Properties

Label 9702.2.a.dp.1.2
Level $9702$
Weight $2$
Character 9702.1
Self dual yes
Analytic conductor $77.471$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9702.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(77.4708600410\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 9702.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.41421 q^{5} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.41421 q^{5} +1.00000 q^{8} +1.41421 q^{10} +1.00000 q^{11} +1.41421 q^{13} +1.00000 q^{16} -1.41421 q^{17} -2.82843 q^{19} +1.41421 q^{20} +1.00000 q^{22} -4.00000 q^{23} -3.00000 q^{25} +1.41421 q^{26} -6.00000 q^{29} -2.82843 q^{31} +1.00000 q^{32} -1.41421 q^{34} -8.00000 q^{37} -2.82843 q^{38} +1.41421 q^{40} -7.07107 q^{41} -8.00000 q^{43} +1.00000 q^{44} -4.00000 q^{46} -8.48528 q^{47} -3.00000 q^{50} +1.41421 q^{52} +1.41421 q^{55} -6.00000 q^{58} -7.07107 q^{61} -2.82843 q^{62} +1.00000 q^{64} +2.00000 q^{65} -8.00000 q^{67} -1.41421 q^{68} -8.00000 q^{71} +1.41421 q^{73} -8.00000 q^{74} -2.82843 q^{76} +4.00000 q^{79} +1.41421 q^{80} -7.07107 q^{82} -2.82843 q^{83} -2.00000 q^{85} -8.00000 q^{86} +1.00000 q^{88} +9.89949 q^{89} -4.00000 q^{92} -8.48528 q^{94} -4.00000 q^{95} +15.5563 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 2 q^{11} + 2 q^{16} + 2 q^{22} - 8 q^{23} - 6 q^{25} - 12 q^{29} + 2 q^{32} - 16 q^{37} - 16 q^{43} + 2 q^{44} - 8 q^{46} - 6 q^{50} - 12 q^{58} + 2 q^{64} + 4 q^{65} - 16 q^{67} - 16 q^{71} - 16 q^{74} + 8 q^{79} - 4 q^{85} - 16 q^{86} + 2 q^{88} - 8 q^{92} - 8 q^{95} + O(q^{100}) \)

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
\(5\) 1.41421 0.632456 0.316228 0.948683i \(-0.397584\pi\)
0.316228 + 0.948683i \(0.397584\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 1.41421 0.447214
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 1.41421 0.392232 0.196116 0.980581i \(-0.437167\pi\)
0.196116 + 0.980581i \(0.437167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −1.41421 −0.342997 −0.171499 0.985184i \(-0.554861\pi\)
−0.171499 + 0.985184i \(0.554861\pi\)
\(18\) 0 0
\(19\) −2.82843 −0.648886 −0.324443 0.945905i \(-0.605177\pi\)
−0.324443 + 0.945905i \(0.605177\pi\)
\(20\) 1.41421 0.316228
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 1.41421 0.277350
\(27\) 0 0
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −2.82843 −0.508001 −0.254000 0.967204i \(-0.581746\pi\)
−0.254000 + 0.967204i \(0.581746\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −1.41421 −0.242536
\(35\) 0 0
\(36\) 0 0
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) −2.82843 −0.458831
\(39\) 0 0
\(40\) 1.41421 0.223607
\(41\) −7.07107 −1.10432 −0.552158 0.833740i \(-0.686195\pi\)
−0.552158 + 0.833740i \(0.686195\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) −8.48528 −1.23771 −0.618853 0.785507i \(-0.712402\pi\)
−0.618853 + 0.785507i \(0.712402\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −3.00000 −0.424264
\(51\) 0 0
\(52\) 1.41421 0.196116
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 1.41421 0.190693
\(56\) 0 0
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −7.07107 −0.905357 −0.452679 0.891674i \(-0.649532\pi\)
−0.452679 + 0.891674i \(0.649532\pi\)
\(62\) −2.82843 −0.359211
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) −1.41421 −0.171499
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 1.41421 0.165521 0.0827606 0.996569i \(-0.473626\pi\)
0.0827606 + 0.996569i \(0.473626\pi\)
\(74\) −8.00000 −0.929981
\(75\) 0 0
\(76\) −2.82843 −0.324443
\(77\) 0 0
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 1.41421 0.158114
\(81\) 0 0
\(82\) −7.07107 −0.780869
\(83\) −2.82843 −0.310460 −0.155230 0.987878i \(-0.549612\pi\)
−0.155230 + 0.987878i \(0.549612\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) −8.00000 −0.862662
