Properties

Label 5390.2.a.ca.1.1
Level $5390$
Weight $2$
Character 5390.1
Self dual yes
Analytic conductor $43.039$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5390 = 2 \cdot 5 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5390.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.0393666895\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.81361\) of defining polynomial
Character \(\chi\) \(=\) 5390.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -3.10278 q^{3} +1.00000 q^{4} -1.00000 q^{5} -3.10278 q^{6} +1.00000 q^{8} +6.62721 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -3.10278 q^{3} +1.00000 q^{4} -1.00000 q^{5} -3.10278 q^{6} +1.00000 q^{8} +6.62721 q^{9} -1.00000 q^{10} +1.00000 q^{11} -3.10278 q^{12} +3.62721 q^{13} +3.10278 q^{15} +1.00000 q^{16} +4.20555 q^{17} +6.62721 q^{18} +8.15165 q^{19} -1.00000 q^{20} +1.00000 q^{22} +0.897225 q^{23} -3.10278 q^{24} +1.00000 q^{25} +3.62721 q^{26} -11.2544 q^{27} -7.30833 q^{29} +3.10278 q^{30} +3.42166 q^{31} +1.00000 q^{32} -3.10278 q^{33} +4.20555 q^{34} +6.62721 q^{36} -1.10278 q^{37} +8.15165 q^{38} -11.2544 q^{39} -1.00000 q^{40} -12.3572 q^{41} +10.6761 q^{43} +1.00000 q^{44} -6.62721 q^{45} +0.897225 q^{46} +1.15667 q^{47} -3.10278 q^{48} +1.00000 q^{50} -13.0489 q^{51} +3.62721 q^{52} +11.3083 q^{53} -11.2544 q^{54} -1.00000 q^{55} -25.2927 q^{57} -7.30833 q^{58} -7.25443 q^{59} +3.10278 q^{60} +3.15667 q^{61} +3.42166 q^{62} +1.00000 q^{64} -3.62721 q^{65} -3.10278 q^{66} -8.41110 q^{67} +4.20555 q^{68} -2.78389 q^{69} -13.0489 q^{71} +6.62721 q^{72} -0.205550 q^{73} -1.10278 q^{74} -3.10278 q^{75} +8.15165 q^{76} -11.2544 q^{78} +12.1517 q^{79} -1.00000 q^{80} +15.0383 q^{81} -12.3572 q^{82} -2.20555 q^{83} -4.20555 q^{85} +10.6761 q^{86} +22.6761 q^{87} +1.00000 q^{88} +7.45998 q^{89} -6.62721 q^{90} +0.897225 q^{92} -10.6167 q^{93} +1.15667 q^{94} -8.15165 q^{95} -3.10278 q^{96} -2.25945 q^{97} +6.62721 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} - 2q^{3} + 3q^{4} - 3q^{5} - 2q^{6} + 3q^{8} + 7q^{9} + O(q^{10}) \) \( 3q + 3q^{2} - 2q^{3} + 3q^{4} - 3q^{5} - 2q^{6} + 3q^{8} + 7q^{9} - 3q^{10} + 3q^{11} - 2q^{12} - 2q^{13} + 2q^{15} + 3q^{16} - 2q^{17} + 7q^{18} + 6q^{19} - 3q^{20} + 3q^{22} + 10q^{23} - 2q^{24} + 3q^{25} - 2q^{26} - 8q^{27} + 2q^{30} + 12q^{31} + 3q^{32} - 2q^{33} - 2q^{34} + 7q^{36} + 4q^{37} + 6q^{38} - 8q^{39} - 3q^{40} - 4q^{41} + 8q^{43} + 3q^{44} - 7q^{45} + 10q^{46} - 2q^{48} + 3q^{50} - 28q^{51} - 2q^{52} + 12q^{53} - 8q^{54} - 3q^{55} - 8q^{57} + 4q^{59} + 2q^{60} + 6q^{61} + 12q^{62} + 3q^{64} + 2q^{65} - 2q^{66} + 4q^{67} - 2q^{68} + 8q^{69} - 28q^{71} + 7q^{72} + 14q^{73} + 4q^{74} - 2q^{75} + 6q^{76} - 8q^{78} + 18q^{79} - 3q^{80} + 3q^{81} - 4q^{82} + 8q^{83} + 2q^{85} + 8q^{86} + 44q^{87} + 3q^{88} - 18q^{89} - 7q^{90} + 10q^{92} + 12q^{93} - 6q^{95} - 2q^{96} + 4q^{97} + 7q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −3.10278 −1.79139 −0.895694 0.444671i \(-0.853321\pi\)
−0.895694 + 0.444671i \(0.853321\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −3.10278 −1.26670
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 6.62721 2.20907
\(10\) −1.00000 −0.316228
\(11\) 1.00000 0.301511
\(12\) −3.10278 −0.895694
\(13\) 3.62721 1.00601 0.503004 0.864284i \(-0.332228\pi\)
0.503004 + 0.864284i \(0.332228\pi\)
\(14\) 0 0
\(15\) 3.10278 0.801133
\(16\) 1.00000 0.250000
\(17\) 4.20555 1.02000 0.509998 0.860176i \(-0.329646\pi\)
0.509998 + 0.860176i \(0.329646\pi\)
\(18\) 6.62721 1.56205
\(19\) 8.15165 1.87012 0.935058 0.354493i \(-0.115347\pi\)
0.935058 + 0.354493i \(0.115347\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 0.897225 0.187084 0.0935422 0.995615i \(-0.470181\pi\)
0.0935422 + 0.995615i \(0.470181\pi\)
\(24\) −3.10278 −0.633351
\(25\) 1.00000 0.200000
\(26\) 3.62721 0.711355
\(27\) −11.2544 −2.16592
\(28\) 0 0
\(29\) −7.30833 −1.35712 −0.678561 0.734544i \(-0.737396\pi\)
−0.678561 + 0.734544i \(0.737396\pi\)
\(30\) 3.10278 0.566487
\(31\) 3.42166 0.614549 0.307274 0.951621i \(-0.400583\pi\)
0.307274 + 0.951621i \(0.400583\pi\)
\(32\) 1.00000 0.176777
\(33\) −3.10278 −0.540124
\(34\) 4.20555 0.721246
\(35\) 0 0
\(36\) 6.62721 1.10454
\(37\) −1.10278 −0.181295 −0.0906476 0.995883i \(-0.528894\pi\)
−0.0906476 + 0.995883i \(0.528894\pi\)
\(38\) 8.15165 1.32237
\(39\) −11.2544 −1.80215
\(40\) −1.00000 −0.158114
\(41\) −12.3572 −1.92987 −0.964935 0.262488i \(-0.915457\pi\)
−0.964935 + 0.262488i \(0.915457\pi\)
\(42\) 0 0
\(43\) 10.6761 1.62809 0.814044 0.580803i \(-0.197261\pi\)
0.814044 + 0.580803i \(0.197261\pi\)
\(44\) 1.00000 0.150756
\(45\) −6.62721 −0.987927
\(46\) 0.897225 0.132289
\(47\) 1.15667 0.168718 0.0843591 0.996435i \(-0.473116\pi\)
0.0843591 + 0.996435i \(0.473116\pi\)
\(48\) −3.10278 −0.447847
\(49\) 0 0
\(50\) 1.00000 0.141421
\(51\) −13.0489 −1.82721
\(52\) 3.62721 0.503004
\(53\) 11.3083 1.55332 0.776659 0.629921i \(-0.216913\pi\)
