Properties

Label 91.2.a.d.1.1
Level $91$
Weight $2$
Character 91.1
Self dual yes
Analytic conductor $0.727$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 91.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.726638658394\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.81361\) of defining polynomial
Character \(\chi\) \(=\) 91.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.81361 q^{2} -3.10278 q^{3} +1.28917 q^{4} +2.81361 q^{5} +5.62721 q^{6} -1.00000 q^{7} +1.28917 q^{8} +6.62721 q^{9} +O(q^{10})\) \(q-1.81361 q^{2} -3.10278 q^{3} +1.28917 q^{4} +2.81361 q^{5} +5.62721 q^{6} -1.00000 q^{7} +1.28917 q^{8} +6.62721 q^{9} -5.10278 q^{10} +3.10278 q^{11} -4.00000 q^{12} +1.00000 q^{13} +1.81361 q^{14} -8.72999 q^{15} -4.91638 q^{16} -0.524438 q^{17} -12.0192 q^{18} +0.813607 q^{19} +3.62721 q^{20} +3.10278 q^{21} -5.62721 q^{22} +7.33804 q^{23} -4.00000 q^{24} +2.91638 q^{25} -1.81361 q^{26} -11.2544 q^{27} -1.28917 q^{28} +8.28917 q^{29} +15.8328 q^{30} +1.39194 q^{31} +6.33804 q^{32} -9.62721 q^{33} +0.951124 q^{34} -2.81361 q^{35} +8.54359 q^{36} -6.15165 q^{37} -1.47556 q^{38} -3.10278 q^{39} +3.62721 q^{40} -4.20555 q^{41} -5.62721 q^{42} +6.75971 q^{43} +4.00000 q^{44} +18.6464 q^{45} -13.3083 q^{46} -5.97028 q^{47} +15.2544 q^{48} +1.00000 q^{49} -5.28917 q^{50} +1.62721 q^{51} +1.28917 q^{52} -2.49472 q^{53} +20.4111 q^{54} +8.72999 q^{55} -1.28917 q^{56} -2.52444 q^{57} -15.0333 q^{58} -4.47054 q^{59} -11.2544 q^{60} -2.00000 q^{61} -2.52444 q^{62} -6.62721 q^{63} -1.66196 q^{64} +2.81361 q^{65} +17.4600 q^{66} +10.0383 q^{67} -0.676089 q^{68} -22.7683 q^{69} +5.10278 q^{70} -8.72999 q^{71} +8.54359 q^{72} -2.34307 q^{73} +11.1567 q^{74} -9.04888 q^{75} +1.04888 q^{76} -3.10278 q^{77} +5.62721 q^{78} -13.5436 q^{79} -13.8328 q^{80} +15.0383 q^{81} +7.62721 q^{82} +16.4791 q^{83} +4.00000 q^{84} -1.47556 q^{85} -12.2594 q^{86} -25.7194 q^{87} +4.00000 q^{88} -10.6464 q^{89} -33.8172 q^{90} -1.00000 q^{91} +9.45998 q^{92} -4.31889 q^{93} +10.8277 q^{94} +2.28917 q^{95} -19.6655 q^{96} -1.18639 q^{97} -1.81361 q^{98} +20.5628 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + q^{2} - 2q^{3} + 3q^{4} + 2q^{5} + 4q^{6} - 3q^{7} + 3q^{8} + 7q^{9} + O(q^{10}) \) \( 3q + q^{2} - 2q^{3} + 3q^{4} + 2q^{5} + 4q^{6} - 3q^{7} + 3q^{8} + 7q^{9} - 8q^{10} + 2q^{11} - 12q^{12} + 3q^{13} - q^{14} - 6q^{15} - q^{16} + 4q^{17} - 15q^{18} - 4q^{19} - 2q^{20} + 2q^{21} - 4q^{22} + 10q^{23} - 12q^{24} - 5q^{25} + q^{26} - 8q^{27} - 3q^{28} + 24q^{29} + 20q^{30} - 4q^{31} + 7q^{32} - 16q^{33} + 14q^{34} - 2q^{35} - q^{36} - 10q^{38} - 2q^{39} - 2q^{40} + 2q^{41} - 4q^{42} + 10q^{43} + 12q^{44} + 22q^{45} - 18q^{46} - 8q^{47} + 20q^{48} + 3q^{49} - 15q^{50} - 8q^{51} + 3q^{52} + 8q^{53} + 32q^{54} + 6q^{55} - 3q^{56} - 2q^{57} + 12q^{58} - 4q^{59} - 8q^{60} - 6q^{61} - 2q^{62} - 7q^{63} - 17q^{64} + 2q^{65} + 12q^{66} - 12q^{67} + 22q^{68} - 6q^{69} + 8q^{70} - 6q^{71} - q^{72} - 10q^{73} + 30q^{74} - 16q^{75} - 8q^{76} - 2q^{77} + 4q^{78} - 14q^{79} - 14q^{80} + 3q^{81} + 10q^{82} - 12q^{83} + 12q^{84} - 10q^{85} - 26q^{86} - 26q^{87} + 12q^{88} + 2q^{89} - 28q^{90} - 3q^{91} - 12q^{92} - 22q^{93} - 10q^{94} + 6q^{95} - 4q^{96} - 10q^{97} + q^{98} + 14q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.81361 −1.28241 −0.641207 0.767368i \(-0.721566\pi\)
−0.641207 + 0.767368i \(0.721566\pi\)
\(3\) −3.10278 −1.79139 −0.895694 0.444671i \(-0.853321\pi\)
−0.895694 + 0.444671i \(0.853321\pi\)
\(4\) 1.28917 0.644584
\(5\) 2.81361 1.25828 0.629142 0.777291i \(-0.283407\pi\)
0.629142 + 0.777291i \(0.283407\pi\)
\(6\) 5.62721 2.29730
\(7\) −1.00000 −0.377964
\(8\) 1.28917 0.455790
\(9\) 6.62721 2.20907
\(10\) −5.10278 −1.61364
\(11\) 3.10278 0.935522 0.467761 0.883855i \(-0.345061\pi\)
0.467761 + 0.883855i \(0.345061\pi\)
\(12\) −4.00000 −1.15470
\(13\) 1.00000 0.277350
\(14\) 1.81361 0.484707
\(15\) −8.72999 −2.25407
\(16\) −4.91638 −1.22910
\(17\) −0.524438 −0.127195 −0.0635974 0.997976i \(-0.520257\pi\)
−0.0635974 + 0.997976i \(0.520257\pi\)
\(18\) −12.0192 −2.83294
\(19\) 0.813607 0.186654 0.0933271 0.995636i \(-0.470250\pi\)
0.0933271 + 0.995636i \(0.470250\pi\)
\(20\) 3.62721 0.811069
\(21\) 3.10278 0.677081
\(22\) −5.62721 −1.19973
\(23\) 7.33804 1.53009 0.765044 0.643978i \(-0.222717\pi\)
0.765044 + 0.643978i \(0.222717\pi\)
\(24\) −4.00000 −0.816497
\(25\) 2.91638 0.583276
\(26\) −1.81361 −0.355677
\(27\) −11.2544 −2.16592
\(28\) −1.28917 −0.243630
\(29\) 8.28917 1.53926 0.769630 0.638490i \(-0.220441\pi\)
0.769630 + 0.638490i \(0.220441\pi\)
\(30\) 15.8328 2.89065
\(31\) 1.39194 0.250000 0.125000 0.992157i \(-0.460107\pi\)
0.125000 + 0.992157i \(0.460107\pi\)
\(32\) 6.33804 1.12042
\(33\) −9.62721 −1.67588
\(34\) 0.951124 0.163116
\(35\) −2.81361 −0.475586
\(36\) 8.54359 1.42393
\(37\) −6.15165 −1.01133 −0.505663 0.862731i \(-0.668752\pi\)
−0.505663 + 0.862731i \(0.668752\pi\)
\(38\) −1.47556 −0.239368
\(39\) −3.10278 −0.496842
\(40\) 3.62721 0.573513
\(41\) −4.20555 −0.656797 −0.328398 0.944539i \(-0.606509\pi\)
−0.328398 + 0.944539i \(0.606509\pi\)
\(42\) −5.62721 −0.868298
\(43\) 6.75971 1.03085 0.515423 0.856936i \(-0.327635\pi\)
0.515423 + 0.856936i \(0.327635\pi\)
\(44\) 4.00000 0.603023
\(45\) 18.6464 2.77964
\(46\) −13.3083 −1.96221
