Properties

Label 5070.2.a.bl.1.3
Level $5070$
Weight $2$
Character 5070.1
Self dual yes
Analytic conductor $40.484$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.4841538248\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.24698\) of defining polynomial
Character \(\chi\) \(=\) 5070.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +4.24698 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +4.24698 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.91185 q^{11} -1.00000 q^{12} -4.24698 q^{14} -1.00000 q^{15} +1.00000 q^{16} -3.33513 q^{17} -1.00000 q^{18} +4.85086 q^{19} +1.00000 q^{20} -4.24698 q^{21} +1.91185 q^{22} -0.445042 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} +4.24698 q^{28} -8.56465 q^{29} +1.00000 q^{30} -5.29590 q^{31} -1.00000 q^{32} +1.91185 q^{33} +3.33513 q^{34} +4.24698 q^{35} +1.00000 q^{36} -1.75302 q^{37} -4.85086 q^{38} -1.00000 q^{40} -3.24698 q^{41} +4.24698 q^{42} -1.97823 q^{43} -1.91185 q^{44} +1.00000 q^{45} +0.445042 q^{46} -10.3840 q^{47} -1.00000 q^{48} +11.0368 q^{49} -1.00000 q^{50} +3.33513 q^{51} -1.14914 q^{53} +1.00000 q^{54} -1.91185 q^{55} -4.24698 q^{56} -4.85086 q^{57} +8.56465 q^{58} -13.4601 q^{59} -1.00000 q^{60} +1.13169 q^{61} +5.29590 q^{62} +4.24698 q^{63} +1.00000 q^{64} -1.91185 q^{66} -13.5308 q^{67} -3.33513 q^{68} +0.445042 q^{69} -4.24698 q^{70} +2.14675 q^{71} -1.00000 q^{72} -5.15883 q^{73} +1.75302 q^{74} -1.00000 q^{75} +4.85086 q^{76} -8.11960 q^{77} -14.5526 q^{79} +1.00000 q^{80} +1.00000 q^{81} +3.24698 q^{82} +9.49157 q^{83} -4.24698 q^{84} -3.33513 q^{85} +1.97823 q^{86} +8.56465 q^{87} +1.91185 q^{88} +1.25667 q^{89} -1.00000 q^{90} -0.445042 q^{92} +5.29590 q^{93} +10.3840 q^{94} +4.85086 q^{95} +1.00000 q^{96} +5.01507 q^{97} -11.0368 q^{98} -1.91185 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} - 3q^{3} + 3q^{4} + 3q^{5} + 3q^{6} + 8q^{7} - 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{2} - 3q^{3} + 3q^{4} + 3q^{5} + 3q^{6} + 8q^{7} - 3q^{8} + 3q^{9} - 3q^{10} - 2q^{11} - 3q^{12} - 8q^{14} - 3q^{15} + 3q^{16} - 9q^{17} - 3q^{18} + q^{19} + 3q^{20} - 8q^{21} + 2q^{22} - q^{23} + 3q^{24} + 3q^{25} - 3q^{27} + 8q^{28} - 4q^{29} + 3q^{30} - 2q^{31} - 3q^{32} + 2q^{33} + 9q^{34} + 8q^{35} + 3q^{36} - 10q^{37} - q^{38} - 3q^{40} - 5q^{41} + 8q^{42} - 9q^{43} - 2q^{44} + 3q^{45} + q^{46} - 21q^{47} - 3q^{48} + 5q^{49} - 3q^{50} + 9q^{51} - 17q^{53} + 3q^{54} - 2q^{55} - 8q^{56} - q^{57} + 4q^{58} - 15q^{59} - 3q^{60} + q^{61} + 2q^{62} + 8q^{63} + 3q^{64} - 2q^{66} - 3q^{67} - 9q^{68} + q^{69} - 8q^{70} - 21q^{71} - 3q^{72} - 7q^{73} + 10q^{74} - 3q^{75} + q^{76} - 3q^{77} - 3q^{79} + 3q^{80} + 3q^{81} + 5q^{82} - 22q^{83} - 8q^{84} - 9q^{85} + 9q^{86} + 4q^{87} + 2q^{88} - 23q^{89} - 3q^{90} - q^{92} + 2q^{93} + 21q^{94} + q^{95} + 3q^{96} - 10q^{97} - 5q^{98} - 2q^{99} + O(q^{100}) \)

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 1.00000 0.408248
\(7\) 4.24698 1.60521 0.802604 0.596513i \(-0.203447\pi\)
0.802604 + 0.596513i \(0.203447\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −1.91185 −0.576446 −0.288223 0.957563i \(-0.593064\pi\)
−0.288223 + 0.957563i \(0.593064\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) −4.24698 −1.13505
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −3.33513 −0.808887 −0.404443 0.914563i \(-0.632535\pi\)
−0.404443 + 0.914563i \(0.632535\pi\)
\(18\) −1.00000 −0.235702
\(19\) 4.85086 1.11286 0.556431 0.830894i \(-0.312170\pi\)
0.556431 + 0.830894i \(0.312170\pi\)
\(20\) 1.00000 0.223607
\(21\) −4.24698 −0.926767
\(22\) 1.91185 0.407609
\(23\) −0.445042 −0.0927976 −0.0463988 0.998923i \(-0.514775\pi\)
−0.0463988 + 0.998923i \(0.514775\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 4.24698 0.802604
\(29\) −8.56465 −1.59041 −0.795207 0.606337i \(-0.792638\pi\)
−0.795207 + 0.606337i \(0.792638\pi\)
\(30\) 1.00000 0.182574
\(31\) −5.29590 −0.951171 −0.475586 0.879669i \(-0.657764\pi\)
−0.475586 + 0.879669i \(0.657764\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.91185 0.332811
\(34\) 3.33513 0.571969
\(35\) 4.24698 0.717871
\(36\) 1.00000 0.166667
\(37\) −1.75302 −0.288195 −0.144097 0.989564i \(-0.546028\pi\)
−0.144097 + 0.989564i \(0.546028\pi\)
\(38\) −4.85086 −0.786913
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −3.24698 −0.507093 −0.253547 0.967323i \(-0.581597\pi\)
−0.253547 + 0.967323i \(0.581597\pi\)
\(42\) 4.24698 0.655323
\(43\) −1.97823 −0.301677 −0.150839 0.988558i \(-0.548197\pi\)
−0.150839 + 0.988558i \(0.548197\pi\)
\(44\) −1.91185 −0.288223
\(45\) 1.00000 0.149071
\(46\) 0.445042 0.0656178
\(47\) −10.3840 −1.51467 −0.757334 0.653028i \(-0.773499\pi\)
−0.757334 + 0.653028i \(0.773499\pi\)
\(48\) −1.00000 −0.144338
\(49\) 11.0368 1.57669
\(50\) −1.00000 −0.141421
\(51\) 3.33513 0.467011
\(52\) 0 0
\(53\) −1.14914 −0.157847 −0.0789236 0.996881i \(-0.525148\pi\)
−0.0789236 + 0.996881i \(0.525148\pi\)
\(54\) 1.00000 0.136083
\(55\) −1.91185 −0.257794
\(56\) −4.24698 −0.567527
\(57\) −4.85086 −0.642511
\(58\) 8.56465 1.12459
