Properties

Label 2070.2.a.z.1.2
Level $2070$
Weight $2$
Character 2070.1
Self dual yes
Analytic conductor $16.529$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(16.5290332184\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1101.1
Defining polynomial: \(x^{3} - x^{2} - 9 x + 12\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(\chi\) \(=\) 2070.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +3.08719 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +3.08719 q^{7} -1.00000 q^{8} -1.00000 q^{10} +6.46926 q^{11} +3.95044 q^{13} -3.08719 q^{14} +1.00000 q^{16} +3.43163 q^{17} +3.08719 q^{19} +1.00000 q^{20} -6.46926 q^{22} +1.00000 q^{23} +1.00000 q^{25} -3.95044 q^{26} +3.08719 q^{28} -0.863254 q^{29} -5.95044 q^{31} -1.00000 q^{32} -3.43163 q^{34} +3.08719 q^{35} -7.03763 q^{37} -3.08719 q^{38} -1.00000 q^{40} -5.60601 q^{41} +8.00000 q^{43} +6.46926 q^{44} -1.00000 q^{46} -3.90089 q^{47} +2.53074 q^{49} -1.00000 q^{50} +3.95044 q^{52} +6.00000 q^{53} +6.46926 q^{55} -3.08719 q^{56} +0.863254 q^{58} -6.86325 q^{59} -13.5069 q^{61} +5.95044 q^{62} +1.00000 q^{64} +3.95044 q^{65} -10.0753 q^{67} +3.43163 q^{68} -3.08719 q^{70} -2.56837 q^{71} +5.90089 q^{73} +7.03763 q^{74} +3.08719 q^{76} +19.9718 q^{77} +15.8018 q^{79} +1.00000 q^{80} +5.60601 q^{82} -9.03763 q^{83} +3.43163 q^{85} -8.00000 q^{86} -6.46926 q^{88} -16.7641 q^{89} +12.1958 q^{91} +1.00000 q^{92} +3.90089 q^{94} +3.08719 q^{95} -14.2949 q^{97} -2.53074 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} + 3q^{4} + 3q^{5} + 3q^{7} - 3q^{8} + O(q^{10}) \) \( 3q - 3q^{2} + 3q^{4} + 3q^{5} + 3q^{7} - 3q^{8} - 3q^{10} - 3q^{11} - q^{13} - 3q^{14} + 3q^{16} + 7q^{17} + 3q^{19} + 3q^{20} + 3q^{22} + 3q^{23} + 3q^{25} + q^{26} + 3q^{28} + 4q^{29} - 5q^{31} - 3q^{32} - 7q^{34} + 3q^{35} - 2q^{37} - 3q^{38} - 3q^{40} - q^{41} + 24q^{43} - 3q^{44} - 3q^{46} + 14q^{47} + 30q^{49} - 3q^{50} - q^{52} + 18q^{53} - 3q^{55} - 3q^{56} - 4q^{58} - 14q^{59} + q^{61} + 5q^{62} + 3q^{64} - q^{65} + 8q^{67} + 7q^{68} - 3q^{70} - 11q^{71} - 8q^{73} + 2q^{74} + 3q^{76} + 24q^{77} - 4q^{79} + 3q^{80} + q^{82} - 8q^{83} + 7q^{85} - 24q^{86} + 3q^{88} - 18q^{89} + q^{91} + 3q^{92} - 14q^{94} + 3q^{95} - 33q^{97} - 30q^{98} + O(q^{100}) \)

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 3.08719 1.16685 0.583424 0.812168i \(-0.301713\pi\)
0.583424 + 0.812168i \(0.301713\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) 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 \(0.315455\pi\)
\(14\) −3.08719 −0.825086
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.43163 0.832292 0.416146 0.909298i \(-0.363381\pi\)
0.416146 + 0.909298i \(0.363381\pi\)
\(18\) 0 0
\(19\) 3.08719 0.708250 0.354125 0.935198i \(-0.384779\pi\)
0.354125 + 0.935198i \(0.384779\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −6.46926 −1.37925
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −3.95044 −0.774746
\(27\) 0 0
\(28\) 3.08719 0.583424
\(29\) −0.863254 −0.160302 −0.0801511 0.996783i \(-0.525540\pi\)
−0.0801511 + 0.996783i \(0.525540\pi\)
\(30\) 0 0
\(31\) −5.95044 −1.06873 −0.534366 0.845253i \(-0.679449\pi\)
−0.534366 + 0.845253i \(0.679449\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −3.43163 −0.588519
\(35\) 3.08719 0.521830
\(36\) 0 0
\(37\) −7.03763 −1.15698 −0.578490 0.815690i \(-0.696358\pi\)
−0.578490 + 0.815690i \(0.696358\pi\)
\(38\) −3.08719 −0.500808
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −5.60601 −0.875511 −0.437756 0.899094i \(-0.644226\pi\)
−0.437756 + 0.899094i \(0.644226\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 6.46926 0.975278
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) −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
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 3.95044 0.547828
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 6.46926 0.872315
\(56\) −3.08719 −0.412543
\(57\) 0 0
\(58\) 0.863254 0.113351
\(59\) −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 \(0.832485\pi\)