\(47\) −5.97028 −0.870855 −0.435427 0.900224i \(-0.643403\pi\)
−0.435427 + 0.900224i \(0.643403\pi\)
\(48\) 15.2544 2.20179
\(49\) 1.00000 0.142857
\(50\) −5.28917 −0.748001
\(51\) 1.62721 0.227855
\(52\) 1.28917 0.178776
\(53\) −2.49472 −0.342676 −0.171338 0.985212i \(-0.554809\pi\)
−0.171338 + 0.985212i \(0.554809\pi\)
\(54\) 20.4111 2.77760
\(55\) 8.72999 1.17715
\(56\) −1.28917 −0.172272
\(57\) −2.52444 −0.334370
\(58\) −15.0333 −1.97397
\(59\) −4.47054 −0.582015 −0.291007 0.956721i \(-0.593990\pi\)
−0.291007 + 0.956721i \(0.593990\pi\)
\(60\) −11.2544 −1.45294
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −2.52444 −0.320604
\(63\) −6.62721 −0.834950
\(64\) −1.66196 −0.207744
\(65\) 2.81361 0.348985
\(66\) 17.4600 2.14917
\(67\) 10.0383 1.22638 0.613188 0.789937i \(-0.289887\pi\)
0.613188 + 0.789937i \(0.289887\pi\)
\(68\) −0.676089 −0.0819878
\(69\) −22.7683 −2.74098
\(70\) 5.10278 0.609898
\(71\) −8.72999 −1.03606 −0.518029 0.855363i \(-0.673334\pi\)
−0.518029 + 0.855363i \(0.673334\pi\)
\(72\) 8.54359 1.00687
\(73\) −2.34307 −0.274235 −0.137118 0.990555i \(-0.543784\pi\)
−0.137118 + 0.990555i \(0.543784\pi\)
\(74\) 11.1567 1.29694
\(75\) −9.04888 −1.04487
\(76\) 1.04888 0.120314
\(77\) −3.10278 −0.353594
\(78\) 5.62721 0.637156
\(79\) −13.5436 −1.52377 −0.761887 0.647710i \(-0.775727\pi\)
−0.761887 + 0.647710i \(0.775727\pi\)
\(80\) −13.8328 −1.54655
\(81\) 15.0383 1.67092
\(82\) 7.62721 0.842285
\(83\) 16.4791 1.80882 0.904410 0.426665i \(-0.140312\pi\)
0.904410 + 0.426665i \(0.140312\pi\)
\(84\) 4.00000 0.436436
\(85\) −1.47556 −0.160047
\(86\) −12.2594 −1.32197
\(87\) −25.7194 −2.75741
\(88\) 4.00000 0.426401
\(89\) −10.6464 −1.12851 −0.564256 0.825600i \(-0.690837\pi\)
−0.564256 + 0.825600i \(0.690837\pi\)
\(90\) −33.8172 −3.56464
\(91\) −1.00000 −0.104828
\(92\) 9.45998 0.986271
\(93\) −4.31889 −0.447848
\(94\) 10.8277 1.11680
\(95\) 2.28917 0.234864
\(96\) −19.6655 −2.00710
\(97\) −1.18639 −0.120460 −0.0602300 0.998185i \(-0.519183\pi\)
−0.0602300 + 0.998185i \(0.519183\pi\)
\(98\) −1.81361 −0.183202
\(99\) 20.5628 2.06663
\(100\) 3.75971 0.375971
\(101\) 13.1028 1.30377 0.651887 0.758316i \(-0.273977\pi\)
0.651887 + 0.758316i \(0.273977\pi\)
\(102\) −2.95112 −0.292205
\(103\) 4.41110 0.434639 0.217319 0.976101i \(-0.430269\pi\)
0.217319 + 0.976101i \(0.430269\pi\)
\(104\) 1.28917 0.126413
\(105\) 8.72999 0.851960
\(106\) 4.52444 0.439452
\(107\) 0.578337 0.0559100 0.0279550 0.999609i \(-0.491100\pi\)
0.0279550 + 0.999609i \(0.491100\pi\)
\(108\) −14.5089 −1.39611
\(109\) 5.57331 0.533827 0.266913 0.963721i \(-0.413996\pi\)
0.266913 + 0.963721i \(0.413996\pi\)
\(110\) −15.8328 −1.50959
\(111\) 19.0872 1.81168
\(112\) 4.91638 0.464554
\(113\) 5.44584 0.512302 0.256151 0.966637i \(-0.417546\pi\)
0.256151 + 0.966637i \(0.417546\pi\)
\(114\) 4.57834 0.428801
\(115\) 20.6464 1.92528
\(116\) 10.6861 0.992183
\(117\) 6.62721 0.612686
\(118\) 8.10780 0.746383
\(119\) 0.524438 0.0480751
\(120\) −11.2544 −1.02738
\(121\) −1.37279 −0.124799
\(122\) 3.62721 0.328392
\(123\) 13.0489 1.17658
\(124\) 1.79445 0.161146
\(125\) −5.86248 −0.524356
\(126\) 12.0192 1.07075
\(127\) −12.8816 −1.14306 −0.571530 0.820581i \(-0.693650\pi\)
−0.571530 + 0.820581i \(0.693650\pi\)
\(128\) −9.66196 −0.854004
\(129\) −20.9739 −1.84664
\(130\) −5.10278 −0.447543
\(131\) 9.04888 0.790604 0.395302 0.918551i \(-0.370640\pi\)
0.395302 + 0.918551i \(0.370640\pi\)
\(132\) −12.4111 −1.08025
\(133\) −0.813607 −0.0705486
\(134\) −18.2056 −1.57272
\(135\) −31.6655 −2.72533
\(136\) −0.676089 −0.0579741
\(137\) −6.25945 −0.534781 −0.267390 0.963588i \(-0.586161\pi\)
−0.267390 + 0.963588i \(0.586161\pi\)
\(138\) 41.2927 3.51507
\(139\) −11.5733 −0.981636 −0.490818 0.871262i \(-0.663302\pi\)
−0.490818 + 0.871262i \(0.663302\pi\)
\(140\) −3.62721 −0.306555
\(141\) 18.5244 1.56004
\(142\) 15.8328 1.32866
\(143\) 3.10278 0.259467
\(144\) −32.5819 −2.71516
\(145\) 23.3225 1.93682
\(146\) 4.24940 0.351683
\(147\) −3.10278 −0.255913
\(148\) −7.93051 −0.651884
\(149\) 8.52444 0.698349 0.349175 0.937058i \(-0.386462\pi\)
0.349175 + 0.937058i \(0.386462\pi\)
\(150\) 16.4111 1.33996
\(151\) −11.9844 −0.975278 −0.487639 0.873045i \(-0.662142\pi\)
−0.487639 + 0.873045i \(0.662142\pi\)
\(152\) 1.04888 0.0850751
\(153\) −3.47556 −0.280983
\(154\) 5.62721 0.453454
\(155\) 3.91638 0.314571
\(156\) −4.00000 −0.320256
\(157\) −12.8277 −1.02377 −0.511883 0.859055i \(-0.671052\pi\)
−0.511883 + 0.859055i \(0.671052\pi\)
\(158\) 24.5628 1.95411
\(159\) 7.74055 0.613866
\(160\) 17.8328 1.40980
\(161\) −7.33804 −0.578319
\(162\) −27.2736 −2.14282
\(163\) −13.4600 −1.05427 −0.527133 0.849783i \(-0.676733\pi\)
−0.527133 + 0.849783i \(0.676733\pi\)
\(164\) −5.42166 −0.423361
\(165\) −27.0872 −2.10873
\(166\) −29.8867 −2.31965
\(167\) −2.02972 −0.157064 −0.0785322 0.996912i \(-0.525023\pi\)
−0.0785322 + 0.996912i \(0.525023\pi\)
\(168\) 4.00000 0.308607
\(169\) 1.00000 0.0769231
\(170\) 2.67609 0.205247
\(171\) 5.39194 0.412332
\(172\) 8.71440 0.664467
\(173\) 20.2978 1.54321 0.771605 0.636102i \(-0.219454\pi\)
0.771605 + 0.636102i \(0.219454\pi\)
\(174\) 46.6449 3.53614
\(175\) −2.91638 −0.220458
\(176\) −15.2544 −1.14985
\(177\) 13.8711 1.04261
\(178\) 19.3083 1.44722
\(179\) −11.0036 −0.822445 −0.411223 0.911535i \(-0.634898\pi\)
