Properties

Label 9.94.a.b
Level $9$
Weight $94$
Character orbit 9.a
Self dual yes
Analytic conductor $492.953$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 94 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(492.952887545\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - x^{6} - 160477500301516091326739 x^{5} + 877016488484326647371325741724874 x^{4} + 7260529465737129707868752892581169765229378456 x^{3} - 20781038399188480098606854392326662967337072615105929280 x^{2} - 71309214652872234197294752847774640455181142633761719353245451878000 x - 1353216958878139720025204995487184336935523797943751976847532373756765247900000\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: multiple of \( 2^{88}\cdot 3^{47}\cdot 5^{10}\cdot 7^{6}\cdot 13^{2}\cdot 19\cdot 23\cdot 31^{2} \)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q +(-6247918101970 - \beta_{1}) q^{2} +(\)\(53\!\cdots\!63\)\( + 7774038789800 \beta_{1} + \beta_{2} + \beta_{3}) q^{4} +(\)\(34\!\cdots\!14\)\( - 546350849336486824 \beta_{1} + 7000 \beta_{2} - 133741 \beta_{3} - \beta_{4}) q^{5} +(-\)\(13\!\cdots\!05\)\( - \)\(77\!\cdots\!53\)\( \beta_{1} + 17728695310 \beta_{2} - 247821250976 \beta_{3} - 1347465 \beta_{4} + 5 \beta_{5} + \beta_{6}) q^{7} +(-\)\(89\!\cdots\!80\)\( - \)\(34\!\cdots\!28\)\( \beta_{1} + 4555802753680 \beta_{2} - 155142065626688 \beta_{3} - 603912480 \beta_{4} - 320 \beta_{5} + 432 \beta_{6}) q^{8} +O(q^{10})\) \( q +(-6247918101970 - \beta_{1}) q^{2} +(\)\(53\!\cdots\!63\)\( + 7774038789800 \beta_{1} + \beta_{2} + \beta_{3}) q^{4} +(\)\(34\!\cdots\!14\)\( - 546350849336486824 \beta_{1} + 7000 \beta_{2} - 133741 \beta_{3} - \beta_{4}) q^{5} +(-\)\(13\!\cdots\!05\)\( - \)\(77\!\cdots\!53\)\( \beta_{1} + 17728695310 \beta_{2} - 247821250976 \beta_{3} - 1347465 \beta_{4} + 5 \beta_{5} + \beta_{6}) q^{7} +(-\)\(89\!\cdots\!80\)\( - \)\(34\!\cdots\!28\)\( \beta_{1} + 4555802753680 \beta_{2} - 155142065626688 \beta_{3} - 603912480 \beta_{4} - 320 \beta_{5} + 432 \beta_{6}) q^{8} +(\)\(80\!\cdots\!52\)\( - \)\(80\!\cdots\!82\)\( \beta_{1} + 1221375550027936100 \beta_{2} + 7867627157681475912 \beta_{3} - 29250595664868 \beta_{4} + 761680700 \beta_{5} + 13148800 \beta_{6}) q^{10} +(-\)\(15\!\cdots\!22\)\( - \)\(31\!\cdots\!70\)\( \beta_{1} - 6948721937616726828 \beta_{2} + \)\(69\!\cdots\!59\)\( \beta_{3} + 4354693919113802 \beta_{4} + 13194774494 \beta_{5} + 846185670 \beta_{6}) q^{11} +(\)\(27\!\cdots\!10\)\( + \)\(73\!\cdots\!36\)\( \beta_{1} - \)\(98\!\cdots\!60\)\( \beta_{2} + \)\(16\!\cdots\!05\)\( \beta_{3} + 67994208078837105 \beta_{4} + 92713892893960 \beta_{5} - 744622284952 \beta_{6}) q^{13} +(\)\(11\!\cdots\!88\)\( + \)\(19\!\cdots\!40\)\( \beta_{1} + \)\(19\!\cdots\!82\)\( \beta_{2} - \)\(10\!\cdots\!68\)\( \beta_{3} - \)\(41\!\cdots\!50\)\( \beta_{4} + 1637175777514310 \beta_{5} - 9315909757440 \beta_{6}) q^{14} +(-\)\(26\!\cdots\!12\)\( - \)\(78\!\cdots\!80\)\( \beta_{1} - \)\(13\!\cdots\!52\)\( \beta_{2} + \)\(40\!\cdots\!64\)\( \beta_{3} - \)\(14\!\cdots\!64\)\( \beta_{4} + 825393271759040512 \beta_{5} - 6225619981419520 \beta_{6}) q^{16} +(-\)\(11\!\cdots\!90\)\( - \)\(71\!\cdots\!00\)\( \beta_{1} - \)\(50\!\cdots\!40\)\( \beta_{2} - \)\(46\!\cdots\!62\)\( \beta_{3} - \)\(21\!\cdots\!90\)\( \beta_{4} + 19907570924128813380 \beta_{5} - 182593265909684364 \beta_{6}) q^{17} +(-\)\(70\!\cdots\!46\)\( - \)\(37\!\cdots\!10\)\( \beta_{1} - \)\(93\!\cdots\!00\)\( \beta_{2} + \)\(13\!\cdots\!51\)\( \beta_{3} - \)\(56\!\cdots\!54\)\( \beta_{4} - \)\(18\!\cdots\!38\)\( \beta_{5} + 56835316641090318310 \beta_{6}) q^{19} +(\)\(83\!\cdots\!98\)\( - \)\(12\!\cdots\!68\)\( \beta_{1} + \)\(20\!\cdots\!50\)\( \beta_{2} - \)\(38\!\cdots\!62\)\( \beta_{3} - \)\(55\!\cdots\!32\)\( \beta_{4} + \)\(18\!\cdots\!00\)\( \beta_{5} + \)\(72\!\cdots\!00\)\( \beta_{6}) q^{20} +(\)\(48\!\cdots\!90\)\( + \)\(16\!\cdots\!54\)\( \beta_{1} + \)\(10\!\cdots\!85\)\( \beta_{2} - \)\(11\!\cdots\!50\)\( \beta_{3} - \)\(31\!\cdots\!05\)\( \beta_{4} - \)\(31\!\cdots\!85\)\( \beta_{5} - \)\(69\!\cdots\!68\)\( \beta_{6}) q^{22} +(\)\(37\!\cdots\!35\)\( + \)\(58\!\cdots\!07\)\( \beta_{1} + \)\(67\!\cdots\!50\)\( \beta_{2} + \)\(92\!\cdots\!56\)\( \beta_{3} - \)\(24\!\cdots\!25\)\( \beta_{4} - \)\(25\!\cdots\!75\)\( \beta_{5} - \)\(51\!\cdots\!15\)\( \beta_{6}) q^{23} +(\)\(26\!\cdots\!75\)\( - \)\(30\!