Properties

Label 3380.2.a.i.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

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.00000 q^{9} -3.00000 q^{11} +1.00000 q^{15} -3.00000 q^{17} -5.00000 q^{19} +1.00000 q^{21} +9.00000 q^{23} +1.00000 q^{25} -5.00000 q^{27} -9.00000 q^{29} -8.00000 q^{31} -3.00000 q^{33} +1.00000 q^{35} +7.00000 q^{37} -3.00000 q^{41} -1.00000 q^{43} -2.00000 q^{45} -6.00000 q^{49} -3.00000 q^{51} +6.00000 q^{53} -3.00000 q^{55} -5.00000 q^{57} -9.00000 q^{59} -1.00000 q^{61} -2.00000 q^{63} -5.00000 q^{67} +9.00000 q^{69} -9.00000 q^{71} -2.00000 q^{73} +1.00000 q^{75} -3.00000 q^{77} +8.00000 q^{79} +1.00000 q^{81} -3.00000 q^{85} -9.00000 q^{87} -3.00000 q^{89} -8.00000 q^{93} -5.00000 q^{95} -17.0000 q^{97} +6.00000 q^{99} +O(q^{100})\)

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
\(3\) 1.00000 0.577350 0.288675 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) −5.00000 −1.14708 −0.573539 0.819178i \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 9.00000 1.87663 0.938315 0.345782i \(-0.112386\pi\)
0.938315 + 0.345782i \(0.112386\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) −9.00000 −1.67126 −0.835629 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) −3.00000 −0.522233
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) 7.00000 1.15079 0.575396 0.817875i \(-0.304848\pi\)
0.575396 + 0.817875i \(0.304848\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.00000 −0.468521 −0.234261 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 0 0
\(45\) −2.00000 −0.298142
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) −3.00000 −0.420084
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) −5.00000 −0.662266
\(58\) 0 0
\(59\) −9.00000 −1.17170 −0.585850 0.810419i \(-0.699239\pi\)
−0.585850 + 0.810419i \(0.699239\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 0 0
\(63\) −2.00000 −0.251976
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −5.00000 −0.610847 −0.305424 0.952217i \(-0.598798\pi\)
−0.305424 + 0.952217i \(0.598798\pi\)
\(68\) 0 0
\(69\) 9.00000 1.08347
\(70\) 0 0
\(71\) −9.00000 −1.06810 −0.534052 0.845452i \(-0.679331\pi\)
−0.534052 + 0.845452i \(0.679331\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −3.00000 −0.341882
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −3.00000 −0.325396
\(86\) 0 0
\(87\) −9.00000 −0.964901
\(88\) 0 0
\(89\) −3.00000 −0.317999 −0.159000 0.987279i \(-0.550827\pi\)
−0.159000 + 0.987279i \(0.550827\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −8.00000 −0.829561
\(94\) 0 0
\(95\) −5.00000 −0.512989
\(96\) 0 0
\(97\) −17.0000 −1.72609 −0.863044 0.505128i \(-0.831445\pi\)
−0.863044 + 0.505128i \(0.831445\pi\)
\(98\) 0 0
\(99\) 6.00000 0.603023
\(100\) 0 0
\(101\) 15.0000 1.49256 0.746278 0.665635i \(-0.231839\pi\)
0.746278 + 0.665635i \(0.231839\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) 1.00000 0.0975900
\(106\) 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
\(111\) 7.00000 0.664411
\(112\) 0 0
\(113\) −15.0000 −1.41108 −0.705541 0.708669i \(-0.749296\pi\)
−0.705541 + 0.708669i \(0.749296\pi\)
\(114\) 0 0
\(115\) 9.00000 0.839254
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) −3.00000 −0.270501
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 17.0000 1.50851 0.754253 0.656584i \(-0.227999\pi\)
0.754253 + 0.656584i \(0.227999\pi\)
\(128\) 0 0
\(129\) −1.00000 −0.0880451
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −5.00000 −0.433555
\(134\) 0 0
\(135\) −5.00000 −0.430331
