Properties

Label 1900.2.c.g.1749.3
Level $1900$
Weight $2$
Character 1900.1749
Analytic conductor $15.172$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1900.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.1715763840\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.4227136.2
Defining polynomial: \(x^{6} + 9 x^{4} + 22 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1749.3
Root \(0.713538i\) of defining polynomial
Character \(\chi\) \(=\) 1900.1749
Dual form 1900.2.c.g.1749.4

$q$-expansion

\(f(q)\) \(=\) \(q-0.286462i q^{3} +0.286462i q^{7} +2.91794 q^{9} +O(q^{10})\) \(q-0.286462i q^{3} +0.286462i q^{7} +2.91794 q^{9} +4.26819 q^{11} -3.20440i q^{13} +0.286462i q^{17} +1.00000 q^{19} +0.0820605 q^{21} -0.936212i q^{23} -1.69527i q^{27} -2.26819 q^{29} -4.18613 q^{31} -1.22267i q^{33} +8.67699i q^{37} -0.917939 q^{39} +1.08206 q^{41} -4.42708i q^{43} -0.759053i q^{47} +6.91794 q^{49} +0.0820605 q^{51} -4.42708i q^{53} -0.286462i q^{57} +4.70050 q^{59} +10.1861 q^{61} +0.835879i q^{63} +9.82284i q^{67} -0.268189 q^{69} +4.83588 q^{71} -10.4726i q^{73} +1.22267i q^{77} -5.10407 q^{79} +8.26819 q^{81} -15.7355i q^{83} +0.649750i q^{87} +9.75382 q^{89} +0.917939 q^{91} +1.19917i q^{93} -9.61320i q^{97} +12.4543 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 2q^{9} + O(q^{10}) \) \( 6q - 2q^{9} - 2q^{11} + 6q^{19} + 20q^{21} + 14q^{29} + 22q^{31} + 14q^{39} + 26q^{41} + 22q^{49} + 20q^{51} + 12q^{59} + 14q^{61} + 26q^{69} - 10q^{71} + 36q^{79} + 22q^{81} - 14q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) − 0.286462i − 0.165389i −0.996575 0.0826945i \(-0.973647\pi\)
0.996575 0.0826945i \(-0.0263526\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.286462i 0.108272i 0.998534 + 0.0541362i \(0.0172405\pi\)
−0.998534 + 0.0541362i \(0.982759\pi\)
\(8\) 0 0
\(9\) 2.91794 0.972646
\(10\) 0 0
\(11\) 4.26819 1.28691 0.643454 0.765485i \(-0.277501\pi\)
0.643454 + 0.765485i \(0.277501\pi\)
\(12\) 0 0
\(13\) − 3.20440i − 0.888741i −0.895843 0.444371i \(-0.853427\pi\)
0.895843 0.444371i \(-0.146573\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.286462i 0.0694773i 0.999396 + 0.0347386i \(0.0110599\pi\)
−0.999396 + 0.0347386i \(0.988940\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0.0820605 0.0179071
\(22\) 0 0
\(23\) − 0.936212i − 0.195214i −0.995225 0.0976069i \(-0.968881\pi\)
0.995225 0.0976069i \(-0.0311188\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 1.69527i − 0.326254i
\(28\) 0 0
\(29\) −2.26819 −0.421192 −0.210596 0.977573i \(-0.567540\pi\)
−0.210596 + 0.977573i \(0.567540\pi\)
\(30\) 0 0
\(31\) −4.18613 −0.751851 −0.375925 0.926650i \(-0.622675\pi\)
−0.375925 + 0.926650i \(0.622675\pi\)
\(32\) 0 0
\(33\) − 1.22267i − 0.212840i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.67699i 1.42649i 0.700915 + 0.713244i \(0.252775\pi\)
−0.700915 + 0.713244i \(0.747225\pi\)
\(38\) 0 0
\(39\) −0.917939 −0.146988
\(40\) 0 0
\(41\) 1.08206 0.168989 0.0844947 0.996424i \(-0.473072\pi\)
0.0844947 + 0.996424i \(0.473072\pi\)
\(42\) 0 0
\(43\) − 4.42708i − 0.675123i −0.941304 0.337561i \(-0.890398\pi\)
0.941304 0.337561i \(-0.109602\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 0.759053i − 0.110719i −0.998466 0.0553596i \(-0.982369\pi\)
0.998466 0.0553596i \(-0.0176305\pi\)
\(48\) 0 0
\(49\) 6.91794 0.988277
\(50\) 0 0
\(51\) 0.0820605 0.0114908
\(52\) 0 0
\(53\) − 4.42708i − 0.608106i −0.952655 0.304053i \(-0.901660\pi\)
0.952655 0.304053i \(-0.0983399\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 0.286462i − 0.0379428i
\(58\) 0 0
\(59\) 4.70050 0.611953 0.305976 0.952039i \(-0.401017\pi\)
0.305976 + 0.952039i \(0.401017\pi\)
\(60\) 0 0
\(61\) 10.1861 1.30420 0.652100 0.758133i \(-0.273888\pi\)
0.652100 + 0.758133i \(0.273888\pi\)
\(62\) 0 0
\(63\) 0.835879i 0.105311i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 9.82284i 1.20005i 0.799981 + 0.600025i \(0.204843\pi\)
−0.799981 + 0.600025i \(0.795157\pi\)
\(68\) 0 0
\(69\) −0.268189 −0.0322862
\(70\) 0 0
\(71\) 4.83588 0.573913 0.286957 0.957944i \(-0.407356\pi\)
0.286957 + 0.957944i \(0.407356\pi\)
\(72\) 0 0
\(73\) − 10.4726i − 1.22572i −0.790190 0.612862i \(-0.790018\pi\)
