Properties

Label 6240.2.a.bv.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 $1$

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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: \(\Q(\zeta_{10})^+\)
Defining polynomial: \( x^{2} - x - 1 \) 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 \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 6240.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} +5.23607 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{5} +5.23607 q^{7} +1.00000 q^{9} -2.47214 q^{11} -1.00000 q^{13} +1.00000 q^{15} +0.763932 q^{17} +5.23607 q^{19} +5.23607 q^{21} +2.76393 q^{23} +1.00000 q^{25} +1.00000 q^{27} +4.76393 q^{29} -8.94427 q^{31} -2.47214 q^{33} +5.23607 q^{35} -8.47214 q^{37} -1.00000 q^{39} -3.52786 q^{41} +4.94427 q^{43} +1.00000 q^{45} +12.9443 q^{47} +20.4164 q^{49} +0.763932 q^{51} -8.47214 q^{53} -2.47214 q^{55} +5.23607 q^{57} +2.47214 q^{59} +10.9443 q^{61} +5.23607 q^{63} -1.00000 q^{65} -8.00000 q^{67} +2.76393 q^{69} -4.00000 q^{71} +9.70820 q^{73} +1.00000 q^{75} -12.9443 q^{77} +4.94427 q^{79} +1.00000 q^{81} +8.00000 q^{83} +0.763932 q^{85} +4.76393 q^{87} -14.9443 q^{89} -5.23607 q^{91} -8.94427 q^{93} +5.23607 q^{95} +14.6525 q^{97} -2.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9} + 4 q^{11} - 2 q^{13} + 2 q^{15} + 6 q^{17} + 6 q^{19} + 6 q^{21} + 10 q^{23} + 2 q^{25} + 2 q^{27} + 14 q^{29} + 4 q^{33} + 6 q^{35} - 8 q^{37} - 2 q^{39} - 16 q^{41} - 8 q^{43} + 2 q^{45} + 8 q^{47} + 14 q^{49} + 6 q^{51} - 8 q^{53} + 4 q^{55} + 6 q^{57} - 4 q^{59} + 4 q^{61} + 6 q^{63} - 2 q^{65} - 16 q^{67} + 10 q^{69} - 8 q^{71} + 6 q^{73} + 2 q^{75} - 8 q^{77} - 8 q^{79} + 2 q^{81} + 16 q^{83} + 6 q^{85} + 14 q^{87} - 12 q^{89} - 6 q^{91} + 6 q^{95} - 2 q^{97} + 4 q^{99}+O(q^{100}) \) 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
\(7\) 5.23607 1.97905 0.989524 0.144370i \(-0.0461154\pi\)
0.989524 + 0.144370i \(0.0461154\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.47214 −0.745377 −0.372689 0.927957i \(-0.621564\pi\)
−0.372689 + 0.927957i \(0.621564\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 0.763932 0.185281 0.0926404 0.995700i \(-0.470469\pi\)
0.0926404 + 0.995700i \(0.470469\pi\)
\(18\) 0 0
\(19\) 5.23607 1.20124 0.600618 0.799536i \(-0.294921\pi\)
0.600618 + 0.799536i \(0.294921\pi\)
\(20\) 0 0
\(21\) 5.23607 1.14260
\(22\) 0 0
\(23\) 2.76393 0.576320 0.288160 0.957582i \(-0.406957\pi\)
0.288160 + 0.957582i \(0.406957\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 4.76393 0.884640 0.442320 0.896857i \(-0.354156\pi\)
0.442320 + 0.896857i \(0.354156\pi\)
\(30\) 0 0
\(31\) −8.94427 −1.60644 −0.803219 0.595683i \(-0.796881\pi\)
−0.803219 + 0.595683i \(0.796881\pi\)
\(32\) 0 0
\(33\) −2.47214 −0.430344
\(34\) 0 0
\(35\) 5.23607 0.885057
\(36\) 0 0
\(37\) −8.47214 −1.39281 −0.696405 0.717649i \(-0.745218\pi\)
−0.696405 + 0.717649i \(0.745218\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −3.52786 −0.550960 −0.275480 0.961307i \(-0.588837\pi\)
−0.275480 + 0.961307i \(0.588837\pi\)
\(42\) 0 0
\(43\) 4.94427 0.753994 0.376997 0.926214i \(-0.376957\pi\)
0.376997 + 0.926214i \(0.376957\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 12.9443 1.88812 0.944058 0.329779i \(-0.106974\pi\)
0.944058 + 0.329779i \(0.106974\pi\)
\(48\) 0 0
\(49\) 20.4164 2.91663
\(50\) 0 0
\(51\) 0.763932 0.106972
\(52\) 0 0
\(53\) −8.47214 −1.16374 −0.581869 0.813283i \(-0.697678\pi\)
−0.581869 + 0.813283i \(0.697678\pi\)
\(54\) 0 0
\(55\) −2.47214 −0.333343
\(56\) 0 0
\(57\) 5.23607 0.693534
\(58\) 0 0
\(59\) 2.47214 0.321845 0.160922 0.986967i \(-0.448553\pi\)
0.160922 + 0.986967i \(0.448553\pi\)
\(60\) 0 0
\(61\) 10.9443 1.40127 0.700635 0.713520i \(-0.252900\pi\)
0.700635 + 0.713520i \(0.252900\pi\)
\(62\) 0 0
\(63\) 5.23607 0.659683
\(64\) 0 0
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) 2.76393 0.332738
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 0 0
\(73\) 9.70820 1.13626 0.568130 0.822939i \(-0.307667\pi\)
