Properties

Label 1078.2.a.e.1.1
Level $1078$
Weight $2$
Character 1078.1
Self dual yes
Analytic conductor $8.608$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1078.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.60787333789\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 154)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1078.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} -2.00000 q^{9} -1.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} +1.00000 q^{16} -6.00000 q^{17} +2.00000 q^{18} +2.00000 q^{19} +1.00000 q^{22} -6.00000 q^{23} -1.00000 q^{24} -5.00000 q^{25} +1.00000 q^{26} -5.00000 q^{27} +9.00000 q^{29} -4.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} +6.00000 q^{34} -2.00000 q^{36} +2.00000 q^{37} -2.00000 q^{38} -1.00000 q^{39} -6.00000 q^{41} -4.00000 q^{43} -1.00000 q^{44} +6.00000 q^{46} -6.00000 q^{47} +1.00000 q^{48} +5.00000 q^{50} -6.00000 q^{51} -1.00000 q^{52} +5.00000 q^{54} +2.00000 q^{57} -9.00000 q^{58} -3.00000 q^{59} +11.0000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +1.00000 q^{66} +11.0000 q^{67} -6.00000 q^{68} -6.00000 q^{69} +2.00000 q^{72} +2.00000 q^{73} -2.00000 q^{74} -5.00000 q^{75} +2.00000 q^{76} +1.00000 q^{78} +5.00000 q^{79} +1.00000 q^{81} +6.00000 q^{82} -6.00000 q^{83} +4.00000 q^{86} +9.00000 q^{87} +1.00000 q^{88} -18.0000 q^{89} -6.00000 q^{92} -4.00000 q^{93} +6.00000 q^{94} -1.00000 q^{96} -13.0000 q^{97} +2.00000 q^{99} +O(q^{100})\)

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 0.288675 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 1.00000 0.288675
\(13\) −1.00000 −0.277350 −0.138675 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 2.00000 0.471405
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) −1.00000 −0.204124
\(25\) −5.00000 −1.00000
\(26\) 1.00000 0.196116
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) 9.00000 1.67126 0.835629 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.00000 −0.174078
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) −2.00000 −0.333333
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −2.00000 −0.324443
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 5.00000 0.707107
\(51\) −6.00000 −0.840168
\(52\) −1.00000 −0.138675
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 5.00000 0.680414
\(55\) 0 0
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) −9.00000 −1.18176
\(59\) −3.00000 −0.390567 −0.195283 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) 0 0
\(61\) 11.0000 1.40841 0.704203 0.709999i \(-0.251305\pi\)
0.704203 + 0.709999i \(0.251305\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 1.00000 0.123091
\(67\) 11.0000 1.34386 0.671932 0.740613i \(-0.265465\pi\)
0.671932 + 0.740613i \(0.265465\pi\)
\(68\) −6.00000 −0.727607
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 2.00000 0.235702
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) −2.00000 −0.232495
\(75\) −5.00000 −0.577350
\(76\) 2.00000 0.229416
\(77\) 0 0
\(78\) 1.00000 0.113228
\(79\) 5.00000 0.562544 0.281272 0.959628i \(-0.409244\pi\)
0.281272 + 0.959628i \(0.409244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 9.00000 0.964901
\(88\) 1.00000 0.106600
\(89\) −18.0000 −1.90800 −0.953998 0.299813i \(-0.903076\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −6.00000 −0.625543
\(93\) −4.00000 −0.414781
\(94\) 6.00000 0.618853
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −13.0000 −1.31995 −0.659975 0.751288i \(-0.729433\pi\)
−0.659975 + 0.751288i \(0.729433\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
\(100\) −5.00000 −0.500000
\(101\) −15.0000 −1.49256 −0.746278 0.665635i \(-0.768161\pi\)
−0.746278 + 0.665635i \(0.768161\pi\)
\(102\) 6.00000 0.594089
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) −5.00000 −0.481125
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) 9.00000 0.846649 0.423324 0.905978i \(-0.360863\pi\)
