Properties

Label 4400.2.a.ca.1.1
Level $4400$
Weight $2$
Character 4400.1
Self dual yes
Analytic conductor $35.134$
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] = [4400,2,Mod(1,4400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4400, 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("4400.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4400.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: \(35.1341768894\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.837.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 1 \) 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 2200)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.52892\) of defining polynomial
Character \(\chi\) \(=\) 4400.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.52892 q^{3} -1.39543 q^{7} -0.662410 q^{9} +O(q^{10})\) \(q-1.52892 q^{3} -1.39543 q^{7} -0.662410 q^{9} +1.00000 q^{11} +6.05784 q^{13} +3.26193 q^{17} -1.00000 q^{19} +2.13349 q^{21} -0.528918 q^{23} +5.59952 q^{27} +1.60457 q^{29} -3.79085 q^{31} -1.52892 q^{33} -8.92434 q^{37} -9.26193 q^{39} +1.86651 q^{41} -0.866508 q^{43} -1.19133 q^{47} -5.05279 q^{49} -4.98723 q^{51} +10.7202 q^{53} +1.52892 q^{57} -13.1913 q^{59} -3.12844 q^{61} +0.924344 q^{63} +2.26698 q^{67} +0.808672 q^{69} -10.0400 q^{71} +13.3954 q^{73} -1.39543 q^{77} +2.58675 q^{79} -6.57398 q^{81} -0.128442 q^{83} -2.45326 q^{87} +4.12844 q^{89} -8.45326 q^{91} +5.79590 q^{93} -11.4354 q^{97} -0.662410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 3 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} + 3 q^{7} + 6 q^{9} + 3 q^{11} + 3 q^{13} + 3 q^{17} - 3 q^{19} + 6 q^{21} + 6 q^{23} + 18 q^{27} + 12 q^{29} + 3 q^{31} + 3 q^{33} - 12 q^{37} - 21 q^{39} + 6 q^{41} - 3 q^{43} + 12 q^{47} + 6 q^{49} + 9 q^{51} + 9 q^{53} - 3 q^{57} - 24 q^{59} - 3 q^{61} - 12 q^{63} + 6 q^{67} + 18 q^{69} + 15 q^{71} + 33 q^{73} + 3 q^{77} - 15 q^{79} + 27 q^{81} + 6 q^{83} + 15 q^{87} + 6 q^{89} - 3 q^{91} + 9 q^{93} + 18 q^{97} + 6 q^{99}+O(q^{100}) \) 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
\(3\) −1.52892 −0.882721 −0.441361 0.897330i \(-0.645504\pi\)
−0.441361 + 0.897330i \(0.645504\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.39543 −0.527421 −0.263711 0.964602i \(-0.584946\pi\)
−0.263711 + 0.964602i \(0.584946\pi\)
\(8\) 0 0
\(9\) −0.662410 −0.220803
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 6.05784 1.68014 0.840071 0.542477i \(-0.182513\pi\)
0.840071 + 0.542477i \(0.182513\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.26193 0.791135 0.395568 0.918437i \(-0.370548\pi\)
0.395568 + 0.918437i \(0.370548\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) 2.13349 0.465566
\(22\) 0 0
\(23\) −0.528918 −0.110287 −0.0551435 0.998478i \(-0.517562\pi\)
−0.0551435 + 0.998478i \(0.517562\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.59952 1.07763
\(28\) 0 0
\(29\) 1.60457 0.297962 0.148981 0.988840i \(-0.452401\pi\)
0.148981 + 0.988840i \(0.452401\pi\)
\(30\) 0 0
\(31\) −3.79085 −0.680857 −0.340429 0.940270i \(-0.610572\pi\)
−0.340429 + 0.940270i \(0.610572\pi\)
\(32\) 0 0
\(33\) −1.52892 −0.266150
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8.92434 −1.46715 −0.733577 0.679607i \(-0.762150\pi\)
−0.733577 + 0.679607i \(0.762150\pi\)
\(38\) 0 0