\(62\) 5.95044 0.755707
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 3.95044 0.489992
\(66\) 0 0
\(67\) −10.0753 −1.23089 −0.615445 0.788180i \(-0.711024\pi\)
−0.615445 + 0.788180i \(0.711024\pi\)
\(68\) 3.43163 0.416146
\(69\) 0 0
\(70\) −3.08719 −0.368990
\(71\) −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 \(0.387769\pi\)
\(74\) 7.03763 0.818108
\(75\) 0 0
\(76\) 3.08719 0.354125
\(77\) 19.9718 2.27600
\(78\) 0 0
\(79\) 15.8018 1.77784 0.888919 0.458064i \(-0.151457\pi\)
0.888919 + 0.458064i \(0.151457\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 5.60601 0.619080
\(83\) −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
\(86\) −8.00000 −0.862662
\(87\) 0 0
\(88\) −6.46926 −0.689625
\(89\) −16.7641 −1.77700 −0.888498 0.458881i \(-0.848250\pi\)
−0.888498 + 0.458881i \(0.848250\pi\)
\(90\) 0 0
\(91\) 12.1958 1.27846
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) 3.90089 0.402346
\(95\) 3.08719 0.316739
\(96\) 0 0
\(97\) −14.2949 −1.45143 −0.725713 0.687998i \(-0.758490\pi\)
−0.725713 + 0.687998i \(0.758490\pi\)
\(98\) −2.53074 −0.255643
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −0.863254 −0.0858970 −0.0429485 0.999077i \(-0.513675\pi\)
−0.0429485 + 0.999077i \(0.513675\pi\)
\(102\) 0 0
\(103\) 1.53074 0.150828 0.0754141 0.997152i \(-0.475972\pi\)
0.0754141 + 0.997152i \(0.475972\pi\)
\(104\) −3.95044 −0.387373
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −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 \(0.544549\pi\)
\(110\) −6.46926 −0.616820
\(111\) 0 0
\(112\) 3.08719 0.291712
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) −0.863254 −0.0801511
\(117\) 0 0
\(118\) 6.86325 0.631814
\(119\) 10.5941 0.971158
\(120\) 0 0
\(121\) 30.8513 2.80467
\(122\) 13.5069 1.22286
\(123\) 0 0
\(124\) −5.95044 −0.534366
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 20.9385 1.85799 0.928997 0.370088i \(-0.120673\pi\)
0.928997 + 0.370088i \(0.120673\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −3.95044 −0.346477
\(131\) −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
\(134\) 10.0753 0.870370
\(135\) 0 0
\(136\) −3.43163 −0.294260
\(137\) −3.26157 −0.278655 −0.139327 0.990246i \(-0.544494\pi\)
−0.139327 + 0.990246i \(0.544494\pi\)
\(138\) 0 0
\(139\) −16.9385 −1.43671 −0.718353 0.695678i \(-0.755104\pi\)
−0.718353 + 0.695678i \(0.755104\pi\)
\(140\) 3.08719 0.260915
\(141\) 0 0
\(142\) 2.56837 0.215533
\(143\) 25.5565 2.13714
\(144\) 0 0
\(145\) −0.863254 −0.0716894
\(146\) −5.90089 −0.488361
\(147\) 0 0
\(148\) −7.03763 −0.578490
\(149\) −3.26157 −0.267198 −0.133599 0.991035i \(-0.542653\pi\)
−0.133599 + 0.991035i \(0.542653\pi\)
\(150\) 0 0
\(151\) 0.294881 0.0239971 0.0119986 0.999928i \(-0.496181\pi\)
0.0119986 + 0.999928i \(0.496181\pi\)
\(152\) −3.08719 −0.250404
\(153\) 0 0
\(154\) −19.9718 −1.60938
\(155\) −5.95044 −0.477951
\(156\) 0 0
\(157\) 7.13675 0.569574 0.284787 0.958591i \(-0.408077\pi\)
0.284787 + 0.958591i \(0.408077\pi\)
\(158\) −15.8018 −1.25712
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 3.08719 0.243305
\(162\) 0 0
\(163\) −8.12482 −0.636385 −0.318193 0.948026i \(-0.603076\pi\)
−0.318193 + 0.948026i \(0.603076\pi\)
\(164\) −5.60601 −0.437756
\(165\) 0 0
\(166\) 9.03763 0.701456
\(167\) 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
\(170\) −3.43163 −0.263194
\(171\) 0 0
\(172\) 8.00000 0.609994
\(173\) 19.3325 1.46982 0.734912 0.678163i \(-0.237223\pi\)
0.734912 + 0.678163i \(0.237223\pi\)
\(174\) 0 0
\(175\) 3.08719 0.233370
\(176\) 6.46926 0.487639
\(177\) 0 0
\(178\) 16.7641 1.25653
\(179\) −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 \(0.586982\pi\)
\(182\) −12.1958 −0.904011
\(183\) 0 0
\(184\) −1.00000 −0.0737210
\(185\) −7.03763 −0.517417
\(186\) 0 0
\(187\) 22.2001 1.62343
\(188\) −3.90089 −0.284501
\(189\) 0 0
\(190\) −3.08719 −0.223968
\(191\) −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 \(0.471920\pi\)
\(194\) 14.2949 1.02631
\(195\) 0 0
\(196\) 2.53074 0.180767
