Properties

Label 6930.2.a.ce.1.1
Level $6930$
Weight $2$
Character 6930.1
Self dual yes
Analytic conductor $55.336$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6930 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6930.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.3363286007\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.81361\) of defining polynomial
Character \(\chi\) \(=\) 6930.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} -1.00000 q^{11} -3.62721 q^{13} +1.00000 q^{14} +1.00000 q^{16} +4.20555 q^{17} -8.15165 q^{19} -1.00000 q^{20} +1.00000 q^{22} -0.897225 q^{23} +1.00000 q^{25} +3.62721 q^{26} -1.00000 q^{28} +7.30833 q^{29} -3.42166 q^{31} -1.00000 q^{32} -4.20555 q^{34} +1.00000 q^{35} -1.10278 q^{37} +8.15165 q^{38} +1.00000 q^{40} -12.3572 q^{41} +10.6761 q^{43} -1.00000 q^{44} +0.897225 q^{46} +1.15667 q^{47} +1.00000 q^{49} -1.00000 q^{50} -3.62721 q^{52} -11.3083 q^{53} +1.00000 q^{55} +1.00000 q^{56} -7.30833 q^{58} -7.25443 q^{59} -3.15667 q^{61} +3.42166 q^{62} +1.00000 q^{64} +3.62721 q^{65} -8.41110 q^{67} +4.20555 q^{68} -1.00000 q^{70} +13.0489 q^{71} +0.205550 q^{73} +1.10278 q^{74} -8.15165 q^{76} +1.00000 q^{77} +12.1517 q^{79} -1.00000 q^{80} +12.3572 q^{82} -2.20555 q^{83} -4.20555 q^{85} -10.6761 q^{86} +1.00000 q^{88} +7.45998 q^{89} +3.62721 q^{91} -0.897225 q^{92} -1.15667 q^{94} +8.15165 q^{95} +2.25945 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} + 3q^{4} - 3q^{5} - 3q^{7} - 3q^{8} + O(q^{10}) \) \( 3q - 3q^{2} + 3q^{4} - 3q^{5} - 3q^{7} - 3q^{8} + 3q^{10} - 3q^{11} + 2q^{13} + 3q^{14} + 3q^{16} - 2q^{17} - 6q^{19} - 3q^{20} + 3q^{22} - 10q^{23} + 3q^{25} - 2q^{26} - 3q^{28} - 12q^{31} - 3q^{32} + 2q^{34} + 3q^{35} + 4q^{37} + 6q^{38} + 3q^{40} - 4q^{41} + 8q^{43} - 3q^{44} + 10q^{46} + 3q^{49} - 3q^{50} + 2q^{52} - 12q^{53} + 3q^{55} + 3q^{56} + 4q^{59} - 6q^{61} + 12q^{62} + 3q^{64} - 2q^{65} + 4q^{67} - 2q^{68} - 3q^{70} + 28q^{71} - 14q^{73} - 4q^{74} - 6q^{76} + 3q^{77} + 18q^{79} - 3q^{80} + 4q^{82} + 8q^{83} + 2q^{85} - 8q^{86} + 3q^{88} - 18q^{89} - 2q^{91} - 10q^{92} + 6q^{95} - 4q^{97} - 3q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −3.62721 −1.00601 −0.503004 0.864284i \(-0.667772\pi\)
−0.503004 + 0.864284i \(0.667772\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.20555 1.02000 0.509998 0.860176i \(-0.329646\pi\)
0.509998 + 0.860176i \(0.329646\pi\)
\(18\) 0 0
\(19\) −8.15165 −1.87012 −0.935058 0.354493i \(-0.884653\pi\)
−0.935058 + 0.354493i \(0.884653\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −0.897225 −0.187084 −0.0935422 0.995615i \(-0.529819\pi\)
−0.0935422 + 0.995615i \(0.529819\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 3.62721 0.711355
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) 7.30833 1.35712 0.678561 0.734544i \(-0.262604\pi\)
0.678561 + 0.734544i \(0.262604\pi\)
\(30\) 0 0
\(31\) −3.42166 −0.614549 −0.307274 0.951621i \(-0.599417\pi\)
−0.307274 + 0.951621i \(0.599417\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −4.20555 −0.721246
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −1.10278 −0.181295 −0.0906476 0.995883i \(-0.528894\pi\)
−0.0906476 + 0.995883i \(0.528894\pi\)
\(38\) 8.15165 1.32237
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −12.3572 −1.92987 −0.964935 0.262488i \(-0.915457\pi\)
−0.964935 + 0.262488i \(0.915457\pi\)
\(42\) 0 0
\(43\) 10.6761 1.62809 0.814044 0.580803i \(-0.197261\pi\)
0.814044 + 0.580803i \(0.197261\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 0.897225 0.132289
\(47\) 1.15667 0.168718 0.0843591 0.996435i \(-0.473116\pi\)
0.0843591 + 0.996435i \(0.473116\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −3.62721 −0.503004
\(53\) −11.3083 −1.55332 −0.776659 0.629921i \(-0.783087\pi\)
−0.776659 + 0.629921i \(0.783087\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −7.30833 −0.959630
\(59\) −7.25443 −0.944446 −0.472223 0.881479i \(-0.656548\pi\)
−0.472223 + 0.881479i \(0.656548\pi\)
\(60\) 0 0
\(61\) −3.15667 −0.404171 −0.202085 0.979368i \(-0.564772\pi\)
−0.202085 + 0.979368i \(0.564772\pi\)
\(62\) 3.42166 0.434552
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 3.62721 0.449900
\(66\) 0 0
\(67\) −8.41110 −1.02758 −0.513790 0.857916i \(-0.671759\pi\)
−0.513790 + 0.857916i \(0.671759\pi\)
\(68\) 4.20555 0.509998
\(69\) 0 0
\(70\) −1.00000 −0.119523
\(71\) 13.0489 1.54862 0.774308 0.632809i \(-0.218098\pi\)
0.774308 + 0.632809i \(0.218098\pi\)
\(72\) 0 0
\(73\) 0.205550 0.0240578 0.0120289 0.999928i \(-0.496171\pi\)
