Properties

Label 7623.2.a.cz.1.3
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 1
Dimension 12
CM no
Inner twists 2

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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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.09771\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.09771 q^{2} +2.40037 q^{4} -2.53213 q^{5} -1.00000 q^{7} -0.839859 q^{8} +O(q^{10})\) \(q-2.09771 q^{2} +2.40037 q^{4} -2.53213 q^{5} -1.00000 q^{7} -0.839859 q^{8} +5.31167 q^{10} -5.69822 q^{13} +2.09771 q^{14} -3.03896 q^{16} +0.996504 q^{17} -6.02990 q^{19} -6.07805 q^{20} -1.25785 q^{23} +1.41169 q^{25} +11.9532 q^{26} -2.40037 q^{28} +1.33880 q^{29} -1.85065 q^{31} +8.05457 q^{32} -2.09037 q^{34} +2.53213 q^{35} +9.65942 q^{37} +12.6490 q^{38} +2.12663 q^{40} +10.0523 q^{41} +5.04597 q^{43} +2.63859 q^{46} -1.03023 q^{47} +1.00000 q^{49} -2.96131 q^{50} -13.6778 q^{52} +7.43975 q^{53} +0.839859 q^{56} -2.80840 q^{58} +6.55446 q^{59} -3.76506 q^{61} +3.88212 q^{62} -10.8182 q^{64} +14.4286 q^{65} +12.0485 q^{67} +2.39198 q^{68} -5.31167 q^{70} +15.8526 q^{71} -6.41645 q^{73} -20.2626 q^{74} -14.4740 q^{76} -10.8036 q^{79} +7.69506 q^{80} -21.0868 q^{82} -14.2491 q^{83} -2.52328 q^{85} -10.5850 q^{86} -15.4991 q^{89} +5.69822 q^{91} -3.01930 q^{92} +2.16112 q^{94} +15.2685 q^{95} -2.05898 q^{97} -2.09771 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 20q^{4} - 12q^{7} + O(q^{10}) \) \( 12q + 20q^{4} - 12q^{7} - 20q^{10} - 20q^{13} + 28q^{16} - 12q^{19} + 32q^{25} - 20q^{28} - 16q^{31} - 24q^{34} + 4q^{37} - 48q^{40} - 16q^{43} - 24q^{46} + 12q^{49} - 96q^{52} + 20q^{58} - 44q^{61} + 76q^{64} + 20q^{70} - 52q^{73} + 8q^{79} - 68q^{82} - 72q^{85} + 20q^{91} - 20q^{94} - 32q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.09771 −1.48330 −0.741651 0.670786i \(-0.765957\pi\)
−0.741651 + 0.670786i \(0.765957\pi\)
\(3\) 0 0
\(4\) 2.40037 1.20019
\(5\) −2.53213 −1.13240 −0.566202 0.824267i \(-0.691588\pi\)
−0.566202 + 0.824267i \(0.691588\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −0.839859 −0.296935
\(9\) 0 0
\(10\) 5.31167 1.67970
\(11\) 0 0
\(12\) 0 0
\(13\) −5.69822 −1.58040 −0.790201 0.612848i \(-0.790024\pi\)
−0.790201 + 0.612848i \(0.790024\pi\)
\(14\) 2.09771 0.560635
\(15\) 0 0
\(16\) −3.03896 −0.759741
\(17\) 0.996504 0.241688 0.120844 0.992672i \(-0.461440\pi\)
0.120844 + 0.992672i \(0.461440\pi\)
\(18\) 0 0
\(19\) −6.02990 −1.38335 −0.691677 0.722207i \(-0.743128\pi\)
−0.691677 + 0.722207i \(0.743128\pi\)
\(20\) −6.07805 −1.35909
\(21\) 0 0
\(22\) 0 0
\(23\) −1.25785 −0.262279 −0.131140 0.991364i \(-0.541864\pi\)
−0.131140 + 0.991364i \(0.541864\pi\)
\(24\) 0 0
\(25\) 1.41169 0.282338
\(26\) 11.9532 2.34421
\(27\) 0 0
\(28\) −2.40037 −0.453627
\(29\) 1.33880 0.248608 0.124304 0.992244i \(-0.460330\pi\)
0.124304 + 0.992244i \(0.460330\pi\)
\(30\) 0 0
\(31\) −1.85065 −0.332387 −0.166193 0.986093i \(-0.553148\pi\)
−0.166193 + 0.986093i \(0.553148\pi\)
\(32\) 8.05457 1.42386
\(33\) 0 0
\(34\) −2.09037 −0.358496
\(35\) 2.53213 0.428008
\(36\) 0 0
\(37\) 9.65942 1.58800 0.794000 0.607918i \(-0.207995\pi\)
0.794000 + 0.607918i \(0.207995\pi\)
\(38\) 12.6490 2.05193
\(39\) 0 0
\(40\) 2.12663 0.336250
\(41\) 10.0523 1.56991 0.784953 0.619555i \(-0.212687\pi\)
0.784953 + 0.619555i \(0.212687\pi\)
\(42\) 0 0
\(43\) 5.04597 0.769504 0.384752 0.923020i \(-0.374287\pi\)
0.384752 + 0.923020i \(0.374287\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 2.63859 0.389039
\(47\) −1.03023 −0.150274 −0.0751371 0.997173i \(-0.523939\pi\)
−0.0751371 + 0.997173i \(0.523939\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −2.96131 −0.418793
\(51\) 0 0
\(52\) −13.6778 −1.89677
\(53\) 7.43975 1.02193 0.510964 0.859602i \(-0.329288\pi\)
0.510964 + 0.859602i \(0.329288\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0.839859 0.112231
\(57\) 0 0
\(58\) −2.80840 −0.368761
\(59\) 6.55446 0.853318 0.426659 0.904413i \(-0.359690\pi\)
0.426659 + 0.904413i \(0.359690\pi\)
\(60\) 0 0
\(61\) −3.76506 −0.482066 −0.241033 0.970517i \(-0.577486\pi\)
−0.241033 + 0.970517i \(0.577486\pi\)
\(62\) 3.88212 0.493030
\(63\) 0 0
\(64\) −10.8182 −1.35227
\(65\) 14.4286 1.78965
\(66\) 0 0
\(67\) 12.0485 1.47196 0.735978 0.677006i \(-0.236723\pi\)
0.735978 + 0.677006i \(0.236723\pi\)
\(68\) 2.39198 0.290070
\(69\) 0 0
\(70\) −5.31167 −0.634866
