Properties

Label 3822.2.a.n.1.1
Level $3822$
Weight $2$
Character 3822.1
Self dual yes
Analytic conductor $30.519$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3822.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.5188236525\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3822.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} +1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} -7.00000 q^{19} +1.00000 q^{20} +1.00000 q^{22} +3.00000 q^{23} -1.00000 q^{24} -4.00000 q^{25} +1.00000 q^{26} +1.00000 q^{27} -3.00000 q^{29} -1.00000 q^{30} -8.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} +7.00000 q^{37} +7.00000 q^{38} -1.00000 q^{39} -1.00000 q^{40} -8.00000 q^{41} +7.00000 q^{43} -1.00000 q^{44} +1.00000 q^{45} -3.00000 q^{46} -8.00000 q^{47} +1.00000 q^{48} +4.00000 q^{50} +1.00000 q^{51} -1.00000 q^{52} -10.0000 q^{53} -1.00000 q^{54} -1.00000 q^{55} -7.00000 q^{57} +3.00000 q^{58} -4.00000 q^{59} +1.00000 q^{60} -7.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} -1.00000 q^{65} +1.00000 q^{66} +2.00000 q^{67} +1.00000 q^{68} +3.00000 q^{69} +4.00000 q^{71} -1.00000 q^{72} +1.00000 q^{73} -7.00000 q^{74} -4.00000 q^{75} -7.00000 q^{76} +1.00000 q^{78} +2.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +8.00000 q^{82} +6.00000 q^{83} +1.00000 q^{85} -7.00000 q^{86} -3.00000 q^{87} +1.00000 q^{88} -14.0000 q^{89} -1.00000 q^{90} +3.00000 q^{92} -8.00000 q^{93} +8.00000 q^{94} -7.00000 q^{95} -1.00000 q^{96} +14.0000 q^{97} -1.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 1.00000 0.288675
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536 0.121268 0.992620i \(-0.461304\pi\)
0.121268 + 0.992620i \(0.461304\pi\)
\(18\) −1.00000 −0.235702
\(19\) −7.00000 −1.60591 −0.802955 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 3.00000 0.625543 0.312772 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) −1.00000 −0.204124
\(25\) −4.00000 −0.800000
\(26\) 1.00000 0.196116
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) −1.00000 −0.182574
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.00000 −0.174078
\(34\) −1.00000 −0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 7.00000 1.15079 0.575396 0.817875i \(-0.304848\pi\)
0.575396 + 0.817875i \(0.304848\pi\)
\(38\) 7.00000 1.13555
\(39\) −1.00000 −0.160128
\(40\) −1.00000 −0.158114
\(41\) −8.00000 −1.24939 −0.624695 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) 7.00000 1.06749 0.533745 0.845645i \(-0.320784\pi\)
0.533745 + 0.845645i \(0.320784\pi\)
\(44\) −1.00000 −0.150756
\(45\) 1.00000 0.149071
\(46\) −3.00000 −0.442326
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 4.00000 0.565685
\(51\) 1.00000 0.140028
\(52\) −1.00000 −0.138675
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) −1.00000 −0.136083
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) −7.00000 −0.927173
\(58\) 3.00000 0.393919
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 1.00000 0.129099
\(61\) −7.00000 −0.896258 −0.448129 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.00000 −0.124035
\(66\) 1.00000 0.123091
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 1.00000 0.121268
\(69\) 3.00000 0.361158
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) −1.00000 −0.117851
\(73\) 1.00000 0.117041 0.0585206 0.998286i \(-0.481362\pi\)
0.0585206 + 0.998286i \(0.481362\pi\)
\(74\) −7.00000 −0.813733
\(75\) −4.00000 −0.461880
\(76\) −7.00000 −0.802955
\(77\) 0 0
\(78\) 1.00000 0.113228
\(79\) 2.00000 0.225018 0.112509 0.993651i \(-0.464111\pi\)
0.112509 + 0.993651i \(0.464111\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 8.00000 0.883452
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) 1.00000 0.108465
\(86\) −7.00000 −0.754829
\(87\) −3.00000 −0.321634
\(88\) 1.00000 0.106600
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) 3.00000 0.312772
\(93\) −8.00000 −0.829561
\(94\) 8.00000 0.825137
\(95\) −7.00000 −0.718185
\(96\) −1.00000 −0.102062
\(97\) 14.0000 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) −4.00000 −0.400000
\(101\) −4.00000 −0.398015 −0.199007 0.979998i \(-0.563772\pi\)
−0.199007 + 0.979998i \(0.563772\pi\)
\(102\) −1.00000 −0.0990148
\(103\) 5.00000 0.492665 0.246332 0.969185i \(-0.420775\pi\)
0.246332 + 0.969185i \(0.420775\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) 10.0000 0.971286
\(107\) −18.0000 −1.74013 −0.870063 0.492941i \(-0.835922\pi\)
