Properties

Label 7623.2.a.cl.1.2
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: 4.4.725.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.35567\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.162147 q^{2} -1.97371 q^{4} +4.38705 q^{5} +1.00000 q^{7} -0.644326 q^{8} +O(q^{10})\) \(q+0.162147 q^{2} -1.97371 q^{4} +4.38705 q^{5} +1.00000 q^{7} -0.644326 q^{8} +0.711349 q^{10} -1.67571 q^{13} +0.162147 q^{14} +3.84294 q^{16} +6.18663 q^{17} +3.47528 q^{19} -8.65877 q^{20} -0.568595 q^{23} +14.2462 q^{25} -0.271711 q^{26} -1.97371 q^{28} -8.86742 q^{29} +4.33447 q^{31} +1.91177 q^{32} +1.00314 q^{34} +4.38705 q^{35} +0.969445 q^{37} +0.563507 q^{38} -2.82669 q^{40} +5.77725 q^{41} +5.04388 q^{43} -0.0921961 q^{46} -4.67256 q^{47} +1.00000 q^{49} +2.30999 q^{50} +3.30735 q^{52} +2.37882 q^{53} -0.644326 q^{56} -1.43783 q^{58} +2.84901 q^{59} -9.29883 q^{61} +0.702822 q^{62} -7.37589 q^{64} -7.35141 q^{65} -7.14275 q^{67} -12.2106 q^{68} +0.711349 q^{70} +0.794487 q^{71} -2.38705 q^{73} +0.157193 q^{74} -6.85919 q^{76} +0.670617 q^{79} +16.8592 q^{80} +0.936765 q^{82} -9.28503 q^{83} +27.1411 q^{85} +0.817850 q^{86} +12.2106 q^{89} -1.67571 q^{91} +1.12224 q^{92} -0.757643 q^{94} +15.2462 q^{95} +15.0596 q^{97} +0.162147 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + q^{2} + 3q^{4} + 4q^{5} + 4q^{7} - 9q^{8} + O(q^{10}) \) \( 4q + q^{2} + 3q^{4} + 4q^{5} + 4q^{7} - 9q^{8} - 10q^{10} - 6q^{13} + q^{14} - 3q^{16} + 8q^{17} + 10q^{19} + 10q^{23} + 12q^{25} + 20q^{26} + 3q^{28} - 18q^{31} - 2q^{32} + 18q^{34} + 4q^{35} - 2q^{37} + 8q^{38} - 6q^{40} + 10q^{41} + 4q^{43} - 11q^{46} - 4q^{47} + 4q^{49} - 9q^{50} - 20q^{52} - 9q^{56} + 14q^{58} + 16q^{59} - 14q^{61} - 11q^{64} - 28q^{65} - 28q^{67} - 16q^{68} - 10q^{70} + 18q^{71} + 4q^{73} - 41q^{74} + 4q^{76} + 20q^{79} + 36q^{80} - 24q^{82} + 6q^{83} + 20q^{85} + 20q^{86} + 16q^{89} - 6q^{91} + 22q^{92} + 16q^{94} + 16q^{95} + 32q^{97} + q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.162147 0.114655 0.0573277 0.998355i \(-0.481742\pi\)
0.0573277 + 0.998355i \(0.481742\pi\)
\(3\) 0 0
\(4\) −1.97371 −0.986854
\(5\) 4.38705 1.96195 0.980975 0.194133i \(-0.0621895\pi\)
0.980975 + 0.194133i \(0.0621895\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −0.644326 −0.227804
\(9\) 0 0
\(10\) 0.711349 0.224948
\(11\) 0 0
\(12\) 0 0
\(13\) −1.67571 −0.464757 −0.232379 0.972625i \(-0.574651\pi\)
−0.232379 + 0.972625i \(0.574651\pi\)
\(14\) 0.162147 0.0433357
\(15\) 0 0
\(16\) 3.84294 0.960735
\(17\) 6.18663 1.50048 0.750239 0.661167i \(-0.229938\pi\)
0.750239 + 0.661167i \(0.229938\pi\)
\(18\) 0 0
\(19\) 3.47528 0.797284 0.398642 0.917107i \(-0.369482\pi\)
0.398642 + 0.917107i \(0.369482\pi\)
\(20\) −8.65877 −1.93616
\(21\) 0 0
\(22\) 0 0
\(23\) −0.568595 −0.118560 −0.0592801 0.998241i \(-0.518881\pi\)
−0.0592801 + 0.998241i \(0.518881\pi\)
\(24\) 0 0
\(25\) 14.2462 2.84925
\(26\) −0.271711 −0.0532869
\(27\) 0 0
\(28\) −1.97371 −0.372996
\(29\) −8.86742 −1.64664 −0.823320 0.567578i \(-0.807880\pi\)
−0.823320 + 0.567578i \(0.807880\pi\)
\(30\) 0 0
\(31\) 4.33447 0.778494 0.389247 0.921133i \(-0.372735\pi\)
0.389247 + 0.921133i \(0.372735\pi\)
\(32\) 1.91177 0.337957
\(33\) 0 0
\(34\) 1.00314 0.172038
\(35\) 4.38705 0.741547
\(36\) 0 0
\(37\) 0.969445 0.159376 0.0796879 0.996820i \(-0.474608\pi\)
0.0796879 + 0.996820i \(0.474608\pi\)
\(38\) 0.563507 0.0914129
\(39\) 0 0
\(40\) −2.82669 −0.446939
\(41\) 5.77725 0.902255 0.451128 0.892459i \(-0.351022\pi\)
0.451128 + 0.892459i \(0.351022\pi\)
\(42\) 0 0
\(43\) 5.04388 0.769184 0.384592 0.923087i \(-0.374342\pi\)
0.384592 + 0.923087i \(0.374342\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −0.0921961 −0.0135936
\(47\) −4.67256 −0.681563 −0.340782 0.940143i \(-0.610692\pi\)
−0.340782 + 0.940143i \(0.610692\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 2.30999 0.326682
\(51\) 0 0
\(52\) 3.30735 0.458647
\(53\) 2.37882 0.326756 0.163378 0.986564i \(-0.447761\pi\)
0.163378 + 0.986564i \(0.447761\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −0.644326 −0.0861016
\(57\) 0 0
\(58\) −1.43783 −0.188796
\(59\) 2.84901 0.370910 0.185455 0.982653i \(-0.440624\pi\)
0.185455 + 0.982653i \(0.440624\pi\)
\(60\) 0 0
\(61\) −9.29883 −1.19059 −0.595296 0.803506i \(-0.702965\pi\)
−0.595296 + 0.803506i \(0.702965\pi\)
\(62\) 0.702822 0.0892585
\(63\) 0 0
\(64\) −7.37589 −0.921987
\(65\) −7.35141 −0.911830
\(66\) 0 0