0.776659 + 0.629921i \(0.216913\pi\)
\(54\) −11.2544 −1.53153
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) −25.2927 −3.35011
\(58\) −7.30833 −0.959630
\(59\) −7.25443 −0.944446 −0.472223 0.881479i \(-0.656548\pi\)
−0.472223 + 0.881479i \(0.656548\pi\)
\(60\) 3.10278 0.400567
\(61\) 3.15667 0.404171 0.202085 0.979368i \(-0.435228\pi\)
0.202085 + 0.979368i \(0.435228\pi\)
\(62\) 3.42166 0.434552
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −3.62721 −0.449900
\(66\) −3.10278 −0.381925
\(67\) −8.41110 −1.02758 −0.513790 0.857916i \(-0.671759\pi\)
−0.513790 + 0.857916i \(0.671759\pi\)
\(68\) 4.20555 0.509998
\(69\) −2.78389 −0.335141
\(70\) 0 0
\(71\) −13.0489 −1.54862 −0.774308 0.632809i \(-0.781902\pi\)
−0.774308 + 0.632809i \(0.781902\pi\)
\(72\) 6.62721 0.781025
\(73\) −0.205550 −0.0240578 −0.0120289 0.999928i \(-0.503829\pi\)
−0.0120289 + 0.999928i \(0.503829\pi\)
\(74\) −1.10278 −0.128195
\(75\) −3.10278 −0.358278
\(76\) 8.15165 0.935058
\(77\) 0 0
\(78\) −11.2544 −1.27431
\(79\) 12.1517 1.36717 0.683584 0.729872i \(-0.260420\pi\)
0.683584 + 0.729872i \(0.260420\pi\)
\(80\) −1.00000 −0.111803
\(81\) 15.0383 1.67092
\(82\) −12.3572 −1.36462
\(83\) −2.20555 −0.242091 −0.121045 0.992647i \(-0.538625\pi\)
−0.121045 + 0.992647i \(0.538625\pi\)
\(84\) 0 0
\(85\) −4.20555 −0.456156
\(86\) 10.6761 1.15123
\(87\) 22.6761 2.43113
\(88\) 1.00000 0.106600
\(89\) 7.45998 0.790756 0.395378 0.918519i \(-0.370614\pi\)
0.395378 + 0.918519i \(0.370614\pi\)
\(90\) −6.62721 −0.698570
\(91\) 0 0
\(92\) 0.897225 0.0935422
\(93\) −10.6167 −1.10090
\(94\) 1.15667 0.119302
\(95\) −8.15165 −0.836342
\(96\) −3.10278 −0.316676
\(97\) −2.25945 −0.229412 −0.114706 0.993399i \(-0.536593\pi\)
−0.114706 + 0.993399i \(0.536593\pi\)
\(98\) 0 0
\(99\) 6.62721 0.666060
\(100\) 1.00000 0.100000
\(101\) −13.6655 −1.35977 −0.679885 0.733318i \(-0.737970\pi\)
−0.679885 + 0.733318i \(0.737970\pi\)
\(102\) −13.0489 −1.29203
\(103\) 5.04888 0.497481 0.248740 0.968570i \(-0.419983\pi\)
0.248740 + 0.968570i \(0.419983\pi\)
\(104\) 3.62721 0.355677
\(105\) 0 0
\(106\) 11.3083 1.09836
\(107\) 12.9894 1.25574 0.627868 0.778320i \(-0.283928\pi\)
0.627868 + 0.778320i \(0.283928\pi\)
\(108\) −11.2544 −1.08296
\(109\) 5.10278 0.488757 0.244379 0.969680i \(-0.421416\pi\)
0.244379 + 0.969680i \(0.421416\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 3.42166 0.324770
\(112\) 0 0
\(113\) −10.4111 −0.979394 −0.489697 0.871893i \(-0.662893\pi\)
−0.489697 + 0.871893i \(0.662893\pi\)
\(114\) −25.2927 −2.36888
\(115\) −0.897225 −0.0836667
\(116\) −7.30833 −0.678561
\(117\) 24.0383 2.22234
\(118\) −7.25443 −0.667824
\(119\) 0 0
\(120\) 3.10278 0.283243
\(121\) 1.00000 0.0909091
\(122\) 3.15667 0.285792
\(123\) 38.3416 3.45715
\(124\) 3.42166 0.307274
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 3.25443 0.288784 0.144392 0.989521i \(-0.453877\pi\)
0.144392 + 0.989521i \(0.453877\pi\)
\(128\) 1.00000 0.0883883
\(129\) −33.1255 −2.91654
\(130\) −3.62721 −0.318128
\(131\) 17.3083 1.51224 0.756118 0.654436i \(-0.227094\pi\)
0.756118 + 0.654436i \(0.227094\pi\)
\(132\) −3.10278 −0.270062
\(133\) 0 0
\(134\) −8.41110 −0.726608
\(135\) 11.2544 0.968627
\(136\) 4.20555 0.360623
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) −2.78389 −0.236980
\(139\) −13.2005 −1.11965 −0.559827 0.828609i \(-0.689132\pi\)
−0.559827 + 0.828609i \(0.689132\pi\)
\(140\) 0 0
\(141\) −3.58890 −0.302240
\(142\) −13.0489 −1.09504
\(143\) 3.62721 0.303323
\(144\) 6.62721 0.552268
\(145\) 7.30833 0.606923
\(146\) −0.205550 −0.0170114
\(147\) 0 0
\(148\) −1.10278 −0.0906476
\(149\) −23.6116 −1.93434 −0.967170 0.254131i \(-0.918211\pi\)
−0.967170 + 0.254131i \(0.918211\pi\)
\(150\) −3.10278 −0.253341
\(151\) 11.5139 0.936986 0.468493 0.883467i \(-0.344797\pi\)
0.468493 + 0.883467i \(0.344797\pi\)
\(152\) 8.15165 0.661186
\(153\) 27.8711 2.25324
\(154\) 0 0
\(155\) −3.42166 −0.274835
\(156\) −11.2544 −0.901075
\(157\) 16.2056 1.29334 0.646672 0.762768i \(-0.276160\pi\)
0.646672 + 0.762768i \(0.276160\pi\)
\(158\) 12.1517 0.966733
\(159\) −35.0872 −2.78260
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 15.0383 1.18152
\(163\) 6.31335 0.494500 0.247250 0.968952i \(-0.420473\pi\)
0.247250 + 0.968952i \(0.420473\pi\)
\(164\) −12.3572 −0.964935
\(165\) 3.10278 0.241551
\(166\) −2.20555 −0.171184
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) 0.156674 0.0120519
\(170\) −4.20555 −0.322551
\(171\) 54.0227 4.13122
\(172\) 10.6761 0.814044
\(173\) −9.25443 −0.703601 −0.351800 0.936075i \(-0.614430\pi\)
−0.351800 + 0.936075i \(0.614430\pi\)
\(174\) 22.6761 1.71907
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) 22.5089 1.69187
\(178\) 7.45998 0.559149
\(179\) 4.98944 0.372928 0.186464 0.982462i \(-0.440297\pi\)
0.186464 + 0.982462i \(0.440297\pi\)
\(180\) −6.62721 −0.493963
\(181\) 9.66553 0.718433 0.359216 0.933254i \(-0.383044\pi\)
0.359216 + 0.933254i \(0.383044\pi\)
