Properties

Label 7623.2.a.ct.1.3
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 1
Dimension 8
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 7623.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.8699614608\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.11447\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.11447 q^{2} -0.757964 q^{4} -3.45608 q^{5} +1.00000 q^{7} +3.07366 q^{8} +O(q^{10})\) \(q-1.11447 q^{2} -0.757964 q^{4} -3.45608 q^{5} +1.00000 q^{7} +3.07366 q^{8} +3.85168 q^{10} -2.05965 q^{13} -1.11447 q^{14} -1.90956 q^{16} -1.93373 q^{17} +1.62296 q^{19} +2.61958 q^{20} +0.807136 q^{23} +6.94447 q^{25} +2.29541 q^{26} -0.757964 q^{28} -7.97368 q^{29} +0.788420 q^{31} -4.01918 q^{32} +2.15508 q^{34} -3.45608 q^{35} +10.0618 q^{37} -1.80873 q^{38} -10.6228 q^{40} +2.12613 q^{41} +3.08043 q^{43} -0.899526 q^{46} -7.56632 q^{47} +1.00000 q^{49} -7.73938 q^{50} +1.56114 q^{52} -10.8224 q^{53} +3.07366 q^{56} +8.88640 q^{58} +3.29664 q^{59} -1.07663 q^{61} -0.878667 q^{62} +8.29836 q^{64} +7.11832 q^{65} +2.40314 q^{67} +1.46570 q^{68} +3.85168 q^{70} +3.18859 q^{71} +1.22628 q^{73} -11.2135 q^{74} -1.23015 q^{76} -9.48182 q^{79} +6.59959 q^{80} -2.36950 q^{82} +16.0694 q^{83} +6.68312 q^{85} -3.43303 q^{86} +4.43830 q^{89} -2.05965 q^{91} -0.611780 q^{92} +8.43241 q^{94} -5.60908 q^{95} +6.46807 q^{97} -1.11447 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - q^{2} + 7q^{4} - 10q^{5} + 8q^{7} + O(q^{10}) \) \( 8q - q^{2} + 7q^{4} - 10q^{5} + 8q^{7} + 6q^{10} - 6q^{13} - q^{14} + q^{16} + 5q^{17} - 13q^{19} - 23q^{20} - 16q^{23} + 16q^{25} + 6q^{26} + 7q^{28} - 9q^{29} + 9q^{31} - 16q^{32} - 12q^{34} - 10q^{35} + 7q^{37} + 10q^{38} + 5q^{40} + 10q^{41} - 4q^{43} + 4q^{46} - 16q^{47} + 8q^{49} - 6q^{50} - 41q^{52} - 37q^{53} - 15q^{58} - q^{59} + 19q^{61} + 18q^{62} - 4q^{64} + 4q^{65} - 19q^{67} - 9q^{68} + 6q^{70} - 13q^{71} - 25q^{73} - 33q^{74} + 26q^{76} - 4q^{80} - 13q^{82} + 25q^{83} + 3q^{85} - 4q^{86} - 37q^{89} - 6q^{91} - 35q^{92} - 42q^{94} - 21q^{95} + 15q^{97} - q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.11447 −0.788047 −0.394023 0.919100i \(-0.628917\pi\)
−0.394023 + 0.919100i \(0.628917\pi\)
\(3\) 0 0
\(4\) −0.757964 −0.378982
\(5\) −3.45608 −1.54561 −0.772803 0.634647i \(-0.781146\pi\)
−0.772803 + 0.634647i \(0.781146\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 3.07366 1.08670
\(9\) 0 0
\(10\) 3.85168 1.21801
\(11\) 0 0
\(12\) 0 0
\(13\) −2.05965 −0.571245 −0.285622 0.958342i \(-0.592200\pi\)
−0.285622 + 0.958342i \(0.592200\pi\)
\(14\) −1.11447 −0.297854
\(15\) 0 0
\(16\) −1.90956 −0.477390
\(17\) −1.93373 −0.468998 −0.234499 0.972116i \(-0.575345\pi\)
−0.234499 + 0.972116i \(0.575345\pi\)
\(18\) 0 0
\(19\) 1.62296 0.372333 0.186166 0.982518i \(-0.440394\pi\)
0.186166 + 0.982518i \(0.440394\pi\)
\(20\) 2.61958 0.585757
\(21\) 0 0
\(22\) 0 0
\(23\) 0.807136 0.168299 0.0841497 0.996453i \(-0.473183\pi\)
0.0841497 + 0.996453i \(0.473183\pi\)
\(24\) 0 0
\(25\) 6.94447 1.38889
\(26\) 2.29541 0.450168
\(27\) 0 0
\(28\) −0.757964 −0.143242
\(29\) −7.97368 −1.48067 −0.740337 0.672235i \(-0.765334\pi\)
−0.740337 + 0.672235i \(0.765334\pi\)
\(30\) 0 0
\(31\) 0.788420 0.141604 0.0708022 0.997490i \(-0.477444\pi\)
0.0708022 + 0.997490i \(0.477444\pi\)
\(32\) −4.01918 −0.710497
\(33\) 0 0
\(34\) 2.15508 0.369592
\(35\) −3.45608 −0.584184
\(36\) 0 0
\(37\) 10.0618 1.65414 0.827072 0.562096i \(-0.190005\pi\)
0.827072 + 0.562096i \(0.190005\pi\)
\(38\) −1.80873 −0.293415
\(39\) 0 0
\(40\) −10.6228 −1.67961
\(41\) 2.12613 0.332046 0.166023 0.986122i \(-0.446907\pi\)
0.166023 + 0.986122i \(0.446907\pi\)
\(42\) 0 0
\(43\) 3.08043 0.469761 0.234880 0.972024i \(-0.424530\pi\)
0.234880 + 0.972024i \(0.424530\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −0.899526 −0.132628
\(47\) −7.56632 −1.10366 −0.551831 0.833956i \(-0.686070\pi\)
−0.551831 + 0.833956i \(0.686070\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −7.73938 −1.09451
\(51\) 0 0
\(52\) 1.56114 0.216492
\(53\) −10.8224 −1.48658 −0.743289 0.668971i \(-0.766735\pi\)
−0.743289 + 0.668971i \(0.766735\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 3.07366 0.410735
\(57\) 0 0
\(58\) 8.88640 1.16684
\(59\) 3.29664 0.429186 0.214593 0.976704i \(-0.431158\pi\)
0.214593 + 0.976704i \(0.431158\pi\)
\(60\) 0 0
\(61\) −1.07663 −0.137848 −0.0689240 0.997622i \(-0.521957\pi\)
−0.0689240 + 0.997622i \(0.521957\pi\)
\(62\) −0.878667 −0.111591
\(63\) 0 0
\(64\) 8.29836 1.03729
\(65\) 7.11832 0.882919
\(66\) 0 0
\(67\) 2.40314 0.293590 0.146795 0.989167i \(-0.453104\pi\)
