Properties

Label 2280.2.a.o.1.1
Level $2280$
Weight $2$
Character 2280.1
Self dual yes
Analytic conductor $18.206$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2280.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(18.2058916609\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 2280.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} -3.41421 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{5} -3.41421 q^{7} +1.00000 q^{9} -1.41421 q^{11} +4.24264 q^{13} -1.00000 q^{15} +2.82843 q^{17} -1.00000 q^{19} -3.41421 q^{21} -4.82843 q^{23} +1.00000 q^{25} +1.00000 q^{27} +2.24264 q^{29} -8.82843 q^{31} -1.41421 q^{33} +3.41421 q^{35} -7.07107 q^{37} +4.24264 q^{39} +2.24264 q^{41} -1.75736 q^{43} -1.00000 q^{45} +4.82843 q^{47} +4.65685 q^{49} +2.82843 q^{51} -12.4853 q^{53} +1.41421 q^{55} -1.00000 q^{57} -2.82843 q^{59} -8.00000 q^{61} -3.41421 q^{63} -4.24264 q^{65} -11.3137 q^{67} -4.82843 q^{69} -5.17157 q^{71} -3.65685 q^{73} +1.00000 q^{75} +4.82843 q^{77} -2.34315 q^{79} +1.00000 q^{81} -6.00000 q^{83} -2.82843 q^{85} +2.24264 q^{87} -13.5563 q^{89} -14.4853 q^{91} -8.82843 q^{93} +1.00000 q^{95} +9.89949 q^{97} -1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} - 4 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{3} - 2 q^{5} - 4 q^{7} + 2 q^{9} - 2 q^{15} - 2 q^{19} - 4 q^{21} - 4 q^{23} + 2 q^{25} + 2 q^{27} - 4 q^{29} - 12 q^{31} + 4 q^{35} - 4 q^{41} - 12 q^{43} - 2 q^{45} + 4 q^{47} - 2 q^{49} - 8 q^{53} - 2 q^{57} - 16 q^{61} - 4 q^{63} - 4 q^{69} - 16 q^{71} + 4 q^{73} + 2 q^{75} + 4 q^{77} - 16 q^{79} + 2 q^{81} - 12 q^{83} - 4 q^{87} + 4 q^{89} - 12 q^{91} - 12 q^{93} + 2 q^{95} + O(q^{100}) \)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −3.41421 −1.29045 −0.645226 0.763992i \(-0.723237\pi\)
−0.645226 + 0.763992i \(0.723237\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.41421 −0.426401 −0.213201 0.977008i \(-0.568389\pi\)
−0.213201 + 0.977008i \(0.568389\pi\)
\(12\) 0 0
\(13\) 4.24264 1.17670 0.588348 0.808608i \(-0.299778\pi\)
0.588348 + 0.808608i \(0.299778\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 2.82843 0.685994 0.342997 0.939336i \(-0.388558\pi\)
0.342997 + 0.939336i \(0.388558\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −3.41421 −0.745042
\(22\) 0 0
\(23\) −4.82843 −1.00680 −0.503398 0.864054i \(-0.667917\pi\)
−0.503398 + 0.864054i \(0.667917\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 2.24264 0.416448 0.208224 0.978081i \(-0.433232\pi\)
0.208224 + 0.978081i \(0.433232\pi\)
\(30\) 0 0
\(31\) −8.82843 −1.58563 −0.792816 0.609461i \(-0.791386\pi\)
−0.792816 + 0.609461i \(0.791386\pi\)
\(32\) 0 0
\(33\) −1.41421 −0.246183
\(34\) 0 0
\(35\) 3.41421 0.577107
\(36\) 0 0
\(37\) −7.07107 −1.16248 −0.581238 0.813733i \(-0.697432\pi\)
−0.581238 + 0.813733i \(0.697432\pi\)
\(38\) 0 0
\(39\) 4.24264 0.679366
\(40\) 0 0
\(41\) 2.24264 0.350242 0.175121 0.984547i \(-0.443968\pi\)
0.175121 + 0.984547i \(0.443968\pi\)
\(42\) 0 0
\(43\) −1.75736 −0.267995 −0.133997 0.990982i \(-0.542781\pi\)
−0.133997 + 0.990982i \(0.542781\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 4.82843 0.704298 0.352149 0.935944i \(-0.385451\pi\)
0.352149 + 0.935944i \(0.385451\pi\)
\(48\) 0 0
\(49\) 4.65685 0.665265
\(50\) 0 0
\(51\) 2.82843 0.396059
\(52\) 0 0
\(53\) −12.4853 −1.71499 −0.857493 0.514496i \(-0.827979\pi\)
−0.857493 + 0.514496i \(0.827979\pi\)
\(54\) 0 0
\(55\) 1.41421 0.190693
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) −2.82843 −0.368230 −0.184115 0.982905i \(-0.558942\pi\)
−0.184115 + 0.982905i \(0.558942\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 0 0
\(63\) −3.41421 −0.430150
\(64\) 0 0
\(65\) −4.24264 −0.526235
\(66\) 0 0
\(67\) −11.3137 −1.38219 −0.691095 0.722764i \(-0.742871\pi\)
−0.691095 + 0.722764i \(0.742871\pi\)
\(68\) 0 0
\(69\) −4.82843 −0.581274
\(70\) 0 0
\(71\) −5.17157 −0.613753 −0.306876 0.951749i \(-0.599284\pi\)
−0.306876 + 0.951749i \(0.599284\pi\)
\(72\) 0 0
\(73\) −3.65685 −0.428002 −0.214001 0.976833i \(-0.568650\pi\)
