Properties

Label 7623.2.a.cp.1.6
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 1
Dimension 6
CM no
Inner twists 1

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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: \(6\)
Coefficient field: 6.6.7674048.1
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 1.6
Root \(2.38595\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.38595 q^{2} -0.0791355 q^{4} +0.133004 q^{5} +1.00000 q^{7} -2.88158 q^{8} +O(q^{10})\) \(q+1.38595 q^{2} -0.0791355 q^{4} +0.133004 q^{5} +1.00000 q^{7} -2.88158 q^{8} +0.184338 q^{10} -0.641436 q^{13} +1.38595 q^{14} -3.83547 q^{16} +1.42482 q^{17} +7.18297 q^{19} -0.0105254 q^{20} -1.66655 q^{23} -4.98231 q^{25} -0.889000 q^{26} -0.0791355 q^{28} -4.47657 q^{29} -6.83182 q^{31} +0.447392 q^{32} +1.97473 q^{34} +0.133004 q^{35} +3.76864 q^{37} +9.95526 q^{38} -0.383263 q^{40} -6.17286 q^{41} -1.03970 q^{43} -2.30976 q^{46} -9.48937 q^{47} +1.00000 q^{49} -6.90524 q^{50} +0.0507604 q^{52} +0.666549 q^{53} -2.88158 q^{56} -6.20431 q^{58} -8.18695 q^{59} +9.49807 q^{61} -9.46858 q^{62} +8.29100 q^{64} -0.0853139 q^{65} +12.0398 q^{67} -0.112754 q^{68} +0.184338 q^{70} -4.83418 q^{71} -8.85748 q^{73} +5.22316 q^{74} -0.568428 q^{76} +11.4907 q^{79} -0.510134 q^{80} -8.55529 q^{82} -9.57998 q^{83} +0.189507 q^{85} -1.44098 q^{86} +17.7001 q^{89} -0.641436 q^{91} +0.131883 q^{92} -13.1518 q^{94} +0.955367 q^{95} +6.58139 q^{97} +1.38595 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 4q^{2} + 4q^{4} + 4q^{5} + 6q^{7} - 12q^{8} + O(q^{10}) \) \( 6q - 4q^{2} + 4q^{4} + 4q^{5} + 6q^{7} - 12q^{8} - 8q^{10} + 4q^{13} - 4q^{14} + 8q^{16} - 22q^{17} + 6q^{19} - 2q^{20} - 2q^{23} + 4q^{25} - 6q^{26} + 4q^{28} - 12q^{29} - 2q^{31} - 8q^{32} + 24q^{34} + 4q^{35} + 14q^{37} + 22q^{38} + 18q^{40} - 26q^{41} - 4q^{43} + 12q^{46} + 16q^{47} + 6q^{49} + 4q^{50} + 12q^{52} - 4q^{53} - 12q^{56} - 2q^{58} + 4q^{59} - 8q^{61} - 20q^{62} + 26q^{64} - 24q^{65} + 6q^{67} - 12q^{68} - 8q^{70} - 22q^{71} + 14q^{73} - 44q^{74} - 30q^{76} - 28q^{79} + 4q^{80} - 4q^{82} - 22q^{83} - 24q^{85} + 30q^{86} + 4q^{91} - 10q^{92} - 38q^{94} + 24q^{95} - 4q^{97} - 4q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.38595 0.980016 0.490008 0.871718i \(-0.336994\pi\)
0.490008 + 0.871718i \(0.336994\pi\)
\(3\) 0 0
\(4\) −0.0791355 −0.0395678
\(5\) 0.133004 0.0594814 0.0297407 0.999558i \(-0.490532\pi\)
0.0297407 + 0.999558i \(0.490532\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −2.88158 −1.01879
\(9\) 0 0
\(10\) 0.184338 0.0582928
\(11\) 0 0
\(12\) 0 0
\(13\) −0.641436 −0.177902 −0.0889512 0.996036i \(-0.528352\pi\)
−0.0889512 + 0.996036i \(0.528352\pi\)
\(14\) 1.38595 0.370411
\(15\) 0 0
\(16\) −3.83547 −0.958867
\(17\) 1.42482 0.345570 0.172785 0.984960i \(-0.444723\pi\)
0.172785 + 0.984960i \(0.444723\pi\)
\(18\) 0 0
\(19\) 7.18297 1.64789 0.823943 0.566672i \(-0.191769\pi\)
0.823943 + 0.566672i \(0.191769\pi\)
\(20\) −0.0105254 −0.00235355
\(21\) 0 0
\(22\) 0 0
\(23\) −1.66655 −0.347500 −0.173750 0.984790i \(-0.555588\pi\)
−0.173750 + 0.984790i \(0.555588\pi\)
\(24\) 0 0
\(25\) −4.98231 −0.996462
\(26\) −0.889000 −0.174347
\(27\) 0 0
\(28\) −0.0791355 −0.0149552
\(29\) −4.47657 −0.831278 −0.415639 0.909530i \(-0.636442\pi\)
−0.415639 + 0.909530i \(0.636442\pi\)
\(30\) 0 0
\(31\) −6.83182 −1.22703 −0.613516 0.789682i \(-0.710245\pi\)
−0.613516 + 0.789682i \(0.710245\pi\)
\(32\) 0.447392 0.0790885
\(33\) 0 0
\(34\) 1.97473 0.338664
\(35\) 0.133004 0.0224819
\(36\) 0 0
\(37\) 3.76864 0.619561 0.309781 0.950808i \(-0.399744\pi\)
0.309781 + 0.950808i \(0.399744\pi\)
\(38\) 9.95526 1.61496
\(39\) 0 0
\(40\) −0.383263 −0.0605993
\(41\) −6.17286 −0.964039 −0.482019 0.876161i \(-0.660096\pi\)
−0.482019 + 0.876161i \(0.660096\pi\)
\(42\) 0 0
\(43\) −1.03970 −0.158553 −0.0792764 0.996853i \(-0.525261\pi\)
−0.0792764 + 0.996853i \(0.525261\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −2.30976 −0.340555
\(47\) −9.48937 −1.38417 −0.692084 0.721817i \(-0.743307\pi\)
−0.692084 + 0.721817i \(0.743307\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −6.90524 −0.976549
\(51\) 0 0
\(52\) 0.0507604 0.00703920
\(53\) 0.666549 0.0915576 0.0457788 0.998952i \(-0.485423\pi\)
0.0457788 + 0.998952i \(0.485423\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.88158 −0.385068
\(57\) 0 0
\(58\) −6.20431 −0.814666
\(59\) −8.18695 −1.06585 −0.532925 0.846162i \(-0.678907\pi\)
−0.532925 + 0.846162i \(0.678907\pi\)
\(60\) 0 0
\(61\) 9.49807 1.21610 0.608051 0.793898i \(-0.291951\pi\)
0.608051 + 0.793898i \(0.291951\pi\)
