Properties

Label 4368.2.a.s.1.1
Level $4368$
Weight $2$
Character 4368.1
Self dual yes
Analytic conductor $34.879$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4368.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(34.8786556029\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4368.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} -1.00000 q^{15} -1.00000 q^{17} -7.00000 q^{19} +1.00000 q^{21} -3.00000 q^{23} -4.00000 q^{25} +1.00000 q^{27} -3.00000 q^{29} -8.00000 q^{31} +1.00000 q^{33} -1.00000 q^{35} +7.00000 q^{37} +1.00000 q^{39} +8.00000 q^{41} -7.00000 q^{43} -1.00000 q^{45} -8.00000 q^{47} +1.00000 q^{49} -1.00000 q^{51} -10.0000 q^{53} -1.00000 q^{55} -7.00000 q^{57} -4.00000 q^{59} +7.00000 q^{61} +1.00000 q^{63} -1.00000 q^{65} -2.00000 q^{67} -3.00000 q^{69} -4.00000 q^{71} -1.00000 q^{73} -4.00000 q^{75} +1.00000 q^{77} -2.00000 q^{79} +1.00000 q^{81} +6.00000 q^{83} +1.00000 q^{85} -3.00000 q^{87} +14.0000 q^{89} +1.00000 q^{91} -8.00000 q^{93} +7.00000 q^{95} -14.0000 q^{97} +1.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −1.00000 −0.242536 −0.121268 0.992620i \(-0.538696\pi\)
−0.121268 + 0.992620i \(0.538696\pi\)
\(18\) 0 0
\(19\) −7.00000 −1.60591 −0.802955 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) 1.00000 0.174078
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) 7.00000 1.15079 0.575396 0.817875i \(-0.304848\pi\)
0.575396 + 0.817875i \(0.304848\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 8.00000 1.24939 0.624695 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 0 0
\(43\) −7.00000 −1.06749 −0.533745 0.845645i \(-0.679216\pi\)
−0.533745 + 0.845645i \(0.679216\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −1.00000 −0.140028
\(52\) 0 0
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) −7.00000 −0.927173
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 7.00000 0.896258 0.448129 0.893969i \(-0.352090\pi\)
0.448129 + 0.893969i \(0.352090\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) 0 0
\(69\) −3.00000 −0.361158
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 0 0
\(73\) −1.00000 −0.117041 −0.0585206 0.998286i \(-0.518638\pi\)
−0.0585206 + 0.998286i \(0.518638\pi\)
\(74\) 0 0
\(75\) −4.00000 −0.461880
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) −2.00000 −0.225018 −0.112509 0.993651i \(-0.535889\pi\)
−0.112509 + 0.993651i \(0.535889\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) 1.00000 0.108465
\(86\) 0 0
\(87\) −3.00000 −0.321634
\(88\) 0 0
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) −8.00000 −0.829561
\(94\) 0 0
\(95\) 7.00000 0.718185
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) 4.00000 0.398015 0.199007 0.979998i \(-0.436228\pi\)
0.199007 + 0.979998i \(0.436228\pi\)
\(102\) 0 0
\(103\) 5.00000 0.492665 0.246332 0.969185i \(-0.420775\pi\)
0.246332 + 0.969185i \(0.420775\pi\)
\(104\) 0 0
\(105\) −1.00000 −0.0975900
\(106\) 0 0
\(107\) 18.0000 1.74013 0.870063 0.492941i \(-0.164078\pi\)
0.870063 + 0.492941i \(0.164078\pi\)
\(108\) 0 0
\(109\) −11.0000 −1.05361 −0.526804 0.849987i \(-0.676610\pi\)
−0.526804 + 0.849987i \(0.676610\pi\)
\(110\) 0 0
\(111\) 7.00000 0.664411
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 3.00000 0.279751
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) −1.00000 −0.0916698
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 0 0
\(123\) 8.00000 0.721336
\(124\) 0 0
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) −18.0000 −1.59724 −0.798621 0.601834i \(-0.794437\pi\)
−0.798621 + 0.601834i \(0.794437\pi\)
