Properties

Label 4704.2.a.bm.1.2
Level $4704$
Weight $2$
Character 4704.1
Self dual yes
Analytic conductor $37.562$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4704 = 2^{5} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4704.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.5616291108\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 672)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 4704.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +3.46410 q^{5} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +3.46410 q^{5} +1.00000 q^{9} +5.46410 q^{11} -2.00000 q^{13} -3.46410 q^{15} -7.46410 q^{17} +6.92820 q^{19} +5.46410 q^{23} +7.00000 q^{25} -1.00000 q^{27} +8.92820 q^{29} -2.92820 q^{31} -5.46410 q^{33} -2.00000 q^{37} +2.00000 q^{39} -4.53590 q^{41} -8.00000 q^{43} +3.46410 q^{45} +2.92820 q^{47} +7.46410 q^{51} -2.00000 q^{53} +18.9282 q^{55} -6.92820 q^{57} +14.9282 q^{59} -4.92820 q^{61} -6.92820 q^{65} +10.9282 q^{67} -5.46410 q^{69} +2.53590 q^{71} +0.928203 q^{73} -7.00000 q^{75} +2.92820 q^{79} +1.00000 q^{81} +4.00000 q^{83} -25.8564 q^{85} -8.92820 q^{87} +3.46410 q^{89} +2.92820 q^{93} +24.0000 q^{95} -4.92820 q^{97} +5.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{9} + 4q^{11} - 4q^{13} - 8q^{17} + 4q^{23} + 14q^{25} - 2q^{27} + 4q^{29} + 8q^{31} - 4q^{33} - 4q^{37} + 4q^{39} - 16q^{41} - 16q^{43} - 8q^{47} + 8q^{51} - 4q^{53} + 24q^{55} + 16q^{59} + 4q^{61} + 8q^{67} - 4q^{69} + 12q^{71} - 12q^{73} - 14q^{75} - 8q^{79} + 2q^{81} + 8q^{83} - 24q^{85} - 4q^{87} - 8q^{93} + 48q^{95} + 4q^{97} + 4q^{99} + O(q^{100}) \)

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
\(5\) 3.46410 1.54919 0.774597 0.632456i \(-0.217953\pi\)
0.774597 + 0.632456i \(0.217953\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.46410 1.64749 0.823744 0.566961i \(-0.191881\pi\)
0.823744 + 0.566961i \(0.191881\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) −3.46410 −0.894427
\(16\) 0 0
\(17\) −7.46410 −1.81031 −0.905155 0.425081i \(-0.860246\pi\)
−0.905155 + 0.425081i \(0.860246\pi\)
\(18\) 0 0
\(19\) 6.92820 1.58944 0.794719 0.606977i \(-0.207618\pi\)
0.794719 + 0.606977i \(0.207618\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.46410 1.13934 0.569672 0.821872i \(-0.307070\pi\)
0.569672 + 0.821872i \(0.307070\pi\)
\(24\) 0 0
\(25\) 7.00000 1.40000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 8.92820 1.65793 0.828963 0.559304i \(-0.188931\pi\)
0.828963 + 0.559304i \(0.188931\pi\)
\(30\) 0 0
\(31\) −2.92820 −0.525921 −0.262960 0.964807i \(-0.584699\pi\)
−0.262960 + 0.964807i \(0.584699\pi\)
\(32\) 0 0
\(33\) −5.46410 −0.951178
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −4.53590 −0.708388 −0.354194 0.935172i \(-0.615245\pi\)
−0.354194 + 0.935172i \(0.615245\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 3.46410 0.516398
\(46\) 0 0
\(47\) 2.92820 0.427122 0.213561 0.976930i \(-0.431494\pi\)
0.213561 + 0.976930i \(0.431494\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 7.46410 1.04518
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) 18.9282 2.55228
\(56\) 0 0
\(57\) −6.92820 −0.917663
\(58\) 0 0
\(59\) 14.9282 1.94349 0.971743 0.236040i \(-0.0758497\pi\)
0.971743 + 0.236040i \(0.0758497\pi\)
\(60\) 0 0
\(61\) −4.92820 −0.630992 −0.315496 0.948927i \(-0.602171\pi\)
−0.315496 + 0.948927i \(0.602171\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −6.92820 −0.859338
\(66\) 0 0
\(67\) 10.9282 1.33509 0.667546 0.744568i \(-0.267345\pi\)
0.667546 + 0.744568i \(0.267345\pi\)
\(68\) 0 0
\(69\) −5.46410 −0.657801
\(70\) 0 0
\(71\) 2.53590 0.300956 0.150478 0.988613i \(-0.451919\pi\)
0.150478 + 0.988613i \(0.451919\pi\)
\(72\) 0 0
\(73\) 0.928203 0.108638 0.0543190 0.998524i \(-0.482701\pi\)
0.0543190 + 0.998524i \(0.482701\pi\)
\(74\) 0 0
\(75\) −7.00000 −0.808290
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 2.92820 0.329449 0.164724 0.986340i \(-0.447327\pi\)