\(39\) −9.26193 −1.48310
\(40\) 0 0
\(41\) 1.86651 0.291500 0.145750 0.989321i \(-0.453441\pi\)
0.145750 + 0.989321i \(0.453441\pi\)
\(42\) 0 0
\(43\) −0.866508 −0.132141 −0.0660706 0.997815i \(-0.521046\pi\)
−0.0660706 + 0.997815i \(0.521046\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.19133 −0.173773 −0.0868865 0.996218i \(-0.527692\pi\)
−0.0868865 + 0.996218i \(0.527692\pi\)
\(48\) 0 0
\(49\) −5.05279 −0.721827
\(50\) 0 0
\(51\) −4.98723 −0.698352
\(52\) 0 0
\(53\) 10.7202 1.47254 0.736270 0.676688i \(-0.236586\pi\)
0.736270 + 0.676688i \(0.236586\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.52892 0.202510
\(58\) 0 0
\(59\) −13.1913 −1.71736 −0.858682 0.512508i \(-0.828716\pi\)
−0.858682 + 0.512508i \(0.828716\pi\)
\(60\) 0 0
\(61\) −3.12844 −0.400556 −0.200278 0.979739i \(-0.564185\pi\)
−0.200278 + 0.979739i \(0.564185\pi\)
\(62\) 0 0
\(63\) 0.924344 0.116456
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2.26698 0.276956 0.138478 0.990365i \(-0.455779\pi\)
0.138478 + 0.990365i \(0.455779\pi\)
\(68\) 0 0
\(69\) 0.808672 0.0973527
\(70\) 0 0
\(71\) −10.0400 −1.19153 −0.595765 0.803159i \(-0.703151\pi\)
−0.595765 + 0.803159i \(0.703151\pi\)
\(72\) 0 0
\(73\) 13.3954 1.56782 0.783908 0.620877i \(-0.213223\pi\)
0.783908 + 0.620877i \(0.213223\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.39543 −0.159024
\(78\) 0 0
\(79\) 2.58675 0.291033 0.145516 0.989356i \(-0.453516\pi\)
0.145516 + 0.989356i \(0.453516\pi\)
\(80\) 0 0
\(81\) −6.57398 −0.730443
\(82\) 0 0
\(83\) −0.128442 −0.0140984 −0.00704918 0.999975i \(-0.502244\pi\)
−0.00704918 + 0.999975i \(0.502244\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −2.45326 −0.263017
\(88\) 0 0
\(89\) 4.12844 0.437614 0.218807 0.975768i \(-0.429783\pi\)
0.218807 + 0.975768i \(0.429783\pi\)
\(90\) 0 0
\(91\) −8.45326 −0.886143
\(92\) 0 0
\(93\) 5.79590 0.601007
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −11.4354 −1.16109 −0.580547 0.814227i \(-0.697161\pi\)
−0.580547 + 0.814227i \(0.697161\pi\)
\(98\) 0 0
\(99\) −0.662410 −0.0665747
\(100\) 0 0
\(101\) 14.2619 1.41912 0.709558 0.704647i \(-0.248895\pi\)
0.709558 + 0.704647i \(0.248895\pi\)
\(102\) 0 0
\(103\) 18.3598 1.80904 0.904522 0.426428i \(-0.140228\pi\)
0.904522 + 0.426428i \(0.140228\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 5.11567 0.494551 0.247275 0.968945i \(-0.420465\pi\)
0.247275 + 0.968945i \(0.420465\pi\)
\(108\) 0 0
\(109\) 8.24411 0.789643 0.394821 0.918758i \(-0.370806\pi\)
0.394821 + 0.918758i \(0.370806\pi\)
\(110\) 0 0
\(111\) 13.6446 1.29509
\(112\) 0 0
\(113\) 10.9243 1.02768 0.513838 0.857887i \(-0.328223\pi\)
0.513838 + 0.857887i \(0.328223\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −4.01277 −0.370981
\(118\) 0 0
\(119\) −4.55179 −0.417262
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −2.85374 −0.257313
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 8.47108 0.751687 0.375843 0.926683i \(-0.377353\pi\)
0.375843 + 0.926683i \(0.377353\pi\)
\(128\) 0 0
\(129\) 1.32482 0.116644
\(130\) 0 0
\(131\) 11.4354 0.999119 0.499560 0.866280i \(-0.333495\pi\)
0.499560 + 0.866280i \(0.333495\pi\)
\(132\) 0 0
\(133\) 1.39543 0.120999
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.66241 0.569208 0.284604 0.958645i \(-0.408138\pi\)
0.284604 + 0.958645i \(0.408138\pi\)