0.790190 0.612862i \(-0.209982\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.22267i 0.139337i
\(78\) 0 0
\(79\) −5.10407 −0.574253 −0.287126 0.957893i \(-0.592700\pi\)
−0.287126 + 0.957893i \(0.592700\pi\)
\(80\) 0 0
\(81\) 8.26819 0.918688
\(82\) 0 0
\(83\) − 15.7355i − 1.72720i −0.504177 0.863600i \(-0.668204\pi\)
0.504177 0.863600i \(-0.331796\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0.649750i 0.0696605i
\(88\) 0 0
\(89\) 9.75382 1.03390 0.516951 0.856015i \(-0.327067\pi\)
0.516951 + 0.856015i \(0.327067\pi\)
\(90\) 0 0
\(91\) 0.917939 0.0962262
\(92\) 0 0
\(93\) 1.19917i 0.124348i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 9.61320i − 0.976073i −0.872823 0.488037i \(-0.837713\pi\)
0.872823 0.488037i \(-0.162287\pi\)
\(98\) 0 0
\(99\) 12.4543 1.25171
\(100\) 0 0
\(101\) 0.731811 0.0728179 0.0364089 0.999337i \(-0.488408\pi\)
0.0364089 + 0.999337i \(0.488408\pi\)
\(102\) 0 0
\(103\) 3.10407i 0.305853i 0.988238 + 0.152926i \(0.0488698\pi\)
−0.988238 + 0.152926i \(0.951130\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 14.1731i 1.37016i 0.728466 + 0.685082i \(0.240234\pi\)
−0.728466 + 0.685082i \(0.759766\pi\)
\(108\) 0 0
\(109\) 9.61844 0.921279 0.460640 0.887587i \(-0.347620\pi\)
0.460640 + 0.887587i \(0.347620\pi\)
\(110\) 0 0
\(111\) 2.48563 0.235925
\(112\) 0 0
\(113\) − 8.77733i − 0.825701i −0.910799 0.412851i \(-0.864533\pi\)
0.910799 0.412851i \(-0.135467\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 9.35025i − 0.864431i
\(118\) 0 0
\(119\) −0.0820605 −0.00752248
\(120\) 0 0
\(121\) 7.21744 0.656131
\(122\) 0 0
\(123\) − 0.309969i − 0.0279490i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 6.77209i 0.600926i 0.953793 + 0.300463i \(0.0971412\pi\)
−0.953793 + 0.300463i \(0.902859\pi\)
\(128\) 0 0
\(129\) −1.26819 −0.111658
\(130\) 0 0
\(131\) −2.91794 −0.254942 −0.127471 0.991842i \(-0.540686\pi\)
−0.127471 + 0.991842i \(0.540686\pi\)
\(132\) 0 0
\(133\) 0.286462i 0.0248394i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 14.4360i − 1.23335i −0.787216 0.616677i \(-0.788478\pi\)
0.787216 0.616677i \(-0.211522\pi\)
\(138\) 0 0
\(139\) 2.29950 0.195041 0.0975205 0.995234i \(-0.468909\pi\)
0.0975205 + 0.995234i \(0.468909\pi\)
\(140\) 0 0
\(141\) −0.217440 −0.0183117
\(142\) 0 0
\(143\) − 13.6770i − 1.14373i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 1.98173i − 0.163450i
\(148\) 0 0
\(149\) −6.75382 −0.553294 −0.276647 0.960972i \(-0.589223\pi\)
−0.276647 + 0.960972i \(0.589223\pi\)
\(150\) 0 0
\(151\) −11.8866 −0.967320 −0.483660 0.875256i \(-0.660693\pi\)
−0.483660 + 0.875256i \(0.660693\pi\)
\(152\) 0 0
\(153\) 0.835879i 0.0675768i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 14.1731i − 1.13114i −0.824702 0.565568i \(-0.808657\pi\)
0.824702 0.565568i \(-0.191343\pi\)
\(158\) 0 0
\(159\) −1.26819 −0.100574
\(160\) 0 0
\(161\) 0.268189 0.0211363
\(162\) 0 0
\(163\) − 7.89443i − 0.618340i −0.951007 0.309170i \(-0.899949\pi\)
0.951007 0.309170i \(-0.100051\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 20.4726i 1.58422i 0.610381 + 0.792108i \(0.291017\pi\)
−0.610381 + 0.792108i \(0.708983\pi\)
\(168\) 0 0
\(169\) 2.73181 0.210139
\(170\) 0 0
\(171\) 2.91794 0.223140
\(172\) 0 0
\(173\) 24.0492i 1.82843i 0.405229 + 0.914215i \(0.367192\pi\)
−0.405229 + 0.914215i \(0.632808\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 1.34651i − 0.101210i
\(178\) 0 0
\(179\) −18.6718 −1.39559 −0.697796 0.716296i \(-0.745836\pi\)
−0.697796 + 0.716296i \(0.745836\pi\)
\(180\) 0 0
\(181\) −3.53638 −0.262857 −0.131428 0.991326i \(-0.541956\pi\)
−0.131428 + 0.991326i \(0.541956\pi\)
\(182\) 0 0
\(183\) − 2.91794i − 0.215700i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 1.22267i 0.0894108i
\(188\) 0 0
\(189\) 0.485629 0.0353243
\(190\) 0 0
\(191\) −14.2682 −1.03241 −0.516205 0.856465i \(-0.672656\pi\)
−0.516205 + 0.856465i \(0.672656\pi\)
\(192\) 0 0
\(193\) 8.18089i 0.588874i 0.955671 + 0.294437i \(0.0951321\pi\)
−0.955671 + 0.294437i \(0.904868\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.6404i 0.900595i 0.892879 + 0.450297i \(0.148682\pi\)
−0.892879 + 0.450297i \(0.851318\pi\)
