Properties

Label 6800.2.a.bh.1.1
Level $6800$
Weight $2$
Character 6800.1
Self dual yes
Analytic conductor $54.298$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6800,2,Mod(1,6800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6800.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6800 = 2^{4} \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6800.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(54.2982733745\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 68)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 6800.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.732051 q^{3} +0.732051 q^{7} -2.46410 q^{9} +4.73205 q^{11} +1.46410 q^{13} +1.00000 q^{17} -5.46410 q^{19} -0.535898 q^{21} -4.73205 q^{23} +4.00000 q^{27} -3.46410 q^{29} +6.19615 q^{31} -3.46410 q^{33} -11.4641 q^{37} -1.07180 q^{39} -6.00000 q^{41} +12.3923 q^{43} +6.92820 q^{47} -6.46410 q^{49} -0.732051 q^{51} +0.928203 q^{53} +4.00000 q^{57} -9.46410 q^{59} -7.46410 q^{61} -1.80385 q^{63} +1.07180 q^{67} +3.46410 q^{69} -2.19615 q^{71} -2.00000 q^{73} +3.46410 q^{77} +1.80385 q^{79} +4.46410 q^{81} -9.46410 q^{83} +2.53590 q^{87} +9.46410 q^{89} +1.07180 q^{91} -4.53590 q^{93} -8.92820 q^{97} -11.6603 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9} + 6 q^{11} - 4 q^{13} + 2 q^{17} - 4 q^{19} - 8 q^{21} - 6 q^{23} + 8 q^{27} + 2 q^{31} - 16 q^{37} - 16 q^{39} - 12 q^{41} + 4 q^{43} - 6 q^{49} + 2 q^{51} - 12 q^{53}+ \cdots - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −0.732051 −0.422650 −0.211325 0.977416i \(-0.567778\pi\)
−0.211325 + 0.977416i \(0.567778\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.732051 0.276689 0.138345 0.990384i \(-0.455822\pi\)
0.138345 + 0.990384i \(0.455822\pi\)
\(8\) 0 0
\(9\) −2.46410 −0.821367
\(10\) 0 0
\(11\) 4.73205 1.42677 0.713384 0.700774i \(-0.247162\pi\)
0.713384 + 0.700774i \(0.247162\pi\)
\(12\) 0 0
\(13\) 1.46410 0.406069 0.203034 0.979172i \(-0.434920\pi\)
0.203034 + 0.979172i \(0.434920\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.00000 0.242536
\(18\) 0 0
\(19\) −5.46410 −1.25355 −0.626775 0.779200i \(-0.715626\pi\)
−0.626775 + 0.779200i \(0.715626\pi\)
\(20\) 0 0
\(21\) −0.535898 −0.116943
\(22\) 0 0
\(23\) −4.73205 −0.986701 −0.493350 0.869831i \(-0.664228\pi\)
−0.493350 + 0.869831i \(0.664228\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) −3.46410 −0.643268 −0.321634 0.946864i \(-0.604232\pi\)
−0.321634 + 0.946864i \(0.604232\pi\)
\(30\) 0 0
\(31\) 6.19615 1.11286 0.556431 0.830894i \(-0.312170\pi\)
0.556431 + 0.830894i \(0.312170\pi\)
\(32\) 0 0
\(33\) −3.46410 −0.603023
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −11.4641 −1.88469 −0.942343 0.334648i \(-0.891383\pi\)
−0.942343 + 0.334648i \(0.891383\pi\)
\(38\) 0 0
\(39\) −1.07180 −0.171625
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 12.3923 1.88981 0.944904 0.327346i \(-0.106154\pi\)
0.944904 + 0.327346i \(0.106154\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.92820 1.01058 0.505291 0.862949i \(-0.331385\pi\)
0.505291 + 0.862949i \(0.331385\pi\)
\(48\) 0 0
\(49\) −6.46410 −0.923443
\(50\) 0 0
\(51\) −0.732051 −0.102508
\(52\) 0 0
\(53\) 0.928203 0.127499 0.0637493 0.997966i \(-0.479694\pi\)
0.0637493 + 0.997966i \(0.479694\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 0 0
\(59\) −9.46410 −1.23212 −0.616061 0.787699i \(-0.711272\pi\)
−0.616061 + 0.787699i \(0.711272\pi\)
\(60\) 0 0
\(61\) −7.46410 −0.955680 −0.477840 0.878447i \(-0.658580\pi\)
−0.477840 + 0.878447i \(0.658580\pi\)
\(62\) 0 0
\(63\) −1.80385 −0.227263
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 1.07180 0.130941 0.0654704 0.997855i \(-0.479145\pi\)
0.0654704 + 0.997855i \(0.479145\pi\)
\(68\) 0 0
\(69\) 3.46410 0.417029
\(70\) 0 0
\(71\) −2.19615 −0.260635 −0.130318 0.991472i \(-0.541600\pi\)
−0.130318 + 0.991472i \(0.541600\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.46410 0.394771
\(78\) 0 0
\(79\) 1.80385 0.202949 0.101474 0.994838i \(-0.467644\pi\)
0.101474 + 0.994838i \(0.467644\pi\)
\(80\) 0 0
\(81\) 4.46410 0.496011
\(82\) 0 0
\(83\) −9.46410 −1.03882 −0.519410 0.854525i \(-0.673848\pi\)