\(67\) −7.14275 −0.872626 −0.436313 0.899795i \(-0.643716\pi\)
−0.436313 + 0.899795i \(0.643716\pi\)
\(68\) −12.2106 −1.48075
\(69\) 0 0
\(70\) 0.711349 0.0850224
\(71\) 0.794487 0.0942882 0.0471441 0.998888i \(-0.484988\pi\)
0.0471441 + 0.998888i \(0.484988\pi\)
\(72\) 0 0
\(73\) −2.38705 −0.279384 −0.139692 0.990195i \(-0.544611\pi\)
−0.139692 + 0.990195i \(0.544611\pi\)
\(74\) 0.157193 0.0182733
\(75\) 0 0
\(76\) −6.85919 −0.786803
\(77\) 0 0
\(78\) 0 0
\(79\) 0.670617 0.0754504 0.0377252 0.999288i \(-0.487989\pi\)
0.0377252 + 0.999288i \(0.487989\pi\)
\(80\) 16.8592 1.88491
\(81\) 0 0
\(82\) 0.936765 0.103448
\(83\) −9.28503 −1.01916 −0.509582 0.860422i \(-0.670200\pi\)
−0.509582 + 0.860422i \(0.670200\pi\)
\(84\) 0 0
\(85\) 27.1411 2.94386
\(86\) 0.817850 0.0881911
\(87\) 0 0
\(88\) 0 0
\(89\) 12.2106 1.29432 0.647161 0.762354i \(-0.275956\pi\)
0.647161 + 0.762354i \(0.275956\pi\)
\(90\) 0 0
\(91\) −1.67571 −0.175662
\(92\) 1.12224 0.117002
\(93\) 0 0
\(94\) −0.757643 −0.0781449
\(95\) 15.2462 1.56423
\(96\) 0 0
\(97\) 15.0596 1.52907 0.764536 0.644581i \(-0.222968\pi\)
0.764536 + 0.644581i \(0.222968\pi\)
\(98\) 0.162147 0.0163793
\(99\) 0 0
\(100\) −28.1179 −2.81179
\(101\) 5.76079 0.573220 0.286610 0.958047i \(-0.407472\pi\)
0.286610 + 0.958047i \(0.407472\pi\)
\(102\) 0 0
\(103\) 7.71449 0.760132 0.380066 0.924959i \(-0.375901\pi\)
0.380066 + 0.924959i \(0.375901\pi\)
\(104\) 1.07970 0.105873
\(105\) 0 0
\(106\) 0.385719 0.0374644
\(107\) −16.4634 −1.59158 −0.795790 0.605573i \(-0.792944\pi\)
−0.795790 + 0.605573i \(0.792944\pi\)
\(108\) 0 0
\(109\) −9.74355 −0.933263 −0.466632 0.884452i \(-0.654533\pi\)
−0.466632 + 0.884452i \(0.654533\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 3.84294 0.363124
\(113\) 2.85725 0.268787 0.134394 0.990928i \(-0.457091\pi\)
0.134394 + 0.990928i \(0.457091\pi\)
\(114\) 0 0
\(115\) −2.49446 −0.232609
\(116\) 17.5017 1.62499
\(117\) 0 0
\(118\) 0.461960 0.0425268
\(119\) 6.18663 0.567127
\(120\) 0 0
\(121\) 0 0
\(122\) −1.50778 −0.136508
\(123\) 0 0
\(124\) −8.55498 −0.768260
\(125\) 40.5638 3.62813
\(126\) 0 0
\(127\) 16.0514 1.42433 0.712165 0.702012i \(-0.247715\pi\)
0.712165 + 0.702012i \(0.247715\pi\)
\(128\) −5.01953 −0.443668
\(129\) 0 0
\(130\) −1.19201 −0.104546
\(131\) 0.750136 0.0655397 0.0327699 0.999463i \(-0.489567\pi\)
0.0327699 + 0.999463i \(0.489567\pi\)
\(132\) 0 0
\(133\) 3.47528 0.301345
\(134\) −1.15818 −0.100051
\(135\) 0 0
\(136\) −3.98620 −0.341814
\(137\) 13.0082 1.11137 0.555684 0.831393i \(-0.312456\pi\)
0.555684 + 0.831393i \(0.312456\pi\)
\(138\) 0 0
\(139\) −3.90112 −0.330889 −0.165444 0.986219i \(-0.552906\pi\)
−0.165444 + 0.986219i \(0.552906\pi\)
\(140\) −8.65877 −0.731799
\(141\) 0 0
\(142\) 0.128824 0.0108107
\(143\) 0 0
\(144\) 0 0
\(145\) −38.9019 −3.23062
\(146\) −0.387054 −0.0320328
\(147\) 0 0
\(148\) −1.91340 −0.157281
\(149\) 18.5434 1.51914 0.759568 0.650428i \(-0.225410\pi\)
0.759568 + 0.650428i \(0.225410\pi\)
\(150\) 0 0
\(151\) −20.7979 −1.69251 −0.846255 0.532779i \(-0.821148\pi\)
−0.846255 + 0.532779i \(0.821148\pi\)
\(152\) −2.23921 −0.181624
\(153\) 0 0
\(154\) 0 0
\(155\) 19.0156 1.52737
\(156\) 0 0
\(157\) 5.80975 0.463669 0.231834 0.972755i \(-0.425527\pi\)
0.231834 + 0.972755i \(0.425527\pi\)
\(158\) 0.108739 0.00865079
\(159\) 0 0
\(160\) 8.38705 0.663055
\(161\) −0.568595 −0.0448116
\(162\) 0 0
\(163\) −8.90307 −0.697342 −0.348671 0.937245i \(-0.613367\pi\)
−0.348671 + 0.937245i \(0.613367\pi\)
\(164\) −11.4026 −0.890394
\(165\) 0 0
\(166\) −1.50554 −0.116853
\(167\) −3.82355 −0.295875 −0.147937 0.988997i \(-0.547263\pi\)
−0.147937 + 0.988997i \(0.547263\pi\)
\(168\) 0 0
\(169\) −10.1920 −0.784001
\(170\) 4.40085 0.337530
\(171\) 0 0
\(172\) −9.95514 −0.759072
\(173\) 1.75061 0.133096 0.0665482 0.997783i \(-0.478801\pi\)
0.0665482 + 0.997783i \(0.478801\pi\)
\(174\) 0 0
\(175\) 14.2462 1.07691
\(176\) 0 0
\(177\) 0 0
\(178\) 1.97991 0.148401
\(179\) 12.8541 0.960761 0.480380 0.877060i \(-0.340499\pi\)
0.480380 + 0.877060i \(0.340499\pi\)
\(180\) 0 0
\(181\) −22.0831 −1.64142 −0.820712 0.571342i \(-0.806423\pi\)
−0.820712 + 0.571342i \(0.806423\pi\)
\(182\) −0.271711 −0.0201406
\(183\) 0 0
\(184\) 0.366361 0.0270085
\(185\) 4.25301 0.312687
\(186\) 0 0
\(187\) 0 0
\(188\) 9.22227 0.672603
\(189\) 0 0
\(190\) 2.47214 0.179348
\(191\) −14.7160 −1.06481 −0.532405 0.846490i \(-0.678712\pi\)
−0.532405 + 0.846490i \(0.678712\pi\)
\(192\) 0 0
\(193\) −8.51287 −0.612770 −0.306385 0.951908i \(-0.599119\pi\)
