Properties

Label 7623.2.a.bz.1.3
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 1
Dimension 3
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 7623.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.8699614608\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 847)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(3.12489\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.12489 q^{2} +2.51514 q^{4} -0.484862 q^{5} +1.00000 q^{7} +1.09461 q^{8} +O(q^{10})\) \(q+2.12489 q^{2} +2.51514 q^{4} -0.484862 q^{5} +1.00000 q^{7} +1.09461 q^{8} -1.03028 q^{10} +5.60975 q^{13} +2.12489 q^{14} -2.70436 q^{16} -5.60975 q^{17} -5.28005 q^{19} -1.21949 q^{20} -2.48486 q^{23} -4.76491 q^{25} +11.9201 q^{26} +2.51514 q^{28} -5.28005 q^{29} -7.12489 q^{31} -7.93567 q^{32} -11.9201 q^{34} -0.484862 q^{35} -0.235091 q^{37} -11.2195 q^{38} -0.530734 q^{40} -2.39025 q^{41} +1.03028 q^{43} -5.28005 q^{46} +1.60975 q^{47} +1.00000 q^{49} -10.1249 q^{50} +14.1093 q^{52} +3.03028 q^{53} +1.09461 q^{56} -11.2195 q^{58} -3.12489 q^{59} -2.39025 q^{61} -15.1396 q^{62} -11.4537 q^{64} -2.71995 q^{65} +10.0147 q^{67} -14.1093 q^{68} -1.03028 q^{70} -12.0752 q^{71} +2.39025 q^{73} -0.499542 q^{74} -13.2800 q^{76} +9.03028 q^{79} +1.31124 q^{80} -5.07901 q^{82} +3.21949 q^{83} +2.71995 q^{85} +2.18922 q^{86} +1.26537 q^{89} +5.60975 q^{91} -6.24977 q^{92} +3.42053 q^{94} +2.56009 q^{95} -8.79518 q^{97} +2.12489 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 2q^{2} + 8q^{4} - q^{5} + 3q^{7} - 6q^{8} + O(q^{10}) \) \( 3q - 2q^{2} + 8q^{4} - q^{5} + 3q^{7} - 6q^{8} - 4q^{10} + 8q^{13} - 2q^{14} + 10q^{16} - 8q^{17} + 14q^{20} - 7q^{23} + 2q^{25} + 12q^{26} + 8q^{28} - 13q^{31} - 34q^{32} - 12q^{34} - q^{35} - 17q^{37} - 16q^{38} - 36q^{40} - 16q^{41} + 4q^{43} - 4q^{47} + 3q^{49} - 22q^{50} + 10q^{53} - 6q^{56} - 16q^{58} - q^{59} - 16q^{61} - 4q^{62} + 34q^{64} - 24q^{65} - 3q^{67} - 4q^{70} - 5q^{71} + 16q^{73} + 32q^{74} - 24q^{76} + 28q^{79} + 56q^{80} + 28q^{82} - 8q^{83} + 24q^{85} - 12q^{86} + 21q^{89} + 8q^{91} - 2q^{92} + 20q^{94} - 24q^{95} - 11q^{97} - 2q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.12489 1.50252 0.751260 0.660006i \(-0.229446\pi\)
0.751260 + 0.660006i \(0.229446\pi\)
\(3\) 0 0
\(4\) 2.51514 1.25757
\(5\) −0.484862 −0.216837 −0.108418 0.994105i \(-0.534579\pi\)
−0.108418 + 0.994105i \(0.534579\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 1.09461 0.387003
\(9\) 0 0
\(10\) −1.03028 −0.325802
\(11\) 0 0
\(12\) 0 0
\(13\) 5.60975 1.55586 0.777932 0.628348i \(-0.216269\pi\)
0.777932 + 0.628348i \(0.216269\pi\)
\(14\) 2.12489 0.567900
\(15\) 0 0
\(16\) −2.70436 −0.676089
\(17\) −5.60975 −1.36056 −0.680282 0.732951i \(-0.738143\pi\)
−0.680282 + 0.732951i \(0.738143\pi\)
\(18\) 0 0
\(19\) −5.28005 −1.21133 −0.605663 0.795721i \(-0.707092\pi\)
−0.605663 + 0.795721i \(0.707092\pi\)
\(20\) −1.21949 −0.272687
\(21\) 0 0
\(22\) 0 0
\(23\) −2.48486 −0.518130 −0.259065 0.965860i \(-0.583414\pi\)
−0.259065 + 0.965860i \(0.583414\pi\)
\(24\) 0 0
\(25\) −4.76491 −0.952982
\(26\) 11.9201 2.33772
\(27\) 0 0
\(28\) 2.51514 0.475316
\(29\) −5.28005 −0.980480 −0.490240 0.871587i \(-0.663091\pi\)
−0.490240 + 0.871587i \(0.663091\pi\)
\(30\) 0 0
\(31\) −7.12489 −1.27967 −0.639834 0.768513i \(-0.720997\pi\)
−0.639834 + 0.768513i \(0.720997\pi\)
\(32\) −7.93567 −1.40284
\(33\) 0 0
\(34\) −11.9201 −2.04428
\(35\) −0.484862 −0.0819566
\(36\) 0 0
\(37\) −0.235091 −0.0386487 −0.0193244 0.999813i \(-0.506152\pi\)
−0.0193244 + 0.999813i \(0.506152\pi\)
\(38\) −11.2195 −1.82004
\(39\) 0 0
\(40\) −0.530734 −0.0839165
\(41\) −2.39025 −0.373295 −0.186647 0.982427i \(-0.559762\pi\)
−0.186647 + 0.982427i \(0.559762\pi\)
\(42\) 0 0
\(43\) 1.03028 0.157116 0.0785578 0.996910i \(-0.474968\pi\)
0.0785578 + 0.996910i \(0.474968\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −5.28005 −0.778500
\(47\) 1.60975 0.234806 0.117403 0.993084i \(-0.462543\pi\)
0.117403 + 0.993084i \(0.462543\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −10.1249 −1.43188
\(51\) 0 0
\(52\) 14.1093 1.95661
\(53\) 3.03028 0.416240 0.208120 0.978103i \(-0.433265\pi\)
0.208120 + 0.978103i \(0.433265\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.09461 0.146273
\(57\) 0 0
\(58\) −11.2195 −1.47319
\(59\) −3.12489 −0.406825 −0.203413 0.979093i \(-0.565203\pi\)
−0.203413 + 0.979093i \(0.565203\pi\)
\(60\) 0 0
\(61\) −2.39025 −0.306040 −0.153020 0.988223i \(-0.548900\pi\)
−0.153020 + 0.988223i \(0.548900\pi\)