\(71\) 15.8526 1.88136 0.940678 0.339301i \(-0.110191\pi\)
0.940678 + 0.339301i \(0.110191\pi\)
\(72\) 0 0
\(73\) −6.41645 −0.750988 −0.375494 0.926825i \(-0.622527\pi\)
−0.375494 + 0.926825i \(0.622527\pi\)
\(74\) −20.2626 −2.35548
\(75\) 0 0
\(76\) −14.4740 −1.66028
\(77\) 0 0
\(78\) 0 0
\(79\) −10.8036 −1.21550 −0.607751 0.794128i \(-0.707928\pi\)
−0.607751 + 0.794128i \(0.707928\pi\)
\(80\) 7.69506 0.860333
\(81\) 0 0
\(82\) −21.0868 −2.32865
\(83\) −14.2491 −1.56404 −0.782020 0.623253i \(-0.785811\pi\)
−0.782020 + 0.623253i \(0.785811\pi\)
\(84\) 0 0
\(85\) −2.52328 −0.273688
\(86\) −10.5850 −1.14141
\(87\) 0 0
\(88\) 0 0
\(89\) −15.4991 −1.64290 −0.821451 0.570279i \(-0.806835\pi\)
−0.821451 + 0.570279i \(0.806835\pi\)
\(90\) 0 0
\(91\) 5.69822 0.597335
\(92\) −3.01930 −0.314784
\(93\) 0 0
\(94\) 2.16112 0.222902
\(95\) 15.2685 1.56651
\(96\) 0 0
\(97\) −2.05898 −0.209057 −0.104529 0.994522i \(-0.533333\pi\)
−0.104529 + 0.994522i \(0.533333\pi\)
\(98\) −2.09771 −0.211900
\(99\) 0 0
\(100\) 3.38858 0.338858
\(101\) 4.79295 0.476916 0.238458 0.971153i \(-0.423358\pi\)
0.238458 + 0.971153i \(0.423358\pi\)
\(102\) 0 0
\(103\) −7.54804 −0.743730 −0.371865 0.928287i \(-0.621282\pi\)
−0.371865 + 0.928287i \(0.621282\pi\)
\(104\) 4.78570 0.469276
\(105\) 0 0
\(106\) −15.6064 −1.51583
\(107\) 0.498937 0.0482341 0.0241170 0.999709i \(-0.492323\pi\)
0.0241170 + 0.999709i \(0.492323\pi\)
\(108\) 0 0
\(109\) −5.80710 −0.556220 −0.278110 0.960549i \(-0.589708\pi\)
−0.278110 + 0.960549i \(0.589708\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 3.03896 0.287155
\(113\) 13.2820 1.24947 0.624733 0.780839i \(-0.285208\pi\)
0.624733 + 0.780839i \(0.285208\pi\)
\(114\) 0 0
\(115\) 3.18503 0.297006
\(116\) 3.21361 0.298376
\(117\) 0 0
\(118\) −13.7493 −1.26573
\(119\) −0.996504 −0.0913494
\(120\) 0 0
\(121\) 0 0
\(122\) 7.89798 0.715049
\(123\) 0 0
\(124\) −4.44225 −0.398926
\(125\) 9.08607 0.812683
\(126\) 0 0
\(127\) 6.00223 0.532611 0.266306 0.963889i \(-0.414197\pi\)
0.266306 + 0.963889i \(0.414197\pi\)
\(128\) 6.58425 0.581971
\(129\) 0 0
\(130\) −30.2670 −2.65459
\(131\) 12.5395 1.09558 0.547792 0.836615i \(-0.315469\pi\)
0.547792 + 0.836615i \(0.315469\pi\)
\(132\) 0 0
\(133\) 6.02990 0.522858
\(134\) −25.2742 −2.18335
\(135\) 0 0
\(136\) −0.836923 −0.0717656
\(137\) −16.0274 −1.36931 −0.684657 0.728866i \(-0.740048\pi\)
−0.684657 + 0.728866i \(0.740048\pi\)
\(138\) 0 0
\(139\) −8.31663 −0.705407 −0.352704 0.935735i \(-0.614738\pi\)
−0.352704 + 0.935735i \(0.614738\pi\)
\(140\) 6.07805 0.513689
\(141\) 0 0
\(142\) −33.2541 −2.79062
\(143\) 0 0
\(144\) 0 0
\(145\) −3.39001 −0.281525
\(146\) 13.4598 1.11394
\(147\) 0 0
\(148\) 23.1862 1.90589
\(149\) −9.91993 −0.812673 −0.406336 0.913724i \(-0.633194\pi\)
−0.406336 + 0.913724i \(0.633194\pi\)
\(150\) 0 0
\(151\) 15.9065 1.29445 0.647227 0.762298i \(-0.275929\pi\)
0.647227 + 0.762298i \(0.275929\pi\)
\(152\) 5.06426 0.410766
\(153\) 0 0
\(154\) 0 0
\(155\) 4.68609 0.376396
\(156\) 0 0
\(157\) 10.9327 0.872525 0.436262 0.899819i \(-0.356302\pi\)
0.436262 + 0.899819i \(0.356302\pi\)
\(158\) 22.6628 1.80296
\(159\) 0 0
\(160\) −20.3952 −1.61238
\(161\) 1.25785 0.0991322
\(162\) 0 0
\(163\) 5.77018 0.451956 0.225978 0.974132i \(-0.427442\pi\)
0.225978 + 0.974132i \(0.427442\pi\)
\(164\) 24.1293 1.88418
\(165\) 0 0
\(166\) 29.8904 2.31995
\(167\) −3.43598 −0.265884 −0.132942 0.991124i \(-0.542442\pi\)
−0.132942 + 0.991124i \(0.542442\pi\)
\(168\) 0 0
\(169\) 19.4697 1.49767
\(170\) 5.29310 0.405962
\(171\) 0 0
\(172\) 12.1122 0.923547
\(173\) −16.6405 −1.26515 −0.632577 0.774498i \(-0.718003\pi\)
−0.632577 + 0.774498i \(0.718003\pi\)
\(174\) 0 0
\(175\) −1.41169 −0.106714
\(176\) 0 0
\(177\) 0 0
\(178\) 32.5126 2.43692
\(179\) 8.87894 0.663643 0.331821 0.943342i \(-0.392337\pi\)
0.331821 + 0.943342i \(0.392337\pi\)
\(180\) 0 0
\(181\) 0.982518 0.0730300 0.0365150 0.999333i \(-0.488374\pi\)
0.0365150 + 0.999333i \(0.488374\pi\)
\(182\) −11.9532 −0.886029
\(183\) 0 0
\(184\) 1.05641 0.0778799
\(185\) −24.4589 −1.79826
\(186\) 0 0
\(187\) 0 0
\(188\) −2.47293 −0.180357
\(189\) 0 0
\(190\) −32.0288 −2.32361
\(191\) 19.0702 1.37987 0.689937 0.723869i \(-0.257638\pi\)
0.689937 + 0.723869i \(0.257638\pi\)
\(192\) 0 0
\(193\) −4.47300 −0.321973 −0.160987 0.986957i \(-0.551468\pi\)
−0.160987 + 0.986957i \(0.551468\pi\)
\(194\) 4.31913 0.310095
\(195\) 0 0
