Properties

Label 1840.2.a.v.1.2
Level $1840$
Weight $2$
Character 1840.1
Self dual yes
Analytic conductor $14.692$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.6924739719\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.13955077.1
Defining polynomial: \(x^{5} - 14 x^{3} - x^{2} + 32 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.93283\) of defining polynomial
Character \(\chi\) \(=\) 1840.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.93283 q^{3} -1.00000 q^{5} -2.38236 q^{7} +0.735829 q^{9} +O(q^{10})\) \(q-1.93283 q^{3} -1.00000 q^{5} -2.38236 q^{7} +0.735829 q^{9} -5.33368 q^{11} -4.53752 q^{13} +1.93283 q^{15} +1.81464 q^{17} -7.00233 q^{19} +4.60469 q^{21} +1.00000 q^{23} +1.00000 q^{25} +4.37626 q^{27} -0.118188 q^{29} +0.884147 q^{31} +10.3091 q^{33} +2.38236 q^{35} +7.51903 q^{37} +8.77026 q^{39} -1.45186 q^{41} -0.735829 q^{45} -10.4389 q^{47} -1.32437 q^{49} -3.50739 q^{51} -9.42167 q^{53} +5.33368 q^{55} +13.5343 q^{57} -7.79239 q^{59} -2.80533 q^{61} -1.75301 q^{63} +4.53752 q^{65} +3.11134 q^{67} -1.93283 q^{69} +13.5909 q^{71} +12.4389 q^{73} -1.93283 q^{75} +12.7067 q^{77} -6.80169 q^{79} -10.6660 q^{81} +13.5190 q^{83} -1.81464 q^{85} +0.228437 q^{87} +2.89906 q^{89} +10.8100 q^{91} -1.70890 q^{93} +7.00233 q^{95} -1.97774 q^{97} -3.92468 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 5q^{5} + 2q^{7} + 13q^{9} + O(q^{10}) \) \( 5q - 5q^{5} + 2q^{7} + 13q^{9} + q^{11} + 4q^{13} + 4q^{17} - 7q^{19} + 6q^{21} + 5q^{23} + 5q^{25} - 3q^{27} + 4q^{29} - 19q^{31} + 17q^{33} - 2q^{35} + 15q^{37} - 19q^{39} + 25q^{41} - 13q^{45} + 11q^{47} + 25q^{49} - 19q^{51} + 3q^{53} - q^{55} + 48q^{57} + q^{59} - 5q^{61} + 41q^{63} - 4q^{65} - 9q^{67} - q^{71} - q^{73} + 18q^{77} + 2q^{79} + 57q^{81} + 45q^{83} - 4q^{85} + 9q^{87} + 6q^{89} - 11q^{91} - 39q^{93} + 7q^{95} + 25q^{97} + 65q^{99} + O(q^{100}) \)

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
\(3\) −1.93283 −1.11592 −0.557960 0.829868i \(-0.688416\pi\)
−0.557960 + 0.829868i \(0.688416\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.38236 −0.900447 −0.450223 0.892916i \(-0.648656\pi\)
−0.450223 + 0.892916i \(0.648656\pi\)
\(8\) 0 0
\(9\) 0.735829 0.245276
\(10\) 0 0
\(11\) −5.33368 −1.60816 −0.804082 0.594519i \(-0.797343\pi\)
−0.804082 + 0.594519i \(0.797343\pi\)
\(12\) 0 0
\(13\) −4.53752 −1.25848 −0.629241 0.777210i \(-0.716634\pi\)
−0.629241 + 0.777210i \(0.716634\pi\)
\(14\) 0 0
\(15\) 1.93283 0.499054
\(16\) 0 0
\(17\) 1.81464 0.440115 0.220058 0.975487i \(-0.429375\pi\)
0.220058 + 0.975487i \(0.429375\pi\)
\(18\) 0 0
\(19\) −7.00233 −1.60645 −0.803223 0.595679i \(-0.796883\pi\)
−0.803223 + 0.595679i \(0.796883\pi\)
\(20\) 0 0
\(21\) 4.60469 1.00483
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 4.37626 0.842211
\(28\) 0 0
\(29\) −0.118188 −0.0219469 −0.0109735 0.999940i \(-0.503493\pi\)
−0.0109735 + 0.999940i \(0.503493\pi\)
\(30\) 0 0
\(31\) 0.884147 0.158797 0.0793987 0.996843i \(-0.474700\pi\)
0.0793987 + 0.996843i \(0.474700\pi\)
\(32\) 0 0
\(33\) 10.3091 1.79458
\(34\) 0 0
\(35\) 2.38236 0.402692
\(36\) 0 0
\(37\) 7.51903 1.23612 0.618061 0.786130i \(-0.287919\pi\)
0.618061 + 0.786130i \(0.287919\pi\)
\(38\) 0 0
\(39\) 8.77026 1.40436
\(40\) 0 0
\(41\) −1.45186 −0.226743 −0.113371 0.993553i \(-0.536165\pi\)
−0.113371 + 0.993553i \(0.536165\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) −0.735829 −0.109691
\(46\) 0 0
\(47\) −10.4389 −1.52267 −0.761336 0.648357i \(-0.775456\pi\)
−0.761336 + 0.648357i \(0.775456\pi\)
\(48\) 0 0
\(49\) −1.32437 −0.189195
\(50\) 0 0
\(51\) −3.50739 −0.491133
\(52\) 0 0
\(53\) −9.42167 −1.29417 −0.647083 0.762420i \(-0.724011\pi\)
−0.647083 + 0.762420i \(0.724011\pi\)
\(54\) 0 0
\(55\) 5.33368 0.719193
\(56\) 0 0
\(57\) 13.5343 1.79266
\(58\) 0 0
\(59\) −7.79239 −1.01448 −0.507241 0.861804i \(-0.669335\pi\)
−0.507241 + 0.861804i \(0.669335\pi\)
\(60\) 0 0
\(61\) −2.80533 −0.359186 −0.179593 0.983741i \(-0.557478\pi\)
−0.179593 + 0.983741i \(0.557478\pi\)
\(62\) 0 0
\(63\) −1.75301 −0.220858
\(64\) 0 0
\(65\) 4.53752 0.562810
\(66\) 0 0
\(67\) 3.11134 0.380111 0.190055 0.981773i \(-0.439133\pi\)
0.190055 + 0.981773i \(0.439133\pi\)
