Properties

Label 495.2.a.c.1.1
Level $495$
Weight $2$
Character 495.1
Self dual yes
Analytic conductor $3.953$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 495.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.95259490005\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 495.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.73205 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} +1.73205 q^{8} +O(q^{10})\) \(q-1.73205 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} +1.73205 q^{8} -1.73205 q^{10} +1.00000 q^{11} -1.46410 q^{13} -3.46410 q^{14} -5.00000 q^{16} -1.46410 q^{19} +1.00000 q^{20} -1.73205 q^{22} +6.92820 q^{23} +1.00000 q^{25} +2.53590 q^{26} +2.00000 q^{28} -3.46410 q^{29} +2.92820 q^{31} +5.19615 q^{32} +2.00000 q^{35} +8.92820 q^{37} +2.53590 q^{38} +1.73205 q^{40} +3.46410 q^{41} +8.92820 q^{43} +1.00000 q^{44} -12.0000 q^{46} -6.92820 q^{47} -3.00000 q^{49} -1.73205 q^{50} -1.46410 q^{52} +12.9282 q^{53} +1.00000 q^{55} +3.46410 q^{56} +6.00000 q^{58} -6.92820 q^{59} +2.00000 q^{61} -5.07180 q^{62} +1.00000 q^{64} -1.46410 q^{65} +8.00000 q^{67} -3.46410 q^{70} +13.8564 q^{71} +12.3923 q^{73} -15.4641 q^{74} -1.46410 q^{76} +2.00000 q^{77} -13.4641 q^{79} -5.00000 q^{80} -6.00000 q^{82} -15.4641 q^{83} -15.4641 q^{86} +1.73205 q^{88} +12.9282 q^{89} -2.92820 q^{91} +6.92820 q^{92} +12.0000 q^{94} -1.46410 q^{95} -10.0000 q^{97} +5.19615 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{4} + 2q^{5} + 4q^{7} + O(q^{10}) \) \( 2q + 2q^{4} + 2q^{5} + 4q^{7} + 2q^{11} + 4q^{13} - 10q^{16} + 4q^{19} + 2q^{20} + 2q^{25} + 12q^{26} + 4q^{28} - 8q^{31} + 4q^{35} + 4q^{37} + 12q^{38} + 4q^{43} + 2q^{44} - 24q^{46} - 6q^{49} + 4q^{52} + 12q^{53} + 2q^{55} + 12q^{58} + 4q^{61} - 24q^{62} + 2q^{64} + 4q^{65} + 16q^{67} + 4q^{73} - 24q^{74} + 4q^{76} + 4q^{77} - 20q^{79} - 10q^{80} - 12q^{82} - 24q^{83} - 24q^{86} + 12q^{89} + 8q^{91} + 24q^{94} + 4q^{95} - 20q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.73205 −1.22474 −0.612372 0.790569i \(-0.709785\pi\)
−0.612372 + 0.790569i \(0.709785\pi\)
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 1.73205 0.612372
\(9\) 0 0
\(10\) −1.73205 −0.547723
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −1.46410 −0.406069 −0.203034 0.979172i \(-0.565080\pi\)
−0.203034 + 0.979172i \(0.565080\pi\)
\(14\) −3.46410 −0.925820
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −1.46410 −0.335888 −0.167944 0.985797i \(-0.553713\pi\)
−0.167944 + 0.985797i \(0.553713\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −1.73205 −0.369274
\(23\) 6.92820 1.44463 0.722315 0.691564i \(-0.243078\pi\)
0.722315 + 0.691564i \(0.243078\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 2.53590 0.497331
\(27\) 0 0
\(28\) 2.00000 0.377964
\(29\) −3.46410 −0.643268 −0.321634 0.946864i \(-0.604232\pi\)
−0.321634 + 0.946864i \(0.604232\pi\)
\(30\) 0 0
\(31\) 2.92820 0.525921 0.262960 0.964807i \(-0.415301\pi\)
0.262960 + 0.964807i \(0.415301\pi\)
\(32\) 5.19615 0.918559
\(33\) 0 0
\(34\) 0 0
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) 8.92820 1.46779 0.733894 0.679264i \(-0.237701\pi\)
0.733894 + 0.679264i \(0.237701\pi\)
\(38\) 2.53590 0.411377
\(39\) 0 0
\(40\) 1.73205 0.273861
\(41\) 3.46410 0.541002 0.270501 0.962720i \(-0.412811\pi\)
0.270501 + 0.962720i \(0.412811\pi\)
\(42\) 0 0
\(43\) 8.92820 1.36154 0.680769 0.732498i \(-0.261646\pi\)
0.680769 + 0.732498i \(0.261646\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −12.0000 −1.76930
\(47\) −6.92820 −1.01058 −0.505291 0.862949i \(-0.668615\pi\)
−0.505291 + 0.862949i \(0.668615\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) −1.73205 −0.244949
\(51\) 0 0
\(52\) −1.46410 −0.203034
\(53\) 12.9282 1.77583 0.887913 0.460012i \(-0.152155\pi\)
0.887913 + 0.460012i \(0.152155\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 3.46410 0.462910
\(57\) 0 0
\(58\) 6.00000 0.787839
\(59\) −6.92820 −0.901975 −0.450988 0.892530i \(-0.648928\pi\)
−0.450988 + 0.892530i \(0.648928\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −5.07180 −0.644119
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.46410 −0.181599
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) −3.46410 −0.414039
\(71\) 13.8564 1.64445 0.822226 0.569160i \(-0.192732\pi\)
0.822226 + 0.569160i \(0.192732\pi\)
\(72\) 0 0
