Properties

Label 4560.2.a.be.1.2
Level $4560$
Weight $2$
Character 4560.1
Self dual yes
Analytic conductor $36.412$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4560.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.4117833217\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2280)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 4560.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} -0.763932 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} -0.763932 q^{7} +1.00000 q^{9} -3.23607 q^{11} +5.23607 q^{13} +1.00000 q^{15} -4.47214 q^{17} -1.00000 q^{19} +0.763932 q^{21} +6.47214 q^{23} +1.00000 q^{25} -1.00000 q^{27} +5.23607 q^{29} -8.94427 q^{31} +3.23607 q^{33} +0.763932 q^{35} -1.23607 q^{37} -5.23607 q^{39} +3.70820 q^{41} -0.763932 q^{43} -1.00000 q^{45} +6.47214 q^{47} -6.41641 q^{49} +4.47214 q^{51} +8.47214 q^{53} +3.23607 q^{55} +1.00000 q^{57} +1.52786 q^{59} -4.47214 q^{61} -0.763932 q^{63} -5.23607 q^{65} -10.4721 q^{67} -6.47214 q^{69} +3.52786 q^{73} -1.00000 q^{75} +2.47214 q^{77} +1.00000 q^{81} -4.00000 q^{83} +4.47214 q^{85} -5.23607 q^{87} +11.7082 q^{89} -4.00000 q^{91} +8.94427 q^{93} +1.00000 q^{95} -9.23607 q^{97} -3.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{5} - 6 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{3} - 2 q^{5} - 6 q^{7} + 2 q^{9} - 2 q^{11} + 6 q^{13} + 2 q^{15} - 2 q^{19} + 6 q^{21} + 4 q^{23} + 2 q^{25} - 2 q^{27} + 6 q^{29} + 2 q^{33} + 6 q^{35} + 2 q^{37} - 6 q^{39} - 6 q^{41} - 6 q^{43} - 2 q^{45} + 4 q^{47} + 14 q^{49} + 8 q^{53} + 2 q^{55} + 2 q^{57} + 12 q^{59} - 6 q^{63} - 6 q^{65} - 12 q^{67} - 4 q^{69} + 16 q^{73} - 2 q^{75} - 4 q^{77} + 2 q^{81} - 8 q^{83} - 6 q^{87} + 10 q^{89} - 8 q^{91} + 2 q^{95} - 14 q^{97} - 2 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −0.763932 −0.288739 −0.144370 0.989524i \(-0.546115\pi\)
−0.144370 + 0.989524i \(0.546115\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.23607 −0.975711 −0.487856 0.872924i \(-0.662221\pi\)
−0.487856 + 0.872924i \(0.662221\pi\)
\(12\) 0 0
\(13\) 5.23607 1.45222 0.726112 0.687576i \(-0.241325\pi\)
0.726112 + 0.687576i \(0.241325\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −4.47214 −1.08465 −0.542326 0.840168i \(-0.682456\pi\)
−0.542326 + 0.840168i \(0.682456\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0.763932 0.166704
\(22\) 0 0
\(23\) 6.47214 1.34953 0.674767 0.738031i \(-0.264244\pi\)
0.674767 + 0.738031i \(0.264244\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 5.23607 0.972313 0.486157 0.873872i \(-0.338398\pi\)
0.486157 + 0.873872i \(0.338398\pi\)
\(30\) 0 0
\(31\) −8.94427 −1.60644 −0.803219 0.595683i \(-0.796881\pi\)
−0.803219 + 0.595683i \(0.796881\pi\)
\(32\) 0 0
\(33\) 3.23607 0.563327
\(34\) 0 0
\(35\) 0.763932 0.129128
\(36\) 0 0
\(37\) −1.23607 −0.203208 −0.101604 0.994825i \(-0.532398\pi\)
−0.101604 + 0.994825i \(0.532398\pi\)
\(38\) 0 0
\(39\) −5.23607 −0.838442
\(40\) 0 0
\(41\) 3.70820 0.579124 0.289562 0.957159i \(-0.406490\pi\)
0.289562 + 0.957159i \(0.406490\pi\)
\(42\) 0 0
\(43\) −0.763932 −0.116499 −0.0582493 0.998302i \(-0.518552\pi\)
−0.0582493 + 0.998302i \(0.518552\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 6.47214 0.944058 0.472029 0.881583i \(-0.343522\pi\)
0.472029 + 0.881583i \(0.343522\pi\)
\(48\) 0 0
\(49\) −6.41641 −0.916630
\(50\) 0 0
\(51\) 4.47214 0.626224
\(52\) 0 0
\(53\) 8.47214 1.16374 0.581869 0.813283i \(-0.302322\pi\)
0.581869 + 0.813283i \(0.302322\pi\)
\(54\) 0 0
\(55\) 3.23607 0.436351
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 0 0
\(59\) 1.52786 0.198911 0.0994555 0.995042i \(-0.468290\pi\)
0.0994555 + 0.995042i \(0.468290\pi\)
\(60\) 0 0
\(61\) −4.47214 −0.572598 −0.286299 0.958140i \(-0.592425\pi\)
−0.286299 + 0.958140i \(0.592425\pi\)
\(62\) 0 0
\(63\) −0.763932 −0.0962464
\(64\) 0 0
\(65\) −5.23607 −0.649454
\(66\) 0 0
\(67\) −10.4721 −1.27938 −0.639688 0.768635i \(-0.720936\pi\)
−0.639688 + 0.768635i \(0.720936\pi\)
\(68\) 0 0
\(69\) −6.47214 −0.779154
