Properties

Label 6930.2.a.d.1.1
Level $6930$
Weight $2$
Character 6930.1
Self dual yes
Analytic conductor $55.336$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6930 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6930.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.3363286007\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2310)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 6930.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} +1.00000 q^{11} -2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} -4.00000 q^{19} -1.00000 q^{20} -1.00000 q^{22} -8.00000 q^{23} +1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{28} -6.00000 q^{29} +8.00000 q^{31} -1.00000 q^{32} +2.00000 q^{34} -1.00000 q^{35} -2.00000 q^{37} +4.00000 q^{38} +1.00000 q^{40} -2.00000 q^{41} +4.00000 q^{43} +1.00000 q^{44} +8.00000 q^{46} -8.00000 q^{47} +1.00000 q^{49} -1.00000 q^{50} -2.00000 q^{52} +10.0000 q^{53} -1.00000 q^{55} -1.00000 q^{56} +6.00000 q^{58} +12.0000 q^{59} -2.00000 q^{61} -8.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} -4.00000 q^{67} -2.00000 q^{68} +1.00000 q^{70} -8.00000 q^{71} +2.00000 q^{73} +2.00000 q^{74} -4.00000 q^{76} +1.00000 q^{77} +16.0000 q^{79} -1.00000 q^{80} +2.00000 q^{82} +4.00000 q^{83} +2.00000 q^{85} -4.00000 q^{86} -1.00000 q^{88} +14.0000 q^{89} -2.00000 q^{91} -8.00000 q^{92} +8.00000 q^{94} +4.00000 q^{95} +10.0000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) 10.0000 1.37361 0.686803 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 6.00000 0.787839
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −8.00000 −1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0 0
\(70\) 1.00000 0.119523
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 16.0000 1.80014 0.900070 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 2.00000 0.220863
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) −8.00000 −0.834058
\(93\) 0 0
\(94\) 8.00000 0.825137
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) −10.0000 −0.971286
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 1.00000 0.0953463
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 0 0
\(115\) 8.00000 0.746004
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) −12.0000 −1.10469
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 2.00000 0.181071
\(123\) 0 0
\(124\) 8.00000 0.718421
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −2.00000 −0.175412
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) −4.00000 −0.346844
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) −10.0000 −0.854358 −0.427179 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 0 0
\(142\) 8.00000 0.671345
\(143\) −2.00000 −0.167248
\(144\) 0 0
\(145\) 6.00000 0.498273
\(146\) −2.00000 −0.165521
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) −14.0000 −1.14692 −0.573462 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) 0 0
\(151\) 24.0000 1.95309 0.976546 0.215308i \(-0.0690756\pi\)
0.976546 + 0.215308i \(0.0690756\pi\)
\(152\) 4.00000 0.324443
\(153\) 0 0
\(154\) −1.00000 −0.0805823
\(155\) −8.00000 −0.642575
\(156\) 0 0
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) −16.0000 −1.27289
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) −8.00000 −0.630488
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −2.00000 −0.153393
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) −14.0000 −1.04934
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) 22.0000 1.63525 0.817624 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) 2.00000 0.148250