\(138\) 0 0
\(139\) −12.8487 −1.08981 −0.544906 0.838497i \(-0.683435\pi\)
−0.544906 + 0.838497i \(0.683435\pi\)
\(140\) 0 0
\(141\) 1.82144 0.153393
\(142\) 0 0
\(143\) 6.05784 0.506582
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 7.72530 0.637172
\(148\) 0 0
\(149\) −1.73302 −0.141974 −0.0709871 0.997477i \(-0.522615\pi\)
−0.0709871 + 0.997477i \(0.522615\pi\)
\(150\) 0 0
\(151\) 12.9243 1.05177 0.525884 0.850556i \(-0.323735\pi\)
0.525884 + 0.850556i \(0.323735\pi\)
\(152\) 0 0
\(153\) −2.16074 −0.174685
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −12.6496 −1.00955 −0.504776 0.863251i \(-0.668425\pi\)
−0.504776 + 0.863251i \(0.668425\pi\)
\(158\) 0 0
\(159\) −16.3904 −1.29984
\(160\) 0 0
\(161\) 0.738066 0.0581677
\(162\) 0 0
\(163\) 10.9771 0.859795 0.429898 0.902878i \(-0.358550\pi\)
0.429898 + 0.902878i \(0.358550\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2.66746 0.206414 0.103207 0.994660i \(-0.467090\pi\)
0.103207 + 0.994660i \(0.467090\pi\)
\(168\) 0 0
\(169\) 23.6974 1.82288
\(170\) 0 0
\(171\) 0.662410 0.0506557
\(172\) 0 0
\(173\) −15.1157 −1.14922 −0.574612 0.818426i \(-0.694847\pi\)
−0.574612 + 0.818426i \(0.694847\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 20.1685 1.51595
\(178\) 0 0
\(179\) −13.6624 −1.02118 −0.510588 0.859825i \(-0.670572\pi\)
−0.510588 + 0.859825i \(0.670572\pi\)
\(180\) 0 0
\(181\) 8.98723 0.668016 0.334008 0.942570i \(-0.391599\pi\)
0.334008 + 0.942570i \(0.391599\pi\)
\(182\) 0 0
\(183\) 4.78313 0.353579
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 3.26193 0.238536
\(188\) 0 0
\(189\) −7.81372 −0.568365
\(190\) 0 0
\(191\) 4.24411 0.307093 0.153547 0.988141i \(-0.450930\pi\)
0.153547 + 0.988141i \(0.450930\pi\)
\(192\) 0 0
\(193\) 4.53397 0.326362 0.163181 0.986596i \(-0.447825\pi\)
0.163181 + 0.986596i \(0.447825\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3.72797 0.265607 0.132803 0.991142i \(-0.457602\pi\)
0.132803 + 0.991142i \(0.457602\pi\)
\(198\) 0 0
\(199\) 12.5111 0.886888 0.443444 0.896302i \(-0.353756\pi\)
0.443444 + 0.896302i \(0.353756\pi\)
\(200\) 0 0
\(201\) −3.46603 −0.244475
\(202\) 0 0
\(203\) −2.23906 −0.157152
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0.350360 0.0243517
\(208\) 0 0
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) 23.0400 1.58614 0.793070 0.609130i \(-0.208481\pi\)
0.793070 + 0.609130i \(0.208481\pi\)
\(212\) 0 0
\(213\) 15.3504 1.05179
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 5.28985 0.359099
\(218\) 0 0
\(219\) −20.4805 −1.38394
\(220\) 0 0
\(221\) 19.7603 1.32922
\(222\) 0 0
\(223\) −9.19133 −0.615497 −0.307748 0.951468i \(-0.599576\pi\)
−0.307748 + 0.951468i \(0.599576\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 15.0350 0.997906 0.498953 0.866629i \(-0.333718\pi\)
0.498953 + 0.866629i \(0.333718\pi\)
\(228\) 0 0
\(229\) −4.58675 −0.303101 −0.151551 0.988449i \(-0.548427\pi\)
−0.151551 + 0.988449i \(0.548427\pi\)
\(230\) 0 0
\(231\) 2.13349 0.140373
\(232\) 0 0
\(233\) 0.337590 0.0221163 0.0110581 0.999939i \(-0.496480\pi\)
0.0110581 + 0.999939i \(0.496480\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −3.95493 −0.256901
\(238\) 0 0
\(239\) 25.5111 1.65018 0.825088 0.565004i \(-0.191126\pi\)
0.825088 + 0.565004i \(0.191126\pi\)
\(240\) 0 0