\(128\) 0 0
\(129\) −7.00000 −0.616316
\(130\) 0 0
\(131\) 5.00000 0.436852 0.218426 0.975854i \(-0.429908\pi\)
0.218426 + 0.975854i \(0.429908\pi\)
\(132\) 0 0
\(133\) −7.00000 −0.606977
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −19.0000 −1.62328 −0.811640 0.584158i \(-0.801425\pi\)
−0.811640 + 0.584158i \(0.801425\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) 1.00000 0.0836242
\(144\) 0 0
\(145\) 3.00000 0.249136
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 9.00000 0.732410 0.366205 0.930534i \(-0.380657\pi\)
0.366205 + 0.930534i \(0.380657\pi\)
\(152\) 0 0
\(153\) −1.00000 −0.0808452
\(154\) 0 0
\(155\) 8.00000 0.642575
\(156\) 0 0
\(157\) −13.0000 −1.03751 −0.518756 0.854922i \(-0.673605\pi\)
−0.518756 + 0.854922i \(0.673605\pi\)
\(158\) 0 0
\(159\) −10.0000 −0.793052
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) 0 0
\(163\) −2.00000 −0.156652 −0.0783260 0.996928i \(-0.524958\pi\)
−0.0783260 + 0.996928i \(0.524958\pi\)
\(164\) 0 0
\(165\) −1.00000 −0.0778499
\(166\) 0 0
\(167\) −15.0000 −1.16073 −0.580367 0.814355i \(-0.697091\pi\)
−0.580367 + 0.814355i \(0.697091\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −7.00000 −0.535303
\(172\) 0 0
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) 0 0
\(177\) −4.00000 −0.300658
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) 0 0
\(183\) 7.00000 0.517455
\(184\) 0 0
\(185\) −7.00000 −0.514650
\(186\) 0 0
\(187\) −1.00000 −0.0731272
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −19.0000 −1.37479 −0.687396 0.726283i \(-0.741246\pi\)
−0.687396 + 0.726283i \(0.741246\pi\)
\(192\) 0 0
\(193\) 16.0000 1.15171 0.575853 0.817554i \(-0.304670\pi\)
0.575853 + 0.817554i \(0.304670\pi\)
\(194\) 0 0
\(195\) −1.00000 −0.0716115
\(196\) 0 0
\(197\) −12.0000 −0.854965 −0.427482 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) 0 0
\(199\) −1.00000 −0.0708881 −0.0354441 0.999372i \(-0.511285\pi\)
−0.0354441 + 0.999372i \(0.511285\pi\)
\(200\) 0 0
\(201\) −2.00000 −0.141069
\(202\) 0 0
\(203\) −3.00000 −0.210559
\(204\) 0 0
\(205\) −8.00000 −0.558744
\(206\) 0 0
\(207\) −3.00000 −0.208514
\(208\) 0 0
\(209\) −7.00000 −0.484200
\(210\) 0 0
\(211\) 3.00000 0.206529 0.103264 0.994654i \(-0.467071\pi\)
0.103264 + 0.994654i \(0.467071\pi\)
\(212\) 0 0
\(213\) −4.00000 −0.274075
\(214\) 0 0
\(215\) 7.00000 0.477396
\(216\) 0 0
\(217\) −8.00000 −0.543075
\(218\) 0 0
\(219\) −1.00000 −0.0675737
\(220\) 0 0
\(221\) −1.00000 −0.0672673
\(222\) 0 0
\(223\) −14.0000 −0.937509 −0.468755 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) 0 0
\(225\) −4.00000 −0.266667
\(226\) 0 0
\(227\) 22.0000 1.46019 0.730096 0.683345i \(-0.239475\pi\)
0.730096 + 0.683345i \(0.239475\pi\)
\(228\) 0 0
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) 1.00000 0.0657952
\(232\) 0 0
\(233\) 12.0000 0.786146 0.393073 0.919507i \(-0.371412\pi\)
0.393073 + 0.919507i \(0.371412\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) 0 0
\(237\) −2.00000 −0.129914
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) −7.00000 −0.445399
\(248\) 0 0
\(249\) 6.00000 0.380235
\(250\) 0 0
\(251\) −7.00000 −0.441836 −0.220918 0.975292i \(-0.570905\pi\)
−0.220918 + 0.975292i \(0.570905\pi\)
\(252\) 0 0
\(253\) −3.00000 −0.188608
\(254\) 0 0
\(255\) 1.00000 0.0626224
\(256\) 0 0
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) 7.00000 0.434959
\(260\) 0 0
\(261\) −3.00000 −0.185695
\(262\) 0 0
\(263\) 4.00000 0.246651 0.123325 0.992366i \(-0.460644\pi\)
0.123325 + 0.992366i \(0.460644\pi\)
\(264\) 0 0
\(265\) 10.0000 0.614295
