Properties

Label 7623.2.a.cy.1.3
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 0
Dimension 10
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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.09767\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.09767 q^{2} +2.40021 q^{4} -3.15947 q^{5} +1.00000 q^{7} -0.839503 q^{8} +O(q^{10})\) \(q-2.09767 q^{2} +2.40021 q^{4} -3.15947 q^{5} +1.00000 q^{7} -0.839503 q^{8} +6.62751 q^{10} -2.12862 q^{13} -2.09767 q^{14} -3.03942 q^{16} -4.79430 q^{17} -1.53847 q^{19} -7.58338 q^{20} -5.11509 q^{23} +4.98224 q^{25} +4.46514 q^{26} +2.40021 q^{28} -0.958246 q^{29} -7.22037 q^{31} +8.05469 q^{32} +10.0569 q^{34} -3.15947 q^{35} +2.39901 q^{37} +3.22720 q^{38} +2.65238 q^{40} +0.266914 q^{41} -9.28678 q^{43} +10.7298 q^{46} -10.3245 q^{47} +1.00000 q^{49} -10.4511 q^{50} -5.10914 q^{52} +0.945600 q^{53} -0.839503 q^{56} +2.01008 q^{58} +9.55077 q^{59} -8.55756 q^{61} +15.1459 q^{62} -10.8172 q^{64} +6.72532 q^{65} +13.2710 q^{67} -11.5073 q^{68} +6.62751 q^{70} -14.1564 q^{71} -7.72198 q^{73} -5.03232 q^{74} -3.69265 q^{76} -9.31226 q^{79} +9.60295 q^{80} -0.559897 q^{82} -16.8536 q^{83} +15.1475 q^{85} +19.4806 q^{86} +11.2670 q^{89} -2.12862 q^{91} -12.2773 q^{92} +21.6574 q^{94} +4.86075 q^{95} +7.08789 q^{97} -2.09767 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 18q^{4} - 5q^{5} + 10q^{7} + 3q^{8} + O(q^{10}) \) \( 10q + 18q^{4} - 5q^{5} + 10q^{7} + 3q^{8} - 6q^{10} + 6q^{13} + 38q^{16} - 8q^{17} - 7q^{20} + 31q^{25} - q^{26} + 18q^{28} + 14q^{29} + 26q^{31} + 41q^{32} + 21q^{34} - 5q^{35} + 24q^{37} - 8q^{38} - 5q^{40} - 19q^{41} - 6q^{43} - q^{46} - 15q^{47} + 10q^{49} + q^{50} - 25q^{52} + q^{53} + 3q^{56} + 11q^{58} - 23q^{59} - 11q^{62} + 53q^{64} + 29q^{65} + 38q^{67} - 87q^{68} - 6q^{70} - 26q^{71} - q^{73} + 39q^{74} - 2q^{76} + 5q^{79} - 6q^{80} + 5q^{82} - 6q^{83} - q^{85} + 41q^{86} + 9q^{89} + 6q^{91} + 48q^{92} + 42q^{95} + 24q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.09767 −1.48327 −0.741637 0.670801i \(-0.765950\pi\)
−0.741637 + 0.670801i \(0.765950\pi\)
\(3\) 0 0
\(4\) 2.40021 1.20010
\(5\) −3.15947 −1.41296 −0.706479 0.707734i \(-0.749717\pi\)
−0.706479 + 0.707734i \(0.749717\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −0.839503 −0.296809
\(9\) 0 0
\(10\) 6.62751 2.09580
\(11\) 0 0
\(12\) 0 0
\(13\) −2.12862 −0.590374 −0.295187 0.955440i \(-0.595382\pi\)
−0.295187 + 0.955440i \(0.595382\pi\)
\(14\) −2.09767 −0.560625
\(15\) 0 0
\(16\) −3.03942 −0.759855
\(17\) −4.79430 −1.16279 −0.581395 0.813622i \(-0.697493\pi\)
−0.581395 + 0.813622i \(0.697493\pi\)
\(18\) 0 0
\(19\) −1.53847 −0.352949 −0.176475 0.984305i \(-0.556469\pi\)
−0.176475 + 0.984305i \(0.556469\pi\)
\(20\) −7.58338 −1.69570
\(21\) 0 0
\(22\) 0 0
\(23\) −5.11509 −1.06657 −0.533285 0.845935i \(-0.679043\pi\)
−0.533285 + 0.845935i \(0.679043\pi\)
\(24\) 0 0
\(25\) 4.98224 0.996449
\(26\) 4.46514 0.875687
\(27\) 0 0
\(28\) 2.40021 0.453597
\(29\) −0.958246 −0.177942 −0.0889709 0.996034i \(-0.528358\pi\)
−0.0889709 + 0.996034i \(0.528358\pi\)
\(30\) 0 0
\(31\) −7.22037 −1.29682 −0.648409 0.761292i \(-0.724565\pi\)
−0.648409 + 0.761292i \(0.724565\pi\)
\(32\) 8.05469 1.42388
\(33\) 0 0
\(34\) 10.0569 1.72474
\(35\) −3.15947 −0.534048
\(36\) 0 0
\(37\) 2.39901 0.394394 0.197197 0.980364i \(-0.436816\pi\)
0.197197 + 0.980364i \(0.436816\pi\)
\(38\) 3.22720 0.523521
\(39\) 0 0
\(40\) 2.65238 0.419378
\(41\) 0.266914 0.0416850 0.0208425 0.999783i \(-0.493365\pi\)
0.0208425 + 0.999783i \(0.493365\pi\)
\(42\) 0 0
\(43\) −9.28678 −1.41622 −0.708110 0.706102i \(-0.750452\pi\)
−0.708110 + 0.706102i \(0.750452\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 10.7298 1.58202
\(47\) −10.3245 −1.50599 −0.752993 0.658029i \(-0.771391\pi\)
−0.752993 + 0.658029i \(0.771391\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −10.4511 −1.47801
\(51\) 0 0
\(52\) −5.10914 −0.708510
\(53\) 0.945600 0.129888 0.0649441 0.997889i \(-0.479313\pi\)
0.0649441 + 0.997889i \(0.479313\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −0.839503 −0.112183
\(57\) 0 0
\(58\) 2.01008 0.263937
\(59\) 9.55077 1.24340 0.621702 0.783254i \(-0.286442\pi\)
0.621702 + 0.783254i \(0.286442\pi\)
\(60\) 0 0
\(61\) −8.55756 −1.09568 −0.547841 0.836582i \(-0.684550\pi\)
−0.547841 + 0.836582i \(0.684550\pi\)
\(62\) 15.1459 1.92354
\(63\) 0 0
\(64\) −10.8172 −1.35215
\(65\) 6.72532 0.834173
\(66\) 0 0