\cdots\!00\)\( \beta_{1} + \)\(57\!\cdots\!00\)\( \beta_{2} - \)\(27\!\cdots\!00\)\( \beta_{3} - \)\(14\!\cdots\!00\)\( \beta_{4} + \)\(32\!\cdots\!00\)\( \beta_{5} + \)\(13\!\cdots\!00\)\( \beta_{6}) q^{25} +(-\)\(11\!\cdots\!80\)\( + \)\(38\!\cdots\!10\)\( \beta_{1} - \)\(14\!\cdots\!36\)\( \beta_{2} - \)\(65\!\cdots\!24\)\( \beta_{3} - \)\(92\!\cdots\!48\)\( \beta_{4} - \)\(44\!\cdots\!36\)\( \beta_{5} + \)\(21\!\cdots\!40\)\( \beta_{6}) q^{26} +(\)\(27\!\cdots\!40\)\( - \)\(19\!\cdots\!36\)\( \beta_{1} + \)\(23\!\cdots\!80\)\( \beta_{2} - \)\(87\!\cdots\!44\)\( \beta_{3} - \)\(27\!\cdots\!80\)\( \beta_{4} + \)\(18\!\cdots\!80\)\( \beta_{5} + \)\(38\!\cdots\!52\)\( \beta_{6}) q^{28} +(-\)\(16\!\cdots\!26\)\( + \)\(69\!\cdots\!40\)\( \beta_{1} - \)\(34\!\cdots\!68\)\( \beta_{2} + \)\(37\!\cdots\!15\)\( \beta_{3} - \)\(16\!\cdots\!57\)\( \beta_{4} + \)\(34\!\cdots\!76\)\( \beta_{5} + \)\(29\!\cdots\!60\)\( \beta_{6}) q^{29} +(-\)\(16\!\cdots\!28\)\( - \)\(93\!\cdots\!20\)\( \beta_{1} - \)\(76\!\cdots\!96\)\( \beta_{2} + \)\(14\!\cdots\!48\)\( \beta_{3} - \)\(15\!\cdots\!76\)\( \beta_{4} + \)\(17\!\cdots\!08\)\( \beta_{5} - \)\(77\!\cdots\!80\)\( \beta_{6}) q^{31} +(\)\(10\!\cdots\!00\)\( + \)\(44\!\cdots\!00\)\( \beta_{1} - \)\(23\!\cdots\!40\)\( \beta_{2} - \)\(34\!\cdots\!20\)\( \beta_{3} - \)\(60\!\cdots\!40\)\( \beta_{4} + \)\(51\!\cdots\!80\)\( \beta_{5} - \)\(28\!\cdots\!64\)\( \beta_{6}) q^{32} +(\)\(11\!\cdots\!96\)\( + \)\(55\!\cdots\!90\)\( \beta_{1} + \)\(29\!\cdots\!08\)\( \beta_{2} - \)\(96\!\cdots\!36\)\( \beta_{3} - \)\(21\!\cdots\!24\)\( \beta_{4} + \)\(17\!\cdots\!72\)\( \beta_{5} + \)\(14\!\cdots\!60\)\( \beta_{6}) q^{34} +(\)\(23\!\cdots\!88\)\( - \)\(42\!\cdots\!08\)\( \beta_{1} + \)\(39\!\cdots\!00\)\( \beta_{2} + \)\(15\!\cdots\!28\)\( \beta_{3} - \)\(35\!\cdots\!92\)\( \beta_{4} + \)\(90\!\cdots\!00\)\( \beta_{5} + \)\(96\!\cdots\!00\)\( \beta_{6}) q^{35} +(\)\(15\!\cdots\!70\)\( + \)\(86\!\cdots\!72\)\( \beta_{1} + \)\(31\!\cdots\!40\)\( \beta_{2} + \)\(85\!\cdots\!13\)\( \beta_{3} - \)\(11\!\cdots\!15\)\( \beta_{4} - \)\(37\!\cdots\!60\)\( \beta_{5} - \)\(60\!\cdots\!24\)\( \beta_{6}) q^{37} +(\)\(60\!\cdots\!30\)\( + \)\(13\!\cdots\!74\)\( \beta_{1} + \)\(94\!\cdots\!25\)\( \beta_{2} + \)\(92\!\cdots\!78\)\( \beta_{3} - \)\(75\!\cdots\!85\)\( \beta_{4} - \)\(28\!\cdots\!85\)\( \beta_{5} - \)\(10\!\cdots\!56\)\( \beta_{6}) q^{38} +(\)\(10\!\cdots\!80\)\( - \)\(14\!\cdots\!80\)\( \beta_{1} + \)\(17\!\cdots\!00\)\( \beta_{2} + \)\(74\!\cdots\!80\)\( \beta_{3} - \)\(13\!\cdots\!20\)\( \beta_{4} + \)\(40\!\cdots\!00\)\( \beta_{5} - \)\(71\!\cdots\!00\)\( \beta_{6}) q^{40} +(\)\(72\!\cdots\!38\)\( + \)\(21\!\cdots\!00\)\( \beta_{1} - \)\(12\!\cdots\!04\)\( \beta_{2} - \)\(28\!\cdots\!08\)\( \beta_{3} + \)\(13\!\cdots\!16\)\( \beta_{4} + \)\(17\!\cdots\!92\)\( \beta_{5} - \)\(11\!\cdots\!00\)\( \beta_{6}) q^{41} +(-\)\(10\!\cdots\!40\)\( + \)\(30\!\cdots\!68\)\( \beta_{1} - \)\(14\!\cdots\!60\)\( \beta_{2} - \)\(50\!\cdots\!83\)\( \beta_{3} - \)\(10\!\cdots\!00\)\( \beta_{4} - \)\(50\!\cdots\!20\)\( \beta_{5} + \)\(13\!\cdots\!00\)\( \beta_{6}) q^{43} +(-\)\(13\!\cdots\!36\)\( - \)\(90\!\cdots\!20\)\( \beta_{1} - \)\(60\!\cdots\!48\)\( \beta_{2} + \)\(17\!\cdots\!76\)\( \beta_{3} + \)\(61\!\cdots\!04\)\( \beta_{4} + \)\(26\!\cdots\!68\)\( \beta_{5} + \)\(82\!\cdots\!20\)\( \beta_{6}) q^{44} +(-\)\(91\!\cdots\!68\)\( - \)\(97\!\cdots\!00\)\( \beta_{1} - \)\(11\!\cdots\!66\)\( \beta_{2} + \)\(11\!\cdots\!88\)\( \beta_{3} + \)\(34\!\cdots\!34\)\( \beta_{4} - \)\(91\!\cdots\!42\)\( \beta_{5} + \)\(59\!\cdots\!00\)\( \beta_{6}) q^{46} +(\)\(53\!\cdots\!30\)\( + \)\(28\!\cdots\!74\)\( \beta_{1} + \)\(43\!\cdots\!80\)\( \beta_{2} + \)\(16\!\cdots\!64\)\( \beta_{3} + \)\(12\!\cdots\!50\)\( \beta_{4} + \)\(67\!\cdots\!10\)\( \beta_{5} + \)\(12\!\cdots\!50\)\( \beta_{6}) q^{47} +(\)\(36\!\cdots\!77\)\( - \)\(66\!\cdots\!00\)\( \beta_{1} + \)\(19\!\cdots\!76\)\( \beta_{2} - \)\(15\!\cdots\!08\)\( \beta_{3} - \)\(15\!\cdots\!64\)\( \beta_{4} + \)\(33\!\cdots\!32\)\( \beta_{5} + \)\(10\!\cdots\!00\)\( \beta_{6}) q^{49} +(\)\(44\!\cdots\!50\)\( - \)\(71\!\cdots\!75\)\( \beta_{1} + \)\(59\!\cdots\!00\)\( \beta_{2} - \)\(88\!\cdots\!00\)\( \beta_{3} - \)\(12\!\cdots\!00\)\( \beta_{4} + \)\(85\!\cdots\!00\)\( \beta_{5} + \)\(35\!\cdots\!00\)\( \beta_{6}) q^{50} +(-\)\(78\!