\(197\) 10.7428 0.765389 0.382695 0.923875i \(-0.374996\pi\)
0.382695 + 0.923875i \(0.374996\pi\)
\(198\) 0 0
\(199\) −11.3111 −0.801824 −0.400912 0.916116i \(-0.631307\pi\)
−0.400912 + 0.916116i \(0.631307\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) 0.863254 0.0607384
\(203\) −2.66503 −0.187048
\(204\) 0 0
\(205\) −5.60601 −0.391540
\(206\) −1.53074 −0.106652
\(207\) 0 0
\(208\) 3.95044 0.273914
\(209\) 19.9718 1.38148
\(210\) 0 0
\(211\) 23.1129 1.59116 0.795579 0.605850i \(-0.207167\pi\)
0.795579 + 0.605850i \(0.207167\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) 17.6274 1.20498
\(215\) 8.00000 0.545595
\(216\) 0 0
\(217\) −18.3701 −1.24705
\(218\) 2.91281 0.197280
\(219\) 0 0
\(220\) 6.46926 0.436157
\(221\) 13.5565 0.911906
\(222\) 0 0
\(223\) −5.72651 −0.383475 −0.191738 0.981446i \(-0.561412\pi\)
−0.191738 + 0.981446i \(0.561412\pi\)
\(224\) −3.08719 −0.206272
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 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 \(0.678237\pi\)
\(230\) −1.00000 −0.0659380
\(231\) 0 0
\(232\) 0.863254 0.0566754
\(233\) 6.44787 0.422414 0.211207 0.977441i \(-0.432261\pi\)
0.211207 + 0.977441i \(0.432261\pi\)
\(234\) 0 0
\(235\) −3.90089 −0.254466
\(236\) −6.86325 −0.446760
\(237\) 0 0
\(238\) −10.5941 −0.686712
\(239\) −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 \(0.492165\pi\)
\(242\) −30.8513 −1.98320
\(243\) 0 0
\(244\) −13.5069 −0.864690
\(245\) 2.53074 0.161683
\(246\) 0 0
\(247\) 12.1958 0.775998
\(248\) 5.95044 0.377854
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 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
\(254\) −20.9385 −1.31380
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 9.90089 0.617600 0.308800 0.951127i \(-0.400073\pi\)
0.308800 + 0.951127i \(0.400073\pi\)
\(258\) 0 0
\(259\) −21.7265 −1.35002
\(260\) 3.95044 0.244996
\(261\) 0 0
\(262\) 10.7641 0.665011
\(263\) 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
\(266\) −9.53074 −0.584367
\(267\) 0 0
\(268\) −10.0753 −0.615445
\(269\) 6.93852 0.423049 0.211525 0.977373i \(-0.432157\pi\)
0.211525 + 0.977373i \(0.432157\pi\)
\(270\) 0 0
\(271\) −10.2992 −0.625632 −0.312816 0.949814i \(-0.601272\pi\)
−0.312816 + 0.949814i \(0.601272\pi\)
\(272\) 3.43163 0.208073
\(273\) 0 0
\(274\) 3.26157 0.197039
\(275\) 6.46926 0.390111
\(276\) 0 0
\(277\) −17.8018 −1.06961 −0.534803 0.844977i \(-0.679614\pi\)
−0.534803 + 0.844977i \(0.679614\pi\)
\(278\) 16.9385 1.01590
\(279\) 0 0
\(280\) −3.08719 −0.184495
\(281\) 18.9385 1.12978 0.564889 0.825167i \(-0.308919\pi\)
0.564889 + 0.825167i \(0.308919\pi\)
\(282\) 0 0
\(283\) 12.6889 0.754275 0.377138 0.926157i \(-0.376908\pi\)
0.377138 + 0.926157i \(0.376908\pi\)
\(284\) −2.56837 −0.152405
\(285\) 0 0
\(286\) −25.5565 −1.51118
\(287\) −17.3068 −1.02159
\(288\) 0 0
\(289\) −5.22394 −0.307290
\(290\) 0.863254 0.0506920
\(291\) 0 0
\(292\) 5.90089 0.345323
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) −6.86325 −0.399594
\(296\) 7.03763 0.409054
\(297\) 0 0
\(298\) 3.26157 0.188938
\(299\) 3.95044 0.228460
\(300\) 0 0
\(301\) 24.6975 1.42354
\(302\) −0.294881 −0.0169685
\(303\) 0 0
\(304\) 3.08719 0.177062
\(305\) −13.5069 −0.773402
\(306\) 0 0
\(307\) −22.9599 −1.31039 −0.655196 0.755459i \(-0.727414\pi\)
−0.655196 + 0.755459i \(0.727414\pi\)
\(308\) 19.9718 1.13800
\(309\) 0 0
\(310\) 5.95044 0.337962
\(311\) −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 \(0.359036\pi\)
\(314\) −7.13675 −0.402750
\(315\) 0 0
\(316\) 15.8018 0.888919
\(317\) −14.0257 −0.787762 −0.393881 0.919161i \(-0.628868\pi\)
−0.393881 + 0.919161i \(0.628868\pi\)
\(318\) 0 0
\(319\) −5.58462 −0.312678
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) −3.08719 −0.172042
\(323\) 10.5941 0.589471
\(324\) 0 0
\(325\) 3.95044 0.219131
\(326\) 8.12482 0.449992
\(327\) 0 0
\(328\) 5.60601 0.309540
\(329\) −12.0428 −0.663940
\(330\) 0 0
\(331\) −24.7403 −1.35985 −0.679925 0.733282i \(-0.737988\pi\)
−0.679925 + 0.733282i \(0.737988\pi\)
\(332\) −9.03763 −0.496005
