Properties

Label 3920.2.a.bf.1.1
Level $3920$
Weight $2$
Character 3920.1
Self dual yes
Analytic conductor $31.301$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3920.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.3013575923\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3920.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{3} -1.00000 q^{5} +1.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{3} -1.00000 q^{5} +1.00000 q^{9} +1.00000 q^{11} -3.00000 q^{13} -2.00000 q^{15} -2.00000 q^{17} +5.00000 q^{19} -7.00000 q^{23} +1.00000 q^{25} -4.00000 q^{27} -6.00000 q^{29} -4.00000 q^{31} +2.00000 q^{33} -5.00000 q^{37} -6.00000 q^{39} -5.00000 q^{41} -6.00000 q^{43} -1.00000 q^{45} +9.00000 q^{47} -4.00000 q^{51} +11.0000 q^{53} -1.00000 q^{55} +10.0000 q^{57} -8.00000 q^{59} -12.0000 q^{61} +3.00000 q^{65} +4.00000 q^{67} -14.0000 q^{69} +4.00000 q^{71} +12.0000 q^{73} +2.00000 q^{75} -14.0000 q^{79} -11.0000 q^{81} +4.00000 q^{83} +2.00000 q^{85} -12.0000 q^{87} +6.00000 q^{89} -8.00000 q^{93} -5.00000 q^{95} +6.00000 q^{97} +1.00000 q^{99} +O(q^{100})\)

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
\(3\) 2.00000 1.15470 0.577350 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0 0
\(13\) −3.00000 −0.832050 −0.416025 0.909353i \(-0.636577\pi\)
−0.416025 + 0.909353i \(0.636577\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 5.00000 1.14708 0.573539 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 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
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 2.00000 0.348155
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.00000 −0.821995 −0.410997 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) 0 0
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) −5.00000 −0.780869 −0.390434 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 9.00000 1.31278 0.656392 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) 11.0000 1.51097 0.755483 0.655168i \(-0.227402\pi\)
0.755483 + 0.655168i \(0.227402\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) 10.0000 1.32453
\(58\) 0 0
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) −12.0000 −1.53644 −0.768221 0.640184i \(-0.778858\pi\)
−0.768221 + 0.640184i \(0.778858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.00000 0.372104
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) −14.0000 −1.68540
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 0 0
\(73\) 12.0000 1.40449 0.702247 0.711934i \(-0.252180\pi\)
0.702247 + 0.711934i \(0.252180\pi\)
\(74\) 0 0
\(75\) 2.00000 0.230940
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −14.0000 −1.57512 −0.787562 0.616236i \(-0.788657\pi\)
−0.787562 + 0.616236i \(0.788657\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 0 0
\(87\) −12.0000 −1.28654
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −8.00000 −0.829561
\(94\) 0 0
\(95\) −5.00000 −0.512989
\(96\) 0 0
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 0 0
\(103\) −20.0000 −1.97066 −0.985329 0.170664i \(-0.945409\pi\)
−0.985329 + 0.170664i \(0.945409\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) 0 0
\(113\) 20.0000 1.88144 0.940721 0.339182i \(-0.110150\pi\)
0.940721 + 0.339182i \(0.110150\pi\)
\(114\) 0 0
\(115\) 7.00000 0.652753
\(116\) 0 0
\(117\) −3.00000 −0.277350
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 0 0
\(123\) −10.0000 −0.901670
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −17.0000 −1.50851 −0.754253 0.656584i \(-0.772001\pi\)
