Properties

Label 8016.2.a.o.1.2
Level $8016$
Weight $2$
Character 8016.1
Self dual yes
Analytic conductor $64.008$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8016.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.0080822603\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.2777.1
Defining polynomial: \(x^{4} - x^{3} - 4 x^{2} + x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1002)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.679643\) of defining polynomial
Character \(\chi\) \(=\) 8016.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +0.416566 q^{5} -4.65960 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +0.416566 q^{5} -4.65960 q^{7} +1.00000 q^{9} +0.283116 q^{13} +0.416566 q^{15} +0.640714 q^{17} +5.88544 q^{19} -4.65960 q^{21} -5.88544 q^{23} -4.82647 q^{25} +1.00000 q^{27} +5.16687 q^{29} -1.22584 q^{31} -1.94103 q^{35} -6.82816 q^{37} +0.283116 q^{39} +12.4116 q^{41} +3.07617 q^{43} +0.416566 q^{45} -7.82647 q^{47} +14.7119 q^{49} +0.640714 q^{51} -5.46888 q^{53} +5.88544 q^{57} -3.20962 q^{59} -1.16687 q^{61} -4.65960 q^{63} +0.117936 q^{65} +2.29863 q^{67} -5.88544 q^{69} +8.60401 q^{71} +2.26690 q^{73} -4.82647 q^{75} +7.69302 q^{79} +1.00000 q^{81} -9.27646 q^{83} +0.266899 q^{85} +5.16687 q^{87} +3.94103 q^{89} -1.31921 q^{91} -1.22584 q^{93} +2.45167 q^{95} +8.99334 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + 5q^{5} - q^{7} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + 5q^{5} - q^{7} + 4q^{9} + 8q^{13} + 5q^{15} + 10q^{17} + 2q^{19} - q^{21} - 2q^{23} + 5q^{25} + 4q^{27} + 14q^{29} - q^{31} - 5q^{35} + 5q^{37} + 8q^{39} + 14q^{41} - 2q^{43} + 5q^{45} - 7q^{47} + 13q^{49} + 10q^{51} + 3q^{53} + 2q^{57} + 5q^{59} + 2q^{61} - q^{63} + 6q^{65} + 7q^{67} - 2q^{69} - 2q^{71} + 2q^{73} + 5q^{75} + 10q^{79} + 4q^{81} - 13q^{83} - 6q^{85} + 14q^{87} + 13q^{89} + 30q^{91} - q^{93} + 2q^{95} + 5q^{97} + O(q^{100}) \)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 0.416566 0.186294 0.0931469 0.995652i \(-0.470307\pi\)
0.0931469 + 0.995652i \(0.470307\pi\)
\(6\) 0 0
\(7\) −4.65960 −1.76116 −0.880582 0.473893i \(-0.842848\pi\)
−0.880582 + 0.473893i \(0.842848\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 0.283116 0.0785223 0.0392611 0.999229i \(-0.487500\pi\)
0.0392611 + 0.999229i \(0.487500\pi\)
\(14\) 0 0
\(15\) 0.416566 0.107557
\(16\) 0 0
\(17\) 0.640714 0.155396 0.0776979 0.996977i \(-0.475243\pi\)
0.0776979 + 0.996977i \(0.475243\pi\)
\(18\) 0 0
\(19\) 5.88544 1.35021 0.675106 0.737720i \(-0.264098\pi\)
0.675106 + 0.737720i \(0.264098\pi\)
\(20\) 0 0
\(21\) −4.65960 −1.01681
\(22\) 0 0
\(23\) −5.88544 −1.22720 −0.613600 0.789617i \(-0.710279\pi\)
−0.613600 + 0.789617i \(0.710279\pi\)
\(24\) 0 0
\(25\) −4.82647 −0.965295
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 5.16687 0.959463 0.479732 0.877415i \(-0.340734\pi\)
0.479732 + 0.877415i \(0.340734\pi\)
\(30\) 0 0
\(31\) −1.22584 −0.220167 −0.110083 0.993922i \(-0.535112\pi\)
−0.110083 + 0.993922i \(0.535112\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.94103 −0.328094
\(36\) 0 0
\(37\) −6.82816 −1.12254 −0.561271 0.827632i \(-0.689688\pi\)
−0.561271 + 0.827632i \(0.689688\pi\)
\(38\) 0 0
\(39\) 0.283116 0.0453349
\(40\) 0 0
\(41\) 12.4116 1.93837 0.969183 0.246343i \(-0.0792288\pi\)
0.969183 + 0.246343i \(0.0792288\pi\)
\(42\) 0 0
\(43\) 3.07617 0.469112 0.234556 0.972103i \(-0.424636\pi\)
0.234556 + 0.972103i \(0.424636\pi\)
\(44\) 0 0
\(45\) 0.416566 0.0620980
\(46\) 0 0
\(47\) −7.82647 −1.14161 −0.570804 0.821086i \(-0.693368\pi\)
−0.570804 + 0.821086i \(0.693368\pi\)
\(48\) 0 0
\(49\) 14.7119 2.10170
\(50\) 0 0
\(51\) 0.640714 0.0897179
\(52\) 0 0
\(53\) −5.46888 −0.751208 −0.375604 0.926780i \(-0.622565\pi\)
−0.375604 + 0.926780i \(0.622565\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 5.88544 0.779546
\(58\) 0 0
\(59\) −3.20962 −0.417857 −0.208928 0.977931i \(-0.566998\pi\)
−0.208928 + 0.977931i \(0.566998\pi\)
\(60\) 0 0
\(61\) −1.16687 −0.149402 −0.0747011 0.997206i \(-0.523800\pi\)
−0.0747011 + 0.997206i \(0.523800\pi\)
\(62\) 0 0
\(63\) −4.65960 −0.587055
\(64\) 0 0
