Properties

Label 3380.2.a.l.1.1
Level $3380$
Weight $2$
Character 3380.1
Self dual yes
Analytic conductor $26.989$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.80194\) of defining polynomial
Character \(\chi\) \(=\) 3380.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.80194 q^{3} +1.00000 q^{5} +1.80194 q^{7} +0.246980 q^{9} +O(q^{10})\) \(q-1.80194 q^{3} +1.00000 q^{5} +1.80194 q^{7} +0.246980 q^{9} -1.55496 q^{11} -1.80194 q^{15} +0.911854 q^{17} -1.69202 q^{19} -3.24698 q^{21} +0.246980 q^{23} +1.00000 q^{25} +4.96077 q^{27} -5.29590 q^{29} +1.08815 q^{31} +2.80194 q^{33} +1.80194 q^{35} -3.00000 q^{37} -6.96077 q^{41} +2.26875 q^{43} +0.246980 q^{45} -8.52111 q^{47} -3.75302 q^{49} -1.64310 q^{51} +6.92692 q^{53} -1.55496 q^{55} +3.04892 q^{57} +14.7017 q^{59} -9.55496 q^{61} +0.445042 q^{63} -5.21983 q^{67} -0.445042 q^{69} +5.74094 q^{71} -14.5918 q^{73} -1.80194 q^{75} -2.80194 q^{77} +7.75063 q^{79} -9.67994 q^{81} -4.94869 q^{83} +0.911854 q^{85} +9.54288 q^{87} +11.9095 q^{89} -1.96077 q^{93} -1.69202 q^{95} +9.17390 q^{97} -0.384043 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{3} + 3 q^{5} + q^{7} - 4 q^{9} + O(q^{10}) \) \( 3 q - q^{3} + 3 q^{5} + q^{7} - 4 q^{9} - 5 q^{11} - q^{15} - q^{17} - 5 q^{21} - 4 q^{23} + 3 q^{25} + 2 q^{27} - 2 q^{29} + 7 q^{31} + 4 q^{33} + q^{35} - 9 q^{37} - 8 q^{41} - q^{43} - 4 q^{45} - 10 q^{47} - 16 q^{49} - 9 q^{51} - 8 q^{53} - 5 q^{55} + 17 q^{59} - 29 q^{61} + q^{63} - 17 q^{67} - q^{69} + 3 q^{71} - 16 q^{73} - q^{75} - 4 q^{77} - 13 q^{79} - 5 q^{81} + 17 q^{83} - q^{85} + 10 q^{87} - 9 q^{89} + 7 q^{93} - 6 q^{97} + 9 q^{99} + O(q^{100}) \)

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
\(3\) −1.80194 −1.04035 −0.520175 0.854060i \(-0.674133\pi\)
−0.520175 + 0.854060i \(0.674133\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.80194 0.681068 0.340534 0.940232i \(-0.389392\pi\)
0.340534 + 0.940232i \(0.389392\pi\)
\(8\) 0 0
\(9\) 0.246980 0.0823265
\(10\) 0 0
\(11\) −1.55496 −0.468838 −0.234419 0.972136i \(-0.575319\pi\)
−0.234419 + 0.972136i \(0.575319\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) −1.80194 −0.465258
\(16\) 0 0
\(17\) 0.911854 0.221157 0.110579 0.993867i \(-0.464730\pi\)
0.110579 + 0.993867i \(0.464730\pi\)
\(18\) 0 0
\(19\) −1.69202 −0.388176 −0.194088 0.980984i \(-0.562175\pi\)
−0.194088 + 0.980984i \(0.562175\pi\)
\(20\) 0 0
\(21\) −3.24698 −0.708549
\(22\) 0 0
\(23\) 0.246980 0.0514988 0.0257494 0.999668i \(-0.491803\pi\)
0.0257494 + 0.999668i \(0.491803\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 4.96077 0.954701
\(28\) 0 0
\(29\) −5.29590 −0.983423 −0.491712 0.870758i \(-0.663629\pi\)
−0.491712 + 0.870758i \(0.663629\pi\)
\(30\) 0 0
\(31\) 1.08815 0.195437 0.0977184 0.995214i \(-0.468846\pi\)
0.0977184 + 0.995214i \(0.468846\pi\)
\(32\) 0 0
\(33\) 2.80194 0.487755
\(34\) 0 0
\(35\) 1.80194 0.304583
\(36\) 0 0
\(37\) −3.00000 −0.493197 −0.246598 0.969118i \(-0.579313\pi\)
−0.246598 + 0.969118i \(0.579313\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.96077 −1.08709 −0.543545 0.839380i \(-0.682918\pi\)
−0.543545 + 0.839380i \(0.682918\pi\)
\(42\) 0 0
\(43\) 2.26875 0.345981 0.172991 0.984923i \(-0.444657\pi\)
0.172991 + 0.984923i \(0.444657\pi\)
\(44\) 0 0
\(45\) 0.246980 0.0368175
\(46\) 0 0
\(47\) −8.52111 −1.24293 −0.621466 0.783441i \(-0.713462\pi\)
−0.621466 + 0.783441i \(0.713462\pi\)
\(48\) 0 0
\(49\) −3.75302 −0.536146
\(50\) 0 0
\(51\) −1.64310 −0.230081
\(52\) 0 0
\(53\) 6.92692 0.951486 0.475743 0.879584i \(-0.342179\pi\)
0.475743 + 0.879584i \(0.342179\pi\)
\(54\) 0 0
\(55\) −1.55496 −0.209671
\(56\) 0 0
\(57\) 3.04892 0.403839
\(58\) 0 0
\(59\) 14.7017 1.91400 0.957000 0.290089i \(-0.0936848\pi\)
0.957000 + 0.290089i \(0.0936848\pi\)
\(60\) 0 0
\(61\) −9.55496 −1.22339 −0.611694 0.791095i \(-0.709511\pi\)
−0.611694 + 0.791095i \(0.709511\pi\)
\(62\) 0 0
\(63\) 0.445042 0.0560700
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −5.21983 −0.637704 −0.318852 0.947805i \(-0.603297\pi\)
−0.318852 + 0.947805i \(0.603297\pi\)
\(68\) 0 0
\(69\) −0.445042 −0.0535767
\(70\) 0 0