0.423324 + 0.905978i \(0.360863\pi\)
\(114\) −2.00000 −0.187317
\(115\) 0 0
\(116\) 9.00000 0.835629
\(117\) 2.00000 0.184900
\(118\) 3.00000 0.276172
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −11.0000 −0.995893
\(123\) −6.00000 −0.541002
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) 0 0
\(127\) −7.00000 −0.621150 −0.310575 0.950549i \(-0.600522\pi\)
−0.310575 + 0.950549i \(0.600522\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 18.0000 1.57267 0.786334 0.617802i \(-0.211977\pi\)
0.786334 + 0.617802i \(0.211977\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 0 0
\(134\) −11.0000 −0.950255
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) −9.00000 −0.768922 −0.384461 0.923141i \(-0.625613\pi\)
−0.384461 + 0.923141i \(0.625613\pi\)
\(138\) 6.00000 0.510754
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 0 0
\(143\) 1.00000 0.0836242
\(144\) −2.00000 −0.166667
\(145\) 0 0
\(146\) −2.00000 −0.165521
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 5.00000 0.408248
\(151\) −19.0000 −1.54620 −0.773099 0.634285i \(-0.781294\pi\)
−0.773099 + 0.634285i \(0.781294\pi\)
\(152\) −2.00000 −0.162221
\(153\) 12.0000 0.970143
\(154\) 0 0
\(155\) 0 0
\(156\) −1.00000 −0.0800641
\(157\) −4.00000 −0.319235 −0.159617 0.987179i \(-0.551026\pi\)
−0.159617 + 0.987179i \(0.551026\pi\)
\(158\) −5.00000 −0.397779
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 17.0000 1.33154 0.665771 0.746156i \(-0.268103\pi\)
0.665771 + 0.746156i \(0.268103\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 6.00000 0.465690
\(167\) 3.00000 0.232147 0.116073 0.993241i \(-0.462969\pi\)
0.116073 + 0.993241i \(0.462969\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) −4.00000 −0.304997
\(173\) −21.0000 −1.59660 −0.798300 0.602260i \(-0.794267\pi\)
−0.798300 + 0.602260i \(0.794267\pi\)
\(174\) −9.00000 −0.682288
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) −3.00000 −0.225494
\(178\) 18.0000 1.34916
\(179\) −15.0000 −1.12115 −0.560576 0.828103i \(-0.689420\pi\)
−0.560576 + 0.828103i \(0.689420\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 11.0000 0.813143
\(184\) 6.00000 0.442326
\(185\) 0 0
\(186\) 4.00000 0.293294
\(187\) 6.00000 0.438763
\(188\) −6.00000 −0.437595
\(189\) 0 0
\(190\) 0 0
\(191\) 6.00000 0.434145 0.217072 0.976156i \(-0.430349\pi\)
0.217072 + 0.976156i \(0.430349\pi\)
\(192\) 1.00000 0.0721688
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 13.0000 0.933346
\(195\) 0 0
\(196\) 0 0
\(197\) −3.00000 −0.213741 −0.106871 0.994273i \(-0.534083\pi\)
−0.106871 + 0.994273i \(0.534083\pi\)
\(198\) −2.00000 −0.142134
\(199\) 14.0000 0.992434 0.496217 0.868199i \(-0.334722\pi\)
0.496217 + 0.868199i \(0.334722\pi\)
\(200\) 5.00000 0.353553
\(201\) 11.0000 0.775880
\(202\) 15.0000 1.05540
\(203\) 0 0
\(204\) −6.00000 −0.420084
\(205\) 0 0
\(206\) 16.0000 1.11477
\(207\) 12.0000 0.834058
\(208\) −1.00000 −0.0693375
\(209\) −2.00000 −0.138343
\(210\) 0 0
\(211\) −10.0000 −0.688428 −0.344214 0.938891i \(-0.611855\pi\)
−0.344214 + 0.938891i \(0.611855\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 5.00000 0.340207
\(217\) 0 0
\(218\) −2.00000 −0.135457
\(219\) 2.00000 0.135147
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) −2.00000 −0.134231
\(223\) 26.0000 1.74109 0.870544 0.492090i \(-0.163767\pi\)
0.870544 + 0.492090i \(0.163767\pi\)
\(224\) 0 0
\(225\) 10.0000 0.666667
\(226\) −9.00000 −0.598671
\(227\) 18.0000 1.19470 0.597351 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) 2.00000 0.132453
\(229\) −16.0000 −1.05731 −0.528655 0.848837i \(-0.677303\pi\)
−0.528655 + 0.848837i \(0.677303\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −9.00000 −0.590879
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) −3.00000 −0.195283
\(237\) 5.00000 0.324785
\(238\) 0 0
\(239\) −9.00000 −0.582162 −0.291081 0.956698i \(-0.594015\pi\)
−0.291081 + 0.956698i \(0.594015\pi\)