\(196\) 2.40037 0.171455
\(197\) −9.54397 −0.679980 −0.339990 0.940429i \(-0.610424\pi\)
−0.339990 + 0.940429i \(0.610424\pi\)
\(198\) 0 0
\(199\) 22.7626 1.61360 0.806798 0.590828i \(-0.201199\pi\)
0.806798 + 0.590828i \(0.201199\pi\)
\(200\) −1.18562 −0.0838361
\(201\) 0 0
\(202\) −10.0542 −0.707410
\(203\) −1.33880 −0.0939651
\(204\) 0 0
\(205\) −25.4538 −1.77777
\(206\) 15.8336 1.10318
\(207\) 0 0
\(208\) 17.3167 1.20070
\(209\) 0 0
\(210\) 0 0
\(211\) −11.8991 −0.819170 −0.409585 0.912272i \(-0.634326\pi\)
−0.409585 + 0.912272i \(0.634326\pi\)
\(212\) 17.8582 1.22650
\(213\) 0 0
\(214\) −1.04662 −0.0715457
\(215\) −12.7771 −0.871389
\(216\) 0 0
\(217\) 1.85065 0.125630
\(218\) 12.1816 0.825042
\(219\) 0 0
\(220\) 0 0
\(221\) −5.67830 −0.381964
\(222\) 0 0
\(223\) −13.9876 −0.936681 −0.468340 0.883548i \(-0.655148\pi\)
−0.468340 + 0.883548i \(0.655148\pi\)
\(224\) −8.05457 −0.538169
\(225\) 0 0
\(226\) −27.8617 −1.85334
\(227\) 19.1843 1.27331 0.636654 0.771149i \(-0.280318\pi\)
0.636654 + 0.771149i \(0.280318\pi\)
\(228\) 0 0
\(229\) 11.1268 0.735280 0.367640 0.929968i \(-0.380166\pi\)
0.367640 + 0.929968i \(0.380166\pi\)
\(230\) −6.68127 −0.440550
\(231\) 0 0
\(232\) −1.12440 −0.0738205
\(233\) 17.2671 1.13120 0.565602 0.824678i \(-0.308644\pi\)
0.565602 + 0.824678i \(0.308644\pi\)
\(234\) 0 0
\(235\) 2.60867 0.170171
\(236\) 15.7331 1.02414
\(237\) 0 0
\(238\) 2.09037 0.135499
\(239\) 5.85774 0.378906 0.189453 0.981890i \(-0.439329\pi\)
0.189453 + 0.981890i \(0.439329\pi\)
\(240\) 0 0
\(241\) −7.53402 −0.485309 −0.242655 0.970113i \(-0.578018\pi\)
−0.242655 + 0.970113i \(0.578018\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −9.03753 −0.578568
\(245\) −2.53213 −0.161772
\(246\) 0 0
\(247\) 34.3597 2.18625
\(248\) 1.55429 0.0986972
\(249\) 0 0
\(250\) −19.0599 −1.20545
\(251\) 1.32063 0.0833572 0.0416786 0.999131i \(-0.486729\pi\)
0.0416786 + 0.999131i \(0.486729\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −12.5909 −0.790024
\(255\) 0 0
\(256\) 7.82457 0.489036
\(257\) −0.240009 −0.0149713 −0.00748567 0.999972i \(-0.502383\pi\)
−0.00748567 + 0.999972i \(0.502383\pi\)
\(258\) 0 0
\(259\) −9.65942 −0.600207
\(260\) 34.6341 2.14791
\(261\) 0 0
\(262\) −26.3043 −1.62508
\(263\) −25.8712 −1.59528 −0.797642 0.603131i \(-0.793920\pi\)
−0.797642 + 0.603131i \(0.793920\pi\)
\(264\) 0 0
\(265\) −18.8384 −1.15724
\(266\) −12.6490 −0.775557
\(267\) 0 0
\(268\) 28.9208 1.76662
\(269\) 19.5864 1.19420 0.597102 0.802165i \(-0.296319\pi\)
0.597102 + 0.802165i \(0.296319\pi\)
\(270\) 0 0
\(271\) 20.2033 1.22727 0.613633 0.789592i \(-0.289708\pi\)
0.613633 + 0.789592i \(0.289708\pi\)
\(272\) −3.02834 −0.183620
\(273\) 0 0
\(274\) 33.6208 2.03111
\(275\) 0 0
\(276\) 0 0
\(277\) 25.6837 1.54318 0.771592 0.636118i \(-0.219461\pi\)
0.771592 + 0.636118i \(0.219461\pi\)
\(278\) 17.4458 1.04633
\(279\) 0 0
\(280\) −2.12663 −0.127091
\(281\) −28.3537 −1.69144 −0.845719 0.533628i \(-0.820828\pi\)
−0.845719 + 0.533628i \(0.820828\pi\)
\(282\) 0 0
\(283\) 3.88445 0.230907 0.115453 0.993313i \(-0.463168\pi\)
0.115453 + 0.993313i \(0.463168\pi\)
\(284\) 38.0521 2.25797
\(285\) 0 0
\(286\) 0 0
\(287\) −10.0523 −0.593369
\(288\) 0 0
\(289\) −16.0070 −0.941587
\(290\) 7.11124 0.417586
\(291\) 0 0
\(292\) −15.4018 −0.901325
\(293\) 12.4915 0.729759 0.364879 0.931055i \(-0.381110\pi\)
0.364879 + 0.931055i \(0.381110\pi\)
\(294\) 0 0
\(295\) −16.5968 −0.966301
\(296\) −8.11255 −0.471533
\(297\) 0 0
\(298\) 20.8091 1.20544
\(299\) 7.16748 0.414506
\(300\) 0 0
\(301\) −5.04597 −0.290845
\(302\) −33.3672 −1.92007
\(303\) 0 0
\(304\) 18.3246 1.05099
\(305\) 9.53362 0.545893
\(306\) 0 0
\(307\) −10.6108 −0.605592 −0.302796 0.953055i \(-0.597920\pi\)
−0.302796 + 0.953055i \(0.597920\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −9.83004 −0.558309
\(311\) −25.6921 −1.45686 −0.728432 0.685118i \(-0.759751\pi\)
−0.728432 + 0.685118i \(0.759751\pi\)
\(312\) 0 0
\(313\) −23.7556 −1.34274 −0.671372 0.741121i \(-0.734295\pi\)
−0.671372 + 0.741121i \(0.734295\pi\)
\(314\) −22.9336 −1.29422
\(315\) 0 0
\(316\) −25.9327 −1.45883
\(317\) 27.9316 1.56880 0.784398 0.620258i \(-0.212972\pi\)
0.784398 + 0.620258i \(0.212972\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 27.3931 1.53132
\(321\) 0 0
\(322\) −2.63859 −0.147043
\(323\) −6.00882 −0.334340
\(324\) 0 0
\(325\) −8.04412 −0.446208
\(326\) −12.1042 −0.670387
