Properties

Label 4400.2.a.be.1.1
Level $4400$
Weight $2$
Character 4400.1
Self dual yes
Analytic conductor $35.134$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4400.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(35.1341768894\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4400.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{3} +1.00000 q^{7} +6.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} +1.00000 q^{7} +6.00000 q^{9} +1.00000 q^{11} +6.00000 q^{13} -3.00000 q^{17} +5.00000 q^{19} +3.00000 q^{21} -2.00000 q^{23} +9.00000 q^{27} -5.00000 q^{29} -5.00000 q^{31} +3.00000 q^{33} +1.00000 q^{37} +18.0000 q^{39} -2.00000 q^{41} +12.0000 q^{43} -2.00000 q^{47} -6.00000 q^{49} -9.00000 q^{51} +13.0000 q^{53} +15.0000 q^{57} -2.00000 q^{59} +1.00000 q^{61} +6.00000 q^{63} +16.0000 q^{67} -6.00000 q^{69} -15.0000 q^{71} -10.0000 q^{73} +1.00000 q^{77} -2.00000 q^{79} +9.00000 q^{81} -14.0000 q^{83} -15.0000 q^{87} +9.00000 q^{89} +6.00000 q^{91} -15.0000 q^{93} +16.0000 q^{97} +6.00000 q^{99} +O(q^{100})\)

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
\(3\) 3.00000 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000 0.377964 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 0 0
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) 5.00000 1.14708 0.573539 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) 0 0
\(21\) 3.00000 0.654654
\(22\) 0 0
\(23\) −2.00000 −0.417029 −0.208514 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 9.00000 1.73205
\(28\) 0 0
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) −5.00000 −0.898027 −0.449013 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) 0 0
\(33\) 3.00000 0.522233
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.00000 0.164399 0.0821995 0.996616i \(-0.473806\pi\)
0.0821995 + 0.996616i \(0.473806\pi\)
\(38\) 0 0
\(39\) 18.0000 2.88231
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 12.0000 1.82998 0.914991 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.00000 −0.291730 −0.145865 0.989305i \(-0.546597\pi\)
−0.145865 + 0.989305i \(0.546597\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) −9.00000 −1.26025
\(52\) 0 0
\(53\) 13.0000 1.78569 0.892844 0.450367i \(-0.148707\pi\)
0.892844 + 0.450367i \(0.148707\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 15.0000 1.98680
\(58\) 0 0
\(59\) −2.00000 −0.260378 −0.130189 0.991489i \(-0.541558\pi\)
−0.130189 + 0.991489i \(0.541558\pi\)
\(60\) 0 0
\(61\) 1.00000 0.128037 0.0640184 0.997949i \(-0.479608\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) 0 0
\(63\) 6.00000 0.755929
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 16.0000 1.95471 0.977356 0.211604i \(-0.0678686\pi\)
0.977356 + 0.211604i \(0.0678686\pi\)
\(68\) 0 0
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) −15.0000 −1.78017 −0.890086 0.455792i \(-0.849356\pi\)
−0.890086 + 0.455792i \(0.849356\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 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
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) −14.0000 −1.53670 −0.768350 0.640030i \(-0.778922\pi\)
−0.768350 + 0.640030i \(0.778922\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −15.0000 −1.60817
\(88\) 0 0
\(89\) 9.00000 0.953998 0.476999 0.878904i \(-0.341725\pi\)
0.476999 + 0.878904i \(0.341725\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) 0 0
\(93\) −15.0000 −1.55543
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 16.0000 1.62455 0.812277 0.583272i \(-0.198228\pi\)
0.812277 + 0.583272i \(0.198228\pi\)