\(62\) −9.46858 −1.20251
\(63\) 0 0
\(64\) 8.29100 1.03637
\(65\) −0.0853139 −0.0105819
\(66\) 0 0
\(67\) 12.0398 1.47089 0.735446 0.677583i \(-0.236973\pi\)
0.735446 + 0.677583i \(0.236973\pi\)
\(68\) −0.112754 −0.0136734
\(69\) 0 0
\(70\) 0.184338 0.0220326
\(71\) −4.83418 −0.573711 −0.286856 0.957974i \(-0.592610\pi\)
−0.286856 + 0.957974i \(0.592610\pi\)
\(72\) 0 0
\(73\) −8.85748 −1.03669 −0.518345 0.855172i \(-0.673452\pi\)
−0.518345 + 0.855172i \(0.673452\pi\)
\(74\) 5.22316 0.607180
\(75\) 0 0
\(76\) −0.568428 −0.0652032
\(77\) 0 0
\(78\) 0 0
\(79\) 11.4907 1.29280 0.646400 0.762998i \(-0.276274\pi\)
0.646400 + 0.762998i \(0.276274\pi\)
\(80\) −0.510134 −0.0570347
\(81\) 0 0
\(82\) −8.55529 −0.944774
\(83\) −9.57998 −1.05154 −0.525770 0.850627i \(-0.676223\pi\)
−0.525770 + 0.850627i \(0.676223\pi\)
\(84\) 0 0
\(85\) 0.189507 0.0205550
\(86\) −1.44098 −0.155384
\(87\) 0 0
\(88\) 0 0
\(89\) 17.7001 1.87621 0.938104 0.346353i \(-0.112580\pi\)
0.938104 + 0.346353i \(0.112580\pi\)
\(90\) 0 0
\(91\) −0.641436 −0.0672408
\(92\) 0.131883 0.0137498
\(93\) 0 0
\(94\) −13.1518 −1.35651
\(95\) 0.955367 0.0980186
\(96\) 0 0
\(97\) 6.58139 0.668239 0.334119 0.942531i \(-0.391561\pi\)
0.334119 + 0.942531i \(0.391561\pi\)
\(98\) 1.38595 0.140002
\(99\) 0 0
\(100\) 0.394278 0.0394278
\(101\) −18.6204 −1.85280 −0.926400 0.376541i \(-0.877113\pi\)
−0.926400 + 0.376541i \(0.877113\pi\)
\(102\) 0 0
\(103\) −6.92356 −0.682199 −0.341099 0.940027i \(-0.610799\pi\)
−0.341099 + 0.940027i \(0.610799\pi\)
\(104\) 1.84835 0.181246
\(105\) 0 0
\(106\) 0.923806 0.0897279
\(107\) 5.44446 0.526336 0.263168 0.964750i \(-0.415233\pi\)
0.263168 + 0.964750i \(0.415233\pi\)
\(108\) 0 0
\(109\) −9.22316 −0.883419 −0.441709 0.897158i \(-0.645628\pi\)
−0.441709 + 0.897158i \(0.645628\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −3.83547 −0.362418
\(113\) −9.38223 −0.882606 −0.441303 0.897358i \(-0.645484\pi\)
−0.441303 + 0.897358i \(0.645484\pi\)
\(114\) 0 0
\(115\) −0.221659 −0.0206698
\(116\) 0.354256 0.0328918
\(117\) 0 0
\(118\) −11.3467 −1.04455
\(119\) 1.42482 0.130613
\(120\) 0 0
\(121\) 0 0
\(122\) 13.1639 1.19180
\(123\) 0 0
\(124\) 0.540640 0.0485509
\(125\) −1.32769 −0.118752
\(126\) 0 0
\(127\) 15.4560 1.37150 0.685749 0.727838i \(-0.259475\pi\)
0.685749 + 0.727838i \(0.259475\pi\)
\(128\) 10.5961 0.936576
\(129\) 0 0
\(130\) −0.118241 −0.0103704
\(131\) −17.6675 −1.54362 −0.771810 0.635854i \(-0.780648\pi\)
−0.771810 + 0.635854i \(0.780648\pi\)
\(132\) 0 0
\(133\) 7.18297 0.622843
\(134\) 16.6866 1.44150
\(135\) 0 0
\(136\) −4.10574 −0.352064
\(137\) −19.0323 −1.62604 −0.813021 0.582235i \(-0.802178\pi\)
−0.813021 + 0.582235i \(0.802178\pi\)
\(138\) 0 0
\(139\) −14.0665 −1.19310 −0.596552 0.802574i \(-0.703463\pi\)
−0.596552 + 0.802574i \(0.703463\pi\)
\(140\) −0.0105254 −0.000889557 0
\(141\) 0 0
\(142\) −6.69994 −0.562247
\(143\) 0 0
\(144\) 0 0
\(145\) −0.595404 −0.0494456
\(146\) −12.2760 −1.01597
\(147\) 0 0
\(148\) −0.298234 −0.0245147
\(149\) 11.0367 0.904160 0.452080 0.891978i \(-0.350682\pi\)
0.452080 + 0.891978i \(0.350682\pi\)
\(150\) 0 0
\(151\) −16.1350 −1.31305 −0.656523 0.754306i \(-0.727974\pi\)
−0.656523 + 0.754306i \(0.727974\pi\)
\(152\) −20.6983 −1.67886
\(153\) 0 0
\(154\) 0 0
\(155\) −0.908663 −0.0729856
\(156\) 0 0
\(157\) −14.8699 −1.18674 −0.593372 0.804928i \(-0.702204\pi\)
−0.593372 + 0.804928i \(0.702204\pi\)
\(158\) 15.9255 1.26697
\(159\) 0 0
\(160\) 0.0595051 0.00470429
\(161\) −1.66655 −0.131342
\(162\) 0 0
\(163\) −16.4539 −1.28877 −0.644385 0.764701i \(-0.722887\pi\)
−0.644385 + 0.764701i \(0.722887\pi\)
\(164\) 0.488493 0.0381449
\(165\) 0 0
\(166\) −13.2774 −1.03053
\(167\) 19.1519 1.48201 0.741007 0.671497i \(-0.234348\pi\)
0.741007 + 0.671497i \(0.234348\pi\)
\(168\) 0 0
\(169\) −12.5886 −0.968351
\(170\) 0.262648 0.0201442
\(171\) 0 0
\(172\) 0.0822773 0.00627358
\(173\) −15.2246 −1.15750 −0.578752 0.815504i \(-0.696460\pi\)
−0.578752 + 0.815504i \(0.696460\pi\)
\(174\) 0 0
\(175\) −4.98231 −0.376627
\(176\) 0 0
\(177\) 0 0
\(178\) 24.5315 1.83872
\(179\) −1.12862 −0.0843574 −0.0421787 0.999110i \(-0.513430\pi\)
−0.0421787 + 0.999110i \(0.513430\pi\)
\(180\) 0 0
\(181\) 18.4775 1.37342 0.686711 0.726931i \(-0.259054\pi\)
0.686711 + 0.726931i \(0.259054\pi\)
\(182\) −0.889000 −0.0658971
\(183\) 0 0
\(184\) 4.80230 0.354030
\(185\) 0.501246 0.0368524
\(186\) 0 0
\(187\) 0 0
\(188\) 0.750947 0.0547684
\(189\) 0 0
\(190\) 1.32409 0.0960599
\(191\) 2.42981 0.175815 0.0879076 0.996129i \(-0.471982\pi\)
0.0879076 + 0.996129i \(0.471982\pi\)
