Properties

Label 9200.2.a.bs.1.2
Level $9200$
Weight $2$
Character 9200.1
Self dual yes
Analytic conductor $73.462$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9200.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(73.4623698596\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
Defining polynomial: \(x^{2} - x - 5\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.79129\) of defining polynomial
Character \(\chi\) \(=\) 9200.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.79129 q^{3} +2.79129 q^{7} +0.208712 q^{9} +O(q^{10})\) \(q+1.79129 q^{3} +2.79129 q^{7} +0.208712 q^{9} -3.79129 q^{11} -1.20871 q^{13} +3.79129 q^{17} -1.20871 q^{19} +5.00000 q^{21} +1.00000 q^{23} -5.00000 q^{27} -1.58258 q^{29} -10.3739 q^{31} -6.79129 q^{33} +4.00000 q^{37} -2.16515 q^{39} -2.20871 q^{41} -7.16515 q^{43} -13.5826 q^{47} +0.791288 q^{49} +6.79129 q^{51} -6.00000 q^{53} -2.16515 q^{57} +4.41742 q^{59} -3.37386 q^{61} +0.582576 q^{63} -7.16515 q^{67} +1.79129 q^{69} +5.37386 q^{71} +14.7477 q^{73} -10.5826 q^{77} -8.00000 q^{79} -9.58258 q^{81} -6.00000 q^{83} -2.83485 q^{87} -3.16515 q^{89} -3.37386 q^{91} -18.5826 q^{93} -14.9564 q^{97} -0.791288 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} + q^{7} + 5q^{9} + O(q^{10}) \) \( 2q - q^{3} + q^{7} + 5q^{9} - 3q^{11} - 7q^{13} + 3q^{17} - 7q^{19} + 10q^{21} + 2q^{23} - 10q^{27} + 6q^{29} - 7q^{31} - 9q^{33} + 8q^{37} + 14q^{39} - 9q^{41} + 4q^{43} - 18q^{47} - 3q^{49} + 9q^{51} - 12q^{53} + 14q^{57} + 18q^{59} + 7q^{61} - 8q^{63} + 4q^{67} - q^{69} - 3q^{71} + 2q^{73} - 12q^{77} - 16q^{79} - 10q^{81} - 12q^{83} - 24q^{87} + 12q^{89} + 7q^{91} - 28q^{93} - 7q^{97} + 3q^{99} + O(q^{100}) \)

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
\(3\) 1.79129 1.03420 0.517100 0.855925i \(-0.327011\pi\)
0.517100 + 0.855925i \(0.327011\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.79129 1.05501 0.527504 0.849553i \(-0.323128\pi\)
0.527504 + 0.849553i \(0.323128\pi\)
\(8\) 0 0
\(9\) 0.208712 0.0695707
\(10\) 0 0
\(11\) −3.79129 −1.14312 −0.571558 0.820562i \(-0.693661\pi\)
−0.571558 + 0.820562i \(0.693661\pi\)
\(12\) 0 0
\(13\) −1.20871 −0.335236 −0.167618 0.985852i \(-0.553608\pi\)
−0.167618 + 0.985852i \(0.553608\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.79129 0.919522 0.459761 0.888043i \(-0.347935\pi\)
0.459761 + 0.888043i \(0.347935\pi\)
\(18\) 0 0
\(19\) −1.20871 −0.277298 −0.138649 0.990342i \(-0.544276\pi\)
−0.138649 + 0.990342i \(0.544276\pi\)
\(20\) 0 0
\(21\) 5.00000 1.09109
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) −1.58258 −0.293877 −0.146938 0.989146i \(-0.546942\pi\)
−0.146938 + 0.989146i \(0.546942\pi\)
\(30\) 0 0
\(31\) −10.3739 −1.86320 −0.931600 0.363484i \(-0.881587\pi\)
−0.931600 + 0.363484i \(0.881587\pi\)
\(32\) 0 0
\(33\) −6.79129 −1.18221
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 0 0
\(39\) −2.16515 −0.346702
\(40\) 0 0
\(41\) −2.20871 −0.344943 −0.172471 0.985015i \(-0.555175\pi\)
−0.172471 + 0.985015i \(0.555175\pi\)
\(42\) 0 0
\(43\) −7.16515 −1.09268 −0.546338 0.837565i \(-0.683978\pi\)
−0.546338 + 0.837565i \(0.683978\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −13.5826 −1.98122 −0.990611 0.136710i \(-0.956347\pi\)
−0.990611 + 0.136710i \(0.956347\pi\)
\(48\) 0 0
\(49\) 0.791288 0.113041
\(50\) 0 0
\(51\) 6.79129 0.950971
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2.16515 −0.286781
\(58\) 0 0
\(59\) 4.41742 0.575100 0.287550 0.957766i \(-0.407159\pi\)
0.287550 + 0.957766i \(0.407159\pi\)
\(60\) 0 0
\(61\) −3.37386 −0.431979 −0.215989 0.976396i \(-0.569298\pi\)
−0.215989 + 0.976396i \(0.569298\pi\)
\(62\) 0 0
\(63\) 0.582576 0.0733976
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −7.16515 −0.875363 −0.437681 0.899130i \(-0.644200\pi\)
−0.437681 + 0.899130i \(0.644200\pi\)
\(68\) 0 0
\(69\) 1.79129 0.215646
\(70\) 0 0
\(71\) 5.37386 0.637760 0.318880 0.947795i \(-0.396693\pi\)
0.318880 + 0.947795i \(0.396693\pi\)
\(72\) 0 0
\(73\) 14.7477 1.72609 0.863045 0.505126i \(-0.168554\pi\)
