Properties

Label 9408.2.a.em.1.4
Level 9408
Weight 2
Character 9408.1
Self dual yes
Analytic conductor 75.123
Analytic rank 0
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 9408 = 2^{6} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9408.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(75.1232582216\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.2624.1
Defining polynomial: \(x^{4} - 2 x^{3} - 3 x^{2} + 2 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 4704)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.77462\) of defining polynomial
Character \(\chi\) \(=\) 9408.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +2.13503 q^{5} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +2.13503 q^{5} +1.00000 q^{9} -0.298573 q^{11} -6.43361 q^{13} +2.13503 q^{15} +1.11564 q^{17} -1.01939 q^{19} -4.29857 q^{23} -0.441637 q^{25} +1.00000 q^{27} +0.422246 q^{29} +10.6762 q^{31} -0.298573 q^{33} +5.65685 q^{37} -6.43361 q^{39} +8.32600 q^{41} +7.09849 q^{43} +2.13503 q^{45} -8.11788 q^{47} +1.11564 q^{51} +3.44164 q^{53} -0.637463 q^{55} -1.01939 q^{57} +6.46103 q^{59} -5.83646 q^{61} -13.7360 q^{65} -5.05971 q^{67} -4.29857 q^{69} +1.35828 q^{71} -2.85585 q^{73} -0.441637 q^{75} +7.69564 q^{79} +1.00000 q^{81} +6.03878 q^{83} +2.38193 q^{85} +0.422246 q^{87} +15.3696 q^{89} +10.6762 q^{93} -2.17643 q^{95} -0.512705 q^{97} -0.298573 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} - 4q^{5} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} - 4q^{5} + 4q^{9} + 4q^{11} - 8q^{13} - 4q^{15} + 4q^{17} + 8q^{19} - 12q^{23} + 12q^{25} + 4q^{27} + 8q^{31} + 4q^{33} - 8q^{39} + 20q^{41} - 8q^{43} - 4q^{45} + 16q^{47} + 4q^{51} + 8q^{55} + 8q^{57} - 16q^{61} - 8q^{65} - 8q^{67} - 12q^{69} - 12q^{71} + 8q^{73} + 12q^{75} - 16q^{79} + 4q^{81} + 8q^{85} + 28q^{89} + 8q^{93} - 24q^{95} + 40q^{97} + 4q^{99} + O(q^{100}) \)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 2.13503 0.954815 0.477408 0.878682i \(-0.341576\pi\)
0.477408 + 0.878682i \(0.341576\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −0.298573 −0.0900231 −0.0450116 0.998986i \(-0.514332\pi\)
−0.0450116 + 0.998986i \(0.514332\pi\)
\(12\) 0 0
\(13\) −6.43361 −1.78436 −0.892181 0.451679i \(-0.850825\pi\)
−0.892181 + 0.451679i \(0.850825\pi\)
\(14\) 0 0
\(15\) 2.13503 0.551263
\(16\) 0 0
\(17\) 1.11564 0.270583 0.135291 0.990806i \(-0.456803\pi\)
0.135291 + 0.990806i \(0.456803\pi\)
\(18\) 0 0
\(19\) −1.01939 −0.233864 −0.116932 0.993140i \(-0.537306\pi\)
−0.116932 + 0.993140i \(0.537306\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.29857 −0.896314 −0.448157 0.893955i \(-0.647920\pi\)
−0.448157 + 0.893955i \(0.647920\pi\)
\(24\) 0 0
\(25\) −0.441637 −0.0883275
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 0.422246 0.0784091 0.0392045 0.999231i \(-0.487518\pi\)
0.0392045 + 0.999231i \(0.487518\pi\)
\(30\) 0 0
\(31\) 10.6762 1.91751 0.958755 0.284233i \(-0.0917390\pi\)
0.958755 + 0.284233i \(0.0917390\pi\)
\(32\) 0 0
\(33\) −0.298573 −0.0519749
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.65685 0.929981 0.464991 0.885316i \(-0.346058\pi\)
0.464991 + 0.885316i \(0.346058\pi\)
\(38\) 0 0
\(39\) −6.43361 −1.03020
\(40\) 0 0
\(41\) 8.32600 1.30030 0.650151 0.759805i \(-0.274706\pi\)
0.650151 + 0.759805i \(0.274706\pi\)
\(42\) 0 0
\(43\) 7.09849 1.08251 0.541255 0.840859i \(-0.317949\pi\)
0.541255 + 0.840859i \(0.317949\pi\)
\(44\) 0 0
\(45\) 2.13503 0.318272
\(46\) 0 0
\(47\) −8.11788 −1.18411 −0.592057 0.805896i \(-0.701684\pi\)
−0.592057 + 0.805896i \(0.701684\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 1.11564 0.156221
\(52\) 0 0
\(53\) 3.44164 0.472745 0.236373 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(54\) 0 0
\(55\) −0.637463 −0.0859555
\(56\) 0 0
\(57\) −1.01939 −0.135022
\(58\) 0 0
\(59\) 6.46103 0.841154 0.420577 0.907257i \(-0.361828\pi\)
0.420577 + 0.907257i \(0.361828\pi\)
\(60\) 0 0
\(61\) −5.83646 −0.747282 −0.373641 0.927573i \(-0.621891\pi\)
−0.373641 + 0.927573i \(0.621891\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −13.7360 −1.70374
\(66\) 0 0
\(67\) −5.05971 −0.618142 −0.309071 0.951039i \(-0.600018\pi\)