\(98\) 0 0
\(99\) 6.00000 0.603023
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 3.00000 0.284747
\(112\) 0 0
\(113\) 16.0000 1.50515 0.752577 0.658505i \(-0.228811\pi\)
0.752577 + 0.658505i \(0.228811\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 36.0000 3.32820
\(118\) 0 0
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −6.00000 −0.541002
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) 36.0000 3.16962
\(130\) 0 0
\(131\) −7.00000 −0.611593 −0.305796 0.952097i \(-0.598923\pi\)
−0.305796 + 0.952097i \(0.598923\pi\)
\(132\) 0 0
\(133\) 5.00000 0.433555
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 0 0
\(143\) 6.00000 0.501745
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −18.0000 −1.48461
\(148\) 0 0
\(149\) −15.0000 −1.22885 −0.614424 0.788976i \(-0.710612\pi\)
−0.614424 + 0.788976i \(0.710612\pi\)
\(150\) 0 0
\(151\) −18.0000 −1.46482 −0.732410 0.680864i \(-0.761604\pi\)
−0.732410 + 0.680864i \(0.761604\pi\)
\(152\) 0 0
\(153\) −18.0000 −1.45521
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −7.00000 −0.558661 −0.279330 0.960195i \(-0.590112\pi\)
−0.279330 + 0.960195i \(0.590112\pi\)
\(158\) 0 0
\(159\) 39.0000 3.09290
\(160\) 0 0
\(161\) −2.00000 −0.157622
\(162\) 0 0
\(163\) −1.00000 −0.0783260 −0.0391630 0.999233i \(-0.512469\pi\)
−0.0391630 + 0.999233i \(0.512469\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.00000 0.0773823 0.0386912 0.999251i \(-0.487681\pi\)
0.0386912 + 0.999251i \(0.487681\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) 30.0000 2.29416
\(172\) 0 0
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −6.00000 −0.450988
\(178\) 0 0
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) 22.0000 1.63525 0.817624 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) 0 0
\(183\) 3.00000 0.221766
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −3.00000 −0.219382
\(188\) 0 0
\(189\) 9.00000 0.654654
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) −7.00000 −0.503871 −0.251936 0.967744i \(-0.581067\pi\)
−0.251936 + 0.967744i \(0.581067\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 24.0000 1.70993 0.854965 0.518686i \(-0.173579\pi\)
0.854965 + 0.518686i \(0.173579\pi\)
\(198\) 0 0
\(199\) 1.00000 0.0708881 0.0354441 0.999372i \(-0.488715\pi\)
0.0354441 + 0.999372i \(0.488715\pi\)
\(200\) 0 0
\(201\) 48.0000 3.38566
\(202\) 0 0
\(203\) −5.00000 −0.350931
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −12.0000 −0.834058
\(208\) 0 0
\(209\) 5.00000 0.345857
\(210\) 0 0
\(211\) 1.00000 0.0688428 0.0344214 0.999407i \(-0.489041\pi\)
0.0344214 + 0.999407i \(0.489041\pi\)
\(212\) 0 0
\(213\) −45.0000 −3.08335
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −5.00000 −0.339422
\(218\) 0 0
\(219\) −30.0000 −2.02721
\(220\) 0 0
\(221\) −18.0000 −1.21081
\(222\) 0 0
\(223\) 26.0000 1.74109 0.870544 0.492090i \(-0.163767\pi\)
0.870544 + 0.492090i \(0.163767\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 18.0000 1.19470 0.597351 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 0 0
\(231\) 3.00000 0.197386
\(232\) 0 0
\(233\) −11.0000 −0.720634 −0.360317 0.932830i \(-0.617331\pi\)
−0.360317 + 0.932830i \(0.617331\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −6.00000 −0.389742
\(238\) 0 0
\(239\) 2.00000 0.129369 0.0646846 0.997906i \(-0.479396\pi\)
0.0646846 + 0.997906i \(0.479396\pi\)
\(240\) 0 0
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 30.0000 1.90885
\(248\) 0 0
\(249\) −42.0000 −2.66164
\(250\) 0 0
\(251\) 10.0000 0.631194 0.315597 0.948893i \(-0.397795\pi\)