−0.411223 + 0.911535i \(0.634898\pi\)
\(180\) 24.0383 1.79171
\(181\) 0.691675 0.0514118 0.0257059 0.999670i \(-0.491817\pi\)
0.0257059 + 0.999670i \(0.491817\pi\)
\(182\) 1.81361 0.134433
\(183\) 6.20555 0.458727
\(184\) 9.45998 0.697399
\(185\) −17.3083 −1.27253
\(186\) 7.83276 0.574326
\(187\) −1.62721 −0.118994
\(188\) −7.69670 −0.561339
\(189\) 11.2544 0.818639
\(190\) −4.15165 −0.301192
\(191\) 7.83276 0.566759 0.283379 0.959008i \(-0.408544\pi\)
0.283379 + 0.959008i \(0.408544\pi\)
\(192\) 5.15667 0.372151
\(193\) −12.2056 −0.878575 −0.439287 0.898347i \(-0.644769\pi\)
−0.439287 + 0.898347i \(0.644769\pi\)
\(194\) 2.15165 0.154480
\(195\) −8.72999 −0.625167
\(196\) 1.28917 0.0920835
\(197\) −18.8222 −1.34103 −0.670513 0.741898i \(-0.733926\pi\)
−0.670513 + 0.741898i \(0.733926\pi\)
\(198\) −37.2927 −2.65028
\(199\) 21.6116 1.53201 0.766004 0.642836i \(-0.222242\pi\)
0.766004 + 0.642836i \(0.222242\pi\)
\(200\) 3.75971 0.265851
\(201\) −31.1466 −2.19691
\(202\) −23.7633 −1.67198
\(203\) −8.28917 −0.581786
\(204\) 2.09775 0.146872
\(205\) −11.8328 −0.826436
\(206\) −8.00000 −0.557386
\(207\) 48.6308 3.38007
\(208\) −4.91638 −0.340890
\(209\) 2.52444 0.174619
\(210\) −15.8328 −1.09256
\(211\) −17.3764 −1.19624 −0.598119 0.801407i \(-0.704085\pi\)
−0.598119 + 0.801407i \(0.704085\pi\)
\(212\) −3.21611 −0.220884
\(213\) 27.0872 1.85598
\(214\) −1.04888 −0.0716997
\(215\) 19.0192 1.29710
\(216\) −14.5089 −0.987202
\(217\) −1.39194 −0.0944913
\(218\) −10.1078 −0.684586
\(219\) 7.27001 0.491262
\(220\) 11.2544 0.758773
\(221\) −0.524438 −0.0352775
\(222\) −34.6167 −2.32332
\(223\) −10.5486 −0.706388 −0.353194 0.935550i \(-0.614904\pi\)
−0.353194 + 0.935550i \(0.614904\pi\)
\(224\) −6.33804 −0.423478
\(225\) 19.3275 1.28850
\(226\) −9.87662 −0.656983
\(227\) −6.95112 −0.461362 −0.230681 0.973029i \(-0.574095\pi\)
−0.230681 + 0.973029i \(0.574095\pi\)
\(228\) −3.25443 −0.215530
\(229\) −21.0872 −1.39348 −0.696740 0.717323i \(-0.745367\pi\)
−0.696740 + 0.717323i \(0.745367\pi\)
\(230\) −37.4444 −2.46901
\(231\) 9.62721 0.633424
\(232\) 10.6861 0.701579
\(233\) 6.08362 0.398551 0.199276 0.979943i \(-0.436141\pi\)
0.199276 + 0.979943i \(0.436141\pi\)
\(234\) −12.0192 −0.785717
\(235\) −16.7980 −1.09578
\(236\) −5.76328 −0.375157
\(237\) 42.0227 2.72967
\(238\) −0.951124 −0.0616522
\(239\) −14.2056 −0.918881 −0.459440 0.888209i \(-0.651950\pi\)
−0.459440 + 0.888209i \(0.651950\pi\)
\(240\) 42.9200 2.77047
\(241\) 8.44082 0.543721 0.271860 0.962337i \(-0.412361\pi\)
0.271860 + 0.962337i \(0.412361\pi\)
\(242\) 2.48970 0.160044
\(243\) −12.8972 −0.827357
\(244\) −2.57834 −0.165061
\(245\) 2.81361 0.179755
\(246\) −23.6655 −1.50886
\(247\) 0.813607 0.0517685
\(248\) 1.79445 0.113948
\(249\) −51.1310 −3.24030
\(250\) 10.6322 0.672442
\(251\) −23.9844 −1.51388 −0.756941 0.653483i \(-0.773307\pi\)
−0.756941 + 0.653483i \(0.773307\pi\)
\(252\) −8.54359 −0.538196
\(253\) 22.7683 1.43143
\(254\) 23.3622 1.46588
\(255\) 4.57834 0.286707
\(256\) 20.8469 1.30293
\(257\) 15.6116 0.973827 0.486913 0.873450i \(-0.338123\pi\)
0.486913 + 0.873450i \(0.338123\pi\)
\(258\) 38.0383 2.36816
\(259\) 6.15165 0.382245
\(260\) 3.62721 0.224950
\(261\) 54.9341 3.40033
\(262\) −16.4111 −1.01388
\(263\) −15.1708 −0.935472 −0.467736 0.883868i \(-0.654930\pi\)
−0.467736 + 0.883868i \(0.654930\pi\)
\(264\) −12.4111 −0.763850
\(265\) −7.01916 −0.431183
\(266\) 1.47556 0.0904725
\(267\) 33.0333 2.02160
\(268\) 12.9411 0.790502
\(269\) 23.2927 1.42018 0.710092 0.704109i \(-0.248653\pi\)
0.710092 + 0.704109i \(0.248653\pi\)
\(270\) 57.4288 3.49501
\(271\) 12.7456 0.774238 0.387119 0.922030i \(-0.373470\pi\)
0.387119 + 0.922030i \(0.373470\pi\)
\(272\) 2.57834 0.156335
\(273\) 3.10278 0.187788
\(274\) 11.3522 0.685810
\(275\) 9.04888 0.545668
\(276\) −29.3522 −1.76679
\(277\) 8.12193 0.488000 0.244000 0.969775i \(-0.421540\pi\)
0.244000 + 0.969775i \(0.421540\pi\)
\(278\) 20.9894 1.25886
\(279\) 9.22471 0.552269
\(280\) −3.62721 −0.216767
\(281\) 19.0333 1.13543 0.567715 0.823225i \(-0.307827\pi\)
0.567715 + 0.823225i \(0.307827\pi\)
\(282\) −33.5960 −2.00062
\(283\) 11.1466 0.662598 0.331299 0.943526i \(-0.392513\pi\)
0.331299 + 0.943526i \(0.392513\pi\)
\(284\) −11.2544 −0.667827
\(285\) −7.10278 −0.420732
\(286\) −5.62721 −0.332744
\(287\) 4.20555 0.248246
\(288\) 42.0036 2.47508
\(289\) −16.7250 −0.983821
\(290\) −42.2978 −2.48381
\(291\) 3.68111 0.215791
\(292\) −3.02061 −0.176768
\(293\) −14.1758 −0.828161 −0.414080 0.910240i \(-0.635897\pi\)
−0.414080 + 0.910240i \(0.635897\pi\)
\(294\) 5.62721 0.328186
\(295\) −12.5783 −0.732339
\(296\) −7.93051 −0.460952
\(297\) −34.9200 −2.02626
\(298\) −15.4600 −0.895572
\(299\) 7.33804 0.424370
\(300\) −11.6655 −0.673509
\(301\) −6.75971 −0.389623
\(302\) 21.7350 1.25071
\(303\) −40.6550 −2.33557
\(304\) −4.00000 −0.229416
\(305\) −5.62721 −0.322213
\(306\) 6.30330 0.360336
\(307\) 13.5592 0.773863 0.386932 0.922108i \(-0.373535\pi\)
0.386932 + 0.922108i \(0.373535\pi\)
\(308\) −4.00000 −0.227921
\(309\) −13.6867 −0.778606
\(310\) −7.10278 −0.403411
\(311\) 0.426686 0.0241952 0.0120976 0.999927i \(-0.496149\pi\)
0.0120976 + 0.999927i \(0.496149\pi\)
\(312\) −4.00000 −0.226455
\(313\) −18.1517 −1.02599 −0.512996 0.858391i \(-0.671464\pi\)
−0.512996 + 0.858391i \(0.671464\pi\)