\(136\) 0 0
\(137\) 3.00000 0.256307 0.128154 0.991754i \(-0.459095\pi\)
0.128154 + 0.991754i \(0.459095\pi\)
\(138\) 0 0
\(139\) 5.00000 0.424094 0.212047 0.977259i \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −9.00000 −0.747409
\(146\) 0 0
\(147\) −6.00000 −0.494872
\(148\) 0 0
\(149\) 9.00000 0.737309 0.368654 0.929567i \(-0.379819\pi\)
0.368654 + 0.929567i \(0.379819\pi\)
\(150\) 0 0
\(151\) −20.0000 −1.62758 −0.813788 0.581161i \(-0.802599\pi\)
−0.813788 + 0.581161i \(0.802599\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) −8.00000 −0.642575
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 9.00000 0.709299
\(162\) 0 0
\(163\) 1.00000 0.0783260 0.0391630 0.999233i \(-0.487531\pi\)
0.0391630 + 0.999233i \(0.487531\pi\)
\(164\) 0 0
\(165\) −3.00000 −0.233550
\(166\) 0 0
\(167\) 15.0000 1.16073 0.580367 0.814355i \(-0.302909\pi\)
0.580367 + 0.814355i \(0.302909\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 10.0000 0.764719
\(172\) 0 0
\(173\) 21.0000 1.59660 0.798300 0.602260i \(-0.205733\pi\)
0.798300 + 0.602260i \(0.205733\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −9.00000 −0.676481
\(178\) 0 0
\(179\) 15.0000 1.12115 0.560576 0.828103i \(-0.310580\pi\)
0.560576 + 0.828103i \(0.310580\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) −1.00000 −0.0739221
\(184\) 0 0
\(185\) 7.00000 0.514650
\(186\) 0 0
\(187\) 9.00000 0.658145
\(188\) 0 0
\(189\) −5.00000 −0.363696
\(190\) 0 0
\(191\) −3.00000 −0.217072 −0.108536 0.994092i \(-0.534616\pi\)
−0.108536 + 0.994092i \(0.534616\pi\)
\(192\) 0 0
\(193\) −5.00000 −0.359908 −0.179954 0.983675i \(-0.557595\pi\)
−0.179954 + 0.983675i \(0.557595\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3.00000 0.213741 0.106871 0.994273i \(-0.465917\pi\)
0.106871 + 0.994273i \(0.465917\pi\)
\(198\) 0 0
\(199\) −7.00000 −0.496217 −0.248108 0.968732i \(-0.579809\pi\)
−0.248108 + 0.968732i \(0.579809\pi\)
\(200\) 0 0
\(201\) −5.00000 −0.352673
\(202\) 0 0
\(203\) −9.00000 −0.631676
\(204\) 0 0
\(205\) −3.00000 −0.209529
\(206\) 0 0
\(207\) −18.0000 −1.25109
\(208\) 0 0
\(209\) 15.0000 1.03757
\(210\) 0 0
\(211\) −25.0000 −1.72107 −0.860535 0.509390i \(-0.829871\pi\)
−0.860535 + 0.509390i \(0.829871\pi\)
\(212\) 0 0
\(213\) −9.00000 −0.616670
\(214\) 0 0
\(215\) −1.00000 −0.0681994
\(216\) 0 0
\(217\) −8.00000 −0.543075
\(218\) 0 0
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 19.0000 1.27233 0.636167 0.771551i \(-0.280519\pi\)
0.636167 + 0.771551i \(0.280519\pi\)
\(224\) 0 0
\(225\) −2.00000 −0.133333
\(226\) 0 0
\(227\) −15.0000 −0.995585 −0.497792 0.867296i \(-0.665856\pi\)
−0.497792 + 0.867296i \(0.665856\pi\)
\(228\) 0 0
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 0 0
\(231\) −3.00000 −0.197386
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) −23.0000 −1.48156 −0.740780 0.671748i \(-0.765544\pi\)
−0.740780 + 0.671748i \(0.765544\pi\)
\(242\) 0 0
\(243\) 16.0000 1.02640
\(244\) 0 0
\(245\) −6.00000 −0.383326
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 9.00000 0.568075 0.284037 0.958813i \(-0.408326\pi\)
0.284037 + 0.958813i \(0.408326\pi\)
\(252\) 0 0
\(253\) −27.0000 −1.69748
\(254\) 0 0
\(255\) −3.00000 −0.187867
\(256\) 0 0
\(257\) −27.0000 −1.68421 −0.842107 0.539311i \(-0.818685\pi\)
−0.842107 + 0.539311i \(0.818685\pi\)
\(258\) 0 0
\(259\) 7.00000 0.434959
\(260\) 0 0
\(261\) 18.0000 1.11417
\(262\) 0 0
\(263\) −3.00000 −0.184988 −0.0924940 0.995713i \(-0.529484\pi\)
−0.0924940 + 0.995713i \(0.529484\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) −3.00000 −0.183597