\(240\) 0 0
\(241\) 26.0000 1.67481 0.837404 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 16.0000 1.02640
\(244\) 11.0000 0.704203
\(245\) 0 0
\(246\) 6.00000 0.382546
\(247\) −2.00000 −0.127257
\(248\) 4.00000 0.254000
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 6.00000 0.377217
\(254\) 7.00000 0.439219
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 3.00000 0.187135 0.0935674 0.995613i \(-0.470173\pi\)
0.0935674 + 0.995613i \(0.470173\pi\)
\(258\) 4.00000 0.249029
\(259\) 0 0
\(260\) 0 0
\(261\) −18.0000 −1.11417
\(262\) −18.0000 −1.11204
\(263\) −9.00000 −0.554964 −0.277482 0.960731i \(-0.589500\pi\)
−0.277482 + 0.960731i \(0.589500\pi\)
\(264\) 1.00000 0.0615457
\(265\) 0 0
\(266\) 0 0
\(267\) −18.0000 −1.10158
\(268\) 11.0000 0.671932
\(269\) 12.0000 0.731653 0.365826 0.930683i \(-0.380786\pi\)
0.365826 + 0.930683i \(0.380786\pi\)
\(270\) 0 0
\(271\) 29.0000 1.76162 0.880812 0.473466i \(-0.156997\pi\)
0.880812 + 0.473466i \(0.156997\pi\)
\(272\) −6.00000 −0.363803
\(273\) 0 0
\(274\) 9.00000 0.543710
\(275\) 5.00000 0.301511
\(276\) −6.00000 −0.361158
\(277\) 29.0000 1.74244 0.871221 0.490892i \(-0.163329\pi\)
0.871221 + 0.490892i \(0.163329\pi\)
\(278\) −8.00000 −0.479808
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 6.00000 0.357295
\(283\) −10.0000 −0.594438 −0.297219 0.954809i \(-0.596059\pi\)
−0.297219 + 0.954809i \(0.596059\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −1.00000 −0.0591312
\(287\) 0 0
\(288\) 2.00000 0.117851
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −13.0000 −0.762073
\(292\) 2.00000 0.117041
\(293\) −30.0000 −1.75262 −0.876309 0.481749i \(-0.840002\pi\)
−0.876309 + 0.481749i \(0.840002\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) 5.00000 0.290129
\(298\) −6.00000 −0.347571
\(299\) 6.00000 0.346989
\(300\) −5.00000 −0.288675
\(301\) 0 0
\(302\) 19.0000 1.09333
\(303\) −15.0000 −0.861727
\(304\) 2.00000 0.114708
\(305\) 0 0
\(306\) −12.0000 −0.685994
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 0 0
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 1.00000 0.0566139
\(313\) 17.0000 0.960897 0.480448 0.877023i \(-0.340474\pi\)
0.480448 + 0.877023i \(0.340474\pi\)
\(314\) 4.00000 0.225733
\(315\) 0 0
\(316\) 5.00000 0.281272
\(317\) 12.0000 0.673987 0.336994 0.941507i \(-0.390590\pi\)
0.336994 + 0.941507i \(0.390590\pi\)
\(318\) 0 0
\(319\) −9.00000 −0.503903
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) −12.0000 −0.667698
\(324\) 1.00000 0.0555556
\(325\) 5.00000 0.277350
\(326\) −17.0000 −0.941543
\(327\) 2.00000 0.110600
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) 35.0000 1.92377 0.961887 0.273447i \(-0.0881639\pi\)
0.961887 + 0.273447i \(0.0881639\pi\)
\(332\) −6.00000 −0.329293
\(333\) −4.00000 −0.219199
\(334\) −3.00000 −0.164153
\(335\) 0 0
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 12.0000 0.652714
\(339\) 9.00000 0.488813
\(340\) 0 0
\(341\) 4.00000 0.216612
\(342\) 4.00000 0.216295
\(343\) 0 0
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 21.0000 1.12897
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 9.00000 0.482451
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) 5.00000 0.266880
\(352\) 1.00000 0.0533002
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 3.00000 0.159448
\(355\) 0 0
\(356\) −18.0000 −0.953998
\(357\) 0 0
\(358\) 15.0000 0.792775
\(359\) −3.00000 −0.158334 −0.0791670 0.996861i \(-0.525226\pi\)
−0.0791670 + 0.996861i \(0.525226\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) −2.00000 −0.105118
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) 0 0
\(366\) −11.0000 −0.574979
\(367\) −10.0000 −0.521996 −0.260998 0.965339i \(-0.584052\pi\)
−0.260998 + 0.965339i \(0.584052\pi\)
\(368\) −6.00000 −0.312772
\(369\) 12.0000 0.624695
\(370\) 0 0
\(371\) 0 0
\(372\) −4.00000 −0.207390
\(373\) −31.0000 −1.60512 −0.802560 0.596572i \(-0.796529\pi\)
