Properties

Label 6240.2.a.br.1.2
Level $6240$
Weight $2$
Character 6240.1
Self dual yes
Analytic conductor $49.827$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6240.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(49.8266508613\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 6240.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} +2.82843 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{5} +2.82843 q^{7} +1.00000 q^{9} +2.82843 q^{11} +1.00000 q^{13} -1.00000 q^{15} +7.65685 q^{17} +6.82843 q^{19} +2.82843 q^{21} -4.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +3.65685 q^{29} +1.17157 q^{31} +2.82843 q^{33} -2.82843 q^{35} -2.00000 q^{37} +1.00000 q^{39} -2.00000 q^{41} -1.65685 q^{43} -1.00000 q^{45} +1.17157 q^{47} +1.00000 q^{49} +7.65685 q^{51} -2.00000 q^{53} -2.82843 q^{55} +6.82843 q^{57} -8.48528 q^{59} +6.00000 q^{61} +2.82843 q^{63} -1.00000 q^{65} +2.82843 q^{67} -4.00000 q^{69} -10.8284 q^{71} +4.34315 q^{73} +1.00000 q^{75} +8.00000 q^{77} -13.6569 q^{79} +1.00000 q^{81} +6.82843 q^{83} -7.65685 q^{85} +3.65685 q^{87} -5.31371 q^{89} +2.82843 q^{91} +1.17157 q^{93} -6.82843 q^{95} +7.65685 q^{97} +2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9} + 2 q^{13} - 2 q^{15} + 4 q^{17} + 8 q^{19} - 8 q^{23} + 2 q^{25} + 2 q^{27} - 4 q^{29} + 8 q^{31} - 4 q^{37} + 2 q^{39} - 4 q^{41} + 8 q^{43} - 2 q^{45} + 8 q^{47} + 2 q^{49} + 4 q^{51} - 4 q^{53} + 8 q^{57} + 12 q^{61} - 2 q^{65} - 8 q^{69} - 16 q^{71} + 20 q^{73} + 2 q^{75} + 16 q^{77} - 16 q^{79} + 2 q^{81} + 8 q^{83} - 4 q^{85} - 4 q^{87} + 12 q^{89} + 8 q^{93} - 8 q^{95} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(7\) 2.82843 1.06904 0.534522 0.845154i \(-0.320491\pi\)
0.534522 + 0.845154i \(0.320491\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.82843 0.852803 0.426401 0.904534i \(-0.359781\pi\)
0.426401 + 0.904534i \(0.359781\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 7.65685 1.85706 0.928530 0.371257i \(-0.121073\pi\)
0.928530 + 0.371257i \(0.121073\pi\)
\(18\) 0 0
\(19\) 6.82843 1.56655 0.783274 0.621676i \(-0.213548\pi\)
0.783274 + 0.621676i \(0.213548\pi\)
\(20\) 0 0
\(21\) 2.82843 0.617213
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 3.65685 0.679061 0.339530 0.940595i \(-0.389732\pi\)
0.339530 + 0.940595i \(0.389732\pi\)
\(30\) 0 0
\(31\) 1.17157 0.210421 0.105210 0.994450i \(-0.466448\pi\)
0.105210 + 0.994450i \(0.466448\pi\)
\(32\) 0 0
\(33\) 2.82843 0.492366
\(34\) 0 0
\(35\) −2.82843 −0.478091
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −1.65685 −0.252668 −0.126334 0.991988i \(-0.540321\pi\)
−0.126334 + 0.991988i \(0.540321\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 1.17157 0.170891 0.0854457 0.996343i \(-0.472769\pi\)
0.0854457 + 0.996343i \(0.472769\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 7.65685 1.07217
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) −2.82843 −0.381385
\(56\) 0 0
\(57\) 6.82843 0.904447
\(58\) 0 0
\(59\) −8.48528 −1.10469 −0.552345 0.833616i \(-0.686267\pi\)
−0.552345 + 0.833616i \(0.686267\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) 2.82843 0.356348
\(64\) 0 0
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) 2.82843 0.345547 0.172774 0.984962i \(-0.444727\pi\)
0.172774 + 0.984962i \(0.444727\pi\)
\(68\) 0 0
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) −10.8284 −1.28510 −0.642549 0.766245i \(-0.722123\pi\)
−0.642549 + 0.766245i \(0.722123\pi\)
\(72\) 0 0
\(73\) 4.34315 0.508327 0.254163 0.967161i \(-0.418200\pi\)
0.254163 + 0.967161i \(0.418200\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 8.00000 0.911685
\(78\) 0 0
\(79\) −13.6569 −1.53652 −0.768258 0.640140i \(-0.778876\pi\)
−0.768258 + 0.640140i \(0.778876\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.82843 0.749517 0.374759 0.927122i \(-0.377726\pi\)
0.374759 + 0.927122i \(0.377726\pi\)
\(84\) 0 0
\(85\) −7.65685 −0.830502
\(86\) 0 0
\(87\) 3.65685 0.392056