\(62\) −15.1396 −1.92273
\(63\) 0 0
\(64\) −11.4537 −1.43171
\(65\) −2.71995 −0.337369
\(66\) 0 0
\(67\) 10.0147 1.22349 0.611744 0.791056i \(-0.290468\pi\)
0.611744 + 0.791056i \(0.290468\pi\)
\(68\) −14.1093 −1.71100
\(69\) 0 0
\(70\) −1.03028 −0.123142
\(71\) −12.0752 −1.43307 −0.716533 0.697553i \(-0.754272\pi\)
−0.716533 + 0.697553i \(0.754272\pi\)
\(72\) 0 0
\(73\) 2.39025 0.279758 0.139879 0.990169i \(-0.455329\pi\)
0.139879 + 0.990169i \(0.455329\pi\)
\(74\) −0.499542 −0.0580705
\(75\) 0 0
\(76\) −13.2800 −1.52333
\(77\) 0 0
\(78\) 0 0
\(79\) 9.03028 1.01599 0.507993 0.861361i \(-0.330388\pi\)
0.507993 + 0.861361i \(0.330388\pi\)
\(80\) 1.31124 0.146601
\(81\) 0 0
\(82\) −5.07901 −0.560883
\(83\) 3.21949 0.353385 0.176693 0.984266i \(-0.443460\pi\)
0.176693 + 0.984266i \(0.443460\pi\)
\(84\) 0 0
\(85\) 2.71995 0.295020
\(86\) 2.18922 0.236070
\(87\) 0 0
\(88\) 0 0
\(89\) 1.26537 0.134129 0.0670643 0.997749i \(-0.478637\pi\)
0.0670643 + 0.997749i \(0.478637\pi\)
\(90\) 0 0
\(91\) 5.60975 0.588061
\(92\) −6.24977 −0.651584
\(93\) 0 0
\(94\) 3.42053 0.352801
\(95\) 2.56009 0.262660
\(96\) 0 0
\(97\) −8.79518 −0.893016 −0.446508 0.894780i \(-0.647333\pi\)
−0.446508 + 0.894780i \(0.647333\pi\)
\(98\) 2.12489 0.214646
\(99\) 0 0
\(100\) −11.9844 −1.19844
\(101\) 12.9503 1.28861 0.644304 0.764770i \(-0.277147\pi\)
0.644304 + 0.764770i \(0.277147\pi\)
\(102\) 0 0
\(103\) 12.1698 1.19913 0.599565 0.800326i \(-0.295340\pi\)
0.599565 + 0.800326i \(0.295340\pi\)
\(104\) 6.14048 0.602124
\(105\) 0 0
\(106\) 6.43899 0.625410
\(107\) −10.5601 −1.02088 −0.510441 0.859913i \(-0.670518\pi\)
−0.510441 + 0.859913i \(0.670518\pi\)
\(108\) 0 0
\(109\) −7.34060 −0.703102 −0.351551 0.936169i \(-0.614346\pi\)
−0.351551 + 0.936169i \(0.614346\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −2.70436 −0.255538
\(113\) −13.7649 −1.29489 −0.647447 0.762111i \(-0.724163\pi\)
−0.647447 + 0.762111i \(0.724163\pi\)
\(114\) 0 0
\(115\) 1.20482 0.112350
\(116\) −13.2800 −1.23302
\(117\) 0 0
\(118\) −6.64002 −0.611264
\(119\) −5.60975 −0.514245
\(120\) 0 0
\(121\) 0 0
\(122\) −5.07901 −0.459832
\(123\) 0 0
\(124\) −17.9201 −1.60927
\(125\) 4.73463 0.423478
\(126\) 0 0
\(127\) 2.56009 0.227172 0.113586 0.993528i \(-0.463766\pi\)
0.113586 + 0.993528i \(0.463766\pi\)
\(128\) −8.46640 −0.748331
\(129\) 0 0
\(130\) −5.77959 −0.506903
\(131\) 2.71995 0.237643 0.118822 0.992916i \(-0.462088\pi\)
0.118822 + 0.992916i \(0.462088\pi\)
\(132\) 0 0
\(133\) −5.28005 −0.457838
\(134\) 21.2800 1.83832
\(135\) 0 0
\(136\) −6.14048 −0.526542
\(137\) 9.70436 0.829099 0.414550 0.910027i \(-0.363939\pi\)
0.414550 + 0.910027i \(0.363939\pi\)
\(138\) 0 0
\(139\) −19.7190 −1.67255 −0.836273 0.548313i \(-0.815270\pi\)
−0.836273 + 0.548313i \(0.815270\pi\)
\(140\) −1.21949 −0.103066
\(141\) 0 0
\(142\) −25.6585 −2.15321
\(143\) 0 0
\(144\) 0 0
\(145\) 2.56009 0.212604
\(146\) 5.07901 0.420342
\(147\) 0 0
\(148\) −0.591287 −0.0486035
\(149\) −10.5601 −0.865117 −0.432558 0.901606i \(-0.642389\pi\)
−0.432558 + 0.901606i \(0.642389\pi\)
\(150\) 0 0
\(151\) 15.4693 1.25887 0.629435 0.777053i \(-0.283286\pi\)
0.629435 + 0.777053i \(0.283286\pi\)
\(152\) −5.77959 −0.468787
\(153\) 0 0
\(154\) 0 0
\(155\) 3.45459 0.277479
\(156\) 0 0
\(157\) 9.10551 0.726699 0.363349 0.931653i \(-0.381633\pi\)
0.363349 + 0.931653i \(0.381633\pi\)
\(158\) 19.1883 1.52654
\(159\) 0 0
\(160\) 3.84770 0.304188
\(161\) −2.48486 −0.195835
\(162\) 0 0
\(163\) −13.2800 −1.04017 −0.520087 0.854113i \(-0.674100\pi\)
−0.520087 + 0.854113i \(0.674100\pi\)
\(164\) −6.01182 −0.469444
\(165\) 0 0
\(166\) 6.84106 0.530969
\(167\) −10.0606 −0.778509 −0.389254 0.921130i \(-0.627267\pi\)
−0.389254 + 0.921130i \(0.627267\pi\)
\(168\) 0 0
\(169\) 18.4693 1.42071
\(170\) 5.77959 0.443274
\(171\) 0 0
\(172\) 2.59129 0.197584
\(173\) 8.16984 0.621142 0.310571 0.950550i \(-0.399480\pi\)
0.310571 + 0.950550i \(0.399480\pi\)
\(174\) 0 0
\(175\) −4.76491 −0.360193
\(176\) 0 0
\(177\) 0 0
\(178\) 2.68876 0.201531
\(179\) 12.2645 0.916688 0.458344 0.888775i \(-0.348443\pi\)
0.458344 + 0.888775i \(0.348443\pi\)
\(180\) 0 0
\(181\) −7.04496 −0.523647 −0.261824 0.965116i \(-0.584324\pi\)
−0.261824 + 0.965116i \(0.584324\pi\)
\(182\) 11.9201 0.883574
\(183\) 0 0
\(184\) −2.71995 −0.200518
\(185\) 0.113987 0.00838047
\(186\) 0 0
\(187\) 0 0
\(188\) 4.04874 0.295284
\(189\) 0 0
\(190\) 5.43991 0.394652
\(191\) −18.7952 −1.35997 −0.679986 0.733225i \(-0.738014\pi\)
−0.679986 + 0.733225i \(0.738014\pi\)
