Properties

Label 5520.2.a.bj.1.1
Level $5520$
Weight $2$
Character 5520.1
Self dual yes
Analytic conductor $44.077$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5520.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.0774219157\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{15}) \)
Defining polynomial: \( x^{2} - 15 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1380)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.87298\) of defining polynomial
Character \(\chi\) \(=\) 5520.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} -4.87298 q^{11} -2.87298 q^{13} -1.00000 q^{15} +3.87298 q^{17} +2.87298 q^{19} +3.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -5.87298 q^{29} -3.00000 q^{31} +4.87298 q^{33} -3.00000 q^{35} +1.00000 q^{37} +2.87298 q^{39} -5.87298 q^{41} -3.74597 q^{43} +1.00000 q^{45} +6.87298 q^{47} +2.00000 q^{49} -3.87298 q^{51} -3.87298 q^{53} -4.87298 q^{55} -2.87298 q^{57} -5.87298 q^{59} +8.87298 q^{61} -3.00000 q^{63} -2.87298 q^{65} -10.7460 q^{67} -1.00000 q^{69} +13.6190 q^{71} -2.87298 q^{73} -1.00000 q^{75} +14.6190 q^{77} -4.00000 q^{79} +1.00000 q^{81} +0.127017 q^{83} +3.87298 q^{85} +5.87298 q^{87} +1.74597 q^{89} +8.61895 q^{91} +3.00000 q^{93} +2.87298 q^{95} +8.00000 q^{97} -4.87298 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{5} - 6 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{5} - 6 q^{7} + 2 q^{9} - 2 q^{11} + 2 q^{13} - 2 q^{15} - 2 q^{19} + 6 q^{21} + 2 q^{23} + 2 q^{25} - 2 q^{27} - 4 q^{29} - 6 q^{31} + 2 q^{33} - 6 q^{35} + 2 q^{37} - 2 q^{39} - 4 q^{41} + 8 q^{43} + 2 q^{45} + 6 q^{47} + 4 q^{49} - 2 q^{55} + 2 q^{57} - 4 q^{59} + 10 q^{61} - 6 q^{63} + 2 q^{65} - 6 q^{67} - 2 q^{69} + 4 q^{71} + 2 q^{73} - 2 q^{75} + 6 q^{77} - 8 q^{79} + 2 q^{81} + 8 q^{83} + 4 q^{87} - 12 q^{89} - 6 q^{91} + 6 q^{93} - 2 q^{95} + 16 q^{97} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.87298 −1.46926 −0.734630 0.678468i \(-0.762644\pi\)
−0.734630 + 0.678468i \(0.762644\pi\)
\(12\) 0 0
\(13\) −2.87298 −0.796822 −0.398411 0.917207i \(-0.630438\pi\)
−0.398411 + 0.917207i \(0.630438\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 3.87298 0.939336 0.469668 0.882843i \(-0.344374\pi\)
0.469668 + 0.882843i \(0.344374\pi\)
\(18\) 0 0
\(19\) 2.87298 0.659108 0.329554 0.944137i \(-0.393102\pi\)
0.329554 + 0.944137i \(0.393102\pi\)
\(20\) 0 0
\(21\) 3.00000 0.654654
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −5.87298 −1.09059 −0.545293 0.838246i \(-0.683582\pi\)
−0.545293 + 0.838246i \(0.683582\pi\)
\(30\) 0 0
\(31\) −3.00000 −0.538816 −0.269408 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) 0 0
\(33\) 4.87298 0.848278
\(34\) 0 0
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) 1.00000 0.164399 0.0821995 0.996616i \(-0.473806\pi\)
0.0821995 + 0.996616i \(0.473806\pi\)
\(38\) 0 0
\(39\) 2.87298 0.460046
\(40\) 0 0
\(41\) −5.87298 −0.917206 −0.458603 0.888641i \(-0.651650\pi\)
−0.458603 + 0.888641i \(0.651650\pi\)
\(42\) 0 0
\(43\) −3.74597 −0.571255 −0.285627 0.958341i \(-0.592202\pi\)
−0.285627 + 0.958341i \(0.592202\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 6.87298 1.00253 0.501264 0.865295i \(-0.332869\pi\)
0.501264 + 0.865295i \(0.332869\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) −3.87298 −0.542326
\(52\) 0 0
\(53\) −3.87298 −0.531995 −0.265998 0.963974i \(-0.585701\pi\)
−0.265998 + 0.963974i \(0.585701\pi\)
\(54\) 0 0
\(55\) −4.87298 −0.657073
\(56\) 0 0
\(57\) −2.87298 −0.380536
\(58\) 0 0
\(59\) −5.87298 −0.764597 −0.382299 0.924039i \(-0.624867\pi\)
−0.382299 + 0.924039i \(0.624867\pi\)
\(60\) 0 0
\(61\) 8.87298 1.13607 0.568035 0.823005i \(-0.307704\pi\)
0.568035 + 0.823005i \(0.307704\pi\)
\(62\) 0 0
\(63\) −3.00000 −0.377964
\(64\) 0 0
\(65\) −2.87298 −0.356350
\(66\) 0 0
\(67\) −10.7460 −1.31283 −0.656414 0.754401i \(-0.727928\pi\)
−0.656414 + 0.754401i \(0.727928\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 13.6190 1.61627 0.808136 0.588996i \(-0.200477\pi\)
0.808136 + 0.588996i \(0.200477\pi\)
\(72\) 0 0
\(73\) −2.87298 −0.336257 −0.168129 0.985765i \(-0.553772\pi\)
