Properties

Label 7623.2.a.ch.1.4
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 0
Dimension 4
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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.2525.1
Coefficient ring: \(\Z[a_1, a_2]\)
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 1.4
Root \(-1.46673\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.46673 q^{2} +0.151302 q^{4} -0.466732 q^{5} -1.00000 q^{7} -2.71154 q^{8} +O(q^{10})\) \(q+1.46673 q^{2} +0.151302 q^{4} -0.466732 q^{5} -1.00000 q^{7} -2.71154 q^{8} -0.684570 q^{10} -1.58232 q^{13} -1.46673 q^{14} -4.27971 q^{16} -5.22732 q^{17} +4.22192 q^{19} -0.0706175 q^{20} +1.80505 q^{23} -4.78216 q^{25} -2.32083 q^{26} -0.151302 q^{28} -2.71947 q^{29} +1.29386 q^{31} -0.854102 q^{32} -7.66708 q^{34} +0.466732 q^{35} -1.94221 q^{37} +6.19242 q^{38} +1.26556 q^{40} +1.04112 q^{41} +8.70820 q^{43} +2.64753 q^{46} +6.39530 q^{47} +1.00000 q^{49} -7.01415 q^{50} -0.239408 q^{52} +13.2044 q^{53} +2.71154 q^{56} -3.98873 q^{58} +8.60389 q^{59} -15.2401 q^{61} +1.89775 q^{62} +7.30669 q^{64} +0.738517 q^{65} -4.67583 q^{67} -0.790906 q^{68} +0.684570 q^{70} -9.74310 q^{71} -13.3200 q^{73} -2.84870 q^{74} +0.638786 q^{76} +3.58232 q^{79} +1.99748 q^{80} +1.52705 q^{82} +17.2589 q^{83} +2.43976 q^{85} +12.7726 q^{86} +8.91982 q^{89} +1.58232 q^{91} +0.273109 q^{92} +9.38018 q^{94} -1.97050 q^{95} -2.70362 q^{97} +1.46673 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} + 4q^{4} + 6q^{5} - 4q^{7} - 9q^{8} + O(q^{10}) \) \( 4q - 2q^{2} + 4q^{4} + 6q^{5} - 4q^{7} - 9q^{8} - 14q^{10} + 2q^{14} - 4q^{16} + 3q^{17} + 3q^{19} + 17q^{20} + 8q^{23} + 12q^{26} - 4q^{28} - 3q^{29} - 3q^{31} + 10q^{32} - 12q^{34} - 6q^{35} - 7q^{37} + 20q^{38} - 13q^{40} - 4q^{41} + 8q^{43} - 3q^{46} + 14q^{47} + 4q^{49} - 33q^{50} - 17q^{52} + 9q^{53} + 9q^{56} + 3q^{58} + 25q^{59} - 19q^{61} - 10q^{62} + 3q^{64} - 12q^{65} - 15q^{67} + q^{68} + 14q^{70} + 7q^{71} - 11q^{73} - 8q^{74} - 26q^{76} + 8q^{79} + 4q^{80} + 3q^{82} + q^{83} + 15q^{85} - 4q^{86} + 17q^{89} + 17q^{92} - 20q^{94} - 17q^{95} - 15q^{97} - 2q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.46673 1.03714 0.518568 0.855036i \(-0.326465\pi\)
0.518568 + 0.855036i \(0.326465\pi\)
\(3\) 0 0
\(4\) 0.151302 0.0756511
\(5\) −0.466732 −0.208729 −0.104364 0.994539i \(-0.533281\pi\)
−0.104364 + 0.994539i \(0.533281\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −2.71154 −0.958676
\(9\) 0 0
\(10\) −0.684570 −0.216480
\(11\) 0 0
\(12\) 0 0
\(13\) −1.58232 −0.438856 −0.219428 0.975629i \(-0.570419\pi\)
−0.219428 + 0.975629i \(0.570419\pi\)
\(14\) −1.46673 −0.392001
\(15\) 0 0
\(16\) −4.27971 −1.06993
\(17\) −5.22732 −1.26781 −0.633906 0.773410i \(-0.718549\pi\)
−0.633906 + 0.773410i \(0.718549\pi\)
\(18\) 0 0
\(19\) 4.22192 0.968575 0.484287 0.874909i \(-0.339079\pi\)
0.484287 + 0.874909i \(0.339079\pi\)
\(20\) −0.0706175 −0.0157906
\(21\) 0 0
\(22\) 0 0
\(23\) 1.80505 0.376380 0.188190 0.982133i \(-0.439738\pi\)
0.188190 + 0.982133i \(0.439738\pi\)
\(24\) 0 0
\(25\) −4.78216 −0.956432
\(26\) −2.32083 −0.455153
\(27\) 0 0
\(28\) −0.151302 −0.0285934
\(29\) −2.71947 −0.504993 −0.252496 0.967598i \(-0.581252\pi\)
−0.252496 + 0.967598i \(0.581252\pi\)
\(30\) 0 0
\(31\) 1.29386 0.232384 0.116192 0.993227i \(-0.462931\pi\)
0.116192 + 0.993227i \(0.462931\pi\)
\(32\) −0.854102 −0.150985
\(33\) 0 0
\(34\) −7.66708 −1.31489
\(35\) 0.466732 0.0788921
\(36\) 0 0
\(37\) −1.94221 −0.319297 −0.159648 0.987174i \(-0.551036\pi\)
−0.159648 + 0.987174i \(0.551036\pi\)
\(38\) 6.19242 1.00454
\(39\) 0 0
\(40\) 1.26556 0.200103
\(41\) 1.04112 0.162596 0.0812980 0.996690i \(-0.474093\pi\)
0.0812980 + 0.996690i \(0.474093\pi\)
\(42\) 0 0
\(43\) 8.70820 1.32799 0.663994 0.747738i \(-0.268860\pi\)
0.663994 + 0.747738i \(0.268860\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 2.64753 0.390357
\(47\) 6.39530 0.932850 0.466425 0.884561i \(-0.345542\pi\)
0.466425 + 0.884561i \(0.345542\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −7.01415 −0.991950
\(51\) 0 0
\(52\) −0.239408 −0.0331999
\(53\) 13.2044 1.81377 0.906884 0.421380i \(-0.138454\pi\)
0.906884 + 0.421380i \(0.138454\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.71154 0.362345
\(57\) 0 0
\(58\) −3.98873 −0.523746
\(59\) 8.60389 1.12013 0.560065 0.828448i \(-0.310776\pi\)
0.560065 + 0.828448i \(0.310776\pi\)
\(60\) 0 0
\(61\) −15.2401 −1.95130 −0.975651 0.219331i \(-0.929613\pi\)
−0.975651 + 0.219331i \(0.929613\pi\)
\(62\) 1.89775 0.241014
\(63\) 0 0
\(64\) 7.30669 0.913336
\(65\) 0.738517 0.0916018
\(66\) 0 0
\(67\) −4.67583 −0.571243 −0.285622 0.958342i \(-0.592200\pi\)