0.146795 + 0.989167i \(0.453104\pi\)
\(68\) 1.46570 0.177742
\(69\) 0 0
\(70\) 3.85168 0.460364
\(71\) 3.18859 0.378417 0.189208 0.981937i \(-0.439408\pi\)
0.189208 + 0.981937i \(0.439408\pi\)
\(72\) 0 0
\(73\) 1.22628 0.143525 0.0717624 0.997422i \(-0.477138\pi\)
0.0717624 + 0.997422i \(0.477138\pi\)
\(74\) −11.2135 −1.30354
\(75\) 0 0
\(76\) −1.23015 −0.141107
\(77\) 0 0
\(78\) 0 0
\(79\) −9.48182 −1.06679 −0.533394 0.845867i \(-0.679084\pi\)
−0.533394 + 0.845867i \(0.679084\pi\)
\(80\) 6.59959 0.737857
\(81\) 0 0
\(82\) −2.36950 −0.261668
\(83\) 16.0694 1.76385 0.881923 0.471394i \(-0.156249\pi\)
0.881923 + 0.471394i \(0.156249\pi\)
\(84\) 0 0
\(85\) 6.68312 0.724886
\(86\) −3.43303 −0.370193
\(87\) 0 0
\(88\) 0 0
\(89\) 4.43830 0.470459 0.235230 0.971940i \(-0.424416\pi\)
0.235230 + 0.971940i \(0.424416\pi\)
\(90\) 0 0
\(91\) −2.05965 −0.215910
\(92\) −0.611780 −0.0637825
\(93\) 0 0
\(94\) 8.43241 0.869737
\(95\) −5.60908 −0.575479
\(96\) 0 0
\(97\) 6.46807 0.656733 0.328366 0.944550i \(-0.393502\pi\)
0.328366 + 0.944550i \(0.393502\pi\)
\(98\) −1.11447 −0.112578
\(99\) 0 0
\(100\) −5.26366 −0.526366
\(101\) 15.4449 1.53682 0.768412 0.639955i \(-0.221047\pi\)
0.768412 + 0.639955i \(0.221047\pi\)
\(102\) 0 0
\(103\) 8.90812 0.877743 0.438872 0.898550i \(-0.355378\pi\)
0.438872 + 0.898550i \(0.355378\pi\)
\(104\) −6.33067 −0.620773
\(105\) 0 0
\(106\) 12.0613 1.17149
\(107\) −3.51219 −0.339536 −0.169768 0.985484i \(-0.554302\pi\)
−0.169768 + 0.985484i \(0.554302\pi\)
\(108\) 0 0
\(109\) 3.87655 0.371306 0.185653 0.982615i \(-0.440560\pi\)
0.185653 + 0.982615i \(0.440560\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.90956 −0.180437
\(113\) −10.6539 −1.00224 −0.501118 0.865379i \(-0.667078\pi\)
−0.501118 + 0.865379i \(0.667078\pi\)
\(114\) 0 0
\(115\) −2.78952 −0.260124
\(116\) 6.04376 0.561149
\(117\) 0 0
\(118\) −3.67399 −0.338218
\(119\) −1.93373 −0.177265
\(120\) 0 0
\(121\) 0 0
\(122\) 1.19987 0.108631
\(123\) 0 0
\(124\) −0.597594 −0.0536655
\(125\) −6.72025 −0.601078
\(126\) 0 0
\(127\) 19.4509 1.72599 0.862994 0.505214i \(-0.168586\pi\)
0.862994 + 0.505214i \(0.168586\pi\)
\(128\) −1.20989 −0.106941
\(129\) 0 0
\(130\) −7.93313 −0.695782
\(131\) −5.11284 −0.446711 −0.223355 0.974737i \(-0.571701\pi\)
−0.223355 + 0.974737i \(0.571701\pi\)
\(132\) 0 0
\(133\) 1.62296 0.140728
\(134\) −2.67822 −0.231363
\(135\) 0 0
\(136\) −5.94362 −0.509661
\(137\) −9.10052 −0.777510 −0.388755 0.921341i \(-0.627095\pi\)
−0.388755 + 0.921341i \(0.627095\pi\)
\(138\) 0 0
\(139\) −13.0166 −1.10405 −0.552026 0.833827i \(-0.686145\pi\)
−0.552026 + 0.833827i \(0.686145\pi\)
\(140\) 2.61958 0.221395
\(141\) 0 0
\(142\) −3.55358 −0.298210
\(143\) 0 0
\(144\) 0 0
\(145\) 27.5577 2.28854
\(146\) −1.36664 −0.113104
\(147\) 0 0
\(148\) −7.62646 −0.626891
\(149\) 3.14650 0.257771 0.128886 0.991659i \(-0.458860\pi\)
0.128886 + 0.991659i \(0.458860\pi\)
\(150\) 0 0
\(151\) 2.86696 0.233310 0.116655 0.993172i \(-0.462783\pi\)
0.116655 + 0.993172i \(0.462783\pi\)
\(152\) 4.98843 0.404615
\(153\) 0 0
\(154\) 0 0
\(155\) −2.72484 −0.218864
\(156\) 0 0
\(157\) −21.4895 −1.71505 −0.857524 0.514443i \(-0.827999\pi\)
−0.857524 + 0.514443i \(0.827999\pi\)
\(158\) 10.5672 0.840679
\(159\) 0 0
\(160\) 13.8906 1.09815
\(161\) 0.807136 0.0636112
\(162\) 0 0
\(163\) −8.22245 −0.644032 −0.322016 0.946734i \(-0.604361\pi\)
−0.322016 + 0.946734i \(0.604361\pi\)
\(164\) −1.61153 −0.125840
\(165\) 0 0
\(166\) −17.9088 −1.38999
\(167\) 21.7086 1.67986 0.839930 0.542695i \(-0.182596\pi\)
0.839930 + 0.542695i \(0.182596\pi\)
\(168\) 0 0
\(169\) −8.75783 −0.673679
\(170\) −7.44811 −0.571244
\(171\) 0 0
\(172\) −2.33485 −0.178031
\(173\) 8.04563 0.611698 0.305849 0.952080i \(-0.401060\pi\)
0.305849 + 0.952080i \(0.401060\pi\)
\(174\) 0 0
\(175\) 6.94447 0.524953
\(176\) 0 0
\(177\) 0 0
\(178\) −4.94634 −0.370744
\(179\) 3.62091 0.270640 0.135320 0.990802i \(-0.456794\pi\)
0.135320 + 0.990802i \(0.456794\pi\)
\(180\) 0 0
\(181\) 15.8179 1.17574 0.587868 0.808957i \(-0.299967\pi\)
0.587868 + 0.808957i \(0.299967\pi\)
\(182\) 2.29541 0.170147
\(183\) 0 0
\(184\) 2.48086 0.182891
\(185\) −34.7743 −2.55665
\(186\) 0 0
\(187\) 0 0
\(188\) 5.73500 0.418268
\(189\) 0 0
\(190\) 6.25113 0.453504
\(191\) −0.429081 −0.0310472 −0.0155236 0.999880i \(-0.504942\pi\)
−0.0155236 + 0.999880i \(0.504942\pi\)
\(192\) 0 0
\(193\) −15.1748 −1.09231 −0.546154 0.837685i \(-0.683909\pi\)
−0.546154 + 0.837685i \(0.683909\pi\)
\(194\) −7.20845 −0.517536
\(195\) 0 0
\(196\) −0.757964 −0.0541403
\(197\) 20.8082 1.48252 0.741262 0.671216i \(-0.234228\pi\)
0.741262 + 0.671216i \(0.234228\pi\)
