Properties

Label 4730.2.a.be.1.6
Level 4730
Weight 2
Character 4730.1
Self dual yes
Analytic conductor 37.769
Analytic rank 0
Dimension 12
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4730 = 2 \cdot 5 \cdot 11 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4730.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.7692401561\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-0.342987\)
Character \(\chi\) = 4730.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -0.342987 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.342987 q^{6} -3.87285 q^{7} +1.00000 q^{8} -2.88236 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -0.342987 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.342987 q^{6} -3.87285 q^{7} +1.00000 q^{8} -2.88236 q^{9} +1.00000 q^{10} -1.00000 q^{11} -0.342987 q^{12} +5.08439 q^{13} -3.87285 q^{14} -0.342987 q^{15} +1.00000 q^{16} -1.18024 q^{17} -2.88236 q^{18} -2.78382 q^{19} +1.00000 q^{20} +1.32834 q^{21} -1.00000 q^{22} -3.08439 q^{23} -0.342987 q^{24} +1.00000 q^{25} +5.08439 q^{26} +2.01757 q^{27} -3.87285 q^{28} +1.90897 q^{29} -0.342987 q^{30} +2.67757 q^{31} +1.00000 q^{32} +0.342987 q^{33} -1.18024 q^{34} -3.87285 q^{35} -2.88236 q^{36} +5.97555 q^{37} -2.78382 q^{38} -1.74388 q^{39} +1.00000 q^{40} +8.98944 q^{41} +1.32834 q^{42} +1.00000 q^{43} -1.00000 q^{44} -2.88236 q^{45} -3.08439 q^{46} +9.62693 q^{47} -0.342987 q^{48} +7.99899 q^{49} +1.00000 q^{50} +0.404809 q^{51} +5.08439 q^{52} -7.16428 q^{53} +2.01757 q^{54} -1.00000 q^{55} -3.87285 q^{56} +0.954814 q^{57} +1.90897 q^{58} +4.37336 q^{59} -0.342987 q^{60} +9.48586 q^{61} +2.67757 q^{62} +11.1630 q^{63} +1.00000 q^{64} +5.08439 q^{65} +0.342987 q^{66} +8.49930 q^{67} -1.18024 q^{68} +1.05791 q^{69} -3.87285 q^{70} -10.8233 q^{71} -2.88236 q^{72} -4.68226 q^{73} +5.97555 q^{74} -0.342987 q^{75} -2.78382 q^{76} +3.87285 q^{77} -1.74388 q^{78} -4.94508 q^{79} +1.00000 q^{80} +7.95508 q^{81} +8.98944 q^{82} -10.5687 q^{83} +1.32834 q^{84} -1.18024 q^{85} +1.00000 q^{86} -0.654754 q^{87} -1.00000 q^{88} +4.58827 q^{89} -2.88236 q^{90} -19.6911 q^{91} -3.08439 q^{92} -0.918373 q^{93} +9.62693 q^{94} -2.78382 q^{95} -0.342987 q^{96} +16.4470 q^{97} +7.99899 q^{98} +2.88236 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 12q^{2} + 3q^{3} + 12q^{4} + 12q^{5} + 3q^{6} + 8q^{7} + 12q^{8} + 25q^{9} + O(q^{10}) \) \( 12q + 12q^{2} + 3q^{3} + 12q^{4} + 12q^{5} + 3q^{6} + 8q^{7} + 12q^{8} + 25q^{9} + 12q^{10} - 12q^{11} + 3q^{12} + 16q^{13} + 8q^{14} + 3q^{15} + 12q^{16} + 18q^{17} + 25q^{18} - 4q^{19} + 12q^{20} + 4q^{21} - 12q^{22} + 8q^{23} + 3q^{24} + 12q^{25} + 16q^{26} + 6q^{27} + 8q^{28} + 20q^{29} + 3q^{30} + 5q^{31} + 12q^{32} - 3q^{33} + 18q^{34} + 8q^{35} + 25q^{36} + 19q^{37} - 4q^{38} + 6q^{39} + 12q^{40} + 16q^{41} + 4q^{42} + 12q^{43} - 12q^{44} + 25q^{45} + 8q^{46} - q^{47} + 3q^{48} + 52q^{49} + 12q^{50} + q^{51} + 16q^{52} + 11q^{53} + 6q^{54} - 12q^{55} + 8q^{56} + 9q^{57} + 20q^{58} - 11q^{59} + 3q^{60} + 18q^{61} + 5q^{62} + 15q^{63} + 12q^{64} + 16q^{65} - 3q^{66} - 10q^{67} + 18q^{68} + 8q^{70} - 2q^{71} + 25q^{72} + 29q^{73} + 19q^{74} + 3q^{75} - 4q^{76} - 8q^{77} + 6q^{78} + 2q^{79} + 12q^{80} - 8q^{81} + 16q^{82} + 26q^{83} + 4q^{84} + 18q^{85} + 12q^{86} - 4q^{87} - 12q^{88} + 41q^{89} + 25q^{90} - 4q^{91} + 8q^{92} + 5q^{93} - q^{94} - 4q^{95} + 3q^{96} - 7q^{97} + 52q^{98} - 25q^{99} + O(q^{100}) \)

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
\(3\) −0.342987 −0.198024 −0.0990119 0.995086i \(-0.531568\pi\)
−0.0990119 + 0.995086i \(0.531568\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −0.342987 −0.140024
\(7\) −3.87285 −1.46380 −0.731900 0.681412i \(-0.761366\pi\)
−0.731900 + 0.681412i \(0.761366\pi\)
\(8\) 1.00000 0.353553
\(9\) −2.88236 −0.960787
\(10\) 1.00000 0.316228
\(11\) −1.00000 −0.301511
\(12\) −0.342987 −0.0990119
\(13\) 5.08439 1.41016 0.705078 0.709130i \(-0.250912\pi\)
0.705078 + 0.709130i \(0.250912\pi\)
\(14\) −3.87285 −1.03506
\(15\) −0.342987 −0.0885589
\(16\) 1.00000 0.250000
\(17\) −1.18024 −0.286251 −0.143126 0.989705i \(-0.545715\pi\)
−0.143126 + 0.989705i \(0.545715\pi\)
\(18\) −2.88236 −0.679379
\(19\) −2.78382 −0.638652 −0.319326 0.947645i \(-0.603456\pi\)
−0.319326 + 0.947645i \(0.603456\pi\)
\(20\) 1.00000 0.223607
\(21\) 1.32834 0.289867
\(22\) −1.00000 −0.213201
\(23\) −3.08439 −0.643139 −0.321570 0.946886i \(-0.604210\pi\)
−0.321570 + 0.946886i \(0.604210\pi\)
\(24\) −0.342987 −0.0700120
\(25\) 1.00000 0.200000
\(26\) 5.08439 0.997130
\(27\) 2.01757 0.388282
\(28\) −3.87285 −0.731900
\(29\) 1.90897 0.354488 0.177244 0.984167i \(-0.443282\pi\)
0.177244 + 0.984167i \(0.443282\pi\)
\(30\) −0.342987 −0.0626206
\(31\) 2.67757 0.480906 0.240453 0.970661i \(-0.422704\pi\)
0.240453 + 0.970661i \(0.422704\pi\)
\(32\) 1.00000 0.176777
\(33\) 0.342987 0.0597064
\(34\) −1.18024 −0.202410
\(35\) −3.87285 −0.654632
\(36\) −2.88236 −0.480393
\(37\) 5.97555 0.982374 0.491187 0.871054i \(-0.336563\pi\)
0.491187 + 0.871054i \(0.336563\pi\)
\(38\) −2.78382 −0.451595
\(39\) −1.74388 −0.279244
\(40\) 1.00000 0.158114
\(41\) 8.98944 1.40392 0.701958 0.712219i \(-0.252310\pi\)
0.701958 + 0.712219i \(0.252310\pi\)
\(42\) 1.32834 0.204967
\(43\) 1.00000 0.152499