\(68\) 0 0
\(69\) −1.93283 −0.232685
\(70\) 0 0
\(71\) 13.5909 1.61294 0.806470 0.591275i \(-0.201375\pi\)
0.806470 + 0.591275i \(0.201375\pi\)
\(72\) 0 0
\(73\) 12.4389 1.45586 0.727932 0.685649i \(-0.240481\pi\)
0.727932 + 0.685649i \(0.240481\pi\)
\(74\) 0 0
\(75\) −1.93283 −0.223184
\(76\) 0 0
\(77\) 12.7067 1.44807
\(78\) 0 0
\(79\) −6.80169 −0.765250 −0.382625 0.923904i \(-0.624980\pi\)
−0.382625 + 0.923904i \(0.624980\pi\)
\(80\) 0 0
\(81\) −10.6660 −1.18512
\(82\) 0 0
\(83\) 13.5190 1.48391 0.741953 0.670451i \(-0.233900\pi\)
0.741953 + 0.670451i \(0.233900\pi\)
\(84\) 0 0
\(85\) −1.81464 −0.196826
\(86\) 0 0
\(87\) 0.228437 0.0244910
\(88\) 0 0
\(89\) 2.89906 0.307300 0.153650 0.988125i \(-0.450897\pi\)
0.153650 + 0.988125i \(0.450897\pi\)
\(90\) 0 0
\(91\) 10.8100 1.13320
\(92\) 0 0
\(93\) −1.70890 −0.177205
\(94\) 0 0
\(95\) 7.00233 0.718424
\(96\) 0 0
\(97\) −1.97774 −0.200809 −0.100405 0.994947i \(-0.532014\pi\)
−0.100405 + 0.994947i \(0.532014\pi\)
\(98\) 0 0
\(99\) −3.92468 −0.394445
\(100\) 0 0
\(101\) −12.2838 −1.22228 −0.611139 0.791523i \(-0.709289\pi\)
−0.611139 + 0.791523i \(0.709289\pi\)
\(102\) 0 0
\(103\) 12.5061 1.23226 0.616131 0.787644i \(-0.288699\pi\)
0.616131 + 0.787644i \(0.288699\pi\)
\(104\) 0 0
\(105\) −4.60469 −0.449372
\(106\) 0 0
\(107\) 17.3847 1.68064 0.840321 0.542089i \(-0.182367\pi\)
0.840321 + 0.542089i \(0.182367\pi\)
\(108\) 0 0
\(109\) 4.30908 0.412735 0.206368 0.978475i \(-0.433836\pi\)
0.206368 + 0.978475i \(0.433836\pi\)
\(110\) 0 0
\(111\) −14.5330 −1.37941
\(112\) 0 0
\(113\) −12.1494 −1.14292 −0.571460 0.820630i \(-0.693623\pi\)
−0.571460 + 0.820630i \(0.693623\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) 0 0
\(117\) −3.33884 −0.308676
\(118\) 0 0
\(119\) −4.32313 −0.396300
\(120\) 0 0
\(121\) 17.4481 1.58619
\(122\) 0 0
\(123\) 2.80620 0.253027
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 1.22953 0.109103 0.0545515 0.998511i \(-0.482627\pi\)
0.0545515 + 0.998511i \(0.482627\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1.97345 −0.172421 −0.0862104 0.996277i \(-0.527476\pi\)
−0.0862104 + 0.996277i \(0.527476\pi\)
\(132\) 0 0
\(133\) 16.6821 1.44652
\(134\) 0 0
\(135\) −4.37626 −0.376648
\(136\) 0 0
\(137\) 13.7671 1.17620 0.588099 0.808789i \(-0.299876\pi\)
0.588099 + 0.808789i \(0.299876\pi\)
\(138\) 0 0
\(139\) −5.30280 −0.449778 −0.224889 0.974384i \(-0.572202\pi\)
−0.224889 + 0.974384i \(0.572202\pi\)
\(140\) 0 0
\(141\) 20.1766 1.69918
\(142\) 0 0
\(143\) 24.2017 2.02385
\(144\) 0 0
\(145\) 0.118188 0.00981497
\(146\) 0 0
\(147\) 2.55978 0.211127
\(148\) 0 0
\(149\) 7.73256 0.633476 0.316738 0.948513i \(-0.397412\pi\)
0.316738 + 0.948513i \(0.397412\pi\)
\(150\) 0 0
\(151\) −21.4156 −1.74277 −0.871387 0.490596i \(-0.836779\pi\)
−0.871387 + 0.490596i \(0.836779\pi\)
\(152\) 0 0
\(153\) 1.33527 0.107950
\(154\) 0 0
\(155\) −0.884147 −0.0710164
\(156\) 0 0
\(157\) −4.04738 −0.323016 −0.161508 0.986871i \(-0.551636\pi\)
−0.161508 + 0.986871i \(0.551636\pi\)
\(158\) 0 0
\(159\) 18.2105 1.44418
\(160\) 0 0
\(161\) −2.38236 −0.187756
\(162\) 0 0
\(163\) −9.04725 −0.708635 −0.354318 0.935125i \(-0.615287\pi\)
−0.354318 + 0.935125i \(0.615287\pi\)
\(164\) 0 0
\(165\) −10.3091 −0.802561
\(166\) 0 0
\(167\) 12.5467 0.970893 0.485446 0.874266i \(-0.338657\pi\)
0.485446 + 0.874266i \(0.338657\pi\)
\(168\) 0 0
\(169\) 7.58910 0.583777
\(170\) 0 0
\(171\) −5.15252 −0.394023
\(172\) 0 0
\(173\) −14.1354 −1.07469 −0.537346 0.843362i \(-0.680573\pi\)
−0.537346 + 0.843362i \(0.680573\pi\)
\(174\) 0 0
\(175\) −2.38236 −0.180089
\(176\) 0 0
\(177\) 15.0614 1.13208
\(178\) 0 0
\(179\) 26.2442 1.96158 0.980791 0.195061i \(-0.0624904\pi\)
0.980791 + 0.195061i \(0.0624904\pi\)
\(180\) 0 0
\(181\) −18.3461 −1.36365 −0.681826 0.731514i \(-0.738814\pi\)
−0.681826 + 0.731514i \(0.738814\pi\)
\(182\) 0 0
\(183\) 5.42223 0.400823
\(184\) 0 0
\(185\) −7.51903 −0.552810
\(186\) 0 0
\(187\) −9.67871 −0.707777
\(188\) 0 0
\(189\) −10.4258 −0.758366
\(190\) 0 0
\(191\) 3.86566 0.279709 0.139855 0.990172i \(-0.455336\pi\)
0.139855 + 0.990172i \(0.455336\pi\)
\(192\) 0 0
\(193\) −23.8599 −1.71747 −0.858737 0.512417i \(-0.828750\pi\)
−0.858737 + 0.512417i \(0.828750\pi\)
