Properties

Label 2151.2.a.e.1.5
Level $2151$
Weight $2$
Character 2151.1
Self dual yes
Analytic conductor $17.176$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2151 = 3^{2} \cdot 239 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2151.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.1758214748\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.1767625.1
Defining polynomial: \(x^{6} - 7 x^{4} - x^{3} + 11 x^{2} + x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 717)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-0.360520\) of defining polynomial
Character \(\chi\) \(=\) 2151.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.15574 q^{2} +2.64721 q^{4} -3.41325 q^{5} +0.201367 q^{7} +1.39522 q^{8} +O(q^{10})\) \(q+2.15574 q^{2} +2.64721 q^{4} -3.41325 q^{5} +0.201367 q^{7} +1.39522 q^{8} -7.35809 q^{10} +5.35035 q^{11} -5.86183 q^{13} +0.434095 q^{14} -2.28669 q^{16} -4.05336 q^{17} -2.42759 q^{19} -9.03561 q^{20} +11.5340 q^{22} +4.58506 q^{23} +6.65030 q^{25} -12.6366 q^{26} +0.533062 q^{28} -3.44243 q^{29} -7.68267 q^{31} -7.71995 q^{32} -8.73799 q^{34} -0.687317 q^{35} -0.781965 q^{37} -5.23326 q^{38} -4.76224 q^{40} -5.87100 q^{41} -8.05607 q^{43} +14.1635 q^{44} +9.88419 q^{46} +3.12717 q^{47} -6.95945 q^{49} +14.3363 q^{50} -15.5175 q^{52} +11.6536 q^{53} -18.2621 q^{55} +0.280952 q^{56} -7.42099 q^{58} -9.93358 q^{59} +5.67948 q^{61} -16.5618 q^{62} -12.0688 q^{64} +20.0079 q^{65} -6.84647 q^{67} -10.7301 q^{68} -1.48168 q^{70} -12.8677 q^{71} -7.37793 q^{73} -1.68571 q^{74} -6.42636 q^{76} +1.07739 q^{77} +14.6391 q^{79} +7.80506 q^{80} -12.6564 q^{82} +15.8846 q^{83} +13.8351 q^{85} -17.3668 q^{86} +7.46492 q^{88} +5.12607 q^{89} -1.18038 q^{91} +12.1376 q^{92} +6.74135 q^{94} +8.28599 q^{95} -11.8563 q^{97} -15.0028 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 2q^{2} + 4q^{4} - 5q^{5} - 9q^{7} + 3q^{8} + O(q^{10}) \) \( 6q + 2q^{2} + 4q^{4} - 5q^{5} - 9q^{7} + 3q^{8} - 11q^{10} + 13q^{11} - q^{13} - 4q^{16} - 11q^{17} - 22q^{19} + q^{20} - 2q^{22} + 12q^{23} - q^{25} - 12q^{26} - 16q^{28} - 18q^{31} - 7q^{32} - 3q^{34} + 9q^{35} - 8q^{37} + 5q^{38} - 11q^{40} - 10q^{41} - 14q^{43} + 4q^{44} - 18q^{46} + 9q^{47} + 5q^{49} - 4q^{50} - 16q^{52} + 8q^{53} - 20q^{55} - 11q^{56} - 15q^{58} + 10q^{59} - 12q^{61} + 13q^{62} - 31q^{64} + 11q^{65} - 36q^{67} - 22q^{68} + q^{70} + 3q^{71} - 32q^{73} - 9q^{74} - 4q^{76} - 6q^{77} - q^{79} + 7q^{80} + 7q^{82} + 7q^{83} - 14q^{85} - 45q^{86} - 15q^{88} - 17q^{89} - 23q^{91} + 12q^{92} + 50q^{94} - 28q^{97} - 13q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.15574 1.52434 0.762169 0.647378i \(-0.224135\pi\)
0.762169 + 0.647378i \(0.224135\pi\)
\(3\) 0 0
\(4\) 2.64721 1.32361
\(5\) −3.41325 −1.52645 −0.763227 0.646131i \(-0.776386\pi\)
−0.763227 + 0.646131i \(0.776386\pi\)
\(6\) 0 0
\(7\) 0.201367 0.0761096 0.0380548 0.999276i \(-0.487884\pi\)
0.0380548 + 0.999276i \(0.487884\pi\)
\(8\) 1.39522 0.493285
\(9\) 0 0
\(10\) −7.35809 −2.32683
\(11\) 5.35035 1.61319 0.806596 0.591103i \(-0.201307\pi\)
0.806596 + 0.591103i \(0.201307\pi\)
\(12\) 0 0
\(13\) −5.86183 −1.62578 −0.812890 0.582417i \(-0.802107\pi\)
−0.812890 + 0.582417i \(0.802107\pi\)
\(14\) 0.434095 0.116017
\(15\) 0 0
\(16\) −2.28669 −0.571673
\(17\) −4.05336 −0.983084 −0.491542 0.870854i \(-0.663567\pi\)
−0.491542 + 0.870854i \(0.663567\pi\)
\(18\) 0 0
\(19\) −2.42759 −0.556928 −0.278464 0.960447i \(-0.589825\pi\)
−0.278464 + 0.960447i \(0.589825\pi\)
\(20\) −9.03561 −2.02042
\(21\) 0 0
\(22\) 11.5340 2.45905
\(23\) 4.58506 0.956050 0.478025 0.878346i \(-0.341353\pi\)
0.478025 + 0.878346i \(0.341353\pi\)
\(24\) 0 0
\(25\) 6.65030 1.33006
\(26\) −12.6366 −2.47824
\(27\) 0 0
\(28\) 0.533062 0.100739
\(29\) −3.44243 −0.639244 −0.319622 0.947545i \(-0.603556\pi\)
−0.319622 + 0.947545i \(0.603556\pi\)
\(30\) 0 0
\(31\) −7.68267 −1.37985 −0.689924 0.723881i \(-0.742356\pi\)
−0.689924 + 0.723881i \(0.742356\pi\)
\(32\) −7.71995 −1.36471
\(33\) 0 0
\(34\) −8.73799 −1.49855
\(35\) −0.687317 −0.116178
\(36\) 0 0
\(37\) −0.781965 −0.128554 −0.0642771 0.997932i \(-0.520474\pi\)
−0.0642771 + 0.997932i \(0.520474\pi\)
\(38\) −5.23326 −0.848947
\(39\) 0 0
\(40\) −4.76224 −0.752977
\(41\) −5.87100 −0.916897 −0.458448 0.888721i \(-0.651595\pi\)
−0.458448 + 0.888721i \(0.651595\pi\)
\(42\) 0 0
\(43\) −8.05607 −1.22854 −0.614270 0.789096i \(-0.710549\pi\)
−0.614270 + 0.789096i \(0.710549\pi\)
\(44\) 14.1635 2.13523
\(45\) 0 0
\(46\) 9.88419 1.45734
\(47\) 3.12717 0.456144 0.228072 0.973644i \(-0.426758\pi\)