\(182\) 0 0
\(183\) −9.79445 −0.724027
\(184\) 0.897225 0.0661443
\(185\) 1.10278 0.0810776
\(186\) −10.6167 −0.778451
\(187\) 4.20555 0.307540
\(188\) 1.15667 0.0843591
\(189\) 0 0
\(190\) −8.15165 −0.591383
\(191\) −9.15667 −0.662554 −0.331277 0.943534i \(-0.607479\pi\)
−0.331277 + 0.943534i \(0.607479\pi\)
\(192\) −3.10278 −0.223924
\(193\) 1.52946 0.110093 0.0550465 0.998484i \(-0.482469\pi\)
0.0550465 + 0.998484i \(0.482469\pi\)
\(194\) −2.25945 −0.162219
\(195\) 11.2544 0.805946
\(196\) 0 0
\(197\) 14.3033 1.01907 0.509534 0.860451i \(-0.329818\pi\)
0.509534 + 0.860451i \(0.329818\pi\)
\(198\) 6.62721 0.470976
\(199\) 10.0978 0.715811 0.357905 0.933758i \(-0.383491\pi\)
0.357905 + 0.933758i \(0.383491\pi\)
\(200\) 1.00000 0.0707107
\(201\) 26.0978 1.84079
\(202\) −13.6655 −0.961503
\(203\) 0 0
\(204\) −13.0489 −0.913604
\(205\) 12.3572 0.863064
\(206\) 5.04888 0.351772
\(207\) 5.94610 0.413283
\(208\) 3.62721 0.251502
\(209\) 8.15165 0.563861
\(210\) 0 0
\(211\) 10.8433 0.746485 0.373243 0.927734i \(-0.378246\pi\)
0.373243 + 0.927734i \(0.378246\pi\)
\(212\) 11.3083 0.776659
\(213\) 40.4877 2.77417
\(214\) 12.9894 0.887940
\(215\) −10.6761 −0.728103
\(216\) −11.2544 −0.765767
\(217\) 0 0
\(218\) 5.10278 0.345604
\(219\) 0.637776 0.0430969
\(220\) −1.00000 −0.0674200
\(221\) 15.2544 1.02612
\(222\) 3.42166 0.229647
\(223\) 9.45998 0.633487 0.316743 0.948511i \(-0.397411\pi\)
0.316743 + 0.948511i \(0.397411\pi\)
\(224\) 0 0
\(225\) 6.62721 0.441814
\(226\) −10.4111 −0.692536
\(227\) 17.5678 1.16601 0.583007 0.812467i \(-0.301876\pi\)
0.583007 + 0.812467i \(0.301876\pi\)
\(228\) −25.2927 −1.67505
\(229\) −25.7250 −1.69995 −0.849977 0.526820i \(-0.823384\pi\)
−0.849977 + 0.526820i \(0.823384\pi\)
\(230\) −0.897225 −0.0591613
\(231\) 0 0
\(232\) −7.30833 −0.479815
\(233\) 19.1567 1.25500 0.627498 0.778618i \(-0.284079\pi\)
0.627498 + 0.778618i \(0.284079\pi\)
\(234\) 24.0383 1.57143
\(235\) −1.15667 −0.0754531
\(236\) −7.25443 −0.472223
\(237\) −37.7038 −2.44913
\(238\) 0 0
\(239\) −18.0539 −1.16781 −0.583905 0.811822i \(-0.698476\pi\)
−0.583905 + 0.811822i \(0.698476\pi\)
\(240\) 3.10278 0.200283
\(241\) −6.15165 −0.396263 −0.198131 0.980175i \(-0.563487\pi\)
−0.198131 + 0.980175i \(0.563487\pi\)
\(242\) 1.00000 0.0642824
\(243\) −12.8972 −0.827357
\(244\) 3.15667 0.202085
\(245\) 0 0
\(246\) 38.3416 2.44457
\(247\) 29.5678 1.88135
\(248\) 3.42166 0.217276
\(249\) 6.84333 0.433678
\(250\) −1.00000 −0.0632456
\(251\) 29.9789 1.89225 0.946125 0.323802i \(-0.104961\pi\)
0.946125 + 0.323802i \(0.104961\pi\)
\(252\) 0 0
\(253\) 0.897225 0.0564080
\(254\) 3.25443 0.204201
\(255\) 13.0489 0.817152
\(256\) 1.00000 0.0625000
\(257\) −4.05390 −0.252875 −0.126438 0.991975i \(-0.540354\pi\)
−0.126438 + 0.991975i \(0.540354\pi\)
\(258\) −33.1255 −2.06230
\(259\) 0 0
\(260\) −3.62721 −0.224950
\(261\) −48.4338 −2.99798
\(262\) 17.3083 1.06931
\(263\) −14.5089 −0.894654 −0.447327 0.894370i \(-0.647624\pi\)
−0.447327 + 0.894370i \(0.647624\pi\)
\(264\) −3.10278 −0.190963
\(265\) −11.3083 −0.694665
\(266\) 0 0
\(267\) −23.1466 −1.41655
\(268\) −8.41110 −0.513790
\(269\) −4.37279 −0.266614 −0.133307 0.991075i \(-0.542560\pi\)
−0.133307 + 0.991075i \(0.542560\pi\)
\(270\) 11.2544 0.684923
\(271\) 10.6167 0.644916 0.322458 0.946584i \(-0.395491\pi\)
0.322458 + 0.946584i \(0.395491\pi\)
\(272\) 4.20555 0.254999
\(273\) 0 0
\(274\) 10.0000 0.604122
\(275\) 1.00000 0.0603023
\(276\) −2.78389 −0.167570
\(277\) −19.7633 −1.18746 −0.593730 0.804664i \(-0.702345\pi\)
−0.593730 + 0.804664i \(0.702345\pi\)
\(278\) −13.2005 −0.791715
\(279\) 22.6761 1.35758
\(280\) 0 0
\(281\) −27.0489 −1.61360 −0.806800 0.590824i \(-0.798803\pi\)
−0.806800 + 0.590824i \(0.798803\pi\)
\(282\) −3.58890 −0.213716
\(283\) −27.6655 −1.64454 −0.822272 0.569094i \(-0.807294\pi\)
−0.822272 + 0.569094i \(0.807294\pi\)
\(284\) −13.0489 −0.774308
\(285\) 25.2927 1.49821
\(286\) 3.62721 0.214482
\(287\) 0 0
\(288\) 6.62721 0.390512
\(289\) 0.686652 0.0403913
\(290\) 7.30833 0.429160
\(291\) 7.01056 0.410966
\(292\) −0.205550 −0.0120289
\(293\) 8.37279 0.489143 0.244572 0.969631i \(-0.421353\pi\)
0.244572 + 0.969631i \(0.421353\pi\)
\(294\) 0 0
\(295\) 7.25443 0.422369
\(296\) −1.10278 −0.0640975
\(297\) −11.2544 −0.653048
\(298\) −23.6116 −1.36778
\(299\) 3.25443 0.188208
\(300\) −3.10278 −0.179139
\(301\) 0 0
\(302\) 11.5139 0.662549
\(303\) 42.4011 2.43588
\(304\) 8.15165 0.467529
\(305\) −3.15667 −0.180751
\(306\) 27.8711 1.59328
\(307\) 10.2056 0.582462 0.291231 0.956653i \(-0.405935\pi\)
0.291231 + 0.956653i \(0.405935\pi\)
\(308\) 0 0
\(309\) −15.6655 −0.891181
\(310\) −3.42166 −0.194337
\(311\) 8.16724 0.463122 0.231561 0.972820i \(-0.425617\pi\)
0.231561 + 0.972820i \(0.425617\pi\)
\(312\) −11.2544 −0.637156
\(313\) −34.5628 −1.95360 −0.976801 0.214149i \(-0.931302\pi\)
−0.976801 + 0.214149i \(0.931302\pi\)
\(314\) 16.2056 0.914532
\(315\) 0 0
\(316\) 12.1517 0.683584
