Properties

Label 1900.2.c.g.1749.1
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 1749.1
Root \(-1.91223i\) of defining polynomial
Character \(\chi\) \(=\) 1900.1749
Dual form 1900.2.c.g.1749.6

$q$-expansion

\(f(q)\) \(=\) \(q-2.91223i q^{3} +2.91223i q^{7} -5.48108 q^{9} +O(q^{10})\) \(q-2.91223i q^{3} +2.91223i q^{7} -5.48108 q^{9} +0.598988 q^{11} +2.56885i q^{13} +2.91223i q^{17} +1.00000 q^{19} +8.48108 q^{21} +1.16784i q^{23} +7.22547i q^{27} +1.40101 q^{29} +7.88209 q^{31} -1.74439i q^{33} -6.53871i q^{37} +7.48108 q^{39} +9.48108 q^{41} +0.824458i q^{43} +6.05763i q^{47} -1.48108 q^{49} +8.48108 q^{51} +0.824458i q^{53} -2.91223i q^{57} +14.1601 q^{59} -1.88209 q^{61} -15.9622i q^{63} +5.11021i q^{67} +3.40101 q^{69} -11.9622 q^{71} -1.03014i q^{73} +1.74439i q^{77} +15.3632 q^{79} +4.59899 q^{81} +15.7565i q^{83} -4.08007i q^{87} -15.4432 q^{89} -7.48108 q^{91} -22.9545i q^{93} +7.70655i q^{97} -3.28310 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 2q^{9} + O(q^{10}) \) \( 6q - 2q^{9} - 2q^{11} + 6q^{19} + 20q^{21} + 14q^{29} + 22q^{31} + 14q^{39} + 26q^{41} + 22q^{49} + 20q^{51} + 12q^{59} + 14q^{61} + 26q^{69} - 10q^{71} + 36q^{79} + 22q^{81} - 14q^{91} + O(q^{100}) \)

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
\(3\) − 2.91223i − 1.68138i −0.541520 0.840688i \(-0.682151\pi\)
0.541520 0.840688i \(-0.317849\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.91223i 1.10072i 0.834928 + 0.550360i \(0.185509\pi\)
−0.834928 + 0.550360i \(0.814491\pi\)
\(8\) 0 0
\(9\) −5.48108 −1.82703
\(10\) 0 0
\(11\) 0.598988 0.180602 0.0903009 0.995915i \(-0.471217\pi\)
0.0903009 + 0.995915i \(0.471217\pi\)
\(12\) 0 0
\(13\) 2.56885i 0.712471i 0.934396 + 0.356235i \(0.115940\pi\)
−0.934396 + 0.356235i \(0.884060\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.91223i 0.706319i 0.935563 + 0.353160i \(0.114893\pi\)
−0.935563 + 0.353160i \(0.885107\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 8.48108 1.85072
\(22\) 0 0
\(23\) 1.16784i 0.243511i 0.992560 + 0.121756i \(0.0388524\pi\)
−0.992560 + 0.121756i \(0.961148\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 7.22547i 1.39054i
\(28\) 0 0
\(29\) 1.40101 0.260161 0.130081 0.991503i \(-0.458476\pi\)
0.130081 + 0.991503i \(0.458476\pi\)
\(30\) 0 0
\(31\) 7.88209 1.41567 0.707833 0.706380i \(-0.249673\pi\)
0.707833 + 0.706380i \(0.249673\pi\)
\(32\) 0 0
\(33\) − 1.74439i − 0.303660i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 6.53871i − 1.07496i −0.843277 0.537479i \(-0.819377\pi\)
0.843277 0.537479i \(-0.180623\pi\)
\(38\) 0 0
\(39\) 7.48108 1.19793
\(40\) 0 0
\(41\) 9.48108 1.48070 0.740348 0.672224i \(-0.234661\pi\)
0.740348 + 0.672224i \(0.234661\pi\)
\(42\) 0 0
\(43\) 0.824458i 0.125729i 0.998022 + 0.0628644i \(0.0200236\pi\)
−0.998022 + 0.0628644i \(0.979976\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.05763i 0.883596i 0.897114 + 0.441798i \(0.145659\pi\)
−0.897114 + 0.441798i \(0.854341\pi\)
\(48\) 0 0
\(49\) −1.48108 −0.211583
\(50\) 0 0
\(51\) 8.48108 1.18759
\(52\) 0 0
\(53\) 0.824458i 0.113248i 0.998396 + 0.0566240i \(0.0180336\pi\)
−0.998396 + 0.0566240i \(0.981966\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 2.91223i − 0.385734i
\(58\) 0 0
\(59\) 14.1601 1.84349 0.921746 0.387794i \(-0.126763\pi\)
0.921746 + 0.387794i \(0.126763\pi\)
\(60\) 0 0
\(61\) −1.88209 −0.240977 −0.120488 0.992715i \(-0.538446\pi\)
−0.120488 + 0.992715i \(0.538446\pi\)
\(62\) 0 0
\(63\) − 15.9622i − 2.01104i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 5.11021i 0.624311i 0.950031 + 0.312156i \(0.101051\pi\)
−0.950031 + 0.312156i \(0.898949\pi\)
\(68\) 0 0
\(69\) 3.40101 0.409434
\(70\) 0 0
\(71\) −11.9622 −1.41965 −0.709823 0.704380i \(-0.751225\pi\)
−0.709823 + 0.704380i \(0.751225\pi\)
\(72\) 0 0
\(73\) − 1.03014i − 0.120569i −0.998181 0.0602843i \(-0.980799\pi\)
0.998181 0.0602843i \(-0.0192007\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.74439i 0.198792i
\(78\) 0 0
\(79\) 15.3632 1.72849 0.864246 0.503070i \(-0.167796\pi\)
0.864246 + 0.503070i \(0.167796\pi\)
\(80\) 0 0
