta_{6} - 1735 \beta_{7} ) q^{13} + ( -2816 + 100069120 \beta_{1} - 10624 \beta_{2} - 7808 \beta_{3} + 1408 \beta_{4} - 1408 \beta_{5} + 896 \beta_{6} + 6528 \beta_{7} ) q^{14} + ( -622113276 + 855881664 \beta_{1} + 8495 \beta_{2} + 3654 \beta_{3} + 7500 \beta_{4} + 2631 \beta_{5} - 1326 \beta_{6} - 5625 \beta_{7} ) q^{15} -1073741824 q^{16} + ( 673268295 + 673290571 \beta_{1} + 111248 \beta_{2} + 11138 \beta_{3} - 10825 \beta_{4} + 313 \beta_{5} - 11451 \beta_{6} + 21963 \beta_{7} ) q^{17} + ( -530263680 + 530268288 \beta_{1} + 2304 \beta_{2} - 631808 \beta_{3} - 4224 \beta_{4} + 2688 \beta_{5} + 384 \beta_{6} - 1920 \beta_{7} ) q^{18} + ( -302 - 3762221656 \beta_{1} - 1727424 \beta_{2} - 1727122 \beta_{3} + 151 \beta_{4} - 151 \beta_{5} - 3833 \beta_{6} - 3229 \beta_{7} ) q^{19} + ( -1490649088 - 886210560 \beta_{1} + 884736 \beta_{2} + 557056 \beta_{3} + 32768 \beta_{5} ) q^{20} + ( 1768608018 + 37416 \beta_{1} - 1083563 \beta_{2} + 1120979 \beta_{3} + 61437 \beta_{4} + 136269 \beta_{5} - 18708 \beta_{6} + 18708 \beta_{7} ) q^{21} + ( 8062308352 + 8062309888 \beta_{1} - 4275712 \beta_{2} + 768 \beta_{3} + 71680 \beta_{4} + 72448 \beta_{5} - 73216 \beta_{6} - 70912 \beta_{7} ) q^{22} + ( -16030141218 + 16030216432 \beta_{1} + 37607 \beta_{2} - 9207832 \beta_{3} + 44233 \beta_{4} + 157054 \beta_{5} + 119447 \beta_{6} + 81840 \beta_{7} ) q^{23} + ( -10947133440 \beta_{1} - 4194304 \beta_{2} - 4194304 \beta_{3} ) q^{24} + ( -66688657725 + 32615060150 \beta_{1} - 7379625 \beta_{2} + 4250025 \beta_{3} - 240625 \beta_{4} + 725 \beta_{5} - 11350 \beta_{6} + 512500 \beta_{7} ) q^{25} + ( 38964541696 + 397568 \beta_{1} + 8831744 \beta_{2} - 8434176 \beta_{3} - 642944 \beta_{4} + 152192 \beta_{5} - 198784 \beta_{6} + 198784 \beta_{7} ) q^{26} + ( 25949150112 + 25949138160 \beta_{1} - 39893980 \beta_{2} - 5976 \beta_{3} + 12300 \beta_{4} + 6324 \beta_{5} - 348 \beta_{6} - 18276 \beta_{7} ) q^{27} + ( -12809207808 + 12808486912 \beta_{1} - 360448 \beta_{2} - 2359296 \beta_{3} + 1015808 \beta_{4} - 65536 \beta_{5} + 294912 \beta_{6} + 655360 \beta_{7} ) q^{28} + ( 456438 - 89760059392 \beta_{1} + 53176094 \beta_{2} + 52719656 \beta_{3} - 228219 \beta_{4} + 228219 \beta_{5} + 128257 \beta_{6} - 784619 \beta_{7} ) q^{29} + ( -189183352320 + 29922353664 \beta_{1} + 619648 \beta_{2} + 1555072 \beta_{3} + 240000 \beta_{4} + 167040 \beta_{5} - 506496 \beta_{6} - 1680000 \beta_{7} ) q^{30} + ( 390753459278 - 1948534 \beta_{1} + 76624907 \beta_{2} - 78573441 \beta_{3} + 2046847 \beta_{4} - 1850221 \beta_{5} + 974267 \beta_{6} - 974267 \beta_{7} ) q^{31} + ( -137438953472 - 137438953472 \beta_{1} ) q^{32} + ( 507115101918 - 507113138166 \beta_{1} + 981876 \beta_{2} + 32744770 \beta_{3} - 4938861 \beta_{4} - 1993233 \beta_{5} - 2975109 \beta_{6} - 3956985 \beta_{7} ) q^{33} + ( -2851328 + 172359534848 \beta_{1} + 12814080 \beta_{2} + 15665408 \beta_{3} + 1425664 \beta_{4} - 1425664 \beta_{5} - 1505792 \beta_{6} + 4196864 \beta_{7} ) q^{34} + ( -543298158979 - 43792988961 \beta_{1} + 160075431 \beta_{2} + 408705270 \beta_{3} + 2625625 \beta_{4} - 3101161 \beta_{5} + 3358009 \beta_{6} - 66875 \beta_{7} ) q^{35} + ( -135748091904 + 589824 \beta_{1} + 81166336 \beta_{2} - 80576512 \beta_{3} - 786432 \beta_{4} + 393216 \beta_{5} - 294912 \beta_{6} + 294912 \beta_{7} ) q^{36} + ( 198727346030 + 198723465556 \beta_{1} - 329641927 \beta_{2} - 1940237 \beta_{3} - 4600590 \beta_{4} - 6540827 \beta_{5} + 8481064 \beta_{6} + 2660353 \beta_{7} ) q^{37} + ( 481564333312 - 481564410624 \beta_{1} - 38656 \beta_{2} - 442181888 \beta_{3} - 393984 \beta_{4} - 509952 \beta_{5} - 471296 \beta_{6} - 432640 \beta_{7} ) q^{38} + ( 6445950 + 2772399118914 \beta_{1} + 361707027 \beta_{2} + 355261077 \beta_{3} - 3222975 \beta_{4} + 3222975 \beta_{5} + 7928325 \beta_{6} - 4963575 \beta_{7} ) q^{39} + ( -77368131584 - 304238034944 \beta_{1} + 41943040 \beta_{2} + 184549376 \beta_{3} + 4194304 \beta_{5} - 4194304 \beta_{6} ) q^{40} + ( -2388752863134 - 1719578 \beta_{1} - 199798067 \beta_{2} + 198078489 \beta_{3} - 7248646 \beta_{4} - 10687802 \beta_{5} + 859789 \beta_{6} - 859789 \beta_{7} ) q^{41} + ( 226377037056 + 226386615552 \beta_{1} - 282181376 \beta_{2} + 4789248 \beta_{3} + 10258560 \beta_{4} + 15047808 \beta_{5} - 19837056 \beta_{6} - 5469312 \beta_{7} ) q^{42} + ( 6465938431089 - 6465951610525 \beta_{1} - 6589718 \beta_{2} - 206806765 \beta_{3} + 27156498 \beta_{4} + 7387344 \beta_{5} + 13977062 \beta_{6} + 20566780 \beta_{7} ) q^{43} + ( -196608 + 2063951134720 \beta_{1} - 547389440 \beta_{2} - 547192832 \beta_{3} + 98304 \beta_{4} - 98304 \beta_{5} - 18644992 \beta_{6} - 18251776 \beta_{7} ) q^{44} + ( -2663674352769 - 3271116625065 \beta_{1} + 304169048 \beta_{2} - 67373102 \beta_{3} - 35126250 \beta_{4} + 369309 \beta_{5} - 6513885 \beta_{6} + 1243125 \beta_{7} ) q^{45} + ( -4103725779200 + 9627392 \beta_{1} + 1183416192 \beta_{2} - 1173788800 \beta_{3} + 16137344 \beta_{4} + 35392128 \beta_{5} - 4813696 \beta_{6} + 4813696 \beta_{7} ) q^{46} + ( -124821703342 - 124851297668 \beta_{1} - 1370776678 \beta_{2} - 14797163 \beta_{3} + 19362660 \beta_{4} + 4565497 \beta_{5} + 10231666 \beta_{6} - 34159823 \beta_{7} ) q^{47} + ( 1401233080320 - 1401233080320 \beta_{1} - 1073741824 \beta_{3} ) q^{48} + ( -13934174 + 30146905242017 \beta_{1} + 82243023 \beta_{2} + 96177197 \beta_{3} + 6967087 \beta_{4} - 6967087 \beta_{5} + 12045334 \beta_{6} + 39913682 \beta_{7} ) q^{49} + ( -12710875888000 - 4361420489600 \beta_{1} - 1488595200 \beta_{2} - 400588800 \beta_{3} + 34800000 \beta_{4} - 1360000 \beta_{5} - 1545600 \beta_{6} + 96400000 \beta_{7} ) q^{50} + ( -5590577217618 + 56740416 \beta_{1} - 1714319005 \beta_{2} + 1771059421 \beta_{3} - 67578708 \beta_{4} + 45902124 \beta_{5} - 28370208 \beta_{6} + 28370208 \beta_{7} ) q^{51} + ( 4987410448384 + 4987512225792 \beta_{1} + 2210037760 \beta_{2} + 50888704 \beta_{3} - 56852480 \beta_{4} - 5963776 \beta_{5} - 44924928 \beta_{6} + 107741184 \beta_{7} ) q^{52} + ( 17711232163842 - 17711153700880 \beta_{1} + 39231481 \beta_{2} + 2416659013 \beta_{3} - 77913181 \beta_{4} + 39781262 \beta_{5} + 549781 \beta_{6} - 38681700 \beta_{7} ) q^{53} + ( 1529856 + 6642980898816 \beta_{1} - 5105664512 \beta_{2} - 5107194368 \beta_{3} - 764928 \beta_{4} + 764928 \beta_{5} - 854016 \beta_{6} - 3913728 \beta_{7} ) q^{54} + ( -37063586441760 - 54380776764458 \beta_{1} - 4151146481 \beta_{2} - 8260083159 \beta_{3} + 153688750 \beta_{4} + 86625420 \beta_{5} + 66109062 \beta_{6} - 222540000 \beta_{7} ) q^{55} + ( -3279064924160 - 92274688 \beta_{1} + 255852544 \beta_{2} - 348127232 \beta_{3} + 213909504 \beta_{4} + 29360128 \beta_{5} + 46137344 \beta_{6} - 46137344 \beta_{7} ) q^{56} + ( 66202500759342 + 66202573174050 \beta_{1} + 7701780350 \beta_{2} + 36207354 \beta_{3} - 50454990 \beta_{4} - 14247636 \beta_{5} - 21959718 \beta_{6} + 86662344 \beta_{7} ) q^{57} + ( 11489346026240 - 11489229178112 \beta_{1} + 58424064 \beta_{2} + 13554656000 \beta_{3} - 129643264 \beta_{4} + 45628928 \beta_{5} - 12795136 \beta_{6} - 71219200 \beta_{7} ) q^{58} + ( -19412394 - 235662265304 \beta_{1} + 7198898628 \beta_{2} + 7218311022 \beta_{3} + 9706197 \beta_{4} - 9706197 \beta_{5} + 89087229 \beta_{6} + 127912017 \beta_{7} ) q^{59} + ( -28045530365952 - 20385407827968 \beta_{1} - 119734272 \beta_{2} + 278364160 \beta_{3} - 184320000 \beta_{4} - 43450368 \beta_{5} - 86212608 \beta_{6} - 245760000 \beta_{7} ) q^{60} + ( -34493904948686 - 212968298 \beta_{1} - 13302700291 \beta_{2} + 13089731993 \beta_{3} + 10210494 \beta_{4} - 415726102 \beta_{5} + 106484149 \beta_{6} - 106484149 \beta_{7} ) q^{61} + ( 