Properties

Label 3450.2.a.d.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

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} +1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} -4.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} -2.00000 q^{26} -1.00000 q^{27} -2.00000 q^{29} -1.00000 q^{32} -4.00000 q^{33} -6.00000 q^{34} +1.00000 q^{36} +2.00000 q^{37} -4.00000 q^{38} -2.00000 q^{39} +10.0000 q^{41} +4.00000 q^{43} +4.00000 q^{44} -1.00000 q^{46} -1.00000 q^{48} -7.00000 q^{49} -6.00000 q^{51} +2.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} -4.00000 q^{57} +2.00000 q^{58} -4.00000 q^{59} -10.0000 q^{61} +1.00000 q^{64} +4.00000 q^{66} +12.0000 q^{67} +6.00000 q^{68} -1.00000 q^{69} -8.00000 q^{71} -1.00000 q^{72} -10.0000 q^{73} -2.00000 q^{74} +4.00000 q^{76} +2.00000 q^{78} -8.00000 q^{79} +1.00000 q^{81} -10.0000 q^{82} +4.00000 q^{83} -4.00000 q^{86} +2.00000 q^{87} -4.00000 q^{88} +18.0000 q^{89} +1.00000 q^{92} +1.00000 q^{96} -2.00000 q^{97} +7.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) −1.00000 −0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 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
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) 1.00000 0.208514
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.00000 −0.696311
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −4.00000 −0.648886
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 4.00000 0.603023
\(45\) 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
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 2.00000 0.277350
\(53\) −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
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 2.00000 0.262613
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 6.00000 0.727607
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) −1.00000 −0.117851
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 2.00000 0.226455
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −10.0000 −1.10432
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 2.00000 0.214423
\(88\) −4.00000 −0.426401
\(89\) 18.0000 1.90800 0.953998 0.299813i \(-0.0969242\pi\)
0.953998 + 0.299813i \(0.0969242\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 7.00000 0.707107
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 6.00000 0.594089
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) −2.00000 −0.185695
\(117\) 2.00000 0.184900
\(118\) 4.00000 0.368230
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 10.0000 0.905357
\(123\) −10.0000 −0.901670
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 20.0000 1.74741 0.873704 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) −4.00000 −0.348155
\(133\) 0 0
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 1.00000 0.0851257
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 8.00000 0.671345
\(143\) 8.00000 0.668994
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 10.0000 0.827606
\(147\) 7.00000 0.577350
\(148\) 2.00000 0.164399
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) −4.00000 −0.324443
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) −2.00000 −0.160128
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 8.00000 0.636446
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −20.0000 −1.56652 −0.783260 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) 10.0000 0.780869
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 4.00000 0.304997
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) −2.00000 −0.151620
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) 4.00000 0.300658
\(178\) −18.0000 −1.34916