\(314\) 23.2645 1.31289
\(315\) −18.6464 −1.05060
\(316\) −17.4600 −0.982200
\(317\) 9.42166 0.529173 0.264587 0.964362i \(-0.414764\pi\)
0.264587 + 0.964362i \(0.414764\pi\)
\(318\) −14.0383 −0.787230
\(319\) 25.7194 1.44001
\(320\) −4.67609 −0.261401
\(321\) −1.79445 −0.100156
\(322\) 13.3083 0.741644
\(323\) −0.426686 −0.0237415
\(324\) 19.3869 1.07705
\(325\) 2.91638 0.161772
\(326\) 24.4111 1.35201
\(327\) −17.2927 −0.956291
\(328\) −5.42166 −0.299361
\(329\) 5.97028 0.329152
\(330\) 49.1255 2.70427
\(331\) −17.4005 −0.956420 −0.478210 0.878246i \(-0.658714\pi\)
−0.478210 + 0.878246i \(0.658714\pi\)
\(332\) 21.2444 1.16594
\(333\) −40.7683 −2.23409
\(334\) 3.68111 0.201421
\(335\) 28.2439 1.54313
\(336\) −15.2544 −0.832197
\(337\) 22.0524 1.20127 0.600637 0.799522i \(-0.294914\pi\)
0.600637 + 0.799522i \(0.294914\pi\)
\(338\) −1.81361 −0.0986472
\(339\) −16.8972 −0.917731
\(340\) −1.90225 −0.103164
\(341\) 4.31889 0.233881
\(342\) −9.77886 −0.528780
\(343\) −1.00000 −0.0539949
\(344\) 8.71440 0.469849
\(345\) −64.0610 −3.44893
\(346\) −36.8122 −1.97903
\(347\) 25.3522 1.36098 0.680488 0.732759i \(-0.261768\pi\)
0.680488 + 0.732759i \(0.261768\pi\)
\(348\) −33.1567 −1.77738
\(349\) −5.70529 −0.305397 −0.152699 0.988273i \(-0.548796\pi\)
−0.152699 + 0.988273i \(0.548796\pi\)
\(350\) 5.28917 0.282718
\(351\) −11.2544 −0.600717
\(352\) 19.6655 1.04818
\(353\) −28.6761 −1.52627 −0.763137 0.646237i \(-0.776342\pi\)
−0.763137 + 0.646237i \(0.776342\pi\)
\(354\) −25.1567 −1.33706
\(355\) −24.5628 −1.30366
\(356\) −13.7250 −0.727422
\(357\) −1.62721 −0.0861212
\(358\) 19.9561 1.05472
\(359\) 11.0433 0.582845 0.291423 0.956594i \(-0.405871\pi\)
0.291423 + 0.956594i \(0.405871\pi\)
\(360\) 24.0383 1.26693
\(361\) −18.3380 −0.965160
\(362\) −1.25443 −0.0659312
\(363\) 4.25945 0.223563
\(364\) −1.28917 −0.0675708
\(365\) −6.59247 −0.345066
\(366\) −11.2544 −0.588278
\(367\) 27.3466 1.42748 0.713741 0.700409i \(-0.246999\pi\)
0.713741 + 0.700409i \(0.246999\pi\)
\(368\) −36.0766 −1.88062
\(369\) −27.8711 −1.45091
\(370\) 31.3905 1.63191
\(371\) 2.49472 0.129519
\(372\) −5.56777 −0.288676
\(373\) 16.1461 0.836014 0.418007 0.908444i \(-0.362729\pi\)
0.418007 + 0.908444i \(0.362729\pi\)
\(374\) 2.95112 0.152599
\(375\) 18.1900 0.939326
\(376\) −7.69670 −0.396927
\(377\) 8.28917 0.426914
\(378\) −20.4111 −1.04983
\(379\) 26.1305 1.34223 0.671117 0.741351i \(-0.265815\pi\)
0.671117 + 0.741351i \(0.265815\pi\)
\(380\) 2.95112 0.151389
\(381\) 39.9688 2.04767
\(382\) −14.2056 −0.726819
\(383\) −21.0489 −1.07555 −0.537774 0.843089i \(-0.680735\pi\)
−0.537774 + 0.843089i \(0.680735\pi\)
\(384\) 29.9789 1.52985
\(385\) −8.72999 −0.444921
\(386\) 22.1361 1.12670
\(387\) 44.7980 2.27721
\(388\) −1.52946 −0.0776466
\(389\) −21.6061 −1.09547 −0.547736 0.836651i \(-0.684510\pi\)
−0.547736 + 0.836651i \(0.684510\pi\)
\(390\) 15.8328 0.801723
\(391\) −3.84835 −0.194619
\(392\) 1.28917 0.0651128
\(393\) −28.0766 −1.41628
\(394\) 34.1361 1.71975
\(395\) −38.1063 −1.91734
\(396\) 26.5089 1.33212
\(397\) −27.6952 −1.38998 −0.694992 0.719017i \(-0.744592\pi\)
−0.694992 + 0.719017i \(0.744592\pi\)
\(398\) −39.1950 −1.96467
\(399\) 2.52444 0.126380
\(400\) −14.3380 −0.716902
\(401\) −2.57834 −0.128756 −0.0643780 0.997926i \(-0.520506\pi\)
−0.0643780 + 0.997926i \(0.520506\pi\)
\(402\) 56.4877 2.81735
\(403\) 1.39194 0.0693376
\(404\) 16.8917 0.840393
\(405\) 42.3119 2.10250
\(406\) 15.0333 0.746090
\(407\) −19.0872 −0.946117
\(408\) 2.09775 0.103854
\(409\) 15.1169 0.747483 0.373742 0.927533i \(-0.378075\pi\)
0.373742 + 0.927533i \(0.378075\pi\)
\(410\) 21.4600 1.05983
\(411\) 19.4217 0.958000
\(412\) 5.68665 0.280161
\(413\) 4.47054 0.219981
\(414\) −88.1971 −4.33465
\(415\) 46.3658 2.27601
\(416\) 6.33804 0.310748
\(417\) 35.9094 1.75849
\(418\) −4.57834 −0.223934
\(419\) −9.99446 −0.488261 −0.244131 0.969742i \(-0.578503\pi\)
−0.244131 + 0.969742i \(0.578503\pi\)
\(420\) 11.2544 0.549160
\(421\) −25.9250 −1.26351 −0.631753 0.775170i \(-0.717664\pi\)
−0.631753 + 0.775170i \(0.717664\pi\)
\(422\) 31.5139 1.53407
\(423\) −39.5663 −1.92378
\(424\) −3.21611 −0.156188
\(425\) −1.52946 −0.0741898
\(426\) −49.1255 −2.38014
\(427\) 2.00000 0.0967868
\(428\) 0.745574 0.0360387
\(429\) −9.62721 −0.464806
\(430\) −34.4933 −1.66341
\(431\) 30.6761 1.47762 0.738808 0.673916i \(-0.235389\pi\)
0.738808 + 0.673916i \(0.235389\pi\)
\(432\) 55.3311 2.66212
\(433\) −3.51941 −0.169132 −0.0845661 0.996418i \(-0.526950\pi\)
−0.0845661 + 0.996418i \(0.526950\pi\)
\(434\) 2.52444 0.121177
\(435\) −72.3643 −3.46960
\(436\) 7.18494 0.344096
\(437\) 5.97028 0.285597
\(438\) −13.1849 −0.630001
\(439\) 32.3517 1.54406 0.772030 0.635586i \(-0.219241\pi\)
0.772030 + 0.635586i \(0.219241\pi\)
\(440\) 11.2544 0.536534
\(441\) 6.62721 0.315582
\(442\) 0.951124 0.0452404
\(443\) 15.4458 0.733854 0.366927 0.930250i \(-0.380410\pi\)
0.366927 + 0.930250i \(0.380410\pi\)
\(444\) 24.6066 1.16778
\(445\) −29.9547 −1.41999
\(446\) 19.1310 0.905881
\(447\) −26.4494 −1.25101
\(448\) 1.66196 0.0785200
\(449\) −14.4705 −0.682907 −0.341453 0.939899i \(-0.610919\pi\)
−0.341453 + 0.939899i \(0.610919\pi\)
\(450\) −35.0524 −1.65239
\(451\) −13.0489 −0.614448
\(452\) 7.02061 0.330222
\(453\) 37.1849 1.74710
\(454\) 12.6066 0.591657