\(192\) 0 0
\(193\) −0.263324 −0.0189545 −0.00947725 0.999955i \(-0.503017\pi\)
−0.00947725 + 0.999955i \(0.503017\pi\)
\(194\) 9.12149 0.654885
\(195\) 0 0
\(196\) −0.0791355 −0.00565254
\(197\) −17.4681 −1.24455 −0.622276 0.782798i \(-0.713792\pi\)
−0.622276 + 0.782798i \(0.713792\pi\)
\(198\) 0 0
\(199\) −20.1415 −1.42780 −0.713898 0.700250i \(-0.753072\pi\)
−0.713898 + 0.700250i \(0.753072\pi\)
\(200\) 14.3569 1.01519
\(201\) 0 0
\(202\) −25.8070 −1.81577
\(203\) −4.47657 −0.314193
\(204\) 0 0
\(205\) −0.821018 −0.0573424
\(206\) −9.59572 −0.668566
\(207\) 0 0
\(208\) 2.46021 0.170585
\(209\) 0 0
\(210\) 0 0
\(211\) −14.0802 −0.969320 −0.484660 0.874702i \(-0.661057\pi\)
−0.484660 + 0.874702i \(0.661057\pi\)
\(212\) −0.0527477 −0.00362273
\(213\) 0 0
\(214\) 7.54576 0.515817
\(215\) −0.138285 −0.00943095
\(216\) 0 0
\(217\) −6.83182 −0.463774
\(218\) −12.7829 −0.865765
\(219\) 0 0
\(220\) 0 0
\(221\) −0.913931 −0.0614777
\(222\) 0 0
\(223\) −9.63364 −0.645116 −0.322558 0.946550i \(-0.604543\pi\)
−0.322558 + 0.946550i \(0.604543\pi\)
\(224\) 0.447392 0.0298926
\(225\) 0 0
\(226\) −13.0033 −0.864969
\(227\) 10.4092 0.690880 0.345440 0.938441i \(-0.387730\pi\)
0.345440 + 0.938441i \(0.387730\pi\)
\(228\) 0 0
\(229\) −9.32529 −0.616232 −0.308116 0.951349i \(-0.599699\pi\)
−0.308116 + 0.951349i \(0.599699\pi\)
\(230\) −0.307208 −0.0202567
\(231\) 0 0
\(232\) 12.8996 0.846900
\(233\) −16.9560 −1.11082 −0.555412 0.831575i \(-0.687440\pi\)
−0.555412 + 0.831575i \(0.687440\pi\)
\(234\) 0 0
\(235\) −1.26213 −0.0823322
\(236\) 0.647879 0.0421733
\(237\) 0 0
\(238\) 1.97473 0.128003
\(239\) −8.27964 −0.535565 −0.267783 0.963479i \(-0.586291\pi\)
−0.267783 + 0.963479i \(0.586291\pi\)
\(240\) 0 0
\(241\) 9.31212 0.599846 0.299923 0.953963i \(-0.403039\pi\)
0.299923 + 0.953963i \(0.403039\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −0.751635 −0.0481185
\(245\) 0.133004 0.00849734
\(246\) 0 0
\(247\) −4.60742 −0.293163
\(248\) 19.6865 1.25009
\(249\) 0 0
\(250\) −1.84012 −0.116379
\(251\) 4.93532 0.311514 0.155757 0.987795i \(-0.450218\pi\)
0.155757 + 0.987795i \(0.450218\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 21.4213 1.34409
\(255\) 0 0
\(256\) −1.89624 −0.118515
\(257\) 16.8159 1.04895 0.524474 0.851426i \(-0.324262\pi\)
0.524474 + 0.851426i \(0.324262\pi\)
\(258\) 0 0
\(259\) 3.76864 0.234172
\(260\) 0.00675136 0.000418702 0
\(261\) 0 0
\(262\) −24.4864 −1.51277
\(263\) −4.11162 −0.253533 −0.126767 0.991933i \(-0.540460\pi\)
−0.126767 + 0.991933i \(0.540460\pi\)
\(264\) 0 0
\(265\) 0.0886540 0.00544597
\(266\) 9.95526 0.610396
\(267\) 0 0
\(268\) −0.952774 −0.0581999
\(269\) 23.9684 1.46138 0.730690 0.682710i \(-0.239199\pi\)
0.730690 + 0.682710i \(0.239199\pi\)
\(270\) 0 0
\(271\) −6.00791 −0.364954 −0.182477 0.983210i \(-0.558412\pi\)
−0.182477 + 0.983210i \(0.558412\pi\)
\(272\) −5.46485 −0.331355
\(273\) 0 0
\(274\) −26.3779 −1.59355
\(275\) 0 0
\(276\) 0 0
\(277\) −9.16206 −0.550495 −0.275247 0.961373i \(-0.588760\pi\)
−0.275247 + 0.961373i \(0.588760\pi\)
\(278\) −19.4955 −1.16926
\(279\) 0 0
\(280\) −0.383263 −0.0229044
\(281\) −12.9579 −0.773006 −0.386503 0.922288i \(-0.626317\pi\)
−0.386503 + 0.922288i \(0.626317\pi\)
\(282\) 0 0
\(283\) −3.78781 −0.225162 −0.112581 0.993643i \(-0.535912\pi\)
−0.112581 + 0.993643i \(0.535912\pi\)
\(284\) 0.382555 0.0227005
\(285\) 0 0
\(286\) 0 0
\(287\) −6.17286 −0.364372
\(288\) 0 0
\(289\) −14.9699 −0.880582
\(290\) −0.825201 −0.0484575
\(291\) 0 0
\(292\) 0.700941 0.0410195
\(293\) −21.3690 −1.24839 −0.624195 0.781269i \(-0.714573\pi\)
−0.624195 + 0.781269i \(0.714573\pi\)
\(294\) 0 0
\(295\) −1.08890 −0.0633983
\(296\) −10.8597 −0.631205
\(297\) 0 0
\(298\) 15.2963 0.886091
\(299\) 1.06898 0.0618210
\(300\) 0 0
\(301\) −1.03970 −0.0599274
\(302\) −22.3623 −1.28681
\(303\) 0 0
\(304\) −27.5500 −1.58010
\(305\) 1.26329 0.0723355
\(306\) 0 0
\(307\) −8.44677 −0.482082 −0.241041 0.970515i \(-0.577489\pi\)
−0.241041 + 0.970515i \(0.577489\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −1.25936 −0.0715271
\(311\) −18.0190 −1.02176 −0.510882 0.859651i \(-0.670681\pi\)
−0.510882 + 0.859651i \(0.670681\pi\)
\(312\) 0 0
\(313\) 0.236340 0.0133587 0.00667937 0.999978i \(-0.497874\pi\)
0.00667937 + 0.999978i \(0.497874\pi\)
\(314\) −20.6089 −1.16303
\(315\) 0 0
\(316\) −0.909320 −0.0511532
\(317\) −17.0931 −0.960044 −0.480022 0.877256i \(-0.659371\pi\)
−0.480022 + 0.877256i \(0.659371\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 1.10274 0.0616450
\(321\) 0 0
\(322\) −2.30976 −0.128718
