Properties

Label 7406.2.a.f.1.1
Level $7406$
Weight $2$
Character 7406.1
Self dual yes
Analytic conductor $59.137$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7406 = 2 \cdot 7 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7406.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(59.1372077370\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 322)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 7406.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +2.00000 q^{11} -2.00000 q^{12} -4.00000 q^{13} +1.00000 q^{14} -4.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} +2.00000 q^{20} -2.00000 q^{21} +2.00000 q^{22} -2.00000 q^{24} -1.00000 q^{25} -4.00000 q^{26} +4.00000 q^{27} +1.00000 q^{28} -2.00000 q^{29} -4.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} -4.00000 q^{33} +6.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} +8.00000 q^{39} +2.00000 q^{40} +6.00000 q^{41} -2.00000 q^{42} -6.00000 q^{43} +2.00000 q^{44} +2.00000 q^{45} -2.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -12.0000 q^{51} -4.00000 q^{52} +12.0000 q^{53} +4.00000 q^{54} +4.00000 q^{55} +1.00000 q^{56} -2.00000 q^{58} -10.0000 q^{59} -4.00000 q^{60} -2.00000 q^{61} +4.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -8.00000 q^{65} -4.00000 q^{66} +2.00000 q^{67} +6.00000 q^{68} +2.00000 q^{70} +8.00000 q^{71} +1.00000 q^{72} +2.00000 q^{73} +2.00000 q^{75} +2.00000 q^{77} +8.00000 q^{78} -8.00000 q^{79} +2.00000 q^{80} -11.0000 q^{81} +6.00000 q^{82} +16.0000 q^{83} -2.00000 q^{84} +12.0000 q^{85} -6.00000 q^{86} +4.00000 q^{87} +2.00000 q^{88} -6.00000 q^{89} +2.00000 q^{90} -4.00000 q^{91} -8.00000 q^{93} -2.00000 q^{96} +2.00000 q^{97} +1.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) −2.00000 −0.816497
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) −2.00000 −0.577350
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 1.00000 0.267261
\(15\) −4.00000 −1.03280
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 1.00000 0.235702
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 2.00000 0.447214
\(21\) −2.00000 −0.436436
\(22\) 2.00000 0.426401
\(23\) 0 0
\(24\) −2.00000 −0.408248
\(25\) −1.00000 −0.200000
\(26\) −4.00000 −0.784465
\(27\) 4.00000 0.769800
\(28\) 1.00000 0.188982
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) −4.00000 −0.730297
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 1.00000 0.176777
\(33\) −4.00000 −0.696311
\(34\) 6.00000 1.02899
\(35\) 2.00000 0.338062
\(36\) 1.00000 0.166667
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 8.00000 1.28103
\(40\) 2.00000 0.316228
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) −2.00000 −0.308607
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 2.00000 0.301511
\(45\) 2.00000 0.298142
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −2.00000 −0.288675
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −12.0000 −1.68034
\(52\) −4.00000 −0.554700
\(53\) 12.0000 1.64833 0.824163 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) 4.00000 0.544331
\(55\) 4.00000 0.539360
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −2.00000 −0.262613
\(59\) −10.0000 −1.30189 −0.650945 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) −4.00000 −0.516398
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 4.00000 0.508001
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) −8.00000 −0.992278
\(66\) −4.00000 −0.492366
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 6.00000 0.727607
\(69\) 0 0
\(70\) 2.00000 0.239046
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 1.00000 0.117851
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) 2.00000 0.230940
\(76\) 0 0
\(77\) 2.00000 0.227921
\(78\) 8.00000 0.905822
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 2.00000 0.223607
\(81\) −11.0000 −1.22222
\(82\) 6.00000 0.662589
\(83\) 16.0000 1.75623 0.878114 0.478451i \(-0.158802\pi\)
0.878114 + 0.478451i \(0.158802\pi\)
\(84\) −2.00000 −0.218218
\(85\) 12.0000 1.30158
\(86\) −6.00000 −0.646997
\(87\) 4.00000 0.428845
\(88\) 2.00000 0.213201
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 2.00000 0.210819
\(91\) −4.00000 −0.419314
