Properties

Label 9800.2.a.ce.1.2
Level $9800$
Weight $2$
Character 9800.1
Self dual yes
Analytic conductor $78.253$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(78.2533939809\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1944.1
Defining polynomial: \(x^{3} - 9 x - 6\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.705720\) of defining polynomial
Character \(\chi\) \(=\) 9800.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.705720 q^{3} -2.50196 q^{9} +O(q^{10})\) \(q-0.705720 q^{3} -2.50196 q^{9} -4.50196 q^{11} -5.09052 q^{13} -2.00000 q^{17} -3.09052 q^{19} -5.79624 q^{23} +3.88284 q^{27} +9.50196 q^{29} +5.41144 q^{31} +3.17712 q^{33} -7.09052 q^{37} +3.59248 q^{39} -6.59248 q^{41} -4.70572 q^{43} -10.0944 q^{47} +1.41144 q^{51} -9.91340 q^{53} +2.18104 q^{57} +8.00000 q^{59} -8.91340 q^{61} +0.117159 q^{67} +4.09052 q^{69} -8.18104 q^{73} -14.1810 q^{79} +4.76568 q^{81} +10.7057 q^{83} -6.70572 q^{87} +4.09052 q^{89} -3.81896 q^{93} +2.00000 q^{97} +11.2637 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 9q^{9} + O(q^{10}) \) \( 3q + 9q^{9} + 3q^{11} - 3q^{13} - 6q^{17} + 3q^{19} - 3q^{23} + 18q^{27} + 12q^{29} + 12q^{31} + 18q^{33} - 9q^{37} - 18q^{39} + 9q^{41} - 12q^{43} + 15q^{47} - 9q^{53} - 18q^{57} + 24q^{59} - 6q^{61} - 6q^{67} - 18q^{79} + 27q^{81} + 30q^{83} - 18q^{87} - 36q^{93} + 6q^{97} + 63q^{99} + O(q^{100}) \)

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
\(3\) −0.705720 −0.407447 −0.203724 0.979028i \(-0.565304\pi\)
−0.203724 + 0.979028i \(0.565304\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −2.50196 −0.833987
\(10\) 0 0
\(11\) −4.50196 −1.35739 −0.678696 0.734419i \(-0.737455\pi\)
−0.678696 + 0.734419i \(0.737455\pi\)
\(12\) 0 0
\(13\) −5.09052 −1.41186 −0.705928 0.708283i \(-0.749470\pi\)
−0.705928 + 0.708283i \(0.749470\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −3.09052 −0.709014 −0.354507 0.935053i \(-0.615351\pi\)
−0.354507 + 0.935053i \(0.615351\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5.79624 −1.20860 −0.604300 0.796757i \(-0.706547\pi\)
−0.604300 + 0.796757i \(0.706547\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 3.88284 0.747253
\(28\) 0 0
\(29\) 9.50196 1.76447 0.882235 0.470810i \(-0.156038\pi\)
0.882235 + 0.470810i \(0.156038\pi\)
\(30\) 0 0
\(31\) 5.41144 0.971923 0.485962 0.873980i \(-0.338469\pi\)
0.485962 + 0.873980i \(0.338469\pi\)
\(32\) 0 0
\(33\) 3.17712 0.553066
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −7.09052 −1.16567 −0.582837 0.812589i \(-0.698057\pi\)
−0.582837 + 0.812589i \(0.698057\pi\)
\(38\) 0 0
\(39\) 3.59248 0.575257
\(40\) 0 0
\(41\) −6.59248 −1.02957 −0.514786 0.857319i \(-0.672129\pi\)
−0.514786 + 0.857319i \(0.672129\pi\)
\(42\) 0 0
\(43\) −4.70572 −0.717616 −0.358808 0.933411i \(-0.616817\pi\)
−0.358808 + 0.933411i \(0.616817\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −10.0944 −1.47243 −0.736213 0.676750i \(-0.763388\pi\)
−0.736213 + 0.676750i \(0.763388\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 1.41144 0.197641
\(52\) 0 0
\(53\) −9.91340 −1.36171 −0.680855 0.732418i \(-0.738392\pi\)
−0.680855 + 0.732418i \(0.738392\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.18104 0.288886
\(58\) 0 0
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) −8.91340 −1.14124 −0.570622 0.821213i \(-0.693298\pi\)
−0.570622 + 0.821213i \(0.693298\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0.117159 0.0143132 0.00715662 0.999974i \(-0.497722\pi\)
0.00715662 + 0.999974i \(0.497722\pi\)
\(68\) 0 0
\(69\) 4.09052 0.492441
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −8.18104 −0.957518 −0.478759 0.877946i \(-0.658913\pi\)
−0.478759 + 0.877946i \(0.658913\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −14.1810 −1.59549 −0.797746 0.602994i \(-0.793974\pi\)
−0.797746 + 0.602994i \(0.793974\pi\)
\(80\) 0 0
\(81\) 4.76568 0.529520
\(82\) 0 0