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) 9.89949 1.04934 0.524672 0.851304i \(-0.324188\pi\)
0.524672 + 0.851304i \(0.324188\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −4.00000 −0.417029
\(93\) 0 0
\(94\) −8.48528 −0.875190
\(95\) −4.00000 −0.410391
\(96\) 0 0
\(97\) 15.5563 1.57951 0.789754 0.613424i \(-0.210208\pi\)
0.789754 + 0.613424i \(0.210208\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −3.00000 −0.300000
\(101\) 15.5563 1.54791 0.773957 0.633238i \(-0.218274\pi\)
0.773957 + 0.633238i \(0.218274\pi\)
\(102\) 0 0
\(103\) 8.48528 0.836080 0.418040 0.908429i \(-0.362717\pi\)
0.418040 + 0.908429i \(0.362717\pi\)
\(104\) 1.41421 0.138675
\(105\) 0 0
\(106\) 0 0
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 0 0
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 1.41421 0.134840
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) −5.65685 −0.527504
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −7.07107 −0.640184
\(123\) 0 0
\(124\) −2.82843 −0.254000
\(125\) −11.3137 −1.01193
\(126\) 0 0
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 2.00000 0.175412
\(131\) 2.82843 0.247121 0.123560 0.992337i \(-0.460569\pi\)
0.123560 + 0.992337i \(0.460569\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) −1.41421 −0.121268
\(137\) −10.0000 −0.854358 −0.427179 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 0 0
\(139\) 2.82843 0.239904 0.119952 0.992780i \(-0.461726\pi\)
0.119952 + 0.992780i \(0.461726\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −8.00000 −0.671345
\(143\) 1.41421 0.118262
\(144\) 0 0
\(145\) −8.48528 −0.704664
\(146\) 1.41421 0.117041
\(147\) 0 0
\(148\) −8.00000 −0.657596
\(149\) 4.00000 0.327693 0.163846 0.986486i \(-0.447610\pi\)
0.163846 + 0.986486i \(0.447610\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) −2.82843 −0.229416
\(153\) 0 0
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) −1.41421 −0.112867 −0.0564333 0.998406i \(-0.517973\pi\)
−0.0564333 + 0.998406i \(0.517973\pi\)
\(158\) 4.00000 0.318223
\(159\) 0 0
\(160\) 1.41421 0.111803
\(161\) 0 0
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) −7.07107 −0.552158
\(165\) 0 0
\(166\) −2.82843 −0.219529
\(167\) −11.3137 −0.875481 −0.437741 0.899101i \(-0.644221\pi\)
−0.437741 + 0.899101i \(0.644221\pi\)
\(168\) 0 0
\(169\) −11.0000 −0.846154
\(170\) −2.00000 −0.153393
\(171\) 0 0
\(172\) −8.00000 −0.609994
\(173\) 1.41421 0.107521 0.0537603 0.998554i \(-0.482879\pi\)
0.0537603 + 0.998554i \(0.482879\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) 9.89949 0.741999
\(179\) 16.0000 1.19590 0.597948 0.801535i \(-0.295983\pi\)
0.597948 + 0.801535i \(0.295983\pi\)
\(180\) 0 0
\(181\) 12.7279 0.946059 0.473029 0.881047i \(-0.343160\pi\)
0.473029 + 0.881047i \(0.343160\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −4.00000 −0.294884
\(185\) −11.3137 −0.831800
\(186\) 0 0
\(187\) −1.41421 −0.103418
\(188\) −8.48528 −0.618853
\(189\) 0 0
\(190\) −4.00000 −0.290191
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) −18.0000 −1.29567 −0.647834 0.761781i \(-0.724325\pi\)
−0.647834 + 0.761781i \(0.724325\pi\)
\(194\) 15.5563 1.11688
\(195\) 0 0
\(196\) 0 0
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) 25.4558 1.80452 0.902258 0.431196i \(-0.141908\pi\)
0.902258 + 0.431196i \(0.141908\pi\)
\(200\) −3.00000 −0.212132
\(201\) 0 0
\(202\) 15.5563 1.09454
\(203\) 0 0
\(204\) 0 0
\(205\) −10.0000 −0.698430
\(206\) 8.48528 0.591198
\(207\) 0 0
\(208\) 1.41421 0.0980581
\(209\) −2.82843 −0.195646
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 8.00000 0.546869
\(215\) −11.3137 −0.771589
\(216\) 0 0
\(217\) 0 0
\(218\) 4.00000 0.270914
\(219\) 0 0
\(220\) 1.41421 0.0953463
\(221\) −2.00000 −0.134535
\(222\) 0 0
\(223\) 2.82843 0.189405 0.0947027 0.995506i \(-0.469810\pi\)