−0.802560 + 0.596572i \(0.796529\pi\)
\(374\) −6.00000 −0.310253
\(375\) 0 0
\(376\) 6.00000 0.309426
\(377\) −9.00000 −0.463524
\(378\) 0 0
\(379\) 23.0000 1.18143 0.590715 0.806880i \(-0.298846\pi\)
0.590715 + 0.806880i \(0.298846\pi\)
\(380\) 0 0
\(381\) −7.00000 −0.358621
\(382\) −6.00000 −0.306987
\(383\) −36.0000 −1.83951 −0.919757 0.392488i \(-0.871614\pi\)
−0.919757 + 0.392488i \(0.871614\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) 8.00000 0.406663
\(388\) −13.0000 −0.659975
\(389\) 12.0000 0.608424 0.304212 0.952604i \(-0.401607\pi\)
0.304212 + 0.952604i \(0.401607\pi\)
\(390\) 0 0
\(391\) 36.0000 1.82060
\(392\) 0 0
\(393\) 18.0000 0.907980
\(394\) 3.00000 0.151138
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) −14.0000 −0.701757
\(399\) 0 0
\(400\) −5.00000 −0.250000
\(401\) −33.0000 −1.64794 −0.823971 0.566632i \(-0.808246\pi\)
−0.823971 + 0.566632i \(0.808246\pi\)
\(402\) −11.0000 −0.548630
\(403\) 4.00000 0.199254
\(404\) −15.0000 −0.746278
\(405\) 0 0
\(406\) 0 0
\(407\) −2.00000 −0.0991363
\(408\) 6.00000 0.297044
\(409\) −4.00000 −0.197787 −0.0988936 0.995098i \(-0.531530\pi\)
−0.0988936 + 0.995098i \(0.531530\pi\)
\(410\) 0 0
\(411\) −9.00000 −0.443937
\(412\) −16.0000 −0.788263
\(413\) 0 0
\(414\) −12.0000 −0.589768
\(415\) 0 0
\(416\) 1.00000 0.0490290
\(417\) 8.00000 0.391762
\(418\) 2.00000 0.0978232
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) −28.0000 −1.36464 −0.682318 0.731055i \(-0.739028\pi\)
−0.682318 + 0.731055i \(0.739028\pi\)
\(422\) 10.0000 0.486792
\(423\) 12.0000 0.583460
\(424\) 0 0
\(425\) 30.0000 1.45521
\(426\) 0 0
\(427\) 0 0
\(428\) 12.0000 0.580042
\(429\) 1.00000 0.0482805
\(430\) 0 0
\(431\) −3.00000 −0.144505 −0.0722525 0.997386i \(-0.523019\pi\)
−0.0722525 + 0.997386i \(0.523019\pi\)
\(432\) −5.00000 −0.240563
\(433\) −34.0000 −1.63394 −0.816968 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) −12.0000 −0.574038
\(438\) −2.00000 −0.0955637
\(439\) −1.00000 −0.0477274 −0.0238637 0.999715i \(-0.507597\pi\)
−0.0238637 + 0.999715i \(0.507597\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −6.00000 −0.285391
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 2.00000 0.0949158
\(445\) 0 0
\(446\) −26.0000 −1.23114
\(447\) 6.00000 0.283790
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) −10.0000 −0.471405
\(451\) 6.00000 0.282529
\(452\) 9.00000 0.423324
\(453\) −19.0000 −0.892698
\(454\) −18.0000 −0.844782
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 16.0000 0.747631
\(459\) 30.0000 1.40028
\(460\) 0 0
\(461\) 21.0000 0.978068 0.489034 0.872265i \(-0.337349\pi\)
0.489034 + 0.872265i \(0.337349\pi\)
\(462\) 0 0
\(463\) −22.0000 −1.02243 −0.511213 0.859454i \(-0.670804\pi\)
−0.511213 + 0.859454i \(0.670804\pi\)
\(464\) 9.00000 0.417815
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 2.00000 0.0924500
\(469\) 0 0
\(470\) 0 0
\(471\) −4.00000 −0.184310
\(472\) 3.00000 0.138086
\(473\) 4.00000 0.183920
\(474\) −5.00000 −0.229658
\(475\) −10.0000 −0.458831
\(476\) 0 0
\(477\) 0 0
\(478\) 9.00000 0.411650
\(479\) 15.0000 0.685367 0.342684 0.939451i \(-0.388664\pi\)
0.342684 + 0.939451i \(0.388664\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) −26.0000 −1.18427
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) −16.0000 −0.725775
\(487\) 20.0000 0.906287 0.453143 0.891438i \(-0.350303\pi\)
0.453143 + 0.891438i \(0.350303\pi\)
\(488\) −11.0000 −0.497947
\(489\) 17.0000 0.768767
\(490\) 0 0
\(491\) 42.0000 1.89543 0.947717 0.319113i \(-0.103385\pi\)
0.947717 + 0.319113i \(0.103385\pi\)
\(492\) −6.00000 −0.270501
\(493\) −54.0000 −2.43204
\(494\) 2.00000 0.0899843
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) 6.00000 0.268866
\(499\) −40.0000 −1.79065 −0.895323 0.445418i \(-0.853055\pi\)
−0.895323 + 0.445418i \(0.853055\pi\)
\(500\) 0 0
\(501\) 3.00000 0.134030
\(502\) 12.0000 0.535586
\(503\) −33.0000 −1.47140 −0.735699 0.677309i \(-0.763146\pi\)
