Properties

Label 9.102.a.b
Level $9$
Weight $102$
Character orbit 9.a
Self dual yes
Analytic conductor $581.406$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 102 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(581.406281043\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - x^{7} - \)\(15\!\cdots\!64\)\( x^{6} - \)\(13\!\cdots\!76\)\( x^{5} + \)\(79\!\cdots\!56\)\( x^{4} + \)\(16\!\cdots\!20\)\( x^{3} - \)\(12\!\cdots\!00\)\( x^{2} - \)\(46\!\cdots\!00\)\( x + \)\(14\!\cdots\!00\)\(\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: multiple of \( 2^{119}\cdot 3^{56}\cdot 5^{14}\cdot 7^{7}\cdot 11^{2}\cdot 13^{2}\cdot 17^{2} \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q +(54373636474380 + \beta_{1}) q^{2} +(\)\(11\!\cdots\!12\)\( + 229422224422231 \beta_{1} + \beta_{2} + \beta_{3}) q^{4} +(-\)\(47\!\cdots\!50\)\( - 12781604289557109609 \beta_{1} - 41036 \beta_{2} - 30556 \beta_{3} + \beta_{4}) q^{5} +(-\)\(72\!\cdots\!00\)\( + \)\(17\!\cdots\!53\)\( \beta_{1} - 260900639867 \beta_{2} + 311630903907 \beta_{3} + 4702697 \beta_{4} + \beta_{5}) q^{7} +(\)\(76\!\cdots\!40\)\( + \)\(40\!\cdots\!45\)\( \beta_{1} + 881331779136682 \beta_{2} - 124289729727659 \beta_{3} - 1912855272 \beta_{4} + 183 \beta_{5} + 76 \beta_{6} + \beta_{7}) q^{8} +O(q^{10})\) \( q +(54373636474380 + \beta_{1}) q^{2} +(\)\(11\!\cdots\!12\)\( + 229422224422231 \beta_{1} + \beta_{2} + \beta_{3}) q^{4} +(-\)\(47\!\cdots\!50\)\( - 12781604289557109609 \beta_{1} - 41036 \beta_{2} - 30556 \beta_{3} + \beta_{4}) q^{5} +(-\)\(72\!\cdots\!00\)\( + \)\(17\!\cdots\!53\)\( \beta_{1} - 260900639867 \beta_{2} + 311630903907 \beta_{3} + 4702697 \beta_{4} + \beta_{5}) q^{7} +(\)\(76\!\cdots\!40\)\( + \)\(40\!\cdots\!45\)\( \beta_{1} + 881331779136682 \beta_{2} - 124289729727659 \beta_{3} - 1912855272 \beta_{4} + 183 \beta_{5} + 76 \beta_{6} + \beta_{7}) q^{8} +(-\)\(47\!\cdots\!00\)\( - \)\(62\!\cdots\!42\)\( \beta_{1} - \)\(10\!\cdots\!68\)\( \beta_{2} - 11902232897206577588 \beta_{3} + 2448426379668 \beta_{4} - 72318680 \beta_{5} - 24187580 \beta_{6} - 98280 \beta_{7}) q^{10} +(-\)\(57\!\cdots\!12\)\( + \)\(21\!\cdots\!27\)\( \beta_{1} + \)\(58\!\cdots\!05\)\( \beta_{2} + \)\(37\!\cdots\!58\)\( \beta_{3} - 22306725857809514 \beta_{4} - 1900924986 \beta_{5} + 6014867840 \beta_{6} - 7198816 \beta_{7}) q^{11} +(\)\(31\!\cdots\!10\)\( + \)\(28\!\cdots\!93\)\( \beta_{1} + \)\(15\!\cdots\!76\)\( \beta_{2} - \)\(22\!\cdots\!52\)\( \beta_{3} + \)\(39\!\cdots\!15\)\( \beta_{4} - 22175840367016 \beta_{5} + 14281371887296 \beta_{6} + 11181327696 \beta_{7}) q^{13} +(\)\(60\!\cdots\!36\)\( - \)\(19\!\cdots\!28\)\( \beta_{1} + \)\(55\!\cdots\!52\)\( \beta_{2} + \)\(61\!\cdots\!26\)\( \beta_{3} - \)\(35\!\cdots\!22\)\( \beta_{4} - 765440512740768 \beta_{5} - 821654712140038 \beta_{6} + 259232530592 \beta_{7}) q^{14} +(-\)\(13\!\cdots\!84\)\( + \)\(12\!\cdots\!48\)\( \beta_{1} - \)\(12\!\cdots\!48\)\( \beta_{2} - \)\(66\!\cdots\!76\)\( \beta_{3} - \)\(20\!\cdots\!56\)\( \beta_{4} + 36199878363452576 \beta_{5} + 242190600571739264 \beta_{6} - 3581416918944 \beta_{7}) q^{16} +(\)\(49\!\cdots\!90\)\( - \)\(41\!\cdots\!18\)\( \beta_{1} + \)\(20\!\cdots\!24\)\( \beta_{2} + \)\(34\!\cdots\!72\)\( \beta_{3} + \)\(21\!\cdots\!42\)\( \beta_{4} - 5303318752936720404 \beta_{5} - 10563708499835168160 \beta_{6} + 1680372172971240 \beta_{7}) q^{17} +(-\)\(26\!\cdots\!60\)\( + \)\(24\!\cdots\!75\)\( \beta_{1} + \)\(74\!\cdots\!69\)\( \beta_{2} + \)\(10\!\cdots\!18\)\( \beta_{3} + \)\(17\!\cdots\!38\)\( \beta_{4} + \)\(34\!\cdots\!62\)\( \beta_{5} + \)\(33\!\cdots\!00\)\( \beta_{6} - 1448443710124252128 \beta_{7}) q^{19} +(-\)\(21\!\cdots\!00\)\( - \)\(55\!\cdots\!78\)\( \beta_{1} - \)\(39\!\cdots\!62\)\( \beta_{2} - \)\(14\!\cdots\!02\)\( \beta_{3} + \)\(17\!\cdots\!92\)\( \beta_{4} + \)\(47\!\cdots\!00\)\( \beta_{5} - \)\(10\!\cdots\!00\)\( \beta_{6} - 26979311695698549600 \beta_{7}) q^{20} +(\)\(76\!\cdots\!40\)\( + \)\(65\!\cdots\!12\)\( \beta_{1} + \)\(73\!\cdots\!34\)\( \beta_{2} - \)\(10\!\cdots\!83\)\( \beta_{3} + \)\(55\!\cdots\!25\)\( \beta_{4} - \)\(31\!\cdots\!64\)\( \beta_{5} + \)\(74\!\cdots\!29\)\( \beta_{6} + \)\(56\!\cdots\!04\)\( \beta_{7}) q^{22} +(\)\(17\!\cdots\!60\)\( + \)\(34\!\cdots\!61\)\( \beta_{1} - \)\(58\!\cdots\!31\)\( \beta_{2} + \)\(91\!\cdots\!15\)\( \beta_{3} + \)\(10\!\cdots\!85\)\( \beta_{4} - \)\(49\!\cdots\!55\)\( \beta_{5} + \)\(50\!\cdots\!60\)\( \beta_{6} + \)\(66\!\cdots\!60\)\( \beta_{7}) q^{23} +(\)\(96\!\cdots\!75\)\( + \)\(12\!\cdots\!00\)\( \beta_{1} + \)\(23\!\cdots\!00\)\( \beta_{2} + \)\(75\!\cdots\!00\)\( \beta_{3} + \)\(84\!\cdots\!00\)\( \beta_{4} - \)\(32\!\cdots\!00\)\( \beta_{5} + \)\(27\!\cdots\!00\)\( \beta_{6} - \)\(64\!\cdots\!00\)\( \beta_{7}) q^{25} +(\)\(12\!\cdots\!68\)\( - \)\(36\!\cdots\!66\)\( \beta_{1} + \)\(13\!