\(327\) 0 0
\(328\) −8.44252 −0.466160
\(329\) 1.03023 0.0567983
\(330\) 0 0
\(331\) −10.6516 −0.585462 −0.292731 0.956195i \(-0.594564\pi\)
−0.292731 + 0.956195i \(0.594564\pi\)
\(332\) −34.2031 −1.87714
\(333\) 0 0
\(334\) 7.20768 0.394386
\(335\) −30.5083 −1.66685
\(336\) 0 0
\(337\) 17.6414 0.960990 0.480495 0.876997i \(-0.340457\pi\)
0.480495 + 0.876997i \(0.340457\pi\)
\(338\) −40.8417 −2.22149
\(339\) 0 0
\(340\) −6.05681 −0.328476
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −4.23791 −0.228493
\(345\) 0 0
\(346\) 34.9069 1.87660
\(347\) −31.7274 −1.70322 −0.851608 0.524180i \(-0.824372\pi\)
−0.851608 + 0.524180i \(0.824372\pi\)
\(348\) 0 0
\(349\) −24.2636 −1.29880 −0.649401 0.760447i \(-0.724980\pi\)
−0.649401 + 0.760447i \(0.724980\pi\)
\(350\) 2.96131 0.158289
\(351\) 0 0
\(352\) 0 0
\(353\) −35.1150 −1.86898 −0.934491 0.355986i \(-0.884145\pi\)
−0.934491 + 0.355986i \(0.884145\pi\)
\(354\) 0 0
\(355\) −40.1408 −2.13045
\(356\) −37.2036 −1.97179
\(357\) 0 0
\(358\) −18.6254 −0.984383
\(359\) −31.7004 −1.67308 −0.836541 0.547904i \(-0.815426\pi\)
−0.836541 + 0.547904i \(0.815426\pi\)
\(360\) 0 0
\(361\) 17.3597 0.913667
\(362\) −2.06103 −0.108325
\(363\) 0 0
\(364\) 13.6778 0.716913
\(365\) 16.2473 0.850422
\(366\) 0 0
\(367\) −6.48862 −0.338703 −0.169352 0.985556i \(-0.554167\pi\)
−0.169352 + 0.985556i \(0.554167\pi\)
\(368\) 3.82255 0.199264
\(369\) 0 0
\(370\) 51.3077 2.66736
\(371\) −7.43975 −0.386253
\(372\) 0 0
\(373\) −24.8809 −1.28828 −0.644142 0.764906i \(-0.722785\pi\)
−0.644142 + 0.764906i \(0.722785\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0.865246 0.0446217
\(377\) −7.62875 −0.392901
\(378\) 0 0
\(379\) −28.6289 −1.47057 −0.735284 0.677759i \(-0.762951\pi\)
−0.735284 + 0.677759i \(0.762951\pi\)
\(380\) 36.6500 1.88011
\(381\) 0 0
\(382\) −40.0038 −2.04677
\(383\) −8.85605 −0.452523 −0.226261 0.974067i \(-0.572650\pi\)
−0.226261 + 0.974067i \(0.572650\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 9.38303 0.477584
\(387\) 0 0
\(388\) −4.94230 −0.250908
\(389\) −2.53908 −0.128736 −0.0643682 0.997926i \(-0.520503\pi\)
−0.0643682 + 0.997926i \(0.520503\pi\)
\(390\) 0 0
\(391\) −1.25345 −0.0633897
\(392\) −0.839859 −0.0424193
\(393\) 0 0
\(394\) 20.0205 1.00862
\(395\) 27.3562 1.37644
\(396\) 0 0
\(397\) −26.9155 −1.35085 −0.675426 0.737428i \(-0.736040\pi\)
−0.675426 + 0.737428i \(0.736040\pi\)
\(398\) −47.7492 −2.39345
\(399\) 0 0
\(400\) −4.29008 −0.214504
\(401\) 28.1770 1.40709 0.703546 0.710650i \(-0.251599\pi\)
0.703546 + 0.710650i \(0.251599\pi\)
\(402\) 0 0
\(403\) 10.5454 0.525304
\(404\) 11.5048 0.572387
\(405\) 0 0
\(406\) 2.80840 0.139379
\(407\) 0 0
\(408\) 0 0
\(409\) 16.1243 0.797297 0.398648 0.917104i \(-0.369479\pi\)
0.398648 + 0.917104i \(0.369479\pi\)
\(410\) 53.3945 2.63697
\(411\) 0 0
\(412\) −18.1181 −0.892614
\(413\) −6.55446 −0.322524
\(414\) 0 0
\(415\) 36.0806 1.77113
\(416\) −45.8967 −2.25027
\(417\) 0 0
\(418\) 0 0
\(419\) 6.80618 0.332503 0.166252 0.986083i \(-0.446834\pi\)
0.166252 + 0.986083i \(0.446834\pi\)
\(420\) 0 0
\(421\) −33.7446 −1.64461 −0.822305 0.569047i \(-0.807312\pi\)
−0.822305 + 0.569047i \(0.807312\pi\)
\(422\) 24.9609 1.21508
\(423\) 0 0
\(424\) −6.24834 −0.303446
\(425\) 1.40676 0.0682377
\(426\) 0 0
\(427\) 3.76506 0.182204
\(428\) 1.19763 0.0578898
\(429\) 0 0
\(430\) 26.8025 1.29253
\(431\) 30.0834 1.44907 0.724534 0.689239i \(-0.242055\pi\)
0.724534 + 0.689239i \(0.242055\pi\)
\(432\) 0 0
\(433\) 17.7479 0.852909 0.426455 0.904509i \(-0.359762\pi\)
0.426455 + 0.904509i \(0.359762\pi\)
\(434\) −3.88212 −0.186348
\(435\) 0 0
\(436\) −13.9392 −0.667566
\(437\) 7.58469 0.362825
\(438\) 0 0
\(439\) 30.8212 1.47102 0.735508 0.677516i \(-0.236944\pi\)
0.735508 + 0.677516i \(0.236944\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 11.9114 0.566568
\(443\) −13.1717 −0.625806 −0.312903 0.949785i \(-0.601302\pi\)
−0.312903 + 0.949785i \(0.601302\pi\)
\(444\) 0 0
\(445\) 39.2458 1.86043
\(446\) 29.3419 1.38938
\(447\) 0 0
\(448\) 10.8182 0.511111
\(449\) 13.4014 0.632452 0.316226 0.948684i \(-0.397584\pi\)
0.316226 + 0.948684i \(0.397584\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 31.8817 1.49959
\(453\) 0 0
\(454\) −40.2431 −1.88870
\(455\) −14.4286 −0.676425
\(456\) 0 0
\(457\) −5.20237 −0.243356 −0.121678 0.992570i \(-0.538828\pi\)
−0.121678 + 0.992570i \(0.538828\pi\)