0.164724 + 0.986340i \(0.447327\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) −25.8564 −2.80452
\(86\) 0 0
\(87\) −8.92820 −0.957204
\(88\) 0 0
\(89\) 3.46410 0.367194 0.183597 0.983002i \(-0.441226\pi\)
0.183597 + 0.983002i \(0.441226\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 2.92820 0.303641
\(94\) 0 0
\(95\) 24.0000 2.46235
\(96\) 0 0
\(97\) −4.92820 −0.500383 −0.250192 0.968196i \(-0.580494\pi\)
−0.250192 + 0.968196i \(0.580494\pi\)
\(98\) 0 0
\(99\) 5.46410 0.549163
\(100\) 0 0
\(101\) 8.53590 0.849354 0.424677 0.905345i \(-0.360388\pi\)
0.424677 + 0.905345i \(0.360388\pi\)
\(102\) 0 0
\(103\) −2.92820 −0.288524 −0.144262 0.989539i \(-0.546081\pi\)
−0.144262 + 0.989539i \(0.546081\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −8.39230 −0.811315 −0.405657 0.914025i \(-0.632957\pi\)
−0.405657 + 0.914025i \(0.632957\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) 7.85641 0.739069 0.369534 0.929217i \(-0.379517\pi\)
0.369534 + 0.929217i \(0.379517\pi\)
\(114\) 0 0
\(115\) 18.9282 1.76506
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 18.8564 1.71422
\(122\) 0 0
\(123\) 4.53590 0.408988
\(124\) 0 0
\(125\) 6.92820 0.619677
\(126\) 0 0
\(127\) −10.9282 −0.969721 −0.484861 0.874591i \(-0.661130\pi\)
−0.484861 + 0.874591i \(0.661130\pi\)
\(128\) 0 0
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −3.46410 −0.298142
\(136\) 0 0
\(137\) 4.92820 0.421045 0.210522 0.977589i \(-0.432484\pi\)
0.210522 + 0.977589i \(0.432484\pi\)
\(138\) 0 0
\(139\) −9.85641 −0.836009 −0.418005 0.908445i \(-0.637270\pi\)
−0.418005 + 0.908445i \(0.637270\pi\)
\(140\) 0 0
\(141\) −2.92820 −0.246599
\(142\) 0 0
\(143\) −10.9282 −0.913862
\(144\) 0 0
\(145\) 30.9282 2.56845
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) −5.85641 −0.476588 −0.238294 0.971193i \(-0.576588\pi\)
−0.238294 + 0.971193i \(0.576588\pi\)
\(152\) 0 0
\(153\) −7.46410 −0.603437
\(154\) 0 0
\(155\) −10.1436 −0.814753
\(156\) 0 0
\(157\) −12.9282 −1.03178 −0.515891 0.856654i \(-0.672539\pi\)
−0.515891 + 0.856654i \(0.672539\pi\)
\(158\) 0 0
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −2.92820 −0.229355 −0.114677 0.993403i \(-0.536583\pi\)
−0.114677 + 0.993403i \(0.536583\pi\)
\(164\) 0 0
\(165\) −18.9282 −1.47356
\(166\) 0 0
\(167\) −2.92820 −0.226591 −0.113296 0.993561i \(-0.536141\pi\)
−0.113296 + 0.993561i \(0.536141\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 6.92820 0.529813
\(172\) 0 0
\(173\) −1.60770 −0.122231 −0.0611154 0.998131i \(-0.519466\pi\)
−0.0611154 + 0.998131i \(0.519466\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −14.9282 −1.12207
\(178\) 0 0
\(179\) 2.53590 0.189542 0.0947710 0.995499i \(-0.469788\pi\)
0.0947710 + 0.995499i \(0.469788\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) 4.92820 0.364303
\(184\) 0 0
\(185\) −6.92820 −0.509372
\(186\) 0 0
\(187\) −40.7846 −2.98247
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 2.53590 0.183491 0.0917456 0.995782i \(-0.470755\pi\)
0.0917456 + 0.995782i \(0.470755\pi\)
\(192\) 0 0
\(193\) −11.8564 −0.853443 −0.426721 0.904383i \(-0.640332\pi\)
−0.426721 + 0.904383i \(0.640332\pi\)
\(194\) 0 0
\(195\) 6.92820 0.496139
\(196\) 0 0
\(197\) 19.8564 1.41471 0.707355 0.706858i \(-0.249888\pi\)
0.707355 + 0.706858i \(0.249888\pi\)
\(198\) 0 0
\(199\) 21.8564 1.54936 0.774680 0.632354i \(-0.217911\pi\)
0.774680 + 0.632354i \(0.217911\pi\)
\(200\) 0 0
\(201\) −10.9282 −0.770816
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −15.7128 −1.09743
\(206\) 0 0
\(207\) 5.46410 0.379781
\(208\) 0 0
\(209\) 37.8564 2.61858
\(210\) 0 0
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) 0 0
\(213\) −2.53590 −0.173757
\(214\) 0 0
\(215\) −27.7128 −1.89000
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −0.928203 −0.0627222
\(220\) 0 0
\(221\) 14.9282 1.00418
\(222\) 0 0
\(223\) 24.0000 1.60716 0.803579 0.595198i \(-0.202926\pi\)
0.803579 + 0.595198i \(0.202926\pi\)
\(224\) 0 0
\(225\) 7.00000 0.466667
\(226\) 0 0
