Properties

Label 3380.2.a.b.1.1
Level $3380$
Weight $2$
Character 3380.1
Self dual yes
Analytic conductor $26.989$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3380.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.9894358832\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 260)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3380.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} +1.00000 q^{5} +3.00000 q^{7} +6.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} +1.00000 q^{5} +3.00000 q^{7} +6.00000 q^{9} +3.00000 q^{11} -3.00000 q^{15} -7.00000 q^{17} +1.00000 q^{19} -9.00000 q^{21} -7.00000 q^{23} +1.00000 q^{25} -9.00000 q^{27} -5.00000 q^{29} -4.00000 q^{31} -9.00000 q^{33} +3.00000 q^{35} -3.00000 q^{37} +7.00000 q^{41} -9.00000 q^{43} +6.00000 q^{45} +8.00000 q^{47} +2.00000 q^{49} +21.0000 q^{51} -6.00000 q^{53} +3.00000 q^{55} -3.00000 q^{57} +5.00000 q^{59} -5.00000 q^{61} +18.0000 q^{63} +13.0000 q^{67} +21.0000 q^{69} -3.00000 q^{71} -14.0000 q^{73} -3.00000 q^{75} +9.00000 q^{77} -8.00000 q^{79} +9.00000 q^{81} +12.0000 q^{83} -7.00000 q^{85} +15.0000 q^{87} +7.00000 q^{89} +12.0000 q^{93} +1.00000 q^{95} -11.0000 q^{97} +18.0000 q^{99} +O(q^{100})\)

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
\(3\) −3.00000 −1.73205 −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 0 0
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) −3.00000 −0.774597
\(16\) 0 0
\(17\) −7.00000 −1.69775 −0.848875 0.528594i \(-0.822719\pi\)
−0.848875 + 0.528594i \(0.822719\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416 0.114708 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) −9.00000 −1.96396
\(22\) 0 0
\(23\) −7.00000 −1.45960 −0.729800 0.683660i \(-0.760387\pi\)
−0.729800 + 0.683660i \(0.760387\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −9.00000 −1.73205
\(28\) 0 0
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) −9.00000 −1.56670
\(34\) 0 0
\(35\) 3.00000 0.507093
\(36\) 0 0
\(37\) −3.00000 −0.493197 −0.246598 0.969118i \(-0.579313\pi\)
−0.246598 + 0.969118i \(0.579313\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.00000 1.09322 0.546608 0.837389i \(-0.315919\pi\)
0.546608 + 0.837389i \(0.315919\pi\)
\(42\) 0 0
\(43\) −9.00000 −1.37249 −0.686244 0.727372i \(-0.740742\pi\)
−0.686244 + 0.727372i \(0.740742\pi\)
\(44\) 0 0
\(45\) 6.00000 0.894427
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) 21.0000 2.94059
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) 0 0
\(57\) −3.00000 −0.397360
\(58\) 0 0
\(59\) 5.00000 0.650945 0.325472 0.945552i \(-0.394477\pi\)
0.325472 + 0.945552i \(0.394477\pi\)
\(60\) 0 0
\(61\) −5.00000 −0.640184 −0.320092 0.947386i \(-0.603714\pi\)
−0.320092 + 0.947386i \(0.603714\pi\)
\(62\) 0 0
\(63\) 18.0000 2.26779
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 13.0000 1.58820 0.794101 0.607785i \(-0.207942\pi\)
0.794101 + 0.607785i \(0.207942\pi\)
\(68\) 0 0
\(69\) 21.0000 2.52810
\(70\) 0 0
\(71\) −3.00000 −0.356034 −0.178017 0.984027i \(-0.556968\pi\)
−0.178017 + 0.984027i \(0.556968\pi\)
\(72\) 0 0
\(73\) −14.0000 −1.63858 −0.819288 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 0 0
\(75\) −3.00000 −0.346410
\(76\) 0 0
\(77\) 9.00000 1.02565
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) −7.00000 −0.759257
\(86\) 0 0
\(87\) 15.0000 1.60817
\(88\) 0 0
\(89\) 7.00000 0.741999 0.370999 0.928633i \(-0.379015\pi\)
0.370999 + 0.928633i \(0.379015\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 12.0000 1.24434
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −11.0000 −1.11688 −0.558440 0.829545i \(-0.688600\pi\)
−0.558440 + 0.829545i \(0.688600\pi\)
\(98\) 0 0
\(99\) 18.0000 1.80907
\(100\) 0 0
\(101\) −9.00000 −0.895533 −0.447767 0.894150i \(-0.647781\pi\)
−0.447767 + 0.894150i \(0.647781\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 0 0
