Properties

Label 9408.2.a.en.1.1
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 1.1
Root \(-0.360409\) 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 \(-0.658424\pi\)
−0.477408 + 0.878682i \(0.658424\pi\)
\(6\) 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 \(-0.485668\pi\)
0.0450116 + 0.998986i \(0.485668\pi\)
\(12\) 0 0
\(13\) 6.43361 1.78436 0.892181 0.451679i \(-0.149175\pi\)
0.892181 + 0.451679i \(0.149175\pi\)
\(14\) 0 0
\(15\) −2.13503 −0.551263
\(16\) 0 0
\(17\) −1.11564 −0.270583 −0.135291 0.990806i \(-0.543197\pi\)
−0.135291 + 0.990806i \(0.543197\pi\)
\(18\) 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 \(-0.352080\pi\)
0.448157 + 0.893955i \(0.352080\pi\)
\(24\) 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 \(-0.725294\pi\)
−0.650151 + 0.759805i \(0.725294\pi\)
\(42\) 0 0
\(43\) −7.09849 −1.08251 −0.541255 0.840859i \(-0.682051\pi\)
−0.541255 + 0.840859i \(0.682051\pi\)
\(44\) 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 \(-0.378109\pi\)
0.373641 + 0.927573i \(0.378109\pi\)
\(62\) 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 \(-0.399982\pi\)
0.309071 + 0.951039i \(0.399982\pi\)
\(68\) 0 0
\(69\) 4.29857 0.517487
\(70\) 0 0
\(71\) −1.35828 −0.161198 −0.0805992 0.996747i \(-0.525683\pi\)
−0.0805992 + 0.996747i \(0.525683\pi\)
\(72\) 0 0
\(73\) 2.85585 0.334252 0.167126 0.985936i \(-0.446551\pi\)
0.167126 + 0.985936i \(0.446551\pi\)
\(74\) 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 \(-0.642514\pi\)
−0.432913 + 0.901436i \(0.642514\pi\)
\(80\) 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 \(-0.803038\pi\)
−0.814589 + 0.580038i \(0.803038\pi\)
\(90\) 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 \(-0.491714\pi\)
0.0260287 + 0.999661i \(0.491714\pi\)
\(98\) 0 0
\(99\) 0.298573 0.0300077
\(100\) 0 0
\(101\) −4.96346 −0.493883 −0.246941 0.969030i \(-0.579425\pi\)
−0.246941 + 0.969030i \(0.579425\pi\)
\(102\) 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 \(-0.802676\pi\)
−0.813929 + 0.580964i \(0.802676\pi\)
\(108\) 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 \(-0.774969\pi\)
−0.760344 + 0.649521i \(0.774969\pi\)
\(128\) 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 \(-0.438019\pi\)
0.193492 + 0.981102i \(0.438019\pi\)
\(152\) 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 \(-0.347555\pi\)
0.460822 + 0.887493i \(0.347555\pi\)
\(158\) 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 \(-0.654473\pi\)
−0.466465 + 0.884540i \(0.654473\pi\)
\(164\) 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 \(-0.548677\pi\)
−0.152327 + 0.988330i \(0.548677\pi\)
\(174\) 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 \(-0.217524\pi\)
0.775447 + 0.631412i \(0.217524\pi\)
\(180\) 0 0
\(181\) 21.7246 1.61478 0.807388 0.590021i \(-0.200880\pi\)
0.807388 + 0.590021i \(0.200880\pi\)
\(182\) 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 \(-0.768931\pi\)
−0.747886 + 0.663827i \(0.768931\pi\)
\(192\) 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 \(-0.179775\pi\)
0.844706 + 0.535230i \(0.179775\pi\)
\(212\) 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 \(-0.317496\pi\)
0.542451 + 0.840088i \(0.317496\pi\)
\(230\) 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 \(-0.455603\pi\)
0.139026 + 0.990289i \(0.455603\pi\)
\(240\) 0 0
\(241\) −7.12808 −0.459160 −0.229580 0.973290i \(-0.573735\pi\)
−0.229580 + 0.973290i \(0.573735\pi\)
\(242\) 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 \(-0.720604\pi\)
−0.638885 + 0.769303i \(0.720604\pi\)
\(258\) 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 \(-0.241266\pi\)
0.726240 + 0.687441i \(0.241266\pi\)
\(264\) 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 \(-0.371255\pi\)