−0.754253 + 0.656584i \(0.772001\pi\)
\(128\) 0 0
\(129\) −12.0000 −1.05654
\(130\) 0 0
\(131\) −7.00000 −0.611593 −0.305796 0.952097i \(-0.598923\pi\)
−0.305796 + 0.952097i \(0.598923\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 4.00000 0.344265
\(136\) 0 0
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 18.0000 1.51587
\(142\) 0 0
\(143\) −3.00000 −0.250873
\(144\) 0 0
\(145\) 6.00000 0.498273
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) 0 0
\(157\) 5.00000 0.399043 0.199522 0.979893i \(-0.436061\pi\)
0.199522 + 0.979893i \(0.436061\pi\)
\(158\) 0 0
\(159\) 22.0000 1.74471
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 0 0
\(165\) −2.00000 −0.155700
\(166\) 0 0
\(167\) −5.00000 −0.386912 −0.193456 0.981109i \(-0.561970\pi\)
−0.193456 + 0.981109i \(0.561970\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) 0 0
\(171\) 5.00000 0.382360
\(172\) 0 0
\(173\) 19.0000 1.44454 0.722272 0.691609i \(-0.243098\pi\)
0.722272 + 0.691609i \(0.243098\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −16.0000 −1.20263
\(178\) 0 0
\(179\) −9.00000 −0.672692 −0.336346 0.941739i \(-0.609191\pi\)
−0.336346 + 0.941739i \(0.609191\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) −24.0000 −1.77413
\(184\) 0 0
\(185\) 5.00000 0.367607
\(186\) 0 0
\(187\) −2.00000 −0.146254
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) −20.0000 −1.43963 −0.719816 0.694165i \(-0.755774\pi\)
−0.719816 + 0.694165i \(0.755774\pi\)
\(194\) 0 0
\(195\) 6.00000 0.429669
\(196\) 0 0
\(197\) −27.0000 −1.92367 −0.961835 0.273629i \(-0.911776\pi\)
−0.961835 + 0.273629i \(0.911776\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 5.00000 0.349215
\(206\) 0 0
\(207\) −7.00000 −0.486534
\(208\) 0 0
\(209\) 5.00000 0.345857
\(210\) 0 0
\(211\) 13.0000 0.894957 0.447478 0.894295i \(-0.352322\pi\)
0.447478 + 0.894295i \(0.352322\pi\)
\(212\) 0 0
\(213\) 8.00000 0.548151
\(214\) 0 0
\(215\) 6.00000 0.409197
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 24.0000 1.62177
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) 0 0
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 8.00000 0.530979 0.265489 0.964114i \(-0.414466\pi\)
0.265489 + 0.964114i \(0.414466\pi\)
\(228\) 0 0
\(229\) −28.0000 −1.85029 −0.925146 0.379611i \(-0.876058\pi\)
−0.925146 + 0.379611i \(0.876058\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2.00000 0.131024 0.0655122 0.997852i \(-0.479132\pi\)
0.0655122 + 0.997852i \(0.479132\pi\)
\(234\) 0 0
\(235\) −9.00000 −0.587095
\(236\) 0 0
\(237\) −28.0000 −1.81880
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) 23.0000 1.48156 0.740780 0.671748i \(-0.234456\pi\)
0.740780 + 0.671748i \(0.234456\pi\)
\(242\) 0 0
\(243\) −10.0000 −0.641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −15.0000 −0.954427
\(248\) 0 0
\(249\) 8.00000 0.506979
\(250\) 0 0
\(251\) −29.0000 −1.83046 −0.915232 0.402928i \(-0.867993\pi\)
−0.915232 + 0.402928i \(0.867993\pi\)
\(252\) 0 0
\(253\) −7.00000 −0.440086
\(254\) 0 0
\(255\) 4.00000 0.250490
\(256\) 0 0
\(257\) 12.0000 0.748539 0.374270 0.927320i \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) −8.00000 −0.493301 −0.246651 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) 0 0
\(265\) −11.0000 −0.675725
\(266\) 0 0
\(267\) 12.0000 0.734388
\(268\) 0 0
\(269\) −12.0000 −0.731653 −0.365826 0.930683i \(-0.619214\pi\)