0.863045 + 0.505126i \(0.168554\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −10.5826 −1.20600
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) −9.58258 −1.06473
\(82\) 0 0
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −2.83485 −0.303928
\(88\) 0 0
\(89\) −3.16515 −0.335505 −0.167753 0.985829i \(-0.553651\pi\)
−0.167753 + 0.985829i \(0.553651\pi\)
\(90\) 0 0
\(91\) −3.37386 −0.353677
\(92\) 0 0
\(93\) −18.5826 −1.92692
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −14.9564 −1.51860 −0.759298 0.650743i \(-0.774458\pi\)
−0.759298 + 0.650743i \(0.774458\pi\)
\(98\) 0 0
\(99\) −0.791288 −0.0795274
\(100\) 0 0
\(101\) 13.5826 1.35152 0.675758 0.737123i \(-0.263816\pi\)
0.675758 + 0.737123i \(0.263816\pi\)
\(102\) 0 0
\(103\) 7.37386 0.726568 0.363284 0.931678i \(-0.381655\pi\)
0.363284 + 0.931678i \(0.381655\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 13.5826 1.31308 0.656539 0.754292i \(-0.272020\pi\)
0.656539 + 0.754292i \(0.272020\pi\)
\(108\) 0 0
\(109\) 10.3739 0.993636 0.496818 0.867855i \(-0.334502\pi\)
0.496818 + 0.867855i \(0.334502\pi\)
\(110\) 0 0
\(111\) 7.16515 0.680086
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −0.252273 −0.0233226
\(118\) 0 0
\(119\) 10.5826 0.970103
\(120\) 0 0
\(121\) 3.37386 0.306715
\(122\) 0 0
\(123\) −3.95644 −0.356740
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −14.7477 −1.30865 −0.654325 0.756214i \(-0.727047\pi\)
−0.654325 + 0.756214i \(0.727047\pi\)
\(128\) 0 0
\(129\) −12.8348 −1.13005
\(130\) 0 0
\(131\) −9.16515 −0.800763 −0.400381 0.916349i \(-0.631122\pi\)
−0.400381 + 0.916349i \(0.631122\pi\)
\(132\) 0 0
\(133\) −3.37386 −0.292551
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0.791288 0.0676043 0.0338021 0.999429i \(-0.489238\pi\)
0.0338021 + 0.999429i \(0.489238\pi\)
\(138\) 0 0
\(139\) 14.7477 1.25089 0.625443 0.780270i \(-0.284918\pi\)
0.625443 + 0.780270i \(0.284918\pi\)
\(140\) 0 0
\(141\) −24.3303 −2.04898
\(142\) 0 0
\(143\) 4.58258 0.383214
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1.41742 0.116907
\(148\) 0 0
\(149\) 12.7913 1.04790 0.523952 0.851748i \(-0.324457\pi\)
0.523952 + 0.851748i \(0.324457\pi\)
\(150\) 0 0
\(151\) 6.20871 0.505258 0.252629 0.967563i \(-0.418705\pi\)
0.252629 + 0.967563i \(0.418705\pi\)
\(152\) 0 0
\(153\) 0.791288 0.0639718
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −12.7477 −1.01738 −0.508690 0.860950i \(-0.669870\pi\)
−0.508690 + 0.860950i \(0.669870\pi\)
\(158\) 0 0
\(159\) −10.7477 −0.852350
\(160\) 0 0
\(161\) 2.79129 0.219984
\(162\) 0 0
\(163\) 22.3739 1.75246 0.876228 0.481897i \(-0.160052\pi\)
0.876228 + 0.481897i \(0.160052\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −18.3303 −1.41844 −0.709221 0.704987i \(-0.750953\pi\)
−0.709221 + 0.704987i \(0.750953\pi\)
\(168\) 0 0
\(169\) −11.5390 −0.887617
\(170\) 0 0
\(171\) −0.252273 −0.0192918
\(172\) 0 0
\(173\) 14.2087 1.08027 0.540134 0.841579i \(-0.318373\pi\)
0.540134 + 0.841579i \(0.318373\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 7.91288 0.594768
\(178\) 0 0
\(179\) 16.7477 1.25178 0.625892 0.779910i \(-0.284735\pi\)
0.625892 + 0.779910i \(0.284735\pi\)
\(180\) 0 0
\(181\) 13.5390 1.00635 0.503174 0.864185i \(-0.332166\pi\)
0.503174 + 0.864185i \(0.332166\pi\)
\(182\) 0 0
\(183\) −6.04356 −0.446753
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −14.3739 −1.05112
\(188\) 0 0
\(189\) −13.9564 −1.01518
\(190\) 0 0
\(191\) −16.4174 −1.18792 −0.593962 0.804493i \(-0.702437\pi\)
−0.593962 + 0.804493i \(0.702437\pi\)
\(192\) 0 0
\(193\) −6.74773 −0.485712 −0.242856 0.970062i \(-0.578084\pi\)
−0.242856 + 0.970062i \(0.578084\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −20.5390 −1.46334 −0.731672 0.681657i \(-0.761260\pi\)
−0.731672 + 0.681657i \(0.761260\pi\)
\(198\) 0 0
\(199\) −20.3303 −1.44118 −0.720588 0.693363i \(-0.756128\pi\)
−0.720588 + 0.693363i \(0.756128\pi\)
\(200\) 0 0
\(201\) −12.8348 −0.905300
\(202\) 0 0
\(203\) −4.41742 −0.310042