\(241\) −1.93711 −0.124781 −0.0623903 0.998052i \(-0.519872\pi\)
−0.0623903 + 0.998052i \(0.519872\pi\)
\(242\) 0 0
\(243\) −6.74749 −0.432852
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −6.05784 −0.385451
\(248\) 0 0
\(249\) 0.196378 0.0124449
\(250\) 0 0
\(251\) −17.3776 −1.09686 −0.548432 0.836195i \(-0.684775\pi\)
−0.548432 + 0.836195i \(0.684775\pi\)
\(252\) 0 0
\(253\) −0.528918 −0.0332528
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −2.98218 −0.186023 −0.0930116 0.995665i \(-0.529649\pi\)
−0.0930116 + 0.995665i \(0.529649\pi\)
\(258\) 0 0
\(259\) 12.4533 0.773808
\(260\) 0 0
\(261\) −1.06289 −0.0657910
\(262\) 0 0
\(263\) 22.7730 1.40425 0.702123 0.712056i \(-0.252236\pi\)
0.702123 + 0.712056i \(0.252236\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −6.31205 −0.386291
\(268\) 0 0
\(269\) −25.4933 −1.55435 −0.777176 0.629283i \(-0.783349\pi\)
−0.777176 + 0.629283i \(0.783349\pi\)
\(270\) 0 0
\(271\) −9.18123 −0.557720 −0.278860 0.960332i \(-0.589957\pi\)
−0.278860 + 0.960332i \(0.589957\pi\)
\(272\) 0 0
\(273\) 12.9243 0.782217
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 24.0979 1.44790 0.723950 0.689853i \(-0.242325\pi\)
0.723950 + 0.689853i \(0.242325\pi\)
\(278\) 0 0
\(279\) 2.51110 0.150335
\(280\) 0 0
\(281\) 27.4126 1.63530 0.817648 0.575718i \(-0.195277\pi\)
0.817648 + 0.575718i \(0.195277\pi\)
\(282\) 0 0
\(283\) 17.6446 1.04886 0.524431 0.851453i \(-0.324278\pi\)
0.524431 + 0.851453i \(0.324278\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.60457 −0.153743
\(288\) 0 0
\(289\) −6.35979 −0.374105
\(290\) 0 0
\(291\) 17.4839 1.02492
\(292\) 0 0
\(293\) −25.1634 −1.47006 −0.735031 0.678033i \(-0.762832\pi\)
−0.735031 + 0.678033i \(0.762832\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 5.59952 0.324917
\(298\) 0 0
\(299\) −3.20410 −0.185298
\(300\) 0 0
\(301\) 1.20915 0.0696941
\(302\) 0 0
\(303\) −21.8053 −1.25268
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 29.6089 1.68987 0.844936 0.534867i \(-0.179638\pi\)
0.844936 + 0.534867i \(0.179638\pi\)
\(308\) 0 0
\(309\) −28.0706 −1.59688
\(310\) 0 0
\(311\) −10.7330 −0.608614 −0.304307 0.952574i \(-0.598425\pi\)
−0.304307 + 0.952574i \(0.598425\pi\)
\(312\) 0 0
\(313\) 33.2892 1.88162 0.940808 0.338940i \(-0.110069\pi\)
0.940808 + 0.338940i \(0.110069\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −18.5511 −1.04193 −0.520967 0.853577i \(-0.674429\pi\)
−0.520967 + 0.853577i \(0.674429\pi\)
\(318\) 0 0
\(319\) 1.60457 0.0898389
\(320\) 0 0
\(321\) −7.82144 −0.436550
\(322\) 0 0
\(323\) −3.26193 −0.181499
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −12.6046 −0.697035
\(328\) 0 0
\(329\) 1.66241 0.0916516
\(330\) 0 0
\(331\) 5.86651 0.322452 0.161226 0.986917i \(-0.448455\pi\)
0.161226 + 0.986917i \(0.448455\pi\)
\(332\) 0 0
\(333\) 5.91157 0.323952
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 7.58170 0.413002 0.206501 0.978446i \(-0.433792\pi\)
0.206501 + 0.978446i \(0.433792\pi\)
\(338\) 0 0
\(339\) −16.7024 −0.907151
\(340\) 0 0
\(341\) −3.79085 −0.205286
\(342\) 0 0
\(343\) 16.8188 0.908128
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −25.3954 −1.36330 −0.681649 0.731679i \(-0.738737\pi\)
−0.681649 + 0.731679i \(0.738737\pi\)
\(348\) 0 0
\(349\) −21.2670 −1.13840 −0.569198 0.822201i \(-0.692746\pi\)
