Properties

Label 2960.2.a.s.1.2
Level $2960$
Weight $2$
Character 2960.1
Self dual yes
Analytic conductor $23.636$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2960.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.6357189983\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \(x^{3} - x^{2} - 6 x - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1480)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.363328\) of defining polynomial
Character \(\chi\) \(=\) 2960.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.363328 q^{3} +1.00000 q^{5} -0.363328 q^{7} -2.86799 q^{9} +O(q^{10})\) \(q+0.363328 q^{3} +1.00000 q^{5} -0.363328 q^{7} -2.86799 q^{9} +1.14134 q^{11} +0.363328 q^{15} -4.77801 q^{19} -0.132007 q^{21} -4.00000 q^{23} +1.00000 q^{25} -2.13201 q^{27} +4.28267 q^{29} -5.50466 q^{31} +0.414680 q^{33} -0.363328 q^{35} +1.00000 q^{37} +3.86799 q^{41} +2.28267 q^{43} -2.86799 q^{45} -12.9287 q^{47} -6.86799 q^{49} -6.15066 q^{53} +1.14134 q^{55} -1.73599 q^{57} -7.78734 q^{59} -0.546687 q^{61} +1.04202 q^{63} -0.212663 q^{67} -1.45331 q^{69} -3.58532 q^{71} +8.69735 q^{73} +0.363328 q^{75} -0.414680 q^{77} +7.06068 q^{79} +7.82936 q^{81} +0.466031 q^{83} +1.55602 q^{87} -16.0187 q^{89} -2.00000 q^{93} -4.77801 q^{95} +6.00000 q^{97} -3.27334 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - q^{3} + 3q^{5} + q^{7} + 4q^{9} + O(q^{10}) \) \( 3q - q^{3} + 3q^{5} + q^{7} + 4q^{9} - 5q^{11} - q^{15} - 8q^{19} - 13q^{21} - 12q^{23} + 3q^{25} - 19q^{27} - 4q^{29} - 6q^{31} - 3q^{33} + q^{35} + 3q^{37} - q^{41} - 10q^{43} + 4q^{45} - 3q^{47} - 8q^{49} + 11q^{53} - 5q^{55} + 20q^{57} + 4q^{59} - 10q^{61} + 22q^{63} - 28q^{67} + 4q^{69} - 15q^{71} + 5q^{73} - q^{75} + 3q^{77} - 2q^{79} + 15q^{81} - 5q^{83} - 8q^{87} - 6q^{89} - 6q^{93} - 8q^{95} + 18q^{97} - 14q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.363328 0.209768 0.104884 0.994484i \(-0.466553\pi\)
0.104884 + 0.994484i \(0.466553\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −0.363328 −0.137325 −0.0686626 0.997640i \(-0.521873\pi\)
−0.0686626 + 0.997640i \(0.521873\pi\)
\(8\) 0 0
\(9\) −2.86799 −0.955998
\(10\) 0 0
\(11\) 1.14134 0.344126 0.172063 0.985086i \(-0.444957\pi\)
0.172063 + 0.985086i \(0.444957\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0.363328 0.0938109
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −4.77801 −1.09615 −0.548075 0.836429i \(-0.684639\pi\)
−0.548075 + 0.836429i \(0.684639\pi\)
\(20\) 0 0
\(21\) −0.132007 −0.0288064
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −2.13201 −0.410305
\(28\) 0 0
\(29\) 4.28267 0.795272 0.397636 0.917543i \(-0.369831\pi\)
0.397636 + 0.917543i \(0.369831\pi\)
\(30\) 0 0
\(31\) −5.50466 −0.988667 −0.494333 0.869272i \(-0.664588\pi\)
−0.494333 + 0.869272i \(0.664588\pi\)
\(32\) 0 0
\(33\) 0.414680 0.0721865
\(34\) 0 0
\(35\) −0.363328 −0.0614137
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.86799 0.604079 0.302039 0.953295i \(-0.402333\pi\)
0.302039 + 0.953295i \(0.402333\pi\)
\(42\) 0 0
\(43\) 2.28267 0.348104 0.174052 0.984736i \(-0.444314\pi\)
0.174052 + 0.984736i \(0.444314\pi\)
\(44\) 0 0
\(45\) −2.86799 −0.427535
\(46\) 0 0
\(47\) −12.9287 −1.88584 −0.942920 0.333018i \(-0.891933\pi\)
−0.942920 + 0.333018i \(0.891933\pi\)
\(48\) 0 0
\(49\) −6.86799 −0.981142
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −6.15066 −0.844859 −0.422429 0.906396i \(-0.638823\pi\)
−0.422429 + 0.906396i \(0.638823\pi\)
\(54\) 0 0
\(55\) 1.14134 0.153898
\(56\) 0 0
\(57\) −1.73599 −0.229937
\(58\) 0 0
\(59\) −7.78734 −1.01382 −0.506912 0.861998i \(-0.669213\pi\)
−0.506912 + 0.861998i \(0.669213\pi\)
\(60\) 0 0
\(61\) −0.546687 −0.0699961 −0.0349981 0.999387i \(-0.511143\pi\)
−0.0349981 + 0.999387i \(0.511143\pi\)
\(62\) 0 0
\(63\) 1.04202 0.131283
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −0.212663 −0.0259810 −0.0129905 0.999916i \(-0.504135\pi\)
−0.0129905 + 0.999916i \(0.504135\pi\)
\(68\) 0 0
\(69\) −1.45331 −0.174958
\(70\) 0 0
\(71\) −3.58532 −0.425499 −0.212750 0.977107i \(-0.568242\pi\)