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0.208712 0.0145065
\(208\) 0 0
\(209\) 4.58258 0.316983
\(210\) 0 0
\(211\) 10.0000 0.688428 0.344214 0.938891i \(-0.388145\pi\)
0.344214 + 0.938891i \(0.388145\pi\)
\(212\) 0 0
\(213\) 9.62614 0.659572
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −28.9564 −1.96569
\(218\) 0 0
\(219\) 26.4174 1.78512
\(220\) 0 0
\(221\) −4.58258 −0.308257
\(222\) 0 0
\(223\) 11.1652 0.747674 0.373837 0.927494i \(-0.378042\pi\)
0.373837 + 0.927494i \(0.378042\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 4.74773 0.315118 0.157559 0.987510i \(-0.449638\pi\)
0.157559 + 0.987510i \(0.449638\pi\)
\(228\) 0 0
\(229\) −16.3303 −1.07914 −0.539568 0.841942i \(-0.681413\pi\)
−0.539568 + 0.841942i \(0.681413\pi\)
\(230\) 0 0
\(231\) −18.9564 −1.24724
\(232\) 0 0
\(233\) −7.58258 −0.496751 −0.248376 0.968664i \(-0.579897\pi\)
−0.248376 + 0.968664i \(0.579897\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −14.3303 −0.930853
\(238\) 0 0
\(239\) −3.16515 −0.204737 −0.102368 0.994747i \(-0.532642\pi\)
−0.102368 + 0.994747i \(0.532642\pi\)
\(240\) 0 0
\(241\) −28.0000 −1.80364 −0.901819 0.432113i \(-0.857768\pi\)
−0.901819 + 0.432113i \(0.857768\pi\)
\(242\) 0 0
\(243\) −2.16515 −0.138895
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1.46099 0.0929603
\(248\) 0 0
\(249\) −10.7477 −0.681110
\(250\) 0 0
\(251\) −30.7913 −1.94353 −0.971764 0.235953i \(-0.924179\pi\)
−0.971764 + 0.235953i \(0.924179\pi\)
\(252\) 0 0
\(253\) −3.79129 −0.238356
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −22.7477 −1.41896 −0.709482 0.704723i \(-0.751071\pi\)
−0.709482 + 0.704723i \(0.751071\pi\)
\(258\) 0 0
\(259\) 11.1652 0.693769
\(260\) 0 0
\(261\) −0.330303 −0.0204452
\(262\) 0 0
\(263\) 15.7913 0.973733 0.486866 0.873477i \(-0.338140\pi\)
0.486866 + 0.873477i \(0.338140\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −5.66970 −0.346980
\(268\) 0 0
\(269\) 16.7477 1.02113 0.510563 0.859840i \(-0.329437\pi\)
0.510563 + 0.859840i \(0.329437\pi\)
\(270\) 0 0
\(271\) 23.1216 1.40454 0.702268 0.711912i \(-0.252171\pi\)
0.702268 + 0.711912i \(0.252171\pi\)
\(272\) 0 0
\(273\) −6.04356 −0.365773
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1.16515 0.0700072 0.0350036 0.999387i \(-0.488856\pi\)
0.0350036 + 0.999387i \(0.488856\pi\)
\(278\) 0 0
\(279\) −2.16515 −0.129624
\(280\) 0 0
\(281\) −16.7477 −0.999086 −0.499543 0.866289i \(-0.666499\pi\)
−0.499543 + 0.866289i \(0.666499\pi\)
\(282\) 0 0
\(283\) −28.3303 −1.68406 −0.842031 0.539429i \(-0.818640\pi\)
−0.842031 + 0.539429i \(0.818640\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −6.16515 −0.363917
\(288\) 0 0
\(289\) −2.62614 −0.154479
\(290\) 0 0
\(291\) −26.7913 −1.57053
\(292\) 0 0
\(293\) 27.4955 1.60630 0.803151 0.595776i \(-0.203155\pi\)
0.803151 + 0.595776i \(0.203155\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 18.9564 1.09996
\(298\) 0 0
\(299\) −1.20871 −0.0699016
\(300\) 0 0
\(301\) −20.0000 −1.15278
\(302\) 0 0
\(303\) 24.3303 1.39774
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 16.5390 0.943931 0.471966 0.881617i \(-0.343545\pi\)
0.471966 + 0.881617i \(0.343545\pi\)
\(308\) 0 0
\(309\) 13.2087 0.751417
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) 18.3739 1.03855 0.519276 0.854607i \(-0.326202\pi\)
0.519276 + 0.854607i \(0.326202\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −5.20871 −0.292550 −0.146275 0.989244i \(-0.546729\pi\)
−0.146275 + 0.989244i \(0.546729\pi\)
\(318\) 0 0
\(319\) 6.00000 0.335936
\(320\) 0 0
\(321\) 24.3303 1.35799
\(322\) 0 0
\(323\) −4.58258 −0.254981
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 18.5826 1.02762
\(328\) 0 0
\(329\) −37.9129 −2.09020
\(330\) 0 0
\(331\) −6.74773 −0.370889 −0.185444 0.982655i \(-0.559372\pi\)
−0.185444 + 0.982655i \(0.559372\pi\)
\(332\) 0 0
\(333\) 0.834849 0.0457494
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 16.7913 0.914680 0.457340 0.889292i \(-0.348802\pi\)
0.457340 + 0.889292i \(0.348802\pi\)
