Properties

Label 2736.2.a.bf.1.3
Level $2736$
Weight $2$
Character 2736.1
Self dual yes
Analytic conductor $21.847$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.13068.1
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 171)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.548230\) of defining polynomial
Character \(\chi\) \(=\) 2736.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.15121 q^{5} -3.37228 q^{7} +O(q^{10})\) \(q+2.15121 q^{5} -3.37228 q^{7} +4.70285 q^{11} +2.00000 q^{13} +2.15121 q^{17} -1.00000 q^{19} +6.85407 q^{23} -0.372281 q^{25} -6.85407 q^{29} +6.74456 q^{31} -7.25450 q^{35} -0.744563 q^{37} -2.55164 q^{41} -6.11684 q^{43} -9.00528 q^{47} +4.37228 q^{49} +11.9574 q^{53} +10.1168 q^{55} -5.10328 q^{59} +12.1168 q^{61} +4.30243 q^{65} +4.00000 q^{67} +13.7081 q^{71} +12.1168 q^{73} -15.8593 q^{77} +4.00000 q^{79} -1.75079 q^{83} +4.62772 q^{85} +11.9574 q^{89} -6.74456 q^{91} -2.15121 q^{95} -15.4891 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{7} + 8 q^{13} - 4 q^{19} + 10 q^{25} + 4 q^{31} + 20 q^{37} + 10 q^{43} + 6 q^{49} + 6 q^{55} + 14 q^{61} + 16 q^{67} + 14 q^{73} + 16 q^{79} + 30 q^{85} - 4 q^{91} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) 2.15121 0.962052 0.481026 0.876706i \(-0.340264\pi\)
0.481026 + 0.876706i \(0.340264\pi\)
\(6\) 0 0
\(7\) −3.37228 −1.27460 −0.637301 0.770615i \(-0.719949\pi\)
−0.637301 + 0.770615i \(0.719949\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.70285 1.41796 0.708982 0.705227i \(-0.249155\pi\)
0.708982 + 0.705227i \(0.249155\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.15121 0.521746 0.260873 0.965373i \(-0.415990\pi\)
0.260873 + 0.965373i \(0.415990\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.85407 1.42917 0.714586 0.699548i \(-0.246615\pi\)
0.714586 + 0.699548i \(0.246615\pi\)
\(24\) 0 0
\(25\) −0.372281 −0.0744563
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.85407 −1.27277 −0.636384 0.771372i \(-0.719571\pi\)
−0.636384 + 0.771372i \(0.719571\pi\)
\(30\) 0 0
\(31\) 6.74456 1.21136 0.605680 0.795709i \(-0.292901\pi\)
0.605680 + 0.795709i \(0.292901\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −7.25450 −1.22623
\(36\) 0 0
\(37\) −0.744563 −0.122405 −0.0612027 0.998125i \(-0.519494\pi\)
−0.0612027 + 0.998125i \(0.519494\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.55164 −0.398499 −0.199250 0.979949i \(-0.563850\pi\)
−0.199250 + 0.979949i \(0.563850\pi\)
\(42\) 0 0
\(43\) −6.11684 −0.932810 −0.466405 0.884571i \(-0.654451\pi\)
−0.466405 + 0.884571i \(0.654451\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −9.00528 −1.31356 −0.656778 0.754084i \(-0.728081\pi\)
−0.656778 + 0.754084i \(0.728081\pi\)
\(48\) 0 0
\(49\) 4.37228 0.624612
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 11.9574 1.64247 0.821234 0.570591i \(-0.193286\pi\)
0.821234 + 0.570591i \(0.193286\pi\)
\(54\) 0 0
\(55\) 10.1168 1.36415
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −5.10328 −0.664391 −0.332195 0.943211i \(-0.607789\pi\)
−0.332195 + 0.943211i \(0.607789\pi\)
\(60\) 0 0
\(61\) 12.1168 1.55140 0.775701 0.631100i \(-0.217396\pi\)
0.775701 + 0.631100i \(0.217396\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.30243 0.533650
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 13.7081 1.62686 0.813428 0.581665i \(-0.197599\pi\)
0.813428 + 0.581665i \(0.197599\pi\)
\(72\) 0 0
\(73\) 12.1168 1.41817 0.709085 0.705123i \(-0.249108\pi\)
0.709085 + 0.705123i \(0.249108\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −15.8593 −1.80734
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −1.75079 −0.192174 −0.0960868 0.995373i \(-0.530633\pi\)
−0.0960868 + 0.995373i \(0.530633\pi\)
\(84\) 0 0
\(85\) 4.62772 0.501947
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 11.9574 1.26748 0.633738 0.773547i \(-0.281520\pi\)