−0.285622 + 0.958342i \(0.592200\pi\)
\(68\) −0.790906 −0.0959114
\(69\) 0 0
\(70\) 0.684570 0.0818218
\(71\) −9.74310 −1.15629 −0.578147 0.815933i \(-0.696224\pi\)
−0.578147 + 0.815933i \(0.696224\pi\)
\(72\) 0 0
\(73\) −13.3200 −1.55899 −0.779495 0.626408i \(-0.784524\pi\)
−0.779495 + 0.626408i \(0.784524\pi\)
\(74\) −2.84870 −0.331154
\(75\) 0 0
\(76\) 0.638786 0.0732737
\(77\) 0 0
\(78\) 0 0
\(79\) 3.58232 0.403042 0.201521 0.979484i \(-0.435412\pi\)
0.201521 + 0.979484i \(0.435412\pi\)
\(80\) 1.99748 0.223325
\(81\) 0 0
\(82\) 1.52705 0.168634
\(83\) 17.2589 1.89441 0.947204 0.320631i \(-0.103895\pi\)
0.947204 + 0.320631i \(0.103895\pi\)
\(84\) 0 0
\(85\) 2.43976 0.264629
\(86\) 12.7726 1.37730
\(87\) 0 0
\(88\) 0 0
\(89\) 8.91982 0.945499 0.472750 0.881197i \(-0.343262\pi\)
0.472750 + 0.881197i \(0.343262\pi\)
\(90\) 0 0
\(91\) 1.58232 0.165872
\(92\) 0.273109 0.0284735
\(93\) 0 0
\(94\) 9.38018 0.967492
\(95\) −1.97050 −0.202169
\(96\) 0 0
\(97\) −2.70362 −0.274511 −0.137255 0.990536i \(-0.543828\pi\)
−0.137255 + 0.990536i \(0.543828\pi\)
\(98\) 1.46673 0.148162
\(99\) 0 0
\(100\) −0.723551 −0.0723551
\(101\) −0.178781 −0.0177894 −0.00889469 0.999960i \(-0.502831\pi\)
−0.00889469 + 0.999960i \(0.502831\pi\)
\(102\) 0 0
\(103\) 16.8772 1.66296 0.831481 0.555553i \(-0.187493\pi\)
0.831481 + 0.555553i \(0.187493\pi\)
\(104\) 4.29052 0.420720
\(105\) 0 0
\(106\) 19.3674 1.88112
\(107\) 15.4762 1.49614 0.748071 0.663618i \(-0.230980\pi\)
0.748071 + 0.663618i \(0.230980\pi\)
\(108\) 0 0
\(109\) 11.0349 1.05695 0.528476 0.848948i \(-0.322764\pi\)
0.528476 + 0.848948i \(0.322764\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 4.27971 0.404395
\(113\) −1.77008 −0.166515 −0.0832574 0.996528i \(-0.526532\pi\)
−0.0832574 + 0.996528i \(0.526532\pi\)
\(114\) 0 0
\(115\) −0.842476 −0.0785613
\(116\) −0.411462 −0.0382033
\(117\) 0 0
\(118\) 12.6196 1.16173
\(119\) 5.22732 0.479188
\(120\) 0 0
\(121\) 0 0
\(122\) −22.3532 −2.02376
\(123\) 0 0
\(124\) 0.195764 0.0175801
\(125\) 4.56565 0.408364
\(126\) 0 0
\(127\) −8.54023 −0.757823 −0.378911 0.925433i \(-0.623702\pi\)
−0.378911 + 0.925433i \(0.623702\pi\)
\(128\) 12.4252 1.09824
\(129\) 0 0
\(130\) 1.08321 0.0950035
\(131\) 9.66708 0.844617 0.422308 0.906452i \(-0.361220\pi\)
0.422308 + 0.906452i \(0.361220\pi\)
\(132\) 0 0
\(133\) −4.22192 −0.366087
\(134\) −6.85818 −0.592457
\(135\) 0 0
\(136\) 14.1741 1.21542
\(137\) 14.0108 1.19702 0.598512 0.801114i \(-0.295759\pi\)
0.598512 + 0.801114i \(0.295759\pi\)
\(138\) 0 0
\(139\) 9.57765 0.812366 0.406183 0.913792i \(-0.366860\pi\)
0.406183 + 0.913792i \(0.366860\pi\)
\(140\) 0.0706175 0.00596827
\(141\) 0 0
\(142\) −14.2905 −1.19923
\(143\) 0 0
\(144\) 0 0
\(145\) 1.26926 0.105407
\(146\) −19.5369 −1.61688
\(147\) 0 0
\(148\) −0.293860 −0.0241552
\(149\) −14.9625 −1.22578 −0.612888 0.790170i \(-0.709992\pi\)
−0.612888 + 0.790170i \(0.709992\pi\)
\(150\) 0 0
\(151\) −2.87233 −0.233747 −0.116874 0.993147i \(-0.537287\pi\)
−0.116874 + 0.993147i \(0.537287\pi\)
\(152\) −11.4479 −0.928549
\(153\) 0 0
\(154\) 0 0
\(155\) −0.603886 −0.0485053
\(156\) 0 0
\(157\) −18.8823 −1.50697 −0.753487 0.657463i \(-0.771630\pi\)
−0.753487 + 0.657463i \(0.771630\pi\)
\(158\) 5.25430 0.418009
\(159\) 0 0
\(160\) 0.398637 0.0315150
\(161\) −1.80505 −0.142258
\(162\) 0 0
\(163\) 11.5951 0.908202 0.454101 0.890950i \(-0.349961\pi\)
0.454101 + 0.890950i \(0.349961\pi\)
\(164\) 0.157524 0.0123006
\(165\) 0 0
\(166\) 25.3142 1.96476
\(167\) −6.32491 −0.489437 −0.244718 0.969594i \(-0.578695\pi\)
−0.244718 + 0.969594i \(0.578695\pi\)
\(168\) 0 0
\(169\) −10.4963 −0.807406
\(170\) 3.57847 0.274456
\(171\) 0 0
\(172\) 1.31757 0.100464
\(173\) −1.33906 −0.101807 −0.0509035 0.998704i \(-0.516210\pi\)
−0.0509035 + 0.998704i \(0.516210\pi\)
\(174\) 0 0
\(175\) 4.78216 0.361497
\(176\) 0 0
\(177\) 0 0
\(178\) 13.0830 0.980611
\(179\) −17.7888 −1.32960 −0.664799 0.747022i \(-0.731483\pi\)
−0.664799 + 0.747022i \(0.731483\pi\)
\(180\) 0 0
\(181\) −0.963777 −0.0716370 −0.0358185 0.999358i \(-0.511404\pi\)
−0.0358185 + 0.999358i \(0.511404\pi\)
\(182\) 2.32083 0.172032
\(183\) 0 0
\(184\) −4.89448 −0.360826
\(185\) 0.906490 0.0666465
\(186\) 0 0
\(187\) 0 0
\(188\) 0.967622 0.0705711
\(189\) 0 0
\(190\) −2.89020 −0.209677
\(191\) 16.0888 1.16415 0.582074 0.813136i \(-0.302241\pi\)
0.582074 + 0.813136i \(0.302241\pi\)
\(192\) 0 0
\(193\) 12.1475 0.874393 0.437197 0.899366i \(-0.355971\pi\)
0.437197 + 0.899366i \(0.355971\pi\)
\(194\) −3.96548 −0.284705
\(195\) 0 0
\(196\) 0.151302 0.0108073
\(197\) 2.30179 0.163996 0.0819978 0.996633i \(-0.473870\pi\)