\(73\) 12.3923 1.45041 0.725205 0.688533i \(-0.241745\pi\)
0.725205 + 0.688533i \(0.241745\pi\)
\(74\) −15.4641 −1.79767
\(75\) 0 0
\(76\) −1.46410 −0.167944
\(77\) 2.00000 0.227921
\(78\) 0 0
\(79\) −13.4641 −1.51483 −0.757415 0.652934i \(-0.773538\pi\)
−0.757415 + 0.652934i \(0.773538\pi\)
\(80\) −5.00000 −0.559017
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) −15.4641 −1.69741 −0.848703 0.528870i \(-0.822616\pi\)
−0.848703 + 0.528870i \(0.822616\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −15.4641 −1.66754
\(87\) 0 0
\(88\) 1.73205 0.184637
\(89\) 12.9282 1.37039 0.685193 0.728361i \(-0.259718\pi\)
0.685193 + 0.728361i \(0.259718\pi\)
\(90\) 0 0
\(91\) −2.92820 −0.306959
\(92\) 6.92820 0.722315
\(93\) 0 0
\(94\) 12.0000 1.23771
\(95\) −1.46410 −0.150214
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 5.19615 0.524891
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −10.3923 −1.03407 −0.517036 0.855963i \(-0.672965\pi\)
−0.517036 + 0.855963i \(0.672965\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −2.53590 −0.248665
\(105\) 0 0
\(106\) −22.3923 −2.17493
\(107\) −15.4641 −1.49497 −0.747486 0.664278i \(-0.768739\pi\)
−0.747486 + 0.664278i \(0.768739\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) −1.73205 −0.165145
\(111\) 0 0
\(112\) −10.0000 −0.944911
\(113\) −0.928203 −0.0873180 −0.0436590 0.999046i \(-0.513902\pi\)
−0.0436590 + 0.999046i \(0.513902\pi\)
\(114\) 0 0
\(115\) 6.92820 0.646058
\(116\) −3.46410 −0.321634
\(117\) 0 0
\(118\) 12.0000 1.10469
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −3.46410 −0.313625
\(123\) 0 0
\(124\) 2.92820 0.262960
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −4.92820 −0.437307 −0.218654 0.975803i \(-0.570166\pi\)
−0.218654 + 0.975803i \(0.570166\pi\)
\(128\) −12.1244 −1.07165
\(129\) 0 0
\(130\) 2.53590 0.222413
\(131\) −5.07180 −0.443125 −0.221562 0.975146i \(-0.571116\pi\)
−0.221562 + 0.975146i \(0.571116\pi\)
\(132\) 0 0
\(133\) −2.92820 −0.253907
\(134\) −13.8564 −1.19701
\(135\) 0 0
\(136\) 0 0
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 0 0
\(139\) −8.39230 −0.711826 −0.355913 0.934519i \(-0.615830\pi\)
−0.355913 + 0.934519i \(0.615830\pi\)
\(140\) 2.00000 0.169031
\(141\) 0 0
\(142\) −24.0000 −2.01404
\(143\) −1.46410 −0.122434
\(144\) 0 0
\(145\) −3.46410 −0.287678
\(146\) −21.4641 −1.77638
\(147\) 0 0
\(148\) 8.92820 0.733894
\(149\) 8.53590 0.699288 0.349644 0.936883i \(-0.386303\pi\)
0.349644 + 0.936883i \(0.386303\pi\)
\(150\) 0 0
\(151\) 0.392305 0.0319253 0.0159627 0.999873i \(-0.494919\pi\)
0.0159627 + 0.999873i \(0.494919\pi\)
\(152\) −2.53590 −0.205689
\(153\) 0 0
\(154\) −3.46410 −0.279145
\(155\) 2.92820 0.235199
\(156\) 0 0
\(157\) −16.9282 −1.35102 −0.675509 0.737352i \(-0.736076\pi\)
−0.675509 + 0.737352i \(0.736076\pi\)
\(158\) 23.3205 1.85528
\(159\) 0 0
\(160\) 5.19615 0.410792
\(161\) 13.8564 1.09204
\(162\) 0 0
\(163\) −17.8564 −1.39862 −0.699311 0.714818i \(-0.746510\pi\)
−0.699311 + 0.714818i \(0.746510\pi\)
\(164\) 3.46410 0.270501
\(165\) 0 0
\(166\) 26.7846 2.07889
\(167\) −10.3923 −0.804181 −0.402090 0.915600i \(-0.631716\pi\)
−0.402090 + 0.915600i \(0.631716\pi\)
\(168\) 0 0
\(169\) −10.8564 −0.835108
\(170\) 0 0
\(171\) 0 0
\(172\) 8.92820 0.680769
\(173\) 12.0000 0.912343 0.456172 0.889892i \(-0.349220\pi\)
0.456172 + 0.889892i \(0.349220\pi\)
\(174\) 0 0
\(175\) 2.00000 0.151186
\(176\) −5.00000 −0.376889
\(177\) 0 0
\(178\) −22.3923 −1.67837
\(179\) 6.92820 0.517838 0.258919 0.965899i \(-0.416634\pi\)
0.258919 + 0.965899i \(0.416634\pi\)
\(180\) 0 0
\(181\) −11.8564 −0.881280 −0.440640 0.897684i \(-0.645248\pi\)
−0.440640 + 0.897684i \(0.645248\pi\)
\(182\) 5.07180 0.375947
\(183\) 0 0
\(184\) 12.0000 0.884652
\(185\) 8.92820 0.656415
\(186\) 0 0
\(187\) 0 0
\(188\) −6.92820 −0.505291
\(189\) 0 0
\(190\) 2.53590 0.183973
\(191\) 5.07180 0.366982 0.183491 0.983021i \(-0.441260\pi\)
0.183491 + 0.983021i \(0.441260\pi\)
\(192\) 0 0
\(193\) 3.60770 0.259688 0.129844 0.991534i \(-0.458552\pi\)
0.129844 + 0.991534i \(0.458552\pi\)
\(194\) 17.3205 1.24354
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) 16.7846 1.18983 0.594915 0.803789i \(-0.297186\pi\)
0.594915 + 0.803789i \(0.297186\pi\)
\(200\) 1.73205 0.122474
\(201\) 0 0
\(202\) 18.0000 1.26648
\(203\) −6.92820 −0.486265
\(204\) 0 0
\(205\) 3.46410 0.241943
\(206\) −13.8564 −0.965422
