Properties

Label 7200.2.o.m.7199.4
Level $7200$
Weight $2$
Character 7200.7199
Analytic conductor $57.492$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7200.o (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(57.4922894553\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1440)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 7199.4
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 7200.7199
Dual form 7200.2.o.m.7199.3

$q$-expansion

\(f(q)\) \(=\) \(q+3.41421 q^{7} +O(q^{10})\) \(q+3.41421 q^{7} +2.58579 q^{11} +3.41421i q^{13} +1.17157 q^{17} -4.82843i q^{23} +6.00000i q^{29} -6.48528i q^{31} -9.07107i q^{37} -11.0711i q^{41} +6.82843 q^{43} -5.65685i q^{47} +4.65685 q^{49} -1.17157 q^{53} -6.58579 q^{59} +12.8284 q^{61} +8.00000 q^{67} +5.65685 q^{71} +10.4853i q^{73} +8.82843 q^{77} +14.4853i q^{79} -9.17157i q^{83} -4.24264i q^{89} +11.6569i q^{91} +2.48528i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{7} + O(q^{10}) \) \( 4 q + 8 q^{7} + 16 q^{11} + 16 q^{17} + 16 q^{43} - 4 q^{49} - 16 q^{53} - 32 q^{59} + 40 q^{61} + 32 q^{67} + 24 q^{77} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7200\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(6401\) \(6751\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.41421 1.29045 0.645226 0.763992i \(-0.276763\pi\)
0.645226 + 0.763992i \(0.276763\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.58579 0.779644 0.389822 0.920890i \(-0.372537\pi\)
0.389822 + 0.920890i \(0.372537\pi\)
\(12\) 0 0
\(13\) 3.41421i 0.946932i 0.880812 + 0.473466i \(0.156997\pi\)
−0.880812 + 0.473466i \(0.843003\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.17157 0.284148 0.142074 0.989856i \(-0.454623\pi\)
0.142074 + 0.989856i \(0.454623\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 4.82843i − 1.00680i −0.864054 0.503398i \(-0.832083\pi\)
0.864054 0.503398i \(-0.167917\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.00000i 1.11417i 0.830455 + 0.557086i \(0.188081\pi\)
−0.830455 + 0.557086i \(0.811919\pi\)
\(30\) 0 0
\(31\) − 6.48528i − 1.16479i −0.812906 0.582395i \(-0.802116\pi\)
0.812906 0.582395i \(-0.197884\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 9.07107i − 1.49127i −0.666352 0.745637i \(-0.732145\pi\)
0.666352 0.745637i \(-0.267855\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 11.0711i − 1.72901i −0.502624 0.864505i \(-0.667632\pi\)
0.502624 0.864505i \(-0.332368\pi\)
\(42\) 0 0
\(43\) 6.82843 1.04133 0.520663 0.853762i \(-0.325685\pi\)
0.520663 + 0.853762i \(0.325685\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 5.65685i − 0.825137i −0.910927 0.412568i \(-0.864632\pi\)
0.910927 0.412568i \(-0.135368\pi\)
\(48\) 0 0
\(49\) 4.65685 0.665265
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.17157 −0.160928 −0.0804640 0.996758i \(-0.525640\pi\)
−0.0804640 + 0.996758i \(0.525640\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −6.58579 −0.857396 −0.428698 0.903448i \(-0.641028\pi\)
−0.428698 + 0.903448i \(0.641028\pi\)
\(60\) 0 0
\(61\) 12.8284 1.64251 0.821256 0.570560i \(-0.193274\pi\)
0.821256 + 0.570560i \(0.193274\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 5.65685 0.671345 0.335673 0.941979i \(-0.391036\pi\)
0.335673 + 0.941979i \(0.391036\pi\)
\(72\) 0 0
\(73\) 10.4853i 1.22721i 0.789613 + 0.613605i \(0.210281\pi\)
−0.789613 + 0.613605i \(0.789719\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8.82843 1.00609
\(78\) 0 0
\(79\) 14.4853i 1.62972i 0.579657 + 0.814861i \(0.303187\pi\)
−0.579657 + 0.814861i \(0.696813\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 9.17157i − 1.00671i −0.864079 0.503355i \(-0.832099\pi\)
