Properties

Label 3380.2.a.r.1.1
Level $3380$
Weight $2$
Character 3380.1
Self dual yes
Analytic conductor $26.989$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3380.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.9894358832\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - x^{8} - 19 x^{7} + 16 x^{6} + 106 x^{5} - 87 x^{4} - 153 x^{3} + 149 x^{2} - 26 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.40622\) of defining polynomial
Character \(\chi\) \(=\) 3380.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.40622 q^{3} -1.00000 q^{5} -2.33790 q^{7} +8.60234 q^{9} +O(q^{10})\) \(q-3.40622 q^{3} -1.00000 q^{5} -2.33790 q^{7} +8.60234 q^{9} +6.16565 q^{11} +3.40622 q^{15} +3.07520 q^{17} +0.0902681 q^{19} +7.96342 q^{21} +4.81954 q^{23} +1.00000 q^{25} -19.0828 q^{27} +5.63201 q^{29} -8.34078 q^{31} -21.0016 q^{33} +2.33790 q^{35} -0.794425 q^{37} +3.95907 q^{41} -8.75108 q^{43} -8.60234 q^{45} +3.67871 q^{47} -1.53420 q^{49} -10.4748 q^{51} +8.08456 q^{53} -6.16565 q^{55} -0.307473 q^{57} -0.379577 q^{59} -4.96367 q^{61} -20.1114 q^{63} -0.376774 q^{67} -16.4164 q^{69} +9.04130 q^{71} -5.48133 q^{73} -3.40622 q^{75} -14.4147 q^{77} -4.79283 q^{79} +39.1932 q^{81} -11.6189 q^{83} -3.07520 q^{85} -19.1839 q^{87} +11.7118 q^{89} +28.4105 q^{93} -0.0902681 q^{95} -2.43658 q^{97} +53.0390 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q - q^{3} - 9q^{5} - q^{7} + 12q^{9} + O(q^{10}) \) \( 9q - q^{3} - 9q^{5} - q^{7} + 12q^{9} + 7q^{11} + q^{15} + 13q^{17} + 4q^{19} + 3q^{21} + 12q^{23} + 9q^{25} - 4q^{27} + 16q^{29} - 13q^{31} - 34q^{33} + q^{35} - q^{37} + 6q^{41} + q^{43} - 12q^{45} + 2q^{47} + 20q^{49} + 11q^{51} + 30q^{53} - 7q^{55} - 38q^{57} - 15q^{59} + 21q^{61} + 17q^{63} + 7q^{67} + 15q^{69} + 7q^{71} + 28q^{73} - q^{75} + 46q^{77} + 31q^{79} + 41q^{81} - 45q^{83} - 13q^{85} + 28q^{87} + 41q^{89} + 11q^{93} - 4q^{95} - 8q^{97} + 81q^{99} + O(q^{100}) \)

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
\(3\) −3.40622 −1.96658 −0.983291 0.182040i \(-0.941730\pi\)
−0.983291 + 0.182040i \(0.941730\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.33790 −0.883645 −0.441822 0.897103i \(-0.645668\pi\)
−0.441822 + 0.897103i \(0.645668\pi\)
\(8\) 0 0
\(9\) 8.60234 2.86745
\(10\) 0 0
\(11\) 6.16565 1.85901 0.929507 0.368806i \(-0.120233\pi\)
0.929507 + 0.368806i \(0.120233\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) 3.40622 0.879482
\(16\) 0 0
\(17\) 3.07520 0.745844 0.372922 0.927863i \(-0.378356\pi\)
0.372922 + 0.927863i \(0.378356\pi\)
\(18\) 0 0
\(19\) 0.0902681 0.0207089 0.0103545 0.999946i \(-0.496704\pi\)
0.0103545 + 0.999946i \(0.496704\pi\)
\(20\) 0 0
\(21\) 7.96342 1.73776
\(22\) 0 0
\(23\) 4.81954 1.00494 0.502472 0.864594i \(-0.332424\pi\)
0.502472 + 0.864594i \(0.332424\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −19.0828 −3.67249
\(28\) 0 0
\(29\) 5.63201 1.04584 0.522919 0.852382i \(-0.324843\pi\)
0.522919 + 0.852382i \(0.324843\pi\)
\(30\) 0 0
\(31\) −8.34078 −1.49805 −0.749024 0.662543i \(-0.769477\pi\)
−0.749024 + 0.662543i \(0.769477\pi\)
\(32\) 0 0
\(33\) −21.0016 −3.65590
\(34\) 0 0
\(35\) 2.33790 0.395178
\(36\) 0 0
\(37\) −0.794425 −0.130603 −0.0653014 0.997866i \(-0.520801\pi\)
−0.0653014 + 0.997866i \(0.520801\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.95907 0.618302 0.309151 0.951013i \(-0.399955\pi\)
0.309151 + 0.951013i \(0.399955\pi\)
\(42\) 0 0
\(43\) −8.75108 −1.33453 −0.667264 0.744822i \(-0.732535\pi\)
−0.667264 + 0.744822i \(0.732535\pi\)
\(44\) 0 0
\(45\) −8.60234 −1.28236
\(46\) 0 0
\(47\) 3.67871 0.536595 0.268298 0.963336i \(-0.413539\pi\)
0.268298 + 0.963336i \(0.413539\pi\)
\(48\) 0 0
\(49\) −1.53420 −0.219172
\(50\) 0 0
\(51\) −10.4748 −1.46676
\(52\) 0 0
\(53\) 8.08456 1.11050 0.555250 0.831683i \(-0.312623\pi\)
0.555250 + 0.831683i \(0.312623\pi\)
\(54\) 0 0
\(55\) −6.16565 −0.831376
\(56\) 0 0
\(57\) −0.307473 −0.0407258
\(58\) 0 0
\(59\) −0.379577 −0.0494167 −0.0247084 0.999695i \(-0.507866\pi\)
−0.0247084 + 0.999695i \(0.507866\pi\)
\(60\) 0 0
\(61\) −4.96367 −0.635533 −0.317766 0.948169i \(-0.602933\pi\)
−0.317766 + 0.948169i \(0.602933\pi\)