\(458\) −23.3408 −1.09064
\(459\) 0 0
\(460\) 7.64526 0.356462
\(461\) 28.2415 1.31534 0.657669 0.753307i \(-0.271543\pi\)
0.657669 + 0.753307i \(0.271543\pi\)
\(462\) 0 0
\(463\) −1.83204 −0.0851423 −0.0425712 0.999093i \(-0.513555\pi\)
−0.0425712 + 0.999093i \(0.513555\pi\)
\(464\) −4.06855 −0.188878
\(465\) 0 0
\(466\) −36.2213 −1.67792
\(467\) −8.10568 −0.375086 −0.187543 0.982256i \(-0.560052\pi\)
−0.187543 + 0.982256i \(0.560052\pi\)
\(468\) 0 0
\(469\) −12.0485 −0.556347
\(470\) −5.47223 −0.252415
\(471\) 0 0
\(472\) −5.50482 −0.253380
\(473\) 0 0
\(474\) 0 0
\(475\) −8.51235 −0.390574
\(476\) −2.39198 −0.109636
\(477\) 0 0
\(478\) −12.2878 −0.562032
\(479\) 20.8348 0.951965 0.475982 0.879455i \(-0.342093\pi\)
0.475982 + 0.879455i \(0.342093\pi\)
\(480\) 0 0
\(481\) −55.0415 −2.50968
\(482\) 15.8042 0.719860
\(483\) 0 0
\(484\) 0 0
\(485\) 5.21360 0.236737
\(486\) 0 0
\(487\) −20.2679 −0.918426 −0.459213 0.888326i \(-0.651868\pi\)
−0.459213 + 0.888326i \(0.651868\pi\)
\(488\) 3.16212 0.143142
\(489\) 0 0
\(490\) 5.31167 0.239957
\(491\) −36.0577 −1.62726 −0.813631 0.581382i \(-0.802512\pi\)
−0.813631 + 0.581382i \(0.802512\pi\)
\(492\) 0 0
\(493\) 1.33412 0.0600856
\(494\) −72.0765 −3.24287
\(495\) 0 0
\(496\) 5.62406 0.252528
\(497\) −15.8526 −0.711085
\(498\) 0 0
\(499\) 13.7325 0.614750 0.307375 0.951588i \(-0.400549\pi\)
0.307375 + 0.951588i \(0.400549\pi\)
\(500\) 21.8099 0.975370
\(501\) 0 0
\(502\) −2.77029 −0.123644
\(503\) 20.0298 0.893083 0.446542 0.894763i \(-0.352655\pi\)
0.446542 + 0.894763i \(0.352655\pi\)
\(504\) 0 0
\(505\) −12.1364 −0.540061
\(506\) 0 0
\(507\) 0 0
\(508\) 14.4076 0.639232
\(509\) 20.5334 0.910126 0.455063 0.890459i \(-0.349617\pi\)
0.455063 + 0.890459i \(0.349617\pi\)
\(510\) 0 0
\(511\) 6.41645 0.283847
\(512\) −29.5821 −1.30736
\(513\) 0 0
\(514\) 0.503468 0.0222070
\(515\) 19.1126 0.842203
\(516\) 0 0
\(517\) 0 0
\(518\) 20.2626 0.890289
\(519\) 0 0
\(520\) −12.1180 −0.531410
\(521\) 16.5662 0.725777 0.362888 0.931833i \(-0.381791\pi\)
0.362888 + 0.931833i \(0.381791\pi\)
\(522\) 0 0
\(523\) −10.4119 −0.455282 −0.227641 0.973745i \(-0.573101\pi\)
−0.227641 + 0.973745i \(0.573101\pi\)
\(524\) 30.0995 1.31490
\(525\) 0 0
\(526\) 54.2701 2.36629
\(527\) −1.84418 −0.0803338
\(528\) 0 0
\(529\) −21.4178 −0.931210
\(530\) 39.5175 1.71653
\(531\) 0 0
\(532\) 14.4740 0.627527
\(533\) −57.2802 −2.48108
\(534\) 0 0
\(535\) −1.26337 −0.0546204
\(536\) −10.1190 −0.437075
\(537\) 0 0
\(538\) −41.0865 −1.77137
\(539\) 0 0
\(540\) 0 0
\(541\) 40.2319 1.72970 0.864852 0.502027i \(-0.167412\pi\)
0.864852 + 0.502027i \(0.167412\pi\)
\(542\) −42.3807 −1.82041
\(543\) 0 0
\(544\) 8.02641 0.344130
\(545\) 14.7044 0.629865
\(546\) 0 0
\(547\) 10.4385 0.446320 0.223160 0.974782i \(-0.428363\pi\)
0.223160 + 0.974782i \(0.428363\pi\)
\(548\) −38.4717 −1.64343
\(549\) 0 0
\(550\) 0 0
\(551\) −8.07280 −0.343913
\(552\) 0 0
\(553\) 10.8036 0.459416
\(554\) −53.8768 −2.28901
\(555\) 0 0
\(556\) −19.9630 −0.846619
\(557\) −8.50681 −0.360445 −0.180223 0.983626i \(-0.557682\pi\)
−0.180223 + 0.983626i \(0.557682\pi\)
\(558\) 0 0
\(559\) −28.7531 −1.21612
\(560\) −7.69506 −0.325175
\(561\) 0 0
\(562\) 59.4777 2.50891
\(563\) 45.2115 1.90544 0.952718 0.303855i \(-0.0982738\pi\)
0.952718 + 0.303855i \(0.0982738\pi\)
\(564\) 0 0
\(565\) −33.6318 −1.41490
\(566\) −8.14844 −0.342505
\(567\) 0 0
\(568\) −13.3139 −0.558640
\(569\) 44.7713 1.87691 0.938454 0.345403i \(-0.112258\pi\)
0.938454 + 0.345403i \(0.112258\pi\)
\(570\) 0 0
\(571\) −43.0895 −1.80324 −0.901619 0.432532i \(-0.857620\pi\)
−0.901619 + 0.432532i \(0.857620\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 21.0868 0.880146
\(575\) −1.77569 −0.0740515
\(576\) 0 0
\(577\) −31.0429 −1.29233 −0.646166 0.763197i \(-0.723628\pi\)
−0.646166 + 0.763197i \(0.723628\pi\)
\(578\) 33.5779 1.39666
\(579\) 0 0
\(580\) −8.13727 −0.337882
\(581\) 14.2491 0.591152
\(582\) 0 0
\(583\) 0 0
\(584\) 5.38891 0.222995
\(585\) 0 0
\(586\) −26.2034 −1.08245
\(587\) −12.7914 −0.527958 −0.263979 0.964528i \(-0.585035\pi\)
−0.263979 + 0.964528i \(0.585035\pi\)
\(588\) 0 0
\(589\) 11.1592 0.459808
\(590\) 34.8151 1.43332
\(591\) 0 0
\(592\) −29.3546 −1.20647
\(593\) −19.1567 −0.786669 −0.393335 0.919395i \(-0.628679\pi\)
−0.393335 + 0.919395i \(0.628679\pi\)
\(594\) 0 0
\(595\) 2.52328 0.103444
\(596\) −23.8115 −0.975358
\(597\) 0 0