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 3.52786 0.412905 0.206453 0.978457i \(-0.433808\pi\)
0.206453 + 0.978457i \(0.433808\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 2.47214 0.281726
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 4.47214 0.485071
\(86\) 0 0
\(87\) −5.23607 −0.561365
\(88\) 0 0
\(89\) 11.7082 1.24107 0.620534 0.784180i \(-0.286916\pi\)
0.620534 + 0.784180i \(0.286916\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) 0 0
\(93\) 8.94427 0.927478
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −9.23607 −0.937781 −0.468890 0.883256i \(-0.655346\pi\)
−0.468890 + 0.883256i \(0.655346\pi\)
\(98\) 0 0
\(99\) −3.23607 −0.325237
\(100\) 0 0
\(101\) 13.4164 1.33498 0.667491 0.744618i \(-0.267368\pi\)
0.667491 + 0.744618i \(0.267368\pi\)
\(102\) 0 0
\(103\) −18.4721 −1.82011 −0.910057 0.414484i \(-0.863962\pi\)
−0.910057 + 0.414484i \(0.863962\pi\)
\(104\) 0 0
\(105\) −0.763932 −0.0745521
\(106\) 0 0
\(107\) −8.94427 −0.864675 −0.432338 0.901712i \(-0.642311\pi\)
−0.432338 + 0.901712i \(0.642311\pi\)
\(108\) 0 0
\(109\) 4.47214 0.428353 0.214176 0.976795i \(-0.431293\pi\)
0.214176 + 0.976795i \(0.431293\pi\)
\(110\) 0 0
\(111\) 1.23607 0.117322
\(112\) 0 0
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) −6.47214 −0.603530
\(116\) 0 0
\(117\) 5.23607 0.484075
\(118\) 0 0
\(119\) 3.41641 0.313182
\(120\) 0 0
\(121\) −0.527864 −0.0479876
\(122\) 0 0
\(123\) −3.70820 −0.334357
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −20.0000 −1.77471 −0.887357 0.461084i \(-0.847461\pi\)
−0.887357 + 0.461084i \(0.847461\pi\)
\(128\) 0 0
\(129\) 0.763932 0.0672605
\(130\) 0 0
\(131\) 8.18034 0.714720 0.357360 0.933967i \(-0.383677\pi\)
0.357360 + 0.933967i \(0.383677\pi\)
\(132\) 0 0
\(133\) 0.763932 0.0662413
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −19.8885 −1.69919 −0.849596 0.527433i \(-0.823154\pi\)
−0.849596 + 0.527433i \(0.823154\pi\)
\(138\) 0 0
\(139\) 5.52786 0.468867 0.234434 0.972132i \(-0.424676\pi\)
0.234434 + 0.972132i \(0.424676\pi\)
\(140\) 0 0
\(141\) −6.47214 −0.545052
\(142\) 0 0
\(143\) −16.9443 −1.41695
\(144\) 0 0
\(145\) −5.23607 −0.434832
\(146\) 0 0
\(147\) 6.41641 0.529216
\(148\) 0 0
\(149\) −9.41641 −0.771422 −0.385711 0.922620i \(-0.626044\pi\)
−0.385711 + 0.922620i \(0.626044\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 0 0
\(153\) −4.47214 −0.361551
\(154\) 0 0
\(155\) 8.94427 0.718421
\(156\) 0 0
\(157\) 5.41641 0.432276 0.216138 0.976363i \(-0.430654\pi\)
0.216138 + 0.976363i \(0.430654\pi\)
\(158\) 0 0
\(159\) −8.47214 −0.671884
\(160\) 0 0
\(161\) −4.94427 −0.389663
\(162\) 0 0
\(163\) 10.6525 0.834366 0.417183 0.908822i \(-0.363017\pi\)
0.417183 + 0.908822i \(0.363017\pi\)
\(164\) 0 0
\(165\) −3.23607 −0.251928
\(166\) 0 0
\(167\) 21.8885 1.69379 0.846893 0.531763i \(-0.178470\pi\)
0.846893 + 0.531763i \(0.178470\pi\)
\(168\) 0 0
\(169\) 14.4164 1.10895
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 0 0
\(173\) −22.0000 −1.67263 −0.836315 0.548250i \(-0.815294\pi\)
−0.836315 + 0.548250i \(0.815294\pi\)
\(174\) 0 0
\(175\) −0.763932 −0.0577478
\(176\) 0 0
\(177\) −1.52786 −0.114841
\(178\) 0 0
\(179\) −11.4164 −0.853302 −0.426651 0.904416i \(-0.640307\pi\)
−0.426651 + 0.904416i \(0.640307\pi\)
\(180\) 0 0
\(181\) −0.472136 −0.0350936 −0.0175468 0.999846i \(-0.505586\pi\)
−0.0175468 + 0.999846i \(0.505586\pi\)
\(182\) 0 0
\(183\) 4.47214 0.330590
\(184\) 0 0
\(185\) 1.23607 0.0908775
\(186\) 0 0
\(187\) 14.4721 1.05831
\(188\) 0 0
\(189\) 0.763932 0.0555679
\(190\) 0 0
\(191\) −16.1803 −1.17077 −0.585384 0.810756i \(-0.699056\pi\)
−0.585384 + 0.810756i \(0.699056\pi\)
\(192\) 0 0
\(193\) −7.70820 −0.554849 −0.277424 0.960747i \(-0.589481\pi\)
−0.277424 + 0.960747i \(0.589481\pi\)
\(194\) 0 0
\(195\) 5.23607 0.374963
\(196\) 0 0
\(197\) −3.52786 −0.251350 −0.125675 0.992071i \(-0.540110\pi\)
−0.125675 + 0.992071i \(0.540110\pi\)
\(198\) 0 0
\(199\) −9.52786 −0.675412 −0.337706 0.941252i \(-0.609651\pi\)
−0.337706 + 0.941252i \(0.609651\pi\)