\(207\) 0 0
\(208\) 7.32051 0.507586
\(209\) −1.46410 −0.101274
\(210\) 0 0
\(211\) 12.3923 0.853121 0.426561 0.904459i \(-0.359725\pi\)
0.426561 + 0.904459i \(0.359725\pi\)
\(212\) 12.9282 0.887913
\(213\) 0 0
\(214\) 26.7846 1.83096
\(215\) 8.92820 0.608898
\(216\) 0 0
\(217\) 5.85641 0.397559
\(218\) 17.3205 1.17309
\(219\) 0 0
\(220\) 1.00000 0.0674200
\(221\) 0 0
\(222\) 0 0
\(223\) −17.8564 −1.19575 −0.597877 0.801588i \(-0.703989\pi\)
−0.597877 + 0.801588i \(0.703989\pi\)
\(224\) 10.3923 0.694365
\(225\) 0 0
\(226\) 1.60770 0.106942
\(227\) −8.53590 −0.566547 −0.283274 0.959039i \(-0.591420\pi\)
−0.283274 + 0.959039i \(0.591420\pi\)
\(228\) 0 0
\(229\) 3.85641 0.254839 0.127419 0.991849i \(-0.459331\pi\)
0.127419 + 0.991849i \(0.459331\pi\)
\(230\) −12.0000 −0.791257
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) −12.0000 −0.786146 −0.393073 0.919507i \(-0.628588\pi\)
−0.393073 + 0.919507i \(0.628588\pi\)
\(234\) 0 0
\(235\) −6.92820 −0.451946
\(236\) −6.92820 −0.450988
\(237\) 0 0
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) 27.8564 1.79439 0.897194 0.441636i \(-0.145602\pi\)
0.897194 + 0.441636i \(0.145602\pi\)
\(242\) −1.73205 −0.111340
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) −3.00000 −0.191663
\(246\) 0 0
\(247\) 2.14359 0.136394
\(248\) 5.07180 0.322059
\(249\) 0 0
\(250\) −1.73205 −0.109545
\(251\) −25.8564 −1.63204 −0.816021 0.578022i \(-0.803825\pi\)
−0.816021 + 0.578022i \(0.803825\pi\)
\(252\) 0 0
\(253\) 6.92820 0.435572
\(254\) 8.53590 0.535590
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) −7.85641 −0.490069 −0.245035 0.969514i \(-0.578799\pi\)
−0.245035 + 0.969514i \(0.578799\pi\)
\(258\) 0 0
\(259\) 17.8564 1.10954
\(260\) −1.46410 −0.0907997
\(261\) 0 0
\(262\) 8.78461 0.542715
\(263\) −27.4641 −1.69351 −0.846755 0.531984i \(-0.821447\pi\)
−0.846755 + 0.531984i \(0.821447\pi\)
\(264\) 0 0
\(265\) 12.9282 0.794173
\(266\) 5.07180 0.310972
\(267\) 0 0
\(268\) 8.00000 0.488678
\(269\) 7.85641 0.479014 0.239507 0.970895i \(-0.423014\pi\)
0.239507 + 0.970895i \(0.423014\pi\)
\(270\) 0 0
\(271\) −32.3923 −1.96769 −0.983846 0.179016i \(-0.942709\pi\)
−0.983846 + 0.179016i \(0.942709\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −31.1769 −1.88347
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) 22.5359 1.35405 0.677025 0.735960i \(-0.263269\pi\)
0.677025 + 0.735960i \(0.263269\pi\)
\(278\) 14.5359 0.871805
\(279\) 0 0
\(280\) 3.46410 0.207020
\(281\) 3.46410 0.206651 0.103325 0.994648i \(-0.467052\pi\)
0.103325 + 0.994648i \(0.467052\pi\)
\(282\) 0 0
\(283\) 8.92820 0.530727 0.265363 0.964148i \(-0.414508\pi\)
0.265363 + 0.964148i \(0.414508\pi\)
\(284\) 13.8564 0.822226
\(285\) 0 0
\(286\) 2.53590 0.149951
\(287\) 6.92820 0.408959
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 6.00000 0.352332
\(291\) 0 0
\(292\) 12.3923 0.725205
\(293\) −13.8564 −0.809500 −0.404750 0.914427i \(-0.632641\pi\)
−0.404750 + 0.914427i \(0.632641\pi\)
\(294\) 0 0
\(295\) −6.92820 −0.403376
\(296\) 15.4641 0.898833
\(297\) 0 0
\(298\) −14.7846 −0.856449
\(299\) −10.1436 −0.586619
\(300\) 0 0
\(301\) 17.8564 1.02923
\(302\) −0.679492 −0.0391004
\(303\) 0 0
\(304\) 7.32051 0.419860
\(305\) 2.00000 0.114520
\(306\) 0 0
\(307\) 14.0000 0.799022 0.399511 0.916728i \(-0.369180\pi\)
0.399511 + 0.916728i \(0.369180\pi\)
\(308\) 2.00000 0.113961
\(309\) 0 0
\(310\) −5.07180 −0.288059
\(311\) −18.9282 −1.07332 −0.536660 0.843799i \(-0.680314\pi\)
−0.536660 + 0.843799i \(0.680314\pi\)
\(312\) 0 0
\(313\) 7.07180 0.399722 0.199861 0.979824i \(-0.435951\pi\)
0.199861 + 0.979824i \(0.435951\pi\)
\(314\) 29.3205 1.65465
\(315\) 0 0
\(316\) −13.4641 −0.757415
\(317\) 11.0718 0.621854 0.310927 0.950434i \(-0.399361\pi\)
0.310927 + 0.950434i \(0.399361\pi\)
\(318\) 0 0
\(319\) −3.46410 −0.193952
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) −24.0000 −1.33747
\(323\) 0 0
\(324\) 0 0
\(325\) −1.46410 −0.0812137
\(326\) 30.9282 1.71295
\(327\) 0 0
\(328\) 6.00000 0.331295
\(329\) −13.8564 −0.763928
\(330\) 0 0
\(331\) −17.8564 −0.981477 −0.490738 0.871307i \(-0.663273\pi\)
−0.490738 + 0.871307i \(0.663273\pi\)
\(332\) −15.4641 −0.848703
\(333\) 0 0
\(334\) 18.0000 0.984916
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) −29.1769 −1.58937 −0.794684 0.607023i \(-0.792363\pi\)
−0.794684 + 0.607023i \(0.792363\pi\)
