Properties

Label 1216.2.a.w.1.4
Level $1216$
Weight $2$
Character 1216.1
Self dual yes
Analytic conductor $9.710$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.70980888579\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.15317.1
Defining polynomial: \(x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 608)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.329727\) of defining polynomial
Character \(\chi\) \(=\) 1216.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.56155 q^{3} +0.844614 q^{5} -5.06562 q^{7} -0.561553 q^{9} +O(q^{10})\) \(q+1.56155 q^{3} +0.844614 q^{5} -5.06562 q^{7} -0.561553 q^{9} -2.84461 q^{11} -2.90210 q^{13} +1.31891 q^{15} +7.72508 q^{17} -1.00000 q^{19} -7.91023 q^{21} +3.91023 q^{23} -4.28663 q^{25} -5.56155 q^{27} -2.90210 q^{29} -7.00814 q^{31} -4.44201 q^{33} -4.27849 q^{35} -9.00814 q^{37} -4.53178 q^{39} -6.81233 q^{41} -5.96772 q^{43} -0.474295 q^{45} -1.04042 q^{47} +18.6605 q^{49} +12.0631 q^{51} +9.03334 q^{53} -2.40260 q^{55} -1.56155 q^{57} -0.749220 q^{59} -2.27849 q^{61} +2.84461 q^{63} -2.45115 q^{65} -10.3739 q^{67} +6.10604 q^{69} +2.13124 q^{71} +4.60197 q^{73} -6.69380 q^{75} +14.4097 q^{77} +3.88503 q^{79} -7.00000 q^{81} -8.44201 q^{83} +6.52470 q^{85} -4.53178 q^{87} +16.0667 q^{89} +14.7009 q^{91} -10.9436 q^{93} -0.844614 q^{95} -3.68923 q^{97} +1.59740 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{3} - q^{5} - q^{7} + 6q^{9} + O(q^{10}) \) \( 4q - 2q^{3} - q^{5} - q^{7} + 6q^{9} - 7q^{11} - 10q^{13} - 8q^{15} + 5q^{17} - 4q^{19} - 8q^{21} - 8q^{23} + 17q^{25} - 14q^{27} - 10q^{29} - 6q^{31} + 12q^{33} - 5q^{35} - 14q^{37} - 12q^{39} - 2q^{41} - 3q^{43} + 7q^{45} - 3q^{47} + 7q^{49} + 6q^{51} - 4q^{53} - 35q^{55} + 2q^{57} - 20q^{59} + 3q^{61} + 7q^{63} - 12q^{65} - 8q^{67} + 4q^{69} - 30q^{71} + 9q^{73} - 34q^{75} + 7q^{77} + 10q^{79} - 28q^{81} - 4q^{83} - 19q^{85} - 12q^{87} - 16q^{89} - 10q^{91} + 20q^{93} + q^{95} - 6q^{97} - 19q^{99} + O(q^{100}) \)

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
\(3\) 1.56155 0.901563 0.450781 0.892634i \(-0.351145\pi\)
0.450781 + 0.892634i \(0.351145\pi\)
\(4\) 0 0
\(5\) 0.844614 0.377723 0.188861 0.982004i \(-0.439520\pi\)
0.188861 + 0.982004i \(0.439520\pi\)
\(6\) 0 0
\(7\) −5.06562 −1.91462 −0.957312 0.289056i \(-0.906659\pi\)
−0.957312 + 0.289056i \(0.906659\pi\)
\(8\) 0 0
\(9\) −0.561553 −0.187184
\(10\) 0 0
\(11\) −2.84461 −0.857683 −0.428842 0.903380i \(-0.641078\pi\)
−0.428842 + 0.903380i \(0.641078\pi\)
\(12\) 0 0
\(13\) −2.90210 −0.804897 −0.402449 0.915443i \(-0.631841\pi\)
−0.402449 + 0.915443i \(0.631841\pi\)
\(14\) 0 0
\(15\) 1.31891 0.340541
\(16\) 0 0
\(17\) 7.72508 1.87361 0.936803 0.349857i \(-0.113770\pi\)
0.936803 + 0.349857i \(0.113770\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −7.91023 −1.72615
\(22\) 0 0
\(23\) 3.91023 0.815340 0.407670 0.913129i \(-0.366341\pi\)
0.407670 + 0.913129i \(0.366341\pi\)
\(24\) 0 0
\(25\) −4.28663 −0.857326
\(26\) 0 0
\(27\) −5.56155 −1.07032
\(28\) 0 0
\(29\) −2.90210 −0.538906 −0.269453 0.963014i \(-0.586843\pi\)
−0.269453 + 0.963014i \(0.586843\pi\)
\(30\) 0 0
\(31\) −7.00814 −1.25870 −0.629349 0.777123i \(-0.716678\pi\)
−0.629349 + 0.777123i \(0.716678\pi\)
\(32\) 0 0
\(33\) −4.44201 −0.773255
\(34\) 0 0
\(35\) −4.27849 −0.723197
\(36\) 0 0
\(37\) −9.00814 −1.48093 −0.740464 0.672096i \(-0.765394\pi\)
−0.740464 + 0.672096i \(0.765394\pi\)
\(38\) 0 0
\(39\) −4.53178 −0.725666
\(40\) 0 0
\(41\) −6.81233 −1.06391 −0.531954 0.846773i \(-0.678542\pi\)
−0.531954 + 0.846773i \(0.678542\pi\)
\(42\) 0 0
\(43\) −5.96772 −0.910069 −0.455034 0.890474i \(-0.650373\pi\)
−0.455034 + 0.890474i \(0.650373\pi\)
\(44\) 0 0
\(45\) −0.474295 −0.0707037
\(46\) 0 0
\(47\) −1.04042 −0.151760 −0.0758802 0.997117i \(-0.524177\pi\)
−0.0758802 + 0.997117i \(0.524177\pi\)
\(48\) 0 0
\(49\) 18.6605 2.66579
\(50\) 0 0
\(51\) 12.0631 1.68917
\(52\) 0 0
\(53\) 9.03334 1.24082 0.620412 0.784276i \(-0.286965\pi\)
0.620412 + 0.784276i \(0.286965\pi\)
\(54\) 0 0
\(55\) −2.40260 −0.323966
\(56\) 0 0
\(57\) −1.56155 −0.206833
\(58\) 0 0
\(59\) −0.749220 −0.0975401 −0.0487701 0.998810i \(-0.515530\pi\)
−0.0487701 + 0.998810i \(0.515530\pi\)
\(60\) 0 0
