Properties

Label 5040.2.a.bt.1.2
Level $5040$
Weight $2$
Character 5040.1
Self dual yes
Analytic conductor $40.245$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5040.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.2446026187\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 5040.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{5} +1.00000 q^{7} +O(q^{10})\) \(q-1.00000 q^{5} +1.00000 q^{7} +2.56155 q^{11} +4.56155 q^{13} +4.56155 q^{17} -1.12311 q^{19} -5.12311 q^{23} +1.00000 q^{25} +5.68466 q^{29} -1.00000 q^{35} +6.00000 q^{37} +3.12311 q^{41} -9.12311 q^{43} +3.68466 q^{47} +1.00000 q^{49} -3.12311 q^{53} -2.56155 q^{55} -4.00000 q^{59} -9.36932 q^{61} -4.56155 q^{65} +6.24621 q^{67} +8.00000 q^{71} +4.24621 q^{73} +2.56155 q^{77} +6.56155 q^{79} +4.00000 q^{83} -4.56155 q^{85} -7.12311 q^{89} +4.56155 q^{91} +1.12311 q^{95} -14.8078 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{5} + 2q^{7} + O(q^{10}) \) \( 2q - 2q^{5} + 2q^{7} + q^{11} + 5q^{13} + 5q^{17} + 6q^{19} - 2q^{23} + 2q^{25} - q^{29} - 2q^{35} + 12q^{37} - 2q^{41} - 10q^{43} - 5q^{47} + 2q^{49} + 2q^{53} - q^{55} - 8q^{59} + 6q^{61} - 5q^{65} - 4q^{67} + 16q^{71} - 8q^{73} + q^{77} + 9q^{79} + 8q^{83} - 5q^{85} - 6q^{89} + 5q^{91} - 6q^{95} - 9q^{97} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.56155 0.772337 0.386169 0.922428i \(-0.373798\pi\)
0.386169 + 0.922428i \(0.373798\pi\)
\(12\) 0 0
\(13\) 4.56155 1.26515 0.632574 0.774500i \(-0.281999\pi\)
0.632574 + 0.774500i \(0.281999\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.56155 1.10634 0.553170 0.833069i \(-0.313418\pi\)
0.553170 + 0.833069i \(0.313418\pi\)
\(18\) 0 0
\(19\) −1.12311 −0.257658 −0.128829 0.991667i \(-0.541122\pi\)
−0.128829 + 0.991667i \(0.541122\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5.12311 −1.06824 −0.534121 0.845408i \(-0.679357\pi\)
−0.534121 + 0.845408i \(0.679357\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.68466 1.05561 0.527807 0.849364i \(-0.323014\pi\)
0.527807 + 0.849364i \(0.323014\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.12311 0.487747 0.243874 0.969807i \(-0.421582\pi\)
0.243874 + 0.969807i \(0.421582\pi\)
\(42\) 0 0
\(43\) −9.12311 −1.39126 −0.695630 0.718400i \(-0.744875\pi\)
−0.695630 + 0.718400i \(0.744875\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.68466 0.537463 0.268731 0.963215i \(-0.413396\pi\)
0.268731 + 0.963215i \(0.413396\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −3.12311 −0.428992 −0.214496 0.976725i \(-0.568811\pi\)
−0.214496 + 0.976725i \(0.568811\pi\)
\(54\) 0 0
\(55\) −2.56155 −0.345400
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −9.36932 −1.19962 −0.599809 0.800143i \(-0.704757\pi\)
−0.599809 + 0.800143i \(0.704757\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.56155 −0.565791
\(66\) 0 0
\(67\) 6.24621 0.763096 0.381548 0.924349i \(-0.375391\pi\)
0.381548 + 0.924349i \(0.375391\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) 4.24621 0.496981 0.248491 0.968634i \(-0.420065\pi\)
0.248491 + 0.968634i \(0.420065\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.56155 0.291916
\(78\) 0 0
\(79\) 6.56155 0.738232 0.369116 0.929383i \(-0.379660\pi\)
0.369116 + 0.929383i \(0.379660\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) −4.56155 −0.494770
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −7.12311 −0.755048 −0.377524 0.926000i \(-0.623224\pi\)
−0.377524 + 0.926000i \(0.623224\pi\)
\(90\) 0 0
\(91\) 4.56155 0.478181
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.12311 0.115228
\(96\) 0 0
\(97\) −14.8078 −1.50350 −0.751750 0.659448i \(-0.770790\pi\)
−0.751750 + 0.659448i \(0.770790\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −0.246211 −0.0244989 −0.0122495 0.999925i \(-0.503899\pi\)
−0.0122495 + 0.999925i \(0.503899\pi\)
\(102\) 0 0