\(105\) −9.00000 −0.878310
\(106\) 0 0
\(107\) −3.00000 −0.290021 −0.145010 0.989430i \(-0.546322\pi\)
−0.145010 + 0.989430i \(0.546322\pi\)
\(108\) 0 0
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) 9.00000 0.854242
\(112\) 0 0
\(113\) 13.0000 1.22294 0.611469 0.791269i \(-0.290579\pi\)
0.611469 + 0.791269i \(0.290579\pi\)
\(114\) 0 0
\(115\) −7.00000 −0.652753
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −21.0000 −1.92507
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) −21.0000 −1.89351
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 1.00000 0.0887357 0.0443678 0.999015i \(-0.485873\pi\)
0.0443678 + 0.999015i \(0.485873\pi\)
\(128\) 0 0
\(129\) 27.0000 2.37722
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) 3.00000 0.260133
\(134\) 0 0
\(135\) −9.00000 −0.774597
\(136\) 0 0
\(137\) −3.00000 −0.256307 −0.128154 0.991754i \(-0.540905\pi\)
−0.128154 + 0.991754i \(0.540905\pi\)
\(138\) 0 0
\(139\) 13.0000 1.10265 0.551323 0.834292i \(-0.314123\pi\)
0.551323 + 0.834292i \(0.314123\pi\)
\(140\) 0 0
\(141\) −24.0000 −2.02116
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −5.00000 −0.415227
\(146\) 0 0
\(147\) −6.00000 −0.494872
\(148\) 0 0
\(149\) 3.00000 0.245770 0.122885 0.992421i \(-0.460785\pi\)
0.122885 + 0.992421i \(0.460785\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) −42.0000 −3.39550
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) 6.00000 0.478852 0.239426 0.970915i \(-0.423041\pi\)
0.239426 + 0.970915i \(0.423041\pi\)
\(158\) 0 0
\(159\) 18.0000 1.42749
\(160\) 0 0
\(161\) −21.0000 −1.65503
\(162\) 0 0
\(163\) 11.0000 0.861586 0.430793 0.902451i \(-0.358234\pi\)
0.430793 + 0.902451i \(0.358234\pi\)
\(164\) 0 0
\(165\) −9.00000 −0.700649
\(166\) 0 0
\(167\) 1.00000 0.0773823 0.0386912 0.999251i \(-0.487681\pi\)
0.0386912 + 0.999251i \(0.487681\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 6.00000 0.458831
\(172\) 0 0
\(173\) −15.0000 −1.14043 −0.570214 0.821496i \(-0.693140\pi\)
−0.570214 + 0.821496i \(0.693140\pi\)
\(174\) 0 0
\(175\) 3.00000 0.226779
\(176\) 0 0
\(177\) −15.0000 −1.12747
\(178\) 0 0
\(179\) 19.0000 1.42013 0.710063 0.704138i \(-0.248666\pi\)
0.710063 + 0.704138i \(0.248666\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 0 0
\(183\) 15.0000 1.10883
\(184\) 0 0
\(185\) −3.00000 −0.220564
\(186\) 0 0
\(187\) −21.0000 −1.53567
\(188\) 0 0
\(189\) −27.0000 −1.96396
\(190\) 0 0
\(191\) −3.00000 −0.217072 −0.108536 0.994092i \(-0.534616\pi\)
−0.108536 + 0.994092i \(0.534616\pi\)
\(192\) 0 0
\(193\) −15.0000 −1.07972 −0.539862 0.841754i \(-0.681524\pi\)
−0.539862 + 0.841754i \(0.681524\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −23.0000 −1.63868 −0.819341 0.573306i \(-0.805660\pi\)
−0.819341 + 0.573306i \(0.805660\pi\)
\(198\) 0 0
\(199\) 9.00000 0.637993 0.318997 0.947756i \(-0.396654\pi\)
0.318997 + 0.947756i \(0.396654\pi\)
\(200\) 0 0
\(201\) −39.0000 −2.75085
\(202\) 0 0
\(203\) −15.0000 −1.05279
\(204\) 0 0
\(205\) 7.00000 0.488901
\(206\) 0 0
\(207\) −42.0000 −2.91920
\(208\) 0 0
\(209\) 3.00000 0.207514
\(210\) 0 0
\(211\) −5.00000 −0.344214 −0.172107 0.985078i \(-0.555058\pi\)
−0.172107 + 0.985078i \(0.555058\pi\)
\(212\) 0 0
\(213\) 9.00000 0.616670
\(214\) 0 0
\(215\) −9.00000 −0.613795
\(216\) 0 0
\(217\) −12.0000 −0.814613
\(218\) 0 0
\(219\) 42.0000 2.83810
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −23.0000 −1.54019 −0.770097 0.637927i \(-0.779792\pi\)
−0.770097 + 0.637927i \(0.779792\pi\)
\(224\) 0 0
\(225\) 6.00000 0.400000
\(226\) 0 0
\(227\) −1.00000 −0.0663723 −0.0331862 0.999449i \(-0.510565\pi\)
−0.0331862 + 0.999449i \(0.510565\pi\)
\(228\) 0 0
\(229\) 26.0000 1.71813 0.859064 0.511868i \(-0.171046\pi\)
0.859064 + 0.511868i \(0.171046\pi\)
\(230\) 0 0
\(231\) −27.0000 −1.77647
\(232\) 0 0
