Properties

Label 1840.2.a.r.1.2
Level $1840$
Weight $2$
Character 1840.1
Self dual yes
Analytic conductor $14.692$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.6924739719\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.1101.1
Defining polynomial: \(x^{3} - x^{2} - 9 x + 12\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.43163\) of defining polynomial
Character \(\chi\) \(=\) 1840.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.43163 q^{3} -1.00000 q^{5} -3.08719 q^{7} -0.950444 q^{9} +O(q^{10})\) \(q-1.43163 q^{3} -1.00000 q^{5} -3.08719 q^{7} -0.950444 q^{9} +6.46926 q^{11} +3.95044 q^{13} +1.43163 q^{15} -3.43163 q^{17} -3.08719 q^{19} +4.41970 q^{21} +1.00000 q^{23} +1.00000 q^{25} +5.65556 q^{27} +0.863254 q^{29} +5.95044 q^{31} -9.26157 q^{33} +3.08719 q^{35} -7.03763 q^{37} -5.65556 q^{39} +5.60601 q^{41} -8.00000 q^{43} +0.950444 q^{45} -3.90089 q^{47} +2.53074 q^{49} +4.91281 q^{51} -6.00000 q^{53} -6.46926 q^{55} +4.41970 q^{57} -6.86325 q^{59} -13.5069 q^{61} +2.93420 q^{63} -3.95044 q^{65} +10.0753 q^{67} -1.43163 q^{69} -2.56837 q^{71} +5.90089 q^{73} -1.43163 q^{75} -19.9718 q^{77} -15.8018 q^{79} -5.24533 q^{81} -9.03763 q^{83} +3.43163 q^{85} -1.23586 q^{87} +16.7641 q^{89} -12.1958 q^{91} -8.51882 q^{93} +3.08719 q^{95} -14.2949 q^{97} -6.14867 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - q^{3} - 3q^{5} - 3q^{7} + 10q^{9} + O(q^{10}) \) \( 3q - q^{3} - 3q^{5} - 3q^{7} + 10q^{9} - 3q^{11} - q^{13} + q^{15} - 7q^{17} - 3q^{19} - 22q^{21} + 3q^{23} + 3q^{25} + 14q^{27} - 4q^{29} + 5q^{31} - 9q^{33} + 3q^{35} - 2q^{37} - 14q^{39} + q^{41} - 24q^{43} - 10q^{45} + 14q^{47} + 30q^{49} + 21q^{51} - 18q^{53} + 3q^{55} - 22q^{57} - 14q^{59} + q^{61} - 8q^{63} + q^{65} - 8q^{67} - q^{69} - 11q^{71} - 8q^{73} - q^{75} - 24q^{77} + 4q^{79} + 7q^{81} - 8q^{83} + 7q^{85} - 36q^{87} + 18q^{89} - q^{91} - 16q^{93} + 3q^{95} - 33q^{97} - 57q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.43163 −0.826550 −0.413275 0.910606i \(-0.635615\pi\)
−0.413275 + 0.910606i \(0.635615\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −3.08719 −1.16685 −0.583424 0.812168i \(-0.698287\pi\)
−0.583424 + 0.812168i \(0.698287\pi\)
\(8\) 0 0
\(9\) −0.950444 −0.316815
\(10\) 0 0
\(11\) 6.46926 1.95056 0.975278 0.220983i \(-0.0709265\pi\)
0.975278 + 0.220983i \(0.0709265\pi\)
\(12\) 0 0
\(13\) 3.95044 1.09566 0.547828 0.836591i \(-0.315455\pi\)
0.547828 + 0.836591i \(0.315455\pi\)
\(14\) 0 0
\(15\) 1.43163 0.369645
\(16\) 0 0
\(17\) −3.43163 −0.832292 −0.416146 0.909298i \(-0.636619\pi\)
−0.416146 + 0.909298i \(0.636619\pi\)
\(18\) 0 0
\(19\) −3.08719 −0.708250 −0.354125 0.935198i \(-0.615221\pi\)
−0.354125 + 0.935198i \(0.615221\pi\)
\(20\) 0 0
\(21\) 4.41970 0.964459
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.65556 1.08841
\(28\) 0 0
\(29\) 0.863254 0.160302 0.0801511 0.996783i \(-0.474460\pi\)
0.0801511 + 0.996783i \(0.474460\pi\)
\(30\) 0 0
\(31\) 5.95044 1.06873 0.534366 0.845253i \(-0.320551\pi\)
0.534366 + 0.845253i \(0.320551\pi\)
\(32\) 0 0
\(33\) −9.26157 −1.61223
\(34\) 0 0
\(35\) 3.08719 0.521830
\(36\) 0 0
\(37\) −7.03763 −1.15698 −0.578490 0.815690i \(-0.696358\pi\)
−0.578490 + 0.815690i \(0.696358\pi\)
\(38\) 0 0
\(39\) −5.65556 −0.905615
\(40\) 0 0
\(41\) 5.60601 0.875511 0.437756 0.899094i \(-0.355774\pi\)
0.437756 + 0.899094i \(0.355774\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 0.950444 0.141684
\(46\) 0 0
\(47\) −3.90089 −0.569003 −0.284501 0.958676i \(-0.591828\pi\)
−0.284501 + 0.958676i \(0.591828\pi\)
\(48\) 0 0
\(49\) 2.53074 0.361534
\(50\) 0 0
\(51\) 4.91281 0.687931
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) −6.46926 −0.872315
\(56\) 0 0
\(57\) 4.41970 0.585404
\(58\) 0 0
\(59\) −6.86325 −0.893520 −0.446760 0.894654i \(-0.647422\pi\)
−0.446760 + 0.894654i \(0.647422\pi\)
\(60\) 0 0
\(61\) −13.5069 −1.72938 −0.864690 0.502305i \(-0.832485\pi\)
−0.864690 + 0.502305i \(0.832485\pi\)
\(62\) 0 0
\(63\) 2.93420 0.369674
\(64\) 0 0
\(65\) −3.95044 −0.489992
\(66\) 0 0
\(67\) 10.0753 1.23089 0.615445 0.788180i \(-0.288976\pi\)
0.615445 + 0.788180i \(0.288976\pi\)
\(68\) 0 0
\(69\) −1.43163 −0.172348