\(192\) 0 0
\(193\) 16.4995 1.18766 0.593831 0.804589i \(-0.297615\pi\)
0.593831 + 0.804589i \(0.297615\pi\)
\(194\) −18.6888 −1.34177
\(195\) 0 0
\(196\) 2.51514 0.179653
\(197\) −24.4995 −1.74552 −0.872760 0.488149i \(-0.837672\pi\)
−0.872760 + 0.488149i \(0.837672\pi\)
\(198\) 0 0
\(199\) −15.3893 −1.09092 −0.545461 0.838136i \(-0.683645\pi\)
−0.545461 + 0.838136i \(0.683645\pi\)
\(200\) −5.21571 −0.368807
\(201\) 0 0
\(202\) 27.5180 1.93616
\(203\) −5.28005 −0.370587
\(204\) 0 0
\(205\) 1.15894 0.0809441
\(206\) 25.8595 1.80172
\(207\) 0 0
\(208\) −15.1708 −1.05190
\(209\) 0 0
\(210\) 0 0
\(211\) −14.4390 −0.994021 −0.497011 0.867745i \(-0.665569\pi\)
−0.497011 + 0.867745i \(0.665569\pi\)
\(212\) 7.62156 0.523451
\(213\) 0 0
\(214\) −22.4390 −1.53390
\(215\) −0.499542 −0.0340685
\(216\) 0 0
\(217\) −7.12489 −0.483669
\(218\) −15.5979 −1.05643
\(219\) 0 0
\(220\) 0 0
\(221\) −31.4693 −2.11685
\(222\) 0 0
\(223\) 17.2536 1.15538 0.577692 0.816255i \(-0.303954\pi\)
0.577692 + 0.816255i \(0.303954\pi\)
\(224\) −7.93567 −0.530224
\(225\) 0 0
\(226\) −29.2489 −1.94560
\(227\) −10.7200 −0.711508 −0.355754 0.934580i \(-0.615776\pi\)
−0.355754 + 0.934580i \(0.615776\pi\)
\(228\) 0 0
\(229\) 5.45459 0.360449 0.180225 0.983625i \(-0.442318\pi\)
0.180225 + 0.983625i \(0.442318\pi\)
\(230\) 2.56009 0.168808
\(231\) 0 0
\(232\) −5.77959 −0.379449
\(233\) −29.7796 −1.95093 −0.975463 0.220164i \(-0.929341\pi\)
−0.975463 + 0.220164i \(0.929341\pi\)
\(234\) 0 0
\(235\) −0.780505 −0.0509145
\(236\) −7.85952 −0.511611
\(237\) 0 0
\(238\) −11.9201 −0.772663
\(239\) −2.56009 −0.165599 −0.0827994 0.996566i \(-0.526386\pi\)
−0.0827994 + 0.996566i \(0.526386\pi\)
\(240\) 0 0
\(241\) 27.3893 1.76430 0.882151 0.470966i \(-0.156095\pi\)
0.882151 + 0.470966i \(0.156095\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −6.01182 −0.384867
\(245\) −0.484862 −0.0309767
\(246\) 0 0
\(247\) −29.6197 −1.88466
\(248\) −7.79897 −0.495235
\(249\) 0 0
\(250\) 10.0606 0.636285
\(251\) −9.84484 −0.621401 −0.310700 0.950508i \(-0.600564\pi\)
−0.310700 + 0.950508i \(0.600564\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 5.43991 0.341330
\(255\) 0 0
\(256\) 4.91721 0.307325
\(257\) 22.4995 1.40348 0.701741 0.712432i \(-0.252406\pi\)
0.701741 + 0.712432i \(0.252406\pi\)
\(258\) 0 0
\(259\) −0.235091 −0.0146079
\(260\) −6.84106 −0.424264
\(261\) 0 0
\(262\) 5.77959 0.357064
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 0 0
\(265\) −1.46927 −0.0902563
\(266\) −11.2195 −0.687911
\(267\) 0 0
\(268\) 25.1883 1.53862
\(269\) −9.21949 −0.562123 −0.281061 0.959690i \(-0.590686\pi\)
−0.281061 + 0.959690i \(0.590686\pi\)
\(270\) 0 0
\(271\) 10.5601 0.641480 0.320740 0.947167i \(-0.396068\pi\)
0.320740 + 0.947167i \(0.396068\pi\)
\(272\) 15.1708 0.919862
\(273\) 0 0
\(274\) 20.6206 1.24574
\(275\) 0 0
\(276\) 0 0
\(277\) −18.5601 −1.11517 −0.557584 0.830121i \(-0.688272\pi\)
−0.557584 + 0.830121i \(0.688272\pi\)
\(278\) −41.9007 −2.51304
\(279\) 0 0
\(280\) −0.530734 −0.0317174
\(281\) −25.6585 −1.53066 −0.765328 0.643640i \(-0.777423\pi\)
−0.765328 + 0.643640i \(0.777423\pi\)
\(282\) 0 0
\(283\) 30.2791 1.79991 0.899954 0.435985i \(-0.143600\pi\)
0.899954 + 0.435985i \(0.143600\pi\)
\(284\) −30.3709 −1.80218
\(285\) 0 0
\(286\) 0 0
\(287\) −2.39025 −0.141092
\(288\) 0 0
\(289\) 14.4693 0.851133
\(290\) 5.43991 0.319442
\(291\) 0 0
\(292\) 6.01182 0.351815
\(293\) −3.04965 −0.178163 −0.0890813 0.996024i \(-0.528393\pi\)
−0.0890813 + 0.996024i \(0.528393\pi\)
\(294\) 0 0
\(295\) 1.51514 0.0882147
\(296\) −0.257333 −0.0149572
\(297\) 0 0
\(298\) −22.4390 −1.29986
\(299\) −13.9394 −0.806139
\(300\) 0 0
\(301\) 1.03028 0.0593841
\(302\) 32.8704 1.89148
\(303\) 0 0
\(304\) 14.2791 0.818964
\(305\) 1.15894 0.0663609
\(306\) 0 0
\(307\) 3.71904 0.212257 0.106128 0.994352i \(-0.466155\pi\)
0.106128 + 0.994352i \(0.466155\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 7.34060 0.416918
\(311\) −4.16984 −0.236450 −0.118225 0.992987i \(-0.537720\pi\)
−0.118225 + 0.992987i \(0.537720\pi\)
\(312\) 0 0
\(313\) −11.7044 −0.661569 −0.330785 0.943706i \(-0.607313\pi\)
−0.330785 + 0.943706i \(0.607313\pi\)
\(314\) 19.3482 1.09188
\(315\) 0 0
\(316\) 22.7124 1.27767
\(317\) 2.23509 0.125535 0.0627676 0.998028i \(-0.480007\pi\)
0.0627676 + 0.998028i \(0.480007\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 5.55345 0.310447
\(321\) 0 0
\(322\) −5.28005 −0.294246
\(323\) 29.6197 1.64809
\(324\) 0 0
\(325\) −26.7299 −1.48271