\cdots\!70\)\( + \)\(19\!\cdots\!56\)\( \beta_{1} + \)\(14\!\cdots\!70\)\( \beta_{2} + \)\(42\!\cdots\!54\)\( \beta_{3} + \)\(46\!\cdots\!00\)\( \beta_{4} + \)\(94\!\cdots\!40\)\( \beta_{5} + \)\(79\!\cdots\!00\)\( \beta_{6}) q^{52} +(\)\(51\!\cdots\!50\)\( - \)\(98\!\cdots\!56\)\( \beta_{1} - \)\(35\!\cdots\!20\)\( \beta_{2} + \)\(30\!\cdots\!99\)\( \beta_{3} - \)\(22\!\cdots\!05\)\( \beta_{4} + \)\(23\!\cdots\!80\)\( \beta_{5} + \)\(51\!\cdots\!12\)\( \beta_{6}) q^{53} +(-\)\(50\!\cdots\!73\)\( - \)\(33\!\cdots\!57\)\( \beta_{1} + \)\(29\!\cdots\!50\)\( \beta_{2} + \)\(86\!\cdots\!12\)\( \beta_{3} + \)\(41\!\cdots\!07\)\( \beta_{4} + \)\(70\!\cdots\!25\)\( \beta_{5} - \)\(19\!\cdots\!75\)\( \beta_{6}) q^{55} +(\)\(18\!\cdots\!36\)\( - \)\(14\!\cdots\!60\)\( \beta_{1} + \)\(74\!\cdots\!88\)\( \beta_{2} - \)\(48\!\cdots\!80\)\( \beta_{3} - \)\(18\!\cdots\!28\)\( \beta_{4} + \)\(16\!\cdots\!24\)\( \beta_{5} + \)\(42\!\cdots\!60\)\( \beta_{6}) q^{56} +(-\)\(10\!\cdots\!40\)\( + \)\(40\!\cdots\!26\)\( \beta_{1} - \)\(36\!\cdots\!20\)\( \beta_{2} - \)\(36\!\cdots\!52\)\( \beta_{3} - \)\(17\!\cdots\!40\)\( \beta_{4} - \)\(22\!\cdots\!80\)\( \beta_{5} - \)\(64\!\cdots\!64\)\( \beta_{6}) q^{58} +(\)\(17\!\cdots\!68\)\( - \)\(86\!\cdots\!80\)\( \beta_{1} - \)\(32\!\cdots\!72\)\( \beta_{2} + \)\(61\!\cdots\!43\)\( \beta_{3} - \)\(52\!\cdots\!60\)\( \beta_{4} - \)\(13\!\cdots\!40\)\( \beta_{5} + \)\(21\!\cdots\!80\)\( \beta_{6}) q^{59} +(-\)\(46\!\cdots\!98\)\( - \)\(35\!\cdots\!60\)\( \beta_{1} + \)\(84\!\cdots\!20\)\( \beta_{2} + \)\(23\!\cdots\!85\)\( \beta_{3} - \)\(69\!\cdots\!35\)\( \beta_{4} - \)\(16\!\cdots\!60\)\( \beta_{5} + \)\(44\!\cdots\!60\)\( \beta_{6}) q^{61} +(\)\(14\!\cdots\!40\)\( + \)\(69\!\cdots\!32\)\( \beta_{1} + \)\(21\!\cdots\!40\)\( \beta_{2} - \)\(14\!\cdots\!80\)\( \beta_{3} - \)\(16\!\cdots\!60\)\( \beta_{4} - \)\(29\!\cdots\!80\)\( \beta_{5} + \)\(15\!\cdots\!24\)\( \beta_{6}) q^{62} +(-\)\(67\!\cdots\!56\)\( - \)\(59\!\cdots\!00\)\( \beta_{1} - \)\(54\!\cdots\!96\)\( \beta_{2} - \)\(26\!\cdots\!64\)\( \beta_{3} + \)\(15\!\cdots\!72\)\( \beta_{4} + \)\(49\!\cdots\!64\)\( \beta_{5} + \)\(22\!\cdots\!00\)\( \beta_{6}) q^{64} +(-\)\(34\!\cdots\!56\)\( - \)\(10\!\cdots\!04\)\( \beta_{1} - \)\(67\!\cdots\!00\)\( \beta_{2} + \)\(42\!\cdots\!64\)\( \beta_{3} + \)\(54\!\cdots\!04\)\( \beta_{4} - \)\(11\!\cdots\!00\)\( \beta_{5} + \)\(30\!\cdots\!00\)\( \beta_{6}) q^{65} +(\)\(13\!\cdots\!30\)\( - \)\(23\!\cdots\!50\)\( \beta_{1} - \)\(19\!\cdots\!00\)\( \beta_{2} + \)\(36\!\cdots\!87\)\( \beta_{3} + \)\(31\!\cdots\!90\)\( \beta_{4} - \)\(13\!\cdots\!10\)\( \beta_{5} + \)\(31\!\cdots\!54\)\( \beta_{6}) q^{67} +(-\)\(73\!\cdots\!10\)\( - \)\(13\!\cdots\!84\)\( \beta_{1} + \)\(37\!\cdots\!70\)\( \beta_{2} - \)\(26\!\cdots\!82\)\( \beta_{3} - \)\(92\!\cdots\!20\)\( \beta_{4} + \)\(11\!\cdots\!20\)\( \beta_{5} + \)\(44\!\cdots\!08\)\( \beta_{6}) q^{68} +(\)\(62\!\cdots\!84\)\( - \)\(55\!\cdots\!44\)\( \beta_{1} + \)\(52\!\cdots\!00\)\( \beta_{2} - \)\(63\!\cdots\!96\)\( \beta_{3} - \)\(15\!\cdots\!56\)\( \beta_{4} - \)\(12\!\cdots\!00\)\( \beta_{5} + \)\(24\!\cdots\!00\)\( \beta_{6}) q^{70} +(-\)\(60\!\cdots\!47\)\( - \)\(18\!\cdots\!95\)\( \beta_{1} + \)\(39\!\cdots\!90\)\( \beta_{2} + \)\(75\!\cdots\!20\)\( \beta_{3} - \)\(10\!\cdots\!95\)\( \beta_{4} + \)\(15\!\cdots\!55\)\( \beta_{5} + \)\(40\!\cdots\!95\)\( \beta_{6}) q^{71} +(\)\(35\!\cdots\!70\)\( + \)\(16\!\cdots\!84\)\( \beta_{1} - \)\(20\!\cdots\!00\)\( \beta_{2} + \)\(26\!\cdots\!34\)\( \beta_{3} - \)\(28\!\cdots\!90\)\( \beta_{4} - \)\(26\!\cdots\!40\)\( \beta_{5} + \)\(96\!\cdots\!16\)\( \beta_{6}) q^{73} +(-\)\(14\!\cdots\!64\)\( - \)\(41\!\cdots\!90\)\( \beta_{1} - \)\(17\!\cdots\!64\)\( \beta_{2} + \)\(19\!\cdots\!92\)\( \beta_{3} - \)\(42\!\cdots\!24\)\( \beta_{4} + \)\(41\!\cdots\!52\)\( \beta_{5} + \)\(24\!\cdots\!40\)\( \beta_{6}) q^{74} +(-\)\(14\!\cdots\!72\)\( - \)\(10\!\cdots\!80\)\( \beta_{1} - \)\(47\!\cdots\!56\)\( \beta_{2} - \)\(17\!\cdots\!00\)\( \beta_{3} + \)\(52\!\cdots\!76\)\( \beta_{4} + \)\(31\!\cdots\!92\)\( \beta_{5} - \)\(56\!\cdots\!20\)\( \beta_{6}) q^{76} +(\)\(23\!\cdots\!20\)\( + \)\(11\!\cdots\!64\)\( \beta_{1} - \)\(24\!\cdots\!40\)\( \beta_{2} - \)\(27\!\cdots\!28\)\( \beta_{3} - \)\(11\!\cdots\!40\)\( \beta_{4} - \)\(29\!\cdots\!20\)\( \beta_{5} + \)\(10\!