\(62\) 0 0
\(63\) −20.1114 −2.53380
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −0.376774 −0.0460302 −0.0230151 0.999735i \(-0.507327\pi\)
−0.0230151 + 0.999735i \(0.507327\pi\)
\(68\) 0 0
\(69\) −16.4164 −1.97630
\(70\) 0 0
\(71\) 9.04130 1.07300 0.536502 0.843899i \(-0.319745\pi\)
0.536502 + 0.843899i \(0.319745\pi\)
\(72\) 0 0
\(73\) −5.48133 −0.641541 −0.320770 0.947157i \(-0.603942\pi\)
−0.320770 + 0.947157i \(0.603942\pi\)
\(74\) 0 0
\(75\) −3.40622 −0.393316
\(76\) 0 0
\(77\) −14.4147 −1.64271
\(78\) 0 0
\(79\) −4.79283 −0.539235 −0.269617 0.962967i \(-0.586897\pi\)
−0.269617 + 0.962967i \(0.586897\pi\)
\(80\) 0 0
\(81\) 39.1932 4.35480
\(82\) 0 0
\(83\) −11.6189 −1.27534 −0.637670 0.770310i \(-0.720102\pi\)
−0.637670 + 0.770310i \(0.720102\pi\)
\(84\) 0 0
\(85\) −3.07520 −0.333552
\(86\) 0 0
\(87\) −19.1839 −2.05673
\(88\) 0 0
\(89\) 11.7118 1.24145 0.620724 0.784029i \(-0.286839\pi\)
0.620724 + 0.784029i \(0.286839\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 28.4105 2.94603
\(94\) 0 0
\(95\) −0.0902681 −0.00926131
\(96\) 0 0
\(97\) −2.43658 −0.247397 −0.123698 0.992320i \(-0.539476\pi\)
−0.123698 + 0.992320i \(0.539476\pi\)
\(98\) 0 0
\(99\) 53.0390 5.33062
\(100\) 0 0
\(101\) −5.52834 −0.550090 −0.275045 0.961431i \(-0.588693\pi\)
−0.275045 + 0.961431i \(0.588693\pi\)
\(102\) 0 0
\(103\) 8.46999 0.834573 0.417287 0.908775i \(-0.362981\pi\)
0.417287 + 0.908775i \(0.362981\pi\)
\(104\) 0 0
\(105\) −7.96342 −0.777150
\(106\) 0 0
\(107\) 11.3891 1.10103 0.550514 0.834826i \(-0.314432\pi\)
0.550514 + 0.834826i \(0.314432\pi\)
\(108\) 0 0
\(109\) 9.04624 0.866473 0.433236 0.901280i \(-0.357372\pi\)
0.433236 + 0.901280i \(0.357372\pi\)
\(110\) 0 0
\(111\) 2.70599 0.256841
\(112\) 0 0
\(113\) −2.99157 −0.281423 −0.140712 0.990051i \(-0.544939\pi\)
−0.140712 + 0.990051i \(0.544939\pi\)
\(114\) 0 0
\(115\) −4.81954 −0.449425
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −7.18951 −0.659061
\(120\) 0 0
\(121\) 27.0152 2.45593
\(122\) 0 0
\(123\) −13.4855 −1.21594
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 10.6776 0.947482 0.473741 0.880664i \(-0.342903\pi\)
0.473741 + 0.880664i \(0.342903\pi\)
\(128\) 0 0
\(129\) 29.8081 2.62446
\(130\) 0 0
\(131\) −20.6072 −1.80046 −0.900231 0.435412i \(-0.856603\pi\)
−0.900231 + 0.435412i \(0.856603\pi\)
\(132\) 0 0
\(133\) −0.211038 −0.0182993
\(134\) 0 0
\(135\) 19.0828 1.64239
\(136\) 0 0
\(137\) 0.0850150 0.00726332 0.00363166 0.999993i \(-0.498844\pi\)
0.00363166 + 0.999993i \(0.498844\pi\)
\(138\) 0 0
\(139\) 11.0101 0.933864 0.466932 0.884293i \(-0.345359\pi\)
0.466932 + 0.884293i \(0.345359\pi\)
\(140\) 0 0
\(141\) −12.5305 −1.05526
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −5.63201 −0.467713
\(146\) 0 0
\(147\) 5.22584 0.431020
\(148\) 0 0
\(149\) −4.41512 −0.361701 −0.180850 0.983511i \(-0.557885\pi\)
−0.180850 + 0.983511i \(0.557885\pi\)
\(150\) 0 0
\(151\) −15.7555 −1.28216 −0.641081 0.767473i \(-0.721514\pi\)
−0.641081 + 0.767473i \(0.721514\pi\)
\(152\) 0 0
\(153\) 26.4539 2.13867
\(154\) 0 0
\(155\) 8.34078 0.669947
\(156\) 0 0
\(157\) 3.40120 0.271445 0.135722 0.990747i \(-0.456664\pi\)
0.135722 + 0.990747i \(0.456664\pi\)
\(158\) 0 0
\(159\) −27.5378 −2.18389
\(160\) 0 0
\(161\) −11.2676 −0.888013
\(162\) 0 0
\(163\) −14.4332 −1.13050 −0.565249 0.824920i \(-0.691220\pi\)
−0.565249 + 0.824920i \(0.691220\pi\)
\(164\) 0 0
\(165\) 21.0016 1.63497
\(166\) 0 0
\(167\) 18.9815 1.46883 0.734417 0.678698i \(-0.237456\pi\)
0.734417 + 0.678698i \(0.237456\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 0.776517 0.0593817
\(172\) 0 0
\(173\) 11.0970 0.843691 0.421845 0.906668i \(-0.361383\pi\)
0.421845 + 0.906668i \(0.361383\pi\)
\(174\) 0 0
\(175\) −2.33790 −0.176729
\(176\) 0 0
\(177\) 1.29292 0.0971821
\(178\) 0 0
\(179\) 5.27165 0.394021 0.197011 0.980401i \(-0.436877\pi\)
0.197011 + 0.980401i \(0.436877\pi\)
\(180\) 0 0
\(181\) 16.1744 1.20223 0.601117 0.799161i \(-0.294723\pi\)
0.601117 + 0.799161i \(0.294723\pi\)
\(182\) 0 0
\(183\) 16.9074 1.24983
\(184\) 0 0
\(185\) 0.794425 0.0584073
\(186\) 0 0
\(187\) 18.9606 1.38653
\(188\) 0 0
\(189\) 44.6138 3.24517
