Properties

Label 9702.2.a.cz.1.1
Level $9702$
Weight $2$
Character 9702.1
Self dual yes
Analytic conductor $77.471$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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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: \(\Q(\sqrt{7}) \)
Defining polynomial: \(x^{2} - 7\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 154)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.64575\) of defining polynomial
Character \(\chi\) \(=\) 9702.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -3.64575 q^{5} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -3.64575 q^{5} +1.00000 q^{8} -3.64575 q^{10} -1.00000 q^{11} +5.00000 q^{13} +1.00000 q^{16} -6.00000 q^{17} -0.354249 q^{19} -3.64575 q^{20} -1.00000 q^{22} +3.64575 q^{23} +8.29150 q^{25} +5.00000 q^{26} +4.29150 q^{29} -4.00000 q^{31} +1.00000 q^{32} -6.00000 q^{34} -1.64575 q^{37} -0.354249 q^{38} -3.64575 q^{40} +4.93725 q^{41} -4.00000 q^{43} -1.00000 q^{44} +3.64575 q^{46} -13.2915 q^{47} +8.29150 q^{50} +5.00000 q^{52} +3.64575 q^{53} +3.64575 q^{55} +4.29150 q^{58} +0.645751 q^{59} +3.70850 q^{61} -4.00000 q^{62} +1.00000 q^{64} -18.2288 q^{65} +3.93725 q^{67} -6.00000 q^{68} -9.64575 q^{71} +5.64575 q^{73} -1.64575 q^{74} -0.354249 q^{76} +2.64575 q^{79} -3.64575 q^{80} +4.93725 q^{82} -13.2915 q^{83} +21.8745 q^{85} -4.00000 q^{86} -1.00000 q^{88} +14.5830 q^{89} +3.64575 q^{92} -13.2915 q^{94} +1.29150 q^{95} -5.70850 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{8} - 2 q^{10} - 2 q^{11} + 10 q^{13} + 2 q^{16} - 12 q^{17} - 6 q^{19} - 2 q^{20} - 2 q^{22} + 2 q^{23} + 6 q^{25} + 10 q^{26} - 2 q^{29} - 8 q^{31} + 2 q^{32} - 12 q^{34} + 2 q^{37} - 6 q^{38} - 2 q^{40} - 6 q^{41} - 8 q^{43} - 2 q^{44} + 2 q^{46} - 16 q^{47} + 6 q^{50} + 10 q^{52} + 2 q^{53} + 2 q^{55} - 2 q^{58} - 4 q^{59} + 18 q^{61} - 8 q^{62} + 2 q^{64} - 10 q^{65} - 8 q^{67} - 12 q^{68} - 14 q^{71} + 6 q^{73} + 2 q^{74} - 6 q^{76} - 2 q^{80} - 6 q^{82} - 16 q^{83} + 12 q^{85} - 8 q^{86} - 2 q^{88} + 8 q^{89} + 2 q^{92} - 16 q^{94} - 8 q^{95} - 22 q^{97} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −3.64575 −1.63043 −0.815215 0.579159i \(-0.803381\pi\)
−0.815215 + 0.579159i \(0.803381\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −3.64575 −1.15289
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 5.00000 1.38675 0.693375 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) −0.354249 −0.0812702 −0.0406351 0.999174i \(-0.512938\pi\)
−0.0406351 + 0.999174i \(0.512938\pi\)
\(20\) −3.64575 −0.815215
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) 3.64575 0.760192 0.380096 0.924947i \(-0.375891\pi\)
0.380096 + 0.924947i \(0.375891\pi\)
\(24\) 0 0
\(25\) 8.29150 1.65830
\(26\) 5.00000 0.980581
\(27\) 0 0
\(28\) 0 0
\(29\) 4.29150 0.796912 0.398456 0.917187i \(-0.369546\pi\)
0.398456 + 0.917187i \(0.369546\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 0 0
\(37\) −1.64575 −0.270560 −0.135280 0.990807i \(-0.543193\pi\)
−0.135280 + 0.990807i \(0.543193\pi\)
\(38\) −0.354249 −0.0574667
\(39\) 0 0
\(40\) −3.64575 −0.576444
\(41\) 4.93725 0.771070 0.385535 0.922693i \(-0.374017\pi\)
0.385535 + 0.922693i \(0.374017\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 3.64575 0.537537
\(47\) −13.2915 −1.93876 −0.969382 0.245556i \(-0.921030\pi\)
−0.969382 + 0.245556i \(0.921030\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 8.29150 1.17260
\(51\) 0 0
\(52\) 5.00000 0.693375
\(53\) 3.64575 0.500782 0.250391 0.968145i \(-0.419441\pi\)
0.250391 + 0.968145i \(0.419441\pi\)
\(54\) 0 0
\(55\) 3.64575 0.491593
\(56\) 0 0
\(57\) 0 0
\(58\) 4.29150 0.563502
\(59\) 0.645751 0.0840697 0.0420348 0.999116i \(-0.486616\pi\)
0.0420348 + 0.999116i \(0.486616\pi\)
\(60\) 0 0
\(61\) 3.70850 0.474824 0.237412 0.971409i \(-0.423701\pi\)
0.237412 + 0.971409i \(0.423701\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −18.2288 −2.26100
\(66\) 0 0
\(67\) 3.93725 0.481012 0.240506 0.970648i \(-0.422687\pi\)
0.240506 + 0.970648i \(0.422687\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) 0 0
\(71\) −9.64575 −1.14474 −0.572370 0.819995i \(-0.693976\pi\)
−0.572370 + 0.819995i \(0.693976\pi\)
\(72\) 0 0
\(73\) 5.64575 0.660785 0.330393 0.943844i \(-0.392819\pi\)
0.330393 + 0.943844i \(0.392819\pi\)
\(74\) −1.64575 −0.191315
\(75\) 0 0