\(198\) 0 0
\(199\) 4.10407 0.290930 0.145465 0.989363i \(-0.453532\pi\)
0.145465 + 0.989363i \(0.453532\pi\)
\(200\) 0 0
\(201\) 2.81387 0.198475
\(202\) 0 0
\(203\) − 0.649750i − 0.0456035i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 2.73181i − 0.189874i
\(208\) 0 0
\(209\) 4.26819 0.295237
\(210\) 0 0
\(211\) 1.53638 0.105769 0.0528843 0.998601i \(-0.483159\pi\)
0.0528843 + 0.998601i \(0.483159\pi\)
\(212\) 0 0
\(213\) − 1.38530i − 0.0949189i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 1.19917i − 0.0814048i
\(218\) 0 0
\(219\) −3.00000 −0.202721
\(220\) 0 0
\(221\) 0.917939 0.0617473
\(222\) 0 0
\(223\) − 1.40880i − 0.0943404i −0.998887 0.0471702i \(-0.984980\pi\)
0.998887 0.0471702i \(-0.0150203\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 22.9179i 1.52112i 0.649269 + 0.760559i \(0.275075\pi\)
−0.649269 + 0.760559i \(0.724925\pi\)
\(228\) 0 0
\(229\) −3.78513 −0.250128 −0.125064 0.992149i \(-0.539914\pi\)
−0.125064 + 0.992149i \(0.539914\pi\)
\(230\) 0 0
\(231\) 0.350250 0.0230447
\(232\) 0 0
\(233\) − 2.26819i − 0.148594i −0.997236 0.0742970i \(-0.976329\pi\)
0.997236 0.0742970i \(-0.0236713\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1.46212i 0.0949750i
\(238\) 0 0
\(239\) 2.18613 0.141409 0.0707045 0.997497i \(-0.477475\pi\)
0.0707045 + 0.997497i \(0.477475\pi\)
\(240\) 0 0
\(241\) 19.9907 1.28771 0.643857 0.765146i \(-0.277333\pi\)
0.643857 + 0.765146i \(0.277333\pi\)
\(242\) 0 0
\(243\) − 7.45432i − 0.478195i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 3.20440i − 0.203891i
\(248\) 0 0
\(249\) −4.50764 −0.285660
\(250\) 0 0
\(251\) 1.35025 0.0852270 0.0426135 0.999092i \(-0.486432\pi\)
0.0426135 + 0.999092i \(0.486432\pi\)
\(252\) 0 0
\(253\) − 3.99593i − 0.251222i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 11.9582i − 0.745933i −0.927845 0.372967i \(-0.878341\pi\)
0.927845 0.372967i \(-0.121659\pi\)
\(258\) 0 0
\(259\) −2.48563 −0.154449
\(260\) 0 0
\(261\) −6.61844 −0.409671
\(262\) 0 0
\(263\) 8.31370i 0.512645i 0.966591 + 0.256322i \(0.0825109\pi\)
−0.966591 + 0.256322i \(0.917489\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 2.79410i − 0.170996i
\(268\) 0 0
\(269\) 5.02201 0.306197 0.153099 0.988211i \(-0.451075\pi\)
0.153099 + 0.988211i \(0.451075\pi\)
\(270\) 0 0
\(271\) 6.96869 0.423318 0.211659 0.977344i \(-0.432113\pi\)
0.211659 + 0.977344i \(0.432113\pi\)
\(272\) 0 0
\(273\) − 0.262955i − 0.0159148i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 12.7083i − 0.763568i −0.924252 0.381784i \(-0.875310\pi\)
0.924252 0.381784i \(-0.124690\pi\)
\(278\) 0 0
\(279\) −12.2149 −0.731285
\(280\) 0 0
\(281\) −16.5364 −0.986478 −0.493239 0.869894i \(-0.664187\pi\)
−0.493239 + 0.869894i \(0.664187\pi\)
\(282\) 0 0
\(283\) 26.1548i 1.55474i 0.629042 + 0.777371i \(0.283447\pi\)
−0.629042 + 0.777371i \(0.716553\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0.309969i 0.0182969i
\(288\) 0 0
\(289\) 16.9179 0.995173
\(290\) 0 0
\(291\) −2.75382 −0.161432
\(292\) 0 0
\(293\) 3.95449i 0.231023i 0.993306 + 0.115512i \(0.0368508\pi\)
−0.993306 + 0.115512i \(0.963149\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 7.23571i − 0.419859i
\(298\) 0 0
\(299\) −3.00000 −0.173494
\(300\) 0 0
\(301\) 1.26819 0.0730972
\(302\) 0 0
\(303\) − 0.209636i − 0.0120433i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 20.7355i − 1.18344i −0.806144 0.591720i \(-0.798449\pi\)
0.806144 0.591720i \(-0.201551\pi\)
\(308\) 0 0
\(309\) 0.889198 0.0505847
\(310\) 0 0
\(311\) −1.97799 −0.112162 −0.0560808 0.998426i \(-0.517860\pi\)
−0.0560808 + 0.998426i \(0.517860\pi\)
\(312\) 0 0
\(313\) 8.67699i 0.490453i 0.969466 + 0.245226i \(0.0788623\pi\)
−0.969466 + 0.245226i \(0.921138\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 2.34502i − 0.131709i −0.997829 0.0658546i \(-0.979023\pi\)
0.997829 0.0658546i \(-0.0209773\pi\)
\(318\) 0 0
\(319\) −9.68106 −0.542035
\(320\) 0 0
\(321\) 4.06005 0.226610
\(322\) 0 0
\(323\) 0.286462i 0.0159392i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 2.75532i − 0.152369i
\(328\) 0 0
\(329\) 0.217440 0.0119878
\(330\) 0 0
\(331\) −22.8579 −1.25638 −0.628192 0.778059i \(-0.716205\pi\)