\(338\) 18.8038 1.02279
\(339\) 0 0
\(340\) 0 0
\(341\) 2.92820 0.158571
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 15.4641 0.833768
\(345\) 0 0
\(346\) −20.7846 −1.11739
\(347\) −1.60770 −0.0863056 −0.0431528 0.999068i \(-0.513740\pi\)
−0.0431528 + 0.999068i \(0.513740\pi\)
\(348\) 0 0
\(349\) −35.8564 −1.91935 −0.959675 0.281113i \(-0.909296\pi\)
−0.959675 + 0.281113i \(0.909296\pi\)
\(350\) −3.46410 −0.185164
\(351\) 0 0
\(352\) 5.19615 0.276956
\(353\) 0.928203 0.0494033 0.0247016 0.999695i \(-0.492136\pi\)
0.0247016 + 0.999695i \(0.492136\pi\)
\(354\) 0 0
\(355\) 13.8564 0.735422
\(356\) 12.9282 0.685193
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) −20.7846 −1.09697 −0.548485 0.836160i \(-0.684795\pi\)
−0.548485 + 0.836160i \(0.684795\pi\)
\(360\) 0 0
\(361\) −16.8564 −0.887179
\(362\) 20.5359 1.07934
\(363\) 0 0
\(364\) −2.92820 −0.153480
\(365\) 12.3923 0.648643
\(366\) 0 0
\(367\) 20.0000 1.04399 0.521996 0.852948i \(-0.325188\pi\)
0.521996 + 0.852948i \(0.325188\pi\)
\(368\) −34.6410 −1.80579
\(369\) 0 0
\(370\) −15.4641 −0.803940
\(371\) 25.8564 1.34240
\(372\) 0 0
\(373\) 0.392305 0.0203128 0.0101564 0.999948i \(-0.496767\pi\)
0.0101564 + 0.999948i \(0.496767\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 5.07180 0.261211
\(378\) 0 0
\(379\) 9.85641 0.506290 0.253145 0.967428i \(-0.418535\pi\)
0.253145 + 0.967428i \(0.418535\pi\)
\(380\) −1.46410 −0.0751068
\(381\) 0 0
\(382\) −8.78461 −0.449460
\(383\) −13.8564 −0.708029 −0.354015 0.935240i \(-0.615184\pi\)
−0.354015 + 0.935240i \(0.615184\pi\)
\(384\) 0 0
\(385\) 2.00000 0.101929
\(386\) −6.24871 −0.318051
\(387\) 0 0
\(388\) −10.0000 −0.507673
\(389\) −24.9282 −1.26391 −0.631955 0.775005i \(-0.717747\pi\)
−0.631955 + 0.775005i \(0.717747\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −5.19615 −0.262445
\(393\) 0 0
\(394\) −20.7846 −1.04711
\(395\) −13.4641 −0.677452
\(396\) 0 0
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) −29.0718 −1.45724
\(399\) 0 0
\(400\) −5.00000 −0.250000
\(401\) −19.8564 −0.991582 −0.495791 0.868442i \(-0.665122\pi\)
−0.495791 + 0.868442i \(0.665122\pi\)
\(402\) 0 0
\(403\) −4.28719 −0.213560
\(404\) −10.3923 −0.517036
\(405\) 0 0
\(406\) 12.0000 0.595550
\(407\) 8.92820 0.442555
\(408\) 0 0
\(409\) 34.7846 1.71999 0.859994 0.510304i \(-0.170467\pi\)
0.859994 + 0.510304i \(0.170467\pi\)
\(410\) −6.00000 −0.296319
\(411\) 0 0
\(412\) 8.00000 0.394132
\(413\) −13.8564 −0.681829
\(414\) 0 0
\(415\) −15.4641 −0.759103
\(416\) −7.60770 −0.372998
\(417\) 0 0
\(418\) 2.53590 0.124035
\(419\) −17.0718 −0.834012 −0.417006 0.908904i \(-0.636921\pi\)
−0.417006 + 0.908904i \(0.636921\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) −21.4641 −1.04486
\(423\) 0 0
\(424\) 22.3923 1.08747
\(425\) 0 0
\(426\) 0 0
\(427\) 4.00000 0.193574
\(428\) −15.4641 −0.747486
\(429\) 0 0
\(430\) −15.4641 −0.745745
\(431\) 32.7846 1.57918 0.789590 0.613635i \(-0.210294\pi\)
0.789590 + 0.613635i \(0.210294\pi\)
\(432\) 0 0
\(433\) 27.8564 1.33869 0.669347 0.742950i \(-0.266574\pi\)
0.669347 + 0.742950i \(0.266574\pi\)
\(434\) −10.1436 −0.486908
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) −10.1436 −0.485234
\(438\) 0 0
\(439\) −29.1769 −1.39254 −0.696269 0.717781i \(-0.745158\pi\)
−0.696269 + 0.717781i \(0.745158\pi\)
\(440\) 1.73205 0.0825723
\(441\) 0 0
\(442\) 0 0
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 0 0
\(445\) 12.9282 0.612856
\(446\) 30.9282 1.46449
\(447\) 0 0
\(448\) 2.00000 0.0944911
\(449\) −14.7846 −0.697729 −0.348864 0.937173i \(-0.613433\pi\)
−0.348864 + 0.937173i \(0.613433\pi\)
\(450\) 0 0
\(451\) 3.46410 0.163118
\(452\) −0.928203 −0.0436590
\(453\) 0 0
\(454\) 14.7846 0.693876
\(455\) −2.92820 −0.137276
\(456\) 0 0
\(457\) −8.39230 −0.392575 −0.196288 0.980546i \(-0.562889\pi\)
−0.196288 + 0.980546i \(0.562889\pi\)
\(458\) −6.67949 −0.312112
\(459\) 0 0
\(460\) 6.92820 0.323029
\(461\) 12.2487 0.570479 0.285240 0.958456i \(-0.407927\pi\)
0.285240 + 0.958456i \(0.407927\pi\)
\(462\) 0 0
\(463\) −28.0000 −1.30127 −0.650635 0.759390i \(-0.725497\pi\)
−0.650635 + 0.759390i \(0.725497\pi\)
\(464\) 17.3205 0.804084
\(465\) 0 0
\(466\) 20.7846 0.962828
\(467\) −18.9282 −0.875893 −0.437946 0.899001i \(-0.644294\pi\)
−0.437946 + 0.899001i \(0.644294\pi\)
\(468\) 0 0
\(469\) 16.0000 0.738811
\(470\) 12.0000 0.553519