−0.212750 + 0.977107i \(0.568242\pi\)
\(72\) 0 0
\(73\) 8.69735 1.01795 0.508974 0.860782i \(-0.330025\pi\)
0.508974 + 0.860782i \(0.330025\pi\)
\(74\) 0 0
\(75\) 0.363328 0.0419535
\(76\) 0 0
\(77\) −0.414680 −0.0472571
\(78\) 0 0
\(79\) 7.06068 0.794389 0.397194 0.917735i \(-0.369984\pi\)
0.397194 + 0.917735i \(0.369984\pi\)
\(80\) 0 0
\(81\) 7.82936 0.869929
\(82\) 0 0
\(83\) 0.466031 0.0511536 0.0255768 0.999673i \(-0.491858\pi\)
0.0255768 + 0.999673i \(0.491858\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1.55602 0.166822
\(88\) 0 0
\(89\) −16.0187 −1.69797 −0.848987 0.528414i \(-0.822787\pi\)
−0.848987 + 0.528414i \(0.822787\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −2.00000 −0.207390
\(94\) 0 0
\(95\) −4.77801 −0.490213
\(96\) 0 0
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 0 0
\(99\) −3.27334 −0.328983
\(100\) 0 0
\(101\) 13.6040 1.35365 0.676823 0.736146i \(-0.263356\pi\)
0.676823 + 0.736146i \(0.263356\pi\)
\(102\) 0 0
\(103\) −6.17997 −0.608930 −0.304465 0.952523i \(-0.598478\pi\)
−0.304465 + 0.952523i \(0.598478\pi\)
\(104\) 0 0
\(105\) −0.132007 −0.0128826
\(106\) 0 0
\(107\) −18.7967 −1.81714 −0.908571 0.417730i \(-0.862826\pi\)
−0.908571 + 0.417730i \(0.862826\pi\)
\(108\) 0 0
\(109\) −4.54669 −0.435494 −0.217747 0.976005i \(-0.569871\pi\)
−0.217747 + 0.976005i \(0.569871\pi\)
\(110\) 0 0
\(111\) 0.363328 0.0344856
\(112\) 0 0
\(113\) −8.82936 −0.830596 −0.415298 0.909685i \(-0.636323\pi\)
−0.415298 + 0.909685i \(0.636323\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.373002
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −9.69735 −0.881577
\(122\) 0 0
\(123\) 1.40535 0.126716
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −10.7487 −0.953793 −0.476897 0.878959i \(-0.658238\pi\)
−0.476897 + 0.878959i \(0.658238\pi\)
\(128\) 0 0
\(129\) 0.829359 0.0730210
\(130\) 0 0
\(131\) 5.50466 0.480945 0.240472 0.970656i \(-0.422698\pi\)
0.240472 + 0.970656i \(0.422698\pi\)
\(132\) 0 0
\(133\) 1.73599 0.150529
\(134\) 0 0
\(135\) −2.13201 −0.183494
\(136\) 0 0
\(137\) 3.45331 0.295036 0.147518 0.989059i \(-0.452871\pi\)
0.147518 + 0.989059i \(0.452871\pi\)
\(138\) 0 0
\(139\) −10.0187 −0.849771 −0.424886 0.905247i \(-0.639686\pi\)
−0.424886 + 0.905247i \(0.639686\pi\)
\(140\) 0 0
\(141\) −4.69735 −0.395588
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 4.28267 0.355657
\(146\) 0 0
\(147\) −2.49534 −0.205812
\(148\) 0 0
\(149\) −1.86799 −0.153032 −0.0765160 0.997068i \(-0.524380\pi\)
−0.0765160 + 0.997068i \(0.524380\pi\)
\(150\) 0 0
\(151\) 4.46264 0.363165 0.181582 0.983376i \(-0.441878\pi\)
0.181582 + 0.983376i \(0.441878\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −5.50466 −0.442145
\(156\) 0 0
\(157\) 20.1693 1.60969 0.804844 0.593487i \(-0.202249\pi\)
0.804844 + 0.593487i \(0.202249\pi\)
\(158\) 0 0
\(159\) −2.23471 −0.177224
\(160\) 0 0
\(161\) 1.45331 0.114537
\(162\) 0 0
\(163\) −9.55602 −0.748485 −0.374242 0.927331i \(-0.622097\pi\)
−0.374242 + 0.927331i \(0.622097\pi\)
\(164\) 0 0
\(165\) 0.414680 0.0322828
\(166\) 0 0
\(167\) 15.8387 1.22563 0.612817 0.790225i \(-0.290036\pi\)
0.612817 + 0.790225i \(0.290036\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 13.7033 1.04792
\(172\) 0 0
\(173\) 15.8680 1.20642 0.603211 0.797582i \(-0.293888\pi\)
0.603211 + 0.797582i \(0.293888\pi\)
\(174\) 0 0
\(175\) −0.363328 −0.0274650
\(176\) 0 0
\(177\) −2.82936 −0.212668
\(178\) 0 0
\(179\) 9.34335 0.698355 0.349177 0.937057i \(-0.386461\pi\)
0.349177 + 0.937057i \(0.386461\pi\)
\(180\) 0 0
\(181\) −19.2627 −1.43178 −0.715892 0.698211i \(-0.753980\pi\)
−0.715892 + 0.698211i \(0.753980\pi\)
\(182\) 0 0
\(183\) −0.198627 −0.0146829
\(184\) 0 0
\(185\) 1.00000 0.0735215
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0.774618 0.0563452
\(190\) 0 0
\(191\) −3.32469 −0.240566 −0.120283 0.992740i \(-0.538380\pi\)
−0.120283 + 0.992740i \(0.538380\pi\)
\(192\) 0 0
\(193\) −1.73599 −0.124959 −0.0624795 0.998046i \(-0.519901\pi\)
−0.0624795 + 0.998046i \(0.519901\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2.41468 −0.172039 −0.0860194 0.996293i \(-0.527415\pi\)