\(44\) −1.00000 −0.150756
\(45\) −2.88236 −0.429677
\(46\) −3.08439 −0.454768
\(47\) 9.62693 1.40423 0.702116 0.712062i \(-0.252239\pi\)
0.702116 + 0.712062i \(0.252239\pi\)
\(48\) −0.342987 −0.0495059
\(49\) 7.99899 1.14271
\(50\) 1.00000 0.141421
\(51\) 0.404809 0.0566845
\(52\) 5.08439 0.705078
\(53\) −7.16428 −0.984090 −0.492045 0.870570i \(-0.663750\pi\)
−0.492045 + 0.870570i \(0.663750\pi\)
\(54\) 2.01757 0.274557
\(55\) −1.00000 −0.134840
\(56\) −3.87285 −0.517532
\(57\) 0.954814 0.126468
\(58\) 1.90897 0.250661
\(59\) 4.37336 0.569363 0.284682 0.958622i \(-0.408112\pi\)
0.284682 + 0.958622i \(0.408112\pi\)
\(60\) −0.342987 −0.0442795
\(61\) 9.48586 1.21454 0.607270 0.794496i \(-0.292265\pi\)
0.607270 + 0.794496i \(0.292265\pi\)
\(62\) 2.67757 0.340052
\(63\) 11.1630 1.40640
\(64\) 1.00000 0.125000
\(65\) 5.08439 0.630641
\(66\) 0.342987 0.0422188
\(67\) 8.49930 1.03835 0.519177 0.854666i \(-0.326238\pi\)
0.519177 + 0.854666i \(0.326238\pi\)
\(68\) −1.18024 −0.143126
\(69\) 1.05791 0.127357
\(70\) −3.87285 −0.462894
\(71\) −10.8233 −1.28449 −0.642247 0.766498i \(-0.721998\pi\)
−0.642247 + 0.766498i \(0.721998\pi\)
\(72\) −2.88236 −0.339689
\(73\) −4.68226 −0.548017 −0.274009 0.961727i \(-0.588350\pi\)
−0.274009 + 0.961727i \(0.588350\pi\)
\(74\) 5.97555 0.694643
\(75\) −0.342987 −0.0396048
\(76\) −2.78382 −0.319326
\(77\) 3.87285 0.441353
\(78\) −1.74388 −0.197456
\(79\) −4.94508 −0.556365 −0.278183 0.960528i \(-0.589732\pi\)
−0.278183 + 0.960528i \(0.589732\pi\)
\(80\) 1.00000 0.111803
\(81\) 7.95508 0.883897
\(82\) 8.98944 0.992718
\(83\) −10.5687 −1.16007 −0.580035 0.814592i \(-0.696961\pi\)
−0.580035 + 0.814592i \(0.696961\pi\)
\(84\) 1.32834 0.144934
\(85\) −1.18024 −0.128015
\(86\) 1.00000 0.107833
\(87\) −0.654754 −0.0701970
\(88\) −1.00000 −0.106600
\(89\) 4.58827 0.486356 0.243178 0.969982i \(-0.421810\pi\)
0.243178 + 0.969982i \(0.421810\pi\)
\(90\) −2.88236 −0.303827
\(91\) −19.6911 −2.06419
\(92\) −3.08439 −0.321570
\(93\) −0.918373 −0.0952308
\(94\) 9.62693 0.992942
\(95\) −2.78382 −0.285614
\(96\) −0.342987 −0.0350060
\(97\) 16.4470 1.66994 0.834969 0.550297i \(-0.185486\pi\)
0.834969 + 0.550297i \(0.185486\pi\)
\(98\) 7.99899 0.808020
\(99\) 2.88236 0.289688
\(100\) 1.00000 0.100000
\(101\) 18.4869 1.83951 0.919756 0.392490i \(-0.128386\pi\)
0.919756 + 0.392490i \(0.128386\pi\)
\(102\) 0.404809 0.0400820
\(103\) 3.66411 0.361036 0.180518 0.983572i \(-0.442223\pi\)
0.180518 + 0.983572i \(0.442223\pi\)
\(104\) 5.08439 0.498565
\(105\) 1.32834 0.129633
\(106\) −7.16428 −0.695857
\(107\) 9.37944 0.906745 0.453372 0.891321i \(-0.350221\pi\)
0.453372 + 0.891321i \(0.350221\pi\)
\(108\) 2.01757 0.194141
\(109\) 0.816833 0.0782384 0.0391192 0.999235i \(-0.487545\pi\)
0.0391192 + 0.999235i \(0.487545\pi\)
\(110\) −1.00000 −0.0953463
\(111\) −2.04954 −0.194533
\(112\) −3.87285 −0.365950
\(113\) −12.3567 −1.16242 −0.581212 0.813752i \(-0.697421\pi\)
−0.581212 + 0.813752i \(0.697421\pi\)
\(114\) 0.954814 0.0894265
\(115\) −3.08439 −0.287621
\(116\) 1.90897 0.177244
\(117\) −14.6550 −1.35486
\(118\) 4.37336 0.402601
\(119\) 4.57091 0.419015
\(120\) −0.342987 −0.0313103
\(121\) 1.00000 0.0909091
\(122\) 9.48586 0.858809
\(123\) −3.08326 −0.278009
\(124\) 2.67757 0.240453
\(125\) 1.00000 0.0894427
\(126\) 11.1630 0.994475
\(127\) 20.4964 1.81876 0.909382 0.415963i \(-0.136555\pi\)
0.909382 + 0.415963i \(0.136555\pi\)
\(128\) 1.00000 0.0883883
\(129\) −0.342987 −0.0301983
\(130\) 5.08439 0.445930
\(131\) 7.50200 0.655453 0.327726 0.944773i \(-0.393718\pi\)
0.327726 + 0.944773i \(0.393718\pi\)
\(132\) 0.342987 0.0298532
\(133\) 10.7813 0.934859
\(134\) 8.49930 0.734228
\(135\) 2.01757 0.173645
\(136\) −1.18024 −0.101205
\(137\) 0.602218 0.0514510 0.0257255 0.999669i \(-0.491810\pi\)
0.0257255 + 0.999669i \(0.491810\pi\)
\(138\) 1.05791 0.0900549
\(139\) 12.3783 1.04991 0.524955 0.851130i \(-0.324082\pi\)
0.524955 + 0.851130i \(0.324082\pi\)
\(140\) −3.87285 −0.327316
\(141\) −3.30192 −0.278071
\(142\) −10.8233 −0.908275
\(143\) −5.08439 −0.425178
\(144\) −2.88236 −0.240197
\(145\) 1.90897 0.158532
\(146\) −4.68226 −0.387507
\(147\) −2.74355 −0.226284
\(148\) 5.97555 0.491187
\(149\) −9.45473 −0.774562 −0.387281 0.921962i \(-0.626586\pi\)
−0.387281 + 0.921962i \(0.626586\pi\)
\(150\) −0.342987 −0.0280048
\(151\) −11.5779 −0.942198 −0.471099 0.882080i \(-0.656142\pi\)
−0.471099 + 0.882080i \(0.656142\pi\)
\(152\) −2.78382 −0.225797
\(153\) 3.40189 0.275026
\(154\) 3.87285 0.312083
\(155\) 2.67757 0.215068
\(156\) −1.74388 −0.139622
\(157\) −8.78374 −0.701019 −0.350510 0.936559i \(-0.613992\pi\)
−0.350510 + 0.936559i \(0.613992\pi\)
\(158\) −4.94508 −0.393410
\(159\) 2.45726 0.194873
\(160\) 1.00000 0.0790569
\(161\) 11.9454 0.941427
\(162\) 7.95508 0.625010
\(163\) 8.63082 0.676018 0.338009 0.941143i \(-0.390247\pi\)
0.338009 + 0.941143i \(0.390247\pi\)
\(164\) 8.98944 0.701958
\(165\) 0.342987 0.0267015
\(166\) −10.5687 −0.820293
\(167\) −9.74131 −0.753805 −0.376903 0.926253i \(-0.623011\pi\)
−0.376903 + 0.926253i \(0.623011\pi\)
\(168\) 1.32834 0.102484
\(169\) 12.8510 0.988538
\(170\) −1.18024 −0.0905206
\(171\) 8.02397 0.613608
\(172\) 1.00000 0.0762493
\(173\) 11.4785 0.872691 0.436346 0.899779i \(-0.356272\pi\)
