Properties

Label 9216.2.a.be.1.1
Level $9216$
Weight $2$
Character 9216.1
Self dual yes
Analytic conductor $73.590$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 9216 = 2^{10} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9216.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(73.5901305028\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
Defining polynomial: \(x^{4} - 4 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 2304)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.517638\) of defining polynomial
Character \(\chi\) \(=\) 9216.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.46410 q^{5} -1.41421 q^{7} +O(q^{10})\) \(q-3.46410 q^{5} -1.41421 q^{7} -4.89898 q^{11} -1.41421 q^{13} +4.89898 q^{17} -6.00000 q^{19} +6.92820 q^{23} +7.00000 q^{25} +3.46410 q^{29} +1.41421 q^{31} +4.89898 q^{35} +9.89949 q^{37} +4.89898 q^{41} -6.00000 q^{43} +6.92820 q^{47} -5.00000 q^{49} -3.46410 q^{53} +16.9706 q^{55} -9.79796 q^{59} +7.07107 q^{61} +4.89898 q^{65} +8.00000 q^{67} +13.8564 q^{71} -12.0000 q^{73} +6.92820 q^{77} -15.5563 q^{79} +14.6969 q^{83} -16.9706 q^{85} +9.79796 q^{89} +2.00000 q^{91} +20.7846 q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 24q^{19} + 28q^{25} - 24q^{43} - 20q^{49} + 32q^{67} - 48q^{73} + 8q^{91} + O(q^{100}) \)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −3.46410 −1.54919 −0.774597 0.632456i \(-0.782047\pi\)
−0.774597 + 0.632456i \(0.782047\pi\)
\(6\) 0 0
\(7\) −1.41421 −0.534522 −0.267261 0.963624i \(-0.586119\pi\)
−0.267261 + 0.963624i \(0.586119\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.89898 −1.47710 −0.738549 0.674200i \(-0.764489\pi\)
−0.738549 + 0.674200i \(0.764489\pi\)
\(12\) 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 0
\(16\) 0 0
\(17\) 4.89898 1.18818 0.594089 0.804400i \(-0.297513\pi\)
0.594089 + 0.804400i \(0.297513\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.92820 1.44463 0.722315 0.691564i \(-0.243078\pi\)
0.722315 + 0.691564i \(0.243078\pi\)
\(24\) 0 0
\(25\) 7.00000 1.40000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.46410 0.643268 0.321634 0.946864i \(-0.395768\pi\)
0.321634 + 0.946864i \(0.395768\pi\)
\(30\) 0 0
\(31\) 1.41421 0.254000 0.127000 0.991903i \(-0.459465\pi\)
0.127000 + 0.991903i \(0.459465\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.89898 0.828079
\(36\) 0 0
\(37\) 9.89949 1.62747 0.813733 0.581238i \(-0.197432\pi\)
0.813733 + 0.581238i \(0.197432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.89898 0.765092 0.382546 0.923936i \(-0.375047\pi\)
0.382546 + 0.923936i \(0.375047\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.92820 1.01058 0.505291 0.862949i \(-0.331385\pi\)
0.505291 + 0.862949i \(0.331385\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −3.46410 −0.475831 −0.237915 0.971286i \(-0.576464\pi\)
−0.237915 + 0.971286i \(0.576464\pi\)
\(54\) 0 0
\(55\) 16.9706 2.28831
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −9.79796 −1.27559 −0.637793 0.770208i \(-0.720152\pi\)
−0.637793 + 0.770208i \(0.720152\pi\)
\(60\) 0 0
\(61\) 7.07107 0.905357 0.452679 0.891674i \(-0.350468\pi\)
0.452679 + 0.891674i \(0.350468\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.89898 0.607644
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 13.8564 1.64445 0.822226 0.569160i \(-0.192732\pi\)
0.822226 + 0.569160i \(0.192732\pi\)
\(72\) 0 0
\(73\) −12.0000 −1.40449 −0.702247 0.711934i \(-0.747820\pi\)
−0.702247 + 0.711934i \(0.747820\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.92820 0.789542
\(78\) 0 0
\(79\) −15.5563 −1.75023 −0.875113 0.483919i \(-0.839213\pi\)
−0.875113 + 0.483919i \(0.839213\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 14.6969 1.61320 0.806599 0.591099i \(-0.201306\pi\)
0.806599 + 0.591099i \(0.201306\pi\)
\(84\) 0 0
\(85\) −16.9706 −1.84072
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 9.79796 1.03858 0.519291 0.854598i \(-0.326196\pi\)