\(455\) −2.81361 −0.131904
\(456\) −3.25443 −0.152402
\(457\) −34.6705 −1.62182 −0.810910 0.585171i \(-0.801027\pi\)
−0.810910 + 0.585171i \(0.801027\pi\)
\(458\) 38.2439 1.78702
\(459\) 5.90225 0.275493
\(460\) 26.6167 1.24101
\(461\) 12.5400 0.584047 0.292024 0.956411i \(-0.405671\pi\)
0.292024 + 0.956411i \(0.405671\pi\)
\(462\) −17.4600 −0.812312
\(463\) 12.1517 0.564735 0.282368 0.959306i \(-0.408880\pi\)
0.282368 + 0.959306i \(0.408880\pi\)
\(464\) −40.7527 −1.89190
\(465\) −12.1517 −0.563519
\(466\) −11.0333 −0.511107
\(467\) −37.0333 −1.71370 −0.856848 0.515569i \(-0.827581\pi\)
−0.856848 + 0.515569i \(0.827581\pi\)
\(468\) 8.54359 0.394928
\(469\) −10.0383 −0.463526
\(470\) 30.4650 1.40525
\(471\) 39.8016 1.83396
\(472\) −5.76328 −0.265276
\(473\) 20.9739 0.964379
\(474\) −76.2127 −3.50056
\(475\) 2.37279 0.108871
\(476\) 0.676089 0.0309885
\(477\) −16.5330 −0.756996
\(478\) 25.7633 1.17838
\(479\) 12.0086 0.548687 0.274343 0.961632i \(-0.411540\pi\)
0.274343 + 0.961632i \(0.411540\pi\)
\(480\) −55.3311 −2.52551
\(481\) −6.15165 −0.280491
\(482\) −15.3083 −0.697275
\(483\) 22.7683 1.03599
\(484\) −1.76975 −0.0804434
\(485\) −3.33804 −0.151573
\(486\) 23.3905 1.06101
\(487\) 11.1184 0.503821 0.251911 0.967751i \(-0.418941\pi\)
0.251911 + 0.967751i \(0.418941\pi\)
\(488\) −2.57834 −0.116716
\(489\) 41.7633 1.88860
\(490\) −5.10278 −0.230520
\(491\) 0.0594386 0.00268243 0.00134121 0.999999i \(-0.499573\pi\)
0.00134121 + 0.999999i \(0.499573\pi\)
\(492\) 16.8222 0.758403
\(493\) −4.34715 −0.195786
\(494\) −1.47556 −0.0663887
\(495\) 57.8555 2.60041
\(496\) −6.84333 −0.307274
\(497\) 8.72999 0.391593
\(498\) 92.7316 4.15540
\(499\) 10.2978 0.460991 0.230496 0.973073i \(-0.425965\pi\)
0.230496 + 0.973073i \(0.425965\pi\)
\(500\) −7.55773 −0.337992
\(501\) 6.29776 0.281363
\(502\) 43.4983 1.94142
\(503\) −9.32391 −0.415733 −0.207866 0.978157i \(-0.566652\pi\)
−0.207866 + 0.978157i \(0.566652\pi\)
\(504\) −8.54359 −0.380562
\(505\) 36.8661 1.64052
\(506\) −41.2927 −1.83569
\(507\) −3.10278 −0.137799
\(508\) −16.6066 −0.736799
\(509\) 39.6952 1.75946 0.879730 0.475473i \(-0.157723\pi\)
0.879730 + 0.475473i \(0.157723\pi\)
\(510\) −8.30330 −0.367676
\(511\) 2.34307 0.103651
\(512\) −18.4842 −0.816892
\(513\) −9.15667 −0.404277
\(514\) −28.3133 −1.24885
\(515\) 12.4111 0.546898
\(516\) −27.0388 −1.19032
\(517\) −18.5244 −0.814704
\(518\) −11.1567 −0.490196
\(519\) −62.9794 −2.76449
\(520\) 3.62721 0.159064
\(521\) 22.3627 0.979729 0.489865 0.871798i \(-0.337046\pi\)
0.489865 + 0.871798i \(0.337046\pi\)
\(522\) −99.6288 −4.36063
\(523\) −20.6550 −0.903178 −0.451589 0.892226i \(-0.649143\pi\)
−0.451589 + 0.892226i \(0.649143\pi\)
\(524\) 11.6655 0.509611
\(525\) 9.04888 0.394925
\(526\) 27.5139 1.19966
\(527\) −0.729988 −0.0317988
\(528\) 47.3311 2.05982
\(529\) 30.8469 1.34117
\(530\) 12.7300 0.552955
\(531\) −29.6272 −1.28571
\(532\) −1.04888 −0.0454745
\(533\) −4.20555 −0.182163
\(534\) −59.9094 −2.59253
\(535\) 1.62721 0.0703506
\(536\) 12.9411 0.558969
\(537\) 34.1416 1.47332
\(538\) −42.2439 −1.82126
\(539\) 3.10278 0.133646
\(540\) −40.8222 −1.75671
\(541\) 5.62167 0.241695 0.120847 0.992671i \(-0.461439\pi\)
0.120847 + 0.992671i \(0.461439\pi\)
\(542\) −23.1155 −0.992894
\(543\) −2.14611 −0.0920985
\(544\) −3.32391 −0.142512
\(545\) 15.6811 0.671705
\(546\) −5.62721 −0.240822
\(547\) −10.3970 −0.444542 −0.222271 0.974985i \(-0.571347\pi\)
−0.222271 + 0.974985i \(0.571347\pi\)
\(548\) −8.06949 −0.344711
\(549\) −13.2544 −0.565685
\(550\) −16.4111 −0.699772
\(551\) 6.74412 0.287309
\(552\) −29.3522 −1.24931
\(553\) 13.5436 0.575932
\(554\) −14.7300 −0.625817
\(555\) 53.7038 2.27960
\(556\) −14.9200 −0.632747
\(557\) 14.6550 0.620951 0.310475 0.950581i \(-0.399512\pi\)
0.310475 + 0.950581i \(0.399512\pi\)
\(558\) −16.7300 −0.708237
\(559\) 6.75971 0.285905
\(560\) 13.8328 0.584541
\(561\) 5.04888 0.213164
\(562\) −34.5189 −1.45609
\(563\) −24.7456 −1.04290 −0.521451 0.853281i \(-0.674609\pi\)
−0.521451 + 0.853281i \(0.674609\pi\)
\(564\) 23.8811 1.00558
\(565\) 15.3225 0.644621
\(566\) −20.2156 −0.849725
\(567\) −15.0383 −0.631550
\(568\) −11.2544 −0.472225
\(569\) −20.5330 −0.860789 −0.430395 0.902641i \(-0.641626\pi\)
−0.430395 + 0.902641i \(0.641626\pi\)
\(570\) 12.8816 0.539552
\(571\) −41.8953 −1.75326 −0.876631 0.481163i \(-0.840214\pi\)
−0.876631 + 0.481163i \(0.840214\pi\)
\(572\) 4.00000 0.167248
\(573\) −24.3033 −1.01529
\(574\) −7.62721 −0.318354
\(575\) 21.4005 0.892464
\(576\) −11.0141 −0.458922
\(577\) −20.1744 −0.839870 −0.419935 0.907554i \(-0.637947\pi\)
−0.419935 + 0.907554i \(0.637947\pi\)
\(578\) 30.3325 1.26167
\(579\) 37.8711 1.57387
\(580\) 30.0666 1.24845
\(581\) −16.4791 −0.683670
\(582\) −6.67609 −0.276733
\(583\) −7.74055 −0.320581
\(584\) −3.02061 −0.124994
\(585\) 18.6464 0.770933
\(586\) 25.7094 1.06204
\(587\) −18.7441 −0.773653 −0.386826 0.922153i \(-0.626429\pi\)
−0.386826 + 0.922153i \(0.626429\pi\)
\(588\) −4.00000 −0.164957
\(589\) 1.13249 0.0466636
\(590\) 22.8122 0.939162
\(591\) 58.4011 2.40230
\(592\) 30.2439 1.24302
\(593\) −2.98084 −0.122409 −0.0612043 0.998125i \(-0.519494\pi\)
−0.0612043 + 0.998125i \(0.519494\pi\)
\(594\) 63.3311 2.59850
\(595\) 1.47556 0.0604921
\(596\) 10.9894 0.450145
\(597\) −67.0560 −2.74442
\(598\) −13.3083 −0.544218