\cdots\!12\)\( \beta_{2} + \)\(51\!\cdots\!48\)\( \beta_{3} + \)\(39\!\cdots\!12\)\( \beta_{4} - \)\(31\!\cdots\!92\)\( \beta_{5} + \)\(25\!\cdots\!04\)\( \beta_{6} - \)\(54\!\cdots\!52\)\( \beta_{7}) q^{26} +(\)\(11\!\cdots\!60\)\( + \)\(12\!\cdots\!00\)\( \beta_{1} - \)\(71\!\cdots\!92\)\( \beta_{2} - \)\(79\!\cdots\!04\)\( \beta_{3} + \)\(25\!\cdots\!08\)\( \beta_{4} + \)\(11\!\cdots\!48\)\( \beta_{5} + \)\(48\!\cdots\!76\)\( \beta_{6} + \)\(27\!\cdots\!76\)\( \beta_{7}) q^{28} +(-\)\(19\!\cdots\!10\)\( + \)\(10\!\cdots\!47\)\( \beta_{1} - \)\(62\!\cdots\!96\)\( \beta_{2} - \)\(15\!\cdots\!72\)\( \beta_{3} + \)\(51\!\cdots\!33\)\( \beta_{4} + \)\(35\!\cdots\!72\)\( \beta_{5} + \)\(75\!\cdots\!56\)\( \beta_{6} + \)\(16\!\cdots\!32\)\( \beta_{7}) q^{29} +(-\)\(82\!\cdots\!68\)\( + \)\(21\!\cdots\!64\)\( \beta_{1} + \)\(29\!\cdots\!00\)\( \beta_{2} - \)\(45\!\cdots\!72\)\( \beta_{3} + \)\(59\!\cdots\!56\)\( \beta_{4} + \)\(85\!\cdots\!24\)\( \beta_{5} - \)\(83\!\cdots\!64\)\( \beta_{6} - \)\(42\!\cdots\!56\)\( \beta_{7}) q^{31} +(-\)\(15\!\cdots\!20\)\( - \)\(34\!\cdots\!32\)\( \beta_{1} + \)\(19\!\cdots\!92\)\( \beta_{2} + \)\(85\!\cdots\!72\)\( \beta_{3} + \)\(19\!\cdots\!32\)\( \beta_{4} + \)\(56\!\cdots\!96\)\( \beta_{5} + \)\(47\!\cdots\!60\)\( \beta_{6} - \)\(17\!\cdots\!40\)\( \beta_{7}) q^{32} +(\)\(11\!\cdots\!64\)\( + \)\(10\!\cdots\!38\)\( \beta_{1} - \)\(12\!\cdots\!28\)\( \beta_{2} - \)\(13\!\cdots\!84\)\( \beta_{3} + \)\(26\!\cdots\!28\)\( \beta_{4} + \)\(45\!\cdots\!72\)\( \beta_{5} - \)\(10\!\cdots\!80\)\( \beta_{6} + \)\(17\!\cdots\!32\)\( \beta_{7}) q^{34} +(\)\(14\!\cdots\!00\)\( - \)\(44\!\cdots\!28\)\( \beta_{1} - \)\(90\!\cdots\!12\)\( \beta_{2} + \)\(25\!\cdots\!08\)\( \beta_{3} - \)\(11\!\cdots\!88\)\( \beta_{4} + \)\(78\!\cdots\!80\)\( \beta_{5} + \)\(32\!\cdots\!80\)\( \beta_{6} + \)\(32\!\cdots\!80\)\( \beta_{7}) q^{35} +(\)\(49\!\cdots\!30\)\( + \)\(48\!\cdots\!65\)\( \beta_{1} + \)\(63\!\cdots\!64\)\( \beta_{2} + \)\(19\!\cdots\!88\)\( \beta_{3} - \)\(10\!\cdots\!01\)\( \beta_{4} - \)\(20\!\cdots\!56\)\( \beta_{5} - \)\(90\!\cdots\!72\)\( \beta_{6} + \)\(25\!\cdots\!28\)\( \beta_{7}) q^{37} +(\)\(88\!\cdots\!60\)\( + \)\(22\!\cdots\!92\)\( \beta_{1} + \)\(45\!\cdots\!70\)\( \beta_{2} + \)\(16\!\cdots\!05\)\( \beta_{3} - \)\(16\!\cdots\!23\)\( \beta_{4} - \)\(59\!\cdots\!40\)\( \beta_{5} + \)\(20\!\cdots\!41\)\( \beta_{6} + \)\(11\!\cdots\!16\)\( \beta_{7}) q^{38} +(-\)\(95\!\cdots\!00\)\( - \)\(35\!\cdots\!30\)\( \beta_{1} - \)\(95\!\cdots\!20\)\( \beta_{2} - \)\(71\!\cdots\!70\)\( \beta_{3} - \)\(92\!\cdots\!80\)\( \beta_{4} + \)\(14\!\cdots\!50\)\( \beta_{5} - \)\(74\!\cdots\!00\)\( \beta_{6} - \)\(18\!\cdots\!50\)\( \beta_{7}) q^{40} +(-\)\(70\!\cdots\!42\)\( + \)\(10\!\cdots\!96\)\( \beta_{1} - \)\(23\!\cdots\!40\)\( \beta_{2} - \)\(33\!\cdots\!60\)\( \beta_{3} - \)\(43\!\cdots\!80\)\( \beta_{4} + \)\(64\!\cdots\!20\)\( \beta_{5} - \)\(46\!\cdots\!92\)\( \beta_{6} - \)\(51\!\cdots\!80\)\( \beta_{7}) q^{41} +(-\)\(35\!\cdots\!00\)\( - \)\(22\!\cdots\!37\)\( \beta_{1} - \)\(95\!\cdots\!27\)\( \beta_{2} - \)\(40\!\cdots\!92\)\( \beta_{3} - \)\(75\!\cdots\!04\)\( \beta_{4} - \)\(24\!\cdots\!76\)\( \beta_{5} + \)\(42\!\cdots\!84\)\( \beta_{6} + \)\(78\!\cdots\!84\)\( \beta_{7}) q^{43} +(\)\(25\!\cdots\!56\)\( - \)\(74\!\cdots\!28\)\( \beta_{1} + \)\(10\!\cdots\!88\)\( \beta_{2} + \)\(83\!\cdots\!76\)\( \beta_{3} - \)\(82\!\cdots\!24\)\( \beta_{4} - \)\(57\!\cdots\!96\)\( \beta_{5} + \)\(11\!\cdots\!56\)\( \beta_{6} + \)\(30\!\cdots\!24\)\( \beta_{7}) q^{44} +(\)\(13\!\cdots\!72\)\( + \)\(27\!\cdots\!64\)\( \beta_{1} + \)\(39\!\cdots\!20\)\( \beta_{2} + \)\(20\!\cdots\!10\)\( \beta_{3} - \)\(11\!\cdots\!30\)\( \beta_{4} - \)\(93\!\cdots\!60\)\( \beta_{5} - \)\(13\!\cdots\!58\)\( \beta_{6} + \)\(16\!\cdots\!40\)\( \beta_{7}) q^{46} +(\)\(57\!\cdots\!60\)\( - \)\(60\!\cdots\!62\)\( \beta_{1} - \)\(15\!\cdots\!54\)\( \beta_{2} + \)\(93\!\cdots\!06\)\( \beta_{3} + \)\(14\!\cdots\!22\)\( \beta_{4} - \)\(37\!\cdots\!82\)\( \beta_{5} - \)\(11\!\cdots\!12\)\( \beta_{6} + \)\(15\!\cdots\!88\)\( \beta_{7}) q^{47} +(\)\(15\!\cdots\!57\)\( - \)\(69\!\cdots\!84\)\( \beta_{1} - \)\(11\!\cdots\!80\)\( \beta_{2} + \)\(21\!\cdots\!60\)\( \beta_{3} + \)\(39\!\cdots\!40\)\( \beta_{4} - \)\(18\!\cdots\!00\)\( \beta_{5} + \)\(13\!\cdots\!68\)\( \beta_{6} - \)\(94\!\cdots\!00\)\( \beta_{7}) q^{49} +(\)\(50\!\cdots\!00\)\( + \)\(24\!\cdots\!75\)\( \beta_{1} + \)\(33\!\cdots\!00\)\( \beta_{2} - \)\(12\!\cdots\!00\)\( \beta_{3} + \)\(11\!\cdots\!00\)\( \beta_{4} - \)\(57\!\cdots\!00\)\( \beta_{5} + \)\(71\!\cdots\!00\)\( \beta_{6} + \)\(16\!\cdots\!00\)\( \beta_{7}) q^{50} +(-\)\(92\!\cdots\!00\)\( + \)\(26\!\cdots\!66\)\( \beta_{1} + \)\(24\!\cdots\!66\)\( \beta_{2} - \)\(40\!\cdots\!06\)\( \beta_{3} + \)\(22\!\cdots\!28\)\( \beta_{4} + \)\(17\!