−0.309071 + 0.951039i \(0.600018\pi\)
\(68\) 0 0
\(69\) −4.29857 −0.517487
\(70\) 0 0
\(71\) 1.35828 0.161198 0.0805992 0.996747i \(-0.474317\pi\)
0.0805992 + 0.996747i \(0.474317\pi\)
\(72\) 0 0
\(73\) −2.85585 −0.334252 −0.167126 0.985936i \(-0.553449\pi\)
−0.167126 + 0.985936i \(0.553449\pi\)
\(74\) 0 0
\(75\) −0.441637 −0.0509959
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 7.69564 0.865827 0.432913 0.901436i \(-0.357486\pi\)
0.432913 + 0.901436i \(0.357486\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.03878 0.662843 0.331421 0.943483i \(-0.392472\pi\)
0.331421 + 0.943483i \(0.392472\pi\)
\(84\) 0 0
\(85\) 2.38193 0.258356
\(86\) 0 0
\(87\) 0.422246 0.0452695
\(88\) 0 0
\(89\) 15.3696 1.62918 0.814589 0.580038i \(-0.196962\pi\)
0.814589 + 0.580038i \(0.196962\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 10.6762 1.10708
\(94\) 0 0
\(95\) −2.17643 −0.223297
\(96\) 0 0
\(97\) −0.512705 −0.0520573 −0.0260287 0.999661i \(-0.508286\pi\)
−0.0260287 + 0.999661i \(0.508286\pi\)
\(98\) 0 0
\(99\) −0.298573 −0.0300077
\(100\) 0 0
\(101\) 4.96346 0.493883 0.246941 0.969030i \(-0.420575\pi\)
0.246941 + 0.969030i \(0.420575\pi\)
\(102\) 0 0
\(103\) 9.48195 0.934285 0.467142 0.884182i \(-0.345284\pi\)
0.467142 + 0.884182i \(0.345284\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 16.8387 1.62786 0.813929 0.580964i \(-0.197324\pi\)
0.813929 + 0.580964i \(0.197324\pi\)
\(108\) 0 0
\(109\) −3.09849 −0.296782 −0.148391 0.988929i \(-0.547409\pi\)
−0.148391 + 0.988929i \(0.547409\pi\)
\(110\) 0 0
\(111\) 5.65685 0.536925
\(112\) 0 0
\(113\) 9.31371 0.876160 0.438080 0.898936i \(-0.355659\pi\)
0.438080 + 0.898936i \(0.355659\pi\)
\(114\) 0 0
\(115\) −9.17759 −0.855815
\(116\) 0 0
\(117\) −6.43361 −0.594787
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −10.9109 −0.991896
\(122\) 0 0
\(123\) 8.32600 0.750730
\(124\) 0 0
\(125\) −11.6181 −1.03915
\(126\) 0 0
\(127\) 17.1373 1.52069 0.760344 0.649521i \(-0.225031\pi\)
0.760344 + 0.649521i \(0.225031\pi\)
\(128\) 0 0
\(129\) 7.09849 0.624987
\(130\) 0 0
\(131\) 16.1582 1.41175 0.705874 0.708337i \(-0.250554\pi\)
0.705874 + 0.708337i \(0.250554\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 2.13503 0.183754
\(136\) 0 0
\(137\) 21.7747 1.86034 0.930171 0.367127i \(-0.119659\pi\)
0.930171 + 0.367127i \(0.119659\pi\)
\(138\) 0 0
\(139\) 15.3137 1.29889 0.649446 0.760408i \(-0.275001\pi\)
0.649446 + 0.760408i \(0.275001\pi\)
\(140\) 0 0
\(141\) −8.11788 −0.683649
\(142\) 0 0
\(143\) 1.92090 0.160634
\(144\) 0 0
\(145\) 0.901508 0.0748662
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3.87207 0.317212 0.158606 0.987342i \(-0.449300\pi\)
0.158606 + 0.987342i \(0.449300\pi\)
\(150\) 0 0
\(151\) −4.75535 −0.386985 −0.193492 0.981102i \(-0.561981\pi\)
−0.193492 + 0.981102i \(0.561981\pi\)
\(152\) 0 0
\(153\) 1.11564 0.0901942
\(154\) 0 0
\(155\) 22.7941 1.83087
\(156\) 0 0
\(157\) −11.5482 −0.921644 −0.460822 0.887493i \(-0.652445\pi\)
−0.460822 + 0.887493i \(0.652445\pi\)
\(158\) 0 0
\(159\) 3.44164 0.272940
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 11.9109 0.932930 0.466465 0.884540i \(-0.345527\pi\)
0.466465 + 0.884540i \(0.345527\pi\)
\(164\) 0 0
\(165\) −0.637463 −0.0496264
\(166\) 0 0
\(167\) 15.2734 1.18189 0.590945 0.806712i \(-0.298755\pi\)
0.590945 + 0.806712i \(0.298755\pi\)
\(168\) 0 0
\(169\) 28.3913 2.18394
\(170\) 0 0
\(171\) −1.01939 −0.0779548
\(172\) 0 0
\(173\) 4.00710 0.304654 0.152327 0.988330i \(-0.451323\pi\)
0.152327 + 0.988330i \(0.451323\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 6.46103 0.485641
\(178\) 0 0
\(179\) −20.7496 −1.55089 −0.775447 0.631412i \(-0.782476\pi\)
−0.775447 + 0.631412i \(0.782476\pi\)
\(180\) 0 0
\(181\) −21.7246 −1.61478 −0.807388 0.590021i \(-0.799120\pi\)
−0.807388 + 0.590021i \(0.799120\pi\)
\(182\) 0 0
\(183\) −5.83646 −0.431443
\(184\) 0 0
\(185\) 12.0776 0.887960
\(186\) 0 0
\(187\) −0.333100 −0.0243587
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 20.6720 1.49577 0.747886 0.663827i \(-0.231069\pi\)
0.747886 + 0.663827i \(0.231069\pi\)
\(192\) 0 0