\(88\) 0 0
\(89\) −5.31371 −0.563252 −0.281626 0.959524i \(-0.590874\pi\)
−0.281626 + 0.959524i \(0.590874\pi\)
\(90\) 0 0
\(91\) 2.82843 0.296500
\(92\) 0 0
\(93\) 1.17157 0.121486
\(94\) 0 0
\(95\) −6.82843 −0.700582
\(96\) 0 0
\(97\) 7.65685 0.777436 0.388718 0.921357i \(-0.372918\pi\)
0.388718 + 0.921357i \(0.372918\pi\)
\(98\) 0 0
\(99\) 2.82843 0.284268
\(100\) 0 0
\(101\) 3.65685 0.363871 0.181935 0.983311i \(-0.441764\pi\)
0.181935 + 0.983311i \(0.441764\pi\)
\(102\) 0 0
\(103\) −1.65685 −0.163255 −0.0816274 0.996663i \(-0.526012\pi\)
−0.0816274 + 0.996663i \(0.526012\pi\)
\(104\) 0 0
\(105\) −2.82843 −0.276026
\(106\) 0 0
\(107\) −7.31371 −0.707043 −0.353521 0.935426i \(-0.615016\pi\)
−0.353521 + 0.935426i \(0.615016\pi\)
\(108\) 0 0
\(109\) −15.6569 −1.49965 −0.749827 0.661634i \(-0.769863\pi\)
−0.749827 + 0.661634i \(0.769863\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) −11.6569 −1.09658 −0.548292 0.836287i \(-0.684722\pi\)
−0.548292 + 0.836287i \(0.684722\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) 21.6569 1.98528
\(120\) 0 0
\(121\) −3.00000 −0.272727
\(122\) 0 0
\(123\) −2.00000 −0.180334
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 9.65685 0.856907 0.428454 0.903564i \(-0.359059\pi\)
0.428454 + 0.903564i \(0.359059\pi\)
\(128\) 0 0
\(129\) −1.65685 −0.145878
\(130\) 0 0
\(131\) 16.0000 1.39793 0.698963 0.715158i \(-0.253645\pi\)
0.698963 + 0.715158i \(0.253645\pi\)
\(132\) 0 0
\(133\) 19.3137 1.67471
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) 16.9706 1.43942 0.719712 0.694273i \(-0.244274\pi\)
0.719712 + 0.694273i \(0.244274\pi\)
\(140\) 0 0
\(141\) 1.17157 0.0986642
\(142\) 0 0
\(143\) 2.82843 0.236525
\(144\) 0 0
\(145\) −3.65685 −0.303685
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 13.3137 1.09070 0.545351 0.838208i \(-0.316396\pi\)
0.545351 + 0.838208i \(0.316396\pi\)
\(150\) 0 0
\(151\) −1.17157 −0.0953412 −0.0476706 0.998863i \(-0.515180\pi\)
−0.0476706 + 0.998863i \(0.515180\pi\)
\(152\) 0 0
\(153\) 7.65685 0.619020
\(154\) 0 0
\(155\) −1.17157 −0.0941030
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 0 0
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) −11.3137 −0.891645
\(162\) 0 0
\(163\) −6.14214 −0.481089 −0.240545 0.970638i \(-0.577326\pi\)
−0.240545 + 0.970638i \(0.577326\pi\)
\(164\) 0 0
\(165\) −2.82843 −0.220193
\(166\) 0 0
\(167\) 12.4853 0.966140 0.483070 0.875582i \(-0.339522\pi\)
0.483070 + 0.875582i \(0.339522\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 6.82843 0.522183
\(172\) 0 0
\(173\) −24.6274 −1.87239 −0.936194 0.351484i \(-0.885677\pi\)
−0.936194 + 0.351484i \(0.885677\pi\)
\(174\) 0 0
\(175\) 2.82843 0.213809
\(176\) 0 0
\(177\) −8.48528 −0.637793
\(178\) 0 0
\(179\) 2.34315 0.175135 0.0875675 0.996159i \(-0.472091\pi\)
0.0875675 + 0.996159i \(0.472091\pi\)
\(180\) 0 0
\(181\) 17.3137 1.28692 0.643459 0.765481i \(-0.277499\pi\)
0.643459 + 0.765481i \(0.277499\pi\)
\(182\) 0 0
\(183\) 6.00000 0.443533
\(184\) 0 0
\(185\) 2.00000 0.147043
\(186\) 0 0
\(187\) 21.6569 1.58371
\(188\) 0 0
\(189\) 2.82843 0.205738
\(190\) 0 0
\(191\) −5.65685 −0.409316 −0.204658 0.978834i \(-0.565608\pi\)
−0.204658 + 0.978834i \(0.565608\pi\)
\(192\) 0 0
\(193\) 4.34315 0.312626 0.156313 0.987708i \(-0.450039\pi\)
0.156313 + 0.987708i \(0.450039\pi\)
\(194\) 0 0
\(195\) −1.00000 −0.0716115
\(196\) 0 0
\(197\) −22.0000 −1.56744 −0.783718 0.621117i \(-0.786679\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 2.82843 0.199502
\(202\) 0 0
\(203\) 10.3431 0.725947
\(204\) 0 0
\(205\) 2.00000 0.139686
\(206\) 0 0
\(207\) −4.00000 −0.278019
\(208\) 0 0
\(209\) 19.3137 1.33596
\(210\) 0 0
\(211\) 5.65685 0.389434 0.194717 0.980859i \(-0.437621\pi\)
0.194717 + 0.980859i \(0.437621\pi\)
\(212\) 0 0
\(213\) −10.8284 −0.741952
\(214\) 0 0
\(215\) 1.65685 0.112997
\(216\) 0 0
\(217\) 3.31371 0.224949
\(218\) 0 0
\(219\) 4.34315 0.293483
\(220\) 0 0
\(221\) 7.65685 0.515056
\(222\) 0 0
