Properties

Label 9800.2.a.cl.1.2
Level $9800$
Weight $2$
Character 9800.1
Self dual yes
Analytic conductor $78.253$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.16448.2
Defining polynomial: \(x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 14\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1960)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.18398\) of defining polynomial
Character \(\chi\) \(=\) 9800.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.18398 q^{3} +1.76977 q^{9} +O(q^{10})\) \(q-2.18398 q^{3} +1.76977 q^{9} +3.59819 q^{11} -5.85838 q^{13} +6.36121 q^{17} +4.50283 q^{19} -2.62814 q^{23} +2.68681 q^{27} +2.05866 q^{29} -6.62814 q^{31} -7.85838 q^{33} +5.91704 q^{37} +12.7946 q^{39} +7.22348 q^{41} +5.34880 q^{43} -2.31885 q^{47} -13.8927 q^{51} +4.10777 q^{53} -9.83408 q^{57} -5.07107 q^{59} -7.37361 q^{61} -3.39667 q^{67} +5.73981 q^{69} +16.7224 q^{71} -14.7184 q^{73} -3.25505 q^{79} -11.1772 q^{81} -10.6936 q^{83} -4.49607 q^{87} +11.1540 q^{89} +14.4757 q^{93} -17.1896 q^{97} +6.36796 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{3} + 6q^{9} + O(q^{10}) \) \( 4q - 2q^{3} + 6q^{9} + 2q^{11} - 10q^{13} - 6q^{17} + 4q^{23} - 14q^{27} - 2q^{29} - 12q^{31} - 18q^{33} + 14q^{39} + 12q^{41} + 8q^{43} + 2q^{47} + 2q^{51} + 4q^{53} + 8q^{57} + 8q^{59} + 20q^{61} + 8q^{67} + 24q^{69} + 4q^{71} - 16q^{73} + 22q^{79} - 20q^{81} - 36q^{83} + 18q^{87} + 40q^{89} + 32q^{93} - 26q^{97} + 12q^{99} + O(q^{100}) \)

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
\(3\) −2.18398 −1.26092 −0.630461 0.776221i \(-0.717134\pi\)
−0.630461 + 0.776221i \(0.717134\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.76977 0.589922
\(10\) 0 0
\(11\) 3.59819 1.08490 0.542448 0.840089i \(-0.317498\pi\)
0.542448 + 0.840089i \(0.317498\pi\)
\(12\) 0 0
\(13\) −5.85838 −1.62482 −0.812411 0.583085i \(-0.801845\pi\)
−0.812411 + 0.583085i \(0.801845\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.36121 1.54282 0.771410 0.636339i \(-0.219552\pi\)
0.771410 + 0.636339i \(0.219552\pi\)
\(18\) 0 0
\(19\) 4.50283 1.03302 0.516510 0.856281i \(-0.327231\pi\)
0.516510 + 0.856281i \(0.327231\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.62814 −0.548006 −0.274003 0.961729i \(-0.588348\pi\)
−0.274003 + 0.961729i \(0.588348\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 2.68681 0.517076
\(28\) 0 0
\(29\) 2.05866 0.382284 0.191142 0.981562i \(-0.438781\pi\)
0.191142 + 0.981562i \(0.438781\pi\)
\(30\) 0 0
\(31\) −6.62814 −1.19045 −0.595225 0.803559i \(-0.702937\pi\)
−0.595225 + 0.803559i \(0.702937\pi\)
\(32\) 0 0
\(33\) −7.85838 −1.36797
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.91704 0.972755 0.486378 0.873749i \(-0.338318\pi\)
0.486378 + 0.873749i \(0.338318\pi\)
\(38\) 0 0
\(39\) 12.7946 2.04877
\(40\) 0 0
\(41\) 7.22348 1.12812 0.564059 0.825734i \(-0.309239\pi\)
0.564059 + 0.825734i \(0.309239\pi\)
\(42\) 0 0
\(43\) 5.34880 0.815684 0.407842 0.913052i \(-0.366281\pi\)
0.407842 + 0.913052i \(0.366281\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.31885 −0.338239 −0.169119 0.985596i \(-0.554092\pi\)
−0.169119 + 0.985596i \(0.554092\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −13.8927 −1.94537
\(52\) 0 0
\(53\) 4.10777 0.564246 0.282123 0.959378i \(-0.408961\pi\)
0.282123 + 0.959378i \(0.408961\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −9.83408 −1.30256
\(58\) 0 0
\(59\) −5.07107 −0.660197 −0.330098 0.943947i \(-0.607082\pi\)
−0.330098 + 0.943947i \(0.607082\pi\)
\(60\) 0 0
\(61\) −7.37361 −0.944094 −0.472047 0.881573i \(-0.656485\pi\)
−0.472047 + 0.881573i \(0.656485\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −3.39667 −0.414969 −0.207485 0.978238i \(-0.566528\pi\)
−0.207485 + 0.978238i \(0.566528\pi\)
\(68\) 0 0
\(69\) 5.73981 0.690992
\(70\) 0 0
\(71\) 16.7224 1.98459 0.992293 0.123917i \(-0.0395457\pi\)
0.992293 + 0.123917i \(0.0395457\pi\)