0.228072 + 0.973644i \(0.426758\pi\)
\(48\) 0 0
\(49\) −6.95945 −0.994207
\(50\) 14.3363 2.02746
\(51\) 0 0
\(52\) −15.5175 −2.15189
\(53\) 11.6536 1.60075 0.800375 0.599499i \(-0.204634\pi\)
0.800375 + 0.599499i \(0.204634\pi\)
\(54\) 0 0
\(55\) −18.2621 −2.46246
\(56\) 0.280952 0.0375437
\(57\) 0 0
\(58\) −7.42099 −0.974423
\(59\) −9.93358 −1.29324 −0.646621 0.762811i \(-0.723818\pi\)
−0.646621 + 0.762811i \(0.723818\pi\)
\(60\) 0 0
\(61\) 5.67948 0.727183 0.363591 0.931559i \(-0.381550\pi\)
0.363591 + 0.931559i \(0.381550\pi\)
\(62\) −16.5618 −2.10336
\(63\) 0 0
\(64\) −12.0688 −1.50860
\(65\) 20.0079 2.48168
\(66\) 0 0
\(67\) −6.84647 −0.836429 −0.418215 0.908348i \(-0.637344\pi\)
−0.418215 + 0.908348i \(0.637344\pi\)
\(68\) −10.7301 −1.30122
\(69\) 0 0
\(70\) −1.48168 −0.177094
\(71\) −12.8677 −1.52711 −0.763555 0.645743i \(-0.776548\pi\)
−0.763555 + 0.645743i \(0.776548\pi\)
\(72\) 0 0
\(73\) −7.37793 −0.863521 −0.431761 0.901988i \(-0.642107\pi\)
−0.431761 + 0.901988i \(0.642107\pi\)
\(74\) −1.68571 −0.195960
\(75\) 0 0
\(76\) −6.42636 −0.737154
\(77\) 1.07739 0.122779
\(78\) 0 0
\(79\) 14.6391 1.64702 0.823512 0.567299i \(-0.192012\pi\)
0.823512 + 0.567299i \(0.192012\pi\)
\(80\) 7.80506 0.872632
\(81\) 0 0
\(82\) −12.6564 −1.39766
\(83\) 15.8846 1.74357 0.871783 0.489893i \(-0.162964\pi\)
0.871783 + 0.489893i \(0.162964\pi\)
\(84\) 0 0
\(85\) 13.8351 1.50063
\(86\) −17.3668 −1.87271
\(87\) 0 0
\(88\) 7.46492 0.795763
\(89\) 5.12607 0.543362 0.271681 0.962387i \(-0.412420\pi\)
0.271681 + 0.962387i \(0.412420\pi\)
\(90\) 0 0
\(91\) −1.18038 −0.123738
\(92\) 12.1376 1.26543
\(93\) 0 0
\(94\) 6.74135 0.695318
\(95\) 8.28599 0.850125
\(96\) 0 0
\(97\) −11.8563 −1.20382 −0.601911 0.798563i \(-0.705594\pi\)
−0.601911 + 0.798563i \(0.705594\pi\)
\(98\) −15.0028 −1.51551
\(99\) 0 0
\(100\) 17.6048 1.76048
\(101\) 15.8321 1.57535 0.787677 0.616089i \(-0.211284\pi\)
0.787677 + 0.616089i \(0.211284\pi\)
\(102\) 0 0
\(103\) 16.7153 1.64701 0.823505 0.567310i \(-0.192016\pi\)
0.823505 + 0.567310i \(0.192016\pi\)
\(104\) −8.17855 −0.801973
\(105\) 0 0
\(106\) 25.1222 2.44008
\(107\) 3.25084 0.314271 0.157135 0.987577i \(-0.449774\pi\)
0.157135 + 0.987577i \(0.449774\pi\)
\(108\) 0 0
\(109\) −2.91908 −0.279597 −0.139799 0.990180i \(-0.544646\pi\)
−0.139799 + 0.990180i \(0.544646\pi\)
\(110\) −39.3684 −3.75363
\(111\) 0 0
\(112\) −0.460465 −0.0435098
\(113\) −2.24561 −0.211249 −0.105625 0.994406i \(-0.533684\pi\)
−0.105625 + 0.994406i \(0.533684\pi\)
\(114\) 0 0
\(115\) −15.6500 −1.45937
\(116\) −9.11285 −0.846107
\(117\) 0 0
\(118\) −21.4142 −1.97134
\(119\) −0.816214 −0.0748222
\(120\) 0 0
\(121\) 17.6263 1.60239
\(122\) 12.2435 1.10847
\(123\) 0 0
\(124\) −20.3377 −1.82638
\(125\) −5.63289 −0.503821
\(126\) 0 0
\(127\) −8.75323 −0.776724 −0.388362 0.921507i \(-0.626959\pi\)
−0.388362 + 0.921507i \(0.626959\pi\)
\(128\) −10.5773 −0.934913
\(129\) 0 0
\(130\) 43.1319 3.78292
\(131\) −8.69454 −0.759646 −0.379823 0.925059i \(-0.624015\pi\)
−0.379823 + 0.925059i \(0.624015\pi\)
\(132\) 0 0
\(133\) −0.488838 −0.0423876
\(134\) −14.7592 −1.27500
\(135\) 0 0
\(136\) −5.65533 −0.484941
\(137\) 8.49089 0.725426 0.362713 0.931901i \(-0.381851\pi\)
0.362713 + 0.931901i \(0.381851\pi\)
\(138\) 0 0
\(139\) −1.97577 −0.167582 −0.0837912 0.996483i \(-0.526703\pi\)
−0.0837912 + 0.996483i \(0.526703\pi\)
\(140\) −1.81947 −0.153774
\(141\) 0 0
\(142\) −27.7393 −2.32783
\(143\) −31.3629 −2.62270
\(144\) 0 0
\(145\) 11.7499 0.975775
\(146\) −15.9049 −1.31630
\(147\) 0 0
\(148\) −2.07003 −0.170155
\(149\) −3.31898 −0.271901 −0.135951 0.990716i \(-0.543409\pi\)
−0.135951 + 0.990716i \(0.543409\pi\)
\(150\) 0 0
\(151\) −13.3688 −1.08794 −0.543968 0.839106i \(-0.683079\pi\)
−0.543968 + 0.839106i \(0.683079\pi\)
\(152\) −3.38703 −0.274724
\(153\) 0 0
\(154\) 2.32256 0.187157
\(155\) 26.2229 2.10628
\(156\) 0 0
\(157\) 3.64757 0.291108 0.145554 0.989350i \(-0.453504\pi\)
0.145554 + 0.989350i \(0.453504\pi\)
\(158\) 31.5580 2.51062
\(159\) 0 0
\(160\) 26.3502 2.08316
\(161\) 0.923280 0.0727646
\(162\) 0 0
\(163\) −14.8589 −1.16384 −0.581920 0.813246i \(-0.697698\pi\)
−0.581920 + 0.813246i \(0.697698\pi\)
\(164\) −15.5418 −1.21361
\(165\) 0 0
\(166\) 34.2431 2.65778
\(167\) −12.1352 −0.939049 −0.469524 0.882919i \(-0.655575\pi\)
−0.469524 + 0.882919i \(0.655575\pi\)
\(168\) 0 0
\(169\) 21.3611 1.64316
\(170\) 29.8250 2.28747
\(171\) 0 0
\(172\) −21.3261 −1.62610
\(173\) 13.0426 0.991610 0.495805 0.868434i \(-0.334873\pi\)
0.495805 + 0.868434i \(0.334873\pi\)