−0.628192 + 0.778059i \(0.716205\pi\)
\(332\) 0 0
\(333\) 25.3189i 1.38747i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 0.386795i − 0.0210701i −0.999945 0.0105350i \(-0.996647\pi\)
0.999945 0.0105350i \(-0.00335347\pi\)
\(338\) 0 0
\(339\) −2.51437 −0.136562
\(340\) 0 0
\(341\) −17.8672 −0.967563
\(342\) 0 0
\(343\) 3.98696i 0.215276i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 14.1406i 0.759108i 0.925170 + 0.379554i \(0.123923\pi\)
−0.925170 + 0.379554i \(0.876077\pi\)
\(348\) 0 0
\(349\) 2.81387 0.150623 0.0753115 0.997160i \(-0.476005\pi\)
0.0753115 + 0.997160i \(0.476005\pi\)
\(350\) 0 0
\(351\) −5.43231 −0.289955
\(352\) 0 0
\(353\) 13.3137i 0.708617i 0.935129 + 0.354308i \(0.115284\pi\)
−0.935129 + 0.354308i \(0.884716\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0.0235072i 0.00124413i
\(358\) 0 0
\(359\) −14.1861 −0.748715 −0.374358 0.927284i \(-0.622137\pi\)
−0.374358 + 0.927284i \(0.622137\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) − 2.06752i − 0.108517i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 28.0272i − 1.46301i −0.681836 0.731505i \(-0.738818\pi\)
0.681836 0.731505i \(-0.261182\pi\)
\(368\) 0 0
\(369\) 3.15739 0.164367
\(370\) 0 0
\(371\) 1.26819 0.0658411
\(372\) 0 0
\(373\) 24.3502i 1.26081i 0.776267 + 0.630404i \(0.217111\pi\)
−0.776267 + 0.630404i \(0.782889\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 7.26819i 0.374331i
\(378\) 0 0
\(379\) −28.8866 −1.48381 −0.741903 0.670507i \(-0.766077\pi\)
−0.741903 + 0.670507i \(0.766077\pi\)
\(380\) 0 0
\(381\) 1.93995 0.0993865
\(382\) 0 0
\(383\) 10.3398i 0.528338i 0.964476 + 0.264169i \(0.0850977\pi\)
−0.964476 + 0.264169i \(0.914902\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 12.9179i − 0.656656i
\(388\) 0 0
\(389\) −33.9086 −1.71924 −0.859618 0.510937i \(-0.829298\pi\)
−0.859618 + 0.510937i \(0.829298\pi\)
\(390\) 0 0
\(391\) 0.268189 0.0135629
\(392\) 0 0
\(393\) 0.835879i 0.0421645i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 33.8866i − 1.70072i −0.526200 0.850361i \(-0.676384\pi\)
0.526200 0.850361i \(-0.323616\pi\)
\(398\) 0 0
\(399\) 0.0820605 0.00410816
\(400\) 0 0
\(401\) −10.5051 −0.524598 −0.262299 0.964987i \(-0.584481\pi\)
−0.262299 + 0.964987i \(0.584481\pi\)
\(402\) 0 0
\(403\) 13.4140i 0.668201i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 37.0350i 1.83576i
\(408\) 0 0
\(409\) 6.78256 0.335376 0.167688 0.985840i \(-0.446370\pi\)
0.167688 + 0.985840i \(0.446370\pi\)
\(410\) 0 0
\(411\) −4.13538 −0.203983
\(412\) 0 0
\(413\) 1.34651i 0.0662577i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 0.658720i − 0.0322576i
\(418\) 0 0
\(419\) −18.6498 −0.911100 −0.455550 0.890210i \(-0.650557\pi\)
−0.455550 + 0.890210i \(0.650557\pi\)
\(420\) 0 0
\(421\) 9.64975 0.470300 0.235150 0.971959i \(-0.424442\pi\)
0.235150 + 0.971959i \(0.424442\pi\)
\(422\) 0 0
\(423\) − 2.21487i − 0.107691i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 2.91794i 0.141209i
\(428\) 0 0
\(429\) −3.91794 −0.189160
\(430\) 0 0
\(431\) −33.7758 −1.62692 −0.813462 0.581618i \(-0.802420\pi\)
−0.813462 + 0.581618i \(0.802420\pi\)
\(432\) 0 0
\(433\) − 1.08986i − 0.0523755i −0.999657 0.0261878i \(-0.991663\pi\)
0.999657 0.0261878i \(-0.00833678\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 0.936212i − 0.0447851i
\(438\) 0 0
\(439\) 12.9687 0.618962 0.309481 0.950906i \(-0.399845\pi\)
0.309481 + 0.950906i \(0.399845\pi\)
\(440\) 0 0
\(441\) 20.1861 0.961244
\(442\) 0 0
\(443\) 31.9437i 1.51769i 0.651271 + 0.758845i \(0.274236\pi\)
−0.651271 + 0.758845i \(0.725764\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 1.93471i 0.0915088i
\(448\) 0 0
\(449\) 5.05075 0.238360 0.119180 0.992873i \(-0.461974\pi\)
0.119180 + 0.992873i \(0.461974\pi\)
\(450\) 0 0
\(451\) 4.61844 0.217474
\(452\) 0 0
\(453\) 3.40507i 0.159984i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 35.0948i 1.64166i 0.571170 + 0.820832i \(0.306490\pi\)
−0.571170 + 0.820832i \(0.693510\pi\)
\(458\) 0 0
\(459\) 0.485629 0.0226672
\(460\) 0 0
\(461\) −27.8866 −1.29881 −0.649405 0.760443i \(-0.724982\pi\)
−0.649405 + 0.760443i \(0.724982\pi\)
