Properties

Label 3380.2.a.k.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 $1$

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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: \(1\)
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 \(-0.610608\pi\)
−0.340534 + 0.940232i \(0.610608\pi\)
\(8\) 0 0
\(9\) 0.246980 0.0823265
\(10\) 0 0
\(11\) 1.55496 0.468838 0.234419 0.972136i \(-0.424681\pi\)
0.234419 + 0.972136i \(0.424681\pi\)
\(12\) 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 \(-0.437825\pi\)
0.194088 + 0.980984i \(0.437825\pi\)
\(20\) 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 \(-0.531154\pi\)
−0.0977184 + 0.995214i \(0.531154\pi\)
\(32\) 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 \(-0.420687\pi\)
0.246598 + 0.969118i \(0.420687\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.96077 1.08709 0.543545 0.839380i \(-0.317082\pi\)
0.543545 + 0.839380i \(0.317082\pi\)
\(42\) 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 \(-0.286538\pi\)
0.621466 + 0.783441i \(0.286538\pi\)
\(48\) 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 \(-0.906315\pi\)
−0.957000 + 0.290089i \(0.906315\pi\)
\(60\) 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 \(-0.396703\pi\)
0.318852 + 0.947805i \(0.396703\pi\)
\(68\) 0 0
\(69\) −0.445042 −0.0535767
\(70\) 0 0
\(71\) −5.74094 −0.681324 −0.340662 0.940186i \(-0.610651\pi\)
−0.340662 + 0.940186i \(0.610651\pi\)
\(72\) 0 0
\(73\) 14.5918 1.70784 0.853920 0.520404i \(-0.174219\pi\)
0.853920 + 0.520404i \(0.174219\pi\)
\(74\) 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 \(-0.412449\pi\)
0.271595 + 0.962412i \(0.412449\pi\)
\(84\) 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 \(-0.717437\pi\)
−0.631200 + 0.775620i \(0.717437\pi\)
\(90\) 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 \(-0.654210\pi\)
−0.465734 + 0.884925i \(0.654210\pi\)
\(98\) 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 \(-0.670729\pi\)
−0.511012 + 0.859573i \(0.670729\pi\)
\(110\) 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 \(-0.518922\pi\)
−0.0594098 + 0.998234i \(0.518922\pi\)
\(138\) 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 \(-0.402787\pi\)
0.300679 + 0.953725i \(0.402787\pi\)
\(150\) 0 0
\(151\) 8.46011 0.688474 0.344237 0.938883i \(-0.388138\pi\)
0.344237 + 0.938883i \(0.388138\pi\)
\(152\) 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 \(-0.349492\pi\)
0.455412 + 0.890281i \(0.349492\pi\)
\(164\) 0 0
\(165\) 2.80194 0.218131
\(166\) 0 0
\(167\) 9.20237 0.712101 0.356051 0.934467i \(-0.384123\pi\)
0.356051 + 0.934467i \(0.384123\pi\)
\(168\) 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 \(-0.546551\pi\)
−0.145724 + 0.989325i \(0.546551\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 5.20775 0.371037 0.185518 0.982641i \(-0.440604\pi\)
0.185518 + 0.982641i \(0.440604\pi\)
\(198\) 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 \(-0.681111\pi\)
−0.538771 + 0.842452i \(0.681111\pi\)
\(224\) 0 0
\(225\) 0.246980 0.0164653
\(226\) 0 0
\(227\) −1.08575 −0.0720640 −0.0360320 0.999351i \(-0.511472\pi\)
−0.0360320 + 0.999351i \(0.511472\pi\)
\(228\) 0 0
\(229\) −25.2620 −1.66936 −0.834681 0.550733i \(-0.814348\pi\)
−0.834681 + 0.550733i \(0.814348\pi\)
\(230\) 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 \(-0.459870\pi\)
0.125737 + 0.992064i \(0.459870\pi\)
\(240\) 0 0
\(241\) 24.5894 1.58394 0.791971 0.610558i \(-0.209055\pi\)
0.791971 + 0.610558i \(0.209055\pi\)
\(242\) 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 \(-0.335998\pi\)
0.492733 + 0.870181i \(0.335998\pi\)