\(323\) 10.2344 0.569460
\(324\) 0 0
\(325\) 3.19583 0.177273
\(326\) −22.8043 −1.26302
\(327\) 0 0
\(328\) 17.7876 0.982156
\(329\) −9.48937 −0.523166
\(330\) 0 0
\(331\) 4.41186 0.242498 0.121249 0.992622i \(-0.461310\pi\)
0.121249 + 0.992622i \(0.461310\pi\)
\(332\) 0.758117 0.0416071
\(333\) 0 0
\(334\) 26.5436 1.45240
\(335\) 1.60134 0.0874908
\(336\) 0 0
\(337\) 27.7879 1.51370 0.756852 0.653586i \(-0.226736\pi\)
0.756852 + 0.653586i \(0.226736\pi\)
\(338\) −17.4471 −0.949000
\(339\) 0 0
\(340\) −0.0149968 −0.000813315 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 2.99598 0.161533
\(345\) 0 0
\(346\) −21.1006 −1.13437
\(347\) −31.2515 −1.67767 −0.838834 0.544388i \(-0.816762\pi\)
−0.838834 + 0.544388i \(0.816762\pi\)
\(348\) 0 0
\(349\) −0.543438 −0.0290896 −0.0145448 0.999894i \(-0.504630\pi\)
−0.0145448 + 0.999894i \(0.504630\pi\)
\(350\) −6.90524 −0.369101
\(351\) 0 0
\(352\) 0 0
\(353\) 17.8517 0.950148 0.475074 0.879946i \(-0.342421\pi\)
0.475074 + 0.879946i \(0.342421\pi\)
\(354\) 0 0
\(355\) −0.642967 −0.0341252
\(356\) −1.40071 −0.0742374
\(357\) 0 0
\(358\) −1.56422 −0.0826716
\(359\) 11.9909 0.632854 0.316427 0.948617i \(-0.397517\pi\)
0.316427 + 0.948617i \(0.397517\pi\)
\(360\) 0 0
\(361\) 32.5951 1.71553
\(362\) 25.6089 1.34598
\(363\) 0 0
\(364\) 0.0507604 0.00266057
\(365\) −1.17808 −0.0616637
\(366\) 0 0
\(367\) 29.5958 1.54489 0.772444 0.635083i \(-0.219034\pi\)
0.772444 + 0.635083i \(0.219034\pi\)
\(368\) 6.39199 0.333206
\(369\) 0 0
\(370\) 0.694704 0.0361159
\(371\) 0.666549 0.0346055
\(372\) 0 0
\(373\) 10.5209 0.544753 0.272377 0.962191i \(-0.412190\pi\)
0.272377 + 0.962191i \(0.412190\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 27.3444 1.41018
\(377\) 2.87143 0.147886
\(378\) 0 0
\(379\) −8.34913 −0.428866 −0.214433 0.976739i \(-0.568790\pi\)
−0.214433 + 0.976739i \(0.568790\pi\)
\(380\) −0.0756035 −0.00387838
\(381\) 0 0
\(382\) 3.36761 0.172302
\(383\) 18.4737 0.943965 0.471982 0.881608i \(-0.343539\pi\)
0.471982 + 0.881608i \(0.343539\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −0.364955 −0.0185757
\(387\) 0 0
\(388\) −0.520822 −0.0264407
\(389\) −23.2560 −1.17913 −0.589564 0.807722i \(-0.700700\pi\)
−0.589564 + 0.807722i \(0.700700\pi\)
\(390\) 0 0
\(391\) −2.37453 −0.120085
\(392\) −2.88158 −0.145542
\(393\) 0 0
\(394\) −24.2100 −1.21968
\(395\) 1.52831 0.0768976
\(396\) 0 0
\(397\) 21.6794 1.08806 0.544029 0.839066i \(-0.316898\pi\)
0.544029 + 0.839066i \(0.316898\pi\)
\(398\) −27.9152 −1.39926
\(399\) 0 0
\(400\) 19.1095 0.955474
\(401\) 34.8075 1.73821 0.869103 0.494631i \(-0.164697\pi\)
0.869103 + 0.494631i \(0.164697\pi\)
\(402\) 0 0
\(403\) 4.38218 0.218292
\(404\) 1.47354 0.0733112
\(405\) 0 0
\(406\) −6.20431 −0.307915
\(407\) 0 0
\(408\) 0 0
\(409\) 16.2927 0.805620 0.402810 0.915284i \(-0.368033\pi\)
0.402810 + 0.915284i \(0.368033\pi\)
\(410\) −1.13789 −0.0561965
\(411\) 0 0
\(412\) 0.547900 0.0269931
\(413\) −8.18695 −0.402853
\(414\) 0 0
\(415\) −1.27418 −0.0625471
\(416\) −0.286973 −0.0140700
\(417\) 0 0
\(418\) 0 0
\(419\) 22.6536 1.10670 0.553350 0.832949i \(-0.313349\pi\)
0.553350 + 0.832949i \(0.313349\pi\)
\(420\) 0 0
\(421\) −15.2061 −0.741102 −0.370551 0.928812i \(-0.620831\pi\)
−0.370551 + 0.928812i \(0.620831\pi\)
\(422\) −19.5145 −0.949950
\(423\) 0 0
\(424\) −1.92072 −0.0932783
\(425\) −7.09890 −0.344347
\(426\) 0 0
\(427\) 9.49807 0.459644
\(428\) −0.430850 −0.0208259
\(429\) 0 0
\(430\) −0.191656 −0.00924249
\(431\) 3.20691 0.154471 0.0772356 0.997013i \(-0.475391\pi\)
0.0772356 + 0.997013i \(0.475391\pi\)
\(432\) 0 0
\(433\) 5.99246 0.287979 0.143990 0.989579i \(-0.454007\pi\)
0.143990 + 0.989579i \(0.454007\pi\)
\(434\) −9.46858 −0.454507
\(435\) 0 0
\(436\) 0.729880 0.0349549
\(437\) −11.9708 −0.572640
\(438\) 0 0
\(439\) −7.68712 −0.366886 −0.183443 0.983030i \(-0.558724\pi\)
−0.183443 + 0.983030i \(0.558724\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −1.26667 −0.0602491
\(443\) −37.2499 −1.76980 −0.884898 0.465784i \(-0.845772\pi\)
−0.884898 + 0.465784i \(0.845772\pi\)
\(444\) 0 0
\(445\) 2.35419 0.111600
\(446\) −13.3518 −0.632224
\(447\) 0 0
\(448\) 8.29100 0.391713
\(449\) −11.0724 −0.522539 −0.261269 0.965266i \(-0.584141\pi\)
−0.261269 + 0.965266i \(0.584141\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0.742468 0.0349228
\(453\) 0 0
\(454\) 14.4266 0.677074
\(455\) −0.0853139 −0.00399958
\(456\) 0 0
\(457\) 28.3961 1.32831 0.664157 0.747593i \(-0.268791\pi\)
0.664157 + 0.747593i \(0.268791\pi\)
\(458\) −12.9244 −0.603918
\(459\) 0 0