\(83\) 10.7057 1.17511 0.587553 0.809186i \(-0.300091\pi\)
0.587553 + 0.809186i \(0.300091\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −6.70572 −0.718929
\(88\) 0 0
\(89\) 4.09052 0.433594 0.216797 0.976217i \(-0.430439\pi\)
0.216797 + 0.976217i \(0.430439\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −3.81896 −0.396008
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) 11.2637 1.13205
\(100\) 0 0
\(101\) −3.32092 −0.330444 −0.165222 0.986256i \(-0.552834\pi\)
−0.165222 + 0.986256i \(0.552834\pi\)
\(102\) 0 0
\(103\) 10.8868 1.07270 0.536352 0.843994i \(-0.319802\pi\)
0.536352 + 0.843994i \(0.319802\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −3.88284 −0.375368 −0.187684 0.982229i \(-0.560098\pi\)
−0.187684 + 0.982229i \(0.560098\pi\)
\(108\) 0 0
\(109\) −10.2716 −0.983837 −0.491919 0.870641i \(-0.663704\pi\)
−0.491919 + 0.870641i \(0.663704\pi\)
\(110\) 0 0
\(111\) 5.00392 0.474951
\(112\) 0 0
\(113\) −2.82288 −0.265554 −0.132777 0.991146i \(-0.542389\pi\)
−0.132777 + 0.991146i \(0.542389\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 12.7363 1.17747
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 9.26764 0.842513
\(122\) 0 0
\(123\) 4.65244 0.419497
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 1.73236 0.153722 0.0768610 0.997042i \(-0.475510\pi\)
0.0768610 + 0.997042i \(0.475510\pi\)
\(128\) 0 0
\(129\) 3.32092 0.292391
\(130\) 0 0
\(131\) −12.7363 −1.11277 −0.556387 0.830923i \(-0.687813\pi\)
−0.556387 + 0.830923i \(0.687813\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −4.00000 −0.341743 −0.170872 0.985293i \(-0.554658\pi\)
−0.170872 + 0.985293i \(0.554658\pi\)
\(138\) 0 0
\(139\) −1.41144 −0.119717 −0.0598584 0.998207i \(-0.519065\pi\)
−0.0598584 + 0.998207i \(0.519065\pi\)
\(140\) 0 0
\(141\) 7.12384 0.599936
\(142\) 0 0
\(143\) 22.9173 1.91644
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 23.0944 1.89197 0.945985 0.324210i \(-0.105098\pi\)
0.945985 + 0.324210i \(0.105098\pi\)
\(150\) 0 0
\(151\) −0.407520 −0.0331635 −0.0165817 0.999863i \(-0.505278\pi\)
−0.0165817 + 0.999863i \(0.505278\pi\)
\(152\) 0 0
\(153\) 5.00392 0.404543
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 14.0944 1.12486 0.562429 0.826845i \(-0.309867\pi\)
0.562429 + 0.826845i \(0.309867\pi\)
\(158\) 0 0
\(159\) 6.99608 0.554825
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 16.1810 1.26740 0.633698 0.773580i \(-0.281536\pi\)
0.633698 + 0.773580i \(0.281536\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −7.79624 −0.603291 −0.301646 0.953420i \(-0.597536\pi\)
−0.301646 + 0.953420i \(0.597536\pi\)
\(168\) 0 0
\(169\) 12.9134 0.993338
\(170\) 0 0
\(171\) 7.73236 0.591308
\(172\) 0 0
\(173\) 8.90948 0.677375 0.338688 0.940899i \(-0.390017\pi\)
0.338688 + 0.940899i \(0.390017\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −5.64576 −0.424361
\(178\) 0 0
\(179\) 1.67908 0.125500 0.0627502 0.998029i \(-0.480013\pi\)
0.0627502 + 0.998029i \(0.480013\pi\)
\(180\) 0 0
\(181\) 12.5059 0.929555 0.464777 0.885428i \(-0.346134\pi\)
0.464777 + 0.885428i \(0.346134\pi\)
\(182\) 0 0
\(183\) 6.29036 0.464997
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 9.00392 0.658432
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 6.58856 0.476732 0.238366 0.971175i \(-0.423388\pi\)
0.238366 + 0.971175i \(0.423388\pi\)
\(192\) 0 0
\(193\) −17.8268 −1.28320 −0.641601 0.767039i \(-0.721729\pi\)
−0.641601 + 0.767039i \(0.721729\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −5.09052 −0.362685 −0.181342 0.983420i \(-0.558044\pi\)
−0.181342 + 0.983420i \(0.558044\pi\)
\(198\) 0 0
\(199\) 24.3621 1.72698 0.863491 0.504364i \(-0.168273\pi\)
0.863491 + 0.504364i \(0.168273\pi\)
\(200\) 0 0
\(201\) −0.0826814 −0.00583189
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 14.5020 1.00796
\(208\) 0 0
\(209\) 13.9134 0.962410
\(210\) 0 0
\(211\) −8.26764 −0.569168 −0.284584 0.958651i \(-0.591855\pi\)