50016692199936 + 50016193375232 \beta_{1} + 19865388544 \beta_{2} - 249412352 \beta_{3} + 137290240 \beta_{4} - 112122112 \beta_{5} + 361534464 \beta_{6} - 386702592 \beta_{7} ) q^{62} + ( 19445237677506 - 19445673657612 \beta_{1} - 217990053 \beta_{2} + 931159372 \beta_{3} + 232806993 \beta_{4} - 421163166 \beta_{5} - 203173113 \beta_{6} + 14816940 \beta_{7} ) q^{63} -35184372088832 \beta_{1} q^{64} + ( -96463754659881 - 92774107380611 \beta_{1} + 34480467705 \beta_{2} - 21676914111 \beta_{3} - 332453750 \beta_{4} - 596066964 \beta_{5} - 163076501 \beta_{6} + 1031098125 \beta_{7} ) q^{65} + ( 129821214730752 + 251360256 \beta_{1} - 4065650432 \beta_{2} + 4317010688 \beta_{3} - 1138668288 \beta_{4} - 635947776 \beta_{5} - 125680128 \beta_{6} + 125680128 \beta_{7} ) q^{66} + ( -87524444427195 - 87524645316791 \beta_{1} + 12400564525 \beta_{2} - 100444798 \beta_{3} + 128810760 \beta_{4} + 28365962 \beta_{5} + 72078836 \beta_{6} - 229255558 \beta_{7} ) q^{67} + ( -22062385430528 + 22061655490560 \beta_{1} - 364969984 \beta_{2} + 3645374464 \beta_{3} + 719683584 \beta_{4} - 375226368 \beta_{5} - 10256384 \beta_{6} + 354713600 \beta_{7} ) q^{68} + ( 311055624 + 286138833587262 \beta_{1} - 297101573 \beta_{2} - 608157197 \beta_{3} - 155527812 \beta_{4} + 155527812 \beta_{5} - 1079731719 \beta_{6} - 1701842967 \beta_{7} ) q^{69} + ( -63936661762304 - 75147666936320 \beta_{1} - 31824619392 \beta_{2} + 72803929728 \beta_{3} + 327520000 \beta_{4} + 32876544 \beta_{5} + 826773760 \beta_{6} - 344640000 \beta_{7} ) q^{70} + ( 240244973185614 + 105561786 \beta_{1} - 2316045661 \beta_{2} + 2421607447 \beta_{3} + 1639776287 \beta_{4} + 1850899859 \beta_{5} - 52780893 \beta_{6} + 52780893 \beta_{7} ) q^{71} + ( -17375831261184 - 17375680266240 \beta_{1} + 20703084544 \beta_{2} + 75497472 \beta_{3} - 62914560 \beta_{4} + 12582912 \beta_{5} - 88080384 \beta_{6} + 138412032 \beta_{7} ) q^{72} + ( -101086463449193 + 101087705605125 \beta_{1} + 621077966 \beta_{2} - 74674230842 \beta_{3} - 547915836 \beta_{4} + 1315318062 \beta_{5} + 694240096 \beta_{6} + 73162130 \beta_{7} ) q^{73} + ( 496700672 + 50873703883008 \beta_{1} - 41945816320 \beta_{2} - 42442516992 \beta_{3} - 248350336 \beta_{4} + 248350336 \beta_{5} + 1922802048 \beta_{6} + 929400704 \beta_{7} ) q^{74} + ( 310294048154325 - 273276948202125 \beta_{1} - 39249448650 \beta_{2} - 63143139025 \beta_{3} - 78637500 \beta_{4} + 1985645550 \beta_{5} - 1006435500 \beta_{6} - 860943750 \beta_{7} ) q^{75} + ( 123280479223808 - 9895936 \beta_{1} + 56594333696 \beta_{2} - 56604229632 \beta_{3} - 105807872 \beta_{4} - 125599744 \beta_{5} + 4947968 \beta_{6} - 4947968 \beta_{7} ) q^{76} + ( -668288121645456 - 668285150918460 \beta_{1} - 284858737112 \beta_{2} + 1485363498 \beta_{3} - 259130845 \beta_{4} + 1226232653 \beta_{5} - 2711596151 \beta_{6} + 1744494343 \beta_{7} ) q^{77} + ( -354866262139392 + 354867912302592 \beta_{1} + 825081600 \beta_{2} + 91771917312 \beta_{3} - 1047878400 \beta_{4} + 1427366400 \beta_{5} + 602284800 \beta_{6} - 222796800 \beta_{7} ) q^{78} + ( -2356617784 + 934849735744624 \beta_{1} + 79220216344 \beta_{2} + 81576834128 \beta_{3} + 1178308892 \beta_{4} - 1178308892 \beta_{5} + 943455304 \beta_{6} + 5656690872 \beta_{7} ) q^{79} + ( 29039347630080 - 48845589315584 \beta_{1} - 18253611008 \beta_{2} + 28991029248 \beta_{3} - 1073741824 \beta_{6} ) q^{80} + ( 1169953315427265 + 1476683658 \beta_{1} + 127836629573 \beta_{2} - 126359945915 \beta_{3} - 2045317704 \beta_{4} + 908049612 \beta_{5} - 738341829 \beta_{6} + 738341829 \beta_{7} ) q^{81} + ( -305760146375168 - 305760586587136 \beta_{1} - 50928199168 \beta_{2} - 220105984 \beta_{3} - 1037879680 \beta_{4} - 1257985664 \beta_{5} + 1478091648 \beta_{6} + 817773696 \beta_{7} ) q^{82} + ( -1647069827184865 + 1647070391165697 \beta_{1} + 