\(92\) 0 0
\(93\) −8.00000 −0.829561
\(94\) 0 0
\(95\) 0 0
\(96\) −2.00000 −0.204124
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 1.00000 0.101015
\(99\) 2.00000 0.201008
\(100\) −1.00000 −0.100000
\(101\) −8.00000 −0.796030 −0.398015 0.917379i \(-0.630301\pi\)
−0.398015 + 0.917379i \(0.630301\pi\)
\(102\) −12.0000 −1.18818
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) −4.00000 −0.392232
\(105\) −4.00000 −0.390360
\(106\) 12.0000 1.16554
\(107\) 18.0000 1.74013 0.870063 0.492941i \(-0.164078\pi\)
0.870063 + 0.492941i \(0.164078\pi\)
\(108\) 4.00000 0.384900
\(109\) 20.0000 1.91565 0.957826 0.287348i \(-0.0927736\pi\)
0.957826 + 0.287348i \(0.0927736\pi\)
\(110\) 4.00000 0.381385
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −2.00000 −0.185695
\(117\) −4.00000 −0.369800
\(118\) −10.0000 −0.920575
\(119\) 6.00000 0.550019
\(120\) −4.00000 −0.365148
\(121\) −7.00000 −0.636364
\(122\) −2.00000 −0.181071
\(123\) −12.0000 −1.08200
\(124\) 4.00000 0.359211
\(125\) −12.0000 −1.07331
\(126\) 1.00000 0.0890871
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 1.00000 0.0883883
\(129\) 12.0000 1.05654
\(130\) −8.00000 −0.701646
\(131\) 22.0000 1.92215 0.961074 0.276289i \(-0.0891049\pi\)
0.961074 + 0.276289i \(0.0891049\pi\)
\(132\) −4.00000 −0.348155
\(133\) 0 0
\(134\) 2.00000 0.172774
\(135\) 8.00000 0.688530
\(136\) 6.00000 0.514496
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 0 0
\(139\) −14.0000 −1.18746 −0.593732 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 2.00000 0.169031
\(141\) 0 0
\(142\) 8.00000 0.671345
\(143\) −8.00000 −0.668994
\(144\) 1.00000 0.0833333
\(145\) −4.00000 −0.332182
\(146\) 2.00000 0.165521
\(147\) −2.00000 −0.164957
\(148\) 0 0
\(149\) 12.0000 0.983078 0.491539 0.870855i \(-0.336434\pi\)
0.491539 + 0.870855i \(0.336434\pi\)
\(150\) 2.00000 0.163299
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 2.00000 0.161165
\(155\) 8.00000 0.642575
\(156\) 8.00000 0.640513
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) −8.00000 −0.636446
\(159\) −24.0000 −1.90332
\(160\) 2.00000 0.158114
\(161\) 0 0
\(162\) −11.0000 −0.864242
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 6.00000 0.468521
\(165\) −8.00000 −0.622799
\(166\) 16.0000 1.24184
\(167\) −16.0000 −1.23812 −0.619059 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) −2.00000 −0.154303
\(169\) 3.00000 0.230769
\(170\) 12.0000 0.920358
\(171\) 0 0
\(172\) −6.00000 −0.457496
\(173\) −24.0000 −1.82469 −0.912343 0.409426i \(-0.865729\pi\)
−0.912343 + 0.409426i \(0.865729\pi\)
\(174\) 4.00000 0.303239
\(175\) −1.00000 −0.0755929
\(176\) 2.00000 0.150756
\(177\) 20.0000 1.50329
\(178\) −6.00000 −0.449719
\(179\) 16.0000 1.19590 0.597948 0.801535i \(-0.295983\pi\)
0.597948 + 0.801535i \(0.295983\pi\)
\(180\) 2.00000 0.149071
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) −4.00000 −0.296500
\(183\) 4.00000 0.295689
\(184\) 0 0
\(185\) 0 0
\(186\) −8.00000 −0.586588
\(187\) 12.0000 0.877527
\(188\) 0 0
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) 16.0000 1.15772 0.578860 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) −2.00000 −0.144338
\(193\) −18.0000 −1.29567 −0.647834 0.761781i \(-0.724325\pi\)
−0.647834 + 0.761781i \(0.724325\pi\)
\(194\) 2.00000 0.143592
\(195\) 16.0000 1.14578
\(196\) 1.00000 0.0714286
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 2.00000 0.142134
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −4.00000 −0.282138
\(202\) −8.00000 −0.562878
\(203\) −2.00000 −0.140372
\(204\) −12.0000 −0.840168
\(205\) 12.0000 0.838116
\(206\) 0 0
\(207\) 0 0
\(208\) −4.00000 −0.277350
\(209\) 0 0
\(210\) −4.00000 −0.276026
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 12.0000 0.824163
\(213\) −16.0000 −1.09630
\(214\) 18.0000 1.23045
\(215\) −12.0000 −0.818393
\(216\) 4.00000 0.272166
\(217\) 4.00000 0.271538
\(218\) 20.0000 1.35457
\(219\) −4.00000 −0.270295
\(220\) 4.00000 0.269680
\(221\) −24.0000 −1.61441
\(222\) 0 0
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 1.00000 0.0668153
\(225\) −1.00000 −0.0666667
\(226\) 6.00000 0.399114
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) −2.00000 −0.131306