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 10.0000 0.739221
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) 0 0
\(187\) 24.0000 1.75505
\(188\) 0 0
\(189\) 0 0
\(190\) 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 \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) −4.00000 −0.284268
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) −12.0000 −0.846415
\(202\) −6.00000 −0.422159
\(203\) 0 0
\(204\) −6.00000 −0.420084
\(205\) 0 0
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 2.00000 0.138675
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) −6.00000 −0.412082
\(213\) 8.00000 0.548151
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 10.0000 0.677285
\(219\) 10.0000 0.675737
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 2.00000 0.134231
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 20.0000 1.32745 0.663723 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) −4.00000 −0.264906
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 2.00000 0.131306
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) −4.00000 −0.260378
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 0 0
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) −5.00000 −0.321412
\(243\) −1.00000 −0.0641500
\(244\) −10.0000 −0.640184
\(245\) 0 0
\(246\) 10.0000 0.637577
\(247\) 8.00000 0.509028
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) −16.0000 −1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) 4.00000 0.249029
\(259\) 0 0
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) −20.0000 −1.23560
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 4.00000 0.246183
\(265\) 0 0
\(266\) 0 0
\(267\) −18.0000 −1.10158
\(268\) 12.0000 0.733017
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) 2.00000 0.120824
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 20.0000 1.19952
\(279\) 0 0
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) −8.00000 −0.473050
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 2.00000 0.117242
\(292\) −10.0000 −0.585206
\(293\) −22.0000 −1.28525 −0.642627 0.766179i \(-0.722155\pi\)
−0.642627 + 0.766179i \(0.722155\pi\)
\(294\) −7.00000 −0.408248
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) −4.00000 −0.232104
\(298\) 10.0000 0.579284
\(299\) 2.00000 0.115663
\(300\) 0 0
\(301\) 0 0
\(302\) −8.00000 −0.460348
\(303\) −6.00000 −0.344691
\(304\) 4.00000 0.229416
\(305\) 0 0
\(306\) −6.00000 −0.342997
\(307\) −4.00000 −0.228292 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 2.00000 0.113228
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) −10.0000 −0.564333
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) −30.0000 −1.68497 −0.842484 0.538721i \(-0.818908\pi\)
−0.842484 + 0.538721i \(0.818908\pi\)
\(318\) −6.00000 −0.336463
\(319\) −8.00000 −0.447914
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 24.0000 1.33540
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 20.0000 1.10770
\(327\) 10.0000 0.553001
\(328\) −10.0000 −0.552158
\(329\) 0 0
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) 4.00000 0.219529
\(333\) 2.00000 0.109599
\(334\) 8.00000 0.437741
\(335\) 0 0
\(336\) 0 0
\(337\) −2.00000 −0.108947 −0.0544735 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) 9.00000 0.489535
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 0 0
\(342\) −4.00000 −0.216295
\(343\) 0 0
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 2.00000 0.107211
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) −4.00000 −0.213201
\(353\) −34.0000 −1.80964 −0.904819 0.425797i \(-0.859994\pi\)
−0.904819 + 0.425797i \(0.859994\pi\)
\(354\) −4.00000 −0.212598
