Properties

Label 3450.2.a.l.1.1
Level $3450$
Weight $2$
Character 3450.1
Self dual yes
Analytic conductor $27.548$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.5483886973\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 690)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3450.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{11} +1.00000 q^{12} +6.00000 q^{13} -2.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} +2.00000 q^{21} +2.00000 q^{22} -1.00000 q^{23} -1.00000 q^{24} -6.00000 q^{26} +1.00000 q^{27} +2.00000 q^{28} +2.00000 q^{29} -1.00000 q^{32} -2.00000 q^{33} -4.00000 q^{34} +1.00000 q^{36} +8.00000 q^{37} +6.00000 q^{39} -6.00000 q^{41} -2.00000 q^{42} +4.00000 q^{43} -2.00000 q^{44} +1.00000 q^{46} +1.00000 q^{48} -3.00000 q^{49} +4.00000 q^{51} +6.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} -2.00000 q^{56} -2.00000 q^{58} -8.00000 q^{61} +2.00000 q^{63} +1.00000 q^{64} +2.00000 q^{66} +4.00000 q^{67} +4.00000 q^{68} -1.00000 q^{69} +16.0000 q^{71} -1.00000 q^{72} -6.00000 q^{73} -8.00000 q^{74} -4.00000 q^{77} -6.00000 q^{78} +14.0000 q^{79} +1.00000 q^{81} +6.00000 q^{82} -14.0000 q^{83} +2.00000 q^{84} -4.00000 q^{86} +2.00000 q^{87} +2.00000 q^{88} -8.00000 q^{89} +12.0000 q^{91} -1.00000 q^{92} -1.00000 q^{96} +6.00000 q^{97} +3.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 1.00000 0.288675
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) −1.00000 −0.235702
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 2.00000 0.426401
\(23\) −1.00000 −0.208514
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −6.00000 −1.17670
\(27\) 1.00000 0.192450
\(28\) 2.00000 0.377964
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.00000 −0.348155
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 0 0
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) −2.00000 −0.308607
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) 6.00000 0.832050
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) 0 0
\(58\) −2.00000 −0.262613
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 0 0
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 4.00000 0.485071
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 16.0000 1.89885 0.949425 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) −1.00000 −0.117851
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) −8.00000 −0.929981
\(75\) 0 0
\(76\) 0 0
\(77\) −4.00000 −0.455842
\(78\) −6.00000 −0.679366
\(79\) 14.0000 1.57512 0.787562 0.616236i \(-0.211343\pi\)
0.787562 + 0.616236i \(0.211343\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) −14.0000 −1.53670 −0.768350 0.640030i \(-0.778922\pi\)
−0.768350 + 0.640030i \(0.778922\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 2.00000 0.214423
\(88\) 2.00000 0.213201
\(89\) −8.00000 −0.847998 −0.423999 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) 0 0
\(91\) 12.0000 1.25794
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 3.00000 0.303046
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) −4.00000 −0.396059
\(103\) 2.00000 0.197066 0.0985329 0.995134i \(-0.468585\pi\)
0.0985329 + 0.995134i \(0.468585\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 18.0000 1.74013 0.870063 0.492941i \(-0.164078\pi\)
0.870063 + 0.492941i \(0.164078\pi\)
\(108\) 1.00000 0.0962250
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) 8.00000 0.759326
\(112\) 2.00000 0.188982
\(113\) −12.0000 −1.12887 −0.564433 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) 6.00000 0.554700
\(118\) 0 0
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 8.00000 0.724286
\(123\) −6.00000 −0.541002
\(124\) 0 0
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) −2.00000 −0.174078
