Properties

Label 3800.2.a.bc.1.2
Level $3800$
Weight $2$
Character 3800.1
Self dual yes
Analytic conductor $30.343$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.3431527681\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - 2 x^{5} - 10 x^{4} + 16 x^{3} + 15 x^{2} - 14 x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.08999\) of defining polynomial
Character \(\chi\) \(=\) 3800.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.08999 q^{3} +4.19727 q^{7} -1.81192 q^{9} +O(q^{10})\) \(q-1.08999 q^{3} +4.19727 q^{7} -1.81192 q^{9} -6.43052 q^{11} +2.24614 q^{13} -7.84744 q^{17} -1.00000 q^{19} -4.57497 q^{21} -0.859601 q^{23} +5.24494 q^{27} +8.38284 q^{29} +1.24541 q^{31} +7.00919 q^{33} -6.79977 q^{37} -2.44827 q^{39} +5.92480 q^{41} +6.81073 q^{43} -6.00919 q^{47} +10.6171 q^{49} +8.55363 q^{51} +13.7594 q^{53} +1.08999 q^{57} +6.88976 q^{59} -1.31884 q^{61} -7.60513 q^{63} +4.73266 q^{67} +0.936955 q^{69} +10.2546 q^{71} -9.86150 q^{73} -26.9906 q^{77} +6.47056 q^{79} -0.281158 q^{81} +5.42133 q^{83} -9.13720 q^{87} +3.10288 q^{89} +9.42766 q^{91} -1.35748 q^{93} +6.76032 q^{97} +11.6516 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{3} + 2 q^{7} + 6 q^{9} + O(q^{10}) \) \( 6 q + 2 q^{3} + 2 q^{7} + 6 q^{9} + 3 q^{11} - q^{13} - 14 q^{17} - 6 q^{19} + 15 q^{21} + 12 q^{23} + 8 q^{27} + 9 q^{29} + 5 q^{31} + 2 q^{33} - 8 q^{37} + 12 q^{39} + 3 q^{41} + 15 q^{43} + 4 q^{47} + 22 q^{49} + 33 q^{51} + 13 q^{53} - 2 q^{57} + 9 q^{61} + 21 q^{63} - 3 q^{67} - 11 q^{69} + 19 q^{71} + 3 q^{73} - 36 q^{77} - 16 q^{79} + 26 q^{81} + 31 q^{83} - 25 q^{87} + 14 q^{89} + 42 q^{91} + 39 q^{93} - 11 q^{97} - 6 q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.08999 −0.629305 −0.314653 0.949207i \(-0.601888\pi\)
−0.314653 + 0.949207i \(0.601888\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.19727 1.58642 0.793209 0.608949i \(-0.208409\pi\)
0.793209 + 0.608949i \(0.208409\pi\)
\(8\) 0 0
\(9\) −1.81192 −0.603975
\(10\) 0 0
\(11\) −6.43052 −1.93887 −0.969437 0.245341i \(-0.921100\pi\)
−0.969437 + 0.245341i \(0.921100\pi\)
\(12\) 0 0
\(13\) 2.24614 0.622967 0.311484 0.950251i \(-0.399174\pi\)
0.311484 + 0.950251i \(0.399174\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −7.84744 −1.90328 −0.951642 0.307208i \(-0.900605\pi\)
−0.951642 + 0.307208i \(0.900605\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −4.57497 −0.998341
\(22\) 0 0
\(23\) −0.859601 −0.179239 −0.0896195 0.995976i \(-0.528565\pi\)
−0.0896195 + 0.995976i \(0.528565\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.24494 1.00939
\(28\) 0 0
\(29\) 8.38284 1.55665 0.778327 0.627859i \(-0.216068\pi\)
0.778327 + 0.627859i \(0.216068\pi\)
\(30\) 0 0
\(31\) 1.24541 0.223681 0.111841 0.993726i \(-0.464325\pi\)
0.111841 + 0.993726i \(0.464325\pi\)
\(32\) 0 0
\(33\) 7.00919 1.22014
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.79977 −1.11787 −0.558937 0.829210i \(-0.688791\pi\)
−0.558937 + 0.829210i \(0.688791\pi\)
\(38\) 0 0
\(39\) −2.44827 −0.392037
\(40\) 0 0
\(41\) 5.92480 0.925298 0.462649 0.886542i \(-0.346899\pi\)
0.462649 + 0.886542i \(0.346899\pi\)
\(42\) 0 0
\(43\) 6.81073 1.03863 0.519313 0.854584i \(-0.326188\pi\)
0.519313 + 0.854584i \(0.326188\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.00919 −0.876531 −0.438265 0.898846i \(-0.644407\pi\)
−0.438265 + 0.898846i \(0.644407\pi\)
\(48\) 0 0
\(49\) 10.6171 1.51672
\(50\) 0 0
\(51\) 8.55363 1.19775
\(52\) 0 0
\(53\) 13.7594 1.88999 0.944996 0.327082i \(-0.106065\pi\)
0.944996 + 0.327082i \(0.106065\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.08999 0.144373
\(58\) 0 0
\(59\) 6.88976 0.896970 0.448485 0.893790i \(-0.351964\pi\)
0.448485 + 0.893790i \(0.351964\pi\)
\(60\) 0 0
\(61\) −1.31884 −0.168860 −0.0844301 0.996429i \(-0.526907\pi\)
−0.0844301 + 0.996429i \(0.526907\pi\)
\(62\) 0 0
\(63\) −7.60513 −0.958156
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.73266 0.578187 0.289093 0.957301i \(-0.406646\pi\)
0.289093 + 0.957301i \(0.406646\pi\)
\(68\) 0 0
\(69\) 0.936955 0.112796
\(70\) 0 0