\(72\) 0 0
\(73\) −14.7184 −1.72266 −0.861328 0.508050i \(-0.830367\pi\)
−0.861328 + 0.508050i \(0.830367\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −3.25505 −0.366221 −0.183111 0.983092i \(-0.558617\pi\)
−0.183111 + 0.983092i \(0.558617\pi\)
\(80\) 0 0
\(81\) −11.1772 −1.24191
\(82\) 0 0
\(83\) −10.6936 −1.17377 −0.586885 0.809670i \(-0.699646\pi\)
−0.586885 + 0.809670i \(0.699646\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −4.49607 −0.482030
\(88\) 0 0
\(89\) 11.1540 1.18232 0.591162 0.806553i \(-0.298669\pi\)
0.591162 + 0.806553i \(0.298669\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 14.4757 1.50106
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −17.1896 −1.74534 −0.872671 0.488308i \(-0.837614\pi\)
−0.872671 + 0.488308i \(0.837614\pi\)
\(98\) 0 0
\(99\) 6.36796 0.640004
\(100\) 0 0
\(101\) 3.44131 0.342423 0.171212 0.985234i \(-0.445232\pi\)
0.171212 + 0.985234i \(0.445232\pi\)
\(102\) 0 0
\(103\) 11.9757 1.18000 0.590000 0.807403i \(-0.299128\pi\)
0.590000 + 0.807403i \(0.299128\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 10.6760 1.03209 0.516045 0.856562i \(-0.327404\pi\)
0.516045 + 0.856562i \(0.327404\pi\)
\(108\) 0 0
\(109\) −8.33411 −0.798263 −0.399131 0.916894i \(-0.630688\pi\)
−0.399131 + 0.916894i \(0.630688\pi\)
\(110\) 0 0
\(111\) −12.9227 −1.22657
\(112\) 0 0
\(113\) 12.1542 1.14337 0.571684 0.820474i \(-0.306290\pi\)
0.571684 + 0.820474i \(0.306290\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −10.3680 −0.958518
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.94699 0.176999
\(122\) 0 0
\(123\) −15.7759 −1.42247
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 9.82453 0.871786 0.435893 0.899998i \(-0.356433\pi\)
0.435893 + 0.899998i \(0.356433\pi\)
\(128\) 0 0
\(129\) −11.6817 −1.02851
\(130\) 0 0
\(131\) 4.95375 0.432811 0.216405 0.976304i \(-0.430567\pi\)
0.216405 + 0.976304i \(0.430567\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 11.3137 0.966595 0.483298 0.875456i \(-0.339439\pi\)
0.483298 + 0.875456i \(0.339439\pi\)
\(138\) 0 0
\(139\) 8.70311 0.738188 0.369094 0.929392i \(-0.379668\pi\)
0.369094 + 0.929392i \(0.379668\pi\)
\(140\) 0 0
\(141\) 5.06431 0.426492
\(142\) 0 0
\(143\) −21.0796 −1.76276
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −6.45268 −0.528624 −0.264312 0.964437i \(-0.585145\pi\)
−0.264312 + 0.964437i \(0.585145\pi\)
\(150\) 0 0
\(151\) −4.02606 −0.327636 −0.163818 0.986491i \(-0.552381\pi\)
−0.163818 + 0.986491i \(0.552381\pi\)
\(152\) 0 0
\(153\) 11.2578 0.910143
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −15.5260 −1.23911 −0.619556 0.784953i \(-0.712687\pi\)
−0.619556 + 0.784953i \(0.712687\pi\)
\(158\) 0 0
\(159\) −8.97129 −0.711470
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 24.9610 1.95510 0.977549 0.210710i \(-0.0675774\pi\)
0.977549 + 0.210710i \(0.0675774\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 16.0739 1.24384 0.621919 0.783082i \(-0.286353\pi\)
0.621919 + 0.783082i \(0.286353\pi\)
\(168\) 0 0
\(169\) 21.3206 1.64005
\(170\) 0 0
\(171\) 7.96895 0.609401
\(172\) 0 0
\(173\) 0.0356054 0.00270703 0.00135351 0.999999i \(-0.499569\pi\)
0.00135351 + 0.999999i \(0.499569\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 11.0751 0.832456
\(178\) 0 0
\(179\) −4.45268 −0.332809 −0.166404 0.986058i \(-0.553216\pi\)
−0.166404 + 0.986058i \(0.553216\pi\)
\(180\) 0 0
\(181\) −22.7377 −1.69008 −0.845039 0.534705i \(-0.820423\pi\)
−0.845039 + 0.534705i \(0.820423\pi\)
\(182\) 0 0
\(183\) 16.1038 1.19043
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 22.8888 1.67380
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −25.5521 −1.84888 −0.924442 0.381323i \(-0.875469\pi\)
−0.924442 + 0.381323i \(0.875469\pi\)
\(192\) 0 0
\(193\) −21.0904 −1.51812 −0.759059 0.651022i \(-0.774341\pi\)
−0.759059 + 0.651022i \(0.774341\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −16.4622 −1.17288 −0.586442 0.809991i \(-0.699472\pi\)