0.0120289 + 0.999928i \(0.496171\pi\)
\(74\) 1.10278 0.128195
\(75\) 0 0
\(76\) −8.15165 −0.935058
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 12.1517 1.36717 0.683584 0.729872i \(-0.260420\pi\)
0.683584 + 0.729872i \(0.260420\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 12.3572 1.36462
\(83\) −2.20555 −0.242091 −0.121045 0.992647i \(-0.538625\pi\)
−0.121045 + 0.992647i \(0.538625\pi\)
\(84\) 0 0
\(85\) −4.20555 −0.456156
\(86\) −10.6761 −1.15123
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) 7.45998 0.790756 0.395378 0.918519i \(-0.370614\pi\)
0.395378 + 0.918519i \(0.370614\pi\)
\(90\) 0 0
\(91\) 3.62721 0.380235
\(92\) −0.897225 −0.0935422
\(93\) 0 0
\(94\) −1.15667 −0.119302
\(95\) 8.15165 0.836342
\(96\) 0 0
\(97\) 2.25945 0.229412 0.114706 0.993399i \(-0.463407\pi\)
0.114706 + 0.993399i \(0.463407\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −13.6655 −1.35977 −0.679885 0.733318i \(-0.737970\pi\)
−0.679885 + 0.733318i \(0.737970\pi\)
\(102\) 0 0
\(103\) −5.04888 −0.497481 −0.248740 0.968570i \(-0.580017\pi\)
−0.248740 + 0.968570i \(0.580017\pi\)
\(104\) 3.62721 0.355677
\(105\) 0 0
\(106\) 11.3083 1.09836
\(107\) −12.9894 −1.25574 −0.627868 0.778320i \(-0.716072\pi\)
−0.627868 + 0.778320i \(0.716072\pi\)
\(108\) 0 0
\(109\) 5.10278 0.488757 0.244379 0.969680i \(-0.421416\pi\)
0.244379 + 0.969680i \(0.421416\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) 10.4111 0.979394 0.489697 0.871893i \(-0.337107\pi\)
0.489697 + 0.871893i \(0.337107\pi\)
\(114\) 0 0
\(115\) 0.897225 0.0836667
\(116\) 7.30833 0.678561
\(117\) 0 0
\(118\) 7.25443 0.667824
\(119\) −4.20555 −0.385522
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 3.15667 0.285792
\(123\) 0 0
\(124\) −3.42166 −0.307274
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 3.25443 0.288784 0.144392 0.989521i \(-0.453877\pi\)
0.144392 + 0.989521i \(0.453877\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −3.62721 −0.318128
\(131\) 17.3083 1.51224 0.756118 0.654436i \(-0.227094\pi\)
0.756118 + 0.654436i \(0.227094\pi\)
\(132\) 0 0
\(133\) 8.15165 0.706838
\(134\) 8.41110 0.726608
\(135\) 0 0
\(136\) −4.20555 −0.360623
\(137\) −10.0000 −0.854358 −0.427179 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 0 0
\(139\) 13.2005 1.11965 0.559827 0.828609i \(-0.310868\pi\)
0.559827 + 0.828609i \(0.310868\pi\)
\(140\) 1.00000 0.0845154
\(141\) 0 0
\(142\) −13.0489 −1.09504
\(143\) 3.62721 0.303323
\(144\) 0 0
\(145\) −7.30833 −0.606923
\(146\) −0.205550 −0.0170114
\(147\) 0 0
\(148\) −1.10278 −0.0906476
\(149\) 23.6116 1.93434 0.967170 0.254131i \(-0.0817893\pi\)
0.967170 + 0.254131i \(0.0817893\pi\)
\(150\) 0 0
\(151\) 11.5139 0.936986 0.468493 0.883467i \(-0.344797\pi\)
0.468493 + 0.883467i \(0.344797\pi\)
\(152\) 8.15165 0.661186
\(153\) 0 0
\(154\) −1.00000 −0.0805823
\(155\) 3.42166 0.274835
\(156\) 0 0
\(157\) −16.2056 −1.29334 −0.646672 0.762768i \(-0.723840\pi\)
−0.646672 + 0.762768i \(0.723840\pi\)
\(158\) −12.1517 −0.966733
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) 0.897225 0.0707112
\(162\) 0 0
\(163\) 6.31335 0.494500 0.247250 0.968952i \(-0.420473\pi\)
0.247250 + 0.968952i \(0.420473\pi\)
\(164\) −12.3572 −0.964935
\(165\) 0 0
\(166\) 2.20555 0.171184
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) 0.156674 0.0120519
\(170\) 4.20555 0.322551
\(171\) 0 0
\(172\) 10.6761 0.814044
\(173\) −9.25443 −0.703601 −0.351800 0.936075i \(-0.614430\pi\)
−0.351800 + 0.936075i \(0.614430\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) −7.45998 −0.559149
\(179\) −4.98944 −0.372928 −0.186464 0.982462i \(-0.559703\pi\)
−0.186464 + 0.982462i \(0.559703\pi\)
\(180\) 0 0
\(181\) −9.66553 −0.718433 −0.359216 0.933254i \(-0.616956\pi\)
−0.359216 + 0.933254i \(0.616956\pi\)
\(182\) −3.62721 −0.268867
\(183\) 0 0
\(184\) 0.897225 0.0661443
\(185\) 1.10278 0.0810776
\(186\) 0 0
\(187\) −4.20555 −0.307540
\(188\) 1.15667 0.0843591
\(189\) 0 0
\(190\) −8.15165 −0.591383
\(191\) 9.15667 0.662554 0.331277 0.943534i \(-0.392521\pi\)
0.331277 + 0.943534i \(0.392521\pi\)
\(192\) 0 0
\(193\) 1.52946 0.110093 0.0550465 0.998484i \(-0.482469\pi\)
0.0550465 + 0.998484i \(0.482469\pi\)
\(194\) −2.25945 −0.162219
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −14.3033 −1.01907 −0.509534 0.860451i \(-0.670182\pi\)
−0.509534 + 0.860451i \(0.670182\pi\)
\(198\) 0 0
\(199\) −10.0978 −0.715811 −0.357905 0.933758i \(-0.616509\pi\)