\(223\) −10.8284 −0.725125 −0.362563 0.931959i \(-0.618098\pi\)
−0.362563 + 0.931959i \(0.618098\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 22.8284 1.51518 0.757588 0.652733i \(-0.226378\pi\)
0.757588 + 0.652733i \(0.226378\pi\)
\(228\) 0 0
\(229\) 19.6569 1.29896 0.649481 0.760378i \(-0.274986\pi\)
0.649481 + 0.760378i \(0.274986\pi\)
\(230\) 0 0
\(231\) 8.00000 0.526361
\(232\) 0 0
\(233\) −22.9706 −1.50485 −0.752426 0.658677i \(-0.771116\pi\)
−0.752426 + 0.658677i \(0.771116\pi\)
\(234\) 0 0
\(235\) −1.17157 −0.0764250
\(236\) 0 0
\(237\) −13.6569 −0.887108
\(238\) 0 0
\(239\) 24.4853 1.58382 0.791911 0.610637i \(-0.209087\pi\)
0.791911 + 0.610637i \(0.209087\pi\)
\(240\) 0 0
\(241\) −1.31371 −0.0846234 −0.0423117 0.999104i \(-0.513472\pi\)
−0.0423117 + 0.999104i \(0.513472\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 6.82843 0.434482
\(248\) 0 0
\(249\) 6.82843 0.432734
\(250\) 0 0
\(251\) −24.9706 −1.57613 −0.788064 0.615593i \(-0.788916\pi\)
−0.788064 + 0.615593i \(0.788916\pi\)
\(252\) 0 0
\(253\) −11.3137 −0.711287
\(254\) 0 0
\(255\) −7.65685 −0.479491
\(256\) 0 0
\(257\) 18.9706 1.18335 0.591676 0.806176i \(-0.298467\pi\)
0.591676 + 0.806176i \(0.298467\pi\)
\(258\) 0 0
\(259\) −5.65685 −0.351500
\(260\) 0 0
\(261\) 3.65685 0.226354
\(262\) 0 0
\(263\) 9.65685 0.595467 0.297734 0.954649i \(-0.403769\pi\)
0.297734 + 0.954649i \(0.403769\pi\)
\(264\) 0 0
\(265\) 2.00000 0.122859
\(266\) 0 0
\(267\) −5.31371 −0.325194
\(268\) 0 0
\(269\) −7.65685 −0.466847 −0.233423 0.972375i \(-0.574993\pi\)
−0.233423 + 0.972375i \(0.574993\pi\)
\(270\) 0 0
\(271\) −4.48528 −0.272461 −0.136231 0.990677i \(-0.543499\pi\)
−0.136231 + 0.990677i \(0.543499\pi\)
\(272\) 0 0
\(273\) 2.82843 0.171184
\(274\) 0 0
\(275\) 2.82843 0.170561
\(276\) 0 0
\(277\) −29.3137 −1.76129 −0.880645 0.473777i \(-0.842890\pi\)
−0.880645 + 0.473777i \(0.842890\pi\)
\(278\) 0 0
\(279\) 1.17157 0.0701402
\(280\) 0 0
\(281\) −24.6274 −1.46915 −0.734574 0.678528i \(-0.762618\pi\)
−0.734574 + 0.678528i \(0.762618\pi\)
\(282\) 0 0
\(283\) 17.6569 1.04959 0.524796 0.851228i \(-0.324142\pi\)
0.524796 + 0.851228i \(0.324142\pi\)
\(284\) 0 0
\(285\) −6.82843 −0.404481
\(286\) 0 0
\(287\) −5.65685 −0.333914
\(288\) 0 0
\(289\) 41.6274 2.44867
\(290\) 0 0
\(291\) 7.65685 0.448853
\(292\) 0 0
\(293\) −9.31371 −0.544113 −0.272056 0.962281i \(-0.587704\pi\)
−0.272056 + 0.962281i \(0.587704\pi\)
\(294\) 0 0
\(295\) 8.48528 0.494032
\(296\) 0 0
\(297\) 2.82843 0.164122
\(298\) 0 0
\(299\) −4.00000 −0.231326
\(300\) 0 0
\(301\) −4.68629 −0.270113
\(302\) 0 0
\(303\) 3.65685 0.210081
\(304\) 0 0
\(305\) −6.00000 −0.343559
\(306\) 0 0
\(307\) 21.1716 1.20833 0.604163 0.796861i \(-0.293508\pi\)
0.604163 + 0.796861i \(0.293508\pi\)
\(308\) 0 0
\(309\) −1.65685 −0.0942551
\(310\) 0 0
\(311\) −4.68629 −0.265735 −0.132868 0.991134i \(-0.542419\pi\)
−0.132868 + 0.991134i \(0.542419\pi\)
\(312\) 0 0
\(313\) −14.0000 −0.791327 −0.395663 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) 0 0
\(315\) −2.82843 −0.159364
\(316\) 0 0
\(317\) −2.68629 −0.150877 −0.0754386 0.997150i \(-0.524036\pi\)
−0.0754386 + 0.997150i \(0.524036\pi\)
\(318\) 0 0
\(319\) 10.3431 0.579105
\(320\) 0 0
\(321\) −7.31371 −0.408211
\(322\) 0 0
\(323\) 52.2843 2.90917
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) 0 0
\(327\) −15.6569 −0.865826
\(328\) 0 0
\(329\) 3.31371 0.182691
\(330\) 0 0
\(331\) 18.1421 0.997182 0.498591 0.866837i \(-0.333851\pi\)
0.498591 + 0.866837i \(0.333851\pi\)
\(332\) 0 0
\(333\) −2.00000 −0.109599
\(334\) 0 0
\(335\) −2.82843 −0.154533
\(336\) 0 0
\(337\) −6.00000 −0.326841 −0.163420 0.986557i \(-0.552253\pi\)
−0.163420 + 0.986557i \(0.552253\pi\)
\(338\) 0 0
\(339\) −11.6569 −0.633113
\(340\) 0 0
\(341\) 3.31371 0.179447
\(342\) 0 0
\(343\) −16.9706 −0.916324
\(344\) 0 0
\(345\) 4.00000 0.215353
\(346\) 0 0
\(347\) −10.6274 −0.570510 −0.285255 0.958452i \(-0.592078\pi\)
−0.285255 + 0.958452i \(0.592078\pi\)