\(174\) 0 0
\(175\) 1.33915 0.101230
\(176\) −12.2346 −0.922219
\(177\) 0 0
\(178\) 11.0505 0.828267
\(179\) −5.36036 −0.400652 −0.200326 0.979729i \(-0.564200\pi\)
−0.200326 + 0.979729i \(0.564200\pi\)
\(180\) 0 0
\(181\) 5.12883 0.381223 0.190611 0.981666i \(-0.438953\pi\)
0.190611 + 0.981666i \(0.438953\pi\)
\(182\) −2.54459 −0.188618
\(183\) 0 0
\(184\) 6.39716 0.471605
\(185\) 2.66905 0.196232
\(186\) 0 0
\(187\) −21.6869 −1.58590
\(188\) 8.27827 0.603755
\(189\) 0 0
\(190\) 17.8624 1.29588
\(191\) 12.0807 0.874130 0.437065 0.899430i \(-0.356018\pi\)
0.437065 + 0.899430i \(0.356018\pi\)
\(192\) 0 0
\(193\) 25.6877 1.84904 0.924519 0.381135i \(-0.124467\pi\)
0.924519 + 0.381135i \(0.124467\pi\)
\(194\) −25.5590 −1.83503
\(195\) 0 0
\(196\) −18.4231 −1.31594
\(197\) 8.44159 0.601439 0.300719 0.953713i \(-0.402773\pi\)
0.300719 + 0.953713i \(0.402773\pi\)
\(198\) 0 0
\(199\) 4.76581 0.337839 0.168920 0.985630i \(-0.445972\pi\)
0.168920 + 0.985630i \(0.445972\pi\)
\(200\) 9.27864 0.656099
\(201\) 0 0
\(202\) 34.1299 2.40137
\(203\) −0.693193 −0.0486526
\(204\) 0 0
\(205\) 20.0392 1.39960
\(206\) 36.0339 2.51060
\(207\) 0 0
\(208\) 13.4042 0.929415
\(209\) −12.9885 −0.898432
\(210\) 0 0
\(211\) 7.98569 0.549758 0.274879 0.961479i \(-0.411362\pi\)
0.274879 + 0.961479i \(0.411362\pi\)
\(212\) 30.8497 2.11876
\(213\) 0 0
\(214\) 7.00797 0.479055
\(215\) 27.4974 1.87531
\(216\) 0 0
\(217\) −1.54704 −0.105020
\(218\) −6.29278 −0.426201
\(219\) 0 0
\(220\) −48.3437 −3.25933
\(221\) 23.7601 1.59828
\(222\) 0 0
\(223\) −24.9471 −1.67058 −0.835291 0.549808i \(-0.814701\pi\)
−0.835291 + 0.549808i \(0.814701\pi\)
\(224\) −1.55455 −0.103867
\(225\) 0 0
\(226\) −4.84095 −0.322015
\(227\) −25.4484 −1.68907 −0.844536 0.535499i \(-0.820123\pi\)
−0.844536 + 0.535499i \(0.820123\pi\)
\(228\) 0 0
\(229\) −8.84963 −0.584800 −0.292400 0.956296i \(-0.594454\pi\)
−0.292400 + 0.956296i \(0.594454\pi\)
\(230\) −33.7372 −2.22457
\(231\) 0 0
\(232\) −4.80295 −0.315329
\(233\) −23.9242 −1.56733 −0.783665 0.621184i \(-0.786652\pi\)
−0.783665 + 0.621184i \(0.786652\pi\)
\(234\) 0 0
\(235\) −10.6738 −0.696283
\(236\) −26.2963 −1.71174
\(237\) 0 0
\(238\) −1.75954 −0.114054
\(239\) 1.00000 0.0646846
\(240\) 0 0
\(241\) −5.57115 −0.358870 −0.179435 0.983770i \(-0.557427\pi\)
−0.179435 + 0.983770i \(0.557427\pi\)
\(242\) 37.9977 2.44258
\(243\) 0 0
\(244\) 15.0348 0.962503
\(245\) 23.7544 1.51761
\(246\) 0 0
\(247\) 14.2302 0.905443
\(248\) −10.7190 −0.680659
\(249\) 0 0
\(250\) −12.1430 −0.767994
\(251\) 24.0752 1.51961 0.759806 0.650150i \(-0.225294\pi\)
0.759806 + 0.650150i \(0.225294\pi\)
\(252\) 0 0
\(253\) 24.5317 1.54229
\(254\) −18.8697 −1.18399
\(255\) 0 0
\(256\) 1.33568 0.0834801
\(257\) −1.65551 −0.103268 −0.0516339 0.998666i \(-0.516443\pi\)
−0.0516339 + 0.998666i \(0.516443\pi\)
\(258\) 0 0
\(259\) −0.157462 −0.00978422
\(260\) 52.9652 3.28476
\(261\) 0 0
\(262\) −18.7432 −1.15796
\(263\) 13.7969 0.850751 0.425375 0.905017i \(-0.360142\pi\)
0.425375 + 0.905017i \(0.360142\pi\)
\(264\) 0 0
\(265\) −39.7768 −2.44347
\(266\) −1.05381 −0.0646130
\(267\) 0 0
\(268\) −18.1241 −1.10710
\(269\) 2.37746 0.144956 0.0724782 0.997370i \(-0.476909\pi\)
0.0724782 + 0.997370i \(0.476909\pi\)
\(270\) 0 0
\(271\) 3.47333 0.210990 0.105495 0.994420i \(-0.466357\pi\)
0.105495 + 0.994420i \(0.466357\pi\)
\(272\) 9.26879 0.562003
\(273\) 0 0
\(274\) 18.3042 1.10579
\(275\) 35.5815 2.14564
\(276\) 0 0
\(277\) −2.25803 −0.135672 −0.0678359 0.997696i \(-0.521609\pi\)
−0.0678359 + 0.997696i \(0.521609\pi\)
\(278\) −4.25924 −0.255452
\(279\) 0 0
\(280\) −0.958959 −0.0573088
\(281\) 0.502713 0.0299894 0.0149947 0.999888i \(-0.495227\pi\)
0.0149947 + 0.999888i \(0.495227\pi\)
\(282\) 0 0
\(283\) 21.6234 1.28538 0.642690 0.766127i \(-0.277818\pi\)
0.642690 + 0.766127i \(0.277818\pi\)
\(284\) −34.0634 −2.02129
\(285\) 0 0
\(286\) −67.6102 −3.99787
\(287\) −1.18223 −0.0697847
\(288\) 0 0
\(289\) −0.570276 −0.0335456
\(290\) 25.3297 1.48741
\(291\) 0 0
\(292\) −19.5309 −1.14296
\(293\) −18.7307 −1.09426 −0.547129 0.837048i \(-0.684279\pi\)
−0.547129 + 0.837048i \(0.684279\pi\)
\(294\) 0 0
\(295\) 33.9058 1.97407
\(296\) −1.09101 −0.0634139
\(297\) 0 0
\(298\) −7.15485 −0.414469
\(299\) −26.8768 −1.55433
\(300\) 0 0
\(301\) −1.62223 −0.0935037
\(302\) −28.8196 −1.65838
\(303\) 0 0
\(304\) 5.55116 0.318381
\(305\) −19.3855 −1.11001
\(306\) 0 0
\(307\) 4.73274 0.270112 0.135056 0.990838i \(-0.456879\pi\)
0.135056 + 0.990838i \(0.456879\pi\)
\(308\) 2.85207 0.162512
\(309\) 0 0