0.436346 + 0.899779i \(0.356272\pi\)
\(174\) −0.654754 −0.0496368
\(175\) −3.87285 −0.292760
\(176\) −1.00000 −0.0753778
\(177\) −1.50001 −0.112747
\(178\) 4.58827 0.343905
\(179\) −14.4576 −1.08061 −0.540305 0.841469i \(-0.681691\pi\)
−0.540305 + 0.841469i \(0.681691\pi\)
\(180\) −2.88236 −0.214838
\(181\) 4.30048 0.319652 0.159826 0.987145i \(-0.448907\pi\)
0.159826 + 0.987145i \(0.448907\pi\)
\(182\) −19.6911 −1.45960
\(183\) −3.25353 −0.240508
\(184\) −3.08439 −0.227384
\(185\) 5.97555 0.439331
\(186\) −0.918373 −0.0673384
\(187\) 1.18024 0.0863080
\(188\) 9.62693 0.702116
\(189\) −7.81377 −0.568368
\(190\) −2.78382 −0.201959
\(191\) −4.74647 −0.343443 −0.171721 0.985146i \(-0.554933\pi\)
−0.171721 + 0.985146i \(0.554933\pi\)
\(192\) −0.342987 −0.0247530
\(193\) −19.1907 −1.38138 −0.690690 0.723151i \(-0.742693\pi\)
−0.690690 + 0.723151i \(0.742693\pi\)
\(194\) 16.4470 1.18082
\(195\) −1.74388 −0.124882
\(196\) 7.99899 0.571356
\(197\) 21.1138 1.50429 0.752147 0.658996i \(-0.229018\pi\)
0.752147 + 0.658996i \(0.229018\pi\)
\(198\) 2.88236 0.204840
\(199\) 0.404858 0.0286996 0.0143498 0.999897i \(-0.495432\pi\)
0.0143498 + 0.999897i \(0.495432\pi\)
\(200\) 1.00000 0.0707107
\(201\) −2.91515 −0.205619
\(202\) 18.4869 1.30073
\(203\) −7.39317 −0.518899
\(204\) 0.404809 0.0283423
\(205\) 8.98944 0.627850
\(206\) 3.66411 0.255291
\(207\) 8.89031 0.617919
\(208\) 5.08439 0.352539
\(209\) 2.78382 0.192561
\(210\) 1.32834 0.0916641
\(211\) −0.606821 −0.0417753 −0.0208876 0.999782i \(-0.506649\pi\)
−0.0208876 + 0.999782i \(0.506649\pi\)
\(212\) −7.16428 −0.492045
\(213\) 3.71227 0.254360
\(214\) 9.37944 0.641166
\(215\) 1.00000 0.0681994
\(216\) 2.01757 0.137279
\(217\) −10.3698 −0.703950
\(218\) 0.816833 0.0553229
\(219\) 1.60596 0.108520
\(220\) −1.00000 −0.0674200
\(221\) −6.00082 −0.403659
\(222\) −2.04954 −0.137556
\(223\) −23.9828 −1.60601 −0.803003 0.595975i \(-0.796766\pi\)
−0.803003 + 0.595975i \(0.796766\pi\)
\(224\) −3.87285 −0.258766
\(225\) −2.88236 −0.192157
\(226\) −12.3567 −0.821959
\(227\) −0.142439 −0.00945401 −0.00472701 0.999989i \(-0.501505\pi\)
−0.00472701 + 0.999989i \(0.501505\pi\)
\(228\) 0.954814 0.0632341
\(229\) 15.6290 1.03279 0.516397 0.856349i \(-0.327273\pi\)
0.516397 + 0.856349i \(0.327273\pi\)
\(230\) −3.08439 −0.203378
\(231\) −1.32834 −0.0873983
\(232\) 1.90897 0.125330
\(233\) 17.0333 1.11589 0.557946 0.829877i \(-0.311590\pi\)
0.557946 + 0.829877i \(0.311590\pi\)
\(234\) −14.6550 −0.958029
\(235\) 9.62693 0.627992
\(236\) 4.37336 0.284682
\(237\) 1.69610 0.110174
\(238\) 4.57091 0.296288
\(239\) −7.32611 −0.473887 −0.236943 0.971523i \(-0.576146\pi\)
−0.236943 + 0.971523i \(0.576146\pi\)
\(240\) −0.342987 −0.0221397
\(241\) 9.79369 0.630867 0.315433 0.948948i \(-0.397850\pi\)
0.315433 + 0.948948i \(0.397850\pi\)
\(242\) 1.00000 0.0642824
\(243\) −8.78121 −0.563315
\(244\) 9.48586 0.607270
\(245\) 7.99899 0.511037
\(246\) −3.08326 −0.196582
\(247\) −14.1540 −0.900598
\(248\) 2.67757 0.170026
\(249\) 3.62494 0.229721
\(250\) 1.00000 0.0632456
\(251\) 0.467866 0.0295315 0.0147657 0.999891i \(-0.495300\pi\)
0.0147657 + 0.999891i \(0.495300\pi\)
\(252\) 11.1630 0.703200
\(253\) 3.08439 0.193914
\(254\) 20.4964 1.28606
\(255\) 0.404809 0.0253501
\(256\) 1.00000 0.0625000
\(257\) 16.3447 1.01956 0.509779 0.860306i \(-0.329727\pi\)
0.509779 + 0.860306i \(0.329727\pi\)
\(258\) −0.342987 −0.0213535
\(259\) −23.1424 −1.43800
\(260\) 5.08439 0.315320
\(261\) −5.50235 −0.340587
\(262\) 7.50200 0.463475
\(263\) 19.5709 1.20680 0.603398 0.797440i \(-0.293813\pi\)
0.603398 + 0.797440i \(0.293813\pi\)
\(264\) 0.342987 0.0211094
\(265\) −7.16428 −0.440098
\(266\) 10.7813 0.661045
\(267\) −1.57372 −0.0963100
\(268\) 8.49930 0.519177
\(269\) 6.15925 0.375536 0.187768 0.982213i \(-0.439875\pi\)
0.187768 + 0.982213i \(0.439875\pi\)
\(270\) 2.01757 0.122786
\(271\) 3.09310 0.187893 0.0939464 0.995577i \(-0.470052\pi\)
0.0939464 + 0.995577i \(0.470052\pi\)
\(272\) −1.18024 −0.0715628
\(273\) 6.75379 0.408758
\(274\) 0.602218 0.0363813
\(275\) −1.00000 −0.0603023
\(276\) 1.05791 0.0636784
\(277\) −13.6889 −0.822484 −0.411242 0.911526i \(-0.634905\pi\)
−0.411242 + 0.911526i \(0.634905\pi\)
\(278\) 12.3783 0.742399
\(279\) −7.71772 −0.462048
\(280\) −3.87285 −0.231447
\(281\) −9.84644 −0.587390 −0.293695 0.955899i \(-0.594885\pi\)
−0.293695 + 0.955899i \(0.594885\pi\)
\(282\) −3.30192 −0.196626
\(283\) −8.73821 −0.519433 −0.259716 0.965685i \(-0.583629\pi\)
−0.259716 + 0.965685i \(0.583629\pi\)
\(284\) −10.8233 −0.642247
\(285\) 0.954814 0.0565583
\(286\) −5.08439 −0.300646
\(287\) −34.8148 −2.05505
\(288\) −2.88236 −0.169845
\(289\) −15.6070 −0.918060
\(290\) 1.90897 0.112099
\(291\) −5.64110 −0.330687
\(292\) −4.68226 −0.274009
\(293\) −8.02450 −0.468796 −0.234398 0.972141i \(-0.575312\pi\)
−0.234398 + 0.972141i \(0.575312\pi\)
\(294\) −2.74355 −0.160007
\(295\) 4.37336 0.254627
\(296\) 5.97555 0.347322
\(297\) −2.01757 −0.117072
\(298\) −9.45473 −0.547698
\(299\) −15.6822 −0.906926
\(300\) −0.342987 −0.0198024
\(301\) −3.87285 −0.223228
\(302\) −11.5779 −0.666234
\(303\) −6.34076 −0.364267
\(304\) −2.78382 −0.159663
\(305\) 9.48586 0.543159
\(306\) 3.40189 0.194473
\(307\) 23.1615 1.32190 0.660948 0.750432i \(-0.270155\pi\)