\(471\) 0 0
\(472\) −12.0000 −0.552345
\(473\) 8.92820 0.410519
\(474\) 0 0
\(475\) −1.46410 −0.0671776
\(476\) 0 0
\(477\) 0 0
\(478\) −20.7846 −0.950666
\(479\) −12.0000 −0.548294 −0.274147 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) 0 0
\(481\) −13.0718 −0.596023
\(482\) −48.2487 −2.19767
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −10.0000 −0.454077
\(486\) 0 0
\(487\) 23.7128 1.07453 0.537265 0.843413i \(-0.319457\pi\)
0.537265 + 0.843413i \(0.319457\pi\)
\(488\) 3.46410 0.156813
\(489\) 0 0
\(490\) 5.19615 0.234738
\(491\) 17.0718 0.770439 0.385220 0.922825i \(-0.374126\pi\)
0.385220 + 0.922825i \(0.374126\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −3.71281 −0.167047
\(495\) 0 0
\(496\) −14.6410 −0.657401
\(497\) 27.7128 1.24309
\(498\) 0 0
\(499\) −12.7846 −0.572318 −0.286159 0.958182i \(-0.592379\pi\)
−0.286159 + 0.958182i \(0.592379\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 44.7846 1.99883
\(503\) 31.1769 1.39011 0.695055 0.718957i \(-0.255380\pi\)
0.695055 + 0.718957i \(0.255380\pi\)
\(504\) 0 0
\(505\) −10.3923 −0.462451
\(506\) −12.0000 −0.533465
\(507\) 0 0
\(508\) −4.92820 −0.218654
\(509\) 7.85641 0.348229 0.174115 0.984725i \(-0.444294\pi\)
0.174115 + 0.984725i \(0.444294\pi\)
\(510\) 0 0
\(511\) 24.7846 1.09641
\(512\) −8.66025 −0.382733
\(513\) 0 0
\(514\) 13.6077 0.600210
\(515\) 8.00000 0.352522
\(516\) 0 0
\(517\) −6.92820 −0.304702
\(518\) −30.9282 −1.35891
\(519\) 0 0
\(520\) −2.53590 −0.111207
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) −22.0000 −0.961993 −0.480996 0.876723i \(-0.659725\pi\)
−0.480996 + 0.876723i \(0.659725\pi\)
\(524\) −5.07180 −0.221562
\(525\) 0 0
\(526\) 47.5692 2.07412
\(527\) 0 0
\(528\) 0 0
\(529\) 25.0000 1.08696
\(530\) −22.3923 −0.972660
\(531\) 0 0
\(532\) −2.92820 −0.126954
\(533\) −5.07180 −0.219684
\(534\) 0 0
\(535\) −15.4641 −0.668571
\(536\) 13.8564 0.598506
\(537\) 0 0
\(538\) −13.6077 −0.586669
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) 0.143594 0.00617357 0.00308678 0.999995i \(-0.499017\pi\)
0.00308678 + 0.999995i \(0.499017\pi\)
\(542\) 56.1051 2.40992
\(543\) 0 0
\(544\) 0 0
\(545\) −10.0000 −0.428353
\(546\) 0 0
\(547\) 2.00000 0.0855138 0.0427569 0.999086i \(-0.486386\pi\)
0.0427569 + 0.999086i \(0.486386\pi\)
\(548\) 18.0000 0.768922
\(549\) 0 0
\(550\) −1.73205 −0.0738549
\(551\) 5.07180 0.216066
\(552\) 0 0
\(553\) −26.9282 −1.14510
\(554\) −39.0333 −1.65837
\(555\) 0 0
\(556\) −8.39230 −0.355913
\(557\) −44.7846 −1.89758 −0.948792 0.315900i \(-0.897694\pi\)
−0.948792 + 0.315900i \(0.897694\pi\)
\(558\) 0 0
\(559\) −13.0718 −0.552878
\(560\) −10.0000 −0.422577
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) 10.3923 0.437983 0.218992 0.975727i \(-0.429723\pi\)
0.218992 + 0.975727i \(0.429723\pi\)
\(564\) 0 0
\(565\) −0.928203 −0.0390498
\(566\) −15.4641 −0.650005
\(567\) 0 0
\(568\) 24.0000 1.00702
\(569\) 29.3205 1.22918 0.614590 0.788847i \(-0.289322\pi\)
0.614590 + 0.788847i \(0.289322\pi\)
\(570\) 0 0
\(571\) 24.3923 1.02079 0.510393 0.859941i \(-0.329500\pi\)
0.510393 + 0.859941i \(0.329500\pi\)
\(572\) −1.46410 −0.0612172
\(573\) 0 0
\(574\) −12.0000 −0.500870
\(575\) 6.92820 0.288926
\(576\) 0 0
\(577\) 22.7846 0.948536 0.474268 0.880381i \(-0.342713\pi\)
0.474268 + 0.880381i \(0.342713\pi\)
\(578\) 29.4449 1.22474
\(579\) 0 0
\(580\) −3.46410 −0.143839
\(581\) −30.9282 −1.28312
\(582\) 0 0
\(583\) 12.9282 0.535431
\(584\) 21.4641 0.888191
\(585\) 0 0
\(586\) 24.0000 0.991431
\(587\) 5.07180 0.209335 0.104668 0.994507i \(-0.466622\pi\)
0.104668 + 0.994507i \(0.466622\pi\)
\(588\) 0 0
\(589\) −4.28719 −0.176650
\(590\) 12.0000 0.494032
\(591\) 0 0
\(592\) −44.6410 −1.83473
\(593\) 32.7846 1.34630 0.673151 0.739505i \(-0.264940\pi\)
0.673151 + 0.739505i \(0.264940\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 8.53590 0.349644
\(597\) 0 0
\(598\) 17.5692 0.718459
\(599\) 10.1436 0.414456 0.207228 0.978293i \(-0.433556\pi\)
0.207228 + 0.978293i \(0.433556\pi\)
\(600\) 0 0
\(601\) 36.6410 1.49462 0.747309 0.664477i \(-0.231345\pi\)
0.747309 + 0.664477i \(0.231345\pi\)
\(602\) −30.9282 −1.26054
\(603\) 0 0
\(604\) 0.392305 0.0159627
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) 22.7846 0.924799 0.462399 0.886672i \(-0.346989\pi\)
0.462399 + 0.886672i \(0.346989\pi\)
\(608\) −7.60770 −0.308533
