Properties

Label 4410.2.a.bs.1.1
Level $4410$
Weight $2$
Character 4410.1
Self dual yes
Analytic conductor $35.214$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4410.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(35.2140272914\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 4410.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{8} -1.00000 q^{10} +0.585786 q^{11} +6.82843 q^{13} +1.00000 q^{16} -0.585786 q^{17} -0.828427 q^{19} +1.00000 q^{20} -0.585786 q^{22} +7.65685 q^{23} +1.00000 q^{25} -6.82843 q^{26} +2.58579 q^{29} +4.58579 q^{31} -1.00000 q^{32} +0.585786 q^{34} -0.585786 q^{37} +0.828427 q^{38} -1.00000 q^{40} -7.65685 q^{41} +7.07107 q^{43} +0.585786 q^{44} -7.65685 q^{46} -5.41421 q^{47} -1.00000 q^{50} +6.82843 q^{52} -10.4853 q^{53} +0.585786 q^{55} -2.58579 q^{58} -4.82843 q^{59} +9.65685 q^{61} -4.58579 q^{62} +1.00000 q^{64} +6.82843 q^{65} -13.8995 q^{67} -0.585786 q^{68} +6.48528 q^{71} +7.17157 q^{73} +0.585786 q^{74} -0.828427 q^{76} +2.34315 q^{79} +1.00000 q^{80} +7.65685 q^{82} +16.4853 q^{83} -0.585786 q^{85} -7.07107 q^{86} -0.585786 q^{88} +2.48528 q^{89} +7.65685 q^{92} +5.41421 q^{94} -0.828427 q^{95} -2.48528 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} + 2q^{5} - 2q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} + 2q^{5} - 2q^{8} - 2q^{10} + 4q^{11} + 8q^{13} + 2q^{16} - 4q^{17} + 4q^{19} + 2q^{20} - 4q^{22} + 4q^{23} + 2q^{25} - 8q^{26} + 8q^{29} + 12q^{31} - 2q^{32} + 4q^{34} - 4q^{37} - 4q^{38} - 2q^{40} - 4q^{41} + 4q^{44} - 4q^{46} - 8q^{47} - 2q^{50} + 8q^{52} - 4q^{53} + 4q^{55} - 8q^{58} - 4q^{59} + 8q^{61} - 12q^{62} + 2q^{64} + 8q^{65} - 8q^{67} - 4q^{68} - 4q^{71} + 20q^{73} + 4q^{74} + 4q^{76} + 16q^{79} + 2q^{80} + 4q^{82} + 16q^{83} - 4q^{85} - 4q^{88} - 12q^{89} + 4q^{92} + 8q^{94} + 4q^{95} + 12q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) 0.585786 0.176621 0.0883106 0.996093i \(-0.471853\pi\)
0.0883106 + 0.996093i \(0.471853\pi\)
\(12\) 0 0
\(13\) 6.82843 1.89386 0.946932 0.321433i \(-0.104164\pi\)
0.946932 + 0.321433i \(0.104164\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −0.585786 −0.142074 −0.0710370 0.997474i \(-0.522631\pi\)
−0.0710370 + 0.997474i \(0.522631\pi\)
\(18\) 0 0
\(19\) −0.828427 −0.190054 −0.0950271 0.995475i \(-0.530294\pi\)
−0.0950271 + 0.995475i \(0.530294\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −0.585786 −0.124890
\(23\) 7.65685 1.59656 0.798282 0.602284i \(-0.205742\pi\)
0.798282 + 0.602284i \(0.205742\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −6.82843 −1.33916
\(27\) 0 0
\(28\) 0 0
\(29\) 2.58579 0.480168 0.240084 0.970752i \(-0.422825\pi\)
0.240084 + 0.970752i \(0.422825\pi\)
\(30\) 0 0
\(31\) 4.58579 0.823632 0.411816 0.911267i \(-0.364895\pi\)
0.411816 + 0.911267i \(0.364895\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 0.585786 0.100462
\(35\) 0 0
\(36\) 0 0
\(37\) −0.585786 −0.0963027 −0.0481513 0.998840i \(-0.515333\pi\)
−0.0481513 + 0.998840i \(0.515333\pi\)
\(38\) 0.828427 0.134389
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −7.65685 −1.19580 −0.597900 0.801571i \(-0.703998\pi\)
−0.597900 + 0.801571i \(0.703998\pi\)
\(42\) 0 0
\(43\) 7.07107 1.07833 0.539164 0.842201i \(-0.318740\pi\)
0.539164 + 0.842201i \(0.318740\pi\)
\(44\) 0.585786 0.0883106
\(45\) 0 0
\(46\) −7.65685 −1.12894
\(47\) −5.41421 −0.789744 −0.394872 0.918736i \(-0.629211\pi\)
−0.394872 + 0.918736i \(0.629211\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 6.82843 0.946932
\(53\) −10.4853 −1.44026 −0.720132 0.693837i \(-0.755919\pi\)
−0.720132 + 0.693837i \(0.755919\pi\)
\(54\) 0 0
\(55\) 0.585786 0.0789874
\(56\) 0 0
\(57\) 0 0
\(58\) −2.58579 −0.339530
\(59\) −4.82843 −0.628608 −0.314304 0.949322i \(-0.601771\pi\)
−0.314304 + 0.949322i \(0.601771\pi\)
\(60\) 0 0
\(61\) 9.65685 1.23643 0.618217 0.786008i \(-0.287855\pi\)
0.618217 + 0.786008i \(0.287855\pi\)
\(62\) −4.58579 −0.582395
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 6.82843 0.846962
\(66\) 0 0
\(67\) −13.8995 −1.69809 −0.849047 0.528318i \(-0.822823\pi\)
−0.849047 + 0.528318i \(0.822823\pi\)
\(68\) −0.585786 −0.0710370
\(69\) 0 0
\(70\) 0 0
\(71\) 6.48528 0.769661 0.384831 0.922987i \(-0.374260\pi\)
0.384831 + 0.922987i \(0.374260\pi\)
\(72\) 0 0
\(73\) 7.17157 0.839369 0.419685 0.907670i \(-0.362141\pi\)
