Properties

Label 9702.2.a.dm.1.1
Level $9702$
Weight $2$
Character 9702.1
Self dual yes
Analytic conductor $77.471$
Analytic rank $0$
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: \(0\)
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 462)
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

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.64575 q^{5} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.64575 q^{5} +1.00000 q^{8} -2.64575 q^{10} -1.00000 q^{11} +4.00000 q^{13} +1.00000 q^{16} -3.00000 q^{17} +5.29150 q^{19} -2.64575 q^{20} -1.00000 q^{22} -2.64575 q^{23} +2.00000 q^{25} +4.00000 q^{26} -2.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} -3.00000 q^{34} -9.29150 q^{37} +5.29150 q^{38} -2.64575 q^{40} +9.00000 q^{41} -1.29150 q^{43} -1.00000 q^{44} -2.64575 q^{46} +11.9373 q^{47} +2.00000 q^{50} +4.00000 q^{52} -4.00000 q^{53} +2.64575 q^{55} -2.00000 q^{58} -14.5830 q^{59} +3.93725 q^{61} +4.00000 q^{62} +1.00000 q^{64} -10.5830 q^{65} -13.5830 q^{67} -3.00000 q^{68} +13.2915 q^{71} +4.70850 q^{73} -9.29150 q^{74} +5.29150 q^{76} +5.35425 q^{79} -2.64575 q^{80} +9.00000 q^{82} -5.58301 q^{83} +7.93725 q^{85} -1.29150 q^{86} -1.00000 q^{88} +13.2915 q^{89} -2.64575 q^{92} +11.9373 q^{94} -14.0000 q^{95} -9.58301 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{11} + 8 q^{13} + 2 q^{16} - 6 q^{17} - 2 q^{22} + 4 q^{25} + 8 q^{26} - 4 q^{29} + 8 q^{31} + 2 q^{32} - 6 q^{34} - 8 q^{37} + 18 q^{41} + 8 q^{43} - 2 q^{44} + 8 q^{47} + 4 q^{50} + 8 q^{52} - 8 q^{53} - 4 q^{58} - 8 q^{59} - 8 q^{61} + 8 q^{62} + 2 q^{64} - 6 q^{67} - 6 q^{68} + 16 q^{71} + 20 q^{73} - 8 q^{74} + 16 q^{79} + 18 q^{82} + 10 q^{83} + 8 q^{86} - 2 q^{88} + 16 q^{89} + 8 q^{94} - 28 q^{95} + 2 q^{97} + O(q^{100}) \)

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
\(5\) −2.64575 −1.18322 −0.591608 0.806226i \(-0.701507\pi\)
−0.591608 + 0.806226i \(0.701507\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −2.64575 −0.836660
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) 5.29150 1.21395 0.606977 0.794719i \(-0.292382\pi\)
0.606977 + 0.794719i \(0.292382\pi\)
\(20\) −2.64575 −0.591608
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) −2.64575 −0.551677 −0.275839 0.961204i \(-0.588956\pi\)
−0.275839 + 0.961204i \(0.588956\pi\)
\(24\) 0 0
\(25\) 2.00000 0.400000
\(26\) 4.00000 0.784465
\(27\) 0 0
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −3.00000 −0.514496
\(35\) 0 0
\(36\) 0 0
\(37\) −9.29150 −1.52751 −0.763757 0.645504i \(-0.776647\pi\)
−0.763757 + 0.645504i \(0.776647\pi\)
\(38\) 5.29150 0.858395
\(39\) 0 0
\(40\) −2.64575 −0.418330
\(41\) 9.00000 1.40556 0.702782 0.711405i \(-0.251941\pi\)
0.702782 + 0.711405i \(0.251941\pi\)
\(42\) 0 0
\(43\) −1.29150 −0.196952 −0.0984762 0.995139i \(-0.531397\pi\)
−0.0984762 + 0.995139i \(0.531397\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −2.64575 −0.390095
\(47\) 11.9373 1.74123 0.870614 0.491967i \(-0.163722\pi\)
0.870614 + 0.491967i \(0.163722\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 2.00000 0.282843
\(51\) 0 0
\(52\) 4.00000 0.554700
\(53\) −4.00000 −0.549442 −0.274721 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 0 0
\(55\) 2.64575 0.356753
\(56\) 0 0
\(57\) 0 0
\(58\) −2.00000 −0.262613
\(59\) −14.5830 −1.89855 −0.949273 0.314454i \(-0.898179\pi\)
−0.949273 + 0.314454i \(0.898179\pi\)
\(60\) 0 0
\(61\) 3.93725 0.504114 0.252057 0.967712i \(-0.418893\pi\)
0.252057 + 0.967712i \(0.418893\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −10.5830 −1.31266
\(66\) 0 0
\(67\) −13.5830 −1.65943 −0.829714 0.558189i \(-0.811497\pi\)
−0.829714 + 0.558189i \(0.811497\pi\)
\(68\) −3.00000 −0.363803
\(69\) 0 0
\(70\) 0 0
\(71\) 13.2915 1.57741 0.788706 0.614771i \(-0.210752\pi\)
0.788706 + 0.614771i \(0.210752\pi\)
\(72\) 0 0
\(73\) 4.70850 0.551088 0.275544 0.961288i \(-0.411142\pi\)
0.275544 + 0.961288i \(0.411142\pi\)
\(74\) −9.29150 −1.08012
\(75\) 0 0
\(76\) 5.29150 0.606977
\(77\) 0 0
\(78\) 0 0
\(79\) 5.35425 0.602400 0.301200 0.953561i \(-0.402613\pi\)
0.301200 + 0.953561i \(0.402613\pi\)
\(80\) −2.64575 −0.295804
\(81\) 0 0
\(82\) 9.00000 0.993884
\(83\) −5.58301 −0.612814 −0.306407 0.951901i \(-0.599127\pi\)
−0.306407 + 0.951901i \(0.599127\pi\)
\(84\) 0 0