\(310\) 56.5298 3.21067
\(311\) 21.6404 1.22711 0.613556 0.789651i \(-0.289738\pi\)
0.613556 + 0.789651i \(0.289738\pi\)
\(312\) 0 0
\(313\) 24.4364 1.38123 0.690615 0.723223i \(-0.257340\pi\)
0.690615 + 0.723223i \(0.257340\pi\)
\(314\) 7.86321 0.443747
\(315\) 0 0
\(316\) 38.7527 2.18001
\(317\) −2.93779 −0.165003 −0.0825013 0.996591i \(-0.526291\pi\)
−0.0825013 + 0.996591i \(0.526291\pi\)
\(318\) 0 0
\(319\) −18.4182 −1.03122
\(320\) 41.1940 2.30281
\(321\) 0 0
\(322\) 1.99035 0.110918
\(323\) 9.83991 0.547507
\(324\) 0 0
\(325\) −38.9830 −2.16239
\(326\) −32.0319 −1.77408
\(327\) 0 0
\(328\) −8.19135 −0.452291
\(329\) 0.629709 0.0347170
\(330\) 0 0
\(331\) −13.2517 −0.728379 −0.364190 0.931325i \(-0.618654\pi\)
−0.364190 + 0.931325i \(0.618654\pi\)
\(332\) 42.0500 2.30779
\(333\) 0 0
\(334\) −26.1603 −1.43143
\(335\) 23.3687 1.27677
\(336\) 0 0
\(337\) −6.76419 −0.368469 −0.184234 0.982882i \(-0.558981\pi\)
−0.184234 + 0.982882i \(0.558981\pi\)
\(338\) 46.0490 2.50473
\(339\) 0 0
\(340\) 36.6246 1.98625
\(341\) −41.1050 −2.22596
\(342\) 0 0
\(343\) −2.81098 −0.151778
\(344\) −11.2400 −0.606020
\(345\) 0 0
\(346\) 28.1164 1.51155
\(347\) −28.8872 −1.55075 −0.775374 0.631503i \(-0.782439\pi\)
−0.775374 + 0.631503i \(0.782439\pi\)
\(348\) 0 0
\(349\) 8.86947 0.474772 0.237386 0.971415i \(-0.423709\pi\)
0.237386 + 0.971415i \(0.423709\pi\)
\(350\) 2.88686 0.154309
\(351\) 0 0
\(352\) −41.3045 −2.20154
\(353\) −1.97699 −0.105224 −0.0526122 0.998615i \(-0.516755\pi\)
−0.0526122 + 0.998615i \(0.516755\pi\)
\(354\) 0 0
\(355\) 43.9206 2.33106
\(356\) 13.5698 0.719197
\(357\) 0 0
\(358\) −11.5555 −0.610730
\(359\) −16.3404 −0.862413 −0.431206 0.902253i \(-0.641912\pi\)
−0.431206 + 0.902253i \(0.641912\pi\)
\(360\) 0 0
\(361\) −13.1068 −0.689831
\(362\) 11.0564 0.581112
\(363\) 0 0
\(364\) −3.12472 −0.163780
\(365\) 25.1827 1.31812
\(366\) 0 0
\(367\) −32.2353 −1.68267 −0.841333 0.540517i \(-0.818229\pi\)
−0.841333 + 0.540517i \(0.818229\pi\)
\(368\) −10.4846 −0.546548
\(369\) 0 0
\(370\) 5.75377 0.299124
\(371\) 2.34666 0.121833
\(372\) 0 0
\(373\) −0.382479 −0.0198040 −0.00990201 0.999951i \(-0.503152\pi\)
−0.00990201 + 0.999951i \(0.503152\pi\)
\(374\) −46.7513 −2.41745
\(375\) 0 0
\(376\) 4.36309 0.225009
\(377\) 20.1790 1.03927
\(378\) 0 0
\(379\) −3.06644 −0.157512 −0.0787562 0.996894i \(-0.525095\pi\)
−0.0787562 + 0.996894i \(0.525095\pi\)
\(380\) 21.9348 1.12523
\(381\) 0 0
\(382\) 26.0429 1.33247
\(383\) 7.87059 0.402168 0.201084 0.979574i \(-0.435554\pi\)
0.201084 + 0.979574i \(0.435554\pi\)
\(384\) 0 0
\(385\) −3.67739 −0.187417
\(386\) 55.3759 2.81856
\(387\) 0 0
\(388\) −31.3861 −1.59339
\(389\) −9.22062 −0.467504 −0.233752 0.972296i \(-0.575100\pi\)
−0.233752 + 0.972296i \(0.575100\pi\)
\(390\) 0 0
\(391\) −18.5849 −0.939878
\(392\) −9.70997 −0.490428
\(393\) 0 0
\(394\) 18.1979 0.916796
\(395\) −49.9669 −2.51411
\(396\) 0 0
\(397\) 29.5873 1.48495 0.742473 0.669876i \(-0.233653\pi\)
0.742473 + 0.669876i \(0.233653\pi\)
\(398\) 10.2738 0.514981
\(399\) 0 0
\(400\) −15.2072 −0.760360
\(401\) −33.1480 −1.65533 −0.827666 0.561221i \(-0.810332\pi\)
−0.827666 + 0.561221i \(0.810332\pi\)
\(402\) 0 0
\(403\) 45.0346 2.24333
\(404\) 41.9109 2.08515
\(405\) 0 0
\(406\) −1.49434 −0.0741630
\(407\) −4.18379 −0.207383
\(408\) 0 0
\(409\) −27.5969 −1.36458 −0.682290 0.731081i \(-0.739016\pi\)
−0.682290 + 0.731081i \(0.739016\pi\)
\(410\) 43.1993 2.13346
\(411\) 0 0
\(412\) 44.2490 2.17999
\(413\) −2.00030 −0.0984282
\(414\) 0 0
\(415\) −54.2183 −2.66147
\(416\) 45.2531 2.21872
\(417\) 0 0
\(418\) −27.9998 −1.36951
\(419\) 23.6207 1.15394 0.576972 0.816764i \(-0.304234\pi\)
0.576972 + 0.816764i \(0.304234\pi\)
\(420\) 0 0
\(421\) −19.6014 −0.955315 −0.477657 0.878546i \(-0.658514\pi\)
−0.477657 + 0.878546i \(0.658514\pi\)
\(422\) 17.2151 0.838017
\(423\) 0 0
\(424\) 16.2594 0.789626
\(425\) −26.9561 −1.30756
\(426\) 0 0
\(427\) 1.14366 0.0553456
\(428\) 8.60567 0.415971
\(429\) 0 0
\(430\) 59.2773 2.85860
\(431\) −34.8589 −1.67910 −0.839548 0.543286i \(-0.817180\pi\)
−0.839548 + 0.543286i \(0.817180\pi\)
\(432\) 0 0
\(433\) −11.7316 −0.563787 −0.281893 0.959446i \(-0.590962\pi\)
−0.281893 + 0.959446i \(0.590962\pi\)
\(434\) −3.33501 −0.160086
\(435\) 0 0
\(436\) −7.72743 −0.370077
\(437\) −11.1307 −0.532451
\(438\) 0 0
\(439\) 37.1023 1.77079 0.885397 0.464835i \(-0.153886\pi\)
0.885397 + 0.464835i \(0.153886\pi\)
\(440\) −25.4797 −1.21470
\(441\) 0 0
\(442\) 51.2206 2.43632
\(443\) −5.25818 −0.249824 −0.124912 0.992168i \(-0.539865\pi\)