0.633738 + 0.773547i \(0.281520\pi\)
\(90\) 0 0
\(91\) −6.74456 −0.707022
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.15121 −0.220710
\(96\) 0 0
\(97\) −15.4891 −1.57268 −0.786341 0.617792i \(-0.788027\pi\)
−0.786341 + 0.617792i \(0.788027\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −8.60485 −0.856215 −0.428107 0.903728i \(-0.640820\pi\)
−0.428107 + 0.903728i \(0.640820\pi\)
\(102\) 0 0
\(103\) 18.7446 1.84696 0.923478 0.383651i \(-0.125333\pi\)
0.923478 + 0.383651i \(0.125333\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −9.40571 −0.909284 −0.454642 0.890674i \(-0.650233\pi\)
−0.454642 + 0.890674i \(0.650233\pi\)
\(108\) 0 0
\(109\) 16.7446 1.60384 0.801919 0.597433i \(-0.203812\pi\)
0.801919 + 0.597433i \(0.203812\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.75079 0.164700 0.0823500 0.996603i \(-0.473757\pi\)
0.0823500 + 0.996603i \(0.473757\pi\)
\(114\) 0 0
\(115\) 14.7446 1.37494
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −7.25450 −0.665019
\(120\) 0 0
\(121\) 11.1168 1.01062
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −11.5569 −1.03368
\(126\) 0 0
\(127\) 1.25544 0.111402 0.0557010 0.998447i \(-0.482261\pi\)
0.0557010 + 0.998447i \(0.482261\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −3.90200 −0.340919 −0.170460 0.985365i \(-0.554525\pi\)
−0.170460 + 0.985365i \(0.554525\pi\)
\(132\) 0 0
\(133\) 3.37228 0.292414
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 16.6602 1.42338 0.711689 0.702495i \(-0.247931\pi\)
0.711689 + 0.702495i \(0.247931\pi\)
\(138\) 0 0
\(139\) 14.1168 1.19738 0.598688 0.800983i \(-0.295689\pi\)
0.598688 + 0.800983i \(0.295689\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 9.40571 0.786545
\(144\) 0 0
\(145\) −14.7446 −1.22447
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1.35036 −0.110626 −0.0553128 0.998469i \(-0.517616\pi\)
−0.0553128 + 0.998469i \(0.517616\pi\)
\(150\) 0 0
\(151\) 4.00000 0.325515 0.162758 0.986666i \(-0.447961\pi\)
0.162758 + 0.986666i \(0.447961\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 14.5090 1.16539
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −23.1138 −1.82163
\(162\) 0 0
\(163\) −1.48913 −0.116637 −0.0583186 0.998298i \(-0.518574\pi\)
−0.0583186 + 0.998298i \(0.518574\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −4.30243 −0.332932 −0.166466 0.986047i \(-0.553236\pi\)
−0.166466 + 0.986047i \(0.553236\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 6.85407 0.521105 0.260553 0.965460i \(-0.416095\pi\)
0.260553 + 0.965460i \(0.416095\pi\)
\(174\) 0 0
\(175\) 1.25544 0.0949021
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 9.40571 0.703016 0.351508 0.936185i \(-0.385669\pi\)
0.351508 + 0.936185i \(0.385669\pi\)
\(180\) 0 0
\(181\) −3.48913 −0.259345 −0.129672 0.991557i \(-0.541393\pi\)
−0.129672 + 0.991557i \(0.541393\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.60171 −0.117760
\(186\) 0 0
\(187\) 10.1168 0.739817
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −8.20442 −0.593651 −0.296826 0.954932i \(-0.595928\pi\)
−0.296826 + 0.954932i \(0.595928\pi\)
\(192\) 0 0
\(193\) −18.2337 −1.31249 −0.656245 0.754548i \(-0.727856\pi\)
−0.656245 + 0.754548i \(0.727856\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3.50157 0.249477 0.124738 0.992190i \(-0.460191\pi\)
0.124738 + 0.992190i \(0.460191\pi\)
\(198\) 0 0
\(199\) −18.1168 −1.28427 −0.642135 0.766592i \(-0.721951\pi\)
−0.642135 + 0.766592i \(0.721951\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 23.1138 1.62227
\(204\) 0 0
\(205\) −5.48913 −0.383377
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −4.70285 −0.325303
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −13.1586 −0.897412
\(216\) 0 0
\(217\) −22.7446 −1.54400
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 4.30243 0.289413
