Properties

Label 5200.2.a.ci.1.3
Level $5200$
Weight $2$
Character 5200.1
Self dual yes
Analytic conductor $41.522$
Analytic rank $0$
Dimension $3$
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] = [5200,2,Mod(1,5200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5200, 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("5200.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5200 = 2^{4} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5200.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: \(41.5222090511\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.564.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 260)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.51414\) of defining polynomial
Character \(\chi\) \(=\) 5200.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.32088 q^{3} -5.02827 q^{7} +8.02827 q^{9} +O(q^{10})\) \(q+3.32088 q^{3} -5.02827 q^{7} +8.02827 q^{9} -1.70739 q^{11} -1.00000 q^{13} +4.64177 q^{17} +4.34916 q^{19} -16.6983 q^{21} -0.679116 q^{23} +16.6983 q^{27} -1.02827 q^{29} +2.29261 q^{31} -5.67004 q^{33} +1.61350 q^{37} -3.32088 q^{39} +4.64177 q^{41} -3.32088 q^{43} +1.02827 q^{47} +18.2835 q^{49} +15.4148 q^{51} +9.41478 q^{53} +14.4431 q^{57} +8.93438 q^{59} +9.02827 q^{61} -40.3684 q^{63} +5.61350 q^{67} -2.25526 q^{69} -1.70739 q^{71} -12.4431 q^{73} +8.58522 q^{77} +2.64177 q^{79} +31.3684 q^{81} -8.25526 q^{83} -3.41478 q^{87} +1.22699 q^{89} +5.02827 q^{91} +7.61350 q^{93} +0.0565477 q^{97} -13.7074 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{3} - 2 q^{7} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{3} - 2 q^{7} + 11 q^{9} - 3 q^{13} - 2 q^{17} - 8 q^{19} - 8 q^{21} - 10 q^{23} + 8 q^{27} + 10 q^{29} + 12 q^{31} + 12 q^{33} + 2 q^{37} - 2 q^{39} - 2 q^{41} - 2 q^{43} - 10 q^{47} + 23 q^{49} + 36 q^{51} + 18 q^{53} + 20 q^{57} + 16 q^{59} + 14 q^{61} - 50 q^{63} + 14 q^{67} + 12 q^{69} - 14 q^{73} + 36 q^{77} - 8 q^{79} + 23 q^{81} - 6 q^{83} - 2 q^{89} + 2 q^{91} + 20 q^{93} - 26 q^{97} - 36 q^{99}+O(q^{100}) \) 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
\(3\) 3.32088 1.91731 0.958657 0.284565i \(-0.0918491\pi\)
0.958657 + 0.284565i \(0.0918491\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −5.02827 −1.90051 −0.950254 0.311475i \(-0.899177\pi\)
−0.950254 + 0.311475i \(0.899177\pi\)
\(8\) 0 0
\(9\) 8.02827 2.67609
\(10\) 0 0
\(11\) −1.70739 −0.514797 −0.257399 0.966305i \(-0.582865\pi\)
−0.257399 + 0.966305i \(0.582865\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.64177 1.12579 0.562897 0.826527i \(-0.309687\pi\)
0.562897 + 0.826527i \(0.309687\pi\)
\(18\) 0 0
\(19\) 4.34916 0.997765 0.498883 0.866670i \(-0.333744\pi\)
0.498883 + 0.866670i \(0.333744\pi\)
\(20\) 0 0
\(21\) −16.6983 −3.64387
\(22\) 0 0
\(23\) −0.679116 −0.141605 −0.0708027 0.997490i \(-0.522556\pi\)
−0.0708027 + 0.997490i \(0.522556\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 16.6983 3.21359
\(28\) 0 0
\(29\) −1.02827 −0.190946 −0.0954728 0.995432i \(-0.530436\pi\)
−0.0954728 + 0.995432i \(0.530436\pi\)
\(30\) 0 0
\(31\) 2.29261 0.411765 0.205883 0.978577i \(-0.433994\pi\)
0.205883 + 0.978577i \(0.433994\pi\)
\(32\) 0 0
\(33\) −5.67004 −0.987028
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.61350 0.265257 0.132628 0.991166i \(-0.457658\pi\)
0.132628 + 0.991166i \(0.457658\pi\)
\(38\) 0 0
\(39\) −3.32088 −0.531767
\(40\) 0 0
\(41\) 4.64177 0.724923 0.362461 0.931999i \(-0.381937\pi\)
0.362461 + 0.931999i \(0.381937\pi\)
\(42\) 0 0
\(43\) −3.32088 −0.506430 −0.253215 0.967410i \(-0.581488\pi\)
−0.253215 + 0.967410i \(0.581488\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.02827 0.149989 0.0749946 0.997184i \(-0.476106\pi\)
0.0749946 + 0.997184i \(0.476106\pi\)
\(48\) 0 0
\(49\) 18.2835 2.61193
\(50\) 0 0
\(51\) 15.4148 2.15850
\(52\) 0 0
\(53\) 9.41478 1.29322 0.646610 0.762821i \(-0.276186\pi\)
0.646610 + 0.762821i \(0.276186\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 14.4431 1.91303
\(58\) 0 0
\(59\) 8.93438 1.16316 0.581579 0.813490i \(-0.302435\pi\)
0.581579 + 0.813490i \(0.302435\pi\)
\(60\) 0 0