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) −4.00000 −0.261488
\(235\) 0 0
\(236\) −10.0000 −0.650945
\(237\) 16.0000 1.03931
\(238\) 6.00000 0.388922
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) −4.00000 −0.258199
\(241\) −14.0000 −0.901819 −0.450910 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) −7.00000 −0.449977
\(243\) 10.0000 0.641500
\(244\) −2.00000 −0.128037
\(245\) 2.00000 0.127775
\(246\) −12.0000 −0.765092
\(247\) 0 0
\(248\) 4.00000 0.254000
\(249\) −32.0000 −2.02792
\(250\) −12.0000 −0.758947
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 1.00000 0.0629941
\(253\) 0 0
\(254\) 16.0000 1.00393
\(255\) −24.0000 −1.50294
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 12.0000 0.747087
\(259\) 0 0
\(260\) −8.00000 −0.496139
\(261\) −2.00000 −0.123797
\(262\) 22.0000 1.35916
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) −4.00000 −0.246183
\(265\) 24.0000 1.47431
\(266\) 0 0
\(267\) 12.0000 0.734388
\(268\) 2.00000 0.122169
\(269\) 24.0000 1.46331 0.731653 0.681677i \(-0.238749\pi\)
0.731653 + 0.681677i \(0.238749\pi\)
\(270\) 8.00000 0.486864
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 6.00000 0.363803
\(273\) 8.00000 0.484182
\(274\) 2.00000 0.120824
\(275\) −2.00000 −0.120605
\(276\) 0 0
\(277\) 26.0000 1.56219 0.781094 0.624413i \(-0.214662\pi\)
0.781094 + 0.624413i \(0.214662\pi\)
\(278\) −14.0000 −0.839664
\(279\) 4.00000 0.239474
\(280\) 2.00000 0.119523
\(281\) 22.0000 1.31241 0.656205 0.754583i \(-0.272161\pi\)
0.656205 + 0.754583i \(0.272161\pi\)
\(282\) 0 0
\(283\) 12.0000 0.713326 0.356663 0.934233i \(-0.383914\pi\)
0.356663 + 0.934233i \(0.383914\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) −8.00000 −0.473050
\(287\) 6.00000 0.354169
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) −4.00000 −0.234888
\(291\) −4.00000 −0.234484
\(292\) 2.00000 0.117041
\(293\) −14.0000 −0.817889 −0.408944 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) −2.00000 −0.116642
\(295\) −20.0000 −1.16445
\(296\) 0 0
\(297\) 8.00000 0.464207
\(298\) 12.0000 0.695141
\(299\) 0 0
\(300\) 2.00000 0.115470
\(301\) −6.00000 −0.345834
\(302\) 0 0
\(303\) 16.0000 0.919176
\(304\) 0 0
\(305\) −4.00000 −0.229039
\(306\) 6.00000 0.342997
\(307\) 10.0000 0.570730 0.285365 0.958419i \(-0.407885\pi\)
0.285365 + 0.958419i \(0.407885\pi\)
\(308\) 2.00000 0.113961
\(309\) 0 0
\(310\) 8.00000 0.454369
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 8.00000 0.452911
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) −18.0000 −1.01580
\(315\) 2.00000 0.112687
\(316\) −8.00000 −0.450035
\(317\) 2.00000 0.112331 0.0561656 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) −24.0000 −1.34585
\(319\) −4.00000 −0.223957
\(320\) 2.00000 0.111803
\(321\) −36.0000 −2.00932
\(322\) 0 0
\(323\) 0 0
\(324\) −11.0000 −0.611111
\(325\) 4.00000 0.221880
\(326\) −4.00000 −0.221540
\(327\) −40.0000 −2.21201
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) −8.00000 −0.440386
\(331\) 16.0000 0.879440 0.439720 0.898135i \(-0.355078\pi\)
0.439720 + 0.898135i \(0.355078\pi\)
\(332\) 16.0000 0.878114
\(333\) 0 0
\(334\) −16.0000 −0.875481
\(335\) 4.00000 0.218543
\(336\) −2.00000 −0.109109
\(337\) −26.0000 −1.41631 −0.708155 0.706057i \(-0.750472\pi\)
−0.708155 + 0.706057i \(0.750472\pi\)
\(338\) 3.00000 0.163178
\(339\) −12.0000 −0.651751
\(340\) 12.0000 0.650791
\(341\) 8.00000 0.433224
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −6.00000 −0.323498
\(345\) 0 0
\(346\) −24.0000 −1.29025
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 4.00000 0.214423
\(349\) 24.0000 1.28469 0.642345 0.766415i \(-0.277962\pi\)
0.642345 + 0.766415i \(0.277962\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −16.0000 −0.854017
\(352\) 2.00000 0.106600
\(353\) 10.0000 0.532246 0.266123 0.963939i \(-0.414257\pi\)
0.266123 + 0.963939i \(0.414257\pi\)
\(354\) 20.0000 1.06299
\(355\) 16.0000 0.849192
\(356\) −6.00000 −0.317999
\(357\) −12.0000 −0.635107
\(358\) 16.0000 0.845626
\(359\) 4.00000 0.211112 0.105556 0.994413i \(-0.466338\pi\)
0.105556 + 0.994413i \(0.466338\pi\)
\(360\) 2.00000 0.105409
\(361\) −19.0000 −1.00000
\(362\) 2.00000 0.105118