\(338\) 0 0
\(339\) −10.7477 −0.583736
\(340\) 0 0
\(341\) 39.3303 2.12986
\(342\) 0 0
\(343\) −17.3303 −0.935748
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 9.79129 0.525624 0.262812 0.964847i \(-0.415350\pi\)
0.262812 + 0.964847i \(0.415350\pi\)
\(348\) 0 0
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 0 0
\(351\) 6.04356 0.322581
\(352\) 0 0
\(353\) 15.1652 0.807160 0.403580 0.914944i \(-0.367766\pi\)
0.403580 + 0.914944i \(0.367766\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 18.9564 1.00328
\(358\) 0 0
\(359\) 9.16515 0.483718 0.241859 0.970311i \(-0.422243\pi\)
0.241859 + 0.970311i \(0.422243\pi\)
\(360\) 0 0
\(361\) −17.5390 −0.923106
\(362\) 0 0
\(363\) 6.04356 0.317205
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −0.834849 −0.0435787 −0.0217894 0.999763i \(-0.506936\pi\)
−0.0217894 + 0.999763i \(0.506936\pi\)
\(368\) 0 0
\(369\) −0.460985 −0.0239979
\(370\) 0 0
\(371\) −16.7477 −0.869499
\(372\) 0 0
\(373\) 14.7477 0.763608 0.381804 0.924243i \(-0.375303\pi\)
0.381804 + 0.924243i \(0.375303\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.91288 0.0985183
\(378\) 0 0
\(379\) −7.37386 −0.378770 −0.189385 0.981903i \(-0.560649\pi\)
−0.189385 + 0.981903i \(0.560649\pi\)
\(380\) 0 0
\(381\) −26.4174 −1.35341
\(382\) 0 0
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1.49545 −0.0760182
\(388\) 0 0
\(389\) 29.7042 1.50606 0.753031 0.657986i \(-0.228591\pi\)
0.753031 + 0.657986i \(0.228591\pi\)
\(390\) 0 0
\(391\) 3.79129 0.191734
\(392\) 0 0
\(393\) −16.4174 −0.828150
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −16.5390 −0.830069 −0.415035 0.909806i \(-0.636231\pi\)
−0.415035 + 0.909806i \(0.636231\pi\)
\(398\) 0 0
\(399\) −6.04356 −0.302556
\(400\) 0 0
\(401\) 22.7477 1.13597 0.567984 0.823040i \(-0.307724\pi\)
0.567984 + 0.823040i \(0.307724\pi\)
\(402\) 0 0
\(403\) 12.5390 0.624613
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −15.1652 −0.751709
\(408\) 0 0
\(409\) −22.7913 −1.12696 −0.563478 0.826131i \(-0.690537\pi\)
−0.563478 + 0.826131i \(0.690537\pi\)
\(410\) 0 0
\(411\) 1.41742 0.0699164
\(412\) 0 0
\(413\) 12.3303 0.606735
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 26.4174 1.29367
\(418\) 0 0
\(419\) −39.1652 −1.91334 −0.956671 0.291170i \(-0.905956\pi\)
−0.956671 + 0.291170i \(0.905956\pi\)
\(420\) 0 0
\(421\) −23.1216 −1.12688 −0.563439 0.826158i \(-0.690522\pi\)
−0.563439 + 0.826158i \(0.690522\pi\)
\(422\) 0 0
\(423\) −2.83485 −0.137835
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −9.41742 −0.455741
\(428\) 0 0
\(429\) 8.20871 0.396320
\(430\) 0 0
\(431\) 19.9129 0.959170 0.479585 0.877496i \(-0.340787\pi\)
0.479585 + 0.877496i \(0.340787\pi\)
\(432\) 0 0
\(433\) −1.53901 −0.0739603 −0.0369802 0.999316i \(-0.511774\pi\)
−0.0369802 + 0.999316i \(0.511774\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1.20871 −0.0578205
\(438\) 0 0
\(439\) −25.5390 −1.21891 −0.609455 0.792820i \(-0.708612\pi\)
−0.609455 + 0.792820i \(0.708612\pi\)
\(440\) 0 0
\(441\) 0.165151 0.00786435
\(442\) 0 0
\(443\) −35.2087 −1.67282 −0.836408 0.548107i \(-0.815349\pi\)
−0.836408 + 0.548107i \(0.815349\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 22.9129 1.08374
\(448\) 0 0
\(449\) 25.1216 1.18556 0.592781 0.805364i \(-0.298030\pi\)
0.592781 + 0.805364i \(0.298030\pi\)
\(450\) 0 0
\(451\) 8.37386 0.394310
\(452\) 0 0
\(453\) 11.1216 0.522538
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 0 0
\(459\) −18.9564 −0.884811
\(460\) 0 0
\(461\) −1.25227 −0.0583242 −0.0291621 0.999575i \(-0.509284\pi\)
−0.0291621 + 0.999575i \(0.509284\pi\)
\(462\) 0 0
\(463\) −10.0000 −0.464739 −0.232370 0.972628i \(-0.574648\pi\)
−0.232370 + 0.972628i \(0.574648\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 25.9129 1.19911 0.599553 0.800335i \(-0.295345\pi\)
0.599553 + 0.800335i \(0.295345\pi\)
\(468\) 0 0
\(469\) −20.0000 −0.923514
\(470\) 0 0
\(471\) −22.8348 −1.05217
\(472\) 0 0
\(473\) 27.1652 1.24905
