Properties

Label 9408.2.a.ds.1.1
Level $9408$
Weight $2$
Character 9408.1
Self dual yes
Analytic conductor $75.123$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1176)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 9408.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +0.585786 q^{5} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +0.585786 q^{5} +1.00000 q^{9} -0.828427 q^{11} -1.41421 q^{13} -0.585786 q^{15} -2.24264 q^{17} -6.82843 q^{19} -4.82843 q^{23} -4.65685 q^{25} -1.00000 q^{27} -8.48528 q^{29} +5.17157 q^{31} +0.828427 q^{33} -1.65685 q^{37} +1.41421 q^{39} +0.585786 q^{41} -8.00000 q^{43} +0.585786 q^{45} +6.82843 q^{47} +2.24264 q^{51} +13.3137 q^{53} -0.485281 q^{55} +6.82843 q^{57} -5.17157 q^{59} +13.8995 q^{61} -0.828427 q^{65} -8.00000 q^{67} +4.82843 q^{69} +0.828427 q^{71} +11.0711 q^{73} +4.65685 q^{75} +2.34315 q^{79} +1.00000 q^{81} -15.3137 q^{83} -1.31371 q^{85} +8.48528 q^{87} +10.7279 q^{89} -5.17157 q^{93} -4.00000 q^{95} +7.75736 q^{97} -0.828427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 4q^{5} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 4q^{5} + 2q^{9} + 4q^{11} - 4q^{15} + 4q^{17} - 8q^{19} - 4q^{23} + 2q^{25} - 2q^{27} + 16q^{31} - 4q^{33} + 8q^{37} + 4q^{41} - 16q^{43} + 4q^{45} + 8q^{47} - 4q^{51} + 4q^{53} + 16q^{55} + 8q^{57} - 16q^{59} + 8q^{61} + 4q^{65} - 16q^{67} + 4q^{69} - 4q^{71} + 8q^{73} - 2q^{75} + 16q^{79} + 2q^{81} - 8q^{83} + 20q^{85} - 4q^{89} - 16q^{93} - 8q^{95} + 24q^{97} + 4q^{99} + O(q^{100}) \)

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
\(5\) 0.585786 0.261972 0.130986 0.991384i \(-0.458186\pi\)
0.130986 + 0.991384i \(0.458186\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −0.828427 −0.249780 −0.124890 0.992171i \(-0.539858\pi\)
−0.124890 + 0.992171i \(0.539858\pi\)
\(12\) 0 0
\(13\) −1.41421 −0.392232 −0.196116 0.980581i \(-0.562833\pi\)
−0.196116 + 0.980581i \(0.562833\pi\)
\(14\) 0 0
\(15\) −0.585786 −0.151249
\(16\) 0 0
\(17\) −2.24264 −0.543920 −0.271960 0.962309i \(-0.587672\pi\)
−0.271960 + 0.962309i \(0.587672\pi\)
\(18\) 0 0
\(19\) −6.82843 −1.56655 −0.783274 0.621676i \(-0.786452\pi\)
−0.783274 + 0.621676i \(0.786452\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.82843 −1.00680 −0.503398 0.864054i \(-0.667917\pi\)
−0.503398 + 0.864054i \(0.667917\pi\)
\(24\) 0 0
\(25\) −4.65685 −0.931371
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −8.48528 −1.57568 −0.787839 0.615882i \(-0.788800\pi\)
−0.787839 + 0.615882i \(0.788800\pi\)
\(30\) 0 0
\(31\) 5.17157 0.928842 0.464421 0.885615i \(-0.346262\pi\)
0.464421 + 0.885615i \(0.346262\pi\)
\(32\) 0 0
\(33\) 0.828427 0.144211
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.65685 −0.272385 −0.136193 0.990682i \(-0.543487\pi\)
−0.136193 + 0.990682i \(0.543487\pi\)
\(38\) 0 0
\(39\) 1.41421 0.226455
\(40\) 0 0
\(41\) 0.585786 0.0914845 0.0457422 0.998953i \(-0.485435\pi\)
0.0457422 + 0.998953i \(0.485435\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 0.585786 0.0873239
\(46\) 0 0
\(47\) 6.82843 0.996028 0.498014 0.867169i \(-0.334063\pi\)
0.498014 + 0.867169i \(0.334063\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 2.24264 0.314033
\(52\) 0 0
\(53\) 13.3137 1.82878 0.914389 0.404836i \(-0.132671\pi\)
0.914389 + 0.404836i \(0.132671\pi\)
\(54\) 0 0
\(55\) −0.485281 −0.0654353
\(56\) 0 0
\(57\) 6.82843 0.904447
\(58\) 0 0
\(59\) −5.17157 −0.673281 −0.336641 0.941633i \(-0.609291\pi\)
−0.336641 + 0.941633i \(0.609291\pi\)
\(60\) 0 0
\(61\) 13.8995 1.77965 0.889824 0.456304i \(-0.150827\pi\)
0.889824 + 0.456304i \(0.150827\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.828427 −0.102754
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) 4.82843 0.581274
\(70\) 0 0
\(71\) 0.828427 0.0983162 0.0491581 0.998791i \(-0.484346\pi\)
0.0491581 + 0.998791i \(0.484346\pi\)
\(72\) 0 0
\(73\) 11.0711 1.29577 0.647885 0.761738i \(-0.275654\pi\)
0.647885 + 0.761738i \(0.275654\pi\)
\(74\) 0 0
\(75\) 4.65685 0.537727
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 2.34315 0.263624 0.131812 0.991275i \(-0.457920\pi\)