\(609\) 0 0
\(610\) −3.46410 −0.140257
\(611\) 10.1436 0.410366
\(612\) 0 0
\(613\) 0.392305 0.0158450 0.00792252 0.999969i \(-0.497478\pi\)
0.00792252 + 0.999969i \(0.497478\pi\)
\(614\) −24.2487 −0.978598
\(615\) 0 0
\(616\) 3.46410 0.139573
\(617\) 23.0718 0.928836 0.464418 0.885616i \(-0.346264\pi\)
0.464418 + 0.885616i \(0.346264\pi\)
\(618\) 0 0
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 2.92820 0.117599
\(621\) 0 0
\(622\) 32.7846 1.31454
\(623\) 25.8564 1.03592
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −12.2487 −0.489557
\(627\) 0 0
\(628\) −16.9282 −0.675509
\(629\) 0 0
\(630\) 0 0
\(631\) −34.9282 −1.39047 −0.695235 0.718783i \(-0.744700\pi\)
−0.695235 + 0.718783i \(0.744700\pi\)
\(632\) −23.3205 −0.927640
\(633\) 0 0
\(634\) −19.1769 −0.761613
\(635\) −4.92820 −0.195570
\(636\) 0 0
\(637\) 4.39230 0.174029
\(638\) 6.00000 0.237542
\(639\) 0 0
\(640\) −12.1244 −0.479257
\(641\) −0.928203 −0.0366618 −0.0183309 0.999832i \(-0.505835\pi\)
−0.0183309 + 0.999832i \(0.505835\pi\)
\(642\) 0 0
\(643\) −45.5692 −1.79707 −0.898537 0.438897i \(-0.855369\pi\)
−0.898537 + 0.438897i \(0.855369\pi\)
\(644\) 13.8564 0.546019
\(645\) 0 0
\(646\) 0 0
\(647\) 27.7128 1.08950 0.544752 0.838597i \(-0.316624\pi\)
0.544752 + 0.838597i \(0.316624\pi\)
\(648\) 0 0
\(649\) −6.92820 −0.271956
\(650\) 2.53590 0.0994661
\(651\) 0 0
\(652\) −17.8564 −0.699311
\(653\) 7.85641 0.307445 0.153722 0.988114i \(-0.450874\pi\)
0.153722 + 0.988114i \(0.450874\pi\)
\(654\) 0 0
\(655\) −5.07180 −0.198171
\(656\) −17.3205 −0.676252
\(657\) 0 0
\(658\) 24.0000 0.935617
\(659\) −39.7128 −1.54699 −0.773496 0.633801i \(-0.781494\pi\)
−0.773496 + 0.633801i \(0.781494\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) 30.9282 1.20206
\(663\) 0 0
\(664\) −26.7846 −1.03944
\(665\) −2.92820 −0.113551
\(666\) 0 0
\(667\) −24.0000 −0.929284
\(668\) −10.3923 −0.402090
\(669\) 0 0
\(670\) −13.8564 −0.535320
\(671\) 2.00000 0.0772091
\(672\) 0 0
\(673\) 31.3205 1.20732 0.603658 0.797243i \(-0.293709\pi\)
0.603658 + 0.797243i \(0.293709\pi\)
\(674\) 50.5359 1.94657
\(675\) 0 0
\(676\) −10.8564 −0.417554
\(677\) 32.7846 1.26001 0.630007 0.776589i \(-0.283052\pi\)
0.630007 + 0.776589i \(0.283052\pi\)
\(678\) 0 0
\(679\) −20.0000 −0.767530
\(680\) 0 0
\(681\) 0 0
\(682\) −5.07180 −0.194209
\(683\) −8.78461 −0.336134 −0.168067 0.985776i \(-0.553752\pi\)
−0.168067 + 0.985776i \(0.553752\pi\)
\(684\) 0 0
\(685\) 18.0000 0.687745
\(686\) 34.6410 1.32260
\(687\) 0 0
\(688\) −44.6410 −1.70192
\(689\) −18.9282 −0.721107
\(690\) 0 0
\(691\) −7.71281 −0.293409 −0.146705 0.989180i \(-0.546867\pi\)
−0.146705 + 0.989180i \(0.546867\pi\)
\(692\) 12.0000 0.456172
\(693\) 0 0
\(694\) 2.78461 0.105702
\(695\) −8.39230 −0.318338
\(696\) 0 0
\(697\) 0 0
\(698\) 62.1051 2.35071
\(699\) 0 0
\(700\) 2.00000 0.0755929
\(701\) −32.5359 −1.22886 −0.614432 0.788970i \(-0.710615\pi\)
−0.614432 + 0.788970i \(0.710615\pi\)
\(702\) 0 0
\(703\) −13.0718 −0.493012
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) −1.60770 −0.0605064
\(707\) −20.7846 −0.781686
\(708\) 0 0
\(709\) 15.8564 0.595500 0.297750 0.954644i \(-0.403764\pi\)
0.297750 + 0.954644i \(0.403764\pi\)
\(710\) −24.0000 −0.900704
\(711\) 0 0
\(712\) 22.3923 0.839187
\(713\) 20.2872 0.759761
\(714\) 0 0
\(715\) −1.46410 −0.0547543
\(716\) 6.92820 0.258919
\(717\) 0 0
\(718\) 36.0000 1.34351
\(719\) 18.9282 0.705903 0.352951 0.935642i \(-0.385178\pi\)
0.352951 + 0.935642i \(0.385178\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 29.1962 1.08657
\(723\) 0 0
\(724\) −11.8564 −0.440640
\(725\) −3.46410 −0.128654
\(726\) 0 0
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) −5.07180 −0.187973
\(729\) 0 0
\(730\) −21.4641 −0.794422
\(731\) 0 0
\(732\) 0 0
\(733\) −49.9615 −1.84537 −0.922686 0.385553i \(-0.874011\pi\)
−0.922686 + 0.385553i \(0.874011\pi\)
\(734\) −34.6410 −1.27862
\(735\) 0 0
\(736\) 36.0000 1.32698
\(737\) 8.00000 0.294684
\(738\) 0 0
\(739\) 10.5359 0.387569 0.193785 0.981044i \(-0.437924\pi\)
0.193785 + 0.981044i \(0.437924\pi\)
\(740\) 8.92820 0.328207
\(741\) 0 0
\(742\) −44.7846 −1.64409
\(743\) −46.3923 −1.70197 −0.850984 0.525191i \(-0.823994\pi\)
−0.850984 + 0.525191i \(0.823994\pi\)
\(744\) 0 0
\(745\) 8.53590 0.312731
\(746\) −0.679492 −0.0248780
\(747\) 0 0
\(748\) 0 0
\(749\) −30.9282 −1.13009
\(750\) 0 0