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) 8.00000 0.683486 0.341743 0.939793i \(-0.388983\pi\)
0.341743 + 0.939793i \(0.388983\pi\)
\(138\) 1.00000 0.0851257
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −16.0000 −1.34269
\(143\) −12.0000 −1.00349
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 6.00000 0.496564
\(147\) −3.00000 −0.247436
\(148\) 8.00000 0.657596
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 0 0
\(153\) 4.00000 0.323381
\(154\) 4.00000 0.322329
\(155\) 0 0
\(156\) 6.00000 0.480384
\(157\) 16.0000 1.27694 0.638470 0.769647i \(-0.279568\pi\)
0.638470 + 0.769647i \(0.279568\pi\)
\(158\) −14.0000 −1.11378
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) −2.00000 −0.157622
\(162\) −1.00000 −0.0785674
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 14.0000 1.08661
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) −2.00000 −0.154303
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) −2.00000 −0.151620
\(175\) 0 0
\(176\) −2.00000 −0.150756
\(177\) 0 0
\(178\) 8.00000 0.599625
\(179\) −20.0000 −1.49487 −0.747435 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 0 0
\(181\) −12.0000 −0.891953 −0.445976 0.895045i \(-0.647144\pi\)
−0.445976 + 0.895045i \(0.647144\pi\)
\(182\) −12.0000 −0.889499
\(183\) −8.00000 −0.591377
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) 0 0
\(187\) −8.00000 −0.585018
\(188\) 0 0
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1.00000 0.0721688
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) −6.00000 −0.430775
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 2.00000 0.142134
\(199\) −2.00000 −0.141776 −0.0708881 0.997484i \(-0.522583\pi\)
−0.0708881 + 0.997484i \(0.522583\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) −6.00000 −0.422159
\(203\) 4.00000 0.280745
\(204\) 4.00000 0.280056
\(205\) 0 0
\(206\) −2.00000 −0.139347
\(207\) −1.00000 −0.0695048
\(208\) 6.00000 0.416025
\(209\) 0 0
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) −6.00000 −0.412082
\(213\) 16.0000 1.09630
\(214\) −18.0000 −1.23045
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 4.00000 0.270914
\(219\) −6.00000 −0.405442
\(220\) 0 0
\(221\) 24.0000 1.61441
\(222\) −8.00000 −0.536925
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) −2.00000 −0.133631
\(225\) 0 0
\(226\) 12.0000 0.798228
\(227\) 18.0000 1.19470 0.597351 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) 0 0
\(229\) −20.0000 −1.32164 −0.660819 0.750546i \(-0.729791\pi\)
−0.660819 + 0.750546i \(0.729791\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) −2.00000 −0.131306
\(233\) 26.0000 1.70332 0.851658 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) −6.00000 −0.392232
\(235\) 0 0
\(236\) 0 0
\(237\) 14.0000 0.909398
\(238\) −8.00000 −0.518563
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 0 0
\(241\) −14.0000 −0.901819 −0.450910 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 7.00000 0.449977
\(243\) 1.00000 0.0641500
\(244\) −8.00000 −0.512148
\(245\) 0 0
\(246\) 6.00000 0.382546
\(247\) 0 0
\(248\) 0 0
\(249\) −14.0000 −0.887214
\(250\) 0 0
\(251\) 6.00000 0.378717 0.189358 0.981908i \(-0.439359\pi\)
0.189358 + 0.981908i \(0.439359\pi\)
\(252\) 2.00000 0.125988
\(253\) 2.00000 0.125739
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) −4.00000 −0.249029
\(259\) 16.0000 0.994192
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) −12.0000 −0.741362
\(263\) −4.00000 −0.246651 −0.123325 0.992366i \(-0.539356\pi\)
−0.123325 + 0.992366i \(0.539356\pi\)
\(264\) 2.00000 0.123091
\(265\) 0 0
\(266\) 0 0
\(267\) −8.00000 −0.489592
\(268\) 4.00000 0.244339
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 4.00000 0.242536