\(190\) 0 0
\(191\) −6.75365 −0.488677 −0.244338 0.969690i \(-0.578571\pi\)
−0.244338 + 0.969690i \(0.578571\pi\)
\(192\) 0 0
\(193\) −0.481519 −0.0346605 −0.0173302 0.999850i \(-0.505517\pi\)
−0.0173302 + 0.999850i \(0.505517\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −21.8881 −1.55946 −0.779732 0.626113i \(-0.784645\pi\)
−0.779732 + 0.626113i \(0.784645\pi\)
\(198\) 0 0
\(199\) 23.5567 1.66989 0.834947 0.550331i \(-0.185498\pi\)
0.834947 + 0.550331i \(0.185498\pi\)
\(200\) 0 0
\(201\) 1.28337 0.0905223
\(202\) 0 0
\(203\) −13.1671 −0.924149
\(204\) 0 0
\(205\) −3.95907 −0.276513
\(206\) 0 0
\(207\) 41.4593 2.88162
\(208\) 0 0
\(209\) 0.556561 0.0384981
\(210\) 0 0
\(211\) −11.0661 −0.761824 −0.380912 0.924611i \(-0.624390\pi\)
−0.380912 + 0.924611i \(0.624390\pi\)
\(212\) 0 0
\(213\) −30.7967 −2.11015
\(214\) 0 0
\(215\) 8.75108 0.596819
\(216\) 0 0
\(217\) 19.4999 1.32374
\(218\) 0 0
\(219\) 18.6706 1.26164
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −24.3465 −1.63036 −0.815182 0.579205i \(-0.803363\pi\)
−0.815182 + 0.579205i \(0.803363\pi\)
\(224\) 0 0
\(225\) 8.60234 0.573489
\(226\) 0 0
\(227\) −0.100791 −0.00668971 −0.00334485 0.999994i \(-0.501065\pi\)
−0.00334485 + 0.999994i \(0.501065\pi\)
\(228\) 0 0
\(229\) 5.60506 0.370393 0.185196 0.982702i \(-0.440708\pi\)
0.185196 + 0.982702i \(0.440708\pi\)
\(230\) 0 0
\(231\) 49.0996 3.23052
\(232\) 0 0
\(233\) 13.2785 0.869902 0.434951 0.900454i \(-0.356766\pi\)
0.434951 + 0.900454i \(0.356766\pi\)
\(234\) 0 0
\(235\) −3.67871 −0.239973
\(236\) 0 0
\(237\) 16.3254 1.06045
\(238\) 0 0
\(239\) 12.9817 0.839715 0.419858 0.907590i \(-0.362080\pi\)
0.419858 + 0.907590i \(0.362080\pi\)
\(240\) 0 0
\(241\) −1.49222 −0.0961223 −0.0480612 0.998844i \(-0.515304\pi\)
−0.0480612 + 0.998844i \(0.515304\pi\)
\(242\) 0 0
\(243\) −76.2524 −4.89159
\(244\) 0 0
\(245\) 1.53420 0.0980167
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 39.5765 2.50806
\(250\) 0 0
\(251\) −11.2640 −0.710975 −0.355487 0.934681i \(-0.615685\pi\)
−0.355487 + 0.934681i \(0.615685\pi\)
\(252\) 0 0
\(253\) 29.7156 1.86820
\(254\) 0 0
\(255\) 10.4748 0.655957
\(256\) 0 0
\(257\) 15.0148 0.936600 0.468300 0.883569i \(-0.344867\pi\)
0.468300 + 0.883569i \(0.344867\pi\)
\(258\) 0 0
\(259\) 1.85729 0.115406
\(260\) 0 0
\(261\) 48.4484 2.99888
\(262\) 0 0
\(263\) 11.4305 0.704837 0.352418 0.935843i \(-0.385359\pi\)
0.352418 + 0.935843i \(0.385359\pi\)
\(264\) 0 0
\(265\) −8.08456 −0.496631
\(266\) 0 0
\(267\) −39.8930 −2.44141
\(268\) 0 0
\(269\) 25.3906 1.54809 0.774047 0.633129i \(-0.218230\pi\)
0.774047 + 0.633129i \(0.218230\pi\)
\(270\) 0 0
\(271\) −10.6327 −0.645892 −0.322946 0.946417i \(-0.604673\pi\)
−0.322946 + 0.946417i \(0.604673\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 6.16565 0.371803
\(276\) 0 0
\(277\) 25.9901 1.56159 0.780797 0.624785i \(-0.214813\pi\)
0.780797 + 0.624785i \(0.214813\pi\)
\(278\) 0 0
\(279\) −71.7502 −4.29557
\(280\) 0 0
\(281\) 7.65920 0.456909 0.228455 0.973555i \(-0.426633\pi\)
0.228455 + 0.973555i \(0.426633\pi\)
\(282\) 0 0
\(283\) 19.5025 1.15930 0.579651 0.814865i \(-0.303189\pi\)
0.579651 + 0.814865i \(0.303189\pi\)
\(284\) 0 0
\(285\) 0.307473 0.0182131
\(286\) 0 0
\(287\) −9.25592 −0.546360
\(288\) 0 0
\(289\) −7.54317 −0.443716
\(290\) 0 0
\(291\) 8.29952 0.486526
\(292\) 0 0
\(293\) 12.7879 0.747078 0.373539 0.927614i \(-0.378144\pi\)
0.373539 + 0.927614i \(0.378144\pi\)
\(294\) 0 0
\(295\) 0.379577 0.0220998
\(296\) 0 0
\(297\) −117.658 −6.82720
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 20.4592 1.17925
\(302\) 0 0
\(303\) 18.8307 1.08180
\(304\) 0 0
\(305\) 4.96367 0.284219
\(306\) 0 0
\(307\) −32.5882 −1.85990 −0.929952 0.367680i \(-0.880152\pi\)
−0.929952 + 0.367680i \(0.880152\pi\)
\(308\) 0 0
\(309\) −28.8507 −1.64126
\(310\) 0 0
\(311\) 4.51961 0.256284 0.128142 0.991756i \(-0.459099\pi\)
0.128142 + 0.991756i \(0.459099\pi\)
\(312\) 0 0
\(313\) −2.40976 −0.136208 −0.0681038 0.997678i \(-0.521695\pi\)
−0.0681038 + 0.997678i \(0.521695\pi\)
\(314\) 0 0
\(315\) 20.1114 1.13315
\(316\) 0 0
\(317\) −11.6751 −0.655739 −0.327870 0.944723i \(-0.606331\pi\)