\(348\) 0 0
\(349\) −2.97056 −0.159011 −0.0795053 0.996834i \(-0.525334\pi\)
−0.0795053 + 0.996834i \(0.525334\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) 17.3137 0.921516 0.460758 0.887526i \(-0.347578\pi\)
0.460758 + 0.887526i \(0.347578\pi\)
\(354\) 0 0
\(355\) 10.8284 0.574713
\(356\) 0 0
\(357\) 21.6569 1.14620
\(358\) 0 0
\(359\) −16.4853 −0.870060 −0.435030 0.900416i \(-0.643262\pi\)
−0.435030 + 0.900416i \(0.643262\pi\)
\(360\) 0 0
\(361\) 27.6274 1.45407
\(362\) 0 0
\(363\) −3.00000 −0.157459
\(364\) 0 0
\(365\) −4.34315 −0.227331
\(366\) 0 0
\(367\) 8.68629 0.453421 0.226710 0.973962i \(-0.427203\pi\)
0.226710 + 0.973962i \(0.427203\pi\)
\(368\) 0 0
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) −5.65685 −0.293689
\(372\) 0 0
\(373\) 28.6274 1.48227 0.741136 0.671355i \(-0.234287\pi\)
0.741136 + 0.671355i \(0.234287\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 3.65685 0.188338
\(378\) 0 0
\(379\) 18.1421 0.931899 0.465949 0.884811i \(-0.345713\pi\)
0.465949 + 0.884811i \(0.345713\pi\)
\(380\) 0 0
\(381\) 9.65685 0.494736
\(382\) 0 0
\(383\) 14.8284 0.757697 0.378849 0.925459i \(-0.376320\pi\)
0.378849 + 0.925459i \(0.376320\pi\)
\(384\) 0 0
\(385\) −8.00000 −0.407718
\(386\) 0 0
\(387\) −1.65685 −0.0842226
\(388\) 0 0
\(389\) −23.6569 −1.19945 −0.599725 0.800206i \(-0.704723\pi\)
−0.599725 + 0.800206i \(0.704723\pi\)
\(390\) 0 0
\(391\) −30.6274 −1.54890
\(392\) 0 0
\(393\) 16.0000 0.807093
\(394\) 0 0
\(395\) 13.6569 0.687151
\(396\) 0 0
\(397\) −16.6274 −0.834506 −0.417253 0.908790i \(-0.637007\pi\)
−0.417253 + 0.908790i \(0.637007\pi\)
\(398\) 0 0
\(399\) 19.3137 0.966895
\(400\) 0 0
\(401\) −6.68629 −0.333897 −0.166949 0.985966i \(-0.553391\pi\)
−0.166949 + 0.985966i \(0.553391\pi\)
\(402\) 0 0
\(403\) 1.17157 0.0583602
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −5.65685 −0.280400
\(408\) 0 0
\(409\) −20.6274 −1.01996 −0.509980 0.860186i \(-0.670347\pi\)
−0.509980 + 0.860186i \(0.670347\pi\)
\(410\) 0 0
\(411\) −18.0000 −0.887875
\(412\) 0 0
\(413\) −24.0000 −1.18096
\(414\) 0 0
\(415\) −6.82843 −0.335194
\(416\) 0 0
\(417\) 16.9706 0.831052
\(418\) 0 0
\(419\) −14.6274 −0.714596 −0.357298 0.933990i \(-0.616302\pi\)
−0.357298 + 0.933990i \(0.616302\pi\)
\(420\) 0 0
\(421\) −18.9706 −0.924569 −0.462284 0.886732i \(-0.652970\pi\)
−0.462284 + 0.886732i \(0.652970\pi\)
\(422\) 0 0
\(423\) 1.17157 0.0569638
\(424\) 0 0
\(425\) 7.65685 0.371412
\(426\) 0 0
\(427\) 16.9706 0.821263
\(428\) 0 0
\(429\) 2.82843 0.136558
\(430\) 0 0
\(431\) −30.1421 −1.45190 −0.725948 0.687750i \(-0.758599\pi\)
−0.725948 + 0.687750i \(0.758599\pi\)
\(432\) 0 0
\(433\) −9.31371 −0.447588 −0.223794 0.974636i \(-0.571844\pi\)
−0.223794 + 0.974636i \(0.571844\pi\)
\(434\) 0 0
\(435\) −3.65685 −0.175333
\(436\) 0 0
\(437\) −27.3137 −1.30659
\(438\) 0 0
\(439\) 14.6274 0.698129 0.349064 0.937099i \(-0.386499\pi\)
0.349064 + 0.937099i \(0.386499\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 18.6274 0.885015 0.442508 0.896765i \(-0.354089\pi\)
0.442508 + 0.896765i \(0.354089\pi\)
\(444\) 0 0
\(445\) 5.31371 0.251894
\(446\) 0 0
\(447\) 13.3137 0.629717
\(448\) 0 0
\(449\) 36.6274 1.72855 0.864277 0.503016i \(-0.167776\pi\)
0.864277 + 0.503016i \(0.167776\pi\)
\(450\) 0 0
\(451\) −5.65685 −0.266371
\(452\) 0 0
\(453\) −1.17157 −0.0550453
\(454\) 0 0
\(455\) −2.82843 −0.132599
\(456\) 0 0
\(457\) 22.2843 1.04241 0.521207 0.853430i \(-0.325482\pi\)
0.521207 + 0.853430i \(0.325482\pi\)
\(458\) 0 0
\(459\) 7.65685 0.357391
\(460\) 0 0
\(461\) −9.31371 −0.433783 −0.216891 0.976196i \(-0.569592\pi\)
−0.216891 + 0.976196i \(0.569592\pi\)
\(462\) 0 0
\(463\) 22.1421 1.02903 0.514516 0.857481i \(-0.327972\pi\)
0.514516 + 0.857481i \(0.327972\pi\)
\(464\) 0 0
\(465\) −1.17157 −0.0543304
\(466\) 0 0
\(467\) 28.9706 1.34060 0.670299 0.742091i \(-0.266166\pi\)
0.670299 + 0.742091i \(0.266166\pi\)
\(468\) 0 0