\cdots\!36\)\( \beta_{6}) q^{77} +(-\)\(61\!\cdots\!94\)\( + \)\(59\!\cdots\!70\)\( \beta_{1} + \)\(69\!\cdots\!92\)\( \beta_{2} + \)\(16\!\cdots\!28\)\( \beta_{3} - \)\(37\!\cdots\!94\)\( \beta_{4} - \)\(44\!\cdots\!98\)\( \beta_{5} - \)\(26\!\cdots\!70\)\( \beta_{6}) q^{79} +(\)\(12\!\cdots\!24\)\( - \)\(12\!\cdots\!84\)\( \beta_{1} - \)\(33\!\cdots\!00\)\( \beta_{2} + \)\(88\!\cdots\!44\)\( \beta_{3} - \)\(10\!\cdots\!16\)\( \beta_{4} - \)\(10\!\cdots\!00\)\( \beta_{5} + \)\(36\!\cdots\!00\)\( \beta_{6}) q^{80} +(-\)\(33\!\cdots\!20\)\( + \)\(28\!\cdots\!78\)\( \beta_{1} - \)\(34\!\cdots\!60\)\( \beta_{2} - \)\(11\!\cdots\!40\)\( \beta_{3} - \)\(14\!\cdots\!40\)\( \beta_{4} - \)\(68\!\cdots\!60\)\( \beta_{5} - \)\(78\!\cdots\!24\)\( \beta_{6}) q^{82} +(\)\(29\!\cdots\!80\)\( + \)\(73\!\cdots\!72\)\( \beta_{1} + \)\(79\!\cdots\!00\)\( \beta_{2} - \)\(28\!\cdots\!63\)\( \beta_{3} + \)\(68\!\cdots\!00\)\( \beta_{4} + \)\(20\!\cdots\!00\)\( \beta_{5} + \)\(13\!\cdots\!00\)\( \beta_{6}) q^{83} +(\)\(23\!\cdots\!32\)\( - \)\(20\!\cdots\!12\)\( \beta_{1} + \)\(34\!\cdots\!00\)\( \beta_{2} + \)\(27\!\cdots\!42\)\( \beta_{3} + \)\(11\!\cdots\!62\)\( \beta_{4} - \)\(26\!\cdots\!00\)\( \beta_{5} + \)\(60\!\cdots\!00\)\( \beta_{6}) q^{85} +(-\)\(46\!\cdots\!46\)\( + \)\(24\!\cdots\!10\)\( \beta_{1} - \)\(95\!\cdots\!81\)\( \beta_{2} + \)\(25\!\cdots\!62\)\( \beta_{3} - \)\(43\!\cdots\!47\)\( \beta_{4} + \)\(16\!\cdots\!01\)\( \beta_{5} + \)\(21\!\cdots\!40\)\( \beta_{6}) q^{86} +(\)\(90\!\cdots\!40\)\( + \)\(34\!\cdots\!76\)\( \beta_{1} - \)\(23\!\cdots\!40\)\( \beta_{2} - \)\(11\!\cdots\!64\)\( \beta_{3} - \)\(23\!\cdots\!40\)\( \beta_{4} - \)\(12\!\cdots\!20\)\( \beta_{5} + \)\(47\!\cdots\!36\)\( \beta_{6}) q^{88} +(-\)\(79\!\cdots\!78\)\( - \)\(12\!\cdots\!40\)\( \beta_{1} + \)\(39\!\cdots\!96\)\( \beta_{2} + \)\(12\!\cdots\!10\)\( \beta_{3} - \)\(18\!\cdots\!06\)\( \beta_{4} + \)\(12\!\cdots\!68\)\( \beta_{5} - \)\(74\!\cdots\!60\)\( \beta_{6}) q^{89} +(-\)\(25\!\cdots\!04\)\( - \)\(13\!\cdots\!40\)\( \beta_{1} + \)\(16\!\cdots\!32\)\( \beta_{2} - \)\(31\!\cdots\!40\)\( \beta_{3} - \)\(11\!\cdots\!12\)\( \beta_{4} - \)\(14\!\cdots\!04\)\( \beta_{5} - \)\(23\!\cdots\!60\)\( \beta_{6}) q^{91} +(\)\(11\!\cdots\!00\)\( + \)\(11\!\cdots\!48\)\( \beta_{1} + \)\(51\!\cdots\!60\)\( \beta_{2} - \)\(66\!\cdots\!52\)\( \beta_{3} + \)\(31\!\cdots\!40\)\( \beta_{4} - \)\(15\!\cdots\!40\)\( \beta_{5} - \)\(56\!\cdots\!36\)\( \beta_{6}) q^{92} +(-\)\(43\!\cdots\!72\)\( - \)\(92\!\cdots\!80\)\( \beta_{1} - \)\(20\!\cdots\!20\)\( \beta_{2} - \)\(11\!\cdots\!64\)\( \beta_{3} - \)\(12\!\cdots\!24\)\( \beta_{4} - \)\(55\!\cdots\!08\)\( \beta_{5} - \)\(14\!\cdots\!20\)\( \beta_{6}) q^{94} +(\)\(30\!\cdots\!15\)\( + \)\(23\!\cdots\!35\)\( \beta_{1} - \)\(42\!\cdots\!50\)\( \beta_{2} + \)\(45\!\cdots\!40\)\( \beta_{3} - \)\(52\!\cdots\!85\)\( \beta_{4} + \)\(55\!\cdots\!25\)\( \beta_{5} - \)\(10\!\cdots\!75\)\( \beta_{6}) q^{95} +(\)\(62\!\cdots\!10\)\( + \)\(13\!\cdots\!84\)\( \beta_{1} + \)\(26\!\cdots\!20\)\( \beta_{2} + \)\(25\!\cdots\!86\)\( \beta_{3} - \)\(38\!\cdots\!70\)\( \beta_{4} + \)\(16\!\cdots\!20\)\( \beta_{5} - \)\(44\!\cdots\!12\)\( \beta_{6}) q^{97} +(\)\(98\!\cdots\!50\)\( - \)\(17\!\cdots\!61\)\( \beta_{1} + \)\(24\!\cdots\!40\)\( \beta_{2} - \)\(15\!\cdots\!20\)\( \beta_{3} - \)\(86\!\cdots\!40\)\( \beta_{4} + \)\(47\!\cdots\!40\)\( \beta_{5} + \)\(76\!\cdots\!16\)\( \beta_{6}) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q - 43735426713792q^{2} + \)\(37\!\cdots\!44\)\(q^{4} + \)\(24\!\cdots\!50\)\(q^{5} - \)\(92\!\cdots\!08\)\(q^{7} - \)\(62\!\cdots\!60\)\(q^{8} + O(q^{10}) \) \( 7q - 43735426713792q^{2} + \)\(37\!\cdots\!44\)\(q^{4} + \)\(24\!\cdots\!50\)\(q^{5} - \)\(92\!\cdots\!08\)\(q^{7} - \)\(62\!\cdots\!60\)\(q^{8} + \)\(56\!\cdots\!00\)\(q^{10} - \)\(10\!\cdots\!24\)\(q^{11} + \)\(19\!\cdots\!26\)\(q^{13} + \)\(82\!\cdots\!32\)\(q^{14} - \)\(18\!\cdots\!48\)\(q^{16} - \)\(80\!\cdots\!42\)\(q^{17} - \)\(49\!\cdots\!00\)\(q^{19} + \)\(58\!\cdots\!00\)\(q^{20} + \)\(34\!\cdots\!44\)\(q^{22} + \)\(25\!\cdots\!64\)\(q^{23} + \)\(18\!\cdots\!25\)\(q^{25} - \)\(79\!\cdots\!64\)\(q^{26} + \)\(19\!\cdots\!24\)\(q^{28} - \)\(11\!\cdots\!50\)\(q^{29} - \)\(11\!\cdots\!56\)\(q^{31} + \)\(70\!\cdots\!88\)\(q^{32} + \)\(80\!\cdots\!28\)\(q^{34} + \)\(16\!