\(61\) 9.02827 1.15595 0.577976 0.816054i \(-0.303843\pi\)
0.577976 + 0.816054i \(0.303843\pi\)
\(62\) 0 0
\(63\) −40.3684 −5.08594
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 5.61350 0.685798 0.342899 0.939372i \(-0.388591\pi\)
0.342899 + 0.939372i \(0.388591\pi\)
\(68\) 0 0
\(69\) −2.25526 −0.271502
\(70\) 0 0
\(71\) −1.70739 −0.202630 −0.101315 0.994854i \(-0.532305\pi\)
−0.101315 + 0.994854i \(0.532305\pi\)
\(72\) 0 0
\(73\) −12.4431 −1.45635 −0.728175 0.685392i \(-0.759631\pi\)
−0.728175 + 0.685392i \(0.759631\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8.58522 0.978377
\(78\) 0 0
\(79\) 2.64177 0.297222 0.148611 0.988896i \(-0.452520\pi\)
0.148611 + 0.988896i \(0.452520\pi\)
\(80\) 0 0
\(81\) 31.3684 3.48537
\(82\) 0 0
\(83\) −8.25526 −0.906133 −0.453066 0.891477i \(-0.649670\pi\)
−0.453066 + 0.891477i \(0.649670\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −3.41478 −0.366103
\(88\) 0 0
\(89\) 1.22699 0.130061 0.0650304 0.997883i \(-0.479286\pi\)
0.0650304 + 0.997883i \(0.479286\pi\)
\(90\) 0 0
\(91\) 5.02827 0.527106
\(92\) 0 0
\(93\) 7.61350 0.789483
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.0565477 0.00574155 0.00287078 0.999996i \(-0.499086\pi\)
0.00287078 + 0.999996i \(0.499086\pi\)
\(98\) 0 0
\(99\) −13.7074 −1.37764
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) 9.37743 0.923986 0.461993 0.886884i \(-0.347135\pi\)
0.461993 + 0.886884i \(0.347135\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −13.3774 −1.29325 −0.646623 0.762810i \(-0.723819\pi\)
−0.646623 + 0.762810i \(0.723819\pi\)
\(108\) 0 0
\(109\) 11.2835 1.08077 0.540383 0.841419i \(-0.318279\pi\)
0.540383 + 0.841419i \(0.318279\pi\)
\(110\) 0 0
\(111\) 5.35823 0.508581
\(112\) 0 0
\(113\) 18.6983 1.75899 0.879495 0.475908i \(-0.157881\pi\)
0.879495 + 0.475908i \(0.157881\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −8.02827 −0.742214
\(118\) 0 0
\(119\) −23.3401 −2.13958
\(120\) 0 0
\(121\) −8.08482 −0.734984
\(122\) 0 0
\(123\) 15.4148 1.38990
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 10.7357 0.952636 0.476318 0.879273i \(-0.341971\pi\)
0.476318 + 0.879273i \(0.341971\pi\)
\(128\) 0 0
\(129\) −11.0283 −0.970985
\(130\) 0 0
\(131\) −7.22699 −0.631425 −0.315713 0.948855i \(-0.602244\pi\)
−0.315713 + 0.948855i \(0.602244\pi\)
\(132\) 0 0
\(133\) −21.8688 −1.89626
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −17.3401 −1.48146 −0.740732 0.671801i \(-0.765521\pi\)
−0.740732 + 0.671801i \(0.765521\pi\)
\(138\) 0 0
\(139\) −9.35823 −0.793755 −0.396877 0.917872i \(-0.629906\pi\)
−0.396877 + 0.917872i \(0.629906\pi\)
\(140\) 0 0
\(141\) 3.41478 0.287576
\(142\) 0 0
\(143\) 1.70739 0.142779
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 60.7175 5.00790
\(148\) 0 0
\(149\) 17.3401 1.42056 0.710278 0.703922i \(-0.248569\pi\)
0.710278 + 0.703922i \(0.248569\pi\)
\(150\) 0 0
\(151\) −3.57615 −0.291023 −0.145511 0.989357i \(-0.546483\pi\)
−0.145511 + 0.989357i \(0.546483\pi\)
\(152\) 0 0
\(153\) 37.2654 3.01273
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 7.28354 0.581290 0.290645 0.956831i \(-0.406130\pi\)
0.290645 + 0.956831i \(0.406130\pi\)
\(158\) 0 0
\(159\) 31.2654 2.47951
\(160\) 0 0
\(161\) 3.41478 0.269122
\(162\) 0 0
\(163\) 0.840485 0.0658319 0.0329159 0.999458i \(-0.489521\pi\)
0.0329159 + 0.999458i \(0.489521\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −13.0283 −1.00816 −0.504079 0.863658i \(-0.668168\pi\)
−0.504079 + 0.863658i \(0.668168\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 34.9162 2.67011
\(172\) 0 0
\(173\) −25.9253 −1.97106 −0.985532 0.169488i \(-0.945789\pi\)
−0.985532 + 0.169488i \(0.945789\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 29.6700 2.23014
\(178\) 0 0
\(179\) 4.77301 0.356751 0.178376 0.983962i \(-0.442916\pi\)
0.178376 + 0.983962i \(0.442916\pi\)
\(180\) 0 0
\(181\) −14.3118 −1.06379 −0.531894 0.846811i \(-0.678520\pi\)
−0.531894 + 0.846811i \(0.678520\pi\)
\(182\) 0 0
\(183\) 29.9819 2.21632
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −7.92531 −0.579556
\(188\) 0 0
\(189\) −83.9637 −6.10746