−0.0860194 + 0.996293i \(0.527415\pi\)
\(198\) 0 0
\(199\) −4.05135 −0.287193 −0.143596 0.989636i \(-0.545867\pi\)
−0.143596 + 0.989636i \(0.545867\pi\)
\(200\) 0 0
\(201\) −0.0772666 −0.00544997
\(202\) 0 0
\(203\) −1.55602 −0.109211
\(204\) 0 0
\(205\) 3.86799 0.270152
\(206\) 0 0
\(207\) 11.4720 0.797357
\(208\) 0 0
\(209\) −5.45331 −0.377214
\(210\) 0 0
\(211\) 1.56666 0.107854 0.0539268 0.998545i \(-0.482826\pi\)
0.0539268 + 0.998545i \(0.482826\pi\)
\(212\) 0 0
\(213\) −1.30265 −0.0892560
\(214\) 0 0
\(215\) 2.28267 0.155677
\(216\) 0 0
\(217\) 2.00000 0.135769
\(218\) 0 0
\(219\) 3.15999 0.213533
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 17.0900 1.14443 0.572215 0.820104i \(-0.306084\pi\)
0.572215 + 0.820104i \(0.306084\pi\)
\(224\) 0 0
\(225\) −2.86799 −0.191200
\(226\) 0 0
\(227\) −4.99067 −0.331242 −0.165621 0.986189i \(-0.552963\pi\)
−0.165621 + 0.986189i \(0.552963\pi\)
\(228\) 0 0
\(229\) −28.1693 −1.86148 −0.930741 0.365680i \(-0.880836\pi\)
−0.930741 + 0.365680i \(0.880836\pi\)
\(230\) 0 0
\(231\) −0.150665 −0.00991302
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) −12.9287 −0.843374
\(236\) 0 0
\(237\) 2.56534 0.166637
\(238\) 0 0
\(239\) 20.9766 1.35687 0.678433 0.734662i \(-0.262659\pi\)
0.678433 + 0.734662i \(0.262659\pi\)
\(240\) 0 0
\(241\) 2.30133 0.148242 0.0741208 0.997249i \(-0.476385\pi\)
0.0741208 + 0.997249i \(0.476385\pi\)
\(242\) 0 0
\(243\) 9.24065 0.592788
\(244\) 0 0
\(245\) −6.86799 −0.438780
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0.169322 0.0107304
\(250\) 0 0
\(251\) −26.7967 −1.69139 −0.845695 0.533666i \(-0.820814\pi\)
−0.845695 + 0.533666i \(0.820814\pi\)
\(252\) 0 0
\(253\) −4.56534 −0.287021
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 16.8294 1.04979 0.524893 0.851168i \(-0.324105\pi\)
0.524893 + 0.851168i \(0.324105\pi\)
\(258\) 0 0
\(259\) −0.363328 −0.0225761
\(260\) 0 0
\(261\) −12.2827 −0.760278
\(262\) 0 0
\(263\) −1.91934 −0.118352 −0.0591759 0.998248i \(-0.518847\pi\)
−0.0591759 + 0.998248i \(0.518847\pi\)
\(264\) 0 0
\(265\) −6.15066 −0.377832
\(266\) 0 0
\(267\) −5.82003 −0.356180
\(268\) 0 0
\(269\) 12.3013 0.750025 0.375013 0.927020i \(-0.377638\pi\)
0.375013 + 0.927020i \(0.377638\pi\)
\(270\) 0 0
\(271\) 27.8867 1.69399 0.846997 0.531598i \(-0.178408\pi\)
0.846997 + 0.531598i \(0.178408\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.14134 0.0688252
\(276\) 0 0
\(277\) −16.3013 −0.979452 −0.489726 0.871876i \(-0.662903\pi\)
−0.489726 + 0.871876i \(0.662903\pi\)
\(278\) 0 0
\(279\) 15.7873 0.945163
\(280\) 0 0
\(281\) −8.90663 −0.531325 −0.265662 0.964066i \(-0.585591\pi\)
−0.265662 + 0.964066i \(0.585591\pi\)
\(282\) 0 0
\(283\) 15.5747 0.925818 0.462909 0.886406i \(-0.346806\pi\)
0.462909 + 0.886406i \(0.346806\pi\)
\(284\) 0 0
\(285\) −1.73599 −0.102831
\(286\) 0 0
\(287\) −1.40535 −0.0829552
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 2.17997 0.127792
\(292\) 0 0
\(293\) −10.0000 −0.584206 −0.292103 0.956387i \(-0.594355\pi\)
−0.292103 + 0.956387i \(0.594355\pi\)
\(294\) 0 0
\(295\) −7.78734 −0.453396
\(296\) 0 0
\(297\) −2.43334 −0.141197
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −0.829359 −0.0478035
\(302\) 0 0
\(303\) 4.94271 0.283951
\(304\) 0 0
\(305\) −0.546687 −0.0313032
\(306\) 0 0
\(307\) −11.3727 −0.649072 −0.324536 0.945873i \(-0.605208\pi\)
−0.324536 + 0.945873i \(0.605208\pi\)
\(308\) 0 0
\(309\) −2.24536 −0.127734
\(310\) 0 0
\(311\) −1.24065 −0.0703508 −0.0351754 0.999381i \(-0.511199\pi\)
−0.0351754 + 0.999381i \(0.511199\pi\)
\(312\) 0 0
\(313\) −12.9066 −0.729526 −0.364763 0.931100i \(-0.618850\pi\)
−0.364763 + 0.931100i \(0.618850\pi\)
\(314\) 0 0
\(315\) 1.04202 0.0587113
\(316\) 0 0
\(317\) −0.0186574 −0.00104790 −0.000523952 1.00000i \(-0.500167\pi\)
−0.000523952 1.00000i \(0.500167\pi\)
\(318\) 0 0
\(319\) 4.88797 0.273674
\(320\) 0 0
\(321\) −6.82936 −0.381178
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −1.65194 −0.0913525
\(328\) 0 0
\(329\) 4.69735 0.258973