0.131812 + 0.991275i \(0.457920\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −15.3137 −1.68090 −0.840449 0.541891i \(-0.817709\pi\)
−0.840449 + 0.541891i \(0.817709\pi\)
\(84\) 0 0
\(85\) −1.31371 −0.142492
\(86\) 0 0
\(87\) 8.48528 0.909718
\(88\) 0 0
\(89\) 10.7279 1.13716 0.568579 0.822629i \(-0.307493\pi\)
0.568579 + 0.822629i \(0.307493\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −5.17157 −0.536267
\(94\) 0 0
\(95\) −4.00000 −0.410391
\(96\) 0 0
\(97\) 7.75736 0.787641 0.393820 0.919187i \(-0.371153\pi\)
0.393820 + 0.919187i \(0.371153\pi\)
\(98\) 0 0
\(99\) −0.828427 −0.0832601
\(100\) 0 0
\(101\) 3.41421 0.339727 0.169863 0.985468i \(-0.445667\pi\)
0.169863 + 0.985468i \(0.445667\pi\)
\(102\) 0 0
\(103\) 10.8284 1.06696 0.533478 0.845814i \(-0.320885\pi\)
0.533478 + 0.845814i \(0.320885\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 14.4853 1.40035 0.700173 0.713974i \(-0.253106\pi\)
0.700173 + 0.713974i \(0.253106\pi\)
\(108\) 0 0
\(109\) 11.3137 1.08366 0.541828 0.840489i \(-0.317732\pi\)
0.541828 + 0.840489i \(0.317732\pi\)
\(110\) 0 0
\(111\) 1.65685 0.157262
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) −2.82843 −0.263752
\(116\) 0 0
\(117\) −1.41421 −0.130744
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −10.3137 −0.937610
\(122\) 0 0
\(123\) −0.585786 −0.0528186
\(124\) 0 0
\(125\) −5.65685 −0.505964
\(126\) 0 0
\(127\) 15.3137 1.35887 0.679436 0.733735i \(-0.262225\pi\)
0.679436 + 0.733735i \(0.262225\pi\)
\(128\) 0 0
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) 7.31371 0.639002 0.319501 0.947586i \(-0.396485\pi\)
0.319501 + 0.947586i \(0.396485\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −0.585786 −0.0504165
\(136\) 0 0
\(137\) 4.48528 0.383203 0.191602 0.981473i \(-0.438632\pi\)
0.191602 + 0.981473i \(0.438632\pi\)
\(138\) 0 0
\(139\) 1.65685 0.140533 0.0702663 0.997528i \(-0.477615\pi\)
0.0702663 + 0.997528i \(0.477615\pi\)
\(140\) 0 0
\(141\) −6.82843 −0.575057
\(142\) 0 0
\(143\) 1.17157 0.0979718
\(144\) 0 0
\(145\) −4.97056 −0.412783
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) −9.65685 −0.785864 −0.392932 0.919568i \(-0.628539\pi\)
−0.392932 + 0.919568i \(0.628539\pi\)
\(152\) 0 0
\(153\) −2.24264 −0.181307
\(154\) 0 0
\(155\) 3.02944 0.243330
\(156\) 0 0
\(157\) −13.8995 −1.10930 −0.554650 0.832084i \(-0.687148\pi\)
−0.554650 + 0.832084i \(0.687148\pi\)
\(158\) 0 0
\(159\) −13.3137 −1.05585
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −13.6569 −1.06969 −0.534844 0.844951i \(-0.679630\pi\)
−0.534844 + 0.844951i \(0.679630\pi\)
\(164\) 0 0
\(165\) 0.485281 0.0377791
\(166\) 0 0
\(167\) 1.17157 0.0906590 0.0453295 0.998972i \(-0.485566\pi\)
0.0453295 + 0.998972i \(0.485566\pi\)
\(168\) 0 0
\(169\) −11.0000 −0.846154
\(170\) 0 0
\(171\) −6.82843 −0.522183
\(172\) 0 0
\(173\) −3.41421 −0.259578 −0.129789 0.991542i \(-0.541430\pi\)
−0.129789 + 0.991542i \(0.541430\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 5.17157 0.388719
\(178\) 0 0
\(179\) 17.7990 1.33036 0.665179 0.746684i \(-0.268355\pi\)
0.665179 + 0.746684i \(0.268355\pi\)
\(180\) 0 0
\(181\) −9.89949 −0.735824 −0.367912 0.929861i \(-0.619927\pi\)
−0.367912 + 0.929861i \(0.619927\pi\)
\(182\) 0 0
\(183\) −13.8995 −1.02748
\(184\) 0 0
\(185\) −0.970563 −0.0713572
\(186\) 0 0
\(187\) 1.85786 0.135860
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −15.1716 −1.09778 −0.548888 0.835896i \(-0.684949\pi\)
−0.548888 + 0.835896i \(0.684949\pi\)
\(192\) 0 0
\(193\) 24.6274 1.77272 0.886360 0.462996i \(-0.153226\pi\)
0.886360 + 0.462996i \(0.153226\pi\)
\(194\) 0 0
\(195\) 0.828427 0.0593249
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) 5.65685 0.401004 0.200502 0.979693i \(-0.435743\pi\)
0.200502 + 0.979693i \(0.435743\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0.343146 0.0239663
\(206\) 0 0
\(207\) −4.82843 −0.335599
\(208\) 0 0
\(209\) 5.65685 0.391293
\(210\) 0 0
\(211\) −14.3431 −0.987423 −0.493711 0.869626i \(-0.664360\pi\)
−0.493711 + 0.869626i \(0.664360\pi\)
\(212\) 0 0
\(213\) −0.828427 −0.0567629
\(214\) 0 0
\(215\) −4.68629 −0.319602
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −11.0711 −0.748113