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) −4.05655 −0.291997 −0.145998 0.989285i \(-0.546639\pi\)
−0.145998 + 0.989285i \(0.546639\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 15.2835 1.08891 0.544453 0.838791i \(-0.316737\pi\)
0.544453 + 0.838791i \(0.316737\pi\)
\(198\) 0 0
\(199\) 8.77301 0.621902 0.310951 0.950426i \(-0.399352\pi\)
0.310951 + 0.950426i \(0.399352\pi\)
\(200\) 0 0
\(201\) 18.6418 1.31489
\(202\) 0 0
\(203\) 5.17044 0.362894
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −5.45213 −0.378949
\(208\) 0 0
\(209\) −7.42571 −0.513647
\(210\) 0 0
\(211\) −0.773010 −0.0532162 −0.0266081 0.999646i \(-0.508471\pi\)
−0.0266081 + 0.999646i \(0.508471\pi\)
\(212\) 0 0
\(213\) −5.67004 −0.388505
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −11.5279 −0.782563
\(218\) 0 0
\(219\) −41.3219 −2.79228
\(220\) 0 0
\(221\) −4.64177 −0.312239
\(222\) 0 0
\(223\) −14.3118 −0.958390 −0.479195 0.877709i \(-0.659071\pi\)
−0.479195 + 0.877709i \(0.659071\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −13.7266 −0.911066 −0.455533 0.890219i \(-0.650551\pi\)
−0.455533 + 0.890219i \(0.650551\pi\)
\(228\) 0 0
\(229\) 21.9253 1.44887 0.724433 0.689346i \(-0.242102\pi\)
0.724433 + 0.689346i \(0.242102\pi\)
\(230\) 0 0
\(231\) 28.5105 1.87586
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 8.77301 0.569868
\(238\) 0 0
\(239\) 15.7639 1.01968 0.509842 0.860268i \(-0.329704\pi\)
0.509842 + 0.860268i \(0.329704\pi\)
\(240\) 0 0
\(241\) 9.92531 0.639345 0.319673 0.947528i \(-0.396427\pi\)
0.319673 + 0.947528i \(0.396427\pi\)
\(242\) 0 0
\(243\) 54.0757 3.46896
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −4.34916 −0.276730
\(248\) 0 0
\(249\) −27.4148 −1.73734
\(250\) 0 0
\(251\) −24.6983 −1.55894 −0.779472 0.626437i \(-0.784513\pi\)
−0.779472 + 0.626437i \(0.784513\pi\)
\(252\) 0 0
\(253\) 1.15951 0.0728981
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 20.0565 1.25109 0.625547 0.780187i \(-0.284876\pi\)
0.625547 + 0.780187i \(0.284876\pi\)
\(258\) 0 0
\(259\) −8.11310 −0.504123
\(260\) 0 0
\(261\) −8.25526 −0.510988
\(262\) 0 0
\(263\) 26.0757 1.60790 0.803950 0.594697i \(-0.202728\pi\)
0.803950 + 0.594697i \(0.202728\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 4.07469 0.249367
\(268\) 0 0
\(269\) −12.8296 −0.782232 −0.391116 0.920341i \(-0.627911\pi\)
−0.391116 + 0.920341i \(0.627911\pi\)
\(270\) 0 0
\(271\) 17.7074 1.07565 0.537824 0.843057i \(-0.319247\pi\)
0.537824 + 0.843057i \(0.319247\pi\)
\(272\) 0 0
\(273\) 16.6983 1.01063
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 15.4713 0.929582 0.464791 0.885420i \(-0.346129\pi\)
0.464791 + 0.885420i \(0.346129\pi\)
\(278\) 0 0
\(279\) 18.4057 1.10192
\(280\) 0 0
\(281\) 7.35823 0.438955 0.219478 0.975618i \(-0.429565\pi\)
0.219478 + 0.975618i \(0.429565\pi\)
\(282\) 0 0
\(283\) −0.604422 −0.0359292 −0.0179646 0.999839i \(-0.505719\pi\)
−0.0179646 + 0.999839i \(0.505719\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −23.3401 −1.37772
\(288\) 0 0
\(289\) 4.54602 0.267413
\(290\) 0 0
\(291\) 0.187788 0.0110084
\(292\) 0 0
\(293\) −23.0101 −1.34427 −0.672133 0.740430i \(-0.734622\pi\)
−0.672133 + 0.740430i \(0.734622\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −28.5105 −1.65435
\(298\) 0 0
\(299\) 0.679116 0.0392743
\(300\) 0 0
\(301\) 16.6983 0.962475
\(302\) 0 0
\(303\) 19.9253 1.14468
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 20.3684 1.16248 0.581242 0.813731i \(-0.302567\pi\)
0.581242 + 0.813731i \(0.302567\pi\)
\(308\) 0 0
\(309\) 31.1414 1.77157
\(310\) 0 0
\(311\) 19.9253 1.12986 0.564930 0.825139i \(-0.308903\pi\)
0.564930 + 0.825139i \(0.308903\pi\)
\(312\) 0 0
\(313\) −31.4713 −1.77886 −0.889432 0.457067i \(-0.848900\pi\)
−0.889432 + 0.457067i \(0.848900\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.72659 −0.0969750 −0.0484875 0.998824i \(-0.515440\pi\)
−0.0484875 + 0.998824i \(0.515440\pi\)
\(318\) 0 0
\(319\) 1.75566 0.0982983
\(320\) 0 0
\(321\) −44.4249 −2.47956
\(322\) 0 0
\(323\) 20.1878 1.12328
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 37.4713 2.07217
\(328\) 0 0
\(329\) −5.17044 −0.285056
\(330\) 0 0
