Properties

Label 3380.2.f.j.3041.1
Level $3380$
Weight $2$
Character 3380.3041
Analytic conductor $26.989$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3380.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(26.9894358832\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
Defining polynomial: \( x^{18} + 39x^{16} + 605x^{14} + 4764x^{12} + 20080x^{10} + 43783x^{8} + 43791x^{6} + 14071x^{4} + 378x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3041.1
Root \(-3.40622i\) of defining polynomial
Character \(\chi\) \(=\) 3380.3041
Dual form 3380.2.f.j.3041.2

$q$-expansion

\(f(q)\) \(=\) \(q-3.40622 q^{3} -1.00000i q^{5} +2.33790i q^{7} +8.60234 q^{9} +O(q^{10})\) \(q-3.40622 q^{3} -1.00000i q^{5} +2.33790i q^{7} +8.60234 q^{9} -6.16565i q^{11} +3.40622i q^{15} -3.07520 q^{17} +0.0902681i q^{19} -7.96342i q^{21} -4.81954 q^{23} -1.00000 q^{25} -19.0828 q^{27} +5.63201 q^{29} -8.34078i q^{31} +21.0016i q^{33} +2.33790 q^{35} +0.794425i q^{37} +3.95907i q^{41} +8.75108 q^{43} -8.60234i q^{45} -3.67871i q^{47} +1.53420 q^{49} +10.4748 q^{51} +8.08456 q^{53} -6.16565 q^{55} -0.307473i q^{57} +0.379577i q^{59} -4.96367 q^{61} +20.1114i q^{63} -0.376774i q^{67} +16.4164 q^{69} +9.04130i q^{71} +5.48133i q^{73} +3.40622 q^{75} +14.4147 q^{77} -4.79283 q^{79} +39.1932 q^{81} -11.6189i q^{83} +3.07520i q^{85} -19.1839 q^{87} -11.7118i q^{89} +28.4105i q^{93} +0.0902681 q^{95} -2.43658i q^{97} -53.0390i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 2 q^{3} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 2 q^{3} + 24 q^{9} - 26 q^{17} - 24 q^{23} - 18 q^{25} - 8 q^{27} + 32 q^{29} + 2 q^{35} - 2 q^{43} - 40 q^{49} - 22 q^{51} + 60 q^{53} - 14 q^{55} + 42 q^{61} - 30 q^{69} + 2 q^{75} - 92 q^{77} + 62 q^{79} + 82 q^{81} + 56 q^{87} + 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1691\) \(1861\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(5\) − 1.00000i − 0.447214i
\(6\) 0 0
\(7\) 2.33790i 0.883645i 0.897103 + 0.441822i \(0.145668\pi\)
−0.897103 + 0.441822i \(0.854332\pi\)
\(8\) 0 0
\(9\) 8.60234 2.86745
\(10\) 0 0
\(11\) − 6.16565i − 1.85901i −0.368806 0.929507i \(-0.620233\pi\)
0.368806 0.929507i \(-0.379767\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) 3.40622i 0.879482i
\(16\) 0 0
\(17\) −3.07520 −0.745844 −0.372922 0.927863i \(-0.621644\pi\)
−0.372922 + 0.927863i \(0.621644\pi\)
\(18\) 0 0
\(19\) 0.0902681i 0.0207089i 0.999946 + 0.0103545i \(0.00329599\pi\)
−0.999946 + 0.0103545i \(0.996704\pi\)
\(20\) 0 0
\(21\) − 7.96342i − 1.73776i
\(22\) 0 0
\(23\) −4.81954 −1.00494 −0.502472 0.864594i \(-0.667576\pi\)
−0.502472 + 0.864594i \(0.667576\pi\)
\(24\) 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
\(31\) − 8.34078i − 1.49805i −0.662543 0.749024i \(-0.730523\pi\)
0.662543 0.749024i \(-0.269477\pi\)
\(32\) 0 0
\(33\) 21.0016i 3.65590i
\(34\) 0 0
\(35\) 2.33790 0.395178
\(36\) 0 0
\(37\) 0.794425i 0.130603i 0.997866 + 0.0653014i \(0.0208009\pi\)
−0.997866 + 0.0653014i \(0.979199\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.95907i 0.618302i 0.951013 + 0.309151i \(0.100045\pi\)
−0.951013 + 0.309151i \(0.899955\pi\)
\(42\) 0 0
\(43\) 8.75108 1.33453 0.667264 0.744822i \(-0.267465\pi\)
0.667264 + 0.744822i \(0.267465\pi\)
\(44\) 0 0
\(45\) − 8.60234i − 1.28236i
\(46\) 0 0
\(47\) − 3.67871i − 0.536595i −0.963336 0.268298i \(-0.913539\pi\)
0.963336 0.268298i \(-0.0864611\pi\)
\(48\) 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
\(57\) − 0.307473i − 0.0407258i
\(58\) 0 0
\(59\) 0.379577i 0.0494167i 0.999695 + 0.0247084i \(0.00786572\pi\)
−0.999695 + 0.0247084i \(0.992134\pi\)
\(60\) 0 0
\(61\) −4.96367 −0.635533 −0.317766 0.948169i \(-0.602933\pi\)