−0.284584 + 0.958651i \(0.591855\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 5.77352 0.390138
\(220\) 0 0
\(221\) 10.1810 0.684851
\(222\) 0 0
\(223\) 17.8268 1.19377 0.596885 0.802327i \(-0.296405\pi\)
0.596885 + 0.802327i \(0.296405\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −5.35816 −0.355634 −0.177817 0.984064i \(-0.556903\pi\)
−0.177817 + 0.984064i \(0.556903\pi\)
\(228\) 0 0
\(229\) 24.1810 1.59793 0.798964 0.601379i \(-0.205382\pi\)
0.798964 + 0.601379i \(0.205382\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 10.0078 0.650079
\(238\) 0 0
\(239\) −12.7696 −0.825997 −0.412998 0.910732i \(-0.635519\pi\)
−0.412998 + 0.910732i \(0.635519\pi\)
\(240\) 0 0
\(241\) −22.0944 −1.42323 −0.711614 0.702571i \(-0.752035\pi\)
−0.711614 + 0.702571i \(0.752035\pi\)
\(242\) 0 0
\(243\) −15.0118 −0.963005
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 15.7324 1.00103
\(248\) 0 0
\(249\) −7.55524 −0.478794
\(250\) 0 0
\(251\) −13.5059 −0.852484 −0.426242 0.904609i \(-0.640163\pi\)
−0.426242 + 0.904609i \(0.640163\pi\)
\(252\) 0 0
\(253\) 26.0944 1.64054
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 16.3621 1.02064 0.510319 0.859985i \(-0.329527\pi\)
0.510319 + 0.859985i \(0.329527\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −23.7735 −1.47154
\(262\) 0 0
\(263\) −9.11324 −0.561946 −0.280973 0.959716i \(-0.590657\pi\)
−0.280973 + 0.959716i \(0.590657\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −2.88676 −0.176667
\(268\) 0 0
\(269\) 8.91340 0.543460 0.271730 0.962374i \(-0.412404\pi\)
0.271730 + 0.962374i \(0.412404\pi\)
\(270\) 0 0
\(271\) 16.3621 0.993926 0.496963 0.867772i \(-0.334449\pi\)
0.496963 + 0.867772i \(0.334449\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −11.1771 −0.671568 −0.335784 0.941939i \(-0.609001\pi\)
−0.335784 + 0.941939i \(0.609001\pi\)
\(278\) 0 0
\(279\) −13.5392 −0.810571
\(280\) 0 0
\(281\) −28.2755 −1.68677 −0.843387 0.537307i \(-0.819442\pi\)
−0.843387 + 0.537307i \(0.819442\pi\)
\(282\) 0 0
\(283\) 12.8229 0.762241 0.381121 0.924525i \(-0.375538\pi\)
0.381121 + 0.924525i \(0.375538\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −1.41144 −0.0827400
\(292\) 0 0
\(293\) −8.90948 −0.520497 −0.260249 0.965542i \(-0.583805\pi\)
−0.260249 + 0.965542i \(0.583805\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −17.4804 −1.01432
\(298\) 0 0
\(299\) 29.5059 1.70637
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 2.34364 0.134638
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 9.12108 0.520567 0.260284 0.965532i \(-0.416184\pi\)
0.260284 + 0.965532i \(0.416184\pi\)
\(308\) 0 0
\(309\) −7.68300 −0.437071
\(310\) 0 0
\(311\) −13.1771 −0.747206 −0.373603 0.927589i \(-0.621878\pi\)
−0.373603 + 0.927589i \(0.621878\pi\)
\(312\) 0 0
\(313\) 17.5392 0.991374 0.495687 0.868501i \(-0.334916\pi\)
0.495687 + 0.868501i \(0.334916\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 13.1850 0.740541 0.370271 0.928924i \(-0.379265\pi\)
0.370271 + 0.928924i \(0.379265\pi\)
\(318\) 0 0
\(319\) −42.7774 −2.39508
\(320\) 0 0
\(321\) 2.74020 0.152943
\(322\) 0 0
\(323\) 6.18104 0.343922
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 7.24884 0.400862
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 8.50196 0.467310 0.233655 0.972320i \(-0.424931\pi\)
0.233655 + 0.972320i \(0.424931\pi\)
\(332\) 0 0
\(333\) 17.7402 0.972157
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −28.1889 −1.53555 −0.767773 0.640722i \(-0.778635\pi\)
−0.767773 + 0.640722i \(0.778635\pi\)
\(338\) 0 0
\(339\) 1.99216 0.108199
\(340\) 0 0
\(341\) −24.3621 −1.31928
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −2.70572 −0.145251 −0.0726253 0.997359i \(-0.523138\pi\)
−0.0726253 + 0.997359i \(0.523138\pi\)
\(348\) 0 0
\(349\) −13.6830 −0.732434 −0.366217 0.930529i \(-0.619347\pi\)
−0.366217 + 0.930529i \(0.619347\pi\)
\(350\) 0 0
\(351\) −19.7657 −1.05501
\(352\) 0 0