\(67\) 13.2710 1.62131 0.810657 0.585521i \(-0.199110\pi\)
0.810657 + 0.585521i \(0.199110\pi\)
\(68\) −11.5073 −1.39547
\(69\) 0 0
\(70\) 6.62751 0.792139
\(71\) −14.1564 −1.68006 −0.840030 0.542540i \(-0.817463\pi\)
−0.840030 + 0.542540i \(0.817463\pi\)
\(72\) 0 0
\(73\) −7.72198 −0.903789 −0.451895 0.892071i \(-0.649252\pi\)
−0.451895 + 0.892071i \(0.649252\pi\)
\(74\) −5.03232 −0.584995
\(75\) 0 0
\(76\) −3.69265 −0.423576
\(77\) 0 0
\(78\) 0 0
\(79\) −9.31226 −1.04771 −0.523856 0.851807i \(-0.675507\pi\)
−0.523856 + 0.851807i \(0.675507\pi\)
\(80\) 9.60295 1.07364
\(81\) 0 0
\(82\) −0.559897 −0.0618302
\(83\) −16.8536 −1.84992 −0.924961 0.380061i \(-0.875903\pi\)
−0.924961 + 0.380061i \(0.875903\pi\)
\(84\) 0 0
\(85\) 15.1475 1.64297
\(86\) 19.4806 2.10064
\(87\) 0 0
\(88\) 0 0
\(89\) 11.2670 1.19430 0.597150 0.802130i \(-0.296300\pi\)
0.597150 + 0.802130i \(0.296300\pi\)
\(90\) 0 0
\(91\) −2.12862 −0.223140
\(92\) −12.2773 −1.28000
\(93\) 0 0
\(94\) 21.6574 2.23379
\(95\) 4.86075 0.498702
\(96\) 0 0
\(97\) 7.08789 0.719666 0.359833 0.933017i \(-0.382834\pi\)
0.359833 + 0.933017i \(0.382834\pi\)
\(98\) −2.09767 −0.211896
\(99\) 0 0
\(100\) 11.9584 1.19584
\(101\) 8.34609 0.830467 0.415234 0.909715i \(-0.363700\pi\)
0.415234 + 0.909715i \(0.363700\pi\)
\(102\) 0 0
\(103\) −7.53555 −0.742500 −0.371250 0.928533i \(-0.621071\pi\)
−0.371250 + 0.928533i \(0.621071\pi\)
\(104\) 1.78698 0.175228
\(105\) 0 0
\(106\) −1.98355 −0.192660
\(107\) 15.3887 1.48768 0.743841 0.668357i \(-0.233002\pi\)
0.743841 + 0.668357i \(0.233002\pi\)
\(108\) 0 0
\(109\) −20.2581 −1.94038 −0.970188 0.242353i \(-0.922081\pi\)
−0.970188 + 0.242353i \(0.922081\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −3.03942 −0.287198
\(113\) 18.1019 1.70288 0.851440 0.524452i \(-0.175730\pi\)
0.851440 + 0.524452i \(0.175730\pi\)
\(114\) 0 0
\(115\) 16.1610 1.50702
\(116\) −2.29999 −0.213549
\(117\) 0 0
\(118\) −20.0343 −1.84431
\(119\) −4.79430 −0.439493
\(120\) 0 0
\(121\) 0 0
\(122\) 17.9509 1.62520
\(123\) 0 0
\(124\) −17.3304 −1.55632
\(125\) 0.0560984 0.00501759
\(126\) 0 0
\(127\) 15.3711 1.36396 0.681981 0.731369i \(-0.261118\pi\)
0.681981 + 0.731369i \(0.261118\pi\)
\(128\) 6.58156 0.581733
\(129\) 0 0
\(130\) −14.1075 −1.23731
\(131\) −10.7418 −0.938516 −0.469258 0.883061i \(-0.655478\pi\)
−0.469258 + 0.883061i \(0.655478\pi\)
\(132\) 0 0
\(133\) −1.53847 −0.133402
\(134\) −27.8382 −2.40485
\(135\) 0 0
\(136\) 4.02483 0.345126
\(137\) 4.42612 0.378149 0.189075 0.981963i \(-0.439451\pi\)
0.189075 + 0.981963i \(0.439451\pi\)
\(138\) 0 0
\(139\) −4.52505 −0.383810 −0.191905 0.981414i \(-0.561467\pi\)
−0.191905 + 0.981414i \(0.561467\pi\)
\(140\) −7.58338 −0.640913
\(141\) 0 0
\(142\) 29.6955 2.49199
\(143\) 0 0
\(144\) 0 0
\(145\) 3.02755 0.251424
\(146\) 16.1981 1.34057
\(147\) 0 0
\(148\) 5.75812 0.473314
\(149\) 2.99471 0.245336 0.122668 0.992448i \(-0.460855\pi\)
0.122668 + 0.992448i \(0.460855\pi\)
\(150\) 0 0
\(151\) −8.81260 −0.717159 −0.358579 0.933499i \(-0.616739\pi\)
−0.358579 + 0.933499i \(0.616739\pi\)
\(152\) 1.29155 0.104759
\(153\) 0 0
\(154\) 0 0
\(155\) 22.8125 1.83235
\(156\) 0 0
\(157\) 1.56655 0.125024 0.0625122 0.998044i \(-0.480089\pi\)
0.0625122 + 0.998044i \(0.480089\pi\)
\(158\) 19.5340 1.55404
\(159\) 0 0
\(160\) −25.4486 −2.01188
\(161\) −5.11509 −0.403126
\(162\) 0 0
\(163\) −6.80568 −0.533062 −0.266531 0.963826i \(-0.585877\pi\)
−0.266531 + 0.963826i \(0.585877\pi\)
\(164\) 0.640649 0.0500263
\(165\) 0 0
\(166\) 35.3532 2.74394
\(167\) −12.2497 −0.947913 −0.473957 0.880548i \(-0.657175\pi\)
−0.473957 + 0.880548i \(0.657175\pi\)
\(168\) 0 0
\(169\) −8.46896 −0.651459
\(170\) −31.7743 −2.43698
\(171\) 0 0
\(172\) −22.2902 −1.69961
\(173\) −19.8500 −1.50917 −0.754583 0.656204i \(-0.772161\pi\)
−0.754583 + 0.656204i \(0.772161\pi\)
\(174\) 0 0
\(175\) 4.98224 0.376622
\(176\) 0 0
\(177\) 0 0
\(178\) −23.6344 −1.77147
\(179\) −8.22231 −0.614564 −0.307282 0.951618i \(-0.599420\pi\)
−0.307282 + 0.951618i \(0.599420\pi\)
\(180\) 0 0
\(181\) 10.4317 0.775381 0.387691 0.921790i \(-0.373273\pi\)
0.387691 + 0.921790i \(0.373273\pi\)
\(182\) 4.46514 0.330978
\(183\) 0 0
\(184\) 4.29413 0.316568
\(185\) −7.57959 −0.557263
\(186\) 0 0
\(187\) 0 0
\(188\) −24.7810 −1.80734
\(189\) 0 0
\(190\) −10.1962 −0.739713
\(191\) −14.2162 −1.02865 −0.514325 0.857595i \(-0.671957\pi\)
−0.514325 + 0.857595i \(0.671957\pi\)
\(192\) 0 0
\(193\) 7.01633 0.505046 0.252523 0.967591i \(-0.418740\pi\)