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) 0 0
\(237\) 24.0000 1.55897
\(238\) 0 0
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) −1.00000 −0.0644157 −0.0322078 0.999481i \(-0.510254\pi\)
−0.0322078 + 0.999481i \(0.510254\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.00000 0.127775
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −36.0000 −2.28141
\(250\) 0 0
\(251\) 5.00000 0.315597 0.157799 0.987471i \(-0.449560\pi\)
0.157799 + 0.987471i \(0.449560\pi\)
\(252\) 0 0
\(253\) −21.0000 −1.32026
\(254\) 0 0
\(255\) 21.0000 1.31507
\(256\) 0 0
\(257\) −19.0000 −1.18519 −0.592594 0.805502i \(-0.701896\pi\)
−0.592594 + 0.805502i \(0.701896\pi\)
\(258\) 0 0
\(259\) −9.00000 −0.559233
\(260\) 0 0
\(261\) −30.0000 −1.85695
\(262\) 0 0
\(263\) −7.00000 −0.431638 −0.215819 0.976433i \(-0.569242\pi\)
−0.215819 + 0.976433i \(0.569242\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) 0 0
\(267\) −21.0000 −1.28518
\(268\) 0 0
\(269\) 3.00000 0.182913 0.0914566 0.995809i \(-0.470848\pi\)
0.0914566 + 0.995809i \(0.470848\pi\)
\(270\) 0 0
\(271\) 23.0000 1.39715 0.698575 0.715537i \(-0.253818\pi\)
0.698575 + 0.715537i \(0.253818\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.00000 0.180907
\(276\) 0 0
\(277\) −23.0000 −1.38194 −0.690968 0.722885i \(-0.742815\pi\)
−0.690968 + 0.722885i \(0.742815\pi\)
\(278\) 0 0
\(279\) −24.0000 −1.43684
\(280\) 0 0
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 0 0
\(283\) 1.00000 0.0594438 0.0297219 0.999558i \(-0.490538\pi\)
0.0297219 + 0.999558i \(0.490538\pi\)
\(284\) 0 0
\(285\) −3.00000 −0.177705
\(286\) 0 0
\(287\) 21.0000 1.23959
\(288\) 0 0
\(289\) 32.0000 1.88235
\(290\) 0 0
\(291\) 33.0000 1.93449
\(292\) 0 0
\(293\) 9.00000 0.525786 0.262893 0.964825i \(-0.415323\pi\)
0.262893 + 0.964825i \(0.415323\pi\)
\(294\) 0 0
\(295\) 5.00000 0.291111
\(296\) 0 0
\(297\) −27.0000 −1.56670
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −27.0000 −1.55625
\(302\) 0 0
\(303\) 27.0000 1.55111
\(304\) 0 0
\(305\) −5.00000 −0.286299
\(306\) 0 0
\(307\) 28.0000 1.59804 0.799022 0.601302i \(-0.205351\pi\)
0.799022 + 0.601302i \(0.205351\pi\)
\(308\) 0 0
\(309\) 48.0000 2.73062
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 0 0
\(315\) 18.0000 1.01419
\(316\) 0 0
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) 0 0
\(319\) −15.0000 −0.839839
\(320\) 0 0
\(321\) 9.00000 0.502331
\(322\) 0 0
\(323\) −7.00000 −0.389490
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 42.0000 2.32261
\(328\) 0 0
\(329\) 24.0000 1.32316
\(330\) 0 0
\(331\) 13.0000 0.714545 0.357272 0.934000i \(-0.383707\pi\)
0.357272 + 0.934000i \(0.383707\pi\)
\(332\) 0 0
\(333\) −18.0000 −0.986394
\(334\) 0 0
\(335\) 13.0000 0.710266
\(336\) 0 0
\(337\) −18.0000 −0.980522 −0.490261 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) 0 0
\(339\) −39.0000 −2.11819
\(340\) 0 0
\(341\) −12.0000 −0.649836
\(342\) 0 0
\(343\) −15.0000 −0.809924
\(344\) 0 0
\(345\) 21.0000 1.13060
\(346\) 0 0
\(347\) −13.0000 −0.697877 −0.348938 0.937146i \(-0.613458\pi\)
−0.348938 + 0.937146i \(0.613458\pi\)
\(348\) 0 0
\(349\) −25.0000 −1.33822 −0.669110 0.743164i \(-0.733324\pi\)
−0.669110 + 0.743164i \(0.733324\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 21.0000 1.11772 0.558859 0.829263i \(-0.311239\pi\)
0.558859 + 0.829263i \(0.311239\pi\)
\(354\) 0 0
\(355\) −3.00000 −0.159223
\(356\) 0 0
\(357\) 63.0000 3.33431
\(358\) 0 0
\(359\) −8.00000 −0.422224 −0.211112 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) 6.00000 0.314918
\(364\) 0 0
\(365\) −14.0000 −0.732793
\(366\) 0 0
\(367\) 9.00000 0.469796 0.234898 0.972020i \(-0.424524\pi\)
0.234898 + 0.972020i \(0.424524\pi\)
\(368\) 0 0
\(369\) 42.0000 2.18643
\(370\) 0 0
\(371\) −18.0000 −0.934513
\(372\) 0 0
\(373\) −27.0000 −1.39801 −0.699004 0.715118i \(-0.746373\pi\)