−0.586442 + 0.809991i \(0.699472\pi\)
\(198\) 0 0
\(199\) −25.9419 −1.83897 −0.919485 0.393126i \(-0.871394\pi\)
−0.919485 + 0.393126i \(0.871394\pi\)
\(200\) 0 0
\(201\) 7.41825 0.523243
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −4.65120 −0.323281
\(208\) 0 0
\(209\) 16.2020 1.12072
\(210\) 0 0
\(211\) 8.84877 0.609174 0.304587 0.952484i \(-0.401481\pi\)
0.304587 + 0.952484i \(0.401481\pi\)
\(212\) 0 0
\(213\) −36.5214 −2.50241
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 32.1446 2.17213
\(220\) 0 0
\(221\) −37.2664 −2.50681
\(222\) 0 0
\(223\) −4.06042 −0.271906 −0.135953 0.990715i \(-0.543410\pi\)
−0.135953 + 0.990715i \(0.543410\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 11.9606 0.793856 0.396928 0.917850i \(-0.370076\pi\)
0.396928 + 0.917850i \(0.370076\pi\)
\(228\) 0 0
\(229\) 11.3042 0.747000 0.373500 0.927630i \(-0.378158\pi\)
0.373500 + 0.927630i \(0.378158\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −21.3059 −1.39580 −0.697898 0.716197i \(-0.745881\pi\)
−0.697898 + 0.716197i \(0.745881\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 7.10896 0.461776
\(238\) 0 0
\(239\) −22.5361 −1.45774 −0.728871 0.684651i \(-0.759955\pi\)
−0.728871 + 0.684651i \(0.759955\pi\)
\(240\) 0 0
\(241\) −12.8477 −0.827595 −0.413798 0.910369i \(-0.635798\pi\)
−0.413798 + 0.910369i \(0.635798\pi\)
\(242\) 0 0
\(243\) 16.3504 1.04888
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −26.3793 −1.67847
\(248\) 0 0
\(249\) 23.3545 1.48003
\(250\) 0 0
\(251\) 23.7361 1.49821 0.749103 0.662453i \(-0.230485\pi\)
0.749103 + 0.662453i \(0.230485\pi\)
\(252\) 0 0
\(253\) −9.45657 −0.594530
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −13.2139 −0.824262 −0.412131 0.911125i \(-0.635215\pi\)
−0.412131 + 0.911125i \(0.635215\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 3.64335 0.225518
\(262\) 0 0
\(263\) 28.8532 1.77917 0.889584 0.456773i \(-0.150995\pi\)
0.889584 + 0.456773i \(0.150995\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −24.3602 −1.49082
\(268\) 0 0
\(269\) 2.57073 0.156740 0.0783700 0.996924i \(-0.475028\pi\)
0.0783700 + 0.996924i \(0.475028\pi\)
\(270\) 0 0
\(271\) 6.74722 0.409865 0.204932 0.978776i \(-0.434303\pi\)
0.204932 + 0.978776i \(0.434303\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 3.25453 0.195546 0.0977730 0.995209i \(-0.468828\pi\)
0.0977730 + 0.995209i \(0.468828\pi\)
\(278\) 0 0
\(279\) −11.7303 −0.702273
\(280\) 0 0
\(281\) −2.55422 −0.152372 −0.0761860 0.997094i \(-0.524274\pi\)
−0.0761860 + 0.997094i \(0.524274\pi\)
\(282\) 0 0
\(283\) 12.5136 0.743857 0.371929 0.928261i \(-0.378697\pi\)
0.371929 + 0.928261i \(0.378697\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 23.4649 1.38029
\(290\) 0 0
\(291\) 37.5418 2.20074
\(292\) 0 0
\(293\) 25.6574 1.49892 0.749460 0.662050i \(-0.230313\pi\)
0.749460 + 0.662050i \(0.230313\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 9.66765 0.560974
\(298\) 0 0
\(299\) 15.3967 0.890412
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −7.51575 −0.431768
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 11.9391 0.681398 0.340699 0.940172i \(-0.389336\pi\)
0.340699 + 0.940172i \(0.389336\pi\)
\(308\) 0 0
\(309\) −26.1547 −1.48789
\(310\) 0 0
\(311\) 8.49879 0.481922 0.240961 0.970535i \(-0.422537\pi\)
0.240961 + 0.970535i \(0.422537\pi\)
\(312\) 0 0
\(313\) 1.75612 0.0992616 0.0496308 0.998768i \(-0.484196\pi\)
0.0496308 + 0.998768i \(0.484196\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −5.96491 −0.335023 −0.167511 0.985870i \(-0.553573\pi\)
−0.167511 + 0.985870i \(0.553573\pi\)
\(318\) 0 0
\(319\) 7.40746 0.414738
\(320\) 0 0
\(321\) −23.3162 −1.30138
\(322\) 0 0
\(323\) 28.6434 1.59376
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 18.2015 1.00655
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −14.1038 −0.775216 −0.387608 0.921824i \(-0.626699\pi\)
