Properties

Label 1521.2.b.k.1351.4
Level $1521$
Weight $2$
Character 1521.1351
Analytic conductor $12.145$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1521.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.1452461474\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
Defining polynomial: \(x^{6} + 5 x^{4} + 6 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 507)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1351.4
Root \(1.24698i\) of defining polynomial
Character \(\chi\) \(=\) 1521.1351
Dual form 1521.2.b.k.1351.3

$q$-expansion

\(f(q)\) \(=\) \(q+2.04892i q^{2} -2.19806 q^{4} -3.35690i q^{5} -2.24698i q^{7} -0.405813i q^{8} +O(q^{10})\) \(q+2.04892i q^{2} -2.19806 q^{4} -3.35690i q^{5} -2.24698i q^{7} -0.405813i q^{8} +6.87800 q^{10} +4.93900i q^{11} +4.60388 q^{14} -3.56465 q^{16} +0.911854 q^{17} -3.80194i q^{19} +7.37867i q^{20} -10.1196 q^{22} +2.02715 q^{23} -6.26875 q^{25} +4.93900i q^{28} +3.93900 q^{29} -8.82908i q^{31} -8.11529i q^{32} +1.86831i q^{34} -7.54288 q^{35} -8.80194i q^{37} +7.78986 q^{38} -1.36227 q^{40} -6.93900i q^{41} +2.28621 q^{43} -10.8562i q^{44} +4.15346i q^{46} +3.80194i q^{47} +1.95108 q^{49} -12.8442i q^{50} -0.542877 q^{53} +16.5797 q^{55} -0.911854 q^{56} +8.07069i q^{58} -4.71379i q^{59} +3.67994 q^{61} +18.0901 q^{62} +9.49827 q^{64} -1.52111i q^{67} -2.00431 q^{68} -15.4547i q^{70} -2.37867i q^{71} -7.41119i q^{73} +18.0344 q^{74} +8.35690i q^{76} +11.0978 q^{77} -3.74094 q^{79} +11.9661i q^{80} +14.2174 q^{82} -2.30798i q^{83} -3.06100i q^{85} +4.68425i q^{86} +2.00431 q^{88} -10.0586i q^{89} -4.45580 q^{92} -7.78986 q^{94} -12.7627 q^{95} -16.1293i q^{97} +3.99761i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 22q^{4} + O(q^{10}) \) \( 6q - 22q^{4} + 2q^{10} + 10q^{14} + 22q^{16} - 2q^{17} - 18q^{22} - 22q^{25} + 4q^{29} - 8q^{35} + 6q^{40} + 30q^{43} + 30q^{49} + 34q^{53} + 6q^{55} + 2q^{56} - 26q^{61} - 4q^{62} + 26q^{68} - 30q^{74} + 30q^{77} + 6q^{79} + 6q^{82} - 26q^{88} - 14q^{92} - 42q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1521\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.04892i 1.44880i 0.689378 + 0.724402i \(0.257884\pi\)
−0.689378 + 0.724402i \(0.742116\pi\)
\(3\) 0 0
\(4\) −2.19806 −1.09903
\(5\) − 3.35690i − 1.50125i −0.660729 0.750625i \(-0.729753\pi\)
0.660729 0.750625i \(-0.270247\pi\)
\(6\) 0 0
\(7\) − 2.24698i − 0.849278i −0.905363 0.424639i \(-0.860401\pi\)
0.905363 0.424639i \(-0.139599\pi\)
\(8\) − 0.405813i − 0.143477i
\(9\) 0 0
\(10\) 6.87800 2.17502
\(11\) 4.93900i 1.48916i 0.667531 + 0.744582i \(0.267351\pi\)
−0.667531 + 0.744582i \(0.732649\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 4.60388 1.23044
\(15\) 0 0
\(16\) −3.56465 −0.891162
\(17\) 0.911854 0.221157 0.110579 0.993867i \(-0.464730\pi\)
0.110579 + 0.993867i \(0.464730\pi\)
\(18\) 0 0
\(19\) − 3.80194i − 0.872224i −0.899892 0.436112i \(-0.856355\pi\)
0.899892 0.436112i \(-0.143645\pi\)
\(20\) 7.37867i 1.64992i
\(21\) 0 0
\(22\) −10.1196 −2.15751
\(23\) 2.02715 0.422689 0.211345 0.977412i \(-0.432216\pi\)
0.211345 + 0.977412i \(0.432216\pi\)
\(24\) 0 0
\(25\) −6.26875 −1.25375
\(26\) 0 0
\(27\) 0 0
\(28\) 4.93900i 0.933383i
\(29\) 3.93900 0.731454 0.365727 0.930722i \(-0.380820\pi\)
0.365727 + 0.930722i \(0.380820\pi\)
\(30\) 0 0
\(31\) − 8.82908i − 1.58575i −0.609384 0.792875i \(-0.708583\pi\)
0.609384 0.792875i \(-0.291417\pi\)
\(32\) − 8.11529i − 1.43459i
\(33\) 0 0
\(34\) 1.86831i 0.320413i
\(35\) −7.54288 −1.27498
\(36\) 0 0
\(37\) − 8.80194i − 1.44703i −0.690309 0.723515i \(-0.742525\pi\)
0.690309 0.723515i \(-0.257475\pi\)
\(38\) 7.78986 1.26368
\(39\) 0 0
\(40\) −1.36227 −0.215394
\(41\) − 6.93900i − 1.08369i −0.840478 0.541845i \(-0.817726\pi\)
0.840478 0.541845i \(-0.182274\pi\)
\(42\) 0 0
\(43\) 2.28621 0.348643 0.174322 0.984689i \(-0.444227\pi\)
0.174322 + 0.984689i \(0.444227\pi\)
\(44\) − 10.8562i − 1.63664i
\(45\) 0 0
\(46\) 4.15346i 0.612394i
\(47\) 3.80194i 0.554570i 0.960788 + 0.277285i \(0.0894346\pi\)
−0.960788 + 0.277285i \(0.910565\pi\)
\(48\) 0 0
\(49\) 1.95108 0.278726
\(50\) − 12.8442i − 1.81644i
\(51\) 0 0
\(52\) 0 0
\(53\) −0.542877 −0.0745698 −0.0372849 0.999305i \(-0.511871\pi\)
−0.0372849 + 0.999305i \(0.511871\pi\)
\(54\) 0 0
\(55\) 16.5797 2.23561
\(56\) −0.911854 −0.121852
\(57\) 0 0
\(58\) 8.07069i 1.05973i
\(59\) − 4.71379i − 0.613683i −0.951761 0.306842i \(-0.900728\pi\)
0.951761 0.306842i \(-0.0992722\pi\)
\(60\) 0 0
\(61\) 3.67994 0.471168 0.235584 0.971854i \(-0.424300\pi\)
0.235584 + 0.971854i \(0.424300\pi\)
\(62\) 18.0901 2.29744
\(63\) 0 0
\(64\) 9.49827 1.18728
\(65\) 0 0
\(66\) 0 0
\(67\) − 1.52111i − 0.185833i −0.995674 0.0929164i \(-0.970381\pi\)