−0.569198 + 0.822201i \(0.692746\pi\)
\(350\) 0 0
\(351\) 33.9210 1.81057
\(352\) 0 0
\(353\) −8.88433 −0.472865 −0.236433 0.971648i \(-0.575978\pi\)
−0.236433 + 0.971648i \(0.575978\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 6.95931 0.368326
\(358\) 0 0
\(359\) −11.1990 −0.591063 −0.295532 0.955333i \(-0.595497\pi\)
−0.295532 + 0.955333i \(0.595497\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) −1.52892 −0.0802474
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −1.58675 −0.0828279 −0.0414139 0.999142i \(-0.513186\pi\)
−0.0414139 + 0.999142i \(0.513186\pi\)
\(368\) 0 0
\(369\) −1.23639 −0.0643641
\(370\) 0 0
\(371\) −14.9593 −0.776649
\(372\) 0 0
\(373\) −33.2714 −1.72273 −0.861363 0.507990i \(-0.830389\pi\)
−0.861363 + 0.507990i \(0.830389\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 9.72025 0.500618
\(378\) 0 0
\(379\) 24.0979 1.23782 0.618912 0.785461i \(-0.287574\pi\)
0.618912 + 0.785461i \(0.287574\pi\)
\(380\) 0 0
\(381\) −12.9516 −0.663530
\(382\) 0 0
\(383\) 15.3648 0.785106 0.392553 0.919729i \(-0.371592\pi\)
0.392553 + 0.919729i \(0.371592\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0.573984 0.0291772
\(388\) 0 0
\(389\) −1.88433 −0.0955392 −0.0477696 0.998858i \(-0.515211\pi\)
−0.0477696 + 0.998858i \(0.515211\pi\)
\(390\) 0 0
\(391\) −1.72530 −0.0872519
\(392\) 0 0
\(393\) −17.4839 −0.881944
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 7.27203 0.364973 0.182486 0.983208i \(-0.441585\pi\)
0.182486 + 0.983208i \(0.441585\pi\)
\(398\) 0 0
\(399\) −2.13349 −0.106808
\(400\) 0 0
\(401\) 12.8665 0.642523 0.321261 0.946991i \(-0.395893\pi\)
0.321261 + 0.946991i \(0.395893\pi\)
\(402\) 0 0
\(403\) −22.9644 −1.14394
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −8.92434 −0.442363
\(408\) 0 0
\(409\) 5.15569 0.254932 0.127466 0.991843i \(-0.459316\pi\)
0.127466 + 0.991843i \(0.459316\pi\)
\(410\) 0 0
\(411\) −10.1863 −0.502452
\(412\) 0 0
\(413\) 18.4075 0.905775
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 19.6446 0.962000
\(418\) 0 0
\(419\) 39.0145 1.90598 0.952991 0.302999i \(-0.0979878\pi\)
0.952991 + 0.302999i \(0.0979878\pi\)
\(420\) 0 0
\(421\) 28.9415 1.41052 0.705261 0.708948i \(-0.250830\pi\)
0.705261 + 0.708948i \(0.250830\pi\)
\(422\) 0 0
\(423\) 0.789147 0.0383697
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 4.36551 0.211262
\(428\) 0 0
\(429\) −9.26193 −0.447170
\(430\) 0 0
\(431\) 17.2969 0.833162 0.416581 0.909099i \(-0.363228\pi\)
0.416581 + 0.909099i \(0.363228\pi\)
\(432\) 0 0
\(433\) 17.8010 0.855459 0.427730 0.903907i \(-0.359314\pi\)
0.427730 + 0.903907i \(0.359314\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0.528918 0.0253016
\(438\) 0 0
\(439\) 28.9694 1.38263 0.691317 0.722551i \(-0.257031\pi\)
0.691317 + 0.722551i \(0.257031\pi\)
\(440\) 0 0
\(441\) 3.34702 0.159382
\(442\) 0 0
\(443\) −5.70748 −0.271170 −0.135585 0.990766i \(-0.543291\pi\)
−0.135585 + 0.990766i \(0.543291\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 2.64964 0.125324
\(448\) 0 0
\(449\) 21.8436 1.03087 0.515433 0.856930i \(-0.327631\pi\)
0.515433 + 0.856930i \(0.327631\pi\)
\(450\) 0 0
\(451\) 1.86651 0.0878904
\(452\) 0 0
\(453\) −19.7603 −0.928418
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 19.2441 0.900202 0.450101 0.892978i \(-0.351388\pi\)