\(183\) 0 0
\(184\) 8.00000 0.589768
\(185\) 2.00000 0.147043
\(186\) 0 0
\(187\) −2.00000 −0.146254
\(188\) −8.00000 −0.583460
\(189\) 0 0
\(190\) −4.00000 −0.290191
\(191\) 16.0000 1.15772 0.578860 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) 0 0
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 26.0000 1.85242 0.926212 0.377004i \(-0.123046\pi\)
0.926212 + 0.377004i \(0.123046\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) 6.00000 0.422159
\(203\) −6.00000 −0.421117
\(204\) 0 0
\(205\) 2.00000 0.139686
\(206\) −8.00000 −0.557386
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 10.0000 0.686803
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) 8.00000 0.543075
\(218\) −14.0000 −0.948200
\(219\) 0 0
\(220\) −1.00000 −0.0674200
\(221\) 4.00000 0.269069
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −14.0000 −0.931266
\(227\) 4.00000 0.265489 0.132745 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) −8.00000 −0.527504
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) 12.0000 0.781133
\(237\) 0 0
\(238\) 2.00000 0.129641
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) −2.00000 −0.128037
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 8.00000 0.509028
\(248\) −8.00000 −0.508001
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) −8.00000 −0.502956
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) 0 0
\(259\) −2.00000 −0.124274
\(260\) 2.00000 0.124035
\(261\) 0 0
\(262\) 12.0000 0.741362
\(263\) 8.00000 0.493301 0.246651 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(264\) 0 0
\(265\) −10.0000 −0.614295
\(266\) 4.00000 0.245256
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) 2.00000 0.121942 0.0609711 0.998140i \(-0.480580\pi\)
0.0609711 + 0.998140i \(0.480580\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) 10.0000 0.604122
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 12.0000 0.719712
\(279\) 0 0
\(280\) 1.00000 0.0597614
\(281\) 22.0000 1.31241 0.656205 0.754583i \(-0.272161\pi\)
0.656205 + 0.754583i \(0.272161\pi\)
\(282\) 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
\(286\) 2.00000 0.118262
\(287\) −2.00000 −0.118056
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) −6.00000 −0.352332
\(291\) 0 0
\(292\) 2.00000 0.117041
\(293\) 26.0000 1.51894 0.759468 0.650545i \(-0.225459\pi\)
0.759468 + 0.650545i \(0.225459\pi\)
\(294\) 0 0
\(295\) −12.0000 −0.698667
\(296\) 2.00000 0.116248
\(297\) 0 0
\(298\) 14.0000 0.810998
\(299\) 16.0000 0.925304
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) −24.0000 −1.38104
\(303\) 0 0
\(304\) −4.00000 −0.229416
\(305\) 2.00000 0.114520
\(306\) 0 0
\(307\) −20.0000 −1.14146 −0.570730 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 1.00000 0.0569803
\(309\) 0 0
\(310\) 8.00000 0.454369
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) −30.0000 −1.69570 −0.847850 0.530236i \(-0.822103\pi\)
−0.847850 + 0.530236i \(0.822103\pi\)
\(314\) 18.0000 1.01580
\(315\) 0 0
\(316\) 16.0000 0.900070
\(317\) −30.0000 −1.68497 −0.842484 0.538721i \(-0.818908\pi\)
−0.842484 + 0.538721i \(0.818908\pi\)
\(318\) 0 0
\(319\) −6.00000 −0.335936
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) 8.00000 0.445823
\(323\) 8.00000 0.445132
\(324\) 0 0
\(325\) −2.00000 −0.110940
\(326\) 4.00000 0.221540
\(327\) 0 0
\(328\) 2.00000 0.110432
\(329\) −8.00000 −0.441054
\(330\) 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
\(333\) 0 0
\(334\) 0 0
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 9.00000 0.489535
\(339\) 0 0