−0.387608 + 0.921824i \(0.626699\pi\)
\(332\) 0 0
\(333\) 10.4718 0.573850
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 22.8876 1.24677 0.623384 0.781916i \(-0.285758\pi\)
0.623384 + 0.781916i \(0.285758\pi\)
\(338\) 0 0
\(339\) −26.5445 −1.44170
\(340\) 0 0
\(341\) −23.8493 −1.29151
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 23.6586 1.27006 0.635030 0.772487i \(-0.280988\pi\)
0.635030 + 0.772487i \(0.280988\pi\)
\(348\) 0 0
\(349\) 27.7912 1.48763 0.743814 0.668386i \(-0.233015\pi\)
0.743814 + 0.668386i \(0.233015\pi\)
\(350\) 0 0
\(351\) −15.7403 −0.840157
\(352\) 0 0
\(353\) −21.1680 −1.12666 −0.563331 0.826231i \(-0.690480\pi\)
−0.563331 + 0.826231i \(0.690480\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 36.2843 1.91501 0.957505 0.288416i \(-0.0931285\pi\)
0.957505 + 0.288416i \(0.0931285\pi\)
\(360\) 0 0
\(361\) 1.27545 0.0671289
\(362\) 0 0
\(363\) −4.25219 −0.223182
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 31.1586 1.62646 0.813232 0.581939i \(-0.197706\pi\)
0.813232 + 0.581939i \(0.197706\pi\)
\(368\) 0 0
\(369\) 12.7839 0.665502
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 6.67279 0.345504 0.172752 0.984965i \(-0.444734\pi\)
0.172752 + 0.984965i \(0.444734\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −12.0604 −0.621143
\(378\) 0 0
\(379\) 21.3984 1.09916 0.549582 0.835440i \(-0.314787\pi\)
0.549582 + 0.835440i \(0.314787\pi\)
\(380\) 0 0
\(381\) −21.4566 −1.09925
\(382\) 0 0
\(383\) −3.63453 −0.185716 −0.0928578 0.995679i \(-0.529600\pi\)
−0.0928578 + 0.995679i \(0.529600\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 9.46612 0.481190
\(388\) 0 0
\(389\) 21.8952 1.11013 0.555066 0.831806i \(-0.312693\pi\)
0.555066 + 0.831806i \(0.312693\pi\)
\(390\) 0 0
\(391\) −16.7182 −0.845474
\(392\) 0 0
\(393\) −10.8189 −0.545740
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 17.9757 0.902175 0.451087 0.892480i \(-0.351036\pi\)
0.451087 + 0.892480i \(0.351036\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 27.6343 1.37999 0.689996 0.723813i \(-0.257612\pi\)
0.689996 + 0.723813i \(0.257612\pi\)
\(402\) 0 0
\(403\) 38.8302 1.93427
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 21.2907 1.05534
\(408\) 0 0
\(409\) −39.9261 −1.97422 −0.987108 0.160053i \(-0.948833\pi\)
−0.987108 + 0.160053i \(0.948833\pi\)
\(410\) 0 0
\(411\) −24.7089 −1.21880
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −19.0074 −0.930797
\(418\) 0 0
\(419\) 2.89384 0.141373 0.0706867 0.997499i \(-0.477481\pi\)
0.0706867 + 0.997499i \(0.477481\pi\)
\(420\) 0 0
\(421\) −33.5744 −1.63632 −0.818158 0.574993i \(-0.805005\pi\)
−0.818158 + 0.574993i \(0.805005\pi\)
\(422\) 0 0
\(423\) −4.10382 −0.199534
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 46.0374 2.22270
\(430\) 0 0
\(431\) 0.330882 0.0159380 0.00796902 0.999968i \(-0.497463\pi\)
0.00796902 + 0.999968i \(0.497463\pi\)
\(432\) 0 0
\(433\) 0.573752 0.0275727 0.0137864 0.999905i \(-0.495612\pi\)
0.0137864 + 0.999905i \(0.495612\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −11.8341 −0.566101
\(438\) 0 0
\(439\) −27.5165 −1.31329 −0.656645 0.754200i \(-0.728025\pi\)
−0.656645 + 0.754200i \(0.728025\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 19.7168 0.936771 0.468386 0.883524i \(-0.344836\pi\)
0.468386 + 0.883524i \(0.344836\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 14.0925 0.666553
\(448\) 0 0
\(449\) 32.6751 1.54203 0.771016 0.636816i \(-0.219749\pi\)
0.771016 + 0.636816i \(0.219749\pi\)
\(450\) 0 0
\(451\) 25.9915 1.22389
\(452\) 0 0
\(453\) 8.79282 0.413123
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −3.75112 −0.175470 −0.0877350 0.996144i \(-0.527963\pi\)
−0.0877350 + 0.996144i \(0.527963\pi\)
\(458\) 0 0
\(459\) 17.0913 0.797755
\(460\) 0 0
\(461\) 8.77983 0.408917 0.204459 0.978875i \(-0.434457\pi\)
0.204459 + 0.978875i \(0.434457\pi\)
\(462\) 0 0