\(460\) 0.0175411 0.000817856 0
\(461\) 25.7420 1.19892 0.599462 0.800403i \(-0.295381\pi\)
0.599462 + 0.800403i \(0.295381\pi\)
\(462\) 0 0
\(463\) 37.6543 1.74995 0.874973 0.484172i \(-0.160879\pi\)
0.874973 + 0.484172i \(0.160879\pi\)
\(464\) 17.1697 0.797084
\(465\) 0 0
\(466\) −23.5002 −1.08863
\(467\) 15.3875 0.712046 0.356023 0.934477i \(-0.384132\pi\)
0.356023 + 0.934477i \(0.384132\pi\)
\(468\) 0 0
\(469\) 12.0398 0.555945
\(470\) −1.74925 −0.0806869
\(471\) 0 0
\(472\) 23.5914 1.08588
\(473\) 0 0
\(474\) 0 0
\(475\) −35.7878 −1.64206
\(476\) −0.112754 −0.00516807
\(477\) 0 0
\(478\) −11.4752 −0.524863
\(479\) 0.600178 0.0274228 0.0137114 0.999906i \(-0.495635\pi\)
0.0137114 + 0.999906i \(0.495635\pi\)
\(480\) 0 0
\(481\) −2.41734 −0.110221
\(482\) 12.9062 0.587859
\(483\) 0 0
\(484\) 0 0
\(485\) 0.875354 0.0397478
\(486\) 0 0
\(487\) 10.2991 0.466698 0.233349 0.972393i \(-0.425032\pi\)
0.233349 + 0.972393i \(0.425032\pi\)
\(488\) −27.3695 −1.23896
\(489\) 0 0
\(490\) 0.184338 0.00832754
\(491\) 10.0925 0.455466 0.227733 0.973724i \(-0.426869\pi\)
0.227733 + 0.973724i \(0.426869\pi\)
\(492\) 0 0
\(493\) −6.37830 −0.287264
\(494\) −6.38566 −0.287304
\(495\) 0 0
\(496\) 26.2032 1.17656
\(497\) −4.83418 −0.216842
\(498\) 0 0
\(499\) 19.7955 0.886170 0.443085 0.896480i \(-0.353884\pi\)
0.443085 + 0.896480i \(0.353884\pi\)
\(500\) 0.105068 0.00469877
\(501\) 0 0
\(502\) 6.84012 0.305289
\(503\) 14.3034 0.637757 0.318879 0.947796i \(-0.396694\pi\)
0.318879 + 0.947796i \(0.396694\pi\)
\(504\) 0 0
\(505\) −2.47660 −0.110207
\(506\) 0 0
\(507\) 0 0
\(508\) −1.22312 −0.0542671
\(509\) 42.0975 1.86594 0.932970 0.359953i \(-0.117207\pi\)
0.932970 + 0.359953i \(0.117207\pi\)
\(510\) 0 0
\(511\) −8.85748 −0.391832
\(512\) −23.8204 −1.05272
\(513\) 0 0
\(514\) 23.3061 1.02799
\(515\) −0.920864 −0.0405781
\(516\) 0 0
\(517\) 0 0
\(518\) 5.22316 0.229492
\(519\) 0 0
\(520\) 0.245839 0.0107808
\(521\) 21.4472 0.939619 0.469810 0.882768i \(-0.344323\pi\)
0.469810 + 0.882768i \(0.344323\pi\)
\(522\) 0 0
\(523\) −35.5946 −1.55644 −0.778222 0.627989i \(-0.783878\pi\)
−0.778222 + 0.627989i \(0.783878\pi\)
\(524\) 1.39813 0.0610776
\(525\) 0 0
\(526\) −5.69851 −0.248467
\(527\) −9.73412 −0.424025
\(528\) 0 0
\(529\) −20.2226 −0.879244
\(530\) 0.122870 0.00533714
\(531\) 0 0
\(532\) −0.568428 −0.0246445
\(533\) 3.95949 0.171505
\(534\) 0 0
\(535\) 0.724137 0.0313072
\(536\) −34.6936 −1.49854
\(537\) 0 0
\(538\) 33.2191 1.43218
\(539\) 0 0
\(540\) 0 0
\(541\) 16.9350 0.728092 0.364046 0.931381i \(-0.381395\pi\)
0.364046 + 0.931381i \(0.381395\pi\)
\(542\) −8.32667 −0.357661
\(543\) 0 0
\(544\) 0.637453 0.0273306
\(545\) −1.22672 −0.0525470
\(546\) 0 0
\(547\) −8.79082 −0.375868 −0.187934 0.982182i \(-0.560179\pi\)
−0.187934 + 0.982182i \(0.560179\pi\)
\(548\) 1.50613 0.0643388
\(549\) 0 0
\(550\) 0 0
\(551\) −32.1551 −1.36985
\(552\) 0 0
\(553\) 11.4907 0.488633
\(554\) −12.6982 −0.539494
\(555\) 0 0
\(556\) 1.11316 0.0472085
\(557\) −21.7276 −0.920629 −0.460315 0.887756i \(-0.652263\pi\)
−0.460315 + 0.887756i \(0.652263\pi\)
\(558\) 0 0
\(559\) 0.666902 0.0282069
\(560\) −0.510134 −0.0215571
\(561\) 0 0
\(562\) −17.9591 −0.757559
\(563\) 1.82134 0.0767605 0.0383803 0.999263i \(-0.487780\pi\)
0.0383803 + 0.999263i \(0.487780\pi\)
\(564\) 0 0
\(565\) −1.24788 −0.0524987
\(566\) −5.24972 −0.220662
\(567\) 0 0
\(568\) 13.9301 0.584493
\(569\) −32.9418 −1.38099 −0.690496 0.723336i \(-0.742608\pi\)
−0.690496 + 0.723336i \(0.742608\pi\)
\(570\) 0 0
\(571\) −23.4394 −0.980908 −0.490454 0.871467i \(-0.663169\pi\)
−0.490454 + 0.871467i \(0.663169\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −8.55529 −0.357091
\(575\) 8.30326 0.346270
\(576\) 0 0
\(577\) 34.4539 1.43433 0.717166 0.696902i \(-0.245439\pi\)
0.717166 + 0.696902i \(0.245439\pi\)
\(578\) −20.7476 −0.862984
\(579\) 0 0
\(580\) 0.0471176 0.00195645
\(581\) −9.57998 −0.397445
\(582\) 0 0
\(583\) 0 0
\(584\) 25.5236 1.05617
\(585\) 0 0
\(586\) −29.6164 −1.22344
\(587\) 20.8670 0.861274 0.430637 0.902525i \(-0.358289\pi\)
0.430637 + 0.902525i \(0.358289\pi\)
\(588\) 0 0
\(589\) −49.0728 −2.02201
\(590\) −1.50916 −0.0621313
\(591\) 0 0
\(592\) −14.4545 −0.594076
\(593\) −26.6381 −1.09389 −0.546947 0.837167i \(-0.684210\pi\)
−0.546947 + 0.837167i \(0.684210\pi\)
\(594\) 0 0
\(595\) 0.189507 0.00776905
\(596\) −0.873393 −0.0357756
\(597\) 0 0
\(598\) 1.48156 0.0605856
\(599\) 20.6712 0.844602 0.422301 0.906456i \(-0.361223\pi\)
0.422301 + 0.906456i \(0.361223\pi\)
\(600\) 0 0
\(601\) −4.94812 −0.201838 −0.100919 0.994895i \(-0.532178\pi\)