\(266\) 0 0
\(267\) 14.0000 0.856786
\(268\) 0 0
\(269\) −2.00000 −0.121942 −0.0609711 0.998140i \(-0.519420\pi\)
−0.0609711 + 0.998140i \(0.519420\pi\)
\(270\) 0 0
\(271\) −2.00000 −0.121491 −0.0607457 0.998153i \(-0.519348\pi\)
−0.0607457 + 0.998153i \(0.519348\pi\)
\(272\) 0 0
\(273\) 1.00000 0.0605228
\(274\) 0 0
\(275\) −4.00000 −0.241209
\(276\) 0 0
\(277\) −14.0000 −0.841178 −0.420589 0.907251i \(-0.638177\pi\)
−0.420589 + 0.907251i \(0.638177\pi\)
\(278\) 0 0
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) −32.0000 −1.90220 −0.951101 0.308879i \(-0.900046\pi\)
−0.951101 + 0.308879i \(0.900046\pi\)
\(284\) 0 0
\(285\) 7.00000 0.414644
\(286\) 0 0
\(287\) 8.00000 0.472225
\(288\) 0 0
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) −14.0000 −0.820695
\(292\) 0 0
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) 0 0
\(297\) 1.00000 0.0580259
\(298\) 0 0
\(299\) −3.00000 −0.173494
\(300\) 0 0
\(301\) −7.00000 −0.403473
\(302\) 0 0
\(303\) 4.00000 0.229794
\(304\) 0 0
\(305\) −7.00000 −0.400819
\(306\) 0 0
\(307\) 32.0000 1.82634 0.913168 0.407583i \(-0.133628\pi\)
0.913168 + 0.407583i \(0.133628\pi\)
\(308\) 0 0
\(309\) 5.00000 0.284440
\(310\) 0 0
\(311\) −30.0000 −1.70114 −0.850572 0.525859i \(-0.823744\pi\)
−0.850572 + 0.525859i \(0.823744\pi\)
\(312\) 0 0
\(313\) −24.0000 −1.35656 −0.678280 0.734803i \(-0.737274\pi\)
−0.678280 + 0.734803i \(0.737274\pi\)
\(314\) 0 0
\(315\) −1.00000 −0.0563436
\(316\) 0 0
\(317\) 12.0000 0.673987 0.336994 0.941507i \(-0.390590\pi\)
0.336994 + 0.941507i \(0.390590\pi\)
\(318\) 0 0
\(319\) −3.00000 −0.167968
\(320\) 0 0
\(321\) 18.0000 1.00466
\(322\) 0 0
\(323\) 7.00000 0.389490
\(324\) 0 0
\(325\) −4.00000 −0.221880
\(326\) 0 0
\(327\) −11.0000 −0.608301
\(328\) 0 0
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) 10.0000 0.549650 0.274825 0.961494i \(-0.411380\pi\)
0.274825 + 0.961494i \(0.411380\pi\)
\(332\) 0 0
\(333\) 7.00000 0.383598
\(334\) 0 0
\(335\) 2.00000 0.109272
\(336\) 0 0
\(337\) −21.0000 −1.14394 −0.571971 0.820274i \(-0.693821\pi\)
−0.571971 + 0.820274i \(0.693821\pi\)
\(338\) 0 0
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) −8.00000 −0.433224
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 3.00000 0.161515
\(346\) 0 0
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) −18.0000 −0.963518 −0.481759 0.876304i \(-0.660002\pi\)
−0.481759 + 0.876304i \(0.660002\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 0 0
\(355\) 4.00000 0.212298
\(356\) 0 0
\(357\) −1.00000 −0.0529256
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) 0 0
\(363\) −10.0000 −0.524864
\(364\) 0 0
\(365\) 1.00000 0.0523424
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 0 0
\(369\) 8.00000 0.416463
\(370\) 0 0
\(371\) −10.0000 −0.519174
\(372\) 0 0
\(373\) −12.0000 −0.621336 −0.310668 0.950518i \(-0.600553\pi\)
−0.310668 + 0.950518i \(0.600553\pi\)
\(374\) 0 0
\(375\) 9.00000 0.464758
\(376\) 0 0
\(377\) −3.00000 −0.154508
\(378\) 0 0
\(379\) −12.0000 −0.616399 −0.308199 0.951322i \(-0.599726\pi\)
−0.308199 + 0.951322i \(0.599726\pi\)
\(380\) 0 0
\(381\) −18.0000 −0.922168
\(382\) 0 0
\(383\) 9.00000 0.459879 0.229939 0.973205i \(-0.426147\pi\)
0.229939 + 0.973205i \(0.426147\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) 0 0
\(387\) −7.00000 −0.355830
\(388\) 0 0
\(389\) −2.00000 −0.101404 −0.0507020 0.998714i \(-0.516146\pi\)
−0.0507020 + 0.998714i \(0.516146\pi\)
\(390\) 0 0
\(391\) 3.00000 0.151717
\(392\) 0 0
\(393\) 5.00000 0.252217
\(394\) 0 0
\(395\) 2.00000 0.100631
\(396\) 0 0