0.568130 + 0.822939i \(0.307667\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −12.9443 −1.47514
\(78\) 0 0
\(79\) 4.94427 0.556274 0.278137 0.960541i \(-0.410283\pi\)
0.278137 + 0.960541i \(0.410283\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 8.00000 0.878114 0.439057 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) 0 0
\(85\) 0.763932 0.0828601
\(86\) 0 0
\(87\) 4.76393 0.510747
\(88\) 0 0
\(89\) −14.9443 −1.58409 −0.792045 0.610463i \(-0.790983\pi\)
−0.792045 + 0.610463i \(0.790983\pi\)
\(90\) 0 0
\(91\) −5.23607 −0.548889
\(92\) 0 0
\(93\) −8.94427 −0.927478
\(94\) 0 0
\(95\) 5.23607 0.537209
\(96\) 0 0
\(97\) 14.6525 1.48773 0.743867 0.668328i \(-0.232990\pi\)
0.743867 + 0.668328i \(0.232990\pi\)
\(98\) 0 0
\(99\) −2.47214 −0.248459
\(100\) 0 0
\(101\) 2.29180 0.228042 0.114021 0.993478i \(-0.463627\pi\)
0.114021 + 0.993478i \(0.463627\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) 5.23607 0.510988
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) −12.1803 −1.16666 −0.583332 0.812233i \(-0.698252\pi\)
−0.583332 + 0.812233i \(0.698252\pi\)
\(110\) 0 0
\(111\) −8.47214 −0.804140
\(112\) 0 0
\(113\) 11.2361 1.05700 0.528500 0.848933i \(-0.322755\pi\)
0.528500 + 0.848933i \(0.322755\pi\)
\(114\) 0 0
\(115\) 2.76393 0.257738
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) −4.88854 −0.444413
\(122\) 0 0
\(123\) −3.52786 −0.318097
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 0 0
\(129\) 4.94427 0.435319
\(130\) 0 0
\(131\) −20.6525 −1.80442 −0.902208 0.431302i \(-0.858054\pi\)
−0.902208 + 0.431302i \(0.858054\pi\)
\(132\) 0 0
\(133\) 27.4164 2.37730
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −0.472136 −0.0403373 −0.0201686 0.999797i \(-0.506420\pi\)
−0.0201686 + 0.999797i \(0.506420\pi\)
\(138\) 0 0
\(139\) 6.47214 0.548959 0.274480 0.961593i \(-0.411494\pi\)
0.274480 + 0.961593i \(0.411494\pi\)
\(140\) 0 0
\(141\) 12.9443 1.09010
\(142\) 0 0
\(143\) 2.47214 0.206730
\(144\) 0 0
\(145\) 4.76393 0.395623
\(146\) 0 0
\(147\) 20.4164 1.68392
\(148\) 0 0
\(149\) −4.47214 −0.366372 −0.183186 0.983078i \(-0.558641\pi\)
−0.183186 + 0.983078i \(0.558641\pi\)
\(150\) 0 0
\(151\) 16.9443 1.37891 0.689453 0.724331i \(-0.257851\pi\)
0.689453 + 0.724331i \(0.257851\pi\)
\(152\) 0 0
\(153\) 0.763932 0.0617602
\(154\) 0 0
\(155\) −8.94427 −0.718421
\(156\) 0 0
\(157\) 13.4164 1.07075 0.535373 0.844616i \(-0.320171\pi\)
0.535373 + 0.844616i \(0.320171\pi\)
\(158\) 0 0
\(159\) −8.47214 −0.671884
\(160\) 0 0
\(161\) 14.4721 1.14056
\(162\) 0 0
\(163\) −10.4721 −0.820241 −0.410120 0.912031i \(-0.634513\pi\)
−0.410120 + 0.912031i \(0.634513\pi\)
\(164\) 0 0
\(165\) −2.47214 −0.192456
\(166\) 0 0
\(167\) −23.4164 −1.81202 −0.906008 0.423261i \(-0.860886\pi\)
−0.906008 + 0.423261i \(0.860886\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 5.23607 0.400412
\(172\) 0 0
\(173\) 4.47214 0.340010 0.170005 0.985443i \(-0.445622\pi\)
0.170005 + 0.985443i \(0.445622\pi\)
\(174\) 0 0
\(175\) 5.23607 0.395810
\(176\) 0 0
\(177\) 2.47214 0.185817
\(178\) 0 0
\(179\) −21.2361 −1.58726 −0.793629 0.608402i \(-0.791811\pi\)
−0.793629 + 0.608402i \(0.791811\pi\)
\(180\) 0 0
\(181\) 11.5279 0.856859 0.428430 0.903575i \(-0.359067\pi\)
0.428430 + 0.903575i \(0.359067\pi\)
\(182\) 0 0
\(183\) 10.9443 0.809024
\(184\) 0 0
\(185\) −8.47214 −0.622884
\(186\) 0 0
\(187\) −1.88854 −0.138104
\(188\) 0 0
\(189\) 5.23607 0.380868
\(190\) 0 0
\(191\) −3.05573 −0.221105 −0.110552 0.993870i \(-0.535262\pi\)
−0.110552 + 0.993870i \(0.535262\pi\)
\(192\) 0 0
\(193\) 20.1803 1.45261 0.726306 0.687371i \(-0.241235\pi\)
0.726306 + 0.687371i \(0.241235\pi\)
\(194\) 0 0
\(195\) −1.00000 −0.0716115
\(196\) 0 0
\(197\) 17.4164 1.24087 0.620434 0.784259i \(-0.286957\pi\)
0.620434 + 0.784259i \(0.286957\pi\)
\(198\) 0 0
\(199\) 25.8885 1.83519 0.917595 0.397516i \(-0.130128\pi\)
0.917595 + 0.397516i \(0.130128\pi\)
\(200\) 0 0
\(201\) −8.00000 −0.564276
\(202\) 0 0
\(203\) 24.9443 1.75074
\(204\) 0 0
\(205\) −3.52786 −0.246397