\(751\) 13.0718 0.476997 0.238498 0.971143i \(-0.423345\pi\)
0.238498 + 0.971143i \(0.423345\pi\)
\(752\) 34.6410 1.26323
\(753\) 0 0
\(754\) −8.78461 −0.319917
\(755\) 0.392305 0.0142774
\(756\) 0 0
\(757\) −6.78461 −0.246591 −0.123295 0.992370i \(-0.539346\pi\)
−0.123295 + 0.992370i \(0.539346\pi\)
\(758\) −17.0718 −0.620076
\(759\) 0 0
\(760\) −2.53590 −0.0919867
\(761\) 39.4641 1.43057 0.715286 0.698832i \(-0.246296\pi\)
0.715286 + 0.698832i \(0.246296\pi\)
\(762\) 0 0
\(763\) −20.0000 −0.724049
\(764\) 5.07180 0.183491
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 10.1436 0.366264
\(768\) 0 0
\(769\) −46.4974 −1.67674 −0.838370 0.545102i \(-0.816491\pi\)
−0.838370 + 0.545102i \(0.816491\pi\)
\(770\) −3.46410 −0.124838
\(771\) 0 0
\(772\) 3.60770 0.129844
\(773\) 31.8564 1.14580 0.572898 0.819627i \(-0.305819\pi\)
0.572898 + 0.819627i \(0.305819\pi\)
\(774\) 0 0
\(775\) 2.92820 0.105184
\(776\) −17.3205 −0.621770
\(777\) 0 0
\(778\) 43.1769 1.54797
\(779\) −5.07180 −0.181716
\(780\) 0 0
\(781\) 13.8564 0.495821
\(782\) 0 0
\(783\) 0 0
\(784\) 15.0000 0.535714
\(785\) −16.9282 −0.604193
\(786\) 0 0
\(787\) −18.7846 −0.669599 −0.334800 0.942289i \(-0.608669\pi\)
−0.334800 + 0.942289i \(0.608669\pi\)
\(788\) 12.0000 0.427482
\(789\) 0 0
\(790\) 23.3205 0.829706
\(791\) −1.85641 −0.0660062
\(792\) 0 0
\(793\) −2.92820 −0.103984
\(794\) −3.46410 −0.122936
\(795\) 0 0
\(796\) 16.7846 0.594915
\(797\) −16.6410 −0.589455 −0.294728 0.955581i \(-0.595229\pi\)
−0.294728 + 0.955581i \(0.595229\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 5.19615 0.183712
\(801\) 0 0
\(802\) 34.3923 1.21443
\(803\) 12.3923 0.437315
\(804\) 0 0
\(805\) 13.8564 0.488374
\(806\) 7.42563 0.261557
\(807\) 0 0
\(808\) −18.0000 −0.633238
\(809\) 8.53590 0.300106 0.150053 0.988678i \(-0.452056\pi\)
0.150053 + 0.988678i \(0.452056\pi\)
\(810\) 0 0
\(811\) −8.39230 −0.294694 −0.147347 0.989085i \(-0.547073\pi\)
−0.147347 + 0.989085i \(0.547073\pi\)
\(812\) −6.92820 −0.243132
\(813\) 0 0
\(814\) −15.4641 −0.542016
\(815\) −17.8564 −0.625483
\(816\) 0 0
\(817\) −13.0718 −0.457324
\(818\) −60.2487 −2.10655
\(819\) 0 0
\(820\) 3.46410 0.120972
\(821\) −27.4641 −0.958504 −0.479252 0.877677i \(-0.659092\pi\)
−0.479252 + 0.877677i \(0.659092\pi\)
\(822\) 0 0
\(823\) 49.5692 1.72787 0.863937 0.503600i \(-0.167991\pi\)
0.863937 + 0.503600i \(0.167991\pi\)
\(824\) 13.8564 0.482711
\(825\) 0 0
\(826\) 24.0000 0.835067
\(827\) −1.60770 −0.0559050 −0.0279525 0.999609i \(-0.508899\pi\)
−0.0279525 + 0.999609i \(0.508899\pi\)
\(828\) 0 0
\(829\) −25.7128 −0.893043 −0.446521 0.894773i \(-0.647337\pi\)
−0.446521 + 0.894773i \(0.647337\pi\)
\(830\) 26.7846 0.929707
\(831\) 0 0
\(832\) −1.46410 −0.0507586
\(833\) 0 0
\(834\) 0 0
\(835\) −10.3923 −0.359641
\(836\) −1.46410 −0.0506370
\(837\) 0 0
\(838\) 29.5692 1.02145
\(839\) −15.2154 −0.525294 −0.262647 0.964892i \(-0.584595\pi\)
−0.262647 + 0.964892i \(0.584595\pi\)
\(840\) 0 0
\(841\) −17.0000 −0.586207
\(842\) −3.46410 −0.119381
\(843\) 0 0
\(844\) 12.3923 0.426561
\(845\) −10.8564 −0.373472
\(846\) 0 0
\(847\) 2.00000 0.0687208
\(848\) −64.6410 −2.21978
\(849\) 0 0
\(850\) 0 0
\(851\) 61.8564 2.12041
\(852\) 0 0
\(853\) 24.3923 0.835177 0.417588 0.908636i \(-0.362875\pi\)
0.417588 + 0.908636i \(0.362875\pi\)
\(854\) −6.92820 −0.237078
\(855\) 0 0
\(856\) −26.7846 −0.915479
\(857\) 10.1436 0.346499 0.173249 0.984878i \(-0.444573\pi\)
0.173249 + 0.984878i \(0.444573\pi\)
\(858\) 0 0
\(859\) 47.7128 1.62794 0.813970 0.580907i \(-0.197302\pi\)
0.813970 + 0.580907i \(0.197302\pi\)
\(860\) 8.92820 0.304449
\(861\) 0 0
\(862\) −56.7846 −1.93409
\(863\) 10.1436 0.345292 0.172646 0.984984i \(-0.444768\pi\)
0.172646 + 0.984984i \(0.444768\pi\)
\(864\) 0 0
\(865\) 12.0000 0.408012
\(866\) −48.2487 −1.63956
\(867\) 0 0
\(868\) 5.85641 0.198779
\(869\) −13.4641 −0.456738
\(870\) 0 0
\(871\) −11.7128 −0.396874
\(872\) −17.3205 −0.586546
\(873\) 0 0
\(874\) 17.5692 0.594288
\(875\) 2.00000 0.0676123
\(876\) 0 0
\(877\) 14.2487 0.481145 0.240572 0.970631i \(-0.422665\pi\)
0.240572 + 0.970631i \(0.422665\pi\)
\(878\) 50.5359 1.70550
\(879\) 0 0
\(880\) −5.00000 −0.168550
\(881\) −12.9282 −0.435562 −0.217781 0.975998i \(-0.569882\pi\)
−0.217781 + 0.975998i \(0.569882\pi\)
\(882\) 0 0
\(883\) −4.00000 −0.134611 −0.0673054 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 20.7846 0.698273