0.419685 + 0.907670i \(0.362141\pi\)
\(74\) 0.585786 0.0680963
\(75\) 0 0
\(76\) −0.828427 −0.0950271
\(77\) 0 0
\(78\) 0 0
\(79\) 2.34315 0.263624 0.131812 0.991275i \(-0.457920\pi\)
0.131812 + 0.991275i \(0.457920\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 7.65685 0.845558
\(83\) 16.4853 1.80949 0.904747 0.425949i \(-0.140060\pi\)
0.904747 + 0.425949i \(0.140060\pi\)
\(84\) 0 0
\(85\) −0.585786 −0.0635375
\(86\) −7.07107 −0.762493
\(87\) 0 0
\(88\) −0.585786 −0.0624450
\(89\) 2.48528 0.263439 0.131720 0.991287i \(-0.457950\pi\)
0.131720 + 0.991287i \(0.457950\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 7.65685 0.798282
\(93\) 0 0
\(94\) 5.41421 0.558433
\(95\) −0.828427 −0.0849948
\(96\) 0 0
\(97\) −2.48528 −0.252342 −0.126171 0.992009i \(-0.540269\pi\)
−0.126171 + 0.992009i \(0.540269\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 1.65685 0.164863 0.0824316 0.996597i \(-0.473731\pi\)
0.0824316 + 0.996597i \(0.473731\pi\)
\(102\) 0 0
\(103\) −5.65685 −0.557386 −0.278693 0.960380i \(-0.589901\pi\)
−0.278693 + 0.960380i \(0.589901\pi\)
\(104\) −6.82843 −0.669582
\(105\) 0 0
\(106\) 10.4853 1.01842
\(107\) −5.17157 −0.499955 −0.249977 0.968252i \(-0.580423\pi\)
−0.249977 + 0.968252i \(0.580423\pi\)
\(108\) 0 0
\(109\) −2.48528 −0.238047 −0.119023 0.992891i \(-0.537976\pi\)
−0.119023 + 0.992891i \(0.537976\pi\)
\(110\) −0.585786 −0.0558525
\(111\) 0 0
\(112\) 0 0
\(113\) −15.6569 −1.47287 −0.736436 0.676507i \(-0.763493\pi\)
−0.736436 + 0.676507i \(0.763493\pi\)
\(114\) 0 0
\(115\) 7.65685 0.714005
\(116\) 2.58579 0.240084
\(117\) 0 0
\(118\) 4.82843 0.444493
\(119\) 0 0
\(120\) 0 0
\(121\) −10.6569 −0.968805
\(122\) −9.65685 −0.874291
\(123\) 0 0
\(124\) 4.58579 0.411816
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 14.1421 1.25491 0.627456 0.778652i \(-0.284096\pi\)
0.627456 + 0.778652i \(0.284096\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −6.82843 −0.598893
\(131\) −10.4853 −0.916103 −0.458052 0.888926i \(-0.651453\pi\)
−0.458052 + 0.888926i \(0.651453\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 13.8995 1.20073
\(135\) 0 0
\(136\) 0.585786 0.0502308
\(137\) −12.4853 −1.06669 −0.533345 0.845898i \(-0.679065\pi\)
−0.533345 + 0.845898i \(0.679065\pi\)
\(138\) 0 0
\(139\) 21.3137 1.80781 0.903903 0.427738i \(-0.140690\pi\)
0.903903 + 0.427738i \(0.140690\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −6.48528 −0.544233
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) 2.58579 0.214738
\(146\) −7.17157 −0.593524
\(147\) 0 0
\(148\) −0.585786 −0.0481513
\(149\) 12.7279 1.04271 0.521356 0.853339i \(-0.325426\pi\)
0.521356 + 0.853339i \(0.325426\pi\)
\(150\) 0 0
\(151\) 8.82843 0.718447 0.359224 0.933252i \(-0.383042\pi\)
0.359224 + 0.933252i \(0.383042\pi\)
\(152\) 0.828427 0.0671943
\(153\) 0 0
\(154\) 0 0
\(155\) 4.58579 0.368339
\(156\) 0 0
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) −2.34315 −0.186411
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 0 0
\(163\) −7.07107 −0.553849 −0.276924 0.960892i \(-0.589315\pi\)
−0.276924 + 0.960892i \(0.589315\pi\)
\(164\) −7.65685 −0.597900
\(165\) 0 0
\(166\) −16.4853 −1.27951
\(167\) 9.41421 0.728494 0.364247 0.931302i \(-0.381326\pi\)
0.364247 + 0.931302i \(0.381326\pi\)
\(168\) 0 0
\(169\) 33.6274 2.58672
\(170\) 0.585786 0.0449278
\(171\) 0 0
\(172\) 7.07107 0.539164
\(173\) −8.14214 −0.619035 −0.309518 0.950894i \(-0.600168\pi\)
−0.309518 + 0.950894i \(0.600168\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0.585786 0.0441553
\(177\) 0 0
\(178\) −2.48528 −0.186280
\(179\) −14.7279 −1.10082 −0.550408 0.834896i \(-0.685528\pi\)
−0.550408 + 0.834896i \(0.685528\pi\)
\(180\) 0 0
\(181\) 20.4853 1.52266 0.761329 0.648365i \(-0.224547\pi\)
0.761329 + 0.648365i \(0.224547\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −7.65685 −0.564471
\(185\) −0.585786 −0.0430679
\(186\) 0 0
\(187\) −0.343146 −0.0250933
\(188\) −5.41421 −0.394872
\(189\) 0 0
\(190\) 0.828427 0.0601004
\(191\) 13.3137 0.963346 0.481673 0.876351i \(-0.340029\pi\)
0.481673 + 0.876351i \(0.340029\pi\)
\(192\) 0 0
\(193\) −8.14214 −0.586084 −0.293042 0.956100i \(-0.594668\pi\)
−0.293042 + 0.956100i \(0.594668\pi\)
\(194\) 2.48528 0.178433
\(195\) 0 0
\(196\) 0 0
\(197\) −11.6569 −0.830516 −0.415258 0.909704i \(-0.636309\pi\)
−0.415258 + 0.909704i \(0.636309\pi\)