\cdots\!32\)\( \beta_{5} + \)\(29\!\cdots\!12\)\( \beta_{6} - \)\(52\!\cdots\!88\)\( \beta_{7}) q^{52} +(-\)\(16\!\cdots\!30\)\( - \)\(22\!\cdots\!01\)\( \beta_{1} - \)\(12\!\cdots\!20\)\( \beta_{2} + \)\(78\!\cdots\!56\)\( \beta_{3} - \)\(30\!\cdots\!67\)\( \beta_{4} - \)\(28\!\cdots\!72\)\( \beta_{5} + \)\(16\!\cdots\!96\)\( \beta_{6} - \)\(68\!\cdots\!04\)\( \beta_{7}) q^{53} +(-\)\(18\!\cdots\!00\)\( + \)\(77\!\cdots\!33\)\( \beta_{1} - \)\(70\!\cdots\!43\)\( \beta_{2} - \)\(43\!\cdots\!53\)\( \beta_{3} + \)\(22\!\cdots\!13\)\( \beta_{4} - \)\(11\!\cdots\!75\)\( \beta_{5} - \)\(12\!\cdots\!00\)\( \beta_{6} - \)\(12\!\cdots\!00\)\( \beta_{7}) q^{55} +(\)\(29\!\cdots\!60\)\( - \)\(16\!\cdots\!12\)\( \beta_{1} + \)\(31\!\cdots\!16\)\( \beta_{2} + \)\(33\!\cdots\!40\)\( \beta_{3} + \)\(13\!\cdots\!48\)\( \beta_{4} - \)\(68\!\cdots\!08\)\( \beta_{5} - \)\(23\!\cdots\!72\)\( \beta_{6} - \)\(30\!\cdots\!48\)\( \beta_{7}) q^{56} +(\)\(36\!\cdots\!60\)\( - \)\(28\!\cdots\!14\)\( \beta_{1} + \)\(96\!\cdots\!64\)\( \beta_{2} + \)\(27\!\cdots\!16\)\( \beta_{3} - \)\(43\!\cdots\!80\)\( \beta_{4} - \)\(10\!\cdots\!72\)\( \beta_{5} + \)\(56\!\cdots\!52\)\( \beta_{6} - \)\(15\!\cdots\!48\)\( \beta_{7}) q^{58} +(-\)\(26\!\cdots\!20\)\( + \)\(21\!\cdots\!13\)\( \beta_{1} - \)\(78\!\cdots\!97\)\( \beta_{2} - \)\(18\!\cdots\!60\)\( \beta_{3} - \)\(45\!\cdots\!36\)\( \beta_{4} - \)\(28\!\cdots\!84\)\( \beta_{5} - \)\(18\!\cdots\!24\)\( \beta_{6} - \)\(62\!\cdots\!04\)\( \beta_{7}) q^{59} +(-\)\(42\!\cdots\!38\)\( + \)\(34\!\cdots\!05\)\( \beta_{1} - \)\(52\!\cdots\!20\)\( \beta_{2} - \)\(36\!\cdots\!36\)\( \beta_{3} - \)\(20\!\cdots\!77\)\( \beta_{4} - \)\(15\!\cdots\!88\)\( \beta_{5} + \)\(14\!\cdots\!12\)\( \beta_{6} - \)\(79\!\cdots\!28\)\( \beta_{7}) q^{61} +(\)\(73\!\cdots\!60\)\( - \)\(15\!\cdots\!40\)\( \beta_{1} - \)\(13\!\cdots\!92\)\( \beta_{2} - \)\(91\!\cdots\!72\)\( \beta_{3} + \)\(35\!\cdots\!68\)\( \beta_{4} - \)\(48\!\cdots\!96\)\( \beta_{5} - \)\(99\!\cdots\!60\)\( \beta_{6} - \)\(81\!\cdots\!60\)\( \beta_{7}) q^{62} +(-\)\(93\!\cdots\!88\)\( - \)\(84\!\cdots\!56\)\( \beta_{1} + \)\(29\!\cdots\!12\)\( \beta_{2} - \)\(40\!\cdots\!68\)\( \beta_{3} + \)\(45\!\cdots\!60\)\( \beta_{4} - \)\(96\!\cdots\!80\)\( \beta_{5} - \)\(27\!\cdots\!04\)\( \beta_{6} + \)\(10\!\cdots\!20\)\( \beta_{7}) q^{64} +(\)\(20\!\cdots\!00\)\( + \)\(33\!\cdots\!96\)\( \beta_{1} - \)\(27\!\cdots\!16\)\( \beta_{2} - \)\(22\!\cdots\!56\)\( \beta_{3} + \)\(97\!\cdots\!16\)\( \beta_{4} - \)\(36\!\cdots\!60\)\( \beta_{5} - \)\(12\!\cdots\!60\)\( \beta_{6} - \)\(71\!\cdots\!60\)\( \beta_{7}) q^{65} +(-\)\(77\!\cdots\!40\)\( - \)\(33\!\cdots\!97\)\( \beta_{1} - \)\(12\!\cdots\!19\)\( \beta_{2} + \)\(34\!\cdots\!10\)\( \beta_{3} - \)\(81\!\cdots\!38\)\( \beta_{4} + \)\(11\!\cdots\!50\)\( \beta_{5} - \)\(20\!\cdots\!44\)\( \beta_{6} + \)\(28\!\cdots\!56\)\( \beta_{7}) q^{67} +(\)\(27\!\cdots\!20\)\( - \)\(13\!\cdots\!78\)\( \beta_{1} - \)\(12\!\cdots\!42\)\( \beta_{2} + \)\(45\!\cdots\!34\)\( \beta_{3} - \)\(64\!\cdots\!88\)\( \beta_{4} + \)\(26\!\cdots\!92\)\( \beta_{5} - \)\(18\!\cdots\!56\)\( \beta_{6} - \)\(36\!\cdots\!56\)\( \beta_{7}) q^{68} +(-\)\(16\!\cdots\!00\)\( + \)\(47\!\cdots\!36\)\( \beta_{1} + \)\(12\!\cdots\!44\)\( \beta_{2} - \)\(45\!\cdots\!76\)\( \beta_{3} - \)\(37\!\cdots\!04\)\( \beta_{4} + \)\(59\!\cdots\!00\)\( \beta_{5} - \)\(60\!\cdots\!00\)\( \beta_{6} + \)\(35\!\cdots\!00\)\( \beta_{7}) q^{70} +(\)\(19\!\cdots\!28\)\( - \)\(59\!\cdots\!65\)\( \beta_{1} + \)\(86\!\cdots\!35\)\( \beta_{2} - \)\(23\!\cdots\!87\)\( \beta_{3} - \)\(36\!\cdots\!09\)\( \beta_{4} + \)\(31\!\cdots\!79\)\( \beta_{5} + \)\(31\!\cdots\!04\)\( \beta_{6} + \)\(57\!\cdots\!24\)\( \beta_{7}) q^{71} +(-\)\(25\!\cdots\!10\)\( - \)\(29\!\cdots\!54\)\( \beta_{1} + \)\(51\!\cdots\!48\)\( \beta_{2} + \)\(23\!\cdots\!80\)\( \beta_{3} + \)\(64\!\cdots\!38\)\( \beta_{4} + \)\(49\!\cdots\!20\)\( \beta_{5} + \)\(15\!\cdots\!24\)\( \beta_{6} + \)\(28\!\cdots\!24\)\( \beta_{7}) q^{73} +(\)\(17\!\cdots\!36\)\( + \)\(96\!\cdots\!66\)\( \beta_{1} - \)\(36\!\cdots\!60\)\( \beta_{2} + \)\(37\!\cdots\!56\)\( \beta_{3} + \)\(76\!\cdots\!12\)\( \beta_{4} + \)\(29\!\cdots\!68\)\( \beta_{5} + \)\(19\!\cdots\!76\)\( \beta_{6} + \)\(20\!\cdots\!08\)\( \beta_{7}) q^{74} +(\)\(80\!\cdots\!80\)\( + \)\(99\!\cdots\!64\)\( \beta_{1} + \)\(30\!\cdots\!08\)\( \beta_{2} - \)\(18\!\cdots\!40\)\( \beta_{3} + \)\(86\!\cdots\!04\)\( \beta_{4} + \)\(37\!\cdots\!16\)\( \beta_{5} + \)\(11\!\cdots\!44\)\( \beta_{6} + \)\(14\!\cdots\!96\)\( \beta_{7}) q^{76} +(-\)\(31\!\cdots\!00\)\( - \)\(24\!\cdots\!60\)\( \beta_{1} + \)\(63\!\cdots\!40\)\( \beta_{2} - \)\(20\!\cdots\!88\)\( \beta_{3} + \)\(61\!\cdots\!32\)\( \beta_{4} + \)\(11\!\cdots\!16\)\( \beta_{5} + \)\(49\!\cdots\!40\)\( \beta_{6} + \)\(20\!\cdots\!40\)\( \beta_{7}) q^{77} +(\)\(18\!