0.0947027 + 0.995506i \(0.469810\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 14.1421 0.938647 0.469323 0.883026i \(-0.344498\pi\)
0.469323 + 0.883026i \(0.344498\pi\)
\(228\) 0 0
\(229\) −21.2132 −1.40181 −0.700904 0.713256i \(-0.747220\pi\)
−0.700904 + 0.713256i \(0.747220\pi\)
\(230\) −5.65685 −0.373002
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) −12.0000 −0.782794
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) 9.89949 0.637683 0.318841 0.947808i \(-0.396706\pi\)
0.318841 + 0.947808i \(0.396706\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) −7.07107 −0.452679
\(245\) 0 0
\(246\) 0 0
\(247\) −4.00000 −0.254514
\(248\) −2.82843 −0.179605
\(249\) 0 0
\(250\) −11.3137 −0.715542
\(251\) −11.3137 −0.714115 −0.357057 0.934082i \(-0.616220\pi\)
−0.357057 + 0.934082i \(0.616220\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) −12.0000 −0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 7.07107 0.441081 0.220541 0.975378i \(-0.429218\pi\)
0.220541 + 0.975378i \(0.429218\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 2.00000 0.124035
\(261\) 0 0
\(262\) 2.82843 0.174741
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −8.00000 −0.488678
\(269\) 9.89949 0.603583 0.301791 0.953374i \(-0.402415\pi\)
0.301791 + 0.953374i \(0.402415\pi\)
\(270\) 0 0
\(271\) −16.9706 −1.03089 −0.515444 0.856923i \(-0.672373\pi\)
−0.515444 + 0.856923i \(0.672373\pi\)
\(272\) −1.41421 −0.0857493
\(273\) 0 0
\(274\) −10.0000 −0.604122
\(275\) −3.00000 −0.180907
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 2.82843 0.169638
\(279\) 0 0
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 8.48528 0.504398 0.252199 0.967675i \(-0.418846\pi\)
0.252199 + 0.967675i \(0.418846\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) 1.41421 0.0836242
\(287\) 0 0
\(288\) 0 0
\(289\) −15.0000 −0.882353
\(290\) −8.48528 −0.498273
\(291\) 0 0
\(292\) 1.41421 0.0827606
\(293\) 21.2132 1.23929 0.619644 0.784883i \(-0.287277\pi\)
0.619644 + 0.784883i \(0.287277\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −8.00000 −0.464991
\(297\) 0 0
\(298\) 4.00000 0.231714
\(299\) −5.65685 −0.327144
\(300\) 0 0
\(301\) 0 0
\(302\) 16.0000 0.920697
\(303\) 0 0
\(304\) −2.82843 −0.162221
\(305\) −10.0000 −0.572598
\(306\) 0 0
\(307\) 2.82843 0.161427 0.0807134 0.996737i \(-0.474280\pi\)
0.0807134 + 0.996737i \(0.474280\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −4.00000 −0.227185
\(311\) 19.7990 1.12270 0.561349 0.827579i \(-0.310283\pi\)
0.561349 + 0.827579i \(0.310283\pi\)
\(312\) 0 0
\(313\) 7.07107 0.399680 0.199840 0.979829i \(-0.435958\pi\)
0.199840 + 0.979829i \(0.435958\pi\)
\(314\) −1.41421 −0.0798087
\(315\) 0 0
\(316\) 4.00000 0.225018
\(317\) 8.00000 0.449325 0.224662 0.974437i \(-0.427872\pi\)
0.224662 + 0.974437i \(0.427872\pi\)
\(318\) 0 0
\(319\) −6.00000 −0.335936
\(320\) 1.41421 0.0790569
\(321\) 0 0
\(322\) 0 0
\(323\) 4.00000 0.222566
\(324\) 0 0
\(325\) −4.24264 −0.235339
\(326\) −4.00000 −0.221540
\(327\) 0 0
\(328\) −7.07107 −0.390434
\(329\) 0 0
\(330\) 0 0
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) −2.82843 −0.155230
\(333\) 0 0
\(334\) −11.3137 −0.619059
\(335\) −11.3137 −0.618134
\(336\) 0 0
\(337\) −20.0000 −1.08947 −0.544735 0.838608i \(-0.683370\pi\)
−0.544735 + 0.838608i \(0.683370\pi\)
\(338\) −11.0000 −0.598321
\(339\) 0 0
\(340\) −2.00000 −0.108465
\(341\) −2.82843 −0.153168
\(342\) 0 0
\(343\) 0 0
\(344\) −8.00000 −0.431331
\(345\) 0 0
\(346\) 1.41421 0.0760286
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 18.3848 0.984115 0.492057 0.870563i \(-0.336245\pi\)
0.492057 + 0.870563i \(0.336245\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) 26.8701 1.43015 0.715074 0.699048i \(-0.246393\pi\)
0.715074 + 0.699048i \(0.246393\pi\)
\(354\) 0 0