\(326\) −28.2186 −1.56288
\(327\) 0 0
\(328\) −2.61639 −0.144466
\(329\) 1.60975 0.0887482
\(330\) 0 0
\(331\) 22.3250 1.22709 0.613547 0.789659i \(-0.289742\pi\)
0.613547 + 0.789659i \(0.289742\pi\)
\(332\) 8.09747 0.444407
\(333\) 0 0
\(334\) −21.3775 −1.16973
\(335\) −4.85574 −0.265297
\(336\) 0 0
\(337\) −13.2800 −0.723410 −0.361705 0.932293i \(-0.617805\pi\)
−0.361705 + 0.932293i \(0.617805\pi\)
\(338\) 39.2451 2.13465
\(339\) 0 0
\(340\) 6.84106 0.371008
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 1.12775 0.0608042
\(345\) 0 0
\(346\) 17.3600 0.933278
\(347\) 18.0294 0.967867 0.483933 0.875105i \(-0.339208\pi\)
0.483933 + 0.875105i \(0.339208\pi\)
\(348\) 0 0
\(349\) −21.9494 −1.17493 −0.587463 0.809251i \(-0.699873\pi\)
−0.587463 + 0.809251i \(0.699873\pi\)
\(350\) −10.1249 −0.541198
\(351\) 0 0
\(352\) 0 0
\(353\) −18.8557 −1.00359 −0.501795 0.864987i \(-0.667327\pi\)
−0.501795 + 0.864987i \(0.667327\pi\)
\(354\) 0 0
\(355\) 5.85482 0.310742
\(356\) 3.18257 0.168676
\(357\) 0 0
\(358\) 26.0606 1.37734
\(359\) −1.52982 −0.0807407 −0.0403703 0.999185i \(-0.512854\pi\)
−0.0403703 + 0.999185i \(0.512854\pi\)
\(360\) 0 0
\(361\) 8.87890 0.467310
\(362\) −14.9697 −0.786791
\(363\) 0 0
\(364\) 14.1093 0.739528
\(365\) −1.15894 −0.0606618
\(366\) 0 0
\(367\) 14.6547 0.764969 0.382485 0.923962i \(-0.375068\pi\)
0.382485 + 0.923962i \(0.375068\pi\)
\(368\) 6.71995 0.350302
\(369\) 0 0
\(370\) 0.242209 0.0125918
\(371\) 3.03028 0.157324
\(372\) 0 0
\(373\) 10.0606 0.520916 0.260458 0.965485i \(-0.416127\pi\)
0.260458 + 0.965485i \(0.416127\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 1.76204 0.0908705
\(377\) −29.6197 −1.52549
\(378\) 0 0
\(379\) −17.8936 −0.919131 −0.459566 0.888144i \(-0.651995\pi\)
−0.459566 + 0.888144i \(0.651995\pi\)
\(380\) 6.43899 0.330313
\(381\) 0 0
\(382\) −39.9376 −2.04339
\(383\) 18.9650 0.969068 0.484534 0.874773i \(-0.338989\pi\)
0.484534 + 0.874773i \(0.338989\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 35.0596 1.78449
\(387\) 0 0
\(388\) −22.1211 −1.12303
\(389\) −34.2645 −1.73728 −0.868638 0.495447i \(-0.835004\pi\)
−0.868638 + 0.495447i \(0.835004\pi\)
\(390\) 0 0
\(391\) 13.9394 0.704948
\(392\) 1.09461 0.0552861
\(393\) 0 0
\(394\) −52.0587 −2.62268
\(395\) −4.37844 −0.220303
\(396\) 0 0
\(397\) 16.2791 0.817026 0.408513 0.912752i \(-0.366047\pi\)
0.408513 + 0.912752i \(0.366047\pi\)
\(398\) −32.7006 −1.63913
\(399\) 0 0
\(400\) 12.8860 0.644301
\(401\) −28.9385 −1.44512 −0.722561 0.691308i \(-0.757035\pi\)
−0.722561 + 0.691308i \(0.757035\pi\)
\(402\) 0 0
\(403\) −39.9688 −1.99099
\(404\) 32.5719 1.62051
\(405\) 0 0
\(406\) −11.2195 −0.556814
\(407\) 0 0
\(408\) 0 0
\(409\) 15.5104 0.766942 0.383471 0.923553i \(-0.374729\pi\)
0.383471 + 0.923553i \(0.374729\pi\)
\(410\) 2.46262 0.121620
\(411\) 0 0
\(412\) 30.6088 1.50799
\(413\) −3.12489 −0.153766
\(414\) 0 0
\(415\) −1.56101 −0.0766270
\(416\) −44.5171 −2.18263
\(417\) 0 0
\(418\) 0 0
\(419\) 12.9503 0.632666 0.316333 0.948648i \(-0.397548\pi\)
0.316333 + 0.948648i \(0.397548\pi\)
\(420\) 0 0
\(421\) −17.0908 −0.832956 −0.416478 0.909146i \(-0.636736\pi\)
−0.416478 + 0.909146i \(0.636736\pi\)
\(422\) −30.6812 −1.49354
\(423\) 0 0
\(424\) 3.31697 0.161086
\(425\) 26.7299 1.29659
\(426\) 0 0
\(427\) −2.39025 −0.115672
\(428\) −26.5601 −1.28383
\(429\) 0 0
\(430\) −1.06147 −0.0511886
\(431\) −19.3482 −0.931968 −0.465984 0.884793i \(-0.654300\pi\)
−0.465984 + 0.884793i \(0.654300\pi\)
\(432\) 0 0
\(433\) 15.8255 0.760523 0.380262 0.924879i \(-0.375834\pi\)
0.380262 + 0.924879i \(0.375834\pi\)
\(434\) −15.1396 −0.726722
\(435\) 0 0
\(436\) −18.4626 −0.884199
\(437\) 13.1202 0.627624
\(438\) 0 0
\(439\) −11.0596 −0.527848 −0.263924 0.964544i \(-0.585017\pi\)
−0.263924 + 0.964544i \(0.585017\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −66.8686 −3.18061
\(443\) 34.7034 1.64881 0.824405 0.566000i \(-0.191510\pi\)
0.824405 + 0.566000i \(0.191510\pi\)
\(444\) 0 0
\(445\) −0.613528 −0.0290840
\(446\) 36.6618 1.73599
\(447\) 0 0
\(448\) −11.4537 −0.541135
\(449\) 36.2938 1.71281 0.856405 0.516304i \(-0.172693\pi\)
0.856405 + 0.516304i \(0.172693\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −34.6206 −1.62842
\(453\) 0 0
\(454\) −22.7787 −1.06906
\(455\) −2.71995 −0.127513
\(456\) 0 0
\(457\) −2.06055 −0.0963886 −0.0481943 0.998838i \(-0.515347\pi\)
−0.0481943 + 0.998838i \(0.515347\pi\)
\(458\) 11.5904 0.541582
\(459\) 0 0
\(460\) 3.03028 0.141287
\(461\) 7.17076 0.333975 0.166988 0.985959i \(-0.446596\pi\)