\(200\) 0 0
\(201\) 10.4721 0.738648
\(202\) 0 0
\(203\) −4.00000 −0.280745
\(204\) 0 0
\(205\) −3.70820 −0.258992
\(206\) 0 0
\(207\) 6.47214 0.449845
\(208\) 0 0
\(209\) 3.23607 0.223844
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0.763932 0.0520997
\(216\) 0 0
\(217\) 6.83282 0.463842
\(218\) 0 0
\(219\) −3.52786 −0.238391
\(220\) 0 0
\(221\) −23.4164 −1.57516
\(222\) 0 0
\(223\) −13.5279 −0.905893 −0.452946 0.891538i \(-0.649627\pi\)
−0.452946 + 0.891538i \(0.649627\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 5.41641 0.357926 0.178963 0.983856i \(-0.442726\pi\)
0.178963 + 0.983856i \(0.442726\pi\)
\(230\) 0 0
\(231\) −2.47214 −0.162655
\(232\) 0 0
\(233\) −11.8885 −0.778844 −0.389422 0.921059i \(-0.627325\pi\)
−0.389422 + 0.921059i \(0.627325\pi\)
\(234\) 0 0
\(235\) −6.47214 −0.422196
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1.70820 −0.110495 −0.0552473 0.998473i \(-0.517595\pi\)
−0.0552473 + 0.998473i \(0.517595\pi\)
\(240\) 0 0
\(241\) −10.9443 −0.704983 −0.352491 0.935815i \(-0.614665\pi\)
−0.352491 + 0.935815i \(0.614665\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 6.41641 0.409929
\(246\) 0 0
\(247\) −5.23607 −0.333163
\(248\) 0 0
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) 9.70820 0.612776 0.306388 0.951907i \(-0.400879\pi\)
0.306388 + 0.951907i \(0.400879\pi\)
\(252\) 0 0
\(253\) −20.9443 −1.31676
\(254\) 0 0
\(255\) −4.47214 −0.280056
\(256\) 0 0
\(257\) −5.05573 −0.315368 −0.157684 0.987490i \(-0.550403\pi\)
−0.157684 + 0.987490i \(0.550403\pi\)
\(258\) 0 0
\(259\) 0.944272 0.0586742
\(260\) 0 0
\(261\) 5.23607 0.324104
\(262\) 0 0
\(263\) −20.9443 −1.29148 −0.645740 0.763558i \(-0.723451\pi\)
−0.645740 + 0.763558i \(0.723451\pi\)
\(264\) 0 0
\(265\) −8.47214 −0.520439
\(266\) 0 0
\(267\) −11.7082 −0.716530
\(268\) 0 0
\(269\) −7.70820 −0.469977 −0.234989 0.971998i \(-0.575505\pi\)
−0.234989 + 0.971998i \(0.575505\pi\)
\(270\) 0 0
\(271\) −6.47214 −0.393154 −0.196577 0.980488i \(-0.562983\pi\)
−0.196577 + 0.980488i \(0.562983\pi\)
\(272\) 0 0
\(273\) 4.00000 0.242091
\(274\) 0 0
\(275\) −3.23607 −0.195142
\(276\) 0 0
\(277\) −7.52786 −0.452306 −0.226153 0.974092i \(-0.572615\pi\)
−0.226153 + 0.974092i \(0.572615\pi\)
\(278\) 0 0
\(279\) −8.94427 −0.535480
\(280\) 0 0
\(281\) −10.7639 −0.642122 −0.321061 0.947058i \(-0.604040\pi\)
−0.321061 + 0.947058i \(0.604040\pi\)
\(282\) 0 0
\(283\) −30.0689 −1.78741 −0.893705 0.448655i \(-0.851903\pi\)
−0.893705 + 0.448655i \(0.851903\pi\)
\(284\) 0 0
\(285\) −1.00000 −0.0592349
\(286\) 0 0
\(287\) −2.83282 −0.167216
\(288\) 0 0
\(289\) 3.00000 0.176471
\(290\) 0 0
\(291\) 9.23607 0.541428
\(292\) 0 0
\(293\) −31.8885 −1.86295 −0.931474 0.363807i \(-0.881477\pi\)
−0.931474 + 0.363807i \(0.881477\pi\)
\(294\) 0 0
\(295\) −1.52786 −0.0889557
\(296\) 0 0
\(297\) 3.23607 0.187776
\(298\) 0 0
\(299\) 33.8885 1.95983
\(300\) 0 0
\(301\) 0.583592 0.0336377
\(302\) 0 0
\(303\) −13.4164 −0.770752
\(304\) 0 0
\(305\) 4.47214 0.256074
\(306\) 0 0
\(307\) 26.4721 1.51084 0.755422 0.655238i \(-0.227432\pi\)
0.755422 + 0.655238i \(0.227432\pi\)
\(308\) 0 0
\(309\) 18.4721 1.05084
\(310\) 0 0
\(311\) −6.29180 −0.356775 −0.178388 0.983960i \(-0.557088\pi\)
−0.178388 + 0.983960i \(0.557088\pi\)
\(312\) 0 0
\(313\) 11.8885 0.671980 0.335990 0.941866i \(-0.390929\pi\)
0.335990 + 0.941866i \(0.390929\pi\)
\(314\) 0 0
\(315\) 0.763932 0.0430427
\(316\) 0 0
\(317\) 3.52786 0.198145 0.0990723 0.995080i \(-0.468412\pi\)
0.0990723 + 0.995080i \(0.468412\pi\)
\(318\) 0 0
\(319\) −16.9443 −0.948697
\(320\) 0 0
\(321\) 8.94427 0.499221
\(322\) 0 0
\(323\) 4.47214 0.248836
\(324\) 0 0
\(325\) 5.23607 0.290445
\(326\) 0 0
\(327\) −4.47214 −0.247310
\(328\) 0 0
\(329\) −4.94427 −0.272587
\(330\) 0 0
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 0 0
\(333\) −1.23607 −0.0677361
\(334\) 0 0
\(335\) 10.4721 0.572154
\(336\) 0 0
\(337\) −33.2361 −1.81048 −0.905242 0.424896i \(-0.860311\pi\)
−0.905242 + 0.424896i \(0.860311\pi\)