−0.327870 + 0.944723i \(0.606331\pi\)
\(318\) 0 0
\(319\) 34.7250 1.94423
\(320\) 0 0
\(321\) −38.7939 −2.16526
\(322\) 0 0
\(323\) 0.277592 0.0154456
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −30.8135 −1.70399
\(328\) 0 0
\(329\) −8.60048 −0.474160
\(330\) 0 0
\(331\) −22.3827 −1.23026 −0.615131 0.788425i \(-0.710897\pi\)
−0.615131 + 0.788425i \(0.710897\pi\)
\(332\) 0 0
\(333\) −6.83392 −0.374496
\(334\) 0 0
\(335\) 0.376774 0.0205854
\(336\) 0 0
\(337\) 12.6686 0.690102 0.345051 0.938584i \(-0.387862\pi\)
0.345051 + 0.938584i \(0.387862\pi\)
\(338\) 0 0
\(339\) 10.1900 0.553442
\(340\) 0 0
\(341\) −51.4263 −2.78489
\(342\) 0 0
\(343\) 19.9522 1.07731
\(344\) 0 0
\(345\) 16.4164 0.883830
\(346\) 0 0
\(347\) 7.56502 0.406111 0.203056 0.979167i \(-0.434913\pi\)
0.203056 + 0.979167i \(0.434913\pi\)
\(348\) 0 0
\(349\) 18.3617 0.982879 0.491439 0.870912i \(-0.336471\pi\)
0.491439 + 0.870912i \(0.336471\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1.27380 0.0677976 0.0338988 0.999425i \(-0.489208\pi\)
0.0338988 + 0.999425i \(0.489208\pi\)
\(354\) 0 0
\(355\) −9.04130 −0.479862
\(356\) 0 0
\(357\) 24.4891 1.29610
\(358\) 0 0
\(359\) 11.0008 0.580598 0.290299 0.956936i \(-0.406245\pi\)
0.290299 + 0.956936i \(0.406245\pi\)
\(360\) 0 0
\(361\) −18.9919 −0.999571
\(362\) 0 0
\(363\) −92.0198 −4.82979
\(364\) 0 0
\(365\) 5.48133 0.286906
\(366\) 0 0
\(367\) 5.95825 0.311018 0.155509 0.987834i \(-0.450298\pi\)
0.155509 + 0.987834i \(0.450298\pi\)
\(368\) 0 0
\(369\) 34.0572 1.77295
\(370\) 0 0
\(371\) −18.9009 −0.981288
\(372\) 0 0
\(373\) −21.7186 −1.12455 −0.562274 0.826951i \(-0.690073\pi\)
−0.562274 + 0.826951i \(0.690073\pi\)
\(374\) 0 0
\(375\) 3.40622 0.175896
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −29.5331 −1.51701 −0.758507 0.651665i \(-0.774071\pi\)
−0.758507 + 0.651665i \(0.774071\pi\)
\(380\) 0 0
\(381\) −36.3702 −1.86330
\(382\) 0 0
\(383\) 26.8889 1.37396 0.686980 0.726676i \(-0.258936\pi\)
0.686980 + 0.726676i \(0.258936\pi\)
\(384\) 0 0
\(385\) 14.4147 0.734641
\(386\) 0 0
\(387\) −75.2798 −3.82669
\(388\) 0 0
\(389\) −3.11775 −0.158076 −0.0790380 0.996872i \(-0.525185\pi\)
−0.0790380 + 0.996872i \(0.525185\pi\)
\(390\) 0 0
\(391\) 14.8210 0.749532
\(392\) 0 0
\(393\) 70.1928 3.54076
\(394\) 0 0
\(395\) 4.79283 0.241153
\(396\) 0 0
\(397\) 21.2851 1.06827 0.534133 0.845400i \(-0.320638\pi\)
0.534133 + 0.845400i \(0.320638\pi\)
\(398\) 0 0
\(399\) 0.718842 0.0359871
\(400\) 0 0
\(401\) −7.30134 −0.364611 −0.182306 0.983242i \(-0.558356\pi\)
−0.182306 + 0.983242i \(0.558356\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −39.1932 −1.94753
\(406\) 0 0
\(407\) −4.89815 −0.242792
\(408\) 0 0
\(409\) 4.26732 0.211005 0.105503 0.994419i \(-0.466355\pi\)
0.105503 + 0.994419i \(0.466355\pi\)
\(410\) 0 0
\(411\) −0.289580 −0.0142839
\(412\) 0 0
\(413\) 0.887415 0.0436668
\(414\) 0 0
\(415\) 11.6189 0.570349
\(416\) 0 0
\(417\) −37.5028 −1.83652
\(418\) 0 0
\(419\) −8.14300 −0.397811 −0.198906 0.980019i \(-0.563739\pi\)
−0.198906 + 0.980019i \(0.563739\pi\)
\(420\) 0 0
\(421\) −7.36448 −0.358923 −0.179461 0.983765i \(-0.557436\pi\)
−0.179461 + 0.983765i \(0.557436\pi\)
\(422\) 0 0
\(423\) 31.6455 1.53866
\(424\) 0 0
\(425\) 3.07520 0.149169
\(426\) 0 0
\(427\) 11.6046 0.561585
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 20.1661 0.971369 0.485684 0.874134i \(-0.338570\pi\)
0.485684 + 0.874134i \(0.338570\pi\)
\(432\) 0 0
\(433\) 11.0657 0.531783 0.265892 0.964003i \(-0.414334\pi\)
0.265892 + 0.964003i \(0.414334\pi\)
\(434\) 0 0
\(435\) 19.1839 0.919796
\(436\) 0 0
\(437\) 0.435051 0.0208113
\(438\) 0 0
\(439\) 34.5470 1.64884 0.824419 0.565980i \(-0.191502\pi\)
0.824419 + 0.565980i \(0.191502\pi\)
\(440\) 0 0
\(441\) −13.1977 −0.628464
\(442\) 0 0
\(443\) −1.22275 −0.0580947 −0.0290473 0.999578i \(-0.509247\pi\)
−0.0290473 + 0.999578i \(0.509247\pi\)
\(444\) 0 0
\(445\) −11.7118 −0.555193
\(446\) 0 0
\(447\) 15.0389 0.711315
\(448\) 0 0
\(449\) −18.2491 −0.861228 −0.430614 0.902536i \(-0.641703\pi\)
−0.430614 + 0.902536i \(0.641703\pi\)
\(450\) 0 0
\(451\) 24.4102 1.14943
\(452\) 0 0
\(453\) 53.6666 2.52148