0.166988 + 0.985959i \(0.446596\pi\)
\(462\) 0 0
\(463\) 3.45459 0.160548 0.0802741 0.996773i \(-0.474420\pi\)
0.0802741 + 0.996773i \(0.474420\pi\)
\(464\) 14.2791 0.662892
\(465\) 0 0
\(466\) −63.2782 −2.93131
\(467\) 13.4958 0.624509 0.312255 0.949998i \(-0.398916\pi\)
0.312255 + 0.949998i \(0.398916\pi\)
\(468\) 0 0
\(469\) 10.0147 0.462435
\(470\) −1.65848 −0.0765002
\(471\) 0 0
\(472\) −3.42053 −0.157443
\(473\) 0 0
\(474\) 0 0
\(475\) 25.1589 1.15437
\(476\) −14.1093 −0.646698
\(477\) 0 0
\(478\) −5.43991 −0.248816
\(479\) −35.0596 −1.60192 −0.800958 0.598721i \(-0.795676\pi\)
−0.800958 + 0.598721i \(0.795676\pi\)
\(480\) 0 0
\(481\) −1.31880 −0.0601322
\(482\) 58.1992 2.65090
\(483\) 0 0
\(484\) 0 0
\(485\) 4.26445 0.193639
\(486\) 0 0
\(487\) 29.6656 1.34428 0.672138 0.740426i \(-0.265376\pi\)
0.672138 + 0.740426i \(0.265376\pi\)
\(488\) −2.61639 −0.118439
\(489\) 0 0
\(490\) −1.03028 −0.0465431
\(491\) −25.5298 −1.15214 −0.576072 0.817399i \(-0.695415\pi\)
−0.576072 + 0.817399i \(0.695415\pi\)
\(492\) 0 0
\(493\) 29.6197 1.33401
\(494\) −62.9385 −2.83174
\(495\) 0 0
\(496\) 19.2682 0.865169
\(497\) −12.0752 −0.541648
\(498\) 0 0
\(499\) −25.6585 −1.14863 −0.574316 0.818634i \(-0.694732\pi\)
−0.574316 + 0.818634i \(0.694732\pi\)
\(500\) 11.9083 0.532553
\(501\) 0 0
\(502\) −20.9192 −0.933668
\(503\) −2.56009 −0.114149 −0.0570745 0.998370i \(-0.518177\pi\)
−0.0570745 + 0.998370i \(0.518177\pi\)
\(504\) 0 0
\(505\) −6.27913 −0.279418
\(506\) 0 0
\(507\) 0 0
\(508\) 6.43899 0.285684
\(509\) −13.4546 −0.596364 −0.298182 0.954509i \(-0.596380\pi\)
−0.298182 + 0.954509i \(0.596380\pi\)
\(510\) 0 0
\(511\) 2.39025 0.105739
\(512\) 27.3813 1.21009
\(513\) 0 0
\(514\) 47.8089 2.10876
\(515\) −5.90069 −0.260016
\(516\) 0 0
\(517\) 0 0
\(518\) −0.499542 −0.0219486
\(519\) 0 0
\(520\) −2.97729 −0.130563
\(521\) 32.2333 1.41216 0.706082 0.708130i \(-0.250461\pi\)
0.706082 + 0.708130i \(0.250461\pi\)
\(522\) 0 0
\(523\) −32.3397 −1.41412 −0.707058 0.707156i \(-0.749978\pi\)
−0.707058 + 0.707156i \(0.749978\pi\)
\(524\) 6.84106 0.298853
\(525\) 0 0
\(526\) 33.9982 1.48239
\(527\) 39.9688 1.74107
\(528\) 0 0
\(529\) −16.8255 −0.731542
\(530\) −3.12202 −0.135612
\(531\) 0 0
\(532\) −13.2800 −0.575763
\(533\) −13.4087 −0.580796
\(534\) 0 0
\(535\) 5.12019 0.221365
\(536\) 10.9622 0.473493
\(537\) 0 0
\(538\) −19.5904 −0.844601
\(539\) 0 0
\(540\) 0 0
\(541\) −23.5005 −1.01036 −0.505182 0.863013i \(-0.668575\pi\)
−0.505182 + 0.863013i \(0.668575\pi\)
\(542\) 22.4390 0.963837
\(543\) 0 0
\(544\) 44.5171 1.90865
\(545\) 3.55918 0.152458
\(546\) 0 0
\(547\) −4.12110 −0.176206 −0.0881028 0.996111i \(-0.528080\pi\)
−0.0881028 + 0.996111i \(0.528080\pi\)
\(548\) 24.4078 1.04265
\(549\) 0 0
\(550\) 0 0
\(551\) 27.8789 1.18768
\(552\) 0 0
\(553\) 9.03028 0.384006
\(554\) −39.4381 −1.67556
\(555\) 0 0
\(556\) −49.5961 −2.10334
\(557\) 13.9394 0.590633 0.295317 0.955399i \(-0.404575\pi\)
0.295317 + 0.955399i \(0.404575\pi\)
\(558\) 0 0
\(559\) 5.77959 0.244451
\(560\) 1.31124 0.0554100
\(561\) 0 0
\(562\) −54.5213 −2.29984
\(563\) −2.71995 −0.114632 −0.0573162 0.998356i \(-0.518254\pi\)
−0.0573162 + 0.998356i \(0.518254\pi\)
\(564\) 0 0
\(565\) 6.67408 0.280781
\(566\) 64.3397 2.70440
\(567\) 0 0
\(568\) −13.2177 −0.554601
\(569\) 27.7190 1.16204 0.581021 0.813888i \(-0.302653\pi\)
0.581021 + 0.813888i \(0.302653\pi\)
\(570\) 0 0
\(571\) 38.1193 1.59524 0.797621 0.603159i \(-0.206092\pi\)
0.797621 + 0.603159i \(0.206092\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −5.07901 −0.211994
\(575\) 11.8401 0.493768
\(576\) 0 0
\(577\) 23.2048 0.966029 0.483015 0.875612i \(-0.339542\pi\)
0.483015 + 0.875612i \(0.339542\pi\)
\(578\) 30.7455 1.27885
\(579\) 0 0
\(580\) 6.43899 0.267364
\(581\) 3.21949 0.133567
\(582\) 0 0
\(583\) 0 0
\(584\) 2.61639 0.108267
\(585\) 0 0
\(586\) −6.48016 −0.267693
\(587\) 11.0497 0.456068 0.228034 0.973653i \(-0.426770\pi\)
0.228034 + 0.973653i \(0.426770\pi\)
\(588\) 0 0
\(589\) 37.6197 1.55009
\(590\) 3.21949 0.132545
\(591\) 0 0
\(592\) 0.635770 0.0261300
\(593\) 36.3884 1.49429 0.747147 0.664659i \(-0.231423\pi\)
0.747147 + 0.664659i \(0.231423\pi\)
\(594\) 0 0
\(595\) 2.71995 0.111507
\(596\) −26.5601 −1.08794
\(597\) 0 0
\(598\) −29.6197 −1.21124
\(599\) −29.6585 −1.21181 −0.605906 0.795536i \(-0.707189\pi\)
−0.605906 + 0.795536i \(0.707189\pi\)
\(600\) 0 0
\(601\) 1.39117 0.0567470 0.0283735 0.999597i \(-0.490967\pi\)
0.0283735 + 0.999597i \(0.490967\pi\)