\(206\) 0 0
\(207\) 2.76393 0.192107
\(208\) 0 0
\(209\) −12.9443 −0.895374
\(210\) 0 0
\(211\) −16.9443 −1.16649 −0.583246 0.812296i \(-0.698218\pi\)
−0.583246 + 0.812296i \(0.698218\pi\)
\(212\) 0 0
\(213\) −4.00000 −0.274075
\(214\) 0 0
\(215\) 4.94427 0.337197
\(216\) 0 0
\(217\) −46.8328 −3.17922
\(218\) 0 0
\(219\) 9.70820 0.656020
\(220\) 0 0
\(221\) −0.763932 −0.0513876
\(222\) 0 0
\(223\) −10.1803 −0.681726 −0.340863 0.940113i \(-0.610719\pi\)
−0.340863 + 0.940113i \(0.610719\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 13.8885 0.921815 0.460908 0.887448i \(-0.347524\pi\)
0.460908 + 0.887448i \(0.347524\pi\)
\(228\) 0 0
\(229\) −7.23607 −0.478173 −0.239086 0.970998i \(-0.576848\pi\)
−0.239086 + 0.970998i \(0.576848\pi\)
\(230\) 0 0
\(231\) −12.9443 −0.851671
\(232\) 0 0
\(233\) −15.2361 −0.998148 −0.499074 0.866559i \(-0.666326\pi\)
−0.499074 + 0.866559i \(0.666326\pi\)
\(234\) 0 0
\(235\) 12.9443 0.844391
\(236\) 0 0
\(237\) 4.94427 0.321165
\(238\) 0 0
\(239\) −28.9443 −1.87225 −0.936125 0.351668i \(-0.885614\pi\)
−0.936125 + 0.351668i \(0.885614\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 20.4164 1.30436
\(246\) 0 0
\(247\) −5.23607 −0.333163
\(248\) 0 0
\(249\) 8.00000 0.506979
\(250\) 0 0
\(251\) −0.291796 −0.0184180 −0.00920900 0.999958i \(-0.502931\pi\)
−0.00920900 + 0.999958i \(0.502931\pi\)
\(252\) 0 0
\(253\) −6.83282 −0.429575
\(254\) 0 0
\(255\) 0.763932 0.0478393
\(256\) 0 0
\(257\) 6.29180 0.392471 0.196236 0.980557i \(-0.437128\pi\)
0.196236 + 0.980557i \(0.437128\pi\)
\(258\) 0 0
\(259\) −44.3607 −2.75644
\(260\) 0 0
\(261\) 4.76393 0.294880
\(262\) 0 0
\(263\) −12.6525 −0.780185 −0.390093 0.920776i \(-0.627557\pi\)
−0.390093 + 0.920776i \(0.627557\pi\)
\(264\) 0 0
\(265\) −8.47214 −0.520439
\(266\) 0 0
\(267\) −14.9443 −0.914575
\(268\) 0 0
\(269\) 20.1803 1.23042 0.615209 0.788364i \(-0.289072\pi\)
0.615209 + 0.788364i \(0.289072\pi\)
\(270\) 0 0
\(271\) −13.8885 −0.843669 −0.421834 0.906673i \(-0.638614\pi\)
−0.421834 + 0.906673i \(0.638614\pi\)
\(272\) 0 0
\(273\) −5.23607 −0.316901
\(274\) 0 0
\(275\) −2.47214 −0.149075
\(276\) 0 0
\(277\) −12.4721 −0.749378 −0.374689 0.927151i \(-0.622251\pi\)
−0.374689 + 0.927151i \(0.622251\pi\)
\(278\) 0 0
\(279\) −8.94427 −0.535480
\(280\) 0 0
\(281\) −27.8885 −1.66369 −0.831846 0.555007i \(-0.812715\pi\)
−0.831846 + 0.555007i \(0.812715\pi\)
\(282\) 0 0
\(283\) −16.0000 −0.951101 −0.475551 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) 0 0
\(285\) 5.23607 0.310158
\(286\) 0 0
\(287\) −18.4721 −1.09038
\(288\) 0 0
\(289\) −16.4164 −0.965671
\(290\) 0 0
\(291\) 14.6525 0.858943
\(292\) 0 0
\(293\) −3.52786 −0.206100 −0.103050 0.994676i \(-0.532860\pi\)
−0.103050 + 0.994676i \(0.532860\pi\)
\(294\) 0 0
\(295\) 2.47214 0.143933
\(296\) 0 0
\(297\) −2.47214 −0.143448
\(298\) 0 0
\(299\) −2.76393 −0.159842
\(300\) 0 0
\(301\) 25.8885 1.49219
\(302\) 0 0
\(303\) 2.29180 0.131660
\(304\) 0 0
\(305\) 10.9443 0.626667
\(306\) 0 0
\(307\) −20.9443 −1.19535 −0.597676 0.801737i \(-0.703909\pi\)
−0.597676 + 0.801737i \(0.703909\pi\)
\(308\) 0 0
\(309\) 4.00000 0.227552
\(310\) 0 0
\(311\) 3.05573 0.173274 0.0866372 0.996240i \(-0.472388\pi\)
0.0866372 + 0.996240i \(0.472388\pi\)
\(312\) 0 0
\(313\) −5.41641 −0.306153 −0.153077 0.988214i \(-0.548918\pi\)
−0.153077 + 0.988214i \(0.548918\pi\)
\(314\) 0 0
\(315\) 5.23607 0.295019
\(316\) 0 0
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) 0 0
\(319\) −11.7771 −0.659390
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 4.00000 0.222566
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) 0 0
\(327\) −12.1803 −0.673574
\(328\) 0 0
\(329\) 67.7771 3.73667
\(330\) 0 0
\(331\) 0.291796 0.0160386 0.00801928 0.999968i \(-0.497447\pi\)
0.00801928 + 0.999968i \(0.497447\pi\)
\(332\) 0 0
\(333\) −8.47214 −0.464270
\(334\) 0 0
\(335\) −8.00000 −0.437087
\(336\) 0 0
\(337\) 23.5279 1.28164 0.640822 0.767689i \(-0.278594\pi\)
0.640822 + 0.767689i \(0.278594\pi\)
\(338\) 0 0