\(61\) −2.27849 −0.291731 −0.145866 0.989304i \(-0.546597\pi\)
−0.145866 + 0.989304i \(0.546597\pi\)
\(62\) 0 0
\(63\) 2.84461 0.358388
\(64\) 0 0
\(65\) −2.45115 −0.304028
\(66\) 0 0
\(67\) −10.3739 −1.26737 −0.633686 0.773590i \(-0.718459\pi\)
−0.633686 + 0.773590i \(0.718459\pi\)
\(68\) 0 0
\(69\) 6.10604 0.735081
\(70\) 0 0
\(71\) 2.13124 0.252932 0.126466 0.991971i \(-0.459637\pi\)
0.126466 + 0.991971i \(0.459637\pi\)
\(72\) 0 0
\(73\) 4.60197 0.538620 0.269310 0.963054i \(-0.413204\pi\)
0.269310 + 0.963054i \(0.413204\pi\)
\(74\) 0 0
\(75\) −6.69380 −0.772933
\(76\) 0 0
\(77\) 14.4097 1.64214
\(78\) 0 0
\(79\) 3.88503 0.437100 0.218550 0.975826i \(-0.429867\pi\)
0.218550 + 0.975826i \(0.429867\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) −8.44201 −0.926631 −0.463316 0.886193i \(-0.653340\pi\)
−0.463316 + 0.886193i \(0.653340\pi\)
\(84\) 0 0
\(85\) 6.52470 0.707703
\(86\) 0 0
\(87\) −4.53178 −0.485858
\(88\) 0 0
\(89\) 16.0667 1.70306 0.851532 0.524302i \(-0.175674\pi\)
0.851532 + 0.524302i \(0.175674\pi\)
\(90\) 0 0
\(91\) 14.7009 1.54108
\(92\) 0 0
\(93\) −10.9436 −1.13480
\(94\) 0 0
\(95\) −0.844614 −0.0866555
\(96\) 0 0
\(97\) −3.68923 −0.374584 −0.187292 0.982304i \(-0.559971\pi\)
−0.187292 + 0.982304i \(0.559971\pi\)
\(98\) 0 0
\(99\) 1.59740 0.160545
\(100\) 0 0
\(101\) 16.2462 1.61656 0.808279 0.588799i \(-0.200399\pi\)
0.808279 + 0.588799i \(0.200399\pi\)
\(102\) 0 0
\(103\) −0.812333 −0.0800415 −0.0400208 0.999199i \(-0.512742\pi\)
−0.0400208 + 0.999199i \(0.512742\pi\)
\(104\) 0 0
\(105\) −6.68109 −0.652008
\(106\) 0 0
\(107\) −9.88860 −0.955967 −0.477983 0.878369i \(-0.658632\pi\)
−0.477983 + 0.878369i \(0.658632\pi\)
\(108\) 0 0
\(109\) 10.8375 1.03805 0.519024 0.854760i \(-0.326296\pi\)
0.519024 + 0.854760i \(0.326296\pi\)
\(110\) 0 0
\(111\) −14.0667 −1.33515
\(112\) 0 0
\(113\) −0.310773 −0.0292351 −0.0146175 0.999893i \(-0.504653\pi\)
−0.0146175 + 0.999893i \(0.504653\pi\)
\(114\) 0 0
\(115\) 3.30264 0.307972
\(116\) 0 0
\(117\) 1.62968 0.150664
\(118\) 0 0
\(119\) −39.1323 −3.58725
\(120\) 0 0
\(121\) −2.90817 −0.264379
\(122\) 0 0
\(123\) −10.6378 −0.959180
\(124\) 0 0
\(125\) −7.84361 −0.701554
\(126\) 0 0
\(127\) 18.6882 1.65831 0.829156 0.559017i \(-0.188822\pi\)
0.829156 + 0.559017i \(0.188822\pi\)
\(128\) 0 0
\(129\) −9.31891 −0.820484
\(130\) 0 0
\(131\) 16.6055 1.45083 0.725416 0.688310i \(-0.241647\pi\)
0.725416 + 0.688310i \(0.241647\pi\)
\(132\) 0 0
\(133\) 5.06562 0.439245
\(134\) 0 0
\(135\) −4.69736 −0.404285
\(136\) 0 0
\(137\) −2.91274 −0.248852 −0.124426 0.992229i \(-0.539709\pi\)
−0.124426 + 0.992229i \(0.539709\pi\)
\(138\) 0 0
\(139\) 0.278492 0.0236214 0.0118107 0.999930i \(-0.496240\pi\)
0.0118107 + 0.999930i \(0.496240\pi\)
\(140\) 0 0
\(141\) −1.62467 −0.136822
\(142\) 0 0
\(143\) 8.25535 0.690347
\(144\) 0 0
\(145\) −2.45115 −0.203557
\(146\) 0 0
\(147\) 29.1394 2.40338
\(148\) 0 0
\(149\) 8.08269 0.662160 0.331080 0.943603i \(-0.392587\pi\)
0.331080 + 0.943603i \(0.392587\pi\)
\(150\) 0 0
\(151\) 15.0585 1.22545 0.612723 0.790297i \(-0.290074\pi\)
0.612723 + 0.790297i \(0.290074\pi\)
\(152\) 0 0
\(153\) −4.33804 −0.350710
\(154\) 0 0
\(155\) −5.91917 −0.475439
\(156\) 0 0
\(157\) −10.9273 −0.872094 −0.436047 0.899924i \(-0.643622\pi\)
−0.436047 + 0.899924i \(0.643622\pi\)
\(158\) 0 0
\(159\) 14.1060 1.11868
\(160\) 0 0
\(161\) −19.8078 −1.56107
\(162\) 0 0
\(163\) −0.697363 −0.0546217 −0.0273108 0.999627i \(-0.508694\pi\)
−0.0273108 + 0.999627i \(0.508694\pi\)
\(164\) 0 0
\(165\) −3.75179 −0.292076
\(166\) 0 0
\(167\) 6.24621 0.483346 0.241673 0.970358i \(-0.422304\pi\)
0.241673 + 0.970358i \(0.422304\pi\)
\(168\) 0 0
\(169\) −4.57782 −0.352140
\(170\) 0 0
\(171\) 0.561553 0.0429430
\(172\) 0 0
\(173\) −6.69736 −0.509191 −0.254596 0.967048i \(-0.581942\pi\)
−0.254596 + 0.967048i \(0.581942\pi\)
\(174\) 0 0
\(175\) 21.7144 1.64146
\(176\) 0 0
\(177\) −1.16995 −0.0879386
\(178\) 0 0
\(179\) −18.2462 −1.36379 −0.681893 0.731452i \(-0.738843\pi\)
−0.681893 + 0.731452i \(0.738843\pi\)
\(180\) 0 0
\(181\) −17.5651 −1.30561 −0.652803 0.757528i \(-0.726407\pi\)
−0.652803 + 0.757528i \(0.726407\pi\)
\(182\) 0 0
\(183\) −3.55799 −0.263014
\(184\) 0 0
\(185\) −7.60839 −0.559380