0.252523 + 0.967591i \(0.418740\pi\)
\(194\) −14.8680 −1.06746
\(195\) 0 0
\(196\) 2.40021 0.171443
\(197\) −3.60302 −0.256705 −0.128352 0.991729i \(-0.540969\pi\)
−0.128352 + 0.991729i \(0.540969\pi\)
\(198\) 0 0
\(199\) −3.85136 −0.273015 −0.136508 0.990639i \(-0.543588\pi\)
−0.136508 + 0.990639i \(0.543588\pi\)
\(200\) −4.18261 −0.295755
\(201\) 0 0
\(202\) −17.5073 −1.23181
\(203\) −0.958246 −0.0672557
\(204\) 0 0
\(205\) −0.843306 −0.0588991
\(206\) 15.8071 1.10133
\(207\) 0 0
\(208\) 6.46978 0.448598
\(209\) 0 0
\(210\) 0 0
\(211\) −11.1325 −0.766391 −0.383196 0.923667i \(-0.625176\pi\)
−0.383196 + 0.923667i \(0.625176\pi\)
\(212\) 2.26964 0.155879
\(213\) 0 0
\(214\) −32.2804 −2.20664
\(215\) 29.3413 2.00106
\(216\) 0 0
\(217\) −7.22037 −0.490151
\(218\) 42.4948 2.87811
\(219\) 0 0
\(220\) 0 0
\(221\) 10.2053 0.686481
\(222\) 0 0
\(223\) −5.27119 −0.352985 −0.176493 0.984302i \(-0.556475\pi\)
−0.176493 + 0.984302i \(0.556475\pi\)
\(224\) 8.05469 0.538177
\(225\) 0 0
\(226\) −37.9717 −2.52584
\(227\) −14.8204 −0.983661 −0.491831 0.870691i \(-0.663672\pi\)
−0.491831 + 0.870691i \(0.663672\pi\)
\(228\) 0 0
\(229\) 7.60478 0.502538 0.251269 0.967917i \(-0.419152\pi\)
0.251269 + 0.967917i \(0.419152\pi\)
\(230\) −33.9004 −2.23532
\(231\) 0 0
\(232\) 0.804450 0.0528147
\(233\) −1.50544 −0.0986248 −0.0493124 0.998783i \(-0.515703\pi\)
−0.0493124 + 0.998783i \(0.515703\pi\)
\(234\) 0 0
\(235\) 32.6200 2.12789
\(236\) 22.9238 1.49221
\(237\) 0 0
\(238\) 10.0569 0.651889
\(239\) −5.02336 −0.324934 −0.162467 0.986714i \(-0.551945\pi\)
−0.162467 + 0.986714i \(0.551945\pi\)
\(240\) 0 0
\(241\) −16.1502 −1.04033 −0.520164 0.854066i \(-0.674129\pi\)
−0.520164 + 0.854066i \(0.674129\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −20.5399 −1.31493
\(245\) −3.15947 −0.201851
\(246\) 0 0
\(247\) 3.27482 0.208372
\(248\) 6.06152 0.384907
\(249\) 0 0
\(250\) −0.117676 −0.00744246
\(251\) −25.7765 −1.62700 −0.813500 0.581565i \(-0.802440\pi\)
−0.813500 + 0.581565i \(0.802440\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −32.2434 −2.02313
\(255\) 0 0
\(256\) 7.82853 0.489283
\(257\) −17.6678 −1.10208 −0.551042 0.834477i \(-0.685770\pi\)
−0.551042 + 0.834477i \(0.685770\pi\)
\(258\) 0 0
\(259\) 2.39901 0.149067
\(260\) 16.1422 1.00109
\(261\) 0 0
\(262\) 22.5327 1.39208
\(263\) 18.6325 1.14893 0.574465 0.818529i \(-0.305210\pi\)
0.574465 + 0.818529i \(0.305210\pi\)
\(264\) 0 0
\(265\) −2.98759 −0.183526
\(266\) 3.22720 0.197872
\(267\) 0 0
\(268\) 31.8532 1.94575
\(269\) 20.7126 1.26287 0.631435 0.775429i \(-0.282466\pi\)
0.631435 + 0.775429i \(0.282466\pi\)
\(270\) 0 0
\(271\) −14.8438 −0.901694 −0.450847 0.892601i \(-0.648878\pi\)
−0.450847 + 0.892601i \(0.648878\pi\)
\(272\) 14.5719 0.883551
\(273\) 0 0
\(274\) −9.28453 −0.560899
\(275\) 0 0
\(276\) 0 0
\(277\) 3.07309 0.184644 0.0923221 0.995729i \(-0.470571\pi\)
0.0923221 + 0.995729i \(0.470571\pi\)
\(278\) 9.49206 0.569296
\(279\) 0 0
\(280\) 2.65238 0.158510
\(281\) 4.09698 0.244405 0.122203 0.992505i \(-0.461004\pi\)
0.122203 + 0.992505i \(0.461004\pi\)
\(282\) 0 0
\(283\) 7.39800 0.439766 0.219883 0.975526i \(-0.429432\pi\)
0.219883 + 0.975526i \(0.429432\pi\)
\(284\) −33.9784 −2.01625
\(285\) 0 0
\(286\) 0 0
\(287\) 0.266914 0.0157554
\(288\) 0 0
\(289\) 5.98536 0.352080
\(290\) −6.35079 −0.372931
\(291\) 0 0
\(292\) −18.5344 −1.08464
\(293\) −10.5489 −0.616273 −0.308137 0.951342i \(-0.599705\pi\)
−0.308137 + 0.951342i \(0.599705\pi\)
\(294\) 0 0
\(295\) −30.1754 −1.75688
\(296\) −2.01397 −0.117060
\(297\) 0 0
\(298\) −6.28190 −0.363901
\(299\) 10.8881 0.629675
\(300\) 0 0
\(301\) −9.28678 −0.535281
\(302\) 18.4859 1.06374
\(303\) 0 0
\(304\) 4.67605 0.268190
\(305\) 27.0373 1.54815
\(306\) 0 0
\(307\) −1.74955 −0.0998522 −0.0499261 0.998753i \(-0.515899\pi\)
−0.0499261 + 0.998753i \(0.515899\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −47.8531 −2.71788
\(311\) 23.0075 1.30463 0.652317 0.757946i \(-0.273797\pi\)
0.652317 + 0.757946i \(0.273797\pi\)
\(312\) 0 0
\(313\) 18.2568 1.03193 0.515967 0.856608i \(-0.327433\pi\)
0.515967 + 0.856608i \(0.327433\pi\)
\(314\) −3.28610 −0.185446
\(315\) 0 0
\(316\) −22.3514 −1.25736
\(317\) −32.6206 −1.83216 −0.916079 0.400999i \(-0.868663\pi\)
−0.916079 + 0.400999i \(0.868663\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 34.1767 1.91054
\(321\) 0 0
\(322\) 10.7298 0.597946
\(323\) 7.37589 0.410406