−0.306385 + 0.951908i \(0.599119\pi\)
\(194\) 2.44187 0.175316
\(195\) 0 0
\(196\) −1.97371 −0.140979
\(197\) −4.72273 −0.336480 −0.168240 0.985746i \(-0.553808\pi\)
−0.168240 + 0.985746i \(0.553808\pi\)
\(198\) 0 0
\(199\) 16.7620 1.18822 0.594112 0.804382i \(-0.297504\pi\)
0.594112 + 0.804382i \(0.297504\pi\)
\(200\) −9.17922 −0.649069
\(201\) 0 0
\(202\) 0.934096 0.0657227
\(203\) −8.86742 −0.622371
\(204\) 0 0
\(205\) 25.3451 1.77018
\(206\) 1.25088 0.0871532
\(207\) 0 0
\(208\) −6.43964 −0.446509
\(209\) 0 0
\(210\) 0 0
\(211\) 10.1454 0.698440 0.349220 0.937041i \(-0.386447\pi\)
0.349220 + 0.937041i \(0.386447\pi\)
\(212\) −4.69510 −0.322461
\(213\) 0 0
\(214\) −2.66950 −0.182483
\(215\) 22.1278 1.50910
\(216\) 0 0
\(217\) 4.33447 0.294243
\(218\) −1.57989 −0.107004
\(219\) 0 0
\(220\) 0 0
\(221\) −10.3670 −0.697358
\(222\) 0 0
\(223\) −6.11172 −0.409271 −0.204636 0.978838i \(-0.565601\pi\)
−0.204636 + 0.978838i \(0.565601\pi\)
\(224\) 1.91177 0.127736
\(225\) 0 0
\(226\) 0.463295 0.0308179
\(227\) 14.2090 0.943081 0.471541 0.881844i \(-0.343698\pi\)
0.471541 + 0.881844i \(0.343698\pi\)
\(228\) 0 0
\(229\) −2.72781 −0.180259 −0.0901295 0.995930i \(-0.528728\pi\)
−0.0901295 + 0.995930i \(0.528728\pi\)
\(230\) −0.404469 −0.0266699
\(231\) 0 0
\(232\) 5.71351 0.375110
\(233\) −17.7897 −1.16544 −0.582720 0.812673i \(-0.698012\pi\)
−0.582720 + 0.812673i \(0.698012\pi\)
\(234\) 0 0
\(235\) −20.4988 −1.33719
\(236\) −5.62312 −0.366034
\(237\) 0 0
\(238\) 1.00314 0.0650242
\(239\) 12.7058 0.821869 0.410934 0.911665i \(-0.365202\pi\)
0.410934 + 0.911665i \(0.365202\pi\)
\(240\) 0 0
\(241\) 18.6604 1.20202 0.601012 0.799240i \(-0.294765\pi\)
0.601012 + 0.799240i \(0.294765\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 18.3532 1.17494
\(245\) 4.38705 0.280279
\(246\) 0 0
\(247\) −5.82355 −0.370543
\(248\) −2.79281 −0.177344
\(249\) 0 0
\(250\) 6.57730 0.415985
\(251\) −18.8493 −1.18976 −0.594878 0.803816i \(-0.702800\pi\)
−0.594878 + 0.803816i \(0.702800\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 2.60269 0.163307
\(255\) 0 0
\(256\) 13.9379 0.871118
\(257\) 10.3345 0.644647 0.322323 0.946630i \(-0.395536\pi\)
0.322323 + 0.946630i \(0.395536\pi\)
\(258\) 0 0
\(259\) 0.969445 0.0602384
\(260\) 14.5095 0.899844
\(261\) 0 0
\(262\) 0.121632 0.00751448
\(263\) −4.14979 −0.255887 −0.127943 0.991781i \(-0.540838\pi\)
−0.127943 + 0.991781i \(0.540838\pi\)
\(264\) 0 0
\(265\) 10.4360 0.641079
\(266\) 0.563507 0.0345508
\(267\) 0 0
\(268\) 14.0977 0.861155
\(269\) 5.06228 0.308653 0.154326 0.988020i \(-0.450679\pi\)
0.154326 + 0.988020i \(0.450679\pi\)
\(270\) 0 0
\(271\) 26.3160 1.59859 0.799293 0.600942i \(-0.205208\pi\)
0.799293 + 0.600942i \(0.205208\pi\)
\(272\) 23.7749 1.44156
\(273\) 0 0
\(274\) 2.10925 0.127424
\(275\) 0 0
\(276\) 0 0
\(277\) −25.8187 −1.55130 −0.775648 0.631165i \(-0.782577\pi\)
−0.775648 + 0.631165i \(0.782577\pi\)
\(278\) −0.632556 −0.0379382
\(279\) 0 0
\(280\) −2.82669 −0.168927
\(281\) −5.43844 −0.324430 −0.162215 0.986755i \(-0.551864\pi\)
−0.162215 + 0.986755i \(0.551864\pi\)
\(282\) 0 0
\(283\) −25.3839 −1.50892 −0.754458 0.656348i \(-0.772100\pi\)
−0.754458 + 0.656348i \(0.772100\pi\)
\(284\) −1.56809 −0.0930487
\(285\) 0 0
\(286\) 0 0
\(287\) 5.77725 0.341020
\(288\) 0 0
\(289\) 21.2744 1.25143
\(290\) −6.30783 −0.370408
\(291\) 0 0
\(292\) 4.71135 0.275711
\(293\) 6.75402 0.394574 0.197287 0.980346i \(-0.436787\pi\)
0.197287 + 0.980346i \(0.436787\pi\)
\(294\) 0 0
\(295\) 12.4988 0.727707
\(296\) −0.624638 −0.0363064
\(297\) 0 0
\(298\) 3.00676 0.174177
\(299\) 0.952798 0.0551017
\(300\) 0 0
\(301\) 5.04388 0.290724
\(302\) −3.37232 −0.194055
\(303\) 0 0
\(304\) 13.3553 0.765979
\(305\) −40.7945 −2.33588
\(306\) 0 0
\(307\) 17.6396 1.00674 0.503372 0.864070i \(-0.332092\pi\)
0.503372 + 0.864070i \(0.332092\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 3.08332 0.175121
\(311\) 12.2414 0.694149 0.347074 0.937838i \(-0.387175\pi\)
0.347074 + 0.937838i \(0.387175\pi\)
\(312\) 0 0
\(313\) 3.15951 0.178586 0.0892931 0.996005i \(-0.471539\pi\)
0.0892931 + 0.996005i \(0.471539\pi\)
\(314\) 0.942035 0.0531621
\(315\) 0 0
\(316\) −1.32360 −0.0744585
\(317\) −8.43364 −0.473681 −0.236840 0.971549i \(-0.576112\pi\)
−0.236840 + 0.971549i \(0.576112\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −32.3584 −1.80889
\(321\) 0 0
\(322\) −0.0921961 −0.00513789
\(323\) 21.5003 1.19631
\(324\) 0 0