0.519291 + 0.854598i \(0.326196\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 20.7846 2.13246
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 10.3923 1.03407 0.517036 0.855963i \(-0.327035\pi\)
0.517036 + 0.855963i \(0.327035\pi\)
\(102\) 0 0
\(103\) −7.07107 −0.696733 −0.348367 0.937358i \(-0.613264\pi\)
−0.348367 + 0.937358i \(0.613264\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 1.41421 0.135457 0.0677285 0.997704i \(-0.478425\pi\)
0.0677285 + 0.997704i \(0.478425\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −19.5959 −1.84343 −0.921714 0.387869i \(-0.873211\pi\)
−0.921714 + 0.387869i \(0.873211\pi\)
\(114\) 0 0
\(115\) −24.0000 −2.23801
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −6.92820 −0.635107
\(120\) 0 0
\(121\) 13.0000 1.18182
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −6.92820 −0.619677
\(126\) 0 0
\(127\) −9.89949 −0.878438 −0.439219 0.898380i \(-0.644745\pi\)
−0.439219 + 0.898380i \(0.644745\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 9.79796 0.856052 0.428026 0.903767i \(-0.359209\pi\)
0.428026 + 0.903767i \(0.359209\pi\)
\(132\) 0 0
\(133\) 8.48528 0.735767
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −14.6969 −1.25564 −0.627822 0.778357i \(-0.716053\pi\)
−0.627822 + 0.778357i \(0.716053\pi\)
\(138\) 0 0
\(139\) −16.0000 −1.35710 −0.678551 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 6.92820 0.579365
\(144\) 0 0
\(145\) −12.0000 −0.996546
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3.46410 0.283790 0.141895 0.989882i \(-0.454680\pi\)
0.141895 + 0.989882i \(0.454680\pi\)
\(150\) 0 0
\(151\) 15.5563 1.26596 0.632979 0.774169i \(-0.281832\pi\)
0.632979 + 0.774169i \(0.281832\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −4.89898 −0.393496
\(156\) 0 0
\(157\) 9.89949 0.790066 0.395033 0.918667i \(-0.370733\pi\)
0.395033 + 0.918667i \(0.370733\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −9.79796 −0.772187
\(162\) 0 0
\(163\) −6.00000 −0.469956 −0.234978 0.972001i \(-0.575502\pi\)
−0.234978 + 0.972001i \(0.575502\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −6.92820 −0.536120 −0.268060 0.963402i \(-0.586383\pi\)
−0.268060 + 0.963402i \(0.586383\pi\)
\(168\) 0 0
\(169\) −11.0000 −0.846154
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −17.3205 −1.31685 −0.658427 0.752645i \(-0.728778\pi\)
−0.658427 + 0.752645i \(0.728778\pi\)
\(174\) 0 0
\(175\) −9.89949 −0.748331
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 1.41421 0.105118 0.0525588 0.998618i \(-0.483262\pi\)
0.0525588 + 0.998618i \(0.483262\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −34.2929 −2.52126
\(186\) 0 0
\(187\) −24.0000 −1.75505
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −20.7846 −1.50392 −0.751961 0.659208i \(-0.770892\pi\)
−0.751961 + 0.659208i \(0.770892\pi\)
\(192\) 0 0
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −3.46410 −0.246807 −0.123404 0.992357i \(-0.539381\pi\)
−0.123404 + 0.992357i \(0.539381\pi\)
\(198\) 0 0
\(199\) 7.07107 0.501255 0.250627 0.968084i \(-0.419363\pi\)
0.250627 + 0.968084i \(0.419363\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −4.89898 −0.343841
\(204\) 0 0
\(205\) −16.9706 −1.18528
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 29.3939 2.03322
\(210\) 0 0
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 20.7846 1.41750
\(216\) 0 0
\(217\) −2.00000 −0.135769
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −6.92820 −0.466041
\(222\) 0 0
\(223\) −7.07107 −0.473514 −0.236757 0.971569i \(-0.576084\pi\)
−0.236757 + 0.971569i \(0.576084\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −4.89898 −0.325157 −0.162578 0.986696i \(-0.551981\pi\)
−0.162578 + 0.986696i \(0.551981\pi\)
\(228\) 0 0
\(229\) −1.41421 −0.0934539 −0.0467269 0.998908i \(-0.514879\pi\)
−0.0467269 + 0.998908i \(0.514879\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 9.79796 0.641886 0.320943 0.947099i \(-0.396000\pi\)
