Properties

Label 4730.2.a.be.1.4
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 1.4
Root \(-1.94615\)
Character \(\chi\) = 4730.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.94615 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.94615 q^{6} -1.80392 q^{7} +1.00000 q^{8} +0.787492 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.94615 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.94615 q^{6} -1.80392 q^{7} +1.00000 q^{8} +0.787492 q^{9} +1.00000 q^{10} -1.00000 q^{11} -1.94615 q^{12} -3.07092 q^{13} -1.80392 q^{14} -1.94615 q^{15} +1.00000 q^{16} -2.66744 q^{17} +0.787492 q^{18} -7.67236 q^{19} +1.00000 q^{20} +3.51070 q^{21} -1.00000 q^{22} +5.07092 q^{23} -1.94615 q^{24} +1.00000 q^{25} -3.07092 q^{26} +4.30587 q^{27} -1.80392 q^{28} +6.04298 q^{29} -1.94615 q^{30} -6.12573 q^{31} +1.00000 q^{32} +1.94615 q^{33} -2.66744 q^{34} -1.80392 q^{35} +0.787492 q^{36} +6.64486 q^{37} -7.67236 q^{38} +5.97647 q^{39} +1.00000 q^{40} +3.25559 q^{41} +3.51070 q^{42} +1.00000 q^{43} -1.00000 q^{44} +0.787492 q^{45} +5.07092 q^{46} -9.42291 q^{47} -1.94615 q^{48} -3.74586 q^{49} +1.00000 q^{50} +5.19123 q^{51} -3.07092 q^{52} +8.46618 q^{53} +4.30587 q^{54} -1.00000 q^{55} -1.80392 q^{56} +14.9316 q^{57} +6.04298 q^{58} +4.12176 q^{59} -1.94615 q^{60} +8.30029 q^{61} -6.12573 q^{62} -1.42058 q^{63} +1.00000 q^{64} -3.07092 q^{65} +1.94615 q^{66} +6.38267 q^{67} -2.66744 q^{68} -9.86877 q^{69} -1.80392 q^{70} +7.02540 q^{71} +0.787492 q^{72} -6.73478 q^{73} +6.64486 q^{74} -1.94615 q^{75} -7.67236 q^{76} +1.80392 q^{77} +5.97647 q^{78} +15.7403 q^{79} +1.00000 q^{80} -10.7423 q^{81} +3.25559 q^{82} +8.30389 q^{83} +3.51070 q^{84} -2.66744 q^{85} +1.00000 q^{86} -11.7605 q^{87} -1.00000 q^{88} +0.597136 q^{89} +0.787492 q^{90} +5.53971 q^{91} +5.07092 q^{92} +11.9216 q^{93} -9.42291 q^{94} -7.67236 q^{95} -1.94615 q^{96} +11.1598 q^{97} -3.74586 q^{98} -0.787492 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 12q^{2} + 3q^{3} + 12q^{4} + 12q^{5} + 3q^{6} + 8q^{7} + 12q^{8} + 25q^{9} + O(q^{10}) \) \( 12q + 12q^{2} + 3q^{3} + 12q^{4} + 12q^{5} + 3q^{6} + 8q^{7} + 12q^{8} + 25q^{9} + 12q^{10} - 12q^{11} + 3q^{12} + 16q^{13} + 8q^{14} + 3q^{15} + 12q^{16} + 18q^{17} + 25q^{18} - 4q^{19} + 12q^{20} + 4q^{21} - 12q^{22} + 8q^{23} + 3q^{24} + 12q^{25} + 16q^{26} + 6q^{27} + 8q^{28} + 20q^{29} + 3q^{30} + 5q^{31} + 12q^{32} - 3q^{33} + 18q^{34} + 8q^{35} + 25q^{36} + 19q^{37} - 4q^{38} + 6q^{39} + 12q^{40} + 16q^{41} + 4q^{42} + 12q^{43} - 12q^{44} + 25q^{45} + 8q^{46} - q^{47} + 3q^{48} + 52q^{49} + 12q^{50} + q^{51} + 16q^{52} + 11q^{53} + 6q^{54} - 12q^{55} + 8q^{56} + 9q^{57} + 20q^{58} - 11q^{59} + 3q^{60} + 18q^{61} + 5q^{62} + 15q^{63} + 12q^{64} + 16q^{65} - 3q^{66} - 10q^{67} + 18q^{68} + 8q^{70} - 2q^{71} + 25q^{72} + 29q^{73} + 19q^{74} + 3q^{75} - 4q^{76} - 8q^{77} + 6q^{78} + 2q^{79} + 12q^{80} - 8q^{81} + 16q^{82} + 26q^{83} + 4q^{84} + 18q^{85} + 12q^{86} - 4q^{87} - 12q^{88} + 41q^{89} + 25q^{90} - 4q^{91} + 8q^{92} + 5q^{93} - q^{94} - 4q^{95} + 3q^{96} - 7q^{97} + 52q^{98} - 25q^{99} + O(q^{100}) \)

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
\(3\) −1.94615 −1.12361 −0.561805 0.827270i \(-0.689893\pi\)
−0.561805 + 0.827270i \(0.689893\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.94615 −0.794512
\(7\) −1.80392 −0.681819 −0.340910 0.940096i \(-0.610735\pi\)
−0.340910 + 0.940096i \(0.610735\pi\)
\(8\) 1.00000 0.353553
\(9\) 0.787492 0.262497
\(10\) 1.00000 0.316228
\(11\) −1.00000 −0.301511
\(12\) −1.94615 −0.561805
\(13\) −3.07092 −0.851721 −0.425860 0.904789i \(-0.640029\pi\)
−0.425860 + 0.904789i \(0.640029\pi\)
\(14\) −1.80392 −0.482119
\(15\) −1.94615 −0.502493
\(16\) 1.00000 0.250000
\(17\) −2.66744 −0.646949 −0.323474 0.946237i \(-0.604851\pi\)
−0.323474 + 0.946237i \(0.604851\pi\)
\(18\) 0.787492 0.185614
\(19\) −7.67236 −1.76016 −0.880081 0.474824i \(-0.842512\pi\)
−0.880081 + 0.474824i \(0.842512\pi\)
\(20\) 1.00000 0.223607
\(21\) 3.51070 0.766098
\(22\) −1.00000 −0.213201
\(23\) 5.07092 1.05736 0.528680 0.848821i \(-0.322687\pi\)
0.528680 + 0.848821i \(0.322687\pi\)
\(24\) −1.94615 −0.397256
\(25\) 1.00000 0.200000
\(26\) −3.07092 −0.602258
\(27\) 4.30587 0.828665
\(28\) −1.80392 −0.340910
\(29\) 6.04298 1.12215 0.561077 0.827764i \(-0.310387\pi\)
0.561077 + 0.827764i \(0.310387\pi\)
\(30\) −1.94615 −0.355316
\(31\) −6.12573 −1.10021 −0.550107 0.835094i \(-0.685413\pi\)
−0.550107 + 0.835094i \(0.685413\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.94615 0.338781
\(34\) −2.66744 −0.457462
\(35\) −1.80392 −0.304919
\(36\) 0.787492 0.131249
\(37\) 6.64486 1.09241 0.546204 0.837652i \(-0.316072\pi\)
0.546204 + 0.837652i \(0.316072\pi\)
\(38\) −7.67236 −1.24462
\(39\) 5.97647 0.957001
\(40\) 1.00000 0.158114
\(41\) 3.25559 0.508437 0.254219 0.967147i \(-0.418182\pi\)
0.254219 + 0.967147i \(0.418182\pi\)
\(42\) 3.51070 0.541713
\(43\) 1.00000 0.152499
\(44\) −1.00000 −0.150756
\(45\) 0.787492 0.117392
\(46\) 5.07092 0.747667
\(47\) −9.42291 −1.37447 −0.687236 0.726434i \(-0.741176\pi\)
−0.687236 + 0.726434i \(0.741176\pi\)
\(48\) −1.94615 −0.280902
\(49\) −3.74586 −0.535122
\(50\) 1.00000 0.141421