\(70\) 0 0
\(71\) −2.56837 −0.304810 −0.152405 0.988318i \(-0.548702\pi\)
−0.152405 + 0.988318i \(0.548702\pi\)
\(72\) 0 0
\(73\) 5.90089 0.690647 0.345323 0.938484i \(-0.387769\pi\)
0.345323 + 0.938484i \(0.387769\pi\)
\(74\) 0 0
\(75\) −1.43163 −0.165310
\(76\) 0 0
\(77\) −19.9718 −2.27600
\(78\) 0 0
\(79\) −15.8018 −1.77784 −0.888919 0.458064i \(-0.848543\pi\)
−0.888919 + 0.458064i \(0.848543\pi\)
\(80\) 0 0
\(81\) −5.24533 −0.582814
\(82\) 0 0
\(83\) −9.03763 −0.992009 −0.496005 0.868320i \(-0.665200\pi\)
−0.496005 + 0.868320i \(0.665200\pi\)
\(84\) 0 0
\(85\) 3.43163 0.372212
\(86\) 0 0
\(87\) −1.23586 −0.132498
\(88\) 0 0
\(89\) 16.7641 1.77700 0.888498 0.458881i \(-0.151750\pi\)
0.888498 + 0.458881i \(0.151750\pi\)
\(90\) 0 0
\(91\) −12.1958 −1.27846
\(92\) 0 0
\(93\) −8.51882 −0.883360
\(94\) 0 0
\(95\) 3.08719 0.316739
\(96\) 0 0
\(97\) −14.2949 −1.45143 −0.725713 0.687998i \(-0.758490\pi\)
−0.725713 + 0.687998i \(0.758490\pi\)
\(98\) 0 0
\(99\) −6.14867 −0.617964
\(100\) 0 0
\(101\) 0.863254 0.0858970 0.0429485 0.999077i \(-0.486325\pi\)
0.0429485 + 0.999077i \(0.486325\pi\)
\(102\) 0 0
\(103\) −1.53074 −0.150828 −0.0754141 0.997152i \(-0.524028\pi\)
−0.0754141 + 0.997152i \(0.524028\pi\)
\(104\) 0 0
\(105\) −4.41970 −0.431319
\(106\) 0 0
\(107\) −17.6274 −1.70410 −0.852052 0.523457i \(-0.824642\pi\)
−0.852052 + 0.523457i \(0.824642\pi\)
\(108\) 0 0
\(109\) −2.91281 −0.278997 −0.139498 0.990222i \(-0.544549\pi\)
−0.139498 + 0.990222i \(0.544549\pi\)
\(110\) 0 0
\(111\) 10.0753 0.956302
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) 0 0
\(117\) −3.75467 −0.347120
\(118\) 0 0
\(119\) 10.5941 0.971158
\(120\) 0 0
\(121\) 30.8513 2.80467
\(122\) 0 0
\(123\) −8.02571 −0.723654
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −20.9385 −1.85799 −0.928997 0.370088i \(-0.879327\pi\)
−0.928997 + 0.370088i \(0.879327\pi\)
\(128\) 0 0
\(129\) 11.4530 1.00838
\(130\) 0 0
\(131\) −10.7641 −0.940467 −0.470234 0.882542i \(-0.655830\pi\)
−0.470234 + 0.882542i \(0.655830\pi\)
\(132\) 0 0
\(133\) 9.53074 0.826420
\(134\) 0 0
\(135\) −5.65556 −0.486753
\(136\) 0 0
\(137\) 3.26157 0.278655 0.139327 0.990246i \(-0.455506\pi\)
0.139327 + 0.990246i \(0.455506\pi\)
\(138\) 0 0
\(139\) 16.9385 1.43671 0.718353 0.695678i \(-0.244896\pi\)
0.718353 + 0.695678i \(0.244896\pi\)
\(140\) 0 0
\(141\) 5.58462 0.470310
\(142\) 0 0
\(143\) 25.5565 2.13714
\(144\) 0 0
\(145\) −0.863254 −0.0716894
\(146\) 0 0
\(147\) −3.62308 −0.298826
\(148\) 0 0
\(149\) 3.26157 0.267198 0.133599 0.991035i \(-0.457347\pi\)
0.133599 + 0.991035i \(0.457347\pi\)
\(150\) 0 0
\(151\) −0.294881 −0.0239971 −0.0119986 0.999928i \(-0.503819\pi\)
−0.0119986 + 0.999928i \(0.503819\pi\)
\(152\) 0 0
\(153\) 3.26157 0.263682
\(154\) 0 0
\(155\) −5.95044 −0.477951
\(156\) 0 0
\(157\) 7.13675 0.569574 0.284787 0.958591i \(-0.408077\pi\)
0.284787 + 0.958591i \(0.408077\pi\)
\(158\) 0 0
\(159\) 8.58976 0.681212
\(160\) 0 0
\(161\) −3.08719 −0.243305
\(162\) 0 0
\(163\) 8.12482 0.636385 0.318193 0.948026i \(-0.396924\pi\)
0.318193 + 0.948026i \(0.396924\pi\)
\(164\) 0 0
\(165\) 9.26157 0.721012
\(166\) 0 0
\(167\) 7.80178 0.603719 0.301860 0.953352i \(-0.402393\pi\)
0.301860 + 0.953352i \(0.402393\pi\)
\(168\) 0 0
\(169\) 2.60601 0.200462
\(170\) 0 0
\(171\) 2.93420 0.224384
\(172\) 0 0
\(173\) −19.3325 −1.46982 −0.734912 0.678163i \(-0.762777\pi\)
−0.734912 + 0.678163i \(0.762777\pi\)
\(174\) 0 0
\(175\) −3.08719 −0.233370
\(176\) 0 0
\(177\) 9.82562 0.738539
\(178\) 0 0
\(179\) −2.17438 −0.162521 −0.0812604 0.996693i \(-0.525895\pi\)
−0.0812604 + 0.996693i \(0.525895\pi\)
\(180\) 0 0
\(181\) −7.26157 −0.539748 −0.269874 0.962896i \(-0.586982\pi\)
−0.269874 + 0.962896i \(0.586982\pi\)
\(182\) 0 0
\(183\) 19.3368 1.42942
\(184\) 0 0
\(185\) 7.03763 0.517417
\(186\) 0 0
\(187\) −22.2001 −1.62343
\(188\) 0 0
\(189\) −17.4598 −1.27001
\(190\) 0 0
\(191\) −8.58976 −0.621533 −0.310767 0.950486i \(-0.600586\pi\)
−0.310767 + 0.950486i \(0.600586\pi\)
\(192\) 0 0
\(193\) 2.44787 0.176202 0.0881008 0.996112i \(-0.471920\pi\)
0.0881008 + 0.996112i \(0.471920\pi\)
\(194\) 0 0
\(195\) 5.65556 0.405003
\(196\) 0 0