0.320943 + 0.947099i \(0.396000\pi\)
\(234\) 0 0
\(235\) −24.0000 −1.56559
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −6.92820 −0.448148 −0.224074 0.974572i \(-0.571936\pi\)
−0.224074 + 0.974572i \(0.571936\pi\)
\(240\) 0 0
\(241\) 22.0000 1.41714 0.708572 0.705638i \(-0.249340\pi\)
0.708572 + 0.705638i \(0.249340\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 17.3205 1.10657
\(246\) 0 0
\(247\) 8.48528 0.539906
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 24.4949 1.54610 0.773052 0.634343i \(-0.218729\pi\)
0.773052 + 0.634343i \(0.218729\pi\)
\(252\) 0 0
\(253\) −33.9411 −2.13386
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −9.79796 −0.611180 −0.305590 0.952163i \(-0.598854\pi\)
−0.305590 + 0.952163i \(0.598854\pi\)
\(258\) 0 0
\(259\) −14.0000 −0.869918
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −13.8564 −0.854423 −0.427211 0.904152i \(-0.640504\pi\)
−0.427211 + 0.904152i \(0.640504\pi\)
\(264\) 0 0
\(265\) 12.0000 0.737154
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −10.3923 −0.633630 −0.316815 0.948487i \(-0.602613\pi\)
−0.316815 + 0.948487i \(0.602613\pi\)
\(270\) 0 0
\(271\) −7.07107 −0.429537 −0.214768 0.976665i \(-0.568900\pi\)
−0.214768 + 0.976665i \(0.568900\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −34.2929 −2.06794
\(276\) 0 0
\(277\) −1.41421 −0.0849719 −0.0424859 0.999097i \(-0.513528\pi\)
−0.0424859 + 0.999097i \(0.513528\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 29.3939 1.75349 0.876746 0.480954i \(-0.159710\pi\)
0.876746 + 0.480954i \(0.159710\pi\)
\(282\) 0 0
\(283\) −8.00000 −0.475551 −0.237775 0.971320i \(-0.576418\pi\)
−0.237775 + 0.971320i \(0.576418\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −6.92820 −0.408959
\(288\) 0 0
\(289\) 7.00000 0.411765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −17.3205 −1.01187 −0.505937 0.862570i \(-0.668853\pi\)
−0.505937 + 0.862570i \(0.668853\pi\)
\(294\) 0 0
\(295\) 33.9411 1.97613
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −9.79796 −0.566631
\(300\) 0 0
\(301\) 8.48528 0.489083
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −24.4949 −1.40257
\(306\) 0 0
\(307\) 8.00000 0.456584 0.228292 0.973593i \(-0.426686\pi\)
0.228292 + 0.973593i \(0.426686\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −13.8564 −0.785725 −0.392862 0.919597i \(-0.628515\pi\)
−0.392862 + 0.919597i \(0.628515\pi\)
\(312\) 0 0
\(313\) −14.0000 −0.791327 −0.395663 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 10.3923 0.583690 0.291845 0.956466i \(-0.405731\pi\)
0.291845 + 0.956466i \(0.405731\pi\)
\(318\) 0 0
\(319\) −16.9706 −0.950169
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −29.3939 −1.63552
\(324\) 0 0
\(325\) −9.89949 −0.549125
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −9.79796 −0.540179
\(330\) 0 0
\(331\) 16.0000 0.879440 0.439720 0.898135i \(-0.355078\pi\)
0.439720 + 0.898135i \(0.355078\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −27.7128 −1.51411
\(336\) 0 0
\(337\) 12.0000 0.653682 0.326841 0.945079i \(-0.394016\pi\)
0.326841 + 0.945079i \(0.394016\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −6.92820 −0.375183
\(342\) 0 0
\(343\) 16.9706 0.916324
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 4.89898 0.262991 0.131495 0.991317i \(-0.458022\pi\)
0.131495 + 0.991317i \(0.458022\pi\)
\(348\) 0 0
\(349\) 26.8701 1.43832 0.719161 0.694844i \(-0.244527\pi\)
0.719161 + 0.694844i \(0.244527\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 9.79796 0.521493 0.260746 0.965407i \(-0.416031\pi\)
0.260746 + 0.965407i \(0.416031\pi\)
\(354\) 0 0
\(355\) −48.0000 −2.54758
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 41.5692 2.17583
\(366\) 0 0
\(367\) −32.5269 −1.69789 −0.848945 0.528480i \(-0.822762\pi\)
−0.848945 + 0.528480i \(0.822762\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 4.89898 0.254342