−0.365826 + 0.930683i \(0.619214\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 0 0
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) −3.00000 −0.178965 −0.0894825 0.995988i \(-0.528521\pi\)
−0.0894825 + 0.995988i \(0.528521\pi\)
\(282\) 0 0
\(283\) 22.0000 1.30776 0.653882 0.756596i \(-0.273139\pi\)
0.653882 + 0.756596i \(0.273139\pi\)
\(284\) 0 0
\(285\) −10.0000 −0.592349
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 12.0000 0.703452
\(292\) 0 0
\(293\) 21.0000 1.22683 0.613417 0.789760i \(-0.289795\pi\)
0.613417 + 0.789760i \(0.289795\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) 0 0
\(297\) −4.00000 −0.232104
\(298\) 0 0
\(299\) 21.0000 1.21446
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 24.0000 1.37876
\(304\) 0 0
\(305\) 12.0000 0.687118
\(306\) 0 0
\(307\) 6.00000 0.342438 0.171219 0.985233i \(-0.445229\pi\)
0.171219 + 0.985233i \(0.445229\pi\)
\(308\) 0 0
\(309\) −40.0000 −2.27552
\(310\) 0 0
\(311\) 4.00000 0.226819 0.113410 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) 0 0
\(313\) 16.0000 0.904373 0.452187 0.891923i \(-0.350644\pi\)
0.452187 + 0.891923i \(0.350644\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.00000 0.112331 0.0561656 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) 0 0
\(319\) −6.00000 −0.335936
\(320\) 0 0
\(321\) −24.0000 −1.33955
\(322\) 0 0
\(323\) −10.0000 −0.556415
\(324\) 0 0
\(325\) −3.00000 −0.166410
\(326\) 0 0
\(327\) −8.00000 −0.442401
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 27.0000 1.48405 0.742027 0.670370i \(-0.233865\pi\)
0.742027 + 0.670370i \(0.233865\pi\)
\(332\) 0 0
\(333\) −5.00000 −0.273998
\(334\) 0 0
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) −26.0000 −1.41631 −0.708155 0.706057i \(-0.750472\pi\)
−0.708155 + 0.706057i \(0.750472\pi\)
\(338\) 0 0
\(339\) 40.0000 2.17250
\(340\) 0 0
\(341\) −4.00000 −0.216612
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 14.0000 0.753735
\(346\) 0 0
\(347\) −8.00000 −0.429463 −0.214731 0.976673i \(-0.568888\pi\)
−0.214731 + 0.976673i \(0.568888\pi\)
\(348\) 0 0
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) 12.0000 0.640513
\(352\) 0 0
\(353\) 20.0000 1.06449 0.532246 0.846590i \(-0.321348\pi\)
0.532246 + 0.846590i \(0.321348\pi\)
\(354\) 0 0
\(355\) −4.00000 −0.212298
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −30.0000 −1.58334 −0.791670 0.610949i \(-0.790788\pi\)
−0.791670 + 0.610949i \(0.790788\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 0 0
\(363\) −20.0000 −1.04973
\(364\) 0 0
\(365\) −12.0000 −0.628109
\(366\) 0 0
\(367\) −19.0000 −0.991792 −0.495896 0.868382i \(-0.665160\pi\)
−0.495896 + 0.868382i \(0.665160\pi\)
\(368\) 0 0
\(369\) −5.00000 −0.260290
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 0 0
\(375\) −2.00000 −0.103280
\(376\) 0 0
\(377\) 18.0000 0.927047
\(378\) 0 0
\(379\) 21.0000 1.07870 0.539349 0.842082i \(-0.318670\pi\)
0.539349 + 0.842082i \(0.318670\pi\)
\(380\) 0 0
\(381\) −34.0000 −1.74187
\(382\) 0 0
\(383\) 21.0000 1.07305 0.536525 0.843884i \(-0.319737\pi\)
0.536525 + 0.843884i \(0.319737\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −6.00000 −0.304997
\(388\) 0 0
\(389\) −16.0000 −0.811232 −0.405616 0.914044i \(-0.632943\pi\)
−0.405616 + 0.914044i \(0.632943\pi\)
\(390\) 0 0
\(391\) 14.0000 0.708010
\(392\) 0 0
\(393\) −14.0000 −0.706207
\(394\) 0 0
\(395\) 14.0000 0.704416
\(396\) 0 0
\(397\) 18.0000 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 13.0000 0.649189 0.324595 0.945853i \(-0.394772\pi\)