\(85\) 7.93725 0.860916
\(86\) −1.29150 −0.139266
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) 13.2915 1.40890 0.704448 0.709755i \(-0.251195\pi\)
0.704448 + 0.709755i \(0.251195\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −2.64575 −0.275839
\(93\) 0 0
\(94\) 11.9373 1.23123
\(95\) −14.0000 −1.43637
\(96\) 0 0
\(97\) −9.58301 −0.973007 −0.486503 0.873679i \(-0.661728\pi\)
−0.486503 + 0.873679i \(0.661728\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 2.00000 0.200000
\(101\) −3.29150 −0.327517 −0.163758 0.986500i \(-0.552362\pi\)
−0.163758 + 0.986500i \(0.552362\pi\)
\(102\) 0 0
\(103\) 10.0000 0.985329 0.492665 0.870219i \(-0.336023\pi\)
0.492665 + 0.870219i \(0.336023\pi\)
\(104\) 4.00000 0.392232
\(105\) 0 0
\(106\) −4.00000 −0.388514
\(107\) −9.00000 −0.870063 −0.435031 0.900415i \(-0.643263\pi\)
−0.435031 + 0.900415i \(0.643263\pi\)
\(108\) 0 0
\(109\) 11.9373 1.14338 0.571691 0.820469i \(-0.306288\pi\)
0.571691 + 0.820469i \(0.306288\pi\)
\(110\) 2.64575 0.252262
\(111\) 0 0
\(112\) 0 0
\(113\) 8.58301 0.807421 0.403711 0.914887i \(-0.367720\pi\)
0.403711 + 0.914887i \(0.367720\pi\)
\(114\) 0 0
\(115\) 7.00000 0.652753
\(116\) −2.00000 −0.185695
\(117\) 0 0
\(118\) −14.5830 −1.34247
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 3.93725 0.356462
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) 7.93725 0.709930
\(126\) 0 0
\(127\) −2.64575 −0.234772 −0.117386 0.993086i \(-0.537452\pi\)
−0.117386 + 0.993086i \(0.537452\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −10.5830 −0.928191
\(131\) 2.58301 0.225678 0.112839 0.993613i \(-0.464006\pi\)
0.112839 + 0.993613i \(0.464006\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −13.5830 −1.17339
\(135\) 0 0
\(136\) −3.00000 −0.257248
\(137\) 2.70850 0.231403 0.115701 0.993284i \(-0.463088\pi\)
0.115701 + 0.993284i \(0.463088\pi\)
\(138\) 0 0
\(139\) −8.58301 −0.728001 −0.364001 0.931399i \(-0.618589\pi\)
−0.364001 + 0.931399i \(0.618589\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 13.2915 1.11540
\(143\) −4.00000 −0.334497
\(144\) 0 0
\(145\) 5.29150 0.439435
\(146\) 4.70850 0.389678
\(147\) 0 0
\(148\) −9.29150 −0.763757
\(149\) 11.8745 0.972798 0.486399 0.873737i \(-0.338310\pi\)
0.486399 + 0.873737i \(0.338310\pi\)
\(150\) 0 0
\(151\) 17.3542 1.41227 0.706134 0.708078i \(-0.250437\pi\)
0.706134 + 0.708078i \(0.250437\pi\)
\(152\) 5.29150 0.429198
\(153\) 0 0
\(154\) 0 0
\(155\) −10.5830 −0.850047
\(156\) 0 0
\(157\) 11.2915 0.901160 0.450580 0.892736i \(-0.351217\pi\)
0.450580 + 0.892736i \(0.351217\pi\)
\(158\) 5.35425 0.425961
\(159\) 0 0
\(160\) −2.64575 −0.209165
\(161\) 0 0
\(162\) 0 0
\(163\) 15.5830 1.22056 0.610278 0.792188i \(-0.291058\pi\)
0.610278 + 0.792188i \(0.291058\pi\)
\(164\) 9.00000 0.702782
\(165\) 0 0
\(166\) −5.58301 −0.433325
\(167\) 19.2915 1.49282 0.746411 0.665486i \(-0.231776\pi\)
0.746411 + 0.665486i \(0.231776\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 7.93725 0.608760
\(171\) 0 0
\(172\) −1.29150 −0.0984762
\(173\) 4.00000 0.304114 0.152057 0.988372i \(-0.451410\pi\)
0.152057 + 0.988372i \(0.451410\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) 13.2915 0.996240
\(179\) −15.2915 −1.14294 −0.571470 0.820623i \(-0.693627\pi\)
−0.571470 + 0.820623i \(0.693627\pi\)
\(180\) 0 0
\(181\) −5.29150 −0.393314 −0.196657 0.980472i \(-0.563009\pi\)
−0.196657 + 0.980472i \(0.563009\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −2.64575 −0.195047
\(185\) 24.5830 1.80738
\(186\) 0 0
\(187\) 3.00000 0.219382
\(188\) 11.9373 0.870614
\(189\) 0 0
\(190\) −14.0000 −1.01567
\(191\) −17.2915 −1.25117 −0.625585 0.780156i \(-0.715139\pi\)
−0.625585 + 0.780156i \(0.715139\pi\)
\(192\) 0 0
\(193\) 15.8745 1.14267 0.571336 0.820716i \(-0.306425\pi\)
0.571336 + 0.820716i \(0.306425\pi\)
\(194\) −9.58301 −0.688020
\(195\) 0 0
\(196\) 0 0
\(197\) 9.87451 0.703530 0.351765 0.936088i \(-0.385582\pi\)
0.351765 + 0.936088i \(0.385582\pi\)
\(198\) 0 0
\(199\) −23.1660 −1.64219 −0.821097 0.570788i \(-0.806638\pi\)
−0.821097 + 0.570788i \(0.806638\pi\)
\(200\) 2.00000 0.141421
\(201\) 0 0
\(202\) −3.29150 −0.231589
\(203\) 0 0
\(204\) 0 0
\(205\) −23.8118 −1.66309
\(206\) 10.0000 0.696733
\(207\) 0 0
\(208\) 4.00000 0.277350
\(209\) −5.29150 −0.366021