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) 14.0000 0.645086
\(472\) 0 0
\(473\) −4.68629 −0.215476
\(474\) 0 0
\(475\) 6.82843 0.313310
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) 0 0
\(479\) −24.4853 −1.11876 −0.559381 0.828911i \(-0.688961\pi\)
−0.559381 + 0.828911i \(0.688961\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) 0 0
\(483\) −11.3137 −0.514792
\(484\) 0 0
\(485\) −7.65685 −0.347680
\(486\) 0 0
\(487\) −25.4558 −1.15351 −0.576757 0.816916i \(-0.695682\pi\)
−0.576757 + 0.816916i \(0.695682\pi\)
\(488\) 0 0
\(489\) −6.14214 −0.277757
\(490\) 0 0
\(491\) −13.6569 −0.616325 −0.308163 0.951334i \(-0.599714\pi\)
−0.308163 + 0.951334i \(0.599714\pi\)
\(492\) 0 0
\(493\) 28.0000 1.26106
\(494\) 0 0
\(495\) −2.82843 −0.127128
\(496\) 0 0
\(497\) −30.6274 −1.37383
\(498\) 0 0
\(499\) 30.8284 1.38007 0.690035 0.723776i \(-0.257595\pi\)
0.690035 + 0.723776i \(0.257595\pi\)
\(500\) 0 0
\(501\) 12.4853 0.557801
\(502\) 0 0
\(503\) −1.65685 −0.0738755 −0.0369377 0.999318i \(-0.511760\pi\)
−0.0369377 + 0.999318i \(0.511760\pi\)
\(504\) 0 0
\(505\) −3.65685 −0.162728
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) 29.3137 1.29931 0.649654 0.760230i \(-0.274914\pi\)
0.649654 + 0.760230i \(0.274914\pi\)
\(510\) 0 0
\(511\) 12.2843 0.543424
\(512\) 0 0
\(513\) 6.82843 0.301482
\(514\) 0 0
\(515\) 1.65685 0.0730097
\(516\) 0 0
\(517\) 3.31371 0.145737
\(518\) 0 0
\(519\) −24.6274 −1.08102
\(520\) 0 0
\(521\) 0.627417 0.0274876 0.0137438 0.999906i \(-0.495625\pi\)
0.0137438 + 0.999906i \(0.495625\pi\)
\(522\) 0 0
\(523\) 7.31371 0.319806 0.159903 0.987133i \(-0.448882\pi\)
0.159903 + 0.987133i \(0.448882\pi\)
\(524\) 0 0
\(525\) 2.82843 0.123443
\(526\) 0 0
\(527\) 8.97056 0.390764
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −8.48528 −0.368230
\(532\) 0 0
\(533\) −2.00000 −0.0866296
\(534\) 0 0
\(535\) 7.31371 0.316199
\(536\) 0 0
\(537\) 2.34315 0.101114
\(538\) 0 0
\(539\) 2.82843 0.121829
\(540\) 0 0
\(541\) 22.9706 0.987582 0.493791 0.869581i \(-0.335611\pi\)
0.493791 + 0.869581i \(0.335611\pi\)
\(542\) 0 0
\(543\) 17.3137 0.743002
\(544\) 0 0
\(545\) 15.6569 0.670666
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) 0 0
\(549\) 6.00000 0.256074
\(550\) 0 0
\(551\) 24.9706 1.06378
\(552\) 0 0
\(553\) −38.6274 −1.64260
\(554\) 0 0
\(555\) 2.00000 0.0848953
\(556\) 0 0
\(557\) −39.9411 −1.69236 −0.846180 0.532897i \(-0.821103\pi\)
−0.846180 + 0.532897i \(0.821103\pi\)
\(558\) 0 0
\(559\) −1.65685 −0.0700775
\(560\) 0 0
\(561\) 21.6569 0.914353
\(562\) 0 0
\(563\) −3.02944 −0.127676 −0.0638378 0.997960i \(-0.520334\pi\)
−0.0638378 + 0.997960i \(0.520334\pi\)
\(564\) 0 0
\(565\) 11.6569 0.490408
\(566\) 0 0
\(567\) 2.82843 0.118783
\(568\) 0 0
\(569\) −25.3137 −1.06121 −0.530603 0.847621i \(-0.678034\pi\)
−0.530603 + 0.847621i \(0.678034\pi\)
\(570\) 0 0
\(571\) −30.6274 −1.28172 −0.640859 0.767659i \(-0.721422\pi\)
−0.640859 + 0.767659i \(0.721422\pi\)
\(572\) 0 0
\(573\) −5.65685 −0.236318
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) −38.9706 −1.62237 −0.811183 0.584793i \(-0.801176\pi\)
−0.811183 + 0.584793i \(0.801176\pi\)
\(578\) 0 0
\(579\) 4.34315 0.180495
\(580\) 0 0
\(581\) 19.3137 0.801268
\(582\) 0 0
\(583\) −5.65685 −0.234283
\(584\) 0 0
\(585\) −1.00000 −0.0413449
\(586\) 0 0
\(587\) 26.1421 1.07900 0.539501 0.841985i \(-0.318613\pi\)
0.539501 + 0.841985i \(0.318613\pi\)
\(588\) 0 0
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) −22.0000 −0.904959
\(592\) 0 0
\(593\) −6.68629 −0.274573 −0.137287 0.990531i \(-0.543838\pi\)
−0.137287 + 0.990531i \(0.543838\pi\)
\(594\) 0 0
\(595\) −21.6569 −0.887844
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 37.6569 1.53862 0.769309 0.638877i \(-0.220601\pi\)
0.769309 + 0.638877i \(0.220601\pi\)
\(600\) 0 0
\(601\) −14.0000 −0.571072 −0.285536 0.958368i \(-0.592172\pi\)
−0.285536 + 0.958368i \(0.592172\pi\)
\(602\) 0 0
\(603\) 2.82843 0.115182
\(604\) 0 0
\(605\) 3.00000 0.121967
\(606\) 0 0