−0.317766 + 0.948169i \(0.602933\pi\)
\(62\) 0 0
\(63\) 20.1114i 2.53380i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 0.376774i − 0.0460302i −0.999735 0.0230151i \(-0.992673\pi\)
0.999735 0.0230151i \(-0.00732659\pi\)
\(68\) 0 0
\(69\) 16.4164 1.97630
\(70\) 0 0
\(71\) 9.04130i 1.07300i 0.843899 + 0.536502i \(0.180255\pi\)
−0.843899 + 0.536502i \(0.819745\pi\)
\(72\) 0 0
\(73\) 5.48133i 0.641541i 0.947157 + 0.320770i \(0.103942\pi\)
−0.947157 + 0.320770i \(0.896058\pi\)
\(74\) 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
\(83\) − 11.6189i − 1.27534i −0.770310 0.637670i \(-0.779898\pi\)
0.770310 0.637670i \(-0.220102\pi\)
\(84\) 0 0
\(85\) 3.07520i 0.333552i
\(86\) 0 0
\(87\) −19.1839 −2.05673
\(88\) 0 0
\(89\) − 11.7118i − 1.24145i −0.784029 0.620724i \(-0.786839\pi\)
0.784029 0.620724i \(-0.213161\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 28.4105i 2.94603i
\(94\) 0 0
\(95\) 0.0902681 0.00926131
\(96\) 0 0
\(97\) − 2.43658i − 0.247397i −0.992320 0.123698i \(-0.960524\pi\)
0.992320 0.123698i \(-0.0394755\pi\)
\(98\) 0 0
\(99\) − 53.0390i − 5.33062i
\(100\) 0 0
\(101\) 5.52834 0.550090 0.275045 0.961431i \(-0.411307\pi\)
0.275045 + 0.961431i \(0.411307\pi\)
\(102\) 0 0
\(103\) −8.46999 −0.834573 −0.417287 0.908775i \(-0.637019\pi\)
−0.417287 + 0.908775i \(0.637019\pi\)
\(104\) 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
\(109\) 9.04624i 0.866473i 0.901280 + 0.433236i \(0.142628\pi\)
−0.901280 + 0.433236i \(0.857372\pi\)
\(110\) 0 0
\(111\) − 2.70599i − 0.256841i
\(112\) 0 0
\(113\) −2.99157 −0.281423 −0.140712 0.990051i \(-0.544939\pi\)
−0.140712 + 0.990051i \(0.544939\pi\)
\(114\) 0 0
\(115\) 4.81954i 0.449425i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 7.18951i − 0.659061i
\(120\) 0 0
\(121\) −27.0152 −2.45593
\(122\) 0 0
\(123\) − 13.4855i − 1.21594i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) −10.6776 −0.947482 −0.473741 0.880664i \(-0.657097\pi\)
−0.473741 + 0.880664i \(0.657097\pi\)
\(128\) 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
\(135\) 19.0828i 1.64239i
\(136\) 0 0
\(137\) − 0.0850150i − 0.00726332i −0.999993 0.00363166i \(-0.998844\pi\)
0.999993 0.00363166i \(-0.00115600\pi\)
\(138\) 0 0
\(139\) 11.0101 0.933864 0.466932 0.884293i \(-0.345359\pi\)
0.466932 + 0.884293i \(0.345359\pi\)
\(140\) 0 0
\(141\) 12.5305i 1.05526i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) − 5.63201i − 0.467713i
\(146\) 0 0
\(147\) −5.22584 −0.431020
\(148\) 0 0
\(149\) − 4.41512i − 0.361701i −0.983511 0.180850i \(-0.942115\pi\)
0.983511 0.180850i \(-0.0578850\pi\)
\(150\) 0 0
\(151\) 15.7555i 1.28216i 0.767473 + 0.641081i \(0.221514\pi\)
−0.767473 + 0.641081i \(0.778486\pi\)
\(152\) 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
\(161\) − 11.2676i − 0.888013i
\(162\) 0 0
\(163\) 14.4332i 1.13050i 0.824920 + 0.565249i \(0.191220\pi\)
−0.824920 + 0.565249i \(0.808780\pi\)
\(164\) 0 0
\(165\) 21.0016 1.63497
\(166\) 0 0
\(167\) − 18.9815i − 1.46883i −0.678698 0.734417i \(-0.737456\pi\)
0.678698 0.734417i \(-0.262544\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 0.776517i 0.0593817i
\(172\) 0 0
\(173\) −11.0970 −0.843691 −0.421845 0.906668i \(-0.638617\pi\)
−0.421845 + 0.906668i \(0.638617\pi\)
\(174\) 0 0
\(175\) − 2.33790i − 0.176729i
\(176\) 0 0
\(177\) − 1.29292i − 0.0971821i
\(178\) 0 0
\(179\) −5.27165 −0.394021 −0.197011 0.980401i \(-0.563123\pi\)
−0.197011 + 0.980401i \(0.563123\pi\)
\(180\) 0 0
\(181\) −16.1744 −1.20223 −0.601117 0.799161i \(-0.705277\pi\)
−0.601117 + 0.799161i \(0.705277\pi\)
\(182\) 0 0
\(183\) 16.9074 1.24983
\(184\) 0 0
\(185\) 0.794425 0.0584073
\(186\) 0 0
\(187\) 18.9606i 1.38653i
\(188\) 0 0
\(189\) − 44.6138i − 3.24517i