\(227\) 9.07180 0.602116 0.301058 0.953606i \(-0.402660\pi\)
0.301058 + 0.953606i \(0.402660\pi\)
\(228\) 0 0
\(229\) 0.143594 0.00948893 0.00474446 0.999989i \(-0.498490\pi\)
0.00474446 + 0.999989i \(0.498490\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −0.928203 −0.0608086 −0.0304043 0.999538i \(-0.509679\pi\)
−0.0304043 + 0.999538i \(0.509679\pi\)
\(234\) 0 0
\(235\) 10.1436 0.661695
\(236\) 0 0
\(237\) −2.92820 −0.190207
\(238\) 0 0
\(239\) 21.4641 1.38840 0.694199 0.719783i \(-0.255759\pi\)
0.694199 + 0.719783i \(0.255759\pi\)
\(240\) 0 0
\(241\) 16.9282 1.09044 0.545221 0.838293i \(-0.316446\pi\)
0.545221 + 0.838293i \(0.316446\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −13.8564 −0.881662
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) 22.9282 1.44722 0.723608 0.690212i \(-0.242483\pi\)
0.723608 + 0.690212i \(0.242483\pi\)
\(252\) 0 0
\(253\) 29.8564 1.87706
\(254\) 0 0
\(255\) 25.8564 1.61919
\(256\) 0 0
\(257\) −4.53590 −0.282942 −0.141471 0.989942i \(-0.545183\pi\)
−0.141471 + 0.989942i \(0.545183\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 8.92820 0.552642
\(262\) 0 0
\(263\) −8.39230 −0.517492 −0.258746 0.965945i \(-0.583309\pi\)
−0.258746 + 0.965945i \(0.583309\pi\)
\(264\) 0 0
\(265\) −6.92820 −0.425596
\(266\) 0 0
\(267\) −3.46410 −0.212000
\(268\) 0 0
\(269\) 3.46410 0.211210 0.105605 0.994408i \(-0.466322\pi\)
0.105605 + 0.994408i \(0.466322\pi\)
\(270\) 0 0
\(271\) 10.9282 0.663841 0.331921 0.943307i \(-0.392303\pi\)
0.331921 + 0.943307i \(0.392303\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 38.2487 2.30648
\(276\) 0 0
\(277\) 27.8564 1.67373 0.836865 0.547410i \(-0.184386\pi\)
0.836865 + 0.547410i \(0.184386\pi\)
\(278\) 0 0
\(279\) −2.92820 −0.175307
\(280\) 0 0
\(281\) −0.928203 −0.0553720 −0.0276860 0.999617i \(-0.508814\pi\)
−0.0276860 + 0.999617i \(0.508814\pi\)
\(282\) 0 0
\(283\) 1.07180 0.0637117 0.0318559 0.999492i \(-0.489858\pi\)
0.0318559 + 0.999492i \(0.489858\pi\)
\(284\) 0 0
\(285\) −24.0000 −1.42164
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 38.7128 2.27722
\(290\) 0 0
\(291\) 4.92820 0.288896
\(292\) 0 0
\(293\) −23.4641 −1.37079 −0.685394 0.728173i \(-0.740370\pi\)
−0.685394 + 0.728173i \(0.740370\pi\)
\(294\) 0 0
\(295\) 51.7128 3.01084
\(296\) 0 0
\(297\) −5.46410 −0.317059
\(298\) 0 0
\(299\) −10.9282 −0.631994
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −8.53590 −0.490375
\(304\) 0 0
\(305\) −17.0718 −0.977528
\(306\) 0 0
\(307\) −9.07180 −0.517755 −0.258877 0.965910i \(-0.583353\pi\)
−0.258877 + 0.965910i \(0.583353\pi\)
\(308\) 0 0
\(309\) 2.92820 0.166580
\(310\) 0 0
\(311\) −5.07180 −0.287595 −0.143798 0.989607i \(-0.545931\pi\)
−0.143798 + 0.989607i \(0.545931\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 27.8564 1.56457 0.782286 0.622920i \(-0.214054\pi\)
0.782286 + 0.622920i \(0.214054\pi\)
\(318\) 0 0
\(319\) 48.7846 2.73141
\(320\) 0 0
\(321\) 8.39230 0.468413
\(322\) 0 0
\(323\) −51.7128 −2.87738
\(324\) 0 0
\(325\) −14.0000 −0.776580
\(326\) 0 0
\(327\) 2.00000 0.110600
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 24.0000 1.31916 0.659580 0.751635i \(-0.270734\pi\)
0.659580 + 0.751635i \(0.270734\pi\)
\(332\) 0 0
\(333\) −2.00000 −0.109599
\(334\) 0 0
\(335\) 37.8564 2.06832
\(336\) 0 0
\(337\) −19.8564 −1.08165 −0.540824 0.841136i \(-0.681887\pi\)
−0.540824 + 0.841136i \(0.681887\pi\)
\(338\) 0 0
\(339\) −7.85641 −0.426701
\(340\) 0 0
\(341\) −16.0000 −0.866449
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −18.9282 −1.01906
\(346\) 0 0
\(347\) −11.3205 −0.607717 −0.303858 0.952717i \(-0.598275\pi\)
−0.303858 + 0.952717i \(0.598275\pi\)
\(348\) 0 0
\(349\) 30.7846 1.64786 0.823931 0.566690i \(-0.191776\pi\)
0.823931 + 0.566690i \(0.191776\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) −15.4641 −0.823071 −0.411536 0.911394i \(-0.635007\pi\)
−0.411536 + 0.911394i \(0.635007\pi\)
\(354\) 0 0
\(355\) 8.78461 0.466239