−0.735699 + 0.677309i \(0.763146\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −6.00000 −0.266733
\(507\) −12.0000 −0.532939
\(508\) −7.00000 −0.310575
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −10.0000 −0.441511
\(514\) −3.00000 −0.132324
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) 6.00000 0.263880
\(518\) 0 0
\(519\) −21.0000 −0.921798
\(520\) 0 0
\(521\) −42.0000 −1.84005 −0.920027 0.391856i \(-0.871833\pi\)
−0.920027 + 0.391856i \(0.871833\pi\)
\(522\) 18.0000 0.787839
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) 18.0000 0.786334
\(525\) 0 0
\(526\) 9.00000 0.392419
\(527\) 24.0000 1.04546
\(528\) −1.00000 −0.0435194
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) 0 0
\(533\) 6.00000 0.259889
\(534\) 18.0000 0.778936
\(535\) 0 0
\(536\) −11.0000 −0.475128
\(537\) −15.0000 −0.647298
\(538\) −12.0000 −0.517357
\(539\) 0 0
\(540\) 0 0
\(541\) −25.0000 −1.07483 −0.537417 0.843317i \(-0.680600\pi\)
−0.537417 + 0.843317i \(0.680600\pi\)
\(542\) −29.0000 −1.24566
\(543\) 2.00000 0.0858282
\(544\) 6.00000 0.257248
\(545\) 0 0
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) −9.00000 −0.384461
\(549\) −22.0000 −0.938937
\(550\) −5.00000 −0.213201
\(551\) 18.0000 0.766826
\(552\) 6.00000 0.255377
\(553\) 0 0
\(554\) −29.0000 −1.23209
\(555\) 0 0
\(556\) 8.00000 0.339276
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) −8.00000 −0.338667
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) 6.00000 0.253320
\(562\) −18.0000 −0.759284
\(563\) −18.0000 −0.758610 −0.379305 0.925272i \(-0.623837\pi\)
−0.379305 + 0.925272i \(0.623837\pi\)
\(564\) −6.00000 −0.252646
\(565\) 0 0
\(566\) 10.0000 0.420331
\(567\) 0 0
\(568\) 0 0
\(569\) −36.0000 −1.50920 −0.754599 0.656186i \(-0.772169\pi\)
−0.754599 + 0.656186i \(0.772169\pi\)
\(570\) 0 0
\(571\) 32.0000 1.33916 0.669579 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) 1.00000 0.0418121
\(573\) 6.00000 0.250654
\(574\) 0 0
\(575\) 30.0000 1.25109
\(576\) −2.00000 −0.0833333
\(577\) −7.00000 −0.291414 −0.145707 0.989328i \(-0.546546\pi\)
−0.145707 + 0.989328i \(0.546546\pi\)
\(578\) −19.0000 −0.790296
\(579\) 14.0000 0.581820
\(580\) 0 0
\(581\) 0 0
\(582\) 13.0000 0.538867
\(583\) 0 0
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) 30.0000 1.23929
\(587\) 9.00000 0.371470 0.185735 0.982600i \(-0.440533\pi\)
0.185735 + 0.982600i \(0.440533\pi\)
\(588\) 0 0
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) −3.00000 −0.123404
\(592\) 2.00000 0.0821995
\(593\) 36.0000 1.47834 0.739171 0.673517i \(-0.235217\pi\)
0.739171 + 0.673517i \(0.235217\pi\)
\(594\) −5.00000 −0.205152
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) 14.0000 0.572982
\(598\) −6.00000 −0.245358
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 5.00000 0.204124
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 0 0
\(603\) −22.0000 −0.895909
\(604\) −19.0000 −0.773099
\(605\) 0 0
\(606\) 15.0000 0.609333
\(607\) −40.0000 −1.62355 −0.811775 0.583970i \(-0.801498\pi\)
−0.811775 + 0.583970i \(0.801498\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 0 0
\(610\) 0 0
\(611\) 6.00000 0.242734
\(612\) 12.0000 0.485071
\(613\) −10.0000 −0.403896 −0.201948 0.979396i \(-0.564727\pi\)
−0.201948 + 0.979396i \(0.564727\pi\)
\(614\) −20.0000 −0.807134
\(615\) 0 0
\(616\) 0 0
\(617\) 21.0000 0.845428 0.422714 0.906263i \(-0.361077\pi\)
0.422714 + 0.906263i \(0.361077\pi\)
\(618\) 16.0000 0.643614
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 0 0
\(621\) 30.0000 1.20386
\(622\) −24.0000 −0.962312
\(623\) 0 0
\(624\) −1.00000 −0.0400320
\(625\) 25.0000 1.00000
\(626\) −17.0000 −0.679457
\(627\) −2.00000 −0.0798723
\(628\) −4.00000 −0.159617
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) −5.00000 −0.198889
\(633\) −10.0000 −0.397464
\(634\) −12.0000 −0.476581
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 9.00000 0.356313