−0.124912 + 0.992168i \(0.539865\pi\)
\(444\) 0 0
\(445\) −17.4966 −0.829417
\(446\) −53.7794 −2.54653
\(447\) 0 0
\(448\) −2.43026 −0.114819
\(449\) −21.0799 −0.994822 −0.497411 0.867515i \(-0.665716\pi\)
−0.497411 + 0.867515i \(0.665716\pi\)
\(450\) 0 0
\(451\) −31.4119 −1.47913
\(452\) −5.94460 −0.279611
\(453\) 0 0
\(454\) −54.8602 −2.57472
\(455\) 4.02894 0.188880
\(456\) 0 0
\(457\) −31.7180 −1.48371 −0.741853 0.670563i \(-0.766053\pi\)
−0.741853 + 0.670563i \(0.766053\pi\)
\(458\) −19.0775 −0.891433
\(459\) 0 0
\(460\) −41.4288 −1.93163
\(461\) −14.1119 −0.657258 −0.328629 0.944459i \(-0.606587\pi\)
−0.328629 + 0.944459i \(0.606587\pi\)
\(462\) 0 0
\(463\) 5.51783 0.256435 0.128218 0.991746i \(-0.459074\pi\)
0.128218 + 0.991746i \(0.459074\pi\)
\(464\) 7.87178 0.365438
\(465\) 0 0
\(466\) −51.5744 −2.38914
\(467\) 11.0132 0.509630 0.254815 0.966990i \(-0.417985\pi\)
0.254815 + 0.966990i \(0.417985\pi\)
\(468\) 0 0
\(469\) −1.37865 −0.0636603
\(470\) −23.0100 −1.06137
\(471\) 0 0
\(472\) −13.8595 −0.637937
\(473\) −43.1028 −1.98187
\(474\) 0 0
\(475\) −16.1442 −0.740748
\(476\) −2.16069 −0.0990351
\(477\) 0 0
\(478\) 2.15574 0.0986012
\(479\) −18.8603 −0.861747 −0.430874 0.902412i \(-0.641795\pi\)
−0.430874 + 0.902412i \(0.641795\pi\)
\(480\) 0 0
\(481\) 4.58375 0.209001
\(482\) −12.0100 −0.547039
\(483\) 0 0
\(484\) 46.6605 2.12093
\(485\) 40.4685 1.83758
\(486\) 0 0
\(487\) 30.7402 1.39297 0.696485 0.717571i \(-0.254746\pi\)
0.696485 + 0.717571i \(0.254746\pi\)
\(488\) 7.92412 0.358708
\(489\) 0 0
\(490\) 51.2082 2.31335
\(491\) −31.9405 −1.44146 −0.720728 0.693218i \(-0.756192\pi\)
−0.720728 + 0.693218i \(0.756192\pi\)
\(492\) 0 0
\(493\) 13.9534 0.628430
\(494\) 30.6765 1.38020
\(495\) 0 0
\(496\) 17.5679 0.788822
\(497\) −2.59112 −0.116228
\(498\) 0 0
\(499\) −21.7380 −0.973127 −0.486564 0.873645i \(-0.661750\pi\)
−0.486564 + 0.873645i \(0.661750\pi\)
\(500\) −14.9115 −0.666861
\(501\) 0 0
\(502\) 51.8998 2.31640
\(503\) 13.3905 0.597052 0.298526 0.954401i \(-0.403505\pi\)
0.298526 + 0.954401i \(0.403505\pi\)
\(504\) 0 0
\(505\) −54.0390 −2.40470
\(506\) 52.8839 2.35098
\(507\) 0 0
\(508\) −23.1717 −1.02808
\(509\) 15.7825 0.699547 0.349774 0.936834i \(-0.386259\pi\)
0.349774 + 0.936834i \(0.386259\pi\)
\(510\) 0 0
\(511\) −1.48567 −0.0657223
\(512\) 24.0340 1.06216
\(513\) 0 0
\(514\) −3.56884 −0.157415
\(515\) −57.0536 −2.51408
\(516\) 0 0
\(517\) 16.7314 0.735848
\(518\) −0.339447 −0.0149145
\(519\) 0 0
\(520\) 27.9155 1.22417
\(521\) 3.01206 0.131961 0.0659803 0.997821i \(-0.478983\pi\)
0.0659803 + 0.997821i \(0.478983\pi\)
\(522\) 0 0
\(523\) 21.0704 0.921342 0.460671 0.887571i \(-0.347609\pi\)
0.460671 + 0.887571i \(0.347609\pi\)
\(524\) −23.0163 −1.00547
\(525\) 0 0
\(526\) 29.7424 1.29683
\(527\) 31.1406 1.35651
\(528\) 0 0
\(529\) −1.97726 −0.0859679
\(530\) −85.7485 −3.72468
\(531\) 0 0
\(532\) −1.29406 −0.0561045
\(533\) 34.4148 1.49067
\(534\) 0 0
\(535\) −11.0960 −0.479720
\(536\) −9.55234 −0.412598
\(537\) 0 0
\(538\) 5.12519 0.220963
\(539\) −37.2355 −1.60385
\(540\) 0 0
\(541\) 34.6808 1.49104 0.745522 0.666481i \(-0.232200\pi\)
0.745522 + 0.666481i \(0.232200\pi\)
\(542\) 7.48760 0.321620
\(543\) 0 0
\(544\) 31.2917 1.34162
\(545\) 9.96357 0.426792
\(546\) 0 0
\(547\) −26.6304 −1.13863 −0.569317 0.822118i \(-0.692792\pi\)
−0.569317 + 0.822118i \(0.692792\pi\)
\(548\) 22.4772 0.960178
\(549\) 0 0
\(550\) 76.7043 3.27068
\(551\) 8.35683 0.356013
\(552\) 0 0
\(553\) 2.94783 0.125354
\(554\) −4.86772 −0.206810
\(555\) 0 0
\(556\) −5.23027 −0.221813
\(557\) 29.6425 1.25599 0.627996 0.778217i \(-0.283875\pi\)
0.627996 + 0.778217i \(0.283875\pi\)
\(558\) 0 0
\(559\) 47.2234 1.99734
\(560\) 1.57168 0.0664157
\(561\) 0 0
\(562\) 1.08372 0.0457139
\(563\) 11.2332 0.473422 0.236711 0.971580i \(-0.423931\pi\)
0.236711 + 0.971580i \(0.423931\pi\)
\(564\) 0 0
\(565\) 7.66483 0.322462
\(566\) 46.6145 1.95935
\(567\) 0 0
\(568\) −17.9532 −0.753300
\(569\) −41.8311 −1.75365 −0.876825 0.480810i \(-0.840343\pi\)
−0.876825 + 0.480810i \(0.840343\pi\)
\(570\) 0 0
\(571\) 10.8247 0.453001 0.226501 0.974011i \(-0.427271\pi\)
0.226501 + 0.974011i \(0.427271\pi\)
\(572\) −83.0242 −3.47142
\(573\) 0 0
\(574\) −2.54857 −0.106375
\(575\) 30.4920 1.27160
\(576\) 0 0
\(577\) 10.1277 0.421621 0.210810 0.977527i \(-0.432390\pi\)
0.210810 + 0.977527i \(0.432390\pi\)
\(578\) −1.22937 −0.0511349
\(579\) 0 0
\(580\) 31.1045 1.29154
\(581\) 3.19864 0.132702
\(582\) 0 0
\(583\) 62.3511 2.58232
\(584\) −10.2938 −0.425962
\(585\) 0 0
