Properties

Label 9.34.a.b.1.1
Level $9$
Weight $34$
Character 9.1
Self dual yes
Analytic conductor $62.085$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 34 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(62.0845459929\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Defining polynomial: \(x^{2} - x - 589050\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(767.996\) of defining polynomial
Character \(\chi\) \(=\) 9.1

$q$-expansion

\(f(q)\) \(=\) \(q-49679.4 q^{2} -6.12189e9 q^{4} -2.04307e10 q^{5} -1.22168e14 q^{7} +7.30875e14 q^{8} +O(q^{10})\) \(q-49679.4 q^{2} -6.12189e9 q^{4} -2.04307e10 q^{5} -1.22168e14 q^{7} +7.30875e14 q^{8} +1.01499e15 q^{10} -2.15763e17 q^{11} -1.07762e18 q^{13} +6.06926e18 q^{14} +1.62772e19 q^{16} -2.54532e20 q^{17} -1.22020e21 q^{19} +1.25075e20 q^{20} +1.07190e22 q^{22} +5.11280e21 q^{23} -1.15998e23 q^{25} +5.35354e22 q^{26} +7.47901e23 q^{28} +1.64733e23 q^{29} -6.75706e24 q^{31} -7.08681e24 q^{32} +1.26450e25 q^{34} +2.49599e24 q^{35} -7.46520e25 q^{37} +6.06190e25 q^{38} -1.49323e25 q^{40} -4.96453e26 q^{41} -1.99347e26 q^{43} +1.32088e27 q^{44} -2.54001e26 q^{46} -2.16452e27 q^{47} +7.19411e27 q^{49} +5.76271e27 q^{50} +6.59706e27 q^{52} +3.60439e28 q^{53} +4.40819e27 q^{55} -8.92898e28 q^{56} -8.18385e27 q^{58} +1.87520e29 q^{59} +4.18340e27 q^{61} +3.35687e29 q^{62} +2.12249e29 q^{64} +2.20165e28 q^{65} +4.85975e29 q^{67} +1.55822e30 q^{68} -1.23999e29 q^{70} +3.42819e30 q^{71} +7.01467e30 q^{73} +3.70867e30 q^{74} +7.46995e30 q^{76} +2.63594e31 q^{77} -2.95630e30 q^{79} -3.32554e29 q^{80} +2.46635e31 q^{82} +1.23020e31 q^{83} +5.20028e30 q^{85} +9.90343e30 q^{86} -1.57696e32 q^{88} -7.05623e31 q^{89} +1.31651e32 q^{91} -3.13000e31 q^{92} +1.07532e32 q^{94} +2.49297e31 q^{95} -7.71791e32 q^{97} -3.57399e32 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 121680q^{2} + 14652233984q^{4} + 181061536500q^{5} - 67153080066800q^{7} + 2818750585098240q^{8} + O(q^{10}) \) \( 2q + 121680q^{2} + 14652233984q^{4} + 181061536500q^{5} - 67153080066800q^{7} + 2818750585098240q^{8} + 35542592216532000q^{10} - 133871815441914264q^{11} - 2981610478259443940q^{13} + 15496641262468340352q^{14} + \)\(19\!\cdots\!72\)\(q^{16} + 79361149261175525340q^{17} - \)\(13\!\cdots\!00\)\(q^{19} + \)\(43\!\cdots\!00\)\(q^{20} + \)\(24\!\cdots\!40\)\(q^{22} - \)\(26\!\cdots\!40\)\(q^{23} - \)\(19\!\cdots\!50\)\(q^{25} - \)\(27\!\cdots\!04\)\(q^{26} + \)\(18\!\cdots\!60\)\(q^{28} + \)\(16\!\cdots\!00\)\(q^{29} - \)\(62\!\cdots\!16\)\(q^{31} + \)\(57\!\cdots\!80\)\(q^{32} + \)\(69\!\cdots\!08\)\(q^{34} + \)\(13\!\cdots\!00\)\(q^{35} - \)\(10\!\cdots\!20\)\(q^{37} + \)\(36\!\cdots\!60\)\(q^{38} + \)\(40\!\cdots\!00\)\(q^{40} - \)\(27\!\cdots\!44\)\(q^{41} + \)\(15\!\cdots\!00\)\(q^{43} + \)\(30\!\cdots\!12\)\(q^{44} - \)\(56\!\cdots\!76\)\(q^{46} - \)\(54\!\cdots\!40\)\(q^{47} + \)\(24\!\cdots\!14\)\(q^{49} - \)\(72\!\cdots\!00\)\(q^{50} - \)\(32\!\cdots\!00\)\(q^{52} + \)\(26\!\cdots\!20\)\(q^{53} + \)\(20\!\cdots\!00\)\(q^{55} + \)\(25\!\cdots\!00\)\(q^{56} + \)\(25\!\cdots\!60\)\(q^{58} + \)\(30\!\cdots\!00\)\(q^{59} - \)\(57\!\cdots\!36\)\(q^{61} + \)\(42\!\cdots\!60\)\(q^{62} + \)\(86\!\cdots\!24\)\(q^{64} - \)\(36\!\cdots\!00\)\(q^{65} + \)\(15\!\cdots\!60\)\(q^{67} + \)\(84\!\cdots\!20\)\(q^{68} + \)\(17\!\cdots\!00\)\(q^{70} + \)\(26\!\cdots\!76\)\(q^{71} + \)\(94\!\cdots\!40\)\(q^{73} - \)\(14\!\cdots\!28\)\(q^{74} + \)\(45\!\cdots\!00\)\(q^{76} + \)\(30\!\cdots\!00\)\(q^{77} - \)\(85\!\cdots\!00\)\(q^{79} + \)\(35\!\cdots\!00\)\(q^{80} + \)\(62\!\cdots\!40\)\(q^{82} - \)\(29\!\cdots\!20\)\(q^{83} + \)\(72\!\cdots\!00\)\(q^{85} + \)\(31\!\cdots\!36\)\(q^{86} + \)\(13\!\cdots\!20\)\(q^{88} - \)\(13\!\cdots\!00\)\(q^{89} + \)\(26\!\cdots\!44\)\(q^{91} - \)\(68\!\cdots\!60\)\(q^{92} - \)\(45\!\cdots\!12\)\(q^{94} - \)\(33\!\cdots\!00\)\(q^{95} - \)\(36\!\cdots\!60\)\(q^{97} - \)\(11\!\cdots\!40\)\(q^{98} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −49679.4 −0.536021 −0.268010 0.963416i \(-0.586366\pi\)
−0.268010 + 0.963416i \(0.586366\pi\)
\(3\) 0 0
\(4\) −6.12189e9 −0.712682
\(5\) −2.04307e10 −0.0598796 −0.0299398 0.999552i \(-0.509532\pi\)
−0.0299398 + 0.999552i \(0.509532\pi\)
\(6\) 0 0
\(7\) −1.22168e14 −1.38944 −0.694722 0.719278i \(-0.744473\pi\)
−0.694722 + 0.719278i \(0.744473\pi\)
\(8\) 7.30875e14 0.918033
\(9\) 0 0
\(10\) 1.01499e15 0.0320967
\(11\) −2.15763e17 −1.41578 −0.707892 0.706321i \(-0.750354\pi\)
−0.707892 + 0.706321i \(0.750354\pi\)
\(12\) 0 0
\(13\) −1.07762e18 −0.449158 −0.224579 0.974456i \(-0.572101\pi\)
−0.224579 + 0.974456i \(0.572101\pi\)
\(14\) 6.06926e18 0.744771
\(15\) 0 0
\(16\) 1.62772e19 0.220597
\(17\) −2.54532e20 −1.26863 −0.634316 0.773074i \(-0.718718\pi\)
−0.634316 + 0.773074i \(0.718718\pi\)
\(18\) 0 0
\(19\) −1.22020e21 −0.970505 −0.485252 0.874374i \(-0.661272\pi\)
−0.485252 + 0.874374i \(0.661272\pi\)
\(20\) 1.25075e20 0.0426751
\(21\) 0 0
\(22\) 1.07190e22 0.758890
\(23\) 5.11280e21 0.173840 0.0869199 0.996215i \(-0.472298\pi\)
