Properties

Label 7623.2.a.cy.1.2
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 1.2
Root \(-2.39396\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.39396 q^{2} +3.73106 q^{4} +3.93829 q^{5} +1.00000 q^{7} -4.14409 q^{8} +O(q^{10})\) \(q-2.39396 q^{2} +3.73106 q^{4} +3.93829 q^{5} +1.00000 q^{7} -4.14409 q^{8} -9.42812 q^{10} +2.99890 q^{13} -2.39396 q^{14} +2.45868 q^{16} -6.60457 q^{17} +5.90310 q^{19} +14.6940 q^{20} +6.02551 q^{23} +10.5101 q^{25} -7.17925 q^{26} +3.73106 q^{28} -1.52075 q^{29} +8.46902 q^{31} +2.40219 q^{32} +15.8111 q^{34} +3.93829 q^{35} -0.607840 q^{37} -14.1318 q^{38} -16.3206 q^{40} -1.70333 q^{41} +3.23364 q^{43} -14.4249 q^{46} +4.80237 q^{47} +1.00000 q^{49} -25.1609 q^{50} +11.1891 q^{52} +6.12834 q^{53} -4.14409 q^{56} +3.64062 q^{58} -6.23883 q^{59} +2.08899 q^{61} -20.2745 q^{62} -10.6681 q^{64} +11.8105 q^{65} -0.599719 q^{67} -24.6420 q^{68} -9.42812 q^{70} -1.40968 q^{71} -7.08851 q^{73} +1.45515 q^{74} +22.0248 q^{76} +1.02327 q^{79} +9.68299 q^{80} +4.07770 q^{82} +3.08729 q^{83} -26.0107 q^{85} -7.74120 q^{86} +2.48531 q^{89} +2.99890 q^{91} +22.4815 q^{92} -11.4967 q^{94} +23.2481 q^{95} +2.55296 q^{97} -2.39396 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 18q^{4} - 5q^{5} + 10q^{7} + 3q^{8} + O(q^{10}) \) \( 10q + 18q^{4} - 5q^{5} + 10q^{7} + 3q^{8} - 6q^{10} + 6q^{13} + 38q^{16} - 8q^{17} - 7q^{20} + 31q^{25} - q^{26} + 18q^{28} + 14q^{29} + 26q^{31} + 41q^{32} + 21q^{34} - 5q^{35} + 24q^{37} - 8q^{38} - 5q^{40} - 19q^{41} - 6q^{43} - q^{46} - 15q^{47} + 10q^{49} + q^{50} - 25q^{52} + q^{53} + 3q^{56} + 11q^{58} - 23q^{59} - 11q^{62} + 53q^{64} + 29q^{65} + 38q^{67} - 87q^{68} - 6q^{70} - 26q^{71} - q^{73} + 39q^{74} - 2q^{76} + 5q^{79} - 6q^{80} + 5q^{82} - 6q^{83} - q^{85} + 41q^{86} + 9q^{89} + 6q^{91} + 48q^{92} + 42q^{95} + 24q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.39396 −1.69279 −0.846394 0.532558i \(-0.821231\pi\)
−0.846394 + 0.532558i \(0.821231\pi\)
\(3\) 0 0
\(4\) 3.73106 1.86553
\(5\) 3.93829 1.76126 0.880629 0.473807i \(-0.157121\pi\)
0.880629 + 0.473807i \(0.157121\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −4.14409 −1.46516
\(9\) 0 0
\(10\) −9.42812 −2.98143
\(11\) 0 0
\(12\) 0 0
\(13\) 2.99890 0.831744 0.415872 0.909423i \(-0.363476\pi\)
0.415872 + 0.909423i \(0.363476\pi\)
\(14\) −2.39396 −0.639813
\(15\) 0 0
\(16\) 2.45868 0.614669
\(17\) −6.60457 −1.60184 −0.800922 0.598769i \(-0.795657\pi\)
−0.800922 + 0.598769i \(0.795657\pi\)
\(18\) 0 0
\(19\) 5.90310 1.35426 0.677132 0.735861i \(-0.263222\pi\)
0.677132 + 0.735861i \(0.263222\pi\)
\(20\) 14.6940 3.28568
\(21\) 0 0
\(22\) 0 0
\(23\) 6.02551 1.25641 0.628203 0.778049i \(-0.283791\pi\)
0.628203 + 0.778049i \(0.283791\pi\)
\(24\) 0 0
\(25\) 10.5101 2.10203
\(26\) −7.17925 −1.40797
\(27\) 0 0
\(28\) 3.73106 0.705104
\(29\) −1.52075 −0.282397 −0.141198 0.989981i \(-0.545096\pi\)
−0.141198 + 0.989981i \(0.545096\pi\)
\(30\) 0 0
\(31\) 8.46902 1.52108 0.760541 0.649290i \(-0.224934\pi\)
0.760541 + 0.649290i \(0.224934\pi\)
\(32\) 2.40219 0.424652
\(33\) 0 0
\(34\) 15.8111 2.71158
\(35\) 3.93829 0.665693
\(36\) 0 0
\(37\) −0.607840 −0.0999283 −0.0499641 0.998751i \(-0.515911\pi\)
−0.0499641 + 0.998751i \(0.515911\pi\)
\(38\) −14.1318 −2.29248
\(39\) 0 0
\(40\) −16.3206 −2.58052
\(41\) −1.70333 −0.266015 −0.133007 0.991115i \(-0.542463\pi\)
−0.133007 + 0.991115i \(0.542463\pi\)
\(42\) 0 0
\(43\) 3.23364 0.493125 0.246562 0.969127i \(-0.420699\pi\)
0.246562 + 0.969127i \(0.420699\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −14.4249 −2.12683
\(47\) 4.80237 0.700498 0.350249 0.936657i \(-0.386097\pi\)
0.350249 + 0.936657i \(0.386097\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −25.1609 −3.55829
\(51\) 0 0
\(52\) 11.1891 1.55164
\(53\) 6.12834 0.841793 0.420896 0.907109i \(-0.361716\pi\)
0.420896 + 0.907109i \(0.361716\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −4.14409 −0.553777
\(57\) 0 0
\(58\) 3.64062 0.478037
\(59\) −6.23883 −0.812227 −0.406113 0.913823i \(-0.633116\pi\)
−0.406113 + 0.913823i \(0.633116\pi\)
\(60\) 0 0
\(61\) 2.08899 0.267467 0.133734 0.991017i \(-0.457303\pi\)
0.133734 + 0.991017i \(0.457303\pi\)
\(62\) −20.2745 −2.57487
\(63\) 0 0
\(64\) −10.6681 −1.33351
\(65\) 11.8105 1.46492
\(66\) 0 0
\(67\) −0.599719 −0.0732673 −0.0366337 0.999329i \(-0.511663\pi\)
−0.0366337 + 0.999329i \(0.511663\pi\)
\(68\) −24.6420 −2.98829