0.0819978 + 0.996633i \(0.473870\pi\)
\(198\) 0 0
\(199\) 20.2797 1.43759 0.718795 0.695222i \(-0.244694\pi\)
0.718795 + 0.695222i \(0.244694\pi\)
\(200\) 12.9670 0.916908
\(201\) 0 0
\(202\) −0.262224 −0.0184500
\(203\) 2.71947 0.190869
\(204\) 0 0
\(205\) −0.485925 −0.0339384
\(206\) 24.7544 1.72472
\(207\) 0 0
\(208\) 6.77186 0.469544
\(209\) 0 0
\(210\) 0 0
\(211\) 5.36530 0.369362 0.184681 0.982799i \(-0.440875\pi\)
0.184681 + 0.982799i \(0.440875\pi\)
\(212\) 1.99786 0.137214
\(213\) 0 0
\(214\) 22.6995 1.55170
\(215\) −4.06440 −0.277189
\(216\) 0 0
\(217\) −1.29386 −0.0878330
\(218\) 16.1852 1.09620
\(219\) 0 0
\(220\) 0 0
\(221\) 8.27128 0.556387
\(222\) 0 0
\(223\) −25.4230 −1.70245 −0.851225 0.524800i \(-0.824140\pi\)
−0.851225 + 0.524800i \(0.824140\pi\)
\(224\) 0.854102 0.0570671
\(225\) 0 0
\(226\) −2.59623 −0.172699
\(227\) 21.7098 1.44093 0.720464 0.693493i \(-0.243929\pi\)
0.720464 + 0.693493i \(0.243929\pi\)
\(228\) 0 0
\(229\) 20.5307 1.35670 0.678352 0.734737i \(-0.262694\pi\)
0.678352 + 0.734737i \(0.262694\pi\)
\(230\) −1.23569 −0.0814787
\(231\) 0 0
\(232\) 7.37396 0.484124
\(233\) −0.694056 −0.0454691 −0.0227345 0.999742i \(-0.507237\pi\)
−0.0227345 + 0.999742i \(0.507237\pi\)
\(234\) 0 0
\(235\) −2.98489 −0.194713
\(236\) 1.30179 0.0847391
\(237\) 0 0
\(238\) 7.66708 0.496983
\(239\) −0.346561 −0.0224171 −0.0112086 0.999937i \(-0.503568\pi\)
−0.0112086 + 0.999937i \(0.503568\pi\)
\(240\) 0 0
\(241\) −10.4372 −0.672317 −0.336158 0.941806i \(-0.609128\pi\)
−0.336158 + 0.941806i \(0.609128\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −2.30587 −0.147618
\(245\) −0.466732 −0.0298184
\(246\) 0 0
\(247\) −6.68041 −0.425064
\(248\) −3.50836 −0.222781
\(249\) 0 0
\(250\) 6.69658 0.423529
\(251\) 6.99502 0.441522 0.220761 0.975328i \(-0.429146\pi\)
0.220761 + 0.975328i \(0.429146\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −12.5262 −0.785965
\(255\) 0 0
\(256\) 3.61099 0.225687
\(257\) 9.94843 0.620566 0.310283 0.950644i \(-0.399576\pi\)
0.310283 + 0.950644i \(0.399576\pi\)
\(258\) 0 0
\(259\) 1.94221 0.120683
\(260\) 0.111739 0.00692978
\(261\) 0 0
\(262\) 14.1790 0.875983
\(263\) −14.1803 −0.874397 −0.437199 0.899365i \(-0.644029\pi\)
−0.437199 + 0.899365i \(0.644029\pi\)
\(264\) 0 0
\(265\) −6.16293 −0.378586
\(266\) −6.19242 −0.379682
\(267\) 0 0
\(268\) −0.707463 −0.0432152
\(269\) 18.4031 1.12206 0.561028 0.827797i \(-0.310406\pi\)
0.561028 + 0.827797i \(0.310406\pi\)
\(270\) 0 0
\(271\) 0.730591 0.0443802 0.0221901 0.999754i \(-0.492936\pi\)
0.0221901 + 0.999754i \(0.492936\pi\)
\(272\) 22.3714 1.35647
\(273\) 0 0
\(274\) 20.5501 1.24148
\(275\) 0 0
\(276\) 0 0
\(277\) 15.0644 0.905132 0.452566 0.891731i \(-0.350509\pi\)
0.452566 + 0.891731i \(0.350509\pi\)
\(278\) 14.0478 0.842534
\(279\) 0 0
\(280\) −1.26556 −0.0756319
\(281\) 10.6961 0.638077 0.319039 0.947742i \(-0.396640\pi\)
0.319039 + 0.947742i \(0.396640\pi\)
\(282\) 0 0
\(283\) 9.10890 0.541468 0.270734 0.962654i \(-0.412734\pi\)
0.270734 + 0.962654i \(0.412734\pi\)
\(284\) −1.47415 −0.0874749
\(285\) 0 0
\(286\) 0 0
\(287\) −1.04112 −0.0614555
\(288\) 0 0
\(289\) 10.3249 0.607348
\(290\) 1.86167 0.109321
\(291\) 0 0
\(292\) −2.01535 −0.117939
\(293\) −11.8890 −0.694563 −0.347281 0.937761i \(-0.612895\pi\)
−0.347281 + 0.937761i \(0.612895\pi\)
\(294\) 0 0
\(295\) −4.01571 −0.233804
\(296\) 5.26638 0.306102
\(297\) 0 0
\(298\) −21.9460 −1.27130
\(299\) −2.85617 −0.165176
\(300\) 0 0
\(301\) −8.70820 −0.501933
\(302\) −4.21294 −0.242427
\(303\) 0 0
\(304\) −18.0686 −1.03631
\(305\) 7.11306 0.407293
\(306\) 0 0
\(307\) 2.22072 0.126743 0.0633716 0.997990i \(-0.479815\pi\)
0.0633716 + 0.997990i \(0.479815\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −0.885738 −0.0503066
\(311\) 21.4126 1.21420 0.607098 0.794627i \(-0.292334\pi\)
0.607098 + 0.794627i \(0.292334\pi\)
\(312\) 0 0
\(313\) −31.5548 −1.78358 −0.891790 0.452449i \(-0.850550\pi\)
−0.891790 + 0.452449i \(0.850550\pi\)
\(314\) −27.6953 −1.56294
\(315\) 0 0
\(316\) 0.542012 0.0304906
\(317\) 13.2007 0.741423 0.370712 0.928748i \(-0.379114\pi\)
0.370712 + 0.928748i \(0.379114\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −3.41026 −0.190639
\(321\) 0 0
\(322\) −2.64753 −0.147541
\(323\) −22.0693 −1.22797
\(324\) 0 0
\(325\) 7.56689 0.419736
\(326\) 17.0070 0.941929
\(327\) 0 0
\(328\) −2.82305 −0.155877
\(329\) −6.39530 −0.352584
\(330\) 0 0
\(331\) 9.47653 0.520877 0.260439 0.965490i \(-0.416133\pi\)
0.260439 + 0.965490i \(0.416133\pi\)
\(332\) 2.61131 0.143314
\(333\) 0 0
\(334\) −9.27695 −0.507612
\(335\) 2.18236 0.119235
\(336\) 0 0