\(186\) 0 0
\(187\) −21.9749 −1.60696
\(188\) 0 0
\(189\) 28.1727 2.04926
\(190\) 0 0
\(191\) 7.19686 0.520747 0.260373 0.965508i \(-0.416154\pi\)
0.260373 + 0.965508i \(0.416154\pi\)
\(192\) 0 0
\(193\) −2.92730 −0.210712 −0.105356 0.994435i \(-0.533598\pi\)
−0.105356 + 0.994435i \(0.533598\pi\)
\(194\) 0 0
\(195\) −3.82760 −0.274100
\(196\) 0 0
\(197\) 11.1394 0.793648 0.396824 0.917895i \(-0.370112\pi\)
0.396824 + 0.917895i \(0.370112\pi\)
\(198\) 0 0
\(199\) 5.57220 0.395003 0.197501 0.980303i \(-0.436717\pi\)
0.197501 + 0.980303i \(0.436717\pi\)
\(200\) 0 0
\(201\) −16.1994 −1.14262
\(202\) 0 0
\(203\) 14.7009 1.03180
\(204\) 0 0
\(205\) −5.75379 −0.401862
\(206\) 0 0
\(207\) −2.19580 −0.152619
\(208\) 0 0
\(209\) 2.84461 0.196766
\(210\) 0 0
\(211\) 1.23451 0.0849871 0.0424935 0.999097i \(-0.486470\pi\)
0.0424935 + 0.999097i \(0.486470\pi\)
\(212\) 0 0
\(213\) 3.32805 0.228034
\(214\) 0 0
\(215\) −5.04042 −0.343754
\(216\) 0 0
\(217\) 35.5006 2.40993
\(218\) 0 0
\(219\) 7.18622 0.485600
\(220\) 0 0
\(221\) −22.4189 −1.50806
\(222\) 0 0
\(223\) −5.76092 −0.385780 −0.192890 0.981220i \(-0.561786\pi\)
−0.192890 + 0.981220i \(0.561786\pi\)
\(224\) 0 0
\(225\) 2.40717 0.160478
\(226\) 0 0
\(227\) 11.1862 0.742455 0.371228 0.928542i \(-0.378937\pi\)
0.371228 + 0.928542i \(0.378937\pi\)
\(228\) 0 0
\(229\) −25.3624 −1.67600 −0.837999 0.545672i \(-0.816274\pi\)
−0.837999 + 0.545672i \(0.816274\pi\)
\(230\) 0 0
\(231\) 22.5016 1.48049
\(232\) 0 0
\(233\) 8.33804 0.546243 0.273122 0.961980i \(-0.411944\pi\)
0.273122 + 0.961980i \(0.411944\pi\)
\(234\) 0 0
\(235\) −0.878750 −0.0573233
\(236\) 0 0
\(237\) 6.06668 0.394073
\(238\) 0 0
\(239\) −16.4441 −1.06368 −0.531839 0.846845i \(-0.678499\pi\)
−0.531839 + 0.846845i \(0.678499\pi\)
\(240\) 0 0
\(241\) −12.8286 −0.826363 −0.413182 0.910649i \(-0.635583\pi\)
−0.413182 + 0.910649i \(0.635583\pi\)
\(242\) 0 0
\(243\) 5.75379 0.369106
\(244\) 0 0
\(245\) 15.7609 1.00693
\(246\) 0 0
\(247\) 2.90210 0.184656
\(248\) 0 0
\(249\) −13.1827 −0.835417
\(250\) 0 0
\(251\) −17.8527 −1.12686 −0.563428 0.826165i \(-0.690518\pi\)
−0.563428 + 0.826165i \(0.690518\pi\)
\(252\) 0 0
\(253\) −11.1231 −0.699304
\(254\) 0 0
\(255\) 10.1887 0.638039
\(256\) 0 0
\(257\) 1.49342 0.0931572 0.0465786 0.998915i \(-0.485168\pi\)
0.0465786 + 0.998915i \(0.485168\pi\)
\(258\) 0 0
\(259\) 45.6318 2.83542
\(260\) 0 0
\(261\) 1.62968 0.100875
\(262\) 0 0
\(263\) 0.483433 0.0298097 0.0149049 0.999889i \(-0.495255\pi\)
0.0149049 + 0.999889i \(0.495255\pi\)
\(264\) 0 0
\(265\) 7.62968 0.468688
\(266\) 0 0
\(267\) 25.0890 1.53542
\(268\) 0 0
\(269\) −8.45115 −0.515276 −0.257638 0.966242i \(-0.582944\pi\)
−0.257638 + 0.966242i \(0.582944\pi\)
\(270\) 0 0
\(271\) −16.5822 −1.00730 −0.503648 0.863909i \(-0.668009\pi\)
−0.503648 + 0.863909i \(0.668009\pi\)
\(272\) 0 0
\(273\) 22.9563 1.38938
\(274\) 0 0
\(275\) 12.1938 0.735314
\(276\) 0 0
\(277\) −15.7124 −0.944065 −0.472032 0.881581i \(-0.656479\pi\)
−0.472032 + 0.881581i \(0.656479\pi\)
\(278\) 0 0
\(279\) 3.93544 0.235609
\(280\) 0 0
\(281\) 3.13938 0.187280 0.0936398 0.995606i \(-0.470150\pi\)
0.0936398 + 0.995606i \(0.470150\pi\)
\(282\) 0 0
\(283\) 26.3884 1.56863 0.784315 0.620363i \(-0.213015\pi\)
0.784315 + 0.620363i \(0.213015\pi\)
\(284\) 0 0
\(285\) −1.31891 −0.0781254
\(286\) 0 0
\(287\) 34.5087 2.03698
\(288\) 0 0
\(289\) 42.6768 2.51040
\(290\) 0 0
\(291\) −5.76092 −0.337711
\(292\) 0 0
\(293\) −13.5903 −0.793955 −0.396978 0.917828i \(-0.629941\pi\)
−0.396978 + 0.917828i \(0.629941\pi\)
\(294\) 0 0
\(295\) −0.632801 −0.0368431
\(296\) 0 0
\(297\) 15.8205 0.917997
\(298\) 0 0
\(299\) −11.3479 −0.656265
\(300\) 0 0
\(301\) 30.2302 1.74244
\(302\) 0 0
\(303\) 25.3693 1.45743
\(304\) 0 0
\(305\) −1.92445 −0.110193
\(306\) 0 0
\(307\) 3.44302 0.196503 0.0982516 0.995162i \(-0.468675\pi\)
0.0982516 + 0.995162i \(0.468675\pi\)
\(308\) 0 0
\(309\) −1.26850 −0.0721625
\(310\) 0 0
\(311\) −29.4935 −1.67242 −0.836211 0.548408i \(-0.815234\pi\)
−0.836211 + 0.548408i \(0.815234\pi\)
\(312\) 0 0
\(313\) −10.0631 −0.568801 −0.284400 0.958706i \(-0.591794\pi\)
−0.284400 + 0.958706i \(0.591794\pi\)
\(314\) 0 0
\(315\) 2.40260 0.135371
\(316\) 0 0