281990416 \beta_{2} - 228256562979 \beta_{3} - 1691237096 \beta_{4} - 845265848 \beta_{5} - 1127256264 \beta_{6} - 1409246680 \beta_{7} ) q^{83} + ( -1226047488 + 57953747533824 \beta_{1} - 36732239872 \beta_{2} - 35506192384 \beta_{3} + 613023744 \beta_{4} - 613023744 \beta_{5} - 4465262592 \beta_{6} - 2013167616 \beta_{7} ) q^{84} + ( 494403155418546 - 1128156709134976 \beta_{1} + 155570193926 \beta_{2} + 216175583150 \beta_{3} + 5632618125 \beta_{4} - 1333936636 \beta_{5} + 3246859594 \beta_{6} - 766170625 \beta_{7} ) q^{85} + ( 1655281925326592 - 1686967808 \beta_{1} + 25627782016 \beta_{2} - 27314749824 \beta_{3} + 6108579584 \beta_{4} + 2734643968 \beta_{5} + 843483904 \beta_{6} - 843483904 \beta_{7} ) q^{86} + ( -1765493874833538 - 1765497922975518 \beta_{1} + 3511212398 \beta_{2} - 2024070990 \beta_{3} + 895173660 \beta_{4} - 1128897330 \beta_{5} + 3152968320 \beta_{6} - 2919244650 \beta_{7} ) q^{87} + ( -264185770409984 + 264185720078336 \beta_{1} - 25165824 \beta_{2} - 140106530816 \beta_{3} - 2323644416 \beta_{4} - 2399141888 \beta_{5} - 2373976064 \beta_{6} - 2348810240 \beta_{7} ) q^{88} + ( 4968787700 - 348584708395840 \beta_{1} + 221472295580 \beta_{2} + 216503507880 \beta_{3} - 2484393850 \beta_{4} + 2484393850 \beta_{5} + 1104393030 \beta_{6} - 8833182370 \beta_{7} ) q^{89} + ( 77752610853888 - 759653245162752 \beta_{1} + 47557395200 \beta_{2} + 30309881088 \beta_{3} - 4337040000 \beta_{4} - 786505728 \beta_{5} - 881048832 \beta_{6} + 4655280000 \beta_{7} ) q^{90} + ( 5921913309432286 - 2220590492 \beta_{1} - 361211696617 \beta_{2} + 358991106125 \beta_{3} - 7534592574 \beta_{4} - 11975773558 \beta_{5} + 1110295246 \beta_{6} - 1110295246 \beta_{7} ) q^{91} + ( -525278132043776 - 525275667431424 \beta_{1} + 301722238976 \beta_{2} + 1232306176 \beta_{3} + 2681733120 \beta_{4} + 3914039296 \beta_{5} - 5146345472 \beta_{6} - 1449426944 \beta_{7} ) q^{92} + ( -3269445619720578 + 3269430617941434 \beta_{1} - 7500889572 \beta_{2} + 395039918674 \beta_{3} + 17603736657 \beta_{4} - 4898932059 \beta_{5} + 2601957513 \beta_{6} + 10102847085 \beta_{7} ) q^{93} + ( 3788073728 - 31958144129280 \beta_{1} - 173565377920 \beta_{2} - 177353451648 \beta_{3} - 1894036864 \beta_{4} + 1894036864 \beta_{5} + 725269632 \beta_{6} - 6850877824 \beta_{7} ) q^{94} + ( -8417785958410 - 2643811520601860 \beta_{1} + 34349449500 \beta_{2} - 14010923410 \beta_{3} - 22651829375 \beta_{4} - 5917609665 \beta_{5} - 2622600635 \beta_{6} - 4111131875 \beta_{7} ) q^{95} + ( 358715668561920 + 137438953472 \beta_{2} - 137438953472 \beta_{3} ) q^{96} + ( -1351883066510263 - 1351901172734839 \beta_{1} + 770400561608 \beta_{2} - 9053112288 \beta_{3} + 1074411730 \beta_{4} - 7978700558 \beta_{5} + 17031812846 \beta_{6} - 10127524018 \beta_{7} ) q^{97} + ( -3858805654552448 + 3858802087403904 \beta_{1} - 1783574272 \beta_{2} + 22837788160 \beta_{3} + 6000738432 \beta_{4} + 650015616 \beta_{5} + 2433589888 \beta_{6} + 4217164160 \beta_{7} ) q^{98} + ( -1552720626 + 2824089147740298 \beta_{1} - 210828745871 \beta_{2} - 209276025245 \beta_{3} + 776360313 \beta_{4} - 776360313 \beta_{5} + 8691321081 \beta_{6} + 11796762333 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 1024q^{2} - 10438q^{3} - 216450q^{5} - 2672128q^{6} + 3127282q^{7} - 33554432q^{8} + O(q^{10})$$ $$8q + 1024q^{2} - 10438q^{3} - 216450q^{5} - 2672128q^{6} + 3127282q^{7} - 33554432q^{8} - 74291200q^{10} + 503961116q^{11} - 342032384q^{12} + 1217503932q^{13} - 4976914750q^{15} - 8589934592q^{16} + 5385990692q^{17} - 4243396096q^{18} - 11925913600q^{20} + 14153198396q^{21} + 64507022848q^{22} - 128259921478q^{23} - 533485437500q^{25} + 311681006592q^{26} + 207672953000q^{27} - 102474776576q^{28} - 1513467148800q^{30} + 3125721174596q^{31} - 1099511627776q^{32} + 4056976486124q^{33} - 4345883408350q^{35} - 