\cdots\!00\)\(q^{35} + \)\(11\!\cdots\!42\)\(q^{37} + \)\(42\!\cdots\!60\)\(q^{38} + \)\(76\!\cdots\!00\)\(q^{40} + \)\(50\!\cdots\!46\)\(q^{41} - \)\(72\!\cdots\!44\)\(q^{43} - \)\(94\!\cdots\!08\)\(q^{44} - \)\(63\!\cdots\!16\)\(q^{46} + \)\(37\!\cdots\!08\)\(q^{47} + \)\(25\!\cdots\!99\)\(q^{49} + \)\(31\!\cdots\!00\)\(q^{50} - \)\(55\!\cdots\!28\)\(q^{52} + \)\(36\!\cdots\!34\)\(q^{53} - \)\(35\!\cdots\!00\)\(q^{55} + \)\(12\!\cdots\!00\)\(q^{56} - \)\(73\!\cdots\!40\)\(q^{58} + \)\(11\!\cdots\!00\)\(q^{59} - \)\(32\!\cdots\!26\)\(q^{61} + \)\(99\!\cdots\!36\)\(q^{62} - \)\(47\!\cdots\!16\)\(q^{64} - \)\(24\!\cdots\!00\)\(q^{65} + \)\(97\!\cdots\!92\)\(q^{67} - \)\(51\!\cdots\!24\)\(q^{68} + \)\(43\!\cdots\!00\)\(q^{70} - \)\(42\!\cdots\!84\)\(q^{71} + \)\(24\!\cdots\!86\)\(q^{73} - \)\(98\!\cdots\!48\)\(q^{74} - \)\(98\!\cdots\!00\)\(q^{76} + \)\(16\!\cdots\!56\)\(q^{77} - \)\(43\!\cdots\!00\)\(q^{79} + \)\(88\!\cdots\!00\)\(q^{80} - \)\(23\!\cdots\!76\)\(q^{82} + \)\(20\!\cdots\!04\)\(q^{83} + \)\(16\!\cdots\!00\)\(q^{85} - \)\(32\!\cdots\!24\)\(q^{86} + \)\(63\!\cdots\!20\)\(q^{88} - \)\(55\!\cdots\!50\)\(q^{89} - \)\(18\!\cdots\!96\)\(q^{91} + \)\(81\!\cdots\!08\)\(q^{92} - \)\(30\!\cdots\!92\)\(q^{94} + \)\(21\!\cdots\!00\)\(q^{95} + \)\(43\!\cdots\!42\)\(q^{97} + \)\(69\!\cdots\!56\)\(q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - x^{6} - 160477500301516091326739 x^{5} + 877016488484326647371325741724874 x^{4} + 7260529465737129707868752892581169765229378456 x^{3} - 20781038399188480098606854392326662967337072615105929280 x^{2} - 71309214652872234197294752847774640455181142633761719353245451878000 x - 1353216958878139720025204995487184336935523797943751976847532373756765247900000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 576 \nu - 82 \)
\(\beta_{2}\)\(=\)\((\)\(\)\(42\!\cdots\!39\)\( \nu^{6} - \)\(20\!\cdots\!89\)\( \nu^{5} - \)\(39\!\cdots\!43\)\( \nu^{4} + \)\(20\!\cdots\!60\)\( \nu^{3} + \)\(78\!\cdots\!80\)\( \nu^{2} - \)\(30\!\cdots\!00\)\( \nu - \)\(25\!\cdots\!36\)\(\)\()/ \)\(15\!\cdots\!56\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(42\!\cdots\!39\)\( \nu^{6} + \)\(20\!\cdots\!89\)\( \nu^{5} + \)\(39\!\cdots\!43\)\( \nu^{4} - \)\(20\!\cdots\!60\)\( \nu^{3} - \)\(28\!\cdots\!24\)\( \nu^{2} + \)\(34\!\cdots\!56\)\( \nu + \)\(22\!\cdots\!20\)\(\)\()/ \)\(15\!\cdots\!56\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(13\!\cdots\!91\)\( \nu^{6} + \)\(15\!\cdots\!83\)\( \nu^{5} - \)\(19\!\cdots\!27\)\( \nu^{4} - \)\(11\!\cdots\!24\)\( \nu^{3} + \)\(75\!\cdots\!40\)\( \nu^{2} + \)\(21\!\cdots\!00\)\( \nu - \)\(43\!\cdots\!40\)\(\)\()/ \)\(76\!\cdots\!80\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(12\!\cdots\!81\)\( \nu^{6} + \)\(12\!\cdots\!33\)\( \nu^{5} - \)\(13\!\cdots\!37\)\( \nu^{4} - \)\(80\!\cdots\!24\)\( \nu^{3} + \)\(29\!\cdots\!20\)\( \nu^{2} - \)\(17\!\cdots\!40\)\( \nu - \)\(44\!\cdots\!20\)\(\)\()/ \)\(38\!\cdots\!40\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(58\!\cdots\!23\)\( \nu^{6} + \)\(17\!\cdots\!31\)\( \nu^{5} - \)\(63\!\cdots\!71\)\( \nu^{4} - \)\(25\!\cdots\!88\)\( \nu^{3} + \)\(12\!\cdots\!96\)\( \nu^{2} + \)\(86\!\cdots\!12\)\( \nu + \)\(85\!\cdots\!36\)\(\)\()/ \)\(21\!\cdots\!08\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 82\)\()/576\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + \beta_{2} - 4721797413976 \beta_{1} + 15212166611438802125090782379\)\()/331776\)
\(\nu^{3}\)\(=\)\((\)\(-27 \beta_{6} + 20 \beta_{5} + 37744530 \beta_{4} + 8524894457564 \beta_{3} - 1456222316209 \beta_{2} + 1451331883308276940739978409 \beta_{1} - 4489298060351043874647971219875128277487\)\()/11943936\)
\(\nu^{4}\)\(=\)\((\)\(4463654783831 \beta_{6} + 798238219324828 \beta_{5} - 155399058601565452026 \beta_{4} + 29377675416162273888592472 \beta_{3} + 28081700575634744158301577 \beta_{2} - 348763612539248792892703285607081928733 \beta_{1} + 344967225271537957345817803555730324565438833409493895\)\()/ 107495424 \)
\(\nu^{5}\)\(=\)\((\)\(-108861376894669650366687411 \beta_{6} - 494413471742663815676737900 \beta_{5} + 264589821563581979080459601072770 \beta_{4} + 65328708508147712239093572183235676082 \beta_{3} - 8003402270895747784200920079840996515 \beta_{2} + 4460806892794388635602542693550737166985734847100897 \beta_{1} - 41448829546297321210104497334548292109691185965473610528068207829\)\()/ 483729408 \)