−0.357905 + 0.933758i \(0.616509\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) 13.6655 0.961503
\(203\) −7.30833 −0.512944
\(204\) 0 0
\(205\) 12.3572 0.863064
\(206\) 5.04888 0.351772
\(207\) 0 0
\(208\) −3.62721 −0.251502
\(209\) 8.15165 0.563861
\(210\) 0 0
\(211\) 10.8433 0.746485 0.373243 0.927734i \(-0.378246\pi\)
0.373243 + 0.927734i \(0.378246\pi\)
\(212\) −11.3083 −0.776659
\(213\) 0 0
\(214\) 12.9894 0.887940
\(215\) −10.6761 −0.728103
\(216\) 0 0
\(217\) 3.42166 0.232278
\(218\) −5.10278 −0.345604
\(219\) 0 0
\(220\) 1.00000 0.0674200
\(221\) −15.2544 −1.02612
\(222\) 0 0
\(223\) −9.45998 −0.633487 −0.316743 0.948511i \(-0.602589\pi\)
−0.316743 + 0.948511i \(0.602589\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −10.4111 −0.692536
\(227\) 17.5678 1.16601 0.583007 0.812467i \(-0.301876\pi\)
0.583007 + 0.812467i \(0.301876\pi\)
\(228\) 0 0
\(229\) 25.7250 1.69995 0.849977 0.526820i \(-0.176616\pi\)
0.849977 + 0.526820i \(0.176616\pi\)
\(230\) −0.897225 −0.0591613
\(231\) 0 0
\(232\) −7.30833 −0.479815
\(233\) −19.1567 −1.25500 −0.627498 0.778618i \(-0.715921\pi\)
−0.627498 + 0.778618i \(0.715921\pi\)
\(234\) 0 0
\(235\) −1.15667 −0.0754531
\(236\) −7.25443 −0.472223
\(237\) 0 0
\(238\) 4.20555 0.272605
\(239\) 18.0539 1.16781 0.583905 0.811822i \(-0.301524\pi\)
0.583905 + 0.811822i \(0.301524\pi\)
\(240\) 0 0
\(241\) 6.15165 0.396263 0.198131 0.980175i \(-0.436513\pi\)
0.198131 + 0.980175i \(0.436513\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) −3.15667 −0.202085
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 29.5678 1.88135
\(248\) 3.42166 0.217276
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 29.9789 1.89225 0.946125 0.323802i \(-0.104961\pi\)
0.946125 + 0.323802i \(0.104961\pi\)
\(252\) 0 0
\(253\) 0.897225 0.0564080
\(254\) −3.25443 −0.204201
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −4.05390 −0.252875 −0.126438 0.991975i \(-0.540354\pi\)
−0.126438 + 0.991975i \(0.540354\pi\)
\(258\) 0 0
\(259\) 1.10278 0.0685231
\(260\) 3.62721 0.224950
\(261\) 0 0
\(262\) −17.3083 −1.06931
\(263\) 14.5089 0.894654 0.447327 0.894370i \(-0.352376\pi\)
0.447327 + 0.894370i \(0.352376\pi\)
\(264\) 0 0
\(265\) 11.3083 0.694665
\(266\) −8.15165 −0.499810
\(267\) 0 0
\(268\) −8.41110 −0.513790
\(269\) −4.37279 −0.266614 −0.133307 0.991075i \(-0.542560\pi\)
−0.133307 + 0.991075i \(0.542560\pi\)
\(270\) 0 0
\(271\) −10.6167 −0.644916 −0.322458 0.946584i \(-0.604509\pi\)
−0.322458 + 0.946584i \(0.604509\pi\)
\(272\) 4.20555 0.254999
\(273\) 0 0
\(274\) 10.0000 0.604122
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) −19.7633 −1.18746 −0.593730 0.804664i \(-0.702345\pi\)
−0.593730 + 0.804664i \(0.702345\pi\)
\(278\) −13.2005 −0.791715
\(279\) 0 0
\(280\) −1.00000 −0.0597614
\(281\) 27.0489 1.61360 0.806800 0.590824i \(-0.201197\pi\)
0.806800 + 0.590824i \(0.201197\pi\)
\(282\) 0 0
\(283\) 27.6655 1.64454 0.822272 0.569094i \(-0.192706\pi\)
0.822272 + 0.569094i \(0.192706\pi\)
\(284\) 13.0489 0.774308
\(285\) 0 0
\(286\) −3.62721 −0.214482
\(287\) 12.3572 0.729423
\(288\) 0 0
\(289\) 0.686652 0.0403913
\(290\) 7.30833 0.429160
\(291\) 0 0
\(292\) 0.205550 0.0120289
\(293\) 8.37279 0.489143 0.244572 0.969631i \(-0.421353\pi\)
0.244572 + 0.969631i \(0.421353\pi\)
\(294\) 0 0
\(295\) 7.25443 0.422369
\(296\) 1.10278 0.0640975
\(297\) 0 0
\(298\) −23.6116 −1.36778
\(299\) 3.25443 0.188208
\(300\) 0 0
\(301\) −10.6761 −0.615360
\(302\) −11.5139 −0.662549
\(303\) 0 0
\(304\) −8.15165 −0.467529
\(305\) 3.15667 0.180751
\(306\) 0 0
\(307\) −10.2056 −0.582462 −0.291231 0.956653i \(-0.594065\pi\)
−0.291231 + 0.956653i \(0.594065\pi\)
\(308\) 1.00000 0.0569803
\(309\) 0 0
\(310\) −3.42166 −0.194337
\(311\) 8.16724 0.463122 0.231561 0.972820i \(-0.425617\pi\)
0.231561 + 0.972820i \(0.425617\pi\)
\(312\) 0 0
\(313\) 34.5628 1.95360 0.976801 0.214149i \(-0.0686977\pi\)
0.976801 + 0.214149i \(0.0686977\pi\)
\(314\) 16.2056 0.914532
\(315\) 0 0
\(316\) 12.1517 0.683584
\(317\) −9.21057 −0.517317 −0.258659 0.965969i \(-0.583280\pi\)
−0.258659 + 0.965969i \(0.583280\pi\)
\(318\) 0 0
\(319\) −7.30833 −0.409188
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) −0.897225 −0.0500004
\(323\) −34.2822 −1.90751
\(324\) 0 0
\(325\) −3.62721 −0.201202
\(326\) −6.31335 −0.349664
\(327\) 0 0
\(328\) 12.3572 0.682312
\(329\) −1.15667 −0.0637695
\(330\) 0 0
\(331\) 16.2439 0.892843 0.446422 0.894823i \(-0.352698\pi\)
0.446422 + 0.894823i \(0.352698\pi\)
\(332\) −2.20555 −0.121045
\(333\) 0 0
\(334\) −8.00000 −0.437741