\(325\) −23.8725 −1.32421
\(326\) −1.44361 −0.0799540
\(327\) 0 0
\(328\) −3.72243 −0.205537
\(329\) −4.67256 −0.257607
\(330\) 0 0
\(331\) 6.13284 0.337092 0.168546 0.985694i \(-0.446093\pi\)
0.168546 + 0.985694i \(0.446093\pi\)
\(332\) 18.3259 1.00577
\(333\) 0 0
\(334\) −0.619977 −0.0339237
\(335\) −31.3356 −1.71205
\(336\) 0 0
\(337\) 29.8792 1.62763 0.813813 0.581127i \(-0.197388\pi\)
0.813813 + 0.581127i \(0.197388\pi\)
\(338\) −1.65261 −0.0898899
\(339\) 0 0
\(340\) −53.5686 −2.90516
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −3.24990 −0.175223
\(345\) 0 0
\(346\) 0.283857 0.0152602
\(347\) −29.1716 −1.56602 −0.783008 0.622012i \(-0.786315\pi\)
−0.783008 + 0.622012i \(0.786315\pi\)
\(348\) 0 0
\(349\) −13.3924 −0.716881 −0.358440 0.933553i \(-0.616691\pi\)
−0.358440 + 0.933553i \(0.616691\pi\)
\(350\) 2.30999 0.123474
\(351\) 0 0
\(352\) 0 0
\(353\) 21.3175 1.13462 0.567309 0.823505i \(-0.307985\pi\)
0.567309 + 0.823505i \(0.307985\pi\)
\(354\) 0 0
\(355\) 3.48546 0.184989
\(356\) −24.1002 −1.27731
\(357\) 0 0
\(358\) 2.08426 0.110156
\(359\) −20.6234 −1.08846 −0.544231 0.838935i \(-0.683179\pi\)
−0.544231 + 0.838935i \(0.683179\pi\)
\(360\) 0 0
\(361\) −6.92242 −0.364338
\(362\) −3.58072 −0.188198
\(363\) 0 0
\(364\) 3.30735 0.173352
\(365\) −10.4721 −0.548137
\(366\) 0 0
\(367\) 32.8546 1.71499 0.857497 0.514489i \(-0.172018\pi\)
0.857497 + 0.514489i \(0.172018\pi\)
\(368\) −2.18508 −0.113905
\(369\) 0 0
\(370\) 0.689613 0.0358513
\(371\) 2.37882 0.123502
\(372\) 0 0
\(373\) −26.0979 −1.35130 −0.675650 0.737223i \(-0.736137\pi\)
−0.675650 + 0.737223i \(0.736137\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 3.01065 0.155262
\(377\) 14.8592 0.765287
\(378\) 0 0
\(379\) −11.6733 −0.599616 −0.299808 0.953999i \(-0.596923\pi\)
−0.299808 + 0.953999i \(0.596923\pi\)
\(380\) −30.0916 −1.54367
\(381\) 0 0
\(382\) −2.38615 −0.122086
\(383\) 15.3701 0.785376 0.392688 0.919672i \(-0.371545\pi\)
0.392688 + 0.919672i \(0.371545\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −1.38034 −0.0702573
\(387\) 0 0
\(388\) −29.7233 −1.50897
\(389\) 9.64424 0.488983 0.244491 0.969651i \(-0.421379\pi\)
0.244491 + 0.969651i \(0.421379\pi\)
\(390\) 0 0
\(391\) −3.51769 −0.177897
\(392\) −0.644326 −0.0325434
\(393\) 0 0
\(394\) −0.765777 −0.0385793
\(395\) 2.94203 0.148030
\(396\) 0 0
\(397\) 20.2855 1.01810 0.509050 0.860737i \(-0.329997\pi\)
0.509050 + 0.860737i \(0.329997\pi\)
\(398\) 2.71791 0.136236
\(399\) 0 0
\(400\) 54.7475 2.73737
\(401\) 8.15655 0.407319 0.203659 0.979042i \(-0.434717\pi\)
0.203659 + 0.979042i \(0.434717\pi\)
\(402\) 0 0
\(403\) −7.26330 −0.361811
\(404\) −11.3701 −0.565684
\(405\) 0 0
\(406\) −1.43783 −0.0713582
\(407\) 0 0
\(408\) 0 0
\(409\) 9.67209 0.478254 0.239127 0.970988i \(-0.423139\pi\)
0.239127 + 0.970988i \(0.423139\pi\)
\(410\) 4.10964 0.202961
\(411\) 0 0
\(412\) −15.2262 −0.750139
\(413\) 2.84901 0.140191
\(414\) 0 0
\(415\) −40.7339 −1.99955
\(416\) −3.20357 −0.157068
\(417\) 0 0
\(418\) 0 0
\(419\) 18.9357 0.925072 0.462536 0.886601i \(-0.346940\pi\)
0.462536 + 0.886601i \(0.346940\pi\)
\(420\) 0 0
\(421\) −13.0950 −0.638210 −0.319105 0.947719i \(-0.603382\pi\)
−0.319105 + 0.947719i \(0.603382\pi\)
\(422\) 1.64505 0.0800799
\(423\) 0 0
\(424\) −1.53274 −0.0744362
\(425\) 88.1362 4.27524
\(426\) 0 0
\(427\) −9.29883 −0.450002
\(428\) 32.4940 1.57066
\(429\) 0 0
\(430\) 3.58795 0.173027
\(431\) −16.3034 −0.785309 −0.392655 0.919686i \(-0.628443\pi\)
−0.392655 + 0.919686i \(0.628443\pi\)
\(432\) 0 0
\(433\) 11.6842 0.561508 0.280754 0.959780i \(-0.409415\pi\)
0.280754 + 0.959780i \(0.409415\pi\)
\(434\) 0.702822 0.0337366
\(435\) 0 0
\(436\) 19.2309 0.920995
\(437\) −1.97603 −0.0945262
\(438\) 0 0
\(439\) −11.7172 −0.559230 −0.279615 0.960112i \(-0.590207\pi\)
−0.279615 + 0.960112i \(0.590207\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −1.68098 −0.0799558
\(443\) 12.5671 0.597081 0.298540 0.954397i \(-0.403500\pi\)
0.298540 + 0.954397i \(0.403500\pi\)
\(444\) 0 0
\(445\) 53.5686 2.53939
\(446\) −0.990999 −0.0469252
\(447\) 0 0
\(448\) −7.37589 −0.348478
\(449\) 15.9360 0.752068 0.376034 0.926606i \(-0.377288\pi\)
0.376034 + 0.926606i \(0.377288\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −5.63937 −0.265254
\(453\) 0 0
\(454\) 2.30394 0.108129
\(455\) −7.35141 −0.344639
\(456\) 0 0
\(457\) 23.7945 1.11306 0.556530 0.830828i \(-0.312133\pi\)
0.556530 + 0.830828i \(0.312133\pi\)