\(198\) 0 0
\(199\) 9.75736 0.691681 0.345840 0.938293i \(-0.387594\pi\)
0.345840 + 0.938293i \(0.387594\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −1.65685 −0.116576
\(203\) 0 0
\(204\) 0 0
\(205\) −7.65685 −0.534778
\(206\) 5.65685 0.394132
\(207\) 0 0
\(208\) 6.82843 0.473466
\(209\) −0.485281 −0.0335676
\(210\) 0 0
\(211\) 15.7990 1.08765 0.543824 0.839200i \(-0.316976\pi\)
0.543824 + 0.839200i \(0.316976\pi\)
\(212\) −10.4853 −0.720132
\(213\) 0 0
\(214\) 5.17157 0.353521
\(215\) 7.07107 0.482243
\(216\) 0 0
\(217\) 0 0
\(218\) 2.48528 0.168324
\(219\) 0 0
\(220\) 0.585786 0.0394937
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) −5.17157 −0.346314 −0.173157 0.984894i \(-0.555397\pi\)
−0.173157 + 0.984894i \(0.555397\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 15.6569 1.04148
\(227\) −8.97056 −0.595397 −0.297699 0.954660i \(-0.596219\pi\)
−0.297699 + 0.954660i \(0.596219\pi\)
\(228\) 0 0
\(229\) −10.8284 −0.715563 −0.357781 0.933805i \(-0.616467\pi\)
−0.357781 + 0.933805i \(0.616467\pi\)
\(230\) −7.65685 −0.504878
\(231\) 0 0
\(232\) −2.58579 −0.169765
\(233\) −12.3431 −0.808626 −0.404313 0.914621i \(-0.632489\pi\)
−0.404313 + 0.914621i \(0.632489\pi\)
\(234\) 0 0
\(235\) −5.41421 −0.353184
\(236\) −4.82843 −0.314304
\(237\) 0 0
\(238\) 0 0
\(239\) 8.34315 0.539673 0.269837 0.962906i \(-0.413030\pi\)
0.269837 + 0.962906i \(0.413030\pi\)
\(240\) 0 0
\(241\) 4.92893 0.317500 0.158750 0.987319i \(-0.449254\pi\)
0.158750 + 0.987319i \(0.449254\pi\)
\(242\) 10.6569 0.685049
\(243\) 0 0
\(244\) 9.65685 0.618217
\(245\) 0 0
\(246\) 0 0
\(247\) −5.65685 −0.359937
\(248\) −4.58579 −0.291198
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 25.6569 1.61945 0.809723 0.586812i \(-0.199617\pi\)
0.809723 + 0.586812i \(0.199617\pi\)
\(252\) 0 0
\(253\) 4.48528 0.281987
\(254\) −14.1421 −0.887357
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 15.8995 0.991783 0.495892 0.868384i \(-0.334841\pi\)
0.495892 + 0.868384i \(0.334841\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 6.82843 0.423481
\(261\) 0 0
\(262\) 10.4853 0.647783
\(263\) 10.0000 0.616626 0.308313 0.951285i \(-0.400236\pi\)
0.308313 + 0.951285i \(0.400236\pi\)
\(264\) 0 0
\(265\) −10.4853 −0.644106
\(266\) 0 0
\(267\) 0 0
\(268\) −13.8995 −0.849047
\(269\) −18.9706 −1.15666 −0.578328 0.815804i \(-0.696295\pi\)
−0.578328 + 0.815804i \(0.696295\pi\)
\(270\) 0 0
\(271\) 10.2426 0.622196 0.311098 0.950378i \(-0.399303\pi\)
0.311098 + 0.950378i \(0.399303\pi\)
\(272\) −0.585786 −0.0355185
\(273\) 0 0
\(274\) 12.4853 0.754263
\(275\) 0.585786 0.0353243
\(276\) 0 0
\(277\) −28.5858 −1.71755 −0.858777 0.512350i \(-0.828775\pi\)
−0.858777 + 0.512350i \(0.828775\pi\)
\(278\) −21.3137 −1.27831
\(279\) 0 0
\(280\) 0 0
\(281\) −29.4558 −1.75719 −0.878594 0.477569i \(-0.841518\pi\)
−0.878594 + 0.477569i \(0.841518\pi\)
\(282\) 0 0
\(283\) 18.9706 1.12768 0.563841 0.825883i \(-0.309323\pi\)
0.563841 + 0.825883i \(0.309323\pi\)
\(284\) 6.48528 0.384831
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) 0 0
\(288\) 0 0
\(289\) −16.6569 −0.979815
\(290\) −2.58579 −0.151843
\(291\) 0 0
\(292\) 7.17157 0.419685
\(293\) −2.68629 −0.156935 −0.0784674 0.996917i \(-0.525003\pi\)
−0.0784674 + 0.996917i \(0.525003\pi\)
\(294\) 0 0
\(295\) −4.82843 −0.281122
\(296\) 0.585786 0.0340481
\(297\) 0 0
\(298\) −12.7279 −0.737309
\(299\) 52.2843 3.02368
\(300\) 0 0
\(301\) 0 0
\(302\) −8.82843 −0.508019
\(303\) 0 0
\(304\) −0.828427 −0.0475136
\(305\) 9.65685 0.552950
\(306\) 0 0
\(307\) −18.6274 −1.06312 −0.531561 0.847020i \(-0.678395\pi\)
−0.531561 + 0.847020i \(0.678395\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −4.58579 −0.260455
\(311\) 29.4558 1.67029 0.835144 0.550032i \(-0.185384\pi\)
0.835144 + 0.550032i \(0.185384\pi\)
\(312\) 0 0
\(313\) −3.65685 −0.206698 −0.103349 0.994645i \(-0.532956\pi\)
−0.103349 + 0.994645i \(0.532956\pi\)
\(314\) −10.0000 −0.564333
\(315\) 0 0
\(316\) 2.34315 0.131812
\(317\) 18.9706 1.06549 0.532746 0.846275i \(-0.321160\pi\)
0.532746 + 0.846275i \(0.321160\pi\)
\(318\) 0 0
\(319\) 1.51472 0.0848080
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 0 0
\(323\) 0.485281 0.0270018
\(324\) 0 0
\(325\) 6.82843 0.378773
\(326\) 7.07107 0.391630
\(327\) 0 0
\(328\) 7.65685 0.422779
\(329\) 0 0
\(330\) 0 0