\(69\) 0 0
\(70\) −9.42812 −1.12688
\(71\) −1.40968 −0.167298 −0.0836489 0.996495i \(-0.526657\pi\)
−0.0836489 + 0.996495i \(0.526657\pi\)
\(72\) 0 0
\(73\) −7.08851 −0.829648 −0.414824 0.909902i \(-0.636157\pi\)
−0.414824 + 0.909902i \(0.636157\pi\)
\(74\) 1.45515 0.169157
\(75\) 0 0
\(76\) 22.0248 2.52642
\(77\) 0 0
\(78\) 0 0
\(79\) 1.02327 0.115126 0.0575632 0.998342i \(-0.481667\pi\)
0.0575632 + 0.998342i \(0.481667\pi\)
\(80\) 9.68299 1.08259
\(81\) 0 0
\(82\) 4.07770 0.450307
\(83\) 3.08729 0.338874 0.169437 0.985541i \(-0.445805\pi\)
0.169437 + 0.985541i \(0.445805\pi\)
\(84\) 0 0
\(85\) −26.0107 −2.82126
\(86\) −7.74120 −0.834755
\(87\) 0 0
\(88\) 0 0
\(89\) 2.48531 0.263442 0.131721 0.991287i \(-0.457950\pi\)
0.131721 + 0.991287i \(0.457950\pi\)
\(90\) 0 0
\(91\) 2.99890 0.314370
\(92\) 22.4815 2.34386
\(93\) 0 0
\(94\) −11.4967 −1.18579
\(95\) 23.2481 2.38521
\(96\) 0 0
\(97\) 2.55296 0.259214 0.129607 0.991565i \(-0.458628\pi\)
0.129607 + 0.991565i \(0.458628\pi\)
\(98\) −2.39396 −0.241827
\(99\) 0 0
\(100\) 39.2139 3.92139
\(101\) −15.1573 −1.50821 −0.754106 0.656753i \(-0.771929\pi\)
−0.754106 + 0.656753i \(0.771929\pi\)
\(102\) 0 0
\(103\) −13.8364 −1.36334 −0.681670 0.731660i \(-0.738746\pi\)
−0.681670 + 0.731660i \(0.738746\pi\)
\(104\) −12.4277 −1.21864
\(105\) 0 0
\(106\) −14.6710 −1.42498
\(107\) 0.445651 0.0430827 0.0215413 0.999768i \(-0.493143\pi\)
0.0215413 + 0.999768i \(0.493143\pi\)
\(108\) 0 0
\(109\) −11.0988 −1.06307 −0.531536 0.847036i \(-0.678385\pi\)
−0.531536 + 0.847036i \(0.678385\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 2.45868 0.232323
\(113\) 8.08938 0.760985 0.380492 0.924784i \(-0.375755\pi\)
0.380492 + 0.924784i \(0.375755\pi\)
\(114\) 0 0
\(115\) 23.7302 2.21285
\(116\) −5.67401 −0.526819
\(117\) 0 0
\(118\) 14.9355 1.37493
\(119\) −6.60457 −0.605440
\(120\) 0 0
\(121\) 0 0
\(122\) −5.00096 −0.452765
\(123\) 0 0
\(124\) 31.5984 2.83762
\(125\) 21.7005 1.94095
\(126\) 0 0
\(127\) −0.179659 −0.0159422 −0.00797109 0.999968i \(-0.502537\pi\)
−0.00797109 + 0.999968i \(0.502537\pi\)
\(128\) 20.7347 1.83271
\(129\) 0 0
\(130\) −28.2740 −2.47979
\(131\) 9.38232 0.819737 0.409868 0.912145i \(-0.365575\pi\)
0.409868 + 0.912145i \(0.365575\pi\)
\(132\) 0 0
\(133\) 5.90310 0.511864
\(134\) 1.43570 0.124026
\(135\) 0 0
\(136\) 27.3699 2.34695
\(137\) −0.893617 −0.0763468 −0.0381734 0.999271i \(-0.512154\pi\)
−0.0381734 + 0.999271i \(0.512154\pi\)
\(138\) 0 0
\(139\) −2.12281 −0.180054 −0.0900271 0.995939i \(-0.528695\pi\)
−0.0900271 + 0.995939i \(0.528695\pi\)
\(140\) 14.6940 1.24187
\(141\) 0 0
\(142\) 3.37471 0.283199
\(143\) 0 0
\(144\) 0 0
\(145\) −5.98916 −0.497373
\(146\) 16.9696 1.40442
\(147\) 0 0
\(148\) −2.26789 −0.186419
\(149\) 14.3220 1.17331 0.586653 0.809838i \(-0.300445\pi\)
0.586653 + 0.809838i \(0.300445\pi\)
\(150\) 0 0
\(151\) 13.7837 1.12170 0.560850 0.827918i \(-0.310475\pi\)
0.560850 + 0.827918i \(0.310475\pi\)
\(152\) −24.4630 −1.98421
\(153\) 0 0
\(154\) 0 0
\(155\) 33.3535 2.67902
\(156\) 0 0
\(157\) −11.4608 −0.914668 −0.457334 0.889295i \(-0.651196\pi\)
−0.457334 + 0.889295i \(0.651196\pi\)
\(158\) −2.44966 −0.194884
\(159\) 0 0
\(160\) 9.46054 0.747922
\(161\) 6.02551 0.474877
\(162\) 0 0
\(163\) 4.94262 0.387136 0.193568 0.981087i \(-0.437994\pi\)
0.193568 + 0.981087i \(0.437994\pi\)
\(164\) −6.35521 −0.496258
\(165\) 0 0
\(166\) −7.39086 −0.573642
\(167\) 8.53421 0.660397 0.330198 0.943912i \(-0.392884\pi\)
0.330198 + 0.943912i \(0.392884\pi\)
\(168\) 0 0
\(169\) −4.00662 −0.308201
\(170\) 62.2687 4.77579
\(171\) 0 0
\(172\) 12.0649 0.919939
\(173\) 13.0876 0.995031 0.497515 0.867455i \(-0.334246\pi\)
0.497515 + 0.867455i \(0.334246\pi\)
\(174\) 0 0
\(175\) 10.5101 0.794492
\(176\) 0 0
\(177\) 0 0
\(178\) −5.94973 −0.445951
\(179\) −25.8622 −1.93303 −0.966514 0.256612i \(-0.917394\pi\)
−0.966514 + 0.256612i \(0.917394\pi\)
\(180\) 0 0
\(181\) 21.4202 1.59215 0.796077 0.605195i \(-0.206905\pi\)
0.796077 + 0.605195i \(0.206905\pi\)
\(182\) −7.17925 −0.532161
\(183\) 0 0
\(184\) −24.9703 −1.84083
\(185\) −2.39385 −0.175999
\(186\) 0 0
\(187\) 0 0
\(188\) 17.9179 1.30680
\(189\) 0 0
\(190\) −55.6552 −4.03765
\(191\) 3.73988 0.270608 0.135304 0.990804i \(-0.456799\pi\)
0.135304 + 0.990804i \(0.456799\pi\)
\(192\) 0 0
\(193\) −6.07932 −0.437599 −0.218799 0.975770i \(-0.570214\pi\)
−0.218799 + 0.975770i \(0.570214\pi\)
\(194\) −6.11169 −0.438794