0.315597 + 0.948893i \(0.397795\pi\)
\(252\) 0 0
\(253\) −2.00000 −0.125739
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 22.0000 1.37232 0.686161 0.727450i \(-0.259294\pi\)
0.686161 + 0.727450i \(0.259294\pi\)
\(258\) 0 0
\(259\) 1.00000 0.0621370
\(260\) 0 0
\(261\) −30.0000 −1.85695
\(262\) 0 0
\(263\) −13.0000 −0.801614 −0.400807 0.916162i \(-0.631270\pi\)
−0.400807 + 0.916162i \(0.631270\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 27.0000 1.65237
\(268\) 0 0
\(269\) 20.0000 1.21942 0.609711 0.792624i \(-0.291286\pi\)
0.609711 + 0.792624i \(0.291286\pi\)
\(270\) 0 0
\(271\) −24.0000 −1.45790 −0.728948 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(272\) 0 0
\(273\) 18.0000 1.08941
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −16.0000 −0.961347 −0.480673 0.876900i \(-0.659608\pi\)
−0.480673 + 0.876900i \(0.659608\pi\)
\(278\) 0 0
\(279\) −30.0000 −1.79605
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) 10.0000 0.594438 0.297219 0.954809i \(-0.403941\pi\)
0.297219 + 0.954809i \(0.403941\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.00000 −0.118056
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 48.0000 2.81381
\(292\) 0 0
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 9.00000 0.522233
\(298\) 0 0
\(299\) −12.0000 −0.693978
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) 0 0
\(303\) −30.0000 −1.72345
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −22.0000 −1.25561 −0.627803 0.778372i \(-0.716046\pi\)
−0.627803 + 0.778372i \(0.716046\pi\)
\(308\) 0 0
\(309\) −48.0000 −2.73062
\(310\) 0 0
\(311\) −21.0000 −1.19080 −0.595400 0.803429i \(-0.703007\pi\)
−0.595400 + 0.803429i \(0.703007\pi\)
\(312\) 0 0
\(313\) 30.0000 1.69570 0.847850 0.530236i \(-0.177897\pi\)
0.847850 + 0.530236i \(0.177897\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3.00000 0.168497 0.0842484 0.996445i \(-0.473151\pi\)
0.0842484 + 0.996445i \(0.473151\pi\)
\(318\) 0 0
\(319\) −5.00000 −0.279946
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) −15.0000 −0.834622
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 30.0000 1.65900
\(328\) 0 0
\(329\) −2.00000 −0.110264
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 0 0
\(333\) 6.00000 0.328798
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −29.0000 −1.57973 −0.789865 0.613280i \(-0.789850\pi\)
−0.789865 + 0.613280i \(0.789850\pi\)
\(338\) 0 0
\(339\) 48.0000 2.60700
\(340\) 0 0
\(341\) −5.00000 −0.270765
\(342\) 0 0
\(343\) −13.0000 −0.701934
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −2.00000 −0.107366 −0.0536828 0.998558i \(-0.517096\pi\)
−0.0536828 + 0.998558i \(0.517096\pi\)
\(348\) 0 0
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) 54.0000 2.88231
\(352\) 0 0
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −9.00000 −0.476331
\(358\) 0 0
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 0 0
\(363\) 3.00000 0.157459
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −12.0000 −0.626395 −0.313197 0.949688i \(-0.601400\pi\)
−0.313197 + 0.949688i \(0.601400\pi\)
\(368\) 0 0
\(369\) −12.0000 −0.624695
\(370\) 0 0
\(371\) 13.0000 0.674926
\(372\) 0 0
\(373\) −14.0000 −0.724893 −0.362446 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −30.0000 −1.54508
\(378\) 0 0
\(379\) 2.00000 0.102733 0.0513665 0.998680i \(-0.483642\pi\)
0.0513665 + 0.998680i \(0.483642\pi\)
\(380\) 0 0
\(381\) −24.0000 −1.22956
\(382\) 0 0
\(383\) −14.0000 −0.715367 −0.357683 0.933843i \(-0.616433\pi\)