\(353\) 20.3621 1.08376 0.541882 0.840454i \(-0.317712\pi\)
0.541882 + 0.840454i \(0.317712\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −18.4154 −0.971925 −0.485963 0.873980i \(-0.661531\pi\)
−0.485963 + 0.873980i \(0.661531\pi\)
\(360\) 0 0
\(361\) −9.44868 −0.497299
\(362\) 0 0
\(363\) −6.54036 −0.343280
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 25.6230 1.33751 0.668756 0.743482i \(-0.266827\pi\)
0.668756 + 0.743482i \(0.266827\pi\)
\(368\) 0 0
\(369\) 16.4941 0.858650
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −28.1889 −1.45956 −0.729782 0.683679i \(-0.760379\pi\)
−0.729782 + 0.683679i \(0.760379\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −48.3699 −2.49118
\(378\) 0 0
\(379\) 21.7402 1.11672 0.558359 0.829599i \(-0.311431\pi\)
0.558359 + 0.829599i \(0.311431\pi\)
\(380\) 0 0
\(381\) −1.22256 −0.0626336
\(382\) 0 0
\(383\) 20.2116 1.03276 0.516382 0.856358i \(-0.327278\pi\)
0.516382 + 0.856358i \(0.327278\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 11.7735 0.598482
\(388\) 0 0
\(389\) 13.8268 0.701046 0.350523 0.936554i \(-0.386004\pi\)
0.350523 + 0.936554i \(0.386004\pi\)
\(390\) 0 0
\(391\) 11.5925 0.586257
\(392\) 0 0
\(393\) 8.98824 0.453397
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −36.3699 −1.82535 −0.912677 0.408682i \(-0.865989\pi\)
−0.912677 + 0.408682i \(0.865989\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −6.81896 −0.340523 −0.170261 0.985399i \(-0.554461\pi\)
−0.170261 + 0.985399i \(0.554461\pi\)
\(402\) 0 0
\(403\) −27.5470 −1.37222
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 31.9212 1.58228
\(408\) 0 0
\(409\) −17.8640 −0.883320 −0.441660 0.897182i \(-0.645610\pi\)
−0.441660 + 0.897182i \(0.645610\pi\)
\(410\) 0 0
\(411\) 2.82288 0.139242
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0.996080 0.0487783
\(418\) 0 0
\(419\) 18.4487 0.901277 0.450639 0.892706i \(-0.351196\pi\)
0.450639 + 0.892706i \(0.351196\pi\)
\(420\) 0 0
\(421\) −10.7324 −0.523063 −0.261532 0.965195i \(-0.584228\pi\)
−0.261532 + 0.965195i \(0.584228\pi\)
\(422\) 0 0
\(423\) 25.2559 1.22798
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −16.1732 −0.780850
\(430\) 0 0
\(431\) 6.58856 0.317360 0.158680 0.987330i \(-0.449276\pi\)
0.158680 + 0.987330i \(0.449276\pi\)
\(432\) 0 0
\(433\) −5.17712 −0.248797 −0.124398 0.992232i \(-0.539700\pi\)
−0.124398 + 0.992232i \(0.539700\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 17.9134 0.856914
\(438\) 0 0
\(439\) 31.0118 1.48011 0.740055 0.672546i \(-0.234799\pi\)
0.740055 + 0.672546i \(0.234799\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −4.87892 −0.231805 −0.115902 0.993261i \(-0.536976\pi\)
−0.115902 + 0.993261i \(0.536976\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −16.2982 −0.770878
\(448\) 0 0
\(449\) 27.7696 1.31053 0.655264 0.755400i \(-0.272557\pi\)
0.655264 + 0.755400i \(0.272557\pi\)
\(450\) 0 0
\(451\) 29.6791 1.39753
\(452\) 0 0
\(453\) 0.287595 0.0135124
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 25.0039 1.16963 0.584817 0.811165i \(-0.301166\pi\)
0.584817 + 0.811165i \(0.301166\pi\)
\(458\) 0 0
\(459\) −7.76568 −0.362471
\(460\) 0 0
\(461\) −4.00784 −0.186664 −0.0933318 0.995635i \(-0.529752\pi\)
−0.0933318 + 0.995635i \(0.529752\pi\)
\(462\) 0 0
\(463\) 36.8002 1.71025 0.855124 0.518423i \(-0.173481\pi\)
0.855124 + 0.518423i \(0.173481\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 23.4753 1.08631 0.543154 0.839633i \(-0.317230\pi\)
0.543154 + 0.839633i \(0.317230\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −9.94672 −0.458321
\(472\) 0 0
\(473\) 21.1850 0.974086
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 24.8029 1.13565
\(478\) 0 0
\(479\) −37.7735 −1.72592 −0.862958 0.505275i \(-0.831391\pi\)
−0.862958 + 0.505275i \(0.831391\pi\)
\(480\) 0 0
\(481\) 36.0944 1.64576
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 6.17320 0.279734 0.139867 0.990170i \(-0.455332\pi\)