\(335\) 8.41110 0.459547
\(336\) 0 0
\(337\) −1.25443 −0.0683329 −0.0341665 0.999416i \(-0.510878\pi\)
−0.0341665 + 0.999416i \(0.510878\pi\)
\(338\) −0.156674 −0.00852195
\(339\) 0 0
\(340\) −4.20555 −0.228078
\(341\) 3.42166 0.185293
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −10.6761 −0.575616
\(345\) 0 0
\(346\) 9.25443 0.497521
\(347\) −7.42166 −0.398416 −0.199208 0.979957i \(-0.563837\pi\)
−0.199208 + 0.979957i \(0.563837\pi\)
\(348\) 0 0
\(349\) −18.3033 −0.979753 −0.489877 0.871792i \(-0.662958\pi\)
−0.489877 + 0.871792i \(0.662958\pi\)
\(350\) 1.00000 0.0534522
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) −28.8761 −1.53692 −0.768460 0.639898i \(-0.778977\pi\)
−0.768460 + 0.639898i \(0.778977\pi\)
\(354\) 0 0
\(355\) −13.0489 −0.692562
\(356\) 7.45998 0.395378
\(357\) 0 0
\(358\) 4.98944 0.263700
\(359\) −4.15165 −0.219116 −0.109558 0.993980i \(-0.534943\pi\)
−0.109558 + 0.993980i \(0.534943\pi\)
\(360\) 0 0
\(361\) 47.4494 2.49734
\(362\) 9.66553 0.508009
\(363\) 0 0
\(364\) 3.62721 0.190118
\(365\) −0.205550 −0.0107590
\(366\) 0 0
\(367\) −5.26447 −0.274803 −0.137402 0.990515i \(-0.543875\pi\)
−0.137402 + 0.990515i \(0.543875\pi\)
\(368\) −0.897225 −0.0467711
\(369\) 0 0
\(370\) −1.10278 −0.0573305
\(371\) 11.3083 0.587099
\(372\) 0 0
\(373\) 35.8711 1.85733 0.928667 0.370915i \(-0.120956\pi\)
0.928667 + 0.370915i \(0.120956\pi\)
\(374\) 4.20555 0.217464
\(375\) 0 0
\(376\) −1.15667 −0.0596509
\(377\) −26.5089 −1.36528
\(378\) 0 0
\(379\) −33.0177 −1.69601 −0.848003 0.529992i \(-0.822195\pi\)
−0.848003 + 0.529992i \(0.822195\pi\)
\(380\) 8.15165 0.418171
\(381\) 0 0
\(382\) −9.15667 −0.468496
\(383\) −2.95112 −0.150795 −0.0753977 0.997154i \(-0.524023\pi\)
−0.0753977 + 0.997154i \(0.524023\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) −1.52946 −0.0778475
\(387\) 0 0
\(388\) 2.25945 0.114706
\(389\) −1.89220 −0.0959384 −0.0479692 0.998849i \(-0.515275\pi\)
−0.0479692 + 0.998849i \(0.515275\pi\)
\(390\) 0 0
\(391\) −3.77332 −0.190825
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) 14.3033 0.720590
\(395\) −12.1517 −0.611416
\(396\) 0 0
\(397\) 5.36222 0.269122 0.134561 0.990905i \(-0.457038\pi\)
0.134561 + 0.990905i \(0.457038\pi\)
\(398\) 10.0978 0.506155
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −23.7038 −1.18371 −0.591857 0.806043i \(-0.701605\pi\)
−0.591857 + 0.806043i \(0.701605\pi\)
\(402\) 0 0
\(403\) 12.4111 0.618241
\(404\) −13.6655 −0.679885
\(405\) 0 0
\(406\) 7.30833 0.362706
\(407\) 1.10278 0.0546625
\(408\) 0 0
\(409\) 27.0816 1.33910 0.669551 0.742766i \(-0.266487\pi\)
0.669551 + 0.742766i \(0.266487\pi\)
\(410\) −12.3572 −0.610279
\(411\) 0 0
\(412\) −5.04888 −0.248740
\(413\) 7.25443 0.356967
\(414\) 0 0
\(415\) 2.20555 0.108266
\(416\) 3.62721 0.177839
\(417\) 0 0
\(418\) −8.15165 −0.398710
\(419\) 3.05892 0.149438 0.0747191 0.997205i \(-0.476194\pi\)
0.0747191 + 0.997205i \(0.476194\pi\)
\(420\) 0 0
\(421\) 21.6655 1.05591 0.527957 0.849271i \(-0.322958\pi\)
0.527957 + 0.849271i \(0.322958\pi\)
\(422\) −10.8433 −0.527845
\(423\) 0 0
\(424\) 11.3083 0.549181
\(425\) 4.20555 0.203999
\(426\) 0 0
\(427\) 3.15667 0.152762
\(428\) −12.9894 −0.627868
\(429\) 0 0
\(430\) 10.6761 0.514847
\(431\) 34.3260 1.65343 0.826713 0.562623i \(-0.190208\pi\)
0.826713 + 0.562623i \(0.190208\pi\)
\(432\) 0 0
\(433\) −0.0538991 −0.00259023 −0.00129511 0.999999i \(-0.500412\pi\)
−0.00129511 + 0.999999i \(0.500412\pi\)
\(434\) −3.42166 −0.164245
\(435\) 0 0
\(436\) 5.10278 0.244379
\(437\) 7.31386 0.349870
\(438\) 0 0
\(439\) −7.14663 −0.341090 −0.170545 0.985350i \(-0.554553\pi\)
−0.170545 + 0.985350i \(0.554553\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 0 0
\(442\) 15.2544 0.725579
\(443\) −8.74557 −0.415515 −0.207757 0.978180i \(-0.566616\pi\)
−0.207757 + 0.978180i \(0.566616\pi\)
\(444\) 0 0
\(445\) −7.45998 −0.353637
\(446\) 9.45998 0.447943
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) −39.7038 −1.87374 −0.936870 0.349678i \(-0.886291\pi\)
−0.936870 + 0.349678i \(0.886291\pi\)
\(450\) 0 0
\(451\) 12.3572 0.581878
\(452\) 10.4111 0.489697
\(453\) 0 0
\(454\) −17.5678 −0.824497
\(455\) −3.62721 −0.170046
\(456\) 0 0
\(457\) 12.3133 0.575994 0.287997 0.957631i \(-0.407011\pi\)
0.287997 + 0.957631i \(0.407011\pi\)
\(458\) −25.7250 −1.20205
\(459\) 0 0
\(460\) 0.897225 0.0418333
\(461\) 15.0489 0.700896 0.350448 0.936582i \(-0.386029\pi\)
0.350448 + 0.936582i \(0.386029\pi\)
\(462\) 0 0
\(463\) 31.4061 1.45956 0.729782 0.683680i \(-0.239622\pi\)