−0.357683 + 0.933843i \(0.616433\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 72.0000 3.65997
\(388\) 0 0
\(389\) −2.00000 −0.101404 −0.0507020 0.998714i \(-0.516146\pi\)
−0.0507020 + 0.998714i \(0.516146\pi\)
\(390\) 0 0
\(391\) 6.00000 0.303433
\(392\) 0 0
\(393\) −21.0000 −1.05931
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) 0 0
\(399\) 15.0000 0.750939
\(400\) 0 0
\(401\) −21.0000 −1.04869 −0.524345 0.851506i \(-0.675690\pi\)
−0.524345 + 0.851506i \(0.675690\pi\)
\(402\) 0 0
\(403\) −30.0000 −1.49441
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.00000 0.0495682
\(408\) 0 0
\(409\) −32.0000 −1.58230 −0.791149 0.611623i \(-0.790517\pi\)
−0.791149 + 0.611623i \(0.790517\pi\)
\(410\) 0 0
\(411\) −36.0000 −1.77575
\(412\) 0 0
\(413\) −2.00000 −0.0984136
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 12.0000 0.587643
\(418\) 0 0
\(419\) 4.00000 0.195413 0.0977064 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 0 0
\(421\) −36.0000 −1.75453 −0.877266 0.480004i \(-0.840635\pi\)
−0.877266 + 0.480004i \(0.840635\pi\)
\(422\) 0 0
\(423\) −12.0000 −0.583460
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1.00000 0.0483934
\(428\) 0 0
\(429\) 18.0000 0.869048
\(430\) 0 0
\(431\) 8.00000 0.385346 0.192673 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) 0 0
\(433\) −4.00000 −0.192228 −0.0961139 0.995370i \(-0.530641\pi\)
−0.0961139 + 0.995370i \(0.530641\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −10.0000 −0.478365
\(438\) 0 0
\(439\) −16.0000 −0.763638 −0.381819 0.924237i \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) 0 0
\(441\) −36.0000 −1.71429
\(442\) 0 0
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −45.0000 −2.12843
\(448\) 0 0
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) −2.00000 −0.0941763
\(452\) 0 0
\(453\) −54.0000 −2.53714
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −33.0000 −1.54367 −0.771837 0.635820i \(-0.780662\pi\)
−0.771837 + 0.635820i \(0.780662\pi\)
\(458\) 0 0
\(459\) −27.0000 −1.26025
\(460\) 0 0
\(461\) 5.00000 0.232873 0.116437 0.993198i \(-0.462853\pi\)
0.116437 + 0.993198i \(0.462853\pi\)
\(462\) 0 0
\(463\) 6.00000 0.278844 0.139422 0.990233i \(-0.455476\pi\)
0.139422 + 0.990233i \(0.455476\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 27.0000 1.24941 0.624705 0.780860i \(-0.285219\pi\)
0.624705 + 0.780860i \(0.285219\pi\)
\(468\) 0 0
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) −21.0000 −0.967629
\(472\) 0 0
\(473\) 12.0000 0.551761
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 78.0000 3.57137
\(478\) 0 0
\(479\) −20.0000 −0.913823 −0.456912 0.889512i \(-0.651044\pi\)
−0.456912 + 0.889512i \(0.651044\pi\)
\(480\) 0 0
\(481\) 6.00000 0.273576
\(482\) 0 0
\(483\) −6.00000 −0.273009
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −12.0000 −0.543772 −0.271886 0.962329i \(-0.587647\pi\)
−0.271886 + 0.962329i \(0.587647\pi\)
\(488\) 0 0
\(489\) −3.00000 −0.135665
\(490\) 0 0
\(491\) 13.0000 0.586682 0.293341 0.956008i \(-0.405233\pi\)
0.293341 + 0.956008i \(0.405233\pi\)
\(492\) 0 0
\(493\) 15.0000 0.675566
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −15.0000 −0.672842
\(498\) 0 0
\(499\) 8.00000 0.358129 0.179065 0.983837i \(-0.442693\pi\)
0.179065 + 0.983837i \(0.442693\pi\)
\(500\) 0 0
\(501\) 3.00000 0.134030
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 69.0000 3.06440
\(508\) 0 0
\(509\) 8.00000 0.354594 0.177297 0.984157i \(-0.443265\pi\)
0.177297 + 0.984157i \(0.443265\pi\)
\(510\) 0 0
\(511\) −10.0000 −0.442374
\(512\) 0 0
\(513\) 45.0000 1.98680
