Properties

Label 7623.2.a.dc.1.7
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 0
Dimension 16
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: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 693)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-0.759363\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.759363 q^{2} -1.42337 q^{4} +2.86525 q^{5} +1.00000 q^{7} +2.59958 q^{8} +O(q^{10})\) \(q-0.759363 q^{2} -1.42337 q^{4} +2.86525 q^{5} +1.00000 q^{7} +2.59958 q^{8} -2.17577 q^{10} -6.96619 q^{13} -0.759363 q^{14} +0.872709 q^{16} +6.65872 q^{17} -5.75106 q^{19} -4.07830 q^{20} -0.724581 q^{23} +3.20966 q^{25} +5.28987 q^{26} -1.42337 q^{28} +7.56740 q^{29} +2.81968 q^{31} -5.86186 q^{32} -5.05639 q^{34} +2.86525 q^{35} -1.72211 q^{37} +4.36715 q^{38} +7.44844 q^{40} +1.12086 q^{41} -11.7092 q^{43} +0.550220 q^{46} +5.18956 q^{47} +1.00000 q^{49} -2.43729 q^{50} +9.91544 q^{52} -6.25771 q^{53} +2.59958 q^{56} -5.74641 q^{58} +3.98228 q^{59} -5.97074 q^{61} -2.14116 q^{62} +2.70587 q^{64} -19.9599 q^{65} +10.3611 q^{67} -9.47781 q^{68} -2.17577 q^{70} -0.995183 q^{71} +3.53684 q^{73} +1.30771 q^{74} +8.18588 q^{76} -0.669794 q^{79} +2.50053 q^{80} -0.851138 q^{82} +13.4240 q^{83} +19.0789 q^{85} +8.89157 q^{86} +16.6006 q^{89} -6.96619 q^{91} +1.03135 q^{92} -3.94076 q^{94} -16.4782 q^{95} +10.1121 q^{97} -0.759363 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 20q^{4} + 16q^{7} + O(q^{10}) \) \( 16q + 20q^{4} + 16q^{7} - 6q^{10} + 32q^{16} + 10q^{19} + 44q^{25} + 20q^{28} + 30q^{31} + 12q^{34} + 38q^{37} - 68q^{40} + 16q^{43} + 80q^{46} + 16q^{49} + 2q^{52} + 18q^{58} - 28q^{61} + 34q^{64} + 52q^{67} - 6q^{70} - 14q^{73} - 14q^{76} + 54q^{79} + 64q^{82} + 30q^{85} - 60q^{94} + 108q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.759363 −0.536951 −0.268475 0.963287i \(-0.586520\pi\)
−0.268475 + 0.963287i \(0.586520\pi\)
\(3\) 0 0
\(4\) −1.42337 −0.711684
\(5\) 2.86525 1.28138 0.640689 0.767800i \(-0.278649\pi\)
0.640689 + 0.767800i \(0.278649\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 2.59958 0.919090
\(9\) 0 0
\(10\) −2.17577 −0.688038
\(11\) 0 0
\(12\) 0 0
\(13\) −6.96619 −1.93207 −0.966036 0.258407i \(-0.916803\pi\)
−0.966036 + 0.258407i \(0.916803\pi\)
\(14\) −0.759363 −0.202948
\(15\) 0 0
\(16\) 0.872709 0.218177
\(17\) 6.65872 1.61498 0.807489 0.589883i \(-0.200826\pi\)
0.807489 + 0.589883i \(0.200826\pi\)
\(18\) 0 0
\(19\) −5.75106 −1.31938 −0.659692 0.751536i \(-0.729313\pi\)
−0.659692 + 0.751536i \(0.729313\pi\)
\(20\) −4.07830 −0.911936
\(21\) 0 0
\(22\) 0 0
\(23\) −0.724581 −0.151086 −0.0755428 0.997143i \(-0.524069\pi\)
−0.0755428 + 0.997143i \(0.524069\pi\)
\(24\) 0 0
\(25\) 3.20966 0.641931
\(26\) 5.28987 1.03743
\(27\) 0 0
\(28\) −1.42337 −0.268991
\(29\) 7.56740 1.40523 0.702616 0.711570i \(-0.252015\pi\)
0.702616 + 0.711570i \(0.252015\pi\)
\(30\) 0 0
\(31\) 2.81968 0.506430 0.253215 0.967410i \(-0.418512\pi\)
0.253215 + 0.967410i \(0.418512\pi\)
\(32\) −5.86186 −1.03624
\(33\) 0 0
\(34\) −5.05639 −0.867164
\(35\) 2.86525 0.484316
\(36\) 0 0
\(37\) −1.72211 −0.283113 −0.141557 0.989930i \(-0.545211\pi\)
−0.141557 + 0.989930i \(0.545211\pi\)
\(38\) 4.36715 0.708445
\(39\) 0 0
\(40\) 7.44844 1.17770
\(41\) 1.12086 0.175048 0.0875242 0.996162i \(-0.472104\pi\)
0.0875242 + 0.996162i \(0.472104\pi\)
\(42\) 0 0
\(43\) −11.7092 −1.78564 −0.892821 0.450411i \(-0.851277\pi\)
−0.892821 + 0.450411i \(0.851277\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0.550220 0.0811256
\(47\) 5.18956 0.756975 0.378488 0.925606i \(-0.376444\pi\)
0.378488 + 0.925606i \(0.376444\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −2.43729 −0.344686
\(51\) 0 0
\(52\) 9.91544 1.37502
\(53\) −6.25771 −0.859562 −0.429781 0.902933i \(-0.641409\pi\)
−0.429781 + 0.902933i \(0.641409\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.59958 0.347383
\(57\) 0 0
\(58\) −5.74641 −0.754540
\(59\) 3.98228 0.518448 0.259224 0.965817i \(-0.416533\pi\)
0.259224 + 0.965817i \(0.416533\pi\)
\(60\) 0 0
\(61\) −5.97074 −0.764475 −0.382237 0.924064i \(-0.624846\pi\)
−0.382237 + 0.924064i \(0.624846\pi\)
\(62\) −2.14116 −0.271928
\(63\) 0 0
\(64\) 2.70587 0.338233
\(65\) −19.9599 −2.47572
\(66\) 0 0
\(67\) 10.3611 1.26581 0.632906 0.774229i \(-0.281862\pi\)
0.632906 + 0.774229i \(0.281862\pi\)
\(68\) −9.47781 −1.14935
\(69\) 0 0
\(70\) −2.17577 −0.260054
\(71\) −0.995183 −0.118107 −0.0590533 0.998255i \(-0.518808\pi\)