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −1.25227 −0.0573376
\(478\) 0 0
\(479\) 39.4955 1.80459 0.902297 0.431116i \(-0.141880\pi\)
0.902297 + 0.431116i \(0.141880\pi\)
\(480\) 0 0
\(481\) −4.83485 −0.220450
\(482\) 0 0
\(483\) 5.00000 0.227508
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 15.5826 0.706114 0.353057 0.935602i \(-0.385142\pi\)
0.353057 + 0.935602i \(0.385142\pi\)
\(488\) 0 0
\(489\) 40.0780 1.81239
\(490\) 0 0
\(491\) 16.7477 0.755814 0.377907 0.925843i \(-0.376644\pi\)
0.377907 + 0.925843i \(0.376644\pi\)
\(492\) 0 0
\(493\) −6.00000 −0.270226
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 15.0000 0.672842
\(498\) 0 0
\(499\) −23.1652 −1.03701 −0.518507 0.855073i \(-0.673512\pi\)
−0.518507 + 0.855073i \(0.673512\pi\)
\(500\) 0 0
\(501\) −32.8348 −1.46695
\(502\) 0 0
\(503\) 18.7913 0.837862 0.418931 0.908018i \(-0.362405\pi\)
0.418931 + 0.908018i \(0.362405\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −20.6697 −0.917973
\(508\) 0 0
\(509\) 7.25227 0.321451 0.160726 0.986999i \(-0.448617\pi\)
0.160726 + 0.986999i \(0.448617\pi\)
\(510\) 0 0
\(511\) 41.1652 1.82104
\(512\) 0 0
\(513\) 6.04356 0.266830
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 51.4955 2.26477
\(518\) 0 0
\(519\) 25.4519 1.11721
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 0 0
\(523\) −1.16515 −0.0509485 −0.0254743 0.999675i \(-0.508110\pi\)
−0.0254743 + 0.999675i \(0.508110\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −39.3303 −1.71325
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 0.921970 0.0400101
\(532\) 0 0
\(533\) 2.66970 0.115637
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 30.0000 1.29460
\(538\) 0 0
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) 38.3303 1.64795 0.823974 0.566627i \(-0.191752\pi\)
0.823974 + 0.566627i \(0.191752\pi\)
\(542\) 0 0
\(543\) 24.2523 1.04076
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 15.1216 0.646553 0.323276 0.946305i \(-0.395216\pi\)
0.323276 + 0.946305i \(0.395216\pi\)
\(548\) 0 0
\(549\) −0.704166 −0.0300531
\(550\) 0 0
\(551\) 1.91288 0.0814914
\(552\) 0 0
\(553\) −22.3303 −0.949581
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −30.3303 −1.28514 −0.642568 0.766229i \(-0.722131\pi\)
−0.642568 + 0.766229i \(0.722131\pi\)
\(558\) 0 0
\(559\) 8.66061 0.366305
\(560\) 0 0
\(561\) −25.7477 −1.08707
\(562\) 0 0
\(563\) 3.16515 0.133395 0.0666976 0.997773i \(-0.478754\pi\)
0.0666976 + 0.997773i \(0.478754\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −26.7477 −1.12330
\(568\) 0 0
\(569\) 15.4955 0.649603 0.324802 0.945782i \(-0.394702\pi\)
0.324802 + 0.945782i \(0.394702\pi\)
\(570\) 0 0
\(571\) −30.1216 −1.26055 −0.630275 0.776372i \(-0.717058\pi\)
−0.630275 + 0.776372i \(0.717058\pi\)
\(572\) 0 0
\(573\) −29.4083 −1.22855
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −22.8348 −0.950627 −0.475314 0.879816i \(-0.657665\pi\)
−0.475314 + 0.879816i \(0.657665\pi\)
\(578\) 0 0
\(579\) −12.0871 −0.502324
\(580\) 0 0
\(581\) −16.7477 −0.694813
\(582\) 0 0
\(583\) 22.7477 0.942115
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −26.2087 −1.08175 −0.540875 0.841103i \(-0.681907\pi\)
−0.540875 + 0.841103i \(0.681907\pi\)
\(588\) 0 0
\(589\) 12.5390 0.516661
\(590\) 0 0
\(591\) −36.7913 −1.51339
\(592\) 0 0
\(593\) −13.9129 −0.571333 −0.285667 0.958329i \(-0.592215\pi\)
−0.285667 + 0.958329i \(0.592215\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −36.4174 −1.49047
\(598\) 0 0
\(599\) 40.1216 1.63932 0.819662 0.572848i \(-0.194161\pi\)
0.819662 + 0.572848i \(0.194161\pi\)
\(600\) 0 0
\(601\) −22.7913 −0.929676 −0.464838 0.885396i \(-0.653887\pi\)
−0.464838 + 0.885396i \(0.653887\pi\)
\(602\) 0 0
\(603\) −1.49545 −0.0608996
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −28.0000 −1.13648 −0.568242 0.822861i \(-0.692376\pi\)
−0.568242 + 0.822861i \(0.692376\pi\)
\(608\) 0 0
\(609\) −7.91288 −0.320646
\(610\) 0 0
\(611\) 16.4174 0.664178
\(612\) 0 0
\(613\) −14.0000 −0.565455 −0.282727 0.959200i \(-0.591239\pi\)