\(65\) 0.117936 0.0146282
\(66\) 0 0
\(67\) 2.29863 0.280822 0.140411 0.990093i \(-0.455158\pi\)
0.140411 + 0.990093i \(0.455158\pi\)
\(68\) 0 0
\(69\) −5.88544 −0.708524
\(70\) 0 0
\(71\) 8.60401 1.02111 0.510554 0.859846i \(-0.329440\pi\)
0.510554 + 0.859846i \(0.329440\pi\)
\(72\) 0 0
\(73\) 2.26690 0.265321 0.132660 0.991162i \(-0.457648\pi\)
0.132660 + 0.991162i \(0.457648\pi\)
\(74\) 0 0
\(75\) −4.82647 −0.557313
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 7.69302 0.865533 0.432766 0.901506i \(-0.357538\pi\)
0.432766 + 0.901506i \(0.357538\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −9.27646 −1.01822 −0.509112 0.860700i \(-0.670026\pi\)
−0.509112 + 0.860700i \(0.670026\pi\)
\(84\) 0 0
\(85\) 0.266899 0.0289493
\(86\) 0 0
\(87\) 5.16687 0.553946
\(88\) 0 0
\(89\) 3.94103 0.417749 0.208874 0.977943i \(-0.433020\pi\)
0.208874 + 0.977943i \(0.433020\pi\)
\(90\) 0 0
\(91\) −1.31921 −0.138291
\(92\) 0 0
\(93\) −1.22584 −0.127113
\(94\) 0 0
\(95\) 2.45167 0.251536
\(96\) 0 0
\(97\) 8.99334 0.913135 0.456568 0.889689i \(-0.349079\pi\)
0.456568 + 0.889689i \(0.349079\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −12.3364 −1.22752 −0.613759 0.789493i \(-0.710344\pi\)
−0.613759 + 0.789493i \(0.710344\pi\)
\(102\) 0 0
\(103\) 12.3448 1.21637 0.608183 0.793797i \(-0.291899\pi\)
0.608183 + 0.793797i \(0.291899\pi\)
\(104\) 0 0
\(105\) −1.94103 −0.189425
\(106\) 0 0
\(107\) 19.0867 1.84518 0.922591 0.385779i \(-0.126067\pi\)
0.922591 + 0.385779i \(0.126067\pi\)
\(108\) 0 0
\(109\) 17.2553 1.65276 0.826378 0.563116i \(-0.190398\pi\)
0.826378 + 0.563116i \(0.190398\pi\)
\(110\) 0 0
\(111\) −6.82816 −0.648100
\(112\) 0 0
\(113\) 2.91099 0.273843 0.136921 0.990582i \(-0.456279\pi\)
0.136921 + 0.990582i \(0.456279\pi\)
\(114\) 0 0
\(115\) −2.45167 −0.228620
\(116\) 0 0
\(117\) 0.283116 0.0261741
\(118\) 0 0
\(119\) −2.98547 −0.273678
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 12.4116 1.11912
\(124\) 0 0
\(125\) −4.09337 −0.366122
\(126\) 0 0
\(127\) 21.2636 1.88684 0.943421 0.331599i \(-0.107588\pi\)
0.943421 + 0.331599i \(0.107588\pi\)
\(128\) 0 0
\(129\) 3.07617 0.270842
\(130\) 0 0
\(131\) 14.2030 1.24092 0.620459 0.784239i \(-0.286946\pi\)
0.620459 + 0.784239i \(0.286946\pi\)
\(132\) 0 0
\(133\) −27.4238 −2.37795
\(134\) 0 0
\(135\) 0.416566 0.0358523
\(136\) 0 0
\(137\) −10.5484 −0.901213 −0.450606 0.892723i \(-0.648792\pi\)
−0.450606 + 0.892723i \(0.648792\pi\)
\(138\) 0 0
\(139\) −11.9550 −1.01401 −0.507003 0.861944i \(-0.669247\pi\)
−0.507003 + 0.861944i \(0.669247\pi\)
\(140\) 0 0
\(141\) −7.82647 −0.659108
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 2.15234 0.178742
\(146\) 0 0
\(147\) 14.7119 1.21342
\(148\) 0 0
\(149\) 3.58343 0.293566 0.146783 0.989169i \(-0.453108\pi\)
0.146783 + 0.989169i \(0.453108\pi\)
\(150\) 0 0
\(151\) 3.74074 0.304417 0.152209 0.988348i \(-0.451361\pi\)
0.152209 + 0.988348i \(0.451361\pi\)
\(152\) 0 0
\(153\) 0.640714 0.0517986
\(154\) 0 0
\(155\) −0.510642 −0.0410157
\(156\) 0 0
\(157\) −13.7709 −1.09904 −0.549518 0.835482i \(-0.685189\pi\)
−0.549518 + 0.835482i \(0.685189\pi\)
\(158\) 0 0
\(159\) −5.46888 −0.433710
\(160\) 0 0
\(161\) 27.4238 2.16130
\(162\) 0 0
\(163\) −11.2398 −0.880366 −0.440183 0.897908i \(-0.645086\pi\)
−0.440183 + 0.897908i \(0.645086\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.00000 0.0773823
\(168\) 0 0
\(169\) −12.9198 −0.993834
\(170\) 0 0
\(171\) 5.88544 0.450071
\(172\) 0 0
\(173\) 1.16687 0.0887154 0.0443577 0.999016i \(-0.485876\pi\)
0.0443577 + 0.999016i \(0.485876\pi\)
\(174\) 0 0
\(175\) 22.4895 1.70004
\(176\) 0 0
\(177\) −3.20962 −0.241250
\(178\) 0 0
\(179\) −7.77088 −0.580823 −0.290412 0.956902i \(-0.593792\pi\)
−0.290412 + 0.956902i \(0.593792\pi\)
\(180\) 0 0
\(181\) 10.2570 0.762394 0.381197 0.924494i \(-0.375512\pi\)
0.381197 + 0.924494i \(0.375512\pi\)
\(182\) 0 0
\(183\) −1.16687 −0.0862574
\(184\) 0 0
\(185\) −2.84438 −0.209123
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −4.65960 −0.338936
\(190\) 0 0
\(191\) 13.5630 0.981381 0.490690 0.871334i \(-0.336745\pi\)
0.490690 + 0.871334i \(0.336745\pi\)