\(190\) 0 0
\(191\) −6.75365 −0.488677 −0.244338 0.969690i \(-0.578571\pi\)
−0.244338 + 0.969690i \(0.578571\pi\)
\(192\) 0 0
\(193\) 0.481519i 0.0346605i 0.999850 + 0.0173302i \(0.00551666\pi\)
−0.999850 + 0.0173302i \(0.994483\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 21.8881i − 1.55946i −0.626113 0.779732i \(-0.715355\pi\)
0.626113 0.779732i \(-0.284645\pi\)
\(198\) 0 0
\(199\) −23.5567 −1.66989 −0.834947 0.550331i \(-0.814502\pi\)
−0.834947 + 0.550331i \(0.814502\pi\)
\(200\) 0 0
\(201\) 1.28337i 0.0905223i
\(202\) 0 0
\(203\) 13.1671i 0.924149i
\(204\) 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
\(213\) − 30.7967i − 2.11015i
\(214\) 0 0
\(215\) − 8.75108i − 0.596819i
\(216\) 0 0
\(217\) 19.4999 1.32374
\(218\) 0 0
\(219\) − 18.6706i − 1.26164i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) − 24.3465i − 1.63036i −0.579205 0.815182i \(-0.696637\pi\)
0.579205 0.815182i \(-0.303363\pi\)
\(224\) 0 0
\(225\) −8.60234 −0.573489
\(226\) 0 0
\(227\) − 0.100791i − 0.00668971i −0.999994 0.00334485i \(-0.998935\pi\)
0.999994 0.00334485i \(-0.00106470\pi\)
\(228\) 0 0
\(229\) − 5.60506i − 0.370393i −0.982702 0.185196i \(-0.940708\pi\)
0.982702 0.185196i \(-0.0592921\pi\)
\(230\) 0 0
\(231\) −49.0996 −3.23052
\(232\) 0 0
\(233\) −13.2785 −0.869902 −0.434951 0.900454i \(-0.643234\pi\)
−0.434951 + 0.900454i \(0.643234\pi\)
\(234\) 0 0
\(235\) −3.67871 −0.239973
\(236\) 0 0
\(237\) 16.3254 1.06045
\(238\) 0 0
\(239\) 12.9817i 0.839715i 0.907590 + 0.419858i \(0.137920\pi\)
−0.907590 + 0.419858i \(0.862080\pi\)
\(240\) 0 0
\(241\) 1.49222i 0.0961223i 0.998844 + 0.0480612i \(0.0153042\pi\)
−0.998844 + 0.0480612i \(0.984696\pi\)
\(242\) 0 0
\(243\) −76.2524 −4.89159
\(244\) 0 0
\(245\) − 1.53420i − 0.0980167i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 39.5765i 2.50806i
\(250\) 0 0
\(251\) 11.2640 0.710975 0.355487 0.934681i \(-0.384315\pi\)
0.355487 + 0.934681i \(0.384315\pi\)
\(252\) 0 0
\(253\) 29.7156i 1.86820i
\(254\) 0 0
\(255\) − 10.4748i − 0.655957i
\(256\) 0 0
\(257\) −15.0148 −0.936600 −0.468300 0.883569i \(-0.655133\pi\)
−0.468300 + 0.883569i \(0.655133\pi\)
\(258\) 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
\(265\) − 8.08456i − 0.496631i
\(266\) 0 0
\(267\) 39.8930i 2.44141i
\(268\) 0 0
\(269\) 25.3906 1.54809 0.774047 0.633129i \(-0.218230\pi\)
0.774047 + 0.633129i \(0.218230\pi\)
\(270\) 0 0
\(271\) 10.6327i 0.645892i 0.946417 + 0.322946i \(0.104673\pi\)
−0.946417 + 0.322946i \(0.895327\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 6.16565i 0.371803i
\(276\) 0 0
\(277\) −25.9901 −1.56159 −0.780797 0.624785i \(-0.785187\pi\)
−0.780797 + 0.624785i \(0.785187\pi\)
\(278\) 0 0
\(279\) − 71.7502i − 4.29557i
\(280\) 0 0
\(281\) − 7.65920i − 0.456909i −0.973555 0.228455i \(-0.926633\pi\)
0.973555 0.228455i \(-0.0733673\pi\)
\(282\) 0 0
\(283\) −19.5025 −1.15930 −0.579651 0.814865i \(-0.696811\pi\)
−0.579651 + 0.814865i \(0.696811\pi\)
\(284\) 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
\(291\) 8.29952i 0.486526i
\(292\) 0 0
\(293\) − 12.7879i − 0.747078i −0.927614 0.373539i \(-0.878144\pi\)
0.927614 0.373539i \(-0.121856\pi\)
\(294\) 0 0
\(295\) 0.379577 0.0220998
\(296\) 0 0
\(297\) 117.658i 6.82720i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 20.4592i 1.17925i
\(302\) 0 0
\(303\) −18.8307 −1.08180
\(304\) 0 0
\(305\) 4.96367i 0.284219i
\(306\) 0 0
\(307\) 32.5882i 1.85990i 0.367680 + 0.929952i \(0.380152\pi\)
−0.367680 + 0.929952i \(0.619848\pi\)
\(308\) 0 0
\(309\) 28.8507 1.64126
\(310\) 0 0
\(311\) −4.51961 −0.256284 −0.128142 0.991756i \(-0.540901\pi\)
−0.128142 + 0.991756i \(0.540901\pi\)
\(312\) 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