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −27.3205 −1.44192 −0.720961 0.692976i \(-0.756299\pi\)
−0.720961 + 0.692976i \(0.756299\pi\)
\(360\) 0 0
\(361\) 29.0000 1.52632
\(362\) 0 0
\(363\) −18.8564 −0.989705
\(364\) 0 0
\(365\) 3.21539 0.168301
\(366\) 0 0
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 0 0
\(369\) −4.53590 −0.236129
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 30.0000 1.55334 0.776671 0.629907i \(-0.216907\pi\)
0.776671 + 0.629907i \(0.216907\pi\)
\(374\) 0 0
\(375\) −6.92820 −0.357771
\(376\) 0 0
\(377\) −17.8564 −0.919652
\(378\) 0 0
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 0 0
\(381\) 10.9282 0.559869
\(382\) 0 0
\(383\) 19.7128 1.00728 0.503639 0.863914i \(-0.331994\pi\)
0.503639 + 0.863914i \(0.331994\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −8.00000 −0.406663
\(388\) 0 0
\(389\) −26.7846 −1.35803 −0.679017 0.734123i \(-0.737594\pi\)
−0.679017 + 0.734123i \(0.737594\pi\)
\(390\) 0 0
\(391\) −40.7846 −2.06257
\(392\) 0 0
\(393\) 12.0000 0.605320
\(394\) 0 0
\(395\) 10.1436 0.510380
\(396\) 0 0
\(397\) 30.7846 1.54504 0.772518 0.634993i \(-0.218997\pi\)
0.772518 + 0.634993i \(0.218997\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −8.92820 −0.445853 −0.222927 0.974835i \(-0.571561\pi\)
−0.222927 + 0.974835i \(0.571561\pi\)
\(402\) 0 0
\(403\) 5.85641 0.291728
\(404\) 0 0
\(405\) 3.46410 0.172133
\(406\) 0 0
\(407\) −10.9282 −0.541691
\(408\) 0 0
\(409\) 8.92820 0.441471 0.220736 0.975334i \(-0.429154\pi\)
0.220736 + 0.975334i \(0.429154\pi\)
\(410\) 0 0
\(411\) −4.92820 −0.243090
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 13.8564 0.680184
\(416\) 0 0
\(417\) 9.85641 0.482670
\(418\) 0 0
\(419\) −30.9282 −1.51094 −0.755471 0.655182i \(-0.772592\pi\)
−0.755471 + 0.655182i \(0.772592\pi\)
\(420\) 0 0
\(421\) −31.8564 −1.55259 −0.776293 0.630372i \(-0.782902\pi\)
−0.776293 + 0.630372i \(0.782902\pi\)
\(422\) 0 0
\(423\) 2.92820 0.142374
\(424\) 0 0
\(425\) −52.2487 −2.53443
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 10.9282 0.527619
\(430\) 0 0
\(431\) 38.2487 1.84238 0.921188 0.389118i \(-0.127220\pi\)
0.921188 + 0.389118i \(0.127220\pi\)
\(432\) 0 0
\(433\) 0.143594 0.00690067 0.00345033 0.999994i \(-0.498902\pi\)
0.00345033 + 0.999994i \(0.498902\pi\)
\(434\) 0 0
\(435\) −30.9282 −1.48289
\(436\) 0 0
\(437\) 37.8564 1.81092
\(438\) 0 0
\(439\) 13.8564 0.661330 0.330665 0.943748i \(-0.392727\pi\)
0.330665 + 0.943748i \(0.392727\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −7.60770 −0.361453 −0.180726 0.983533i \(-0.557845\pi\)
−0.180726 + 0.983533i \(0.557845\pi\)
\(444\) 0 0
\(445\) 12.0000 0.568855
\(446\) 0 0
\(447\) 18.0000 0.851371
\(448\) 0 0
\(449\) −19.8564 −0.937082 −0.468541 0.883442i \(-0.655220\pi\)
−0.468541 + 0.883442i \(0.655220\pi\)
\(450\) 0 0
\(451\) −24.7846 −1.16706
\(452\) 0 0
\(453\) 5.85641 0.275158
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −17.7128 −0.828570 −0.414285 0.910147i \(-0.635968\pi\)
−0.414285 + 0.910147i \(0.635968\pi\)
\(458\) 0 0
\(459\) 7.46410 0.348394
\(460\) 0 0
\(461\) −35.1769 −1.63835 −0.819176 0.573542i \(-0.805569\pi\)
−0.819176 + 0.573542i \(0.805569\pi\)
\(462\) 0 0
\(463\) −8.78461 −0.408255 −0.204128 0.978944i \(-0.565436\pi\)
−0.204128 + 0.978944i \(0.565436\pi\)
\(464\) 0 0
\(465\) 10.1436 0.470398
\(466\) 0 0
\(467\) 20.7846 0.961797 0.480899 0.876776i \(-0.340311\pi\)
0.480899 + 0.876776i \(0.340311\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 12.9282 0.595700
\(472\) 0 0
\(473\) −43.7128 −2.00992
\(474\) 0 0
\(475\) 48.4974 2.22521
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) 0 0
\(479\) 18.9282 0.864852 0.432426 0.901670i \(-0.357658\pi\)
0.432426 + 0.901670i \(0.357658\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −17.0718 −0.775190
\(486\) 0 0
\(487\) 27.7128 1.25579 0.627894 0.778299i \(-0.283917\pi\)
0.627894 + 0.778299i \(0.283917\pi\)
\(488\) 0 0
\(489\) 2.92820 0.132418
\(490\) 0 0