\(317\) −4.14931 −0.233049 −0.116524 0.993188i \(-0.537175\pi\)
−0.116524 + 0.993188i \(0.537175\pi\)
\(318\) 0 0
\(319\) 8.25535 0.462211
\(320\) 0 0
\(321\) −15.4416 −0.861864
\(322\) 0 0
\(323\) −7.72508 −0.429835
\(324\) 0 0
\(325\) 12.4402 0.690059
\(326\) 0 0
\(327\) 16.9234 0.935865
\(328\) 0 0
\(329\) 5.27036 0.290564
\(330\) 0 0
\(331\) −25.8886 −1.42297 −0.711483 0.702703i \(-0.751976\pi\)
−0.711483 + 0.702703i \(0.751976\pi\)
\(332\) 0 0
\(333\) 5.05854 0.277207
\(334\) 0 0
\(335\) −8.76192 −0.478715
\(336\) 0 0
\(337\) 31.5097 1.71644 0.858221 0.513280i \(-0.171570\pi\)
0.858221 + 0.513280i \(0.171570\pi\)
\(338\) 0 0
\(339\) −0.485288 −0.0263572
\(340\) 0 0
\(341\) 19.9354 1.07956
\(342\) 0 0
\(343\) −59.0677 −3.18936
\(344\) 0 0
\(345\) 5.15724 0.277657
\(346\) 0 0
\(347\) 13.9173 0.747120 0.373560 0.927606i \(-0.378137\pi\)
0.373560 + 0.927606i \(0.378137\pi\)
\(348\) 0 0
\(349\) −6.34305 −0.339536 −0.169768 0.985484i \(-0.554302\pi\)
−0.169768 + 0.985484i \(0.554302\pi\)
\(350\) 0 0
\(351\) 16.1402 0.861499
\(352\) 0 0
\(353\) −32.0794 −1.70741 −0.853707 0.520754i \(-0.825651\pi\)
−0.853707 + 0.520754i \(0.825651\pi\)
\(354\) 0 0
\(355\) 1.80008 0.0955381
\(356\) 0 0
\(357\) −61.1072 −3.23413
\(358\) 0 0
\(359\) −1.86670 −0.0985205 −0.0492603 0.998786i \(-0.515686\pi\)
−0.0492603 + 0.998786i \(0.515686\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −4.54127 −0.238355
\(364\) 0 0
\(365\) 3.88689 0.203449
\(366\) 0 0
\(367\) 20.8840 1.09014 0.545069 0.838391i \(-0.316504\pi\)
0.545069 + 0.838391i \(0.316504\pi\)
\(368\) 0 0
\(369\) 3.82548 0.199147
\(370\) 0 0
\(371\) −45.7595 −2.37571
\(372\) 0 0
\(373\) −25.9819 −1.34529 −0.672647 0.739964i \(-0.734843\pi\)
−0.672647 + 0.739964i \(0.734843\pi\)
\(374\) 0 0
\(375\) −12.2482 −0.632495
\(376\) 0 0
\(377\) 8.42218 0.433764
\(378\) 0 0
\(379\) −2.50301 −0.128571 −0.0642855 0.997932i \(-0.520477\pi\)
−0.0642855 + 0.997932i \(0.520477\pi\)
\(380\) 0 0
\(381\) 29.1827 1.49507
\(382\) 0 0
\(383\) −21.8155 −1.11472 −0.557359 0.830272i \(-0.688185\pi\)
−0.557359 + 0.830272i \(0.688185\pi\)
\(384\) 0 0
\(385\) 12.1707 0.620274
\(386\) 0 0
\(387\) 3.35119 0.170351
\(388\) 0 0
\(389\) −35.2383 −1.78665 −0.893327 0.449407i \(-0.851635\pi\)
−0.893327 + 0.449407i \(0.851635\pi\)
\(390\) 0 0
\(391\) 30.2069 1.52763
\(392\) 0 0
\(393\) 25.9304 1.30802
\(394\) 0 0
\(395\) 3.28135 0.165103
\(396\) 0 0
\(397\) 27.0263 1.35641 0.678205 0.734873i \(-0.262758\pi\)
0.678205 + 0.734873i \(0.262758\pi\)
\(398\) 0 0
\(399\) 7.91023 0.396007
\(400\) 0 0
\(401\) −24.8932 −1.24311 −0.621553 0.783372i \(-0.713498\pi\)
−0.621553 + 0.783372i \(0.713498\pi\)
\(402\) 0 0
\(403\) 20.3383 1.01312
\(404\) 0 0
\(405\) −5.91230 −0.293784
\(406\) 0 0
\(407\) 25.6247 1.27017
\(408\) 0 0
\(409\) −14.8336 −0.733475 −0.366738 0.930324i \(-0.619525\pi\)
−0.366738 + 0.930324i \(0.619525\pi\)
\(410\) 0 0
\(411\) −4.54840 −0.224356
\(412\) 0 0
\(413\) 3.79526 0.186753
\(414\) 0 0
\(415\) −7.13024 −0.350010
\(416\) 0 0
\(417\) 0.434880 0.0212962
\(418\) 0 0
\(419\) 6.05955 0.296028 0.148014 0.988985i \(-0.452712\pi\)
0.148014 + 0.988985i \(0.452712\pi\)
\(420\) 0 0
\(421\) −13.8670 −0.675834 −0.337917 0.941176i \(-0.609722\pi\)
−0.337917 + 0.941176i \(0.609722\pi\)
\(422\) 0 0
\(423\) 0.584249 0.0284072
\(424\) 0 0
\(425\) −33.1145 −1.60629
\(426\) 0 0
\(427\) 11.5420 0.558555
\(428\) 0 0
\(429\) 12.8912 0.622391
\(430\) 0 0
\(431\) −11.5580 −0.556729 −0.278364 0.960476i \(-0.589792\pi\)
−0.278364 + 0.960476i \(0.589792\pi\)
\(432\) 0 0
\(433\) −31.7468 −1.52565 −0.762826 0.646604i \(-0.776189\pi\)
−0.762826 + 0.646604i \(0.776189\pi\)
\(434\) 0 0
\(435\) −3.82760 −0.183520
\(436\) 0 0
\(437\) −3.91023 −0.187052
\(438\) 0 0
\(439\) −24.8840 −1.18765 −0.593825 0.804594i \(-0.702383\pi\)
−0.593825 + 0.804594i \(0.702383\pi\)
\(440\) 0 0
\(441\) −10.4789 −0.498994
\(442\) 0 0
\(443\) 23.9911 1.13985 0.569926 0.821696i \(-0.306972\pi\)
0.569926 + 0.821696i \(0.306972\pi\)
\(444\) 0 0
\(445\) 13.5701 0.643286
\(446\) 0 0
\(447\) 12.6215 0.596979
\(448\) 0 0
\(449\) 32.1384 1.51670 0.758352 0.651845i \(-0.226005\pi\)
0.758352 + 0.651845i \(0.226005\pi\)
\(450\) 0 0
\(451\) 19.3785 0.912496
\(452\) 0 0
\(453\) 23.5147 1.10482