−0.214001 + 0.976833i \(0.568650\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 4.82843 0.550250
\(78\) 0 0
\(79\) −2.34315 −0.263624 −0.131812 0.991275i \(-0.542080\pi\)
−0.131812 + 0.991275i \(0.542080\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) −2.82843 −0.306786
\(86\) 0 0
\(87\) 2.24264 0.240436
\(88\) 0 0
\(89\) −13.5563 −1.43697 −0.718485 0.695542i \(-0.755164\pi\)
−0.718485 + 0.695542i \(0.755164\pi\)
\(90\) 0 0
\(91\) −14.4853 −1.51847
\(92\) 0 0
\(93\) −8.82843 −0.915465
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) 9.89949 1.00514 0.502571 0.864536i \(-0.332388\pi\)
0.502571 + 0.864536i \(0.332388\pi\)
\(98\) 0 0
\(99\) −1.41421 −0.142134
\(100\) 0 0
\(101\) −12.8284 −1.27648 −0.638238 0.769839i \(-0.720336\pi\)
−0.638238 + 0.769839i \(0.720336\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 0 0
\(105\) 3.41421 0.333193
\(106\) 0 0
\(107\) 13.6569 1.32026 0.660129 0.751152i \(-0.270502\pi\)
0.660129 + 0.751152i \(0.270502\pi\)
\(108\) 0 0
\(109\) 12.1421 1.16301 0.581503 0.813544i \(-0.302465\pi\)
0.581503 + 0.813544i \(0.302465\pi\)
\(110\) 0 0
\(111\) −7.07107 −0.671156
\(112\) 0 0
\(113\) 9.65685 0.908440 0.454220 0.890889i \(-0.349918\pi\)
0.454220 + 0.890889i \(0.349918\pi\)
\(114\) 0 0
\(115\) 4.82843 0.450253
\(116\) 0 0
\(117\) 4.24264 0.392232
\(118\) 0 0
\(119\) −9.65685 −0.885242
\(120\) 0 0
\(121\) −9.00000 −0.818182
\(122\) 0 0
\(123\) 2.24264 0.202212
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 16.9706 1.50589 0.752947 0.658081i \(-0.228632\pi\)
0.752947 + 0.658081i \(0.228632\pi\)
\(128\) 0 0
\(129\) −1.75736 −0.154727
\(130\) 0 0
\(131\) 4.92893 0.430643 0.215321 0.976543i \(-0.430920\pi\)
0.215321 + 0.976543i \(0.430920\pi\)
\(132\) 0 0
\(133\) 3.41421 0.296050
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −14.9706 −1.27902 −0.639511 0.768782i \(-0.720863\pi\)
−0.639511 + 0.768782i \(0.720863\pi\)
\(138\) 0 0
\(139\) −12.4853 −1.05899 −0.529494 0.848314i \(-0.677618\pi\)
−0.529494 + 0.848314i \(0.677618\pi\)
\(140\) 0 0
\(141\) 4.82843 0.406627
\(142\) 0 0
\(143\) −6.00000 −0.501745
\(144\) 0 0
\(145\) −2.24264 −0.186241
\(146\) 0 0
\(147\) 4.65685 0.384091
\(148\) 0 0
\(149\) 11.6569 0.954967 0.477483 0.878641i \(-0.341549\pi\)
0.477483 + 0.878641i \(0.341549\pi\)
\(150\) 0 0
\(151\) −11.1716 −0.909130 −0.454565 0.890714i \(-0.650205\pi\)
−0.454565 + 0.890714i \(0.650205\pi\)
\(152\) 0 0
\(153\) 2.82843 0.228665
\(154\) 0 0
\(155\) 8.82843 0.709116
\(156\) 0 0
\(157\) −8.14214 −0.649813 −0.324907 0.945746i \(-0.605333\pi\)
−0.324907 + 0.945746i \(0.605333\pi\)
\(158\) 0 0
\(159\) −12.4853 −0.990147
\(160\) 0 0
\(161\) 16.4853 1.29922
\(162\) 0 0
\(163\) −0.100505 −0.00787216 −0.00393608 0.999992i \(-0.501253\pi\)
−0.00393608 + 0.999992i \(0.501253\pi\)
\(164\) 0 0
\(165\) 1.41421 0.110096
\(166\) 0 0
\(167\) 15.6569 1.21156 0.605782 0.795631i \(-0.292860\pi\)
0.605782 + 0.795631i \(0.292860\pi\)
\(168\) 0 0
\(169\) 5.00000 0.384615
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 0 0
\(173\) 13.6569 1.03831 0.519156 0.854680i \(-0.326246\pi\)
0.519156 + 0.854680i \(0.326246\pi\)
\(174\) 0 0
\(175\) −3.41421 −0.258090
\(176\) 0 0
\(177\) −2.82843 −0.212598
\(178\) 0 0
\(179\) 23.7990 1.77882 0.889410 0.457110i \(-0.151116\pi\)
0.889410 + 0.457110i \(0.151116\pi\)
\(180\) 0 0
\(181\) 4.82843 0.358894 0.179447 0.983768i \(-0.442569\pi\)
0.179447 + 0.983768i \(0.442569\pi\)
\(182\) 0 0
\(183\) −8.00000 −0.591377
\(184\) 0 0
\(185\) 7.07107 0.519875
\(186\) 0 0
\(187\) −4.00000 −0.292509
\(188\) 0 0
\(189\) −3.41421 −0.248347
\(190\) 0 0
\(191\) 13.4142 0.970618 0.485309 0.874343i \(-0.338707\pi\)
0.485309 + 0.874343i \(0.338707\pi\)
\(192\) 0 0
\(193\) 23.0711 1.66069 0.830346 0.557248i \(-0.188143\pi\)
0.830346 + 0.557248i \(0.188143\pi\)
\(194\) 0 0
\(195\) −4.24264 −0.303822
\(196\) 0 0
\(197\) 8.48528 0.604551 0.302276 0.953221i \(-0.402254\pi\)
0.302276 + 0.953221i \(0.402254\pi\)
\(198\) 0 0
\(199\) −14.1421 −1.00251 −0.501255 0.865300i \(-0.667128\pi\)
−0.501255 + 0.865300i \(0.667128\pi\)
\(200\) 0 0
\(201\) −11.3137 −0.798007