\(76\) −0.354249 −0.0406351
\(77\) 0 0
\(78\) 0 0
\(79\) 2.64575 0.297670 0.148835 0.988862i \(-0.452448\pi\)
0.148835 + 0.988862i \(0.452448\pi\)
\(80\) −3.64575 −0.407607
\(81\) 0 0
\(82\) 4.93725 0.545228
\(83\) −13.2915 −1.45893 −0.729466 0.684017i \(-0.760231\pi\)
−0.729466 + 0.684017i \(0.760231\pi\)
\(84\) 0 0
\(85\) 21.8745 2.37262
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) 14.5830 1.54580 0.772898 0.634531i \(-0.218807\pi\)
0.772898 + 0.634531i \(0.218807\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 3.64575 0.380096
\(93\) 0 0
\(94\) −13.2915 −1.37091
\(95\) 1.29150 0.132505
\(96\) 0 0
\(97\) −5.70850 −0.579610 −0.289805 0.957086i \(-0.593590\pi\)
−0.289805 + 0.957086i \(0.593590\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 8.29150 0.829150
\(101\) −3.00000 −0.298511 −0.149256 0.988799i \(-0.547688\pi\)
−0.149256 + 0.988799i \(0.547688\pi\)
\(102\) 0 0
\(103\) 12.9373 1.27475 0.637373 0.770556i \(-0.280021\pi\)
0.637373 + 0.770556i \(0.280021\pi\)
\(104\) 5.00000 0.490290
\(105\) 0 0
\(106\) 3.64575 0.354107
\(107\) −4.93725 −0.477302 −0.238651 0.971105i \(-0.576705\pi\)
−0.238651 + 0.971105i \(0.576705\pi\)
\(108\) 0 0
\(109\) 10.5830 1.01367 0.506834 0.862044i \(-0.330816\pi\)
0.506834 + 0.862044i \(0.330816\pi\)
\(110\) 3.64575 0.347609
\(111\) 0 0
\(112\) 0 0
\(113\) 7.70850 0.725154 0.362577 0.931954i \(-0.381897\pi\)
0.362577 + 0.931954i \(0.381897\pi\)
\(114\) 0 0
\(115\) −13.2915 −1.23944
\(116\) 4.29150 0.398456
\(117\) 0 0
\(118\) 0.645751 0.0594462
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 3.70850 0.335752
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 0.0627461 0.00556781 0.00278391 0.999996i \(-0.499114\pi\)
0.00278391 + 0.999996i \(0.499114\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −18.2288 −1.59877
\(131\) −15.6458 −1.36698 −0.683488 0.729962i \(-0.739538\pi\)
−0.683488 + 0.729962i \(0.739538\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 3.93725 0.340127
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) −18.8745 −1.61256 −0.806279 0.591535i \(-0.798522\pi\)
−0.806279 + 0.591535i \(0.798522\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −9.64575 −0.809453
\(143\) −5.00000 −0.418121
\(144\) 0 0
\(145\) −15.6458 −1.29931
\(146\) 5.64575 0.467246
\(147\) 0 0
\(148\) −1.64575 −0.135280
\(149\) 4.70850 0.385735 0.192868 0.981225i \(-0.438221\pi\)
0.192868 + 0.981225i \(0.438221\pi\)
\(150\) 0 0
\(151\) −3.35425 −0.272965 −0.136482 0.990642i \(-0.543580\pi\)
−0.136482 + 0.990642i \(0.543580\pi\)
\(152\) −0.354249 −0.0287334
\(153\) 0 0
\(154\) 0 0
\(155\) 14.5830 1.17134
\(156\) 0 0
\(157\) −21.1660 −1.68923 −0.844616 0.535373i \(-0.820171\pi\)
−0.844616 + 0.535373i \(0.820171\pi\)
\(158\) 2.64575 0.210485
\(159\) 0 0
\(160\) −3.64575 −0.288222
\(161\) 0 0
\(162\) 0 0
\(163\) −4.64575 −0.363883 −0.181942 0.983309i \(-0.558238\pi\)
−0.181942 + 0.983309i \(0.558238\pi\)
\(164\) 4.93725 0.385535
\(165\) 0 0
\(166\) −13.2915 −1.03162
\(167\) −15.2288 −1.17844 −0.589218 0.807974i \(-0.700564\pi\)
−0.589218 + 0.807974i \(0.700564\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 21.8745 1.67770
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) −10.2915 −0.782448 −0.391224 0.920295i \(-0.627948\pi\)
−0.391224 + 0.920295i \(0.627948\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) 14.5830 1.09304
\(179\) 4.06275 0.303664 0.151832 0.988406i \(-0.451483\pi\)
0.151832 + 0.988406i \(0.451483\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 3.64575 0.268768
\(185\) 6.00000 0.441129
\(186\) 0 0
\(187\) 6.00000 0.438763
\(188\) −13.2915 −0.969382
\(189\) 0 0
\(190\) 1.29150 0.0936954
\(191\) 13.2915 0.961739 0.480870 0.876792i \(-0.340321\pi\)
0.480870 + 0.876792i \(0.340321\pi\)
\(192\) 0 0
\(193\) −11.5203 −0.829246 −0.414623 0.909993i \(-0.636087\pi\)
−0.414623 + 0.909993i \(0.636087\pi\)
\(194\) −5.70850 −0.409846
\(195\) 0 0
\(196\) 0 0
\(197\) −18.8745 −1.34475 −0.672377 0.740209i \(-0.734726\pi\)
−0.672377 + 0.740209i \(0.734726\pi\)
\(198\) 0 0
\(199\) −22.2288 −1.57575 −0.787877 0.615832i \(-0.788820\pi\)
−0.787877 + 0.615832i \(0.788820\pi\)
\(200\) 8.29150 0.586298
\(201\) 0 0
\(202\) −3.00000 −0.211079
\(203\) 0 0
\(204\) 0 0
\(205\) −18.0000 −1.25717
\(206\) 12.9373 0.901381
\(207\) 0 0
\(208\) 5.00000 0.346688