−0.870063 + 0.492941i \(0.835922\pi\)
\(108\) 1.00000 0.0962250
\(109\) −11.0000 −1.05361 −0.526804 0.849987i \(-0.676610\pi\)
−0.526804 + 0.849987i \(0.676610\pi\)
\(110\) 1.00000 0.0953463
\(111\) 7.00000 0.664411
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 7.00000 0.655610
\(115\) 3.00000 0.279751
\(116\) −3.00000 −0.278543
\(117\) −1.00000 −0.0924500
\(118\) 4.00000 0.368230
\(119\) 0 0
\(120\) −1.00000 −0.0912871
\(121\) −10.0000 −0.909091
\(122\) 7.00000 0.633750
\(123\) −8.00000 −0.721336
\(124\) −8.00000 −0.718421
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) 18.0000 1.59724 0.798621 0.601834i \(-0.205563\pi\)
0.798621 + 0.601834i \(0.205563\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 7.00000 0.616316
\(130\) 1.00000 0.0877058
\(131\) 5.00000 0.436852 0.218426 0.975854i \(-0.429908\pi\)
0.218426 + 0.975854i \(0.429908\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 0 0
\(134\) −2.00000 −0.172774
\(135\) 1.00000 0.0860663
\(136\) −1.00000 −0.0857493
\(137\) −19.0000 −1.62328 −0.811640 0.584158i \(-0.801425\pi\)
−0.811640 + 0.584158i \(0.801425\pi\)
\(138\) −3.00000 −0.255377
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) −4.00000 −0.335673
\(143\) 1.00000 0.0836242
\(144\) 1.00000 0.0833333
\(145\) −3.00000 −0.249136
\(146\) −1.00000 −0.0827606
\(147\) 0 0
\(148\) 7.00000 0.575396
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 4.00000 0.326599
\(151\) −9.00000 −0.732410 −0.366205 0.930534i \(-0.619343\pi\)
−0.366205 + 0.930534i \(0.619343\pi\)
\(152\) 7.00000 0.567775
\(153\) 1.00000 0.0808452
\(154\) 0 0
\(155\) −8.00000 −0.642575
\(156\) −1.00000 −0.0800641
\(157\) 13.0000 1.03751 0.518756 0.854922i \(-0.326395\pi\)
0.518756 + 0.854922i \(0.326395\pi\)
\(158\) −2.00000 −0.159111
\(159\) −10.0000 −0.793052
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 2.00000 0.156652 0.0783260 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) −8.00000 −0.624695
\(165\) −1.00000 −0.0778499
\(166\) −6.00000 −0.465690
\(167\) −15.0000 −1.16073 −0.580367 0.814355i \(-0.697091\pi\)
−0.580367 + 0.814355i \(0.697091\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −1.00000 −0.0766965
\(171\) −7.00000 −0.535303
\(172\) 7.00000 0.533745
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 3.00000 0.227429
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) −4.00000 −0.300658
\(178\) 14.0000 1.04934
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 1.00000 0.0745356
\(181\) 22.0000 1.63525 0.817624 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) 0 0
\(183\) −7.00000 −0.517455
\(184\) −3.00000 −0.221163
\(185\) 7.00000 0.514650
\(186\) 8.00000 0.586588
\(187\) −1.00000 −0.0731272
\(188\) −8.00000 −0.583460
\(189\) 0 0
\(190\) 7.00000 0.507833
\(191\) 19.0000 1.37479 0.687396 0.726283i \(-0.258754\pi\)
0.687396 + 0.726283i \(0.258754\pi\)
\(192\) 1.00000 0.0721688
\(193\) 16.0000 1.15171 0.575853 0.817554i \(-0.304670\pi\)
0.575853 + 0.817554i \(0.304670\pi\)
\(194\) −14.0000 −1.00514
\(195\) −1.00000 −0.0716115
\(196\) 0 0
\(197\) −12.0000 −0.854965 −0.427482 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) 1.00000 0.0710669
\(199\) −1.00000 −0.0708881 −0.0354441 0.999372i \(-0.511285\pi\)
−0.0354441 + 0.999372i \(0.511285\pi\)
\(200\) 4.00000 0.282843
\(201\) 2.00000 0.141069
\(202\) 4.00000 0.281439
\(203\) 0 0
\(204\) 1.00000 0.0700140
\(205\) −8.00000 −0.558744
\(206\) −5.00000 −0.348367
\(207\) 3.00000 0.208514
\(208\) −1.00000 −0.0693375
\(209\) 7.00000 0.484200
\(210\) 0 0
\(211\) −3.00000 −0.206529 −0.103264 0.994654i \(-0.532929\pi\)
−0.103264 + 0.994654i \(0.532929\pi\)
\(212\) −10.0000 −0.686803
\(213\) 4.00000 0.274075
\(214\) 18.0000 1.23045
\(215\) 7.00000 0.477396
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 11.0000 0.745014
\(219\) 1.00000 0.0675737
\(220\) −1.00000 −0.0674200
\(221\) −1.00000 −0.0672673
\(222\) −7.00000 −0.469809
\(223\) −14.0000 −0.937509 −0.468755 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) 0 0
\(225\) −4.00000 −0.266667
\(226\) −6.00000 −0.399114
\(227\) 22.0000 1.46019 0.730096 0.683345i \(-0.239475\pi\)
0.730096 + 0.683345i \(0.239475\pi\)
\(228\) −7.00000 −0.463586
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) −3.00000 −0.197814
\(231\) 0 0
\(232\) 3.00000 0.196960
\(233\) 12.0000 0.786146 0.393073 0.919507i \(-0.371412\pi\)