\(324\) 0 0
\(325\) −10.6053 −0.588277
\(326\) 14.2760 0.790677
\(327\) 0 0
\(328\) −0.224075 −0.0123725
\(329\) −10.3245 −0.569209
\(330\) 0 0
\(331\) 6.80537 0.374057 0.187029 0.982354i \(-0.440114\pi\)
0.187029 + 0.982354i \(0.440114\pi\)
\(332\) −40.4521 −2.22010
\(333\) 0 0
\(334\) 25.6959 1.40602
\(335\) −41.9294 −2.29085
\(336\) 0 0
\(337\) −0.372041 −0.0202664 −0.0101332 0.999949i \(-0.503226\pi\)
−0.0101332 + 0.999949i \(0.503226\pi\)
\(338\) 17.7651 0.966292
\(339\) 0 0
\(340\) 36.3570 1.97174
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 7.79627 0.420347
\(345\) 0 0
\(346\) 41.6387 2.23851
\(347\) 5.75075 0.308716 0.154358 0.988015i \(-0.450669\pi\)
0.154358 + 0.988015i \(0.450669\pi\)
\(348\) 0 0
\(349\) 0.646523 0.0346076 0.0173038 0.999850i \(-0.494492\pi\)
0.0173038 + 0.999850i \(0.494492\pi\)
\(350\) −10.4511 −0.558634
\(351\) 0 0
\(352\) 0 0
\(353\) 18.3601 0.977209 0.488604 0.872505i \(-0.337506\pi\)
0.488604 + 0.872505i \(0.337506\pi\)
\(354\) 0 0
\(355\) 44.7268 2.37385
\(356\) 27.0431 1.43328
\(357\) 0 0
\(358\) 17.2477 0.911567
\(359\) 26.9753 1.42370 0.711852 0.702330i \(-0.247857\pi\)
0.711852 + 0.702330i \(0.247857\pi\)
\(360\) 0 0
\(361\) −16.6331 −0.875427
\(362\) −21.8822 −1.15010
\(363\) 0 0
\(364\) −5.10914 −0.267792
\(365\) 24.3974 1.27702
\(366\) 0 0
\(367\) 28.3215 1.47837 0.739186 0.673501i \(-0.235210\pi\)
0.739186 + 0.673501i \(0.235210\pi\)
\(368\) 15.5469 0.810439
\(369\) 0 0
\(370\) 15.8995 0.826574
\(371\) 0.945600 0.0490931
\(372\) 0 0
\(373\) 12.9215 0.669050 0.334525 0.942387i \(-0.391424\pi\)
0.334525 + 0.942387i \(0.391424\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 8.66746 0.446990
\(377\) 2.03974 0.105052
\(378\) 0 0
\(379\) 20.9480 1.07603 0.538014 0.842936i \(-0.319175\pi\)
0.538014 + 0.842936i \(0.319175\pi\)
\(380\) 11.6668 0.598495
\(381\) 0 0
\(382\) 29.8209 1.52577
\(383\) −24.9764 −1.27623 −0.638117 0.769939i \(-0.720286\pi\)
−0.638117 + 0.769939i \(0.720286\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −14.7179 −0.749123
\(387\) 0 0
\(388\) 17.0124 0.863674
\(389\) 13.5780 0.688430 0.344215 0.938891i \(-0.388145\pi\)
0.344215 + 0.938891i \(0.388145\pi\)
\(390\) 0 0
\(391\) 24.5233 1.24020
\(392\) −0.839503 −0.0424013
\(393\) 0 0
\(394\) 7.55794 0.380763
\(395\) 29.4218 1.48037
\(396\) 0 0
\(397\) −4.26450 −0.214029 −0.107014 0.994257i \(-0.534129\pi\)
−0.107014 + 0.994257i \(0.534129\pi\)
\(398\) 8.07886 0.404957
\(399\) 0 0
\(400\) −15.1431 −0.757156
\(401\) 32.4423 1.62009 0.810045 0.586367i \(-0.199442\pi\)
0.810045 + 0.586367i \(0.199442\pi\)
\(402\) 0 0
\(403\) 15.3695 0.765607
\(404\) 20.0324 0.996647
\(405\) 0 0
\(406\) 2.01008 0.0997587
\(407\) 0 0
\(408\) 0 0
\(409\) 24.8877 1.23062 0.615308 0.788287i \(-0.289032\pi\)
0.615308 + 0.788287i \(0.289032\pi\)
\(410\) 1.76898 0.0873635
\(411\) 0 0
\(412\) −18.0869 −0.891077
\(413\) 9.55077 0.469963
\(414\) 0 0
\(415\) 53.2484 2.61386
\(416\) −17.1454 −0.840623
\(417\) 0 0
\(418\) 0 0
\(419\) −10.5948 −0.517588 −0.258794 0.965933i \(-0.583325\pi\)
−0.258794 + 0.965933i \(0.583325\pi\)
\(420\) 0 0
\(421\) −37.8632 −1.84534 −0.922670 0.385591i \(-0.873998\pi\)
−0.922670 + 0.385591i \(0.873998\pi\)
\(422\) 23.3522 1.13677
\(423\) 0 0
\(424\) −0.793833 −0.0385520
\(425\) −23.8864 −1.15866
\(426\) 0 0
\(427\) −8.55756 −0.414129
\(428\) 36.9361 1.78537
\(429\) 0 0
\(430\) −61.5483 −2.96812
\(431\) 5.64489 0.271905 0.135952 0.990715i \(-0.456591\pi\)
0.135952 + 0.990715i \(0.456591\pi\)
\(432\) 0 0
\(433\) −2.69528 −0.129527 −0.0647635 0.997901i \(-0.520629\pi\)
−0.0647635 + 0.997901i \(0.520629\pi\)
\(434\) 15.1459 0.727028
\(435\) 0 0
\(436\) −48.6237 −2.32865
\(437\) 7.86942 0.376445
\(438\) 0 0
\(439\) −30.5668 −1.45888 −0.729438 0.684047i \(-0.760218\pi\)
−0.729438 + 0.684047i \(0.760218\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −21.4073 −1.01824
\(443\) 24.6371 1.17054 0.585272 0.810837i \(-0.300988\pi\)
0.585272 + 0.810837i \(0.300988\pi\)
\(444\) 0 0
\(445\) −35.5977 −1.68749
\(446\) 11.0572 0.523574
\(447\) 0 0
\(448\) −10.8172 −0.511066
\(449\) 15.3755 0.725617 0.362808 0.931864i \(-0.381818\pi\)
0.362808 + 0.931864i \(0.381818\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 43.4482 2.04363
\(453\) 0 0
\(454\) 31.0882 1.45904
\(455\) 6.72532 0.315288
\(456\) 0 0
\(457\) −14.2362 −0.665943 −0.332972 0.942937i \(-0.608051\pi\)
−0.332972 + 0.942937i \(0.608051\pi\)
\(458\) −15.9523 −0.745402