\(338\) 0 0
\(339\) 2.00000 0.108625
\(340\) 0 0
\(341\) 28.9443 1.56742
\(342\) 0 0
\(343\) 10.2492 0.553406
\(344\) 0 0
\(345\) 6.47214 0.348448
\(346\) 0 0
\(347\) 20.0000 1.07366 0.536828 0.843692i \(-0.319622\pi\)
0.536828 + 0.843692i \(0.319622\pi\)
\(348\) 0 0
\(349\) −6.94427 −0.371718 −0.185859 0.982576i \(-0.559507\pi\)
−0.185859 + 0.982576i \(0.559507\pi\)
\(350\) 0 0
\(351\) −5.23607 −0.279481
\(352\) 0 0
\(353\) 2.94427 0.156708 0.0783539 0.996926i \(-0.475034\pi\)
0.0783539 + 0.996926i \(0.475034\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −3.41641 −0.180815
\(358\) 0 0
\(359\) −0.180340 −0.00951798 −0.00475899 0.999989i \(-0.501515\pi\)
−0.00475899 + 0.999989i \(0.501515\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0.527864 0.0277057
\(364\) 0 0
\(365\) −3.52786 −0.184657
\(366\) 0 0
\(367\) 24.7639 1.29267 0.646333 0.763055i \(-0.276302\pi\)
0.646333 + 0.763055i \(0.276302\pi\)
\(368\) 0 0
\(369\) 3.70820 0.193041
\(370\) 0 0
\(371\) −6.47214 −0.336017
\(372\) 0 0
\(373\) −2.76393 −0.143111 −0.0715555 0.997437i \(-0.522796\pi\)
−0.0715555 + 0.997437i \(0.522796\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 27.4164 1.41202
\(378\) 0 0
\(379\) 33.8885 1.74074 0.870369 0.492400i \(-0.163880\pi\)
0.870369 + 0.492400i \(0.163880\pi\)
\(380\) 0 0
\(381\) 20.0000 1.02463
\(382\) 0 0
\(383\) −25.8885 −1.32284 −0.661421 0.750014i \(-0.730046\pi\)
−0.661421 + 0.750014i \(0.730046\pi\)
\(384\) 0 0
\(385\) −2.47214 −0.125992
\(386\) 0 0
\(387\) −0.763932 −0.0388328
\(388\) 0 0
\(389\) −20.8328 −1.05627 −0.528133 0.849162i \(-0.677108\pi\)
−0.528133 + 0.849162i \(0.677108\pi\)
\(390\) 0 0
\(391\) −28.9443 −1.46377
\(392\) 0 0
\(393\) −8.18034 −0.412644
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −33.4164 −1.67712 −0.838561 0.544808i \(-0.816602\pi\)
−0.838561 + 0.544808i \(0.816602\pi\)
\(398\) 0 0
\(399\) −0.763932 −0.0382444
\(400\) 0 0
\(401\) −18.7639 −0.937026 −0.468513 0.883457i \(-0.655210\pi\)
−0.468513 + 0.883457i \(0.655210\pi\)
\(402\) 0 0
\(403\) −46.8328 −2.33291
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 4.00000 0.198273
\(408\) 0 0
\(409\) −4.47214 −0.221133 −0.110566 0.993869i \(-0.535266\pi\)
−0.110566 + 0.993869i \(0.535266\pi\)
\(410\) 0 0
\(411\) 19.8885 0.981030
\(412\) 0 0
\(413\) −1.16718 −0.0574334
\(414\) 0 0
\(415\) 4.00000 0.196352
\(416\) 0 0
\(417\) −5.52786 −0.270701
\(418\) 0 0
\(419\) −0.180340 −0.00881018 −0.00440509 0.999990i \(-0.501402\pi\)
−0.00440509 + 0.999990i \(0.501402\pi\)
\(420\) 0 0
\(421\) 28.8328 1.40523 0.702613 0.711572i \(-0.252017\pi\)
0.702613 + 0.711572i \(0.252017\pi\)
\(422\) 0 0
\(423\) 6.47214 0.314686
\(424\) 0 0
\(425\) −4.47214 −0.216930
\(426\) 0 0
\(427\) 3.41641 0.165332
\(428\) 0 0
\(429\) 16.9443 0.818077
\(430\) 0 0
\(431\) 37.3050 1.79692 0.898458 0.439059i \(-0.144688\pi\)
0.898458 + 0.439059i \(0.144688\pi\)
\(432\) 0 0
\(433\) 5.23607 0.251629 0.125815 0.992054i \(-0.459846\pi\)
0.125815 + 0.992054i \(0.459846\pi\)
\(434\) 0 0
\(435\) 5.23607 0.251050
\(436\) 0 0
\(437\) −6.47214 −0.309604
\(438\) 0 0
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) −6.41641 −0.305543
\(442\) 0 0
\(443\) 7.41641 0.352364 0.176182 0.984358i \(-0.443625\pi\)
0.176182 + 0.984358i \(0.443625\pi\)
\(444\) 0 0
\(445\) −11.7082 −0.555022
\(446\) 0 0
\(447\) 9.41641 0.445381
\(448\) 0 0
\(449\) 3.70820 0.175001 0.0875005 0.996164i \(-0.472112\pi\)
0.0875005 + 0.996164i \(0.472112\pi\)
\(450\) 0 0
\(451\) −12.0000 −0.565058
\(452\) 0 0
\(453\) 12.0000 0.563809
\(454\) 0 0
\(455\) 4.00000 0.187523
\(456\) 0 0
\(457\) −36.8328 −1.72297 −0.861483 0.507786i \(-0.830464\pi\)
−0.861483 + 0.507786i \(0.830464\pi\)
\(458\) 0 0
\(459\) 4.47214 0.208741
\(460\) 0 0
\(461\) 0.472136 0.0219896 0.0109948 0.999940i \(-0.496500\pi\)
0.0109948 + 0.999940i \(0.496500\pi\)
\(462\) 0 0
\(463\) 23.5967 1.09663 0.548317 0.836271i \(-0.315269\pi\)
0.548317 + 0.836271i \(0.315269\pi\)
\(464\) 0 0
\(465\) −8.94427 −0.414781
\(466\) 0 0