\(268\) 0 0
\(269\) −21.0000 −1.28039 −0.640196 0.768211i \(-0.721147\pi\)
−0.640196 + 0.768211i \(0.721147\pi\)
\(270\) 0 0
\(271\) 13.0000 0.789694 0.394847 0.918747i \(-0.370798\pi\)
0.394847 + 0.918747i \(0.370798\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −3.00000 −0.180907
\(276\) 0 0
\(277\) −19.0000 −1.14160 −0.570800 0.821089i \(-0.693367\pi\)
−0.570800 + 0.821089i \(0.693367\pi\)
\(278\) 0 0
\(279\) 16.0000 0.957895
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) 17.0000 1.01055 0.505273 0.862960i \(-0.331392\pi\)
0.505273 + 0.862960i \(0.331392\pi\)
\(284\) 0 0
\(285\) −5.00000 −0.296174
\(286\) 0 0
\(287\) −3.00000 −0.177084
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −17.0000 −0.996558
\(292\) 0 0
\(293\) 3.00000 0.175262 0.0876309 0.996153i \(-0.472070\pi\)
0.0876309 + 0.996153i \(0.472070\pi\)
\(294\) 0 0
\(295\) −9.00000 −0.524000
\(296\) 0 0
\(297\) 15.0000 0.870388
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −1.00000 −0.0576390
\(302\) 0 0
\(303\) 15.0000 0.861727
\(304\) 0 0
\(305\) −1.00000 −0.0572598
\(306\) 0 0
\(307\) 4.00000 0.228292 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 0 0
\(315\) −2.00000 −0.112687
\(316\) 0 0
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) 27.0000 1.51171
\(320\) 0 0
\(321\) −3.00000 −0.167444
\(322\) 0 0
\(323\) 15.0000 0.834622
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −14.0000 −0.774202
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 19.0000 1.04433 0.522167 0.852843i \(-0.325124\pi\)
0.522167 + 0.852843i \(0.325124\pi\)
\(332\) 0 0
\(333\) −14.0000 −0.767195
\(334\) 0 0
\(335\) −5.00000 −0.273179
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 0 0
\(339\) −15.0000 −0.814688
\(340\) 0 0
\(341\) 24.0000 1.29967
\(342\) 0 0
\(343\) −13.0000 −0.701934
\(344\) 0 0
\(345\) 9.00000 0.484544
\(346\) 0 0
\(347\) −9.00000 −0.483145 −0.241573 0.970383i \(-0.577663\pi\)
−0.241573 + 0.970383i \(0.577663\pi\)
\(348\) 0 0
\(349\) −35.0000 −1.87351 −0.936754 0.349990i \(-0.886185\pi\)
−0.936754 + 0.349990i \(0.886185\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 27.0000 1.43706 0.718532 0.695493i \(-0.244814\pi\)
0.718532 + 0.695493i \(0.244814\pi\)
\(354\) 0 0
\(355\) −9.00000 −0.477670
\(356\) 0 0
\(357\) −3.00000 −0.158777
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 0 0
\(363\) −2.00000 −0.104973
\(364\) 0 0
\(365\) −2.00000 −0.104685
\(366\) 0 0
\(367\) 17.0000 0.887393 0.443696 0.896177i \(-0.353667\pi\)
0.443696 + 0.896177i \(0.353667\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 6.00000 0.311504
\(372\) 0 0
\(373\) 5.00000 0.258890 0.129445 0.991587i \(-0.458680\pi\)
0.129445 + 0.991587i \(0.458680\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −11.0000 −0.565032 −0.282516 0.959263i \(-0.591169\pi\)
−0.282516 + 0.959263i \(0.591169\pi\)
\(380\) 0 0
\(381\) 17.0000 0.870936
\(382\) 0 0
\(383\) 21.0000 1.07305 0.536525 0.843884i \(-0.319737\pi\)
0.536525 + 0.843884i \(0.319737\pi\)
\(384\) 0 0
\(385\) −3.00000 −0.152894
\(386\) 0 0
\(387\) 2.00000 0.101666
\(388\) 0 0
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) −27.0000 −1.36545
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) −29.0000 −1.45547 −0.727734 0.685859i \(-0.759427\pi\)
−0.727734 + 0.685859i \(0.759427\pi\)
\(398\) 0 0
\(399\) −5.00000 −0.250313
\(400\) 0 0
\(401\) −15.0000 −0.749064 −0.374532 0.927214i \(-0.622197\pi\)
−0.374532 + 0.927214i \(0.622197\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −21.0000 −1.04093