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 23.2180 1.08609 0.543046 0.839703i \(-0.317271\pi\)
0.543046 + 0.839703i \(0.317271\pi\)
\(458\) 0 0
\(459\) −58.6833 −2.73910
\(460\) 0 0
\(461\) −29.6014 −1.37868 −0.689338 0.724440i \(-0.742099\pi\)
−0.689338 + 0.724440i \(0.742099\pi\)
\(462\) 0 0
\(463\) 30.4362 1.41449 0.707244 0.706969i \(-0.249938\pi\)
0.707244 + 0.706969i \(0.249938\pi\)
\(464\) 0 0
\(465\) −28.4105 −1.31751
\(466\) 0 0
\(467\) 15.6000 0.721880 0.360940 0.932589i \(-0.382456\pi\)
0.360940 + 0.932589i \(0.382456\pi\)
\(468\) 0 0
\(469\) 0.880861 0.0406744
\(470\) 0 0
\(471\) −11.5852 −0.533819
\(472\) 0 0
\(473\) −53.9561 −2.48090
\(474\) 0 0
\(475\) 0.0902681 0.00414178
\(476\) 0 0
\(477\) 69.5462 3.18430
\(478\) 0 0
\(479\) 41.5259 1.89737 0.948684 0.316227i \(-0.102416\pi\)
0.948684 + 0.316227i \(0.102416\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 38.3800 1.74635
\(484\) 0 0
\(485\) 2.43658 0.110639
\(486\) 0 0
\(487\) 9.84163 0.445967 0.222983 0.974822i \(-0.428420\pi\)
0.222983 + 0.974822i \(0.428420\pi\)
\(488\) 0 0
\(489\) 49.1628 2.22322
\(490\) 0 0
\(491\) 29.1322 1.31472 0.657359 0.753577i \(-0.271673\pi\)
0.657359 + 0.753577i \(0.271673\pi\)
\(492\) 0 0
\(493\) 17.3195 0.780032
\(494\) 0 0
\(495\) −53.0390 −2.38393
\(496\) 0 0
\(497\) −21.1377 −0.948155
\(498\) 0 0
\(499\) −33.1347 −1.48331 −0.741656 0.670781i \(-0.765959\pi\)
−0.741656 + 0.670781i \(0.765959\pi\)
\(500\) 0 0
\(501\) −64.6553 −2.88858
\(502\) 0 0
\(503\) 24.3323 1.08492 0.542462 0.840080i \(-0.317492\pi\)
0.542462 + 0.840080i \(0.317492\pi\)
\(504\) 0 0
\(505\) 5.52834 0.246008
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 16.9484 0.751223 0.375611 0.926777i \(-0.377433\pi\)
0.375611 + 0.926777i \(0.377433\pi\)
\(510\) 0 0
\(511\) 12.8148 0.566894
\(512\) 0 0
\(513\) −1.72257 −0.0760532
\(514\) 0 0
\(515\) −8.46999 −0.373232
\(516\) 0 0
\(517\) 22.6817 0.997538
\(518\) 0 0
\(519\) −37.7989 −1.65919
\(520\) 0 0
\(521\) −16.0211 −0.701896 −0.350948 0.936395i \(-0.614141\pi\)
−0.350948 + 0.936395i \(0.614141\pi\)
\(522\) 0 0
\(523\) −5.05699 −0.221127 −0.110563 0.993869i \(-0.535266\pi\)
−0.110563 + 0.993869i \(0.535266\pi\)
\(524\) 0 0
\(525\) 7.96342 0.347552
\(526\) 0 0
\(527\) −25.6495 −1.11731
\(528\) 0 0
\(529\) 0.227981 0.00991221
\(530\) 0 0
\(531\) −3.26525 −0.141700
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −11.3891 −0.492395
\(536\) 0 0
\(537\) −17.9564 −0.774876
\(538\) 0 0
\(539\) −9.45936 −0.407444
\(540\) 0 0
\(541\) −44.1770 −1.89932 −0.949659 0.313286i \(-0.898570\pi\)
−0.949659 + 0.313286i \(0.898570\pi\)
\(542\) 0 0
\(543\) −55.0936 −2.36429
\(544\) 0 0
\(545\) −9.04624 −0.387498
\(546\) 0 0
\(547\) 16.3624 0.699605 0.349803 0.936823i \(-0.386249\pi\)
0.349803 + 0.936823i \(0.386249\pi\)
\(548\) 0 0
\(549\) −42.6992 −1.82236
\(550\) 0 0
\(551\) 0.508391 0.0216582
\(552\) 0 0
\(553\) 11.2052 0.476492
\(554\) 0 0
\(555\) −2.70599 −0.114863
\(556\) 0 0
\(557\) 29.8749 1.26584 0.632919 0.774218i \(-0.281857\pi\)
0.632919 + 0.774218i \(0.281857\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −64.5839 −2.72673
\(562\) 0 0
\(563\) 16.9922 0.716135 0.358067 0.933696i \(-0.383436\pi\)
0.358067 + 0.933696i \(0.383436\pi\)
\(564\) 0 0
\(565\) 2.99157 0.125856
\(566\) 0 0
\(567\) −91.6300 −3.84810
\(568\) 0 0
\(569\) 16.8326 0.705660 0.352830 0.935687i \(-0.385219\pi\)
0.352830 + 0.935687i \(0.385219\pi\)
\(570\) 0 0
\(571\) 1.09953 0.0460138 0.0230069 0.999735i \(-0.492676\pi\)
0.0230069 + 0.999735i \(0.492676\pi\)
\(572\) 0 0
\(573\) 23.0044 0.961023
\(574\) 0 0
\(575\) 4.81954 0.200989
\(576\) 0 0
\(577\) −21.1153 −0.879042 −0.439521 0.898232i \(-0.644852\pi\)
−0.439521 + 0.898232i \(0.644852\pi\)
\(578\) 0 0
\(579\) 1.64016 0.0681627
\(580\) 0 0
\(581\) 27.1639 1.12695
\(582\) 0 0
\(583\) 49.8466 2.06443
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 32.2938 1.33291 0.666453 0.745547i \(-0.267812\pi\)
0.666453 + 0.745547i \(0.267812\pi\)
\(588\) 0 0
\(589\) −0.752906 −0.0310229
\(590\) 0 0
\(591\) 74.5558 3.06681
\(592\) 0 0
\(593\) −12.5827 −0.516711 −0.258356 0.966050i \(-0.583181\pi\)
−0.258356 + 0.966050i \(0.583181\pi\)