\(71\) 5.74094 0.681324 0.340662 0.940186i \(-0.389349\pi\)
0.340662 + 0.940186i \(0.389349\pi\)
\(72\) 0 0
\(73\) −14.5918 −1.70784 −0.853920 0.520404i \(-0.825781\pi\)
−0.853920 + 0.520404i \(0.825781\pi\)
\(74\) 0 0
\(75\) −1.80194 −0.208070
\(76\) 0 0
\(77\) −2.80194 −0.319310
\(78\) 0 0
\(79\) 7.75063 0.872014 0.436007 0.899943i \(-0.356392\pi\)
0.436007 + 0.899943i \(0.356392\pi\)
\(80\) 0 0
\(81\) −9.67994 −1.07555
\(82\) 0 0
\(83\) −4.94869 −0.543189 −0.271595 0.962412i \(-0.587551\pi\)
−0.271595 + 0.962412i \(0.587551\pi\)
\(84\) 0 0
\(85\) 0.911854 0.0989045
\(86\) 0 0
\(87\) 9.54288 1.02310
\(88\) 0 0
\(89\) 11.9095 1.26240 0.631200 0.775620i \(-0.282563\pi\)
0.631200 + 0.775620i \(0.282563\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −1.96077 −0.203323
\(94\) 0 0
\(95\) −1.69202 −0.173598
\(96\) 0 0
\(97\) 9.17390 0.931468 0.465734 0.884925i \(-0.345790\pi\)
0.465734 + 0.884925i \(0.345790\pi\)
\(98\) 0 0
\(99\) −0.384043 −0.0385978
\(100\) 0 0
\(101\) −2.18060 −0.216978 −0.108489 0.994098i \(-0.534601\pi\)
−0.108489 + 0.994098i \(0.534601\pi\)
\(102\) 0 0
\(103\) −9.58211 −0.944153 −0.472076 0.881558i \(-0.656495\pi\)
−0.472076 + 0.881558i \(0.656495\pi\)
\(104\) 0 0
\(105\) −3.24698 −0.316873
\(106\) 0 0
\(107\) 6.03684 0.583603 0.291801 0.956479i \(-0.405745\pi\)
0.291801 + 0.956479i \(0.405745\pi\)
\(108\) 0 0
\(109\) 10.6703 1.02202 0.511012 0.859573i \(-0.329271\pi\)
0.511012 + 0.859573i \(0.329271\pi\)
\(110\) 0 0
\(111\) 5.40581 0.513097
\(112\) 0 0
\(113\) −17.6136 −1.65694 −0.828472 0.560030i \(-0.810790\pi\)
−0.828472 + 0.560030i \(0.810790\pi\)
\(114\) 0 0
\(115\) 0.246980 0.0230310
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1.64310 0.150623
\(120\) 0 0
\(121\) −8.58211 −0.780191
\(122\) 0 0
\(123\) 12.5429 1.13095
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −21.4644 −1.90466 −0.952330 0.305071i \(-0.901320\pi\)
−0.952330 + 0.305071i \(0.901320\pi\)
\(128\) 0 0
\(129\) −4.08815 −0.359941
\(130\) 0 0
\(131\) 7.94869 0.694480 0.347240 0.937776i \(-0.387119\pi\)
0.347240 + 0.937776i \(0.387119\pi\)
\(132\) 0 0
\(133\) −3.04892 −0.264375
\(134\) 0 0
\(135\) 4.96077 0.426955
\(136\) 0 0
\(137\) 1.39075 0.118820 0.0594098 0.998234i \(-0.481078\pi\)
0.0594098 + 0.998234i \(0.481078\pi\)
\(138\) 0 0
\(139\) 20.1008 1.70493 0.852465 0.522785i \(-0.175107\pi\)
0.852465 + 0.522785i \(0.175107\pi\)
\(140\) 0 0
\(141\) 15.3545 1.29308
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −5.29590 −0.439800
\(146\) 0 0
\(147\) 6.76271 0.557779
\(148\) 0 0
\(149\) −7.34050 −0.601357 −0.300679 0.953725i \(-0.597213\pi\)
−0.300679 + 0.953725i \(0.597213\pi\)
\(150\) 0 0
\(151\) −8.46011 −0.688474 −0.344237 0.938883i \(-0.611862\pi\)
−0.344237 + 0.938883i \(0.611862\pi\)
\(152\) 0 0
\(153\) 0.225209 0.0182071
\(154\) 0 0
\(155\) 1.08815 0.0874020
\(156\) 0 0
\(157\) −17.7681 −1.41805 −0.709024 0.705185i \(-0.750864\pi\)
−0.709024 + 0.705185i \(0.750864\pi\)
\(158\) 0 0
\(159\) −12.4819 −0.989877
\(160\) 0 0
\(161\) 0.445042 0.0350742
\(162\) 0 0
\(163\) −11.6286 −0.910825 −0.455412 0.890281i \(-0.650508\pi\)
−0.455412 + 0.890281i \(0.650508\pi\)
\(164\) 0 0
\(165\) 2.80194 0.218131
\(166\) 0 0
\(167\) −9.20237 −0.712101 −0.356051 0.934467i \(-0.615877\pi\)
−0.356051 + 0.934467i \(0.615877\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) −0.417895 −0.0319572
\(172\) 0 0
\(173\) −15.4601 −1.17541 −0.587705 0.809075i \(-0.699968\pi\)
−0.587705 + 0.809075i \(0.699968\pi\)
\(174\) 0 0
\(175\) 1.80194 0.136214
\(176\) 0 0
\(177\) −26.4916 −1.99123
\(178\) 0 0
\(179\) −21.0519 −1.57349 −0.786746 0.617276i \(-0.788236\pi\)
−0.786746 + 0.617276i \(0.788236\pi\)
\(180\) 0 0
\(181\) −14.9608 −1.11203 −0.556013 0.831174i \(-0.687669\pi\)
−0.556013 + 0.831174i \(0.687669\pi\)
\(182\) 0 0
\(183\) 17.2174 1.27275
\(184\) 0 0
\(185\) −3.00000 −0.220564
\(186\) 0 0
\(187\) −1.41789 −0.103687
\(188\) 0 0
\(189\) 8.93900 0.650217
\(190\) 0 0
\(191\) 3.15452 0.228253 0.114127 0.993466i \(-0.463593\pi\)
0.114127 + 0.993466i \(0.463593\pi\)
\(192\) 0 0
\(193\) 4.04892 0.291447 0.145724 0.989325i \(-0.453449\pi\)