\(209\) 0.354249 0.0245039
\(210\) 0 0
\(211\) −14.9373 −1.02832 −0.514161 0.857693i \(-0.671897\pi\)
−0.514161 + 0.857693i \(0.671897\pi\)
\(212\) 3.64575 0.250391
\(213\) 0 0
\(214\) −4.93725 −0.337504
\(215\) 14.5830 0.994553
\(216\) 0 0
\(217\) 0 0
\(218\) 10.5830 0.716772
\(219\) 0 0
\(220\) 3.64575 0.245797
\(221\) −30.0000 −2.01802
\(222\) 0 0
\(223\) −12.3542 −0.827302 −0.413651 0.910436i \(-0.635747\pi\)
−0.413651 + 0.910436i \(0.635747\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 7.70850 0.512762
\(227\) 13.2915 0.882188 0.441094 0.897461i \(-0.354591\pi\)
0.441094 + 0.897461i \(0.354591\pi\)
\(228\) 0 0
\(229\) −16.0000 −1.05731 −0.528655 0.848837i \(-0.677303\pi\)
−0.528655 + 0.848837i \(0.677303\pi\)
\(230\) −13.2915 −0.876416
\(231\) 0 0
\(232\) 4.29150 0.281751
\(233\) −16.9373 −1.10960 −0.554798 0.831985i \(-0.687205\pi\)
−0.554798 + 0.831985i \(0.687205\pi\)
\(234\) 0 0
\(235\) 48.4575 3.16102
\(236\) 0.645751 0.0420348
\(237\) 0 0
\(238\) 0 0
\(239\) −9.22876 −0.596959 −0.298479 0.954416i \(-0.596479\pi\)
−0.298479 + 0.954416i \(0.596479\pi\)
\(240\) 0 0
\(241\) 22.8118 1.46943 0.734717 0.678373i \(-0.237315\pi\)
0.734717 + 0.678373i \(0.237315\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) 3.70850 0.237412
\(245\) 0 0
\(246\) 0 0
\(247\) −1.77124 −0.112702
\(248\) −4.00000 −0.254000
\(249\) 0 0
\(250\) −12.0000 −0.758947
\(251\) −7.29150 −0.460236 −0.230118 0.973163i \(-0.573911\pi\)
−0.230118 + 0.973163i \(0.573911\pi\)
\(252\) 0 0
\(253\) −3.64575 −0.229206
\(254\) 0.0627461 0.00393704
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 0.416995 0.0260114 0.0130057 0.999915i \(-0.495860\pi\)
0.0130057 + 0.999915i \(0.495860\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −18.2288 −1.13050
\(261\) 0 0
\(262\) −15.6458 −0.966598
\(263\) −4.06275 −0.250520 −0.125260 0.992124i \(-0.539976\pi\)
−0.125260 + 0.992124i \(0.539976\pi\)
\(264\) 0 0
\(265\) −13.2915 −0.816491
\(266\) 0 0
\(267\) 0 0
\(268\) 3.93725 0.240506
\(269\) −26.5830 −1.62079 −0.810397 0.585881i \(-0.800749\pi\)
−0.810397 + 0.585881i \(0.800749\pi\)
\(270\) 0 0
\(271\) −17.9373 −1.08961 −0.544805 0.838563i \(-0.683396\pi\)
−0.544805 + 0.838563i \(0.683396\pi\)
\(272\) −6.00000 −0.363803
\(273\) 0 0
\(274\) −18.8745 −1.14025
\(275\) −8.29150 −0.499996
\(276\) 0 0
\(277\) −11.7085 −0.703495 −0.351748 0.936095i \(-0.614413\pi\)
−0.351748 + 0.936095i \(0.614413\pi\)
\(278\) −4.00000 −0.239904
\(279\) 0 0
\(280\) 0 0
\(281\) 7.06275 0.421328 0.210664 0.977559i \(-0.432437\pi\)
0.210664 + 0.977559i \(0.432437\pi\)
\(282\) 0 0
\(283\) −7.64575 −0.454493 −0.227246 0.973837i \(-0.572972\pi\)
−0.227246 + 0.973837i \(0.572972\pi\)
\(284\) −9.64575 −0.572370
\(285\) 0 0
\(286\) −5.00000 −0.295656
\(287\) 0 0
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) −15.6458 −0.918750
\(291\) 0 0
\(292\) 5.64575 0.330393
\(293\) −12.0000 −0.701047 −0.350524 0.936554i \(-0.613996\pi\)
−0.350524 + 0.936554i \(0.613996\pi\)
\(294\) 0 0
\(295\) −2.35425 −0.137070
\(296\) −1.64575 −0.0956574
\(297\) 0 0
\(298\) 4.70850 0.272756
\(299\) 18.2288 1.05420
\(300\) 0 0
\(301\) 0 0
\(302\) −3.35425 −0.193015
\(303\) 0 0
\(304\) −0.354249 −0.0203176
\(305\) −13.5203 −0.774168
\(306\) 0 0
\(307\) −4.22876 −0.241348 −0.120674 0.992692i \(-0.538506\pi\)
−0.120674 + 0.992692i \(0.538506\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 14.5830 0.828259
\(311\) −16.9373 −0.960424 −0.480212 0.877153i \(-0.659440\pi\)
−0.480212 + 0.877153i \(0.659440\pi\)
\(312\) 0 0
\(313\) 2.41699 0.136617 0.0683083 0.997664i \(-0.478240\pi\)
0.0683083 + 0.997664i \(0.478240\pi\)
\(314\) −21.1660 −1.19447
\(315\) 0 0
\(316\) 2.64575 0.148835
\(317\) −12.0000 −0.673987 −0.336994 0.941507i \(-0.609410\pi\)
−0.336994 + 0.941507i \(0.609410\pi\)
\(318\) 0 0
\(319\) −4.29150 −0.240278
\(320\) −3.64575 −0.203804
\(321\) 0 0
\(322\) 0 0
\(323\) 2.12549 0.118266
\(324\) 0 0
\(325\) 41.4575 2.29965
\(326\) −4.64575 −0.257304
\(327\) 0 0
\(328\) 4.93725 0.272614
\(329\) 0 0
\(330\) 0 0
\(331\) −15.3542 −0.843946 −0.421973 0.906608i \(-0.638662\pi\)
−0.421973 + 0.906608i \(0.638662\pi\)
\(332\) −13.2915 −0.729466
\(333\) 0 0
\(334\) −15.2288 −0.833280
\(335\) −14.3542 −0.784256
\(336\) 0 0
\(337\) 24.9373 1.35842 0.679209 0.733945i \(-0.262323\pi\)
0.679209 + 0.733945i \(0.262323\pi\)