\(408\) 0 0
\(409\) 25.0000 1.23617 0.618085 0.786111i \(-0.287909\pi\)
0.618085 + 0.786111i \(0.287909\pi\)
\(410\) 0 0
\(411\) 3.00000 0.147979
\(412\) 0 0
\(413\) −9.00000 −0.442861
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 5.00000 0.244851
\(418\) 0 0
\(419\) −21.0000 −1.02592 −0.512959 0.858413i \(-0.671451\pi\)
−0.512959 + 0.858413i \(0.671451\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3.00000 −0.145521
\(426\) 0 0
\(427\) −1.00000 −0.0483934
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −3.00000 −0.144505 −0.0722525 0.997386i \(-0.523019\pi\)
−0.0722525 + 0.997386i \(0.523019\pi\)
\(432\) 0 0
\(433\) 5.00000 0.240285 0.120142 0.992757i \(-0.461665\pi\)
0.120142 + 0.992757i \(0.461665\pi\)
\(434\) 0 0
\(435\) −9.00000 −0.431517
\(436\) 0 0
\(437\) −45.0000 −2.15264
\(438\) 0 0
\(439\) −1.00000 −0.0477274 −0.0238637 0.999715i \(-0.507597\pi\)
−0.0238637 + 0.999715i \(0.507597\pi\)
\(440\) 0 0
\(441\) 12.0000 0.571429
\(442\) 0 0
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 0 0
\(445\) −3.00000 −0.142214
\(446\) 0 0
\(447\) 9.00000 0.425685
\(448\) 0 0
\(449\) −15.0000 −0.707894 −0.353947 0.935266i \(-0.615161\pi\)
−0.353947 + 0.935266i \(0.615161\pi\)
\(450\) 0 0
\(451\) 9.00000 0.423793
\(452\) 0 0
\(453\) −20.0000 −0.939682
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −41.0000 −1.91790 −0.958950 0.283577i \(-0.908479\pi\)
−0.958950 + 0.283577i \(0.908479\pi\)
\(458\) 0 0
\(459\) 15.0000 0.700140
\(460\) 0 0
\(461\) −15.0000 −0.698620 −0.349310 0.937007i \(-0.613584\pi\)
−0.349310 + 0.937007i \(0.613584\pi\)
\(462\) 0 0
\(463\) 40.0000 1.85896 0.929479 0.368875i \(-0.120257\pi\)
0.929479 + 0.368875i \(0.120257\pi\)
\(464\) 0 0
\(465\) −8.00000 −0.370991
\(466\) 0 0
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 0 0
\(469\) −5.00000 −0.230879
\(470\) 0 0
\(471\) 14.0000 0.645086
\(472\) 0 0
\(473\) 3.00000 0.137940
\(474\) 0 0
\(475\) −5.00000 −0.229416
\(476\) 0 0
\(477\) −12.0000 −0.549442
\(478\) 0 0
\(479\) −15.0000 −0.685367 −0.342684 0.939451i \(-0.611336\pi\)
−0.342684 + 0.939451i \(0.611336\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 9.00000 0.409514
\(484\) 0 0
\(485\) −17.0000 −0.771930
\(486\) 0 0
\(487\) −35.0000 −1.58600 −0.793001 0.609221i \(-0.791482\pi\)
−0.793001 + 0.609221i \(0.791482\pi\)
\(488\) 0 0
\(489\) 1.00000 0.0452216
\(490\) 0 0
\(491\) 15.0000 0.676941 0.338470 0.940977i \(-0.390091\pi\)
0.338470 + 0.940977i \(0.390091\pi\)
\(492\) 0 0
\(493\) 27.0000 1.21602
\(494\) 0 0
\(495\) 6.00000 0.269680
\(496\) 0 0
\(497\) −9.00000 −0.403705
\(498\) 0 0
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) 15.0000 0.670151
\(502\) 0 0
\(503\) 39.0000 1.73892 0.869462 0.494000i \(-0.164466\pi\)
0.869462 + 0.494000i \(0.164466\pi\)
\(504\) 0 0
\(505\) 15.0000 0.667491
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 33.0000 1.46270 0.731350 0.682003i \(-0.238891\pi\)
0.731350 + 0.682003i \(0.238891\pi\)
\(510\) 0 0
\(511\) −2.00000 −0.0884748
\(512\) 0 0
\(513\) 25.0000 1.10378
\(514\) 0 0
\(515\) 8.00000 0.352522
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 21.0000 0.921798
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) 5.00000 0.218635 0.109317 0.994007i \(-0.465134\pi\)
0.109317 + 0.994007i \(0.465134\pi\)
\(524\) 0 0
\(525\) 1.00000 0.0436436
\(526\) 0 0
\(527\) 24.0000 1.04546
\(528\) 0 0
\(529\) 58.0000 2.52174
\(530\) 0 0
\(531\) 18.0000 0.781133
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −3.00000 −0.129701