\(317\) 9.21057 0.517317 0.258659 0.965969i \(-0.416720\pi\)
0.258659 + 0.965969i \(0.416720\pi\)
\(318\) −35.0872 −1.96759
\(319\) −7.30833 −0.409188
\(320\) −1.00000 −0.0559017
\(321\) −40.3033 −2.24951
\(322\) 0 0
\(323\) 34.2822 1.90751
\(324\) 15.0383 0.835462
\(325\) 3.62721 0.201202
\(326\) 6.31335 0.349664
\(327\) −15.8328 −0.875554
\(328\) −12.3572 −0.682312
\(329\) 0 0
\(330\) 3.10278 0.170802
\(331\) 16.2439 0.892843 0.446422 0.894823i \(-0.352698\pi\)
0.446422 + 0.894823i \(0.352698\pi\)
\(332\) −2.20555 −0.121045
\(333\) −7.30833 −0.400494
\(334\) 8.00000 0.437741
\(335\) 8.41110 0.459547
\(336\) 0 0
\(337\) −1.25443 −0.0683329 −0.0341665 0.999416i \(-0.510878\pi\)
−0.0341665 + 0.999416i \(0.510878\pi\)
\(338\) 0.156674 0.00852195
\(339\) 32.3033 1.75447
\(340\) −4.20555 −0.228078
\(341\) 3.42166 0.185293
\(342\) 54.0227 2.92121
\(343\) 0 0
\(344\) 10.6761 0.575616
\(345\) 2.78389 0.149879
\(346\) −9.25443 −0.497521
\(347\) 7.42166 0.398416 0.199208 0.979957i \(-0.436163\pi\)
0.199208 + 0.979957i \(0.436163\pi\)
\(348\) 22.6761 1.21557
\(349\) 18.3033 0.979753 0.489877 0.871792i \(-0.337042\pi\)
0.489877 + 0.871792i \(0.337042\pi\)
\(350\) 0 0
\(351\) −40.8222 −2.17893
\(352\) 1.00000 0.0533002
\(353\) −28.8761 −1.53692 −0.768460 0.639898i \(-0.778977\pi\)
−0.768460 + 0.639898i \(0.778977\pi\)
\(354\) 22.5089 1.19633
\(355\) 13.0489 0.692562
\(356\) 7.45998 0.395378
\(357\) 0 0
\(358\) 4.98944 0.263700
\(359\) 4.15165 0.219116 0.109558 0.993980i \(-0.465057\pi\)
0.109558 + 0.993980i \(0.465057\pi\)
\(360\) −6.62721 −0.349285
\(361\) 47.4494 2.49734
\(362\) 9.66553 0.508009
\(363\) −3.10278 −0.162853
\(364\) 0 0
\(365\) 0.205550 0.0107590
\(366\) −9.79445 −0.511964
\(367\) 5.26447 0.274803 0.137402 0.990515i \(-0.456125\pi\)
0.137402 + 0.990515i \(0.456125\pi\)
\(368\) 0.897225 0.0467711
\(369\) −81.8938 −4.26322
\(370\) 1.10278 0.0573305
\(371\) 0 0
\(372\) −10.6167 −0.550448
\(373\) 35.8711 1.85733 0.928667 0.370915i \(-0.120956\pi\)
0.928667 + 0.370915i \(0.120956\pi\)
\(374\) 4.20555 0.217464
\(375\) 3.10278 0.160227
\(376\) 1.15667 0.0596509
\(377\) −26.5089 −1.36528
\(378\) 0 0
\(379\) −33.0177 −1.69601 −0.848003 0.529992i \(-0.822195\pi\)
−0.848003 + 0.529992i \(0.822195\pi\)
\(380\) −8.15165 −0.418171
\(381\) −10.0978 −0.517323
\(382\) −9.15667 −0.468496
\(383\) −2.95112 −0.150795 −0.0753977 0.997154i \(-0.524023\pi\)
−0.0753977 + 0.997154i \(0.524023\pi\)
\(384\) −3.10278 −0.158338
\(385\) 0 0
\(386\) 1.52946 0.0778475
\(387\) 70.7527 3.59656
\(388\) −2.25945 −0.114706
\(389\) 1.89220 0.0959384 0.0479692 0.998849i \(-0.484725\pi\)
0.0479692 + 0.998849i \(0.484725\pi\)
\(390\) 11.2544 0.569890
\(391\) 3.77332 0.190825
\(392\) 0 0
\(393\) −53.7038 −2.70900
\(394\) 14.3033 0.720590
\(395\) −12.1517 −0.611416
\(396\) 6.62721 0.333030
\(397\) −5.36222 −0.269122 −0.134561 0.990905i \(-0.542962\pi\)
−0.134561 + 0.990905i \(0.542962\pi\)
\(398\) 10.0978 0.506155
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 23.7038 1.18371 0.591857 0.806043i \(-0.298395\pi\)
0.591857 + 0.806043i \(0.298395\pi\)
\(402\) 26.0978 1.30164
\(403\) 12.4111 0.618241
\(404\) −13.6655 −0.679885
\(405\) −15.0383 −0.747260
\(406\) 0 0
\(407\) −1.10278 −0.0546625
\(408\) −13.0489 −0.646016
\(409\) −27.0816 −1.33910 −0.669551 0.742766i \(-0.733513\pi\)
−0.669551 + 0.742766i \(0.733513\pi\)
\(410\) 12.3572 0.610279
\(411\) −31.0278 −1.53049
\(412\) 5.04888 0.248740
\(413\) 0 0
\(414\) 5.94610 0.292235
\(415\) 2.20555 0.108266
\(416\) 3.62721 0.177839
\(417\) 40.9583 2.00573
\(418\) 8.15165 0.398710
\(419\) 3.05892 0.149438 0.0747191 0.997205i \(-0.476194\pi\)
0.0747191 + 0.997205i \(0.476194\pi\)
\(420\) 0 0
\(421\) 21.6655 1.05591 0.527957 0.849271i \(-0.322958\pi\)
0.527957 + 0.849271i \(0.322958\pi\)
\(422\) 10.8433 0.527845
\(423\) 7.66553 0.372711
\(424\) 11.3083 0.549181
\(425\) 4.20555 0.203999
\(426\) 40.4877 1.96164
\(427\) 0 0
\(428\) 12.9894 0.627868
\(429\) −11.2544 −0.543369
\(430\) −10.6761 −0.514847
\(431\) −34.3260 −1.65343 −0.826713 0.562623i \(-0.809792\pi\)
−0.826713 + 0.562623i \(0.809792\pi\)
\(432\) −11.2544 −0.541479
\(433\) 0.0538991 0.00259023 0.00129511 0.999999i \(-0.499588\pi\)
0.00129511 + 0.999999i \(0.499588\pi\)
\(434\) 0 0
\(435\) −22.6761 −1.08724
\(436\) 5.10278 0.244379
\(437\) 7.31386 0.349870
\(438\) 0.637776 0.0304741
\(439\) 7.14663 0.341090 0.170545 0.985350i \(-0.445447\pi\)
0.170545 + 0.985350i \(0.445447\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 0 0
\(442\) 15.2544 0.725579
\(443\) 8.74557 0.415515 0.207757 0.978180i \(-0.433384\pi\)
0.207757 + 0.978180i \(0.433384\pi\)
\(444\) 3.42166 0.162385
\(445\) −7.45998 −0.353637
\(446\) 9.45998 0.447943
\(447\) 73.2616 3.46515
\(448\) 0 0
\(449\) 39.7038 1.87374 0.936870 0.349678i \(-0.113709\pi\)
0.936870 + 0.349678i \(0.113709\pi\)
\(450\) 6.62721 0.312410
\(451\) −12.3572 −0.581878
\(452\) −10.4111 −0.489697
\(453\) −35.7250 −1.67851