\(363\) 14.0000 0.734809
\(364\) −4.00000 −0.209657
\(365\) 4.00000 0.209370
\(366\) 4.00000 0.209083
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) −8.00000 −0.414781
\(373\) −4.00000 −0.207112 −0.103556 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) 12.0000 0.620505
\(375\) 24.0000 1.23935
\(376\) 0 0
\(377\) 8.00000 0.412021
\(378\) 4.00000 0.205738
\(379\) −2.00000 −0.102733 −0.0513665 0.998680i \(-0.516358\pi\)
−0.0513665 + 0.998680i \(0.516358\pi\)
\(380\) 0 0
\(381\) −32.0000 −1.63941
\(382\) 16.0000 0.818631
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) −2.00000 −0.102062
\(385\) 4.00000 0.203859
\(386\) −18.0000 −0.916176
\(387\) −6.00000 −0.304997
\(388\) 2.00000 0.101535
\(389\) 28.0000 1.41966 0.709828 0.704375i \(-0.248773\pi\)
0.709828 + 0.704375i \(0.248773\pi\)
\(390\) 16.0000 0.810191
\(391\) 0 0
\(392\) 1.00000 0.0505076
\(393\) −44.0000 −2.21951
\(394\) 18.0000 0.906827
\(395\) −16.0000 −0.805047
\(396\) 2.00000 0.100504
\(397\) −20.0000 −1.00377 −0.501886 0.864934i \(-0.667360\pi\)
−0.501886 + 0.864934i \(0.667360\pi\)
\(398\) −24.0000 −1.20301
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) −4.00000 −0.199502
\(403\) −16.0000 −0.797017
\(404\) −8.00000 −0.398015
\(405\) −22.0000 −1.09319
\(406\) −2.00000 −0.0992583
\(407\) 0 0
\(408\) −12.0000 −0.594089
\(409\) −34.0000 −1.68119 −0.840596 0.541663i \(-0.817795\pi\)
−0.840596 + 0.541663i \(0.817795\pi\)
\(410\) 12.0000 0.592638
\(411\) −4.00000 −0.197305
\(412\) 0 0
\(413\) −10.0000 −0.492068
\(414\) 0 0
\(415\) 32.0000 1.57082
\(416\) −4.00000 −0.196116
\(417\) 28.0000 1.37117
\(418\) 0 0
\(419\) −4.00000 −0.195413 −0.0977064 0.995215i \(-0.531151\pi\)
−0.0977064 + 0.995215i \(0.531151\pi\)
\(420\) −4.00000 −0.195180
\(421\) 20.0000 0.974740 0.487370 0.873195i \(-0.337956\pi\)
0.487370 + 0.873195i \(0.337956\pi\)
\(422\) 4.00000 0.194717
\(423\) 0 0
\(424\) 12.0000 0.582772
\(425\) −6.00000 −0.291043
\(426\) −16.0000 −0.775203
\(427\) −2.00000 −0.0967868
\(428\) 18.0000 0.870063
\(429\) 16.0000 0.772487
\(430\) −12.0000 −0.578691
\(431\) −28.0000 −1.34871 −0.674356 0.738406i \(-0.735579\pi\)
−0.674356 + 0.738406i \(0.735579\pi\)
\(432\) 4.00000 0.192450
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 4.00000 0.192006
\(435\) 8.00000 0.383571
\(436\) 20.0000 0.957826
\(437\) 0 0
\(438\) −4.00000 −0.191127
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 4.00000 0.190693
\(441\) 1.00000 0.0476190
\(442\) −24.0000 −1.14156
\(443\) −16.0000 −0.760183 −0.380091 0.924949i \(-0.624107\pi\)
−0.380091 + 0.924949i \(0.624107\pi\)
\(444\) 0 0
\(445\) −12.0000 −0.568855
\(446\) −16.0000 −0.757622
\(447\) −24.0000 −1.13516
\(448\) 1.00000 0.0472456
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 12.0000 0.565058
\(452\) 6.00000 0.282216
\(453\) 0 0
\(454\) 12.0000 0.563188
\(455\) −8.00000 −0.375046
\(456\) 0 0
\(457\) 18.0000 0.842004 0.421002 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) −6.00000 −0.280362
\(459\) 24.0000 1.12022
\(460\) 0 0
\(461\) 8.00000 0.372597 0.186299 0.982493i \(-0.440351\pi\)
0.186299 + 0.982493i \(0.440351\pi\)
\(462\) −4.00000 −0.186097
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) −2.00000 −0.0928477
\(465\) −16.0000 −0.741982
\(466\) 10.0000 0.463241
\(467\) 40.0000 1.85098 0.925490 0.378773i \(-0.123654\pi\)
0.925490 + 0.378773i \(0.123654\pi\)
\(468\) −4.00000 −0.184900
\(469\) 2.00000 0.0923514
\(470\) 0 0
\(471\) 36.0000 1.65879
\(472\) −10.0000 −0.460287
\(473\) −12.0000 −0.551761
\(474\) 16.0000 0.734904
\(475\) 0 0
\(476\) 6.00000 0.275010
\(477\) 12.0000 0.549442
\(478\) −16.0000 −0.731823
\(479\) 8.00000 0.365529 0.182765 0.983157i \(-0.441495\pi\)
0.182765 + 0.983157i \(0.441495\pi\)
\(480\) −4.00000 −0.182574
\(481\) 0 0
\(482\) −14.0000 −0.637683
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 4.00000 0.181631
\(486\) 10.0000 0.453609
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 8.00000 0.361773
\(490\) 2.00000 0.0903508
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) −12.0000 −0.541002
\(493\) −12.0000 −0.540453
\(494\) 0 0
\(495\) 4.00000 0.179787
\(496\) 4.00000 0.179605
\(497\) 8.00000 0.358849