\(71\) 10.2546 1.21700 0.608498 0.793555i \(-0.291772\pi\)
0.608498 + 0.793555i \(0.291772\pi\)
\(72\) 0 0
\(73\) −9.86150 −1.15420 −0.577100 0.816673i \(-0.695816\pi\)
−0.577100 + 0.816673i \(0.695816\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −26.9906 −3.07586
\(78\) 0 0
\(79\) 6.47056 0.727995 0.363997 0.931400i \(-0.381412\pi\)
0.363997 + 0.931400i \(0.381412\pi\)
\(80\) 0 0
\(81\) −0.281158 −0.0312397
\(82\) 0 0
\(83\) 5.42133 0.595068 0.297534 0.954711i \(-0.403836\pi\)
0.297534 + 0.954711i \(0.403836\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −9.13720 −0.979611
\(88\) 0 0
\(89\) 3.10288 0.328905 0.164452 0.986385i \(-0.447414\pi\)
0.164452 + 0.986385i \(0.447414\pi\)
\(90\) 0 0
\(91\) 9.42766 0.988287
\(92\) 0 0
\(93\) −1.35748 −0.140764
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 6.76032 0.686406 0.343203 0.939261i \(-0.388488\pi\)
0.343203 + 0.939261i \(0.388488\pi\)
\(98\) 0 0
\(99\) 11.6516 1.17103
\(100\) 0 0
\(101\) −3.05247 −0.303732 −0.151866 0.988401i \(-0.548528\pi\)
−0.151866 + 0.988401i \(0.548528\pi\)
\(102\) 0 0
\(103\) 7.06366 0.696003 0.348002 0.937494i \(-0.386860\pi\)
0.348002 + 0.937494i \(0.386860\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −5.05281 −0.488474 −0.244237 0.969716i \(-0.578537\pi\)
−0.244237 + 0.969716i \(0.578537\pi\)
\(108\) 0 0
\(109\) −6.39334 −0.612371 −0.306185 0.951972i \(-0.599053\pi\)
−0.306185 + 0.951972i \(0.599053\pi\)
\(110\) 0 0
\(111\) 7.41167 0.703485
\(112\) 0 0
\(113\) −19.5837 −1.84228 −0.921140 0.389231i \(-0.872741\pi\)
−0.921140 + 0.389231i \(0.872741\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −4.06984 −0.376257
\(118\) 0 0
\(119\) −32.9378 −3.01941
\(120\) 0 0
\(121\) 30.3515 2.75923
\(122\) 0 0
\(123\) −6.45796 −0.582295
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 6.40323 0.568195 0.284098 0.958795i \(-0.408306\pi\)
0.284098 + 0.958795i \(0.408306\pi\)
\(128\) 0 0
\(129\) −7.42362 −0.653613
\(130\) 0 0
\(131\) 12.1913 1.06516 0.532581 0.846379i \(-0.321222\pi\)
0.532581 + 0.846379i \(0.321222\pi\)
\(132\) 0 0
\(133\) −4.19727 −0.363949
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −9.71228 −0.829776 −0.414888 0.909872i \(-0.636179\pi\)
−0.414888 + 0.909872i \(0.636179\pi\)
\(138\) 0 0
\(139\) 20.0687 1.70220 0.851102 0.525000i \(-0.175935\pi\)
0.851102 + 0.525000i \(0.175935\pi\)
\(140\) 0 0
\(141\) 6.54995 0.551605
\(142\) 0 0
\(143\) −14.4438 −1.20786
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −11.5725 −0.954481
\(148\) 0 0
\(149\) −14.8593 −1.21732 −0.608659 0.793432i \(-0.708292\pi\)
−0.608659 + 0.793432i \(0.708292\pi\)
\(150\) 0 0
\(151\) −1.29571 −0.105444 −0.0527218 0.998609i \(-0.516790\pi\)
−0.0527218 + 0.998609i \(0.516790\pi\)
\(152\) 0 0
\(153\) 14.2190 1.14954
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 4.05355 0.323508 0.161754 0.986831i \(-0.448285\pi\)
0.161754 + 0.986831i \(0.448285\pi\)
\(158\) 0 0
\(159\) −14.9975 −1.18938
\(160\) 0 0
\(161\) −3.60797 −0.284348
\(162\) 0 0
\(163\) 8.09764 0.634256 0.317128 0.948383i \(-0.397281\pi\)
0.317128 + 0.948383i \(0.397281\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −18.6059 −1.43976 −0.719882 0.694096i \(-0.755804\pi\)
−0.719882 + 0.694096i \(0.755804\pi\)
\(168\) 0 0
\(169\) −7.95485 −0.611911
\(170\) 0 0
\(171\) 1.81192 0.138561
\(172\) 0 0
\(173\) 8.35972 0.635578 0.317789 0.948161i \(-0.397060\pi\)
0.317789 + 0.948161i \(0.397060\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −7.50976 −0.564468
\(178\) 0 0
\(179\) 14.5971 1.09104 0.545521 0.838097i \(-0.316332\pi\)
0.545521 + 0.838097i \(0.316332\pi\)
\(180\) 0 0
\(181\) 0.131566 0.00977923 0.00488961 0.999988i \(-0.498444\pi\)
0.00488961 + 0.999988i \(0.498444\pi\)
\(182\) 0 0
\(183\) 1.43752 0.106265
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 50.4631 3.69023
\(188\) 0 0
\(189\) 22.0144 1.60131
\(190\) 0 0
\(191\) −22.5544 −1.63198 −0.815989 0.578067i \(-0.803807\pi\)
−0.815989 + 0.578067i \(0.803807\pi\)
\(192\) 0 0
\(193\) 4.93863 0.355490 0.177745 0.984077i \(-0.443120\pi\)