\(59\) −13.4601 −1.75236 −0.876178 0.481987i \(-0.839915\pi\)
−0.876178 + 0.481987i \(0.839915\pi\)
\(60\) −1.00000 −0.129099
\(61\) 1.13169 0.144898 0.0724488 0.997372i \(-0.476919\pi\)
0.0724488 + 0.997372i \(0.476919\pi\)
\(62\) 5.29590 0.672580
\(63\) 4.24698 0.535069
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.91185 −0.235333
\(67\) −13.5308 −1.65305 −0.826525 0.562900i \(-0.809686\pi\)
−0.826525 + 0.562900i \(0.809686\pi\)
\(68\) −3.33513 −0.404443
\(69\) 0.445042 0.0535767
\(70\) −4.24698 −0.507611
\(71\) 2.14675 0.254773 0.127386 0.991853i \(-0.459341\pi\)
0.127386 + 0.991853i \(0.459341\pi\)
\(72\) −1.00000 −0.117851
\(73\) −5.15883 −0.603796 −0.301898 0.953340i \(-0.597620\pi\)
−0.301898 + 0.953340i \(0.597620\pi\)
\(74\) 1.75302 0.203784
\(75\) −1.00000 −0.115470
\(76\) 4.85086 0.556431
\(77\) −8.11960 −0.925315
\(78\) 0 0
\(79\) −14.5526 −1.63729 −0.818646 0.574299i \(-0.805275\pi\)
−0.818646 + 0.574299i \(0.805275\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 3.24698 0.358569
\(83\) 9.49157 1.04183 0.520917 0.853607i \(-0.325590\pi\)
0.520917 + 0.853607i \(0.325590\pi\)
\(84\) −4.24698 −0.463383
\(85\) −3.33513 −0.361745
\(86\) 1.97823 0.213318
\(87\) 8.56465 0.918227
\(88\) 1.91185 0.203804
\(89\) 1.25667 0.133207 0.0666033 0.997780i \(-0.478784\pi\)
0.0666033 + 0.997780i \(0.478784\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) −0.445042 −0.0463988
\(93\) 5.29590 0.549159
\(94\) 10.3840 1.07103
\(95\) 4.85086 0.497687
\(96\) 1.00000 0.102062
\(97\) 5.01507 0.509203 0.254601 0.967046i \(-0.418056\pi\)
0.254601 + 0.967046i \(0.418056\pi\)
\(98\) −11.0368 −1.11489
\(99\) −1.91185 −0.192149
\(100\) 1.00000 0.100000
\(101\) 14.8116 1.47381 0.736906 0.675995i \(-0.236286\pi\)
0.736906 + 0.675995i \(0.236286\pi\)
\(102\) −3.33513 −0.330227
\(103\) −13.6896 −1.34888 −0.674440 0.738330i \(-0.735615\pi\)
−0.674440 + 0.738330i \(0.735615\pi\)
\(104\) 0 0
\(105\) −4.24698 −0.414463
\(106\) 1.14914 0.111615
\(107\) −12.8334 −1.24065 −0.620326 0.784344i \(-0.713000\pi\)
−0.620326 + 0.784344i \(0.713000\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −10.8726 −1.04141 −0.520704 0.853737i \(-0.674331\pi\)
−0.520704 + 0.853737i \(0.674331\pi\)
\(110\) 1.91185 0.182288
\(111\) 1.75302 0.166389
\(112\) 4.24698 0.401302
\(113\) −11.2295 −1.05638 −0.528192 0.849125i \(-0.677130\pi\)
−0.528192 + 0.849125i \(0.677130\pi\)
\(114\) 4.85086 0.454324
\(115\) −0.445042 −0.0415004
\(116\) −8.56465 −0.795207
\(117\) 0 0
\(118\) 13.4601 1.23910
\(119\) −14.1642 −1.29843
\(120\) 1.00000 0.0912871
\(121\) −7.34481 −0.667710
\(122\) −1.13169 −0.102458
\(123\) 3.24698 0.292770
\(124\) −5.29590 −0.475586
\(125\) 1.00000 0.0894427
\(126\) −4.24698 −0.378351
\(127\) 20.8267 1.84807 0.924035 0.382308i \(-0.124871\pi\)
0.924035 + 0.382308i \(0.124871\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 1.97823 0.174173
\(130\) 0 0
\(131\) 17.5036 1.52930 0.764650 0.644445i \(-0.222912\pi\)
0.764650 + 0.644445i \(0.222912\pi\)
\(132\) 1.91185 0.166406
\(133\) 20.6015 1.78638
\(134\) 13.5308 1.16888
\(135\) −1.00000 −0.0860663
\(136\) 3.33513 0.285985
\(137\) 0.929312 0.0793965 0.0396983 0.999212i \(-0.487360\pi\)
0.0396983 + 0.999212i \(0.487360\pi\)
\(138\) −0.445042 −0.0378845
\(139\) 3.12200 0.264804 0.132402 0.991196i \(-0.457731\pi\)
0.132402 + 0.991196i \(0.457731\pi\)
\(140\) 4.24698 0.358935
\(141\) 10.3840 0.874494
\(142\) −2.14675 −0.180151
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −8.56465 −0.711255
\(146\) 5.15883 0.426948
\(147\) −11.0368 −0.910303
\(148\) −1.75302 −0.144097
\(149\) −6.57673 −0.538787 −0.269393 0.963030i \(-0.586823\pi\)
−0.269393 + 0.963030i \(0.586823\pi\)
\(150\) 1.00000 0.0816497
\(151\) −6.29590 −0.512353 −0.256176 0.966630i \(-0.582463\pi\)
−0.256176 + 0.966630i \(0.582463\pi\)
\(152\) −4.85086 −0.393456
\(153\) −3.33513 −0.269629
\(154\) 8.11960 0.654296
\(155\) −5.29590 −0.425377
\(156\) 0 0
\(157\) 19.9148 1.58938 0.794689 0.607017i \(-0.207634\pi\)
0.794689 + 0.607017i \(0.207634\pi\)
\(158\) 14.5526 1.15774
\(159\) 1.14914 0.0911331
\(160\) −1.00000 −0.0790569
\(161\) −1.89008 −0.148959
\(162\) −1.00000 −0.0785674
\(163\) −13.1075 −1.02666 −0.513330 0.858191i \(-0.671588\pi\)
−0.513330 + 0.858191i \(0.671588\pi\)
\(164\) −3.24698 −0.253547
\(165\) 1.91185 0.148838
\(166\) −9.49157 −0.736688
\(167\) −21.5351 −1.66644 −0.833218 0.552944i \(-0.813504\pi\)
−0.833218 + 0.552944i \(0.813504\pi\)
\(168\) 4.24698 0.327662
\(169\) 0 0
\(170\) 3.33513 0.255792
\(171\) 4.85086 0.370954
\(172\) −1.97823 −0.150839
\(173\) −13.9541 −1.06091 −0.530454 0.847714i \(-0.677979\pi\)
−0.530454 + 0.847714i \(0.677979\pi\)
\(174\) −8.56465 −0.649284
\(175\) 4.24698 0.321041
\(176\) −1.91185 −0.144111
\(177\) 13.4601 1.01172
\(178\) −1.25667 −0.0941913
\(179\) 8.32975 0.622595 0.311297 0.950313i \(-0.399236\pi\)
0.311297 + 0.950313i \(0.399236\pi\)
\(180\) 1.00000 0.0745356
\(181\) 17.1371 1.27379 0.636894 0.770951i \(-0.280219\pi\)
0.636894 + 0.770951i \(0.280219\pi\)
\(182\) 0 0
\(183\) −1.13169 −0.0836567
\(184\) 0.445042 0.0328089