\(193\) 15.5998 1.12290 0.561450 0.827510i \(-0.310244\pi\)
0.561450 + 0.827510i \(0.310244\pi\)
\(194\) 0 0
\(195\) −13.7360 −0.983652
\(196\) 0 0
\(197\) −25.6386 −1.82668 −0.913338 0.407202i \(-0.866504\pi\)
−0.913338 + 0.407202i \(0.866504\pi\)
\(198\) 0 0
\(199\) −2.88327 −0.204390 −0.102195 0.994764i \(-0.532587\pi\)
−0.102195 + 0.994764i \(0.532587\pi\)
\(200\) 0 0
\(201\) −5.05971 −0.356884
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 17.7763 1.24155
\(206\) 0 0
\(207\) −4.29857 −0.298771
\(208\) 0 0
\(209\) 0.304363 0.0210532
\(210\) 0 0
\(211\) −24.5401 −1.68941 −0.844706 0.535230i \(-0.820225\pi\)
−0.844706 + 0.535230i \(0.820225\pi\)
\(212\) 0 0
\(213\) 1.35828 0.0930679
\(214\) 0 0
\(215\) 15.1555 1.03360
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −2.85585 −0.192981
\(220\) 0 0
\(221\) −7.17759 −0.482817
\(222\) 0 0
\(223\) −20.5080 −1.37332 −0.686659 0.726980i \(-0.740923\pi\)
−0.686659 + 0.726980i \(0.740923\pi\)
\(224\) 0 0
\(225\) −0.441637 −0.0294425
\(226\) 0 0
\(227\) 13.6972 0.909113 0.454557 0.890718i \(-0.349798\pi\)
0.454557 + 0.890718i \(0.349798\pi\)
\(228\) 0 0
\(229\) −16.4175 −1.08490 −0.542451 0.840088i \(-0.682504\pi\)
−0.542451 + 0.840088i \(0.682504\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −10.8915 −0.713523 −0.356762 0.934195i \(-0.616119\pi\)
−0.356762 + 0.934195i \(0.616119\pi\)
\(234\) 0 0
\(235\) −17.3319 −1.13061
\(236\) 0 0
\(237\) 7.69564 0.499885
\(238\) 0 0
\(239\) −4.29857 −0.278052 −0.139026 0.990289i \(-0.544397\pi\)
−0.139026 + 0.990289i \(0.544397\pi\)
\(240\) 0 0
\(241\) 7.12808 0.459160 0.229580 0.973290i \(-0.426265\pi\)
0.229580 + 0.973290i \(0.426265\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 6.55836 0.417299
\(248\) 0 0
\(249\) 6.03878 0.382692
\(250\) 0 0
\(251\) −18.9690 −1.19731 −0.598657 0.801005i \(-0.704299\pi\)
−0.598657 + 0.801005i \(0.704299\pi\)
\(252\) 0 0
\(253\) 1.28344 0.0806890
\(254\) 0 0
\(255\) 2.38193 0.149162
\(256\) 0 0
\(257\) 20.4842 1.27777 0.638885 0.769303i \(-0.279396\pi\)
0.638885 + 0.769303i \(0.279396\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0.422246 0.0261364
\(262\) 0 0
\(263\) −23.5553 −1.45248 −0.726240 0.687441i \(-0.758734\pi\)
−0.726240 + 0.687441i \(0.758734\pi\)
\(264\) 0 0
\(265\) 7.34801 0.451384
\(266\) 0 0
\(267\) 15.3696 0.940607
\(268\) 0 0
\(269\) −12.9086 −0.787052 −0.393526 0.919313i \(-0.628745\pi\)
−0.393526 + 0.919313i \(0.628745\pi\)
\(270\) 0 0
\(271\) 19.8705 1.20705 0.603525 0.797344i \(-0.293763\pi\)
0.603525 + 0.797344i \(0.293763\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0.131861 0.00795151
\(276\) 0 0
\(277\) −25.9884 −1.56149 −0.780746 0.624848i \(-0.785161\pi\)
−0.780746 + 0.624848i \(0.785161\pi\)
\(278\) 0 0
\(279\) 10.6762 0.639170
\(280\) 0 0
\(281\) −19.6553 −1.17254 −0.586269 0.810116i \(-0.699404\pi\)
−0.586269 + 0.810116i \(0.699404\pi\)
\(282\) 0 0
\(283\) 19.0582 1.13289 0.566445 0.824099i \(-0.308318\pi\)
0.566445 + 0.824099i \(0.308318\pi\)
\(284\) 0 0
\(285\) −2.17643 −0.128921
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −15.7553 −0.926785
\(290\) 0 0
\(291\) −0.512705 −0.0300553
\(292\) 0 0
\(293\) −7.84890 −0.458538 −0.229269 0.973363i \(-0.573634\pi\)
−0.229269 + 0.973363i \(0.573634\pi\)
\(294\) 0 0
\(295\) 13.7945 0.803147
\(296\) 0 0
\(297\) −0.298573 −0.0173250
\(298\) 0 0
\(299\) 27.6553 1.59935
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 4.96346 0.285143
\(304\) 0 0
\(305\) −12.4610 −0.713516
\(306\) 0 0
\(307\) 26.5301 1.51415 0.757076 0.653327i \(-0.226627\pi\)
0.757076 + 0.653327i \(0.226627\pi\)
\(308\) 0 0
\(309\) 9.48195 0.539410
\(310\) 0 0
\(311\) 0.726608 0.0412022 0.0206011 0.999788i \(-0.493442\pi\)
0.0206011 + 0.999788i \(0.493442\pi\)
\(312\) 0 0
\(313\) 15.1259 0.854967 0.427484 0.904023i \(-0.359400\pi\)
0.427484 + 0.904023i \(0.359400\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 19.9496 1.12048 0.560242 0.828329i \(-0.310708\pi\)
0.560242 + 0.828329i \(0.310708\pi\)
\(318\) 0 0
\(319\) −0.126071 −0.00705863
\(320\) 0 0
\(321\) 16.8387 0.939845
\(322\) 0 0
\(323\) −1.13727 −0.0632797
\(324\) 0 0
\(325\) 2.84132 0.157608