0.177745 + 0.984077i \(0.443120\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 26.2422 1.86968 0.934840 0.355069i \(-0.115543\pi\)
0.934840 + 0.355069i \(0.115543\pi\)
\(198\) 0 0
\(199\) 15.1880 1.07665 0.538323 0.842738i \(-0.319058\pi\)
0.538323 + 0.842738i \(0.319058\pi\)
\(200\) 0 0
\(201\) −5.15855 −0.363856
\(202\) 0 0
\(203\) 35.1850 2.46950
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1.55753 0.108256
\(208\) 0 0
\(209\) 6.43052 0.444808
\(210\) 0 0
\(211\) 20.3632 1.40186 0.700929 0.713231i \(-0.252769\pi\)
0.700929 + 0.713231i \(0.252769\pi\)
\(212\) 0 0
\(213\) −11.1774 −0.765863
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 5.22730 0.354852
\(218\) 0 0
\(219\) 10.7489 0.726345
\(220\) 0 0
\(221\) −17.6265 −1.18568
\(222\) 0 0
\(223\) 15.3764 1.02968 0.514841 0.857285i \(-0.327851\pi\)
0.514841 + 0.857285i \(0.327851\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −20.0187 −1.32868 −0.664342 0.747428i \(-0.731288\pi\)
−0.664342 + 0.747428i \(0.731288\pi\)
\(228\) 0 0
\(229\) −12.6175 −0.833789 −0.416895 0.908955i \(-0.636882\pi\)
−0.416895 + 0.908955i \(0.636882\pi\)
\(230\) 0 0
\(231\) 29.4195 1.93566
\(232\) 0 0
\(233\) 21.9230 1.43622 0.718112 0.695927i \(-0.245007\pi\)
0.718112 + 0.695927i \(0.245007\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −7.05284 −0.458131
\(238\) 0 0
\(239\) 3.66866 0.237306 0.118653 0.992936i \(-0.462142\pi\)
0.118653 + 0.992936i \(0.462142\pi\)
\(240\) 0 0
\(241\) −6.03281 −0.388608 −0.194304 0.980941i \(-0.562245\pi\)
−0.194304 + 0.980941i \(0.562245\pi\)
\(242\) 0 0
\(243\) −15.4284 −0.989731
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −2.24614 −0.142919
\(248\) 0 0
\(249\) −5.90918 −0.374479
\(250\) 0 0
\(251\) 0.145763 0.00920048 0.00460024 0.999989i \(-0.498536\pi\)
0.00460024 + 0.999989i \(0.498536\pi\)
\(252\) 0 0
\(253\) 5.52768 0.347522
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −14.7514 −0.920164 −0.460082 0.887876i \(-0.652180\pi\)
−0.460082 + 0.887876i \(0.652180\pi\)
\(258\) 0 0
\(259\) −28.5404 −1.77342
\(260\) 0 0
\(261\) −15.1891 −0.940180
\(262\) 0 0
\(263\) 30.2967 1.86817 0.934087 0.357046i \(-0.116216\pi\)
0.934087 + 0.357046i \(0.116216\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −3.38211 −0.206981
\(268\) 0 0
\(269\) 20.3460 1.24052 0.620259 0.784397i \(-0.287027\pi\)
0.620259 + 0.784397i \(0.287027\pi\)
\(270\) 0 0
\(271\) 1.01349 0.0615651 0.0307825 0.999526i \(-0.490200\pi\)
0.0307825 + 0.999526i \(0.490200\pi\)
\(272\) 0 0
\(273\) −10.2760 −0.621934
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −10.4346 −0.626953 −0.313476 0.949596i \(-0.601494\pi\)
−0.313476 + 0.949596i \(0.601494\pi\)
\(278\) 0 0
\(279\) −2.25658 −0.135098
\(280\) 0 0
\(281\) −8.12732 −0.484835 −0.242418 0.970172i \(-0.577940\pi\)
−0.242418 + 0.970172i \(0.577940\pi\)
\(282\) 0 0
\(283\) 11.6397 0.691907 0.345953 0.938252i \(-0.387556\pi\)
0.345953 + 0.938252i \(0.387556\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 24.8680 1.46791
\(288\) 0 0
\(289\) 44.5824 2.62249
\(290\) 0 0
\(291\) −7.36867 −0.431959
\(292\) 0 0
\(293\) −2.21896 −0.129633 −0.0648164 0.997897i \(-0.520646\pi\)
−0.0648164 + 0.997897i \(0.520646\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −33.7277 −1.95708
\(298\) 0 0
\(299\) −1.93078 −0.111660
\(300\) 0 0
\(301\) 28.5864 1.64770
\(302\) 0 0
\(303\) 3.32716 0.191140
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 17.6712 1.00855 0.504274 0.863544i \(-0.331760\pi\)
0.504274 + 0.863544i \(0.331760\pi\)
\(308\) 0 0
\(309\) −7.69931 −0.437999
\(310\) 0 0
\(311\) 12.3628 0.701031 0.350516 0.936557i \(-0.386006\pi\)
0.350516 + 0.936557i \(0.386006\pi\)
\(312\) 0 0
\(313\) 8.35545 0.472278 0.236139 0.971719i \(-0.424118\pi\)
0.236139 + 0.971719i \(0.424118\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −19.3526 −1.08695 −0.543475 0.839425i \(-0.682892\pi\)
−0.543475 + 0.839425i \(0.682892\pi\)
\(318\) 0 0
\(319\) −53.9060 −3.01816
\(320\) 0 0
\(321\) 5.50751 0.307399
\(322\) 0 0
\(323\) 7.84744 0.436643
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 6.96867 0.385368
\(328\) 0 0