\(602\) 2.18922 0.0892259
\(603\) 0 0
\(604\) 38.9073 1.58312
\(605\) 0 0
\(606\) 0 0
\(607\) 30.9385 1.25576 0.627878 0.778312i \(-0.283924\pi\)
0.627878 + 0.778312i \(0.283924\pi\)
\(608\) 41.9007 1.69930
\(609\) 0 0
\(610\) 2.46262 0.0997086
\(611\) 9.03028 0.365326
\(612\) 0 0
\(613\) 8.15986 0.329574 0.164787 0.986329i \(-0.447306\pi\)
0.164787 + 0.986329i \(0.447306\pi\)
\(614\) 7.90253 0.318920
\(615\) 0 0
\(616\) 0 0
\(617\) 27.5298 1.10831 0.554154 0.832414i \(-0.313042\pi\)
0.554154 + 0.832414i \(0.313042\pi\)
\(618\) 0 0
\(619\) 19.9735 0.802803 0.401401 0.915902i \(-0.368523\pi\)
0.401401 + 0.915902i \(0.368523\pi\)
\(620\) 8.68876 0.348949
\(621\) 0 0
\(622\) −8.86043 −0.355271
\(623\) 1.26537 0.0506959
\(624\) 0 0
\(625\) 21.5289 0.861156
\(626\) −24.8704 −0.994022
\(627\) 0 0
\(628\) 22.9016 0.913874
\(629\) 1.31880 0.0525841
\(630\) 0 0
\(631\) −17.2342 −0.686082 −0.343041 0.939320i \(-0.611457\pi\)
−0.343041 + 0.939320i \(0.611457\pi\)
\(632\) 9.88462 0.393189
\(633\) 0 0
\(634\) 4.74931 0.188619
\(635\) −1.24129 −0.0492592
\(636\) 0 0
\(637\) 5.60975 0.222266
\(638\) 0 0
\(639\) 0 0
\(640\) 4.10504 0.162266
\(641\) 44.4149 1.75428 0.877142 0.480231i \(-0.159447\pi\)
0.877142 + 0.480231i \(0.159447\pi\)
\(642\) 0 0
\(643\) −22.5336 −0.888638 −0.444319 0.895869i \(-0.646554\pi\)
−0.444319 + 0.895869i \(0.646554\pi\)
\(644\) −6.24977 −0.246275
\(645\) 0 0
\(646\) 62.9385 2.47628
\(647\) 18.7758 0.738153 0.369077 0.929399i \(-0.379674\pi\)
0.369077 + 0.929399i \(0.379674\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −56.7980 −2.22780
\(651\) 0 0
\(652\) −33.4012 −1.30809
\(653\) −31.6126 −1.23710 −0.618549 0.785747i \(-0.712279\pi\)
−0.618549 + 0.785747i \(0.712279\pi\)
\(654\) 0 0
\(655\) −1.31880 −0.0515298
\(656\) 6.46410 0.252381
\(657\) 0 0
\(658\) 3.42053 0.133346
\(659\) −19.8477 −0.773157 −0.386578 0.922257i \(-0.626343\pi\)
−0.386578 + 0.922257i \(0.626343\pi\)
\(660\) 0 0
\(661\) 31.4234 1.22223 0.611114 0.791542i \(-0.290722\pi\)
0.611114 + 0.791542i \(0.290722\pi\)
\(662\) 47.4381 1.84373
\(663\) 0 0
\(664\) 3.52409 0.136761
\(665\) 2.56009 0.0992762
\(666\) 0 0
\(667\) 13.1202 0.508016
\(668\) −25.3037 −0.979029
\(669\) 0 0
\(670\) −10.3179 −0.398615
\(671\) 0 0
\(672\) 0 0
\(673\) −19.2195 −0.740857 −0.370429 0.928861i \(-0.620789\pi\)
−0.370429 + 0.928861i \(0.620789\pi\)
\(674\) −28.2186 −1.08694
\(675\) 0 0
\(676\) 46.4528 1.78664
\(677\) −25.7309 −0.988917 −0.494458 0.869201i \(-0.664634\pi\)
−0.494458 + 0.869201i \(0.664634\pi\)
\(678\) 0 0
\(679\) −8.79518 −0.337528
\(680\) 2.97729 0.114174
\(681\) 0 0
\(682\) 0 0
\(683\) 11.0596 0.423185 0.211593 0.977358i \(-0.432135\pi\)
0.211593 + 0.977358i \(0.432135\pi\)
\(684\) 0 0
\(685\) −4.70527 −0.179779
\(686\) 2.12489 0.0811285
\(687\) 0 0
\(688\) −2.78623 −0.106224
\(689\) 16.9991 0.647613
\(690\) 0 0
\(691\) 45.0256 1.71285 0.856427 0.516268i \(-0.172679\pi\)
0.856427 + 0.516268i \(0.172679\pi\)
\(692\) 20.5483 0.781128
\(693\) 0 0
\(694\) 38.3103 1.45424
\(695\) 9.56101 0.362670
\(696\) 0 0
\(697\) 13.4087 0.507891
\(698\) −46.6400 −1.76535
\(699\) 0 0
\(700\) −11.9844 −0.452968
\(701\) −46.7787 −1.76681 −0.883403 0.468614i \(-0.844754\pi\)
−0.883403 + 0.468614i \(0.844754\pi\)
\(702\) 0 0
\(703\) 1.24129 0.0468162
\(704\) 0 0
\(705\) 0 0
\(706\) −40.0663 −1.50791
\(707\) 12.9503 0.487048
\(708\) 0 0
\(709\) −9.14426 −0.343420 −0.171710 0.985148i \(-0.554929\pi\)
−0.171710 + 0.985148i \(0.554929\pi\)
\(710\) 12.4408 0.466896
\(711\) 0 0
\(712\) 1.38508 0.0519082
\(713\) 17.7044 0.663033
\(714\) 0 0
\(715\) 0 0
\(716\) 30.8468 1.15280
\(717\) 0 0
\(718\) −3.25069 −0.121315
\(719\) 30.4655 1.13617 0.568085 0.822970i \(-0.307684\pi\)
0.568085 + 0.822970i \(0.307684\pi\)
\(720\) 0 0
\(721\) 12.1698 0.453229
\(722\) 18.8666 0.702143
\(723\) 0 0
\(724\) −17.7190 −0.658523
\(725\) 25.1589 0.934380
\(726\) 0 0
\(727\) −6.12580 −0.227193 −0.113597 0.993527i \(-0.536237\pi\)
−0.113597 + 0.993527i \(0.536237\pi\)
\(728\) 6.14048 0.227581
\(729\) 0 0
\(730\) −2.46262 −0.0911457
\(731\) −5.77959 −0.213766
\(732\) 0 0
\(733\) −42.0705 −1.55391 −0.776955 0.629556i \(-0.783237\pi\)
−0.776955 + 0.629556i \(0.783237\pi\)
\(734\) 31.1396 1.14938
\(735\) 0 0
\(736\) 19.7190 0.726853
\(737\) 0 0
\(738\) 0 0
\(739\) −28.8780 −1.06229 −0.531147 0.847280i \(-0.678239\pi\)
−0.531147 + 0.847280i \(0.678239\pi\)
\(740\) 0.286692 0.0105390
\(741\) 0 0
\(742\) 6.43899 0.236383
\(743\) 14.6812 0.538601 0.269300 0.963056i \(-0.413208\pi\)