1086309400576q^{36} + 1590466410672q^{37} + 3851630689280q^{38} - 618659840000q^{40} - 19109223712804q^{41} + 1811609394688q^{42} + 51727159732362q^{43} - 21310193383550q^{45} - 32834539898368q^{46} - 995920856358q^{47} + 11207717158912q^{48} - 101684562880000q^{50} - 44717760464924q^{51} + 39895168843776q^{52} + 141694298313952q^{53} - 296517352578900q^{55} - 26233542803456q^{56} + 529604819758160q^{57} + 91941293281280q^{58} - 224364047564800q^{60} - 275898028788324q^{61} + 400092310348288q^{62} + 155565943639322q^{63} - 771819436467000q^{65} + 1038585980447744q^{66} - 700220959215398q^{67} - 176488142995456q^{68} - 511285790540800q^{70} + 1921969049667556q^{71} - 139047603273728q^{72} - 808847266834888q^{73} + 2482300760271250q^{75} + 986017456455680q^{76} - 5345726343508436q^{77} - 2838754804096512q^{78} + 232411417804800q^{80} + 9359115176899828q^{81} - 2445980635238912q^{82} - 13177017950509038q^{83} + 3955352361175900q^{85} + 13242152891484672q^{86} - 14123970165519040q^{87} - 2113766124683264q^{88} + 621990363392000q^{90} + 47376751322244756q^{91} - 4202821106991104q^{92} - 26154699869031556q^{93} - 67475453915000q^{95} + 2869175592681472q^{96} - 10816659651879048q^{97} - 30870381725100544q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 1418558 x^{6} + 548432140705 x^{4} + 58849481441870112 x^{2} + 514279003991950266624$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} + 1418558 \nu^{5} + 525754420273 \nu^{3} + 42764650571581584 \nu$$$$)/ 3704169421431180288$$ $$\beta_{2}$$ $$=$$ $$($$$$-366819931 \nu^{7} + 32126770612 \nu^{6} - 485802060981274 \nu^{5} + 33789648479257048 \nu^{4} - 163101706184429735259 \nu^{3} + 2888700794886463721268 \nu^{2} - 13106142846055842139730736 \nu - 1109244641576126054194254528$$$$)/$$$$23\!\cdots\!00$$ $$\beta_{3}$$ $$=$$ $$($$$$-366819931 \nu^{7} - 32126770612 \nu^{6} - 485802060981274 \nu^{5} - 33789648479257048 \nu^{4} - 163101706184429735259 \nu^{3} - 2888700794886463721268 \nu^{2} - 13106142846055842139730736 \nu + 1109244641576126054194254528$$$$)/$$$$23\!\cdots\!00$$ $$\beta_{4}$$ $$=$$ $$($$$$1796724051 \nu^{7} - 1838785165028 \nu^{6} + 3180021861523754 \nu^{5} - 2803615474514587512 \nu^{4} + 1745200462931022569939 \nu^{3} - 1058523763529478882446692 \nu^{2} + 283564018530776027080353456 \nu - 54245356829648560610123362368$$$$)/$$$$23\!\cdots\!00$$ $$\beta_{5}$$ $$=$$ $$($$$$-1796724051 \nu^{7} + 6737172778340 \nu^{6} - 3180021861523754 \nu^{5} + 7969078540880330360 \nu^{4} - 1745200462931022569939 \nu^{3} + 2061286291565165287948260 \nu^{2} - 283564018530776027080353456 \nu + 81450490966040234087734655040$$$$)/$$$$23\!\cdots\!00$$ $$\beta_{6}$$ $$=$$ $$($$$$-7367028365 \nu^{7} - 6480158613444 \nu^{6} - 16961217170822710 \nu^{5} - 8130204984265073976 \nu^{4} - 11792463242731346421485 \nu^{3} - 2344190592513312823378116 \nu^{2} - 2026121404718230660828399440 \nu - 100108371243518620583127027264$$$$)/$$$$70\!\cdots\!00$$ $$\beta_{7}$$ $$=$$ $$($$$$5464733111 \nu^{7} + 2160052871148 \nu^{6} + 8037772590676994 \nu^{5} + 2710068328088357992 \nu^{4} + 3376117499996862984279 \nu^{3} + 781396864171104274459372 \nu^{2} + 485089134259291410060504816 \nu + 33369457081172873527709009088$$$$)/$$$$23\!\cdots\!00$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$7 \beta_{7} + 3 \beta_{6} - \beta_{5} + \beta_{4} + 48 \beta_{3} + 46 \beta_{2} + 24 \beta_{1} - 2$$$$)/3000$$ $$\nu^{2}$$ $$=$$ $$($$$$-4 \beta_{7} + 4 \beta_{6} + 1287 \beta_{5} + 1303 \beta_{4} + 97384 \beta_{3} - 97392 \beta_{2} - 8 \beta_{1} - 1063967196$$$$)/3000$$ $$\nu^{3}$$ $$=$$ $$($$$$-3798781 \beta_{7} - 2411649 \beta_{6} + 346783 \beta_{5} - 346783 \beta_{4} - 54039984 \beta_{3} - 53346418 \beta_{2} - 367541083992 \beta_{1} + 693566$$$$)/3000$$ $$\nu^{4}$$ $$=$$ $$($$$$6647068 \beta_{7} - 6647068 \beta_{6} - 56114433 \beta_{5} - 82702705 \beta_{4} - 3063441880 \beta_{3} + 3076736016 \beta_{2} + 13294136 \beta_{1} + 27464813906628$$$$)/120$$ $$\nu^{5}$$ $$=$$ $$($$$$2748483023827 \beta_{7} + 1852716874383 \beta_{6} - 223941537361 \beta_{5} + 223941537361 \beta_{4} + 53148863995728 \beta_{3} + 52700980921006 \beta_{2} + 434476587431757864 \beta_{1} - 447883074722$$$$)/3000$$ $$\nu^{6}$$ $$=$$ $$($$$$-174418323675844 \beta_{7} + 174418323675844 \beta_{6} + 1359751241313207 \beta_{5} + 2057424536016583 \beta_{4} + 60826009427573224 \beta_{3} - 61174846074924912 \beta_{2} - 348836647351688 \beta_{1} - 522911050714435420956$$$$)/3000$$ $$\nu^{7}$$ $$=$$ $$($$$$-2201009232915965341 \beta_{7} - 1488545173708784289 \beta_{6} + 178116014801795263 \beta_{5} - 178116014801795263 \beta_{4} - 49035688980005640624 \beta_{3} - 48679456950402050098 \beta_{2} - 411982407460615788564312 \beta_{1} + 356232029603590526$$$$)/3000$$

## 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
 623.844i − 926.880i 400.872i − 97.8350i − 623.844i 926.880i − 400.872i 97.8350i
128.000 128.000i −5522.10 5522.10i 32768.0i 105678. + 376059.i −1.41366e6 −5.16013e6 + 5.16013e6i −4.19430e6 4.19430e6i 1.79404e7i 6.16623e7 + 3.46087e7i
3.2 128.000 128.000i −5464.81 5464.81i 32768.0i 187349. 342765.i −1.39899e6 6.16251e6 6.16251e6i −4.19430e6 4.19430e6i 1.66815e7i −1.98933e7 6.78547e7i
3.3 128.000 128.000i 2246.18 + 2246.18i 32768.0i −50567.5 387338.i 575022. −5.28707e6 + 5.28707e6i −4.19430e6 4.19430e6i 3.29561e7i −5.60519e7 4.31066e7i
3.4 128.000 128.000i 3521.72 + 3521.72i 32768.0i −350685. + 172070.i 901561. 5.84833e6 5.84833e6i −4.19430e6 4.19430e6i 1.82416e7i −2.28627e7 + 6.69126e7i
7.1 128.000 + 128.000i −5522.10 + 5522.10i 32768.0i 105678. 376059.i −1.41366e6 −5.16013e6 5.16013e6i −4.19430e6 + 4.19430e6i 1.79404e7i 6.16623e7 3.46087e7i
7.2 128.000 + 128.000i −5464.81 + 5464.81i 32768.0i 187349. + 342765.i −1.39899e6 6.16251e6 + 6.16251e6i −4.19430e6 + 4.19430e6i 1.66815e7i −1.98933e7 + 6.78547e7i
7.3 128.000 + 128.000i 2246.18 2246.18i 32768.0i −50567.5 + 387338.i 575022. −5.28707e6 5.28707e6i −4.19430e6 + 4.19430e6i 3.29561e7i −5.60519e7 + 4.31066e7i
7.4 128.000 + 128.000i 3521.72 3521.72i 32768.0i −350685. 172070.i 901561. 5.84833e6 + 5.84833e6i −4.19430e6 + 4.19430e6i 1.82416e7i −2.28627e7 6.69126e7i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 7.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.b 8
3.b odd 2 1 90.17.g.a 8
4.b odd 2 1 80.17.p.b 8
5.b even 2 1 50.17.c.c 8
5.c odd 4 1 inner 10.17.c.b 8
5.c odd 4 1 50.17.c.c 8
15.e even 4 1 90.17.g.a 8
20.e even 4 1 80.17.p.b 8

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

$p$ $F_p(T)$
$2$ $$( 1 - 256 T + 32768 T^{2} )^{4}$$
$3$ $$1 + 10438 T + 54475922 T^{2} + 270089464878 T^{3} - 647631811521792 T^{4} - 2052256065916987422 T^{5} +$$$$50\!\cdots\!30$$$$T^{6} +$$$$96\!\cdots\!50$$$$T^{7} +$$$$11\!\cdots\!26$$$$T^{8} +$$$$41\!\cdots\!50$$$$T^{9} +$$$$93\!\cdots\!30$$$$T^{10} -$$$$16\!\cdots\!42$$$$T^{11} -$$$$22\!\cdots\!52$$$$T^{12} +$$$$39\!\cdots\!78$$$$T^{13} +$$$$34\!\cdots\!62$$$$T^{14} +$$$$28\!\cdots\!58$$$$T^{15} +$$$$11\!\cdots\!61$$$$T^{16}$$
$5$ $$1 + 216450 T + 290168020000 T^{2} + 121066521011718750 T^{3} +$$$$47\!\cdots\!50$$$$T^{4} +$$$$18\!\cdots\!50$$$$T^{5} +$$$$67\!\cdots\!00$$$$T^{6} +$$$$76\!\cdots\!50$$$$T^{7} +$$$$54\!\cdots\!25$$$$T^{8}$$
$7$ $$1 - 3127282 T + 4889946353762 T^{2} + 89470664313069764998 T^{3} -$$$$45\!\cdots\!12$$$$T^{4} +$$$$10\!\cdots\!98$$$$T^{5} -$$$$53\!\cdots\!90$$$$T^{6} -$$$$31\!\cdots\!70$$$$T^{7} +$$$$75\!\cdots\!66$$$$T^{8} -$$$$10\!\cdots\!70$$$$T^{9} -$$$$58\!\cdots\!90$$$$T^{10} +$$$$36\!\cdots\!98$$$$T^{11} -$$$$55\!\cdots\!12$$$$T^{12} +$$$$36\!\cdots\!98$$$$T^{13} +$$$$65\!\cdots\!62$$$$T^{14} -$$$$14\!\cdots\!82$$$$T^{15} +$$$$14\!\cdots\!01$$$$T^{16}$$