\(491\) −13.4641 −0.607626 −0.303813 0.952732i \(-0.598260\pi\)
−0.303813 + 0.952732i \(0.598260\pi\)
\(492\) 0 0
\(493\) −66.6410 −3.00136
\(494\) 0 0
\(495\) 18.9282 0.850759
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 5.85641 0.262169 0.131084 0.991371i \(-0.458154\pi\)
0.131084 + 0.991371i \(0.458154\pi\)
\(500\) 0 0
\(501\) 2.92820 0.130822
\(502\) 0 0
\(503\) 16.0000 0.713405 0.356702 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) 0 0
\(505\) 29.5692 1.31581
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) 0 0
\(509\) −8.24871 −0.365618 −0.182809 0.983148i \(-0.558519\pi\)
−0.182809 + 0.983148i \(0.558519\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −6.92820 −0.305888
\(514\) 0 0
\(515\) −10.1436 −0.446980
\(516\) 0 0
\(517\) 16.0000 0.703679
\(518\) 0 0
\(519\) 1.60770 0.0705700
\(520\) 0 0
\(521\) −9.60770 −0.420921 −0.210460 0.977602i \(-0.567496\pi\)
−0.210460 + 0.977602i \(0.567496\pi\)
\(522\) 0 0
\(523\) −28.0000 −1.22435 −0.612177 0.790721i \(-0.709706\pi\)
−0.612177 + 0.790721i \(0.709706\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 21.8564 0.952080
\(528\) 0 0
\(529\) 6.85641 0.298105
\(530\) 0 0
\(531\) 14.9282 0.647829
\(532\) 0 0
\(533\) 9.07180 0.392943
\(534\) 0 0
\(535\) −29.0718 −1.25688
\(536\) 0 0
\(537\) −2.53590 −0.109432
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 19.8564 0.853694 0.426847 0.904324i \(-0.359624\pi\)
0.426847 + 0.904324i \(0.359624\pi\)
\(542\) 0 0
\(543\) −6.00000 −0.257485
\(544\) 0 0
\(545\) −6.92820 −0.296772
\(546\) 0 0
\(547\) −2.92820 −0.125201 −0.0626005 0.998039i \(-0.519939\pi\)
−0.0626005 + 0.998039i \(0.519939\pi\)
\(548\) 0 0
\(549\) −4.92820 −0.210331
\(550\) 0 0
\(551\) 61.8564 2.63517
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 6.92820 0.294086
\(556\) 0 0
\(557\) −31.8564 −1.34980 −0.674900 0.737910i \(-0.735813\pi\)
−0.674900 + 0.737910i \(0.735813\pi\)
\(558\) 0 0
\(559\) 16.0000 0.676728
\(560\) 0 0
\(561\) 40.7846 1.72193
\(562\) 0 0
\(563\) −46.9282 −1.97779 −0.988894 0.148623i \(-0.952516\pi\)
−0.988894 + 0.148623i \(0.952516\pi\)
\(564\) 0 0
\(565\) 27.2154 1.14496
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −11.0718 −0.464154 −0.232077 0.972697i \(-0.574552\pi\)
−0.232077 + 0.972697i \(0.574552\pi\)
\(570\) 0 0
\(571\) −24.7846 −1.03720 −0.518602 0.855016i \(-0.673547\pi\)
−0.518602 + 0.855016i \(0.673547\pi\)
\(572\) 0 0
\(573\) −2.53590 −0.105939
\(574\) 0 0
\(575\) 38.2487 1.59508
\(576\) 0 0
\(577\) 3.85641 0.160544 0.0802722 0.996773i \(-0.474421\pi\)
0.0802722 + 0.996773i \(0.474421\pi\)
\(578\) 0 0
\(579\) 11.8564 0.492735
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −10.9282 −0.452600
\(584\) 0 0
\(585\) −6.92820 −0.286446
\(586\) 0 0
\(587\) −20.7846 −0.857873 −0.428936 0.903335i \(-0.641112\pi\)
−0.428936 + 0.903335i \(0.641112\pi\)
\(588\) 0 0
\(589\) −20.2872 −0.835919
\(590\) 0 0
\(591\) −19.8564 −0.816783
\(592\) 0 0
\(593\) −15.4641 −0.635035 −0.317517 0.948252i \(-0.602849\pi\)
−0.317517 + 0.948252i \(0.602849\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −21.8564 −0.894523
\(598\) 0 0
\(599\) −7.60770 −0.310842 −0.155421 0.987848i \(-0.549673\pi\)
−0.155421 + 0.987848i \(0.549673\pi\)
\(600\) 0 0
\(601\) 39.5692 1.61406 0.807031 0.590509i \(-0.201073\pi\)
0.807031 + 0.590509i \(0.201073\pi\)
\(602\) 0 0
\(603\) 10.9282 0.445031
\(604\) 0 0
\(605\) 65.3205 2.65566
\(606\) 0 0
\(607\) 13.8564 0.562414 0.281207 0.959647i \(-0.409265\pi\)
0.281207 + 0.959647i \(0.409265\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −5.85641 −0.236925
\(612\) 0 0
\(613\) 0.143594 0.00579969 0.00289984 0.999996i \(-0.499077\pi\)
0.00289984 + 0.999996i \(0.499077\pi\)
\(614\) 0 0
\(615\) 15.7128 0.633602
\(616\) 0 0
\(617\) −22.7846 −0.917274 −0.458637 0.888624i \(-0.651662\pi\)
−0.458637 + 0.888624i \(0.651662\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) −5.46410 −0.219267
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) −37.8564 −1.51184