\(273\) 12.0000 0.726273
\(274\) −8.00000 −0.483298
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 4.00000 0.239904
\(279\) 0 0
\(280\) 0 0
\(281\) 8.00000 0.477240 0.238620 0.971113i \(-0.423305\pi\)
0.238620 + 0.971113i \(0.423305\pi\)
\(282\) 0 0
\(283\) 16.0000 0.951101 0.475551 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) 16.0000 0.949425
\(285\) 0 0
\(286\) 12.0000 0.709575
\(287\) −12.0000 −0.708338
\(288\) −1.00000 −0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 6.00000 0.351726
\(292\) −6.00000 −0.351123
\(293\) 10.0000 0.584206 0.292103 0.956387i \(-0.405645\pi\)
0.292103 + 0.956387i \(0.405645\pi\)
\(294\) 3.00000 0.174964
\(295\) 0 0
\(296\) −8.00000 −0.464991
\(297\) −2.00000 −0.116052
\(298\) −10.0000 −0.579284
\(299\) −6.00000 −0.346989
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) −12.0000 −0.690522
\(303\) 6.00000 0.344691
\(304\) 0 0
\(305\) 0 0
\(306\) −4.00000 −0.228665
\(307\) −28.0000 −1.59804 −0.799022 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) −4.00000 −0.227921
\(309\) 2.00000 0.113776
\(310\) 0 0
\(311\) −32.0000 −1.81455 −0.907277 0.420534i \(-0.861843\pi\)
−0.907277 + 0.420534i \(0.861843\pi\)
\(312\) −6.00000 −0.339683
\(313\) 34.0000 1.92179 0.960897 0.276907i \(-0.0893093\pi\)
0.960897 + 0.276907i \(0.0893093\pi\)
\(314\) −16.0000 −0.902932
\(315\) 0 0
\(316\) 14.0000 0.787562
\(317\) 22.0000 1.23564 0.617822 0.786318i \(-0.288015\pi\)
0.617822 + 0.786318i \(0.288015\pi\)
\(318\) 6.00000 0.336463
\(319\) −4.00000 −0.223957
\(320\) 0 0
\(321\) 18.0000 1.00466
\(322\) 2.00000 0.111456
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) −4.00000 −0.221201
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −14.0000 −0.768350
\(333\) 8.00000 0.438397
\(334\) 0 0
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) 34.0000 1.85210 0.926049 0.377403i \(-0.123183\pi\)
0.926049 + 0.377403i \(0.123183\pi\)
\(338\) −23.0000 −1.25104
\(339\) −12.0000 −0.651751
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 18.0000 0.967686
\(347\) 8.00000 0.429463 0.214731 0.976673i \(-0.431112\pi\)
0.214731 + 0.976673i \(0.431112\pi\)
\(348\) 2.00000 0.107211
\(349\) −6.00000 −0.321173 −0.160586 0.987022i \(-0.551338\pi\)
−0.160586 + 0.987022i \(0.551338\pi\)
\(350\) 0 0
\(351\) 6.00000 0.320256
\(352\) 2.00000 0.106600
\(353\) −26.0000 −1.38384 −0.691920 0.721974i \(-0.743235\pi\)
−0.691920 + 0.721974i \(0.743235\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −8.00000 −0.423999
\(357\) 8.00000 0.423405
\(358\) 20.0000 1.05703
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 12.0000 0.630706
\(363\) −7.00000 −0.367405
\(364\) 12.0000 0.628971
\(365\) 0 0
\(366\) 8.00000 0.418167
\(367\) −10.0000 −0.521996 −0.260998 0.965339i \(-0.584052\pi\)
−0.260998 + 0.965339i \(0.584052\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) −12.0000 −0.623009
\(372\) 0 0
\(373\) −32.0000 −1.65690 −0.828449 0.560065i \(-0.810776\pi\)
−0.828449 + 0.560065i \(0.810776\pi\)
\(374\) 8.00000 0.413670
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) −2.00000 −0.102869
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 12.0000 0.613171 0.306586 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) 4.00000 0.203331
\(388\) 6.00000 0.304604
\(389\) 10.0000 0.507020 0.253510 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) 3.00000 0.151523
\(393\) 12.0000 0.605320
\(394\) 6.00000 0.302276
\(395\) 0 0
\(396\) −2.00000 −0.100504
\(397\) 22.0000 1.10415 0.552074 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) 2.00000 0.100251
\(399\) 0 0
\(400\) 0 0
\(401\) −24.0000 −1.19850 −0.599251 0.800561i \(-0.704535\pi\)