\(331\) −10.4057 −0.571949 −0.285975 0.958237i \(-0.592317\pi\)
−0.285975 + 0.958237i \(0.592317\pi\)
\(332\) 0 0
\(333\) 12.9536 0.709852
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 10.6983 0.582774 0.291387 0.956605i \(-0.405883\pi\)
0.291387 + 0.956605i \(0.405883\pi\)
\(338\) 0 0
\(339\) 62.0950 3.37253
\(340\) 0 0
\(341\) −3.91438 −0.211976
\(342\) 0 0
\(343\) −56.7367 −3.06349
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −32.6044 −1.75030 −0.875149 0.483854i \(-0.839236\pi\)
−0.875149 + 0.483854i \(0.839236\pi\)
\(348\) 0 0
\(349\) −13.4148 −0.718077 −0.359038 0.933323i \(-0.616895\pi\)
−0.359038 + 0.933323i \(0.616895\pi\)
\(350\) 0 0
\(351\) −16.6983 −0.891290
\(352\) 0 0
\(353\) 35.0101 1.86340 0.931701 0.363227i \(-0.118325\pi\)
0.931701 + 0.363227i \(0.118325\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −77.5097 −4.10225
\(358\) 0 0
\(359\) −20.9344 −1.10487 −0.552437 0.833555i \(-0.686302\pi\)
−0.552437 + 0.833555i \(0.686302\pi\)
\(360\) 0 0
\(361\) −0.0848216 −0.00446429
\(362\) 0 0
\(363\) −26.8488 −1.40919
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 11.4340 0.596849 0.298424 0.954433i \(-0.403539\pi\)
0.298424 + 0.954433i \(0.403539\pi\)
\(368\) 0 0
\(369\) 37.2654 1.93996
\(370\) 0 0
\(371\) −47.3401 −2.45777
\(372\) 0 0
\(373\) 14.1131 0.730748 0.365374 0.930861i \(-0.380941\pi\)
0.365374 + 0.930861i \(0.380941\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.02827 0.0529588
\(378\) 0 0
\(379\) 1.89518 0.0973487 0.0486744 0.998815i \(-0.484500\pi\)
0.0486744 + 0.998815i \(0.484500\pi\)
\(380\) 0 0
\(381\) 35.6519 1.82650
\(382\) 0 0
\(383\) −22.3118 −1.14008 −0.570040 0.821617i \(-0.693072\pi\)
−0.570040 + 0.821617i \(0.693072\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −26.6610 −1.35525
\(388\) 0 0
\(389\) −28.6802 −1.45414 −0.727071 0.686562i \(-0.759119\pi\)
−0.727071 + 0.686562i \(0.759119\pi\)
\(390\) 0 0
\(391\) −3.15230 −0.159419
\(392\) 0 0
\(393\) −24.0000 −1.21064
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −15.5569 −0.780781 −0.390390 0.920649i \(-0.627660\pi\)
−0.390390 + 0.920649i \(0.627660\pi\)
\(398\) 0 0
\(399\) −72.6236 −3.63573
\(400\) 0 0
\(401\) −24.1696 −1.20697 −0.603487 0.797373i \(-0.706223\pi\)
−0.603487 + 0.797373i \(0.706223\pi\)
\(402\) 0 0
\(403\) −2.29261 −0.114203
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −2.75486 −0.136554
\(408\) 0 0
\(409\) −29.9253 −1.47971 −0.739856 0.672766i \(-0.765106\pi\)
−0.739856 + 0.672766i \(0.765106\pi\)
\(410\) 0 0
\(411\) −57.5844 −2.84043
\(412\) 0 0
\(413\) −44.9245 −2.21059
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −31.0776 −1.52188
\(418\) 0 0
\(419\) 8.58522 0.419416 0.209708 0.977764i \(-0.432749\pi\)
0.209708 + 0.977764i \(0.432749\pi\)
\(420\) 0 0
\(421\) −21.7375 −1.05942 −0.529711 0.848178i \(-0.677700\pi\)
−0.529711 + 0.848178i \(0.677700\pi\)
\(422\) 0 0
\(423\) 8.25526 0.401385
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −45.3966 −2.19690
\(428\) 0 0
\(429\) 5.67004 0.273752
\(430\) 0 0
\(431\) −13.7074 −0.660262 −0.330131 0.943935i \(-0.607093\pi\)
−0.330131 + 0.943935i \(0.607093\pi\)
\(432\) 0 0
\(433\) 40.5671 1.94953 0.974765 0.223235i \(-0.0716618\pi\)
0.974765 + 0.223235i \(0.0716618\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −2.95358 −0.141289
\(438\) 0 0
\(439\) −26.5671 −1.26798 −0.633989 0.773342i \(-0.718583\pi\)
−0.633989 + 0.773342i \(0.718583\pi\)
\(440\) 0 0
\(441\) 146.785 6.98977
\(442\) 0 0
\(443\) −16.7922 −0.797822 −0.398911 0.916990i \(-0.630612\pi\)
−0.398911 + 0.916990i \(0.630612\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 57.5844 2.72365
\(448\) 0 0
\(449\) −16.2443 −0.766618 −0.383309 0.923620i \(-0.625215\pi\)
−0.383309 + 0.923620i \(0.625215\pi\)
\(450\) 0 0
\(451\) −7.92531 −0.373188
\(452\) 0 0
\(453\) −11.8760 −0.557982
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −6.77301 −0.316828 −0.158414 0.987373i \(-0.550638\pi\)
−0.158414 + 0.987373i \(0.550638\pi\)
\(458\) 0 0
\(459\) 77.5097 3.61784
\(460\) 0 0
\(461\) −22.8114 −1.06243 −0.531217 0.847236i \(-0.678265\pi\)
−0.531217 + 0.847236i \(0.678265\pi\)
\(462\) 0 0