\cdots\!60\)\( + \)\(12\!\cdots\!22\)\( \beta_{1} - \)\(11\!\cdots\!34\)\( \beta_{2} - \)\(10\!\cdots\!42\)\( \beta_{3} - \)\(12\!\cdots\!66\)\( \beta_{4} + \)\(33\!\cdots\!86\)\( \beta_{5} - \)\(41\!\cdots\!56\)\( \beta_{6} + \)\(42\!\cdots\!16\)\( \beta_{7}) q^{79} +(-\)\(75\!\cdots\!00\)\( - \)\(38\!\cdots\!84\)\( \beta_{1} - \)\(21\!\cdots\!36\)\( \beta_{2} + \)\(52\!\cdots\!44\)\( \beta_{3} - \)\(35\!\cdots\!24\)\( \beta_{4} + \)\(12\!\cdots\!00\)\( \beta_{5} - \)\(74\!\cdots\!00\)\( \beta_{6} + \)\(12\!\cdots\!00\)\( \beta_{7}) q^{80} +(\)\(37\!\cdots\!40\)\( - \)\(13\!\cdots\!10\)\( \beta_{1} - \)\(21\!\cdots\!08\)\( \beta_{2} + \)\(11\!\cdots\!48\)\( \beta_{3} + \)\(13\!\cdots\!64\)\( \beta_{4} + \)\(23\!\cdots\!24\)\( \beta_{5} - \)\(52\!\cdots\!32\)\( \beta_{6} - \)\(29\!\cdots\!32\)\( \beta_{7}) q^{82} +(-\)\(42\!\cdots\!20\)\( + \)\(18\!\cdots\!99\)\( \beta_{1} + \)\(83\!\cdots\!13\)\( \beta_{2} - \)\(49\!\cdots\!00\)\( \beta_{3} + \)\(11\!\cdots\!00\)\( \beta_{4} + \)\(73\!\cdots\!00\)\( \beta_{5} - \)\(23\!\cdots\!00\)\( \beta_{6} + \)\(44\!\cdots\!00\)\( \beta_{7}) q^{83} +(\)\(21\!\cdots\!00\)\( + \)\(28\!\cdots\!18\)\( \beta_{1} - \)\(79\!\cdots\!28\)\( \beta_{2} - \)\(63\!\cdots\!48\)\( \beta_{3} + \)\(78\!\cdots\!78\)\( \beta_{4} + \)\(11\!\cdots\!20\)\( \beta_{5} + \)\(12\!\cdots\!20\)\( \beta_{6} - \)\(70\!\cdots\!80\)\( \beta_{7}) q^{85} +(-\)\(84\!\cdots\!52\)\( - \)\(11\!\cdots\!36\)\( \beta_{1} + \)\(16\!\cdots\!66\)\( \beta_{2} - \)\(11\!\cdots\!53\)\( \beta_{3} + \)\(87\!\cdots\!87\)\( \beta_{4} + \)\(52\!\cdots\!48\)\( \beta_{5} + \)\(17\!\cdots\!67\)\( \beta_{6} + \)\(34\!\cdots\!88\)\( \beta_{7}) q^{86} +(-\)\(45\!\cdots\!80\)\( + \)\(32\!\cdots\!00\)\( \beta_{1} - \)\(55\!\cdots\!84\)\( \beta_{2} + \)\(20\!\cdots\!12\)\( \beta_{3} - \)\(26\!\cdots\!08\)\( \beta_{4} + \)\(70\!\cdots\!16\)\( \beta_{5} + \)\(20\!\cdots\!20\)\( \beta_{6} - \)\(12\!\cdots\!80\)\( \beta_{7}) q^{88} +(\)\(77\!\cdots\!70\)\( + \)\(36\!\cdots\!86\)\( \beta_{1} - \)\(91\!\cdots\!08\)\( \beta_{2} + \)\(26\!\cdots\!08\)\( \beta_{3} - \)\(17\!\cdots\!58\)\( \beta_{4} + \)\(56\!\cdots\!08\)\( \beta_{5} - \)\(14\!\cdots\!20\)\( \beta_{6} - \)\(10\!\cdots\!52\)\( \beta_{7}) q^{89} +(-\)\(46\!\cdots\!68\)\( - \)\(15\!\cdots\!88\)\( \beta_{1} + \)\(43\!\cdots\!04\)\( \beta_{2} + \)\(14\!\cdots\!20\)\( \beta_{3} - \)\(12\!\cdots\!68\)\( \beta_{4} + \)\(56\!\cdots\!28\)\( \beta_{5} - \)\(80\!\cdots\!68\)\( \beta_{6} + \)\(19\!\cdots\!68\)\( \beta_{7}) q^{91} +(\)\(57\!\cdots\!40\)\( + \)\(16\!\cdots\!28\)\( \beta_{1} + \)\(35\!\cdots\!40\)\( \beta_{2} + \)\(65\!\cdots\!32\)\( \beta_{3} + \)\(20\!\cdots\!56\)\( \beta_{4} + \)\(25\!\cdots\!16\)\( \beta_{5} - \)\(67\!\cdots\!48\)\( \beta_{6} - \)\(11\!\cdots\!48\)\( \beta_{7}) q^{92} +(-\)\(21\!\cdots\!36\)\( + \)\(46\!\cdots\!00\)\( \beta_{1} - \)\(28\!\cdots\!56\)\( \beta_{2} - \)\(75\!\cdots\!64\)\( \beta_{3} + \)\(14\!\cdots\!84\)\( \beta_{4} + \)\(23\!\cdots\!36\)\( \beta_{5} - \)\(29\!\cdots\!76\)\( \beta_{6} - \)\(12\!\cdots\!84\)\( \beta_{7}) q^{94} +(\)\(33\!\cdots\!00\)\( - \)\(22\!\cdots\!15\)\( \beta_{1} - \)\(36\!\cdots\!35\)\( \beta_{2} + \)\(93\!\cdots\!15\)\( \beta_{3} + \)\(45\!\cdots\!85\)\( \beta_{4} + \)\(13\!\cdots\!25\)\( \beta_{5} - \)\(15\!\cdots\!00\)\( \beta_{6} - \)\(55\!\cdots\!00\)\( \beta_{7}) q^{95} +(\)\(80\!\cdots\!90\)\( + \)\(36\!\cdots\!58\)\( \beta_{1} + \)\(39\!\cdots\!60\)\( \beta_{2} + \)\(11\!\cdots\!04\)\( \beta_{3} - \)\(10\!\cdots\!38\)\( \beta_{4} - \)\(37\!\cdots\!48\)\( \beta_{5} + \)\(15\!\cdots\!84\)\( \beta_{6} - \)\(15\!\cdots\!16\)\( \beta_{7}) q^{97} +(-\)\(25\!\cdots\!40\)\( + \)\(27\!\cdots\!09\)\( \beta_{1} + \)\(19\!\cdots\!32\)\( \beta_{2} - \)\(10\!\cdots\!92\)\( \beta_{3} + \)\(31\!\cdots\!24\)\( \beta_{4} - \)\(34\!\cdots\!96\)\( \beta_{5} - \)\(27\!\cdots\!32\)\( \beta_{6} - \)\(38\!\cdots\!32\)\( \beta_{7}) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 434989091795040q^{2} + \)\(90\!\cdots\!96\)\(q^{4} - \)\(38\!\cdots\!00\)\(q^{5} - \)\(57\!\cdots\!00\)\(q^{7} + \)\(61\!\cdots\!20\)\(q^{8} + O(q^{10}) \) \( 8q + 434989091795040q^{2} + \)\(90\!\cdots\!96\)\(q^{4} - \)\(38\!\cdots\!00\)\(q^{5} - \)\(57\!\cdots\!00\)\(q^{7} + \)\(61\!\cdots\!20\)\(q^{8} - \)\(37\!\cdots\!00\)\(q^{10} - \)\(46\!\cdots\!96\)\(q^{11} + \)\(25\!\cdots\!80\)\(q^{13} + \)\(48\!\cdots\!88\)\(q^{14} - \)\(10\!\cdots\!72\)\(q^{16} + \)\(39\!\cdots\!20\)\(q^{17} - \)\(21\!\cdots\!80\)\(q^{19} - \)\(17\!\cdots\!00\)\(q^{20} + \)\(61\!\cdots\!20\)\(q^{22} + \)\(13\!\cdots\!80\)\(q^{23} + \)\(77\!\cdots\!00\)\(q^{25} + \)\(97\!\cdots\!44\)\(q^{26} + \)\(92\!\cdots\!80\)\(q^{28} - \)\(15\!\cdots\!80\)\(q^{29} - \)\(65\!\cdots\!44\)\(q^{31} - \)\(12\!\cdots\!60\)\(q^{32} + \)\(95\!\cdots\!12\)\(q^{34} + \)\(11\!\cdots\!00\)\(q^{35} + \)\(39\!