0.660948 + 0.750432i \(0.270155\pi\)
\(308\) 3.87285 0.220676
\(309\) −1.25674 −0.0714937
\(310\) 2.67757 0.152076
\(311\) −26.5655 −1.50639 −0.753195 0.657798i \(-0.771488\pi\)
−0.753195 + 0.657798i \(0.771488\pi\)
\(312\) −1.74388 −0.0987278
\(313\) 6.12755 0.346350 0.173175 0.984891i \(-0.444597\pi\)
0.173175 + 0.984891i \(0.444597\pi\)
\(314\) −8.78374 −0.495695
\(315\) 11.1630 0.628961
\(316\) −4.94508 −0.278183
\(317\) 14.7661 0.829347 0.414673 0.909970i \(-0.363896\pi\)
0.414673 + 0.909970i \(0.363896\pi\)
\(318\) 2.45726 0.137796
\(319\) −1.90897 −0.106882
\(320\) 1.00000 0.0559017
\(321\) −3.21703 −0.179557
\(322\) 11.9454 0.665690
\(323\) 3.28558 0.182815
\(324\) 7.95508 0.441949
\(325\) 5.08439 0.282031
\(326\) 8.63082 0.478017
\(327\) −0.280163 −0.0154931
\(328\) 8.98944 0.496359
\(329\) −37.2837 −2.05552
\(330\) 0.342987 0.0188808
\(331\) 33.5866 1.84609 0.923043 0.384697i \(-0.125694\pi\)
0.923043 + 0.384697i \(0.125694\pi\)
\(332\) −10.5687 −0.580035
\(333\) −17.2237 −0.943852
\(334\) −9.74131 −0.533021
\(335\) 8.49930 0.464366
\(336\) 1.32834 0.0724668
\(337\) −15.6502 −0.852522 −0.426261 0.904600i \(-0.640170\pi\)
−0.426261 + 0.904600i \(0.640170\pi\)
\(338\) 12.8510 0.699002
\(339\) 4.23821 0.230188
\(340\) −1.18024 −0.0640077
\(341\) −2.67757 −0.144999
\(342\) 8.02397 0.433886
\(343\) −3.86893 −0.208902
\(344\) 1.00000 0.0539164
\(345\) 1.05791 0.0569557
\(346\) 11.4785 0.617086
\(347\) −9.10091 −0.488563 −0.244281 0.969704i \(-0.578552\pi\)
−0.244281 + 0.969704i \(0.578552\pi\)
\(348\) −0.654754 −0.0350985
\(349\) −11.9911 −0.641872 −0.320936 0.947101i \(-0.603997\pi\)
−0.320936 + 0.947101i \(0.603997\pi\)
\(350\) −3.87285 −0.207013
\(351\) 10.2581 0.547538
\(352\) −1.00000 −0.0533002
\(353\) 16.3446 0.869933 0.434966 0.900447i \(-0.356760\pi\)
0.434966 + 0.900447i \(0.356760\pi\)
\(354\) −1.50001 −0.0797245
\(355\) −10.8233 −0.574443
\(356\) 4.58827 0.243178
\(357\) −1.56776 −0.0829749
\(358\) −14.4576 −0.764107
\(359\) −35.5278 −1.87508 −0.937542 0.347872i \(-0.886904\pi\)
−0.937542 + 0.347872i \(0.886904\pi\)
\(360\) −2.88236 −0.151914
\(361\) −11.2504 −0.592124
\(362\) 4.30048 0.226028
\(363\) −0.342987 −0.0180022
\(364\) −19.6911 −1.03209
\(365\) −4.68226 −0.245081
\(366\) −3.25353 −0.170065
\(367\) 2.58400 0.134884 0.0674419 0.997723i \(-0.478516\pi\)
0.0674419 + 0.997723i \(0.478516\pi\)
\(368\) −3.08439 −0.160785
\(369\) −25.9108 −1.34886
\(370\) 5.97555 0.310654
\(371\) 27.7462 1.44051
\(372\) −0.918373 −0.0476154
\(373\) 17.5403 0.908202 0.454101 0.890950i \(-0.349960\pi\)
0.454101 + 0.890950i \(0.349960\pi\)
\(374\) 1.18024 0.0610289
\(375\) −0.342987 −0.0177118
\(376\) 9.62693 0.496471
\(377\) 9.70596 0.499882
\(378\) −7.81377 −0.401897
\(379\) 12.8839 0.661801 0.330901 0.943666i \(-0.392648\pi\)
0.330901 + 0.943666i \(0.392648\pi\)
\(380\) −2.78382 −0.142807
\(381\) −7.03001 −0.360158
\(382\) −4.74647 −0.242851
\(383\) 14.6945 0.750856 0.375428 0.926852i \(-0.377496\pi\)
0.375428 + 0.926852i \(0.377496\pi\)
\(384\) −0.342987 −0.0175030
\(385\) 3.87285 0.197379
\(386\) −19.1907 −0.976783
\(387\) −2.88236 −0.146519
\(388\) 16.4470 0.834969
\(389\) −12.3657 −0.626965 −0.313482 0.949594i \(-0.601496\pi\)
−0.313482 + 0.949594i \(0.601496\pi\)
\(390\) −1.74388 −0.0883048
\(391\) 3.64033 0.184099
\(392\) 7.99899 0.404010
\(393\) −2.57309 −0.129795
\(394\) 21.1138 1.06370
\(395\) −4.94508 −0.248814
\(396\) 2.88236 0.144844
\(397\) 6.55812 0.329142 0.164571 0.986365i \(-0.447376\pi\)
0.164571 + 0.986365i \(0.447376\pi\)
\(398\) 0.404858 0.0202937
\(399\) −3.69785 −0.185124
\(400\) 1.00000 0.0500000
\(401\) −9.12400 −0.455631 −0.227815 0.973704i \(-0.573158\pi\)
−0.227815 + 0.973704i \(0.573158\pi\)
\(402\) −2.91515 −0.145395
\(403\) 13.6138 0.678152
\(404\) 18.4869 0.919756
\(405\) 7.95508 0.395291
\(406\) −7.39317 −0.366917
\(407\) −5.97555 −0.296197
\(408\) 0.404809 0.0200410
\(409\) 5.90843 0.292153 0.146076 0.989273i \(-0.453335\pi\)
0.146076 + 0.989273i \(0.453335\pi\)
\(410\) 8.98944 0.443957
\(411\) −0.206553 −0.0101885
\(412\) 3.66411 0.180518
\(413\) −16.9374 −0.833434
\(414\) 8.89031 0.436935
\(415\) −10.5687 −0.518799
\(416\) 5.08439 0.249283
\(417\) −4.24558 −0.207907
\(418\) 2.78382 0.136161
\(419\) −20.8979 −1.02093 −0.510465 0.859898i \(-0.670527\pi\)
−0.510465 + 0.859898i \(0.670527\pi\)
\(420\) 1.32834 0.0648163
\(421\) 1.44685 0.0705149 0.0352575 0.999378i \(-0.488775\pi\)
0.0352575 + 0.999378i \(0.488775\pi\)
\(422\) −0.606821 −0.0295396
\(423\) −27.7483 −1.34917
\(424\) −7.16428 −0.347928
\(425\) −1.18024 −0.0572502
\(426\) 3.71227 0.179860
\(427\) −36.7373 −1.77784
\(428\) 9.37944 0.453372
\(429\) 1.74388 0.0841953
\(430\) 1.00000 0.0482243
\(431\) −12.0796 −0.581856 −0.290928 0.956745i \(-0.593964\pi\)
−0.290928 + 0.956745i \(0.593964\pi\)
\(432\) 2.01757 0.0970706
\(433\) −27.3298 −1.31339 −0.656693 0.754158i \(-0.728045\pi\)
−0.656693 + 0.754158i \(0.728045\pi\)
\(434\) −10.3698 −0.497768
\(435\) −0.654754 −0.0313930
\(436\) 0.816833 0.0391192
\(437\) 8.58637 0.410742
\(438\) 1.60596 0.0767355
\(439\) −6.28995 −0.300203 −0.150101 0.988671i \(-0.547960\pi\)
−0.150101 + 0.988671i \(0.547960\pi\)
\(440\) −1.00000 −0.0476731
\(441\) −23.0560 −1.09790
\(442\) −6.00082 −0.285430