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −2.00000 −0.0879599
\(518\) 0 0
\(519\) 18.0000 0.790112
\(520\) 0 0
\(521\) −26.0000 −1.13908 −0.569540 0.821963i \(-0.692879\pi\)
−0.569540 + 0.821963i \(0.692879\pi\)
\(522\) 0 0
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 15.0000 0.653410
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −12.0000 −0.517838
\(538\) 0 0
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) 15.0000 0.644900 0.322450 0.946586i \(-0.395494\pi\)
0.322450 + 0.946586i \(0.395494\pi\)
\(542\) 0 0
\(543\) 66.0000 2.83233
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −36.0000 −1.53925 −0.769624 0.638497i \(-0.779557\pi\)
−0.769624 + 0.638497i \(0.779557\pi\)
\(548\) 0 0
\(549\) 6.00000 0.256074
\(550\) 0 0
\(551\) −25.0000 −1.06504
\(552\) 0 0
\(553\) −2.00000 −0.0850487
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 30.0000 1.27114 0.635570 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) 0 0
\(559\) 72.0000 3.04528
\(560\) 0 0
\(561\) −9.00000 −0.379980
\(562\) 0 0
\(563\) −18.0000 −0.758610 −0.379305 0.925272i \(-0.623837\pi\)
−0.379305 + 0.925272i \(0.623837\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 9.00000 0.377964
\(568\) 0 0
\(569\) 36.0000 1.50920 0.754599 0.656186i \(-0.227831\pi\)
0.754599 + 0.656186i \(0.227831\pi\)
\(570\) 0 0
\(571\) 19.0000 0.795125 0.397563 0.917575i \(-0.369856\pi\)
0.397563 + 0.917575i \(0.369856\pi\)
\(572\) 0 0
\(573\) −36.0000 −1.50392
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 14.0000 0.582828 0.291414 0.956597i \(-0.405874\pi\)
0.291414 + 0.956597i \(0.405874\pi\)
\(578\) 0 0
\(579\) −21.0000 −0.872730
\(580\) 0 0
\(581\) −14.0000 −0.580818
\(582\) 0 0
\(583\) 13.0000 0.538405
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −39.0000 −1.60970 −0.804851 0.593477i \(-0.797755\pi\)
−0.804851 + 0.593477i \(0.797755\pi\)
\(588\) 0 0
\(589\) −25.0000 −1.03011
\(590\) 0 0
\(591\) 72.0000 2.96168
\(592\) 0 0
\(593\) 22.0000 0.903432 0.451716 0.892162i \(-0.350812\pi\)
0.451716 + 0.892162i \(0.350812\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 3.00000 0.122782
\(598\) 0 0
\(599\) 27.0000 1.10319 0.551595 0.834112i \(-0.314019\pi\)
0.551595 + 0.834112i \(0.314019\pi\)
\(600\) 0 0
\(601\) 14.0000 0.571072 0.285536 0.958368i \(-0.407828\pi\)
0.285536 + 0.958368i \(0.407828\pi\)
\(602\) 0 0
\(603\) 96.0000 3.90942
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −21.0000 −0.852364 −0.426182 0.904638i \(-0.640142\pi\)
−0.426182 + 0.904638i \(0.640142\pi\)
\(608\) 0 0
\(609\) −15.0000 −0.607831
\(610\) 0 0
\(611\) −12.0000 −0.485468
\(612\) 0 0
\(613\) 10.0000 0.403896 0.201948 0.979396i \(-0.435273\pi\)
0.201948 + 0.979396i \(0.435273\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −12.0000 −0.483102 −0.241551 0.970388i \(-0.577656\pi\)
−0.241551 + 0.970388i \(0.577656\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) −18.0000 −0.722315
\(622\) 0 0
\(623\) 9.00000 0.360577
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 15.0000 0.599042
\(628\) 0 0
\(629\) −3.00000 −0.119618
\(630\) 0 0
\(631\) −15.0000 −0.597141 −0.298570 0.954388i \(-0.596510\pi\)
−0.298570 + 0.954388i \(0.596510\pi\)
\(632\) 0 0
\(633\) 3.00000 0.119239
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −36.0000 −1.42637
\(638\) 0 0
\(639\) −90.0000 −3.56034
\(640\) 0 0
\(641\) −9.00000 −0.355479 −0.177739 0.984078i \(-0.556878\pi\)
−0.177739 + 0.984078i \(0.556878\pi\)
\(642\) 0 0
\(643\) 7.00000 0.276053 0.138027 0.990429i \(-0.455924\pi\)