0.145724 + 0.989325i \(0.453449\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −5.20775 −0.371037 −0.185518 0.982641i \(-0.559396\pi\)
−0.185518 + 0.982641i \(0.559396\pi\)
\(198\) 0 0
\(199\) −22.5013 −1.59507 −0.797536 0.603272i \(-0.793864\pi\)
−0.797536 + 0.603272i \(0.793864\pi\)
\(200\) 0 0
\(201\) 9.40581 0.663435
\(202\) 0 0
\(203\) −9.54288 −0.669779
\(204\) 0 0
\(205\) −6.96077 −0.486161
\(206\) 0 0
\(207\) 0.0609989 0.00423972
\(208\) 0 0
\(209\) 2.63102 0.181992
\(210\) 0 0
\(211\) −0.0435405 −0.00299745 −0.00149873 0.999999i \(-0.500477\pi\)
−0.00149873 + 0.999999i \(0.500477\pi\)
\(212\) 0 0
\(213\) −10.3448 −0.708815
\(214\) 0 0
\(215\) 2.26875 0.154727
\(216\) 0 0
\(217\) 1.96077 0.133106
\(218\) 0 0
\(219\) 26.2935 1.77675
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 16.0911 1.07754 0.538771 0.842452i \(-0.318889\pi\)
0.538771 + 0.842452i \(0.318889\pi\)
\(224\) 0 0
\(225\) 0.246980 0.0164653
\(226\) 0 0
\(227\) 1.08575 0.0720640 0.0360320 0.999351i \(-0.488528\pi\)
0.0360320 + 0.999351i \(0.488528\pi\)
\(228\) 0 0
\(229\) 25.2620 1.66936 0.834681 0.550733i \(-0.185652\pi\)
0.834681 + 0.550733i \(0.185652\pi\)
\(230\) 0 0
\(231\) 5.04892 0.332194
\(232\) 0 0
\(233\) −23.9879 −1.57150 −0.785750 0.618544i \(-0.787723\pi\)
−0.785750 + 0.618544i \(0.787723\pi\)
\(234\) 0 0
\(235\) −8.52111 −0.555856
\(236\) 0 0
\(237\) −13.9661 −0.907199
\(238\) 0 0
\(239\) −3.88769 −0.251474 −0.125737 0.992064i \(-0.540130\pi\)
−0.125737 + 0.992064i \(0.540130\pi\)
\(240\) 0 0
\(241\) −24.5894 −1.58394 −0.791971 0.610558i \(-0.790945\pi\)
−0.791971 + 0.610558i \(0.790945\pi\)
\(242\) 0 0
\(243\) 2.56033 0.164246
\(244\) 0 0
\(245\) −3.75302 −0.239772
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 8.91723 0.565107
\(250\) 0 0
\(251\) −14.5090 −0.915802 −0.457901 0.889003i \(-0.651399\pi\)
−0.457901 + 0.889003i \(0.651399\pi\)
\(252\) 0 0
\(253\) −0.384043 −0.0241446
\(254\) 0 0
\(255\) −1.64310 −0.102895
\(256\) 0 0
\(257\) −5.19700 −0.324180 −0.162090 0.986776i \(-0.551823\pi\)
−0.162090 + 0.986776i \(0.551823\pi\)
\(258\) 0 0
\(259\) −5.40581 −0.335901
\(260\) 0 0
\(261\) −1.30798 −0.0809618
\(262\) 0 0
\(263\) 7.86294 0.484849 0.242425 0.970170i \(-0.422057\pi\)
0.242425 + 0.970170i \(0.422057\pi\)
\(264\) 0 0
\(265\) 6.92692 0.425517
\(266\) 0 0
\(267\) −21.4601 −1.31334
\(268\) 0 0
\(269\) 6.81700 0.415640 0.207820 0.978167i \(-0.433363\pi\)
0.207820 + 0.978167i \(0.433363\pi\)
\(270\) 0 0
\(271\) −16.2228 −0.985466 −0.492733 0.870181i \(-0.664002\pi\)
−0.492733 + 0.870181i \(0.664002\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.55496 −0.0937675
\(276\) 0 0
\(277\) 2.15644 0.129568 0.0647840 0.997899i \(-0.479364\pi\)
0.0647840 + 0.997899i \(0.479364\pi\)
\(278\) 0 0
\(279\) 0.268750 0.0160896
\(280\) 0 0
\(281\) −10.7235 −0.639709 −0.319855 0.947467i \(-0.603634\pi\)
−0.319855 + 0.947467i \(0.603634\pi\)
\(282\) 0 0
\(283\) −5.50902 −0.327478 −0.163739 0.986504i \(-0.552355\pi\)
−0.163739 + 0.986504i \(0.552355\pi\)
\(284\) 0 0
\(285\) 3.04892 0.180602
\(286\) 0 0
\(287\) −12.5429 −0.740383
\(288\) 0 0
\(289\) −16.1685 −0.951090
\(290\) 0 0
\(291\) −16.5308 −0.969052
\(292\) 0 0
\(293\) −1.80864 −0.105662 −0.0528310 0.998603i \(-0.516824\pi\)
−0.0528310 + 0.998603i \(0.516824\pi\)
\(294\) 0 0
\(295\) 14.7017 0.855967
\(296\) 0 0
\(297\) −7.71379 −0.447600
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 4.08815 0.235637
\(302\) 0 0
\(303\) 3.92931 0.225733
\(304\) 0 0
\(305\) −9.55496 −0.547115
\(306\) 0 0
\(307\) −10.9414 −0.624458 −0.312229 0.950007i \(-0.601076\pi\)
−0.312229 + 0.950007i \(0.601076\pi\)
\(308\) 0 0
\(309\) 17.2664 0.982249
\(310\) 0 0
\(311\) −2.95646 −0.167645 −0.0838227 0.996481i \(-0.526713\pi\)
−0.0838227 + 0.996481i \(0.526713\pi\)
\(312\) 0 0
\(313\) −1.17523 −0.0664278 −0.0332139 0.999448i \(-0.510574\pi\)
−0.0332139 + 0.999448i \(0.510574\pi\)
\(314\) 0 0
\(315\) 0.445042 0.0250753
\(316\) 0 0
\(317\) 17.4620 0.980765 0.490383 0.871507i \(-0.336857\pi\)
0.490383 + 0.871507i \(0.336857\pi\)
\(318\) 0 0
\(319\) 8.23490 0.461066
\(320\) 0 0
\(321\) −10.8780 −0.607151
\(322\) 0 0
\(323\) −1.54288 −0.0858479