\(598\) −15.0353 −0.614838
\(599\) −18.2676 −0.746395 −0.373198 0.927752i \(-0.621739\pi\)
−0.373198 + 0.927752i \(0.621739\pi\)
\(600\) 0 0
\(601\) 3.22883 0.131707 0.0658534 0.997829i \(-0.479023\pi\)
0.0658534 + 0.997829i \(0.479023\pi\)
\(602\) 10.5850 0.431411
\(603\) 0 0
\(604\) 38.1815 1.55358
\(605\) 0 0
\(606\) 0 0
\(607\) −28.9919 −1.17675 −0.588373 0.808590i \(-0.700231\pi\)
−0.588373 + 0.808590i \(0.700231\pi\)
\(608\) −48.5682 −1.96970
\(609\) 0 0
\(610\) −19.9987 −0.809725
\(611\) 5.87046 0.237493
\(612\) 0 0
\(613\) 23.3561 0.943345 0.471672 0.881774i \(-0.343651\pi\)
0.471672 + 0.881774i \(0.343651\pi\)
\(614\) 22.2584 0.898275
\(615\) 0 0
\(616\) 0 0
\(617\) −33.0024 −1.32863 −0.664313 0.747454i \(-0.731276\pi\)
−0.664313 + 0.747454i \(0.731276\pi\)
\(618\) 0 0
\(619\) −4.67984 −0.188099 −0.0940494 0.995568i \(-0.529981\pi\)
−0.0940494 + 0.995568i \(0.529981\pi\)
\(620\) 11.2484 0.451745
\(621\) 0 0
\(622\) 53.8944 2.16097
\(623\) 15.4991 0.620959
\(624\) 0 0
\(625\) −30.0656 −1.20262
\(626\) 49.8322 1.99169
\(627\) 0 0
\(628\) 26.2425 1.04719
\(629\) 9.62566 0.383800
\(630\) 0 0
\(631\) 48.1057 1.91506 0.957530 0.288335i \(-0.0931017\pi\)
0.957530 + 0.288335i \(0.0931017\pi\)
\(632\) 9.07351 0.360925
\(633\) 0 0
\(634\) −58.5923 −2.32700
\(635\) −15.1984 −0.603131
\(636\) 0 0
\(637\) −5.69822 −0.225772
\(638\) 0 0
\(639\) 0 0
\(640\) −16.6722 −0.659026
\(641\) −19.9494 −0.787954 −0.393977 0.919120i \(-0.628901\pi\)
−0.393977 + 0.919120i \(0.628901\pi\)
\(642\) 0 0
\(643\) 9.21828 0.363534 0.181767 0.983342i \(-0.441818\pi\)
0.181767 + 0.983342i \(0.441818\pi\)
\(644\) 3.01930 0.118977
\(645\) 0 0
\(646\) 12.6047 0.495927
\(647\) −47.0275 −1.84884 −0.924421 0.381373i \(-0.875451\pi\)
−0.924421 + 0.381373i \(0.875451\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 16.8742 0.661861
\(651\) 0 0
\(652\) 13.8506 0.542431
\(653\) −29.7075 −1.16254 −0.581272 0.813709i \(-0.697445\pi\)
−0.581272 + 0.813709i \(0.697445\pi\)
\(654\) 0 0
\(655\) −31.7517 −1.24064
\(656\) −30.5486 −1.19272
\(657\) 0 0
\(658\) −2.16112 −0.0842490
\(659\) −11.4989 −0.447932 −0.223966 0.974597i \(-0.571900\pi\)
−0.223966 + 0.974597i \(0.571900\pi\)
\(660\) 0 0
\(661\) −29.7497 −1.15713 −0.578565 0.815636i \(-0.696387\pi\)
−0.578565 + 0.815636i \(0.696387\pi\)
\(662\) 22.3438 0.868417
\(663\) 0 0
\(664\) 11.9672 0.464418
\(665\) −15.2685 −0.592087
\(666\) 0 0
\(667\) −1.68400 −0.0652048
\(668\) −8.24762 −0.319110
\(669\) 0 0
\(670\) 63.9975 2.47244
\(671\) 0 0
\(672\) 0 0
\(673\) −20.2823 −0.781826 −0.390913 0.920428i \(-0.627841\pi\)
−0.390913 + 0.920428i \(0.627841\pi\)
\(674\) −37.0065 −1.42544
\(675\) 0 0
\(676\) 46.7344 1.79748
\(677\) 45.6199 1.75331 0.876657 0.481115i \(-0.159768\pi\)
0.876657 + 0.481115i \(0.159768\pi\)
\(678\) 0 0
\(679\) 2.05898 0.0790162
\(680\) 2.11920 0.0812676
\(681\) 0 0
\(682\) 0 0
\(683\) −12.9279 −0.494674 −0.247337 0.968930i \(-0.579555\pi\)
−0.247337 + 0.968930i \(0.579555\pi\)
\(684\) 0 0
\(685\) 40.5835 1.55062
\(686\) 2.09771 0.0800908
\(687\) 0 0
\(688\) −15.3345 −0.584623
\(689\) −42.3933 −1.61506
\(690\) 0 0
\(691\) 22.2368 0.845928 0.422964 0.906147i \(-0.360990\pi\)
0.422964 + 0.906147i \(0.360990\pi\)
\(692\) −39.9433 −1.51842
\(693\) 0 0
\(694\) 66.5547 2.52638
\(695\) 21.0588 0.798806
\(696\) 0 0
\(697\) 10.0172 0.379427
\(698\) 50.8979 1.92651
\(699\) 0 0
\(700\) −3.38858 −0.128076
\(701\) 12.3613 0.466880 0.233440 0.972371i \(-0.425002\pi\)
0.233440 + 0.972371i \(0.425002\pi\)
\(702\) 0 0
\(703\) −58.2453 −2.19676
\(704\) 0 0
\(705\) 0 0
\(706\) 73.6609 2.77227
\(707\) −4.79295 −0.180257
\(708\) 0 0
\(709\) 46.7101 1.75423 0.877117 0.480277i \(-0.159464\pi\)
0.877117 + 0.480277i \(0.159464\pi\)
\(710\) 84.2037 3.16011
\(711\) 0 0
\(712\) 13.0171 0.487835
\(713\) 2.32784 0.0871781
\(714\) 0 0
\(715\) 0 0
\(716\) 21.3127 0.796494
\(717\) 0 0
\(718\) 66.4981 2.48169
\(719\) 27.4387 1.02329 0.511646 0.859196i \(-0.329036\pi\)
0.511646 + 0.859196i \(0.329036\pi\)
\(720\) 0 0
\(721\) 7.54804 0.281104
\(722\) −36.4155 −1.35524
\(723\) 0 0
\(724\) 2.35841 0.0876495
\(725\) 1.88997 0.0701916
\(726\) 0 0
\(727\) −34.8207 −1.29143 −0.645714 0.763579i \(-0.723440\pi\)
−0.645714 + 0.763579i \(0.723440\pi\)
\(728\) −4.78570 −0.177370
\(729\) 0 0
\(730\) −34.0820 −1.26143
\(731\) 5.02834 0.185980
\(732\) 0 0
\(733\) −30.1355 −1.11308 −0.556541 0.830820i \(-0.687872\pi\)