\(337\) −19.2011 −1.04595 −0.522975 0.852348i \(-0.675178\pi\)
−0.522975 + 0.852348i \(0.675178\pi\)
\(338\) −15.3952 −0.837390
\(339\) 0 0
\(340\) 0.369141 0.0200195
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −23.6127 −1.27311
\(345\) 0 0
\(346\) −1.96405 −0.105588
\(347\) −3.04831 −0.163642 −0.0818208 0.996647i \(-0.526074\pi\)
−0.0818208 + 0.996647i \(0.526074\pi\)
\(348\) 0 0
\(349\) 19.3961 1.03825 0.519125 0.854698i \(-0.326258\pi\)
0.519125 + 0.854698i \(0.326258\pi\)
\(350\) 7.01415 0.374922
\(351\) 0 0
\(352\) 0 0
\(353\) 10.7585 0.572619 0.286309 0.958137i \(-0.407572\pi\)
0.286309 + 0.958137i \(0.407572\pi\)
\(354\) 0 0
\(355\) 4.54742 0.241352
\(356\) 1.34959 0.0715280
\(357\) 0 0
\(358\) −26.0914 −1.37897
\(359\) −0.607226 −0.0320481 −0.0160241 0.999872i \(-0.505101\pi\)
−0.0160241 + 0.999872i \(0.505101\pi\)
\(360\) 0 0
\(361\) −1.17539 −0.0618628
\(362\) −1.41360 −0.0742973
\(363\) 0 0
\(364\) 0.239408 0.0125484
\(365\) 6.21688 0.325406
\(366\) 0 0
\(367\) 27.6628 1.44399 0.721994 0.691899i \(-0.243226\pi\)
0.721994 + 0.691899i \(0.243226\pi\)
\(368\) −7.72511 −0.402699
\(369\) 0 0
\(370\) 1.32958 0.0691215
\(371\) −13.2044 −0.685540
\(372\) 0 0
\(373\) −29.4513 −1.52493 −0.762465 0.647029i \(-0.776011\pi\)
−0.762465 + 0.647029i \(0.776011\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −17.3411 −0.894300
\(377\) 4.30306 0.221619
\(378\) 0 0
\(379\) 25.3436 1.30182 0.650908 0.759157i \(-0.274388\pi\)
0.650908 + 0.759157i \(0.274388\pi\)
\(380\) −0.298142 −0.0152943
\(381\) 0 0
\(382\) 23.5980 1.20738
\(383\) 31.9322 1.63166 0.815829 0.578293i \(-0.196281\pi\)
0.815829 + 0.578293i \(0.196281\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 17.8171 0.906865
\(387\) 0 0
\(388\) −0.409063 −0.0207670
\(389\) −17.7517 −0.900047 −0.450024 0.893017i \(-0.648584\pi\)
−0.450024 + 0.893017i \(0.648584\pi\)
\(390\) 0 0
\(391\) −9.43560 −0.477179
\(392\) −2.71154 −0.136954
\(393\) 0 0
\(394\) 3.37610 0.170086
\(395\) −1.67198 −0.0841265
\(396\) 0 0
\(397\) −13.3047 −0.667742 −0.333871 0.942619i \(-0.608355\pi\)
−0.333871 + 0.942619i \(0.608355\pi\)
\(398\) 29.7449 1.49098
\(399\) 0 0
\(400\) 20.4663 1.02331
\(401\) 3.48962 0.174264 0.0871318 0.996197i \(-0.472230\pi\)
0.0871318 + 0.996197i \(0.472230\pi\)
\(402\) 0 0
\(403\) −2.04730 −0.101983
\(404\) −0.0270500 −0.00134579
\(405\) 0 0
\(406\) 3.98873 0.197958
\(407\) 0 0
\(408\) 0 0
\(409\) −29.6255 −1.46489 −0.732443 0.680828i \(-0.761620\pi\)
−0.732443 + 0.680828i \(0.761620\pi\)
\(410\) −0.712721 −0.0351988
\(411\) 0 0
\(412\) 2.55356 0.125805
\(413\) −8.60389 −0.423370
\(414\) 0 0
\(415\) −8.05527 −0.395418
\(416\) 1.35146 0.0662608
\(417\) 0 0
\(418\) 0 0
\(419\) 11.6452 0.568907 0.284454 0.958690i \(-0.408188\pi\)
0.284454 + 0.958690i \(0.408188\pi\)
\(420\) 0 0
\(421\) 19.8848 0.969128 0.484564 0.874756i \(-0.338978\pi\)
0.484564 + 0.874756i \(0.338978\pi\)
\(422\) 7.86945 0.383079
\(423\) 0 0
\(424\) −35.8044 −1.73882
\(425\) 24.9979 1.21258
\(426\) 0 0
\(427\) 15.2401 0.737523
\(428\) 2.34159 0.113185
\(429\) 0 0
\(430\) −5.96138 −0.287483
\(431\) −30.2464 −1.45692 −0.728458 0.685090i \(-0.759763\pi\)
−0.728458 + 0.685090i \(0.759763\pi\)
\(432\) 0 0
\(433\) −5.70719 −0.274270 −0.137135 0.990552i \(-0.543789\pi\)
−0.137135 + 0.990552i \(0.543789\pi\)
\(434\) −1.89775 −0.0910947
\(435\) 0 0
\(436\) 1.66960 0.0799596
\(437\) 7.62079 0.364552
\(438\) 0 0
\(439\) 6.84875 0.326873 0.163436 0.986554i \(-0.447742\pi\)
0.163436 + 0.986554i \(0.447742\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 12.1317 0.577048
\(443\) −0.100695 −0.00478417 −0.00239209 0.999997i \(-0.500761\pi\)
−0.00239209 + 0.999997i \(0.500761\pi\)
\(444\) 0 0
\(445\) −4.16316 −0.197353
\(446\) −37.2887 −1.76567
\(447\) 0 0
\(448\) −7.30669 −0.345208
\(449\) 30.8047 1.45377 0.726883 0.686762i \(-0.240968\pi\)
0.726883 + 0.686762i \(0.240968\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −0.267817 −0.0125970
\(453\) 0 0
\(454\) 31.8424 1.49444
\(455\) −0.738517 −0.0346222
\(456\) 0 0
\(457\) 23.1356 1.08224 0.541118 0.840947i \(-0.318001\pi\)
0.541118 + 0.840947i \(0.318001\pi\)
\(458\) 30.1130 1.40709
\(459\) 0 0
\(460\) −0.127468 −0.00594325
\(461\) 2.77839 0.129403 0.0647013 0.997905i \(-0.479391\pi\)
0.0647013 + 0.997905i \(0.479391\pi\)
\(462\) 0 0
\(463\) −26.0950 −1.21274 −0.606369 0.795184i \(-0.707374\pi\)
−0.606369 + 0.795184i \(0.707374\pi\)
\(464\) 11.6385 0.540306
\(465\) 0 0
\(466\) −1.01799 −0.0471576
\(467\) 2.65829 0.123011 0.0615055 0.998107i \(-0.480410\pi\)
0.0615055 + 0.998107i \(0.480410\pi\)
\(468\) 0 0
\(469\) 4.67583 0.215910
\(470\) −4.37803 −0.201943
\(471\) 0 0