\(339\) 11.2361 0.610259
\(340\) 0 0
\(341\) 22.1115 1.19740
\(342\) 0 0
\(343\) 70.2492 3.79310
\(344\) 0 0
\(345\) 2.76393 0.148805
\(346\) 0 0
\(347\) −11.4164 −0.612865 −0.306432 0.951892i \(-0.599135\pi\)
−0.306432 + 0.951892i \(0.599135\pi\)
\(348\) 0 0
\(349\) −10.2918 −0.550907 −0.275454 0.961314i \(-0.588828\pi\)
−0.275454 + 0.961314i \(0.588828\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) 4.47214 0.238028 0.119014 0.992893i \(-0.462027\pi\)
0.119014 + 0.992893i \(0.462027\pi\)
\(354\) 0 0
\(355\) −4.00000 −0.212298
\(356\) 0 0
\(357\) 4.00000 0.211702
\(358\) 0 0
\(359\) 7.05573 0.372387 0.186194 0.982513i \(-0.440385\pi\)
0.186194 + 0.982513i \(0.440385\pi\)
\(360\) 0 0
\(361\) 8.41641 0.442969
\(362\) 0 0
\(363\) −4.88854 −0.256582
\(364\) 0 0
\(365\) 9.70820 0.508151
\(366\) 0 0
\(367\) 18.8328 0.983065 0.491532 0.870859i \(-0.336437\pi\)
0.491532 + 0.870859i \(0.336437\pi\)
\(368\) 0 0
\(369\) −3.52786 −0.183653
\(370\) 0 0
\(371\) −44.3607 −2.30309
\(372\) 0 0
\(373\) 21.4164 1.10890 0.554450 0.832217i \(-0.312929\pi\)
0.554450 + 0.832217i \(0.312929\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −4.76393 −0.245355
\(378\) 0 0
\(379\) −37.2361 −1.91269 −0.956344 0.292243i \(-0.905598\pi\)
−0.956344 + 0.292243i \(0.905598\pi\)
\(380\) 0 0
\(381\) 4.00000 0.204926
\(382\) 0 0
\(383\) 25.3050 1.29302 0.646511 0.762904i \(-0.276227\pi\)
0.646511 + 0.762904i \(0.276227\pi\)
\(384\) 0 0
\(385\) −12.9443 −0.659701
\(386\) 0 0
\(387\) 4.94427 0.251331
\(388\) 0 0
\(389\) 7.23607 0.366883 0.183442 0.983031i \(-0.441276\pi\)
0.183442 + 0.983031i \(0.441276\pi\)
\(390\) 0 0
\(391\) 2.11146 0.106781
\(392\) 0 0
\(393\) −20.6525 −1.04178
\(394\) 0 0
\(395\) 4.94427 0.248773
\(396\) 0 0
\(397\) 4.47214 0.224450 0.112225 0.993683i \(-0.464202\pi\)
0.112225 + 0.993683i \(0.464202\pi\)
\(398\) 0 0
\(399\) 27.4164 1.37254
\(400\) 0 0
\(401\) −21.4164 −1.06948 −0.534742 0.845015i \(-0.679591\pi\)
−0.534742 + 0.845015i \(0.679591\pi\)
\(402\) 0 0
\(403\) 8.94427 0.445546
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 20.9443 1.03817
\(408\) 0 0
\(409\) 7.52786 0.372229 0.186114 0.982528i \(-0.440410\pi\)
0.186114 + 0.982528i \(0.440410\pi\)
\(410\) 0 0
\(411\) −0.472136 −0.0232887
\(412\) 0 0
\(413\) 12.9443 0.636946
\(414\) 0 0
\(415\) 8.00000 0.392705
\(416\) 0 0
\(417\) 6.47214 0.316942
\(418\) 0 0
\(419\) 15.1246 0.738886 0.369443 0.929253i \(-0.379549\pi\)
0.369443 + 0.929253i \(0.379549\pi\)
\(420\) 0 0
\(421\) 14.2918 0.696540 0.348270 0.937394i \(-0.386769\pi\)
0.348270 + 0.937394i \(0.386769\pi\)
\(422\) 0 0
\(423\) 12.9443 0.629372
\(424\) 0 0
\(425\) 0.763932 0.0370561
\(426\) 0 0
\(427\) 57.3050 2.77318
\(428\) 0 0
\(429\) 2.47214 0.119356
\(430\) 0 0
\(431\) 9.88854 0.476314 0.238157 0.971227i \(-0.423457\pi\)
0.238157 + 0.971227i \(0.423457\pi\)
\(432\) 0 0
\(433\) −6.58359 −0.316387 −0.158194 0.987408i \(-0.550567\pi\)
−0.158194 + 0.987408i \(0.550567\pi\)
\(434\) 0 0
\(435\) 4.76393 0.228413
\(436\) 0 0
\(437\) 14.4721 0.692296
\(438\) 0 0
\(439\) −21.8885 −1.04468 −0.522342 0.852736i \(-0.674941\pi\)
−0.522342 + 0.852736i \(0.674941\pi\)
\(440\) 0 0
\(441\) 20.4164 0.972210
\(442\) 0 0
\(443\) −30.4721 −1.44777 −0.723887 0.689918i \(-0.757647\pi\)
−0.723887 + 0.689918i \(0.757647\pi\)
\(444\) 0 0
\(445\) −14.9443 −0.708426
\(446\) 0 0
\(447\) −4.47214 −0.211525
\(448\) 0 0
\(449\) 28.8328 1.36070 0.680352 0.732885i \(-0.261827\pi\)
0.680352 + 0.732885i \(0.261827\pi\)
\(450\) 0 0
\(451\) 8.72136 0.410673
\(452\) 0 0
\(453\) 16.9443 0.796111
\(454\) 0 0
\(455\) −5.23607 −0.245471
\(456\) 0 0
\(457\) 33.7082 1.57680 0.788402 0.615161i \(-0.210909\pi\)
0.788402 + 0.615161i \(0.210909\pi\)
\(458\) 0 0
\(459\) 0.763932 0.0356573
\(460\) 0 0
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 0 0
\(463\) 7.70820 0.358231 0.179115 0.983828i \(-0.442676\pi\)
0.179115 + 0.983828i \(0.442676\pi\)
\(464\) 0 0
\(465\) −8.94427 −0.414781
\(466\) 0 0
\(467\) 8.94427 0.413892 0.206946 0.978352i \(-0.433648\pi\)