0.864079 0.503355i \(-0.167901\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 4.24264i − 0.449719i −0.974391 0.224860i \(-0.927808\pi\)
0.974391 0.224860i \(-0.0721923\pi\)
\(90\) 0 0
\(91\) 11.6569i 1.22197i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.48528i 0.252342i 0.992009 + 0.126171i \(0.0402688\pi\)
−0.992009 + 0.126171i \(0.959731\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 4.82843i − 0.480446i −0.970718 0.240223i \(-0.922779\pi\)
0.970718 0.240223i \(-0.0772206\pi\)
\(102\) 0 0
\(103\) −17.5563 −1.72988 −0.864939 0.501877i \(-0.832643\pi\)
−0.864939 + 0.501877i \(0.832643\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 14.8284i 1.43352i 0.697321 + 0.716759i \(0.254375\pi\)
−0.697321 + 0.716759i \(0.745625\pi\)
\(108\) 0 0
\(109\) 7.17157 0.686912 0.343456 0.939169i \(-0.388402\pi\)
0.343456 + 0.939169i \(0.388402\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.65685 0.155864 0.0779319 0.996959i \(-0.475168\pi\)
0.0779319 + 0.996959i \(0.475168\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) −4.31371 −0.392155
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 10.7279 0.951949 0.475975 0.879459i \(-0.342095\pi\)
0.475975 + 0.879459i \(0.342095\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 12.7279 1.11204 0.556022 0.831168i \(-0.312327\pi\)
0.556022 + 0.831168i \(0.312327\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −19.3137 −1.65008 −0.825041 0.565073i \(-0.808848\pi\)
−0.825041 + 0.565073i \(0.808848\pi\)
\(138\) 0 0
\(139\) − 3.65685i − 0.310170i −0.987901 0.155085i \(-0.950435\pi\)
0.987901 0.155085i \(-0.0495652\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 8.82843i 0.738270i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 8.82843i − 0.723253i −0.932323 0.361626i \(-0.882222\pi\)
0.932323 0.361626i \(-0.117778\pi\)
\(150\) 0 0
\(151\) 15.6569i 1.27414i 0.770807 + 0.637068i \(0.219853\pi\)
−0.770807 + 0.637068i \(0.780147\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 9.75736i − 0.778722i −0.921085 0.389361i \(-0.872696\pi\)
0.921085 0.389361i \(-0.127304\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 16.4853i − 1.29922i
\(162\) 0 0
\(163\) 0.485281 0.0380102 0.0190051 0.999819i \(-0.493950\pi\)
0.0190051 + 0.999819i \(0.493950\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 21.7990i − 1.68686i −0.537242 0.843428i \(-0.680534\pi\)
0.537242 0.843428i \(-0.319466\pi\)
\(168\) 0 0
\(169\) 1.34315 0.103319
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −8.48528 −0.645124 −0.322562 0.946548i \(-0.604544\pi\)
−0.322562 + 0.946548i \(0.604544\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 24.0416 1.79696 0.898478 0.439019i \(-0.144674\pi\)
0.898478 + 0.439019i \(0.144674\pi\)
\(180\) 0 0
\(181\) 1.51472 0.112588 0.0562941 0.998414i \(-0.482072\pi\)
0.0562941 + 0.998414i \(0.482072\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 3.02944 0.221534
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −3.51472 −0.254316 −0.127158 0.991882i \(-0.540586\pi\)
−0.127158 + 0.991882i \(0.540586\pi\)
\(192\) 0 0
\(193\) − 18.9706i − 1.36553i −0.730638 0.682765i \(-0.760777\pi\)
0.730638 0.682765i \(-0.239223\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −21.3137 −1.51854 −0.759269 0.650776i \(-0.774444\pi\)
−0.759269 + 0.650776i \(0.774444\pi\)
\(198\) 0 0
\(199\) 15.6569i 1.10988i 0.831889 + 0.554942i \(0.187260\pi\)
−0.831889 + 0.554942i \(0.812740\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 20.4853i 1.43778i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 26.0000i 1.78991i 0.446153 + 0.894957i \(0.352794\pi\)