\(192\) 0 0
\(193\) −2.56623 −0.184721 −0.0923607 0.995726i \(-0.529441\pi\)
−0.0923607 + 0.995726i \(0.529441\pi\)
\(194\) 0 0
\(195\) 0.117936 0.00844561
\(196\) 0 0
\(197\) −2.92383 −0.208314 −0.104157 0.994561i \(-0.533214\pi\)
−0.104157 + 0.994561i \(0.533214\pi\)
\(198\) 0 0
\(199\) 3.43377 0.243413 0.121707 0.992566i \(-0.461163\pi\)
0.121707 + 0.992566i \(0.461163\pi\)
\(200\) 0 0
\(201\) 2.29863 0.162133
\(202\) 0 0
\(203\) −24.0756 −1.68977
\(204\) 0 0
\(205\) 5.17025 0.361106
\(206\) 0 0
\(207\) −5.88544 −0.409066
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 21.8854 1.50666 0.753328 0.657645i \(-0.228447\pi\)
0.753328 + 0.657645i \(0.228447\pi\)
\(212\) 0 0
\(213\) 8.60401 0.589537
\(214\) 0 0
\(215\) 1.28143 0.0873926
\(216\) 0 0
\(217\) 5.71191 0.387750
\(218\) 0 0
\(219\) 2.26690 0.153183
\(220\) 0 0
\(221\) 0.181396 0.0122020
\(222\) 0 0
\(223\) 4.65960 0.312030 0.156015 0.987755i \(-0.450135\pi\)
0.156015 + 0.987755i \(0.450135\pi\)
\(224\) 0 0
\(225\) −4.82647 −0.321765
\(226\) 0 0
\(227\) 22.4467 1.48984 0.744920 0.667154i \(-0.232488\pi\)
0.744920 + 0.667154i \(0.232488\pi\)
\(228\) 0 0
\(229\) 12.3715 0.817533 0.408766 0.912639i \(-0.365959\pi\)
0.408766 + 0.912639i \(0.365959\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 19.5940 1.28364 0.641822 0.766854i \(-0.278179\pi\)
0.641822 + 0.766854i \(0.278179\pi\)
\(234\) 0 0
\(235\) −3.26024 −0.212675
\(236\) 0 0
\(237\) 7.69302 0.499716
\(238\) 0 0
\(239\) −3.43377 −0.222112 −0.111056 0.993814i \(-0.535423\pi\)
−0.111056 + 0.993814i \(0.535423\pi\)
\(240\) 0 0
\(241\) 14.4861 0.933130 0.466565 0.884487i \(-0.345491\pi\)
0.466565 + 0.884487i \(0.345491\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 6.12848 0.391534
\(246\) 0 0
\(247\) 1.66626 0.106022
\(248\) 0 0
\(249\) −9.27646 −0.587872
\(250\) 0 0
\(251\) 26.8576 1.69524 0.847618 0.530607i \(-0.178036\pi\)
0.847618 + 0.530607i \(0.178036\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0.266899 0.0167139
\(256\) 0 0
\(257\) −3.57846 −0.223218 −0.111609 0.993752i \(-0.535600\pi\)
−0.111609 + 0.993752i \(0.535600\pi\)
\(258\) 0 0
\(259\) 31.8165 1.97698
\(260\) 0 0
\(261\) 5.16687 0.319821
\(262\) 0 0
\(263\) −5.52517 −0.340697 −0.170348 0.985384i \(-0.554489\pi\)
−0.170348 + 0.985384i \(0.554489\pi\)
\(264\) 0 0
\(265\) −2.27815 −0.139945
\(266\) 0 0
\(267\) 3.94103 0.241187
\(268\) 0 0
\(269\) 29.0039 1.76840 0.884199 0.467110i \(-0.154705\pi\)
0.884199 + 0.467110i \(0.154705\pi\)
\(270\) 0 0
\(271\) 24.0301 1.45973 0.729863 0.683593i \(-0.239584\pi\)
0.729863 + 0.683593i \(0.239584\pi\)
\(272\) 0 0
\(273\) −1.31921 −0.0798422
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2.99382 0.179881 0.0899406 0.995947i \(-0.471332\pi\)
0.0899406 + 0.995947i \(0.471332\pi\)
\(278\) 0 0
\(279\) −1.22584 −0.0733889
\(280\) 0 0
\(281\) 1.72523 0.102919 0.0514593 0.998675i \(-0.483613\pi\)
0.0514593 + 0.998675i \(0.483613\pi\)
\(282\) 0 0
\(283\) −18.5372 −1.10192 −0.550960 0.834531i \(-0.685738\pi\)
−0.550960 + 0.834531i \(0.685738\pi\)
\(284\) 0 0
\(285\) 2.45167 0.145225
\(286\) 0 0
\(287\) −57.8331 −3.41378
\(288\) 0 0
\(289\) −16.5895 −0.975852
\(290\) 0 0
\(291\) 8.99334 0.527199
\(292\) 0 0
\(293\) 14.8331 0.866561 0.433280 0.901259i \(-0.357356\pi\)
0.433280 + 0.901259i \(0.357356\pi\)
\(294\) 0 0
\(295\) −1.33702 −0.0778442
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.66626 −0.0963625
\(300\) 0 0
\(301\) −14.3337 −0.826183
\(302\) 0 0
\(303\) −12.3364 −0.708708
\(304\) 0 0
\(305\) −0.486077 −0.0278327
\(306\) 0 0
\(307\) 31.9927 1.82592 0.912961 0.408047i \(-0.133790\pi\)
0.912961 + 0.408047i \(0.133790\pi\)
\(308\) 0 0
\(309\) 12.3448 0.702269
\(310\) 0 0
\(311\) 7.07884 0.401404 0.200702 0.979652i \(-0.435678\pi\)
0.200702 + 0.979652i \(0.435678\pi\)
\(312\) 0 0
\(313\) 23.6496 1.33675 0.668376 0.743823i \(-0.266990\pi\)
0.668376 + 0.743823i \(0.266990\pi\)
\(314\) 0 0
\(315\) −1.94103 −0.109365
\(316\) 0 0
\(317\) −13.7007 −0.769506 −0.384753 0.923020i \(-0.625713\pi\)
−0.384753 + 0.923020i \(0.625713\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 19.0867 1.06532
