Properties

Label 4600.2.e.u.4049.1
Level $4600$
Weight $2$
Character 4600.4049
Analytic conductor $36.731$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.7311849298\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Defining polynomial: \(x^{10} + 28 x^{8} + 260 x^{6} + 897 x^{4} + 1056 x^{2} + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.1
Root \(-3.36002i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.u.4049.10

$q$-expansion

\(f(q)\) \(=\) \(q-3.36002i q^{3} +1.90754i q^{7} -8.28974 q^{9} +O(q^{10})\) \(q-3.36002i q^{3} +1.90754i q^{7} -8.28974 q^{9} -5.48021 q^{11} -1.04937i q^{13} +6.74222i q^{17} +1.55049 q^{19} +6.40939 q^{21} -1.00000i q^{23} +17.7736i q^{27} +3.38219 q^{29} +10.9327 q^{31} +18.4136i q^{33} -5.26201i q^{37} -3.52589 q^{39} +6.09801 q^{41} -0.403830i q^{47} +3.36128 q^{49} +22.6540 q^{51} +5.88332i q^{53} -5.20968i q^{57} -9.60111 q^{59} -7.09927 q^{61} -15.8130i q^{63} -13.7971i q^{67} -3.36002 q^{69} +0.478950 q^{71} +2.40383i q^{73} -10.4537i q^{77} +4.24037 q^{79} +34.8505 q^{81} -11.2620i q^{83} -11.3642i q^{87} -4.90495 q^{89} +2.00171 q^{91} -36.7340i q^{93} +12.3433i q^{97} +45.4295 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 26q^{9} + O(q^{10}) \) \( 10q - 26q^{9} - 2q^{11} - 14q^{19} + 12q^{21} - 8q^{29} + 38q^{31} - 38q^{39} + 50q^{41} - 50q^{49} + 38q^{51} + 2q^{59} - 10q^{61} + 2q^{71} + 4q^{79} + 114q^{81} - 12q^{89} + 22q^{91} + 130q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times\).

\(n\) \(1151\) \(1201\) \(2301\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) − 3.36002i − 1.93991i −0.243287 0.969954i \(-0.578226\pi\)
0.243287 0.969954i \(-0.421774\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.90754i 0.720984i 0.932762 + 0.360492i \(0.117391\pi\)
−0.932762 + 0.360492i \(0.882609\pi\)
\(8\) 0 0
\(9\) −8.28974 −2.76325
\(10\) 0 0
\(11\) −5.48021 −1.65234 −0.826172 0.563418i \(-0.809486\pi\)
−0.826172 + 0.563418i \(0.809486\pi\)
\(12\) 0 0
\(13\) − 1.04937i − 0.291041i −0.989355 0.145521i \(-0.953514\pi\)
0.989355 0.145521i \(-0.0464858\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.74222i 1.63523i 0.575767 + 0.817614i \(0.304703\pi\)
−0.575767 + 0.817614i \(0.695297\pi\)
\(18\) 0 0
\(19\) 1.55049 0.355707 0.177853 0.984057i \(-0.443085\pi\)
0.177853 + 0.984057i \(0.443085\pi\)
\(20\) 0 0
\(21\) 6.40939 1.39864
\(22\) 0 0
\(23\) − 1.00000i − 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 17.7736i 3.42054i
\(28\) 0 0
\(29\) 3.38219 0.628058 0.314029 0.949413i \(-0.398321\pi\)
0.314029 + 0.949413i \(0.398321\pi\)
\(30\) 0 0
\(31\) 10.9327 1.96357 0.981784 0.190000i \(-0.0608489\pi\)
0.981784 + 0.190000i \(0.0608489\pi\)
\(32\) 0 0
\(33\) 18.4136i 3.20540i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 5.26201i − 0.865069i −0.901617 0.432534i \(-0.857619\pi\)
0.901617 0.432534i \(-0.142381\pi\)
\(38\) 0 0
\(39\) −3.52589 −0.564594
\(40\) 0 0
\(41\) 6.09801 0.952350 0.476175 0.879351i \(-0.342023\pi\)
0.476175 + 0.879351i \(0.342023\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 0.403830i − 0.0589046i −0.999566 0.0294523i \(-0.990624\pi\)
0.999566 0.0294523i \(-0.00937631\pi\)
\(48\) 0 0
\(49\) 3.36128 0.480183
\(50\) 0 0
\(51\) 22.6540 3.17219
\(52\) 0 0
\(53\) 5.88332i 0.808136i 0.914729 + 0.404068i \(0.132404\pi\)
−0.914729 + 0.404068i \(0.867596\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 5.20968i − 0.690039i
\(58\) 0 0
\(59\) −9.60111 −1.24996 −0.624979 0.780641i \(-0.714893\pi\)
−0.624979 + 0.780641i \(0.714893\pi\)
\(60\) 0 0
\(61\) −7.09927 −0.908968 −0.454484 0.890755i \(-0.650176\pi\)
−0.454484 + 0.890755i \(0.650176\pi\)
\(62\) 0 0
\(63\) − 15.8130i − 1.99226i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 13.7971i − 1.68559i −0.538236 0.842794i \(-0.680909\pi\)
0.538236 0.842794i \(-0.319091\pi\)
\(68\) 0 0
\(69\) −3.36002 −0.404499
\(70\) 0 0
\(71\) 0.478950 0.0568409 0.0284205 0.999596i \(-0.490952\pi\)
0.0284205 + 0.999596i \(0.490952\pi\)
\(72\) 0 0
\(73\) 2.40383i 0.281347i 0.990056 + 0.140673i \(0.0449268\pi\)
−0.990056 + 0.140673i \(0.955073\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 10.4537i − 1.19131i