0.139867 + 0.990170i \(0.455332\pi\)
\(488\) 0 0
\(489\) −11.4193 −0.516398
\(490\) 0 0
\(491\) 39.7814 1.79531 0.897654 0.440701i \(-0.145270\pi\)
0.897654 + 0.440701i \(0.145270\pi\)
\(492\) 0 0
\(493\) −19.0039 −0.855893
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 3.76568 0.168575 0.0842875 0.996441i \(-0.473139\pi\)
0.0842875 + 0.996441i \(0.473139\pi\)
\(500\) 0 0
\(501\) 5.50196 0.245809
\(502\) 0 0
\(503\) −18.7057 −0.834047 −0.417023 0.908896i \(-0.636927\pi\)
−0.417023 + 0.908896i \(0.636927\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −9.11324 −0.404733
\(508\) 0 0
\(509\) 12.3248 0.546289 0.273144 0.961973i \(-0.411936\pi\)
0.273144 + 0.961973i \(0.411936\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −12.0000 −0.529813
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 45.4448 1.99866
\(518\) 0 0
\(519\) −6.28759 −0.275995
\(520\) 0 0
\(521\) −29.5592 −1.29501 −0.647505 0.762061i \(-0.724188\pi\)
−0.647505 + 0.762061i \(0.724188\pi\)
\(522\) 0 0
\(523\) 43.0039 1.88043 0.940215 0.340581i \(-0.110624\pi\)
0.940215 + 0.340581i \(0.110624\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −10.8229 −0.471452
\(528\) 0 0
\(529\) 10.5964 0.460713
\(530\) 0 0
\(531\) −20.0157 −0.868606
\(532\) 0 0
\(533\) 33.5592 1.45361
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −1.18496 −0.0511348
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −4.96668 −0.213534 −0.106767 0.994284i \(-0.534050\pi\)
−0.106767 + 0.994284i \(0.534050\pi\)
\(542\) 0 0
\(543\) −8.82564 −0.378745
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 22.1250 0.945997 0.472998 0.881063i \(-0.343172\pi\)
0.472998 + 0.881063i \(0.343172\pi\)
\(548\) 0 0
\(549\) 22.3010 0.951782
\(550\) 0 0
\(551\) −29.3660 −1.25103
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −5.55916 −0.235549 −0.117775 0.993040i \(-0.537576\pi\)
−0.117775 + 0.993040i \(0.537576\pi\)
\(558\) 0 0
\(559\) 23.9546 1.01317
\(560\) 0 0
\(561\) −6.35424 −0.268276
\(562\) 0 0
\(563\) −9.48316 −0.399668 −0.199834 0.979830i \(-0.564040\pi\)
−0.199834 + 0.979830i \(0.564040\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −34.2833 −1.43723 −0.718616 0.695407i \(-0.755224\pi\)
−0.718616 + 0.695407i \(0.755224\pi\)
\(570\) 0 0
\(571\) −31.5470 −1.32020 −0.660101 0.751177i \(-0.729487\pi\)
−0.660101 + 0.751177i \(0.729487\pi\)
\(572\) 0 0
\(573\) −4.64968 −0.194243
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 10.6418 0.443025 0.221513 0.975157i \(-0.428901\pi\)
0.221513 + 0.975157i \(0.428901\pi\)
\(578\) 0 0
\(579\) 12.5807 0.522837
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 44.6297 1.84837
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −6.64184 −0.274138 −0.137069 0.990562i \(-0.543768\pi\)
−0.137069 + 0.990562i \(0.543768\pi\)
\(588\) 0 0
\(589\) −16.7242 −0.689107
\(590\) 0 0
\(591\) 3.59248 0.147775
\(592\) 0 0
\(593\) −13.5392 −0.555988 −0.277994 0.960583i \(-0.589670\pi\)
−0.277994 + 0.960583i \(0.589670\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −17.1928 −0.703654
\(598\) 0 0
\(599\) 26.1810 1.06973 0.534864 0.844938i \(-0.320363\pi\)
0.534864 + 0.844938i \(0.320363\pi\)
\(600\) 0 0
\(601\) −36.0078 −1.46879 −0.734395 0.678722i \(-0.762534\pi\)
−0.734395 + 0.678722i \(0.762534\pi\)
\(602\) 0 0
\(603\) −0.293127 −0.0119371
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −23.7430 −0.963697 −0.481849 0.876255i \(-0.660034\pi\)
−0.481849 + 0.876255i \(0.660034\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 51.3860 2.07885
\(612\) 0 0
\(613\) 9.55916 0.386091 0.193045 0.981190i \(-0.438164\pi\)
0.193045 + 0.981190i \(0.438164\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 19.8268 0.798197 0.399098 0.916908i \(-0.369323\pi\)
0.399098 + 0.916908i \(0.369323\pi\)
\(618\) 0 0
\(619\) 27.1516 1.09132 0.545658 0.838008i \(-0.316280\pi\)
0.545658 + 0.838008i \(0.316280\pi\)