\(639\) 0 0
\(640\) 0 0
\(641\) 9.00000 0.355479 0.177739 0.984078i \(-0.443122\pi\)
0.177739 + 0.984078i \(0.443122\pi\)
\(642\) −12.0000 −0.473602
\(643\) 5.00000 0.197181 0.0985904 0.995128i \(-0.468567\pi\)
0.0985904 + 0.995128i \(0.468567\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 3.00000 0.117760
\(650\) −5.00000 −0.196116
\(651\) 0 0
\(652\) 17.0000 0.665771
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) −2.00000 −0.0782062
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) −4.00000 −0.156055
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) −35.0000 −1.36031
\(663\) 6.00000 0.233021
\(664\) 6.00000 0.232845
\(665\) 0 0
\(666\) 4.00000 0.154997
\(667\) −54.0000 −2.09089
\(668\) 3.00000 0.116073
\(669\) 26.0000 1.00522
\(670\) 0 0
\(671\) −11.0000 −0.424650
\(672\) 0 0
\(673\) −28.0000 −1.07932 −0.539660 0.841883i \(-0.681447\pi\)
−0.539660 + 0.841883i \(0.681447\pi\)
\(674\) −14.0000 −0.539260
\(675\) 25.0000 0.962250
\(676\) −12.0000 −0.461538
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) −9.00000 −0.345643
\(679\) 0 0
\(680\) 0 0
\(681\) 18.0000 0.689761
\(682\) −4.00000 −0.153168
\(683\) −21.0000 −0.803543 −0.401771 0.915740i \(-0.631605\pi\)
−0.401771 + 0.915740i \(0.631605\pi\)
\(684\) −4.00000 −0.152944
\(685\) 0 0
\(686\) 0 0
\(687\) −16.0000 −0.610438
\(688\) −4.00000 −0.152499
\(689\) 0 0
\(690\) 0 0
\(691\) 11.0000 0.418460 0.209230 0.977866i \(-0.432904\pi\)
0.209230 + 0.977866i \(0.432904\pi\)
\(692\) −21.0000 −0.798300
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) −9.00000 −0.341144
\(697\) 36.0000 1.36360
\(698\) −2.00000 −0.0757011
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) 39.0000 1.47301 0.736505 0.676432i \(-0.236475\pi\)
0.736505 + 0.676432i \(0.236475\pi\)
\(702\) −5.00000 −0.188713
\(703\) 4.00000 0.150863
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) 0 0
\(708\) −3.00000 −0.112747
\(709\) 26.0000 0.976450 0.488225 0.872718i \(-0.337644\pi\)
0.488225 + 0.872718i \(0.337644\pi\)
\(710\) 0 0
\(711\) −10.0000 −0.375029
\(712\) 18.0000 0.674579
\(713\) 24.0000 0.898807
\(714\) 0 0
\(715\) 0 0
\(716\) −15.0000 −0.560576
\(717\) −9.00000 −0.336111
\(718\) 3.00000 0.111959
\(719\) −42.0000 −1.56634 −0.783168 0.621810i \(-0.786397\pi\)
−0.783168 + 0.621810i \(0.786397\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 15.0000 0.558242
\(723\) 26.0000 0.966950
\(724\) 2.00000 0.0743294
\(725\) −45.0000 −1.67126
\(726\) −1.00000 −0.0371135
\(727\) 14.0000 0.519231 0.259616 0.965712i \(-0.416404\pi\)
0.259616 + 0.965712i \(0.416404\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) 11.0000 0.406572
\(733\) −25.0000 −0.923396 −0.461698 0.887037i \(-0.652760\pi\)
−0.461698 + 0.887037i \(0.652760\pi\)
\(734\) 10.0000 0.369107
\(735\) 0 0
\(736\) 6.00000 0.221163
\(737\) −11.0000 −0.405190
\(738\) −12.0000 −0.441726
\(739\) 50.0000 1.83928 0.919640 0.392763i \(-0.128481\pi\)
0.919640 + 0.392763i \(0.128481\pi\)
\(740\) 0 0
\(741\) −2.00000 −0.0734718
\(742\) 0 0
\(743\) −36.0000 −1.32071 −0.660356 0.750953i \(-0.729595\pi\)
−0.660356 + 0.750953i \(0.729595\pi\)
\(744\) 4.00000 0.146647
\(745\) 0 0
\(746\) 31.0000 1.13499
\(747\) 12.0000 0.439057
\(748\) 6.00000 0.219382
\(749\) 0 0
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) −6.00000 −0.218797
\(753\) −12.0000 −0.437304
\(754\) 9.00000 0.327761
\(755\) 0 0
\(756\) 0 0
\(757\) −46.0000 −1.67190 −0.835949 0.548807i \(-0.815082\pi\)
−0.835949 + 0.548807i \(0.815082\pi\)
\(758\) −23.0000 −0.835398
\(759\) 6.00000 0.217786
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 7.00000 0.253583
\(763\) 0 0
\(764\) 6.00000 0.217072
\(765\) 0 0
\(766\) 36.0000 1.30073
\(767\) 3.00000 0.108324
\(768\) 1.00000 0.0360844
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) 0 0
\(771\) 3.00000 0.108042
\(772\) 14.0000 0.503871
\(773\) 24.0000 0.863220 0.431610 0.902060i \(-0.357946\pi\)