\(397\) 18.0000 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) 0 0
\(399\) −7.00000 −0.350438
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 0 0
\(403\) −8.00000 −0.398508
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 7.00000 0.346977
\(408\) 0 0
\(409\) 5.00000 0.247234 0.123617 0.992330i \(-0.460551\pi\)
0.123617 + 0.992330i \(0.460551\pi\)
\(410\) 0 0
\(411\) −19.0000 −0.937201
\(412\) 0 0
\(413\) −4.00000 −0.196827
\(414\) 0 0
\(415\) −6.00000 −0.294528
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −1.00000 −0.0488532 −0.0244266 0.999702i \(-0.507776\pi\)
−0.0244266 + 0.999702i \(0.507776\pi\)
\(420\) 0 0
\(421\) 34.0000 1.65706 0.828529 0.559946i \(-0.189178\pi\)
0.828529 + 0.559946i \(0.189178\pi\)
\(422\) 0 0
\(423\) −8.00000 −0.388973
\(424\) 0 0
\(425\) 4.00000 0.194029
\(426\) 0 0
\(427\) 7.00000 0.338754
\(428\) 0 0
\(429\) 1.00000 0.0482805
\(430\) 0 0
\(431\) 38.0000 1.83040 0.915198 0.403005i \(-0.132034\pi\)
0.915198 + 0.403005i \(0.132034\pi\)
\(432\) 0 0
\(433\) 12.0000 0.576683 0.288342 0.957528i \(-0.406896\pi\)
0.288342 + 0.957528i \(0.406896\pi\)
\(434\) 0 0
\(435\) 3.00000 0.143839
\(436\) 0 0
\(437\) 21.0000 1.00457
\(438\) 0 0
\(439\) 17.0000 0.811366 0.405683 0.914014i \(-0.367034\pi\)
0.405683 + 0.914014i \(0.367034\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 10.0000 0.475114 0.237557 0.971374i \(-0.423653\pi\)
0.237557 + 0.971374i \(0.423653\pi\)
\(444\) 0 0
\(445\) −14.0000 −0.663664
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −39.0000 −1.84052 −0.920262 0.391303i \(-0.872024\pi\)
−0.920262 + 0.391303i \(0.872024\pi\)
\(450\) 0 0
\(451\) 8.00000 0.376705
\(452\) 0 0
\(453\) 9.00000 0.422857
\(454\) 0 0
\(455\) −1.00000 −0.0468807
\(456\) 0 0
\(457\) −12.0000 −0.561336 −0.280668 0.959805i \(-0.590556\pi\)
−0.280668 + 0.959805i \(0.590556\pi\)
\(458\) 0 0
\(459\) −1.00000 −0.0466760
\(460\) 0 0
\(461\) 21.0000 0.978068 0.489034 0.872265i \(-0.337349\pi\)
0.489034 + 0.872265i \(0.337349\pi\)
\(462\) 0 0
\(463\) 17.0000 0.790057 0.395029 0.918669i \(-0.370735\pi\)
0.395029 + 0.918669i \(0.370735\pi\)
\(464\) 0 0
\(465\) 8.00000 0.370991
\(466\) 0 0
\(467\) 25.0000 1.15686 0.578431 0.815731i \(-0.303665\pi\)
0.578431 + 0.815731i \(0.303665\pi\)
\(468\) 0 0
\(469\) −2.00000 −0.0923514
\(470\) 0 0
\(471\) −13.0000 −0.599008
\(472\) 0 0
\(473\) −7.00000 −0.321860
\(474\) 0 0
\(475\) 28.0000 1.28473
\(476\) 0 0
\(477\) −10.0000 −0.457869
\(478\) 0 0
\(479\) −5.00000 −0.228456 −0.114228 0.993455i \(-0.536439\pi\)
−0.114228 + 0.993455i \(0.536439\pi\)
\(480\) 0 0
\(481\) 7.00000 0.319173
\(482\) 0 0
\(483\) −3.00000 −0.136505
\(484\) 0 0
\(485\) 14.0000 0.635707
\(486\) 0 0
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) 0 0
\(489\) −2.00000 −0.0904431
\(490\) 0 0
\(491\) 30.0000 1.35388 0.676941 0.736038i \(-0.263305\pi\)
0.676941 + 0.736038i \(0.263305\pi\)
\(492\) 0 0
\(493\) 3.00000 0.135113
\(494\) 0 0
\(495\) −1.00000 −0.0449467
\(496\) 0 0
\(497\) −4.00000 −0.179425
\(498\) 0 0
\(499\) −12.0000 −0.537194 −0.268597 0.963253i \(-0.586560\pi\)
−0.268597 + 0.963253i \(0.586560\pi\)
\(500\) 0 0
\(501\) −15.0000 −0.670151
\(502\) 0 0
\(503\) −34.0000 −1.51599 −0.757993 0.652263i \(-0.773820\pi\)
−0.757993 + 0.652263i \(0.773820\pi\)
\(504\) 0 0
\(505\) −4.00000 −0.177998
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) −3.00000 −0.132973 −0.0664863 0.997787i \(-0.521179\pi\)
−0.0664863 + 0.997787i \(0.521179\pi\)
\(510\) 0 0
\(511\) −1.00000 −0.0442374
\(512\) 0 0
\(513\) −7.00000 −0.309058
\(514\) 0 0
\(515\) −5.00000 −0.220326