\(103\) −1.43845 −0.141734 −0.0708672 0.997486i \(-0.522577\pi\)
−0.0708672 + 0.997486i \(0.522577\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −11.3693 −1.09911 −0.549557 0.835456i \(-0.685203\pi\)
−0.549557 + 0.835456i \(0.685203\pi\)
\(108\) 0 0
\(109\) 17.6847 1.69388 0.846942 0.531686i \(-0.178441\pi\)
0.846942 + 0.531686i \(0.178441\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 0 0
\(115\) 5.12311 0.477732
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 4.56155 0.418157
\(120\) 0 0
\(121\) −4.43845 −0.403495
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −10.2462 −0.909204 −0.454602 0.890695i \(-0.650219\pi\)
−0.454602 + 0.890695i \(0.650219\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −9.12311 −0.797089 −0.398545 0.917149i \(-0.630485\pi\)
−0.398545 + 0.917149i \(0.630485\pi\)
\(132\) 0 0
\(133\) −1.12311 −0.0973856
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 8.87689 0.758404 0.379202 0.925314i \(-0.376199\pi\)
0.379202 + 0.925314i \(0.376199\pi\)
\(138\) 0 0
\(139\) 6.87689 0.583291 0.291645 0.956527i \(-0.405797\pi\)
0.291645 + 0.956527i \(0.405797\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 11.6847 0.977120
\(144\) 0 0
\(145\) −5.68466 −0.472085
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 4.24621 0.347863 0.173932 0.984758i \(-0.444353\pi\)
0.173932 + 0.984758i \(0.444353\pi\)
\(150\) 0 0
\(151\) −21.9309 −1.78471 −0.892354 0.451335i \(-0.850948\pi\)
−0.892354 + 0.451335i \(0.850948\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 3.75379 0.299585 0.149792 0.988717i \(-0.452139\pi\)
0.149792 + 0.988717i \(0.452139\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −5.12311 −0.403757
\(162\) 0 0
\(163\) −1.12311 −0.0879684 −0.0439842 0.999032i \(-0.514005\pi\)
−0.0439842 + 0.999032i \(0.514005\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 21.9309 1.69706 0.848531 0.529146i \(-0.177488\pi\)
0.848531 + 0.529146i \(0.177488\pi\)
\(168\) 0 0
\(169\) 7.80776 0.600597
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 8.56155 0.650923 0.325461 0.945555i \(-0.394480\pi\)
0.325461 + 0.945555i \(0.394480\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 0 0
\(181\) 23.6155 1.75533 0.877664 0.479276i \(-0.159101\pi\)
0.877664 + 0.479276i \(0.159101\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −6.00000 −0.441129
\(186\) 0 0
\(187\) 11.6847 0.854467
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −9.43845 −0.682942 −0.341471 0.939892i \(-0.610925\pi\)
−0.341471 + 0.939892i \(0.610925\pi\)
\(192\) 0 0
\(193\) −5.36932 −0.386492 −0.193246 0.981150i \(-0.561902\pi\)
−0.193246 + 0.981150i \(0.561902\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 7.12311 0.507500 0.253750 0.967270i \(-0.418336\pi\)
0.253750 + 0.967270i \(0.418336\pi\)
\(198\) 0 0
\(199\) 18.2462 1.29344 0.646720 0.762728i \(-0.276140\pi\)
0.646720 + 0.762728i \(0.276140\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 5.68466 0.398985
\(204\) 0 0
\(205\) −3.12311 −0.218127
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −2.87689 −0.198999
\(210\) 0 0
\(211\) 23.0540 1.58710 0.793551 0.608504i \(-0.208230\pi\)
0.793551 + 0.608504i \(0.208230\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 9.12311 0.622191
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 20.8078 1.39968
\(222\) 0 0
\(223\) 6.56155 0.439394 0.219697 0.975568i \(-0.429493\pi\)
0.219697 + 0.975568i \(0.429493\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 23.6847 1.57201 0.786003 0.618223i \(-0.212147\pi\)
0.786003 + 0.618223i \(0.212147\pi\)
\(228\) 0 0
\(229\) 19.1231 1.26369 0.631845 0.775095i \(-0.282298\pi\)
0.631845 + 0.775095i \(0.282298\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3.12311 0.204601 0.102301 0.994754i \(-0.467380\pi\)
0.102301 + 0.994754i \(0.467380\pi\)
\(234\) 0 0