\(586\) −40.3785 −1.66802
\(587\) −10.6488 −0.439521 −0.219761 0.975554i \(-0.570528\pi\)
−0.219761 + 0.975554i \(0.570528\pi\)
\(588\) 0 0
\(589\) 18.6504 0.768477
\(590\) 73.0921 3.00915
\(591\) 0 0
\(592\) 1.78811 0.0734910
\(593\) 16.3794 0.672620 0.336310 0.941751i \(-0.390821\pi\)
0.336310 + 0.941751i \(0.390821\pi\)
\(594\) 0 0
\(595\) 2.78594 0.114213
\(596\) −8.78603 −0.359890
\(597\) 0 0
\(598\) −57.9395 −2.36932
\(599\) −36.6709 −1.49833 −0.749167 0.662381i \(-0.769546\pi\)
−0.749167 + 0.662381i \(0.769546\pi\)
\(600\) 0 0
\(601\) −31.4184 −1.28158 −0.640792 0.767714i \(-0.721394\pi\)
−0.640792 + 0.767714i \(0.721394\pi\)
\(602\) −3.49710 −0.142531
\(603\) 0 0
\(604\) −35.3900 −1.44000
\(605\) −60.1630 −2.44597
\(606\) 0 0
\(607\) −25.9196 −1.05204 −0.526021 0.850471i \(-0.676317\pi\)
−0.526021 + 0.850471i \(0.676317\pi\)
\(608\) 18.7409 0.760044
\(609\) 0 0
\(610\) −41.7901 −1.69203
\(611\) −18.3309 −0.741590
\(612\) 0 0
\(613\) 42.5971 1.72048 0.860240 0.509890i \(-0.170314\pi\)
0.860240 + 0.509890i \(0.170314\pi\)
\(614\) 10.2026 0.411741
\(615\) 0 0
\(616\) 1.50319 0.0605653
\(617\) 31.0455 1.24984 0.624922 0.780687i \(-0.285131\pi\)
0.624922 + 0.780687i \(0.285131\pi\)
\(618\) 0 0
\(619\) 2.71096 0.108963 0.0544813 0.998515i \(-0.482649\pi\)
0.0544813 + 0.998515i \(0.482649\pi\)
\(620\) 69.4176 2.78788
\(621\) 0 0
\(622\) 46.6510 1.87053
\(623\) 1.03222 0.0413551
\(624\) 0 0
\(625\) −14.0250 −0.561000
\(626\) 52.6786 2.10546
\(627\) 0 0
\(628\) 9.65589 0.385312
\(629\) 3.16959 0.126380
\(630\) 0 0
\(631\) −22.5598 −0.898093 −0.449047 0.893508i \(-0.648236\pi\)
−0.449047 + 0.893508i \(0.648236\pi\)
\(632\) 20.4247 0.812452
\(633\) 0 0
\(634\) −6.33311 −0.251520
\(635\) 29.8770 1.18563
\(636\) 0 0
\(637\) 40.7951 1.61636
\(638\) −39.7049 −1.57193
\(639\) 0 0
\(640\) 36.1031 1.42710
\(641\) −45.7883 −1.80853 −0.904265 0.426971i \(-0.859580\pi\)
−0.904265 + 0.426971i \(0.859580\pi\)
\(642\) 0 0
\(643\) −16.8011 −0.662571 −0.331285 0.943531i \(-0.607482\pi\)
−0.331285 + 0.943531i \(0.607482\pi\)
\(644\) 2.44412 0.0963117
\(645\) 0 0
\(646\) 21.2123 0.834586
\(647\) −30.7434 −1.20865 −0.604323 0.796739i \(-0.706557\pi\)
−0.604323 + 0.796739i \(0.706557\pi\)
\(648\) 0 0
\(649\) −53.1482 −2.08625
\(650\) −84.0371 −3.29621
\(651\) 0 0
\(652\) −39.3347 −1.54046
\(653\) −18.9143 −0.740172 −0.370086 0.928997i \(-0.620672\pi\)
−0.370086 + 0.928997i \(0.620672\pi\)
\(654\) 0 0
\(655\) 29.6767 1.15956
\(656\) 13.4252 0.524165
\(657\) 0 0
\(658\) 1.35749 0.0529204
\(659\) 29.5539 1.15126 0.575629 0.817711i \(-0.304757\pi\)
0.575629 + 0.817711i \(0.304757\pi\)
\(660\) 0 0
\(661\) 16.6977 0.649465 0.324733 0.945806i \(-0.394726\pi\)
0.324733 + 0.945806i \(0.394726\pi\)
\(662\) −28.5672 −1.11030
\(663\) 0 0
\(664\) 22.1626 0.860075
\(665\) 1.66853 0.0647027
\(666\) 0 0
\(667\) −15.7837 −0.611149
\(668\) −32.1244 −1.24293
\(669\) 0 0
\(670\) 50.3769 1.94623
\(671\) 30.3872 1.17309
\(672\) 0 0
\(673\) 45.9061 1.76955 0.884775 0.466018i \(-0.154312\pi\)
0.884775 + 0.466018i \(0.154312\pi\)
\(674\) −14.5818 −0.561671
\(675\) 0 0
\(676\) 56.5473 2.17490
\(677\) −12.4084 −0.476892 −0.238446 0.971156i \(-0.576638\pi\)
−0.238446 + 0.971156i \(0.576638\pi\)
\(678\) 0 0
\(679\) −2.38746 −0.0916225
\(680\) 19.3031 0.740239
\(681\) 0 0
\(682\) −88.6117 −3.39312
\(683\) 12.9318 0.494820 0.247410 0.968911i \(-0.420421\pi\)
0.247410 + 0.968911i \(0.420421\pi\)
\(684\) 0 0
\(685\) −28.9816 −1.10733
\(686\) −6.05973 −0.231362
\(687\) 0 0
\(688\) 18.4218 0.702323
\(689\) −68.3117 −2.60247
\(690\) 0 0
\(691\) 7.42986 0.282645 0.141323 0.989964i \(-0.454865\pi\)
0.141323 + 0.989964i \(0.454865\pi\)
\(692\) 34.5265 1.31250
\(693\) 0 0
\(694\) −62.2733 −2.36386
\(695\) 6.74379 0.255807
\(696\) 0 0
\(697\) 23.7973 0.901387
\(698\) 19.1203 0.723713
\(699\) 0 0
\(700\) 3.54502 0.133989
\(701\) 13.2033 0.498682 0.249341 0.968416i \(-0.419786\pi\)
0.249341 + 0.968416i \(0.419786\pi\)
\(702\) 0 0
\(703\) 1.89829 0.0715955
\(704\) −64.5725 −2.43367
\(705\) 0 0
\(706\) −4.26187 −0.160398
\(707\) 3.18807 0.119900
\(708\) 0 0
\(709\) −38.7599 −1.45566 −0.727830 0.685758i \(-0.759471\pi\)
−0.727830 + 0.685758i \(0.759471\pi\)
\(710\) 94.6813 3.55333
\(711\) 0 0
\(712\) 7.15199 0.268032
\(713\) −35.2255 −1.31920
\(714\) 0 0
\(715\) 107.049 4.00342
\(716\) −14.1900 −0.530306
\(717\) 0 0
\(718\) −35.2256 −1.31461
\(719\) −37.8102 −1.41008 −0.705041 0.709167i \(-0.749071\pi\)
−0.705041 + 0.709167i \(0.749071\pi\)
\(720\) 0 0
\(721\) 3.36592 0.125353
\(722\) −28.2548 −1.05154