0.431610 + 0.902060i \(0.357946\pi\)
\(774\) −8.00000 −0.287554
\(775\) 20.0000 0.718421
\(776\) 13.0000 0.466673
\(777\) 0 0
\(778\) −12.0000 −0.430221
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) 0 0
\(782\) −36.0000 −1.28736
\(783\) −45.0000 −1.60817
\(784\) 0 0
\(785\) 0 0
\(786\) −18.0000 −0.642039
\(787\) 14.0000 0.499046 0.249523 0.968369i \(-0.419726\pi\)
0.249523 + 0.968369i \(0.419726\pi\)
\(788\) −3.00000 −0.106871
\(789\) −9.00000 −0.320408
\(790\) 0 0
\(791\) 0 0
\(792\) −2.00000 −0.0710669
\(793\) −11.0000 −0.390621
\(794\) 22.0000 0.780751
\(795\) 0 0
\(796\) 14.0000 0.496217
\(797\) −18.0000 −0.637593 −0.318796 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) 0 0
\(799\) 36.0000 1.27359
\(800\) 5.00000 0.176777
\(801\) 36.0000 1.27200
\(802\) 33.0000 1.16527
\(803\) −2.00000 −0.0705785
\(804\) 11.0000 0.387940
\(805\) 0 0
\(806\) −4.00000 −0.140894
\(807\) 12.0000 0.422420
\(808\) 15.0000 0.527698
\(809\) −48.0000 −1.68759 −0.843795 0.536666i \(-0.819684\pi\)
−0.843795 + 0.536666i \(0.819684\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 0 0
\(813\) 29.0000 1.01707
\(814\) 2.00000 0.0701000
\(815\) 0 0
\(816\) −6.00000 −0.210042
\(817\) −8.00000 −0.279885
\(818\) 4.00000 0.139857
\(819\) 0 0
\(820\) 0 0
\(821\) −15.0000 −0.523504 −0.261752 0.965135i \(-0.584300\pi\)
−0.261752 + 0.965135i \(0.584300\pi\)
\(822\) 9.00000 0.313911
\(823\) −10.0000 −0.348578 −0.174289 0.984695i \(-0.555763\pi\)
−0.174289 + 0.984695i \(0.555763\pi\)
\(824\) 16.0000 0.557386
\(825\) 5.00000 0.174078
\(826\) 0 0
\(827\) 30.0000 1.04320 0.521601 0.853189i \(-0.325335\pi\)
0.521601 + 0.853189i \(0.325335\pi\)
\(828\) 12.0000 0.417029
\(829\) −52.0000 −1.80603 −0.903017 0.429604i \(-0.858653\pi\)
−0.903017 + 0.429604i \(0.858653\pi\)
\(830\) 0 0
\(831\) 29.0000 1.00600
\(832\) −1.00000 −0.0346688
\(833\) 0 0
\(834\) −8.00000 −0.277017
\(835\) 0 0
\(836\) −2.00000 −0.0691714
\(837\) 20.0000 0.691301
\(838\) 12.0000 0.414533
\(839\) 6.00000 0.207143 0.103572 0.994622i \(-0.466973\pi\)
0.103572 + 0.994622i \(0.466973\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 28.0000 0.964944
\(843\) 18.0000 0.619953
\(844\) −10.0000 −0.344214
\(845\) 0 0
\(846\) −12.0000 −0.412568
\(847\) 0 0
\(848\) 0 0
\(849\) −10.0000 −0.343199
\(850\) −30.0000 −1.02899
\(851\) −12.0000 −0.411355
\(852\) 0 0
\(853\) −46.0000 −1.57501 −0.787505 0.616308i \(-0.788628\pi\)
−0.787505 + 0.616308i \(0.788628\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) 36.0000 1.22974 0.614868 0.788630i \(-0.289209\pi\)
0.614868 + 0.788630i \(0.289209\pi\)
\(858\) −1.00000 −0.0341394
\(859\) −37.0000 −1.26242 −0.631212 0.775610i \(-0.717442\pi\)
−0.631212 + 0.775610i \(0.717442\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 3.00000 0.102180
\(863\) 30.0000 1.02121 0.510606 0.859815i \(-0.329421\pi\)
0.510606 + 0.859815i \(0.329421\pi\)
\(864\) 5.00000 0.170103
\(865\) 0 0
\(866\) 34.0000 1.15537
\(867\) 19.0000 0.645274
\(868\) 0 0
\(869\) −5.00000 −0.169613
\(870\) 0 0
\(871\) −11.0000 −0.372721
\(872\) −2.00000 −0.0677285
\(873\) 26.0000 0.879967
\(874\) 12.0000 0.405906
\(875\) 0 0
\(876\) 2.00000 0.0675737
\(877\) 17.0000 0.574049 0.287025 0.957923i \(-0.407334\pi\)
0.287025 + 0.957923i \(0.407334\pi\)
\(878\) 1.00000 0.0337484
\(879\) −30.0000 −1.01187
\(880\) 0 0
\(881\) 33.0000 1.11180 0.555899 0.831250i \(-0.312374\pi\)
0.555899 + 0.831250i \(0.312374\pi\)
\(882\) 0 0
\(883\) 29.0000 0.975928 0.487964 0.872864i \(-0.337740\pi\)
0.487964 + 0.872864i \(0.337740\pi\)
\(884\) 6.00000 0.201802
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) 3.00000 0.100730 0.0503651 0.998731i \(-0.483962\pi\)
0.0503651 + 0.998731i \(0.483962\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 0 0
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 26.0000 0.870544
\(893\) −12.0000 −0.401565
\(894\) −6.00000 −0.200670
\(895\) 0 0
\(896\) 0 0
\(897\) 6.00000 0.200334