\(462\) 0 0
\(463\) 28.0037i 1.30144i 0.759316 + 0.650722i \(0.225534\pi\)
−0.759316 + 0.650722i \(0.774466\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 17.4543i 0.807690i 0.914828 + 0.403845i \(0.132326\pi\)
−0.914828 + 0.403845i \(0.867674\pi\)
\(468\) 0 0
\(469\) −2.81387 −0.129933
\(470\) 0 0
\(471\) −4.06005 −0.187077
\(472\) 0 0
\(473\) − 18.8956i − 0.868821i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 12.9179i − 0.591472i
\(478\) 0 0
\(479\) −26.0440 −1.18998 −0.594991 0.803733i \(-0.702844\pi\)
−0.594991 + 0.803733i \(0.702844\pi\)
\(480\) 0 0
\(481\) 27.8046 1.26778
\(482\) 0 0
\(483\) − 0.0768261i − 0.00349571i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 21.9907i − 0.996494i −0.867035 0.498247i \(-0.833977\pi\)
0.867035 0.498247i \(-0.166023\pi\)
\(488\) 0 0
\(489\) −2.26146 −0.102267
\(490\) 0 0
\(491\) −33.9907 −1.53398 −0.766989 0.641660i \(-0.778246\pi\)
−0.766989 + 0.641660i \(0.778246\pi\)
\(492\) 0 0
\(493\) − 0.649750i − 0.0292633i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1.38530i 0.0621390i
\(498\) 0 0
\(499\) −6.83331 −0.305901 −0.152950 0.988234i \(-0.548877\pi\)
−0.152950 + 0.988234i \(0.548877\pi\)
\(500\) 0 0
\(501\) 5.86462 0.262012
\(502\) 0 0
\(503\) − 34.4685i − 1.53688i −0.639925 0.768438i \(-0.721034\pi\)
0.639925 0.768438i \(-0.278966\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 0.782560i − 0.0347547i
\(508\) 0 0
\(509\) −18.9179 −0.838523 −0.419261 0.907866i \(-0.637711\pi\)
−0.419261 + 0.907866i \(0.637711\pi\)
\(510\) 0 0
\(511\) 3.00000 0.132712
\(512\) 0 0
\(513\) − 1.69527i − 0.0748478i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 3.23978i − 0.142485i
\(518\) 0 0
\(519\) 6.88920 0.302402
\(520\) 0 0
\(521\) −1.62774 −0.0713127 −0.0356563 0.999364i \(-0.511352\pi\)
−0.0356563 + 0.999364i \(0.511352\pi\)
\(522\) 0 0
\(523\) 1.16669i 0.0510158i 0.999675 + 0.0255079i \(0.00812030\pi\)
−0.999675 + 0.0255079i \(0.991880\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 1.19917i − 0.0522365i
\(528\) 0 0
\(529\) 22.1235 0.961892
\(530\) 0 0
\(531\) 13.7158 0.595214
\(532\) 0 0
\(533\) − 3.46736i − 0.150188i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 5.34875i 0.230816i
\(538\) 0 0
\(539\) 29.5271 1.27182
\(540\) 0 0
\(541\) −23.9399 −1.02926 −0.514629 0.857413i \(-0.672070\pi\)
−0.514629 + 0.857413i \(0.672070\pi\)
\(542\) 0 0
\(543\) 1.01304i 0.0434736i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 14.8139i 0.633395i 0.948527 + 0.316698i \(0.102574\pi\)
−0.948527 + 0.316698i \(0.897426\pi\)
\(548\) 0 0
\(549\) 29.7225 1.26853
\(550\) 0 0
\(551\) −2.26819 −0.0966281
\(552\) 0 0
\(553\) − 1.46212i − 0.0621757i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 39.3592i − 1.66770i −0.551988 0.833852i \(-0.686131\pi\)
0.551988 0.833852i \(-0.313869\pi\)
\(558\) 0 0
\(559\) −14.1861 −0.600009
\(560\) 0 0
\(561\) 0.350250 0.0147876
\(562\) 0 0
\(563\) − 9.55722i − 0.402789i −0.979510 0.201394i \(-0.935453\pi\)
0.979510 0.201394i \(-0.0645473\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 2.36852i 0.0994686i
\(568\) 0 0
\(569\) −21.2995 −0.892922 −0.446461 0.894803i \(-0.647316\pi\)
−0.446461 + 0.894803i \(0.647316\pi\)
\(570\) 0 0
\(571\) −17.4543 −0.730440 −0.365220 0.930921i \(-0.619006\pi\)
−0.365220 + 0.930921i \(0.619006\pi\)
\(572\) 0 0
\(573\) 4.08729i 0.170749i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 37.2861i − 1.55224i −0.630584 0.776121i \(-0.717185\pi\)
0.630584 0.776121i \(-0.282815\pi\)
\(578\) 0 0
\(579\) 2.34352 0.0973932
\(580\) 0 0
\(581\) 4.50764 0.187008
\(582\) 0 0
\(583\) − 18.8956i − 0.782576i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 7.73181i − 0.319126i −0.987188 0.159563i \(-0.948991\pi\)
0.987188 0.159563i \(-0.0510085\pi\)
\(588\) 0 0
\(589\) −4.18613 −0.172486
\(590\) 0 0
\(591\) 3.62101 0.148948
\(592\) 0 0
\(593\) 6.74858i 0.277131i 0.990353 + 0.138566i \(0.0442492\pi\)
−0.990353 + 0.138566i \(0.955751\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 1.17566i − 0.0481166i
\(598\) 0 0
\(599\) 16.9687 0.693322 0.346661 0.937991i \(-0.387315\pi\)
0.346661 + 0.937991i \(0.387315\pi\)
\(600\) 0 0