−0.100919 + 0.994895i \(0.532178\pi\)
\(602\) −1.44098 −0.0587298
\(603\) 0 0
\(604\) 1.27685 0.0519543
\(605\) 0 0
\(606\) 0 0
\(607\) −27.6086 −1.12060 −0.560300 0.828290i \(-0.689314\pi\)
−0.560300 + 0.828290i \(0.689314\pi\)
\(608\) 3.21360 0.130329
\(609\) 0 0
\(610\) 1.75085 0.0708900
\(611\) 6.08683 0.246247
\(612\) 0 0
\(613\) 7.18919 0.290368 0.145184 0.989405i \(-0.453623\pi\)
0.145184 + 0.989405i \(0.453623\pi\)
\(614\) −11.7068 −0.472449
\(615\) 0 0
\(616\) 0 0
\(617\) −21.1215 −0.850320 −0.425160 0.905118i \(-0.639782\pi\)
−0.425160 + 0.905118i \(0.639782\pi\)
\(618\) 0 0
\(619\) 2.77412 0.111501 0.0557506 0.998445i \(-0.482245\pi\)
0.0557506 + 0.998445i \(0.482245\pi\)
\(620\) 0.0719076 0.00288788
\(621\) 0 0
\(622\) −24.9735 −1.00134
\(623\) 17.7001 0.709140
\(624\) 0 0
\(625\) 24.7350 0.989398
\(626\) 0.327556 0.0130918
\(627\) 0 0
\(628\) 1.17673 0.0469568
\(629\) 5.36964 0.214102
\(630\) 0 0
\(631\) −24.2090 −0.963743 −0.481872 0.876242i \(-0.660043\pi\)
−0.481872 + 0.876242i \(0.660043\pi\)
\(632\) −33.1113 −1.31710
\(633\) 0 0
\(634\) −23.6902 −0.940859
\(635\) 2.05572 0.0815786
\(636\) 0 0
\(637\) −0.641436 −0.0254146
\(638\) 0 0
\(639\) 0 0
\(640\) 1.40933 0.0557088
\(641\) −43.4897 −1.71774 −0.858871 0.512192i \(-0.828834\pi\)
−0.858871 + 0.512192i \(0.828834\pi\)
\(642\) 0 0
\(643\) −17.1517 −0.676397 −0.338199 0.941075i \(-0.609818\pi\)
−0.338199 + 0.941075i \(0.609818\pi\)
\(644\) 0.131883 0.00519693
\(645\) 0 0
\(646\) 14.1845 0.558080
\(647\) −14.2909 −0.561833 −0.280916 0.959732i \(-0.590638\pi\)
−0.280916 + 0.959732i \(0.590638\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 4.42927 0.173730
\(651\) 0 0
\(652\) 1.30209 0.0509938
\(653\) 37.8518 1.48126 0.740628 0.671915i \(-0.234528\pi\)
0.740628 + 0.671915i \(0.234528\pi\)
\(654\) 0 0
\(655\) −2.34986 −0.0918166
\(656\) 23.6758 0.924384
\(657\) 0 0
\(658\) −13.1518 −0.512711
\(659\) −8.13829 −0.317023 −0.158511 0.987357i \(-0.550669\pi\)
−0.158511 + 0.987357i \(0.550669\pi\)
\(660\) 0 0
\(661\) −5.33161 −0.207375 −0.103688 0.994610i \(-0.533064\pi\)
−0.103688 + 0.994610i \(0.533064\pi\)
\(662\) 6.11463 0.237652
\(663\) 0 0
\(664\) 27.6055 1.07130
\(665\) 0.955367 0.0370476
\(666\) 0 0
\(667\) 7.46042 0.288869
\(668\) −1.51559 −0.0586400
\(669\) 0 0
\(670\) 2.21939 0.0857424
\(671\) 0 0
\(672\) 0 0
\(673\) −19.4495 −0.749724 −0.374862 0.927081i \(-0.622310\pi\)
−0.374862 + 0.927081i \(0.622310\pi\)
\(674\) 38.5127 1.48345
\(675\) 0 0
\(676\) 0.996203 0.0383155
\(677\) 36.3042 1.39528 0.697642 0.716446i \(-0.254232\pi\)
0.697642 + 0.716446i \(0.254232\pi\)
\(678\) 0 0
\(679\) 6.58139 0.252571
\(680\) −0.546082 −0.0209413
\(681\) 0 0
\(682\) 0 0
\(683\) 20.7805 0.795142 0.397571 0.917571i \(-0.369853\pi\)
0.397571 + 0.917571i \(0.369853\pi\)
\(684\) 0 0
\(685\) −2.53138 −0.0967192
\(686\) 1.38595 0.0529159
\(687\) 0 0
\(688\) 3.98774 0.152031
\(689\) −0.427549 −0.0162883
\(690\) 0 0
\(691\) −50.1050 −1.90608 −0.953042 0.302838i \(-0.902066\pi\)
−0.953042 + 0.302838i \(0.902066\pi\)
\(692\) 1.20481 0.0457998
\(693\) 0 0
\(694\) −43.3131 −1.64414
\(695\) −1.87091 −0.0709675
\(696\) 0 0
\(697\) −8.79521 −0.333143
\(698\) −0.753179 −0.0285083
\(699\) 0 0
\(700\) 0.394278 0.0149023
\(701\) 16.3229 0.616508 0.308254 0.951304i \(-0.400255\pi\)
0.308254 + 0.951304i \(0.400255\pi\)
\(702\) 0 0
\(703\) 27.0701 1.02097
\(704\) 0 0
\(705\) 0 0
\(706\) 24.7416 0.931161
\(707\) −18.6204 −0.700292
\(708\) 0 0
\(709\) 33.2299 1.24798 0.623988 0.781434i \(-0.285511\pi\)
0.623988 + 0.781434i \(0.285511\pi\)
\(710\) −0.891122 −0.0334432
\(711\) 0 0
\(712\) −51.0044 −1.91147
\(713\) 11.3856 0.426393
\(714\) 0 0
\(715\) 0 0
\(716\) 0.0893143 0.00333783
\(717\) 0 0
\(718\) 16.6188 0.620207
\(719\) 25.2455 0.941498 0.470749 0.882267i \(-0.343984\pi\)
0.470749 + 0.882267i \(0.343984\pi\)
\(720\) 0 0
\(721\) −6.92356 −0.257847
\(722\) 45.1752 1.68125
\(723\) 0 0
\(724\) −1.46223 −0.0543432
\(725\) 22.3036 0.828337
\(726\) 0 0
\(727\) 1.86242 0.0690733 0.0345366 0.999403i \(-0.489004\pi\)
0.0345366 + 0.999403i \(0.489004\pi\)
\(728\) 1.84835 0.0685045
\(729\) 0 0
\(730\) −1.63277 −0.0604315
\(731\) −1.48139 −0.0547911
\(732\) 0 0
\(733\) 20.0651 0.741123 0.370561 0.928808i \(-0.379165\pi\)
0.370561 + 0.928808i \(0.379165\pi\)
\(734\) 41.0184 1.51402
\(735\) 0 0
\(736\) −0.745601 −0.0274832
\(737\) 0 0
\(738\) 0 0
\(739\) −22.3970 −0.823886 −0.411943 0.911210i \(-0.635150\pi\)
−0.411943 + 0.911210i \(0.635150\pi\)
\(740\) −0.0396664 −0.00145817
\(741\) 0 0
\(742\) 0.923806 0.0339140