\(443\) 32.8673 1.56157 0.780785 0.624799i \(-0.214819\pi\)
0.780785 + 0.624799i \(0.214819\pi\)
\(444\) −2.04954 −0.0972667
\(445\) 4.58827 0.217505
\(446\) −23.9828 −1.13562
\(447\) 3.24285 0.153382
\(448\) −3.87285 −0.182975
\(449\) −38.0446 −1.79544 −0.897718 0.440570i \(-0.854776\pi\)
−0.897718 + 0.440570i \(0.854776\pi\)
\(450\) −2.88236 −0.135876
\(451\) −8.98944 −0.423296
\(452\) −12.3567 −0.581212
\(453\) 3.97108 0.186578
\(454\) −0.142439 −0.00668500
\(455\) −19.6911 −0.923132
\(456\) 0.954814 0.0447133
\(457\) 10.3079 0.482184 0.241092 0.970502i \(-0.422494\pi\)
0.241092 + 0.970502i \(0.422494\pi\)
\(458\) 15.6290 0.730296
\(459\) −2.38123 −0.111146
\(460\) −3.08439 −0.143810
\(461\) −25.5297 −1.18904 −0.594519 0.804082i \(-0.702657\pi\)
−0.594519 + 0.804082i \(0.702657\pi\)
\(462\) −1.32834 −0.0617999
\(463\) 19.6089 0.911305 0.455652 0.890158i \(-0.349406\pi\)
0.455652 + 0.890158i \(0.349406\pi\)
\(464\) 1.90897 0.0886219
\(465\) −0.918373 −0.0425885
\(466\) 17.0333 0.789054
\(467\) 16.8330 0.778938 0.389469 0.921040i \(-0.372659\pi\)
0.389469 + 0.921040i \(0.372659\pi\)
\(468\) −14.6550 −0.677429
\(469\) −32.9165 −1.51994
\(470\) 9.62693 0.444057
\(471\) 3.01271 0.138818
\(472\) 4.37336 0.201300
\(473\) −1.00000 −0.0459800
\(474\) 1.69610 0.0779044
\(475\) −2.78382 −0.127730
\(476\) 4.57091 0.209507
\(477\) 20.6500 0.945500
\(478\) −7.32611 −0.335089
\(479\) 17.2296 0.787240 0.393620 0.919273i \(-0.371223\pi\)
0.393620 + 0.919273i \(0.371223\pi\)
\(480\) −0.342987 −0.0156552
\(481\) 30.3820 1.38530
\(482\) 9.79369 0.446090
\(483\) −4.09711 −0.186425
\(484\) 1.00000 0.0454545
\(485\) 16.4470 0.746819
\(486\) −8.78121 −0.398324
\(487\) −2.57189 −0.116543 −0.0582716 0.998301i \(-0.518559\pi\)
−0.0582716 + 0.998301i \(0.518559\pi\)
\(488\) 9.48586 0.429405
\(489\) −2.96026 −0.133868
\(490\) 7.99899 0.361357
\(491\) 39.5104 1.78308 0.891539 0.452943i \(-0.149626\pi\)
0.891539 + 0.452943i \(0.149626\pi\)
\(492\) −3.08326 −0.139004
\(493\) −2.25305 −0.101472
\(494\) −14.1540 −0.636819
\(495\) 2.88236 0.129552
\(496\) 2.67757 0.120226
\(497\) 41.9172 1.88024
\(498\) 3.62494 0.162437
\(499\) −15.7140 −0.703457 −0.351728 0.936102i \(-0.614406\pi\)
−0.351728 + 0.936102i \(0.614406\pi\)
\(500\) 1.00000 0.0447214
\(501\) 3.34115 0.149271
\(502\) 0.467866 0.0208819
\(503\) 10.4555 0.466189 0.233095 0.972454i \(-0.425115\pi\)
0.233095 + 0.972454i \(0.425115\pi\)
\(504\) 11.1630 0.497237
\(505\) 18.4869 0.822655
\(506\) 3.08439 0.137118
\(507\) −4.40773 −0.195754
\(508\) 20.4964 0.909382
\(509\) 24.0528 1.06612 0.533061 0.846076i \(-0.321041\pi\)
0.533061 + 0.846076i \(0.321041\pi\)
\(510\) 0.404809 0.0179252
\(511\) 18.1337 0.802188
\(512\) 1.00000 0.0441942
\(513\) −5.61656 −0.247977
\(514\) 16.3447 0.720936
\(515\) 3.66411 0.161460
\(516\) −0.342987 −0.0150992
\(517\) −9.62693 −0.423392
\(518\) −23.1424 −1.01682
\(519\) −3.93697 −0.172814
\(520\) 5.08439 0.222965
\(521\) 31.6055 1.38466 0.692331 0.721580i \(-0.256584\pi\)
0.692331 + 0.721580i \(0.256584\pi\)
\(522\) −5.50235 −0.240831
\(523\) 8.44666 0.369347 0.184673 0.982800i \(-0.440877\pi\)
0.184673 + 0.982800i \(0.440877\pi\)
\(524\) 7.50200 0.327726
\(525\) 1.32834 0.0579735
\(526\) 19.5709 0.853334
\(527\) −3.16019 −0.137660
\(528\) 0.342987 0.0149266
\(529\) −13.4866 −0.586372
\(530\) −7.16428 −0.311197
\(531\) −12.6056 −0.547037
\(532\) 10.7813 0.467429
\(533\) 45.7058 1.97974
\(534\) −1.57372 −0.0681014
\(535\) 9.37944 0.405509
\(536\) 8.49930 0.367114
\(537\) 4.95877 0.213987
\(538\) 6.15925 0.265544
\(539\) −7.99899 −0.344541
\(540\) 2.01757 0.0868226
\(541\) −39.3323 −1.69103 −0.845513 0.533954i \(-0.820705\pi\)
−0.845513 + 0.533954i \(0.820705\pi\)
\(542\) 3.09310 0.132860
\(543\) −1.47501 −0.0632987
\(544\) −1.18024 −0.0506025
\(545\) 0.816833 0.0349893
\(546\) 6.75379 0.289036
\(547\) 2.16850 0.0927182 0.0463591 0.998925i \(-0.485238\pi\)
0.0463591 + 0.998925i \(0.485238\pi\)
\(548\) 0.602218 0.0257255
\(549\) −27.3417 −1.16691
\(550\) −1.00000 −0.0426401
\(551\) −5.31424 −0.226394
\(552\) 1.05791 0.0450274
\(553\) 19.1516 0.814408
\(554\) −13.6889 −0.581584
\(555\) −2.04954 −0.0869980
\(556\) 12.3783 0.524955
\(557\) 33.8051 1.43237 0.716184 0.697912i \(-0.245887\pi\)
0.716184 + 0.697912i \(0.245887\pi\)
\(558\) −7.71772 −0.326717
\(559\) 5.08439 0.215047
\(560\) −3.87285 −0.163658
\(561\) −0.404809 −0.0170910
\(562\) −9.84644 −0.415347
\(563\) −11.3117 −0.476731 −0.238366 0.971176i \(-0.576612\pi\)
−0.238366 + 0.971176i \(0.576612\pi\)
\(564\) −3.30192 −0.139036
\(565\) −12.3567 −0.519852
\(566\) −8.73821 −0.367294
\(567\) −30.8088 −1.29385
\(568\) −10.8233 −0.454137
\(569\) −10.7096 −0.448969 −0.224485 0.974478i \(-0.572070\pi\)
−0.224485 + 0.974478i \(0.572070\pi\)
\(570\) 0.954814 0.0399928
\(571\) 3.34474 0.139973 0.0699865 0.997548i \(-0.477704\pi\)
0.0699865 + 0.997548i \(0.477704\pi\)
\(572\) −5.08439 −0.212589
\(573\) 1.62798 0.0680098
\(574\) −34.8148 −1.45314
\(575\) −3.08439 −0.128628
\(576\) −2.88236 −0.120098
\(577\) 5.16292 0.214935 0.107468 0.994209i \(-0.465726\pi\)
0.107468 + 0.994209i \(0.465726\pi\)
\(578\) −15.6070 −0.649167
\(579\) 6.58218 0.273546
\(580\) 1.90897 0.0792658
\(581\) 40.9311 1.69811
\(582\) −5.64110 −0.233831
\(583\) 7.16428 0.296714