\(516\) 0 0
\(517\) −8.00000 −0.351840
\(518\) 0 0
\(519\) 14.0000 0.614532
\(520\) 0 0
\(521\) −35.0000 −1.53338 −0.766689 0.642019i \(-0.778097\pi\)
−0.766689 + 0.642019i \(0.778097\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 0 0
\(525\) −4.00000 −0.174574
\(526\) 0 0
\(527\) 8.00000 0.348485
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) 8.00000 0.346518
\(534\) 0 0
\(535\) −18.0000 −0.778208
\(536\) 0 0
\(537\) −12.0000 −0.517838
\(538\) 0 0
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) −43.0000 −1.84871 −0.924357 0.381528i \(-0.875398\pi\)
−0.924357 + 0.381528i \(0.875398\pi\)
\(542\) 0 0
\(543\) −22.0000 −0.944110
\(544\) 0 0
\(545\) 11.0000 0.471188
\(546\) 0 0
\(547\) 4.00000 0.171028 0.0855138 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) 0 0
\(549\) 7.00000 0.298753
\(550\) 0 0
\(551\) 21.0000 0.894630
\(552\) 0 0
\(553\) −2.00000 −0.0850487
\(554\) 0 0
\(555\) −7.00000 −0.297133
\(556\) 0 0
\(557\) −24.0000 −1.01691 −0.508456 0.861088i \(-0.669784\pi\)
−0.508456 + 0.861088i \(0.669784\pi\)
\(558\) 0 0
\(559\) −7.00000 −0.296068
\(560\) 0 0
\(561\) −1.00000 −0.0422200
\(562\) 0 0
\(563\) −19.0000 −0.800755 −0.400377 0.916350i \(-0.631121\pi\)
−0.400377 + 0.916350i \(0.631121\pi\)
\(564\) 0 0
\(565\) −6.00000 −0.252422
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) −16.0000 −0.669579 −0.334790 0.942293i \(-0.608665\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) 0 0
\(573\) −19.0000 −0.793736
\(574\) 0 0
\(575\) 12.0000 0.500435
\(576\) 0 0
\(577\) 42.0000 1.74848 0.874241 0.485491i \(-0.161359\pi\)
0.874241 + 0.485491i \(0.161359\pi\)
\(578\) 0 0
\(579\) 16.0000 0.664937
\(580\) 0 0
\(581\) 6.00000 0.248922
\(582\) 0 0
\(583\) −10.0000 −0.414158
\(584\) 0 0
\(585\) −1.00000 −0.0413449
\(586\) 0 0
\(587\) 38.0000 1.56843 0.784214 0.620491i \(-0.213066\pi\)
0.784214 + 0.620491i \(0.213066\pi\)
\(588\) 0 0
\(589\) 56.0000 2.30744
\(590\) 0 0
\(591\) −12.0000 −0.493614
\(592\) 0 0
\(593\) 30.0000 1.23195 0.615976 0.787765i \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) 0 0
\(595\) 1.00000 0.0409960
\(596\) 0 0
\(597\) −1.00000 −0.0409273
\(598\) 0 0
\(599\) 3.00000 0.122577 0.0612883 0.998120i \(-0.480479\pi\)
0.0612883 + 0.998120i \(0.480479\pi\)
\(600\) 0 0
\(601\) −28.0000 −1.14214 −0.571072 0.820900i \(-0.693472\pi\)
−0.571072 + 0.820900i \(0.693472\pi\)
\(602\) 0 0
\(603\) −2.00000 −0.0814463
\(604\) 0 0
\(605\) 10.0000 0.406558
\(606\) 0 0
\(607\) 1.00000 0.0405887 0.0202944 0.999794i \(-0.493540\pi\)
0.0202944 + 0.999794i \(0.493540\pi\)
\(608\) 0 0
\(609\) −3.00000 −0.121566
\(610\) 0 0
\(611\) −8.00000 −0.323645
\(612\) 0 0
\(613\) 37.0000 1.49442 0.747208 0.664590i \(-0.231394\pi\)
0.747208 + 0.664590i \(0.231394\pi\)
\(614\) 0 0
\(615\) −8.00000 −0.322591
\(616\) 0 0
\(617\) −33.0000 −1.32853 −0.664265 0.747497i \(-0.731255\pi\)
−0.664265 + 0.747497i \(0.731255\pi\)
\(618\) 0 0
\(619\) 1.00000 0.0401934 0.0200967 0.999798i \(-0.493603\pi\)
0.0200967 + 0.999798i \(0.493603\pi\)
\(620\) 0 0
\(621\) −3.00000 −0.120386
\(622\) 0 0
\(623\) 14.0000 0.560898
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) −7.00000 −0.279553
\(628\) 0 0
\(629\) −7.00000 −0.279108
\(630\) 0 0
\(631\) −27.0000 −1.07485 −0.537427 0.843311i \(-0.680603\pi\)
−0.537427 + 0.843311i \(0.680603\pi\)
\(632\) 0 0
\(633\) 3.00000 0.119239
\(634\) 0 0
\(635\) 18.0000 0.714308
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 0 0
\(639\) −4.00000 −0.158238
\(640\) 0 0
\(641\) 50.0000 1.97488 0.987441 0.157991i \(-0.0505015\pi\)