0.393526 + 0.919313i \(0.371255\pi\)
\(270\) 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 \(-0.426366\pi\)
0.229269 + 0.973363i \(0.426366\pi\)
\(294\) 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 \(-0.640600\pi\)
−0.427484 + 0.904023i \(0.640600\pi\)
\(314\) 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 \(-0.343337\pi\)
0.472542 + 0.881308i \(0.343337\pi\)
\(332\) 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 \(-0.479326\pi\)
0.0649022 + 0.997892i \(0.479326\pi\)
\(348\) 0 0
\(349\) −6.26689 −0.335459 −0.167730 0.985833i \(-0.553644\pi\)
−0.167730 + 0.985833i \(0.553644\pi\)
\(350\) 0 0
\(351\) 6.43361 0.343400
\(352\) 0 0
\(353\) 10.3099 0.548742 0.274371 0.961624i \(-0.411530\pi\)
0.274371 + 0.961624i \(0.411530\pi\)
\(354\) 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 \(-0.343416\pi\)
0.472321 + 0.881427i \(0.343416\pi\)
\(360\) 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 \(-0.427734\pi\)
0.225085 + 0.974339i \(0.427734\pi\)
\(380\) 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 \(-0.455668\pi\)
0.138824 + 0.990317i \(0.455668\pi\)
\(398\) 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 \(-0.363117\pi\)
0.416899 + 0.908953i \(0.363117\pi\)
\(410\) 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 \(-0.296183\pi\)
0.597445 + 0.801910i \(0.296183\pi\)
\(432\) 0 0
\(433\) −13.4918 −0.648374 −0.324187 0.945993i \(-0.605091\pi\)
−0.324187 + 0.945993i \(0.605091\pi\)
\(434\) 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 \(-0.242721\pi\)
0.723089 + 0.690755i \(0.242721\pi\)
\(444\) 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 \(-0.147829\pi\)
0.894082 + 0.447903i \(0.147829\pi\)
\(462\) 0 0
\(463\) 5.59984 0.260247 0.130123 0.991498i \(-0.458463\pi\)
0.130123 + 0.991498i \(0.458463\pi\)
\(464\) 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 \(-0.582000\pi\)
−0.254772 + 0.967001i \(0.582000\pi\)
\(488\) 0 0
\(489\) −11.9109 −0.538627
\(490\) 0 0
\(491\) 13.5250 0.610374 0.305187 0.952292i \(-0.401281\pi\)
0.305187 + 0.952292i \(0.401281\pi\)
\(492\) 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 \(-0.570921\pi\)
−0.220967 + 0.975281i \(0.570921\pi\)
\(500\) 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 \(-0.680356\pi\)
−0.536770 + 0.843729i \(0.680356\pi\)
\(510\) 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 \(-0.338635\pi\)
0.485506 + 0.874233i \(0.338635\pi\)
\(522\) 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 \(-0.712576\pi\)
−0.619280 + 0.785170i \(0.712576\pi\)
\(548\) 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 \(-0.334869\pi\)
0.495816 + 0.868428i \(0.334869\pi\)
\(572\) 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 \(-0.565693\pi\)
−0.204919 + 0.978779i \(0.565693\pi\)
\(578\) 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 \(-0.433694\pi\)
0.206804 + 0.978382i \(0.433694\pi\)
\(594\) 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 \(-0.275580\pi\)
0.648062 + 0.761587i \(0.275580\pi\)
\(600\) 0 0
\(601\) −29.7533 −1.21366 −0.606832 0.794830i \(-0.707560\pi\)
−0.606832 + 0.794830i \(0.707560\pi\)
\(602\) 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 \(-0.684226\pi\)
−0.546989 + 0.837140i \(0.684226\pi\)
\(632\) 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 \(-0.529916\pi\)
−0.0938443 + 0.995587i \(0.529916\pi\)
\(660\) 0 0
\(661\) −4.38324 −0.170488 −0.0852442 0.996360i \(-0.527167\pi\)
−0.0852442 + 0.996360i \(0.527167\pi\)
\(662\) 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 \(-0.493632\pi\)
0.0200049 + 0.999800i \(0.493632\pi\)
\(678\) 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 \(-0.739513\pi\)
−0.683432 + 0.730014i \(0.739513\pi\)
\(684\) 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 \(-0.213021\pi\)