\(330\) 0 0
\(331\) −9.24065 −0.507912 −0.253956 0.967216i \(-0.581732\pi\)
−0.253956 + 0.967216i \(0.581732\pi\)
\(332\) 0 0
\(333\) −2.86799 −0.157165
\(334\) 0 0
\(335\) −0.212663 −0.0116190
\(336\) 0 0
\(337\) 25.6226 1.39575 0.697877 0.716218i \(-0.254128\pi\)
0.697877 + 0.716218i \(0.254128\pi\)
\(338\) 0 0
\(339\) −3.20796 −0.174232
\(340\) 0 0
\(341\) −6.28267 −0.340226
\(342\) 0 0
\(343\) 5.03863 0.272061
\(344\) 0 0
\(345\) −1.45331 −0.0782437
\(346\) 0 0
\(347\) −9.45331 −0.507480 −0.253740 0.967272i \(-0.581661\pi\)
−0.253740 + 0.967272i \(0.581661\pi\)
\(348\) 0 0
\(349\) −16.2640 −0.870593 −0.435296 0.900287i \(-0.643356\pi\)
−0.435296 + 0.900287i \(0.643356\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 5.43466 0.289258 0.144629 0.989486i \(-0.453801\pi\)
0.144629 + 0.989486i \(0.453801\pi\)
\(354\) 0 0
\(355\) −3.58532 −0.190289
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 2.85866 0.150875 0.0754373 0.997151i \(-0.475965\pi\)
0.0754373 + 0.997151i \(0.475965\pi\)
\(360\) 0 0
\(361\) 3.82936 0.201545
\(362\) 0 0
\(363\) −3.52332 −0.184926
\(364\) 0 0
\(365\) 8.69735 0.455240
\(366\) 0 0
\(367\) 4.77801 0.249410 0.124705 0.992194i \(-0.460202\pi\)
0.124705 + 0.992194i \(0.460202\pi\)
\(368\) 0 0
\(369\) −11.0934 −0.577498
\(370\) 0 0
\(371\) 2.23471 0.116020
\(372\) 0 0
\(373\) 0.433337 0.0224373 0.0112187 0.999937i \(-0.496429\pi\)
0.0112187 + 0.999937i \(0.496429\pi\)
\(374\) 0 0
\(375\) 0.363328 0.0187622
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −14.8587 −0.763238 −0.381619 0.924320i \(-0.624633\pi\)
−0.381619 + 0.924320i \(0.624633\pi\)
\(380\) 0 0
\(381\) −3.90531 −0.200075
\(382\) 0 0
\(383\) 7.73599 0.395290 0.197645 0.980274i \(-0.436671\pi\)
0.197645 + 0.980274i \(0.436671\pi\)
\(384\) 0 0
\(385\) −0.414680 −0.0211340
\(386\) 0 0
\(387\) −6.54669 −0.332787
\(388\) 0 0
\(389\) −15.7546 −0.798792 −0.399396 0.916778i \(-0.630780\pi\)
−0.399396 + 0.916778i \(0.630780\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 2.00000 0.100887
\(394\) 0 0
\(395\) 7.06068 0.355261
\(396\) 0 0
\(397\) 5.58532 0.280319 0.140160 0.990129i \(-0.455238\pi\)
0.140160 + 0.990129i \(0.455238\pi\)
\(398\) 0 0
\(399\) 0.630732 0.0315761
\(400\) 0 0
\(401\) −1.11203 −0.0555322 −0.0277661 0.999614i \(-0.508839\pi\)
−0.0277661 + 0.999614i \(0.508839\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 7.82936 0.389044
\(406\) 0 0
\(407\) 1.14134 0.0565739
\(408\) 0 0
\(409\) −16.8480 −0.833081 −0.416540 0.909117i \(-0.636758\pi\)
−0.416540 + 0.909117i \(0.636758\pi\)
\(410\) 0 0
\(411\) 1.25469 0.0618891
\(412\) 0 0
\(413\) 2.82936 0.139224
\(414\) 0 0
\(415\) 0.466031 0.0228766
\(416\) 0 0
\(417\) −3.64006 −0.178255
\(418\) 0 0
\(419\) −26.8960 −1.31395 −0.656977 0.753910i \(-0.728165\pi\)
−0.656977 + 0.753910i \(0.728165\pi\)
\(420\) 0 0
\(421\) 16.8480 0.821122 0.410561 0.911833i \(-0.365333\pi\)
0.410561 + 0.911833i \(0.365333\pi\)
\(422\) 0 0
\(423\) 37.0793 1.80286
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0.198627 0.00961223
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 3.11929 0.150251 0.0751254 0.997174i \(-0.476064\pi\)
0.0751254 + 0.997174i \(0.476064\pi\)
\(432\) 0 0
\(433\) 8.43334 0.405280 0.202640 0.979253i \(-0.435048\pi\)
0.202640 + 0.979253i \(0.435048\pi\)
\(434\) 0 0
\(435\) 1.55602 0.0746052
\(436\) 0 0
\(437\) 19.1120 0.914252
\(438\) 0 0
\(439\) 25.3434 1.20957 0.604786 0.796388i \(-0.293259\pi\)
0.604786 + 0.796388i \(0.293259\pi\)
\(440\) 0 0
\(441\) 19.6974 0.937969
\(442\) 0 0
\(443\) 26.8446 1.27543 0.637713 0.770274i \(-0.279880\pi\)
0.637713 + 0.770274i \(0.279880\pi\)
\(444\) 0 0
\(445\) −16.0187 −0.759357
\(446\) 0 0
\(447\) −0.678694 −0.0321011
\(448\) 0 0
\(449\) 14.0000 0.660701 0.330350 0.943858i \(-0.392833\pi\)
0.330350 + 0.943858i \(0.392833\pi\)
\(450\) 0 0
\(451\) 4.41468 0.207879
\(452\) 0 0
\(453\) 1.62140 0.0761802
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −19.6587 −0.919596 −0.459798 0.888024i \(-0.652078\pi\)