\(326\) 0 0
\(327\) −3.09849 −0.171347
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −17.1943 −0.945084 −0.472542 0.881308i \(-0.656663\pi\)
−0.472542 + 0.881308i \(0.656663\pi\)
\(332\) 0 0
\(333\) 5.65685 0.309994
\(334\) 0 0
\(335\) −10.8026 −0.590211
\(336\) 0 0
\(337\) 10.0388 0.546847 0.273424 0.961894i \(-0.411844\pi\)
0.273424 + 0.961894i \(0.411844\pi\)
\(338\) 0 0
\(339\) 9.31371 0.505851
\(340\) 0 0
\(341\) −3.18764 −0.172620
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −9.17759 −0.494105
\(346\) 0 0
\(347\) −2.41799 −0.129804 −0.0649022 0.997892i \(-0.520674\pi\)
−0.0649022 + 0.997892i \(0.520674\pi\)
\(348\) 0 0
\(349\) 6.26689 0.335459 0.167730 0.985833i \(-0.446356\pi\)
0.167730 + 0.985833i \(0.446356\pi\)
\(350\) 0 0
\(351\) −6.43361 −0.343400
\(352\) 0 0
\(353\) −10.3099 −0.548742 −0.274371 0.961624i \(-0.588470\pi\)
−0.274371 + 0.961624i \(0.588470\pi\)
\(354\) 0 0
\(355\) 2.89997 0.153915
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −17.8984 −0.944642 −0.472321 0.881427i \(-0.656584\pi\)
−0.472321 + 0.881427i \(0.656584\pi\)
\(360\) 0 0
\(361\) −17.9608 −0.945307
\(362\) 0 0
\(363\) −10.9109 −0.572671
\(364\) 0 0
\(365\) −6.09733 −0.319149
\(366\) 0 0
\(367\) 23.0415 1.20276 0.601378 0.798965i \(-0.294619\pi\)
0.601378 + 0.798965i \(0.294619\pi\)
\(368\) 0 0
\(369\) 8.32600 0.433434
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −3.68898 −0.191008 −0.0955042 0.995429i \(-0.530446\pi\)
−0.0955042 + 0.995429i \(0.530446\pi\)
\(374\) 0 0
\(375\) −11.6181 −0.599955
\(376\) 0 0
\(377\) −2.71656 −0.139910
\(378\) 0 0
\(379\) −8.76386 −0.450169 −0.225085 0.974339i \(-0.572266\pi\)
−0.225085 + 0.974339i \(0.572266\pi\)
\(380\) 0 0
\(381\) 17.1373 0.877969
\(382\) 0 0
\(383\) 21.7665 1.11222 0.556109 0.831109i \(-0.312294\pi\)
0.556109 + 0.831109i \(0.312294\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 7.09849 0.360837
\(388\) 0 0
\(389\) −6.38346 −0.323654 −0.161827 0.986819i \(-0.551739\pi\)
−0.161827 + 0.986819i \(0.551739\pi\)
\(390\) 0 0
\(391\) −4.79566 −0.242527
\(392\) 0 0
\(393\) 16.1582 0.815073
\(394\) 0 0
\(395\) 16.4304 0.826705
\(396\) 0 0
\(397\) −5.53210 −0.277648 −0.138824 0.990317i \(-0.544332\pi\)
−0.138824 + 0.990317i \(0.544332\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −15.5778 −0.777916 −0.388958 0.921256i \(-0.627165\pi\)
−0.388958 + 0.921256i \(0.627165\pi\)
\(402\) 0 0
\(403\) −68.6867 −3.42153
\(404\) 0 0
\(405\) 2.13503 0.106091
\(406\) 0 0
\(407\) −1.68898 −0.0837198
\(408\) 0 0
\(409\) −16.8625 −0.833797 −0.416899 0.908953i \(-0.636883\pi\)
−0.416899 + 0.908953i \(0.636883\pi\)
\(410\) 0 0
\(411\) 21.7747 1.07407
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 12.8930 0.632892
\(416\) 0 0
\(417\) 15.3137 0.749916
\(418\) 0 0
\(419\) −27.8942 −1.36272 −0.681359 0.731949i \(-0.738611\pi\)
−0.681359 + 0.731949i \(0.738611\pi\)
\(420\) 0 0
\(421\) 31.6774 1.54386 0.771931 0.635706i \(-0.219291\pi\)
0.771931 + 0.635706i \(0.219291\pi\)
\(422\) 0 0
\(423\) −8.11788 −0.394705
\(424\) 0 0
\(425\) −0.492709 −0.0238999
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 1.92090 0.0927419
\(430\) 0 0
\(431\) −24.8066 −1.19489 −0.597445 0.801910i \(-0.703817\pi\)
−0.597445 + 0.801910i \(0.703817\pi\)
\(432\) 0 0
\(433\) 13.4918 0.648374 0.324187 0.945993i \(-0.394909\pi\)
0.324187 + 0.945993i \(0.394909\pi\)
\(434\) 0 0
\(435\) 0.901508 0.0432240
\(436\) 0 0
\(437\) 4.38193 0.209616
\(438\) 0 0
\(439\) 35.3137 1.68543 0.842716 0.538359i \(-0.180956\pi\)
0.842716 + 0.538359i \(0.180956\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −30.4385 −1.44618 −0.723089 0.690755i \(-0.757279\pi\)
−0.723089 + 0.690755i \(0.757279\pi\)
\(444\) 0 0
\(445\) 32.8147 1.55556
\(446\) 0 0
\(447\) 3.87207 0.183143
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) −2.48592 −0.117057
\(452\) 0 0
\(453\) −4.75535 −0.223426
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 15.1943 0.710759 0.355379 0.934722i \(-0.384352\pi\)
0.355379 + 0.934722i \(0.384352\pi\)
\(458\) 0 0
\(459\) 1.11564 0.0520736
\(460\) 0 0
\(461\) −38.3935 −1.78816 −0.894082 0.447903i \(-0.852171\pi\)