\(467\) 31.4164 1.45378 0.726889 0.686755i \(-0.240965\pi\)
0.726889 + 0.686755i \(0.240965\pi\)
\(468\) 0 0
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) −5.41641 −0.249575
\(472\) 0 0
\(473\) 2.47214 0.113669
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) 8.47214 0.387912
\(478\) 0 0
\(479\) −8.18034 −0.373769 −0.186885 0.982382i \(-0.559839\pi\)
−0.186885 + 0.982382i \(0.559839\pi\)
\(480\) 0 0
\(481\) −6.47214 −0.295104
\(482\) 0 0
\(483\) 4.94427 0.224972
\(484\) 0 0
\(485\) 9.23607 0.419388
\(486\) 0 0
\(487\) −31.4164 −1.42361 −0.711807 0.702375i \(-0.752123\pi\)
−0.711807 + 0.702375i \(0.752123\pi\)
\(488\) 0 0
\(489\) −10.6525 −0.481722
\(490\) 0 0
\(491\) 1.34752 0.0608129 0.0304065 0.999538i \(-0.490320\pi\)
0.0304065 + 0.999538i \(0.490320\pi\)
\(492\) 0 0
\(493\) −23.4164 −1.05462
\(494\) 0 0
\(495\) 3.23607 0.145450
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −15.4164 −0.690133 −0.345067 0.938578i \(-0.612144\pi\)
−0.345067 + 0.938578i \(0.612144\pi\)
\(500\) 0 0
\(501\) −21.8885 −0.977908
\(502\) 0 0
\(503\) −4.94427 −0.220454 −0.110227 0.993906i \(-0.535158\pi\)
−0.110227 + 0.993906i \(0.535158\pi\)
\(504\) 0 0
\(505\) −13.4164 −0.597022
\(506\) 0 0
\(507\) −14.4164 −0.640255
\(508\) 0 0
\(509\) 43.7082 1.93733 0.968666 0.248367i \(-0.0798939\pi\)
0.968666 + 0.248367i \(0.0798939\pi\)
\(510\) 0 0
\(511\) −2.69505 −0.119222
\(512\) 0 0
\(513\) 1.00000 0.0441511
\(514\) 0 0
\(515\) 18.4721 0.813980
\(516\) 0 0
\(517\) −20.9443 −0.921128
\(518\) 0 0
\(519\) 22.0000 0.965693
\(520\) 0 0
\(521\) −39.7082 −1.73965 −0.869824 0.493362i \(-0.835768\pi\)
−0.869824 + 0.493362i \(0.835768\pi\)
\(522\) 0 0
\(523\) −16.9443 −0.740921 −0.370461 0.928848i \(-0.620800\pi\)
−0.370461 + 0.928848i \(0.620800\pi\)
\(524\) 0 0
\(525\) 0.763932 0.0333407
\(526\) 0 0
\(527\) 40.0000 1.74243
\(528\) 0 0
\(529\) 18.8885 0.821241
\(530\) 0 0
\(531\) 1.52786 0.0663037
\(532\) 0 0
\(533\) 19.4164 0.841018
\(534\) 0 0
\(535\) 8.94427 0.386695
\(536\) 0 0
\(537\) 11.4164 0.492654
\(538\) 0 0
\(539\) 20.7639 0.894366
\(540\) 0 0
\(541\) 41.7771 1.79614 0.898069 0.439855i \(-0.144970\pi\)
0.898069 + 0.439855i \(0.144970\pi\)
\(542\) 0 0
\(543\) 0.472136 0.0202613
\(544\) 0 0
\(545\) −4.47214 −0.191565
\(546\) 0 0
\(547\) −25.3050 −1.08196 −0.540981 0.841035i \(-0.681947\pi\)
−0.540981 + 0.841035i \(0.681947\pi\)
\(548\) 0 0
\(549\) −4.47214 −0.190866
\(550\) 0 0
\(551\) −5.23607 −0.223064
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −1.23607 −0.0524682
\(556\) 0 0
\(557\) −20.8328 −0.882715 −0.441357 0.897331i \(-0.645503\pi\)
−0.441357 + 0.897331i \(0.645503\pi\)
\(558\) 0 0
\(559\) −4.00000 −0.169182
\(560\) 0 0
\(561\) −14.4721 −0.611014
\(562\) 0 0
\(563\) −25.8885 −1.09107 −0.545536 0.838087i \(-0.683674\pi\)
−0.545536 + 0.838087i \(0.683674\pi\)
\(564\) 0 0
\(565\) 2.00000 0.0841406
\(566\) 0 0
\(567\) −0.763932 −0.0320821
\(568\) 0 0
\(569\) 34.5410 1.44803 0.724017 0.689782i \(-0.242293\pi\)
0.724017 + 0.689782i \(0.242293\pi\)
\(570\) 0 0
\(571\) −41.3050 −1.72856 −0.864279 0.503012i \(-0.832225\pi\)
−0.864279 + 0.503012i \(0.832225\pi\)
\(572\) 0 0
\(573\) 16.1803 0.675943
\(574\) 0 0
\(575\) 6.47214 0.269907
\(576\) 0 0
\(577\) −26.9443 −1.12170 −0.560852 0.827916i \(-0.689526\pi\)
−0.560852 + 0.827916i \(0.689526\pi\)
\(578\) 0 0
\(579\) 7.70820 0.320342
\(580\) 0 0
\(581\) 3.05573 0.126773
\(582\) 0 0
\(583\) −27.4164 −1.13547
\(584\) 0 0
\(585\) −5.23607 −0.216485
\(586\) 0 0
\(587\) −15.4164 −0.636303 −0.318152 0.948040i \(-0.603062\pi\)
−0.318152 + 0.948040i \(0.603062\pi\)
\(588\) 0 0
\(589\) 8.94427 0.368542
\(590\) 0 0
\(591\) 3.52786 0.145117
\(592\) 0 0
\(593\) −2.00000 −0.0821302 −0.0410651 0.999156i \(-0.513075\pi\)
−0.0410651 + 0.999156i \(0.513075\pi\)
\(594\) 0 0
\(595\) −3.41641 −0.140059
\(596\) 0 0
\(597\) 9.52786 0.389950
\(598\) 0 0
\(599\) −20.5836 −0.841023 −0.420511 0.907287i \(-0.638149\pi\)
−0.420511 + 0.907287i \(0.638149\pi\)
\(600\) 0 0
\(601\) 34.3607 1.40160 0.700801 0.713357i \(-0.252826\pi\)
