Properties

Label 1216.2.a.x.1.2
Level $1216$
Weight $2$
Character 1216.1
Self dual yes
Analytic conductor $9.710$
Analytic rank $0$
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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.15317.1
Defining polynomial: \( x^{4} - 2x^{3} - 4x^{2} + 5x + 2 \) Copy content Toggle raw display
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 1.2
Root \(1.32973\) of defining polynomial
Character \(\chi\) \(=\) 1216.1

$q$-expansion

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

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 \(-0.648855\pi\)
−0.450781 + 0.892634i \(0.648855\pi\)
\(4\) 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 \(-0.0933412\pi\)
0.957312 + 0.289056i \(0.0933412\pi\)
\(8\) 0 0
\(9\) −0.561553 −0.187184
\(10\) 0 0
\(11\) 2.84461 0.857683 0.428842 0.903380i \(-0.358922\pi\)
0.428842 + 0.903380i \(0.358922\pi\)
\(12\) 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 \(-0.633659\pi\)
−0.407670 + 0.913129i \(0.633659\pi\)
\(24\) 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 \(-0.283322\pi\)
0.629349 + 0.777123i \(0.283322\pi\)
\(32\) 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 \(-0.349627\pi\)
0.455034 + 0.890474i \(0.349627\pi\)
\(44\) 0 0
\(45\) −0.474295 −0.0707037
\(46\) 0 0
\(47\) 1.04042 0.151760 0.0758802 0.997117i \(-0.475823\pi\)
0.0758802 + 0.997117i \(0.475823\pi\)
\(48\) 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 \(-0.484470\pi\)
0.0487701 + 0.998810i \(0.484470\pi\)
\(60\) 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 \(-0.281541\pi\)
0.633686 + 0.773590i \(0.281541\pi\)
\(68\) 0 0
\(69\) 6.10604 0.735081
\(70\) 0 0
\(71\) −2.13124 −0.252932 −0.126466 0.991971i \(-0.540363\pi\)
−0.126466 + 0.991971i \(0.540363\pi\)
\(72\) 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 \(-0.570133\pi\)
−0.218550 + 0.975826i \(0.570133\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) 8.44201 0.926631 0.463316 0.886193i \(-0.346660\pi\)
0.463316 + 0.886193i \(0.346660\pi\)
\(84\) 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 \(-0.487258\pi\)
0.0400208 + 0.999199i \(0.487258\pi\)
\(104\) 0 0
\(105\) −6.68109 −0.652008
\(106\) 0 0
\(107\) 9.88860 0.955967 0.477983 0.878369i \(-0.341368\pi\)
0.477983 + 0.878369i \(0.341368\pi\)
\(108\) 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 \(-0.811178\pi\)
−0.829156 + 0.559017i \(0.811178\pi\)
\(128\) 0 0
\(129\) −9.31891 −0.820484
\(130\) 0 0
\(131\) −16.6055 −1.45083 −0.725416 0.688310i \(-0.758353\pi\)
−0.725416 + 0.688310i \(0.758353\pi\)
\(132\) 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 \(-0.503760\pi\)
−0.0118107 + 0.999930i \(0.503760\pi\)
\(140\) 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 \(-0.709926\pi\)
−0.612723 + 0.790297i \(0.709926\pi\)
\(152\) 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 \(-0.491306\pi\)
0.0273108 + 0.999627i \(0.491306\pi\)
\(164\) 0 0
\(165\) −3.75179 −0.292076
\(166\) 0 0
\(167\) −6.24621 −0.483346 −0.241673 0.970358i \(-0.577696\pi\)
−0.241673 + 0.970358i \(0.577696\pi\)
\(168\) 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 \(-0.261157\pi\)
0.681893 + 0.731452i \(0.261157\pi\)
\(180\) 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 \(-0.583846\pi\)
−0.260373 + 0.965508i \(0.583846\pi\)
\(192\) 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 \(-0.563283\pi\)
−0.197501 + 0.980303i \(0.563283\pi\)
\(200\) 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 \(-0.513530\pi\)
−0.0424935 + 0.999097i \(0.513530\pi\)
\(212\) 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 \(-0.438214\pi\)
0.192890 + 0.981220i \(0.438214\pi\)
\(224\) 0 0
\(225\) 2.40717 0.160478
\(226\) 0 0
\(227\) −11.1862 −0.742455 −0.371228 0.928542i \(-0.621063\pi\)
−0.371228 + 0.928542i \(0.621063\pi\)
\(228\) 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 \(-0.321501\pi\)