−0.168129 + 0.985765i \(0.553772\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 14.6190 1.66598
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0.127017 0.0139419 0.00697094 0.999976i \(-0.497781\pi\)
0.00697094 + 0.999976i \(0.497781\pi\)
\(84\) 0 0
\(85\) 3.87298 0.420084
\(86\) 0 0
\(87\) 5.87298 0.629650
\(88\) 0 0
\(89\) 1.74597 0.185072 0.0925360 0.995709i \(-0.470503\pi\)
0.0925360 + 0.995709i \(0.470503\pi\)
\(90\) 0 0
\(91\) 8.61895 0.903511
\(92\) 0 0
\(93\) 3.00000 0.311086
\(94\) 0 0
\(95\) 2.87298 0.294762
\(96\) 0 0
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 0 0
\(99\) −4.87298 −0.489753
\(100\) 0 0
\(101\) −3.87298 −0.385376 −0.192688 0.981260i \(-0.561721\pi\)
−0.192688 + 0.981260i \(0.561721\pi\)
\(102\) 0 0
\(103\) 6.00000 0.591198 0.295599 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) 0 0
\(105\) 3.00000 0.292770
\(106\) 0 0
\(107\) 5.61895 0.543204 0.271602 0.962410i \(-0.412447\pi\)
0.271602 + 0.962410i \(0.412447\pi\)
\(108\) 0 0
\(109\) 16.6190 1.59181 0.795903 0.605424i \(-0.206996\pi\)
0.795903 + 0.605424i \(0.206996\pi\)
\(110\) 0 0
\(111\) −1.00000 −0.0949158
\(112\) 0 0
\(113\) −3.87298 −0.364340 −0.182170 0.983267i \(-0.558312\pi\)
−0.182170 + 0.983267i \(0.558312\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 0 0
\(117\) −2.87298 −0.265607
\(118\) 0 0
\(119\) −11.6190 −1.06511
\(120\) 0 0
\(121\) 12.7460 1.15872
\(122\) 0 0
\(123\) 5.87298 0.529549
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 6.61895 0.587337 0.293668 0.955907i \(-0.405124\pi\)
0.293668 + 0.955907i \(0.405124\pi\)
\(128\) 0 0
\(129\) 3.74597 0.329814
\(130\) 0 0
\(131\) −11.4919 −1.00405 −0.502027 0.864852i \(-0.667412\pi\)
−0.502027 + 0.864852i \(0.667412\pi\)
\(132\) 0 0
\(133\) −8.61895 −0.747358
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 8.00000 0.683486 0.341743 0.939793i \(-0.388983\pi\)
0.341743 + 0.939793i \(0.388983\pi\)
\(138\) 0 0
\(139\) 20.4919 1.73810 0.869052 0.494722i \(-0.164730\pi\)
0.869052 + 0.494722i \(0.164730\pi\)
\(140\) 0 0
\(141\) −6.87298 −0.578810
\(142\) 0 0
\(143\) 14.0000 1.17074
\(144\) 0 0
\(145\) −5.87298 −0.487725
\(146\) 0 0
\(147\) −2.00000 −0.164957
\(148\) 0 0
\(149\) −10.8730 −0.890750 −0.445375 0.895344i \(-0.646930\pi\)
−0.445375 + 0.895344i \(0.646930\pi\)
\(150\) 0 0
\(151\) −3.74597 −0.304842 −0.152421 0.988316i \(-0.548707\pi\)
−0.152421 + 0.988316i \(0.548707\pi\)
\(152\) 0 0
\(153\) 3.87298 0.313112
\(154\) 0 0
\(155\) −3.00000 −0.240966
\(156\) 0 0
\(157\) 17.0000 1.35675 0.678374 0.734717i \(-0.262685\pi\)
0.678374 + 0.734717i \(0.262685\pi\)
\(158\) 0 0
\(159\) 3.87298 0.307148
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) 0 0
\(163\) 22.0000 1.72317 0.861586 0.507611i \(-0.169471\pi\)
0.861586 + 0.507611i \(0.169471\pi\)
\(164\) 0 0
\(165\) 4.87298 0.379361
\(166\) 0 0
\(167\) 9.12702 0.706270 0.353135 0.935572i \(-0.385116\pi\)
0.353135 + 0.935572i \(0.385116\pi\)
\(168\) 0 0
\(169\) −4.74597 −0.365074
\(170\) 0 0
\(171\) 2.87298 0.219703
\(172\) 0 0
\(173\) 23.4919 1.78606 0.893029 0.449999i \(-0.148575\pi\)
0.893029 + 0.449999i \(0.148575\pi\)
\(174\) 0 0
\(175\) −3.00000 −0.226779
\(176\) 0 0
\(177\) 5.87298 0.441440
\(178\) 0 0
\(179\) 15.7460 1.17691 0.588454 0.808530i \(-0.299737\pi\)
0.588454 + 0.808530i \(0.299737\pi\)
\(180\) 0 0
\(181\) −21.7460 −1.61636 −0.808182 0.588932i \(-0.799549\pi\)
−0.808182 + 0.588932i \(0.799549\pi\)
\(182\) 0 0
\(183\) −8.87298 −0.655910
\(184\) 0 0
\(185\) 1.00000 0.0735215
\(186\) 0 0
\(187\) −18.8730 −1.38013
\(188\) 0 0
\(189\) 3.00000 0.218218
\(190\) 0 0
\(191\) 18.8730 1.36560 0.682801 0.730605i \(-0.260762\pi\)
0.682801 + 0.730605i \(0.260762\pi\)
\(192\) 0 0
\(193\) 9.74597 0.701530 0.350765 0.936464i \(-0.385922\pi\)
0.350765 + 0.936464i \(0.385922\pi\)
\(194\) 0 0
\(195\) 2.87298 0.205739
\(196\) 0 0
\(197\) 4.25403 0.303087 0.151544 0.988451i \(-0.451576\pi\)
0.151544 + 0.988451i \(0.451576\pi\)
\(198\) 0 0
\(199\) −19.2379 −1.36374 −0.681869 0.731474i \(-0.738833\pi\)
−0.681869 + 0.731474i \(0.738833\pi\)
\(200\) 0 0
\(201\) 10.7460 0.757962
\(202\) 0 0