\(322\) 0 0
\(323\) 3.77088 0.209818
\(324\) 0 0
\(325\) −1.36645 −0.0757971
\(326\) 0 0
\(327\) 17.2553 0.954219
\(328\) 0 0
\(329\) 36.4683 2.01056
\(330\) 0 0
\(331\) 9.40991 0.517215 0.258608 0.965982i \(-0.416736\pi\)
0.258608 + 0.965982i \(0.416736\pi\)
\(332\) 0 0
\(333\) −6.82816 −0.374181
\(334\) 0 0
\(335\) 0.957530 0.0523155
\(336\) 0 0
\(337\) 21.9345 1.19485 0.597423 0.801926i \(-0.296191\pi\)
0.597423 + 0.801926i \(0.296191\pi\)
\(338\) 0 0
\(339\) 2.91099 0.158103
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −35.9345 −1.94028
\(344\) 0 0
\(345\) −2.45167 −0.131994
\(346\) 0 0
\(347\) −16.6468 −0.893645 −0.446823 0.894623i \(-0.647444\pi\)
−0.446823 + 0.894623i \(0.647444\pi\)
\(348\) 0 0
\(349\) 4.49442 0.240581 0.120291 0.992739i \(-0.461617\pi\)
0.120291 + 0.992739i \(0.461617\pi\)
\(350\) 0 0
\(351\) 0.283116 0.0151116
\(352\) 0 0
\(353\) 9.72523 0.517622 0.258811 0.965928i \(-0.416669\pi\)
0.258811 + 0.965928i \(0.416669\pi\)
\(354\) 0 0
\(355\) 3.58414 0.190226
\(356\) 0 0
\(357\) −2.98547 −0.158008
\(358\) 0 0
\(359\) 9.61854 0.507647 0.253824 0.967251i \(-0.418312\pi\)
0.253824 + 0.967251i \(0.418312\pi\)
\(360\) 0 0
\(361\) 15.6384 0.823075
\(362\) 0 0
\(363\) −11.0000 −0.577350
\(364\) 0 0
\(365\) 0.944313 0.0494276
\(366\) 0 0
\(367\) −14.1523 −0.738746 −0.369373 0.929281i \(-0.620428\pi\)
−0.369373 + 0.929281i \(0.620428\pi\)
\(368\) 0 0
\(369\) 12.4116 0.646122
\(370\) 0 0
\(371\) 25.4828 1.32300
\(372\) 0 0
\(373\) −28.1149 −1.45574 −0.727868 0.685717i \(-0.759489\pi\)
−0.727868 + 0.685717i \(0.759489\pi\)
\(374\) 0 0
\(375\) −4.09337 −0.211381
\(376\) 0 0
\(377\) 1.46282 0.0753392
\(378\) 0 0
\(379\) −16.5212 −0.848636 −0.424318 0.905513i \(-0.639486\pi\)
−0.424318 + 0.905513i \(0.639486\pi\)
\(380\) 0 0
\(381\) 21.2636 1.08937
\(382\) 0 0
\(383\) 15.6907 0.801759 0.400879 0.916131i \(-0.368705\pi\)
0.400879 + 0.916131i \(0.368705\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 3.07617 0.156371
\(388\) 0 0
\(389\) −21.1842 −1.07408 −0.537040 0.843557i \(-0.680458\pi\)
−0.537040 + 0.843557i \(0.680458\pi\)
\(390\) 0 0
\(391\) −3.77088 −0.190702
\(392\) 0 0
\(393\) 14.2030 0.716445
\(394\) 0 0
\(395\) 3.20465 0.161243
\(396\) 0 0
\(397\) −2.03778 −0.102273 −0.0511367 0.998692i \(-0.516284\pi\)
−0.0511367 + 0.998692i \(0.516284\pi\)
\(398\) 0 0
\(399\) −27.4238 −1.37291
\(400\) 0 0
\(401\) −11.3971 −0.569142 −0.284571 0.958655i \(-0.591851\pi\)
−0.284571 + 0.958655i \(0.591851\pi\)
\(402\) 0 0
\(403\) −0.347054 −0.0172880
\(404\) 0 0
\(405\) 0.416566 0.0206993
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −18.1012 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(410\) 0 0
\(411\) −10.5484 −0.520315
\(412\) 0 0
\(413\) 14.9556 0.735915
\(414\) 0 0
\(415\) −3.86425 −0.189689
\(416\) 0 0
\(417\) −11.9550 −0.585437
\(418\) 0 0
\(419\) −15.2047 −0.742796 −0.371398 0.928474i \(-0.621121\pi\)
−0.371398 + 0.928474i \(0.621121\pi\)
\(420\) 0 0
\(421\) 16.2914 0.793992 0.396996 0.917820i \(-0.370053\pi\)
0.396996 + 0.917820i \(0.370053\pi\)
\(422\) 0 0
\(423\) −7.82647 −0.380536
\(424\) 0 0
\(425\) −3.09239 −0.150003
\(426\) 0 0
\(427\) 5.43715 0.263122
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −9.53052 −0.459069 −0.229534 0.973301i \(-0.573720\pi\)
−0.229534 + 0.973301i \(0.573720\pi\)
\(432\) 0 0
\(433\) −28.5027 −1.36975 −0.684876 0.728660i \(-0.740143\pi\)
−0.684876 + 0.728660i \(0.740143\pi\)
\(434\) 0 0
\(435\) 2.15234 0.103197
\(436\) 0 0
\(437\) −34.6384 −1.65698
\(438\) 0 0
\(439\) 38.0222 1.81470 0.907350 0.420377i \(-0.138102\pi\)
0.907350 + 0.420377i \(0.138102\pi\)
\(440\) 0 0
\(441\) 14.7119 0.700567
\(442\) 0 0
\(443\) −6.29633 −0.299148 −0.149574 0.988751i \(-0.547790\pi\)
−0.149574 + 0.988751i \(0.547790\pi\)
\(444\) 0 0
\(445\) 1.64170 0.0778240
\(446\) 0 0
\(447\) 3.58343 0.169491
\(448\) 0 0
\(449\) 20.0967 0.948424 0.474212 0.880411i \(-0.342733\pi\)
0.474212 + 0.880411i \(0.342733\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 3.74074 0.175756
\(454\) 0 0
\(455\) −0.549537 −0.0257627
\(456\) 0 0
\(457\) 30.0278 1.40464 0.702322 0.711860i \(-0.252147\pi\)