−0.894082 + 0.447903i \(0.852171\pi\)
\(462\) 0 0
\(463\) −5.59984 −0.260247 −0.130123 0.991498i \(-0.541537\pi\)
−0.130123 + 0.991498i \(0.541537\pi\)
\(464\) 0 0
\(465\) 22.7941 1.05705
\(466\) 0 0
\(467\) −23.9717 −1.10928 −0.554639 0.832091i \(-0.687144\pi\)
−0.554639 + 0.832091i \(0.687144\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −11.5482 −0.532111
\(472\) 0 0
\(473\) −2.11942 −0.0974509
\(474\) 0 0
\(475\) 0.450201 0.0206567
\(476\) 0 0
\(477\) 3.44164 0.157582
\(478\) 0 0
\(479\) −5.23461 −0.239175 −0.119588 0.992824i \(-0.538157\pi\)
−0.119588 + 0.992824i \(0.538157\pi\)
\(480\) 0 0
\(481\) −36.3940 −1.65942
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −1.09464 −0.0497051
\(486\) 0 0
\(487\) 11.2447 0.509544 0.254772 0.967001i \(-0.418000\pi\)
0.254772 + 0.967001i \(0.418000\pi\)
\(488\) 0 0
\(489\) 11.9109 0.538627
\(490\) 0 0
\(491\) −13.5250 −0.610374 −0.305187 0.952292i \(-0.598719\pi\)
−0.305187 + 0.952292i \(0.598719\pi\)
\(492\) 0 0
\(493\) 0.471075 0.0212161
\(494\) 0 0
\(495\) −0.637463 −0.0286518
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 9.87207 0.441935 0.220967 0.975281i \(-0.429079\pi\)
0.220967 + 0.975281i \(0.429079\pi\)
\(500\) 0 0
\(501\) 15.2734 0.682365
\(502\) 0 0
\(503\) 39.0415 1.74077 0.870387 0.492369i \(-0.163869\pi\)
0.870387 + 0.492369i \(0.163869\pi\)
\(504\) 0 0
\(505\) 10.5971 0.471567
\(506\) 0 0
\(507\) 28.3913 1.26090
\(508\) 0 0
\(509\) 24.2202 1.07354 0.536770 0.843729i \(-0.319644\pi\)
0.536770 + 0.843729i \(0.319644\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −1.01939 −0.0450072
\(514\) 0 0
\(515\) 20.2443 0.892069
\(516\) 0 0
\(517\) 2.42378 0.106598
\(518\) 0 0
\(519\) 4.00710 0.175892
\(520\) 0 0
\(521\) −22.1638 −0.971012 −0.485506 0.874233i \(-0.661365\pi\)
−0.485506 + 0.874233i \(0.661365\pi\)
\(522\) 0 0
\(523\) 35.8053 1.56566 0.782829 0.622237i \(-0.213776\pi\)
0.782829 + 0.622237i \(0.213776\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 11.9109 0.518845
\(528\) 0 0
\(529\) −4.52227 −0.196620
\(530\) 0 0
\(531\) 6.46103 0.280385
\(532\) 0 0
\(533\) −53.5662 −2.32021
\(534\) 0 0
\(535\) 35.9512 1.55430
\(536\) 0 0
\(537\) −20.7496 −0.895409
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −12.7941 −0.550063 −0.275031 0.961435i \(-0.588688\pi\)
−0.275031 + 0.961435i \(0.588688\pi\)
\(542\) 0 0
\(543\) −21.7246 −0.932292
\(544\) 0 0
\(545\) −6.61538 −0.283372
\(546\) 0 0
\(547\) 28.9675 1.23856 0.619280 0.785170i \(-0.287424\pi\)
0.619280 + 0.785170i \(0.287424\pi\)
\(548\) 0 0
\(549\) −5.83646 −0.249094
\(550\) 0 0
\(551\) −0.430434 −0.0183371
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 12.0776 0.512664
\(556\) 0 0
\(557\) −9.75265 −0.413233 −0.206617 0.978422i \(-0.566245\pi\)
−0.206617 + 0.978422i \(0.566245\pi\)
\(558\) 0 0
\(559\) −45.6689 −1.93159
\(560\) 0 0
\(561\) −0.333100 −0.0140635
\(562\) 0 0
\(563\) −10.6192 −0.447547 −0.223774 0.974641i \(-0.571838\pi\)
−0.223774 + 0.974641i \(0.571838\pi\)
\(564\) 0 0
\(565\) 19.8851 0.836571
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 37.1660 1.55808 0.779040 0.626974i \(-0.215707\pi\)
0.779040 + 0.626974i \(0.215707\pi\)
\(570\) 0 0
\(571\) −23.6956 −0.991632 −0.495816 0.868428i \(-0.665131\pi\)
−0.495816 + 0.868428i \(0.665131\pi\)
\(572\) 0 0
\(573\) 20.6720 0.863585
\(574\) 0 0
\(575\) 1.89841 0.0791692
\(576\) 0 0
\(577\) 9.84465 0.409838 0.204919 0.978779i \(-0.434307\pi\)
0.204919 + 0.978779i \(0.434307\pi\)
\(578\) 0 0
\(579\) 15.5998 0.648307
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −1.02758 −0.0425580
\(584\) 0 0
\(585\) −13.7360 −0.567912
\(586\) 0 0
\(587\) 30.4610 1.25726 0.628631 0.777704i \(-0.283616\pi\)
0.628631 + 0.777704i \(0.283616\pi\)
\(588\) 0 0
\(589\) −10.8833 −0.448438
\(590\) 0 0
\(591\) −25.6386 −1.05463
\(592\) 0 0
\(593\) −10.0720 −0.413607 −0.206804 0.978382i \(-0.566306\pi\)
−0.206804 + 0.978382i \(0.566306\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −2.88327 −0.118005
\(598\) 0 0
\(599\) −31.7220 −1.29612 −0.648062 0.761587i \(-0.724420\pi\)
−0.648062 + 0.761587i \(0.724420\pi\)
\(600\) 0 0