\(202\) 0 0
\(203\) −7.65685 −0.537406
\(204\) 0 0
\(205\) −2.24264 −0.156633
\(206\) 0 0
\(207\) −4.82843 −0.335599
\(208\) 0 0
\(209\) 1.41421 0.0978232
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 0 0
\(213\) −5.17157 −0.354350
\(214\) 0 0
\(215\) 1.75736 0.119851
\(216\) 0 0
\(217\) 30.1421 2.04618
\(218\) 0 0
\(219\) −3.65685 −0.247107
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) −27.3137 −1.82906 −0.914531 0.404517i \(-0.867440\pi\)
−0.914531 + 0.404517i \(0.867440\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −5.31371 −0.352683 −0.176342 0.984329i \(-0.556426\pi\)
−0.176342 + 0.984329i \(0.556426\pi\)
\(228\) 0 0
\(229\) −18.6274 −1.23093 −0.615467 0.788163i \(-0.711033\pi\)
−0.615467 + 0.788163i \(0.711033\pi\)
\(230\) 0 0
\(231\) 4.82843 0.317687
\(232\) 0 0
\(233\) 7.65685 0.501617 0.250809 0.968037i \(-0.419304\pi\)
0.250809 + 0.968037i \(0.419304\pi\)
\(234\) 0 0
\(235\) −4.82843 −0.314972
\(236\) 0 0
\(237\) −2.34315 −0.152204
\(238\) 0 0
\(239\) −12.2426 −0.791911 −0.395955 0.918270i \(-0.629586\pi\)
−0.395955 + 0.918270i \(0.629586\pi\)
\(240\) 0 0
\(241\) −14.9706 −0.964339 −0.482169 0.876078i \(-0.660151\pi\)
−0.482169 + 0.876078i \(0.660151\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −4.65685 −0.297516
\(246\) 0 0
\(247\) −4.24264 −0.269953
\(248\) 0 0
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) 24.7279 1.56081 0.780406 0.625273i \(-0.215012\pi\)
0.780406 + 0.625273i \(0.215012\pi\)
\(252\) 0 0
\(253\) 6.82843 0.429300
\(254\) 0 0
\(255\) −2.82843 −0.177123
\(256\) 0 0
\(257\) 17.6569 1.10140 0.550702 0.834702i \(-0.314360\pi\)
0.550702 + 0.834702i \(0.314360\pi\)
\(258\) 0 0
\(259\) 24.1421 1.50012
\(260\) 0 0
\(261\) 2.24264 0.138816
\(262\) 0 0
\(263\) −19.6569 −1.21209 −0.606047 0.795429i \(-0.707246\pi\)
−0.606047 + 0.795429i \(0.707246\pi\)
\(264\) 0 0
\(265\) 12.4853 0.766965
\(266\) 0 0
\(267\) −13.5563 −0.829635
\(268\) 0 0
\(269\) 4.10051 0.250012 0.125006 0.992156i \(-0.460105\pi\)
0.125006 + 0.992156i \(0.460105\pi\)
\(270\) 0 0
\(271\) 2.14214 0.130125 0.0650627 0.997881i \(-0.479275\pi\)
0.0650627 + 0.997881i \(0.479275\pi\)
\(272\) 0 0
\(273\) −14.4853 −0.876689
\(274\) 0 0
\(275\) −1.41421 −0.0852803
\(276\) 0 0
\(277\) −5.31371 −0.319270 −0.159635 0.987176i \(-0.551032\pi\)
−0.159635 + 0.987176i \(0.551032\pi\)
\(278\) 0 0
\(279\) −8.82843 −0.528544
\(280\) 0 0
\(281\) 14.2426 0.849645 0.424822 0.905277i \(-0.360337\pi\)
0.424822 + 0.905277i \(0.360337\pi\)
\(282\) 0 0
\(283\) −4.10051 −0.243750 −0.121875 0.992545i \(-0.538891\pi\)
−0.121875 + 0.992545i \(0.538891\pi\)
\(284\) 0 0
\(285\) 1.00000 0.0592349
\(286\) 0 0
\(287\) −7.65685 −0.451970
\(288\) 0 0
\(289\) −9.00000 −0.529412
\(290\) 0 0
\(291\) 9.89949 0.580319
\(292\) 0 0
\(293\) 4.00000 0.233682 0.116841 0.993151i \(-0.462723\pi\)
0.116841 + 0.993151i \(0.462723\pi\)
\(294\) 0 0
\(295\) 2.82843 0.164677
\(296\) 0 0
\(297\) −1.41421 −0.0820610
\(298\) 0 0
\(299\) −20.4853 −1.18469
\(300\) 0 0
\(301\) 6.00000 0.345834
\(302\) 0 0
\(303\) −12.8284 −0.736974
\(304\) 0 0
\(305\) 8.00000 0.458079
\(306\) 0 0
\(307\) 1.17157 0.0668652 0.0334326 0.999441i \(-0.489356\pi\)
0.0334326 + 0.999441i \(0.489356\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) −9.41421 −0.533831 −0.266916 0.963720i \(-0.586004\pi\)
−0.266916 + 0.963720i \(0.586004\pi\)
\(312\) 0 0
\(313\) 29.7990 1.68434 0.842169 0.539213i \(-0.181278\pi\)
0.842169 + 0.539213i \(0.181278\pi\)
\(314\) 0 0
\(315\) 3.41421 0.192369
\(316\) 0 0
\(317\) 8.48528 0.476581 0.238290 0.971194i \(-0.423413\pi\)
0.238290 + 0.971194i \(0.423413\pi\)
\(318\) 0 0
\(319\) −3.17157 −0.177574
\(320\) 0 0
\(321\) 13.6569 0.762251
\(322\) 0 0
\(323\) −2.82843 −0.157378
\(324\) 0 0
\(325\) 4.24264 0.235339
\(326\) 0 0
\(327\) 12.1421 0.671462
\(328\) 0 0
\(329\) −16.4853 −0.908863
\(330\) 0 0
\(331\) −12.8284 −0.705114 −0.352557 0.935790i \(-0.614688\pi\)
−0.352557 + 0.935790i \(0.614688\pi\)
\(332\) 0 0
\(333\) −7.07107 −0.387492
\(334\) 0 0
\(335\) 11.3137 0.618134
\(336\) 0 0