0.987441 + 0.157991i \(0.0505015\pi\)
\(642\) 0 0
\(643\) −17.0000 −0.670415 −0.335207 0.942144i \(-0.608806\pi\)
−0.335207 + 0.942144i \(0.608806\pi\)
\(644\) 0 0
\(645\) 7.00000 0.275625
\(646\) 0 0
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 0 0
\(649\) −4.00000 −0.157014
\(650\) 0 0
\(651\) −8.00000 −0.313545
\(652\) 0 0
\(653\) 35.0000 1.36966 0.684828 0.728705i \(-0.259877\pi\)
0.684828 + 0.728705i \(0.259877\pi\)
\(654\) 0 0
\(655\) −5.00000 −0.195366
\(656\) 0 0
\(657\) −1.00000 −0.0390137
\(658\) 0 0
\(659\) −6.00000 −0.233727 −0.116863 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 0 0
\(663\) −1.00000 −0.0388368
\(664\) 0 0
\(665\) 7.00000 0.271448
\(666\) 0 0
\(667\) 9.00000 0.348481
\(668\) 0 0
\(669\) −14.0000 −0.541271
\(670\) 0 0
\(671\) 7.00000 0.270232
\(672\) 0 0
\(673\) 7.00000 0.269830 0.134915 0.990857i \(-0.456924\pi\)
0.134915 + 0.990857i \(0.456924\pi\)
\(674\) 0 0
\(675\) −4.00000 −0.153960
\(676\) 0 0
\(677\) −22.0000 −0.845529 −0.422764 0.906240i \(-0.638940\pi\)
−0.422764 + 0.906240i \(0.638940\pi\)
\(678\) 0 0
\(679\) −14.0000 −0.537271
\(680\) 0 0
\(681\) 22.0000 0.843042
\(682\) 0 0
\(683\) 19.0000 0.727015 0.363507 0.931591i \(-0.381579\pi\)
0.363507 + 0.931591i \(0.381579\pi\)
\(684\) 0 0
\(685\) 19.0000 0.725953
\(686\) 0 0
\(687\) 10.0000 0.381524
\(688\) 0 0
\(689\) −10.0000 −0.380970
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 0 0
\(693\) 1.00000 0.0379869
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −8.00000 −0.303022
\(698\) 0 0
\(699\) 12.0000 0.453882
\(700\) 0 0
\(701\) 10.0000 0.377695 0.188847 0.982006i \(-0.439525\pi\)
0.188847 + 0.982006i \(0.439525\pi\)
\(702\) 0 0
\(703\) −49.0000 −1.84807
\(704\) 0 0
\(705\) 8.00000 0.301297
\(706\) 0 0
\(707\) 4.00000 0.150435
\(708\) 0 0
\(709\) −6.00000 −0.225335 −0.112667 0.993633i \(-0.535939\pi\)
−0.112667 + 0.993633i \(0.535939\pi\)
\(710\) 0 0
\(711\) −2.00000 −0.0750059
\(712\) 0 0
\(713\) 24.0000 0.898807
\(714\) 0 0
\(715\) −1.00000 −0.0373979
\(716\) 0 0
\(717\) −24.0000 −0.896296
\(718\) 0 0
\(719\) 42.0000 1.56634 0.783168 0.621810i \(-0.213603\pi\)
0.783168 + 0.621810i \(0.213603\pi\)
\(720\) 0 0
\(721\) 5.00000 0.186210
\(722\) 0 0
\(723\) 10.0000 0.371904
\(724\) 0 0
\(725\) 12.0000 0.445669
\(726\) 0 0
\(727\) −23.0000 −0.853023 −0.426511 0.904482i \(-0.640258\pi\)
−0.426511 + 0.904482i \(0.640258\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 7.00000 0.258904
\(732\) 0 0
\(733\) −32.0000 −1.18195 −0.590973 0.806691i \(-0.701256\pi\)
−0.590973 + 0.806691i \(0.701256\pi\)
\(734\) 0 0
\(735\) −1.00000 −0.0368856
\(736\) 0 0
\(737\) −2.00000 −0.0736709
\(738\) 0 0
\(739\) −22.0000 −0.809283 −0.404642 0.914475i \(-0.632604\pi\)
−0.404642 + 0.914475i \(0.632604\pi\)
\(740\) 0 0
\(741\) −7.00000 −0.257151
\(742\) 0 0
\(743\) −6.00000 −0.220119 −0.110059 0.993925i \(-0.535104\pi\)
−0.110059 + 0.993925i \(0.535104\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 6.00000 0.219529
\(748\) 0 0
\(749\) 18.0000 0.657706
\(750\) 0 0
\(751\) 28.0000 1.02173 0.510867 0.859660i \(-0.329324\pi\)
0.510867 + 0.859660i \(0.329324\pi\)
\(752\) 0 0
\(753\) −7.00000 −0.255094
\(754\) 0 0
\(755\) −9.00000 −0.327544
\(756\) 0 0
\(757\) 18.0000 0.654221 0.327111 0.944986i \(-0.393925\pi\)
0.327111 + 0.944986i \(0.393925\pi\)
\(758\) 0 0
\(759\) −3.00000 −0.108893
\(760\) 0 0
\(761\) 16.0000 0.580000 0.290000 0.957027i \(-0.406345\pi\)
0.290000 + 0.957027i \(0.406345\pi\)
\(762\) 0 0
\(763\) −11.0000 −0.398227
\(764\) 0 0
\(765\) 1.00000 0.0361551
\(766\) 0 0