\(454\) 17.5678 0.824497
\(455\) 0 0
\(456\) −25.2927 −1.18444
\(457\) 12.3133 0.575994 0.287997 0.957631i \(-0.407011\pi\)
0.287997 + 0.957631i \(0.407011\pi\)
\(458\) −25.7250 −1.20205
\(459\) −47.3311 −2.20922
\(460\) −0.897225 −0.0418333
\(461\) 15.0489 0.700896 0.350448 0.936582i \(-0.386029\pi\)
0.350448 + 0.936582i \(0.386029\pi\)
\(462\) 0 0
\(463\) 31.4061 1.45956 0.729782 0.683680i \(-0.239622\pi\)
0.729782 + 0.683680i \(0.239622\pi\)
\(464\) −7.30833 −0.339280
\(465\) 10.6167 0.492335
\(466\) 19.1567 0.887416
\(467\) 19.0716 0.882529 0.441264 0.897377i \(-0.354530\pi\)
0.441264 + 0.897377i \(0.354530\pi\)
\(468\) 24.0383 1.11117
\(469\) 0 0
\(470\) −1.15667 −0.0533534
\(471\) −50.2822 −2.31688
\(472\) −7.25443 −0.333912
\(473\) 10.6761 0.490887
\(474\) −37.7038 −1.73179
\(475\) 8.15165 0.374023
\(476\) 0 0
\(477\) 74.9427 3.43139
\(478\) −18.0539 −0.825766
\(479\) 9.49115 0.433662 0.216831 0.976209i \(-0.430428\pi\)
0.216831 + 0.976209i \(0.430428\pi\)
\(480\) 3.10278 0.141622
\(481\) −4.00000 −0.182384
\(482\) −6.15165 −0.280200
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 2.25945 0.102596
\(486\) −12.8972 −0.585030
\(487\) 0.0438527 0.00198715 0.000993577 1.00000i \(-0.499684\pi\)
0.000993577 1.00000i \(0.499684\pi\)
\(488\) 3.15667 0.142896
\(489\) −19.5889 −0.885841
\(490\) 0 0
\(491\) 1.56777 0.0707527 0.0353763 0.999374i \(-0.488737\pi\)
0.0353763 + 0.999374i \(0.488737\pi\)
\(492\) 38.3416 1.72857
\(493\) −30.7355 −1.38426
\(494\) 29.5678 1.33032
\(495\) −6.62721 −0.297871
\(496\) 3.42166 0.153637
\(497\) 0 0
\(498\) 6.84333 0.306657
\(499\) −42.3416 −1.89547 −0.947736 0.319057i \(-0.896634\pi\)
−0.947736 + 0.319057i \(0.896634\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −24.8222 −1.10897
\(502\) 29.9789 1.33802
\(503\) 36.4111 1.62349 0.811745 0.584012i \(-0.198518\pi\)
0.811745 + 0.584012i \(0.198518\pi\)
\(504\) 0 0
\(505\) 13.6655 0.608108
\(506\) 0.897225 0.0398865
\(507\) −0.486125 −0.0215896
\(508\) 3.25443 0.144392
\(509\) 30.0766 1.33312 0.666562 0.745450i \(-0.267765\pi\)
0.666562 + 0.745450i \(0.267765\pi\)
\(510\) 13.0489 0.577814
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −91.7422 −4.05051
\(514\) −4.05390 −0.178810
\(515\) −5.04888 −0.222480
\(516\) −33.1255 −1.45827
\(517\) 1.15667 0.0508705
\(518\) 0 0
\(519\) 28.7144 1.26042
\(520\) −3.62721 −0.159064
\(521\) 33.2233 1.45554 0.727769 0.685823i \(-0.240557\pi\)
0.727769 + 0.685823i \(0.240557\pi\)
\(522\) −48.4338 −2.11989
\(523\) 15.5577 0.680292 0.340146 0.940373i \(-0.389524\pi\)
0.340146 + 0.940373i \(0.389524\pi\)
\(524\) 17.3083 0.756118
\(525\) 0 0
\(526\) −14.5089 −0.632616
\(527\) 14.3900 0.626837
\(528\) −3.10278 −0.135031
\(529\) −22.1950 −0.964999
\(530\) −11.3083 −0.491202
\(531\) −48.0766 −2.08635
\(532\) 0 0
\(533\) −44.8222 −1.94147
\(534\) −23.1466 −1.00165
\(535\) −12.9894 −0.561582
\(536\) −8.41110 −0.363304
\(537\) −15.4811 −0.668059
\(538\) −4.37279 −0.188524
\(539\) 0 0
\(540\) 11.2544 0.484313
\(541\) 23.4161 1.00674 0.503369 0.864072i \(-0.332094\pi\)
0.503369 + 0.864072i \(0.332094\pi\)
\(542\) 10.6167 0.456024
\(543\) −29.9900 −1.28699
\(544\) 4.20555 0.180311
\(545\) −5.10278 −0.218579
\(546\) 0 0
\(547\) −8.41110 −0.359633 −0.179816 0.983700i \(-0.557550\pi\)
−0.179816 + 0.983700i \(0.557550\pi\)
\(548\) 10.0000 0.427179
\(549\) 20.9200 0.892842
\(550\) 1.00000 0.0426401
\(551\) −59.5749 −2.53798
\(552\) −2.78389 −0.118490
\(553\) 0 0
\(554\) −19.7633 −0.839661
\(555\) −3.42166 −0.145242
\(556\) −13.2005 −0.559827
\(557\) 37.0278 1.56892 0.784458 0.620182i \(-0.212941\pi\)
0.784458 + 0.620182i \(0.212941\pi\)
\(558\) 22.6761 0.959955
\(559\) 38.7244 1.63787
\(560\) 0 0
\(561\) −13.0489 −0.550924
\(562\) −27.0489 −1.14099
\(563\) −22.9200 −0.965961 −0.482980 0.875631i \(-0.660446\pi\)
−0.482980 + 0.875631i \(0.660446\pi\)
\(564\) −3.58890 −0.151120
\(565\) 10.4111 0.437998
\(566\) −27.6655 −1.16287
\(567\) 0 0
\(568\) −13.0489 −0.547519
\(569\) −22.0000 −0.922288 −0.461144 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) 25.2927 1.05940
\(571\) 20.3033 0.849667 0.424833 0.905272i \(-0.360333\pi\)
0.424833 + 0.905272i \(0.360333\pi\)
\(572\) 3.62721 0.151661
\(573\) 28.4111 1.18689
\(574\) 0 0
\(575\) 0.897225 0.0374169
\(576\) 6.62721 0.276134
\(577\) −16.4650 −0.685447 −0.342723 0.939436i \(-0.611349\pi\)
−0.342723 + 0.939436i \(0.611349\pi\)
\(578\) 0.686652 0.0285609
\(579\) −4.74557 −0.197219
\(580\) 7.30833 0.303462
\(581\) 0 0
\(582\) 7.01056 0.290597
\(583\) 11.3083 0.468343
\(584\) −0.205550 −0.00850572
\(585\) −24.0383 −0.993862
\(586\) 8.37279 0.345877
\(587\) 30.6605 1.26549 0.632747 0.774358i \(-0.281927\pi\)
0.632747 + 0.774358i \(0.281927\pi\)
\(588\) 0 0
\(589\) 27.8922 1.14928
\(590\) 7.25443 0.298660
\(591\) −44.3799 −1.82555
\(592\) −1.10278 −0.0453238
\(593\) −18.3033 −0.751627 −0.375813 0.926695i \(-0.622637\pi\)
−0.375813 + 0.926695i \(0.622637\pi\)