\(195\) 0 0
\(196\) 3.73106 0.266504
\(197\) 21.5458 1.53508 0.767538 0.641003i \(-0.221482\pi\)
0.767538 + 0.641003i \(0.221482\pi\)
\(198\) 0 0
\(199\) −14.8965 −1.05598 −0.527992 0.849249i \(-0.677055\pi\)
−0.527992 + 0.849249i \(0.677055\pi\)
\(200\) −43.5549 −3.07980
\(201\) 0 0
\(202\) 36.2861 2.55308
\(203\) −1.52075 −0.106736
\(204\) 0 0
\(205\) −6.70819 −0.468521
\(206\) 33.1238 2.30784
\(207\) 0 0
\(208\) 7.37332 0.511248
\(209\) 0 0
\(210\) 0 0
\(211\) −16.9594 −1.16753 −0.583766 0.811922i \(-0.698421\pi\)
−0.583766 + 0.811922i \(0.698421\pi\)
\(212\) 22.8652 1.57039
\(213\) 0 0
\(214\) −1.06687 −0.0729298
\(215\) 12.7350 0.868520
\(216\) 0 0
\(217\) 8.46902 0.574915
\(218\) 26.5701 1.79955
\(219\) 0 0
\(220\) 0 0
\(221\) −19.8064 −1.33232
\(222\) 0 0
\(223\) −13.8125 −0.924954 −0.462477 0.886631i \(-0.653039\pi\)
−0.462477 + 0.886631i \(0.653039\pi\)
\(224\) 2.40219 0.160503
\(225\) 0 0
\(226\) −19.3657 −1.28819
\(227\) −7.25345 −0.481428 −0.240714 0.970596i \(-0.577382\pi\)
−0.240714 + 0.970596i \(0.577382\pi\)
\(228\) 0 0
\(229\) −0.327136 −0.0216178 −0.0108089 0.999942i \(-0.503441\pi\)
−0.0108089 + 0.999942i \(0.503441\pi\)
\(230\) −56.8093 −3.74589
\(231\) 0 0
\(232\) 6.30213 0.413755
\(233\) 16.2452 1.06426 0.532128 0.846664i \(-0.321393\pi\)
0.532128 + 0.846664i \(0.321393\pi\)
\(234\) 0 0
\(235\) 18.9131 1.23376
\(236\) −23.2774 −1.51523
\(237\) 0 0
\(238\) 15.8111 1.02488
\(239\) −29.0009 −1.87591 −0.937955 0.346756i \(-0.887283\pi\)
−0.937955 + 0.346756i \(0.887283\pi\)
\(240\) 0 0
\(241\) 29.3358 1.88968 0.944841 0.327530i \(-0.106216\pi\)
0.944841 + 0.327530i \(0.106216\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 7.79413 0.498968
\(245\) 3.93829 0.251608
\(246\) 0 0
\(247\) 17.7028 1.12640
\(248\) −35.0964 −2.22862
\(249\) 0 0
\(250\) −51.9502 −3.28562
\(251\) −12.8316 −0.809925 −0.404963 0.914333i \(-0.632715\pi\)
−0.404963 + 0.914333i \(0.632715\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0.430097 0.0269867
\(255\) 0 0
\(256\) −28.3018 −1.76887
\(257\) −19.5210 −1.21769 −0.608844 0.793290i \(-0.708367\pi\)
−0.608844 + 0.793290i \(0.708367\pi\)
\(258\) 0 0
\(259\) −0.607840 −0.0377693
\(260\) 44.0658 2.73284
\(261\) 0 0
\(262\) −22.4609 −1.38764
\(263\) 23.9606 1.47747 0.738736 0.673995i \(-0.235423\pi\)
0.738736 + 0.673995i \(0.235423\pi\)
\(264\) 0 0
\(265\) 24.1352 1.48261
\(266\) −14.1318 −0.866477
\(267\) 0 0
\(268\) −2.23759 −0.136682
\(269\) −11.3260 −0.690558 −0.345279 0.938500i \(-0.612216\pi\)
−0.345279 + 0.938500i \(0.612216\pi\)
\(270\) 0 0
\(271\) −6.12278 −0.371932 −0.185966 0.982556i \(-0.559541\pi\)
−0.185966 + 0.982556i \(0.559541\pi\)
\(272\) −16.2385 −0.984604
\(273\) 0 0
\(274\) 2.13929 0.129239
\(275\) 0 0
\(276\) 0 0
\(277\) −0.600165 −0.0360604 −0.0180302 0.999837i \(-0.505740\pi\)
−0.0180302 + 0.999837i \(0.505740\pi\)
\(278\) 5.08192 0.304793
\(279\) 0 0
\(280\) −16.3206 −0.975344
\(281\) 23.8022 1.41992 0.709960 0.704242i \(-0.248713\pi\)
0.709960 + 0.704242i \(0.248713\pi\)
\(282\) 0 0
\(283\) 17.3400 1.03076 0.515379 0.856962i \(-0.327651\pi\)
0.515379 + 0.856962i \(0.327651\pi\)
\(284\) −5.25958 −0.312099
\(285\) 0 0
\(286\) 0 0
\(287\) −1.70333 −0.100544
\(288\) 0 0
\(289\) 26.6204 1.56590
\(290\) 14.3378 0.841947
\(291\) 0 0
\(292\) −26.4477 −1.54773
\(293\) −14.9122 −0.871179 −0.435589 0.900145i \(-0.643460\pi\)
−0.435589 + 0.900145i \(0.643460\pi\)
\(294\) 0 0
\(295\) −24.5703 −1.43054
\(296\) 2.51894 0.146411
\(297\) 0 0
\(298\) −34.2864 −1.98616
\(299\) 18.0699 1.04501
\(300\) 0 0
\(301\) 3.23364 0.186384
\(302\) −32.9976 −1.89880
\(303\) 0 0
\(304\) 14.5138 0.832425
\(305\) 8.22704 0.471079
\(306\) 0 0
\(307\) −29.0453 −1.65771 −0.828853 0.559467i \(-0.811006\pi\)
−0.828853 + 0.559467i \(0.811006\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −79.8470 −4.53500
\(311\) 20.3651 1.15480 0.577399 0.816462i \(-0.304068\pi\)
0.577399 + 0.816462i \(0.304068\pi\)
\(312\) 0 0
\(313\) −29.4807 −1.66635 −0.833174 0.553010i \(-0.813479\pi\)
−0.833174 + 0.553010i \(0.813479\pi\)
\(314\) 27.4366 1.54834
\(315\) 0 0
\(316\) 3.81786 0.214772
\(317\) −7.62530 −0.428280 −0.214140 0.976803i \(-0.568695\pi\)
−0.214140 + 0.976803i \(0.568695\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −42.0142 −2.34866
\(321\) 0 0
\(322\) −14.4249 −0.803865
\(323\) −38.9875 −2.16932
\(324\) 0 0
\(325\) 31.5188 1.74835
\(326\) −11.8324 −0.655338