\(458\) −0.442307 −0.0206677
\(459\) 0 0
\(460\) 4.92333 0.229552
\(461\) −22.6035 −1.05275 −0.526375 0.850252i \(-0.676449\pi\)
−0.526375 + 0.850252i \(0.676449\pi\)
\(462\) 0 0
\(463\) 3.03759 0.141169 0.0705843 0.997506i \(-0.477514\pi\)
0.0705843 + 0.997506i \(0.477514\pi\)
\(464\) −34.0770 −1.58198
\(465\) 0 0
\(466\) −2.88454 −0.133624
\(467\) 23.0051 1.06455 0.532274 0.846572i \(-0.321338\pi\)
0.532274 + 0.846572i \(0.321338\pi\)
\(468\) 0 0
\(469\) −7.14275 −0.329822
\(470\) −3.32382 −0.153316
\(471\) 0 0
\(472\) −1.83569 −0.0844946
\(473\) 0 0
\(474\) 0 0
\(475\) 49.5097 2.27166
\(476\) −12.2106 −0.559672
\(477\) 0 0
\(478\) 2.06021 0.0942317
\(479\) 8.17283 0.373426 0.186713 0.982414i \(-0.440217\pi\)
0.186713 + 0.982414i \(0.440217\pi\)
\(480\) 0 0
\(481\) −1.62450 −0.0740710
\(482\) 3.02573 0.137818
\(483\) 0 0
\(484\) 0 0
\(485\) 66.0673 2.99996
\(486\) 0 0
\(487\) −0.999232 −0.0452795 −0.0226398 0.999744i \(-0.507207\pi\)
−0.0226398 + 0.999744i \(0.507207\pi\)
\(488\) 5.99147 0.271221
\(489\) 0 0
\(490\) 0.711349 0.0321355
\(491\) 24.2957 1.09645 0.548224 0.836331i \(-0.315304\pi\)
0.548224 + 0.836331i \(0.315304\pi\)
\(492\) 0 0
\(493\) −54.8595 −2.47075
\(494\) −0.944272 −0.0424848
\(495\) 0 0
\(496\) 16.6571 0.747927
\(497\) 0.794487 0.0356376
\(498\) 0 0
\(499\) −41.0807 −1.83902 −0.919512 0.393061i \(-0.871416\pi\)
−0.919512 + 0.393061i \(0.871416\pi\)
\(500\) −80.0611 −3.58044
\(501\) 0 0
\(502\) −3.05636 −0.136412
\(503\) −10.3270 −0.460457 −0.230228 0.973137i \(-0.573947\pi\)
−0.230228 + 0.973137i \(0.573947\pi\)
\(504\) 0 0
\(505\) 25.2729 1.12463
\(506\) 0 0
\(507\) 0 0
\(508\) −31.6807 −1.40561
\(509\) 40.6105 1.80003 0.900014 0.435860i \(-0.143556\pi\)
0.900014 + 0.435860i \(0.143556\pi\)
\(510\) 0 0
\(511\) −2.38705 −0.105597
\(512\) 12.2990 0.543546
\(513\) 0 0
\(514\) 1.67571 0.0739122
\(515\) 33.8439 1.49134
\(516\) 0 0
\(517\) 0 0
\(518\) 0.157193 0.00690666
\(519\) 0 0
\(520\) 4.73670 0.207718
\(521\) 22.2145 0.973234 0.486617 0.873615i \(-0.338231\pi\)
0.486617 + 0.873615i \(0.338231\pi\)
\(522\) 0 0
\(523\) 22.6653 0.991085 0.495543 0.868584i \(-0.334969\pi\)
0.495543 + 0.868584i \(0.334969\pi\)
\(524\) −1.48055 −0.0646781
\(525\) 0 0
\(526\) −0.672876 −0.0293388
\(527\) 26.8158 1.16811
\(528\) 0 0
\(529\) −22.6767 −0.985943
\(530\) 1.69217 0.0735032
\(531\) 0 0
\(532\) −6.85919 −0.297384
\(533\) −9.68098 −0.419330
\(534\) 0 0
\(535\) −72.2260 −3.12260
\(536\) 4.60226 0.198787
\(537\) 0 0
\(538\) 0.820835 0.0353887
\(539\) 0 0
\(540\) 0 0
\(541\) 6.26662 0.269423 0.134712 0.990885i \(-0.456989\pi\)
0.134712 + 0.990885i \(0.456989\pi\)
\(542\) 4.26707 0.183286
\(543\) 0 0
\(544\) 11.8274 0.507097
\(545\) −42.7455 −1.83102
\(546\) 0 0
\(547\) −34.0916 −1.45765 −0.728827 0.684698i \(-0.759934\pi\)
−0.728827 + 0.684698i \(0.759934\pi\)
\(548\) −25.6745 −1.09676
\(549\) 0 0
\(550\) 0 0
\(551\) −30.8168 −1.31284
\(552\) 0 0
\(553\) 0.670617 0.0285176
\(554\) −4.18643 −0.177865
\(555\) 0 0
\(556\) 7.69968 0.326539
\(557\) −26.3992 −1.11857 −0.559284 0.828976i \(-0.688924\pi\)
−0.559284 + 0.828976i \(0.688924\pi\)
\(558\) 0 0
\(559\) −8.45205 −0.357484
\(560\) 16.8592 0.712431
\(561\) 0 0
\(562\) −0.881827 −0.0371976
\(563\) −6.85125 −0.288746 −0.144373 0.989523i \(-0.546116\pi\)
−0.144373 + 0.989523i \(0.546116\pi\)
\(564\) 0 0
\(565\) 12.5349 0.527347
\(566\) −4.11593 −0.173005
\(567\) 0 0
\(568\) −0.511908 −0.0214792
\(569\) 18.4008 0.771404 0.385702 0.922623i \(-0.373959\pi\)
0.385702 + 0.922623i \(0.373959\pi\)
\(570\) 0 0
\(571\) −0.336889 −0.0140984 −0.00704918 0.999975i \(-0.502244\pi\)
−0.00704918 + 0.999975i \(0.502244\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0.936765 0.0390998
\(575\) −8.10035 −0.337808
\(576\) 0 0
\(577\) −25.8347 −1.07551 −0.537757 0.843100i \(-0.680728\pi\)
−0.537757 + 0.843100i \(0.680728\pi\)
\(578\) 3.44958 0.143484
\(579\) 0 0
\(580\) 76.7809 3.18815
\(581\) −9.28503 −0.385208
\(582\) 0 0
\(583\) 0 0
\(584\) 1.53804 0.0636446
\(585\) 0 0
\(586\) 1.09515 0.0452401
\(587\) 3.73415 0.154125 0.0770623 0.997026i \(-0.475446\pi\)
0.0770623 + 0.997026i \(0.475446\pi\)
\(588\) 0 0
\(589\) 15.0635 0.620681
\(590\) 2.02664 0.0834355
\(591\) 0 0
\(592\) 3.72552 0.153118
\(593\) 17.5647 0.721295 0.360648 0.932702i \(-0.382556\pi\)
0.360648 + 0.932702i \(0.382556\pi\)
\(594\) 0 0
\(595\) 27.1411 1.11268
\(596\) −36.5993 −1.49917
\(597\) 0 0
\(598\) 0.154494 0.00631771
\(599\) −11.3088 −0.462066 −0.231033 0.972946i \(-0.574211\pi\)