0.702322 + 0.711860i \(0.252147\pi\)
\(458\) 0 0
\(459\) 0.640714 0.0299060
\(460\) 0 0
\(461\) −1.21459 −0.0565691 −0.0282845 0.999600i \(-0.509004\pi\)
−0.0282845 + 0.999600i \(0.509004\pi\)
\(462\) 0 0
\(463\) −6.80637 −0.316319 −0.158159 0.987414i \(-0.550556\pi\)
−0.158159 + 0.987414i \(0.550556\pi\)
\(464\) 0 0
\(465\) −0.510642 −0.0236804
\(466\) 0 0
\(467\) −16.5563 −0.766134 −0.383067 0.923721i \(-0.625132\pi\)
−0.383067 + 0.923721i \(0.625132\pi\)
\(468\) 0 0
\(469\) −10.7107 −0.494574
\(470\) 0 0
\(471\) −13.7709 −0.634529
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −28.4059 −1.30335
\(476\) 0 0
\(477\) −5.46888 −0.250403
\(478\) 0 0
\(479\) −36.6629 −1.67517 −0.837585 0.546307i \(-0.816033\pi\)
−0.837585 + 0.546307i \(0.816033\pi\)
\(480\) 0 0
\(481\) −1.93316 −0.0881446
\(482\) 0 0
\(483\) 27.4238 1.24783
\(484\) 0 0
\(485\) 3.74632 0.170112
\(486\) 0 0
\(487\) −9.01223 −0.408383 −0.204192 0.978931i \(-0.565457\pi\)
−0.204192 + 0.978931i \(0.565457\pi\)
\(488\) 0 0
\(489\) −11.2398 −0.508279
\(490\) 0 0
\(491\) 34.5430 1.55890 0.779451 0.626463i \(-0.215498\pi\)
0.779451 + 0.626463i \(0.215498\pi\)
\(492\) 0 0
\(493\) 3.31048 0.149097
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −40.0913 −1.79834
\(498\) 0 0
\(499\) 35.3331 1.58173 0.790864 0.611992i \(-0.209631\pi\)
0.790864 + 0.611992i \(0.209631\pi\)
\(500\) 0 0
\(501\) 1.00000 0.0446767
\(502\) 0 0
\(503\) 14.0245 0.625320 0.312660 0.949865i \(-0.398780\pi\)
0.312660 + 0.949865i \(0.398780\pi\)
\(504\) 0 0
\(505\) −5.13893 −0.228679
\(506\) 0 0
\(507\) −12.9198 −0.573790
\(508\) 0 0
\(509\) −7.01453 −0.310913 −0.155457 0.987843i \(-0.549685\pi\)
−0.155457 + 0.987843i \(0.549685\pi\)
\(510\) 0 0
\(511\) −10.5629 −0.467273
\(512\) 0 0
\(513\) 5.88544 0.259849
\(514\) 0 0
\(515\) 5.14240 0.226601
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 1.16687 0.0512198
\(520\) 0 0
\(521\) 3.70634 0.162378 0.0811889 0.996699i \(-0.474128\pi\)
0.0811889 + 0.996699i \(0.474128\pi\)
\(522\) 0 0
\(523\) −34.4417 −1.50603 −0.753016 0.658002i \(-0.771402\pi\)
−0.753016 + 0.658002i \(0.771402\pi\)
\(524\) 0 0
\(525\) 22.4895 0.981520
\(526\) 0 0
\(527\) −0.785410 −0.0342130
\(528\) 0 0
\(529\) 11.6384 0.506018
\(530\) 0 0
\(531\) −3.20962 −0.139286
\(532\) 0 0
\(533\) 3.51392 0.152205
\(534\) 0 0
\(535\) 7.95087 0.343746
\(536\) 0 0
\(537\) −7.77088 −0.335338
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 2.07378 0.0891587 0.0445793 0.999006i \(-0.485805\pi\)
0.0445793 + 0.999006i \(0.485805\pi\)
\(542\) 0 0
\(543\) 10.2570 0.440168
\(544\) 0 0
\(545\) 7.18796 0.307898
\(546\) 0 0
\(547\) −37.0928 −1.58597 −0.792986 0.609240i \(-0.791475\pi\)
−0.792986 + 0.609240i \(0.791475\pi\)
\(548\) 0 0
\(549\) −1.16687 −0.0498007
\(550\) 0 0
\(551\) 30.4093 1.29548
\(552\) 0 0
\(553\) −35.8464 −1.52435
\(554\) 0 0
\(555\) −2.84438 −0.120737
\(556\) 0 0
\(557\) 8.75635 0.371019 0.185509 0.982643i \(-0.440607\pi\)
0.185509 + 0.982643i \(0.440607\pi\)
\(558\) 0 0
\(559\) 0.870913 0.0368357
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 2.41924 0.101959 0.0509794 0.998700i \(-0.483766\pi\)
0.0509794 + 0.998700i \(0.483766\pi\)
\(564\) 0 0
\(565\) 1.21262 0.0510153
\(566\) 0 0
\(567\) −4.65960 −0.195685
\(568\) 0 0
\(569\) −1.68627 −0.0706920 −0.0353460 0.999375i \(-0.511253\pi\)
−0.0353460 + 0.999375i \(0.511253\pi\)
\(570\) 0 0
\(571\) 1.66359 0.0696190 0.0348095 0.999394i \(-0.488918\pi\)
0.0348095 + 0.999394i \(0.488918\pi\)
\(572\) 0 0
\(573\) 13.5630 0.566600
\(574\) 0 0
\(575\) 28.4059 1.18461
\(576\) 0 0
\(577\) −31.7020 −1.31977 −0.659885 0.751366i \(-0.729395\pi\)
−0.659885 + 0.751366i \(0.729395\pi\)
\(578\) 0 0
\(579\) −2.56623 −0.106649
\(580\) 0 0
\(581\) 43.2246 1.79326
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0.117936 0.00487607
\(586\) 0 0
\(587\) −23.0817 −0.952686 −0.476343 0.879260i \(-0.658038\pi\)
−0.476343 + 0.879260i \(0.658038\pi\)
\(588\) 0 0
\(589\) −7.21459 −0.297272
\(590\) 0 0
\(591\) −2.92383 −0.120270
\(592\) 0 0
\(593\) 21.0156 0.863008 0.431504 0.902111i \(-0.357983\pi\)
0.431504 + 0.902111i \(0.357983\pi\)
\(594\) 0 0