\(210\) 0 0
\(211\) 10.7085 0.737203 0.368602 0.929587i \(-0.379837\pi\)
0.368602 + 0.929587i \(0.379837\pi\)
\(212\) −4.00000 −0.274721
\(213\) 0 0
\(214\) −9.00000 −0.615227
\(215\) 3.41699 0.233037
\(216\) 0 0
\(217\) 0 0
\(218\) 11.9373 0.808493
\(219\) 0 0
\(220\) 2.64575 0.178377
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) 24.4575 1.63780 0.818898 0.573939i \(-0.194585\pi\)
0.818898 + 0.573939i \(0.194585\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 8.58301 0.570933
\(227\) 23.5830 1.56526 0.782630 0.622488i \(-0.213878\pi\)
0.782630 + 0.622488i \(0.213878\pi\)
\(228\) 0 0
\(229\) −17.2915 −1.14265 −0.571327 0.820722i \(-0.693571\pi\)
−0.571327 + 0.820722i \(0.693571\pi\)
\(230\) 7.00000 0.461566
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) 22.1660 1.45214 0.726072 0.687619i \(-0.241344\pi\)
0.726072 + 0.687619i \(0.241344\pi\)
\(234\) 0 0
\(235\) −31.5830 −2.06025
\(236\) −14.5830 −0.949273
\(237\) 0 0
\(238\) 0 0
\(239\) −13.2915 −0.859756 −0.429878 0.902887i \(-0.641443\pi\)
−0.429878 + 0.902887i \(0.641443\pi\)
\(240\) 0 0
\(241\) −15.1660 −0.976929 −0.488464 0.872584i \(-0.662443\pi\)
−0.488464 + 0.872584i \(0.662443\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) 3.93725 0.252057
\(245\) 0 0
\(246\) 0 0
\(247\) 21.1660 1.34676
\(248\) 4.00000 0.254000
\(249\) 0 0
\(250\) 7.93725 0.501996
\(251\) 12.7085 0.802153 0.401077 0.916045i \(-0.368636\pi\)
0.401077 + 0.916045i \(0.368636\pi\)
\(252\) 0 0
\(253\) 2.64575 0.166337
\(254\) −2.64575 −0.166009
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −1.29150 −0.0805617 −0.0402809 0.999188i \(-0.512825\pi\)
−0.0402809 + 0.999188i \(0.512825\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −10.5830 −0.656330
\(261\) 0 0
\(262\) 2.58301 0.159579
\(263\) 9.87451 0.608888 0.304444 0.952530i \(-0.401529\pi\)
0.304444 + 0.952530i \(0.401529\pi\)
\(264\) 0 0
\(265\) 10.5830 0.650109
\(266\) 0 0
\(267\) 0 0
\(268\) −13.5830 −0.829714
\(269\) −5.35425 −0.326454 −0.163227 0.986589i \(-0.552190\pi\)
−0.163227 + 0.986589i \(0.552190\pi\)
\(270\) 0 0
\(271\) 1.29150 0.0784532 0.0392266 0.999230i \(-0.487511\pi\)
0.0392266 + 0.999230i \(0.487511\pi\)
\(272\) −3.00000 −0.181902
\(273\) 0 0
\(274\) 2.70850 0.163626
\(275\) −2.00000 −0.120605
\(276\) 0 0
\(277\) 25.1660 1.51208 0.756040 0.654526i \(-0.227132\pi\)
0.756040 + 0.654526i \(0.227132\pi\)
\(278\) −8.58301 −0.514774
\(279\) 0 0
\(280\) 0 0
\(281\) 3.00000 0.178965 0.0894825 0.995988i \(-0.471479\pi\)
0.0894825 + 0.995988i \(0.471479\pi\)
\(282\) 0 0
\(283\) 22.4575 1.33496 0.667480 0.744627i \(-0.267373\pi\)
0.667480 + 0.744627i \(0.267373\pi\)
\(284\) 13.2915 0.788706
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) 0 0
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 5.29150 0.310728
\(291\) 0 0
\(292\) 4.70850 0.275544
\(293\) 6.58301 0.384583 0.192292 0.981338i \(-0.438408\pi\)
0.192292 + 0.981338i \(0.438408\pi\)
\(294\) 0 0
\(295\) 38.5830 2.24639
\(296\) −9.29150 −0.540058
\(297\) 0 0
\(298\) 11.8745 0.687872
\(299\) −10.5830 −0.612031
\(300\) 0 0
\(301\) 0 0
\(302\) 17.3542 0.998625
\(303\) 0 0
\(304\) 5.29150 0.303488
\(305\) −10.4170 −0.596475
\(306\) 0 0
\(307\) 32.5830 1.85961 0.929805 0.368052i \(-0.119975\pi\)
0.929805 + 0.368052i \(0.119975\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −10.5830 −0.601074
\(311\) −14.5203 −0.823368 −0.411684 0.911327i \(-0.635059\pi\)
−0.411684 + 0.911327i \(0.635059\pi\)
\(312\) 0 0
\(313\) −0.583005 −0.0329534 −0.0164767 0.999864i \(-0.505245\pi\)
−0.0164767 + 0.999864i \(0.505245\pi\)
\(314\) 11.2915 0.637216
\(315\) 0 0
\(316\) 5.35425 0.301200
\(317\) 17.2288 0.967663 0.483832 0.875161i \(-0.339245\pi\)
0.483832 + 0.875161i \(0.339245\pi\)
\(318\) 0 0
\(319\) 2.00000 0.111979
\(320\) −2.64575 −0.147902
\(321\) 0 0
\(322\) 0 0
\(323\) −15.8745 −0.883281
\(324\) 0 0
\(325\) 8.00000 0.443760
\(326\) 15.5830 0.863063
\(327\) 0 0
\(328\) 9.00000 0.496942
\(329\) 0 0
\(330\) 0 0
\(331\) 13.0000 0.714545 0.357272 0.934000i \(-0.383707\pi\)
0.357272 + 0.934000i \(0.383707\pi\)
\(332\) −5.58301 −0.306407
\(333\) 0 0
\(334\) 19.2915 1.05558
\(335\) 35.9373 1.96346
\(336\) 0 0
\(337\) 17.2915 0.941928 0.470964 0.882152i \(-0.343906\pi\)
0.470964 + 0.882152i \(0.343906\pi\)