\(220\) 0 0
\(221\) 3.17157 0.213343
\(222\) 0 0
\(223\) −13.6569 −0.914531 −0.457265 0.889330i \(-0.651171\pi\)
−0.457265 + 0.889330i \(0.651171\pi\)
\(224\) 0 0
\(225\) −4.65685 −0.310457
\(226\) 0 0
\(227\) 19.7990 1.31411 0.657053 0.753845i \(-0.271803\pi\)
0.657053 + 0.753845i \(0.271803\pi\)
\(228\) 0 0
\(229\) 15.0711 0.995924 0.497962 0.867199i \(-0.334082\pi\)
0.497962 + 0.867199i \(0.334082\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −28.4853 −1.86613 −0.933066 0.359704i \(-0.882878\pi\)
−0.933066 + 0.359704i \(0.882878\pi\)
\(234\) 0 0
\(235\) 4.00000 0.260931
\(236\) 0 0
\(237\) −2.34315 −0.152204
\(238\) 0 0
\(239\) 3.17157 0.205152 0.102576 0.994725i \(-0.467292\pi\)
0.102576 + 0.994725i \(0.467292\pi\)
\(240\) 0 0
\(241\) 21.8995 1.41067 0.705335 0.708874i \(-0.250796\pi\)
0.705335 + 0.708874i \(0.250796\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 9.65685 0.614451
\(248\) 0 0
\(249\) 15.3137 0.970467
\(250\) 0 0
\(251\) −8.48528 −0.535586 −0.267793 0.963476i \(-0.586294\pi\)
−0.267793 + 0.963476i \(0.586294\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 0 0
\(255\) 1.31371 0.0822676
\(256\) 0 0
\(257\) 30.2426 1.88648 0.943242 0.332106i \(-0.107759\pi\)
0.943242 + 0.332106i \(0.107759\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −8.48528 −0.525226
\(262\) 0 0
\(263\) −24.8284 −1.53099 −0.765493 0.643444i \(-0.777505\pi\)
−0.765493 + 0.643444i \(0.777505\pi\)
\(264\) 0 0
\(265\) 7.79899 0.479088
\(266\) 0 0
\(267\) −10.7279 −0.656538
\(268\) 0 0
\(269\) 30.0416 1.83167 0.915835 0.401554i \(-0.131530\pi\)
0.915835 + 0.401554i \(0.131530\pi\)
\(270\) 0 0
\(271\) −13.1716 −0.800116 −0.400058 0.916490i \(-0.631010\pi\)
−0.400058 + 0.916490i \(0.631010\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.85786 0.232638
\(276\) 0 0
\(277\) 6.00000 0.360505 0.180253 0.983620i \(-0.442309\pi\)
0.180253 + 0.983620i \(0.442309\pi\)
\(278\) 0 0
\(279\) 5.17157 0.309614
\(280\) 0 0
\(281\) −12.4853 −0.744809 −0.372405 0.928070i \(-0.621467\pi\)
−0.372405 + 0.928070i \(0.621467\pi\)
\(282\) 0 0
\(283\) 14.8284 0.881458 0.440729 0.897640i \(-0.354720\pi\)
0.440729 + 0.897640i \(0.354720\pi\)
\(284\) 0 0
\(285\) 4.00000 0.236940
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −11.9706 −0.704151
\(290\) 0 0
\(291\) −7.75736 −0.454744
\(292\) 0 0
\(293\) 1.07107 0.0625724 0.0312862 0.999510i \(-0.490040\pi\)
0.0312862 + 0.999510i \(0.490040\pi\)
\(294\) 0 0
\(295\) −3.02944 −0.176381
\(296\) 0 0
\(297\) 0.828427 0.0480702
\(298\) 0 0
\(299\) 6.82843 0.394898
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −3.41421 −0.196141
\(304\) 0 0
\(305\) 8.14214 0.466217
\(306\) 0 0
\(307\) 11.5147 0.657180 0.328590 0.944473i \(-0.393427\pi\)
0.328590 + 0.944473i \(0.393427\pi\)
\(308\) 0 0
\(309\) −10.8284 −0.616008
\(310\) 0 0
\(311\) −26.1421 −1.48238 −0.741192 0.671293i \(-0.765739\pi\)
−0.741192 + 0.671293i \(0.765739\pi\)
\(312\) 0 0
\(313\) 17.4142 0.984310 0.492155 0.870508i \(-0.336209\pi\)
0.492155 + 0.870508i \(0.336209\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −21.3137 −1.19710 −0.598549 0.801087i \(-0.704256\pi\)
−0.598549 + 0.801087i \(0.704256\pi\)
\(318\) 0 0
\(319\) 7.02944 0.393573
\(320\) 0 0
\(321\) −14.4853 −0.808490
\(322\) 0 0
\(323\) 15.3137 0.852078
\(324\) 0 0
\(325\) 6.58579 0.365314
\(326\) 0 0
\(327\) −11.3137 −0.625650
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −8.68629 −0.477442 −0.238721 0.971088i \(-0.576728\pi\)
−0.238721 + 0.971088i \(0.576728\pi\)
\(332\) 0 0
\(333\) −1.65685 −0.0907951
\(334\) 0 0
\(335\) −4.68629 −0.256039
\(336\) 0 0
\(337\) −16.9706 −0.924445 −0.462223 0.886764i \(-0.652948\pi\)
−0.462223 + 0.886764i \(0.652948\pi\)
\(338\) 0 0
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) −4.28427 −0.232006
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 2.82843 0.152277
\(346\) 0 0
\(347\) −4.14214 −0.222361 −0.111181 0.993800i \(-0.535463\pi\)
−0.111181 + 0.993800i \(0.535463\pi\)
\(348\) 0 0
\(349\) 30.3848 1.62646 0.813230 0.581943i \(-0.197707\pi\)
0.813230 + 0.581943i \(0.197707\pi\)
\(350\) 0 0
\(351\) 1.41421 0.0754851