\(331\) −23.7990 −1.30811 −0.654055 0.756447i \(-0.726934\pi\)
−0.654055 + 0.756447i \(0.726934\pi\)
\(332\) 16.4853 0.904747
\(333\) 0 0
\(334\) −9.41421 −0.515123
\(335\) −13.8995 −0.759411
\(336\) 0 0
\(337\) −9.51472 −0.518300 −0.259150 0.965837i \(-0.583442\pi\)
−0.259150 + 0.965837i \(0.583442\pi\)
\(338\) −33.6274 −1.82909
\(339\) 0 0
\(340\) −0.585786 −0.0317687
\(341\) 2.68629 0.145471
\(342\) 0 0
\(343\) 0 0
\(344\) −7.07107 −0.381246
\(345\) 0 0
\(346\) 8.14214 0.437724
\(347\) −6.34315 −0.340518 −0.170259 0.985399i \(-0.554460\pi\)
−0.170259 + 0.985399i \(0.554460\pi\)
\(348\) 0 0
\(349\) −3.51472 −0.188139 −0.0940693 0.995566i \(-0.529988\pi\)
−0.0940693 + 0.995566i \(0.529988\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −0.585786 −0.0312225
\(353\) 7.89949 0.420448 0.210224 0.977653i \(-0.432581\pi\)
0.210224 + 0.977653i \(0.432581\pi\)
\(354\) 0 0
\(355\) 6.48528 0.344203
\(356\) 2.48528 0.131720
\(357\) 0 0
\(358\) 14.7279 0.778395
\(359\) 27.1716 1.43406 0.717030 0.697042i \(-0.245501\pi\)
0.717030 + 0.697042i \(0.245501\pi\)
\(360\) 0 0
\(361\) −18.3137 −0.963879
\(362\) −20.4853 −1.07668
\(363\) 0 0
\(364\) 0 0
\(365\) 7.17157 0.375377
\(366\) 0 0
\(367\) 16.4853 0.860525 0.430262 0.902704i \(-0.358421\pi\)
0.430262 + 0.902704i \(0.358421\pi\)
\(368\) 7.65685 0.399141
\(369\) 0 0
\(370\) 0.585786 0.0304536
\(371\) 0 0
\(372\) 0 0
\(373\) 30.2426 1.56590 0.782952 0.622082i \(-0.213713\pi\)
0.782952 + 0.622082i \(0.213713\pi\)
\(374\) 0.343146 0.0177436
\(375\) 0 0
\(376\) 5.41421 0.279217
\(377\) 17.6569 0.909374
\(378\) 0 0
\(379\) −12.4853 −0.641326 −0.320663 0.947193i \(-0.603906\pi\)
−0.320663 + 0.947193i \(0.603906\pi\)
\(380\) −0.828427 −0.0424974
\(381\) 0 0
\(382\) −13.3137 −0.681189
\(383\) −29.6985 −1.51752 −0.758761 0.651369i \(-0.774195\pi\)
−0.758761 + 0.651369i \(0.774195\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 8.14214 0.414424
\(387\) 0 0
\(388\) −2.48528 −0.126171
\(389\) 32.0416 1.62458 0.812288 0.583257i \(-0.198222\pi\)
0.812288 + 0.583257i \(0.198222\pi\)
\(390\) 0 0
\(391\) −4.48528 −0.226830
\(392\) 0 0
\(393\) 0 0
\(394\) 11.6569 0.587264
\(395\) 2.34315 0.117896
\(396\) 0 0
\(397\) 19.6569 0.986549 0.493275 0.869874i \(-0.335800\pi\)
0.493275 + 0.869874i \(0.335800\pi\)
\(398\) −9.75736 −0.489092
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 28.9706 1.44672 0.723360 0.690471i \(-0.242597\pi\)
0.723360 + 0.690471i \(0.242597\pi\)
\(402\) 0 0
\(403\) 31.3137 1.55985
\(404\) 1.65685 0.0824316
\(405\) 0 0
\(406\) 0 0
\(407\) −0.343146 −0.0170091
\(408\) 0 0
\(409\) −26.8701 −1.32864 −0.664319 0.747449i \(-0.731279\pi\)
−0.664319 + 0.747449i \(0.731279\pi\)
\(410\) 7.65685 0.378145
\(411\) 0 0
\(412\) −5.65685 −0.278693
\(413\) 0 0
\(414\) 0 0
\(415\) 16.4853 0.809231
\(416\) −6.82843 −0.334791
\(417\) 0 0
\(418\) 0.485281 0.0237359
\(419\) −16.2843 −0.795539 −0.397769 0.917485i \(-0.630216\pi\)
−0.397769 + 0.917485i \(0.630216\pi\)
\(420\) 0 0
\(421\) 3.17157 0.154573 0.0772865 0.997009i \(-0.475374\pi\)
0.0772865 + 0.997009i \(0.475374\pi\)
\(422\) −15.7990 −0.769083
\(423\) 0 0
\(424\) 10.4853 0.509210
\(425\) −0.585786 −0.0284148
\(426\) 0 0
\(427\) 0 0
\(428\) −5.17157 −0.249977
\(429\) 0 0
\(430\) −7.07107 −0.340997
\(431\) −12.1421 −0.584866 −0.292433 0.956286i \(-0.594465\pi\)
−0.292433 + 0.956286i \(0.594465\pi\)
\(432\) 0 0
\(433\) 27.1716 1.30578 0.652891 0.757452i \(-0.273556\pi\)
0.652891 + 0.757452i \(0.273556\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −2.48528 −0.119023
\(437\) −6.34315 −0.303434
\(438\) 0 0
\(439\) 5.75736 0.274784 0.137392 0.990517i \(-0.456128\pi\)
0.137392 + 0.990517i \(0.456128\pi\)
\(440\) −0.585786 −0.0279263
\(441\) 0 0
\(442\) 4.00000 0.190261
\(443\) −32.9706 −1.56648 −0.783239 0.621720i \(-0.786434\pi\)
−0.783239 + 0.621720i \(0.786434\pi\)
\(444\) 0 0
\(445\) 2.48528 0.117814
\(446\) 5.17157 0.244881
\(447\) 0 0
\(448\) 0 0
\(449\) 33.1716 1.56546 0.782732 0.622359i \(-0.213826\pi\)
0.782732 + 0.622359i \(0.213826\pi\)
\(450\) 0 0
\(451\) −4.48528 −0.211204
\(452\) −15.6569 −0.736436
\(453\) 0 0
\(454\) 8.97056 0.421009
\(455\) 0 0
\(456\) 0 0
\(457\) −28.1421 −1.31643 −0.658217 0.752828i \(-0.728689\pi\)
−0.658217 + 0.752828i \(0.728689\pi\)
\(458\) 10.8284 0.505979
\(459\) 0 0
\(460\) 7.65685 0.357003
\(461\) 25.3137 1.17898 0.589488 0.807777i \(-0.299329\pi\)