0.0869199 + 0.996215i \(0.472298\pi\)
\(24\) 0 0
\(25\) −1.15998e23 −0.996414
\(26\) 5.35354e22 0.240758
\(27\) 0 0
\(28\) 7.47901e23 0.990231
\(29\) 1.64733e23 0.122240 0.0611201 0.998130i \(-0.480533\pi\)
0.0611201 + 0.998130i \(0.480533\pi\)
\(30\) 0 0
\(31\) −6.75706e24 −1.66836 −0.834181 0.551491i \(-0.814059\pi\)
−0.834181 + 0.551491i \(0.814059\pi\)
\(32\) −7.08681e24 −1.03628
\(33\) 0 0
\(34\) 1.26450e25 0.680013
\(35\) 2.49599e24 0.0831994
\(36\) 0 0
\(37\) −7.46520e25 −0.994748 −0.497374 0.867536i \(-0.665702\pi\)
−0.497374 + 0.867536i \(0.665702\pi\)
\(38\) 6.06190e25 0.520211
\(39\) 0 0
\(40\) −1.49323e25 −0.0549715
\(41\) −4.96453e26 −1.21603 −0.608016 0.793925i \(-0.708034\pi\)
−0.608016 + 0.793925i \(0.708034\pi\)
\(42\) 0 0
\(43\) −1.99347e26 −0.222525 −0.111263 0.993791i \(-0.535489\pi\)
−0.111263 + 0.993791i \(0.535489\pi\)
\(44\) 1.32088e27 1.00900
\(45\) 0 0
\(46\) −2.54001e26 −0.0931818
\(47\) −2.16452e27 −0.556862 −0.278431 0.960456i \(-0.589814\pi\)
−0.278431 + 0.960456i \(0.589814\pi\)
\(48\) 0 0
\(49\) 7.19411e27 0.930555
\(50\) 5.76271e27 0.534099
\(51\) 0 0
\(52\) 6.59706e27 0.320107
\(53\) 3.60439e28 1.27726 0.638631 0.769513i \(-0.279501\pi\)
0.638631 + 0.769513i \(0.279501\pi\)
\(54\) 0 0
\(55\) 4.40819e27 0.0847766
\(56\) −8.92898e28 −1.27556
\(57\) 0 0
\(58\) −8.18385e27 −0.0655233
\(59\) 1.87520e29 1.13237 0.566186 0.824278i \(-0.308418\pi\)
0.566186 + 0.824278i \(0.308418\pi\)
\(60\) 0 0
\(61\) 4.18340e27 0.0145743 0.00728715 0.999973i \(-0.497680\pi\)
0.00728715 + 0.999973i \(0.497680\pi\)
\(62\) 3.35687e29 0.894276
\(63\) 0 0
\(64\) 2.12249e29 0.334870
\(65\) 2.20165e28 0.0268954
\(66\) 0 0
\(67\) 4.85975e29 0.360064 0.180032 0.983661i \(-0.442380\pi\)
0.180032 + 0.983661i \(0.442380\pi\)
\(68\) 1.55822e30 0.904130
\(69\) 0 0
\(70\) −1.23999e29 −0.0445966
\(71\) 3.42819e30 0.975667 0.487834 0.872937i \(-0.337787\pi\)
0.487834 + 0.872937i \(0.337787\pi\)
\(72\) 0 0
\(73\) 7.01467e30 1.26235 0.631175 0.775640i \(-0.282573\pi\)
0.631175 + 0.775640i \(0.282573\pi\)
\(74\) 3.70867e30 0.533206
\(75\) 0 0
\(76\) 7.46995e30 0.691661
\(77\) 2.63594e31 1.96715
\(78\) 0 0
\(79\) −2.95630e30 −0.144511 −0.0722555 0.997386i \(-0.523020\pi\)
−0.0722555 + 0.997386i \(0.523020\pi\)
\(80\) −3.32554e29 −0.0132092
\(81\) 0 0
\(82\) 2.46635e31 0.651818
\(83\) 1.23020e31 0.266187 0.133094 0.991103i \(-0.457509\pi\)
0.133094 + 0.991103i \(0.457509\pi\)
\(84\) 0 0
\(85\) 5.20028e30 0.0759652
\(86\) 9.90343e30 0.119278
\(87\) 0 0
\(88\) −1.57696e32 −1.29974
\(89\) −7.05623e31 −0.482656 −0.241328 0.970444i \(-0.577583\pi\)
−0.241328 + 0.970444i \(0.577583\pi\)
\(90\) 0 0
\(91\) 1.31651e32 0.624080
\(92\) −3.13000e31 −0.123892
\(93\) 0 0
\(94\) 1.07532e32 0.298490
\(95\) 2.49297e31 0.0581135
\(96\) 0 0
\(97\) −7.71791e32 −1.27575 −0.637875 0.770140i \(-0.720187\pi\)
−0.637875 + 0.770140i \(0.720187\pi\)
\(98\) −3.57399e32 −0.498797
\(99\) 0 0
\(100\) 7.10126e32 0.710126
\(101\) −1.88601e33 −1.60045 −0.800224 0.599701i \(-0.795286\pi\)
−0.800224 + 0.599701i \(0.795286\pi\)
\(102\) 0 0
\(103\) −4.05317e32 −0.248874 −0.124437 0.992227i \(-0.539713\pi\)
−0.124437 + 0.992227i \(0.539713\pi\)
\(104\) −7.87604e32 −0.412342
\(105\) 0 0
\(106\) −1.79064e33 −0.684640
\(107\) 1.13905e33 0.373003 0.186501 0.982455i \(-0.440285\pi\)
0.186501 + 0.982455i \(0.440285\pi\)
\(108\) 0 0
\(109\) 5.47789e32 0.132153 0.0660764 0.997815i \(-0.478952\pi\)
0.0660764 + 0.997815i \(0.478952\pi\)
\(110\) −2.18997e32 −0.0454421
\(111\) 0 0
\(112\) −1.98855e33 −0.306507
\(113\) −7.05689e33 −0.939332 −0.469666 0.882844i \(-0.655626\pi\)
−0.469666 + 0.882844i \(0.655626\pi\)
\(114\) 0 0
\(115\) −1.04458e32 −0.0104095
\(116\) −1.00848e33 −0.0871183
\(117\) 0 0
\(118\) −9.31588e33 −0.606975
\(119\) 3.10958e34 1.76269
\(120\) 0 0
\(121\) 2.33284e34 1.00444
\(122\) −2.07829e32 −0.00781213
\(123\) 0 0
\(124\) 4.13660e34 1.18901
\(125\) 4.74838e33 0.119545
\(126\) 0 0
\(127\) −1.88228e34 −0.364689 −0.182344 0.983235i \(-0.558369\pi\)
−0.182344 + 0.983235i \(0.558369\pi\)
\(128\) 5.03308e34 0.856780
\(129\) 0 0
\(130\) −1.09377e33 −0.0144165
\(131\) 1.26378e35 1.46790 0.733949 0.679204i \(-0.237675\pi\)
0.733949 + 0.679204i \(0.237675\pi\)
\(132\) 0 0
\(133\) 1.49070e35 1.34846
\(134\) −2.41430e34 −0.193002
\(135\) 0 0
\(136\) −1.86031e35 −1.16465
\(137\) −2.95759e35 −1.64077 −0.820387 0.571809i \(-0.806242\pi\)
−0.820387 + 0.571809i \(0.806242\pi\)
\(138\) 0 0
\(139\) −8.22498e34 −0.359245 −0.179622 0.983736i \(-0.557488\pi\)
−0.179622 + 0.983736i \(0.557488\pi\)
\(140\) −1.52802e34 −0.0592947
\(141\) 0 0
\(142\) −1.70310e35 −0.522978
\(143\) 2.32510e35 0.635911
\(144\) 0 0
\(145\) −3.36562e33 −0.00731970
\(146\) −3.48485e35 −0.676646
\(147\) 0 0
\(148\) 4.57011e35 0.708939
\(149\) −5.29045e35 −0.734378 −0.367189 0.930146i \(-0.619680\pi\)
−0.367189 + 0.930146i \(0.619680\pi\)
\(150\) 0 0
\(151\) −5.18728e35 −0.577856 −0.288928 0.957351i \(-0.593299\pi\)
−0.288928 + 0.957351i \(0.593299\pi\)
\(152\) −8.91816e35 −0.890955