\(333\) 0 0
\(334\) −7.80178 −0.426894
\(335\) −10.0753 −0.550471
\(336\) 0 0
\(337\) 14.7146 0.801555 0.400777 0.916176i \(-0.368740\pi\)
0.400777 + 0.916176i \(0.368740\pi\)
\(338\) −2.60601 −0.141748
\(339\) 0 0
\(340\) 3.43163 0.186106
\(341\) −38.4950 −2.08462
\(342\) 0 0
\(343\) −13.7975 −0.744993
\(344\) −8.00000 −0.431331
\(345\) 0 0
\(346\) −19.3325 −1.03932
\(347\) −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 \(0.336597\pi\)
\(350\) −3.08719 −0.165017
\(351\) 0 0
\(352\) −6.46926 −0.344813
\(353\) 33.1129 1.76242 0.881211 0.472723i \(-0.156729\pi\)
0.881211 + 0.472723i \(0.156729\pi\)
\(354\) 0 0
\(355\) −2.56837 −0.136315
\(356\) −16.7641 −0.888498
\(357\) 0 0
\(358\) 2.17438 0.114920
\(359\) −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
\(362\) 7.26157 0.381660
\(363\) 0 0
\(364\) 12.1958 0.639232
\(365\) 5.90089 0.308867
\(366\) 0 0
\(367\) −2.27349 −0.118675 −0.0593376 0.998238i \(-0.518899\pi\)
−0.0593376 + 0.998238i \(0.518899\pi\)
\(368\) 1.00000 0.0521286
\(369\) 0 0
\(370\) 7.03763 0.365869
\(371\) 18.5231 0.961673
\(372\) 0 0
\(373\) 23.9762 1.24144 0.620719 0.784033i \(-0.286841\pi\)
0.620719 + 0.784033i \(0.286841\pi\)
\(374\) −22.2001 −1.14794
\(375\) 0 0
\(376\) 3.90089 0.201173
\(377\) −3.41024 −0.175636
\(378\) 0 0
\(379\) −13.8795 −0.712942 −0.356471 0.934306i \(-0.616020\pi\)
−0.356471 + 0.934306i \(0.616020\pi\)
\(380\) 3.08719 0.158369
\(381\) 0 0
\(382\) 8.58976 0.439490
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 19.9718 1.01786
\(386\) −2.44787 −0.124593
\(387\) 0 0
\(388\) −14.2949 −0.725713
\(389\) 26.6436 1.35089 0.675443 0.737412i \(-0.263952\pi\)
0.675443 + 0.737412i \(0.263952\pi\)
\(390\) 0 0
\(391\) 3.43163 0.173545
\(392\) −2.53074 −0.127822
\(393\) 0 0
\(394\) −10.7428 −0.541212
\(395\) 15.8018 0.795074
\(396\) 0 0
\(397\) −8.66749 −0.435009 −0.217504 0.976059i \(-0.569792\pi\)
−0.217504 + 0.976059i \(0.569792\pi\)
\(398\) 11.3111 0.566975
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −39.9762 −1.99631 −0.998157 0.0606854i \(-0.980671\pi\)
−0.998157 + 0.0606854i \(0.980671\pi\)
\(402\) 0 0
\(403\) −23.5069 −1.17096
\(404\) −0.863254 −0.0429485
\(405\) 0 0
\(406\) 2.66503 0.132263
\(407\) −45.5283 −2.25675
\(408\) 0 0
\(409\) 30.1248 1.48958 0.744788 0.667301i \(-0.232550\pi\)
0.744788 + 0.667301i \(0.232550\pi\)
\(410\) 5.60601 0.276861
\(411\) 0 0
\(412\) 1.53074 0.0754141
\(413\) −21.1882 −1.04260
\(414\) 0 0
\(415\) −9.03763 −0.443640
\(416\) −3.95044 −0.193686
\(417\) 0 0
\(418\) −19.9718 −0.976854
\(419\) −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 \(0.584085\pi\)
\(422\) −23.1129 −1.12512
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 3.43163 0.166458
\(426\) 0 0
\(427\) −41.6983 −2.01792
\(428\) −17.6274 −0.852052
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) −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 \(0.334827\pi\)
\(434\) 18.3701 0.881795
\(435\) 0 0
\(436\) −2.91281 −0.139498
\(437\) 3.08719 0.147680
\(438\) 0 0
\(439\) −16.2239 −0.774326 −0.387163 0.922011i \(-0.626545\pi\)
−0.387163 + 0.922011i \(0.626545\pi\)
\(440\) −6.46926 −0.308410
\(441\) 0 0
\(442\) −13.5565 −0.644815
\(443\) 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
\(446\) 5.72651 0.271158
\(447\) 0 0
\(448\) 3.08719 0.145856
\(449\) −22.5727 −1.06527 −0.532636 0.846345i \(-0.678798\pi\)
−0.532636 + 0.846345i \(0.678798\pi\)
\(450\) 0 0
\(451\) −36.2667 −1.70773
\(452\) 6.00000 0.282216
\(453\) 0 0
\(454\) −17.6274 −0.827295
\(455\) 12.1958 0.571746
\(456\) 0 0
\(457\) −1.45302 −0.0679693 −0.0339846 0.999422i \(-0.510820\pi\)
−0.0339846 + 0.999422i \(0.510820\pi\)
\(458\) 16.0753 0.751148
\(459\) 0 0
\(460\) 1.00000 0.0466252
\(461\) 29.7027 1.38339 0.691695 0.722189i \(-0.256864\pi\)
0.691695 + 0.722189i \(0.256864\pi\)
\(462\) 0 0
\(463\) 22.9624 1.06715 0.533576 0.845752i \(-0.320848\pi\)
0.533576 + 0.845752i \(0.320848\pi\)
\(464\) −0.863254 −0.0400756
\(465\) 0 0
\(466\) −6.44787 −0.298692