\(203\) 17.6190 1.23661
\(204\) 0 0
\(205\) −5.87298 −0.410187
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −14.0000 −0.968400
\(210\) 0 0
\(211\) −1.00000 −0.0688428 −0.0344214 0.999407i \(-0.510959\pi\)
−0.0344214 + 0.999407i \(0.510959\pi\)
\(212\) 0 0
\(213\) −13.6190 −0.933155
\(214\) 0 0
\(215\) −3.74597 −0.255473
\(216\) 0 0
\(217\) 9.00000 0.610960
\(218\) 0 0
\(219\) 2.87298 0.194138
\(220\) 0 0
\(221\) −11.1270 −0.748484
\(222\) 0 0
\(223\) 2.00000 0.133930 0.0669650 0.997755i \(-0.478668\pi\)
0.0669650 + 0.997755i \(0.478668\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −7.49193 −0.497257 −0.248629 0.968599i \(-0.579980\pi\)
−0.248629 + 0.968599i \(0.579980\pi\)
\(228\) 0 0
\(229\) −5.74597 −0.379704 −0.189852 0.981813i \(-0.560801\pi\)
−0.189852 + 0.981813i \(0.560801\pi\)
\(230\) 0 0
\(231\) −14.6190 −0.961856
\(232\) 0 0
\(233\) 15.4919 1.01491 0.507455 0.861678i \(-0.330586\pi\)
0.507455 + 0.861678i \(0.330586\pi\)
\(234\) 0 0
\(235\) 6.87298 0.448344
\(236\) 0 0
\(237\) 4.00000 0.259828
\(238\) 0 0
\(239\) 17.8730 1.15611 0.578054 0.815999i \(-0.303812\pi\)
0.578054 + 0.815999i \(0.303812\pi\)
\(240\) 0 0
\(241\) −5.12702 −0.330260 −0.165130 0.986272i \(-0.552804\pi\)
−0.165130 + 0.986272i \(0.552804\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 2.00000 0.127775
\(246\) 0 0
\(247\) −8.25403 −0.525192
\(248\) 0 0
\(249\) −0.127017 −0.00804935
\(250\) 0 0
\(251\) −2.00000 −0.126239 −0.0631194 0.998006i \(-0.520105\pi\)
−0.0631194 + 0.998006i \(0.520105\pi\)
\(252\) 0 0
\(253\) −4.87298 −0.306362
\(254\) 0 0
\(255\) −3.87298 −0.242536
\(256\) 0 0
\(257\) −16.6190 −1.03666 −0.518331 0.855180i \(-0.673446\pi\)
−0.518331 + 0.855180i \(0.673446\pi\)
\(258\) 0 0
\(259\) −3.00000 −0.186411
\(260\) 0 0
\(261\) −5.87298 −0.363529
\(262\) 0 0
\(263\) 21.8730 1.34875 0.674373 0.738391i \(-0.264414\pi\)
0.674373 + 0.738391i \(0.264414\pi\)
\(264\) 0 0
\(265\) −3.87298 −0.237915
\(266\) 0 0
\(267\) −1.74597 −0.106851
\(268\) 0 0
\(269\) −23.3649 −1.42458 −0.712292 0.701883i \(-0.752343\pi\)
−0.712292 + 0.701883i \(0.752343\pi\)
\(270\) 0 0
\(271\) 7.25403 0.440651 0.220326 0.975426i \(-0.429288\pi\)
0.220326 + 0.975426i \(0.429288\pi\)
\(272\) 0 0
\(273\) −8.61895 −0.521643
\(274\) 0 0
\(275\) −4.87298 −0.293852
\(276\) 0 0
\(277\) 14.0000 0.841178 0.420589 0.907251i \(-0.361823\pi\)
0.420589 + 0.907251i \(0.361823\pi\)
\(278\) 0 0
\(279\) −3.00000 −0.179605
\(280\) 0 0
\(281\) −16.3649 −0.976249 −0.488125 0.872774i \(-0.662319\pi\)
−0.488125 + 0.872774i \(0.662319\pi\)
\(282\) 0 0
\(283\) −22.2379 −1.32191 −0.660953 0.750427i \(-0.729848\pi\)
−0.660953 + 0.750427i \(0.729848\pi\)
\(284\) 0 0
\(285\) −2.87298 −0.170181
\(286\) 0 0
\(287\) 17.6190 1.04001
\(288\) 0 0
\(289\) −2.00000 −0.117647
\(290\) 0 0
\(291\) −8.00000 −0.468968
\(292\) 0 0
\(293\) −13.6190 −0.795628 −0.397814 0.917466i \(-0.630231\pi\)
−0.397814 + 0.917466i \(0.630231\pi\)
\(294\) 0 0
\(295\) −5.87298 −0.341938
\(296\) 0 0
\(297\) 4.87298 0.282759
\(298\) 0 0
\(299\) −2.87298 −0.166149
\(300\) 0 0
\(301\) 11.2379 0.647742
\(302\) 0 0
\(303\) 3.87298 0.222497
\(304\) 0 0
\(305\) 8.87298 0.508066
\(306\) 0 0
\(307\) −9.12702 −0.520906 −0.260453 0.965486i \(-0.583872\pi\)
−0.260453 + 0.965486i \(0.583872\pi\)
\(308\) 0 0
\(309\) −6.00000 −0.341328
\(310\) 0 0
\(311\) 4.00000 0.226819 0.113410 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) 0 0
\(313\) 7.00000 0.395663 0.197832 0.980236i \(-0.436610\pi\)
0.197832 + 0.980236i \(0.436610\pi\)
\(314\) 0 0
\(315\) −3.00000 −0.169031
\(316\) 0 0
\(317\) 22.3649 1.25614 0.628069 0.778157i \(-0.283845\pi\)
0.628069 + 0.778157i \(0.283845\pi\)
\(318\) 0 0
\(319\) 28.6190 1.60235
\(320\) 0 0
\(321\) −5.61895 −0.313619
\(322\) 0 0
\(323\) 11.1270 0.619124
\(324\) 0 0
\(325\) −2.87298 −0.159364
\(326\) 0 0
\(327\) −16.6190 −0.919030
\(328\) 0 0
\(329\) −20.6190 −1.13676
\(330\) 0 0
\(331\) 22.2379 1.22231 0.611153 0.791513i \(-0.290706\pi\)
0.611153 + 0.791513i \(0.290706\pi\)
\(332\) 0 0
\(333\) 1.00000 0.0547997
\(334\) 0 0
\(335\) −10.7460 −0.587115
\(336\) 0 0