−0.519410 + 0.854525i \(0.673848\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.53590 0.271877
\(88\) 0 0
\(89\) 9.46410 1.00319 0.501596 0.865102i \(-0.332746\pi\)
0.501596 + 0.865102i \(0.332746\pi\)
\(90\) 0 0
\(91\) 1.07180 0.112355
\(92\) 0 0
\(93\) −4.53590 −0.470351
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −8.92820 −0.906522 −0.453261 0.891378i \(-0.649739\pi\)
−0.453261 + 0.891378i \(0.649739\pi\)
\(98\) 0 0
\(99\) −11.6603 −1.17190
\(100\) 0 0
\(101\) −9.46410 −0.941713 −0.470857 0.882210i \(-0.656055\pi\)
−0.470857 + 0.882210i \(0.656055\pi\)
\(102\) 0 0
\(103\) 2.92820 0.288524 0.144262 0.989539i \(-0.453919\pi\)
0.144262 + 0.989539i \(0.453919\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.26795 0.702619 0.351310 0.936259i \(-0.385736\pi\)
0.351310 + 0.936259i \(0.385736\pi\)
\(108\) 0 0
\(109\) 6.39230 0.612272 0.306136 0.951988i \(-0.400964\pi\)
0.306136 + 0.951988i \(0.400964\pi\)
\(110\) 0 0
\(111\) 8.39230 0.796562
\(112\) 0 0
\(113\) 7.85641 0.739069 0.369534 0.929217i \(-0.379517\pi\)
0.369534 + 0.929217i \(0.379517\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −3.60770 −0.333532
\(118\) 0 0
\(119\) 0.732051 0.0671070
\(120\) 0 0
\(121\) 11.3923 1.03566
\(122\) 0 0
\(123\) 4.39230 0.396041
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −20.3923 −1.80952 −0.904762 0.425917i \(-0.859952\pi\)
−0.904762 + 0.425917i \(0.859952\pi\)
\(128\) 0 0
\(129\) −9.07180 −0.798727
\(130\) 0 0
\(131\) 11.6603 1.01876 0.509381 0.860541i \(-0.329875\pi\)
0.509381 + 0.860541i \(0.329875\pi\)
\(132\) 0 0
\(133\) −4.00000 −0.346844
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −9.46410 −0.808573 −0.404286 0.914632i \(-0.632480\pi\)
−0.404286 + 0.914632i \(0.632480\pi\)
\(138\) 0 0
\(139\) 11.2679 0.955735 0.477867 0.878432i \(-0.341410\pi\)
0.477867 + 0.878432i \(0.341410\pi\)
\(140\) 0 0
\(141\) −5.07180 −0.427122
\(142\) 0 0
\(143\) 6.92820 0.579365
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 4.73205 0.390293
\(148\) 0 0
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 6.53590 0.531884 0.265942 0.963989i \(-0.414317\pi\)
0.265942 + 0.963989i \(0.414317\pi\)
\(152\) 0 0
\(153\) −2.46410 −0.199211
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 0 0
\(159\) −0.679492 −0.0538872
\(160\) 0 0
\(161\) −3.46410 −0.273009
\(162\) 0 0
\(163\) −10.5885 −0.829352 −0.414676 0.909969i \(-0.636105\pi\)
−0.414676 + 0.909969i \(0.636105\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −11.6603 −0.902298 −0.451149 0.892449i \(-0.648986\pi\)
−0.451149 + 0.892449i \(0.648986\pi\)
\(168\) 0 0
\(169\) −10.8564 −0.835108
\(170\) 0 0
\(171\) 13.4641 1.02963
\(172\) 0 0
\(173\) 8.53590 0.648972 0.324486 0.945890i \(-0.394809\pi\)
0.324486 + 0.945890i \(0.394809\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 6.92820 0.520756
\(178\) 0 0
\(179\) −21.4641 −1.60430 −0.802151 0.597121i \(-0.796311\pi\)
−0.802151 + 0.597121i \(0.796311\pi\)
\(180\) 0 0
\(181\) 11.4641 0.852120 0.426060 0.904695i \(-0.359901\pi\)
0.426060 + 0.904695i \(0.359901\pi\)
\(182\) 0 0
\(183\) 5.46410 0.403918
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 4.73205 0.346042
\(188\) 0 0
\(189\) 2.92820 0.212995
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) −7.07180 −0.509039 −0.254520 0.967068i \(-0.581917\pi\)
−0.254520 + 0.967068i \(0.581917\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3.46410 0.246807 0.123404 0.992357i \(-0.460619\pi\)
0.123404 + 0.992357i \(0.460619\pi\)
\(198\) 0 0
\(199\) 6.19615 0.439234 0.219617 0.975586i \(-0.429519\pi\)
0.219617 + 0.975586i \(0.429519\pi\)
\(200\) 0 0
\(201\) −0.784610 −0.0553421
\(202\) 0 0
\(203\) −2.53590 −0.177985
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 11.6603 0.810444
\(208\) 0 0
\(209\) −25.8564 −1.78853
\(210\) 0 0
\(211\) −19.6603 −1.35347 −0.676734 0.736228i \(-0.736605\pi\)
−0.676734 + 0.736228i \(0.736605\pi\)
\(212\) 0 0
\(213\) 1.60770 0.110157
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 4.53590 0.307917
\(218\) 0 0
\(219\) 1.46410 0.0989348
\(220\) 0 0
\(221\) 1.46410 0.0984861
\(222\) 0 0