\(51\) 5.19123 0.726917
\(52\) −3.07092 −0.425860
\(53\) 8.46618 1.16292 0.581460 0.813575i \(-0.302482\pi\)
0.581460 + 0.813575i \(0.302482\pi\)
\(54\) 4.30587 0.585954
\(55\) −1.00000 −0.134840
\(56\) −1.80392 −0.241060
\(57\) 14.9316 1.97773
\(58\) 6.04298 0.793483
\(59\) 4.12176 0.536607 0.268304 0.963334i \(-0.413537\pi\)
0.268304 + 0.963334i \(0.413537\pi\)
\(60\) −1.94615 −0.251247
\(61\) 8.30029 1.06274 0.531372 0.847139i \(-0.321677\pi\)
0.531372 + 0.847139i \(0.321677\pi\)
\(62\) −6.12573 −0.777968
\(63\) −1.42058 −0.178976
\(64\) 1.00000 0.125000
\(65\) −3.07092 −0.380901
\(66\) 1.94615 0.239554
\(67\) 6.38267 0.779767 0.389883 0.920864i \(-0.372515\pi\)
0.389883 + 0.920864i \(0.372515\pi\)
\(68\) −2.66744 −0.323474
\(69\) −9.86877 −1.18806
\(70\) −1.80392 −0.215610
\(71\) 7.02540 0.833762 0.416881 0.908961i \(-0.363123\pi\)
0.416881 + 0.908961i \(0.363123\pi\)
\(72\) 0.787492 0.0928068
\(73\) −6.73478 −0.788246 −0.394123 0.919058i \(-0.628952\pi\)
−0.394123 + 0.919058i \(0.628952\pi\)
\(74\) 6.64486 0.772449
\(75\) −1.94615 −0.224722
\(76\) −7.67236 −0.880081
\(77\) 1.80392 0.205576
\(78\) 5.97647 0.676702
\(79\) 15.7403 1.77093 0.885464 0.464708i \(-0.153841\pi\)
0.885464 + 0.464708i \(0.153841\pi\)
\(80\) 1.00000 0.111803
\(81\) −10.7423 −1.19359
\(82\) 3.25559 0.359519
\(83\) 8.30389 0.911470 0.455735 0.890115i \(-0.349376\pi\)
0.455735 + 0.890115i \(0.349376\pi\)
\(84\) 3.51070 0.383049
\(85\) −2.66744 −0.289324
\(86\) 1.00000 0.107833
\(87\) −11.7605 −1.26086
\(88\) −1.00000 −0.106600
\(89\) 0.597136 0.0632963 0.0316482 0.999499i \(-0.489924\pi\)
0.0316482 + 0.999499i \(0.489924\pi\)
\(90\) 0.787492 0.0830089
\(91\) 5.53971 0.580720
\(92\) 5.07092 0.528680
\(93\) 11.9216 1.23621
\(94\) −9.42291 −0.971899
\(95\) −7.67236 −0.787168
\(96\) −1.94615 −0.198628
\(97\) 11.1598 1.13311 0.566553 0.824025i \(-0.308277\pi\)
0.566553 + 0.824025i \(0.308277\pi\)
\(98\) −3.74586 −0.378389
\(99\) −0.787492 −0.0791459
\(100\) 1.00000 0.100000
\(101\) 2.12892 0.211835 0.105918 0.994375i \(-0.466222\pi\)
0.105918 + 0.994375i \(0.466222\pi\)
\(102\) 5.19123 0.514008
\(103\) −14.9767 −1.47570 −0.737850 0.674965i \(-0.764159\pi\)
−0.737850 + 0.674965i \(0.764159\pi\)
\(104\) −3.07092 −0.301129
\(105\) 3.51070 0.342610
\(106\) 8.46618 0.822308
\(107\) 8.15661 0.788529 0.394265 0.918997i \(-0.370999\pi\)
0.394265 + 0.918997i \(0.370999\pi\)
\(108\) 4.30587 0.414332
\(109\) 1.25351 0.120064 0.0600320 0.998196i \(-0.480880\pi\)
0.0600320 + 0.998196i \(0.480880\pi\)
\(110\) −1.00000 −0.0953463
\(111\) −12.9319 −1.22744
\(112\) −1.80392 −0.170455
\(113\) 11.4892 1.08082 0.540408 0.841403i \(-0.318270\pi\)
0.540408 + 0.841403i \(0.318270\pi\)
\(114\) 14.9316 1.39847
\(115\) 5.07092 0.472866
\(116\) 6.04298 0.561077
\(117\) −2.41833 −0.223574
\(118\) 4.12176 0.379439
\(119\) 4.81185 0.441102
\(120\) −1.94615 −0.177658
\(121\) 1.00000 0.0909091
\(122\) 8.30029 0.751473
\(123\) −6.33585 −0.571285
\(124\) −6.12573 −0.550107
\(125\) 1.00000 0.0894427
\(126\) −1.42058 −0.126555
\(127\) −19.9054 −1.76632 −0.883159 0.469073i \(-0.844588\pi\)
−0.883159 + 0.469073i \(0.844588\pi\)
\(128\) 1.00000 0.0883883
\(129\) −1.94615 −0.171349
\(130\) −3.07092 −0.269338
\(131\) −2.04693 −0.178841 −0.0894206 0.995994i \(-0.528502\pi\)
−0.0894206 + 0.995994i \(0.528502\pi\)
\(132\) 1.94615 0.169390
\(133\) 13.8404 1.20011
\(134\) 6.38267 0.551379
\(135\) 4.30587 0.370590
\(136\) −2.66744 −0.228731
\(137\) −2.08422 −0.178067 −0.0890335 0.996029i \(-0.528378\pi\)
−0.0890335 + 0.996029i \(0.528378\pi\)
\(138\) −9.86877 −0.840085
\(139\) −20.3274 −1.72415 −0.862075 0.506781i \(-0.830835\pi\)
−0.862075 + 0.506781i \(0.830835\pi\)
\(140\) −1.80392 −0.152459
\(141\) 18.3384 1.54437
\(142\) 7.02540 0.589559
\(143\) 3.07092 0.256804
\(144\) 0.787492 0.0656243
\(145\) 6.04298 0.501842
\(146\) −6.73478 −0.557374
\(147\) 7.28999 0.601268
\(148\) 6.64486 0.546204
\(149\) 17.2922 1.41663 0.708316 0.705895i \(-0.249455\pi\)
0.708316 + 0.705895i \(0.249455\pi\)
\(150\) −1.94615 −0.158902
\(151\) −2.41168 −0.196260 −0.0981298 0.995174i \(-0.531286\pi\)
−0.0981298 + 0.995174i \(0.531286\pi\)
\(152\) −7.67236 −0.622311
\(153\) −2.10059 −0.169822
\(154\) 1.80392 0.145364
\(155\) −6.12573 −0.492030
\(156\) 5.97647 0.478501
\(157\) −12.7552 −1.01798 −0.508989 0.860773i \(-0.669981\pi\)
−0.508989 + 0.860773i \(0.669981\pi\)
\(158\) 15.7403 1.25223
\(159\) −16.4764 −1.30667
\(160\) 1.00000 0.0790569
\(161\) −9.14756 −0.720929
\(162\) −10.7423 −0.843997
\(163\) −1.86939 −0.146422 −0.0732108 0.997316i \(-0.523325\pi\)
−0.0732108 + 0.997316i \(0.523325\pi\)
\(164\) 3.25559 0.254219
\(165\) 1.94615 0.151507
\(166\) 8.30389 0.644507
\(167\) 25.2636 1.95496 0.977480 0.211029i \(-0.0676814\pi\)
0.977480 + 0.211029i \(0.0676814\pi\)
\(168\) 3.51070 0.270857
\(169\) −3.56943 −0.274572
\(170\) −2.66744 −0.204583
\(171\) −6.04193 −0.462038
\(172\) 1.00000 0.0762493
\(173\) 22.5713 1.71606 0.858032 0.513596i \(-0.171687\pi\)
0.858032 + 0.513596i \(0.171687\pi\)
\(174\) −11.7605 −0.891564
\(175\) −1.80392 −0.136364
\(176\) −1.00000 −0.0753778
\(177\) −8.02155 −0.602937
\(178\) 0.597136 0.0447572
\(179\) −17.0575 −1.27494 −0.637469 0.770476i \(-0.720019\pi\)