\(222\) 0 0
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.9073 0.856686 0.428343 0.903616i \(-0.359097\pi\)
0.428343 + 0.903616i \(0.359097\pi\)
\(228\) 0 0
\(229\) 21.3723 1.41232 0.706160 0.708052i \(-0.250426\pi\)
0.706160 + 0.708052i \(0.250426\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −7.25450 −0.475258 −0.237629 0.971356i \(-0.576370\pi\)
−0.237629 + 0.971356i \(0.576370\pi\)
\(234\) 0 0
\(235\) −19.3723 −1.26371
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −27.0158 −1.74751 −0.873755 0.486367i \(-0.838322\pi\)
−0.873755 + 0.486367i \(0.838322\pi\)
\(240\) 0 0
\(241\) 10.2337 0.659210 0.329605 0.944119i \(-0.393084\pi\)
0.329605 + 0.944119i \(0.393084\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 9.40571 0.600909
\(246\) 0 0
\(247\) −2.00000 −0.127257
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −14.9094 −0.941074 −0.470537 0.882380i \(-0.655940\pi\)
−0.470537 + 0.882380i \(0.655940\pi\)
\(252\) 0 0
\(253\) 32.2337 2.02651
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.55164 0.159167 0.0795835 0.996828i \(-0.474641\pi\)
0.0795835 + 0.996828i \(0.474641\pi\)
\(258\) 0 0
\(259\) 2.51087 0.156018
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −9.80614 −0.604672 −0.302336 0.953201i \(-0.597767\pi\)
−0.302336 + 0.953201i \(0.597767\pi\)
\(264\) 0 0
\(265\) 25.7228 1.58014
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 25.6655 1.56485 0.782426 0.622743i \(-0.213982\pi\)
0.782426 + 0.622743i \(0.213982\pi\)
\(270\) 0 0
\(271\) −2.51087 −0.152525 −0.0762624 0.997088i \(-0.524299\pi\)
−0.0762624 + 0.997088i \(0.524299\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.75079 −0.105576
\(276\) 0 0
\(277\) −8.11684 −0.487694 −0.243847 0.969814i \(-0.578409\pi\)
−0.243847 + 0.969814i \(0.578409\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −10.3556 −0.617766 −0.308883 0.951100i \(-0.599955\pi\)
−0.308883 + 0.951100i \(0.599955\pi\)
\(282\) 0 0
\(283\) −0.627719 −0.0373140 −0.0186570 0.999826i \(-0.505939\pi\)
−0.0186570 + 0.999826i \(0.505939\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 8.60485 0.507928
\(288\) 0 0
\(289\) −12.3723 −0.727781
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −26.4663 −1.54618 −0.773090 0.634296i \(-0.781290\pi\)
−0.773090 + 0.634296i \(0.781290\pi\)
\(294\) 0 0
\(295\) −10.9783 −0.639178
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 13.7081 0.792762
\(300\) 0 0
\(301\) 20.6277 1.18896
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 26.0659 1.49253
\(306\) 0 0
\(307\) −2.51087 −0.143303 −0.0716516 0.997430i \(-0.522827\pi\)
−0.0716516 + 0.997430i \(0.522827\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 27.0158 1.53193 0.765964 0.642883i \(-0.222262\pi\)
0.765964 + 0.642883i \(0.222262\pi\)
\(312\) 0 0
\(313\) −15.4891 −0.875497 −0.437749 0.899097i \(-0.644224\pi\)
−0.437749 + 0.899097i \(0.644224\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −35.0712 −1.96979 −0.984897 0.173139i \(-0.944609\pi\)
−0.984897 + 0.173139i \(0.944609\pi\)
\(318\) 0 0
\(319\) −32.2337 −1.80474
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −2.15121 −0.119697
\(324\) 0 0
\(325\) −0.744563 −0.0413009
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 30.3683 1.67426
\(330\) 0 0
\(331\) 26.9783 1.48286 0.741429 0.671031i \(-0.234148\pi\)
0.741429 + 0.671031i \(0.234148\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 8.60485 0.470133
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 31.7187 1.71766
\(342\) 0 0
\(343\) 8.86141 0.478471
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 3.10114 0.166478 0.0832390 0.996530i \(-0.473474\pi\)
0.0832390 + 0.996530i \(0.473474\pi\)
\(348\) 0 0
\(349\) −10.8614 −0.581398 −0.290699 0.956815i \(-0.593888\pi\)