\(620\) 0 0
\(621\) −22.5059 −0.903130
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −9.81896 −0.392131
\(628\) 0 0
\(629\) 14.1810 0.565435
\(630\) 0 0
\(631\) −49.7735 −1.98145 −0.990726 0.135873i \(-0.956616\pi\)
−0.990726 + 0.135873i \(0.956616\pi\)
\(632\) 0 0
\(633\) 5.83464 0.231906
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 20.2304 0.799053 0.399526 0.916722i \(-0.369175\pi\)
0.399526 + 0.916722i \(0.369175\pi\)
\(642\) 0 0
\(643\) 25.1850 0.993198 0.496599 0.867980i \(-0.334582\pi\)
0.496599 + 0.867980i \(0.334582\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −19.5698 −0.769367 −0.384683 0.923049i \(-0.625689\pi\)
−0.384683 + 0.923049i \(0.625689\pi\)
\(648\) 0 0
\(649\) −36.0157 −1.41374
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −18.2676 −0.714868 −0.357434 0.933938i \(-0.616348\pi\)
−0.357434 + 0.933938i \(0.616348\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 20.4686 0.798558
\(658\) 0 0
\(659\) 46.0078 1.79221 0.896105 0.443841i \(-0.146385\pi\)
0.896105 + 0.443841i \(0.146385\pi\)
\(660\) 0 0
\(661\) −48.4604 −1.88489 −0.942446 0.334357i \(-0.891481\pi\)
−0.942446 + 0.334357i \(0.891481\pi\)
\(662\) 0 0
\(663\) −7.18496 −0.279041
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −55.0756 −2.13254
\(668\) 0 0
\(669\) −12.5807 −0.486399
\(670\) 0 0
\(671\) 40.1278 1.54912
\(672\) 0 0
\(673\) 15.0118 0.578661 0.289330 0.957229i \(-0.406567\pi\)
0.289330 + 0.957229i \(0.406567\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −7.91340 −0.304137 −0.152068 0.988370i \(-0.548593\pi\)
−0.152068 + 0.988370i \(0.548593\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 3.78136 0.144902
\(682\) 0 0
\(683\) −29.7096 −1.13681 −0.568404 0.822750i \(-0.692439\pi\)
−0.568404 + 0.822750i \(0.692439\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −17.0650 −0.651072
\(688\) 0 0
\(689\) 50.4644 1.92254
\(690\) 0 0
\(691\) −21.7735 −0.828304 −0.414152 0.910208i \(-0.635922\pi\)
−0.414152 + 0.910208i \(0.635922\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 13.1850 0.499416
\(698\) 0 0
\(699\) 12.7030 0.480470
\(700\) 0 0
\(701\) 24.0905 0.909886 0.454943 0.890520i \(-0.349660\pi\)
0.454943 + 0.890520i \(0.349660\pi\)
\(702\) 0 0
\(703\) 21.9134 0.826479
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −24.0372 −0.902738 −0.451369 0.892337i \(-0.649064\pi\)
−0.451369 + 0.892337i \(0.649064\pi\)
\(710\) 0 0
\(711\) 35.4804 1.33062
\(712\) 0 0
\(713\) −31.3660 −1.17467
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 9.01176 0.336550
\(718\) 0 0
\(719\) −0.234318 −0.00873858 −0.00436929 0.999990i \(-0.501391\pi\)
−0.00436929 + 0.999990i \(0.501391\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 15.5925 0.579891
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −9.44200 −0.350184 −0.175092 0.984552i \(-0.556022\pi\)
−0.175092 + 0.984552i \(0.556022\pi\)
\(728\) 0 0
\(729\) −3.70295 −0.137146
\(730\) 0 0
\(731\) 9.41144 0.348095
\(732\) 0 0
\(733\) 28.7441 1.06169 0.530844 0.847469i \(-0.321875\pi\)
0.530844 + 0.847469i \(0.321875\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −0.527445 −0.0194287
\(738\) 0 0
\(739\) −26.1477 −0.961859 −0.480930 0.876759i \(-0.659701\pi\)
−0.480930 + 0.876759i \(0.659701\pi\)
\(740\) 0 0
\(741\) −11.1026 −0.407865
\(742\) 0 0
\(743\) −32.7547 −1.20165 −0.600827 0.799379i \(-0.705162\pi\)
−0.600827 + 0.799379i \(0.705162\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −26.7853 −0.980022
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 26.1810 0.955360 0.477680 0.878534i \(-0.341478\pi\)
0.477680 + 0.878534i \(0.341478\pi\)
\(752\) 0 0
\(753\) 9.53136 0.347342
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −16.8229 −0.611438 −0.305719 0.952122i \(-0.598897\pi\)
−0.305719 + 0.952122i \(0.598897\pi\)
\(758\) 0 0
\(759\) −18.4154 −0.668435
\(760\) 0 0
\(761\) −5.90556 −0.214076 −0.107038 0.994255i \(-0.534137\pi\)