\(601\) 31.7251 1.29409 0.647046 0.762451i \(-0.276004\pi\)
0.647046 + 0.762451i \(0.276004\pi\)
\(602\) 0 0
\(603\) 28.6625i 1.16723i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 3.84518i 0.156071i 0.996951 + 0.0780356i \(0.0248648\pi\)
−0.996951 + 0.0780356i \(0.975135\pi\)
\(608\) 0 0
\(609\) −0.186129 −0.00754232
\(610\) 0 0
\(611\) −2.43231 −0.0984007
\(612\) 0 0
\(613\) 32.1208i 1.29735i 0.761066 + 0.648674i \(0.224676\pi\)
−0.761066 + 0.648674i \(0.775324\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 35.2630i 1.41963i 0.704387 + 0.709817i \(0.251222\pi\)
−0.704387 + 0.709817i \(0.748778\pi\)
\(618\) 0 0
\(619\) −9.29020 −0.373405 −0.186702 0.982417i \(-0.559780\pi\)
−0.186702 + 0.982417i \(0.559780\pi\)
\(620\) 0 0
\(621\) −1.58713 −0.0636893
\(622\) 0 0
\(623\) 2.79410i 0.111943i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 1.22267i − 0.0488289i
\(628\) 0 0
\(629\) −2.48563 −0.0991085
\(630\) 0 0
\(631\) −26.5271 −1.05603 −0.528013 0.849236i \(-0.677063\pi\)
−0.528013 + 0.849236i \(0.677063\pi\)
\(632\) 0 0
\(633\) − 0.440114i − 0.0174930i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 22.1679i − 0.878322i
\(638\) 0 0
\(639\) 14.1108 0.558215
\(640\) 0 0
\(641\) −7.86462 −0.310634 −0.155317 0.987865i \(-0.549640\pi\)
−0.155317 + 0.987865i \(0.549640\pi\)
\(642\) 0 0
\(643\) 1.67176i 0.0659277i 0.999457 + 0.0329638i \(0.0104946\pi\)
−0.999457 + 0.0329638i \(0.989505\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 38.4909i 1.51323i 0.653859 + 0.756616i \(0.273149\pi\)
−0.653859 + 0.756616i \(0.726851\pi\)
\(648\) 0 0
\(649\) 20.0626 0.787527
\(650\) 0 0
\(651\) −0.343516 −0.0134634
\(652\) 0 0
\(653\) 25.9985i 1.01740i 0.860944 + 0.508700i \(0.169874\pi\)
−0.860944 + 0.508700i \(0.830126\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 30.5584i − 1.19220i
\(658\) 0 0
\(659\) −46.9594 −1.82928 −0.914639 0.404272i \(-0.867525\pi\)
−0.914639 + 0.404272i \(0.867525\pi\)
\(660\) 0 0
\(661\) −35.7225 −1.38944 −0.694722 0.719278i \(-0.744473\pi\)
−0.694722 + 0.719278i \(0.744473\pi\)
\(662\) 0 0
\(663\) − 0.262955i − 0.0102123i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 2.12351i 0.0822225i
\(668\) 0 0
\(669\) −0.403569 −0.0156029
\(670\) 0 0
\(671\) 43.4763 1.67838
\(672\) 0 0
\(673\) − 27.3450i − 1.05407i −0.849843 0.527036i \(-0.823303\pi\)
0.849843 0.527036i \(-0.176697\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 39.9944i − 1.53711i −0.639783 0.768555i \(-0.720976\pi\)
0.639783 0.768555i \(-0.279024\pi\)
\(678\) 0 0
\(679\) 2.75382 0.105682
\(680\) 0 0
\(681\) 6.56512 0.251576
\(682\) 0 0
\(683\) − 21.0130i − 0.804042i −0.915631 0.402021i \(-0.868308\pi\)
0.915631 0.402021i \(-0.131692\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 1.08430i 0.0413685i
\(688\) 0 0
\(689\) −14.1861 −0.540448
\(690\) 0 0
\(691\) −10.7445 −0.408741 −0.204370 0.978894i \(-0.565515\pi\)
−0.204370 + 0.978894i \(0.565515\pi\)
\(692\) 0 0
\(693\) 3.56769i 0.135525i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0.309969i 0.0117409i
\(698\) 0 0
\(699\) −0.649750 −0.0245758
\(700\) 0 0
\(701\) 42.3029 1.59776 0.798879 0.601491i \(-0.205427\pi\)
0.798879 + 0.601491i \(0.205427\pi\)
\(702\) 0 0
\(703\) 8.67699i 0.327259i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0.209636i 0.00788417i
\(708\) 0 0
\(709\) 40.5584 1.52320 0.761601 0.648046i \(-0.224414\pi\)
0.761601 + 0.648046i \(0.224414\pi\)
\(710\) 0 0
\(711\) −14.8934 −0.558545
\(712\) 0 0
\(713\) 3.91911i 0.146772i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 0.626243i − 0.0233875i
\(718\) 0 0
\(719\) −16.8866 −0.629765 −0.314882 0.949131i \(-0.601965\pi\)
−0.314882 + 0.949131i \(0.601965\pi\)
\(720\) 0 0
\(721\) −0.889198 −0.0331155
\(722\) 0 0
\(723\) − 5.72658i − 0.212974i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 27.4398i 1.01769i 0.860860 + 0.508843i \(0.169926\pi\)
−0.860860 + 0.508843i \(0.830074\pi\)
\(728\) 0 0
\(729\) 22.6692 0.839600
\(730\) 0 0
\(731\) 1.26819 0.0469057
\(732\) 0 0
\(733\) 14.0078i 0.517390i 0.965959 + 0.258695i \(0.0832925\pi\)
−0.965959 + 0.258695i \(0.916708\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 41.9257i 1.54435i