\(743\) 36.2705 1.33063 0.665317 0.746561i \(-0.268296\pi\)
0.665317 + 0.746561i \(0.268296\pi\)
\(744\) 0 0
\(745\) 1.46793 0.0537807
\(746\) 14.5815 0.533867
\(747\) 0 0
\(748\) 0 0
\(749\) 5.44446 0.198936
\(750\) 0 0
\(751\) 44.6237 1.62834 0.814170 0.580626i \(-0.197192\pi\)
0.814170 + 0.580626i \(0.197192\pi\)
\(752\) 36.3962 1.32723
\(753\) 0 0
\(754\) 3.97967 0.144931
\(755\) −2.14603 −0.0781019
\(756\) 0 0
\(757\) 8.99964 0.327097 0.163549 0.986535i \(-0.447706\pi\)
0.163549 + 0.986535i \(0.447706\pi\)
\(758\) −11.5715 −0.420296
\(759\) 0 0
\(760\) −2.75297 −0.0998607
\(761\) −5.15155 −0.186743 −0.0933717 0.995631i \(-0.529765\pi\)
−0.0933717 + 0.995631i \(0.529765\pi\)
\(762\) 0 0
\(763\) −9.22316 −0.333901
\(764\) −0.192285 −0.00695661
\(765\) 0 0
\(766\) 25.6037 0.925101
\(767\) 5.25140 0.189617
\(768\) 0 0
\(769\) −38.5242 −1.38922 −0.694608 0.719388i \(-0.744422\pi\)
−0.694608 + 0.719388i \(0.744422\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0.0208383 0.000749987 0
\(773\) 6.99024 0.251421 0.125711 0.992067i \(-0.459879\pi\)
0.125711 + 0.992067i \(0.459879\pi\)
\(774\) 0 0
\(775\) 34.0383 1.22269
\(776\) −18.9648 −0.680797
\(777\) 0 0
\(778\) −32.2317 −1.15556
\(779\) −44.3395 −1.58863
\(780\) 0 0
\(781\) 0 0
\(782\) −3.29099 −0.117686
\(783\) 0 0
\(784\) −3.83547 −0.136981
\(785\) −1.97776 −0.0705892
\(786\) 0 0
\(787\) 31.3859 1.11879 0.559393 0.828903i \(-0.311034\pi\)
0.559393 + 0.828903i \(0.311034\pi\)
\(788\) 1.38235 0.0492441
\(789\) 0 0
\(790\) 2.11816 0.0753609
\(791\) −9.38223 −0.333594
\(792\) 0 0
\(793\) −6.09240 −0.216348
\(794\) 30.0466 1.06631
\(795\) 0 0
\(796\) 1.59391 0.0564947
\(797\) −26.2972 −0.931496 −0.465748 0.884917i \(-0.654215\pi\)
−0.465748 + 0.884917i \(0.654215\pi\)
\(798\) 0 0
\(799\) −13.5207 −0.478326
\(800\) −2.22904 −0.0788086
\(801\) 0 0
\(802\) 48.2416 1.70347
\(803\) 0 0
\(804\) 0 0
\(805\) −0.221659 −0.00781244
\(806\) 6.07349 0.213930
\(807\) 0 0
\(808\) 53.6562 1.88762
\(809\) −11.7634 −0.413578 −0.206789 0.978386i \(-0.566301\pi\)
−0.206789 + 0.978386i \(0.566301\pi\)
\(810\) 0 0
\(811\) −8.27304 −0.290506 −0.145253 0.989395i \(-0.546400\pi\)
−0.145253 + 0.989395i \(0.546400\pi\)
\(812\) 0.354256 0.0124319
\(813\) 0 0
\(814\) 0 0
\(815\) −2.18844 −0.0766579
\(816\) 0 0
\(817\) −7.46814 −0.261277
\(818\) 22.5809 0.789521
\(819\) 0 0
\(820\) 0.0649717 0.00226891
\(821\) 7.38513 0.257743 0.128871 0.991661i \(-0.458865\pi\)
0.128871 + 0.991661i \(0.458865\pi\)
\(822\) 0 0
\(823\) 49.4433 1.72349 0.861743 0.507345i \(-0.169373\pi\)
0.861743 + 0.507345i \(0.169373\pi\)
\(824\) 19.9508 0.695019
\(825\) 0 0
\(826\) −11.3467 −0.394803
\(827\) −23.2339 −0.807922 −0.403961 0.914776i \(-0.632367\pi\)
−0.403961 + 0.914776i \(0.632367\pi\)
\(828\) 0 0
\(829\) 30.1938 1.04867 0.524336 0.851511i \(-0.324314\pi\)
0.524336 + 0.851511i \(0.324314\pi\)
\(830\) −1.76595 −0.0612972
\(831\) 0 0
\(832\) −5.31814 −0.184374
\(833\) 1.42482 0.0493671
\(834\) 0 0
\(835\) 2.54728 0.0881523
\(836\) 0 0
\(837\) 0 0
\(838\) 31.3968 1.08458
\(839\) −16.6125 −0.573529 −0.286764 0.958001i \(-0.592580\pi\)
−0.286764 + 0.958001i \(0.592580\pi\)
\(840\) 0 0
\(841\) −8.96035 −0.308977
\(842\) −21.0750 −0.726292
\(843\) 0 0
\(844\) 1.11424 0.0383539
\(845\) −1.67433 −0.0575989
\(846\) 0 0
\(847\) 0 0
\(848\) −2.55653 −0.0877915
\(849\) 0 0
\(850\) −9.83873 −0.337466
\(851\) −6.28063 −0.215297
\(852\) 0 0
\(853\) −40.0507 −1.37131 −0.685654 0.727927i \(-0.740484\pi\)
−0.685654 + 0.727927i \(0.740484\pi\)
\(854\) 13.1639 0.450458
\(855\) 0 0
\(856\) −15.6887 −0.536227
\(857\) 31.7070 1.08309 0.541546 0.840671i \(-0.317839\pi\)
0.541546 + 0.840671i \(0.317839\pi\)
\(858\) 0 0
\(859\) 1.63654 0.0558381 0.0279190 0.999610i \(-0.491112\pi\)
0.0279190 + 0.999610i \(0.491112\pi\)
\(860\) 0.0109432 0.000373162 0
\(861\) 0 0
\(862\) 4.44462 0.151384
\(863\) 17.4055 0.592489 0.296244 0.955112i \(-0.404266\pi\)
0.296244 + 0.955112i \(0.404266\pi\)
\(864\) 0 0
\(865\) −2.02494 −0.0688500
\(866\) 8.30526 0.282224
\(867\) 0 0
\(868\) 0.540640 0.0183505
\(869\) 0 0
\(870\) 0 0
\(871\) −7.72275 −0.261675
\(872\) 26.5773 0.900021
\(873\) 0 0
\(874\) −16.5909 −0.561196
\(875\) −1.32769 −0.0448842
\(876\) 0 0
\(877\) −48.1667 −1.62647 −0.813237 0.581933i \(-0.802297\pi\)
−0.813237 + 0.581933i \(0.802297\pi\)
\(878\) −10.6540 −0.359555
\(879\) 0 0
\(880\) 0 0
\(881\) 30.0141 1.01120 0.505601 0.862767i \(-0.331271\pi\)
0.505601 + 0.862767i \(0.331271\pi\)
\(882\) 0 0
\(883\) 56.3355 1.89584 0.947921 0.318506i \(-0.103181\pi\)
0.947921 + 0.318506i \(0.103181\pi\)