\(355\) −11.3137 −0.600469
\(356\) 9.89949 0.524672
\(357\) 0 0
\(358\) 16.0000 0.845626
\(359\) 4.00000 0.211112 0.105556 0.994413i \(-0.466338\pi\)
0.105556 + 0.994413i \(0.466338\pi\)
\(360\) 0 0
\(361\) −11.0000 −0.578947
\(362\) 12.7279 0.668965
\(363\) 0 0
\(364\) 0 0
\(365\) 2.00000 0.104685
\(366\) 0 0
\(367\) −8.48528 −0.442928 −0.221464 0.975169i \(-0.571084\pi\)
−0.221464 + 0.975169i \(0.571084\pi\)
\(368\) −4.00000 −0.208514
\(369\) 0 0
\(370\) −11.3137 −0.588172
\(371\) 0 0
\(372\) 0 0
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) −1.41421 −0.0731272
\(375\) 0 0
\(376\) −8.48528 −0.437595
\(377\) −8.48528 −0.437014
\(378\) 0 0
\(379\) −12.0000 −0.616399 −0.308199 0.951322i \(-0.599726\pi\)
−0.308199 + 0.951322i \(0.599726\pi\)
\(380\) −4.00000 −0.205196
\(381\) 0 0
\(382\) 12.0000 0.613973
\(383\) 14.1421 0.722629 0.361315 0.932444i \(-0.382328\pi\)
0.361315 + 0.932444i \(0.382328\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −18.0000 −0.916176
\(387\) 0 0
\(388\) 15.5563 0.789754
\(389\) 10.0000 0.507020 0.253510 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) 0 0
\(391\) 5.65685 0.286079
\(392\) 0 0
\(393\) 0 0
\(394\) 12.0000 0.604551
\(395\) 5.65685 0.284627
\(396\) 0 0
\(397\) −9.89949 −0.496841 −0.248421 0.968652i \(-0.579912\pi\)
−0.248421 + 0.968652i \(0.579912\pi\)
\(398\) 25.4558 1.27599
\(399\) 0 0
\(400\) −3.00000 −0.150000
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 0 0
\(403\) −4.00000 −0.199254
\(404\) 15.5563 0.773957
\(405\) 0 0
\(406\) 0 0
\(407\) −8.00000 −0.396545
\(408\) 0 0
\(409\) 24.0416 1.18878 0.594391 0.804176i \(-0.297393\pi\)
0.594391 + 0.804176i \(0.297393\pi\)
\(410\) −10.0000 −0.493865
\(411\) 0 0
\(412\) 8.48528 0.418040
\(413\) 0 0
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) 1.41421 0.0693375
\(417\) 0 0
\(418\) −2.82843 −0.138343
\(419\) −16.9706 −0.829066 −0.414533 0.910034i \(-0.636055\pi\)
−0.414533 + 0.910034i \(0.636055\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) −8.00000 −0.389434
\(423\) 0 0
\(424\) 0 0
\(425\) 4.24264 0.205798
\(426\) 0 0
\(427\) 0 0
\(428\) 8.00000 0.386695
\(429\) 0 0
\(430\) −11.3137 −0.545595
\(431\) −28.0000 −1.34871 −0.674356 0.738406i \(-0.735579\pi\)
−0.674356 + 0.738406i \(0.735579\pi\)
\(432\) 0 0
\(433\) 4.24264 0.203888 0.101944 0.994790i \(-0.467494\pi\)
0.101944 + 0.994790i \(0.467494\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 4.00000 0.191565
\(437\) 11.3137 0.541208
\(438\) 0 0
\(439\) −33.9411 −1.61992 −0.809961 0.586484i \(-0.800512\pi\)
−0.809961 + 0.586484i \(0.800512\pi\)
\(440\) 1.41421 0.0674200
\(441\) 0 0
\(442\) −2.00000 −0.0951303
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 0 0
\(445\) 14.0000 0.663664
\(446\) 2.82843 0.133930
\(447\) 0 0
\(448\) 0 0
\(449\) 24.0000 1.13263 0.566315 0.824189i \(-0.308369\pi\)
0.566315 + 0.824189i \(0.308369\pi\)
\(450\) 0 0
\(451\) −7.07107 −0.332964
\(452\) 0 0
\(453\) 0 0
\(454\) 14.1421 0.663723
\(455\) 0 0
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) −21.2132 −0.991228
\(459\) 0 0
\(460\) −5.65685 −0.263752
\(461\) −26.8701 −1.25146 −0.625732 0.780038i \(-0.715200\pi\)
−0.625732 + 0.780038i \(0.715200\pi\)
\(462\) 0 0
\(463\) −32.0000 −1.48717 −0.743583 0.668644i \(-0.766875\pi\)
−0.743583 + 0.668644i \(0.766875\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) −16.9706 −0.785304 −0.392652 0.919687i \(-0.628442\pi\)
−0.392652 + 0.919687i \(0.628442\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −12.0000 −0.553519
\(471\) 0 0
\(472\) 0 0
\(473\) −8.00000 −0.367840
\(474\) 0 0
\(475\) 8.48528 0.389331
\(476\) 0 0
\(477\) 0 0
\(478\) −8.00000 −0.365911
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −11.3137 −0.515861
\(482\) 9.89949 0.450910
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 22.0000 0.998969