\(767\) −4.00000 −0.144432
\(768\) 0 0
\(769\) 5.00000 0.180305 0.0901523 0.995928i \(-0.471265\pi\)
0.0901523 + 0.995928i \(0.471265\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) 0 0
\(773\) −27.0000 −0.971123 −0.485561 0.874203i \(-0.661385\pi\)
−0.485561 + 0.874203i \(0.661385\pi\)
\(774\) 0 0
\(775\) 32.0000 1.14947
\(776\) 0 0
\(777\) 7.00000 0.251124
\(778\) 0 0
\(779\) −56.0000 −2.00641
\(780\) 0 0
\(781\) −4.00000 −0.143131
\(782\) 0 0
\(783\) −3.00000 −0.107211
\(784\) 0 0
\(785\) 13.0000 0.463990
\(786\) 0 0
\(787\) −55.0000 −1.96054 −0.980269 0.197667i \(-0.936663\pi\)
−0.980269 + 0.197667i \(0.936663\pi\)
\(788\) 0 0
\(789\) 4.00000 0.142404
\(790\) 0 0
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) 7.00000 0.248577
\(794\) 0 0
\(795\) 10.0000 0.354663
\(796\) 0 0
\(797\) 16.0000 0.566749 0.283375 0.959009i \(-0.408546\pi\)
0.283375 + 0.959009i \(0.408546\pi\)
\(798\) 0 0
\(799\) 8.00000 0.283020
\(800\) 0 0
\(801\) 14.0000 0.494666
\(802\) 0 0
\(803\) −1.00000 −0.0352892
\(804\) 0 0
\(805\) 3.00000 0.105736
\(806\) 0 0
\(807\) −2.00000 −0.0704033
\(808\) 0 0
\(809\) −10.0000 −0.351581 −0.175791 0.984428i \(-0.556248\pi\)
−0.175791 + 0.984428i \(0.556248\pi\)
\(810\) 0 0
\(811\) 45.0000 1.58016 0.790082 0.613001i \(-0.210038\pi\)
0.790082 + 0.613001i \(0.210038\pi\)
\(812\) 0 0
\(813\) −2.00000 −0.0701431
\(814\) 0 0
\(815\) 2.00000 0.0700569
\(816\) 0 0
\(817\) 49.0000 1.71429
\(818\) 0 0
\(819\) 1.00000 0.0349428
\(820\) 0 0
\(821\) −24.0000 −0.837606 −0.418803 0.908077i \(-0.637550\pi\)
−0.418803 + 0.908077i \(0.637550\pi\)
\(822\) 0 0
\(823\) −20.0000 −0.697156 −0.348578 0.937280i \(-0.613335\pi\)
−0.348578 + 0.937280i \(0.613335\pi\)
\(824\) 0 0
\(825\) −4.00000 −0.139262
\(826\) 0 0
\(827\) 41.0000 1.42571 0.712855 0.701312i \(-0.247402\pi\)
0.712855 + 0.701312i \(0.247402\pi\)
\(828\) 0 0
\(829\) 7.00000 0.243120 0.121560 0.992584i \(-0.461210\pi\)
0.121560 + 0.992584i \(0.461210\pi\)
\(830\) 0 0
\(831\) −14.0000 −0.485655
\(832\) 0 0
\(833\) −1.00000 −0.0346479
\(834\) 0 0
\(835\) 15.0000 0.519096
\(836\) 0 0
\(837\) −8.00000 −0.276520
\(838\) 0 0
\(839\) 4.00000 0.138095 0.0690477 0.997613i \(-0.478004\pi\)
0.0690477 + 0.997613i \(0.478004\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) 10.0000 0.344418
\(844\) 0 0
\(845\) −1.00000 −0.0344010
\(846\) 0 0
\(847\) −10.0000 −0.343604
\(848\) 0 0
\(849\) −32.0000 −1.09824
\(850\) 0 0
\(851\) −21.0000 −0.719871
\(852\) 0 0
\(853\) −44.0000 −1.50653 −0.753266 0.657716i \(-0.771523\pi\)
−0.753266 + 0.657716i \(0.771523\pi\)
\(854\) 0 0
\(855\) 7.00000 0.239395
\(856\) 0 0
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) 44.0000 1.50126 0.750630 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) 0 0
\(861\) 8.00000 0.272639
\(862\) 0 0
\(863\) −46.0000 −1.56586 −0.782929 0.622111i \(-0.786275\pi\)
−0.782929 + 0.622111i \(0.786275\pi\)
\(864\) 0 0
\(865\) −14.0000 −0.476014
\(866\) 0 0
\(867\) −16.0000 −0.543388
\(868\) 0 0
\(869\) −2.00000 −0.0678454
\(870\) 0 0
\(871\) −2.00000 −0.0677674
\(872\) 0 0
\(873\) −14.0000 −0.473828
\(874\) 0 0
\(875\) 9.00000 0.304256
\(876\) 0 0
\(877\) 34.0000 1.14810 0.574049 0.818821i \(-0.305372\pi\)
0.574049 + 0.818821i \(0.305372\pi\)
\(878\) 0 0
\(879\) 14.0000 0.472208
\(880\) 0 0
\(881\) −49.0000 −1.65085 −0.825426 0.564510i \(-0.809065\pi\)
−0.825426 + 0.564510i \(0.809065\pi\)
\(882\) 0 0
\(883\) −31.0000 −1.04323 −0.521617 0.853180i \(-0.674671\pi\)
−0.521617 + 0.853180i \(0.674671\pi\)
\(884\) 0 0
\(885\) 4.00000 0.134459
\(886\) 0 0