\(372\) 0 0
\(373\) 26.8701 1.39128 0.695639 0.718391i \(-0.255121\pi\)
0.695639 + 0.718391i \(0.255121\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −4.89898 −0.252310
\(378\) 0 0
\(379\) −6.00000 −0.308199 −0.154100 0.988055i \(-0.549248\pi\)
−0.154100 + 0.988055i \(0.549248\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −6.92820 −0.354015 −0.177007 0.984210i \(-0.556642\pi\)
−0.177007 + 0.984210i \(0.556642\pi\)
\(384\) 0 0
\(385\) −24.0000 −1.22315
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 24.2487 1.22946 0.614729 0.788738i \(-0.289265\pi\)
0.614729 + 0.788738i \(0.289265\pi\)
\(390\) 0 0
\(391\) 33.9411 1.71648
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 53.8888 2.71144
\(396\) 0 0
\(397\) 9.89949 0.496841 0.248421 0.968652i \(-0.420088\pi\)
0.248421 + 0.968652i \(0.420088\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 14.6969 0.733930 0.366965 0.930235i \(-0.380397\pi\)
0.366965 + 0.930235i \(0.380397\pi\)
\(402\) 0 0
\(403\) −2.00000 −0.0996271
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −48.4974 −2.40393
\(408\) 0 0
\(409\) −24.0000 −1.18672 −0.593362 0.804936i \(-0.702200\pi\)
−0.593362 + 0.804936i \(0.702200\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 13.8564 0.681829
\(414\) 0 0
\(415\) −50.9117 −2.49916
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −14.6969 −0.717992 −0.358996 0.933339i \(-0.616881\pi\)
−0.358996 + 0.933339i \(0.616881\pi\)
\(420\) 0 0
\(421\) −35.3553 −1.72311 −0.861557 0.507661i \(-0.830510\pi\)
−0.861557 + 0.507661i \(0.830510\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 34.2929 1.66345
\(426\) 0 0
\(427\) −10.0000 −0.483934
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −6.92820 −0.333720 −0.166860 0.985981i \(-0.553363\pi\)
−0.166860 + 0.985981i \(0.553363\pi\)
\(432\) 0 0
\(433\) −24.0000 −1.15337 −0.576683 0.816968i \(-0.695653\pi\)
−0.576683 + 0.816968i \(0.695653\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −41.5692 −1.98853
\(438\) 0 0
\(439\) −26.8701 −1.28244 −0.641219 0.767358i \(-0.721571\pi\)
−0.641219 + 0.767358i \(0.721571\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −14.6969 −0.698273 −0.349136 0.937072i \(-0.613525\pi\)
−0.349136 + 0.937072i \(0.613525\pi\)
\(444\) 0 0
\(445\) −33.9411 −1.60896
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 34.2929 1.61838 0.809190 0.587547i \(-0.199906\pi\)
0.809190 + 0.587547i \(0.199906\pi\)
\(450\) 0 0
\(451\) −24.0000 −1.13012
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −6.92820 −0.324799
\(456\) 0 0
\(457\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3.46410 0.161339 0.0806696 0.996741i \(-0.474294\pi\)
0.0806696 + 0.996741i \(0.474294\pi\)
\(462\) 0 0
\(463\) 24.0416 1.11731 0.558655 0.829400i \(-0.311318\pi\)
0.558655 + 0.829400i \(0.311318\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −4.89898 −0.226698 −0.113349 0.993555i \(-0.536158\pi\)
−0.113349 + 0.993555i \(0.536158\pi\)
\(468\) 0 0
\(469\) −11.3137 −0.522419
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 29.3939 1.35153
\(474\) 0 0
\(475\) −42.0000 −1.92709
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −27.7128 −1.26623 −0.633115 0.774057i \(-0.718224\pi\)
−0.633115 + 0.774057i \(0.718224\pi\)
\(480\) 0 0
\(481\) −14.0000 −0.638345
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 35.3553 1.60210 0.801052 0.598595i \(-0.204274\pi\)
0.801052 + 0.598595i \(0.204274\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 29.3939 1.32653 0.663264 0.748386i \(-0.269171\pi\)
0.663264 + 0.748386i \(0.269171\pi\)
\(492\) 0 0
\(493\) 16.9706 0.764316
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −19.5959 −0.878997
\(498\) 0 0
\(499\) −8.00000 −0.358129 −0.179065 0.983837i \(-0.557307\pi\)
−0.179065 + 0.983837i \(0.557307\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) −36.0000 −1.60198