\(338\) 3.00000 0.163178
\(339\) 0 0
\(340\) 7.93725 0.430458
\(341\) −4.00000 −0.216612
\(342\) 0 0
\(343\) 0 0
\(344\) −1.29150 −0.0696332
\(345\) 0 0
\(346\) 4.00000 0.215041
\(347\) −8.41699 −0.451848 −0.225924 0.974145i \(-0.572540\pi\)
−0.225924 + 0.974145i \(0.572540\pi\)
\(348\) 0 0
\(349\) −13.2288 −0.708119 −0.354060 0.935223i \(-0.615199\pi\)
−0.354060 + 0.935223i \(0.615199\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) −12.7085 −0.676405 −0.338203 0.941073i \(-0.609819\pi\)
−0.338203 + 0.941073i \(0.609819\pi\)
\(354\) 0 0
\(355\) −35.1660 −1.86642
\(356\) 13.2915 0.704448
\(357\) 0 0
\(358\) −15.2915 −0.808181
\(359\) 0.583005 0.0307698 0.0153849 0.999882i \(-0.495103\pi\)
0.0153849 + 0.999882i \(0.495103\pi\)
\(360\) 0 0
\(361\) 9.00000 0.473684
\(362\) −5.29150 −0.278115
\(363\) 0 0
\(364\) 0 0
\(365\) −12.4575 −0.652056
\(366\) 0 0
\(367\) −21.2915 −1.11141 −0.555704 0.831380i \(-0.687551\pi\)
−0.555704 + 0.831380i \(0.687551\pi\)
\(368\) −2.64575 −0.137919
\(369\) 0 0
\(370\) 24.5830 1.27801
\(371\) 0 0
\(372\) 0 0
\(373\) 31.8118 1.64715 0.823575 0.567207i \(-0.191976\pi\)
0.823575 + 0.567207i \(0.191976\pi\)
\(374\) 3.00000 0.155126
\(375\) 0 0
\(376\) 11.9373 0.615617
\(377\) −8.00000 −0.412021
\(378\) 0 0
\(379\) −6.41699 −0.329619 −0.164809 0.986325i \(-0.552701\pi\)
−0.164809 + 0.986325i \(0.552701\pi\)
\(380\) −14.0000 −0.718185
\(381\) 0 0
\(382\) −17.2915 −0.884710
\(383\) 29.2915 1.49673 0.748363 0.663289i \(-0.230840\pi\)
0.748363 + 0.663289i \(0.230840\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 15.8745 0.807991
\(387\) 0 0
\(388\) −9.58301 −0.486503
\(389\) −16.0627 −0.814414 −0.407207 0.913336i \(-0.633497\pi\)
−0.407207 + 0.913336i \(0.633497\pi\)
\(390\) 0 0
\(391\) 7.93725 0.401404
\(392\) 0 0
\(393\) 0 0
\(394\) 9.87451 0.497471
\(395\) −14.1660 −0.712769
\(396\) 0 0
\(397\) −6.00000 −0.301131 −0.150566 0.988600i \(-0.548110\pi\)
−0.150566 + 0.988600i \(0.548110\pi\)
\(398\) −23.1660 −1.16121
\(399\) 0 0
\(400\) 2.00000 0.100000
\(401\) 4.12549 0.206017 0.103009 0.994680i \(-0.467153\pi\)
0.103009 + 0.994680i \(0.467153\pi\)
\(402\) 0 0
\(403\) 16.0000 0.797017
\(404\) −3.29150 −0.163758
\(405\) 0 0
\(406\) 0 0
\(407\) 9.29150 0.460563
\(408\) 0 0
\(409\) −4.12549 −0.203992 −0.101996 0.994785i \(-0.532523\pi\)
−0.101996 + 0.994785i \(0.532523\pi\)
\(410\) −23.8118 −1.17598
\(411\) 0 0
\(412\) 10.0000 0.492665
\(413\) 0 0
\(414\) 0 0
\(415\) 14.7712 0.725092
\(416\) 4.00000 0.196116
\(417\) 0 0
\(418\) −5.29150 −0.258816
\(419\) 21.8745 1.06864 0.534320 0.845282i \(-0.320568\pi\)
0.534320 + 0.845282i \(0.320568\pi\)
\(420\) 0 0
\(421\) 37.8745 1.84589 0.922945 0.384931i \(-0.125775\pi\)
0.922945 + 0.384931i \(0.125775\pi\)
\(422\) 10.7085 0.521281
\(423\) 0 0
\(424\) −4.00000 −0.194257
\(425\) −6.00000 −0.291043
\(426\) 0 0
\(427\) 0 0
\(428\) −9.00000 −0.435031
\(429\) 0 0
\(430\) 3.41699 0.164782
\(431\) 6.70850 0.323137 0.161568 0.986862i \(-0.448345\pi\)
0.161568 + 0.986862i \(0.448345\pi\)
\(432\) 0 0
\(433\) 19.5830 0.941099 0.470550 0.882374i \(-0.344056\pi\)
0.470550 + 0.882374i \(0.344056\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 11.9373 0.571691
\(437\) −14.0000 −0.669711
\(438\) 0 0
\(439\) 14.6458 0.699004 0.349502 0.936936i \(-0.386351\pi\)
0.349502 + 0.936936i \(0.386351\pi\)
\(440\) 2.64575 0.126131
\(441\) 0 0
\(442\) −12.0000 −0.570782
\(443\) −11.1660 −0.530513 −0.265257 0.964178i \(-0.585457\pi\)
−0.265257 + 0.964178i \(0.585457\pi\)
\(444\) 0 0
\(445\) −35.1660 −1.66703
\(446\) 24.4575 1.15810
\(447\) 0 0
\(448\) 0 0
\(449\) 22.4575 1.05984 0.529918 0.848049i \(-0.322223\pi\)
0.529918 + 0.848049i \(0.322223\pi\)
\(450\) 0 0
\(451\) −9.00000 −0.423793
\(452\) 8.58301 0.403711
\(453\) 0 0
\(454\) 23.5830 1.10681
\(455\) 0 0
\(456\) 0 0
\(457\) −19.4170 −0.908289 −0.454144 0.890928i \(-0.650055\pi\)
−0.454144 + 0.890928i \(0.650055\pi\)
\(458\) −17.2915 −0.807979
\(459\) 0 0
\(460\) 7.00000 0.326377
\(461\) −3.41699 −0.159145 −0.0795727 0.996829i \(-0.525356\pi\)
−0.0795727 + 0.996829i \(0.525356\pi\)
\(462\) 0 0
\(463\) −35.0405 −1.62847 −0.814235 0.580535i \(-0.802844\pi\)
−0.814235 + 0.580535i \(0.802844\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) 22.1660 1.02682