\(887\) 36.2487 1.21711 0.608556 0.793511i \(-0.291749\pi\)
0.608556 + 0.793511i \(0.291749\pi\)
\(888\) 0 0
\(889\) −9.85641 −0.330573
\(890\) −22.3923 −0.750592
\(891\) 0 0
\(892\) −17.8564 −0.597877
\(893\) 10.1436 0.339442
\(894\) 0 0
\(895\) 6.92820 0.231584
\(896\) −24.2487 −0.810093
\(897\) 0 0
\(898\) 25.6077 0.854540
\(899\) −10.1436 −0.338308
\(900\) 0 0
\(901\) 0 0
\(902\) −6.00000 −0.199778
\(903\) 0 0
\(904\) −1.60770 −0.0534711
\(905\) −11.8564 −0.394120
\(906\) 0 0
\(907\) 45.8564 1.52264 0.761318 0.648378i \(-0.224552\pi\)
0.761318 + 0.648378i \(0.224552\pi\)
\(908\) −8.53590 −0.283274
\(909\) 0 0
\(910\) 5.07180 0.168128
\(911\) −5.07180 −0.168036 −0.0840181 0.996464i \(-0.526775\pi\)
−0.0840181 + 0.996464i \(0.526775\pi\)
\(912\) 0 0
\(913\) −15.4641 −0.511787
\(914\) 14.5359 0.480805
\(915\) 0 0
\(916\) 3.85641 0.127419
\(917\) −10.1436 −0.334971
\(918\) 0 0
\(919\) −11.6077 −0.382903 −0.191451 0.981502i \(-0.561319\pi\)
−0.191451 + 0.981502i \(0.561319\pi\)
\(920\) 12.0000 0.395628
\(921\) 0 0
\(922\) −21.2154 −0.698692
\(923\) −20.2872 −0.667761
\(924\) 0 0
\(925\) 8.92820 0.293558
\(926\) 48.4974 1.59372
\(927\) 0 0
\(928\) −18.0000 −0.590879
\(929\) 38.7846 1.27248 0.636241 0.771490i \(-0.280488\pi\)
0.636241 + 0.771490i \(0.280488\pi\)
\(930\) 0 0
\(931\) 4.39230 0.143952
\(932\) −12.0000 −0.393073
\(933\) 0 0
\(934\) 32.7846 1.07275
\(935\) 0 0
\(936\) 0 0
\(937\) 0.392305 0.0128160 0.00640802 0.999979i \(-0.497960\pi\)
0.00640802 + 0.999979i \(0.497960\pi\)
\(938\) −27.7128 −0.904855
\(939\) 0 0
\(940\) −6.92820 −0.225973
\(941\) −20.5359 −0.669451 −0.334726 0.942316i \(-0.608644\pi\)
−0.334726 + 0.942316i \(0.608644\pi\)
\(942\) 0 0
\(943\) 24.0000 0.781548
\(944\) 34.6410 1.12747
\(945\) 0 0
\(946\) −15.4641 −0.502781
\(947\) 5.07180 0.164811 0.0824056 0.996599i \(-0.473740\pi\)
0.0824056 + 0.996599i \(0.473740\pi\)
\(948\) 0 0
\(949\) −18.1436 −0.588966
\(950\) 2.53590 0.0822754
\(951\) 0 0
\(952\) 0 0
\(953\) 44.7846 1.45072 0.725358 0.688372i \(-0.241674\pi\)
0.725358 + 0.688372i \(0.241674\pi\)
\(954\) 0 0
\(955\) 5.07180 0.164119
\(956\) 12.0000 0.388108
\(957\) 0 0
\(958\) 20.7846 0.671520
\(959\) 36.0000 1.16250
\(960\) 0 0
\(961\) −22.4256 −0.723407
\(962\) 22.6410 0.729976
\(963\) 0 0
\(964\) 27.8564 0.897194
\(965\) 3.60770 0.116136
\(966\) 0 0
\(967\) −18.7846 −0.604072 −0.302036 0.953296i \(-0.597666\pi\)
−0.302036 + 0.953296i \(0.597666\pi\)
\(968\) 1.73205 0.0556702
\(969\) 0 0
\(970\) 17.3205 0.556128
\(971\) 1.85641 0.0595749 0.0297875 0.999556i \(-0.490517\pi\)
0.0297875 + 0.999556i \(0.490517\pi\)
\(972\) 0 0
\(973\) −16.7846 −0.538090
\(974\) −41.0718 −1.31603
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) 35.5692 1.13796 0.568980 0.822351i \(-0.307338\pi\)
0.568980 + 0.822351i \(0.307338\pi\)
\(978\) 0 0
\(979\) 12.9282 0.413187
\(980\) −3.00000 −0.0958315
\(981\) 0 0
\(982\) −29.5692 −0.943592
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 0 0
\(985\) 12.0000 0.382352
\(986\) 0 0
\(987\) 0 0
\(988\) 2.14359 0.0681968
\(989\) 61.8564 1.96692
\(990\) 0 0
\(991\) −48.7846 −1.54969 −0.774847 0.632149i \(-0.782173\pi\)
−0.774847 + 0.632149i \(0.782173\pi\)
\(992\) 15.2154 0.483089
\(993\) 0 0
\(994\) −48.0000 −1.52247
\(995\) 16.7846 0.532108
\(996\) 0 0
\(997\) 48.3923 1.53260 0.766300 0.642483i \(-0.222096\pi\)
0.766300 + 0.642483i \(0.222096\pi\)
\(998\) 22.1436 0.700943
\(999\) 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 495.2.a.c.1.1 2
3.2 odd 2 165.2.a.b.1.2 2
4.3 odd 2 7920.2.a.bz.1.1 2
5.2 odd 4 2475.2.c.n.199.2 4
5.3 odd 4 2475.2.c.n.199.3 4
5.4 even 2 2475.2.a.r.1.2 2
11.10 odd 2 5445.2.a.s.1.2 2
12.11 even 2 2640.2.a.x.1.1 2
15.2 even 4 825.2.c.c.199.3 4
15.8 even 4 825.2.c.c.199.2 4
15.14 odd 2 825.2.a.e.1.1 2
21.20 even 2 8085.2.a.bd.1.2 2
33.32 even 2 1815.2.a.i.1.1 2
165.164 even 2 9075.2.a.bh.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.b.1.2 2 3.2 odd 2
495.2.a.c.1.1 2 1.1 even 1 trivial
825.2.a.e.1.1 2 15.14 odd 2
825.2.c.c.199.2 4 15.8 even 4
825.2.c.c.199.3 4 15.2 even 4
1815.2.a.i.1.1 2 33.32 even 2
2475.2.a.r.1.2 2 5.4 even 2
2475.2.c.n.199.2 4 5.2 odd 4
2475.2.c.n.199.3 4 5.3 odd 4
2640.2.a.x.1.1 2 12.11 even 2
5445.2.a.s.1.2 2 11.10 odd 2
7920.2.a.bz.1.1 2 4.3 odd 2
8085.2.a.bd.1.2 2 21.20 even 2
9075.2.a.bh.1.2 2 165.164 even 2