−0.599251 + 0.800561i \(0.704535\pi\)
\(402\) −4.00000 −0.199502
\(403\) 0 0
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) −4.00000 −0.198517
\(407\) −16.0000 −0.793091
\(408\) −4.00000 −0.198030
\(409\) 38.0000 1.87898 0.939490 0.342578i \(-0.111300\pi\)
0.939490 + 0.342578i \(0.111300\pi\)
\(410\) 0 0
\(411\) 8.00000 0.394611
\(412\) 2.00000 0.0985329
\(413\) 0 0
\(414\) 1.00000 0.0491473
\(415\) 0 0
\(416\) −6.00000 −0.294174
\(417\) −4.00000 −0.195881
\(418\) 0 0
\(419\) −14.0000 −0.683945 −0.341972 0.939710i \(-0.611095\pi\)
−0.341972 + 0.939710i \(0.611095\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(422\) 20.0000 0.973585
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) −16.0000 −0.775203
\(427\) −16.0000 −0.774294
\(428\) 18.0000 0.870063
\(429\) −12.0000 −0.579365
\(430\) 0 0
\(431\) −36.0000 −1.73406 −0.867029 0.498257i \(-0.833974\pi\)
−0.867029 + 0.498257i \(0.833974\pi\)
\(432\) 1.00000 0.0481125
\(433\) −30.0000 −1.44171 −0.720854 0.693087i \(-0.756250\pi\)
−0.720854 + 0.693087i \(0.756250\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −4.00000 −0.191565
\(437\) 0 0
\(438\) 6.00000 0.286691
\(439\) 32.0000 1.52728 0.763638 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) −24.0000 −1.14156
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 8.00000 0.379663
\(445\) 0 0
\(446\) −16.0000 −0.757622
\(447\) 10.0000 0.472984
\(448\) 2.00000 0.0944911
\(449\) −22.0000 −1.03824 −0.519122 0.854700i \(-0.673741\pi\)
−0.519122 + 0.854700i \(0.673741\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) −12.0000 −0.564433
\(453\) 12.0000 0.563809
\(454\) −18.0000 −0.844782
\(455\) 0 0
\(456\) 0 0
\(457\) −18.0000 −0.842004 −0.421002 0.907060i \(-0.638322\pi\)
−0.421002 + 0.907060i \(0.638322\pi\)
\(458\) 20.0000 0.934539
\(459\) 4.00000 0.186704
\(460\) 0 0
\(461\) −10.0000 −0.465746 −0.232873 0.972507i \(-0.574813\pi\)
−0.232873 + 0.972507i \(0.574813\pi\)
\(462\) 4.00000 0.186097
\(463\) 36.0000 1.67306 0.836531 0.547920i \(-0.184580\pi\)
0.836531 + 0.547920i \(0.184580\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) −26.0000 −1.20443
\(467\) 30.0000 1.38823 0.694117 0.719862i \(-0.255795\pi\)
0.694117 + 0.719862i \(0.255795\pi\)
\(468\) 6.00000 0.277350
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) 16.0000 0.737241
\(472\) 0 0
\(473\) −8.00000 −0.367840
\(474\) −14.0000 −0.643041
\(475\) 0 0
\(476\) 8.00000 0.366679
\(477\) −6.00000 −0.274721
\(478\) −16.0000 −0.731823
\(479\) 8.00000 0.365529 0.182765 0.983157i \(-0.441495\pi\)
0.182765 + 0.983157i \(0.441495\pi\)
\(480\) 0 0
\(481\) 48.0000 2.18861
\(482\) 14.0000 0.637683
\(483\) −2.00000 −0.0910032
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −12.0000 −0.543772 −0.271886 0.962329i \(-0.587647\pi\)
−0.271886 + 0.962329i \(0.587647\pi\)
\(488\) 8.00000 0.362143
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) −6.00000 −0.270501
\(493\) 8.00000 0.360302
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 32.0000 1.43540
\(498\) 14.0000 0.627355
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −6.00000 −0.267793
\(503\) −4.00000 −0.178351 −0.0891756 0.996016i \(-0.528423\pi\)
−0.0891756 + 0.996016i \(0.528423\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 0 0
\(506\) −2.00000 −0.0889108
\(507\) 23.0000 1.02147
\(508\) 0 0
\(509\) −14.0000 −0.620539 −0.310270 0.950649i \(-0.600419\pi\)
−0.310270 + 0.950649i \(0.600419\pi\)
\(510\) 0 0
\(511\) −12.0000 −0.530849
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 14.0000 0.617514
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) −16.0000 −0.703000
\(519\) −18.0000 −0.790112
\(520\) 0 0
\(521\) −4.00000 −0.175243 −0.0876216 0.996154i \(-0.527927\pi\)