\(153\) 0 0
\(154\) −1.30952e36 −1.05444
\(155\) 1.38052e35 0.0999009
\(156\) 0 0
\(157\) 1.31075e36 0.767669 0.383835 0.923402i \(-0.374603\pi\)
0.383835 + 0.923402i \(0.374603\pi\)
\(158\) 1.46867e35 0.0774609
\(159\) 0 0
\(160\) 1.44789e35 0.0620519
\(161\) −6.24622e35 −0.241541
\(162\) 0 0
\(163\) 1.61518e36 0.509478 0.254739 0.967010i \(-0.418010\pi\)
0.254739 + 0.967010i \(0.418010\pi\)
\(164\) 3.03923e36 0.866643
\(165\) 0 0
\(166\) −6.11157e35 −0.142682
\(167\) 6.45682e35 0.136520 0.0682601 0.997668i \(-0.478255\pi\)
0.0682601 + 0.997668i \(0.478255\pi\)
\(168\) 0 0
\(169\) −4.59487e36 −0.798257
\(170\) −2.58347e35 −0.0407189
\(171\) 0 0
\(172\) 1.22038e36 0.158590
\(173\) 6.58049e36 0.777137 0.388569 0.921420i \(-0.372970\pi\)
0.388569 + 0.921420i \(0.372970\pi\)
\(174\) 0 0
\(175\) 1.41713e37 1.38446
\(176\) −3.51200e36 −0.312317
\(177\) 0 0
\(178\) 3.50549e36 0.258714
\(179\) 2.08823e37 1.40508 0.702542 0.711642i \(-0.252048\pi\)
0.702542 + 0.711642i \(0.252048\pi\)
\(180\) 0 0
\(181\) −2.42658e37 −1.35925 −0.679624 0.733561i \(-0.737857\pi\)
−0.679624 + 0.733561i \(0.737857\pi\)
\(182\) −6.54034e36 −0.334520
\(183\) 0 0
\(184\) 3.73682e36 0.159591
\(185\) 1.52520e36 0.0595652
\(186\) 0 0
\(187\) 5.49185e37 1.79611
\(188\) 1.32510e37 0.396865
\(189\) 0 0
\(190\) −1.23849e36 −0.0311500
\(191\) −4.06968e37 −0.938664 −0.469332 0.883022i \(-0.655505\pi\)
−0.469332 + 0.883022i \(0.655505\pi\)
\(192\) 0 0
\(193\) −5.23242e37 −1.01627 −0.508133 0.861279i \(-0.669664\pi\)
−0.508133 + 0.861279i \(0.669664\pi\)
\(194\) 3.83421e37 0.683829
\(195\) 0 0
\(196\) −4.40416e37 −0.663189
\(197\) −1.35006e38 −1.86922 −0.934611 0.355672i \(-0.884252\pi\)
−0.934611 + 0.355672i \(0.884252\pi\)
\(198\) 0 0
\(199\) −9.91562e37 −1.16210 −0.581051 0.813867i \(-0.697358\pi\)
−0.581051 + 0.813867i \(0.697358\pi\)
\(200\) −8.47800e37 −0.914741
\(201\) 0 0
\(202\) 9.36960e37 0.857874
\(203\) −2.01252e37 −0.169846
\(204\) 0 0
\(205\) 1.01429e37 0.0728155
\(206\) 2.01359e37 0.133402
\(207\) 0 0
\(208\) −1.75406e37 −0.0990828
\(209\) 2.63274e38 1.37403
\(210\) 0 0
\(211\) −2.07128e38 −0.923802 −0.461901 0.886931i \(-0.652832\pi\)
−0.461901 + 0.886931i \(0.652832\pi\)
\(212\) −2.20657e38 −0.910282
\(213\) 0 0
\(214\) −5.65876e37 −0.199937
\(215\) 4.07280e36 0.0133247
\(216\) 0 0
\(217\) 8.25499e38 2.31809
\(218\) −2.72139e37 −0.0708367
\(219\) 0 0
\(220\) −2.69865e37 −0.0604187
\(221\) 2.74288e38 0.569816
\(222\) 0 0
\(223\) −4.51334e38 −0.808106 −0.404053 0.914736i \(-0.632399\pi\)
−0.404053 + 0.914736i \(0.632399\pi\)
\(224\) 8.65784e38 1.43985
\(225\) 0 0
\(226\) 3.50582e38 0.503501
\(227\) 2.43217e38 0.324764 0.162382 0.986728i \(-0.448082\pi\)
0.162382 + 0.986728i \(0.448082\pi\)
\(228\) 0 0
\(229\) 3.41418e38 0.394458 0.197229 0.980357i \(-0.436806\pi\)
0.197229 + 0.980357i \(0.436806\pi\)
\(230\) 5.18943e36 0.00557969
\(231\) 0 0
\(232\) 1.20399e38 0.112221
\(233\) 6.99608e37 0.0607410 0.0303705 0.999539i \(-0.490331\pi\)
0.0303705 + 0.999539i \(0.490331\pi\)
\(234\) 0 0
\(235\) 4.42228e37 0.0333447
\(236\) −1.14798e39 −0.807020
\(237\) 0 0
\(238\) −1.54482e39 −0.944840
\(239\) 1.63523e39 0.933282 0.466641 0.884447i \(-0.345464\pi\)
0.466641 + 0.884447i \(0.345464\pi\)
\(240\) 0 0
\(241\) −3.64560e38 −0.181338 −0.0906689 0.995881i \(-0.528900\pi\)
−0.0906689 + 0.995881i \(0.528900\pi\)
\(242\) −1.15894e39 −0.538403
\(243\) 0 0
\(244\) −2.56103e37 −0.0103868
\(245\) −1.46981e38 −0.0557213
\(246\) 0 0
\(247\) 1.31491e39 0.435910
\(248\) −4.93857e39 −1.53161
\(249\) 0 0
\(250\) −2.35897e38 −0.0640784
\(251\) −1.94693e39 −0.495147 −0.247573 0.968869i \(-0.579633\pi\)
−0.247573 + 0.968869i \(0.579633\pi\)
\(252\) 0 0
\(253\) −1.10315e39 −0.246120
\(254\) 9.35105e38 0.195481
\(255\) 0 0
\(256\) −4.32361e39 −0.794122
\(257\) 5.64464e39 0.972164 0.486082 0.873913i \(-0.338426\pi\)
0.486082 + 0.873913i \(0.338426\pi\)
\(258\) 0 0
\(259\) 9.12012e39 1.38215
\(260\) −1.34783e38 −0.0191679
\(261\) 0 0
\(262\) −6.27840e39 −0.786824
\(263\) −9.75930e39 −1.14855 −0.574274 0.818663i \(-0.694715\pi\)
−0.574274 + 0.818663i \(0.694715\pi\)
\(264\) 0 0
\(265\) −7.36404e38 −0.0764820
\(266\) −7.40572e39 −0.722804
\(267\) 0 0
\(268\) −2.97508e39 −0.256611
\(269\) 1.08142e40 0.877168 0.438584 0.898690i \(-0.355480\pi\)
0.438584 + 0.898690i \(0.355480\pi\)
\(270\) 0 0
\(271\) −6.19121e39 −0.444408 −0.222204 0.975000i \(-0.571325\pi\)
−0.222204 + 0.975000i \(0.571325\pi\)
\(272\) −4.14306e39 −0.279856
\(273\) 0 0
\(274\) 1.46931e40 0.879489
\(275\) 2.50280e40 1.41071
\(276\) 0 0
\(277\) −5.15662e39 −0.257898 −0.128949 0.991651i \(-0.541160\pi\)
−0.128949 + 0.991651i \(0.541160\pi\)
\(278\) 4.08612e39 0.192563
\(279\) 0 0
\(280\) 1.82426e39 0.0763798
\(281\) 1.28386e40 0.506832 0.253416 0.967357i \(-0.418446\pi\)
0.253416 + 0.967357i \(0.418446\pi\)
\(282\) 0 0
\(283\) 3.83806e40 1.34783 0.673915 0.738809i \(-0.264611\pi\)
0.673915 + 0.738809i \(0.264611\pi\)
\(284\) −2.09870e40 −0.695340
\(285\) 0 0
\(286\) −1.15510e40 −0.340862
\(287\) 6.06509e40 1.68961