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 10.3923 0.460631 0.230315 0.973116i \(-0.426024\pi\)
0.230315 + 0.973116i \(0.426024\pi\)
\(510\) 0 0
\(511\) 16.9706 0.750733
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 24.4949 1.07937
\(516\) 0 0
\(517\) −33.9411 −1.49273
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −14.6969 −0.643885 −0.321942 0.946759i \(-0.604336\pi\)
−0.321942 + 0.946759i \(0.604336\pi\)
\(522\) 0 0
\(523\) −30.0000 −1.31181 −0.655904 0.754844i \(-0.727712\pi\)
−0.655904 + 0.754844i \(0.727712\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.92820 0.301797
\(528\) 0 0
\(529\) 25.0000 1.08696
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −6.92820 −0.300094
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 24.4949 1.05507
\(540\) 0 0
\(541\) 15.5563 0.668820 0.334410 0.942428i \(-0.391463\pi\)
0.334410 + 0.942428i \(0.391463\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −4.89898 −0.209849
\(546\) 0 0
\(547\) −18.0000 −0.769624 −0.384812 0.922995i \(-0.625734\pi\)
−0.384812 + 0.922995i \(0.625734\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −20.7846 −0.885454
\(552\) 0 0
\(553\) 22.0000 0.935535
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −10.3923 −0.440336 −0.220168 0.975462i \(-0.570661\pi\)
−0.220168 + 0.975462i \(0.570661\pi\)
\(558\) 0 0
\(559\) 8.48528 0.358889
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 4.89898 0.206467 0.103234 0.994657i \(-0.467081\pi\)
0.103234 + 0.994657i \(0.467081\pi\)
\(564\) 0 0
\(565\) 67.8823 2.85583
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −14.6969 −0.616128 −0.308064 0.951366i \(-0.599681\pi\)
−0.308064 + 0.951366i \(0.599681\pi\)
\(570\) 0 0
\(571\) −40.0000 −1.67395 −0.836974 0.547243i \(-0.815677\pi\)
−0.836974 + 0.547243i \(0.815677\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 48.4974 2.02248
\(576\) 0 0
\(577\) 12.0000 0.499567 0.249783 0.968302i \(-0.419641\pi\)
0.249783 + 0.968302i \(0.419641\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −20.7846 −0.862291
\(582\) 0 0
\(583\) 16.9706 0.702849
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −39.1918 −1.61762 −0.808810 0.588070i \(-0.799888\pi\)
−0.808810 + 0.588070i \(0.799888\pi\)
\(588\) 0 0
\(589\) −8.48528 −0.349630
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 39.1918 1.60942 0.804708 0.593671i \(-0.202322\pi\)
0.804708 + 0.593671i \(0.202322\pi\)
\(594\) 0 0
\(595\) 24.0000 0.983904
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −6.92820 −0.283079 −0.141539 0.989933i \(-0.545205\pi\)
−0.141539 + 0.989933i \(0.545205\pi\)
\(600\) 0 0
\(601\) 12.0000 0.489490 0.244745 0.969587i \(-0.421296\pi\)
0.244745 + 0.969587i \(0.421296\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −45.0333 −1.83086
\(606\) 0 0
\(607\) −9.89949 −0.401808 −0.200904 0.979611i \(-0.564388\pi\)
−0.200904 + 0.979611i \(0.564388\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −9.79796 −0.396383
\(612\) 0 0
\(613\) 26.8701 1.08527 0.542636 0.839968i \(-0.317426\pi\)
0.542636 + 0.839968i \(0.317426\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −19.5959 −0.788902 −0.394451 0.918917i \(-0.629065\pi\)
−0.394451 + 0.918917i \(0.629065\pi\)
\(618\) 0 0
\(619\) 16.0000 0.643094 0.321547 0.946894i \(-0.395797\pi\)
0.321547 + 0.946894i \(0.395797\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −13.8564 −0.555145
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 48.4974 1.93372
\(630\) 0 0
\(631\) 1.41421 0.0562990 0.0281495 0.999604i \(-0.491039\pi\)
0.0281495 + 0.999604i \(0.491039\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 34.2929 1.36087
\(636\) 0 0
\(637\) 7.07107 0.280166
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −14.6969 −0.580494 −0.290247 0.956952i \(-0.593737\pi\)
−0.290247 + 0.956952i \(0.593737\pi\)
\(642\) 0 0
\(643\) −42.0000 −1.65632 −0.828159 0.560493i \(-0.810612\pi\)