\(327\) 0 0
\(328\) 7.05873 0.389753
\(329\) 4.80237 0.264763
\(330\) 0 0
\(331\) 20.6731 1.13630 0.568148 0.822926i \(-0.307660\pi\)
0.568148 + 0.822926i \(0.307660\pi\)
\(332\) 11.5189 0.632180
\(333\) 0 0
\(334\) −20.4306 −1.11791
\(335\) −2.36187 −0.129043
\(336\) 0 0
\(337\) 5.46255 0.297564 0.148782 0.988870i \(-0.452465\pi\)
0.148782 + 0.988870i \(0.452465\pi\)
\(338\) 9.59169 0.521719
\(339\) 0 0
\(340\) −97.0475 −5.26314
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −13.4005 −0.722505
\(345\) 0 0
\(346\) −31.3312 −1.68438
\(347\) 9.58110 0.514341 0.257170 0.966366i \(-0.417210\pi\)
0.257170 + 0.966366i \(0.417210\pi\)
\(348\) 0 0
\(349\) −9.36045 −0.501054 −0.250527 0.968110i \(-0.580604\pi\)
−0.250527 + 0.968110i \(0.580604\pi\)
\(350\) −25.1609 −1.34491
\(351\) 0 0
\(352\) 0 0
\(353\) 27.6629 1.47235 0.736173 0.676794i \(-0.236631\pi\)
0.736173 + 0.676794i \(0.236631\pi\)
\(354\) 0 0
\(355\) −5.55171 −0.294654
\(356\) 9.27282 0.491459
\(357\) 0 0
\(358\) 61.9131 3.27221
\(359\) −2.60258 −0.137359 −0.0686796 0.997639i \(-0.521879\pi\)
−0.0686796 + 0.997639i \(0.521879\pi\)
\(360\) 0 0
\(361\) 15.8466 0.834033
\(362\) −51.2793 −2.69518
\(363\) 0 0
\(364\) 11.1891 0.586466
\(365\) −27.9166 −1.46122
\(366\) 0 0
\(367\) −10.3394 −0.539710 −0.269855 0.962901i \(-0.586976\pi\)
−0.269855 + 0.962901i \(0.586976\pi\)
\(368\) 14.8148 0.772274
\(369\) 0 0
\(370\) 5.73079 0.297930
\(371\) 6.12834 0.318168
\(372\) 0 0
\(373\) 5.15587 0.266961 0.133480 0.991051i \(-0.457385\pi\)
0.133480 + 0.991051i \(0.457385\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −19.9015 −1.02634
\(377\) −4.56058 −0.234882
\(378\) 0 0
\(379\) −35.4223 −1.81952 −0.909761 0.415133i \(-0.863735\pi\)
−0.909761 + 0.415133i \(0.863735\pi\)
\(380\) 86.7401 4.44968
\(381\) 0 0
\(382\) −8.95313 −0.458082
\(383\) 31.4076 1.60486 0.802428 0.596750i \(-0.203541\pi\)
0.802428 + 0.596750i \(0.203541\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 14.5537 0.740762
\(387\) 0 0
\(388\) 9.52524 0.483571
\(389\) −20.0789 −1.01804 −0.509020 0.860755i \(-0.669992\pi\)
−0.509020 + 0.860755i \(0.669992\pi\)
\(390\) 0 0
\(391\) −39.7959 −2.01257
\(392\) −4.14409 −0.209308
\(393\) 0 0
\(394\) −51.5799 −2.59856
\(395\) 4.02992 0.202767
\(396\) 0 0
\(397\) 3.63351 0.182361 0.0911804 0.995834i \(-0.470936\pi\)
0.0911804 + 0.995834i \(0.470936\pi\)
\(398\) 35.6616 1.78756
\(399\) 0 0
\(400\) 25.8410 1.29205
\(401\) −37.6309 −1.87920 −0.939598 0.342279i \(-0.888801\pi\)
−0.939598 + 0.342279i \(0.888801\pi\)
\(402\) 0 0
\(403\) 25.3977 1.26515
\(404\) −56.5529 −2.81361
\(405\) 0 0
\(406\) 3.64062 0.180681
\(407\) 0 0
\(408\) 0 0
\(409\) −16.1059 −0.796386 −0.398193 0.917302i \(-0.630363\pi\)
−0.398193 + 0.917302i \(0.630363\pi\)
\(410\) 16.0592 0.793106
\(411\) 0 0
\(412\) −51.6243 −2.54335
\(413\) −6.23883 −0.306993
\(414\) 0 0
\(415\) 12.1587 0.596845
\(416\) 7.20393 0.353202
\(417\) 0 0
\(418\) 0 0
\(419\) 17.7128 0.865325 0.432662 0.901556i \(-0.357574\pi\)
0.432662 + 0.901556i \(0.357574\pi\)
\(420\) 0 0
\(421\) 30.2721 1.47537 0.737686 0.675144i \(-0.235919\pi\)
0.737686 + 0.675144i \(0.235919\pi\)
\(422\) 40.6001 1.97638
\(423\) 0 0
\(424\) −25.3964 −1.23336
\(425\) −69.4149 −3.36712
\(426\) 0 0
\(427\) 2.08899 0.101093
\(428\) 1.66275 0.0803720
\(429\) 0 0
\(430\) −30.4871 −1.47022
\(431\) 19.1918 0.924438 0.462219 0.886766i \(-0.347053\pi\)
0.462219 + 0.886766i \(0.347053\pi\)
\(432\) 0 0
\(433\) 21.7538 1.04542 0.522711 0.852510i \(-0.324921\pi\)
0.522711 + 0.852510i \(0.324921\pi\)
\(434\) −20.2745 −0.973208
\(435\) 0 0
\(436\) −41.4102 −1.98319
\(437\) 35.5692 1.70151
\(438\) 0 0
\(439\) −2.70519 −0.129112 −0.0645558 0.997914i \(-0.520563\pi\)
−0.0645558 + 0.997914i \(0.520563\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 47.4159 2.25534
\(443\) −38.2274 −1.81624 −0.908118 0.418713i \(-0.862481\pi\)
−0.908118 + 0.418713i \(0.862481\pi\)
\(444\) 0 0
\(445\) 9.78786 0.463989
\(446\) 33.0666 1.56575
\(447\) 0 0
\(448\) −10.6681 −0.504021
\(449\) −22.7112 −1.07181 −0.535904 0.844279i \(-0.680029\pi\)
−0.535904 + 0.844279i \(0.680029\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 30.1819 1.41964
\(453\) 0 0
\(454\) 17.3645 0.814956
\(455\) 11.8105 0.553686
\(456\) 0 0
\(457\) 1.57183 0.0735271 0.0367636 0.999324i \(-0.488295\pi\)
0.0367636 + 0.999324i \(0.488295\pi\)
\(458\) 0.783152 0.0365943
\(459\) 0 0
\(460\) 88.5388 4.12814