\cdots\!40\)\(q^{37} + \)\(70\!\cdots\!80\)\(q^{38} - \)\(76\!\cdots\!00\)\(q^{40} - \)\(56\!\cdots\!36\)\(q^{41} - \)\(28\!\cdots\!00\)\(q^{43} + \)\(20\!\cdots\!48\)\(q^{44} + \)\(10\!\cdots\!76\)\(q^{46} + \)\(45\!\cdots\!80\)\(q^{47} + \)\(12\!\cdots\!56\)\(q^{49} + \)\(40\!\cdots\!00\)\(q^{50} - \)\(73\!\cdots\!00\)\(q^{52} - \)\(13\!\cdots\!40\)\(q^{53} - \)\(14\!\cdots\!00\)\(q^{55} + \)\(23\!\cdots\!80\)\(q^{56} + \)\(29\!\cdots\!80\)\(q^{58} - \)\(21\!\cdots\!60\)\(q^{59} - \)\(33\!\cdots\!04\)\(q^{61} + \)\(58\!\cdots\!80\)\(q^{62} - \)\(74\!\cdots\!04\)\(q^{64} + \)\(16\!\cdots\!00\)\(q^{65} - \)\(61\!\cdots\!20\)\(q^{67} + \)\(21\!\cdots\!60\)\(q^{68} - \)\(12\!\cdots\!00\)\(q^{70} + \)\(15\!\cdots\!24\)\(q^{71} - \)\(20\!\cdots\!80\)\(q^{73} + \)\(14\!\cdots\!88\)\(q^{74} + \)\(64\!\cdots\!40\)\(q^{76} - \)\(25\!\cdots\!00\)\(q^{77} + \)\(14\!\cdots\!80\)\(q^{79} - \)\(60\!\cdots\!00\)\(q^{80} + \)\(30\!\cdots\!20\)\(q^{82} - \)\(33\!\cdots\!60\)\(q^{83} + \)\(17\!\cdots\!00\)\(q^{85} - \)\(67\!\cdots\!16\)\(q^{86} - \)\(36\!\cdots\!40\)\(q^{88} + \)\(62\!\cdots\!60\)\(q^{89} - \)\(36\!\cdots\!44\)\(q^{91} + \)\(46\!\cdots\!20\)\(q^{92} - \)\(17\!\cdots\!88\)\(q^{94} + \)\(26\!\cdots\!00\)\(q^{95} + \)\(64\!\cdots\!20\)\(q^{97} - \)\(20\!\cdots\!20\)\(q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} - \)\(15\!\cdots\!64\)\( x^{6} - \)\(13\!\cdots\!76\)\( x^{5} + \)\(79\!\cdots\!56\)\( x^{4} + \)\(16\!\cdots\!20\)\( x^{3} - \)\(12\!\cdots\!00\)\( x^{2} - \)\(46\!\cdots\!00\)\( x + \)\(14\!\cdots\!00\)\(\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 96 \nu - 12 \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(19\!\cdots\!41\)\( \nu^{7} + \)\(51\!\cdots\!09\)\( \nu^{6} + \)\(32\!\cdots\!16\)\( \nu^{5} - \)\(58\!\cdots\!80\)\( \nu^{4} - \)\(16\!\cdots\!76\)\( \nu^{3} + \)\(16\!\cdots\!48\)\( \nu^{2} + \)\(21\!\cdots\!12\)\( \nu - \)\(45\!\cdots\!28\)\(\)\()/ \)\(68\!\cdots\!48\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(19\!\cdots\!41\)\( \nu^{7} - \)\(51\!\cdots\!09\)\( \nu^{6} - \)\(32\!\cdots\!16\)\( \nu^{5} + \)\(58\!\cdots\!80\)\( \nu^{4} + \)\(16\!\cdots\!76\)\( \nu^{3} - \)\(10\!\cdots\!80\)\( \nu^{2} - \)\(22\!\cdots\!72\)\( \nu - \)\(20\!\cdots\!36\)\(\)\()/ \)\(68\!\cdots\!48\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(25\!\cdots\!97\)\( \nu^{7} + \)\(12\!\cdots\!43\)\( \nu^{6} - \)\(27\!\cdots\!48\)\( \nu^{5} - \)\(45\!\cdots\!32\)\( \nu^{4} + \)\(68\!\cdots\!92\)\( \nu^{3} + \)\(27\!\cdots\!40\)\( \nu^{2} + \)\(12\!\cdots\!40\)\( \nu - \)\(20\!\cdots\!60\)\(\)\()/ \)\(86\!\cdots\!60\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(14\!\cdots\!23\)\( \nu^{7} - \)\(25\!\cdots\!37\)\( \nu^{6} + \)\(23\!\cdots\!52\)\( \nu^{5} + \)\(28\!\cdots\!88\)\( \nu^{4} - \)\(11\!\cdots\!88\)\( \nu^{3} - \)\(77\!\cdots\!40\)\( \nu^{2} + \)\(12\!\cdots\!00\)\( \nu + \)\(38\!\cdots\!40\)\(\)\()/ \)\(10\!\cdots\!20\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(33\!\cdots\!91\)\( \nu^{7} - \)\(26\!\cdots\!67\)\( \nu^{6} - \)\(41\!\cdots\!32\)\( \nu^{5} + \)\(48\!\cdots\!84\)\( \nu^{4} + \)\(14\!\cdots\!96\)\( \nu^{3} - \)\(33\!\cdots\!92\)\( \nu^{2} - \)\(11\!\cdots\!28\)\( \nu + \)\(23\!\cdots\!56\)\(\)\()/ \)\(98\!\cdots\!64\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(16\!\cdots\!67\)\( \nu^{7} - \)\(13\!\cdots\!67\)\( \nu^{6} - \)\(30\!\cdots\!28\)\( \nu^{5} + \)\(11\!\cdots\!88\)\( \nu^{4} + \)\(20\!\cdots\!12\)\( \nu^{3} - \)\(17\!\cdots\!20\)\( \nu^{2} - \)\(47\!\cdots\!20\)\( \nu - \)\(10\!\cdots\!60\)\(\)\()/ \)\(49\!\cdots\!20\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 12\)\()/96\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + \beta_{2} + 120674951473495 \beta_{1} + 3658465598508525014815957089408\)\()/9216\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} + 76 \beta_{6} + 183 \beta_{5} - 1912855272 \beta_{4} - 287410639150763 \beta_{3} + 718210869713578 \beta_{2} + 5446795118301596829918048967197 \beta_{1} + 441485158567510392680440580527129275362783232\)\()/884736\)
\(\nu^{4}\)\(=\)\((\)\(-6908623838013 \beta_{7} + 7051906721360356 \beta_{6} - 112550735493275 \beta_{5} - 638684132547279383731896 \beta_{4} + 218388789875692846046305083071 \beta_{3} + 228269717013415875570472550014 \beta_{2} + 21463000708385357333083247401597115690391831 \beta_{1} + 622716017575955242197588480382533197831239774225040133525504\)\()/2654208\)
\(\nu^{5}\)\(=\)\((\)\(8218298097895220964852696257 \beta_{7} + 1156585995142168567475078277452 \beta_{6} + 2358026885524086501532101010935 \beta_{5} - 11545372157851571850082181315091523816 \beta_{4} - 2246920757075568879843551674768733432034795 \beta_{3} + 8696522114483835805367226101306901146392362 \beta_{2} + 31676186309574918996782507971840122174574904332702728439517 \beta_{1} + 2453801554137476819559674420827512093324808121982095371414910936356466688\)\()/7962624\)