\(288\) 0 0
\(289\) 2.45321e40 0.609426
\(290\) 1.67202e38 0.00392351
\(291\) 0 0
\(292\) −4.29430e40 −0.899654
\(293\) −1.90344e40 −0.376897 −0.188449 0.982083i \(-0.560346\pi\)
−0.188449 + 0.982083i \(0.560346\pi\)
\(294\) 0 0
\(295\) −3.83117e39 −0.0678060
\(296\) −5.45613e40 −0.913212
\(297\) 0 0
\(298\) 2.62827e40 0.393642
\(299\) −5.50964e39 −0.0780816
\(300\) 0 0
\(301\) 2.43538e40 0.309186
\(302\) 2.57701e40 0.309743
\(303\) 0 0
\(304\) −1.98614e40 −0.214090
\(305\) −8.54701e37 −0.000872703 0
\(306\) 0 0
\(307\) −1.48955e41 −1.36544 −0.682718 0.730682i \(-0.739202\pi\)
−0.682718 + 0.730682i \(0.739202\pi\)
\(308\) −1.61369e41 −1.40195
\(309\) 0 0
\(310\) −6.85834e39 −0.0535489
\(311\) 7.37928e39 0.0546345 0.0273173 0.999627i \(-0.491304\pi\)
0.0273173 + 0.999627i \(0.491304\pi\)
\(312\) 0 0
\(313\) −2.63324e40 −0.175391 −0.0876957 0.996147i \(-0.527950\pi\)
−0.0876957 + 0.996147i \(0.527950\pi\)
\(314\) −6.51171e40 −0.411487
\(315\) 0 0
\(316\) 1.80981e40 0.102990
\(317\) −1.79226e41 −0.968109 −0.484054 0.875038i \(-0.660836\pi\)
−0.484054 + 0.875038i \(0.660836\pi\)
\(318\) 0 0
\(319\) −3.55433e40 −0.173066
\(320\) −4.33641e39 −0.0200519
\(321\) 0 0
\(322\) 3.10309e40 0.129471
\(323\) 3.10581e41 1.23121
\(324\) 0 0
\(325\) 1.25001e41 0.447548
\(326\) −8.02413e40 −0.273091
\(327\) 0 0
\(328\) −3.62845e41 −1.11636
\(329\) 2.64436e41 0.773728
\(330\) 0 0
\(331\) 6.16181e41 1.63135 0.815674 0.578512i \(-0.196366\pi\)
0.815674 + 0.578512i \(0.196366\pi\)
\(332\) −7.53115e40 −0.189707
\(333\) 0 0
\(334\) −3.20771e40 −0.0731777
\(335\) −9.92883e39 −0.0215605
\(336\) 0 0
\(337\) 4.51357e41 0.888437 0.444218 0.895919i \(-0.353481\pi\)
0.444218 + 0.895919i \(0.353481\pi\)
\(338\) 2.28271e41 0.427882
\(339\) 0 0
\(340\) −3.18355e40 −0.0541390
\(341\) 1.45792e42 2.36204
\(342\) 0 0
\(343\) 6.55899e40 0.0964903
\(344\) −1.45697e41 −0.204286
\(345\) 0 0
\(346\) −3.26915e41 −0.416562
\(347\) −8.17900e41 −0.993720 −0.496860 0.867831i \(-0.665514\pi\)
−0.496860 + 0.867831i \(0.665514\pi\)
\(348\) 0 0
\(349\) −1.22969e42 −1.35886 −0.679432 0.733739i \(-0.737774\pi\)
−0.679432 + 0.733739i \(0.737774\pi\)
\(350\) −7.04021e41 −0.742101
\(351\) 0 0
\(352\) 1.52907e42 1.46715
\(353\) −1.62570e41 −0.148853 −0.0744266 0.997226i \(-0.523713\pi\)
−0.0744266 + 0.997226i \(0.523713\pi\)
\(354\) 0 0
\(355\) −7.00404e40 −0.0584226
\(356\) 4.31974e41 0.343980
\(357\) 0 0
\(358\) −1.03742e42 −0.753155
\(359\) −3.64036e41 −0.252398 −0.126199 0.992005i \(-0.540278\pi\)
−0.126199 + 0.992005i \(0.540278\pi\)
\(360\) 0 0
\(361\) −9.18755e40 −0.0581207
\(362\) 1.20551e42 0.728585
\(363\) 0 0
\(364\) −8.05951e41 −0.444771
\(365\) −1.43315e41 −0.0755891
\(366\) 0 0
\(367\) 1.35581e42 0.653442 0.326721 0.945121i \(-0.394056\pi\)
0.326721 + 0.945121i \(0.394056\pi\)
\(368\) 8.32218e40 0.0383485
\(369\) 0 0
\(370\) −7.57709e40 −0.0319282
\(371\) −4.40343e42 −1.77469
\(372\) 0 0
\(373\) 1.93457e42 0.713493 0.356746 0.934201i \(-0.383886\pi\)
0.356746 + 0.934201i \(0.383886\pi\)
\(374\) −2.72832e42 −0.962752
\(375\) 0 0
\(376\) −1.58200e42 −0.511218
\(377\) −1.77519e41 −0.0549052
\(378\) 0 0
\(379\) −5.45789e42 −1.54696 −0.773478 0.633823i \(-0.781485\pi\)
−0.773478 + 0.633823i \(0.781485\pi\)
\(380\) −1.52617e41 −0.0414164
\(381\) 0 0
\(382\) 2.02180e42 0.503143
\(383\) 5.06942e42 1.20831 0.604155 0.796867i \(-0.293511\pi\)
0.604155 + 0.796867i \(0.293511\pi\)
\(384\) 0 0
\(385\) −5.38542e41 −0.117792
\(386\) 2.59944e42 0.544740
\(387\) 0 0
\(388\) 4.72482e42 0.909204
\(389\) −7.51028e42 −1.38512 −0.692560 0.721360i \(-0.743517\pi\)
−0.692560 + 0.721360i \(0.743517\pi\)
\(390\) 0 0
\(391\) −1.30137e42 −0.220539
\(392\) 5.25800e42 0.854280
\(393\) 0 0
\(394\) 6.70703e42 1.00194
\(395\) 6.03993e40 0.00865327
\(396\) 0 0
\(397\) −1.26926e43 −1.67304 −0.836519 0.547938i \(-0.815413\pi\)
−0.836519 + 0.547938i \(0.815413\pi\)
\(398\) 4.92603e42 0.622911
\(399\) 0 0
\(400\) −1.88812e42 −0.219806
\(401\) −5.29553e42 −0.591598 −0.295799 0.955250i \(-0.595586\pi\)
−0.295799 + 0.955250i \(0.595586\pi\)
\(402\) 0 0
\(403\) 7.28153e42 0.749358
\(404\) 1.15459e43 1.14061
\(405\) 0 0
\(406\) 9.99807e41 0.0910410
\(407\) 1.61071e43 1.40835
\(408\) 0 0
\(409\) 1.80907e43 1.45888 0.729442 0.684042i \(-0.239780\pi\)
0.729442 + 0.684042i \(0.239780\pi\)
\(410\) −5.03894e41 −0.0390306
\(411\) 0 0
\(412\) 2.48130e42 0.177368
\(413\) −2.29090e43 −1.57337
\(414\) 0 0
\(415\) −2.51339e41 −0.0159392
\(416\) 7.63687e42 0.465453
\(417\) 0 0
\(418\) −1.30793e43 −0.736506
\(419\) −9.59306e42 −0.519309 −0.259654 0.965702i \(-0.583609\pi\)
−0.259654 + 0.965702i \(0.583609\pi\)
\(420\) 0 0
\(421\) −1.95406e43 −0.977875 −0.488938 0.872319i \(-0.662615\pi\)
−0.488938 + 0.872319i \(0.662615\pi\)
\(422\) 1.02900e43 0.495177
\(423\) 0 0
\(424\) 2.63436e43 1.17257
\(425\) 2.95252e43 1.26408
\(426\) 0 0
\(427\) −5.11080e41 −0.0202502
\(428\) −6.97316e42 −0.265832
\(429\) 0 0
\(430\) −2.02334e41 −0.00714233
\(431\) −4.01382e43 −1.36359 −0.681794 0.731544i \(-0.738800\pi\)