\(317\) − 11.6751i − 0.655739i −0.944723 0.327870i \(-0.893669\pi\)
0.944723 0.327870i \(-0.106331\pi\)
\(318\) 0 0
\(319\) − 34.7250i − 1.94423i
\(320\) 0 0
\(321\) −38.7939 −2.16526
\(322\) 0 0
\(323\) − 0.277592i − 0.0154456i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 30.8135i − 1.70399i
\(328\) 0 0
\(329\) 8.60048 0.474160
\(330\) 0 0
\(331\) − 22.3827i − 1.23026i −0.788425 0.615131i \(-0.789103\pi\)
0.788425 0.615131i \(-0.210897\pi\)
\(332\) 0 0
\(333\) 6.83392i 0.374496i
\(334\) 0 0
\(335\) −0.376774 −0.0205854
\(336\) 0 0
\(337\) −12.6686 −0.690102 −0.345051 0.938584i \(-0.612138\pi\)
−0.345051 + 0.938584i \(0.612138\pi\)
\(338\) 0 0
\(339\) 10.1900 0.553442
\(340\) 0 0
\(341\) −51.4263 −2.78489
\(342\) 0 0
\(343\) 19.9522i 1.07731i
\(344\) 0 0
\(345\) − 16.4164i − 0.883830i
\(346\) 0 0
\(347\) 7.56502 0.406111 0.203056 0.979167i \(-0.434913\pi\)
0.203056 + 0.979167i \(0.434913\pi\)
\(348\) 0 0
\(349\) − 18.3617i − 0.982879i −0.870912 0.491439i \(-0.836471\pi\)
0.870912 0.491439i \(-0.163529\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1.27380i 0.0677976i 0.999425 + 0.0338988i \(0.0107924\pi\)
−0.999425 + 0.0338988i \(0.989208\pi\)
\(354\) 0 0
\(355\) 9.04130 0.479862
\(356\) 0 0
\(357\) 24.4891i 1.29610i
\(358\) 0 0
\(359\) − 11.0008i − 0.580598i −0.956936 0.290299i \(-0.906245\pi\)
0.956936 0.290299i \(-0.0937548\pi\)
\(360\) 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
\(369\) 34.0572i 1.77295i
\(370\) 0 0
\(371\) 18.9009i 0.981288i
\(372\) 0 0
\(373\) −21.7186 −1.12455 −0.562274 0.826951i \(-0.690073\pi\)
−0.562274 + 0.826951i \(0.690073\pi\)
\(374\) 0 0
\(375\) − 3.40622i − 0.175896i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) − 29.5331i − 1.51701i −0.651665 0.758507i \(-0.725929\pi\)
0.651665 0.758507i \(-0.274071\pi\)
\(380\) 0 0
\(381\) 36.3702 1.86330
\(382\) 0 0
\(383\) 26.8889i 1.37396i 0.726676 + 0.686980i \(0.241064\pi\)
−0.726676 + 0.686980i \(0.758936\pi\)
\(384\) 0 0
\(385\) − 14.4147i − 0.734641i
\(386\) 0 0
\(387\) 75.2798 3.82669
\(388\) 0 0
\(389\) 3.11775 0.158076 0.0790380 0.996872i \(-0.474815\pi\)
0.0790380 + 0.996872i \(0.474815\pi\)
\(390\) 0 0
\(391\) 14.8210 0.749532
\(392\) 0 0
\(393\) 70.1928 3.54076
\(394\) 0 0
\(395\) 4.79283i 0.241153i
\(396\) 0 0
\(397\) − 21.2851i − 1.06827i −0.845400 0.534133i \(-0.820638\pi\)
0.845400 0.534133i \(-0.179362\pi\)
\(398\) 0 0
\(399\) 0.718842 0.0359871
\(400\) 0 0
\(401\) 7.30134i 0.364611i 0.983242 + 0.182306i \(0.0583560\pi\)
−0.983242 + 0.182306i \(0.941644\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) − 39.1932i − 1.94753i
\(406\) 0 0
\(407\) 4.89815 0.242792
\(408\) 0 0
\(409\) 4.26732i 0.211005i 0.994419 + 0.105503i \(0.0336451\pi\)
−0.994419 + 0.105503i \(0.966355\pi\)
\(410\) 0 0
\(411\) 0.289580i 0.0142839i
\(412\) 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
\(421\) − 7.36448i − 0.358923i −0.983765 0.179461i \(-0.942564\pi\)
0.983765 0.179461i \(-0.0574355\pi\)
\(422\) 0 0
\(423\) − 31.6455i − 1.53866i
\(424\) 0 0
\(425\) 3.07520 0.149169
\(426\) 0 0
\(427\) − 11.6046i − 0.561585i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 20.1661i 0.971369i 0.874134 + 0.485684i \(0.161430\pi\)
−0.874134 + 0.485684i \(0.838570\pi\)
\(432\) 0 0
\(433\) −11.0657 −0.531783 −0.265892 0.964003i \(-0.585666\pi\)
−0.265892 + 0.964003i \(0.585666\pi\)
\(434\) 0 0
\(435\) 19.1839i 0.919796i
\(436\) 0 0
\(437\) − 0.435051i − 0.0208113i
\(438\) 0 0
\(439\) −34.5470 −1.64884 −0.824419 0.565980i \(-0.808498\pi\)
−0.824419 + 0.565980i \(0.808498\pi\)
\(440\) 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
\(447\) 15.0389i 0.711315i
\(448\) 0 0