\(197\) −10.7428 −0.765389 −0.382695 0.923875i \(-0.625004\pi\)
−0.382695 + 0.923875i \(0.625004\pi\)
\(198\) 0 0
\(199\) 11.3111 0.801824 0.400912 0.916116i \(-0.368693\pi\)
0.400912 + 0.916116i \(0.368693\pi\)
\(200\) 0 0
\(201\) −14.4240 −1.01739
\(202\) 0 0
\(203\) −2.66503 −0.187048
\(204\) 0 0
\(205\) −5.60601 −0.391540
\(206\) 0 0
\(207\) −0.950444 −0.0660604
\(208\) 0 0
\(209\) −19.9718 −1.38148
\(210\) 0 0
\(211\) −23.1129 −1.59116 −0.795579 0.605850i \(-0.792833\pi\)
−0.795579 + 0.605850i \(0.792833\pi\)
\(212\) 0 0
\(213\) 3.67695 0.251941
\(214\) 0 0
\(215\) 8.00000 0.545595
\(216\) 0 0
\(217\) −18.3701 −1.24705
\(218\) 0 0
\(219\) −8.44787 −0.570854
\(220\) 0 0
\(221\) −13.5565 −0.911906
\(222\) 0 0
\(223\) 5.72651 0.383475 0.191738 0.981446i \(-0.438588\pi\)
0.191738 + 0.981446i \(0.438588\pi\)
\(224\) 0 0
\(225\) −0.950444 −0.0633629
\(226\) 0 0
\(227\) 17.6274 1.16997 0.584986 0.811044i \(-0.301100\pi\)
0.584986 + 0.811044i \(0.301100\pi\)
\(228\) 0 0
\(229\) −16.0753 −1.06228 −0.531142 0.847283i \(-0.678237\pi\)
−0.531142 + 0.847283i \(0.678237\pi\)
\(230\) 0 0
\(231\) 28.5922 1.88123
\(232\) 0 0
\(233\) −6.44787 −0.422414 −0.211207 0.977441i \(-0.567739\pi\)
−0.211207 + 0.977441i \(0.567739\pi\)
\(234\) 0 0
\(235\) 3.90089 0.254466
\(236\) 0 0
\(237\) 22.6222 1.46947
\(238\) 0 0
\(239\) −4.34876 −0.281298 −0.140649 0.990060i \(-0.544919\pi\)
−0.140649 + 0.990060i \(0.544919\pi\)
\(240\) 0 0
\(241\) 0.764142 0.0492227 0.0246114 0.999697i \(-0.492165\pi\)
0.0246114 + 0.999697i \(0.492165\pi\)
\(242\) 0 0
\(243\) −9.45734 −0.606688
\(244\) 0 0
\(245\) −2.53074 −0.161683
\(246\) 0 0
\(247\) −12.1958 −0.775998
\(248\) 0 0
\(249\) 12.9385 0.819945
\(250\) 0 0
\(251\) 22.5402 1.42273 0.711363 0.702825i \(-0.248078\pi\)
0.711363 + 0.702825i \(0.248078\pi\)
\(252\) 0 0
\(253\) 6.46926 0.406719
\(254\) 0 0
\(255\) −4.91281 −0.307652
\(256\) 0 0
\(257\) −9.90089 −0.617600 −0.308800 0.951127i \(-0.599927\pi\)
−0.308800 + 0.951127i \(0.599927\pi\)
\(258\) 0 0
\(259\) 21.7265 1.35002
\(260\) 0 0
\(261\) −0.820475 −0.0507861
\(262\) 0 0
\(263\) 7.25725 0.447501 0.223751 0.974646i \(-0.428170\pi\)
0.223751 + 0.974646i \(0.428170\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) −24.0000 −1.46878
\(268\) 0 0
\(269\) −6.93852 −0.423049 −0.211525 0.977373i \(-0.567843\pi\)
−0.211525 + 0.977373i \(0.567843\pi\)
\(270\) 0 0
\(271\) 10.2992 0.625632 0.312816 0.949814i \(-0.398728\pi\)
0.312816 + 0.949814i \(0.398728\pi\)
\(272\) 0 0
\(273\) 17.4598 1.05671
\(274\) 0 0
\(275\) 6.46926 0.390111
\(276\) 0 0
\(277\) −17.8018 −1.06961 −0.534803 0.844977i \(-0.679614\pi\)
−0.534803 + 0.844977i \(0.679614\pi\)
\(278\) 0 0
\(279\) −5.65556 −0.338590
\(280\) 0 0
\(281\) −18.9385 −1.12978 −0.564889 0.825167i \(-0.691081\pi\)
−0.564889 + 0.825167i \(0.691081\pi\)
\(282\) 0 0
\(283\) −12.6889 −0.754275 −0.377138 0.926157i \(-0.623092\pi\)
−0.377138 + 0.926157i \(0.623092\pi\)
\(284\) 0 0
\(285\) −4.41970 −0.261801
\(286\) 0 0
\(287\) −17.3068 −1.02159
\(288\) 0 0
\(289\) −5.22394 −0.307290
\(290\) 0 0
\(291\) 20.4649 1.19968
\(292\) 0 0
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) 6.86325 0.399594
\(296\) 0 0
\(297\) 36.5873 2.12301
\(298\) 0 0
\(299\) 3.95044 0.228460
\(300\) 0 0
\(301\) 24.6975 1.42354
\(302\) 0 0
\(303\) −1.23586 −0.0709982
\(304\) 0 0
\(305\) 13.5069 0.773402
\(306\) 0 0
\(307\) 22.9599 1.31039 0.655196 0.755459i \(-0.272586\pi\)
0.655196 + 0.755459i \(0.272586\pi\)
\(308\) 0 0
\(309\) 2.19145 0.124667
\(310\) 0 0
\(311\) −18.0753 −1.02495 −0.512477 0.858701i \(-0.671272\pi\)
−0.512477 + 0.858701i \(0.671272\pi\)
\(312\) 0 0
\(313\) 15.1625 0.857033 0.428516 0.903534i \(-0.359036\pi\)
0.428516 + 0.903534i \(0.359036\pi\)
\(314\) 0 0
\(315\) −2.93420 −0.165323
\(316\) 0 0
\(317\) 14.0257 0.787762 0.393881 0.919161i \(-0.371132\pi\)
0.393881 + 0.919161i \(0.371132\pi\)
\(318\) 0 0
\(319\) 5.58462 0.312678
\(320\) 0 0
\(321\) 25.2359 1.40853
\(322\) 0 0
\(323\) 10.5941 0.589471
\(324\) 0 0
\(325\) 3.95044 0.219131
\(326\) 0 0
\(327\) 4.17006 0.230605
\(328\) 0 0
\(329\) 12.0428 0.663940
\(330\) 0 0
\(331\) 24.7403 1.35985 0.679925 0.733282i \(-0.262012\pi\)