−0.681794 + 0.731544i \(0.738800\pi\)
\(432\) 0 0
\(433\) −1.24146e43 −0.390735 −0.195367 0.980730i \(-0.562590\pi\)
−0.195367 + 0.980730i \(0.562590\pi\)
\(434\) −4.10103e43 −1.24255
\(435\) 0 0
\(436\) −3.35350e42 −0.0941829
\(437\) −6.23865e42 −0.168712
\(438\) 0 0
\(439\) −6.07044e43 −1.52249 −0.761247 0.648462i \(-0.775413\pi\)
−0.761247 + 0.648462i \(0.775413\pi\)
\(440\) 3.22184e42 0.0778278
\(441\) 0 0
\(442\) −1.36265e43 −0.305433
\(443\) −6.61294e43 −1.42802 −0.714009 0.700137i \(-0.753122\pi\)
−0.714009 + 0.700137i \(0.753122\pi\)
\(444\) 0 0
\(445\) 1.44164e42 0.0289013
\(446\) 2.24220e43 0.433162
\(447\) 0 0
\(448\) −2.59301e43 −0.465283
\(449\) 1.54504e43 0.267225 0.133612 0.991034i \(-0.457342\pi\)
0.133612 + 0.991034i \(0.457342\pi\)
\(450\) 0 0
\(451\) 1.07116e44 1.72164
\(452\) 4.32015e43 0.669444
\(453\) 0 0
\(454\) −1.20829e43 −0.174080
\(455\) −2.68972e42 −0.0373697
\(456\) 0 0
\(457\) 7.12331e43 0.920586 0.460293 0.887767i \(-0.347744\pi\)
0.460293 + 0.887767i \(0.347744\pi\)
\(458\) −1.69615e43 −0.211438
\(459\) 0 0
\(460\) 6.39482e41 0.00741864
\(461\) 1.34004e44 1.49986 0.749932 0.661515i \(-0.230086\pi\)
0.749932 + 0.661515i \(0.230086\pi\)
\(462\) 0 0
\(463\) −5.53073e43 −0.576363 −0.288182 0.957576i \(-0.593051\pi\)
−0.288182 + 0.957576i \(0.593051\pi\)
\(464\) 2.68139e42 0.0269658
\(465\) 0 0
\(466\) −3.47561e42 −0.0325584
\(467\) 3.51798e43 0.318101 0.159050 0.987270i \(-0.449157\pi\)
0.159050 + 0.987270i \(0.449157\pi\)
\(468\) 0 0
\(469\) −5.93707e43 −0.500288
\(470\) −2.19696e42 −0.0178734
\(471\) 0 0
\(472\) 1.37054e44 1.03955
\(473\) 4.30116e43 0.315048
\(474\) 0 0
\(475\) 1.41541e44 0.967025
\(476\) −1.90365e44 −1.25624
\(477\) 0 0
\(478\) −8.12372e43 −0.500258
\(479\) 2.90520e44 1.72838 0.864192 0.503162i \(-0.167830\pi\)
0.864192 + 0.503162i \(0.167830\pi\)
\(480\) 0 0
\(481\) 8.04464e43 0.446799
\(482\) 1.81112e43 0.0972008
\(483\) 0 0
\(484\) −1.42814e44 −0.715849
\(485\) 1.57683e43 0.0763915
\(486\) 0 0
\(487\) −1.45706e44 −0.659551 −0.329776 0.944059i \(-0.606973\pi\)
−0.329776 + 0.944059i \(0.606973\pi\)
\(488\) 3.05755e42 0.0133797
\(489\) 0 0
\(490\) 7.30194e42 0.0298678
\(491\) −1.43984e44 −0.569469 −0.284735 0.958606i \(-0.591905\pi\)
−0.284735 + 0.958606i \(0.591905\pi\)
\(492\) 0 0
\(493\) −4.19299e43 −0.155078
\(494\) −6.53241e43 −0.233657
\(495\) 0 0
\(496\) −1.09986e44 −0.368035
\(497\) −4.18816e44 −1.35564
\(498\) 0 0
\(499\) 1.76176e44 0.533687 0.266843 0.963740i \(-0.414019\pi\)
0.266843 + 0.963740i \(0.414019\pi\)
\(500\) −2.90690e43 −0.0851972
\(501\) 0 0
\(502\) 9.67224e43 0.265409
\(503\) 2.43829e44 0.647460 0.323730 0.946149i \(-0.395063\pi\)
0.323730 + 0.946149i \(0.395063\pi\)
\(504\) 0 0
\(505\) 3.85326e43 0.0958343
\(506\) 5.48039e43 0.131925
\(507\) 0 0
\(508\) 1.15231e44 0.259907
\(509\) −3.49786e44 −0.763763 −0.381882 0.924211i \(-0.624724\pi\)
−0.381882 + 0.924211i \(0.624724\pi\)
\(510\) 0 0
\(511\) −8.56970e44 −1.75396
\(512\) −2.17544e44 −0.431114
\(513\) 0 0
\(514\) −2.80422e44 −0.521100
\(515\) 8.28092e42 0.0149025
\(516\) 0 0
\(517\) 4.67023e44 0.788396
\(518\) −4.53082e44 −0.740860
\(519\) 0 0
\(520\) 1.60913e43 0.0246909
\(521\) 3.31981e44 0.493505 0.246752 0.969079i \(-0.420637\pi\)
0.246752 + 0.969079i \(0.420637\pi\)
\(522\) 0 0
\(523\) 1.02645e45 1.43238 0.716190 0.697905i \(-0.245884\pi\)
0.716190 + 0.697905i \(0.245884\pi\)
\(524\) −7.73674e44 −1.04614
\(525\) 0 0
\(526\) 4.84837e44 0.615645
\(527\) 1.71989e45 2.11654
\(528\) 0 0
\(529\) −8.38864e44 −0.969780
\(530\) 3.65842e43 0.0409960
\(531\) 0 0
\(532\) −9.12591e44 −0.961024
\(533\) 5.34987e44 0.546191
\(534\) 0 0
\(535\) −2.32717e43 −0.0223353
\(536\) 3.55187e44 0.330550
\(537\) 0 0
\(538\) −5.37245e44 −0.470180
\(539\) −1.55222e45 −1.31746
\(540\) 0 0
\(541\) 4.08746e44 0.326361 0.163180 0.986596i \(-0.447825\pi\)
0.163180 + 0.986596i \(0.447825\pi\)
\(542\) 3.07576e44 0.238212
\(543\) 0 0
\(544\) 1.80382e45 1.31465
\(545\) −1.11917e43 −0.00791326
\(546\) 0 0
\(547\) 7.54748e44 0.502355 0.251178 0.967941i \(-0.419182\pi\)
0.251178 + 0.967941i \(0.419182\pi\)
\(548\) 1.81060e45 1.16935
\(549\) 0 0
\(550\) −1.24338e45 −0.756169
\(551\) −2.01008e44 −0.118635
\(552\) 0 0
\(553\) 3.61166e44 0.200790
\(554\) 2.56178e44 0.138239
\(555\) 0 0
\(556\) 5.03524e44 0.256027
\(557\) 3.77046e45 1.86116 0.930581 0.366087i \(-0.119303\pi\)
0.930581 + 0.366087i \(0.119303\pi\)
\(558\) 0 0
\(559\) 2.14819e44 0.0999491
\(560\) 4.06276e43 0.0183535
\(561\) 0 0
\(562\) −6.37816e44 −0.271673
\(563\) −1.99523e45 −0.825284 −0.412642 0.910893i \(-0.635394\pi\)
−0.412642 + 0.910893i \(0.635394\pi\)
\(564\) 0 0
\(565\) 1.44177e44 0.0562468
\(566\) −1.90672e45 −0.722465
\(567\) 0 0
\(568\) 2.50558e45 0.895695
\(569\) −2.38895e45 −0.829574 −0.414787 0.909919i \(-0.636144\pi\)
−0.414787 + 0.909919i \(0.636144\pi\)
\(570\) 0 0
\(571\) 9.25875e44 0.303429 0.151714 0.988424i \(-0.451521\pi\)
0.151714 + 0.988424i \(0.451521\pi\)
\(572\) −1.42340e45 −0.453202
\(573\) 0 0
\(574\) −3.01310e45 −0.905665