\(461\) −28.1276 −1.31003 −0.655017 0.755614i \(-0.727339\pi\)
−0.655017 + 0.755614i \(0.727339\pi\)
\(462\) 0 0
\(463\) −4.69202 −0.218057 −0.109028 0.994039i \(-0.534774\pi\)
−0.109028 + 0.994039i \(0.534774\pi\)
\(464\) −3.73904 −0.173580
\(465\) 0 0
\(466\) −38.8903 −1.80156
\(467\) 17.6135 0.815054 0.407527 0.913193i \(-0.366391\pi\)
0.407527 + 0.913193i \(0.366391\pi\)
\(468\) 0 0
\(469\) −0.599719 −0.0276924
\(470\) −45.2773 −2.08849
\(471\) 0 0
\(472\) 25.8543 1.19004
\(473\) 0 0
\(474\) 0 0
\(475\) 62.0424 2.84670
\(476\) −24.6420 −1.12947
\(477\) 0 0
\(478\) 69.4270 3.17552
\(479\) −10.7070 −0.489215 −0.244608 0.969622i \(-0.578659\pi\)
−0.244608 + 0.969622i \(0.578659\pi\)
\(480\) 0 0
\(481\) −1.82285 −0.0831148
\(482\) −70.2287 −3.19883
\(483\) 0 0
\(484\) 0 0
\(485\) 10.0543 0.456542
\(486\) 0 0
\(487\) 38.7602 1.75639 0.878197 0.478299i \(-0.158746\pi\)
0.878197 + 0.478299i \(0.158746\pi\)
\(488\) −8.65695 −0.391882
\(489\) 0 0
\(490\) −9.42812 −0.425919
\(491\) −6.00061 −0.270804 −0.135402 0.990791i \(-0.543233\pi\)
−0.135402 + 0.990791i \(0.543233\pi\)
\(492\) 0 0
\(493\) 10.0439 0.452355
\(494\) −42.3798 −1.90676
\(495\) 0 0
\(496\) 20.8226 0.934962
\(497\) −1.40968 −0.0632326
\(498\) 0 0
\(499\) −28.9776 −1.29722 −0.648608 0.761123i \(-0.724648\pi\)
−0.648608 + 0.761123i \(0.724648\pi\)
\(500\) 80.9659 3.62091
\(501\) 0 0
\(502\) 30.7184 1.37103
\(503\) −12.8657 −0.573653 −0.286827 0.957982i \(-0.592600\pi\)
−0.286827 + 0.957982i \(0.592600\pi\)
\(504\) 0 0
\(505\) −59.6940 −2.65635
\(506\) 0 0
\(507\) 0 0
\(508\) −0.670319 −0.0297406
\(509\) −19.7361 −0.874788 −0.437394 0.899270i \(-0.644098\pi\)
−0.437394 + 0.899270i \(0.644098\pi\)
\(510\) 0 0
\(511\) −7.08851 −0.313577
\(512\) 26.2842 1.16161
\(513\) 0 0
\(514\) 46.7326 2.06129
\(515\) −54.4917 −2.40119
\(516\) 0 0
\(517\) 0 0
\(518\) 1.45515 0.0639355
\(519\) 0 0
\(520\) −48.9439 −2.14633
\(521\) 23.3926 1.02485 0.512423 0.858733i \(-0.328748\pi\)
0.512423 + 0.858733i \(0.328748\pi\)
\(522\) 0 0
\(523\) 10.7796 0.471361 0.235680 0.971831i \(-0.424268\pi\)
0.235680 + 0.971831i \(0.424268\pi\)
\(524\) 35.0060 1.52924
\(525\) 0 0
\(526\) −57.3607 −2.50105
\(527\) −55.9343 −2.43653
\(528\) 0 0
\(529\) 13.3068 0.578556
\(530\) −57.7788 −2.50975
\(531\) 0 0
\(532\) 22.0248 0.954897
\(533\) −5.10810 −0.221256
\(534\) 0 0
\(535\) 1.75510 0.0758797
\(536\) 2.48529 0.107348
\(537\) 0 0
\(538\) 27.1140 1.16897
\(539\) 0 0
\(540\) 0 0
\(541\) 6.42155 0.276084 0.138042 0.990426i \(-0.455919\pi\)
0.138042 + 0.990426i \(0.455919\pi\)
\(542\) 14.6577 0.629602
\(543\) 0 0
\(544\) −15.8655 −0.680226
\(545\) −43.7103 −1.87234
\(546\) 0 0
\(547\) 28.0529 1.19946 0.599728 0.800204i \(-0.295276\pi\)
0.599728 + 0.800204i \(0.295276\pi\)
\(548\) −3.33414 −0.142427
\(549\) 0 0
\(550\) 0 0
\(551\) −8.97715 −0.382440
\(552\) 0 0
\(553\) 1.02327 0.0435137
\(554\) 1.43677 0.0610427
\(555\) 0 0
\(556\) −7.92031 −0.335896
\(557\) 23.1102 0.979211 0.489606 0.871944i \(-0.337141\pi\)
0.489606 + 0.871944i \(0.337141\pi\)
\(558\) 0 0
\(559\) 9.69734 0.410154
\(560\) 9.68299 0.409181
\(561\) 0 0
\(562\) −56.9815 −2.40362
\(563\) −28.8436 −1.21561 −0.607806 0.794086i \(-0.707950\pi\)
−0.607806 + 0.794086i \(0.707950\pi\)
\(564\) 0 0
\(565\) 31.8583 1.34029
\(566\) −41.5114 −1.74485
\(567\) 0 0
\(568\) 5.84182 0.245117
\(569\) −17.7760 −0.745208 −0.372604 0.927991i \(-0.621535\pi\)
−0.372604 + 0.927991i \(0.621535\pi\)
\(570\) 0 0
\(571\) −17.8981 −0.749012 −0.374506 0.927225i \(-0.622188\pi\)
−0.374506 + 0.927225i \(0.622188\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 4.07770 0.170200
\(575\) 63.3289 2.64100
\(576\) 0 0
\(577\) 21.0769 0.877441 0.438721 0.898624i \(-0.355432\pi\)
0.438721 + 0.898624i \(0.355432\pi\)
\(578\) −63.7281 −2.65074
\(579\) 0 0
\(580\) −22.3459 −0.927864
\(581\) 3.08729 0.128083
\(582\) 0 0
\(583\) 0 0
\(584\) 29.3754 1.21556
\(585\) 0 0
\(586\) 35.6992 1.47472
\(587\) −17.3136 −0.714609 −0.357305 0.933988i \(-0.616304\pi\)
−0.357305 + 0.933988i \(0.616304\pi\)
\(588\) 0 0
\(589\) 49.9935 2.05995
\(590\) 58.8205 2.42160
\(591\) 0 0
\(592\) −1.49448 −0.0614228
\(593\) −26.3881 −1.08363 −0.541814 0.840498i \(-0.682262\pi\)
−0.541814 + 0.840498i \(0.682262\pi\)
\(594\) 0 0
\(595\) −26.0107 −1.06634
\(596\) 53.4363 2.18884
\(597\) 0 0
\(598\) −43.2586 −1.76898
\(599\) 28.2593 1.15465 0.577323 0.816516i \(-0.304097\pi\)
0.577323 + 0.816516i \(0.304097\pi\)