$11$ $$( 1 - 251980558 T + 69276306028444768 T^{2} +$$$$64\!\cdots\!54$$$$T^{3} -$$$$99\!\cdots\!30$$$$T^{4} +$$$$29\!\cdots\!94$$$$T^{5} +$$$$14\!\cdots\!28$$$$T^{6} -$$$$24\!\cdots\!98$$$$T^{7} +$$$$44\!\cdots\!41$$$$T^{8} )^{2}$$
$13$ $$1 - 1217503932 T + 741157912217730312 T^{2} -$$$$17\!\cdots\!32$$$$T^{3} +$$$$11\!\cdots\!28$$$$T^{4} +$$$$65\!\cdots\!28$$$$T^{5} +$$$$49\!\cdots\!80$$$$T^{6} -$$$$17\!\cdots\!00$$$$T^{7} -$$$$50\!\cdots\!94$$$$T^{8} -$$$$11\!\cdots\!00$$$$T^{9} +$$$$22\!\cdots\!80$$$$T^{10} +$$$$19\!\cdots\!88$$$$T^{11} +$$$$23\!\cdots\!08$$$$T^{12} -$$$$22\!\cdots\!32$$$$T^{13} +$$$$64\!\cdots\!92$$$$T^{14} -$$$$70\!\cdots\!92$$$$T^{15} +$$$$38\!\cdots\!21$$$$T^{16}$$
$17$ $$1 - 5385990692 T + 14504447867155319432 T^{2} -$$$$35\!\cdots\!72$$$$T^{3} +$$$$11\!\cdots\!68$$$$T^{4} +$$$$15\!\cdots\!88$$$$T^{5} -$$$$34\!\cdots\!80$$$$T^{6} +$$$$71\!\cdots\!40$$$$T^{7} -$$$$14\!\cdots\!74$$$$T^{8} +$$$$34\!\cdots\!40$$$$T^{9} -$$$$81\!\cdots\!80$$$$T^{10} +$$$$17\!\cdots\!08$$$$T^{11} +$$$$61\!\cdots\!28$$$$T^{12} -$$$$97\!\cdots\!72$$$$T^{13} +$$$$19\!\cdots\!92$$$$T^{14} -$$$$34\!\cdots\!12$$$$T^{15} +$$$$31\!\cdots\!41$$$$T^{16}$$
$19$ $$1 -$$$$13\!\cdots\!48$$$$T^{2} +$$$$10\!\cdots\!08$$$$T^{4} -$$$$49\!\cdots\!96$$$$T^{6} +$$$$16\!\cdots\!70$$$$T^{8} -$$$$41\!\cdots\!56$$$$T^{10} +$$$$71\!\cdots\!68$$$$T^{12} -$$$$80\!\cdots\!88$$$$T^{14} +$$$$47\!\cdots\!41$$$$T^{16}$$
$23$ $$1 + 128259921478 T +$$$$82\!\cdots\!42$$$$T^{2} +$$$$11\!\cdots\!38$$$$T^{3} +$$$$71\!\cdots\!48$$$$T^{4} -$$$$24\!\cdots\!62$$$$T^{5} -$$$$18\!\cdots\!30$$$$T^{6} -$$$$37\!\cdots\!10$$$$T^{7} -$$$$62\!\cdots\!54$$$$T^{8} -$$$$23\!\cdots\!10$$$$T^{9} -$$$$69\!\cdots\!30$$$$T^{10} -$$$$56\!\cdots\!22$$$$T^{11} +$$$$10\!\cdots\!68$$$$T^{12} +$$$$10\!\cdots\!38$$$$T^{13} +$$$$43\!\cdots\!62$$$$T^{14} +$$$$41\!\cdots\!38$$$$T^{15} +$$$$20\!\cdots\!81$$$$T^{16}$$
$29$ $$1 -$$$$10\!\cdots\!68$$$$T^{2} +$$$$50\!\cdots\!48$$$$T^{4} -$$$$16\!\cdots\!16$$$$T^{6} +$$$$42\!\cdots\!70$$$$T^{8} -$$$$10\!\cdots\!56$$$$T^{10} +$$$$19\!\cdots\!88$$$$T^{12} -$$$$25\!\cdots\!28$$$$T^{14} +$$$$15\!\cdots\!61$$$$T^{16}$$
$31$ $$( 1 - 1562860587298 T +$$$$14\!\cdots\!88$$$$T^{2} -$$$$64\!\cdots\!26$$$$T^{3} -$$$$25\!\cdots\!30$$$$T^{4} -$$$$47\!\cdots\!06$$$$T^{5} +$$$$75\!\cdots\!68$$$$T^{6} -$$$$60\!\cdots\!18$$$$T^{7} +$$$$27\!\cdots\!21$$$$T^{8} )^{2}$$
$37$ $$1 - 1590466410672 T +$$$$12\!\cdots\!92$$$$T^{2} -$$$$11\!\cdots\!72$$$$T^{3} +$$$$21\!\cdots\!28$$$$T^{4} -$$$$42\!\cdots\!52$$$$T^{5} +$$$$59\!\cdots\!60$$$$T^{6} -$$$$16\!\cdots\!20$$$$T^{7} +$$$$31\!\cdots\!26$$$$T^{8} -$$$$20\!\cdots\!20$$$$T^{9} +$$$$91\!\cdots\!60$$$$T^{10} -$$$$79\!\cdots\!92$$$$T^{11} +$$$$50\!\cdots\!08$$$$T^{12} -$$$$31\!\cdots\!72$$$$T^{13} +$$$$44\!\cdots\!72$$$$T^{14} -$$$$69\!\cdots\!32$$$$T^{15} +$$$$53\!\cdots\!21$$$$T^{16}$$
$41$ $$( 1 + 9554611856402 T +$$$$25\!\cdots\!28$$$$T^{2} +$$$$17\!\cdots\!34$$$$T^{3} +$$$$24\!\cdots\!70$$$$T^{4} +$$$$11\!\cdots\!94$$$$T^{5} +$$$$10\!\cdots\!68$$$$T^{6} +$$$$24\!\cdots\!42$$$$T^{7} +$$$$16\!\cdots\!61$$$$T^{8} )^{2}$$
$43$ $$1 - 51727159732362 T +$$$$13\!\cdots\!22$$$$T^{2} -$$$$24\!\cdots\!82$$$$T^{3} +$$$$39\!\cdots\!88$$$$T^{4} -$$$$60\!\cdots\!62$$$$T^{5} +$$$$86\!\cdots\!70$$$$T^{6} -$$$$11\!\cdots\!10$$$$T^{7} +$$$$13\!\cdots\!06$$$$T^{8} -$$$$15\!\cdots\!10$$$$T^{9} +$$$$16\!\cdots\!70$$$$T^{10} -$$$$15\!\cdots\!62$$$$T^{11} +$$$$13\!\cdots\!88$$$$T^{12} -$$$$11\!\cdots\!82$$$$T^{13} +$$$$86\!\cdots\!22$$$$T^{14} -$$$$45\!\cdots\!62$$$$T^{15} +$$$$12\!\cdots\!01$$$$T^{16}$$
$47$ $$1 + 995920856358 T +$$$$49\!\cdots\!82$$$$T^{2} +$$$$24\!\cdots\!98$$$$T^{3} +$$$$72\!\cdots\!08$$$$T^{4} -$$$$22\!\cdots\!82$$$$T^{5} +$$$$37\!\cdots\!90$$$$T^{6} -$$$$67\!\cdots\!90$$$$T^{7} +$$$$24\!\cdots\!66$$$$T^{8} -$$$$38\!\cdots\!90$$$$T^{9} +$$$$12\!\cdots\!90$$$$T^{10} -$$$$41\!\cdots\!02$$$$T^{11} +$$$$74\!\cdots\!48$$$$T^{12} +$$$$14\!\cdots\!98$$$$T^{13} +$$$$16\!\cdots\!22$$$$T^{14} +$$$$18\!\cdots\!78$$$$T^{15} +$$$$10\!\cdots\!61$$$$T^{16}$$