\(463\) 4.78203 0.222240 0.111120 0.993807i \(-0.464556\pi\)
0.111120 + 0.993807i \(0.464556\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −21.9198 −1.01433 −0.507165 0.861849i \(-0.669306\pi\)
−0.507165 + 0.861849i \(0.669306\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 33.9085 1.56242
\(472\) 0 0
\(473\) 19.2460 0.884933
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 7.26980 0.332861
\(478\) 0 0
\(479\) 5.08861 0.232505 0.116252 0.993220i \(-0.462912\pi\)
0.116252 + 0.993220i \(0.462912\pi\)
\(480\) 0 0
\(481\) −34.6643 −1.58055
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −8.05035 −0.364796 −0.182398 0.983225i \(-0.558386\pi\)
−0.182398 + 0.983225i \(0.558386\pi\)
\(488\) 0 0
\(489\) −54.5143 −2.46522
\(490\) 0 0
\(491\) 12.8081 0.578021 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(492\) 0 0
\(493\) 13.0956 0.589795
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −28.5284 −1.27710 −0.638552 0.769578i \(-0.720466\pi\)
−0.638552 + 0.769578i \(0.720466\pi\)
\(500\) 0 0
\(501\) −35.1051 −1.56838
\(502\) 0 0
\(503\) 6.56948 0.292919 0.146459 0.989217i \(-0.453212\pi\)
0.146459 + 0.989217i \(0.453212\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −46.5638 −2.06797
\(508\) 0 0
\(509\) 24.1574 1.07076 0.535379 0.844612i \(-0.320169\pi\)
0.535379 + 0.844612i \(0.320169\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 12.0982 0.534150
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −8.34366 −0.366954
\(518\) 0 0
\(519\) −0.0777615 −0.00341335
\(520\) 0 0
\(521\) 20.8630 0.914024 0.457012 0.889460i \(-0.348920\pi\)
0.457012 + 0.889460i \(0.348920\pi\)
\(522\) 0 0
\(523\) 12.8708 0.562800 0.281400 0.959591i \(-0.409201\pi\)
0.281400 + 0.959591i \(0.409201\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −42.1630 −1.83665
\(528\) 0 0
\(529\) −16.0929 −0.699689
\(530\) 0 0
\(531\) −8.97460 −0.389465
\(532\) 0 0
\(533\) −42.3179 −1.83299
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 9.72455 0.419645
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −24.5361 −1.05489 −0.527446 0.849589i \(-0.676850\pi\)
−0.527446 + 0.849589i \(0.676850\pi\)
\(542\) 0 0
\(543\) 49.6586 2.13105
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 27.5781 1.17916 0.589578 0.807711i \(-0.299294\pi\)
0.589578 + 0.807711i \(0.299294\pi\)
\(548\) 0 0
\(549\) −13.0496 −0.556942
\(550\) 0 0
\(551\) 9.26980 0.394907
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 6.60260 0.279761 0.139881 0.990168i \(-0.455328\pi\)
0.139881 + 0.990168i \(0.455328\pi\)
\(558\) 0 0
\(559\) −31.3353 −1.32534
\(560\) 0 0
\(561\) −49.9888 −2.11053
\(562\) 0 0
\(563\) 1.97466 0.0832221 0.0416110 0.999134i \(-0.486751\pi\)
0.0416110 + 0.999134i \(0.486751\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −11.7281 −0.491666 −0.245833 0.969312i \(-0.579061\pi\)
−0.245833 + 0.969312i \(0.579061\pi\)
\(570\) 0 0
\(571\) 5.69974 0.238527 0.119263 0.992863i \(-0.461947\pi\)
0.119263 + 0.992863i \(0.461947\pi\)
\(572\) 0 0
\(573\) 55.8052 2.33130
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −23.3343 −0.971418 −0.485709 0.874121i \(-0.661438\pi\)
−0.485709 + 0.874121i \(0.661438\pi\)
\(578\) 0 0
\(579\) 46.0609 1.91423
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 14.7806 0.612148
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 22.8470 0.942997 0.471498 0.881867i \(-0.343713\pi\)
0.471498 + 0.881867i \(0.343713\pi\)
\(588\) 0 0
\(589\) −29.8454 −1.22976
\(590\) 0 0
\(591\) 35.9532 1.47892
\(592\) 0 0
\(593\) 14.4810 0.594664 0.297332 0.954774i \(-0.403903\pi\)
0.297332 + 0.954774i \(0.403903\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 56.6565 2.31880
\(598\) 0 0
\(599\) −3.13524 −0.128102 −0.0640512 0.997947i \(-0.520402\pi\)
−0.0640512 + 0.997947i \(0.520402\pi\)
\(600\) 0 0
\(601\) 32.2118 1.31395 0.656973 0.753914i \(-0.271836\pi\)
0.656973 + 0.753914i \(0.271836\pi\)
\(602\) 0 0