\(337\) −1.49193 −0.0812708 −0.0406354 0.999174i \(-0.512938\pi\)
−0.0406354 + 0.999174i \(0.512938\pi\)
\(338\) 0 0
\(339\) 3.87298 0.210352
\(340\) 0 0
\(341\) 14.6190 0.791661
\(342\) 0 0
\(343\) 15.0000 0.809924
\(344\) 0 0
\(345\) −1.00000 −0.0538382
\(346\) 0 0
\(347\) 25.7460 1.38212 0.691058 0.722799i \(-0.257145\pi\)
0.691058 + 0.722799i \(0.257145\pi\)
\(348\) 0 0
\(349\) −14.7460 −0.789334 −0.394667 0.918824i \(-0.629140\pi\)
−0.394667 + 0.918824i \(0.629140\pi\)
\(350\) 0 0
\(351\) 2.87298 0.153349
\(352\) 0 0
\(353\) −1.12702 −0.0599850 −0.0299925 0.999550i \(-0.509548\pi\)
−0.0299925 + 0.999550i \(0.509548\pi\)
\(354\) 0 0
\(355\) 13.6190 0.722819
\(356\) 0 0
\(357\) 11.6190 0.614940
\(358\) 0 0
\(359\) 21.1270 1.11504 0.557521 0.830163i \(-0.311753\pi\)
0.557521 + 0.830163i \(0.311753\pi\)
\(360\) 0 0
\(361\) −10.7460 −0.565577
\(362\) 0 0
\(363\) −12.7460 −0.668990
\(364\) 0 0
\(365\) −2.87298 −0.150379
\(366\) 0 0
\(367\) −25.9839 −1.35635 −0.678173 0.734902i \(-0.737228\pi\)
−0.678173 + 0.734902i \(0.737228\pi\)
\(368\) 0 0
\(369\) −5.87298 −0.305735
\(370\) 0 0
\(371\) 11.6190 0.603226
\(372\) 0 0
\(373\) 1.74597 0.0904027 0.0452014 0.998978i \(-0.485607\pi\)
0.0452014 + 0.998978i \(0.485607\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 16.8730 0.869003
\(378\) 0 0
\(379\) 12.0000 0.616399 0.308199 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) 0 0
\(381\) −6.61895 −0.339099
\(382\) 0 0
\(383\) 25.6190 1.30907 0.654534 0.756033i \(-0.272865\pi\)
0.654534 + 0.756033i \(0.272865\pi\)
\(384\) 0 0
\(385\) 14.6190 0.745051
\(386\) 0 0
\(387\) −3.74597 −0.190418
\(388\) 0 0
\(389\) 3.74597 0.189928 0.0949640 0.995481i \(-0.469726\pi\)
0.0949640 + 0.995481i \(0.469726\pi\)
\(390\) 0 0
\(391\) 3.87298 0.195865
\(392\) 0 0
\(393\) 11.4919 0.579691
\(394\) 0 0
\(395\) −4.00000 −0.201262
\(396\) 0 0
\(397\) 19.4919 0.978272 0.489136 0.872208i \(-0.337312\pi\)
0.489136 + 0.872208i \(0.337312\pi\)
\(398\) 0 0
\(399\) 8.61895 0.431487
\(400\) 0 0
\(401\) 27.2379 1.36020 0.680098 0.733121i \(-0.261937\pi\)
0.680098 + 0.733121i \(0.261937\pi\)
\(402\) 0 0
\(403\) 8.61895 0.429340
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −4.87298 −0.241545
\(408\) 0 0
\(409\) 22.2379 1.09959 0.549797 0.835299i \(-0.314705\pi\)
0.549797 + 0.835299i \(0.314705\pi\)
\(410\) 0 0
\(411\) −8.00000 −0.394611
\(412\) 0 0
\(413\) 17.6190 0.866972
\(414\) 0 0
\(415\) 0.127017 0.00623500
\(416\) 0 0
\(417\) −20.4919 −1.00349
\(418\) 0 0
\(419\) −2.61895 −0.127944 −0.0639720 0.997952i \(-0.520377\pi\)
−0.0639720 + 0.997952i \(0.520377\pi\)
\(420\) 0 0
\(421\) −38.8730 −1.89455 −0.947277 0.320417i \(-0.896177\pi\)
−0.947277 + 0.320417i \(0.896177\pi\)
\(422\) 0 0
\(423\) 6.87298 0.334176
\(424\) 0 0
\(425\) 3.87298 0.187867
\(426\) 0 0
\(427\) −26.6190 −1.28818
\(428\) 0 0
\(429\) −14.0000 −0.675926
\(430\) 0 0
\(431\) −5.74597 −0.276773 −0.138387 0.990378i \(-0.544192\pi\)
−0.138387 + 0.990378i \(0.544192\pi\)
\(432\) 0 0
\(433\) 19.0000 0.913082 0.456541 0.889702i \(-0.349088\pi\)
0.456541 + 0.889702i \(0.349088\pi\)
\(434\) 0 0
\(435\) 5.87298 0.281588
\(436\) 0 0
\(437\) 2.87298 0.137433
\(438\) 0 0
\(439\) 26.0000 1.24091 0.620456 0.784241i \(-0.286947\pi\)
0.620456 + 0.784241i \(0.286947\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 0 0
\(443\) 24.6190 1.16968 0.584841 0.811148i \(-0.301157\pi\)
0.584841 + 0.811148i \(0.301157\pi\)
\(444\) 0 0
\(445\) 1.74597 0.0827668
\(446\) 0 0
\(447\) 10.8730 0.514274
\(448\) 0 0
\(449\) −21.8730 −1.03225 −0.516125 0.856513i \(-0.672626\pi\)
−0.516125 + 0.856513i \(0.672626\pi\)
\(450\) 0 0
\(451\) 28.6190 1.34761
\(452\) 0 0
\(453\) 3.74597 0.176001
\(454\) 0 0
\(455\) 8.61895 0.404063
\(456\) 0 0
\(457\) −17.0000 −0.795226 −0.397613 0.917553i \(-0.630161\pi\)
−0.397613 + 0.917553i \(0.630161\pi\)
\(458\) 0 0
\(459\) −3.87298 −0.180775
\(460\) 0 0
\(461\) 31.7460 1.47856 0.739279 0.673400i \(-0.235167\pi\)
0.739279 + 0.673400i \(0.235167\pi\)
\(462\) 0 0
\(463\) −34.8730 −1.62068 −0.810342 0.585957i \(-0.800719\pi\)
−0.810342 + 0.585957i \(0.800719\pi\)
\(464\) 0 0