−0.556541 + 0.830820i \(0.687872\pi\)
\(734\) 13.6112 0.502399
\(735\) 0 0
\(736\) −10.1314 −0.373449
\(737\) 0 0
\(738\) 0 0
\(739\) 13.9266 0.512298 0.256149 0.966637i \(-0.417546\pi\)
0.256149 + 0.966637i \(0.417546\pi\)
\(740\) −58.7105 −2.15824
\(741\) 0 0
\(742\) 15.6064 0.572929
\(743\) −45.8625 −1.68253 −0.841267 0.540620i \(-0.818190\pi\)
−0.841267 + 0.540620i \(0.818190\pi\)
\(744\) 0 0
\(745\) 25.1186 0.920273
\(746\) 52.1928 1.91091
\(747\) 0 0
\(748\) 0 0
\(749\) −0.498937 −0.0182308
\(750\) 0 0
\(751\) 21.6057 0.788402 0.394201 0.919024i \(-0.371021\pi\)
0.394201 + 0.919024i \(0.371021\pi\)
\(752\) 3.13082 0.114169
\(753\) 0 0
\(754\) 16.0029 0.582790
\(755\) −40.2774 −1.46584
\(756\) 0 0
\(757\) 14.8957 0.541392 0.270696 0.962665i \(-0.412746\pi\)
0.270696 + 0.962665i \(0.412746\pi\)
\(758\) 60.0550 2.18130
\(759\) 0 0
\(760\) −12.8234 −0.465153
\(761\) −19.2120 −0.696435 −0.348217 0.937414i \(-0.613213\pi\)
−0.348217 + 0.937414i \(0.613213\pi\)
\(762\) 0 0
\(763\) 5.80710 0.210231
\(764\) 45.7756 1.65610
\(765\) 0 0
\(766\) 18.5774 0.671228
\(767\) −37.3487 −1.34858
\(768\) 0 0
\(769\) 19.5580 0.705280 0.352640 0.935759i \(-0.385284\pi\)
0.352640 + 0.935759i \(0.385284\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −10.7368 −0.386428
\(773\) −1.14034 −0.0410153 −0.0205076 0.999790i \(-0.506528\pi\)
−0.0205076 + 0.999790i \(0.506528\pi\)
\(774\) 0 0
\(775\) −2.61255 −0.0938455
\(776\) 1.72925 0.0620764
\(777\) 0 0
\(778\) 5.32624 0.190955
\(779\) −60.6144 −2.17174
\(780\) 0 0
\(781\) 0 0
\(782\) 2.62937 0.0940261
\(783\) 0 0
\(784\) −3.03896 −0.108534
\(785\) −27.6831 −0.988051
\(786\) 0 0
\(787\) 14.0116 0.499458 0.249729 0.968316i \(-0.419658\pi\)
0.249729 + 0.968316i \(0.419658\pi\)
\(788\) −22.9091 −0.816102
\(789\) 0 0
\(790\) −57.3852 −2.04167
\(791\) −13.2820 −0.472254
\(792\) 0 0
\(793\) 21.4541 0.761858
\(794\) 56.4609 2.00372
\(795\) 0 0
\(796\) 54.6386 1.93661
\(797\) 6.62662 0.234727 0.117364 0.993089i \(-0.462556\pi\)
0.117364 + 0.993089i \(0.462556\pi\)
\(798\) 0 0
\(799\) −1.02663 −0.0363194
\(800\) 11.3706 0.402010
\(801\) 0 0
\(802\) −59.1070 −2.08714
\(803\) 0 0
\(804\) 0 0
\(805\) −3.18503 −0.112258
\(806\) −22.1212 −0.779185
\(807\) 0 0
\(808\) −4.02540 −0.141613
\(809\) 13.1145 0.461080 0.230540 0.973063i \(-0.425951\pi\)
0.230540 + 0.973063i \(0.425951\pi\)
\(810\) 0 0
\(811\) −19.2816 −0.677068 −0.338534 0.940954i \(-0.609931\pi\)
−0.338534 + 0.940954i \(0.609931\pi\)
\(812\) −3.21361 −0.112775
\(813\) 0 0
\(814\) 0 0
\(815\) −14.6109 −0.511796
\(816\) 0 0
\(817\) −30.4267 −1.06450
\(818\) −33.8241 −1.18263
\(819\) 0 0
\(820\) −61.0985 −2.13365
\(821\) −23.7058 −0.827337 −0.413668 0.910428i \(-0.635753\pi\)
−0.413668 + 0.910428i \(0.635753\pi\)
\(822\) 0 0
\(823\) −5.47655 −0.190900 −0.0954502 0.995434i \(-0.530429\pi\)
−0.0954502 + 0.995434i \(0.530429\pi\)
\(824\) 6.33929 0.220840
\(825\) 0 0
\(826\) 13.7493 0.478400
\(827\) −23.6878 −0.823707 −0.411854 0.911250i \(-0.635118\pi\)
−0.411854 + 0.911250i \(0.635118\pi\)
\(828\) 0 0
\(829\) 37.6550 1.30781 0.653906 0.756576i \(-0.273129\pi\)
0.653906 + 0.756576i \(0.273129\pi\)
\(830\) −75.6864 −2.62711
\(831\) 0 0
\(832\) 61.6444 2.13714
\(833\) 0.996504 0.0345268
\(834\) 0 0
\(835\) 8.70035 0.301088
\(836\) 0 0
\(837\) 0 0
\(838\) −14.2774 −0.493203
\(839\) 21.6679 0.748059 0.374030 0.927417i \(-0.377976\pi\)
0.374030 + 0.927417i \(0.377976\pi\)
\(840\) 0 0
\(841\) −27.2076 −0.938194
\(842\) 70.7862 2.43945
\(843\) 0 0
\(844\) −28.5623 −0.983155
\(845\) −49.2998 −1.69596
\(846\) 0 0
\(847\) 0 0
\(848\) −22.6091 −0.776401
\(849\) 0 0
\(850\) −2.95096 −0.101217
\(851\) −12.1501 −0.416499
\(852\) 0 0
\(853\) −13.1638 −0.450721 −0.225360 0.974275i \(-0.572356\pi\)
−0.225360 + 0.974275i \(0.572356\pi\)
\(854\) −7.89798 −0.270263
\(855\) 0 0
\(856\) −0.419037 −0.0143224
\(857\) 46.2746 1.58071 0.790355 0.612649i \(-0.209896\pi\)
0.790355 + 0.612649i \(0.209896\pi\)
\(858\) 0 0
\(859\) −40.4096 −1.37876 −0.689379 0.724401i \(-0.742117\pi\)
−0.689379 + 0.724401i \(0.742117\pi\)
\(860\) −30.6697 −1.04583
\(861\) 0 0
\(862\) −63.1062 −2.14941
\(863\) 25.4564 0.866545 0.433273 0.901263i \(-0.357359\pi\)
0.433273 + 0.901263i \(0.357359\pi\)
\(864\) 0 0
\(865\) 42.1359 1.43266
\(866\) −37.2299 −1.26512
\(867\) 0 0
\(868\) 4.44225 0.150780
\(869\) 0 0
\(870\) 0 0
\(871\) −68.6548 −2.32628