\(337\) −1.89949 −0.103472 −0.0517360 0.998661i \(-0.516475\pi\)
−0.0517360 + 0.998661i \(0.516475\pi\)
\(338\) 0 0
\(339\) 9.65685 0.524488
\(340\) 0 0
\(341\) 12.4853 0.676116
\(342\) 0 0
\(343\) 8.00000 0.431959
\(344\) 0 0
\(345\) 4.82843 0.259954
\(346\) 0 0
\(347\) 0.343146 0.0184210 0.00921051 0.999958i \(-0.497068\pi\)
0.00921051 + 0.999958i \(0.497068\pi\)
\(348\) 0 0
\(349\) −22.0000 −1.17763 −0.588817 0.808267i \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) 0 0
\(351\) 4.24264 0.226455
\(352\) 0 0
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 0 0
\(355\) 5.17157 0.274479
\(356\) 0 0
\(357\) −9.65685 −0.511095
\(358\) 0 0
\(359\) 20.0416 1.05776 0.528878 0.848698i \(-0.322613\pi\)
0.528878 + 0.848698i \(0.322613\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −9.00000 −0.472377
\(364\) 0 0
\(365\) 3.65685 0.191408
\(366\) 0 0
\(367\) −5.07107 −0.264708 −0.132354 0.991203i \(-0.542254\pi\)
−0.132354 + 0.991203i \(0.542254\pi\)
\(368\) 0 0
\(369\) 2.24264 0.116747
\(370\) 0 0
\(371\) 42.6274 2.21311
\(372\) 0 0
\(373\) −28.0416 −1.45194 −0.725970 0.687726i \(-0.758609\pi\)
−0.725970 + 0.687726i \(0.758609\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 9.51472 0.490033
\(378\) 0 0
\(379\) −10.4853 −0.538593 −0.269296 0.963057i \(-0.586791\pi\)
−0.269296 + 0.963057i \(0.586791\pi\)
\(380\) 0 0
\(381\) 16.9706 0.869428
\(382\) 0 0
\(383\) −23.3137 −1.19127 −0.595637 0.803253i \(-0.703100\pi\)
−0.595637 + 0.803253i \(0.703100\pi\)
\(384\) 0 0
\(385\) −4.82843 −0.246079
\(386\) 0 0
\(387\) −1.75736 −0.0893316
\(388\) 0 0
\(389\) −23.6569 −1.19945 −0.599725 0.800206i \(-0.704723\pi\)
−0.599725 + 0.800206i \(0.704723\pi\)
\(390\) 0 0
\(391\) −13.6569 −0.690657
\(392\) 0 0
\(393\) 4.92893 0.248632
\(394\) 0 0
\(395\) 2.34315 0.117896
\(396\) 0 0
\(397\) 26.0000 1.30490 0.652451 0.757831i \(-0.273741\pi\)
0.652451 + 0.757831i \(0.273741\pi\)
\(398\) 0 0
\(399\) 3.41421 0.170924
\(400\) 0 0
\(401\) −11.2132 −0.559961 −0.279980 0.960006i \(-0.590328\pi\)
−0.279980 + 0.960006i \(0.590328\pi\)
\(402\) 0 0
\(403\) −37.4558 −1.86581
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 10.0000 0.495682
\(408\) 0 0
\(409\) −36.1421 −1.78711 −0.893557 0.448950i \(-0.851798\pi\)
−0.893557 + 0.448950i \(0.851798\pi\)
\(410\) 0 0
\(411\) −14.9706 −0.738443
\(412\) 0 0
\(413\) 9.65685 0.475183
\(414\) 0 0
\(415\) 6.00000 0.294528
\(416\) 0 0
\(417\) −12.4853 −0.611407
\(418\) 0 0
\(419\) 28.2426 1.37974 0.689872 0.723932i \(-0.257667\pi\)
0.689872 + 0.723932i \(0.257667\pi\)
\(420\) 0 0
\(421\) 25.3137 1.23371 0.616857 0.787075i \(-0.288406\pi\)
0.616857 + 0.787075i \(0.288406\pi\)
\(422\) 0 0
\(423\) 4.82843 0.234766
\(424\) 0 0
\(425\) 2.82843 0.137199
\(426\) 0 0
\(427\) 27.3137 1.32180
\(428\) 0 0
\(429\) −6.00000 −0.289683
\(430\) 0 0
\(431\) −34.1421 −1.64457 −0.822284 0.569077i \(-0.807301\pi\)
−0.822284 + 0.569077i \(0.807301\pi\)
\(432\) 0 0
\(433\) 11.7574 0.565023 0.282511 0.959264i \(-0.408833\pi\)
0.282511 + 0.959264i \(0.408833\pi\)
\(434\) 0 0
\(435\) −2.24264 −0.107526
\(436\) 0 0
\(437\) 4.82843 0.230975
\(438\) 0 0
\(439\) −4.68629 −0.223664 −0.111832 0.993727i \(-0.535672\pi\)
−0.111832 + 0.993727i \(0.535672\pi\)
\(440\) 0 0
\(441\) 4.65685 0.221755
\(442\) 0 0
\(443\) 4.14214 0.196799 0.0983994 0.995147i \(-0.468628\pi\)
0.0983994 + 0.995147i \(0.468628\pi\)
\(444\) 0 0
\(445\) 13.5563 0.642633
\(446\) 0 0
\(447\) 11.6569 0.551350
\(448\) 0 0
\(449\) 26.5269 1.25188 0.625941 0.779870i \(-0.284715\pi\)
0.625941 + 0.779870i \(0.284715\pi\)
\(450\) 0 0
\(451\) −3.17157 −0.149344
\(452\) 0 0
\(453\) −11.1716 −0.524886
\(454\) 0 0
\(455\) 14.4853 0.679080
\(456\) 0 0
\(457\) 20.1421 0.942209 0.471105 0.882077i \(-0.343855\pi\)
0.471105 + 0.882077i \(0.343855\pi\)
\(458\) 0 0
\(459\) 2.82843 0.132020
\(460\) 0 0
\(461\) −20.6274 −0.960715 −0.480357 0.877073i \(-0.659493\pi\)
−0.480357 + 0.877073i \(0.659493\pi\)
\(462\) 0 0
\(463\) −25.7574 −1.19705 −0.598523 0.801106i \(-0.704245\pi\)
−0.598523 + 0.801106i \(0.704245\pi\)
\(464\) 0 0
\(465\) 8.82843 0.409409