−0.0876216 + 0.996154i \(0.527927\pi\)
\(522\) −2.00000 −0.0875376
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 4.00000 0.174408
\(527\) 0 0
\(528\) −2.00000 −0.0870388
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −36.0000 −1.55933
\(534\) 8.00000 0.346194
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) −20.0000 −0.863064
\(538\) 18.0000 0.776035
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) 26.0000 1.11783 0.558914 0.829226i \(-0.311218\pi\)
0.558914 + 0.829226i \(0.311218\pi\)
\(542\) −20.0000 −0.859074
\(543\) −12.0000 −0.514969
\(544\) −4.00000 −0.171499
\(545\) 0 0
\(546\) −12.0000 −0.513553
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 8.00000 0.341743
\(549\) −8.00000 −0.341432
\(550\) 0 0
\(551\) 0 0
\(552\) 1.00000 0.0425628
\(553\) 28.0000 1.19068
\(554\) −10.0000 −0.424859
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 22.0000 0.932170 0.466085 0.884740i \(-0.345664\pi\)
0.466085 + 0.884740i \(0.345664\pi\)
\(558\) 0 0
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) −8.00000 −0.337760
\(562\) −8.00000 −0.337460
\(563\) 10.0000 0.421450 0.210725 0.977545i \(-0.432418\pi\)
0.210725 + 0.977545i \(0.432418\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −16.0000 −0.672530
\(567\) 2.00000 0.0839921
\(568\) −16.0000 −0.671345
\(569\) 24.0000 1.00613 0.503066 0.864248i \(-0.332205\pi\)
0.503066 + 0.864248i \(0.332205\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) −12.0000 −0.501745
\(573\) 0 0
\(574\) 12.0000 0.500870
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −2.00000 −0.0832611 −0.0416305 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) 1.00000 0.0415945
\(579\) 2.00000 0.0831172
\(580\) 0 0
\(581\) −28.0000 −1.16164
\(582\) −6.00000 −0.248708
\(583\) 12.0000 0.496989
\(584\) 6.00000 0.248282
\(585\) 0 0
\(586\) −10.0000 −0.413096
\(587\) −8.00000 −0.330195 −0.165098 0.986277i \(-0.552794\pi\)
−0.165098 + 0.986277i \(0.552794\pi\)
\(588\) −3.00000 −0.123718
\(589\) 0 0
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 8.00000 0.328798
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) −2.00000 −0.0818546
\(598\) 6.00000 0.245358
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) −8.00000 −0.326056
\(603\) 4.00000 0.162893
\(604\) 12.0000 0.488273
\(605\) 0 0
\(606\) −6.00000 −0.243733
\(607\) −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
\(610\) 0 0
\(611\) 0 0
\(612\) 4.00000 0.161690
\(613\) 4.00000 0.161558 0.0807792 0.996732i \(-0.474259\pi\)
0.0807792 + 0.996732i \(0.474259\pi\)
\(614\) 28.0000 1.12999
\(615\) 0 0
\(616\) 4.00000 0.161165
\(617\) −28.0000 −1.12724 −0.563619 0.826035i \(-0.690591\pi\)
−0.563619 + 0.826035i \(0.690591\pi\)
\(618\) −2.00000 −0.0804518
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 32.0000 1.28308
\(623\) −16.0000 −0.641026
\(624\) 6.00000 0.240192
\(625\) 0 0
\(626\) −34.0000 −1.35891
\(627\) 0 0
\(628\) 16.0000 0.638470
\(629\) 32.0000 1.27592
\(630\) 0 0
\(631\) −22.0000 −0.875806 −0.437903 0.899022i \(-0.644279\pi\)
−0.437903 + 0.899022i \(0.644279\pi\)
\(632\) −14.0000 −0.556890
\(633\) −20.0000 −0.794929
\(634\) −22.0000 −0.873732
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) −18.0000 −0.713186
\(638\) 4.00000 0.158362
\(639\) 16.0000 0.632950
\(640\) 0 0
\(641\) 16.0000 0.631962 0.315981 0.948766i \(-0.397666\pi\)
0.315981 + 0.948766i \(0.397666\pi\)
\(642\) −18.0000 −0.710403
\(643\) 28.0000 1.10421 0.552106 0.833774i \(-0.313824\pi\)
0.552106 + 0.833774i \(0.313824\pi\)
\(644\) −2.00000 −0.0788110
\(645\) 0 0
\(646\) 0 0
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 4.00000 0.156652
\(653\) −22.0000 −0.860927 −0.430463 0.902608i \(-0.641650\pi\)