\(465\) 3.00000 0.139122
\(466\) 0 0
\(467\) 5.61895 0.260014 0.130007 0.991513i \(-0.458500\pi\)
0.130007 + 0.991513i \(0.458500\pi\)
\(468\) 0 0
\(469\) 32.2379 1.48861
\(470\) 0 0
\(471\) −17.0000 −0.783319
\(472\) 0 0
\(473\) 18.2540 0.839321
\(474\) 0 0
\(475\) 2.87298 0.131822
\(476\) 0 0
\(477\) −3.87298 −0.177332
\(478\) 0 0
\(479\) 34.8730 1.59339 0.796694 0.604383i \(-0.206580\pi\)
0.796694 + 0.604383i \(0.206580\pi\)
\(480\) 0 0
\(481\) −2.87298 −0.130997
\(482\) 0 0
\(483\) 3.00000 0.136505
\(484\) 0 0
\(485\) 8.00000 0.363261
\(486\) 0 0
\(487\) 26.1109 1.18320 0.591599 0.806233i \(-0.298497\pi\)
0.591599 + 0.806233i \(0.298497\pi\)
\(488\) 0 0
\(489\) −22.0000 −0.994874
\(490\) 0 0
\(491\) −28.1270 −1.26935 −0.634677 0.772777i \(-0.718867\pi\)
−0.634677 + 0.772777i \(0.718867\pi\)
\(492\) 0 0
\(493\) −22.7460 −1.02443
\(494\) 0 0
\(495\) −4.87298 −0.219024
\(496\) 0 0
\(497\) −40.8569 −1.83268
\(498\) 0 0
\(499\) −3.25403 −0.145671 −0.0728353 0.997344i \(-0.523205\pi\)
−0.0728353 + 0.997344i \(0.523205\pi\)
\(500\) 0 0
\(501\) −9.12702 −0.407765
\(502\) 0 0
\(503\) 37.1109 1.65469 0.827346 0.561692i \(-0.189849\pi\)
0.827346 + 0.561692i \(0.189849\pi\)
\(504\) 0 0
\(505\) −3.87298 −0.172345
\(506\) 0 0
\(507\) 4.74597 0.210776
\(508\) 0 0
\(509\) 20.0000 0.886484 0.443242 0.896402i \(-0.353828\pi\)
0.443242 + 0.896402i \(0.353828\pi\)
\(510\) 0 0
\(511\) 8.61895 0.381280
\(512\) 0 0
\(513\) −2.87298 −0.126845
\(514\) 0 0
\(515\) 6.00000 0.264392
\(516\) 0 0
\(517\) −33.4919 −1.47297
\(518\) 0 0
\(519\) −23.4919 −1.03118
\(520\) 0 0
\(521\) 40.8730 1.79068 0.895339 0.445385i \(-0.146933\pi\)
0.895339 + 0.445385i \(0.146933\pi\)
\(522\) 0 0
\(523\) 18.9839 0.830107 0.415053 0.909797i \(-0.363763\pi\)
0.415053 + 0.909797i \(0.363763\pi\)
\(524\) 0 0
\(525\) 3.00000 0.130931
\(526\) 0 0
\(527\) −11.6190 −0.506129
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −5.87298 −0.254866
\(532\) 0 0
\(533\) 16.8730 0.730850
\(534\) 0 0
\(535\) 5.61895 0.242928
\(536\) 0 0
\(537\) −15.7460 −0.679489
\(538\) 0 0
\(539\) −9.74597 −0.419789
\(540\) 0 0
\(541\) −40.0000 −1.71973 −0.859867 0.510518i \(-0.829454\pi\)
−0.859867 + 0.510518i \(0.829454\pi\)
\(542\) 0 0
\(543\) 21.7460 0.933209
\(544\) 0 0
\(545\) 16.6190 0.711878
\(546\) 0 0
\(547\) −16.0000 −0.684111 −0.342055 0.939680i \(-0.611123\pi\)
−0.342055 + 0.939680i \(0.611123\pi\)
\(548\) 0 0
\(549\) 8.87298 0.378690
\(550\) 0 0
\(551\) −16.8730 −0.718813
\(552\) 0 0
\(553\) 12.0000 0.510292
\(554\) 0 0
\(555\) −1.00000 −0.0424476
\(556\) 0 0
\(557\) 28.1270 1.19178 0.595890 0.803066i \(-0.296799\pi\)
0.595890 + 0.803066i \(0.296799\pi\)
\(558\) 0 0
\(559\) 10.7621 0.455188
\(560\) 0 0
\(561\) 18.8730 0.796818
\(562\) 0 0
\(563\) −33.1109 −1.39546 −0.697729 0.716362i \(-0.745806\pi\)
−0.697729 + 0.716362i \(0.745806\pi\)
\(564\) 0 0
\(565\) −3.87298 −0.162938
\(566\) 0 0
\(567\) −3.00000 −0.125988
\(568\) 0 0
\(569\) 4.25403 0.178338 0.0891692 0.996016i \(-0.471579\pi\)
0.0891692 + 0.996016i \(0.471579\pi\)
\(570\) 0 0
\(571\) −36.3649 −1.52182 −0.760912 0.648855i \(-0.775248\pi\)
−0.760912 + 0.648855i \(0.775248\pi\)
\(572\) 0 0
\(573\) −18.8730 −0.788430
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) 8.00000 0.333044 0.166522 0.986038i \(-0.446746\pi\)
0.166522 + 0.986038i \(0.446746\pi\)
\(578\) 0 0
\(579\) −9.74597 −0.405029
\(580\) 0 0
\(581\) −0.381050 −0.0158086
\(582\) 0 0
\(583\) 18.8730 0.781639
\(584\) 0 0
\(585\) −2.87298 −0.118783
\(586\) 0 0
\(587\) −13.4919 −0.556872 −0.278436 0.960455i \(-0.589816\pi\)
−0.278436 + 0.960455i \(0.589816\pi\)
\(588\) 0 0
\(589\) −8.61895 −0.355138
\(590\) 0 0
\(591\) −4.25403 −0.174988
\(592\) 0 0
\(593\) 44.6190 1.83228 0.916140 0.400858i \(-0.131288\pi\)
0.916140 + 0.400858i \(0.131288\pi\)
\(594\) 0 0
\(595\) −11.6190 −0.476331
\(596\) 0 0
\(597\) 19.2379 0.787355
\(598\) 0 0
\(599\) −21.7460 −0.888516 −0.444258 0.895899i \(-0.646533\pi\)
−0.444258 + 0.895899i \(0.646533\pi\)
\(600\) 0 0
\(601\) −12.7460 −0.519919 −0.259959 0.965620i \(-0.583709\pi\)
−0.259959 + 0.965620i \(0.583709\pi\)
\(602\) 0 0