\(329\) −25.2222 −1.39054
\(330\) 0 0
\(331\) 33.8929 1.86292 0.931461 0.363840i \(-0.118535\pi\)
0.931461 + 0.363840i \(0.118535\pi\)
\(332\) 0 0
\(333\) 12.3207 0.675168
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −28.6728 −1.56191 −0.780954 0.624588i \(-0.785267\pi\)
−0.780954 + 0.624588i \(0.785267\pi\)
\(338\) 0 0
\(339\) 21.3460 1.15936
\(340\) 0 0
\(341\) −8.00860 −0.433690
\(342\) 0 0
\(343\) 15.1818 0.819738
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 15.9590 0.856725 0.428362 0.903607i \(-0.359091\pi\)
0.428362 + 0.903607i \(0.359091\pi\)
\(348\) 0 0
\(349\) 29.4645 1.57720 0.788599 0.614907i \(-0.210807\pi\)
0.788599 + 0.614907i \(0.210807\pi\)
\(350\) 0 0
\(351\) 11.7809 0.628817
\(352\) 0 0
\(353\) −28.3655 −1.50974 −0.754871 0.655873i \(-0.772301\pi\)
−0.754871 + 0.655873i \(0.772301\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 35.9019 1.90013
\(358\) 0 0
\(359\) 0.0301128 0.00158929 0.000794647 1.00000i \(-0.499747\pi\)
0.000794647 1.00000i \(0.499747\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −33.0829 −1.73640
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −1.31478 −0.0686310 −0.0343155 0.999411i \(-0.510925\pi\)
−0.0343155 + 0.999411i \(0.510925\pi\)
\(368\) 0 0
\(369\) −10.7353 −0.558857
\(370\) 0 0
\(371\) 57.7517 2.99832
\(372\) 0 0
\(373\) −3.63316 −0.188118 −0.0940589 0.995567i \(-0.529984\pi\)
−0.0940589 + 0.995567i \(0.529984\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 18.8290 0.969745
\(378\) 0 0
\(379\) 24.4908 1.25801 0.629004 0.777402i \(-0.283463\pi\)
0.629004 + 0.777402i \(0.283463\pi\)
\(380\) 0 0
\(381\) −6.97945 −0.357568
\(382\) 0 0
\(383\) 3.23421 0.165261 0.0826303 0.996580i \(-0.473668\pi\)
0.0826303 + 0.996580i \(0.473668\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −12.3405 −0.627304
\(388\) 0 0
\(389\) 22.9240 1.16229 0.581147 0.813799i \(-0.302604\pi\)
0.581147 + 0.813799i \(0.302604\pi\)
\(390\) 0 0
\(391\) 6.74567 0.341143
\(392\) 0 0
\(393\) −13.2884 −0.670312
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −14.1567 −0.710507 −0.355253 0.934770i \(-0.615605\pi\)
−0.355253 + 0.934770i \(0.615605\pi\)
\(398\) 0 0
\(399\) 4.57497 0.229035
\(400\) 0 0
\(401\) −6.06391 −0.302817 −0.151409 0.988471i \(-0.548381\pi\)
−0.151409 + 0.988471i \(0.548381\pi\)
\(402\) 0 0
\(403\) 2.79736 0.139346
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 43.7260 2.16742
\(408\) 0 0
\(409\) 3.95218 0.195423 0.0977113 0.995215i \(-0.468848\pi\)
0.0977113 + 0.995215i \(0.468848\pi\)
\(410\) 0 0
\(411\) 10.5863 0.522183
\(412\) 0 0
\(413\) 28.9182 1.42297
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −21.8746 −1.07121
\(418\) 0 0
\(419\) −12.6137 −0.616221 −0.308111 0.951351i \(-0.599697\pi\)
−0.308111 + 0.951351i \(0.599697\pi\)
\(420\) 0 0
\(421\) −9.15060 −0.445973 −0.222986 0.974822i \(-0.571581\pi\)
−0.222986 + 0.974822i \(0.571581\pi\)
\(422\) 0 0
\(423\) 10.8882 0.529402
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −5.53553 −0.267883
\(428\) 0 0
\(429\) 15.7436 0.760110
\(430\) 0 0
\(431\) −27.2560 −1.31288 −0.656439 0.754379i \(-0.727938\pi\)
−0.656439 + 0.754379i \(0.727938\pi\)
\(432\) 0 0
\(433\) 2.33860 0.112386 0.0561929 0.998420i \(-0.482104\pi\)
0.0561929 + 0.998420i \(0.482104\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0.859601 0.0411203
\(438\) 0 0
\(439\) 37.6773 1.79824 0.899119 0.437703i \(-0.144208\pi\)
0.899119 + 0.437703i \(0.144208\pi\)
\(440\) 0 0
\(441\) −19.2373 −0.916062
\(442\) 0 0
\(443\) 6.39870 0.304011 0.152006 0.988380i \(-0.451427\pi\)
0.152006 + 0.988380i \(0.451427\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 16.1964 0.766065
\(448\) 0 0
\(449\) −30.1784 −1.42421 −0.712103 0.702075i \(-0.752257\pi\)
−0.712103 + 0.702075i \(0.752257\pi\)
\(450\) 0 0
\(451\) −38.0995 −1.79404
\(452\) 0 0
\(453\) 1.41231 0.0663562
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 7.21464 0.337487 0.168743 0.985660i \(-0.446029\pi\)
0.168743 + 0.985660i \(0.446029\pi\)
\(458\) 0 0
\(459\) −41.1594 −1.92116
\(460\) 0 0