−0.430463 + 0.902608i \(0.641650\pi\)
\(654\) 4.00000 0.156412
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) −6.00000 −0.234082
\(658\) 0 0
\(659\) 6.00000 0.233727 0.116863 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) −20.0000 −0.777322
\(663\) 24.0000 0.932083
\(664\) 14.0000 0.543305
\(665\) 0 0
\(666\) −8.00000 −0.309994
\(667\) −2.00000 −0.0774403
\(668\) 0 0
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) 16.0000 0.617673
\(672\) −2.00000 −0.0771517
\(673\) −42.0000 −1.61898 −0.809491 0.587133i \(-0.800257\pi\)
−0.809491 + 0.587133i \(0.800257\pi\)
\(674\) −34.0000 −1.30963
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) −2.00000 −0.0768662 −0.0384331 0.999261i \(-0.512237\pi\)
−0.0384331 + 0.999261i \(0.512237\pi\)
\(678\) 12.0000 0.460857
\(679\) 12.0000 0.460518
\(680\) 0 0
\(681\) 18.0000 0.689761
\(682\) 0 0
\(683\) −48.0000 −1.83667 −0.918334 0.395805i \(-0.870466\pi\)
−0.918334 + 0.395805i \(0.870466\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 20.0000 0.763604
\(687\) −20.0000 −0.763048
\(688\) 4.00000 0.152499
\(689\) −36.0000 −1.37149
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) −18.0000 −0.684257
\(693\) −4.00000 −0.151947
\(694\) −8.00000 −0.303676
\(695\) 0 0
\(696\) −2.00000 −0.0758098
\(697\) −24.0000 −0.909065
\(698\) 6.00000 0.227103
\(699\) 26.0000 0.983410
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) −6.00000 −0.226455
\(703\) 0 0
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) 26.0000 0.978523
\(707\) 12.0000 0.451306
\(708\) 0 0
\(709\) 16.0000 0.600893 0.300446 0.953799i \(-0.402864\pi\)
0.300446 + 0.953799i \(0.402864\pi\)
\(710\) 0 0
\(711\) 14.0000 0.525041
\(712\) 8.00000 0.299813
\(713\) 0 0
\(714\) −8.00000 −0.299392
\(715\) 0 0
\(716\) −20.0000 −0.747435
\(717\) 16.0000 0.597531
\(718\) −24.0000 −0.895672
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) 19.0000 0.707107
\(723\) −14.0000 −0.520666
\(724\) −12.0000 −0.445976
\(725\) 0 0
\(726\) 7.00000 0.259794
\(727\) −38.0000 −1.40934 −0.704671 0.709534i \(-0.748905\pi\)
−0.704671 + 0.709534i \(0.748905\pi\)
\(728\) −12.0000 −0.444750
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 16.0000 0.591781
\(732\) −8.00000 −0.295689
\(733\) −36.0000 −1.32969 −0.664845 0.746981i \(-0.731502\pi\)
−0.664845 + 0.746981i \(0.731502\pi\)
\(734\) 10.0000 0.369107
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) −8.00000 −0.294684
\(738\) 6.00000 0.220863
\(739\) 12.0000 0.441427 0.220714 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 12.0000 0.440534
\(743\) −4.00000 −0.146746 −0.0733729 0.997305i \(-0.523376\pi\)
−0.0733729 + 0.997305i \(0.523376\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 32.0000 1.17160
\(747\) −14.0000 −0.512233
\(748\) −8.00000 −0.292509
\(749\) 36.0000 1.31541
\(750\) 0 0
\(751\) −50.0000 −1.82453 −0.912263 0.409605i \(-0.865667\pi\)
−0.912263 + 0.409605i \(0.865667\pi\)
\(752\) 0 0
\(753\) 6.00000 0.218652
\(754\) −12.0000 −0.437014
\(755\) 0 0
\(756\) 2.00000 0.0727393
\(757\) −40.0000 −1.45382 −0.726912 0.686730i \(-0.759045\pi\)
−0.726912 + 0.686730i \(0.759045\pi\)
\(758\) −16.0000 −0.581146
\(759\) 2.00000 0.0725954
\(760\) 0 0
\(761\) −10.0000 −0.362500 −0.181250 0.983437i \(-0.558014\pi\)
−0.181250 + 0.983437i \(0.558014\pi\)
\(762\) 0 0
\(763\) −8.00000 −0.289619
\(764\) 0 0
\(765\) 0 0
\(766\) −12.0000 −0.433578
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 6.00000 0.216366 0.108183 0.994131i \(-0.465497\pi\)
0.108183 + 0.994131i \(0.465497\pi\)
\(770\) 0 0
\(771\) −14.0000 −0.504198
\(772\) 2.00000 0.0719816
\(773\) 18.0000 0.647415 0.323708 0.946157i \(-0.395071\pi\)