\(78\) 0 0
\(79\) 4.24037 0.477079 0.238540 0.971133i \(-0.423331\pi\)
0.238540 + 0.971133i \(0.423331\pi\)
\(80\) 0 0
\(81\) 34.8505 3.87228
\(82\) 0 0
\(83\) − 11.2620i − 1.23617i −0.786113 0.618083i \(-0.787910\pi\)
0.786113 0.618083i \(-0.212090\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 11.3642i − 1.21837i
\(88\) 0 0
\(89\) −4.90495 −0.519924 −0.259962 0.965619i \(-0.583710\pi\)
−0.259962 + 0.965619i \(0.583710\pi\)
\(90\) 0 0
\(91\) 2.00171 0.209836
\(92\) 0 0
\(93\) − 36.7340i − 3.80914i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 12.3433i 1.25327i 0.779311 + 0.626637i \(0.215569\pi\)
−0.779311 + 0.626637i \(0.784431\pi\)
\(98\) 0 0
\(99\) 45.4295 4.56583
\(100\) 0 0
\(101\) −1.44692 −0.143974 −0.0719870 0.997406i \(-0.522934\pi\)
−0.0719870 + 0.997406i \(0.522934\pi\)
\(102\) 0 0
\(103\) − 7.76385i − 0.764995i −0.923957 0.382497i \(-0.875064\pi\)
0.923957 0.382497i \(-0.124936\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.54197i 0.439089i 0.975603 + 0.219544i \(0.0704570\pi\)
−0.975603 + 0.219544i \(0.929543\pi\)
\(108\) 0 0
\(109\) −12.4136 −1.18901 −0.594504 0.804093i \(-0.702652\pi\)
−0.594504 + 0.804093i \(0.702652\pi\)
\(110\) 0 0
\(111\) −17.6805 −1.67815
\(112\) 0 0
\(113\) 9.27312i 0.872342i 0.899864 + 0.436171i \(0.143666\pi\)
−0.899864 + 0.436171i \(0.856334\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 8.69896i 0.804219i
\(118\) 0 0
\(119\) −12.8611 −1.17897
\(120\) 0 0
\(121\) 19.0327 1.73024
\(122\) 0 0
\(123\) − 20.4894i − 1.84747i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 12.4149i − 1.10165i −0.834621 0.550824i \(-0.814314\pi\)
0.834621 0.550824i \(-0.185686\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 9.55434 0.834766 0.417383 0.908731i \(-0.362947\pi\)
0.417383 + 0.908731i \(0.362947\pi\)
\(132\) 0 0
\(133\) 2.95763i 0.256459i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 3.36558i 0.287541i 0.989611 + 0.143770i \(0.0459226\pi\)
−0.989611 + 0.143770i \(0.954077\pi\)
\(138\) 0 0
\(139\) 20.7361 1.75881 0.879406 0.476072i \(-0.157940\pi\)
0.879406 + 0.476072i \(0.157940\pi\)
\(140\) 0 0
\(141\) −1.35688 −0.114270
\(142\) 0 0
\(143\) 5.75074i 0.480901i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 11.2940i − 0.931510i
\(148\) 0 0
\(149\) 19.9399 1.63354 0.816769 0.576965i \(-0.195763\pi\)
0.816769 + 0.576965i \(0.195763\pi\)
\(150\) 0 0
\(151\) 23.7979 1.93664 0.968321 0.249709i \(-0.0803349\pi\)
0.968321 + 0.249709i \(0.0803349\pi\)
\(152\) 0 0
\(153\) − 55.8912i − 4.51854i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 13.3175i − 1.06285i −0.847105 0.531425i \(-0.821657\pi\)
0.847105 0.531425i \(-0.178343\pi\)
\(158\) 0 0
\(159\) 19.7681 1.56771
\(160\) 0 0
\(161\) 1.90754 0.150335
\(162\) 0 0
\(163\) 16.1529i 1.26519i 0.774483 + 0.632595i \(0.218010\pi\)
−0.774483 + 0.632595i \(0.781990\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 20.6782i 1.60013i 0.599915 + 0.800064i \(0.295201\pi\)
−0.599915 + 0.800064i \(0.704799\pi\)
\(168\) 0 0
\(169\) 11.8988 0.915295
\(170\) 0 0
\(171\) −12.8532 −0.982905
\(172\) 0 0
\(173\) 7.72058i 0.586985i 0.955961 + 0.293492i \(0.0948176\pi\)
−0.955961 + 0.293492i \(0.905182\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 32.2599i 2.42480i
\(178\) 0 0
\(179\) −9.88683 −0.738976 −0.369488 0.929236i \(-0.620467\pi\)
−0.369488 + 0.929236i \(0.620467\pi\)
\(180\) 0 0
\(181\) −23.9883 −1.78304 −0.891519 0.452983i \(-0.850360\pi\)
−0.891519 + 0.452983i \(0.850360\pi\)
\(182\) 0 0
\(183\) 23.8537i 1.76332i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 36.9487i − 2.70196i
\(188\) 0 0
\(189\) −33.9040 −2.46615
\(190\) 0 0
\(191\) 6.72004 0.486245 0.243123 0.969996i \(-0.421828\pi\)
0.243123 + 0.969996i \(0.421828\pi\)
\(192\) 0 0
\(193\) 8.95007i 0.644240i 0.946699 + 0.322120i \(0.104395\pi\)
−0.946699 + 0.322120i \(0.895605\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 6.65109i 0.473871i 0.971525 + 0.236935i \(0.0761429\pi\)
−0.971525 + 0.236935i \(0.923857\pi\)
\(198\) 0 0
\(199\) 10.9050 0.773032 0.386516 0.922283i \(-0.373678\pi\)