0.700801 + 0.713357i \(0.252826\pi\)
\(602\) 0 0
\(603\) −10.4721 −0.426458
\(604\) 0 0
\(605\) 0.527864 0.0214607
\(606\) 0 0
\(607\) 42.4721 1.72389 0.861945 0.507001i \(-0.169246\pi\)
0.861945 + 0.507001i \(0.169246\pi\)
\(608\) 0 0
\(609\) 4.00000 0.162088
\(610\) 0 0
\(611\) 33.8885 1.37098
\(612\) 0 0
\(613\) −44.4721 −1.79621 −0.898106 0.439778i \(-0.855057\pi\)
−0.898106 + 0.439778i \(0.855057\pi\)
\(614\) 0 0
\(615\) 3.70820 0.149529
\(616\) 0 0
\(617\) 23.3050 0.938222 0.469111 0.883139i \(-0.344574\pi\)
0.469111 + 0.883139i \(0.344574\pi\)
\(618\) 0 0
\(619\) 12.3607 0.496818 0.248409 0.968655i \(-0.420092\pi\)
0.248409 + 0.968655i \(0.420092\pi\)
\(620\) 0 0
\(621\) −6.47214 −0.259718
\(622\) 0 0
\(623\) −8.94427 −0.358345
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −3.23607 −0.129236
\(628\) 0 0
\(629\) 5.52786 0.220410
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 0 0
\(633\) −12.0000 −0.476957
\(634\) 0 0
\(635\) 20.0000 0.793676
\(636\) 0 0
\(637\) −33.5967 −1.33115
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −49.2361 −1.94471 −0.972354 0.233512i \(-0.924978\pi\)
−0.972354 + 0.233512i \(0.924978\pi\)
\(642\) 0 0
\(643\) 31.5967 1.24605 0.623027 0.782200i \(-0.285903\pi\)
0.623027 + 0.782200i \(0.285903\pi\)
\(644\) 0 0
\(645\) −0.763932 −0.0300798
\(646\) 0 0
\(647\) 38.8328 1.52668 0.763338 0.646000i \(-0.223559\pi\)
0.763338 + 0.646000i \(0.223559\pi\)
\(648\) 0 0
\(649\) −4.94427 −0.194080
\(650\) 0 0
\(651\) −6.83282 −0.267799
\(652\) 0 0
\(653\) −40.4721 −1.58380 −0.791899 0.610653i \(-0.790907\pi\)
−0.791899 + 0.610653i \(0.790907\pi\)
\(654\) 0 0
\(655\) −8.18034 −0.319632
\(656\) 0 0
\(657\) 3.52786 0.137635
\(658\) 0 0
\(659\) 22.8328 0.889440 0.444720 0.895670i \(-0.353303\pi\)
0.444720 + 0.895670i \(0.353303\pi\)
\(660\) 0 0
\(661\) −39.3050 −1.52879 −0.764393 0.644751i \(-0.776961\pi\)
−0.764393 + 0.644751i \(0.776961\pi\)
\(662\) 0 0
\(663\) 23.4164 0.909418
\(664\) 0 0
\(665\) −0.763932 −0.0296240
\(666\) 0 0
\(667\) 33.8885 1.31217
\(668\) 0 0
\(669\) 13.5279 0.523017
\(670\) 0 0
\(671\) 14.4721 0.558691
\(672\) 0 0
\(673\) 21.5967 0.832493 0.416247 0.909252i \(-0.363345\pi\)
0.416247 + 0.909252i \(0.363345\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 24.4721 0.940541 0.470270 0.882522i \(-0.344156\pi\)
0.470270 + 0.882522i \(0.344156\pi\)
\(678\) 0 0
\(679\) 7.05573 0.270774
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 39.7771 1.52203 0.761014 0.648735i \(-0.224702\pi\)
0.761014 + 0.648735i \(0.224702\pi\)
\(684\) 0 0
\(685\) 19.8885 0.759902
\(686\) 0 0
\(687\) −5.41641 −0.206649
\(688\) 0 0
\(689\) 44.3607 1.69001
\(690\) 0 0
\(691\) 34.4721 1.31138 0.655691 0.755029i \(-0.272377\pi\)
0.655691 + 0.755029i \(0.272377\pi\)
\(692\) 0 0
\(693\) 2.47214 0.0939087
\(694\) 0 0
\(695\) −5.52786 −0.209684
\(696\) 0 0
\(697\) −16.5836 −0.628148
\(698\) 0 0
\(699\) 11.8885 0.449666
\(700\) 0 0
\(701\) −41.4164 −1.56428 −0.782138 0.623105i \(-0.785871\pi\)
−0.782138 + 0.623105i \(0.785871\pi\)
\(702\) 0 0
\(703\) 1.23607 0.0466192
\(704\) 0 0
\(705\) 6.47214 0.243755
\(706\) 0 0
\(707\) −10.2492 −0.385462
\(708\) 0 0
\(709\) 31.3050 1.17568 0.587841 0.808976i \(-0.299978\pi\)
0.587841 + 0.808976i \(0.299978\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −57.8885 −2.16794
\(714\) 0 0
\(715\) 16.9443 0.633680
\(716\) 0 0
\(717\) 1.70820 0.0637940
\(718\) 0 0
\(719\) 3.23607 0.120685 0.0603425 0.998178i \(-0.480781\pi\)
0.0603425 + 0.998178i \(0.480781\pi\)
\(720\) 0 0
\(721\) 14.1115 0.525538
\(722\) 0 0
\(723\) 10.9443 0.407022
\(724\) 0 0
\(725\) 5.23607 0.194463
\(726\) 0 0
\(727\) 10.6525 0.395078 0.197539 0.980295i \(-0.436705\pi\)
0.197539 + 0.980295i \(0.436705\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 3.41641 0.126360
\(732\) 0 0
\(733\) 16.8328 0.621734 0.310867 0.950453i \(-0.399381\pi\)
0.310867 + 0.950453i \(0.399381\pi\)
\(734\) 0 0
\(735\) −6.41641 −0.236673
\(736\) 0 0
\(737\) 33.8885 1.24830
\(738\) 0 0