0.269300 + 0.963056i \(0.413208\pi\)
\(744\) 0 0
\(745\) 5.12019 0.187589
\(746\) 21.3775 0.782687
\(747\) 0 0
\(748\) 0 0
\(749\) −10.5601 −0.385857
\(750\) 0 0
\(751\) −6.60597 −0.241055 −0.120528 0.992710i \(-0.538459\pi\)
−0.120528 + 0.992710i \(0.538459\pi\)
\(752\) −4.35333 −0.158750
\(753\) 0 0
\(754\) −62.9385 −2.29209
\(755\) −7.50046 −0.272970
\(756\) 0 0
\(757\) 2.49954 0.0908474 0.0454237 0.998968i \(-0.485536\pi\)
0.0454237 + 0.998968i \(0.485536\pi\)
\(758\) −38.0218 −1.38101
\(759\) 0 0
\(760\) 2.80230 0.101650
\(761\) 12.0487 0.436766 0.218383 0.975863i \(-0.429922\pi\)
0.218383 + 0.975863i \(0.429922\pi\)
\(762\) 0 0
\(763\) −7.34060 −0.265748
\(764\) −47.2725 −1.71026
\(765\) 0 0
\(766\) 40.2985 1.45604
\(767\) −17.5298 −0.632965
\(768\) 0 0
\(769\) 0.489560 0.0176540 0.00882699 0.999961i \(-0.497190\pi\)
0.00882699 + 0.999961i \(0.497190\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 41.4986 1.49357
\(773\) 46.7181 1.68033 0.840167 0.542328i \(-0.182457\pi\)
0.840167 + 0.542328i \(0.182457\pi\)
\(774\) 0 0
\(775\) 33.9494 1.21950
\(776\) −9.62729 −0.345600
\(777\) 0 0
\(778\) −72.8080 −2.61029
\(779\) 12.6206 0.452182
\(780\) 0 0
\(781\) 0 0
\(782\) 29.6197 1.05920
\(783\) 0 0
\(784\) −2.70436 −0.0965842
\(785\) −4.41491 −0.157575
\(786\) 0 0
\(787\) −8.33968 −0.297278 −0.148639 0.988892i \(-0.547489\pi\)
−0.148639 + 0.988892i \(0.547489\pi\)
\(788\) −61.6197 −2.19511
\(789\) 0 0
\(790\) −9.30368 −0.331010
\(791\) −13.7649 −0.489424
\(792\) 0 0
\(793\) −13.4087 −0.476157
\(794\) 34.5913 1.22760
\(795\) 0 0
\(796\) −38.7063 −1.37191
\(797\) 31.5151 1.11632 0.558162 0.829732i \(-0.311507\pi\)
0.558162 + 0.829732i \(0.311507\pi\)
\(798\) 0 0
\(799\) −9.03028 −0.319468
\(800\) 37.8127 1.33688
\(801\) 0 0
\(802\) −61.4911 −2.17132
\(803\) 0 0
\(804\) 0 0
\(805\) 1.20482 0.0424641
\(806\) −84.9291 −2.99150
\(807\) 0 0
\(808\) 14.1756 0.498695
\(809\) −15.5005 −0.544967 −0.272484 0.962160i \(-0.587845\pi\)
−0.272484 + 0.962160i \(0.587845\pi\)
\(810\) 0 0
\(811\) −48.3397 −1.69744 −0.848718 0.528846i \(-0.822625\pi\)
−0.848718 + 0.528846i \(0.822625\pi\)
\(812\) −13.2800 −0.466038
\(813\) 0 0
\(814\) 0 0
\(815\) 6.43899 0.225548
\(816\) 0 0
\(817\) −5.43991 −0.190318
\(818\) 32.9579 1.15235
\(819\) 0 0
\(820\) 2.91490 0.101793
\(821\) 10.5601 0.368550 0.184275 0.982875i \(-0.441006\pi\)
0.184275 + 0.982875i \(0.441006\pi\)
\(822\) 0 0
\(823\) 19.3241 0.673595 0.336798 0.941577i \(-0.390656\pi\)
0.336798 + 0.941577i \(0.390656\pi\)
\(824\) 13.3212 0.464067
\(825\) 0 0
\(826\) −6.64002 −0.231036
\(827\) 0.530734 0.0184554 0.00922772 0.999957i \(-0.497063\pi\)
0.00922772 + 0.999957i \(0.497063\pi\)
\(828\) 0 0
\(829\) −47.1954 −1.63916 −0.819582 0.572961i \(-0.805794\pi\)
−0.819582 + 0.572961i \(0.805794\pi\)
\(830\) −3.31697 −0.115134
\(831\) 0 0
\(832\) −64.2522 −2.22754
\(833\) −5.60975 −0.194366
\(834\) 0 0
\(835\) 4.87798 0.168809
\(836\) 0 0
\(837\) 0 0
\(838\) 27.5180 0.950594
\(839\) 37.9036 1.30858 0.654288 0.756245i \(-0.272968\pi\)
0.654288 + 0.756245i \(0.272968\pi\)
\(840\) 0 0
\(841\) −1.12110 −0.0386588
\(842\) −36.3161 −1.25153
\(843\) 0 0
\(844\) −36.3161 −1.25005
\(845\) −8.95504 −0.308063
\(846\) 0 0
\(847\) 0 0
\(848\) −8.19495 −0.281416
\(849\) 0 0
\(850\) 56.7980 1.94816
\(851\) 0.584169 0.0200251
\(852\) 0 0
\(853\) −6.26915 −0.214652 −0.107326 0.994224i \(-0.534229\pi\)
−0.107326 + 0.994224i \(0.534229\pi\)
\(854\) −5.07901 −0.173800
\(855\) 0 0
\(856\) −11.5592 −0.395085
\(857\) −36.9503 −1.26220 −0.631100 0.775702i \(-0.717396\pi\)
−0.631100 + 0.775702i \(0.717396\pi\)
\(858\) 0 0
\(859\) 8.90447 0.303817 0.151908 0.988395i \(-0.451458\pi\)
0.151908 + 0.988395i \(0.451458\pi\)
\(860\) −1.25642 −0.0428434
\(861\) 0 0
\(862\) −41.1126 −1.40030
\(863\) 42.1193 1.43376 0.716878 0.697198i \(-0.245570\pi\)
0.716878 + 0.697198i \(0.245570\pi\)
\(864\) 0 0
\(865\) −3.96125 −0.134686
\(866\) 33.6273 1.14270
\(867\) 0 0
\(868\) −17.9201 −0.608247
\(869\) 0 0
\(870\) 0 0
\(871\) 56.1798 1.90358
\(872\) −8.03509 −0.272102
\(873\) 0 0
\(874\) 27.8789 0.943018
\(875\) 4.73463 0.160060
\(876\) 0 0
\(877\) 24.3397 0.821893 0.410946 0.911660i \(-0.365198\pi\)
0.410946 + 0.911660i \(0.365198\pi\)
\(878\) −23.5005 −0.793102
\(879\) 0 0
\(880\) 0 0
\(881\) −5.64380 −0.190145 −0.0950723 0.995470i \(-0.530308\pi\)
−0.0950723 + 0.995470i \(0.530308\pi\)
\(882\) 0 0
\(883\) 15.1807 0.510873 0.255436 0.966826i \(-0.417781\pi\)
0.255436 + 0.966826i \(0.417781\pi\)