0.995674 0.0929164i \(-0.0296189\pi\)
\(68\) −2.00431 −0.243059
\(69\) 0 0
\(70\) − 15.4547i − 1.84719i
\(71\) − 2.37867i − 0.282296i −0.989989 0.141148i \(-0.954921\pi\)
0.989989 0.141148i \(-0.0450793\pi\)
\(72\) 0 0
\(73\) − 7.41119i − 0.867414i −0.901054 0.433707i \(-0.857205\pi\)
0.901054 0.433707i \(-0.142795\pi\)
\(74\) 18.0344 2.09646
\(75\) 0 0
\(76\) 8.35690i 0.958602i
\(77\) 11.0978 1.26472
\(78\) 0 0
\(79\) −3.74094 −0.420888 −0.210444 0.977606i \(-0.567491\pi\)
−0.210444 + 0.977606i \(0.567491\pi\)
\(80\) 11.9661i 1.33786i
\(81\) 0 0
\(82\) 14.2174 1.57005
\(83\) − 2.30798i − 0.253334i −0.991945 0.126667i \(-0.959572\pi\)
0.991945 0.126667i \(-0.0404279\pi\)
\(84\) 0 0
\(85\) − 3.06100i − 0.332012i
\(86\) 4.68425i 0.505116i
\(87\) 0 0
\(88\) 2.00431 0.213660
\(89\) − 10.0586i − 1.06621i −0.846049 0.533105i \(-0.821025\pi\)
0.846049 0.533105i \(-0.178975\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −4.45580 −0.464549
\(93\) 0 0
\(94\) −7.78986 −0.803462
\(95\) −12.7627 −1.30943
\(96\) 0 0
\(97\) − 16.1293i − 1.63768i −0.574021 0.818841i \(-0.694617\pi\)
0.574021 0.818841i \(-0.305383\pi\)
\(98\) 3.99761i 0.403819i
\(99\) 0 0
\(100\) 13.7791 1.37791
\(101\) −9.94869 −0.989932 −0.494966 0.868912i \(-0.664819\pi\)
−0.494966 + 0.868912i \(0.664819\pi\)
\(102\) 0 0
\(103\) 10.9879 1.08267 0.541336 0.840806i \(-0.317919\pi\)
0.541336 + 0.840806i \(0.317919\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) − 1.11231i − 0.108037i
\(107\) 9.87263 0.954423 0.477211 0.878789i \(-0.341648\pi\)
0.477211 + 0.878789i \(0.341648\pi\)
\(108\) 0 0
\(109\) 20.2446i 1.93908i 0.244934 + 0.969540i \(0.421234\pi\)
−0.244934 + 0.969540i \(0.578766\pi\)
\(110\) 33.9705i 3.23896i
\(111\) 0 0
\(112\) 8.00969i 0.756844i
\(113\) −9.69202 −0.911749 −0.455874 0.890044i \(-0.650673\pi\)
−0.455874 + 0.890044i \(0.650673\pi\)
\(114\) 0 0
\(115\) − 6.80492i − 0.634562i
\(116\) −8.65817 −0.803891
\(117\) 0 0
\(118\) 9.65817 0.889107
\(119\) − 2.04892i − 0.187824i
\(120\) 0 0
\(121\) −13.3937 −1.21761
\(122\) 7.53989i 0.682630i
\(123\) 0 0
\(124\) 19.4069i 1.74279i
\(125\) 4.25906i 0.380942i
\(126\) 0 0
\(127\) −13.8116 −1.22558 −0.612792 0.790244i \(-0.709954\pi\)
−0.612792 + 0.790244i \(0.709954\pi\)
\(128\) 3.23059i 0.285546i
\(129\) 0 0
\(130\) 0 0
\(131\) −2.99462 −0.261641 −0.130821 0.991406i \(-0.541761\pi\)
−0.130821 + 0.991406i \(0.541761\pi\)
\(132\) 0 0
\(133\) −8.54288 −0.740761
\(134\) 3.11662 0.269235
\(135\) 0 0
\(136\) − 0.370042i − 0.0317309i
\(137\) 23.0194i 1.96668i 0.181781 + 0.983339i \(0.441814\pi\)
−0.181781 + 0.983339i \(0.558186\pi\)
\(138\) 0 0
\(139\) 0.982542 0.0833381 0.0416690 0.999131i \(-0.486732\pi\)
0.0416690 + 0.999131i \(0.486732\pi\)
\(140\) 16.5797 1.40124
\(141\) 0 0
\(142\) 4.87369 0.408991
\(143\) 0 0
\(144\) 0 0
\(145\) − 13.2228i − 1.09810i
\(146\) 15.1849 1.25671
\(147\) 0 0
\(148\) 19.3472i 1.59033i
\(149\) − 10.2591i − 0.840455i −0.907419 0.420228i \(-0.861950\pi\)
0.907419 0.420228i \(-0.138050\pi\)
\(150\) 0 0
\(151\) 20.1685i 1.64129i 0.571438 + 0.820646i \(0.306386\pi\)
−0.571438 + 0.820646i \(0.693614\pi\)
\(152\) −1.54288 −0.125144
\(153\) 0 0
\(154\) 22.7385i 1.83232i
\(155\) −29.6383 −2.38061
\(156\) 0 0
\(157\) 10.4383 0.833070 0.416535 0.909120i \(-0.363244\pi\)
0.416535 + 0.909120i \(0.363244\pi\)
\(158\) − 7.66487i − 0.609785i
\(159\) 0 0
\(160\) −27.2422 −2.15368
\(161\) − 4.55496i − 0.358981i
\(162\) 0 0
\(163\) 11.0465i 0.865231i 0.901579 + 0.432615i \(0.142409\pi\)
−0.901579 + 0.432615i \(0.857591\pi\)
\(164\) 15.2524i 1.19101i
\(165\) 0 0
\(166\) 4.72886 0.367031
\(167\) 8.10992i 0.627564i 0.949495 + 0.313782i \(0.101596\pi\)
−0.949495 + 0.313782i \(0.898404\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 6.27173 0.481020
\(171\) 0 0
\(172\) −5.02523 −0.383170
\(173\) −18.0562 −1.37279 −0.686394 0.727230i \(-0.740808\pi\)
−0.686394 + 0.727230i \(0.740808\pi\)
\(174\) 0 0
\(175\) 14.0858i 1.06478i
\(176\) − 17.6058i − 1.32709i
\(177\) 0 0
\(178\) 20.6093 1.54473
\(179\) −19.8702 −1.48517 −0.742585 0.669751i \(-0.766401\pi\)
−0.742585 + 0.669751i \(0.766401\pi\)
\(180\) 0 0
\(181\) 10.0828 0.749446 0.374723 0.927137i \(-0.377738\pi\)
0.374723 + 0.927137i \(0.377738\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) − 0.822643i − 0.0606461i
\(185\) −29.5472 −2.17235
\(186\) 0 0
\(187\) 4.50365i 0.329339i
\(188\) − 8.35690i − 0.609489i
\(189\) 0 0
\(190\) − 26.1497i − 1.89710i
\(191\) 6.58748 0.476653 0.238327 0.971185i \(-0.423401\pi\)
0.238327 + 0.971185i \(0.423401\pi\)
\(192\) 0 0
\(193\) − 10.8672i − 0.782242i −0.920339 0.391121i \(-0.872087\pi\)