−0.446153 + 0.894957i \(0.647206\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 22.1421i − 1.50311i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 4.00000i 0.269069i
\(222\) 0 0
\(223\) 2.72792 0.182675 0.0913376 0.995820i \(-0.470886\pi\)
0.0913376 + 0.995820i \(0.470886\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5.65685i 0.375459i 0.982221 + 0.187729i \(0.0601128\pi\)
−0.982221 + 0.187729i \(0.939887\pi\)
\(228\) 0 0
\(229\) 14.9706 0.989283 0.494641 0.869097i \(-0.335299\pi\)
0.494641 + 0.869097i \(0.335299\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2.14214 −0.140336 −0.0701680 0.997535i \(-0.522354\pi\)
−0.0701680 + 0.997535i \(0.522354\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −25.4558 −1.64660 −0.823301 0.567605i \(-0.807870\pi\)
−0.823301 + 0.567605i \(0.807870\pi\)
\(240\) 0 0
\(241\) 28.9706 1.86616 0.933079 0.359671i \(-0.117111\pi\)
0.933079 + 0.359671i \(0.117111\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 13.8995 0.877328 0.438664 0.898651i \(-0.355452\pi\)
0.438664 + 0.898651i \(0.355452\pi\)
\(252\) 0 0
\(253\) − 12.4853i − 0.784943i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) − 30.9706i − 1.92442i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 9.51472i 0.586703i 0.956005 + 0.293351i \(0.0947706\pi\)
−0.956005 + 0.293351i \(0.905229\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 7.65685i − 0.466847i −0.972375 0.233423i \(-0.925007\pi\)
0.972375 0.233423i \(-0.0749928\pi\)
\(270\) 0 0
\(271\) 22.0000i 1.33640i 0.743980 + 0.668202i \(0.232936\pi\)
−0.743980 + 0.668202i \(0.767064\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 5.27208i − 0.316768i −0.987378 0.158384i \(-0.949372\pi\)
0.987378 0.158384i \(-0.0506285\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 22.5858i 1.34736i 0.739025 + 0.673678i \(0.235286\pi\)
−0.739025 + 0.673678i \(0.764714\pi\)
\(282\) 0 0
\(283\) −7.31371 −0.434755 −0.217377 0.976088i \(-0.569750\pi\)
−0.217377 + 0.976088i \(0.569750\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 37.7990i − 2.23120i
\(288\) 0 0
\(289\) −15.6274 −0.919260
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 29.3137 1.71253 0.856263 0.516541i \(-0.172781\pi\)
0.856263 + 0.516541i \(0.172781\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 16.4853 0.953368
\(300\) 0 0
\(301\) 23.3137 1.34378
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 24.0000 1.36975 0.684876 0.728659i \(-0.259856\pi\)
0.684876 + 0.728659i \(0.259856\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 26.1421 1.48238 0.741192 0.671293i \(-0.234261\pi\)
0.741192 + 0.671293i \(0.234261\pi\)
\(312\) 0 0
\(313\) − 7.65685i − 0.432791i −0.976306 0.216395i \(-0.930570\pi\)
0.976306 0.216395i \(-0.0694301\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −16.6274 −0.933889 −0.466944 0.884287i \(-0.654645\pi\)
−0.466944 + 0.884287i \(0.654645\pi\)
\(318\) 0 0
\(319\) 15.5147i 0.868657i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 19.3137i − 1.06480i
\(330\) 0 0
\(331\) 28.9706i 1.59237i 0.605056 + 0.796183i \(0.293151\pi\)
−0.605056 + 0.796183i \(0.706849\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 26.4853i − 1.44275i −0.692547 0.721373i \(-0.743512\pi\)
0.692547 0.721373i \(-0.256488\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 16.7696i − 0.908122i
\(342\) 0 0
\(343\) −8.00000 −0.431959
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 22.3431i − 1.19944i −0.800209 0.599721i \(-0.795278\pi\)
0.800209 0.599721i \(-0.204722\pi\)
\(348\) 0 0
\(349\) −31.4558 −1.68379 −0.841896 0.539639i \(-0.818561\pi\)