\(272\) 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 \(-0.396366\pi\)
0.319855 + 0.947467i \(0.396366\pi\)
\(282\) 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 \(-0.483176\pi\)
0.0528310 + 0.998603i \(0.483176\pi\)
\(294\) 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 \(-0.398924\pi\)
0.312229 + 0.950007i \(0.398924\pi\)
\(308\) 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 \(-0.663143\pi\)
−0.490383 + 0.871507i \(0.663143\pi\)
\(318\) 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 \(-0.604469\pi\)
−0.322338 + 0.946625i \(0.604469\pi\)
\(332\) 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 \(-0.625219\pi\)
−0.383319 + 0.923616i \(0.625219\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 4.57242 0.243365 0.121683 0.992569i \(-0.461171\pi\)
0.121683 + 0.992569i \(0.461171\pi\)
\(354\) 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 \(-0.777811\pi\)
−0.766111 + 0.642708i \(0.777811\pi\)
\(360\) 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 \(-0.533240\pi\)
−0.104238 + 0.994552i \(0.533240\pi\)
\(380\) 0 0
\(381\) 38.6775 1.98151
\(382\) 0 0
\(383\) −12.9269 −0.660535 −0.330267 0.943887i \(-0.607139\pi\)
−0.330267 + 0.943887i \(0.607139\pi\)
\(384\) 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 \(-0.376559\pi\)
0.378155 + 0.925742i \(0.376559\pi\)
\(398\) 0 0
\(399\) 5.49396 0.275042
\(400\) 0 0
\(401\) 1.25129 0.0624865 0.0312433 0.999512i \(-0.490053\pi\)
0.0312433 + 0.999512i \(0.490053\pi\)
\(402\) 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 \(-0.391807\pi\)
0.333392 + 0.942788i \(0.391807\pi\)
\(410\) 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 \(-0.465854\pi\)
0.107069 + 0.994252i \(0.465854\pi\)
\(422\) 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 \(-0.640379\pi\)
−0.426855 + 0.904320i \(0.640379\pi\)
\(432\) 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 \(-0.873843\pi\)
−0.922483 + 0.386039i \(0.873843\pi\)
\(450\) 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 \(-0.348393\pi\)
0.458484 + 0.888703i \(0.348393\pi\)
\(458\) 0 0
\(459\) 4.52350 0.211139
\(460\) 0 0
\(461\) 40.5435 1.88830 0.944149 0.329520i \(-0.106887\pi\)
0.944149 + 0.329520i \(0.106887\pi\)
\(462\) 0 0
\(463\) −31.4959 −1.46374 −0.731869 0.681446i \(-0.761352\pi\)
−0.731869 + 0.681446i \(0.761352\pi\)
\(464\) 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 \(-0.505081\pi\)
−0.0159628 + 0.999873i \(0.505081\pi\)
\(480\) 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 \(-0.569000\pi\)
−0.215076 + 0.976597i \(0.569000\pi\)
\(488\) 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 \(-0.387138\pi\)
0.347183 + 0.937798i \(0.387138\pi\)
\(500\) 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 \(-0.651287\pi\)
−0.457590 + 0.889163i \(0.651287\pi\)
\(510\) 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 \(-0.696828\pi\)
−0.579695 + 0.814833i \(0.696828\pi\)
\(542\) 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 \(-0.154349\pi\)
0.884721 + 0.466122i \(0.154349\pi\)
\(558\) 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 \(-0.778207\pi\)
−0.766911 + 0.641753i \(0.778207\pi\)
\(578\) 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 \(-0.130377\pi\)
0.917283 + 0.398236i \(0.130377\pi\)
\(588\) 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 \(-0.360789\pi\)
0.423536 + 0.905879i \(0.360789\pi\)
\(594\) 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 \(-0.759150\pi\)
−0.727139 + 0.686490i \(0.759150\pi\)
\(614\) 0 0
\(615\) 12.5429 0.505778
\(616\) 0 0
\(617\) 3.14808 0.126737 0.0633685 0.997990i \(-0.479816\pi\)