\(235\) −3.68466 −0.240361
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −0.807764 −0.0522499 −0.0261250 0.999659i \(-0.508317\pi\)
−0.0261250 + 0.999659i \(0.508317\pi\)
\(240\) 0 0
\(241\) 12.2462 0.788848 0.394424 0.918929i \(-0.370944\pi\)
0.394424 + 0.918929i \(0.370944\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) −5.12311 −0.325975
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −17.1231 −1.08080 −0.540400 0.841408i \(-0.681727\pi\)
−0.540400 + 0.841408i \(0.681727\pi\)
\(252\) 0 0
\(253\) −13.1231 −0.825043
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −22.4924 −1.40304 −0.701519 0.712650i \(-0.747495\pi\)
−0.701519 + 0.712650i \(0.747495\pi\)
\(258\) 0 0
\(259\) 6.00000 0.372822
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −21.1231 −1.30251 −0.651253 0.758860i \(-0.725756\pi\)
−0.651253 + 0.758860i \(0.725756\pi\)
\(264\) 0 0
\(265\) 3.12311 0.191851
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −28.7386 −1.75223 −0.876113 0.482106i \(-0.839872\pi\)
−0.876113 + 0.482106i \(0.839872\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.56155 0.154467
\(276\) 0 0
\(277\) 16.2462 0.976140 0.488070 0.872804i \(-0.337701\pi\)
0.488070 + 0.872804i \(0.337701\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −16.5616 −0.987979 −0.493990 0.869468i \(-0.664462\pi\)
−0.493990 + 0.869468i \(0.664462\pi\)
\(282\) 0 0
\(283\) 23.6847 1.40791 0.703953 0.710246i \(-0.251416\pi\)
0.703953 + 0.710246i \(0.251416\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3.12311 0.184351
\(288\) 0 0
\(289\) 3.80776 0.223986
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −9.68466 −0.565784 −0.282892 0.959152i \(-0.591294\pi\)
−0.282892 + 0.959152i \(0.591294\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −23.3693 −1.35148
\(300\) 0 0
\(301\) −9.12311 −0.525847
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 9.36932 0.536486
\(306\) 0 0
\(307\) 31.6847 1.80834 0.904169 0.427174i \(-0.140491\pi\)
0.904169 + 0.427174i \(0.140491\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −9.61553 −0.545247 −0.272623 0.962121i \(-0.587891\pi\)
−0.272623 + 0.962121i \(0.587891\pi\)
\(312\) 0 0
\(313\) 31.3002 1.76919 0.884596 0.466359i \(-0.154434\pi\)
0.884596 + 0.466359i \(0.154434\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 22.4924 1.26330 0.631650 0.775254i \(-0.282378\pi\)
0.631650 + 0.775254i \(0.282378\pi\)
\(318\) 0 0
\(319\) 14.5616 0.815290
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −5.12311 −0.285057
\(324\) 0 0
\(325\) 4.56155 0.253029
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 3.68466 0.203142
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −6.24621 −0.341267
\(336\) 0 0
\(337\) −34.4924 −1.87892 −0.939461 0.342656i \(-0.888674\pi\)
−0.939461 + 0.342656i \(0.888674\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −1.12311 −0.0602915 −0.0301457 0.999546i \(-0.509597\pi\)
−0.0301457 + 0.999546i \(0.509597\pi\)
\(348\) 0 0
\(349\) −22.4924 −1.20399 −0.601996 0.798499i \(-0.705628\pi\)
−0.601996 + 0.798499i \(0.705628\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 14.8078 0.788138 0.394069 0.919081i \(-0.371067\pi\)
0.394069 + 0.919081i \(0.371067\pi\)
\(354\) 0 0
\(355\) −8.00000 −0.424596
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 8.00000 0.422224 0.211112 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) 0 0
\(361\) −17.7386 −0.933612
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −4.24621 −0.222257
\(366\) 0 0
\(367\) −3.68466 −0.192338 −0.0961688 0.995365i \(-0.530659\pi\)
−0.0961688 + 0.995365i \(0.530659\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −3.12311 −0.162144
\(372\) 0 0
\(373\) 29.3693 1.52069 0.760343 0.649522i \(-0.225031\pi\)
0.760343 + 0.649522i \(0.225031\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 25.9309 1.33551