\(355\) 0 0
\(356\) 18.0000 0.953998
\(357\) 0 0
\(358\) −4.00000 −0.211407
\(359\) 8.00000 0.422224 0.211112 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 2.00000 0.105118
\(363\) −5.00000 −0.262432
\(364\) 0 0
\(365\) 0 0
\(366\) −10.0000 −0.522708
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 1.00000 0.0521286
\(369\) 10.0000 0.520579
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 18.0000 0.932005 0.466002 0.884783i \(-0.345694\pi\)
0.466002 + 0.884783i \(0.345694\pi\)
\(374\) −24.0000 −1.24101
\(375\) 0 0
\(376\) 0 0
\(377\) −4.00000 −0.206010
\(378\) 0 0
\(379\) 12.0000 0.616399 0.308199 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) 0 0
\(383\) 32.0000 1.63512 0.817562 0.575841i \(-0.195325\pi\)
0.817562 + 0.575841i \(0.195325\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 2.00000 0.101797
\(387\) 4.00000 0.203331
\(388\) −2.00000 −0.101535
\(389\) 38.0000 1.92668 0.963338 0.268290i \(-0.0864585\pi\)
0.963338 + 0.268290i \(0.0864585\pi\)
\(390\) 0 0
\(391\) 6.00000 0.303433
\(392\) 7.00000 0.353553
\(393\) −20.0000 −1.00887
\(394\) 6.00000 0.302276
\(395\) 0 0
\(396\) 4.00000 0.201008
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 16.0000 0.802008
\(399\) 0 0
\(400\) 0 0
\(401\) 10.0000 0.499376 0.249688 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) 12.0000 0.598506
\(403\) 0 0
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) 6.00000 0.297044
\(409\) 26.0000 1.28562 0.642809 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) 0 0
\(411\) 2.00000 0.0986527
\(412\) 0 0
\(413\) 0 0
\(414\) −1.00000 −0.0491473
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) 20.0000 0.979404
\(418\) −16.0000 −0.782586
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) −4.00000 −0.194717
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) −8.00000 −0.387601
\(427\) 0 0
\(428\) 12.0000 0.580042
\(429\) −8.00000 −0.386244
\(430\) 0 0
\(431\) −32.0000 −1.54139 −0.770693 0.637207i \(-0.780090\pi\)
−0.770693 + 0.637207i \(0.780090\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −34.0000 −1.63394 −0.816968 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) 4.00000 0.191346
\(438\) −10.0000 −0.477818
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) −12.0000 −0.570782
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 0 0
\(446\) 16.0000 0.757622
\(447\) 10.0000 0.472984
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) 40.0000 1.88353
\(452\) 6.00000 0.282216
\(453\) −8.00000 −0.375873
\(454\) −20.0000 −0.938647
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) −14.0000 −0.654177
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 2.00000 0.0924500
\(469\) 0 0
\(470\) 0 0
\(471\) −10.0000 −0.460776
\(472\) 4.00000 0.184115
\(473\) 16.0000 0.735681
\(474\) −8.00000 −0.367452
\(475\) 0 0
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) −16.0000 −0.731823
\(479\) 32.0000 1.46212 0.731059 0.682315i \(-0.239027\pi\)
0.731059 + 0.682315i \(0.239027\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) −18.0000 −0.819878
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −40.0000 −1.81257 −0.906287 0.422664i \(-0.861095\pi\)
−0.906287 + 0.422664i \(0.861095\pi\)
\(488\) 10.0000 0.452679
\(489\) 20.0000 0.904431
\(490\) 0 0
\(491\) −4.00000 −0.180517 −0.0902587 0.995918i \(-0.528769\pi\)
−0.0902587 + 0.995918i \(0.528769\pi\)
\(492\) −10.0000 −0.450835
\(493\) −12.0000 −0.540453
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 4.00000 0.179244
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 0 0
\(501\) 8.00000 0.357414
\(502\) −20.0000 −0.892644
\(503\) 40.0000 1.78351 0.891756 0.452517i \(-0.149474\pi\)