0.138027 + 0.990429i \(0.455924\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −6.00000 −0.235884 −0.117942 0.993020i \(-0.537630\pi\)
−0.117942 + 0.993020i \(0.537630\pi\)
\(648\) 0 0
\(649\) −2.00000 −0.0785069
\(650\) 0 0
\(651\) −15.0000 −0.587896
\(652\) 0 0
\(653\) 19.0000 0.743527 0.371764 0.928327i \(-0.378753\pi\)
0.371764 + 0.928327i \(0.378753\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −60.0000 −2.34082
\(658\) 0 0
\(659\) −49.0000 −1.90877 −0.954384 0.298580i \(-0.903487\pi\)
−0.954384 + 0.298580i \(0.903487\pi\)
\(660\) 0 0
\(661\) −22.0000 −0.855701 −0.427850 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) 0 0
\(663\) −54.0000 −2.09719
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 10.0000 0.387202
\(668\) 0 0
\(669\) 78.0000 3.01565
\(670\) 0 0
\(671\) 1.00000 0.0386046
\(672\) 0 0
\(673\) 1.00000 0.0385472 0.0192736 0.999814i \(-0.493865\pi\)
0.0192736 + 0.999814i \(0.493865\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 16.0000 0.614024
\(680\) 0 0
\(681\) 54.0000 2.06928
\(682\) 0 0
\(683\) −19.0000 −0.727015 −0.363507 0.931591i \(-0.618421\pi\)
−0.363507 + 0.931591i \(0.618421\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 18.0000 0.686743
\(688\) 0 0
\(689\) 78.0000 2.97156
\(690\) 0 0
\(691\) −34.0000 −1.29342 −0.646710 0.762736i \(-0.723856\pi\)
−0.646710 + 0.762736i \(0.723856\pi\)
\(692\) 0 0
\(693\) 6.00000 0.227921
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 6.00000 0.227266
\(698\) 0 0
\(699\) −33.0000 −1.24817
\(700\) 0 0
\(701\) −31.0000 −1.17085 −0.585427 0.810725i \(-0.699073\pi\)
−0.585427 + 0.810725i \(0.699073\pi\)
\(702\) 0 0
\(703\) 5.00000 0.188579
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −10.0000 −0.376089
\(708\) 0 0
\(709\) 46.0000 1.72757 0.863783 0.503864i \(-0.168089\pi\)
0.863783 + 0.503864i \(0.168089\pi\)
\(710\) 0 0
\(711\) −12.0000 −0.450035
\(712\) 0 0
\(713\) 10.0000 0.374503
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 6.00000 0.224074
\(718\) 0 0
\(719\) −1.00000 −0.0372937 −0.0186469 0.999826i \(-0.505936\pi\)
−0.0186469 + 0.999826i \(0.505936\pi\)
\(720\) 0 0
\(721\) −16.0000 −0.595871
\(722\) 0 0
\(723\) 54.0000 2.00828
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −26.0000 −0.964287 −0.482143 0.876092i \(-0.660142\pi\)
−0.482143 + 0.876092i \(0.660142\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −36.0000 −1.33151
\(732\) 0 0
\(733\) 32.0000 1.18195 0.590973 0.806691i \(-0.298744\pi\)
0.590973 + 0.806691i \(0.298744\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 16.0000 0.589368
\(738\) 0 0
\(739\) 8.00000 0.294285 0.147142 0.989115i \(-0.452992\pi\)
0.147142 + 0.989115i \(0.452992\pi\)
\(740\) 0 0
\(741\) 90.0000 3.30623
\(742\) 0 0
\(743\) 33.0000 1.21065 0.605326 0.795977i \(-0.293043\pi\)
0.605326 + 0.795977i \(0.293043\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −84.0000 −3.07340
\(748\) 0 0
\(749\) −4.00000 −0.146157
\(750\) 0 0
\(751\) 7.00000 0.255434 0.127717 0.991811i \(-0.459235\pi\)
0.127717 + 0.991811i \(0.459235\pi\)
\(752\) 0 0
\(753\) 30.0000 1.09326
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 0 0
\(759\) −6.00000 −0.217786
\(760\) 0 0
\(761\) −30.0000 −1.08750 −0.543750 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 0 0
\(763\) 10.0000 0.362024
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −12.0000 −0.433295
\(768\) 0 0
\(769\) 10.0000 0.360609 0.180305 0.983611i \(-0.442292\pi\)
0.180305 + 0.983611i \(0.442292\pi\)
\(770\) 0 0
\(771\) 66.0000 2.37693
\(772\) 0 0
\(773\) 9.00000 0.323708 0.161854 0.986815i \(-0.448253\pi\)