0.729782 + 0.683680i \(0.239622\pi\)
\(464\) 7.30833 0.339280
\(465\) 0 0
\(466\) 19.1567 0.887416
\(467\) 19.0716 0.882529 0.441264 0.897377i \(-0.354530\pi\)
0.441264 + 0.897377i \(0.354530\pi\)
\(468\) 0 0
\(469\) 8.41110 0.388389
\(470\) 1.15667 0.0533534
\(471\) 0 0
\(472\) 7.25443 0.333912
\(473\) −10.6761 −0.490887
\(474\) 0 0
\(475\) −8.15165 −0.374023
\(476\) −4.20555 −0.192761
\(477\) 0 0
\(478\) −18.0539 −0.825766
\(479\) 9.49115 0.433662 0.216831 0.976209i \(-0.430428\pi\)
0.216831 + 0.976209i \(0.430428\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) −6.15165 −0.280200
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −2.25945 −0.102596
\(486\) 0 0
\(487\) 0.0438527 0.00198715 0.000993577 1.00000i \(-0.499684\pi\)
0.000993577 1.00000i \(0.499684\pi\)
\(488\) 3.15667 0.142896
\(489\) 0 0
\(490\) 1.00000 0.0451754
\(491\) −1.56777 −0.0707527 −0.0353763 0.999374i \(-0.511263\pi\)
−0.0353763 + 0.999374i \(0.511263\pi\)
\(492\) 0 0
\(493\) 30.7355 1.38426
\(494\) −29.5678 −1.33032
\(495\) 0 0
\(496\) −3.42166 −0.153637
\(497\) −13.0489 −0.585322
\(498\) 0 0
\(499\) −42.3416 −1.89547 −0.947736 0.319057i \(-0.896634\pi\)
−0.947736 + 0.319057i \(0.896634\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −29.9789 −1.33802
\(503\) 36.4111 1.62349 0.811745 0.584012i \(-0.198518\pi\)
0.811745 + 0.584012i \(0.198518\pi\)
\(504\) 0 0
\(505\) 13.6655 0.608108
\(506\) −0.897225 −0.0398865
\(507\) 0 0
\(508\) 3.25443 0.144392
\(509\) 30.0766 1.33312 0.666562 0.745450i \(-0.267765\pi\)
0.666562 + 0.745450i \(0.267765\pi\)
\(510\) 0 0
\(511\) −0.205550 −0.00909300
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 4.05390 0.178810
\(515\) 5.04888 0.222480
\(516\) 0 0
\(517\) −1.15667 −0.0508705
\(518\) −1.10278 −0.0484532
\(519\) 0 0
\(520\) −3.62721 −0.159064
\(521\) 33.2233 1.45554 0.727769 0.685823i \(-0.240557\pi\)
0.727769 + 0.685823i \(0.240557\pi\)
\(522\) 0 0
\(523\) −15.5577 −0.680292 −0.340146 0.940373i \(-0.610476\pi\)
−0.340146 + 0.940373i \(0.610476\pi\)
\(524\) 17.3083 0.756118
\(525\) 0 0
\(526\) −14.5089 −0.632616
\(527\) −14.3900 −0.626837
\(528\) 0 0
\(529\) −22.1950 −0.964999
\(530\) −11.3083 −0.491202
\(531\) 0 0
\(532\) 8.15165 0.353419
\(533\) 44.8222 1.94147
\(534\) 0 0
\(535\) 12.9894 0.561582
\(536\) 8.41110 0.363304
\(537\) 0 0
\(538\) 4.37279 0.188524
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) 23.4161 1.00674 0.503369 0.864072i \(-0.332094\pi\)
0.503369 + 0.864072i \(0.332094\pi\)
\(542\) 10.6167 0.456024
\(543\) 0 0
\(544\) −4.20555 −0.180311
\(545\) −5.10278 −0.218579
\(546\) 0 0
\(547\) −8.41110 −0.359633 −0.179816 0.983700i \(-0.557550\pi\)
−0.179816 + 0.983700i \(0.557550\pi\)
\(548\) −10.0000 −0.427179
\(549\) 0 0
\(550\) 1.00000 0.0426401
\(551\) −59.5749 −2.53798
\(552\) 0 0
\(553\) −12.1517 −0.516741
\(554\) 19.7633 0.839661
\(555\) 0 0
\(556\) 13.2005 0.559827
\(557\) −37.0278 −1.56892 −0.784458 0.620182i \(-0.787059\pi\)
−0.784458 + 0.620182i \(0.787059\pi\)
\(558\) 0 0
\(559\) −38.7244 −1.63787
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −27.0489 −1.14099
\(563\) −22.9200 −0.965961 −0.482980 0.875631i \(-0.660446\pi\)
−0.482980 + 0.875631i \(0.660446\pi\)
\(564\) 0 0
\(565\) −10.4111 −0.437998
\(566\) −27.6655 −1.16287
\(567\) 0 0
\(568\) −13.0489 −0.547519
\(569\) 22.0000 0.922288 0.461144 0.887325i \(-0.347439\pi\)
0.461144 + 0.887325i \(0.347439\pi\)
\(570\) 0 0
\(571\) 20.3033 0.849667 0.424833 0.905272i \(-0.360333\pi\)
0.424833 + 0.905272i \(0.360333\pi\)
\(572\) 3.62721 0.151661
\(573\) 0 0
\(574\) −12.3572 −0.515780
\(575\) −0.897225 −0.0374169
\(576\) 0 0
\(577\) 16.4650 0.685447 0.342723 0.939436i \(-0.388651\pi\)
0.342723 + 0.939436i \(0.388651\pi\)
\(578\) −0.686652 −0.0285609
\(579\) 0 0
\(580\) −7.30833 −0.303462
\(581\) 2.20555 0.0915016
\(582\) 0 0
\(583\) 11.3083 0.468343
\(584\) −0.205550 −0.00850572
\(585\) 0 0
\(586\) −8.37279 −0.345877
\(587\) 30.6605 1.26549 0.632747 0.774358i \(-0.281927\pi\)
0.632747 + 0.774358i \(0.281927\pi\)
\(588\) 0 0
\(589\) 27.8922 1.14928
\(590\) −7.25443 −0.298660
\(591\) 0 0
\(592\) −1.10278 −0.0453238
\(593\) −18.3033 −0.751627 −0.375813 0.926695i \(-0.622637\pi\)
−0.375813 + 0.926695i \(0.622637\pi\)
\(594\) 0 0
\(595\) 4.20555 0.172411
\(596\) 23.6116 0.967170
\(597\) 0 0
\(598\) −3.25443 −0.133083
\(599\) −29.6555 −1.21169 −0.605845 0.795583i \(-0.707165\pi\)
−0.605845 + 0.795583i \(0.707165\pi\)
\(600\) 0 0
\(601\) 18.7783 0.765985 0.382992 0.923751i \(-0.374894\pi\)