0.891756 + 0.452517i \(0.149474\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −4.00000 −0.177822
\(507\) 9.00000 0.399704
\(508\) 16.0000 0.709885
\(509\) −34.0000 −1.50702 −0.753512 0.657434i \(-0.771642\pi\)
−0.753512 + 0.657434i \(0.771642\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −4.00000 −0.176604
\(514\) −14.0000 −0.617514
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) 0 0
\(519\) 14.0000 0.614532
\(520\) 0 0
\(521\) −14.0000 −0.613351 −0.306676 0.951814i \(-0.599217\pi\)
−0.306676 + 0.951814i \(0.599217\pi\)
\(522\) 2.00000 0.0875376
\(523\) 36.0000 1.57417 0.787085 0.616844i \(-0.211589\pi\)
0.787085 + 0.616844i \(0.211589\pi\)
\(524\) 20.0000 0.873704
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) 0 0
\(528\) −4.00000 −0.174078
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) 20.0000 0.866296
\(534\) 18.0000 0.778936
\(535\) 0 0
\(536\) −12.0000 −0.518321
\(537\) −4.00000 −0.172613
\(538\) −14.0000 −0.603583
\(539\) −28.0000 −1.20605
\(540\) 0 0
\(541\) 14.0000 0.601907 0.300954 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) 0 0
\(543\) 2.00000 0.0858282
\(544\) −6.00000 −0.257248
\(545\) 0 0
\(546\) 0 0
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) −2.00000 −0.0854358
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) −8.00000 −0.340811
\(552\) 1.00000 0.0425628
\(553\) 0 0
\(554\) −10.0000 −0.424859
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) −14.0000 −0.593199 −0.296600 0.955002i \(-0.595853\pi\)
−0.296600 + 0.955002i \(0.595853\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) −24.0000 −1.01328
\(562\) −18.0000 −0.759284
\(563\) 36.0000 1.51722 0.758610 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) 8.00000 0.335673
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 8.00000 0.334497
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −2.00000 −0.0832611 −0.0416305 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) −19.0000 −0.790296
\(579\) 2.00000 0.0831172
\(580\) 0 0
\(581\) 0 0
\(582\) −2.00000 −0.0829027
\(583\) −24.0000 −0.993978
\(584\) 10.0000 0.413803
\(585\) 0 0
\(586\) 22.0000 0.908812
\(587\) −28.0000 −1.15568 −0.577842 0.816149i \(-0.696105\pi\)
−0.577842 + 0.816149i \(0.696105\pi\)
\(588\) 7.00000 0.288675
\(589\) 0 0
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) 2.00000 0.0821995
\(593\) 30.0000 1.23195 0.615976 0.787765i \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) 16.0000 0.654836
\(598\) −2.00000 −0.0817861
\(599\) −40.0000 −1.63436 −0.817178 0.576386i \(-0.804463\pi\)
−0.817178 + 0.576386i \(0.804463\pi\)
\(600\) 0 0
\(601\) −38.0000 −1.55005 −0.775026 0.631929i \(-0.782263\pi\)
−0.775026 + 0.631929i \(0.782263\pi\)
\(602\) 0 0
\(603\) 12.0000 0.488678
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) 6.00000 0.243733
\(607\) 16.0000 0.649420 0.324710 0.945814i \(-0.394733\pi\)
0.324710 + 0.945814i \(0.394733\pi\)
\(608\) −4.00000 −0.162221
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 6.00000 0.242536
\(613\) 18.0000 0.727013 0.363507 0.931592i \(-0.381579\pi\)
0.363507 + 0.931592i \(0.381579\pi\)
\(614\) 4.00000 0.161427
\(615\) 0 0
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 0 0
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) −8.00000 −0.320771
\(623\) 0 0
\(624\) −2.00000 −0.0800641
\(625\) 0 0
\(626\) 10.0000 0.399680
\(627\) −16.0000 −0.638978
\(628\) 10.0000 0.399043
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) 8.00000 0.318223
\(633\) −4.00000 −0.158986
\(634\) 30.0000 1.19145
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) −14.0000 −0.554700
\(638\) 8.00000 0.316723