\(338\) 12.0000 0.652714
\(339\) 0 0
\(340\) 21.8745 1.18631
\(341\) 4.00000 0.216612
\(342\) 0 0
\(343\) 0 0
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) −10.2915 −0.553275
\(347\) −26.8118 −1.43933 −0.719665 0.694321i \(-0.755705\pi\)
−0.719665 + 0.694321i \(0.755705\pi\)
\(348\) 0 0
\(349\) 29.8745 1.59915 0.799573 0.600569i \(-0.205059\pi\)
0.799573 + 0.600569i \(0.205059\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) 35.1660 1.87170 0.935849 0.352401i \(-0.114635\pi\)
0.935849 + 0.352401i \(0.114635\pi\)
\(354\) 0 0
\(355\) 35.1660 1.86642
\(356\) 14.5830 0.772898
\(357\) 0 0
\(358\) 4.06275 0.214723
\(359\) 10.0627 0.531091 0.265546 0.964098i \(-0.414448\pi\)
0.265546 + 0.964098i \(0.414448\pi\)
\(360\) 0 0
\(361\) −18.8745 −0.993395
\(362\) −10.0000 −0.525588
\(363\) 0 0
\(364\) 0 0
\(365\) −20.5830 −1.07736
\(366\) 0 0
\(367\) −9.77124 −0.510055 −0.255027 0.966934i \(-0.582084\pi\)
−0.255027 + 0.966934i \(0.582084\pi\)
\(368\) 3.64575 0.190048
\(369\) 0 0
\(370\) 6.00000 0.311925
\(371\) 0 0
\(372\) 0 0
\(373\) −10.8745 −0.563061 −0.281530 0.959552i \(-0.590842\pi\)
−0.281530 + 0.959552i \(0.590842\pi\)
\(374\) 6.00000 0.310253
\(375\) 0 0
\(376\) −13.2915 −0.685457
\(377\) 21.4575 1.10512
\(378\) 0 0
\(379\) 21.9373 1.12684 0.563421 0.826170i \(-0.309485\pi\)
0.563421 + 0.826170i \(0.309485\pi\)
\(380\) 1.29150 0.0662527
\(381\) 0 0
\(382\) 13.2915 0.680052
\(383\) 34.1033 1.74260 0.871298 0.490755i \(-0.163279\pi\)
0.871298 + 0.490755i \(0.163279\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −11.5203 −0.586366
\(387\) 0 0
\(388\) −5.70850 −0.289805
\(389\) 20.8118 1.05520 0.527599 0.849493i \(-0.323092\pi\)
0.527599 + 0.849493i \(0.323092\pi\)
\(390\) 0 0
\(391\) −21.8745 −1.10624
\(392\) 0 0
\(393\) 0 0
\(394\) −18.8745 −0.950884
\(395\) −9.64575 −0.485330
\(396\) 0 0
\(397\) 31.1660 1.56418 0.782089 0.623167i \(-0.214154\pi\)
0.782089 + 0.623167i \(0.214154\pi\)
\(398\) −22.2288 −1.11423
\(399\) 0 0
\(400\) 8.29150 0.414575
\(401\) −0.416995 −0.0208237 −0.0104119 0.999946i \(-0.503314\pi\)
−0.0104119 + 0.999946i \(0.503314\pi\)
\(402\) 0 0
\(403\) −20.0000 −0.996271
\(404\) −3.00000 −0.149256
\(405\) 0 0
\(406\) 0 0
\(407\) 1.64575 0.0815769
\(408\) 0 0
\(409\) 18.9373 0.936387 0.468193 0.883626i \(-0.344905\pi\)
0.468193 + 0.883626i \(0.344905\pi\)
\(410\) −18.0000 −0.888957
\(411\) 0 0
\(412\) 12.9373 0.637373
\(413\) 0 0
\(414\) 0 0
\(415\) 48.4575 2.37869
\(416\) 5.00000 0.245145
\(417\) 0 0
\(418\) 0.354249 0.0173269
\(419\) −21.8745 −1.06864 −0.534320 0.845282i \(-0.679432\pi\)
−0.534320 + 0.845282i \(0.679432\pi\)
\(420\) 0 0
\(421\) −33.1660 −1.61641 −0.808206 0.588900i \(-0.799561\pi\)
−0.808206 + 0.588900i \(0.799561\pi\)
\(422\) −14.9373 −0.727134
\(423\) 0 0
\(424\) 3.64575 0.177053
\(425\) −49.7490 −2.41318
\(426\) 0 0
\(427\) 0 0
\(428\) −4.93725 −0.238651
\(429\) 0 0
\(430\) 14.5830 0.703255
\(431\) −2.77124 −0.133486 −0.0667430 0.997770i \(-0.521261\pi\)
−0.0667430 + 0.997770i \(0.521261\pi\)
\(432\) 0 0
\(433\) −16.0000 −0.768911 −0.384455 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 10.5830 0.506834
\(437\) −1.29150 −0.0617809
\(438\) 0 0
\(439\) −11.9373 −0.569734 −0.284867 0.958567i \(-0.591949\pi\)
−0.284867 + 0.958567i \(0.591949\pi\)
\(440\) 3.64575 0.173804
\(441\) 0 0
\(442\) −30.0000 −1.42695
\(443\) 18.4575 0.876943 0.438471 0.898745i \(-0.355520\pi\)
0.438471 + 0.898745i \(0.355520\pi\)
\(444\) 0 0
\(445\) −53.1660 −2.52031
\(446\) −12.3542 −0.584991
\(447\) 0 0
\(448\) 0 0
\(449\) −9.87451 −0.466007 −0.233003 0.972476i \(-0.574855\pi\)
−0.233003 + 0.972476i \(0.574855\pi\)
\(450\) 0 0
\(451\) −4.93725 −0.232486
\(452\) 7.70850 0.362577
\(453\) 0 0
\(454\) 13.2915 0.623801
\(455\) 0 0
\(456\) 0 0
\(457\) −39.1660 −1.83211 −0.916054 0.401054i \(-0.868644\pi\)
−0.916054 + 0.401054i \(0.868644\pi\)
\(458\) −16.0000 −0.747631
\(459\) 0 0
\(460\) −13.2915 −0.619720
\(461\) 32.1660 1.49812 0.749060 0.662502i \(-0.230505\pi\)
0.749060 + 0.662502i \(0.230505\pi\)
\(462\) 0 0
\(463\) −22.4575 −1.04369 −0.521845 0.853041i \(-0.674756\pi\)
−0.521845 + 0.853041i \(0.674756\pi\)
\(464\) 4.29150 0.199228
\(465\) 0 0
\(466\) −16.9373 −0.784603
\(467\) −10.7085 −0.495530 −0.247765 0.968820i \(-0.579696\pi\)
−0.247765 + 0.968820i \(0.579696\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 48.4575 2.23518