−0.231033 + 0.972946i \(0.574211\pi\)
\(600\) 0 0
\(601\) 28.9821 1.18220 0.591102 0.806596i \(-0.298693\pi\)
0.591102 + 0.806596i \(0.298693\pi\)
\(602\) 0.817850 0.0333331
\(603\) 0 0
\(604\) 41.0490 1.67026
\(605\) 0 0
\(606\) 0 0
\(607\) −7.34347 −0.298062 −0.149031 0.988832i \(-0.547615\pi\)
−0.149031 + 0.988832i \(0.547615\pi\)
\(608\) 6.64395 0.269448
\(609\) 0 0
\(610\) −6.61471 −0.267822
\(611\) 7.82984 0.316761
\(612\) 0 0
\(613\) −18.9535 −0.765526 −0.382763 0.923847i \(-0.625027\pi\)
−0.382763 + 0.923847i \(0.625027\pi\)
\(614\) 2.86021 0.115429
\(615\) 0 0
\(616\) 0 0
\(617\) −25.2571 −1.01681 −0.508407 0.861117i \(-0.669766\pi\)
−0.508407 + 0.861117i \(0.669766\pi\)
\(618\) 0 0
\(619\) −15.5725 −0.625912 −0.312956 0.949768i \(-0.601319\pi\)
−0.312956 + 0.949768i \(0.601319\pi\)
\(620\) −37.5312 −1.50729
\(621\) 0 0
\(622\) 1.98492 0.0795879
\(623\) 12.2106 0.489207
\(624\) 0 0
\(625\) 106.724 4.26897
\(626\) 0.512306 0.0204759
\(627\) 0 0
\(628\) −11.4668 −0.457573
\(629\) 5.99760 0.239140
\(630\) 0 0
\(631\) −36.0169 −1.43381 −0.716905 0.697170i \(-0.754442\pi\)
−0.716905 + 0.697170i \(0.754442\pi\)
\(632\) −0.432096 −0.0171879
\(633\) 0 0
\(634\) −1.36749 −0.0543100
\(635\) 70.4183 2.79446
\(636\) 0 0
\(637\) −1.67571 −0.0663939
\(638\) 0 0
\(639\) 0 0
\(640\) −22.0209 −0.870454
\(641\) 26.7410 1.05620 0.528102 0.849181i \(-0.322904\pi\)
0.528102 + 0.849181i \(0.322904\pi\)
\(642\) 0 0
\(643\) 7.31364 0.288422 0.144211 0.989547i \(-0.453936\pi\)
0.144211 + 0.989547i \(0.453936\pi\)
\(644\) 1.12224 0.0442225
\(645\) 0 0
\(646\) 3.48621 0.137163
\(647\) 35.0221 1.37686 0.688431 0.725302i \(-0.258300\pi\)
0.688431 + 0.725302i \(0.258300\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −3.87086 −0.151828
\(651\) 0 0
\(652\) 17.5721 0.688175
\(653\) 4.05138 0.158543 0.0792714 0.996853i \(-0.474741\pi\)
0.0792714 + 0.996853i \(0.474741\pi\)
\(654\) 0 0
\(655\) 3.29089 0.128586
\(656\) 22.2016 0.866828
\(657\) 0 0
\(658\) −0.757643 −0.0295360
\(659\) 28.6360 1.11550 0.557750 0.830009i \(-0.311665\pi\)
0.557750 + 0.830009i \(0.311665\pi\)
\(660\) 0 0
\(661\) 45.2741 1.76096 0.880479 0.474085i \(-0.157221\pi\)
0.880479 + 0.474085i \(0.157221\pi\)
\(662\) 0.994424 0.0386494
\(663\) 0 0
\(664\) 5.98258 0.232169
\(665\) 15.2462 0.591224
\(666\) 0 0
\(667\) 5.04197 0.195226
\(668\) 7.54657 0.291985
\(669\) 0 0
\(670\) −5.08099 −0.196296
\(671\) 0 0
\(672\) 0 0
\(673\) −0.0509075 −0.00196234 −0.000981170 1.00000i \(-0.500312\pi\)
−0.000981170 1.00000i \(0.500312\pi\)
\(674\) 4.84484 0.186616
\(675\) 0 0
\(676\) 20.1161 0.773694
\(677\) 39.1437 1.50442 0.752208 0.658925i \(-0.228989\pi\)
0.752208 + 0.658925i \(0.228989\pi\)
\(678\) 0 0
\(679\) 15.0596 0.577935
\(680\) −17.4877 −0.670622
\(681\) 0 0
\(682\) 0 0
\(683\) −15.4140 −0.589800 −0.294900 0.955528i \(-0.595286\pi\)
−0.294900 + 0.955528i \(0.595286\pi\)
\(684\) 0 0
\(685\) 57.0678 2.18045
\(686\) 0.162147 0.00619081
\(687\) 0 0
\(688\) 19.3833 0.738982
\(689\) −3.98620 −0.151862
\(690\) 0 0
\(691\) −38.2746 −1.45603 −0.728017 0.685559i \(-0.759558\pi\)
−0.728017 + 0.685559i \(0.759558\pi\)
\(692\) −3.45520 −0.131347
\(693\) 0 0
\(694\) −4.73010 −0.179552
\(695\) −17.1144 −0.649188
\(696\) 0 0
\(697\) 35.7417 1.35381
\(698\) −2.17155 −0.0821942
\(699\) 0 0
\(700\) −28.1179 −1.06276
\(701\) −1.38873 −0.0524516 −0.0262258 0.999656i \(-0.508349\pi\)
−0.0262258 + 0.999656i \(0.508349\pi\)
\(702\) 0 0
\(703\) 3.36909 0.127068
\(704\) 0 0
\(705\) 0 0
\(706\) 3.45658 0.130090
\(707\) 5.76079 0.216657
\(708\) 0 0
\(709\) −12.8361 −0.482071 −0.241036 0.970516i \(-0.577487\pi\)
−0.241036 + 0.970516i \(0.577487\pi\)
\(710\) 0.565157 0.0212100
\(711\) 0 0
\(712\) −7.86760 −0.294851
\(713\) −2.46456 −0.0922985
\(714\) 0 0
\(715\) 0 0
\(716\) −25.3702 −0.948131
\(717\) 0 0
\(718\) −3.34403 −0.124798
\(719\) 21.5055 0.802021 0.401011 0.916073i \(-0.368659\pi\)
0.401011 + 0.916073i \(0.368659\pi\)
\(720\) 0 0
\(721\) 7.71449 0.287303
\(722\) −1.12245 −0.0417733
\(723\) 0 0
\(724\) 43.5856 1.61985
\(725\) −126.327 −4.69168
\(726\) 0 0
\(727\) −5.35770 −0.198706 −0.0993531 0.995052i \(-0.531677\pi\)
−0.0993531 + 0.995052i \(0.531677\pi\)
\(728\) 1.07970 0.0400164
\(729\) 0 0
\(730\) −1.69803 −0.0628468
\(731\) 31.2046 1.15414
\(732\) 0 0
\(733\) −41.4506 −1.53101 −0.765506 0.643429i \(-0.777511\pi\)
−0.765506 + 0.643429i \(0.777511\pi\)
\(734\) 5.32727 0.196633
\(735\) 0 0