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −31.5830 −1.45682
\(471\) 0 0
\(472\) −14.5830 −0.671237
\(473\) 1.29150 0.0593834
\(474\) 0 0
\(475\) 10.5830 0.485582
\(476\) 0 0
\(477\) 0 0
\(478\) −13.2915 −0.607939
\(479\) −2.12549 −0.0971162 −0.0485581 0.998820i \(-0.515463\pi\)
−0.0485581 + 0.998820i \(0.515463\pi\)
\(480\) 0 0
\(481\) −37.1660 −1.69462
\(482\) −15.1660 −0.690793
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 25.3542 1.15128
\(486\) 0 0
\(487\) −9.87451 −0.447457 −0.223728 0.974652i \(-0.571823\pi\)
−0.223728 + 0.974652i \(0.571823\pi\)
\(488\) 3.93725 0.178231
\(489\) 0 0
\(490\) 0 0
\(491\) 1.00000 0.0451294 0.0225647 0.999745i \(-0.492817\pi\)
0.0225647 + 0.999745i \(0.492817\pi\)
\(492\) 0 0
\(493\) 6.00000 0.270226
\(494\) 21.1660 0.952304
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) 0 0
\(498\) 0 0
\(499\) 2.58301 0.115631 0.0578156 0.998327i \(-0.481586\pi\)
0.0578156 + 0.998327i \(0.481586\pi\)
\(500\) 7.93725 0.354965
\(501\) 0 0
\(502\) 12.7085 0.567208
\(503\) 9.29150 0.414288 0.207144 0.978311i \(-0.433583\pi\)
0.207144 + 0.978311i \(0.433583\pi\)
\(504\) 0 0
\(505\) 8.70850 0.387523
\(506\) 2.64575 0.117618
\(507\) 0 0
\(508\) −2.64575 −0.117386
\(509\) −17.1660 −0.760870 −0.380435 0.924808i \(-0.624226\pi\)
−0.380435 + 0.924808i \(0.624226\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −1.29150 −0.0569657
\(515\) −26.4575 −1.16586
\(516\) 0 0
\(517\) −11.9373 −0.525000
\(518\) 0 0
\(519\) 0 0
\(520\) −10.5830 −0.464095
\(521\) 42.0000 1.84005 0.920027 0.391856i \(-0.128167\pi\)
0.920027 + 0.391856i \(0.128167\pi\)
\(522\) 0 0
\(523\) −13.2915 −0.581197 −0.290598 0.956845i \(-0.593854\pi\)
−0.290598 + 0.956845i \(0.593854\pi\)
\(524\) 2.58301 0.112839
\(525\) 0 0
\(526\) 9.87451 0.430549
\(527\) −12.0000 −0.522728
\(528\) 0 0
\(529\) −16.0000 −0.695652
\(530\) 10.5830 0.459696
\(531\) 0 0
\(532\) 0 0
\(533\) 36.0000 1.55933
\(534\) 0 0
\(535\) 23.8118 1.02947
\(536\) −13.5830 −0.586696
\(537\) 0 0
\(538\) −5.35425 −0.230838
\(539\) 0 0
\(540\) 0 0
\(541\) −43.8118 −1.88361 −0.941807 0.336153i \(-0.890874\pi\)
−0.941807 + 0.336153i \(0.890874\pi\)
\(542\) 1.29150 0.0554748
\(543\) 0 0
\(544\) −3.00000 −0.128624
\(545\) −31.5830 −1.35287
\(546\) 0 0
\(547\) 25.2915 1.08139 0.540693 0.841220i \(-0.318162\pi\)
0.540693 + 0.841220i \(0.318162\pi\)
\(548\) 2.70850 0.115701
\(549\) 0 0
\(550\) −2.00000 −0.0852803
\(551\) −10.5830 −0.450851
\(552\) 0 0
\(553\) 0 0
\(554\) 25.1660 1.06920
\(555\) 0 0
\(556\) −8.58301 −0.364001
\(557\) 8.70850 0.368991 0.184495 0.982833i \(-0.440935\pi\)
0.184495 + 0.982833i \(0.440935\pi\)
\(558\) 0 0
\(559\) −5.16601 −0.218499
\(560\) 0 0
\(561\) 0 0
\(562\) 3.00000 0.126547
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) −22.7085 −0.955354
\(566\) 22.4575 0.943960
\(567\) 0 0
\(568\) 13.2915 0.557699
\(569\) −27.1660 −1.13886 −0.569429 0.822040i \(-0.692836\pi\)
−0.569429 + 0.822040i \(0.692836\pi\)
\(570\) 0 0
\(571\) 33.7490 1.41235 0.706176 0.708036i \(-0.250419\pi\)
0.706176 + 0.708036i \(0.250419\pi\)
\(572\) −4.00000 −0.167248
\(573\) 0 0
\(574\) 0 0
\(575\) −5.29150 −0.220671
\(576\) 0 0
\(577\) 40.7490 1.69640 0.848202 0.529673i \(-0.177685\pi\)
0.848202 + 0.529673i \(0.177685\pi\)
\(578\) −8.00000 −0.332756
\(579\) 0 0
\(580\) 5.29150 0.219718
\(581\) 0 0
\(582\) 0 0
\(583\) 4.00000 0.165663
\(584\) 4.70850 0.194839
\(585\) 0 0
\(586\) 6.58301 0.271941
\(587\) −21.8745 −0.902858 −0.451429 0.892307i \(-0.649086\pi\)
−0.451429 + 0.892307i \(0.649086\pi\)
\(588\) 0 0
\(589\) 21.1660 0.872130
\(590\) 38.5830 1.58844
\(591\) 0 0
\(592\) −9.29150 −0.381878
\(593\) 37.7490 1.55017 0.775083 0.631859i \(-0.217708\pi\)
0.775083 + 0.631859i \(0.217708\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 11.8745 0.486399
\(597\) 0 0
\(598\) −10.5830 −0.432771
\(599\) −34.6458 −1.41559 −0.707794 0.706419i \(-0.750309\pi\)
−0.707794 + 0.706419i \(0.750309\pi\)
\(600\) 0 0
\(601\) 17.4170 0.710454 0.355227 0.934780i \(-0.384404\pi\)
0.355227 + 0.934780i \(0.384404\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 17.3542 0.706134
\(605\) −2.64575 −0.107565
\(606\) 0 0
\(607\) −45.1033 −1.83069 −0.915343 0.402676i \(-0.868080\pi\)
−0.915343 + 0.402676i \(0.868080\pi\)
\(608\) 5.29150 0.214599
\(609\) 0 0