\(185\) −1.75302 −0.128885
\(186\) −5.29590 −0.388314
\(187\) 6.37627 0.466279
\(188\) −10.3840 −0.757334
\(189\) −4.24698 −0.308922
\(190\) −4.85086 −0.351918
\(191\) 23.1685 1.67642 0.838208 0.545351i \(-0.183604\pi\)
0.838208 + 0.545351i \(0.183604\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −6.38942 −0.459920 −0.229960 0.973200i \(-0.573860\pi\)
−0.229960 + 0.973200i \(0.573860\pi\)
\(194\) −5.01507 −0.360061
\(195\) 0 0
\(196\) 11.0368 0.788345
\(197\) 14.8605 1.05877 0.529385 0.848382i \(-0.322423\pi\)
0.529385 + 0.848382i \(0.322423\pi\)
\(198\) 1.91185 0.135870
\(199\) 26.3860 1.87045 0.935226 0.354052i \(-0.115197\pi\)
0.935226 + 0.354052i \(0.115197\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 13.5308 0.954389
\(202\) −14.8116 −1.04214
\(203\) −36.3739 −2.55295
\(204\) 3.33513 0.233505
\(205\) −3.24698 −0.226779
\(206\) 13.6896 0.953802
\(207\) −0.445042 −0.0309325
\(208\) 0 0
\(209\) −9.27413 −0.641505
\(210\) 4.24698 0.293069
\(211\) 17.4383 1.20050 0.600252 0.799811i \(-0.295067\pi\)
0.600252 + 0.799811i \(0.295067\pi\)
\(212\) −1.14914 −0.0789236
\(213\) −2.14675 −0.147093
\(214\) 12.8334 0.877273
\(215\) −1.97823 −0.134914
\(216\) 1.00000 0.0680414
\(217\) −22.4916 −1.52683
\(218\) 10.8726 0.736387
\(219\) 5.15883 0.348602
\(220\) −1.91185 −0.128897
\(221\) 0 0
\(222\) −1.75302 −0.117655
\(223\) 9.95838 0.666862 0.333431 0.942774i \(-0.391794\pi\)
0.333431 + 0.942774i \(0.391794\pi\)
\(224\) −4.24698 −0.283763
\(225\) 1.00000 0.0666667
\(226\) 11.2295 0.746977
\(227\) 19.6112 1.30164 0.650820 0.759232i \(-0.274425\pi\)
0.650820 + 0.759232i \(0.274425\pi\)
\(228\) −4.85086 −0.321256
\(229\) 8.72886 0.576819 0.288410 0.957507i \(-0.406874\pi\)
0.288410 + 0.957507i \(0.406874\pi\)
\(230\) 0.445042 0.0293452
\(231\) 8.11960 0.534231
\(232\) 8.56465 0.562297
\(233\) 0.975837 0.0639292 0.0319646 0.999489i \(-0.489824\pi\)
0.0319646 + 0.999489i \(0.489824\pi\)
\(234\) 0 0
\(235\) −10.3840 −0.677380
\(236\) −13.4601 −0.876178
\(237\) 14.5526 0.945291
\(238\) 14.1642 0.918129
\(239\) 7.39373 0.478261 0.239130 0.970987i \(-0.423138\pi\)
0.239130 + 0.970987i \(0.423138\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 2.11529 0.136258 0.0681290 0.997677i \(-0.478297\pi\)
0.0681290 + 0.997677i \(0.478297\pi\)
\(242\) 7.34481 0.472143
\(243\) −1.00000 −0.0641500
\(244\) 1.13169 0.0724488
\(245\) 11.0368 0.705118
\(246\) −3.24698 −0.207020
\(247\) 0 0
\(248\) 5.29590 0.336290
\(249\) −9.49157 −0.601504
\(250\) −1.00000 −0.0632456
\(251\) −1.62863 −0.102798 −0.0513991 0.998678i \(-0.516368\pi\)
−0.0513991 + 0.998678i \(0.516368\pi\)
\(252\) 4.24698 0.267535
\(253\) 0.850855 0.0534928
\(254\) −20.8267 −1.30678
\(255\) 3.33513 0.208854
\(256\) 1.00000 0.0625000
\(257\) −8.87263 −0.553459 −0.276730 0.960948i \(-0.589251\pi\)
−0.276730 + 0.960948i \(0.589251\pi\)
\(258\) −1.97823 −0.123159
\(259\) −7.44504 −0.462612
\(260\) 0 0
\(261\) −8.56465 −0.530138
\(262\) −17.5036 −1.08138
\(263\) 22.4185 1.38238 0.691192 0.722672i \(-0.257086\pi\)
0.691192 + 0.722672i \(0.257086\pi\)
\(264\) −1.91185 −0.117666
\(265\) −1.14914 −0.0705914
\(266\) −20.6015 −1.26316
\(267\) −1.25667 −0.0769068
\(268\) −13.5308 −0.826525
\(269\) 26.9071 1.64055 0.820276 0.571967i \(-0.193820\pi\)
0.820276 + 0.571967i \(0.193820\pi\)
\(270\) 1.00000 0.0608581
\(271\) −31.7265 −1.92725 −0.963623 0.267266i \(-0.913880\pi\)
−0.963623 + 0.267266i \(0.913880\pi\)
\(272\) −3.33513 −0.202222
\(273\) 0 0
\(274\) −0.929312 −0.0561418
\(275\) −1.91185 −0.115289
\(276\) 0.445042 0.0267884
\(277\) −30.6069 −1.83899 −0.919494 0.393104i \(-0.871401\pi\)
−0.919494 + 0.393104i \(0.871401\pi\)
\(278\) −3.12200 −0.187245
\(279\) −5.29590 −0.317057
\(280\) −4.24698 −0.253806
\(281\) −16.6649 −0.994143 −0.497072 0.867710i \(-0.665591\pi\)
−0.497072 + 0.867710i \(0.665591\pi\)
\(282\) −10.3840 −0.618361
\(283\) −4.32304 −0.256978 −0.128489 0.991711i \(-0.541013\pi\)
−0.128489 + 0.991711i \(0.541013\pi\)
\(284\) 2.14675 0.127386
\(285\) −4.85086 −0.287340
\(286\) 0 0
\(287\) −13.7899 −0.813989
\(288\) −1.00000 −0.0589256
\(289\) −5.87694 −0.345702
\(290\) 8.56465 0.502933
\(291\) −5.01507 −0.293988
\(292\) −5.15883 −0.301898
\(293\) −5.62804 −0.328794 −0.164397 0.986394i \(-0.552568\pi\)
−0.164397 + 0.986394i \(0.552568\pi\)
\(294\) 11.0368 0.643681
\(295\) −13.4601 −0.783678
\(296\) 1.75302 0.101892
\(297\) 1.91185 0.110937
\(298\) 6.57673 0.380980
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) −8.40150 −0.484254
\(302\) 6.29590 0.362288
\(303\) −14.8116 −0.850906
\(304\) 4.85086 0.278216
\(305\) 1.13169 0.0648002
\(306\) 3.33513 0.190656
\(307\) −12.6635 −0.722747 −0.361373 0.932421i \(-0.617692\pi\)
−0.361373 + 0.932421i \(0.617692\pi\)
\(308\) −8.11960 −0.462657
\(309\) 13.6896 0.778776
\(310\) 5.29590 0.300787
\(311\) 8.49635 0.481784 0.240892 0.970552i \(-0.422560\pi\)
0.240892 + 0.970552i \(0.422560\pi\)
\(312\) 0 0
\(313\) −25.2610 −1.42784 −0.713918 0.700230i \(-0.753081\pi\)
−0.713918 + 0.700230i \(0.753081\pi\)
\(314\) −19.9148 −1.12386
\(315\) 4.24698 0.239290