\(471\) 0 0
\(472\) 0.645751 0.0297231
\(473\) 4.00000 0.183920
\(474\) 0 0
\(475\) −2.93725 −0.134770
\(476\) 0 0
\(477\) 0 0
\(478\) −9.22876 −0.422113
\(479\) 25.9373 1.18510 0.592552 0.805532i \(-0.298121\pi\)
0.592552 + 0.805532i \(0.298121\pi\)
\(480\) 0 0
\(481\) −8.22876 −0.375199
\(482\) 22.8118 1.03905
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 20.8118 0.945013
\(486\) 0 0
\(487\) −30.5830 −1.38585 −0.692924 0.721011i \(-0.743678\pi\)
−0.692924 + 0.721011i \(0.743678\pi\)
\(488\) 3.70850 0.167876
\(489\) 0 0
\(490\) 0 0
\(491\) −10.7085 −0.483268 −0.241634 0.970367i \(-0.577683\pi\)
−0.241634 + 0.970367i \(0.577683\pi\)
\(492\) 0 0
\(493\) −25.7490 −1.15968
\(494\) −1.77124 −0.0796920
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) 0 0
\(499\) 17.8745 0.800173 0.400086 0.916477i \(-0.368980\pi\)
0.400086 + 0.916477i \(0.368980\pi\)
\(500\) −12.0000 −0.536656
\(501\) 0 0
\(502\) −7.29150 −0.325436
\(503\) 19.9373 0.888958 0.444479 0.895789i \(-0.353389\pi\)
0.444479 + 0.895789i \(0.353389\pi\)
\(504\) 0 0
\(505\) 10.9373 0.486701
\(506\) −3.64575 −0.162073
\(507\) 0 0
\(508\) 0.0627461 0.00278391
\(509\) −20.5830 −0.912326 −0.456163 0.889896i \(-0.650777\pi\)
−0.456163 + 0.889896i \(0.650777\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 0.416995 0.0183929
\(515\) −47.1660 −2.07838
\(516\) 0 0
\(517\) 13.2915 0.584560
\(518\) 0 0
\(519\) 0 0
\(520\) −18.2288 −0.799384
\(521\) 2.12549 0.0931195 0.0465598 0.998916i \(-0.485174\pi\)
0.0465598 + 0.998916i \(0.485174\pi\)
\(522\) 0 0
\(523\) 15.5203 0.678654 0.339327 0.940669i \(-0.389801\pi\)
0.339327 + 0.940669i \(0.389801\pi\)
\(524\) −15.6458 −0.683488
\(525\) 0 0
\(526\) −4.06275 −0.177144
\(527\) 24.0000 1.04546
\(528\) 0 0
\(529\) −9.70850 −0.422109
\(530\) −13.2915 −0.577346
\(531\) 0 0
\(532\) 0 0
\(533\) 24.6863 1.06928
\(534\) 0 0
\(535\) 18.0000 0.778208
\(536\) 3.93725 0.170063
\(537\) 0 0
\(538\) −26.5830 −1.14607
\(539\) 0 0
\(540\) 0 0
\(541\) −8.29150 −0.356480 −0.178240 0.983987i \(-0.557040\pi\)
−0.178240 + 0.983987i \(0.557040\pi\)
\(542\) −17.9373 −0.770471
\(543\) 0 0
\(544\) −6.00000 −0.257248
\(545\) −38.5830 −1.65271
\(546\) 0 0
\(547\) 27.5203 1.17668 0.588341 0.808613i \(-0.299781\pi\)
0.588341 + 0.808613i \(0.299781\pi\)
\(548\) −18.8745 −0.806279
\(549\) 0 0
\(550\) −8.29150 −0.353551
\(551\) −1.52026 −0.0647652
\(552\) 0 0
\(553\) 0 0
\(554\) −11.7085 −0.497446
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) −31.7490 −1.34525 −0.672624 0.739984i \(-0.734833\pi\)
−0.672624 + 0.739984i \(0.734833\pi\)
\(558\) 0 0
\(559\) −20.0000 −0.845910
\(560\) 0 0
\(561\) 0 0
\(562\) 7.06275 0.297924
\(563\) −13.0627 −0.550529 −0.275265 0.961369i \(-0.588765\pi\)
−0.275265 + 0.961369i \(0.588765\pi\)
\(564\) 0 0
\(565\) −28.1033 −1.18231
\(566\) −7.64575 −0.321375
\(567\) 0 0
\(568\) −9.64575 −0.404727
\(569\) 41.1660 1.72577 0.862884 0.505401i \(-0.168656\pi\)
0.862884 + 0.505401i \(0.168656\pi\)
\(570\) 0 0
\(571\) −44.9373 −1.88057 −0.940283 0.340394i \(-0.889439\pi\)
−0.940283 + 0.340394i \(0.889439\pi\)
\(572\) −5.00000 −0.209061
\(573\) 0 0
\(574\) 0 0
\(575\) 30.2288 1.26063
\(576\) 0 0
\(577\) −25.4575 −1.05981 −0.529905 0.848057i \(-0.677772\pi\)
−0.529905 + 0.848057i \(0.677772\pi\)
\(578\) 19.0000 0.790296
\(579\) 0 0
\(580\) −15.6458 −0.649654
\(581\) 0 0
\(582\) 0 0
\(583\) −3.64575 −0.150992
\(584\) 5.64575 0.233623
\(585\) 0 0
\(586\) −12.0000 −0.495715
\(587\) 7.93725 0.327606 0.163803 0.986493i \(-0.447624\pi\)
0.163803 + 0.986493i \(0.447624\pi\)
\(588\) 0 0
\(589\) 1.41699 0.0583863
\(590\) −2.35425 −0.0969229
\(591\) 0 0
\(592\) −1.64575 −0.0676400
\(593\) −22.9373 −0.941920 −0.470960 0.882155i \(-0.656092\pi\)
−0.470960 + 0.882155i \(0.656092\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 4.70850 0.192868
\(597\) 0 0
\(598\) 18.2288 0.745429
\(599\) 19.7490 0.806923 0.403461 0.914997i \(-0.367807\pi\)
0.403461 + 0.914997i \(0.367807\pi\)
\(600\) 0 0
\(601\) −24.5830 −1.00276 −0.501381 0.865227i \(-0.667174\pi\)
−0.501381 + 0.865227i \(0.667174\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −3.35425 −0.136482
\(605\) −3.64575 −0.148221
\(606\) 0 0
\(607\) 21.2915 0.864195 0.432098 0.901827i \(-0.357774\pi\)
0.432098 + 0.901827i \(0.357774\pi\)
\(608\) −0.354249 −0.0143667