\(467\) −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
\(470\) 3.90089 0.179935
\(471\) 0 0
\(472\) 6.86325 0.315907
\(473\) 51.7541 2.37966
\(474\) 0 0
\(475\) 3.08719 0.141650
\(476\) 10.5941 0.485579
\(477\) 0 0
\(478\) 4.34876 0.198908
\(479\) 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
\(482\) −0.764142 −0.0348057
\(483\) 0 0
\(484\) 30.8513 1.40233
\(485\) −14.2949 −0.649097
\(486\) 0 0
\(487\) −21.1367 −0.957797 −0.478899 0.877870i \(-0.658964\pi\)
−0.478899 + 0.877870i \(0.658964\pi\)
\(488\) 13.5069 0.611428
\(489\) 0 0
\(490\) −2.53074 −0.114327
\(491\) 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
\(494\) −12.1958 −0.548714
\(495\) 0 0
\(496\) −5.95044 −0.267183
\(497\) −7.92905 −0.355667
\(498\) 0 0
\(499\) 3.80178 0.170191 0.0850954 0.996373i \(-0.472880\pi\)
0.0850954 + 0.996373i \(0.472880\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) −22.5402 −1.00602
\(503\) −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
\(506\) −6.46926 −0.287594
\(507\) 0 0
\(508\) 20.9385 0.928997
\(509\) 9.41024 0.417101 0.208551 0.978012i \(-0.433125\pi\)
0.208551 + 0.978012i \(0.433125\pi\)
\(510\) 0 0
\(511\) 18.2172 0.805880
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −9.90089 −0.436709
\(515\) 1.53074 0.0674524
\(516\) 0 0
\(517\) −25.2359 −1.10987
\(518\) 21.7265 0.954608
\(519\) 0 0
\(520\) −3.95044 −0.173238
\(521\) −25.8018 −1.13040 −0.565198 0.824955i \(-0.691200\pi\)
−0.565198 + 0.824955i \(0.691200\pi\)
\(522\) 0 0
\(523\) −19.1129 −0.835749 −0.417874 0.908505i \(-0.637225\pi\)
−0.417874 + 0.908505i \(0.637225\pi\)
\(524\) −10.7641 −0.470234
\(525\) 0 0
\(526\) −7.25725 −0.316431
\(527\) −20.4197 −0.889496
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −6.00000 −0.260623
\(531\) 0 0
\(532\) 9.53074 0.413210
\(533\) −22.1462 −0.959259
\(534\) 0 0
\(535\) −17.6274 −0.762099
\(536\) 10.0753 0.435185
\(537\) 0 0
\(538\) −6.93852 −0.299141
\(539\) 16.3720 0.705193
\(540\) 0 0
\(541\) 30.8394 1.32589 0.662945 0.748668i \(-0.269306\pi\)
0.662945 + 0.748668i \(0.269306\pi\)
\(542\) 10.2992 0.442389
\(543\) 0 0
\(544\) −3.43163 −0.147130
\(545\) −2.91281 −0.124771
\(546\) 0 0
\(547\) 13.8513 0.592240 0.296120 0.955151i \(-0.404307\pi\)
0.296120 + 0.955151i \(0.404307\pi\)
\(548\) −3.26157 −0.139327
\(549\) 0 0
\(550\) −6.46926 −0.275850
\(551\) −2.66503 −0.113534
\(552\) 0 0
\(553\) 48.7831 2.07447
\(554\) 17.8018 0.756325
\(555\) 0 0
\(556\) −16.9385 −0.718353
\(557\) 28.7641 1.21878 0.609388 0.792872i \(-0.291415\pi\)
0.609388 + 0.792872i \(0.291415\pi\)
\(558\) 0 0
\(559\) 31.6036 1.33669
\(560\) 3.08719 0.130458
\(561\) 0 0
\(562\) −18.9385 −0.798873
\(563\) −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
\(566\) −12.6889 −0.533353
\(567\) 0 0
\(568\) 2.56837 0.107767
\(569\) 10.3488 0.433843 0.216921 0.976189i \(-0.430399\pi\)
0.216921 + 0.976189i \(0.430399\pi\)
\(570\) 0 0
\(571\) 19.6060 0.820486 0.410243 0.911976i \(-0.365444\pi\)
0.410243 + 0.911976i \(0.365444\pi\)
\(572\) 25.5565 1.06857
\(573\) 0 0
\(574\) 17.3068 0.722372
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) −34.1505 −1.42171 −0.710853 0.703341i \(-0.751691\pi\)
−0.710853 + 0.703341i \(0.751691\pi\)
\(578\) 5.22394 0.217287
\(579\) 0 0
\(580\) −0.863254 −0.0358447
\(581\) −27.9009 −1.15752
\(582\) 0 0
\(583\) 38.8156 1.60758
\(584\) −5.90089 −0.244180
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) 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
\(590\) 6.86325 0.282556
\(591\) 0 0
\(592\) −7.03763 −0.289245
\(593\) 2.09911 0.0862002 0.0431001 0.999071i \(-0.486277\pi\)
0.0431001 + 0.999071i \(0.486277\pi\)
\(594\) 0 0
\(595\) 10.5941 0.434315
\(596\) −3.26157 −0.133599
\(597\) 0 0
\(598\) −3.95044 −0.161546
\(599\) −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 \(0.437418\pi\)
\(602\) −24.6975 −1.00660
\(603\) 0 0
\(604\) 0.294881 0.0119986
\(605\) 30.8513 1.25428
\(606\) 0 0
\(607\) −24.6975 −1.00244 −0.501221 0.865320i \(-0.667115\pi\)
−0.501221 + 0.865320i \(0.667115\pi\)
\(608\) −3.08719 −0.125202