\(603\) −6.01131 −0.244799
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −11.1586 −0.452913 −0.226456 0.974021i \(-0.572714\pi\)
−0.226456 + 0.974021i \(0.572714\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 13.5847 0.549578
\(612\) 0 0
\(613\) −7.53705 −0.304418 −0.152209 0.988348i \(-0.548639\pi\)
−0.152209 + 0.988348i \(0.548639\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 22.7728 0.916797 0.458399 0.888747i \(-0.348423\pi\)
0.458399 + 0.888747i \(0.348423\pi\)
\(618\) 0 0
\(619\) −0.964766 −0.0387772 −0.0193886 0.999812i \(-0.506172\pi\)
−0.0193886 + 0.999812i \(0.506172\pi\)
\(620\) 0 0
\(621\) −7.06132 −0.283361
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −35.3849 −1.41314
\(628\) 0 0
\(629\) 37.6395 1.50079
\(630\) 0 0
\(631\) 6.90059 0.274708 0.137354 0.990522i \(-0.456140\pi\)
0.137354 + 0.990522i \(0.456140\pi\)
\(632\) 0 0
\(633\) −19.3255 −0.768121
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 29.5948 1.17075
\(640\) 0 0
\(641\) −2.14433 −0.0846961 −0.0423481 0.999103i \(-0.513484\pi\)
−0.0423481 + 0.999103i \(0.513484\pi\)
\(642\) 0 0
\(643\) 27.4980 1.08441 0.542207 0.840245i \(-0.317589\pi\)
0.542207 + 0.840245i \(0.317589\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −30.9482 −1.21670 −0.608350 0.793669i \(-0.708168\pi\)
−0.608350 + 0.793669i \(0.708168\pi\)
\(648\) 0 0
\(649\) −18.2467 −0.716245
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 39.4448 1.54360 0.771798 0.635868i \(-0.219358\pi\)
0.771798 + 0.635868i \(0.219358\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −26.0481 −1.01623
\(658\) 0 0
\(659\) 8.80237 0.342892 0.171446 0.985194i \(-0.445156\pi\)
0.171446 + 0.985194i \(0.445156\pi\)
\(660\) 0 0
\(661\) 28.1095 1.09333 0.546667 0.837350i \(-0.315896\pi\)
0.546667 + 0.837350i \(0.315896\pi\)
\(662\) 0 0
\(663\) 81.3889 3.16088
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −5.41046 −0.209494
\(668\) 0 0
\(669\) 8.86787 0.342852
\(670\) 0 0
\(671\) −26.5317 −1.02424
\(672\) 0 0
\(673\) 14.7933 0.570241 0.285121 0.958492i \(-0.407966\pi\)
0.285121 + 0.958492i \(0.407966\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 2.71162 0.104216 0.0521080 0.998641i \(-0.483406\pi\)
0.0521080 + 0.998641i \(0.483406\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −26.1218 −1.00099
\(682\) 0 0
\(683\) 1.62633 0.0622297 0.0311149 0.999516i \(-0.490094\pi\)
0.0311149 + 0.999516i \(0.490094\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −24.6880 −0.941908
\(688\) 0 0
\(689\) −24.0649 −0.916799
\(690\) 0 0
\(691\) 50.8821 1.93565 0.967823 0.251632i \(-0.0809672\pi\)
0.967823 + 0.251632i \(0.0809672\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 45.9501 1.74048
\(698\) 0 0
\(699\) 46.5317 1.75999
\(700\) 0 0
\(701\) 7.93106 0.299552 0.149776 0.988720i \(-0.452145\pi\)
0.149776 + 0.988720i \(0.452145\pi\)
\(702\) 0 0
\(703\) 26.6434 1.00488
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 17.4922 0.656933 0.328466 0.944516i \(-0.393468\pi\)
0.328466 + 0.944516i \(0.393468\pi\)
\(710\) 0 0
\(711\) −5.76067 −0.216042
\(712\) 0 0
\(713\) 17.4197 0.652374
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 49.2185 1.83810
\(718\) 0 0
\(719\) 18.1294 0.676111 0.338055 0.941126i \(-0.390231\pi\)
0.338055 + 0.941126i \(0.390231\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 28.0592 1.04353
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −2.42538 −0.0899523 −0.0449761 0.998988i \(-0.514321\pi\)
−0.0449761 + 0.998988i \(0.514321\pi\)
\(728\) 0 0
\(729\) −2.17729 −0.0806402
\(730\) 0 0
\(731\) 34.0248 1.25845
\(732\) 0 0
\(733\) 19.7659 0.730069 0.365035 0.930994i \(-0.381057\pi\)
0.365035 + 0.930994i \(0.381057\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −12.2219 −0.450198
\(738\) 0 0
\(739\) 39.9065 1.46799 0.733993 0.679157i \(-0.237655\pi\)
0.733993 + 0.679157i \(0.237655\pi\)
\(740\) 0 0
\(741\) 57.6118 2.11642