\(352\) 0 0
\(353\) 15.8995 0.846245 0.423122 0.906073i \(-0.360934\pi\)
0.423122 + 0.906073i \(0.360934\pi\)
\(354\) 0 0
\(355\) 0.485281 0.0257561
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 35.4558 1.87129 0.935644 0.352945i \(-0.114820\pi\)
0.935644 + 0.352945i \(0.114820\pi\)
\(360\) 0 0
\(361\) 27.6274 1.45407
\(362\) 0 0
\(363\) 10.3137 0.541329
\(364\) 0 0
\(365\) 6.48528 0.339455
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) 0.585786 0.0304948
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −14.6863 −0.760427 −0.380214 0.924899i \(-0.624150\pi\)
−0.380214 + 0.924899i \(0.624150\pi\)
\(374\) 0 0
\(375\) 5.65685 0.292119
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) −0.686292 −0.0352524 −0.0176262 0.999845i \(-0.505611\pi\)
−0.0176262 + 0.999845i \(0.505611\pi\)
\(380\) 0 0
\(381\) −15.3137 −0.784545
\(382\) 0 0
\(383\) 24.9706 1.27594 0.637968 0.770063i \(-0.279775\pi\)
0.637968 + 0.770063i \(0.279775\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −8.00000 −0.406663
\(388\) 0 0
\(389\) 14.1421 0.717035 0.358517 0.933523i \(-0.383282\pi\)
0.358517 + 0.933523i \(0.383282\pi\)
\(390\) 0 0
\(391\) 10.8284 0.547617
\(392\) 0 0
\(393\) −7.31371 −0.368928
\(394\) 0 0
\(395\) 1.37258 0.0690621
\(396\) 0 0
\(397\) 7.27208 0.364975 0.182488 0.983208i \(-0.441585\pi\)
0.182488 + 0.983208i \(0.441585\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −12.4853 −0.623485 −0.311743 0.950167i \(-0.600913\pi\)
−0.311743 + 0.950167i \(0.600913\pi\)
\(402\) 0 0
\(403\) −7.31371 −0.364322
\(404\) 0 0
\(405\) 0.585786 0.0291080
\(406\) 0 0
\(407\) 1.37258 0.0680364
\(408\) 0 0
\(409\) −16.2426 −0.803147 −0.401573 0.915827i \(-0.631537\pi\)
−0.401573 + 0.915827i \(0.631537\pi\)
\(410\) 0 0
\(411\) −4.48528 −0.221243
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −8.97056 −0.440348
\(416\) 0 0
\(417\) −1.65685 −0.0811365
\(418\) 0 0
\(419\) −10.8284 −0.529003 −0.264502 0.964385i \(-0.585207\pi\)
−0.264502 + 0.964385i \(0.585207\pi\)
\(420\) 0 0
\(421\) −6.68629 −0.325870 −0.162935 0.986637i \(-0.552096\pi\)
−0.162935 + 0.986637i \(0.552096\pi\)
\(422\) 0 0
\(423\) 6.82843 0.332009
\(424\) 0 0
\(425\) 10.4437 0.506591
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −1.17157 −0.0565641
\(430\) 0 0
\(431\) 30.4853 1.46842 0.734212 0.678920i \(-0.237552\pi\)
0.734212 + 0.678920i \(0.237552\pi\)
\(432\) 0 0
\(433\) −26.3848 −1.26797 −0.633986 0.773345i \(-0.718582\pi\)
−0.633986 + 0.773345i \(0.718582\pi\)
\(434\) 0 0
\(435\) 4.97056 0.238320
\(436\) 0 0
\(437\) 32.9706 1.57720
\(438\) 0 0
\(439\) −3.31371 −0.158155 −0.0790773 0.996868i \(-0.525197\pi\)
−0.0790773 + 0.996868i \(0.525197\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 38.4853 1.82849 0.914245 0.405161i \(-0.132784\pi\)
0.914245 + 0.405161i \(0.132784\pi\)
\(444\) 0 0
\(445\) 6.28427 0.297903
\(446\) 0 0
\(447\) −10.0000 −0.472984
\(448\) 0 0
\(449\) −10.0000 −0.471929 −0.235965 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 0 0
\(451\) −0.485281 −0.0228510
\(452\) 0 0
\(453\) 9.65685 0.453719
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 20.6274 0.964910 0.482455 0.875921i \(-0.339745\pi\)
0.482455 + 0.875921i \(0.339745\pi\)
\(458\) 0 0
\(459\) 2.24264 0.104678
\(460\) 0 0
\(461\) 18.2426 0.849644 0.424822 0.905277i \(-0.360337\pi\)
0.424822 + 0.905277i \(0.360337\pi\)
\(462\) 0 0
\(463\) −20.9706 −0.974585 −0.487292 0.873239i \(-0.662015\pi\)
−0.487292 + 0.873239i \(0.662015\pi\)
\(464\) 0 0
\(465\) −3.02944 −0.140487
\(466\) 0 0
\(467\) 10.8284 0.501080 0.250540 0.968106i \(-0.419392\pi\)
0.250540 + 0.968106i \(0.419392\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 13.8995 0.640455
\(472\) 0 0
\(473\) 6.62742 0.304729
\(474\) 0 0
\(475\) 31.7990 1.45904
\(476\) 0 0
\(477\) 13.3137 0.609593
\(478\) 0 0
\(479\) −43.1127 −1.96987 −0.984935 0.172926i \(-0.944678\pi\)
−0.984935 + 0.172926i \(0.944678\pi\)
\(480\) 0 0
\(481\) 2.34315 0.106838
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 4.54416 0.206339
\(486\) 0 0
\(487\) 20.9706 0.950267 0.475133 0.879914i \(-0.342400\pi\)
0.475133 + 0.879914i \(0.342400\pi\)