0.920339 0.391121i \(-0.127913\pi\)
\(194\) 33.0476 2.37268
\(195\) 0 0
\(196\) −4.28860 −0.306329
\(197\) − 9.24160i − 0.658437i −0.944254 0.329218i \(-0.893215\pi\)
0.944254 0.329218i \(-0.106785\pi\)
\(198\) 0 0
\(199\) 1.56465 0.110915 0.0554574 0.998461i \(-0.482338\pi\)
0.0554574 + 0.998461i \(0.482338\pi\)
\(200\) 2.54394i 0.179884i
\(201\) 0 0
\(202\) − 20.3840i − 1.43422i
\(203\) − 8.85086i − 0.621208i
\(204\) 0 0
\(205\) −23.2935 −1.62689
\(206\) 22.5133i 1.56858i
\(207\) 0 0
\(208\) 0 0
\(209\) 18.7778 1.29889
\(210\) 0 0
\(211\) 23.2446 1.60022 0.800112 0.599851i \(-0.204774\pi\)
0.800112 + 0.599851i \(0.204774\pi\)
\(212\) 1.19328 0.0819546
\(213\) 0 0
\(214\) 20.2282i 1.38277i
\(215\) − 7.67456i − 0.523401i
\(216\) 0 0
\(217\) −19.8388 −1.34674
\(218\) −41.4795 −2.80935
\(219\) 0 0
\(220\) −36.4432 −2.45700
\(221\) 0 0
\(222\) 0 0
\(223\) 10.7385i 0.719106i 0.933125 + 0.359553i \(0.117071\pi\)
−0.933125 + 0.359553i \(0.882929\pi\)
\(224\) −18.2349 −1.21837
\(225\) 0 0
\(226\) − 19.8582i − 1.32094i
\(227\) 6.97584i 0.463003i 0.972835 + 0.231501i \(0.0743637\pi\)
−0.972835 + 0.231501i \(0.925636\pi\)
\(228\) 0 0
\(229\) 16.8049i 1.11050i 0.831683 + 0.555250i \(0.187378\pi\)
−0.831683 + 0.555250i \(0.812622\pi\)
\(230\) 13.9427 0.919356
\(231\) 0 0
\(232\) − 1.59850i − 0.104947i
\(233\) 8.69202 0.569433 0.284717 0.958612i \(-0.408100\pi\)
0.284717 + 0.958612i \(0.408100\pi\)
\(234\) 0 0
\(235\) 12.7627 0.832547
\(236\) 10.3612i 0.674457i
\(237\) 0 0
\(238\) 4.19806 0.272120
\(239\) 22.9191i 1.48252i 0.671220 + 0.741258i \(0.265771\pi\)
−0.671220 + 0.741258i \(0.734229\pi\)
\(240\) 0 0
\(241\) − 21.9801i − 1.41587i −0.706280 0.707933i \(-0.749628\pi\)
0.706280 0.707933i \(-0.250372\pi\)
\(242\) − 27.4426i − 1.76408i
\(243\) 0 0
\(244\) −8.08874 −0.517828
\(245\) − 6.54958i − 0.418437i
\(246\) 0 0
\(247\) 0 0
\(248\) −3.58296 −0.227518
\(249\) 0 0
\(250\) −8.72646 −0.551910
\(251\) 26.6437 1.68174 0.840868 0.541241i \(-0.182045\pi\)
0.840868 + 0.541241i \(0.182045\pi\)
\(252\) 0 0
\(253\) 10.0121i 0.629454i
\(254\) − 28.2989i − 1.77563i
\(255\) 0 0
\(256\) 12.3773 0.773584
\(257\) 23.1444 1.44371 0.721853 0.692047i \(-0.243291\pi\)
0.721853 + 0.692047i \(0.243291\pi\)
\(258\) 0 0
\(259\) −19.7778 −1.22893
\(260\) 0 0
\(261\) 0 0
\(262\) − 6.13574i − 0.379067i
\(263\) 18.5284 1.14251 0.571255 0.820773i \(-0.306457\pi\)
0.571255 + 0.820773i \(0.306457\pi\)
\(264\) 0 0
\(265\) 1.82238i 0.111948i
\(266\) − 17.5036i − 1.07322i
\(267\) 0 0
\(268\) 3.34349i 0.204236i
\(269\) −9.75840 −0.594980 −0.297490 0.954725i \(-0.596149\pi\)
−0.297490 + 0.954725i \(0.596149\pi\)
\(270\) 0 0
\(271\) 22.8019i 1.38512i 0.721361 + 0.692560i \(0.243517\pi\)
−0.721361 + 0.692560i \(0.756483\pi\)
\(272\) −3.25044 −0.197087
\(273\) 0 0
\(274\) −47.1648 −2.84933
\(275\) − 30.9614i − 1.86704i
\(276\) 0 0
\(277\) 23.9705 1.44025 0.720123 0.693847i \(-0.244086\pi\)
0.720123 + 0.693847i \(0.244086\pi\)
\(278\) 2.01315i 0.120741i
\(279\) 0 0
\(280\) 3.06100i 0.182930i
\(281\) − 4.12498i − 0.246076i −0.992402 0.123038i \(-0.960736\pi\)
0.992402 0.123038i \(-0.0392637\pi\)
\(282\) 0 0
\(283\) 15.6558 0.930639 0.465320 0.885143i \(-0.345939\pi\)
0.465320 + 0.885143i \(0.345939\pi\)
\(284\) 5.22846i 0.310252i
\(285\) 0 0
\(286\) 0 0
\(287\) −15.5918 −0.920354
\(288\) 0 0
\(289\) −16.1685 −0.951090
\(290\) 27.0925 1.59092
\(291\) 0 0
\(292\) 16.2903i 0.953315i
\(293\) 22.5948i 1.32000i 0.751265 + 0.660001i \(0.229444\pi\)
−0.751265 + 0.660001i \(0.770556\pi\)
\(294\) 0 0
\(295\) −15.8237 −0.921292
\(296\) −3.57194 −0.207615
\(297\) 0 0
\(298\) 21.0200 1.21765
\(299\) 0 0
\(300\) 0 0
\(301\) − 5.13706i − 0.296095i
\(302\) −41.3236 −2.37791
\(303\) 0 0
\(304\) 13.5526i 0.777293i
\(305\) − 12.3532i − 0.707341i
\(306\) 0 0
\(307\) 6.55496i 0.374111i 0.982349 + 0.187056i \(0.0598945\pi\)
−0.982349 + 0.187056i \(0.940106\pi\)
\(308\) −24.3937 −1.38996
\(309\) 0 0
\(310\) − 60.7265i − 3.44903i
\(311\) −12.0392 −0.682682 −0.341341 0.939940i \(-0.610881\pi\)
−0.341341 + 0.939940i \(0.610881\pi\)
\(312\) 0 0
\(313\) −33.8950 −1.91586 −0.957929 0.287005i \(-0.907340\pi\)
−0.957929 + 0.287005i \(0.907340\pi\)
\(314\) 21.3873i 1.20695i
\(315\) 0 0
\(316\) 8.22282 0.462570
\(317\) 4.49827i 0.252648i 0.991989 + 0.126324i \(0.0403179\pi\)
−0.991989 + 0.126324i \(0.959682\pi\)
\(318\) 0 0
\(319\) 19.4547i 1.08926i
\(320\) − 31.8847i − 1.78241i
\(321\) 0 0
\(322\) 9.33273 0.520093
\(323\) − 3.46681i − 0.192899i
\(324\) 0 0
\(325\) 0 0
\(326\) −22.6334 −1.25355
\(327\) 0 0
\(328\) −2.81594 −0.155484
\(329\) 8.54288 0.470984
\(330\) 0 0
\(331\) − 11.2131i − 0.616329i −0.951333 0.308165i \(-0.900285\pi\)