\(198\) 0 0
\(199\) 8.44567 0.598698 0.299349 0.954144i \(-0.403231\pi\)
0.299349 + 0.954144i \(0.403231\pi\)
\(200\) 21.3449 1.50932
\(201\) 0 0
\(202\) −17.2128 −1.21109
\(203\) −7.97368 −0.559642
\(204\) 0 0
\(205\) −7.34808 −0.513212
\(206\) −9.92781 −0.691703
\(207\) 0 0
\(208\) 3.93303 0.272707
\(209\) 0 0
\(210\) 0 0
\(211\) −9.85927 −0.678740 −0.339370 0.940653i \(-0.610214\pi\)
−0.339370 + 0.940653i \(0.610214\pi\)
\(212\) 8.20303 0.563386
\(213\) 0 0
\(214\) 3.91422 0.267570
\(215\) −10.6462 −0.726065
\(216\) 0 0
\(217\) 0.788420 0.0535214
\(218\) −4.32028 −0.292606
\(219\) 0 0
\(220\) 0 0
\(221\) 3.98281 0.267913
\(222\) 0 0
\(223\) 17.3959 1.16491 0.582457 0.812861i \(-0.302091\pi\)
0.582457 + 0.812861i \(0.302091\pi\)
\(224\) −4.01918 −0.268542
\(225\) 0 0
\(226\) 11.8734 0.789809
\(227\) 12.6315 0.838381 0.419190 0.907898i \(-0.362314\pi\)
0.419190 + 0.907898i \(0.362314\pi\)
\(228\) 0 0
\(229\) 4.57341 0.302220 0.151110 0.988517i \(-0.451715\pi\)
0.151110 + 0.988517i \(0.451715\pi\)
\(230\) 3.10883 0.204990
\(231\) 0 0
\(232\) −24.5084 −1.60905
\(233\) −23.8569 −1.56292 −0.781458 0.623957i \(-0.785524\pi\)
−0.781458 + 0.623957i \(0.785524\pi\)
\(234\) 0 0
\(235\) 26.1498 1.70582
\(236\) −2.49873 −0.162654
\(237\) 0 0
\(238\) 2.15508 0.139693
\(239\) −8.83814 −0.571692 −0.285846 0.958276i \(-0.592275\pi\)
−0.285846 + 0.958276i \(0.592275\pi\)
\(240\) 0 0
\(241\) 18.9464 1.22045 0.610224 0.792229i \(-0.291079\pi\)
0.610224 + 0.792229i \(0.291079\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0.816045 0.0522419
\(245\) −3.45608 −0.220801
\(246\) 0 0
\(247\) −3.34273 −0.212693
\(248\) 2.42333 0.153882
\(249\) 0 0
\(250\) 7.48950 0.473678
\(251\) 2.86691 0.180958 0.0904790 0.995898i \(-0.471160\pi\)
0.0904790 + 0.995898i \(0.471160\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −21.6774 −1.36016
\(255\) 0 0
\(256\) −15.2483 −0.953021
\(257\) −22.4159 −1.39826 −0.699132 0.714993i \(-0.746430\pi\)
−0.699132 + 0.714993i \(0.746430\pi\)
\(258\) 0 0
\(259\) 10.0618 0.625208
\(260\) −5.39543 −0.334611
\(261\) 0 0
\(262\) 5.69808 0.352029
\(263\) −0.990706 −0.0610895 −0.0305448 0.999533i \(-0.509724\pi\)
−0.0305448 + 0.999533i \(0.509724\pi\)
\(264\) 0 0
\(265\) 37.4032 2.29766
\(266\) −1.80873 −0.110901
\(267\) 0 0
\(268\) −1.82149 −0.111265
\(269\) 7.19036 0.438404 0.219202 0.975679i \(-0.429655\pi\)
0.219202 + 0.975679i \(0.429655\pi\)
\(270\) 0 0
\(271\) −27.1643 −1.65011 −0.825056 0.565050i \(-0.808857\pi\)
−0.825056 + 0.565050i \(0.808857\pi\)
\(272\) 3.69257 0.223895
\(273\) 0 0
\(274\) 10.1422 0.612714
\(275\) 0 0
\(276\) 0 0
\(277\) 20.9856 1.26090 0.630451 0.776229i \(-0.282870\pi\)
0.630451 + 0.776229i \(0.282870\pi\)
\(278\) 14.5065 0.870045
\(279\) 0 0
\(280\) −10.6228 −0.634834
\(281\) 28.1580 1.67977 0.839883 0.542768i \(-0.182624\pi\)
0.839883 + 0.542768i \(0.182624\pi\)
\(282\) 0 0
\(283\) −26.1917 −1.55694 −0.778469 0.627684i \(-0.784003\pi\)
−0.778469 + 0.627684i \(0.784003\pi\)
\(284\) −2.41684 −0.143413
\(285\) 0 0
\(286\) 0 0
\(287\) 2.12613 0.125502
\(288\) 0 0
\(289\) −13.2607 −0.780041
\(290\) −30.7121 −1.80348
\(291\) 0 0
\(292\) −0.929473 −0.0543933
\(293\) 4.46385 0.260781 0.130391 0.991463i \(-0.458377\pi\)
0.130391 + 0.991463i \(0.458377\pi\)
\(294\) 0 0
\(295\) −11.3934 −0.663351
\(296\) 30.9264 1.79756
\(297\) 0 0
\(298\) −3.50667 −0.203136
\(299\) −1.66242 −0.0961402
\(300\) 0 0
\(301\) 3.08043 0.177553
\(302\) −3.19513 −0.183859
\(303\) 0 0
\(304\) −3.09914 −0.177748
\(305\) 3.72091 0.213059
\(306\) 0 0
\(307\) 12.8841 0.735334 0.367667 0.929957i \(-0.380157\pi\)
0.367667 + 0.929957i \(0.380157\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 3.03674 0.172475
\(311\) −26.8199 −1.52081 −0.760407 0.649447i \(-0.775001\pi\)
−0.760407 + 0.649447i \(0.775001\pi\)
\(312\) 0 0
\(313\) 3.58869 0.202845 0.101422 0.994843i \(-0.467661\pi\)
0.101422 + 0.994843i \(0.467661\pi\)
\(314\) 23.9493 1.35154
\(315\) 0 0
\(316\) 7.18688 0.404294
\(317\) 16.9181 0.950213 0.475106 0.879928i \(-0.342410\pi\)
0.475106 + 0.879928i \(0.342410\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −28.6798 −1.60325
\(321\) 0 0
\(322\) −0.899526 −0.0501286
\(323\) −3.13836 −0.174623
\(324\) 0 0
\(325\) −14.3032 −0.793399
\(326\) 9.16365 0.507528
\(327\) 0 0
\(328\) 6.53501 0.360836
\(329\) −7.56632 −0.417145
\(330\) 0 0
\(331\) −1.23826 −0.0680610 −0.0340305 0.999421i \(-0.510834\pi\)
−0.0340305 + 0.999421i \(0.510834\pi\)
\(332\) −12.1800 −0.668466
\(333\) 0 0
\(334\) −24.1935 −1.32381
\(335\) −8.30544 −0.453774
\(336\) 0 0
\(337\) −20.4806 −1.11565 −0.557824 0.829959i \(-0.688364\pi\)