\(\nu^{6}\)\(=\)\((\)\(-17866385029142170477162005989303517043911 \beta_{7} + 24372615119029306000442302086123253741171948 \beta_{6} - 13134770359675329303231654456438487452275265 \beta_{5} - 2182010788982862386537895916497298008882033645073832 \beta_{4} + 466616763367924761775789451186610010648109688706644637677 \beta_{3} + 569073053817864568549690429742770550561702730391970175706 \beta_{2} + 34953337675822992992132362385505771109221405328621145161303203632550517 \beta_{1} + 1207148311515901739164637113231032212904420303199482676292001939010982743072127047053312\)\()/7962624\)
\(\nu^{7}\)\(=\)\((\)\(\)\(19\!\cdots\!67\)\( \beta_{7} + \)\(40\!\cdots\!20\)\( \beta_{6} + \)\(66\!\cdots\!49\)\( \beta_{5} - \)\(13\!\cdots\!48\)\( \beta_{4} - \)\(52\!\cdots\!49\)\( \beta_{3} + \)\(23\!\cdots\!82\)\( \beta_{2} + \)\(64\!\cdots\!15\)\( \beta_{1} + \)\(39\!\cdots\!64\)\(\)\()/23887872\)

Embeddings

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

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

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.72721e13
−2.32579e13
−1.40661e13
−6.74954e12
2.10926e12
1.78363e13
2.42719e13
2.71282e13
−2.56375e15 0 4.03750e30 −2.88877e35 0 −6.25978e42 −3.85125e45 0 7.40607e50
1.2 −2.17839e15 0 2.21006e30 2.56275e35 0 8.01618e42 7.08493e44 0 −5.58265e50
1.3 −1.29597e15 0 −8.55769e29 1.52987e35 0 −2.30272e42 4.39472e45 0 −1.98266e50
1.4 −5.93582e14 0 −2.18296e30 −2.07725e34 0 −6.15512e42 2.80068e45 0 1.23302e49
1.5 2.56863e14 0 −2.46932e30 −1.16158e35 0 2.79663e42 −1.28550e45 0 −2.98366e49
1.6 1.76666e15 0 5.85773e29 3.44135e35 0 −5.35995e42 −3.44415e45 0 6.07967e50
1.7 2.38447e15 0 3.15040e30 −7.99157e34 0 3.14769e42 1.46670e45 0 −1.90557e50
1.8 2.65868e15 0 4.53328e30 −2.85912e35 0 3.30273e41 5.31200e45 0 −7.60148e50
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9.102.a.b 8
3.b odd 2 1 1.102.a.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.102.a.a 8 3.b odd 2 1
9.102.a.b 8 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{8} - \cdots\) acting on \(S_{102}^{\mathrm{new}}(\Gamma_0(9))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 434989091795040 T + \)\(57\!\cdots\!60\)\( T^{2} - \)\(41\!\cdots\!80\)\( T^{3} + \)\(25\!\cdots\!16\)\( T^{4} - \)\(14\!\cdots\!80\)\( T^{5} + \)\(90\!\cdots\!20\)\( T^{6} - \)\(52\!\cdots\!60\)\( T^{7} + \)\(24\!\cdots\!96\)\( T^{8} - \)\(13\!\cdots\!20\)\( T^{9} + \)\(58\!\cdots\!80\)\( T^{10} - \)\(23\!\cdots\!40\)\( T^{11} + \)\(10\!\cdots\!56\)\( T^{12} - \)\(43\!\cdots\!60\)\( T^{13} + \)\(15\!\cdots\!40\)\( T^{14} - \)\(29\!\cdots\!20\)\( T^{15} + \)\(17\!\cdots\!56\)\( T^{16} \)
$3$ 1
$5$ \( 1 + \)\(38\!\cdots\!00\)\( T + \)\(11\!\cdots\!00\)\( T^{2} - \)\(81\!\cdots\!00\)\( T^{3} + \)\(80\!\cdots\!00\)\( T^{4} - \)\(22\!\cdots\!00\)\( T^{5} + \)\(43\!\cdots\!00\)\( T^{6} - \)\(17\!\cdots\!00\)\( T^{7} + \)\(18\!\cdots\!50\)\( T^{8} - \)\(69\!\cdots\!00\)\( T^{9} + \)\(67\!\cdots\!00\)\( T^{10} - \)\(13\!\cdots\!00\)\( T^{11} + \)\(19\!\cdots\!00\)\( T^{12} - \)\(77\!\cdots\!00\)\( T^{13} + \)\(45\!\cdots\!00\)\( T^{14} + \)\(56\!\cdots\!00\)\( T^{15} + \)\(58\!\cdots\!25\)\( T^{16} \)
$7$ \( 1 + \)\(57\!\cdots\!00\)\( T + \)\(10\!\cdots\!00\)\( T^{2} + \)\(41\!\cdots\!00\)\( T^{3} + \)\(47\!\cdots\!96\)\( T^{4} + \)\(13\!\cdots\!00\)\( T^{5} + \)\(13\!\cdots\!00\)\( T^{6} + \)\(29\!\cdots\!00\)\( T^{7} + \)\(32\!\cdots\!06\)\( T^{8} + \)\(66\!\cdots\!00\)\( T^{9} + \)\(70\!\cdots\!00\)\( T^{10} + \)\(15\!\cdots\!00\)\( T^{11} + \)\(12\!\cdots\!96\)\( T^{12} + \)\(24\!\cdots\!00\)\( T^{13} + \)\(13\!\cdots\!00\)\( T^{14} + \)\(17\!\cdots\!00\)\( T^{15} + \)\(69\!\cdots\!01\)\( T^{16} \)
$11$ \( 1 + \)\(46\!\cdots\!96\)\( T + \)\(53\!\cdots\!20\)\( T^{2} + \)\(16\!\cdots\!60\)\( T^{3} + \)\(13\!\cdots\!20\)\( T^{4} + \)\(68\!\cdots\!68\)\( T^{5} + \)\(32\!\cdots\!48\)\( T^{6} + \)\(12\!\cdots\!40\)\( T^{7} + \)\(64\!\cdots\!70\)\( T^{8} + \)\(19\!\cdots\!40\)\( T^{9} + \)\(75\!\cdots\!08\)\( T^{10} + \)\(23\!\cdots\!08\)\( T^{11} + \)\(71\!\cdots\!20\)\( T^{12} + \)\(13\!\cdots\!60\)\( T^{13} + \)\(64\!\cdots\!20\)\( T^{14} + \)\(85\!\cdots\!16\)\( T^{15} + \)\(27\!\cdots\!81\)\( T^{16} \)
$13$ \( 1 - \)\(25\!\cdots\!80\)\( T + \)\(17\!\cdots\!40\)\( T^{2} - \)\(39\!\cdots\!40\)\( T^{3} + \)\(15\!\cdots\!76\)\( T^{4} - \)\(30\!\cdots\!60\)\( T^{5} + \)\(83\!\cdots\!80\)\( T^{6} - \)\(14\!\cdots\!80\)\( T^{7} + \)\(31\!\cdots\!66\)\( T^{8} - \)\(47\!\cdots\!40\)\( T^{9} + \)\(86\!\cdots\!20\)\( T^{10} - \)\(10\!\cdots\!20\)\( T^{11} + \)\(16\!\cdots\!36\)\( T^{12} - \)\(13\!\cdots\!20\)\( T^{13} + \)\(19\!\cdots\!60\)\( T^{14} - \)\(90\!\cdots\!60\)\( T^{15} + \)\(11\!\cdots\!21\)\( T^{16} \)