−0.459798 + 0.888024i \(0.652078\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 38.6027 1.79791 0.898953 0.438046i \(-0.144329\pi\)
0.898953 + 0.438046i \(0.144329\pi\)
\(462\) 0 0
\(463\) 37.7546 1.75461 0.877304 0.479936i \(-0.159340\pi\)
0.877304 + 0.479936i \(0.159340\pi\)
\(464\) 0 0
\(465\) −2.00000 −0.0927478
\(466\) 0 0
\(467\) −14.9507 −0.691837 −0.345918 0.938265i \(-0.612433\pi\)
−0.345918 + 0.938265i \(0.612433\pi\)
\(468\) 0 0
\(469\) 0.0772666 0.00356784
\(470\) 0 0
\(471\) 7.32808 0.337660
\(472\) 0 0
\(473\) 2.60530 0.119792
\(474\) 0 0
\(475\) −4.77801 −0.219230
\(476\) 0 0
\(477\) 17.6401 0.807683
\(478\) 0 0
\(479\) −8.21266 −0.375246 −0.187623 0.982241i \(-0.560078\pi\)
−0.187623 + 0.982241i \(0.560078\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0.528030 0.0240262
\(484\) 0 0
\(485\) 6.00000 0.272446
\(486\) 0 0
\(487\) 27.5747 1.24953 0.624764 0.780814i \(-0.285195\pi\)
0.624764 + 0.780814i \(0.285195\pi\)
\(488\) 0 0
\(489\) −3.47197 −0.157008
\(490\) 0 0
\(491\) 25.4906 1.15038 0.575188 0.818021i \(-0.304929\pi\)
0.575188 + 0.818021i \(0.304929\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −3.27334 −0.147126
\(496\) 0 0
\(497\) 1.30265 0.0584318
\(498\) 0 0
\(499\) −21.4020 −0.958083 −0.479042 0.877792i \(-0.659016\pi\)
−0.479042 + 0.877792i \(0.659016\pi\)
\(500\) 0 0
\(501\) 5.75464 0.257098
\(502\) 0 0
\(503\) 5.45331 0.243151 0.121576 0.992582i \(-0.461205\pi\)
0.121576 + 0.992582i \(0.461205\pi\)
\(504\) 0 0
\(505\) 13.6040 0.605369
\(506\) 0 0
\(507\) −4.72327 −0.209768
\(508\) 0 0
\(509\) −36.4333 −1.61488 −0.807440 0.589950i \(-0.799147\pi\)
−0.807440 + 0.589950i \(0.799147\pi\)
\(510\) 0 0
\(511\) −3.15999 −0.139790
\(512\) 0 0
\(513\) 10.1867 0.449756
\(514\) 0 0
\(515\) −6.17997 −0.272322
\(516\) 0 0
\(517\) −14.7560 −0.648966
\(518\) 0 0
\(519\) 5.76529 0.253068
\(520\) 0 0
\(521\) 17.6040 0.771244 0.385622 0.922657i \(-0.373987\pi\)
0.385622 + 0.922657i \(0.373987\pi\)
\(522\) 0 0
\(523\) 5.71733 0.250001 0.125001 0.992157i \(-0.460107\pi\)
0.125001 + 0.992157i \(0.460107\pi\)
\(524\) 0 0
\(525\) −0.132007 −0.00576128
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 22.3340 0.969214
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −18.7967 −0.812651
\(536\) 0 0
\(537\) 3.39470 0.146492
\(538\) 0 0
\(539\) −7.83869 −0.337636
\(540\) 0 0
\(541\) −8.90663 −0.382926 −0.191463 0.981500i \(-0.561323\pi\)
−0.191463 + 0.981500i \(0.561323\pi\)
\(542\) 0 0
\(543\) −6.99868 −0.300342
\(544\) 0 0
\(545\) −4.54669 −0.194759
\(546\) 0 0
\(547\) −21.8133 −0.932667 −0.466334 0.884609i \(-0.654425\pi\)
−0.466334 + 0.884609i \(0.654425\pi\)
\(548\) 0 0
\(549\) 1.56789 0.0669161
\(550\) 0 0
\(551\) −20.4626 −0.871738
\(552\) 0 0
\(553\) −2.56534 −0.109090
\(554\) 0 0
\(555\) 0.363328 0.0154224
\(556\) 0 0
\(557\) 41.1307 1.74276 0.871382 0.490606i \(-0.163224\pi\)
0.871382 + 0.490606i \(0.163224\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −14.5467 −0.613070 −0.306535 0.951859i \(-0.599170\pi\)
−0.306535 + 0.951859i \(0.599170\pi\)
\(564\) 0 0
\(565\) −8.82936 −0.371454
\(566\) 0 0
\(567\) −2.84463 −0.119463
\(568\) 0 0
\(569\) 24.5467 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(570\) 0 0
\(571\) 18.1693 0.760362 0.380181 0.924912i \(-0.375862\pi\)
0.380181 + 0.924912i \(0.375862\pi\)
\(572\) 0 0
\(573\) −1.20796 −0.0504631
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) 9.65872 0.402098 0.201049 0.979581i \(-0.435565\pi\)
0.201049 + 0.979581i \(0.435565\pi\)
\(578\) 0 0
\(579\) −0.630732 −0.0262123
\(580\) 0 0
\(581\) −0.169322 −0.00702467
\(582\) 0 0
\(583\) −7.01998 −0.290738
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 30.3200 1.25144 0.625720 0.780048i \(-0.284805\pi\)
0.625720 + 0.780048i \(0.284805\pi\)
\(588\) 0 0
\(589\) 26.3013 1.08373
\(590\) 0 0
\(591\) −0.877321 −0.0360882
\(592\) 0 0
\(593\) 29.8280 1.22489 0.612445 0.790513i \(-0.290186\pi\)
0.612445 + 0.790513i \(0.290186\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −1.47197 −0.0602437
\(598\) 0 0
\(599\) 7.95204 0.324911 0.162456 0.986716i \(-0.448059\pi\)