\(594\) −11.2544 −0.461775
\(595\) 0 0
\(596\) −23.6116 −0.967170
\(597\) −31.3311 −1.28229
\(598\) 3.25443 0.133083
\(599\) 29.6555 1.21169 0.605845 0.795583i \(-0.292835\pi\)
0.605845 + 0.795583i \(0.292835\pi\)
\(600\) −3.10278 −0.126670
\(601\) −18.7783 −0.765985 −0.382992 0.923751i \(-0.625106\pi\)
−0.382992 + 0.923751i \(0.625106\pi\)
\(602\) 0 0
\(603\) −55.7422 −2.27000
\(604\) 11.5139 0.468493
\(605\) −1.00000 −0.0406558
\(606\) 42.4011 1.72243
\(607\) −14.6761 −0.595684 −0.297842 0.954615i \(-0.596267\pi\)
−0.297842 + 0.954615i \(0.596267\pi\)
\(608\) 8.15165 0.330593
\(609\) 0 0
\(610\) −3.15667 −0.127810
\(611\) 4.19550 0.169732
\(612\) 27.8711 1.12662
\(613\) −5.28560 −0.213483 −0.106742 0.994287i \(-0.534042\pi\)
−0.106742 + 0.994287i \(0.534042\pi\)
\(614\) 10.2056 0.411862
\(615\) −38.3416 −1.54608
\(616\) 0 0
\(617\) −44.6933 −1.79928 −0.899642 0.436629i \(-0.856172\pi\)
−0.899642 + 0.436629i \(0.856172\pi\)
\(618\) −15.6655 −0.630160
\(619\) 17.5678 0.706108 0.353054 0.935603i \(-0.385143\pi\)
0.353054 + 0.935603i \(0.385143\pi\)
\(620\) −3.42166 −0.137417
\(621\) −10.0978 −0.405209
\(622\) 8.16724 0.327476
\(623\) 0 0
\(624\) −11.2544 −0.450538
\(625\) 1.00000 0.0400000
\(626\) −34.5628 −1.38141
\(627\) −25.2927 −1.01009
\(628\) 16.2056 0.646672
\(629\) −4.63778 −0.184920
\(630\) 0 0
\(631\) 4.74557 0.188918 0.0944592 0.995529i \(-0.469888\pi\)
0.0944592 + 0.995529i \(0.469888\pi\)
\(632\) 12.1517 0.483367
\(633\) −33.6444 −1.33724
\(634\) 9.21057 0.365799
\(635\) −3.25443 −0.129148
\(636\) −35.0872 −1.39130
\(637\) 0 0
\(638\) −7.30833 −0.289339
\(639\) −86.4777 −3.42100
\(640\) −1.00000 −0.0395285
\(641\) −2.52998 −0.0999281 −0.0499641 0.998751i \(-0.515911\pi\)
−0.0499641 + 0.998751i \(0.515911\pi\)
\(642\) −40.3033 −1.59064
\(643\) 0.670549 0.0264439 0.0132219 0.999913i \(-0.495791\pi\)
0.0132219 + 0.999913i \(0.495791\pi\)
\(644\) 0 0
\(645\) 33.1255 1.30432
\(646\) 34.2822 1.34881
\(647\) −11.8922 −0.467531 −0.233765 0.972293i \(-0.575105\pi\)
−0.233765 + 0.972293i \(0.575105\pi\)
\(648\) 15.0383 0.590761
\(649\) −7.25443 −0.284761
\(650\) 3.62721 0.142271
\(651\) 0 0
\(652\) 6.31335 0.247250
\(653\) −11.4161 −0.446747 −0.223374 0.974733i \(-0.571707\pi\)
−0.223374 + 0.974733i \(0.571707\pi\)
\(654\) −15.8328 −0.619110
\(655\) −17.3083 −0.676292
\(656\) −12.3572 −0.482468
\(657\) −1.36222 −0.0531454
\(658\) 0 0
\(659\) 1.87108 0.0728868 0.0364434 0.999336i \(-0.488397\pi\)
0.0364434 + 0.999336i \(0.488397\pi\)
\(660\) 3.10278 0.120775
\(661\) −33.1355 −1.28882 −0.644412 0.764679i \(-0.722898\pi\)
−0.644412 + 0.764679i \(0.722898\pi\)
\(662\) 16.2439 0.631336
\(663\) −47.3311 −1.83819
\(664\) −2.20555 −0.0855919
\(665\) 0 0
\(666\) −7.30833 −0.283192
\(667\) −6.55721 −0.253896
\(668\) 8.00000 0.309529
\(669\) −29.3522 −1.13482
\(670\) 8.41110 0.324949
\(671\) 3.15667 0.121862
\(672\) 0 0
\(673\) −18.1955 −0.701385 −0.350693 0.936491i \(-0.614054\pi\)
−0.350693 + 0.936491i \(0.614054\pi\)
\(674\) −1.25443 −0.0483187
\(675\) −11.2544 −0.433183
\(676\) 0.156674 0.00602593
\(677\) −15.9789 −0.614118 −0.307059 0.951690i \(-0.599345\pi\)
−0.307059 + 0.951690i \(0.599345\pi\)
\(678\) 32.3033 1.24060
\(679\) 0 0
\(680\) −4.20555 −0.161275
\(681\) −54.5089 −2.08878
\(682\) 3.42166 0.131022
\(683\) −17.8711 −0.683818 −0.341909 0.939733i \(-0.611073\pi\)
−0.341909 + 0.939733i \(0.611073\pi\)
\(684\) 54.0227 2.06561
\(685\) −10.0000 −0.382080
\(686\) 0 0
\(687\) 79.8188 3.04528
\(688\) 10.6761 0.407022
\(689\) 41.0177 1.56265
\(690\) 2.78389 0.105981
\(691\) 2.20555 0.0839031 0.0419515 0.999120i \(-0.486642\pi\)
0.0419515 + 0.999120i \(0.486642\pi\)
\(692\) −9.25443 −0.351800
\(693\) 0 0
\(694\) 7.42166 0.281722
\(695\) 13.2005 0.500725
\(696\) 22.6761 0.859535
\(697\) −51.9688 −1.96846
\(698\) 18.3033 0.692790
\(699\) −59.4389 −2.24818
\(700\) 0 0
\(701\) −25.6217 −0.967717 −0.483859 0.875146i \(-0.660765\pi\)
−0.483859 + 0.875146i \(0.660765\pi\)
\(702\) −40.8222 −1.54073
\(703\) −8.98944 −0.339043
\(704\) 1.00000 0.0376889
\(705\) 3.58890 0.135166
\(706\) −28.8761 −1.08677
\(707\) 0 0
\(708\) 22.5089 0.845934
\(709\) 5.48110 0.205847 0.102924 0.994689i \(-0.467180\pi\)
0.102924 + 0.994689i \(0.467180\pi\)
\(710\) 13.0489 0.489716
\(711\) 80.5316 3.02017
\(712\) 7.45998 0.279574
\(713\) 3.07000 0.114972
\(714\) 0 0
\(715\) −3.62721 −0.135650
\(716\) 4.98944 0.186464
\(717\) 56.0172 2.09200
\(718\) 4.15165 0.154938
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) −6.62721 −0.246982
\(721\) 0 0
\(722\) 47.4494 1.76588
\(723\) 19.0872 0.709860
\(724\) 9.66553 0.359216
\(725\) −7.30833 −0.271424
\(726\) −3.10278 −0.115155
\(727\) −18.5855 −0.689297 −0.344649 0.938732i \(-0.612002\pi\)
−0.344649 + 0.938732i \(0.612002\pi\)
\(728\) 0 0
\(729\) −5.09775 −0.188806
\(730\) 0.205550 0.00760775
\(731\) 44.8988 1.66064
\(732\) −9.79445 −0.362013
\(733\) 42.0000 1.55131 0.775653 0.631160i \(-0.217421\pi\)