\(316\) −14.5526 −0.818646
\(317\) −4.96615 −0.278927 −0.139463 0.990227i \(-0.544538\pi\)
−0.139463 + 0.990227i \(0.544538\pi\)
\(318\) −1.14914 −0.0644408
\(319\) 16.3744 0.916788
\(320\) 1.00000 0.0559017
\(321\) 12.8334 0.716290
\(322\) 1.89008 0.105330
\(323\) −16.1782 −0.900180
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 13.1075 0.725959
\(327\) 10.8726 0.601258
\(328\) 3.24698 0.179284
\(329\) −44.1008 −2.43136
\(330\) −1.91185 −0.105244
\(331\) 35.3749 1.94438 0.972191 0.234188i \(-0.0752430\pi\)
0.972191 + 0.234188i \(0.0752430\pi\)
\(332\) 9.49157 0.520917
\(333\) −1.75302 −0.0960649
\(334\) 21.5351 1.17835
\(335\) −13.5308 −0.739266
\(336\) −4.24698 −0.231692
\(337\) −4.70171 −0.256118 −0.128059 0.991767i \(-0.540875\pi\)
−0.128059 + 0.991767i \(0.540875\pi\)
\(338\) 0 0
\(339\) 11.2295 0.609904
\(340\) −3.33513 −0.180873
\(341\) 10.1250 0.548299
\(342\) −4.85086 −0.262304
\(343\) 17.1444 0.925708
\(344\) 1.97823 0.106659
\(345\) 0.445042 0.0239602
\(346\) 13.9541 0.750175
\(347\) 14.9028 0.800022 0.400011 0.916510i \(-0.369006\pi\)
0.400011 + 0.916510i \(0.369006\pi\)
\(348\) 8.56465 0.459113
\(349\) −35.2669 −1.88780 −0.943898 0.330237i \(-0.892871\pi\)
−0.943898 + 0.330237i \(0.892871\pi\)
\(350\) −4.24698 −0.227011
\(351\) 0 0
\(352\) 1.91185 0.101902
\(353\) −10.5574 −0.561911 −0.280956 0.959721i \(-0.590651\pi\)
−0.280956 + 0.959721i \(0.590651\pi\)
\(354\) −13.4601 −0.715397
\(355\) 2.14675 0.113938
\(356\) 1.25667 0.0666033
\(357\) 14.1642 0.749650
\(358\) −8.32975 −0.440241
\(359\) −28.7724 −1.51855 −0.759275 0.650770i \(-0.774446\pi\)
−0.759275 + 0.650770i \(0.774446\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 4.53079 0.238463
\(362\) −17.1371 −0.900704
\(363\) 7.34481 0.385503
\(364\) 0 0
\(365\) −5.15883 −0.270026
\(366\) 1.13169 0.0591542
\(367\) −24.1511 −1.26068 −0.630338 0.776321i \(-0.717084\pi\)
−0.630338 + 0.776321i \(0.717084\pi\)
\(368\) −0.445042 −0.0231994
\(369\) −3.24698 −0.169031
\(370\) 1.75302 0.0911352
\(371\) −4.88040 −0.253377
\(372\) 5.29590 0.274579
\(373\) −21.0664 −1.09078 −0.545388 0.838184i \(-0.683618\pi\)
−0.545388 + 0.838184i \(0.683618\pi\)
\(374\) −6.37627 −0.329709
\(375\) −1.00000 −0.0516398
\(376\) 10.3840 0.535516
\(377\) 0 0
\(378\) 4.24698 0.218441
\(379\) 8.42327 0.432674 0.216337 0.976319i \(-0.430589\pi\)
0.216337 + 0.976319i \(0.430589\pi\)
\(380\) 4.85086 0.248844
\(381\) −20.8267 −1.06698
\(382\) −23.1685 −1.18540
\(383\) −10.6377 −0.543562 −0.271781 0.962359i \(-0.587613\pi\)
−0.271781 + 0.962359i \(0.587613\pi\)
\(384\) 1.00000 0.0510310
\(385\) −8.11960 −0.413813
\(386\) 6.38942 0.325213
\(387\) −1.97823 −0.100559
\(388\) 5.01507 0.254601
\(389\) −25.7429 −1.30521 −0.652607 0.757696i \(-0.726325\pi\)
−0.652607 + 0.757696i \(0.726325\pi\)
\(390\) 0 0
\(391\) 1.48427 0.0750628
\(392\) −11.0368 −0.557444
\(393\) −17.5036 −0.882942
\(394\) −14.8605 −0.748663
\(395\) −14.5526 −0.732219
\(396\) −1.91185 −0.0960743
\(397\) −11.4373 −0.574020 −0.287010 0.957928i \(-0.592661\pi\)
−0.287010 + 0.957928i \(0.592661\pi\)
\(398\) −26.3860 −1.32261
\(399\) −20.6015 −1.03136
\(400\) 1.00000 0.0500000
\(401\) −22.1511 −1.10617 −0.553086 0.833124i \(-0.686550\pi\)
−0.553086 + 0.833124i \(0.686550\pi\)
\(402\) −13.5308 −0.674855
\(403\) 0 0
\(404\) 14.8116 0.736906
\(405\) 1.00000 0.0496904
\(406\) 36.3739 1.80521
\(407\) 3.35152 0.166129
\(408\) −3.33513 −0.165113
\(409\) 22.0965 1.09260 0.546301 0.837589i \(-0.316035\pi\)
0.546301 + 0.837589i \(0.316035\pi\)
\(410\) 3.24698 0.160357
\(411\) −0.929312 −0.0458396
\(412\) −13.6896 −0.674440
\(413\) −57.1648 −2.81290
\(414\) 0.445042 0.0218726
\(415\) 9.49157 0.465923
\(416\) 0 0
\(417\) −3.12200 −0.152885
\(418\) 9.27413 0.453612
\(419\) 9.03444 0.441361 0.220681 0.975346i \(-0.429172\pi\)
0.220681 + 0.975346i \(0.429172\pi\)
\(420\) −4.24698 −0.207231
\(421\) −7.17928 −0.349896 −0.174948 0.984578i \(-0.555976\pi\)
−0.174948 + 0.984578i \(0.555976\pi\)
\(422\) −17.4383 −0.848885
\(423\) −10.3840 −0.504889
\(424\) 1.14914 0.0558074
\(425\) −3.33513 −0.161777
\(426\) 2.14675 0.104010
\(427\) 4.80625 0.232591
\(428\) −12.8334 −0.620326
\(429\) 0 0
\(430\) 1.97823 0.0953987
\(431\) −8.36360 −0.402860 −0.201430 0.979503i \(-0.564559\pi\)
−0.201430 + 0.979503i \(0.564559\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −12.2851 −0.590386 −0.295193 0.955438i \(-0.595384\pi\)
−0.295193 + 0.955438i \(0.595384\pi\)
\(434\) 22.4916 1.07963
\(435\) 8.56465 0.410643
\(436\) −10.8726 −0.520704
\(437\) −2.15883 −0.103271
\(438\) −5.15883 −0.246499
\(439\) 20.4849 0.977689 0.488845 0.872371i \(-0.337419\pi\)
0.488845 + 0.872371i \(0.337419\pi\)
\(440\) 1.91185 0.0911441
\(441\) 11.0368 0.525564
\(442\) 0 0
\(443\) −0.928247 −0.0441024 −0.0220512 0.999757i \(-0.507020\pi\)
−0.0220512 + 0.999757i \(0.507020\pi\)
\(444\) 1.75302 0.0831947
\(445\) 1.25667 0.0595718
\(446\) −9.95838 −0.471543
\(447\) 6.57673 0.311069
\(448\) 4.24698 0.200651
\(449\) 22.9355 1.08240 0.541198 0.840895i \(-0.317971\pi\)