0.589488 + 0.807777i \(0.299329\pi\)
\(462\) 0 0
\(463\) 36.0000 1.67306 0.836531 0.547920i \(-0.184580\pi\)
0.836531 + 0.547920i \(0.184580\pi\)
\(464\) 2.58579 0.120042
\(465\) 0 0
\(466\) 12.3431 0.571785
\(467\) 33.1716 1.53500 0.767499 0.641051i \(-0.221501\pi\)
0.767499 + 0.641051i \(0.221501\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 5.41421 0.249739
\(471\) 0 0
\(472\) 4.82843 0.222246
\(473\) 4.14214 0.190456
\(474\) 0 0
\(475\) −0.828427 −0.0380108
\(476\) 0 0
\(477\) 0 0
\(478\) −8.34315 −0.381607
\(479\) 25.1716 1.15012 0.575059 0.818112i \(-0.304979\pi\)
0.575059 + 0.818112i \(0.304979\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) −4.92893 −0.224507
\(483\) 0 0
\(484\) −10.6569 −0.484402
\(485\) −2.48528 −0.112851
\(486\) 0 0
\(487\) −14.8284 −0.671940 −0.335970 0.941873i \(-0.609064\pi\)
−0.335970 + 0.941873i \(0.609064\pi\)
\(488\) −9.65685 −0.437145
\(489\) 0 0
\(490\) 0 0
\(491\) −33.0711 −1.49248 −0.746238 0.665679i \(-0.768142\pi\)
−0.746238 + 0.665679i \(0.768142\pi\)
\(492\) 0 0
\(493\) −1.51472 −0.0682195
\(494\) 5.65685 0.254514
\(495\) 0 0
\(496\) 4.58579 0.205908
\(497\) 0 0
\(498\) 0 0
\(499\) 31.1127 1.39280 0.696398 0.717656i \(-0.254785\pi\)
0.696398 + 0.717656i \(0.254785\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) −25.6569 −1.14512
\(503\) 22.5858 1.00705 0.503525 0.863981i \(-0.332036\pi\)
0.503525 + 0.863981i \(0.332036\pi\)
\(504\) 0 0
\(505\) 1.65685 0.0737290
\(506\) −4.48528 −0.199395
\(507\) 0 0
\(508\) 14.1421 0.627456
\(509\) 43.5980 1.93245 0.966223 0.257707i \(-0.0829668\pi\)
0.966223 + 0.257707i \(0.0829668\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −15.8995 −0.701297
\(515\) −5.65685 −0.249271
\(516\) 0 0
\(517\) −3.17157 −0.139486
\(518\) 0 0
\(519\) 0 0
\(520\) −6.82843 −0.299446
\(521\) 15.1716 0.664679 0.332339 0.943160i \(-0.392162\pi\)
0.332339 + 0.943160i \(0.392162\pi\)
\(522\) 0 0
\(523\) −20.6274 −0.901974 −0.450987 0.892531i \(-0.648928\pi\)
−0.450987 + 0.892531i \(0.648928\pi\)
\(524\) −10.4853 −0.458052
\(525\) 0 0
\(526\) −10.0000 −0.436021
\(527\) −2.68629 −0.117017
\(528\) 0 0
\(529\) 35.6274 1.54902
\(530\) 10.4853 0.455452
\(531\) 0 0
\(532\) 0 0
\(533\) −52.2843 −2.26468
\(534\) 0 0
\(535\) −5.17157 −0.223587
\(536\) 13.8995 0.600367
\(537\) 0 0
\(538\) 18.9706 0.817879
\(539\) 0 0
\(540\) 0 0
\(541\) 23.9411 1.02931 0.514655 0.857398i \(-0.327920\pi\)
0.514655 + 0.857398i \(0.327920\pi\)
\(542\) −10.2426 −0.439959
\(543\) 0 0
\(544\) 0.585786 0.0251154
\(545\) −2.48528 −0.106458
\(546\) 0 0
\(547\) −32.7279 −1.39934 −0.699672 0.714464i \(-0.746671\pi\)
−0.699672 + 0.714464i \(0.746671\pi\)
\(548\) −12.4853 −0.533345
\(549\) 0 0
\(550\) −0.585786 −0.0249780
\(551\) −2.14214 −0.0912580
\(552\) 0 0
\(553\) 0 0
\(554\) 28.5858 1.21449
\(555\) 0 0
\(556\) 21.3137 0.903903
\(557\) 29.3137 1.24206 0.621031 0.783786i \(-0.286714\pi\)
0.621031 + 0.783786i \(0.286714\pi\)
\(558\) 0 0
\(559\) 48.2843 2.04221
\(560\) 0 0
\(561\) 0 0
\(562\) 29.4558 1.24252
\(563\) 23.7990 1.00301 0.501504 0.865155i \(-0.332780\pi\)
0.501504 + 0.865155i \(0.332780\pi\)
\(564\) 0 0
\(565\) −15.6569 −0.658689
\(566\) −18.9706 −0.797392
\(567\) 0 0
\(568\) −6.48528 −0.272116
\(569\) 23.3137 0.977362 0.488681 0.872463i \(-0.337478\pi\)
0.488681 + 0.872463i \(0.337478\pi\)
\(570\) 0 0
\(571\) −27.3137 −1.14304 −0.571522 0.820587i \(-0.693647\pi\)
−0.571522 + 0.820587i \(0.693647\pi\)
\(572\) 4.00000 0.167248
\(573\) 0 0
\(574\) 0 0
\(575\) 7.65685 0.319313
\(576\) 0 0
\(577\) 8.62742 0.359164 0.179582 0.983743i \(-0.442525\pi\)
0.179582 + 0.983743i \(0.442525\pi\)
\(578\) 16.6569 0.692834
\(579\) 0 0
\(580\) 2.58579 0.107369
\(581\) 0 0
\(582\) 0 0
\(583\) −6.14214 −0.254381
\(584\) −7.17157 −0.296762
\(585\) 0 0
\(586\) 2.68629 0.110970
\(587\) 18.1421 0.748806 0.374403 0.927266i \(-0.377848\pi\)
0.374403 + 0.927266i \(0.377848\pi\)
\(588\) 0 0
\(589\) −3.79899 −0.156535
\(590\) 4.82843 0.198783
\(591\) 0 0
\(592\) −0.585786 −0.0240757
\(593\) −19.6985 −0.808920 −0.404460 0.914556i \(-0.632540\pi\)
−0.404460 + 0.914556i \(0.632540\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 12.7279 0.521356
\(597\) 0 0
\(598\) −52.2843 −2.13806
\(599\) −12.6274 −0.515942 −0.257971 0.966153i \(-0.583054\pi\)
−0.257971 + 0.966153i \(0.583054\pi\)
\(600\) 0 0