0.206946 + 0.978352i \(0.433648\pi\)
\(468\) 0 0
\(469\) −41.8885 −1.93423
\(470\) 0 0
\(471\) 13.4164 0.618195
\(472\) 0 0
\(473\) −12.2229 −0.562010
\(474\) 0 0
\(475\) 5.23607 0.240247
\(476\) 0 0
\(477\) −8.47214 −0.387912
\(478\) 0 0
\(479\) 22.8328 1.04326 0.521629 0.853172i \(-0.325325\pi\)
0.521629 + 0.853172i \(0.325325\pi\)
\(480\) 0 0
\(481\) 8.47214 0.386296
\(482\) 0 0
\(483\) 14.4721 0.658505
\(484\) 0 0
\(485\) 14.6525 0.665335
\(486\) 0 0
\(487\) 26.1803 1.18634 0.593172 0.805076i \(-0.297875\pi\)
0.593172 + 0.805076i \(0.297875\pi\)
\(488\) 0 0
\(489\) −10.4721 −0.473566
\(490\) 0 0
\(491\) −31.7082 −1.43097 −0.715486 0.698627i \(-0.753795\pi\)
−0.715486 + 0.698627i \(0.753795\pi\)
\(492\) 0 0
\(493\) 3.63932 0.163907
\(494\) 0 0
\(495\) −2.47214 −0.111114
\(496\) 0 0
\(497\) −20.9443 −0.939479
\(498\) 0 0
\(499\) 10.7639 0.481860 0.240930 0.970543i \(-0.422548\pi\)
0.240930 + 0.970543i \(0.422548\pi\)
\(500\) 0 0
\(501\) −23.4164 −1.04617
\(502\) 0 0
\(503\) −33.5967 −1.49800 −0.749002 0.662567i \(-0.769467\pi\)
−0.749002 + 0.662567i \(0.769467\pi\)
\(504\) 0 0
\(505\) 2.29180 0.101984
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) −3.88854 −0.172357 −0.0861783 0.996280i \(-0.527465\pi\)
−0.0861783 + 0.996280i \(0.527465\pi\)
\(510\) 0 0
\(511\) 50.8328 2.24871
\(512\) 0 0
\(513\) 5.23607 0.231178
\(514\) 0 0
\(515\) 4.00000 0.176261
\(516\) 0 0
\(517\) −32.0000 −1.40736
\(518\) 0 0
\(519\) 4.47214 0.196305
\(520\) 0 0
\(521\) −18.3607 −0.804396 −0.402198 0.915553i \(-0.631754\pi\)
−0.402198 + 0.915553i \(0.631754\pi\)
\(522\) 0 0
\(523\) −42.8328 −1.87295 −0.936474 0.350737i \(-0.885931\pi\)
−0.936474 + 0.350737i \(0.885931\pi\)
\(524\) 0 0
\(525\) 5.23607 0.228521
\(526\) 0 0
\(527\) −6.83282 −0.297642
\(528\) 0 0
\(529\) −15.3607 −0.667856
\(530\) 0 0
\(531\) 2.47214 0.107282
\(532\) 0 0
\(533\) 3.52786 0.152809
\(534\) 0 0
\(535\) −12.0000 −0.518805
\(536\) 0 0
\(537\) −21.2361 −0.916404
\(538\) 0 0
\(539\) −50.4721 −2.17399
\(540\) 0 0
\(541\) 29.1246 1.25216 0.626082 0.779757i \(-0.284657\pi\)
0.626082 + 0.779757i \(0.284657\pi\)
\(542\) 0 0
\(543\) 11.5279 0.494708
\(544\) 0 0
\(545\) −12.1803 −0.521748
\(546\) 0 0
\(547\) −5.88854 −0.251776 −0.125888 0.992044i \(-0.540178\pi\)
−0.125888 + 0.992044i \(0.540178\pi\)
\(548\) 0 0
\(549\) 10.9443 0.467090
\(550\) 0 0
\(551\) 24.9443 1.06266
\(552\) 0 0
\(553\) 25.8885 1.10089
\(554\) 0 0
\(555\) −8.47214 −0.359622
\(556\) 0 0
\(557\) −3.88854 −0.164763 −0.0823814 0.996601i \(-0.526253\pi\)
−0.0823814 + 0.996601i \(0.526253\pi\)
\(558\) 0 0
\(559\) −4.94427 −0.209120
\(560\) 0 0
\(561\) −1.88854 −0.0797344
\(562\) 0 0
\(563\) 37.3050 1.57222 0.786108 0.618089i \(-0.212093\pi\)
0.786108 + 0.618089i \(0.212093\pi\)
\(564\) 0 0
\(565\) 11.2361 0.472705
\(566\) 0 0
\(567\) 5.23607 0.219894
\(568\) 0 0
\(569\) −26.3607 −1.10510 −0.552549 0.833481i \(-0.686345\pi\)
−0.552549 + 0.833481i \(0.686345\pi\)
\(570\) 0 0
\(571\) 37.3050 1.56116 0.780582 0.625054i \(-0.214923\pi\)
0.780582 + 0.625054i \(0.214923\pi\)
\(572\) 0 0
\(573\) −3.05573 −0.127655
\(574\) 0 0
\(575\) 2.76393 0.115264
\(576\) 0 0
\(577\) −7.59675 −0.316257 −0.158128 0.987419i \(-0.550546\pi\)
−0.158128 + 0.987419i \(0.550546\pi\)
\(578\) 0 0
\(579\) 20.1803 0.838666
\(580\) 0 0
\(581\) 41.8885 1.73783
\(582\) 0 0
\(583\) 20.9443 0.867423
\(584\) 0 0
\(585\) −1.00000 −0.0413449
\(586\) 0 0
\(587\) −16.9443 −0.699365 −0.349682 0.936868i \(-0.613711\pi\)
−0.349682 + 0.936868i \(0.613711\pi\)
\(588\) 0 0
\(589\) −46.8328 −1.92971
\(590\) 0 0
\(591\) 17.4164 0.716415
\(592\) 0 0
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) 4.00000 0.163984
\(596\) 0 0
\(597\) 25.8885 1.05955
\(598\) 0 0
\(599\) −16.0000 −0.653742 −0.326871 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) 0 0
\(601\) 42.3607 1.72793 0.863964 0.503553i \(-0.167974\pi\)
0.863964 + 0.503553i \(0.167974\pi\)
\(602\) 0 0
\(603\) −8.00000 −0.325785
\(604\) 0 0
\(605\) −4.88854 −0.198748
\(606\) 0 0