\(536\) 0 0
\(537\) 15.0000 0.647298
\(538\) 0 0
\(539\) 18.0000 0.775315
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 0 0
\(543\) −10.0000 −0.429141
\(544\) 0 0
\(545\) −14.0000 −0.599694
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 0 0
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) 45.0000 1.91706
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) 0 0
\(555\) 7.00000 0.297133
\(556\) 0 0
\(557\) 39.0000 1.65248 0.826242 0.563316i \(-0.190475\pi\)
0.826242 + 0.563316i \(0.190475\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 9.00000 0.379980
\(562\) 0 0
\(563\) −33.0000 −1.39078 −0.695392 0.718631i \(-0.744769\pi\)
−0.695392 + 0.718631i \(0.744769\pi\)
\(564\) 0 0
\(565\) −15.0000 −0.631055
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −33.0000 −1.38343 −0.691716 0.722170i \(-0.743145\pi\)
−0.691716 + 0.722170i \(0.743145\pi\)
\(570\) 0 0
\(571\) 44.0000 1.84134 0.920671 0.390339i \(-0.127642\pi\)
0.920671 + 0.390339i \(0.127642\pi\)
\(572\) 0 0
\(573\) −3.00000 −0.125327
\(574\) 0 0
\(575\) 9.00000 0.375326
\(576\) 0 0
\(577\) 22.0000 0.915872 0.457936 0.888985i \(-0.348589\pi\)
0.457936 + 0.888985i \(0.348589\pi\)
\(578\) 0 0
\(579\) −5.00000 −0.207793
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −18.0000 −0.745484
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 27.0000 1.11441 0.557205 0.830375i \(-0.311874\pi\)
0.557205 + 0.830375i \(0.311874\pi\)
\(588\) 0 0
\(589\) 40.0000 1.64817
\(590\) 0 0
\(591\) 3.00000 0.123404
\(592\) 0 0
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 0 0
\(595\) −3.00000 −0.122988
\(596\) 0 0
\(597\) −7.00000 −0.286491
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −1.00000 −0.0407909 −0.0203954 0.999792i \(-0.506493\pi\)
−0.0203954 + 0.999792i \(0.506493\pi\)
\(602\) 0 0
\(603\) 10.0000 0.407231
\(604\) 0 0
\(605\) −2.00000 −0.0813116
\(606\) 0 0
\(607\) 23.0000 0.933541 0.466771 0.884378i \(-0.345417\pi\)
0.466771 + 0.884378i \(0.345417\pi\)
\(608\) 0 0
\(609\) −9.00000 −0.364698
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −29.0000 −1.17130 −0.585649 0.810564i \(-0.699160\pi\)
−0.585649 + 0.810564i \(0.699160\pi\)
\(614\) 0 0
\(615\) −3.00000 −0.120972
\(616\) 0 0
\(617\) 27.0000 1.08698 0.543490 0.839416i \(-0.317103\pi\)
0.543490 + 0.839416i \(0.317103\pi\)
\(618\) 0 0
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 0 0
\(621\) −45.0000 −1.80579
\(622\) 0 0
\(623\) −3.00000 −0.120192
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 15.0000 0.599042
\(628\) 0 0
\(629\) −21.0000 −0.837325
\(630\) 0 0
\(631\) 7.00000 0.278666 0.139333 0.990246i \(-0.455504\pi\)
0.139333 + 0.990246i \(0.455504\pi\)
\(632\) 0 0
\(633\) −25.0000 −0.993661
\(634\) 0 0
\(635\) 17.0000 0.674624
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 18.0000 0.712069
\(640\) 0 0
\(641\) 3.00000 0.118493 0.0592464 0.998243i \(-0.481130\pi\)
0.0592464 + 0.998243i \(0.481130\pi\)
\(642\) 0 0
\(643\) 25.0000 0.985904 0.492952 0.870057i \(-0.335918\pi\)
0.492952 + 0.870057i \(0.335918\pi\)
\(644\) 0 0
\(645\) −1.00000 −0.0393750
\(646\) 0 0
\(647\) −27.0000 −1.06148 −0.530740 0.847535i \(-0.678086\pi\)
−0.530740 + 0.847535i \(0.678086\pi\)
\(648\) 0 0
\(649\) 27.0000 1.05984
\(650\) 0 0
\(651\) −8.00000 −0.313545
\(652\) 0 0
\(653\) −27.0000 −1.05659 −0.528296 0.849060i \(-0.677169\pi\)
−0.528296 + 0.849060i \(0.677169\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 4.00000 0.156055
\(658\) 0 0