−0.557824 + 0.829959i \(0.688364\pi\)
\(338\) 9.76031 0.530891
\(339\) 0 0
\(340\) −5.06556 −0.274719
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 9.46818 0.510490
\(345\) 0 0
\(346\) −8.96659 −0.482047
\(347\) −27.2699 −1.46392 −0.731961 0.681346i \(-0.761395\pi\)
−0.731961 + 0.681346i \(0.761395\pi\)
\(348\) 0 0
\(349\) 7.98434 0.427392 0.213696 0.976900i \(-0.431450\pi\)
0.213696 + 0.976900i \(0.431450\pi\)
\(350\) −7.73938 −0.413688
\(351\) 0 0
\(352\) 0 0
\(353\) −5.93472 −0.315873 −0.157937 0.987449i \(-0.550484\pi\)
−0.157937 + 0.987449i \(0.550484\pi\)
\(354\) 0 0
\(355\) −11.0200 −0.584883
\(356\) −3.36407 −0.178296
\(357\) 0 0
\(358\) −4.03538 −0.213277
\(359\) −28.4203 −1.49996 −0.749982 0.661458i \(-0.769938\pi\)
−0.749982 + 0.661458i \(0.769938\pi\)
\(360\) 0 0
\(361\) −16.3660 −0.861368
\(362\) −17.6285 −0.926535
\(363\) 0 0
\(364\) 1.56114 0.0818261
\(365\) −4.23810 −0.221832
\(366\) 0 0
\(367\) −30.4014 −1.58694 −0.793471 0.608608i \(-0.791728\pi\)
−0.793471 + 0.608608i \(0.791728\pi\)
\(368\) −1.54128 −0.0803445
\(369\) 0 0
\(370\) 38.7547 2.01476
\(371\) −10.8224 −0.561873
\(372\) 0 0
\(373\) 14.4226 0.746772 0.373386 0.927676i \(-0.378197\pi\)
0.373386 + 0.927676i \(0.378197\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −23.2563 −1.19935
\(377\) 16.4230 0.845828
\(378\) 0 0
\(379\) −22.4072 −1.15098 −0.575490 0.817809i \(-0.695189\pi\)
−0.575490 + 0.817809i \(0.695189\pi\)
\(380\) 4.25148 0.218096
\(381\) 0 0
\(382\) 0.478196 0.0244667
\(383\) −33.6785 −1.72089 −0.860446 0.509541i \(-0.829815\pi\)
−0.860446 + 0.509541i \(0.829815\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 16.9118 0.860790
\(387\) 0 0
\(388\) −4.90256 −0.248890
\(389\) −2.42783 −0.123096 −0.0615479 0.998104i \(-0.519604\pi\)
−0.0615479 + 0.998104i \(0.519604\pi\)
\(390\) 0 0
\(391\) −1.56078 −0.0789321
\(392\) 3.07366 0.155243
\(393\) 0 0
\(394\) −23.1900 −1.16830
\(395\) 32.7699 1.64883
\(396\) 0 0
\(397\) −5.89696 −0.295960 −0.147980 0.988990i \(-0.547277\pi\)
−0.147980 + 0.988990i \(0.547277\pi\)
\(398\) −9.41242 −0.471802
\(399\) 0 0
\(400\) −13.2609 −0.663045
\(401\) −11.2396 −0.561278 −0.280639 0.959813i \(-0.590546\pi\)
−0.280639 + 0.959813i \(0.590546\pi\)
\(402\) 0 0
\(403\) −1.62387 −0.0808908
\(404\) −11.7067 −0.582429
\(405\) 0 0
\(406\) 8.88640 0.441025
\(407\) 0 0
\(408\) 0 0
\(409\) 29.3344 1.45049 0.725246 0.688490i \(-0.241726\pi\)
0.725246 + 0.688490i \(0.241726\pi\)
\(410\) 8.18919 0.404435
\(411\) 0 0
\(412\) −6.75204 −0.332649
\(413\) 3.29664 0.162217
\(414\) 0 0
\(415\) −55.5371 −2.72621
\(416\) 8.27811 0.405868
\(417\) 0 0
\(418\) 0 0
\(419\) 20.2858 0.991027 0.495514 0.868600i \(-0.334980\pi\)
0.495514 + 0.868600i \(0.334980\pi\)
\(420\) 0 0
\(421\) −3.06003 −0.149137 −0.0745683 0.997216i \(-0.523758\pi\)
−0.0745683 + 0.997216i \(0.523758\pi\)
\(422\) 10.9878 0.534879
\(423\) 0 0
\(424\) −33.2645 −1.61547
\(425\) −13.4287 −0.651389
\(426\) 0 0
\(427\) −1.07663 −0.0521017
\(428\) 2.66211 0.128678
\(429\) 0 0
\(430\) 11.8648 0.572173
\(431\) −7.54197 −0.363284 −0.181642 0.983365i \(-0.558141\pi\)
−0.181642 + 0.983365i \(0.558141\pi\)
\(432\) 0 0
\(433\) −32.1616 −1.54559 −0.772794 0.634657i \(-0.781141\pi\)
−0.772794 + 0.634657i \(0.781141\pi\)
\(434\) −0.878667 −0.0421774
\(435\) 0 0
\(436\) −2.93828 −0.140718
\(437\) 1.30995 0.0626633
\(438\) 0 0
\(439\) −4.66725 −0.222756 −0.111378 0.993778i \(-0.535526\pi\)
−0.111378 + 0.993778i \(0.535526\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −4.43871 −0.211128
\(443\) −17.2772 −0.820862 −0.410431 0.911892i \(-0.634622\pi\)
−0.410431 + 0.911892i \(0.634622\pi\)
\(444\) 0 0
\(445\) −15.3391 −0.727144
\(446\) −19.3871 −0.918007
\(447\) 0 0
\(448\) 8.29836 0.392061
\(449\) 16.7401 0.790013 0.395007 0.918678i \(-0.370742\pi\)
0.395007 + 0.918678i \(0.370742\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 8.07529 0.379829
\(453\) 0 0
\(454\) −14.0774 −0.660683
\(455\) 7.11832 0.333712
\(456\) 0 0
\(457\) −21.6974 −1.01496 −0.507481 0.861663i \(-0.669423\pi\)
−0.507481 + 0.861663i \(0.669423\pi\)
\(458\) −5.09692 −0.238163
\(459\) 0 0
\(460\) 2.11436 0.0985825
\(461\) 6.07778 0.283070 0.141535 0.989933i \(-0.454796\pi\)
0.141535 + 0.989933i \(0.454796\pi\)
\(462\) 0 0
\(463\) −5.14719 −0.239210 −0.119605 0.992822i \(-0.538163\pi\)
−0.119605 + 0.992822i \(0.538163\pi\)
\(464\) 15.2262 0.706860
\(465\) 0 0
\(466\) 26.5877 1.23165
\(467\) 3.91927 0.181362 0.0906812 0.995880i \(-0.471096\pi\)
0.0906812 + 0.995880i \(0.471096\pi\)
\(468\) 0 0
\(469\) 2.40314 0.110967
\(470\) −29.1431 −1.34427
\(471\) 0 0
\(472\) 10.1327 0.466397