0.393073 + 0.919507i \(0.371412\pi\)
\(234\) 1.00000 0.0653720
\(235\) −8.00000 −0.521862
\(236\) −4.00000 −0.260378
\(237\) 2.00000 0.129914
\(238\) 0 0
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 1.00000 0.0645497
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 10.0000 0.642824
\(243\) 1.00000 0.0641500
\(244\) −7.00000 −0.448129
\(245\) 0 0
\(246\) 8.00000 0.510061
\(247\) 7.00000 0.445399
\(248\) 8.00000 0.508001
\(249\) 6.00000 0.380235
\(250\) 9.00000 0.569210
\(251\) −7.00000 −0.441836 −0.220918 0.975292i \(-0.570905\pi\)
−0.220918 + 0.975292i \(0.570905\pi\)
\(252\) 0 0
\(253\) −3.00000 −0.188608
\(254\) −18.0000 −1.12942
\(255\) 1.00000 0.0626224
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) −7.00000 −0.435801
\(259\) 0 0
\(260\) −1.00000 −0.0620174
\(261\) −3.00000 −0.185695
\(262\) −5.00000 −0.308901
\(263\) −4.00000 −0.246651 −0.123325 0.992366i \(-0.539356\pi\)
−0.123325 + 0.992366i \(0.539356\pi\)
\(264\) 1.00000 0.0615457
\(265\) −10.0000 −0.614295
\(266\) 0 0
\(267\) −14.0000 −0.856786
\(268\) 2.00000 0.122169
\(269\) 2.00000 0.121942 0.0609711 0.998140i \(-0.480580\pi\)
0.0609711 + 0.998140i \(0.480580\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −2.00000 −0.121491 −0.0607457 0.998153i \(-0.519348\pi\)
−0.0607457 + 0.998153i \(0.519348\pi\)
\(272\) 1.00000 0.0606339
\(273\) 0 0
\(274\) 19.0000 1.14783
\(275\) 4.00000 0.241209
\(276\) 3.00000 0.180579
\(277\) −14.0000 −0.841178 −0.420589 0.907251i \(-0.638177\pi\)
−0.420589 + 0.907251i \(0.638177\pi\)
\(278\) 0 0
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 8.00000 0.476393
\(283\) −32.0000 −1.90220 −0.951101 0.308879i \(-0.900046\pi\)
−0.951101 + 0.308879i \(0.900046\pi\)
\(284\) 4.00000 0.237356
\(285\) −7.00000 −0.414644
\(286\) −1.00000 −0.0591312
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −16.0000 −0.941176
\(290\) 3.00000 0.176166
\(291\) 14.0000 0.820695
\(292\) 1.00000 0.0585206
\(293\) −14.0000 −0.817889 −0.408944 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) 0 0
\(295\) −4.00000 −0.232889
\(296\) −7.00000 −0.406867
\(297\) −1.00000 −0.0580259
\(298\) 0 0
\(299\) −3.00000 −0.173494
\(300\) −4.00000 −0.230940
\(301\) 0 0
\(302\) 9.00000 0.517892
\(303\) −4.00000 −0.229794
\(304\) −7.00000 −0.401478
\(305\) −7.00000 −0.400819
\(306\) −1.00000 −0.0571662
\(307\) 32.0000 1.82634 0.913168 0.407583i \(-0.133628\pi\)
0.913168 + 0.407583i \(0.133628\pi\)
\(308\) 0 0
\(309\) 5.00000 0.284440
\(310\) 8.00000 0.454369
\(311\) −30.0000 −1.70114 −0.850572 0.525859i \(-0.823744\pi\)
−0.850572 + 0.525859i \(0.823744\pi\)
\(312\) 1.00000 0.0566139
\(313\) 24.0000 1.35656 0.678280 0.734803i \(-0.262726\pi\)
0.678280 + 0.734803i \(0.262726\pi\)
\(314\) −13.0000 −0.733632
\(315\) 0 0
\(316\) 2.00000 0.112509
\(317\) 12.0000 0.673987 0.336994 0.941507i \(-0.390590\pi\)
0.336994 + 0.941507i \(0.390590\pi\)
\(318\) 10.0000 0.560772
\(319\) 3.00000 0.167968
\(320\) 1.00000 0.0559017
\(321\) −18.0000 −1.00466
\(322\) 0 0
\(323\) −7.00000 −0.389490
\(324\) 1.00000 0.0555556
\(325\) 4.00000 0.221880
\(326\) −2.00000 −0.110770
\(327\) −11.0000 −0.608301
\(328\) 8.00000 0.441726
\(329\) 0 0
\(330\) 1.00000 0.0550482
\(331\) −10.0000 −0.549650 −0.274825 0.961494i \(-0.588620\pi\)
−0.274825 + 0.961494i \(0.588620\pi\)
\(332\) 6.00000 0.329293
\(333\) 7.00000 0.383598
\(334\) 15.0000 0.820763
\(335\) 2.00000 0.109272
\(336\) 0 0
\(337\) −21.0000 −1.14394 −0.571971 0.820274i \(-0.693821\pi\)
−0.571971 + 0.820274i \(0.693821\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 6.00000 0.325875
\(340\) 1.00000 0.0542326
\(341\) 8.00000 0.433224
\(342\) 7.00000 0.378517
\(343\) 0 0
\(344\) −7.00000 −0.377415
\(345\) 3.00000 0.161515
\(346\) 14.0000 0.752645
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) −3.00000 −0.160817
\(349\) 18.0000 0.963518 0.481759 0.876304i \(-0.339998\pi\)
0.481759 + 0.876304i \(0.339998\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 1.00000 0.0533002
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 4.00000 0.212598
\(355\) 4.00000 0.212298
\(356\) −14.0000 −0.741999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 30.0000 1.57895
\(362\) −22.0000 −1.15629
\(363\) −10.0000 −0.524864
\(364\) 0 0
\(365\) 1.00000 0.0523424
\(366\) 7.00000 0.365896
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 3.00000 0.156386
\(369\) −8.00000 −0.416463