\(449\) 18.2491i 0.861228i 0.902536 + 0.430614i \(0.141703\pi\)
−0.902536 + 0.430614i \(0.858297\pi\)
\(450\) 0 0
\(451\) 24.4102 1.14943
\(452\) 0 0
\(453\) − 53.6666i − 2.52148i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 23.2180i 1.08609i 0.839703 + 0.543046i \(0.182729\pi\)
−0.839703 + 0.543046i \(0.817271\pi\)
\(458\) 0 0
\(459\) 58.6833 2.73910
\(460\) 0 0
\(461\) − 29.6014i − 1.37868i −0.724440 0.689338i \(-0.757901\pi\)
0.724440 0.689338i \(-0.242099\pi\)
\(462\) 0 0
\(463\) − 30.4362i − 1.41449i −0.706969 0.707244i \(-0.749938\pi\)
0.706969 0.707244i \(-0.250062\pi\)
\(464\) 0 0
\(465\) 28.4105 1.31751
\(466\) 0 0
\(467\) −15.6000 −0.721880 −0.360940 0.932589i \(-0.617544\pi\)
−0.360940 + 0.932589i \(0.617544\pi\)
\(468\) 0 0
\(469\) 0.880861 0.0406744
\(470\) 0 0
\(471\) −11.5852 −0.533819
\(472\) 0 0
\(473\) − 53.9561i − 2.48090i
\(474\) 0 0
\(475\) − 0.0902681i − 0.00414178i
\(476\) 0 0
\(477\) 69.5462 3.18430
\(478\) 0 0
\(479\) − 41.5259i − 1.89737i −0.316227 0.948684i \(-0.602416\pi\)
0.316227 0.948684i \(-0.397584\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 38.3800i 1.74635i
\(484\) 0 0
\(485\) −2.43658 −0.110639
\(486\) 0 0
\(487\) 9.84163i 0.445967i 0.974822 + 0.222983i \(0.0715796\pi\)
−0.974822 + 0.222983i \(0.928420\pi\)
\(488\) 0 0
\(489\) − 49.1628i − 2.22322i
\(490\) 0 0
\(491\) −29.1322 −1.31472 −0.657359 0.753577i \(-0.728327\pi\)
−0.657359 + 0.753577i \(0.728327\pi\)
\(492\) 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
\(499\) − 33.1347i − 1.48331i −0.670781 0.741656i \(-0.734041\pi\)
0.670781 0.741656i \(-0.265959\pi\)
\(500\) 0 0
\(501\) 64.6553i 2.88858i
\(502\) 0 0
\(503\) 24.3323 1.08492 0.542462 0.840080i \(-0.317492\pi\)
0.542462 + 0.840080i \(0.317492\pi\)
\(504\) 0 0
\(505\) − 5.52834i − 0.246008i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 16.9484i 0.751223i 0.926777 + 0.375611i \(0.122567\pi\)
−0.926777 + 0.375611i \(0.877433\pi\)
\(510\) 0 0
\(511\) −12.8148 −0.566894
\(512\) 0 0
\(513\) − 1.72257i − 0.0760532i
\(514\) 0 0
\(515\) 8.46999i 0.373232i
\(516\) 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
\(525\) 7.96342i 0.347552i
\(526\) 0 0
\(527\) 25.6495i 1.11731i
\(528\) 0 0
\(529\) 0.227981 0.00991221
\(530\) 0 0
\(531\) 3.26525i 0.141700i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) − 11.3891i − 0.492395i
\(536\) 0 0
\(537\) 17.9564 0.774876
\(538\) 0 0
\(539\) − 9.45936i − 0.407444i
\(540\) 0 0
\(541\) 44.1770i 1.89932i 0.313286 + 0.949659i \(0.398570\pi\)
−0.313286 + 0.949659i \(0.601430\pi\)
\(542\) 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
\(551\) 0.508391i 0.0216582i
\(552\) 0 0
\(553\) − 11.2052i − 0.476492i
\(554\) 0 0
\(555\) −2.70599 −0.114863
\(556\) 0 0
\(557\) − 29.8749i − 1.26584i −0.774218 0.632919i \(-0.781857\pi\)
0.774218 0.632919i \(-0.218143\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) − 64.5839i − 2.72673i
\(562\) 0 0
\(563\) −16.9922 −0.716135 −0.358067 0.933696i \(-0.616564\pi\)
−0.358067 + 0.933696i \(0.616564\pi\)
\(564\) 0 0
\(565\) 2.99157i 0.125856i
\(566\) 0 0
\(567\) 91.6300i 3.84810i
\(568\) 0 0
\(569\) −16.8326 −0.705660 −0.352830 0.935687i \(-0.614781\pi\)
−0.352830 + 0.935687i \(0.614781\pi\)
\(570\) 0 0
\(571\) −1.09953 −0.0460138 −0.0230069 0.999735i \(-0.507324\pi\)
−0.0230069 + 0.999735i \(0.507324\pi\)
\(572\) 0 0
\(573\) 23.0044 0.961023
\(574\) 0 0
\(575\) 4.81954 0.200989
\(576\) 0 0
\(577\) − 21.1153i − 0.879042i −0.898232 0.439521i \(-0.855148\pi\)
0.898232 0.439521i \(-0.144852\pi\)
\(578\) 0 0
\(579\) − 1.64016i − 0.0681627i
\(580\) 0 0
\(581\) 27.1639 1.12695
\(582\) 0 0
\(583\) − 49.8466i − 2.06443i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 32.2938i 1.33291i 0.745547 + 0.666453i \(0.232188\pi\)