\(340\) 2.00000 0.108465
\(341\) 8.00000 0.433224
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) 30.0000 1.59674 0.798369 0.602168i \(-0.205696\pi\)
0.798369 + 0.602168i \(0.205696\pi\)
\(354\) 0 0
\(355\) 8.00000 0.424596
\(356\) 14.0000 0.741999
\(357\) 0 0
\(358\) −4.00000 −0.211407
\(359\) −8.00000 −0.422224 −0.211112 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) −22.0000 −1.15629
\(363\) 0 0
\(364\) −2.00000 −0.104828
\(365\) −2.00000 −0.104685
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) −8.00000 −0.417029
\(369\) 0 0
\(370\) −2.00000 −0.103975
\(371\) 10.0000 0.519174
\(372\) 0 0
\(373\) 30.0000 1.55334 0.776671 0.629907i \(-0.216907\pi\)
0.776671 + 0.629907i \(0.216907\pi\)
\(374\) 2.00000 0.103418
\(375\) 0 0
\(376\) 8.00000 0.412568
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) 28.0000 1.43826 0.719132 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) 4.00000 0.205196
\(381\) 0 0
\(382\) −16.0000 −0.818631
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) 14.0000 0.712581
\(387\) 0 0
\(388\) 10.0000 0.507673
\(389\) 2.00000 0.101404 0.0507020 0.998714i \(-0.483854\pi\)
0.0507020 + 0.998714i \(0.483854\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) −26.0000 −1.30986
\(395\) −16.0000 −0.805047
\(396\) 0 0
\(397\) 30.0000 1.50566 0.752828 0.658217i \(-0.228689\pi\)
0.752828 + 0.658217i \(0.228689\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) −16.0000 −0.797017
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) 6.00000 0.297775
\(407\) −2.00000 −0.0991363
\(408\) 0 0
\(409\) 26.0000 1.28562 0.642809 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) −2.00000 −0.0987730
\(411\) 0 0
\(412\) 8.00000 0.394132
\(413\) 12.0000 0.590481
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) 2.00000 0.0980581
\(417\) 0 0
\(418\) 4.00000 0.195646
\(419\) 4.00000 0.195413 0.0977064 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) 12.0000 0.584151
\(423\) 0 0
\(424\) −10.0000 −0.485643
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) −2.00000 −0.0967868
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) −16.0000 −0.770693 −0.385346 0.922772i \(-0.625918\pi\)
−0.385346 + 0.922772i \(0.625918\pi\)
\(432\) 0 0
\(433\) −38.0000 −1.82616 −0.913082 0.407777i \(-0.866304\pi\)
−0.913082 + 0.407777i \(0.866304\pi\)
\(434\) −8.00000 −0.384012
\(435\) 0 0
\(436\) 14.0000 0.670478
\(437\) 32.0000 1.53077
\(438\) 0 0
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 1.00000 0.0476731
\(441\) 0 0
\(442\) −4.00000 −0.190261
\(443\) 36.0000 1.71041 0.855206 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) 0 0
\(445\) −14.0000 −0.663664
\(446\) 0 0
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) −2.00000 −0.0941763
\(452\) 14.0000 0.658505
\(453\) 0 0
\(454\) −4.00000 −0.187729
\(455\) 2.00000 0.0937614
\(456\) 0 0
\(457\) −38.0000 −1.77757 −0.888783 0.458329i \(-0.848448\pi\)
−0.888783 + 0.458329i \(0.848448\pi\)
\(458\) −6.00000 −0.280362
\(459\) 0 0
\(460\) 8.00000 0.373002
\(461\) 34.0000 1.58354 0.791769 0.610821i \(-0.209160\pi\)
0.791769 + 0.610821i \(0.209160\pi\)
\(462\) 0 0
\(463\) −32.0000 −1.48717 −0.743583 0.668644i \(-0.766875\pi\)
−0.743583 + 0.668644i \(0.766875\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 10.0000 0.463241
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) −8.00000 −0.369012
\(471\) 0 0
\(472\) −12.0000 −0.552345
\(473\) 4.00000 0.183920
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) −2.00000 −0.0916698
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) −2.00000 −0.0910975