\(594\) 0 0
\(595\) 7.18951 0.294741
\(596\) 0 0
\(597\) −80.2395 −3.28398
\(598\) 0 0
\(599\) 9.46110 0.386570 0.193285 0.981143i \(-0.438086\pi\)
0.193285 + 0.981143i \(0.438086\pi\)
\(600\) 0 0
\(601\) 8.02016 0.327149 0.163574 0.986531i \(-0.447698\pi\)
0.163574 + 0.986531i \(0.447698\pi\)
\(602\) 0 0
\(603\) −3.24114 −0.131989
\(604\) 0 0
\(605\) −27.0152 −1.09833
\(606\) 0 0
\(607\) −26.5832 −1.07898 −0.539490 0.841992i \(-0.681383\pi\)
−0.539490 + 0.841992i \(0.681383\pi\)
\(608\) 0 0
\(609\) 44.8500 1.81742
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −44.8547 −1.81166 −0.905831 0.423639i \(-0.860753\pi\)
−0.905831 + 0.423639i \(0.860753\pi\)
\(614\) 0 0
\(615\) 13.4855 0.543786
\(616\) 0 0
\(617\) 20.9169 0.842082 0.421041 0.907042i \(-0.361665\pi\)
0.421041 + 0.907042i \(0.361665\pi\)
\(618\) 0 0
\(619\) −9.55179 −0.383919 −0.191959 0.981403i \(-0.561484\pi\)
−0.191959 + 0.981403i \(0.561484\pi\)
\(620\) 0 0
\(621\) −91.9704 −3.69064
\(622\) 0 0
\(623\) −27.3811 −1.09700
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −1.89577 −0.0757098
\(628\) 0 0
\(629\) −2.44301 −0.0974093
\(630\) 0 0
\(631\) 4.38539 0.174580 0.0872899 0.996183i \(-0.472179\pi\)
0.0872899 + 0.996183i \(0.472179\pi\)
\(632\) 0 0
\(633\) 37.6937 1.49819
\(634\) 0 0
\(635\) −10.6776 −0.423727
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 77.7763 3.07678
\(640\) 0 0
\(641\) −19.1325 −0.755690 −0.377845 0.925869i \(-0.623335\pi\)
−0.377845 + 0.925869i \(0.623335\pi\)
\(642\) 0 0
\(643\) 17.7043 0.698190 0.349095 0.937087i \(-0.386489\pi\)
0.349095 + 0.937087i \(0.386489\pi\)
\(644\) 0 0
\(645\) −29.8081 −1.17369
\(646\) 0 0
\(647\) −30.0594 −1.18176 −0.590879 0.806761i \(-0.701219\pi\)
−0.590879 + 0.806761i \(0.701219\pi\)
\(648\) 0 0
\(649\) −2.34034 −0.0918663
\(650\) 0 0
\(651\) −66.4211 −2.60325
\(652\) 0 0
\(653\) −19.3677 −0.757915 −0.378958 0.925414i \(-0.623717\pi\)
−0.378958 + 0.925414i \(0.623717\pi\)
\(654\) 0 0
\(655\) 20.6072 0.805191
\(656\) 0 0
\(657\) −47.1522 −1.83958
\(658\) 0 0
\(659\) 8.47410 0.330104 0.165052 0.986285i \(-0.447221\pi\)
0.165052 + 0.986285i \(0.447221\pi\)
\(660\) 0 0
\(661\) 14.9713 0.582318 0.291159 0.956675i \(-0.405959\pi\)
0.291159 + 0.956675i \(0.405959\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0.211038 0.00818371
\(666\) 0 0
\(667\) 27.1437 1.05101
\(668\) 0 0
\(669\) 82.9296 3.20625
\(670\) 0 0
\(671\) −30.6043 −1.18146
\(672\) 0 0
\(673\) 35.3418 1.36233 0.681163 0.732132i \(-0.261475\pi\)
0.681163 + 0.732132i \(0.261475\pi\)
\(674\) 0 0
\(675\) −19.0828 −0.734497
\(676\) 0 0
\(677\) −31.1737 −1.19810 −0.599051 0.800711i \(-0.704455\pi\)
−0.599051 + 0.800711i \(0.704455\pi\)
\(678\) 0 0
\(679\) 5.69648 0.218611
\(680\) 0 0
\(681\) 0.343315 0.0131559
\(682\) 0 0
\(683\) −48.5452 −1.85753 −0.928766 0.370667i \(-0.879129\pi\)
−0.928766 + 0.370667i \(0.879129\pi\)
\(684\) 0 0
\(685\) −0.0850150 −0.00324826
\(686\) 0 0
\(687\) −19.0921 −0.728408
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 17.9719 0.683684 0.341842 0.939757i \(-0.388949\pi\)
0.341842 + 0.939757i \(0.388949\pi\)
\(692\) 0 0
\(693\) −124.000 −4.71037
\(694\) 0 0
\(695\) −11.0101 −0.417637
\(696\) 0 0
\(697\) 12.1749 0.461157
\(698\) 0 0
\(699\) −45.2294 −1.71073
\(700\) 0 0
\(701\) 13.0567 0.493147 0.246573 0.969124i \(-0.420695\pi\)
0.246573 + 0.969124i \(0.420695\pi\)
\(702\) 0 0
\(703\) −0.0717113 −0.00270464
\(704\) 0 0
\(705\) 12.5305 0.471926
\(706\) 0 0
\(707\) 12.9247 0.486084
\(708\) 0 0
\(709\) 14.9427 0.561186 0.280593 0.959827i \(-0.409469\pi\)
0.280593 + 0.959827i \(0.409469\pi\)
\(710\) 0 0
\(711\) −41.2295 −1.54623
\(712\) 0 0
\(713\) −40.1987 −1.50545
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −44.2185 −1.65137
\(718\) 0 0
\(719\) 10.7997 0.402761 0.201381 0.979513i \(-0.435457\pi\)
0.201381 + 0.979513i \(0.435457\pi\)
\(720\) 0 0
\(721\) −19.8020 −0.737466
\(722\) 0 0
\(723\) 5.08283 0.189032
\(724\) 0 0
\(725\) 5.63201 0.209168
\(726\) 0 0
\(727\) −25.1935 −0.934376 −0.467188 0.884158i \(-0.654733\pi\)
−0.467188 + 0.884158i \(0.654733\pi\)
\(728\) 0 0
\(729\) 142.153 5.26491
\(730\) 0 0
\(731\) −26.9113 −0.995350