\(459\) 0 0
\(460\) 38.7897 1.80858
\(461\) −36.7148 −1.70998 −0.854990 0.518645i \(-0.826437\pi\)
−0.854990 + 0.518645i \(0.826437\pi\)
\(462\) 0 0
\(463\) −34.7682 −1.61582 −0.807908 0.589308i \(-0.799400\pi\)
−0.807908 + 0.589308i \(0.799400\pi\)
\(464\) 2.91251 0.135210
\(465\) 0 0
\(466\) 3.15792 0.146288
\(467\) −13.5536 −0.627184 −0.313592 0.949558i \(-0.601532\pi\)
−0.313592 + 0.949558i \(0.601532\pi\)
\(468\) 0 0
\(469\) 13.2710 0.612799
\(470\) −68.4259 −3.15625
\(471\) 0 0
\(472\) −8.01789 −0.369053
\(473\) 0 0
\(474\) 0 0
\(475\) −7.66503 −0.351696
\(476\) −11.5073 −0.527437
\(477\) 0 0
\(478\) 10.5373 0.481967
\(479\) 27.5973 1.26095 0.630476 0.776208i \(-0.282860\pi\)
0.630476 + 0.776208i \(0.282860\pi\)
\(480\) 0 0
\(481\) −5.10658 −0.232840
\(482\) 33.8778 1.54309
\(483\) 0 0
\(484\) 0 0
\(485\) −22.3940 −1.01686
\(486\) 0 0
\(487\) −20.4778 −0.927936 −0.463968 0.885852i \(-0.653575\pi\)
−0.463968 + 0.885852i \(0.653575\pi\)
\(488\) 7.18409 0.325208
\(489\) 0 0
\(490\) 6.62751 0.299401
\(491\) 28.8664 1.30272 0.651361 0.758768i \(-0.274198\pi\)
0.651361 + 0.758768i \(0.274198\pi\)
\(492\) 0 0
\(493\) 4.59412 0.206909
\(494\) −6.86949 −0.309073
\(495\) 0 0
\(496\) 21.9457 0.985393
\(497\) −14.1564 −0.635003
\(498\) 0 0
\(499\) −21.4940 −0.962202 −0.481101 0.876665i \(-0.659763\pi\)
−0.481101 + 0.876665i \(0.659763\pi\)
\(500\) 0.134648 0.00602163
\(501\) 0 0
\(502\) 54.0706 2.41329
\(503\) −8.60198 −0.383543 −0.191772 0.981440i \(-0.561423\pi\)
−0.191772 + 0.981440i \(0.561423\pi\)
\(504\) 0 0
\(505\) −26.3692 −1.17342
\(506\) 0 0
\(507\) 0 0
\(508\) 36.8938 1.63690
\(509\) −11.0437 −0.489505 −0.244752 0.969586i \(-0.578707\pi\)
−0.244752 + 0.969586i \(0.578707\pi\)
\(510\) 0 0
\(511\) −7.72198 −0.341600
\(512\) −29.5848 −1.30748
\(513\) 0 0
\(514\) 37.0611 1.63469
\(515\) 23.8083 1.04912
\(516\) 0 0
\(517\) 0 0
\(518\) −5.03232 −0.221107
\(519\) 0 0
\(520\) −5.64592 −0.247590
\(521\) 12.6215 0.552960 0.276480 0.961020i \(-0.410832\pi\)
0.276480 + 0.961020i \(0.410832\pi\)
\(522\) 0 0
\(523\) 43.9100 1.92005 0.960025 0.279913i \(-0.0903055\pi\)
0.960025 + 0.279913i \(0.0903055\pi\)
\(524\) −25.7826 −1.12632
\(525\) 0 0
\(526\) −39.0848 −1.70418
\(527\) 34.6167 1.50793
\(528\) 0 0
\(529\) 3.16418 0.137573
\(530\) 6.26698 0.272220
\(531\) 0 0
\(532\) −3.69265 −0.160097
\(533\) −0.568159 −0.0246097
\(534\) 0 0
\(535\) −48.6201 −2.10203
\(536\) −11.1411 −0.481221
\(537\) 0 0
\(538\) −43.4482 −1.87318
\(539\) 0 0
\(540\) 0 0
\(541\) 27.3128 1.17427 0.587134 0.809490i \(-0.300256\pi\)
0.587134 + 0.809490i \(0.300256\pi\)
\(542\) 31.1373 1.33746
\(543\) 0 0
\(544\) −38.6167 −1.65568
\(545\) 64.0049 2.74167
\(546\) 0 0
\(547\) −18.3092 −0.782845 −0.391423 0.920211i \(-0.628017\pi\)
−0.391423 + 0.920211i \(0.628017\pi\)
\(548\) 10.6236 0.453818
\(549\) 0 0
\(550\) 0 0
\(551\) 1.47423 0.0628044
\(552\) 0 0
\(553\) −9.31226 −0.395998
\(554\) −6.44632 −0.273878
\(555\) 0 0
\(556\) −10.8611 −0.460612
\(557\) −3.82733 −0.162169 −0.0810845 0.996707i \(-0.525838\pi\)
−0.0810845 + 0.996707i \(0.525838\pi\)
\(558\) 0 0
\(559\) 19.7681 0.836099
\(560\) 9.60295 0.405799
\(561\) 0 0
\(562\) −8.59411 −0.362520
\(563\) −36.1468 −1.52340 −0.761702 0.647927i \(-0.775636\pi\)
−0.761702 + 0.647927i \(0.775636\pi\)
\(564\) 0 0
\(565\) −57.1923 −2.40610
\(566\) −15.5186 −0.652293
\(567\) 0 0
\(568\) 11.8844 0.498657
\(569\) −9.55602 −0.400609 −0.200305 0.979734i \(-0.564193\pi\)
−0.200305 + 0.979734i \(0.564193\pi\)
\(570\) 0 0
\(571\) 31.8446 1.33265 0.666327 0.745659i \(-0.267865\pi\)
0.666327 + 0.745659i \(0.267865\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −0.559897 −0.0233696
\(575\) −25.4846 −1.06278
\(576\) 0 0
\(577\) 8.65987 0.360515 0.180258 0.983619i \(-0.442307\pi\)
0.180258 + 0.983619i \(0.442307\pi\)
\(578\) −12.5553 −0.522231
\(579\) 0 0
\(580\) 7.26674 0.301735
\(581\) −16.8536 −0.699205
\(582\) 0 0
\(583\) 0 0
\(584\) 6.48262 0.268253
\(585\) 0 0
\(586\) 22.1281 0.914103
\(587\) 5.32917 0.219959 0.109979 0.993934i \(-0.464922\pi\)
0.109979 + 0.993934i \(0.464922\pi\)
\(588\) 0 0
\(589\) 11.1083 0.457711
\(590\) 63.2978 2.60593
\(591\) 0 0
\(592\) −7.29159 −0.299682
\(593\) 11.0464 0.453622 0.226811 0.973939i \(-0.427170\pi\)
0.226811 + 0.973939i \(0.427170\pi\)
\(594\) 0 0
\(595\) 15.1475 0.620985
\(596\) 7.18792 0.294429
\(597\) 0 0
\(598\) −22.8396 −0.933982
\(599\) −26.2832 −1.07390 −0.536951 0.843614i \(-0.680424\pi\)