\(884\) 0.0723244 0.00243253
\(885\) 0 0
\(886\) −51.6266 −1.73443
\(887\) 4.53095 0.152135 0.0760673 0.997103i \(-0.475764\pi\)
0.0760673 + 0.997103i \(0.475764\pi\)
\(888\) 0 0
\(889\) 15.4560 0.518377
\(890\) 3.26280 0.109369
\(891\) 0 0
\(892\) 0.762363 0.0255258
\(893\) −68.1619 −2.28095
\(894\) 0 0
\(895\) −0.150112 −0.00501770
\(896\) 10.5961 0.353992
\(897\) 0 0
\(898\) −15.3458 −0.512097
\(899\) 30.5831 1.02000
\(900\) 0 0
\(901\) 0.949713 0.0316395
\(902\) 0 0
\(903\) 0 0
\(904\) 27.0357 0.899193
\(905\) 2.45759 0.0816930
\(906\) 0 0
\(907\) −20.7609 −0.689355 −0.344678 0.938721i \(-0.612012\pi\)
−0.344678 + 0.938721i \(0.612012\pi\)
\(908\) −0.823735 −0.0273366
\(909\) 0 0
\(910\) −0.118241 −0.00391965
\(911\) 15.3952 0.510066 0.255033 0.966932i \(-0.417914\pi\)
0.255033 + 0.966932i \(0.417914\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 39.3557 1.30177
\(915\) 0 0
\(916\) 0.737962 0.0243829
\(917\) −17.6675 −0.583433
\(918\) 0 0
\(919\) 21.3581 0.704540 0.352270 0.935898i \(-0.385410\pi\)
0.352270 + 0.935898i \(0.385410\pi\)
\(920\) 0.638727 0.0210582
\(921\) 0 0
\(922\) 35.6772 1.17496
\(923\) 3.10082 0.102065
\(924\) 0 0
\(925\) −18.7765 −0.617369
\(926\) 52.1871 1.71498
\(927\) 0 0
\(928\) −2.00278 −0.0657445
\(929\) −16.1336 −0.529325 −0.264662 0.964341i \(-0.585261\pi\)
−0.264662 + 0.964341i \(0.585261\pi\)
\(930\) 0 0
\(931\) 7.18297 0.235412
\(932\) 1.34182 0.0439529
\(933\) 0 0
\(934\) 21.3263 0.697817
\(935\) 0 0
\(936\) 0 0
\(937\) −4.56306 −0.149069 −0.0745343 0.997218i \(-0.523747\pi\)
−0.0745343 + 0.997218i \(0.523747\pi\)
\(938\) 16.6866 0.544835
\(939\) 0 0
\(940\) 0.0998793 0.00325770
\(941\) −12.5504 −0.409130 −0.204565 0.978853i \(-0.565578\pi\)
−0.204565 + 0.978853i \(0.565578\pi\)
\(942\) 0 0
\(943\) 10.2874 0.335003
\(944\) 31.4008 1.02201
\(945\) 0 0
\(946\) 0 0
\(947\) −51.6790 −1.67934 −0.839672 0.543094i \(-0.817253\pi\)
−0.839672 + 0.543094i \(0.817253\pi\)
\(948\) 0 0
\(949\) 5.68151 0.184429
\(950\) −49.6002 −1.60924
\(951\) 0 0
\(952\) −4.10574 −0.133068
\(953\) −1.32251 −0.0428403 −0.0214202 0.999771i \(-0.506819\pi\)
−0.0214202 + 0.999771i \(0.506819\pi\)
\(954\) 0 0
\(955\) 0.323176 0.0104577
\(956\) 0.655213 0.0211911
\(957\) 0 0
\(958\) 0.831818 0.0268748
\(959\) −19.0323 −0.614586
\(960\) 0 0
\(961\) 15.6738 0.505607
\(962\) −3.35032 −0.108019
\(963\) 0 0
\(964\) −0.736919 −0.0237346
\(965\) −0.0350233 −0.00112744
\(966\) 0 0
\(967\) −44.1214 −1.41885 −0.709425 0.704781i \(-0.751045\pi\)
−0.709425 + 0.704781i \(0.751045\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 1.21320 0.0389535
\(971\) 1.31170 0.0420944 0.0210472 0.999778i \(-0.493300\pi\)
0.0210472 + 0.999778i \(0.493300\pi\)
\(972\) 0 0
\(973\) −14.0665 −0.450951
\(974\) 14.2741 0.457371
\(975\) 0 0
\(976\) −36.4295 −1.16608
\(977\) 12.4102 0.397036 0.198518 0.980097i \(-0.436387\pi\)
0.198518 + 0.980097i \(0.436387\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −0.0105254 −0.000336221 0
\(981\) 0 0
\(982\) 13.9877 0.446364
\(983\) 31.9565 1.01925 0.509627 0.860395i \(-0.329783\pi\)
0.509627 + 0.860395i \(0.329783\pi\)
\(984\) 0 0
\(985\) −2.32334 −0.0740277
\(986\) −8.84003 −0.281524
\(987\) 0 0
\(988\) 0.364610 0.0115998
\(989\) 1.73271 0.0550970
\(990\) 0 0
\(991\) −9.17926 −0.291589 −0.145794 0.989315i \(-0.546574\pi\)
−0.145794 + 0.989315i \(0.546574\pi\)
\(992\) −3.05650 −0.0970440
\(993\) 0 0
\(994\) −6.69994 −0.212509
\(995\) −2.67891 −0.0849273
\(996\) 0 0
\(997\) 14.3428 0.454242 0.227121 0.973867i \(-0.427069\pi\)
0.227121 + 0.973867i \(0.427069\pi\)
\(998\) 27.4357 0.868461
\(999\) 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.cp.1.6 6
3.2 odd 2 847.2.a.n.1.1 yes 6
11.10 odd 2 7623.2.a.cs.1.1 6
21.20 even 2 5929.2.a.bm.1.1 6
33.2 even 10 847.2.f.z.323.1 24
33.5 odd 10 847.2.f.y.729.6 24
33.8 even 10 847.2.f.z.372.6 24
33.14 odd 10 847.2.f.y.372.1 24
33.17 even 10 847.2.f.z.729.1 24
33.20 odd 10 847.2.f.y.323.6 24
33.26 odd 10 847.2.f.y.148.1 24
33.29 even 10 847.2.f.z.148.6 24
33.32 even 2 847.2.a.m.1.6 6
231.230 odd 2 5929.2.a.bj.1.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
847.2.a.m.1.6 6 33.32 even 2
847.2.a.n.1.1 yes 6 3.2 odd 2
847.2.f.y.148.1 24 33.26 odd 10
847.2.f.y.323.6 24 33.20 odd 10
847.2.f.y.372.1 24 33.14 odd 10
847.2.f.y.729.6 24 33.5 odd 10
847.2.f.z.148.6 24 33.29 even 10
847.2.f.z.323.1 24 33.2 even 10
847.2.f.z.372.6 24 33.8 even 10
847.2.f.z.729.1 24 33.17 even 10
5929.2.a.bj.1.6 6 231.230 odd 2
5929.2.a.bm.1.1 6 21.20 even 2
7623.2.a.cp.1.6 6 1.1 even 1 trivial
7623.2.a.cs.1.1 6 11.10 odd 2