\(486\) 0 0
\(487\) 28.0000 1.26880 0.634401 0.773004i \(-0.281247\pi\)
0.634401 + 0.773004i \(0.281247\pi\)
\(488\) −7.07107 −0.320092
\(489\) 0 0
\(490\) 0 0
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 0 0
\(493\) 8.48528 0.382158
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) −2.82843 −0.127000
\(497\) 0 0
\(498\) 0 0
\(499\) −24.0000 −1.07439 −0.537194 0.843459i \(-0.680516\pi\)
−0.537194 + 0.843459i \(0.680516\pi\)
\(500\) −11.3137 −0.505964
\(501\) 0 0
\(502\) −11.3137 −0.504956
\(503\) −5.65685 −0.252227 −0.126113 0.992016i \(-0.540250\pi\)
−0.126113 + 0.992016i \(0.540250\pi\)
\(504\) 0 0
\(505\) 22.0000 0.978987
\(506\) −4.00000 −0.177822
\(507\) 0 0
\(508\) −12.0000 −0.532414
\(509\) −12.7279 −0.564155 −0.282078 0.959392i \(-0.591024\pi\)
−0.282078 + 0.959392i \(0.591024\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 7.07107 0.311891
\(515\) 12.0000 0.528783
\(516\) 0 0
\(517\) −8.48528 −0.373182
\(518\) 0 0
\(519\) 0 0
\(520\) 2.00000 0.0877058
\(521\) −7.07107 −0.309789 −0.154895 0.987931i \(-0.549504\pi\)
−0.154895 + 0.987931i \(0.549504\pi\)
\(522\) 0 0
\(523\) −42.4264 −1.85518 −0.927589 0.373603i \(-0.878122\pi\)
−0.927589 + 0.373603i \(0.878122\pi\)
\(524\) 2.82843 0.123560
\(525\) 0 0
\(526\) −12.0000 −0.523225
\(527\) 4.00000 0.174243
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −10.0000 −0.433148
\(534\) 0 0
\(535\) 11.3137 0.489134
\(536\) −8.00000 −0.345547
\(537\) 0 0
\(538\) 9.89949 0.426798
\(539\) 0 0
\(540\) 0 0
\(541\) 30.0000 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) −16.9706 −0.728948
\(543\) 0 0
\(544\) −1.41421 −0.0606339
\(545\) 5.65685 0.242313
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) −10.0000 −0.427179
\(549\) 0 0
\(550\) −3.00000 −0.127920
\(551\) 16.9706 0.722970
\(552\) 0 0
\(553\) 0 0
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) 2.82843 0.119952
\(557\) 36.0000 1.52537 0.762684 0.646771i \(-0.223881\pi\)
0.762684 + 0.646771i \(0.223881\pi\)
\(558\) 0 0
\(559\) −11.3137 −0.478519
\(560\) 0 0
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) 2.82843 0.119204 0.0596020 0.998222i \(-0.481017\pi\)
0.0596020 + 0.998222i \(0.481017\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 8.48528 0.356663
\(567\) 0 0
\(568\) −8.00000 −0.335673
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 1.41421 0.0591312
\(573\) 0 0
\(574\) 0 0
\(575\) 12.0000 0.500435
\(576\) 0 0
\(577\) 38.1838 1.58961 0.794805 0.606864i \(-0.207573\pi\)
0.794805 + 0.606864i \(0.207573\pi\)
\(578\) −15.0000 −0.623918
\(579\) 0 0
\(580\) −8.48528 −0.352332
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 1.41421 0.0585206
\(585\) 0 0
\(586\) 21.2132 0.876309
\(587\) 45.2548 1.86787 0.933933 0.357447i \(-0.116353\pi\)
0.933933 + 0.357447i \(0.116353\pi\)
\(588\) 0 0
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) 0 0
\(592\) −8.00000 −0.328798
\(593\) −18.3848 −0.754972 −0.377486 0.926015i \(-0.623211\pi\)
−0.377486 + 0.926015i \(0.623211\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 4.00000 0.163846
\(597\) 0 0
\(598\) −5.65685 −0.231326
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −7.07107 −0.288435 −0.144217 0.989546i \(-0.546066\pi\)
−0.144217 + 0.989546i \(0.546066\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 16.0000 0.651031
\(605\) 1.41421 0.0574960
\(606\) 0 0
\(607\) −16.9706 −0.688814 −0.344407 0.938820i \(-0.611920\pi\)
−0.344407 + 0.938820i \(0.611920\pi\)
\(608\) −2.82843 −0.114708
\(609\) 0 0
\(610\) −10.0000 −0.404888
\(611\) −12.0000 −0.485468
\(612\) 0 0
\(613\) −28.0000 −1.13091 −0.565455 0.824779i \(-0.691299\pi\)
−0.565455 + 0.824779i \(0.691299\pi\)
\(614\) 2.82843 0.114146
\(615\) 0 0
\(616\) 0 0
\(617\) 30.0000 1.20775 0.603877 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 0 0