\(723\) 0 0
\(724\) 13.5771 0.504589
\(725\) −22.8932 −0.850232
\(726\) 0 0
\(727\) −7.34986 −0.272591 −0.136296 0.990668i \(-0.543520\pi\)
−0.136296 + 0.990668i \(0.543520\pi\)
\(728\) −1.64689 −0.0610379
\(729\) 0 0
\(730\) 54.2874 2.00927
\(731\) 32.6542 1.20776
\(732\) 0 0
\(733\) −30.6192 −1.13095 −0.565473 0.824767i \(-0.691306\pi\)
−0.565473 + 0.824767i \(0.691306\pi\)
\(734\) −69.4908 −2.56495
\(735\) 0 0
\(736\) −35.3964 −1.30473
\(737\) −36.6310 −1.34932
\(738\) 0 0
\(739\) 17.9779 0.661328 0.330664 0.943749i \(-0.392727\pi\)
0.330664 + 0.943749i \(0.392727\pi\)
\(740\) 7.06553 0.259734
\(741\) 0 0
\(742\) 5.05879 0.185714
\(743\) 15.7997 0.579634 0.289817 0.957082i \(-0.406405\pi\)
0.289817 + 0.957082i \(0.406405\pi\)
\(744\) 0 0
\(745\) 11.3285 0.415044
\(746\) −0.824525 −0.0301880
\(747\) 0 0
\(748\) −57.4098 −2.09911
\(749\) 0.654613 0.0239190
\(750\) 0 0
\(751\) −18.1005 −0.660495 −0.330248 0.943894i \(-0.607132\pi\)
−0.330248 + 0.943894i \(0.607132\pi\)
\(752\) −7.15087 −0.260765
\(753\) 0 0
\(754\) 43.5006 1.58420
\(755\) 45.6310 1.66068
\(756\) 0 0
\(757\) −29.8230 −1.08394 −0.541968 0.840399i \(-0.682321\pi\)
−0.541968 + 0.840399i \(0.682321\pi\)
\(758\) −6.61044 −0.240102
\(759\) 0 0
\(760\) 11.5608 0.419354
\(761\) 13.1033 0.474996 0.237498 0.971388i \(-0.423673\pi\)
0.237498 + 0.971388i \(0.423673\pi\)
\(762\) 0 0
\(763\) −0.587807 −0.0212801
\(764\) 31.9802 1.15700
\(765\) 0 0
\(766\) 16.9669 0.613040
\(767\) 58.2290 2.10253
\(768\) 0 0
\(769\) −23.6108 −0.851428 −0.425714 0.904858i \(-0.639977\pi\)
−0.425714 + 0.904858i \(0.639977\pi\)
\(770\) −7.92749 −0.285687
\(771\) 0 0
\(772\) 68.0007 2.44740
\(773\) 35.3954 1.27308 0.636542 0.771242i \(-0.280364\pi\)
0.636542 + 0.771242i \(0.280364\pi\)
\(774\) 0 0
\(775\) −51.0921 −1.83528
\(776\) −16.5421 −0.593828
\(777\) 0 0
\(778\) −19.8773 −0.712634
\(779\) 14.2524 0.510646
\(780\) 0 0
\(781\) −68.8465 −2.46352
\(782\) −40.0642 −1.43269
\(783\) 0 0
\(784\) 15.9141 0.568362
\(785\) −12.4501 −0.444362
\(786\) 0 0
\(787\) 8.17427 0.291381 0.145691 0.989330i \(-0.453460\pi\)
0.145691 + 0.989330i \(0.453460\pi\)
\(788\) 22.3467 0.796068
\(789\) 0 0
\(790\) −107.716 −3.83235
\(791\) −0.452192 −0.0160781
\(792\) 0 0
\(793\) −33.2922 −1.18224
\(794\) 63.7826 2.26356
\(795\) 0 0
\(796\) 12.6161 0.447166
\(797\) 1.26503 0.0448098 0.0224049 0.999749i \(-0.492868\pi\)
0.0224049 + 0.999749i \(0.492868\pi\)
\(798\) 0 0
\(799\) −12.6755 −0.448428
\(800\) −51.3400 −1.81514
\(801\) 0 0
\(802\) −71.4584 −2.52328
\(803\) −39.4745 −1.39303
\(804\) 0 0
\(805\) −3.15139 −0.111072
\(806\) 97.0828 3.41959
\(807\) 0 0
\(808\) 22.0893 0.777098
\(809\) 43.0690 1.51423 0.757113 0.653284i \(-0.226609\pi\)
0.757113 + 0.653284i \(0.226609\pi\)
\(810\) 0 0
\(811\) −21.0119 −0.737829 −0.368914 0.929463i \(-0.620270\pi\)
−0.368914 + 0.929463i \(0.620270\pi\)
\(812\) −1.83503 −0.0643969
\(813\) 0 0
\(814\) −9.01916 −0.316121
\(815\) 50.7172 1.77655
\(816\) 0 0
\(817\) 19.5569 0.684208
\(818\) −59.4918 −2.08008
\(819\) 0 0
\(820\) 53.0481 1.85252
\(821\) 39.9137 1.39300 0.696498 0.717559i \(-0.254740\pi\)
0.696498 + 0.717559i \(0.254740\pi\)
\(822\) 0 0
\(823\) −40.1020 −1.39787 −0.698934 0.715186i \(-0.746342\pi\)
−0.698934 + 0.715186i \(0.746342\pi\)
\(824\) 23.3216 0.812445
\(825\) 0 0
\(826\) −4.31212 −0.150038
\(827\) 12.4405 0.432599 0.216299 0.976327i \(-0.430601\pi\)
0.216299 + 0.976327i \(0.430601\pi\)
\(828\) 0 0
\(829\) −37.5224 −1.30321 −0.651603 0.758560i \(-0.725903\pi\)
−0.651603 + 0.758560i \(0.725903\pi\)
\(830\) −116.881 −4.05698
\(831\) 0 0
\(832\) 70.7454 2.45266
\(833\) 28.2092 0.977389
\(834\) 0 0
\(835\) 41.4205 1.43341
\(836\) −34.3833 −1.18917
\(837\) 0 0
\(838\) 50.9200 1.75900
\(839\) 16.9559 0.585381 0.292691 0.956207i \(-0.405449\pi\)
0.292691 + 0.956207i \(0.405449\pi\)
\(840\) 0 0
\(841\) −17.1497 −0.591368
\(842\) −42.2556 −1.45622
\(843\) 0 0
\(844\) 21.1398 0.727663
\(845\) −72.9108 −2.50821
\(846\) 0 0
\(847\) 3.54935 0.121957
\(848\) −26.6483 −0.915106
\(849\) 0 0
\(850\) −58.1102 −1.99316
\(851\) −3.58535 −0.122904
\(852\) 0 0
\(853\) 7.61916 0.260875 0.130438 0.991457i \(-0.458362\pi\)
0.130438 + 0.991457i \(0.458362\pi\)
\(854\) 2.46543 0.0843654
\(855\) 0 0
\(856\) 4.53564 0.155025
\(857\) 39.0722 1.33468 0.667341 0.744753i \(-0.267433\pi\)
0.667341 + 0.744753i \(0.267433\pi\)
\(858\) 0 0
\(859\) 19.9870 0.681949 0.340974 0.940073i \(-0.389243\pi\)
0.340974 + 0.940073i \(0.389243\pi\)
\(860\) 72.7915 2.48217
\(861\) 0 0
\(862\) −75.1467 −2.55951