\(473\) 0 0
\(474\) 0 0
\(475\) 11.2706 0.517131
\(476\) 1.46570 0.0671801
\(477\) 0 0
\(478\) 9.84982 0.450520
\(479\) 15.1129 0.690527 0.345263 0.938506i \(-0.387790\pi\)
0.345263 + 0.938506i \(0.387790\pi\)
\(480\) 0 0
\(481\) −20.7237 −0.944922
\(482\) −21.1152 −0.961770
\(483\) 0 0
\(484\) 0 0
\(485\) −22.3541 −1.01505
\(486\) 0 0
\(487\) 25.0768 1.13634 0.568169 0.822912i \(-0.307652\pi\)
0.568169 + 0.822912i \(0.307652\pi\)
\(488\) −3.30919 −0.149800
\(489\) 0 0
\(490\) 3.85168 0.174001
\(491\) −37.2208 −1.67975 −0.839875 0.542780i \(-0.817372\pi\)
−0.839875 + 0.542780i \(0.817372\pi\)
\(492\) 0 0
\(493\) 15.4189 0.694434
\(494\) 3.72536 0.167612
\(495\) 0 0
\(496\) −1.50554 −0.0676006
\(497\) 3.18859 0.143028
\(498\) 0 0
\(499\) −31.8134 −1.42416 −0.712081 0.702098i \(-0.752247\pi\)
−0.712081 + 0.702098i \(0.752247\pi\)
\(500\) 5.09371 0.227798
\(501\) 0 0
\(502\) −3.19508 −0.142603
\(503\) 19.2058 0.856346 0.428173 0.903697i \(-0.359157\pi\)
0.428173 + 0.903697i \(0.359157\pi\)
\(504\) 0 0
\(505\) −53.3788 −2.37532
\(506\) 0 0
\(507\) 0 0
\(508\) −14.7431 −0.654119
\(509\) −2.51549 −0.111497 −0.0557485 0.998445i \(-0.517754\pi\)
−0.0557485 + 0.998445i \(0.517754\pi\)
\(510\) 0 0
\(511\) 1.22628 0.0542472
\(512\) 19.4135 0.857966
\(513\) 0 0
\(514\) 24.9817 1.10190
\(515\) −30.7872 −1.35664
\(516\) 0 0
\(517\) 0 0
\(518\) −11.2135 −0.492693
\(519\) 0 0
\(520\) 21.8793 0.959470
\(521\) −14.8968 −0.652639 −0.326320 0.945260i \(-0.605809\pi\)
−0.326320 + 0.945260i \(0.605809\pi\)
\(522\) 0 0
\(523\) −9.90502 −0.433116 −0.216558 0.976270i \(-0.569483\pi\)
−0.216558 + 0.976270i \(0.569483\pi\)
\(524\) 3.87535 0.169295
\(525\) 0 0
\(526\) 1.10411 0.0481414
\(527\) −1.52459 −0.0664122
\(528\) 0 0
\(529\) −22.3485 −0.971675
\(530\) −41.6846 −1.81066
\(531\) 0 0
\(532\) −1.23015 −0.0533336
\(533\) −4.37910 −0.189680
\(534\) 0 0
\(535\) 12.1384 0.524789
\(536\) 7.38643 0.319045
\(537\) 0 0
\(538\) −8.01342 −0.345483
\(539\) 0 0
\(540\) 0 0
\(541\) −22.0084 −0.946214 −0.473107 0.881005i \(-0.656868\pi\)
−0.473107 + 0.881005i \(0.656868\pi\)
\(542\) 30.2737 1.30037
\(543\) 0 0
\(544\) 7.77199 0.333222
\(545\) −13.3977 −0.573892
\(546\) 0 0
\(547\) −10.8643 −0.464523 −0.232261 0.972653i \(-0.574612\pi\)
−0.232261 + 0.972653i \(0.574612\pi\)
\(548\) 6.89787 0.294662
\(549\) 0 0
\(550\) 0 0
\(551\) −12.9410 −0.551303
\(552\) 0 0
\(553\) −9.48182 −0.403208
\(554\) −23.3878 −0.993650
\(555\) 0 0
\(556\) 9.86610 0.418416
\(557\) −33.6126 −1.42421 −0.712106 0.702072i \(-0.752258\pi\)
−0.712106 + 0.702072i \(0.752258\pi\)
\(558\) 0 0
\(559\) −6.34461 −0.268348
\(560\) 6.59959 0.278884
\(561\) 0 0
\(562\) −31.3811 −1.32373
\(563\) 2.05348 0.0865440 0.0432720 0.999063i \(-0.486222\pi\)
0.0432720 + 0.999063i \(0.486222\pi\)
\(564\) 0 0
\(565\) 36.8208 1.54906
\(566\) 29.1898 1.22694
\(567\) 0 0
\(568\) 9.80065 0.411226
\(569\) −3.27416 −0.137260 −0.0686300 0.997642i \(-0.521863\pi\)
−0.0686300 + 0.997642i \(0.521863\pi\)
\(570\) 0 0
\(571\) −43.8897 −1.83673 −0.918363 0.395738i \(-0.870489\pi\)
−0.918363 + 0.395738i \(0.870489\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −2.36950 −0.0989012
\(575\) 5.60513 0.233750
\(576\) 0 0
\(577\) −43.8904 −1.82718 −0.913591 0.406635i \(-0.866702\pi\)
−0.913591 + 0.406635i \(0.866702\pi\)
\(578\) 14.7786 0.614709
\(579\) 0 0
\(580\) −20.8877 −0.867315
\(581\) 16.0694 0.666671
\(582\) 0 0
\(583\) 0 0
\(584\) 3.76915 0.155969
\(585\) 0 0
\(586\) −4.97481 −0.205508
\(587\) −2.79166 −0.115224 −0.0576121 0.998339i \(-0.518349\pi\)
−0.0576121 + 0.998339i \(0.518349\pi\)
\(588\) 0 0
\(589\) 1.27957 0.0527239
\(590\) 12.6976 0.522752
\(591\) 0 0
\(592\) −19.2136 −0.789673
\(593\) −23.2526 −0.954871 −0.477435 0.878667i \(-0.658434\pi\)
−0.477435 + 0.878667i \(0.658434\pi\)
\(594\) 0 0
\(595\) 6.68312 0.273981
\(596\) −2.38493 −0.0976907
\(597\) 0 0
\(598\) 1.85271 0.0757630
\(599\) −10.4595 −0.427362 −0.213681 0.976904i \(-0.568545\pi\)
−0.213681 + 0.976904i \(0.568545\pi\)
\(600\) 0 0
\(601\) 22.3096 0.910029 0.455015 0.890484i \(-0.349634\pi\)
0.455015 + 0.890484i \(0.349634\pi\)
\(602\) −3.43303 −0.139920
\(603\) 0 0
\(604\) −2.17306 −0.0884204
\(605\) 0 0
\(606\) 0 0
\(607\) 18.9884 0.770716 0.385358 0.922767i \(-0.374078\pi\)
0.385358 + 0.922767i \(0.374078\pi\)
\(608\) −6.52296 −0.264541
\(609\) 0 0
\(610\) −4.14683 −0.167900
\(611\) 15.5840 0.630461
\(612\) 0 0
\(613\) 20.1617 0.814323 0.407161 0.913356i \(-0.366519\pi\)
0.407161 + 0.913356i \(0.366519\pi\)
\(614\) −14.3589 −0.579478
\(615\) 0 0
\(616\) 0 0
\(617\) −7.03919 −0.283387 −0.141694 0.989911i \(-0.545255\pi\)