\(498\) −32.0000 −1.43395
\(499\) −24.0000 −1.07439 −0.537194 0.843459i \(-0.680516\pi\)
−0.537194 + 0.843459i \(0.680516\pi\)
\(500\) −12.0000 −0.536656
\(501\) 32.0000 1.42965
\(502\) 12.0000 0.535586
\(503\) 32.0000 1.42681 0.713405 0.700752i \(-0.247152\pi\)
0.713405 + 0.700752i \(0.247152\pi\)
\(504\) 1.00000 0.0445435
\(505\) −16.0000 −0.711991
\(506\) 0 0
\(507\) −6.00000 −0.266469
\(508\) 16.0000 0.709885
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) −24.0000 −1.06274
\(511\) 2.00000 0.0884748
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −6.00000 −0.264649
\(515\) 0 0
\(516\) 12.0000 0.528271
\(517\) 0 0
\(518\) 0 0
\(519\) 48.0000 2.10697
\(520\) −8.00000 −0.350823
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) −2.00000 −0.0875376
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) 22.0000 0.961074
\(525\) 2.00000 0.0872872
\(526\) 12.0000 0.523225
\(527\) 24.0000 1.04546
\(528\) −4.00000 −0.174078
\(529\) 0 0
\(530\) 24.0000 1.04249
\(531\) −10.0000 −0.433963
\(532\) 0 0
\(533\) −24.0000 −1.03956
\(534\) 12.0000 0.519291
\(535\) 36.0000 1.55642
\(536\) 2.00000 0.0863868
\(537\) −32.0000 −1.38090
\(538\) 24.0000 1.03471
\(539\) 2.00000 0.0861461
\(540\) 8.00000 0.344265
\(541\) −6.00000 −0.257960 −0.128980 0.991647i \(-0.541170\pi\)
−0.128980 + 0.991647i \(0.541170\pi\)
\(542\) 20.0000 0.859074
\(543\) −4.00000 −0.171656
\(544\) 6.00000 0.257248
\(545\) 40.0000 1.71341
\(546\) 8.00000 0.342368
\(547\) −40.0000 −1.71028 −0.855138 0.518400i \(-0.826528\pi\)
−0.855138 + 0.518400i \(0.826528\pi\)
\(548\) 2.00000 0.0854358
\(549\) −2.00000 −0.0853579
\(550\) −2.00000 −0.0852803
\(551\) 0 0
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) 26.0000 1.10463
\(555\) 0 0
\(556\) −14.0000 −0.593732
\(557\) −32.0000 −1.35588 −0.677942 0.735116i \(-0.737128\pi\)
−0.677942 + 0.735116i \(0.737128\pi\)
\(558\) 4.00000 0.169334
\(559\) 24.0000 1.01509
\(560\) 2.00000 0.0845154
\(561\) −24.0000 −1.01328
\(562\) 22.0000 0.928014
\(563\) −24.0000 −1.01148 −0.505740 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 0 0
\(565\) 12.0000 0.504844
\(566\) 12.0000 0.504398
\(567\) −11.0000 −0.461957
\(568\) 8.00000 0.335673
\(569\) 42.0000 1.76073 0.880366 0.474295i \(-0.157297\pi\)
0.880366 + 0.474295i \(0.157297\pi\)
\(570\) 0 0
\(571\) 38.0000 1.59025 0.795125 0.606445i \(-0.207405\pi\)
0.795125 + 0.606445i \(0.207405\pi\)
\(572\) −8.00000 −0.334497
\(573\) −32.0000 −1.33682
\(574\) 6.00000 0.250435
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −38.0000 −1.58196 −0.790980 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) 19.0000 0.790296
\(579\) 36.0000 1.49611
\(580\) −4.00000 −0.166091
\(581\) 16.0000 0.663792
\(582\) −4.00000 −0.165805
\(583\) 24.0000 0.993978
\(584\) 2.00000 0.0827606
\(585\) −8.00000 −0.330759
\(586\) −14.0000 −0.578335
\(587\) 18.0000 0.742940 0.371470 0.928445i \(-0.378854\pi\)
0.371470 + 0.928445i \(0.378854\pi\)
\(588\) −2.00000 −0.0824786
\(589\) 0 0
\(590\) −20.0000 −0.823387
\(591\) −36.0000 −1.48084
\(592\) 0 0
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 8.00000 0.328244
\(595\) 12.0000 0.491952
\(596\) 12.0000 0.491539
\(597\) 48.0000 1.96451
\(598\) 0 0
\(599\) 40.0000 1.63436 0.817178 0.576386i \(-0.195537\pi\)
0.817178 + 0.576386i \(0.195537\pi\)
\(600\) 2.00000 0.0816497
\(601\) −46.0000 −1.87638 −0.938190 0.346122i \(-0.887498\pi\)
−0.938190 + 0.346122i \(0.887498\pi\)
\(602\) −6.00000 −0.244542
\(603\) 2.00000 0.0814463
\(604\) 0 0
\(605\) −14.0000 −0.569181
\(606\) 16.0000 0.649956
\(607\) −40.0000 −1.62355 −0.811775 0.583970i \(-0.801498\pi\)
−0.811775 + 0.583970i \(0.801498\pi\)
\(608\) 0 0
\(609\) 4.00000 0.162088
\(610\) −4.00000 −0.161955
\(611\) 0 0
\(612\) 6.00000 0.242536
\(613\) −44.0000 −1.77714 −0.888572 0.458738i \(-0.848302\pi\)
−0.888572 + 0.458738i \(0.848302\pi\)
\(614\) 10.0000 0.403567
\(615\) −24.0000 −0.967773
\(616\) 2.00000 0.0805823
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 0 0
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 8.00000 0.321288
\(621\) 0 0
\(622\) 24.0000 0.962312
\(623\) −6.00000 −0.240385
\(624\) 8.00000 0.320256
\(625\) −19.0000 −0.760000
\(626\) 10.0000 0.399680
\(627\) 0 0