\(472\) −23.3298 −1.07384
\(473\) 0 0
\(474\) 0 0
\(475\) −20.1899 −0.926376
\(476\) 0.790906 0.0362511
\(477\) 0 0
\(478\) −0.508312 −0.0232496
\(479\) 8.28223 0.378425 0.189212 0.981936i \(-0.439407\pi\)
0.189212 + 0.981936i \(0.439407\pi\)
\(480\) 0 0
\(481\) 3.07319 0.140125
\(482\) −15.3085 −0.697284
\(483\) 0 0
\(484\) 0 0
\(485\) 1.26186 0.0572983
\(486\) 0 0
\(487\) −19.5956 −0.887960 −0.443980 0.896037i \(-0.646434\pi\)
−0.443980 + 0.896037i \(0.646434\pi\)
\(488\) 41.3243 1.87066
\(489\) 0 0
\(490\) −0.684570 −0.0309257
\(491\) 28.6817 1.29439 0.647193 0.762327i \(-0.275943\pi\)
0.647193 + 0.762327i \(0.275943\pi\)
\(492\) 0 0
\(493\) 14.2156 0.640236
\(494\) −9.79837 −0.440850
\(495\) 0 0
\(496\) −5.53735 −0.248634
\(497\) 9.74310 0.437038
\(498\) 0 0
\(499\) 27.9499 1.25121 0.625605 0.780140i \(-0.284852\pi\)
0.625605 + 0.780140i \(0.284852\pi\)
\(500\) 0.690792 0.0308932
\(501\) 0 0
\(502\) 10.2598 0.457918
\(503\) 8.09736 0.361043 0.180522 0.983571i \(-0.442221\pi\)
0.180522 + 0.983571i \(0.442221\pi\)
\(504\) 0 0
\(505\) 0.0834428 0.00371316
\(506\) 0 0
\(507\) 0 0
\(508\) −1.29216 −0.0573301
\(509\) −16.3002 −0.722492 −0.361246 0.932471i \(-0.617648\pi\)
−0.361246 + 0.932471i \(0.617648\pi\)
\(510\) 0 0
\(511\) 13.3200 0.589243
\(512\) −19.5539 −0.864170
\(513\) 0 0
\(514\) 14.5917 0.643611
\(515\) −7.87714 −0.347108
\(516\) 0 0
\(517\) 0 0
\(518\) 2.84870 0.125165
\(519\) 0 0
\(520\) −2.00252 −0.0878164
\(521\) −7.68605 −0.336732 −0.168366 0.985725i \(-0.553849\pi\)
−0.168366 + 0.985725i \(0.553849\pi\)
\(522\) 0 0
\(523\) 26.1229 1.14228 0.571138 0.820854i \(-0.306502\pi\)
0.571138 + 0.820854i \(0.306502\pi\)
\(524\) 1.46265 0.0638962
\(525\) 0 0
\(526\) −20.7988 −0.906869
\(527\) −6.76343 −0.294619
\(528\) 0 0
\(529\) −19.7418 −0.858338
\(530\) −9.03936 −0.392645
\(531\) 0 0
\(532\) −0.638786 −0.0276949
\(533\) −1.64738 −0.0713561
\(534\) 0 0
\(535\) −7.22324 −0.312288
\(536\) 12.6787 0.547637
\(537\) 0 0
\(538\) 26.9924 1.16372
\(539\) 0 0
\(540\) 0 0
\(541\) −25.8777 −1.11257 −0.556284 0.830992i \(-0.687773\pi\)
−0.556284 + 0.830992i \(0.687773\pi\)
\(542\) 1.07158 0.0460283
\(543\) 0 0
\(544\) 4.46467 0.191421
\(545\) −5.15034 −0.220616
\(546\) 0 0
\(547\) −38.0968 −1.62890 −0.814451 0.580232i \(-0.802962\pi\)
−0.814451 + 0.580232i \(0.802962\pi\)
\(548\) 2.11987 0.0905562
\(549\) 0 0
\(550\) 0 0
\(551\) −11.4814 −0.489123
\(552\) 0 0
\(553\) −3.58232 −0.152336
\(554\) 22.0954 0.938745
\(555\) 0 0
\(556\) 1.44912 0.0614564
\(557\) −34.5422 −1.46360 −0.731799 0.681520i \(-0.761319\pi\)
−0.731799 + 0.681520i \(0.761319\pi\)
\(558\) 0 0
\(559\) −13.7791 −0.582795
\(560\) −1.99748 −0.0844088
\(561\) 0 0
\(562\) 15.6883 0.661773
\(563\) −19.4819 −0.821066 −0.410533 0.911846i \(-0.634657\pi\)
−0.410533 + 0.911846i \(0.634657\pi\)
\(564\) 0 0
\(565\) 0.826151 0.0347565
\(566\) 13.3603 0.561576
\(567\) 0 0
\(568\) 26.4189 1.10851
\(569\) −17.1288 −0.718075 −0.359038 0.933323i \(-0.616895\pi\)
−0.359038 + 0.933323i \(0.616895\pi\)
\(570\) 0 0
\(571\) −3.85581 −0.161360 −0.0806802 0.996740i \(-0.525709\pi\)
−0.0806802 + 0.996740i \(0.525709\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −1.52705 −0.0637377
\(575\) −8.63206 −0.359982
\(576\) 0 0
\(577\) −9.78185 −0.407224 −0.203612 0.979052i \(-0.565268\pi\)
−0.203612 + 0.979052i \(0.565268\pi\)
\(578\) 15.1439 0.629902
\(579\) 0 0
\(580\) 0.192042 0.00797412
\(581\) −17.2589 −0.716019
\(582\) 0 0
\(583\) 0 0
\(584\) 36.1178 1.49457
\(585\) 0 0
\(586\) −17.4380 −0.720356
\(587\) 6.09891 0.251729 0.125865 0.992047i \(-0.459830\pi\)
0.125865 + 0.992047i \(0.459830\pi\)
\(588\) 0 0
\(589\) 5.46257 0.225081
\(590\) −5.88997 −0.242486
\(591\) 0 0
\(592\) 8.31209 0.341625
\(593\) −13.2330 −0.543413 −0.271706 0.962380i \(-0.587588\pi\)
−0.271706 + 0.962380i \(0.587588\pi\)
\(594\) 0 0
\(595\) −2.43976 −0.100020
\(596\) −2.26386 −0.0927313
\(597\) 0 0
\(598\) −4.18923 −0.171310
\(599\) 5.92515 0.242095 0.121048 0.992647i \(-0.461375\pi\)
0.121048 + 0.992647i \(0.461375\pi\)
\(600\) 0 0
\(601\) 12.7408 0.519708 0.259854 0.965648i \(-0.416326\pi\)
0.259854 + 0.965648i \(0.416326\pi\)
\(602\) −12.7726 −0.520572
\(603\) 0 0
\(604\) −0.434590 −0.0176832
\(605\) 0 0
\(606\) 0 0
\(607\) 8.36141 0.339379 0.169690 0.985498i \(-0.445724\pi\)
0.169690 + 0.985498i \(0.445724\pi\)
\(608\) −3.60595 −0.146241
\(609\) 0 0
\(610\) 10.4330 0.422418
\(611\) −10.1194 −0.409386
\(612\) 0 0
\(613\) −6.68294 −0.269921 −0.134961 0.990851i \(-0.543091\pi\)
−0.134961 + 0.990851i \(0.543091\pi\)
\(614\) 3.25720 0.131450
\(615\) 0 0
\(616\) 0 0
\(617\) −11.8669 −0.477741 −0.238871 0.971051i \(-0.576777\pi\)