−0.141694 + 0.989911i \(0.545255\pi\)
\(618\) 0 0
\(619\) −31.0831 −1.24933 −0.624667 0.780891i \(-0.714765\pi\)
−0.624667 + 0.780891i \(0.714765\pi\)
\(620\) 2.06533 0.0829457
\(621\) 0 0
\(622\) 29.8898 1.19847
\(623\) 4.43830 0.177817
\(624\) 0 0
\(625\) −11.4966 −0.459866
\(626\) −3.99947 −0.159851
\(627\) 0 0
\(628\) 16.2883 0.649973
\(629\) −19.4567 −0.775791
\(630\) 0 0
\(631\) 21.9720 0.874690 0.437345 0.899294i \(-0.355919\pi\)
0.437345 + 0.899294i \(0.355919\pi\)
\(632\) −29.1439 −1.15928
\(633\) 0 0
\(634\) −18.8546 −0.748812
\(635\) −67.2238 −2.66770
\(636\) 0 0
\(637\) −2.05965 −0.0816064
\(638\) 0 0
\(639\) 0 0
\(640\) 4.18149 0.165288
\(641\) 13.2909 0.524961 0.262480 0.964937i \(-0.415460\pi\)
0.262480 + 0.964937i \(0.415460\pi\)
\(642\) 0 0
\(643\) 14.6904 0.579333 0.289666 0.957128i \(-0.406456\pi\)
0.289666 + 0.957128i \(0.406456\pi\)
\(644\) −0.611780 −0.0241075
\(645\) 0 0
\(646\) 3.49760 0.137611
\(647\) 6.14523 0.241594 0.120797 0.992677i \(-0.461455\pi\)
0.120797 + 0.992677i \(0.461455\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 15.9404 0.625236
\(651\) 0 0
\(652\) 6.23232 0.244077
\(653\) 40.6691 1.59151 0.795753 0.605622i \(-0.207076\pi\)
0.795753 + 0.605622i \(0.207076\pi\)
\(654\) 0 0
\(655\) 17.6704 0.690438
\(656\) −4.05998 −0.158516
\(657\) 0 0
\(658\) 8.43241 0.328730
\(659\) −18.0090 −0.701531 −0.350765 0.936463i \(-0.614079\pi\)
−0.350765 + 0.936463i \(0.614079\pi\)
\(660\) 0 0
\(661\) −17.1420 −0.666745 −0.333373 0.942795i \(-0.608187\pi\)
−0.333373 + 0.942795i \(0.608187\pi\)
\(662\) 1.38000 0.0536353
\(663\) 0 0
\(664\) 49.3919 1.91678
\(665\) −5.60908 −0.217511
\(666\) 0 0
\(667\) −6.43584 −0.249197
\(668\) −16.4543 −0.636637
\(669\) 0 0
\(670\) 9.25613 0.357596
\(671\) 0 0
\(672\) 0 0
\(673\) −23.1926 −0.894008 −0.447004 0.894532i \(-0.647509\pi\)
−0.447004 + 0.894532i \(0.647509\pi\)
\(674\) 22.8249 0.879183
\(675\) 0 0
\(676\) 6.63812 0.255312
\(677\) 27.5705 1.05962 0.529811 0.848116i \(-0.322263\pi\)
0.529811 + 0.848116i \(0.322263\pi\)
\(678\) 0 0
\(679\) 6.46807 0.248222
\(680\) 20.5416 0.787735
\(681\) 0 0
\(682\) 0 0
\(683\) −21.9351 −0.839322 −0.419661 0.907681i \(-0.637851\pi\)
−0.419661 + 0.907681i \(0.637851\pi\)
\(684\) 0 0
\(685\) 31.4521 1.20172
\(686\) −1.11447 −0.0425505
\(687\) 0 0
\(688\) −5.88227 −0.224259
\(689\) 22.2905 0.849200
\(690\) 0 0
\(691\) −25.6666 −0.976404 −0.488202 0.872731i \(-0.662347\pi\)
−0.488202 + 0.872731i \(0.662347\pi\)
\(692\) −6.09830 −0.231823
\(693\) 0 0
\(694\) 30.3913 1.15364
\(695\) 44.9863 1.70643
\(696\) 0 0
\(697\) −4.11137 −0.155729
\(698\) −8.89828 −0.336805
\(699\) 0 0
\(700\) −5.26366 −0.198948
\(701\) 18.4135 0.695467 0.347733 0.937593i \(-0.386951\pi\)
0.347733 + 0.937593i \(0.386951\pi\)
\(702\) 0 0
\(703\) 16.3298 0.615892
\(704\) 0 0
\(705\) 0 0
\(706\) 6.61404 0.248923
\(707\) 15.4449 0.580865
\(708\) 0 0
\(709\) 23.6621 0.888647 0.444323 0.895866i \(-0.353444\pi\)
0.444323 + 0.895866i \(0.353444\pi\)
\(710\) 12.2815 0.460915
\(711\) 0 0
\(712\) 13.6418 0.511249
\(713\) 0.636362 0.0238319
\(714\) 0 0
\(715\) 0 0
\(716\) −2.74452 −0.102568
\(717\) 0 0
\(718\) 31.6734 1.18204
\(719\) −11.1222 −0.414790 −0.207395 0.978257i \(-0.566498\pi\)
−0.207395 + 0.978257i \(0.566498\pi\)
\(720\) 0 0
\(721\) 8.90812 0.331756
\(722\) 18.2394 0.678799
\(723\) 0 0
\(724\) −11.9894 −0.445583
\(725\) −55.3730 −2.05650
\(726\) 0 0
\(727\) 42.4803 1.57551 0.787753 0.615991i \(-0.211244\pi\)
0.787753 + 0.615991i \(0.211244\pi\)
\(728\) −6.33067 −0.234630
\(729\) 0 0
\(730\) 4.72323 0.174814
\(731\) −5.95671 −0.220317
\(732\) 0 0
\(733\) 22.1884 0.819548 0.409774 0.912187i \(-0.365607\pi\)
0.409774 + 0.912187i \(0.365607\pi\)
\(734\) 33.8814 1.25058
\(735\) 0 0
\(736\) −3.24402 −0.119576
\(737\) 0 0
\(738\) 0 0
\(739\) 29.4481 1.08327 0.541633 0.840615i \(-0.317806\pi\)
0.541633 + 0.840615i \(0.317806\pi\)
\(740\) 26.3576 0.968926
\(741\) 0 0
\(742\) 12.0613 0.442783
\(743\) −16.9059 −0.620217 −0.310109 0.950701i \(-0.600365\pi\)
−0.310109 + 0.950701i \(0.600365\pi\)
\(744\) 0 0
\(745\) −10.8745 −0.398412
\(746\) −16.0735 −0.588491
\(747\) 0 0
\(748\) 0 0
\(749\) −3.51219 −0.128333
\(750\) 0 0
\(751\) −1.55147 −0.0566138 −0.0283069 0.999599i \(-0.509012\pi\)
−0.0283069 + 0.999599i \(0.509012\pi\)
\(752\) 14.4484 0.526877
\(753\) 0 0
\(754\) −18.3029 −0.666552
\(755\) −9.90845 −0.360605
\(756\) 0 0
\(757\) 12.5467 0.456016 0.228008 0.973659i \(-0.426779\pi\)
0.228008 + 0.973659i \(0.426779\pi\)
\(758\) 24.9721 0.907027
\(759\) 0 0
\(760\) −17.2404 −0.625375
\(761\) 9.03436 0.327495 0.163748 0.986502i \(-0.447642\pi\)