−0.282727 + 0.959200i \(0.591239\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −44.8693 −1.80637 −0.903185 0.429251i \(-0.858778\pi\)
−0.903185 + 0.429251i \(0.858778\pi\)
\(618\) 0 0
\(619\) −2.79129 −0.112191 −0.0560957 0.998425i \(-0.517865\pi\)
−0.0560957 + 0.998425i \(0.517865\pi\)
\(620\) 0 0
\(621\) −5.00000 −0.200643
\(622\) 0 0
\(623\) −8.83485 −0.353961
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 8.20871 0.327824
\(628\) 0 0
\(629\) 15.1652 0.604674
\(630\) 0 0
\(631\) 17.9129 0.713100 0.356550 0.934276i \(-0.383953\pi\)
0.356550 + 0.934276i \(0.383953\pi\)
\(632\) 0 0
\(633\) 17.9129 0.711973
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −0.956439 −0.0378955
\(638\) 0 0
\(639\) 1.12159 0.0443694
\(640\) 0 0
\(641\) −3.16515 −0.125016 −0.0625080 0.998044i \(-0.519910\pi\)
−0.0625080 + 0.998044i \(0.519910\pi\)
\(642\) 0 0
\(643\) −20.7477 −0.818210 −0.409105 0.912487i \(-0.634159\pi\)
−0.409105 + 0.912487i \(0.634159\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 2.83485 0.111449 0.0557247 0.998446i \(-0.482253\pi\)
0.0557247 + 0.998446i \(0.482253\pi\)
\(648\) 0 0
\(649\) −16.7477 −0.657406
\(650\) 0 0
\(651\) −51.8693 −2.03292
\(652\) 0 0
\(653\) −35.5390 −1.39075 −0.695375 0.718647i \(-0.744762\pi\)
−0.695375 + 0.718647i \(0.744762\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 3.07803 0.120085
\(658\) 0 0
\(659\) 27.1652 1.05820 0.529102 0.848558i \(-0.322529\pi\)
0.529102 + 0.848558i \(0.322529\pi\)
\(660\) 0 0
\(661\) −39.3739 −1.53147 −0.765733 0.643159i \(-0.777624\pi\)
−0.765733 + 0.643159i \(0.777624\pi\)
\(662\) 0 0
\(663\) −8.20871 −0.318800
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −1.58258 −0.0612776
\(668\) 0 0
\(669\) 20.0000 0.773245
\(670\) 0 0
\(671\) 12.7913 0.493802
\(672\) 0 0
\(673\) −38.0000 −1.46479 −0.732396 0.680879i \(-0.761598\pi\)
−0.732396 + 0.680879i \(0.761598\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 30.6606 1.17838 0.589191 0.807993i \(-0.299446\pi\)
0.589191 + 0.807993i \(0.299446\pi\)
\(678\) 0 0
\(679\) −41.7477 −1.60213
\(680\) 0 0
\(681\) 8.50455 0.325895
\(682\) 0 0
\(683\) 2.37386 0.0908334 0.0454167 0.998968i \(-0.485538\pi\)
0.0454167 + 0.998968i \(0.485538\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −29.2523 −1.11604
\(688\) 0 0
\(689\) 7.25227 0.276290
\(690\) 0 0
\(691\) −15.2523 −0.580224 −0.290112 0.956993i \(-0.593693\pi\)
−0.290112 + 0.956993i \(0.593693\pi\)
\(692\) 0 0
\(693\) −2.20871 −0.0839020
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −8.37386 −0.317183
\(698\) 0 0
\(699\) −13.5826 −0.513740
\(700\) 0 0
\(701\) 9.62614 0.363574 0.181787 0.983338i \(-0.441812\pi\)
0.181787 + 0.983338i \(0.441812\pi\)
\(702\) 0 0
\(703\) −4.83485 −0.182350
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 37.9129 1.42586
\(708\) 0 0
\(709\) 34.5390 1.29714 0.648570 0.761155i \(-0.275367\pi\)
0.648570 + 0.761155i \(0.275367\pi\)
\(710\) 0 0
\(711\) −1.66970 −0.0626185
\(712\) 0 0
\(713\) −10.3739 −0.388504
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −5.66970 −0.211739
\(718\) 0 0
\(719\) 29.5390 1.10162 0.550810 0.834631i \(-0.314319\pi\)
0.550810 + 0.834631i \(0.314319\pi\)
\(720\) 0 0
\(721\) 20.5826 0.766535
\(722\) 0 0
\(723\) −50.1561 −1.86532
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −2.12159 −0.0786854 −0.0393427 0.999226i \(-0.512526\pi\)
−0.0393427 + 0.999226i \(0.512526\pi\)
\(728\) 0 0
\(729\) 24.8693 0.921086
\(730\) 0 0
\(731\) −27.1652 −1.00474
\(732\) 0 0
\(733\) −26.0000 −0.960332 −0.480166 0.877178i \(-0.659424\pi\)
−0.480166 + 0.877178i \(0.659424\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 27.1652 1.00064
\(738\) 0 0
\(739\) −8.00000 −0.294285 −0.147142 0.989115i \(-0.547008\pi\)
−0.147142 + 0.989115i \(0.547008\pi\)
\(740\) 0 0
\(741\) 2.61704 0.0961395
\(742\) 0 0
\(743\) 9.95644 0.365266 0.182633 0.983181i \(-0.441538\pi\)
0.182633 + 0.983181i \(0.441538\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −1.25227 −0.0458183
\(748\) 0 0