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −19.2271 −1.06326
\(328\) 0 0
\(329\) −15.3545 −0.846521
\(330\) 0 0
\(331\) 11.7289 0.644676 0.322338 0.946625i \(-0.395531\pi\)
0.322338 + 0.946625i \(0.395531\pi\)
\(332\) 0 0
\(333\) −0.740939 −0.0406032
\(334\) 0 0
\(335\) −5.21983 −0.285190
\(336\) 0 0
\(337\) −28.1879 −1.53549 −0.767746 0.640754i \(-0.778622\pi\)
−0.767746 + 0.640754i \(0.778622\pi\)
\(338\) 0 0
\(339\) 31.7385 1.72380
\(340\) 0 0
\(341\) −1.69202 −0.0916281
\(342\) 0 0
\(343\) −19.3763 −1.04622
\(344\) 0 0
\(345\) −0.445042 −0.0239602
\(346\) 0 0
\(347\) 4.92394 0.264331 0.132165 0.991228i \(-0.457807\pi\)
0.132165 + 0.991228i \(0.457807\pi\)
\(348\) 0 0
\(349\) 14.3220 0.766638 0.383319 0.923616i \(-0.374781\pi\)
0.383319 + 0.923616i \(0.374781\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −4.57242 −0.243365 −0.121683 0.992569i \(-0.538829\pi\)
−0.121683 + 0.992569i \(0.538829\pi\)
\(354\) 0 0
\(355\) 5.74094 0.304697
\(356\) 0 0
\(357\) −2.96077 −0.156701
\(358\) 0 0
\(359\) 29.0315 1.53222 0.766111 0.642708i \(-0.222189\pi\)
0.766111 + 0.642708i \(0.222189\pi\)
\(360\) 0 0
\(361\) −16.1371 −0.849319
\(362\) 0 0
\(363\) 15.4644 0.811672
\(364\) 0 0
\(365\) −14.5918 −0.763769
\(366\) 0 0
\(367\) 27.8297 1.45270 0.726349 0.687326i \(-0.241216\pi\)
0.726349 + 0.687326i \(0.241216\pi\)
\(368\) 0 0
\(369\) −1.71917 −0.0894963
\(370\) 0 0
\(371\) 12.4819 0.648027
\(372\) 0 0
\(373\) −16.1588 −0.836673 −0.418336 0.908292i \(-0.637387\pi\)
−0.418336 + 0.908292i \(0.637387\pi\)
\(374\) 0 0
\(375\) −1.80194 −0.0930517
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 4.05861 0.208477 0.104238 0.994552i \(-0.466760\pi\)
0.104238 + 0.994552i \(0.466760\pi\)
\(380\) 0 0
\(381\) 38.6775 1.98151
\(382\) 0 0
\(383\) 12.9269 0.660535 0.330267 0.943887i \(-0.392861\pi\)
0.330267 + 0.943887i \(0.392861\pi\)
\(384\) 0 0
\(385\) −2.80194 −0.142800
\(386\) 0 0
\(387\) 0.560335 0.0284834
\(388\) 0 0
\(389\) 10.2784 0.521138 0.260569 0.965455i \(-0.416090\pi\)
0.260569 + 0.965455i \(0.416090\pi\)
\(390\) 0 0
\(391\) 0.225209 0.0113893
\(392\) 0 0
\(393\) −14.3230 −0.722502
\(394\) 0 0
\(395\) 7.75063 0.389976
\(396\) 0 0
\(397\) −15.0694 −0.756309 −0.378155 0.925742i \(-0.623441\pi\)
−0.378155 + 0.925742i \(0.623441\pi\)
\(398\) 0 0
\(399\) 5.49396 0.275042
\(400\) 0 0
\(401\) −1.25129 −0.0624865 −0.0312433 0.999512i \(-0.509947\pi\)
−0.0312433 + 0.999512i \(0.509947\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −9.67994 −0.481000
\(406\) 0 0
\(407\) 4.66487 0.231229
\(408\) 0 0
\(409\) −13.4849 −0.666783 −0.333392 0.942788i \(-0.608193\pi\)
−0.333392 + 0.942788i \(0.608193\pi\)
\(410\) 0 0
\(411\) −2.50604 −0.123614
\(412\) 0 0
\(413\) 26.4916 1.30356
\(414\) 0 0
\(415\) −4.94869 −0.242922
\(416\) 0 0
\(417\) −36.2204 −1.77372
\(418\) 0 0
\(419\) −8.84846 −0.432276 −0.216138 0.976363i \(-0.569346\pi\)
−0.216138 + 0.976363i \(0.569346\pi\)
\(420\) 0 0
\(421\) −4.39373 −0.214137 −0.107069 0.994252i \(-0.534146\pi\)
−0.107069 + 0.994252i \(0.534146\pi\)
\(422\) 0 0
\(423\) −2.10454 −0.102326
\(424\) 0 0
\(425\) 0.911854 0.0442314
\(426\) 0 0
\(427\) −17.2174 −0.833210
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 17.7235 0.853710 0.426855 0.904320i \(-0.359621\pi\)
0.426855 + 0.904320i \(0.359621\pi\)
\(432\) 0 0
\(433\) −7.81402 −0.375518 −0.187759 0.982215i \(-0.560122\pi\)
−0.187759 + 0.982215i \(0.560122\pi\)
\(434\) 0 0
\(435\) 9.54288 0.457546
\(436\) 0 0
\(437\) −0.417895 −0.0199906
\(438\) 0 0
\(439\) −5.64178 −0.269267 −0.134634 0.990895i \(-0.542986\pi\)
−0.134634 + 0.990895i \(0.542986\pi\)
\(440\) 0 0
\(441\) −0.926919 −0.0441390
\(442\) 0 0
\(443\) 26.3491 1.25188 0.625942 0.779869i \(-0.284715\pi\)
0.625942 + 0.779869i \(0.284715\pi\)
\(444\) 0 0
\(445\) 11.9095 0.564563
\(446\) 0 0
\(447\) 13.2271 0.625622
\(448\) 0 0
\(449\) 39.0941 1.84497 0.922483 0.386039i \(-0.126157\pi\)
0.922483 + 0.386039i \(0.126157\pi\)
\(450\) 0 0
\(451\) 10.8237 0.509669
\(452\) 0 0
\(453\) 15.2446 0.716253
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −19.6025 −0.916968 −0.458484 0.888703i \(-0.651607\pi\)