0.163748 + 0.986502i \(0.447642\pi\)
\(762\) 0 0
\(763\) 3.87655 0.140340
\(764\) 0.325228 0.0117663
\(765\) 0 0
\(766\) 37.5336 1.35614
\(767\) −6.78993 −0.245170
\(768\) 0 0
\(769\) −16.1383 −0.581963 −0.290981 0.956729i \(-0.593982\pi\)
−0.290981 + 0.956729i \(0.593982\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 11.5020 0.413965
\(773\) 18.4134 0.662285 0.331143 0.943581i \(-0.392566\pi\)
0.331143 + 0.943581i \(0.392566\pi\)
\(774\) 0 0
\(775\) 5.47516 0.196674
\(776\) 19.8806 0.713673
\(777\) 0 0
\(778\) 2.70573 0.0970053
\(779\) 3.45063 0.123632
\(780\) 0 0
\(781\) 0 0
\(782\) 1.73944 0.0622022
\(783\) 0 0
\(784\) −1.90956 −0.0681986
\(785\) 74.2694 2.65079
\(786\) 0 0
\(787\) 46.9870 1.67491 0.837453 0.546509i \(-0.184043\pi\)
0.837453 + 0.546509i \(0.184043\pi\)
\(788\) −15.7719 −0.561850
\(789\) 0 0
\(790\) −36.5210 −1.29936
\(791\) −10.6539 −0.378810
\(792\) 0 0
\(793\) 2.21748 0.0787450
\(794\) 6.57197 0.233230
\(795\) 0 0
\(796\) −6.40152 −0.226896
\(797\) 3.12454 0.110677 0.0553385 0.998468i \(-0.482376\pi\)
0.0553385 + 0.998468i \(0.482376\pi\)
\(798\) 0 0
\(799\) 14.6312 0.517615
\(800\) −27.9111 −0.986805
\(801\) 0 0
\(802\) 12.5261 0.442314
\(803\) 0 0
\(804\) 0 0
\(805\) −2.78952 −0.0983178
\(806\) 1.80975 0.0637457
\(807\) 0 0
\(808\) 47.4724 1.67007
\(809\) 32.7257 1.15057 0.575286 0.817952i \(-0.304891\pi\)
0.575286 + 0.817952i \(0.304891\pi\)
\(810\) 0 0
\(811\) 32.6613 1.14689 0.573447 0.819243i \(-0.305606\pi\)
0.573447 + 0.819243i \(0.305606\pi\)
\(812\) 6.04376 0.212095
\(813\) 0 0
\(814\) 0 0
\(815\) 28.4174 0.995419
\(816\) 0 0
\(817\) 4.99941 0.174907
\(818\) −32.6922 −1.14306
\(819\) 0 0
\(820\) 5.56958 0.194498
\(821\) 7.43142 0.259358 0.129679 0.991556i \(-0.458605\pi\)
0.129679 + 0.991556i \(0.458605\pi\)
\(822\) 0 0
\(823\) 6.55866 0.228621 0.114310 0.993445i \(-0.463534\pi\)
0.114310 + 0.993445i \(0.463534\pi\)
\(824\) 27.3805 0.953846
\(825\) 0 0
\(826\) −3.67399 −0.127835
\(827\) −23.8538 −0.829479 −0.414739 0.909940i \(-0.636127\pi\)
−0.414739 + 0.909940i \(0.636127\pi\)
\(828\) 0 0
\(829\) −26.1431 −0.907989 −0.453994 0.891005i \(-0.650001\pi\)
−0.453994 + 0.891005i \(0.650001\pi\)
\(830\) 61.8942 2.14838
\(831\) 0 0
\(832\) −17.0917 −0.592549
\(833\) −1.93373 −0.0669997
\(834\) 0 0
\(835\) −75.0265 −2.59640
\(836\) 0 0
\(837\) 0 0
\(838\) −22.6079 −0.780976
\(839\) −34.2890 −1.18379 −0.591893 0.806016i \(-0.701619\pi\)
−0.591893 + 0.806016i \(0.701619\pi\)
\(840\) 0 0
\(841\) 34.5795 1.19240
\(842\) 3.41030 0.117527
\(843\) 0 0
\(844\) 7.47297 0.257230
\(845\) 30.2677 1.04124
\(846\) 0 0
\(847\) 0 0
\(848\) 20.6661 0.709678
\(849\) 0 0
\(850\) 14.9659 0.513325
\(851\) 8.12121 0.278392
\(852\) 0 0
\(853\) 21.3842 0.732181 0.366090 0.930579i \(-0.380696\pi\)
0.366090 + 0.930579i \(0.380696\pi\)
\(854\) 1.19987 0.0410585
\(855\) 0 0
\(856\) −10.7953 −0.368975
\(857\) −42.8697 −1.46440 −0.732200 0.681090i \(-0.761506\pi\)
−0.732200 + 0.681090i \(0.761506\pi\)
\(858\) 0 0
\(859\) −30.3915 −1.03695 −0.518473 0.855094i \(-0.673499\pi\)
−0.518473 + 0.855094i \(0.673499\pi\)
\(860\) 8.06944 0.275165
\(861\) 0 0
\(862\) 8.40527 0.286285
\(863\) −11.8184 −0.402304 −0.201152 0.979560i \(-0.564468\pi\)
−0.201152 + 0.979560i \(0.564468\pi\)
\(864\) 0 0
\(865\) −27.8063 −0.945444
\(866\) 35.8430 1.21800
\(867\) 0 0
\(868\) −0.597594 −0.0202837
\(869\) 0 0
\(870\) 0 0
\(871\) −4.94963 −0.167712
\(872\) 11.9152 0.403499
\(873\) 0 0
\(874\) −1.45989 −0.0493816
\(875\) −6.72025 −0.227186
\(876\) 0 0
\(877\) −9.20488 −0.310827 −0.155413 0.987850i \(-0.549671\pi\)
−0.155413 + 0.987850i \(0.549671\pi\)
\(878\) 5.20149 0.175542
\(879\) 0 0
\(880\) 0 0
\(881\) 41.9030 1.41175 0.705874 0.708338i \(-0.250555\pi\)
0.705874 + 0.708338i \(0.250555\pi\)
\(882\) 0 0
\(883\) −16.0478 −0.540053 −0.270026 0.962853i \(-0.587032\pi\)
−0.270026 + 0.962853i \(0.587032\pi\)
\(884\) −3.01883 −0.101534
\(885\) 0 0
\(886\) 19.2548 0.646878
\(887\) 33.8483 1.13652 0.568258 0.822850i \(-0.307618\pi\)
0.568258 + 0.822850i \(0.307618\pi\)
\(888\) 0 0
\(889\) 19.4509 0.652362
\(890\) 17.0949 0.573024
\(891\) 0 0
\(892\) −13.1855 −0.441482
\(893\) −12.2798 −0.410929
\(894\) 0 0
\(895\) −12.5141 −0.418302
\(896\) −1.20989 −0.0404197
\(897\) 0 0
\(898\) −18.6563 −0.622568
\(899\) −6.28660 −0.209670
\(900\) 0 0
\(901\) 20.9277 0.697202
\(902\) 0 0
\(903\) 0 0
\(904\) −32.7465 −1.08913
\(905\) −54.6679 −1.81722
\(906\) 0 0
\(907\) −44.1013 −1.46436 −0.732181 0.681111i \(-0.761497\pi\)
−0.732181 + 0.681111i \(0.761497\pi\)
\(908\) −9.57421 −0.317731
\(909\) 0 0
\(910\) −7.93313 −0.262981
\(911\) −49.5756 −1.64251 −0.821257 0.570558i \(-0.806727\pi\)