\(194\) 0 0
\(195\) −8.77026 −0.628051
\(196\) 0 0
\(197\) 11.7356 0.836128 0.418064 0.908418i \(-0.362709\pi\)
0.418064 + 0.908418i \(0.362709\pi\)
\(198\) 0 0
\(199\) 8.89906 0.630838 0.315419 0.948953i \(-0.397855\pi\)
0.315419 + 0.948953i \(0.397855\pi\)
\(200\) 0 0
\(201\) −6.01369 −0.424173
\(202\) 0 0
\(203\) 0.281566 0.0197620
\(204\) 0 0
\(205\) 1.45186 0.101403
\(206\) 0 0
\(207\) 0.735829 0.0511437
\(208\) 0 0
\(209\) 37.3482 2.58343
\(210\) 0 0
\(211\) −23.1224 −1.59181 −0.795906 0.605420i \(-0.793005\pi\)
−0.795906 + 0.605420i \(0.793005\pi\)
\(212\) 0 0
\(213\) −26.2688 −1.79991
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −2.10635 −0.142989
\(218\) 0 0
\(219\) −24.0423 −1.62463
\(220\) 0 0
\(221\) −8.23398 −0.553877
\(222\) 0 0
\(223\) −1.50863 −0.101026 −0.0505128 0.998723i \(-0.516086\pi\)
−0.0505128 + 0.998723i \(0.516086\pi\)
\(224\) 0 0
\(225\) 0.735829 0.0490553
\(226\) 0 0
\(227\) 22.0047 1.46050 0.730251 0.683179i \(-0.239403\pi\)
0.730251 + 0.683179i \(0.239403\pi\)
\(228\) 0 0
\(229\) −11.2931 −0.746266 −0.373133 0.927778i \(-0.621717\pi\)
−0.373133 + 0.927778i \(0.621717\pi\)
\(230\) 0 0
\(231\) −24.5599 −1.61593
\(232\) 0 0
\(233\) −17.6789 −1.15818 −0.579091 0.815263i \(-0.696592\pi\)
−0.579091 + 0.815263i \(0.696592\pi\)
\(234\) 0 0
\(235\) 10.4389 0.680960
\(236\) 0 0
\(237\) 13.1465 0.853958
\(238\) 0 0
\(239\) 25.2583 1.63382 0.816911 0.576763i \(-0.195684\pi\)
0.816911 + 0.576763i \(0.195684\pi\)
\(240\) 0 0
\(241\) 0.790615 0.0509280 0.0254640 0.999676i \(-0.491894\pi\)
0.0254640 + 0.999676i \(0.491894\pi\)
\(242\) 0 0
\(243\) 7.48688 0.480283
\(244\) 0 0
\(245\) 1.32437 0.0846108
\(246\) 0 0
\(247\) 31.7732 2.02168
\(248\) 0 0
\(249\) −26.1300 −1.65592
\(250\) 0 0
\(251\) −12.3461 −0.779276 −0.389638 0.920968i \(-0.627400\pi\)
−0.389638 + 0.920968i \(0.627400\pi\)
\(252\) 0 0
\(253\) −5.33368 −0.335325
\(254\) 0 0
\(255\) 3.50739 0.219641
\(256\) 0 0
\(257\) 6.40442 0.399497 0.199748 0.979847i \(-0.435988\pi\)
0.199748 + 0.979847i \(0.435988\pi\)
\(258\) 0 0
\(259\) −17.9130 −1.11306
\(260\) 0 0
\(261\) −0.0869661 −0.00538307
\(262\) 0 0
\(263\) −19.6914 −1.21422 −0.607111 0.794617i \(-0.707672\pi\)
−0.607111 + 0.794617i \(0.707672\pi\)
\(264\) 0 0
\(265\) 9.42167 0.578768
\(266\) 0 0
\(267\) −5.60338 −0.342922
\(268\) 0 0
\(269\) −3.42494 −0.208822 −0.104411 0.994534i \(-0.533296\pi\)
−0.104411 + 0.994534i \(0.533296\pi\)
\(270\) 0 0
\(271\) −13.7197 −0.833411 −0.416705 0.909042i \(-0.636815\pi\)
−0.416705 + 0.909042i \(0.636815\pi\)
\(272\) 0 0
\(273\) −20.8939 −1.26456
\(274\) 0 0
\(275\) −5.33368 −0.321633
\(276\) 0 0
\(277\) 0.636129 0.0382213 0.0191107 0.999817i \(-0.493917\pi\)
0.0191107 + 0.999817i \(0.493917\pi\)
\(278\) 0 0
\(279\) 0.650581 0.0389493
\(280\) 0 0
\(281\) 18.4124 1.09839 0.549195 0.835694i \(-0.314935\pi\)
0.549195 + 0.835694i \(0.314935\pi\)
\(282\) 0 0
\(283\) 7.95729 0.473012 0.236506 0.971630i \(-0.423998\pi\)
0.236506 + 0.971630i \(0.423998\pi\)
\(284\) 0 0
\(285\) −13.5343 −0.801704
\(286\) 0 0
\(287\) 3.45886 0.204170
\(288\) 0 0
\(289\) −13.7071 −0.806299
\(290\) 0 0
\(291\) 3.82264 0.224087
\(292\) 0 0
\(293\) −20.9166 −1.22196 −0.610981 0.791645i \(-0.709225\pi\)
−0.610981 + 0.791645i \(0.709225\pi\)
\(294\) 0 0
\(295\) 7.79239 0.453690
\(296\) 0 0
\(297\) −23.3415 −1.35441
\(298\) 0 0
\(299\) −4.53752 −0.262412
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 23.7424 1.36396
\(304\) 0 0
\(305\) 2.80533 0.160633
\(306\) 0 0
\(307\) 5.78807 0.330342 0.165171 0.986265i \(-0.447182\pi\)
0.165171 + 0.986265i \(0.447182\pi\)
\(308\) 0 0
\(309\) −24.1721 −1.37510
\(310\) 0 0
\(311\) −18.3674 −1.04152 −0.520761 0.853702i \(-0.674352\pi\)
−0.520761 + 0.853702i \(0.674352\pi\)
\(312\) 0 0
\(313\) 9.85869 0.557246 0.278623 0.960401i \(-0.410122\pi\)
0.278623 + 0.960401i \(0.410122\pi\)
\(314\) 0 0
\(315\) 1.75301 0.0987709
\(316\) 0 0
\(317\) −32.9488 −1.85059 −0.925294 0.379250i \(-0.876182\pi\)
−0.925294 + 0.379250i \(0.876182\pi\)
\(318\) 0 0
\(319\) 0.630376 0.0352943
\(320\) 0 0
\(321\) −33.6016 −1.87546
\(322\) 0 0
\(323\) −12.7067 −0.707021
\(324\) 0 0
\(325\) −4.53752 −0.251696
\(326\) 0 0