0.323708 + 0.946157i \(0.395071\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) −6.00000 −0.215387
\(777\) 16.0000 0.573997
\(778\) −10.0000 −0.358517
\(779\) 0 0
\(780\) 0 0
\(781\) −32.0000 −1.14505
\(782\) 4.00000 0.143040
\(783\) 2.00000 0.0714742
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) −12.0000 −0.428026
\(787\) 4.00000 0.142585 0.0712923 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) −6.00000 −0.213741
\(789\) −4.00000 −0.142404
\(790\) 0 0
\(791\) −24.0000 −0.853342
\(792\) 2.00000 0.0710669
\(793\) −48.0000 −1.70453
\(794\) −22.0000 −0.780751
\(795\) 0 0
\(796\) −2.00000 −0.0708881
\(797\) 46.0000 1.62940 0.814702 0.579880i \(-0.196901\pi\)
0.814702 + 0.579880i \(0.196901\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −8.00000 −0.282666
\(802\) 24.0000 0.847469
\(803\) 12.0000 0.423471
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) 0 0
\(807\) −18.0000 −0.633630
\(808\) −6.00000 −0.211079
\(809\) −18.0000 −0.632846 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) 0 0
\(811\) −44.0000 −1.54505 −0.772524 0.634985i \(-0.781006\pi\)
−0.772524 + 0.634985i \(0.781006\pi\)
\(812\) 4.00000 0.140372
\(813\) 20.0000 0.701431
\(814\) 16.0000 0.560800
\(815\) 0 0
\(816\) 4.00000 0.140028
\(817\) 0 0
\(818\) −38.0000 −1.32864
\(819\) 12.0000 0.419314
\(820\) 0 0
\(821\) −10.0000 −0.349002 −0.174501 0.984657i \(-0.555831\pi\)
−0.174501 + 0.984657i \(0.555831\pi\)
\(822\) −8.00000 −0.279032
\(823\) 4.00000 0.139431 0.0697156 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) −2.00000 −0.0696733
\(825\) 0 0
\(826\) 0 0
\(827\) 22.0000 0.765015 0.382507 0.923952i \(-0.375061\pi\)
0.382507 + 0.923952i \(0.375061\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −38.0000 −1.31979 −0.659897 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 0 0
\(831\) 10.0000 0.346896
\(832\) 6.00000 0.208013
\(833\) −12.0000 −0.415775
\(834\) 4.00000 0.138509
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 14.0000 0.483622
\(839\) −20.0000 −0.690477 −0.345238 0.938515i \(-0.612202\pi\)
−0.345238 + 0.938515i \(0.612202\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 8.00000 0.275535
\(844\) −20.0000 −0.688428
\(845\) 0 0
\(846\) 0 0
\(847\) −14.0000 −0.481046
\(848\) −6.00000 −0.206041
\(849\) 16.0000 0.549119
\(850\) 0 0
\(851\) −8.00000 −0.274236
\(852\) 16.0000 0.548151
\(853\) −26.0000 −0.890223 −0.445112 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) 16.0000 0.547509
\(855\) 0 0
\(856\) −18.0000 −0.615227
\(857\) −30.0000 −1.02478 −0.512390 0.858753i \(-0.671240\pi\)
−0.512390 + 0.858753i \(0.671240\pi\)
\(858\) 12.0000 0.409673
\(859\) −12.0000 −0.409435 −0.204717 0.978821i \(-0.565628\pi\)
−0.204717 + 0.978821i \(0.565628\pi\)
\(860\) 0 0
\(861\) −12.0000 −0.408959
\(862\) 36.0000 1.22616
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 30.0000 1.01944
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) −28.0000 −0.949835
\(870\) 0 0
\(871\) 24.0000 0.813209
\(872\) 4.00000 0.135457
\(873\) 6.00000 0.203069
\(874\) 0 0
\(875\) 0 0
\(876\) −6.00000 −0.202721
\(877\) 2.00000 0.0675352 0.0337676 0.999430i \(-0.489249\pi\)
0.0337676 + 0.999430i \(0.489249\pi\)
\(878\) −32.0000 −1.07995
\(879\) 10.0000 0.337292
\(880\) 0 0
\(881\) −56.0000 −1.88669 −0.943344 0.331816i \(-0.892339\pi\)
−0.943344 + 0.331816i \(0.892339\pi\)
\(882\) 3.00000 0.101015
\(883\) 36.0000 1.21150 0.605748 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) 24.0000 0.807207
\(885\) 0 0
\(886\) 4.00000 0.134383
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) −8.00000 −0.268462
\(889\) 0 0
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) 16.0000 0.535720
\(893\) 0 0
\(894\) −10.0000 −0.334450
\(895\) 0 0