\(603\) −10.7460 −0.437610
\(604\) 0 0
\(605\) 12.7460 0.518197
\(606\) 0 0
\(607\) −31.1270 −1.26341 −0.631703 0.775210i \(-0.717644\pi\)
−0.631703 + 0.775210i \(0.717644\pi\)
\(608\) 0 0
\(609\) −17.6190 −0.713956
\(610\) 0 0
\(611\) −19.7460 −0.798836
\(612\) 0 0
\(613\) 48.9839 1.97844 0.989220 0.146438i \(-0.0467808\pi\)
0.989220 + 0.146438i \(0.0467808\pi\)
\(614\) 0 0
\(615\) 5.87298 0.236822
\(616\) 0 0
\(617\) −17.6190 −0.709312 −0.354656 0.934997i \(-0.615402\pi\)
−0.354656 + 0.934997i \(0.615402\pi\)
\(618\) 0 0
\(619\) 16.0000 0.643094 0.321547 0.946894i \(-0.395797\pi\)
0.321547 + 0.946894i \(0.395797\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) −5.23790 −0.209852
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 14.0000 0.559106
\(628\) 0 0
\(629\) 3.87298 0.154426
\(630\) 0 0
\(631\) −32.3649 −1.28843 −0.644213 0.764846i \(-0.722815\pi\)
−0.644213 + 0.764846i \(0.722815\pi\)
\(632\) 0 0
\(633\) 1.00000 0.0397464
\(634\) 0 0
\(635\) 6.61895 0.262665
\(636\) 0 0
\(637\) −5.74597 −0.227663
\(638\) 0 0
\(639\) 13.6190 0.538757
\(640\) 0 0
\(641\) −10.1109 −0.399356 −0.199678 0.979862i \(-0.563990\pi\)
−0.199678 + 0.979862i \(0.563990\pi\)
\(642\) 0 0
\(643\) 22.4919 0.886995 0.443498 0.896276i \(-0.353737\pi\)
0.443498 + 0.896276i \(0.353737\pi\)
\(644\) 0 0
\(645\) 3.74597 0.147497
\(646\) 0 0
\(647\) −27.8569 −1.09517 −0.547583 0.836751i \(-0.684452\pi\)
−0.547583 + 0.836751i \(0.684452\pi\)
\(648\) 0 0
\(649\) 28.6190 1.12339
\(650\) 0 0
\(651\) −9.00000 −0.352738
\(652\) 0 0
\(653\) −10.1109 −0.395669 −0.197835 0.980235i \(-0.563391\pi\)
−0.197835 + 0.980235i \(0.563391\pi\)
\(654\) 0 0
\(655\) −11.4919 −0.449027
\(656\) 0 0
\(657\) −2.87298 −0.112086
\(658\) 0 0
\(659\) −28.8730 −1.12473 −0.562366 0.826889i \(-0.690109\pi\)
−0.562366 + 0.826889i \(0.690109\pi\)
\(660\) 0 0
\(661\) −43.2379 −1.68176 −0.840880 0.541222i \(-0.817962\pi\)
−0.840880 + 0.541222i \(0.817962\pi\)
\(662\) 0 0
\(663\) 11.1270 0.432138
\(664\) 0 0
\(665\) −8.61895 −0.334229
\(666\) 0 0
\(667\) −5.87298 −0.227403
\(668\) 0 0
\(669\) −2.00000 −0.0773245
\(670\) 0 0
\(671\) −43.2379 −1.66918
\(672\) 0 0
\(673\) 19.1270 0.737292 0.368646 0.929570i \(-0.379821\pi\)
0.368646 + 0.929570i \(0.379821\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −7.11088 −0.273293 −0.136647 0.990620i \(-0.543633\pi\)
−0.136647 + 0.990620i \(0.543633\pi\)
\(678\) 0 0
\(679\) −24.0000 −0.921035
\(680\) 0 0
\(681\) 7.49193 0.287092
\(682\) 0 0
\(683\) −17.1270 −0.655347 −0.327674 0.944791i \(-0.606265\pi\)
−0.327674 + 0.944791i \(0.606265\pi\)
\(684\) 0 0
\(685\) 8.00000 0.305664
\(686\) 0 0
\(687\) 5.74597 0.219222
\(688\) 0 0
\(689\) 11.1270 0.423906
\(690\) 0 0
\(691\) −46.7298 −1.77769 −0.888843 0.458211i \(-0.848490\pi\)
−0.888843 + 0.458211i \(0.848490\pi\)
\(692\) 0 0
\(693\) 14.6190 0.555328
\(694\) 0 0
\(695\) 20.4919 0.777303
\(696\) 0 0
\(697\) −22.7460 −0.861565
\(698\) 0 0
\(699\) −15.4919 −0.585959
\(700\) 0 0
\(701\) 29.8569 1.12768 0.563839 0.825885i \(-0.309324\pi\)
0.563839 + 0.825885i \(0.309324\pi\)
\(702\) 0 0
\(703\) 2.87298 0.108357
\(704\) 0 0
\(705\) −6.87298 −0.258852
\(706\) 0 0
\(707\) 11.6190 0.436976
\(708\) 0 0
\(709\) −6.61895 −0.248580 −0.124290 0.992246i \(-0.539665\pi\)
−0.124290 + 0.992246i \(0.539665\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 0 0
\(713\) −3.00000 −0.112351
\(714\) 0 0
\(715\) 14.0000 0.523570
\(716\) 0 0
\(717\) −17.8730 −0.667479
\(718\) 0 0
\(719\) 39.3649 1.46806 0.734032 0.679115i \(-0.237636\pi\)
0.734032 + 0.679115i \(0.237636\pi\)
\(720\) 0 0
\(721\) −18.0000 −0.670355
\(722\) 0 0
\(723\) 5.12702 0.190676
\(724\) 0 0
\(725\) −5.87298 −0.218117
\(726\) 0 0
\(727\) −11.2540 −0.417389 −0.208694 0.977981i \(-0.566921\pi\)
−0.208694 + 0.977981i \(0.566921\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −14.5081 −0.536600
\(732\) 0 0
\(733\) 22.7460 0.840141 0.420071 0.907491i \(-0.362005\pi\)
0.420071 + 0.907491i \(0.362005\pi\)
\(734\) 0 0
\(735\) −2.00000 −0.0737711
\(736\) 0 0
\(737\) 52.3649 1.92889
\(738\) 0 0
\(739\) −13.2540 −0.487557 −0.243779 0.969831i \(-0.578387\pi\)