\(\nu^{6}\)\(=\)\((\)\(36059619174145705934575197716465475543 \beta_{6} + 3442514373728979593752929087482644458972 \beta_{5} - 539187986013260379481353444036585312505076154 \beta_{4} + 38053089683300542705171376852432808343516150260380 \beta_{3} + 62488481450642421860209148047838152762788129717645 \beta_{2} - 1267123614351128959251451075940738131354705291299847534628033405 \beta_{1} + 706859767444163502813301862011591983439332501377631587115301081642467091815283\)\()/ 2902376448 \)

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
2.74671e11
2.59428e11
1.30998e11
−1.98834e10
−9.86534e10
−2.45391e11
−3.01170e11
−1.64459e14 0 1.71431e28 −3.18897e32 0 −2.05341e39 −1.19061e42 0 5.24454e46
1.2 −1.55678e14 0 1.43322e28 3.85879e32 0 −1.93687e39 −6.89447e41 0 −6.00730e46
1.3 −8.17029e13 0 −3.22816e27 −2.56302e32 0 2.42990e39 1.07290e42 0 2.09406e46
1.4 5.20495e12 0 −9.87643e27 3.34454e32 0 −2.46426e38 −1.02954e41 0 1.74082e45
1.5 5.05764e13 0 −7.34554e27 −1.20105e32 0 −1.85159e39 −8.72396e41 0 −6.07451e45
1.6 1.35097e14 0 8.34770e27 −3.52234e32 0 −7.46408e38 −2.10186e41 0 −4.75858e46
1.7 1.67226e14 0 1.80609e28 5.69707e32 0 3.48367e39 1.36413e42 0 9.52696e46
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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 9.94.a.b 7
3.b odd 2 1 1.94.a.a 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.94.a.a 7 3.b odd 2 1
9.94.a.b 7 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} + \)43735426713792

'>\(43\!\cdots\!92\)\( T_{2}^{6} - \)52422817047248315187327578112
'>\(52\!\cdots\!12\)\( T_{2}^{5} - \)1822340570225776508355098661880437883797504'>\(18\!\cdots\!04\)\( T_{2}^{4} + \)774285750205728873399065770796719479780298817622452994048'>\(77\!\cdots\!48\)\( T_{2}^{3} + \)16128779241049475574046621035320990254797119395978775780995372381372416'>\(16\!\cdots\!16\)\( T_{2}^{2} - \)2494747230911430666923213738457194545718007575218901948993472855205244009258327998464'>\(24\!\cdots\!64\)\( T_{2} + \)12440427443768479410827757573485089561768038630319505068700371236404014284338312788954309787123712'>\(12\!\cdots\!12\)\( \) acting on \(S_{94}^{\mathrm{new}}(\Gamma_0(9))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 43735426713792 T + \)\(16\!\cdots\!32\)\( T^{2} + \)\(77\!\cdots\!80\)\( T^{3} + \)\(23\!\cdots\!72\)\( T^{4} + \)\(82\!\cdots\!64\)\( T^{5} + \)\(30\!\cdots\!84\)\( T^{6} + \)\(10\!\cdots\!40\)\( T^{7} + \)\(30\!\cdots\!28\)\( T^{8} + \)\(81\!\cdots\!96\)\( T^{9} + \)\(23\!\cdots\!36\)\( T^{10} + \)\(74\!\cdots\!80\)\( T^{11} + \)\(16\!\cdots\!24\)\( T^{12} + \)\(41\!\cdots\!48\)\( T^{13} + \)\(93\!\cdots\!48\)\( T^{14} \)
$3$ 1
$5$ \( 1 - \)\(24\!\cdots\!50\)\( T + \)\(29\!\cdots\!75\)\( T^{2} - \)\(10\!\cdots\!00\)\( T^{3} + \)\(62\!\cdots\!25\)\( T^{4} - \)\(18\!\cdots\!50\)\( T^{5} + \)\(86\!\cdots\!75\)\( T^{6} - \)\(24\!\cdots\!00\)\( T^{7} + \)\(87\!\cdots\!75\)\( T^{8} - \)\(19\!\cdots\!50\)\( T^{9} + \)\(64\!\cdots\!25\)\( T^{10} - \)\(10\!\cdots\!00\)\( T^{11} + \)\(30\!\cdots\!75\)\( T^{12} - \)\(25\!\cdots\!50\)\( T^{13} + \)\(10\!\cdots\!25\)\( T^{14} \)
$7$ \( 1 + \)\(92\!\cdots\!08\)\( T + \)\(12\!\cdots\!57\)\( T^{2} - \)\(33\!\cdots\!00\)\( T^{3} + \)\(76\!\cdots\!97\)\( T^{4} - \)\(67\!\cdots\!24\)\( T^{5} + \)\(40\!\cdots\!29\)\( T^{6} - \)\(30\!\cdots\!00\)\( T^{7} + \)\(15\!\cdots\!03\)\( T^{8} - \)\(10\!\cdots\!76\)\( T^{9} + \)\(46\!\cdots\!71\)\( T^{10} - \)\(78\!\cdots\!00\)\( T^{11} + \)\(12\!\cdots\!99\)\( T^{12} + \)\(33\!\cdots\!92\)\( T^{13} + \)\(14\!\cdots\!43\)\( T^{14} \)
$11$ \( 1 + \)\(10\!\cdots\!24\)\( T + \)\(32\!\cdots\!21\)\( T^{2} + \)\(41\!\cdots\!44\)\( T^{3} + \)\(48\!\cdots\!61\)\( T^{4} + \)\(65\!\cdots\!52\)\( T^{5} + \)\(47\!\cdots\!73\)\( T^{6} + \)\(59\!\cdots\!92\)\( T^{7} + \)\(33\!\cdots\!63\)\( T^{8} + \)\(32\!\cdots\!72\)\( T^{9} + \)\(17\!\cdots\!51\)\( T^{10} + \)\(10\!\cdots\!24\)\( T^{11} + \)\(56\!\cdots\!71\)\( T^{12} + \)\(13\!\cdots\!44\)\( T^{13} + \)\(88\!\cdots\!11\)\( T^{14} \)
$13$ \( 1 - \)\(19\!\cdots\!26\)\( T + \)\(17\!\cdots\!63\)\( T^{2} - \)\(37\!\cdots\!60\)\( T^{3} + \)\(15\!\cdots\!57\)\( T^{4} - \)\(33\!\cdots\!02\)\( T^{5} + \)\(85\!\cdots\!11\)\( T^{6} - \)\(17\!\cdots\!80\)\( T^{7} + \)\(33\!\cdots\!83\)\( T^{8} - \)\(52\!\cdots\!18\)\( T^{9} + \)\(93\!\cdots\!89\)\( T^{10} - \)\(91\!\cdots\!60\)\( T^{11} + \)\(16\!\cdots\!59\)\( T^{12} - \)\(73\!\cdots\!54\)\( T^{13} + \)\(15\!\cdots\!37\)\( T^{14} \)