\(628\) 0 0
\(629\) 14.9282 0.595226
\(630\) 0 0
\(631\) 13.0718 0.520380 0.260190 0.965557i \(-0.416215\pi\)
0.260190 + 0.965557i \(0.416215\pi\)
\(632\) 0 0
\(633\) 16.0000 0.635943
\(634\) 0 0
\(635\) −37.8564 −1.50229
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 2.53590 0.100319
\(640\) 0 0
\(641\) 34.7846 1.37391 0.686955 0.726700i \(-0.258947\pi\)
0.686955 + 0.726700i \(0.258947\pi\)
\(642\) 0 0
\(643\) −28.7846 −1.13515 −0.567577 0.823320i \(-0.692119\pi\)
−0.567577 + 0.823320i \(0.692119\pi\)
\(644\) 0 0
\(645\) 27.7128 1.09119
\(646\) 0 0
\(647\) 18.9282 0.744144 0.372072 0.928204i \(-0.378647\pi\)
0.372072 + 0.928204i \(0.378647\pi\)
\(648\) 0 0
\(649\) 81.5692 3.20187
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −7.07180 −0.276741 −0.138370 0.990381i \(-0.544186\pi\)
−0.138370 + 0.990381i \(0.544186\pi\)
\(654\) 0 0
\(655\) −41.5692 −1.62424
\(656\) 0 0
\(657\) 0.928203 0.0362127
\(658\) 0 0
\(659\) 21.4641 0.836123 0.418061 0.908419i \(-0.362710\pi\)
0.418061 + 0.908419i \(0.362710\pi\)
\(660\) 0 0
\(661\) 14.7846 0.575055 0.287527 0.957772i \(-0.407167\pi\)
0.287527 + 0.957772i \(0.407167\pi\)
\(662\) 0 0
\(663\) −14.9282 −0.579763
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 48.7846 1.88895
\(668\) 0 0
\(669\) −24.0000 −0.927894
\(670\) 0 0
\(671\) −26.9282 −1.03955
\(672\) 0 0
\(673\) 4.14359 0.159724 0.0798619 0.996806i \(-0.474552\pi\)
0.0798619 + 0.996806i \(0.474552\pi\)
\(674\) 0 0
\(675\) −7.00000 −0.269430
\(676\) 0 0
\(677\) 22.3923 0.860606 0.430303 0.902684i \(-0.358407\pi\)
0.430303 + 0.902684i \(0.358407\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −9.07180 −0.347632
\(682\) 0 0
\(683\) 30.2487 1.15743 0.578717 0.815528i \(-0.303553\pi\)
0.578717 + 0.815528i \(0.303553\pi\)
\(684\) 0 0
\(685\) 17.0718 0.652280
\(686\) 0 0
\(687\) −0.143594 −0.00547844
\(688\) 0 0
\(689\) 4.00000 0.152388
\(690\) 0 0
\(691\) 14.1436 0.538048 0.269024 0.963134i \(-0.413299\pi\)
0.269024 + 0.963134i \(0.413299\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −34.1436 −1.29514
\(696\) 0 0
\(697\) 33.8564 1.28240
\(698\) 0 0
\(699\) 0.928203 0.0351079
\(700\) 0 0
\(701\) −40.6410 −1.53499 −0.767495 0.641055i \(-0.778497\pi\)
−0.767495 + 0.641055i \(0.778497\pi\)
\(702\) 0 0
\(703\) −13.8564 −0.522604
\(704\) 0 0
\(705\) −10.1436 −0.382030
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −37.7128 −1.41633 −0.708167 0.706045i \(-0.750478\pi\)
−0.708167 + 0.706045i \(0.750478\pi\)
\(710\) 0 0
\(711\) 2.92820 0.109816
\(712\) 0 0
\(713\) −16.0000 −0.599205
\(714\) 0 0
\(715\) −37.8564 −1.41575
\(716\) 0 0
\(717\) −21.4641 −0.801592
\(718\) 0 0
\(719\) −13.8564 −0.516757 −0.258378 0.966044i \(-0.583188\pi\)
−0.258378 + 0.966044i \(0.583188\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −16.9282 −0.629567
\(724\) 0 0
\(725\) 62.4974 2.32110
\(726\) 0 0
\(727\) −34.9282 −1.29542 −0.647708 0.761889i \(-0.724272\pi\)
−0.647708 + 0.761889i \(0.724272\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 59.7128 2.20856
\(732\) 0 0
\(733\) 49.7128 1.83618 0.918092 0.396367i \(-0.129729\pi\)
0.918092 + 0.396367i \(0.129729\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 59.7128 2.19955
\(738\) 0 0
\(739\) −50.9282 −1.87342 −0.936712 0.350101i \(-0.886147\pi\)
−0.936712 + 0.350101i \(0.886147\pi\)
\(740\) 0 0
\(741\) 13.8564 0.509028
\(742\) 0 0
\(743\) 32.3923 1.18836 0.594179 0.804333i \(-0.297477\pi\)
0.594179 + 0.804333i \(0.297477\pi\)
\(744\) 0 0
\(745\) −62.3538 −2.28447
\(746\) 0 0
\(747\) 4.00000 0.146352
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −27.7128 −1.01125 −0.505627 0.862752i \(-0.668739\pi\)
−0.505627 + 0.862752i \(0.668739\pi\)
\(752\) 0 0
\(753\) −22.9282 −0.835550
\(754\) 0 0
\(755\) −20.2872 −0.738326
\(756\) 0 0
\(757\) −26.0000 −0.944986 −0.472493 0.881334i \(-0.656646\pi\)
−0.472493 + 0.881334i \(0.656646\pi\)
\(758\) 0 0
\(759\) −29.8564 −1.08372
\(760\) 0 0