−0.0590533 + 0.998255i \(0.518808\pi\)
\(72\) 0 0
\(73\) 3.53684 0.413956 0.206978 0.978346i \(-0.433637\pi\)
0.206978 + 0.978346i \(0.433637\pi\)
\(74\) 1.30771 0.152018
\(75\) 0 0
\(76\) 8.18588 0.938984
\(77\) 0 0
\(78\) 0 0
\(79\) −0.669794 −0.0753578 −0.0376789 0.999290i \(-0.511996\pi\)
−0.0376789 + 0.999290i \(0.511996\pi\)
\(80\) 2.50053 0.279568
\(81\) 0 0
\(82\) −0.851138 −0.0939924
\(83\) 13.4240 1.47348 0.736739 0.676177i \(-0.236365\pi\)
0.736739 + 0.676177i \(0.236365\pi\)
\(84\) 0 0
\(85\) 19.0789 2.06940
\(86\) 8.89157 0.958803
\(87\) 0 0
\(88\) 0 0
\(89\) 16.6006 1.75966 0.879828 0.475292i \(-0.157657\pi\)
0.879828 + 0.475292i \(0.157657\pi\)
\(90\) 0 0
\(91\) −6.96619 −0.730255
\(92\) 1.03135 0.107525
\(93\) 0 0
\(94\) −3.94076 −0.406459
\(95\) −16.4782 −1.69063
\(96\) 0 0
\(97\) 10.1121 1.02672 0.513362 0.858172i \(-0.328400\pi\)
0.513362 + 0.858172i \(0.328400\pi\)
\(98\) −0.759363 −0.0767073
\(99\) 0 0
\(100\) −4.56852 −0.456852
\(101\) 6.42353 0.639165 0.319582 0.947559i \(-0.396457\pi\)
0.319582 + 0.947559i \(0.396457\pi\)
\(102\) 0 0
\(103\) −8.95787 −0.882645 −0.441323 0.897348i \(-0.645491\pi\)
−0.441323 + 0.897348i \(0.645491\pi\)
\(104\) −18.1092 −1.77575
\(105\) 0 0
\(106\) 4.75188 0.461543
\(107\) 4.52891 0.437826 0.218913 0.975744i \(-0.429749\pi\)
0.218913 + 0.975744i \(0.429749\pi\)
\(108\) 0 0
\(109\) −1.02744 −0.0984106 −0.0492053 0.998789i \(-0.515669\pi\)
−0.0492053 + 0.998789i \(0.515669\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0.872709 0.0824632
\(113\) 13.2380 1.24533 0.622664 0.782490i \(-0.286050\pi\)
0.622664 + 0.782490i \(0.286050\pi\)
\(114\) 0 0
\(115\) −2.07611 −0.193598
\(116\) −10.7712 −1.00008
\(117\) 0 0
\(118\) −3.02399 −0.278381
\(119\) 6.65872 0.610404
\(120\) 0 0
\(121\) 0 0
\(122\) 4.53396 0.410486
\(123\) 0 0
\(124\) −4.01344 −0.360418
\(125\) −5.12978 −0.458822
\(126\) 0 0
\(127\) 16.2934 1.44581 0.722903 0.690949i \(-0.242807\pi\)
0.722903 + 0.690949i \(0.242807\pi\)
\(128\) 9.66899 0.854626
\(129\) 0 0
\(130\) 15.1568 1.32934
\(131\) 6.19831 0.541549 0.270774 0.962643i \(-0.412720\pi\)
0.270774 + 0.962643i \(0.412720\pi\)
\(132\) 0 0
\(133\) −5.75106 −0.498680
\(134\) −7.86785 −0.679679
\(135\) 0 0
\(136\) 17.3099 1.48431
\(137\) 16.0095 1.36779 0.683893 0.729582i \(-0.260286\pi\)
0.683893 + 0.729582i \(0.260286\pi\)
\(138\) 0 0
\(139\) 4.61401 0.391355 0.195678 0.980668i \(-0.437309\pi\)
0.195678 + 0.980668i \(0.437309\pi\)
\(140\) −4.07830 −0.344679
\(141\) 0 0
\(142\) 0.755706 0.0634174
\(143\) 0 0
\(144\) 0 0
\(145\) 21.6825 1.80063
\(146\) −2.68575 −0.222274
\(147\) 0 0
\(148\) 2.45119 0.201487
\(149\) −7.21841 −0.591355 −0.295678 0.955288i \(-0.595545\pi\)
−0.295678 + 0.955288i \(0.595545\pi\)
\(150\) 0 0
\(151\) 0.626349 0.0509716 0.0254858 0.999675i \(-0.491887\pi\)
0.0254858 + 0.999675i \(0.491887\pi\)
\(152\) −14.9503 −1.21263
\(153\) 0 0
\(154\) 0 0
\(155\) 8.07909 0.648928
\(156\) 0 0
\(157\) −6.14752 −0.490626 −0.245313 0.969444i \(-0.578891\pi\)
−0.245313 + 0.969444i \(0.578891\pi\)
\(158\) 0.508617 0.0404634
\(159\) 0 0
\(160\) −16.7957 −1.32782
\(161\) −0.724581 −0.0571050
\(162\) 0 0
\(163\) 21.6415 1.69510 0.847548 0.530719i \(-0.178078\pi\)
0.847548 + 0.530719i \(0.178078\pi\)
\(164\) −1.59539 −0.124579
\(165\) 0 0
\(166\) −10.1937 −0.791186
\(167\) −23.9947 −1.85677 −0.928384 0.371622i \(-0.878802\pi\)
−0.928384 + 0.371622i \(0.878802\pi\)
\(168\) 0 0
\(169\) 35.5278 2.73290
\(170\) −14.4878 −1.11117
\(171\) 0 0
\(172\) 16.6666 1.27081
\(173\) −10.8109 −0.821935 −0.410968 0.911650i \(-0.634809\pi\)
−0.410968 + 0.911650i \(0.634809\pi\)
\(174\) 0 0
\(175\) 3.20966 0.242627
\(176\) 0 0
\(177\) 0 0
\(178\) −12.6059 −0.944849
\(179\) 5.92839 0.443109 0.221554 0.975148i \(-0.428887\pi\)
0.221554 + 0.975148i \(0.428887\pi\)
\(180\) 0 0
\(181\) −4.84617 −0.360213 −0.180106 0.983647i \(-0.557644\pi\)
−0.180106 + 0.983647i \(0.557644\pi\)
\(182\) 5.28987 0.392111
\(183\) 0 0
\(184\) −1.88361 −0.138861
\(185\) −4.93427 −0.362775
\(186\) 0 0
\(187\) 0 0
\(188\) −7.38665 −0.538727
\(189\) 0 0
\(190\) 12.5130 0.907786
\(191\) −15.6923 −1.13546 −0.567729 0.823216i \(-0.692178\pi\)
−0.567729 + 0.823216i \(0.692178\pi\)
\(192\) 0 0
\(193\) −22.7548 −1.63793 −0.818963 0.573846i \(-0.805451\pi\)
−0.818963 + 0.573846i \(0.805451\pi\)
\(194\) −7.67873 −0.551301
\(195\) 0 0
\(196\) −1.42337 −0.101669
\(197\) 15.3025 1.09026 0.545129 0.838352i \(-0.316481\pi\)