\(466\) 0 0
\(467\) −20.1421 −0.932067 −0.466033 0.884767i \(-0.654317\pi\)
−0.466033 + 0.884767i \(0.654317\pi\)
\(468\) 0 0
\(469\) 38.6274 1.78365
\(470\) 0 0
\(471\) −8.14214 −0.375170
\(472\) 0 0
\(473\) 2.48528 0.114273
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) −12.4853 −0.571662
\(478\) 0 0
\(479\) 1.61522 0.0738015 0.0369007 0.999319i \(-0.488251\pi\)
0.0369007 + 0.999319i \(0.488251\pi\)
\(480\) 0 0
\(481\) −30.0000 −1.36788
\(482\) 0 0
\(483\) 16.4853 0.750106
\(484\) 0 0
\(485\) −9.89949 −0.449513
\(486\) 0 0
\(487\) 18.8284 0.853197 0.426599 0.904441i \(-0.359712\pi\)
0.426599 + 0.904441i \(0.359712\pi\)
\(488\) 0 0
\(489\) −0.100505 −0.00454500
\(490\) 0 0
\(491\) 31.7574 1.43319 0.716595 0.697490i \(-0.245700\pi\)
0.716595 + 0.697490i \(0.245700\pi\)
\(492\) 0 0
\(493\) 6.34315 0.285681
\(494\) 0 0
\(495\) 1.41421 0.0635642
\(496\) 0 0
\(497\) 17.6569 0.792018
\(498\) 0 0
\(499\) 14.8284 0.663812 0.331906 0.943313i \(-0.392308\pi\)
0.331906 + 0.943313i \(0.392308\pi\)
\(500\) 0 0
\(501\) 15.6569 0.699497
\(502\) 0 0
\(503\) −26.2843 −1.17196 −0.585979 0.810326i \(-0.699290\pi\)
−0.585979 + 0.810326i \(0.699290\pi\)
\(504\) 0 0
\(505\) 12.8284 0.570858
\(506\) 0 0
\(507\) 5.00000 0.222058
\(508\) 0 0
\(509\) 13.0711 0.579365 0.289682 0.957123i \(-0.406450\pi\)
0.289682 + 0.957123i \(0.406450\pi\)
\(510\) 0 0
\(511\) 12.4853 0.552316
\(512\) 0 0
\(513\) −1.00000 −0.0441511
\(514\) 0 0
\(515\) 8.00000 0.352522
\(516\) 0 0
\(517\) −6.82843 −0.300314
\(518\) 0 0
\(519\) 13.6569 0.599469
\(520\) 0 0
\(521\) −20.3848 −0.893073 −0.446537 0.894765i \(-0.647343\pi\)
−0.446537 + 0.894765i \(0.647343\pi\)
\(522\) 0 0
\(523\) 38.1421 1.66784 0.833920 0.551886i \(-0.186092\pi\)
0.833920 + 0.551886i \(0.186092\pi\)
\(524\) 0 0
\(525\) −3.41421 −0.149008
\(526\) 0 0
\(527\) −24.9706 −1.08773
\(528\) 0 0
\(529\) 0.313708 0.0136395
\(530\) 0 0
\(531\) −2.82843 −0.122743
\(532\) 0 0
\(533\) 9.51472 0.412128
\(534\) 0 0
\(535\) −13.6569 −0.590437
\(536\) 0 0
\(537\) 23.7990 1.02700
\(538\) 0 0
\(539\) −6.58579 −0.283670
\(540\) 0 0
\(541\) 18.6863 0.803386 0.401693 0.915774i \(-0.368422\pi\)
0.401693 + 0.915774i \(0.368422\pi\)
\(542\) 0 0
\(543\) 4.82843 0.207208
\(544\) 0 0
\(545\) −12.1421 −0.520112
\(546\) 0 0
\(547\) 28.4853 1.21794 0.608971 0.793192i \(-0.291582\pi\)
0.608971 + 0.793192i \(0.291582\pi\)
\(548\) 0 0
\(549\) −8.00000 −0.341432
\(550\) 0 0
\(551\) −2.24264 −0.0955397
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) 0 0
\(555\) 7.07107 0.300150
\(556\) 0 0
\(557\) 10.9706 0.464838 0.232419 0.972616i \(-0.425336\pi\)
0.232419 + 0.972616i \(0.425336\pi\)
\(558\) 0 0
\(559\) −7.45584 −0.315349
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) 0 0
\(563\) 33.3137 1.40401 0.702003 0.712174i \(-0.252289\pi\)
0.702003 + 0.712174i \(0.252289\pi\)
\(564\) 0 0
\(565\) −9.65685 −0.406267
\(566\) 0 0
\(567\) −3.41421 −0.143383
\(568\) 0 0
\(569\) −4.10051 −0.171902 −0.0859511 0.996299i \(-0.527393\pi\)
−0.0859511 + 0.996299i \(0.527393\pi\)
\(570\) 0 0
\(571\) 0.485281 0.0203084 0.0101542 0.999948i \(-0.496768\pi\)
0.0101542 + 0.999948i \(0.496768\pi\)
\(572\) 0 0
\(573\) 13.4142 0.560387
\(574\) 0 0
\(575\) −4.82843 −0.201359
\(576\) 0 0
\(577\) 1.51472 0.0630586 0.0315293 0.999503i \(-0.489962\pi\)
0.0315293 + 0.999503i \(0.489962\pi\)
\(578\) 0 0
\(579\) 23.0711 0.958801
\(580\) 0 0
\(581\) 20.4853 0.849873
\(582\) 0 0
\(583\) 17.6569 0.731272
\(584\) 0 0
\(585\) −4.24264 −0.175412
\(586\) 0 0
\(587\) 7.17157 0.296002 0.148001 0.988987i \(-0.452716\pi\)
0.148001 + 0.988987i \(0.452716\pi\)
\(588\) 0 0
\(589\) 8.82843 0.363769
\(590\) 0 0
\(591\) 8.48528 0.349038
\(592\) 0 0
\(593\) −21.3137 −0.875249 −0.437625 0.899158i \(-0.644180\pi\)
−0.437625 + 0.899158i \(0.644180\pi\)
\(594\) 0 0
\(595\) 9.65685 0.395892
\(596\) 0 0
\(597\) −14.1421 −0.578799
\(598\) 0 0
\(599\) −16.9706 −0.693398 −0.346699 0.937976i \(-0.612698\pi\)
−0.346699 + 0.937976i \(0.612698\pi\)
\(600\) 0 0
\(601\) −40.1421 −1.63743 −0.818716 0.574199i \(-0.805314\pi\)