0.775653 + 0.631160i \(0.217421\pi\)
\(734\) 5.26447 0.194315
\(735\) 0 0
\(736\) 0.897225 0.0330722
\(737\) −8.41110 −0.309827
\(738\) −81.8938 −3.01455
\(739\) 5.97887 0.219936 0.109968 0.993935i \(-0.464925\pi\)
0.109968 + 0.993935i \(0.464925\pi\)
\(740\) 1.10278 0.0405388
\(741\) −91.7422 −3.37023
\(742\) 0 0
\(743\) −50.0978 −1.83791 −0.918954 0.394364i \(-0.870965\pi\)
−0.918954 + 0.394364i \(0.870965\pi\)
\(744\) −10.6167 −0.389225
\(745\) 23.6116 0.865063
\(746\) 35.8711 1.31333
\(747\) −14.6167 −0.534795
\(748\) 4.20555 0.153770
\(749\) 0 0
\(750\) 3.10278 0.113297
\(751\) −8.63778 −0.315197 −0.157598 0.987503i \(-0.550375\pi\)
−0.157598 + 0.987503i \(0.550375\pi\)
\(752\) 1.15667 0.0421796
\(753\) −93.0177 −3.38975
\(754\) −26.5089 −0.965395
\(755\) −11.5139 −0.419033
\(756\) 0 0
\(757\) 25.5139 0.927318 0.463659 0.886014i \(-0.346536\pi\)
0.463659 + 0.886014i \(0.346536\pi\)
\(758\) −33.0177 −1.19926
\(759\) −2.78389 −0.101049
\(760\) −8.15165 −0.295691
\(761\) 32.6605 1.18394 0.591971 0.805959i \(-0.298350\pi\)
0.591971 + 0.805959i \(0.298350\pi\)
\(762\) −10.0978 −0.365803
\(763\) 0 0
\(764\) −9.15667 −0.331277
\(765\) −27.8711 −1.00768
\(766\) −2.95112 −0.106628
\(767\) −26.3133 −0.950120
\(768\) −3.10278 −0.111962
\(769\) −16.7683 −0.604680 −0.302340 0.953200i \(-0.597768\pi\)
−0.302340 + 0.953200i \(0.597768\pi\)
\(770\) 0 0
\(771\) 12.5783 0.452998
\(772\) 1.52946 0.0550465
\(773\) 2.82220 0.101507 0.0507537 0.998711i \(-0.483838\pi\)
0.0507537 + 0.998711i \(0.483838\pi\)
\(774\) 70.7527 2.54315
\(775\) 3.42166 0.122910
\(776\) −2.25945 −0.0811095
\(777\) 0 0
\(778\) 1.89220 0.0678387
\(779\) −100.732 −3.60908
\(780\) 11.2544 0.402973
\(781\) −13.0489 −0.466925
\(782\) 3.77332 0.134934
\(783\) 82.2510 2.93941
\(784\) 0 0
\(785\) −16.2056 −0.578401
\(786\) −53.7038 −1.91555
\(787\) 4.82220 0.171893 0.0859464 0.996300i \(-0.472609\pi\)
0.0859464 + 0.996300i \(0.472609\pi\)
\(788\) 14.3033 0.509534
\(789\) 45.0177 1.60267
\(790\) −12.1517 −0.432336
\(791\) 0 0
\(792\) 6.62721 0.235488
\(793\) 11.4499 0.406599
\(794\) −5.36222 −0.190298
\(795\) 35.0872 1.24441
\(796\) 10.0978 0.357905
\(797\) 0.540024 0.0191286 0.00956431 0.999954i \(-0.496956\pi\)
0.00956431 + 0.999954i \(0.496956\pi\)
\(798\) 0 0
\(799\) 4.86445 0.172092
\(800\) 1.00000 0.0353553
\(801\) 49.4389 1.74684
\(802\) 23.7038 0.837012
\(803\) −0.205550 −0.00725371
\(804\) 26.0978 0.920397
\(805\) 0 0
\(806\) 12.4111 0.437162
\(807\) 13.5678 0.477608
\(808\) −13.6655 −0.480752
\(809\) −17.8922 −0.629056 −0.314528 0.949248i \(-0.601846\pi\)
−0.314528 + 0.949248i \(0.601846\pi\)
\(810\) −15.0383 −0.528392
\(811\) 12.2283 0.429393 0.214696 0.976681i \(-0.431124\pi\)
0.214696 + 0.976681i \(0.431124\pi\)
\(812\) 0 0
\(813\) −32.9411 −1.15529
\(814\) −1.10278 −0.0386522
\(815\) −6.31335 −0.221147
\(816\) −13.0489 −0.456802
\(817\) 87.0278 3.04472
\(818\) −27.0816 −0.946888
\(819\) 0 0
\(820\) 12.3572 0.431532
\(821\) 38.5316 1.34476 0.672381 0.740206i \(-0.265272\pi\)
0.672381 + 0.740206i \(0.265272\pi\)
\(822\) −31.0278 −1.08222
\(823\) 21.9461 0.764993 0.382496 0.923957i \(-0.375064\pi\)
0.382496 + 0.923957i \(0.375064\pi\)
\(824\) 5.04888 0.175886
\(825\) −3.10278 −0.108025
\(826\) 0 0
\(827\) 9.68665 0.336838 0.168419 0.985716i \(-0.446134\pi\)
0.168419 + 0.985716i \(0.446134\pi\)
\(828\) 5.94610 0.206641
\(829\) −31.0771 −1.07935 −0.539677 0.841872i \(-0.681454\pi\)
−0.539677 + 0.841872i \(0.681454\pi\)
\(830\) 2.20555 0.0765558
\(831\) 61.3210 2.12720
\(832\) 3.62721 0.125751
\(833\) 0 0
\(834\) 40.9583 1.41827
\(835\) −8.00000 −0.276851
\(836\) 8.15165 0.281931
\(837\) −38.5089 −1.33106
\(838\) 3.05892 0.105669
\(839\) 24.7738 0.855288 0.427644 0.903947i \(-0.359344\pi\)
0.427644 + 0.903947i \(0.359344\pi\)
\(840\) 0 0
\(841\) 24.4116 0.841780
\(842\) 21.6655 0.746643
\(843\) 83.9266 2.89058
\(844\) 10.8433 0.373243
\(845\) −0.156674 −0.00538976
\(846\) 7.66553 0.263546
\(847\) 0 0
\(848\) 11.3083 0.388329
\(849\) 85.8399 2.94602
\(850\) 4.20555 0.144249
\(851\) −0.989437 −0.0339175
\(852\) 40.4877 1.38709
\(853\) −1.58890 −0.0544029 −0.0272014 0.999630i \(-0.508660\pi\)
−0.0272014 + 0.999630i \(0.508660\pi\)
\(854\) 0 0
\(855\) −54.0227 −1.84754
\(856\) 12.9894 0.443970
\(857\) −19.1255 −0.653315 −0.326657 0.945143i \(-0.605922\pi\)
−0.326657 + 0.945143i \(0.605922\pi\)
\(858\) −11.2544 −0.384220
\(859\) −4.94108 −0.168587 −0.0842937 0.996441i \(-0.526863\pi\)
−0.0842937 + 0.996441i \(0.526863\pi\)
\(860\) −10.6761 −0.364052
\(861\) 0 0
\(862\) −34.3260 −1.16915
\(863\) −19.3295 −0.657982 −0.328991 0.944333i \(-0.606709\pi\)
−0.328991 + 0.944333i \(0.606709\pi\)
\(864\) −11.2544 −0.382883
\(865\) 9.25443 0.314660
\(866\) 0.0538991 0.00183157
\(867\) −2.13053 −0.0723564
\(868\) 0 0
\(869\) 12.1517 0.412217
\(870\) −22.6761 −0.768791
\(871\) −30.5089 −1.03375
\(872\) 5.10278 0.172802
\(873\) −14.9739 −0.506788
\(874\) 7.31386 0.247395