\(223\) −8.39230 −0.561990 −0.280995 0.959709i \(-0.590664\pi\)
−0.280995 + 0.959709i \(0.590664\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −9.80385 −0.650704 −0.325352 0.945593i \(-0.605483\pi\)
−0.325352 + 0.945593i \(0.605483\pi\)
\(228\) 0 0
\(229\) 0.392305 0.0259242 0.0129621 0.999916i \(-0.495874\pi\)
0.0129621 + 0.999916i \(0.495874\pi\)
\(230\) 0 0
\(231\) −2.53590 −0.166850
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −1.32051 −0.0857762
\(238\) 0 0
\(239\) 20.7846 1.34444 0.672222 0.740349i \(-0.265340\pi\)
0.672222 + 0.740349i \(0.265340\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 0 0
\(243\) −15.2679 −0.979439
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −8.00000 −0.509028
\(248\) 0 0
\(249\) 6.92820 0.439057
\(250\) 0 0
\(251\) 6.92820 0.437304 0.218652 0.975803i \(-0.429834\pi\)
0.218652 + 0.975803i \(0.429834\pi\)
\(252\) 0 0
\(253\) −22.3923 −1.40779
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 7.60770 0.474555 0.237277 0.971442i \(-0.423745\pi\)
0.237277 + 0.971442i \(0.423745\pi\)
\(258\) 0 0
\(259\) −8.39230 −0.521472
\(260\) 0 0
\(261\) 8.53590 0.528359
\(262\) 0 0
\(263\) −23.3205 −1.43800 −0.719002 0.695008i \(-0.755401\pi\)
−0.719002 + 0.695008i \(0.755401\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −6.92820 −0.423999
\(268\) 0 0
\(269\) −20.5359 −1.25210 −0.626048 0.779785i \(-0.715329\pi\)
−0.626048 + 0.779785i \(0.715329\pi\)
\(270\) 0 0
\(271\) −14.9282 −0.906824 −0.453412 0.891301i \(-0.649793\pi\)
−0.453412 + 0.891301i \(0.649793\pi\)
\(272\) 0 0
\(273\) −0.784610 −0.0474867
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 14.3923 0.864750 0.432375 0.901694i \(-0.357676\pi\)
0.432375 + 0.901694i \(0.357676\pi\)
\(278\) 0 0
\(279\) −15.2679 −0.914068
\(280\) 0 0
\(281\) 7.85641 0.468674 0.234337 0.972155i \(-0.424708\pi\)
0.234337 + 0.972155i \(0.424708\pi\)
\(282\) 0 0
\(283\) −10.5885 −0.629418 −0.314709 0.949188i \(-0.601907\pi\)
−0.314709 + 0.949188i \(0.601907\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −4.39230 −0.259270
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 6.53590 0.383141
\(292\) 0 0
\(293\) −30.0000 −1.75262 −0.876309 0.481749i \(-0.840002\pi\)
−0.876309 + 0.481749i \(0.840002\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 18.9282 1.09833
\(298\) 0 0
\(299\) −6.92820 −0.400668
\(300\) 0 0
\(301\) 9.07180 0.522890
\(302\) 0 0
\(303\) 6.92820 0.398015
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 16.7846 0.957948 0.478974 0.877829i \(-0.341009\pi\)
0.478974 + 0.877829i \(0.341009\pi\)
\(308\) 0 0
\(309\) −2.14359 −0.121945
\(310\) 0 0
\(311\) −21.1244 −1.19785 −0.598926 0.800804i \(-0.704406\pi\)
−0.598926 + 0.800804i \(0.704406\pi\)
\(312\) 0 0
\(313\) −7.07180 −0.399722 −0.199861 0.979824i \(-0.564049\pi\)
−0.199861 + 0.979824i \(0.564049\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −15.4641 −0.868550 −0.434275 0.900780i \(-0.642995\pi\)
−0.434275 + 0.900780i \(0.642995\pi\)
\(318\) 0 0
\(319\) −16.3923 −0.917793
\(320\) 0 0
\(321\) −5.32051 −0.296962
\(322\) 0 0
\(323\) −5.46410 −0.304031
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −4.67949 −0.258776
\(328\) 0 0
\(329\) 5.07180 0.279617
\(330\) 0 0
\(331\) 15.3205 0.842091 0.421046 0.907039i \(-0.361663\pi\)
0.421046 + 0.907039i \(0.361663\pi\)
\(332\) 0 0
\(333\) 28.2487 1.54802
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −34.7846 −1.89484 −0.947419 0.319995i \(-0.896319\pi\)
−0.947419 + 0.319995i \(0.896319\pi\)
\(338\) 0 0
\(339\) −5.75129 −0.312367
\(340\) 0 0
\(341\) 29.3205 1.58779
\(342\) 0 0
\(343\) −9.85641 −0.532196
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 4.05256 0.217553 0.108776 0.994066i \(-0.465307\pi\)
0.108776 + 0.994066i \(0.465307\pi\)
\(348\) 0 0
\(349\) −30.7846 −1.64786 −0.823931 0.566690i \(-0.808224\pi\)
−0.823931 + 0.566690i \(0.808224\pi\)
\(350\) 0 0
\(351\) 5.85641 0.312592
\(352\) 0 0
\(353\) −7.85641 −0.418154 −0.209077 0.977899i \(-0.567046\pi\)
−0.209077 + 0.977899i \(0.567046\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −0.535898 −0.0283628
\(358\) 0 0
\(359\) −11.3205 −0.597474 −0.298737 0.954336i \(-0.596565\pi\)