−0.699004 + 0.715118i \(0.746373\pi\)
\(374\) 0 0
\(375\) −3.00000 −0.154919
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −9.00000 −0.462299 −0.231149 0.972918i \(-0.574249\pi\)
−0.231149 + 0.972918i \(0.574249\pi\)
\(380\) 0 0
\(381\) −3.00000 −0.153695
\(382\) 0 0
\(383\) −13.0000 −0.664269 −0.332134 0.943232i \(-0.607769\pi\)
−0.332134 + 0.943232i \(0.607769\pi\)
\(384\) 0 0
\(385\) 9.00000 0.458682
\(386\) 0 0
\(387\) −54.0000 −2.74497
\(388\) 0 0
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) 49.0000 2.47804
\(392\) 0 0
\(393\) −12.0000 −0.605320
\(394\) 0 0
\(395\) −8.00000 −0.402524
\(396\) 0 0
\(397\) 33.0000 1.65622 0.828111 0.560564i \(-0.189416\pi\)
0.828111 + 0.560564i \(0.189416\pi\)
\(398\) 0 0
\(399\) −9.00000 −0.450564
\(400\) 0 0
\(401\) 15.0000 0.749064 0.374532 0.927214i \(-0.377803\pi\)
0.374532 + 0.927214i \(0.377803\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 9.00000 0.447214
\(406\) 0 0
\(407\) −9.00000 −0.446113
\(408\) 0 0
\(409\) −1.00000 −0.0494468 −0.0247234 0.999694i \(-0.507871\pi\)
−0.0247234 + 0.999694i \(0.507871\pi\)
\(410\) 0 0
\(411\) 9.00000 0.443937
\(412\) 0 0
\(413\) 15.0000 0.738102
\(414\) 0 0
\(415\) 12.0000 0.589057
\(416\) 0 0
\(417\) −39.0000 −1.90984
\(418\) 0 0
\(419\) −33.0000 −1.61216 −0.806078 0.591810i \(-0.798414\pi\)
−0.806078 + 0.591810i \(0.798414\pi\)
\(420\) 0 0
\(421\) 34.0000 1.65706 0.828529 0.559946i \(-0.189178\pi\)
0.828529 + 0.559946i \(0.189178\pi\)
\(422\) 0 0
\(423\) 48.0000 2.33384
\(424\) 0 0
\(425\) −7.00000 −0.339550
\(426\) 0 0
\(427\) −15.0000 −0.725901
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −9.00000 −0.433515 −0.216757 0.976226i \(-0.569548\pi\)
−0.216757 + 0.976226i \(0.569548\pi\)
\(432\) 0 0
\(433\) 1.00000 0.0480569 0.0240285 0.999711i \(-0.492351\pi\)
0.0240285 + 0.999711i \(0.492351\pi\)
\(434\) 0 0
\(435\) 15.0000 0.719195
\(436\) 0 0
\(437\) −7.00000 −0.334855
\(438\) 0 0
\(439\) 3.00000 0.143182 0.0715911 0.997434i \(-0.477192\pi\)
0.0715911 + 0.997434i \(0.477192\pi\)
\(440\) 0 0
\(441\) 12.0000 0.571429
\(442\) 0 0
\(443\) 24.0000 1.14027 0.570137 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 0 0
\(445\) 7.00000 0.331832
\(446\) 0 0
\(447\) −9.00000 −0.425685
\(448\) 0 0
\(449\) −21.0000 −0.991051 −0.495526 0.868593i \(-0.665025\pi\)
−0.495526 + 0.868593i \(0.665025\pi\)
\(450\) 0 0
\(451\) 21.0000 0.988851
\(452\) 0 0
\(453\) 24.0000 1.12762
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −23.0000 −1.07589 −0.537947 0.842978i \(-0.680800\pi\)
−0.537947 + 0.842978i \(0.680800\pi\)
\(458\) 0 0
\(459\) 63.0000 2.94059
\(460\) 0 0
\(461\) 11.0000 0.512321 0.256161 0.966634i \(-0.417542\pi\)
0.256161 + 0.966634i \(0.417542\pi\)
\(462\) 0 0
\(463\) −28.0000 −1.30127 −0.650635 0.759390i \(-0.725497\pi\)
−0.650635 + 0.759390i \(0.725497\pi\)
\(464\) 0 0
\(465\) 12.0000 0.556487
\(466\) 0 0
\(467\) 20.0000 0.925490 0.462745 0.886492i \(-0.346865\pi\)
0.462745 + 0.886492i \(0.346865\pi\)
\(468\) 0 0
\(469\) 39.0000 1.80085
\(470\) 0 0
\(471\) −18.0000 −0.829396
\(472\) 0 0
\(473\) −27.0000 −1.24146
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) −36.0000 −1.64833
\(478\) 0 0
\(479\) −1.00000 −0.0456912 −0.0228456 0.999739i \(-0.507273\pi\)
−0.0228456 + 0.999739i \(0.507273\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 63.0000 2.86660
\(484\) 0 0
\(485\) −11.0000 −0.499484
\(486\) 0 0
\(487\) −17.0000 −0.770344 −0.385172 0.922845i \(-0.625858\pi\)
−0.385172 + 0.922845i \(0.625858\pi\)
\(488\) 0 0
\(489\) −33.0000 −1.49231
\(490\) 0 0
\(491\) 23.0000 1.03798 0.518988 0.854782i \(-0.326309\pi\)
0.518988 + 0.854782i \(0.326309\pi\)
\(492\) 0 0
\(493\) 35.0000 1.57632
\(494\) 0 0
\(495\) 18.0000 0.809040
\(496\) 0 0
\(497\) −9.00000 −0.403705
\(498\) 0 0
\(499\) −16.0000 −0.716258 −0.358129 0.933672i \(-0.616585\pi\)