\(601\) 29.7533 1.21366 0.606832 0.794830i \(-0.292440\pi\)
0.606832 + 0.794830i \(0.292440\pi\)
\(602\) 0 0
\(603\) −5.05971 −0.206047
\(604\) 0 0
\(605\) −23.2950 −0.947077
\(606\) 0 0
\(607\) 17.6247 0.715366 0.357683 0.933843i \(-0.383567\pi\)
0.357683 + 0.933843i \(0.383567\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 52.2273 2.11289
\(612\) 0 0
\(613\) −12.7068 −0.513224 −0.256612 0.966514i \(-0.582606\pi\)
−0.256612 + 0.966514i \(0.582606\pi\)
\(614\) 0 0
\(615\) 17.7763 0.716808
\(616\) 0 0
\(617\) −40.7386 −1.64008 −0.820038 0.572309i \(-0.806048\pi\)
−0.820038 + 0.572309i \(0.806048\pi\)
\(618\) 0 0
\(619\) −46.9407 −1.88671 −0.943354 0.331788i \(-0.892348\pi\)
−0.943354 + 0.331788i \(0.892348\pi\)
\(620\) 0 0
\(621\) −4.29857 −0.172496
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −22.5968 −0.903871
\(626\) 0 0
\(627\) 0.304363 0.0121551
\(628\) 0 0
\(629\) 6.31102 0.251637
\(630\) 0 0
\(631\) 27.4804 1.09398 0.546989 0.837140i \(-0.315774\pi\)
0.546989 + 0.837140i \(0.315774\pi\)
\(632\) 0 0
\(633\) −24.5401 −0.975383
\(634\) 0 0
\(635\) 36.5886 1.45198
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 1.35828 0.0537328
\(640\) 0 0
\(641\) −48.6580 −1.92188 −0.960938 0.276764i \(-0.910738\pi\)
−0.960938 + 0.276764i \(0.910738\pi\)
\(642\) 0 0
\(643\) 17.6856 0.697452 0.348726 0.937225i \(-0.386614\pi\)
0.348726 + 0.937225i \(0.386614\pi\)
\(644\) 0 0
\(645\) 15.1555 0.596748
\(646\) 0 0
\(647\) −4.04032 −0.158841 −0.0794206 0.996841i \(-0.525307\pi\)
−0.0794206 + 0.996841i \(0.525307\pi\)
\(648\) 0 0
\(649\) −1.92909 −0.0757233
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −30.9690 −1.21191 −0.605956 0.795498i \(-0.707209\pi\)
−0.605956 + 0.795498i \(0.707209\pi\)
\(654\) 0 0
\(655\) 34.4983 1.34796
\(656\) 0 0
\(657\) −2.85585 −0.111417
\(658\) 0 0
\(659\) 4.81815 0.187689 0.0938443 0.995587i \(-0.470084\pi\)
0.0938443 + 0.995587i \(0.470084\pi\)
\(660\) 0 0
\(661\) 4.38324 0.170488 0.0852442 0.996360i \(-0.472833\pi\)
0.0852442 + 0.996360i \(0.472833\pi\)
\(662\) 0 0
\(663\) −7.17759 −0.278755
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −1.81505 −0.0702792
\(668\) 0 0
\(669\) −20.5080 −0.792885
\(670\) 0 0
\(671\) 1.74261 0.0672727
\(672\) 0 0
\(673\) −8.67471 −0.334386 −0.167193 0.985924i \(-0.553470\pi\)
−0.167193 + 0.985924i \(0.553470\pi\)
\(674\) 0 0
\(675\) −0.441637 −0.0169986
\(676\) 0 0
\(677\) −1.04103 −0.0400099 −0.0200049 0.999800i \(-0.506368\pi\)
−0.0200049 + 0.999800i \(0.506368\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 13.6972 0.524877
\(682\) 0 0
\(683\) 35.7220 1.36686 0.683432 0.730014i \(-0.260487\pi\)
0.683432 + 0.730014i \(0.260487\pi\)
\(684\) 0 0
\(685\) 46.4898 1.77628
\(686\) 0 0
\(687\) −16.4175 −0.626368
\(688\) 0 0
\(689\) −22.1421 −0.843548
\(690\) 0 0
\(691\) −16.9221 −0.643745 −0.321873 0.946783i \(-0.604312\pi\)
−0.321873 + 0.946783i \(0.604312\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 32.6953 1.24020
\(696\) 0 0
\(697\) 9.28882 0.351839
\(698\) 0 0
\(699\) −10.8915 −0.411953
\(700\) 0 0
\(701\) 22.2052 0.838678 0.419339 0.907830i \(-0.362262\pi\)
0.419339 + 0.907830i \(0.362262\pi\)
\(702\) 0 0
\(703\) −5.76655 −0.217490
\(704\) 0 0
\(705\) −17.3319 −0.652759
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 7.35435 0.276198 0.138099 0.990418i \(-0.455901\pi\)
0.138099 + 0.990418i \(0.455901\pi\)
\(710\) 0 0
\(711\) 7.69564 0.288609
\(712\) 0 0
\(713\) −45.8926 −1.71869
\(714\) 0 0
\(715\) 4.10118 0.153376
\(716\) 0 0
\(717\) −4.29857 −0.160533
\(718\) 0 0
\(719\) 27.5495 1.02742 0.513711 0.857963i \(-0.328270\pi\)
0.513711 + 0.857963i \(0.328270\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 7.12808 0.265096
\(724\) 0 0
\(725\) −0.186480 −0.00692568
\(726\) 0 0
\(727\) 14.9321 0.553801 0.276901 0.960899i \(-0.410693\pi\)
0.276901 + 0.960899i \(0.410693\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 7.91937 0.292908
\(732\) 0 0
\(733\) −42.4684 −1.56860 −0.784302 0.620379i \(-0.786979\pi\)
−0.784302 + 0.620379i \(0.786979\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.51069 0.0556470
\(738\) 0 0