\(378\) 0 0
\(379\) −16.4924 −0.847159 −0.423579 0.905859i \(-0.639227\pi\)
−0.423579 + 0.905859i \(0.639227\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −10.2462 −0.523557 −0.261778 0.965128i \(-0.584309\pi\)
−0.261778 + 0.965128i \(0.584309\pi\)
\(384\) 0 0
\(385\) −2.56155 −0.130549
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −3.93087 −0.199303 −0.0996515 0.995022i \(-0.531773\pi\)
−0.0996515 + 0.995022i \(0.531773\pi\)
\(390\) 0 0
\(391\) −23.3693 −1.18184
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −6.56155 −0.330148
\(396\) 0 0
\(397\) 23.4384 1.17634 0.588171 0.808737i \(-0.299848\pi\)
0.588171 + 0.808737i \(0.299848\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −27.4384 −1.37021 −0.685105 0.728444i \(-0.740244\pi\)
−0.685105 + 0.728444i \(0.740244\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 15.3693 0.761829
\(408\) 0 0
\(409\) −26.4924 −1.30997 −0.654983 0.755644i \(-0.727324\pi\)
−0.654983 + 0.755644i \(0.727324\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −4.00000 −0.196827
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 9.75379 0.476504 0.238252 0.971203i \(-0.423426\pi\)
0.238252 + 0.971203i \(0.423426\pi\)
\(420\) 0 0
\(421\) 9.68466 0.472001 0.236001 0.971753i \(-0.424163\pi\)
0.236001 + 0.971753i \(0.424163\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 4.56155 0.221268
\(426\) 0 0
\(427\) −9.36932 −0.453413
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0.807764 0.0389086 0.0194543 0.999811i \(-0.493807\pi\)
0.0194543 + 0.999811i \(0.493807\pi\)
\(432\) 0 0
\(433\) −8.24621 −0.396288 −0.198144 0.980173i \(-0.563491\pi\)
−0.198144 + 0.980173i \(0.563491\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 5.75379 0.275241
\(438\) 0 0
\(439\) 15.3693 0.733537 0.366769 0.930312i \(-0.380464\pi\)
0.366769 + 0.930312i \(0.380464\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −27.3693 −1.30036 −0.650178 0.759782i \(-0.725306\pi\)
−0.650178 + 0.759782i \(0.725306\pi\)
\(444\) 0 0
\(445\) 7.12311 0.337668
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −18.8078 −0.887593 −0.443797 0.896128i \(-0.646369\pi\)
−0.443797 + 0.896128i \(0.646369\pi\)
\(450\) 0 0
\(451\) 8.00000 0.376705
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −4.56155 −0.213849
\(456\) 0 0
\(457\) −8.87689 −0.415244 −0.207622 0.978209i \(-0.566572\pi\)
−0.207622 + 0.978209i \(0.566572\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 4.87689 0.227140 0.113570 0.993530i \(-0.463771\pi\)
0.113570 + 0.993530i \(0.463771\pi\)
\(462\) 0 0
\(463\) 20.4924 0.952364 0.476182 0.879347i \(-0.342020\pi\)
0.476182 + 0.879347i \(0.342020\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 26.5616 1.22912 0.614561 0.788869i \(-0.289333\pi\)
0.614561 + 0.788869i \(0.289333\pi\)
\(468\) 0 0
\(469\) 6.24621 0.288423
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −23.3693 −1.07452
\(474\) 0 0
\(475\) −1.12311 −0.0515316
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 13.1231 0.599610 0.299805 0.954001i \(-0.403078\pi\)
0.299805 + 0.954001i \(0.403078\pi\)
\(480\) 0 0
\(481\) 27.3693 1.24793
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 14.8078 0.672386
\(486\) 0 0
\(487\) −5.12311 −0.232150 −0.116075 0.993240i \(-0.537031\pi\)
−0.116075 + 0.993240i \(0.537031\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 4.17708 0.188509 0.0942545 0.995548i \(-0.469953\pi\)
0.0942545 + 0.995548i \(0.469953\pi\)
\(492\) 0 0
\(493\) 25.9309 1.16787
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 8.00000 0.358849
\(498\) 0 0
\(499\) 4.17708 0.186992 0.0934959 0.995620i \(-0.470196\pi\)
0.0934959 + 0.995620i \(0.470196\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 10.0691 0.448960 0.224480 0.974479i \(-0.427932\pi\)
0.224480 + 0.974479i \(0.427932\pi\)
\(504\) 0 0
\(505\) 0.246211 0.0109563