−0.358129 + 0.933672i \(0.616585\pi\)
\(500\) 0 0
\(501\) −3.00000 −0.134030
\(502\) 0 0
\(503\) 11.0000 0.490466 0.245233 0.969464i \(-0.421136\pi\)
0.245233 + 0.969464i \(0.421136\pi\)
\(504\) 0 0
\(505\) −9.00000 −0.400495
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 15.0000 0.664863 0.332432 0.943127i \(-0.392131\pi\)
0.332432 + 0.943127i \(0.392131\pi\)
\(510\) 0 0
\(511\) −42.0000 −1.85797
\(512\) 0 0
\(513\) −9.00000 −0.397360
\(514\) 0 0
\(515\) −16.0000 −0.705044
\(516\) 0 0
\(517\) 24.0000 1.05552
\(518\) 0 0
\(519\) 45.0000 1.97528
\(520\) 0 0
\(521\) −34.0000 −1.48957 −0.744784 0.667306i \(-0.767447\pi\)
−0.744784 + 0.667306i \(0.767447\pi\)
\(522\) 0 0
\(523\) −23.0000 −1.00572 −0.502860 0.864368i \(-0.667719\pi\)
−0.502860 + 0.864368i \(0.667719\pi\)
\(524\) 0 0
\(525\) −9.00000 −0.392792
\(526\) 0 0
\(527\) 28.0000 1.21970
\(528\) 0 0
\(529\) 26.0000 1.13043
\(530\) 0 0
\(531\) 30.0000 1.30189
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −3.00000 −0.129701
\(536\) 0 0
\(537\) −57.0000 −2.45973
\(538\) 0 0
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) 10.0000 0.429934 0.214967 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) 0 0
\(543\) 42.0000 1.80239
\(544\) 0 0
\(545\) −14.0000 −0.599694
\(546\) 0 0
\(547\) −16.0000 −0.684111 −0.342055 0.939680i \(-0.611123\pi\)
−0.342055 + 0.939680i \(0.611123\pi\)
\(548\) 0 0
\(549\) −30.0000 −1.28037
\(550\) 0 0
\(551\) −5.00000 −0.213007
\(552\) 0 0
\(553\) −24.0000 −1.02058
\(554\) 0 0
\(555\) 9.00000 0.382029
\(556\) 0 0
\(557\) −19.0000 −0.805056 −0.402528 0.915408i \(-0.631868\pi\)
−0.402528 + 0.915408i \(0.631868\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 63.0000 2.65986
\(562\) 0 0
\(563\) −45.0000 −1.89652 −0.948262 0.317489i \(-0.897160\pi\)
−0.948262 + 0.317489i \(0.897160\pi\)
\(564\) 0 0
\(565\) 13.0000 0.546914
\(566\) 0 0
\(567\) 27.0000 1.13389
\(568\) 0 0
\(569\) 19.0000 0.796521 0.398261 0.917272i \(-0.369614\pi\)
0.398261 + 0.917272i \(0.369614\pi\)
\(570\) 0 0
\(571\) −40.0000 −1.67395 −0.836974 0.547243i \(-0.815677\pi\)
−0.836974 + 0.547243i \(0.815677\pi\)
\(572\) 0 0
\(573\) 9.00000 0.375980
\(574\) 0 0
\(575\) −7.00000 −0.291920
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 0 0
\(579\) 45.0000 1.87014
\(580\) 0 0
\(581\) 36.0000 1.49353
\(582\) 0 0
\(583\) −18.0000 −0.745484
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 37.0000 1.52715 0.763577 0.645717i \(-0.223441\pi\)
0.763577 + 0.645717i \(0.223441\pi\)
\(588\) 0 0
\(589\) −4.00000 −0.164817
\(590\) 0 0
\(591\) 69.0000 2.83828
\(592\) 0 0
\(593\) 26.0000 1.06769 0.533846 0.845582i \(-0.320746\pi\)
0.533846 + 0.845582i \(0.320746\pi\)
\(594\) 0 0
\(595\) −21.0000 −0.860916
\(596\) 0 0
\(597\) −27.0000 −1.10504
\(598\) 0 0
\(599\) 8.00000 0.326871 0.163436 0.986554i \(-0.447742\pi\)
0.163436 + 0.986554i \(0.447742\pi\)
\(600\) 0 0
\(601\) −13.0000 −0.530281 −0.265141 0.964210i \(-0.585418\pi\)
−0.265141 + 0.964210i \(0.585418\pi\)
\(602\) 0 0
\(603\) 78.0000 3.17641
\(604\) 0 0
\(605\) −2.00000 −0.0813116
\(606\) 0 0
\(607\) −9.00000 −0.365299 −0.182649 0.983178i \(-0.558467\pi\)
−0.182649 + 0.983178i \(0.558467\pi\)
\(608\) 0 0
\(609\) 45.0000 1.82349
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −31.0000 −1.25208 −0.626039 0.779792i \(-0.715325\pi\)
−0.626039 + 0.779792i \(0.715325\pi\)
\(614\) 0 0
\(615\) −21.0000 −0.846802
\(616\) 0 0
\(617\) 29.0000 1.16750 0.583748 0.811935i \(-0.301586\pi\)
0.583748 + 0.811935i \(0.301586\pi\)
\(618\) 0 0
\(619\) 12.0000 0.482321 0.241160 0.970485i \(-0.422472\pi\)
0.241160 + 0.970485i \(0.422472\pi\)
\(620\) 0 0
\(621\) 63.0000 2.52810
\(622\) 0 0
\(623\) 21.0000 0.841347
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −9.00000 −0.359425
\(628\) 0 0
\(629\) 21.0000 0.837325
\(630\) 0 0
\(631\) −15.0000 −0.597141 −0.298570 0.954388i \(-0.596510\pi\)