\(610\) −10.4170 −0.421772
\(611\) 47.7490 1.93172
\(612\) 0 0
\(613\) −39.8118 −1.60798 −0.803991 0.594642i \(-0.797294\pi\)
−0.803991 + 0.594642i \(0.797294\pi\)
\(614\) 32.5830 1.31494
\(615\) 0 0
\(616\) 0 0
\(617\) 25.2915 1.01820 0.509099 0.860708i \(-0.329979\pi\)
0.509099 + 0.860708i \(0.329979\pi\)
\(618\) 0 0
\(619\) 41.0000 1.64793 0.823965 0.566641i \(-0.191757\pi\)
0.823965 + 0.566641i \(0.191757\pi\)
\(620\) −10.5830 −0.425024
\(621\) 0 0
\(622\) −14.5203 −0.582209
\(623\) 0 0
\(624\) 0 0
\(625\) −31.0000 −1.24000
\(626\) −0.583005 −0.0233016
\(627\) 0 0
\(628\) 11.2915 0.450580
\(629\) 27.8745 1.11143
\(630\) 0 0
\(631\) 10.1255 0.403089 0.201545 0.979479i \(-0.435404\pi\)
0.201545 + 0.979479i \(0.435404\pi\)
\(632\) 5.35425 0.212981
\(633\) 0 0
\(634\) 17.2288 0.684241
\(635\) 7.00000 0.277787
\(636\) 0 0
\(637\) 0 0
\(638\) 2.00000 0.0791808
\(639\) 0 0
\(640\) −2.64575 −0.104583
\(641\) 5.41699 0.213958 0.106979 0.994261i \(-0.465882\pi\)
0.106979 + 0.994261i \(0.465882\pi\)
\(642\) 0 0
\(643\) −41.1660 −1.62343 −0.811714 0.584054i \(-0.801465\pi\)
−0.811714 + 0.584054i \(0.801465\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −15.8745 −0.624574
\(647\) 42.3948 1.66671 0.833355 0.552738i \(-0.186417\pi\)
0.833355 + 0.552738i \(0.186417\pi\)
\(648\) 0 0
\(649\) 14.5830 0.572433
\(650\) 8.00000 0.313786
\(651\) 0 0
\(652\) 15.5830 0.610278
\(653\) −13.1033 −0.512770 −0.256385 0.966575i \(-0.582532\pi\)
−0.256385 + 0.966575i \(0.582532\pi\)
\(654\) 0 0
\(655\) −6.83399 −0.267026
\(656\) 9.00000 0.351391
\(657\) 0 0
\(658\) 0 0
\(659\) −18.1660 −0.707647 −0.353824 0.935312i \(-0.615119\pi\)
−0.353824 + 0.935312i \(0.615119\pi\)
\(660\) 0 0
\(661\) −31.2915 −1.21710 −0.608549 0.793516i \(-0.708248\pi\)
−0.608549 + 0.793516i \(0.708248\pi\)
\(662\) 13.0000 0.505259
\(663\) 0 0
\(664\) −5.58301 −0.216663
\(665\) 0 0
\(666\) 0 0
\(667\) 5.29150 0.204888
\(668\) 19.2915 0.746411
\(669\) 0 0
\(670\) 35.9373 1.38838
\(671\) −3.93725 −0.151996
\(672\) 0 0
\(673\) −40.5830 −1.56436 −0.782180 0.623053i \(-0.785892\pi\)
−0.782180 + 0.623053i \(0.785892\pi\)
\(674\) 17.2915 0.666044
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) −39.8745 −1.53250 −0.766251 0.642541i \(-0.777880\pi\)
−0.766251 + 0.642541i \(0.777880\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 7.93725 0.304380
\(681\) 0 0
\(682\) −4.00000 −0.153168
\(683\) −25.1660 −0.962951 −0.481475 0.876460i \(-0.659899\pi\)
−0.481475 + 0.876460i \(0.659899\pi\)
\(684\) 0 0
\(685\) −7.16601 −0.273799
\(686\) 0 0
\(687\) 0 0
\(688\) −1.29150 −0.0492381
\(689\) −16.0000 −0.609551
\(690\) 0 0
\(691\) 28.1660 1.07149 0.535743 0.844381i \(-0.320032\pi\)
0.535743 + 0.844381i \(0.320032\pi\)
\(692\) 4.00000 0.152057
\(693\) 0 0
\(694\) −8.41699 −0.319505
\(695\) 22.7085 0.861382
\(696\) 0 0
\(697\) −27.0000 −1.02270
\(698\) −13.2288 −0.500716
\(699\) 0 0
\(700\) 0 0
\(701\) 10.4575 0.394975 0.197487 0.980305i \(-0.436722\pi\)
0.197487 + 0.980305i \(0.436722\pi\)
\(702\) 0 0
\(703\) −49.1660 −1.85433
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) −12.7085 −0.478291
\(707\) 0 0
\(708\) 0 0
\(709\) −47.7490 −1.79325 −0.896626 0.442789i \(-0.853989\pi\)
−0.896626 + 0.442789i \(0.853989\pi\)
\(710\) −35.1660 −1.31976
\(711\) 0 0
\(712\) 13.2915 0.498120
\(713\) −10.5830 −0.396337
\(714\) 0 0
\(715\) 10.5830 0.395782
\(716\) −15.2915 −0.571470
\(717\) 0 0
\(718\) 0.583005 0.0217576
\(719\) −43.9373 −1.63858 −0.819292 0.573377i \(-0.805633\pi\)
−0.819292 + 0.573377i \(0.805633\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 9.00000 0.334945
\(723\) 0 0
\(724\) −5.29150 −0.196657
\(725\) −4.00000 −0.148556
\(726\) 0 0
\(727\) 2.00000 0.0741759 0.0370879 0.999312i \(-0.488192\pi\)
0.0370879 + 0.999312i \(0.488192\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −12.4575 −0.461073
\(731\) 3.87451 0.143304
\(732\) 0 0
\(733\) 3.93725 0.145426 0.0727129 0.997353i \(-0.476834\pi\)
0.0727129 + 0.997353i \(0.476834\pi\)
\(734\) −21.2915 −0.785884
\(735\) 0 0
\(736\) −2.64575 −0.0975237
\(737\) 13.5830 0.500336
\(738\) 0 0
\(739\) −3.41699 −0.125696 −0.0628481 0.998023i \(-0.520018\pi\)
−0.0628481 + 0.998023i \(0.520018\pi\)
\(740\) 24.5830 0.903689
\(741\) 0 0
\(742\) 0 0