\(739\) −8.94427 −0.329020 −0.164510 0.986375i \(-0.552604\pi\)
−0.164510 + 0.986375i \(0.552604\pi\)
\(740\) 0 0
\(741\) 5.23607 0.192352
\(742\) 0 0
\(743\) 27.0557 0.992578 0.496289 0.868157i \(-0.334696\pi\)
0.496289 + 0.868157i \(0.334696\pi\)
\(744\) 0 0
\(745\) 9.41641 0.344990
\(746\) 0 0
\(747\) −4.00000 −0.146352
\(748\) 0 0
\(749\) 6.83282 0.249666
\(750\) 0 0
\(751\) 8.94427 0.326381 0.163191 0.986595i \(-0.447821\pi\)
0.163191 + 0.986595i \(0.447821\pi\)
\(752\) 0 0
\(753\) −9.70820 −0.353787
\(754\) 0 0
\(755\) 12.0000 0.436725
\(756\) 0 0
\(757\) 14.9443 0.543159 0.271579 0.962416i \(-0.412454\pi\)
0.271579 + 0.962416i \(0.412454\pi\)
\(758\) 0 0
\(759\) 20.9443 0.760229
\(760\) 0 0
\(761\) −17.4164 −0.631344 −0.315672 0.948868i \(-0.602230\pi\)
−0.315672 + 0.948868i \(0.602230\pi\)
\(762\) 0 0
\(763\) −3.41641 −0.123682
\(764\) 0 0
\(765\) 4.47214 0.161690
\(766\) 0 0
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) 43.8885 1.58266 0.791331 0.611388i \(-0.209389\pi\)
0.791331 + 0.611388i \(0.209389\pi\)
\(770\) 0 0
\(771\) 5.05573 0.182078
\(772\) 0 0
\(773\) 48.4721 1.74342 0.871711 0.490021i \(-0.163011\pi\)
0.871711 + 0.490021i \(0.163011\pi\)
\(774\) 0 0
\(775\) −8.94427 −0.321288
\(776\) 0 0
\(777\) −0.944272 −0.0338756
\(778\) 0 0
\(779\) −3.70820 −0.132860
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −5.23607 −0.187122
\(784\) 0 0
\(785\) −5.41641 −0.193320
\(786\) 0 0
\(787\) 42.8328 1.52682 0.763412 0.645911i \(-0.223522\pi\)
0.763412 + 0.645911i \(0.223522\pi\)
\(788\) 0 0
\(789\) 20.9443 0.745636
\(790\) 0 0
\(791\) 1.52786 0.0543246
\(792\) 0 0
\(793\) −23.4164 −0.831541
\(794\) 0 0
\(795\) 8.47214 0.300476
\(796\) 0 0
\(797\) −33.7771 −1.19645 −0.598223 0.801330i \(-0.704126\pi\)
−0.598223 + 0.801330i \(0.704126\pi\)
\(798\) 0 0
\(799\) −28.9443 −1.02397
\(800\) 0 0
\(801\) 11.7082 0.413689
\(802\) 0 0
\(803\) −11.4164 −0.402876
\(804\) 0 0
\(805\) 4.94427 0.174263
\(806\) 0 0
\(807\) 7.70820 0.271342
\(808\) 0 0
\(809\) 21.0557 0.740280 0.370140 0.928976i \(-0.379310\pi\)
0.370140 + 0.928976i \(0.379310\pi\)
\(810\) 0 0
\(811\) 16.9443 0.594994 0.297497 0.954723i \(-0.403848\pi\)
0.297497 + 0.954723i \(0.403848\pi\)
\(812\) 0 0
\(813\) 6.47214 0.226988
\(814\) 0 0
\(815\) −10.6525 −0.373140
\(816\) 0 0
\(817\) 0.763932 0.0267266
\(818\) 0 0
\(819\) −4.00000 −0.139771
\(820\) 0 0
\(821\) 24.4721 0.854083 0.427042 0.904232i \(-0.359556\pi\)
0.427042 + 0.904232i \(0.359556\pi\)
\(822\) 0 0
\(823\) −20.1803 −0.703442 −0.351721 0.936105i \(-0.614403\pi\)
−0.351721 + 0.936105i \(0.614403\pi\)
\(824\) 0 0
\(825\) 3.23607 0.112665
\(826\) 0 0
\(827\) −8.00000 −0.278187 −0.139094 0.990279i \(-0.544419\pi\)
−0.139094 + 0.990279i \(0.544419\pi\)
\(828\) 0 0
\(829\) −15.3050 −0.531563 −0.265781 0.964033i \(-0.585630\pi\)
−0.265781 + 0.964033i \(0.585630\pi\)
\(830\) 0 0
\(831\) 7.52786 0.261139
\(832\) 0 0
\(833\) 28.6950 0.994224
\(834\) 0 0
\(835\) −21.8885 −0.757484
\(836\) 0 0
\(837\) 8.94427 0.309159
\(838\) 0 0
\(839\) −40.0000 −1.38095 −0.690477 0.723355i \(-0.742599\pi\)
−0.690477 + 0.723355i \(0.742599\pi\)
\(840\) 0 0
\(841\) −1.58359 −0.0546066
\(842\) 0 0
\(843\) 10.7639 0.370730
\(844\) 0 0
\(845\) −14.4164 −0.495940
\(846\) 0 0
\(847\) 0.403252 0.0138559
\(848\) 0 0
\(849\) 30.0689 1.03196
\(850\) 0 0
\(851\) −8.00000 −0.274236
\(852\) 0 0
\(853\) −46.7214 −1.59971 −0.799854 0.600194i \(-0.795090\pi\)
−0.799854 + 0.600194i \(0.795090\pi\)
\(854\) 0 0
\(855\) 1.00000 0.0341993
\(856\) 0 0
\(857\) 15.5279 0.530422 0.265211 0.964190i \(-0.414558\pi\)
0.265211 + 0.964190i \(0.414558\pi\)
\(858\) 0 0
\(859\) 42.8328 1.46144 0.730718 0.682679i \(-0.239185\pi\)
0.730718 + 0.682679i \(0.239185\pi\)
\(860\) 0 0
\(861\) 2.83282 0.0965421
\(862\) 0 0
\(863\) 6.83282 0.232592 0.116296 0.993215i \(-0.462898\pi\)
0.116296 + 0.993215i \(0.462898\pi\)
\(864\) 0 0
\(865\) 22.0000 0.748022
\(866\) 0 0
\(867\) −3.00000 −0.101885
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −54.8328 −1.85794
\(872\) 0 0