\(761\) −20.5359 −0.744426 −0.372213 0.928147i \(-0.621401\pi\)
−0.372213 + 0.928147i \(0.621401\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −25.8564 −0.934840
\(766\) 0 0
\(767\) −29.8564 −1.07805
\(768\) 0 0
\(769\) −31.8564 −1.14877 −0.574386 0.818585i \(-0.694759\pi\)
−0.574386 + 0.818585i \(0.694759\pi\)
\(770\) 0 0
\(771\) 4.53590 0.163356
\(772\) 0 0
\(773\) −29.3205 −1.05459 −0.527293 0.849684i \(-0.676793\pi\)
−0.527293 + 0.849684i \(0.676793\pi\)
\(774\) 0 0
\(775\) −20.4974 −0.736289
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −31.4256 −1.12594
\(780\) 0 0
\(781\) 13.8564 0.495821
\(782\) 0 0
\(783\) −8.92820 −0.319068
\(784\) 0 0
\(785\) −44.7846 −1.59843
\(786\) 0 0
\(787\) 31.7128 1.13044 0.565220 0.824940i \(-0.308791\pi\)
0.565220 + 0.824940i \(0.308791\pi\)
\(788\) 0 0
\(789\) 8.39230 0.298774
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 9.85641 0.350011
\(794\) 0 0
\(795\) 6.92820 0.245718
\(796\) 0 0
\(797\) 14.3923 0.509802 0.254901 0.966967i \(-0.417957\pi\)
0.254901 + 0.966967i \(0.417957\pi\)
\(798\) 0 0
\(799\) −21.8564 −0.773224
\(800\) 0 0
\(801\) 3.46410 0.122398
\(802\) 0 0
\(803\) 5.07180 0.178980
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −3.46410 −0.121942
\(808\) 0 0
\(809\) −38.0000 −1.33601 −0.668004 0.744157i \(-0.732851\pi\)
−0.668004 + 0.744157i \(0.732851\pi\)
\(810\) 0 0
\(811\) −45.5692 −1.60015 −0.800076 0.599899i \(-0.795207\pi\)
−0.800076 + 0.599899i \(0.795207\pi\)
\(812\) 0 0
\(813\) −10.9282 −0.383269
\(814\) 0 0
\(815\) −10.1436 −0.355315
\(816\) 0 0
\(817\) −55.4256 −1.93910
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −39.8564 −1.39100 −0.695499 0.718527i \(-0.744817\pi\)
−0.695499 + 0.718527i \(0.744817\pi\)
\(822\) 0 0
\(823\) −34.9282 −1.21752 −0.608760 0.793354i \(-0.708333\pi\)
−0.608760 + 0.793354i \(0.708333\pi\)
\(824\) 0 0
\(825\) −38.2487 −1.33165
\(826\) 0 0
\(827\) 25.1769 0.875487 0.437744 0.899100i \(-0.355778\pi\)
0.437744 + 0.899100i \(0.355778\pi\)
\(828\) 0 0
\(829\) 33.7128 1.17089 0.585447 0.810711i \(-0.300919\pi\)
0.585447 + 0.810711i \(0.300919\pi\)
\(830\) 0 0
\(831\) −27.8564 −0.966328
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −10.1436 −0.351034
\(836\) 0 0
\(837\) 2.92820 0.101214
\(838\) 0 0
\(839\) −2.92820 −0.101093 −0.0505464 0.998722i \(-0.516096\pi\)
−0.0505464 + 0.998722i \(0.516096\pi\)
\(840\) 0 0
\(841\) 50.7128 1.74872
\(842\) 0 0
\(843\) 0.928203 0.0319690
\(844\) 0 0
\(845\) −31.1769 −1.07252
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −1.07180 −0.0367840
\(850\) 0 0
\(851\) −10.9282 −0.374614
\(852\) 0 0
\(853\) −36.9282 −1.26440 −0.632199 0.774806i \(-0.717847\pi\)
−0.632199 + 0.774806i \(0.717847\pi\)
\(854\) 0 0
\(855\) 24.0000 0.820783
\(856\) 0 0
\(857\) 7.17691 0.245159 0.122579 0.992459i \(-0.460883\pi\)
0.122579 + 0.992459i \(0.460883\pi\)
\(858\) 0 0
\(859\) 25.0718 0.855439 0.427719 0.903912i \(-0.359317\pi\)
0.427719 + 0.903912i \(0.359317\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −2.53590 −0.0863230 −0.0431615 0.999068i \(-0.513743\pi\)
−0.0431615 + 0.999068i \(0.513743\pi\)
\(864\) 0 0
\(865\) −5.56922 −0.189359
\(866\) 0 0
\(867\) −38.7128 −1.31476
\(868\) 0 0
\(869\) 16.0000 0.542763
\(870\) 0 0
\(871\) −21.8564 −0.740576
\(872\) 0 0
\(873\) −4.92820 −0.166794
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 35.8564 1.21078 0.605392 0.795927i \(-0.293016\pi\)
0.605392 + 0.795927i \(0.293016\pi\)
\(878\) 0 0
\(879\) 23.4641 0.791425
\(880\) 0 0
\(881\) −21.3205 −0.718306 −0.359153 0.933279i \(-0.616934\pi\)
−0.359153 + 0.933279i \(0.616934\pi\)
\(882\) 0 0
\(883\) 49.5692 1.66814 0.834069 0.551661i \(-0.186006\pi\)
0.834069 + 0.551661i \(0.186006\pi\)
\(884\) 0 0
\(885\) −51.7128 −1.73831
\(886\) 0 0
\(887\) 13.0718 0.438908 0.219454 0.975623i \(-0.429572\pi\)
0.219454 + 0.975623i \(0.429572\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 5.46410 0.183054
\(892\) 0 0
\(893\) 20.2872 0.678885