−0.637469 + 0.770476i \(0.720019\pi\)
\(180\) 0.787492 0.0586962
\(181\) −11.8662 −0.882007 −0.441004 0.897505i \(-0.645377\pi\)
−0.441004 + 0.897505i \(0.645377\pi\)
\(182\) 5.53971 0.410631
\(183\) −16.1536 −1.19411
\(184\) 5.07092 0.373833
\(185\) 6.64486 0.488540
\(186\) 11.9216 0.874132
\(187\) 2.66744 0.195062
\(188\) −9.42291 −0.687236
\(189\) −7.76746 −0.565000
\(190\) −7.67236 −0.556612
\(191\) −9.03514 −0.653760 −0.326880 0.945066i \(-0.605997\pi\)
−0.326880 + 0.945066i \(0.605997\pi\)
\(192\) −1.94615 −0.140451
\(193\) 19.0304 1.36984 0.684918 0.728621i \(-0.259838\pi\)
0.684918 + 0.728621i \(0.259838\pi\)
\(194\) 11.1598 0.801227
\(195\) 5.97647 0.427984
\(196\) −3.74586 −0.267561
\(197\) −8.57050 −0.610623 −0.305311 0.952253i \(-0.598761\pi\)
−0.305311 + 0.952253i \(0.598761\pi\)
\(198\) −0.787492 −0.0559646
\(199\) −7.56214 −0.536066 −0.268033 0.963410i \(-0.586374\pi\)
−0.268033 + 0.963410i \(0.586374\pi\)
\(200\) 1.00000 0.0707107
\(201\) −12.4216 −0.876153
\(202\) 2.12892 0.149790
\(203\) −10.9011 −0.765106
\(204\) 5.19123 0.363459
\(205\) 3.25559 0.227380
\(206\) −14.9767 −1.04348
\(207\) 3.99331 0.277554
\(208\) −3.07092 −0.212930
\(209\) 7.67236 0.530709
\(210\) 3.51070 0.242262
\(211\) 21.6038 1.48727 0.743635 0.668586i \(-0.233100\pi\)
0.743635 + 0.668586i \(0.233100\pi\)
\(212\) 8.46618 0.581460
\(213\) −13.6725 −0.936822
\(214\) 8.15661 0.557574
\(215\) 1.00000 0.0681994
\(216\) 4.30587 0.292977
\(217\) 11.0504 0.750147
\(218\) 1.25351 0.0848981
\(219\) 13.1069 0.885681
\(220\) −1.00000 −0.0674200
\(221\) 8.19150 0.551020
\(222\) −12.9319 −0.867931
\(223\) 12.6642 0.848060 0.424030 0.905648i \(-0.360615\pi\)
0.424030 + 0.905648i \(0.360615\pi\)
\(224\) −1.80392 −0.120530
\(225\) 0.787492 0.0524995
\(226\) 11.4892 0.764252
\(227\) 24.1290 1.60150 0.800751 0.598998i \(-0.204434\pi\)
0.800751 + 0.598998i \(0.204434\pi\)
\(228\) 14.9316 0.988867
\(229\) −24.5929 −1.62515 −0.812573 0.582860i \(-0.801934\pi\)
−0.812573 + 0.582860i \(0.801934\pi\)
\(230\) 5.07092 0.334367
\(231\) −3.51070 −0.230987
\(232\) 6.04298 0.396741
\(233\) 26.9616 1.76631 0.883157 0.469077i \(-0.155413\pi\)
0.883157 + 0.469077i \(0.155413\pi\)
\(234\) −2.41833 −0.158091
\(235\) −9.42291 −0.614683
\(236\) 4.12176 0.268304
\(237\) −30.6330 −1.98983
\(238\) 4.81185 0.311906
\(239\) 15.3401 0.992270 0.496135 0.868245i \(-0.334752\pi\)
0.496135 + 0.868245i \(0.334752\pi\)
\(240\) −1.94615 −0.125623
\(241\) −11.2781 −0.726487 −0.363244 0.931694i \(-0.618331\pi\)
−0.363244 + 0.931694i \(0.618331\pi\)
\(242\) 1.00000 0.0642824
\(243\) 7.98856 0.512467
\(244\) 8.30029 0.531372
\(245\) −3.74586 −0.239314
\(246\) −6.33585 −0.403959
\(247\) 23.5612 1.49917
\(248\) −6.12573 −0.388984
\(249\) −16.1606 −1.02414
\(250\) 1.00000 0.0632456
\(251\) −1.91281 −0.120736 −0.0603678 0.998176i \(-0.519227\pi\)
−0.0603678 + 0.998176i \(0.519227\pi\)
\(252\) −1.42058 −0.0894879
\(253\) −5.07092 −0.318806
\(254\) −19.9054 −1.24898
\(255\) 5.19123 0.325087
\(256\) 1.00000 0.0625000
\(257\) 14.3787 0.896920 0.448460 0.893803i \(-0.351973\pi\)
0.448460 + 0.893803i \(0.351973\pi\)
\(258\) −1.94615 −0.121162
\(259\) −11.9868 −0.744825
\(260\) −3.07092 −0.190451
\(261\) 4.75880 0.294562
\(262\) −2.04693 −0.126460
\(263\) 13.3732 0.824624 0.412312 0.911043i \(-0.364721\pi\)
0.412312 + 0.911043i \(0.364721\pi\)
\(264\) 1.94615 0.119777
\(265\) 8.46618 0.520073
\(266\) 13.8404 0.848607
\(267\) −1.16212 −0.0711203
\(268\) 6.38267 0.389883
\(269\) 3.86207 0.235475 0.117737 0.993045i \(-0.462436\pi\)
0.117737 + 0.993045i \(0.462436\pi\)
\(270\) 4.30587 0.262047
\(271\) 17.8656 1.08526 0.542630 0.839972i \(-0.317429\pi\)
0.542630 + 0.839972i \(0.317429\pi\)
\(272\) −2.66744 −0.161737
\(273\) −10.7811 −0.652502
\(274\) −2.08422 −0.125912
\(275\) −1.00000 −0.0603023
\(276\) −9.86877 −0.594030
\(277\) 3.55770 0.213762 0.106881 0.994272i \(-0.465914\pi\)
0.106881 + 0.994272i \(0.465914\pi\)
\(278\) −20.3274 −1.21916
\(279\) −4.82396 −0.288803
\(280\) −1.80392 −0.107805
\(281\) 18.7205 1.11677 0.558385 0.829582i \(-0.311421\pi\)
0.558385 + 0.829582i \(0.311421\pi\)
\(282\) 18.3384 1.09203
\(283\) 22.2481 1.32251 0.661255 0.750161i \(-0.270024\pi\)
0.661255 + 0.750161i \(0.270024\pi\)
\(284\) 7.02540 0.416881
\(285\) 14.9316 0.884469
\(286\) 3.07092 0.181588
\(287\) −5.87283 −0.346662
\(288\) 0.787492 0.0464034
\(289\) −9.88478 −0.581458
\(290\) 6.04298 0.354856
\(291\) −21.7186 −1.27317
\(292\) −6.73478 −0.394123
\(293\) 16.1883 0.945728 0.472864 0.881135i \(-0.343220\pi\)
0.472864 + 0.881135i \(0.343220\pi\)
\(294\) 7.28999 0.425161
\(295\) 4.12176 0.239978
\(296\) 6.64486 0.386225
\(297\) −4.30587 −0.249852
\(298\) 17.2922 1.00171
\(299\) −15.5724 −0.900576
\(300\) −1.94615 −0.112361
\(301\) −1.80392 −0.103976
\(302\) −2.41168 −0.138777
\(303\) −4.14319 −0.238020
\(304\) −7.67236 −0.440040
\(305\) 8.30029 0.475273
\(306\) −2.10059 −0.120082
\(307\) 9.08961 0.518771 0.259386 0.965774i \(-0.416480\pi\)
0.259386 + 0.965774i \(0.416480\pi\)
\(308\) 1.80392 0.102788
\(309\) 29.1469 1.65811
\(310\) −6.12573 −0.347918
\(311\) 12.2692 0.695724 0.347862 0.937546i \(-0.386908\pi\)
0.347862 + 0.937546i \(0.386908\pi\)