−0.290699 + 0.956815i \(0.593888\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −22.3130 −1.18760 −0.593800 0.804612i \(-0.702373\pi\)
−0.593800 + 0.804612i \(0.702373\pi\)
\(354\) 0 0
\(355\) 29.4891 1.56512
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 27.8167 1.46811 0.734055 0.679090i \(-0.237626\pi\)
0.734055 + 0.679090i \(0.237626\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 26.0659 1.36435
\(366\) 0 0
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −40.3236 −2.09349
\(372\) 0 0
\(373\) 10.2337 0.529880 0.264940 0.964265i \(-0.414648\pi\)
0.264940 + 0.964265i \(0.414648\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −13.7081 −0.706005
\(378\) 0 0
\(379\) −17.2554 −0.886352 −0.443176 0.896435i \(-0.646148\pi\)
−0.443176 + 0.896435i \(0.646148\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0.800857 0.0409219 0.0204609 0.999791i \(-0.493487\pi\)
0.0204609 + 0.999791i \(0.493487\pi\)
\(384\) 0 0
\(385\) −34.1168 −1.73876
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 16.6602 0.844706 0.422353 0.906431i \(-0.361204\pi\)
0.422353 + 0.906431i \(0.361204\pi\)
\(390\) 0 0
\(391\) 14.7446 0.745665
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 8.60485 0.432957
\(396\) 0 0
\(397\) 0.116844 0.00586423 0.00293212 0.999996i \(-0.499067\pi\)
0.00293212 + 0.999996i \(0.499067\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 16.2598 0.811975 0.405987 0.913879i \(-0.366928\pi\)
0.405987 + 0.913879i \(0.366928\pi\)
\(402\) 0 0
\(403\) 13.4891 0.671941
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −3.50157 −0.173566
\(408\) 0 0
\(409\) −15.4891 −0.765888 −0.382944 0.923772i \(-0.625090\pi\)
−0.382944 + 0.923772i \(0.625090\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 17.2097 0.846834
\(414\) 0 0
\(415\) −3.76631 −0.184881
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 1.75079 0.0855314 0.0427657 0.999085i \(-0.486383\pi\)
0.0427657 + 0.999085i \(0.486383\pi\)
\(420\) 0 0
\(421\) 36.9783 1.80221 0.901105 0.433601i \(-0.142757\pi\)
0.901105 + 0.433601i \(0.142757\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −0.800857 −0.0388472
\(426\) 0 0
\(427\) −40.8614 −1.97742
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −7.80400 −0.375905 −0.187953 0.982178i \(-0.560185\pi\)
−0.187953 + 0.982178i \(0.560185\pi\)
\(432\) 0 0
\(433\) −22.0000 −1.05725 −0.528626 0.848855i \(-0.677293\pi\)
−0.528626 + 0.848855i \(0.677293\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −6.85407 −0.327875
\(438\) 0 0
\(439\) −16.2337 −0.774792 −0.387396 0.921913i \(-0.626625\pi\)
−0.387396 + 0.921913i \(0.626625\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −12.5069 −0.594218 −0.297109 0.954843i \(-0.596023\pi\)
−0.297109 + 0.954843i \(0.596023\pi\)
\(444\) 0 0
\(445\) 25.7228 1.21938
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −33.4695 −1.57952 −0.789761 0.613414i \(-0.789796\pi\)
−0.789761 + 0.613414i \(0.789796\pi\)
\(450\) 0 0
\(451\) −12.0000 −0.565058
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −14.5090 −0.680192
\(456\) 0 0
\(457\) −2.62772 −0.122919 −0.0614597 0.998110i \(-0.519576\pi\)
−0.0614597 + 0.998110i \(0.519576\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 5.65278 0.263276 0.131638 0.991298i \(-0.457976\pi\)
0.131638 + 0.991298i \(0.457976\pi\)
\(462\) 0 0
\(463\) −3.37228 −0.156723 −0.0783616 0.996925i \(-0.524969\pi\)
−0.0783616 + 0.996925i \(0.524969\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −30.5174 −1.41218 −0.706089 0.708123i \(-0.749542\pi\)
−0.706089 + 0.708123i \(0.749542\pi\)
\(468\) 0 0
\(469\) −13.4891 −0.622870
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −28.7666 −1.32269
\(474\) 0 0
\(475\) 0.372281 0.0170814
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −8.45578 −0.386355 −0.193177 0.981164i \(-0.561879\pi\)
−0.193177 + 0.981164i \(0.561879\pi\)