\(875\) 0 0
\(876\) 0.637776 0.0215484
\(877\) 16.4011 0.553824 0.276912 0.960895i \(-0.410689\pi\)
0.276912 + 0.960895i \(0.410689\pi\)
\(878\) 7.14663 0.241187
\(879\) −25.9789 −0.876246
\(880\) −1.00000 −0.0337100
\(881\) −39.5678 −1.33307 −0.666536 0.745473i \(-0.732224\pi\)
−0.666536 + 0.745473i \(0.732224\pi\)
\(882\) 0 0
\(883\) 55.4389 1.86567 0.932833 0.360309i \(-0.117329\pi\)
0.932833 + 0.360309i \(0.117329\pi\)
\(884\) 15.2544 0.513062
\(885\) −22.5089 −0.756627
\(886\) 8.74557 0.293813
\(887\) −44.1955 −1.48394 −0.741970 0.670433i \(-0.766108\pi\)
−0.741970 + 0.670433i \(0.766108\pi\)
\(888\) 3.42166 0.114823
\(889\) 0 0
\(890\) −7.45998 −0.250059
\(891\) 15.0383 0.503802
\(892\) 9.45998 0.316743
\(893\) 9.42880 0.315523
\(894\) 73.2616 2.45023
\(895\) −4.98944 −0.166778
\(896\) 0 0
\(897\) −10.0978 −0.337154
\(898\) 39.7038 1.32493
\(899\) −25.0066 −0.834018
\(900\) 6.62721 0.220907
\(901\) 47.5577 1.58438
\(902\) −12.3572 −0.411450
\(903\) 0 0
\(904\) −10.4111 −0.346268
\(905\) −9.66553 −0.321293
\(906\) −35.7250 −1.18688
\(907\) 9.26447 0.307622 0.153811 0.988100i \(-0.450845\pi\)
0.153811 + 0.988100i \(0.450845\pi\)
\(908\) 17.5678 0.583007
\(909\) −90.5644 −3.00383
\(910\) 0 0
\(911\) −37.0489 −1.22748 −0.613742 0.789507i \(-0.710336\pi\)
−0.613742 + 0.789507i \(0.710336\pi\)
\(912\) −25.2927 −0.837526
\(913\) −2.20555 −0.0729931
\(914\) 12.3133 0.407289
\(915\) 9.79445 0.323795
\(916\) −25.7250 −0.849977
\(917\) 0 0
\(918\) −47.3311 −1.56216
\(919\) −13.9149 −0.459011 −0.229506 0.973307i \(-0.573711\pi\)
−0.229506 + 0.973307i \(0.573711\pi\)
\(920\) −0.897225 −0.0295806
\(921\) −31.6655 −1.04341
\(922\) 15.0489 0.495608
\(923\) −47.3311 −1.55792
\(924\) 0 0
\(925\) −1.10278 −0.0362590
\(926\) 31.4061 1.03207
\(927\) 33.4600 1.09897
\(928\) −7.30833 −0.239908
\(929\) 32.0666 1.05207 0.526035 0.850463i \(-0.323678\pi\)
0.526035 + 0.850463i \(0.323678\pi\)
\(930\) 10.6167 0.348134
\(931\) 0 0
\(932\) 19.1567 0.627498
\(933\) −25.3411 −0.829630
\(934\) 19.0716 0.624042
\(935\) −4.20555 −0.137536
\(936\) 24.0383 0.785717
\(937\) −25.4499 −0.831413 −0.415706 0.909499i \(-0.636466\pi\)
−0.415706 + 0.909499i \(0.636466\pi\)
\(938\) 0 0
\(939\) 107.240 3.49966
\(940\) −1.15667 −0.0377266
\(941\) 3.49115 0.113808 0.0569041 0.998380i \(-0.481877\pi\)
0.0569041 + 0.998380i \(0.481877\pi\)
\(942\) −50.2822 −1.63828
\(943\) −11.0872 −0.361049
\(944\) −7.25443 −0.236111
\(945\) 0 0
\(946\) 10.6761 0.347110
\(947\) 30.6167 0.994907 0.497454 0.867491i \(-0.334268\pi\)
0.497454 + 0.867491i \(0.334268\pi\)
\(948\) −37.7038 −1.22456
\(949\) −0.745574 −0.0242024
\(950\) 8.15165 0.264474
\(951\) −28.5783 −0.926716
\(952\) 0 0
\(953\) −33.9406 −1.09944 −0.549721 0.835348i \(-0.685266\pi\)
−0.549721 + 0.835348i \(0.685266\pi\)
\(954\) 74.9427 2.42636
\(955\) 9.15667 0.296303
\(956\) −18.0539 −0.583905
\(957\) 22.6761 0.733014
\(958\) 9.49115 0.306645
\(959\) 0 0
\(960\) 3.10278 0.100142
\(961\) −19.2922 −0.622330
\(962\) −4.00000 −0.128965
\(963\) 86.0838 2.77401
\(964\) −6.15165 −0.198131
\(965\) −1.52946 −0.0492351
\(966\) 0 0
\(967\) −60.8021 −1.95526 −0.977632 0.210323i \(-0.932548\pi\)
−0.977632 + 0.210323i \(0.932548\pi\)
\(968\) 1.00000 0.0321412
\(969\) −106.370 −3.41709
\(970\) 2.25945 0.0725465
\(971\) −50.2933 −1.61399 −0.806994 0.590560i \(-0.798907\pi\)
−0.806994 + 0.590560i \(0.798907\pi\)
\(972\) −12.8972 −0.413679
\(973\) 0 0
\(974\) 0.0438527 0.00140513
\(975\) −11.2544 −0.360430
\(976\) 3.15667 0.101043
\(977\) 28.0978 0.898927 0.449463 0.893299i \(-0.351615\pi\)
0.449463 + 0.893299i \(0.351615\pi\)
\(978\) −19.5889 −0.626384
\(979\) 7.45998 0.238422
\(980\) 0 0
\(981\) 33.8172 1.07970
\(982\) 1.56777 0.0500297
\(983\) 21.0489 0.671355 0.335677 0.941977i \(-0.391035\pi\)
0.335677 + 0.941977i \(0.391035\pi\)
\(984\) 38.3416 1.22229
\(985\) −14.3033 −0.455741
\(986\) −30.7355 −0.978819
\(987\) 0 0
\(988\) 29.5678 0.940676
\(989\) 9.57885 0.304590
\(990\) −6.62721 −0.210627
\(991\) −44.6832 −1.41941 −0.709705 0.704499i \(-0.751172\pi\)
−0.709705 + 0.704499i \(0.751172\pi\)
\(992\) 3.42166 0.108638
\(993\) −50.4011 −1.59943
\(994\) 0 0
\(995\) −10.0978 −0.320120
\(996\) 6.84333 0.216839
\(997\) −9.47002 −0.299919 −0.149959 0.988692i \(-0.547914\pi\)
−0.149959 + 0.988692i \(0.547914\pi\)
\(998\) −42.3416 −1.34030
\(999\) 12.4111 0.392670
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 5390.2.a.ca.1.1 3
7.6 odd 2 770.2.a.m.1.3 3
21.20 even 2 6930.2.a.ce.1.1 3
28.27 even 2 6160.2.a.bf.1.1 3
35.13 even 4 3850.2.c.ba.1849.3 6
35.27 even 4 3850.2.c.ba.1849.4 6
35.34 odd 2 3850.2.a.bt.1.1 3
77.76 even 2 8470.2.a.ci.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.m.1.3 3 7.6 odd 2
3850.2.a.bt.1.1 3 35.34 odd 2
3850.2.c.ba.1849.3 6 35.13 even 4
3850.2.c.ba.1849.4 6 35.27 even 4
5390.2.a.ca.1.1 3 1.1 even 1 trivial
6160.2.a.bf.1.1 3 28.27 even 2
6930.2.a.ce.1.1 3 21.20 even 2
8470.2.a.ci.1.3 3 77.76 even 2