\(584\) −4.68226 −0.193753
\(585\) −14.6550 −0.605911
\(586\) −8.02450 −0.331489
\(587\) 22.9777 0.948393 0.474196 0.880419i \(-0.342739\pi\)
0.474196 + 0.880419i \(0.342739\pi\)
\(588\) −2.74355 −0.113142
\(589\) −7.45387 −0.307131
\(590\) 4.37336 0.180048
\(591\) −7.24175 −0.297886
\(592\) 5.97555 0.245594
\(593\) 38.3036 1.57294 0.786470 0.617629i \(-0.211906\pi\)
0.786470 + 0.617629i \(0.211906\pi\)
\(594\) −2.01757 −0.0827821
\(595\) 4.57091 0.187389
\(596\) −9.45473 −0.387281
\(597\) −0.138861 −0.00568321
\(598\) −15.6822 −0.641294
\(599\) −42.6395 −1.74220 −0.871101 0.491104i \(-0.836594\pi\)
−0.871101 + 0.491104i \(0.836594\pi\)
\(600\) −0.342987 −0.0140024
\(601\) −5.01872 −0.204718 −0.102359 0.994748i \(-0.532639\pi\)
−0.102359 + 0.994748i \(0.532639\pi\)
\(602\) −3.87285 −0.157846
\(603\) −24.4980 −0.997638
\(604\) −11.5779 −0.471099
\(605\) 1.00000 0.0406558
\(606\) −6.34076 −0.257576
\(607\) 16.6739 0.676774 0.338387 0.941007i \(-0.390119\pi\)
0.338387 + 0.941007i \(0.390119\pi\)
\(608\) −2.78382 −0.112899
\(609\) 2.53576 0.102754
\(610\) 9.48586 0.384071
\(611\) 48.9471 1.98019
\(612\) 3.40189 0.137513
\(613\) 47.1966 1.90625 0.953126 0.302572i \(-0.0978453\pi\)
0.953126 + 0.302572i \(0.0978453\pi\)
\(614\) 23.1615 0.934721
\(615\) −3.08326 −0.124329
\(616\) 3.87285 0.156042
\(617\) −25.1261 −1.01154 −0.505770 0.862669i \(-0.668791\pi\)
−0.505770 + 0.862669i \(0.668791\pi\)
\(618\) −1.25674 −0.0505537
\(619\) 42.6966 1.71612 0.858060 0.513549i \(-0.171670\pi\)
0.858060 + 0.513549i \(0.171670\pi\)
\(620\) 2.67757 0.107534
\(621\) −6.22298 −0.249720
\(622\) −26.5655 −1.06518
\(623\) −17.7697 −0.711928
\(624\) −1.74388 −0.0698111
\(625\) 1.00000 0.0400000
\(626\) 6.12755 0.244906
\(627\) −0.954814 −0.0381316
\(628\) −8.78374 −0.350510
\(629\) −7.05260 −0.281206
\(630\) 11.1630 0.444743
\(631\) −49.1073 −1.95493 −0.977464 0.211100i \(-0.932295\pi\)
−0.977464 + 0.211100i \(0.932295\pi\)
\(632\) −4.94508 −0.196705
\(633\) 0.208132 0.00827250
\(634\) 14.7661 0.586437
\(635\) 20.4964 0.813376
\(636\) 2.45726 0.0974366
\(637\) 40.6699 1.61140
\(638\) −1.90897 −0.0755770
\(639\) 31.1968 1.23412
\(640\) 1.00000 0.0395285
\(641\) 14.2281 0.561977 0.280988 0.959711i \(-0.409338\pi\)
0.280988 + 0.959711i \(0.409338\pi\)
\(642\) −3.21703 −0.126966
\(643\) −39.1359 −1.54337 −0.771685 0.636005i \(-0.780586\pi\)
−0.771685 + 0.636005i \(0.780586\pi\)
\(644\) 11.9454 0.470714
\(645\) −0.342987 −0.0135051
\(646\) 3.28558 0.129270
\(647\) −19.6826 −0.773805 −0.386902 0.922121i \(-0.626455\pi\)
−0.386902 + 0.922121i \(0.626455\pi\)
\(648\) 7.95508 0.312505
\(649\) −4.37336 −0.171670
\(650\) 5.08439 0.199426
\(651\) 3.55672 0.139399
\(652\) 8.63082 0.338009
\(653\) 43.2651 1.69309 0.846547 0.532314i \(-0.178678\pi\)
0.846547 + 0.532314i \(0.178678\pi\)
\(654\) −0.280163 −0.0109552
\(655\) 7.50200 0.293127
\(656\) 8.98944 0.350979
\(657\) 13.4960 0.526528
\(658\) −37.2837 −1.45347
\(659\) −37.6626 −1.46713 −0.733563 0.679622i \(-0.762144\pi\)
−0.733563 + 0.679622i \(0.762144\pi\)
\(660\) 0.342987 0.0133508
\(661\) 22.7164 0.883566 0.441783 0.897122i \(-0.354346\pi\)
0.441783 + 0.897122i \(0.354346\pi\)
\(662\) 33.5866 1.30538
\(663\) 2.05820 0.0799340
\(664\) −10.5687 −0.410146
\(665\) 10.7813 0.418082
\(666\) −17.2237 −0.667404
\(667\) −5.88801 −0.227985
\(668\) −9.74131 −0.376903
\(669\) 8.22579 0.318027
\(670\) 8.49930 0.328357
\(671\) −9.48586 −0.366198
\(672\) 1.32834 0.0512418
\(673\) −20.0130 −0.771446 −0.385723 0.922615i \(-0.626048\pi\)
−0.385723 + 0.922615i \(0.626048\pi\)
\(674\) −15.6502 −0.602824
\(675\) 2.01757 0.0776565
\(676\) 12.8510 0.494269
\(677\) 14.0152 0.538650 0.269325 0.963049i \(-0.413200\pi\)
0.269325 + 0.963049i \(0.413200\pi\)
\(678\) 4.23821 0.162767
\(679\) −63.6967 −2.44446
\(680\) −1.18024 −0.0452603
\(681\) 0.0488548 0.00187212
\(682\) −2.67757 −0.102529
\(683\) 21.4336 0.820135 0.410067 0.912055i \(-0.365505\pi\)
0.410067 + 0.912055i \(0.365505\pi\)
\(684\) 8.02397 0.306804
\(685\) 0.602218 0.0230096
\(686\) −3.86893 −0.147716
\(687\) −5.36055 −0.204518
\(688\) 1.00000 0.0381246
\(689\) −36.4260 −1.38772
\(690\) 1.05791 0.0402738
\(691\) −32.2678 −1.22752 −0.613762 0.789491i \(-0.710345\pi\)
−0.613762 + 0.789491i \(0.710345\pi\)
\(692\) 11.4785 0.436346
\(693\) −11.1630 −0.424046
\(694\) −9.10091 −0.345466
\(695\) 12.3783 0.469534
\(696\) −0.654754 −0.0248184
\(697\) −10.6097 −0.401872
\(698\) −11.9911 −0.453872
\(699\) −5.84222 −0.220973
\(700\) −3.87285 −0.146380
\(701\) −24.4051 −0.921769 −0.460884 0.887460i \(-0.652468\pi\)
−0.460884 + 0.887460i \(0.652468\pi\)
\(702\) 10.2581 0.387168
\(703\) −16.6348 −0.627395
\(704\) −1.00000 −0.0376889
\(705\) −3.30192 −0.124357
\(706\) 16.3446 0.615135
\(707\) −71.5969 −2.69268
\(708\) −1.50001 −0.0563737
\(709\) 35.9324 1.34947 0.674734 0.738061i \(-0.264258\pi\)
0.674734 + 0.738061i \(0.264258\pi\)
\(710\) −10.8233 −0.406193
\(711\) 14.2535 0.534548
\(712\) 4.58827 0.171953
\(713\) −8.25866 −0.309289
\(714\) −1.56776 −0.0586721
\(715\) −5.08439 −0.190145
\(716\) −14.4576 −0.540305
\(717\) 2.51276 0.0938409
\(718\) −35.5278 −1.32588
\(719\) −0.377559 −0.0140806 −0.00704028 0.999975i \(-0.502241\pi\)
−0.00704028 + 0.999975i \(0.502241\pi\)
\(720\) −2.88236 −0.107419