−0.107038 + 0.994255i \(0.534137\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −40.7242 −1.47046
\(768\) 0 0
\(769\) 0.275481 0.00993410 0.00496705 0.999988i \(-0.498419\pi\)
0.00496705 + 0.999988i \(0.498419\pi\)
\(770\) 0 0
\(771\) −11.5470 −0.415857
\(772\) 0 0
\(773\) −7.37812 −0.265372 −0.132686 0.991158i \(-0.542360\pi\)
−0.132686 + 0.991158i \(0.542360\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 20.3742 0.729981
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 36.8946 1.31851
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 18.6603 0.665167 0.332584 0.943074i \(-0.392080\pi\)
0.332584 + 0.943074i \(0.392080\pi\)
\(788\) 0 0
\(789\) 6.43139 0.228964
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 45.3738 1.61127
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −9.81896 −0.347805 −0.173903 0.984763i \(-0.555638\pi\)
−0.173903 + 0.984763i \(0.555638\pi\)
\(798\) 0 0
\(799\) 20.1889 0.714231
\(800\) 0 0
\(801\) −10.2343 −0.361612
\(802\) 0 0
\(803\) 36.8307 1.29973
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −6.29036 −0.221431
\(808\) 0 0
\(809\) −32.0039 −1.12520 −0.562599 0.826730i \(-0.690198\pi\)
−0.562599 + 0.826730i \(0.690198\pi\)
\(810\) 0 0
\(811\) 43.5137 1.52797 0.763987 0.645232i \(-0.223239\pi\)
0.763987 + 0.645232i \(0.223239\pi\)
\(812\) 0 0
\(813\) −11.5470 −0.404972
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 14.5431 0.508799
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −34.7242 −1.21188 −0.605941 0.795510i \(-0.707203\pi\)
−0.605941 + 0.795510i \(0.707203\pi\)
\(822\) 0 0
\(823\) 23.3021 0.812261 0.406130 0.913815i \(-0.366878\pi\)
0.406130 + 0.913815i \(0.366878\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −11.2943 −0.392741 −0.196370 0.980530i \(-0.562915\pi\)
−0.196370 + 0.980530i \(0.562915\pi\)
\(828\) 0 0
\(829\) 20.0078 0.694901 0.347450 0.937698i \(-0.387047\pi\)
0.347450 + 0.937698i \(0.387047\pi\)
\(830\) 0 0
\(831\) 7.88791 0.273629
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 21.0118 0.726273
\(838\) 0 0
\(839\) 13.5846 0.468994 0.234497 0.972117i \(-0.424656\pi\)
0.234497 + 0.972117i \(0.424656\pi\)
\(840\) 0 0
\(841\) 61.2872 2.11335
\(842\) 0 0
\(843\) 19.9546 0.687272
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −9.04936 −0.310573
\(850\) 0 0
\(851\) 41.0984 1.40883
\(852\) 0 0
\(853\) 29.6336 1.01464 0.507318 0.861759i \(-0.330637\pi\)
0.507318 + 0.861759i \(0.330637\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 18.8307 0.643245 0.321623 0.946868i \(-0.395772\pi\)
0.321623 + 0.946868i \(0.395772\pi\)
\(858\) 0 0
\(859\) 8.00000 0.272956 0.136478 0.990643i \(-0.456422\pi\)
0.136478 + 0.990643i \(0.456422\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −19.8574 −0.675952 −0.337976 0.941155i \(-0.609742\pi\)
−0.337976 + 0.941155i \(0.609742\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 9.17436 0.311577
\(868\) 0 0
\(869\) 63.8425 2.16571
\(870\) 0 0
\(871\) −0.596400 −0.0202082
\(872\) 0 0
\(873\) −5.00392 −0.169357
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 13.5513 0.457595 0.228798 0.973474i \(-0.426521\pi\)
0.228798 + 0.973474i \(0.426521\pi\)
\(878\) 0 0
\(879\) 6.28759 0.212075
\(880\) 0 0
\(881\) −47.5964 −1.60356 −0.801782 0.597617i \(-0.796114\pi\)
−0.801782 + 0.597617i \(0.796114\pi\)
\(882\) 0 0
\(883\) −36.6497 −1.23336 −0.616680 0.787214i \(-0.711523\pi\)
−0.616680 + 0.787214i \(0.711523\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −46.1250 −1.54873 −0.774363 0.632742i \(-0.781929\pi\)
−0.774363 + 0.632742i \(0.781929\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −21.4549 −0.718767
\(892\) 0 0
\(893\) 31.1971 1.04397
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −20.8229 −0.695256
\(898\) 0 0
\(899\) 51.4193 1.71493
\(900\) 0 0
\(901\) 19.8268 0.660526
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 5.06780 0.168274 0.0841368 0.996454i \(-0.473187\pi\)