−0.745547 + 0.666453i \(0.767812\pi\)
\(588\) 0 0
\(589\) 0.752906 0.0310229
\(590\) 0 0
\(591\) 74.5558i 3.06681i
\(592\) 0 0
\(593\) 12.5827i 0.516711i 0.966050 + 0.258356i \(0.0831806\pi\)
−0.966050 + 0.258356i \(0.916819\pi\)
\(594\) 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
\(603\) − 3.24114i − 0.131989i
\(604\) 0 0
\(605\) 27.0152i 1.09833i
\(606\) 0 0
\(607\) −26.5832 −1.07898 −0.539490 0.841992i \(-0.681383\pi\)
−0.539490 + 0.841992i \(0.681383\pi\)
\(608\) 0 0
\(609\) − 44.8500i − 1.81742i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) − 44.8547i − 1.81166i −0.423639 0.905831i \(-0.639247\pi\)
0.423639 0.905831i \(-0.360753\pi\)
\(614\) 0 0
\(615\) −13.4855 −0.543786
\(616\) 0 0
\(617\) 20.9169i 0.842082i 0.907042 + 0.421041i \(0.138335\pi\)
−0.907042 + 0.421041i \(0.861665\pi\)
\(618\) 0 0
\(619\) 9.55179i 0.383919i 0.981403 + 0.191959i \(0.0614842\pi\)
−0.981403 + 0.191959i \(0.938516\pi\)
\(620\) 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
\(629\) − 2.44301i − 0.0974093i
\(630\) 0 0
\(631\) − 4.38539i − 0.174580i −0.996183 0.0872899i \(-0.972179\pi\)
0.996183 0.0872899i \(-0.0278206\pi\)
\(632\) 0 0
\(633\) 37.6937 1.49819
\(634\) 0 0
\(635\) 10.6776i 0.423727i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 77.7763i 3.07678i
\(640\) 0 0
\(641\) 19.1325 0.755690 0.377845 0.925869i \(-0.376665\pi\)
0.377845 + 0.925869i \(0.376665\pi\)
\(642\) 0 0
\(643\) 17.7043i 0.698190i 0.937087 + 0.349095i \(0.113511\pi\)
−0.937087 + 0.349095i \(0.886489\pi\)
\(644\) 0 0
\(645\) 29.8081i 1.17369i
\(646\) 0 0
\(647\) 30.0594 1.18176 0.590879 0.806761i \(-0.298781\pi\)
0.590879 + 0.806761i \(0.298781\pi\)
\(648\) 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
\(655\) 20.6072i 0.805191i
\(656\) 0 0
\(657\) 47.1522i 1.83958i
\(658\) 0 0
\(659\) 8.47410 0.330104 0.165052 0.986285i \(-0.447221\pi\)
0.165052 + 0.986285i \(0.447221\pi\)
\(660\) 0 0
\(661\) − 14.9713i − 0.582318i −0.956675 0.291159i \(-0.905959\pi\)
0.956675 0.291159i \(-0.0940408\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0.211038i 0.00818371i
\(666\) 0 0
\(667\) −27.1437 −1.05101
\(668\) 0 0
\(669\) 82.9296i 3.20625i
\(670\) 0 0
\(671\) 30.6043i 1.18146i
\(672\) 0 0
\(673\) −35.3418 −1.36233 −0.681163 0.732132i \(-0.738525\pi\)
−0.681163 + 0.732132i \(0.738525\pi\)
\(674\) 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
\(681\) 0.343315i 0.0131559i
\(682\) 0 0
\(683\) 48.5452i 1.85753i 0.370667 + 0.928766i \(0.379129\pi\)
−0.370667 + 0.928766i \(0.620871\pi\)
\(684\) 0 0
\(685\) −0.0850150 −0.00324826
\(686\) 0 0
\(687\) 19.0921i 0.728408i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 17.9719i 0.683684i 0.939757 + 0.341842i \(0.111051\pi\)
−0.939757 + 0.341842i \(0.888949\pi\)
\(692\) 0 0
\(693\) 124.000 4.71037
\(694\) 0 0
\(695\) − 11.0101i − 0.417637i
\(696\) 0 0
\(697\) − 12.1749i − 0.461157i
\(698\) 0 0
\(699\) 45.2294 1.71073
\(700\) 0 0
\(701\) −13.0567 −0.493147 −0.246573 0.969124i \(-0.579305\pi\)
−0.246573 + 0.969124i \(0.579305\pi\)
\(702\) 0 0
\(703\) −0.0717113 −0.00270464
\(704\) 0 0
\(705\) 12.5305 0.471926
\(706\) 0 0
\(707\) 12.9247i 0.486084i
\(708\) 0 0
\(709\) − 14.9427i − 0.561186i −0.959827 0.280593i \(-0.909469\pi\)
0.959827 0.280593i \(-0.0905312\pi\)
\(710\) 0 0
\(711\) −41.2295 −1.54623
\(712\) 0 0
\(713\) 40.1987i 1.50545i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 44.2185i − 1.65137i
\(718\) 0 0
\(719\) −10.7997 −0.402761 −0.201381 0.979513i \(-0.564543\pi\)
−0.201381 + 0.979513i \(0.564543\pi\)
\(720\) 0 0
\(721\) − 19.8020i − 0.737466i
\(722\) 0 0
\(723\) − 5.08283i − 0.189032i
\(724\) 0 0
\(725\) −5.63201 −0.209168