\(370\) −7.00000 −0.363913
\(371\) 0 0
\(372\) −8.00000 −0.414781
\(373\) −12.0000 −0.621336 −0.310668 0.950518i \(-0.600553\pi\)
−0.310668 + 0.950518i \(0.600553\pi\)
\(374\) 1.00000 0.0517088
\(375\) −9.00000 −0.464758
\(376\) 8.00000 0.412568
\(377\) 3.00000 0.154508
\(378\) 0 0
\(379\) 12.0000 0.616399 0.308199 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) −7.00000 −0.359092
\(381\) 18.0000 0.922168
\(382\) −19.0000 −0.972125
\(383\) 9.00000 0.459879 0.229939 0.973205i \(-0.426147\pi\)
0.229939 + 0.973205i \(0.426147\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −16.0000 −0.814379
\(387\) 7.00000 0.355830
\(388\) 14.0000 0.710742
\(389\) −2.00000 −0.101404 −0.0507020 0.998714i \(-0.516146\pi\)
−0.0507020 + 0.998714i \(0.516146\pi\)
\(390\) 1.00000 0.0506370
\(391\) 3.00000 0.151717
\(392\) 0 0
\(393\) 5.00000 0.252217
\(394\) 12.0000 0.604551
\(395\) 2.00000 0.100631
\(396\) −1.00000 −0.0502519
\(397\) −18.0000 −0.903394 −0.451697 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) 1.00000 0.0501255
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) −2.00000 −0.0997509
\(403\) 8.00000 0.398508
\(404\) −4.00000 −0.199007
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −7.00000 −0.346977
\(408\) −1.00000 −0.0495074
\(409\) −5.00000 −0.247234 −0.123617 0.992330i \(-0.539449\pi\)
−0.123617 + 0.992330i \(0.539449\pi\)
\(410\) 8.00000 0.395092
\(411\) −19.0000 −0.937201
\(412\) 5.00000 0.246332
\(413\) 0 0
\(414\) −3.00000 −0.147442
\(415\) 6.00000 0.294528
\(416\) 1.00000 0.0490290
\(417\) 0 0
\(418\) −7.00000 −0.342381
\(419\) −1.00000 −0.0488532 −0.0244266 0.999702i \(-0.507776\pi\)
−0.0244266 + 0.999702i \(0.507776\pi\)
\(420\) 0 0
\(421\) 34.0000 1.65706 0.828529 0.559946i \(-0.189178\pi\)
0.828529 + 0.559946i \(0.189178\pi\)
\(422\) 3.00000 0.146038
\(423\) −8.00000 −0.388973
\(424\) 10.0000 0.485643
\(425\) −4.00000 −0.194029
\(426\) −4.00000 −0.193801
\(427\) 0 0
\(428\) −18.0000 −0.870063
\(429\) 1.00000 0.0482805
\(430\) −7.00000 −0.337570
\(431\) −38.0000 −1.83040 −0.915198 0.403005i \(-0.867966\pi\)
−0.915198 + 0.403005i \(0.867966\pi\)
\(432\) 1.00000 0.0481125
\(433\) −12.0000 −0.576683 −0.288342 0.957528i \(-0.593104\pi\)
−0.288342 + 0.957528i \(0.593104\pi\)
\(434\) 0 0
\(435\) −3.00000 −0.143839
\(436\) −11.0000 −0.526804
\(437\) −21.0000 −1.00457
\(438\) −1.00000 −0.0477818
\(439\) 17.0000 0.811366 0.405683 0.914014i \(-0.367034\pi\)
0.405683 + 0.914014i \(0.367034\pi\)
\(440\) 1.00000 0.0476731
\(441\) 0 0
\(442\) 1.00000 0.0475651
\(443\) −10.0000 −0.475114 −0.237557 0.971374i \(-0.576347\pi\)
−0.237557 + 0.971374i \(0.576347\pi\)
\(444\) 7.00000 0.332205
\(445\) −14.0000 −0.663664
\(446\) 14.0000 0.662919
\(447\) 0 0
\(448\) 0 0
\(449\) −39.0000 −1.84052 −0.920262 0.391303i \(-0.872024\pi\)
−0.920262 + 0.391303i \(0.872024\pi\)
\(450\) 4.00000 0.188562
\(451\) 8.00000 0.376705
\(452\) 6.00000 0.282216
\(453\) −9.00000 −0.422857
\(454\) −22.0000 −1.03251
\(455\) 0 0
\(456\) 7.00000 0.327805
\(457\) −12.0000 −0.561336 −0.280668 0.959805i \(-0.590556\pi\)
−0.280668 + 0.959805i \(0.590556\pi\)
\(458\) 10.0000 0.467269
\(459\) 1.00000 0.0466760
\(460\) 3.00000 0.139876
\(461\) −21.0000 −0.978068 −0.489034 0.872265i \(-0.662651\pi\)
−0.489034 + 0.872265i \(0.662651\pi\)
\(462\) 0 0
\(463\) −17.0000 −0.790057 −0.395029 0.918669i \(-0.629265\pi\)
−0.395029 + 0.918669i \(0.629265\pi\)
\(464\) −3.00000 −0.139272
\(465\) −8.00000 −0.370991
\(466\) −12.0000 −0.555889
\(467\) 25.0000 1.15686 0.578431 0.815731i \(-0.303665\pi\)
0.578431 + 0.815731i \(0.303665\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 0 0
\(470\) 8.00000 0.369012
\(471\) 13.0000 0.599008
\(472\) 4.00000 0.184115
\(473\) −7.00000 −0.321860
\(474\) −2.00000 −0.0918630
\(475\) 28.0000 1.28473
\(476\) 0 0
\(477\) −10.0000 −0.457869
\(478\) −24.0000 −1.09773
\(479\) −5.00000 −0.228456 −0.114228 0.993455i \(-0.536439\pi\)
−0.114228 + 0.993455i \(0.536439\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −7.00000 −0.319173
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) −10.0000 −0.454545
\(485\) 14.0000 0.635707
\(486\) −1.00000 −0.0453609
\(487\) 16.0000 0.725029 0.362515 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) 7.00000 0.316875
\(489\) 2.00000 0.0904431
\(490\) 0 0
\(491\) −30.0000 −1.35388 −0.676941 0.736038i \(-0.736695\pi\)
−0.676941 + 0.736038i \(0.736695\pi\)