\(454\) 0 0
\(455\) 12.4166 0.582099
\(456\) 0 0
\(457\) 29.1590 1.36400 0.681999 0.731353i \(-0.261111\pi\)
0.681999 + 0.731353i \(0.261111\pi\)
\(458\) 0 0
\(459\) −42.9634 −2.00536
\(460\) 0 0
\(461\) −18.6559 −0.868894 −0.434447 0.900697i \(-0.643056\pi\)
−0.434447 + 0.900697i \(0.643056\pi\)
\(462\) 0 0
\(463\) 22.6005 1.05034 0.525168 0.850999i \(-0.324003\pi\)
0.525168 + 0.850999i \(0.324003\pi\)
\(464\) 0 0
\(465\) −9.24309 −0.428638
\(466\) 0 0
\(467\) 17.1666 0.794377 0.397189 0.917737i \(-0.369986\pi\)
0.397189 + 0.917737i \(0.369986\pi\)
\(468\) 0 0
\(469\) 52.5502 2.42654
\(470\) 0 0
\(471\) −17.0636 −0.786247
\(472\) 0 0
\(473\) 16.9759 0.780551
\(474\) 0 0
\(475\) 4.28663 0.196684
\(476\) 0 0
\(477\) −5.07270 −0.232263
\(478\) 0 0
\(479\) −16.2625 −0.743052 −0.371526 0.928423i \(-0.621165\pi\)
−0.371526 + 0.928423i \(0.621165\pi\)
\(480\) 0 0
\(481\) 26.1425 1.19200
\(482\) 0 0
\(483\) −30.9309 −1.40740
\(484\) 0 0
\(485\) −3.11597 −0.141489
\(486\) 0 0
\(487\) 18.2071 0.825041 0.412520 0.910948i \(-0.364649\pi\)
0.412520 + 0.910948i \(0.364649\pi\)
\(488\) 0 0
\(489\) −1.08897 −0.0492449
\(490\) 0 0
\(491\) −39.2493 −1.77130 −0.885649 0.464356i \(-0.846286\pi\)
−0.885649 + 0.464356i \(0.846286\pi\)
\(492\) 0 0
\(493\) −22.4189 −1.00970
\(494\) 0 0
\(495\) 1.34919 0.0606414
\(496\) 0 0
\(497\) −10.7961 −0.484270
\(498\) 0 0
\(499\) 35.7541 1.60057 0.800286 0.599619i \(-0.204681\pi\)
0.800286 + 0.599619i \(0.204681\pi\)
\(500\) 0 0
\(501\) 9.75379 0.435767
\(502\) 0 0
\(503\) 27.5512 1.22845 0.614223 0.789132i \(-0.289470\pi\)
0.614223 + 0.789132i \(0.289470\pi\)
\(504\) 0 0
\(505\) 13.7218 0.610611
\(506\) 0 0
\(507\) −7.14851 −0.317477
\(508\) 0 0
\(509\) 14.6328 0.648588 0.324294 0.945956i \(-0.394873\pi\)
0.324294 + 0.945956i \(0.394873\pi\)
\(510\) 0 0
\(511\) −23.3118 −1.03125
\(512\) 0 0
\(513\) 5.56155 0.245549
\(514\) 0 0
\(515\) −0.686107 −0.0302335
\(516\) 0 0
\(517\) 2.95958 0.130162
\(518\) 0 0
\(519\) −10.4583 −0.459068
\(520\) 0 0
\(521\) −7.59926 −0.332929 −0.166465 0.986047i \(-0.553235\pi\)
−0.166465 + 0.986047i \(0.553235\pi\)
\(522\) 0 0
\(523\) −19.8723 −0.868956 −0.434478 0.900682i \(-0.643067\pi\)
−0.434478 + 0.900682i \(0.643067\pi\)
\(524\) 0 0
\(525\) 33.9082 1.47988
\(526\) 0 0
\(527\) −54.1384 −2.35830
\(528\) 0 0
\(529\) −7.71007 −0.335220
\(530\) 0 0
\(531\) 0.420727 0.0182580
\(532\) 0 0
\(533\) 19.7701 0.856336
\(534\) 0 0
\(535\) −8.35204 −0.361090
\(536\) 0 0
\(537\) −28.4924 −1.22954
\(538\) 0 0
\(539\) −53.0820 −2.28640
\(540\) 0 0
\(541\) −27.6982 −1.19084 −0.595420 0.803415i \(-0.703014\pi\)
−0.595420 + 0.803415i \(0.703014\pi\)
\(542\) 0 0
\(543\) −27.4289 −1.17709
\(544\) 0 0
\(545\) 9.15353 0.392094
\(546\) 0 0
\(547\) −33.8871 −1.44891 −0.724455 0.689322i \(-0.757908\pi\)
−0.724455 + 0.689322i \(0.757908\pi\)
\(548\) 0 0
\(549\) 1.27949 0.0546075
\(550\) 0 0
\(551\) 2.90210 0.123634
\(552\) 0 0
\(553\) −19.6801 −0.836883
\(554\) 0 0
\(555\) −11.8809 −0.504316
\(556\) 0 0
\(557\) 8.28763 0.351158 0.175579 0.984465i \(-0.443820\pi\)
0.175579 + 0.984465i \(0.443820\pi\)
\(558\) 0 0
\(559\) 17.3189 0.732512
\(560\) 0 0
\(561\) −34.3149 −1.44878
\(562\) 0 0
\(563\) 5.11397 0.215528 0.107764 0.994177i \(-0.465631\pi\)
0.107764 + 0.994177i \(0.465631\pi\)
\(564\) 0 0
\(565\) −0.262483 −0.0110427
\(566\) 0 0
\(567\) 35.4593 1.48915
\(568\) 0 0
\(569\) 1.92830 0.0808387 0.0404194 0.999183i \(-0.487131\pi\)
0.0404194 + 0.999183i \(0.487131\pi\)
\(570\) 0 0
\(571\) 23.3785 0.978358 0.489179 0.872183i \(-0.337297\pi\)
0.489179 + 0.872183i \(0.337297\pi\)
\(572\) 0 0
\(573\) 11.2383 0.469486
\(574\) 0 0
\(575\) −16.7617 −0.699012
\(576\) 0 0
\(577\) −26.8807 −1.11906 −0.559530 0.828810i \(-0.689018\pi\)
−0.559530 + 0.828810i \(0.689018\pi\)
\(578\) 0 0
\(579\) −4.57114 −0.189970
\(580\) 0 0
\(581\) 42.7640 1.77415
\(582\) 0 0
\(583\) −25.6964 −1.06423
\(584\) 0 0
\(585\) 1.37645 0.0569093
\(586\) 0 0
\(587\) 36.7922 1.51858 0.759288 0.650754i \(-0.225547\pi\)
0.759288 + 0.650754i \(0.225547\pi\)
\(588\) 0 0
\(589\) 7.00814 0.288765
\(590\) 0 0
\(591\) 17.3947 0.715523
\(592\) 0 0
\(593\) −33.9034 −1.39225 −0.696123 0.717922i \(-0.745093\pi\)
−0.696123 + 0.717922i \(0.745093\pi\)