$17$ \( 1 - \)\(39\!\cdots\!20\)\( T + \)\(16\!\cdots\!80\)\( T^{2} - \)\(41\!\cdots\!40\)\( T^{3} + \)\(10\!\cdots\!56\)\( T^{4} - \)\(18\!\cdots\!40\)\( T^{5} + \)\(35\!\cdots\!60\)\( T^{6} - \)\(53\!\cdots\!80\)\( T^{7} + \)\(81\!\cdots\!26\)\( T^{8} - \)\(10\!\cdots\!60\)\( T^{9} + \)\(12\!\cdots\!40\)\( T^{10} - \)\(12\!\cdots\!20\)\( T^{11} + \)\(12\!\cdots\!76\)\( T^{12} - \)\(97\!\cdots\!80\)\( T^{13} + \)\(74\!\cdots\!20\)\( T^{14} - \)\(33\!\cdots\!60\)\( T^{15} + \)\(15\!\cdots\!41\)\( T^{16} \)
$19$ \( 1 + \)\(21\!\cdots\!80\)\( T + \)\(87\!\cdots\!52\)\( T^{2} + \)\(14\!\cdots\!40\)\( T^{3} + \)\(36\!\cdots\!08\)\( T^{4} + \)\(48\!\cdots\!80\)\( T^{5} + \)\(92\!\cdots\!04\)\( T^{6} + \)\(10\!\cdots\!00\)\( T^{7} + \)\(15\!\cdots\!70\)\( T^{8} + \)\(14\!\cdots\!00\)\( T^{9} + \)\(18\!\cdots\!44\)\( T^{10} + \)\(14\!\cdots\!20\)\( T^{11} + \)\(14\!\cdots\!68\)\( T^{12} + \)\(85\!\cdots\!60\)\( T^{13} + \)\(73\!\cdots\!12\)\( T^{14} + \)\(25\!\cdots\!20\)\( T^{15} + \)\(17\!\cdots\!41\)\( T^{16} \)
$23$ \( 1 - \)\(13\!\cdots\!80\)\( T + \)\(17\!\cdots\!60\)\( T^{2} - \)\(15\!\cdots\!40\)\( T^{3} + \)\(13\!\cdots\!16\)\( T^{4} - \)\(96\!\cdots\!60\)\( T^{5} + \)\(71\!\cdots\!20\)\( T^{6} - \)\(45\!\cdots\!80\)\( T^{7} + \)\(28\!\cdots\!46\)\( T^{8} - \)\(15\!\cdots\!40\)\( T^{9} + \)\(83\!\cdots\!80\)\( T^{10} - \)\(38\!\cdots\!20\)\( T^{11} + \)\(18\!\cdots\!56\)\( T^{12} - \)\(71\!\cdots\!20\)\( T^{13} + \)\(28\!\cdots\!40\)\( T^{14} - \)\(75\!\cdots\!60\)\( T^{15} + \)\(18\!\cdots\!81\)\( T^{16} \)
$29$ \( 1 + \)\(15\!\cdots\!80\)\( T + \)\(17\!\cdots\!32\)\( T^{2} + \)\(72\!\cdots\!40\)\( T^{3} + \)\(13\!\cdots\!48\)\( T^{4} + \)\(97\!\cdots\!80\)\( T^{5} + \)\(74\!\cdots\!84\)\( T^{6} + \)\(75\!\cdots\!00\)\( T^{7} + \)\(35\!\cdots\!70\)\( T^{8} + \)\(38\!\cdots\!00\)\( T^{9} + \)\(18\!\cdots\!44\)\( T^{10} + \)\(12\!\cdots\!20\)\( T^{11} + \)\(89\!\cdots\!88\)\( T^{12} + \)\(23\!\cdots\!60\)\( T^{13} + \)\(27\!\cdots\!72\)\( T^{14} + \)\(12\!\cdots\!20\)\( T^{15} + \)\(41\!\cdots\!61\)\( T^{16} \)
$31$ \( 1 + \)\(65\!\cdots\!44\)\( T + \)\(33\!\cdots\!20\)\( T^{2} + \)\(12\!\cdots\!40\)\( T^{3} + \)\(38\!\cdots\!20\)\( T^{4} + \)\(10\!\cdots\!32\)\( T^{5} + \)\(27\!\cdots\!28\)\( T^{6} + \)\(62\!\cdots\!60\)\( T^{7} + \)\(13\!\cdots\!70\)\( T^{8} + \)\(26\!\cdots\!60\)\( T^{9} + \)\(49\!\cdots\!08\)\( T^{10} + \)\(81\!\cdots\!12\)\( T^{11} + \)\(12\!\cdots\!20\)\( T^{12} + \)\(16\!\cdots\!40\)\( T^{13} + \)\(19\!\cdots\!20\)\( T^{14} + \)\(16\!\cdots\!84\)\( T^{15} + \)\(10\!\cdots\!41\)\( T^{16} \)
$37$ \( 1 - \)\(39\!\cdots\!40\)\( T + \)\(16\!\cdots\!40\)\( T^{2} - \)\(35\!\cdots\!80\)\( T^{3} + \)\(86\!\cdots\!76\)\( T^{4} - \)\(13\!\cdots\!80\)\( T^{5} + \)\(25\!\cdots\!80\)\( T^{6} - \)\(31\!\cdots\!60\)\( T^{7} + \)\(59\!\cdots\!66\)\( T^{8} - \)\(76\!\cdots\!20\)\( T^{9} + \)\(14\!\cdots\!20\)\( T^{10} - \)\(19\!\cdots\!40\)\( T^{11} + \)\(30\!\cdots\!36\)\( T^{12} - \)\(31\!\cdots\!60\)\( T^{13} + \)\(34\!\cdots\!60\)\( T^{14} - \)\(20\!\cdots\!20\)\( T^{15} + \)\(12\!\cdots\!21\)\( T^{16} \)
$41$ \( 1 + \)\(56\!\cdots\!36\)\( T + \)\(41\!\cdots\!20\)\( T^{2} + \)\(17\!\cdots\!60\)\( T^{3} + \)\(83\!\cdots\!20\)\( T^{4} + \)\(28\!\cdots\!68\)\( T^{5} + \)\(10\!\cdots\!68\)\( T^{6} + \)\(30\!\cdots\!40\)\( T^{7} + \)\(96\!\cdots\!70\)\( T^{8} + \)\(23\!\cdots\!40\)\( T^{9} + \)\(64\!\cdots\!08\)\( T^{10} + \)\(13\!\cdots\!28\)\( T^{11} + \)\(30\!\cdots\!20\)\( T^{12} + \)\(49\!\cdots\!60\)\( T^{13} + \)\(91\!\cdots\!20\)\( T^{14} + \)\(97\!\cdots\!16\)\( T^{15} + \)\(13\!\cdots\!21\)\( T^{16} \)
$43$ \( 1 + \)\(28\!\cdots\!00\)\( T + \)\(58\!\cdots\!00\)\( T^{2} + \)\(12\!\cdots\!00\)\( T^{3} + \)\(15\!\cdots\!96\)\( T^{4} + \)\(26\!\cdots\!00\)\( T^{5} + \)\(24\!\cdots\!00\)\( T^{6} + \)\(34\!\cdots\!00\)\( T^{7} + \)\(27\!\cdots\!06\)\( T^{8} + \)\(33\!\cdots\!00\)\( T^{9} + \)\(22\!\cdots\!00\)\( T^{10} + \)\(23\!\cdots\!00\)\( T^{11} + \)\(12\!\cdots\!96\)\( T^{12} + \)\(10\!\cdots\!00\)\( T^{13} + \)\(44\!\cdots\!00\)\( T^{14} + \)\(20\!\cdots\!00\)\( T^{15} + \)\(69\!\cdots\!01\)\( T^{16} \)
$47$ \( 1 - \)\(45\!\cdots\!80\)\( T + \)\(42\!\cdots\!20\)\( T^{2} - \)\(14\!\cdots\!60\)\( T^{3} + \)\(80\!\cdots\!36\)\( T^{4} - \)\(22\!\cdots\!60\)\( T^{5} + \)\(99\!\cdots\!40\)\( T^{6} - \)\(24\!\cdots\!20\)\( T^{7} + \)\(89\!\cdots\!86\)\( T^{8} - \)\(18\!\cdots\!40\)\( T^{9} + \)\(57\!\cdots\!60\)\( T^{10} - \)\(10\!\cdots\!80\)\( T^{11} + \)\(26\!\cdots\!16\)\( T^{12} - \)\(36\!\cdots\!20\)\( T^{13} + \)\(82\!\cdots\!80\)\( T^{14} - \)\(68\!\cdots\!40\)\( T^{15} + \)\(11\!\cdots\!61\)\( T^{16} \)
$53$ \( 1 + \)\(13\!\cdots\!40\)\( T + \)\(52\!\cdots\!20\)\( T^{2} + \)\(30\!\cdots\!20\)\( T^{3} + \)\(15\!\cdots\!36\)\( T^{4} + \)\(10\!\cdots\!80\)\( T^{5} + \)\(35\!\cdots\!40\)\( T^{6} + \)\(22\!\cdots\!40\)\( T^{7} + \)\(58\!\cdots\!86\)\( T^{8} + \)\(31\!\cdots\!20\)\( T^{9} + \)\(71\!\cdots\!60\)\( T^{10} + \)\(30\!\cdots\!60\)\( T^{11} + \)\(63\!\cdots\!16\)\( T^{12} + \)\(17\!\cdots\!60\)\( T^{13} + \)\(42\!\cdots\!80\)\( T^{14} + \)\(15\!\cdots\!80\)\( T^{15} + \)\(16\!\cdots\!61\)\( T^{16} \)