0.784302 + 0.620379i \(0.213021\pi\)
\(734\) 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 \(-0.453979\pi\)
0.144076 + 0.989567i \(0.453979\pi\)
\(740\) 0 0
\(741\) −6.55836 −0.240927
\(742\) 0 0
\(743\) −30.4064 −1.11550 −0.557752 0.830008i \(-0.688336\pi\)
−0.557752 + 0.830008i \(0.688336\pi\)
\(744\) 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 \(-0.672600\pi\)
−0.516054 + 0.856556i \(0.672600\pi\)
\(752\) 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 \(-0.711896\pi\)
−0.617602 + 0.786491i \(0.711896\pi\)
\(762\) 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 \(-0.329199\pi\)
0.511206 + 0.859458i \(0.329199\pi\)
\(770\) 0 0
\(771\) −20.4842 −0.737720
\(772\) 0 0
\(773\) 1.96615 0.0707175 0.0353588 0.999375i \(-0.488743\pi\)
0.0353588 + 0.999375i \(0.488743\pi\)
\(774\) 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 \(-0.680089\pi\)
−0.536064 + 0.844177i \(0.680089\pi\)
\(798\) 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 \(-0.700620\pi\)
−0.589359 + 0.807871i \(0.700620\pi\)
\(824\) 0 0
\(825\) −0.131861 −0.00459081
\(826\) 0 0
\(827\) −48.0748 −1.67173 −0.835863 0.548938i \(-0.815032\pi\)
−0.835863 + 0.548938i \(0.815032\pi\)
\(828\) 0 0
\(829\) −9.26203 −0.321684 −0.160842 0.986980i \(-0.551421\pi\)
−0.160842 + 0.986980i \(0.551421\pi\)
\(830\) 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 \(-0.745876\pi\)
−0.697886 + 0.716208i \(0.745876\pi\)
\(854\) 0 0
\(855\) 2.17643 0.0744325
\(856\) 0 0
\(857\) −12.0559 −0.411823 −0.205911 0.978571i \(-0.566016\pi\)
−0.205911 + 0.978571i \(0.566016\pi\)
\(858\) 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 \(-0.438242\pi\)
0.192805 + 0.981237i \(0.438242\pi\)
\(864\) 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 \(-0.365940\pi\)
0.408821 + 0.912614i \(0.365940\pi\)
\(882\) 0 0
\(883\) 47.1439 1.58652 0.793260 0.608883i \(-0.208382\pi\)
0.793260 + 0.608883i \(0.208382\pi\)
\(884\) 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 \(-0.521109\pi\)
−0.0662676 + 0.997802i \(0.521109\pi\)
\(908\) 0 0
\(909\) −4.96346 −0.164628
\(910\) 0 0
\(911\) 18.9260 0.627046 0.313523 0.949581i \(-0.398491\pi\)
0.313523 + 0.949581i \(0.398491\pi\)
\(912\) 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 \(-0.221859\pi\)
0.766778 + 0.641912i \(0.221859\pi\)
\(920\) 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 \(-0.656169\pi\)
−0.471171 + 0.882042i \(0.656169\pi\)
\(930\) 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 \(-0.387132\pi\)
0.347202 + 0.937790i \(0.387132\pi\)
\(938\) 0 0
\(939\) −15.1259 −0.493616
\(940\) 0 0
\(941\) 19.4100 0.632747 0.316373 0.948635i \(-0.397535\pi\)
0.316373 + 0.948635i \(0.397535\pi\)
\(942\) 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 \(-0.648091\pi\)
−0.448639 + 0.893713i \(0.648091\pi\)
\(948\) 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 \(-0.867027\pi\)
−0.914005 + 0.405703i \(0.867027\pi\)
\(968\) 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 \(-0.240111\pi\)
0.728730 + 0.684802i \(0.240111\pi\)
\(992\) 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 \(-0.687441\pi\)
−0.555417 + 0.831572i \(0.687441\pi\)
\(998\) 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.en.1.1 4
4.3 odd 2 9408.2.a.el.1.1 4
7.6 odd 2 9408.2.a.ek.1.4 4
8.3 odd 2 4704.2.a.by.1.4 yes 4
8.5 even 2 4704.2.a.bw.1.4 4
28.27 even 2 9408.2.a.em.1.4 4
56.13 odd 2 4704.2.a.bz.1.1 yes 4
56.27 even 2 4704.2.a.bx.1.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4704.2.a.bw.1.4 4 8.5 even 2
4704.2.a.bx.1.1 yes 4 56.27 even 2
4704.2.a.by.1.4 yes 4 8.3 odd 2
4704.2.a.bz.1.1 yes 4 56.13 odd 2
9408.2.a.ek.1.4 4 7.6 odd 2
9408.2.a.el.1.1 4 4.3 odd 2
9408.2.a.em.1.4 4 28.27 even 2
9408.2.a.en.1.1 4 1.1 even 1 trivial