−0.298570 + 0.954388i \(0.596510\pi\)
\(632\) 0 0
\(633\) 15.0000 0.596196
\(634\) 0 0
\(635\) 1.00000 0.0396838
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −18.0000 −0.712069
\(640\) 0 0
\(641\) −21.0000 −0.829450 −0.414725 0.909947i \(-0.636122\pi\)
−0.414725 + 0.909947i \(0.636122\pi\)
\(642\) 0 0
\(643\) 7.00000 0.276053 0.138027 0.990429i \(-0.455924\pi\)
0.138027 + 0.990429i \(0.455924\pi\)
\(644\) 0 0
\(645\) 27.0000 1.06312
\(646\) 0 0
\(647\) 17.0000 0.668339 0.334169 0.942513i \(-0.391544\pi\)
0.334169 + 0.942513i \(0.391544\pi\)
\(648\) 0 0
\(649\) 15.0000 0.588802
\(650\) 0 0
\(651\) 36.0000 1.41095
\(652\) 0 0
\(653\) −11.0000 −0.430463 −0.215232 0.976563i \(-0.569051\pi\)
−0.215232 + 0.976563i \(0.569051\pi\)
\(654\) 0 0
\(655\) 4.00000 0.156293
\(656\) 0 0
\(657\) −84.0000 −3.27715
\(658\) 0 0
\(659\) −15.0000 −0.584317 −0.292159 0.956370i \(-0.594373\pi\)
−0.292159 + 0.956370i \(0.594373\pi\)
\(660\) 0 0
\(661\) −17.0000 −0.661223 −0.330612 0.943767i \(-0.607255\pi\)
−0.330612 + 0.943767i \(0.607255\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 3.00000 0.116335
\(666\) 0 0
\(667\) 35.0000 1.35521
\(668\) 0 0
\(669\) 69.0000 2.66769
\(670\) 0 0
\(671\) −15.0000 −0.579069
\(672\) 0 0
\(673\) 13.0000 0.501113 0.250557 0.968102i \(-0.419386\pi\)
0.250557 + 0.968102i \(0.419386\pi\)
\(674\) 0 0
\(675\) −9.00000 −0.346410
\(676\) 0 0
\(677\) 22.0000 0.845529 0.422764 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) 0 0
\(679\) −33.0000 −1.26642
\(680\) 0 0
\(681\) 3.00000 0.114960
\(682\) 0 0
\(683\) 31.0000 1.18618 0.593091 0.805135i \(-0.297907\pi\)
0.593091 + 0.805135i \(0.297907\pi\)
\(684\) 0 0
\(685\) −3.00000 −0.114624
\(686\) 0 0
\(687\) −78.0000 −2.97589
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −25.0000 −0.951045 −0.475522 0.879704i \(-0.657741\pi\)
−0.475522 + 0.879704i \(0.657741\pi\)
\(692\) 0 0
\(693\) 54.0000 2.05129
\(694\) 0 0
\(695\) 13.0000 0.493118
\(696\) 0 0
\(697\) −49.0000 −1.85601
\(698\) 0 0
\(699\) −54.0000 −2.04247
\(700\) 0 0
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) 0 0
\(703\) −3.00000 −0.113147
\(704\) 0 0
\(705\) −24.0000 −0.903892
\(706\) 0 0
\(707\) −27.0000 −1.01544
\(708\) 0 0
\(709\) 35.0000 1.31445 0.657226 0.753693i \(-0.271730\pi\)
0.657226 + 0.753693i \(0.271730\pi\)
\(710\) 0 0
\(711\) −48.0000 −1.80014
\(712\) 0 0
\(713\) 28.0000 1.04861
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 48.0000 1.79259
\(718\) 0 0
\(719\) 1.00000 0.0372937 0.0186469 0.999826i \(-0.494064\pi\)
0.0186469 + 0.999826i \(0.494064\pi\)
\(720\) 0 0
\(721\) −48.0000 −1.78761
\(722\) 0 0
\(723\) 3.00000 0.111571
\(724\) 0 0
\(725\) −5.00000 −0.185695
\(726\) 0 0
\(727\) −28.0000 −1.03846 −0.519231 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 63.0000 2.33014
\(732\) 0 0
\(733\) 14.0000 0.517102 0.258551 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) 0 0
\(735\) −6.00000 −0.221313
\(736\) 0 0
\(737\) 39.0000 1.43658
\(738\) 0 0
\(739\) 39.0000 1.43464 0.717319 0.696745i \(-0.245369\pi\)
0.717319 + 0.696745i \(0.245369\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1.00000 0.0366864 0.0183432 0.999832i \(-0.494161\pi\)
0.0183432 + 0.999832i \(0.494161\pi\)
\(744\) 0 0
\(745\) 3.00000 0.109911
\(746\) 0 0
\(747\) 72.0000 2.63434
\(748\) 0 0
\(749\) −9.00000 −0.328853
\(750\) 0 0
\(751\) −13.0000 −0.474377 −0.237188 0.971464i \(-0.576226\pi\)
−0.237188 + 0.971464i \(0.576226\pi\)
\(752\) 0 0
\(753\) −15.0000 −0.546630
\(754\) 0 0
\(755\) −8.00000 −0.291150
\(756\) 0 0
\(757\) 1.00000 0.0363456 0.0181728 0.999835i \(-0.494215\pi\)
0.0181728 + 0.999835i \(0.494215\pi\)
\(758\) 0 0
\(759\) 63.0000 2.28676
\(760\) 0 0
\(761\) 51.0000 1.84875 0.924374 0.381487i \(-0.124588\pi\)
0.924374 + 0.381487i \(0.124588\pi\)
\(762\) 0 0
\(763\) −42.0000 −1.52050