\(480\) 0 0
\(481\) −1.48913 −0.0678983
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −33.3204 −1.51300
\(486\) 0 0
\(487\) −8.00000 −0.362515 −0.181257 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −6.85407 −0.309320 −0.154660 0.987968i \(-0.549428\pi\)
−0.154660 + 0.987968i \(0.549428\pi\)
\(492\) 0 0
\(493\) −14.7446 −0.664062
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −46.2277 −2.07360
\(498\) 0 0
\(499\) −6.11684 −0.273828 −0.136914 0.990583i \(-0.543718\pi\)
−0.136914 + 0.990583i \(0.543718\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 22.1639 0.988240 0.494120 0.869394i \(-0.335490\pi\)
0.494120 + 0.869394i \(0.335490\pi\)
\(504\) 0 0
\(505\) −18.5109 −0.823723
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −10.3556 −0.459006 −0.229503 0.973308i \(-0.573710\pi\)
−0.229503 + 0.973308i \(0.573710\pi\)
\(510\) 0 0
\(511\) −40.8614 −1.80760
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 40.3236 1.77687
\(516\) 0 0
\(517\) −42.3505 −1.86257
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 7.65492 0.335368 0.167684 0.985841i \(-0.446371\pi\)
0.167684 + 0.985841i \(0.446371\pi\)
\(522\) 0 0
\(523\) −16.2337 −0.709850 −0.354925 0.934895i \(-0.615494\pi\)
−0.354925 + 0.934895i \(0.615494\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 14.5090 0.632022
\(528\) 0 0
\(529\) 23.9783 1.04253
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −5.10328 −0.221048
\(534\) 0 0
\(535\) −20.2337 −0.874779
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 20.5622 0.885677
\(540\) 0 0
\(541\) 3.88316 0.166950 0.0834750 0.996510i \(-0.473398\pi\)
0.0834750 + 0.996510i \(0.473398\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 36.0211 1.54298
\(546\) 0 0
\(547\) 12.2337 0.523075 0.261537 0.965193i \(-0.415771\pi\)
0.261537 + 0.965193i \(0.415771\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 6.85407 0.291993
\(552\) 0 0
\(553\) −13.4891 −0.573616
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.35036 0.0572165 0.0286082 0.999591i \(-0.490892\pi\)
0.0286082 + 0.999591i \(0.490892\pi\)
\(558\) 0 0
\(559\) −12.2337 −0.517430
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −25.8146 −1.08795 −0.543977 0.839100i \(-0.683082\pi\)
−0.543977 + 0.839100i \(0.683082\pi\)
\(564\) 0 0
\(565\) 3.76631 0.158450
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −20.5622 −0.862012 −0.431006 0.902349i \(-0.641841\pi\)
−0.431006 + 0.902349i \(0.641841\pi\)
\(570\) 0 0
\(571\) −25.4891 −1.06669 −0.533343 0.845899i \(-0.679065\pi\)
−0.533343 + 0.845899i \(0.679065\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −2.55164 −0.106411
\(576\) 0 0
\(577\) −23.8832 −0.994269 −0.497134 0.867674i \(-0.665614\pi\)
−0.497134 + 0.867674i \(0.665614\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 5.90414 0.244945
\(582\) 0 0
\(583\) 56.2337 2.32896
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −32.1191 −1.32570 −0.662849 0.748753i \(-0.730653\pi\)
−0.662849 + 0.748753i \(0.730653\pi\)
\(588\) 0 0
\(589\) −6.74456 −0.277905
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −27.4163 −1.12585 −0.562926 0.826508i \(-0.690324\pi\)
−0.562926 + 0.826508i \(0.690324\pi\)
\(594\) 0 0
\(595\) −15.6060 −0.639782
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 8.60485 0.351585 0.175792 0.984427i \(-0.443751\pi\)
0.175792 + 0.984427i \(0.443751\pi\)
\(600\) 0 0
\(601\) −36.7446 −1.49884 −0.749421 0.662094i \(-0.769668\pi\)
−0.749421 + 0.662094i \(0.769668\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 23.9147 0.972271
\(606\) 0 0
\(607\) −17.2554 −0.700377 −0.350188 0.936679i \(-0.613882\pi\)
−0.350188 + 0.936679i \(0.613882\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −18.0106 −0.728629
\(612\) 0 0
\(613\) −37.6060 −1.51889 −0.759445 0.650571i \(-0.774530\pi\)