\(81\) 4.59899 0.510999
\(82\) 0 0
\(83\) 15.7565i 1.72950i 0.502204 + 0.864749i \(0.332523\pi\)
−0.502204 + 0.864749i \(0.667477\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 4.08007i − 0.437429i
\(88\) 0 0
\(89\) −15.4432 −1.63698 −0.818490 0.574521i \(-0.805188\pi\)
−0.818490 + 0.574521i \(0.805188\pi\)
\(90\) 0 0
\(91\) −7.48108 −0.784230
\(92\) 0 0
\(93\) − 22.9545i − 2.38027i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 7.70655i 0.782481i 0.920288 + 0.391241i \(0.127954\pi\)
−0.920288 + 0.391241i \(0.872046\pi\)
\(98\) 0 0
\(99\) −3.28310 −0.329964
\(100\) 0 0
\(101\) 4.40101 0.437917 0.218959 0.975734i \(-0.429734\pi\)
0.218959 + 0.975734i \(0.429734\pi\)
\(102\) 0 0
\(103\) − 17.3632i − 1.71084i −0.517932 0.855422i \(-0.673298\pi\)
0.517932 0.855422i \(-0.326702\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 14.1903i 1.37183i 0.727684 + 0.685913i \(0.240597\pi\)
−0.727684 + 0.685913i \(0.759403\pi\)
\(108\) 0 0
\(109\) 10.6791 1.02287 0.511434 0.859323i \(-0.329114\pi\)
0.511434 + 0.859323i \(0.329114\pi\)
\(110\) 0 0
\(111\) −19.0422 −1.80741
\(112\) 0 0
\(113\) − 8.25561i − 0.776622i −0.921528 0.388311i \(-0.873059\pi\)
0.921528 0.388311i \(-0.126941\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 14.0801i − 1.30170i
\(118\) 0 0
\(119\) −8.48108 −0.777459
\(120\) 0 0
\(121\) −10.6412 −0.967383
\(122\) 0 0
\(123\) − 27.6111i − 2.48961i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 12.1300i − 1.07636i −0.842829 0.538182i \(-0.819111\pi\)
0.842829 0.538182i \(-0.180889\pi\)
\(128\) 0 0
\(129\) 2.40101 0.211397
\(130\) 0 0
\(131\) 5.48108 0.478884 0.239442 0.970911i \(-0.423035\pi\)
0.239442 + 0.970911i \(0.423035\pi\)
\(132\) 0 0
\(133\) 2.91223i 0.252522i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 7.59634i 0.648999i 0.945886 + 0.324500i \(0.105196\pi\)
−0.945886 + 0.324500i \(0.894804\pi\)
\(138\) 0 0
\(139\) −7.16013 −0.607315 −0.303657 0.952781i \(-0.598208\pi\)
−0.303657 + 0.952781i \(0.598208\pi\)
\(140\) 0 0
\(141\) 17.6412 1.48566
\(142\) 0 0
\(143\) 1.53871i 0.128673i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 4.31324i 0.355750i
\(148\) 0 0
\(149\) 18.4432 1.51093 0.755464 0.655190i \(-0.227411\pi\)
0.755464 + 0.655190i \(0.227411\pi\)
\(150\) 0 0
\(151\) −9.27804 −0.755036 −0.377518 0.926002i \(-0.623222\pi\)
−0.377518 + 0.926002i \(0.623222\pi\)
\(152\) 0 0
\(153\) − 15.9622i − 1.29046i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 14.1903i − 1.13251i −0.824231 0.566254i \(-0.808392\pi\)
0.824231 0.566254i \(-0.191608\pi\)
\(158\) 0 0
\(159\) 2.40101 0.190413
\(160\) 0 0
\(161\) −3.40101 −0.268037
\(162\) 0 0
\(163\) 25.1799i 1.97224i 0.166023 + 0.986122i \(0.446907\pi\)
−0.166023 + 0.986122i \(0.553093\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 11.0301i 0.853538i 0.904361 + 0.426769i \(0.140348\pi\)
−0.904361 + 0.426769i \(0.859652\pi\)
\(168\) 0 0
\(169\) 6.40101 0.492386
\(170\) 0 0
\(171\) −5.48108 −0.419149
\(172\) 0 0
\(173\) − 15.3029i − 1.16346i −0.813383 0.581729i \(-0.802377\pi\)
0.813383 0.581729i \(-0.197623\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 41.2376i − 3.09960i
\(178\) 0 0
\(179\) 14.9243 1.11550 0.557748 0.830010i \(-0.311666\pi\)
0.557748 + 0.830010i \(0.311666\pi\)
\(180\) 0 0
\(181\) 3.80202 0.282602 0.141301 0.989967i \(-0.454871\pi\)
0.141301 + 0.989967i \(0.454871\pi\)
\(182\) 0 0
\(183\) 5.48108i 0.405173i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 1.74439i 0.127563i
\(188\) 0 0
\(189\) −21.0422 −1.53060
\(190\) 0 0
\(191\) −10.5990 −0.766916 −0.383458 0.923558i \(-0.625267\pi\)
−0.383458 + 0.923558i \(0.625267\pi\)
\(192\) 0 0
\(193\) − 22.2677i − 1.60286i −0.598086 0.801432i \(-0.704072\pi\)
0.598086 0.801432i \(-0.295928\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 15.1652i − 1.08048i −0.841513 0.540238i \(-0.818334\pi\)
0.841513 0.540238i \(-0.181666\pi\)
\(198\) 0 0
\(199\) −16.3632 −1.15995 −0.579977 0.814633i \(-0.696939\pi\)
−0.579977 + 0.814633i \(0.696939\pi\)
\(200\) 0 0
\(201\) 14.8821 1.04970
\(202\) 0 0