0.679925 + 0.733282i \(0.262012\pi\)
\(332\) 0 0
\(333\) 6.68888 0.366548
\(334\) 0 0
\(335\) −10.0753 −0.550471
\(336\) 0 0
\(337\) 14.7146 0.801555 0.400777 0.916176i \(-0.368740\pi\)
0.400777 + 0.916176i \(0.368740\pi\)
\(338\) 0 0
\(339\) 8.58976 0.466532
\(340\) 0 0
\(341\) 38.4950 2.08462
\(342\) 0 0
\(343\) 13.7975 0.744993
\(344\) 0 0
\(345\) 1.43163 0.0770762
\(346\) 0 0
\(347\) −9.43163 −0.506316 −0.253158 0.967425i \(-0.581469\pi\)
−0.253158 + 0.967425i \(0.581469\pi\)
\(348\) 0 0
\(349\) 18.3488 0.982187 0.491093 0.871107i \(-0.336597\pi\)
0.491093 + 0.871107i \(0.336597\pi\)
\(350\) 0 0
\(351\) 22.3420 1.19253
\(352\) 0 0
\(353\) −33.1129 −1.76242 −0.881211 0.472723i \(-0.843271\pi\)
−0.881211 + 0.472723i \(0.843271\pi\)
\(354\) 0 0
\(355\) 2.56837 0.136315
\(356\) 0 0
\(357\) −15.1668 −0.802711
\(358\) 0 0
\(359\) −33.0376 −1.74366 −0.871830 0.489809i \(-0.837067\pi\)
−0.871830 + 0.489809i \(0.837067\pi\)
\(360\) 0 0
\(361\) −9.46926 −0.498382
\(362\) 0 0
\(363\) −44.1676 −2.31820
\(364\) 0 0
\(365\) −5.90089 −0.308867
\(366\) 0 0
\(367\) 2.27349 0.118675 0.0593376 0.998238i \(-0.481101\pi\)
0.0593376 + 0.998238i \(0.481101\pi\)
\(368\) 0 0
\(369\) −5.32819 −0.277375
\(370\) 0 0
\(371\) 18.5231 0.961673
\(372\) 0 0
\(373\) 23.9762 1.24144 0.620719 0.784033i \(-0.286841\pi\)
0.620719 + 0.784033i \(0.286841\pi\)
\(374\) 0 0
\(375\) 1.43163 0.0739289
\(376\) 0 0
\(377\) 3.41024 0.175636
\(378\) 0 0
\(379\) 13.8795 0.712942 0.356471 0.934306i \(-0.383980\pi\)
0.356471 + 0.934306i \(0.383980\pi\)
\(380\) 0 0
\(381\) 29.9762 1.53572
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 19.9718 1.01786
\(386\) 0 0
\(387\) 7.60355 0.386510
\(388\) 0 0
\(389\) −26.6436 −1.35089 −0.675443 0.737412i \(-0.736048\pi\)
−0.675443 + 0.737412i \(0.736048\pi\)
\(390\) 0 0
\(391\) −3.43163 −0.173545
\(392\) 0 0
\(393\) 15.4102 0.777344
\(394\) 0 0
\(395\) 15.8018 0.795074
\(396\) 0 0
\(397\) −8.66749 −0.435009 −0.217504 0.976059i \(-0.569792\pi\)
−0.217504 + 0.976059i \(0.569792\pi\)
\(398\) 0 0
\(399\) −13.6445 −0.683078
\(400\) 0 0
\(401\) 39.9762 1.99631 0.998157 0.0606854i \(-0.0193286\pi\)
0.998157 + 0.0606854i \(0.0193286\pi\)
\(402\) 0 0
\(403\) 23.5069 1.17096
\(404\) 0 0
\(405\) 5.24533 0.260642
\(406\) 0 0
\(407\) −45.5283 −2.25675
\(408\) 0 0
\(409\) 30.1248 1.48958 0.744788 0.667301i \(-0.232550\pi\)
0.744788 + 0.667301i \(0.232550\pi\)
\(410\) 0 0
\(411\) −4.66935 −0.230322
\(412\) 0 0
\(413\) 21.1882 1.04260
\(414\) 0 0
\(415\) 9.03763 0.443640
\(416\) 0 0
\(417\) −24.2496 −1.18751
\(418\) 0 0
\(419\) −25.8770 −1.26418 −0.632088 0.774897i \(-0.717802\pi\)
−0.632088 + 0.774897i \(0.717802\pi\)
\(420\) 0 0
\(421\) −10.7146 −0.522197 −0.261098 0.965312i \(-0.584085\pi\)
−0.261098 + 0.965312i \(0.584085\pi\)
\(422\) 0 0
\(423\) 3.70757 0.180268
\(424\) 0 0
\(425\) −3.43163 −0.166458
\(426\) 0 0
\(427\) 41.6983 2.01792
\(428\) 0 0
\(429\) −36.5873 −1.76645
\(430\) 0 0
\(431\) −32.2496 −1.55341 −0.776705 0.629864i \(-0.783111\pi\)
−0.776705 + 0.629864i \(0.783111\pi\)
\(432\) 0 0
\(433\) 20.6393 0.991862 0.495931 0.868362i \(-0.334827\pi\)
0.495931 + 0.868362i \(0.334827\pi\)
\(434\) 0 0
\(435\) 1.23586 0.0592549
\(436\) 0 0
\(437\) −3.08719 −0.147680
\(438\) 0 0
\(439\) 16.2239 0.774326 0.387163 0.922011i \(-0.373455\pi\)
0.387163 + 0.922011i \(0.373455\pi\)
\(440\) 0 0
\(441\) −2.40533 −0.114539
\(442\) 0 0
\(443\) 15.6770 0.744834 0.372417 0.928065i \(-0.378529\pi\)
0.372417 + 0.928065i \(0.378529\pi\)
\(444\) 0 0
\(445\) −16.7641 −0.794697
\(446\) 0 0
\(447\) −4.66935 −0.220853
\(448\) 0 0
\(449\) 22.5727 1.06527 0.532636 0.846345i \(-0.321202\pi\)
0.532636 + 0.846345i \(0.321202\pi\)
\(450\) 0 0
\(451\) 36.2667 1.70773
\(452\) 0 0
\(453\) 0.422160 0.0198348
\(454\) 0 0
\(455\) 12.1958 0.571746
\(456\) 0 0
\(457\) −1.45302 −0.0679693 −0.0339846 0.999422i \(-0.510820\pi\)
−0.0339846 + 0.999422i \(0.510820\pi\)
\(458\) 0 0
\(459\) −19.4078 −0.905878
\(460\) 0 0
\(461\) −29.7027 −1.38339 −0.691695 0.722189i \(-0.743136\pi\)
−0.691695 + 0.722189i \(0.743136\pi\)
\(462\) 0 0
\(463\) −22.9624 −1.06715 −0.533576 0.845752i \(-0.679152\pi\)