−0.759445 + 0.650571i \(0.774530\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −2.15121 −0.0866046 −0.0433023 0.999062i \(-0.513788\pi\)
−0.0433023 + 0.999062i \(0.513788\pi\)
\(618\) 0 0
\(619\) 38.9783 1.56667 0.783334 0.621601i \(-0.213517\pi\)
0.783334 + 0.621601i \(0.213517\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −40.3236 −1.61553
\(624\) 0 0
\(625\) −23.0000 −0.920000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −1.60171 −0.0638645
\(630\) 0 0
\(631\) 2.11684 0.0842702 0.0421351 0.999112i \(-0.486584\pi\)
0.0421351 + 0.999112i \(0.486584\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 2.70071 0.107175
\(636\) 0 0
\(637\) 8.74456 0.346472
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 10.3556 0.409023 0.204512 0.978864i \(-0.434439\pi\)
0.204512 + 0.978864i \(0.434439\pi\)
\(642\) 0 0
\(643\) 17.8832 0.705243 0.352621 0.935766i \(-0.385290\pi\)
0.352621 + 0.935766i \(0.385290\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −17.6101 −0.692326 −0.346163 0.938174i \(-0.612516\pi\)
−0.346163 + 0.938174i \(0.612516\pi\)
\(648\) 0 0
\(649\) −24.0000 −0.942082
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 2.95207 0.115523 0.0577617 0.998330i \(-0.481604\pi\)
0.0577617 + 0.998330i \(0.481604\pi\)
\(654\) 0 0
\(655\) −8.39403 −0.327982
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −41.9253 −1.63318 −0.816588 0.577221i \(-0.804137\pi\)
−0.816588 + 0.577221i \(0.804137\pi\)
\(660\) 0 0
\(661\) 4.74456 0.184542 0.0922710 0.995734i \(-0.470587\pi\)
0.0922710 + 0.995734i \(0.470587\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 7.25450 0.281317
\(666\) 0 0
\(667\) −46.9783 −1.81901
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 56.9838 2.19983
\(672\) 0 0
\(673\) −36.7446 −1.41640 −0.708199 0.706012i \(-0.750492\pi\)
−0.708199 + 0.706012i \(0.750492\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 13.5591 0.521117 0.260559 0.965458i \(-0.416093\pi\)
0.260559 + 0.965458i \(0.416093\pi\)
\(678\) 0 0
\(679\) 52.2337 2.00454
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 35.2203 1.34767 0.673833 0.738884i \(-0.264647\pi\)
0.673833 + 0.738884i \(0.264647\pi\)
\(684\) 0 0
\(685\) 35.8397 1.36936
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 23.9147 0.911078
\(690\) 0 0
\(691\) −9.88316 −0.375973 −0.187986 0.982172i \(-0.560196\pi\)
−0.187986 + 0.982172i \(0.560196\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 30.3683 1.15194
\(696\) 0 0
\(697\) −5.48913 −0.207915
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −29.0180 −1.09599 −0.547997 0.836480i \(-0.684610\pi\)
−0.547997 + 0.836480i \(0.684610\pi\)
\(702\) 0 0
\(703\) 0.744563 0.0280817
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 29.0180 1.09133
\(708\) 0 0
\(709\) 38.0000 1.42712 0.713560 0.700594i \(-0.247082\pi\)
0.713560 + 0.700594i \(0.247082\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 46.2277 1.73124
\(714\) 0 0
\(715\) 20.2337 0.756697
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −7.40357 −0.276107 −0.138053 0.990425i \(-0.544085\pi\)
−0.138053 + 0.990425i \(0.544085\pi\)
\(720\) 0 0
\(721\) −63.2119 −2.35414
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 2.55164 0.0947656
\(726\) 0 0
\(727\) −14.3505 −0.532232 −0.266116 0.963941i \(-0.585740\pi\)
−0.266116 + 0.963941i \(0.585740\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −13.1586 −0.486690
\(732\) 0 0
\(733\) −50.4674 −1.86406 −0.932028 0.362387i \(-0.881962\pi\)
−0.932028 + 0.362387i \(0.881962\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 18.8114 0.692928
\(738\) 0 0
\(739\) −9.88316 −0.363558 −0.181779 0.983339i \(-0.558186\pi\)
−0.181779 + 0.983339i \(0.558186\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −42.7261 −1.56747 −0.783735 0.621096i \(-0.786688\pi\)
−0.783735 + 0.621096i \(0.786688\pi\)
\(744\) 0 0