$59$ \( 1 + \)\(21\!\cdots\!60\)\( T + \)\(28\!\cdots\!72\)\( T^{2} + \)\(86\!\cdots\!80\)\( T^{3} + \)\(45\!\cdots\!68\)\( T^{4} + \)\(15\!\cdots\!60\)\( T^{5} + \)\(50\!\cdots\!24\)\( T^{6} + \)\(16\!\cdots\!00\)\( T^{7} + \)\(41\!\cdots\!70\)\( T^{8} + \)\(12\!\cdots\!00\)\( T^{9} + \)\(26\!\cdots\!44\)\( T^{10} + \)\(57\!\cdots\!40\)\( T^{11} + \)\(12\!\cdots\!48\)\( T^{12} + \)\(16\!\cdots\!20\)\( T^{13} + \)\(39\!\cdots\!52\)\( T^{14} + \)\(20\!\cdots\!40\)\( T^{15} + \)\(70\!\cdots\!21\)\( T^{16} \)
$61$ \( 1 + \)\(33\!\cdots\!04\)\( T + \)\(13\!\cdots\!20\)\( T^{2} + \)\(35\!\cdots\!40\)\( T^{3} + \)\(88\!\cdots\!20\)\( T^{4} + \)\(18\!\cdots\!32\)\( T^{5} + \)\(34\!\cdots\!48\)\( T^{6} + \)\(56\!\cdots\!60\)\( T^{7} + \)\(86\!\cdots\!70\)\( T^{8} + \)\(11\!\cdots\!60\)\( T^{9} + \)\(14\!\cdots\!08\)\( T^{10} + \)\(16\!\cdots\!92\)\( T^{11} + \)\(16\!\cdots\!20\)\( T^{12} + \)\(13\!\cdots\!40\)\( T^{13} + \)\(11\!\cdots\!20\)\( T^{14} + \)\(57\!\cdots\!84\)\( T^{15} + \)\(35\!\cdots\!81\)\( T^{16} \)
$67$ \( 1 + \)\(61\!\cdots\!20\)\( T + \)\(25\!\cdots\!80\)\( T^{2} + \)\(85\!\cdots\!40\)\( T^{3} + \)\(23\!\cdots\!56\)\( T^{4} + \)\(57\!\cdots\!40\)\( T^{5} + \)\(12\!\cdots\!60\)\( T^{6} + \)\(23\!\cdots\!80\)\( T^{7} + \)\(41\!\cdots\!26\)\( T^{8} + \)\(64\!\cdots\!60\)\( T^{9} + \)\(90\!\cdots\!40\)\( T^{10} + \)\(11\!\cdots\!20\)\( T^{11} + \)\(12\!\cdots\!76\)\( T^{12} + \)\(12\!\cdots\!80\)\( T^{13} + \)\(10\!\cdots\!20\)\( T^{14} + \)\(67\!\cdots\!60\)\( T^{15} + \)\(29\!\cdots\!41\)\( T^{16} \)
$71$ \( 1 - \)\(15\!\cdots\!24\)\( T + \)\(16\!\cdots\!20\)\( T^{2} - \)\(12\!\cdots\!40\)\( T^{3} + \)\(75\!\cdots\!20\)\( T^{4} - \)\(37\!\cdots\!32\)\( T^{5} + \)\(16\!\cdots\!88\)\( T^{6} - \)\(61\!\cdots\!60\)\( T^{7} + \)\(20\!\cdots\!70\)\( T^{8} - \)\(58\!\cdots\!60\)\( T^{9} + \)\(14\!\cdots\!08\)\( T^{10} - \)\(32\!\cdots\!52\)\( T^{11} + \)\(61\!\cdots\!20\)\( T^{12} - \)\(95\!\cdots\!40\)\( T^{13} + \)\(12\!\cdots\!20\)\( T^{14} - \)\(10\!\cdots\!84\)\( T^{15} + \)\(65\!\cdots\!61\)\( T^{16} \)
$73$ \( 1 + \)\(20\!\cdots\!80\)\( T + \)\(67\!\cdots\!60\)\( T^{2} + \)\(75\!\cdots\!40\)\( T^{3} + \)\(15\!\cdots\!16\)\( T^{4} + \)\(10\!\cdots\!60\)\( T^{5} + \)\(21\!\cdots\!20\)\( T^{6} + \)\(10\!\cdots\!80\)\( T^{7} + \)\(31\!\cdots\!46\)\( T^{8} + \)\(16\!\cdots\!40\)\( T^{9} + \)\(52\!\cdots\!80\)\( T^{10} + \)\(38\!\cdots\!20\)\( T^{11} + \)\(92\!\cdots\!56\)\( T^{12} + \)\(72\!\cdots\!20\)\( T^{13} + \)\(10\!\cdots\!40\)\( T^{14} + \)\(47\!\cdots\!60\)\( T^{15} + \)\(36\!\cdots\!81\)\( T^{16} \)
$79$ \( 1 - \)\(14\!\cdots\!80\)\( T + \)\(34\!\cdots\!32\)\( T^{2} - \)\(37\!\cdots\!40\)\( T^{3} + \)\(51\!\cdots\!48\)\( T^{4} - \)\(44\!\cdots\!80\)\( T^{5} + \)\(44\!\cdots\!84\)\( T^{6} - \)\(31\!\cdots\!00\)\( T^{7} + \)\(25\!\cdots\!70\)\( T^{8} - \)\(14\!\cdots\!00\)\( T^{9} + \)\(93\!\cdots\!44\)\( T^{10} - \)\(42\!\cdots\!20\)\( T^{11} + \)\(22\!\cdots\!88\)\( T^{12} - \)\(74\!\cdots\!60\)\( T^{13} + \)\(31\!\cdots\!72\)\( T^{14} - \)\(61\!\cdots\!20\)\( T^{15} + \)\(19\!\cdots\!61\)\( T^{16} \)
$83$ \( 1 + \)\(33\!\cdots\!60\)\( T + \)\(94\!\cdots\!80\)\( T^{2} + \)\(17\!\cdots\!80\)\( T^{3} + \)\(29\!\cdots\!56\)\( T^{4} + \)\(38\!\cdots\!20\)\( T^{5} + \)\(45\!\cdots\!60\)\( T^{6} + \)\(44\!\cdots\!60\)\( T^{7} + \)\(39\!\cdots\!26\)\( T^{8} + \)\(29\!\cdots\!80\)\( T^{9} + \)\(20\!\cdots\!40\)\( T^{10} + \)\(11\!\cdots\!40\)\( T^{11} + \)\(59\!\cdots\!76\)\( T^{12} + \)\(24\!\cdots\!40\)\( T^{13} + \)\(86\!\cdots\!20\)\( T^{14} + \)\(20\!\cdots\!20\)\( T^{15} + \)\(41\!\cdots\!41\)\( T^{16} \)
$89$ \( 1 - \)\(62\!\cdots\!60\)\( T + \)\(43\!\cdots\!12\)\( T^{2} - \)\(20\!\cdots\!80\)\( T^{3} + \)\(99\!\cdots\!88\)\( T^{4} - \)\(37\!\cdots\!60\)\( T^{5} + \)\(13\!\cdots\!64\)\( T^{6} - \)\(42\!\cdots\!00\)\( T^{7} + \)\(12\!\cdots\!70\)\( T^{8} - \)\(32\!\cdots\!00\)\( T^{9} + \)\(82\!\cdots\!44\)\( T^{10} - \)\(17\!\cdots\!40\)\( T^{11} + \)\(35\!\cdots\!08\)\( T^{12} - \)\(57\!\cdots\!20\)\( T^{13} + \)\(93\!\cdots\!32\)\( T^{14} - \)\(10\!\cdots\!40\)\( T^{15} + \)\(12\!\cdots\!81\)\( T^{16} \)
$97$ \( 1 - \)\(64\!\cdots\!20\)\( T + \)\(50\!\cdots\!20\)\( T^{2} - \)\(21\!\cdots\!40\)\( T^{3} + \)\(94\!\cdots\!36\)\( T^{4} - \)\(29\!\cdots\!40\)\( T^{5} + \)\(94\!\cdots\!40\)\( T^{6} - \)\(22\!\cdots\!80\)\( T^{7} + \)\(55\!\cdots\!86\)\( T^{8} - \)\(10\!\cdots\!60\)\( T^{9} + \)\(20\!\cdots\!60\)\( T^{10} - \)\(28\!\cdots\!20\)\( T^{11} + \)\(42\!\cdots\!16\)\( T^{12} - \)\(44\!\cdots\!80\)\( T^{13} + \)\(48\!\cdots\!80\)\( T^{14} - \)\(28\!\cdots\!60\)\( T^{15} + \)\(20\!\cdots\!61\)\( T^{16} \)
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