$17$ \( 1 + \)\(80\!\cdots\!42\)\( T + \)\(11\!\cdots\!07\)\( T^{2} + \)\(11\!\cdots\!40\)\( T^{3} + \)\(63\!\cdots\!97\)\( T^{4} + \)\(75\!\cdots\!94\)\( T^{5} + \)\(23\!\cdots\!19\)\( T^{6} + \)\(26\!\cdots\!20\)\( T^{7} + \)\(62\!\cdots\!03\)\( T^{8} + \)\(54\!\cdots\!86\)\( T^{9} + \)\(12\!\cdots\!41\)\( T^{10} + \)\(62\!\cdots\!40\)\( T^{11} + \)\(16\!\cdots\!99\)\( T^{12} + \)\(31\!\cdots\!78\)\( T^{13} + \)\(10\!\cdots\!33\)\( T^{14} \)
$19$ \( 1 + \)\(49\!\cdots\!00\)\( T + \)\(40\!\cdots\!13\)\( T^{2} + \)\(12\!\cdots\!00\)\( T^{3} + \)\(53\!\cdots\!01\)\( T^{4} + \)\(97\!\cdots\!00\)\( T^{5} + \)\(34\!\cdots\!65\)\( T^{6} + \)\(47\!\cdots\!00\)\( T^{7} + \)\(28\!\cdots\!35\)\( T^{8} + \)\(68\!\cdots\!00\)\( T^{9} + \)\(31\!\cdots\!79\)\( T^{10} + \)\(61\!\cdots\!00\)\( T^{11} + \)\(16\!\cdots\!87\)\( T^{12} + \)\(17\!\cdots\!00\)\( T^{13} + \)\(29\!\cdots\!19\)\( T^{14} \)
$23$ \( 1 - \)\(25\!\cdots\!64\)\( T + \)\(15\!\cdots\!73\)\( T^{2} - \)\(32\!\cdots\!60\)\( T^{3} + \)\(13\!\cdots\!37\)\( T^{4} - \)\(26\!\cdots\!88\)\( T^{5} + \)\(83\!\cdots\!81\)\( T^{6} - \)\(13\!\cdots\!80\)\( T^{7} + \)\(36\!\cdots\!23\)\( T^{8} - \)\(51\!\cdots\!32\)\( T^{9} + \)\(11\!\cdots\!19\)\( T^{10} - \)\(11\!\cdots\!60\)\( T^{11} + \)\(25\!\cdots\!39\)\( T^{12} - \)\(18\!\cdots\!16\)\( T^{13} + \)\(30\!\cdots\!27\)\( T^{14} \)
$29$ \( 1 + \)\(11\!\cdots\!50\)\( T + \)\(20\!\cdots\!23\)\( T^{2} + \)\(31\!\cdots\!00\)\( T^{3} + \)\(44\!\cdots\!41\)\( T^{4} + \)\(51\!\cdots\!50\)\( T^{5} + \)\(57\!\cdots\!15\)\( T^{6} + \)\(61\!\cdots\!00\)\( T^{7} + \)\(58\!\cdots\!35\)\( T^{8} + \)\(52\!\cdots\!50\)\( T^{9} + \)\(45\!\cdots\!29\)\( T^{10} + \)\(32\!\cdots\!00\)\( T^{11} + \)\(21\!\cdots\!27\)\( T^{12} + \)\(12\!\cdots\!50\)\( T^{13} + \)\(10\!\cdots\!29\)\( T^{14} \)
$31$ \( 1 + \)\(11\!\cdots\!56\)\( T + \)\(11\!\cdots\!81\)\( T^{2} + \)\(32\!\cdots\!96\)\( T^{3} + \)\(77\!\cdots\!81\)\( T^{4} + \)\(22\!\cdots\!48\)\( T^{5} + \)\(64\!\cdots\!13\)\( T^{6} + \)\(90\!\cdots\!88\)\( T^{7} + \)\(32\!\cdots\!83\)\( T^{8} + \)\(55\!\cdots\!88\)\( T^{9} + \)\(95\!\cdots\!51\)\( T^{10} + \)\(20\!\cdots\!56\)\( T^{11} + \)\(35\!\cdots\!31\)\( T^{12} + \)\(17\!\cdots\!96\)\( T^{13} + \)\(75\!\cdots\!31\)\( T^{14} \)
$37$ \( 1 - \)\(11\!\cdots\!42\)\( T + \)\(42\!\cdots\!07\)\( T^{2} - \)\(37\!\cdots\!20\)\( T^{3} + \)\(80\!\cdots\!97\)\( T^{4} - \)\(56\!\cdots\!34\)\( T^{5} + \)\(87\!\cdots\!99\)\( T^{6} - \)\(49\!\cdots\!60\)\( T^{7} + \)\(60\!\cdots\!03\)\( T^{8} - \)\(27\!\cdots\!06\)\( T^{9} + \)\(27\!\cdots\!81\)\( T^{10} - \)\(87\!\cdots\!20\)\( T^{11} + \)\(70\!\cdots\!99\)\( T^{12} - \)\(12\!\cdots\!18\)\( T^{13} + \)\(79\!\cdots\!13\)\( T^{14} \)
$41$ \( 1 - \)\(50\!\cdots\!46\)\( T + \)\(28\!\cdots\!11\)\( T^{2} - \)\(30\!\cdots\!16\)\( T^{3} + \)\(51\!\cdots\!41\)\( T^{4} - \)\(54\!\cdots\!98\)\( T^{5} + \)\(79\!\cdots\!83\)\( T^{6} - \)\(57\!\cdots\!28\)\( T^{7} + \)\(77\!\cdots\!43\)\( T^{8} - \)\(51\!\cdots\!18\)\( T^{9} + \)\(47\!\cdots\!01\)\( T^{10} - \)\(27\!\cdots\!96\)\( T^{11} + \)\(25\!\cdots\!11\)\( T^{12} - \)\(43\!\cdots\!66\)\( T^{13} + \)\(83\!\cdots\!41\)\( T^{14} \)
$43$ \( 1 + \)\(72\!\cdots\!44\)\( T + \)\(39\!\cdots\!93\)\( T^{2} + \)\(29\!\cdots\!00\)\( T^{3} + \)\(73\!\cdots\!97\)\( T^{4} + \)\(55\!\cdots\!68\)\( T^{5} + \)\(87\!\cdots\!21\)\( T^{6} + \)\(58\!\cdots\!00\)\( T^{7} + \)\(71\!\cdots\!03\)\( T^{8} + \)\(36\!\cdots\!32\)\( T^{9} + \)\(40\!\cdots\!79\)\( T^{10} + \)\(13\!\cdots\!00\)\( T^{11} + \)\(14\!\cdots\!99\)\( T^{12} + \)\(21\!\cdots\!56\)\( T^{13} + \)\(24\!\cdots\!07\)\( T^{14} \)
$47$ \( 1 - \)\(37\!\cdots\!08\)\( T + \)\(11\!\cdots\!57\)\( T^{2} - \)\(50\!\cdots\!40\)\( T^{3} + \)\(66\!\cdots\!97\)\( T^{4} - \)\(30\!\cdots\!96\)\( T^{5} + \)\(29\!\cdots\!89\)\( T^{6} - \)\(11\!\cdots\!20\)\( T^{7} + \)\(94\!\cdots\!03\)\( T^{8} - \)\(30\!\cdots\!84\)\( T^{9} + \)\(21\!\cdots\!51\)\( T^{10} - \)\(52\!\cdots\!40\)\( T^{11} + \)\(37\!\cdots\!99\)\( T^{12} - \)\(39\!\cdots\!12\)\( T^{13} + \)\(34\!\cdots\!03\)\( T^{14} \)