\(595\) −1.24365 −0.0509845
\(596\) 0 0
\(597\) 3.43377 0.140535
\(598\) 0 0
\(599\) 0.274769 0.0112267 0.00561337 0.999984i \(-0.498213\pi\)
0.00561337 + 0.999984i \(0.498213\pi\)
\(600\) 0 0
\(601\) −42.1590 −1.71970 −0.859851 0.510546i \(-0.829443\pi\)
−0.859851 + 0.510546i \(0.829443\pi\)
\(602\) 0 0
\(603\) 2.29863 0.0936074
\(604\) 0 0
\(605\) −4.58222 −0.186294
\(606\) 0 0
\(607\) −36.3658 −1.47604 −0.738022 0.674777i \(-0.764240\pi\)
−0.738022 + 0.674777i \(0.764240\pi\)
\(608\) 0 0
\(609\) −24.0756 −0.975591
\(610\) 0 0
\(611\) −2.21580 −0.0896417
\(612\) 0 0
\(613\) −28.3291 −1.14420 −0.572102 0.820183i \(-0.693872\pi\)
−0.572102 + 0.820183i \(0.693872\pi\)
\(614\) 0 0
\(615\) 5.17025 0.208484
\(616\) 0 0
\(617\) 44.7941 1.80334 0.901672 0.432421i \(-0.142340\pi\)
0.901672 + 0.432421i \(0.142340\pi\)
\(618\) 0 0
\(619\) −43.9682 −1.76723 −0.883615 0.468214i \(-0.844898\pi\)
−0.883615 + 0.468214i \(0.844898\pi\)
\(620\) 0 0
\(621\) −5.88544 −0.236175
\(622\) 0 0
\(623\) −18.3636 −0.735724
\(624\) 0 0
\(625\) 22.4272 0.897088
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −4.37490 −0.174439
\(630\) 0 0
\(631\) −15.8265 −0.630042 −0.315021 0.949085i \(-0.602012\pi\)
−0.315021 + 0.949085i \(0.602012\pi\)
\(632\) 0 0
\(633\) 21.8854 0.869868
\(634\) 0 0
\(635\) 8.85770 0.351507
\(636\) 0 0
\(637\) 4.16518 0.165030
\(638\) 0 0
\(639\) 8.60401 0.340370
\(640\) 0 0
\(641\) −25.4792 −1.00637 −0.503184 0.864179i \(-0.667838\pi\)
−0.503184 + 0.864179i \(0.667838\pi\)
\(642\) 0 0
\(643\) −20.6623 −0.814841 −0.407420 0.913241i \(-0.633572\pi\)
−0.407420 + 0.913241i \(0.633572\pi\)
\(644\) 0 0
\(645\) 1.28143 0.0504561
\(646\) 0 0
\(647\) −1.24899 −0.0491030 −0.0245515 0.999699i \(-0.507816\pi\)
−0.0245515 + 0.999699i \(0.507816\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 5.71191 0.223868
\(652\) 0 0
\(653\) 3.27149 0.128023 0.0640116 0.997949i \(-0.479611\pi\)
0.0640116 + 0.997949i \(0.479611\pi\)
\(654\) 0 0
\(655\) 5.91647 0.231176
\(656\) 0 0
\(657\) 2.26690 0.0884402
\(658\) 0 0
\(659\) 20.0805 0.782227 0.391113 0.920343i \(-0.372090\pi\)
0.391113 + 0.920343i \(0.372090\pi\)
\(660\) 0 0
\(661\) −24.6738 −0.959698 −0.479849 0.877351i \(-0.659309\pi\)
−0.479849 + 0.877351i \(0.659309\pi\)
\(662\) 0 0
\(663\) 0.181396 0.00704485
\(664\) 0 0
\(665\) −11.4238 −0.442997
\(666\) 0 0
\(667\) −30.4093 −1.17745
\(668\) 0 0
\(669\) 4.65960 0.180151
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 40.9688 1.57923 0.789615 0.613602i \(-0.210280\pi\)
0.789615 + 0.613602i \(0.210280\pi\)
\(674\) 0 0
\(675\) −4.82647 −0.185771
\(676\) 0 0
\(677\) −0.194714 −0.00748345 −0.00374172 0.999993i \(-0.501191\pi\)
−0.00374172 + 0.999993i \(0.501191\pi\)
\(678\) 0 0
\(679\) −41.9054 −1.60818
\(680\) 0 0
\(681\) 22.4467 0.860160
\(682\) 0 0
\(683\) 41.5156 1.58855 0.794275 0.607558i \(-0.207851\pi\)
0.794275 + 0.607558i \(0.207851\pi\)
\(684\) 0 0
\(685\) −4.39411 −0.167890
\(686\) 0 0
\(687\) 12.3715 0.472003
\(688\) 0 0
\(689\) −1.54833 −0.0589865
\(690\) 0 0
\(691\) −30.0517 −1.14322 −0.571610 0.820525i \(-0.693681\pi\)
−0.571610 + 0.820525i \(0.693681\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −4.98002 −0.188903
\(696\) 0 0
\(697\) 7.95228 0.301214
\(698\) 0 0
\(699\) 19.5940 0.741112
\(700\) 0 0
\(701\) 47.9934 1.81269 0.906344 0.422542i \(-0.138862\pi\)
0.906344 + 0.422542i \(0.138862\pi\)
\(702\) 0 0
\(703\) −40.1867 −1.51567
\(704\) 0 0
\(705\) −3.26024 −0.122788
\(706\) 0 0
\(707\) 57.4828 2.16186
\(708\) 0 0
\(709\) 26.5976 0.998895 0.499448 0.866344i \(-0.333536\pi\)
0.499448 + 0.866344i \(0.333536\pi\)
\(710\) 0 0
\(711\) 7.69302 0.288511
\(712\) 0 0
\(713\) 7.21459 0.270189
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −3.43377 −0.128236
\(718\) 0 0
\(719\) −0.951068 −0.0354689 −0.0177344 0.999843i \(-0.505645\pi\)
−0.0177344 + 0.999843i \(0.505645\pi\)
\(720\) 0 0
\(721\) −57.5217 −2.14222
\(722\) 0 0
\(723\) 14.4861 0.538743
\(724\) 0 0
\(725\) −24.9378 −0.926165
\(726\) 0 0
\(727\) 5.91558 0.219397 0.109698 0.993965i \(-0.465012\pi\)
0.109698 + 0.993965i \(0.465012\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 1.97094 0.0728980