\(609\) 0 0
\(610\) −13.5203 −0.547419
\(611\) −66.4575 −2.68858
\(612\) 0 0
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) −4.22876 −0.170659
\(615\) 0 0
\(616\) 0 0
\(617\) 16.2915 0.655871 0.327936 0.944700i \(-0.393647\pi\)
0.327936 + 0.944700i \(0.393647\pi\)
\(618\) 0 0
\(619\) 34.5830 1.39001 0.695004 0.719006i \(-0.255403\pi\)
0.695004 + 0.719006i \(0.255403\pi\)
\(620\) 14.5830 0.585668
\(621\) 0 0
\(622\) −16.9373 −0.679122
\(623\) 0 0
\(624\) 0 0
\(625\) 2.29150 0.0916601
\(626\) 2.41699 0.0966025
\(627\) 0 0
\(628\) −21.1660 −0.844616
\(629\) 9.87451 0.393722
\(630\) 0 0
\(631\) 34.8118 1.38583 0.692917 0.721017i \(-0.256325\pi\)
0.692917 + 0.721017i \(0.256325\pi\)
\(632\) 2.64575 0.105242
\(633\) 0 0
\(634\) −12.0000 −0.476581
\(635\) −0.228757 −0.00907793
\(636\) 0 0
\(637\) 0 0
\(638\) −4.29150 −0.169902
\(639\) 0 0
\(640\) −3.64575 −0.144111
\(641\) 18.8745 0.745498 0.372749 0.927932i \(-0.378415\pi\)
0.372749 + 0.927932i \(0.378415\pi\)
\(642\) 0 0
\(643\) 6.52026 0.257134 0.128567 0.991701i \(-0.458962\pi\)
0.128567 + 0.991701i \(0.458962\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 2.12549 0.0836264
\(647\) −38.8118 −1.52585 −0.762924 0.646488i \(-0.776237\pi\)
−0.762924 + 0.646488i \(0.776237\pi\)
\(648\) 0 0
\(649\) −0.645751 −0.0253480
\(650\) 41.4575 1.62610
\(651\) 0 0
\(652\) −4.64575 −0.181942
\(653\) −2.35425 −0.0921289 −0.0460644 0.998938i \(-0.514668\pi\)
−0.0460644 + 0.998938i \(0.514668\pi\)
\(654\) 0 0
\(655\) 57.0405 2.22876
\(656\) 4.93725 0.192767
\(657\) 0 0
\(658\) 0 0
\(659\) −14.5830 −0.568073 −0.284037 0.958813i \(-0.591674\pi\)
−0.284037 + 0.958813i \(0.591674\pi\)
\(660\) 0 0
\(661\) 28.5830 1.11175 0.555875 0.831266i \(-0.312383\pi\)
0.555875 + 0.831266i \(0.312383\pi\)
\(662\) −15.3542 −0.596760
\(663\) 0 0
\(664\) −13.2915 −0.515810
\(665\) 0 0
\(666\) 0 0
\(667\) 15.6458 0.605806
\(668\) −15.2288 −0.589218
\(669\) 0 0
\(670\) −14.3542 −0.554553
\(671\) −3.70850 −0.143165
\(672\) 0 0
\(673\) −14.9373 −0.575789 −0.287894 0.957662i \(-0.592955\pi\)
−0.287894 + 0.957662i \(0.592955\pi\)
\(674\) 24.9373 0.960547
\(675\) 0 0
\(676\) 12.0000 0.461538
\(677\) 2.12549 0.0816893 0.0408446 0.999166i \(-0.486995\pi\)
0.0408446 + 0.999166i \(0.486995\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 21.8745 0.838849
\(681\) 0 0
\(682\) 4.00000 0.153168
\(683\) 13.9373 0.533294 0.266647 0.963794i \(-0.414084\pi\)
0.266647 + 0.963794i \(0.414084\pi\)
\(684\) 0 0
\(685\) 68.8118 2.62916
\(686\) 0 0
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) 18.2288 0.694460
\(690\) 0 0
\(691\) −18.7712 −0.714092 −0.357046 0.934087i \(-0.616216\pi\)
−0.357046 + 0.934087i \(0.616216\pi\)
\(692\) −10.2915 −0.391224
\(693\) 0 0
\(694\) −26.8118 −1.01776
\(695\) 14.5830 0.553165
\(696\) 0 0
\(697\) −29.6235 −1.12207
\(698\) 29.8745 1.13077
\(699\) 0 0
\(700\) 0 0
\(701\) 6.87451 0.259647 0.129823 0.991537i \(-0.458559\pi\)
0.129823 + 0.991537i \(0.458559\pi\)
\(702\) 0 0
\(703\) 0.583005 0.0219885
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 35.1660 1.32349
\(707\) 0 0
\(708\) 0 0
\(709\) −6.81176 −0.255821 −0.127911 0.991786i \(-0.540827\pi\)
−0.127911 + 0.991786i \(0.540827\pi\)
\(710\) 35.1660 1.31976
\(711\) 0 0
\(712\) 14.5830 0.546521
\(713\) −14.5830 −0.546138
\(714\) 0 0
\(715\) 18.2288 0.681717
\(716\) 4.06275 0.151832
\(717\) 0 0
\(718\) 10.0627 0.375538
\(719\) 3.87451 0.144495 0.0722474 0.997387i \(-0.476983\pi\)
0.0722474 + 0.997387i \(0.476983\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −18.8745 −0.702436
\(723\) 0 0
\(724\) −10.0000 −0.371647
\(725\) 35.5830 1.32152
\(726\) 0 0
\(727\) −17.2915 −0.641306 −0.320653 0.947197i \(-0.603902\pi\)
−0.320653 + 0.947197i \(0.603902\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −20.5830 −0.761811
\(731\) 24.0000 0.887672
\(732\) 0 0
\(733\) 41.4575 1.53127 0.765634 0.643276i \(-0.222425\pi\)
0.765634 + 0.643276i \(0.222425\pi\)
\(734\) −9.77124 −0.360663
\(735\) 0 0
\(736\) 3.64575 0.134384
\(737\) −3.93725 −0.145031
\(738\) 0 0
\(739\) −7.87451 −0.289668 −0.144834 0.989456i \(-0.546265\pi\)
−0.144834 + 0.989456i \(0.546265\pi\)
\(740\) 6.00000 0.220564
\(741\) 0 0
\(742\) 0 0
\(743\) 34.7085 1.27333 0.636666 0.771140i \(-0.280313\pi\)
0.636666 + 0.771140i \(0.280313\pi\)
\(744\) 0 0
\(745\) −17.1660 −0.628914