\(726\) 0 0
\(727\) 25.1935 0.934376 0.467188 0.884158i \(-0.345267\pi\)
0.467188 + 0.884158i \(0.345267\pi\)
\(728\) 0 0
\(729\) 142.153 5.26491
\(730\) 0 0
\(731\) −26.9113 −0.995350
\(732\) 0 0
\(733\) 32.1714i 1.18828i 0.804363 + 0.594139i \(0.202507\pi\)
−0.804363 + 0.594139i \(0.797493\pi\)
\(734\) 0 0
\(735\) 5.22584i 0.192758i
\(736\) 0 0
\(737\) −2.32306 −0.0855708
\(738\) 0 0
\(739\) 44.1036i 1.62238i 0.584785 + 0.811188i \(0.301179\pi\)
−0.584785 + 0.811188i \(0.698821\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 44.4544i 1.63087i 0.578846 + 0.815437i \(0.303503\pi\)
−0.578846 + 0.815437i \(0.696497\pi\)
\(744\) 0 0
\(745\) −4.41512 −0.161758
\(746\) 0 0
\(747\) − 99.9497i − 3.65697i
\(748\) 0 0
\(749\) 26.6267i 0.972918i
\(750\) 0 0
\(751\) −37.6126 −1.37250 −0.686252 0.727364i \(-0.740745\pi\)
−0.686252 + 0.727364i \(0.740745\pi\)
\(752\) 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
\(759\) − 101.218i − 3.67398i
\(760\) 0 0
\(761\) 27.3170i 0.990240i 0.868825 + 0.495120i \(0.164876\pi\)
−0.868825 + 0.495120i \(0.835124\pi\)
\(762\) 0 0
\(763\) −21.1492 −0.765654
\(764\) 0 0
\(765\) 26.4539i 0.956442i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 23.1958i 0.836460i 0.908341 + 0.418230i \(0.137349\pi\)
−0.908341 + 0.418230i \(0.862651\pi\)
\(770\) 0 0
\(771\) 51.1439 1.84190
\(772\) 0 0
\(773\) − 6.52005i − 0.234510i −0.993102 0.117255i \(-0.962591\pi\)
0.993102 0.117255i \(-0.0374095\pi\)
\(774\) 0 0
\(775\) 8.34078i 0.299610i
\(776\) 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
\(785\) − 3.40120i − 0.121394i
\(786\) 0 0
\(787\) − 40.7311i − 1.45191i −0.687744 0.725954i \(-0.741399\pi\)
0.687744 0.725954i \(-0.258601\pi\)
\(788\) 0 0
\(789\) −38.9349 −1.38612
\(790\) 0 0
\(791\) − 6.99401i − 0.248678i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 27.5378i 0.976665i
\(796\) 0 0
\(797\) −4.48208 −0.158763 −0.0793816 0.996844i \(-0.525295\pi\)
−0.0793816 + 0.996844i \(0.525295\pi\)
\(798\) 0 0
\(799\) 11.3128i 0.400217i
\(800\) 0 0
\(801\) − 100.749i − 3.55979i
\(802\) 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
\(811\) 29.5808i 1.03872i 0.854555 + 0.519361i \(0.173830\pi\)
−0.854555 + 0.519361i \(0.826170\pi\)
\(812\) 0 0
\(813\) − 36.2174i − 1.27020i
\(814\) 0 0
\(815\) 14.4332 0.505574
\(816\) 0 0
\(817\) 0.789943i 0.0276366i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 47.1035i − 1.64392i −0.569543 0.821962i \(-0.692880\pi\)
0.569543 0.821962i \(-0.307120\pi\)
\(822\) 0 0
\(823\) 16.4099 0.572012 0.286006 0.958228i \(-0.407672\pi\)
0.286006 + 0.958228i \(0.407672\pi\)
\(824\) 0 0
\(825\) − 21.0016i − 0.731180i
\(826\) 0 0
\(827\) − 7.42042i − 0.258033i −0.991642 0.129017i \(-0.958818\pi\)
0.991642 0.129017i \(-0.0411821\pi\)
\(828\) 0 0
\(829\) 48.0555 1.66904 0.834518 0.550980i \(-0.185746\pi\)
0.834518 + 0.550980i \(0.185746\pi\)
\(830\) 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
\(837\) 159.165i 5.50156i
\(838\) 0 0
\(839\) 12.5460i 0.433136i 0.976268 + 0.216568i \(0.0694862\pi\)
−0.976268 + 0.216568i \(0.930514\pi\)
\(840\) 0 0
\(841\) 2.71952 0.0937766
\(842\) 0 0
\(843\) 26.0889i 0.898550i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 63.1590i − 2.17017i
\(848\) 0 0
\(849\) 66.4298 2.27986
\(850\) 0 0
\(851\) − 3.82877i − 0.131248i
\(852\) 0 0
\(853\) 8.26494i 0.282986i 0.989939 + 0.141493i \(0.0451903\pi\)
−0.989939 + 0.141493i \(0.954810\pi\)
\(854\) 0 0
\(855\) 0.776517 0.0265563
\(856\) 0 0
\(857\) −15.0212 −0.513115 −0.256558 0.966529i \(-0.582588\pi\)
−0.256558 + 0.966529i \(0.582588\pi\)
\(858\) 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
\(863\) − 4.17657i − 0.142172i −0.997470 0.0710860i \(-0.977354\pi\)