0.386516 + 0.922283i \(0.373678\pi\)
\(200\) 0 0
\(201\) −46.3587 −3.26989
\(202\) 0 0
\(203\) 6.45168i 0.452819i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 8.28974i 0.576177i
\(208\) 0 0
\(209\) −8.49700 −0.587750
\(210\) 0 0
\(211\) −1.21874 −0.0839014 −0.0419507 0.999120i \(-0.513357\pi\)
−0.0419507 + 0.999120i \(0.513357\pi\)
\(212\) 0 0
\(213\) − 1.60928i − 0.110266i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 20.8546i 1.41570i
\(218\) 0 0
\(219\) 8.07692 0.545787
\(220\) 0 0
\(221\) 7.07505 0.475919
\(222\) 0 0
\(223\) 14.1542i 0.947835i 0.880569 + 0.473917i \(0.157160\pi\)
−0.880569 + 0.473917i \(0.842840\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 4.89902i 0.325159i 0.986695 + 0.162580i \(0.0519815\pi\)
−0.986695 + 0.162580i \(0.948019\pi\)
\(228\) 0 0
\(229\) −12.3946 −0.819056 −0.409528 0.912298i \(-0.634307\pi\)
−0.409528 + 0.912298i \(0.634307\pi\)
\(230\) 0 0
\(231\) −35.1248 −2.31104
\(232\) 0 0
\(233\) 0.882062i 0.0577858i 0.999583 + 0.0288929i \(0.00919818\pi\)
−0.999583 + 0.0288929i \(0.990802\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 14.2477i − 0.925490i
\(238\) 0 0
\(239\) 19.9506 1.29049 0.645247 0.763974i \(-0.276754\pi\)
0.645247 + 0.763974i \(0.276754\pi\)
\(240\) 0 0
\(241\) −2.81877 −0.181573 −0.0907865 0.995870i \(-0.528938\pi\)
−0.0907865 + 0.995870i \(0.528938\pi\)
\(242\) 0 0
\(243\) − 63.7777i − 4.09134i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 1.62703i − 0.103525i
\(248\) 0 0
\(249\) −37.8406 −2.39805
\(250\) 0 0
\(251\) 17.9883 1.13541 0.567706 0.823231i \(-0.307831\pi\)
0.567706 + 0.823231i \(0.307831\pi\)
\(252\) 0 0
\(253\) 5.48021i 0.344538i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 14.1705i 0.883930i 0.897032 + 0.441965i \(0.145718\pi\)
−0.897032 + 0.441965i \(0.854282\pi\)
\(258\) 0 0
\(259\) 10.0375 0.623701
\(260\) 0 0
\(261\) −28.0375 −1.73548
\(262\) 0 0
\(263\) − 6.88386i − 0.424477i −0.977218 0.212238i \(-0.931925\pi\)
0.977218 0.212238i \(-0.0680754\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 16.4807i 1.00861i
\(268\) 0 0
\(269\) 23.3463 1.42345 0.711724 0.702459i \(-0.247915\pi\)
0.711724 + 0.702459i \(0.247915\pi\)
\(270\) 0 0
\(271\) 13.9519 0.847517 0.423759 0.905775i \(-0.360711\pi\)
0.423759 + 0.905775i \(0.360711\pi\)
\(272\) 0 0
\(273\) − 6.72579i − 0.407063i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 3.69490i − 0.222005i −0.993820 0.111003i \(-0.964594\pi\)
0.993820 0.111003i \(-0.0354062\pi\)
\(278\) 0 0
\(279\) −90.6291 −5.42582
\(280\) 0 0
\(281\) 15.9582 0.951984 0.475992 0.879450i \(-0.342089\pi\)
0.475992 + 0.879450i \(0.342089\pi\)
\(282\) 0 0
\(283\) − 8.21649i − 0.488420i −0.969722 0.244210i \(-0.921471\pi\)
0.969722 0.244210i \(-0.0785286\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 11.6322i 0.686628i
\(288\) 0 0
\(289\) −28.4575 −1.67397
\(290\) 0 0
\(291\) 41.4738 2.43124
\(292\) 0 0
\(293\) 22.0878i 1.29038i 0.764021 + 0.645191i \(0.223222\pi\)
−0.764021 + 0.645191i \(0.776778\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 97.4032i − 5.65191i
\(298\) 0 0
\(299\) −1.04937 −0.0606863
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 4.86168i 0.279296i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 15.2091i − 0.868030i −0.900906 0.434015i \(-0.857097\pi\)
0.900906 0.434015i \(-0.142903\pi\)
\(308\) 0 0
\(309\) −26.0867 −1.48402
\(310\) 0 0
\(311\) 0.254818 0.0144494 0.00722471 0.999974i \(-0.497700\pi\)
0.00722471 + 0.999974i \(0.497700\pi\)
\(312\) 0 0
\(313\) 3.57095i 0.201842i 0.994894 + 0.100921i \(0.0321790\pi\)
−0.994894 + 0.100921i \(0.967821\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 12.3235i − 0.692155i −0.938206 0.346078i \(-0.887513\pi\)
0.938206 0.346078i \(-0.112487\pi\)
\(318\) 0 0
\(319\) −18.5351 −1.03777
\(320\) 0 0
\(321\) 15.2611 0.851792
\(322\) 0 0
\(323\) 10.4537i 0.581661i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 41.7100i 2.30657i
\(328\) 0 0
\(329\) 0.770323 0.0424693
\(330\) 0 0
\(331\) 26.1082 1.43503 0.717517 0.696541i \(-0.245278\pi\)
0.717517 + 0.696541i \(0.245278\pi\)
\(332\) 0 0