\(203\) 4.08007i 0.286365i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 6.40101i − 0.444901i
\(208\) 0 0
\(209\) 0.598988 0.0414329
\(210\) 0 0
\(211\) −5.80202 −0.399428 −0.199714 0.979854i \(-0.564001\pi\)
−0.199714 + 0.979854i \(0.564001\pi\)
\(212\) 0 0
\(213\) 34.8365i 2.38696i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 22.9545i 1.55825i
\(218\) 0 0
\(219\) −3.00000 −0.202721
\(220\) 0 0
\(221\) −7.48108 −0.503232
\(222\) 0 0
\(223\) 10.1377i 0.678871i 0.940629 + 0.339435i \(0.110236\pi\)
−0.940629 + 0.339435i \(0.889764\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 14.5189i 0.963655i 0.876266 + 0.481827i \(0.160027\pi\)
−0.876266 + 0.481827i \(0.839973\pi\)
\(228\) 0 0
\(229\) 27.2024 1.79758 0.898791 0.438377i \(-0.144446\pi\)
0.898791 + 0.438377i \(0.144446\pi\)
\(230\) 0 0
\(231\) 5.08007 0.334244
\(232\) 0 0
\(233\) 1.40101i 0.0917833i 0.998946 + 0.0458917i \(0.0146129\pi\)
−0.998946 + 0.0458917i \(0.985387\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 44.7411i − 2.90624i
\(238\) 0 0
\(239\) −9.88209 −0.639219 −0.319610 0.947549i \(-0.603552\pi\)
−0.319610 + 0.947549i \(0.603552\pi\)
\(240\) 0 0
\(241\) −3.08513 −0.198730 −0.0993652 0.995051i \(-0.531681\pi\)
−0.0993652 + 0.995051i \(0.531681\pi\)
\(242\) 0 0
\(243\) 8.28310i 0.531361i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 2.56885i 0.163452i
\(248\) 0 0
\(249\) 45.8865 2.90794
\(250\) 0 0
\(251\) 6.08007 0.383770 0.191885 0.981417i \(-0.438540\pi\)
0.191885 + 0.981417i \(0.438540\pi\)
\(252\) 0 0
\(253\) 0.699521i 0.0439785i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 19.0121i 1.18594i 0.805224 + 0.592971i \(0.202045\pi\)
−0.805224 + 0.592971i \(0.797955\pi\)
\(258\) 0 0
\(259\) 19.0422 1.18323
\(260\) 0 0
\(261\) −7.67906 −0.475322
\(262\) 0 0
\(263\) 0.453585i 0.0279693i 0.999902 + 0.0139846i \(0.00445159\pi\)
−0.999902 + 0.0139846i \(0.995548\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 44.9742i 2.75238i
\(268\) 0 0
\(269\) −23.8442 −1.45381 −0.726905 0.686738i \(-0.759042\pi\)
−0.726905 + 0.686738i \(0.759042\pi\)
\(270\) 0 0
\(271\) 12.7591 0.775061 0.387531 0.921857i \(-0.373328\pi\)
0.387531 + 0.921857i \(0.373328\pi\)
\(272\) 0 0
\(273\) 21.7866i 1.31859i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 8.29783i 0.498568i 0.968430 + 0.249284i \(0.0801953\pi\)
−0.968430 + 0.249284i \(0.919805\pi\)
\(278\) 0 0
\(279\) −43.2024 −2.58646
\(280\) 0 0
\(281\) −9.19798 −0.548705 −0.274353 0.961629i \(-0.588464\pi\)
−0.274353 + 0.961629i \(0.588464\pi\)
\(282\) 0 0
\(283\) 19.8770i 1.18157i 0.806830 + 0.590783i \(0.201181\pi\)
−0.806830 + 0.590783i \(0.798819\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 27.6111i 1.62983i
\(288\) 0 0
\(289\) 8.51892 0.501113
\(290\) 0 0
\(291\) 22.4432 1.31565
\(292\) 0 0
\(293\) 8.14540i 0.475860i 0.971282 + 0.237930i \(0.0764688\pi\)
−0.971282 + 0.237930i \(0.923531\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 4.32797i 0.251134i
\(298\) 0 0
\(299\) −3.00000 −0.173494
\(300\) 0 0
\(301\) −2.40101 −0.138392
\(302\) 0 0
\(303\) − 12.8168i − 0.736303i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 10.7565i 0.613905i 0.951725 + 0.306952i \(0.0993092\pi\)
−0.951725 + 0.306952i \(0.900691\pi\)
\(308\) 0 0
\(309\) −50.5655 −2.87657
\(310\) 0 0
\(311\) −30.8442 −1.74902 −0.874508 0.485010i \(-0.838816\pi\)
−0.874508 + 0.485010i \(0.838816\pi\)
\(312\) 0 0
\(313\) − 6.53871i − 0.369590i −0.982777 0.184795i \(-0.940838\pi\)
0.982777 0.184795i \(-0.0591621\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 11.3055i 0.634982i 0.948261 + 0.317491i \(0.102840\pi\)
−0.948261 + 0.317491i \(0.897160\pi\)
\(318\) 0 0
\(319\) 0.839190 0.0469856
\(320\) 0 0
\(321\) 41.3253 2.30655
\(322\) 0 0
\(323\) 2.91223i 0.162041i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 31.0999i − 1.71983i
\(328\) 0 0
\(329\) −17.6412 −0.972592
\(330\) 0 0
\(331\) 22.8064 1.25355 0.626777 0.779199i \(-0.284374\pi\)
0.626777 + 0.779199i \(0.284374\pi\)
\(332\) 0 0
\(333\) 35.8392i 1.96398i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 17.7065i − 0.964537i −0.876023 0.482269i \(-0.839813\pi\)