0.951333 0.308165i \(-0.0997148\pi\)
\(332\) 5.07308i 0.278421i
\(333\) 0 0
\(334\) −16.6165 −0.909217
\(335\) −5.10620 −0.278981
\(336\) 0 0
\(337\) −7.04892 −0.383979 −0.191989 0.981397i \(-0.561494\pi\)
−0.191989 + 0.981397i \(0.561494\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 6.72827i 0.364891i
\(341\) 43.6069 2.36144
\(342\) 0 0
\(343\) − 20.1129i − 1.08599i
\(344\) − 0.927774i − 0.0500222i
\(345\) 0 0
\(346\) − 36.9957i − 1.98890i
\(347\) 2.70410 0.145164 0.0725819 0.997362i \(-0.476876\pi\)
0.0725819 + 0.997362i \(0.476876\pi\)
\(348\) 0 0
\(349\) 0.415502i 0.0222413i 0.999938 + 0.0111207i \(0.00353989\pi\)
−0.999938 + 0.0111207i \(0.996460\pi\)
\(350\) −28.8605 −1.54266
\(351\) 0 0
\(352\) 40.0814 2.13635
\(353\) 32.1672i 1.71209i 0.516904 + 0.856044i \(0.327084\pi\)
−0.516904 + 0.856044i \(0.672916\pi\)
\(354\) 0 0
\(355\) −7.98493 −0.423796
\(356\) 22.1094i 1.17180i
\(357\) 0 0
\(358\) − 40.7125i − 2.15172i
\(359\) 22.3521i 1.17970i 0.807513 + 0.589850i \(0.200813\pi\)
−0.807513 + 0.589850i \(0.799187\pi\)
\(360\) 0 0
\(361\) 4.54527 0.239225
\(362\) 20.6588i 1.08580i
\(363\) 0 0
\(364\) 0 0
\(365\) −24.8786 −1.30221
\(366\) 0 0
\(367\) −2.30260 −0.120195 −0.0600974 0.998193i \(-0.519141\pi\)
−0.0600974 + 0.998193i \(0.519141\pi\)
\(368\) −7.22606 −0.376685
\(369\) 0 0
\(370\) − 60.5397i − 3.14731i
\(371\) 1.21983i 0.0633305i
\(372\) 0 0
\(373\) −19.2760 −0.998076 −0.499038 0.866580i \(-0.666313\pi\)
−0.499038 + 0.866580i \(0.666313\pi\)
\(374\) −9.22760 −0.477148
\(375\) 0 0
\(376\) 1.54288 0.0795678
\(377\) 0 0
\(378\) 0 0
\(379\) − 7.33944i − 0.377002i −0.982073 0.188501i \(-0.939637\pi\)
0.982073 0.188501i \(-0.0603628\pi\)
\(380\) 28.0532 1.43910
\(381\) 0 0
\(382\) 13.4972i 0.690577i
\(383\) − 19.0901i − 0.975457i −0.872995 0.487728i \(-0.837826\pi\)
0.872995 0.487728i \(-0.162174\pi\)
\(384\) 0 0
\(385\) − 37.2543i − 1.89865i
\(386\) 22.2661 1.13331
\(387\) 0 0
\(388\) 35.4532i 1.79986i
\(389\) −23.9879 −1.21624 −0.608118 0.793847i \(-0.708075\pi\)
−0.608118 + 0.793847i \(0.708075\pi\)
\(390\) 0 0
\(391\) 1.84846 0.0934807
\(392\) − 0.791775i − 0.0399907i
\(393\) 0 0
\(394\) 18.9353 0.953946
\(395\) 12.5579i 0.631859i
\(396\) 0 0
\(397\) 4.41789i 0.221728i 0.993836 + 0.110864i \(0.0353618\pi\)
−0.993836 + 0.110864i \(0.964638\pi\)
\(398\) 3.20583i 0.160694i
\(399\) 0 0
\(400\) 22.3459 1.11729
\(401\) − 20.4088i − 1.01917i −0.860421 0.509583i \(-0.829800\pi\)
0.860421 0.509583i \(-0.170200\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 21.8678 1.08797
\(405\) 0 0
\(406\) 18.1347 0.900009
\(407\) 43.4728 2.15487
\(408\) 0 0
\(409\) − 22.1491i − 1.09520i −0.836739 0.547602i \(-0.815541\pi\)
0.836739 0.547602i \(-0.184459\pi\)
\(410\) − 47.7265i − 2.35704i
\(411\) 0 0
\(412\) −24.1521 −1.18989
\(413\) −10.5918 −0.521188
\(414\) 0 0
\(415\) −7.74764 −0.380317
\(416\) 0 0
\(417\) 0 0
\(418\) 38.4741i 1.88183i
\(419\) −14.7560 −0.720878 −0.360439 0.932783i \(-0.617373\pi\)
−0.360439 + 0.932783i \(0.617373\pi\)
\(420\) 0 0
\(421\) 8.47219i 0.412909i 0.978456 + 0.206455i \(0.0661926\pi\)
−0.978456 + 0.206455i \(0.933807\pi\)
\(422\) 47.6262i 2.31841i
\(423\) 0 0
\(424\) 0.220306i 0.0106990i
\(425\) −5.71618 −0.277276
\(426\) 0 0
\(427\) − 8.26875i − 0.400153i
\(428\) −21.7006 −1.04894
\(429\) 0 0
\(430\) 15.7245 0.758305
\(431\) − 2.39181i − 0.115210i −0.998339 0.0576048i \(-0.981654\pi\)
0.998339 0.0576048i \(-0.0183463\pi\)
\(432\) 0 0
\(433\) 10.4286 0.501169 0.250584 0.968095i \(-0.419377\pi\)
0.250584 + 0.968095i \(0.419377\pi\)
\(434\) − 40.6480i − 1.95117i
\(435\) 0 0
\(436\) − 44.4989i − 2.13111i
\(437\) − 7.70709i − 0.368680i
\(438\) 0 0
\(439\) 32.5502 1.55353 0.776767 0.629787i \(-0.216858\pi\)
0.776767 + 0.629787i \(0.216858\pi\)
\(440\) − 6.72827i − 0.320758i
\(441\) 0 0
\(442\) 0 0
\(443\) −9.58211 −0.455260 −0.227630 0.973748i \(-0.573098\pi\)
−0.227630 + 0.973748i \(0.573098\pi\)
\(444\) 0 0
\(445\) −33.7657 −1.60065
\(446\) −22.0024 −1.04184
\(447\) 0 0
\(448\) − 21.3424i − 1.00833i
\(449\) − 28.3937i − 1.33998i −0.742369 0.669992i \(-0.766298\pi\)
0.742369 0.669992i \(-0.233702\pi\)
\(450\) 0 0
\(451\) 34.2717 1.61379
\(452\) 21.3037 1.00204
\(453\) 0 0
\(454\) −14.2929 −0.670800
\(455\) 0 0
\(456\) 0 0
\(457\) 4.99569i 0.233688i 0.993150 + 0.116844i \(0.0372778\pi\)
−0.993150 + 0.116844i \(0.962722\pi\)
\(458\) −34.4319 −1.60890
\(459\) 0 0
\(460\) 14.9576i 0.697404i
\(461\) − 1.35258i − 0.0629961i −0.999504 0.0314981i \(-0.989972\pi\)
0.999504 0.0314981i \(-0.0100278\pi\)
\(462\) 0 0
\(463\) 3.36467i 0.156369i 0.996939 + 0.0781846i \(0.0249124\pi\)
−0.996939 + 0.0781846i \(0.975088\pi\)
\(464\) −14.0411 −0.651844