\(607\) 16.9443 0.687747 0.343873 0.939016i \(-0.388261\pi\)
0.343873 + 0.939016i \(0.388261\pi\)
\(608\) 0 0
\(609\) 24.9443 1.01079
\(610\) 0 0
\(611\) −12.9443 −0.523669
\(612\) 0 0
\(613\) −29.4164 −1.18812 −0.594059 0.804422i \(-0.702475\pi\)
−0.594059 + 0.804422i \(0.702475\pi\)
\(614\) 0 0
\(615\) −3.52786 −0.142257
\(616\) 0 0
\(617\) 17.4164 0.701158 0.350579 0.936533i \(-0.385985\pi\)
0.350579 + 0.936533i \(0.385985\pi\)
\(618\) 0 0
\(619\) −3.34752 −0.134548 −0.0672742 0.997735i \(-0.521430\pi\)
−0.0672742 + 0.997735i \(0.521430\pi\)
\(620\) 0 0
\(621\) 2.76393 0.110913
\(622\) 0 0
\(623\) −78.2492 −3.13499
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −12.9443 −0.516944
\(628\) 0 0
\(629\) −6.47214 −0.258061
\(630\) 0 0
\(631\) −3.41641 −0.136005 −0.0680025 0.997685i \(-0.521663\pi\)
−0.0680025 + 0.997685i \(0.521663\pi\)
\(632\) 0 0
\(633\) −16.9443 −0.673474
\(634\) 0 0
\(635\) 4.00000 0.158735
\(636\) 0 0
\(637\) −20.4164 −0.808928
\(638\) 0 0
\(639\) −4.00000 −0.158238
\(640\) 0 0
\(641\) 26.0000 1.02694 0.513469 0.858108i \(-0.328360\pi\)
0.513469 + 0.858108i \(0.328360\pi\)
\(642\) 0 0
\(643\) −8.00000 −0.315489 −0.157745 0.987480i \(-0.550422\pi\)
−0.157745 + 0.987480i \(0.550422\pi\)
\(644\) 0 0
\(645\) 4.94427 0.194681
\(646\) 0 0
\(647\) 16.2918 0.640497 0.320248 0.947334i \(-0.396234\pi\)
0.320248 + 0.947334i \(0.396234\pi\)
\(648\) 0 0
\(649\) −6.11146 −0.239896
\(650\) 0 0
\(651\) −46.8328 −1.83552
\(652\) 0 0
\(653\) 26.0000 1.01746 0.508729 0.860927i \(-0.330115\pi\)
0.508729 + 0.860927i \(0.330115\pi\)
\(654\) 0 0
\(655\) −20.6525 −0.806959
\(656\) 0 0
\(657\) 9.70820 0.378753
\(658\) 0 0
\(659\) −39.7082 −1.54681 −0.773406 0.633911i \(-0.781449\pi\)
−0.773406 + 0.633911i \(0.781449\pi\)
\(660\) 0 0
\(661\) 13.1246 0.510488 0.255244 0.966877i \(-0.417844\pi\)
0.255244 + 0.966877i \(0.417844\pi\)
\(662\) 0 0
\(663\) −0.763932 −0.0296687
\(664\) 0 0
\(665\) 27.4164 1.06316
\(666\) 0 0
\(667\) 13.1672 0.509835
\(668\) 0 0
\(669\) −10.1803 −0.393595
\(670\) 0 0
\(671\) −27.0557 −1.04447
\(672\) 0 0
\(673\) 22.9443 0.884437 0.442218 0.896907i \(-0.354192\pi\)
0.442218 + 0.896907i \(0.354192\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −30.0000 −1.15299 −0.576497 0.817099i \(-0.695581\pi\)
−0.576497 + 0.817099i \(0.695581\pi\)
\(678\) 0 0
\(679\) 76.7214 2.94430
\(680\) 0 0
\(681\) 13.8885 0.532210
\(682\) 0 0
\(683\) −14.1115 −0.539960 −0.269980 0.962866i \(-0.587017\pi\)
−0.269980 + 0.962866i \(0.587017\pi\)
\(684\) 0 0
\(685\) −0.472136 −0.0180394
\(686\) 0 0
\(687\) −7.23607 −0.276073
\(688\) 0 0
\(689\) 8.47214 0.322763
\(690\) 0 0
\(691\) 8.29180 0.315435 0.157717 0.987484i \(-0.449586\pi\)
0.157717 + 0.987484i \(0.449586\pi\)
\(692\) 0 0
\(693\) −12.9443 −0.491712
\(694\) 0 0
\(695\) 6.47214 0.245502
\(696\) 0 0
\(697\) −2.69505 −0.102082
\(698\) 0 0
\(699\) −15.2361 −0.576281
\(700\) 0 0
\(701\) 23.2361 0.877614 0.438807 0.898581i \(-0.355401\pi\)
0.438807 + 0.898581i \(0.355401\pi\)
\(702\) 0 0
\(703\) −44.3607 −1.67309
\(704\) 0 0
\(705\) 12.9443 0.487509
\(706\) 0 0
\(707\) 12.0000 0.451306
\(708\) 0 0
\(709\) −13.3475 −0.501277 −0.250638 0.968081i \(-0.580641\pi\)
−0.250638 + 0.968081i \(0.580641\pi\)
\(710\) 0 0
\(711\) 4.94427 0.185425
\(712\) 0 0
\(713\) −24.7214 −0.925822
\(714\) 0 0
\(715\) 2.47214 0.0924526
\(716\) 0 0
\(717\) −28.9443 −1.08094
\(718\) 0 0
\(719\) 34.4721 1.28559 0.642797 0.766037i \(-0.277774\pi\)
0.642797 + 0.766037i \(0.277774\pi\)
\(720\) 0 0
\(721\) 20.9443 0.780005
\(722\) 0 0
\(723\) 10.0000 0.371904
\(724\) 0 0
\(725\) 4.76393 0.176928
\(726\) 0 0
\(727\) −14.4721 −0.536742 −0.268371 0.963316i \(-0.586485\pi\)
−0.268371 + 0.963316i \(0.586485\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 3.77709 0.139701
\(732\) 0 0
\(733\) −16.4721 −0.608412 −0.304206 0.952606i \(-0.598391\pi\)
−0.304206 + 0.952606i \(0.598391\pi\)
\(734\) 0 0
\(735\) 20.4164 0.753071
\(736\) 0 0
\(737\) 19.7771 0.728498
\(738\) 0 0
\(739\) −2.18034 −0.0802051 −0.0401025 0.999196i \(-0.512768\pi\)