0.162456 + 0.986716i \(0.448059\pi\)
\(600\) 0 0
\(601\) −25.3947 −1.03587 −0.517936 0.855420i \(-0.673299\pi\)
−0.517936 + 0.855420i \(0.673299\pi\)
\(602\) 0 0
\(603\) 0.609917 0.0248377
\(604\) 0 0
\(605\) −9.69735 −0.394253
\(606\) 0 0
\(607\) 29.4533 1.19547 0.597737 0.801693i \(-0.296067\pi\)
0.597737 + 0.801693i \(0.296067\pi\)
\(608\) 0 0
\(609\) −0.565344 −0.0229089
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1.32131 −0.0533670 −0.0266835 0.999644i \(-0.508495\pi\)
−0.0266835 + 0.999644i \(0.508495\pi\)
\(614\) 0 0
\(615\) 1.40535 0.0566692
\(616\) 0 0
\(617\) 43.6999 1.75929 0.879646 0.475629i \(-0.157780\pi\)
0.879646 + 0.475629i \(0.157780\pi\)
\(618\) 0 0
\(619\) −0.209274 −0.00841143 −0.00420572 0.999991i \(-0.501339\pi\)
−0.00420572 + 0.999991i \(0.501339\pi\)
\(620\) 0 0
\(621\) 8.52803 0.342218
\(622\) 0 0
\(623\) 5.82003 0.233175
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −1.98134 −0.0791272
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 26.0700 1.03783 0.518915 0.854826i \(-0.326336\pi\)
0.518915 + 0.854826i \(0.326336\pi\)
\(632\) 0 0
\(633\) 0.569213 0.0226242
\(634\) 0 0
\(635\) −10.7487 −0.426549
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 10.2827 0.406776
\(640\) 0 0
\(641\) −24.0921 −0.951579 −0.475789 0.879559i \(-0.657838\pi\)
−0.475789 + 0.879559i \(0.657838\pi\)
\(642\) 0 0
\(643\) −29.5933 −1.16705 −0.583524 0.812096i \(-0.698326\pi\)
−0.583524 + 0.812096i \(0.698326\pi\)
\(644\) 0 0
\(645\) 0.829359 0.0326560
\(646\) 0 0
\(647\) −35.7733 −1.40639 −0.703197 0.710995i \(-0.748245\pi\)
−0.703197 + 0.710995i \(0.748245\pi\)
\(648\) 0 0
\(649\) −8.88797 −0.348883
\(650\) 0 0
\(651\) 0.726656 0.0284799
\(652\) 0 0
\(653\) 24.3786 0.954008 0.477004 0.878901i \(-0.341723\pi\)
0.477004 + 0.878901i \(0.341723\pi\)
\(654\) 0 0
\(655\) 5.50466 0.215085
\(656\) 0 0
\(657\) −24.9439 −0.973156
\(658\) 0 0
\(659\) 17.5454 0.683471 0.341735 0.939796i \(-0.388985\pi\)
0.341735 + 0.939796i \(0.388985\pi\)
\(660\) 0 0
\(661\) 32.0560 1.24683 0.623416 0.781890i \(-0.285744\pi\)
0.623416 + 0.781890i \(0.285744\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.73599 0.0673186
\(666\) 0 0
\(667\) −17.1307 −0.663303
\(668\) 0 0
\(669\) 6.20927 0.240064
\(670\) 0 0
\(671\) −0.623954 −0.0240875
\(672\) 0 0
\(673\) 23.2440 0.895992 0.447996 0.894036i \(-0.352138\pi\)
0.447996 + 0.894036i \(0.352138\pi\)
\(674\) 0 0
\(675\) −2.13201 −0.0820610
\(676\) 0 0
\(677\) 20.0734 0.771483 0.385742 0.922607i \(-0.373946\pi\)
0.385742 + 0.922607i \(0.373946\pi\)
\(678\) 0 0
\(679\) −2.17997 −0.0836595
\(680\) 0 0
\(681\) −1.81325 −0.0694840
\(682\) 0 0
\(683\) 25.4533 0.973944 0.486972 0.873418i \(-0.338101\pi\)
0.486972 + 0.873418i \(0.338101\pi\)
\(684\) 0 0
\(685\) 3.45331 0.131944
\(686\) 0 0
\(687\) −10.2347 −0.390479
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −1.19608 −0.0455009 −0.0227505 0.999741i \(-0.507242\pi\)
−0.0227505 + 0.999741i \(0.507242\pi\)
\(692\) 0 0
\(693\) 1.18930 0.0451777
\(694\) 0 0
\(695\) −10.0187 −0.380029
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −2.17997 −0.0824540
\(700\) 0 0
\(701\) 29.9414 1.13087 0.565435 0.824793i \(-0.308708\pi\)
0.565435 + 0.824793i \(0.308708\pi\)
\(702\) 0 0
\(703\) −4.77801 −0.180206
\(704\) 0 0
\(705\) −4.69735 −0.176913
\(706\) 0 0
\(707\) −4.94271 −0.185890
\(708\) 0 0
\(709\) −26.0959 −0.980053 −0.490026 0.871708i \(-0.663013\pi\)
−0.490026 + 0.871708i \(0.663013\pi\)
\(710\) 0 0
\(711\) −20.2500 −0.759434
\(712\) 0 0
\(713\) 22.0187 0.824605
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 7.62140 0.284627
\(718\) 0 0
\(719\) −18.5360 −0.691278 −0.345639 0.938368i \(-0.612338\pi\)
−0.345639 + 0.938368i \(0.612338\pi\)
\(720\) 0 0
\(721\) 2.24536 0.0836215
\(722\) 0 0
\(723\) 0.836138 0.0310963
\(724\) 0 0
\(725\) 4.28267 0.159054
\(726\) 0 0
\(727\) 36.0959 1.33872 0.669362 0.742937i \(-0.266568\pi\)
0.669362 + 0.742937i \(0.266568\pi\)
\(728\) 0 0
\(729\) −20.1307 −0.745581
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 8.13201 0.300363 0.150181 0.988658i \(-0.452014\pi\)