−0.533576 + 0.845752i \(0.679152\pi\)
\(464\) 0 0
\(465\) 8.51882 0.395051
\(466\) 0 0
\(467\) −9.48550 −0.438937 −0.219468 0.975620i \(-0.570432\pi\)
−0.219468 + 0.975620i \(0.570432\pi\)
\(468\) 0 0
\(469\) −31.1043 −1.43626
\(470\) 0 0
\(471\) −10.2172 −0.470782
\(472\) 0 0
\(473\) −51.7541 −2.37966
\(474\) 0 0
\(475\) −3.08719 −0.141650
\(476\) 0 0
\(477\) 5.70266 0.261107
\(478\) 0 0
\(479\) 30.5659 1.39659 0.698296 0.715809i \(-0.253942\pi\)
0.698296 + 0.715809i \(0.253942\pi\)
\(480\) 0 0
\(481\) −27.8018 −1.26765
\(482\) 0 0
\(483\) 4.41970 0.201104
\(484\) 0 0
\(485\) 14.2949 0.649097
\(486\) 0 0
\(487\) 21.1367 0.957797 0.478899 0.877870i \(-0.341036\pi\)
0.478899 + 0.877870i \(0.341036\pi\)
\(488\) 0 0
\(489\) −11.6317 −0.526004
\(490\) 0 0
\(491\) 28.0514 1.26594 0.632971 0.774175i \(-0.281835\pi\)
0.632971 + 0.774175i \(0.281835\pi\)
\(492\) 0 0
\(493\) −2.96237 −0.133418
\(494\) 0 0
\(495\) 6.14867 0.276362
\(496\) 0 0
\(497\) 7.92905 0.355667
\(498\) 0 0
\(499\) −3.80178 −0.170191 −0.0850954 0.996373i \(-0.527120\pi\)
−0.0850954 + 0.996373i \(0.527120\pi\)
\(500\) 0 0
\(501\) −11.1692 −0.499005
\(502\) 0 0
\(503\) −19.0872 −0.851056 −0.425528 0.904945i \(-0.639912\pi\)
−0.425528 + 0.904945i \(0.639912\pi\)
\(504\) 0 0
\(505\) −0.863254 −0.0384143
\(506\) 0 0
\(507\) −3.73083 −0.165692
\(508\) 0 0
\(509\) −9.41024 −0.417101 −0.208551 0.978012i \(-0.566875\pi\)
−0.208551 + 0.978012i \(0.566875\pi\)
\(510\) 0 0
\(511\) −18.2172 −0.805880
\(512\) 0 0
\(513\) −17.4598 −0.770869
\(514\) 0 0
\(515\) 1.53074 0.0674524
\(516\) 0 0
\(517\) −25.2359 −1.10987
\(518\) 0 0
\(519\) 27.6770 1.21488
\(520\) 0 0
\(521\) 25.8018 1.13040 0.565198 0.824955i \(-0.308800\pi\)
0.565198 + 0.824955i \(0.308800\pi\)
\(522\) 0 0
\(523\) 19.1129 0.835749 0.417874 0.908505i \(-0.362775\pi\)
0.417874 + 0.908505i \(0.362775\pi\)
\(524\) 0 0
\(525\) 4.41970 0.192892
\(526\) 0 0
\(527\) −20.4197 −0.889496
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 6.52314 0.283080
\(532\) 0 0
\(533\) 22.1462 0.959259
\(534\) 0 0
\(535\) 17.6274 0.762099
\(536\) 0 0
\(537\) 3.11290 0.134332
\(538\) 0 0
\(539\) 16.3720 0.705193
\(540\) 0 0
\(541\) 30.8394 1.32589 0.662945 0.748668i \(-0.269306\pi\)
0.662945 + 0.748668i \(0.269306\pi\)
\(542\) 0 0
\(543\) 10.3959 0.446129
\(544\) 0 0
\(545\) 2.91281 0.124771
\(546\) 0 0
\(547\) −13.8513 −0.592240 −0.296120 0.955151i \(-0.595693\pi\)
−0.296120 + 0.955151i \(0.595693\pi\)
\(548\) 0 0
\(549\) 12.8375 0.547893
\(550\) 0 0
\(551\) −2.66503 −0.113534
\(552\) 0 0
\(553\) 48.7831 2.07447
\(554\) 0 0
\(555\) −10.0753 −0.427671
\(556\) 0 0
\(557\) −28.7641 −1.21878 −0.609388 0.792872i \(-0.708585\pi\)
−0.609388 + 0.792872i \(0.708585\pi\)
\(558\) 0 0
\(559\) −31.6036 −1.33669
\(560\) 0 0
\(561\) 31.7823 1.34185
\(562\) 0 0
\(563\) −36.1505 −1.52356 −0.761782 0.647834i \(-0.775675\pi\)
−0.761782 + 0.647834i \(0.775675\pi\)
\(564\) 0 0
\(565\) 6.00000 0.252422
\(566\) 0 0
\(567\) 16.1933 0.680055
\(568\) 0 0
\(569\) −10.3488 −0.433843 −0.216921 0.976189i \(-0.569601\pi\)
−0.216921 + 0.976189i \(0.569601\pi\)
\(570\) 0 0
\(571\) −19.6060 −0.820486 −0.410243 0.911976i \(-0.634556\pi\)
−0.410243 + 0.911976i \(0.634556\pi\)
\(572\) 0 0
\(573\) 12.2973 0.513729
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) −34.1505 −1.42171 −0.710853 0.703341i \(-0.751691\pi\)
−0.710853 + 0.703341i \(0.751691\pi\)
\(578\) 0 0
\(579\) −3.50444 −0.145639
\(580\) 0 0
\(581\) 27.9009 1.15752
\(582\) 0 0
\(583\) −38.8156 −1.60758
\(584\) 0 0
\(585\) 3.75467 0.155237
\(586\) 0 0
\(587\) 5.19062 0.214240 0.107120 0.994246i \(-0.465837\pi\)
0.107120 + 0.994246i \(0.465837\pi\)
\(588\) 0 0
\(589\) −18.3701 −0.756929
\(590\) 0 0
\(591\) 15.3796 0.632633
\(592\) 0 0
\(593\) −2.09911 −0.0862002 −0.0431001 0.999071i \(-0.513723\pi\)
−0.0431001 + 0.999071i \(0.513723\pi\)
\(594\) 0 0
\(595\) −10.5941 −0.434315
\(596\) 0 0
\(597\) −16.1933 −0.662748
\(598\) 0 0
\(599\) −11.1581 −0.455909 −0.227955 0.973672i \(-0.573204\pi\)
−0.227955 + 0.973672i \(0.573204\pi\)
\(600\) 0 0
\(601\) 9.57784 0.390688 0.195344 0.980735i \(-0.437418\pi\)
0.195344 + 0.980735i \(0.437418\pi\)
\(602\) 0 0