−0.298737 + 0.954336i \(0.596565\pi\)
\(360\) 0 0
\(361\) 10.8564 0.571390
\(362\) 0 0
\(363\) −8.33975 −0.437723
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 19.6603 1.02626 0.513128 0.858312i \(-0.328486\pi\)
0.513128 + 0.858312i \(0.328486\pi\)
\(368\) 0 0
\(369\) 14.7846 0.769656
\(370\) 0 0
\(371\) 0.679492 0.0352775
\(372\) 0 0
\(373\) −12.3923 −0.641649 −0.320825 0.947139i \(-0.603960\pi\)
−0.320825 + 0.947139i \(0.603960\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −5.07180 −0.261211
\(378\) 0 0
\(379\) 8.73205 0.448535 0.224268 0.974528i \(-0.428001\pi\)
0.224268 + 0.974528i \(0.428001\pi\)
\(380\) 0 0
\(381\) 14.9282 0.764795
\(382\) 0 0
\(383\) −9.46410 −0.483593 −0.241797 0.970327i \(-0.577737\pi\)
−0.241797 + 0.970327i \(0.577737\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −30.5359 −1.55223
\(388\) 0 0
\(389\) −25.1769 −1.27652 −0.638260 0.769821i \(-0.720346\pi\)
−0.638260 + 0.769821i \(0.720346\pi\)
\(390\) 0 0
\(391\) −4.73205 −0.239310
\(392\) 0 0
\(393\) −8.53590 −0.430579
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 26.3923 1.32459 0.662296 0.749242i \(-0.269582\pi\)
0.662296 + 0.749242i \(0.269582\pi\)
\(398\) 0 0
\(399\) 2.92820 0.146594
\(400\) 0 0
\(401\) 4.14359 0.206921 0.103461 0.994634i \(-0.467008\pi\)
0.103461 + 0.994634i \(0.467008\pi\)
\(402\) 0 0
\(403\) 9.07180 0.451898
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −54.2487 −2.68901
\(408\) 0 0
\(409\) −37.7128 −1.86478 −0.932389 0.361456i \(-0.882280\pi\)
−0.932389 + 0.361456i \(0.882280\pi\)
\(410\) 0 0
\(411\) 6.92820 0.341743
\(412\) 0 0
\(413\) −6.92820 −0.340915
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −8.24871 −0.403941
\(418\) 0 0
\(419\) −6.58846 −0.321867 −0.160934 0.986965i \(-0.551450\pi\)
−0.160934 + 0.986965i \(0.551450\pi\)
\(420\) 0 0
\(421\) 26.2487 1.27928 0.639642 0.768673i \(-0.279083\pi\)
0.639642 + 0.768673i \(0.279083\pi\)
\(422\) 0 0
\(423\) −17.0718 −0.830059
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −5.46410 −0.264426
\(428\) 0 0
\(429\) −5.07180 −0.244869
\(430\) 0 0
\(431\) −0.339746 −0.0163650 −0.00818249 0.999967i \(-0.502605\pi\)
−0.00818249 + 0.999967i \(0.502605\pi\)
\(432\) 0 0
\(433\) −24.3923 −1.17222 −0.586110 0.810232i \(-0.699341\pi\)
−0.586110 + 0.810232i \(0.699341\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 25.8564 1.23688
\(438\) 0 0
\(439\) 13.8038 0.658822 0.329411 0.944187i \(-0.393150\pi\)
0.329411 + 0.944187i \(0.393150\pi\)
\(440\) 0 0
\(441\) 15.9282 0.758486
\(442\) 0 0
\(443\) 20.7846 0.987507 0.493753 0.869602i \(-0.335625\pi\)
0.493753 + 0.869602i \(0.335625\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −4.39230 −0.207749
\(448\) 0 0
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) −28.3923 −1.33694
\(452\) 0 0
\(453\) −4.78461 −0.224801
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −0.392305 −0.0183512 −0.00917562 0.999958i \(-0.502921\pi\)
−0.00917562 + 0.999958i \(0.502921\pi\)
\(458\) 0 0
\(459\) 4.00000 0.186704
\(460\) 0 0
\(461\) −14.7846 −0.688588 −0.344294 0.938862i \(-0.611882\pi\)
−0.344294 + 0.938862i \(0.611882\pi\)
\(462\) 0 0
\(463\) −29.8564 −1.38754 −0.693772 0.720194i \(-0.744053\pi\)
−0.693772 + 0.720194i \(0.744053\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 21.4641 0.993240 0.496620 0.867968i \(-0.334574\pi\)
0.496620 + 0.867968i \(0.334574\pi\)
\(468\) 0 0
\(469\) 0.784610 0.0362299
\(470\) 0 0
\(471\) 1.46410 0.0674622
\(472\) 0 0
\(473\) 58.6410 2.69632
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −2.28719 −0.104723
\(478\) 0 0
\(479\) 0.339746 0.0155234 0.00776169 0.999970i \(-0.497529\pi\)
0.00776169 + 0.999970i \(0.497529\pi\)
\(480\) 0 0
\(481\) −16.7846 −0.765312
\(482\) 0 0
\(483\) 2.53590 0.115387
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −32.0526 −1.45244 −0.726220 0.687462i \(-0.758725\pi\)
−0.726220 + 0.687462i \(0.758725\pi\)
\(488\) 0 0
\(489\) 7.75129 0.350525
\(490\) 0 0
\(491\) −25.1769 −1.13622 −0.568109 0.822953i \(-0.692325\pi\)
−0.568109 + 0.822953i \(0.692325\pi\)
\(492\) 0 0
\(493\) −3.46410 −0.156015