\(600\) 0 0
\(601\) 0.440908 0.0179850 0.00899251 0.999960i \(-0.497138\pi\)
0.00899251 + 0.999960i \(0.497138\pi\)
\(602\) −7.74120 −0.315508
\(603\) 0 0
\(604\) 51.4277 2.09256
\(605\) 0 0
\(606\) 0 0
\(607\) −0.559085 −0.0226925 −0.0113463 0.999936i \(-0.503612\pi\)
−0.0113463 + 0.999936i \(0.503612\pi\)
\(608\) 14.1804 0.575091
\(609\) 0 0
\(610\) −19.6952 −0.797436
\(611\) 14.4018 0.582635
\(612\) 0 0
\(613\) −30.3820 −1.22712 −0.613559 0.789649i \(-0.710263\pi\)
−0.613559 + 0.789649i \(0.710263\pi\)
\(614\) 69.5335 2.80614
\(615\) 0 0
\(616\) 0 0
\(617\) −2.35080 −0.0946395 −0.0473198 0.998880i \(-0.515068\pi\)
−0.0473198 + 0.998880i \(0.515068\pi\)
\(618\) 0 0
\(619\) 6.11038 0.245597 0.122798 0.992432i \(-0.460813\pi\)
0.122798 + 0.992432i \(0.460813\pi\)
\(620\) 124.444 4.99778
\(621\) 0 0
\(622\) −48.7532 −1.95483
\(623\) 2.48531 0.0995717
\(624\) 0 0
\(625\) 32.9123 1.31649
\(626\) 70.5757 2.82077
\(627\) 0 0
\(628\) −42.7608 −1.70634
\(629\) 4.01452 0.160069
\(630\) 0 0
\(631\) −28.8822 −1.14978 −0.574891 0.818230i \(-0.694955\pi\)
−0.574891 + 0.818230i \(0.694955\pi\)
\(632\) −4.24050 −0.168678
\(633\) 0 0
\(634\) 18.2547 0.724987
\(635\) −0.707550 −0.0280783
\(636\) 0 0
\(637\) 2.99890 0.118821
\(638\) 0 0
\(639\) 0 0
\(640\) 81.6592 3.22787
\(641\) 39.8256 1.57302 0.786508 0.617579i \(-0.211887\pi\)
0.786508 + 0.617579i \(0.211887\pi\)
\(642\) 0 0
\(643\) 5.85723 0.230987 0.115493 0.993308i \(-0.463155\pi\)
0.115493 + 0.993308i \(0.463155\pi\)
\(644\) 22.4815 0.885896
\(645\) 0 0
\(646\) 93.3345 3.67220
\(647\) −22.5855 −0.887926 −0.443963 0.896045i \(-0.646428\pi\)
−0.443963 + 0.896045i \(0.646428\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −75.4549 −2.95958
\(651\) 0 0
\(652\) 18.4412 0.722213
\(653\) 40.8369 1.59807 0.799035 0.601284i \(-0.205344\pi\)
0.799035 + 0.601284i \(0.205344\pi\)
\(654\) 0 0
\(655\) 36.9503 1.44377
\(656\) −4.18793 −0.163511
\(657\) 0 0
\(658\) −11.4967 −0.448188
\(659\) 32.5869 1.26941 0.634703 0.772756i \(-0.281122\pi\)
0.634703 + 0.772756i \(0.281122\pi\)
\(660\) 0 0
\(661\) 43.6393 1.69737 0.848686 0.528898i \(-0.177395\pi\)
0.848686 + 0.528898i \(0.177395\pi\)
\(662\) −49.4906 −1.92351
\(663\) 0 0
\(664\) −12.7940 −0.496504
\(665\) 23.2481 0.901524
\(666\) 0 0
\(667\) −9.16331 −0.354805
\(668\) 31.8416 1.23199
\(669\) 0 0
\(670\) 5.65422 0.218442
\(671\) 0 0
\(672\) 0 0
\(673\) 23.8718 0.920190 0.460095 0.887870i \(-0.347815\pi\)
0.460095 + 0.887870i \(0.347815\pi\)
\(674\) −13.0771 −0.503712
\(675\) 0 0
\(676\) −14.9489 −0.574958
\(677\) −4.26651 −0.163975 −0.0819877 0.996633i \(-0.526127\pi\)
−0.0819877 + 0.996633i \(0.526127\pi\)
\(678\) 0 0
\(679\) 2.55296 0.0979736
\(680\) 107.791 4.13359
\(681\) 0 0
\(682\) 0 0
\(683\) −5.88217 −0.225075 −0.112537 0.993647i \(-0.535898\pi\)
−0.112537 + 0.993647i \(0.535898\pi\)
\(684\) 0 0
\(685\) −3.51932 −0.134466
\(686\) −2.39396 −0.0914019
\(687\) 0 0
\(688\) 7.95047 0.303109
\(689\) 18.3783 0.700156
\(690\) 0 0
\(691\) −0.237559 −0.00903716 −0.00451858 0.999990i \(-0.501438\pi\)
−0.00451858 + 0.999990i \(0.501438\pi\)
\(692\) 48.8305 1.85626
\(693\) 0 0
\(694\) −22.9368 −0.870669
\(695\) −8.36023 −0.317122
\(696\) 0 0
\(697\) 11.2497 0.426114
\(698\) 22.4086 0.848177
\(699\) 0 0
\(700\) 39.2139 1.48215
\(701\) −10.6036 −0.400494 −0.200247 0.979745i \(-0.564174\pi\)
−0.200247 + 0.979745i \(0.564174\pi\)
\(702\) 0 0
\(703\) −3.58814 −0.135329
\(704\) 0 0
\(705\) 0 0
\(706\) −66.2239 −2.49237
\(707\) −15.1573 −0.570050
\(708\) 0 0
\(709\) −48.3236 −1.81483 −0.907415 0.420235i \(-0.861947\pi\)
−0.907415 + 0.420235i \(0.861947\pi\)
\(710\) 13.2906 0.498787
\(711\) 0 0
\(712\) −10.2993 −0.385984
\(713\) 51.0302 1.91110
\(714\) 0 0
\(715\) 0 0
\(716\) −96.4932 −3.60612
\(717\) 0 0
\(718\) 6.23049 0.232520
\(719\) 36.2637 1.35241 0.676203 0.736715i \(-0.263624\pi\)
0.676203 + 0.736715i \(0.263624\pi\)
\(720\) 0 0
\(721\) −13.8364 −0.515294
\(722\) −37.9362 −1.41184
\(723\) 0 0
\(724\) 79.9202 2.97021
\(725\) −15.9833 −0.593605
\(726\) 0 0
\(727\) 1.15184 0.0427195 0.0213597 0.999772i \(-0.493200\pi\)
0.0213597 + 0.999772i \(0.493200\pi\)
\(728\) −12.4277 −0.460601
\(729\) 0 0
\(730\) 66.8314 2.47354
\(731\) −21.3568 −0.789909
\(732\) 0 0
\(733\) 27.1042 1.00111 0.500557 0.865703i \(-0.333128\pi\)
0.500557 + 0.865703i \(0.333128\pi\)
\(734\) 24.7520 0.913614
\(735\) 0 0
\(736\) 14.4745 0.533535
\(737\) 0 0