\(745\) −2.90491 −0.106428
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 31.7187 1.15898
\(750\) 0 0
\(751\) 44.4674 1.62264 0.811319 0.584604i \(-0.198750\pi\)
0.811319 + 0.584604i \(0.198750\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 8.60485 0.313163
\(756\) 0 0
\(757\) 12.1168 0.440394 0.220197 0.975455i \(-0.429330\pi\)
0.220197 + 0.975455i \(0.429330\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −25.2651 −0.915858 −0.457929 0.888989i \(-0.651409\pi\)
−0.457929 + 0.888989i \(0.651409\pi\)
\(762\) 0 0
\(763\) −56.4674 −2.04426
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −10.2066 −0.368538
\(768\) 0 0
\(769\) 24.1168 0.869676 0.434838 0.900509i \(-0.356806\pi\)
0.434838 + 0.900509i \(0.356806\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 2.55164 0.0917762 0.0458881 0.998947i \(-0.485388\pi\)
0.0458881 + 0.998947i \(0.485388\pi\)
\(774\) 0 0
\(775\) −2.51087 −0.0901933
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 2.55164 0.0914220
\(780\) 0 0
\(781\) 64.4674 2.30682
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 4.30243 0.153560
\(786\) 0 0
\(787\) −41.9565 −1.49559 −0.747794 0.663931i \(-0.768887\pi\)
−0.747794 + 0.663931i \(0.768887\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −5.90414 −0.209927
\(792\) 0 0
\(793\) 24.2337 0.860563
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 11.1565 0.395183 0.197592 0.980284i \(-0.436688\pi\)
0.197592 + 0.980284i \(0.436688\pi\)
\(798\) 0 0
\(799\) −19.3723 −0.685342
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 56.9838 2.01091
\(804\) 0 0
\(805\) −49.7228 −1.75250
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −34.6708 −1.21896 −0.609480 0.792802i \(-0.708622\pi\)
−0.609480 + 0.792802i \(0.708622\pi\)
\(810\) 0 0
\(811\) 25.2554 0.886838 0.443419 0.896314i \(-0.353765\pi\)
0.443419 + 0.896314i \(0.353765\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −3.20343 −0.112211
\(816\) 0 0
\(817\) 6.11684 0.214001
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −17.4611 −0.609395 −0.304698 0.952449i \(-0.598555\pi\)
−0.304698 + 0.952449i \(0.598555\pi\)
\(822\) 0 0
\(823\) 31.6060 1.10171 0.550857 0.834599i \(-0.314301\pi\)
0.550857 + 0.834599i \(0.314301\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 42.7261 1.48573 0.742866 0.669440i \(-0.233466\pi\)
0.742866 + 0.669440i \(0.233466\pi\)
\(828\) 0 0
\(829\) −12.7446 −0.442637 −0.221318 0.975202i \(-0.571036\pi\)
−0.221318 + 0.975202i \(0.571036\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 9.40571 0.325889
\(834\) 0 0
\(835\) −9.25544 −0.320298
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 7.80400 0.269424 0.134712 0.990885i \(-0.456989\pi\)
0.134712 + 0.990885i \(0.456989\pi\)
\(840\) 0 0
\(841\) 17.9783 0.619940
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −19.3609 −0.666036
\(846\) 0 0
\(847\) −37.4891 −1.28814
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −5.10328 −0.174938
\(852\) 0 0
\(853\) 30.4674 1.04318 0.521592 0.853195i \(-0.325339\pi\)
0.521592 + 0.853195i \(0.325339\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 42.0743 1.43723 0.718616 0.695407i \(-0.244776\pi\)
0.718616 + 0.695407i \(0.244776\pi\)
\(858\) 0 0
\(859\) 14.1168 0.481661 0.240830 0.970567i \(-0.422580\pi\)
0.240830 + 0.970567i \(0.422580\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −22.3130 −0.759543 −0.379771 0.925080i \(-0.623997\pi\)
−0.379771 + 0.925080i \(0.623997\pi\)
\(864\) 0 0
\(865\) 14.7446 0.501330
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 18.8114 0.638134
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 38.9732 1.31753
\(876\) 0 0
\(877\) 14.0000 0.472746 0.236373 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 25.2651 0.851201 0.425601 0.904911i \(-0.360063\pi\)
0.425601 + 0.904911i \(0.360063\pi\)