\(732\) 0 0
\(733\) −5.27487 −0.194832 −0.0974158 0.995244i \(-0.531058\pi\)
−0.0974158 + 0.995244i \(0.531058\pi\)
\(734\) 0 0
\(735\) 6.12848 0.226052
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −7.61925 −0.280278 −0.140139 0.990132i \(-0.544755\pi\)
−0.140139 + 0.990132i \(0.544755\pi\)
\(740\) 0 0
\(741\) 1.66626 0.0612117
\(742\) 0 0
\(743\) −35.4470 −1.30042 −0.650212 0.759753i \(-0.725320\pi\)
−0.650212 + 0.759753i \(0.725320\pi\)
\(744\) 0 0
\(745\) 1.49274 0.0546896
\(746\) 0 0
\(747\) −9.27646 −0.339408
\(748\) 0 0
\(749\) −88.9365 −3.24967
\(750\) 0 0
\(751\) 22.2680 0.812570 0.406285 0.913746i \(-0.366824\pi\)
0.406285 + 0.913746i \(0.366824\pi\)
\(752\) 0 0
\(753\) 26.8576 0.978745
\(754\) 0 0
\(755\) 1.55827 0.0567111
\(756\) 0 0
\(757\) 25.1259 0.913216 0.456608 0.889668i \(-0.349064\pi\)
0.456608 + 0.889668i \(0.349064\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 19.0755 0.691485 0.345743 0.938329i \(-0.387627\pi\)
0.345743 + 0.938329i \(0.387627\pi\)
\(762\) 0 0
\(763\) −80.4027 −2.91077
\(764\) 0 0
\(765\) 0.266899 0.00964977
\(766\) 0 0
\(767\) −0.908695 −0.0328111
\(768\) 0 0
\(769\) 7.11577 0.256601 0.128301 0.991735i \(-0.459048\pi\)
0.128301 + 0.991735i \(0.459048\pi\)
\(770\) 0 0
\(771\) −3.57846 −0.128875
\(772\) 0 0
\(773\) 38.4478 1.38287 0.691435 0.722438i \(-0.256979\pi\)
0.691435 + 0.722438i \(0.256979\pi\)
\(774\) 0 0
\(775\) 5.91647 0.212526
\(776\) 0 0
\(777\) 31.8165 1.14141
\(778\) 0 0
\(779\) 73.0477 2.61721
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 5.16687 0.184649
\(784\) 0 0
\(785\) −5.73648 −0.204744
\(786\) 0 0
\(787\) −37.5332 −1.33791 −0.668957 0.743301i \(-0.733259\pi\)
−0.668957 + 0.743301i \(0.733259\pi\)
\(788\) 0 0
\(789\) −5.52517 −0.196701
\(790\) 0 0
\(791\) −13.5641 −0.482283
\(792\) 0 0
\(793\) −0.330359 −0.0117314
\(794\) 0 0
\(795\) −2.27815 −0.0807975
\(796\) 0 0
\(797\) 36.2808 1.28513 0.642566 0.766230i \(-0.277870\pi\)
0.642566 + 0.766230i \(0.277870\pi\)
\(798\) 0 0
\(799\) −5.01453 −0.177401
\(800\) 0 0
\(801\) 3.94103 0.139250
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 11.4238 0.402637
\(806\) 0 0
\(807\) 29.0039 1.02099
\(808\) 0 0
\(809\) 10.4305 0.366716 0.183358 0.983046i \(-0.441303\pi\)
0.183358 + 0.983046i \(0.441303\pi\)
\(810\) 0 0
\(811\) −1.68468 −0.0591570 −0.0295785 0.999562i \(-0.509416\pi\)
−0.0295785 + 0.999562i \(0.509416\pi\)
\(812\) 0 0
\(813\) 24.0301 0.842774
\(814\) 0 0
\(815\) −4.68210 −0.164007
\(816\) 0 0
\(817\) 18.1046 0.633400
\(818\) 0 0
\(819\) −1.31921 −0.0460969
\(820\) 0 0
\(821\) 1.38085 0.0481920 0.0240960 0.999710i \(-0.492329\pi\)
0.0240960 + 0.999710i \(0.492329\pi\)
\(822\) 0 0
\(823\) 12.9488 0.451366 0.225683 0.974201i \(-0.427539\pi\)
0.225683 + 0.974201i \(0.427539\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 25.7924 0.896891 0.448446 0.893810i \(-0.351978\pi\)
0.448446 + 0.893810i \(0.351978\pi\)
\(828\) 0 0
\(829\) −23.6347 −0.820866 −0.410433 0.911891i \(-0.634622\pi\)
−0.410433 + 0.911891i \(0.634622\pi\)
\(830\) 0 0
\(831\) 2.99382 0.103854
\(832\) 0 0
\(833\) 9.42612 0.326596
\(834\) 0 0
\(835\) 0.416566 0.0144159
\(836\) 0 0
\(837\) −1.22584 −0.0423711
\(838\) 0 0
\(839\) 37.1047 1.28100 0.640499 0.767959i \(-0.278728\pi\)
0.640499 + 0.767959i \(0.278728\pi\)
\(840\) 0 0
\(841\) −2.30347 −0.0794300
\(842\) 0 0
\(843\) 1.72523 0.0594201
\(844\) 0 0
\(845\) −5.38197 −0.185145
\(846\) 0 0
\(847\) 51.2556 1.76116
\(848\) 0 0
\(849\) −18.5372 −0.636194
\(850\) 0 0
\(851\) 40.1867 1.37758
\(852\) 0 0
\(853\) 3.93316 0.134669 0.0673345 0.997730i \(-0.478551\pi\)
0.0673345 + 0.997730i \(0.478551\pi\)
\(854\) 0 0
\(855\) 2.45167 0.0838455
\(856\) 0 0
\(857\) −57.5073 −1.96441 −0.982205 0.187810i \(-0.939861\pi\)
−0.982205 + 0.187810i \(0.939861\pi\)
\(858\) 0 0
\(859\) 8.51851 0.290648 0.145324 0.989384i \(-0.453578\pi\)
0.145324 + 0.989384i \(0.453578\pi\)
\(860\) 0 0
\(861\) −57.8331 −1.97095
\(862\) 0 0
\(863\) 27.1894 0.925537 0.462768 0.886479i \(-0.346856\pi\)
0.462768 + 0.886479i \(0.346856\pi\)
\(864\) 0 0
\(865\) 0.486077 0.0165271