−0.828159 + 0.560493i \(0.810612\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −6.92820 −0.272376 −0.136188 0.990683i \(-0.543485\pi\)
−0.136188 + 0.990683i \(0.543485\pi\)
\(648\) 0 0
\(649\) 48.0000 1.88416
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −38.1051 −1.49117 −0.745584 0.666411i \(-0.767829\pi\)
−0.745584 + 0.666411i \(0.767829\pi\)
\(654\) 0 0
\(655\) −33.9411 −1.32619
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 9.79796 0.381674 0.190837 0.981622i \(-0.438880\pi\)
0.190837 + 0.981622i \(0.438880\pi\)
\(660\) 0 0
\(661\) −26.8701 −1.04512 −0.522562 0.852601i \(-0.675024\pi\)
−0.522562 + 0.852601i \(0.675024\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −29.3939 −1.13985
\(666\) 0 0
\(667\) 24.0000 0.929284
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −34.6410 −1.33730
\(672\) 0 0
\(673\) −24.0000 −0.925132 −0.462566 0.886585i \(-0.653071\pi\)
−0.462566 + 0.886585i \(0.653071\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −38.1051 −1.46450 −0.732249 0.681037i \(-0.761529\pi\)
−0.732249 + 0.681037i \(0.761529\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −24.4949 −0.937271 −0.468636 0.883392i \(-0.655254\pi\)
−0.468636 + 0.883392i \(0.655254\pi\)
\(684\) 0 0
\(685\) 50.9117 1.94524
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 4.89898 0.186636
\(690\) 0 0
\(691\) −6.00000 −0.228251 −0.114125 0.993466i \(-0.536407\pi\)
−0.114125 + 0.993466i \(0.536407\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 55.4256 2.10241
\(696\) 0 0
\(697\) 24.0000 0.909065
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 3.46410 0.130837 0.0654187 0.997858i \(-0.479162\pi\)
0.0654187 + 0.997858i \(0.479162\pi\)
\(702\) 0 0
\(703\) −59.3970 −2.24020
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −14.6969 −0.552735
\(708\) 0 0
\(709\) 1.41421 0.0531119 0.0265560 0.999647i \(-0.491546\pi\)
0.0265560 + 0.999647i \(0.491546\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 9.79796 0.366936
\(714\) 0 0
\(715\) −24.0000 −0.897549
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 41.5692 1.55027 0.775135 0.631795i \(-0.217682\pi\)
0.775135 + 0.631795i \(0.217682\pi\)
\(720\) 0 0
\(721\) 10.0000 0.372419
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 24.2487 0.900575
\(726\) 0 0
\(727\) −15.5563 −0.576953 −0.288477 0.957487i \(-0.593149\pi\)
−0.288477 + 0.957487i \(0.593149\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −29.3939 −1.08717
\(732\) 0 0
\(733\) −15.5563 −0.574587 −0.287293 0.957843i \(-0.592755\pi\)
−0.287293 + 0.957843i \(0.592755\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −39.1918 −1.44365
\(738\) 0 0
\(739\) 8.00000 0.294285 0.147142 0.989115i \(-0.452992\pi\)
0.147142 + 0.989115i \(0.452992\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 34.6410 1.27086 0.635428 0.772160i \(-0.280824\pi\)
0.635428 + 0.772160i \(0.280824\pi\)
\(744\) 0 0
\(745\) −12.0000 −0.439646
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −18.3848 −0.670870 −0.335435 0.942063i \(-0.608883\pi\)
−0.335435 + 0.942063i \(0.608883\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −53.8888 −1.96121
\(756\) 0 0
\(757\) 35.3553 1.28501 0.642506 0.766281i \(-0.277895\pi\)
0.642506 + 0.766281i \(0.277895\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −4.89898 −0.177588 −0.0887939 0.996050i \(-0.528301\pi\)
−0.0887939 + 0.996050i \(0.528301\pi\)
\(762\) 0 0
\(763\) −2.00000 −0.0724049
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 13.8564 0.500326
\(768\) 0 0
\(769\) −10.0000 −0.360609 −0.180305 0.983611i \(-0.557708\pi\)
−0.180305 + 0.983611i \(0.557708\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 38.1051 1.37055 0.685273 0.728286i \(-0.259683\pi\)
0.685273 + 0.728286i \(0.259683\pi\)
\(774\) 0 0
\(775\) 9.89949 0.355600
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −29.3939 −1.05314
\(780\) 0 0
\(781\) −67.8823 −2.42902