\(492\) −8.00000 −0.360668
\(493\) −3.00000 −0.135113
\(494\) −7.00000 −0.314945
\(495\) −1.00000 −0.0449467
\(496\) −8.00000 −0.359211
\(497\) 0 0
\(498\) −6.00000 −0.268866
\(499\) 12.0000 0.537194 0.268597 0.963253i \(-0.413440\pi\)
0.268597 + 0.963253i \(0.413440\pi\)
\(500\) −9.00000 −0.402492
\(501\) −15.0000 −0.670151
\(502\) 7.00000 0.312425
\(503\) −34.0000 −1.51599 −0.757993 0.652263i \(-0.773820\pi\)
−0.757993 + 0.652263i \(0.773820\pi\)
\(504\) 0 0
\(505\) −4.00000 −0.177998
\(506\) 3.00000 0.133366
\(507\) 1.00000 0.0444116
\(508\) 18.0000 0.798621
\(509\) 3.00000 0.132973 0.0664863 0.997787i \(-0.478821\pi\)
0.0664863 + 0.997787i \(0.478821\pi\)
\(510\) −1.00000 −0.0442807
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −7.00000 −0.309058
\(514\) 18.0000 0.793946
\(515\) 5.00000 0.220326
\(516\) 7.00000 0.308158
\(517\) 8.00000 0.351840
\(518\) 0 0
\(519\) −14.0000 −0.614532
\(520\) 1.00000 0.0438529
\(521\) 35.0000 1.53338 0.766689 0.642019i \(-0.221903\pi\)
0.766689 + 0.642019i \(0.221903\pi\)
\(522\) 3.00000 0.131306
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 5.00000 0.218426
\(525\) 0 0
\(526\) 4.00000 0.174408
\(527\) −8.00000 −0.348485
\(528\) −1.00000 −0.0435194
\(529\) −14.0000 −0.608696
\(530\) 10.0000 0.434372
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) 8.00000 0.346518
\(534\) 14.0000 0.605839
\(535\) −18.0000 −0.778208
\(536\) −2.00000 −0.0863868
\(537\) 12.0000 0.517838
\(538\) −2.00000 −0.0862261
\(539\) 0 0
\(540\) 1.00000 0.0430331
\(541\) −43.0000 −1.84871 −0.924357 0.381528i \(-0.875398\pi\)
−0.924357 + 0.381528i \(0.875398\pi\)
\(542\) 2.00000 0.0859074
\(543\) 22.0000 0.944110
\(544\) −1.00000 −0.0428746
\(545\) −11.0000 −0.471188
\(546\) 0 0
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) −19.0000 −0.811640
\(549\) −7.00000 −0.298753
\(550\) −4.00000 −0.170561
\(551\) 21.0000 0.894630
\(552\) −3.00000 −0.127688
\(553\) 0 0
\(554\) 14.0000 0.594803
\(555\) 7.00000 0.297133
\(556\) 0 0
\(557\) −24.0000 −1.01691 −0.508456 0.861088i \(-0.669784\pi\)
−0.508456 + 0.861088i \(0.669784\pi\)
\(558\) 8.00000 0.338667
\(559\) −7.00000 −0.296068
\(560\) 0 0
\(561\) −1.00000 −0.0422200
\(562\) −10.0000 −0.421825
\(563\) −19.0000 −0.800755 −0.400377 0.916350i \(-0.631121\pi\)
−0.400377 + 0.916350i \(0.631121\pi\)
\(564\) −8.00000 −0.336861
\(565\) 6.00000 0.252422
\(566\) 32.0000 1.34506
\(567\) 0 0
\(568\) −4.00000 −0.167836
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 7.00000 0.293198
\(571\) 16.0000 0.669579 0.334790 0.942293i \(-0.391335\pi\)
0.334790 + 0.942293i \(0.391335\pi\)
\(572\) 1.00000 0.0418121
\(573\) 19.0000 0.793736
\(574\) 0 0
\(575\) −12.0000 −0.500435
\(576\) 1.00000 0.0416667
\(577\) −42.0000 −1.74848 −0.874241 0.485491i \(-0.838641\pi\)
−0.874241 + 0.485491i \(0.838641\pi\)
\(578\) 16.0000 0.665512
\(579\) 16.0000 0.664937
\(580\) −3.00000 −0.124568
\(581\) 0 0
\(582\) −14.0000 −0.580319
\(583\) 10.0000 0.414158
\(584\) −1.00000 −0.0413803
\(585\) −1.00000 −0.0413449
\(586\) 14.0000 0.578335
\(587\) 38.0000 1.56843 0.784214 0.620491i \(-0.213066\pi\)
0.784214 + 0.620491i \(0.213066\pi\)
\(588\) 0 0
\(589\) 56.0000 2.30744
\(590\) 4.00000 0.164677
\(591\) −12.0000 −0.493614
\(592\) 7.00000 0.287698
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 1.00000 0.0410305
\(595\) 0 0
\(596\) 0 0
\(597\) −1.00000 −0.0409273
\(598\) 3.00000 0.122679
\(599\) −3.00000 −0.122577 −0.0612883 0.998120i \(-0.519521\pi\)
−0.0612883 + 0.998120i \(0.519521\pi\)
\(600\) 4.00000 0.163299
\(601\) 28.0000 1.14214 0.571072 0.820900i \(-0.306528\pi\)
0.571072 + 0.820900i \(0.306528\pi\)
\(602\) 0 0
\(603\) 2.00000 0.0814463
\(604\) −9.00000 −0.366205
\(605\) −10.0000 −0.406558
\(606\) 4.00000 0.162489
\(607\) 1.00000 0.0405887 0.0202944 0.999794i \(-0.493540\pi\)
0.0202944 + 0.999794i \(0.493540\pi\)
\(608\) 7.00000 0.283887
\(609\) 0 0
\(610\) 7.00000 0.283422
\(611\) 8.00000 0.323645
\(612\) 1.00000 0.0404226
\(613\) 37.0000 1.49442 0.747208 0.664590i \(-0.231394\pi\)
0.747208 + 0.664590i \(0.231394\pi\)
\(614\) −32.0000 −1.29141
\(615\) −8.00000 −0.322591
\(616\) 0 0
\(617\) −33.0000 −1.32853 −0.664265 0.747497i \(-0.731255\pi\)
−0.664265 + 0.747497i \(0.731255\pi\)
\(618\) −5.00000 −0.201129
\(619\) 1.00000 0.0401934 0.0200967 0.999798i \(-0.493603\pi\)
0.0200967 + 0.999798i \(0.493603\pi\)
\(620\) −8.00000 −0.321288
\(621\) 3.00000 0.120386