\(333\) 43.6207i 2.39040i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 9.68246i − 0.527437i −0.964600 0.263718i \(-0.915051\pi\)
0.964600 0.263718i \(-0.0849490\pi\)
\(338\) 0 0
\(339\) 31.1579 1.69226
\(340\) 0 0
\(341\) −59.9134 −3.24449
\(342\) 0 0
\(343\) 19.7646i 1.06719i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 24.9747i − 1.34071i −0.742040 0.670355i \(-0.766142\pi\)
0.742040 0.670355i \(-0.233858\pi\)
\(348\) 0 0
\(349\) 35.3019 1.88967 0.944835 0.327547i \(-0.106222\pi\)
0.944835 + 0.327547i \(0.106222\pi\)
\(350\) 0 0
\(351\) 18.6510 0.995518
\(352\) 0 0
\(353\) − 10.0326i − 0.533980i −0.963699 0.266990i \(-0.913971\pi\)
0.963699 0.266990i \(-0.0860290\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 43.2135i 2.28710i
\(358\) 0 0
\(359\) 30.8691 1.62921 0.814603 0.580019i \(-0.196955\pi\)
0.814603 + 0.580019i \(0.196955\pi\)
\(360\) 0 0
\(361\) −16.5960 −0.873473
\(362\) 0 0
\(363\) − 63.9502i − 3.35651i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 33.3234i − 1.73947i −0.493521 0.869734i \(-0.664291\pi\)
0.493521 0.869734i \(-0.335709\pi\)
\(368\) 0 0
\(369\) −50.5509 −2.63158
\(370\) 0 0
\(371\) −11.2227 −0.582653
\(372\) 0 0
\(373\) 4.62023i 0.239227i 0.992821 + 0.119613i \(0.0381655\pi\)
−0.992821 + 0.119613i \(0.961835\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 3.54916i − 0.182791i
\(378\) 0 0
\(379\) −15.0020 −0.770600 −0.385300 0.922791i \(-0.625902\pi\)
−0.385300 + 0.922791i \(0.625902\pi\)
\(380\) 0 0
\(381\) −41.7145 −2.13710
\(382\) 0 0
\(383\) 1.87882i 0.0960033i 0.998847 + 0.0480016i \(0.0152853\pi\)
−0.998847 + 0.0480016i \(0.984715\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 1.48539 0.0753121 0.0376560 0.999291i \(-0.488011\pi\)
0.0376560 + 0.999291i \(0.488011\pi\)
\(390\) 0 0
\(391\) 6.74222 0.340968
\(392\) 0 0
\(393\) − 32.1028i − 1.61937i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 5.26722i − 0.264354i −0.991226 0.132177i \(-0.957803\pi\)
0.991226 0.132177i \(-0.0421968\pi\)
\(398\) 0 0
\(399\) 9.93769 0.497507
\(400\) 0 0
\(401\) 26.1490 1.30582 0.652910 0.757436i \(-0.273548\pi\)
0.652910 + 0.757436i \(0.273548\pi\)
\(402\) 0 0
\(403\) − 11.4724i − 0.571480i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 28.8369i 1.42939i
\(408\) 0 0
\(409\) 14.7503 0.729355 0.364677 0.931134i \(-0.381179\pi\)
0.364677 + 0.931134i \(0.381179\pi\)
\(410\) 0 0
\(411\) 11.3084 0.557803
\(412\) 0 0
\(413\) − 18.3145i − 0.901200i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 69.6737i − 3.41194i
\(418\) 0 0
\(419\) −1.10616 −0.0540394 −0.0270197 0.999635i \(-0.508602\pi\)
−0.0270197 + 0.999635i \(0.508602\pi\)
\(420\) 0 0
\(421\) 9.16362 0.446607 0.223304 0.974749i \(-0.428316\pi\)
0.223304 + 0.974749i \(0.428316\pi\)
\(422\) 0 0
\(423\) 3.34764i 0.162768i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 13.5422i − 0.655351i
\(428\) 0 0
\(429\) 19.3226 0.932904
\(430\) 0 0
\(431\) −32.1712 −1.54963 −0.774817 0.632186i \(-0.782158\pi\)
−0.774817 + 0.632186i \(0.782158\pi\)
\(432\) 0 0
\(433\) − 8.37861i − 0.402650i −0.979524 0.201325i \(-0.935475\pi\)
0.979524 0.201325i \(-0.0645248\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 1.55049i − 0.0741700i
\(438\) 0 0
\(439\) 15.3093 0.730673 0.365336 0.930876i \(-0.380954\pi\)
0.365336 + 0.930876i \(0.380954\pi\)
\(440\) 0 0
\(441\) −27.8641 −1.32686
\(442\) 0 0
\(443\) − 3.24129i − 0.153998i −0.997031 0.0769992i \(-0.975466\pi\)
0.997031 0.0769992i \(-0.0245339\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 66.9984i − 3.16892i
\(448\) 0 0
\(449\) 9.02215 0.425782 0.212891 0.977076i \(-0.431712\pi\)
0.212891 + 0.977076i \(0.431712\pi\)
\(450\) 0 0
\(451\) −33.4184 −1.57361
\(452\) 0 0
\(453\) − 79.9613i − 3.75691i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 7.03526i − 0.329096i −0.986369 0.164548i \(-0.947384\pi\)
0.986369 0.164548i \(-0.0526165\pi\)
\(458\) 0 0
\(459\) −119.834 −5.59336
\(460\) 0 0
\(461\) −1.59617 −0.0743411 −0.0371705 0.999309i \(-0.511834\pi\)
−0.0371705 + 0.999309i \(0.511834\pi\)
\(462\) 0 0
\(463\) 9.61655i 0.446919i 0.974713 + 0.223459i \(0.0717350\pi\)