\(461\) −2.06925 −0.0963748 −0.0481874 0.998838i \(-0.515344\pi\)
−0.0481874 + 0.998838i \(0.515344\pi\)
\(462\) 0 0
\(463\) −4.01966 −0.186809 −0.0934047 0.995628i \(-0.529775\pi\)
−0.0934047 + 0.995628i \(0.529775\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −21.5515 −0.997285 −0.498643 0.866808i \(-0.666168\pi\)
−0.498643 + 0.866808i \(0.666168\pi\)
\(468\) 0 0
\(469\) 19.8643 0.917246
\(470\) 0 0
\(471\) −4.41832 −0.203585
\(472\) 0 0
\(473\) −43.7965 −2.01377
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −24.9309 −1.14151
\(478\) 0 0
\(479\) −31.8315 −1.45442 −0.727209 0.686416i \(-0.759183\pi\)
−0.727209 + 0.686416i \(0.759183\pi\)
\(480\) 0 0
\(481\) −15.2732 −0.696400
\(482\) 0 0
\(483\) 3.93265 0.178942
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −3.54933 −0.160835 −0.0804177 0.996761i \(-0.525625\pi\)
−0.0804177 + 0.996761i \(0.525625\pi\)
\(488\) 0 0
\(489\) −8.82634 −0.399141
\(490\) 0 0
\(491\) −16.5336 −0.746152 −0.373076 0.927801i \(-0.621697\pi\)
−0.373076 + 0.927801i \(0.621697\pi\)
\(492\) 0 0
\(493\) −65.7839 −2.96276
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 43.0413 1.93067
\(498\) 0 0
\(499\) −32.8826 −1.47203 −0.736013 0.676968i \(-0.763294\pi\)
−0.736013 + 0.676968i \(0.763294\pi\)
\(500\) 0 0
\(501\) 20.2802 0.906051
\(502\) 0 0
\(503\) −17.2150 −0.767580 −0.383790 0.923420i \(-0.625381\pi\)
−0.383790 + 0.923420i \(0.625381\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 8.67070 0.385079
\(508\) 0 0
\(509\) 28.4368 1.26044 0.630221 0.776416i \(-0.282964\pi\)
0.630221 + 0.776416i \(0.282964\pi\)
\(510\) 0 0
\(511\) −41.3913 −1.83105
\(512\) 0 0
\(513\) −5.24494 −0.231570
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 38.6422 1.69948
\(518\) 0 0
\(519\) −9.11201 −0.399973
\(520\) 0 0
\(521\) 25.1924 1.10370 0.551850 0.833944i \(-0.313922\pi\)
0.551850 + 0.833944i \(0.313922\pi\)
\(522\) 0 0
\(523\) −17.2647 −0.754934 −0.377467 0.926023i \(-0.623205\pi\)
−0.377467 + 0.926023i \(0.623205\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −9.77325 −0.425730
\(528\) 0 0
\(529\) −22.2611 −0.967873
\(530\) 0 0
\(531\) −12.4837 −0.541747
\(532\) 0 0
\(533\) 13.3079 0.576431
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −15.9107 −0.686598
\(538\) 0 0
\(539\) −68.2732 −2.94073
\(540\) 0 0
\(541\) −9.71009 −0.417469 −0.208735 0.977972i \(-0.566935\pi\)
−0.208735 + 0.977972i \(0.566935\pi\)
\(542\) 0 0
\(543\) −0.143406 −0.00615412
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 5.60962 0.239850 0.119925 0.992783i \(-0.461735\pi\)
0.119925 + 0.992783i \(0.461735\pi\)
\(548\) 0 0
\(549\) 2.38964 0.101987
\(550\) 0 0
\(551\) −8.38284 −0.357121
\(552\) 0 0
\(553\) 27.1587 1.15490
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −17.5323 −0.742868 −0.371434 0.928459i \(-0.621134\pi\)
−0.371434 + 0.928459i \(0.621134\pi\)
\(558\) 0 0
\(559\) 15.2979 0.647030
\(560\) 0 0
\(561\) −55.0042 −2.32228
\(562\) 0 0
\(563\) −8.85461 −0.373177 −0.186589 0.982438i \(-0.559743\pi\)
−0.186589 + 0.982438i \(0.559743\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −1.18009 −0.0495593
\(568\) 0 0
\(569\) 33.0220 1.38435 0.692177 0.721727i \(-0.256652\pi\)
0.692177 + 0.721727i \(0.256652\pi\)
\(570\) 0 0
\(571\) 12.9266 0.540963 0.270482 0.962725i \(-0.412817\pi\)
0.270482 + 0.962725i \(0.412817\pi\)
\(572\) 0 0
\(573\) 24.5840 1.02701
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 3.22539 0.134275 0.0671374 0.997744i \(-0.478613\pi\)
0.0671374 + 0.997744i \(0.478613\pi\)
\(578\) 0 0
\(579\) −5.38305 −0.223712
\(580\) 0 0
\(581\) 22.7548 0.944026
\(582\) 0 0
\(583\) −88.4797 −3.66446
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −7.17382 −0.296095 −0.148048 0.988980i \(-0.547299\pi\)
−0.148048 + 0.988980i \(0.547299\pi\)
\(588\) 0 0
\(589\) −1.24541 −0.0513161
\(590\) 0 0
\(591\) −28.6037 −1.17660
\(592\) 0 0
\(593\) −27.2576 −1.11933 −0.559667 0.828718i \(-0.689071\pi\)
−0.559667 + 0.828718i \(0.689071\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −16.5547 −0.677540
\(598\) 0 0
\(599\) −3.53091 −0.144269 −0.0721344 0.997395i \(-0.522981\pi\)