\(622\) 30.0000 1.20289
\(623\) 0 0
\(624\) −1.00000 −0.0400320
\(625\) 11.0000 0.440000
\(626\) −24.0000 −0.959233
\(627\) 7.00000 0.279553
\(628\) 13.0000 0.518756
\(629\) 7.00000 0.279108
\(630\) 0 0
\(631\) 27.0000 1.07485 0.537427 0.843311i \(-0.319397\pi\)
0.537427 + 0.843311i \(0.319397\pi\)
\(632\) −2.00000 −0.0795557
\(633\) −3.00000 −0.119239
\(634\) −12.0000 −0.476581
\(635\) 18.0000 0.714308
\(636\) −10.0000 −0.396526
\(637\) 0 0
\(638\) −3.00000 −0.118771
\(639\) 4.00000 0.158238
\(640\) −1.00000 −0.0395285
\(641\) 50.0000 1.97488 0.987441 0.157991i \(-0.0505015\pi\)
0.987441 + 0.157991i \(0.0505015\pi\)
\(642\) 18.0000 0.710403
\(643\) −17.0000 −0.670415 −0.335207 0.942144i \(-0.608806\pi\)
−0.335207 + 0.942144i \(0.608806\pi\)
\(644\) 0 0
\(645\) 7.00000 0.275625
\(646\) 7.00000 0.275411
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 4.00000 0.157014
\(650\) −4.00000 −0.156893
\(651\) 0 0
\(652\) 2.00000 0.0783260
\(653\) 35.0000 1.36966 0.684828 0.728705i \(-0.259877\pi\)
0.684828 + 0.728705i \(0.259877\pi\)
\(654\) 11.0000 0.430134
\(655\) 5.00000 0.195366
\(656\) −8.00000 −0.312348
\(657\) 1.00000 0.0390137
\(658\) 0 0
\(659\) 6.00000 0.233727 0.116863 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) −1.00000 −0.0389249
\(661\) −22.0000 −0.855701 −0.427850 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) 10.0000 0.388661
\(663\) −1.00000 −0.0388368
\(664\) −6.00000 −0.232845
\(665\) 0 0
\(666\) −7.00000 −0.271244
\(667\) −9.00000 −0.348481
\(668\) −15.0000 −0.580367
\(669\) −14.0000 −0.541271
\(670\) −2.00000 −0.0772667
\(671\) 7.00000 0.270232
\(672\) 0 0
\(673\) 7.00000 0.269830 0.134915 0.990857i \(-0.456924\pi\)
0.134915 + 0.990857i \(0.456924\pi\)
\(674\) 21.0000 0.808890
\(675\) −4.00000 −0.153960
\(676\) 1.00000 0.0384615
\(677\) 22.0000 0.845529 0.422764 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) −6.00000 −0.230429
\(679\) 0 0
\(680\) −1.00000 −0.0383482
\(681\) 22.0000 0.843042
\(682\) −8.00000 −0.306336
\(683\) −19.0000 −0.727015 −0.363507 0.931591i \(-0.618421\pi\)
−0.363507 + 0.931591i \(0.618421\pi\)
\(684\) −7.00000 −0.267652
\(685\) −19.0000 −0.725953
\(686\) 0 0
\(687\) −10.0000 −0.381524
\(688\) 7.00000 0.266872
\(689\) 10.0000 0.380970
\(690\) −3.00000 −0.114208
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) −14.0000 −0.532200
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) 3.00000 0.113715
\(697\) −8.00000 −0.303022
\(698\) −18.0000 −0.681310
\(699\) 12.0000 0.453882
\(700\) 0 0
\(701\) 10.0000 0.377695 0.188847 0.982006i \(-0.439525\pi\)
0.188847 + 0.982006i \(0.439525\pi\)
\(702\) 1.00000 0.0377426
\(703\) −49.0000 −1.84807
\(704\) −1.00000 −0.0376889
\(705\) −8.00000 −0.301297
\(706\) 14.0000 0.526897
\(707\) 0 0
\(708\) −4.00000 −0.150329
\(709\) −6.00000 −0.225335 −0.112667 0.993633i \(-0.535939\pi\)
−0.112667 + 0.993633i \(0.535939\pi\)
\(710\) −4.00000 −0.150117
\(711\) 2.00000 0.0750059
\(712\) 14.0000 0.524672
\(713\) −24.0000 −0.898807
\(714\) 0 0
\(715\) 1.00000 0.0373979
\(716\) 12.0000 0.448461
\(717\) 24.0000 0.896296
\(718\) 0 0
\(719\) 42.0000 1.56634 0.783168 0.621810i \(-0.213603\pi\)
0.783168 + 0.621810i \(0.213603\pi\)
\(720\) 1.00000 0.0372678
\(721\) 0 0
\(722\) −30.0000 −1.11648
\(723\) −10.0000 −0.371904
\(724\) 22.0000 0.817624
\(725\) 12.0000 0.445669
\(726\) 10.0000 0.371135
\(727\) −23.0000 −0.853023 −0.426511 0.904482i \(-0.640258\pi\)
−0.426511 + 0.904482i \(0.640258\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −1.00000 −0.0370117
\(731\) 7.00000 0.258904
\(732\) −7.00000 −0.258727
\(733\) 32.0000 1.18195 0.590973 0.806691i \(-0.298744\pi\)
0.590973 + 0.806691i \(0.298744\pi\)
\(734\) −8.00000 −0.295285
\(735\) 0 0
\(736\) −3.00000 −0.110581
\(737\) −2.00000 −0.0736709
\(738\) 8.00000 0.294484
\(739\) 22.0000 0.809283 0.404642 0.914475i \(-0.367396\pi\)
0.404642 + 0.914475i \(0.367396\pi\)
\(740\) 7.00000 0.257325
\(741\) 7.00000 0.257151
\(742\) 0 0
\(743\) 6.00000 0.220119 0.110059 0.993925i \(-0.464896\pi\)
0.110059 + 0.993925i \(0.464896\pi\)
\(744\) 8.00000 0.293294
\(745\) 0 0
\(746\) 12.0000 0.439351
\(747\) 6.00000 0.219529
\(748\) −1.00000 −0.0365636
\(749\) 0 0
\(750\) 9.00000 0.328634
\(751\) −28.0000 −1.02173 −0.510867 0.859660i \(-0.670676\pi\)
−0.510867 + 0.859660i \(0.670676\pi\)
\(752\) −8.00000 −0.291730
\(753\) −7.00000 −0.255094
\(754\) −3.00000 −0.109254