−0.821257 + 0.570558i \(0.806727\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 24.1810 0.799837
\(915\) 0 0
\(916\) −3.46648 −0.114536
\(917\) −5.11284 −0.168841
\(918\) 0 0
\(919\) −40.7926 −1.34563 −0.672813 0.739813i \(-0.734914\pi\)
−0.672813 + 0.739813i \(0.734914\pi\)
\(920\) −8.57404 −0.282678
\(921\) 0 0
\(922\) −6.77348 −0.223073
\(923\) −6.56740 −0.216169
\(924\) 0 0
\(925\) 69.8737 2.29743
\(926\) 5.73637 0.188509
\(927\) 0 0
\(928\) 32.0476 1.05201
\(929\) −41.3929 −1.35806 −0.679029 0.734112i \(-0.737599\pi\)
−0.679029 + 0.734112i \(0.737599\pi\)
\(930\) 0 0
\(931\) 1.62296 0.0531904
\(932\) 18.0827 0.592318
\(933\) 0 0
\(934\) −4.36790 −0.142922
\(935\) 0 0
\(936\) 0 0
\(937\) 1.59644 0.0521534 0.0260767 0.999660i \(-0.491699\pi\)
0.0260767 + 0.999660i \(0.491699\pi\)
\(938\) −2.67822 −0.0874469
\(939\) 0 0
\(940\) −19.8206 −0.646477
\(941\) −7.10787 −0.231710 −0.115855 0.993266i \(-0.536961\pi\)
−0.115855 + 0.993266i \(0.536961\pi\)
\(942\) 0 0
\(943\) 1.71608 0.0558832
\(944\) −6.29513 −0.204889
\(945\) 0 0
\(946\) 0 0
\(947\) 2.45986 0.0799347 0.0399674 0.999201i \(-0.487275\pi\)
0.0399674 + 0.999201i \(0.487275\pi\)
\(948\) 0 0
\(949\) −2.52570 −0.0819878
\(950\) −12.5607 −0.407523
\(951\) 0 0
\(952\) −5.94362 −0.192634
\(953\) 28.6000 0.926446 0.463223 0.886242i \(-0.346693\pi\)
0.463223 + 0.886242i \(0.346693\pi\)
\(954\) 0 0
\(955\) 1.48294 0.0479867
\(956\) 6.69900 0.216661
\(957\) 0 0
\(958\) −16.8428 −0.544168
\(959\) −9.10052 −0.293871
\(960\) 0 0
\(961\) −30.3784 −0.979948
\(962\) 23.0959 0.744643
\(963\) 0 0
\(964\) −14.3607 −0.462528
\(965\) 52.4454 1.68828
\(966\) 0 0
\(967\) −0.213338 −0.00686047 −0.00343024 0.999994i \(-0.501092\pi\)
−0.00343024 + 0.999994i \(0.501092\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 24.9129 0.799907
\(971\) 0.828199 0.0265782 0.0132891 0.999912i \(-0.495770\pi\)
0.0132891 + 0.999912i \(0.495770\pi\)
\(972\) 0 0
\(973\) −13.0166 −0.417292
\(974\) −27.9473 −0.895488
\(975\) 0 0
\(976\) 2.05589 0.0658073
\(977\) −9.75714 −0.312159 −0.156079 0.987745i \(-0.549886\pi\)
−0.156079 + 0.987745i \(0.549886\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 2.61958 0.0836795
\(981\) 0 0
\(982\) 41.4813 1.32372
\(983\) −45.1198 −1.43910 −0.719548 0.694442i \(-0.755651\pi\)
−0.719548 + 0.694442i \(0.755651\pi\)
\(984\) 0 0
\(985\) −71.9148 −2.29140
\(986\) −17.1839 −0.547246
\(987\) 0 0
\(988\) 2.53367 0.0806069
\(989\) 2.48632 0.0790604
\(990\) 0 0
\(991\) 53.5405 1.70077 0.850384 0.526162i \(-0.176369\pi\)
0.850384 + 0.526162i \(0.176369\pi\)
\(992\) −3.16880 −0.100609
\(993\) 0 0
\(994\) −3.55358 −0.112713
\(995\) −29.1889 −0.925351
\(996\) 0 0
\(997\) −31.1607 −0.986869 −0.493435 0.869783i \(-0.664259\pi\)
−0.493435 + 0.869783i \(0.664259\pi\)
\(998\) 35.4549 1.12231
\(999\) 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 7623.2.a.ct.1.3 8
3.2 odd 2 847.2.a.p.1.6 8
11.5 even 5 693.2.m.i.190.3 16
11.9 even 5 693.2.m.i.631.3 16
11.10 odd 2 7623.2.a.cw.1.6 8
21.20 even 2 5929.2.a.bt.1.6 8
33.2 even 10 847.2.f.x.323.3 16
33.5 odd 10 77.2.f.b.36.2 yes 16
33.8 even 10 847.2.f.v.372.2 16
33.14 odd 10 847.2.f.w.372.3 16
33.17 even 10 847.2.f.x.729.3 16
33.20 odd 10 77.2.f.b.15.2 16
33.26 odd 10 847.2.f.w.148.3 16
33.29 even 10 847.2.f.v.148.2 16
33.32 even 2 847.2.a.o.1.3 8
231.5 even 30 539.2.q.f.410.3 32
231.20 even 10 539.2.f.e.246.2 16
231.38 even 30 539.2.q.f.520.2 32
231.53 odd 30 539.2.q.g.422.3 32
231.86 odd 30 539.2.q.g.312.2 32
231.104 even 10 539.2.f.e.344.2 16
231.137 odd 30 539.2.q.g.520.2 32
231.152 even 30 539.2.q.f.312.2 32
231.170 odd 30 539.2.q.g.410.3 32
231.185 even 30 539.2.q.f.422.3 32
231.230 odd 2 5929.2.a.bs.1.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.f.b.15.2 16 33.20 odd 10
77.2.f.b.36.2 yes 16 33.5 odd 10
539.2.f.e.246.2 16 231.20 even 10
539.2.f.e.344.2 16 231.104 even 10
539.2.q.f.312.2 32 231.152 even 30
539.2.q.f.410.3 32 231.5 even 30
539.2.q.f.422.3 32 231.185 even 30
539.2.q.f.520.2 32 231.38 even 30
539.2.q.g.312.2 32 231.86 odd 30
539.2.q.g.410.3 32 231.170 odd 30
539.2.q.g.422.3 32 231.53 odd 30
539.2.q.g.520.2 32 231.137 odd 30
693.2.m.i.190.3 16 11.5 even 5
693.2.m.i.631.3 16 11.9 even 5
847.2.a.o.1.3 8 33.32 even 2
847.2.a.p.1.6 8 3.2 odd 2
847.2.f.v.148.2 16 33.29 even 10
847.2.f.v.372.2 16 33.8 even 10
847.2.f.w.148.3 16 33.26 odd 10
847.2.f.w.372.3 16 33.14 odd 10
847.2.f.x.323.3 16 33.2 even 10
847.2.f.x.729.3 16 33.17 even 10
5929.2.a.bs.1.3 8 231.230 odd 2
5929.2.a.bt.1.6 8 21.20 even 2
7623.2.a.ct.1.3 8 1.1 even 1 trivial
7623.2.a.cw.1.6 8 11.10 odd 2