\(742\) 0 0
\(743\) −33.0921 −1.21403 −0.607016 0.794689i \(-0.707634\pi\)
−0.607016 + 0.794689i \(0.707634\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −18.9251 −0.692433
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −6.98531 −0.254898 −0.127449 0.991845i \(-0.540679\pi\)
−0.127449 + 0.991845i \(0.540679\pi\)
\(752\) 0 0
\(753\) −51.8391 −1.88912
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 24.4796 0.889725 0.444863 0.895599i \(-0.353253\pi\)
0.444863 + 0.895599i \(0.353253\pi\)
\(758\) 0 0
\(759\) 20.6530 0.749655
\(760\) 0 0
\(761\) 9.39511 0.340573 0.170286 0.985395i \(-0.445531\pi\)
0.170286 + 0.985395i \(0.445531\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 29.7082 1.07270
\(768\) 0 0
\(769\) 29.7183 1.07167 0.535835 0.844323i \(-0.319997\pi\)
0.535835 + 0.844323i \(0.319997\pi\)
\(770\) 0 0
\(771\) 28.8590 1.03933
\(772\) 0 0
\(773\) 38.2108 1.37435 0.687173 0.726494i \(-0.258852\pi\)
0.687173 + 0.726494i \(0.258852\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 32.5261 1.16537
\(780\) 0 0
\(781\) 60.1705 2.15307
\(782\) 0 0
\(783\) 5.53122 0.197670
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −24.8114 −0.884431 −0.442215 0.896909i \(-0.645807\pi\)
−0.442215 + 0.896909i \(0.645807\pi\)
\(788\) 0 0
\(789\) −63.0149 −2.24339
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 43.1974 1.53399
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 22.9296 0.812210 0.406105 0.913826i \(-0.366887\pi\)
0.406105 + 0.913826i \(0.366887\pi\)
\(798\) 0 0
\(799\) −14.7507 −0.521841
\(800\) 0 0
\(801\) 19.7400 0.697479
\(802\) 0 0
\(803\) −52.9595 −1.86890
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −5.61441 −0.197637
\(808\) 0 0
\(809\) −34.2720 −1.20494 −0.602470 0.798142i \(-0.705817\pi\)
−0.602470 + 0.798142i \(0.705817\pi\)
\(810\) 0 0
\(811\) −37.5925 −1.32005 −0.660025 0.751244i \(-0.729454\pi\)
−0.660025 + 0.751244i \(0.729454\pi\)
\(812\) 0 0
\(813\) −14.7358 −0.516807
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 24.0847 0.842618
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 31.2643 1.09113 0.545565 0.838068i \(-0.316315\pi\)
0.545565 + 0.838068i \(0.316315\pi\)
\(822\) 0 0
\(823\) 18.4452 0.642959 0.321480 0.946916i \(-0.395820\pi\)
0.321480 + 0.946916i \(0.395820\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −14.5931 −0.507450 −0.253725 0.967276i \(-0.581656\pi\)
−0.253725 + 0.967276i \(0.581656\pi\)
\(828\) 0 0
\(829\) 12.1057 0.420448 0.210224 0.977653i \(-0.432581\pi\)
0.210224 + 0.977653i \(0.432581\pi\)
\(830\) 0 0
\(831\) −7.10783 −0.246568
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −17.8085 −0.615553
\(838\) 0 0
\(839\) 13.8245 0.477276 0.238638 0.971109i \(-0.423299\pi\)
0.238638 + 0.971109i \(0.423299\pi\)
\(840\) 0 0
\(841\) −24.7619 −0.853859
\(842\) 0 0
\(843\) 5.57836 0.192129
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −27.3295 −0.937946
\(850\) 0 0
\(851\) −15.5508 −0.533076
\(852\) 0 0
\(853\) 15.7912 0.540680 0.270340 0.962765i \(-0.412864\pi\)
0.270340 + 0.962765i \(0.412864\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 13.4837 0.460593 0.230297 0.973120i \(-0.426030\pi\)
0.230297 + 0.973120i \(0.426030\pi\)
\(858\) 0 0
\(859\) 23.8300 0.813071 0.406535 0.913635i \(-0.366737\pi\)
0.406535 + 0.913635i \(0.366737\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −2.35592 −0.0801966 −0.0400983 0.999196i \(-0.512767\pi\)
−0.0400983 + 0.999196i \(0.512767\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −51.2469 −1.74044
\(868\) 0 0
\(869\) −11.7123 −0.397312
\(870\) 0 0
\(871\) 19.8990 0.674251
\(872\) 0 0
\(873\) −30.4216 −1.02962
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −26.2892 −0.887725 −0.443862 0.896095i \(-0.646392\pi\)
−0.443862 + 0.896095i \(0.646392\pi\)
\(878\) 0 0
\(879\) −56.0352 −1.89002
\(880\) 0 0
\(881\) 46.2476 1.55812 0.779061 0.626948i \(-0.215696\pi\)