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −10.0000 −0.454077
\(486\) 0 0
\(487\) 24.0000 1.08754 0.543772 0.839233i \(-0.316996\pi\)
0.543772 + 0.839233i \(0.316996\pi\)
\(488\) 2.00000 0.0905357
\(489\) 0 0
\(490\) 1.00000 0.0451754
\(491\) −20.0000 −0.902587 −0.451294 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) 0 0
\(493\) 12.0000 0.540453
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) −8.00000 −0.358849
\(498\) 0 0
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −12.0000 −0.535586
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) 8.00000 0.355643
\(507\) 0 0
\(508\) 0 0
\(509\) 18.0000 0.797836 0.398918 0.916987i \(-0.369386\pi\)
0.398918 + 0.916987i \(0.369386\pi\)
\(510\) 0 0
\(511\) 2.00000 0.0884748
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −14.0000 −0.617514
\(515\) −8.00000 −0.352522
\(516\) 0 0
\(517\) −8.00000 −0.351840
\(518\) 2.00000 0.0878750
\(519\) 0 0
\(520\) −2.00000 −0.0877058
\(521\) 14.0000 0.613351 0.306676 0.951814i \(-0.400783\pi\)
0.306676 + 0.951814i \(0.400783\pi\)
\(522\) 0 0
\(523\) −12.0000 −0.524723 −0.262362 0.964970i \(-0.584501\pi\)
−0.262362 + 0.964970i \(0.584501\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) −8.00000 −0.348817
\(527\) −16.0000 −0.696971
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 10.0000 0.434372
\(531\) 0 0
\(532\) −4.00000 −0.173422
\(533\) 4.00000 0.173259
\(534\) 0 0
\(535\) 12.0000 0.518805
\(536\) 4.00000 0.172774
\(537\) 0 0
\(538\) −2.00000 −0.0862261
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 8.00000 0.343629
\(543\) 0 0
\(544\) 2.00000 0.0857493
\(545\) −14.0000 −0.599694
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) −10.0000 −0.427179
\(549\) 0 0
\(550\) −1.00000 −0.0426401
\(551\) 24.0000 1.02243
\(552\) 0 0
\(553\) 16.0000 0.680389
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) −12.0000 −0.508913
\(557\) 2.00000 0.0847427 0.0423714 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −22.0000 −0.928014
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 0 0
\(565\) −14.0000 −0.588984
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) 8.00000 0.335673
\(569\) 38.0000 1.59304 0.796521 0.604610i \(-0.206671\pi\)
0.796521 + 0.604610i \(0.206671\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) −2.00000 −0.0836242
\(573\) 0 0
\(574\) 2.00000 0.0834784
\(575\) −8.00000 −0.333623
\(576\) 0 0
\(577\) −6.00000 −0.249783 −0.124892 0.992170i \(-0.539858\pi\)
−0.124892 + 0.992170i \(0.539858\pi\)
\(578\) 13.0000 0.540729
\(579\) 0 0
\(580\) 6.00000 0.249136
\(581\) 4.00000 0.165948
\(582\) 0 0
\(583\) 10.0000 0.414158
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) −26.0000 −1.07405
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) −32.0000 −1.31854
\(590\) 12.0000 0.494032
\(591\) 0 0
\(592\) −2.00000 −0.0821995
\(593\) 14.0000 0.574911 0.287456 0.957794i \(-0.407191\pi\)
0.287456 + 0.957794i \(0.407191\pi\)
\(594\) 0 0
\(595\) 2.00000 0.0819920
\(596\) −14.0000 −0.573462
\(597\) 0 0
\(598\) −16.0000 −0.654289
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) −4.00000 −0.163028
\(603\) 0 0
\(604\) 24.0000 0.976546
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) 32.0000 1.29884 0.649420 0.760430i \(-0.275012\pi\)
0.649420 + 0.760430i \(0.275012\pi\)
\(608\) 4.00000 0.162221
\(609\) 0 0
\(610\) −2.00000 −0.0809776
\(611\) 16.0000 0.647291
\(612\) 0 0
\(613\) −18.0000 −0.727013 −0.363507 0.931592i \(-0.618421\pi\)
−0.363507 + 0.931592i \(0.618421\pi\)
\(614\) 20.0000 0.807134
\(615\) 0 0