−0.818716 + 0.574199i \(0.805314\pi\)
\(602\) 0 0
\(603\) −11.3137 −0.460730
\(604\) 0 0
\(605\) 9.00000 0.365902
\(606\) 0 0
\(607\) 34.1421 1.38579 0.692893 0.721040i \(-0.256336\pi\)
0.692893 + 0.721040i \(0.256336\pi\)
\(608\) 0 0
\(609\) −7.65685 −0.310271
\(610\) 0 0
\(611\) 20.4853 0.828746
\(612\) 0 0
\(613\) 0.142136 0.00574080 0.00287040 0.999996i \(-0.499086\pi\)
0.00287040 + 0.999996i \(0.499086\pi\)
\(614\) 0 0
\(615\) −2.24264 −0.0904320
\(616\) 0 0
\(617\) −31.1127 −1.25255 −0.626275 0.779602i \(-0.715421\pi\)
−0.626275 + 0.779602i \(0.715421\pi\)
\(618\) 0 0
\(619\) −5.17157 −0.207863 −0.103932 0.994584i \(-0.533142\pi\)
−0.103932 + 0.994584i \(0.533142\pi\)
\(620\) 0 0
\(621\) −4.82843 −0.193758
\(622\) 0 0
\(623\) 46.2843 1.85434
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 1.41421 0.0564782
\(628\) 0 0
\(629\) −20.0000 −0.797452
\(630\) 0 0
\(631\) 8.97056 0.357112 0.178556 0.983930i \(-0.442857\pi\)
0.178556 + 0.983930i \(0.442857\pi\)
\(632\) 0 0
\(633\) −12.0000 −0.476957
\(634\) 0 0
\(635\) −16.9706 −0.673456
\(636\) 0 0
\(637\) 19.7574 0.782815
\(638\) 0 0
\(639\) −5.17157 −0.204584
\(640\) 0 0
\(641\) −34.2426 −1.35250 −0.676251 0.736671i \(-0.736397\pi\)
−0.676251 + 0.736671i \(0.736397\pi\)
\(642\) 0 0
\(643\) 1.75736 0.0693035 0.0346517 0.999399i \(-0.488968\pi\)
0.0346517 + 0.999399i \(0.488968\pi\)
\(644\) 0 0
\(645\) 1.75736 0.0691960
\(646\) 0 0
\(647\) 20.3431 0.799772 0.399886 0.916565i \(-0.369050\pi\)
0.399886 + 0.916565i \(0.369050\pi\)
\(648\) 0 0
\(649\) 4.00000 0.157014
\(650\) 0 0
\(651\) 30.1421 1.18136
\(652\) 0 0
\(653\) −12.4853 −0.488587 −0.244293 0.969701i \(-0.578556\pi\)
−0.244293 + 0.969701i \(0.578556\pi\)
\(654\) 0 0
\(655\) −4.92893 −0.192589
\(656\) 0 0
\(657\) −3.65685 −0.142667
\(658\) 0 0
\(659\) −6.34315 −0.247094 −0.123547 0.992339i \(-0.539427\pi\)
−0.123547 + 0.992339i \(0.539427\pi\)
\(660\) 0 0
\(661\) 25.1127 0.976771 0.488385 0.872628i \(-0.337586\pi\)
0.488385 + 0.872628i \(0.337586\pi\)
\(662\) 0 0
\(663\) 12.0000 0.466041
\(664\) 0 0
\(665\) −3.41421 −0.132398
\(666\) 0 0
\(667\) −10.8284 −0.419278
\(668\) 0 0
\(669\) −27.3137 −1.05601
\(670\) 0 0
\(671\) 11.3137 0.436761
\(672\) 0 0
\(673\) −29.8995 −1.15254 −0.576270 0.817259i \(-0.695492\pi\)
−0.576270 + 0.817259i \(0.695492\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 39.7990 1.52960 0.764800 0.644268i \(-0.222838\pi\)
0.764800 + 0.644268i \(0.222838\pi\)
\(678\) 0 0
\(679\) −33.7990 −1.29709
\(680\) 0 0
\(681\) −5.31371 −0.203622
\(682\) 0 0
\(683\) −20.6863 −0.791539 −0.395769 0.918350i \(-0.629522\pi\)
−0.395769 + 0.918350i \(0.629522\pi\)
\(684\) 0 0
\(685\) 14.9706 0.571996
\(686\) 0 0
\(687\) −18.6274 −0.710680
\(688\) 0 0
\(689\) −52.9706 −2.01802
\(690\) 0 0
\(691\) −31.1127 −1.18358 −0.591791 0.806091i \(-0.701579\pi\)
−0.591791 + 0.806091i \(0.701579\pi\)
\(692\) 0 0
\(693\) 4.82843 0.183417
\(694\) 0 0
\(695\) 12.4853 0.473594
\(696\) 0 0
\(697\) 6.34315 0.240264
\(698\) 0 0
\(699\) 7.65685 0.289609
\(700\) 0 0
\(701\) 0.343146 0.0129604 0.00648022 0.999979i \(-0.497937\pi\)
0.00648022 + 0.999979i \(0.497937\pi\)
\(702\) 0 0
\(703\) 7.07107 0.266690
\(704\) 0 0
\(705\) −4.82843 −0.181849
\(706\) 0 0
\(707\) 43.7990 1.64723
\(708\) 0 0
\(709\) −18.3431 −0.688891 −0.344446 0.938806i \(-0.611933\pi\)
−0.344446 + 0.938806i \(0.611933\pi\)
\(710\) 0 0
\(711\) −2.34315 −0.0878748
\(712\) 0 0
\(713\) 42.6274 1.59641
\(714\) 0 0
\(715\) 6.00000 0.224387
\(716\) 0 0
\(717\) −12.2426 −0.457210
\(718\) 0 0
\(719\) 46.3848 1.72986 0.864930 0.501892i \(-0.167363\pi\)
0.864930 + 0.501892i \(0.167363\pi\)
\(720\) 0 0
\(721\) 27.3137 1.01722
\(722\) 0 0
\(723\) −14.9706 −0.556761
\(724\) 0 0
\(725\) 2.24264 0.0832896
\(726\) 0 0
\(727\) 8.38478 0.310974 0.155487 0.987838i \(-0.450305\pi\)
0.155487 + 0.987838i \(0.450305\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −4.97056 −0.183843
\(732\) 0 0
\(733\) 26.0000 0.960332 0.480166 0.877178i \(-0.340576\pi\)
0.480166 + 0.877178i \(0.340576\pi\)
\(734\) 0 0
\(735\) −4.65685 −0.171771
\(736\) 0 0