\(749\) 37.9129 1.38531
\(750\) 0 0
\(751\) −18.7477 −0.684114 −0.342057 0.939679i \(-0.611124\pi\)
−0.342057 + 0.939679i \(0.611124\pi\)
\(752\) 0 0
\(753\) −55.1561 −2.01000
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −26.3303 −0.956991 −0.478496 0.878090i \(-0.658818\pi\)
−0.478496 + 0.878090i \(0.658818\pi\)
\(758\) 0 0
\(759\) −6.79129 −0.246508
\(760\) 0 0
\(761\) −33.9564 −1.23092 −0.615460 0.788168i \(-0.711030\pi\)
−0.615460 + 0.788168i \(0.711030\pi\)
\(762\) 0 0
\(763\) 28.9564 1.04829
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −5.33939 −0.192794
\(768\) 0 0
\(769\) −3.66970 −0.132333 −0.0661663 0.997809i \(-0.521077\pi\)
−0.0661663 + 0.997809i \(0.521077\pi\)
\(770\) 0 0
\(771\) −40.7477 −1.46749
\(772\) 0 0
\(773\) 21.4955 0.773138 0.386569 0.922261i \(-0.373660\pi\)
0.386569 + 0.922261i \(0.373660\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 20.0000 0.717496
\(778\) 0 0
\(779\) 2.66970 0.0956518
\(780\) 0 0
\(781\) −20.3739 −0.729034
\(782\) 0 0
\(783\) 7.91288 0.282783
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −8.41742 −0.300049 −0.150024 0.988682i \(-0.547935\pi\)
−0.150024 + 0.988682i \(0.547935\pi\)
\(788\) 0 0
\(789\) 28.2867 1.00703
\(790\) 0 0
\(791\) −16.7477 −0.595481
\(792\) 0 0
\(793\) 4.07803 0.144815
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 49.9129 1.76800 0.884002 0.467482i \(-0.154839\pi\)
0.884002 + 0.467482i \(0.154839\pi\)
\(798\) 0 0
\(799\) −51.4955 −1.82178
\(800\) 0 0
\(801\) −0.660606 −0.0233413
\(802\) 0 0
\(803\) −55.9129 −1.97312
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 30.0000 1.05605
\(808\) 0 0
\(809\) 11.0436 0.388271 0.194135 0.980975i \(-0.437810\pi\)
0.194135 + 0.980975i \(0.437810\pi\)
\(810\) 0 0
\(811\) 47.9129 1.68245 0.841224 0.540686i \(-0.181835\pi\)
0.841224 + 0.540686i \(0.181835\pi\)
\(812\) 0 0
\(813\) 41.4174 1.45257
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 8.66061 0.302996
\(818\) 0 0
\(819\) −0.704166 −0.0246056
\(820\) 0 0
\(821\) 2.83485 0.0989369 0.0494684 0.998776i \(-0.484247\pi\)
0.0494684 + 0.998776i \(0.484247\pi\)
\(822\) 0 0
\(823\) 41.1652 1.43493 0.717463 0.696596i \(-0.245303\pi\)
0.717463 + 0.696596i \(0.245303\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 41.0780 1.42842 0.714212 0.699930i \(-0.246785\pi\)
0.714212 + 0.699930i \(0.246785\pi\)
\(828\) 0 0
\(829\) −31.4955 −1.09388 −0.546941 0.837171i \(-0.684208\pi\)
−0.546941 + 0.837171i \(0.684208\pi\)
\(830\) 0 0
\(831\) 2.08712 0.0724014
\(832\) 0 0
\(833\) 3.00000 0.103944
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 51.8693 1.79287
\(838\) 0 0
\(839\) 22.4174 0.773935 0.386968 0.922093i \(-0.373522\pi\)
0.386968 + 0.922093i \(0.373522\pi\)
\(840\) 0 0
\(841\) −26.4955 −0.913636
\(842\) 0 0
\(843\) −30.0000 −1.03325
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 9.41742 0.323587
\(848\) 0 0
\(849\) −50.7477 −1.74166
\(850\) 0 0
\(851\) 4.00000 0.137118
\(852\) 0 0
\(853\) −8.46099 −0.289699 −0.144849 0.989454i \(-0.546270\pi\)
−0.144849 + 0.989454i \(0.546270\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −9.16515 −0.313076 −0.156538 0.987672i \(-0.550033\pi\)
−0.156538 + 0.987672i \(0.550033\pi\)
\(858\) 0 0
\(859\) −0.747727 −0.0255121 −0.0127561 0.999919i \(-0.504060\pi\)
−0.0127561 + 0.999919i \(0.504060\pi\)
\(860\) 0 0
\(861\) −11.0436 −0.376364
\(862\) 0 0
\(863\) −31.5826 −1.07508 −0.537542 0.843237i \(-0.680647\pi\)
−0.537542 + 0.843237i \(0.680647\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −4.70417 −0.159762
\(868\) 0 0
\(869\) 30.3303 1.02889
\(870\) 0 0
\(871\) 8.66061 0.293453
\(872\) 0 0
\(873\) −3.12159 −0.105650
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −7.70417 −0.260151 −0.130076 0.991504i \(-0.541522\pi\)
−0.130076 + 0.991504i \(0.541522\pi\)
\(878\) 0 0
\(879\) 49.2523 1.66124
\(880\) 0 0
\(881\) −6.33030 −0.213273 −0.106637 0.994298i \(-0.534008\pi\)
−0.106637 + 0.994298i \(0.534008\pi\)
\(882\) 0 0
\(883\) −12.0436 −0.405298 −0.202649 0.979251i \(-0.564955\pi\)