0.324595 + 0.945853i \(0.394772\pi\)
\(402\) 0 0
\(403\) 12.0000 0.597763
\(404\) 0 0
\(405\) 11.0000 0.546594
\(406\) 0 0
\(407\) −5.00000 −0.247841
\(408\) 0 0
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 0 0
\(411\) −24.0000 −1.18383
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) 0 0
\(417\) −8.00000 −0.391762
\(418\) 0 0
\(419\) 5.00000 0.244266 0.122133 0.992514i \(-0.461027\pi\)
0.122133 + 0.992514i \(0.461027\pi\)
\(420\) 0 0
\(421\) 30.0000 1.46211 0.731055 0.682318i \(-0.239028\pi\)
0.731055 + 0.682318i \(0.239028\pi\)
\(422\) 0 0
\(423\) 9.00000 0.437595
\(424\) 0 0
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −6.00000 −0.289683
\(430\) 0 0
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 0 0
\(433\) 24.0000 1.15337 0.576683 0.816968i \(-0.304347\pi\)
0.576683 + 0.816968i \(0.304347\pi\)
\(434\) 0 0
\(435\) 12.0000 0.575356
\(436\) 0 0
\(437\) −35.0000 −1.67428
\(438\) 0 0
\(439\) 10.0000 0.477274 0.238637 0.971109i \(-0.423299\pi\)
0.238637 + 0.971109i \(0.423299\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 32.0000 1.52037 0.760183 0.649709i \(-0.225109\pi\)
0.760183 + 0.649709i \(0.225109\pi\)
\(444\) 0 0
\(445\) −6.00000 −0.284427
\(446\) 0 0
\(447\) 20.0000 0.945968
\(448\) 0 0
\(449\) 13.0000 0.613508 0.306754 0.951789i \(-0.400757\pi\)
0.306754 + 0.951789i \(0.400757\pi\)
\(450\) 0 0
\(451\) −5.00000 −0.235441
\(452\) 0 0
\(453\) 20.0000 0.939682
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 0 0
\(459\) 8.00000 0.373408
\(460\) 0 0
\(461\) 14.0000 0.652045 0.326023 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) 0 0
\(463\) −9.00000 −0.418265 −0.209133 0.977887i \(-0.567064\pi\)
−0.209133 + 0.977887i \(0.567064\pi\)
\(464\) 0 0
\(465\) 8.00000 0.370991
\(466\) 0 0
\(467\) 10.0000 0.462745 0.231372 0.972865i \(-0.425678\pi\)
0.231372 + 0.972865i \(0.425678\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 10.0000 0.460776
\(472\) 0 0
\(473\) −6.00000 −0.275880
\(474\) 0 0
\(475\) 5.00000 0.229416
\(476\) 0 0
\(477\) 11.0000 0.503655
\(478\) 0 0
\(479\) 16.0000 0.731059 0.365529 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) 0 0
\(481\) 15.0000 0.683941
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −6.00000 −0.272446
\(486\) 0 0
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 0 0
\(489\) −8.00000 −0.361773
\(490\) 0 0
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 0 0
\(493\) 12.0000 0.540453
\(494\) 0 0
\(495\) −1.00000 −0.0449467
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 0 0
\(501\) −10.0000 −0.446767
\(502\) 0 0
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) 0 0
\(507\) −8.00000 −0.355292
\(508\) 0 0
\(509\) −18.0000 −0.797836 −0.398918 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −20.0000 −0.883022
\(514\) 0 0
\(515\) 20.0000 0.881305
\(516\) 0 0
\(517\) 9.00000 0.395820
\(518\) 0 0
\(519\) 38.0000 1.66801
\(520\) 0 0
\(521\) −15.0000 −0.657162 −0.328581 0.944476i \(-0.606570\pi\)
−0.328581 + 0.944476i \(0.606570\pi\)
\(522\) 0 0
\(523\) −28.0000 −1.22435 −0.612177 0.790721i \(-0.709706\pi\)
−0.612177 + 0.790721i \(0.709706\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.00000 0.348485
\(528\) 0 0
\(529\) 26.0000 1.13043
\(530\) 0 0
\(531\) −8.00000 −0.347170
\(532\) 0 0
\(533\) 15.0000 0.649722
\(534\) 0 0
\(535\) 12.0000 0.518805
\(536\) 0 0
\(537\) −18.0000 −0.776757
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −40.0000 −1.71973 −0.859867 0.510518i \(-0.829454\pi\)