\(488\) 0 0
\(489\) 13.6569 0.617584
\(490\) 0 0
\(491\) 24.1421 1.08952 0.544760 0.838592i \(-0.316621\pi\)
0.544760 + 0.838592i \(0.316621\pi\)
\(492\) 0 0
\(493\) 19.0294 0.857043
\(494\) 0 0
\(495\) −0.485281 −0.0218118
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −28.2843 −1.26618 −0.633089 0.774079i \(-0.718213\pi\)
−0.633089 + 0.774079i \(0.718213\pi\)
\(500\) 0 0
\(501\) −1.17157 −0.0523420
\(502\) 0 0
\(503\) −14.6274 −0.652204 −0.326102 0.945335i \(-0.605735\pi\)
−0.326102 + 0.945335i \(0.605735\pi\)
\(504\) 0 0
\(505\) 2.00000 0.0889988
\(506\) 0 0
\(507\) 11.0000 0.488527
\(508\) 0 0
\(509\) 37.0711 1.64315 0.821573 0.570103i \(-0.193097\pi\)
0.821573 + 0.570103i \(0.193097\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 6.82843 0.301482
\(514\) 0 0
\(515\) 6.34315 0.279512
\(516\) 0 0
\(517\) −5.65685 −0.248788
\(518\) 0 0
\(519\) 3.41421 0.149867
\(520\) 0 0
\(521\) −30.7279 −1.34621 −0.673107 0.739545i \(-0.735041\pi\)
−0.673107 + 0.739545i \(0.735041\pi\)
\(522\) 0 0
\(523\) −9.65685 −0.422265 −0.211132 0.977457i \(-0.567715\pi\)
−0.211132 + 0.977457i \(0.567715\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −11.5980 −0.505216
\(528\) 0 0
\(529\) 0.313708 0.0136395
\(530\) 0 0
\(531\) −5.17157 −0.224427
\(532\) 0 0
\(533\) −0.828427 −0.0358832
\(534\) 0 0
\(535\) 8.48528 0.366851
\(536\) 0 0
\(537\) −17.7990 −0.768083
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −36.6274 −1.57474 −0.787368 0.616483i \(-0.788557\pi\)
−0.787368 + 0.616483i \(0.788557\pi\)
\(542\) 0 0
\(543\) 9.89949 0.424828
\(544\) 0 0
\(545\) 6.62742 0.283887
\(546\) 0 0
\(547\) 28.9706 1.23869 0.619346 0.785118i \(-0.287398\pi\)
0.619346 + 0.785118i \(0.287398\pi\)
\(548\) 0 0
\(549\) 13.8995 0.593216
\(550\) 0 0
\(551\) 57.9411 2.46837
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0.970563 0.0411981
\(556\) 0 0
\(557\) 23.9411 1.01442 0.507209 0.861823i \(-0.330677\pi\)
0.507209 + 0.861823i \(0.330677\pi\)
\(558\) 0 0
\(559\) 11.3137 0.478519
\(560\) 0 0
\(561\) −1.85786 −0.0784391
\(562\) 0 0
\(563\) −21.1716 −0.892275 −0.446138 0.894964i \(-0.647201\pi\)
−0.446138 + 0.894964i \(0.647201\pi\)
\(564\) 0 0
\(565\) 3.51472 0.147865
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1.17157 0.0491149 0.0245574 0.999698i \(-0.492182\pi\)
0.0245574 + 0.999698i \(0.492182\pi\)
\(570\) 0 0
\(571\) −16.2843 −0.681476 −0.340738 0.940158i \(-0.610677\pi\)
−0.340738 + 0.940158i \(0.610677\pi\)
\(572\) 0 0
\(573\) 15.1716 0.633802
\(574\) 0 0
\(575\) 22.4853 0.937701
\(576\) 0 0
\(577\) 6.58579 0.274170 0.137085 0.990559i \(-0.456227\pi\)
0.137085 + 0.990559i \(0.456227\pi\)
\(578\) 0 0
\(579\) −24.6274 −1.02348
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −11.0294 −0.456793
\(584\) 0 0
\(585\) −0.828427 −0.0342512
\(586\) 0 0
\(587\) 21.1716 0.873844 0.436922 0.899499i \(-0.356069\pi\)
0.436922 + 0.899499i \(0.356069\pi\)
\(588\) 0 0
\(589\) −35.3137 −1.45508
\(590\) 0 0
\(591\) 2.00000 0.0822690
\(592\) 0 0
\(593\) −14.9289 −0.613058 −0.306529 0.951861i \(-0.599168\pi\)
−0.306529 + 0.951861i \(0.599168\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −5.65685 −0.231520
\(598\) 0 0
\(599\) 12.1421 0.496114 0.248057 0.968745i \(-0.420208\pi\)
0.248057 + 0.968745i \(0.420208\pi\)
\(600\) 0 0
\(601\) 3.75736 0.153266 0.0766329 0.997059i \(-0.475583\pi\)
0.0766329 + 0.997059i \(0.475583\pi\)
\(602\) 0 0
\(603\) −8.00000 −0.325785
\(604\) 0 0
\(605\) −6.04163 −0.245627
\(606\) 0 0
\(607\) −47.5980 −1.93194 −0.965971 0.258650i \(-0.916722\pi\)
−0.965971 + 0.258650i \(0.916722\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −9.65685 −0.390675
\(612\) 0 0
\(613\) 13.6569 0.551595 0.275798 0.961216i \(-0.411058\pi\)
0.275798 + 0.961216i \(0.411058\pi\)
\(614\) 0 0
\(615\) −0.343146 −0.0138370
\(616\) 0 0
\(617\) −29.4558 −1.18585 −0.592924 0.805259i \(-0.702026\pi\)
−0.592924 + 0.805259i \(0.702026\pi\)
\(618\) 0 0
\(619\) −16.2843 −0.654520 −0.327260 0.944934i \(-0.606125\pi\)
−0.327260 + 0.944934i \(0.606125\pi\)
\(620\) 0 0
\(621\) 4.82843 0.193758
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 19.9706 0.798823