\(721\) −14.1906 −0.528485
\(722\) −11.2504 −0.418695
\(723\) −3.35911 −0.124927
\(724\) 4.30048 0.159826
\(725\) 1.90897 0.0708975
\(726\) −0.342987 −0.0127295
\(727\) −45.9459 −1.70404 −0.852020 0.523509i \(-0.824623\pi\)
−0.852020 + 0.523509i \(0.824623\pi\)
\(728\) −19.6911 −0.729800
\(729\) −20.8534 −0.772348
\(730\) −4.68226 −0.173298
\(731\) −1.18024 −0.0436529
\(732\) −3.25353 −0.120254
\(733\) −21.4198 −0.791158 −0.395579 0.918432i \(-0.629456\pi\)
−0.395579 + 0.918432i \(0.629456\pi\)
\(734\) 2.58400 0.0953772
\(735\) −2.74355 −0.101197
\(736\) −3.08439 −0.113692
\(737\) −8.49930 −0.313076
\(738\) −25.9108 −0.953790
\(739\) 44.7591 1.64649 0.823245 0.567686i \(-0.192161\pi\)
0.823245 + 0.567686i \(0.192161\pi\)
\(740\) 5.97555 0.219666
\(741\) 4.85465 0.178340
\(742\) 27.7462 1.01860
\(743\) −4.13324 −0.151634 −0.0758169 0.997122i \(-0.524156\pi\)
−0.0758169 + 0.997122i \(0.524156\pi\)
\(744\) −0.918373 −0.0336692
\(745\) −9.45473 −0.346395
\(746\) 17.5403 0.642196
\(747\) 30.4629 1.11458
\(748\) 1.18024 0.0431540
\(749\) −36.3252 −1.32729
\(750\) −0.342987 −0.0125241
\(751\) −40.9647 −1.49482 −0.747412 0.664360i \(-0.768704\pi\)
−0.747412 + 0.664360i \(0.768704\pi\)
\(752\) 9.62693 0.351058
\(753\) −0.160472 −0.00584793
\(754\) 9.70596 0.353470
\(755\) −11.5779 −0.421364
\(756\) −7.81377 −0.284184
\(757\) −27.6766 −1.00592 −0.502962 0.864309i \(-0.667756\pi\)
−0.502962 + 0.864309i \(0.667756\pi\)
\(758\) 12.8839 0.467964
\(759\) −1.05791 −0.0383995
\(760\) −2.78382 −0.100980
\(761\) 50.2490 1.82153 0.910763 0.412930i \(-0.135495\pi\)
0.910763 + 0.412930i \(0.135495\pi\)
\(762\) −7.03001 −0.254670
\(763\) −3.16347 −0.114525
\(764\) −4.74647 −0.171721
\(765\) 3.40189 0.122995
\(766\) 14.6945 0.530935
\(767\) 22.2359 0.802891
\(768\) −0.342987 −0.0123765
\(769\) 22.3996 0.807749 0.403874 0.914814i \(-0.367663\pi\)
0.403874 + 0.914814i \(0.367663\pi\)
\(770\) 3.87285 0.139568
\(771\) −5.60604 −0.201897
\(772\) −19.1907 −0.690690
\(773\) 18.6315 0.670128 0.335064 0.942195i \(-0.391242\pi\)
0.335064 + 0.942195i \(0.391242\pi\)
\(774\) −2.88236 −0.103604
\(775\) 2.67757 0.0961812
\(776\) 16.4470 0.590412
\(777\) 7.93755 0.284758
\(778\) −12.3657 −0.443331
\(779\) −25.0250 −0.896613
\(780\) −1.74388 −0.0624409
\(781\) 10.8233 0.387290
\(782\) 3.64033 0.130178
\(783\) 3.85150 0.137641
\(784\) 7.99899 0.285678
\(785\) −8.78374 −0.313505
\(786\) −2.57309 −0.0917791
\(787\) −25.2041 −0.898428 −0.449214 0.893424i \(-0.648296\pi\)
−0.449214 + 0.893424i \(0.648296\pi\)
\(788\) 21.1138 0.752147
\(789\) −6.71259 −0.238974
\(790\) −4.94508 −0.175938
\(791\) 47.8559 1.70156
\(792\) 2.88236 0.102420
\(793\) 48.2298 1.71269
\(794\) 6.55812 0.232739
\(795\) 2.45726 0.0871500
\(796\) 0.404858 0.0143498
\(797\) −26.6689 −0.944660 −0.472330 0.881422i \(-0.656587\pi\)
−0.472330 + 0.881422i \(0.656587\pi\)
\(798\) −3.69785 −0.130903
\(799\) −11.3621 −0.401963
\(800\) 1.00000 0.0353553
\(801\) −13.2250 −0.467284
\(802\) −9.12400 −0.322179
\(803\) 4.68226 0.165233
\(804\) −2.91515 −0.102809
\(805\) 11.9454 0.421019
\(806\) 13.6138 0.479526
\(807\) −2.11255 −0.0743651
\(808\) 18.4869 0.650366
\(809\) 12.0456 0.423499 0.211750 0.977324i \(-0.432084\pi\)
0.211750 + 0.977324i \(0.432084\pi\)
\(810\) 7.95508 0.279513
\(811\) 28.4369 0.998554 0.499277 0.866442i \(-0.333599\pi\)
0.499277 + 0.866442i \(0.333599\pi\)
\(812\) −7.39317 −0.259450
\(813\) −1.06090 −0.0372072
\(814\) −5.97555 −0.209443
\(815\) 8.63082 0.302324
\(816\) 0.404809 0.0141711
\(817\) −2.78382 −0.0973935
\(818\) 5.90843 0.206583
\(819\) 56.7568 1.98324
\(820\) 8.98944 0.313925
\(821\) −15.8847 −0.554382 −0.277191 0.960815i \(-0.589403\pi\)
−0.277191 + 0.960815i \(0.589403\pi\)
\(822\) −0.206553 −0.00720437
\(823\) 16.8109 0.585990 0.292995 0.956114i \(-0.405348\pi\)
0.292995 + 0.956114i \(0.405348\pi\)
\(824\) 3.66411 0.127645
\(825\) 0.342987 0.0119413
\(826\) −16.9374 −0.589327
\(827\) 25.1473 0.874457 0.437228 0.899350i \(-0.355960\pi\)
0.437228 + 0.899350i \(0.355960\pi\)
\(828\) 8.89031 0.308960
\(829\) −35.8253 −1.24426 −0.622132 0.782912i \(-0.713733\pi\)
−0.622132 + 0.782912i \(0.713733\pi\)
\(830\) −10.5687 −0.366846
\(831\) 4.69511 0.162871
\(832\) 5.08439 0.176269
\(833\) −9.44075 −0.327103
\(834\) −4.24558 −0.147013
\(835\) −9.74131 −0.337112
\(836\) 2.78382 0.0962804
\(837\) 5.40220 0.186727
\(838\) −20.8979 −0.721907
\(839\) −22.1553 −0.764886 −0.382443 0.923979i \(-0.624917\pi\)
−0.382443 + 0.923979i \(0.624917\pi\)
\(840\) 1.32834 0.0458321
\(841\) −25.3558 −0.874339
\(842\) 1.44685 0.0498616
\(843\) 3.37720 0.116317
\(844\) −0.606821 −0.0208876
\(845\) 12.8510 0.442088
\(846\) −27.7483 −0.954006
\(847\) −3.87285 −0.133073
\(848\) −7.16428 −0.246022
\(849\) 2.99710 0.102860
\(850\) −1.18024 −0.0404820
\(851\) −18.4309 −0.631803
\(852\) 3.71227 0.127180
\(853\) −46.0026 −1.57510 −0.787550 0.616250i \(-0.788651\pi\)
−0.787550 + 0.616250i \(0.788651\pi\)
\(854\) −36.7373 −1.25713
\(855\) 8.02397 0.274414
\(856\) 9.37944 0.320583
\(857\) 8.42733 0.287872 0.143936 0.989587i \(-0.454024\pi\)
0.143936 + 0.989587i \(0.454024\pi\)
\(858\) 1.74388 0.0595351
\(859\) 9.83910 0.335706 0.167853 0.985812i \(-0.446317\pi\)
0.167853 + 0.985812i \(0.446317\pi\)
\(860\) 1.00000 0.0340997