−0.536951 + 0.843614i \(0.680424\pi\)
\(600\) 0 0
\(601\) −12.9669 −0.528929 −0.264465 0.964395i \(-0.585195\pi\)
−0.264465 + 0.964395i \(0.585195\pi\)
\(602\) 19.4806 0.793969
\(603\) 0 0
\(604\) −21.1521 −0.860665
\(605\) 0 0
\(606\) 0 0
\(607\) 22.6090 0.917672 0.458836 0.888521i \(-0.348266\pi\)
0.458836 + 0.888521i \(0.348266\pi\)
\(608\) −12.3919 −0.502558
\(609\) 0 0
\(610\) −56.7153 −2.29634
\(611\) 21.9770 0.889095
\(612\) 0 0
\(613\) 22.1412 0.894276 0.447138 0.894465i \(-0.352443\pi\)
0.447138 + 0.894465i \(0.352443\pi\)
\(614\) 3.66998 0.148108
\(615\) 0 0
\(616\) 0 0
\(617\) 20.4747 0.824281 0.412140 0.911120i \(-0.364781\pi\)
0.412140 + 0.911120i \(0.364781\pi\)
\(618\) 0 0
\(619\) 40.1013 1.61181 0.805903 0.592047i \(-0.201680\pi\)
0.805903 + 0.592047i \(0.201680\pi\)
\(620\) 54.7549 2.19901
\(621\) 0 0
\(622\) −48.2620 −1.93513
\(623\) 11.2670 0.451403
\(624\) 0 0
\(625\) −25.0885 −1.00354
\(626\) −38.2967 −1.53064
\(627\) 0 0
\(628\) 3.76005 0.150042
\(629\) −11.5016 −0.458598
\(630\) 0 0
\(631\) 29.1072 1.15874 0.579370 0.815064i \(-0.303298\pi\)
0.579370 + 0.815064i \(0.303298\pi\)
\(632\) 7.81767 0.310970
\(633\) 0 0
\(634\) 68.4272 2.71759
\(635\) −48.5645 −1.92722
\(636\) 0 0
\(637\) −2.12862 −0.0843391
\(638\) 0 0
\(639\) 0 0
\(640\) −20.7942 −0.821964
\(641\) −3.75720 −0.148400 −0.0742001 0.997243i \(-0.523640\pi\)
−0.0742001 + 0.997243i \(0.523640\pi\)
\(642\) 0 0
\(643\) −9.96068 −0.392811 −0.196405 0.980523i \(-0.562927\pi\)
−0.196405 + 0.980523i \(0.562927\pi\)
\(644\) −12.2773 −0.483793
\(645\) 0 0
\(646\) −15.4722 −0.608744
\(647\) 11.5017 0.452180 0.226090 0.974106i \(-0.427406\pi\)
0.226090 + 0.974106i \(0.427406\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 22.2464 0.872577
\(651\) 0 0
\(652\) −16.3350 −0.639729
\(653\) 5.08048 0.198815 0.0994073 0.995047i \(-0.468305\pi\)
0.0994073 + 0.995047i \(0.468305\pi\)
\(654\) 0 0
\(655\) 33.9384 1.32608
\(656\) −0.811263 −0.0316745
\(657\) 0 0
\(658\) 21.6574 0.844293
\(659\) −42.3777 −1.65080 −0.825401 0.564547i \(-0.809051\pi\)
−0.825401 + 0.564547i \(0.809051\pi\)
\(660\) 0 0
\(661\) −12.7151 −0.494559 −0.247280 0.968944i \(-0.579537\pi\)
−0.247280 + 0.968944i \(0.579537\pi\)
\(662\) −14.2754 −0.554829
\(663\) 0 0
\(664\) 14.1486 0.549074
\(665\) 4.86075 0.188492
\(666\) 0 0
\(667\) 4.90152 0.189788
\(668\) −29.4019 −1.13759
\(669\) 0 0
\(670\) 87.9539 3.39796
\(671\) 0 0
\(672\) 0 0
\(673\) 28.0236 1.08023 0.540115 0.841591i \(-0.318381\pi\)
0.540115 + 0.841591i \(0.318381\pi\)
\(674\) 0.780419 0.0300606
\(675\) 0 0
\(676\) −20.3273 −0.781818
\(677\) 12.3532 0.474773 0.237387 0.971415i \(-0.423709\pi\)
0.237387 + 0.971415i \(0.423709\pi\)
\(678\) 0 0
\(679\) 7.08789 0.272008
\(680\) −12.7163 −0.487649
\(681\) 0 0
\(682\) 0 0
\(683\) −4.43874 −0.169844 −0.0849219 0.996388i \(-0.527064\pi\)
−0.0849219 + 0.996388i \(0.527064\pi\)
\(684\) 0 0
\(685\) −13.9842 −0.534309
\(686\) −2.09767 −0.0800893
\(687\) 0 0
\(688\) 28.2264 1.07612
\(689\) −2.01283 −0.0766825
\(690\) 0 0
\(691\) 33.1423 1.26079 0.630395 0.776274i \(-0.282893\pi\)
0.630395 + 0.776274i \(0.282893\pi\)
\(692\) −47.6441 −1.81116
\(693\) 0 0
\(694\) −12.0632 −0.457911
\(695\) 14.2968 0.542307
\(696\) 0 0
\(697\) −1.27967 −0.0484708
\(698\) −1.35619 −0.0513325
\(699\) 0 0
\(700\) 11.9584 0.451986
\(701\) 4.62594 0.174719 0.0873596 0.996177i \(-0.472157\pi\)
0.0873596 + 0.996177i \(0.472157\pi\)
\(702\) 0 0
\(703\) −3.69080 −0.139201
\(704\) 0 0
\(705\) 0 0
\(706\) −38.5133 −1.44947
\(707\) 8.34609 0.313887
\(708\) 0 0
\(709\) 14.9159 0.560178 0.280089 0.959974i \(-0.409636\pi\)
0.280089 + 0.959974i \(0.409636\pi\)
\(710\) −93.8220 −3.52108
\(711\) 0 0
\(712\) −9.45868 −0.354479
\(713\) 36.9329 1.38315
\(714\) 0 0
\(715\) 0 0
\(716\) −19.7352 −0.737541
\(717\) 0 0
\(718\) −56.5852 −2.11174
\(719\) −33.0569 −1.23282 −0.616408 0.787427i \(-0.711413\pi\)
−0.616408 + 0.787427i \(0.711413\pi\)
\(720\) 0 0
\(721\) −7.53555 −0.280639
\(722\) 34.8907 1.29850
\(723\) 0 0
\(724\) 25.0382 0.930538
\(725\) −4.77422 −0.177310
\(726\) 0 0
\(727\) 6.06384 0.224895 0.112448 0.993658i \(-0.464131\pi\)
0.112448 + 0.993658i \(0.464131\pi\)
\(728\) 1.78698 0.0662301
\(729\) 0 0
\(730\) −51.1775 −1.89417
\(731\) 44.5236 1.64677
\(732\) 0 0
\(733\) 0.926026 0.0342035 0.0171018 0.999854i \(-0.494556\pi\)
0.0171018 + 0.999854i \(0.494556\pi\)
\(734\) −59.4092 −2.19283
\(735\) 0 0
\(736\) −41.2005 −1.51867