\(732\) 0 0
\(733\) 32.1714 1.18828 0.594139 0.804363i \(-0.297493\pi\)
0.594139 + 0.804363i \(0.297493\pi\)
\(734\) 0 0
\(735\) −5.22584 −0.192758
\(736\) 0 0
\(737\) −2.32306 −0.0855708
\(738\) 0 0
\(739\) −44.1036 −1.62238 −0.811188 0.584785i \(-0.801179\pi\)
−0.811188 + 0.584785i \(0.801179\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 44.4544 1.63087 0.815437 0.578846i \(-0.196497\pi\)
0.815437 + 0.578846i \(0.196497\pi\)
\(744\) 0 0
\(745\) 4.41512 0.161758
\(746\) 0 0
\(747\) −99.9497 −3.65697
\(748\) 0 0
\(749\) −26.6267 −0.972918
\(750\) 0 0
\(751\) 37.6126 1.37250 0.686252 0.727364i \(-0.259255\pi\)
0.686252 + 0.727364i \(0.259255\pi\)
\(752\) 0 0
\(753\) 38.3675 1.39819
\(754\) 0 0
\(755\) 15.7555 0.573400
\(756\) 0 0
\(757\) 35.3294 1.28407 0.642034 0.766676i \(-0.278091\pi\)
0.642034 + 0.766676i \(0.278091\pi\)
\(758\) 0 0
\(759\) −101.218 −3.67398
\(760\) 0 0
\(761\) −27.3170 −0.990240 −0.495120 0.868825i \(-0.664876\pi\)
−0.495120 + 0.868825i \(0.664876\pi\)
\(762\) 0 0
\(763\) −21.1492 −0.765654
\(764\) 0 0
\(765\) −26.4539 −0.956442
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 23.1958 0.836460 0.418230 0.908341i \(-0.362651\pi\)
0.418230 + 0.908341i \(0.362651\pi\)
\(770\) 0 0
\(771\) −51.1439 −1.84190
\(772\) 0 0
\(773\) −6.52005 −0.234510 −0.117255 0.993102i \(-0.537409\pi\)
−0.117255 + 0.993102i \(0.537409\pi\)
\(774\) 0 0
\(775\) −8.34078 −0.299610
\(776\) 0 0
\(777\) −6.32634 −0.226956
\(778\) 0 0
\(779\) 0.357377 0.0128044
\(780\) 0 0
\(781\) 55.7455 1.99473
\(782\) 0 0
\(783\) −107.475 −3.84083
\(784\) 0 0
\(785\) −3.40120 −0.121394
\(786\) 0 0
\(787\) 40.7311 1.45191 0.725954 0.687744i \(-0.241399\pi\)
0.725954 + 0.687744i \(0.241399\pi\)
\(788\) 0 0
\(789\) −38.9349 −1.38612
\(790\) 0 0
\(791\) 6.99401 0.248678
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 27.5378 0.976665
\(796\) 0 0
\(797\) 4.48208 0.158763 0.0793816 0.996844i \(-0.474705\pi\)
0.0793816 + 0.996844i \(0.474705\pi\)
\(798\) 0 0
\(799\) 11.3128 0.400217
\(800\) 0 0
\(801\) 100.749 3.55979
\(802\) 0 0
\(803\) −33.7959 −1.19263
\(804\) 0 0
\(805\) 11.2676 0.397132
\(806\) 0 0
\(807\) −86.4860 −3.04445
\(808\) 0 0
\(809\) 32.7884 1.15278 0.576390 0.817175i \(-0.304461\pi\)
0.576390 + 0.817175i \(0.304461\pi\)
\(810\) 0 0
\(811\) 29.5808 1.03872 0.519361 0.854555i \(-0.326170\pi\)
0.519361 + 0.854555i \(0.326170\pi\)
\(812\) 0 0
\(813\) 36.2174 1.27020
\(814\) 0 0
\(815\) 14.4332 0.505574
\(816\) 0 0
\(817\) −0.789943 −0.0276366
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −47.1035 −1.64392 −0.821962 0.569543i \(-0.807120\pi\)
−0.821962 + 0.569543i \(0.807120\pi\)
\(822\) 0 0
\(823\) −16.4099 −0.572012 −0.286006 0.958228i \(-0.592328\pi\)
−0.286006 + 0.958228i \(0.592328\pi\)
\(824\) 0 0
\(825\) −21.0016 −0.731180
\(826\) 0 0
\(827\) 7.42042 0.258033 0.129017 0.991642i \(-0.458818\pi\)
0.129017 + 0.991642i \(0.458818\pi\)
\(828\) 0 0
\(829\) −48.0555 −1.66904 −0.834518 0.550980i \(-0.814254\pi\)
−0.834518 + 0.550980i \(0.814254\pi\)
\(830\) 0 0
\(831\) −88.5280 −3.07100
\(832\) 0 0
\(833\) −4.71798 −0.163468
\(834\) 0 0
\(835\) −18.9815 −0.656883
\(836\) 0 0
\(837\) 159.165 5.50156
\(838\) 0 0
\(839\) −12.5460 −0.433136 −0.216568 0.976268i \(-0.569486\pi\)
−0.216568 + 0.976268i \(0.569486\pi\)
\(840\) 0 0
\(841\) 2.71952 0.0937766
\(842\) 0 0
\(843\) −26.0889 −0.898550
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −63.1590 −2.17017
\(848\) 0 0
\(849\) −66.4298 −2.27986
\(850\) 0 0
\(851\) −3.82877 −0.131248
\(852\) 0 0
\(853\) −8.26494 −0.282986 −0.141493 0.989939i \(-0.545190\pi\)
−0.141493 + 0.989939i \(0.545190\pi\)
\(854\) 0 0
\(855\) −0.776517 −0.0265563
\(856\) 0 0
\(857\) 15.0212 0.513115 0.256558 0.966529i \(-0.417412\pi\)
0.256558 + 0.966529i \(0.417412\pi\)
\(858\) 0 0
\(859\) 22.3863 0.763810 0.381905 0.924202i \(-0.375268\pi\)
0.381905 + 0.924202i \(0.375268\pi\)
\(860\) 0 0
\(861\) 31.5277 1.07446
\(862\) 0 0
\(863\) −4.17657 −0.142172 −0.0710860 0.997470i \(-0.522646\pi\)
−0.0710860 + 0.997470i \(0.522646\pi\)
\(864\) 0 0
\(865\) −11.0970 −0.377310
\(866\) 0 0
\(867\) 25.6937 0.872604
\(868\) 0 0