\(736\) −1.08703 −0.0400683
\(737\) 0 0
\(738\) 0 0
\(739\) −42.6086 −1.56738 −0.783691 0.621151i \(-0.786665\pi\)
−0.783691 + 0.621151i \(0.786665\pi\)
\(740\) −8.39420 −0.308577
\(741\) 0 0
\(742\) 0.385719 0.0141602
\(743\) −30.4479 −1.11702 −0.558512 0.829496i \(-0.688628\pi\)
−0.558512 + 0.829496i \(0.688628\pi\)
\(744\) 0 0
\(745\) 81.3510 2.98047
\(746\) −4.23171 −0.154934
\(747\) 0 0
\(748\) 0 0
\(749\) −16.4634 −0.601561
\(750\) 0 0
\(751\) 9.06635 0.330836 0.165418 0.986224i \(-0.447103\pi\)
0.165418 + 0.986224i \(0.447103\pi\)
\(752\) −17.9564 −0.654802
\(753\) 0 0
\(754\) 2.40938 0.0877443
\(755\) −91.2415 −3.32062
\(756\) 0 0
\(757\) 7.83076 0.284614 0.142307 0.989823i \(-0.454548\pi\)
0.142307 + 0.989823i \(0.454548\pi\)
\(758\) −1.89279 −0.0687493
\(759\) 0 0
\(760\) −9.82355 −0.356338
\(761\) 26.4983 0.960564 0.480282 0.877114i \(-0.340534\pi\)
0.480282 + 0.877114i \(0.340534\pi\)
\(762\) 0 0
\(763\) −9.74355 −0.352740
\(764\) 29.0450 1.05081
\(765\) 0 0
\(766\) 2.49222 0.0900476
\(767\) −4.77411 −0.172383
\(768\) 0 0
\(769\) 26.3995 0.951989 0.475995 0.879448i \(-0.342088\pi\)
0.475995 + 0.879448i \(0.342088\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 16.8019 0.604714
\(773\) −34.1566 −1.22853 −0.614264 0.789100i \(-0.710547\pi\)
−0.614264 + 0.789100i \(0.710547\pi\)
\(774\) 0 0
\(775\) 61.7499 2.21812
\(776\) −9.70330 −0.348328
\(777\) 0 0
\(778\) 1.56379 0.0560645
\(779\) 20.0776 0.719354
\(780\) 0 0
\(781\) 0 0
\(782\) −0.570383 −0.0203969
\(783\) 0 0
\(784\) 3.84294 0.137248
\(785\) 25.4877 0.909695
\(786\) 0 0
\(787\) −11.7503 −0.418853 −0.209426 0.977824i \(-0.567160\pi\)
−0.209426 + 0.977824i \(0.567160\pi\)
\(788\) 9.32128 0.332057
\(789\) 0 0
\(790\) 0.477043 0.0169724
\(791\) 2.85725 0.101592
\(792\) 0 0
\(793\) 15.5821 0.553337
\(794\) 3.28924 0.116731
\(795\) 0 0
\(796\) −33.0832 −1.17260
\(797\) −15.3027 −0.542050 −0.271025 0.962572i \(-0.587363\pi\)
−0.271025 + 0.962572i \(0.587363\pi\)
\(798\) 0 0
\(799\) −28.9074 −1.02267
\(800\) 27.2356 0.962924
\(801\) 0 0
\(802\) 1.32256 0.0467013
\(803\) 0 0
\(804\) 0 0
\(805\) −2.49446 −0.0879181
\(806\) −1.17772 −0.0414835
\(807\) 0 0
\(808\) −3.71182 −0.130581
\(809\) 18.2311 0.640973 0.320486 0.947253i \(-0.396154\pi\)
0.320486 + 0.947253i \(0.396154\pi\)
\(810\) 0 0
\(811\) −6.23292 −0.218868 −0.109434 0.993994i \(-0.534904\pi\)
−0.109434 + 0.993994i \(0.534904\pi\)
\(812\) 17.5017 0.614189
\(813\) 0 0
\(814\) 0 0
\(815\) −39.0582 −1.36815
\(816\) 0 0
\(817\) 17.5289 0.613258
\(818\) 1.56830 0.0548344
\(819\) 0 0
\(820\) −50.0239 −1.74691
\(821\) −29.6399 −1.03444 −0.517219 0.855853i \(-0.673033\pi\)
−0.517219 + 0.855853i \(0.673033\pi\)
\(822\) 0 0
\(823\) −19.7302 −0.687753 −0.343876 0.939015i \(-0.611740\pi\)
−0.343876 + 0.939015i \(0.611740\pi\)
\(824\) −4.97065 −0.173161
\(825\) 0 0
\(826\) 0.461960 0.0160736
\(827\) 41.1710 1.43166 0.715828 0.698276i \(-0.246049\pi\)
0.715828 + 0.698276i \(0.246049\pi\)
\(828\) 0 0
\(829\) 22.7961 0.791742 0.395871 0.918306i \(-0.370443\pi\)
0.395871 + 0.918306i \(0.370443\pi\)
\(830\) −6.60489 −0.229259
\(831\) 0 0
\(832\) 12.3598 0.428500
\(833\) 6.18663 0.214354
\(834\) 0 0
\(835\) −16.7741 −0.580492
\(836\) 0 0
\(837\) 0 0
\(838\) 3.07038 0.106064
\(839\) 47.2725 1.63203 0.816013 0.578033i \(-0.196180\pi\)
0.816013 + 0.578033i \(0.196180\pi\)
\(840\) 0 0
\(841\) 49.6312 1.71142
\(842\) −2.12331 −0.0731742
\(843\) 0 0
\(844\) −20.0241 −0.689258
\(845\) −44.7129 −1.53817
\(846\) 0 0
\(847\) 0 0
\(848\) 9.14167 0.313926
\(849\) 0 0
\(850\) 14.2910 0.490179
\(851\) −0.551222 −0.0188956
\(852\) 0 0
\(853\) −10.8549 −0.371664 −0.185832 0.982582i \(-0.559498\pi\)
−0.185832 + 0.982582i \(0.559498\pi\)
\(854\) −1.50778 −0.0515951
\(855\) 0 0
\(856\) 10.6078 0.362567
\(857\) 28.5431 0.975014 0.487507 0.873119i \(-0.337906\pi\)
0.487507 + 0.873119i \(0.337906\pi\)
\(858\) 0 0
\(859\) −36.8034 −1.25572 −0.627858 0.778328i \(-0.716068\pi\)
−0.627858 + 0.778328i \(0.716068\pi\)
\(860\) −43.6737 −1.48926
\(861\) 0 0
\(862\) −2.64356 −0.0900399
\(863\) 45.1551 1.53710 0.768549 0.639790i \(-0.220979\pi\)
0.768549 + 0.639790i \(0.220979\pi\)
\(864\) 0 0
\(865\) 7.68003 0.261129
\(866\) 1.89457 0.0643800
\(867\) 0 0
\(868\) −8.55498 −0.290375
\(869\) 0 0
\(870\) 0 0
\(871\) 11.9692 0.405559
\(872\) 6.27802 0.212601
\(873\) 0 0
\(874\) −0.320407 −0.0108379
\(875\) 40.5638 1.37131
\(876\) 0 0
\(877\) −34.8517 −1.17686 −0.588428 0.808549i \(-0.700253\pi\)