\(739\) −7.83329 −0.288152 −0.144076 0.989567i \(-0.546021\pi\)
−0.144076 + 0.989567i \(0.546021\pi\)
\(740\) 0 0
\(741\) 6.55836 0.240927
\(742\) 0 0
\(743\) 30.4064 1.11550 0.557752 0.830008i \(-0.311664\pi\)
0.557752 + 0.830008i \(0.311664\pi\)
\(744\) 0 0
\(745\) 8.26700 0.302879
\(746\) 0 0
\(747\) 6.03878 0.220948
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 28.2843 1.03211 0.516054 0.856556i \(-0.327400\pi\)
0.516054 + 0.856556i \(0.327400\pi\)
\(752\) 0 0
\(753\) −18.9690 −0.691270
\(754\) 0 0
\(755\) −10.1528 −0.369499
\(756\) 0 0
\(757\) 19.5289 0.709791 0.354895 0.934906i \(-0.384516\pi\)
0.354895 + 0.934906i \(0.384516\pi\)
\(758\) 0 0
\(759\) 1.28344 0.0465858
\(760\) 0 0
\(761\) 34.0746 1.23520 0.617602 0.786491i \(-0.288104\pi\)
0.617602 + 0.786491i \(0.288104\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 2.38193 0.0861188
\(766\) 0 0
\(767\) −41.5677 −1.50092
\(768\) 0 0
\(769\) −28.3523 −1.02241 −0.511206 0.859458i \(-0.670801\pi\)
−0.511206 + 0.859458i \(0.670801\pi\)
\(770\) 0 0
\(771\) 20.4842 0.737720
\(772\) 0 0
\(773\) −1.96615 −0.0707175 −0.0353588 0.999375i \(-0.511257\pi\)
−0.0353588 + 0.999375i \(0.511257\pi\)
\(774\) 0 0
\(775\) −4.71503 −0.169369
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −8.48745 −0.304094
\(780\) 0 0
\(781\) −0.405546 −0.0145116
\(782\) 0 0
\(783\) 0.422246 0.0150898
\(784\) 0 0
\(785\) −24.6557 −0.880000
\(786\) 0 0
\(787\) −2.39165 −0.0852531 −0.0426266 0.999091i \(-0.513573\pi\)
−0.0426266 + 0.999091i \(0.513573\pi\)
\(788\) 0 0
\(789\) −23.5553 −0.838590
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 37.5495 1.33342
\(794\) 0 0
\(795\) 7.34801 0.260607
\(796\) 0 0
\(797\) 30.2674 1.07213 0.536064 0.844177i \(-0.319911\pi\)
0.536064 + 0.844177i \(0.319911\pi\)
\(798\) 0 0
\(799\) −9.05664 −0.320401
\(800\) 0 0
\(801\) 15.3696 0.543060
\(802\) 0 0
\(803\) 0.852680 0.0300904
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −12.9086 −0.454405
\(808\) 0 0
\(809\) 28.7692 1.01147 0.505736 0.862688i \(-0.331221\pi\)
0.505736 + 0.862688i \(0.331221\pi\)
\(810\) 0 0
\(811\) 51.6270 1.81287 0.906435 0.422345i \(-0.138793\pi\)
0.906435 + 0.422345i \(0.138793\pi\)
\(812\) 0 0
\(813\) 19.8705 0.696890
\(814\) 0 0
\(815\) 25.4301 0.890776
\(816\) 0 0
\(817\) −7.23614 −0.253161
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −44.1885 −1.54219 −0.771094 0.636721i \(-0.780290\pi\)
−0.771094 + 0.636721i \(0.780290\pi\)
\(822\) 0 0
\(823\) 33.8151 1.17872 0.589359 0.807871i \(-0.299380\pi\)
0.589359 + 0.807871i \(0.299380\pi\)
\(824\) 0 0
\(825\) 0.131861 0.00459081
\(826\) 0 0
\(827\) 48.0748 1.67173 0.835863 0.548938i \(-0.184968\pi\)
0.835863 + 0.548938i \(0.184968\pi\)
\(828\) 0 0
\(829\) 9.26203 0.321684 0.160842 0.986980i \(-0.448579\pi\)
0.160842 + 0.986980i \(0.448579\pi\)
\(830\) 0 0
\(831\) −25.9884 −0.901528
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 32.6092 1.12849
\(836\) 0 0
\(837\) 10.6762 0.369025
\(838\) 0 0
\(839\) 18.1567 0.626838 0.313419 0.949615i \(-0.398526\pi\)
0.313419 + 0.949615i \(0.398526\pi\)
\(840\) 0 0
\(841\) −28.8217 −0.993852
\(842\) 0 0
\(843\) −19.6553 −0.676965
\(844\) 0 0
\(845\) 60.6163 2.08526
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 19.0582 0.654075
\(850\) 0 0
\(851\) −24.3164 −0.833555
\(852\) 0 0
\(853\) 40.7652 1.39577 0.697886 0.716208i \(-0.254124\pi\)
0.697886 + 0.716208i \(0.254124\pi\)
\(854\) 0 0
\(855\) −2.17643 −0.0744325
\(856\) 0 0
\(857\) 12.0559 0.411823 0.205911 0.978571i \(-0.433984\pi\)
0.205911 + 0.978571i \(0.433984\pi\)
\(858\) 0 0
\(859\) −4.25247 −0.145092 −0.0725461 0.997365i \(-0.523112\pi\)
−0.0725461 + 0.997365i \(0.523112\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −11.3280 −0.385610 −0.192805 0.981237i \(-0.561758\pi\)
−0.192805 + 0.981237i \(0.561758\pi\)
\(864\) 0 0
\(865\) 8.55529 0.290889
\(866\) 0 0
\(867\) −15.7553 −0.535080
\(868\) 0 0
\(869\) −2.29771 −0.0779444
\(870\) 0 0
\(871\) 32.5522 1.10299
\(872\) 0 0
\(873\) −0.512705 −0.0173524
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −31.6871 −1.07000 −0.534999 0.844853i \(-0.679688\pi\)
−0.534999 + 0.844853i \(0.679688\pi\)