\(866\) 0 0
\(867\) −16.5895 −0.563408
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0.650779 0.0220508
\(872\) 0 0
\(873\) 8.99334 0.304378
\(874\) 0 0
\(875\) 19.0735 0.644802
\(876\) 0 0
\(877\) −40.8576 −1.37966 −0.689831 0.723970i \(-0.742315\pi\)
−0.689831 + 0.723970i \(0.742315\pi\)
\(878\) 0 0
\(879\) 14.8331 0.500309
\(880\) 0 0
\(881\) −10.9645 −0.369404 −0.184702 0.982795i \(-0.559132\pi\)
−0.184702 + 0.982795i \(0.559132\pi\)
\(882\) 0 0
\(883\) 38.3081 1.28917 0.644584 0.764533i \(-0.277030\pi\)
0.644584 + 0.764533i \(0.277030\pi\)
\(884\) 0 0
\(885\) −1.33702 −0.0449434
\(886\) 0 0
\(887\) 54.4403 1.82793 0.913964 0.405796i \(-0.133006\pi\)
0.913964 + 0.405796i \(0.133006\pi\)
\(888\) 0 0
\(889\) −99.0801 −3.32304
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −46.0622 −1.54141
\(894\) 0 0
\(895\) −3.23708 −0.108204
\(896\) 0 0
\(897\) −1.66626 −0.0556349
\(898\) 0 0
\(899\) −6.33374 −0.211242
\(900\) 0 0
\(901\) −3.50398 −0.116735
\(902\) 0 0
\(903\) −14.3337 −0.476997
\(904\) 0 0
\(905\) 4.27270 0.142029
\(906\) 0 0
\(907\) 1.05231 0.0349414 0.0174707 0.999847i \(-0.494439\pi\)
0.0174707 + 0.999847i \(0.494439\pi\)
\(908\) 0 0
\(909\) −12.3364 −0.409173
\(910\) 0 0
\(911\) −59.5240 −1.97212 −0.986058 0.166400i \(-0.946786\pi\)
−0.986058 + 0.166400i \(0.946786\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −0.486077 −0.0160692
\(916\) 0 0
\(917\) −66.1802 −2.18546
\(918\) 0 0
\(919\) −13.2091 −0.435729 −0.217865 0.975979i \(-0.569909\pi\)
−0.217865 + 0.975979i \(0.569909\pi\)
\(920\) 0 0
\(921\) 31.9927 1.05420
\(922\) 0 0
\(923\) 2.43593 0.0801798
\(924\) 0 0
\(925\) 32.9559 1.08358
\(926\) 0 0
\(927\) 12.3448 0.405455
\(928\) 0 0
\(929\) 34.0834 1.11824 0.559121 0.829086i \(-0.311139\pi\)
0.559121 + 0.829086i \(0.311139\pi\)
\(930\) 0 0
\(931\) 86.5861 2.83774
\(932\) 0 0
\(933\) 7.07884 0.231751
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 20.5272 0.670596 0.335298 0.942112i \(-0.391163\pi\)
0.335298 + 0.942112i \(0.391163\pi\)
\(938\) 0 0
\(939\) 23.6496 0.771774
\(940\) 0 0
\(941\) −32.9415 −1.07386 −0.536932 0.843626i \(-0.680417\pi\)
−0.536932 + 0.843626i \(0.680417\pi\)
\(942\) 0 0
\(943\) −73.0477 −2.37876
\(944\) 0 0
\(945\) −1.94103 −0.0631418
\(946\) 0 0
\(947\) 53.9867 1.75433 0.877166 0.480188i \(-0.159431\pi\)
0.877166 + 0.480188i \(0.159431\pi\)
\(948\) 0 0
\(949\) 0.641796 0.0208336
\(950\) 0 0
\(951\) −13.7007 −0.444275
\(952\) 0 0
\(953\) −21.7843 −0.705664 −0.352832 0.935687i \(-0.614781\pi\)
−0.352832 + 0.935687i \(0.614781\pi\)
\(954\) 0 0
\(955\) 5.64986 0.182825
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 49.1515 1.58718
\(960\) 0 0
\(961\) −29.4973 −0.951527
\(962\) 0 0
\(963\) 19.0867 0.615061
\(964\) 0 0
\(965\) −1.06900 −0.0344125
\(966\) 0 0
\(967\) −25.9385 −0.834126 −0.417063 0.908878i \(-0.636941\pi\)
−0.417063 + 0.908878i \(0.636941\pi\)
\(968\) 0 0
\(969\) 3.77088 0.121138
\(970\) 0 0
\(971\) −27.4566 −0.881126 −0.440563 0.897722i \(-0.645221\pi\)
−0.440563 + 0.897722i \(0.645221\pi\)
\(972\) 0 0
\(973\) 55.7054 1.78583
\(974\) 0 0
\(975\) −1.36645 −0.0437615
\(976\) 0 0
\(977\) 45.3036 1.44939 0.724695 0.689070i \(-0.241981\pi\)
0.724695 + 0.689070i \(0.241981\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 17.2553 0.550918
\(982\) 0 0
\(983\) −34.6987 −1.10672 −0.553358 0.832943i \(-0.686654\pi\)
−0.553358 + 0.832943i \(0.686654\pi\)
\(984\) 0 0
\(985\) −1.21797 −0.0388077
\(986\) 0 0
\(987\) 36.4683 1.16080
\(988\) 0 0
\(989\) −18.1046 −0.575693
\(990\) 0 0
\(991\) −33.6320 −1.06836 −0.534178 0.845372i \(-0.679379\pi\)
−0.534178 + 0.845372i \(0.679379\pi\)
\(992\) 0 0
\(993\) 9.40991 0.298614
\(994\) 0 0
\(995\) 1.43039 0.0453464
\(996\) 0 0
\(997\) 36.8954 1.16849 0.584244 0.811578i \(-0.301391\pi\)
0.584244 + 0.811578i \(0.301391\pi\)
\(998\) 0 0
\(999\) −6.82816 −0.216033
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 8016.2.a.o.1.2 4
4.3 odd 2 1002.2.a.i.1.2 4
12.11 even 2 3006.2.a.s.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1002.2.a.i.1.2 4 4.3 odd 2
3006.2.a.s.1.3 4 12.11 even 2
8016.2.a.o.1.2 4 1.1 even 1 trivial