\(626\) 0 0
\(627\) −5.65685 −0.225913
\(628\) 0 0
\(629\) 3.71573 0.148156
\(630\) 0 0
\(631\) −48.2843 −1.92217 −0.961083 0.276259i \(-0.910905\pi\)
−0.961083 + 0.276259i \(0.910905\pi\)
\(632\) 0 0
\(633\) 14.3431 0.570089
\(634\) 0 0
\(635\) 8.97056 0.355986
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0.828427 0.0327721
\(640\) 0 0
\(641\) 30.8284 1.21765 0.608825 0.793305i \(-0.291641\pi\)
0.608825 + 0.793305i \(0.291641\pi\)
\(642\) 0 0
\(643\) −36.4853 −1.43884 −0.719420 0.694576i \(-0.755592\pi\)
−0.719420 + 0.694576i \(0.755592\pi\)
\(644\) 0 0
\(645\) 4.68629 0.184523
\(646\) 0 0
\(647\) 9.17157 0.360572 0.180286 0.983614i \(-0.442298\pi\)
0.180286 + 0.983614i \(0.442298\pi\)
\(648\) 0 0
\(649\) 4.28427 0.168172
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 2.82843 0.110685 0.0553425 0.998467i \(-0.482375\pi\)
0.0553425 + 0.998467i \(0.482375\pi\)
\(654\) 0 0
\(655\) 4.28427 0.167400
\(656\) 0 0
\(657\) 11.0711 0.431923
\(658\) 0 0
\(659\) 42.4853 1.65499 0.827496 0.561472i \(-0.189765\pi\)
0.827496 + 0.561472i \(0.189765\pi\)
\(660\) 0 0
\(661\) 47.3553 1.84191 0.920955 0.389670i \(-0.127411\pi\)
0.920955 + 0.389670i \(0.127411\pi\)
\(662\) 0 0
\(663\) −3.17157 −0.123174
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 40.9706 1.58639
\(668\) 0 0
\(669\) 13.6569 0.528004
\(670\) 0 0
\(671\) −11.5147 −0.444521
\(672\) 0 0
\(673\) −7.31371 −0.281923 −0.140961 0.990015i \(-0.545019\pi\)
−0.140961 + 0.990015i \(0.545019\pi\)
\(674\) 0 0
\(675\) 4.65685 0.179242
\(676\) 0 0
\(677\) 29.5563 1.13594 0.567971 0.823048i \(-0.307728\pi\)
0.567971 + 0.823048i \(0.307728\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −19.7990 −0.758699
\(682\) 0 0
\(683\) −16.1421 −0.617662 −0.308831 0.951117i \(-0.599938\pi\)
−0.308831 + 0.951117i \(0.599938\pi\)
\(684\) 0 0
\(685\) 2.62742 0.100388
\(686\) 0 0
\(687\) −15.0711 −0.574997
\(688\) 0 0
\(689\) −18.8284 −0.717306
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0.970563 0.0368155
\(696\) 0 0
\(697\) −1.31371 −0.0497603
\(698\) 0 0
\(699\) 28.4853 1.07741
\(700\) 0 0
\(701\) 5.17157 0.195328 0.0976638 0.995219i \(-0.468863\pi\)
0.0976638 + 0.995219i \(0.468863\pi\)
\(702\) 0 0
\(703\) 11.3137 0.426705
\(704\) 0 0
\(705\) −4.00000 −0.150649
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 8.00000 0.300446 0.150223 0.988652i \(-0.452001\pi\)
0.150223 + 0.988652i \(0.452001\pi\)
\(710\) 0 0
\(711\) 2.34315 0.0878748
\(712\) 0 0
\(713\) −24.9706 −0.935155
\(714\) 0 0
\(715\) 0.686292 0.0256658
\(716\) 0 0
\(717\) −3.17157 −0.118445
\(718\) 0 0
\(719\) −4.68629 −0.174769 −0.0873846 0.996175i \(-0.527851\pi\)
−0.0873846 + 0.996175i \(0.527851\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −21.8995 −0.814451
\(724\) 0 0
\(725\) 39.5147 1.46754
\(726\) 0 0
\(727\) 25.4558 0.944105 0.472052 0.881570i \(-0.343513\pi\)
0.472052 + 0.881570i \(0.343513\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 17.9411 0.663576
\(732\) 0 0
\(733\) 11.7574 0.434268 0.217134 0.976142i \(-0.430329\pi\)
0.217134 + 0.976142i \(0.430329\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 6.62742 0.244124
\(738\) 0 0
\(739\) 20.2843 0.746169 0.373084 0.927797i \(-0.378300\pi\)
0.373084 + 0.927797i \(0.378300\pi\)
\(740\) 0 0
\(741\) −9.65685 −0.354753
\(742\) 0 0
\(743\) 11.1716 0.409845 0.204923 0.978778i \(-0.434306\pi\)
0.204923 + 0.978778i \(0.434306\pi\)
\(744\) 0 0
\(745\) 5.85786 0.214616
\(746\) 0 0
\(747\) −15.3137 −0.560299
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −29.6569 −1.08219 −0.541097 0.840960i \(-0.681991\pi\)
−0.541097 + 0.840960i \(0.681991\pi\)
\(752\) 0 0
\(753\) 8.48528 0.309221
\(754\) 0 0
\(755\) −5.65685 −0.205874
\(756\) 0 0
\(757\) 11.3137 0.411204 0.205602 0.978636i \(-0.434085\pi\)
0.205602 + 0.978636i \(0.434085\pi\)
\(758\) 0 0
\(759\) −4.00000 −0.145191
\(760\) 0 0
\(761\) 9.75736 0.353704 0.176852 0.984237i \(-0.443409\pi\)
0.176852 + 0.984237i \(0.443409\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −1.31371 −0.0474972
\(766\) 0 0
\(767\) 7.31371 0.264083
\(768\) 0 0
\(769\) −15.0711 −0.543477 −0.271738 0.962371i \(-0.587599\pi\)