\(312\) 5.97647 0.338351
\(313\) 29.4204 1.66294 0.831471 0.555568i \(-0.187499\pi\)
0.831471 + 0.555568i \(0.187499\pi\)
\(314\) −12.7552 −0.719819
\(315\) −1.42058 −0.0800404
\(316\) 15.7403 0.885464
\(317\) −31.2392 −1.75457 −0.877285 0.479969i \(-0.840648\pi\)
−0.877285 + 0.479969i \(0.840648\pi\)
\(318\) −16.4764 −0.923953
\(319\) −6.04298 −0.338342
\(320\) 1.00000 0.0559017
\(321\) −15.8740 −0.885998
\(322\) −9.14756 −0.509774
\(323\) 20.4656 1.13873
\(324\) −10.7423 −0.596796
\(325\) −3.07092 −0.170344
\(326\) −1.86939 −0.103536
\(327\) −2.43951 −0.134905
\(328\) 3.25559 0.179760
\(329\) 16.9982 0.937142
\(330\) 1.94615 0.107132
\(331\) −0.985673 −0.0541775 −0.0270887 0.999633i \(-0.508624\pi\)
−0.0270887 + 0.999633i \(0.508624\pi\)
\(332\) 8.30389 0.455735
\(333\) 5.23277 0.286754
\(334\) 25.2636 1.38237
\(335\) 6.38267 0.348722
\(336\) 3.51070 0.191525
\(337\) 21.8560 1.19057 0.595287 0.803513i \(-0.297038\pi\)
0.595287 + 0.803513i \(0.297038\pi\)
\(338\) −3.56943 −0.194151
\(339\) −22.3597 −1.21441
\(340\) −2.66744 −0.144662
\(341\) 6.12573 0.331727
\(342\) −6.04193 −0.326710
\(343\) 19.3847 1.04668
\(344\) 1.00000 0.0539164
\(345\) −9.86877 −0.531317
\(346\) 22.5713 1.21344
\(347\) −21.7521 −1.16771 −0.583856 0.811857i \(-0.698457\pi\)
−0.583856 + 0.811857i \(0.698457\pi\)
\(348\) −11.7605 −0.630431
\(349\) −34.1659 −1.82886 −0.914429 0.404746i \(-0.867360\pi\)
−0.914429 + 0.404746i \(0.867360\pi\)
\(350\) −1.80392 −0.0964238
\(351\) −13.2230 −0.705791
\(352\) −1.00000 −0.0533002
\(353\) −0.428704 −0.0228176 −0.0114088 0.999935i \(-0.503632\pi\)
−0.0114088 + 0.999935i \(0.503632\pi\)
\(354\) −8.02155 −0.426341
\(355\) 7.02540 0.372870
\(356\) 0.597136 0.0316482
\(357\) −9.36458 −0.495626
\(358\) −17.0575 −0.901517
\(359\) −23.1249 −1.22049 −0.610243 0.792214i \(-0.708928\pi\)
−0.610243 + 0.792214i \(0.708928\pi\)
\(360\) 0.787492 0.0415045
\(361\) 39.8652 2.09817
\(362\) −11.8662 −0.623673
\(363\) −1.94615 −0.102146
\(364\) 5.53971 0.290360
\(365\) −6.73478 −0.352514
\(366\) −16.1536 −0.844362
\(367\) −1.79699 −0.0938023 −0.0469011 0.998900i \(-0.514935\pi\)
−0.0469011 + 0.998900i \(0.514935\pi\)
\(368\) 5.07092 0.264340
\(369\) 2.56375 0.133463
\(370\) 6.64486 0.345450
\(371\) −15.2723 −0.792901
\(372\) 11.9216 0.618105
\(373\) 21.7123 1.12422 0.562110 0.827063i \(-0.309990\pi\)
0.562110 + 0.827063i \(0.309990\pi\)
\(374\) 2.66744 0.137930
\(375\) −1.94615 −0.100499
\(376\) −9.42291 −0.485949
\(377\) −18.5575 −0.955762
\(378\) −7.76746 −0.399515
\(379\) 13.5210 0.694525 0.347262 0.937768i \(-0.387111\pi\)
0.347262 + 0.937768i \(0.387111\pi\)
\(380\) −7.67236 −0.393584
\(381\) 38.7389 1.98465
\(382\) −9.03514 −0.462278
\(383\) 10.8496 0.554389 0.277194 0.960814i \(-0.410595\pi\)
0.277194 + 0.960814i \(0.410595\pi\)
\(384\) −1.94615 −0.0993139
\(385\) 1.80392 0.0919365
\(386\) 19.0304 0.968620
\(387\) 0.787492 0.0400305
\(388\) 11.1598 0.566553
\(389\) 15.3610 0.778834 0.389417 0.921062i \(-0.372677\pi\)
0.389417 + 0.921062i \(0.372677\pi\)
\(390\) 5.97647 0.302630
\(391\) −13.5264 −0.684058
\(392\) −3.74586 −0.189194
\(393\) 3.98363 0.200948
\(394\) −8.57050 −0.431775
\(395\) 15.7403 0.791983
\(396\) −0.787492 −0.0395730
\(397\) −10.9909 −0.551618 −0.275809 0.961212i \(-0.588946\pi\)
−0.275809 + 0.961212i \(0.588946\pi\)
\(398\) −7.56214 −0.379056
\(399\) −26.9354 −1.34846
\(400\) 1.00000 0.0500000
\(401\) 19.2341 0.960503 0.480251 0.877131i \(-0.340545\pi\)
0.480251 + 0.877131i \(0.340545\pi\)
\(402\) −12.4216 −0.619534
\(403\) 18.8116 0.937075
\(404\) 2.12892 0.105918
\(405\) −10.7423 −0.533791
\(406\) −10.9011 −0.541012
\(407\) −6.64486 −0.329373
\(408\) 5.19123 0.257004
\(409\) 8.25725 0.408295 0.204147 0.978940i \(-0.434558\pi\)
0.204147 + 0.978940i \(0.434558\pi\)
\(410\) 3.25559 0.160782
\(411\) 4.05620 0.200078
\(412\) −14.9767 −0.737850
\(413\) −7.43534 −0.365869
\(414\) 3.99331 0.196261
\(415\) 8.30389 0.407622
\(416\) −3.07092 −0.150564
\(417\) 39.5602 1.93727
\(418\) 7.67236 0.375268
\(419\) −22.5936 −1.10377 −0.551884 0.833921i \(-0.686091\pi\)
−0.551884 + 0.833921i \(0.686091\pi\)
\(420\) 3.51070 0.171305
\(421\) −1.06066 −0.0516933 −0.0258467 0.999666i \(-0.508228\pi\)
−0.0258467 + 0.999666i \(0.508228\pi\)
\(422\) 21.6038 1.05166
\(423\) −7.42047 −0.360795
\(424\) 8.46618 0.411154
\(425\) −2.66744 −0.129390
\(426\) −13.6725 −0.662433
\(427\) −14.9731 −0.724599
\(428\) 8.15661 0.394265
\(429\) −5.97647 −0.288547
\(430\) 1.00000 0.0482243
\(431\) −29.3695 −1.41468 −0.707338 0.706875i \(-0.750104\pi\)
−0.707338 + 0.706875i \(0.750104\pi\)
\(432\) 4.30587 0.207166
\(433\) −28.5671 −1.37285 −0.686424 0.727202i \(-0.740821\pi\)
−0.686424 + 0.727202i \(0.740821\pi\)
\(434\) 11.0504 0.530434
\(435\) −11.7605 −0.563875
\(436\) 1.25351 0.0600320
\(437\) −38.9060 −1.86113
\(438\) 13.1069 0.626271
\(439\) −1.69956 −0.0811158 −0.0405579 0.999177i \(-0.512914\pi\)
−0.0405579 + 0.999177i \(0.512914\pi\)
\(440\) −1.00000 −0.0476731
\(441\) −2.94983 −0.140468
\(442\) 8.19150 0.389630
\(443\) −31.1897 −1.48187 −0.740933 0.671579i \(-0.765616\pi\)
−0.740933 + 0.671579i \(0.765616\pi\)
\(444\) −12.9319 −0.613720
\(445\) 0.597136 0.0283070
\(446\) 12.6642 0.599669