−0.458484 + 0.888703i \(0.651607\pi\)
\(458\) 0 0
\(459\) 4.52350 0.211139
\(460\) 0 0
\(461\) −40.5435 −1.88830 −0.944149 0.329520i \(-0.893113\pi\)
−0.944149 + 0.329520i \(0.893113\pi\)
\(462\) 0 0
\(463\) 31.4959 1.46374 0.731869 0.681446i \(-0.238648\pi\)
0.731869 + 0.681446i \(0.238648\pi\)
\(464\) 0 0
\(465\) −1.96077 −0.0909286
\(466\) 0 0
\(467\) 35.6189 1.64825 0.824124 0.566409i \(-0.191668\pi\)
0.824124 + 0.566409i \(0.191668\pi\)
\(468\) 0 0
\(469\) −9.40581 −0.434320
\(470\) 0 0
\(471\) 32.0170 1.47526
\(472\) 0 0
\(473\) −3.52781 −0.162209
\(474\) 0 0
\(475\) −1.69202 −0.0776353
\(476\) 0 0
\(477\) 1.71081 0.0783325
\(478\) 0 0
\(479\) 0.698726 0.0319256 0.0159628 0.999873i \(-0.494919\pi\)
0.0159628 + 0.999873i \(0.494919\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −0.801938 −0.0364894
\(484\) 0 0
\(485\) 9.17390 0.416565
\(486\) 0 0
\(487\) 9.49263 0.430152 0.215076 0.976597i \(-0.431000\pi\)
0.215076 + 0.976597i \(0.431000\pi\)
\(488\) 0 0
\(489\) 20.9541 0.947576
\(490\) 0 0
\(491\) 23.0261 1.03915 0.519576 0.854424i \(-0.326090\pi\)
0.519576 + 0.854424i \(0.326090\pi\)
\(492\) 0 0
\(493\) −4.82908 −0.217491
\(494\) 0 0
\(495\) −0.384043 −0.0172614
\(496\) 0 0
\(497\) 10.3448 0.464028
\(498\) 0 0
\(499\) −15.5109 −0.694365 −0.347183 0.937798i \(-0.612862\pi\)
−0.347183 + 0.937798i \(0.612862\pi\)
\(500\) 0 0
\(501\) 16.5821 0.740834
\(502\) 0 0
\(503\) −27.8538 −1.24194 −0.620971 0.783834i \(-0.713261\pi\)
−0.620971 + 0.783834i \(0.713261\pi\)
\(504\) 0 0
\(505\) −2.18060 −0.0970356
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 20.6474 0.915181 0.457590 0.889163i \(-0.348713\pi\)
0.457590 + 0.889163i \(0.348713\pi\)
\(510\) 0 0
\(511\) −26.2935 −1.16316
\(512\) 0 0
\(513\) −8.39373 −0.370592
\(514\) 0 0
\(515\) −9.58211 −0.422238
\(516\) 0 0
\(517\) 13.2500 0.582733
\(518\) 0 0
\(519\) 27.8582 1.22284
\(520\) 0 0
\(521\) 27.5700 1.20786 0.603932 0.797036i \(-0.293600\pi\)
0.603932 + 0.797036i \(0.293600\pi\)
\(522\) 0 0
\(523\) 14.2427 0.622788 0.311394 0.950281i \(-0.399204\pi\)
0.311394 + 0.950281i \(0.399204\pi\)
\(524\) 0 0
\(525\) −3.24698 −0.141710
\(526\) 0 0
\(527\) 0.992230 0.0432222
\(528\) 0 0
\(529\) −22.9390 −0.997348
\(530\) 0 0
\(531\) 3.63102 0.157573
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 6.03684 0.260995
\(536\) 0 0
\(537\) 37.9342 1.63698
\(538\) 0 0
\(539\) 5.83579 0.251365
\(540\) 0 0
\(541\) 26.9667 1.15939 0.579695 0.814833i \(-0.303172\pi\)
0.579695 + 0.814833i \(0.303172\pi\)
\(542\) 0 0
\(543\) 26.9584 1.15689
\(544\) 0 0
\(545\) 10.6703 0.457063
\(546\) 0 0
\(547\) −32.7294 −1.39941 −0.699705 0.714432i \(-0.746685\pi\)
−0.699705 + 0.714432i \(0.746685\pi\)
\(548\) 0 0
\(549\) −2.35988 −0.100717
\(550\) 0 0
\(551\) 8.96077 0.381742
\(552\) 0 0
\(553\) 13.9661 0.593901
\(554\) 0 0
\(555\) 5.40581 0.229464
\(556\) 0 0
\(557\) −41.7603 −1.76944 −0.884721 0.466122i \(-0.845651\pi\)
−0.884721 + 0.466122i \(0.845651\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 2.55496 0.107870
\(562\) 0 0
\(563\) 24.8834 1.04871 0.524355 0.851500i \(-0.324307\pi\)
0.524355 + 0.851500i \(0.324307\pi\)
\(564\) 0 0
\(565\) −17.6136 −0.741008
\(566\) 0 0
\(567\) −17.4426 −0.732522
\(568\) 0 0
\(569\) 28.2271 1.18334 0.591671 0.806179i \(-0.298468\pi\)
0.591671 + 0.806179i \(0.298468\pi\)
\(570\) 0 0
\(571\) 22.4316 0.938735 0.469367 0.883003i \(-0.344482\pi\)
0.469367 + 0.883003i \(0.344482\pi\)
\(572\) 0 0
\(573\) −5.68425 −0.237463
\(574\) 0 0
\(575\) 0.246980 0.0102998
\(576\) 0 0
\(577\) 36.8437 1.53382 0.766911 0.641753i \(-0.221793\pi\)
0.766911 + 0.641753i \(0.221793\pi\)
\(578\) 0 0
\(579\) −7.29590 −0.303207
\(580\) 0 0
\(581\) −8.91723 −0.369949
\(582\) 0 0
\(583\) −10.7711 −0.446092
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −44.4480 −1.83457 −0.917283 0.398236i \(-0.869623\pi\)
−0.917283 + 0.398236i \(0.869623\pi\)
\(588\) 0 0
\(589\) −1.84117 −0.0758639
\(590\) 0 0
\(591\) 9.38404 0.386008
\(592\) 0 0
\(593\) −20.6276 −0.847073 −0.423536 0.905879i \(-0.639211\pi\)
−0.423536 + 0.905879i \(0.639211\pi\)
\(594\) 0 0
\(595\) 1.64310 0.0673607
\(596\) 0 0
\(597\) 40.5459 1.65943