\(603\) −9.57597 −0.389964
\(604\) 0 0
\(605\) −30.8513 −1.25428
\(606\) 0 0
\(607\) 24.6975 1.00244 0.501221 0.865320i \(-0.332885\pi\)
0.501221 + 0.865320i \(0.332885\pi\)
\(608\) 0 0
\(609\) 3.81533 0.154605
\(610\) 0 0
\(611\) −15.4102 −0.623431
\(612\) 0 0
\(613\) −26.3488 −1.06422 −0.532108 0.846676i \(-0.678600\pi\)
−0.532108 + 0.846676i \(0.678600\pi\)
\(614\) 0 0
\(615\) 8.02571 0.323628
\(616\) 0 0
\(617\) 5.94612 0.239382 0.119691 0.992811i \(-0.461810\pi\)
0.119691 + 0.992811i \(0.461810\pi\)
\(618\) 0 0
\(619\) −39.6856 −1.59510 −0.797549 0.603254i \(-0.793871\pi\)
−0.797549 + 0.603254i \(0.793871\pi\)
\(620\) 0 0
\(621\) 5.65556 0.226950
\(622\) 0 0
\(623\) −51.7541 −2.07348
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 28.5922 1.14186
\(628\) 0 0
\(629\) 24.1505 0.962945
\(630\) 0 0
\(631\) 23.0138 0.916164 0.458082 0.888910i \(-0.348537\pi\)
0.458082 + 0.888910i \(0.348537\pi\)
\(632\) 0 0
\(633\) 33.0891 1.31517
\(634\) 0 0
\(635\) 20.9385 0.830920
\(636\) 0 0
\(637\) 9.99754 0.396117
\(638\) 0 0
\(639\) 2.44109 0.0965682
\(640\) 0 0
\(641\) 25.4617 1.00568 0.502838 0.864381i \(-0.332289\pi\)
0.502838 + 0.864381i \(0.332289\pi\)
\(642\) 0 0
\(643\) 16.4479 0.648641 0.324320 0.945947i \(-0.394864\pi\)
0.324320 + 0.945947i \(0.394864\pi\)
\(644\) 0 0
\(645\) −11.4530 −0.450962
\(646\) 0 0
\(647\) 34.9147 1.37264 0.686319 0.727301i \(-0.259226\pi\)
0.686319 + 0.727301i \(0.259226\pi\)
\(648\) 0 0
\(649\) −44.4002 −1.74286
\(650\) 0 0
\(651\) 26.2992 1.03075
\(652\) 0 0
\(653\) −34.2754 −1.34130 −0.670649 0.741775i \(-0.733984\pi\)
−0.670649 + 0.741775i \(0.733984\pi\)
\(654\) 0 0
\(655\) 10.7641 0.420590
\(656\) 0 0
\(657\) −5.60846 −0.218807
\(658\) 0 0
\(659\) −35.2548 −1.37333 −0.686666 0.726973i \(-0.740926\pi\)
−0.686666 + 0.726973i \(0.740926\pi\)
\(660\) 0 0
\(661\) 28.5684 1.11118 0.555590 0.831456i \(-0.312492\pi\)
0.555590 + 0.831456i \(0.312492\pi\)
\(662\) 0 0
\(663\) 19.4078 0.753736
\(664\) 0 0
\(665\) −9.53074 −0.369586
\(666\) 0 0
\(667\) 0.863254 0.0334253
\(668\) 0 0
\(669\) −8.19822 −0.316962
\(670\) 0 0
\(671\) −87.3796 −3.37325
\(672\) 0 0
\(673\) −23.8770 −0.920392 −0.460196 0.887817i \(-0.652221\pi\)
−0.460196 + 0.887817i \(0.652221\pi\)
\(674\) 0 0
\(675\) 5.65556 0.217683
\(676\) 0 0
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 0 0
\(679\) 44.1310 1.69359
\(680\) 0 0
\(681\) −25.2359 −0.967040
\(682\) 0 0
\(683\) −34.6480 −1.32577 −0.662884 0.748722i \(-0.730668\pi\)
−0.662884 + 0.748722i \(0.730668\pi\)
\(684\) 0 0
\(685\) −3.26157 −0.124618
\(686\) 0 0
\(687\) 23.0138 0.878031
\(688\) 0 0
\(689\) −23.7027 −0.903000
\(690\) 0 0
\(691\) 18.1744 0.691386 0.345693 0.938348i \(-0.387644\pi\)
0.345693 + 0.938348i \(0.387644\pi\)
\(692\) 0 0
\(693\) 18.9821 0.721071
\(694\) 0 0
\(695\) −16.9385 −0.642515
\(696\) 0 0
\(697\) −19.2377 −0.728681
\(698\) 0 0
\(699\) 9.23095 0.349146
\(700\) 0 0
\(701\) −40.6907 −1.53687 −0.768434 0.639929i \(-0.778964\pi\)
−0.768434 + 0.639929i \(0.778964\pi\)
\(702\) 0 0
\(703\) 21.7265 0.819431
\(704\) 0 0
\(705\) −5.58462 −0.210329
\(706\) 0 0
\(707\) −2.66503 −0.100229
\(708\) 0 0
\(709\) −26.4454 −0.993178 −0.496589 0.867986i \(-0.665414\pi\)
−0.496589 + 0.867986i \(0.665414\pi\)
\(710\) 0 0
\(711\) 15.0187 0.563245
\(712\) 0 0
\(713\) 5.95044 0.222846
\(714\) 0 0
\(715\) −25.5565 −0.955757
\(716\) 0 0
\(717\) 6.22580 0.232507
\(718\) 0 0
\(719\) 2.24778 0.0838281 0.0419140 0.999121i \(-0.486654\pi\)
0.0419140 + 0.999121i \(0.486654\pi\)
\(720\) 0 0
\(721\) 4.72568 0.175994
\(722\) 0 0
\(723\) −1.09397 −0.0406850
\(724\) 0 0
\(725\) 0.863254 0.0320605
\(726\) 0 0
\(727\) −21.4830 −0.796762 −0.398381 0.917220i \(-0.630428\pi\)
−0.398381 + 0.917220i \(0.630428\pi\)
\(728\) 0 0
\(729\) 29.2754 1.08427
\(730\) 0 0
\(731\) 27.4530 1.01539
\(732\) 0 0
\(733\) 11.3778 0.420247 0.210123 0.977675i \(-0.432613\pi\)
0.210123 + 0.977675i \(0.432613\pi\)
\(734\) 0 0
\(735\) 3.62308 0.133639
\(736\) 0 0
\(737\) 65.1795 2.40092
\(738\) 0 0
\(739\) −21.8770 −0.804760 −0.402380 0.915473i \(-0.631817\pi\)
−0.402380 + 0.915473i \(0.631817\pi\)
\(740\) 0 0
\(741\) 17.4598 0.641402
\(742\) 0 0