\(896\) −2.00000 −0.0668153
\(897\) −6.00000 −0.200334
\(898\) 22.0000 0.734150
\(899\) 0 0
\(900\) 0 0
\(901\) −24.0000 −0.799556
\(902\) −12.0000 −0.399556
\(903\) 8.00000 0.266223
\(904\) 12.0000 0.399114
\(905\) 0 0
\(906\) −12.0000 −0.398673
\(907\) 24.0000 0.796907 0.398453 0.917189i \(-0.369547\pi\)
0.398453 + 0.917189i \(0.369547\pi\)
\(908\) 18.0000 0.597351
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) 36.0000 1.19273 0.596367 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) 0 0
\(913\) 28.0000 0.926665
\(914\) 18.0000 0.595387
\(915\) 0 0
\(916\) −20.0000 −0.660819
\(917\) 24.0000 0.792550
\(918\) −4.00000 −0.132020
\(919\) −2.00000 −0.0659739 −0.0329870 0.999456i \(-0.510502\pi\)
−0.0329870 + 0.999456i \(0.510502\pi\)
\(920\) 0 0
\(921\) −28.0000 −0.922631
\(922\) 10.0000 0.329332
\(923\) 96.0000 3.15988
\(924\) −4.00000 −0.131590
\(925\) 0 0
\(926\) −36.0000 −1.18303
\(927\) 2.00000 0.0656886
\(928\) −2.00000 −0.0656532
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 26.0000 0.851658
\(933\) −32.0000 −1.04763
\(934\) −30.0000 −0.981630
\(935\) 0 0
\(936\) −6.00000 −0.196116
\(937\) −42.0000 −1.37208 −0.686040 0.727564i \(-0.740653\pi\)
−0.686040 + 0.727564i \(0.740653\pi\)
\(938\) −8.00000 −0.261209
\(939\) 34.0000 1.10955
\(940\) 0 0
\(941\) −22.0000 −0.717180 −0.358590 0.933495i \(-0.616742\pi\)
−0.358590 + 0.933495i \(0.616742\pi\)
\(942\) −16.0000 −0.521308
\(943\) 6.00000 0.195387
\(944\) 0 0
\(945\) 0 0
\(946\) 8.00000 0.260102
\(947\) −8.00000 −0.259965 −0.129983 0.991516i \(-0.541492\pi\)
−0.129983 + 0.991516i \(0.541492\pi\)
\(948\) 14.0000 0.454699
\(949\) −36.0000 −1.16861
\(950\) 0 0
\(951\) 22.0000 0.713399
\(952\) −8.00000 −0.259281
\(953\) −44.0000 −1.42530 −0.712650 0.701520i \(-0.752505\pi\)
−0.712650 + 0.701520i \(0.752505\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) 16.0000 0.517477
\(957\) −4.00000 −0.129302
\(958\) −8.00000 −0.258468
\(959\) 16.0000 0.516667
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) −48.0000 −1.54758
\(963\) 18.0000 0.580042
\(964\) −14.0000 −0.450910
\(965\) 0 0
\(966\) 2.00000 0.0643489
\(967\) −52.0000 −1.67221 −0.836104 0.548572i \(-0.815172\pi\)
−0.836104 + 0.548572i \(0.815172\pi\)
\(968\) 7.00000 0.224989
\(969\) 0 0
\(970\) 0 0
\(971\) −46.0000 −1.47621 −0.738105 0.674686i \(-0.764279\pi\)
−0.738105 + 0.674686i \(0.764279\pi\)
\(972\) 1.00000 0.0320750
\(973\) −8.00000 −0.256468
\(974\) 12.0000 0.384505
\(975\) 0 0
\(976\) −8.00000 −0.256074
\(977\) −24.0000 −0.767828 −0.383914 0.923369i \(-0.625424\pi\)
−0.383914 + 0.923369i \(0.625424\pi\)
\(978\) −4.00000 −0.127906
\(979\) 16.0000 0.511362
\(980\) 0 0
\(981\) −4.00000 −0.127710
\(982\) 12.0000 0.382935
\(983\) 12.0000 0.382741 0.191370 0.981518i \(-0.438707\pi\)
0.191370 + 0.981518i \(0.438707\pi\)
\(984\) 6.00000 0.191273
\(985\) 0 0
\(986\) −8.00000 −0.254772
\(987\) 0 0
\(988\) 0 0
\(989\) −4.00000 −0.127193
\(990\) 0 0
\(991\) −32.0000 −1.01651 −0.508257 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) 0 0
\(993\) 20.0000 0.634681
\(994\) −32.0000 −1.01498
\(995\) 0 0
\(996\) −14.0000 −0.443607
\(997\) 2.00000 0.0633406 0.0316703 0.999498i \(-0.489917\pi\)
0.0316703 + 0.999498i \(0.489917\pi\)
\(998\) 28.0000 0.886325
\(999\) 8.00000 0.253109
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 3450.2.a.l.1.1 1
5.2 odd 4 3450.2.d.d.2899.1 2
5.3 odd 4 3450.2.d.d.2899.2 2
5.4 even 2 690.2.a.g.1.1 1
15.14 odd 2 2070.2.a.e.1.1 1
20.19 odd 2 5520.2.a.z.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.g.1.1 1 5.4 even 2
2070.2.a.e.1.1 1 15.14 odd 2
3450.2.a.l.1.1 1 1.1 even 1 trivial
3450.2.d.d.2899.1 2 5.2 odd 4
3450.2.d.d.2899.2 2 5.3 odd 4
5520.2.a.z.1.1 1 20.19 odd 2