\(898\) 18.0000 0.600668
\(899\) −36.0000 −1.20067
\(900\) 10.0000 0.333333
\(901\) 0 0
\(902\) −6.00000 −0.199778
\(903\) 0 0
\(904\) −9.00000 −0.299336
\(905\) 0 0
\(906\) 19.0000 0.631233
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) 18.0000 0.597351
\(909\) 30.0000 0.995037
\(910\) 0 0
\(911\) −6.00000 −0.198789 −0.0993944 0.995048i \(-0.531691\pi\)
−0.0993944 + 0.995048i \(0.531691\pi\)
\(912\) 2.00000 0.0662266
\(913\) 6.00000 0.198571
\(914\) 22.0000 0.727695
\(915\) 0 0
\(916\) −16.0000 −0.528655
\(917\) 0 0
\(918\) −30.0000 −0.990148
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 0 0
\(921\) 20.0000 0.659022
\(922\) −21.0000 −0.691598
\(923\) 0 0
\(924\) 0 0
\(925\) −10.0000 −0.328798
\(926\) 22.0000 0.722965
\(927\) 32.0000 1.05102
\(928\) −9.00000 −0.295439
\(929\) 39.0000 1.27955 0.639774 0.768563i \(-0.279028\pi\)
0.639774 + 0.768563i \(0.279028\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −6.00000 −0.196537
\(933\) 24.0000 0.785725
\(934\) −12.0000 −0.392652
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) 8.00000 0.261349 0.130674 0.991425i \(-0.458286\pi\)
0.130674 + 0.991425i \(0.458286\pi\)
\(938\) 0 0
\(939\) 17.0000 0.554774
\(940\) 0 0
\(941\) 15.0000 0.488986 0.244493 0.969651i \(-0.421378\pi\)
0.244493 + 0.969651i \(0.421378\pi\)
\(942\) 4.00000 0.130327
\(943\) 36.0000 1.17232
\(944\) −3.00000 −0.0976417
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) 24.0000 0.779895 0.389948 0.920837i \(-0.372493\pi\)
0.389948 + 0.920837i \(0.372493\pi\)
\(948\) 5.00000 0.162392
\(949\) −2.00000 −0.0649227
\(950\) 10.0000 0.324443
\(951\) 12.0000 0.389127
\(952\) 0 0
\(953\) −36.0000 −1.16615 −0.583077 0.812417i \(-0.698151\pi\)
−0.583077 + 0.812417i \(0.698151\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −9.00000 −0.291081
\(957\) −9.00000 −0.290929
\(958\) −15.0000 −0.484628
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 2.00000 0.0644826
\(963\) −24.0000 −0.773389
\(964\) 26.0000 0.837404
\(965\) 0 0
\(966\) 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
\(969\) −12.0000 −0.385496
\(970\) 0 0
\(971\) 15.0000 0.481373 0.240686 0.970603i \(-0.422627\pi\)
0.240686 + 0.970603i \(0.422627\pi\)
\(972\) 16.0000 0.513200
\(973\) 0 0
\(974\) −20.0000 −0.640841
\(975\) 5.00000 0.160128
\(976\) 11.0000 0.352101
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) −17.0000 −0.543600
\(979\) 18.0000 0.575282
\(980\) 0 0
\(981\) −4.00000 −0.127710
\(982\) −42.0000 −1.34027
\(983\) −6.00000 −0.191370 −0.0956851 0.995412i \(-0.530504\pi\)
−0.0956851 + 0.995412i \(0.530504\pi\)
\(984\) 6.00000 0.191273
\(985\) 0 0
\(986\) 54.0000 1.71971
\(987\) 0 0
\(988\) −2.00000 −0.0636285
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) 4.00000 0.127000
\(993\) 35.0000 1.11069
\(994\) 0 0
\(995\) 0 0
\(996\) −6.00000 −0.190117
\(997\) −34.0000 −1.07679 −0.538395 0.842692i \(-0.680969\pi\)
−0.538395 + 0.842692i \(0.680969\pi\)
\(998\) 40.0000 1.26618
\(999\) −10.0000 −0.316386
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 1078.2.a.e.1.1 1
3.2 odd 2 9702.2.a.br.1.1 1
4.3 odd 2 8624.2.a.k.1.1 1
7.2 even 3 154.2.e.c.67.1 yes 2
7.3 odd 6 1078.2.e.k.177.1 2
7.4 even 3 154.2.e.c.23.1 2
7.5 odd 6 1078.2.e.k.67.1 2
7.6 odd 2 1078.2.a.c.1.1 1
21.2 odd 6 1386.2.k.e.991.1 2
21.11 odd 6 1386.2.k.e.793.1 2
21.20 even 2 9702.2.a.bs.1.1 1
28.11 odd 6 1232.2.q.d.177.1 2
28.23 odd 6 1232.2.q.d.529.1 2
28.27 even 2 8624.2.a.u.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.e.c.23.1 2 7.4 even 3
154.2.e.c.67.1 yes 2 7.2 even 3
1078.2.a.c.1.1 1 7.6 odd 2
1078.2.a.e.1.1 1 1.1 even 1 trivial
1078.2.e.k.67.1 2 7.5 odd 6
1078.2.e.k.177.1 2 7.3 odd 6
1232.2.q.d.177.1 2 28.11 odd 6
1232.2.q.d.529.1 2 28.23 odd 6
1386.2.k.e.793.1 2 21.11 odd 6
1386.2.k.e.991.1 2 21.2 odd 6
8624.2.a.k.1.1 1 4.3 odd 2
8624.2.a.u.1.1 1 28.27 even 2
9702.2.a.br.1.1 1 3.2 odd 2
9702.2.a.bs.1.1 1 21.20 even 2