\(869\) −29.5509 −1.00244
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −20.9603 −0.709397
\(874\) 0 0
\(875\) 2.33790 0.0790356
\(876\) 0 0
\(877\) 39.2315 1.32475 0.662377 0.749171i \(-0.269548\pi\)
0.662377 + 0.749171i \(0.269548\pi\)
\(878\) 0 0
\(879\) −43.5585 −1.46919
\(880\) 0 0
\(881\) 8.53832 0.287663 0.143832 0.989602i \(-0.454058\pi\)
0.143832 + 0.989602i \(0.454058\pi\)
\(882\) 0 0
\(883\) 44.7515 1.50601 0.753004 0.658016i \(-0.228604\pi\)
0.753004 + 0.658016i \(0.228604\pi\)
\(884\) 0 0
\(885\) −1.29292 −0.0434611
\(886\) 0 0
\(887\) 37.1035 1.24581 0.622906 0.782296i \(-0.285952\pi\)
0.622906 + 0.782296i \(0.285952\pi\)
\(888\) 0 0
\(889\) −24.9632 −0.837237
\(890\) 0 0
\(891\) 241.652 8.09563
\(892\) 0 0
\(893\) 0.332070 0.0111123
\(894\) 0 0
\(895\) −5.27165 −0.176212
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −46.9753 −1.56671
\(900\) 0 0
\(901\) 24.8616 0.828260
\(902\) 0 0
\(903\) −69.6885 −2.31909
\(904\) 0 0
\(905\) −16.1744 −0.537655
\(906\) 0 0
\(907\) −37.6342 −1.24962 −0.624812 0.780776i \(-0.714824\pi\)
−0.624812 + 0.780776i \(0.714824\pi\)
\(908\) 0 0
\(909\) −47.5566 −1.57735
\(910\) 0 0
\(911\) 39.8183 1.31924 0.659619 0.751600i \(-0.270718\pi\)
0.659619 + 0.751600i \(0.270718\pi\)
\(912\) 0 0
\(913\) −71.6380 −2.37087
\(914\) 0 0
\(915\) −16.9074 −0.558940
\(916\) 0 0
\(917\) 48.1777 1.59097
\(918\) 0 0
\(919\) −34.1947 −1.12798 −0.563989 0.825782i \(-0.690734\pi\)
−0.563989 + 0.825782i \(0.690734\pi\)
\(920\) 0 0
\(921\) 111.002 3.65766
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −0.794425 −0.0261205
\(926\) 0 0
\(927\) 72.8617 2.39309
\(928\) 0 0
\(929\) 30.5866 1.00351 0.501757 0.865009i \(-0.332687\pi\)
0.501757 + 0.865009i \(0.332687\pi\)
\(930\) 0 0
\(931\) −0.138490 −0.00453881
\(932\) 0 0
\(933\) −15.3948 −0.504003
\(934\) 0 0
\(935\) −18.9606 −0.620077
\(936\) 0 0
\(937\) −39.0385 −1.27533 −0.637666 0.770313i \(-0.720100\pi\)
−0.637666 + 0.770313i \(0.720100\pi\)
\(938\) 0 0
\(939\) 8.20816 0.267863
\(940\) 0 0
\(941\) 6.92575 0.225773 0.112886 0.993608i \(-0.463990\pi\)
0.112886 + 0.993608i \(0.463990\pi\)
\(942\) 0 0
\(943\) 19.0809 0.621359
\(944\) 0 0
\(945\) −44.6138 −1.45129
\(946\) 0 0
\(947\) −15.2145 −0.494405 −0.247202 0.968964i \(-0.579511\pi\)
−0.247202 + 0.968964i \(0.579511\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 39.7680 1.28957
\(952\) 0 0
\(953\) −5.06168 −0.163964 −0.0819819 0.996634i \(-0.526125\pi\)
−0.0819819 + 0.996634i \(0.526125\pi\)
\(954\) 0 0
\(955\) 6.75365 0.218543
\(956\) 0 0
\(957\) −118.281 −3.82348
\(958\) 0 0
\(959\) −0.198757 −0.00641820
\(960\) 0 0
\(961\) 38.5686 1.24415
\(962\) 0 0
\(963\) 97.9732 3.15714
\(964\) 0 0
\(965\) 0.481519 0.0155006
\(966\) 0 0
\(967\) −48.3231 −1.55397 −0.776983 0.629522i \(-0.783251\pi\)
−0.776983 + 0.629522i \(0.783251\pi\)
\(968\) 0 0
\(969\) −0.945539 −0.0303751
\(970\) 0 0
\(971\) 45.8463 1.47128 0.735638 0.677374i \(-0.236882\pi\)
0.735638 + 0.677374i \(0.236882\pi\)
\(972\) 0 0
\(973\) −25.7405 −0.825204
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −43.2717 −1.38438 −0.692192 0.721714i \(-0.743355\pi\)
−0.692192 + 0.721714i \(0.743355\pi\)
\(978\) 0 0
\(979\) 72.2108 2.30787
\(980\) 0 0
\(981\) 77.8188 2.48456
\(982\) 0 0
\(983\) −0.695243 −0.0221748 −0.0110874 0.999939i \(-0.503529\pi\)
−0.0110874 + 0.999939i \(0.503529\pi\)
\(984\) 0 0
\(985\) 21.8881 0.697414
\(986\) 0 0
\(987\) 29.2951 0.932474
\(988\) 0 0
\(989\) −42.1762 −1.34113
\(990\) 0 0
\(991\) 21.1973 0.673355 0.336677 0.941620i \(-0.390697\pi\)
0.336677 + 0.941620i \(0.390697\pi\)
\(992\) 0 0
\(993\) 76.2403 2.41941
\(994\) 0 0
\(995\) −23.5567 −0.746799
\(996\) 0 0
\(997\) −36.7860 −1.16503 −0.582513 0.812822i \(-0.697930\pi\)
−0.582513 + 0.812822i \(0.697930\pi\)
\(998\) 0 0
\(999\) 15.1599 0.479637
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 3380.2.a.r.1.1 9
13.5 odd 4 3380.2.f.j.3041.2 18
13.8 odd 4 3380.2.f.j.3041.1 18
13.12 even 2 3380.2.a.s.1.1 yes 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3380.2.a.r.1.1 9 1.1 even 1 trivial
3380.2.a.s.1.1 yes 9 13.12 even 2
3380.2.f.j.3041.1 18 13.8 odd 4
3380.2.f.j.3041.2 18 13.5 odd 4