\(465\) 0 0
\(466\) 17.8092i 0.824997i
\(467\) 6.91079 0.319793 0.159897 0.987134i \(-0.448884\pi\)
0.159897 + 0.987134i \(0.448884\pi\)
\(468\) 0 0
\(469\) −3.41789 −0.157824
\(470\) 26.1497i 1.20620i
\(471\) 0 0
\(472\) −1.91292 −0.0880492
\(473\) 11.2916i 0.519188i
\(474\) 0 0
\(475\) 23.8334i 1.09355i
\(476\) 4.50365i 0.206424i
\(477\) 0 0
\(478\) −46.9594 −2.14787
\(479\) 3.51573i 0.160638i 0.996769 + 0.0803189i \(0.0255939\pi\)
−0.996769 + 0.0803189i \(0.974406\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 45.0355 2.05131
\(483\) 0 0
\(484\) 29.4403 1.33819
\(485\) −54.1444 −2.45857
\(486\) 0 0
\(487\) − 12.8479i − 0.582193i −0.956694 0.291096i \(-0.905980\pi\)
0.956694 0.291096i \(-0.0940200\pi\)
\(488\) − 1.49337i − 0.0676016i
\(489\) 0 0
\(490\) 13.4196 0.606234
\(491\) 28.6708 1.29390 0.646948 0.762534i \(-0.276045\pi\)
0.646948 + 0.762534i \(0.276045\pi\)
\(492\) 0 0
\(493\) 3.59179 0.161766
\(494\) 0 0
\(495\) 0 0
\(496\) 31.4726i 1.41316i
\(497\) −5.34481 −0.239748
\(498\) 0 0
\(499\) − 33.5555i − 1.50215i −0.660215 0.751076i \(-0.729535\pi\)
0.660215 0.751076i \(-0.270465\pi\)
\(500\) − 9.36168i − 0.418667i
\(501\) 0 0
\(502\) 54.5907i 2.43650i
\(503\) −21.5633 −0.961461 −0.480730 0.876868i \(-0.659628\pi\)
−0.480730 + 0.876868i \(0.659628\pi\)
\(504\) 0 0
\(505\) 33.3967i 1.48613i
\(506\) −20.5139 −0.911955
\(507\) 0 0
\(508\) 30.3588 1.34695
\(509\) 8.89008i 0.394046i 0.980399 + 0.197023i \(0.0631274\pi\)
−0.980399 + 0.197023i \(0.936873\pi\)
\(510\) 0 0
\(511\) −16.6528 −0.736676
\(512\) 31.8213i 1.40632i
\(513\) 0 0
\(514\) 47.4209i 2.09165i
\(515\) − 36.8853i − 1.62536i
\(516\) 0 0
\(517\) −18.7778 −0.825846
\(518\) − 40.5230i − 1.78048i
\(519\) 0 0
\(520\) 0 0
\(521\) −19.3478 −0.847642 −0.423821 0.905746i \(-0.639312\pi\)
−0.423821 + 0.905746i \(0.639312\pi\)
\(522\) 0 0
\(523\) −12.5948 −0.550731 −0.275366 0.961340i \(-0.588799\pi\)
−0.275366 + 0.961340i \(0.588799\pi\)
\(524\) 6.58237 0.287552
\(525\) 0 0
\(526\) 37.9632i 1.65527i
\(527\) − 8.05084i − 0.350700i
\(528\) 0 0
\(529\) −18.8907 −0.821334
\(530\) −3.73391 −0.162191
\(531\) 0 0
\(532\) 18.7778 0.814120
\(533\) 0 0
\(534\) 0 0
\(535\) − 33.1414i − 1.43283i
\(536\) −0.617285 −0.0266627
\(537\) 0 0
\(538\) − 19.9941i − 0.862009i
\(539\) 9.63640i 0.415069i
\(540\) 0 0
\(541\) − 29.0019i − 1.24689i −0.781867 0.623445i \(-0.785733\pi\)
0.781867 0.623445i \(-0.214267\pi\)
\(542\) −46.7193 −2.00677
\(543\) 0 0
\(544\) − 7.39996i − 0.317271i
\(545\) 67.9590 2.91104
\(546\) 0 0
\(547\) 27.7006 1.18439 0.592197 0.805793i \(-0.298261\pi\)
0.592197 + 0.805793i \(0.298261\pi\)
\(548\) − 50.5980i − 2.16144i
\(549\) 0 0
\(550\) 63.4373 2.70497
\(551\) − 14.9758i − 0.637992i
\(552\) 0 0
\(553\) 8.40581i 0.357452i
\(554\) 49.1135i 2.08663i
\(555\) 0 0
\(556\) −2.15969 −0.0915912
\(557\) 6.80971i 0.288537i 0.989539 + 0.144268i \(0.0460828\pi\)
−0.989539 + 0.144268i \(0.953917\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 26.8877 1.13621
\(561\) 0 0
\(562\) 8.45175 0.356515
\(563\) −19.2524 −0.811390 −0.405695 0.914008i \(-0.632970\pi\)
−0.405695 + 0.914008i \(0.632970\pi\)
\(564\) 0 0
\(565\) 32.5351i 1.36876i
\(566\) 32.0774i 1.34831i
\(567\) 0 0
\(568\) −0.965294 −0.0405028
\(569\) 7.84846 0.329025 0.164512 0.986375i \(-0.447395\pi\)
0.164512 + 0.986375i \(0.447395\pi\)
\(570\) 0 0
\(571\) −29.8568 −1.24947 −0.624735 0.780837i \(-0.714793\pi\)
−0.624735 + 0.780837i \(0.714793\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) − 31.9463i − 1.33341i
\(575\) −12.7077 −0.529947
\(576\) 0 0
\(577\) 8.97823i 0.373769i 0.982382 + 0.186884i \(0.0598389\pi\)
−0.982382 + 0.186884i \(0.940161\pi\)
\(578\) − 33.1280i − 1.37794i
\(579\) 0 0
\(580\) 29.0646i 1.20684i
\(581\) −5.18598 −0.215151
\(582\) 0 0
\(583\) − 2.68127i − 0.111047i
\(584\) −3.00756 −0.124454
\(585\) 0 0
\(586\) −46.2948 −1.91242
\(587\) − 23.8538i − 0.984553i −0.870439 0.492277i \(-0.836165\pi\)
0.870439 0.492277i \(-0.163835\pi\)
\(588\) 0 0
\(589\) −33.5676 −1.38313
\(590\) − 32.4215i − 1.33477i
\(591\) 0 0
\(592\) 31.3758i 1.28954i
\(593\) 11.9866i 0.492230i 0.969241 + 0.246115i \(0.0791541\pi\)
−0.969241 + 0.246115i \(0.920846\pi\)
\(594\) 0 0
\(595\) −6.87800 −0.281971
\(596\) 22.5501i 0.923686i
\(597\) 0 0
\(598\) 0 0
\(599\) −29.1142 −1.18958 −0.594788 0.803883i \(-0.702764\pi\)
−0.594788 + 0.803883i \(0.702764\pi\)
\(600\) 0 0
\(601\) 37.1366 1.51483 0.757417 0.652932i \(-0.226461\pi\)
0.757417 + 0.652932i \(0.226461\pi\)
\(602\) 10.5254 0.428984
\(603\) 0 0
\(604\) − 44.3317i − 1.80383i
\(605\) 44.9614i 1.82794i
\(606\) 0 0
\(607\) −14.2325 −0.577680 −0.288840 0.957377i \(-0.593269\pi\)