0.541198 + 0.840895i \(0.317971\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 6.20775 0.292312
\(452\) −11.2295 −0.528192
\(453\) 6.29590 0.295807
\(454\) −19.6112 −0.920398
\(455\) 0 0
\(456\) 4.85086 0.227162
\(457\) −10.3773 −0.485431 −0.242716 0.970097i \(-0.578038\pi\)
−0.242716 + 0.970097i \(0.578038\pi\)
\(458\) −8.72886 −0.407873
\(459\) 3.33513 0.155670
\(460\) −0.445042 −0.0207502
\(461\) −7.69740 −0.358504 −0.179252 0.983803i \(-0.557368\pi\)
−0.179252 + 0.983803i \(0.557368\pi\)
\(462\) −8.11960 −0.377758
\(463\) 39.0640 1.81546 0.907729 0.419558i \(-0.137815\pi\)
0.907729 + 0.419558i \(0.137815\pi\)
\(464\) −8.56465 −0.397604
\(465\) 5.29590 0.245591
\(466\) −0.975837 −0.0452048
\(467\) 0.916166 0.0423951 0.0211976 0.999775i \(-0.493252\pi\)
0.0211976 + 0.999775i \(0.493252\pi\)
\(468\) 0 0
\(469\) −57.4650 −2.65349
\(470\) 10.3840 0.478980
\(471\) −19.9148 −0.917627
\(472\) 13.4601 0.619552
\(473\) 3.78209 0.173901
\(474\) −14.5526 −0.668421
\(475\) 4.85086 0.222572
\(476\) −14.1642 −0.649216
\(477\) −1.14914 −0.0526157
\(478\) −7.39373 −0.338181
\(479\) 23.1715 1.05873 0.529367 0.848393i \(-0.322430\pi\)
0.529367 + 0.848393i \(0.322430\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) −2.11529 −0.0963490
\(483\) 1.89008 0.0860018
\(484\) −7.34481 −0.333855
\(485\) 5.01507 0.227722
\(486\) 1.00000 0.0453609
\(487\) 11.9812 0.542921 0.271460 0.962450i \(-0.412493\pi\)
0.271460 + 0.962450i \(0.412493\pi\)
\(488\) −1.13169 −0.0512290
\(489\) 13.1075 0.592743
\(490\) −11.0368 −0.498593
\(491\) −24.8745 −1.12257 −0.561286 0.827622i \(-0.689693\pi\)
−0.561286 + 0.827622i \(0.689693\pi\)
\(492\) 3.24698 0.146385
\(493\) 28.5642 1.28647
\(494\) 0 0
\(495\) −1.91185 −0.0859314
\(496\) −5.29590 −0.237793
\(497\) 9.11721 0.408963
\(498\) 9.49157 0.425327
\(499\) 42.0737 1.88348 0.941738 0.336347i \(-0.109192\pi\)
0.941738 + 0.336347i \(0.109192\pi\)
\(500\) 1.00000 0.0447214
\(501\) 21.5351 0.962118
\(502\) 1.62863 0.0726893
\(503\) −20.0151 −0.892428 −0.446214 0.894926i \(-0.647228\pi\)
−0.446214 + 0.894926i \(0.647228\pi\)
\(504\) −4.24698 −0.189176
\(505\) 14.8116 0.659109
\(506\) −0.850855 −0.0378251
\(507\) 0 0
\(508\) 20.8267 0.924035
\(509\) −25.6907 −1.13872 −0.569360 0.822088i \(-0.692809\pi\)
−0.569360 + 0.822088i \(0.692809\pi\)
\(510\) −3.33513 −0.147682
\(511\) −21.9095 −0.969217
\(512\) −1.00000 −0.0441942
\(513\) −4.85086 −0.214170
\(514\) 8.87263 0.391355
\(515\) −13.6896 −0.603237
\(516\) 1.97823 0.0870867
\(517\) 19.8528 0.873124
\(518\) 7.44504 0.327116
\(519\) 13.9541 0.612516
\(520\) 0 0
\(521\) −24.7342 −1.08363 −0.541813 0.840499i \(-0.682262\pi\)
−0.541813 + 0.840499i \(0.682262\pi\)
\(522\) 8.56465 0.374864
\(523\) −18.7439 −0.819615 −0.409807 0.912172i \(-0.634404\pi\)
−0.409807 + 0.912172i \(0.634404\pi\)
\(524\) 17.5036 0.764650
\(525\) −4.24698 −0.185353
\(526\) −22.4185 −0.977492
\(527\) 17.6625 0.769390
\(528\) 1.91185 0.0832028
\(529\) −22.8019 −0.991389
\(530\) 1.14914 0.0499157
\(531\) −13.4601 −0.584119
\(532\) 20.6015 0.893188
\(533\) 0 0
\(534\) 1.25667 0.0543814
\(535\) −12.8334 −0.554836
\(536\) 13.5308 0.584441
\(537\) −8.32975 −0.359455
\(538\) −26.9071 −1.16005
\(539\) −21.1008 −0.908877
\(540\) −1.00000 −0.0430331
\(541\) 43.7434 1.88068 0.940339 0.340239i \(-0.110508\pi\)
0.940339 + 0.340239i \(0.110508\pi\)
\(542\) 31.7265 1.36277
\(543\) −17.1371 −0.735422
\(544\) 3.33513 0.142992
\(545\) −10.8726 −0.465732
\(546\) 0 0
\(547\) 17.9681 0.768259 0.384130 0.923279i \(-0.374502\pi\)
0.384130 + 0.923279i \(0.374502\pi\)
\(548\) 0.929312 0.0396983
\(549\) 1.13169 0.0482992
\(550\) 1.91185 0.0815217
\(551\) −41.5459 −1.76991
\(552\) −0.445042 −0.0189422
\(553\) −61.8044 −2.62819
\(554\) 30.6069 1.30036
\(555\) 1.75302 0.0744116
\(556\) 3.12200 0.132402
\(557\) 8.26981 0.350403 0.175202 0.984533i \(-0.443942\pi\)
0.175202 + 0.984533i \(0.443942\pi\)
\(558\) 5.29590 0.224193
\(559\) 0 0
\(560\) 4.24698 0.179468
\(561\) −6.37627 −0.269206
\(562\) 16.6649 0.702965
\(563\) 19.8984 0.838619 0.419310 0.907843i \(-0.362272\pi\)
0.419310 + 0.907843i \(0.362272\pi\)
\(564\) 10.3840 0.437247
\(565\) −11.2295 −0.472430
\(566\) 4.32304 0.181711
\(567\) 4.24698 0.178356
\(568\) −2.14675 −0.0900757
\(569\) 14.6528 0.614277 0.307139 0.951665i \(-0.400629\pi\)
0.307139 + 0.951665i \(0.400629\pi\)
\(570\) 4.85086 0.203180
\(571\) −9.47517 −0.396524 −0.198262 0.980149i \(-0.563530\pi\)
−0.198262 + 0.980149i \(0.563530\pi\)
\(572\) 0 0
\(573\) −23.1685 −0.967879
\(574\) 13.7899 0.575577
\(575\) −0.445042 −0.0185595
\(576\) 1.00000 0.0416667
\(577\) 8.79225 0.366026 0.183013 0.983110i \(-0.441415\pi\)
0.183013 + 0.983110i \(0.441415\pi\)
\(578\) 5.87694 0.244448
\(579\) 6.38942 0.265535
\(580\) −8.56465 −0.355628
\(581\) 40.3105 1.67236
\(582\) 5.01507 0.207881
\(583\) 2.19700 0.0909903
\(584\) 5.15883 0.213474
\(585\) 0 0
\(586\) 5.62804 0.232492
\(587\) 2.38942 0.0986219 0.0493110 0.998783i \(-0.484297\pi\)
0.0493110 + 0.998783i \(0.484297\pi\)
\(588\) −11.0368 −0.455151
\(589\) −25.6896 −1.05852
\(590\) 13.4601 0.554144
\(591\) −14.8605 −0.611281