\(743\) 29.2915 1.07460 0.537301 0.843391i \(-0.319444\pi\)
0.537301 + 0.843391i \(0.319444\pi\)
\(744\) 0 0
\(745\) −31.4170 −1.15103
\(746\) 31.8118 1.16471
\(747\) 0 0
\(748\) 3.00000 0.109691
\(749\) 0 0
\(750\) 0 0
\(751\) −18.7085 −0.682683 −0.341341 0.939939i \(-0.610881\pi\)
−0.341341 + 0.939939i \(0.610881\pi\)
\(752\) 11.9373 0.435307
\(753\) 0 0
\(754\) −8.00000 −0.291343
\(755\) −45.9150 −1.67102
\(756\) 0 0
\(757\) −11.4170 −0.414958 −0.207479 0.978240i \(-0.566526\pi\)
−0.207479 + 0.978240i \(0.566526\pi\)
\(758\) −6.41699 −0.233076
\(759\) 0 0
\(760\) −14.0000 −0.507833
\(761\) 26.4170 0.957615 0.478808 0.877920i \(-0.341069\pi\)
0.478808 + 0.877920i \(0.341069\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −17.2915 −0.625585
\(765\) 0 0
\(766\) 29.2915 1.05835
\(767\) −58.3320 −2.10625
\(768\) 0 0
\(769\) 41.2915 1.48901 0.744505 0.667617i \(-0.232686\pi\)
0.744505 + 0.667617i \(0.232686\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 15.8745 0.571336
\(773\) −23.9373 −0.860963 −0.430482 0.902599i \(-0.641656\pi\)
−0.430482 + 0.902599i \(0.641656\pi\)
\(774\) 0 0
\(775\) 8.00000 0.287368
\(776\) −9.58301 −0.344010
\(777\) 0 0
\(778\) −16.0627 −0.575877
\(779\) 47.6235 1.70629
\(780\) 0 0
\(781\) −13.2915 −0.475607
\(782\) 7.93725 0.283836
\(783\) 0 0
\(784\) 0 0
\(785\) −29.8745 −1.06627
\(786\) 0 0
\(787\) 1.29150 0.0460371 0.0230185 0.999735i \(-0.492672\pi\)
0.0230185 + 0.999735i \(0.492672\pi\)
\(788\) 9.87451 0.351765
\(789\) 0 0
\(790\) −14.1660 −0.504004
\(791\) 0 0
\(792\) 0 0
\(793\) 15.7490 0.559264
\(794\) −6.00000 −0.212932
\(795\) 0 0
\(796\) −23.1660 −0.821097
\(797\) −13.1033 −0.464141 −0.232071 0.972699i \(-0.574550\pi\)
−0.232071 + 0.972699i \(0.574550\pi\)
\(798\) 0 0
\(799\) −35.8118 −1.26693
\(800\) 2.00000 0.0707107
\(801\) 0 0
\(802\) 4.12549 0.145676
\(803\) −4.70850 −0.166159
\(804\) 0 0
\(805\) 0 0
\(806\) 16.0000 0.563576
\(807\) 0 0
\(808\) −3.29150 −0.115795
\(809\) 39.3320 1.38284 0.691420 0.722453i \(-0.256985\pi\)
0.691420 + 0.722453i \(0.256985\pi\)
\(810\) 0 0
\(811\) 4.58301 0.160931 0.0804655 0.996757i \(-0.474359\pi\)
0.0804655 + 0.996757i \(0.474359\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 9.29150 0.325667
\(815\) −41.2288 −1.44418
\(816\) 0 0
\(817\) −6.83399 −0.239091
\(818\) −4.12549 −0.144244
\(819\) 0 0
\(820\) −23.8118 −0.831543
\(821\) −54.3320 −1.89620 −0.948100 0.317971i \(-0.896998\pi\)
−0.948100 + 0.317971i \(0.896998\pi\)
\(822\) 0 0
\(823\) −43.8745 −1.52937 −0.764685 0.644405i \(-0.777105\pi\)
−0.764685 + 0.644405i \(0.777105\pi\)
\(824\) 10.0000 0.348367
\(825\) 0 0
\(826\) 0 0
\(827\) 49.3320 1.71544 0.857721 0.514115i \(-0.171880\pi\)
0.857721 + 0.514115i \(0.171880\pi\)
\(828\) 0 0
\(829\) 11.8745 0.412419 0.206209 0.978508i \(-0.433887\pi\)
0.206209 + 0.978508i \(0.433887\pi\)
\(830\) 14.7712 0.512717
\(831\) 0 0
\(832\) 4.00000 0.138675
\(833\) 0 0
\(834\) 0 0
\(835\) −51.0405 −1.76633
\(836\) −5.29150 −0.183010
\(837\) 0 0
\(838\) 21.8745 0.755642
\(839\) 21.1033 0.728566 0.364283 0.931288i \(-0.381314\pi\)
0.364283 + 0.931288i \(0.381314\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 37.8745 1.30524
\(843\) 0 0
\(844\) 10.7085 0.368602
\(845\) −7.93725 −0.273050
\(846\) 0 0
\(847\) 0 0
\(848\) −4.00000 −0.137361
\(849\) 0 0
\(850\) −6.00000 −0.205798
\(851\) 24.5830 0.842695
\(852\) 0 0
\(853\) 22.5203 0.771079 0.385539 0.922691i \(-0.374015\pi\)
0.385539 + 0.922691i \(0.374015\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −9.00000 −0.307614
\(857\) 52.1660 1.78196 0.890978 0.454046i \(-0.150020\pi\)
0.890978 + 0.454046i \(0.150020\pi\)
\(858\) 0 0
\(859\) 14.7490 0.503230 0.251615 0.967827i \(-0.419038\pi\)
0.251615 + 0.967827i \(0.419038\pi\)
\(860\) 3.41699 0.116519
\(861\) 0 0
\(862\) 6.70850 0.228492
\(863\) −25.1033 −0.854525 −0.427263 0.904128i \(-0.640522\pi\)
−0.427263 + 0.904128i \(0.640522\pi\)
\(864\) 0 0
\(865\) −10.5830 −0.359833
\(866\) 19.5830 0.665458
\(867\) 0 0
\(868\) 0 0
\(869\) −5.35425 −0.181630
\(870\) 0 0
\(871\) −54.3320 −1.84097
\(872\) 11.9373 0.404246
\(873\) 0 0
\(874\) −14.0000 −0.473557
\(875\) 0 0
\(876\) 0 0
\(877\) 37.2288 1.25713 0.628563 0.777759i \(-0.283643\pi\)
0.628563 + 0.777759i \(0.283643\pi\)
\(878\) 14.6458 0.494270
\(879\) 0 0
\(880\) 2.64575 0.0891883