−0.288840 + 0.957377i \(0.593269\pi\)
\(608\) −30.8538 −1.25129
\(609\) 0 0
\(610\) 25.3106 1.02480
\(611\) 0 0
\(612\) 0 0
\(613\) 30.3139i 1.22437i 0.790715 + 0.612184i \(0.209709\pi\)
−0.790715 + 0.612184i \(0.790291\pi\)
\(614\) −13.4306 −0.542014
\(615\) 0 0
\(616\) − 4.50365i − 0.181457i
\(617\) 24.9638i 1.00500i 0.864576 + 0.502501i \(0.167587\pi\)
−0.864576 + 0.502501i \(0.832413\pi\)
\(618\) 0 0
\(619\) − 35.6122i − 1.43138i −0.698420 0.715688i \(-0.746113\pi\)
0.698420 0.715688i \(-0.253887\pi\)
\(620\) 65.1469 2.61636
\(621\) 0 0
\(622\) − 24.6674i − 0.989072i
\(623\) −22.6015 −0.905509
\(624\) 0 0
\(625\) −17.0465 −0.681861
\(626\) − 69.4480i − 2.77570i
\(627\) 0 0
\(628\) −22.9441 −0.915570
\(629\) − 8.02608i − 0.320021i
\(630\) 0 0
\(631\) − 23.8829i − 0.950763i −0.879780 0.475382i \(-0.842310\pi\)
0.879780 0.475382i \(-0.157690\pi\)
\(632\) 1.51812i 0.0603877i
\(633\) 0 0
\(634\) −9.21659 −0.366037
\(635\) 46.3642i 1.83991i
\(636\) 0 0
\(637\) 0 0
\(638\) −39.8611 −1.57812
\(639\) 0 0
\(640\) 10.8447 0.428676
\(641\) −24.6577 −0.973920 −0.486960 0.873424i \(-0.661894\pi\)
−0.486960 + 0.873424i \(0.661894\pi\)
\(642\) 0 0
\(643\) 47.8165i 1.88570i 0.333218 + 0.942850i \(0.391866\pi\)
−0.333218 + 0.942850i \(0.608134\pi\)
\(644\) 10.0121i 0.394531i
\(645\) 0 0
\(646\) 7.10321 0.279472
\(647\) −5.38942 −0.211880 −0.105940 0.994373i \(-0.533785\pi\)
−0.105940 + 0.994373i \(0.533785\pi\)
\(648\) 0 0
\(649\) 23.2814 0.913876
\(650\) 0 0
\(651\) 0 0
\(652\) − 24.2809i − 0.950915i
\(653\) 5.03790 0.197148 0.0985741 0.995130i \(-0.468572\pi\)
0.0985741 + 0.995130i \(0.468572\pi\)
\(654\) 0 0
\(655\) 10.0526i 0.392789i
\(656\) 24.7351i 0.965743i
\(657\) 0 0
\(658\) 17.5036i 0.682363i
\(659\) 43.9812 1.71326 0.856632 0.515927i \(-0.172553\pi\)
0.856632 + 0.515927i \(0.172553\pi\)
\(660\) 0 0
\(661\) 38.2194i 1.48656i 0.668980 + 0.743280i \(0.266731\pi\)
−0.668980 + 0.743280i \(0.733269\pi\)
\(662\) 22.9748 0.892940
\(663\) 0 0
\(664\) −0.936608 −0.0363474
\(665\) 28.6775i 1.11207i
\(666\) 0 0
\(667\) 7.98493 0.309178
\(668\) − 17.8261i − 0.689713i
\(669\) 0 0
\(670\) − 10.4622i − 0.404189i
\(671\) 18.1752i 0.701647i
\(672\) 0 0
\(673\) 6.66487 0.256912 0.128456 0.991715i \(-0.458998\pi\)
0.128456 + 0.991715i \(0.458998\pi\)
\(674\) − 14.4426i − 0.556310i
\(675\) 0 0
\(676\) 0 0
\(677\) −4.80194 −0.184553 −0.0922767 0.995733i \(-0.529414\pi\)
−0.0922767 + 0.995733i \(0.529414\pi\)
\(678\) 0 0
\(679\) −36.2422 −1.39085
\(680\) −1.24219 −0.0476360
\(681\) 0 0
\(682\) 89.3469i 3.42127i
\(683\) 11.2591i 0.430816i 0.976524 + 0.215408i \(0.0691081\pi\)
−0.976524 + 0.215408i \(0.930892\pi\)
\(684\) 0 0
\(685\) 77.2737 2.95247
\(686\) 41.2097 1.57339
\(687\) 0 0
\(688\) −8.14952 −0.310698
\(689\) 0 0
\(690\) 0 0
\(691\) − 24.7144i − 0.940179i −0.882619 0.470090i \(-0.844222\pi\)
0.882619 0.470090i \(-0.155778\pi\)
\(692\) 39.6887 1.50874
\(693\) 0 0
\(694\) 5.54048i 0.210314i
\(695\) − 3.29829i − 0.125111i
\(696\) 0 0
\(697\) − 6.32736i − 0.239666i
\(698\) −0.851329 −0.0322233
\(699\) 0 0
\(700\) − 30.9614i − 1.17023i
\(701\) 25.8920 0.977927 0.488964 0.872304i \(-0.337375\pi\)
0.488964 + 0.872304i \(0.337375\pi\)
\(702\) 0 0
\(703\) −33.4644 −1.26213
\(704\) 46.9120i 1.76806i
\(705\) 0 0
\(706\) −65.9079 −2.48048
\(707\) 22.3545i 0.840728i
\(708\) 0 0
\(709\) − 0.0851621i − 0.00319833i −0.999999 0.00159916i \(-0.999491\pi\)
0.999999 0.00159916i \(-0.000509030\pi\)
\(710\) − 16.3605i − 0.613998i
\(711\) 0 0
\(712\) −4.08192 −0.152976
\(713\) − 17.8979i − 0.670280i
\(714\) 0 0
\(715\) 0 0
\(716\) 43.6760 1.63225
\(717\) 0 0
\(718\) −45.7976 −1.70915
\(719\) 27.0508 1.00883 0.504413 0.863463i \(-0.331709\pi\)
0.504413 + 0.863463i \(0.331709\pi\)
\(720\) 0 0
\(721\) − 24.6896i − 0.919490i
\(722\) 9.31288i 0.346590i
\(723\) 0 0
\(724\) −22.1626 −0.823665
\(725\) −24.6926 −0.917061
\(726\) 0 0
\(727\) 47.1584 1.74901 0.874503 0.485019i \(-0.161187\pi\)
0.874503 + 0.485019i \(0.161187\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) − 50.9742i − 1.88664i
\(731\) 2.08469 0.0771050
\(732\) 0 0
\(733\) − 9.68186i − 0.357608i −0.983885 0.178804i \(-0.942777\pi\)
0.983885 0.178804i \(-0.0572227\pi\)
\(734\) − 4.71784i − 0.174139i
\(735\) 0 0
\(736\) − 16.4509i − 0.606388i
\(737\) 7.51275 0.276736
\(738\) 0 0
\(739\) − 36.3139i − 1.33583i −0.744238 0.667915i \(-0.767187\pi\)
0.744238 0.667915i \(-0.232813\pi\)
\(740\) 64.9466 2.38748
\(741\) 0 0
\(742\) −2.49934 −0.0917535
\(743\) 41.7536i 1.53179i 0.642965 + 0.765896i \(0.277704\pi\)
−0.642965 + 0.765896i \(0.722296\pi\)
\(744\) 0 0
\(745\) −34.4386 −1.26173
\(746\) − 39.4950i − 1.44602i
\(747\) 0 0
\(748\) − 9.89930i − 0.361954i