\(873\) −9.23607 −0.312594
\(874\) 0 0
\(875\) 0.763932 0.0258256
\(876\) 0 0
\(877\) 24.2918 0.820276 0.410138 0.912024i \(-0.365481\pi\)
0.410138 + 0.912024i \(0.365481\pi\)
\(878\) 0 0
\(879\) 31.8885 1.07557
\(880\) 0 0
\(881\) 7.30495 0.246110 0.123055 0.992400i \(-0.460731\pi\)
0.123055 + 0.992400i \(0.460731\pi\)
\(882\) 0 0
\(883\) 23.5967 0.794094 0.397047 0.917798i \(-0.370035\pi\)
0.397047 + 0.917798i \(0.370035\pi\)
\(884\) 0 0
\(885\) 1.52786 0.0513586
\(886\) 0 0
\(887\) −9.88854 −0.332025 −0.166012 0.986124i \(-0.553089\pi\)
−0.166012 + 0.986124i \(0.553089\pi\)
\(888\) 0 0
\(889\) 15.2786 0.512429
\(890\) 0 0
\(891\) −3.23607 −0.108412
\(892\) 0 0
\(893\) −6.47214 −0.216582
\(894\) 0 0
\(895\) 11.4164 0.381608
\(896\) 0 0
\(897\) −33.8885 −1.13151
\(898\) 0 0
\(899\) −46.8328 −1.56196
\(900\) 0 0
\(901\) −37.8885 −1.26225
\(902\) 0 0
\(903\) −0.583592 −0.0194207
\(904\) 0 0
\(905\) 0.472136 0.0156943
\(906\) 0 0
\(907\) −45.8885 −1.52370 −0.761852 0.647751i \(-0.775710\pi\)
−0.761852 + 0.647751i \(0.775710\pi\)
\(908\) 0 0
\(909\) 13.4164 0.444994
\(910\) 0 0
\(911\) −27.7771 −0.920296 −0.460148 0.887842i \(-0.652204\pi\)
−0.460148 + 0.887842i \(0.652204\pi\)
\(912\) 0 0
\(913\) 12.9443 0.428393
\(914\) 0 0
\(915\) −4.47214 −0.147844
\(916\) 0 0
\(917\) −6.24922 −0.206368
\(918\) 0 0
\(919\) 24.0000 0.791687 0.395843 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(920\) 0 0
\(921\) −26.4721 −0.872287
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −1.23607 −0.0406417
\(926\) 0 0
\(927\) −18.4721 −0.606705
\(928\) 0 0
\(929\) −7.88854 −0.258815 −0.129407 0.991592i \(-0.541307\pi\)
−0.129407 + 0.991592i \(0.541307\pi\)
\(930\) 0 0
\(931\) 6.41641 0.210289
\(932\) 0 0
\(933\) 6.29180 0.205984
\(934\) 0 0
\(935\) −14.4721 −0.473289
\(936\) 0 0
\(937\) −10.5836 −0.345751 −0.172875 0.984944i \(-0.555306\pi\)
−0.172875 + 0.984944i \(0.555306\pi\)
\(938\) 0 0
\(939\) −11.8885 −0.387968
\(940\) 0 0
\(941\) 23.1246 0.753841 0.376920 0.926246i \(-0.376983\pi\)
0.376920 + 0.926246i \(0.376983\pi\)
\(942\) 0 0
\(943\) 24.0000 0.781548
\(944\) 0 0
\(945\) −0.763932 −0.0248507
\(946\) 0 0
\(947\) −4.00000 −0.129983 −0.0649913 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) 0 0
\(949\) 18.4721 0.599631
\(950\) 0 0
\(951\) −3.52786 −0.114399
\(952\) 0 0
\(953\) 25.0557 0.811635 0.405817 0.913954i \(-0.366987\pi\)
0.405817 + 0.913954i \(0.366987\pi\)
\(954\) 0 0
\(955\) 16.1803 0.523584
\(956\) 0 0
\(957\) 16.9443 0.547731
\(958\) 0 0
\(959\) 15.1935 0.490624
\(960\) 0 0
\(961\) 49.0000 1.58065
\(962\) 0 0
\(963\) −8.94427 −0.288225
\(964\) 0 0
\(965\) 7.70820 0.248136
\(966\) 0 0
\(967\) −4.18034 −0.134431 −0.0672153 0.997738i \(-0.521411\pi\)
−0.0672153 + 0.997738i \(0.521411\pi\)
\(968\) 0 0
\(969\) −4.47214 −0.143666
\(970\) 0 0
\(971\) −14.8328 −0.476008 −0.238004 0.971264i \(-0.576493\pi\)
−0.238004 + 0.971264i \(0.576493\pi\)
\(972\) 0 0
\(973\) −4.22291 −0.135380
\(974\) 0 0
\(975\) −5.23607 −0.167688
\(976\) 0 0
\(977\) 57.4164 1.83691 0.918457 0.395521i \(-0.129436\pi\)
0.918457 + 0.395521i \(0.129436\pi\)
\(978\) 0 0
\(979\) −37.8885 −1.21092
\(980\) 0 0
\(981\) 4.47214 0.142784
\(982\) 0 0
\(983\) −4.00000 −0.127580 −0.0637901 0.997963i \(-0.520319\pi\)
−0.0637901 + 0.997963i \(0.520319\pi\)
\(984\) 0 0
\(985\) 3.52786 0.112407
\(986\) 0 0
\(987\) 4.94427 0.157378
\(988\) 0 0
\(989\) −4.94427 −0.157219
\(990\) 0 0
\(991\) −11.0557 −0.351197 −0.175598 0.984462i \(-0.556186\pi\)
−0.175598 + 0.984462i \(0.556186\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 9.52786 0.302054
\(996\) 0 0
\(997\) −25.4164 −0.804946 −0.402473 0.915432i \(-0.631849\pi\)
−0.402473 + 0.915432i \(0.631849\pi\)
\(998\) 0 0
\(999\) 1.23607 0.0391075
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 4560.2.a.be.1.2 2
4.3 odd 2 2280.2.a.p.1.1 2
12.11 even 2 6840.2.a.bd.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2280.2.a.p.1.1 2 4.3 odd 2
4560.2.a.be.1.2 2 1.1 even 1 trivial
6840.2.a.bd.1.1 2 12.11 even 2