−0.238871 + 0.971051i \(0.576777\pi\)
\(618\) 0 0
\(619\) 20.6206 0.828814 0.414407 0.910092i \(-0.363989\pi\)
0.414407 + 0.910092i \(0.363989\pi\)
\(620\) −0.0913692 −0.00366948
\(621\) 0 0
\(622\) 31.4065 1.25929
\(623\) −8.91982 −0.357365
\(624\) 0 0
\(625\) 21.7799 0.871195
\(626\) −46.2824 −1.84982
\(627\) 0 0
\(628\) −2.85694 −0.114004
\(629\) 10.1525 0.404809
\(630\) 0 0
\(631\) −15.4795 −0.616228 −0.308114 0.951349i \(-0.599698\pi\)
−0.308114 + 0.951349i \(0.599698\pi\)
\(632\) −9.71361 −0.386387
\(633\) 0 0
\(634\) 19.3618 0.768957
\(635\) 3.98600 0.158179
\(636\) 0 0
\(637\) −1.58232 −0.0626937
\(638\) 0 0
\(639\) 0 0
\(640\) −5.79921 −0.229234
\(641\) 23.5785 0.931294 0.465647 0.884971i \(-0.345822\pi\)
0.465647 + 0.884971i \(0.345822\pi\)
\(642\) 0 0
\(643\) 28.6806 1.13105 0.565527 0.824730i \(-0.308673\pi\)
0.565527 + 0.824730i \(0.308673\pi\)
\(644\) −0.273109 −0.0107620
\(645\) 0 0
\(646\) −32.3698 −1.27357
\(647\) −5.42763 −0.213382 −0.106691 0.994292i \(-0.534026\pi\)
−0.106691 + 0.994292i \(0.534026\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 11.0986 0.435323
\(651\) 0 0
\(652\) 1.75437 0.0687064
\(653\) 12.3421 0.482983 0.241492 0.970403i \(-0.422363\pi\)
0.241492 + 0.970403i \(0.422363\pi\)
\(654\) 0 0
\(655\) −4.51193 −0.176296
\(656\) −4.45570 −0.173966
\(657\) 0 0
\(658\) −9.38018 −0.365678
\(659\) −16.2115 −0.631512 −0.315756 0.948840i \(-0.602258\pi\)
−0.315756 + 0.948840i \(0.602258\pi\)
\(660\) 0 0
\(661\) 43.7050 1.69993 0.849964 0.526840i \(-0.176623\pi\)
0.849964 + 0.526840i \(0.176623\pi\)
\(662\) 13.8995 0.540220
\(663\) 0 0
\(664\) −46.7982 −1.81612
\(665\) 1.97050 0.0764129
\(666\) 0 0
\(667\) −4.90879 −0.190069
\(668\) −0.956973 −0.0370264
\(669\) 0 0
\(670\) 3.20093 0.123663
\(671\) 0 0
\(672\) 0 0
\(673\) −5.86102 −0.225926 −0.112963 0.993599i \(-0.536034\pi\)
−0.112963 + 0.993599i \(0.536034\pi\)
\(674\) −28.1629 −1.08479
\(675\) 0 0
\(676\) −1.58811 −0.0610811
\(677\) 20.5279 0.788952 0.394476 0.918906i \(-0.370926\pi\)
0.394476 + 0.918906i \(0.370926\pi\)
\(678\) 0 0
\(679\) 2.70362 0.103755
\(680\) −6.61551 −0.253693
\(681\) 0 0
\(682\) 0 0
\(683\) −38.7055 −1.48103 −0.740513 0.672042i \(-0.765417\pi\)
−0.740513 + 0.672042i \(0.765417\pi\)
\(684\) 0 0
\(685\) −6.53929 −0.249853
\(686\) −1.46673 −0.0560001
\(687\) 0 0
\(688\) −37.2686 −1.42085
\(689\) −20.8936 −0.795982
\(690\) 0 0
\(691\) −23.1300 −0.879907 −0.439954 0.898020i \(-0.645005\pi\)
−0.439954 + 0.898020i \(0.645005\pi\)
\(692\) −0.202603 −0.00770182
\(693\) 0 0
\(694\) −4.47105 −0.169719
\(695\) −4.47020 −0.169564
\(696\) 0 0
\(697\) −5.44228 −0.206141
\(698\) 28.4489 1.07681
\(699\) 0 0
\(700\) 0.723551 0.0273477
\(701\) −35.5107 −1.34122 −0.670610 0.741810i \(-0.733968\pi\)
−0.670610 + 0.741810i \(0.733968\pi\)
\(702\) 0 0
\(703\) −8.19985 −0.309263
\(704\) 0 0
\(705\) 0 0
\(706\) 15.7799 0.593883
\(707\) 0.178781 0.00672375
\(708\) 0 0
\(709\) 43.8045 1.64511 0.822556 0.568684i \(-0.192547\pi\)
0.822556 + 0.568684i \(0.192547\pi\)
\(710\) 6.66984 0.250315
\(711\) 0 0
\(712\) −24.1865 −0.906427
\(713\) 2.33549 0.0874647
\(714\) 0 0
\(715\) 0 0
\(716\) −2.69149 −0.100586
\(717\) 0 0
\(718\) −0.890637 −0.0332383
\(719\) −15.8605 −0.591496 −0.295748 0.955266i \(-0.595569\pi\)
−0.295748 + 0.955266i \(0.595569\pi\)
\(720\) 0 0
\(721\) −16.8772 −0.628541
\(722\) −1.72399 −0.0641602
\(723\) 0 0
\(724\) −0.145822 −0.00541942
\(725\) 13.0049 0.482992
\(726\) 0 0
\(727\) 13.7719 0.510770 0.255385 0.966839i \(-0.417798\pi\)
0.255385 + 0.966839i \(0.417798\pi\)
\(728\) −4.29052 −0.159017
\(729\) 0 0
\(730\) 9.11849 0.337490
\(731\) −45.5206 −1.68364
\(732\) 0 0
\(733\) −19.5677 −0.722750 −0.361375 0.932421i \(-0.617693\pi\)
−0.361375 + 0.932421i \(0.617693\pi\)
\(734\) 40.5740 1.49761
\(735\) 0 0
\(736\) −1.54170 −0.0568278
\(737\) 0 0
\(738\) 0 0
\(739\) −11.1542 −0.410313 −0.205157 0.978729i \(-0.565770\pi\)
−0.205157 + 0.978729i \(0.565770\pi\)
\(740\) 0.137154 0.00504188
\(741\) 0 0
\(742\) −19.3674 −0.710998
\(743\) −20.6550 −0.757757 −0.378878 0.925446i \(-0.623690\pi\)
−0.378878 + 0.925446i \(0.623690\pi\)
\(744\) 0 0
\(745\) 6.98348 0.255855
\(746\) −43.1972 −1.58156
\(747\) 0 0
\(748\) 0 0
\(749\) −15.4762 −0.565489
\(750\) 0 0
\(751\) 26.5991 0.970614 0.485307 0.874344i \(-0.338708\pi\)
0.485307 + 0.874344i \(0.338708\pi\)
\(752\) −27.3700 −0.998082
\(753\) 0 0
\(754\) 6.31144 0.229849
\(755\) 1.34061 0.0487897
\(756\) 0 0
\(757\) 21.0999 0.766890 0.383445 0.923564i \(-0.374738\pi\)
0.383445 + 0.923564i \(0.374738\pi\)
\(758\) 37.1723 1.35016
\(759\) 0 0
\(760\) 5.34311 0.193815