0.150181 + 0.988658i \(0.452014\pi\)
\(734\) 0 0
\(735\) −2.49534 −0.0920418
\(736\) 0 0
\(737\) −0.242720 −0.00894072
\(738\) 0 0
\(739\) −0.942709 −0.0346781 −0.0173391 0.999850i \(-0.505519\pi\)
−0.0173391 + 0.999850i \(0.505519\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 34.8446 1.27833 0.639163 0.769072i \(-0.279281\pi\)
0.639163 + 0.769072i \(0.279281\pi\)
\(744\) 0 0
\(745\) −1.86799 −0.0684380
\(746\) 0 0
\(747\) −1.33657 −0.0489027
\(748\) 0 0
\(749\) 6.82936 0.249539
\(750\) 0 0
\(751\) 10.8587 0.396238 0.198119 0.980178i \(-0.436517\pi\)
0.198119 + 0.980178i \(0.436517\pi\)
\(752\) 0 0
\(753\) −9.73599 −0.354799
\(754\) 0 0
\(755\) 4.46264 0.162412
\(756\) 0 0
\(757\) −38.3386 −1.39344 −0.696721 0.717342i \(-0.745358\pi\)
−0.696721 + 0.717342i \(0.745358\pi\)
\(758\) 0 0
\(759\) −1.65872 −0.0602077
\(760\) 0 0
\(761\) −24.4333 −0.885708 −0.442854 0.896594i \(-0.646034\pi\)
−0.442854 + 0.896594i \(0.646034\pi\)
\(762\) 0 0
\(763\) 1.65194 0.0598042
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 8.84802 0.319068 0.159534 0.987192i \(-0.449001\pi\)
0.159534 + 0.987192i \(0.449001\pi\)
\(770\) 0 0
\(771\) 6.11458 0.220211
\(772\) 0 0
\(773\) 5.58532 0.200890 0.100445 0.994943i \(-0.467973\pi\)
0.100445 + 0.994943i \(0.467973\pi\)
\(774\) 0 0
\(775\) −5.50466 −0.197733
\(776\) 0 0
\(777\) −0.132007 −0.00473574
\(778\) 0 0
\(779\) −18.4813 −0.662161
\(780\) 0 0
\(781\) −4.09206 −0.146425
\(782\) 0 0
\(783\) −9.13069 −0.326304
\(784\) 0 0
\(785\) 20.1693 0.719874
\(786\) 0 0
\(787\) −27.4754 −0.979391 −0.489695 0.871894i \(-0.662892\pi\)
−0.489695 + 0.871894i \(0.662892\pi\)
\(788\) 0 0
\(789\) −0.697352 −0.0248264
\(790\) 0 0
\(791\) 3.20796 0.114062
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −2.23471 −0.0792570
\(796\) 0 0
\(797\) −2.90663 −0.102958 −0.0514790 0.998674i \(-0.516394\pi\)
−0.0514790 + 0.998674i \(0.516394\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 45.9414 1.62326
\(802\) 0 0
\(803\) 9.92660 0.350302
\(804\) 0 0
\(805\) 1.45331 0.0512226
\(806\) 0 0
\(807\) 4.46942 0.157331
\(808\) 0 0
\(809\) 2.82936 0.0994750 0.0497375 0.998762i \(-0.484162\pi\)
0.0497375 + 0.998762i \(0.484162\pi\)
\(810\) 0 0
\(811\) −37.9053 −1.33104 −0.665518 0.746382i \(-0.731789\pi\)
−0.665518 + 0.746382i \(0.731789\pi\)
\(812\) 0 0
\(813\) 10.1320 0.355345
\(814\) 0 0
\(815\) −9.55602 −0.334733
\(816\) 0 0
\(817\) −10.9066 −0.381575
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −14.2093 −0.495907 −0.247954 0.968772i \(-0.579758\pi\)
−0.247954 + 0.968772i \(0.579758\pi\)
\(822\) 0 0
\(823\) −26.3713 −0.919247 −0.459624 0.888114i \(-0.652016\pi\)
−0.459624 + 0.888114i \(0.652016\pi\)
\(824\) 0 0
\(825\) 0.414680 0.0144373
\(826\) 0 0
\(827\) −8.10270 −0.281759 −0.140879 0.990027i \(-0.544993\pi\)
−0.140879 + 0.990027i \(0.544993\pi\)
\(828\) 0 0
\(829\) −5.20796 −0.180880 −0.0904399 0.995902i \(-0.528827\pi\)
−0.0904399 + 0.995902i \(0.528827\pi\)
\(830\) 0 0
\(831\) −5.92273 −0.205457
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 15.8387 0.548120
\(836\) 0 0
\(837\) 11.7360 0.405655
\(838\) 0 0
\(839\) −9.98134 −0.344594 −0.172297 0.985045i \(-0.555119\pi\)
−0.172297 + 0.985045i \(0.555119\pi\)
\(840\) 0 0
\(841\) −10.6587 −0.367542
\(842\) 0 0
\(843\) −3.23603 −0.111455
\(844\) 0 0
\(845\) −13.0000 −0.447214
\(846\) 0 0
\(847\) 3.52332 0.121063
\(848\) 0 0
\(849\) 5.65872 0.194207
\(850\) 0 0
\(851\) −4.00000 −0.137118
\(852\) 0 0
\(853\) 19.4320 0.665340 0.332670 0.943043i \(-0.392051\pi\)
0.332670 + 0.943043i \(0.392051\pi\)
\(854\) 0 0
\(855\) 13.7033 0.468643
\(856\) 0 0
\(857\) 18.5653 0.634180 0.317090 0.948395i \(-0.397294\pi\)
0.317090 + 0.948395i \(0.397294\pi\)
\(858\) 0 0
\(859\) −17.7687 −0.606260 −0.303130 0.952949i \(-0.598032\pi\)
−0.303130 + 0.952949i \(0.598032\pi\)
\(860\) 0 0
\(861\) −0.510604 −0.0174013
\(862\) 0 0
\(863\) 39.4647 1.34339 0.671697 0.740826i \(-0.265566\pi\)
0.671697 + 0.740826i \(0.265566\pi\)
\(864\) 0 0
\(865\) 15.8680 0.539528
\(866\) 0 0
\(867\) −6.17658 −0.209768