0.382992 + 0.923751i \(0.374894\pi\)
\(602\) 10.6761 0.435125
\(603\) 0 0
\(604\) 11.5139 0.468493
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) 14.6761 0.595684 0.297842 0.954615i \(-0.403733\pi\)
0.297842 + 0.954615i \(0.403733\pi\)
\(608\) 8.15165 0.330593
\(609\) 0 0
\(610\) −3.15667 −0.127810
\(611\) −4.19550 −0.169732
\(612\) 0 0
\(613\) −5.28560 −0.213483 −0.106742 0.994287i \(-0.534042\pi\)
−0.106742 + 0.994287i \(0.534042\pi\)
\(614\) 10.2056 0.411862
\(615\) 0 0
\(616\) −1.00000 −0.0402911
\(617\) 44.6933 1.79928 0.899642 0.436629i \(-0.143828\pi\)
0.899642 + 0.436629i \(0.143828\pi\)
\(618\) 0 0
\(619\) −17.5678 −0.706108 −0.353054 0.935603i \(-0.614857\pi\)
−0.353054 + 0.935603i \(0.614857\pi\)
\(620\) 3.42166 0.137417
\(621\) 0 0
\(622\) −8.16724 −0.327476
\(623\) −7.45998 −0.298878
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −34.5628 −1.38141
\(627\) 0 0
\(628\) −16.2056 −0.646672
\(629\) −4.63778 −0.184920
\(630\) 0 0
\(631\) 4.74557 0.188918 0.0944592 0.995529i \(-0.469888\pi\)
0.0944592 + 0.995529i \(0.469888\pi\)
\(632\) −12.1517 −0.483367
\(633\) 0 0
\(634\) 9.21057 0.365799
\(635\) −3.25443 −0.129148
\(636\) 0 0
\(637\) −3.62721 −0.143715
\(638\) 7.30833 0.289339
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 2.52998 0.0999281 0.0499641 0.998751i \(-0.484089\pi\)
0.0499641 + 0.998751i \(0.484089\pi\)
\(642\) 0 0
\(643\) −0.670549 −0.0264439 −0.0132219 0.999913i \(-0.504209\pi\)
−0.0132219 + 0.999913i \(0.504209\pi\)
\(644\) 0.897225 0.0353556
\(645\) 0 0
\(646\) 34.2822 1.34881
\(647\) −11.8922 −0.467531 −0.233765 0.972293i \(-0.575105\pi\)
−0.233765 + 0.972293i \(0.575105\pi\)
\(648\) 0 0
\(649\) 7.25443 0.284761
\(650\) 3.62721 0.142271
\(651\) 0 0
\(652\) 6.31335 0.247250
\(653\) 11.4161 0.446747 0.223374 0.974733i \(-0.428293\pi\)
0.223374 + 0.974733i \(0.428293\pi\)
\(654\) 0 0
\(655\) −17.3083 −0.676292
\(656\) −12.3572 −0.482468
\(657\) 0 0
\(658\) 1.15667 0.0450919
\(659\) −1.87108 −0.0728868 −0.0364434 0.999336i \(-0.511603\pi\)
−0.0364434 + 0.999336i \(0.511603\pi\)
\(660\) 0 0
\(661\) 33.1355 1.28882 0.644412 0.764679i \(-0.277102\pi\)
0.644412 + 0.764679i \(0.277102\pi\)
\(662\) −16.2439 −0.631336
\(663\) 0 0
\(664\) 2.20555 0.0855919
\(665\) −8.15165 −0.316107
\(666\) 0 0
\(667\) −6.55721 −0.253896
\(668\) 8.00000 0.309529
\(669\) 0 0
\(670\) −8.41110 −0.324949
\(671\) 3.15667 0.121862
\(672\) 0 0
\(673\) −18.1955 −0.701385 −0.350693 0.936491i \(-0.614054\pi\)
−0.350693 + 0.936491i \(0.614054\pi\)
\(674\) 1.25443 0.0483187
\(675\) 0 0
\(676\) 0.156674 0.00602593
\(677\) −15.9789 −0.614118 −0.307059 0.951690i \(-0.599345\pi\)
−0.307059 + 0.951690i \(0.599345\pi\)
\(678\) 0 0
\(679\) −2.25945 −0.0867097
\(680\) 4.20555 0.161275
\(681\) 0 0
\(682\) −3.42166 −0.131022
\(683\) 17.8711 0.683818 0.341909 0.939733i \(-0.388927\pi\)
0.341909 + 0.939733i \(0.388927\pi\)
\(684\) 0 0
\(685\) 10.0000 0.382080
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) 10.6761 0.407022
\(689\) 41.0177 1.56265
\(690\) 0 0
\(691\) −2.20555 −0.0839031 −0.0419515 0.999120i \(-0.513358\pi\)
−0.0419515 + 0.999120i \(0.513358\pi\)
\(692\) −9.25443 −0.351800
\(693\) 0 0
\(694\) 7.42166 0.281722
\(695\) −13.2005 −0.500725
\(696\) 0 0
\(697\) −51.9688 −1.96846
\(698\) 18.3033 0.692790
\(699\) 0 0
\(700\) −1.00000 −0.0377964
\(701\) 25.6217 0.967717 0.483859 0.875146i \(-0.339235\pi\)
0.483859 + 0.875146i \(0.339235\pi\)
\(702\) 0 0
\(703\) 8.98944 0.339043
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 28.8761 1.08677
\(707\) 13.6655 0.513945
\(708\) 0 0
\(709\) 5.48110 0.205847 0.102924 0.994689i \(-0.467180\pi\)
0.102924 + 0.994689i \(0.467180\pi\)
\(710\) 13.0489 0.489716
\(711\) 0 0
\(712\) −7.45998 −0.279574
\(713\) 3.07000 0.114972
\(714\) 0 0
\(715\) −3.62721 −0.135650
\(716\) −4.98944 −0.186464
\(717\) 0 0
\(718\) 4.15165 0.154938
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) 0 0
\(721\) 5.04888 0.188030
\(722\) −47.4494 −1.76588
\(723\) 0 0
\(724\) −9.66553 −0.359216
\(725\) 7.30833 0.271424
\(726\) 0 0
\(727\) 18.5855 0.689297 0.344649 0.938732i \(-0.387998\pi\)
0.344649 + 0.938732i \(0.387998\pi\)
\(728\) −3.62721 −0.134433
\(729\) 0 0
\(730\) 0.205550 0.00760775
\(731\) 44.8988 1.66064
\(732\) 0 0
\(733\) −42.0000 −1.55131 −0.775653 0.631160i \(-0.782579\pi\)
−0.775653 + 0.631160i \(0.782579\pi\)
\(734\) 5.26447 0.194315
\(735\) 0 0
\(736\) 0.897225 0.0330722
\(737\) 8.41110 0.309827
\(738\) 0 0
\(739\) 5.97887 0.219936 0.109968 0.993935i \(-0.464925\pi\)