\(738\) 0 0
\(739\) 23.8018 0.875563 0.437782 0.899081i \(-0.355764\pi\)
0.437782 + 0.899081i \(0.355764\pi\)
\(740\) −8.93160 −0.328332
\(741\) 0 0
\(742\) −14.6710 −0.538590
\(743\) 15.0656 0.552704 0.276352 0.961056i \(-0.410874\pi\)
0.276352 + 0.961056i \(0.410874\pi\)
\(744\) 0 0
\(745\) 56.4043 2.06650
\(746\) −12.3430 −0.451908
\(747\) 0 0
\(748\) 0 0
\(749\) 0.445651 0.0162837
\(750\) 0 0
\(751\) 10.6107 0.387191 0.193596 0.981081i \(-0.437985\pi\)
0.193596 + 0.981081i \(0.437985\pi\)
\(752\) 11.8075 0.430575
\(753\) 0 0
\(754\) 10.9179 0.397605
\(755\) 54.2841 1.97560
\(756\) 0 0
\(757\) −35.0862 −1.27523 −0.637615 0.770355i \(-0.720079\pi\)
−0.637615 + 0.770355i \(0.720079\pi\)
\(758\) 84.7997 3.08006
\(759\) 0 0
\(760\) −96.3423 −3.49470
\(761\) 37.2895 1.35174 0.675872 0.737019i \(-0.263767\pi\)
0.675872 + 0.737019i \(0.263767\pi\)
\(762\) 0 0
\(763\) −11.0988 −0.401803
\(764\) 13.9537 0.504827
\(765\) 0 0
\(766\) −75.1887 −2.71668
\(767\) −18.7096 −0.675565
\(768\) 0 0
\(769\) −21.8827 −0.789110 −0.394555 0.918872i \(-0.629101\pi\)
−0.394555 + 0.918872i \(0.629101\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −22.6823 −0.816353
\(773\) 34.4742 1.23995 0.619975 0.784622i \(-0.287143\pi\)
0.619975 + 0.784622i \(0.287143\pi\)
\(774\) 0 0
\(775\) 89.0106 3.19735
\(776\) −10.5797 −0.379789
\(777\) 0 0
\(778\) 48.0681 1.72332
\(779\) −10.0549 −0.360254
\(780\) 0 0
\(781\) 0 0
\(782\) 95.2699 3.40685
\(783\) 0 0
\(784\) 2.45868 0.0878099
\(785\) −45.1358 −1.61097
\(786\) 0 0
\(787\) −11.6668 −0.415878 −0.207939 0.978142i \(-0.566676\pi\)
−0.207939 + 0.978142i \(0.566676\pi\)
\(788\) 80.3887 2.86373
\(789\) 0 0
\(790\) −9.64747 −0.343242
\(791\) 8.08938 0.287625
\(792\) 0 0
\(793\) 6.26466 0.222464
\(794\) −8.69849 −0.308698
\(795\) 0 0
\(796\) −55.5797 −1.96997
\(797\) 35.1322 1.24445 0.622223 0.782840i \(-0.286230\pi\)
0.622223 + 0.782840i \(0.286230\pi\)
\(798\) 0 0
\(799\) −31.7176 −1.12209
\(800\) 25.2474 0.892630
\(801\) 0 0
\(802\) 90.0869 3.18108
\(803\) 0 0
\(804\) 0 0
\(805\) 23.7302 0.836380
\(806\) −60.8012 −2.14163
\(807\) 0 0
\(808\) 62.8133 2.20977
\(809\) −14.5235 −0.510620 −0.255310 0.966859i \(-0.582178\pi\)
−0.255310 + 0.966859i \(0.582178\pi\)
\(810\) 0 0
\(811\) −1.76417 −0.0619483 −0.0309741 0.999520i \(-0.509861\pi\)
−0.0309741 + 0.999520i \(0.509861\pi\)
\(812\) −5.67401 −0.199119
\(813\) 0 0
\(814\) 0 0
\(815\) 19.4655 0.681846
\(816\) 0 0
\(817\) 19.0885 0.667821
\(818\) 38.5570 1.34811
\(819\) 0 0
\(820\) −25.0287 −0.874039
\(821\) −9.76074 −0.340652 −0.170326 0.985388i \(-0.554482\pi\)
−0.170326 + 0.985388i \(0.554482\pi\)
\(822\) 0 0
\(823\) −45.6322 −1.59064 −0.795319 0.606191i \(-0.792697\pi\)
−0.795319 + 0.606191i \(0.792697\pi\)
\(824\) 57.3392 1.99750
\(825\) 0 0
\(826\) 14.9355 0.519674
\(827\) 20.7517 0.721608 0.360804 0.932642i \(-0.382502\pi\)
0.360804 + 0.932642i \(0.382502\pi\)
\(828\) 0 0
\(829\) 16.3868 0.569137 0.284568 0.958656i \(-0.408150\pi\)
0.284568 + 0.958656i \(0.408150\pi\)
\(830\) −29.1074 −1.01033
\(831\) 0 0
\(832\) −31.9926 −1.10914
\(833\) −6.60457 −0.228835
\(834\) 0 0
\(835\) 33.6102 1.16313
\(836\) 0 0
\(837\) 0 0
\(838\) −42.4037 −1.46481
\(839\) −35.0935 −1.21156 −0.605782 0.795631i \(-0.707140\pi\)
−0.605782 + 0.795631i \(0.707140\pi\)
\(840\) 0 0
\(841\) −26.6873 −0.920252
\(842\) −72.4703 −2.49749
\(843\) 0 0
\(844\) −63.2764 −2.17806
\(845\) −15.7792 −0.542822
\(846\) 0 0
\(847\) 0 0
\(848\) 15.0676 0.517424
\(849\) 0 0
\(850\) 166.177 5.69982
\(851\) −3.66255 −0.125550
\(852\) 0 0
\(853\) 11.3030 0.387007 0.193504 0.981100i \(-0.438015\pi\)
0.193504 + 0.981100i \(0.438015\pi\)
\(854\) −5.00096 −0.171129
\(855\) 0 0
\(856\) −1.84682 −0.0631229
\(857\) 13.8531 0.473214 0.236607 0.971605i \(-0.423965\pi\)
0.236607 + 0.971605i \(0.423965\pi\)
\(858\) 0 0
\(859\) −5.23831 −0.178729 −0.0893645 0.995999i \(-0.528484\pi\)
−0.0893645 + 0.995999i \(0.528484\pi\)
\(860\) 47.5150 1.62025
\(861\) 0 0
\(862\) −45.9445 −1.56488
\(863\) 41.1289 1.40004 0.700022 0.714121i \(-0.253174\pi\)
0.700022 + 0.714121i \(0.253174\pi\)
\(864\) 0 0
\(865\) 51.5427 1.75250
\(866\) −52.0779 −1.76968
\(867\) 0 0
\(868\) 31.5984 1.07252
\(869\) 0 0
\(870\) 0 0
\(871\) −1.79849 −0.0609397
\(872\) 45.9944 1.55757
\(873\) 0 0
\(874\) −85.1514 −2.88029
\(875\) 21.7005 0.733612
\(876\) 0 0
\(877\) 5.41488 0.182848 0.0914238 0.995812i \(-0.470858\pi\)