\(607\) −34.6274 −1.40548 −0.702742 0.711445i \(-0.748041\pi\)
−0.702742 + 0.711445i \(0.748041\pi\)
\(608\) 0 0
\(609\) 10.3431 0.419125
\(610\) 0 0
\(611\) 1.17157 0.0473968
\(612\) 0 0
\(613\) 18.6863 0.754732 0.377366 0.926064i \(-0.376830\pi\)
0.377366 + 0.926064i \(0.376830\pi\)
\(614\) 0 0
\(615\) 2.00000 0.0806478
\(616\) 0 0
\(617\) 30.0000 1.20775 0.603877 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 0 0
\(619\) 23.7990 0.956562 0.478281 0.878207i \(-0.341260\pi\)
0.478281 + 0.878207i \(0.341260\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 0 0
\(623\) −15.0294 −0.602142
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 19.3137 0.771315
\(628\) 0 0
\(629\) −15.3137 −0.610598
\(630\) 0 0
\(631\) −5.85786 −0.233198 −0.116599 0.993179i \(-0.537199\pi\)
−0.116599 + 0.993179i \(0.537199\pi\)
\(632\) 0 0
\(633\) 5.65685 0.224840
\(634\) 0 0
\(635\) −9.65685 −0.383221
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 0 0
\(639\) −10.8284 −0.428366
\(640\) 0 0
\(641\) −10.6863 −0.422083 −0.211042 0.977477i \(-0.567686\pi\)
−0.211042 + 0.977477i \(0.567686\pi\)
\(642\) 0 0
\(643\) −0.485281 −0.0191376 −0.00956881 0.999954i \(-0.503046\pi\)
−0.00956881 + 0.999954i \(0.503046\pi\)
\(644\) 0 0
\(645\) 1.65685 0.0652386
\(646\) 0 0
\(647\) −4.00000 −0.157256 −0.0786281 0.996904i \(-0.525054\pi\)
−0.0786281 + 0.996904i \(0.525054\pi\)
\(648\) 0 0
\(649\) −24.0000 −0.942082
\(650\) 0 0
\(651\) 3.31371 0.129874
\(652\) 0 0
\(653\) 1.31371 0.0514094 0.0257047 0.999670i \(-0.491817\pi\)
0.0257047 + 0.999670i \(0.491817\pi\)
\(654\) 0 0
\(655\) −16.0000 −0.625172
\(656\) 0 0
\(657\) 4.34315 0.169442
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −10.9706 −0.426705 −0.213353 0.976975i \(-0.568438\pi\)
−0.213353 + 0.976975i \(0.568438\pi\)
\(662\) 0 0
\(663\) 7.65685 0.297368
\(664\) 0 0
\(665\) −19.3137 −0.748953
\(666\) 0 0
\(667\) −14.6274 −0.566376
\(668\) 0 0
\(669\) −10.8284 −0.418651
\(670\) 0 0
\(671\) 16.9706 0.655141
\(672\) 0 0
\(673\) 5.31371 0.204828 0.102414 0.994742i \(-0.467343\pi\)
0.102414 + 0.994742i \(0.467343\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −35.9411 −1.38133 −0.690665 0.723175i \(-0.742682\pi\)
−0.690665 + 0.723175i \(0.742682\pi\)
\(678\) 0 0
\(679\) 21.6569 0.831114
\(680\) 0 0
\(681\) 22.8284 0.874787
\(682\) 0 0
\(683\) 9.17157 0.350940 0.175470 0.984485i \(-0.443855\pi\)
0.175470 + 0.984485i \(0.443855\pi\)
\(684\) 0 0
\(685\) 18.0000 0.687745
\(686\) 0 0
\(687\) 19.6569 0.749956
\(688\) 0 0
\(689\) −2.00000 −0.0761939
\(690\) 0 0
\(691\) −47.7990 −1.81836 −0.909180 0.416404i \(-0.863290\pi\)
−0.909180 + 0.416404i \(0.863290\pi\)
\(692\) 0 0
\(693\) 8.00000 0.303895
\(694\) 0 0
\(695\) −16.9706 −0.643730
\(696\) 0 0
\(697\) −15.3137 −0.580048
\(698\) 0 0
\(699\) −22.9706 −0.868826
\(700\) 0 0
\(701\) −15.6569 −0.591351 −0.295676 0.955288i \(-0.595545\pi\)
−0.295676 + 0.955288i \(0.595545\pi\)
\(702\) 0 0
\(703\) −13.6569 −0.515078
\(704\) 0 0
\(705\) −1.17157 −0.0441240
\(706\) 0 0
\(707\) 10.3431 0.388994
\(708\) 0 0
\(709\) −12.3431 −0.463557 −0.231778 0.972769i \(-0.574454\pi\)
−0.231778 + 0.972769i \(0.574454\pi\)
\(710\) 0 0
\(711\) −13.6569 −0.512172
\(712\) 0 0
\(713\) −4.68629 −0.175503
\(714\) 0 0
\(715\) −2.82843 −0.105777
\(716\) 0 0
\(717\) 24.4853 0.914420
\(718\) 0 0
\(719\) 3.31371 0.123580 0.0617902 0.998089i \(-0.480319\pi\)
0.0617902 + 0.998089i \(0.480319\pi\)
\(720\) 0 0
\(721\) −4.68629 −0.174527
\(722\) 0 0
\(723\) −1.31371 −0.0488573
\(724\) 0 0
\(725\) 3.65685 0.135812
\(726\) 0 0
\(727\) 4.97056 0.184348 0.0921740 0.995743i \(-0.470618\pi\)
0.0921740 + 0.995743i \(0.470618\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −12.6863 −0.469219
\(732\) 0 0
\(733\) 41.3137 1.52596 0.762978 0.646424i \(-0.223736\pi\)
0.762978 + 0.646424i \(0.223736\pi\)
\(734\) 0 0
\(735\) −1.00000 −0.0368856
\(736\) 0 0
\(737\) 8.00000 0.294684
\(738\) 0 0
\(739\) −32.7696 −1.20545 −0.602724 0.797950i \(-0.705918\pi\)