0.531839 + 0.846845i \(0.321501\pi\)
\(240\) 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 \(-0.309482\pi\)
0.563428 + 0.826165i \(0.309482\pi\)
\(252\) 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 \(-0.504745\pi\)
−0.0149049 + 0.999889i \(0.504745\pi\)
\(264\) 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 \(-0.331991\pi\)
0.503648 + 0.863909i \(0.331991\pi\)
\(272\) 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 \(-0.786985\pi\)
−0.784315 + 0.620363i \(0.786985\pi\)
\(284\) 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 \(-0.531325\pi\)
−0.0982516 + 0.995162i \(0.531325\pi\)
\(308\) 0 0
\(309\) −1.26850 −0.0721625
\(310\) 0 0
\(311\) 29.4935 1.67242 0.836211 0.548408i \(-0.184766\pi\)
0.836211 + 0.548408i \(0.184766\pi\)
\(312\) 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 \(-0.248024\pi\)
0.711483 + 0.702703i \(0.248024\pi\)
\(332\) 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 \(-0.621863\pi\)
−0.373560 + 0.927606i \(0.621863\pi\)
\(348\) 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 \(-0.484314\pi\)
0.0492603 + 0.998786i \(0.484314\pi\)
\(360\) 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 \(-0.683496\pi\)
−0.545069 + 0.838391i \(0.683496\pi\)
\(368\) 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 \(-0.479523\pi\)
0.0642855 + 0.997932i \(0.479523\pi\)
\(380\) 0 0
\(381\) 29.1827 1.49507
\(382\) 0 0
\(383\) 21.8155 1.11472 0.557359 0.830272i \(-0.311815\pi\)
0.557359 + 0.830272i \(0.311815\pi\)
\(384\) 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 \(-0.547288\pi\)
−0.148014 + 0.988985i \(0.547288\pi\)
\(420\) 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 \(-0.410208\pi\)
0.278364 + 0.960476i \(0.410208\pi\)
\(432\) 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 \(-0.297617\pi\)
0.593825 + 0.804594i \(0.297617\pi\)
\(440\) 0 0
\(441\) −10.4789 −0.498994
\(442\) 0 0
\(443\) −23.9911 −1.13985 −0.569926 0.821696i \(-0.693028\pi\)
−0.569926 + 0.821696i \(0.693028\pi\)
\(444\) 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 \(-0.675997\pi\)
−0.525168 + 0.850999i \(0.675997\pi\)
\(464\) 0 0
\(465\) −9.24309 −0.428638
\(466\) 0 0
\(467\) −17.1666 −0.794377 −0.397189 0.917737i \(-0.630014\pi\)
−0.397189 + 0.917737i \(0.630014\pi\)
\(468\) 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 \(-0.378835\pi\)
0.371526 + 0.928423i \(0.378835\pi\)
\(480\) 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 \(-0.635351\pi\)
−0.412520 + 0.910948i \(0.635351\pi\)
\(488\) 0 0
\(489\) −1.08897 −0.0492449
\(490\) 0 0
\(491\) 39.2493 1.77130 0.885649 0.464356i \(-0.153714\pi\)
0.885649 + 0.464356i \(0.153714\pi\)
\(492\) 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 \(-0.795319\pi\)
−0.800286 + 0.599619i \(0.795319\pi\)
\(500\) 0 0
\(501\) 9.75379 0.435767
\(502\) 0 0
\(503\) −27.5512 −1.22845 −0.614223 0.789132i \(-0.710530\pi\)
−0.614223 + 0.789132i \(0.710530\pi\)
\(504\) 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 \(-0.356933\pi\)
0.434478 + 0.900682i \(0.356933\pi\)
\(524\) 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 \(-0.242092\pi\)
0.724455 + 0.689322i \(0.242092\pi\)
\(548\) 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 \(-0.534369\pi\)
−0.107764 + 0.994177i \(0.534369\pi\)
\(564\) 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 \(-0.662703\pi\)
−0.489179 + 0.872183i \(0.662703\pi\)
\(572\) 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 \(-0.774453\pi\)
−0.759288 + 0.650754i \(0.774453\pi\)
\(588\) 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 \(-0.473538\pi\)
0.0830367 + 0.996546i \(0.473538\pi\)
\(600\) 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 \(-0.673701\pi\)
−0.519014 + 0.854766i \(0.673701\pi\)
\(608\) 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 \(-0.719421\pi\)
−0.636022 + 0.771671i \(0.719421\pi\)
\(620\) 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 \(-0.475507\pi\)