\(598\) 0 0
\(599\) 31.2674 1.27755 0.638776 0.769393i \(-0.279441\pi\)
0.638776 + 0.769393i \(0.279441\pi\)
\(600\) 0 0
\(601\) −7.94438 −0.324058 −0.162029 0.986786i \(-0.551804\pi\)
−0.162029 + 0.986786i \(0.551804\pi\)
\(602\) 0 0
\(603\) −1.28919 −0.0525000
\(604\) 0 0
\(605\) −8.58211 −0.348912
\(606\) 0 0
\(607\) −3.97690 −0.161417 −0.0807087 0.996738i \(-0.525718\pi\)
−0.0807087 + 0.996738i \(0.525718\pi\)
\(608\) 0 0
\(609\) 17.1957 0.696804
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 36.0062 1.45428 0.727139 0.686490i \(-0.240850\pi\)
0.727139 + 0.686490i \(0.240850\pi\)
\(614\) 0 0
\(615\) 12.5429 0.505778
\(616\) 0 0
\(617\) −3.14808 −0.126737 −0.0633685 0.997990i \(-0.520184\pi\)
−0.0633685 + 0.997990i \(0.520184\pi\)
\(618\) 0 0
\(619\) 24.9594 1.00320 0.501602 0.865098i \(-0.332744\pi\)
0.501602 + 0.865098i \(0.332744\pi\)
\(620\) 0 0
\(621\) 1.22521 0.0491660
\(622\) 0 0
\(623\) 21.4601 0.859781
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −4.74094 −0.189335
\(628\) 0 0
\(629\) −2.73556 −0.109074
\(630\) 0 0
\(631\) −1.34960 −0.0537267 −0.0268634 0.999639i \(-0.508552\pi\)
−0.0268634 + 0.999639i \(0.508552\pi\)
\(632\) 0 0
\(633\) 0.0784573 0.00311840
\(634\) 0 0
\(635\) −21.4644 −0.851789
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 1.41789 0.0560911
\(640\) 0 0
\(641\) 30.7808 1.21577 0.607883 0.794026i \(-0.292019\pi\)
0.607883 + 0.794026i \(0.292019\pi\)
\(642\) 0 0
\(643\) −0.528402 −0.0208381 −0.0104191 0.999946i \(-0.503317\pi\)
−0.0104191 + 0.999946i \(0.503317\pi\)
\(644\) 0 0
\(645\) −4.08815 −0.160971
\(646\) 0 0
\(647\) −12.7114 −0.499737 −0.249868 0.968280i \(-0.580387\pi\)
−0.249868 + 0.968280i \(0.580387\pi\)
\(648\) 0 0
\(649\) −22.8605 −0.897355
\(650\) 0 0
\(651\) −3.53319 −0.138477
\(652\) 0 0
\(653\) 16.2241 0.634900 0.317450 0.948275i \(-0.397173\pi\)
0.317450 + 0.948275i \(0.397173\pi\)
\(654\) 0 0
\(655\) 7.94869 0.310581
\(656\) 0 0
\(657\) −3.60388 −0.140601
\(658\) 0 0
\(659\) −36.5478 −1.42370 −0.711850 0.702332i \(-0.752142\pi\)
−0.711850 + 0.702332i \(0.752142\pi\)
\(660\) 0 0
\(661\) 7.94810 0.309145 0.154573 0.987981i \(-0.450600\pi\)
0.154573 + 0.987981i \(0.450600\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −3.04892 −0.118232
\(666\) 0 0
\(667\) −1.30798 −0.0506451
\(668\) 0 0
\(669\) −28.9952 −1.12102
\(670\) 0 0
\(671\) 14.8576 0.573570
\(672\) 0 0
\(673\) −10.0771 −0.388445 −0.194222 0.980958i \(-0.562218\pi\)
−0.194222 + 0.980958i \(0.562218\pi\)
\(674\) 0 0
\(675\) 4.96077 0.190940
\(676\) 0 0
\(677\) −37.3236 −1.43446 −0.717232 0.696835i \(-0.754591\pi\)
−0.717232 + 0.696835i \(0.754591\pi\)
\(678\) 0 0
\(679\) 16.5308 0.634394
\(680\) 0 0
\(681\) −1.95646 −0.0749717
\(682\) 0 0
\(683\) −10.8237 −0.414158 −0.207079 0.978324i \(-0.566396\pi\)
−0.207079 + 0.978324i \(0.566396\pi\)
\(684\) 0 0
\(685\) 1.39075 0.0531377
\(686\) 0 0
\(687\) −45.5206 −1.73672
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 24.4838 0.931408 0.465704 0.884941i \(-0.345801\pi\)
0.465704 + 0.884941i \(0.345801\pi\)
\(692\) 0 0
\(693\) −0.692021 −0.0262877
\(694\) 0 0
\(695\) 20.1008 0.762468
\(696\) 0 0
\(697\) −6.34721 −0.240418
\(698\) 0 0
\(699\) 43.2247 1.63491
\(700\) 0 0
\(701\) 30.9554 1.16917 0.584585 0.811333i \(-0.301258\pi\)
0.584585 + 0.811333i \(0.301258\pi\)
\(702\) 0 0
\(703\) 5.07606 0.191447
\(704\) 0 0
\(705\) 15.3545 0.578284
\(706\) 0 0
\(707\) −3.92931 −0.147777
\(708\) 0 0
\(709\) −21.6396 −0.812694 −0.406347 0.913719i \(-0.633198\pi\)
−0.406347 + 0.913719i \(0.633198\pi\)
\(710\) 0 0
\(711\) 1.91425 0.0717899
\(712\) 0 0
\(713\) 0.268750 0.0100648
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 7.00538 0.261621
\(718\) 0 0
\(719\) −11.2556 −0.419763 −0.209882 0.977727i \(-0.567308\pi\)
−0.209882 + 0.977727i \(0.567308\pi\)
\(720\) 0 0
\(721\) −17.2664 −0.643033
\(722\) 0 0
\(723\) 44.3086 1.64785
\(724\) 0 0
\(725\) −5.29590 −0.196685
\(726\) 0 0
\(727\) 18.6485 0.691634 0.345817 0.938302i \(-0.387602\pi\)
0.345817 + 0.938302i \(0.387602\pi\)
\(728\) 0 0
\(729\) 24.4263 0.904676
\(730\) 0 0
\(731\) 2.06877 0.0765162
\(732\) 0 0