0.109968 + 0.993935i \(0.464925\pi\)
\(740\) 1.10278 0.0405388
\(741\) 0 0
\(742\) −11.3083 −0.415142
\(743\) 50.0978 1.83791 0.918954 0.394364i \(-0.129035\pi\)
0.918954 + 0.394364i \(0.129035\pi\)
\(744\) 0 0
\(745\) −23.6116 −0.865063
\(746\) −35.8711 −1.31333
\(747\) 0 0
\(748\) −4.20555 −0.153770
\(749\) 12.9894 0.474624
\(750\) 0 0
\(751\) −8.63778 −0.315197 −0.157598 0.987503i \(-0.550375\pi\)
−0.157598 + 0.987503i \(0.550375\pi\)
\(752\) 1.15667 0.0421796
\(753\) 0 0
\(754\) 26.5089 0.965395
\(755\) −11.5139 −0.419033
\(756\) 0 0
\(757\) 25.5139 0.927318 0.463659 0.886014i \(-0.346536\pi\)
0.463659 + 0.886014i \(0.346536\pi\)
\(758\) 33.0177 1.19926
\(759\) 0 0
\(760\) −8.15165 −0.295691
\(761\) 32.6605 1.18394 0.591971 0.805959i \(-0.298350\pi\)
0.591971 + 0.805959i \(0.298350\pi\)
\(762\) 0 0
\(763\) −5.10278 −0.184733
\(764\) 9.15667 0.331277
\(765\) 0 0
\(766\) 2.95112 0.106628
\(767\) 26.3133 0.950120
\(768\) 0 0
\(769\) 16.7683 0.604680 0.302340 0.953200i \(-0.402232\pi\)
0.302340 + 0.953200i \(0.402232\pi\)
\(770\) 1.00000 0.0360375
\(771\) 0 0
\(772\) 1.52946 0.0550465
\(773\) 2.82220 0.101507 0.0507537 0.998711i \(-0.483838\pi\)
0.0507537 + 0.998711i \(0.483838\pi\)
\(774\) 0 0
\(775\) −3.42166 −0.122910
\(776\) −2.25945 −0.0811095
\(777\) 0 0
\(778\) 1.89220 0.0678387
\(779\) 100.732 3.60908
\(780\) 0 0
\(781\) −13.0489 −0.466925
\(782\) 3.77332 0.134934
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 16.2056 0.578401
\(786\) 0 0
\(787\) −4.82220 −0.171893 −0.0859464 0.996300i \(-0.527391\pi\)
−0.0859464 + 0.996300i \(0.527391\pi\)
\(788\) −14.3033 −0.509534
\(789\) 0 0
\(790\) 12.1517 0.432336
\(791\) −10.4111 −0.370176
\(792\) 0 0
\(793\) 11.4499 0.406599
\(794\) −5.36222 −0.190298
\(795\) 0 0
\(796\) −10.0978 −0.357905
\(797\) 0.540024 0.0191286 0.00956431 0.999954i \(-0.496956\pi\)
0.00956431 + 0.999954i \(0.496956\pi\)
\(798\) 0 0
\(799\) 4.86445 0.172092
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 23.7038 0.837012
\(803\) −0.205550 −0.00725371
\(804\) 0 0
\(805\) −0.897225 −0.0316230
\(806\) −12.4111 −0.437162
\(807\) 0 0
\(808\) 13.6655 0.480752
\(809\) 17.8922 0.629056 0.314528 0.949248i \(-0.398154\pi\)
0.314528 + 0.949248i \(0.398154\pi\)
\(810\) 0 0
\(811\) −12.2283 −0.429393 −0.214696 0.976681i \(-0.568876\pi\)
−0.214696 + 0.976681i \(0.568876\pi\)
\(812\) −7.30833 −0.256472
\(813\) 0 0
\(814\) −1.10278 −0.0386522
\(815\) −6.31335 −0.221147
\(816\) 0 0
\(817\) −87.0278 −3.04472
\(818\) −27.0816 −0.946888
\(819\) 0 0
\(820\) 12.3572 0.431532
\(821\) −38.5316 −1.34476 −0.672381 0.740206i \(-0.734728\pi\)
−0.672381 + 0.740206i \(0.734728\pi\)
\(822\) 0 0
\(823\) 21.9461 0.764993 0.382496 0.923957i \(-0.375064\pi\)
0.382496 + 0.923957i \(0.375064\pi\)
\(824\) 5.04888 0.175886
\(825\) 0 0
\(826\) −7.25443 −0.252414
\(827\) −9.68665 −0.336838 −0.168419 0.985716i \(-0.553866\pi\)
−0.168419 + 0.985716i \(0.553866\pi\)
\(828\) 0 0
\(829\) 31.0771 1.07935 0.539677 0.841872i \(-0.318546\pi\)
0.539677 + 0.841872i \(0.318546\pi\)
\(830\) −2.20555 −0.0765558
\(831\) 0 0
\(832\) −3.62721 −0.125751
\(833\) 4.20555 0.145714
\(834\) 0 0
\(835\) −8.00000 −0.276851
\(836\) 8.15165 0.281931
\(837\) 0 0
\(838\) −3.05892 −0.105669
\(839\) 24.7738 0.855288 0.427644 0.903947i \(-0.359344\pi\)
0.427644 + 0.903947i \(0.359344\pi\)
\(840\) 0 0
\(841\) 24.4116 0.841780
\(842\) −21.6655 −0.746643
\(843\) 0 0
\(844\) 10.8433 0.373243
\(845\) −0.156674 −0.00538976
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) −11.3083 −0.388329
\(849\) 0 0
\(850\) −4.20555 −0.144249
\(851\) 0.989437 0.0339175
\(852\) 0 0
\(853\) 1.58890 0.0544029 0.0272014 0.999630i \(-0.491340\pi\)
0.0272014 + 0.999630i \(0.491340\pi\)
\(854\) −3.15667 −0.108019
\(855\) 0 0
\(856\) 12.9894 0.443970
\(857\) −19.1255 −0.653315 −0.326657 0.945143i \(-0.605922\pi\)
−0.326657 + 0.945143i \(0.605922\pi\)
\(858\) 0 0
\(859\) 4.94108 0.168587 0.0842937 0.996441i \(-0.473137\pi\)
0.0842937 + 0.996441i \(0.473137\pi\)
\(860\) −10.6761 −0.364052
\(861\) 0 0
\(862\) −34.3260 −1.16915
\(863\) 19.3295 0.657982 0.328991 0.944333i \(-0.393291\pi\)
0.328991 + 0.944333i \(0.393291\pi\)
\(864\) 0 0
\(865\) 9.25443 0.314660
\(866\) 0.0538991 0.00183157
\(867\) 0 0
\(868\) 3.42166 0.116139
\(869\) −12.1517 −0.412217
\(870\) 0 0
\(871\) 30.5089 1.03375
\(872\) −5.10278 −0.172802
\(873\) 0 0
\(874\) −7.31386 −0.247395
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) 16.4011 0.553824 0.276912 0.960895i \(-0.410689\pi\)