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1.60770 −0.0721150
\(498\) 0 0
\(499\) 26.9808 1.20782 0.603912 0.797051i \(-0.293608\pi\)
0.603912 + 0.797051i \(0.293608\pi\)
\(500\) 0 0
\(501\) 8.53590 0.381356
\(502\) 0 0
\(503\) 2.87564 0.128219 0.0641093 0.997943i \(-0.479579\pi\)
0.0641093 + 0.997943i \(0.479579\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 7.94744 0.352958
\(508\) 0 0
\(509\) 33.7128 1.49429 0.747147 0.664659i \(-0.231423\pi\)
0.747147 + 0.664659i \(0.231423\pi\)
\(510\) 0 0
\(511\) −1.46410 −0.0647680
\(512\) 0 0
\(513\) −21.8564 −0.964984
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 32.7846 1.44187
\(518\) 0 0
\(519\) −6.24871 −0.274288
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) 28.7846 1.25866 0.629332 0.777137i \(-0.283329\pi\)
0.629332 + 0.777137i \(0.283329\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.19615 0.269909
\(528\) 0 0
\(529\) −0.607695 −0.0264215
\(530\) 0 0
\(531\) 23.3205 1.01202
\(532\) 0 0
\(533\) −8.78461 −0.380504
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 15.7128 0.678058
\(538\) 0 0
\(539\) −30.5885 −1.31754
\(540\) 0 0
\(541\) −2.39230 −0.102853 −0.0514266 0.998677i \(-0.516377\pi\)
−0.0514266 + 0.998677i \(0.516377\pi\)
\(542\) 0 0
\(543\) −8.39230 −0.360148
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 39.2679 1.67898 0.839488 0.543378i \(-0.182855\pi\)
0.839488 + 0.543378i \(0.182855\pi\)
\(548\) 0 0
\(549\) 18.3923 0.784964
\(550\) 0 0
\(551\) 18.9282 0.806369
\(552\) 0 0
\(553\) 1.32051 0.0561537
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 33.4641 1.41792 0.708960 0.705249i \(-0.249165\pi\)
0.708960 + 0.705249i \(0.249165\pi\)
\(558\) 0 0
\(559\) 18.1436 0.767392
\(560\) 0 0
\(561\) −3.46410 −0.146254
\(562\) 0 0
\(563\) −0.679492 −0.0286372 −0.0143186 0.999897i \(-0.504558\pi\)
−0.0143186 + 0.999897i \(0.504558\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 3.26795 0.137241
\(568\) 0 0
\(569\) 19.8564 0.832424 0.416212 0.909268i \(-0.363357\pi\)
0.416212 + 0.909268i \(0.363357\pi\)
\(570\) 0 0
\(571\) 13.1244 0.549237 0.274619 0.961553i \(-0.411448\pi\)
0.274619 + 0.961553i \(0.411448\pi\)
\(572\) 0 0
\(573\) 8.78461 0.366982
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 6.53590 0.272093 0.136047 0.990702i \(-0.456560\pi\)
0.136047 + 0.990702i \(0.456560\pi\)
\(578\) 0 0
\(579\) 5.17691 0.215145
\(580\) 0 0
\(581\) −6.92820 −0.287430
\(582\) 0 0
\(583\) 4.39230 0.181911
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 7.60770 0.314003 0.157002 0.987598i \(-0.449817\pi\)
0.157002 + 0.987598i \(0.449817\pi\)
\(588\) 0 0
\(589\) −33.8564 −1.39503
\(590\) 0 0
\(591\) −2.53590 −0.104313
\(592\) 0 0
\(593\) 31.8564 1.30819 0.654093 0.756414i \(-0.273051\pi\)
0.654093 + 0.756414i \(0.273051\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −4.53590 −0.185642
\(598\) 0 0
\(599\) −27.7128 −1.13231 −0.566157 0.824297i \(-0.691571\pi\)
−0.566157 + 0.824297i \(0.691571\pi\)
\(600\) 0 0
\(601\) −30.7846 −1.25573 −0.627865 0.778322i \(-0.716071\pi\)
−0.627865 + 0.778322i \(0.716071\pi\)
\(602\) 0 0
\(603\) −2.64102 −0.107550
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −35.2679 −1.43148 −0.715741 0.698366i \(-0.753911\pi\)
−0.715741 + 0.698366i \(0.753911\pi\)
\(608\) 0 0
\(609\) 1.85641 0.0752254
\(610\) 0 0
\(611\) 10.1436 0.410366
\(612\) 0 0
\(613\) 20.1436 0.813592 0.406796 0.913519i \(-0.366646\pi\)
0.406796 + 0.913519i \(0.366646\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −47.5692 −1.91506 −0.957532 0.288326i \(-0.906901\pi\)
−0.957532 + 0.288326i \(0.906901\pi\)
\(618\) 0 0
\(619\) −22.1962 −0.892139 −0.446069 0.894998i \(-0.647177\pi\)
−0.446069 + 0.894998i \(0.647177\pi\)
\(620\) 0 0
\(621\) −18.9282 −0.759563
\(622\) 0 0
\(623\) 6.92820 0.277573
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 18.9282 0.755920
\(628\) 0 0
\(629\) −11.4641 −0.457104
\(630\) 0 0
\(631\) −34.5359 −1.37485 −0.687426 0.726254i \(-0.741260\pi\)
−0.687426 + 0.726254i \(0.741260\pi\)
\(632\) 0 0
\(633\) 14.3923 0.572043
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −9.46410 −0.374981
\(638\) 0 0
\(639\) 5.41154 0.214077