\(863\) −24.0312 −0.818032 −0.409016 0.912527i \(-0.634128\pi\)
−0.409016 + 0.912527i \(0.634128\pi\)
\(864\) 0 0
\(865\) −44.5177 −1.51365
\(866\) −25.2904 −0.859402
\(867\) 0 0
\(868\) −4.09534 −0.139005
\(869\) 78.3242 2.65697
\(870\) 0 0
\(871\) 40.1329 1.35985
\(872\) −4.07276 −0.137921
\(873\) 0 0
\(874\) −23.9948 −0.811636
\(875\) −1.13428 −0.0383457
\(876\) 0 0
\(877\) −53.1210 −1.79377 −0.896884 0.442266i \(-0.854175\pi\)
−0.896884 + 0.442266i \(0.854175\pi\)
\(878\) 79.9828 2.69929
\(879\) 0 0
\(880\) 41.7598 1.40772
\(881\) 22.0782 0.743834 0.371917 0.928266i \(-0.378701\pi\)
0.371917 + 0.928266i \(0.378701\pi\)
\(882\) 0 0
\(883\) 34.6218 1.16512 0.582559 0.812789i \(-0.302052\pi\)
0.582559 + 0.812789i \(0.302052\pi\)
\(884\) 62.8981 2.11549
\(885\) 0 0
\(886\) −11.3353 −0.380816
\(887\) 19.0942 0.641121 0.320561 0.947228i \(-0.396129\pi\)
0.320561 + 0.947228i \(0.396129\pi\)
\(888\) 0 0
\(889\) −1.76261 −0.0591162
\(890\) −37.7180 −1.26431
\(891\) 0 0
\(892\) −66.0403 −2.21119
\(893\) −7.59149 −0.254039
\(894\) 0 0
\(895\) 18.2963 0.611577
\(896\) −2.12993 −0.0711559
\(897\) 0 0
\(898\) −45.4428 −1.51644
\(899\) 26.4471 0.882059
\(900\) 0 0
\(901\) −47.2364 −1.57367
\(902\) −67.7160 −2.25470
\(903\) 0 0
\(904\) −3.13312 −0.104206
\(905\) −17.5060 −0.581919
\(906\) 0 0
\(907\) −41.6275 −1.38222 −0.691109 0.722751i \(-0.742877\pi\)
−0.691109 + 0.722751i \(0.742877\pi\)
\(908\) −67.3674 −2.23567
\(909\) 0 0
\(910\) 8.68534 0.287916
\(911\) 21.5111 0.712693 0.356347 0.934354i \(-0.384022\pi\)
0.356347 + 0.934354i \(0.384022\pi\)
\(912\) 0 0
\(913\) 84.9884 2.81271
\(914\) −68.3757 −2.26167
\(915\) 0 0
\(916\) −23.4269 −0.774045
\(917\) −1.75080 −0.0578163
\(918\) 0 0
\(919\) 31.8930 1.05205 0.526027 0.850468i \(-0.323681\pi\)
0.526027 + 0.850468i \(0.323681\pi\)
\(920\) −21.8351 −0.719883
\(921\) 0 0
\(922\) −30.4216 −1.00188
\(923\) 75.4281 2.48275
\(924\) 0 0
\(925\) −5.20030 −0.170985
\(926\) 11.8950 0.390894
\(927\) 0 0
\(928\) 26.5754 0.872381
\(929\) −27.5226 −0.902986 −0.451493 0.892275i \(-0.649108\pi\)
−0.451493 + 0.892275i \(0.649108\pi\)
\(930\) 0 0
\(931\) 16.8947 0.553702
\(932\) −63.3326 −2.07453
\(933\) 0 0
\(934\) 23.7416 0.776848
\(935\) 74.0229 2.42081
\(936\) 0 0
\(937\) −25.9600 −0.848076 −0.424038 0.905644i \(-0.639388\pi\)
−0.424038 + 0.905644i \(0.639388\pi\)
\(938\) −2.97202 −0.0970399
\(939\) 0 0
\(940\) −28.2558 −0.921604
\(941\) −11.4091 −0.371925 −0.185963 0.982557i \(-0.559540\pi\)
−0.185963 + 0.982557i \(0.559540\pi\)
\(942\) 0 0
\(943\) −26.9189 −0.876599
\(944\) 22.7150 0.739312
\(945\) 0 0
\(946\) −92.9185 −3.02104
\(947\) −41.0600 −1.33427 −0.667136 0.744936i \(-0.732480\pi\)
−0.667136 + 0.744936i \(0.732480\pi\)
\(948\) 0 0
\(949\) 43.2482 1.40390
\(950\) −34.8027 −1.12915
\(951\) 0 0
\(952\) −1.13880 −0.0369087
\(953\) 1.67519 0.0542647 0.0271324 0.999632i \(-0.491362\pi\)
0.0271324 + 0.999632i \(0.491362\pi\)
\(954\) 0 0
\(955\) −41.2346 −1.33432
\(956\) 2.64721 0.0856170
\(957\) 0 0
\(958\) −40.6578 −1.31359
\(959\) 1.70979 0.0552119
\(960\) 0 0
\(961\) 28.0235 0.903983
\(962\) 9.88137 0.318588
\(963\) 0 0
\(964\) −14.7480 −0.475002
\(965\) −87.6785 −2.82247
\(966\) 0 0
\(967\) −5.42710 −0.174524 −0.0872618 0.996185i \(-0.527812\pi\)
−0.0872618 + 0.996185i \(0.527812\pi\)
\(968\) 24.5925 0.790434
\(969\) 0 0
\(970\) 87.2395 2.80109
\(971\) −20.7489 −0.665865 −0.332932 0.942951i \(-0.608038\pi\)
−0.332932 + 0.942951i \(0.608038\pi\)
\(972\) 0 0
\(973\) −0.397854 −0.0127546
\(974\) 66.2678 2.12336
\(975\) 0 0
\(976\) −12.9872 −0.415711
\(977\) 4.41802 0.141345 0.0706725 0.997500i \(-0.477485\pi\)
0.0706725 + 0.997500i \(0.477485\pi\)
\(978\) 0 0
\(979\) 27.4263 0.876547
\(980\) 62.8829 2.00872
\(981\) 0 0
\(982\) −68.8554 −2.19727
\(983\) 10.9017 0.347710 0.173855 0.984771i \(-0.444378\pi\)
0.173855 + 0.984771i \(0.444378\pi\)
\(984\) 0 0
\(985\) −28.8133 −0.918068
\(986\) 30.0799 0.957940
\(987\) 0 0
\(988\) 37.6702 1.19845
\(989\) −36.9375 −1.17455
\(990\) 0 0
\(991\) −2.15506 −0.0684576 −0.0342288 0.999414i \(-0.510897\pi\)
−0.0342288 + 0.999414i \(0.510897\pi\)
\(992\) 59.3099 1.88309
\(993\) 0 0
\(994\) −5.58579 −0.177170
\(995\) −16.2669 −0.515696
\(996\) 0 0
\(997\) 36.3803 1.15217 0.576087 0.817388i \(-0.304579\pi\)
0.576087 + 0.817388i \(0.304579\pi\)
\(998\) −46.8615 −1.48338
\(999\) 0 0
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 2151.2.a.e.1.5 6
3.2 odd 2 717.2.a.d.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
717.2.a.d.1.2 6 3.2 odd 2
2151.2.a.e.1.5 6 1.1 even 1 trivial