\(628\) −18.0000 −0.718278
\(629\) 0 0
\(630\) 2.00000 0.0796819
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) −8.00000 −0.318223
\(633\) −8.00000 −0.317971
\(634\) 2.00000 0.0794301
\(635\) 32.0000 1.26988
\(636\) −24.0000 −0.951662
\(637\) −4.00000 −0.158486
\(638\) −4.00000 −0.158362
\(639\) 8.00000 0.316475
\(640\) 2.00000 0.0790569
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) −36.0000 −1.42081
\(643\) −24.0000 −0.946468 −0.473234 0.880937i \(-0.656913\pi\)
−0.473234 + 0.880937i \(0.656913\pi\)
\(644\) 0 0
\(645\) 24.0000 0.944999
\(646\) 0 0
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) −11.0000 −0.432121
\(649\) −20.0000 −0.785069
\(650\) 4.00000 0.156893
\(651\) −8.00000 −0.313545
\(652\) −4.00000 −0.156652
\(653\) 26.0000 1.01746 0.508729 0.860927i \(-0.330115\pi\)
0.508729 + 0.860927i \(0.330115\pi\)
\(654\) −40.0000 −1.56412
\(655\) 44.0000 1.71922
\(656\) 6.00000 0.234261
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) 6.00000 0.233727 0.116863 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) −8.00000 −0.311400
\(661\) 30.0000 1.16686 0.583432 0.812162i \(-0.301709\pi\)
0.583432 + 0.812162i \(0.301709\pi\)
\(662\) 16.0000 0.621858
\(663\) 48.0000 1.86417
\(664\) 16.0000 0.620920
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −16.0000 −0.619059
\(669\) 32.0000 1.23719
\(670\) 4.00000 0.154533
\(671\) −4.00000 −0.154418
\(672\) −2.00000 −0.0771517
\(673\) −50.0000 −1.92736 −0.963679 0.267063i \(-0.913947\pi\)
−0.963679 + 0.267063i \(0.913947\pi\)
\(674\) −26.0000 −1.00148
\(675\) −4.00000 −0.153960
\(676\) 3.00000 0.115385
\(677\) 22.0000 0.845529 0.422764 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) −12.0000 −0.460857
\(679\) 2.00000 0.0767530
\(680\) 12.0000 0.460179
\(681\) −24.0000 −0.919682
\(682\) 8.00000 0.306336
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) 4.00000 0.152832
\(686\) 1.00000 0.0381802
\(687\) 12.0000 0.457829
\(688\) −6.00000 −0.228748
\(689\) −48.0000 −1.82865
\(690\) 0 0
\(691\) 46.0000 1.74992 0.874961 0.484193i \(-0.160887\pi\)
0.874961 + 0.484193i \(0.160887\pi\)
\(692\) −24.0000 −0.912343
\(693\) 2.00000 0.0759737
\(694\) 12.0000 0.455514
\(695\) −28.0000 −1.06210
\(696\) 4.00000 0.151620
\(697\) 36.0000 1.36360
\(698\) 24.0000 0.908413
\(699\) −20.0000 −0.756469
\(700\) −1.00000 −0.0377964
\(701\) −20.0000 −0.755390 −0.377695 0.925930i \(-0.623283\pi\)
−0.377695 + 0.925930i \(0.623283\pi\)
\(702\) −16.0000 −0.603881
\(703\) 0 0
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) 10.0000 0.376355
\(707\) −8.00000 −0.300871
\(708\) 20.0000 0.751646
\(709\) 24.0000 0.901339 0.450669 0.892691i \(-0.351185\pi\)
0.450669 + 0.892691i \(0.351185\pi\)
\(710\) 16.0000 0.600469
\(711\) −8.00000 −0.300023
\(712\) −6.00000 −0.224860
\(713\) 0 0
\(714\) −12.0000 −0.449089
\(715\) −16.0000 −0.598366
\(716\) 16.0000 0.597948
\(717\) 32.0000 1.19506
\(718\) 4.00000 0.149279
\(719\) −20.0000 −0.745874 −0.372937 0.927857i \(-0.621649\pi\)
−0.372937 + 0.927857i \(0.621649\pi\)
\(720\) 2.00000 0.0745356
\(721\) 0 0
\(722\) −19.0000 −0.707107
\(723\) 28.0000 1.04133
\(724\) 2.00000 0.0743294
\(725\) 2.00000 0.0742781
\(726\) 14.0000 0.519589
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) −4.00000 −0.148250
\(729\) 13.0000 0.481481
\(730\) 4.00000 0.148047
\(731\) −36.0000 −1.33151
\(732\) 4.00000 0.147844
\(733\) −38.0000 −1.40356 −0.701781 0.712393i \(-0.747612\pi\)
−0.701781 + 0.712393i \(0.747612\pi\)
\(734\) 0 0
\(735\) −4.00000 −0.147542
\(736\) 0 0
\(737\) 4.00000 0.147342
\(738\) 6.00000 0.220863
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 12.0000 0.440534
\(743\) −48.0000 −1.76095 −0.880475 0.474093i \(-0.842776\pi\)
−0.880475 + 0.474093i \(0.842776\pi\)
\(744\) −8.00000 −0.293294
\(745\) 24.0000 0.879292
\(746\) −4.00000 −0.146450
\(747\) 16.0000 0.585409
\(748\) 12.0000 0.438763
\(749\) 18.0000 0.657706
\(750\) 24.0000 0.876356
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) −24.0000 −0.874609
\(754\) 8.00000 0.291343
\(755\) 0 0
\(756\) 4.00000 0.145479
\(757\) −44.0000 −1.59921 −0.799604 0.600528i \(-0.794957\pi\)
−0.799604 + 0.600528i \(0.794957\pi\)
\(758\) −2.00000 −0.0726433