\(746\) −10.8745 −0.398144
\(747\) 0 0
\(748\) 6.00000 0.219382
\(749\) 0 0
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) −13.2915 −0.484691
\(753\) 0 0
\(754\) 21.4575 0.781437
\(755\) 12.2288 0.445050
\(756\) 0 0
\(757\) 19.1660 0.696600 0.348300 0.937383i \(-0.386759\pi\)
0.348300 + 0.937383i \(0.386759\pi\)
\(758\) 21.9373 0.796797
\(759\) 0 0
\(760\) 1.29150 0.0468477
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 13.2915 0.480870
\(765\) 0 0
\(766\) 34.1033 1.23220
\(767\) 3.22876 0.116584
\(768\) 0 0
\(769\) −15.1660 −0.546900 −0.273450 0.961886i \(-0.588165\pi\)
−0.273450 + 0.961886i \(0.588165\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −11.5203 −0.414623
\(773\) −2.58301 −0.0929042 −0.0464521 0.998921i \(-0.514791\pi\)
−0.0464521 + 0.998921i \(0.514791\pi\)
\(774\) 0 0
\(775\) −33.1660 −1.19136
\(776\) −5.70850 −0.204923
\(777\) 0 0
\(778\) 20.8118 0.746138
\(779\) −1.74902 −0.0626650
\(780\) 0 0
\(781\) 9.64575 0.345152
\(782\) −21.8745 −0.782231
\(783\) 0 0
\(784\) 0 0
\(785\) 77.1660 2.75417
\(786\) 0 0
\(787\) −0.811762 −0.0289362 −0.0144681 0.999895i \(-0.504605\pi\)
−0.0144681 + 0.999895i \(0.504605\pi\)
\(788\) −18.8745 −0.672377
\(789\) 0 0
\(790\) −9.64575 −0.343180
\(791\) 0 0
\(792\) 0 0
\(793\) 18.5425 0.658463
\(794\) 31.1660 1.10604
\(795\) 0 0
\(796\) −22.2288 −0.787877
\(797\) 35.1660 1.24564 0.622822 0.782364i \(-0.285986\pi\)
0.622822 + 0.782364i \(0.285986\pi\)
\(798\) 0 0
\(799\) 79.7490 2.82132
\(800\) 8.29150 0.293149
\(801\) 0 0
\(802\) −0.416995 −0.0147246
\(803\) −5.64575 −0.199234
\(804\) 0 0
\(805\) 0 0
\(806\) −20.0000 −0.704470
\(807\) 0 0
\(808\) −3.00000 −0.105540
\(809\) −50.5830 −1.77840 −0.889202 0.457515i \(-0.848740\pi\)
−0.889202 + 0.457515i \(0.848740\pi\)
\(810\) 0 0
\(811\) 27.7490 0.974400 0.487200 0.873290i \(-0.338018\pi\)
0.487200 + 0.873290i \(0.338018\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 1.64575 0.0576836
\(815\) 16.9373 0.593286
\(816\) 0 0
\(817\) 1.41699 0.0495744
\(818\) 18.9373 0.662126
\(819\) 0 0
\(820\) −18.0000 −0.628587
\(821\) 7.70850 0.269028 0.134514 0.990912i \(-0.457053\pi\)
0.134514 + 0.990912i \(0.457053\pi\)
\(822\) 0 0
\(823\) −43.8745 −1.52937 −0.764685 0.644405i \(-0.777105\pi\)
−0.764685 + 0.644405i \(0.777105\pi\)
\(824\) 12.9373 0.450691
\(825\) 0 0
\(826\) 0 0
\(827\) 35.3948 1.23080 0.615398 0.788216i \(-0.288995\pi\)
0.615398 + 0.788216i \(0.288995\pi\)
\(828\) 0 0
\(829\) 43.3948 1.50716 0.753581 0.657355i \(-0.228325\pi\)
0.753581 + 0.657355i \(0.228325\pi\)
\(830\) 48.4575 1.68198
\(831\) 0 0
\(832\) 5.00000 0.173344
\(833\) 0 0
\(834\) 0 0
\(835\) 55.5203 1.92136
\(836\) 0.354249 0.0122519
\(837\) 0 0
\(838\) −21.8745 −0.755642
\(839\) −27.8745 −0.962335 −0.481167 0.876629i \(-0.659787\pi\)
−0.481167 + 0.876629i \(0.659787\pi\)
\(840\) 0 0
\(841\) −10.5830 −0.364931
\(842\) −33.1660 −1.14298
\(843\) 0 0
\(844\) −14.9373 −0.514161
\(845\) −43.7490 −1.50501
\(846\) 0 0
\(847\) 0 0
\(848\) 3.64575 0.125196
\(849\) 0 0
\(850\) −49.7490 −1.70638
\(851\) −6.00000 −0.205677
\(852\) 0 0
\(853\) −3.16601 −0.108402 −0.0542011 0.998530i \(-0.517261\pi\)
−0.0542011 + 0.998530i \(0.517261\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −4.93725 −0.168752
\(857\) −36.0000 −1.22974 −0.614868 0.788630i \(-0.710791\pi\)
−0.614868 + 0.788630i \(0.710791\pi\)
\(858\) 0 0
\(859\) 37.8118 1.29012 0.645060 0.764132i \(-0.276832\pi\)
0.645060 + 0.764132i \(0.276832\pi\)
\(860\) 14.5830 0.497276
\(861\) 0 0
\(862\) −2.77124 −0.0943889
\(863\) 49.5203 1.68569 0.842845 0.538157i \(-0.180879\pi\)
0.842845 + 0.538157i \(0.180879\pi\)
\(864\) 0 0
\(865\) 37.5203 1.27573
\(866\) −16.0000 −0.543702
\(867\) 0 0
\(868\) 0 0
\(869\) −2.64575 −0.0897510
\(870\) 0 0
\(871\) 19.6863 0.667044
\(872\) 10.5830 0.358386
\(873\) 0 0
\(874\) −1.29150 −0.0436857
\(875\) 0 0
\(876\) 0 0
\(877\) 8.87451 0.299671 0.149835 0.988711i \(-0.452126\pi\)
0.149835 + 0.988711i \(0.452126\pi\)
\(878\) −11.9373 −0.402863
\(879\) 0 0
\(880\) 3.64575 0.122898
\(881\) −6.87451 −0.231608 −0.115804 0.993272i \(-0.536944\pi\)
−0.115804 + 0.993272i \(0.536944\pi\)
\(882\) 0 0
\(883\) 6.06275 0.204028 0.102014 0.994783i \(-0.467471\pi\)
0.102014 + 0.994783i \(0.467471\pi\)
\(884\) −30.0000 −1.00901
\(885\) 0 0
\(886\) 18.4575 0.620092