\(594\) 0 0
\(595\) −33.0517 −1.35499
\(596\) 0 0
\(597\) 8.70128 0.356120
\(598\) 0 0
\(599\) −4.06456 −0.166073 −0.0830367 0.996546i \(-0.526462\pi\)
−0.0830367 + 0.996546i \(0.526462\pi\)
\(600\) 0 0
\(601\) −27.5560 −1.12403 −0.562016 0.827126i \(-0.689974\pi\)
−0.562016 + 0.827126i \(0.689974\pi\)
\(602\) 0 0
\(603\) 5.82548 0.237232
\(604\) 0 0
\(605\) −2.45628 −0.0998621
\(606\) 0 0
\(607\) 25.5743 1.03803 0.519014 0.854766i \(-0.326299\pi\)
0.519014 + 0.854766i \(0.326299\pi\)
\(608\) 0 0
\(609\) 22.9563 0.930235
\(610\) 0 0
\(611\) 3.01939 0.122152
\(612\) 0 0
\(613\) 31.2383 1.26170 0.630852 0.775903i \(-0.282705\pi\)
0.630852 + 0.775903i \(0.282705\pi\)
\(614\) 0 0
\(615\) −8.98485 −0.362304
\(616\) 0 0
\(617\) −5.86189 −0.235991 −0.117995 0.993014i \(-0.537647\pi\)
−0.117995 + 0.993014i \(0.537647\pi\)
\(618\) 0 0
\(619\) 31.6481 1.27204 0.636022 0.771671i \(-0.280579\pi\)
0.636022 + 0.771671i \(0.280579\pi\)
\(620\) 0 0
\(621\) −21.7470 −0.872676
\(622\) 0 0
\(623\) −81.3877 −3.26073
\(624\) 0 0
\(625\) 14.8083 0.592333
\(626\) 0 0
\(627\) 4.44201 0.177397
\(628\) 0 0
\(629\) −69.5885 −2.77468
\(630\) 0 0
\(631\) −3.86189 −0.153739 −0.0768696 0.997041i \(-0.524493\pi\)
−0.0768696 + 0.997041i \(0.524493\pi\)
\(632\) 0 0
\(633\) 1.92775 0.0766212
\(634\) 0 0
\(635\) 15.7843 0.626382
\(636\) 0 0
\(637\) −54.1546 −2.14569
\(638\) 0 0
\(639\) −1.19680 −0.0473449
\(640\) 0 0
\(641\) −14.9919 −0.592143 −0.296072 0.955166i \(-0.595677\pi\)
−0.296072 + 0.955166i \(0.595677\pi\)
\(642\) 0 0
\(643\) 12.4906 0.492580 0.246290 0.969196i \(-0.420788\pi\)
0.246290 + 0.969196i \(0.420788\pi\)
\(644\) 0 0
\(645\) −7.87088 −0.309915
\(646\) 0 0
\(647\) −23.5743 −0.926802 −0.463401 0.886149i \(-0.653371\pi\)
−0.463401 + 0.886149i \(0.653371\pi\)
\(648\) 0 0
\(649\) 2.13124 0.0836585
\(650\) 0 0
\(651\) 55.4360 2.17271
\(652\) 0 0
\(653\) −30.4693 −1.19236 −0.596178 0.802853i \(-0.703315\pi\)
−0.596178 + 0.802853i \(0.703315\pi\)
\(654\) 0 0
\(655\) 14.0253 0.548012
\(656\) 0 0
\(657\) −2.58425 −0.100821
\(658\) 0 0
\(659\) −39.9049 −1.55447 −0.777237 0.629209i \(-0.783379\pi\)
−0.777237 + 0.629209i \(0.783379\pi\)
\(660\) 0 0
\(661\) 22.8156 0.887422 0.443711 0.896170i \(-0.353662\pi\)
0.443711 + 0.896170i \(0.353662\pi\)
\(662\) 0 0
\(663\) −35.0083 −1.35961
\(664\) 0 0
\(665\) 4.27849 0.165913
\(666\) 0 0
\(667\) −11.3479 −0.439392
\(668\) 0 0
\(669\) −8.99599 −0.347805
\(670\) 0 0
\(671\) 6.48143 0.250213
\(672\) 0 0
\(673\) −37.0265 −1.42727 −0.713634 0.700519i \(-0.752952\pi\)
−0.713634 + 0.700519i \(0.752952\pi\)
\(674\) 0 0
\(675\) 23.8403 0.917614
\(676\) 0 0
\(677\) 45.6733 1.75537 0.877683 0.479241i \(-0.159088\pi\)
0.877683 + 0.479241i \(0.159088\pi\)
\(678\) 0 0
\(679\) 18.6882 0.717188
\(680\) 0 0
\(681\) 17.4679 0.669370
\(682\) 0 0
\(683\) −13.9837 −0.535072 −0.267536 0.963548i \(-0.586210\pi\)
−0.267536 + 0.963548i \(0.586210\pi\)
\(684\) 0 0
\(685\) −2.46014 −0.0939972
\(686\) 0 0
\(687\) −39.6048 −1.51102
\(688\) 0 0
\(689\) −26.2156 −0.998736
\(690\) 0 0
\(691\) 0.00185566 7.05926e−5 0 3.52963e−5 1.00000i \(-0.499989\pi\)
3.52963e−5 1.00000i \(0.499989\pi\)
\(692\) 0 0
\(693\) −8.09183 −0.307383
\(694\) 0 0
\(695\) 0.235218 0.00892233
\(696\) 0 0
\(697\) −52.6258 −1.99334
\(698\) 0 0
\(699\) 13.0203 0.492472
\(700\) 0 0
\(701\) 22.0071 0.831198 0.415599 0.909548i \(-0.363572\pi\)
0.415599 + 0.909548i \(0.363572\pi\)
\(702\) 0 0
\(703\) 9.00814 0.339748
\(704\) 0 0
\(705\) −1.37221 −0.0516806
\(706\) 0 0
\(707\) −82.2971 −3.09510
\(708\) 0 0
\(709\) −3.94045 −0.147987 −0.0739934 0.997259i \(-0.523574\pi\)
−0.0739934 + 0.997259i \(0.523574\pi\)
\(710\) 0 0
\(711\) −2.18165 −0.0818183
\(712\) 0 0
\(713\) −27.4035 −1.02627
\(714\) 0 0
\(715\) 6.97258 0.260760
\(716\) 0 0
\(717\) −25.6783 −0.958973
\(718\) 0 0
\(719\) −29.0656 −1.08396 −0.541982 0.840390i \(-0.682326\pi\)
−0.541982 + 0.840390i \(0.682326\pi\)
\(720\) 0 0
\(721\) 4.11497 0.153249
\(722\) 0 0
\(723\) −20.0325 −0.745018
\(724\) 0 0
\(725\) 12.4402 0.462018
\(726\) 0 0
\(727\) 10.1617 0.376877 0.188439 0.982085i \(-0.439657\pi\)
0.188439 + 0.982085i \(0.439657\pi\)
\(728\) 0 0
\(729\) 29.9848 1.11055
\(730\) 0 0
\(731\) −46.1011 −1.70511