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 28.2462 1.25199 0.625996 0.779827i \(-0.284693\pi\)
0.625996 + 0.779827i \(0.284693\pi\)
\(510\) 0 0
\(511\) 4.24621 0.187841
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1.43845 0.0633856
\(516\) 0 0
\(517\) 9.43845 0.415102
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 0 0
\(523\) −7.50758 −0.328283 −0.164142 0.986437i \(-0.552485\pi\)
−0.164142 + 0.986437i \(0.552485\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 3.24621 0.141140
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 14.2462 0.617072
\(534\) 0 0
\(535\) 11.3693 0.491538
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 2.56155 0.110334
\(540\) 0 0
\(541\) −17.1922 −0.739152 −0.369576 0.929201i \(-0.620497\pi\)
−0.369576 + 0.929201i \(0.620497\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −17.6847 −0.757528
\(546\) 0 0
\(547\) −14.2462 −0.609124 −0.304562 0.952493i \(-0.598510\pi\)
−0.304562 + 0.952493i \(0.598510\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −6.38447 −0.271988
\(552\) 0 0
\(553\) 6.56155 0.279026
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 4.87689 0.206641 0.103320 0.994648i \(-0.467053\pi\)
0.103320 + 0.994648i \(0.467053\pi\)
\(558\) 0 0
\(559\) −41.6155 −1.76015
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −28.0000 −1.18006 −0.590030 0.807382i \(-0.700884\pi\)
−0.590030 + 0.807382i \(0.700884\pi\)
\(564\) 0 0
\(565\) −14.0000 −0.588984
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −34.9848 −1.46664 −0.733320 0.679883i \(-0.762031\pi\)
−0.733320 + 0.679883i \(0.762031\pi\)
\(570\) 0 0
\(571\) −7.50758 −0.314182 −0.157091 0.987584i \(-0.550212\pi\)
−0.157091 + 0.987584i \(0.550212\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −5.12311 −0.213648
\(576\) 0 0
\(577\) 13.0540 0.543444 0.271722 0.962376i \(-0.412407\pi\)
0.271722 + 0.962376i \(0.412407\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 4.00000 0.165948
\(582\) 0 0
\(583\) −8.00000 −0.331326
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −9.75379 −0.402582 −0.201291 0.979531i \(-0.564514\pi\)
−0.201291 + 0.979531i \(0.564514\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 23.4384 0.962502 0.481251 0.876583i \(-0.340183\pi\)
0.481251 + 0.876583i \(0.340183\pi\)
\(594\) 0 0
\(595\) −4.56155 −0.187005
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 8.80776 0.359875 0.179938 0.983678i \(-0.442410\pi\)
0.179938 + 0.983678i \(0.442410\pi\)
\(600\) 0 0
\(601\) −26.4924 −1.08065 −0.540324 0.841457i \(-0.681698\pi\)
−0.540324 + 0.841457i \(0.681698\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 4.43845 0.180449
\(606\) 0 0
\(607\) 4.94602 0.200753 0.100376 0.994950i \(-0.467995\pi\)
0.100376 + 0.994950i \(0.467995\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 16.8078 0.679969
\(612\) 0 0
\(613\) −8.73863 −0.352950 −0.176475 0.984305i \(-0.556469\pi\)
−0.176475 + 0.984305i \(0.556469\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −15.7538 −0.634224 −0.317112 0.948388i \(-0.602713\pi\)
−0.317112 + 0.948388i \(0.602713\pi\)
\(618\) 0 0
\(619\) 42.1080 1.69246 0.846231 0.532817i \(-0.178866\pi\)
0.846231 + 0.532817i \(0.178866\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −7.12311 −0.285381
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 27.3693 1.09129
\(630\) 0 0
\(631\) −8.80776 −0.350632 −0.175316 0.984512i \(-0.556095\pi\)
−0.175316 + 0.984512i \(0.556095\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 10.2462 0.406608
\(636\) 0 0
\(637\) 4.56155 0.180735
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) 0 0
\(643\) 2.56155 0.101018 0.0505089 0.998724i \(-0.483916\pi\)
0.0505089 + 0.998724i \(0.483916\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 3.50758 0.137897 0.0689486 0.997620i \(-0.478036\pi\)