0.0841368 + 0.996454i \(0.473187\pi\)
\(908\) 0 0
\(909\) 8.30881 0.275586
\(910\) 0 0
\(911\) −2.41536 −0.0800244 −0.0400122 0.999199i \(-0.512740\pi\)
−0.0400122 + 0.999199i \(0.512740\pi\)
\(912\) 0 0
\(913\) −48.1967 −1.59508
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 36.5510 1.20570 0.602852 0.797853i \(-0.294031\pi\)
0.602852 + 0.797853i \(0.294031\pi\)
\(920\) 0 0
\(921\) −6.43692 −0.212104
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −27.2382 −0.894621
\(928\) 0 0
\(929\) −16.2304 −0.532502 −0.266251 0.963904i \(-0.585785\pi\)
−0.266251 + 0.963904i \(0.585785\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 9.29935 0.304447
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 44.1889 1.44359 0.721794 0.692108i \(-0.243318\pi\)
0.721794 + 0.692108i \(0.243318\pi\)
\(938\) 0 0
\(939\) −12.3778 −0.403933
\(940\) 0 0
\(941\) 23.4726 0.765183 0.382592 0.923918i \(-0.375032\pi\)
0.382592 + 0.923918i \(0.375032\pi\)
\(942\) 0 0
\(943\) 38.2116 1.24434
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −56.0717 −1.82209 −0.911043 0.412311i \(-0.864722\pi\)
−0.911043 + 0.412311i \(0.864722\pi\)
\(948\) 0 0
\(949\) 41.6458 1.35188
\(950\) 0 0
\(951\) −9.30489 −0.301732
\(952\) 0 0
\(953\) 22.7163 0.735854 0.367927 0.929855i \(-0.380068\pi\)
0.367927 + 0.929855i \(0.380068\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 30.1889 0.975868
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −1.71632 −0.0553653
\(962\) 0 0
\(963\) 9.71471 0.313052
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −10.9400 −0.351808 −0.175904 0.984407i \(-0.556285\pi\)
−0.175904 + 0.984407i \(0.556285\pi\)
\(968\) 0 0
\(969\) −4.36208 −0.140130
\(970\) 0 0
\(971\) 25.7402 0.826042 0.413021 0.910721i \(-0.364474\pi\)
0.413021 + 0.910721i \(0.364474\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −16.8229 −0.538212 −0.269106 0.963111i \(-0.586728\pi\)
−0.269106 + 0.963111i \(0.586728\pi\)
\(978\) 0 0
\(979\) −18.4154 −0.588557
\(980\) 0 0
\(981\) 25.6990 0.820507
\(982\) 0 0
\(983\) 43.9851 1.40291 0.701454 0.712715i \(-0.252535\pi\)
0.701454 + 0.712715i \(0.252535\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 27.2755 0.867310
\(990\) 0 0
\(991\) −44.2343 −1.40515 −0.702575 0.711610i \(-0.747966\pi\)
−0.702575 + 0.711610i \(0.747966\pi\)
\(992\) 0 0
\(993\) −6.00000 −0.190404
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −34.1967 −1.08302 −0.541510 0.840694i \(-0.682147\pi\)
−0.541510 + 0.840694i \(0.682147\pi\)
\(998\) 0 0
\(999\) −27.5314 −0.871054
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 9800.2.a.ce.1.2 3
5.4 even 2 1960.2.a.w.1.2 3
7.2 even 3 1400.2.q.j.1201.2 6
7.4 even 3 1400.2.q.j.401.2 6
7.6 odd 2 9800.2.a.cf.1.2 3
20.19 odd 2 3920.2.a.cc.1.2 3
35.2 odd 12 1400.2.bh.i.249.3 12
35.4 even 6 280.2.q.e.121.2 yes 6
35.9 even 6 280.2.q.e.81.2 6
35.18 odd 12 1400.2.bh.i.849.3 12
35.19 odd 6 1960.2.q.w.361.2 6
35.23 odd 12 1400.2.bh.i.249.4 12
35.24 odd 6 1960.2.q.w.961.2 6
35.32 odd 12 1400.2.bh.i.849.4 12
35.34 odd 2 1960.2.a.v.1.2 3
105.44 odd 6 2520.2.bi.q.361.2 6
105.74 odd 6 2520.2.bi.q.1801.2 6
140.39 odd 6 560.2.q.l.401.2 6
140.79 odd 6 560.2.q.l.81.2 6
140.139 even 2 3920.2.a.cb.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.q.e.81.2 6 35.9 even 6
280.2.q.e.121.2 yes 6 35.4 even 6
560.2.q.l.81.2 6 140.79 odd 6
560.2.q.l.401.2 6 140.39 odd 6
1400.2.q.j.401.2 6 7.4 even 3
1400.2.q.j.1201.2 6 7.2 even 3
1400.2.bh.i.249.3 12 35.2 odd 12
1400.2.bh.i.249.4 12 35.23 odd 12
1400.2.bh.i.849.3 12 35.18 odd 12
1400.2.bh.i.849.4 12 35.32 odd 12
1960.2.a.v.1.2 3 35.34 odd 2
1960.2.a.w.1.2 3 5.4 even 2
1960.2.q.w.361.2 6 35.19 odd 6
1960.2.q.w.961.2 6 35.24 odd 6
2520.2.bi.q.361.2 6 105.44 odd 6
2520.2.bi.q.1801.2 6 105.74 odd 6
3920.2.a.cb.1.2 3 140.139 even 2
3920.2.a.cc.1.2 3 20.19 odd 2
9800.2.a.ce.1.2 3 1.1 even 1 trivial
9800.2.a.cf.1.2 3 7.6 odd 2