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) 26.0000 1.02694 0.513469 0.858108i \(-0.328360\pi\)
0.513469 + 0.858108i \(0.328360\pi\)
\(642\) 12.0000 0.473602
\(643\) −20.0000 −0.788723 −0.394362 0.918955i \(-0.629034\pi\)
−0.394362 + 0.918955i \(0.629034\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −24.0000 −0.944267
\(647\) −8.00000 −0.314512 −0.157256 0.987558i \(-0.550265\pi\)
−0.157256 + 0.987558i \(0.550265\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −16.0000 −0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) −20.0000 −0.783260
\(653\) −14.0000 −0.547862 −0.273931 0.961749i \(-0.588324\pi\)
−0.273931 + 0.961749i \(0.588324\pi\)
\(654\) −10.0000 −0.391031
\(655\) 0 0
\(656\) 10.0000 0.390434
\(657\) −10.0000 −0.390137
\(658\) 0 0
\(659\) −20.0000 −0.779089 −0.389545 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) 0 0
\(661\) −50.0000 −1.94477 −0.972387 0.233373i \(-0.925024\pi\)
−0.972387 + 0.233373i \(0.925024\pi\)
\(662\) −28.0000 −1.08825
\(663\) −12.0000 −0.466041
\(664\) −4.00000 −0.155230
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) −2.00000 −0.0774403
\(668\) −8.00000 −0.309529
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) −40.0000 −1.54418
\(672\) 0 0
\(673\) −2.00000 −0.0770943 −0.0385472 0.999257i \(-0.512273\pi\)
−0.0385472 + 0.999257i \(0.512273\pi\)
\(674\) 2.00000 0.0770371
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −38.0000 −1.46046 −0.730229 0.683202i \(-0.760587\pi\)
−0.730229 + 0.683202i \(0.760587\pi\)
\(678\) 6.00000 0.230429
\(679\) 0 0
\(680\) 0 0
\(681\) −20.0000 −0.766402
\(682\) 0 0
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 4.00000 0.152944
\(685\) 0 0
\(686\) 0 0
\(687\) −14.0000 −0.534133
\(688\) 4.00000 0.152499
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) −14.0000 −0.532200
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) −2.00000 −0.0758098
\(697\) 60.0000 2.27266
\(698\) −14.0000 −0.529908
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) −50.0000 −1.88847 −0.944237 0.329267i \(-0.893198\pi\)
−0.944237 + 0.329267i \(0.893198\pi\)
\(702\) 2.00000 0.0754851
\(703\) 8.00000 0.301726
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) 34.0000 1.27961
\(707\) 0 0
\(708\) 4.00000 0.150329
\(709\) −34.0000 −1.27690 −0.638448 0.769665i \(-0.720423\pi\)
−0.638448 + 0.769665i \(0.720423\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) −18.0000 −0.674579
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 4.00000 0.149487
\(717\) −16.0000 −0.597531
\(718\) −8.00000 −0.298557
\(719\) −32.0000 −1.19340 −0.596699 0.802465i \(-0.703521\pi\)
−0.596699 + 0.802465i \(0.703521\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 3.00000 0.111648
\(723\) −18.0000 −0.669427
\(724\) −2.00000 −0.0743294
\(725\) 0 0
\(726\) 5.00000 0.185567
\(727\) −16.0000 −0.593407 −0.296704 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) 10.0000 0.369611
\(733\) −6.00000 −0.221615 −0.110808 0.993842i \(-0.535344\pi\)
−0.110808 + 0.993842i \(0.535344\pi\)
\(734\) −8.00000 −0.295285
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 48.0000 1.76810
\(738\) −10.0000 −0.368105
\(739\) −12.0000 −0.441427 −0.220714 0.975339i \(-0.570839\pi\)
−0.220714 + 0.975339i \(0.570839\pi\)
\(740\) 0 0
\(741\) −8.00000 −0.293887
\(742\) 0 0
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −18.0000 −0.659027
\(747\) 4.00000 0.146352
\(748\) 24.0000 0.877527
\(749\) 0 0
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 0 0
\(753\) −20.0000 −0.728841
\(754\) 4.00000 0.145671
\(755\) 0 0
\(756\) 0 0
\(757\) 18.0000 0.654221 0.327111 0.944986i \(-0.393925\pi\)