\(743\) −5.36068 −0.196664 −0.0983322 0.995154i \(-0.531351\pi\)
−0.0983322 + 0.995154i \(0.531351\pi\)
\(744\) 0 0
\(745\) −3.26157 −0.119495
\(746\) 0 0
\(747\) 8.58976 0.314283
\(748\) 0 0
\(749\) 54.4191 1.98843
\(750\) 0 0
\(751\) −24.3915 −0.890060 −0.445030 0.895516i \(-0.646807\pi\)
−0.445030 + 0.895516i \(0.646807\pi\)
\(752\) 0 0
\(753\) −32.2692 −1.17595
\(754\) 0 0
\(755\) 0.294881 0.0107318
\(756\) 0 0
\(757\) 41.9437 1.52447 0.762234 0.647301i \(-0.224102\pi\)
0.762234 + 0.647301i \(0.224102\pi\)
\(758\) 0 0
\(759\) −9.26157 −0.336174
\(760\) 0 0
\(761\) −18.3745 −0.666074 −0.333037 0.942914i \(-0.608073\pi\)
−0.333037 + 0.942914i \(0.608073\pi\)
\(762\) 0 0
\(763\) 8.99240 0.325547
\(764\) 0 0
\(765\) −3.26157 −0.117922
\(766\) 0 0
\(767\) −27.1129 −0.978990
\(768\) 0 0
\(769\) 42.4993 1.53256 0.766282 0.642505i \(-0.222105\pi\)
0.766282 + 0.642505i \(0.222105\pi\)
\(770\) 0 0
\(771\) 14.1744 0.510478
\(772\) 0 0
\(773\) −15.0376 −0.540866 −0.270433 0.962739i \(-0.587167\pi\)
−0.270433 + 0.962739i \(0.587167\pi\)
\(774\) 0 0
\(775\) 5.95044 0.213746
\(776\) 0 0
\(777\) −31.1043 −1.11586
\(778\) 0 0
\(779\) −17.3068 −0.620081
\(780\) 0 0
\(781\) −16.6155 −0.594548
\(782\) 0 0
\(783\) 4.88219 0.174475
\(784\) 0 0
\(785\) −7.13675 −0.254721
\(786\) 0 0
\(787\) 8.39154 0.299126 0.149563 0.988752i \(-0.452213\pi\)
0.149563 + 0.988752i \(0.452213\pi\)
\(788\) 0 0
\(789\) −10.3897 −0.369882
\(790\) 0 0
\(791\) 18.5231 0.658607
\(792\) 0 0
\(793\) −53.3582 −1.89481
\(794\) 0 0
\(795\) −8.58976 −0.304647
\(796\) 0 0
\(797\) −22.0514 −0.781101 −0.390551 0.920581i \(-0.627715\pi\)
−0.390551 + 0.920581i \(0.627715\pi\)
\(798\) 0 0
\(799\) 13.3864 0.473576
\(800\) 0 0
\(801\) −15.9334 −0.562978
\(802\) 0 0
\(803\) 38.1744 1.34714
\(804\) 0 0
\(805\) 3.08719 0.108809
\(806\) 0 0
\(807\) 9.93337 0.349671
\(808\) 0 0
\(809\) 2.04524 0.0719066 0.0359533 0.999353i \(-0.488553\pi\)
0.0359533 + 0.999353i \(0.488553\pi\)
\(810\) 0 0
\(811\) −4.24965 −0.149225 −0.0746126 0.997213i \(-0.523772\pi\)
−0.0746126 + 0.997213i \(0.523772\pi\)
\(812\) 0 0
\(813\) −14.7446 −0.517116
\(814\) 0 0
\(815\) −8.12482 −0.284600
\(816\) 0 0
\(817\) 24.6975 0.864057
\(818\) 0 0
\(819\) 11.5914 0.405036
\(820\) 0 0
\(821\) −29.2548 −1.02100 −0.510500 0.859878i \(-0.670540\pi\)
−0.510500 + 0.859878i \(0.670540\pi\)
\(822\) 0 0
\(823\) −13.5846 −0.473530 −0.236765 0.971567i \(-0.576087\pi\)
−0.236765 + 0.971567i \(0.576087\pi\)
\(824\) 0 0
\(825\) −9.26157 −0.322446
\(826\) 0 0
\(827\) −24.7880 −0.861963 −0.430981 0.902361i \(-0.641833\pi\)
−0.430981 + 0.902361i \(0.641833\pi\)
\(828\) 0 0
\(829\) 26.1505 0.908246 0.454123 0.890939i \(-0.349953\pi\)
0.454123 + 0.890939i \(0.349953\pi\)
\(830\) 0 0
\(831\) 25.4855 0.884082
\(832\) 0 0
\(833\) −8.68455 −0.300902
\(834\) 0 0
\(835\) −7.80178 −0.269992
\(836\) 0 0
\(837\) 33.6531 1.16322
\(838\) 0 0
\(839\) −6.37260 −0.220007 −0.110003 0.993931i \(-0.535086\pi\)
−0.110003 + 0.993931i \(0.535086\pi\)
\(840\) 0 0
\(841\) −28.2548 −0.974303
\(842\) 0 0
\(843\) 27.1129 0.933818
\(844\) 0 0
\(845\) −2.60601 −0.0896493
\(846\) 0 0
\(847\) −95.2439 −3.27262
\(848\) 0 0
\(849\) 18.1657 0.623447
\(850\) 0 0
\(851\) −7.03763 −0.241247
\(852\) 0 0
\(853\) −33.6293 −1.15144 −0.575722 0.817646i \(-0.695279\pi\)
−0.575722 + 0.817646i \(0.695279\pi\)
\(854\) 0 0
\(855\) −2.93420 −0.100348
\(856\) 0 0
\(857\) 11.1795 0.381885 0.190943 0.981601i \(-0.438846\pi\)
0.190943 + 0.981601i \(0.438846\pi\)
\(858\) 0 0
\(859\) −47.9009 −1.63436 −0.817179 0.576385i \(-0.804463\pi\)
−0.817179 + 0.576385i \(0.804463\pi\)
\(860\) 0 0
\(861\) 24.7769 0.844394
\(862\) 0 0
\(863\) 30.1180 1.02523 0.512615 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(864\) 0 0
\(865\) 19.3325 0.657325
\(866\) 0 0
\(867\) 7.47873 0.253991
\(868\) 0 0
\(869\) −102.226 −3.46777
\(870\) 0 0
\(871\) 39.8018 1.34863
\(872\) 0 0
\(873\) 13.5865 0.459833
\(874\) 0 0
\(875\) 3.08719 0.104366
\(876\) 0 0
\(877\) −22.3940 −0.756191 −0.378096 0.925767i \(-0.623421\pi\)
−0.378096 + 0.925767i \(0.623421\pi\)
\(878\) 0 0
\(879\) 8.58976 0.289726
\(880\) 0 0
\(881\) 21.1129 0.711312 0.355656 0.934617i \(-0.384258\pi\)