\(887\) 36.0000 1.20876 0.604381 0.796696i \(-0.293421\pi\)
0.604381 + 0.796696i \(0.293421\pi\)
\(888\) 0 0
\(889\) −18.0000 −0.603701
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 0 0
\(893\) 56.0000 1.87397
\(894\) 0 0
\(895\) 12.0000 0.401116
\(896\) 0 0
\(897\) −3.00000 −0.100167
\(898\) 0 0
\(899\) 24.0000 0.800445
\(900\) 0 0
\(901\) 10.0000 0.333148
\(902\) 0 0
\(903\) −7.00000 −0.232945
\(904\) 0 0
\(905\) 22.0000 0.731305
\(906\) 0 0
\(907\) −4.00000 −0.132818 −0.0664089 0.997792i \(-0.521154\pi\)
−0.0664089 + 0.997792i \(0.521154\pi\)
\(908\) 0 0
\(909\) 4.00000 0.132672
\(910\) 0 0
\(911\) −13.0000 −0.430709 −0.215355 0.976536i \(-0.569091\pi\)
−0.215355 + 0.976536i \(0.569091\pi\)
\(912\) 0 0
\(913\) 6.00000 0.198571
\(914\) 0 0
\(915\) −7.00000 −0.231413
\(916\) 0 0
\(917\) 5.00000 0.165115
\(918\) 0 0
\(919\) −38.0000 −1.25350 −0.626752 0.779219i \(-0.715616\pi\)
−0.626752 + 0.779219i \(0.715616\pi\)
\(920\) 0 0
\(921\) 32.0000 1.05444
\(922\) 0 0
\(923\) −4.00000 −0.131662
\(924\) 0 0
\(925\) −28.0000 −0.920634
\(926\) 0 0
\(927\) 5.00000 0.164222
\(928\) 0 0
\(929\) 28.0000 0.918650 0.459325 0.888268i \(-0.348091\pi\)
0.459325 + 0.888268i \(0.348091\pi\)
\(930\) 0 0
\(931\) −7.00000 −0.229416
\(932\) 0 0
\(933\) −30.0000 −0.982156
\(934\) 0 0
\(935\) 1.00000 0.0327035
\(936\) 0 0
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 0 0
\(939\) −24.0000 −0.783210
\(940\) 0 0
\(941\) −46.0000 −1.49956 −0.749779 0.661689i \(-0.769840\pi\)
−0.749779 + 0.661689i \(0.769840\pi\)
\(942\) 0 0
\(943\) −24.0000 −0.781548
\(944\) 0 0
\(945\) −1.00000 −0.0325300
\(946\) 0 0
\(947\) 13.0000 0.422443 0.211222 0.977438i \(-0.432256\pi\)
0.211222 + 0.977438i \(0.432256\pi\)
\(948\) 0 0
\(949\) −1.00000 −0.0324614
\(950\) 0 0
\(951\) 12.0000 0.389127
\(952\) 0 0
\(953\) 36.0000 1.16615 0.583077 0.812417i \(-0.301849\pi\)
0.583077 + 0.812417i \(0.301849\pi\)
\(954\) 0 0
\(955\) 19.0000 0.614826
\(956\) 0 0
\(957\) −3.00000 −0.0969762
\(958\) 0 0
\(959\) −19.0000 −0.613542
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 18.0000 0.580042
\(964\) 0 0
\(965\) −16.0000 −0.515058
\(966\) 0 0
\(967\) −13.0000 −0.418052 −0.209026 0.977910i \(-0.567029\pi\)
−0.209026 + 0.977910i \(0.567029\pi\)
\(968\) 0 0
\(969\) 7.00000 0.224872
\(970\) 0 0
\(971\) 40.0000 1.28366 0.641831 0.766846i \(-0.278175\pi\)
0.641831 + 0.766846i \(0.278175\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −4.00000 −0.128103
\(976\) 0 0
\(977\) −45.0000 −1.43968 −0.719839 0.694141i \(-0.755784\pi\)
−0.719839 + 0.694141i \(0.755784\pi\)
\(978\) 0 0
\(979\) 14.0000 0.447442
\(980\) 0 0
\(981\) −11.0000 −0.351203
\(982\) 0 0
\(983\) 3.00000 0.0956851 0.0478426 0.998855i \(-0.484765\pi\)
0.0478426 + 0.998855i \(0.484765\pi\)
\(984\) 0 0
\(985\) 12.0000 0.382352
\(986\) 0 0
\(987\) −8.00000 −0.254643
\(988\) 0 0
\(989\) 21.0000 0.667761
\(990\) 0 0
\(991\) −42.0000 −1.33417 −0.667087 0.744980i \(-0.732459\pi\)
−0.667087 + 0.744980i \(0.732459\pi\)
\(992\) 0 0
\(993\) 10.0000 0.317340
\(994\) 0 0
\(995\) 1.00000 0.0317021
\(996\) 0 0
\(997\) −14.0000 −0.443384 −0.221692 0.975117i \(-0.571158\pi\)
−0.221692 + 0.975117i \(0.571158\pi\)
\(998\) 0 0
\(999\) 7.00000 0.221470
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 4368.2.a.s.1.1 1
4.3 odd 2 546.2.a.a.1.1 1
12.11 even 2 1638.2.a.r.1.1 1
28.27 even 2 3822.2.a.n.1.1 1
52.51 odd 2 7098.2.a.t.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.a.a.1.1 1 4.3 odd 2
1638.2.a.r.1.1 1 12.11 even 2
3822.2.a.n.1.1 1 28.27 even 2
4368.2.a.s.1.1 1 1.1 even 1 trivial
7098.2.a.t.1.1 1 52.51 odd 2