0.779061 + 0.626948i \(0.215696\pi\)
\(882\) 0 0
\(883\) −40.1184 −1.35009 −0.675045 0.737777i \(-0.735876\pi\)
−0.675045 + 0.737777i \(0.735876\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −32.7685 −1.10026 −0.550130 0.835079i \(-0.685422\pi\)
−0.550130 + 0.835079i \(0.685422\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −40.2178 −1.34735
\(892\) 0 0
\(893\) −10.4414 −0.349407
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −33.6260 −1.12274
\(898\) 0 0
\(899\) −13.6451 −0.455090
\(900\) 0 0
\(901\) 26.1304 0.870529
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −5.48710 −0.182196 −0.0910980 0.995842i \(-0.529038\pi\)
−0.0910980 + 0.995842i \(0.529038\pi\)
\(908\) 0 0
\(909\) 6.09031 0.202003
\(910\) 0 0
\(911\) −38.3276 −1.26985 −0.634924 0.772574i \(-0.718969\pi\)
−0.634924 + 0.772574i \(0.718969\pi\)
\(912\) 0 0
\(913\) −38.4775 −1.27342
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −38.6094 −1.27361 −0.636804 0.771026i \(-0.719744\pi\)
−0.636804 + 0.771026i \(0.719744\pi\)
\(920\) 0 0
\(921\) −26.0747 −0.859189
\(922\) 0 0
\(923\) −97.9662 −3.22460
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 21.1942 0.696108
\(928\) 0 0
\(929\) 39.3387 1.29066 0.645331 0.763903i \(-0.276720\pi\)
0.645331 + 0.763903i \(0.276720\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −18.5612 −0.607666
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 18.4619 0.603124 0.301562 0.953447i \(-0.402492\pi\)
0.301562 + 0.953447i \(0.402492\pi\)
\(938\) 0 0
\(939\) −3.83532 −0.125161
\(940\) 0 0
\(941\) −37.6005 −1.22574 −0.612870 0.790184i \(-0.709985\pi\)
−0.612870 + 0.790184i \(0.709985\pi\)
\(942\) 0 0
\(943\) −18.9844 −0.618216
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 39.4215 1.28103 0.640513 0.767947i \(-0.278722\pi\)
0.640513 + 0.767947i \(0.278722\pi\)
\(948\) 0 0
\(949\) 86.2258 2.79901
\(950\) 0 0
\(951\) 13.0272 0.422437
\(952\) 0 0
\(953\) −24.4541 −0.792148 −0.396074 0.918219i \(-0.629628\pi\)
−0.396074 + 0.918219i \(0.629628\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −16.1777 −0.522952
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 12.9323 0.417171
\(962\) 0 0
\(963\) 18.8940 0.608852
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −22.5757 −0.725986 −0.362993 0.931792i \(-0.618245\pi\)
−0.362993 + 0.931792i \(0.618245\pi\)
\(968\) 0 0
\(969\) −62.5566 −2.00961
\(970\) 0 0
\(971\) 35.1636 1.12845 0.564226 0.825620i \(-0.309174\pi\)
0.564226 + 0.825620i \(0.309174\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −26.4910 −0.847522 −0.423761 0.905774i \(-0.639290\pi\)
−0.423761 + 0.905774i \(0.639290\pi\)
\(978\) 0 0
\(979\) 40.1343 1.28270
\(980\) 0 0
\(981\) −14.7494 −0.470913
\(982\) 0 0
\(983\) −27.3380 −0.871947 −0.435974 0.899960i \(-0.643596\pi\)
−0.435974 + 0.899960i \(0.643596\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −14.0574 −0.447000
\(990\) 0 0
\(991\) 59.9900 1.90565 0.952823 0.303527i \(-0.0981644\pi\)
0.952823 + 0.303527i \(0.0981644\pi\)
\(992\) 0 0
\(993\) 30.8024 0.977486
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 22.4645 0.711458 0.355729 0.934589i \(-0.384233\pi\)
0.355729 + 0.934589i \(0.384233\pi\)
\(998\) 0 0
\(999\) 15.8979 0.502989
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.cl.1.2 4
5.4 even 2 1960.2.a.y.1.3 yes 4
7.6 odd 2 9800.2.a.cs.1.3 4
20.19 odd 2 3920.2.a.cd.1.2 4
35.4 even 6 1960.2.q.x.961.2 8
35.9 even 6 1960.2.q.x.361.2 8
35.19 odd 6 1960.2.q.y.361.3 8
35.24 odd 6 1960.2.q.y.961.3 8
35.34 odd 2 1960.2.a.x.1.2 4
140.139 even 2 3920.2.a.ce.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1960.2.a.x.1.2 4 35.34 odd 2
1960.2.a.y.1.3 yes 4 5.4 even 2
1960.2.q.x.361.2 8 35.9 even 6
1960.2.q.x.961.2 8 35.4 even 6
1960.2.q.y.361.3 8 35.19 odd 6
1960.2.q.y.961.3 8 35.24 odd 6
3920.2.a.cd.1.2 4 20.19 odd 2
3920.2.a.ce.1.3 4 140.139 even 2
9800.2.a.cl.1.2 4 1.1 even 1 trivial
9800.2.a.cs.1.3 4 7.6 odd 2