0.545129 + 0.838352i \(0.316481\pi\)
\(198\) 0 0
\(199\) −0.547814 −0.0388335 −0.0194168 0.999811i \(-0.506181\pi\)
−0.0194168 + 0.999811i \(0.506181\pi\)
\(200\) 8.34376 0.589993
\(201\) 0 0
\(202\) −4.87779 −0.343200
\(203\) 7.56740 0.531128
\(204\) 0 0
\(205\) 3.21153 0.224303
\(206\) 6.80228 0.473937
\(207\) 0 0
\(208\) −6.07945 −0.421534
\(209\) 0 0
\(210\) 0 0
\(211\) −13.8726 −0.955030 −0.477515 0.878624i \(-0.658462\pi\)
−0.477515 + 0.878624i \(0.658462\pi\)
\(212\) 8.90702 0.611737
\(213\) 0 0
\(214\) −3.43908 −0.235091
\(215\) −33.5499 −2.28808
\(216\) 0 0
\(217\) 2.81968 0.191412
\(218\) 0.780198 0.0528416
\(219\) 0 0
\(220\) 0 0
\(221\) −46.3859 −3.12025
\(222\) 0 0
\(223\) −7.60179 −0.509053 −0.254527 0.967066i \(-0.581920\pi\)
−0.254527 + 0.967066i \(0.581920\pi\)
\(224\) −5.86186 −0.391662
\(225\) 0 0
\(226\) −10.0525 −0.668680
\(227\) −0.295156 −0.0195902 −0.00979511 0.999952i \(-0.503118\pi\)
−0.00979511 + 0.999952i \(0.503118\pi\)
\(228\) 0 0
\(229\) 7.11072 0.469890 0.234945 0.972009i \(-0.424509\pi\)
0.234945 + 0.972009i \(0.424509\pi\)
\(230\) 1.57652 0.103953
\(231\) 0 0
\(232\) 19.6721 1.29153
\(233\) 10.2537 0.671743 0.335872 0.941908i \(-0.390969\pi\)
0.335872 + 0.941908i \(0.390969\pi\)
\(234\) 0 0
\(235\) 14.8694 0.969972
\(236\) −5.66824 −0.368971
\(237\) 0 0
\(238\) −5.05639 −0.327757
\(239\) −19.8042 −1.28103 −0.640513 0.767947i \(-0.721278\pi\)
−0.640513 + 0.767947i \(0.721278\pi\)
\(240\) 0 0
\(241\) −20.0305 −1.29028 −0.645138 0.764066i \(-0.723200\pi\)
−0.645138 + 0.764066i \(0.723200\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 8.49856 0.544064
\(245\) 2.86525 0.183054
\(246\) 0 0
\(247\) 40.0630 2.54915
\(248\) 7.32999 0.465455
\(249\) 0 0
\(250\) 3.89537 0.246365
\(251\) −2.69309 −0.169986 −0.0849932 0.996382i \(-0.527087\pi\)
−0.0849932 + 0.996382i \(0.527087\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −12.3726 −0.776327
\(255\) 0 0
\(256\) −12.7540 −0.797126
\(257\) 25.3591 1.58186 0.790929 0.611908i \(-0.209598\pi\)
0.790929 + 0.611908i \(0.209598\pi\)
\(258\) 0 0
\(259\) −1.72211 −0.107007
\(260\) 28.4102 1.76193
\(261\) 0 0
\(262\) −4.70677 −0.290785
\(263\) −9.04980 −0.558034 −0.279017 0.960286i \(-0.590009\pi\)
−0.279017 + 0.960286i \(0.590009\pi\)
\(264\) 0 0
\(265\) −17.9299 −1.10142
\(266\) 4.36715 0.267767
\(267\) 0 0
\(268\) −14.7477 −0.900857
\(269\) 2.73289 0.166627 0.0833136 0.996523i \(-0.473450\pi\)
0.0833136 + 0.996523i \(0.473450\pi\)
\(270\) 0 0
\(271\) 17.9520 1.09051 0.545254 0.838271i \(-0.316433\pi\)
0.545254 + 0.838271i \(0.316433\pi\)
\(272\) 5.81113 0.352351
\(273\) 0 0
\(274\) −12.1570 −0.734434
\(275\) 0 0
\(276\) 0 0
\(277\) −11.1503 −0.669958 −0.334979 0.942226i \(-0.608729\pi\)
−0.334979 + 0.942226i \(0.608729\pi\)
\(278\) −3.50371 −0.210139
\(279\) 0 0
\(280\) 7.44844 0.445130
\(281\) 3.96239 0.236377 0.118188 0.992991i \(-0.462291\pi\)
0.118188 + 0.992991i \(0.462291\pi\)
\(282\) 0 0
\(283\) 5.74911 0.341749 0.170875 0.985293i \(-0.445341\pi\)
0.170875 + 0.985293i \(0.445341\pi\)
\(284\) 1.41651 0.0840545
\(285\) 0 0
\(286\) 0 0
\(287\) 1.12086 0.0661621
\(288\) 0 0
\(289\) 27.3386 1.60815
\(290\) −16.4649 −0.966852
\(291\) 0 0
\(292\) −5.03422 −0.294606
\(293\) 23.2438 1.35792 0.678958 0.734177i \(-0.262432\pi\)
0.678958 + 0.734177i \(0.262432\pi\)
\(294\) 0 0
\(295\) 11.4102 0.664328
\(296\) −4.47676 −0.260206
\(297\) 0 0
\(298\) 5.48140 0.317529
\(299\) 5.04757 0.291908
\(300\) 0 0
\(301\) −11.7092 −0.674909
\(302\) −0.475627 −0.0273692
\(303\) 0 0
\(304\) −5.01900 −0.287860
\(305\) −17.1077 −0.979582
\(306\) 0 0
\(307\) 3.87940 0.221409 0.110705 0.993853i \(-0.464689\pi\)
0.110705 + 0.993853i \(0.464689\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −6.13497 −0.348443
\(311\) 3.74658 0.212449 0.106225 0.994342i \(-0.466124\pi\)
0.106225 + 0.994342i \(0.466124\pi\)
\(312\) 0 0
\(313\) 18.0642 1.02105 0.510523 0.859864i \(-0.329452\pi\)
0.510523 + 0.859864i \(0.329452\pi\)
\(314\) 4.66820 0.263442
\(315\) 0 0
\(316\) 0.953364 0.0536309
\(317\) 7.41444 0.416436 0.208218 0.978082i \(-0.433234\pi\)
0.208218 + 0.978082i \(0.433234\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 7.75298 0.433405
\(321\) 0 0
\(322\) 0.550220 0.0306626
\(323\) −38.2947 −2.13078
\(324\) 0 0
\(325\) −22.3591 −1.24026
\(326\) −16.4338 −0.910183
\(327\) 0 0
\(328\) 2.91376 0.160885
\(329\) 5.18956 0.286110
\(330\) 0 0