\(447\) −33.6532 −1.59174
\(448\) −1.80392 −0.0852274
\(449\) 20.9363 0.988047 0.494024 0.869449i \(-0.335526\pi\)
0.494024 + 0.869449i \(0.335526\pi\)
\(450\) 0.787492 0.0371227
\(451\) −3.25559 −0.153300
\(452\) 11.4892 0.540408
\(453\) 4.69348 0.220519
\(454\) 24.1290 1.13243
\(455\) 5.53971 0.259706
\(456\) 14.9316 0.699234
\(457\) 21.5884 1.00986 0.504931 0.863160i \(-0.331518\pi\)
0.504931 + 0.863160i \(0.331518\pi\)
\(458\) −24.5929 −1.14915
\(459\) −11.4856 −0.536103
\(460\) 5.07092 0.236433
\(461\) 40.7731 1.89899 0.949496 0.313779i \(-0.101595\pi\)
0.949496 + 0.313779i \(0.101595\pi\)
\(462\) −3.51070 −0.163333
\(463\) −32.9637 −1.53196 −0.765978 0.642867i \(-0.777745\pi\)
−0.765978 + 0.642867i \(0.777745\pi\)
\(464\) 6.04298 0.280538
\(465\) 11.9216 0.552850
\(466\) 26.9616 1.24897
\(467\) 6.01996 0.278571 0.139285 0.990252i \(-0.455519\pi\)
0.139285 + 0.990252i \(0.455519\pi\)
\(468\) −2.41833 −0.111787
\(469\) −11.5138 −0.531660
\(470\) −9.42291 −0.434646
\(471\) 24.8236 1.14381
\(472\) 4.12176 0.189719
\(473\) −1.00000 −0.0459800
\(474\) −30.6330 −1.40702
\(475\) −7.67236 −0.352032
\(476\) 4.81185 0.220551
\(477\) 6.66705 0.305263
\(478\) 15.3401 0.701641
\(479\) −21.0217 −0.960507 −0.480253 0.877130i \(-0.659455\pi\)
−0.480253 + 0.877130i \(0.659455\pi\)
\(480\) −1.94615 −0.0888291
\(481\) −20.4059 −0.930427
\(482\) −11.2781 −0.513704
\(483\) 17.8025 0.810042
\(484\) 1.00000 0.0454545
\(485\) 11.1598 0.506740
\(486\) 7.98856 0.362369
\(487\) −35.6027 −1.61331 −0.806656 0.591021i \(-0.798725\pi\)
−0.806656 + 0.591021i \(0.798725\pi\)
\(488\) 8.30029 0.375737
\(489\) 3.63810 0.164521
\(490\) −3.74586 −0.169221
\(491\) −39.5063 −1.78290 −0.891448 0.453123i \(-0.850310\pi\)
−0.891448 + 0.453123i \(0.850310\pi\)
\(492\) −6.33585 −0.285642
\(493\) −16.1193 −0.725976
\(494\) 23.5612 1.06007
\(495\) −0.787492 −0.0353951
\(496\) −6.12573 −0.275053
\(497\) −12.6733 −0.568475
\(498\) −16.1606 −0.724173
\(499\) −29.3492 −1.31385 −0.656924 0.753956i \(-0.728143\pi\)
−0.656924 + 0.753956i \(0.728143\pi\)
\(500\) 1.00000 0.0447214
\(501\) −49.1668 −2.19661
\(502\) −1.91281 −0.0853729
\(503\) 41.1545 1.83499 0.917494 0.397749i \(-0.130209\pi\)
0.917494 + 0.397749i \(0.130209\pi\)
\(504\) −1.42058 −0.0632775
\(505\) 2.12892 0.0947355
\(506\) −5.07092 −0.225430
\(507\) 6.94664 0.308511
\(508\) −19.9054 −0.883159
\(509\) −30.5730 −1.35513 −0.677563 0.735465i \(-0.736964\pi\)
−0.677563 + 0.735465i \(0.736964\pi\)
\(510\) 5.19123 0.229871
\(511\) 12.1490 0.537442
\(512\) 1.00000 0.0441942
\(513\) −33.0362 −1.45858
\(514\) 14.3787 0.634218
\(515\) −14.9767 −0.659953
\(516\) −1.94615 −0.0856744
\(517\) 9.42291 0.414419
\(518\) −11.9868 −0.526671
\(519\) −43.9271 −1.92819
\(520\) −3.07092 −0.134669
\(521\) −17.6279 −0.772291 −0.386145 0.922438i \(-0.626194\pi\)
−0.386145 + 0.922438i \(0.626194\pi\)
\(522\) 4.75880 0.208287
\(523\) −32.4427 −1.41862 −0.709309 0.704898i \(-0.750993\pi\)
−0.709309 + 0.704898i \(0.750993\pi\)
\(524\) −2.04693 −0.0894206
\(525\) 3.51070 0.153220
\(526\) 13.3732 0.583097
\(527\) 16.3400 0.711782
\(528\) 1.94615 0.0846952
\(529\) 2.71426 0.118011
\(530\) 8.46618 0.367747
\(531\) 3.24585 0.140858
\(532\) 13.8404 0.600056
\(533\) −9.99765 −0.433046
\(534\) −1.16212 −0.0502896
\(535\) 8.15661 0.352641
\(536\) 6.38267 0.275689
\(537\) 33.1964 1.43253
\(538\) 3.86207 0.166506
\(539\) 3.74586 0.161345
\(540\) 4.30587 0.185295
\(541\) 37.1940 1.59909 0.799547 0.600603i \(-0.205073\pi\)
0.799547 + 0.600603i \(0.205073\pi\)
\(542\) 17.8656 0.767395
\(543\) 23.0934 0.991032
\(544\) −2.66744 −0.114365
\(545\) 1.25351 0.0536943
\(546\) −10.7811 −0.461389
\(547\) −17.0694 −0.729835 −0.364917 0.931040i \(-0.618903\pi\)
−0.364917 + 0.931040i \(0.618903\pi\)
\(548\) −2.08422 −0.0890335
\(549\) 6.53641 0.278967
\(550\) −1.00000 −0.0426401
\(551\) −46.3640 −1.97517
\(552\) −9.86877 −0.420043
\(553\) −28.3944 −1.20745
\(554\) 3.55770 0.151152
\(555\) −12.9319 −0.548928
\(556\) −20.3274 −0.862075
\(557\) −26.4689 −1.12152 −0.560762 0.827977i \(-0.689492\pi\)
−0.560762 + 0.827977i \(0.689492\pi\)
\(558\) −4.82396 −0.204215
\(559\) −3.07092 −0.129886
\(560\) −1.80392 −0.0762297
\(561\) −5.19123 −0.219174
\(562\) 18.7205 0.789675
\(563\) 36.0976 1.52133 0.760665 0.649144i \(-0.224873\pi\)
0.760665 + 0.649144i \(0.224873\pi\)
\(564\) 18.3384 0.772185
\(565\) 11.4892 0.483356
\(566\) 22.2481 0.935155
\(567\) 19.3784 0.813814
\(568\) 7.02540 0.294779
\(569\) −8.84730 −0.370898 −0.185449 0.982654i \(-0.559374\pi\)
−0.185449 + 0.982654i \(0.559374\pi\)
\(570\) 14.9316 0.625414
\(571\) 13.6973 0.573213 0.286606 0.958048i \(-0.407473\pi\)
0.286606 + 0.958048i \(0.407473\pi\)
\(572\) 3.07092 0.128402
\(573\) 17.5837 0.734570
\(574\) −5.87283 −0.245127
\(575\) 5.07092 0.211472
\(576\) 0.787492 0.0328122
\(577\) 30.5780 1.27298 0.636489 0.771286i \(-0.280386\pi\)
0.636489 + 0.771286i \(0.280386\pi\)
\(578\) −9.88478 −0.411153
\(579\) −37.0359 −1.53916
\(580\) 6.04298 0.250921
\(581\) −14.9796 −0.621458
\(582\) −21.7186 −0.900265
\(583\) −8.46618 −0.350633
\(584\) −6.73478 −0.278687
\(585\) −2.41833 −0.0999855
\(586\) 16.1883 0.668731
\(587\) −2.42159 −0.0999496 −0.0499748 0.998750i \(-0.515914\pi\)