−0.841896 + 0.539639i \(0.818561\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1.85786 −0.0988841 −0.0494421 0.998777i \(-0.515744\pi\)
−0.0494421 + 0.998777i \(0.515744\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 18.8284 0.993726 0.496863 0.867829i \(-0.334485\pi\)
0.496863 + 0.867829i \(0.334485\pi\)
\(360\) 0 0
\(361\) 19.0000 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −24.8701 −1.29821 −0.649103 0.760700i \(-0.724856\pi\)
−0.649103 + 0.760700i \(0.724856\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −4.00000 −0.207670
\(372\) 0 0
\(373\) 10.7279i 0.555471i 0.960658 + 0.277735i \(0.0895838\pi\)
−0.960658 + 0.277735i \(0.910416\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −20.4853 −1.05505
\(378\) 0 0
\(379\) 9.31371i 0.478413i 0.970969 + 0.239207i \(0.0768873\pi\)
−0.970969 + 0.239207i \(0.923113\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 10.3431i − 0.528510i −0.964453 0.264255i \(-0.914874\pi\)
0.964453 0.264255i \(-0.0851261\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 12.3431i − 0.625822i −0.949782 0.312911i \(-0.898696\pi\)
0.949782 0.312911i \(-0.101304\pi\)
\(390\) 0 0
\(391\) − 5.65685i − 0.286079i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 14.7279i 0.739173i 0.929196 + 0.369587i \(0.120501\pi\)
−0.929196 + 0.369587i \(0.879499\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 0.727922i − 0.0363507i −0.999835 0.0181753i \(-0.994214\pi\)
0.999835 0.0181753i \(-0.00578571\pi\)
\(402\) 0 0
\(403\) 22.1421 1.10298
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 23.4558i − 1.16266i
\(408\) 0 0
\(409\) 38.2843 1.89304 0.946518 0.322652i \(-0.104574\pi\)
0.946518 + 0.322652i \(0.104574\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −22.4853 −1.10643
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 38.3848 1.87522 0.937610 0.347690i \(-0.113034\pi\)
0.937610 + 0.347690i \(0.113034\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 43.7990 2.11958
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 2.14214 0.103183 0.0515915 0.998668i \(-0.483571\pi\)
0.0515915 + 0.998668i \(0.483571\pi\)
\(432\) 0 0
\(433\) 14.4853i 0.696118i 0.937473 + 0.348059i \(0.113159\pi\)
−0.937473 + 0.348059i \(0.886841\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) − 18.2843i − 0.872661i −0.899787 0.436330i \(-0.856278\pi\)
0.899787 0.436330i \(-0.143722\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 26.6274i 1.26511i 0.774517 + 0.632553i \(0.217993\pi\)
−0.774517 + 0.632553i \(0.782007\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 18.5858i 0.877117i 0.898703 + 0.438559i \(0.144511\pi\)
−0.898703 + 0.438559i \(0.855489\pi\)
\(450\) 0 0
\(451\) − 28.6274i − 1.34801i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 2.48528i 0.116257i 0.998309 + 0.0581283i \(0.0185132\pi\)
−0.998309 + 0.0581283i \(0.981487\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 24.8284i − 1.15638i −0.815904 0.578188i \(-0.803760\pi\)
0.815904 0.578188i \(-0.196240\pi\)
\(462\) 0 0
\(463\) 17.0711 0.793360 0.396680 0.917957i \(-0.370162\pi\)
0.396680 + 0.917957i \(0.370162\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 31.1127i 1.43972i 0.694117 + 0.719862i \(0.255795\pi\)
−0.694117 + 0.719862i \(0.744205\pi\)
\(468\) 0 0
\(469\) 27.3137 1.26123
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 17.6569 0.811863
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −27.3137 −1.24800 −0.623998 0.781426i \(-0.714493\pi\)
−0.623998 + 0.781426i \(0.714493\pi\)
\(480\) 0 0
\(481\) 30.9706 1.41214
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −10.7279 −0.486129 −0.243064 0.970010i \(-0.578153\pi\)