\(599\) 7.47411 0.305384 0.152692 0.988274i \(-0.451206\pi\)
0.152692 + 0.988274i \(0.451206\pi\)
\(600\) −11.6655 −0.476243
\(601\) 21.4700 0.875780 0.437890 0.899028i \(-0.355726\pi\)
0.437890 + 0.899028i \(0.355726\pi\)
\(602\) 12.2594 0.499658
\(603\) 66.5260 2.70915
\(604\) −15.4499 −0.628649
\(605\) −3.86248 −0.157032
\(606\) 73.7321 2.99516
\(607\) −22.9044 −0.929660 −0.464830 0.885400i \(-0.653884\pi\)
−0.464830 + 0.885400i \(0.653884\pi\)
\(608\) 5.15667 0.209131
\(609\) 25.7194 1.04220
\(610\) 10.2056 0.413211
\(611\) −5.97028 −0.241532
\(612\) −4.48059 −0.181117
\(613\) −20.1461 −0.813694 −0.406847 0.913496i \(-0.633372\pi\)
−0.406847 + 0.913496i \(0.633372\pi\)
\(614\) −24.5910 −0.992413
\(615\) 36.7144 1.48047
\(616\) −4.00000 −0.161165
\(617\) −13.7844 −0.554939 −0.277470 0.960734i \(-0.589496\pi\)
−0.277470 + 0.960734i \(0.589496\pi\)
\(618\) 24.8222 0.998495
\(619\) −19.6655 −0.790424 −0.395212 0.918590i \(-0.629329\pi\)
−0.395212 + 0.918590i \(0.629329\pi\)
\(620\) 5.04888 0.202768
\(621\) −82.5855 −3.31404
\(622\) −0.773841 −0.0310282
\(623\) 10.6464 0.426538
\(624\) 15.2544 0.610666
\(625\) −31.0766 −1.24307
\(626\) 32.9200 1.31575
\(627\) −7.83276 −0.312810
\(628\) −16.5371 −0.659903
\(629\) 3.22616 0.128635
\(630\) 33.8172 1.34731
\(631\) −16.1672 −0.643608 −0.321804 0.946806i \(-0.604289\pi\)
−0.321804 + 0.946806i \(0.604289\pi\)
\(632\) −17.4600 −0.694521
\(633\) 53.9149 2.14293
\(634\) −17.0872 −0.678619
\(635\) −36.2439 −1.43829
\(636\) 9.97887 0.395688
\(637\) 1.00000 0.0396214
\(638\) −46.6449 −1.84669
\(639\) −57.8555 −2.28873
\(640\) −27.1849 −1.07458
\(641\) −29.0036 −1.14557 −0.572786 0.819705i \(-0.694137\pi\)
−0.572786 + 0.819705i \(0.694137\pi\)
\(642\) 3.25443 0.128442
\(643\) −39.2233 −1.54681 −0.773407 0.633910i \(-0.781449\pi\)
−0.773407 + 0.633910i \(0.781449\pi\)
\(644\) −9.45998 −0.372775
\(645\) −59.0122 −2.32360
\(646\) 0.773841 0.0304464
\(647\) −11.9844 −0.471156 −0.235578 0.971855i \(-0.575698\pi\)
−0.235578 + 0.971855i \(0.575698\pi\)
\(648\) 19.3869 0.761590
\(649\) −13.8711 −0.544487
\(650\) −5.28917 −0.207458
\(651\) 4.31889 0.169271
\(652\) −17.3522 −0.679564
\(653\) 45.3311 1.77394 0.886971 0.461826i \(-0.152806\pi\)
0.886971 + 0.461826i \(0.152806\pi\)
\(654\) 31.3622 1.22636
\(655\) 25.4600 0.994804
\(656\) 20.6761 0.807266
\(657\) −15.5280 −0.605805
\(658\) −10.8277 −0.422109
\(659\) 6.12193 0.238477 0.119238 0.992866i \(-0.461955\pi\)
0.119238 + 0.992866i \(0.461955\pi\)
\(660\) −34.9200 −1.35926
\(661\) −27.5280 −1.07072 −0.535358 0.844625i \(-0.679823\pi\)
−0.535358 + 0.844625i \(0.679823\pi\)
\(662\) 31.5577 1.22653
\(663\) 1.62721 0.0631957
\(664\) 21.2444 0.824442
\(665\) −2.28917 −0.0887701
\(666\) 73.9377 2.86503
\(667\) 60.8263 2.35520
\(668\) −2.61665 −0.101241
\(669\) 32.7300 1.26541
\(670\) −51.2233 −1.97893
\(671\) −6.20555 −0.239563
\(672\) 19.6655 0.758614
\(673\) 27.9547 1.07757 0.538787 0.842442i \(-0.318883\pi\)
0.538787 + 0.842442i \(0.318883\pi\)
\(674\) −39.9945 −1.54053
\(675\) −32.8222 −1.26333
\(676\) 1.28917 0.0495834
\(677\) −12.6605 −0.486583 −0.243291 0.969953i \(-0.578227\pi\)
−0.243291 + 0.969953i \(0.578227\pi\)
\(678\) 30.6449 1.17691
\(679\) 1.18639 0.0455296
\(680\) −1.90225 −0.0729479
\(681\) 21.5678 0.826479
\(682\) −7.83276 −0.299932
\(683\) −28.3033 −1.08300 −0.541498 0.840702i \(-0.682143\pi\)
−0.541498 + 0.840702i \(0.682143\pi\)
\(684\) 6.95112 0.265783
\(685\) −17.6116 −0.672906
\(686\) 1.81361 0.0692438
\(687\) 65.4288 2.49626
\(688\) −33.2333 −1.26701
\(689\) −2.49472 −0.0950412
\(690\) 116.182 4.42295
\(691\) 12.2353 0.465452 0.232726 0.972542i \(-0.425236\pi\)
0.232726 + 0.972542i \(0.425236\pi\)
\(692\) 26.1672 0.994729
\(693\) −20.5628 −0.781114
\(694\) −45.9789 −1.74533
\(695\) −32.5628 −1.23518
\(696\) −33.1567 −1.25680
\(697\) 2.20555 0.0835412
\(698\) 10.3472 0.391646
\(699\) −18.8761 −0.713960
\(700\) −3.75971 −0.142104
\(701\) 51.0419 1.92783 0.963913 0.266219i \(-0.0857743\pi\)
0.963913 + 0.266219i \(0.0857743\pi\)
\(702\) 20.4111 0.770367
\(703\) −5.00502 −0.188768
\(704\) −5.15667 −0.194349
\(705\) 52.1205 1.96297
\(706\) 52.0071 1.95731
\(707\) −13.1028 −0.492781
\(708\) 17.8822 0.672053
\(709\) 42.5910 1.59954 0.799770 0.600307i \(-0.204955\pi\)
0.799770 + 0.600307i \(0.204955\pi\)
\(710\) 44.5472 1.67183
\(711\) −89.7563 −3.36612
\(712\) −13.7250 −0.514365
\(713\) 10.2141 0.382523
\(714\) 2.95112 0.110443
\(715\) 8.72999 0.326483
\(716\) −14.1855 −0.530135
\(717\) 44.0766 1.64607
\(718\) −20.0283 −0.747448
\(719\) −42.4933 −1.58473 −0.792366 0.610046i \(-0.791151\pi\)
−0.792366 + 0.610046i \(0.791151\pi\)
\(720\) −91.6727 −3.41644
\(721\) −4.41110 −0.164278
\(722\) 33.2580 1.23773
\(723\) −26.1900 −0.974015
\(724\) 0.891685 0.0331392
\(725\) 24.1744 0.897814
\(726\) −7.72496 −0.286700
\(727\) −3.75614 −0.139307 −0.0696537 0.997571i \(-0.522189\pi\)
−0.0696537 + 0.997571i \(0.522189\pi\)
\(728\) −1.28917 −0.0477798
\(729\) −5.09775 −0.188806
\(730\) 11.9561 0.442517
\(731\) −3.54505 −0.131118
\(732\) 8.00000 0.295689
\(733\) 45.7819 1.69099 0.845497 0.533980i \(-0.179304\pi\)
0.845497 + 0.533980i \(0.179304\pi\)
\(734\) −49.5960 −1.83062
\(735\) −8.72999 −0.322010
\(736\) 46.5089 1.71434
\(737\) 31.1466 1.14730
\(738\) 50.5472 1.86067