\(463\) −16.3300 −0.758917 −0.379459 0.925209i \(-0.623890\pi\)
−0.379459 + 0.925209i \(0.623890\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 10.6610 0.493331 0.246665 0.969101i \(-0.420665\pi\)
0.246665 + 0.969101i \(0.420665\pi\)
\(468\) 0 0
\(469\) −28.2262 −1.30336
\(470\) 0 0
\(471\) 24.1878 1.11451
\(472\) 0 0
\(473\) 5.67004 0.260709
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 75.5844 3.46077
\(478\) 0 0
\(479\) −6.48040 −0.296097 −0.148048 0.988980i \(-0.547299\pi\)
−0.148048 + 0.988980i \(0.547299\pi\)
\(480\) 0 0
\(481\) −1.61350 −0.0735690
\(482\) 0 0
\(483\) 11.3401 0.515992
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 23.7831 1.07772 0.538858 0.842396i \(-0.318856\pi\)
0.538858 + 0.842396i \(0.318856\pi\)
\(488\) 0 0
\(489\) 2.79116 0.126220
\(490\) 0 0
\(491\) 6.13124 0.276699 0.138350 0.990383i \(-0.455820\pi\)
0.138350 + 0.990383i \(0.455820\pi\)
\(492\) 0 0
\(493\) −4.77301 −0.214966
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 8.58522 0.385100
\(498\) 0 0
\(499\) −21.0475 −0.942214 −0.471107 0.882076i \(-0.656146\pi\)
−0.471107 + 0.882076i \(0.656146\pi\)
\(500\) 0 0
\(501\) −43.2654 −1.93296
\(502\) 0 0
\(503\) 33.5652 1.49660 0.748300 0.663361i \(-0.230871\pi\)
0.748300 + 0.663361i \(0.230871\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 3.32088 0.147486
\(508\) 0 0
\(509\) −14.5852 −0.646479 −0.323239 0.946317i \(-0.604772\pi\)
−0.323239 + 0.946317i \(0.604772\pi\)
\(510\) 0 0
\(511\) 62.5671 2.76780
\(512\) 0 0
\(513\) 72.6236 3.20641
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −1.75566 −0.0772140
\(518\) 0 0
\(519\) −86.0950 −3.77915
\(520\) 0 0
\(521\) −3.48225 −0.152560 −0.0762802 0.997086i \(-0.524304\pi\)
−0.0762802 + 0.997086i \(0.524304\pi\)
\(522\) 0 0
\(523\) −11.5087 −0.503239 −0.251620 0.967826i \(-0.580963\pi\)
−0.251620 + 0.967826i \(0.580963\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 10.6418 0.463563
\(528\) 0 0
\(529\) −22.5388 −0.979948
\(530\) 0 0
\(531\) 71.7276 3.11271
\(532\) 0 0
\(533\) −4.64177 −0.201057
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 15.8506 0.684004
\(538\) 0 0
\(539\) −31.2171 −1.34462
\(540\) 0 0
\(541\) −13.8122 −0.593833 −0.296917 0.954903i \(-0.595958\pi\)
−0.296917 + 0.954903i \(0.595958\pi\)
\(542\) 0 0
\(543\) −47.5279 −2.03962
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −5.37743 −0.229922 −0.114961 0.993370i \(-0.536674\pi\)
−0.114961 + 0.993370i \(0.536674\pi\)
\(548\) 0 0
\(549\) 72.4815 3.09343
\(550\) 0 0
\(551\) −4.47213 −0.190519
\(552\) 0 0
\(553\) −13.2835 −0.564873
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0.0674757 0.00285904 0.00142952 0.999999i \(-0.499545\pi\)
0.00142952 + 0.999999i \(0.499545\pi\)
\(558\) 0 0
\(559\) 3.32088 0.140458
\(560\) 0 0
\(561\) −26.3190 −1.11119
\(562\) 0 0
\(563\) −7.20779 −0.303772 −0.151886 0.988398i \(-0.548535\pi\)
−0.151886 + 0.988398i \(0.548535\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −157.729 −6.62398
\(568\) 0 0
\(569\) −7.85783 −0.329417 −0.164709 0.986342i \(-0.552668\pi\)
−0.164709 + 0.986342i \(0.552668\pi\)
\(570\) 0 0
\(571\) 23.9253 1.00124 0.500621 0.865666i \(-0.333105\pi\)
0.500621 + 0.865666i \(0.333105\pi\)
\(572\) 0 0
\(573\) −39.8506 −1.66478
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −31.0101 −1.29097 −0.645484 0.763774i \(-0.723344\pi\)
−0.645484 + 0.763774i \(0.723344\pi\)
\(578\) 0 0
\(579\) −13.4713 −0.559849
\(580\) 0 0
\(581\) 41.5097 1.72211
\(582\) 0 0
\(583\) −16.0747 −0.665746
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 47.4076 1.95672 0.978360 0.206911i \(-0.0663411\pi\)
0.978360 + 0.206911i \(0.0663411\pi\)
\(588\) 0 0
\(589\) 9.97093 0.410845
\(590\) 0 0
\(591\) 50.7549 2.08778
\(592\) 0 0
\(593\) 14.8861 0.611299 0.305650 0.952144i \(-0.401126\pi\)
0.305650 + 0.952144i \(0.401126\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 29.1342 1.19238
\(598\) 0 0
\(599\) −42.8680 −1.75154 −0.875769 0.482731i \(-0.839645\pi\)
−0.875769 + 0.482731i \(0.839645\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 0 0
\(603\) 45.0667 1.83526