−0.243064 + 0.970010i \(0.578153\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −31.0711 −1.40222 −0.701109 0.713054i \(-0.747311\pi\)
−0.701109 + 0.713054i \(0.747311\pi\)
\(492\) 0 0
\(493\) 7.02944i 0.316590i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 19.3137 0.866338
\(498\) 0 0
\(499\) 18.6274i 0.833878i 0.908935 + 0.416939i \(0.136897\pi\)
−0.908935 + 0.416939i \(0.863103\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 36.2843i − 1.61784i −0.587922 0.808918i \(-0.700054\pi\)
0.587922 0.808918i \(-0.299946\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 8.14214i − 0.360894i −0.983585 0.180447i \(-0.942246\pi\)
0.983585 0.180447i \(-0.0577544\pi\)
\(510\) 0 0
\(511\) 35.7990i 1.58365i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 14.6274i − 0.643313i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0.727922i 0.0318908i 0.999873 + 0.0159454i \(0.00507580\pi\)
−0.999873 + 0.0159454i \(0.994924\pi\)
\(522\) 0 0
\(523\) 7.51472 0.328596 0.164298 0.986411i \(-0.447464\pi\)
0.164298 + 0.986411i \(0.447464\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 7.59798i − 0.330973i
\(528\) 0 0
\(529\) −0.313708 −0.0136395
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 37.7990 1.63726
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 12.0416 0.518670
\(540\) 0 0
\(541\) 28.8284 1.23943 0.619715 0.784827i \(-0.287248\pi\)
0.619715 + 0.784827i \(0.287248\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 16.4853 0.704860 0.352430 0.935838i \(-0.385356\pi\)
0.352430 + 0.935838i \(0.385356\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 49.4558i 2.10308i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −20.4853 −0.867989 −0.433995 0.900915i \(-0.642896\pi\)
−0.433995 + 0.900915i \(0.642896\pi\)
\(558\) 0 0
\(559\) 23.3137i 0.986065i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 6.34315i 0.267332i 0.991026 + 0.133666i \(0.0426749\pi\)
−0.991026 + 0.133666i \(0.957325\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 9.21320i 0.386238i 0.981175 + 0.193119i \(0.0618603\pi\)
−0.981175 + 0.193119i \(0.938140\pi\)
\(570\) 0 0
\(571\) − 4.97056i − 0.208012i −0.994577 0.104006i \(-0.966834\pi\)
0.994577 0.104006i \(-0.0331660\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 15.4558i 0.643435i 0.946836 + 0.321718i \(0.104260\pi\)
−0.946836 + 0.321718i \(0.895740\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 31.3137i − 1.29911i
\(582\) 0 0
\(583\) −3.02944 −0.125466
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 26.1421i 1.07900i 0.841985 + 0.539501i \(0.181387\pi\)
−0.841985 + 0.539501i \(0.818613\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 10.3431 0.424742 0.212371 0.977189i \(-0.431881\pi\)
0.212371 + 0.977189i \(0.431881\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −42.1421 −1.72188 −0.860940 0.508706i \(-0.830124\pi\)
−0.860940 + 0.508706i \(0.830124\pi\)
\(600\) 0 0
\(601\) 16.6863 0.680648 0.340324 0.940308i \(-0.389463\pi\)
0.340324 + 0.940308i \(0.389463\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 3.21320 0.130420 0.0652100 0.997872i \(-0.479228\pi\)
0.0652100 + 0.997872i \(0.479228\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 19.3137 0.781349
\(612\) 0 0
\(613\) 3.21320i 0.129780i 0.997892 + 0.0648900i \(0.0206697\pi\)
−0.997892 + 0.0648900i \(0.979330\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 26.8284 1.08007 0.540036 0.841642i \(-0.318411\pi\)
0.540036 + 0.841642i \(0.318411\pi\)
\(618\) 0 0
\(619\) 22.9706i 0.923265i 0.887071 + 0.461632i \(0.152736\pi\)
−0.887071 + 0.461632i \(0.847264\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 14.4853i − 0.580341i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 10.6274i − 0.423743i