\(761\) 7.99743 0.289907 0.144953 0.989438i \(-0.453697\pi\)
0.144953 + 0.989438i \(0.453697\pi\)
\(762\) 0 0
\(763\) −11.0349 −0.399490
\(764\) 2.43428 0.0880691
\(765\) 0 0
\(766\) 46.8360 1.69225
\(767\) −13.6141 −0.491576
\(768\) 0 0
\(769\) −52.0476 −1.87689 −0.938443 0.345435i \(-0.887731\pi\)
−0.938443 + 0.345435i \(0.887731\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.83794 0.0661488
\(773\) 1.58099 0.0568644 0.0284322 0.999596i \(-0.490949\pi\)
0.0284322 + 0.999596i \(0.490949\pi\)
\(774\) 0 0
\(775\) −6.18745 −0.222260
\(776\) 7.33098 0.263167
\(777\) 0 0
\(778\) −26.0370 −0.933471
\(779\) 4.39553 0.157486
\(780\) 0 0
\(781\) 0 0
\(782\) −13.8395 −0.494899
\(783\) 0 0
\(784\) −4.27971 −0.152847
\(785\) 8.81298 0.314549
\(786\) 0 0
\(787\) 23.3907 0.833789 0.416894 0.908955i \(-0.363118\pi\)
0.416894 + 0.908955i \(0.363118\pi\)
\(788\) 0.348265 0.0124064
\(789\) 0 0
\(790\) −2.45235 −0.0872506
\(791\) 1.77008 0.0629367
\(792\) 0 0
\(793\) 24.1147 0.856339
\(794\) −19.5144 −0.692539
\(795\) 0 0
\(796\) 3.06836 0.108755
\(797\) 46.1518 1.63478 0.817391 0.576084i \(-0.195420\pi\)
0.817391 + 0.576084i \(0.195420\pi\)
\(798\) 0 0
\(799\) −33.4303 −1.18268
\(800\) 4.08445 0.144407
\(801\) 0 0
\(802\) 5.11834 0.180735
\(803\) 0 0
\(804\) 0 0
\(805\) 0.842476 0.0296934
\(806\) −3.00283 −0.105770
\(807\) 0 0
\(808\) 0.484773 0.0170542
\(809\) −33.7501 −1.18659 −0.593295 0.804985i \(-0.702173\pi\)
−0.593295 + 0.804985i \(0.702173\pi\)
\(810\) 0 0
\(811\) −30.7650 −1.08030 −0.540152 0.841567i \(-0.681633\pi\)
−0.540152 + 0.841567i \(0.681633\pi\)
\(812\) 0.411462 0.0144395
\(813\) 0 0
\(814\) 0 0
\(815\) −5.41182 −0.189568
\(816\) 0 0
\(817\) 36.7653 1.28626
\(818\) −43.4527 −1.51929
\(819\) 0 0
\(820\) −0.0735215 −0.00256748
\(821\) −12.0784 −0.421538 −0.210769 0.977536i \(-0.567597\pi\)
−0.210769 + 0.977536i \(0.567597\pi\)
\(822\) 0 0
\(823\) −24.8187 −0.865124 −0.432562 0.901604i \(-0.642390\pi\)
−0.432562 + 0.901604i \(0.642390\pi\)
\(824\) −45.7634 −1.59424
\(825\) 0 0
\(826\) −12.6196 −0.439092
\(827\) −5.17330 −0.179893 −0.0899466 0.995947i \(-0.528670\pi\)
−0.0899466 + 0.995947i \(0.528670\pi\)
\(828\) 0 0
\(829\) 31.5535 1.09590 0.547949 0.836511i \(-0.315409\pi\)
0.547949 + 0.836511i \(0.315409\pi\)
\(830\) −11.8149 −0.410102
\(831\) 0 0
\(832\) −11.5615 −0.400822
\(833\) −5.22732 −0.181116
\(834\) 0 0
\(835\) 2.95204 0.102160
\(836\) 0 0
\(837\) 0 0
\(838\) 17.0804 0.590034
\(839\) 5.83642 0.201496 0.100748 0.994912i \(-0.467876\pi\)
0.100748 + 0.994912i \(0.467876\pi\)
\(840\) 0 0
\(841\) −21.6045 −0.744982
\(842\) 29.1657 1.00512
\(843\) 0 0
\(844\) 0.811781 0.0279427
\(845\) 4.89895 0.168529
\(846\) 0 0
\(847\) 0 0
\(848\) −56.5112 −1.94060
\(849\) 0 0
\(850\) 36.6652 1.25761
\(851\) −3.50579 −0.120177
\(852\) 0 0
\(853\) −20.3462 −0.696640 −0.348320 0.937376i \(-0.613248\pi\)
−0.348320 + 0.937376i \(0.613248\pi\)
\(854\) 22.3532 0.764911
\(855\) 0 0
\(856\) −41.9644 −1.43432
\(857\) 15.1087 0.516104 0.258052 0.966131i \(-0.416919\pi\)
0.258052 + 0.966131i \(0.416919\pi\)
\(858\) 0 0
\(859\) −33.9641 −1.15884 −0.579420 0.815029i \(-0.696721\pi\)
−0.579420 + 0.815029i \(0.696721\pi\)
\(860\) −0.614952 −0.0209697
\(861\) 0 0
\(862\) −44.3633 −1.51102
\(863\) 2.77734 0.0945417 0.0472709 0.998882i \(-0.484948\pi\)
0.0472709 + 0.998882i \(0.484948\pi\)
\(864\) 0 0
\(865\) 0.624983 0.0212501
\(866\) −8.37092 −0.284456
\(867\) 0 0
\(868\) −0.195764 −0.00664466
\(869\) 0 0
\(870\) 0 0
\(871\) 7.39864 0.250693
\(872\) −29.9216 −1.01327
\(873\) 0 0
\(874\) 11.1777 0.378090
\(875\) −4.56565 −0.154347
\(876\) 0 0
\(877\) 49.6783 1.67752 0.838759 0.544503i \(-0.183282\pi\)
0.838759 + 0.544503i \(0.183282\pi\)
\(878\) 10.0453 0.339012
\(879\) 0 0
\(880\) 0 0
\(881\) 27.3064 0.919975 0.459988 0.887925i \(-0.347854\pi\)
0.459988 + 0.887925i \(0.347854\pi\)
\(882\) 0 0
\(883\) −17.8109 −0.599386 −0.299693 0.954036i \(-0.596884\pi\)
−0.299693 + 0.954036i \(0.596884\pi\)
\(884\) 1.25146 0.0420912
\(885\) 0 0
\(886\) −0.147693 −0.00496184
\(887\) 16.4729 0.553105 0.276553 0.960999i \(-0.410808\pi\)
0.276553 + 0.960999i \(0.410808\pi\)
\(888\) 0 0
\(889\) 8.54023 0.286430
\(890\) −6.10625 −0.204682
\(891\) 0 0
\(892\) −3.84656 −0.128792
\(893\) 27.0004 0.903535
\(894\) 0 0
\(895\) 8.30260 0.277525
\(896\) −12.4252 −0.415095
\(897\) 0 0
\(898\) 45.1823 1.50775
\(899\) −3.51861 −0.117352
\(900\) 0 0
\(901\) −69.0238 −2.29952
\(902\) 0 0
\(903\) 0 0
\(904\) 4.79964 0.159634
\(905\) 0.449825 0.0149527
\(906\) 0 0
\(907\) 28.4877 0.945918 0.472959 0.881084i \(-0.343186\pi\)
0.472959 + 0.881084i \(0.343186\pi\)
\(908\) 3.28473 0.109008