−0.202649 + 0.979251i \(0.564955\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −3.16515 −0.106275 −0.0531377 0.998587i \(-0.516922\pi\)
−0.0531377 + 0.998587i \(0.516922\pi\)
\(888\) 0 0
\(889\) −41.1652 −1.38063
\(890\) 0 0
\(891\) 36.3303 1.21711
\(892\) 0 0
\(893\) 16.4174 0.549388
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −2.16515 −0.0722923
\(898\) 0 0
\(899\) 16.4174 0.547552
\(900\) 0 0
\(901\) −22.7477 −0.757837
\(902\) 0 0
\(903\) −35.8258 −1.19221
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −20.7477 −0.688917 −0.344458 0.938802i \(-0.611937\pi\)
−0.344458 + 0.938802i \(0.611937\pi\)
\(908\) 0 0
\(909\) 2.83485 0.0940260
\(910\) 0 0
\(911\) −13.5826 −0.450011 −0.225005 0.974358i \(-0.572240\pi\)
−0.225005 + 0.974358i \(0.572240\pi\)
\(912\) 0 0
\(913\) 22.7477 0.752840
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −25.5826 −0.844811
\(918\) 0 0
\(919\) 36.8348 1.21507 0.607535 0.794293i \(-0.292159\pi\)
0.607535 + 0.794293i \(0.292159\pi\)
\(920\) 0 0
\(921\) 29.6261 0.976214
\(922\) 0 0
\(923\) −6.49545 −0.213800
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 1.53901 0.0505479
\(928\) 0 0
\(929\) −39.4955 −1.29580 −0.647902 0.761724i \(-0.724353\pi\)
−0.647902 + 0.761724i \(0.724353\pi\)
\(930\) 0 0
\(931\) −0.956439 −0.0313460
\(932\) 0 0
\(933\) −21.4955 −0.703730
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −58.3739 −1.90699 −0.953495 0.301407i \(-0.902544\pi\)
−0.953495 + 0.301407i \(0.902544\pi\)
\(938\) 0 0
\(939\) 32.9129 1.07407
\(940\) 0 0
\(941\) 54.9564 1.79153 0.895764 0.444529i \(-0.146629\pi\)
0.895764 + 0.444529i \(0.146629\pi\)
\(942\) 0 0
\(943\) −2.20871 −0.0719256
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −29.5390 −0.959889 −0.479945 0.877299i \(-0.659343\pi\)
−0.479945 + 0.877299i \(0.659343\pi\)
\(948\) 0 0
\(949\) −17.8258 −0.578649
\(950\) 0 0
\(951\) −9.33030 −0.302556
\(952\) 0 0
\(953\) 26.5390 0.859683 0.429842 0.902904i \(-0.358569\pi\)
0.429842 + 0.902904i \(0.358569\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 10.7477 0.347425
\(958\) 0 0
\(959\) 2.20871 0.0713230
\(960\) 0 0
\(961\) 76.6170 2.47152
\(962\) 0 0
\(963\) 2.83485 0.0913517
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −5.25227 −0.168902 −0.0844509 0.996428i \(-0.526914\pi\)
−0.0844509 + 0.996428i \(0.526914\pi\)
\(968\) 0 0
\(969\) −8.20871 −0.263702
\(970\) 0 0
\(971\) 6.95644 0.223243 0.111621 0.993751i \(-0.464396\pi\)
0.111621 + 0.993751i \(0.464396\pi\)
\(972\) 0 0
\(973\) 41.1652 1.31969
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −7.12159 −0.227840 −0.113920 0.993490i \(-0.536341\pi\)
−0.113920 + 0.993490i \(0.536341\pi\)
\(978\) 0 0
\(979\) 12.0000 0.383522
\(980\) 0 0
\(981\) 2.16515 0.0691280
\(982\) 0 0
\(983\) −0.626136 −0.0199707 −0.00998533 0.999950i \(-0.503178\pi\)
−0.00998533 + 0.999950i \(0.503178\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −67.9129 −2.16169
\(988\) 0 0
\(989\) −7.16515 −0.227839
\(990\) 0 0
\(991\) 37.7913 1.20048 0.600240 0.799820i \(-0.295072\pi\)
0.600240 + 0.799820i \(0.295072\pi\)
\(992\) 0 0
\(993\) −12.0871 −0.383573
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −11.4955 −0.364065 −0.182032 0.983293i \(-0.558268\pi\)
−0.182032 + 0.983293i \(0.558268\pi\)
\(998\) 0 0
\(999\) −20.0000 −0.632772
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 9200.2.a.bs.1.2 2
4.3 odd 2 1150.2.a.o.1.1 2
5.4 even 2 1840.2.a.n.1.1 2
20.3 even 4 1150.2.b.g.599.1 4
20.7 even 4 1150.2.b.g.599.4 4
20.19 odd 2 230.2.a.a.1.2 2
40.19 odd 2 7360.2.a.bq.1.1 2
40.29 even 2 7360.2.a.bk.1.2 2
60.59 even 2 2070.2.a.x.1.2 2
460.459 even 2 5290.2.a.e.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.a.1.2 2 20.19 odd 2
1150.2.a.o.1.1 2 4.3 odd 2
1150.2.b.g.599.1 4 20.3 even 4
1150.2.b.g.599.4 4 20.7 even 4
1840.2.a.n.1.1 2 5.4 even 2
2070.2.a.x.1.2 2 60.59 even 2
5290.2.a.e.1.2 2 460.459 even 2
7360.2.a.bk.1.2 2 40.29 even 2
7360.2.a.bq.1.1 2 40.19 odd 2
9200.2.a.bs.1.2 2 1.1 even 1 trivial