0.997470 0.0710860i \(-0.0226465\pi\)
\(864\) 0 0
\(865\) 11.0970i 0.377310i
\(866\) 0 0
\(867\) 25.6937 0.872604
\(868\) 0 0
\(869\) 29.5509i 1.00244i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) − 20.9603i − 0.709397i
\(874\) 0 0
\(875\) −2.33790 −0.0790356
\(876\) 0 0
\(877\) 39.2315i 1.32475i 0.749171 + 0.662377i \(0.230452\pi\)
−0.749171 + 0.662377i \(0.769548\pi\)
\(878\) 0 0
\(879\) 43.5585i 1.46919i
\(880\) 0 0
\(881\) −8.53832 −0.287663 −0.143832 0.989602i \(-0.545942\pi\)
−0.143832 + 0.989602i \(0.545942\pi\)
\(882\) 0 0
\(883\) −44.7515 −1.50601 −0.753004 0.658016i \(-0.771396\pi\)
−0.753004 + 0.658016i \(0.771396\pi\)
\(884\) 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
\(889\) − 24.9632i − 0.837237i
\(890\) 0 0
\(891\) − 241.652i − 8.09563i
\(892\) 0 0
\(893\) 0.332070 0.0111123
\(894\) 0 0
\(895\) 5.27165i 0.176212i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 46.9753i − 1.56671i
\(900\) 0 0
\(901\) −24.8616 −0.828260
\(902\) 0 0
\(903\) − 69.6885i − 2.31909i
\(904\) 0 0
\(905\) 16.1744i 0.537655i
\(906\) 0 0
\(907\) 37.6342 1.24962 0.624812 0.780776i \(-0.285176\pi\)
0.624812 + 0.780776i \(0.285176\pi\)
\(908\) 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
\(915\) − 16.9074i − 0.558940i
\(916\) 0 0
\(917\) − 48.1777i − 1.59097i
\(918\) 0 0
\(919\) −34.1947 −1.12798 −0.563989 0.825782i \(-0.690734\pi\)
−0.563989 + 0.825782i \(0.690734\pi\)
\(920\) 0 0
\(921\) − 111.002i − 3.65766i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) − 0.794425i − 0.0261205i
\(926\) 0 0
\(927\) −72.8617 −2.39309
\(928\) 0 0
\(929\) 30.5866i 1.00351i 0.865009 + 0.501757i \(0.167313\pi\)
−0.865009 + 0.501757i \(0.832687\pi\)
\(930\) 0 0
\(931\) 0.138490i 0.00453881i
\(932\) 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
\(941\) 6.92575i 0.225773i 0.993608 + 0.112886i \(0.0360096\pi\)
−0.993608 + 0.112886i \(0.963990\pi\)
\(942\) 0 0
\(943\) − 19.0809i − 0.621359i
\(944\) 0 0
\(945\) −44.6138 −1.45129
\(946\) 0 0
\(947\) 15.2145i 0.494405i 0.968964 + 0.247202i \(0.0795113\pi\)
−0.968964 + 0.247202i \(0.920489\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 39.7680i 1.28957i
\(952\) 0 0
\(953\) 5.06168 0.163964 0.0819819 0.996634i \(-0.473875\pi\)
0.0819819 + 0.996634i \(0.473875\pi\)
\(954\) 0 0
\(955\) 6.75365i 0.218543i
\(956\) 0 0
\(957\) 118.281i 3.82348i
\(958\) 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
\(967\) − 48.3231i − 1.55397i −0.629522 0.776983i \(-0.716749\pi\)
0.629522 0.776983i \(-0.283251\pi\)
\(968\) 0 0
\(969\) 0.945539i 0.0303751i
\(970\) 0 0
\(971\) 45.8463 1.47128 0.735638 0.677374i \(-0.236882\pi\)
0.735638 + 0.677374i \(0.236882\pi\)
\(972\) 0 0
\(973\) 25.7405i 0.825204i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 43.2717i − 1.38438i −0.721714 0.692192i \(-0.756645\pi\)
0.721714 0.692192i \(-0.243355\pi\)
\(978\) 0 0
\(979\) −72.2108 −2.30787
\(980\) 0 0
\(981\) 77.8188i 2.48456i
\(982\) 0 0
\(983\) 0.695243i 0.0221748i 0.999939 + 0.0110874i \(0.00352930\pi\)
−0.999939 + 0.0110874i \(0.996471\pi\)
\(984\) 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
\(993\) 76.2403i 2.41941i
\(994\) 0 0
\(995\) 23.5567i 0.746799i
\(996\) 0 0
\(997\) −36.7860 −1.16503 −0.582513 0.812822i \(-0.697930\pi\)
−0.582513 + 0.812822i \(0.697930\pi\)
\(998\) 0 0
\(999\) − 15.1599i − 0.479637i
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.f.j.3041.1 18
13.5 odd 4 3380.2.a.r.1.1 9
13.8 odd 4 3380.2.a.s.1.1 yes 9
13.12 even 2 inner 3380.2.f.j.3041.2 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3380.2.a.r.1.1 9 13.5 odd 4
3380.2.a.s.1.1 yes 9 13.8 odd 4
3380.2.f.j.3041.1 18 1.1 even 1 trivial
3380.2.f.j.3041.2 18 13.12 even 2 inner