\(616\) −1.00000 −0.0402911
\(617\) 22.0000 0.885687 0.442843 0.896599i \(-0.353970\pi\)
0.442843 + 0.896599i \(0.353970\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) −8.00000 −0.321288
\(621\) 0 0
\(622\) −24.0000 −0.962312
\(623\) 14.0000 0.560898
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 30.0000 1.19904
\(627\) 0 0
\(628\) −18.0000 −0.718278
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) −16.0000 −0.636446
\(633\) 0 0
\(634\) 30.0000 1.19145
\(635\) 0 0
\(636\) 0 0
\(637\) −2.00000 −0.0792429
\(638\) 6.00000 0.237542
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 0 0
\(643\) 12.0000 0.473234 0.236617 0.971603i \(-0.423961\pi\)
0.236617 + 0.971603i \(0.423961\pi\)
\(644\) −8.00000 −0.315244
\(645\) 0 0
\(646\) −8.00000 −0.314756
\(647\) −48.0000 −1.88707 −0.943537 0.331266i \(-0.892524\pi\)
−0.943537 + 0.331266i \(0.892524\pi\)
\(648\) 0 0
\(649\) 12.0000 0.471041
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) −14.0000 −0.547862 −0.273931 0.961749i \(-0.588324\pi\)
−0.273931 + 0.961749i \(0.588324\pi\)
\(654\) 0 0
\(655\) 12.0000 0.468879
\(656\) −2.00000 −0.0780869
\(657\) 0 0
\(658\) 8.00000 0.311872
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) −28.0000 −1.08825
\(663\) 0 0
\(664\) −4.00000 −0.155230
\(665\) 4.00000 0.155113
\(666\) 0 0
\(667\) 48.0000 1.85857
\(668\) 0 0
\(669\) 0 0
\(670\) −4.00000 −0.154533
\(671\) −2.00000 −0.0772091
\(672\) 0 0
\(673\) 34.0000 1.31060 0.655302 0.755367i \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) −2.00000 −0.0770371
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 26.0000 0.999261 0.499631 0.866239i \(-0.333469\pi\)
0.499631 + 0.866239i \(0.333469\pi\)
\(678\) 0 0
\(679\) 10.0000 0.383765
\(680\) −2.00000 −0.0766965
\(681\) 0 0
\(682\) −8.00000 −0.306336
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 0 0
\(685\) 10.0000 0.382080
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) −20.0000 −0.761939
\(690\) 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
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) 12.0000 0.455186
\(696\) 0 0
\(697\) 4.00000 0.151511
\(698\) 2.00000 0.0757011
\(699\) 0 0
\(700\) 1.00000 0.0377964
\(701\) −38.0000 −1.43524 −0.717620 0.696435i \(-0.754769\pi\)
−0.717620 + 0.696435i \(0.754769\pi\)
\(702\) 0 0
\(703\) 8.00000 0.301726
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) −30.0000 −1.12906
\(707\) −6.00000 −0.225653
\(708\) 0 0
\(709\) 6.00000 0.225335 0.112667 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) −8.00000 −0.300235
\(711\) 0 0
\(712\) −14.0000 −0.524672
\(713\) −64.0000 −2.39682
\(714\) 0 0
\(715\) 2.00000 0.0747958
\(716\) 4.00000 0.149487
\(717\) 0 0
\(718\) 8.00000 0.298557
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) 3.00000 0.111648
\(723\) 0 0
\(724\) 22.0000 0.817624
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) −24.0000 −0.890111 −0.445055 0.895503i \(-0.646816\pi\)
−0.445055 + 0.895503i \(0.646816\pi\)
\(728\) 2.00000 0.0741249
\(729\) 0 0
\(730\) 2.00000 0.0740233
\(731\) −8.00000 −0.295891
\(732\) 0 0
\(733\) −2.00000 −0.0738717 −0.0369358 0.999318i \(-0.511760\pi\)
−0.0369358 + 0.999318i \(0.511760\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 8.00000 0.294884
\(737\) −4.00000 −0.147342
\(738\) 0 0
\(739\) 4.00000 0.147142 0.0735712 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 2.00000 0.0735215
\(741\) 0 0
\(742\) −10.0000 −0.367112
\(743\) 8.00000 0.293492 0.146746 0.989174i \(-0.453120\pi\)
0.146746 + 0.989174i \(0.453120\pi\)
\(744\) 0 0