\(619\) 16.9706 0.682105 0.341052 0.940044i \(-0.389217\pi\)
0.341052 + 0.940044i \(0.389217\pi\)
\(620\) −4.00000 −0.160644
\(621\) 0 0
\(622\) 19.7990 0.793867
\(623\) 0 0
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 7.07107 0.282617
\(627\) 0 0
\(628\) −1.41421 −0.0564333
\(629\) 11.3137 0.451107
\(630\) 0 0
\(631\) −12.0000 −0.477712 −0.238856 0.971055i \(-0.576772\pi\)
−0.238856 + 0.971055i \(0.576772\pi\)
\(632\) 4.00000 0.159111
\(633\) 0 0
\(634\) 8.00000 0.317721
\(635\) −16.9706 −0.673456
\(636\) 0 0
\(637\) 0 0
\(638\) −6.00000 −0.237542
\(639\) 0 0
\(640\) 1.41421 0.0559017
\(641\) −6.00000 −0.236986 −0.118493 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 4.00000 0.157378
\(647\) 36.7696 1.44556 0.722780 0.691078i \(-0.242864\pi\)
0.722780 + 0.691078i \(0.242864\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −4.24264 −0.166410
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) 0 0
\(655\) 4.00000 0.156293
\(656\) −7.07107 −0.276079
\(657\) 0 0
\(658\) 0 0
\(659\) −8.00000 −0.311636 −0.155818 0.987786i \(-0.549801\pi\)
−0.155818 + 0.987786i \(0.549801\pi\)
\(660\) 0 0
\(661\) −29.6985 −1.15514 −0.577569 0.816342i \(-0.695998\pi\)
−0.577569 + 0.816342i \(0.695998\pi\)
\(662\) −8.00000 −0.310929
\(663\) 0 0
\(664\) −2.82843 −0.109764
\(665\) 0 0
\(666\) 0 0
\(667\) 24.0000 0.929284
\(668\) −11.3137 −0.437741
\(669\) 0 0
\(670\) −11.3137 −0.437087
\(671\) −7.07107 −0.272976
\(672\) 0 0
\(673\) −4.00000 −0.154189 −0.0770943 0.997024i \(-0.524564\pi\)
−0.0770943 + 0.997024i \(0.524564\pi\)
\(674\) −20.0000 −0.770371
\(675\) 0 0
\(676\) −11.0000 −0.423077
\(677\) −15.5563 −0.597879 −0.298940 0.954272i \(-0.596633\pi\)
−0.298940 + 0.954272i \(0.596633\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −2.00000 −0.0766965
\(681\) 0 0
\(682\) −2.82843 −0.108306
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) −14.1421 −0.540343
\(686\) 0 0
\(687\) 0 0
\(688\) −8.00000 −0.304997
\(689\) 0 0
\(690\) 0 0
\(691\) −22.6274 −0.860788 −0.430394 0.902641i \(-0.641625\pi\)
−0.430394 + 0.902641i \(0.641625\pi\)
\(692\) 1.41421 0.0537603
\(693\) 0 0
\(694\) 0 0
\(695\) 4.00000 0.151729
\(696\) 0 0
\(697\) 10.0000 0.378777
\(698\) 18.3848 0.695874
\(699\) 0 0
\(700\) 0 0
\(701\) −2.00000 −0.0755390 −0.0377695 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) 0 0
\(703\) 22.6274 0.853409
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 26.8701 1.01127
\(707\) 0 0
\(708\) 0 0
\(709\) −8.00000 −0.300446 −0.150223 0.988652i \(-0.547999\pi\)
−0.150223 + 0.988652i \(0.547999\pi\)
\(710\) −11.3137 −0.424596
\(711\) 0 0
\(712\) 9.89949 0.370999
\(713\) 11.3137 0.423702
\(714\) 0 0
\(715\) 2.00000 0.0747958
\(716\) 16.0000 0.597948
\(717\) 0 0
\(718\) 4.00000 0.149279
\(719\) −36.7696 −1.37127 −0.685636 0.727944i \(-0.740476\pi\)
−0.685636 + 0.727944i \(0.740476\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −11.0000 −0.409378
\(723\) 0 0
\(724\) 12.7279 0.473029
\(725\) 18.0000 0.668503
\(726\) 0 0
\(727\) 2.82843 0.104901 0.0524503 0.998624i \(-0.483297\pi\)
0.0524503 + 0.998624i \(0.483297\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 2.00000 0.0740233
\(731\) 11.3137 0.418453
\(732\) 0 0
\(733\) −29.6985 −1.09694 −0.548469 0.836171i \(-0.684789\pi\)
−0.548469 + 0.836171i \(0.684789\pi\)
\(734\) −8.48528 −0.313197
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) −8.00000 −0.294684
\(738\) 0 0
\(739\) 36.0000 1.32428 0.662141 0.749380i \(-0.269648\pi\)
0.662141 + 0.749380i \(0.269648\pi\)
\(740\) −11.3137 −0.415900
\(741\) 0 0
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) 5.65685 0.207251
\(746\) 10.0000 0.366126
\(747\) 0 0
\(748\) −1.41421 −0.0517088
\(749\) 0 0
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) −8.48528 −0.309426
\(753\) 0 0