\(909\) 0 0
\(910\) −1.08321 −0.0359080
\(911\) 11.2353 0.372242 0.186121 0.982527i \(-0.440408\pi\)
0.186121 + 0.982527i \(0.440408\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 33.9337 1.12243
\(915\) 0 0
\(916\) 3.10634 0.102636
\(917\) −9.66708 −0.319235
\(918\) 0 0
\(919\) 0.780457 0.0257449 0.0128724 0.999917i \(-0.495902\pi\)
0.0128724 + 0.999917i \(0.495902\pi\)
\(920\) 2.28441 0.0753148
\(921\) 0 0
\(922\) 4.07516 0.134208
\(923\) 15.4167 0.507446
\(924\) 0 0
\(925\) 9.28795 0.305386
\(926\) −38.2744 −1.25777
\(927\) 0 0
\(928\) 2.32270 0.0762465
\(929\) −26.5963 −0.872597 −0.436298 0.899802i \(-0.643711\pi\)
−0.436298 + 0.899802i \(0.643711\pi\)
\(930\) 0 0
\(931\) 4.22192 0.138368
\(932\) −0.105012 −0.00343979
\(933\) 0 0
\(934\) 3.89900 0.127579
\(935\) 0 0
\(936\) 0 0
\(937\) 41.9697 1.37109 0.685544 0.728031i \(-0.259564\pi\)
0.685544 + 0.728031i \(0.259564\pi\)
\(938\) 6.85818 0.223928
\(939\) 0 0
\(940\) −0.451620 −0.0147302
\(941\) 49.0330 1.59843 0.799215 0.601046i \(-0.205249\pi\)
0.799215 + 0.601046i \(0.205249\pi\)
\(942\) 0 0
\(943\) 1.87928 0.0611978
\(944\) −36.8222 −1.19846
\(945\) 0 0
\(946\) 0 0
\(947\) 27.2953 0.886978 0.443489 0.896280i \(-0.353741\pi\)
0.443489 + 0.896280i \(0.353741\pi\)
\(948\) 0 0
\(949\) 21.0765 0.684171
\(950\) −29.6132 −0.960778
\(951\) 0 0
\(952\) −14.1741 −0.459386
\(953\) −19.7408 −0.639466 −0.319733 0.947508i \(-0.603593\pi\)
−0.319733 + 0.947508i \(0.603593\pi\)
\(954\) 0 0
\(955\) −7.50918 −0.242991
\(956\) −0.0524354 −0.00169588
\(957\) 0 0
\(958\) 12.1478 0.392478
\(959\) −14.0108 −0.452433
\(960\) 0 0
\(961\) −29.3259 −0.945998
\(962\) 4.50754 0.145329
\(963\) 0 0
\(964\) −1.57917 −0.0508615
\(965\) −5.66960 −0.182511
\(966\) 0 0
\(967\) −12.6734 −0.407551 −0.203775 0.979018i \(-0.565321\pi\)
−0.203775 + 0.979018i \(0.565321\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 1.85082 0.0594261
\(971\) −16.7036 −0.536045 −0.268022 0.963413i \(-0.586370\pi\)
−0.268022 + 0.963413i \(0.586370\pi\)
\(972\) 0 0
\(973\) −9.57765 −0.307045
\(974\) −28.7414 −0.920935
\(975\) 0 0
\(976\) 65.2234 2.08775
\(977\) −27.3452 −0.874851 −0.437425 0.899255i \(-0.644110\pi\)
−0.437425 + 0.899255i \(0.644110\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −0.0706175 −0.00225579
\(981\) 0 0
\(982\) 42.0683 1.34245
\(983\) 55.0065 1.75443 0.877217 0.480093i \(-0.159397\pi\)
0.877217 + 0.480093i \(0.159397\pi\)
\(984\) 0 0
\(985\) −1.07432 −0.0342306
\(986\) 20.8504 0.664012
\(987\) 0 0
\(988\) −1.01076 −0.0321566
\(989\) 15.7188 0.499828
\(990\) 0 0
\(991\) 53.2327 1.69099 0.845497 0.533980i \(-0.179304\pi\)
0.845497 + 0.533980i \(0.179304\pi\)
\(992\) −1.10509 −0.0350866
\(993\) 0 0
\(994\) 14.2905 0.453268
\(995\) −9.46519 −0.300067
\(996\) 0 0
\(997\) 32.7546 1.03735 0.518674 0.854972i \(-0.326426\pi\)
0.518674 + 0.854972i \(0.326426\pi\)
\(998\) 40.9950 1.29767
\(999\) 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.ch.1.4 4
3.2 odd 2 847.2.a.l.1.1 4
11.5 even 5 693.2.m.g.190.1 8
11.9 even 5 693.2.m.g.631.1 8
11.10 odd 2 7623.2.a.co.1.1 4
21.20 even 2 5929.2.a.bi.1.1 4
33.2 even 10 847.2.f.q.323.1 8
33.5 odd 10 77.2.f.a.36.2 yes 8
33.8 even 10 847.2.f.s.372.2 8
33.14 odd 10 847.2.f.p.372.1 8
33.17 even 10 847.2.f.q.729.1 8
33.20 odd 10 77.2.f.a.15.2 8
33.26 odd 10 847.2.f.p.148.1 8
33.29 even 10 847.2.f.s.148.2 8
33.32 even 2 847.2.a.k.1.4 4
231.5 even 30 539.2.q.b.410.1 16
231.20 even 10 539.2.f.d.246.2 8
231.38 even 30 539.2.q.b.520.2 16
231.53 odd 30 539.2.q.c.422.1 16
231.86 odd 30 539.2.q.c.312.2 16
231.104 even 10 539.2.f.d.344.2 8
231.137 odd 30 539.2.q.c.520.2 16
231.152 even 30 539.2.q.b.312.2 16
231.170 odd 30 539.2.q.c.410.1 16
231.185 even 30 539.2.q.b.422.1 16
231.230 odd 2 5929.2.a.bb.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.f.a.15.2 8 33.20 odd 10
77.2.f.a.36.2 yes 8 33.5 odd 10
539.2.f.d.246.2 8 231.20 even 10
539.2.f.d.344.2 8 231.104 even 10
539.2.q.b.312.2 16 231.152 even 30
539.2.q.b.410.1 16 231.5 even 30
539.2.q.b.422.1 16 231.185 even 30
539.2.q.b.520.2 16 231.38 even 30
539.2.q.c.312.2 16 231.86 odd 30
539.2.q.c.410.1 16 231.170 odd 30
539.2.q.c.422.1 16 231.53 odd 30
539.2.q.c.520.2 16 231.137 odd 30
693.2.m.g.190.1 8 11.5 even 5
693.2.m.g.631.1 8 11.9 even 5
847.2.a.k.1.4 4 33.32 even 2
847.2.a.l.1.1 4 3.2 odd 2
847.2.f.p.148.1 8 33.26 odd 10
847.2.f.p.372.1 8 33.14 odd 10
847.2.f.q.323.1 8 33.2 even 10
847.2.f.q.729.1 8 33.17 even 10
847.2.f.s.148.2 8 33.29 even 10
847.2.f.s.372.2 8 33.8 even 10
5929.2.a.bb.1.4 4 231.230 odd 2
5929.2.a.bi.1.1 4 21.20 even 2
7623.2.a.ch.1.4 4 1.1 even 1 trivial
7623.2.a.co.1.1 4 11.10 odd 2