\(882\) 0 0
\(883\) 14.1168 0.475070 0.237535 0.971379i \(-0.423661\pi\)
0.237535 + 0.971379i \(0.423661\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −10.2066 −0.342703 −0.171351 0.985210i \(-0.554813\pi\)
−0.171351 + 0.985210i \(0.554813\pi\)
\(888\) 0 0
\(889\) −4.23369 −0.141993
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 9.00528 0.301350
\(894\) 0 0
\(895\) 20.2337 0.676338
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −46.2277 −1.54178
\(900\) 0 0
\(901\) 25.7228 0.856951
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −7.50585 −0.249503
\(906\) 0 0
\(907\) −34.7446 −1.15367 −0.576837 0.816859i \(-0.695713\pi\)
−0.576837 + 0.816859i \(0.695713\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 24.7156 0.818863 0.409432 0.912341i \(-0.365727\pi\)
0.409432 + 0.912341i \(0.365727\pi\)
\(912\) 0 0
\(913\) −8.23369 −0.272495
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 13.1586 0.434536
\(918\) 0 0
\(919\) 26.9783 0.889930 0.444965 0.895548i \(-0.353216\pi\)
0.444965 + 0.895548i \(0.353216\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 27.4163 0.902418
\(924\) 0 0
\(925\) 0.277187 0.00911384
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 46.2277 1.51668 0.758341 0.651858i \(-0.226010\pi\)
0.758341 + 0.651858i \(0.226010\pi\)
\(930\) 0 0
\(931\) −4.37228 −0.143296
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 21.7635 0.711742
\(936\) 0 0
\(937\) 12.1168 0.395840 0.197920 0.980218i \(-0.436581\pi\)
0.197920 + 0.980218i \(0.436581\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −10.3556 −0.337584 −0.168792 0.985652i \(-0.553987\pi\)
−0.168792 + 0.985652i \(0.553987\pi\)
\(942\) 0 0
\(943\) −17.4891 −0.569524
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 11.9574 0.388562 0.194281 0.980946i \(-0.437763\pi\)
0.194281 + 0.980946i \(0.437763\pi\)
\(948\) 0 0
\(949\) 24.2337 0.786659
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −24.8646 −0.805444 −0.402722 0.915322i \(-0.631936\pi\)
−0.402722 + 0.915322i \(0.631936\pi\)
\(954\) 0 0
\(955\) −17.6495 −0.571123
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −56.1829 −1.81424
\(960\) 0 0
\(961\) 14.4891 0.467391
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −39.2246 −1.26268
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −34.4194 −1.10457 −0.552286 0.833655i \(-0.686244\pi\)
−0.552286 + 0.833655i \(0.686244\pi\)
\(972\) 0 0
\(973\) −47.6060 −1.52618
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −17.0606 −0.545818 −0.272909 0.962040i \(-0.587986\pi\)
−0.272909 + 0.962040i \(0.587986\pi\)
\(978\) 0 0
\(979\) 56.2337 1.79724
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −39.2246 −1.25107 −0.625534 0.780197i \(-0.715119\pi\)
−0.625534 + 0.780197i \(0.715119\pi\)
\(984\) 0 0
\(985\) 7.53262 0.240009
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −41.9253 −1.33315
\(990\) 0 0
\(991\) −32.0000 −1.01651 −0.508257 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −38.9732 −1.23553
\(996\) 0 0
\(997\) 32.3505 1.02455 0.512276 0.858821i \(-0.328803\pi\)
0.512276 + 0.858821i \(0.328803\pi\)
\(998\) 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 2736.2.a.bf.1.3 4
3.2 odd 2 inner 2736.2.a.bf.1.2 4
4.3 odd 2 171.2.a.e.1.3 yes 4
12.11 even 2 171.2.a.e.1.2 4
20.19 odd 2 4275.2.a.bp.1.2 4
28.27 even 2 8379.2.a.bw.1.3 4
60.59 even 2 4275.2.a.bp.1.3 4
76.75 even 2 3249.2.a.bf.1.2 4
84.83 odd 2 8379.2.a.bw.1.2 4
228.227 odd 2 3249.2.a.bf.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
171.2.a.e.1.2 4 12.11 even 2
171.2.a.e.1.3 yes 4 4.3 odd 2
2736.2.a.bf.1.2 4 3.2 odd 2 inner
2736.2.a.bf.1.3 4 1.1 even 1 trivial
3249.2.a.bf.1.2 4 76.75 even 2
3249.2.a.bf.1.3 4 228.227 odd 2
4275.2.a.bp.1.2 4 20.19 odd 2
4275.2.a.bp.1.3 4 60.59 even 2
8379.2.a.bw.1.2 4 84.83 odd 2
8379.2.a.bw.1.3 4 28.27 even 2