\(601\) −9.61522 −0.392213 −0.196107 0.980583i \(-0.562830\pi\)
−0.196107 + 0.980583i \(0.562830\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 8.82843 0.359224
\(605\) −10.6569 −0.433263
\(606\) 0 0
\(607\) 31.1127 1.26283 0.631413 0.775447i \(-0.282475\pi\)
0.631413 + 0.775447i \(0.282475\pi\)
\(608\) 0.828427 0.0335972
\(609\) 0 0
\(610\) −9.65685 −0.390995
\(611\) −36.9706 −1.49567
\(612\) 0 0
\(613\) 16.5858 0.669894 0.334947 0.942237i \(-0.391282\pi\)
0.334947 + 0.942237i \(0.391282\pi\)
\(614\) 18.6274 0.751741
\(615\) 0 0
\(616\) 0 0
\(617\) −23.1127 −0.930482 −0.465241 0.885184i \(-0.654032\pi\)
−0.465241 + 0.885184i \(0.654032\pi\)
\(618\) 0 0
\(619\) −4.82843 −0.194071 −0.0970354 0.995281i \(-0.530936\pi\)
−0.0970354 + 0.995281i \(0.530936\pi\)
\(620\) 4.58579 0.184170
\(621\) 0 0
\(622\) −29.4558 −1.18107
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 3.65685 0.146157
\(627\) 0 0
\(628\) 10.0000 0.399043
\(629\) 0.343146 0.0136821
\(630\) 0 0
\(631\) −35.4558 −1.41147 −0.705737 0.708473i \(-0.749384\pi\)
−0.705737 + 0.708473i \(0.749384\pi\)
\(632\) −2.34315 −0.0932053
\(633\) 0 0
\(634\) −18.9706 −0.753417
\(635\) 14.1421 0.561214
\(636\) 0 0
\(637\) 0 0
\(638\) −1.51472 −0.0599683
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 26.6274 1.05172 0.525860 0.850571i \(-0.323744\pi\)
0.525860 + 0.850571i \(0.323744\pi\)
\(642\) 0 0
\(643\) −26.9706 −1.06362 −0.531808 0.846865i \(-0.678487\pi\)
−0.531808 + 0.846865i \(0.678487\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −0.485281 −0.0190931
\(647\) 50.3848 1.98083 0.990415 0.138124i \(-0.0441072\pi\)
0.990415 + 0.138124i \(0.0441072\pi\)
\(648\) 0 0
\(649\) −2.82843 −0.111025
\(650\) −6.82843 −0.267833
\(651\) 0 0
\(652\) −7.07107 −0.276924
\(653\) 17.1127 0.669672 0.334836 0.942276i \(-0.391319\pi\)
0.334836 + 0.942276i \(0.391319\pi\)
\(654\) 0 0
\(655\) −10.4853 −0.409694
\(656\) −7.65685 −0.298950
\(657\) 0 0
\(658\) 0 0
\(659\) 31.4142 1.22372 0.611862 0.790965i \(-0.290421\pi\)
0.611862 + 0.790965i \(0.290421\pi\)
\(660\) 0 0
\(661\) −33.6569 −1.30910 −0.654550 0.756019i \(-0.727142\pi\)
−0.654550 + 0.756019i \(0.727142\pi\)
\(662\) 23.7990 0.924974
\(663\) 0 0
\(664\) −16.4853 −0.639753
\(665\) 0 0
\(666\) 0 0
\(667\) 19.7990 0.766620
\(668\) 9.41421 0.364247
\(669\) 0 0
\(670\) 13.8995 0.536984
\(671\) 5.65685 0.218380
\(672\) 0 0
\(673\) −22.9706 −0.885450 −0.442725 0.896657i \(-0.645988\pi\)
−0.442725 + 0.896657i \(0.645988\pi\)
\(674\) 9.51472 0.366493
\(675\) 0 0
\(676\) 33.6274 1.29336
\(677\) −35.4558 −1.36268 −0.681339 0.731968i \(-0.738602\pi\)
−0.681339 + 0.731968i \(0.738602\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0.585786 0.0224639
\(681\) 0 0
\(682\) −2.68629 −0.102863
\(683\) −37.6569 −1.44090 −0.720450 0.693507i \(-0.756065\pi\)
−0.720450 + 0.693507i \(0.756065\pi\)
\(684\) 0 0
\(685\) −12.4853 −0.477038
\(686\) 0 0
\(687\) 0 0
\(688\) 7.07107 0.269582
\(689\) −71.5980 −2.72767
\(690\) 0 0
\(691\) −34.9706 −1.33034 −0.665171 0.746691i \(-0.731642\pi\)
−0.665171 + 0.746691i \(0.731642\pi\)
\(692\) −8.14214 −0.309518
\(693\) 0 0
\(694\) 6.34315 0.240783
\(695\) 21.3137 0.808475
\(696\) 0 0
\(697\) 4.48528 0.169892
\(698\) 3.51472 0.133034
\(699\) 0 0
\(700\) 0 0
\(701\) −16.2426 −0.613476 −0.306738 0.951794i \(-0.599238\pi\)
−0.306738 + 0.951794i \(0.599238\pi\)
\(702\) 0 0
\(703\) 0.485281 0.0183027
\(704\) 0.585786 0.0220777
\(705\) 0 0
\(706\) −7.89949 −0.297301
\(707\) 0 0
\(708\) 0 0
\(709\) −41.1127 −1.54402 −0.772010 0.635611i \(-0.780748\pi\)
−0.772010 + 0.635611i \(0.780748\pi\)
\(710\) −6.48528 −0.243388
\(711\) 0 0
\(712\) −2.48528 −0.0931399
\(713\) 35.1127 1.31498
\(714\) 0 0
\(715\) 4.00000 0.149592
\(716\) −14.7279 −0.550408
\(717\) 0 0
\(718\) −27.1716 −1.01403
\(719\) 17.4558 0.650993 0.325497 0.945543i \(-0.394469\pi\)
0.325497 + 0.945543i \(0.394469\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 18.3137 0.681566
\(723\) 0 0
\(724\) 20.4853 0.761329
\(725\) 2.58579 0.0960337
\(726\) 0 0
\(727\) −37.4558 −1.38916 −0.694580 0.719415i \(-0.744410\pi\)
−0.694580 + 0.719415i \(0.744410\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −7.17157 −0.265432
\(731\) −4.14214 −0.153202
\(732\) 0 0
\(733\) 26.9706 0.996180 0.498090 0.867125i \(-0.334035\pi\)
0.498090 + 0.867125i \(0.334035\pi\)