\(592\) −1.75302 −0.0720487
\(593\) −21.4034 −0.878933 −0.439467 0.898259i \(-0.644833\pi\)
−0.439467 + 0.898259i \(0.644833\pi\)
\(594\) −1.91185 −0.0784443
\(595\) −14.1642 −0.580676
\(596\) −6.57673 −0.269393
\(597\) −26.3860 −1.07991
\(598\) 0 0
\(599\) 12.4765 0.509776 0.254888 0.966971i \(-0.417961\pi\)
0.254888 + 0.966971i \(0.417961\pi\)
\(600\) 1.00000 0.0408248
\(601\) −12.9487 −0.528188 −0.264094 0.964497i \(-0.585073\pi\)
−0.264094 + 0.964497i \(0.585073\pi\)
\(602\) 8.40150 0.342420
\(603\) −13.5308 −0.551017
\(604\) −6.29590 −0.256176
\(605\) −7.34481 −0.298609
\(606\) 14.8116 0.601681
\(607\) −11.5230 −0.467705 −0.233853 0.972272i \(-0.575133\pi\)
−0.233853 + 0.972272i \(0.575133\pi\)
\(608\) −4.85086 −0.196728
\(609\) 36.3739 1.47394
\(610\) −1.13169 −0.0458206
\(611\) 0 0
\(612\) −3.33513 −0.134814
\(613\) 44.9842 1.81689 0.908447 0.417999i \(-0.137269\pi\)
0.908447 + 0.417999i \(0.137269\pi\)
\(614\) 12.6635 0.511059
\(615\) 3.24698 0.130931
\(616\) 8.11960 0.327148
\(617\) −24.9318 −1.00372 −0.501859 0.864950i \(-0.667350\pi\)
−0.501859 + 0.864950i \(0.667350\pi\)
\(618\) −13.6896 −0.550678
\(619\) 11.2929 0.453900 0.226950 0.973906i \(-0.427125\pi\)
0.226950 + 0.973906i \(0.427125\pi\)
\(620\) −5.29590 −0.212688
\(621\) 0.445042 0.0178589
\(622\) −8.49635 −0.340673
\(623\) 5.33704 0.213824
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 25.2610 1.00963
\(627\) 9.27413 0.370373
\(628\) 19.9148 0.794689
\(629\) 5.84654 0.233117
\(630\) −4.24698 −0.169204
\(631\) 20.6165 0.820732 0.410366 0.911921i \(-0.365401\pi\)
0.410366 + 0.911921i \(0.365401\pi\)
\(632\) 14.5526 0.578870
\(633\) −17.4383 −0.693112
\(634\) 4.96615 0.197231
\(635\) 20.8267 0.826482
\(636\) 1.14914 0.0455666
\(637\) 0 0
\(638\) −16.3744 −0.648267
\(639\) 2.14675 0.0849242
\(640\) −1.00000 −0.0395285
\(641\) 47.1487 1.86226 0.931130 0.364687i \(-0.118824\pi\)
0.931130 + 0.364687i \(0.118824\pi\)
\(642\) −12.8334 −0.506494
\(643\) 8.41981 0.332045 0.166023 0.986122i \(-0.446908\pi\)
0.166023 + 0.986122i \(0.446908\pi\)
\(644\) −1.89008 −0.0744797
\(645\) 1.97823 0.0778927
\(646\) 16.1782 0.636523
\(647\) −43.3836 −1.70558 −0.852792 0.522251i \(-0.825093\pi\)
−0.852792 + 0.522251i \(0.825093\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 25.7338 1.01014
\(650\) 0 0
\(651\) 22.4916 0.881514
\(652\) −13.1075 −0.513330
\(653\) −21.2862 −0.832994 −0.416497 0.909137i \(-0.636742\pi\)
−0.416497 + 0.909137i \(0.636742\pi\)
\(654\) −10.8726 −0.425153
\(655\) 17.5036 0.683924
\(656\) −3.24698 −0.126773
\(657\) −5.15883 −0.201265
\(658\) 44.1008 1.71923
\(659\) 24.9075 0.970260 0.485130 0.874442i \(-0.338772\pi\)
0.485130 + 0.874442i \(0.338772\pi\)
\(660\) 1.91185 0.0744188
\(661\) 17.6209 0.685372 0.342686 0.939450i \(-0.388663\pi\)
0.342686 + 0.939450i \(0.388663\pi\)
\(662\) −35.3749 −1.37489
\(663\) 0 0
\(664\) −9.49157 −0.368344
\(665\) 20.6015 0.798891
\(666\) 1.75302 0.0679282
\(667\) 3.81163 0.147587
\(668\) −21.5351 −0.833218
\(669\) −9.95838 −0.385013
\(670\) 13.5308 0.522740
\(671\) −2.16362 −0.0835256
\(672\) 4.24698 0.163831
\(673\) −28.8888 −1.11358 −0.556790 0.830653i \(-0.687967\pi\)
−0.556790 + 0.830653i \(0.687967\pi\)
\(674\) 4.70171 0.181103
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 24.3927 0.937486 0.468743 0.883335i \(-0.344707\pi\)
0.468743 + 0.883335i \(0.344707\pi\)
\(678\) −11.2295 −0.431267
\(679\) 21.2989 0.817376
\(680\) 3.33513 0.127896
\(681\) −19.6112 −0.751502
\(682\) −10.1250 −0.387706
\(683\) −47.6558 −1.82350 −0.911749 0.410748i \(-0.865268\pi\)
−0.911749 + 0.410748i \(0.865268\pi\)
\(684\) 4.85086 0.185477
\(685\) 0.929312 0.0355072
\(686\) −17.1444 −0.654575
\(687\) −8.72886 −0.333027
\(688\) −1.97823 −0.0754193
\(689\) 0 0
\(690\) −0.445042 −0.0169425
\(691\) −50.2180 −1.91038 −0.955192 0.295987i \(-0.904351\pi\)
−0.955192 + 0.295987i \(0.904351\pi\)
\(692\) −13.9541 −0.530454
\(693\) −8.11960 −0.308438
\(694\) −14.9028 −0.565701
\(695\) 3.12200 0.118424
\(696\) −8.56465 −0.324642
\(697\) 10.8291 0.410181
\(698\) 35.2669 1.33487
\(699\) −0.975837 −0.0369095
\(700\) 4.24698 0.160521
\(701\) −12.8140 −0.483979 −0.241989 0.970279i \(-0.577800\pi\)
−0.241989 + 0.970279i \(0.577800\pi\)
\(702\) 0 0
\(703\) −8.50365 −0.320721
\(704\) −1.91185 −0.0720557
\(705\) 10.3840 0.391086
\(706\) 10.5574 0.397331
\(707\) 62.9047 2.36577
\(708\) 13.4601 0.505862
\(709\) 34.5515 1.29761 0.648804 0.760955i \(-0.275269\pi\)
0.648804 + 0.760955i \(0.275269\pi\)
\(710\) −2.14675 −0.0805662
\(711\) −14.5526 −0.545764
\(712\) −1.25667 −0.0470956
\(713\) 2.35690 0.0882664
\(714\) −14.1642 −0.530082
\(715\) 0 0
\(716\) 8.32975 0.311297
\(717\) −7.39373 −0.276124
\(718\) 28.7724 1.07378
\(719\) 20.8528 0.777677 0.388839 0.921306i \(-0.372876\pi\)
0.388839 + 0.921306i \(0.372876\pi\)
\(720\) 1.00000 0.0372678
\(721\) −58.1396 −2.16523
\(722\) −4.53079 −0.168619
\(723\) −2.11529 −0.0786686
\(724\) 17.1371 0.636894
\(725\) −8.56465 −0.318083
\(726\) −7.34481 −0.272592
\(727\) 32.7192 1.21349 0.606743 0.794898i \(-0.292476\pi\)