−0.0499748 + 0.998750i \(0.515914\pi\)
\(588\) 7.28999 0.300634
\(589\) 46.9988 1.93655
\(590\) 4.12176 0.169690
\(591\) 16.6795 0.686101
\(592\) 6.64486 0.273102
\(593\) 22.8268 0.937384 0.468692 0.883362i \(-0.344725\pi\)
0.468692 + 0.883362i \(0.344725\pi\)
\(594\) −4.30587 −0.176672
\(595\) 4.81185 0.197267
\(596\) 17.2922 0.708316
\(597\) 14.7171 0.602329
\(598\) −15.5724 −0.636803
\(599\) −35.7442 −1.46047 −0.730234 0.683198i \(-0.760589\pi\)
−0.730234 + 0.683198i \(0.760589\pi\)
\(600\) −1.94615 −0.0794512
\(601\) 42.0642 1.71583 0.857917 0.513789i \(-0.171758\pi\)
0.857917 + 0.513789i \(0.171758\pi\)
\(602\) −1.80392 −0.0735225
\(603\) 5.02630 0.204687
\(604\) −2.41168 −0.0981298
\(605\) 1.00000 0.0406558
\(606\) −4.14319 −0.168305
\(607\) 6.69034 0.271553 0.135776 0.990740i \(-0.456647\pi\)
0.135776 + 0.990740i \(0.456647\pi\)
\(608\) −7.67236 −0.311155
\(609\) 21.2151 0.859680
\(610\) 8.30029 0.336069
\(611\) 28.9370 1.17067
\(612\) −2.10059 −0.0849111
\(613\) −25.1359 −1.01523 −0.507616 0.861584i \(-0.669473\pi\)
−0.507616 + 0.861584i \(0.669473\pi\)
\(614\) 9.08961 0.366827
\(615\) −6.33585 −0.255486
\(616\) 1.80392 0.0726822
\(617\) 25.6461 1.03247 0.516236 0.856446i \(-0.327333\pi\)
0.516236 + 0.856446i \(0.327333\pi\)
\(618\) 29.1469 1.17246
\(619\) 4.84131 0.194589 0.0972944 0.995256i \(-0.468981\pi\)
0.0972944 + 0.995256i \(0.468981\pi\)
\(620\) −6.12573 −0.246015
\(621\) 21.8347 0.876197
\(622\) 12.2692 0.491951
\(623\) −1.07719 −0.0431566
\(624\) 5.97647 0.239250
\(625\) 1.00000 0.0400000
\(626\) 29.4204 1.17588
\(627\) −14.9316 −0.596309
\(628\) −12.7552 −0.508989
\(629\) −17.7247 −0.706732
\(630\) −1.42058 −0.0565971
\(631\) −39.1491 −1.55850 −0.779250 0.626713i \(-0.784400\pi\)
−0.779250 + 0.626713i \(0.784400\pi\)
\(632\) 15.7403 0.626117
\(633\) −42.0443 −1.67111
\(634\) −31.2392 −1.24067
\(635\) −19.9054 −0.789922
\(636\) −16.4764 −0.653333
\(637\) 11.5032 0.455775
\(638\) −6.04298 −0.239244
\(639\) 5.53245 0.218860
\(640\) 1.00000 0.0395285
\(641\) −14.7411 −0.582238 −0.291119 0.956687i \(-0.594028\pi\)
−0.291119 + 0.956687i \(0.594028\pi\)
\(642\) −15.8740 −0.626495
\(643\) −4.07803 −0.160822 −0.0804109 0.996762i \(-0.525623\pi\)
−0.0804109 + 0.996762i \(0.525623\pi\)
\(644\) −9.14756 −0.360464
\(645\) −1.94615 −0.0766295
\(646\) 20.4656 0.805206
\(647\) 8.68125 0.341295 0.170648 0.985332i \(-0.445414\pi\)
0.170648 + 0.985332i \(0.445414\pi\)
\(648\) −10.7423 −0.421999
\(649\) −4.12176 −0.161793
\(650\) −3.07092 −0.120452
\(651\) −21.5056 −0.842872
\(652\) −1.86939 −0.0732108
\(653\) −27.7927 −1.08761 −0.543807 0.839210i \(-0.683018\pi\)
−0.543807 + 0.839210i \(0.683018\pi\)
\(654\) −2.43951 −0.0953923
\(655\) −2.04693 −0.0799802
\(656\) 3.25559 0.127109
\(657\) −5.30358 −0.206913
\(658\) 16.9982 0.662659
\(659\) 8.71197 0.339370 0.169685 0.985498i \(-0.445725\pi\)
0.169685 + 0.985498i \(0.445725\pi\)
\(660\) 1.94615 0.0757537
\(661\) 40.5867 1.57864 0.789320 0.613982i \(-0.210433\pi\)
0.789320 + 0.613982i \(0.210433\pi\)
\(662\) −0.985673 −0.0383093
\(663\) −15.9419 −0.619131
\(664\) 8.30389 0.322253
\(665\) 13.8404 0.536706
\(666\) 5.23277 0.202766
\(667\) 30.6435 1.18652
\(668\) 25.2636 0.977480
\(669\) −24.6465 −0.952888
\(670\) 6.38267 0.246584
\(671\) −8.30029 −0.320429
\(672\) 3.51070 0.135428
\(673\) −24.1766 −0.931940 −0.465970 0.884801i \(-0.654295\pi\)
−0.465970 + 0.884801i \(0.654295\pi\)
\(674\) 21.8560 0.841863
\(675\) 4.30587 0.165733
\(676\) −3.56943 −0.137286
\(677\) 34.0871 1.31007 0.655036 0.755597i \(-0.272653\pi\)
0.655036 + 0.755597i \(0.272653\pi\)
\(678\) −22.3597 −0.858721
\(679\) −20.1314 −0.772573
\(680\) −2.66744 −0.102292
\(681\) −46.9587 −1.79946
\(682\) 6.12573 0.234566
\(683\) −13.8590 −0.530301 −0.265150 0.964207i \(-0.585422\pi\)
−0.265150 + 0.964207i \(0.585422\pi\)
\(684\) −6.04193 −0.231019
\(685\) −2.08422 −0.0796340
\(686\) 19.3847 0.740112
\(687\) 47.8615 1.82603
\(688\) 1.00000 0.0381246
\(689\) −25.9990 −0.990482
\(690\) −9.86877 −0.375698
\(691\) 15.2366 0.579628 0.289814 0.957083i \(-0.406407\pi\)
0.289814 + 0.957083i \(0.406407\pi\)
\(692\) 22.5713 0.858032
\(693\) 1.42058 0.0539632
\(694\) −21.7521 −0.825697
\(695\) −20.3274 −0.771063
\(696\) −11.7605 −0.445782
\(697\) −8.68407 −0.328933
\(698\) −34.1659 −1.29320
\(699\) −52.4713 −1.98465
\(700\) −1.80392 −0.0681819
\(701\) 20.2220 0.763774 0.381887 0.924209i \(-0.375274\pi\)
0.381887 + 0.924209i \(0.375274\pi\)
\(702\) −13.2230 −0.499070
\(703\) −50.9818 −1.92281
\(704\) −1.00000 −0.0376889
\(705\) 18.3384 0.690663
\(706\) −0.428704 −0.0161345
\(707\) −3.84040 −0.144433
\(708\) −8.02155 −0.301468
\(709\) 9.28908 0.348859 0.174429 0.984670i \(-0.444192\pi\)
0.174429 + 0.984670i \(0.444192\pi\)
\(710\) 7.02540 0.263659
\(711\) 12.3954 0.464864
\(712\) 0.597136 0.0223786
\(713\) −31.0631 −1.16332
\(714\) −9.36458 −0.350461
\(715\) 3.07092 0.114846
\(716\) −17.0575 −0.637469
\(717\) −29.8541 −1.11492
\(718\) −23.1249 −0.863014
\(719\) −6.92118 −0.258117 −0.129058 0.991637i \(-0.541195\pi\)
−0.129058 + 0.991637i \(0.541195\pi\)
\(720\) 0.787492 0.0293481
\(721\) 27.0169 1.00616
\(722\) 39.8652 1.48363
\(723\) 21.9489 0.816287