\(884\) −79.1495 −2.66209
\(885\) 0 0
\(886\) 73.7408 2.47737
\(887\) −26.8174 −0.900441 −0.450221 0.892917i \(-0.648655\pi\)
−0.450221 + 0.892917i \(0.648655\pi\)
\(888\) 0 0
\(889\) 2.56009 0.0858628
\(890\) −1.30368 −0.0436994
\(891\) 0 0
\(892\) 43.3951 1.45297
\(893\) −8.49954 −0.284426
\(894\) 0 0
\(895\) −5.94657 −0.198772
\(896\) −8.46640 −0.282843
\(897\) 0 0
\(898\) 77.1202 2.57353
\(899\) 37.6197 1.25469
\(900\) 0 0
\(901\) −16.9991 −0.566322
\(902\) 0 0
\(903\) 0 0
\(904\) −15.0672 −0.501128
\(905\) 3.41583 0.113546
\(906\) 0 0
\(907\) 32.0975 1.06578 0.532890 0.846185i \(-0.321106\pi\)
0.532890 + 0.846185i \(0.321106\pi\)
\(908\) −26.9622 −0.894771
\(909\) 0 0
\(910\) −5.77959 −0.191591
\(911\) −12.1599 −0.402874 −0.201437 0.979501i \(-0.564561\pi\)
−0.201437 + 0.979501i \(0.564561\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −4.37844 −0.144826
\(915\) 0 0
\(916\) 13.7190 0.453290
\(917\) 2.71995 0.0898208
\(918\) 0 0
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) 1.31880 0.0434796
\(921\) 0 0
\(922\) 15.2370 0.501805
\(923\) −67.7390 −2.22966
\(924\) 0 0
\(925\) 1.12019 0.0368315
\(926\) 7.34060 0.241227
\(927\) 0 0
\(928\) 41.9007 1.37546
\(929\) 21.3794 0.701434 0.350717 0.936482i \(-0.385938\pi\)
0.350717 + 0.936482i \(0.385938\pi\)
\(930\) 0 0
\(931\) −5.28005 −0.173047
\(932\) −74.8998 −2.45342
\(933\) 0 0
\(934\) 28.6769 0.938338
\(935\) 0 0
\(936\) 0 0
\(937\) −31.5104 −1.02940 −0.514701 0.857370i \(-0.672097\pi\)
−0.514701 + 0.857370i \(0.672097\pi\)
\(938\) 21.2800 0.694818
\(939\) 0 0
\(940\) −1.96308 −0.0640286
\(941\) −42.7299 −1.39296 −0.696478 0.717578i \(-0.745251\pi\)
−0.696478 + 0.717578i \(0.745251\pi\)
\(942\) 0 0
\(943\) 5.93945 0.193415
\(944\) 8.45080 0.275050
\(945\) 0 0
\(946\) 0 0
\(947\) −3.29473 −0.107064 −0.0535321 0.998566i \(-0.517048\pi\)
−0.0535321 + 0.998566i \(0.517048\pi\)
\(948\) 0 0
\(949\) 13.4087 0.435265
\(950\) 53.4599 1.73447
\(951\) 0 0
\(952\) −6.14048 −0.199014
\(953\) −38.6812 −1.25301 −0.626503 0.779419i \(-0.715515\pi\)
−0.626503 + 0.779419i \(0.715515\pi\)
\(954\) 0 0
\(955\) 9.11307 0.294892
\(956\) −6.43899 −0.208252
\(957\) 0 0
\(958\) −74.4977 −2.40691
\(959\) 9.70436 0.313370
\(960\) 0 0
\(961\) 19.7640 0.637548
\(962\) −2.80230 −0.0903499
\(963\) 0 0
\(964\) 68.8880 2.21873
\(965\) −8.00000 −0.257529
\(966\) 0 0
\(967\) −3.34816 −0.107670 −0.0538348 0.998550i \(-0.517144\pi\)
−0.0538348 + 0.998550i \(0.517144\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 9.06147 0.290946
\(971\) −58.7134 −1.88420 −0.942102 0.335327i \(-0.891153\pi\)
−0.942102 + 0.335327i \(0.891153\pi\)
\(972\) 0 0
\(973\) −19.7190 −0.632163
\(974\) 63.0360 2.01980
\(975\) 0 0
\(976\) 6.46410 0.206911
\(977\) −13.7649 −0.440378 −0.220189 0.975457i \(-0.570667\pi\)
−0.220189 + 0.975457i \(0.570667\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −1.21949 −0.0389553
\(981\) 0 0
\(982\) −54.2479 −1.73112
\(983\) −50.2139 −1.60157 −0.800787 0.598949i \(-0.795585\pi\)
−0.800787 + 0.598949i \(0.795585\pi\)
\(984\) 0 0
\(985\) 11.8789 0.378493
\(986\) 62.9385 2.00437
\(987\) 0 0
\(988\) −74.4977 −2.37009
\(989\) −2.56009 −0.0814062
\(990\) 0 0
\(991\) −4.65940 −0.148011 −0.0740054 0.997258i \(-0.523578\pi\)
−0.0740054 + 0.997258i \(0.523578\pi\)
\(992\) 56.5407 1.79517
\(993\) 0 0
\(994\) −25.6585 −0.813838
\(995\) 7.46170 0.236552
\(996\) 0 0
\(997\) 3.38934 0.107341 0.0536707 0.998559i \(-0.482908\pi\)
0.0536707 + 0.998559i \(0.482908\pi\)
\(998\) −54.5213 −1.72584
\(999\) 0 0
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 7623.2.a.bz.1.3 3
3.2 odd 2 847.2.a.j.1.1 yes 3
11.10 odd 2 7623.2.a.ce.1.1 3
21.20 even 2 5929.2.a.y.1.1 3
33.2 even 10 847.2.f.u.323.1 12
33.5 odd 10 847.2.f.t.729.3 12
33.8 even 10 847.2.f.u.372.3 12
33.14 odd 10 847.2.f.t.372.1 12
33.17 even 10 847.2.f.u.729.1 12
33.20 odd 10 847.2.f.t.323.3 12
33.26 odd 10 847.2.f.t.148.1 12
33.29 even 10 847.2.f.u.148.3 12
33.32 even 2 847.2.a.i.1.3 3
231.230 odd 2 5929.2.a.t.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
847.2.a.i.1.3 3 33.32 even 2
847.2.a.j.1.1 yes 3 3.2 odd 2
847.2.f.t.148.1 12 33.26 odd 10
847.2.f.t.323.3 12 33.20 odd 10
847.2.f.t.372.1 12 33.14 odd 10
847.2.f.t.729.3 12 33.5 odd 10
847.2.f.u.148.3 12 33.29 even 10
847.2.f.u.323.1 12 33.2 even 10
847.2.f.u.372.3 12 33.8 even 10
847.2.f.u.729.1 12 33.17 even 10
5929.2.a.t.1.3 3 231.230 odd 2
5929.2.a.y.1.1 3 21.20 even 2
7623.2.a.bz.1.3 3 1.1 even 1 trivial
7623.2.a.ce.1.1 3 11.10 odd 2