−0.974713 + 0.223459i \(0.928265\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 24.4764i − 1.13263i −0.824188 0.566317i \(-0.808368\pi\)
0.824188 0.566317i \(-0.191632\pi\)
\(468\) 0 0
\(469\) 26.3186 1.21528
\(470\) 0 0
\(471\) −44.7470 −2.06183
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 48.7712i − 2.23308i
\(478\) 0 0
\(479\) 13.2108 0.603618 0.301809 0.953368i \(-0.402410\pi\)
0.301809 + 0.953368i \(0.402410\pi\)
\(480\) 0 0
\(481\) −5.52177 −0.251771
\(482\) 0 0
\(483\) − 6.40939i − 0.291637i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 23.7078i 1.07431i 0.843485 + 0.537153i \(0.180500\pi\)
−0.843485 + 0.537153i \(0.819500\pi\)
\(488\) 0 0
\(489\) 54.2739 2.45435
\(490\) 0 0
\(491\) 18.5930 0.839091 0.419546 0.907734i \(-0.362189\pi\)
0.419546 + 0.907734i \(0.362189\pi\)
\(492\) 0 0
\(493\) 22.8035i 1.02702i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0.913618i 0.0409814i
\(498\) 0 0
\(499\) −17.9121 −0.801858 −0.400929 0.916109i \(-0.631313\pi\)
−0.400929 + 0.916109i \(0.631313\pi\)
\(500\) 0 0
\(501\) 69.4792 3.10410
\(502\) 0 0
\(503\) 1.24890i 0.0556855i 0.999612 + 0.0278428i \(0.00886377\pi\)
−0.999612 + 0.0278428i \(0.991136\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 39.9803i − 1.77559i
\(508\) 0 0
\(509\) −15.8757 −0.703679 −0.351839 0.936060i \(-0.614444\pi\)
−0.351839 + 0.936060i \(0.614444\pi\)
\(510\) 0 0
\(511\) −4.58541 −0.202847
\(512\) 0 0
\(513\) 27.5578i 1.21671i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 2.21307i 0.0973307i
\(518\) 0 0
\(519\) 25.9413 1.13870
\(520\) 0 0
\(521\) 26.5488 1.16312 0.581561 0.813503i \(-0.302442\pi\)
0.581561 + 0.813503i \(0.302442\pi\)
\(522\) 0 0
\(523\) 31.0258i 1.35666i 0.734757 + 0.678331i \(0.237296\pi\)
−0.734757 + 0.678331i \(0.762704\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 73.7105i 3.21088i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 79.5907 3.45394
\(532\) 0 0
\(533\) − 6.39904i − 0.277173i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 33.2199i 1.43355i
\(538\) 0 0
\(539\) −18.4205 −0.793427
\(540\) 0 0
\(541\) 22.4123 0.963579 0.481790 0.876287i \(-0.339987\pi\)
0.481790 + 0.876287i \(0.339987\pi\)
\(542\) 0 0
\(543\) 80.6013i 3.45893i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 43.3980i 1.85556i 0.373122 + 0.927782i \(0.378287\pi\)
−0.373122 + 0.927782i \(0.621713\pi\)
\(548\) 0 0
\(549\) 58.8511 2.51170
\(550\) 0 0
\(551\) 5.24406 0.223404
\(552\) 0 0
\(553\) 8.08870i 0.343966i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 21.2690i 0.901197i 0.892727 + 0.450599i \(0.148789\pi\)
−0.892727 + 0.450599i \(0.851211\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −124.149 −5.24155
\(562\) 0 0
\(563\) 10.6629i 0.449389i 0.974429 + 0.224694i \(0.0721383\pi\)
−0.974429 + 0.224694i \(0.927862\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 66.4789i 2.79185i
\(568\) 0 0
\(569\) −31.8986 −1.33726 −0.668630 0.743596i \(-0.733119\pi\)
−0.668630 + 0.743596i \(0.733119\pi\)
\(570\) 0 0
\(571\) −7.87477 −0.329549 −0.164774 0.986331i \(-0.552690\pi\)
−0.164774 + 0.986331i \(0.552690\pi\)
\(572\) 0 0
\(573\) − 22.5795i − 0.943271i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 40.7070i − 1.69466i −0.531070 0.847328i \(-0.678210\pi\)
0.531070 0.847328i \(-0.321790\pi\)
\(578\) 0 0
\(579\) 30.0724 1.24977
\(580\) 0 0
\(581\) 21.4828 0.891256
\(582\) 0 0
\(583\) − 32.2418i − 1.33532i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 22.9321i 0.946508i 0.880926 + 0.473254i \(0.156921\pi\)
−0.880926 + 0.473254i \(0.843079\pi\)
\(588\) 0 0
\(589\) 16.9510 0.698454
\(590\) 0 0
\(591\) 22.3478 0.919266
\(592\) 0 0
\(593\) 27.6250i 1.13442i 0.823572 + 0.567211i \(0.191978\pi\)
−0.823572 + 0.567211i \(0.808022\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 36.6409i − 1.49961i
\(598\) 0 0
\(599\) 18.2139 0.744199 0.372099 0.928193i \(-0.378638\pi\)
0.372099 + 0.928193i \(0.378638\pi\)
\(600\) 0 0
\(601\) −24.0407 −0.980640 −0.490320 0.871543i \(-0.663120\pi\)
−0.490320 + 0.871543i \(0.663120\pi\)
\(602\) 0 0