0.0689486 + 0.997620i \(0.478036\pi\)
\(648\) 0 0
\(649\) −10.2462 −0.402199
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −49.2311 −1.92656 −0.963280 0.268499i \(-0.913473\pi\)
−0.963280 + 0.268499i \(0.913473\pi\)
\(654\) 0 0
\(655\) 9.12311 0.356469
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −36.1771 −1.40926 −0.704629 0.709575i \(-0.748887\pi\)
−0.704629 + 0.709575i \(0.748887\pi\)
\(660\) 0 0
\(661\) 3.12311 0.121475 0.0607374 0.998154i \(-0.480655\pi\)
0.0607374 + 0.998154i \(0.480655\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.12311 0.0435522
\(666\) 0 0
\(667\) −29.1231 −1.12765
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −24.0000 −0.926510
\(672\) 0 0
\(673\) −25.8617 −0.996897 −0.498448 0.866919i \(-0.666097\pi\)
−0.498448 + 0.866919i \(0.666097\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 23.9309 0.919738 0.459869 0.887987i \(-0.347896\pi\)
0.459869 + 0.887987i \(0.347896\pi\)
\(678\) 0 0
\(679\) −14.8078 −0.568270
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 42.7386 1.63535 0.817674 0.575681i \(-0.195263\pi\)
0.817674 + 0.575681i \(0.195263\pi\)
\(684\) 0 0
\(685\) −8.87689 −0.339169
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −14.2462 −0.542737
\(690\) 0 0
\(691\) −8.49242 −0.323067 −0.161533 0.986867i \(-0.551644\pi\)
−0.161533 + 0.986867i \(0.551644\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −6.87689 −0.260855
\(696\) 0 0
\(697\) 14.2462 0.539614
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −0.0691303 −0.00261102 −0.00130551 0.999999i \(-0.500416\pi\)
−0.00130551 + 0.999999i \(0.500416\pi\)
\(702\) 0 0
\(703\) −6.73863 −0.254152
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −0.246211 −0.00925973
\(708\) 0 0
\(709\) −18.1771 −0.682655 −0.341327 0.939945i \(-0.610876\pi\)
−0.341327 + 0.939945i \(0.610876\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −11.6847 −0.436981
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 49.6155 1.85035 0.925173 0.379544i \(-0.123919\pi\)
0.925173 + 0.379544i \(0.123919\pi\)
\(720\) 0 0
\(721\) −1.43845 −0.0535706
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 5.68466 0.211123
\(726\) 0 0
\(727\) −19.5076 −0.723496 −0.361748 0.932276i \(-0.617820\pi\)
−0.361748 + 0.932276i \(0.617820\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −41.6155 −1.53921
\(732\) 0 0
\(733\) −5.68466 −0.209968 −0.104984 0.994474i \(-0.533479\pi\)
−0.104984 + 0.994474i \(0.533479\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 16.0000 0.589368
\(738\) 0 0
\(739\) −6.06913 −0.223257 −0.111628 0.993750i \(-0.535607\pi\)
−0.111628 + 0.993750i \(0.535607\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 32.9848 1.21010 0.605048 0.796189i \(-0.293154\pi\)
0.605048 + 0.796189i \(0.293154\pi\)
\(744\) 0 0
\(745\) −4.24621 −0.155569
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −11.3693 −0.415426
\(750\) 0 0
\(751\) −45.9309 −1.67604 −0.838021 0.545639i \(-0.816287\pi\)
−0.838021 + 0.545639i \(0.816287\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 21.9309 0.798146
\(756\) 0 0
\(757\) 14.6307 0.531761 0.265881 0.964006i \(-0.414337\pi\)
0.265881 + 0.964006i \(0.414337\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −31.7538 −1.15107 −0.575537 0.817776i \(-0.695207\pi\)
−0.575537 + 0.817776i \(0.695207\pi\)
\(762\) 0 0
\(763\) 17.6847 0.640228
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −18.2462 −0.658833
\(768\) 0 0
\(769\) −9.50758 −0.342852 −0.171426 0.985197i \(-0.554837\pi\)
−0.171426 + 0.985197i \(0.554837\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −8.06913 −0.290226 −0.145113 0.989415i \(-0.546355\pi\)
−0.145113 + 0.989415i \(0.546355\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −3.50758 −0.125672
\(780\) 0 0
\(781\) 20.4924 0.733277
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −3.75379 −0.133978