0.876023 0.482269i \(-0.160187\pi\)
\(338\) 0 0
\(339\) −24.0422 −1.30579
\(340\) 0 0
\(341\) 4.72128 0.255672
\(342\) 0 0
\(343\) 16.0724i 0.867826i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6.26331i 0.336232i 0.985767 + 0.168116i \(0.0537684\pi\)
−0.985767 + 0.168116i \(0.946232\pi\)
\(348\) 0 0
\(349\) 14.8821 0.796620 0.398310 0.917251i \(-0.369597\pi\)
0.398310 + 0.917251i \(0.369597\pi\)
\(350\) 0 0
\(351\) −18.5611 −0.990721
\(352\) 0 0
\(353\) 5.45359i 0.290265i 0.989412 + 0.145133i \(0.0463609\pi\)
−0.989412 + 0.145133i \(0.953639\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 24.6988i 1.30720i
\(358\) 0 0
\(359\) −2.11791 −0.111779 −0.0558895 0.998437i \(-0.517799\pi\)
−0.0558895 + 0.998437i \(0.517799\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 30.9897i 1.62653i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 17.5414i − 0.915651i −0.889042 0.457826i \(-0.848628\pi\)
0.889042 0.457826i \(-0.151372\pi\)
\(368\) 0 0
\(369\) −51.9665 −2.70527
\(370\) 0 0
\(371\) −2.40101 −0.124654
\(372\) 0 0
\(373\) 29.0801i 1.50571i 0.658187 + 0.752854i \(0.271324\pi\)
−0.658187 + 0.752854i \(0.728676\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3.59899i 0.185357i
\(378\) 0 0
\(379\) −26.2780 −1.34981 −0.674906 0.737904i \(-0.735816\pi\)
−0.674906 + 0.737904i \(0.735816\pi\)
\(380\) 0 0
\(381\) −35.3253 −1.80977
\(382\) 0 0
\(383\) − 21.6911i − 1.10837i −0.832395 0.554183i \(-0.813031\pi\)
0.832395 0.554183i \(-0.186969\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 4.51892i − 0.229710i
\(388\) 0 0
\(389\) −2.43380 −0.123398 −0.0616992 0.998095i \(-0.519652\pi\)
−0.0616992 + 0.998095i \(0.519652\pi\)
\(390\) 0 0
\(391\) −3.40101 −0.171997
\(392\) 0 0
\(393\) − 15.9622i − 0.805184i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 31.2780i − 1.56980i −0.619622 0.784900i \(-0.712714\pi\)
0.619622 0.784900i \(-0.287286\pi\)
\(398\) 0 0
\(399\) 8.48108 0.424585
\(400\) 0 0
\(401\) −8.95710 −0.447296 −0.223648 0.974670i \(-0.571797\pi\)
−0.223648 + 0.974670i \(0.571797\pi\)
\(402\) 0 0
\(403\) 20.2479i 1.00862i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 3.91661i − 0.194139i
\(408\) 0 0
\(409\) 24.6412 1.21843 0.609215 0.793005i \(-0.291485\pi\)
0.609215 + 0.793005i \(0.291485\pi\)
\(410\) 0 0
\(411\) 22.1223 1.09121
\(412\) 0 0
\(413\) 41.2376i 2.02917i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 20.8520i 1.02112i
\(418\) 0 0
\(419\) −13.9199 −0.680033 −0.340017 0.940419i \(-0.610433\pi\)
−0.340017 + 0.940419i \(0.610433\pi\)
\(420\) 0 0
\(421\) 4.91993 0.239783 0.119891 0.992787i \(-0.461745\pi\)
0.119891 + 0.992787i \(0.461745\pi\)
\(422\) 0 0
\(423\) − 33.2024i − 1.61435i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 5.48108i − 0.265248i
\(428\) 0 0
\(429\) 4.48108 0.216349
\(430\) 0 0
\(431\) 20.2875 0.977214 0.488607 0.872504i \(-0.337505\pi\)
0.488607 + 0.872504i \(0.337505\pi\)
\(432\) 0 0
\(433\) 20.9769i 1.00808i 0.863679 + 0.504042i \(0.168155\pi\)
−0.863679 + 0.504042i \(0.831845\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.16784i 0.0558653i
\(438\) 0 0
\(439\) 18.7591 0.895324 0.447662 0.894203i \(-0.352257\pi\)
0.447662 + 0.894203i \(0.352257\pi\)
\(440\) 0 0
\(441\) 8.11791 0.386567
\(442\) 0 0
\(443\) − 40.4828i − 1.92340i −0.274110 0.961698i \(-0.588383\pi\)
0.274110 0.961698i \(-0.411617\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 53.7109i − 2.54044i
\(448\) 0 0
\(449\) 19.2402 0.908001 0.454001 0.891001i \(-0.349996\pi\)
0.454001 + 0.891001i \(0.349996\pi\)
\(450\) 0 0
\(451\) 5.67906 0.267416
\(452\) 0 0
\(453\) 27.0198i 1.26950i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 8.44829i − 0.395195i −0.980283 0.197597i \(-0.936686\pi\)
0.980283 0.197597i \(-0.0633138\pi\)
\(458\) 0 0
\(459\) −21.0422 −0.982167
\(460\) 0 0
\(461\) −25.2780 −1.17732 −0.588658 0.808382i \(-0.700343\pi\)
−0.588658 + 0.808382i \(0.700343\pi\)
\(462\) 0 0
\(463\) − 7.15749i − 0.332637i −0.986072 0.166318i \(-0.946812\pi\)