\(630\) 0 0
\(631\) − 7.17157i − 0.285496i −0.989759 0.142748i \(-0.954406\pi\)
0.989759 0.142748i \(-0.0455938\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 15.8995i 0.629961i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 43.5563i − 1.72037i −0.509980 0.860186i \(-0.670347\pi\)
0.509980 0.860186i \(-0.329653\pi\)
\(642\) 0 0
\(643\) −35.1127 −1.38471 −0.692355 0.721557i \(-0.743427\pi\)
−0.692355 + 0.721557i \(0.743427\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 12.6863i 0.498750i 0.968407 + 0.249375i \(0.0802251\pi\)
−0.968407 + 0.249375i \(0.919775\pi\)
\(648\) 0 0
\(649\) −17.0294 −0.668464
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −19.6569 −0.769232 −0.384616 0.923077i \(-0.625666\pi\)
−0.384616 + 0.923077i \(0.625666\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −6.58579 −0.256546 −0.128273 0.991739i \(-0.540943\pi\)
−0.128273 + 0.991739i \(0.540943\pi\)
\(660\) 0 0
\(661\) −40.4264 −1.57240 −0.786202 0.617969i \(-0.787956\pi\)
−0.786202 + 0.617969i \(0.787956\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 28.9706 1.12174
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 33.1716 1.28057
\(672\) 0 0
\(673\) − 51.4558i − 1.98348i −0.128276 0.991739i \(-0.540944\pi\)
0.128276 0.991739i \(-0.459056\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −6.68629 −0.256975 −0.128488 0.991711i \(-0.541012\pi\)
−0.128488 + 0.991711i \(0.541012\pi\)
\(678\) 0 0
\(679\) 8.48528i 0.325635i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 10.8284i − 0.414338i −0.978305 0.207169i \(-0.933575\pi\)
0.978305 0.207169i \(-0.0664250\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 4.00000i − 0.152388i
\(690\) 0 0
\(691\) − 8.00000i − 0.304334i −0.988355 0.152167i \(-0.951375\pi\)
0.988355 0.152167i \(-0.0486252\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 12.9706i − 0.491295i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 9.51472i − 0.359366i −0.983725 0.179683i \(-0.942493\pi\)
0.983725 0.179683i \(-0.0575072\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 16.4853i − 0.619993i
\(708\) 0 0
\(709\) −6.97056 −0.261785 −0.130892 0.991397i \(-0.541784\pi\)
−0.130892 + 0.991397i \(0.541784\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −31.3137 −1.17271
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 10.8284 0.403832 0.201916 0.979403i \(-0.435283\pi\)
0.201916 + 0.979403i \(0.435283\pi\)
\(720\) 0 0
\(721\) −59.9411 −2.23232
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −26.9289 −0.998739 −0.499369 0.866389i \(-0.666435\pi\)
−0.499369 + 0.866389i \(0.666435\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 8.00000 0.295891
\(732\) 0 0
\(733\) 12.1005i 0.446942i 0.974711 + 0.223471i \(0.0717389\pi\)
−0.974711 + 0.223471i \(0.928261\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 20.6863 0.761989
\(738\) 0 0
\(739\) − 22.3431i − 0.821906i −0.911657 0.410953i \(-0.865196\pi\)
0.911657 0.410953i \(-0.134804\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 48.0000i − 1.76095i −0.474093 0.880475i \(-0.657224\pi\)
0.474093 0.880475i \(-0.342776\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 50.6274i 1.84989i
\(750\) 0 0
\(751\) − 28.8284i − 1.05196i −0.850496 0.525982i \(-0.823698\pi\)
0.850496 0.525982i \(-0.176302\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 52.8701i − 1.92159i −0.277251 0.960797i \(-0.589423\pi\)
0.277251 0.960797i \(-0.410577\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 30.8701i 1.11904i 0.828817 + 0.559519i \(0.189014\pi\)
−0.828817 + 0.559519i \(0.810986\pi\)
\(762\) 0 0
\(763\) 24.4853 0.886427