\(734\) −16.4853 −0.608483
\(735\) 0 0
\(736\) −7.65685 −0.282235
\(737\) −8.14214 −0.299919
\(738\) 0 0
\(739\) 2.82843 0.104045 0.0520227 0.998646i \(-0.483433\pi\)
0.0520227 + 0.998646i \(0.483433\pi\)
\(740\) −0.585786 −0.0215339
\(741\) 0 0
\(742\) 0 0
\(743\) −34.3431 −1.25993 −0.629964 0.776624i \(-0.716930\pi\)
−0.629964 + 0.776624i \(0.716930\pi\)
\(744\) 0 0
\(745\) 12.7279 0.466315
\(746\) −30.2426 −1.10726
\(747\) 0 0
\(748\) −0.343146 −0.0125467
\(749\) 0 0
\(750\) 0 0
\(751\) −2.20101 −0.0803160 −0.0401580 0.999193i \(-0.512786\pi\)
−0.0401580 + 0.999193i \(0.512786\pi\)
\(752\) −5.41421 −0.197436
\(753\) 0 0
\(754\) −17.6569 −0.643025
\(755\) 8.82843 0.321299
\(756\) 0 0
\(757\) −49.3553 −1.79385 −0.896925 0.442182i \(-0.854204\pi\)
−0.896925 + 0.442182i \(0.854204\pi\)
\(758\) 12.4853 0.453486
\(759\) 0 0
\(760\) 0.828427 0.0300502
\(761\) −36.6274 −1.32774 −0.663871 0.747847i \(-0.731088\pi\)
−0.663871 + 0.747847i \(0.731088\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 13.3137 0.481673
\(765\) 0 0
\(766\) 29.6985 1.07305
\(767\) −32.9706 −1.19050
\(768\) 0 0
\(769\) −27.0711 −0.976208 −0.488104 0.872786i \(-0.662311\pi\)
−0.488104 + 0.872786i \(0.662311\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −8.14214 −0.293042
\(773\) 14.9706 0.538454 0.269227 0.963077i \(-0.413232\pi\)
0.269227 + 0.963077i \(0.413232\pi\)
\(774\) 0 0
\(775\) 4.58579 0.164726
\(776\) 2.48528 0.0892164
\(777\) 0 0
\(778\) −32.0416 −1.14875
\(779\) 6.34315 0.227267
\(780\) 0 0
\(781\) 3.79899 0.135939
\(782\) 4.48528 0.160393
\(783\) 0 0
\(784\) 0 0
\(785\) 10.0000 0.356915
\(786\) 0 0
\(787\) −36.9706 −1.31786 −0.658929 0.752205i \(-0.728990\pi\)
−0.658929 + 0.752205i \(0.728990\pi\)
\(788\) −11.6569 −0.415258
\(789\) 0 0
\(790\) −2.34315 −0.0833654
\(791\) 0 0
\(792\) 0 0
\(793\) 65.9411 2.34164
\(794\) −19.6569 −0.697596
\(795\) 0 0
\(796\) 9.75736 0.345840
\(797\) −36.1421 −1.28022 −0.640110 0.768283i \(-0.721111\pi\)
−0.640110 + 0.768283i \(0.721111\pi\)
\(798\) 0 0
\(799\) 3.17157 0.112202
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) −28.9706 −1.02299
\(803\) 4.20101 0.148250
\(804\) 0 0
\(805\) 0 0
\(806\) −31.3137 −1.10298
\(807\) 0 0
\(808\) −1.65685 −0.0582879
\(809\) −0.485281 −0.0170616 −0.00853079 0.999964i \(-0.502715\pi\)
−0.00853079 + 0.999964i \(0.502715\pi\)
\(810\) 0 0
\(811\) 20.6274 0.724327 0.362163 0.932115i \(-0.382038\pi\)
0.362163 + 0.932115i \(0.382038\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0.343146 0.0120273
\(815\) −7.07107 −0.247689
\(816\) 0 0
\(817\) −5.85786 −0.204941
\(818\) 26.8701 0.939490
\(819\) 0 0
\(820\) −7.65685 −0.267389
\(821\) −43.0711 −1.50319 −0.751595 0.659625i \(-0.770715\pi\)
−0.751595 + 0.659625i \(0.770715\pi\)
\(822\) 0 0
\(823\) 21.6569 0.754910 0.377455 0.926028i \(-0.376799\pi\)
0.377455 + 0.926028i \(0.376799\pi\)
\(824\) 5.65685 0.197066
\(825\) 0 0
\(826\) 0 0
\(827\) 15.7990 0.549385 0.274692 0.961532i \(-0.411424\pi\)
0.274692 + 0.961532i \(0.411424\pi\)
\(828\) 0 0
\(829\) −16.0000 −0.555703 −0.277851 0.960624i \(-0.589622\pi\)
−0.277851 + 0.960624i \(0.589622\pi\)
\(830\) −16.4853 −0.572212
\(831\) 0 0
\(832\) 6.82843 0.236733
\(833\) 0 0
\(834\) 0 0
\(835\) 9.41421 0.325792
\(836\) −0.485281 −0.0167838
\(837\) 0 0
\(838\) 16.2843 0.562531
\(839\) −32.2843 −1.11458 −0.557288 0.830319i \(-0.688158\pi\)
−0.557288 + 0.830319i \(0.688158\pi\)
\(840\) 0 0
\(841\) −22.3137 −0.769438
\(842\) −3.17157 −0.109300
\(843\) 0 0
\(844\) 15.7990 0.543824
\(845\) 33.6274 1.15682
\(846\) 0 0
\(847\) 0 0
\(848\) −10.4853 −0.360066
\(849\) 0 0
\(850\) 0.585786 0.0200923
\(851\) −4.48528 −0.153753
\(852\) 0 0
\(853\) −40.3431 −1.38132 −0.690662 0.723178i \(-0.742681\pi\)
−0.690662 + 0.723178i \(0.742681\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 5.17157 0.176761
\(857\) −32.1005 −1.09653 −0.548266 0.836304i \(-0.684712\pi\)
−0.548266 + 0.836304i \(0.684712\pi\)
\(858\) 0 0
\(859\) −2.97056 −0.101354 −0.0506771 0.998715i \(-0.516138\pi\)
−0.0506771 + 0.998715i \(0.516138\pi\)
\(860\) 7.07107 0.241121
\(861\) 0 0
\(862\) 12.1421 0.413563
\(863\) 4.97056 0.169200 0.0846000 0.996415i \(-0.473039\pi\)
0.0846000 + 0.996415i \(0.473039\pi\)
\(864\) 0 0
\(865\) −8.14214 −0.276841
\(866\) −27.1716 −0.923328
\(867\) 0 0
\(868\) 0 0
\(869\) 1.37258 0.0465617
\(870\) 0 0