\(739\) −14.0539 −0.516981 −0.258491 0.966014i \(-0.583225\pi\)
−0.258491 + 0.966014i \(0.583225\pi\)
\(740\) −22.3133 −0.820255
\(741\) −2.52444 −0.0927375
\(742\) −4.52444 −0.166097
\(743\) 4.74557 0.174098 0.0870491 0.996204i \(-0.472256\pi\)
0.0870491 + 0.996204i \(0.472256\pi\)
\(744\) −5.56777 −0.204125
\(745\) 23.9844 0.878721
\(746\) −29.2827 −1.07212
\(747\) 109.211 3.99581
\(748\) −2.09775 −0.0767014
\(749\) −0.578337 −0.0211320
\(750\) −32.9894 −1.20460
\(751\) 36.1008 1.31734 0.658669 0.752433i \(-0.271120\pi\)
0.658669 + 0.752433i \(0.271120\pi\)
\(752\) 29.3522 1.07036
\(753\) 74.4182 2.71195
\(754\) −15.0333 −0.547480
\(755\) −33.7194 −1.22718
\(756\) 14.5089 0.527682
\(757\) −1.03474 −0.0376084 −0.0188042 0.999823i \(-0.505986\pi\)
−0.0188042 + 0.999823i \(0.505986\pi\)
\(758\) −47.3905 −1.72130
\(759\) −70.6449 −2.56425
\(760\) 2.95112 0.107049
\(761\) 29.8414 1.08175 0.540874 0.841104i \(-0.318094\pi\)
0.540874 + 0.841104i \(0.318094\pi\)
\(762\) −72.4877 −2.62595
\(763\) −5.57331 −0.201768
\(764\) 10.0978 0.365324
\(765\) −9.77886 −0.353556
\(766\) 38.1744 1.37930
\(767\) −4.47054 −0.161422
\(768\) −64.6832 −2.33406
\(769\) 23.6358 0.852329 0.426164 0.904646i \(-0.359864\pi\)
0.426164 + 0.904646i \(0.359864\pi\)
\(770\) 15.8328 0.570573
\(771\) −48.4394 −1.74450
\(772\) −15.7350 −0.566315
\(773\) 13.0278 0.468576 0.234288 0.972167i \(-0.424724\pi\)
0.234288 + 0.972167i \(0.424724\pi\)
\(774\) −81.2460 −2.92033
\(775\) 4.05944 0.145819
\(776\) −1.52946 −0.0549045
\(777\) −19.0872 −0.684749
\(778\) 39.1849 1.40485
\(779\) −3.42166 −0.122594
\(780\) −11.2544 −0.402973
\(781\) −27.0872 −0.969256
\(782\) 6.97939 0.249583
\(783\) −93.2898 −3.33391
\(784\) −4.91638 −0.175585
\(785\) −36.0922 −1.28819
\(786\) 50.9200 1.81625
\(787\) 46.2141 1.64736 0.823678 0.567058i \(-0.191918\pi\)
0.823678 + 0.567058i \(0.191918\pi\)
\(788\) −24.2650 −0.864404
\(789\) 47.0716 1.67579
\(790\) 69.1099 2.45882
\(791\) −5.44584 −0.193632
\(792\) 26.5089 0.941951
\(793\) −2.00000 −0.0710221
\(794\) 50.2283 1.78253
\(795\) 21.7789 0.772417
\(796\) 27.8610 0.987508
\(797\) 53.1155 1.88145 0.940723 0.339176i \(-0.110148\pi\)
0.940723 + 0.339176i \(0.110148\pi\)
\(798\) −4.57834 −0.162071
\(799\) 3.13104 0.110768
\(800\) 18.4842 0.653514
\(801\) −70.5558 −2.49297
\(802\) 4.67609 0.165118
\(803\) −7.27001 −0.256553
\(804\) −40.1533 −1.41610
\(805\) −20.6464 −0.727689
\(806\) −2.52444 −0.0889195
\(807\) −72.2721 −2.54410
\(808\) 16.8917 0.594247
\(809\) −54.4635 −1.91484 −0.957418 0.288705i \(-0.906775\pi\)
−0.957418 + 0.288705i \(0.906775\pi\)
\(810\) −76.7371 −2.69627
\(811\) 38.0978 1.33779 0.668897 0.743356i \(-0.266767\pi\)
0.668897 + 0.743356i \(0.266767\pi\)
\(812\) −10.6861 −0.375010
\(813\) −39.5466 −1.38696
\(814\) 34.6167 1.21331
\(815\) −37.8711 −1.32657
\(816\) −8.00000 −0.280056
\(817\) 5.49974 0.192412
\(818\) −27.4161 −0.958582
\(819\) −6.62721 −0.231574
\(820\) −15.2544 −0.532708
\(821\) −2.30330 −0.0803858 −0.0401929 0.999192i \(-0.512797\pi\)
−0.0401929 + 0.999192i \(0.512797\pi\)
\(822\) −35.2233 −1.22855
\(823\) 23.6172 0.823243 0.411621 0.911355i \(-0.364963\pi\)
0.411621 + 0.911355i \(0.364963\pi\)
\(824\) 5.68665 0.198104
\(825\) −28.0766 −0.977503
\(826\) −8.10780 −0.282106
\(827\) −48.1643 −1.67484 −0.837419 0.546562i \(-0.815936\pi\)
−0.837419 + 0.546562i \(0.815936\pi\)
\(828\) 62.6933 2.17874
\(829\) 13.0716 0.453996 0.226998 0.973895i \(-0.427109\pi\)
0.226998 + 0.973895i \(0.427109\pi\)
\(830\) −84.0893 −2.91878
\(831\) −25.2005 −0.874197
\(832\) −1.66196 −0.0576179
\(833\) −0.524438 −0.0181707
\(834\) −65.1255 −2.25511
\(835\) −5.71083 −0.197631
\(836\) 3.25443 0.112557
\(837\) −15.6655 −0.541480
\(838\) 18.1260 0.626153
\(839\) −17.6756 −0.610229 −0.305114 0.952316i \(-0.598695\pi\)
−0.305114 + 0.952316i \(0.598695\pi\)
\(840\) 11.2544 0.388315
\(841\) 39.7103 1.36932
\(842\) 47.0177 1.62034
\(843\) −59.0560 −2.03400
\(844\) −22.4011 −0.771076
\(845\) 2.81361 0.0967910
\(846\) 71.7577 2.46708
\(847\) 1.37279 0.0471695
\(848\) 12.2650 0.421181
\(849\) −34.5855 −1.18697
\(850\) 2.77384 0.0951420
\(851\) −45.1411 −1.54742
\(852\) 34.9200 1.19634
\(853\) −5.48970 −0.187964 −0.0939818 0.995574i \(-0.529960\pi\)
−0.0939818 + 0.995574i \(0.529960\pi\)
\(854\) −3.62721 −0.124121
\(855\) 15.1708 0.518831
\(856\) 0.745574 0.0254832
\(857\) −11.0489 −0.377422 −0.188711 0.982033i \(-0.560431\pi\)
−0.188711 + 0.982033i \(0.560431\pi\)
\(858\) 17.4600 0.596074
\(859\) 45.2616 1.54430 0.772152 0.635437i \(-0.219180\pi\)
0.772152 + 0.635437i \(0.219180\pi\)
\(860\) 24.5189 0.836087
\(861\) −13.0489 −0.444705
\(862\) −55.6344 −1.89491
\(863\) −5.90225 −0.200915 −0.100457 0.994941i \(-0.532031\pi\)
−0.100457 + 0.994941i \(0.532031\pi\)
\(864\) −71.3311 −2.42673
\(865\) 57.1099 1.94180
\(866\) 6.38283 0.216898
\(867\) 51.8938 1.76241
\(868\) −1.79445 −0.0609076
\(869\) −42.0227 −1.42552
\(870\) 131.240 4.44947
\(871\) 10.0383 0.340135
\(872\) 7.18494 0.243313
\(873\) −7.86248 −0.266105
\(874\) −10.8277 −0.366254
\(875\) 5.86248 0.198188
\(876\) 9.37227 0.316660
\(877\) −4.90727 −0.165707 −0.0828534 0.996562i \(-0.526403\pi\)
−0.0828534 + 0.996562i \(0.526403\pi\)
\(878\) −58.6732 −1.98012
\(879\) 43.9844 1.48356
\(880\) −42.9200 −1.44683
\(881\) 44.2822 1.49190 0.745952 0.665999i \(-0.231995\pi\)