\(894\) 0 0
\(895\) 8.78461 0.293637
\(896\) 0 0
\(897\) 10.9282 0.364882
\(898\) 0 0
\(899\) −26.1436 −0.871938
\(900\) 0 0
\(901\) 14.9282 0.497331
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 20.7846 0.690904
\(906\) 0 0
\(907\) 11.7128 0.388918 0.194459 0.980911i \(-0.437705\pi\)
0.194459 + 0.980911i \(0.437705\pi\)
\(908\) 0 0
\(909\) 8.53590 0.283118
\(910\) 0 0
\(911\) −33.1769 −1.09920 −0.549600 0.835428i \(-0.685220\pi\)
−0.549600 + 0.835428i \(0.685220\pi\)
\(912\) 0 0
\(913\) 21.8564 0.723341
\(914\) 0 0
\(915\) 17.0718 0.564376
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) 0 0
\(921\) 9.07180 0.298926
\(922\) 0 0
\(923\) −5.07180 −0.166940
\(924\) 0 0
\(925\) −14.0000 −0.460317
\(926\) 0 0
\(927\) −2.92820 −0.0961748
\(928\) 0 0
\(929\) −34.3923 −1.12837 −0.564187 0.825647i \(-0.690810\pi\)
−0.564187 + 0.825647i \(0.690810\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 5.07180 0.166043
\(934\) 0 0
\(935\) −141.282 −4.62042
\(936\) 0 0
\(937\) −26.0000 −0.849383 −0.424691 0.905338i \(-0.639617\pi\)
−0.424691 + 0.905338i \(0.639617\pi\)
\(938\) 0 0
\(939\) −14.0000 −0.456873
\(940\) 0 0
\(941\) 43.4641 1.41689 0.708445 0.705766i \(-0.249397\pi\)
0.708445 + 0.705766i \(0.249397\pi\)
\(942\) 0 0
\(943\) −24.7846 −0.807098
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −22.2487 −0.722986 −0.361493 0.932375i \(-0.617733\pi\)
−0.361493 + 0.932375i \(0.617733\pi\)
\(948\) 0 0
\(949\) −1.85641 −0.0602615
\(950\) 0 0
\(951\) −27.8564 −0.903306
\(952\) 0 0
\(953\) −22.0000 −0.712650 −0.356325 0.934362i \(-0.615970\pi\)
−0.356325 + 0.934362i \(0.615970\pi\)
\(954\) 0 0
\(955\) 8.78461 0.284263
\(956\) 0 0
\(957\) −48.7846 −1.57698
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −22.4256 −0.723407
\(962\) 0 0
\(963\) −8.39230 −0.270438
\(964\) 0 0
\(965\) −41.0718 −1.32215
\(966\) 0 0
\(967\) 21.0718 0.677623 0.338812 0.940854i \(-0.389975\pi\)
0.338812 + 0.940854i \(0.389975\pi\)
\(968\) 0 0
\(969\) 51.7128 1.66125
\(970\) 0 0
\(971\) 25.8564 0.829772 0.414886 0.909873i \(-0.363822\pi\)
0.414886 + 0.909873i \(0.363822\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 14.0000 0.448359
\(976\) 0 0
\(977\) −58.4974 −1.87150 −0.935749 0.352666i \(-0.885275\pi\)
−0.935749 + 0.352666i \(0.885275\pi\)
\(978\) 0 0
\(979\) 18.9282 0.604948
\(980\) 0 0
\(981\) −2.00000 −0.0638551
\(982\) 0 0
\(983\) −51.7128 −1.64938 −0.824691 0.565583i \(-0.808651\pi\)
−0.824691 + 0.565583i \(0.808651\pi\)
\(984\) 0 0
\(985\) 68.7846 2.19166
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −43.7128 −1.38999
\(990\) 0 0
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) 0 0
\(993\) −24.0000 −0.761617
\(994\) 0 0
\(995\) 75.7128 2.40026
\(996\) 0 0
\(997\) −44.9282 −1.42289 −0.711445 0.702742i \(-0.751959\pi\)
−0.711445 + 0.702742i \(0.751959\pi\)
\(998\) 0 0
\(999\) 2.00000 0.0632772
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 4704.2.a.bm.1.2 2
4.3 odd 2 4704.2.a.bn.1.2 2
7.6 odd 2 672.2.a.j.1.1 yes 2
8.3 odd 2 9408.2.a.do.1.1 2
8.5 even 2 9408.2.a.dx.1.1 2
21.20 even 2 2016.2.a.s.1.2 2
28.27 even 2 672.2.a.i.1.1 2
56.13 odd 2 1344.2.a.u.1.2 2
56.27 even 2 1344.2.a.v.1.2 2
84.83 odd 2 2016.2.a.t.1.2 2
112.13 odd 4 5376.2.c.bn.2689.4 4
112.27 even 4 5376.2.c.bh.2689.3 4
112.69 odd 4 5376.2.c.bn.2689.1 4
112.83 even 4 5376.2.c.bh.2689.2 4
168.83 odd 2 4032.2.a.bs.1.1 2
168.125 even 2 4032.2.a.br.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
672.2.a.i.1.1 2 28.27 even 2
672.2.a.j.1.1 yes 2 7.6 odd 2
1344.2.a.u.1.2 2 56.13 odd 2
1344.2.a.v.1.2 2 56.27 even 2
2016.2.a.s.1.2 2 21.20 even 2
2016.2.a.t.1.2 2 84.83 odd 2
4032.2.a.br.1.1 2 168.125 even 2
4032.2.a.bs.1.1 2 168.83 odd 2
4704.2.a.bm.1.2 2 1.1 even 1 trivial
4704.2.a.bn.1.2 2 4.3 odd 2
5376.2.c.bh.2689.2 4 112.83 even 4
5376.2.c.bh.2689.3 4 112.27 even 4
5376.2.c.bn.2689.1 4 112.69 odd 4
5376.2.c.bn.2689.4 4 112.13 odd 4
9408.2.a.do.1.1 2 8.3 odd 2
9408.2.a.dx.1.1 2 8.5 even 2