\(871\) −94.9117 −3.21596
\(872\) 2.48528 0.0841622
\(873\) 0 0
\(874\) 6.34315 0.214560
\(875\) 0 0
\(876\) 0 0
\(877\) −10.4437 −0.352657 −0.176329 0.984331i \(-0.556422\pi\)
−0.176329 + 0.984331i \(0.556422\pi\)
\(878\) −5.75736 −0.194301
\(879\) 0 0
\(880\) 0.585786 0.0197469
\(881\) −23.8579 −0.803792 −0.401896 0.915685i \(-0.631649\pi\)
−0.401896 + 0.915685i \(0.631649\pi\)
\(882\) 0 0
\(883\) −44.5269 −1.49845 −0.749225 0.662316i \(-0.769574\pi\)
−0.749225 + 0.662316i \(0.769574\pi\)
\(884\) −4.00000 −0.134535
\(885\) 0 0
\(886\) 32.9706 1.10767
\(887\) −26.1005 −0.876369 −0.438185 0.898885i \(-0.644378\pi\)
−0.438185 + 0.898885i \(0.644378\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −2.48528 −0.0833068
\(891\) 0 0
\(892\) −5.17157 −0.173157
\(893\) 4.48528 0.150094
\(894\) 0 0
\(895\) −14.7279 −0.492300
\(896\) 0 0
\(897\) 0 0
\(898\) −33.1716 −1.10695
\(899\) 11.8579 0.395482
\(900\) 0 0
\(901\) 6.14214 0.204624
\(902\) 4.48528 0.149344
\(903\) 0 0
\(904\) 15.6569 0.520739
\(905\) 20.4853 0.680954
\(906\) 0 0
\(907\) 48.3259 1.60464 0.802318 0.596897i \(-0.203600\pi\)
0.802318 + 0.596897i \(0.203600\pi\)
\(908\) −8.97056 −0.297699
\(909\) 0 0
\(910\) 0 0
\(911\) 53.5980 1.77578 0.887890 0.460056i \(-0.152171\pi\)
0.887890 + 0.460056i \(0.152171\pi\)
\(912\) 0 0
\(913\) 9.65685 0.319595
\(914\) 28.1421 0.930859
\(915\) 0 0
\(916\) −10.8284 −0.357781
\(917\) 0 0
\(918\) 0 0
\(919\) 31.4558 1.03763 0.518816 0.854886i \(-0.326373\pi\)
0.518816 + 0.854886i \(0.326373\pi\)
\(920\) −7.65685 −0.252439
\(921\) 0 0
\(922\) −25.3137 −0.833663
\(923\) 44.2843 1.45763
\(924\) 0 0
\(925\) −0.585786 −0.0192605
\(926\) −36.0000 −1.18303
\(927\) 0 0
\(928\) −2.58579 −0.0848826
\(929\) 48.9117 1.60474 0.802370 0.596827i \(-0.203572\pi\)
0.802370 + 0.596827i \(0.203572\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −12.3431 −0.404313
\(933\) 0 0
\(934\) −33.1716 −1.08541
\(935\) −0.343146 −0.0112221
\(936\) 0 0
\(937\) 14.9706 0.489067 0.244533 0.969641i \(-0.421365\pi\)
0.244533 + 0.969641i \(0.421365\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −5.41421 −0.176592
\(941\) −56.0000 −1.82555 −0.912774 0.408465i \(-0.866064\pi\)
−0.912774 + 0.408465i \(0.866064\pi\)
\(942\) 0 0
\(943\) −58.6274 −1.90917
\(944\) −4.82843 −0.157152
\(945\) 0 0
\(946\) −4.14214 −0.134672
\(947\) −6.62742 −0.215362 −0.107681 0.994185i \(-0.534343\pi\)
−0.107681 + 0.994185i \(0.534343\pi\)
\(948\) 0 0
\(949\) 48.9706 1.58965
\(950\) 0.828427 0.0268777
\(951\) 0 0
\(952\) 0 0
\(953\) −50.8284 −1.64649 −0.823247 0.567683i \(-0.807840\pi\)
−0.823247 + 0.567683i \(0.807840\pi\)
\(954\) 0 0
\(955\) 13.3137 0.430821
\(956\) 8.34315 0.269837
\(957\) 0 0
\(958\) −25.1716 −0.813257
\(959\) 0 0
\(960\) 0 0
\(961\) −9.97056 −0.321631
\(962\) 4.00000 0.128965
\(963\) 0 0
\(964\) 4.92893 0.158750
\(965\) −8.14214 −0.262105
\(966\) 0 0
\(967\) 7.02944 0.226051 0.113026 0.993592i \(-0.463946\pi\)
0.113026 + 0.993592i \(0.463946\pi\)
\(968\) 10.6569 0.342524
\(969\) 0 0
\(970\) 2.48528 0.0797976
\(971\) 1.65685 0.0531710 0.0265855 0.999647i \(-0.491537\pi\)
0.0265855 + 0.999647i \(0.491537\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 14.8284 0.475133
\(975\) 0 0
\(976\) 9.65685 0.309108
\(977\) 6.68629 0.213913 0.106957 0.994264i \(-0.465889\pi\)
0.106957 + 0.994264i \(0.465889\pi\)
\(978\) 0 0
\(979\) 1.45584 0.0465290
\(980\) 0 0
\(981\) 0 0
\(982\) 33.0711 1.05534
\(983\) 22.3848 0.713963 0.356982 0.934111i \(-0.383806\pi\)
0.356982 + 0.934111i \(0.383806\pi\)
\(984\) 0 0
\(985\) −11.6569 −0.371418
\(986\) 1.51472 0.0482385
\(987\) 0 0
\(988\) −5.65685 −0.179969
\(989\) 54.1421 1.72162
\(990\) 0 0
\(991\) 16.1421 0.512772 0.256386 0.966574i \(-0.417468\pi\)
0.256386 + 0.966574i \(0.417468\pi\)
\(992\) −4.58579 −0.145599
\(993\) 0 0
\(994\) 0 0
\(995\) 9.75736 0.309329
\(996\) 0 0
\(997\) −19.6569 −0.622539 −0.311269 0.950322i \(-0.600754\pi\)
−0.311269 + 0.950322i \(0.600754\pi\)
\(998\) −31.1127 −0.984855
\(999\) 0 0
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 4410.2.a.bs.1.1 yes 2
3.2 odd 2 4410.2.a.bu.1.2 yes 2
7.6 odd 2 4410.2.a.bp.1.1 2
21.20 even 2 4410.2.a.bx.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4410.2.a.bp.1.1 2 7.6 odd 2
4410.2.a.bs.1.1 yes 2 1.1 even 1 trivial
4410.2.a.bu.1.2 yes 2 3.2 odd 2
4410.2.a.bx.1.2 yes 2 21.20 even 2