\(331\) 14.5172 0.797937 0.398968 0.916965i \(-0.369368\pi\)
0.398968 + 0.916965i \(0.369368\pi\)
\(332\) −19.1073 −1.04865
\(333\) 0 0
\(334\) 18.2207 0.996993
\(335\) 29.6872 1.62198
\(336\) 0 0
\(337\) 10.5543 0.574930 0.287465 0.957791i \(-0.407187\pi\)
0.287465 + 0.957791i \(0.407187\pi\)
\(338\) −26.9785 −1.46744
\(339\) 0 0
\(340\) −27.1563 −1.47276
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −30.4391 −1.64117
\(345\) 0 0
\(346\) 8.20938 0.441339
\(347\) −22.5100 −1.20840 −0.604199 0.796833i \(-0.706507\pi\)
−0.604199 + 0.796833i \(0.706507\pi\)
\(348\) 0 0
\(349\) −20.7206 −1.10915 −0.554574 0.832135i \(-0.687119\pi\)
−0.554574 + 0.832135i \(0.687119\pi\)
\(350\) −2.43729 −0.130279
\(351\) 0 0
\(352\) 0 0
\(353\) 6.01460 0.320125 0.160062 0.987107i \(-0.448830\pi\)
0.160062 + 0.987107i \(0.448830\pi\)
\(354\) 0 0
\(355\) −2.85145 −0.151339
\(356\) −23.6287 −1.25232
\(357\) 0 0
\(358\) −4.50180 −0.237928
\(359\) −18.4700 −0.974810 −0.487405 0.873176i \(-0.662056\pi\)
−0.487405 + 0.873176i \(0.662056\pi\)
\(360\) 0 0
\(361\) 14.0747 0.740775
\(362\) 3.68000 0.193417
\(363\) 0 0
\(364\) 9.91544 0.519710
\(365\) 10.1339 0.530434
\(366\) 0 0
\(367\) −4.02990 −0.210359 −0.105180 0.994453i \(-0.533542\pi\)
−0.105180 + 0.994453i \(0.533542\pi\)
\(368\) −0.632349 −0.0329634
\(369\) 0 0
\(370\) 3.74691 0.194792
\(371\) −6.25771 −0.324884
\(372\) 0 0
\(373\) 5.06561 0.262287 0.131144 0.991363i \(-0.458135\pi\)
0.131144 + 0.991363i \(0.458135\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 13.4907 0.695728
\(377\) −52.7159 −2.71501
\(378\) 0 0
\(379\) 2.19657 0.112830 0.0564152 0.998407i \(-0.482033\pi\)
0.0564152 + 0.998407i \(0.482033\pi\)
\(380\) 23.4546 1.20319
\(381\) 0 0
\(382\) 11.9162 0.609685
\(383\) −24.4857 −1.25116 −0.625579 0.780161i \(-0.715137\pi\)
−0.625579 + 0.780161i \(0.715137\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 17.2792 0.879487
\(387\) 0 0
\(388\) −14.3932 −0.730703
\(389\) 34.3208 1.74013 0.870066 0.492936i \(-0.164076\pi\)
0.870066 + 0.492936i \(0.164076\pi\)
\(390\) 0 0
\(391\) −4.82479 −0.244000
\(392\) 2.59958 0.131299
\(393\) 0 0
\(394\) −11.6202 −0.585415
\(395\) −1.91913 −0.0965618
\(396\) 0 0
\(397\) 8.46742 0.424968 0.212484 0.977165i \(-0.431845\pi\)
0.212484 + 0.977165i \(0.431845\pi\)
\(398\) 0.415990 0.0208517
\(399\) 0 0
\(400\) 2.80110 0.140055
\(401\) 28.9079 1.44359 0.721797 0.692105i \(-0.243316\pi\)
0.721797 + 0.692105i \(0.243316\pi\)
\(402\) 0 0
\(403\) −19.6424 −0.978459
\(404\) −9.14304 −0.454883
\(405\) 0 0
\(406\) −5.74641 −0.285189
\(407\) 0 0
\(408\) 0 0
\(409\) −1.59556 −0.0788955 −0.0394478 0.999222i \(-0.512560\pi\)
−0.0394478 + 0.999222i \(0.512560\pi\)
\(410\) −2.43872 −0.120440
\(411\) 0 0
\(412\) 12.7503 0.628164
\(413\) 3.98228 0.195955
\(414\) 0 0
\(415\) 38.4632 1.88808
\(416\) 40.8348 2.00209
\(417\) 0 0
\(418\) 0 0
\(419\) −0.224640 −0.0109744 −0.00548719 0.999985i \(-0.501747\pi\)
−0.00548719 + 0.999985i \(0.501747\pi\)
\(420\) 0 0
\(421\) 11.5297 0.561924 0.280962 0.959719i \(-0.409346\pi\)
0.280962 + 0.959719i \(0.409346\pi\)
\(422\) 10.5344 0.512804
\(423\) 0 0
\(424\) −16.2674 −0.790015
\(425\) 21.3722 1.03670
\(426\) 0 0
\(427\) −5.97074 −0.288944
\(428\) −6.44630 −0.311593
\(429\) 0 0
\(430\) 25.4766 1.22859
\(431\) 27.8911 1.34347 0.671733 0.740794i \(-0.265550\pi\)
0.671733 + 0.740794i \(0.265550\pi\)
\(432\) 0 0
\(433\) 25.9669 1.24789 0.623944 0.781469i \(-0.285529\pi\)
0.623944 + 0.781469i \(0.285529\pi\)
\(434\) −2.14116 −0.102779
\(435\) 0 0
\(436\) 1.46242 0.0700372
\(437\) 4.16711 0.199340
\(438\) 0 0
\(439\) 23.5551 1.12422 0.562112 0.827061i \(-0.309989\pi\)
0.562112 + 0.827061i \(0.309989\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 35.2238 1.67542
\(443\) 24.8412 1.18024 0.590120 0.807315i \(-0.299080\pi\)
0.590120 + 0.807315i \(0.299080\pi\)
\(444\) 0 0
\(445\) 47.5648 2.25479
\(446\) 5.77252 0.273337
\(447\) 0 0
\(448\) 2.70587 0.127840
\(449\) −18.4127 −0.868947 −0.434473 0.900685i \(-0.643065\pi\)
−0.434473 + 0.900685i \(0.643065\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −18.8426 −0.886279
\(453\) 0 0
\(454\) 0.224131 0.0105190
\(455\) −19.9599 −0.935733
\(456\) 0 0
\(457\) 4.61369 0.215820 0.107910 0.994161i \(-0.465584\pi\)
0.107910 + 0.994161i \(0.465584\pi\)
\(458\) −5.39962 −0.252308
\(459\) 0 0
\(460\) 2.95506 0.137780
\(461\) 33.5148 1.56094 0.780470 0.625193i \(-0.214980\pi\)