0.0768696 + 0.997041i \(0.475507\pi\)
\(632\) 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 \(-0.579212\pi\)
−0.246290 + 0.969196i \(0.579212\pi\)
\(644\) 0 0
\(645\) −7.87088 −0.309915
\(646\) 0 0
\(647\) 23.5743 0.926802 0.463401 0.886149i \(-0.346629\pi\)
0.463401 + 0.886149i \(0.346629\pi\)
\(648\) 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 \(-0.216621\pi\)
0.777237 + 0.629209i \(0.216621\pi\)
\(660\) 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 \(-0.413790\pi\)
0.267536 + 0.963548i \(0.413790\pi\)
\(684\) 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 \(-0.500011\pi\)
−3.52963e−5 1.00000i \(0.500011\pi\)
\(692\) 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 \(-0.317674\pi\)
0.541982 + 0.840390i \(0.317674\pi\)
\(720\) 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 \(-0.560343\pi\)
−0.188439 + 0.982085i \(0.560343\pi\)
\(728\) 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 \(-0.291346\pi\)
0.609560 + 0.792740i \(0.291346\pi\)
\(740\) 0 0
\(741\) 4.53178 0.166479
\(742\) 0 0
\(743\) −35.5006 −1.30239 −0.651195 0.758911i \(-0.725732\pi\)
−0.651195 + 0.758911i \(0.725732\pi\)
\(744\) 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 \(-0.589808\pi\)
−0.278411 + 0.960462i \(0.589808\pi\)
\(752\) 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 \(-0.427409\pi\)
0.226080 + 0.974109i \(0.427409\pi\)
\(788\) 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 \(-0.464810\pi\)
0.110329 + 0.993895i \(0.464810\pi\)
\(812\) 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 \(-0.348753\pi\)
0.457479 + 0.889221i \(0.348753\pi\)
\(824\) 0 0
\(825\) 19.0413 0.662932
\(826\) 0 0
\(827\) −34.7726 −1.20916 −0.604581 0.796543i \(-0.706660\pi\)
−0.604581 + 0.796543i \(0.706660\pi\)
\(828\) 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 \(-0.750332\pi\)
−0.707844 + 0.706368i \(0.750332\pi\)
\(840\) 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 \(-0.287151\pi\)
0.619955 + 0.784637i \(0.287151\pi\)
\(860\) 0 0
\(861\) 53.8871 1.83647
\(862\) 0 0
\(863\) −48.3454 −1.64570 −0.822849 0.568260i \(-0.807617\pi\)
−0.822849 + 0.568260i \(0.807617\pi\)
\(864\) 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 \(-0.776162\pi\)
−0.762772 + 0.646668i \(0.776162\pi\)
\(884\) 0 0
\(885\) −0.988153 −0.0332164
\(886\) 0 0
\(887\) 50.5271 1.69653 0.848267 0.529569i \(-0.177646\pi\)
0.848267 + 0.529569i \(0.177646\pi\)
\(888\) 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 \(-0.618916\pi\)
−0.364955 + 0.931025i \(0.618916\pi\)
\(908\) 0 0
\(909\) −9.12311 −0.302594
\(910\) 0 0
\(911\) 29.9950 0.993778 0.496889 0.867814i \(-0.334476\pi\)
0.496889 + 0.867814i \(0.334476\pi\)
\(912\) 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 \(-0.331919\pi\)
0.503844 + 0.863795i \(0.331919\pi\)
\(920\) 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 \(-0.564075\pi\)
−0.199942 + 0.979808i \(0.564075\pi\)
\(948\) 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 \(-0.473018\pi\)
0.0846652 + 0.996409i \(0.473018\pi\)
\(968\) 0 0
\(969\) −12.0631 −0.387523
\(970\) 0 0
\(971\) −54.7895 −1.75828 −0.879139 0.476566i \(-0.841881\pi\)
−0.879139 + 0.476566i \(0.841881\pi\)
\(972\) 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 \(-0.690331\pi\)
−0.562943 + 0.826495i \(0.690331\pi\)
\(984\) 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 \(-0.457668\pi\)
0.132597 + 0.991170i \(0.457668\pi\)
\(992\) 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.x.1.2 4
4.3 odd 2 1216.2.a.w.1.4 4
8.3 odd 2 608.2.a.j.1.1 yes 4
8.5 even 2 608.2.a.i.1.3 4
24.5 odd 2 5472.2.a.bt.1.3 4
24.11 even 2 5472.2.a.bs.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
608.2.a.i.1.3 4 8.5 even 2
608.2.a.j.1.1 yes 4 8.3 odd 2
1216.2.a.w.1.4 4 4.3 odd 2
1216.2.a.x.1.2 4 1.1 even 1 trivial
5472.2.a.bs.1.3 4 24.11 even 2
5472.2.a.bt.1.3 4 24.5 odd 2