\(609\) 0 0
\(610\) 13.5069 0.546878
\(611\) −15.4102 −0.623431
\(612\) 0 0
\(613\) −26.3488 −1.06422 −0.532108 0.846676i \(-0.678600\pi\)
−0.532108 + 0.846676i \(0.678600\pi\)
\(614\) 22.9599 0.926587
\(615\) 0 0
\(616\) −19.9718 −0.804688
\(617\) −5.94612 −0.239382 −0.119691 0.992811i \(-0.538190\pi\)
−0.119691 + 0.992811i \(0.538190\pi\)
\(618\) 0 0
\(619\) 39.6856 1.59510 0.797549 0.603254i \(-0.206129\pi\)
0.797549 + 0.603254i \(0.206129\pi\)
\(620\) −5.95044 −0.238976
\(621\) 0 0
\(622\) 18.0753 0.724752
\(623\) −51.7541 −2.07348
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −15.1625 −0.606014
\(627\) 0 0
\(628\) 7.13675 0.284787
\(629\) −24.1505 −0.962945
\(630\) 0 0
\(631\) −23.0138 −0.916164 −0.458082 0.888910i \(-0.651463\pi\)
−0.458082 + 0.888910i \(0.651463\pi\)
\(632\) −15.8018 −0.628561
\(633\) 0 0
\(634\) 14.0257 0.557032
\(635\) 20.9385 0.830920
\(636\) 0 0
\(637\) 9.99754 0.396117
\(638\) 5.58462 0.221097
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −25.4617 −1.00568 −0.502838 0.864381i \(-0.667711\pi\)
−0.502838 + 0.864381i \(0.667711\pi\)
\(642\) 0 0
\(643\) −16.4479 −0.648641 −0.324320 0.945947i \(-0.605136\pi\)
−0.324320 + 0.945947i \(0.605136\pi\)
\(644\) 3.08719 0.121652
\(645\) 0 0
\(646\) −10.5941 −0.416819
\(647\) 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
\(650\) −3.95044 −0.154949
\(651\) 0 0
\(652\) −8.12482 −0.318193
\(653\) 34.2754 1.34130 0.670649 0.741775i \(-0.266016\pi\)
0.670649 + 0.741775i \(0.266016\pi\)
\(654\) 0 0
\(655\) −10.7641 −0.420590
\(656\) −5.60601 −0.218878
\(657\) 0 0
\(658\) 12.0428 0.469476
\(659\) −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 \(0.312492\pi\)
\(662\) 24.7403 0.961559
\(663\) 0 0
\(664\) 9.03763 0.350728
\(665\) 9.53074 0.369586
\(666\) 0 0
\(667\) −0.863254 −0.0334253
\(668\) 7.80178 0.301860
\(669\) 0 0
\(670\) 10.0753 0.389242
\(671\) −87.3796 −3.37325
\(672\) 0 0
\(673\) −23.8770 −0.920392 −0.460196 0.887817i \(-0.652221\pi\)
−0.460196 + 0.887817i \(0.652221\pi\)
\(674\) −14.7146 −0.566785
\(675\) 0 0
\(676\) 2.60601 0.100231
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 0 0
\(679\) −44.1310 −1.69359
\(680\) −3.43163 −0.131597
\(681\) 0 0
\(682\) 38.4950 1.47405
\(683\) −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
\(686\) 13.7975 0.526789
\(687\) 0 0
\(688\) 8.00000 0.304997
\(689\) 23.7027 0.903000
\(690\) 0 0
\(691\) −18.1744 −0.691386 −0.345693 0.938348i \(-0.612356\pi\)
−0.345693 + 0.938348i \(0.612356\pi\)
\(692\) 19.3325 0.734912
\(693\) 0 0
\(694\) 9.43163 0.358020
\(695\) −16.9385 −0.642515
\(696\) 0 0
\(697\) −19.2377 −0.728681
\(698\) −18.3488 −0.694511
\(699\) 0 0
\(700\) 3.08719 0.116685
\(701\) 40.6907 1.53687 0.768434 0.639929i \(-0.221036\pi\)
0.768434 + 0.639929i \(0.221036\pi\)
\(702\) 0 0
\(703\) −21.7265 −0.819431
\(704\) 6.46926 0.243819
\(705\) 0 0
\(706\) −33.1129 −1.24622
\(707\) −2.66503 −0.100229
\(708\) 0 0
\(709\) −26.4454 −0.993178 −0.496589 0.867986i \(-0.665414\pi\)
−0.496589 + 0.867986i \(0.665414\pi\)
\(710\) 2.56837 0.0963893
\(711\) 0 0
\(712\) 16.7641 0.628263
\(713\) −5.95044 −0.222846
\(714\) 0 0
\(715\) 25.5565 0.955757
\(716\) −2.17438 −0.0812604
\(717\) 0 0
\(718\) 33.0376 1.23295
\(719\) 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
\(722\) 9.46926 0.352409
\(723\) 0 0
\(724\) −7.26157 −0.269874
\(725\) −0.863254 −0.0320605
\(726\) 0 0
\(727\) 21.4830 0.796762 0.398381 0.917220i \(-0.369572\pi\)
0.398381 + 0.917220i \(0.369572\pi\)
\(728\) −12.1958 −0.452005
\(729\) 0 0
\(730\) −5.90089 −0.218402
\(731\) 27.4530 1.01539
\(732\) 0 0
\(733\) 11.3778 0.420247 0.210123 0.977675i \(-0.432613\pi\)
0.210123 + 0.977675i \(0.432613\pi\)
\(734\) 2.27349 0.0839161
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) −65.1795 −2.40092
\(738\) 0 0
\(739\) 21.8770 0.804760 0.402380 0.915473i \(-0.368183\pi\)
0.402380 + 0.915473i \(0.368183\pi\)
\(740\) −7.03763 −0.258709
\(741\) 0 0
\(742\) −18.5231 −0.680006
\(743\) −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