\(749\) − 22.1836i − 0.810571i
\(750\) 0 0
\(751\) 22.1927 0.809823 0.404911 0.914356i \(-0.367302\pi\)
0.404911 + 0.914356i \(0.367302\pi\)
\(752\) − 13.5526i − 0.494211i
\(753\) 0 0
\(754\) 0 0
\(755\) 67.7036 2.46399
\(756\) 0 0
\(757\) 9.23729 0.335735 0.167868 0.985810i \(-0.446312\pi\)
0.167868 + 0.985810i \(0.446312\pi\)
\(758\) 15.0379 0.546201
\(759\) 0 0
\(760\) 5.17928i 0.187872i
\(761\) − 7.50173i − 0.271937i −0.990713 0.135969i \(-0.956585\pi\)
0.990713 0.135969i \(-0.0434147\pi\)
\(762\) 0 0
\(763\) 45.4892 1.64682
\(764\) −14.4797 −0.523857
\(765\) 0 0
\(766\) 39.1140 1.41325
\(767\) 0 0
\(768\) 0 0
\(769\) 5.14005i 0.185355i 0.995696 + 0.0926774i \(0.0295425\pi\)
−0.995696 + 0.0926774i \(0.970457\pi\)
\(770\) 76.3309 2.75078
\(771\) 0 0
\(772\) 23.8869i 0.859708i
\(773\) 8.50173i 0.305786i 0.988243 + 0.152893i \(0.0488590\pi\)
−0.988243 + 0.152893i \(0.951141\pi\)
\(774\) 0 0
\(775\) 55.3473i 1.98813i
\(776\) −6.54548 −0.234969
\(777\) 0 0
\(778\) − 49.1493i − 1.76209i
\(779\) −26.3817 −0.945221
\(780\) 0 0
\(781\) 11.7482 0.420385
\(782\) 3.78735i 0.135435i
\(783\) 0 0
\(784\) −6.95492 −0.248390
\(785\) − 35.0404i − 1.25065i
\(786\) 0 0
\(787\) 7.78554i 0.277525i 0.990326 + 0.138762i \(0.0443124\pi\)
−0.990326 + 0.138762i \(0.955688\pi\)
\(788\) 20.3136i 0.723643i
\(789\) 0 0
\(790\) −25.7302 −0.915439
\(791\) 21.7778i 0.774329i
\(792\) 0 0
\(793\) 0 0
\(794\) −9.05190 −0.321240
\(795\) 0 0
\(796\) −3.43919 −0.121899
\(797\) 21.3840 0.757462 0.378731 0.925507i \(-0.376361\pi\)
0.378731 + 0.925507i \(0.376361\pi\)
\(798\) 0 0
\(799\) 3.46681i 0.122647i
\(800\) 50.8727i 1.79862i
\(801\) 0 0
\(802\) 41.8159 1.47657
\(803\) 36.6039 1.29172
\(804\) 0 0
\(805\) −15.2905 −0.538920
\(806\) 0 0
\(807\) 0 0
\(808\) 4.03731i 0.142032i
\(809\) −2.28621 −0.0803788 −0.0401894 0.999192i \(-0.512796\pi\)
−0.0401894 + 0.999192i \(0.512796\pi\)
\(810\) 0 0
\(811\) 17.8079i 0.625320i 0.949865 + 0.312660i \(0.101220\pi\)
−0.949865 + 0.312660i \(0.898780\pi\)
\(812\) 19.4547i 0.682727i
\(813\) 0 0
\(814\) 89.0721i 3.12198i
\(815\) 37.0820 1.29893
\(816\) 0 0
\(817\) − 8.69202i − 0.304095i
\(818\) 45.3818 1.58674
\(819\) 0 0
\(820\) 51.2006 1.78800
\(821\) − 13.9054i − 0.485302i −0.970114 0.242651i \(-0.921983\pi\)
0.970114 0.242651i \(-0.0780170\pi\)
\(822\) 0 0
\(823\) 7.24831 0.252660 0.126330 0.991988i \(-0.459680\pi\)
0.126330 + 0.991988i \(0.459680\pi\)
\(824\) − 4.45904i − 0.155338i
\(825\) 0 0
\(826\) − 21.7017i − 0.755099i
\(827\) 54.1191i 1.88191i 0.338536 + 0.940953i \(0.390068\pi\)
−0.338536 + 0.940953i \(0.609932\pi\)
\(828\) 0 0
\(829\) 5.79178 0.201157 0.100578 0.994929i \(-0.467931\pi\)
0.100578 + 0.994929i \(0.467931\pi\)
\(830\) − 15.8743i − 0.551004i
\(831\) 0 0
\(832\) 0 0
\(833\) 1.77910 0.0616422
\(834\) 0 0
\(835\) 27.2241 0.942130
\(836\) −41.2747 −1.42752
\(837\) 0 0
\(838\) − 30.2338i − 1.04441i
\(839\) − 5.29350i − 0.182752i −0.995816 0.0913760i \(-0.970873\pi\)
0.995816 0.0913760i \(-0.0291265\pi\)
\(840\) 0 0
\(841\) −13.4843 −0.464975
\(842\) −17.3588 −0.598224
\(843\) 0 0
\(844\) −51.0930 −1.75870
\(845\) 0 0
\(846\) 0 0
\(847\) 30.0954i 1.03409i
\(848\) 1.93516 0.0664538
\(849\) 0 0
\(850\) − 11.7120i − 0.401718i
\(851\) − 17.8428i − 0.611644i
\(852\) 0 0
\(853\) 13.5961i 0.465522i 0.972534 + 0.232761i \(0.0747760\pi\)
−0.972534 + 0.232761i \(0.925224\pi\)
\(854\) 16.9420 0.579743
\(855\) 0 0
\(856\) − 4.00644i − 0.136937i
\(857\) −23.8323 −0.814097 −0.407048 0.913407i \(-0.633442\pi\)
−0.407048 + 0.913407i \(0.633442\pi\)
\(858\) 0 0
\(859\) −26.9861 −0.920754 −0.460377 0.887723i \(-0.652286\pi\)
−0.460377 + 0.887723i \(0.652286\pi\)
\(860\) 16.8692i 0.575234i
\(861\) 0 0
\(862\) 4.90063 0.166916
\(863\) 27.0291i 0.920080i 0.887898 + 0.460040i \(0.152165\pi\)
−0.887898 + 0.460040i \(0.847835\pi\)
\(864\) 0 0
\(865\) 60.6128i 2.06090i
\(866\) 21.3674i 0.726095i
\(867\) 0 0
\(868\) 43.6069 1.48011
\(869\) − 18.4765i − 0.626772i
\(870\) 0 0
\(871\) 0 0
\(872\) 8.21552 0.278213
\(873\) 0 0
\(874\) 15.7912 0.534145
\(875\) 9.57002 0.323526
\(876\) 0 0
\(877\) 40.8780i 1.38035i 0.723642 + 0.690176i \(0.242467\pi\)
−0.723642 + 0.690176i \(0.757533\pi\)
\(878\) 66.6926i 2.25077i
\(879\) 0 0
\(880\) −59.1008 −1.99229
\(881\) −22.4964 −0.757921 −0.378961 0.925413i \(-0.623718\pi\)
−0.378961 + 0.925413i \(0.623718\pi\)
\(882\) 0 0
\(883\) 4.16315 0.140101 0.0700505 0.997543i \(-0.477684\pi\)
0.0700505 + 0.997543i \(0.477684\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) − 19.6329i − 0.659582i
\(887\) 26.3002 0.883075 0.441537 0.897243i \(-0.354433\pi\)
0.441537 + 0.897243i \(0.354433\pi\)
\(888\) 0 0
\(889\) 31.0344i 1.04086i