\(745\) 14.0000 0.512920
\(746\) −30.0000 −1.09838
\(747\) 0 0
\(748\) −2.00000 −0.0731272
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) −8.00000 −0.291730
\(753\) 0 0
\(754\) −12.0000 −0.437014
\(755\) −24.0000 −0.873449
\(756\) 0 0
\(757\) −34.0000 −1.23575 −0.617876 0.786276i \(-0.712006\pi\)
−0.617876 + 0.786276i \(0.712006\pi\)
\(758\) −28.0000 −1.01701
\(759\) 0 0
\(760\) −4.00000 −0.145095
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 0 0
\(763\) 14.0000 0.506834
\(764\) 16.0000 0.578860
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) −24.0000 −0.866590
\(768\) 0 0
\(769\) 34.0000 1.22607 0.613036 0.790055i \(-0.289948\pi\)
0.613036 + 0.790055i \(0.289948\pi\)
\(770\) 1.00000 0.0360375
\(771\) 0 0
\(772\) −14.0000 −0.503871
\(773\) −38.0000 −1.36677 −0.683383 0.730061i \(-0.739492\pi\)
−0.683383 + 0.730061i \(0.739492\pi\)
\(774\) 0 0
\(775\) 8.00000 0.287368
\(776\) −10.0000 −0.358979
\(777\) 0 0
\(778\) −2.00000 −0.0717035
\(779\) 8.00000 0.286630
\(780\) 0 0
\(781\) −8.00000 −0.286263
\(782\) −16.0000 −0.572159
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 18.0000 0.642448
\(786\) 0 0
\(787\) 12.0000 0.427754 0.213877 0.976861i \(-0.431391\pi\)
0.213877 + 0.976861i \(0.431391\pi\)
\(788\) 26.0000 0.926212
\(789\) 0 0
\(790\) 16.0000 0.569254
\(791\) 14.0000 0.497783
\(792\) 0 0
\(793\) 4.00000 0.142044
\(794\) −30.0000 −1.06466
\(795\) 0 0
\(796\) 0 0
\(797\) 2.00000 0.0708436 0.0354218 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) 0 0
\(799\) 16.0000 0.566039
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 18.0000 0.635602
\(803\) 2.00000 0.0705785
\(804\) 0 0
\(805\) 8.00000 0.281963
\(806\) 16.0000 0.563576
\(807\) 0 0
\(808\) 6.00000 0.211079
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) −6.00000 −0.210559
\(813\) 0 0
\(814\) 2.00000 0.0701000
\(815\) 4.00000 0.140114
\(816\) 0 0
\(817\) −16.0000 −0.559769
\(818\) −26.0000 −0.909069
\(819\) 0 0
\(820\) 2.00000 0.0698430
\(821\) 18.0000 0.628204 0.314102 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 0 0
\(823\) 40.0000 1.39431 0.697156 0.716919i \(-0.254448\pi\)
0.697156 + 0.716919i \(0.254448\pi\)
\(824\) −8.00000 −0.278693
\(825\) 0 0
\(826\) −12.0000 −0.417533
\(827\) 20.0000 0.695468 0.347734 0.937593i \(-0.386951\pi\)
0.347734 + 0.937593i \(0.386951\pi\)
\(828\) 0 0
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 4.00000 0.138842
\(831\) 0 0
\(832\) −2.00000 −0.0693375
\(833\) −2.00000 −0.0692959
\(834\) 0 0
\(835\) 0 0
\(836\) −4.00000 −0.138343
\(837\) 0 0
\(838\) −4.00000 −0.138178
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 26.0000 0.896019
\(843\) 0 0
\(844\) −12.0000 −0.413057
\(845\) 9.00000 0.309609
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 10.0000 0.343401
\(849\) 0 0
\(850\) 2.00000 0.0685994
\(851\) 16.0000 0.548473
\(852\) 0 0
\(853\) −26.0000 −0.890223 −0.445112 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) 2.00000 0.0684386
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) −4.00000 −0.136399
\(861\) 0 0
\(862\) 16.0000 0.544962
\(863\) −16.0000 −0.544646 −0.272323 0.962206i \(-0.587792\pi\)
−0.272323 + 0.962206i \(0.587792\pi\)
\(864\) 0 0
\(865\) −18.0000 −0.612018
\(866\) 38.0000 1.29129
\(867\) 0 0
\(868\) 8.00000 0.271538
\(869\) 16.0000 0.542763
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) −14.0000 −0.474100
\(873\) 0 0
\(874\) −32.0000 −1.08242
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) 38.0000 1.28317 0.641584 0.767052i \(-0.278277\pi\)
0.641584 + 0.767052i \(0.278277\pi\)