−0.0401025 + 0.999196i \(0.512768\pi\)
\(740\) 0 0
\(741\) −5.23607 −0.192352
\(742\) 0 0
\(743\) −47.4164 −1.73954 −0.869770 0.493458i \(-0.835733\pi\)
−0.869770 + 0.493458i \(0.835733\pi\)
\(744\) 0 0
\(745\) −4.47214 −0.163846
\(746\) 0 0
\(747\) 8.00000 0.292705
\(748\) 0 0
\(749\) −62.8328 −2.29586
\(750\) 0 0
\(751\) 20.0000 0.729810 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) 0 0
\(753\) −0.291796 −0.0106336
\(754\) 0 0
\(755\) 16.9443 0.616665
\(756\) 0 0
\(757\) −9.05573 −0.329136 −0.164568 0.986366i \(-0.552623\pi\)
−0.164568 + 0.986366i \(0.552623\pi\)
\(758\) 0 0
\(759\) −6.83282 −0.248015
\(760\) 0 0
\(761\) −29.4164 −1.06634 −0.533172 0.846007i \(-0.679000\pi\)
−0.533172 + 0.846007i \(0.679000\pi\)
\(762\) 0 0
\(763\) −63.7771 −2.30889
\(764\) 0 0
\(765\) 0.763932 0.0276200
\(766\) 0 0
\(767\) −2.47214 −0.0892637
\(768\) 0 0
\(769\) −20.8328 −0.751251 −0.375625 0.926772i \(-0.622572\pi\)
−0.375625 + 0.926772i \(0.622572\pi\)
\(770\) 0 0
\(771\) 6.29180 0.226594
\(772\) 0 0
\(773\) −1.63932 −0.0589623 −0.0294811 0.999565i \(-0.509385\pi\)
−0.0294811 + 0.999565i \(0.509385\pi\)
\(774\) 0 0
\(775\) −8.94427 −0.321288
\(776\) 0 0
\(777\) −44.3607 −1.59143
\(778\) 0 0
\(779\) −18.4721 −0.661833
\(780\) 0 0
\(781\) 9.88854 0.353840
\(782\) 0 0
\(783\) 4.76393 0.170249
\(784\) 0 0
\(785\) 13.4164 0.478852
\(786\) 0 0
\(787\) −12.3607 −0.440611 −0.220305 0.975431i \(-0.570705\pi\)
−0.220305 + 0.975431i \(0.570705\pi\)
\(788\) 0 0
\(789\) −12.6525 −0.450440
\(790\) 0 0
\(791\) 58.8328 2.09185
\(792\) 0 0
\(793\) −10.9443 −0.388642
\(794\) 0 0
\(795\) −8.47214 −0.300476
\(796\) 0 0
\(797\) −45.4164 −1.60873 −0.804366 0.594134i \(-0.797495\pi\)
−0.804366 + 0.594134i \(0.797495\pi\)
\(798\) 0 0
\(799\) 9.88854 0.349832
\(800\) 0 0
\(801\) −14.9443 −0.528030
\(802\) 0 0
\(803\) −24.0000 −0.846942
\(804\) 0 0
\(805\) 14.4721 0.510076
\(806\) 0 0
\(807\) 20.1803 0.710382
\(808\) 0 0
\(809\) 42.7214 1.50200 0.751002 0.660300i \(-0.229571\pi\)
0.751002 + 0.660300i \(0.229571\pi\)
\(810\) 0 0
\(811\) −7.70820 −0.270672 −0.135336 0.990800i \(-0.543211\pi\)
−0.135336 + 0.990800i \(0.543211\pi\)
\(812\) 0 0
\(813\) −13.8885 −0.487092
\(814\) 0 0
\(815\) −10.4721 −0.366823
\(816\) 0 0
\(817\) 25.8885 0.905725
\(818\) 0 0
\(819\) −5.23607 −0.182963
\(820\) 0 0
\(821\) −11.3050 −0.394546 −0.197273 0.980349i \(-0.563208\pi\)
−0.197273 + 0.980349i \(0.563208\pi\)
\(822\) 0 0
\(823\) −23.7771 −0.828817 −0.414409 0.910091i \(-0.636012\pi\)
−0.414409 + 0.910091i \(0.636012\pi\)
\(824\) 0 0
\(825\) −2.47214 −0.0860687
\(826\) 0 0
\(827\) 20.9443 0.728304 0.364152 0.931340i \(-0.381359\pi\)
0.364152 + 0.931340i \(0.381359\pi\)
\(828\) 0 0
\(829\) −8.11146 −0.281723 −0.140861 0.990029i \(-0.544987\pi\)
−0.140861 + 0.990029i \(0.544987\pi\)
\(830\) 0 0
\(831\) −12.4721 −0.432654
\(832\) 0 0
\(833\) 15.5967 0.540395
\(834\) 0 0
\(835\) −23.4164 −0.810358
\(836\) 0 0
\(837\) −8.94427 −0.309159
\(838\) 0 0
\(839\) 17.8885 0.617581 0.308791 0.951130i \(-0.400076\pi\)
0.308791 + 0.951130i \(0.400076\pi\)
\(840\) 0 0
\(841\) −6.30495 −0.217412
\(842\) 0 0
\(843\) −27.8885 −0.960532
\(844\) 0 0
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) −25.5967 −0.879515
\(848\) 0 0
\(849\) −16.0000 −0.549119
\(850\) 0 0
\(851\) −23.4164 −0.802704
\(852\) 0 0
\(853\) 9.41641 0.322412 0.161206 0.986921i \(-0.448462\pi\)
0.161206 + 0.986921i \(0.448462\pi\)
\(854\) 0 0
\(855\) 5.23607 0.179070
\(856\) 0 0
\(857\) 21.1246 0.721603 0.360801 0.932643i \(-0.382503\pi\)
0.360801 + 0.932643i \(0.382503\pi\)
\(858\) 0 0
\(859\) −0.944272 −0.0322181 −0.0161091 0.999870i \(-0.505128\pi\)
−0.0161091 + 0.999870i \(0.505128\pi\)
\(860\) 0 0
\(861\) −18.4721 −0.629529
\(862\) 0 0
\(863\) 0.583592 0.0198657 0.00993285 0.999951i \(-0.496838\pi\)
0.00993285 + 0.999951i \(0.496838\pi\)
\(864\) 0 0
\(865\) 4.47214 0.152057
\(866\) 0 0
\(867\) −16.4164 −0.557530
\(868\) 0 0
\(869\) −12.2229 −0.414634
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) 0 0