−0.0721344 + 0.997395i \(0.522981\pi\)
\(600\) 0 0
\(601\) 23.6488 0.964655 0.482327 0.875991i \(-0.339792\pi\)
0.482327 + 0.875991i \(0.339792\pi\)
\(602\) 0 0
\(603\) −8.57523 −0.349210
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −28.6598 −1.16326 −0.581632 0.813452i \(-0.697586\pi\)
−0.581632 + 0.813452i \(0.697586\pi\)
\(608\) 0 0
\(609\) −38.3513 −1.55407
\(610\) 0 0
\(611\) −13.4975 −0.546050
\(612\) 0 0
\(613\) 4.64372 0.187558 0.0937790 0.995593i \(-0.470105\pi\)
0.0937790 + 0.995593i \(0.470105\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 4.49990 0.181159 0.0905796 0.995889i \(-0.471128\pi\)
0.0905796 + 0.995889i \(0.471128\pi\)
\(618\) 0 0
\(619\) 20.5767 0.827049 0.413525 0.910493i \(-0.364298\pi\)
0.413525 + 0.910493i \(0.364298\pi\)
\(620\) 0 0
\(621\) −4.50856 −0.180922
\(622\) 0 0
\(623\) 13.0236 0.521780
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −7.00919 −0.279920
\(628\) 0 0
\(629\) 53.3608 2.12763
\(630\) 0 0
\(631\) −32.5361 −1.29524 −0.647620 0.761963i \(-0.724236\pi\)
−0.647620 + 0.761963i \(0.724236\pi\)
\(632\) 0 0
\(633\) −22.1956 −0.882196
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 23.8474 0.944869
\(638\) 0 0
\(639\) −18.5806 −0.735035
\(640\) 0 0
\(641\) 13.0922 0.517113 0.258556 0.965996i \(-0.416753\pi\)
0.258556 + 0.965996i \(0.416753\pi\)
\(642\) 0 0
\(643\) −16.8450 −0.664304 −0.332152 0.943226i \(-0.607775\pi\)
−0.332152 + 0.943226i \(0.607775\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 46.9300 1.84501 0.922504 0.385987i \(-0.126139\pi\)
0.922504 + 0.385987i \(0.126139\pi\)
\(648\) 0 0
\(649\) −44.3047 −1.73911
\(650\) 0 0
\(651\) −5.69770 −0.223310
\(652\) 0 0
\(653\) −14.1069 −0.552046 −0.276023 0.961151i \(-0.589017\pi\)
−0.276023 + 0.961151i \(0.589017\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 17.8683 0.697108
\(658\) 0 0
\(659\) 30.5841 1.19139 0.595693 0.803212i \(-0.296878\pi\)
0.595693 + 0.803212i \(0.296878\pi\)
\(660\) 0 0
\(661\) −42.8953 −1.66843 −0.834216 0.551437i \(-0.814080\pi\)
−0.834216 + 0.551437i \(0.814080\pi\)
\(662\) 0 0
\(663\) 19.2127 0.746158
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −7.20589 −0.279013
\(668\) 0 0
\(669\) −16.7602 −0.647985
\(670\) 0 0
\(671\) 8.48082 0.327399
\(672\) 0 0
\(673\) −19.0609 −0.734744 −0.367372 0.930074i \(-0.619742\pi\)
−0.367372 + 0.930074i \(0.619742\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −48.2121 −1.85294 −0.926471 0.376366i \(-0.877173\pi\)
−0.926471 + 0.376366i \(0.877173\pi\)
\(678\) 0 0
\(679\) 28.3749 1.08893
\(680\) 0 0
\(681\) 21.8201 0.836149
\(682\) 0 0
\(683\) −2.59451 −0.0992763 −0.0496381 0.998767i \(-0.515807\pi\)
−0.0496381 + 0.998767i \(0.515807\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 13.7530 0.524708
\(688\) 0 0
\(689\) 30.9054 1.17740
\(690\) 0 0
\(691\) −15.0799 −0.573667 −0.286834 0.957980i \(-0.592603\pi\)
−0.286834 + 0.957980i \(0.592603\pi\)
\(692\) 0 0
\(693\) 48.9049 1.85774
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −46.4945 −1.76111
\(698\) 0 0
\(699\) −23.8958 −0.903824
\(700\) 0 0
\(701\) 27.2335 1.02860 0.514298 0.857612i \(-0.328053\pi\)
0.514298 + 0.857612i \(0.328053\pi\)
\(702\) 0 0
\(703\) 6.79977 0.256458
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −12.8120 −0.481846
\(708\) 0 0
\(709\) −29.9175 −1.12357 −0.561787 0.827282i \(-0.689886\pi\)
−0.561787 + 0.827282i \(0.689886\pi\)
\(710\) 0 0
\(711\) −11.7242 −0.439690
\(712\) 0 0
\(713\) −1.07055 −0.0400925
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −3.99880 −0.149338
\(718\) 0 0
\(719\) 31.7693 1.18479 0.592397 0.805646i \(-0.298182\pi\)
0.592397 + 0.805646i \(0.298182\pi\)
\(720\) 0 0
\(721\) 29.6481 1.10415
\(722\) 0 0
\(723\) 6.57570 0.244553
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −23.7272 −0.879993 −0.439996 0.898000i \(-0.645020\pi\)
−0.439996 + 0.898000i \(0.645020\pi\)
\(728\) 0 0
\(729\) 17.6602 0.654082
\(730\) 0 0
\(731\) −53.4468 −1.97680
\(732\) 0 0
\(733\) 37.4907 1.38475 0.692374 0.721538i \(-0.256565\pi\)