−0.271738 + 0.962371i \(0.587599\pi\)
\(770\) 0 0
\(771\) −30.2426 −1.08916
\(772\) 0 0
\(773\) −3.41421 −0.122801 −0.0614004 0.998113i \(-0.519557\pi\)
−0.0614004 + 0.998113i \(0.519557\pi\)
\(774\) 0 0
\(775\) −24.0833 −0.865096
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −4.00000 −0.143315
\(780\) 0 0
\(781\) −0.686292 −0.0245574
\(782\) 0 0
\(783\) 8.48528 0.303239
\(784\) 0 0
\(785\) −8.14214 −0.290605
\(786\) 0 0
\(787\) 10.6274 0.378827 0.189413 0.981897i \(-0.439341\pi\)
0.189413 + 0.981897i \(0.439341\pi\)
\(788\) 0 0
\(789\) 24.8284 0.883915
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −19.6569 −0.698035
\(794\) 0 0
\(795\) −7.79899 −0.276602
\(796\) 0 0
\(797\) −43.2132 −1.53069 −0.765345 0.643620i \(-0.777432\pi\)
−0.765345 + 0.643620i \(0.777432\pi\)
\(798\) 0 0
\(799\) −15.3137 −0.541760
\(800\) 0 0
\(801\) 10.7279 0.379052
\(802\) 0 0
\(803\) −9.17157 −0.323658
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −30.0416 −1.05752
\(808\) 0 0
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) 20.9706 0.736376 0.368188 0.929751i \(-0.379978\pi\)
0.368188 + 0.929751i \(0.379978\pi\)
\(812\) 0 0
\(813\) 13.1716 0.461947
\(814\) 0 0
\(815\) −8.00000 −0.280228
\(816\) 0 0
\(817\) 54.6274 1.91117
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −10.0000 −0.349002 −0.174501 0.984657i \(-0.555831\pi\)
−0.174501 + 0.984657i \(0.555831\pi\)
\(822\) 0 0
\(823\) 2.34315 0.0816769 0.0408385 0.999166i \(-0.486997\pi\)
0.0408385 + 0.999166i \(0.486997\pi\)
\(824\) 0 0
\(825\) −3.85786 −0.134314
\(826\) 0 0
\(827\) −19.4558 −0.676546 −0.338273 0.941048i \(-0.609843\pi\)
−0.338273 + 0.941048i \(0.609843\pi\)
\(828\) 0 0
\(829\) 3.27208 0.113644 0.0568220 0.998384i \(-0.481903\pi\)
0.0568220 + 0.998384i \(0.481903\pi\)
\(830\) 0 0
\(831\) −6.00000 −0.208138
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0.686292 0.0237501
\(836\) 0 0
\(837\) −5.17157 −0.178756
\(838\) 0 0
\(839\) 49.1716 1.69759 0.848796 0.528721i \(-0.177328\pi\)
0.848796 + 0.528721i \(0.177328\pi\)
\(840\) 0 0
\(841\) 43.0000 1.48276
\(842\) 0 0
\(843\) 12.4853 0.430016
\(844\) 0 0
\(845\) −6.44365 −0.221668
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −14.8284 −0.508910
\(850\) 0 0
\(851\) 8.00000 0.274236
\(852\) 0 0
\(853\) 36.0416 1.23404 0.617021 0.786947i \(-0.288339\pi\)
0.617021 + 0.786947i \(0.288339\pi\)
\(854\) 0 0
\(855\) −4.00000 −0.136797
\(856\) 0 0
\(857\) 43.6985 1.49271 0.746356 0.665547i \(-0.231802\pi\)
0.746356 + 0.665547i \(0.231802\pi\)
\(858\) 0 0
\(859\) 38.8284 1.32481 0.662404 0.749146i \(-0.269536\pi\)
0.662404 + 0.749146i \(0.269536\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 29.5147 1.00469 0.502346 0.864666i \(-0.332470\pi\)
0.502346 + 0.864666i \(0.332470\pi\)
\(864\) 0 0
\(865\) −2.00000 −0.0680020
\(866\) 0 0
\(867\) 11.9706 0.406542
\(868\) 0 0
\(869\) −1.94113 −0.0658482
\(870\) 0 0
\(871\) 11.3137 0.383350
\(872\) 0 0
\(873\) 7.75736 0.262547
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −20.2843 −0.684951 −0.342476 0.939527i \(-0.611265\pi\)
−0.342476 + 0.939527i \(0.611265\pi\)
\(878\) 0 0
\(879\) −1.07107 −0.0361262
\(880\) 0 0
\(881\) 25.0711 0.844666 0.422333 0.906441i \(-0.361211\pi\)
0.422333 + 0.906441i \(0.361211\pi\)
\(882\) 0 0
\(883\) 18.3431 0.617296 0.308648 0.951176i \(-0.400123\pi\)
0.308648 + 0.951176i \(0.400123\pi\)
\(884\) 0 0
\(885\) 3.02944 0.101833
\(886\) 0 0
\(887\) 21.4558 0.720417 0.360208 0.932872i \(-0.382706\pi\)
0.360208 + 0.932872i \(0.382706\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −0.828427 −0.0277534
\(892\) 0 0
\(893\) −46.6274 −1.56033
\(894\) 0 0
\(895\) 10.4264 0.348516
\(896\) 0 0
\(897\) −6.82843 −0.227995
\(898\) 0 0
\(899\) −43.8823 −1.46356
\(900\) 0 0
\(901\) −29.8579 −0.994710
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −5.79899 −0.192765
\(906\) 0 0
\(907\) −7.02944 −0.233409 −0.116704 0.993167i \(-0.537233\pi\)
−0.116704 + 0.993167i \(0.537233\pi\)
\(908\) 0 0
\(909\) 3.41421 0.113242
\(910\) 0 0
\(911\) 54.4853 1.80518 0.902589 0.430503i \(-0.141664\pi\)
0.902589 + 0.430503i \(0.141664\pi\)
\(912\) 0 0