0.780470 + 0.625193i \(0.214980\pi\)
\(462\) 0 0
\(463\) −16.5338 −0.768392 −0.384196 0.923252i \(-0.625521\pi\)
−0.384196 + 0.923252i \(0.625521\pi\)
\(464\) 6.60414 0.306590
\(465\) 0 0
\(466\) −7.78630 −0.360693
\(467\) −19.4446 −0.899788 −0.449894 0.893082i \(-0.648538\pi\)
−0.449894 + 0.893082i \(0.648538\pi\)
\(468\) 0 0
\(469\) 10.3611 0.478432
\(470\) −11.2913 −0.520827
\(471\) 0 0
\(472\) 10.3522 0.476501
\(473\) 0 0
\(474\) 0 0
\(475\) −18.4589 −0.846954
\(476\) −9.47781 −0.434415
\(477\) 0 0
\(478\) 15.0386 0.687848
\(479\) 39.2757 1.79455 0.897276 0.441469i \(-0.145543\pi\)
0.897276 + 0.441469i \(0.145543\pi\)
\(480\) 0 0
\(481\) 11.9965 0.546995
\(482\) 15.2104 0.692815
\(483\) 0 0
\(484\) 0 0
\(485\) 28.9736 1.31562
\(486\) 0 0
\(487\) −17.1008 −0.774911 −0.387455 0.921889i \(-0.626646\pi\)
−0.387455 + 0.921889i \(0.626646\pi\)
\(488\) −15.5214 −0.702621
\(489\) 0 0
\(490\) −2.17577 −0.0982911
\(491\) −34.7859 −1.56986 −0.784932 0.619582i \(-0.787302\pi\)
−0.784932 + 0.619582i \(0.787302\pi\)
\(492\) 0 0
\(493\) 50.3892 2.26942
\(494\) −30.4224 −1.36877
\(495\) 0 0
\(496\) 2.46076 0.110491
\(497\) −0.995183 −0.0446401
\(498\) 0 0
\(499\) 15.6577 0.700935 0.350467 0.936575i \(-0.386023\pi\)
0.350467 + 0.936575i \(0.386023\pi\)
\(500\) 7.30157 0.326536
\(501\) 0 0
\(502\) 2.04504 0.0912744
\(503\) 21.8981 0.976386 0.488193 0.872736i \(-0.337656\pi\)
0.488193 + 0.872736i \(0.337656\pi\)
\(504\) 0 0
\(505\) 18.4050 0.819012
\(506\) 0 0
\(507\) 0 0
\(508\) −23.1915 −1.02896
\(509\) 26.8438 1.18983 0.594915 0.803789i \(-0.297186\pi\)
0.594915 + 0.803789i \(0.297186\pi\)
\(510\) 0 0
\(511\) 3.53684 0.156461
\(512\) −9.65305 −0.426609
\(513\) 0 0
\(514\) −19.2568 −0.849380
\(515\) −25.6665 −1.13100
\(516\) 0 0
\(517\) 0 0
\(518\) 1.30771 0.0574574
\(519\) 0 0
\(520\) −51.8873 −2.27541
\(521\) 15.9670 0.699528 0.349764 0.936838i \(-0.386262\pi\)
0.349764 + 0.936838i \(0.386262\pi\)
\(522\) 0 0
\(523\) 13.2422 0.579040 0.289520 0.957172i \(-0.406504\pi\)
0.289520 + 0.957172i \(0.406504\pi\)
\(524\) −8.82247 −0.385411
\(525\) 0 0
\(526\) 6.87208 0.299637
\(527\) 18.7755 0.817873
\(528\) 0 0
\(529\) −22.4750 −0.977173
\(530\) 13.6153 0.591411
\(531\) 0 0
\(532\) 8.18588 0.354903
\(533\) −7.80810 −0.338206
\(534\) 0 0
\(535\) 12.9764 0.561021
\(536\) 26.9345 1.16339
\(537\) 0 0
\(538\) −2.07526 −0.0894706
\(539\) 0 0
\(540\) 0 0
\(541\) −25.5950 −1.10042 −0.550209 0.835027i \(-0.685452\pi\)
−0.550209 + 0.835027i \(0.685452\pi\)
\(542\) −13.6321 −0.585550
\(543\) 0 0
\(544\) −39.0325 −1.67351
\(545\) −2.94386 −0.126101
\(546\) 0 0
\(547\) 30.1551 1.28934 0.644670 0.764461i \(-0.276995\pi\)
0.644670 + 0.764461i \(0.276995\pi\)
\(548\) −22.7874 −0.973431
\(549\) 0 0
\(550\) 0 0
\(551\) −43.5206 −1.85404
\(552\) 0 0
\(553\) −0.669794 −0.0284826
\(554\) 8.46715 0.359735
\(555\) 0 0
\(556\) −6.56743 −0.278521
\(557\) −0.733579 −0.0310827 −0.0155414 0.999879i \(-0.504947\pi\)
−0.0155414 + 0.999879i \(0.504947\pi\)
\(558\) 0 0
\(559\) 81.5688 3.44999
\(560\) 2.50053 0.105667
\(561\) 0 0
\(562\) −3.00890 −0.126923
\(563\) −20.9036 −0.880982 −0.440491 0.897757i \(-0.645196\pi\)
−0.440491 + 0.897757i \(0.645196\pi\)
\(564\) 0 0
\(565\) 37.9302 1.59574
\(566\) −4.36566 −0.183502
\(567\) 0 0
\(568\) −2.58706 −0.108551
\(569\) 28.2526 1.18441 0.592205 0.805787i \(-0.298258\pi\)
0.592205 + 0.805787i \(0.298258\pi\)
\(570\) 0 0
\(571\) 38.7512 1.62169 0.810843 0.585264i \(-0.199009\pi\)
0.810843 + 0.585264i \(0.199009\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −0.851138 −0.0355258
\(575\) −2.32566 −0.0969866
\(576\) 0 0
\(577\) 12.7178 0.529450 0.264725 0.964324i \(-0.414719\pi\)
0.264725 + 0.964324i \(0.414719\pi\)
\(578\) −20.7599 −0.863499
\(579\) 0 0
\(580\) −30.8622 −1.28148
\(581\) 13.4240 0.556922
\(582\) 0 0
\(583\) 0 0
\(584\) 9.19430 0.380463
\(585\) 0 0
\(586\) −17.6505 −0.729134
\(587\) 7.77003 0.320704 0.160352 0.987060i \(-0.448737\pi\)
0.160352 + 0.987060i \(0.448737\pi\)
\(588\) 0 0
\(589\) −16.2162 −0.668175
\(590\) −8.66450 −0.356712
\(591\) 0 0
\(592\) −1.50290 −0.0617688
\(593\) −25.2949 −1.03874 −0.519368 0.854551i \(-0.673832\pi\)
−0.519368 + 0.854551i \(0.673832\pi\)
\(594\) 0 0
\(595\) 19.0789 0.782159
\(596\) 10.2744 0.420858
\(597\) 0 0
\(598\) −3.83294 −0.156740
\(599\) −11.9194 −0.487014 −0.243507 0.969899i \(-0.578298\pi\)
−0.243507 + 0.969899i \(0.578298\pi\)
\(600\) 0 0
\(601\) 21.0106 0.857041 0.428521 0.903532i \(-0.359035\pi\)