\(764\) 0 0
\(765\) −42.0000 −1.51851
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −5.00000 −0.180305 −0.0901523 0.995928i \(-0.528735\pi\)
−0.0901523 + 0.995928i \(0.528735\pi\)
\(770\) 0 0
\(771\) 57.0000 2.05280
\(772\) 0 0
\(773\) 25.0000 0.899188 0.449594 0.893233i \(-0.351569\pi\)
0.449594 + 0.893233i \(0.351569\pi\)
\(774\) 0 0
\(775\) −4.00000 −0.143684
\(776\) 0 0
\(777\) 27.0000 0.968620
\(778\) 0 0
\(779\) 7.00000 0.250801
\(780\) 0 0
\(781\) −9.00000 −0.322045
\(782\) 0 0
\(783\) 45.0000 1.60817
\(784\) 0 0
\(785\) 6.00000 0.214149
\(786\) 0 0
\(787\) −1.00000 −0.0356462 −0.0178231 0.999841i \(-0.505674\pi\)
−0.0178231 + 0.999841i \(0.505674\pi\)
\(788\) 0 0
\(789\) 21.0000 0.747620
\(790\) 0 0
\(791\) 39.0000 1.38668
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 18.0000 0.638394
\(796\) 0 0
\(797\) −7.00000 −0.247953 −0.123976 0.992285i \(-0.539565\pi\)
−0.123976 + 0.992285i \(0.539565\pi\)
\(798\) 0 0
\(799\) −56.0000 −1.98114
\(800\) 0 0
\(801\) 42.0000 1.48400
\(802\) 0 0
\(803\) −42.0000 −1.48215
\(804\) 0 0
\(805\) −21.0000 −0.740153
\(806\) 0 0
\(807\) −9.00000 −0.316815
\(808\) 0 0
\(809\) −25.0000 −0.878953 −0.439477 0.898254i \(-0.644836\pi\)
−0.439477 + 0.898254i \(0.644836\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) −69.0000 −2.41994
\(814\) 0 0
\(815\) 11.0000 0.385313
\(816\) 0 0
\(817\) −9.00000 −0.314870
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 15.0000 0.523504 0.261752 0.965135i \(-0.415700\pi\)
0.261752 + 0.965135i \(0.415700\pi\)
\(822\) 0 0
\(823\) 47.0000 1.63832 0.819159 0.573567i \(-0.194441\pi\)
0.819159 + 0.573567i \(0.194441\pi\)
\(824\) 0 0
\(825\) −9.00000 −0.313340
\(826\) 0 0
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 0 0
\(829\) −13.0000 −0.451509 −0.225754 0.974184i \(-0.572485\pi\)
−0.225754 + 0.974184i \(0.572485\pi\)
\(830\) 0 0
\(831\) 69.0000 2.39358
\(832\) 0 0
\(833\) −14.0000 −0.485071
\(834\) 0 0
\(835\) 1.00000 0.0346064
\(836\) 0 0
\(837\) 36.0000 1.24434
\(838\) 0 0
\(839\) 13.0000 0.448810 0.224405 0.974496i \(-0.427956\pi\)
0.224405 + 0.974496i \(0.427956\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 0 0
\(843\) 30.0000 1.03325
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −6.00000 −0.206162
\(848\) 0 0
\(849\) −3.00000 −0.102960
\(850\) 0 0
\(851\) 21.0000 0.719871
\(852\) 0 0
\(853\) 10.0000 0.342393 0.171197 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) 0 0
\(855\) 6.00000 0.205196
\(856\) 0 0
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) 44.0000 1.50126 0.750630 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) 0 0
\(861\) −63.0000 −2.14703
\(862\) 0 0
\(863\) 16.0000 0.544646 0.272323 0.962206i \(-0.412208\pi\)
0.272323 + 0.962206i \(0.412208\pi\)
\(864\) 0 0
\(865\) −15.0000 −0.510015
\(866\) 0 0
\(867\) −96.0000 −3.26033
\(868\) 0 0
\(869\) −24.0000 −0.814144
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −66.0000 −2.23376
\(874\) 0 0
\(875\) 3.00000 0.101419
\(876\) 0 0
\(877\) 5.00000 0.168838 0.0844190 0.996430i \(-0.473097\pi\)
0.0844190 + 0.996430i \(0.473097\pi\)
\(878\) 0 0
\(879\) −27.0000 −0.910687
\(880\) 0 0
\(881\) 11.0000 0.370599 0.185300 0.982682i \(-0.440674\pi\)
0.185300 + 0.982682i \(0.440674\pi\)
\(882\) 0 0
\(883\) −44.0000 −1.48072 −0.740359 0.672212i \(-0.765344\pi\)
−0.740359 + 0.672212i \(0.765344\pi\)
\(884\) 0 0
\(885\) −15.0000 −0.504219
\(886\) 0 0
\(887\) −19.0000 −0.637958 −0.318979 0.947762i \(-0.603340\pi\)
−0.318979 + 0.947762i \(0.603340\pi\)
\(888\) 0 0
\(889\) 3.00000 0.100617
\(890\) 0 0
\(891\) 27.0000 0.904534
\(892\) 0 0
\(893\) 8.00000 0.267710
\(894\) 0 0
\(895\) 19.0000 0.635100
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 20.0000 0.667037
\(900\) 0 0
\(901\) 42.0000 1.39922
\(902\) 0 0
\(903\) 81.0000 2.69551
\(904\) 0 0
\(905\) −14.0000 −0.465376