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −34.2929 −1.22396
\(786\) 0 0
\(787\) −18.0000 −0.641631 −0.320815 0.947142i \(-0.603957\pi\)
−0.320815 + 0.947142i \(0.603957\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 27.7128 0.985354
\(792\) 0 0
\(793\) −10.0000 −0.355110
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 51.9615 1.84057 0.920286 0.391247i \(-0.127956\pi\)
0.920286 + 0.391247i \(0.127956\pi\)
\(798\) 0 0
\(799\) 33.9411 1.20075
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 58.7878 2.07457
\(804\) 0 0
\(805\) 33.9411 1.19627
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −4.89898 −0.172239 −0.0861195 0.996285i \(-0.527447\pi\)
−0.0861195 + 0.996285i \(0.527447\pi\)
\(810\) 0 0
\(811\) 6.00000 0.210688 0.105344 0.994436i \(-0.466406\pi\)
0.105344 + 0.994436i \(0.466406\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 20.7846 0.728053
\(816\) 0 0
\(817\) 36.0000 1.25948
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −10.3923 −0.362694 −0.181347 0.983419i \(-0.558046\pi\)
−0.181347 + 0.983419i \(0.558046\pi\)
\(822\) 0 0
\(823\) 35.3553 1.23241 0.616205 0.787586i \(-0.288669\pi\)
0.616205 + 0.787586i \(0.288669\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −48.9898 −1.70354 −0.851771 0.523914i \(-0.824471\pi\)
−0.851771 + 0.523914i \(0.824471\pi\)
\(828\) 0 0
\(829\) −32.5269 −1.12971 −0.564853 0.825191i \(-0.691067\pi\)
−0.564853 + 0.825191i \(0.691067\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −24.4949 −0.848698
\(834\) 0 0
\(835\) 24.0000 0.830554
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −13.8564 −0.478376 −0.239188 0.970973i \(-0.576881\pi\)
−0.239188 + 0.970973i \(0.576881\pi\)
\(840\) 0 0
\(841\) −17.0000 −0.586207
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 38.1051 1.31086
\(846\) 0 0
\(847\) −18.3848 −0.631708
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 68.5857 2.35109
\(852\) 0 0
\(853\) −41.0122 −1.40423 −0.702115 0.712063i \(-0.747761\pi\)
−0.702115 + 0.712063i \(0.747761\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −24.4949 −0.836730 −0.418365 0.908279i \(-0.637397\pi\)
−0.418365 + 0.908279i \(0.637397\pi\)
\(858\) 0 0
\(859\) −18.0000 −0.614152 −0.307076 0.951685i \(-0.599351\pi\)
−0.307076 + 0.951685i \(0.599351\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 6.92820 0.235839 0.117919 0.993023i \(-0.462378\pi\)
0.117919 + 0.993023i \(0.462378\pi\)
\(864\) 0 0
\(865\) 60.0000 2.04006
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 76.2102 2.58526
\(870\) 0 0
\(871\) −11.3137 −0.383350
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 9.79796 0.331231
\(876\) 0 0
\(877\) 7.07107 0.238773 0.119386 0.992848i \(-0.461907\pi\)
0.119386 + 0.992848i \(0.461907\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −9.79796 −0.330102 −0.165051 0.986285i \(-0.552779\pi\)
−0.165051 + 0.986285i \(0.552779\pi\)
\(882\) 0 0
\(883\) 18.0000 0.605748 0.302874 0.953031i \(-0.402054\pi\)
0.302874 + 0.953031i \(0.402054\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 27.7128 0.930505 0.465253 0.885178i \(-0.345963\pi\)
0.465253 + 0.885178i \(0.345963\pi\)
\(888\) 0 0
\(889\) 14.0000 0.469545
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −41.5692 −1.39106
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 4.89898 0.163390
\(900\) 0 0
\(901\) −16.9706 −0.565371
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −4.89898 −0.162848
\(906\) 0 0
\(907\) 30.0000 0.996134 0.498067 0.867139i \(-0.334043\pi\)
0.498067 + 0.867139i \(0.334043\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −34.6410 −1.14771 −0.573854 0.818958i \(-0.694552\pi\)
−0.573854 + 0.818958i \(0.694552\pi\)
\(912\) 0 0
\(913\) −72.0000 −2.38285
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −13.8564 −0.457579
\(918\) 0 0
\(919\) 35.3553 1.16627 0.583133 0.812377i \(-0.301827\pi\)