0.692374 + 0.721538i \(0.256565\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −30.4335 −1.12103
\(738\) 0 0
\(739\) 7.10315 0.261294 0.130647 0.991429i \(-0.458295\pi\)
0.130647 + 0.991429i \(0.458295\pi\)
\(740\) 0 0
\(741\) 2.44827 0.0899394
\(742\) 0 0
\(743\) −10.6047 −0.389049 −0.194525 0.980898i \(-0.562316\pi\)
−0.194525 + 0.980898i \(0.562316\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −9.82303 −0.359406
\(748\) 0 0
\(749\) −21.2080 −0.774923
\(750\) 0 0
\(751\) 32.1380 1.17273 0.586365 0.810047i \(-0.300558\pi\)
0.586365 + 0.810047i \(0.300558\pi\)
\(752\) 0 0
\(753\) −0.158880 −0.00578991
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 36.2721 1.31833 0.659167 0.751997i \(-0.270909\pi\)
0.659167 + 0.751997i \(0.270909\pi\)
\(758\) 0 0
\(759\) −6.02510 −0.218697
\(760\) 0 0
\(761\) 31.2638 1.13331 0.566656 0.823954i \(-0.308237\pi\)
0.566656 + 0.823954i \(0.308237\pi\)
\(762\) 0 0
\(763\) −26.8346 −0.971476
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 15.4754 0.558783
\(768\) 0 0
\(769\) 48.5188 1.74963 0.874816 0.484456i \(-0.160982\pi\)
0.874816 + 0.484456i \(0.160982\pi\)
\(770\) 0 0
\(771\) 16.0788 0.579064
\(772\) 0 0
\(773\) 26.4968 0.953023 0.476512 0.879168i \(-0.341901\pi\)
0.476512 + 0.879168i \(0.341901\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 31.1088 1.11602
\(778\) 0 0
\(779\) −5.92480 −0.212278
\(780\) 0 0
\(781\) −65.9424 −2.35960
\(782\) 0 0
\(783\) 43.9675 1.57127
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −16.9510 −0.604238 −0.302119 0.953270i \(-0.597694\pi\)
−0.302119 + 0.953270i \(0.597694\pi\)
\(788\) 0 0
\(789\) −33.0230 −1.17565
\(790\) 0 0
\(791\) −82.1981 −2.92263
\(792\) 0 0
\(793\) −2.96230 −0.105194
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 53.4354 1.89278 0.946390 0.323025i \(-0.104700\pi\)
0.946390 + 0.323025i \(0.104700\pi\)
\(798\) 0 0
\(799\) 47.1568 1.66829
\(800\) 0 0
\(801\) −5.62218 −0.198650
\(802\) 0 0
\(803\) 63.4145 2.23785
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −22.1769 −0.780665
\(808\) 0 0
\(809\) −3.88632 −0.136636 −0.0683179 0.997664i \(-0.521763\pi\)
−0.0683179 + 0.997664i \(0.521763\pi\)
\(810\) 0 0
\(811\) 10.8136 0.379716 0.189858 0.981812i \(-0.439197\pi\)
0.189858 + 0.981812i \(0.439197\pi\)
\(812\) 0 0
\(813\) −1.10469 −0.0387432
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −6.81073 −0.238277
\(818\) 0 0
\(819\) −17.0822 −0.596900
\(820\) 0 0
\(821\) −43.7227 −1.52593 −0.762966 0.646438i \(-0.776258\pi\)
−0.762966 + 0.646438i \(0.776258\pi\)
\(822\) 0 0
\(823\) −4.51329 −0.157323 −0.0786617 0.996901i \(-0.525065\pi\)
−0.0786617 + 0.996901i \(0.525065\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 9.56467 0.332596 0.166298 0.986076i \(-0.446819\pi\)
0.166298 + 0.986076i \(0.446819\pi\)
\(828\) 0 0
\(829\) 20.0719 0.697126 0.348563 0.937285i \(-0.386670\pi\)
0.348563 + 0.937285i \(0.386670\pi\)
\(830\) 0 0
\(831\) 11.3736 0.394545
\(832\) 0 0
\(833\) −83.3168 −2.88675
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 6.53208 0.225782
\(838\) 0 0
\(839\) 12.4591 0.430135 0.215067 0.976599i \(-0.431003\pi\)
0.215067 + 0.976599i \(0.431003\pi\)
\(840\) 0 0
\(841\) 41.2720 1.42317
\(842\) 0 0
\(843\) 8.85869 0.305110
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 127.394 4.37730
\(848\) 0 0
\(849\) −12.6871 −0.435421
\(850\) 0 0
\(851\) 5.84508 0.200367
\(852\) 0 0
\(853\) −22.0112 −0.753649 −0.376824 0.926285i \(-0.622984\pi\)
−0.376824 + 0.926285i \(0.622984\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 15.8261 0.540608 0.270304 0.962775i \(-0.412876\pi\)
0.270304 + 0.962775i \(0.412876\pi\)
\(858\) 0 0
\(859\) −23.8728 −0.814530 −0.407265 0.913310i \(-0.633517\pi\)
−0.407265 + 0.913310i \(0.633517\pi\)
\(860\) 0 0
\(861\) −27.1058 −0.923763
\(862\) 0 0
\(863\) 29.6871 1.01056 0.505280 0.862956i \(-0.331389\pi\)
0.505280 + 0.862956i \(0.331389\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −48.5943 −1.65035
\(868\) 0 0
\(869\) −41.6090 −1.41149
\(870\) 0 0
\(871\) 10.6302 0.360192
\(872\) 0 0
\(873\) −12.2492 −0.414572