\(878\) −16.0000 −0.539974
\(879\) 0 0
\(880\) −1.00000 −0.0337100
\(881\) 38.0000 1.28025 0.640126 0.768270i \(-0.278882\pi\)
0.640126 + 0.768270i \(0.278882\pi\)
\(882\) 0 0
\(883\) −20.0000 −0.673054 −0.336527 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) 4.00000 0.134535
\(885\) 0 0
\(886\) −36.0000 −1.20944
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 14.0000 0.469281
\(891\) 0 0
\(892\) 0 0
\(893\) 32.0000 1.07084
\(894\) 0 0
\(895\) −4.00000 −0.133705
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 18.0000 0.600668
\(899\) −48.0000 −1.60089
\(900\) 0 0
\(901\) −20.0000 −0.666297
\(902\) 2.00000 0.0665927
\(903\) 0 0
\(904\) −14.0000 −0.465633
\(905\) −22.0000 −0.731305
\(906\) 0 0
\(907\) 4.00000 0.132818 0.0664089 0.997792i \(-0.478846\pi\)
0.0664089 + 0.997792i \(0.478846\pi\)
\(908\) 4.00000 0.132745
\(909\) 0 0
\(910\) −2.00000 −0.0662994
\(911\) 16.0000 0.530104 0.265052 0.964234i \(-0.414611\pi\)
0.265052 + 0.964234i \(0.414611\pi\)
\(912\) 0 0
\(913\) 4.00000 0.132381
\(914\) 38.0000 1.25693
\(915\) 0 0
\(916\) 6.00000 0.198246
\(917\) −12.0000 −0.396275
\(918\) 0 0
\(919\) −24.0000 −0.791687 −0.395843 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) −8.00000 −0.263752
\(921\) 0 0
\(922\) −34.0000 −1.11973
\(923\) 16.0000 0.526646
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) 32.0000 1.05159
\(927\) 0 0
\(928\) 6.00000 0.196960
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) −4.00000 −0.131095
\(932\) −10.0000 −0.327561
\(933\) 0 0
\(934\) 12.0000 0.392652
\(935\) 2.00000 0.0654070
\(936\) 0 0
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 4.00000 0.130605
\(939\) 0 0
\(940\) 8.00000 0.260931
\(941\) −46.0000 −1.49956 −0.749779 0.661689i \(-0.769840\pi\)
−0.749779 + 0.661689i \(0.769840\pi\)
\(942\) 0 0
\(943\) 16.0000 0.521032
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) −52.0000 −1.68977 −0.844886 0.534946i \(-0.820332\pi\)
−0.844886 + 0.534946i \(0.820332\pi\)
\(948\) 0 0
\(949\) −4.00000 −0.129845
\(950\) 4.00000 0.129777
\(951\) 0 0
\(952\) 2.00000 0.0648204
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 0 0
\(955\) −16.0000 −0.517748
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −10.0000 −0.322917
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) −4.00000 −0.128965
\(963\) 0 0
\(964\) 2.00000 0.0644157
\(965\) 14.0000 0.450676
\(966\) 0 0
\(967\) −40.0000 −1.28631 −0.643157 0.765735i \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) 10.0000 0.321081
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 0 0
\(973\) −12.0000 −0.384702
\(974\) −24.0000 −0.769010
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) 46.0000 1.47167 0.735835 0.677161i \(-0.236790\pi\)
0.735835 + 0.677161i \(0.236790\pi\)
\(978\) 0 0
\(979\) 14.0000 0.447442
\(980\) −1.00000 −0.0319438
\(981\) 0 0
\(982\) 20.0000 0.638226
\(983\) 48.0000 1.53096 0.765481 0.643458i \(-0.222501\pi\)
0.765481 + 0.643458i \(0.222501\pi\)
\(984\) 0 0
\(985\) −26.0000 −0.828429
\(986\) −12.0000 −0.382158
\(987\) 0 0
\(988\) 8.00000 0.254514
\(989\) −32.0000 −1.01754
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) −8.00000 −0.254000
\(993\) 0 0
\(994\) 8.00000 0.253745
\(995\) 0 0
\(996\) 0 0
\(997\) −58.0000 −1.83688 −0.918439 0.395562i \(-0.870550\pi\)
−0.918439 + 0.395562i \(0.870550\pi\)
\(998\) −4.00000 −0.126618
\(999\) 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 6930.2.a.d.1.1 1
3.2 odd 2 2310.2.a.o.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2310.2.a.o.1.1 1 3.2 odd 2
6930.2.a.d.1.1 1 1.1 even 1 trivial