\(724\) −11.8662 −0.441004
\(725\) 6.04298 0.224431
\(726\) −1.94615 −0.0722283
\(727\) 41.6626 1.54518 0.772591 0.634904i \(-0.218960\pi\)
0.772591 + 0.634904i \(0.218960\pi\)
\(728\) 5.53971 0.205315
\(729\) 16.6801 0.617780
\(730\) −6.73478 −0.249265
\(731\) −2.66744 −0.0986587
\(732\) −16.1536 −0.597054
\(733\) −47.2017 −1.74344 −0.871718 0.490008i \(-0.836994\pi\)
−0.871718 + 0.490008i \(0.836994\pi\)
\(734\) −1.79699 −0.0663282
\(735\) 7.28999 0.268895
\(736\) 5.07092 0.186917
\(737\) −6.38267 −0.235109
\(738\) 2.56375 0.0943729
\(739\) 40.8354 1.50215 0.751076 0.660215i \(-0.229535\pi\)
0.751076 + 0.660215i \(0.229535\pi\)
\(740\) 6.64486 0.244270
\(741\) −45.8537 −1.68448
\(742\) −15.2723 −0.560665
\(743\) −2.26473 −0.0830851 −0.0415425 0.999137i \(-0.513227\pi\)
−0.0415425 + 0.999137i \(0.513227\pi\)
\(744\) 11.9216 0.437066
\(745\) 17.2922 0.633537
\(746\) 21.7123 0.794943
\(747\) 6.53924 0.239258
\(748\) 2.66744 0.0975312
\(749\) −14.7139 −0.537634
\(750\) −1.94615 −0.0710633
\(751\) −10.0313 −0.366047 −0.183023 0.983109i \(-0.558588\pi\)
−0.183023 + 0.983109i \(0.558588\pi\)
\(752\) −9.42291 −0.343618
\(753\) 3.72261 0.135660
\(754\) −18.5575 −0.675826
\(755\) −2.41168 −0.0877700
\(756\) −7.76746 −0.282500
\(757\) −49.7002 −1.80639 −0.903193 0.429235i \(-0.858783\pi\)
−0.903193 + 0.429235i \(0.858783\pi\)
\(758\) 13.5210 0.491103
\(759\) 9.86877 0.358214
\(760\) −7.67236 −0.278306
\(761\) 2.24241 0.0812871 0.0406436 0.999174i \(-0.487059\pi\)
0.0406436 + 0.999174i \(0.487059\pi\)
\(762\) 38.7389 1.40336
\(763\) −2.26123 −0.0818620
\(764\) −9.03514 −0.326880
\(765\) −2.10059 −0.0759468
\(766\) 10.8496 0.392012
\(767\) −12.6576 −0.457040
\(768\) −1.94615 −0.0702256
\(769\) −30.0095 −1.08217 −0.541086 0.840967i \(-0.681987\pi\)
−0.541086 + 0.840967i \(0.681987\pi\)
\(770\) 1.80392 0.0650089
\(771\) −27.9831 −1.00779
\(772\) 19.0304 0.684918
\(773\) 9.54302 0.343238 0.171619 0.985163i \(-0.445100\pi\)
0.171619 + 0.985163i \(0.445100\pi\)
\(774\) 0.787492 0.0283058
\(775\) −6.12573 −0.220043
\(776\) 11.1598 0.400613
\(777\) 23.3281 0.836892
\(778\) 15.3610 0.550719
\(779\) −24.9780 −0.894931
\(780\) 5.97647 0.213992
\(781\) −7.02540 −0.251389
\(782\) −13.5264 −0.483702
\(783\) 26.0203 0.929889
\(784\) −3.74586 −0.133781
\(785\) −12.7552 −0.455253
\(786\) 3.98363 0.142091
\(787\) 2.67784 0.0954547 0.0477273 0.998860i \(-0.484802\pi\)
0.0477273 + 0.998860i \(0.484802\pi\)
\(788\) −8.57050 −0.305311
\(789\) −26.0261 −0.926555
\(790\) 15.7403 0.560016
\(791\) −20.7257 −0.736921
\(792\) −0.787492 −0.0279823
\(793\) −25.4896 −0.905161
\(794\) −10.9909 −0.390053
\(795\) −16.4764 −0.584359
\(796\) −7.56214 −0.268033
\(797\) 47.7000 1.68962 0.844810 0.535066i \(-0.179713\pi\)
0.844810 + 0.535066i \(0.179713\pi\)
\(798\) −26.9354 −0.953503
\(799\) 25.1350 0.889213
\(800\) 1.00000 0.0353553
\(801\) 0.470240 0.0166151
\(802\) 19.2341 0.679178
\(803\) 6.73478 0.237665
\(804\) −12.4216 −0.438077
\(805\) −9.14756 −0.322409
\(806\) 18.8116 0.662612
\(807\) −7.51617 −0.264582
\(808\) 2.12892 0.0748950
\(809\) −15.3779 −0.540660 −0.270330 0.962768i \(-0.587133\pi\)
−0.270330 + 0.962768i \(0.587133\pi\)
\(810\) −10.7423 −0.377447
\(811\) −36.9989 −1.29921 −0.649603 0.760273i \(-0.725065\pi\)
−0.649603 + 0.760273i \(0.725065\pi\)
\(812\) −10.9011 −0.382553
\(813\) −34.7692 −1.21941
\(814\) −6.64486 −0.232902
\(815\) −1.86939 −0.0654818
\(816\) 5.19123 0.181729
\(817\) −7.67236 −0.268422
\(818\) 8.25725 0.288708
\(819\) 4.36248 0.152437
\(820\) 3.25559 0.113690
\(821\) −8.86426 −0.309365 −0.154682 0.987964i \(-0.549435\pi\)
−0.154682 + 0.987964i \(0.549435\pi\)
\(822\) 4.05620 0.141476
\(823\) 37.0194 1.29042 0.645208 0.764007i \(-0.276771\pi\)
0.645208 + 0.764007i \(0.276771\pi\)
\(824\) −14.9767 −0.521739
\(825\) 1.94615 0.0677562
\(826\) −7.43534 −0.258709
\(827\) 14.6619 0.509844 0.254922 0.966962i \(-0.417950\pi\)
0.254922 + 0.966962i \(0.417950\pi\)
\(828\) 3.99331 0.138777
\(829\) 33.3015 1.15661 0.578305 0.815821i \(-0.303714\pi\)
0.578305 + 0.815821i \(0.303714\pi\)
\(830\) 8.30389 0.288232
\(831\) −6.92382 −0.240185
\(832\) −3.07092 −0.106465
\(833\) 9.99184 0.346197
\(834\) 39.5602 1.36986
\(835\) 25.2636 0.874285
\(836\) 7.67236 0.265354
\(837\) −26.3766 −0.911708
\(838\) −22.5936 −0.780482
\(839\) 32.5997 1.12547 0.562734 0.826638i \(-0.309750\pi\)
0.562734 + 0.826638i \(0.309750\pi\)
\(840\) 3.51070 0.121131
\(841\) 7.51764 0.259229
\(842\) −1.06066 −0.0365527
\(843\) −36.4328 −1.25481
\(844\) 21.6038 0.743635
\(845\) −3.56943 −0.122792
\(846\) −7.42047 −0.255121
\(847\) −1.80392 −0.0619836
\(848\) 8.46618 0.290730
\(849\) −43.2980 −1.48598
\(850\) −2.66744 −0.0914923
\(851\) 33.6956 1.15507
\(852\) −13.6725 −0.468411
\(853\) −4.22582 −0.144689 −0.0723447 0.997380i \(-0.523048\pi\)
−0.0723447 + 0.997380i \(0.523048\pi\)
\(854\) −14.9731 −0.512369
\(855\) −6.04193 −0.206629
\(856\) 8.15661 0.278787
\(857\) 39.9084 1.36324 0.681622 0.731704i \(-0.261275\pi\)
0.681622 + 0.731704i \(0.261275\pi\)
\(858\) −5.97647 −0.204033
\(859\) −29.3527 −1.00150 −0.500751 0.865591i \(-0.666943\pi\)
−0.500751 + 0.865591i \(0.666943\pi\)
\(860\) 1.00000 0.0340997
\(861\) 11.4294 0.389513