\(755\) −9.00000 −0.327544
\(756\) 0 0
\(757\) 18.0000 0.654221 0.327111 0.944986i \(-0.393925\pi\)
0.327111 + 0.944986i \(0.393925\pi\)
\(758\) −12.0000 −0.435860
\(759\) −3.00000 −0.108893
\(760\) 7.00000 0.253917
\(761\) −16.0000 −0.580000 −0.290000 0.957027i \(-0.593655\pi\)
−0.290000 + 0.957027i \(0.593655\pi\)
\(762\) −18.0000 −0.652071
\(763\) 0 0
\(764\) 19.0000 0.687396
\(765\) 1.00000 0.0361551
\(766\) −9.00000 −0.325183
\(767\) 4.00000 0.144432
\(768\) 1.00000 0.0360844
\(769\) −5.00000 −0.180305 −0.0901523 0.995928i \(-0.528735\pi\)
−0.0901523 + 0.995928i \(0.528735\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) 16.0000 0.575853
\(773\) 27.0000 0.971123 0.485561 0.874203i \(-0.338615\pi\)
0.485561 + 0.874203i \(0.338615\pi\)
\(774\) −7.00000 −0.251610
\(775\) 32.0000 1.14947
\(776\) −14.0000 −0.502571
\(777\) 0 0
\(778\) 2.00000 0.0717035
\(779\) 56.0000 2.00641
\(780\) −1.00000 −0.0358057
\(781\) −4.00000 −0.143131
\(782\) −3.00000 −0.107280
\(783\) −3.00000 −0.107211
\(784\) 0 0
\(785\) 13.0000 0.463990
\(786\) −5.00000 −0.178344
\(787\) −55.0000 −1.96054 −0.980269 0.197667i \(-0.936663\pi\)
−0.980269 + 0.197667i \(0.936663\pi\)
\(788\) −12.0000 −0.427482
\(789\) −4.00000 −0.142404
\(790\) −2.00000 −0.0711568
\(791\) 0 0
\(792\) 1.00000 0.0355335
\(793\) 7.00000 0.248577
\(794\) 18.0000 0.638796
\(795\) −10.0000 −0.354663
\(796\) −1.00000 −0.0354441
\(797\) −16.0000 −0.566749 −0.283375 0.959009i \(-0.591454\pi\)
−0.283375 + 0.959009i \(0.591454\pi\)
\(798\) 0 0
\(799\) −8.00000 −0.283020
\(800\) 4.00000 0.141421
\(801\) −14.0000 −0.494666
\(802\) −18.0000 −0.635602
\(803\) −1.00000 −0.0352892
\(804\) 2.00000 0.0705346
\(805\) 0 0
\(806\) −8.00000 −0.281788
\(807\) 2.00000 0.0704033
\(808\) 4.00000 0.140720
\(809\) −10.0000 −0.351581 −0.175791 0.984428i \(-0.556248\pi\)
−0.175791 + 0.984428i \(0.556248\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 45.0000 1.58016 0.790082 0.613001i \(-0.210038\pi\)
0.790082 + 0.613001i \(0.210038\pi\)
\(812\) 0 0
\(813\) −2.00000 −0.0701431
\(814\) 7.00000 0.245350
\(815\) 2.00000 0.0700569
\(816\) 1.00000 0.0350070
\(817\) −49.0000 −1.71429
\(818\) 5.00000 0.174821
\(819\) 0 0
\(820\) −8.00000 −0.279372
\(821\) −24.0000 −0.837606 −0.418803 0.908077i \(-0.637550\pi\)
−0.418803 + 0.908077i \(0.637550\pi\)
\(822\) 19.0000 0.662701
\(823\) 20.0000 0.697156 0.348578 0.937280i \(-0.386665\pi\)
0.348578 + 0.937280i \(0.386665\pi\)
\(824\) −5.00000 −0.174183
\(825\) 4.00000 0.139262
\(826\) 0 0
\(827\) −41.0000 −1.42571 −0.712855 0.701312i \(-0.752598\pi\)
−0.712855 + 0.701312i \(0.752598\pi\)
\(828\) 3.00000 0.104257
\(829\) −7.00000 −0.243120 −0.121560 0.992584i \(-0.538790\pi\)
−0.121560 + 0.992584i \(0.538790\pi\)
\(830\) −6.00000 −0.208263
\(831\) −14.0000 −0.485655
\(832\) −1.00000 −0.0346688
\(833\) 0 0
\(834\) 0 0
\(835\) −15.0000 −0.519096
\(836\) 7.00000 0.242100
\(837\) −8.00000 −0.276520
\(838\) 1.00000 0.0345444
\(839\) 4.00000 0.138095 0.0690477 0.997613i \(-0.478004\pi\)
0.0690477 + 0.997613i \(0.478004\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) −34.0000 −1.17172
\(843\) 10.0000 0.344418
\(844\) −3.00000 −0.103264
\(845\) 1.00000 0.0344010
\(846\) 8.00000 0.275046
\(847\) 0 0
\(848\) −10.0000 −0.343401
\(849\) −32.0000 −1.09824
\(850\) 4.00000 0.137199
\(851\) 21.0000 0.719871
\(852\) 4.00000 0.137038
\(853\) 44.0000 1.50653 0.753266 0.657716i \(-0.228477\pi\)
0.753266 + 0.657716i \(0.228477\pi\)
\(854\) 0 0
\(855\) −7.00000 −0.239395
\(856\) 18.0000 0.615227
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) −1.00000 −0.0341394
\(859\) 44.0000 1.50126 0.750630 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) 7.00000 0.238698
\(861\) 0 0
\(862\) 38.0000 1.29429
\(863\) 46.0000 1.56586 0.782929 0.622111i \(-0.213725\pi\)
0.782929 + 0.622111i \(0.213725\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −14.0000 −0.476014
\(866\) 12.0000 0.407777
\(867\) −16.0000 −0.543388
\(868\) 0 0
\(869\) −2.00000 −0.0678454
\(870\) 3.00000 0.101710
\(871\) −2.00000 −0.0677674
\(872\) 11.0000 0.372507
\(873\) 14.0000 0.473828
\(874\) 21.0000 0.710336
\(875\) 0 0
\(876\) 1.00000 0.0337869
\(877\) 34.0000 1.14810 0.574049 0.818821i \(-0.305372\pi\)
0.574049 + 0.818821i \(0.305372\pi\)
\(878\) −17.0000 −0.573722
\(879\) −14.0000 −0.472208
\(880\) −1.00000 −0.0337100
\(881\) 49.0000 1.65085 0.825426 0.564510i \(-0.190935\pi\)