−0.243779 + 0.969831i \(0.578387\pi\)
\(740\) 0 0
\(741\) 8.25403 0.303219
\(742\) 0 0
\(743\) 36.0000 1.32071 0.660356 0.750953i \(-0.270405\pi\)
0.660356 + 0.750953i \(0.270405\pi\)
\(744\) 0 0
\(745\) −10.8730 −0.398355
\(746\) 0 0
\(747\) 0.127017 0.00464730
\(748\) 0 0
\(749\) −16.8569 −0.615936
\(750\) 0 0
\(751\) 29.8569 1.08949 0.544746 0.838601i \(-0.316626\pi\)
0.544746 + 0.838601i \(0.316626\pi\)
\(752\) 0 0
\(753\) 2.00000 0.0728841
\(754\) 0 0
\(755\) −3.74597 −0.136330
\(756\) 0 0
\(757\) −4.74597 −0.172495 −0.0862475 0.996274i \(-0.527488\pi\)
−0.0862475 + 0.996274i \(0.527488\pi\)
\(758\) 0 0
\(759\) 4.87298 0.176878
\(760\) 0 0
\(761\) −14.8569 −0.538560 −0.269280 0.963062i \(-0.586786\pi\)
−0.269280 + 0.963062i \(0.586786\pi\)
\(762\) 0 0
\(763\) −49.8569 −1.80494
\(764\) 0 0
\(765\) 3.87298 0.140028
\(766\) 0 0
\(767\) 16.8730 0.609248
\(768\) 0 0
\(769\) 28.8730 1.04119 0.520593 0.853805i \(-0.325711\pi\)
0.520593 + 0.853805i \(0.325711\pi\)
\(770\) 0 0
\(771\) 16.6190 0.598517
\(772\) 0 0
\(773\) 4.25403 0.153007 0.0765035 0.997069i \(-0.475624\pi\)
0.0765035 + 0.997069i \(0.475624\pi\)
\(774\) 0 0
\(775\) −3.00000 −0.107763
\(776\) 0 0
\(777\) 3.00000 0.107624
\(778\) 0 0
\(779\) −16.8730 −0.604537
\(780\) 0 0
\(781\) −66.3649 −2.37472
\(782\) 0 0
\(783\) 5.87298 0.209883
\(784\) 0 0
\(785\) 17.0000 0.606756
\(786\) 0 0
\(787\) −25.0000 −0.891154 −0.445577 0.895244i \(-0.647001\pi\)
−0.445577 + 0.895244i \(0.647001\pi\)
\(788\) 0 0
\(789\) −21.8730 −0.778699
\(790\) 0 0
\(791\) 11.6190 0.413122
\(792\) 0 0
\(793\) −25.4919 −0.905245
\(794\) 0 0
\(795\) 3.87298 0.137361
\(796\) 0 0
\(797\) 11.8730 0.420563 0.210281 0.977641i \(-0.432562\pi\)
0.210281 + 0.977641i \(0.432562\pi\)
\(798\) 0 0
\(799\) 26.6190 0.941711
\(800\) 0 0
\(801\) 1.74597 0.0616907
\(802\) 0 0
\(803\) 14.0000 0.494049
\(804\) 0 0
\(805\) −3.00000 −0.105736
\(806\) 0 0
\(807\) 23.3649 0.822484
\(808\) 0 0
\(809\) 24.8569 0.873920 0.436960 0.899481i \(-0.356055\pi\)
0.436960 + 0.899481i \(0.356055\pi\)
\(810\) 0 0
\(811\) −28.4919 −1.00049 −0.500244 0.865885i \(-0.666756\pi\)
−0.500244 + 0.865885i \(0.666756\pi\)
\(812\) 0 0
\(813\) −7.25403 −0.254410
\(814\) 0 0
\(815\) 22.0000 0.770626
\(816\) 0 0
\(817\) −10.7621 −0.376518
\(818\) 0 0
\(819\) 8.61895 0.301170
\(820\) 0 0
\(821\) −3.49193 −0.121869 −0.0609347 0.998142i \(-0.519408\pi\)
−0.0609347 + 0.998142i \(0.519408\pi\)
\(822\) 0 0
\(823\) −44.0000 −1.53374 −0.766872 0.641800i \(-0.778188\pi\)
−0.766872 + 0.641800i \(0.778188\pi\)
\(824\) 0 0
\(825\) 4.87298 0.169656
\(826\) 0 0
\(827\) 15.1109 0.525457 0.262728 0.964870i \(-0.415378\pi\)
0.262728 + 0.964870i \(0.415378\pi\)
\(828\) 0 0
\(829\) 3.50807 0.121840 0.0609201 0.998143i \(-0.480597\pi\)
0.0609201 + 0.998143i \(0.480597\pi\)
\(830\) 0 0
\(831\) −14.0000 −0.485655
\(832\) 0 0
\(833\) 7.74597 0.268382
\(834\) 0 0
\(835\) 9.12702 0.315853
\(836\) 0 0
\(837\) 3.00000 0.103695
\(838\) 0 0
\(839\) −18.0000 −0.621429 −0.310715 0.950503i \(-0.600568\pi\)
−0.310715 + 0.950503i \(0.600568\pi\)
\(840\) 0 0
\(841\) 5.49193 0.189377
\(842\) 0 0
\(843\) 16.3649 0.563638
\(844\) 0 0
\(845\) −4.74597 −0.163266
\(846\) 0 0
\(847\) −38.2379 −1.31387
\(848\) 0 0
\(849\) 22.2379 0.763203
\(850\) 0 0
\(851\) 1.00000 0.0342796
\(852\) 0 0
\(853\) −43.7460 −1.49783 −0.748917 0.662664i \(-0.769426\pi\)
−0.748917 + 0.662664i \(0.769426\pi\)
\(854\) 0 0
\(855\) 2.87298 0.0982540
\(856\) 0 0
\(857\) −14.5081 −0.495586 −0.247793 0.968813i \(-0.579705\pi\)
−0.247793 + 0.968813i \(0.579705\pi\)
\(858\) 0 0
\(859\) −29.0000 −0.989467 −0.494734 0.869045i \(-0.664734\pi\)
−0.494734 + 0.869045i \(0.664734\pi\)
\(860\) 0 0
\(861\) −17.6190 −0.600452
\(862\) 0 0
\(863\) −8.25403 −0.280971 −0.140485 0.990083i \(-0.544866\pi\)
−0.140485 + 0.990083i \(0.544866\pi\)
\(864\) 0 0
\(865\) 23.4919 0.798750
\(866\) 0 0
\(867\) 2.00000 0.0679236
\(868\) 0 0
\(869\) 19.4919 0.661219
\(870\) 0 0
\(871\) 30.8730 1.04609
\(872\) 0 0
\(873\) 8.00000 0.270759
\(874\) 0 0
\(875\) −3.00000 −0.101419
\(876\) 0 0
\(877\) 54.7298 1.84810 0.924048 0.382277i \(-0.124860\pi\)