−0.859867 + 0.510518i \(0.829454\pi\)
\(542\) 0 0
\(543\) 4.00000 0.171656
\(544\) 0 0
\(545\) 4.00000 0.171341
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) 0 0
\(549\) −12.0000 −0.512148
\(550\) 0 0
\(551\) −30.0000 −1.27804
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 10.0000 0.424476
\(556\) 0 0
\(557\) 21.0000 0.889799 0.444899 0.895581i \(-0.353239\pi\)
0.444899 + 0.895581i \(0.353239\pi\)
\(558\) 0 0
\(559\) 18.0000 0.761319
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) 0 0
\(563\) −6.00000 −0.252870 −0.126435 0.991975i \(-0.540353\pi\)
−0.126435 + 0.991975i \(0.540353\pi\)
\(564\) 0 0
\(565\) −20.0000 −0.841406
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −11.0000 −0.461144 −0.230572 0.973055i \(-0.574060\pi\)
−0.230572 + 0.973055i \(0.574060\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 0 0
\(573\) −24.0000 −1.00261
\(574\) 0 0
\(575\) −7.00000 −0.291920
\(576\) 0 0
\(577\) −4.00000 −0.166522 −0.0832611 0.996528i \(-0.526534\pi\)
−0.0832611 + 0.996528i \(0.526534\pi\)
\(578\) 0 0
\(579\) −40.0000 −1.66234
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 11.0000 0.455573
\(584\) 0 0
\(585\) 3.00000 0.124035
\(586\) 0 0
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) −20.0000 −0.824086
\(590\) 0 0
\(591\) −54.0000 −2.22126
\(592\) 0 0
\(593\) −12.0000 −0.492781 −0.246390 0.969171i \(-0.579245\pi\)
−0.246390 + 0.969171i \(0.579245\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 8.00000 0.327418
\(598\) 0 0
\(599\) −34.0000 −1.38920 −0.694601 0.719395i \(-0.744419\pi\)
−0.694601 + 0.719395i \(0.744419\pi\)
\(600\) 0 0
\(601\) −14.0000 −0.571072 −0.285536 0.958368i \(-0.592172\pi\)
−0.285536 + 0.958368i \(0.592172\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) 0 0
\(605\) 10.0000 0.406558
\(606\) 0 0
\(607\) −33.0000 −1.33943 −0.669714 0.742619i \(-0.733583\pi\)
−0.669714 + 0.742619i \(0.733583\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −27.0000 −1.09230
\(612\) 0 0
\(613\) −41.0000 −1.65597 −0.827987 0.560747i \(-0.810514\pi\)
−0.827987 + 0.560747i \(0.810514\pi\)
\(614\) 0 0
\(615\) 10.0000 0.403239
\(616\) 0 0
\(617\) 42.0000 1.69086 0.845428 0.534089i \(-0.179345\pi\)
0.845428 + 0.534089i \(0.179345\pi\)
\(618\) 0 0
\(619\) −35.0000 −1.40677 −0.703384 0.710810i \(-0.748329\pi\)
−0.703384 + 0.710810i \(0.748329\pi\)
\(620\) 0 0
\(621\) 28.0000 1.12360
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 10.0000 0.399362
\(628\) 0 0
\(629\) 10.0000 0.398726
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 0 0
\(633\) 26.0000 1.03341
\(634\) 0 0
\(635\) 17.0000 0.674624
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 4.00000 0.158238
\(640\) 0 0
\(641\) 35.0000 1.38242 0.691208 0.722655i \(-0.257079\pi\)
0.691208 + 0.722655i \(0.257079\pi\)
\(642\) 0 0
\(643\) 34.0000 1.34083 0.670415 0.741987i \(-0.266116\pi\)
0.670415 + 0.741987i \(0.266116\pi\)
\(644\) 0 0
\(645\) 12.0000 0.472500
\(646\) 0 0
\(647\) −23.0000 −0.904223 −0.452112 0.891961i \(-0.649329\pi\)
−0.452112 + 0.891961i \(0.649329\pi\)
\(648\) 0 0
\(649\) −8.00000 −0.314027
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 45.0000 1.76099 0.880493 0.474059i \(-0.157212\pi\)
0.880493 + 0.474059i \(0.157212\pi\)
\(654\) 0 0
\(655\) 7.00000 0.273513
\(656\) 0 0
\(657\) 12.0000 0.468165
\(658\) 0 0
\(659\) 16.0000 0.623272 0.311636 0.950202i \(-0.399123\pi\)
0.311636 + 0.950202i \(0.399123\pi\)
\(660\) 0 0