\(603\) 114.375i 4.65770i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 5.52878i − 0.224406i −0.993685 0.112203i \(-0.964209\pi\)
0.993685 0.112203i \(-0.0357907\pi\)
\(608\) 0 0
\(609\) 21.6778 0.878428
\(610\) 0 0
\(611\) −0.423765 −0.0171437
\(612\) 0 0
\(613\) − 21.1205i − 0.853050i −0.904476 0.426525i \(-0.859738\pi\)
0.904476 0.426525i \(-0.140262\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 19.6523i − 0.791174i −0.918429 0.395587i \(-0.870541\pi\)
0.918429 0.395587i \(-0.129459\pi\)
\(618\) 0 0
\(619\) −40.9931 −1.64765 −0.823826 0.566843i \(-0.808164\pi\)
−0.823826 + 0.566843i \(0.808164\pi\)
\(620\) 0 0
\(621\) 17.7736 0.713231
\(622\) 0 0
\(623\) − 9.35641i − 0.374857i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 28.5501i 1.14018i
\(628\) 0 0
\(629\) 35.4776 1.41458
\(630\) 0 0
\(631\) −45.5595 −1.81369 −0.906847 0.421461i \(-0.861517\pi\)
−0.906847 + 0.421461i \(0.861517\pi\)
\(632\) 0 0
\(633\) 4.09499i 0.162761i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 3.52721i − 0.139753i
\(638\) 0 0
\(639\) −3.97037 −0.157065
\(640\) 0 0
\(641\) 8.97997 0.354688 0.177344 0.984149i \(-0.443250\pi\)
0.177344 + 0.984149i \(0.443250\pi\)
\(642\) 0 0
\(643\) 2.69616i 0.106326i 0.998586 + 0.0531630i \(0.0169303\pi\)
−0.998586 + 0.0531630i \(0.983070\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 9.12638i − 0.358795i −0.983777 0.179398i \(-0.942585\pi\)
0.983777 0.179398i \(-0.0574149\pi\)
\(648\) 0 0
\(649\) 52.6161 2.06536
\(650\) 0 0
\(651\) 70.0718 2.74633
\(652\) 0 0
\(653\) 41.5497i 1.62596i 0.582289 + 0.812982i \(0.302157\pi\)
−0.582289 + 0.812982i \(0.697843\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 19.9271i − 0.777431i
\(658\) 0 0
\(659\) −20.9256 −0.815145 −0.407573 0.913173i \(-0.633625\pi\)
−0.407573 + 0.913173i \(0.633625\pi\)
\(660\) 0 0
\(661\) 5.25688 0.204469 0.102234 0.994760i \(-0.467401\pi\)
0.102234 + 0.994760i \(0.467401\pi\)
\(662\) 0 0
\(663\) − 23.7723i − 0.923240i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 3.38219i − 0.130959i
\(668\) 0 0
\(669\) 47.5584 1.83871
\(670\) 0 0
\(671\) 38.9055 1.50193
\(672\) 0 0
\(673\) 38.1900i 1.47212i 0.676918 + 0.736059i \(0.263315\pi\)
−0.676918 + 0.736059i \(0.736685\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 4.83529i 0.185835i 0.995674 + 0.0929176i \(0.0296193\pi\)
−0.995674 + 0.0929176i \(0.970381\pi\)
\(678\) 0 0
\(679\) −23.5454 −0.903591
\(680\) 0 0
\(681\) 16.4608 0.630780
\(682\) 0 0
\(683\) − 50.3044i − 1.92484i −0.271559 0.962422i \(-0.587539\pi\)
0.271559 0.962422i \(-0.412461\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 41.6460i 1.58889i
\(688\) 0 0
\(689\) 6.17375 0.235201
\(690\) 0 0
\(691\) 20.6298 0.784793 0.392396 0.919796i \(-0.371646\pi\)
0.392396 + 0.919796i \(0.371646\pi\)
\(692\) 0 0
\(693\) 86.6587i 3.29189i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 41.1141i 1.55731i
\(698\) 0 0
\(699\) 2.96375 0.112099
\(700\) 0 0
\(701\) −22.9598 −0.867182 −0.433591 0.901110i \(-0.642754\pi\)
−0.433591 + 0.901110i \(0.642754\pi\)
\(702\) 0 0
\(703\) − 8.15869i − 0.307711i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 2.76006i − 0.103803i
\(708\) 0 0
\(709\) −11.9069 −0.447174 −0.223587 0.974684i \(-0.571777\pi\)
−0.223587 + 0.974684i \(0.571777\pi\)
\(710\) 0 0
\(711\) −35.1516 −1.31829
\(712\) 0 0
\(713\) − 10.9327i − 0.409432i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 67.0343i − 2.50344i
\(718\) 0 0
\(719\) −24.5244 −0.914604 −0.457302 0.889311i \(-0.651184\pi\)
−0.457302 + 0.889311i \(0.651184\pi\)
\(720\) 0 0
\(721\) 14.8099 0.551549
\(722\) 0 0
\(723\) 9.47113i 0.352235i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 11.4034i − 0.422928i −0.977386 0.211464i \(-0.932177\pi\)
0.977386 0.211464i \(-0.0678231\pi\)
\(728\) 0 0
\(729\) −109.743 −4.06454
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 35.8722i 1.32497i 0.749076 + 0.662484i \(0.230498\pi\)
−0.749076 + 0.662484i \(0.769502\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 75.6112i 2.78517i
\(738\) 0 0
\(739\) 17.1503 0.630883 0.315441 0.948945i \(-0.397847\pi\)
0.315441 + 0.948945i \(0.397847\pi\)
\(740\) 0 0