0.986072 0.166318i \(-0.0531879\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 1.71690i 0.0794486i 0.999211 + 0.0397243i \(0.0126480\pi\)
−0.999211 + 0.0397243i \(0.987352\pi\)
\(468\) 0 0
\(469\) −14.8821 −0.687191
\(470\) 0 0
\(471\) −41.3253 −1.90417
\(472\) 0 0
\(473\) 0.493841i 0.0227068i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 4.51892i − 0.206907i
\(478\) 0 0
\(479\) 31.6885 1.44788 0.723942 0.689861i \(-0.242328\pi\)
0.723942 + 0.689861i \(0.242328\pi\)
\(480\) 0 0
\(481\) 16.7970 0.765876
\(482\) 0 0
\(483\) 9.90453i 0.450672i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 1.08513i 0.0491717i 0.999698 + 0.0245859i \(0.00782671\pi\)
−0.999698 + 0.0245859i \(0.992173\pi\)
\(488\) 0 0
\(489\) 73.3297 3.31608
\(490\) 0 0
\(491\) −10.9149 −0.492581 −0.246291 0.969196i \(-0.579212\pi\)
−0.246291 + 0.969196i \(0.579212\pi\)
\(492\) 0 0
\(493\) 4.08007i 0.183757i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 34.8365i − 1.56263i
\(498\) 0 0
\(499\) −38.8814 −1.74057 −0.870286 0.492547i \(-0.836066\pi\)
−0.870286 + 0.492547i \(0.836066\pi\)
\(500\) 0 0
\(501\) 32.1223 1.43512
\(502\) 0 0
\(503\) − 20.3306i − 0.906497i −0.891384 0.453249i \(-0.850265\pi\)
0.891384 0.453249i \(-0.149735\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 18.6412i − 0.827885i
\(508\) 0 0
\(509\) −10.5189 −0.466243 −0.233121 0.972448i \(-0.574894\pi\)
−0.233121 + 0.972448i \(0.574894\pi\)
\(510\) 0 0
\(511\) 3.00000 0.132712
\(512\) 0 0
\(513\) 7.22547i 0.319012i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 3.62845i 0.159579i
\(518\) 0 0
\(519\) −44.5655 −1.95621
\(520\) 0 0
\(521\) −25.7642 −1.12875 −0.564375 0.825519i \(-0.690883\pi\)
−0.564375 + 0.825519i \(0.690883\pi\)
\(522\) 0 0
\(523\) − 30.8814i − 1.35035i −0.737658 0.675175i \(-0.764068\pi\)
0.737658 0.675175i \(-0.235932\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 22.9545i 0.999912i
\(528\) 0 0
\(529\) 21.6362 0.940702
\(530\) 0 0
\(531\) −77.6128 −3.36811
\(532\) 0 0
\(533\) 24.3555i 1.05495i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 43.4630i − 1.87557i
\(538\) 0 0
\(539\) −0.887149 −0.0382122
\(540\) 0 0
\(541\) 13.3253 0.572901 0.286450 0.958095i \(-0.407525\pi\)
0.286450 + 0.958095i \(0.407525\pi\)
\(542\) 0 0
\(543\) − 11.0724i − 0.475161i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 26.8821i 1.14940i 0.818366 + 0.574698i \(0.194880\pi\)
−0.818366 + 0.574698i \(0.805120\pi\)
\(548\) 0 0
\(549\) 10.3159 0.440271
\(550\) 0 0
\(551\) 1.40101 0.0596851
\(552\) 0 0
\(553\) 44.7411i 1.90258i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 27.3082i − 1.15708i −0.815652 0.578542i \(-0.803622\pi\)
0.815652 0.578542i \(-0.196378\pi\)
\(558\) 0 0
\(559\) −2.11791 −0.0895780
\(560\) 0 0
\(561\) 5.08007 0.214481
\(562\) 0 0
\(563\) 40.3324i 1.69981i 0.526939 + 0.849903i \(0.323340\pi\)
−0.526939 + 0.849903i \(0.676660\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 13.3933i 0.562466i
\(568\) 0 0
\(569\) −11.8399 −0.496353 −0.248176 0.968715i \(-0.579831\pi\)
−0.248176 + 0.968715i \(0.579831\pi\)
\(570\) 0 0
\(571\) −1.71690 −0.0718499 −0.0359250 0.999354i \(-0.511438\pi\)
−0.0359250 + 0.999354i \(0.511438\pi\)
\(572\) 0 0
\(573\) 30.8667i 1.28947i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 0.0552190i − 0.00229880i −0.999999 0.00114940i \(-0.999634\pi\)
0.999999 0.00114940i \(-0.000365865\pi\)
\(578\) 0 0
\(579\) −64.8486 −2.69502
\(580\) 0 0
\(581\) −45.8865 −1.90369
\(582\) 0 0
\(583\) 0.493841i 0.0204528i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 11.4010i − 0.470570i −0.971926 0.235285i \(-0.924398\pi\)
0.971926 0.235285i \(-0.0756024\pi\)
\(588\) 0 0
\(589\) 7.88209 0.324776
\(590\) 0 0
\(591\) −44.1645 −1.81669
\(592\) 0 0
\(593\) − 36.8288i − 1.51238i −0.654353 0.756190i \(-0.727059\pi\)
0.654353 0.756190i \(-0.272941\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 47.6533i 1.95032i
\(598\) 0 0
\(599\) 22.7591 0.929913 0.464956 0.885334i \(-0.346070\pi\)
0.464956 + 0.885334i \(0.346070\pi\)
\(600\) 0 0
\(601\) −36.5277 −1.49000 −0.744998 0.667067i \(-0.767549\pi\)
−0.744998 + 0.667067i \(0.767549\pi\)
\(602\) 0 0