0.825426 + 0.564510i \(0.190935\pi\)
\(882\) 0 0
\(883\) 31.0000 1.04323 0.521617 0.853180i \(-0.325329\pi\)
0.521617 + 0.853180i \(0.325329\pi\)
\(884\) −1.00000 −0.0336336
\(885\) −4.00000 −0.134459
\(886\) 10.0000 0.335957
\(887\) 36.0000 1.20876 0.604381 0.796696i \(-0.293421\pi\)
0.604381 + 0.796696i \(0.293421\pi\)
\(888\) −7.00000 −0.234905
\(889\) 0 0
\(890\) 14.0000 0.469281
\(891\) −1.00000 −0.0335013
\(892\) −14.0000 −0.468755
\(893\) 56.0000 1.87397
\(894\) 0 0
\(895\) 12.0000 0.401116
\(896\) 0 0
\(897\) −3.00000 −0.100167
\(898\) 39.0000 1.30145
\(899\) 24.0000 0.800445
\(900\) −4.00000 −0.133333
\(901\) −10.0000 −0.333148
\(902\) −8.00000 −0.266371
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) 22.0000 0.731305
\(906\) 9.00000 0.299005
\(907\) 4.00000 0.132818 0.0664089 0.997792i \(-0.478846\pi\)
0.0664089 + 0.997792i \(0.478846\pi\)
\(908\) 22.0000 0.730096
\(909\) −4.00000 −0.132672
\(910\) 0 0
\(911\) 13.0000 0.430709 0.215355 0.976536i \(-0.430909\pi\)
0.215355 + 0.976536i \(0.430909\pi\)
\(912\) −7.00000 −0.231793
\(913\) −6.00000 −0.198571
\(914\) 12.0000 0.396925
\(915\) −7.00000 −0.231413
\(916\) −10.0000 −0.330409
\(917\) 0 0
\(918\) −1.00000 −0.0330049
\(919\) 38.0000 1.25350 0.626752 0.779219i \(-0.284384\pi\)
0.626752 + 0.779219i \(0.284384\pi\)
\(920\) −3.00000 −0.0989071
\(921\) 32.0000 1.05444
\(922\) 21.0000 0.691598
\(923\) −4.00000 −0.131662
\(924\) 0 0
\(925\) −28.0000 −0.920634
\(926\) 17.0000 0.558655
\(927\) 5.00000 0.164222
\(928\) 3.00000 0.0984798
\(929\) −28.0000 −0.918650 −0.459325 0.888268i \(-0.651909\pi\)
−0.459325 + 0.888268i \(0.651909\pi\)
\(930\) 8.00000 0.262330
\(931\) 0 0
\(932\) 12.0000 0.393073
\(933\) −30.0000 −0.982156
\(934\) −25.0000 −0.818025
\(935\) −1.00000 −0.0327035
\(936\) 1.00000 0.0326860
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 0 0
\(939\) 24.0000 0.783210
\(940\) −8.00000 −0.260931
\(941\) 46.0000 1.49956 0.749779 0.661689i \(-0.230160\pi\)
0.749779 + 0.661689i \(0.230160\pi\)
\(942\) −13.0000 −0.423563
\(943\) −24.0000 −0.781548
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 7.00000 0.227590
\(947\) −13.0000 −0.422443 −0.211222 0.977438i \(-0.567744\pi\)
−0.211222 + 0.977438i \(0.567744\pi\)
\(948\) 2.00000 0.0649570
\(949\) −1.00000 −0.0324614
\(950\) −28.0000 −0.908440
\(951\) 12.0000 0.389127
\(952\) 0 0
\(953\) 36.0000 1.16615 0.583077 0.812417i \(-0.301849\pi\)
0.583077 + 0.812417i \(0.301849\pi\)
\(954\) 10.0000 0.323762
\(955\) 19.0000 0.614826
\(956\) 24.0000 0.776215
\(957\) 3.00000 0.0969762
\(958\) 5.00000 0.161543
\(959\) 0 0
\(960\) 1.00000 0.0322749
\(961\) 33.0000 1.06452
\(962\) 7.00000 0.225689
\(963\) −18.0000 −0.580042
\(964\) −10.0000 −0.322078
\(965\) 16.0000 0.515058
\(966\) 0 0
\(967\) 13.0000 0.418052 0.209026 0.977910i \(-0.432971\pi\)
0.209026 + 0.977910i \(0.432971\pi\)
\(968\) 10.0000 0.321412
\(969\) −7.00000 −0.224872
\(970\) −14.0000 −0.449513
\(971\) 40.0000 1.28366 0.641831 0.766846i \(-0.278175\pi\)
0.641831 + 0.766846i \(0.278175\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −16.0000 −0.512673
\(975\) 4.00000 0.128103
\(976\) −7.00000 −0.224065
\(977\) −45.0000 −1.43968 −0.719839 0.694141i \(-0.755784\pi\)
−0.719839 + 0.694141i \(0.755784\pi\)
\(978\) −2.00000 −0.0639529
\(979\) 14.0000 0.447442
\(980\) 0 0
\(981\) −11.0000 −0.351203
\(982\) 30.0000 0.957338
\(983\) 3.00000 0.0956851 0.0478426 0.998855i \(-0.484765\pi\)
0.0478426 + 0.998855i \(0.484765\pi\)
\(984\) 8.00000 0.255031
\(985\) −12.0000 −0.382352
\(986\) 3.00000 0.0955395
\(987\) 0 0
\(988\) 7.00000 0.222700
\(989\) 21.0000 0.667761
\(990\) 1.00000 0.0317821
\(991\) 42.0000 1.33417 0.667087 0.744980i \(-0.267541\pi\)
0.667087 + 0.744980i \(0.267541\pi\)
\(992\) 8.00000 0.254000
\(993\) −10.0000 −0.317340
\(994\) 0 0
\(995\) −1.00000 −0.0317021
\(996\) 6.00000 0.190117
\(997\) 14.0000 0.443384 0.221692 0.975117i \(-0.428842\pi\)
0.221692 + 0.975117i \(0.428842\pi\)
\(998\) −12.0000 −0.379853
\(999\) 7.00000 0.221470
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 3822.2.a.n.1.1 1
7.6 odd 2 546.2.a.a.1.1 1
21.20 even 2 1638.2.a.r.1.1 1
28.27 even 2 4368.2.a.s.1.1 1
91.90 odd 2 7098.2.a.t.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.a.a.1.1 1 7.6 odd 2
1638.2.a.r.1.1 1 21.20 even 2
3822.2.a.n.1.1 1 1.1 even 1 trivial
4368.2.a.s.1.1 1 28.27 even 2
7098.2.a.t.1.1 1 91.90 odd 2