−0.602724 + 0.797950i \(0.705918\pi\)
\(740\) 0 0
\(741\) 6.82843 0.250849
\(742\) 0 0
\(743\) −15.7990 −0.579609 −0.289804 0.957086i \(-0.593590\pi\)
−0.289804 + 0.957086i \(0.593590\pi\)
\(744\) 0 0
\(745\) −13.3137 −0.487777
\(746\) 0 0
\(747\) 6.82843 0.249839
\(748\) 0 0
\(749\) −20.6863 −0.755861
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 0 0
\(753\) −24.9706 −0.909978
\(754\) 0 0
\(755\) 1.17157 0.0426379
\(756\) 0 0
\(757\) −13.3137 −0.483895 −0.241947 0.970289i \(-0.577786\pi\)
−0.241947 + 0.970289i \(0.577786\pi\)
\(758\) 0 0
\(759\) −11.3137 −0.410662
\(760\) 0 0
\(761\) 18.6863 0.677378 0.338689 0.940898i \(-0.390017\pi\)
0.338689 + 0.940898i \(0.390017\pi\)
\(762\) 0 0
\(763\) −44.2843 −1.60320
\(764\) 0 0
\(765\) −7.65685 −0.276834
\(766\) 0 0
\(767\) −8.48528 −0.306386
\(768\) 0 0
\(769\) 2.00000 0.0721218 0.0360609 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 0 0
\(771\) 18.9706 0.683208
\(772\) 0 0
\(773\) 13.3137 0.478861 0.239430 0.970914i \(-0.423039\pi\)
0.239430 + 0.970914i \(0.423039\pi\)
\(774\) 0 0
\(775\) 1.17157 0.0420841
\(776\) 0 0
\(777\) −5.65685 −0.202939
\(778\) 0 0
\(779\) −13.6569 −0.489308
\(780\) 0 0
\(781\) −30.6274 −1.09594
\(782\) 0 0
\(783\) 3.65685 0.130685
\(784\) 0 0
\(785\) −14.0000 −0.499681
\(786\) 0 0
\(787\) 43.7990 1.56127 0.780633 0.624990i \(-0.214897\pi\)
0.780633 + 0.624990i \(0.214897\pi\)
\(788\) 0 0
\(789\) 9.65685 0.343793
\(790\) 0 0
\(791\) −32.9706 −1.17230
\(792\) 0 0
\(793\) 6.00000 0.213066
\(794\) 0 0
\(795\) 2.00000 0.0709327
\(796\) 0 0
\(797\) −34.0000 −1.20434 −0.602171 0.798367i \(-0.705697\pi\)
−0.602171 + 0.798367i \(0.705697\pi\)
\(798\) 0 0
\(799\) 8.97056 0.317356
\(800\) 0 0
\(801\) −5.31371 −0.187751
\(802\) 0 0
\(803\) 12.2843 0.433503
\(804\) 0 0
\(805\) 11.3137 0.398756
\(806\) 0 0
\(807\) −7.65685 −0.269534
\(808\) 0 0
\(809\) −33.3137 −1.17125 −0.585624 0.810583i \(-0.699150\pi\)
−0.585624 + 0.810583i \(0.699150\pi\)
\(810\) 0 0
\(811\) −46.4264 −1.63025 −0.815126 0.579284i \(-0.803332\pi\)
−0.815126 + 0.579284i \(0.803332\pi\)
\(812\) 0 0
\(813\) −4.48528 −0.157306
\(814\) 0 0
\(815\) 6.14214 0.215150
\(816\) 0 0
\(817\) −11.3137 −0.395817
\(818\) 0 0
\(819\) 2.82843 0.0988332
\(820\) 0 0
\(821\) 13.3137 0.464652 0.232326 0.972638i \(-0.425366\pi\)
0.232326 + 0.972638i \(0.425366\pi\)
\(822\) 0 0
\(823\) −33.6569 −1.17320 −0.586602 0.809875i \(-0.699535\pi\)
−0.586602 + 0.809875i \(0.699535\pi\)
\(824\) 0 0
\(825\) 2.82843 0.0984732
\(826\) 0 0
\(827\) 12.4853 0.434156 0.217078 0.976154i \(-0.430347\pi\)
0.217078 + 0.976154i \(0.430347\pi\)
\(828\) 0 0
\(829\) −32.6274 −1.13320 −0.566599 0.823994i \(-0.691741\pi\)
−0.566599 + 0.823994i \(0.691741\pi\)
\(830\) 0 0
\(831\) −29.3137 −1.01688
\(832\) 0 0
\(833\) 7.65685 0.265294
\(834\) 0 0
\(835\) −12.4853 −0.432071
\(836\) 0 0
\(837\) 1.17157 0.0404955
\(838\) 0 0
\(839\) 46.1421 1.59300 0.796502 0.604636i \(-0.206682\pi\)
0.796502 + 0.604636i \(0.206682\pi\)
\(840\) 0 0
\(841\) −15.6274 −0.538876
\(842\) 0 0
\(843\) −24.6274 −0.848213
\(844\) 0 0
\(845\) −1.00000 −0.0344010
\(846\) 0 0
\(847\) −8.48528 −0.291558
\(848\) 0 0
\(849\) 17.6569 0.605982
\(850\) 0 0
\(851\) 8.00000 0.274236
\(852\) 0 0
\(853\) 43.2548 1.48102 0.740509 0.672047i \(-0.234585\pi\)
0.740509 + 0.672047i \(0.234585\pi\)
\(854\) 0 0
\(855\) −6.82843 −0.233527
\(856\) 0 0
\(857\) −11.6569 −0.398191 −0.199095 0.979980i \(-0.563800\pi\)
−0.199095 + 0.979980i \(0.563800\pi\)
\(858\) 0 0
\(859\) 29.6569 1.01188 0.505939 0.862569i \(-0.331146\pi\)
0.505939 + 0.862569i \(0.331146\pi\)
\(860\) 0 0
\(861\) −5.65685 −0.192785
\(862\) 0 0
\(863\) −35.1127 −1.19525 −0.597625 0.801776i \(-0.703889\pi\)
−0.597625 + 0.801776i \(0.703889\pi\)
\(864\) 0 0
\(865\) 24.6274 0.837357
\(866\) 0 0
\(867\) 41.6274 1.41374
\(868\) 0 0
\(869\) −38.6274 −1.31035
\(870\) 0 0
\(871\) 2.82843 0.0958376
\(872\) 0 0
\(873\) 7.65685 0.259145
\(874\) 0 0