\(603\) − 28.0094i − 1.14063i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 10.1230i 0.410879i 0.978670 + 0.205439i \(0.0658623\pi\)
−0.978670 + 0.205439i \(0.934138\pi\)
\(608\) 0 0
\(609\) 11.8821 0.481487
\(610\) 0 0
\(611\) −15.5611 −0.629537
\(612\) 0 0
\(613\) − 35.5930i − 1.43759i −0.695223 0.718794i \(-0.744694\pi\)
0.695223 0.718794i \(-0.255306\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 13.2134i 0.531951i 0.963980 + 0.265975i \(0.0856940\pi\)
−0.963980 + 0.265975i \(0.914306\pi\)
\(618\) 0 0
\(619\) 23.2453 0.934306 0.467153 0.884177i \(-0.345280\pi\)
0.467153 + 0.884177i \(0.345280\pi\)
\(620\) 0 0
\(621\) −8.43818 −0.338612
\(622\) 0 0
\(623\) − 44.9742i − 1.80186i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 1.74439i − 0.0696643i
\(628\) 0 0
\(629\) 19.0422 0.759263
\(630\) 0 0
\(631\) 3.88715 0.154745 0.0773725 0.997002i \(-0.475347\pi\)
0.0773725 + 0.997002i \(0.475347\pi\)
\(632\) 0 0
\(633\) 16.8968i 0.671588i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 3.80467i − 0.150746i
\(638\) 0 0
\(639\) 65.5655 2.59373
\(640\) 0 0
\(641\) −34.1223 −1.34775 −0.673875 0.738846i \(-0.735371\pi\)
−0.673875 + 0.738846i \(0.735371\pi\)
\(642\) 0 0
\(643\) − 31.9243i − 1.25897i −0.777012 0.629486i \(-0.783265\pi\)
0.777012 0.629486i \(-0.216735\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 35.3434i 1.38949i 0.719255 + 0.694746i \(0.244483\pi\)
−0.719255 + 0.694746i \(0.755517\pi\)
\(648\) 0 0
\(649\) 8.48175 0.332938
\(650\) 0 0
\(651\) 66.8486 2.62000
\(652\) 0 0
\(653\) − 27.5431i − 1.07784i −0.842356 0.538922i \(-0.818832\pi\)
0.842356 0.538922i \(-0.181168\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 5.64627i 0.220282i
\(658\) 0 0
\(659\) −29.6740 −1.15593 −0.577967 0.816060i \(-0.696154\pi\)
−0.577967 + 0.816060i \(0.696154\pi\)
\(660\) 0 0
\(661\) −16.3159 −0.634614 −0.317307 0.948323i \(-0.602779\pi\)
−0.317307 + 0.948323i \(0.602779\pi\)
\(662\) 0 0
\(663\) 21.7866i 0.846122i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1.63615i 0.0633522i
\(668\) 0 0
\(669\) 29.5233 1.14144
\(670\) 0 0
\(671\) −1.12735 −0.0435209
\(672\) 0 0
\(673\) − 13.6945i − 0.527883i −0.964539 0.263941i \(-0.914977\pi\)
0.964539 0.263941i \(-0.0850225\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 18.2426i 0.701121i 0.936540 + 0.350560i \(0.114009\pi\)
−0.936540 + 0.350560i \(0.885991\pi\)
\(678\) 0 0
\(679\) −22.4432 −0.861292
\(680\) 0 0
\(681\) 42.2824 1.62027
\(682\) 0 0
\(683\) − 8.92764i − 0.341607i −0.985305 0.170803i \(-0.945364\pi\)
0.985305 0.170803i \(-0.0546363\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 79.2195i − 3.02241i
\(688\) 0 0
\(689\) −2.11791 −0.0806859
\(690\) 0 0
\(691\) 37.5284 1.42765 0.713823 0.700326i \(-0.246962\pi\)
0.713823 + 0.700326i \(0.246962\pi\)
\(692\) 0 0
\(693\) − 9.56115i − 0.363198i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 27.6111i 1.04584i
\(698\) 0 0
\(699\) 4.08007 0.154322
\(700\) 0 0
\(701\) −42.1746 −1.59291 −0.796457 0.604695i \(-0.793295\pi\)
−0.796457 + 0.604695i \(0.793295\pi\)
\(702\) 0 0
\(703\) − 6.53871i − 0.246612i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 12.8168i 0.482024i
\(708\) 0 0
\(709\) 4.35373 0.163508 0.0817539 0.996653i \(-0.473948\pi\)
0.0817539 + 0.996653i \(0.473948\pi\)
\(710\) 0 0
\(711\) −84.2067 −3.15800
\(712\) 0 0
\(713\) 9.20500i 0.344730i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 28.7789i 1.07477i
\(718\) 0 0
\(719\) −14.2780 −0.532481 −0.266241 0.963907i \(-0.585782\pi\)
−0.266241 + 0.963907i \(0.585782\pi\)
\(720\) 0 0
\(721\) 50.5655 1.88316
\(722\) 0 0
\(723\) 8.98459i 0.334141i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 29.7538i − 1.10351i −0.834007 0.551754i \(-0.813959\pi\)
0.834007 0.551754i \(-0.186041\pi\)
\(728\) 0 0
\(729\) 37.9193 1.40442
\(730\) 0 0
\(731\) −2.40101 −0.0888046
\(732\) 0 0
\(733\) − 16.4580i − 0.607889i −0.952690 0.303944i \(-0.901696\pi\)
0.952690 0.303944i \(-0.0983037\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3.06095i 0.112752i
\(738\) 0 0
\(739\) −11.2024 −0.412085 −0.206043 0.978543i \(-0.566059\pi\)