0.428521 + 0.903532i \(0.359035\pi\)
\(602\) 8.89157 0.362393
\(603\) 0 0
\(604\) −0.891525 −0.0362756
\(605\) 0 0
\(606\) 0 0
\(607\) 25.7817 1.04645 0.523223 0.852196i \(-0.324729\pi\)
0.523223 + 0.852196i \(0.324729\pi\)
\(608\) 33.7119 1.36720
\(609\) 0 0
\(610\) 12.9909 0.525987
\(611\) −36.1514 −1.46253
\(612\) 0 0
\(613\) −31.9173 −1.28913 −0.644563 0.764551i \(-0.722961\pi\)
−0.644563 + 0.764551i \(0.722961\pi\)
\(614\) −2.94588 −0.118886
\(615\) 0 0
\(616\) 0 0
\(617\) 13.9772 0.562701 0.281350 0.959605i \(-0.409218\pi\)
0.281350 + 0.959605i \(0.409218\pi\)
\(618\) 0 0
\(619\) 22.4745 0.903326 0.451663 0.892189i \(-0.350831\pi\)
0.451663 + 0.892189i \(0.350831\pi\)
\(620\) −11.4995 −0.461832
\(621\) 0 0
\(622\) −2.84502 −0.114075
\(623\) 16.6006 0.665088
\(624\) 0 0
\(625\) −30.7464 −1.22986
\(626\) −13.7173 −0.548252
\(627\) 0 0
\(628\) 8.75018 0.349170
\(629\) −11.4671 −0.457221
\(630\) 0 0
\(631\) −41.0383 −1.63371 −0.816854 0.576845i \(-0.804284\pi\)
−0.816854 + 0.576845i \(0.804284\pi\)
\(632\) −1.74118 −0.0692606
\(633\) 0 0
\(634\) −5.63025 −0.223606
\(635\) 46.6847 1.85262
\(636\) 0 0
\(637\) −6.96619 −0.276010
\(638\) 0 0
\(639\) 0 0
\(640\) 27.7041 1.09510
\(641\) 13.8318 0.546323 0.273162 0.961968i \(-0.411931\pi\)
0.273162 + 0.961968i \(0.411931\pi\)
\(642\) 0 0
\(643\) −33.5125 −1.32161 −0.660803 0.750560i \(-0.729784\pi\)
−0.660803 + 0.750560i \(0.729784\pi\)
\(644\) 1.03135 0.0406407
\(645\) 0 0
\(646\) 29.0796 1.14412
\(647\) −8.58616 −0.337557 −0.168778 0.985654i \(-0.553982\pi\)
−0.168778 + 0.985654i \(0.553982\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 16.9787 0.665957
\(651\) 0 0
\(652\) −30.8039 −1.20637
\(653\) 47.4478 1.85678 0.928388 0.371613i \(-0.121195\pi\)
0.928388 + 0.371613i \(0.121195\pi\)
\(654\) 0 0
\(655\) 17.7597 0.693929
\(656\) 0.978182 0.0381916
\(657\) 0 0
\(658\) −3.94076 −0.153627
\(659\) 47.4501 1.84839 0.924197 0.381916i \(-0.124736\pi\)
0.924197 + 0.381916i \(0.124736\pi\)
\(660\) 0 0
\(661\) −24.3627 −0.947599 −0.473800 0.880633i \(-0.657118\pi\)
−0.473800 + 0.880633i \(0.657118\pi\)
\(662\) −11.0238 −0.428453
\(663\) 0 0
\(664\) 34.8968 1.35426
\(665\) −16.4782 −0.638998
\(666\) 0 0
\(667\) −5.48320 −0.212310
\(668\) 34.1533 1.32143
\(669\) 0 0
\(670\) −22.5433 −0.870926
\(671\) 0 0
\(672\) 0 0
\(673\) 31.9467 1.23146 0.615728 0.787959i \(-0.288862\pi\)
0.615728 + 0.787959i \(0.288862\pi\)
\(674\) −8.01456 −0.308709
\(675\) 0 0
\(676\) −50.5690 −1.94496
\(677\) 1.01464 0.0389957 0.0194979 0.999810i \(-0.493793\pi\)
0.0194979 + 0.999810i \(0.493793\pi\)
\(678\) 0 0
\(679\) 10.1121 0.388065
\(680\) 49.5971 1.90196
\(681\) 0 0
\(682\) 0 0
\(683\) −0.904878 −0.0346242 −0.0173121 0.999850i \(-0.505511\pi\)
−0.0173121 + 0.999850i \(0.505511\pi\)
\(684\) 0 0
\(685\) 45.8713 1.75265
\(686\) −0.759363 −0.0289926
\(687\) 0 0
\(688\) −10.2188 −0.389587
\(689\) 43.5924 1.66074
\(690\) 0 0
\(691\) −21.4535 −0.816130 −0.408065 0.912953i \(-0.633796\pi\)
−0.408065 + 0.912953i \(0.633796\pi\)
\(692\) 15.3878 0.584958
\(693\) 0 0
\(694\) 17.0932 0.648851
\(695\) 13.2203 0.501474
\(696\) 0 0
\(697\) 7.46347 0.282699
\(698\) 15.7345 0.595558
\(699\) 0 0
\(700\) −4.56852 −0.172674
\(701\) 19.0750 0.720453 0.360227 0.932865i \(-0.382699\pi\)
0.360227 + 0.932865i \(0.382699\pi\)
\(702\) 0 0
\(703\) 9.90396 0.373535
\(704\) 0 0
\(705\) 0 0
\(706\) −4.56727 −0.171891
\(707\) 6.42353 0.241582
\(708\) 0 0
\(709\) −9.71578 −0.364884 −0.182442 0.983217i \(-0.558400\pi\)
−0.182442 + 0.983217i \(0.558400\pi\)
\(710\) 2.16529 0.0812617
\(711\) 0 0
\(712\) 43.1545 1.61728
\(713\) −2.04309 −0.0765142
\(714\) 0 0
\(715\) 0 0
\(716\) −8.43828 −0.315353
\(717\) 0 0
\(718\) 14.0255 0.523425
\(719\) 12.2204 0.455743 0.227871 0.973691i \(-0.426823\pi\)
0.227871 + 0.973691i \(0.426823\pi\)
\(720\) 0 0
\(721\) −8.95787 −0.333609
\(722\) −10.6878 −0.397760
\(723\) 0 0
\(724\) 6.89788 0.256358
\(725\) 24.2888 0.902062
\(726\) 0 0
\(727\) 0.0225849 0.000837628 0 0.000418814 1.00000i \(-0.499867\pi\)
0.000418814 1.00000i \(0.499867\pi\)
\(728\) −18.1092 −0.671170
\(729\) 0 0
\(730\) −7.69534 −0.284817
\(731\) −77.9686 −2.88377
\(732\) 0 0
\(733\) 5.28222 0.195103 0.0975516 0.995230i \(-0.468899\pi\)
0.0975516 + 0.995230i \(0.468899\pi\)
\(734\) 3.06016 0.112953
\(735\) 0 0
\(736\) 4.24740 0.156561
\(737\) 0 0
\(738\) 0 0
\(739\) −36.0831 −1.32734 −0.663670 0.748025i \(-0.731002\pi\)