−0.588428 + 0.808549i \(0.700253\pi\)
\(878\) −1.89991 −0.0641187
\(879\) 0 0
\(880\) 0 0
\(881\) 35.3563 1.19118 0.595592 0.803287i \(-0.296917\pi\)
0.595592 + 0.803287i \(0.296917\pi\)
\(882\) 0 0
\(883\) −24.8870 −0.837516 −0.418758 0.908098i \(-0.637534\pi\)
−0.418758 + 0.908098i \(0.637534\pi\)
\(884\) 20.4614 0.688190
\(885\) 0 0
\(886\) 2.03772 0.0684586
\(887\) −49.5708 −1.66442 −0.832212 0.554457i \(-0.812926\pi\)
−0.832212 + 0.554457i \(0.812926\pi\)
\(888\) 0 0
\(889\) 16.0514 0.538346
\(890\) 8.68599 0.291155
\(891\) 0 0
\(892\) 12.0628 0.403891
\(893\) −16.2385 −0.543399
\(894\) 0 0
\(895\) 56.3916 1.88496
\(896\) −5.01953 −0.167691
\(897\) 0 0
\(898\) 2.58398 0.0862287
\(899\) −38.4356 −1.28190
\(900\) 0 0
\(901\) 14.7169 0.490291
\(902\) 0 0
\(903\) 0 0
\(904\) −1.84100 −0.0612307
\(905\) −96.8798 −3.22039
\(906\) 0 0
\(907\) −1.42826 −0.0474246 −0.0237123 0.999719i \(-0.507549\pi\)
−0.0237123 + 0.999719i \(0.507549\pi\)
\(908\) −28.0443 −0.930684
\(909\) 0 0
\(910\) −1.19201 −0.0395148
\(911\) −54.1725 −1.79482 −0.897408 0.441202i \(-0.854552\pi\)
−0.897408 + 0.441202i \(0.854552\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 3.85821 0.127618
\(915\) 0 0
\(916\) 5.38391 0.177889
\(917\) 0.750136 0.0247717
\(918\) 0 0
\(919\) 22.9009 0.755432 0.377716 0.925921i \(-0.376710\pi\)
0.377716 + 0.925921i \(0.376710\pi\)
\(920\) 1.60724 0.0529892
\(921\) 0 0
\(922\) −3.66510 −0.120704
\(923\) −1.33133 −0.0438211
\(924\) 0 0
\(925\) 13.8110 0.454101
\(926\) 0.492536 0.0161857
\(927\) 0 0
\(928\) −16.9525 −0.556493
\(929\) 14.5306 0.476733 0.238366 0.971175i \(-0.423388\pi\)
0.238366 + 0.971175i \(0.423388\pi\)
\(930\) 0 0
\(931\) 3.47528 0.113898
\(932\) 35.1116 1.15012
\(933\) 0 0
\(934\) 3.73021 0.122056
\(935\) 0 0
\(936\) 0 0
\(937\) −30.6351 −1.00080 −0.500402 0.865793i \(-0.666814\pi\)
−0.500402 + 0.865793i \(0.666814\pi\)
\(938\) −1.15818 −0.0378158
\(939\) 0 0
\(940\) 40.4586 1.31961
\(941\) 21.0533 0.686319 0.343159 0.939277i \(-0.388503\pi\)
0.343159 + 0.939277i \(0.388503\pi\)
\(942\) 0 0
\(943\) −3.28492 −0.106972
\(944\) 10.9486 0.356346
\(945\) 0 0
\(946\) 0 0
\(947\) −0.729464 −0.0237044 −0.0118522 0.999930i \(-0.503773\pi\)
−0.0118522 + 0.999930i \(0.503773\pi\)
\(948\) 0 0
\(949\) 4.00000 0.129845
\(950\) 8.02786 0.260458
\(951\) 0 0
\(952\) −3.98620 −0.129194
\(953\) 39.0151 1.26382 0.631911 0.775041i \(-0.282271\pi\)
0.631911 + 0.775041i \(0.282271\pi\)
\(954\) 0 0
\(955\) −64.5597 −2.08910
\(956\) −25.0775 −0.811065
\(957\) 0 0
\(958\) 1.32520 0.0428153
\(959\) 13.0082 0.420058
\(960\) 0 0
\(961\) −12.2124 −0.393947
\(962\) −0.263409 −0.00849264
\(963\) 0 0
\(964\) −36.8302 −1.18622
\(965\) −37.3464 −1.20222
\(966\) 0 0
\(967\) −2.46386 −0.0792324 −0.0396162 0.999215i \(-0.512614\pi\)
−0.0396162 + 0.999215i \(0.512614\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 10.7126 0.343962
\(971\) −34.6794 −1.11291 −0.556457 0.830876i \(-0.687840\pi\)
−0.556457 + 0.830876i \(0.687840\pi\)
\(972\) 0 0
\(973\) −3.90112 −0.125064
\(974\) −0.162023 −0.00519154
\(975\) 0 0
\(976\) −35.7348 −1.14384
\(977\) 25.1338 0.804100 0.402050 0.915618i \(-0.368298\pi\)
0.402050 + 0.915618i \(0.368298\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −8.65877 −0.276594
\(981\) 0 0
\(982\) 3.93948 0.125714
\(983\) −30.3900 −0.969290 −0.484645 0.874711i \(-0.661051\pi\)
−0.484645 + 0.874711i \(0.661051\pi\)
\(984\) 0 0
\(985\) −20.7189 −0.660158
\(986\) −8.89531 −0.283284
\(987\) 0 0
\(988\) 11.4940 0.365672
\(989\) −2.86792 −0.0911947
\(990\) 0 0
\(991\) −8.09793 −0.257239 −0.128620 0.991694i \(-0.541055\pi\)
−0.128620 + 0.991694i \(0.541055\pi\)
\(992\) 8.28653 0.263097
\(993\) 0 0
\(994\) 0.128824 0.00408604
\(995\) 73.5356 2.33124
\(996\) 0 0
\(997\) 45.8866 1.45324 0.726621 0.687038i \(-0.241090\pi\)
0.726621 + 0.687038i \(0.241090\pi\)
\(998\) −6.66112 −0.210854
\(999\) 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.cl.1.2 4
3.2 odd 2 2541.2.a.bm.1.3 4
11.7 odd 10 693.2.m.f.379.2 8
11.8 odd 10 693.2.m.f.64.2 8
11.10 odd 2 7623.2.a.ci.1.3 4
33.8 even 10 231.2.j.f.64.1 8
33.29 even 10 231.2.j.f.148.1 yes 8
33.32 even 2 2541.2.a.bn.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.j.f.64.1 8 33.8 even 10
231.2.j.f.148.1 yes 8 33.29 even 10
693.2.m.f.64.2 8 11.8 odd 10
693.2.m.f.379.2 8 11.7 odd 10
2541.2.a.bm.1.3 4 3.2 odd 2
2541.2.a.bn.1.2 4 33.32 even 2
7623.2.a.ci.1.3 4 11.10 odd 2
7623.2.a.cl.1.2 4 1.1 even 1 trivial