−0.206043 + 0.978543i \(0.566059\pi\)
\(740\) 0 0
\(741\) 7.48108 0.274824
\(742\) 0 0
\(743\) 28.5053i 1.04576i 0.852407 + 0.522878i \(0.175142\pi\)
−0.852407 + 0.522878i \(0.824858\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 86.3625i − 3.15984i
\(748\) 0 0
\(749\) −41.3253 −1.50999
\(750\) 0 0
\(751\) −24.9716 −0.911227 −0.455613 0.890178i \(-0.650580\pi\)
−0.455613 + 0.890178i \(0.650580\pi\)
\(752\) 0 0
\(753\) − 17.7065i − 0.645263i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 12.8717i − 0.467831i −0.972257 0.233916i \(-0.924846\pi\)
0.972257 0.233916i \(-0.0751540\pi\)
\(758\) 0 0
\(759\) 2.03717 0.0739445
\(760\) 0 0
\(761\) −26.6791 −0.967115 −0.483558 0.875313i \(-0.660656\pi\)
−0.483558 + 0.875313i \(0.660656\pi\)
\(762\) 0 0
\(763\) 31.0999i 1.12589i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 36.3753i 1.31343i
\(768\) 0 0
\(769\) −29.7541 −1.07296 −0.536479 0.843913i \(-0.680246\pi\)
−0.536479 + 0.843913i \(0.680246\pi\)
\(770\) 0 0
\(771\) 55.3676 1.99401
\(772\) 0 0
\(773\) − 50.5655i − 1.81872i −0.416015 0.909358i \(-0.636574\pi\)
0.416015 0.909358i \(-0.363426\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 55.4553i − 1.98945i
\(778\) 0 0
\(779\) 9.48108 0.339695
\(780\) 0 0
\(781\) −7.16519 −0.256391
\(782\) 0 0
\(783\) 10.1230i 0.361765i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 54.8530i − 1.95530i −0.210242 0.977649i \(-0.567425\pi\)
0.210242 0.977649i \(-0.432575\pi\)
\(788\) 0 0
\(789\) 1.32094 0.0470269
\(790\) 0 0
\(791\) 24.0422 0.854843
\(792\) 0 0
\(793\) − 4.83481i − 0.171689i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 30.3434i 1.07482i 0.843322 + 0.537409i \(0.180597\pi\)
−0.843322 + 0.537409i \(0.819403\pi\)
\(798\) 0 0
\(799\) −17.6412 −0.624101
\(800\) 0 0
\(801\) 84.6456 2.99081
\(802\) 0 0
\(803\) − 0.617041i − 0.0217749i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 69.4399i 2.44440i
\(808\) 0 0
\(809\) 11.0801 0.389554 0.194777 0.980848i \(-0.437602\pi\)
0.194777 + 0.980848i \(0.437602\pi\)
\(810\) 0 0
\(811\) 39.3625 1.38220 0.691102 0.722757i \(-0.257126\pi\)
0.691102 + 0.722757i \(0.257126\pi\)
\(812\) 0 0
\(813\) − 37.1575i − 1.30317i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0.824458i 0.0288441i
\(818\) 0 0
\(819\) 41.0044 1.43281
\(820\) 0 0
\(821\) 34.8493 1.21625 0.608125 0.793842i \(-0.291922\pi\)
0.608125 + 0.793842i \(0.291922\pi\)
\(822\) 0 0
\(823\) 22.7615i 0.793417i 0.917945 + 0.396709i \(0.129848\pi\)
−0.917945 + 0.396709i \(0.870152\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 32.7615i 1.13923i 0.821912 + 0.569615i \(0.192908\pi\)
−0.821912 + 0.569615i \(0.807092\pi\)
\(828\) 0 0
\(829\) −36.4905 −1.26737 −0.633684 0.773592i \(-0.718458\pi\)
−0.633684 + 0.773592i \(0.718458\pi\)
\(830\) 0 0
\(831\) 24.1652 0.838281
\(832\) 0 0
\(833\) − 4.31324i − 0.149445i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 56.9518i 1.96854i
\(838\) 0 0
\(839\) −47.4949 −1.63971 −0.819853 0.572574i \(-0.805945\pi\)
−0.819853 + 0.572574i \(0.805945\pi\)
\(840\) 0 0
\(841\) −27.0372 −0.932316
\(842\) 0 0
\(843\) 26.7866i 0.922580i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 30.9897i − 1.06482i
\(848\) 0 0
\(849\) 57.8865 1.98666
\(850\) 0 0
\(851\) 7.63615 0.261764
\(852\) 0 0
\(853\) − 1.86736i − 0.0639372i −0.999489 0.0319686i \(-0.989822\pi\)
0.999489 0.0319686i \(-0.0101776\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 16.3608i 0.558873i 0.960164 + 0.279436i \(0.0901476\pi\)
−0.960164 + 0.279436i \(0.909852\pi\)
\(858\) 0 0
\(859\) 21.0851 0.719415 0.359708 0.933065i \(-0.382876\pi\)
0.359708 + 0.933065i \(0.382876\pi\)
\(860\) 0 0
\(861\) 80.4098 2.74036
\(862\) 0 0
\(863\) − 25.6868i − 0.874387i −0.899367 0.437194i \(-0.855972\pi\)
0.899367 0.437194i \(-0.144028\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 24.8091i − 0.842560i
\(868\) 0 0
\(869\) 9.20236 0.312169
\(870\) 0 0
\(871\) −13.1274 −0.444803
\(872\) 0 0
\(873\) − 42.2402i − 1.42961i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 43.2978i − 1.46206i −0.682343 0.731032i \(-0.739039\pi\)