\(327\) −8.32873 −0.460580
\(328\) 0 0
\(329\) 24.8692 1.37109
\(330\) 0 0
\(331\) 6.85308 0.376679 0.188340 0.982104i \(-0.439689\pi\)
0.188340 + 0.982104i \(0.439689\pi\)
\(332\) 0 0
\(333\) 5.53273 0.303192
\(334\) 0 0
\(335\) −3.11134 −0.169991
\(336\) 0 0
\(337\) 35.8432 1.95250 0.976251 0.216641i \(-0.0695100\pi\)
0.976251 + 0.216641i \(0.0695100\pi\)
\(338\) 0 0
\(339\) 23.4827 1.27541
\(340\) 0 0
\(341\) −4.71575 −0.255372
\(342\) 0 0
\(343\) 19.8316 1.07081
\(344\) 0 0
\(345\) 1.93283 0.104060
\(346\) 0 0
\(347\) 4.09198 0.219669 0.109835 0.993950i \(-0.464968\pi\)
0.109835 + 0.993950i \(0.464968\pi\)
\(348\) 0 0
\(349\) −11.3229 −0.606101 −0.303051 0.952974i \(-0.598005\pi\)
−0.303051 + 0.952974i \(0.598005\pi\)
\(350\) 0 0
\(351\) −19.8574 −1.05991
\(352\) 0 0
\(353\) 26.1443 1.39152 0.695761 0.718273i \(-0.255067\pi\)
0.695761 + 0.718273i \(0.255067\pi\)
\(354\) 0 0
\(355\) −13.5909 −0.721329
\(356\) 0 0
\(357\) 8.35587 0.442239
\(358\) 0 0
\(359\) 12.2058 0.644198 0.322099 0.946706i \(-0.395612\pi\)
0.322099 + 0.946706i \(0.395612\pi\)
\(360\) 0 0
\(361\) 30.0327 1.58067
\(362\) 0 0
\(363\) −33.7242 −1.77006
\(364\) 0 0
\(365\) −12.4389 −0.651083
\(366\) 0 0
\(367\) 3.15299 0.164585 0.0822923 0.996608i \(-0.473776\pi\)
0.0822923 + 0.996608i \(0.473776\pi\)
\(368\) 0 0
\(369\) −1.06832 −0.0556147
\(370\) 0 0
\(371\) 22.4458 1.16533
\(372\) 0 0
\(373\) 9.59872 0.497003 0.248501 0.968632i \(-0.420062\pi\)
0.248501 + 0.968632i \(0.420062\pi\)
\(374\) 0 0
\(375\) 1.93283 0.0998109
\(376\) 0 0
\(377\) 0.536280 0.0276198
\(378\) 0 0
\(379\) −32.7841 −1.68401 −0.842003 0.539473i \(-0.818624\pi\)
−0.842003 + 0.539473i \(0.818624\pi\)
\(380\) 0 0
\(381\) −2.37647 −0.121750
\(382\) 0 0
\(383\) −34.7378 −1.77502 −0.887508 0.460792i \(-0.847565\pi\)
−0.887508 + 0.460792i \(0.847565\pi\)
\(384\) 0 0
\(385\) −12.7067 −0.647595
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 13.8965 0.704581 0.352290 0.935891i \(-0.385403\pi\)
0.352290 + 0.935891i \(0.385403\pi\)
\(390\) 0 0
\(391\) 1.81464 0.0917704
\(392\) 0 0
\(393\) 3.81434 0.192408
\(394\) 0 0
\(395\) 6.80169 0.342230
\(396\) 0 0
\(397\) −26.1135 −1.31060 −0.655299 0.755370i \(-0.727457\pi\)
−0.655299 + 0.755370i \(0.727457\pi\)
\(398\) 0 0
\(399\) −32.2436 −1.61420
\(400\) 0 0
\(401\) 18.0715 0.902446 0.451223 0.892411i \(-0.350988\pi\)
0.451223 + 0.892411i \(0.350988\pi\)
\(402\) 0 0
\(403\) −4.01183 −0.199844
\(404\) 0 0
\(405\) 10.6660 0.530000
\(406\) 0 0
\(407\) −40.1041 −1.98789
\(408\) 0 0
\(409\) −0.865532 −0.0427978 −0.0213989 0.999771i \(-0.506812\pi\)
−0.0213989 + 0.999771i \(0.506812\pi\)
\(410\) 0 0
\(411\) −26.6094 −1.31254
\(412\) 0 0
\(413\) 18.5643 0.913487
\(414\) 0 0
\(415\) −13.5190 −0.663623
\(416\) 0 0
\(417\) 10.2494 0.501916
\(418\) 0 0
\(419\) 20.5675 1.00479 0.502394 0.864639i \(-0.332453\pi\)
0.502394 + 0.864639i \(0.332453\pi\)
\(420\) 0 0
\(421\) 26.2423 1.27897 0.639485 0.768803i \(-0.279147\pi\)
0.639485 + 0.768803i \(0.279147\pi\)
\(422\) 0 0
\(423\) −7.68126 −0.373476
\(424\) 0 0
\(425\) 1.81464 0.0880230
\(426\) 0 0
\(427\) 6.68331 0.323428
\(428\) 0 0
\(429\) −46.7777 −2.25845
\(430\) 0 0
\(431\) −23.2654 −1.12066 −0.560328 0.828271i \(-0.689325\pi\)
−0.560328 + 0.828271i \(0.689325\pi\)
\(432\) 0 0
\(433\) −34.7365 −1.66933 −0.834665 0.550758i \(-0.814339\pi\)
−0.834665 + 0.550758i \(0.814339\pi\)
\(434\) 0 0
\(435\) −0.228437 −0.0109527
\(436\) 0 0
\(437\) −7.00233 −0.334967
\(438\) 0 0
\(439\) −28.5238 −1.36137 −0.680684 0.732577i \(-0.738317\pi\)
−0.680684 + 0.732577i \(0.738317\pi\)
\(440\) 0 0
\(441\) −0.974509 −0.0464052
\(442\) 0 0
\(443\) −26.3428 −1.25158 −0.625792 0.779990i \(-0.715224\pi\)
−0.625792 + 0.779990i \(0.715224\pi\)
\(444\) 0 0
\(445\) −2.89906 −0.137429
\(446\) 0 0
\(447\) −14.9457 −0.706908
\(448\) 0 0
\(449\) 29.5411 1.39413 0.697065 0.717008i \(-0.254489\pi\)
0.697065 + 0.717008i \(0.254489\pi\)
\(450\) 0 0
\(451\) 7.74377 0.364640
\(452\) 0 0
\(453\) 41.3926 1.94480
\(454\) 0 0
\(455\) −10.8100 −0.506781
\(456\) 0 0
\(457\) 3.16668 0.148131 0.0740655 0.997253i \(-0.476403\pi\)
0.0740655 + 0.997253i \(0.476403\pi\)
\(458\) 0 0
\(459\) 7.94133 0.370670
\(460\) 0 0