\(786\) 0 0
\(787\) 3.82292 0.136272 0.0681362 0.997676i \(-0.478295\pi\)
0.0681362 + 0.997676i \(0.478295\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 14.0000 0.497783
\(792\) 0 0
\(793\) −42.7386 −1.51769
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 13.0540 0.462396 0.231198 0.972907i \(-0.425736\pi\)
0.231198 + 0.972907i \(0.425736\pi\)
\(798\) 0 0
\(799\) 16.8078 0.594616
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 10.8769 0.383837
\(804\) 0 0
\(805\) 5.12311 0.180566
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 53.5464 1.88259 0.941296 0.337584i \(-0.109610\pi\)
0.941296 + 0.337584i \(0.109610\pi\)
\(810\) 0 0
\(811\) 21.6155 0.759024 0.379512 0.925187i \(-0.376092\pi\)
0.379512 + 0.925187i \(0.376092\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1.12311 0.0393407
\(816\) 0 0
\(817\) 10.2462 0.358470
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −40.4233 −1.41078 −0.705391 0.708818i \(-0.749229\pi\)
−0.705391 + 0.708818i \(0.749229\pi\)
\(822\) 0 0
\(823\) 3.50758 0.122266 0.0611332 0.998130i \(-0.480529\pi\)
0.0611332 + 0.998130i \(0.480529\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 19.3693 0.673537 0.336769 0.941587i \(-0.390666\pi\)
0.336769 + 0.941587i \(0.390666\pi\)
\(828\) 0 0
\(829\) 43.1231 1.49773 0.748864 0.662724i \(-0.230600\pi\)
0.748864 + 0.662724i \(0.230600\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 4.56155 0.158048
\(834\) 0 0
\(835\) −21.9309 −0.758949
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −37.1231 −1.28163 −0.640816 0.767695i \(-0.721404\pi\)
−0.640816 + 0.767695i \(0.721404\pi\)
\(840\) 0 0
\(841\) 3.31534 0.114322
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −7.80776 −0.268595
\(846\) 0 0
\(847\) −4.43845 −0.152507
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −30.7386 −1.05371
\(852\) 0 0
\(853\) −56.7386 −1.94269 −0.971347 0.237666i \(-0.923618\pi\)
−0.971347 + 0.237666i \(0.923618\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 32.2462 1.10151 0.550755 0.834667i \(-0.314340\pi\)
0.550755 + 0.834667i \(0.314340\pi\)
\(858\) 0 0
\(859\) −16.4924 −0.562714 −0.281357 0.959603i \(-0.590785\pi\)
−0.281357 + 0.959603i \(0.590785\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −42.2462 −1.43808 −0.719039 0.694970i \(-0.755418\pi\)
−0.719039 + 0.694970i \(0.755418\pi\)
\(864\) 0 0
\(865\) −8.56155 −0.291102
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 16.8078 0.570164
\(870\) 0 0
\(871\) 28.4924 0.965429
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) −23.7538 −0.802108 −0.401054 0.916054i \(-0.631356\pi\)
−0.401054 + 0.916054i \(0.631356\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −45.8617 −1.54512 −0.772561 0.634941i \(-0.781024\pi\)
−0.772561 + 0.634941i \(0.781024\pi\)
\(882\) 0 0
\(883\) −24.4924 −0.824236 −0.412118 0.911131i \(-0.635211\pi\)
−0.412118 + 0.911131i \(0.635211\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −12.4924 −0.419454 −0.209727 0.977760i \(-0.567258\pi\)
−0.209727 + 0.977760i \(0.567258\pi\)
\(888\) 0 0
\(889\) −10.2462 −0.343647
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −4.13826 −0.138482
\(894\) 0 0
\(895\) −20.0000 −0.668526
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −14.2462 −0.474610
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −23.6155 −0.785007
\(906\) 0 0
\(907\) −50.1080 −1.66381 −0.831904 0.554920i \(-0.812749\pi\)
−0.831904 + 0.554920i \(0.812749\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 4.49242 0.148841 0.0744203 0.997227i \(-0.476289\pi\)
0.0744203 + 0.997227i \(0.476289\pi\)
\(912\) 0 0
\(913\) 10.2462 0.339100
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −9.12311 −0.301271
\(918\) 0 0
\(919\) 13.3002 0.438733 0.219366 0.975643i \(-0.429601\pi\)