\(640\) 0 0
\(641\) 0.928203 0.0366618 0.0183309 0.999832i \(-0.494165\pi\)
0.0183309 + 0.999832i \(0.494165\pi\)
\(642\) 0 0
\(643\) 1.41154 0.0556658 0.0278329 0.999613i \(-0.491139\pi\)
0.0278329 + 0.999613i \(0.491139\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 30.9282 1.21591 0.607957 0.793970i \(-0.291989\pi\)
0.607957 + 0.793970i \(0.291989\pi\)
\(648\) 0 0
\(649\) −44.7846 −1.75795
\(650\) 0 0
\(651\) −3.32051 −0.130141
\(652\) 0 0
\(653\) −10.3923 −0.406682 −0.203341 0.979108i \(-0.565180\pi\)
−0.203341 + 0.979108i \(0.565180\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 4.92820 0.192268
\(658\) 0 0
\(659\) −29.0718 −1.13248 −0.566238 0.824242i \(-0.691602\pi\)
−0.566238 + 0.824242i \(0.691602\pi\)
\(660\) 0 0
\(661\) −28.9282 −1.12518 −0.562588 0.826737i \(-0.690194\pi\)
−0.562588 + 0.826737i \(0.690194\pi\)
\(662\) 0 0
\(663\) −1.07180 −0.0416251
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 16.3923 0.634713
\(668\) 0 0
\(669\) 6.14359 0.237525
\(670\) 0 0
\(671\) −35.3205 −1.36353
\(672\) 0 0
\(673\) −27.8564 −1.07379 −0.536893 0.843650i \(-0.680402\pi\)
−0.536893 + 0.843650i \(0.680402\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −24.2487 −0.931954 −0.465977 0.884797i \(-0.654297\pi\)
−0.465977 + 0.884797i \(0.654297\pi\)
\(678\) 0 0
\(679\) −6.53590 −0.250825
\(680\) 0 0
\(681\) 7.17691 0.275020
\(682\) 0 0
\(683\) −7.94744 −0.304100 −0.152050 0.988373i \(-0.548588\pi\)
−0.152050 + 0.988373i \(0.548588\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −0.287187 −0.0109569
\(688\) 0 0
\(689\) 1.35898 0.0517732
\(690\) 0 0
\(691\) 25.8038 0.981625 0.490812 0.871265i \(-0.336700\pi\)
0.490812 + 0.871265i \(0.336700\pi\)
\(692\) 0 0
\(693\) −8.53590 −0.324252
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −6.00000 −0.227266
\(698\) 0 0
\(699\) 4.39230 0.166132
\(700\) 0 0
\(701\) 4.39230 0.165895 0.0829475 0.996554i \(-0.473567\pi\)
0.0829475 + 0.996554i \(0.473567\pi\)
\(702\) 0 0
\(703\) 62.6410 2.36255
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −6.92820 −0.260562
\(708\) 0 0
\(709\) 32.2487 1.21113 0.605563 0.795797i \(-0.292948\pi\)
0.605563 + 0.795797i \(0.292948\pi\)
\(710\) 0 0
\(711\) −4.44486 −0.166695
\(712\) 0 0
\(713\) −29.3205 −1.09806
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −15.2154 −0.568229
\(718\) 0 0
\(719\) 1.51666 0.0565619 0.0282809 0.999600i \(-0.490997\pi\)
0.0282809 + 0.999600i \(0.490997\pi\)
\(720\) 0 0
\(721\) 2.14359 0.0798316
\(722\) 0 0
\(723\) −1.46410 −0.0544505
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −36.7846 −1.36427 −0.682133 0.731228i \(-0.738947\pi\)
−0.682133 + 0.731228i \(0.738947\pi\)
\(728\) 0 0
\(729\) −2.21539 −0.0820515
\(730\) 0 0
\(731\) 12.3923 0.458346
\(732\) 0 0
\(733\) −22.7846 −0.841569 −0.420784 0.907161i \(-0.638245\pi\)
−0.420784 + 0.907161i \(0.638245\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5.07180 0.186822
\(738\) 0 0
\(739\) 8.39230 0.308716 0.154358 0.988015i \(-0.450669\pi\)
0.154358 + 0.988015i \(0.450669\pi\)
\(740\) 0 0
\(741\) 5.85641 0.215140
\(742\) 0 0
\(743\) −41.9090 −1.53749 −0.768745 0.639555i \(-0.779119\pi\)
−0.768745 + 0.639555i \(0.779119\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 23.3205 0.853253
\(748\) 0 0
\(749\) 5.32051 0.194407
\(750\) 0 0
\(751\) −15.2679 −0.557135 −0.278568 0.960417i \(-0.589860\pi\)
−0.278568 + 0.960417i \(0.589860\pi\)
\(752\) 0 0
\(753\) −5.07180 −0.184827
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −10.5359 −0.382934 −0.191467 0.981499i \(-0.561324\pi\)
−0.191467 + 0.981499i \(0.561324\pi\)
\(758\) 0 0
\(759\) 16.3923 0.595003
\(760\) 0 0
\(761\) −4.39230 −0.159221 −0.0796105 0.996826i \(-0.525368\pi\)
−0.0796105 + 0.996826i \(0.525368\pi\)
\(762\) 0 0
\(763\) 4.67949 0.169409
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −13.8564 −0.500326
\(768\) 0 0
\(769\) −20.3923 −0.735365 −0.367683 0.929951i \(-0.619849\pi\)
−0.367683 + 0.929951i \(0.619849\pi\)
\(770\) 0 0
\(771\) −5.56922 −0.200571
\(772\) 0 0
\(773\) 35.3205 1.27039 0.635195 0.772352i \(-0.280920\pi\)