\(461\) 8.43891 0.393039 0.196520 0.980500i \(-0.437036\pi\)
0.196520 + 0.980500i \(0.437036\pi\)
\(462\) 0 0
\(463\) −9.62461 −0.447294 −0.223647 0.974670i \(-0.571796\pi\)
−0.223647 + 0.974670i \(0.571796\pi\)
\(464\) 0 0
\(465\) 1.70890 0.0792485
\(466\) 0 0
\(467\) 23.1041 1.06913 0.534564 0.845128i \(-0.320476\pi\)
0.534564 + 0.845128i \(0.320476\pi\)
\(468\) 0 0
\(469\) −7.41233 −0.342270
\(470\) 0 0
\(471\) 7.82289 0.360460
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −7.00233 −0.321289
\(476\) 0 0
\(477\) −6.93274 −0.317428
\(478\) 0 0
\(479\) −20.5981 −0.941150 −0.470575 0.882360i \(-0.655953\pi\)
−0.470575 + 0.882360i \(0.655953\pi\)
\(480\) 0 0
\(481\) −34.1178 −1.55564
\(482\) 0 0
\(483\) 4.60469 0.209521
\(484\) 0 0
\(485\) 1.97774 0.0898047
\(486\) 0 0
\(487\) −40.1145 −1.81776 −0.908881 0.417056i \(-0.863062\pi\)
−0.908881 + 0.417056i \(0.863062\pi\)
\(488\) 0 0
\(489\) 17.4868 0.790780
\(490\) 0 0
\(491\) 18.4799 0.833985 0.416993 0.908910i \(-0.363084\pi\)
0.416993 + 0.908910i \(0.363084\pi\)
\(492\) 0 0
\(493\) −0.214469 −0.00965918
\(494\) 0 0
\(495\) 3.92468 0.176401
\(496\) 0 0
\(497\) −32.3783 −1.45237
\(498\) 0 0
\(499\) −0.0506452 −0.00226719 −0.00113360 0.999999i \(-0.500361\pi\)
−0.00113360 + 0.999999i \(0.500361\pi\)
\(500\) 0 0
\(501\) −24.2506 −1.08344
\(502\) 0 0
\(503\) 31.1887 1.39064 0.695318 0.718702i \(-0.255263\pi\)
0.695318 + 0.718702i \(0.255263\pi\)
\(504\) 0 0
\(505\) 12.2838 0.546620
\(506\) 0 0
\(507\) −14.6684 −0.651448
\(508\) 0 0
\(509\) 3.42426 0.151778 0.0758888 0.997116i \(-0.475821\pi\)
0.0758888 + 0.997116i \(0.475821\pi\)
\(510\) 0 0
\(511\) −29.6340 −1.31093
\(512\) 0 0
\(513\) −30.6440 −1.35297
\(514\) 0 0
\(515\) −12.5061 −0.551084
\(516\) 0 0
\(517\) 55.6778 2.44871
\(518\) 0 0
\(519\) 27.3213 1.19927
\(520\) 0 0
\(521\) −9.78442 −0.428663 −0.214332 0.976761i \(-0.568757\pi\)
−0.214332 + 0.976761i \(0.568757\pi\)
\(522\) 0 0
\(523\) −6.22884 −0.272368 −0.136184 0.990684i \(-0.543484\pi\)
−0.136184 + 0.990684i \(0.543484\pi\)
\(524\) 0 0
\(525\) 4.60469 0.200965
\(526\) 0 0
\(527\) 1.60441 0.0698892
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −5.73387 −0.248829
\(532\) 0 0
\(533\) 6.58786 0.285352
\(534\) 0 0
\(535\) −17.3847 −0.751606
\(536\) 0 0
\(537\) −50.7255 −2.18897
\(538\) 0 0
\(539\) 7.06375 0.304257
\(540\) 0 0
\(541\) 19.8477 0.853319 0.426660 0.904412i \(-0.359690\pi\)
0.426660 + 0.904412i \(0.359690\pi\)
\(542\) 0 0
\(543\) 35.4598 1.52173
\(544\) 0 0
\(545\) −4.30908 −0.184581
\(546\) 0 0
\(547\) 7.68024 0.328383 0.164192 0.986428i \(-0.447498\pi\)
0.164192 + 0.986428i \(0.447498\pi\)
\(548\) 0 0
\(549\) −2.06425 −0.0880999
\(550\) 0 0
\(551\) 0.827591 0.0352566
\(552\) 0 0
\(553\) 16.2041 0.689067
\(554\) 0 0
\(555\) 14.5330 0.616892
\(556\) 0 0
\(557\) 18.1260 0.768023 0.384012 0.923328i \(-0.374542\pi\)
0.384012 + 0.923328i \(0.374542\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 18.7073 0.789823
\(562\) 0 0
\(563\) 16.8772 0.711287 0.355644 0.934622i \(-0.384262\pi\)
0.355644 + 0.934622i \(0.384262\pi\)
\(564\) 0 0
\(565\) 12.1494 0.511130
\(566\) 0 0
\(567\) 25.4103 1.06713
\(568\) 0 0
\(569\) 36.0022 1.50929 0.754645 0.656133i \(-0.227809\pi\)
0.754645 + 0.656133i \(0.227809\pi\)
\(570\) 0 0
\(571\) −26.6267 −1.11429 −0.557147 0.830414i \(-0.688104\pi\)
−0.557147 + 0.830414i \(0.688104\pi\)
\(572\) 0 0
\(573\) −7.47166 −0.312133
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) 30.0190 1.24971 0.624854 0.780742i \(-0.285158\pi\)
0.624854 + 0.780742i \(0.285158\pi\)
\(578\) 0 0
\(579\) 46.1171 1.91656
\(580\) 0 0
\(581\) −32.2072 −1.33618
\(582\) 0 0
\(583\) 50.2521 2.08123
\(584\) 0 0
\(585\) 3.33884 0.138044
\(586\) 0 0
\(587\) 31.1814 1.28699 0.643497 0.765449i \(-0.277483\pi\)
0.643497 + 0.765449i \(0.277483\pi\)
\(588\) 0 0
\(589\) −6.19109 −0.255099
\(590\) 0 0
\(591\) −22.6829 −0.933051
\(592\) 0 0
\(593\) 15.0334 0.617348 0.308674 0.951168i \(-0.400115\pi\)
0.308674 + 0.951168i \(0.400115\pi\)
\(594\) 0 0
\(595\) 4.32313 0.177231
\(596\) 0 0
\(597\) −17.2004 −0.703964
\(598\) 0 0
\(599\) −20.3534 −0.831617 −0.415808 0.909452i \(-0.636501\pi\)
−0.415808 + 0.909452i \(0.636501\pi\)
\(600\) 0 0