\(737\) 0 0
\(738\) 0 0
\(739\) 30.4962 1.12182 0.560910 0.827877i \(-0.310451\pi\)
0.560910 + 0.827877i \(0.310451\pi\)
\(740\) −18.1926 −0.668773
\(741\) 0 0
\(742\) −1.98355 −0.0728185
\(743\) 17.8520 0.654925 0.327463 0.944864i \(-0.393806\pi\)
0.327463 + 0.944864i \(0.393806\pi\)
\(744\) 0 0
\(745\) −9.46169 −0.346649
\(746\) −27.1050 −0.992386
\(747\) 0 0
\(748\) 0 0
\(749\) 15.3887 0.562291
\(750\) 0 0
\(751\) 35.7837 1.30577 0.652884 0.757458i \(-0.273559\pi\)
0.652884 + 0.757458i \(0.273559\pi\)
\(752\) 31.3805 1.14433
\(753\) 0 0
\(754\) −4.27870 −0.155821
\(755\) 27.8431 1.01332
\(756\) 0 0
\(757\) 42.5125 1.54514 0.772571 0.634928i \(-0.218970\pi\)
0.772571 + 0.634928i \(0.218970\pi\)
\(758\) −43.9420 −1.59605
\(759\) 0 0
\(760\) −4.08061 −0.148019
\(761\) 9.10570 0.330081 0.165041 0.986287i \(-0.447224\pi\)
0.165041 + 0.986287i \(0.447224\pi\)
\(762\) 0 0
\(763\) −20.2581 −0.733393
\(764\) −34.1219 −1.23449
\(765\) 0 0
\(766\) 52.3922 1.89301
\(767\) −20.3300 −0.734073
\(768\) 0 0
\(769\) 6.93312 0.250015 0.125007 0.992156i \(-0.460105\pi\)
0.125007 + 0.992156i \(0.460105\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 16.8406 0.606108
\(773\) 18.2629 0.656872 0.328436 0.944526i \(-0.393478\pi\)
0.328436 + 0.944526i \(0.393478\pi\)
\(774\) 0 0
\(775\) −35.9737 −1.29221
\(776\) −5.95030 −0.213603
\(777\) 0 0
\(778\) −28.4820 −1.02113
\(779\) −0.410639 −0.0147127
\(780\) 0 0
\(781\) 0 0
\(782\) −51.4418 −1.83955
\(783\) 0 0
\(784\) −3.03942 −0.108551
\(785\) −4.94947 −0.176654
\(786\) 0 0
\(787\) 12.2571 0.436918 0.218459 0.975846i \(-0.429897\pi\)
0.218459 + 0.975846i \(0.429897\pi\)
\(788\) −8.64800 −0.308072
\(789\) 0 0
\(790\) −61.7171 −2.19580
\(791\) 18.1019 0.643628
\(792\) 0 0
\(793\) 18.2158 0.646862
\(794\) 8.94549 0.317464
\(795\) 0 0
\(796\) −9.24405 −0.327647
\(797\) 22.8561 0.809606 0.404803 0.914404i \(-0.367340\pi\)
0.404803 + 0.914404i \(0.367340\pi\)
\(798\) 0 0
\(799\) 49.4989 1.75114
\(800\) 40.1305 1.41883
\(801\) 0 0
\(802\) −68.0531 −2.40304
\(803\) 0 0
\(804\) 0 0
\(805\) 16.1610 0.569600
\(806\) −32.2400 −1.13561
\(807\) 0 0
\(808\) −7.00657 −0.246490
\(809\) −43.6342 −1.53410 −0.767048 0.641589i \(-0.778275\pi\)
−0.767048 + 0.641589i \(0.778275\pi\)
\(810\) 0 0
\(811\) 13.8319 0.485705 0.242853 0.970063i \(-0.421917\pi\)
0.242853 + 0.970063i \(0.421917\pi\)
\(812\) −2.29999 −0.0807138
\(813\) 0 0
\(814\) 0 0
\(815\) 21.5023 0.753194
\(816\) 0 0
\(817\) 14.2874 0.499854
\(818\) −52.2060 −1.82534
\(819\) 0 0
\(820\) −2.02411 −0.0706850
\(821\) −19.5987 −0.684000 −0.342000 0.939700i \(-0.611104\pi\)
−0.342000 + 0.939700i \(0.611104\pi\)
\(822\) 0 0
\(823\) 38.0461 1.32620 0.663101 0.748530i \(-0.269240\pi\)
0.663101 + 0.748530i \(0.269240\pi\)
\(824\) 6.32612 0.220381
\(825\) 0 0
\(826\) −20.0343 −0.697084
\(827\) −2.70341 −0.0940068 −0.0470034 0.998895i \(-0.514967\pi\)
−0.0470034 + 0.998895i \(0.514967\pi\)
\(828\) 0 0
\(829\) −17.6913 −0.614444 −0.307222 0.951638i \(-0.599399\pi\)
−0.307222 + 0.951638i \(0.599399\pi\)
\(830\) −111.697 −3.87708
\(831\) 0 0
\(832\) 23.0258 0.798276
\(833\) −4.79430 −0.166113
\(834\) 0 0
\(835\) 38.7027 1.33936
\(836\) 0 0
\(837\) 0 0
\(838\) 22.2243 0.767725
\(839\) −0.168659 −0.00582276 −0.00291138 0.999996i \(-0.500927\pi\)
−0.00291138 + 0.999996i \(0.500927\pi\)
\(840\) 0 0
\(841\) −28.0818 −0.968337
\(842\) 79.4244 2.73715
\(843\) 0 0
\(844\) −26.7203 −0.919749
\(845\) 26.7574 0.920484
\(846\) 0 0
\(847\) 0 0
\(848\) −2.87407 −0.0986961
\(849\) 0 0
\(850\) 50.1057 1.71861
\(851\) −12.2711 −0.420650
\(852\) 0 0
\(853\) 42.9770 1.47151 0.735753 0.677250i \(-0.236829\pi\)
0.735753 + 0.677250i \(0.236829\pi\)
\(854\) 17.9509 0.614267
\(855\) 0 0
\(856\) −12.9189 −0.441557
\(857\) −3.31550 −0.113255 −0.0566276 0.998395i \(-0.518035\pi\)
−0.0566276 + 0.998395i \(0.518035\pi\)
\(858\) 0 0
\(859\) 12.5924 0.429646 0.214823 0.976653i \(-0.431083\pi\)
0.214823 + 0.976653i \(0.431083\pi\)
\(860\) 70.4252 2.40148
\(861\) 0 0
\(862\) −11.8411 −0.403309
\(863\) 55.8374 1.90073 0.950363 0.311143i \(-0.100712\pi\)
0.950363 + 0.311143i \(0.100712\pi\)
\(864\) 0 0
\(865\) 62.7154 2.13239
\(866\) 5.65381 0.192124
\(867\) 0 0
\(868\) −17.3304 −0.588232
\(869\) 0 0
\(870\) 0 0
\(871\) −28.2490 −0.957182
\(872\) 17.0067 0.575921
\(873\) 0 0
\(874\) −16.5074 −0.558372
\(875\) 0.0560984 0.00189647
\(876\) 0 0
\(877\) −8.26736 −0.279169 −0.139584 0.990210i \(-0.544577\pi\)