0.635195 + 0.772352i \(0.280920\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 6.14359 0.220400
\(778\) 0 0
\(779\) 32.7846 1.17463
\(780\) 0 0
\(781\) −10.3923 −0.371866
\(782\) 0 0
\(783\) −13.8564 −0.495188
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −1.12436 −0.0400790 −0.0200395 0.999799i \(-0.506379\pi\)
−0.0200395 + 0.999799i \(0.506379\pi\)
\(788\) 0 0
\(789\) 17.0718 0.607772
\(790\) 0 0
\(791\) 5.75129 0.204492
\(792\) 0 0
\(793\) −10.9282 −0.388072
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −38.7846 −1.37382 −0.686911 0.726742i \(-0.741034\pi\)
−0.686911 + 0.726742i \(0.741034\pi\)
\(798\) 0 0
\(799\) 6.92820 0.245102
\(800\) 0 0
\(801\) −23.3205 −0.823990
\(802\) 0 0
\(803\) −9.46410 −0.333981
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 15.0333 0.529198
\(808\) 0 0
\(809\) 19.8564 0.698114 0.349057 0.937101i \(-0.386502\pi\)
0.349057 + 0.937101i \(0.386502\pi\)
\(810\) 0 0
\(811\) −35.3731 −1.24212 −0.621058 0.783764i \(-0.713297\pi\)
−0.621058 + 0.783764i \(0.713297\pi\)
\(812\) 0 0
\(813\) 10.9282 0.383269
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −67.7128 −2.36897
\(818\) 0 0
\(819\) −2.64102 −0.0922846
\(820\) 0 0
\(821\) −45.0333 −1.57167 −0.785837 0.618434i \(-0.787767\pi\)
−0.785837 + 0.618434i \(0.787767\pi\)
\(822\) 0 0
\(823\) −15.6603 −0.545882 −0.272941 0.962031i \(-0.587996\pi\)
−0.272941 + 0.962031i \(0.587996\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 25.5167 0.887301 0.443651 0.896200i \(-0.353683\pi\)
0.443651 + 0.896200i \(0.353683\pi\)
\(828\) 0 0
\(829\) 12.1436 0.421764 0.210882 0.977511i \(-0.432366\pi\)
0.210882 + 0.977511i \(0.432366\pi\)
\(830\) 0 0
\(831\) −10.5359 −0.365486
\(832\) 0 0
\(833\) −6.46410 −0.223968
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 24.7846 0.856681
\(838\) 0 0
\(839\) −49.5167 −1.70950 −0.854752 0.519036i \(-0.826291\pi\)
−0.854752 + 0.519036i \(0.826291\pi\)
\(840\) 0 0
\(841\) −17.0000 −0.586207
\(842\) 0 0
\(843\) −5.75129 −0.198085
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 8.33975 0.286557
\(848\) 0 0
\(849\) 7.75129 0.266024
\(850\) 0 0
\(851\) 54.2487 1.85962
\(852\) 0 0
\(853\) 40.2487 1.37809 0.689045 0.724719i \(-0.258030\pi\)
0.689045 + 0.724719i \(0.258030\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 52.6410 1.79818 0.899091 0.437761i \(-0.144228\pi\)
0.899091 + 0.437761i \(0.144228\pi\)
\(858\) 0 0
\(859\) −7.32051 −0.249773 −0.124886 0.992171i \(-0.539857\pi\)
−0.124886 + 0.992171i \(0.539857\pi\)
\(860\) 0 0
\(861\) 3.21539 0.109580
\(862\) 0 0
\(863\) 37.8564 1.28865 0.644324 0.764753i \(-0.277139\pi\)
0.644324 + 0.764753i \(0.277139\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −0.732051 −0.0248617
\(868\) 0 0
\(869\) 8.53590 0.289561
\(870\) 0 0
\(871\) 1.56922 0.0531710
\(872\) 0 0
\(873\) 22.0000 0.744587
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −30.3923 −1.02628 −0.513138 0.858306i \(-0.671517\pi\)
−0.513138 + 0.858306i \(0.671517\pi\)
\(878\) 0 0
\(879\) 21.9615 0.740744
\(880\) 0 0
\(881\) 2.78461 0.0938159 0.0469079 0.998899i \(-0.485063\pi\)
0.0469079 + 0.998899i \(0.485063\pi\)
\(882\) 0 0
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 42.5885 1.42998 0.714990 0.699134i \(-0.246431\pi\)
0.714990 + 0.699134i \(0.246431\pi\)
\(888\) 0 0
\(889\) −14.9282 −0.500676
\(890\) 0 0
\(891\) 21.1244 0.707693
\(892\) 0 0
\(893\) −37.8564 −1.26682
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 5.07180 0.169342
\(898\) 0 0
\(899\) −21.4641 −0.715868
\(900\) 0 0
\(901\) 0.928203 0.0309229
\(902\) 0 0
\(903\) −6.64102 −0.220999
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −24.4449 −0.811678 −0.405839 0.913945i \(-0.633021\pi\)
−0.405839 + 0.913945i \(0.633021\pi\)
\(908\) 0 0
\(909\) 23.3205 0.773492
\(910\) 0 0
\(911\) 26.1962 0.867917 0.433959 0.900933i \(-0.357116\pi\)
0.433959 + 0.900933i \(0.357116\pi\)
\(912\) 0 0
\(913\) −44.7846 −1.48215
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 8.53590 0.281880
\(918\) 0 0
\(919\) 52.0000 1.71532 0.857661 0.514216i \(-0.171917\pi\)
0.857661 + 0.514216i \(0.171917\pi\)
\(920\) 0 0
\(921\) −12.2872 −0.404877
\(922\) 0 0
\(923\) −3.21539 −0.105836