\(575\) −5.93074e44 −0.173217
\(576\) 0 0
\(577\) −3.73598e45 −1.03040 −0.515198 0.857071i \(-0.672281\pi\)
−0.515198 + 0.857071i \(0.672281\pi\)
\(578\) −1.21874e45 −0.326665
\(579\) 0 0
\(580\) 2.06040e43 0.00521661
\(581\) −1.50292e45 −0.369853
\(582\) 0 0
\(583\) −7.77694e45 −1.80833
\(584\) 5.12684e45 1.15888
\(585\) 0 0
\(586\) 9.45617e44 0.202025
\(587\) −6.35230e45 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(588\) 0 0
\(589\) 8.24499e45 1.61915
\(590\) 1.90330e44 0.0363454
\(591\) 0 0
\(592\) −1.21512e45 −0.219438
\(593\) −7.20521e45 −1.26545 −0.632724 0.774378i \(-0.718063\pi\)
−0.632724 + 0.774378i \(0.718063\pi\)
\(594\) 0 0
\(595\) −6.35310e44 −0.105549
\(596\) 3.23876e45 0.523377
\(597\) 0 0
\(598\) 2.73716e44 0.0418534
\(599\) 6.37291e45 0.947971 0.473985 0.880533i \(-0.342815\pi\)
0.473985 + 0.880533i \(0.342815\pi\)
\(600\) 0 0
\(601\) −8.13523e45 −1.14536 −0.572679 0.819780i \(-0.694096\pi\)
−0.572679 + 0.819780i \(0.694096\pi\)
\(602\) −1.20989e45 −0.165730
\(603\) 0 0
\(604\) 3.17559e45 0.411827
\(605\) −4.76616e44 −0.0601458
\(606\) 0 0
\(607\) 8.31211e45 0.993339 0.496670 0.867940i \(-0.334556\pi\)
0.496670 + 0.867940i \(0.334556\pi\)
\(608\) 8.64734e45 1.00571
\(609\) 0 0
\(610\) 4.24610e42 0.000467787 0
\(611\) 2.33253e45 0.250119
\(612\) 0 0
\(613\) −2.99559e44 −0.0304358 −0.0152179 0.999884i \(-0.504844\pi\)
−0.0152179 + 0.999884i \(0.504844\pi\)
\(614\) 7.40000e45 0.731903
\(615\) 0 0
\(616\) 1.92654e46 1.80591
\(617\) −9.03503e45 −0.824564 −0.412282 0.911056i \(-0.635268\pi\)
−0.412282 + 0.911056i \(0.635268\pi\)
\(618\) 0 0
\(619\) −1.15706e46 −1.00106 −0.500528 0.865720i \(-0.666861\pi\)
−0.500528 + 0.865720i \(0.666861\pi\)
\(620\) −8.45138e44 −0.0711975
\(621\) 0 0
\(622\) −3.66598e44 −0.0292853
\(623\) 8.62048e45 0.670623
\(624\) 0 0
\(625\) 1.34069e46 0.989256
\(626\) 1.30818e45 0.0940135
\(627\) 0 0
\(628\) −8.02424e45 −0.547104
\(629\) 1.90013e46 1.26197
\(630\) 0 0
\(631\) −1.43742e46 −0.905939 −0.452970 0.891526i \(-0.649635\pi\)
−0.452970 + 0.891526i \(0.649635\pi\)
\(632\) −2.16068e45 −0.132666
\(633\) 0 0
\(634\) 8.90387e45 0.518927
\(635\) 3.84563e44 0.0218374
\(636\) 0 0
\(637\) −7.75250e45 −0.417966
\(638\) 1.76577e45 0.0927668
\(639\) 0 0
\(640\) −1.02830e45 −0.0513037
\(641\) −1.34213e46 −0.652586 −0.326293 0.945269i \(-0.605800\pi\)
−0.326293 + 0.945269i \(0.605800\pi\)
\(642\) 0 0
\(643\) 3.48572e46 1.60995 0.804973 0.593312i \(-0.202180\pi\)
0.804973 + 0.593312i \(0.202180\pi\)
\(644\) 3.82387e45 0.172142
\(645\) 0 0
\(646\) −1.54295e46 −0.659956
\(647\) −3.47978e46 −1.45088 −0.725440 0.688285i \(-0.758364\pi\)
−0.725440 + 0.688285i \(0.758364\pi\)
\(648\) 0 0
\(649\) −4.04598e46 −1.60319
\(650\) −6.21000e45 −0.239895
\(651\) 0 0
\(652\) −9.88796e45 −0.363096
\(653\) −1.19815e46 −0.428986 −0.214493 0.976726i \(-0.568810\pi\)
−0.214493 + 0.976726i \(0.568810\pi\)
\(654\) 0 0
\(655\) −2.58200e45 −0.0878972
\(656\) −8.08085e45 −0.268252
\(657\) 0 0
\(658\) −1.31370e46 −0.414735
\(659\) 2.00047e46 0.615918 0.307959 0.951400i \(-0.400354\pi\)
0.307959 + 0.951400i \(0.400354\pi\)
\(660\) 0 0
\(661\) 3.80588e46 1.11463 0.557316 0.830301i \(-0.311831\pi\)
0.557316 + 0.830301i \(0.311831\pi\)
\(662\) −3.06115e46 −0.874437
\(663\) 0 0
\(664\) 8.99123e45 0.244369
\(665\) −3.04561e45 −0.0807454
\(666\) 0 0
\(667\) 8.42247e44 0.0212502
\(668\) −3.95279e45 −0.0972954
\(669\) 0 0
\(670\) 4.93259e44 0.0115569
\(671\) −9.02622e44 −0.0206341
\(672\) 0 0
\(673\) −5.38328e46 −1.17165 −0.585825 0.810437i \(-0.699230\pi\)
−0.585825 + 0.810437i \(0.699230\pi\)
\(674\) −2.24231e46 −0.476221
\(675\) 0 0
\(676\) 2.81293e46 0.568903
\(677\) 9.38585e46 1.85251 0.926255 0.376898i \(-0.123009\pi\)
0.926255 + 0.376898i \(0.123009\pi\)
\(678\) 0 0
\(679\) 9.42884e46 1.77258
\(680\) 3.80076e45 0.0697386
\(681\) 0 0
\(682\) −7.24288e46 −1.26610
\(683\) −1.22152e46 −0.208430 −0.104215 0.994555i \(-0.533233\pi\)
−0.104215 + 0.994555i \(0.533233\pi\)
\(684\) 0 0
\(685\) 6.04258e45 0.0982489
\(686\) −3.25847e45 −0.0517208
\(687\) 0 0
\(688\) −3.24480e45 −0.0490883
\(689\) −3.88416e46 −0.573693
\(690\) 0 0
\(691\) −1.01396e47 −1.42768 −0.713841 0.700308i \(-0.753046\pi\)
−0.713841 + 0.700308i \(0.753046\pi\)
\(692\) −4.02850e46 −0.553851
\(693\) 0 0
\(694\) 4.06328e46 0.532655
\(695\) 1.68042e45 0.0215115
\(696\) 0 0
\(697\) 1.26363e47 1.54270
\(698\) 6.10903e46 0.728379
\(699\) 0 0
\(700\) −8.67549e46 −0.986681
\(701\) 1.07227e47 1.19112 0.595560 0.803311i \(-0.296930\pi\)
0.595560 + 0.803311i \(0.296930\pi\)
\(702\) 0 0
\(703\) 9.10906e46 0.965408
\(704\) −4.57954e46 −0.474103
\(705\) 0 0
\(706\) 8.07639e45 0.0797884
\(707\) 2.30411e47 2.22373
\(708\) 0 0
\(709\) −3.54939e45 −0.0326957 −0.0163478 0.999866i \(-0.505204\pi\)
−0.0163478 + 0.999866i \(0.505204\pi\)
\(710\) 3.47957e45 0.0313157
\(711\) 0 0
\(712\) −5.15722e46 −0.443094
\(713\) −3.45475e46 −0.290028
\(714\) 0 0
\(715\) −4.75035e45 −0.0380781
\(716\) −1.27839e47 −1.00138
\(717\) 0 0
\(718\) 1.80851e46 0.135290