\(661\) −36.0000 −1.40024 −0.700119 0.714026i \(-0.746870\pi\)
−0.700119 + 0.714026i \(0.746870\pi\)
\(662\) 0 0
\(663\) 12.0000 0.466041
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 42.0000 1.62625
\(668\) 0 0
\(669\) 32.0000 1.23719
\(670\) 0 0
\(671\) −12.0000 −0.463255
\(672\) 0 0
\(673\) −42.0000 −1.61898 −0.809491 0.587133i \(-0.800257\pi\)
−0.809491 + 0.587133i \(0.800257\pi\)
\(674\) 0 0
\(675\) −4.00000 −0.153960
\(676\) 0 0
\(677\) 21.0000 0.807096 0.403548 0.914959i \(-0.367777\pi\)
0.403548 + 0.914959i \(0.367777\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 16.0000 0.613121
\(682\) 0 0
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 0 0
\(685\) 12.0000 0.458496
\(686\) 0 0
\(687\) −56.0000 −2.13653
\(688\) 0 0
\(689\) −33.0000 −1.25720
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 4.00000 0.151729
\(696\) 0 0
\(697\) 10.0000 0.378777
\(698\) 0 0
\(699\) 4.00000 0.151294
\(700\) 0 0
\(701\) 42.0000 1.58632 0.793159 0.609015i \(-0.208435\pi\)
0.793159 + 0.609015i \(0.208435\pi\)
\(702\) 0 0
\(703\) −25.0000 −0.942893
\(704\) 0 0
\(705\) −18.0000 −0.677919
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −18.0000 −0.676004 −0.338002 0.941145i \(-0.609751\pi\)
−0.338002 + 0.941145i \(0.609751\pi\)
\(710\) 0 0
\(711\) −14.0000 −0.525041
\(712\) 0 0
\(713\) 28.0000 1.04861
\(714\) 0 0
\(715\) 3.00000 0.112194
\(716\) 0 0
\(717\) −12.0000 −0.448148
\(718\) 0 0
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 46.0000 1.71076
\(724\) 0 0
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) −41.0000 −1.52061 −0.760303 0.649569i \(-0.774949\pi\)
−0.760303 + 0.649569i \(0.774949\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 12.0000 0.443836
\(732\) 0 0
\(733\) 31.0000 1.14501 0.572506 0.819901i \(-0.305971\pi\)
0.572506 + 0.819901i \(0.305971\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4.00000 0.147342
\(738\) 0 0
\(739\) 41.0000 1.50821 0.754105 0.656754i \(-0.228071\pi\)
0.754105 + 0.656754i \(0.228071\pi\)
\(740\) 0 0
\(741\) −30.0000 −1.10208
\(742\) 0 0
\(743\) 3.00000 0.110059 0.0550297 0.998485i \(-0.482475\pi\)
0.0550297 + 0.998485i \(0.482475\pi\)
\(744\) 0 0
\(745\) −10.0000 −0.366372
\(746\) 0 0
\(747\) 4.00000 0.146352
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −18.0000 −0.656829 −0.328415 0.944534i \(-0.606514\pi\)
−0.328415 + 0.944534i \(0.606514\pi\)
\(752\) 0 0
\(753\) −58.0000 −2.11364
\(754\) 0 0
\(755\) −10.0000 −0.363937
\(756\) 0 0
\(757\) −6.00000 −0.218074 −0.109037 0.994038i \(-0.534777\pi\)
−0.109037 + 0.994038i \(0.534777\pi\)
\(758\) 0 0
\(759\) −14.0000 −0.508168
\(760\) 0 0
\(761\) −3.00000 −0.108750 −0.0543750 0.998521i \(-0.517317\pi\)
−0.0543750 + 0.998521i \(0.517317\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 2.00000 0.0723102
\(766\) 0 0
\(767\) 24.0000 0.866590
\(768\) 0 0
\(769\) −41.0000 −1.47850 −0.739249 0.673432i \(-0.764819\pi\)
−0.739249 + 0.673432i \(0.764819\pi\)
\(770\) 0 0
\(771\) 24.0000 0.864339
\(772\) 0 0
\(773\) 15.0000 0.539513 0.269756 0.962929i \(-0.413057\pi\)
0.269756 + 0.962929i \(0.413057\pi\)
\(774\) 0 0
\(775\) −4.00000 −0.143684
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −25.0000 −0.895718
\(780\) 0 0
\(781\) 4.00000 0.143131
\(782\) 0 0
\(783\) 24.0000 0.857690
\(784\) 0 0
\(785\) −5.00000 −0.178458
\(786\) 0 0
\(787\) 18.0000 0.641631 0.320815 0.947142i \(-0.396043\pi\)
0.320815 + 0.947142i \(0.396043\pi\)
\(788\) 0 0
\(789\) −16.0000 −0.569615