0.606743 + 0.794898i \(0.292476\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 5.15883 0.190937
\(731\) 6.59764 0.244023
\(732\) −1.13169 −0.0418283
\(733\) −52.2737 −1.93077 −0.965386 0.260826i \(-0.916005\pi\)
−0.965386 + 0.260826i \(0.916005\pi\)
\(734\) 24.1511 0.891432
\(735\) −11.0368 −0.407100
\(736\) 0.445042 0.0164045
\(737\) 25.8689 0.952893
\(738\) 3.24698 0.119523
\(739\) 13.4571 0.495028 0.247514 0.968884i \(-0.420386\pi\)
0.247514 + 0.968884i \(0.420386\pi\)
\(740\) −1.75302 −0.0644423
\(741\) 0 0
\(742\) 4.88040 0.179165
\(743\) −22.9989 −0.843749 −0.421875 0.906654i \(-0.638628\pi\)
−0.421875 + 0.906654i \(0.638628\pi\)
\(744\) −5.29590 −0.194157
\(745\) −6.57673 −0.240953
\(746\) 21.0664 0.771295
\(747\) 9.49157 0.347278
\(748\) 6.37627 0.233140
\(749\) −54.5032 −1.99150
\(750\) 1.00000 0.0365148
\(751\) 34.1159 1.24491 0.622453 0.782657i \(-0.286136\pi\)
0.622453 + 0.782657i \(0.286136\pi\)
\(752\) −10.3840 −0.378667
\(753\) 1.62863 0.0593506
\(754\) 0 0
\(755\) −6.29590 −0.229131
\(756\) −4.24698 −0.154461
\(757\) −38.9004 −1.41386 −0.706929 0.707285i \(-0.749920\pi\)
−0.706929 + 0.707285i \(0.749920\pi\)
\(758\) −8.42327 −0.305947
\(759\) −0.850855 −0.0308841
\(760\) −4.85086 −0.175959
\(761\) 36.8528 1.33591 0.667956 0.744201i \(-0.267169\pi\)
0.667956 + 0.744201i \(0.267169\pi\)
\(762\) 20.8267 0.754471
\(763\) −46.1758 −1.67168
\(764\) 23.1685 0.838208
\(765\) −3.33513 −0.120582
\(766\) 10.6377 0.384357
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 23.4198 0.844540 0.422270 0.906470i \(-0.361233\pi\)
0.422270 + 0.906470i \(0.361233\pi\)
\(770\) 8.11960 0.292610
\(771\) 8.87263 0.319540
\(772\) −6.38942 −0.229960
\(773\) −0.275192 −0.00989795 −0.00494898 0.999988i \(-0.501575\pi\)
−0.00494898 + 0.999988i \(0.501575\pi\)
\(774\) 1.97823 0.0711060
\(775\) −5.29590 −0.190234
\(776\) −5.01507 −0.180030
\(777\) 7.44504 0.267089
\(778\) 25.7429 0.922926
\(779\) −15.7506 −0.564325
\(780\) 0 0
\(781\) −4.10428 −0.146863
\(782\) −1.48427 −0.0530774
\(783\) 8.56465 0.306076
\(784\) 11.0368 0.394173
\(785\) 19.9148 0.710791
\(786\) 17.5036 0.624334
\(787\) −27.2543 −0.971510 −0.485755 0.874095i \(-0.661455\pi\)
−0.485755 + 0.874095i \(0.661455\pi\)
\(788\) 14.8605 0.529385
\(789\) −22.4185 −0.798119
\(790\) 14.5526 0.517757
\(791\) −47.6915 −1.69572
\(792\) 1.91185 0.0679348
\(793\) 0 0
\(794\) 11.4373 0.405894
\(795\) 1.14914 0.0407560
\(796\) 26.3860 0.935226
\(797\) 14.2819 0.505891 0.252945 0.967481i \(-0.418601\pi\)
0.252945 + 0.967481i \(0.418601\pi\)
\(798\) 20.6015 0.729285
\(799\) 34.6321 1.22520
\(800\) −1.00000 −0.0353553
\(801\) 1.25667 0.0444022
\(802\) 22.1511 0.782181
\(803\) 9.86294 0.348055
\(804\) 13.5308 0.477194
\(805\) −1.89008 −0.0666167
\(806\) 0 0
\(807\) −26.9071 −0.947174
\(808\) −14.8116 −0.521071
\(809\) −8.82610 −0.310309 −0.155155 0.987890i \(-0.549588\pi\)
−0.155155 + 0.987890i \(0.549588\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −6.37196 −0.223750 −0.111875 0.993722i \(-0.535686\pi\)
−0.111875 + 0.993722i \(0.535686\pi\)
\(812\) −36.3739 −1.27647
\(813\) 31.7265 1.11270
\(814\) −3.35152 −0.117471
\(815\) −13.1075 −0.459137
\(816\) 3.33513 0.116753
\(817\) −9.59611 −0.335725
\(818\) −22.0965 −0.772586
\(819\) 0 0
\(820\) −3.24698 −0.113389
\(821\) −13.7476 −0.479796 −0.239898 0.970798i \(-0.577114\pi\)
−0.239898 + 0.970798i \(0.577114\pi\)
\(822\) 0.929312 0.0324135
\(823\) 47.8587 1.66825 0.834125 0.551575i \(-0.185973\pi\)
0.834125 + 0.551575i \(0.185973\pi\)
\(824\) 13.6896 0.476901
\(825\) 1.91185 0.0665622
\(826\) 57.1648 1.98902
\(827\) 29.1317 1.01301 0.506504 0.862238i \(-0.330938\pi\)
0.506504 + 0.862238i \(0.330938\pi\)
\(828\) −0.445042 −0.0154663
\(829\) −25.7748 −0.895195 −0.447598 0.894235i \(-0.647720\pi\)
−0.447598 + 0.894235i \(0.647720\pi\)
\(830\) −9.49157 −0.329457
\(831\) 30.6069 1.06174
\(832\) 0 0
\(833\) −36.8092 −1.27536
\(834\) 3.12200 0.108106
\(835\) −21.5351 −0.745253
\(836\) −9.27413 −0.320752
\(837\) 5.29590 0.183053
\(838\) −9.03444 −0.312090
\(839\) 2.91292 0.100565 0.0502826 0.998735i \(-0.483988\pi\)
0.0502826 + 0.998735i \(0.483988\pi\)
\(840\) 4.24698 0.146535
\(841\) 44.3532 1.52942
\(842\) 7.17928 0.247414
\(843\) 16.6649 0.573969
\(844\) 17.4383 0.600252
\(845\) 0 0
\(846\) 10.3840 0.357011
\(847\) −31.1933 −1.07181
\(848\) −1.14914 −0.0394618
\(849\) 4.32304 0.148366
\(850\) 3.33513 0.114394
\(851\) 0.780167 0.0267438
\(852\) −2.14675 −0.0735465
\(853\) −12.9661 −0.443952 −0.221976 0.975052i \(-0.571251\pi\)
−0.221976 + 0.975052i \(0.571251\pi\)
\(854\) −4.80625 −0.164466
\(855\) 4.85086 0.165896
\(856\) 12.8334 0.438636
\(857\) −9.85623 −0.336682 −0.168341 0.985729i \(-0.553841\pi\)
−0.168341 + 0.985729i \(0.553841\pi\)
\(858\) 0 0
\(859\) 30.6746 1.04660 0.523301 0.852148i \(-0.324700\pi\)
0.523301 + 0.852148i \(0.324700\pi\)
\(860\) −1.97823 −0.0674571
\(861\) 13.7899 0.469957
\(862\) 8.36360 0.284865
\(863\) −34.1588 −1.16278 −0.581390 0.813625i \(-0.697491\pi\)
−0.581390 + 0.813625i \(0.697491\pi\)
\(864\) 1.00000 0.0340207
\(865\) −13.9541 −0.474452