\(906\) 0 0
\(907\) 17.0000 0.564476 0.282238 0.959344i \(-0.408923\pi\)
0.282238 + 0.959344i \(0.408923\pi\)
\(908\) 0 0
\(909\) −54.0000 −1.79107
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 0 0
\(913\) 36.0000 1.19143
\(914\) 0 0
\(915\) 15.0000 0.495885
\(916\) 0 0
\(917\) 12.0000 0.396275
\(918\) 0 0
\(919\) −43.0000 −1.41844 −0.709220 0.704988i \(-0.750953\pi\)
−0.709220 + 0.704988i \(0.750953\pi\)
\(920\) 0 0
\(921\) −84.0000 −2.76789
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −3.00000 −0.0986394
\(926\) 0 0
\(927\) −96.0000 −3.15305
\(928\) 0 0
\(929\) −53.0000 −1.73887 −0.869437 0.494044i \(-0.835518\pi\)
−0.869437 + 0.494044i \(0.835518\pi\)
\(930\) 0 0
\(931\) 2.00000 0.0655474
\(932\) 0 0
\(933\) 72.0000 2.35717
\(934\) 0 0
\(935\) −21.0000 −0.686773
\(936\) 0 0
\(937\) 50.0000 1.63343 0.816714 0.577042i \(-0.195793\pi\)
0.816714 + 0.577042i \(0.195793\pi\)
\(938\) 0 0
\(939\) 18.0000 0.587408
\(940\) 0 0
\(941\) 42.0000 1.36916 0.684580 0.728937i \(-0.259985\pi\)
0.684580 + 0.728937i \(0.259985\pi\)
\(942\) 0 0
\(943\) −49.0000 −1.59566
\(944\) 0 0
\(945\) −27.0000 −0.878310
\(946\) 0 0
\(947\) −55.0000 −1.78726 −0.893630 0.448805i \(-0.851850\pi\)
−0.893630 + 0.448805i \(0.851850\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 6.00000 0.194563
\(952\) 0 0
\(953\) −31.0000 −1.00419 −0.502094 0.864813i \(-0.667437\pi\)
−0.502094 + 0.864813i \(0.667437\pi\)
\(954\) 0 0
\(955\) −3.00000 −0.0970777
\(956\) 0 0
\(957\) 45.0000 1.45464
\(958\) 0 0
\(959\) −9.00000 −0.290625
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) −18.0000 −0.580042
\(964\) 0 0
\(965\) −15.0000 −0.482867
\(966\) 0 0
\(967\) −48.0000 −1.54358 −0.771788 0.635880i \(-0.780637\pi\)
−0.771788 + 0.635880i \(0.780637\pi\)
\(968\) 0 0
\(969\) 21.0000 0.674617
\(970\) 0 0
\(971\) −43.0000 −1.37994 −0.689968 0.723840i \(-0.742375\pi\)
−0.689968 + 0.723840i \(0.742375\pi\)
\(972\) 0 0
\(973\) 39.0000 1.25028
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −3.00000 −0.0959785 −0.0479893 0.998848i \(-0.515281\pi\)
−0.0479893 + 0.998848i \(0.515281\pi\)
\(978\) 0 0
\(979\) 21.0000 0.671163
\(980\) 0 0
\(981\) −84.0000 −2.68191
\(982\) 0 0
\(983\) 36.0000 1.14822 0.574111 0.818778i \(-0.305348\pi\)
0.574111 + 0.818778i \(0.305348\pi\)
\(984\) 0 0
\(985\) −23.0000 −0.732841
\(986\) 0 0
\(987\) −72.0000 −2.29179
\(988\) 0 0
\(989\) 63.0000 2.00328
\(990\) 0 0
\(991\) −13.0000 −0.412959 −0.206479 0.978451i \(-0.566201\pi\)
−0.206479 + 0.978451i \(0.566201\pi\)
\(992\) 0 0
\(993\) −39.0000 −1.23763
\(994\) 0 0
\(995\) 9.00000 0.285319
\(996\) 0 0
\(997\) 13.0000 0.411714 0.205857 0.978582i \(-0.434002\pi\)
0.205857 + 0.978582i \(0.434002\pi\)
\(998\) 0 0
\(999\) 27.0000 0.854242
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 3380.2.a.b.1.1 1
13.3 even 3 260.2.i.d.61.1 2
13.5 odd 4 3380.2.f.a.3041.1 2
13.8 odd 4 3380.2.f.a.3041.2 2
13.9 even 3 260.2.i.d.81.1 yes 2
13.12 even 2 3380.2.a.a.1.1 1
39.29 odd 6 2340.2.q.a.1621.1 2
39.35 odd 6 2340.2.q.a.2161.1 2
52.3 odd 6 1040.2.q.b.321.1 2
52.35 odd 6 1040.2.q.b.81.1 2
65.3 odd 12 1300.2.bb.e.1049.2 4
65.9 even 6 1300.2.i.a.601.1 2
65.22 odd 12 1300.2.bb.e.549.2 4
65.29 even 6 1300.2.i.a.1101.1 2
65.42 odd 12 1300.2.bb.e.1049.1 4
65.48 odd 12 1300.2.bb.e.549.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
260.2.i.d.61.1 2 13.3 even 3
260.2.i.d.81.1 yes 2 13.9 even 3
1040.2.q.b.81.1 2 52.35 odd 6
1040.2.q.b.321.1 2 52.3 odd 6
1300.2.i.a.601.1 2 65.9 even 6
1300.2.i.a.1101.1 2 65.29 even 6
1300.2.bb.e.549.1 4 65.48 odd 12
1300.2.bb.e.549.2 4 65.22 odd 12
1300.2.bb.e.1049.1 4 65.42 odd 12
1300.2.bb.e.1049.2 4 65.3 odd 12
2340.2.q.a.1621.1 2 39.29 odd 6
2340.2.q.a.2161.1 2 39.35 odd 6
3380.2.a.a.1.1 1 13.12 even 2
3380.2.a.b.1.1 1 1.1 even 1 trivial
3380.2.f.a.3041.1 2 13.5 odd 4
3380.2.f.a.3041.2 2 13.8 odd 4