\(861\) 11.9410 0.406949
\(862\) −12.0796 −0.411434
\(863\) −43.7032 −1.48767 −0.743837 0.668361i \(-0.766996\pi\)
−0.743837 + 0.668361i \(0.766996\pi\)
\(864\) 2.01757 0.0686393
\(865\) 11.4785 0.390280
\(866\) −27.3298 −0.928704
\(867\) 5.35301 0.181798
\(868\) −10.3698 −0.351975
\(869\) 4.94508 0.167750
\(870\) −0.654754 −0.0221982
\(871\) 43.2137 1.46424
\(872\) 0.816833 0.0276614
\(873\) −47.4061 −1.60445
\(874\) 8.58637 0.290438
\(875\) −3.87285 −0.130926
\(876\) 1.60596 0.0542602
\(877\) −1.48322 −0.0500848 −0.0250424 0.999686i \(-0.507972\pi\)
−0.0250424 + 0.999686i \(0.507972\pi\)
\(878\) −6.28995 −0.212275
\(879\) 2.75230 0.0928328
\(880\) −1.00000 −0.0337100
\(881\) 6.49472 0.218813 0.109406 0.993997i \(-0.465105\pi\)
0.109406 + 0.993997i \(0.465105\pi\)
\(882\) −23.0560 −0.776334
\(883\) 27.3086 0.919007 0.459503 0.888176i \(-0.348027\pi\)
0.459503 + 0.888176i \(0.348027\pi\)
\(884\) −6.00082 −0.201829
\(885\) −1.50001 −0.0504222
\(886\) 32.8673 1.10420
\(887\) 3.86410 0.129744 0.0648719 0.997894i \(-0.479336\pi\)
0.0648719 + 0.997894i \(0.479336\pi\)
\(888\) −2.04954 −0.0687780
\(889\) −79.3796 −2.66231
\(890\) 4.58827 0.153799
\(891\) −7.95508 −0.266505
\(892\) −23.9828 −0.803003
\(893\) −26.7996 −0.896816
\(894\) 3.24285 0.108457
\(895\) −14.4576 −0.483264
\(896\) −3.87285 −0.129383
\(897\) 5.37880 0.179593
\(898\) −38.0446 −1.26957
\(899\) 5.11141 0.170475
\(900\) −2.88236 −0.0960787
\(901\) 8.45560 0.281697
\(902\) −8.98944 −0.299316
\(903\) 1.32834 0.0442044
\(904\) −12.3567 −0.410979
\(905\) 4.30048 0.142953
\(906\) 3.97108 0.131930
\(907\) 5.73441 0.190408 0.0952040 0.995458i \(-0.469650\pi\)
0.0952040 + 0.995458i \(0.469650\pi\)
\(908\) −0.142439 −0.00472701
\(909\) −53.2858 −1.76738
\(910\) −19.6911 −0.652753
\(911\) 29.0566 0.962689 0.481345 0.876531i \(-0.340149\pi\)
0.481345 + 0.876531i \(0.340149\pi\)
\(912\) 0.954814 0.0316171
\(913\) 10.5687 0.349774
\(914\) 10.3079 0.340955
\(915\) −3.25353 −0.107558
\(916\) 15.6290 0.516397
\(917\) −29.0541 −0.959452
\(918\) −2.38123 −0.0785923
\(919\) 33.7416 1.11303 0.556516 0.830837i \(-0.312138\pi\)
0.556516 + 0.830837i \(0.312138\pi\)
\(920\) −3.08439 −0.101689
\(921\) −7.94409 −0.261767
\(922\) −25.5297 −0.840776
\(923\) −55.0301 −1.81134
\(924\) −1.32834 −0.0436991
\(925\) 5.97555 0.196475
\(926\) 19.6089 0.644390
\(927\) −10.5613 −0.346878
\(928\) 1.90897 0.0626651
\(929\) −35.4076 −1.16169 −0.580843 0.814016i \(-0.697277\pi\)
−0.580843 + 0.814016i \(0.697277\pi\)
\(930\) −0.918373 −0.0301146
\(931\) −22.2677 −0.729795
\(932\) 17.0333 0.557946
\(933\) 9.11162 0.298301
\(934\) 16.8330 0.550792
\(935\) 1.18024 0.0385981
\(936\) −14.6550 −0.479015
\(937\) 22.6892 0.741224 0.370612 0.928788i \(-0.379148\pi\)
0.370612 + 0.928788i \(0.379148\pi\)
\(938\) −32.9165 −1.07476
\(939\) −2.10167 −0.0685855
\(940\) 9.62693 0.313996
\(941\) 31.0603 1.01254 0.506268 0.862376i \(-0.331025\pi\)
0.506268 + 0.862376i \(0.331025\pi\)
\(942\) 3.01271 0.0981595
\(943\) −27.7269 −0.902913
\(944\) 4.37336 0.142341
\(945\) −7.81377 −0.254182
\(946\) −1.00000 −0.0325128
\(947\) 5.88068 0.191096 0.0955482 0.995425i \(-0.469540\pi\)
0.0955482 + 0.995425i \(0.469540\pi\)
\(948\) 1.69610 0.0550868
\(949\) −23.8064 −0.772789
\(950\) −2.78382 −0.0903190
\(951\) −5.06458 −0.164230
\(952\) 4.57091 0.148144
\(953\) −28.1041 −0.910381 −0.455190 0.890394i \(-0.650429\pi\)
−0.455190 + 0.890394i \(0.650429\pi\)
\(954\) 20.6500 0.668570
\(955\) −4.74647 −0.153592
\(956\) −7.32611 −0.236943
\(957\) 0.654754 0.0211652
\(958\) 17.2296 0.556663
\(959\) −2.33230 −0.0753139
\(960\) −0.342987 −0.0110699
\(961\) −23.8306 −0.768729
\(962\) 30.3820 0.979555
\(963\) −27.0349 −0.871188
\(964\) 9.79369 0.315433
\(965\) −19.1907 −0.617772
\(966\) −4.09711 −0.131822
\(967\) −16.3611 −0.526136 −0.263068 0.964777i \(-0.584734\pi\)
−0.263068 + 0.964777i \(0.584734\pi\)
\(968\) 1.00000 0.0321412
\(969\) −1.12691 −0.0362017
\(970\) 16.4470 0.528081
\(971\) −44.8721 −1.44002 −0.720008 0.693966i \(-0.755862\pi\)
−0.720008 + 0.693966i \(0.755862\pi\)
\(972\) −8.78121 −0.281658
\(973\) −47.9392 −1.53686
\(974\) −2.57189 −0.0824085
\(975\) −1.74388 −0.0558489
\(976\) 9.48586 0.303635
\(977\) −54.2467 −1.73551 −0.867753 0.496995i \(-0.834437\pi\)
−0.867753 + 0.496995i \(0.834437\pi\)
\(978\) −2.96026 −0.0946587
\(979\) −4.58827 −0.146642
\(980\) 7.99899 0.255518
\(981\) −2.35441 −0.0751704
\(982\) 39.5104 1.26083
\(983\) −44.6995 −1.42569 −0.712847 0.701320i \(-0.752595\pi\)
−0.712847 + 0.701320i \(0.752595\pi\)
\(984\) −3.08326 −0.0982909
\(985\) 21.1138 0.672741
\(986\) −2.25305 −0.0717519
\(987\) 12.7878 0.407041
\(988\) −14.1540 −0.450299
\(989\) −3.08439 −0.0980778
\(990\) 2.88236 0.0916074
\(991\) 7.79017 0.247463 0.123731 0.992316i \(-0.460514\pi\)
0.123731 + 0.992316i \(0.460514\pi\)
\(992\) 2.67757 0.0850130
\(993\) −11.5198 −0.365569
\(994\) 41.9172 1.32953
\(995\) 0.404858 0.0128349
\(996\) 3.62494 0.114861
\(997\) 2.00926 0.0636340 0.0318170 0.999494i \(-0.489871\pi\)
0.0318170 + 0.999494i \(0.489871\pi\)
\(998\) −15.7140 −0.497419
\(999\) 12.0561 0.381439
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 4730.2.a.be.1.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4730.2.a.be.1.6 12 1.1 even 1 trivial