−0.663670 + 0.748025i \(0.731002\pi\)
\(740\) 7.02329 0.258181
\(741\) 0 0
\(742\) 4.75188 0.174447
\(743\) 34.2049 1.25486 0.627429 0.778674i \(-0.284107\pi\)
0.627429 + 0.778674i \(0.284107\pi\)
\(744\) 0 0
\(745\) −20.6825 −0.757750
\(746\) −3.84664 −0.140835
\(747\) 0 0
\(748\) 0 0
\(749\) 4.52891 0.165483
\(750\) 0 0
\(751\) 7.33500 0.267658 0.133829 0.991004i \(-0.457273\pi\)
0.133829 + 0.991004i \(0.457273\pi\)
\(752\) 4.52898 0.165155
\(753\) 0 0
\(754\) 40.0306 1.45783
\(755\) 1.79465 0.0653139
\(756\) 0 0
\(757\) −34.8941 −1.26825 −0.634124 0.773231i \(-0.718639\pi\)
−0.634124 + 0.773231i \(0.718639\pi\)
\(758\) −1.66800 −0.0605844
\(759\) 0 0
\(760\) −42.8365 −1.55384
\(761\) −2.28870 −0.0829655 −0.0414827 0.999139i \(-0.513208\pi\)
−0.0414827 + 0.999139i \(0.513208\pi\)
\(762\) 0 0
\(763\) −1.02744 −0.0371957
\(764\) 22.3360 0.808087
\(765\) 0 0
\(766\) 18.5935 0.671811
\(767\) −27.7413 −1.00168
\(768\) 0 0
\(769\) −9.59977 −0.346176 −0.173088 0.984906i \(-0.555375\pi\)
−0.173088 + 0.984906i \(0.555375\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 32.3885 1.16569
\(773\) −13.7761 −0.495492 −0.247746 0.968825i \(-0.579690\pi\)
−0.247746 + 0.968825i \(0.579690\pi\)
\(774\) 0 0
\(775\) 9.05021 0.325093
\(776\) 26.2871 0.943652
\(777\) 0 0
\(778\) −26.0619 −0.934365
\(779\) −6.44612 −0.230956
\(780\) 0 0
\(781\) 0 0
\(782\) 3.66377 0.131016
\(783\) 0 0
\(784\) 0.872709 0.0311682
\(785\) −17.6142 −0.628677
\(786\) 0 0
\(787\) −2.24272 −0.0799443 −0.0399721 0.999201i \(-0.512727\pi\)
−0.0399721 + 0.999201i \(0.512727\pi\)
\(788\) −21.7811 −0.775919
\(789\) 0 0
\(790\) 1.45732 0.0518490
\(791\) 13.2380 0.470689
\(792\) 0 0
\(793\) 41.5933 1.47702
\(794\) −6.42985 −0.228187
\(795\) 0 0
\(796\) 0.779741 0.0276372
\(797\) 18.1909 0.644353 0.322177 0.946680i \(-0.395586\pi\)
0.322177 + 0.946680i \(0.395586\pi\)
\(798\) 0 0
\(799\) 34.5558 1.22250
\(800\) −18.8146 −0.665195
\(801\) 0 0
\(802\) −21.9516 −0.775139
\(803\) 0 0
\(804\) 0 0
\(805\) −2.07611 −0.0731731
\(806\) 14.9157 0.525384
\(807\) 0 0
\(808\) 16.6985 0.587450
\(809\) −31.2009 −1.09696 −0.548482 0.836162i \(-0.684794\pi\)
−0.548482 + 0.836162i \(0.684794\pi\)
\(810\) 0 0
\(811\) −25.2002 −0.884899 −0.442449 0.896793i \(-0.645890\pi\)
−0.442449 + 0.896793i \(0.645890\pi\)
\(812\) −10.7712 −0.377995
\(813\) 0 0
\(814\) 0 0
\(815\) 62.0084 2.17206
\(816\) 0 0
\(817\) 67.3406 2.35595
\(818\) 1.21161 0.0423630
\(819\) 0 0
\(820\) −4.57119 −0.159633
\(821\) −11.8863 −0.414836 −0.207418 0.978252i \(-0.566506\pi\)
−0.207418 + 0.978252i \(0.566506\pi\)
\(822\) 0 0
\(823\) 7.62865 0.265918 0.132959 0.991122i \(-0.457552\pi\)
0.132959 + 0.991122i \(0.457552\pi\)
\(824\) −23.2867 −0.811231
\(825\) 0 0
\(826\) −3.02399 −0.105218
\(827\) −11.4070 −0.396661 −0.198330 0.980135i \(-0.563552\pi\)
−0.198330 + 0.980135i \(0.563552\pi\)
\(828\) 0 0
\(829\) −44.6474 −1.55067 −0.775334 0.631552i \(-0.782418\pi\)
−0.775334 + 0.631552i \(0.782418\pi\)
\(830\) −29.2075 −1.01381
\(831\) 0 0
\(832\) −18.8496 −0.653491
\(833\) 6.65872 0.230711
\(834\) 0 0
\(835\) −68.7509 −2.37922
\(836\) 0 0
\(837\) 0 0
\(838\) 0.170583 0.00589270
\(839\) 10.0848 0.348166 0.174083 0.984731i \(-0.444304\pi\)
0.174083 + 0.984731i \(0.444304\pi\)
\(840\) 0 0
\(841\) 28.2656 0.974675
\(842\) −8.75525 −0.301726
\(843\) 0 0
\(844\) 19.7458 0.679679
\(845\) 101.796 3.50188
\(846\) 0 0
\(847\) 0 0
\(848\) −5.46116 −0.187537
\(849\) 0 0
\(850\) −16.2293 −0.556659
\(851\) 1.24781 0.0427743
\(852\) 0 0
\(853\) −19.0126 −0.650981 −0.325490 0.945545i \(-0.605529\pi\)
−0.325490 + 0.945545i \(0.605529\pi\)
\(854\) 4.53396 0.155149
\(855\) 0 0
\(856\) 11.7733 0.402401
\(857\) −29.8781 −1.02062 −0.510308 0.859992i \(-0.670469\pi\)
−0.510308 + 0.859992i \(0.670469\pi\)
\(858\) 0 0
\(859\) −16.6976 −0.569714 −0.284857 0.958570i \(-0.591946\pi\)
−0.284857 + 0.958570i \(0.591946\pi\)
\(860\) 47.7538 1.62839
\(861\) 0 0
\(862\) −21.1795 −0.721375
\(863\) 16.7819 0.571261 0.285631 0.958340i \(-0.407797\pi\)
0.285631 + 0.958340i \(0.407797\pi\)
\(864\) 0 0
\(865\) −30.9758 −1.05321
\(866\) −19.7183 −0.670055
\(867\) 0 0
\(868\) −4.01344 −0.136225
\(869\) 0 0
\(870\) 0 0
\(871\) −72.1774 −2.44564
\(872\) −2.67090 −0.0904482
\(873\) 0 0
\(874\) −3.16435 −0.107036
\(875\) −5.12978 −0.173418
\(876\) 0 0
\(877\) 5.53815 0.187010 0.0935050 0.995619i \(-0.470193\pi\)
0.0935050 + 0.995619i \(0.470193\pi\)
\(878\) −17.8869 −0.603653