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 22.4853i − 0.811896i
\(768\) 0 0
\(769\) 2.34315 0.0844960 0.0422480 0.999107i \(-0.486548\pi\)
0.0422480 + 0.999107i \(0.486548\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −41.3137 −1.48595 −0.742975 0.669319i \(-0.766586\pi\)
−0.742975 + 0.669319i \(0.766586\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 14.6274 0.523410
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 42.1421 1.50220 0.751102 0.660186i \(-0.229522\pi\)
0.751102 + 0.660186i \(0.229522\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 5.65685 0.201135
\(792\) 0 0
\(793\) 43.7990i 1.55535i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 18.8284 0.666937 0.333469 0.942761i \(-0.391781\pi\)
0.333469 + 0.942761i \(0.391781\pi\)
\(798\) 0 0
\(799\) − 6.62742i − 0.234461i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 27.1127i 0.956786i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 36.0416i − 1.26716i −0.773679 0.633578i \(-0.781586\pi\)
0.773679 0.633578i \(-0.218414\pi\)
\(810\) 0 0
\(811\) 3.37258i 0.118427i 0.998245 + 0.0592137i \(0.0188593\pi\)
−0.998245 + 0.0592137i \(0.981141\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 29.3137i 1.02306i 0.859267 + 0.511528i \(0.170920\pi\)
−0.859267 + 0.511528i \(0.829080\pi\)
\(822\) 0 0
\(823\) 6.24264 0.217605 0.108802 0.994063i \(-0.465298\pi\)
0.108802 + 0.994063i \(0.465298\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 42.6274i 1.48230i 0.671339 + 0.741150i \(0.265719\pi\)
−0.671339 + 0.741150i \(0.734281\pi\)
\(828\) 0 0
\(829\) 19.4558 0.675729 0.337865 0.941195i \(-0.390295\pi\)
0.337865 + 0.941195i \(0.390295\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 5.45584 0.189034
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −42.4264 −1.46472 −0.732361 0.680916i \(-0.761582\pi\)
−0.732361 + 0.680916i \(0.761582\pi\)
\(840\) 0 0
\(841\) −7.00000 −0.241379
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −14.7279 −0.506057
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −43.7990 −1.50141
\(852\) 0 0
\(853\) − 21.7574i − 0.744958i −0.928041 0.372479i \(-0.878508\pi\)
0.928041 0.372479i \(-0.121492\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −35.7990 −1.22287 −0.611435 0.791295i \(-0.709407\pi\)
−0.611435 + 0.791295i \(0.709407\pi\)
\(858\) 0 0
\(859\) − 26.0000i − 0.887109i −0.896248 0.443554i \(-0.853717\pi\)
0.896248 0.443554i \(-0.146283\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 6.20101i 0.211085i 0.994415 + 0.105542i \(0.0336579\pi\)
−0.994415 + 0.105542i \(0.966342\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 37.4558i 1.27060i
\(870\) 0 0
\(871\) 27.3137i 0.925490i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 1.55635i − 0.0525542i −0.999655 0.0262771i \(-0.991635\pi\)
0.999655 0.0262771i \(-0.00836522\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 35.3553i − 1.19115i −0.803299 0.595576i \(-0.796924\pi\)
0.803299 0.595576i \(-0.203076\pi\)
\(882\) 0 0
\(883\) −42.4264 −1.42776 −0.713881 0.700267i \(-0.753064\pi\)
−0.713881 + 0.700267i \(0.753064\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 12.8284i − 0.430736i −0.976533 0.215368i \(-0.930905\pi\)
0.976533 0.215368i \(-0.0690952\pi\)
\(888\) 0 0
\(889\) 36.6274 1.22844
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 38.9117 1.29778
\(900\) 0 0
\(901\) −1.37258 −0.0457274
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1.85786 0.0616894 0.0308447 0.999524i \(-0.490180\pi\)
0.0308447 + 0.999524i \(0.490180\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 18.3431 0.607736 0.303868 0.952714i \(-0.401722\pi\)