\(759\) 0 0
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) −32.0000 −1.15924
\(763\) 20.0000 0.724049
\(764\) 16.0000 0.578860
\(765\) 12.0000 0.433861
\(766\) 24.0000 0.867155
\(767\) 40.0000 1.44432
\(768\) −2.00000 −0.0721688
\(769\) −26.0000 −0.937584 −0.468792 0.883309i \(-0.655311\pi\)
−0.468792 + 0.883309i \(0.655311\pi\)
\(770\) 4.00000 0.144150
\(771\) 12.0000 0.432169
\(772\) −18.0000 −0.647834
\(773\) −10.0000 −0.359675 −0.179838 0.983696i \(-0.557557\pi\)
−0.179838 + 0.983696i \(0.557557\pi\)
\(774\) −6.00000 −0.215666
\(775\) −4.00000 −0.143684
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) 28.0000 1.00385
\(779\) 0 0
\(780\) 16.0000 0.572892
\(781\) 16.0000 0.572525
\(782\) 0 0
\(783\) −8.00000 −0.285897
\(784\) 1.00000 0.0357143
\(785\) −36.0000 −1.28490
\(786\) −44.0000 −1.56943
\(787\) −20.0000 −0.712923 −0.356462 0.934310i \(-0.616017\pi\)
−0.356462 + 0.934310i \(0.616017\pi\)
\(788\) 18.0000 0.641223
\(789\) −24.0000 −0.854423
\(790\) −16.0000 −0.569254
\(791\) 6.00000 0.213335
\(792\) 2.00000 0.0710669
\(793\) 8.00000 0.284088
\(794\) −20.0000 −0.709773
\(795\) −48.0000 −1.70238
\(796\) −24.0000 −0.850657
\(797\) −14.0000 −0.495905 −0.247953 0.968772i \(-0.579758\pi\)
−0.247953 + 0.968772i \(0.579758\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) −6.00000 −0.212000
\(802\) −6.00000 −0.211867
\(803\) 4.00000 0.141157
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) −16.0000 −0.563576
\(807\) −48.0000 −1.68968
\(808\) −8.00000 −0.281439
\(809\) −42.0000 −1.47664 −0.738321 0.674450i \(-0.764381\pi\)
−0.738321 + 0.674450i \(0.764381\pi\)
\(810\) −22.0000 −0.773001
\(811\) −2.00000 −0.0702295 −0.0351147 0.999383i \(-0.511180\pi\)
−0.0351147 + 0.999383i \(0.511180\pi\)
\(812\) −2.00000 −0.0701862
\(813\) −40.0000 −1.40286
\(814\) 0 0
\(815\) −8.00000 −0.280228
\(816\) −12.0000 −0.420084
\(817\) 0 0
\(818\) −34.0000 −1.18878
\(819\) −4.00000 −0.139771
\(820\) 12.0000 0.419058
\(821\) 54.0000 1.88461 0.942306 0.334751i \(-0.108652\pi\)
0.942306 + 0.334751i \(0.108652\pi\)
\(822\) −4.00000 −0.139516
\(823\) 8.00000 0.278862 0.139431 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) 0 0
\(825\) 4.00000 0.139262
\(826\) −10.0000 −0.347945
\(827\) −14.0000 −0.486828 −0.243414 0.969923i \(-0.578267\pi\)
−0.243414 + 0.969923i \(0.578267\pi\)
\(828\) 0 0
\(829\) 12.0000 0.416777 0.208389 0.978046i \(-0.433178\pi\)
0.208389 + 0.978046i \(0.433178\pi\)
\(830\) 32.0000 1.11074
\(831\) −52.0000 −1.80386
\(832\) −4.00000 −0.138675
\(833\) 6.00000 0.207888
\(834\) 28.0000 0.969561
\(835\) −32.0000 −1.10741
\(836\) 0 0
\(837\) 16.0000 0.553041
\(838\) −4.00000 −0.138178
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) −4.00000 −0.138013
\(841\) −25.0000 −0.862069
\(842\) 20.0000 0.689246
\(843\) −44.0000 −1.51544
\(844\) 4.00000 0.137686
\(845\) 6.00000 0.206406
\(846\) 0 0
\(847\) −7.00000 −0.240523
\(848\) 12.0000 0.412082
\(849\) −24.0000 −0.823678
\(850\) −6.00000 −0.205798
\(851\) 0 0
\(852\) −16.0000 −0.548151
\(853\) −44.0000 −1.50653 −0.753266 0.657716i \(-0.771523\pi\)
−0.753266 + 0.657716i \(0.771523\pi\)
\(854\) −2.00000 −0.0684386
\(855\) 0 0
\(856\) 18.0000 0.615227
\(857\) 38.0000 1.29806 0.649028 0.760765i \(-0.275176\pi\)
0.649028 + 0.760765i \(0.275176\pi\)
\(858\) 16.0000 0.546231
\(859\) −42.0000 −1.43302 −0.716511 0.697576i \(-0.754262\pi\)
−0.716511 + 0.697576i \(0.754262\pi\)
\(860\) −12.0000 −0.409197
\(861\) −12.0000 −0.408959
\(862\) −28.0000 −0.953684
\(863\) −8.00000 −0.272323 −0.136162 0.990687i \(-0.543477\pi\)
−0.136162 + 0.990687i \(0.543477\pi\)
\(864\) 4.00000 0.136083
\(865\) −48.0000 −1.63205
\(866\) 14.0000 0.475739
\(867\) −38.0000 −1.29055
\(868\) 4.00000 0.135769
\(869\) −16.0000 −0.542763
\(870\) 8.00000 0.271225
\(871\) −8.00000 −0.271070
\(872\) 20.0000 0.677285
\(873\) 2.00000 0.0676897
\(874\) 0 0
\(875\) −12.0000 −0.405674
\(876\) −4.00000 −0.135147
\(877\) −2.00000 −0.0675352 −0.0337676 0.999430i \(-0.510751\pi\)
−0.0337676 + 0.999430i \(0.510751\pi\)
\(878\) 20.0000 0.674967
\(879\) 28.0000 0.944417
\(880\) 4.00000 0.134840
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 1.00000 0.0336718