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −18.7357 −0.760457 −0.380229 0.924893i \(-0.624155\pi\)
−0.380229 + 0.924893i \(0.624155\pi\)
\(608\) 0 0
\(609\) 17.1704 0.695781
\(610\) 0 0
\(611\) −1.02827 −0.0415995
\(612\) 0 0
\(613\) −36.6802 −1.48150 −0.740749 0.671782i \(-0.765529\pi\)
−0.740749 + 0.671782i \(0.765529\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 8.05655 0.324344 0.162172 0.986762i \(-0.448150\pi\)
0.162172 + 0.986762i \(0.448150\pi\)
\(618\) 0 0
\(619\) 33.1606 1.33284 0.666418 0.745578i \(-0.267827\pi\)
0.666418 + 0.745578i \(0.267827\pi\)
\(620\) 0 0
\(621\) −11.3401 −0.455062
\(622\) 0 0
\(623\) −6.16964 −0.247182
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −24.6599 −0.984822
\(628\) 0 0
\(629\) 7.48947 0.298625
\(630\) 0 0
\(631\) 33.8205 1.34637 0.673186 0.739473i \(-0.264925\pi\)
0.673186 + 0.739473i \(0.264925\pi\)
\(632\) 0 0
\(633\) −2.56708 −0.102032
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −18.2835 −0.724420
\(638\) 0 0
\(639\) −13.7074 −0.542256
\(640\) 0 0
\(641\) 12.8296 0.506737 0.253369 0.967370i \(-0.418461\pi\)
0.253369 + 0.967370i \(0.418461\pi\)
\(642\) 0 0
\(643\) 43.7084 1.72369 0.861846 0.507169i \(-0.169308\pi\)
0.861846 + 0.507169i \(0.169308\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 25.4158 0.999200 0.499600 0.866256i \(-0.333480\pi\)
0.499600 + 0.866256i \(0.333480\pi\)
\(648\) 0 0
\(649\) −15.2545 −0.598790
\(650\) 0 0
\(651\) −38.2827 −1.50042
\(652\) 0 0
\(653\) 3.94345 0.154319 0.0771596 0.997019i \(-0.475415\pi\)
0.0771596 + 0.997019i \(0.475415\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −99.8962 −3.89732
\(658\) 0 0
\(659\) 38.0950 1.48397 0.741984 0.670417i \(-0.233885\pi\)
0.741984 + 0.670417i \(0.233885\pi\)
\(660\) 0 0
\(661\) 18.1131 0.704518 0.352259 0.935903i \(-0.385414\pi\)
0.352259 + 0.935903i \(0.385414\pi\)
\(662\) 0 0
\(663\) −15.4148 −0.598660
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0.698317 0.0270389
\(668\) 0 0
\(669\) −47.5279 −1.83753
\(670\) 0 0
\(671\) −15.4148 −0.595081
\(672\) 0 0
\(673\) −43.2088 −1.66558 −0.832789 0.553590i \(-0.813257\pi\)
−0.832789 + 0.553590i \(0.813257\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 30.6983 1.17983 0.589916 0.807465i \(-0.299161\pi\)
0.589916 + 0.807465i \(0.299161\pi\)
\(678\) 0 0
\(679\) −0.284337 −0.0109119
\(680\) 0 0
\(681\) −45.5844 −1.74680
\(682\) 0 0
\(683\) −32.9536 −1.26093 −0.630467 0.776216i \(-0.717137\pi\)
−0.630467 + 0.776216i \(0.717137\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 72.8114 2.77793
\(688\) 0 0
\(689\) −9.41478 −0.358675
\(690\) 0 0
\(691\) −37.1222 −1.41219 −0.706097 0.708115i \(-0.749546\pi\)
−0.706097 + 0.708115i \(0.749546\pi\)
\(692\) 0 0
\(693\) 68.9245 2.61823
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 21.5460 0.816114
\(698\) 0 0
\(699\) 19.9253 0.753644
\(700\) 0 0
\(701\) −7.39663 −0.279367 −0.139683 0.990196i \(-0.544609\pi\)
−0.139683 + 0.990196i \(0.544609\pi\)
\(702\) 0 0
\(703\) 7.01735 0.264664
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −30.1696 −1.13465
\(708\) 0 0
\(709\) 28.7549 1.07991 0.539956 0.841693i \(-0.318441\pi\)
0.539956 + 0.841693i \(0.318441\pi\)
\(710\) 0 0
\(711\) 21.2088 0.795394
\(712\) 0 0
\(713\) −1.55695 −0.0583081
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 52.3502 1.95505
\(718\) 0 0
\(719\) 39.4532 1.47136 0.735678 0.677332i \(-0.236864\pi\)
0.735678 + 0.677332i \(0.236864\pi\)
\(720\) 0 0
\(721\) −47.1523 −1.75604
\(722\) 0 0
\(723\) 32.9608 1.22583
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −0.866904 −0.0321517 −0.0160758 0.999871i \(-0.505117\pi\)
−0.0160758 + 0.999871i \(0.505117\pi\)
\(728\) 0 0
\(729\) 85.4742 3.16571
\(730\) 0 0
\(731\) −15.4148 −0.570136
\(732\) 0 0
\(733\) 10.0000 0.369358 0.184679 0.982799i \(-0.440875\pi\)
0.184679 + 0.982799i \(0.440875\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −9.58442 −0.353047
\(738\) 0 0
\(739\) 31.5015 1.15880 0.579400 0.815043i \(-0.303287\pi\)
0.579400 + 0.815043i \(0.303287\pi\)
\(740\) 0 0
\(741\) −14.4431 −0.530579
\(742\) 0 0
\(743\) −31.8578 −1.16875 −0.584375 0.811484i \(-0.698660\pi\)