\(887\) 13.1033 0.439965 0.219982 0.975504i \(-0.429400\pi\)
0.219982 + 0.975504i \(0.429400\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −53.1660 −1.78213
\(891\) 0 0
\(892\) −12.3542 −0.413651
\(893\) 4.70850 0.157564
\(894\) 0 0
\(895\) −14.8118 −0.495103
\(896\) 0 0
\(897\) 0 0
\(898\) −9.87451 −0.329517
\(899\) −17.1660 −0.572519
\(900\) 0 0
\(901\) −21.8745 −0.728746
\(902\) −4.93725 −0.164393
\(903\) 0 0
\(904\) 7.70850 0.256381
\(905\) 36.4575 1.21189
\(906\) 0 0
\(907\) −10.4575 −0.347236 −0.173618 0.984813i \(-0.555546\pi\)
−0.173618 + 0.984813i \(0.555546\pi\)
\(908\) 13.2915 0.441094
\(909\) 0 0
\(910\) 0 0
\(911\) 1.29150 0.0427894 0.0213947 0.999771i \(-0.493189\pi\)
0.0213947 + 0.999771i \(0.493189\pi\)
\(912\) 0 0
\(913\) 13.2915 0.439885
\(914\) −39.1660 −1.29550
\(915\) 0 0
\(916\) −16.0000 −0.528655
\(917\) 0 0
\(918\) 0 0
\(919\) −35.2915 −1.16416 −0.582080 0.813132i \(-0.697761\pi\)
−0.582080 + 0.813132i \(0.697761\pi\)
\(920\) −13.2915 −0.438208
\(921\) 0 0
\(922\) 32.1660 1.05933
\(923\) −48.2288 −1.58747
\(924\) 0 0
\(925\) −13.6458 −0.448670
\(926\) −22.4575 −0.738000
\(927\) 0 0
\(928\) 4.29150 0.140875
\(929\) −35.5830 −1.16744 −0.583720 0.811955i \(-0.698404\pi\)
−0.583720 + 0.811955i \(0.698404\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −16.9373 −0.554798
\(933\) 0 0
\(934\) −10.7085 −0.350393
\(935\) −21.8745 −0.715373
\(936\) 0 0
\(937\) −46.6863 −1.52517 −0.762587 0.646886i \(-0.776071\pi\)
−0.762587 + 0.646886i \(0.776071\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 48.4575 1.58051
\(941\) 44.6235 1.45469 0.727343 0.686274i \(-0.240755\pi\)
0.727343 + 0.686274i \(0.240755\pi\)
\(942\) 0 0
\(943\) 18.0000 0.586161
\(944\) 0.645751 0.0210174
\(945\) 0 0
\(946\) 4.00000 0.130051
\(947\) 33.8745 1.10077 0.550387 0.834910i \(-0.314480\pi\)
0.550387 + 0.834910i \(0.314480\pi\)
\(948\) 0 0
\(949\) 28.2288 0.916344
\(950\) −2.93725 −0.0952971
\(951\) 0 0
\(952\) 0 0
\(953\) 25.5203 0.826682 0.413341 0.910576i \(-0.364362\pi\)
0.413341 + 0.910576i \(0.364362\pi\)
\(954\) 0 0
\(955\) −48.4575 −1.56805
\(956\) −9.22876 −0.298479
\(957\) 0 0
\(958\) 25.9373 0.837995
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −8.22876 −0.265306
\(963\) 0 0
\(964\) 22.8118 0.734717
\(965\) 42.0000 1.35203
\(966\) 0 0
\(967\) −26.3320 −0.846781 −0.423390 0.905947i \(-0.639160\pi\)
−0.423390 + 0.905947i \(0.639160\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) 20.8118 0.668225
\(971\) 25.9373 0.832366 0.416183 0.909281i \(-0.363368\pi\)
0.416183 + 0.909281i \(0.363368\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −30.5830 −0.979943
\(975\) 0 0
\(976\) 3.70850 0.118706
\(977\) −36.4575 −1.16638 −0.583190 0.812336i \(-0.698196\pi\)
−0.583190 + 0.812336i \(0.698196\pi\)
\(978\) 0 0
\(979\) −14.5830 −0.466075
\(980\) 0 0
\(981\) 0 0
\(982\) −10.7085 −0.341722
\(983\) 37.2915 1.18941 0.594707 0.803942i \(-0.297268\pi\)
0.594707 + 0.803942i \(0.297268\pi\)
\(984\) 0 0
\(985\) 68.8118 2.19253
\(986\) −25.7490 −0.820016
\(987\) 0 0
\(988\) −1.77124 −0.0563508
\(989\) −14.5830 −0.463713
\(990\) 0 0
\(991\) 56.6863 1.80070 0.900349 0.435168i \(-0.143311\pi\)
0.900349 + 0.435168i \(0.143311\pi\)
\(992\) −4.00000 −0.127000
\(993\) 0 0
\(994\) 0 0
\(995\) 81.0405 2.56916
\(996\) 0 0
\(997\) −42.5830 −1.34862 −0.674309 0.738450i \(-0.735558\pi\)
−0.674309 + 0.738450i \(0.735558\pi\)
\(998\) 17.8745 0.565808
\(999\) 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.cz.1.1 2
3.2 odd 2 1078.2.a.s.1.1 2
7.2 even 3 1386.2.k.s.991.2 4
7.4 even 3 1386.2.k.s.793.2 4
7.6 odd 2 9702.2.a.dr.1.2 2
12.11 even 2 8624.2.a.ca.1.2 2
21.2 odd 6 154.2.e.f.67.2 yes 4
21.5 even 6 1078.2.e.v.67.1 4
21.11 odd 6 154.2.e.f.23.2 4
21.17 even 6 1078.2.e.v.177.1 4
21.20 even 2 1078.2.a.n.1.2 2
84.11 even 6 1232.2.q.g.177.1 4
84.23 even 6 1232.2.q.g.529.1 4
84.83 odd 2 8624.2.a.bk.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.e.f.23.2 4 21.11 odd 6
154.2.e.f.67.2 yes 4 21.2 odd 6
1078.2.a.n.1.2 2 21.20 even 2
1078.2.a.s.1.1 2 3.2 odd 2
1078.2.e.v.67.1 4 21.5 even 6
1078.2.e.v.177.1 4 21.17 even 6
1232.2.q.g.177.1 4 84.11 even 6
1232.2.q.g.529.1 4 84.23 even 6
1386.2.k.s.793.2 4 7.4 even 3
1386.2.k.s.991.2 4 7.2 even 3
8624.2.a.bk.1.1 2 84.83 odd 2
8624.2.a.ca.1.2 2 12.11 even 2
9702.2.a.cz.1.1 2 1.1 even 1 trivial
9702.2.a.dr.1.2 2 7.6 odd 2