\(878\) 0 0
\(879\) −7.84890 −0.264737
\(880\) 0 0
\(881\) −24.2690 −0.817643 −0.408821 0.912614i \(-0.634060\pi\)
−0.408821 + 0.912614i \(0.634060\pi\)
\(882\) 0 0
\(883\) −47.1439 −1.58652 −0.793260 0.608883i \(-0.791618\pi\)
−0.793260 + 0.608883i \(0.791618\pi\)
\(884\) 0 0
\(885\) 13.7945 0.463697
\(886\) 0 0
\(887\) −14.6647 −0.492391 −0.246196 0.969220i \(-0.579181\pi\)
−0.246196 + 0.969220i \(0.579181\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −0.298573 −0.0100026
\(892\) 0 0
\(893\) 8.27530 0.276922
\(894\) 0 0
\(895\) −44.3010 −1.48082
\(896\) 0 0
\(897\) 27.6553 0.923384
\(898\) 0 0
\(899\) 4.50800 0.150350
\(900\) 0 0
\(901\) 3.83963 0.127917
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −46.3827 −1.54181
\(906\) 0 0
\(907\) 3.99149 0.132535 0.0662676 0.997802i \(-0.478891\pi\)
0.0662676 + 0.997802i \(0.478891\pi\)
\(908\) 0 0
\(909\) 4.96346 0.164628
\(910\) 0 0
\(911\) −18.9260 −0.627046 −0.313523 0.949581i \(-0.601509\pi\)
−0.313523 + 0.949581i \(0.601509\pi\)
\(912\) 0 0
\(913\) −1.80302 −0.0596711
\(914\) 0 0
\(915\) −12.4610 −0.411949
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −46.4898 −1.53356 −0.766778 0.641912i \(-0.778141\pi\)
−0.766778 + 0.641912i \(0.778141\pi\)
\(920\) 0 0
\(921\) 26.5301 0.874196
\(922\) 0 0
\(923\) −8.73865 −0.287636
\(924\) 0 0
\(925\) −2.49828 −0.0821429
\(926\) 0 0
\(927\) 9.48195 0.311428
\(928\) 0 0
\(929\) 28.7221 0.942343 0.471171 0.882042i \(-0.343831\pi\)
0.471171 + 0.882042i \(0.343831\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0.726608 0.0237881
\(934\) 0 0
\(935\) −0.711179 −0.0232581
\(936\) 0 0
\(937\) −21.2560 −0.694404 −0.347202 0.937790i \(-0.612868\pi\)
−0.347202 + 0.937790i \(0.612868\pi\)
\(938\) 0 0
\(939\) 15.1259 0.493616
\(940\) 0 0
\(941\) −19.4100 −0.632747 −0.316373 0.948635i \(-0.602465\pi\)
−0.316373 + 0.948635i \(0.602465\pi\)
\(942\) 0 0
\(943\) −35.7899 −1.16548
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 27.6123 0.897279 0.448639 0.893713i \(-0.351909\pi\)
0.448639 + 0.893713i \(0.351909\pi\)
\(948\) 0 0
\(949\) 18.3734 0.596426
\(950\) 0 0
\(951\) 19.9496 0.646911
\(952\) 0 0
\(953\) 14.5080 0.469960 0.234980 0.972000i \(-0.424497\pi\)
0.234980 + 0.972000i \(0.424497\pi\)
\(954\) 0 0
\(955\) 44.1354 1.42819
\(956\) 0 0
\(957\) −0.126071 −0.00407530
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 82.9822 2.67685
\(962\) 0 0
\(963\) 16.8387 0.542620
\(964\) 0 0
\(965\) 33.3062 1.07216
\(966\) 0 0
\(967\) 56.8449 1.82801 0.914005 0.405703i \(-0.132973\pi\)
0.914005 + 0.405703i \(0.132973\pi\)
\(968\) 0 0
\(969\) −1.13727 −0.0365345
\(970\) 0 0
\(971\) −10.1970 −0.327237 −0.163618 0.986524i \(-0.552317\pi\)
−0.163618 + 0.986524i \(0.552317\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 2.84132 0.0909951
\(976\) 0 0
\(977\) −8.53628 −0.273100 −0.136550 0.990633i \(-0.543601\pi\)
−0.136550 + 0.990633i \(0.543601\pi\)
\(978\) 0 0
\(979\) −4.58896 −0.146664
\(980\) 0 0
\(981\) −3.09849 −0.0989272
\(982\) 0 0
\(983\) 38.1970 1.21829 0.609147 0.793057i \(-0.291512\pi\)
0.609147 + 0.793057i \(0.291512\pi\)
\(984\) 0 0
\(985\) −54.7393 −1.74414
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −30.5134 −0.970269
\(990\) 0 0
\(991\) −45.8810 −1.45746 −0.728730 0.684802i \(-0.759889\pi\)
−0.728730 + 0.684802i \(0.759889\pi\)
\(992\) 0 0
\(993\) −17.1943 −0.545644
\(994\) 0 0
\(995\) −6.15588 −0.195155
\(996\) 0 0
\(997\) 35.0749 1.11083 0.555417 0.831572i \(-0.312559\pi\)
0.555417 + 0.831572i \(0.312559\pi\)
\(998\) 0 0
\(999\) 5.65685 0.178975
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 9408.2.a.em.1.4 4
4.3 odd 2 9408.2.a.ek.1.4 4
7.6 odd 2 9408.2.a.el.1.1 4
8.3 odd 2 4704.2.a.bz.1.1 yes 4
8.5 even 2 4704.2.a.bx.1.1 yes 4
28.27 even 2 9408.2.a.en.1.1 4
56.13 odd 2 4704.2.a.by.1.4 yes 4
56.27 even 2 4704.2.a.bw.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4704.2.a.bw.1.4 4 56.27 even 2
4704.2.a.bx.1.1 yes 4 8.5 even 2
4704.2.a.by.1.4 yes 4 56.13 odd 2
4704.2.a.bz.1.1 yes 4 8.3 odd 2
9408.2.a.ek.1.4 4 4.3 odd 2
9408.2.a.el.1.1 4 7.6 odd 2
9408.2.a.em.1.4 4 1.1 even 1 trivial
9408.2.a.en.1.1 4 28.27 even 2