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −20.3979 −0.688790 −0.344395 0.938825i \(-0.611916\pi\)
−0.344395 + 0.938825i \(0.611916\pi\)
\(878\) 0 0
\(879\) 2.41864 0.0815786
\(880\) 0 0
\(881\) −43.2919 −1.45854 −0.729270 0.684226i \(-0.760140\pi\)
−0.729270 + 0.684226i \(0.760140\pi\)
\(882\) 0 0
\(883\) 9.74875 0.328071 0.164036 0.986454i \(-0.447549\pi\)
0.164036 + 0.986454i \(0.447549\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 10.5431 0.354002 0.177001 0.984211i \(-0.443360\pi\)
0.177001 + 0.984211i \(0.443360\pi\)
\(888\) 0 0
\(889\) 26.8761 0.901395
\(890\) 0 0
\(891\) 1.80799 0.0605699
\(892\) 0 0
\(893\) 6.00919 0.201090
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 2.10453 0.0702683
\(898\) 0 0
\(899\) 10.4400 0.348195
\(900\) 0 0
\(901\) −107.976 −3.59719
\(902\) 0 0
\(903\) −31.1589 −1.03690
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 48.2385 1.60173 0.800866 0.598844i \(-0.204373\pi\)
0.800866 + 0.598844i \(0.204373\pi\)
\(908\) 0 0
\(909\) 5.53084 0.183447
\(910\) 0 0
\(911\) −9.86236 −0.326755 −0.163377 0.986564i \(-0.552239\pi\)
−0.163377 + 0.986564i \(0.552239\pi\)
\(912\) 0 0
\(913\) −34.8619 −1.15376
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 51.1703 1.68979
\(918\) 0 0
\(919\) 38.0526 1.25524 0.627620 0.778520i \(-0.284029\pi\)
0.627620 + 0.778520i \(0.284029\pi\)
\(920\) 0 0
\(921\) −19.2614 −0.634684
\(922\) 0 0
\(923\) 23.0333 0.758149
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −12.7988 −0.420368
\(928\) 0 0
\(929\) 47.3243 1.55266 0.776330 0.630327i \(-0.217079\pi\)
0.776330 + 0.630327i \(0.217079\pi\)
\(930\) 0 0
\(931\) −10.6171 −0.347960
\(932\) 0 0
\(933\) −13.4753 −0.441163
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −19.0466 −0.622224 −0.311112 0.950373i \(-0.600701\pi\)
−0.311112 + 0.950373i \(0.600701\pi\)
\(938\) 0 0
\(939\) −9.10735 −0.297207
\(940\) 0 0
\(941\) 37.7416 1.23034 0.615171 0.788393i \(-0.289087\pi\)
0.615171 + 0.788393i \(0.289087\pi\)
\(942\) 0 0
\(943\) −5.09296 −0.165850
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 4.72833 0.153650 0.0768251 0.997045i \(-0.475522\pi\)
0.0768251 + 0.997045i \(0.475522\pi\)
\(948\) 0 0
\(949\) −22.1503 −0.719030
\(950\) 0 0
\(951\) 21.0941 0.684024
\(952\) 0 0
\(953\) 32.9799 1.06832 0.534162 0.845382i \(-0.320627\pi\)
0.534162 + 0.845382i \(0.320627\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 58.7569 1.89934
\(958\) 0 0
\(959\) −40.7650 −1.31637
\(960\) 0 0
\(961\) −29.4490 −0.949967
\(962\) 0 0
\(963\) 9.15531 0.295026
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 3.71736 0.119542 0.0597712 0.998212i \(-0.480963\pi\)
0.0597712 + 0.998212i \(0.480963\pi\)
\(968\) 0 0
\(969\) −8.55363 −0.274782
\(970\) 0 0
\(971\) 18.6887 0.599749 0.299875 0.953979i \(-0.403055\pi\)
0.299875 + 0.953979i \(0.403055\pi\)
\(972\) 0 0
\(973\) 84.2337 2.70041
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −7.44705 −0.238252 −0.119126 0.992879i \(-0.538009\pi\)
−0.119126 + 0.992879i \(0.538009\pi\)
\(978\) 0 0
\(979\) −19.9531 −0.637705
\(980\) 0 0
\(981\) 11.5842 0.369856
\(982\) 0 0
\(983\) −13.7703 −0.439204 −0.219602 0.975590i \(-0.570476\pi\)
−0.219602 + 0.975590i \(0.570476\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 27.4919 0.875077
\(988\) 0 0
\(989\) −5.85450 −0.186162
\(990\) 0 0
\(991\) 32.0927 1.01946 0.509728 0.860335i \(-0.329746\pi\)
0.509728 + 0.860335i \(0.329746\pi\)
\(992\) 0 0
\(993\) −36.9429 −1.17235
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 22.8068 0.722299 0.361149 0.932508i \(-0.382384\pi\)
0.361149 + 0.932508i \(0.382384\pi\)
\(998\) 0 0
\(999\) −35.6644 −1.12837
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 3800.2.a.bc.1.2 yes 6
4.3 odd 2 7600.2.a.ch.1.5 6
5.2 odd 4 3800.2.d.q.3649.9 12
5.3 odd 4 3800.2.d.q.3649.4 12
5.4 even 2 3800.2.a.ba.1.5 6
20.19 odd 2 7600.2.a.cl.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3800.2.a.ba.1.5 6 5.4 even 2
3800.2.a.bc.1.2 yes 6 1.1 even 1 trivial
3800.2.d.q.3649.4 12 5.3 odd 4
3800.2.d.q.3649.9 12 5.2 odd 4
7600.2.a.ch.1.5 6 4.3 odd 2
7600.2.a.cl.1.2 6 20.19 odd 2