\(737\) 16.0000 0.589368
\(738\) 0 0
\(739\) 8.68629 0.319530 0.159765 0.987155i \(-0.448926\pi\)
0.159765 + 0.987155i \(0.448926\pi\)
\(740\) 0 0
\(741\) −4.24264 −0.155857
\(742\) 0 0
\(743\) 45.9411 1.68542 0.842708 0.538371i \(-0.180960\pi\)
0.842708 + 0.538371i \(0.180960\pi\)
\(744\) 0 0
\(745\) −11.6569 −0.427074
\(746\) 0 0
\(747\) −6.00000 −0.219529
\(748\) 0 0
\(749\) −46.6274 −1.70373
\(750\) 0 0
\(751\) −0.142136 −0.00518660 −0.00259330 0.999997i \(-0.500825\pi\)
−0.00259330 + 0.999997i \(0.500825\pi\)
\(752\) 0 0
\(753\) 24.7279 0.901136
\(754\) 0 0
\(755\) 11.1716 0.406575
\(756\) 0 0
\(757\) 46.7696 1.69987 0.849934 0.526889i \(-0.176642\pi\)
0.849934 + 0.526889i \(0.176642\pi\)
\(758\) 0 0
\(759\) 6.82843 0.247856
\(760\) 0 0
\(761\) 14.9706 0.542682 0.271341 0.962483i \(-0.412533\pi\)
0.271341 + 0.962483i \(0.412533\pi\)
\(762\) 0 0
\(763\) −41.4558 −1.50080
\(764\) 0 0
\(765\) −2.82843 −0.102262
\(766\) 0 0
\(767\) −12.0000 −0.433295
\(768\) 0 0
\(769\) −19.6569 −0.708844 −0.354422 0.935086i \(-0.615322\pi\)
−0.354422 + 0.935086i \(0.615322\pi\)
\(770\) 0 0
\(771\) 17.6569 0.635896
\(772\) 0 0
\(773\) 24.4853 0.880674 0.440337 0.897832i \(-0.354859\pi\)
0.440337 + 0.897832i \(0.354859\pi\)
\(774\) 0 0
\(775\) −8.82843 −0.317126
\(776\) 0 0
\(777\) 24.1421 0.866094
\(778\) 0 0
\(779\) −2.24264 −0.0803509
\(780\) 0 0
\(781\) 7.31371 0.261705
\(782\) 0 0
\(783\) 2.24264 0.0801454
\(784\) 0 0
\(785\) 8.14214 0.290605
\(786\) 0 0
\(787\) −8.48528 −0.302468 −0.151234 0.988498i \(-0.548325\pi\)
−0.151234 + 0.988498i \(0.548325\pi\)
\(788\) 0 0
\(789\) −19.6569 −0.699803
\(790\) 0 0
\(791\) −32.9706 −1.17230
\(792\) 0 0
\(793\) −33.9411 −1.20528
\(794\) 0 0
\(795\) 12.4853 0.442807
\(796\) 0 0
\(797\) −40.9706 −1.45125 −0.725626 0.688089i \(-0.758450\pi\)
−0.725626 + 0.688089i \(0.758450\pi\)
\(798\) 0 0
\(799\) 13.6569 0.483145
\(800\) 0 0
\(801\) −13.5563 −0.478990
\(802\) 0 0
\(803\) 5.17157 0.182501
\(804\) 0 0
\(805\) −16.4853 −0.581030
\(806\) 0 0
\(807\) 4.10051 0.144345
\(808\) 0 0
\(809\) 2.68629 0.0944450 0.0472225 0.998884i \(-0.484963\pi\)
0.0472225 + 0.998884i \(0.484963\pi\)
\(810\) 0 0
\(811\) −48.2843 −1.69549 −0.847745 0.530404i \(-0.822040\pi\)
−0.847745 + 0.530404i \(0.822040\pi\)
\(812\) 0 0
\(813\) 2.14214 0.0751280
\(814\) 0 0
\(815\) 0.100505 0.00352054
\(816\) 0 0
\(817\) 1.75736 0.0614822
\(818\) 0 0
\(819\) −14.4853 −0.506157
\(820\) 0 0
\(821\) −22.4853 −0.784742 −0.392371 0.919807i \(-0.628345\pi\)
−0.392371 + 0.919807i \(0.628345\pi\)
\(822\) 0 0
\(823\) 19.2132 0.669730 0.334865 0.942266i \(-0.391309\pi\)
0.334865 + 0.942266i \(0.391309\pi\)
\(824\) 0 0
\(825\) −1.41421 −0.0492366
\(826\) 0 0
\(827\) 26.0000 0.904109 0.452054 0.891990i \(-0.350691\pi\)
0.452054 + 0.891990i \(0.350691\pi\)
\(828\) 0 0
\(829\) −1.79899 −0.0624815 −0.0312408 0.999512i \(-0.509946\pi\)
−0.0312408 + 0.999512i \(0.509946\pi\)
\(830\) 0 0
\(831\) −5.31371 −0.184331
\(832\) 0 0
\(833\) 13.1716 0.456368
\(834\) 0 0
\(835\) −15.6569 −0.541828
\(836\) 0 0
\(837\) −8.82843 −0.305155
\(838\) 0 0
\(839\) −23.5147 −0.811818 −0.405909 0.913913i \(-0.633045\pi\)
−0.405909 + 0.913913i \(0.633045\pi\)
\(840\) 0 0
\(841\) −23.9706 −0.826571
\(842\) 0 0
\(843\) 14.2426 0.490543
\(844\) 0 0
\(845\) −5.00000 −0.172005
\(846\) 0 0
\(847\) 30.7279 1.05582
\(848\) 0 0
\(849\) −4.10051 −0.140729
\(850\) 0 0
\(851\) 34.1421 1.17038
\(852\) 0 0
\(853\) −47.4558 −1.62486 −0.812429 0.583061i \(-0.801855\pi\)
−0.812429 + 0.583061i \(0.801855\pi\)
\(854\) 0 0
\(855\) 1.00000 0.0341993
\(856\) 0 0
\(857\) 44.0833 1.50586 0.752928 0.658103i \(-0.228641\pi\)
0.752928 + 0.658103i \(0.228641\pi\)
\(858\) 0 0
\(859\) 49.9411 1.70397 0.851985 0.523567i \(-0.175399\pi\)
0.851985 + 0.523567i \(0.175399\pi\)
\(860\) 0 0
\(861\) −7.65685 −0.260945
\(862\) 0 0
\(863\) −36.0000 −1.22545 −0.612727 0.790295i \(-0.709928\pi\)
−0.612727 + 0.790295i \(0.709928\pi\)
\(864\) 0 0
\(865\) −13.6569 −0.464347
\(866\) 0 0
\(867\) −9.00000 −0.305656
\(868\) 0 0
\(869\) 3.31371 0.112410
\(870\) 0 0