−0.584375 + 0.811484i \(0.698660\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −66.2755 −2.42489
\(748\) 0 0
\(749\) 67.2654 2.45782
\(750\) 0 0
\(751\) −46.0950 −1.68203 −0.841014 0.541013i \(-0.818041\pi\)
−0.841014 + 0.541013i \(0.818041\pi\)
\(752\) 0 0
\(753\) −82.0203 −2.98898
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −5.01735 −0.182359 −0.0911793 0.995834i \(-0.529064\pi\)
−0.0911793 + 0.995834i \(0.529064\pi\)
\(758\) 0 0
\(759\) 3.85061 0.139768
\(760\) 0 0
\(761\) 24.8296 0.900071 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(762\) 0 0
\(763\) −56.7367 −2.05401
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −8.93438 −0.322602
\(768\) 0 0
\(769\) 36.9427 1.33219 0.666093 0.745869i \(-0.267965\pi\)
0.666093 + 0.745869i \(0.267965\pi\)
\(770\) 0 0
\(771\) 66.6055 2.39874
\(772\) 0 0
\(773\) 16.4431 0.591415 0.295708 0.955278i \(-0.404445\pi\)
0.295708 + 0.955278i \(0.404445\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −26.9427 −0.966562
\(778\) 0 0
\(779\) 20.1878 0.723303
\(780\) 0 0
\(781\) 2.91518 0.104313
\(782\) 0 0
\(783\) −17.1704 −0.613622
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 40.2553 1.43495 0.717473 0.696587i \(-0.245299\pi\)
0.717473 + 0.696587i \(0.245299\pi\)
\(788\) 0 0
\(789\) 86.5946 3.08285
\(790\) 0 0
\(791\) −94.0203 −3.34298
\(792\) 0 0
\(793\) −9.02827 −0.320603
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −27.2835 −0.966432 −0.483216 0.875501i \(-0.660532\pi\)
−0.483216 + 0.875501i \(0.660532\pi\)
\(798\) 0 0
\(799\) 4.77301 0.168857
\(800\) 0 0
\(801\) 9.85061 0.348054
\(802\) 0 0
\(803\) 21.2451 0.749725
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −42.6055 −1.49978
\(808\) 0 0
\(809\) −28.4815 −1.00135 −0.500677 0.865634i \(-0.666916\pi\)
−0.500677 + 0.865634i \(0.666916\pi\)
\(810\) 0 0
\(811\) −31.6892 −1.11276 −0.556380 0.830928i \(-0.687810\pi\)
−0.556380 + 0.830928i \(0.687810\pi\)
\(812\) 0 0
\(813\) 58.8042 2.06235
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −14.4431 −0.505298
\(818\) 0 0
\(819\) 40.3684 1.41058
\(820\) 0 0
\(821\) −6.65991 −0.232433 −0.116216 0.993224i \(-0.537077\pi\)
−0.116216 + 0.993224i \(0.537077\pi\)
\(822\) 0 0
\(823\) −7.79301 −0.271647 −0.135824 0.990733i \(-0.543368\pi\)
−0.135824 + 0.990733i \(0.543368\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 26.4249 0.918884 0.459442 0.888208i \(-0.348049\pi\)
0.459442 + 0.888208i \(0.348049\pi\)
\(828\) 0 0
\(829\) 11.7447 0.407912 0.203956 0.978980i \(-0.434620\pi\)
0.203956 + 0.978980i \(0.434620\pi\)
\(830\) 0 0
\(831\) 51.3785 1.78230
\(832\) 0 0
\(833\) 84.8680 2.94050
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 38.2827 1.32325
\(838\) 0 0
\(839\) 44.5753 1.53891 0.769456 0.638700i \(-0.220527\pi\)
0.769456 + 0.638700i \(0.220527\pi\)
\(840\) 0 0
\(841\) −27.9427 −0.963540
\(842\) 0 0
\(843\) 24.4358 0.841615
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 40.6527 1.39684
\(848\) 0 0
\(849\) −2.00722 −0.0688875
\(850\) 0 0
\(851\) −1.09575 −0.0375618
\(852\) 0 0
\(853\) −29.6135 −1.01395 −0.506973 0.861962i \(-0.669236\pi\)
−0.506973 + 0.861962i \(0.669236\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −22.5105 −0.768945 −0.384472 0.923136i \(-0.625617\pi\)
−0.384472 + 0.923136i \(0.625617\pi\)
\(858\) 0 0
\(859\) 0.585221 0.0199675 0.00998375 0.999950i \(-0.496822\pi\)
0.00998375 + 0.999950i \(0.496822\pi\)
\(860\) 0 0
\(861\) −77.5097 −2.64152
\(862\) 0 0
\(863\) 45.6519 1.55401 0.777004 0.629495i \(-0.216738\pi\)
0.777004 + 0.629495i \(0.216738\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 15.0968 0.512714
\(868\) 0 0
\(869\) −4.51053 −0.153009
\(870\) 0 0
\(871\) −5.61350 −0.190206
\(872\) 0 0
\(873\) 0.453981 0.0153649
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 10.0000 0.337676 0.168838 0.985644i \(-0.445999\pi\)
0.168838 + 0.985644i \(0.445999\pi\)
\(878\) 0 0
\(879\) −76.4140 −2.57738
\(880\) 0 0
\(881\) 42.5380 1.43314 0.716571 0.697514i \(-0.245711\pi\)
0.716571 + 0.697514i \(0.245711\pi\)
\(882\) 0 0
\(883\) 6.22513 0.209492 0.104746 0.994499i \(-0.466597\pi\)