\(873\) 14.6525 0.495911
\(874\) 0 0
\(875\) 5.23607 0.177011
\(876\) 0 0
\(877\) −38.0000 −1.28317 −0.641584 0.767052i \(-0.721723\pi\)
−0.641584 + 0.767052i \(0.721723\pi\)
\(878\) 0 0
\(879\) −3.52786 −0.118992
\(880\) 0 0
\(881\) 33.4164 1.12583 0.562914 0.826516i \(-0.309680\pi\)
0.562914 + 0.826516i \(0.309680\pi\)
\(882\) 0 0
\(883\) −13.8885 −0.467387 −0.233693 0.972310i \(-0.575081\pi\)
−0.233693 + 0.972310i \(0.575081\pi\)
\(884\) 0 0
\(885\) 2.47214 0.0830999
\(886\) 0 0
\(887\) −44.0689 −1.47969 −0.739844 0.672778i \(-0.765101\pi\)
−0.739844 + 0.672778i \(0.765101\pi\)
\(888\) 0 0
\(889\) 20.9443 0.702448
\(890\) 0 0
\(891\) −2.47214 −0.0828197
\(892\) 0 0
\(893\) 67.7771 2.26807
\(894\) 0 0
\(895\) −21.2361 −0.709843
\(896\) 0 0
\(897\) −2.76393 −0.0922850
\(898\) 0 0
\(899\) −42.6099 −1.42112
\(900\) 0 0
\(901\) −6.47214 −0.215618
\(902\) 0 0
\(903\) 25.8885 0.861517
\(904\) 0 0
\(905\) 11.5279 0.383199
\(906\) 0 0
\(907\) 24.0000 0.796907 0.398453 0.917189i \(-0.369547\pi\)
0.398453 + 0.917189i \(0.369547\pi\)
\(908\) 0 0
\(909\) 2.29180 0.0760141
\(910\) 0 0
\(911\) 45.5279 1.50841 0.754203 0.656642i \(-0.228024\pi\)
0.754203 + 0.656642i \(0.228024\pi\)
\(912\) 0 0
\(913\) −19.7771 −0.654526
\(914\) 0 0
\(915\) 10.9443 0.361806
\(916\) 0 0
\(917\) −108.138 −3.57102
\(918\) 0 0
\(919\) −55.7771 −1.83992 −0.919958 0.392017i \(-0.871778\pi\)
−0.919958 + 0.392017i \(0.871778\pi\)
\(920\) 0 0
\(921\) −20.9443 −0.690137
\(922\) 0 0
\(923\) 4.00000 0.131662
\(924\) 0 0
\(925\) −8.47214 −0.278562
\(926\) 0 0
\(927\) 4.00000 0.131377
\(928\) 0 0
\(929\) 12.1115 0.397364 0.198682 0.980064i \(-0.436334\pi\)
0.198682 + 0.980064i \(0.436334\pi\)
\(930\) 0 0
\(931\) 106.902 3.50356
\(932\) 0 0
\(933\) 3.05573 0.100040
\(934\) 0 0
\(935\) −1.88854 −0.0617620
\(936\) 0 0
\(937\) −54.0000 −1.76410 −0.882052 0.471153i \(-0.843838\pi\)
−0.882052 + 0.471153i \(0.843838\pi\)
\(938\) 0 0
\(939\) −5.41641 −0.176758
\(940\) 0 0
\(941\) 24.4721 0.797769 0.398884 0.917001i \(-0.369397\pi\)
0.398884 + 0.917001i \(0.369397\pi\)
\(942\) 0 0
\(943\) −9.75078 −0.317529
\(944\) 0 0
\(945\) 5.23607 0.170329
\(946\) 0 0
\(947\) −33.8885 −1.10123 −0.550615 0.834759i \(-0.685607\pi\)
−0.550615 + 0.834759i \(0.685607\pi\)
\(948\) 0 0
\(949\) −9.70820 −0.315142
\(950\) 0 0
\(951\) −2.00000 −0.0648544
\(952\) 0 0
\(953\) 19.8197 0.642022 0.321011 0.947076i \(-0.395977\pi\)
0.321011 + 0.947076i \(0.395977\pi\)
\(954\) 0 0
\(955\) −3.05573 −0.0988810
\(956\) 0 0
\(957\) −11.7771 −0.380699
\(958\) 0 0
\(959\) −2.47214 −0.0798294
\(960\) 0 0
\(961\) 49.0000 1.58065
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) 0 0
\(965\) 20.1803 0.649628
\(966\) 0 0
\(967\) 5.23607 0.168381 0.0841903 0.996450i \(-0.473170\pi\)
0.0841903 + 0.996450i \(0.473170\pi\)
\(968\) 0 0
\(969\) 4.00000 0.128499
\(970\) 0 0
\(971\) 6.54102 0.209911 0.104956 0.994477i \(-0.466530\pi\)
0.104956 + 0.994477i \(0.466530\pi\)
\(972\) 0 0
\(973\) 33.8885 1.08642
\(974\) 0 0
\(975\) −1.00000 −0.0320256
\(976\) 0 0
\(977\) −57.7771 −1.84845 −0.924226 0.381845i \(-0.875289\pi\)
−0.924226 + 0.381845i \(0.875289\pi\)
\(978\) 0 0
\(979\) 36.9443 1.18074
\(980\) 0 0
\(981\) −12.1803 −0.388888
\(982\) 0 0
\(983\) −7.41641 −0.236547 −0.118273 0.992981i \(-0.537736\pi\)
−0.118273 + 0.992981i \(0.537736\pi\)
\(984\) 0 0
\(985\) 17.4164 0.554933
\(986\) 0 0
\(987\) 67.7771 2.15737
\(988\) 0 0
\(989\) 13.6656 0.434542
\(990\) 0 0
\(991\) −44.9443 −1.42770 −0.713851 0.700298i \(-0.753051\pi\)
−0.713851 + 0.700298i \(0.753051\pi\)
\(992\) 0 0
\(993\) 0.291796 0.00925987
\(994\) 0 0
\(995\) 25.8885 0.820722
\(996\) 0 0
\(997\) −7.52786 −0.238410 −0.119205 0.992870i \(-0.538035\pi\)
−0.119205 + 0.992870i \(0.538035\pi\)
\(998\) 0 0
\(999\) −8.47214 −0.268047
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.bv.1.2 yes 2
4.3 odd 2 6240.2.a.bl.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6240.2.a.bl.1.1 2 4.3 odd 2
6240.2.a.bv.1.2 yes 2 1.1 even 1 trivial