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 36.0000 1.27840
\(794\) 0 0
\(795\) −22.0000 −0.780260
\(796\) 0 0
\(797\) 2.00000 0.0708436 0.0354218 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) 0 0
\(799\) −18.0000 −0.636794
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 0 0
\(803\) 12.0000 0.423471
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −24.0000 −0.844840
\(808\) 0 0
\(809\) 55.0000 1.93370 0.966849 0.255351i \(-0.0821909\pi\)
0.966849 + 0.255351i \(0.0821909\pi\)
\(810\) 0 0
\(811\) 5.00000 0.175574 0.0877869 0.996139i \(-0.472021\pi\)
0.0877869 + 0.996139i \(0.472021\pi\)
\(812\) 0 0
\(813\) 16.0000 0.561144
\(814\) 0 0
\(815\) 4.00000 0.140114
\(816\) 0 0
\(817\) −30.0000 −1.04957
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 0 0
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) 0 0
\(825\) 2.00000 0.0696311
\(826\) 0 0
\(827\) 14.0000 0.486828 0.243414 0.969923i \(-0.421733\pi\)
0.243414 + 0.969923i \(0.421733\pi\)
\(828\) 0 0
\(829\) 38.0000 1.31979 0.659897 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) 0 0
\(831\) 4.00000 0.138758
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 5.00000 0.173032
\(836\) 0 0
\(837\) 16.0000 0.553041
\(838\) 0 0
\(839\) 14.0000 0.483334 0.241667 0.970359i \(-0.422306\pi\)
0.241667 + 0.970359i \(0.422306\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) −6.00000 −0.206651
\(844\) 0 0
\(845\) 4.00000 0.137604
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 44.0000 1.51008
\(850\) 0 0
\(851\) 35.0000 1.19978
\(852\) 0 0
\(853\) 23.0000 0.787505 0.393753 0.919216i \(-0.371177\pi\)
0.393753 + 0.919216i \(0.371177\pi\)
\(854\) 0 0
\(855\) −5.00000 −0.170996
\(856\) 0 0
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 0 0
\(859\) −40.0000 −1.36478 −0.682391 0.730987i \(-0.739060\pi\)
−0.682391 + 0.730987i \(0.739060\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 13.0000 0.442525 0.221263 0.975214i \(-0.428982\pi\)
0.221263 + 0.975214i \(0.428982\pi\)
\(864\) 0 0
\(865\) −19.0000 −0.646019
\(866\) 0 0
\(867\) −26.0000 −0.883006
\(868\) 0 0
\(869\) −14.0000 −0.474917
\(870\) 0 0
\(871\) −12.0000 −0.406604
\(872\) 0 0
\(873\) 6.00000 0.203069
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 9.00000 0.303908 0.151954 0.988388i \(-0.451443\pi\)
0.151954 + 0.988388i \(0.451443\pi\)
\(878\) 0 0
\(879\) 42.0000 1.41662
\(880\) 0 0
\(881\) −7.00000 −0.235836 −0.117918 0.993023i \(-0.537622\pi\)
−0.117918 + 0.993023i \(0.537622\pi\)
\(882\) 0 0
\(883\) −8.00000 −0.269221 −0.134611 0.990899i \(-0.542978\pi\)
−0.134611 + 0.990899i \(0.542978\pi\)
\(884\) 0 0
\(885\) 16.0000 0.537834
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −11.0000 −0.368514
\(892\) 0 0
\(893\) 45.0000 1.50587
\(894\) 0 0
\(895\) 9.00000 0.300837
\(896\) 0 0
\(897\) 42.0000 1.40234
\(898\) 0 0
\(899\) 24.0000 0.800445
\(900\) 0 0
\(901\) −22.0000 −0.732926
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −2.00000 −0.0664822
\(906\) 0 0
\(907\) −22.0000 −0.730498 −0.365249 0.930910i \(-0.619016\pi\)
−0.365249 + 0.930910i \(0.619016\pi\)
\(908\) 0 0
\(909\) 12.0000 0.398015
\(910\) 0 0
\(911\) −42.0000 −1.39152 −0.695761 0.718273i \(-0.744933\pi\)
−0.695761 + 0.718273i \(0.744933\pi\)
\(912\) 0 0
\(913\) 4.00000 0.132381
\(914\) 0 0
\(915\) 24.0000 0.793416
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 18.0000 0.593765 0.296883 0.954914i \(-0.404053\pi\)
0.296883 + 0.954914i \(0.404053\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) 0 0