\(890\) − 69.1831i − 2.31902i
\(891\) 0 0
\(892\) − 23.6040i − 0.790320i
\(893\) 14.4547 0.483709
\(894\) 0 0
\(895\) 66.7023i 2.22961i
\(896\) 7.25906 0.242508
\(897\) 0 0
\(898\) 58.1764 1.94137
\(899\) − 34.7778i − 1.15990i
\(900\) 0 0
\(901\) −0.495024 −0.0164916
\(902\) 70.2199i 2.33807i
\(903\) 0 0
\(904\) 3.93315i 0.130815i
\(905\) − 33.8468i − 1.12511i
\(906\) 0 0
\(907\) 57.9114 1.92292 0.961458 0.274952i \(-0.0886620\pi\)
0.961458 + 0.274952i \(0.0886620\pi\)
\(908\) − 15.3333i − 0.508854i
\(909\) 0 0
\(910\) 0 0
\(911\) 0.286799 0.00950208 0.00475104 0.999989i \(-0.498488\pi\)
0.00475104 + 0.999989i \(0.498488\pi\)
\(912\) 0 0
\(913\) 11.3991 0.377255
\(914\) −10.2358 −0.338569
\(915\) 0 0
\(916\) − 36.9383i − 1.22047i
\(917\) 6.72886i 0.222206i
\(918\) 0 0
\(919\) −31.1239 −1.02668 −0.513342 0.858184i \(-0.671593\pi\)
−0.513342 + 0.858184i \(0.671593\pi\)
\(920\) −2.76153 −0.0910449
\(921\) 0 0
\(922\) 2.77133 0.0912690
\(923\) 0 0
\(924\) 0 0
\(925\) 55.1771i 1.81421i
\(926\) −6.89392 −0.226548
\(927\) 0 0
\(928\) − 31.9661i − 1.04934i
\(929\) − 7.62671i − 0.250224i −0.992143 0.125112i \(-0.960071\pi\)
0.992143 0.125112i \(-0.0399291\pi\)
\(930\) 0 0
\(931\) − 7.41789i − 0.243112i
\(932\) −19.1056 −0.625825
\(933\) 0 0
\(934\) 14.1596i 0.463317i
\(935\) 15.1183 0.494421
\(936\) 0 0
\(937\) −5.67324 −0.185337 −0.0926683 0.995697i \(-0.529540\pi\)
−0.0926683 + 0.995697i \(0.529540\pi\)
\(938\) − 7.00298i − 0.228656i
\(939\) 0 0
\(940\) −28.0532 −0.914995
\(941\) − 41.5394i − 1.35415i −0.735916 0.677073i \(-0.763248\pi\)
0.735916 0.677073i \(-0.236752\pi\)
\(942\) 0 0
\(943\) − 14.0664i − 0.458064i
\(944\) 16.8030i 0.546891i
\(945\) 0 0
\(946\) −23.1355 −0.752201
\(947\) − 47.3110i − 1.53740i −0.639610 0.768700i \(-0.720904\pi\)
0.639610 0.768700i \(-0.279096\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −48.8327 −1.58434
\(951\) 0 0
\(952\) −0.831478 −0.0269483
\(953\) −34.3435 −1.11249 −0.556247 0.831017i \(-0.687759\pi\)
−0.556247 + 0.831017i \(0.687759\pi\)
\(954\) 0 0
\(955\) − 22.1135i − 0.715576i
\(956\) − 50.3777i − 1.62933i
\(957\) 0 0
\(958\) −7.20344 −0.232733
\(959\) 51.7241 1.67026
\(960\) 0 0
\(961\) −46.9527 −1.51460
\(962\) 0 0
\(963\) 0 0
\(964\) 48.3137i 1.55608i
\(965\) −36.4802 −1.17434
\(966\) 0 0
\(967\) 48.5096i 1.55996i 0.625802 + 0.779982i \(0.284772\pi\)
−0.625802 + 0.779982i \(0.715228\pi\)
\(968\) 5.43535i 0.174699i
\(969\) 0 0
\(970\) − 110.937i − 3.56198i
\(971\) −41.4650 −1.33068 −0.665338 0.746542i \(-0.731712\pi\)
−0.665338 + 0.746542i \(0.731712\pi\)
\(972\) 0 0
\(973\) − 2.20775i − 0.0707772i
\(974\) 26.3242 0.843483
\(975\) 0 0
\(976\) −13.1177 −0.419887
\(977\) 2.09677i 0.0670816i 0.999437 + 0.0335408i \(0.0106784\pi\)
−0.999437 + 0.0335408i \(0.989322\pi\)
\(978\) 0 0
\(979\) 49.6795 1.58776
\(980\) 14.3964i 0.459876i
\(981\) 0 0
\(982\) 58.7442i 1.87460i
\(983\) 25.2336i 0.804826i 0.915458 + 0.402413i \(0.131828\pi\)
−0.915458 + 0.402413i \(0.868172\pi\)
\(984\) 0 0
\(985\) −31.0231 −0.988478
\(986\) 7.35929i 0.234367i
\(987\) 0 0
\(988\) 0 0
\(989\) 4.63448 0.147368
\(990\) 0 0
\(991\) 12.0489 0.382746 0.191373 0.981517i \(-0.438706\pi\)
0.191373 + 0.981517i \(0.438706\pi\)
\(992\) −71.6506 −2.27491
\(993\) 0 0
\(994\) − 10.9511i − 0.347347i
\(995\) − 5.25236i − 0.166511i
\(996\) 0 0
\(997\) −1.43403 −0.0454160 −0.0227080 0.999742i \(-0.507229\pi\)
−0.0227080 + 0.999742i \(0.507229\pi\)
\(998\) 68.7525 2.17632
\(999\) 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 1521.2.b.k.1351.4 6
3.2 odd 2 507.2.b.f.337.3 6
13.5 odd 4 1521.2.a.n.1.3 3
13.8 odd 4 1521.2.a.s.1.1 3
13.12 even 2 inner 1521.2.b.k.1351.3 6
39.2 even 12 507.2.e.i.22.3 6
39.5 even 4 507.2.a.l.1.1 yes 3
39.8 even 4 507.2.a.i.1.3 3
39.11 even 12 507.2.e.l.22.1 6
39.17 odd 6 507.2.j.i.361.4 12
39.20 even 12 507.2.e.l.484.1 6
39.23 odd 6 507.2.j.i.316.3 12
39.29 odd 6 507.2.j.i.316.4 12
39.32 even 12 507.2.e.i.484.3 6
39.35 odd 6 507.2.j.i.361.3 12
39.38 odd 2 507.2.b.f.337.4 6
156.47 odd 4 8112.2.a.cg.1.2 3
156.83 odd 4 8112.2.a.cp.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.a.i.1.3 3 39.8 even 4
507.2.a.l.1.1 yes 3 39.5 even 4
507.2.b.f.337.3 6 3.2 odd 2
507.2.b.f.337.4 6 39.38 odd 2
507.2.e.i.22.3 6 39.2 even 12
507.2.e.i.484.3 6 39.32 even 12
507.2.e.l.22.1 6 39.11 even 12
507.2.e.l.484.1 6 39.20 even 12
507.2.j.i.316.3 12 39.23 odd 6
507.2.j.i.316.4 12 39.29 odd 6
507.2.j.i.361.3 12 39.35 odd 6
507.2.j.i.361.4 12 39.17 odd 6
1521.2.a.n.1.3 3 13.5 odd 4
1521.2.a.s.1.1 3 13.8 odd 4
1521.2.b.k.1351.3 6 13.12 even 2 inner
1521.2.b.k.1351.4 6 1.1 even 1 trivial
8112.2.a.cg.1.2 3 156.47 odd 4
8112.2.a.cp.1.2 3 156.83 odd 4