\(732\) 0 0
\(733\) −51.4777 −1.90137 −0.950686 0.310156i \(-0.899619\pi\)
−0.950686 + 0.310156i \(0.899619\pi\)
\(734\) 0 0
\(735\) 24.6115 0.907809
\(736\) 0 0
\(737\) 29.5097 1.08700
\(738\) 0 0
\(739\) −33.1412 −1.21912 −0.609560 0.792740i \(-0.708654\pi\)
−0.609560 + 0.792740i \(0.708654\pi\)
\(740\) 0 0
\(741\) 4.53178 0.166479
\(742\) 0 0
\(743\) 35.5006 1.30239 0.651195 0.758911i \(-0.274268\pi\)
0.651195 + 0.758911i \(0.274268\pi\)
\(744\) 0 0
\(745\) 6.82675 0.250113
\(746\) 0 0
\(747\) 4.74064 0.173451
\(748\) 0 0
\(749\) 50.0919 1.83032
\(750\) 0 0
\(751\) 15.2594 0.556822 0.278411 0.960462i \(-0.410192\pi\)
0.278411 + 0.960462i \(0.410192\pi\)
\(752\) 0 0
\(753\) −27.8780 −1.01593
\(754\) 0 0
\(755\) 12.7187 0.462879
\(756\) 0 0
\(757\) −7.28161 −0.264655 −0.132327 0.991206i \(-0.542245\pi\)
−0.132327 + 0.991206i \(0.542245\pi\)
\(758\) 0 0
\(759\) −17.3693 −0.630466
\(760\) 0 0
\(761\) 9.47886 0.343609 0.171804 0.985131i \(-0.445040\pi\)
0.171804 + 0.985131i \(0.445040\pi\)
\(762\) 0 0
\(763\) −54.8989 −1.98747
\(764\) 0 0
\(765\) −3.66397 −0.132471
\(766\) 0 0
\(767\) 2.17431 0.0785098
\(768\) 0 0
\(769\) 19.2760 0.695112 0.347556 0.937659i \(-0.387012\pi\)
0.347556 + 0.937659i \(0.387012\pi\)
\(770\) 0 0
\(771\) 2.33206 0.0839871
\(772\) 0 0
\(773\) −9.19501 −0.330721 −0.165361 0.986233i \(-0.552879\pi\)
−0.165361 + 0.986233i \(0.552879\pi\)
\(774\) 0 0
\(775\) 30.0413 1.07911
\(776\) 0 0
\(777\) 71.2565 2.55631
\(778\) 0 0
\(779\) 6.81233 0.244077
\(780\) 0 0
\(781\) −6.06256 −0.216935
\(782\) 0 0
\(783\) 16.1402 0.576803
\(784\) 0 0
\(785\) −9.22935 −0.329410
\(786\) 0 0
\(787\) −12.6847 −0.452159 −0.226080 0.974109i \(-0.572591\pi\)
−0.226080 + 0.974109i \(0.572591\pi\)
\(788\) 0 0
\(789\) 0.754905 0.0268753
\(790\) 0 0
\(791\) 1.57426 0.0559741
\(792\) 0 0
\(793\) 6.61241 0.234814
\(794\) 0 0
\(795\) 11.9142 0.422551
\(796\) 0 0
\(797\) −27.4591 −0.972651 −0.486325 0.873778i \(-0.661663\pi\)
−0.486325 + 0.873778i \(0.661663\pi\)
\(798\) 0 0
\(799\) −8.03730 −0.284339
\(800\) 0 0
\(801\) −9.02229 −0.318787
\(802\) 0 0
\(803\) −13.0908 −0.461965
\(804\) 0 0
\(805\) −16.7299 −0.589652
\(806\) 0 0
\(807\) −13.1969 −0.464554
\(808\) 0 0
\(809\) 18.2912 0.643084 0.321542 0.946895i \(-0.395799\pi\)
0.321542 + 0.946895i \(0.395799\pi\)
\(810\) 0 0
\(811\) −6.28391 −0.220658 −0.110329 0.993895i \(-0.535190\pi\)
−0.110329 + 0.993895i \(0.535190\pi\)
\(812\) 0 0
\(813\) −25.8940 −0.908141
\(814\) 0 0
\(815\) −0.589002 −0.0206318
\(816\) 0 0
\(817\) 5.96772 0.208784
\(818\) 0 0
\(819\) −8.25535 −0.288465
\(820\) 0 0
\(821\) 43.9103 1.53248 0.766240 0.642555i \(-0.222125\pi\)
0.766240 + 0.642555i \(0.222125\pi\)
\(822\) 0 0
\(823\) −26.2483 −0.914957 −0.457479 0.889221i \(-0.651247\pi\)
−0.457479 + 0.889221i \(0.651247\pi\)
\(824\) 0 0
\(825\) 19.0413 0.662932
\(826\) 0 0
\(827\) 34.7726 1.20916 0.604581 0.796543i \(-0.293340\pi\)
0.604581 + 0.796543i \(0.293340\pi\)
\(828\) 0 0
\(829\) 20.9184 0.726525 0.363263 0.931687i \(-0.381663\pi\)
0.363263 + 0.931687i \(0.381663\pi\)
\(830\) 0 0
\(831\) −24.5357 −0.851134
\(832\) 0 0
\(833\) 144.154 4.99464
\(834\) 0 0
\(835\) 5.27563 0.182571
\(836\) 0 0
\(837\) 38.9761 1.34721
\(838\) 0 0
\(839\) 41.0061 1.41569 0.707844 0.706368i \(-0.249668\pi\)
0.707844 + 0.706368i \(0.249668\pi\)
\(840\) 0 0
\(841\) −20.5778 −0.709580
\(842\) 0 0
\(843\) 4.90230 0.168844
\(844\) 0 0
\(845\) −3.86649 −0.133011
\(846\) 0 0
\(847\) 14.7317 0.506187
\(848\) 0 0
\(849\) 41.2070 1.41422
\(850\) 0 0
\(851\) −35.2239 −1.20746
\(852\) 0 0
\(853\) −11.5056 −0.393943 −0.196972 0.980409i \(-0.563111\pi\)
−0.196972 + 0.980409i \(0.563111\pi\)
\(854\) 0 0
\(855\) 0.474295 0.0162206
\(856\) 0 0
\(857\) 14.7478 0.503774 0.251887 0.967757i \(-0.418949\pi\)
0.251887 + 0.967757i \(0.418949\pi\)
\(858\) 0 0
\(859\) −36.3402 −1.23991 −0.619955 0.784637i \(-0.712849\pi\)
−0.619955 + 0.784637i \(0.712849\pi\)
\(860\) 0 0
\(861\) 53.8871 1.83647
\(862\) 0 0
\(863\) 48.3454 1.64570 0.822849 0.568260i \(-0.192383\pi\)
0.822849 + 0.568260i \(0.192383\pi\)
\(864\) 0 0
\(865\) −5.65668 −0.192333
\(866\) 0 0
\(867\) 66.6421 2.26328
\(868\) 0 0
\(869\) −11.0514 −0.374893
\(870\) 0 0
\(871\) 30.1060 1.02010
\(872\) 0 0