\(871\) −48.0000 −1.62642
\(872\) 0 0
\(873\) 9.89949 0.335047
\(874\) 0 0
\(875\) 3.41421 0.115421
\(876\) 0 0
\(877\) 37.4142 1.26339 0.631694 0.775218i \(-0.282360\pi\)
0.631694 + 0.775218i \(0.282360\pi\)
\(878\) 0 0
\(879\) 4.00000 0.134917
\(880\) 0 0
\(881\) 33.1127 1.11560 0.557798 0.829977i \(-0.311647\pi\)
0.557798 + 0.829977i \(0.311647\pi\)
\(882\) 0 0
\(883\) 9.27208 0.312030 0.156015 0.987755i \(-0.450135\pi\)
0.156015 + 0.987755i \(0.450135\pi\)
\(884\) 0 0
\(885\) 2.82843 0.0950765
\(886\) 0 0
\(887\) 16.9706 0.569816 0.284908 0.958555i \(-0.408037\pi\)
0.284908 + 0.958555i \(0.408037\pi\)
\(888\) 0 0
\(889\) −57.9411 −1.94328
\(890\) 0 0
\(891\) −1.41421 −0.0473779
\(892\) 0 0
\(893\) −4.82843 −0.161577
\(894\) 0 0
\(895\) −23.7990 −0.795512
\(896\) 0 0
\(897\) −20.4853 −0.683984
\(898\) 0 0
\(899\) −19.7990 −0.660333
\(900\) 0 0
\(901\) −35.3137 −1.17647
\(902\) 0 0
\(903\) 6.00000 0.199667
\(904\) 0 0
\(905\) −4.82843 −0.160502
\(906\) 0 0
\(907\) −13.8579 −0.460143 −0.230071 0.973174i \(-0.573896\pi\)
−0.230071 + 0.973174i \(0.573896\pi\)
\(908\) 0 0
\(909\) −12.8284 −0.425492
\(910\) 0 0
\(911\) −20.0000 −0.662630 −0.331315 0.943520i \(-0.607492\pi\)
−0.331315 + 0.943520i \(0.607492\pi\)
\(912\) 0 0
\(913\) 8.48528 0.280822
\(914\) 0 0
\(915\) 8.00000 0.264472
\(916\) 0 0
\(917\) −16.8284 −0.555724
\(918\) 0 0
\(919\) −17.6569 −0.582446 −0.291223 0.956655i \(-0.594062\pi\)
−0.291223 + 0.956655i \(0.594062\pi\)
\(920\) 0 0
\(921\) 1.17157 0.0386046
\(922\) 0 0
\(923\) −21.9411 −0.722201
\(924\) 0 0
\(925\) −7.07107 −0.232495
\(926\) 0 0
\(927\) −8.00000 −0.262754
\(928\) 0 0
\(929\) −24.8284 −0.814594 −0.407297 0.913296i \(-0.633529\pi\)
−0.407297 + 0.913296i \(0.633529\pi\)
\(930\) 0 0
\(931\) −4.65685 −0.152622
\(932\) 0 0
\(933\) −9.41421 −0.308208
\(934\) 0 0
\(935\) 4.00000 0.130814
\(936\) 0 0
\(937\) −13.1127 −0.428373 −0.214187 0.976793i \(-0.568710\pi\)
−0.214187 + 0.976793i \(0.568710\pi\)
\(938\) 0 0
\(939\) 29.7990 0.972453
\(940\) 0 0
\(941\) 22.0416 0.718537 0.359268 0.933234i \(-0.383026\pi\)
0.359268 + 0.933234i \(0.383026\pi\)
\(942\) 0 0
\(943\) −10.8284 −0.352622
\(944\) 0 0
\(945\) 3.41421 0.111064
\(946\) 0 0
\(947\) 25.3137 0.822585 0.411292 0.911503i \(-0.365077\pi\)
0.411292 + 0.911503i \(0.365077\pi\)
\(948\) 0 0
\(949\) −15.5147 −0.503629
\(950\) 0 0
\(951\) 8.48528 0.275154
\(952\) 0 0
\(953\) −21.9411 −0.710743 −0.355371 0.934725i \(-0.615646\pi\)
−0.355371 + 0.934725i \(0.615646\pi\)
\(954\) 0 0
\(955\) −13.4142 −0.434074
\(956\) 0 0
\(957\) −3.17157 −0.102522
\(958\) 0 0
\(959\) 51.1127 1.65052
\(960\) 0 0
\(961\) 46.9411 1.51423
\(962\) 0 0
\(963\) 13.6569 0.440086
\(964\) 0 0
\(965\) −23.0711 −0.742684
\(966\) 0 0
\(967\) −7.89949 −0.254031 −0.127015 0.991901i \(-0.540540\pi\)
−0.127015 + 0.991901i \(0.540540\pi\)
\(968\) 0 0
\(969\) −2.82843 −0.0908622
\(970\) 0 0
\(971\) 0.686292 0.0220241 0.0110121 0.999939i \(-0.496495\pi\)
0.0110121 + 0.999939i \(0.496495\pi\)
\(972\) 0 0
\(973\) 42.6274 1.36657
\(974\) 0 0
\(975\) 4.24264 0.135873
\(976\) 0 0
\(977\) −20.2010 −0.646288 −0.323144 0.946350i \(-0.604740\pi\)
−0.323144 + 0.946350i \(0.604740\pi\)
\(978\) 0 0
\(979\) 19.1716 0.612726
\(980\) 0 0
\(981\) 12.1421 0.387669
\(982\) 0 0
\(983\) 21.5980 0.688869 0.344434 0.938810i \(-0.388071\pi\)
0.344434 + 0.938810i \(0.388071\pi\)
\(984\) 0 0
\(985\) −8.48528 −0.270364
\(986\) 0 0
\(987\) −16.4853 −0.524732
\(988\) 0 0
\(989\) 8.48528 0.269816
\(990\) 0 0
\(991\) −15.0294 −0.477426 −0.238713 0.971090i \(-0.576725\pi\)
−0.238713 + 0.971090i \(0.576725\pi\)
\(992\) 0 0
\(993\) −12.8284 −0.407098
\(994\) 0 0
\(995\) 14.1421 0.448336
\(996\) 0 0
\(997\) 49.7990 1.57715 0.788575 0.614939i \(-0.210819\pi\)
0.788575 + 0.614939i \(0.210819\pi\)
\(998\) 0 0
\(999\) −7.07107 −0.223719
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 2280.2.a.o.1.1 2
3.2 odd 2 6840.2.a.x.1.1 2
4.3 odd 2 4560.2.a.bg.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2280.2.a.o.1.1 2 1.1 even 1 trivial
4560.2.a.bg.1.2 2 4.3 odd 2
6840.2.a.x.1.1 2 3.2 odd 2