\(659\) 45.0000 1.75295 0.876476 0.481446i \(-0.159888\pi\)
0.876476 + 0.481446i \(0.159888\pi\)
\(660\) 0 0
\(661\) −35.0000 −1.36134 −0.680671 0.732589i \(-0.738312\pi\)
−0.680671 + 0.732589i \(0.738312\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −5.00000 −0.193892
\(666\) 0 0
\(667\) −81.0000 −3.13633
\(668\) 0 0
\(669\) 19.0000 0.734582
\(670\) 0 0
\(671\) 3.00000 0.115814
\(672\) 0 0
\(673\) −19.0000 −0.732396 −0.366198 0.930537i \(-0.619341\pi\)
−0.366198 + 0.930537i \(0.619341\pi\)
\(674\) 0 0
\(675\) −5.00000 −0.192450
\(676\) 0 0
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 0 0
\(679\) −17.0000 −0.652400
\(680\) 0 0
\(681\) −15.0000 −0.574801
\(682\) 0 0
\(683\) 21.0000 0.803543 0.401771 0.915740i \(-0.368395\pi\)
0.401771 + 0.915740i \(0.368395\pi\)
\(684\) 0 0
\(685\) 3.00000 0.114624
\(686\) 0 0
\(687\) 22.0000 0.839352
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −23.0000 −0.874961 −0.437481 0.899228i \(-0.644129\pi\)
−0.437481 + 0.899228i \(0.644129\pi\)
\(692\) 0 0
\(693\) 6.00000 0.227921
\(694\) 0 0
\(695\) 5.00000 0.189661
\(696\) 0 0
\(697\) 9.00000 0.340899
\(698\) 0 0
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 0 0
\(703\) −35.0000 −1.32005
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 15.0000 0.564133
\(708\) 0 0
\(709\) −35.0000 −1.31445 −0.657226 0.753693i \(-0.728270\pi\)
−0.657226 + 0.753693i \(0.728270\pi\)
\(710\) 0 0
\(711\) −16.0000 −0.600047
\(712\) 0 0
\(713\) −72.0000 −2.69642
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −24.0000 −0.896296
\(718\) 0 0
\(719\) 45.0000 1.67822 0.839108 0.543964i \(-0.183077\pi\)
0.839108 + 0.543964i \(0.183077\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) 0 0
\(723\) −23.0000 −0.855379
\(724\) 0 0
\(725\) −9.00000 −0.334252
\(726\) 0 0
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 3.00000 0.110959
\(732\) 0 0
\(733\) 34.0000 1.25582 0.627909 0.778287i \(-0.283911\pi\)
0.627909 + 0.778287i \(0.283911\pi\)
\(734\) 0 0
\(735\) −6.00000 −0.221313
\(736\) 0 0
\(737\) 15.0000 0.552532
\(738\) 0 0
\(739\) −11.0000 −0.404642 −0.202321 0.979319i \(-0.564848\pi\)
−0.202321 + 0.979319i \(0.564848\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −33.0000 −1.21065 −0.605326 0.795977i \(-0.706957\pi\)
−0.605326 + 0.795977i \(0.706957\pi\)
\(744\) 0 0
\(745\) 9.00000 0.329734
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −3.00000 −0.109618
\(750\) 0 0
\(751\) −13.0000 −0.474377 −0.237188 0.971464i \(-0.576226\pi\)
−0.237188 + 0.971464i \(0.576226\pi\)
\(752\) 0 0
\(753\) 9.00000 0.327978
\(754\) 0 0
\(755\) −20.0000 −0.727875
\(756\) 0 0
\(757\) 5.00000 0.181728 0.0908640 0.995863i \(-0.471037\pi\)
0.0908640 + 0.995863i \(0.471037\pi\)
\(758\) 0 0
\(759\) −27.0000 −0.980038
\(760\) 0 0
\(761\) 9.00000 0.326250 0.163125 0.986605i \(-0.447843\pi\)
0.163125 + 0.986605i \(0.447843\pi\)
\(762\) 0 0
\(763\) −14.0000 −0.506834
\(764\) 0 0
\(765\) 6.00000 0.216930
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −11.0000 −0.396670 −0.198335 0.980134i \(-0.563553\pi\)
−0.198335 + 0.980134i \(0.563553\pi\)
\(770\) 0 0
\(771\) −27.0000 −0.972381
\(772\) 0 0
\(773\) 15.0000 0.539513 0.269756 0.962929i \(-0.413057\pi\)
0.269756 + 0.962929i \(0.413057\pi\)
\(774\) 0 0
\(775\) −8.00000 −0.287368
\(776\) 0 0
\(777\) 7.00000 0.251124
\(778\) 0 0
\(779\) 15.0000 0.537431
\(780\) 0 0
\(781\) 27.0000 0.966136
\(782\) 0 0
\(783\) 45.0000 1.60817
\(784\) 0 0
\(785\) 14.0000 0.499681
\(786\) 0 0
\(787\) 49.0000 1.74666 0.873331 0.487128i \(-0.161955\pi\)