0.161854 + 0.986815i \(0.448253\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 3.00000 0.107624
\(778\) 0 0
\(779\) −10.0000 −0.358287
\(780\) 0 0
\(781\) −15.0000 −0.536742
\(782\) 0 0
\(783\) −45.0000 −1.60817
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −48.0000 −1.71102 −0.855508 0.517790i \(-0.826755\pi\)
−0.855508 + 0.517790i \(0.826755\pi\)
\(788\) 0 0
\(789\) −39.0000 −1.38844
\(790\) 0 0
\(791\) 16.0000 0.568895
\(792\) 0 0
\(793\) 6.00000 0.213066
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −34.0000 −1.20434 −0.602171 0.798367i \(-0.705697\pi\)
−0.602171 + 0.798367i \(0.705697\pi\)
\(798\) 0 0
\(799\) 6.00000 0.212265
\(800\) 0 0
\(801\) 54.0000 1.90800
\(802\) 0 0
\(803\) −10.0000 −0.352892
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 60.0000 2.11210
\(808\) 0 0
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) −1.00000 −0.0351147 −0.0175574 0.999846i \(-0.505589\pi\)
−0.0175574 + 0.999846i \(0.505589\pi\)
\(812\) 0 0
\(813\) −72.0000 −2.52515
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 60.0000 2.09913
\(818\) 0 0
\(819\) 36.0000 1.25794
\(820\) 0 0
\(821\) 50.0000 1.74501 0.872506 0.488603i \(-0.162493\pi\)
0.872506 + 0.488603i \(0.162493\pi\)
\(822\) 0 0
\(823\) 44.0000 1.53374 0.766872 0.641800i \(-0.221812\pi\)
0.766872 + 0.641800i \(0.221812\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −52.0000 −1.80822 −0.904109 0.427303i \(-0.859464\pi\)
−0.904109 + 0.427303i \(0.859464\pi\)
\(828\) 0 0
\(829\) 4.00000 0.138926 0.0694629 0.997585i \(-0.477871\pi\)
0.0694629 + 0.997585i \(0.477871\pi\)
\(830\) 0 0
\(831\) −48.0000 −1.66510
\(832\) 0 0
\(833\) 18.0000 0.623663
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −45.0000 −1.55543
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 0 0
\(843\) 54.0000 1.85986
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 0 0
\(849\) 30.0000 1.02960
\(850\) 0 0
\(851\) −2.00000 −0.0685591
\(852\) 0 0
\(853\) −22.0000 −0.753266 −0.376633 0.926363i \(-0.622918\pi\)
−0.376633 + 0.926363i \(0.622918\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 21.0000 0.717346 0.358673 0.933463i \(-0.383229\pi\)
0.358673 + 0.933463i \(0.383229\pi\)
\(858\) 0 0
\(859\) 22.0000 0.750630 0.375315 0.926897i \(-0.377534\pi\)
0.375315 + 0.926897i \(0.377534\pi\)
\(860\) 0 0
\(861\) −6.00000 −0.204479
\(862\) 0 0
\(863\) −46.0000 −1.56586 −0.782929 0.622111i \(-0.786275\pi\)
−0.782929 + 0.622111i \(0.786275\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −24.0000 −0.815083
\(868\) 0 0
\(869\) −2.00000 −0.0678454
\(870\) 0 0
\(871\) 96.0000 3.25284
\(872\) 0 0
\(873\) 96.0000 3.24911
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −2.00000 −0.0675352 −0.0337676 0.999430i \(-0.510751\pi\)
−0.0337676 + 0.999430i \(0.510751\pi\)
\(878\) 0 0
\(879\) −18.0000 −0.607125
\(880\) 0 0
\(881\) 50.0000 1.68454 0.842271 0.539054i \(-0.181218\pi\)
0.842271 + 0.539054i \(0.181218\pi\)
\(882\) 0 0
\(883\) 5.00000 0.168263 0.0841317 0.996455i \(-0.473188\pi\)
0.0841317 + 0.996455i \(0.473188\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −40.0000 −1.34307 −0.671534 0.740973i \(-0.734364\pi\)
−0.671534 + 0.740973i \(0.734364\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) 9.00000 0.301511
\(892\) 0 0
\(893\) −10.0000 −0.334637
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −36.0000 −1.20201
\(898\) 0 0
\(899\) 25.0000 0.833797
\(900\) 0 0
\(901\) −39.0000 −1.29928
\(902\) 0 0
\(903\) 36.0000 1.19800