0.303868 + 0.952714i \(0.401722\pi\)
\(912\) 0 0
\(913\) − 23.7157i − 0.784876i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 43.4558 1.43504
\(918\) 0 0
\(919\) 12.8284i 0.423171i 0.977360 + 0.211585i \(0.0678626\pi\)
−0.977360 + 0.211585i \(0.932137\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 19.3137i 0.635718i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 45.8995i 1.50591i 0.658070 + 0.752957i \(0.271373\pi\)
−0.658070 + 0.752957i \(0.728627\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 14.9706i − 0.489067i −0.969641 0.244533i \(-0.921365\pi\)
0.969641 0.244533i \(-0.0786348\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 42.2843i 1.37843i 0.724558 + 0.689214i \(0.242044\pi\)
−0.724558 + 0.689214i \(0.757956\pi\)
\(942\) 0 0
\(943\) −53.4558 −1.74076
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 51.5980i 1.67671i 0.545125 + 0.838355i \(0.316482\pi\)
−0.545125 + 0.838355i \(0.683518\pi\)
\(948\) 0 0
\(949\) −35.7990 −1.16208
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −45.2548 −1.46595 −0.732974 0.680257i \(-0.761868\pi\)
−0.732974 + 0.680257i \(0.761868\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −65.9411 −2.12935
\(960\) 0 0
\(961\) −11.0589 −0.356738
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −26.7279 −0.859512 −0.429756 0.902945i \(-0.641400\pi\)
−0.429756 + 0.902945i \(0.641400\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 27.0711 0.868752 0.434376 0.900732i \(-0.356969\pi\)
0.434376 + 0.900732i \(0.356969\pi\)
\(972\) 0 0
\(973\) − 12.4853i − 0.400260i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −37.1716 −1.18922 −0.594612 0.804013i \(-0.702694\pi\)
−0.594612 + 0.804013i \(0.702694\pi\)
\(978\) 0 0
\(979\) − 10.9706i − 0.350621i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 39.5980i 1.26298i 0.775384 + 0.631490i \(0.217556\pi\)
−0.775384 + 0.631490i \(0.782444\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 32.9706i − 1.04840i
\(990\) 0 0
\(991\) − 16.3431i − 0.519157i −0.965722 0.259579i \(-0.916416\pi\)
0.965722 0.259579i \(-0.0835837\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 5.27208i 0.166968i 0.996509 + 0.0834842i \(0.0266048\pi\)
−0.996509 + 0.0834842i \(0.973395\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 7200.2.o.m.7199.4 4
3.2 odd 2 7200.2.o.j.7199.4 4
4.3 odd 2 7200.2.o.b.7199.2 4
5.2 odd 4 7200.2.h.i.1151.4 4
5.3 odd 4 1440.2.h.d.1151.3 yes 4
5.4 even 2 7200.2.o.e.7199.1 4
12.11 even 2 7200.2.o.e.7199.2 4
15.2 even 4 7200.2.h.c.1151.4 4
15.8 even 4 1440.2.h.a.1151.1 4
15.14 odd 2 7200.2.o.b.7199.1 4
20.3 even 4 1440.2.h.a.1151.4 yes 4
20.7 even 4 7200.2.h.c.1151.1 4
20.19 odd 2 7200.2.o.j.7199.3 4
40.3 even 4 2880.2.h.d.1151.2 4
40.13 odd 4 2880.2.h.a.1151.1 4
60.23 odd 4 1440.2.h.d.1151.2 yes 4
60.47 odd 4 7200.2.h.i.1151.1 4
60.59 even 2 inner 7200.2.o.m.7199.3 4
120.53 even 4 2880.2.h.d.1151.3 4
120.83 odd 4 2880.2.h.a.1151.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1440.2.h.a.1151.1 4 15.8 even 4
1440.2.h.a.1151.4 yes 4 20.3 even 4
1440.2.h.d.1151.2 yes 4 60.23 odd 4
1440.2.h.d.1151.3 yes 4 5.3 odd 4
2880.2.h.a.1151.1 4 40.13 odd 4
2880.2.h.a.1151.4 4 120.83 odd 4
2880.2.h.d.1151.2 4 40.3 even 4
2880.2.h.d.1151.3 4 120.53 even 4
7200.2.h.c.1151.1 4 20.7 even 4
7200.2.h.c.1151.4 4 15.2 even 4
7200.2.h.i.1151.1 4 60.47 odd 4
7200.2.h.i.1151.4 4 5.2 odd 4
7200.2.o.b.7199.1 4 15.14 odd 2
7200.2.o.b.7199.2 4 4.3 odd 2
7200.2.o.e.7199.1 4 5.4 even 2
7200.2.o.e.7199.2 4 12.11 even 2
7200.2.o.j.7199.3 4 20.19 odd 2
7200.2.o.j.7199.4 4 3.2 odd 2
7200.2.o.m.7199.3 4 60.59 even 2 inner
7200.2.o.m.7199.4 4 1.1 even 1 trivial