0.276912 + 0.960895i \(0.410689\pi\)
\(878\) 7.14663 0.241187
\(879\) 0 0
\(880\) 1.00000 0.0337100
\(881\) −39.5678 −1.33307 −0.666536 0.745473i \(-0.732224\pi\)
−0.666536 + 0.745473i \(0.732224\pi\)
\(882\) 0 0
\(883\) 55.4389 1.86567 0.932833 0.360309i \(-0.117329\pi\)
0.932833 + 0.360309i \(0.117329\pi\)
\(884\) −15.2544 −0.513062
\(885\) 0 0
\(886\) 8.74557 0.293813
\(887\) −44.1955 −1.48394 −0.741970 0.670433i \(-0.766108\pi\)
−0.741970 + 0.670433i \(0.766108\pi\)
\(888\) 0 0
\(889\) −3.25443 −0.109150
\(890\) 7.45998 0.250059
\(891\) 0 0
\(892\) −9.45998 −0.316743
\(893\) −9.42880 −0.315523
\(894\) 0 0
\(895\) 4.98944 0.166778
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 39.7038 1.32493
\(899\) −25.0066 −0.834018
\(900\) 0 0
\(901\) −47.5577 −1.58438
\(902\) −12.3572 −0.411450
\(903\) 0 0
\(904\) −10.4111 −0.346268
\(905\) 9.66553 0.321293
\(906\) 0 0
\(907\) 9.26447 0.307622 0.153811 0.988100i \(-0.450845\pi\)
0.153811 + 0.988100i \(0.450845\pi\)
\(908\) 17.5678 0.583007
\(909\) 0 0
\(910\) 3.62721 0.120241
\(911\) 37.0489 1.22748 0.613742 0.789507i \(-0.289664\pi\)
0.613742 + 0.789507i \(0.289664\pi\)
\(912\) 0 0
\(913\) 2.20555 0.0729931
\(914\) −12.3133 −0.407289
\(915\) 0 0
\(916\) 25.7250 0.849977
\(917\) −17.3083 −0.571571
\(918\) 0 0
\(919\) −13.9149 −0.459011 −0.229506 0.973307i \(-0.573711\pi\)
−0.229506 + 0.973307i \(0.573711\pi\)
\(920\) −0.897225 −0.0295806
\(921\) 0 0
\(922\) −15.0489 −0.495608
\(923\) −47.3311 −1.55792
\(924\) 0 0
\(925\) −1.10278 −0.0362590
\(926\) −31.4061 −1.03207
\(927\) 0 0
\(928\) −7.30833 −0.239908
\(929\) 32.0666 1.05207 0.526035 0.850463i \(-0.323678\pi\)
0.526035 + 0.850463i \(0.323678\pi\)
\(930\) 0 0
\(931\) −8.15165 −0.267160
\(932\) −19.1567 −0.627498
\(933\) 0 0
\(934\) −19.0716 −0.624042
\(935\) 4.20555 0.137536
\(936\) 0 0
\(937\) 25.4499 0.831413 0.415706 0.909499i \(-0.363534\pi\)
0.415706 + 0.909499i \(0.363534\pi\)
\(938\) −8.41110 −0.274632
\(939\) 0 0
\(940\) −1.15667 −0.0377266
\(941\) 3.49115 0.113808 0.0569041 0.998380i \(-0.481877\pi\)
0.0569041 + 0.998380i \(0.481877\pi\)
\(942\) 0 0
\(943\) 11.0872 0.361049
\(944\) −7.25443 −0.236111
\(945\) 0 0
\(946\) 10.6761 0.347110
\(947\) −30.6167 −0.994907 −0.497454 0.867491i \(-0.665732\pi\)
−0.497454 + 0.867491i \(0.665732\pi\)
\(948\) 0 0
\(949\) −0.745574 −0.0242024
\(950\) 8.15165 0.264474
\(951\) 0 0
\(952\) 4.20555 0.136303
\(953\) 33.9406 1.09944 0.549721 0.835348i \(-0.314734\pi\)
0.549721 + 0.835348i \(0.314734\pi\)
\(954\) 0 0
\(955\) −9.15667 −0.296303
\(956\) 18.0539 0.583905
\(957\) 0 0
\(958\) −9.49115 −0.306645
\(959\) 10.0000 0.322917
\(960\) 0 0
\(961\) −19.2922 −0.622330
\(962\) −4.00000 −0.128965
\(963\) 0 0
\(964\) 6.15165 0.198131
\(965\) −1.52946 −0.0492351
\(966\) 0 0
\(967\) −60.8021 −1.95526 −0.977632 0.210323i \(-0.932548\pi\)
−0.977632 + 0.210323i \(0.932548\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) 2.25945 0.0725465
\(971\) −50.2933 −1.61399 −0.806994 0.590560i \(-0.798907\pi\)
−0.806994 + 0.590560i \(0.798907\pi\)
\(972\) 0 0
\(973\) −13.2005 −0.423189
\(974\) −0.0438527 −0.00140513
\(975\) 0 0
\(976\) −3.15667 −0.101043
\(977\) −28.0978 −0.898927 −0.449463 0.893299i \(-0.648385\pi\)
−0.449463 + 0.893299i \(0.648385\pi\)
\(978\) 0 0
\(979\) −7.45998 −0.238422
\(980\) −1.00000 −0.0319438
\(981\) 0 0
\(982\) 1.56777 0.0500297
\(983\) 21.0489 0.671355 0.335677 0.941977i \(-0.391035\pi\)
0.335677 + 0.941977i \(0.391035\pi\)
\(984\) 0 0
\(985\) 14.3033 0.455741
\(986\) −30.7355 −0.978819
\(987\) 0 0
\(988\) 29.5678 0.940676
\(989\) −9.57885 −0.304590
\(990\) 0 0
\(991\) −44.6832 −1.41941 −0.709705 0.704499i \(-0.751172\pi\)
−0.709705 + 0.704499i \(0.751172\pi\)
\(992\) 3.42166 0.108638
\(993\) 0 0
\(994\) 13.0489 0.413885
\(995\) 10.0978 0.320120
\(996\) 0 0
\(997\) 9.47002 0.299919 0.149959 0.988692i \(-0.452086\pi\)
0.149959 + 0.988692i \(0.452086\pi\)
\(998\) 42.3416 1.34030
\(999\) 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 6930.2.a.ce.1.1 3
3.2 odd 2 770.2.a.m.1.3 3
12.11 even 2 6160.2.a.bf.1.1 3
15.2 even 4 3850.2.c.ba.1849.4 6
15.8 even 4 3850.2.c.ba.1849.3 6
15.14 odd 2 3850.2.a.bt.1.1 3
21.20 even 2 5390.2.a.ca.1.1 3
33.32 even 2 8470.2.a.ci.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.m.1.3 3 3.2 odd 2
3850.2.a.bt.1.1 3 15.14 odd 2
3850.2.c.ba.1849.3 6 15.8 even 4
3850.2.c.ba.1849.4 6 15.2 even 4
5390.2.a.ca.1.1 3 21.20 even 2
6160.2.a.bf.1.1 3 12.11 even 2
6930.2.a.ce.1.1 3 1.1 even 1 trivial
8470.2.a.ci.1.3 3 33.32 even 2