0.327111 + 0.944986i \(0.393925\pi\)
\(758\) −12.0000 −0.435860
\(759\) −4.00000 −0.145191
\(760\) 0 0
\(761\) −22.0000 −0.797499 −0.398750 0.917060i \(-0.630556\pi\)
−0.398750 + 0.917060i \(0.630556\pi\)
\(762\) 16.0000 0.579619
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) −32.0000 −1.15621
\(767\) −8.00000 −0.288863
\(768\) −1.00000 −0.0360844
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) −14.0000 −0.504198
\(772\) −2.00000 −0.0719816
\(773\) −54.0000 −1.94225 −0.971123 0.238581i \(-0.923318\pi\)
−0.971123 + 0.238581i \(0.923318\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) −38.0000 −1.36237
\(779\) 40.0000 1.43315
\(780\) 0 0
\(781\) −32.0000 −1.14505
\(782\) −6.00000 −0.214560
\(783\) 2.00000 0.0714742
\(784\) −7.00000 −0.250000
\(785\) 0 0
\(786\) 20.0000 0.713376
\(787\) −4.00000 −0.142585 −0.0712923 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) −6.00000 −0.213741
\(789\) −24.0000 −0.854423
\(790\) 0 0
\(791\) 0 0
\(792\) −4.00000 −0.142134
\(793\) −20.0000 −0.710221
\(794\) −2.00000 −0.0709773
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) 34.0000 1.20434 0.602171 0.798367i \(-0.294303\pi\)
0.602171 + 0.798367i \(0.294303\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 18.0000 0.635999
\(802\) −10.0000 −0.353112
\(803\) −40.0000 −1.41157
\(804\) −12.0000 −0.423207
\(805\) 0 0
\(806\) 0 0
\(807\) −14.0000 −0.492823
\(808\) −6.00000 −0.211079
\(809\) 26.0000 0.914111 0.457056 0.889438i \(-0.348904\pi\)
0.457056 + 0.889438i \(0.348904\pi\)
\(810\) 0 0
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −8.00000 −0.280400
\(815\) 0 0
\(816\) −6.00000 −0.210042
\(817\) 16.0000 0.559769
\(818\) −26.0000 −0.909069
\(819\) 0 0
\(820\) 0 0
\(821\) −42.0000 −1.46581 −0.732905 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) −2.00000 −0.0697580
\(823\) 40.0000 1.39431 0.697156 0.716919i \(-0.254448\pi\)
0.697156 + 0.716919i \(0.254448\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 1.00000 0.0347524
\(829\) 46.0000 1.59765 0.798823 0.601566i \(-0.205456\pi\)
0.798823 + 0.601566i \(0.205456\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) 2.00000 0.0693375
\(833\) −42.0000 −1.45521
\(834\) −20.0000 −0.692543
\(835\) 0 0
\(836\) 16.0000 0.553372
\(837\) 0 0
\(838\) −12.0000 −0.414533
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 2.00000 0.0689246
\(843\) −18.0000 −0.619953
\(844\) 4.00000 0.137686
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) −6.00000 −0.206041
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) 2.00000 0.0685591
\(852\) 8.00000 0.274075
\(853\) −6.00000 −0.205436 −0.102718 0.994711i \(-0.532754\pi\)
−0.102718 + 0.994711i \(0.532754\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 8.00000 0.273115
\(859\) 44.0000 1.50126 0.750630 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 32.0000 1.08992
\(863\) 32.0000 1.08929 0.544646 0.838666i \(-0.316664\pi\)
0.544646 + 0.838666i \(0.316664\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 34.0000 1.15537
\(867\) −19.0000 −0.645274
\(868\) 0 0
\(869\) −32.0000 −1.08553
\(870\) 0 0
\(871\) 24.0000 0.813209
\(872\) 10.0000 0.338643
\(873\) −2.00000 −0.0676897
\(874\) −4.00000 −0.135302
\(875\) 0 0
\(876\) 10.0000 0.337869
\(877\) 50.0000 1.68838 0.844190 0.536044i \(-0.180082\pi\)
0.844190 + 0.536044i \(0.180082\pi\)
\(878\) 8.00000 0.269987
\(879\) 22.0000 0.742042
\(880\) 0 0
\(881\) 10.0000 0.336909 0.168454 0.985709i \(-0.446122\pi\)
0.168454 + 0.985709i \(0.446122\pi\)
\(882\) 7.00000 0.235702
\(883\) −20.0000 −0.673054 −0.336527 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) −4.00000 −0.134383