0.583133 + 0.812377i \(0.301827\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −19.5959 −0.645007
\(924\) 0 0
\(925\) 69.2965 2.27845
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 24.4949 0.803652 0.401826 0.915716i \(-0.368376\pi\)
0.401826 + 0.915716i \(0.368376\pi\)
\(930\) 0 0
\(931\) 30.0000 0.983210
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 83.1384 2.71892
\(936\) 0 0
\(937\) −22.0000 −0.718709 −0.359354 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −51.9615 −1.69390 −0.846949 0.531675i \(-0.821563\pi\)
−0.846949 + 0.531675i \(0.821563\pi\)
\(942\) 0 0
\(943\) 33.9411 1.10528
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 39.1918 1.27356 0.636782 0.771044i \(-0.280265\pi\)
0.636782 + 0.771044i \(0.280265\pi\)
\(948\) 0 0
\(949\) 16.9706 0.550888
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 44.0908 1.42824 0.714121 0.700022i \(-0.246827\pi\)
0.714121 + 0.700022i \(0.246827\pi\)
\(954\) 0 0
\(955\) 72.0000 2.32987
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 20.7846 0.671170
\(960\) 0 0
\(961\) −29.0000 −0.935484
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −34.6410 −1.11513
\(966\) 0 0
\(967\) −24.0416 −0.773127 −0.386563 0.922263i \(-0.626338\pi\)
−0.386563 + 0.922263i \(0.626338\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −14.6969 −0.471647 −0.235824 0.971796i \(-0.575779\pi\)
−0.235824 + 0.971796i \(0.575779\pi\)
\(972\) 0 0
\(973\) 22.6274 0.725402
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −53.8888 −1.72405 −0.862027 0.506862i \(-0.830805\pi\)
−0.862027 + 0.506862i \(0.830805\pi\)
\(978\) 0 0
\(979\) −48.0000 −1.53409
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −55.4256 −1.76780 −0.883901 0.467673i \(-0.845092\pi\)
−0.883901 + 0.467673i \(0.845092\pi\)
\(984\) 0 0
\(985\) 12.0000 0.382352
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −41.5692 −1.32182
\(990\) 0 0
\(991\) 15.5563 0.494164 0.247082 0.968995i \(-0.420528\pi\)
0.247082 + 0.968995i \(0.420528\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −24.4949 −0.776540
\(996\) 0 0
\(997\) 9.89949 0.313520 0.156760 0.987637i \(-0.449895\pi\)
0.156760 + 0.987637i \(0.449895\pi\)
\(998\) 0 0
\(999\) 0 0
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 9216.2.a.be.1.1 4
3.2 odd 2 inner 9216.2.a.be.1.3 4
4.3 odd 2 9216.2.a.bh.1.2 4
8.3 odd 2 inner 9216.2.a.be.1.4 4
8.5 even 2 9216.2.a.bh.1.3 4
12.11 even 2 9216.2.a.bh.1.4 4
24.5 odd 2 9216.2.a.bh.1.1 4
24.11 even 2 inner 9216.2.a.be.1.2 4
32.3 odd 8 2304.2.k.i.577.3 yes 8
32.5 even 8 2304.2.k.h.1729.1 yes 8
32.11 odd 8 2304.2.k.i.1729.4 yes 8
32.13 even 8 2304.2.k.h.577.2 yes 8
32.19 odd 8 2304.2.k.h.577.1 8
32.21 even 8 2304.2.k.i.1729.3 yes 8
32.27 odd 8 2304.2.k.h.1729.2 yes 8
32.29 even 8 2304.2.k.i.577.4 yes 8
96.5 odd 8 2304.2.k.h.1729.3 yes 8
96.11 even 8 2304.2.k.i.1729.2 yes 8
96.29 odd 8 2304.2.k.i.577.2 yes 8
96.35 even 8 2304.2.k.i.577.1 yes 8
96.53 odd 8 2304.2.k.i.1729.1 yes 8
96.59 even 8 2304.2.k.h.1729.4 yes 8
96.77 odd 8 2304.2.k.h.577.4 yes 8
96.83 even 8 2304.2.k.h.577.3 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2304.2.k.h.577.1 8 32.19 odd 8
2304.2.k.h.577.2 yes 8 32.13 even 8
2304.2.k.h.577.3 yes 8 96.83 even 8
2304.2.k.h.577.4 yes 8 96.77 odd 8
2304.2.k.h.1729.1 yes 8 32.5 even 8
2304.2.k.h.1729.2 yes 8 32.27 odd 8
2304.2.k.h.1729.3 yes 8 96.5 odd 8
2304.2.k.h.1729.4 yes 8 96.59 even 8
2304.2.k.i.577.1 yes 8 96.35 even 8
2304.2.k.i.577.2 yes 8 96.29 odd 8
2304.2.k.i.577.3 yes 8 32.3 odd 8
2304.2.k.i.577.4 yes 8 32.29 even 8
2304.2.k.i.1729.1 yes 8 96.53 odd 8
2304.2.k.i.1729.2 yes 8 96.11 even 8
2304.2.k.i.1729.3 yes 8 32.21 even 8
2304.2.k.i.1729.4 yes 8 32.11 odd 8
9216.2.a.be.1.1 4 1.1 even 1 trivial
9216.2.a.be.1.2 4 24.11 even 2 inner
9216.2.a.be.1.3 4 3.2 odd 2 inner
9216.2.a.be.1.4 4 8.3 odd 2 inner
9216.2.a.bh.1.1 4 24.5 odd 2
9216.2.a.bh.1.2 4 4.3 odd 2
9216.2.a.bh.1.3 4 8.5 even 2
9216.2.a.bh.1.4 4 12.11 even 2