\(862\) −29.3695 −1.00033
\(863\) 50.9961 1.73593 0.867964 0.496627i \(-0.165428\pi\)
0.867964 + 0.496627i \(0.165428\pi\)
\(864\) 4.30587 0.146489
\(865\) 22.5713 0.767447
\(866\) −28.5671 −0.970750
\(867\) 19.2372 0.653331
\(868\) 11.0504 0.375073
\(869\) −15.7403 −0.533955
\(870\) −11.7605 −0.398720
\(871\) −19.6007 −0.664144
\(872\) 1.25351 0.0424490
\(873\) 8.78825 0.297437
\(874\) −38.9060 −1.31601
\(875\) −1.80392 −0.0609838
\(876\) 13.1069 0.442840
\(877\) 11.5551 0.390189 0.195095 0.980784i \(-0.437499\pi\)
0.195095 + 0.980784i \(0.437499\pi\)
\(878\) −1.69956 −0.0573575
\(879\) −31.5047 −1.06263
\(880\) −1.00000 −0.0337100
\(881\) 37.7598 1.27216 0.636080 0.771623i \(-0.280555\pi\)
0.636080 + 0.771623i \(0.280555\pi\)
\(882\) −2.94983 −0.0993260
\(883\) 1.39066 0.0467994 0.0233997 0.999726i \(-0.492551\pi\)
0.0233997 + 0.999726i \(0.492551\pi\)
\(884\) 8.19150 0.275510
\(885\) −8.02155 −0.269641
\(886\) −31.1897 −1.04784
\(887\) 4.55277 0.152867 0.0764335 0.997075i \(-0.475647\pi\)
0.0764335 + 0.997075i \(0.475647\pi\)
\(888\) −12.9319 −0.433965
\(889\) 35.9078 1.20431
\(890\) 0.597136 0.0200160
\(891\) 10.7423 0.359882
\(892\) 12.6642 0.424030
\(893\) 72.2960 2.41929
\(894\) −33.6532 −1.12553
\(895\) −17.0575 −0.570170
\(896\) −1.80392 −0.0602649
\(897\) 30.3062 1.01190
\(898\) 20.9363 0.698655
\(899\) −37.0177 −1.23461
\(900\) 0.787492 0.0262497
\(901\) −22.5830 −0.752349
\(902\) −3.25559 −0.108399
\(903\) 3.51070 0.116829
\(904\) 11.4892 0.382126
\(905\) −11.8662 −0.394446
\(906\) 4.69348 0.155931
\(907\) 0.175942 0.00584205 0.00292103 0.999996i \(-0.499070\pi\)
0.00292103 + 0.999996i \(0.499070\pi\)
\(908\) 24.1290 0.800751
\(909\) 1.67650 0.0556061
\(910\) 5.53971 0.183640
\(911\) −6.53460 −0.216501 −0.108251 0.994124i \(-0.534525\pi\)
−0.108251 + 0.994124i \(0.534525\pi\)
\(912\) 14.9316 0.494433
\(913\) −8.30389 −0.274819
\(914\) 21.5884 0.714080
\(915\) −16.1536 −0.534022
\(916\) −24.5929 −0.812573
\(917\) 3.69251 0.121937
\(918\) −11.4856 −0.379082
\(919\) −57.6333 −1.90115 −0.950573 0.310502i \(-0.899503\pi\)
−0.950573 + 0.310502i \(0.899503\pi\)
\(920\) 5.07092 0.167183
\(921\) −17.6897 −0.582896
\(922\) 40.7731 1.34279
\(923\) −21.5745 −0.710132
\(924\) −3.51070 −0.115494
\(925\) 6.64486 0.218482
\(926\) −32.9637 −1.08326
\(927\) −11.7940 −0.387367
\(928\) 6.04298 0.198371
\(929\) 3.38631 0.111101 0.0555506 0.998456i \(-0.482309\pi\)
0.0555506 + 0.998456i \(0.482309\pi\)
\(930\) 11.9216 0.390924
\(931\) 28.7396 0.941902
\(932\) 26.9616 0.883157
\(933\) −23.8777 −0.781722
\(934\) 6.01996 0.196979
\(935\) 2.66744 0.0872345
\(936\) −2.41833 −0.0790455
\(937\) −16.5132 −0.539463 −0.269731 0.962936i \(-0.586935\pi\)
−0.269731 + 0.962936i \(0.586935\pi\)
\(938\) −11.5138 −0.375941
\(939\) −57.2565 −1.86850
\(940\) −9.42291 −0.307341
\(941\) 0.179492 0.00585126 0.00292563 0.999996i \(-0.499069\pi\)
0.00292563 + 0.999996i \(0.499069\pi\)
\(942\) 24.8236 0.808795
\(943\) 16.5088 0.537601
\(944\) 4.12176 0.134152
\(945\) −7.76746 −0.252675
\(946\) −1.00000 −0.0325128
\(947\) −18.5996 −0.604405 −0.302202 0.953244i \(-0.597722\pi\)
−0.302202 + 0.953244i \(0.597722\pi\)
\(948\) −30.6330 −0.994915
\(949\) 20.6820 0.671366
\(950\) −7.67236 −0.248924
\(951\) 60.7962 1.97145
\(952\) 4.81185 0.155953
\(953\) 27.6162 0.894576 0.447288 0.894390i \(-0.352390\pi\)
0.447288 + 0.894390i \(0.352390\pi\)
\(954\) 6.66705 0.215854
\(955\) −9.03514 −0.292370
\(956\) 15.3401 0.496135
\(957\) 11.7605 0.380164
\(958\) −21.0217 −0.679181
\(959\) 3.75978 0.121410
\(960\) −1.94615 −0.0628117
\(961\) 6.52456 0.210470
\(962\) −20.4059 −0.657911
\(963\) 6.42326 0.206987
\(964\) −11.2781 −0.363244
\(965\) 19.0304 0.612609
\(966\) 17.8025 0.572786
\(967\) −7.41969 −0.238601 −0.119301 0.992858i \(-0.538065\pi\)
−0.119301 + 0.992858i \(0.538065\pi\)
\(968\) 1.00000 0.0321412
\(969\) −39.8290 −1.27949
\(970\) 11.1598 0.358319
\(971\) −25.2446 −0.810138 −0.405069 0.914286i \(-0.632752\pi\)
−0.405069 + 0.914286i \(0.632752\pi\)
\(972\) 7.98856 0.256233
\(973\) 36.6691 1.17556
\(974\) −35.6027 −1.14078
\(975\) 5.97647 0.191400
\(976\) 8.30029 0.265686
\(977\) −44.8974 −1.43640 −0.718198 0.695839i \(-0.755033\pi\)
−0.718198 + 0.695839i \(0.755033\pi\)
\(978\) 3.63810 0.116334
\(979\) −0.597136 −0.0190846
\(980\) −3.74586 −0.119657
\(981\) 0.987125 0.0315165
\(982\) −39.5063 −1.26070
\(983\) −35.5289 −1.13320 −0.566598 0.823995i \(-0.691741\pi\)
−0.566598 + 0.823995i \(0.691741\pi\)
\(984\) −6.33585 −0.201980
\(985\) −8.57050 −0.273079
\(986\) −16.1193 −0.513342
\(987\) −33.0810 −1.05298
\(988\) 23.5612 0.749583
\(989\) 5.07092 0.161246
\(990\) −0.787492 −0.0250281
\(991\) 30.8147 0.978860 0.489430 0.872043i \(-0.337205\pi\)
0.489430 + 0.872043i \(0.337205\pi\)
\(992\) −6.12573 −0.194492
\(993\) 1.91827 0.0608743
\(994\) −12.6733 −0.401972
\(995\) −7.56214 −0.239736
\(996\) −16.1606 −0.512068
\(997\) −56.7452 −1.79714 −0.898569 0.438832i \(-0.855392\pi\)
−0.898569 + 0.438832i \(0.855392\pi\)
\(998\) −29.3492 −0.929031
\(999\) 28.6119 0.905240
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.1.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4730.2.a.be.1.4 12 1.1 even 1 trivial