0.0914238 + 0.995812i \(0.470858\pi\)
\(878\) 6.47612 0.218558
\(879\) 0 0
\(880\) 0 0
\(881\) −11.0961 −0.373837 −0.186918 0.982375i \(-0.559850\pi\)
−0.186918 + 0.982375i \(0.559850\pi\)
\(882\) 0 0
\(883\) −51.4596 −1.73175 −0.865877 0.500256i \(-0.833239\pi\)
−0.865877 + 0.500256i \(0.833239\pi\)
\(884\) −73.8989 −2.48549
\(885\) 0 0
\(886\) 91.5149 3.07450
\(887\) −19.9938 −0.671325 −0.335662 0.941982i \(-0.608960\pi\)
−0.335662 + 0.941982i \(0.608960\pi\)
\(888\) 0 0
\(889\) −0.179659 −0.00602558
\(890\) −23.4318 −0.785435
\(891\) 0 0
\(892\) −51.5353 −1.72553
\(893\) 28.3489 0.948659
\(894\) 0 0
\(895\) −101.853 −3.40456
\(896\) 20.7347 0.692697
\(897\) 0 0
\(898\) 54.3697 1.81434
\(899\) −12.8793 −0.429548
\(900\) 0 0
\(901\) −40.4751 −1.34842
\(902\) 0 0
\(903\) 0 0
\(904\) −33.5231 −1.11496
\(905\) 84.3591 2.80419
\(906\) 0 0
\(907\) −5.23855 −0.173943 −0.0869716 0.996211i \(-0.527719\pi\)
−0.0869716 + 0.996211i \(0.527719\pi\)
\(908\) −27.0630 −0.898119
\(909\) 0 0
\(910\) −28.2740 −0.937273
\(911\) 11.0131 0.364882 0.182441 0.983217i \(-0.441600\pi\)
0.182441 + 0.983217i \(0.441600\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −3.76290 −0.124466
\(915\) 0 0
\(916\) −1.22056 −0.0403286
\(917\) 9.38232 0.309831
\(918\) 0 0
\(919\) 28.8457 0.951533 0.475767 0.879572i \(-0.342171\pi\)
0.475767 + 0.879572i \(0.342171\pi\)
\(920\) −98.3401 −3.24218
\(921\) 0 0
\(922\) 67.3365 2.21761
\(923\) −4.22747 −0.139149
\(924\) 0 0
\(925\) −6.38848 −0.210052
\(926\) 11.2325 0.369123
\(927\) 0 0
\(928\) −3.65314 −0.119920
\(929\) −16.2208 −0.532186 −0.266093 0.963947i \(-0.585733\pi\)
−0.266093 + 0.963947i \(0.585733\pi\)
\(930\) 0 0
\(931\) 5.90310 0.193466
\(932\) 60.6116 1.98540
\(933\) 0 0
\(934\) −42.1660 −1.37971
\(935\) 0 0
\(936\) 0 0
\(937\) 27.4672 0.897315 0.448657 0.893704i \(-0.351902\pi\)
0.448657 + 0.893704i \(0.351902\pi\)
\(938\) 1.43570 0.0468774
\(939\) 0 0
\(940\) 70.5660 2.30161
\(941\) 48.7222 1.58830 0.794150 0.607722i \(-0.207916\pi\)
0.794150 + 0.607722i \(0.207916\pi\)
\(942\) 0 0
\(943\) −10.2634 −0.334223
\(944\) −15.3393 −0.499251
\(945\) 0 0
\(946\) 0 0
\(947\) −21.1998 −0.688902 −0.344451 0.938804i \(-0.611935\pi\)
−0.344451 + 0.938804i \(0.611935\pi\)
\(948\) 0 0
\(949\) −21.2577 −0.690055
\(950\) −148.527 −4.81886
\(951\) 0 0
\(952\) 27.3699 0.887064
\(953\) 34.1228 1.10535 0.552673 0.833398i \(-0.313608\pi\)
0.552673 + 0.833398i \(0.313608\pi\)
\(954\) 0 0
\(955\) 14.7287 0.476610
\(956\) −108.204 −3.49957
\(957\) 0 0
\(958\) 25.6322 0.828138
\(959\) −0.893617 −0.0288564
\(960\) 0 0
\(961\) 40.7243 1.31369
\(962\) 4.36383 0.140696
\(963\) 0 0
\(964\) 109.453 3.52526
\(965\) −23.9421 −0.770724
\(966\) 0 0
\(967\) 18.8881 0.607400 0.303700 0.952768i \(-0.401778\pi\)
0.303700 + 0.952768i \(0.401778\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −24.0696 −0.772829
\(971\) −12.4847 −0.400652 −0.200326 0.979729i \(-0.564200\pi\)
−0.200326 + 0.979729i \(0.564200\pi\)
\(972\) 0 0
\(973\) −2.12281 −0.0680541
\(974\) −92.7906 −2.97320
\(975\) 0 0
\(976\) 5.13614 0.164404
\(977\) 13.8747 0.443890 0.221945 0.975059i \(-0.428759\pi\)
0.221945 + 0.975059i \(0.428759\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 14.6940 0.469382
\(981\) 0 0
\(982\) 14.3652 0.458413
\(983\) −43.8326 −1.39804 −0.699021 0.715101i \(-0.746381\pi\)
−0.699021 + 0.715101i \(0.746381\pi\)
\(984\) 0 0
\(985\) 84.8537 2.70366
\(986\) −24.0448 −0.765741
\(987\) 0 0
\(988\) 66.0502 2.10134
\(989\) 19.4843 0.619565
\(990\) 0 0
\(991\) 42.8926 1.36253 0.681264 0.732037i \(-0.261430\pi\)
0.681264 + 0.732037i \(0.261430\pi\)
\(992\) 20.3442 0.645930
\(993\) 0 0
\(994\) 3.37471 0.107039
\(995\) −58.6667 −1.85986
\(996\) 0 0
\(997\) 28.2985 0.896224 0.448112 0.893977i \(-0.352097\pi\)
0.448112 + 0.893977i \(0.352097\pi\)
\(998\) 69.3713 2.19591
\(999\) 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.1.2 10
3.2 odd 2 2541.2.a.br.1.9 10
11.7 odd 10 693.2.m.j.379.5 20
11.8 odd 10 693.2.m.j.64.5 20
11.10 odd 2 7623.2.a.cx.1.9 10
33.8 even 10 231.2.j.g.64.1 20
33.29 even 10 231.2.j.g.148.1 yes 20
33.32 even 2 2541.2.a.bq.1.2 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.j.g.64.1 20 33.8 even 10
231.2.j.g.148.1 yes 20 33.29 even 10
693.2.m.j.64.5 20 11.8 odd 10
693.2.m.j.379.5 20 11.7 odd 10
2541.2.a.bq.1.2 10 33.32 even 2
2541.2.a.br.1.9 10 3.2 odd 2
7623.2.a.cx.1.9 10 11.10 odd 2
7623.2.a.cy.1.2 10 1.1 even 1 trivial