\(733\) −5.01639 −0.185285 −0.0926424 0.995699i \(-0.529531\pi\)
−0.0926424 + 0.995699i \(0.529531\pi\)
\(734\) 0 0
\(735\) 6.76271 0.249446
\(736\) 0 0
\(737\) 8.11662 0.298980
\(738\) 0 0
\(739\) 26.4276 0.972154 0.486077 0.873916i \(-0.338427\pi\)
0.486077 + 0.873916i \(0.338427\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −2.25129 −0.0825919 −0.0412959 0.999147i \(-0.513149\pi\)
−0.0412959 + 0.999147i \(0.513149\pi\)
\(744\) 0 0
\(745\) −7.34050 −0.268935
\(746\) 0 0
\(747\) −1.22223 −0.0447189
\(748\) 0 0
\(749\) 10.8780 0.397474
\(750\) 0 0
\(751\) 29.7375 1.08514 0.542568 0.840012i \(-0.317452\pi\)
0.542568 + 0.840012i \(0.317452\pi\)
\(752\) 0 0
\(753\) 26.1444 0.952753
\(754\) 0 0
\(755\) −8.46011 −0.307895
\(756\) 0 0
\(757\) 24.9764 0.907784 0.453892 0.891057i \(-0.350035\pi\)
0.453892 + 0.891057i \(0.350035\pi\)
\(758\) 0 0
\(759\) 0.692021 0.0251188
\(760\) 0 0
\(761\) −7.37090 −0.267195 −0.133597 0.991036i \(-0.542653\pi\)
−0.133597 + 0.991036i \(0.542653\pi\)
\(762\) 0 0
\(763\) 19.2271 0.696069
\(764\) 0 0
\(765\) 0.225209 0.00814246
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 55.2780 1.99337 0.996687 0.0813304i \(-0.0259169\pi\)
0.996687 + 0.0813304i \(0.0259169\pi\)
\(770\) 0 0
\(771\) 9.36467 0.337260
\(772\) 0 0
\(773\) −54.5120 −1.96066 −0.980330 0.197364i \(-0.936762\pi\)
−0.980330 + 0.197364i \(0.936762\pi\)
\(774\) 0 0
\(775\) 1.08815 0.0390874
\(776\) 0 0
\(777\) 9.74094 0.349454
\(778\) 0 0
\(779\) 11.7778 0.421983
\(780\) 0 0
\(781\) −8.92692 −0.319430
\(782\) 0 0
\(783\) −26.2717 −0.938875
\(784\) 0 0
\(785\) −17.7681 −0.634170
\(786\) 0 0
\(787\) −54.7493 −1.95160 −0.975801 0.218661i \(-0.929831\pi\)
−0.975801 + 0.218661i \(0.929831\pi\)
\(788\) 0 0
\(789\) −14.1685 −0.504413
\(790\) 0 0
\(791\) −31.7385 −1.12849
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −12.4819 −0.442687
\(796\) 0 0
\(797\) 33.9982 1.20428 0.602139 0.798391i \(-0.294315\pi\)
0.602139 + 0.798391i \(0.294315\pi\)
\(798\) 0 0
\(799\) −7.77000 −0.274883
\(800\) 0 0
\(801\) 2.94139 0.103929
\(802\) 0 0
\(803\) 22.6896 0.800700
\(804\) 0 0
\(805\) 0.445042 0.0156857
\(806\) 0 0
\(807\) −12.2838 −0.432411
\(808\) 0 0
\(809\) −11.3532 −0.399156 −0.199578 0.979882i \(-0.563957\pi\)
−0.199578 + 0.979882i \(0.563957\pi\)
\(810\) 0 0
\(811\) −32.6082 −1.14503 −0.572514 0.819895i \(-0.694032\pi\)
−0.572514 + 0.819895i \(0.694032\pi\)
\(812\) 0 0
\(813\) 29.2325 1.02523
\(814\) 0 0
\(815\) −11.6286 −0.407333
\(816\) 0 0
\(817\) −3.83877 −0.134302
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 13.6377 0.475960 0.237980 0.971270i \(-0.423515\pi\)
0.237980 + 0.971270i \(0.423515\pi\)
\(822\) 0 0
\(823\) −33.4077 −1.16452 −0.582260 0.813003i \(-0.697831\pi\)
−0.582260 + 0.813003i \(0.697831\pi\)
\(824\) 0 0
\(825\) 2.80194 0.0975510
\(826\) 0 0
\(827\) 13.9801 0.486137 0.243069 0.970009i \(-0.421846\pi\)
0.243069 + 0.970009i \(0.421846\pi\)
\(828\) 0 0
\(829\) 36.8528 1.27995 0.639975 0.768396i \(-0.278945\pi\)
0.639975 + 0.768396i \(0.278945\pi\)
\(830\) 0 0
\(831\) −3.88577 −0.134796
\(832\) 0 0
\(833\) −3.42221 −0.118572
\(834\) 0 0
\(835\) −9.20237 −0.318461
\(836\) 0 0
\(837\) 5.39804 0.186584
\(838\) 0 0
\(839\) −3.58775 −0.123863 −0.0619314 0.998080i \(-0.519726\pi\)
−0.0619314 + 0.998080i \(0.519726\pi\)
\(840\) 0 0
\(841\) −0.953476 −0.0328785
\(842\) 0 0
\(843\) 19.3230 0.665521
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −15.4644 −0.531364
\(848\) 0 0
\(849\) 9.92692 0.340691
\(850\) 0 0
\(851\) −0.740939 −0.0253991
\(852\) 0 0
\(853\) 47.2723 1.61857 0.809287 0.587414i \(-0.199854\pi\)
0.809287 + 0.587414i \(0.199854\pi\)
\(854\) 0 0
\(855\) −0.417895 −0.0142917
\(856\) 0 0
\(857\) −20.5435 −0.701751 −0.350876 0.936422i \(-0.614116\pi\)
−0.350876 + 0.936422i \(0.614116\pi\)
\(858\) 0 0
\(859\) 20.0597 0.684427 0.342214 0.939622i \(-0.388823\pi\)
0.342214 + 0.939622i \(0.388823\pi\)
\(860\) 0 0
\(861\) 22.6015 0.770256
\(862\) 0 0
\(863\) 43.7103 1.48792 0.743959 0.668226i \(-0.232946\pi\)
0.743959 + 0.668226i \(0.232946\pi\)
\(864\) 0 0
\(865\) −15.4601 −0.525659
\(866\) 0 0
\(867\) 29.1347 0.989465
\(868\) 0 0
\(869\) −12.0519 −0.408833