\(872\) 4.87715 0.165161
\(873\) 0 0
\(874\) −15.9104 −0.538179
\(875\) −9.08607 −0.307165
\(876\) 0 0
\(877\) −37.3143 −1.26002 −0.630008 0.776589i \(-0.716948\pi\)
−0.630008 + 0.776589i \(0.716948\pi\)
\(878\) −64.6538 −2.18196
\(879\) 0 0
\(880\) 0 0
\(881\) 1.73357 0.0584055 0.0292027 0.999574i \(-0.490703\pi\)
0.0292027 + 0.999574i \(0.490703\pi\)
\(882\) 0 0
\(883\) 33.0694 1.11287 0.556437 0.830890i \(-0.312168\pi\)
0.556437 + 0.830890i \(0.312168\pi\)
\(884\) −13.6300 −0.458427
\(885\) 0 0
\(886\) 27.6304 0.928260
\(887\) −27.4806 −0.922708 −0.461354 0.887216i \(-0.652636\pi\)
−0.461354 + 0.887216i \(0.652636\pi\)
\(888\) 0 0
\(889\) −6.00223 −0.201308
\(890\) −82.3261 −2.75958
\(891\) 0 0
\(892\) −33.5755 −1.12419
\(893\) 6.21217 0.207882
\(894\) 0 0
\(895\) −22.4826 −0.751512
\(896\) −6.58425 −0.219964
\(897\) 0 0
\(898\) −28.1122 −0.938117
\(899\) −2.47764 −0.0826340
\(900\) 0 0
\(901\) 7.41374 0.246988
\(902\) 0 0
\(903\) 0 0
\(904\) −11.1550 −0.371010
\(905\) −2.48786 −0.0826994
\(906\) 0 0
\(907\) −13.3271 −0.442521 −0.221260 0.975215i \(-0.571017\pi\)
−0.221260 + 0.975215i \(0.571017\pi\)
\(908\) 46.0495 1.52821
\(909\) 0 0
\(910\) 30.2670 1.00334
\(911\) 10.1287 0.335579 0.167789 0.985823i \(-0.446337\pi\)
0.167789 + 0.985823i \(0.446337\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 10.9130 0.360971
\(915\) 0 0
\(916\) 26.7085 0.882472
\(917\) −12.5395 −0.414092
\(918\) 0 0
\(919\) 24.7309 0.815799 0.407899 0.913027i \(-0.366261\pi\)
0.407899 + 0.913027i \(0.366261\pi\)
\(920\) −2.67498 −0.0881915
\(921\) 0 0
\(922\) −59.2424 −1.95104
\(923\) −90.3315 −2.97330
\(924\) 0 0
\(925\) 13.6361 0.448353
\(926\) 3.84309 0.126292
\(927\) 0 0
\(928\) 10.7834 0.353983
\(929\) −50.4163 −1.65411 −0.827053 0.562124i \(-0.809984\pi\)
−0.827053 + 0.562124i \(0.809984\pi\)
\(930\) 0 0
\(931\) −6.02990 −0.197622
\(932\) 41.4474 1.35765
\(933\) 0 0
\(934\) 17.0033 0.556366
\(935\) 0 0
\(936\) 0 0
\(937\) −46.0471 −1.50429 −0.752147 0.658995i \(-0.770982\pi\)
−0.752147 + 0.658995i \(0.770982\pi\)
\(938\) 25.2742 0.825230
\(939\) 0 0
\(940\) 6.26178 0.204237
\(941\) 41.8983 1.36584 0.682922 0.730491i \(-0.260709\pi\)
0.682922 + 0.730491i \(0.260709\pi\)
\(942\) 0 0
\(943\) −12.6443 −0.411754
\(944\) −19.9188 −0.648301
\(945\) 0 0
\(946\) 0 0
\(947\) −12.6097 −0.409759 −0.204880 0.978787i \(-0.565680\pi\)
−0.204880 + 0.978787i \(0.565680\pi\)
\(948\) 0 0
\(949\) 36.5623 1.18686
\(950\) 17.8564 0.579339
\(951\) 0 0
\(952\) 0.836923 0.0271248
\(953\) 20.2815 0.656984 0.328492 0.944507i \(-0.393460\pi\)
0.328492 + 0.944507i \(0.393460\pi\)
\(954\) 0 0
\(955\) −48.2884 −1.56257
\(956\) 14.0608 0.454757
\(957\) 0 0
\(958\) −43.7052 −1.41205
\(959\) 16.0274 0.517552
\(960\) 0 0
\(961\) −27.5751 −0.889519
\(962\) 115.461 3.72261
\(963\) 0 0
\(964\) −18.0844 −0.582461
\(965\) 11.3262 0.364604
\(966\) 0 0
\(967\) 32.9041 1.05812 0.529062 0.848583i \(-0.322544\pi\)
0.529062 + 0.848583i \(0.322544\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −10.9366 −0.351153
\(971\) −30.0232 −0.963491 −0.481746 0.876311i \(-0.659997\pi\)
−0.481746 + 0.876311i \(0.659997\pi\)
\(972\) 0 0
\(973\) 8.31663 0.266619
\(974\) 42.5161 1.36230
\(975\) 0 0
\(976\) 11.4419 0.366245
\(977\) 22.3977 0.716566 0.358283 0.933613i \(-0.383362\pi\)
0.358283 + 0.933613i \(0.383362\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −6.07805 −0.194156
\(981\) 0 0
\(982\) 75.6384 2.41372
\(983\) 3.91774 0.124956 0.0624782 0.998046i \(-0.480100\pi\)
0.0624782 + 0.998046i \(0.480100\pi\)
\(984\) 0 0
\(985\) 24.1666 0.770012
\(986\) −2.79858 −0.0891250
\(987\) 0 0
\(988\) 82.4759 2.62391
\(989\) −6.34706 −0.201825
\(990\) 0 0
\(991\) −14.7632 −0.468970 −0.234485 0.972120i \(-0.575340\pi\)
−0.234485 + 0.972120i \(0.575340\pi\)
\(992\) −14.9062 −0.473272
\(993\) 0 0
\(994\) 33.2541 1.05475
\(995\) −57.6378 −1.82724
\(996\) 0 0
\(997\) −0.676566 −0.0214271 −0.0107135 0.999943i \(-0.503410\pi\)
−0.0107135 + 0.999943i \(0.503410\pi\)
\(998\) −28.8067 −0.911860
\(999\) 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.cz.1.3 12
3.2 odd 2 inner 7623.2.a.cz.1.10 yes 12
11.10 odd 2 7623.2.a.da.1.10 yes 12
33.32 even 2 7623.2.a.da.1.3 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7623.2.a.cz.1.3 12 1.1 even 1 trivial
7623.2.a.cz.1.10 yes 12 3.2 odd 2 inner
7623.2.a.da.1.3 yes 12 33.32 even 2
7623.2.a.da.1.10 yes 12 11.10 odd 2