\(913\) 12.6863 0.419855
\(914\) 0 0
\(915\) −8.14214 −0.269171
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 49.2548 1.62477 0.812384 0.583123i \(-0.198170\pi\)
0.812384 + 0.583123i \(0.198170\pi\)
\(920\) 0 0
\(921\) −11.5147 −0.379423
\(922\) 0 0
\(923\) −1.17157 −0.0385628
\(924\) 0 0
\(925\) 7.71573 0.253692
\(926\) 0 0
\(927\) 10.8284 0.355652
\(928\) 0 0
\(929\) −60.8701 −1.99708 −0.998541 0.0540006i \(-0.982803\pi\)
−0.998541 + 0.0540006i \(0.982803\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 26.1421 0.855855
\(934\) 0 0
\(935\) 1.08831 0.0355916
\(936\) 0 0
\(937\) 6.10051 0.199295 0.0996474 0.995023i \(-0.468229\pi\)
0.0996474 + 0.995023i \(0.468229\pi\)
\(938\) 0 0
\(939\) −17.4142 −0.568291
\(940\) 0 0
\(941\) −43.8995 −1.43108 −0.715541 0.698570i \(-0.753820\pi\)
−0.715541 + 0.698570i \(0.753820\pi\)
\(942\) 0 0
\(943\) −2.82843 −0.0921063
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 20.1421 0.654531 0.327266 0.944932i \(-0.393873\pi\)
0.327266 + 0.944932i \(0.393873\pi\)
\(948\) 0 0
\(949\) −15.6569 −0.508243
\(950\) 0 0
\(951\) 21.3137 0.691144
\(952\) 0 0
\(953\) −7.37258 −0.238821 −0.119411 0.992845i \(-0.538101\pi\)
−0.119411 + 0.992845i \(0.538101\pi\)
\(954\) 0 0
\(955\) −8.88730 −0.287586
\(956\) 0 0
\(957\) −7.02944 −0.227229
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −4.25483 −0.137253
\(962\) 0 0
\(963\) 14.4853 0.466782
\(964\) 0 0
\(965\) 14.4264 0.464402
\(966\) 0 0
\(967\) 34.6274 1.11354 0.556771 0.830666i \(-0.312040\pi\)
0.556771 + 0.830666i \(0.312040\pi\)
\(968\) 0 0
\(969\) −15.3137 −0.491947
\(970\) 0 0
\(971\) 57.2548 1.83740 0.918698 0.394962i \(-0.129242\pi\)
0.918698 + 0.394962i \(0.129242\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −6.58579 −0.210914
\(976\) 0 0
\(977\) 21.8579 0.699295 0.349648 0.936881i \(-0.386301\pi\)
0.349648 + 0.936881i \(0.386301\pi\)
\(978\) 0 0
\(979\) −8.88730 −0.284039
\(980\) 0 0
\(981\) 11.3137 0.361219
\(982\) 0 0
\(983\) −14.6274 −0.466542 −0.233271 0.972412i \(-0.574943\pi\)
−0.233271 + 0.972412i \(0.574943\pi\)
\(984\) 0 0
\(985\) −1.17157 −0.0373294
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 38.6274 1.22828
\(990\) 0 0
\(991\) −23.3137 −0.740584 −0.370292 0.928915i \(-0.620742\pi\)
−0.370292 + 0.928915i \(0.620742\pi\)
\(992\) 0 0
\(993\) 8.68629 0.275651
\(994\) 0 0
\(995\) 3.31371 0.105052
\(996\) 0 0
\(997\) −34.5858 −1.09534 −0.547671 0.836693i \(-0.684486\pi\)
−0.547671 + 0.836693i \(0.684486\pi\)
\(998\) 0 0
\(999\) 1.65685 0.0524205
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.ds.1.1 2
4.3 odd 2 9408.2.a.ee.1.1 2
7.6 odd 2 9408.2.a.du.1.2 2
8.3 odd 2 1176.2.a.j.1.2 2
8.5 even 2 2352.2.a.bd.1.2 2
24.5 odd 2 7056.2.a.cx.1.1 2
24.11 even 2 3528.2.a.bl.1.1 2
28.27 even 2 9408.2.a.dg.1.2 2
56.3 even 6 1176.2.q.k.961.2 4
56.5 odd 6 2352.2.q.be.1537.2 4
56.11 odd 6 1176.2.q.o.961.1 4
56.13 odd 2 2352.2.a.bb.1.1 2
56.19 even 6 1176.2.q.k.361.2 4
56.27 even 2 1176.2.a.o.1.1 yes 2
56.37 even 6 2352.2.q.bc.1537.1 4
56.45 odd 6 2352.2.q.be.961.2 4
56.51 odd 6 1176.2.q.o.361.1 4
56.53 even 6 2352.2.q.bc.961.1 4
168.11 even 6 3528.2.s.bd.3313.2 4
168.59 odd 6 3528.2.s.bm.3313.1 4
168.83 odd 2 3528.2.a.bb.1.2 2
168.107 even 6 3528.2.s.bd.361.2 4
168.125 even 2 7056.2.a.cg.1.2 2
168.131 odd 6 3528.2.s.bm.361.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1176.2.a.j.1.2 2 8.3 odd 2
1176.2.a.o.1.1 yes 2 56.27 even 2
1176.2.q.k.361.2 4 56.19 even 6
1176.2.q.k.961.2 4 56.3 even 6
1176.2.q.o.361.1 4 56.51 odd 6
1176.2.q.o.961.1 4 56.11 odd 6
2352.2.a.bb.1.1 2 56.13 odd 2
2352.2.a.bd.1.2 2 8.5 even 2
2352.2.q.bc.961.1 4 56.53 even 6
2352.2.q.bc.1537.1 4 56.37 even 6
2352.2.q.be.961.2 4 56.45 odd 6
2352.2.q.be.1537.2 4 56.5 odd 6
3528.2.a.bb.1.2 2 168.83 odd 2
3528.2.a.bl.1.1 2 24.11 even 2
3528.2.s.bd.361.2 4 168.107 even 6
3528.2.s.bd.3313.2 4 168.11 even 6
3528.2.s.bm.361.1 4 168.131 odd 6
3528.2.s.bm.3313.1 4 168.59 odd 6
7056.2.a.cg.1.2 2 168.125 even 2
7056.2.a.cx.1.1 2 24.5 odd 2
9408.2.a.dg.1.2 2 28.27 even 2
9408.2.a.ds.1.1 2 1.1 even 1 trivial
9408.2.a.du.1.2 2 7.6 odd 2
9408.2.a.ee.1.1 2 4.3 odd 2