\(741\) −5.46685 −0.200830
\(742\) 0 0
\(743\) 27.7242i 1.01710i 0.861031 + 0.508552i \(0.169819\pi\)
−0.861031 + 0.508552i \(0.830181\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 93.3591i 3.41583i
\(748\) 0 0
\(749\) −8.66400 −0.316576
\(750\) 0 0
\(751\) −2.80173 −0.102236 −0.0511182 0.998693i \(-0.516279\pi\)
−0.0511182 + 0.998693i \(0.516279\pi\)
\(752\) 0 0
\(753\) − 60.4411i − 2.20260i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 41.6710i 1.51456i 0.653092 + 0.757278i \(0.273471\pi\)
−0.653092 + 0.757278i \(0.726529\pi\)
\(758\) 0 0
\(759\) 18.4136 0.668372
\(760\) 0 0
\(761\) −35.4285 −1.28428 −0.642141 0.766587i \(-0.721954\pi\)
−0.642141 + 0.766587i \(0.721954\pi\)
\(762\) 0 0
\(763\) − 23.6795i − 0.857255i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 10.0751i 0.363790i
\(768\) 0 0
\(769\) −23.8731 −0.860885 −0.430442 0.902618i \(-0.641642\pi\)
−0.430442 + 0.902618i \(0.641642\pi\)
\(770\) 0 0
\(771\) 47.6131 1.71474
\(772\) 0 0
\(773\) 11.3403i 0.407881i 0.978983 + 0.203941i \(0.0653750\pi\)
−0.978983 + 0.203941i \(0.934625\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 33.7262i − 1.20992i
\(778\) 0 0
\(779\) 9.45490 0.338757
\(780\) 0 0
\(781\) −2.62475 −0.0939208
\(782\) 0 0
\(783\) 60.1139i 2.14829i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 26.1906i 0.933595i 0.884364 + 0.466797i \(0.154592\pi\)
−0.884364 + 0.466797i \(0.845408\pi\)
\(788\) 0 0
\(789\) −23.1299 −0.823446
\(790\) 0 0
\(791\) −17.6889 −0.628944
\(792\) 0 0
\(793\) 7.44973i 0.264547i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 4.48133i 0.158737i 0.996845 + 0.0793684i \(0.0252903\pi\)
−0.996845 + 0.0793684i \(0.974710\pi\)
\(798\) 0 0
\(799\) 2.72271 0.0963224
\(800\) 0 0
\(801\) 40.6608 1.43668
\(802\) 0 0
\(803\) − 13.1735i − 0.464882i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 78.4440i − 2.76136i
\(808\) 0 0
\(809\) 32.4608 1.14126 0.570630 0.821207i \(-0.306699\pi\)
0.570630 + 0.821207i \(0.306699\pi\)
\(810\) 0 0
\(811\) 11.3500 0.398554 0.199277 0.979943i \(-0.436141\pi\)
0.199277 + 0.979943i \(0.436141\pi\)
\(812\) 0 0
\(813\) − 46.8786i − 1.64411i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −16.5936 −0.579829
\(820\) 0 0
\(821\) 48.2070 1.68244 0.841218 0.540697i \(-0.181839\pi\)
0.841218 + 0.540697i \(0.181839\pi\)
\(822\) 0 0
\(823\) − 37.7179i − 1.31476i −0.753558 0.657381i \(-0.771664\pi\)
0.753558 0.657381i \(-0.228336\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 8.89736i 0.309392i 0.987962 + 0.154696i \(0.0494397\pi\)
−0.987962 + 0.154696i \(0.950560\pi\)
\(828\) 0 0
\(829\) −30.9058 −1.07340 −0.536702 0.843772i \(-0.680330\pi\)
−0.536702 + 0.843772i \(0.680330\pi\)
\(830\) 0 0
\(831\) −12.4149 −0.430670
\(832\) 0 0
\(833\) 22.6625i 0.785208i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 194.313i 6.71646i
\(838\) 0 0
\(839\) −1.34387 −0.0463954 −0.0231977 0.999731i \(-0.507385\pi\)
−0.0231977 + 0.999731i \(0.507385\pi\)
\(840\) 0 0
\(841\) −17.5608 −0.605543
\(842\) 0 0
\(843\) − 53.6198i − 1.84676i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 36.3056i 1.24748i
\(848\) 0 0
\(849\) −27.6076 −0.947489
\(850\) 0 0
\(851\) −5.26201 −0.180379
\(852\) 0 0
\(853\) − 26.3940i − 0.903713i −0.892091 0.451857i \(-0.850762\pi\)
0.892091 0.451857i \(-0.149238\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 17.5048i 0.597953i 0.954260 + 0.298976i \(0.0966452\pi\)
−0.954260 + 0.298976i \(0.903355\pi\)
\(858\) 0 0
\(859\) −11.7199 −0.399877 −0.199938 0.979808i \(-0.564074\pi\)
−0.199938 + 0.979808i \(0.564074\pi\)
\(860\) 0 0
\(861\) 39.0845 1.33200
\(862\) 0 0
\(863\) 26.6578i 0.907443i 0.891144 + 0.453722i \(0.149904\pi\)
−0.891144 + 0.453722i \(0.850096\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 95.6177i 3.24735i
\(868\) 0 0
\(869\) −23.2381 −0.788299
\(870\) 0 0
\(871\) −14.4782 −0.490576
\(872\) 0 0
\(873\) − 102.323i − 3.46311i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 19.9574i − 0.673912i −0.941520 0.336956i \(-0.890603\pi\)
0.941520 0.336956i \(-0.109397\pi\)
\(878\) 0 0
\(879\) 74.2154 2.50322
\(880\) 0 0