\(601\) 46.5338 1.89816 0.949078 0.315043i \(-0.102019\pi\)
0.949078 + 0.315043i \(0.102019\pi\)
\(602\) 0 0
\(603\) 2.28942 0.0932323
\(604\) 0 0
\(605\) −17.4481 −0.709366
\(606\) 0 0
\(607\) 7.52725 0.305522 0.152761 0.988263i \(-0.451184\pi\)
0.152761 + 0.988263i \(0.451184\pi\)
\(608\) 0 0
\(609\) −0.544219 −0.0220529
\(610\) 0 0
\(611\) 47.3668 1.91626
\(612\) 0 0
\(613\) 32.2643 1.30314 0.651572 0.758587i \(-0.274110\pi\)
0.651572 + 0.758587i \(0.274110\pi\)
\(614\) 0 0
\(615\) −2.80620 −0.113157
\(616\) 0 0
\(617\) 4.52158 0.182032 0.0910161 0.995849i \(-0.470989\pi\)
0.0910161 + 0.995849i \(0.470989\pi\)
\(618\) 0 0
\(619\) −17.7807 −0.714668 −0.357334 0.933977i \(-0.616314\pi\)
−0.357334 + 0.933977i \(0.616314\pi\)
\(620\) 0 0
\(621\) 4.37626 0.175613
\(622\) 0 0
\(623\) −6.90660 −0.276707
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −72.1877 −2.88290
\(628\) 0 0
\(629\) 13.6444 0.544036
\(630\) 0 0
\(631\) −27.4554 −1.09298 −0.546490 0.837466i \(-0.684036\pi\)
−0.546490 + 0.837466i \(0.684036\pi\)
\(632\) 0 0
\(633\) 44.6917 1.77634
\(634\) 0 0
\(635\) −1.22953 −0.0487924
\(636\) 0 0
\(637\) 6.00935 0.238099
\(638\) 0 0
\(639\) 10.0006 0.395616
\(640\) 0 0
\(641\) −48.9270 −1.93250 −0.966250 0.257605i \(-0.917067\pi\)
−0.966250 + 0.257605i \(0.917067\pi\)
\(642\) 0 0
\(643\) −2.89333 −0.114102 −0.0570508 0.998371i \(-0.518170\pi\)
−0.0570508 + 0.998371i \(0.518170\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −15.0877 −0.593157 −0.296578 0.955009i \(-0.595846\pi\)
−0.296578 + 0.955009i \(0.595846\pi\)
\(648\) 0 0
\(649\) 41.5621 1.63145
\(650\) 0 0
\(651\) 4.07122 0.159564
\(652\) 0 0
\(653\) −13.3255 −0.521468 −0.260734 0.965411i \(-0.583965\pi\)
−0.260734 + 0.965411i \(0.583965\pi\)
\(654\) 0 0
\(655\) 1.97345 0.0771090
\(656\) 0 0
\(657\) 9.15292 0.357089
\(658\) 0 0
\(659\) 39.9000 1.55428 0.777142 0.629325i \(-0.216669\pi\)
0.777142 + 0.629325i \(0.216669\pi\)
\(660\) 0 0
\(661\) −37.3060 −1.45104 −0.725518 0.688203i \(-0.758400\pi\)
−0.725518 + 0.688203i \(0.758400\pi\)
\(662\) 0 0
\(663\) 15.9149 0.618082
\(664\) 0 0
\(665\) −16.6821 −0.646903
\(666\) 0 0
\(667\) −0.118188 −0.00457625
\(668\) 0 0
\(669\) 2.91593 0.112736
\(670\) 0 0
\(671\) 14.9627 0.577630
\(672\) 0 0
\(673\) −18.6641 −0.719447 −0.359724 0.933059i \(-0.617129\pi\)
−0.359724 + 0.933059i \(0.617129\pi\)
\(674\) 0 0
\(675\) 4.37626 0.168442
\(676\) 0 0
\(677\) 19.4978 0.749361 0.374681 0.927154i \(-0.377752\pi\)
0.374681 + 0.927154i \(0.377752\pi\)
\(678\) 0 0
\(679\) 4.71169 0.180818
\(680\) 0 0
\(681\) −42.5313 −1.62980
\(682\) 0 0
\(683\) 41.6331 1.59305 0.796523 0.604608i \(-0.206670\pi\)
0.796523 + 0.604608i \(0.206670\pi\)
\(684\) 0 0
\(685\) −13.7671 −0.526012
\(686\) 0 0
\(687\) 21.8276 0.832773
\(688\) 0 0
\(689\) 42.7510 1.62868
\(690\) 0 0
\(691\) 9.53192 0.362611 0.181306 0.983427i \(-0.441968\pi\)
0.181306 + 0.983427i \(0.441968\pi\)
\(692\) 0 0
\(693\) 9.34998 0.355177
\(694\) 0 0
\(695\) 5.30280 0.201147
\(696\) 0 0
\(697\) −2.63461 −0.0997930
\(698\) 0 0
\(699\) 34.1702 1.29244
\(700\) 0 0
\(701\) 27.9897 1.05716 0.528579 0.848884i \(-0.322725\pi\)
0.528579 + 0.848884i \(0.322725\pi\)
\(702\) 0 0
\(703\) −52.6508 −1.98576
\(704\) 0 0
\(705\) −20.1766 −0.759896
\(706\) 0 0
\(707\) 29.2643 1.10060
\(708\) 0 0
\(709\) 27.6832 1.03966 0.519831 0.854269i \(-0.325995\pi\)
0.519831 + 0.854269i \(0.325995\pi\)
\(710\) 0 0
\(711\) −5.00489 −0.187698
\(712\) 0 0
\(713\) 0.884147 0.0331115
\(714\) 0 0
\(715\) −24.2017 −0.905091
\(716\) 0 0
\(717\) −48.8200 −1.82322
\(718\) 0 0
\(719\) −50.8364 −1.89588 −0.947938 0.318454i \(-0.896836\pi\)
−0.947938 + 0.318454i \(0.896836\pi\)
\(720\) 0 0
\(721\) −29.7940 −1.10959
\(722\) 0 0
\(723\) −1.52812 −0.0568316
\(724\) 0 0
\(725\) −0.118188 −0.00438939
\(726\) 0 0
\(727\) −37.5061 −1.39102 −0.695512 0.718514i \(-0.744823\pi\)
−0.695512 + 0.718514i \(0.744823\pi\)
\(728\) 0 0
\(729\) 17.5273 0.649158
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −36.9374 −1.36431 −0.682157 0.731206i \(-0.738958\pi\)
−0.682157 + 0.731206i \(0.738958\pi\)
\(734\) 0 0
\(735\) −2.55978 −0.0944188
\(736\) 0 0
\(737\) −16.5949 −0.611281
\(738\) 0 0