0.873331 + 0.487128i \(0.161955\pi\)
\(788\) 0 0
\(789\) −3.00000 −0.106803
\(790\) 0 0
\(791\) −15.0000 −0.533339
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 6.00000 0.212798
\(796\) 0 0
\(797\) −27.0000 −0.956389 −0.478195 0.878254i \(-0.658709\pi\)
−0.478195 + 0.878254i \(0.658709\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 0 0
\(803\) 6.00000 0.211735
\(804\) 0 0
\(805\) 9.00000 0.317208
\(806\) 0 0
\(807\) −21.0000 −0.739235
\(808\) 0 0
\(809\) 39.0000 1.37117 0.685583 0.727994i \(-0.259547\pi\)
0.685583 + 0.727994i \(0.259547\pi\)
\(810\) 0 0
\(811\) 16.0000 0.561836 0.280918 0.959732i \(-0.409361\pi\)
0.280918 + 0.959732i \(0.409361\pi\)
\(812\) 0 0
\(813\) 13.0000 0.455930
\(814\) 0 0
\(815\) 1.00000 0.0350285
\(816\) 0 0
\(817\) 5.00000 0.174928
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 21.0000 0.732905 0.366453 0.930437i \(-0.380572\pi\)
0.366453 + 0.930437i \(0.380572\pi\)
\(822\) 0 0
\(823\) 47.0000 1.63832 0.819159 0.573567i \(-0.194441\pi\)
0.819159 + 0.573567i \(0.194441\pi\)
\(824\) 0 0
\(825\) −3.00000 −0.104447
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) 11.0000 0.382046 0.191023 0.981586i \(-0.438820\pi\)
0.191023 + 0.981586i \(0.438820\pi\)
\(830\) 0 0
\(831\) −19.0000 −0.659103
\(832\) 0 0
\(833\) 18.0000 0.623663
\(834\) 0 0
\(835\) 15.0000 0.519096
\(836\) 0 0
\(837\) 40.0000 1.38260
\(838\) 0 0
\(839\) 15.0000 0.517858 0.258929 0.965896i \(-0.416631\pi\)
0.258929 + 0.965896i \(0.416631\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 0 0
\(843\) 18.0000 0.619953
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −2.00000 −0.0687208
\(848\) 0 0
\(849\) 17.0000 0.583438
\(850\) 0 0
\(851\) 63.0000 2.15961
\(852\) 0 0
\(853\) −14.0000 −0.479351 −0.239675 0.970853i \(-0.577041\pi\)
−0.239675 + 0.970853i \(0.577041\pi\)
\(854\) 0 0
\(855\) 10.0000 0.341993
\(856\) 0 0
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 0 0
\(859\) 8.00000 0.272956 0.136478 0.990643i \(-0.456422\pi\)
0.136478 + 0.990643i \(0.456422\pi\)
\(860\) 0 0
\(861\) −3.00000 −0.102240
\(862\) 0 0
\(863\) −48.0000 −1.63394 −0.816970 0.576681i \(-0.804348\pi\)
−0.816970 + 0.576681i \(0.804348\pi\)
\(864\) 0 0
\(865\) 21.0000 0.714021
\(866\) 0 0
\(867\) −8.00000 −0.271694
\(868\) 0 0
\(869\) −24.0000 −0.814144
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 34.0000 1.15073
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) −17.0000 −0.574049 −0.287025 0.957923i \(-0.592666\pi\)
−0.287025 + 0.957923i \(0.592666\pi\)
\(878\) 0 0
\(879\) 3.00000 0.101187
\(880\) 0 0
\(881\) 3.00000 0.101073 0.0505363 0.998722i \(-0.483907\pi\)
0.0505363 + 0.998722i \(0.483907\pi\)
\(882\) 0 0
\(883\) 56.0000 1.88455 0.942275 0.334840i \(-0.108682\pi\)
0.942275 + 0.334840i \(0.108682\pi\)
\(884\) 0 0
\(885\) −9.00000 −0.302532
\(886\) 0 0
\(887\) 33.0000 1.10803 0.554016 0.832506i \(-0.313095\pi\)
0.554016 + 0.832506i \(0.313095\pi\)
\(888\) 0 0
\(889\) 17.0000 0.570162
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 15.0000 0.501395
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 72.0000 2.40133
\(900\) 0 0
\(901\) −18.0000 −0.599667
\(902\) 0 0
\(903\) −1.00000 −0.0332779
\(904\) 0 0
\(905\) −10.0000 −0.332411
\(906\) 0 0
\(907\) −55.0000 −1.82625 −0.913123 0.407685i \(-0.866336\pi\)
−0.913123 + 0.407685i \(0.866336\pi\)
\(908\) 0 0
\(909\) −30.0000 −0.995037
\(910\) 0 0
\(911\) −48.0000 −1.59031 −0.795155 0.606406i \(-0.792611\pi\)
−0.795155 + 0.606406i \(0.792611\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −1.00000 −0.0330590