$53$ \( 1 - \)\(36\!\cdots\!34\)\( T + \)\(10\!\cdots\!03\)\( T^{2} - \)\(19\!\cdots\!20\)\( T^{3} + \)\(39\!\cdots\!77\)\( T^{4} - \)\(66\!\cdots\!58\)\( T^{5} + \)\(13\!\cdots\!91\)\( T^{6} - \)\(19\!\cdots\!60\)\( T^{7} + \)\(29\!\cdots\!43\)\( T^{8} - \)\(34\!\cdots\!82\)\( T^{9} + \)\(46\!\cdots\!09\)\( T^{10} - \)\(52\!\cdots\!20\)\( T^{11} + \)\(65\!\cdots\!79\)\( T^{12} - \)\(50\!\cdots\!26\)\( T^{13} + \)\(31\!\cdots\!97\)\( T^{14} \)
$59$ \( 1 - \)\(11\!\cdots\!00\)\( T + \)\(87\!\cdots\!53\)\( T^{2} - \)\(45\!\cdots\!00\)\( T^{3} + \)\(18\!\cdots\!61\)\( T^{4} - \)\(64\!\cdots\!00\)\( T^{5} + \)\(18\!\cdots\!65\)\( T^{6} - \)\(43\!\cdots\!00\)\( T^{7} + \)\(88\!\cdots\!35\)\( T^{8} - \)\(15\!\cdots\!00\)\( T^{9} + \)\(22\!\cdots\!79\)\( T^{10} - \)\(26\!\cdots\!00\)\( T^{11} + \)\(24\!\cdots\!47\)\( T^{12} - \)\(16\!\cdots\!00\)\( T^{13} + \)\(66\!\cdots\!59\)\( T^{14} \)
$61$ \( 1 + \)\(32\!\cdots\!26\)\( T + \)\(80\!\cdots\!71\)\( T^{2} + \)\(14\!\cdots\!56\)\( T^{3} + \)\(21\!\cdots\!61\)\( T^{4} + \)\(27\!\cdots\!98\)\( T^{5} + \)\(31\!\cdots\!23\)\( T^{6} + \)\(33\!\cdots\!08\)\( T^{7} + \)\(34\!\cdots\!63\)\( T^{8} + \)\(31\!\cdots\!78\)\( T^{9} + \)\(27\!\cdots\!01\)\( T^{10} + \)\(19\!\cdots\!76\)\( T^{11} + \)\(12\!\cdots\!71\)\( T^{12} + \)\(53\!\cdots\!06\)\( T^{13} + \)\(17\!\cdots\!61\)\( T^{14} \)
$67$ \( 1 - \)\(97\!\cdots\!92\)\( T + \)\(34\!\cdots\!57\)\( T^{2} - \)\(28\!\cdots\!40\)\( T^{3} + \)\(54\!\cdots\!97\)\( T^{4} - \)\(39\!\cdots\!44\)\( T^{5} + \)\(53\!\cdots\!69\)\( T^{6} - \)\(33\!\cdots\!20\)\( T^{7} + \)\(36\!\cdots\!03\)\( T^{8} - \)\(17\!\cdots\!36\)\( T^{9} + \)\(16\!\cdots\!91\)\( T^{10} - \)\(56\!\cdots\!40\)\( T^{11} + \)\(45\!\cdots\!99\)\( T^{12} - \)\(86\!\cdots\!28\)\( T^{13} + \)\(59\!\cdots\!83\)\( T^{14} \)
$71$ \( 1 + \)\(42\!\cdots\!84\)\( T + \)\(12\!\cdots\!01\)\( T^{2} + \)\(26\!\cdots\!24\)\( T^{3} + \)\(46\!\cdots\!21\)\( T^{4} + \)\(67\!\cdots\!52\)\( T^{5} + \)\(90\!\cdots\!93\)\( T^{6} + \)\(10\!\cdots\!52\)\( T^{7} + \)\(13\!\cdots\!23\)\( T^{8} + \)\(14\!\cdots\!92\)\( T^{9} + \)\(14\!\cdots\!51\)\( T^{10} + \)\(12\!\cdots\!84\)\( T^{11} + \)\(87\!\cdots\!51\)\( T^{12} + \)\(42\!\cdots\!24\)\( T^{13} + \)\(14\!\cdots\!71\)\( T^{14} \)
$73$ \( 1 - \)\(24\!\cdots\!86\)\( T + \)\(71\!\cdots\!23\)\( T^{2} - \)\(23\!\cdots\!40\)\( T^{3} + \)\(29\!\cdots\!37\)\( T^{4} - \)\(91\!\cdots\!62\)\( T^{5} + \)\(81\!\cdots\!31\)\( T^{6} - \)\(22\!\cdots\!20\)\( T^{7} + \)\(15\!\cdots\!23\)\( T^{8} - \)\(34\!\cdots\!18\)\( T^{9} + \)\(21\!\cdots\!69\)\( T^{10} - \)\(33\!\cdots\!40\)\( T^{11} + \)\(19\!\cdots\!39\)\( T^{12} - \)\(13\!\cdots\!34\)\( T^{13} + \)\(10\!\cdots\!77\)\( T^{14} \)
$79$ \( 1 + \)\(43\!\cdots\!00\)\( T + \)\(15\!\cdots\!73\)\( T^{2} + \)\(36\!\cdots\!00\)\( T^{3} + \)\(71\!\cdots\!41\)\( T^{4} + \)\(11\!\cdots\!00\)\( T^{5} + \)\(16\!\cdots\!65\)\( T^{6} + \)\(25\!\cdots\!00\)\( T^{7} + \)\(48\!\cdots\!35\)\( T^{8} + \)\(10\!\cdots\!00\)\( T^{9} + \)\(19\!\cdots\!79\)\( T^{10} + \)\(30\!\cdots\!00\)\( T^{11} + \)\(38\!\cdots\!27\)\( T^{12} + \)\(32\!\cdots\!00\)\( T^{13} + \)\(22\!\cdots\!79\)\( T^{14} \)
$83$ \( 1 - \)\(20\!\cdots\!04\)\( T + \)\(15\!\cdots\!33\)\( T^{2} - \)\(29\!\cdots\!80\)\( T^{3} + \)\(11\!\cdots\!17\)\( T^{4} - \)\(19\!\cdots\!28\)\( T^{5} + \)\(53\!\cdots\!01\)\( T^{6} - \)\(71\!\cdots\!40\)\( T^{7} + \)\(15\!\cdots\!63\)\( T^{8} - \)\(16\!\cdots\!32\)\( T^{9} + \)\(30\!\cdots\!99\)\( T^{10} - \)\(23\!\cdots\!80\)\( T^{11} + \)\(36\!\cdots\!19\)\( T^{12} - \)\(14\!\cdots\!36\)\( T^{13} + \)\(20\!\cdots\!67\)\( T^{14} \)
$89$ \( 1 + \)\(55\!\cdots\!50\)\( T + \)\(12\!\cdots\!83\)\( T^{2} + \)\(62\!\cdots\!00\)\( T^{3} + \)\(72\!\cdots\!81\)\( T^{4} + \)\(29\!\cdots\!50\)\( T^{5} + \)\(23\!\cdots\!15\)\( T^{6} + \)\(77\!\cdots\!00\)\( T^{7} + \)\(46\!\cdots\!35\)\( T^{8} + \)\(11\!\cdots\!50\)\( T^{9} + \)\(55\!\cdots\!29\)\( T^{10} + \)\(93\!\cdots\!00\)\( T^{11} + \)\(37\!\cdots\!67\)\( T^{12} + \)\(32\!\cdots\!50\)\( T^{13} + \)\(11\!\cdots\!89\)\( T^{14} \)
$97$ \( 1 - \)\(43\!\cdots\!42\)\( T + \)\(20\!\cdots\!07\)\( T^{2} - \)\(46\!\cdots\!60\)\( T^{3} + \)\(22\!\cdots\!97\)\( T^{4} + \)\(91\!\cdots\!46\)\( T^{5} + \)\(16\!\cdots\!39\)\( T^{6} + \)\(19\!\cdots\!20\)\( T^{7} + \)\(96\!\cdots\!03\)\( T^{8} + \)\(31\!\cdots\!34\)\( T^{9} + \)\(45\!\cdots\!01\)\( T^{10} - \)\(55\!\cdots\!60\)\( T^{11} + \)\(14\!\cdots\!99\)\( T^{12} - \)\(18\!\cdots\!38\)\( T^{13} + \)\(24\!\cdots\!53\)\( T^{14} \)
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