\(739\) 12.3287 0.453517 0.226759 0.973951i \(-0.427187\pi\)
0.226759 + 0.973951i \(0.427187\pi\)
\(740\) 0 0
\(741\) −61.4123 −2.25604
\(742\) 0 0
\(743\) 29.7534 1.09154 0.545772 0.837933i \(-0.316236\pi\)
0.545772 + 0.837933i \(0.316236\pi\)
\(744\) 0 0
\(745\) −7.73256 −0.283299
\(746\) 0 0
\(747\) 9.94770 0.363967
\(748\) 0 0
\(749\) −41.4166 −1.51333
\(750\) 0 0
\(751\) 41.9834 1.53200 0.765999 0.642842i \(-0.222245\pi\)
0.765999 + 0.642842i \(0.222245\pi\)
\(752\) 0 0
\(753\) 23.8628 0.869610
\(754\) 0 0
\(755\) 21.4156 0.779392
\(756\) 0 0
\(757\) 11.4708 0.416915 0.208458 0.978031i \(-0.433156\pi\)
0.208458 + 0.978031i \(0.433156\pi\)
\(758\) 0 0
\(759\) 10.3091 0.374196
\(760\) 0 0
\(761\) −13.4122 −0.486193 −0.243097 0.970002i \(-0.578163\pi\)
−0.243097 + 0.970002i \(0.578163\pi\)
\(762\) 0 0
\(763\) −10.2658 −0.371646
\(764\) 0 0
\(765\) −1.33527 −0.0482767
\(766\) 0 0
\(767\) 35.3581 1.27671
\(768\) 0 0
\(769\) 2.18327 0.0787307 0.0393654 0.999225i \(-0.487466\pi\)
0.0393654 + 0.999225i \(0.487466\pi\)
\(770\) 0 0
\(771\) −12.3787 −0.445806
\(772\) 0 0
\(773\) 5.73306 0.206204 0.103102 0.994671i \(-0.467123\pi\)
0.103102 + 0.994671i \(0.467123\pi\)
\(774\) 0 0
\(775\) 0.884147 0.0317595
\(776\) 0 0
\(777\) 34.6228 1.24209
\(778\) 0 0
\(779\) 10.1664 0.364250
\(780\) 0 0
\(781\) −72.4893 −2.59387
\(782\) 0 0
\(783\) −0.517220 −0.0184839
\(784\) 0 0
\(785\) 4.04738 0.144457
\(786\) 0 0
\(787\) 8.13502 0.289982 0.144991 0.989433i \(-0.453685\pi\)
0.144991 + 0.989433i \(0.453685\pi\)
\(788\) 0 0
\(789\) 38.0601 1.35497
\(790\) 0 0
\(791\) 28.9442 1.02914
\(792\) 0 0
\(793\) 12.7293 0.452029
\(794\) 0 0
\(795\) −18.2105 −0.645859
\(796\) 0 0
\(797\) −41.5139 −1.47050 −0.735250 0.677797i \(-0.762935\pi\)
−0.735250 + 0.677797i \(0.762935\pi\)
\(798\) 0 0
\(799\) −18.9429 −0.670151
\(800\) 0 0
\(801\) 2.13321 0.0753733
\(802\) 0 0
\(803\) −66.3451 −2.34127
\(804\) 0 0
\(805\) 2.38236 0.0839671
\(806\) 0 0
\(807\) 6.61982 0.233029
\(808\) 0 0
\(809\) −22.8566 −0.803596 −0.401798 0.915728i \(-0.631615\pi\)
−0.401798 + 0.915728i \(0.631615\pi\)
\(810\) 0 0
\(811\) 42.1917 1.48155 0.740776 0.671753i \(-0.234458\pi\)
0.740776 + 0.671753i \(0.234458\pi\)
\(812\) 0 0
\(813\) 26.5178 0.930020
\(814\) 0 0
\(815\) 9.04725 0.316911
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 7.95432 0.277946
\(820\) 0 0
\(821\) 27.0195 0.942985 0.471493 0.881870i \(-0.343715\pi\)
0.471493 + 0.881870i \(0.343715\pi\)
\(822\) 0 0
\(823\) −27.2435 −0.949650 −0.474825 0.880080i \(-0.657489\pi\)
−0.474825 + 0.880080i \(0.657489\pi\)
\(824\) 0 0
\(825\) 10.3091 0.358916
\(826\) 0 0
\(827\) −10.5732 −0.367667 −0.183834 0.982957i \(-0.558851\pi\)
−0.183834 + 0.982957i \(0.558851\pi\)
\(828\) 0 0
\(829\) −1.16445 −0.0404429 −0.0202215 0.999796i \(-0.506437\pi\)
−0.0202215 + 0.999796i \(0.506437\pi\)
\(830\) 0 0
\(831\) −1.22953 −0.0426519
\(832\) 0 0
\(833\) −2.40325 −0.0832678
\(834\) 0 0
\(835\) −12.5467 −0.434197
\(836\) 0 0
\(837\) 3.86925 0.133741
\(838\) 0 0
\(839\) 20.2920 0.700556 0.350278 0.936646i \(-0.386087\pi\)
0.350278 + 0.936646i \(0.386087\pi\)
\(840\) 0 0
\(841\) −28.9860 −0.999518
\(842\) 0 0
\(843\) −35.5880 −1.22571
\(844\) 0 0
\(845\) −7.58910 −0.261073
\(846\) 0 0
\(847\) −41.5676 −1.42828
\(848\) 0 0
\(849\) −15.3801 −0.527843
\(850\) 0 0
\(851\) 7.51903 0.257749
\(852\) 0 0
\(853\) 26.6154 0.911295 0.455648 0.890160i \(-0.349408\pi\)
0.455648 + 0.890160i \(0.349408\pi\)
\(854\) 0 0
\(855\) 5.15252 0.176213
\(856\) 0 0
\(857\) −10.4342 −0.356427 −0.178214 0.983992i \(-0.557032\pi\)
−0.178214 + 0.983992i \(0.557032\pi\)
\(858\) 0 0
\(859\) 24.6623 0.841466 0.420733 0.907185i \(-0.361773\pi\)
0.420733 + 0.907185i \(0.361773\pi\)
\(860\) 0 0
\(861\) −6.68538 −0.227837
\(862\) 0 0
\(863\) −8.48317 −0.288771 −0.144385 0.989522i \(-0.546120\pi\)
−0.144385 + 0.989522i \(0.546120\pi\)
\(864\) 0 0
\(865\) 14.1354 0.480617
\(866\) 0 0
\(867\) 26.4934 0.899764
\(868\) 0 0
\(869\) 36.2780 1.23065
\(870\) 0 0
\(871\) −14.1178 −0.478363
\(872\) 0 0
\(873\) −1.45528 −0.0492538
\(874\) 0 0
\(875\) 2.38236 0.0805384
\(876\) 0 0
\(877\) 3.08796 0.104273 0.0521365 0.998640i \(-0.483397\pi\)
0.0521365 + 0.998640i \(0.483397\pi\)
\(878\) 0 0