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −7.00000 −0.230909 −0.115454 0.993313i \(-0.536832\pi\)
−0.115454 + 0.993313i \(0.536832\pi\)
\(920\) 0 0
\(921\) 4.00000 0.131804
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 7.00000 0.230159
\(926\) 0 0
\(927\) −16.0000 −0.525509
\(928\) 0 0
\(929\) −3.00000 −0.0984268 −0.0492134 0.998788i \(-0.515671\pi\)
−0.0492134 + 0.998788i \(0.515671\pi\)
\(930\) 0 0
\(931\) 30.0000 0.983210
\(932\) 0 0
\(933\) 12.0000 0.392862
\(934\) 0 0
\(935\) 9.00000 0.294331
\(936\) 0 0
\(937\) −22.0000 −0.718709 −0.359354 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) 0 0
\(939\) −10.0000 −0.326338
\(940\) 0 0
\(941\) −18.0000 −0.586783 −0.293392 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) 0 0
\(943\) −27.0000 −0.879241
\(944\) 0 0
\(945\) −5.00000 −0.162650
\(946\) 0 0
\(947\) 51.0000 1.65728 0.828639 0.559784i \(-0.189116\pi\)
0.828639 + 0.559784i \(0.189116\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) 21.0000 0.680257 0.340128 0.940379i \(-0.389529\pi\)
0.340128 + 0.940379i \(0.389529\pi\)
\(954\) 0 0
\(955\) −3.00000 −0.0970777
\(956\) 0 0
\(957\) 27.0000 0.872786
\(958\) 0 0
\(959\) 3.00000 0.0968751
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 6.00000 0.193347
\(964\) 0 0
\(965\) −5.00000 −0.160956
\(966\) 0 0
\(967\) 4.00000 0.128631 0.0643157 0.997930i \(-0.479514\pi\)
0.0643157 + 0.997930i \(0.479514\pi\)
\(968\) 0 0
\(969\) 15.0000 0.481869
\(970\) 0 0
\(971\) −15.0000 −0.481373 −0.240686 0.970603i \(-0.577373\pi\)
−0.240686 + 0.970603i \(0.577373\pi\)
\(972\) 0 0
\(973\) 5.00000 0.160293
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 3.00000 0.0959785 0.0479893 0.998848i \(-0.484719\pi\)
0.0479893 + 0.998848i \(0.484719\pi\)
\(978\) 0 0
\(979\) 9.00000 0.287641
\(980\) 0 0
\(981\) 28.0000 0.893971
\(982\) 0 0
\(983\) 48.0000 1.53096 0.765481 0.643458i \(-0.222501\pi\)
0.765481 + 0.643458i \(0.222501\pi\)
\(984\) 0 0
\(985\) 3.00000 0.0955879
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −9.00000 −0.286183
\(990\) 0 0
\(991\) −49.0000 −1.55654 −0.778268 0.627932i \(-0.783902\pi\)
−0.778268 + 0.627932i \(0.783902\pi\)
\(992\) 0 0
\(993\) 19.0000 0.602947
\(994\) 0 0
\(995\) −7.00000 −0.221915
\(996\) 0 0
\(997\) 5.00000 0.158352 0.0791758 0.996861i \(-0.474771\pi\)
0.0791758 + 0.996861i \(0.474771\pi\)
\(998\) 0 0
\(999\) −35.0000 −1.10735
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.i.1.1 1
13.4 even 6 260.2.i.a.81.1 yes 2
13.5 odd 4 3380.2.f.d.3041.1 2
13.8 odd 4 3380.2.f.d.3041.2 2
13.10 even 6 260.2.i.a.61.1 2
13.12 even 2 3380.2.a.f.1.1 1
39.17 odd 6 2340.2.q.f.2161.1 2
39.23 odd 6 2340.2.q.f.1621.1 2
52.23 odd 6 1040.2.q.i.321.1 2
52.43 odd 6 1040.2.q.i.81.1 2
65.4 even 6 1300.2.i.d.601.1 2
65.17 odd 12 1300.2.bb.b.549.1 4
65.23 odd 12 1300.2.bb.b.1049.1 4
65.43 odd 12 1300.2.bb.b.549.2 4
65.49 even 6 1300.2.i.d.1101.1 2
65.62 odd 12 1300.2.bb.b.1049.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
260.2.i.a.61.1 2 13.10 even 6
260.2.i.a.81.1 yes 2 13.4 even 6
1040.2.q.i.81.1 2 52.43 odd 6
1040.2.q.i.321.1 2 52.23 odd 6
1300.2.i.d.601.1 2 65.4 even 6
1300.2.i.d.1101.1 2 65.49 even 6
1300.2.bb.b.549.1 4 65.17 odd 12
1300.2.bb.b.549.2 4 65.43 odd 12
1300.2.bb.b.1049.1 4 65.23 odd 12
1300.2.bb.b.1049.2 4 65.62 odd 12
2340.2.q.f.1621.1 2 39.23 odd 6
2340.2.q.f.2161.1 2 39.17 odd 6
3380.2.a.f.1.1 1 13.12 even 2
3380.2.a.i.1.1 1 1.1 even 1 trivial
3380.2.f.d.3041.1 2 13.5 odd 4
3380.2.f.d.3041.2 2 13.8 odd 4