0.355656 + 0.934617i \(0.384258\pi\)
\(882\) 0 0
\(883\) −49.1061 −1.65255 −0.826276 0.563265i \(-0.809545\pi\)
−0.826276 + 0.563265i \(0.809545\pi\)
\(884\) 0 0
\(885\) −9.82562 −0.330285
\(886\) 0 0
\(887\) 43.0566 1.44570 0.722849 0.691006i \(-0.242832\pi\)
0.722849 + 0.691006i \(0.242832\pi\)
\(888\) 0 0
\(889\) 64.6412 2.16800
\(890\) 0 0
\(891\) −33.9334 −1.13681
\(892\) 0 0
\(893\) 12.0428 0.402996
\(894\) 0 0
\(895\) 2.17438 0.0726815
\(896\) 0 0
\(897\) −5.65556 −0.188834
\(898\) 0 0
\(899\) 5.13675 0.171320
\(900\) 0 0
\(901\) 20.5898 0.685944
\(902\) 0 0
\(903\) −35.3576 −1.17663
\(904\) 0 0
\(905\) 7.26157 0.241383
\(906\) 0 0
\(907\) 37.9762 1.26098 0.630489 0.776198i \(-0.282854\pi\)
0.630489 + 0.776198i \(0.282854\pi\)
\(908\) 0 0
\(909\) −0.820475 −0.0272134
\(910\) 0 0
\(911\) 15.7504 0.521833 0.260916 0.965361i \(-0.415975\pi\)
0.260916 + 0.965361i \(0.415975\pi\)
\(912\) 0 0
\(913\) −58.4668 −1.93497
\(914\) 0 0
\(915\) −19.3368 −0.639256
\(916\) 0 0
\(917\) 33.2309 1.09738
\(918\) 0 0
\(919\) −30.4240 −1.00360 −0.501798 0.864985i \(-0.667328\pi\)
−0.501798 + 0.864985i \(0.667328\pi\)
\(920\) 0 0
\(921\) −32.8700 −1.08310
\(922\) 0 0
\(923\) −10.1462 −0.333967
\(924\) 0 0
\(925\) −7.03763 −0.231396
\(926\) 0 0
\(927\) 1.45488 0.0477846
\(928\) 0 0
\(929\) −37.8018 −1.24024 −0.620118 0.784509i \(-0.712915\pi\)
−0.620118 + 0.784509i \(0.712915\pi\)
\(930\) 0 0
\(931\) −7.81287 −0.256057
\(932\) 0 0
\(933\) 25.8770 0.847176
\(934\) 0 0
\(935\) 22.2001 0.726021
\(936\) 0 0
\(937\) −32.6907 −1.06796 −0.533980 0.845497i \(-0.679304\pi\)
−0.533980 + 0.845497i \(0.679304\pi\)
\(938\) 0 0
\(939\) −21.7070 −0.708381
\(940\) 0 0
\(941\) −46.4882 −1.51547 −0.757736 0.652561i \(-0.773694\pi\)
−0.757736 + 0.652561i \(0.773694\pi\)
\(942\) 0 0
\(943\) 5.60601 0.182557
\(944\) 0 0
\(945\) 17.4598 0.567967
\(946\) 0 0
\(947\) 37.2052 1.20901 0.604504 0.796602i \(-0.293371\pi\)
0.604504 + 0.796602i \(0.293371\pi\)
\(948\) 0 0
\(949\) 23.3111 0.756711
\(950\) 0 0
\(951\) −20.0796 −0.651125
\(952\) 0 0
\(953\) 49.1104 1.59084 0.795422 0.606056i \(-0.207249\pi\)
0.795422 + 0.606056i \(0.207249\pi\)
\(954\) 0 0
\(955\) 8.58976 0.277958
\(956\) 0 0
\(957\) −7.99509 −0.258445
\(958\) 0 0
\(959\) −10.0691 −0.325148
\(960\) 0 0
\(961\) 4.40778 0.142187
\(962\) 0 0
\(963\) 16.7538 0.539885
\(964\) 0 0
\(965\) −2.44787 −0.0787997
\(966\) 0 0
\(967\) −1.96751 −0.0632709 −0.0316355 0.999499i \(-0.510072\pi\)
−0.0316355 + 0.999499i \(0.510072\pi\)
\(968\) 0 0
\(969\) −15.1668 −0.487227
\(970\) 0 0
\(971\) −26.1205 −0.838247 −0.419123 0.907929i \(-0.637663\pi\)
−0.419123 + 0.907929i \(0.637663\pi\)
\(972\) 0 0
\(973\) −52.2924 −1.67642
\(974\) 0 0
\(975\) −5.65556 −0.181123
\(976\) 0 0
\(977\) −0.639319 −0.0204536 −0.0102268 0.999948i \(-0.503255\pi\)
−0.0102268 + 0.999948i \(0.503255\pi\)
\(978\) 0 0
\(979\) 108.452 3.46613
\(980\) 0 0
\(981\) 2.76846 0.0883902
\(982\) 0 0
\(983\) 6.46926 0.206337 0.103169 0.994664i \(-0.467102\pi\)
0.103169 + 0.994664i \(0.467102\pi\)
\(984\) 0 0
\(985\) 10.7428 0.342293
\(986\) 0 0
\(987\) −17.2408 −0.548780
\(988\) 0 0
\(989\) −8.00000 −0.254385
\(990\) 0 0
\(991\) −37.2334 −1.18276 −0.591379 0.806394i \(-0.701416\pi\)
−0.591379 + 0.806394i \(0.701416\pi\)
\(992\) 0 0
\(993\) −35.4189 −1.12398
\(994\) 0 0
\(995\) −11.3111 −0.358587
\(996\) 0 0
\(997\) 9.65124 0.305658 0.152829 0.988253i \(-0.451162\pi\)
0.152829 + 0.988253i \(0.451162\pi\)
\(998\) 0 0
\(999\) −39.8018 −1.25927
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 1840.2.a.r.1.2 3
4.3 odd 2 230.2.a.d.1.2 3
5.4 even 2 9200.2.a.cf.1.2 3
8.3 odd 2 7360.2.a.bz.1.2 3
8.5 even 2 7360.2.a.ce.1.2 3
12.11 even 2 2070.2.a.z.1.2 3
20.3 even 4 1150.2.b.j.599.2 6
20.7 even 4 1150.2.b.j.599.5 6
20.19 odd 2 1150.2.a.q.1.2 3
92.91 even 2 5290.2.a.r.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.d.1.2 3 4.3 odd 2
1150.2.a.q.1.2 3 20.19 odd 2
1150.2.b.j.599.2 6 20.3 even 4
1150.2.b.j.599.5 6 20.7 even 4
1840.2.a.r.1.2 3 1.1 even 1 trivial
2070.2.a.z.1.2 3 12.11 even 2
5290.2.a.r.1.2 3 92.91 even 2
7360.2.a.bz.1.2 3 8.3 odd 2
7360.2.a.ce.1.2 3 8.5 even 2
9200.2.a.cf.1.2 3 5.4 even 2