\(754\) −8.48528 −0.309016
\(755\) 22.6274 0.823496
\(756\) 0 0
\(757\) 16.0000 0.581530 0.290765 0.956795i \(-0.406090\pi\)
0.290765 + 0.956795i \(0.406090\pi\)
\(758\) −12.0000 −0.435860
\(759\) 0 0
\(760\) −4.00000 −0.145095
\(761\) 12.7279 0.461387 0.230693 0.973026i \(-0.425901\pi\)
0.230693 + 0.973026i \(0.425901\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 12.0000 0.434145
\(765\) 0 0
\(766\) 14.1421 0.510976
\(767\) 0 0
\(768\) 0 0
\(769\) 29.6985 1.07095 0.535477 0.844550i \(-0.320132\pi\)
0.535477 + 0.844550i \(0.320132\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −18.0000 −0.647834
\(773\) −18.3848 −0.661254 −0.330627 0.943761i \(-0.607260\pi\)
−0.330627 + 0.943761i \(0.607260\pi\)
\(774\) 0 0
\(775\) 8.48528 0.304800
\(776\) 15.5563 0.558440
\(777\) 0 0
\(778\) 10.0000 0.358517
\(779\) 20.0000 0.716574
\(780\) 0 0
\(781\) −8.00000 −0.286263
\(782\) 5.65685 0.202289
\(783\) 0 0
\(784\) 0 0
\(785\) −2.00000 −0.0713831
\(786\) 0 0
\(787\) −42.4264 −1.51234 −0.756169 0.654376i \(-0.772931\pi\)
−0.756169 + 0.654376i \(0.772931\pi\)
\(788\) 12.0000 0.427482
\(789\) 0 0
\(790\) 5.65685 0.201262
\(791\) 0 0
\(792\) 0 0
\(793\) −10.0000 −0.355110
\(794\) −9.89949 −0.351320
\(795\) 0 0
\(796\) 25.4558 0.902258
\(797\) −26.8701 −0.951786 −0.475893 0.879503i \(-0.657875\pi\)
−0.475893 + 0.879503i \(0.657875\pi\)
\(798\) 0 0
\(799\) 12.0000 0.424529
\(800\) −3.00000 −0.106066
\(801\) 0 0
\(802\) 18.0000 0.635602
\(803\) 1.41421 0.0499065
\(804\) 0 0
\(805\) 0 0
\(806\) −4.00000 −0.140894
\(807\) 0 0
\(808\) 15.5563 0.547270
\(809\) 36.0000 1.26569 0.632846 0.774277i \(-0.281886\pi\)
0.632846 + 0.774277i \(0.281886\pi\)
\(810\) 0 0
\(811\) 8.48528 0.297959 0.148979 0.988840i \(-0.452401\pi\)
0.148979 + 0.988840i \(0.452401\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −8.00000 −0.280400
\(815\) −5.65685 −0.198151
\(816\) 0 0
\(817\) 22.6274 0.791633
\(818\) 24.0416 0.840596
\(819\) 0 0
\(820\) −10.0000 −0.349215
\(821\) 20.0000 0.698005 0.349002 0.937122i \(-0.386521\pi\)
0.349002 + 0.937122i \(0.386521\pi\)
\(822\) 0 0
\(823\) −20.0000 −0.697156 −0.348578 0.937280i \(-0.613335\pi\)
−0.348578 + 0.937280i \(0.613335\pi\)
\(824\) 8.48528 0.295599
\(825\) 0 0
\(826\) 0 0
\(827\) −52.0000 −1.80822 −0.904109 0.427303i \(-0.859464\pi\)
−0.904109 + 0.427303i \(0.859464\pi\)
\(828\) 0 0
\(829\) 9.89949 0.343824 0.171912 0.985112i \(-0.445006\pi\)
0.171912 + 0.985112i \(0.445006\pi\)
\(830\) −4.00000 −0.138842
\(831\) 0 0
\(832\) 1.41421 0.0490290
\(833\) 0 0
\(834\) 0 0
\(835\) −16.0000 −0.553703
\(836\) −2.82843 −0.0978232
\(837\) 0 0
\(838\) −16.9706 −0.586238
\(839\) 53.7401 1.85531 0.927657 0.373432i \(-0.121819\pi\)
0.927657 + 0.373432i \(0.121819\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −22.0000 −0.758170
\(843\) 0 0
\(844\) −8.00000 −0.275371
\(845\) −15.5563 −0.535155
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 4.24264 0.145521
\(851\) 32.0000 1.09695
\(852\) 0 0
\(853\) 41.0122 1.40423 0.702115 0.712063i \(-0.252239\pi\)
0.702115 + 0.712063i \(0.252239\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 8.00000 0.273434
\(857\) 12.7279 0.434778 0.217389 0.976085i \(-0.430246\pi\)
0.217389 + 0.976085i \(0.430246\pi\)
\(858\) 0 0
\(859\) 16.9706 0.579028 0.289514 0.957174i \(-0.406506\pi\)
0.289514 + 0.957174i \(0.406506\pi\)
\(860\) −11.3137 −0.385794
\(861\) 0 0
\(862\) −28.0000 −0.953684
\(863\) −4.00000 −0.136162 −0.0680808 0.997680i \(-0.521688\pi\)
−0.0680808 + 0.997680i \(0.521688\pi\)
\(864\) 0 0
\(865\) 2.00000 0.0680020
\(866\) 4.24264 0.144171
\(867\) 0 0
\(868\) 0 0
\(869\) 4.00000 0.135691
\(870\) 0 0
\(871\) −11.3137 −0.383350
\(872\) 4.00000 0.135457
\(873\) 0 0
\(874\) 11.3137 0.382692
\(875\) 0 0
\(876\) 0 0
\(877\) −52.0000 −1.75592 −0.877958 0.478738i \(-0.841094\pi\)