\(883\) 12.0000 0.403832 0.201916 0.979403i \(-0.435283\pi\)
0.201916 + 0.979403i \(0.435283\pi\)
\(884\) −24.0000 −0.807207
\(885\) 40.0000 1.34459
\(886\) −16.0000 −0.537531
\(887\) −8.00000 −0.268614 −0.134307 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) 0 0
\(889\) 16.0000 0.536623
\(890\) −12.0000 −0.402241
\(891\) −22.0000 −0.737028
\(892\) −16.0000 −0.535720
\(893\) 0 0
\(894\) −24.0000 −0.802680
\(895\) 32.0000 1.06964
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 30.0000 1.00111
\(899\) −8.00000 −0.266815
\(900\) −1.00000 −0.0333333
\(901\) 72.0000 2.39867
\(902\) 12.0000 0.399556
\(903\) 12.0000 0.399335
\(904\) 6.00000 0.199557
\(905\) 4.00000 0.132964
\(906\) 0 0
\(907\) 46.0000 1.52740 0.763702 0.645568i \(-0.223379\pi\)
0.763702 + 0.645568i \(0.223379\pi\)
\(908\) 12.0000 0.398234
\(909\) −8.00000 −0.265343
\(910\) −8.00000 −0.265197
\(911\) −36.0000 −1.19273 −0.596367 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) 0 0
\(913\) 32.0000 1.05905
\(914\) 18.0000 0.595387
\(915\) 8.00000 0.264472
\(916\) −6.00000 −0.198246
\(917\) 22.0000 0.726504
\(918\) 24.0000 0.792118
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) −20.0000 −0.659022
\(922\) 8.00000 0.263466
\(923\) −32.0000 −1.05329
\(924\) −4.00000 −0.131590
\(925\) 0 0
\(926\) 16.0000 0.525793
\(927\) 0 0
\(928\) −2.00000 −0.0656532
\(929\) −34.0000 −1.11550 −0.557752 0.830008i \(-0.688336\pi\)
−0.557752 + 0.830008i \(0.688336\pi\)
\(930\) −16.0000 −0.524661
\(931\) 0 0
\(932\) 10.0000 0.327561
\(933\) −48.0000 −1.57145
\(934\) 40.0000 1.30884
\(935\) 24.0000 0.784884
\(936\) −4.00000 −0.130744
\(937\) −30.0000 −0.980057 −0.490029 0.871706i \(-0.663014\pi\)
−0.490029 + 0.871706i \(0.663014\pi\)
\(938\) 2.00000 0.0653023
\(939\) −20.0000 −0.652675
\(940\) 0 0
\(941\) 10.0000 0.325991 0.162995 0.986627i \(-0.447884\pi\)
0.162995 + 0.986627i \(0.447884\pi\)
\(942\) 36.0000 1.17294
\(943\) 0 0
\(944\) −10.0000 −0.325472
\(945\) 8.00000 0.260240
\(946\) −12.0000 −0.390154
\(947\) 60.0000 1.94974 0.974869 0.222779i \(-0.0715128\pi\)
0.974869 + 0.222779i \(0.0715128\pi\)
\(948\) 16.0000 0.519656
\(949\) −8.00000 −0.259691
\(950\) 0 0
\(951\) −4.00000 −0.129709
\(952\) 6.00000 0.194461
\(953\) 26.0000 0.842223 0.421111 0.907009i \(-0.361640\pi\)
0.421111 + 0.907009i \(0.361640\pi\)
\(954\) 12.0000 0.388514
\(955\) 32.0000 1.03550
\(956\) −16.0000 −0.517477
\(957\) 8.00000 0.258603
\(958\) 8.00000 0.258468
\(959\) 2.00000 0.0645834
\(960\) −4.00000 −0.129099
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 18.0000 0.580042
\(964\) −14.0000 −0.450910
\(965\) −36.0000 −1.15888
\(966\) 0 0
\(967\) 24.0000 0.771788 0.385894 0.922543i \(-0.373893\pi\)
0.385894 + 0.922543i \(0.373893\pi\)
\(968\) −7.00000 −0.224989
\(969\) 0 0
\(970\) 4.00000 0.128432
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) 10.0000 0.320750
\(973\) −14.0000 −0.448819
\(974\) 8.00000 0.256337
\(975\) −8.00000 −0.256205
\(976\) −2.00000 −0.0640184
\(977\) −30.0000 −0.959785 −0.479893 0.877327i \(-0.659324\pi\)
−0.479893 + 0.877327i \(0.659324\pi\)
\(978\) 8.00000 0.255812
\(979\) −12.0000 −0.383522
\(980\) 2.00000 0.0638877
\(981\) 20.0000 0.638551
\(982\) 0 0
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) −12.0000 −0.382546
\(985\) 36.0000 1.14706
\(986\) −12.0000 −0.382158
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 4.00000 0.127128
\(991\) 40.0000 1.27064 0.635321 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) 4.00000 0.127000
\(993\) −32.0000 −1.01549
\(994\) 8.00000 0.253745
\(995\) −48.0000 −1.52170
\(996\) −32.0000 −1.01396
\(997\) −36.0000 −1.14013 −0.570066 0.821599i \(-0.693082\pi\)
−0.570066 + 0.821599i \(0.693082\pi\)
\(998\) −24.0000 −0.759707
\(999\) 0 0
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 7406.2.a.f.1.1 1
23.22 odd 2 322.2.a.c.1.1 1
69.68 even 2 2898.2.a.g.1.1 1
92.91 even 2 2576.2.a.m.1.1 1
115.114 odd 2 8050.2.a.m.1.1 1
161.160 even 2 2254.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
322.2.a.c.1.1 1 23.22 odd 2
2254.2.a.g.1.1 1 161.160 even 2
2576.2.a.m.1.1 1 92.91 even 2
2898.2.a.g.1.1 1 69.68 even 2
7406.2.a.f.1.1 1 1.1 even 1 trivial
8050.2.a.m.1.1 1 115.114 odd 2