\(923\) −12.0000 −0.394985
\(924\) 0 0
\(925\) −5.00000 −0.164399
\(926\) 0 0
\(927\) −20.0000 −0.656886
\(928\) 0 0
\(929\) −31.0000 −1.01708 −0.508539 0.861039i \(-0.669814\pi\)
−0.508539 + 0.861039i \(0.669814\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 8.00000 0.261908
\(934\) 0 0
\(935\) 2.00000 0.0654070
\(936\) 0 0
\(937\) −38.0000 −1.24141 −0.620703 0.784046i \(-0.713153\pi\)
−0.620703 + 0.784046i \(0.713153\pi\)
\(938\) 0 0
\(939\) 32.0000 1.04428
\(940\) 0 0
\(941\) 24.0000 0.782378 0.391189 0.920310i \(-0.372064\pi\)
0.391189 + 0.920310i \(0.372064\pi\)
\(942\) 0 0
\(943\) 35.0000 1.13976
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −14.0000 −0.454939 −0.227469 0.973785i \(-0.573045\pi\)
−0.227469 + 0.973785i \(0.573045\pi\)
\(948\) 0 0
\(949\) −36.0000 −1.16861
\(950\) 0 0
\(951\) 4.00000 0.129709
\(952\) 0 0
\(953\) −20.0000 −0.647864 −0.323932 0.946080i \(-0.605005\pi\)
−0.323932 + 0.946080i \(0.605005\pi\)
\(954\) 0 0
\(955\) 12.0000 0.388311
\(956\) 0 0
\(957\) −12.0000 −0.387905
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) 0 0
\(965\) 20.0000 0.643823
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) 0 0
\(969\) −20.0000 −0.642493
\(970\) 0 0
\(971\) −7.00000 −0.224641 −0.112320 0.993672i \(-0.535828\pi\)
−0.112320 + 0.993672i \(0.535828\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −6.00000 −0.192154
\(976\) 0 0
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 0 0
\(979\) 6.00000 0.191761
\(980\) 0 0
\(981\) −4.00000 −0.127710
\(982\) 0 0
\(983\) 3.00000 0.0956851 0.0478426 0.998855i \(-0.484765\pi\)
0.0478426 + 0.998855i \(0.484765\pi\)
\(984\) 0 0
\(985\) 27.0000 0.860292
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 42.0000 1.33552
\(990\) 0 0
\(991\) −44.0000 −1.39771 −0.698853 0.715265i \(-0.746306\pi\)
−0.698853 + 0.715265i \(0.746306\pi\)
\(992\) 0 0
\(993\) 54.0000 1.71364
\(994\) 0 0
\(995\) −4.00000 −0.126809
\(996\) 0 0
\(997\) 46.0000 1.45683 0.728417 0.685134i \(-0.240256\pi\)
0.728417 + 0.685134i \(0.240256\pi\)
\(998\) 0 0
\(999\) 20.0000 0.632772
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 3920.2.a.bf.1.1 1
4.3 odd 2 1960.2.a.a.1.1 1
7.2 even 3 560.2.q.c.81.1 2
7.4 even 3 560.2.q.c.401.1 2
7.6 odd 2 3920.2.a.i.1.1 1
20.19 odd 2 9800.2.a.bi.1.1 1
28.3 even 6 1960.2.q.c.961.1 2
28.11 odd 6 280.2.q.c.121.1 yes 2
28.19 even 6 1960.2.q.c.361.1 2
28.23 odd 6 280.2.q.c.81.1 2
28.27 even 2 1960.2.a.m.1.1 1
84.11 even 6 2520.2.bi.e.1801.1 2
84.23 even 6 2520.2.bi.e.361.1 2
140.23 even 12 1400.2.bh.e.249.2 4
140.39 odd 6 1400.2.q.a.401.1 2
140.67 even 12 1400.2.bh.e.849.2 4
140.79 odd 6 1400.2.q.a.1201.1 2
140.107 even 12 1400.2.bh.e.249.1 4
140.123 even 12 1400.2.bh.e.849.1 4
140.139 even 2 9800.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.q.c.81.1 2 28.23 odd 6
280.2.q.c.121.1 yes 2 28.11 odd 6
560.2.q.c.81.1 2 7.2 even 3
560.2.q.c.401.1 2 7.4 even 3
1400.2.q.a.401.1 2 140.39 odd 6
1400.2.q.a.1201.1 2 140.79 odd 6
1400.2.bh.e.249.1 4 140.107 even 12
1400.2.bh.e.249.2 4 140.23 even 12
1400.2.bh.e.849.1 4 140.123 even 12
1400.2.bh.e.849.2 4 140.67 even 12
1960.2.a.a.1.1 1 4.3 odd 2
1960.2.a.m.1.1 1 28.27 even 2
1960.2.q.c.361.1 2 28.19 even 6
1960.2.q.c.961.1 2 28.3 even 6
2520.2.bi.e.361.1 2 84.23 even 6
2520.2.bi.e.1801.1 2 84.11 even 6
3920.2.a.i.1.1 1 7.6 odd 2
3920.2.a.bf.1.1 1 1.1 even 1 trivial
9800.2.a.g.1.1 1 140.139 even 2
9800.2.a.bi.1.1 1 20.19 odd 2