\(719\) −6.94896e46 −0.508035 −0.254017 0.967200i \(-0.581752\pi\)
−0.254017 + 0.967200i \(0.581752\pi\)
\(720\) 0 0
\(721\) 4.95169e46 0.345797
\(722\) 4.56432e45 0.0311539
\(723\) 0 0
\(724\) 1.48553e47 0.968711
\(725\) −1.91087e46 −0.121802
\(726\) 0 0
\(727\) −1.73359e47 −1.05592 −0.527958 0.849270i \(-0.677042\pi\)
−0.527958 + 0.849270i \(0.677042\pi\)
\(728\) 9.62203e46 0.572926
\(729\) 0 0
\(730\) 7.11980e45 0.0405173
\(731\) 5.07401e46 0.282303
\(732\) 0 0
\(733\) 4.59301e46 0.244277 0.122138 0.992513i \(-0.461025\pi\)
0.122138 + 0.992513i \(0.461025\pi\)
\(734\) −6.73556e46 −0.350259
\(735\) 0 0
\(736\) −3.62334e46 −0.180146
\(737\) −1.04855e47 −0.509773
\(738\) 0 0
\(739\) −3.24514e47 −1.50869 −0.754346 0.656477i \(-0.772046\pi\)
−0.754346 + 0.656477i \(0.772046\pi\)
\(740\) −9.33708e45 −0.0424510
\(741\) 0 0
\(742\) 2.18760e47 0.951268
\(743\) −3.63726e47 −1.54689 −0.773443 0.633866i \(-0.781467\pi\)
−0.773443 + 0.633866i \(0.781467\pi\)
\(744\) 0 0
\(745\) 1.08088e46 0.0439743
\(746\) −9.61085e46 −0.382447
\(747\) 0 0
\(748\) −3.36205e47 −1.28005
\(749\) −1.39156e47 −0.518267
\(750\) 0 0
\(751\) 2.28361e47 0.813885 0.406943 0.913454i \(-0.366595\pi\)
0.406943 + 0.913454i \(0.366595\pi\)
\(752\) −3.52323e46 −0.122842
\(753\) 0 0
\(754\) 8.81906e45 0.0294303
\(755\) 1.05980e46 0.0346018
\(756\) 0 0
\(757\) −1.23646e47 −0.386453 −0.193226 0.981154i \(-0.561895\pi\)
−0.193226 + 0.981154i \(0.561895\pi\)
\(758\) 2.71145e47 0.829201
\(759\) 0 0
\(760\) 1.82205e46 0.0533501
\(761\) 1.65073e47 0.472966 0.236483 0.971636i \(-0.424005\pi\)
0.236483 + 0.971636i \(0.424005\pi\)
\(762\) 0 0
\(763\) −6.69225e46 −0.183619
\(764\) 2.49141e47 0.668968
\(765\) 0 0
\(766\) −2.51846e47 −0.647679
\(767\) −2.02075e47 −0.508614
\(768\) 0 0
\(769\) −2.15810e47 −0.520339 −0.260170 0.965563i \(-0.583778\pi\)
−0.260170 + 0.965563i \(0.583778\pi\)
\(770\) 2.67544e46 0.0631392
\(771\) 0 0
\(772\) 3.20323e47 0.724274
\(773\) −5.97126e46 −0.132161 −0.0660806 0.997814i \(-0.521049\pi\)
−0.0660806 + 0.997814i \(0.521049\pi\)
\(774\) 0 0
\(775\) 7.83805e47 1.66238
\(776\) −5.64083e47 −1.17118
\(777\) 0 0
\(778\) 3.73106e47 0.742453
\(779\) 6.05774e47 1.18016
\(780\) 0 0
\(781\) −7.39675e47 −1.38133
\(782\) 6.46514e46 0.118213
\(783\) 0 0
\(784\) 1.17100e47 0.205277
\(785\) −2.67795e46 −0.0459677
\(786\) 0 0
\(787\) 2.40999e47 0.396673 0.198336 0.980134i \(-0.436446\pi\)
0.198336 + 0.980134i \(0.436446\pi\)
\(788\) 8.26493e47 1.33216
\(789\) 0 0
\(790\) −3.00060e45 −0.00463833
\(791\) 8.62128e47 1.30515
\(792\) 0 0
\(793\) −4.50811e45 −0.00654616
\(794\) 6.30559e47 0.896783
\(795\) 0 0
\(796\) 6.07023e47 0.828209
\(797\) −5.30320e47 −0.708722 −0.354361 0.935109i \(-0.615302\pi\)
−0.354361 + 0.935109i \(0.615302\pi\)
\(798\) 0 0
\(799\) 5.50941e47 0.706452
\(800\) 8.22055e47 1.03256
\(801\) 0 0
\(802\) 2.63079e47 0.317109
\(803\) −1.51350e48 −1.78722
\(804\) 0 0
\(805\) 1.27615e46 0.0144634
\(806\) −3.61742e47 −0.401672
\(807\) 0 0
\(808\) −1.37844e48 −1.46926
\(809\) −6.52623e47 −0.681572 −0.340786 0.940141i \(-0.610693\pi\)
−0.340786 + 0.940141i \(0.610693\pi\)
\(810\) 0 0
\(811\) −4.54256e47 −0.455466 −0.227733 0.973724i \(-0.573131\pi\)
−0.227733 + 0.973724i \(0.573131\pi\)
\(812\) 1.23204e47 0.121046
\(813\) 0 0
\(814\) −8.00193e47 −0.754904
\(815\) −3.29994e46 −0.0305074
\(816\) 0 0
\(817\) 2.43243e47 0.215962
\(818\) −8.98736e47 −0.781993
\(819\) 0 0
\(820\) −6.20938e46 −0.0518943
\(821\) 3.88660e47 0.318352 0.159176 0.987250i \(-0.449116\pi\)
0.159176 + 0.987250i \(0.449116\pi\)
\(822\) 0 0
\(823\) −2.16262e48 −1.70170 −0.850849 0.525410i \(-0.823912\pi\)
−0.850849 + 0.525410i \(0.823912\pi\)
\(824\) −2.96236e47 −0.228475
\(825\) 0 0
\(826\) 1.13811e48 0.843358
\(827\) 5.70292e47 0.414244 0.207122 0.978315i \(-0.433590\pi\)
0.207122 + 0.978315i \(0.433590\pi\)
\(828\) 0 0
\(829\) 1.81254e48 1.26514 0.632569 0.774504i \(-0.282001\pi\)
0.632569 + 0.774504i \(0.282001\pi\)
\(830\) 1.24864e46 0.00854375
\(831\) 0 0
\(832\) −2.28723e47 −0.150410
\(833\) −1.83113e48 −1.18053
\(834\) 0 0
\(835\) −1.31918e46 −0.00817478
\(836\) −1.61174e48 −0.979242
\(837\) 0 0
\(838\) 4.76578e47 0.278360
\(839\) 1.88483e47 0.107945 0.0539723 0.998542i \(-0.482812\pi\)
0.0539723 + 0.998542i \(0.482812\pi\)
\(840\) 0 0
\(841\) −1.78894e48 −0.985057
\(842\) 9.70766e47 0.524162
\(843\) 0 0
\(844\) 1.26802e48 0.658377
\(845\) 9.38766e46 0.0477993
\(846\) 0 0
\(847\) −2.84999e48 −1.39562
\(848\) 5.86693e47 0.281760
\(849\) 0 0
\(850\) −1.46680e48 −0.677575
\(851\) −3.81681e47 −0.172927
\(852\) 0 0
\(853\) 3.65000e48 1.59087 0.795434 0.606040i \(-0.207243\pi\)
0.795434 + 0.606040i \(0.207243\pi\)
\(854\) 2.53901e46 0.0108545
\(855\) 0 0
\(856\) 8.32507e47 0.342429
\(857\) 1.58574e48 0.639807 0.319903 0.947450i \(-0.396350\pi\)
0.319903 + 0.947450i \(0.396350\pi\)
\(858\) 0 0
\(859\) 2.88906e48 1.12168 0.560840 0.827924i \(-0.310478\pi\)
0.560840 + 0.827924i \(0.310478\pi\)
\(860\) −2.49332e46 −0.00949629
\(861\) 0 0
\(862\) 1.99404e48 0.730912