0.0633685 + 0.997990i \(0.479816\pi\)
\(618\) 0 0
\(619\) −24.9594 −1.00320 −0.501602 0.865098i \(-0.667256\pi\)
−0.501602 + 0.865098i \(0.667256\pi\)
\(620\) 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 \(-0.491448\pi\)
0.0268634 + 0.999639i \(0.491448\pi\)
\(632\) 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 \(-0.496683\pi\)
0.0104191 + 0.999946i \(0.496683\pi\)
\(644\) 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 \(-0.549400\pi\)
−0.154573 + 0.987981i \(0.549400\pi\)
\(662\) 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 \(-0.433604\pi\)
0.207079 + 0.978324i \(0.433604\pi\)
\(684\) 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 \(-0.654199\pi\)
−0.465704 + 0.884941i \(0.654199\pi\)
\(692\) 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 \(-0.366802\pi\)
0.406347 + 0.913719i \(0.366802\pi\)
\(710\) 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 \(-0.470469\pi\)
0.0926424 + 0.995699i \(0.470469\pi\)
\(734\) 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 \(-0.661573\pi\)
−0.486077 + 0.873916i \(0.661573\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 2.25129 0.0825919 0.0412959 0.999147i \(-0.486851\pi\)
0.0412959 + 0.999147i \(0.486851\pi\)
\(744\) 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 \(-0.457347\pi\)
0.133597 + 0.991036i \(0.457347\pi\)
\(762\) 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 \(-0.974083\pi\)
−0.996687 + 0.0813304i \(0.974083\pi\)
\(770\) 0 0
\(771\) 9.36467 0.337260
\(772\) 0 0
\(773\) 54.5120 1.96066 0.980330 0.197364i \(-0.0632380\pi\)
0.980330 + 0.197364i \(0.0632380\pi\)
\(774\) 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 \(-0.0701689\pi\)
0.975801 + 0.218661i \(0.0701689\pi\)
\(788\) 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 \(-0.305968\pi\)
0.572514 + 0.819895i \(0.305968\pi\)
\(812\) 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 \(-0.576485\pi\)
−0.237980 + 0.971270i \(0.576485\pi\)
\(822\) 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 \(-0.578154\pi\)
−0.243069 + 0.970009i \(0.578154\pi\)
\(828\) 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 \(-0.480274\pi\)
0.0619314 + 0.998080i \(0.480274\pi\)
\(840\) 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 \(-0.800146\pi\)
−0.809287 + 0.587414i \(0.800146\pi\)
\(854\) 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 \(-0.767054\pi\)
−0.743959 + 0.668226i \(0.767054\pi\)
\(864\) 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 \(-0.630838\pi\)
−0.399563 + 0.916706i \(0.630838\pi\)
\(878\) 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 \(-0.549009\pi\)
−0.153358 + 0.988171i \(0.549009\pi\)
\(930\) 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 \(-0.708776\pi\)
−0.609865 + 0.792505i \(0.708776\pi\)
\(942\) 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 \(-0.564896\pi\)
−0.202467 + 0.979289i \(0.564896\pi\)
\(948\) 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 \(-0.235317\pi\)
0.738961 + 0.673748i \(0.235317\pi\)
\(968\) 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 \(-0.562612\pi\)
−0.195435 + 0.980717i \(0.562612\pi\)
\(978\) 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 \(-0.819659\pi\)
−0.843754 + 0.536731i \(0.819659\pi\)
\(984\) 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.k.1.1 3
13.5 odd 4 3380.2.f.g.3041.2 6
13.8 odd 4 3380.2.f.g.3041.1 6
13.12 even 2 3380.2.a.l.1.1 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3380.2.a.k.1.1 3 1.1 even 1 trivial
3380.2.a.l.1.1 yes 3 13.12 even 2
3380.2.f.g.3041.1 6 13.8 odd 4
3380.2.f.g.3041.2 6 13.5 odd 4