\(881\) −20.7297 −0.698402 −0.349201 0.937048i \(-0.613547\pi\)
−0.349201 + 0.937048i \(0.613547\pi\)
\(882\) 0 0
\(883\) 2.31093i 0.0777690i 0.999244 + 0.0388845i \(0.0123804\pi\)
−0.999244 + 0.0388845i \(0.987620\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 11.6924i 0.392592i 0.980545 + 0.196296i \(0.0628913\pi\)
−0.980545 + 0.196296i \(0.937109\pi\)
\(888\) 0 0
\(889\) 23.6820 0.794270
\(890\) 0 0
\(891\) −190.988 −6.39835
\(892\) 0 0
\(893\) − 0.626134i − 0.0209528i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 3.52589i 0.117726i
\(898\) 0 0
\(899\) 36.9765 1.23323
\(900\) 0 0
\(901\) −39.6666 −1.32149
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 32.5061i − 1.07935i −0.841875 0.539673i \(-0.818548\pi\)
0.841875 0.539673i \(-0.181452\pi\)
\(908\) 0 0
\(909\) 11.9946 0.397836
\(910\) 0 0
\(911\) 10.7518 0.356224 0.178112 0.984010i \(-0.443001\pi\)
0.178112 + 0.984010i \(0.443001\pi\)
\(912\) 0 0
\(913\) 61.7181i 2.04257i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 18.2253i 0.601853i
\(918\) 0 0
\(919\) 24.2758 0.800785 0.400392 0.916344i \(-0.368874\pi\)
0.400392 + 0.916344i \(0.368874\pi\)
\(920\) 0 0
\(921\) −51.1029 −1.68390
\(922\) 0 0
\(923\) − 0.502594i − 0.0165431i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 64.3603i 2.11387i
\(928\) 0 0
\(929\) −8.18842 −0.268653 −0.134327 0.990937i \(-0.542887\pi\)
−0.134327 + 0.990937i \(0.542887\pi\)
\(930\) 0 0
\(931\) 5.21163 0.170804
\(932\) 0 0
\(933\) − 0.856194i − 0.0280305i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 26.2312i − 0.856936i −0.903557 0.428468i \(-0.859053\pi\)
0.903557 0.428468i \(-0.140947\pi\)
\(938\) 0 0
\(939\) 11.9985 0.391556
\(940\) 0 0
\(941\) 23.6292 0.770291 0.385145 0.922856i \(-0.374151\pi\)
0.385145 + 0.922856i \(0.374151\pi\)
\(942\) 0 0
\(943\) − 6.09801i − 0.198579i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 15.9026i 0.516765i 0.966043 + 0.258383i \(0.0831895\pi\)
−0.966043 + 0.258383i \(0.916810\pi\)
\(948\) 0 0
\(949\) 2.52249 0.0818836
\(950\) 0 0
\(951\) −41.4071 −1.34272
\(952\) 0 0
\(953\) 20.9451i 0.678478i 0.940700 + 0.339239i \(0.110169\pi\)
−0.940700 + 0.339239i \(0.889831\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 62.2784i 2.01318i
\(958\) 0 0
\(959\) −6.41998 −0.207312
\(960\) 0 0
\(961\) 88.5236 2.85560
\(962\) 0 0
\(963\) − 37.6517i − 1.21331i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 14.0732i 0.452563i 0.974062 + 0.226281i \(0.0726569\pi\)
−0.974062 + 0.226281i \(0.927343\pi\)
\(968\) 0 0
\(969\) 35.1248 1.12837
\(970\) 0 0
\(971\) −11.4404 −0.367139 −0.183569 0.983007i \(-0.558765\pi\)
−0.183569 + 0.983007i \(0.558765\pi\)
\(972\) 0 0
\(973\) 39.5550i 1.26808i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 43.9119i − 1.40487i −0.711750 0.702433i \(-0.752097\pi\)
0.711750 0.702433i \(-0.247903\pi\)
\(978\) 0 0
\(979\) 26.8802 0.859094
\(980\) 0 0
\(981\) 102.906 3.28552
\(982\) 0 0
\(983\) − 43.0420i − 1.37283i −0.727212 0.686413i \(-0.759184\pi\)
0.727212 0.686413i \(-0.240816\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 2.58830i − 0.0823865i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 43.5288 1.38274 0.691369 0.722501i \(-0.257008\pi\)
0.691369 + 0.722501i \(0.257008\pi\)
\(992\) 0 0
\(993\) − 87.7240i − 2.78384i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 26.4448i − 0.837517i −0.908098 0.418758i \(-0.862465\pi\)
0.908098 0.418758i \(-0.137535\pi\)
\(998\) 0 0
\(999\) 93.5250 2.95900
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 4600.2.e.u.4049.1 10
5.2 odd 4 920.2.a.j.1.1 5
5.3 odd 4 4600.2.a.be.1.5 5
5.4 even 2 inner 4600.2.e.u.4049.10 10
15.2 even 4 8280.2.a.bs.1.3 5
20.3 even 4 9200.2.a.cu.1.1 5
20.7 even 4 1840.2.a.v.1.5 5
40.27 even 4 7360.2.a.cp.1.1 5
40.37 odd 4 7360.2.a.co.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.j.1.1 5 5.2 odd 4
1840.2.a.v.1.5 5 20.7 even 4
4600.2.a.be.1.5 5 5.3 odd 4
4600.2.e.u.4049.1 10 1.1 even 1 trivial
4600.2.e.u.4049.10 10 5.4 even 2 inner
7360.2.a.co.1.5 5 40.37 odd 4
7360.2.a.cp.1.1 5 40.27 even 4
8280.2.a.bs.1.3 5 15.2 even 4
9200.2.a.cu.1.1 5 20.3 even 4