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −7.21539 −0.236985
\(928\) 0 0
\(929\) 45.7128 1.49979 0.749894 0.661558i \(-0.230104\pi\)
0.749894 + 0.661558i \(0.230104\pi\)
\(930\) 0 0
\(931\) 35.3205 1.15758
\(932\) 0 0
\(933\) 15.4641 0.506272
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −27.8564 −0.910029 −0.455015 0.890484i \(-0.650366\pi\)
−0.455015 + 0.890484i \(0.650366\pi\)
\(938\) 0 0
\(939\) 5.17691 0.168942
\(940\) 0 0
\(941\) 6.67949 0.217745 0.108873 0.994056i \(-0.465276\pi\)
0.108873 + 0.994056i \(0.465276\pi\)
\(942\) 0 0
\(943\) 28.3923 0.924581
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 52.7321 1.71356 0.856781 0.515681i \(-0.172461\pi\)
0.856781 + 0.515681i \(0.172461\pi\)
\(948\) 0 0
\(949\) −2.92820 −0.0950535
\(950\) 0 0
\(951\) 11.3205 0.367093
\(952\) 0 0
\(953\) 42.2487 1.36857 0.684285 0.729215i \(-0.260114\pi\)
0.684285 + 0.729215i \(0.260114\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 12.0000 0.387905
\(958\) 0 0
\(959\) −6.92820 −0.223723
\(960\) 0 0
\(961\) 7.39230 0.238461
\(962\) 0 0
\(963\) −17.9090 −0.577108
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 48.3923 1.55619 0.778096 0.628146i \(-0.216186\pi\)
0.778096 + 0.628146i \(0.216186\pi\)
\(968\) 0 0
\(969\) 4.00000 0.128499
\(970\) 0 0
\(971\) 56.1051 1.80050 0.900249 0.435374i \(-0.143384\pi\)
0.900249 + 0.435374i \(0.143384\pi\)
\(972\) 0 0
\(973\) 8.24871 0.264442
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 0 0
\(979\) 44.7846 1.43132
\(980\) 0 0
\(981\) −15.7513 −0.502900
\(982\) 0 0
\(983\) 29.9090 0.953948 0.476974 0.878917i \(-0.341734\pi\)
0.476974 + 0.878917i \(0.341734\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −3.71281 −0.118180
\(988\) 0 0
\(989\) −58.6410 −1.86468
\(990\) 0 0
\(991\) −10.1962 −0.323891 −0.161946 0.986800i \(-0.551777\pi\)
−0.161946 + 0.986800i \(0.551777\pi\)
\(992\) 0 0
\(993\) −11.2154 −0.355910
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −42.3923 −1.34258 −0.671289 0.741196i \(-0.734259\pi\)
−0.671289 + 0.741196i \(0.734259\pi\)
\(998\) 0 0
\(999\) −45.8564 −1.45083
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 6800.2.a.bh.1.1 2
4.3 odd 2 1700.2.a.d.1.2 2
5.4 even 2 272.2.a.e.1.2 2
15.14 odd 2 2448.2.a.y.1.1 2
20.3 even 4 1700.2.e.c.749.3 4
20.7 even 4 1700.2.e.c.749.2 4
20.19 odd 2 68.2.a.a.1.1 2
40.19 odd 2 1088.2.a.p.1.2 2
40.29 even 2 1088.2.a.t.1.1 2
60.59 even 2 612.2.a.e.1.1 2
85.84 even 2 4624.2.a.x.1.1 2
120.29 odd 2 9792.2.a.cs.1.2 2
120.59 even 2 9792.2.a.cr.1.2 2
140.139 even 2 3332.2.a.h.1.2 2
220.219 even 2 8228.2.a.k.1.1 2
340.19 odd 8 1156.2.e.d.905.2 8
340.39 even 16 1156.2.h.f.977.3 16
340.59 odd 8 1156.2.e.d.829.3 8
340.79 even 16 1156.2.h.f.733.3 16
340.99 even 16 1156.2.h.f.757.3 16
340.139 even 16 1156.2.h.f.757.2 16
340.159 even 16 1156.2.h.f.733.2 16
340.179 odd 8 1156.2.e.d.829.2 8
340.199 even 16 1156.2.h.f.977.2 16
340.219 odd 8 1156.2.e.d.905.3 8
340.259 odd 4 1156.2.b.c.577.3 4
340.279 even 16 1156.2.h.f.1001.3 16
340.299 even 16 1156.2.h.f.1001.2 16
340.319 odd 4 1156.2.b.c.577.2 4
340.339 odd 2 1156.2.a.a.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
68.2.a.a.1.1 2 20.19 odd 2
272.2.a.e.1.2 2 5.4 even 2
612.2.a.e.1.1 2 60.59 even 2
1088.2.a.p.1.2 2 40.19 odd 2
1088.2.a.t.1.1 2 40.29 even 2
1156.2.a.a.1.2 2 340.339 odd 2
1156.2.b.c.577.2 4 340.319 odd 4
1156.2.b.c.577.3 4 340.259 odd 4
1156.2.e.d.829.2 8 340.179 odd 8
1156.2.e.d.829.3 8 340.59 odd 8
1156.2.e.d.905.2 8 340.19 odd 8
1156.2.e.d.905.3 8 340.219 odd 8
1156.2.h.f.733.2 16 340.159 even 16
1156.2.h.f.733.3 16 340.79 even 16
1156.2.h.f.757.2 16 340.139 even 16
1156.2.h.f.757.3 16 340.99 even 16
1156.2.h.f.977.2 16 340.199 even 16
1156.2.h.f.977.3 16 340.39 even 16
1156.2.h.f.1001.2 16 340.299 even 16
1156.2.h.f.1001.3 16 340.279 even 16
1700.2.a.d.1.2 2 4.3 odd 2
1700.2.e.c.749.2 4 20.7 even 4
1700.2.e.c.749.3 4 20.3 even 4
2448.2.a.y.1.1 2 15.14 odd 2
3332.2.a.h.1.2 2 140.139 even 2
4624.2.a.x.1.1 2 85.84 even 2
6800.2.a.bh.1.1 2 1.1 even 1 trivial
8228.2.a.k.1.1 2 220.219 even 2
9792.2.a.cr.1.2 2 120.59 even 2
9792.2.a.cs.1.2 2 120.29 odd 2