\(746\) −23.9762 −0.877829
\(747\) 0 0
\(748\) 22.2001 0.811716
\(749\) −54.4191 −1.98843
\(750\) 0 0
\(751\) 24.3915 0.890060 0.445030 0.895516i \(-0.353193\pi\)
0.445030 + 0.895516i \(0.353193\pi\)
\(752\) −3.90089 −0.142251
\(753\) 0 0
\(754\) 3.41024 0.124194
\(755\) 0.294881 0.0107318
\(756\) 0 0
\(757\) 41.9437 1.52447 0.762234 0.647301i \(-0.224102\pi\)
0.762234 + 0.647301i \(0.224102\pi\)
\(758\) 13.8795 0.504126
\(759\) 0 0
\(760\) −3.08719 −0.111984
\(761\) 18.3745 0.666074 0.333037 0.942914i \(-0.391927\pi\)
0.333037 + 0.942914i \(0.391927\pi\)
\(762\) 0 0
\(763\) −8.99240 −0.325547
\(764\) −8.58976 −0.310767
\(765\) 0 0
\(766\) 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 \(0.222105\pi\)
\(770\) −19.9718 −0.719735
\(771\) 0 0
\(772\) 2.44787 0.0881008
\(773\) 15.0376 0.540866 0.270433 0.962739i \(-0.412833\pi\)
0.270433 + 0.962739i \(0.412833\pi\)
\(774\) 0 0
\(775\) −5.95044 −0.213746
\(776\) 14.2949 0.513156
\(777\) 0 0
\(778\) −26.6436 −0.955221
\(779\) −17.3068 −0.620081
\(780\) 0 0
\(781\) −16.6155 −0.594548
\(782\) −3.43163 −0.122715
\(783\) 0 0
\(784\) 2.53074 0.0903836
\(785\) 7.13675 0.254721
\(786\) 0 0
\(787\) −8.39154 −0.299126 −0.149563 0.988752i \(-0.547787\pi\)
−0.149563 + 0.988752i \(0.547787\pi\)
\(788\) 10.7428 0.382695
\(789\) 0 0
\(790\) −15.8018 −0.562202
\(791\) 18.5231 0.658607
\(792\) 0 0
\(793\) −53.3582 −1.89481
\(794\) 8.66749 0.307598
\(795\) 0 0
\(796\) −11.3111 −0.400912
\(797\) 22.0514 0.781101 0.390551 0.920581i \(-0.372285\pi\)
0.390551 + 0.920581i \(0.372285\pi\)
\(798\) 0 0
\(799\) −13.3864 −0.473576
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 39.9762 1.41161
\(803\) 38.1744 1.34714
\(804\) 0 0
\(805\) 3.08719 0.108809
\(806\) 23.5069 0.827995
\(807\) 0 0
\(808\) 0.863254 0.0303692
\(809\) −2.04524 −0.0719066 −0.0359533 0.999353i \(-0.511447\pi\)
−0.0359533 + 0.999353i \(0.511447\pi\)
\(810\) 0 0
\(811\) 4.24965 0.149225 0.0746126 0.997213i \(-0.476228\pi\)
0.0746126 + 0.997213i \(0.476228\pi\)
\(812\) −2.66503 −0.0935242
\(813\) 0 0
\(814\) 45.5283 1.59577
\(815\) −8.12482 −0.284600
\(816\) 0 0
\(817\) 24.6975 0.864057
\(818\) −30.1248 −1.05329
\(819\) 0 0
\(820\) −5.60601 −0.195770
\(821\) 29.2548 1.02100 0.510500 0.859878i \(-0.329460\pi\)
0.510500 + 0.859878i \(0.329460\pi\)
\(822\) 0 0
\(823\) 13.5846 0.473530 0.236765 0.971567i \(-0.423913\pi\)
0.236765 + 0.971567i \(0.423913\pi\)
\(824\) −1.53074 −0.0533258
\(825\) 0 0
\(826\) 21.1882 0.737231
\(827\) −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 \(0.349953\pi\)
\(830\) 9.03763 0.313701
\(831\) 0 0
\(832\) 3.95044 0.136957
\(833\) 8.68455 0.300902
\(834\) 0 0
\(835\) 7.80178 0.269992
\(836\) 19.9718 0.690740
\(837\) 0 0
\(838\) 25.8770 0.893908
\(839\) −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
\(842\) 10.7146 0.369249
\(843\) 0 0
\(844\) 23.1129 0.795579
\(845\) 2.60601 0.0896493
\(846\) 0 0
\(847\) 95.2439 3.27262
\(848\) 6.00000 0.206041
\(849\) 0 0
\(850\) −3.43163 −0.117704
\(851\) −7.03763 −0.241247
\(852\) 0 0
\(853\) −33.6293 −1.15144 −0.575722 0.817646i \(-0.695279\pi\)
−0.575722 + 0.817646i \(0.695279\pi\)
\(854\) 41.6983 1.42689
\(855\) 0 0
\(856\) 17.6274 0.602492
\(857\) −11.1795 −0.381885 −0.190943 0.981601i \(-0.561154\pi\)
−0.190943 + 0.981601i \(0.561154\pi\)
\(858\) 0 0
\(859\) 47.9009 1.63436 0.817179 0.576385i \(-0.195537\pi\)
0.817179 + 0.576385i \(0.195537\pi\)
\(860\) 8.00000 0.272798
\(861\) 0 0
\(862\) 32.2496 1.09843
\(863\) 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
\(866\) −20.6393 −0.701353
\(867\) 0 0
\(868\) −18.3701 −0.623523
\(869\) 102.226 3.46777
\(870\) 0 0
\(871\) −39.8018 −1.34863
\(872\) 2.91281 0.0986402
\(873\) 0 0
\(874\) −3.08719 −0.104426
\(875\) 3.08719 0.104366
\(876\) 0 0
\(877\) −22.3940 −0.756191 −0.378096 0.925767i \(-0.623421\pi\)
−0.378096 + 0.925767i \(0.623421\pi\)
\(878\) 16.2239 0.547531
\(879\) 0 0
\(880\) 6.46926 0.218079
\(881\) −21.1129 −0.711312 −0.355656 0.934617i \(-0.615742\pi\)