\(879\) 40.4282 1.36361
\(880\) 0 0
\(881\) 42.5879 1.43482 0.717412 0.696649i \(-0.245327\pi\)
0.717412 + 0.696649i \(0.245327\pi\)
\(882\) 0 0
\(883\) 48.3342 1.62657 0.813287 0.581863i \(-0.197676\pi\)
0.813287 + 0.581863i \(0.197676\pi\)
\(884\) 0 0
\(885\) −15.0614 −0.506282
\(886\) 0 0
\(887\) 2.12173 0.0712408 0.0356204 0.999365i \(-0.488659\pi\)
0.0356204 + 0.999365i \(0.488659\pi\)
\(888\) 0 0
\(889\) −2.92918 −0.0982415
\(890\) 0 0
\(891\) 56.8892 1.90586
\(892\) 0 0
\(893\) 73.0968 2.44609
\(894\) 0 0
\(895\) −26.2442 −0.877246
\(896\) 0 0
\(897\) 8.77026 0.292830
\(898\) 0 0
\(899\) −0.104495 −0.00348512
\(900\) 0 0
\(901\) −17.0970 −0.569582
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 18.3461 0.609844
\(906\) 0 0
\(907\) −28.6914 −0.952684 −0.476342 0.879260i \(-0.658038\pi\)
−0.476342 + 0.879260i \(0.658038\pi\)
\(908\) 0 0
\(909\) −9.03875 −0.299796
\(910\) 0 0
\(911\) 1.14899 0.0380679 0.0190339 0.999819i \(-0.493941\pi\)
0.0190339 + 0.999819i \(0.493941\pi\)
\(912\) 0 0
\(913\) −72.1061 −2.38636
\(914\) 0 0
\(915\) −5.42223 −0.179253
\(916\) 0 0
\(917\) 4.70146 0.155256
\(918\) 0 0
\(919\) 55.4798 1.83011 0.915055 0.403329i \(-0.132147\pi\)
0.915055 + 0.403329i \(0.132147\pi\)
\(920\) 0 0
\(921\) −11.1873 −0.368636
\(922\) 0 0
\(923\) −61.6689 −2.02986
\(924\) 0 0
\(925\) 7.51903 0.247224
\(926\) 0 0
\(927\) 9.20235 0.302245
\(928\) 0 0
\(929\) −4.05780 −0.133132 −0.0665660 0.997782i \(-0.521204\pi\)
−0.0665660 + 0.997782i \(0.521204\pi\)
\(930\) 0 0
\(931\) 9.27367 0.303932
\(932\) 0 0
\(933\) 35.5011 1.16226
\(934\) 0 0
\(935\) 9.67871 0.316528
\(936\) 0 0
\(937\) 16.0588 0.524617 0.262308 0.964984i \(-0.415516\pi\)
0.262308 + 0.964984i \(0.415516\pi\)
\(938\) 0 0
\(939\) −19.0552 −0.621842
\(940\) 0 0
\(941\) 4.73779 0.154448 0.0772238 0.997014i \(-0.475394\pi\)
0.0772238 + 0.997014i \(0.475394\pi\)
\(942\) 0 0
\(943\) −1.45186 −0.0472792
\(944\) 0 0
\(945\) 10.4258 0.339152
\(946\) 0 0
\(947\) 3.38634 0.110041 0.0550207 0.998485i \(-0.482478\pi\)
0.0550207 + 0.998485i \(0.482478\pi\)
\(948\) 0 0
\(949\) −56.4418 −1.83218
\(950\) 0 0
\(951\) 63.6844 2.06511
\(952\) 0 0
\(953\) 29.2967 0.949013 0.474507 0.880252i \(-0.342627\pi\)
0.474507 + 0.880252i \(0.342627\pi\)
\(954\) 0 0
\(955\) −3.86566 −0.125090
\(956\) 0 0
\(957\) −1.21841 −0.0393856
\(958\) 0 0
\(959\) −32.7981 −1.05910
\(960\) 0 0
\(961\) −30.2183 −0.974783
\(962\) 0 0
\(963\) 12.7922 0.412222
\(964\) 0 0
\(965\) 23.8599 0.768078
\(966\) 0 0
\(967\) 15.5743 0.500837 0.250419 0.968138i \(-0.419432\pi\)
0.250419 + 0.968138i \(0.419432\pi\)
\(968\) 0 0
\(969\) 24.5599 0.788979
\(970\) 0 0
\(971\) −51.5951 −1.65577 −0.827883 0.560900i \(-0.810455\pi\)
−0.827883 + 0.560900i \(0.810455\pi\)
\(972\) 0 0
\(973\) 12.6332 0.405001
\(974\) 0 0
\(975\) 8.77026 0.280873
\(976\) 0 0
\(977\) 9.04984 0.289530 0.144765 0.989466i \(-0.453757\pi\)
0.144765 + 0.989466i \(0.453757\pi\)
\(978\) 0 0
\(979\) −15.4626 −0.494188
\(980\) 0 0
\(981\) 3.17075 0.101234
\(982\) 0 0
\(983\) 39.3855 1.25620 0.628101 0.778132i \(-0.283833\pi\)
0.628101 + 0.778132i \(0.283833\pi\)
\(984\) 0 0
\(985\) −11.7356 −0.373928
\(986\) 0 0
\(987\) −48.0680 −1.53002
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 18.9152 0.600860 0.300430 0.953804i \(-0.402870\pi\)
0.300430 + 0.953804i \(0.402870\pi\)
\(992\) 0 0
\(993\) −13.2458 −0.420344
\(994\) 0 0
\(995\) −8.89906 −0.282119
\(996\) 0 0
\(997\) −12.2966 −0.389438 −0.194719 0.980859i \(-0.562380\pi\)
−0.194719 + 0.980859i \(0.562380\pi\)
\(998\) 0 0
\(999\) 32.9052 1.04107
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 1840.2.a.v.1.2 5
4.3 odd 2 920.2.a.j.1.4 5
5.4 even 2 9200.2.a.cu.1.4 5
8.3 odd 2 7360.2.a.co.1.2 5
8.5 even 2 7360.2.a.cp.1.4 5
12.11 even 2 8280.2.a.bs.1.4 5
20.3 even 4 4600.2.e.u.4049.8 10
20.7 even 4 4600.2.e.u.4049.3 10
20.19 odd 2 4600.2.a.be.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.j.1.4 5 4.3 odd 2
1840.2.a.v.1.2 5 1.1 even 1 trivial
4600.2.a.be.1.2 5 20.19 odd 2
4600.2.e.u.4049.3 10 20.7 even 4
4600.2.e.u.4049.8 10 20.3 even 4
7360.2.a.co.1.2 5 8.3 odd 2
7360.2.a.cp.1.4 5 8.5 even 2
8280.2.a.bs.1.4 5 12.11 even 2
9200.2.a.cu.1.4 5 5.4 even 2