0.682343 0.731032i \(-0.260961\pi\)
\(878\) 0 0
\(879\) 23.7213 0.800099
\(880\) 0 0
\(881\) 22.0750 0.743726 0.371863 0.928288i \(-0.378719\pi\)
0.371863 + 0.928288i \(0.378719\pi\)
\(882\) 0 0
\(883\) 0.385604i 0.0129766i 0.999979 + 0.00648831i \(0.00206531\pi\)
−0.999979 + 0.00648831i \(0.997935\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 54.9210i − 1.84407i −0.387111 0.922033i \(-0.626527\pi\)
0.387111 0.922033i \(-0.373473\pi\)
\(888\) 0 0
\(889\) 35.3253 1.18477
\(890\) 0 0
\(891\) 2.75474 0.0922873
\(892\) 0 0
\(893\) 6.05763i 0.202711i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 8.73669i 0.291710i
\(898\) 0 0
\(899\) 11.0429 0.368301
\(900\) 0 0
\(901\) −2.40101 −0.0799893
\(902\) 0 0
\(903\) 6.99230i 0.232689i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 41.4727i 1.37708i 0.725199 + 0.688539i \(0.241748\pi\)
−0.725199 + 0.688539i \(0.758252\pi\)
\(908\) 0 0
\(909\) −24.1223 −0.800086
\(910\) 0 0
\(911\) 41.8909 1.38791 0.693953 0.720020i \(-0.255868\pi\)
0.693953 + 0.720020i \(0.255868\pi\)
\(912\) 0 0
\(913\) 9.43795i 0.312350i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 15.9622i 0.527117i
\(918\) 0 0
\(919\) 36.0044 1.18767 0.593837 0.804585i \(-0.297612\pi\)
0.593837 + 0.804585i \(0.297612\pi\)
\(920\) 0 0
\(921\) 31.3253 1.03220
\(922\) 0 0
\(923\) − 30.7290i − 1.01146i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 95.1689i 3.12576i
\(928\) 0 0
\(929\) −59.4898 −1.95180 −0.975899 0.218222i \(-0.929974\pi\)
−0.975899 + 0.218222i \(0.929974\pi\)
\(930\) 0 0
\(931\) −1.48108 −0.0485404
\(932\) 0 0
\(933\) 89.8255i 2.94076i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 35.7143i − 1.16673i −0.812209 0.583367i \(-0.801735\pi\)
0.812209 0.583367i \(-0.198265\pi\)
\(938\) 0 0
\(939\) −19.0422 −0.621420
\(940\) 0 0
\(941\) 20.2074 0.658743 0.329371 0.944200i \(-0.393163\pi\)
0.329371 + 0.944200i \(0.393163\pi\)
\(942\) 0 0
\(943\) 11.0724i 0.360566i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 51.4982i 1.67347i 0.547610 + 0.836734i \(0.315538\pi\)
−0.547610 + 0.836734i \(0.684462\pi\)
\(948\) 0 0
\(949\) 2.64627 0.0859016
\(950\) 0 0
\(951\) 32.9243 1.06764
\(952\) 0 0
\(953\) − 7.95272i − 0.257614i −0.991670 0.128807i \(-0.958885\pi\)
0.991670 0.128807i \(-0.0411147\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 2.44391i − 0.0790005i
\(958\) 0 0
\(959\) −22.1223 −0.714366
\(960\) 0 0
\(961\) 31.1274 1.00411
\(962\) 0 0
\(963\) − 77.7780i − 2.50636i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 32.5706i 1.04740i 0.851903 + 0.523700i \(0.175449\pi\)
−0.851903 + 0.523700i \(0.824551\pi\)
\(968\) 0 0
\(969\) 8.48108 0.272452
\(970\) 0 0
\(971\) 21.0378 0.675136 0.337568 0.941301i \(-0.390396\pi\)
0.337568 + 0.941301i \(0.390396\pi\)
\(972\) 0 0
\(973\) − 20.8520i − 0.668483i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 46.9648i 1.50254i 0.659997 + 0.751269i \(0.270558\pi\)
−0.659997 + 0.751269i \(0.729442\pi\)
\(978\) 0 0
\(979\) −9.25032 −0.295641
\(980\) 0 0
\(981\) −58.5327 −1.86881
\(982\) 0 0
\(983\) 18.5182i 0.590640i 0.955398 + 0.295320i \(0.0954263\pi\)
−0.955398 + 0.295320i \(0.904574\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 51.3753i 1.63529i
\(988\) 0 0
\(989\) −0.962834 −0.0306163
\(990\) 0 0
\(991\) −5.40539 −0.171708 −0.0858540 0.996308i \(-0.527362\pi\)
−0.0858540 + 0.996308i \(0.527362\pi\)
\(992\) 0 0
\(993\) − 66.4175i − 2.10770i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 8.55676i 0.270995i 0.990778 + 0.135498i \(0.0432633\pi\)
−0.990778 + 0.135498i \(0.956737\pi\)
\(998\) 0 0
\(999\) 47.2453 1.49477
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.1749.1 6
5.2 odd 4 1900.2.a.f.1.1 3
5.3 odd 4 1900.2.a.h.1.3 yes 3
5.4 even 2 inner 1900.2.c.g.1749.6 6
20.3 even 4 7600.2.a.bj.1.1 3
20.7 even 4 7600.2.a.by.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1900.2.a.f.1.1 3 5.2 odd 4
1900.2.a.h.1.3 yes 3 5.3 odd 4
1900.2.c.g.1749.1 6 1.1 even 1 trivial
1900.2.c.g.1749.6 6 5.4 even 2 inner
7600.2.a.bj.1.1 3 20.3 even 4
7600.2.a.by.1.3 3 20.7 even 4