\(868\) 0 0
\(869\) 8.05861 0.273370
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −17.2080 −0.582401
\(874\) 0 0
\(875\) −0.363328 −0.0122827
\(876\) 0 0
\(877\) 10.0373 0.338936 0.169468 0.985536i \(-0.445795\pi\)
0.169468 + 0.985536i \(0.445795\pi\)
\(878\) 0 0
\(879\) −3.63328 −0.122548
\(880\) 0 0
\(881\) 52.8667 1.78112 0.890562 0.454862i \(-0.150312\pi\)
0.890562 + 0.454862i \(0.150312\pi\)
\(882\) 0 0
\(883\) −33.5560 −1.12925 −0.564625 0.825348i \(-0.690979\pi\)
−0.564625 + 0.825348i \(0.690979\pi\)
\(884\) 0 0
\(885\) −2.82936 −0.0951079
\(886\) 0 0
\(887\) 53.3259 1.79051 0.895255 0.445555i \(-0.146994\pi\)
0.895255 + 0.445555i \(0.146994\pi\)
\(888\) 0 0
\(889\) 3.90531 0.130980
\(890\) 0 0
\(891\) 8.93593 0.299365
\(892\) 0 0
\(893\) 61.7733 2.06716
\(894\) 0 0
\(895\) 9.34335 0.312314
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −23.5747 −0.786259
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −0.301330 −0.0100276
\(904\) 0 0
\(905\) −19.2627 −0.640314
\(906\) 0 0
\(907\) 14.2173 0.472077 0.236039 0.971744i \(-0.424151\pi\)
0.236039 + 0.971744i \(0.424151\pi\)
\(908\) 0 0
\(909\) −39.0161 −1.29408
\(910\) 0 0
\(911\) 8.08867 0.267989 0.133995 0.990982i \(-0.457219\pi\)
0.133995 + 0.990982i \(0.457219\pi\)
\(912\) 0 0
\(913\) 0.531898 0.0176033
\(914\) 0 0
\(915\) −0.198627 −0.00656640
\(916\) 0 0
\(917\) −2.00000 −0.0660458
\(918\) 0 0
\(919\) −39.3620 −1.29843 −0.649216 0.760604i \(-0.724903\pi\)
−0.649216 + 0.760604i \(0.724903\pi\)
\(920\) 0 0
\(921\) −4.13201 −0.136154
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 1.00000 0.0328798
\(926\) 0 0
\(927\) 17.7241 0.582136
\(928\) 0 0
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) 32.8153 1.07548
\(932\) 0 0
\(933\) −0.450763 −0.0147573
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −0.0946926 −0.00309347 −0.00154674 0.999999i \(-0.500492\pi\)
−0.00154674 + 0.999999i \(0.500492\pi\)
\(938\) 0 0
\(939\) −4.68934 −0.153031
\(940\) 0 0
\(941\) −20.8667 −0.680234 −0.340117 0.940383i \(-0.610467\pi\)
−0.340117 + 0.940383i \(0.610467\pi\)
\(942\) 0 0
\(943\) −15.4720 −0.503837
\(944\) 0 0
\(945\) 0.774618 0.0251983
\(946\) 0 0
\(947\) −18.4813 −0.600562 −0.300281 0.953851i \(-0.597080\pi\)
−0.300281 + 0.953851i \(0.597080\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −0.00677877 −0.000219817 0
\(952\) 0 0
\(953\) −0.660037 −0.0213807 −0.0106903 0.999943i \(-0.503403\pi\)
−0.0106903 + 0.999943i \(0.503403\pi\)
\(954\) 0 0
\(955\) −3.32469 −0.107585
\(956\) 0 0
\(957\) 1.77594 0.0574079
\(958\) 0 0
\(959\) −1.25469 −0.0405159
\(960\) 0 0
\(961\) −0.698670 −0.0225378
\(962\) 0 0
\(963\) 53.9087 1.73718
\(964\) 0 0
\(965\) −1.73599 −0.0558833
\(966\) 0 0
\(967\) −48.4413 −1.55777 −0.778884 0.627168i \(-0.784214\pi\)
−0.778884 + 0.627168i \(0.784214\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 15.1120 0.484968 0.242484 0.970155i \(-0.422038\pi\)
0.242484 + 0.970155i \(0.422038\pi\)
\(972\) 0 0
\(973\) 3.64006 0.116695
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 10.2640 0.328375 0.164187 0.986429i \(-0.447500\pi\)
0.164187 + 0.986429i \(0.447500\pi\)
\(978\) 0 0
\(979\) −18.2827 −0.584317
\(980\) 0 0
\(981\) 13.0399 0.416331
\(982\) 0 0
\(983\) −7.10864 −0.226730 −0.113365 0.993553i \(-0.536163\pi\)
−0.113365 + 0.993553i \(0.536163\pi\)
\(984\) 0 0
\(985\) −2.41468 −0.0769381
\(986\) 0 0
\(987\) 1.70668 0.0543242
\(988\) 0 0
\(989\) −9.13069 −0.290339
\(990\) 0 0
\(991\) −3.94865 −0.125433 −0.0627165 0.998031i \(-0.519976\pi\)
−0.0627165 + 0.998031i \(0.519976\pi\)
\(992\) 0 0
\(993\) −3.35739 −0.106544
\(994\) 0 0
\(995\) −4.05135 −0.128436
\(996\) 0 0
\(997\) −49.7733 −1.57634 −0.788168 0.615460i \(-0.788970\pi\)
−0.788168 + 0.615460i \(0.788970\pi\)
\(998\) 0 0
\(999\) −2.13201 −0.0674537
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 2960.2.a.s.1.2 3
4.3 odd 2 1480.2.a.f.1.2 3
20.19 odd 2 7400.2.a.m.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1480.2.a.f.1.2 3 4.3 odd 2
2960.2.a.s.1.2 3 1.1 even 1 trivial
7400.2.a.m.1.2 3 20.19 odd 2