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) 2.00000 0.0671156
\(889\) 0 0
\(890\) 0 0
\(891\) 4.00000 0.134005
\(892\) −16.0000 −0.535720
\(893\) 0 0
\(894\) −10.0000 −0.334450
\(895\) 0 0
\(896\) 0 0
\(897\) −2.00000 −0.0667781
\(898\) 30.0000 1.00111
\(899\) 0 0
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) −40.0000 −1.33185
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) 8.00000 0.265782
\(907\) −44.0000 −1.46100 −0.730498 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(908\) 20.0000 0.663723
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) 32.0000 1.06021 0.530104 0.847933i \(-0.322153\pi\)
0.530104 + 0.847933i \(0.322153\pi\)
\(912\) −4.00000 −0.132453
\(913\) 16.0000 0.529523
\(914\) 10.0000 0.330771
\(915\) 0 0
\(916\) 14.0000 0.462573
\(917\) 0 0
\(918\) 6.00000 0.198030
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 0 0
\(921\) 4.00000 0.131804
\(922\) 18.0000 0.592798
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) 0 0
\(926\) 16.0000 0.525793
\(927\) 0 0
\(928\) 2.00000 0.0656532
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) −28.0000 −0.917663
\(932\) 6.00000 0.196537
\(933\) −8.00000 −0.261908
\(934\) 12.0000 0.392652
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) 22.0000 0.718709 0.359354 0.933201i \(-0.382997\pi\)
0.359354 + 0.933201i \(0.382997\pi\)
\(938\) 0 0
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) 14.0000 0.456387 0.228193 0.973616i \(-0.426718\pi\)
0.228193 + 0.973616i \(0.426718\pi\)
\(942\) 10.0000 0.325818
\(943\) 10.0000 0.325645
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) −16.0000 −0.520205
\(947\) −4.00000 −0.129983 −0.0649913 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) 8.00000 0.259828
\(949\) −20.0000 −0.649227
\(950\) 0 0
\(951\) 30.0000 0.972817
\(952\) 0 0
\(953\) −2.00000 −0.0647864 −0.0323932 0.999475i \(-0.510313\pi\)
−0.0323932 + 0.999475i \(0.510313\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) 16.0000 0.517477
\(957\) 8.00000 0.258603
\(958\) −32.0000 −1.03387
\(959\) 0 0
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) −4.00000 −0.128965
\(963\) 12.0000 0.386695
\(964\) 18.0000 0.579741
\(965\) 0 0
\(966\) 0 0
\(967\) 24.0000 0.771788 0.385894 0.922543i \(-0.373893\pi\)
0.385894 + 0.922543i \(0.373893\pi\)
\(968\) −5.00000 −0.160706
\(969\) −24.0000 −0.770991
\(970\) 0 0
\(971\) 52.0000 1.66876 0.834380 0.551190i \(-0.185826\pi\)
0.834380 + 0.551190i \(0.185826\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 40.0000 1.28168
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) −42.0000 −1.34370 −0.671850 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) −20.0000 −0.639529
\(979\) 72.0000 2.30113
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) 4.00000 0.127645
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 10.0000 0.318788
\(985\) 0 0
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) 8.00000 0.254514
\(989\) 4.00000 0.127193
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) −28.0000 −0.888553
\(994\) 0 0
\(995\) 0 0
\(996\) −4.00000 −0.126745
\(997\) −38.0000 −1.20347 −0.601736 0.798695i \(-0.705524\pi\)
−0.601736 + 0.798695i \(0.705524\pi\)
\(998\) −20.0000 −0.633089
\(999\) −2.00000 −0.0632772
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.d.1.1 1
5.2 odd 4 3450.2.d.t.2899.1 2
5.3 odd 4 3450.2.d.t.2899.2 2
5.4 even 2 690.2.a.k.1.1 1
15.14 odd 2 2070.2.a.b.1.1 1
20.19 odd 2 5520.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.k.1.1 1 5.4 even 2
2070.2.a.b.1.1 1 15.14 odd 2
3450.2.a.d.1.1 1 1.1 even 1 trivial
3450.2.d.t.2899.1 2 5.2 odd 4
3450.2.d.t.2899.2 2 5.3 odd 4
5520.2.a.i.1.1 1 20.19 odd 2