0.924048 + 0.382277i \(0.124860\pi\)
\(878\) 0 0
\(879\) 13.6190 0.459356
\(880\) 0 0
\(881\) 4.11088 0.138499 0.0692496 0.997599i \(-0.477940\pi\)
0.0692496 + 0.997599i \(0.477940\pi\)
\(882\) 0 0
\(883\) −55.3488 −1.86263 −0.931317 0.364209i \(-0.881340\pi\)
−0.931317 + 0.364209i \(0.881340\pi\)
\(884\) 0 0
\(885\) 5.87298 0.197418
\(886\) 0 0
\(887\) −19.7460 −0.663005 −0.331502 0.943454i \(-0.607555\pi\)
−0.331502 + 0.943454i \(0.607555\pi\)
\(888\) 0 0
\(889\) −19.8569 −0.665977
\(890\) 0 0
\(891\) −4.87298 −0.163251
\(892\) 0 0
\(893\) 19.7460 0.660774
\(894\) 0 0
\(895\) 15.7460 0.526330
\(896\) 0 0
\(897\) 2.87298 0.0959261
\(898\) 0 0
\(899\) 17.6190 0.587625
\(900\) 0 0
\(901\) −15.0000 −0.499722
\(902\) 0 0
\(903\) −11.2379 −0.373974
\(904\) 0 0
\(905\) −21.7460 −0.722860
\(906\) 0 0
\(907\) −30.7460 −1.02090 −0.510452 0.859907i \(-0.670522\pi\)
−0.510452 + 0.859907i \(0.670522\pi\)
\(908\) 0 0
\(909\) −3.87298 −0.128459
\(910\) 0 0
\(911\) 16.0000 0.530104 0.265052 0.964234i \(-0.414611\pi\)
0.265052 + 0.964234i \(0.414611\pi\)
\(912\) 0 0
\(913\) −0.618950 −0.0204843
\(914\) 0 0
\(915\) −8.87298 −0.293332
\(916\) 0 0
\(917\) 34.4758 1.13849
\(918\) 0 0
\(919\) 44.0000 1.45143 0.725713 0.687998i \(-0.241510\pi\)
0.725713 + 0.687998i \(0.241510\pi\)
\(920\) 0 0
\(921\) 9.12702 0.300745
\(922\) 0 0
\(923\) −39.1270 −1.28788
\(924\) 0 0
\(925\) 1.00000 0.0328798
\(926\) 0 0
\(927\) 6.00000 0.197066
\(928\) 0 0
\(929\) 31.3649 1.02905 0.514525 0.857476i \(-0.327968\pi\)
0.514525 + 0.857476i \(0.327968\pi\)
\(930\) 0 0
\(931\) 5.74597 0.188316
\(932\) 0 0
\(933\) −4.00000 −0.130954
\(934\) 0 0
\(935\) −18.8730 −0.617213
\(936\) 0 0
\(937\) 54.4758 1.77965 0.889823 0.456305i \(-0.150827\pi\)
0.889823 + 0.456305i \(0.150827\pi\)
\(938\) 0 0
\(939\) −7.00000 −0.228436
\(940\) 0 0
\(941\) −16.1109 −0.525200 −0.262600 0.964905i \(-0.584580\pi\)
−0.262600 + 0.964905i \(0.584580\pi\)
\(942\) 0 0
\(943\) −5.87298 −0.191251
\(944\) 0 0
\(945\) 3.00000 0.0975900
\(946\) 0 0
\(947\) 25.2379 0.820122 0.410061 0.912058i \(-0.365507\pi\)
0.410061 + 0.912058i \(0.365507\pi\)
\(948\) 0 0
\(949\) 8.25403 0.267937
\(950\) 0 0
\(951\) −22.3649 −0.725232
\(952\) 0 0
\(953\) 8.98387 0.291016 0.145508 0.989357i \(-0.453518\pi\)
0.145508 + 0.989357i \(0.453518\pi\)
\(954\) 0 0
\(955\) 18.8730 0.610715
\(956\) 0 0
\(957\) −28.6190 −0.925119
\(958\) 0 0
\(959\) −24.0000 −0.775000
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 0 0
\(963\) 5.61895 0.181068
\(964\) 0 0
\(965\) 9.74597 0.313734
\(966\) 0 0
\(967\) −21.3810 −0.687568 −0.343784 0.939049i \(-0.611709\pi\)
−0.343784 + 0.939049i \(0.611709\pi\)
\(968\) 0 0
\(969\) −11.1270 −0.357451
\(970\) 0 0
\(971\) −53.7460 −1.72479 −0.862395 0.506236i \(-0.831037\pi\)
−0.862395 + 0.506236i \(0.831037\pi\)
\(972\) 0 0
\(973\) −61.4758 −1.97082
\(974\) 0 0
\(975\) 2.87298 0.0920091
\(976\) 0 0
\(977\) −51.8730 −1.65956 −0.829782 0.558088i \(-0.811535\pi\)
−0.829782 + 0.558088i \(0.811535\pi\)
\(978\) 0 0
\(979\) −8.50807 −0.271919
\(980\) 0 0
\(981\) 16.6190 0.530602
\(982\) 0 0
\(983\) −7.36492 −0.234904 −0.117452 0.993079i \(-0.537473\pi\)
−0.117452 + 0.993079i \(0.537473\pi\)
\(984\) 0 0
\(985\) 4.25403 0.135545
\(986\) 0 0
\(987\) 20.6190 0.656308
\(988\) 0 0
\(989\) −3.74597 −0.119115
\(990\) 0 0
\(991\) 11.5081 0.365566 0.182783 0.983153i \(-0.441489\pi\)
0.182783 + 0.983153i \(0.441489\pi\)
\(992\) 0 0
\(993\) −22.2379 −0.705698
\(994\) 0 0
\(995\) −19.2379 −0.609882
\(996\) 0 0
\(997\) −41.7460 −1.32211 −0.661054 0.750338i \(-0.729891\pi\)
−0.661054 + 0.750338i \(0.729891\pi\)
\(998\) 0 0
\(999\) −1.00000 −0.0316386
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 5520.2.a.bj.1.1 2
4.3 odd 2 1380.2.a.i.1.2 2
12.11 even 2 4140.2.a.p.1.1 2
20.3 even 4 6900.2.f.o.6349.4 4
20.7 even 4 6900.2.f.o.6349.2 4
20.19 odd 2 6900.2.a.j.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1380.2.a.i.1.2 2 4.3 odd 2
4140.2.a.p.1.1 2 12.11 even 2
5520.2.a.bj.1.1 2 1.1 even 1 trivial
6900.2.a.j.1.2 2 20.19 odd 2
6900.2.f.o.6349.2 4 20.7 even 4
6900.2.f.o.6349.4 4 20.3 even 4