−0.877958 + 0.478738i \(0.841094\pi\)
\(878\) −33.9411 −1.14546
\(879\) 0 0
\(880\) 1.41421 0.0476731
\(881\) −55.1543 −1.85820 −0.929098 0.369833i \(-0.879415\pi\)
−0.929098 + 0.369833i \(0.879415\pi\)
\(882\) 0 0
\(883\) −36.0000 −1.21150 −0.605748 0.795656i \(-0.707126\pi\)
−0.605748 + 0.795656i \(0.707126\pi\)
\(884\) −2.00000 −0.0672673
\(885\) 0 0
\(886\) −24.0000 −0.806296
\(887\) −45.2548 −1.51951 −0.759754 0.650210i \(-0.774681\pi\)
−0.759754 + 0.650210i \(0.774681\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 14.0000 0.469281
\(891\) 0 0
\(892\) 2.82843 0.0947027
\(893\) 24.0000 0.803129
\(894\) 0 0
\(895\) 22.6274 0.756351
\(896\) 0 0
\(897\) 0 0
\(898\) 24.0000 0.800890
\(899\) 16.9706 0.566000
\(900\) 0 0
\(901\) 0 0
\(902\) −7.07107 −0.235441
\(903\) 0 0
\(904\) 0 0
\(905\) 18.0000 0.598340
\(906\) 0 0
\(907\) −8.00000 −0.265636 −0.132818 0.991140i \(-0.542403\pi\)
−0.132818 + 0.991140i \(0.542403\pi\)
\(908\) 14.1421 0.469323
\(909\) 0 0
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 0 0
\(913\) −2.82843 −0.0936073
\(914\) −22.0000 −0.727695
\(915\) 0 0
\(916\) −21.2132 −0.700904
\(917\) 0 0
\(918\) 0 0
\(919\) 8.00000 0.263896 0.131948 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) −5.65685 −0.186501
\(921\) 0 0
\(922\) −26.8701 −0.884918
\(923\) −11.3137 −0.372395
\(924\) 0 0
\(925\) 24.0000 0.789115
\(926\) −32.0000 −1.05159
\(927\) 0 0
\(928\) −6.00000 −0.196960
\(929\) 43.8406 1.43836 0.719182 0.694822i \(-0.244517\pi\)
0.719182 + 0.694822i \(0.244517\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −6.00000 −0.196537
\(933\) 0 0
\(934\) −16.9706 −0.555294
\(935\) −2.00000 −0.0654070
\(936\) 0 0
\(937\) −46.6690 −1.52461 −0.762306 0.647217i \(-0.775933\pi\)
−0.762306 + 0.647217i \(0.775933\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −12.0000 −0.391397
\(941\) −15.5563 −0.507122 −0.253561 0.967319i \(-0.581602\pi\)
−0.253561 + 0.967319i \(0.581602\pi\)
\(942\) 0 0
\(943\) 28.2843 0.921063
\(944\) 0 0
\(945\) 0 0
\(946\) −8.00000 −0.260102
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 0 0
\(949\) 2.00000 0.0649227
\(950\) 8.48528 0.275299
\(951\) 0 0
\(952\) 0 0
\(953\) 36.0000 1.16615 0.583077 0.812417i \(-0.301849\pi\)
0.583077 + 0.812417i \(0.301849\pi\)
\(954\) 0 0
\(955\) 16.9706 0.549155
\(956\) −8.00000 −0.258738
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −23.0000 −0.741935
\(962\) −11.3137 −0.364769
\(963\) 0 0
\(964\) 9.89949 0.318841
\(965\) −25.4558 −0.819453
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) 22.0000 0.706377
\(971\) −28.2843 −0.907685 −0.453843 0.891082i \(-0.649947\pi\)
−0.453843 + 0.891082i \(0.649947\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 28.0000 0.897178
\(975\) 0 0
\(976\) −7.07107 −0.226339
\(977\) 38.0000 1.21573 0.607864 0.794041i \(-0.292027\pi\)
0.607864 + 0.794041i \(0.292027\pi\)
\(978\) 0 0
\(979\) 9.89949 0.316389
\(980\) 0 0
\(981\) 0 0
\(982\) 20.0000 0.638226
\(983\) 2.82843 0.0902128 0.0451064 0.998982i \(-0.485637\pi\)
0.0451064 + 0.998982i \(0.485637\pi\)
\(984\) 0 0
\(985\) 16.9706 0.540727
\(986\) 8.48528 0.270226
\(987\) 0 0
\(988\) −4.00000 −0.127257
\(989\) 32.0000 1.01754
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) −2.82843 −0.0898027
\(993\) 0 0
\(994\) 0 0
\(995\) 36.0000 1.14128
\(996\) 0 0
\(997\) 29.6985 0.940560 0.470280 0.882517i \(-0.344153\pi\)
0.470280 + 0.882517i \(0.344153\pi\)
\(998\) −24.0000 −0.759707
\(999\) 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 9702.2.a.dp.1.2 yes 2
3.2 odd 2 9702.2.a.cl.1.1 2
7.6 odd 2 inner 9702.2.a.dp.1.1 yes 2
21.20 even 2 9702.2.a.cl.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9702.2.a.cl.1.1 2 3.2 odd 2
9702.2.a.cl.1.2 yes 2 21.20 even 2
9702.2.a.dp.1.1 yes 2 7.6 odd 2 inner
9702.2.a.dp.1.2 yes 2 1.1 even 1 trivial