\(873\) 2.07170 0.0701163
\(874\) 0 0
\(875\) 39.7328 1.34321
\(876\) 0 0
\(877\) 4.23226 0.142913 0.0714567 0.997444i \(-0.477235\pi\)
0.0714567 + 0.997444i \(0.477235\pi\)
\(878\) 0 0
\(879\) −21.2220 −0.715801
\(880\) 0 0
\(881\) 32.6651 1.10051 0.550257 0.834995i \(-0.314530\pi\)
0.550257 + 0.834995i \(0.314530\pi\)
\(882\) 0 0
\(883\) 45.3320 1.52554 0.762772 0.646668i \(-0.223838\pi\)
0.762772 + 0.646668i \(0.223838\pi\)
\(884\) 0 0
\(885\) −0.988153 −0.0332164
\(886\) 0 0
\(887\) −50.5271 −1.69653 −0.848267 0.529569i \(-0.822354\pi\)
−0.848267 + 0.529569i \(0.822354\pi\)
\(888\) 0 0
\(889\) −94.6675 −3.17504
\(890\) 0 0
\(891\) 19.9123 0.667087
\(892\) 0 0
\(893\) 1.04042 0.0348162
\(894\) 0 0
\(895\) −15.4110 −0.515133
\(896\) 0 0
\(897\) −17.7203 −0.591664
\(898\) 0 0
\(899\) 20.3383 0.678320
\(900\) 0 0
\(901\) 69.7832 2.32482
\(902\) 0 0
\(903\) 47.2061 1.57092
\(904\) 0 0
\(905\) −14.8357 −0.493157
\(906\) 0 0
\(907\) 21.9823 0.729910 0.364955 0.931025i \(-0.381084\pi\)
0.364955 + 0.931025i \(0.381084\pi\)
\(908\) 0 0
\(909\) −9.12311 −0.302594
\(910\) 0 0
\(911\) −29.9950 −0.993778 −0.496889 0.867814i \(-0.665524\pi\)
−0.496889 + 0.867814i \(0.665524\pi\)
\(912\) 0 0
\(913\) 24.0143 0.794756
\(914\) 0 0
\(915\) −3.00512 −0.0993463
\(916\) 0 0
\(917\) −84.1174 −2.77780
\(918\) 0 0
\(919\) −30.5481 −1.00769 −0.503844 0.863795i \(-0.668081\pi\)
−0.503844 + 0.863795i \(0.668081\pi\)
\(920\) 0 0
\(921\) 5.37645 0.177160
\(922\) 0 0
\(923\) −6.18507 −0.203584
\(924\) 0 0
\(925\) 38.6145 1.26964
\(926\) 0 0
\(927\) 0.456168 0.0149825
\(928\) 0 0
\(929\) −5.67151 −0.186076 −0.0930380 0.995663i \(-0.529658\pi\)
−0.0930380 + 0.995663i \(0.529658\pi\)
\(930\) 0 0
\(931\) −18.6605 −0.611574
\(932\) 0 0
\(933\) −46.0556 −1.50779
\(934\) 0 0
\(935\) −18.5603 −0.606985
\(936\) 0 0
\(937\) 33.6051 1.09783 0.548915 0.835878i \(-0.315041\pi\)
0.548915 + 0.835878i \(0.315041\pi\)
\(938\) 0 0
\(939\) −15.7141 −0.512810
\(940\) 0 0
\(941\) −30.0543 −0.979743 −0.489871 0.871795i \(-0.662956\pi\)
−0.489871 + 0.871795i \(0.662956\pi\)
\(942\) 0 0
\(943\) −26.6378 −0.867447
\(944\) 0 0
\(945\) 23.7951 0.774053
\(946\) 0 0
\(947\) 12.3058 0.399883 0.199942 0.979808i \(-0.435925\pi\)
0.199942 + 0.979808i \(0.435925\pi\)
\(948\) 0 0
\(949\) −13.3554 −0.433534
\(950\) 0 0
\(951\) −6.47937 −0.210108
\(952\) 0 0
\(953\) 30.4674 0.986937 0.493468 0.869764i \(-0.335729\pi\)
0.493468 + 0.869764i \(0.335729\pi\)
\(954\) 0 0
\(955\) 6.07857 0.196698
\(956\) 0 0
\(957\) 12.8912 0.416712
\(958\) 0 0
\(959\) 14.7548 0.476459
\(960\) 0 0
\(961\) 18.1140 0.584322
\(962\) 0 0
\(963\) 5.55297 0.178942
\(964\) 0 0
\(965\) −2.47244 −0.0795906
\(966\) 0 0
\(967\) −5.26560 −0.169330 −0.0846652 0.996409i \(-0.526982\pi\)
−0.0846652 + 0.996409i \(0.526982\pi\)
\(968\) 0 0
\(969\) −12.0631 −0.387523
\(970\) 0 0
\(971\) 54.7895 1.75828 0.879139 0.476566i \(-0.158119\pi\)
0.879139 + 0.476566i \(0.158119\pi\)
\(972\) 0 0
\(973\) −1.41074 −0.0452261
\(974\) 0 0
\(975\) 19.4261 0.622132
\(976\) 0 0
\(977\) 28.9690 0.926800 0.463400 0.886149i \(-0.346629\pi\)
0.463400 + 0.886149i \(0.346629\pi\)
\(978\) 0 0
\(979\) −45.7035 −1.46069
\(980\) 0 0
\(981\) −6.08585 −0.194306
\(982\) 0 0
\(983\) 35.2997 1.12589 0.562943 0.826495i \(-0.309669\pi\)
0.562943 + 0.826495i \(0.309669\pi\)
\(984\) 0 0
\(985\) 9.40847 0.299779
\(986\) 0 0
\(987\) 8.22994 0.261962
\(988\) 0 0
\(989\) −23.3352 −0.742016
\(990\) 0 0
\(991\) −8.34833 −0.265194 −0.132597 0.991170i \(-0.542332\pi\)
−0.132597 + 0.991170i \(0.542332\pi\)
\(992\) 0 0
\(993\) −40.4264 −1.28289
\(994\) 0 0
\(995\) 4.70635 0.149201
\(996\) 0 0
\(997\) −5.81519 −0.184169 −0.0920845 0.995751i \(-0.529353\pi\)
−0.0920845 + 0.995751i \(0.529353\pi\)
\(998\) 0 0
\(999\) 50.0992 1.58507
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 1216.2.a.w.1.4 4
4.3 odd 2 1216.2.a.x.1.2 4
8.3 odd 2 608.2.a.i.1.3 4
8.5 even 2 608.2.a.j.1.1 yes 4
24.5 odd 2 5472.2.a.bs.1.3 4
24.11 even 2 5472.2.a.bt.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
608.2.a.i.1.3 4 8.3 odd 2
608.2.a.j.1.1 yes 4 8.5 even 2
1216.2.a.w.1.4 4 1.1 even 1 trivial
1216.2.a.x.1.2 4 4.3 odd 2
5472.2.a.bs.1.3 4 24.5 odd 2
5472.2.a.bt.1.3 4 24.11 even 2