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −29.0000 −0.962929 −0.481465 0.876466i \(-0.659895\pi\)
−0.481465 + 0.876466i \(0.659895\pi\)
\(908\) 0 0
\(909\) −60.0000 −1.99007
\(910\) 0 0
\(911\) −3.00000 −0.0993944 −0.0496972 0.998764i \(-0.515826\pi\)
−0.0496972 + 0.998764i \(0.515826\pi\)
\(912\) 0 0
\(913\) −14.0000 −0.463332
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −7.00000 −0.231160
\(918\) 0 0
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 0 0
\(921\) −66.0000 −2.17477
\(922\) 0 0
\(923\) −90.0000 −2.96239
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −96.0000 −3.15305
\(928\) 0 0
\(929\) 59.0000 1.93573 0.967864 0.251476i \(-0.0809159\pi\)
0.967864 + 0.251476i \(0.0809159\pi\)
\(930\) 0 0
\(931\) −30.0000 −0.983210
\(932\) 0 0
\(933\) −63.0000 −2.06253
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −10.0000 −0.326686 −0.163343 0.986569i \(-0.552228\pi\)
−0.163343 + 0.986569i \(0.552228\pi\)
\(938\) 0 0
\(939\) 90.0000 2.93704
\(940\) 0 0
\(941\) −41.0000 −1.33656 −0.668281 0.743909i \(-0.732970\pi\)
−0.668281 + 0.743909i \(0.732970\pi\)
\(942\) 0 0
\(943\) 4.00000 0.130258
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 41.0000 1.33232 0.666160 0.745808i \(-0.267937\pi\)
0.666160 + 0.745808i \(0.267937\pi\)
\(948\) 0 0
\(949\) −60.0000 −1.94768
\(950\) 0 0
\(951\) 9.00000 0.291845
\(952\) 0 0
\(953\) −29.0000 −0.939402 −0.469701 0.882826i \(-0.655638\pi\)
−0.469701 + 0.882826i \(0.655638\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −15.0000 −0.484881
\(958\) 0 0
\(959\) −12.0000 −0.387500
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) 0 0
\(963\) −24.0000 −0.773389
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 55.0000 1.76868 0.884340 0.466843i \(-0.154609\pi\)
0.884340 + 0.466843i \(0.154609\pi\)
\(968\) 0 0
\(969\) −45.0000 −1.44561
\(970\) 0 0
\(971\) −44.0000 −1.41203 −0.706014 0.708198i \(-0.749508\pi\)
−0.706014 + 0.708198i \(0.749508\pi\)
\(972\) 0 0
\(973\) 4.00000 0.128234
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 8.00000 0.255943 0.127971 0.991778i \(-0.459153\pi\)
0.127971 + 0.991778i \(0.459153\pi\)
\(978\) 0 0
\(979\) 9.00000 0.287641
\(980\) 0 0
\(981\) 60.0000 1.91565
\(982\) 0 0
\(983\) −14.0000 −0.446531 −0.223265 0.974758i \(-0.571672\pi\)
−0.223265 + 0.974758i \(0.571672\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −6.00000 −0.190982
\(988\) 0 0
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 0 0
\(993\) 60.0000 1.90404
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −8.00000 −0.253363 −0.126681 0.991943i \(-0.540433\pi\)
−0.126681 + 0.991943i \(0.540433\pi\)
\(998\) 0 0
\(999\) 9.00000 0.284747
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 4400.2.a.be.1.1 1
4.3 odd 2 2200.2.a.a.1.1 1
5.2 odd 4 4400.2.b.a.4049.1 2
5.3 odd 4 4400.2.b.a.4049.2 2
5.4 even 2 880.2.a.a.1.1 1
15.14 odd 2 7920.2.a.e.1.1 1
20.3 even 4 2200.2.b.b.1849.1 2
20.7 even 4 2200.2.b.b.1849.2 2
20.19 odd 2 440.2.a.d.1.1 1
40.19 odd 2 3520.2.a.a.1.1 1
40.29 even 2 3520.2.a.bh.1.1 1
55.54 odd 2 9680.2.a.a.1.1 1
60.59 even 2 3960.2.a.f.1.1 1
220.219 even 2 4840.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.a.d.1.1 1 20.19 odd 2
880.2.a.a.1.1 1 5.4 even 2
2200.2.a.a.1.1 1 4.3 odd 2
2200.2.b.b.1849.1 2 20.3 even 4
2200.2.b.b.1849.2 2 20.7 even 4
3520.2.a.a.1.1 1 40.19 odd 2
3520.2.a.bh.1.1 1 40.29 even 2
3960.2.a.f.1.1 1 60.59 even 2
4400.2.a.be.1.1 1 1.1 even 1 trivial
4400.2.b.a.4049.1 2 5.2 odd 4
4400.2.b.a.4049.2 2 5.3 odd 4
4840.2.a.i.1.1 1 220.219 even 2
7920.2.a.e.1.1 1 15.14 odd 2
9680.2.a.a.1.1 1 55.54 odd 2