\(879\) 0 0
\(880\) 0 0
\(881\) −6.77615 −0.228294 −0.114147 0.993464i \(-0.536414\pi\)
−0.114147 + 0.993464i \(0.536414\pi\)
\(882\) 0 0
\(883\) −23.9604 −0.806331 −0.403165 0.915127i \(-0.632090\pi\)
−0.403165 + 0.915127i \(0.632090\pi\)
\(884\) 66.0242 2.22063
\(885\) 0 0
\(886\) −18.8635 −0.633731
\(887\) −27.6881 −0.929676 −0.464838 0.885396i \(-0.653887\pi\)
−0.464838 + 0.885396i \(0.653887\pi\)
\(888\) 0 0
\(889\) 16.2934 0.546463
\(890\) −36.1189 −1.21071
\(891\) 0 0
\(892\) 10.8201 0.362285
\(893\) −29.8455 −0.998741
\(894\) 0 0
\(895\) 16.9863 0.567790
\(896\) 9.66899 0.323018
\(897\) 0 0
\(898\) 13.9819 0.466582
\(899\) 21.3377 0.711651
\(900\) 0 0
\(901\) −41.6684 −1.38817
\(902\) 0 0
\(903\) 0 0
\(904\) 34.4133 1.14457
\(905\) −13.8855 −0.461569
\(906\) 0 0
\(907\) 14.9565 0.496623 0.248311 0.968680i \(-0.420124\pi\)
0.248311 + 0.968680i \(0.420124\pi\)
\(908\) 0.420116 0.0139420
\(909\) 0 0
\(910\) 15.1568 0.502443
\(911\) 25.1916 0.834635 0.417318 0.908761i \(-0.362970\pi\)
0.417318 + 0.908761i \(0.362970\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −3.50347 −0.115884
\(915\) 0 0
\(916\) −10.1212 −0.334413
\(917\) 6.19831 0.204686
\(918\) 0 0
\(919\) 26.7796 0.883378 0.441689 0.897168i \(-0.354379\pi\)
0.441689 + 0.897168i \(0.354379\pi\)
\(920\) −5.39700 −0.177934
\(921\) 0 0
\(922\) −25.4499 −0.838148
\(923\) 6.93263 0.228190
\(924\) 0 0
\(925\) −5.52738 −0.181739
\(926\) 12.5552 0.412589
\(927\) 0 0
\(928\) −44.3591 −1.45616
\(929\) −5.65027 −0.185379 −0.0926896 0.995695i \(-0.529546\pi\)
−0.0926896 + 0.995695i \(0.529546\pi\)
\(930\) 0 0
\(931\) −5.75106 −0.188483
\(932\) −14.5948 −0.478069
\(933\) 0 0
\(934\) 14.7655 0.483142
\(935\) 0 0
\(936\) 0 0
\(937\) −45.2807 −1.47925 −0.739627 0.673017i \(-0.764998\pi\)
−0.739627 + 0.673017i \(0.764998\pi\)
\(938\) −7.86785 −0.256894
\(939\) 0 0
\(940\) −21.1646 −0.690313
\(941\) 32.6876 1.06558 0.532792 0.846246i \(-0.321143\pi\)
0.532792 + 0.846246i \(0.321143\pi\)
\(942\) 0 0
\(943\) −0.812152 −0.0264473
\(944\) 3.47537 0.113114
\(945\) 0 0
\(946\) 0 0
\(947\) 39.8004 1.29334 0.646669 0.762771i \(-0.276162\pi\)
0.646669 + 0.762771i \(0.276162\pi\)
\(948\) 0 0
\(949\) −24.6383 −0.799793
\(950\) 14.0170 0.454773
\(951\) 0 0
\(952\) 17.3099 0.561016
\(953\) 12.5174 0.405479 0.202739 0.979233i \(-0.435016\pi\)
0.202739 + 0.979233i \(0.435016\pi\)
\(954\) 0 0
\(955\) −44.9625 −1.45495
\(956\) 28.1886 0.911686
\(957\) 0 0
\(958\) −29.8245 −0.963587
\(959\) 16.0095 0.516974
\(960\) 0 0
\(961\) −23.0494 −0.743529
\(962\) −9.10973 −0.293710
\(963\) 0 0
\(964\) 28.5107 0.918269
\(965\) −65.1982 −2.09880
\(966\) 0 0
\(967\) 12.7139 0.408850 0.204425 0.978882i \(-0.434468\pi\)
0.204425 + 0.978882i \(0.434468\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −22.0015 −0.706425
\(971\) −22.6126 −0.725673 −0.362837 0.931853i \(-0.618192\pi\)
−0.362837 + 0.931853i \(0.618192\pi\)
\(972\) 0 0
\(973\) 4.61401 0.147918
\(974\) 12.9857 0.416089
\(975\) 0 0
\(976\) −5.21072 −0.166791
\(977\) −17.9694 −0.574893 −0.287446 0.957797i \(-0.592806\pi\)
−0.287446 + 0.957797i \(0.592806\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −4.07830 −0.130277
\(981\) 0 0
\(982\) 26.4151 0.842940
\(983\) 48.2065 1.53755 0.768775 0.639520i \(-0.220867\pi\)
0.768775 + 0.639520i \(0.220867\pi\)
\(984\) 0 0
\(985\) 43.8455 1.39703
\(986\) −38.2637 −1.21857
\(987\) 0 0
\(988\) −57.0243 −1.81419
\(989\) 8.48430 0.269785
\(990\) 0 0
\(991\) −6.62493 −0.210448 −0.105224 0.994449i \(-0.533556\pi\)
−0.105224 + 0.994449i \(0.533556\pi\)
\(992\) −16.5286 −0.524783
\(993\) 0 0
\(994\) 0.755706 0.0239695
\(995\) −1.56962 −0.0497605
\(996\) 0 0
\(997\) −28.1261 −0.890762 −0.445381 0.895341i \(-0.646932\pi\)
−0.445381 + 0.895341i \(0.646932\pi\)
\(998\) −11.8899 −0.376368
\(999\) 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.dc.1.7 16
3.2 odd 2 inner 7623.2.a.dc.1.10 16
11.5 even 5 693.2.m.k.190.5 yes 32
11.9 even 5 693.2.m.k.631.5 yes 32
11.10 odd 2 7623.2.a.db.1.10 16
33.5 odd 10 693.2.m.k.190.4 32
33.20 odd 10 693.2.m.k.631.4 yes 32
33.32 even 2 7623.2.a.db.1.7 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
693.2.m.k.190.4 32 33.5 odd 10
693.2.m.k.190.5 yes 32 11.5 even 5
693.2.m.k.631.4 yes 32 33.20 odd 10
693.2.m.k.631.5 yes 32 11.9 even 5
7623.2.a.db.1.7 16 33.32 even 2
7623.2.a.db.1.10 16 11.10 odd 2
7623.2.a.dc.1.7 16 1.1 even 1 trivial
7623.2.a.dc.1.10 16 3.2 odd 2 inner