\(738\) 0 0
\(739\) 19.7851 0.727808 0.363904 0.931437i \(-0.381444\pi\)
0.363904 + 0.931437i \(0.381444\pi\)
\(740\) 0 0
\(741\) −0.917939 −0.0337213
\(742\) 0 0
\(743\) − 41.8344i − 1.53475i −0.641196 0.767377i \(-0.721561\pi\)
0.641196 0.767377i \(-0.278439\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 45.9154i − 1.67996i
\(748\) 0 0
\(749\) −4.06005 −0.148351
\(750\) 0 0
\(751\) 48.4983 1.76973 0.884865 0.465848i \(-0.154251\pi\)
0.884865 + 0.465848i \(0.154251\pi\)
\(752\) 0 0
\(753\) − 0.386795i − 0.0140956i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 32.2537i 1.17228i 0.810210 + 0.586139i \(0.199353\pi\)
−0.810210 + 0.586139i \(0.800647\pi\)
\(758\) 0 0
\(759\) −1.14468 −0.0415493
\(760\) 0 0
\(761\) −25.6184 −0.928668 −0.464334 0.885660i \(-0.653706\pi\)
−0.464334 + 0.885660i \(0.653706\pi\)
\(762\) 0 0
\(763\) 2.75532i 0.0997492i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 15.0623i − 0.543868i
\(768\) 0 0
\(769\) −42.3096 −1.52572 −0.762862 0.646561i \(-0.776207\pi\)
−0.762862 + 0.646561i \(0.776207\pi\)
\(770\) 0 0
\(771\) −3.42558 −0.123369
\(772\) 0 0
\(773\) 0.889198i 0.0319822i 0.999872 + 0.0159911i \(0.00509035\pi\)
−0.999872 + 0.0159911i \(0.994910\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0.712038i 0.0255442i
\(778\) 0 0
\(779\) 1.08206 0.0387688
\(780\) 0 0
\(781\) 20.6404 0.738573
\(782\) 0 0
\(783\) 3.84518i 0.137416i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 50.6650i 1.80601i 0.429627 + 0.903007i \(0.358645\pi\)
−0.429627 + 0.903007i \(0.641355\pi\)
\(788\) 0 0
\(789\) 2.38156 0.0847858
\(790\) 0 0
\(791\) 2.51437 0.0894007
\(792\) 0 0
\(793\) − 32.6404i − 1.15910i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 33.4909i 1.18631i 0.805089 + 0.593154i \(0.202117\pi\)
−0.805089 + 0.593154i \(0.797883\pi\)
\(798\) 0 0
\(799\) 0.217440 0.00769247
\(800\) 0 0
\(801\) 28.4611 1.00562
\(802\) 0 0
\(803\) − 44.6990i − 1.57739i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 1.43861i − 0.0506416i
\(808\) 0 0
\(809\) 6.35025 0.223263 0.111631 0.993750i \(-0.464392\pi\)
0.111631 + 0.993750i \(0.464392\pi\)
\(810\) 0 0
\(811\) −1.08463 −0.0380865 −0.0190433 0.999819i \(-0.506062\pi\)
−0.0190433 + 0.999819i \(0.506062\pi\)
\(812\) 0 0
\(813\) − 1.99627i − 0.0700121i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 4.42708i − 0.154884i
\(818\) 0 0
\(819\) 2.67849 0.0935941
\(820\) 0 0
\(821\) −12.3630 −0.431470 −0.215735 0.976452i \(-0.569215\pi\)
−0.215735 + 0.976452i \(0.569215\pi\)
\(822\) 0 0
\(823\) − 27.0765i − 0.943827i −0.881645 0.471914i \(-0.843563\pi\)
0.881645 0.471914i \(-0.156437\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 17.0765i − 0.593808i −0.954907 0.296904i \(-0.904046\pi\)
0.954907 0.296904i \(-0.0959541\pi\)
\(828\) 0 0
\(829\) 28.5804 0.992638 0.496319 0.868140i \(-0.334685\pi\)
0.496319 + 0.868140i \(0.334685\pi\)
\(830\) 0 0
\(831\) −3.64045 −0.126286
\(832\) 0 0
\(833\) 1.98173i 0.0686628i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 7.09660i 0.245294i
\(838\) 0 0
\(839\) 55.9019 1.92995 0.964974 0.262346i \(-0.0844960\pi\)
0.964974 + 0.262346i \(0.0844960\pi\)
\(840\) 0 0
\(841\) −23.8553 −0.822597
\(842\) 0 0
\(843\) 4.73705i 0.163153i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 2.06752i 0.0710409i
\(848\) 0 0
\(849\) 7.49236 0.257137
\(850\) 0 0
\(851\) 8.12351 0.278470
\(852\) 0 0
\(853\) 4.93214i 0.168873i 0.996429 + 0.0844367i \(0.0269091\pi\)
−0.996429 + 0.0844367i \(0.973091\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 39.9411i 1.36436i 0.731183 + 0.682181i \(0.238968\pi\)
−0.731183 + 0.682181i \(0.761032\pi\)
\(858\) 0 0
\(859\) −1.99070 −0.0679217 −0.0339608 0.999423i \(-0.510812\pi\)
−0.0339608 + 0.999423i \(0.510812\pi\)
\(860\) 0 0
\(861\) 0.0887944 0.00302611
\(862\) 0 0
\(863\) − 31.9817i − 1.08867i −0.838868 0.544335i \(-0.816782\pi\)
0.838868 0.544335i \(-0.183218\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 4.84635i − 0.164591i
\(868\) 0 0
\(869\) −21.7851 −0.739010
\(870\) 0 0
\(871\) 31.4763 1.06653
\(872\) 0 0
\(873\) − 28.0507i − 0.949374i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 22.2917i − 0.752737i −0.926470 0.376369i \(-0.877173\pi\)