$53$ $$1 - 141694298313952 T +$$$$10\!\cdots\!52$$$$T^{2} -$$$$85\!\cdots\!12$$$$T^{3} +$$$$10\!\cdots\!08$$$$T^{4} -$$$$80\!\cdots\!52$$$$T^{5} +$$$$48\!\cdots\!60$$$$T^{6} -$$$$35\!\cdots\!20$$$$T^{7} +$$$$26\!\cdots\!46$$$$T^{8} -$$$$13\!\cdots\!20$$$$T^{9} +$$$$72\!\cdots\!60$$$$T^{10} -$$$$46\!\cdots\!72$$$$T^{11} +$$$$22\!\cdots\!48$$$$T^{12} -$$$$74\!\cdots\!12$$$$T^{13} +$$$$34\!\cdots\!92$$$$T^{14} -$$$$18\!\cdots\!32$$$$T^{15} +$$$$50\!\cdots\!61$$$$T^{16}$$
$59$ $$1 -$$$$15\!\cdots\!28$$$$T^{2} +$$$$10\!\cdots\!68$$$$T^{4} -$$$$42\!\cdots\!76$$$$T^{6} +$$$$11\!\cdots\!70$$$$T^{8} -$$$$19\!\cdots\!56$$$$T^{10} +$$$$22\!\cdots\!48$$$$T^{12} -$$$$15\!\cdots\!48$$$$T^{14} +$$$$46\!\cdots\!21$$$$T^{16}$$
$61$ $$( 1 + 137949014394162 T +$$$$83\!\cdots\!48$$$$T^{2} +$$$$12\!\cdots\!54$$$$T^{3} +$$$$34\!\cdots\!70$$$$T^{4} +$$$$45\!\cdots\!94$$$$T^{5} +$$$$11\!\cdots\!08$$$$T^{6} +$$$$68\!\cdots\!22$$$$T^{7} +$$$$18\!\cdots\!41$$$$T^{8} )^{2}$$
$67$ $$1 + 700220959215398 T +$$$$24\!\cdots\!02$$$$T^{2} +$$$$16\!\cdots\!18$$$$T^{3} +$$$$16\!\cdots\!68$$$$T^{4} +$$$$68\!\cdots\!18$$$$T^{5} +$$$$21\!\cdots\!90$$$$T^{6} +$$$$12\!\cdots\!10$$$$T^{7} +$$$$74\!\cdots\!06$$$$T^{8} +$$$$20\!\cdots\!10$$$$T^{9} +$$$$58\!\cdots\!90$$$$T^{10} +$$$$30\!\cdots\!38$$$$T^{11} +$$$$12\!\cdots\!28$$$$T^{12} +$$$$19\!\cdots\!18$$$$T^{13} +$$$$49\!\cdots\!62$$$$T^{14} +$$$$23\!\cdots\!78$$$$T^{15} +$$$$54\!\cdots\!41$$$$T^{16}$$
$71$ $$( 1 - 960984524833778 T +$$$$10\!\cdots\!28$$$$T^{2} -$$$$76\!\cdots\!86$$$$T^{3} +$$$$47\!\cdots\!70$$$$T^{4} -$$$$31\!\cdots\!06$$$$T^{5} +$$$$17\!\cdots\!48$$$$T^{6} -$$$$69\!\cdots\!58$$$$T^{7} +$$$$30\!\cdots\!81$$$$T^{8} )^{2}$$
$73$ $$1 + 808847266834888 T +$$$$32\!\cdots\!72$$$$T^{2} +$$$$17\!\cdots\!48$$$$T^{3} +$$$$23\!\cdots\!48$$$$T^{4} +$$$$21\!\cdots\!08$$$$T^{5} +$$$$11\!\cdots\!00$$$$T^{6} -$$$$32\!\cdots\!80$$$$T^{7} -$$$$22\!\cdots\!34$$$$T^{8} -$$$$20\!\cdots\!80$$$$T^{9} +$$$$49\!\cdots\!00$$$$T^{10} +$$$$60\!\cdots\!48$$$$T^{11} +$$$$41\!\cdots\!68$$$$T^{12} +$$$$20\!\cdots\!48$$$$T^{13} +$$$$24\!\cdots\!92$$$$T^{14} +$$$$39\!\cdots\!48$$$$T^{15} +$$$$32\!\cdots\!81$$$$T^{16}$$
$79$ $$1 -$$$$68\!\cdots\!68$$$$T^{2} +$$$$13\!\cdots\!48$$$$T^{4} +$$$$27\!\cdots\!84$$$$T^{6} -$$$$15\!\cdots\!30$$$$T^{8} +$$$$14\!\cdots\!44$$$$T^{10} +$$$$36\!\cdots\!88$$$$T^{12} -$$$$10\!\cdots\!28$$$$T^{14} +$$$$78\!\cdots\!61$$$$T^{16}$$
$83$ $$1 + 13177017950509038 T +$$$$86\!\cdots\!22$$$$T^{2} +$$$$39\!\cdots\!58$$$$T^{3} +$$$$14\!\cdots\!68$$$$T^{4} +$$$$45\!\cdots\!98$$$$T^{5} +$$$$12\!\cdots\!10$$$$T^{6} +$$$$32\!\cdots\!30$$$$T^{7} +$$$$75\!\cdots\!86$$$$T^{8} +$$$$16\!\cdots\!30$$$$T^{9} +$$$$32\!\cdots\!10$$$$T^{10} +$$$$59\!\cdots\!18$$$$T^{11} +$$$$96\!\cdots\!28$$$$T^{12} +$$$$13\!\cdots\!58$$$$T^{13} +$$$$14\!\cdots\!82$$$$T^{14} +$$$$11\!\cdots\!18$$$$T^{15} +$$$$43\!\cdots\!41$$$$T^{16}$$
$89$ $$1 -$$$$92\!\cdots\!88$$$$T^{2} +$$$$39\!\cdots\!88$$$$T^{4} -$$$$10\!\cdots\!36$$$$T^{6} +$$$$19\!\cdots\!70$$$$T^{8} -$$$$25\!\cdots\!56$$$$T^{10} +$$$$22\!\cdots\!08$$$$T^{12} -$$$$12\!\cdots\!68$$$$T^{14} +$$$$33\!\cdots\!81$$$$T^{16}$$
$97$ $$1 + 10816659651879048 T +$$$$58\!\cdots\!52$$$$T^{2} -$$$$46\!\cdots\!12$$$$T^{3} -$$$$21\!\cdots\!92$$$$T^{4} +$$$$22\!\cdots\!48$$$$T^{5} +$$$$48\!\cdots\!60$$$$T^{6} -$$$$81\!\cdots\!20$$$$T^{7} -$$$$89\!\cdots\!54$$$$T^{8} -$$$$49\!\cdots\!20$$$$T^{9} +$$$$18\!\cdots\!60$$$$T^{10} +$$$$52\!\cdots\!28$$$$T^{11} -$$$$31\!\cdots\!52$$$$T^{12} -$$$$40\!\cdots\!12$$$$T^{13} +$$$$31\!\cdots\!92$$$$T^{14} +$$$$35\!\cdots\!68$$$$T^{15} +$$$$20\!\cdots\!61$$$$T^{16}$$