\(866\) 12.2851 0.417466
\(867\) 5.87694 0.199591
\(868\) −22.4916 −0.763414
\(869\) 27.8224 0.943810
\(870\) −8.56465 −0.290369
\(871\) 0 0
\(872\) 10.8726 0.368194
\(873\) 5.01507 0.169734
\(874\) 2.15883 0.0730236
\(875\) 4.24698 0.143574
\(876\) 5.15883 0.174301
\(877\) −20.1588 −0.680715 −0.340358 0.940296i \(-0.610548\pi\)
−0.340358 + 0.940296i \(0.610548\pi\)
\(878\) −20.4849 −0.691331
\(879\) 5.62804 0.189829
\(880\) −1.91185 −0.0644486
\(881\) −2.79550 −0.0941827 −0.0470913 0.998891i \(-0.514995\pi\)
−0.0470913 + 0.998891i \(0.514995\pi\)
\(882\) −11.0368 −0.371630
\(883\) −16.8605 −0.567402 −0.283701 0.958913i \(-0.591562\pi\)
−0.283701 + 0.958913i \(0.591562\pi\)
\(884\) 0 0
\(885\) 13.4601 0.452457
\(886\) 0.928247 0.0311851
\(887\) −8.71379 −0.292580 −0.146290 0.989242i \(-0.546733\pi\)
−0.146290 + 0.989242i \(0.546733\pi\)
\(888\) −1.75302 −0.0588275
\(889\) 88.4505 2.96654
\(890\) −1.25667 −0.0421236
\(891\) −1.91185 −0.0640495
\(892\) 9.95838 0.333431
\(893\) −50.3715 −1.68562
\(894\) −6.57673 −0.219959
\(895\) 8.32975 0.278433
\(896\) −4.24698 −0.141882
\(897\) 0 0
\(898\) −22.9355 −0.765369
\(899\) 45.3575 1.51276
\(900\) 1.00000 0.0333333
\(901\) 3.83254 0.127681
\(902\) −6.20775 −0.206695
\(903\) 8.40150 0.279584
\(904\) 11.2295 0.373488
\(905\) 17.1371 0.569655
\(906\) −6.29590 −0.209167
\(907\) 14.2295 0.472483 0.236242 0.971694i \(-0.424084\pi\)
0.236242 + 0.971694i \(0.424084\pi\)
\(908\) 19.6112 0.650820
\(909\) 14.8116 0.491271
\(910\) 0 0
\(911\) −19.3672 −0.641663 −0.320832 0.947136i \(-0.603962\pi\)
−0.320832 + 0.947136i \(0.603962\pi\)
\(912\) −4.85086 −0.160628
\(913\) −18.1465 −0.600561
\(914\) 10.3773 0.343252
\(915\) −1.13169 −0.0374124
\(916\) 8.72886 0.288410
\(917\) 74.3376 2.45484
\(918\) −3.33513 −0.110076
\(919\) −44.1420 −1.45611 −0.728055 0.685519i \(-0.759575\pi\)
−0.728055 + 0.685519i \(0.759575\pi\)
\(920\) 0.445042 0.0146726
\(921\) 12.6635 0.417278
\(922\) 7.69740 0.253500
\(923\) 0 0
\(924\) 8.11960 0.267115
\(925\) −1.75302 −0.0576390
\(926\) −39.0640 −1.28372
\(927\) −13.6896 −0.449626
\(928\) 8.56465 0.281148
\(929\) −14.8616 −0.487594 −0.243797 0.969826i \(-0.578393\pi\)
−0.243797 + 0.969826i \(0.578393\pi\)
\(930\) −5.29590 −0.173659
\(931\) 53.5381 1.75464
\(932\) 0.975837 0.0319646
\(933\) −8.49635 −0.278158
\(934\) −0.916166 −0.0299779
\(935\) 6.37627 0.208526
\(936\) 0 0
\(937\) 54.3196 1.77454 0.887272 0.461247i \(-0.152598\pi\)
0.887272 + 0.461247i \(0.152598\pi\)
\(938\) 57.4650 1.87630
\(939\) 25.2610 0.824361
\(940\) −10.3840 −0.338690
\(941\) 14.0121 0.456781 0.228390 0.973570i \(-0.426654\pi\)
0.228390 + 0.973570i \(0.426654\pi\)
\(942\) 19.9148 0.648860
\(943\) 1.44504 0.0470570
\(944\) −13.4601 −0.438089
\(945\) −4.24698 −0.138154
\(946\) −3.78209 −0.122966
\(947\) −56.1831 −1.82571 −0.912853 0.408289i \(-0.866126\pi\)
−0.912853 + 0.408289i \(0.866126\pi\)
\(948\) 14.5526 0.472645
\(949\) 0 0
\(950\) −4.85086 −0.157383
\(951\) 4.96615 0.161038
\(952\) 14.1642 0.459065
\(953\) 42.9090 1.38996 0.694979 0.719030i \(-0.255414\pi\)
0.694979 + 0.719030i \(0.255414\pi\)
\(954\) 1.14914 0.0372049
\(955\) 23.1685 0.749716
\(956\) 7.39373 0.239130
\(957\) −16.3744 −0.529308
\(958\) −23.1715 −0.748637
\(959\) 3.94677 0.127448
\(960\) −1.00000 −0.0322749
\(961\) −2.95348 −0.0952734
\(962\) 0 0
\(963\) −12.8334 −0.413550
\(964\) 2.11529 0.0681290
\(965\) −6.38942 −0.205683
\(966\) −1.89008 −0.0608124
\(967\) −2.89977 −0.0932504 −0.0466252 0.998912i \(-0.514847\pi\)
−0.0466252 + 0.998912i \(0.514847\pi\)
\(968\) 7.34481 0.236071
\(969\) 16.1782 0.519719
\(970\) −5.01507 −0.161024
\(971\) 32.8146 1.05307 0.526535 0.850153i \(-0.323491\pi\)
0.526535 + 0.850153i \(0.323491\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 13.2591 0.425066
\(974\) −11.9812 −0.383903
\(975\) 0 0
\(976\) 1.13169 0.0362244
\(977\) 13.1806 0.421685 0.210842 0.977520i \(-0.432379\pi\)
0.210842 + 0.977520i \(0.432379\pi\)
\(978\) −13.1075 −0.419132
\(979\) −2.40257 −0.0767864
\(980\) 11.0368 0.352559
\(981\) −10.8726 −0.347136
\(982\) 24.8745 0.793779
\(983\) −31.3726 −1.00063 −0.500315 0.865844i \(-0.666782\pi\)
−0.500315 + 0.865844i \(0.666782\pi\)
\(984\) −3.24698 −0.103510
\(985\) 14.8605 0.473496
\(986\) −28.5642 −0.909669
\(987\) 44.1008 1.40374
\(988\) 0 0
\(989\) 0.880395 0.0279949
\(990\) 1.91185 0.0607627
\(991\) −44.1842 −1.40356 −0.701778 0.712395i \(-0.747610\pi\)
−0.701778 + 0.712395i \(0.747610\pi\)
\(992\) 5.29590 0.168145
\(993\) −35.3749 −1.12259
\(994\) −9.11721 −0.289180
\(995\) 26.3860 0.836491
\(996\) −9.49157 −0.300752
\(997\) 50.8254 1.60966 0.804828 0.593509i \(-0.202258\pi\)
0.804828 + 0.593509i \(0.202258\pi\)
\(998\) −42.0737 −1.33182
\(999\) 1.75302 0.0554631
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 5070.2.a.bl.1.3 3
13.5 odd 4 5070.2.b.v.1351.6 6
13.8 odd 4 5070.2.b.v.1351.1 6
13.12 even 2 5070.2.a.bs.1.1 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bl.1.3 3 1.1 even 1 trivial
5070.2.a.bs.1.1 yes 3 13.12 even 2
5070.2.b.v.1351.1 6 13.8 odd 4
5070.2.b.v.1351.6 6 13.5 odd 4