−0.139584 + 0.990210i \(0.544577\pi\)
\(878\) 64.1191 2.16391
\(879\) 0 0
\(880\) 0 0
\(881\) −26.0843 −0.878803 −0.439401 0.898291i \(-0.644809\pi\)
−0.439401 + 0.898291i \(0.644809\pi\)
\(882\) 0 0
\(883\) 0.915087 0.0307951 0.0153976 0.999881i \(-0.495099\pi\)
0.0153976 + 0.999881i \(0.495099\pi\)
\(884\) 24.4948 0.823848
\(885\) 0 0
\(886\) −51.6804 −1.73624
\(887\) −56.0572 −1.88222 −0.941108 0.338107i \(-0.890214\pi\)
−0.941108 + 0.338107i \(0.890214\pi\)
\(888\) 0 0
\(889\) 15.3711 0.515530
\(890\) 74.6722 2.50302
\(891\) 0 0
\(892\) −12.6520 −0.423619
\(893\) 15.8840 0.531537
\(894\) 0 0
\(895\) 25.9781 0.868353
\(896\) 6.58156 0.219875
\(897\) 0 0
\(898\) −32.2528 −1.07629
\(899\) 6.91889 0.230758
\(900\) 0 0
\(901\) −4.53349 −0.151033
\(902\) 0 0
\(903\) 0 0
\(904\) −15.1966 −0.505430
\(905\) −32.9586 −1.09558
\(906\) 0 0
\(907\) 9.24637 0.307021 0.153510 0.988147i \(-0.450942\pi\)
0.153510 + 0.988147i \(0.450942\pi\)
\(908\) −35.5719 −1.18050
\(909\) 0 0
\(910\) −14.1075 −0.467658
\(911\) 37.2957 1.23566 0.617831 0.786311i \(-0.288011\pi\)
0.617831 + 0.786311i \(0.288011\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 29.8629 0.987777
\(915\) 0 0
\(916\) 18.2531 0.603098
\(917\) −10.7418 −0.354726
\(918\) 0 0
\(919\) 14.6385 0.482880 0.241440 0.970416i \(-0.422380\pi\)
0.241440 + 0.970416i \(0.422380\pi\)
\(920\) −13.5672 −0.447297
\(921\) 0 0
\(922\) 77.0155 2.53637
\(923\) 30.1337 0.991864
\(924\) 0 0
\(925\) 11.9524 0.392994
\(926\) 72.9322 2.39670
\(927\) 0 0
\(928\) −7.71838 −0.253368
\(929\) 7.80689 0.256136 0.128068 0.991765i \(-0.459122\pi\)
0.128068 + 0.991765i \(0.459122\pi\)
\(930\) 0 0
\(931\) −1.53847 −0.0504213
\(932\) −3.61337 −0.118360
\(933\) 0 0
\(934\) 28.4309 0.930287
\(935\) 0 0
\(936\) 0 0
\(937\) 53.1013 1.73474 0.867372 0.497661i \(-0.165808\pi\)
0.867372 + 0.497661i \(0.165808\pi\)
\(938\) −27.8382 −0.908950
\(939\) 0 0
\(940\) 78.2948 2.55369
\(941\) −13.0587 −0.425702 −0.212851 0.977085i \(-0.568275\pi\)
−0.212851 + 0.977085i \(0.568275\pi\)
\(942\) 0 0
\(943\) −1.36529 −0.0444599
\(944\) −29.0288 −0.944806
\(945\) 0 0
\(946\) 0 0
\(947\) 39.8291 1.29427 0.647136 0.762375i \(-0.275967\pi\)
0.647136 + 0.762375i \(0.275967\pi\)
\(948\) 0 0
\(949\) 16.4372 0.533574
\(950\) 16.0787 0.521662
\(951\) 0 0
\(952\) 4.02483 0.130446
\(953\) 3.79128 0.122812 0.0614058 0.998113i \(-0.480442\pi\)
0.0614058 + 0.998113i \(0.480442\pi\)
\(954\) 0 0
\(955\) 44.9157 1.45344
\(956\) −12.0571 −0.389955
\(957\) 0 0
\(958\) −57.8899 −1.87034
\(959\) 4.42612 0.142927
\(960\) 0 0
\(961\) 21.1338 0.681735
\(962\) 10.7119 0.345366
\(963\) 0 0
\(964\) −38.7639 −1.24850
\(965\) −22.1679 −0.713609
\(966\) 0 0
\(967\) −5.63563 −0.181230 −0.0906149 0.995886i \(-0.528883\pi\)
−0.0906149 + 0.995886i \(0.528883\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 46.9751 1.50828
\(971\) 2.41724 0.0775729 0.0387865 0.999248i \(-0.487651\pi\)
0.0387865 + 0.999248i \(0.487651\pi\)
\(972\) 0 0
\(973\) −4.52505 −0.145067
\(974\) 42.9555 1.37638
\(975\) 0 0
\(976\) 26.0100 0.832560
\(977\) −46.6426 −1.49223 −0.746114 0.665818i \(-0.768083\pi\)
−0.746114 + 0.665818i \(0.768083\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −7.58338 −0.242242
\(981\) 0 0
\(982\) −60.5521 −1.93230
\(983\) −21.6905 −0.691821 −0.345910 0.938268i \(-0.612430\pi\)
−0.345910 + 0.938268i \(0.612430\pi\)
\(984\) 0 0
\(985\) 11.3836 0.362713
\(986\) −9.63694 −0.306903
\(987\) 0 0
\(988\) 7.86026 0.250068
\(989\) 47.5027 1.51050
\(990\) 0 0
\(991\) 33.7715 1.07279 0.536394 0.843968i \(-0.319786\pi\)
0.536394 + 0.843968i \(0.319786\pi\)
\(992\) −58.1579 −1.84652
\(993\) 0 0
\(994\) 29.6955 0.941884
\(995\) 12.1682 0.385759
\(996\) 0 0
\(997\) −0.624922 −0.0197915 −0.00989575 0.999951i \(-0.503150\pi\)
−0.00989575 + 0.999951i \(0.503150\pi\)
\(998\) 45.0872 1.42721
\(999\) 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.cy.1.3 10
3.2 odd 2 2541.2.a.br.1.8 10
11.7 odd 10 693.2.m.j.379.4 20
11.8 odd 10 693.2.m.j.64.4 20
11.10 odd 2 7623.2.a.cx.1.8 10
33.8 even 10 231.2.j.g.64.2 20
33.29 even 10 231.2.j.g.148.2 yes 20
33.32 even 2 2541.2.a.bq.1.3 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.j.g.64.2 20 33.8 even 10
231.2.j.g.148.2 yes 20 33.29 even 10
693.2.m.j.64.4 20 11.8 odd 10
693.2.m.j.379.4 20 11.7 odd 10
2541.2.a.bq.1.3 10 33.32 even 2
2541.2.a.br.1.8 10 3.2 odd 2
7623.2.a.cx.1.8 10 11.10 odd 2
7623.2.a.cy.1.3 10 1.1 even 1 trivial