0.104746 + 0.994499i \(0.466597\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 22.0011 0.738723 0.369362 0.929286i \(-0.379576\pi\)
0.369362 + 0.929286i \(0.379576\pi\)
\(888\) 0 0
\(889\) −53.9819 −1.81049
\(890\) 0 0
\(891\) −53.5580 −1.79426
\(892\) 0 0
\(893\) 4.47213 0.149654
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 2.25526 0.0753011
\(898\) 0 0
\(899\) −2.35743 −0.0786247
\(900\) 0 0
\(901\) 43.7012 1.45590
\(902\) 0 0
\(903\) 55.4532 1.84537
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 28.2070 0.936598 0.468299 0.883570i \(-0.344867\pi\)
0.468299 + 0.883570i \(0.344867\pi\)
\(908\) 0 0
\(909\) 48.1696 1.59769
\(910\) 0 0
\(911\) −40.5105 −1.34217 −0.671087 0.741379i \(-0.734172\pi\)
−0.671087 + 0.741379i \(0.734172\pi\)
\(912\) 0 0
\(913\) 14.0950 0.466475
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 36.3393 1.20003
\(918\) 0 0
\(919\) −0.0746930 −0.00246389 −0.00123195 0.999999i \(-0.500392\pi\)
−0.00123195 + 0.999999i \(0.500392\pi\)
\(920\) 0 0
\(921\) 67.6410 2.22885
\(922\) 0 0
\(923\) 1.70739 0.0561994
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 75.2846 2.47267
\(928\) 0 0
\(929\) 53.6410 1.75990 0.879952 0.475063i \(-0.157575\pi\)
0.879952 + 0.475063i \(0.157575\pi\)
\(930\) 0 0
\(931\) 79.5180 2.60610
\(932\) 0 0
\(933\) 66.1696 2.16630
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −26.0000 −0.849383 −0.424691 0.905338i \(-0.639617\pi\)
−0.424691 + 0.905338i \(0.639617\pi\)
\(938\) 0 0
\(939\) −104.513 −3.41064
\(940\) 0 0
\(941\) −49.9253 −1.62752 −0.813759 0.581202i \(-0.802583\pi\)
−0.813759 + 0.581202i \(0.802583\pi\)
\(942\) 0 0
\(943\) −3.15230 −0.102653
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −5.80128 −0.188516 −0.0942582 0.995548i \(-0.530048\pi\)
−0.0942582 + 0.995548i \(0.530048\pi\)
\(948\) 0 0
\(949\) 12.4431 0.403919
\(950\) 0 0
\(951\) −5.73381 −0.185931
\(952\) 0 0
\(953\) 13.2270 0.428464 0.214232 0.976783i \(-0.431275\pi\)
0.214232 + 0.976783i \(0.431275\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 5.83036 0.188469
\(958\) 0 0
\(959\) 87.1907 2.81553
\(960\) 0 0
\(961\) −25.7439 −0.830450
\(962\) 0 0
\(963\) −107.398 −3.46084
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −1.61350 −0.0518865 −0.0259433 0.999663i \(-0.508259\pi\)
−0.0259433 + 0.999663i \(0.508259\pi\)
\(968\) 0 0
\(969\) 67.0413 2.15368
\(970\) 0 0
\(971\) −16.7730 −0.538271 −0.269136 0.963102i \(-0.586738\pi\)
−0.269136 + 0.963102i \(0.586738\pi\)
\(972\) 0 0
\(973\) 47.0557 1.50854
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 47.6700 1.52510 0.762550 0.646929i \(-0.223947\pi\)
0.762550 + 0.646929i \(0.223947\pi\)
\(978\) 0 0
\(979\) −2.09495 −0.0669549
\(980\) 0 0
\(981\) 90.5873 2.89223
\(982\) 0 0
\(983\) 30.4996 0.972786 0.486393 0.873740i \(-0.338312\pi\)
0.486393 + 0.873740i \(0.338312\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −17.1704 −0.546541
\(988\) 0 0
\(989\) 2.25526 0.0717132
\(990\) 0 0
\(991\) 19.8506 0.630576 0.315288 0.948996i \(-0.397899\pi\)
0.315288 + 0.948996i \(0.397899\pi\)
\(992\) 0 0
\(993\) −34.5561 −1.09661
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 33.6410 1.06542 0.532710 0.846298i \(-0.321174\pi\)
0.532710 + 0.846298i \(0.321174\pi\)
\(998\) 0 0
\(999\) 26.9427 0.852428
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 5200.2.a.ci.1.3 3
4.3 odd 2 1300.2.a.i.1.1 3
5.4 even 2 1040.2.a.o.1.1 3
15.14 odd 2 9360.2.a.da.1.3 3
20.3 even 4 1300.2.c.f.1249.1 6
20.7 even 4 1300.2.c.f.1249.6 6
20.19 odd 2 260.2.a.b.1.3 3
40.19 odd 2 4160.2.a.bo.1.1 3
40.29 even 2 4160.2.a.br.1.3 3
60.59 even 2 2340.2.a.n.1.1 3
260.99 even 4 3380.2.f.h.3041.6 6
260.239 even 4 3380.2.f.h.3041.5 6
260.259 odd 2 3380.2.a.o.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
260.2.a.b.1.3 3 20.19 odd 2
1040.2.a.o.1.1 3 5.4 even 2
1300.2.a.i.1.1 3 4.3 odd 2
1300.2.c.f.1249.1 6 20.3 even 4
1300.2.c.f.1249.6 6 20.7 even 4
2340.2.a.n.1.1 3 60.59 even 2
3380.2.a.o.1.3 3 260.259 odd 2
3380.2.f.h.3041.5 6 260.239 even 4
3380.2.f.h.3041.6 6 260.99 even 4
4160.2.a.bo.1.1 3 40.19 odd 2
4160.2.a.br.1.3 3 40.29 even 2
5200.2.a.ci.1.3 3 1.1 even 1 trivial
9360.2.a.da.1.3 3 15.14 odd 2