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 2.26577 0.0766846
\(874\) 0 0
\(875\) 1.80194 0.0609166
\(876\) 0 0
\(877\) 23.6655 0.799126 0.399563 0.916706i \(-0.369162\pi\)
0.399563 + 0.916706i \(0.369162\pi\)
\(878\) 0 0
\(879\) 3.25906 0.109925
\(880\) 0 0
\(881\) −33.5743 −1.13115 −0.565574 0.824698i \(-0.691345\pi\)
−0.565574 + 0.824698i \(0.691345\pi\)
\(882\) 0 0
\(883\) −8.10513 −0.272759 −0.136380 0.990657i \(-0.543547\pi\)
−0.136380 + 0.990657i \(0.543547\pi\)
\(884\) 0 0
\(885\) −26.4916 −0.890504
\(886\) 0 0
\(887\) 42.9687 1.44275 0.721373 0.692547i \(-0.243511\pi\)
0.721373 + 0.692547i \(0.243511\pi\)
\(888\) 0 0
\(889\) −38.6775 −1.29720
\(890\) 0 0
\(891\) 15.0519 0.504258
\(892\) 0 0
\(893\) 14.4179 0.482476
\(894\) 0 0
\(895\) −21.0519 −0.703687
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −5.76271 −0.192197
\(900\) 0 0
\(901\) 6.31634 0.210428
\(902\) 0 0
\(903\) −7.36658 −0.245145
\(904\) 0 0
\(905\) −14.9608 −0.497313
\(906\) 0 0
\(907\) −26.9536 −0.894979 −0.447490 0.894289i \(-0.647682\pi\)
−0.447490 + 0.894289i \(0.647682\pi\)
\(908\) 0 0
\(909\) −0.538565 −0.0178631
\(910\) 0 0
\(911\) −30.1909 −1.00027 −0.500134 0.865948i \(-0.666716\pi\)
−0.500134 + 0.865948i \(0.666716\pi\)
\(912\) 0 0
\(913\) 7.69501 0.254668
\(914\) 0 0
\(915\) 17.2174 0.569191
\(916\) 0 0
\(917\) 14.3230 0.472989
\(918\) 0 0
\(919\) −28.9336 −0.954432 −0.477216 0.878786i \(-0.658354\pi\)
−0.477216 + 0.878786i \(0.658354\pi\)
\(920\) 0 0
\(921\) 19.7157 0.649655
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −3.00000 −0.0986394
\(926\) 0 0
\(927\) −2.36658 −0.0777288
\(928\) 0 0
\(929\) 9.34854 0.306715 0.153358 0.988171i \(-0.450991\pi\)
0.153358 + 0.988171i \(0.450991\pi\)
\(930\) 0 0
\(931\) 6.35019 0.208119
\(932\) 0 0
\(933\) 5.32736 0.174410
\(934\) 0 0
\(935\) −1.41789 −0.0463701
\(936\) 0 0
\(937\) −40.7808 −1.33225 −0.666125 0.745840i \(-0.732048\pi\)
−0.666125 + 0.745840i \(0.732048\pi\)
\(938\) 0 0
\(939\) 2.11769 0.0691081
\(940\) 0 0
\(941\) 37.4161 1.21973 0.609865 0.792505i \(-0.291224\pi\)
0.609865 + 0.792505i \(0.291224\pi\)
\(942\) 0 0
\(943\) −1.71917 −0.0559838
\(944\) 0 0
\(945\) 8.93900 0.290786
\(946\) 0 0
\(947\) 12.4612 0.404934 0.202467 0.979289i \(-0.435104\pi\)
0.202467 + 0.979289i \(0.435104\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −31.4655 −1.02034
\(952\) 0 0
\(953\) −6.66978 −0.216055 −0.108028 0.994148i \(-0.534454\pi\)
−0.108028 + 0.994148i \(0.534454\pi\)
\(954\) 0 0
\(955\) 3.15452 0.102078
\(956\) 0 0
\(957\) −14.8388 −0.479669
\(958\) 0 0
\(959\) 2.50604 0.0809243
\(960\) 0 0
\(961\) −29.8159 −0.961804
\(962\) 0 0
\(963\) 1.49098 0.0480460
\(964\) 0 0
\(965\) 4.04892 0.130339
\(966\) 0 0
\(967\) −45.9584 −1.47792 −0.738961 0.673748i \(-0.764683\pi\)
−0.738961 + 0.673748i \(0.764683\pi\)
\(968\) 0 0
\(969\) 2.78017 0.0893118
\(970\) 0 0
\(971\) 45.3991 1.45693 0.728463 0.685085i \(-0.240235\pi\)
0.728463 + 0.685085i \(0.240235\pi\)
\(972\) 0 0
\(973\) 36.2204 1.16117
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 12.2174 0.390871 0.195435 0.980717i \(-0.437388\pi\)
0.195435 + 0.980717i \(0.437388\pi\)
\(978\) 0 0
\(979\) −18.5187 −0.591861
\(980\) 0 0
\(981\) 2.63533 0.0841398
\(982\) 0 0
\(983\) 52.9081 1.68751 0.843754 0.536731i \(-0.180341\pi\)
0.843754 + 0.536731i \(0.180341\pi\)
\(984\) 0 0
\(985\) −5.20775 −0.165933
\(986\) 0 0
\(987\) 27.6679 0.880678
\(988\) 0 0
\(989\) 0.560335 0.0178176
\(990\) 0 0
\(991\) −11.9011 −0.378051 −0.189025 0.981972i \(-0.560533\pi\)
−0.189025 + 0.981972i \(0.560533\pi\)
\(992\) 0 0
\(993\) −21.1347 −0.670688
\(994\) 0 0
\(995\) −22.5013 −0.713338
\(996\) 0 0
\(997\) 32.5937 1.03225 0.516127 0.856512i \(-0.327373\pi\)
0.516127 + 0.856512i \(0.327373\pi\)
\(998\) 0 0
\(999\) −14.8823 −0.470856
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.l.1.1 yes 3
13.5 odd 4 3380.2.f.g.3041.1 6
13.8 odd 4 3380.2.f.g.3041.2 6
13.12 even 2 3380.2.a.k.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3380.2.a.k.1.1 3 13.12 even 2
3380.2.a.l.1.1 yes 3 1.1 even 1 trivial
3380.2.f.g.3041.1 6 13.5 odd 4
3380.2.f.g.3041.2 6 13.8 odd 4