−0.355656 + 0.934617i \(0.615742\pi\)
\(882\) 0 0
\(883\) 49.1061 1.65255 0.826276 0.563265i \(-0.190455\pi\)
0.826276 + 0.563265i \(0.190455\pi\)
\(884\) 13.5565 0.455953
\(885\) 0 0
\(886\) −15.6770 −0.526678
\(887\) 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
\(890\) 16.7641 0.561935
\(891\) 0 0
\(892\) −5.72651 −0.191738
\(893\) −12.0428 −0.402996
\(894\) 0 0
\(895\) −2.17438 −0.0726815
\(896\) −3.08719 −0.103136
\(897\) 0 0
\(898\) 22.5727 0.753261
\(899\) 5.13675 0.171320
\(900\) 0 0
\(901\) 20.5898 0.685944
\(902\) 36.2667 1.20755
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) −7.26157 −0.241383
\(906\) 0 0
\(907\) −37.9762 −1.26098 −0.630489 0.776198i \(-0.717146\pi\)
−0.630489 + 0.776198i \(0.717146\pi\)
\(908\) 17.6274 0.584986
\(909\) 0 0
\(910\) −12.1958 −0.404286
\(911\) 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
\(914\) 1.45302 0.0480615
\(915\) 0 0
\(916\) −16.0753 −0.531142
\(917\) −33.2309 −1.09738
\(918\) 0 0
\(919\) 30.4240 1.00360 0.501798 0.864985i \(-0.332672\pi\)
0.501798 + 0.864985i \(0.332672\pi\)
\(920\) −1.00000 −0.0329690
\(921\) 0 0
\(922\) −29.7027 −0.978205
\(923\) −10.1462 −0.333967
\(924\) 0 0
\(925\) −7.03763 −0.231396
\(926\) −22.9624 −0.754590
\(927\) 0 0
\(928\) 0.863254 0.0283377
\(929\) 37.8018 1.24024 0.620118 0.784509i \(-0.287085\pi\)
0.620118 + 0.784509i \(0.287085\pi\)
\(930\) 0 0
\(931\) 7.81287 0.256057
\(932\) 6.44787 0.211207
\(933\) 0 0
\(934\) 9.48550 0.310375
\(935\) 22.2001 0.726021
\(936\) 0 0
\(937\) −32.6907 −1.06796 −0.533980 0.845497i \(-0.679304\pi\)
−0.533980 + 0.845497i \(0.679304\pi\)
\(938\) 31.1043 1.01559
\(939\) 0 0
\(940\) −3.90089 −0.127233
\(941\) 46.4882 1.51547 0.757736 0.652561i \(-0.226306\pi\)
0.757736 + 0.652561i \(0.226306\pi\)
\(942\) 0 0
\(943\) −5.60601 −0.182557
\(944\) −6.86325 −0.223380
\(945\) 0 0
\(946\) −51.7541 −1.68267
\(947\) 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
\(950\) −3.08719 −0.100162
\(951\) 0 0
\(952\) −10.5941 −0.343356
\(953\) −49.1104 −1.59084 −0.795422 0.606056i \(-0.792751\pi\)
−0.795422 + 0.606056i \(0.792751\pi\)
\(954\) 0 0
\(955\) −8.58976 −0.277958
\(956\) −4.34876 −0.140649
\(957\) 0 0
\(958\) −30.5659 −0.987540
\(959\) −10.0691 −0.325148
\(960\) 0 0
\(961\) 4.40778 0.142187
\(962\) 27.8018 0.896365
\(963\) 0 0
\(964\) 0.764142 0.0246114
\(965\) 2.44787 0.0787997
\(966\) 0 0
\(967\) 1.96751 0.0632709 0.0316355 0.999499i \(-0.489928\pi\)
0.0316355 + 0.999499i \(0.489928\pi\)
\(968\) −30.8513 −0.991599
\(969\) 0 0
\(970\) 14.2949 0.458981
\(971\) −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
\(974\) 21.1367 0.677265
\(975\) 0 0
\(976\) −13.5069 −0.432345
\(977\) 0.639319 0.0204536 0.0102268 0.999948i \(-0.496745\pi\)
0.0102268 + 0.999948i \(0.496745\pi\)
\(978\) 0 0
\(979\) −108.452 −3.46613
\(980\) 2.53074 0.0808415
\(981\) 0 0
\(982\) −28.0514 −0.895157
\(983\) 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
\(986\) 2.96237 0.0943410
\(987\) 0 0
\(988\) 12.1958 0.387999
\(989\) 8.00000 0.254385
\(990\) 0 0
\(991\) 37.2334 1.18276 0.591379 0.806394i \(-0.298584\pi\)
0.591379 + 0.806394i \(0.298584\pi\)
\(992\) 5.95044 0.188927
\(993\) 0 0
\(994\) 7.92905 0.251494
\(995\) −11.3111 −0.358587
\(996\) 0 0
\(997\) 9.65124 0.305658 0.152829 0.988253i \(-0.451162\pi\)
0.152829 + 0.988253i \(0.451162\pi\)
\(998\) −3.80178 −0.120343
\(999\) 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 2070.2.a.z.1.2 3
3.2 odd 2 230.2.a.d.1.2 3
12.11 even 2 1840.2.a.r.1.2 3
15.2 even 4 1150.2.b.j.599.5 6
15.8 even 4 1150.2.b.j.599.2 6
15.14 odd 2 1150.2.a.q.1.2 3
24.5 odd 2 7360.2.a.bz.1.2 3
24.11 even 2 7360.2.a.ce.1.2 3
60.59 even 2 9200.2.a.cf.1.2 3
69.68 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 3.2 odd 2
1150.2.a.q.1.2 3 15.14 odd 2
1150.2.b.j.599.2 6 15.8 even 4
1150.2.b.j.599.5 6 15.2 even 4
1840.2.a.r.1.2 3 12.11 even 2
2070.2.a.z.1.2 3 1.1 even 1 trivial
5290.2.a.r.1.2 3 69.68 even 2
7360.2.a.bz.1.2 3 24.5 odd 2
7360.2.a.ce.1.2 3 24.11 even 2
9200.2.a.cf.1.2 3 60.59 even 2