0.219366 + 0.975643i \(0.429601\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 36.4924 1.20116
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 52.1080 1.70961 0.854803 0.518952i \(-0.173678\pi\)
0.854803 + 0.518952i \(0.173678\pi\)
\(930\) 0 0
\(931\) −1.12311 −0.0368083
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −11.6847 −0.382129
\(936\) 0 0
\(937\) 22.6695 0.740580 0.370290 0.928916i \(-0.379258\pi\)
0.370290 + 0.928916i \(0.379258\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 13.8617 0.451880 0.225940 0.974141i \(-0.427455\pi\)
0.225940 + 0.974141i \(0.427455\pi\)
\(942\) 0 0
\(943\) −16.0000 −0.521032
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 4.00000 0.129983 0.0649913 0.997886i \(-0.479298\pi\)
0.0649913 + 0.997886i \(0.479298\pi\)
\(948\) 0 0
\(949\) 19.3693 0.628755
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 24.8769 0.805842 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(954\) 0 0
\(955\) 9.43845 0.305421
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 8.87689 0.286650
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 5.36932 0.172844
\(966\) 0 0
\(967\) 26.8769 0.864303 0.432151 0.901801i \(-0.357755\pi\)
0.432151 + 0.901801i \(0.357755\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −49.4773 −1.58780 −0.793901 0.608048i \(-0.791953\pi\)
−0.793901 + 0.608048i \(0.791953\pi\)
\(972\) 0 0
\(973\) 6.87689 0.220463
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 49.2311 1.57504 0.787521 0.616288i \(-0.211364\pi\)
0.787521 + 0.616288i \(0.211364\pi\)
\(978\) 0 0
\(979\) −18.2462 −0.583151
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −10.4233 −0.332451 −0.166226 0.986088i \(-0.553158\pi\)
−0.166226 + 0.986088i \(0.553158\pi\)
\(984\) 0 0
\(985\) −7.12311 −0.226961
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 46.7386 1.48620
\(990\) 0 0
\(991\) 20.4924 0.650963 0.325482 0.945548i \(-0.394474\pi\)
0.325482 + 0.945548i \(0.394474\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −18.2462 −0.578444
\(996\) 0 0
\(997\) 9.68466 0.306716 0.153358 0.988171i \(-0.450991\pi\)
0.153358 + 0.988171i \(0.450991\pi\)
\(998\) 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 5040.2.a.bt.1.2 2
3.2 odd 2 560.2.a.i.1.2 2
4.3 odd 2 315.2.a.e.1.1 2
12.11 even 2 35.2.a.b.1.2 2
15.2 even 4 2800.2.g.t.449.1 4
15.8 even 4 2800.2.g.t.449.4 4
15.14 odd 2 2800.2.a.bi.1.1 2
20.3 even 4 1575.2.d.e.1324.3 4
20.7 even 4 1575.2.d.e.1324.2 4
20.19 odd 2 1575.2.a.p.1.2 2
21.20 even 2 3920.2.a.bs.1.1 2
24.5 odd 2 2240.2.a.bd.1.1 2
24.11 even 2 2240.2.a.bh.1.2 2
28.27 even 2 2205.2.a.x.1.1 2
60.23 odd 4 175.2.b.b.99.2 4
60.47 odd 4 175.2.b.b.99.3 4
60.59 even 2 175.2.a.f.1.1 2
84.11 even 6 245.2.e.i.226.1 4
84.23 even 6 245.2.e.i.116.1 4
84.47 odd 6 245.2.e.h.116.1 4
84.59 odd 6 245.2.e.h.226.1 4
84.83 odd 2 245.2.a.d.1.2 2
132.131 odd 2 4235.2.a.m.1.1 2
156.155 even 2 5915.2.a.l.1.1 2
420.83 even 4 1225.2.b.f.99.2 4
420.167 even 4 1225.2.b.f.99.3 4
420.419 odd 2 1225.2.a.s.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.a.b.1.2 2 12.11 even 2
175.2.a.f.1.1 2 60.59 even 2
175.2.b.b.99.2 4 60.23 odd 4
175.2.b.b.99.3 4 60.47 odd 4
245.2.a.d.1.2 2 84.83 odd 2
245.2.e.h.116.1 4 84.47 odd 6
245.2.e.h.226.1 4 84.59 odd 6
245.2.e.i.116.1 4 84.23 even 6
245.2.e.i.226.1 4 84.11 even 6
315.2.a.e.1.1 2 4.3 odd 2
560.2.a.i.1.2 2 3.2 odd 2
1225.2.a.s.1.1 2 420.419 odd 2
1225.2.b.f.99.2 4 420.83 even 4
1225.2.b.f.99.3 4 420.167 even 4
1575.2.a.p.1.2 2 20.19 odd 2
1575.2.d.e.1324.2 4 20.7 even 4
1575.2.d.e.1324.3 4 20.3 even 4
2205.2.a.x.1.1 2 28.27 even 2
2240.2.a.bd.1.1 2 24.5 odd 2
2240.2.a.bh.1.2 2 24.11 even 2
2800.2.a.bi.1.1 2 15.14 odd 2
2800.2.g.t.449.1 4 15.2 even 4
2800.2.g.t.449.4 4 15.8 even 4
3920.2.a.bs.1.1 2 21.20 even 2
4235.2.a.m.1.1 2 132.131 odd 2
5040.2.a.bt.1.2 2 1.1 even 1 trivial
5915.2.a.l.1.1 2 156.155 even 2