\(863\) −3.38580e46 −0.0121754 −0.00608770 0.999981i \(-0.501938\pi\)
−0.00608770 + 0.999981i \(0.501938\pi\)
\(864\) 0 0
\(865\) −1.34444e47 −0.0465347
\(866\) 6.16750e47 0.209442
\(867\) 0 0
\(868\) −5.05361e48 −1.65206
\(869\) 6.37858e47 0.204596
\(870\) 0 0
\(871\) −5.23695e47 −0.161726
\(872\) 4.00366e47 0.121321
\(873\) 0 0
\(874\) 3.09933e47 0.0904334
\(875\) −5.80101e47 −0.166100
\(876\) 0 0
\(877\) −4.24252e48 −1.16985 −0.584925 0.811087i \(-0.698876\pi\)
−0.584925 + 0.811087i \(0.698876\pi\)
\(878\) 3.01576e48 0.816089
\(879\) 0 0
\(880\) 7.17528e46 0.0187014
\(881\) 2.17405e48 0.556119 0.278059 0.960564i \(-0.410309\pi\)
0.278059 + 0.960564i \(0.410309\pi\)
\(882\) 0 0
\(883\) 4.77705e48 1.17708 0.588542 0.808466i \(-0.299702\pi\)
0.588542 + 0.808466i \(0.299702\pi\)
\(884\) −1.67916e48 −0.406098
\(885\) 0 0
\(886\) 3.28527e48 0.765447
\(887\) 5.12275e48 1.17156 0.585779 0.810471i \(-0.300789\pi\)
0.585779 + 0.810471i \(0.300789\pi\)
\(888\) 0 0
\(889\) 2.29955e48 0.506714
\(890\) −7.16199e46 −0.0154917
\(891\) 0 0
\(892\) 2.76302e48 0.575922
\(893\) 2.64116e48 0.540437
\(894\) 0 0
\(895\) −4.26640e47 −0.0841359
\(896\) −6.14883e48 −1.19045
\(897\) 0 0
\(898\) −7.67570e47 −0.143238
\(899\) −1.11311e48 −0.203941
\(900\) 0 0
\(901\) −9.17434e48 −1.62038
\(902\) −5.32147e48 −0.922834
\(903\) 0 0
\(904\) −5.15770e48 −0.862338
\(905\) 4.95769e47 0.0813912
\(906\) 0 0
\(907\) 7.71888e48 1.22190 0.610948 0.791671i \(-0.290789\pi\)
0.610948 + 0.791671i \(0.290789\pi\)
\(908\) −1.48895e48 −0.231453
\(909\) 0 0
\(910\) 1.33624e47 0.0200309
\(911\) 1.06274e49 1.56450 0.782250 0.622964i \(-0.214072\pi\)
0.782250 + 0.622964i \(0.214072\pi\)
\(912\) 0 0
\(913\) −2.65432e48 −0.376864
\(914\) −3.53882e48 −0.493454
\(915\) 0 0
\(916\) −2.09012e48 −0.281123
\(917\) −1.54394e49 −2.03956
\(918\) 0 0
\(919\) 8.89051e48 1.13298 0.566488 0.824070i \(-0.308302\pi\)
0.566488 + 0.824070i \(0.308302\pi\)
\(920\) −7.63460e46 −0.00955624
\(921\) 0 0
\(922\) −6.65723e48 −0.803959
\(923\) −3.69428e48 −0.438229
\(924\) 0 0
\(925\) 8.65948e48 0.991182
\(926\) 2.74764e48 0.308943
\(927\) 0 0
\(928\) −1.16743e48 −0.126675
\(929\) 7.35883e48 0.784422 0.392211 0.919875i \(-0.371710\pi\)
0.392211 + 0.919875i \(0.371710\pi\)
\(930\) 0 0
\(931\) −8.77828e48 −0.903108
\(932\) −4.28292e47 −0.0432890
\(933\) 0 0
\(934\) −1.74771e48 −0.170509
\(935\) −1.12203e48 −0.107550
\(936\) 0 0
\(937\) −7.92924e48 −0.733717 −0.366858 0.930277i \(-0.619567\pi\)
−0.366858 + 0.930277i \(0.619567\pi\)
\(938\) 2.94950e48 0.268165
\(939\) 0 0
\(940\) −2.70727e47 −0.0237641
\(941\) −1.44230e49 −1.24402 −0.622008 0.783011i \(-0.713683\pi\)
−0.622008 + 0.783011i \(0.713683\pi\)
\(942\) 0 0
\(943\) −2.53827e48 −0.211395
\(944\) 3.05229e48 0.249797
\(945\) 0 0
\(946\) −2.13679e48 −0.168872
\(947\) −9.83625e48 −0.763932 −0.381966 0.924176i \(-0.624753\pi\)
−0.381966 + 0.924176i \(0.624753\pi\)
\(948\) 0 0
\(949\) −7.55913e48 −0.566995
\(950\) −7.03168e48 −0.518346
\(951\) 0 0
\(952\) 2.27271e49 1.61821
\(953\) 8.19274e48 0.573319 0.286660 0.958033i \(-0.407455\pi\)
0.286660 + 0.958033i \(0.407455\pi\)
\(954\) 0 0
\(955\) 8.31467e47 0.0562068
\(956\) −1.00107e49 −0.665133
\(957\) 0 0
\(958\) −1.44329e49 −0.926450
\(959\) 3.61324e49 2.27976
\(960\) 0 0
\(961\) 2.92544e49 1.78343
\(962\) −3.99653e48 −0.239494
\(963\) 0 0
\(964\) 2.23180e48 0.129236
\(965\) 1.06902e48 0.0608536
\(966\) 0 0
\(967\) −1.65015e49 −0.907791 −0.453895 0.891055i \(-0.649966\pi\)
−0.453895 + 0.891055i \(0.649966\pi\)
\(968\) 1.70501e49 0.922113
\(969\) 0 0
\(970\) −7.83358e47 −0.0409474
\(971\) 3.28602e49 1.68870 0.844352 0.535789i \(-0.179986\pi\)
0.844352 + 0.535789i \(0.179986\pi\)
\(972\) 0 0
\(973\) 1.00483e49 0.499151
\(974\) 7.23857e48 0.353533
\(975\) 0 0
\(976\) 6.80939e46 0.00321504
\(977\) −1.17177e49 −0.543977 −0.271989 0.962300i \(-0.587681\pi\)
−0.271989 + 0.962300i \(0.587681\pi\)
\(978\) 0 0
\(979\) 1.52247e49 0.683336
\(980\) 8.99802e47 0.0397115
\(981\) 0 0
\(982\) 7.15306e48 0.305247
\(983\) −3.61572e49 −1.51726 −0.758632 0.651519i \(-0.774132\pi\)
−0.758632 + 0.651519i \(0.774132\pi\)
\(984\) 0 0
\(985\) 2.75828e48 0.111928
\(986\) 2.08305e48 0.0831249
\(987\) 0 0
\(988\) −8.04975e48 −0.310665
\(989\) −1.01922e48 −0.0386838
\(990\) 0 0
\(991\) 2.27623e49 0.835603 0.417802 0.908538i \(-0.362801\pi\)
0.417802 + 0.908538i \(0.362801\pi\)
\(992\) 4.78860e49 1.72889
\(993\) 0 0
\(994\) 2.08065e49 0.726649
\(995\) 2.02584e48 0.0695863
\(996\) 0 0
\(997\) −5.01160e49 −1.66535 −0.832676 0.553761i \(-0.813192\pi\)
−0.832676 + 0.553761i \(0.813192\pi\)
\(998\) −8.75230e48 −0.286067
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 9.34.a.b.1.1 2
3.2 odd 2 1.34.a.a.1.2 2
12.11 even 2 16.34.a.b.1.2 2
15.2 even 4 25.34.b.a.24.3 4
15.8 even 4 25.34.b.a.24.2 4
15.14 odd 2 25.34.a.a.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.34.a.a.1.2 2 3.2 odd 2
9.34.a.b.1.1 2 1.1 even 1 trivial
16.34.a.b.1.2 2 12.11 even 2
25.34.a.a.1.1 2 15.14 odd 2
25.34.b.a.24.2 4 15.8 even 4
25.34.b.a.24.3 4 15.2 even 4