\(875\) −2.82843 −0.0956183
\(876\) 0 0
\(877\) 38.0000 1.28317 0.641584 0.767052i \(-0.278277\pi\)
0.641584 + 0.767052i \(0.278277\pi\)
\(878\) 0 0
\(879\) −9.31371 −0.314144
\(880\) 0 0
\(881\) −23.9411 −0.806597 −0.403299 0.915068i \(-0.632136\pi\)
−0.403299 + 0.915068i \(0.632136\pi\)
\(882\) 0 0
\(883\) −32.2843 −1.08645 −0.543226 0.839586i \(-0.682797\pi\)
−0.543226 + 0.839586i \(0.682797\pi\)
\(884\) 0 0
\(885\) 8.48528 0.285230
\(886\) 0 0
\(887\) 4.97056 0.166895 0.0834476 0.996512i \(-0.473407\pi\)
0.0834476 + 0.996512i \(0.473407\pi\)
\(888\) 0 0
\(889\) 27.3137 0.916072
\(890\) 0 0
\(891\) 2.82843 0.0947559
\(892\) 0 0
\(893\) 8.00000 0.267710
\(894\) 0 0
\(895\) −2.34315 −0.0783227
\(896\) 0 0
\(897\) −4.00000 −0.133556
\(898\) 0 0
\(899\) 4.28427 0.142888
\(900\) 0 0
\(901\) −15.3137 −0.510174
\(902\) 0 0
\(903\) −4.68629 −0.155950
\(904\) 0 0
\(905\) −17.3137 −0.575527
\(906\) 0 0
\(907\) 36.9706 1.22759 0.613794 0.789466i \(-0.289643\pi\)
0.613794 + 0.789466i \(0.289643\pi\)
\(908\) 0 0
\(909\) 3.65685 0.121290
\(910\) 0 0
\(911\) 39.5980 1.31194 0.655970 0.754787i \(-0.272260\pi\)
0.655970 + 0.754787i \(0.272260\pi\)
\(912\) 0 0
\(913\) 19.3137 0.639190
\(914\) 0 0
\(915\) −6.00000 −0.198354
\(916\) 0 0
\(917\) 45.2548 1.49445
\(918\) 0 0
\(919\) 36.2843 1.19691 0.598454 0.801157i \(-0.295782\pi\)
0.598454 + 0.801157i \(0.295782\pi\)
\(920\) 0 0
\(921\) 21.1716 0.697627
\(922\) 0 0
\(923\) −10.8284 −0.356422
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) 0 0
\(927\) −1.65685 −0.0544182
\(928\) 0 0
\(929\) −16.6274 −0.545528 −0.272764 0.962081i \(-0.587938\pi\)
−0.272764 + 0.962081i \(0.587938\pi\)
\(930\) 0 0
\(931\) 6.82843 0.223793
\(932\) 0 0
\(933\) −4.68629 −0.153422
\(934\) 0 0
\(935\) −21.6569 −0.708255
\(936\) 0 0
\(937\) 0.627417 0.0204968 0.0102484 0.999947i \(-0.496738\pi\)
0.0102484 + 0.999947i \(0.496738\pi\)
\(938\) 0 0
\(939\) −14.0000 −0.456873
\(940\) 0 0
\(941\) 32.6274 1.06362 0.531812 0.846863i \(-0.321511\pi\)
0.531812 + 0.846863i \(0.321511\pi\)
\(942\) 0 0
\(943\) 8.00000 0.260516
\(944\) 0 0
\(945\) −2.82843 −0.0920087
\(946\) 0 0
\(947\) 41.1716 1.33790 0.668948 0.743309i \(-0.266745\pi\)
0.668948 + 0.743309i \(0.266745\pi\)
\(948\) 0 0
\(949\) 4.34315 0.140984
\(950\) 0 0
\(951\) −2.68629 −0.0871090
\(952\) 0 0
\(953\) −0.343146 −0.0111156 −0.00555779 0.999985i \(-0.501769\pi\)
−0.00555779 + 0.999985i \(0.501769\pi\)
\(954\) 0 0
\(955\) 5.65685 0.183052
\(956\) 0 0
\(957\) 10.3431 0.334346
\(958\) 0 0
\(959\) −50.9117 −1.64402
\(960\) 0 0
\(961\) −29.6274 −0.955723
\(962\) 0 0
\(963\) −7.31371 −0.235681
\(964\) 0 0
\(965\) −4.34315 −0.139811
\(966\) 0 0
\(967\) 24.4853 0.787394 0.393697 0.919240i \(-0.371196\pi\)
0.393697 + 0.919240i \(0.371196\pi\)
\(968\) 0 0
\(969\) 52.2843 1.67961
\(970\) 0 0
\(971\) −35.3137 −1.13327 −0.566635 0.823969i \(-0.691755\pi\)
−0.566635 + 0.823969i \(0.691755\pi\)
\(972\) 0 0
\(973\) 48.0000 1.53881
\(974\) 0 0
\(975\) 1.00000 0.0320256
\(976\) 0 0
\(977\) −2.00000 −0.0639857 −0.0319928 0.999488i \(-0.510185\pi\)
−0.0319928 + 0.999488i \(0.510185\pi\)
\(978\) 0 0
\(979\) −15.0294 −0.480343
\(980\) 0 0
\(981\) −15.6569 −0.499885
\(982\) 0 0
\(983\) −48.7696 −1.55551 −0.777754 0.628569i \(-0.783641\pi\)
−0.777754 + 0.628569i \(0.783641\pi\)
\(984\) 0 0
\(985\) 22.0000 0.700978
\(986\) 0 0
\(987\) 3.31371 0.105477
\(988\) 0 0
\(989\) 6.62742 0.210740
\(990\) 0 0
\(991\) 8.97056 0.284959 0.142480 0.989798i \(-0.454492\pi\)
0.142480 + 0.989798i \(0.454492\pi\)
\(992\) 0 0
\(993\) 18.1421 0.575723
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 25.3137 0.801693 0.400847 0.916145i \(-0.368716\pi\)
0.400847 + 0.916145i \(0.368716\pi\)
\(998\) 0 0
\(999\) −2.00000 −0.0632772
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 6240.2.a.br.1.2 yes 2
4.3 odd 2 6240.2.a.bh.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6240.2.a.bh.1.1 2 4.3 odd 2
6240.2.a.br.1.2 yes 2 1.1 even 1 trivial