0.745952 + 0.665999i \(0.231995\pi\)
\(882\) −12.0192 −0.404706
\(883\) −58.8605 −1.98081 −0.990407 0.138181i \(-0.955874\pi\)
−0.990407 + 0.138181i \(0.955874\pi\)
\(884\) −0.676089 −0.0227393
\(885\) 39.0278 1.31190
\(886\) −28.0127 −0.941104
\(887\) −10.1289 −0.340096 −0.170048 0.985436i \(-0.554392\pi\)
−0.170048 + 0.985436i \(0.554392\pi\)
\(888\) 24.6066 0.825744
\(889\) 12.8816 0.432036
\(890\) 54.3260 1.82101
\(891\) 46.6605 1.56319
\(892\) −13.5989 −0.455326
\(893\) −4.85746 −0.162549
\(894\) 47.9688 1.60432
\(895\) −30.9597 −1.03487
\(896\) 9.66196 0.322783
\(897\) −22.7683 −0.760211
\(898\) 26.2439 0.875769
\(899\) 11.5381 0.384816
\(900\) 24.9164 0.830546
\(901\) 1.30833 0.0435866
\(902\) 23.6655 0.787976
\(903\) 20.9739 0.697966
\(904\) 7.02061 0.233502
\(905\) 1.94610 0.0646906
\(906\) −67.4389 −2.24051
\(907\) 37.9547 1.26026 0.630132 0.776488i \(-0.283001\pi\)
0.630132 + 0.776488i \(0.283001\pi\)
\(908\) −8.96117 −0.297387
\(909\) 86.8349 2.88013
\(910\) 5.10278 0.169155
\(911\) 5.57477 0.184700 0.0923501 0.995727i \(-0.470562\pi\)
0.0923501 + 0.995727i \(0.470562\pi\)
\(912\) 12.4111 0.410973
\(913\) 51.1310 1.69219
\(914\) 62.8787 2.07984
\(915\) 17.4600 0.577209
\(916\) −27.1849 −0.898216
\(917\) −9.04888 −0.298820
\(918\) −10.7044 −0.353296
\(919\) 15.7844 0.520679 0.260340 0.965517i \(-0.416165\pi\)
0.260340 + 0.965517i \(0.416165\pi\)
\(920\) 26.6167 0.877525
\(921\) −42.0711 −1.38629
\(922\) −22.7427 −0.748990
\(923\) −8.72999 −0.287351
\(924\) 12.4111 0.408295
\(925\) −17.9406 −0.589882
\(926\) −22.0383 −0.724224
\(927\) 29.2333 0.960148
\(928\) 52.5371 1.72462
\(929\) −45.2630 −1.48503 −0.742516 0.669829i \(-0.766368\pi\)
−0.742516 + 0.669829i \(0.766368\pi\)
\(930\) 22.0383 0.722665
\(931\) 0.813607 0.0266649
\(932\) 7.84281 0.256900
\(933\) −1.32391 −0.0433429
\(934\) 67.1638 2.19767
\(935\) −4.57834 −0.149728
\(936\) 8.54359 0.279256
\(937\) 53.6188 1.75165 0.875824 0.482630i \(-0.160318\pi\)
0.875824 + 0.482630i \(0.160318\pi\)
\(938\) 18.2056 0.594432
\(939\) 56.3205 1.83795
\(940\) −21.6555 −0.706324
\(941\) 20.7753 0.677255 0.338628 0.940920i \(-0.390037\pi\)
0.338628 + 0.940920i \(0.390037\pi\)
\(942\) −72.1844 −2.35190
\(943\) −30.8605 −1.00496
\(944\) 21.9789 0.715351
\(945\) 31.6655 1.03008
\(946\) −38.0383 −1.23673
\(947\) −10.8605 −0.352919 −0.176460 0.984308i \(-0.556465\pi\)
−0.176460 + 0.984308i \(0.556465\pi\)
\(948\) 54.1744 1.75950
\(949\) −2.34307 −0.0760592
\(950\) −4.30330 −0.139618
\(951\) −29.2333 −0.947955
\(952\) 0.676089 0.0219122
\(953\) −25.7180 −0.833087 −0.416543 0.909116i \(-0.636759\pi\)
−0.416543 + 0.909116i \(0.636759\pi\)
\(954\) 29.9844 0.970781
\(955\) 22.0383 0.713143
\(956\) −18.3133 −0.592296
\(957\) −79.8016 −2.57962
\(958\) −21.7789 −0.703643
\(959\) 6.25945 0.202128
\(960\) 14.5089 0.468271
\(961\) −29.0625 −0.937500
\(962\) 11.1567 0.359706
\(963\) 3.83276 0.123509
\(964\) 10.8816 0.350474
\(965\) −34.3416 −1.10550
\(966\) −41.2927 −1.32857
\(967\) 33.5038 1.07741 0.538705 0.842494i \(-0.318914\pi\)
0.538705 + 0.842494i \(0.318914\pi\)
\(968\) −1.76975 −0.0568820
\(969\) 1.32391 0.0425302
\(970\) 6.05390 0.194379
\(971\) 2.03831 0.0654126 0.0327063 0.999465i \(-0.489587\pi\)
0.0327063 + 0.999465i \(0.489587\pi\)
\(972\) −16.6267 −0.533302
\(973\) 11.5733 0.371023
\(974\) −20.1643 −0.646107
\(975\) −9.04888 −0.289796
\(976\) 9.83276 0.314739
\(977\) 15.1411 0.484406 0.242203 0.970226i \(-0.422130\pi\)
0.242203 + 0.970226i \(0.422130\pi\)
\(978\) −75.7422 −2.42197
\(979\) −33.0333 −1.05575
\(980\) 3.62721 0.115867
\(981\) 36.9355 1.17926
\(982\) −0.107798 −0.00343998
\(983\) 49.3124 1.57282 0.786411 0.617704i \(-0.211937\pi\)
0.786411 + 0.617704i \(0.211937\pi\)
\(984\) 16.8222 0.536272
\(985\) −52.9583 −1.68739
\(986\) 7.88403 0.251079
\(987\) −18.5244 −0.589639
\(988\) 1.04888 0.0333692
\(989\) 49.6030 1.57728
\(990\) −104.927 −3.33480
\(991\) 5.43171 0.172544 0.0862720 0.996272i \(-0.472505\pi\)
0.0862720 + 0.996272i \(0.472505\pi\)
\(992\) 8.82220 0.280105
\(993\) 53.9900 1.71332
\(994\) −15.8328 −0.502185
\(995\) 60.8066 1.92770
\(996\) −65.9165 −2.08865
\(997\) −53.6061 −1.69772 −0.848861 0.528616i \(-0.822711\pi\)
−0.848861 + 0.528616i \(0.822711\pi\)
\(998\) −18.6761 −0.591181
\(999\) 69.2333 2.19044
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 91.2.a.d.1.1 3
3.2 odd 2 819.2.a.i.1.3 3
4.3 odd 2 1456.2.a.t.1.3 3
5.4 even 2 2275.2.a.m.1.3 3
7.2 even 3 637.2.e.j.508.3 6
7.3 odd 6 637.2.e.i.79.3 6
7.4 even 3 637.2.e.j.79.3 6
7.5 odd 6 637.2.e.i.508.3 6
7.6 odd 2 637.2.a.j.1.1 3
8.3 odd 2 5824.2.a.bs.1.1 3
8.5 even 2 5824.2.a.by.1.3 3
13.5 odd 4 1183.2.c.f.337.5 6
13.8 odd 4 1183.2.c.f.337.2 6
13.12 even 2 1183.2.a.i.1.3 3
21.20 even 2 5733.2.a.x.1.3 3
91.90 odd 2 8281.2.a.bg.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.a.d.1.1 3 1.1 even 1 trivial
637.2.a.j.1.1 3 7.6 odd 2
637.2.e.i.79.3 6 7.3 odd 6
637.2.e.i.508.3 6 7.5 odd 6
637.2.e.j.79.3 6 7.4 even 3
637.2.e.j.508.3 6 7.2 even 3
819.2.a.i.1.3 3 3.2 odd 2
1183.2.a.i.1.3 3 13.12 even 2
1183.2.c.f.337.2 6 13.8 odd 4
1183.2.c.f.337.5 6 13.5 odd 4
1456.2.a.t.1.3 3 4.3 odd 2
2275.2.a.m.1.3 3 5.4 even 2
5733.2.a.x.1.3 3 21.20 even 2
5824.2.a.bs.1.1 3 8.3 odd 2
5824.2.a.by.1.3 3 8.5 even 2
8281.2.a.bg.1.3 3 91.90 odd 2