\(881\) −19.1660 −0.645719 −0.322860 0.946447i \(-0.604644\pi\)
−0.322860 + 0.946447i \(0.604644\pi\)
\(882\) 0 0
\(883\) −55.5830 −1.87052 −0.935259 0.353965i \(-0.884833\pi\)
−0.935259 + 0.353965i \(0.884833\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) −11.1660 −0.375129
\(887\) 43.0405 1.44516 0.722580 0.691288i \(-0.242956\pi\)
0.722580 + 0.691288i \(0.242956\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −35.1660 −1.17877
\(891\) 0 0
\(892\) 24.4575 0.818898
\(893\) 63.1660 2.11377
\(894\) 0 0
\(895\) 40.4575 1.35235
\(896\) 0 0
\(897\) 0 0
\(898\) 22.4575 0.749417
\(899\) −8.00000 −0.266815
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) −9.00000 −0.299667
\(903\) 0 0
\(904\) 8.58301 0.285467
\(905\) 14.0000 0.465376
\(906\) 0 0
\(907\) −58.1660 −1.93137 −0.965685 0.259715i \(-0.916371\pi\)
−0.965685 + 0.259715i \(0.916371\pi\)
\(908\) 23.5830 0.782630
\(909\) 0 0
\(910\) 0 0
\(911\) −50.3948 −1.66965 −0.834827 0.550513i \(-0.814432\pi\)
−0.834827 + 0.550513i \(0.814432\pi\)
\(912\) 0 0
\(913\) 5.58301 0.184771
\(914\) −19.4170 −0.642257
\(915\) 0 0
\(916\) −17.2915 −0.571327
\(917\) 0 0
\(918\) 0 0
\(919\) 14.6458 0.483119 0.241559 0.970386i \(-0.422341\pi\)
0.241559 + 0.970386i \(0.422341\pi\)
\(920\) 7.00000 0.230783
\(921\) 0 0
\(922\) −3.41699 −0.112533
\(923\) 53.1660 1.74998
\(924\) 0 0
\(925\) −18.5830 −0.611005
\(926\) −35.0405 −1.15150
\(927\) 0 0
\(928\) −2.00000 −0.0656532
\(929\) 18.4575 0.605571 0.302786 0.953059i \(-0.402083\pi\)
0.302786 + 0.953059i \(0.402083\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 22.1660 0.726072
\(933\) 0 0
\(934\) 8.00000 0.261768
\(935\) −7.93725 −0.259576
\(936\) 0 0
\(937\) −50.0000 −1.63343 −0.816714 0.577042i \(-0.804207\pi\)
−0.816714 + 0.577042i \(0.804207\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −31.5830 −1.03012
\(941\) 24.5830 0.801383 0.400692 0.916213i \(-0.368770\pi\)
0.400692 + 0.916213i \(0.368770\pi\)
\(942\) 0 0
\(943\) −23.8118 −0.775418
\(944\) −14.5830 −0.474636
\(945\) 0 0
\(946\) 1.29150 0.0419904
\(947\) 36.5830 1.18879 0.594394 0.804174i \(-0.297392\pi\)
0.594394 + 0.804174i \(0.297392\pi\)
\(948\) 0 0
\(949\) 18.8340 0.611377
\(950\) 10.5830 0.343358
\(951\) 0 0
\(952\) 0 0
\(953\) 36.7490 1.19042 0.595209 0.803571i \(-0.297069\pi\)
0.595209 + 0.803571i \(0.297069\pi\)
\(954\) 0 0
\(955\) 45.7490 1.48040
\(956\) −13.2915 −0.429878
\(957\) 0 0
\(958\) −2.12549 −0.0686715
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −37.1660 −1.19828
\(963\) 0 0
\(964\) −15.1660 −0.488464
\(965\) −42.0000 −1.35203
\(966\) 0 0
\(967\) 55.9373 1.79882 0.899410 0.437105i \(-0.143996\pi\)
0.899410 + 0.437105i \(0.143996\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) 25.3542 0.814076
\(971\) 6.00000 0.192549 0.0962746 0.995355i \(-0.469307\pi\)
0.0962746 + 0.995355i \(0.469307\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −9.87451 −0.316400
\(975\) 0 0
\(976\) 3.93725 0.126028
\(977\) 29.1660 0.933103 0.466552 0.884494i \(-0.345496\pi\)
0.466552 + 0.884494i \(0.345496\pi\)
\(978\) 0 0
\(979\) −13.2915 −0.424798
\(980\) 0 0
\(981\) 0 0
\(982\) 1.00000 0.0319113
\(983\) 5.47974 0.174777 0.0873883 0.996174i \(-0.472148\pi\)
0.0873883 + 0.996174i \(0.472148\pi\)
\(984\) 0 0
\(985\) −26.1255 −0.832427
\(986\) 6.00000 0.191079
\(987\) 0 0
\(988\) 21.1660 0.673380
\(989\) 3.41699 0.108654
\(990\) 0 0
\(991\) −12.1255 −0.385179 −0.192589 0.981279i \(-0.561689\pi\)
−0.192589 + 0.981279i \(0.561689\pi\)
\(992\) 4.00000 0.127000
\(993\) 0 0
\(994\) 0 0
\(995\) 61.2915 1.94307
\(996\) 0 0
\(997\) −49.1660 −1.55710 −0.778552 0.627581i \(-0.784045\pi\)
−0.778552 + 0.627581i \(0.784045\pi\)
\(998\) 2.58301 0.0817636
\(999\) 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.dm.1.1 2
3.2 odd 2 3234.2.a.ba.1.2 2
7.3 odd 6 1386.2.k.r.793.1 4
7.5 odd 6 1386.2.k.r.991.1 4
7.6 odd 2 9702.2.a.db.1.2 2
21.5 even 6 462.2.i.e.67.2 4
21.17 even 6 462.2.i.e.331.2 yes 4
21.20 even 2 3234.2.a.w.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.i.e.67.2 4 21.5 even 6
462.2.i.e.331.2 yes 4 21.17 even 6
1386.2.k.r.793.1 4 7.3 odd 6
1386.2.k.r.991.1 4 7.5 odd 6
3234.2.a.w.1.1 2 21.20 even 2
3234.2.a.ba.1.2 2 3.2 odd 2
9702.2.a.db.1.2 2 7.6 odd 2
9702.2.a.dm.1.1 2 1.1 even 1 trivial