0.926470 0.376369i \(-0.122827\pi\)
\(878\) 0 0
\(879\) 1.13281 0.0382087
\(880\) 0 0
\(881\) 35.6912 1.20247 0.601233 0.799073i \(-0.294676\pi\)
0.601233 + 0.799073i \(0.294676\pi\)
\(882\) 0 0
\(883\) − 17.9948i − 0.605572i −0.953059 0.302786i \(-0.902083\pi\)
0.953059 0.302786i \(-0.0979168\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 24.3566i 0.817813i 0.912576 + 0.408907i \(0.134090\pi\)
−0.912576 + 0.408907i \(0.865910\pi\)
\(888\) 0 0
\(889\) −1.93995 −0.0650637
\(890\) 0 0
\(891\) 35.2902 1.18227
\(892\) 0 0
\(893\) − 0.759053i − 0.0254007i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0.859386i 0.0286941i
\(898\) 0 0
\(899\) 9.49493 0.316674
\(900\) 0 0
\(901\) 1.26819 0.0422495
\(902\) 0 0
\(903\) − 0.363288i − 0.0120895i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 5.73821i 0.190534i 0.995452 + 0.0952671i \(0.0303705\pi\)
−0.995452 + 0.0952671i \(0.969629\pi\)
\(908\) 0 0
\(909\) 2.13538 0.0708261
\(910\) 0 0
\(911\) −46.8291 −1.55152 −0.775759 0.631029i \(-0.782633\pi\)
−0.775759 + 0.631029i \(0.782633\pi\)
\(912\) 0 0
\(913\) − 67.1623i − 2.22275i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 0.835879i − 0.0276032i
\(918\) 0 0
\(919\) −2.32151 −0.0765795 −0.0382897 0.999267i \(-0.512191\pi\)
−0.0382897 + 0.999267i \(0.512191\pi\)
\(920\) 0 0
\(921\) −5.93995 −0.195728
\(922\) 0 0
\(923\) − 15.4961i − 0.510060i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 9.05748i 0.297487i
\(928\) 0 0
\(929\) 25.5610 0.838628 0.419314 0.907841i \(-0.362271\pi\)
0.419314 + 0.907841i \(0.362271\pi\)
\(930\) 0 0
\(931\) 6.91794 0.226726
\(932\) 0 0
\(933\) 0.566620i 0.0185503i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 25.7501i − 0.841219i −0.907242 0.420609i \(-0.861816\pi\)
0.907242 0.420609i \(-0.138184\pi\)
\(938\) 0 0
\(939\) 2.48563 0.0811154
\(940\) 0 0
\(941\) −29.1261 −0.949483 −0.474741 0.880125i \(-0.657458\pi\)
−0.474741 + 0.880125i \(0.657458\pi\)
\(942\) 0 0
\(943\) − 1.01304i − 0.0329891i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 6.21711i − 0.202029i −0.994885 0.101014i \(-0.967791\pi\)
0.994885 0.101014i \(-0.0322088\pi\)
\(948\) 0 0
\(949\) −33.5584 −1.08935
\(950\) 0 0
\(951\) −0.671758 −0.0217832
\(952\) 0 0
\(953\) − 47.8266i − 1.54925i −0.632418 0.774627i \(-0.717937\pi\)
0.632418 0.774627i \(-0.282063\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 2.77326i 0.0896467i
\(958\) 0 0
\(959\) 4.13538 0.133538
\(960\) 0 0
\(961\) −13.4763 −0.434720
\(962\) 0 0
\(963\) 41.3562i 1.33269i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 37.2301i − 1.19724i −0.801033 0.598620i \(-0.795716\pi\)
0.801033 0.598620i \(-0.204284\pi\)
\(968\) 0 0
\(969\) 0.0820605 0.00263616
\(970\) 0 0
\(971\) 37.8359 1.21421 0.607106 0.794621i \(-0.292331\pi\)
0.607106 + 0.794621i \(0.292331\pi\)
\(972\) 0 0
\(973\) 0.658720i 0.0211176i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 55.8684i 1.78739i 0.448678 + 0.893694i \(0.351895\pi\)
−0.448678 + 0.893694i \(0.648105\pi\)
\(978\) 0 0
\(979\) 41.6311 1.33054
\(980\) 0 0
\(981\) 28.0660 0.896079
\(982\) 0 0
\(983\) 6.93738i 0.221268i 0.993861 + 0.110634i \(0.0352881\pi\)
−0.993861 + 0.110634i \(0.964712\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 0.0622883i − 0.00198266i
\(988\) 0 0
\(989\) −4.14468 −0.131793
\(990\) 0 0
\(991\) 36.5897 1.16231 0.581155 0.813793i \(-0.302601\pi\)
0.581155 + 0.813793i \(0.302601\pi\)
\(992\) 0 0
\(993\) 6.54792i 0.207792i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 33.7538i 1.06899i 0.845170 + 0.534497i \(0.179499\pi\)
−0.845170 + 0.534497i \(0.820501\pi\)
\(998\) 0 0
\(999\) 14.7098 0.465398
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 1900.2.c.g.1749.3 6
5.2 odd 4 1900.2.a.f.1.2 3
5.3 odd 4 1900.2.a.h.1.2 yes 3
5.4 even 2 inner 1900.2.c.g.1749.4 6
20.3 even 4 7600.2.a.bj.1.2 3
20.7 even 4 7600.2.a.by.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1900.2.a.f.1.2 3 5.2 odd 4
1900.2.a.h.1.2 yes 3 5.3 odd 4
1900.2.c.g.1749.3 6 1.1 even 1 trivial
1900.2.c.g.1749.4 6 5.4 even 2 inner
7600.2.a.bj.1.2 3 20.3 even 4
7600.2.a.by.1.2 3 20.7 even 4