0.450101 + 0.892978i \(0.351388\pi\)
\(458\) 0 0
\(459\) 18.2653 0.852550
\(460\) 0 0
\(461\) 10.6574 0.496363 0.248181 0.968714i \(-0.420167\pi\)
0.248181 + 0.968714i \(0.420167\pi\)
\(462\) 0 0
\(463\) −6.98990 −0.324848 −0.162424 0.986721i \(-0.551931\pi\)
−0.162424 + 0.986721i \(0.551931\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −10.3904 −0.480809 −0.240405 0.970673i \(-0.577280\pi\)
−0.240405 + 0.970673i \(0.577280\pi\)
\(468\) 0 0
\(469\) −3.16341 −0.146073
\(470\) 0 0
\(471\) 19.3403 0.891152
\(472\) 0 0
\(473\) −0.866508 −0.0398421
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −7.10120 −0.325141
\(478\) 0 0
\(479\) 37.0044 1.69077 0.845387 0.534155i \(-0.179370\pi\)
0.845387 + 0.534155i \(0.179370\pi\)
\(480\) 0 0
\(481\) −54.0622 −2.46502
\(482\) 0 0
\(483\) −1.12844 −0.0513459
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 24.0578 1.09016 0.545082 0.838383i \(-0.316498\pi\)
0.545082 + 0.838383i \(0.316498\pi\)
\(488\) 0 0
\(489\) −16.7831 −0.758959
\(490\) 0 0
\(491\) 0.314720 0.0142031 0.00710156 0.999975i \(-0.497739\pi\)
0.00710156 + 0.999975i \(0.497739\pi\)
\(492\) 0 0
\(493\) 5.23401 0.235728
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 14.0101 0.628439
\(498\) 0 0
\(499\) −39.6268 −1.77394 −0.886969 0.461829i \(-0.847193\pi\)
−0.886969 + 0.461829i \(0.847193\pi\)
\(500\) 0 0
\(501\) −4.07833 −0.182206
\(502\) 0 0
\(503\) 21.2892 0.949238 0.474619 0.880191i \(-0.342586\pi\)
0.474619 + 0.880191i \(0.342586\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −36.2313 −1.60909
\(508\) 0 0
\(509\) 12.3019 0.545274 0.272637 0.962117i \(-0.412104\pi\)
0.272637 + 0.962117i \(0.412104\pi\)
\(510\) 0 0
\(511\) −18.6923 −0.826900
\(512\) 0 0
\(513\) −5.59952 −0.247225
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −1.19133 −0.0523945
\(518\) 0 0
\(519\) 23.1106 1.01444
\(520\) 0 0
\(521\) 36.4227 1.59571 0.797853 0.602852i \(-0.205969\pi\)
0.797853 + 0.602852i \(0.205969\pi\)
\(522\) 0 0
\(523\) −23.6675 −1.03491 −0.517453 0.855712i \(-0.673120\pi\)
−0.517453 + 0.855712i \(0.673120\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −12.3655 −0.538650
\(528\) 0 0
\(529\) −22.7202 −0.987837
\(530\) 0 0
\(531\) 8.73807 0.379200
\(532\) 0 0
\(533\) 11.3070 0.489761
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 20.8887 0.901414
\(538\) 0 0
\(539\) −5.05279 −0.217639
\(540\) 0 0
\(541\) 12.9015 0.554678 0.277339 0.960772i \(-0.410548\pi\)
0.277339 + 0.960772i \(0.410548\pi\)
\(542\) 0 0
\(543\) −13.7407 −0.589671
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −35.2841 −1.50864 −0.754320 0.656507i \(-0.772033\pi\)
−0.754320 + 0.656507i \(0.772033\pi\)
\(548\) 0 0
\(549\) 2.07231 0.0884441
\(550\) 0 0
\(551\) −1.60457 −0.0683571
\(552\) 0 0
\(553\) −3.60962 −0.153497
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −24.9745 −1.05820 −0.529101 0.848559i \(-0.677471\pi\)
−0.529101 + 0.848559i \(0.677471\pi\)
\(558\) 0 0
\(559\) −5.24916 −0.222016
\(560\) 0 0
\(561\) −4.98723 −0.210561
\(562\) 0 0
\(563\) 40.8258 1.72060 0.860302 0.509785i \(-0.170275\pi\)
0.860302 + 0.509785i \(0.170275\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 9.17351 0.385251
\(568\) 0 0
\(569\) −18.6046 −0.779944 −0.389972 0.920827i \(-0.627515\pi\)
−0.389972 + 0.920827i \(0.627515\pi\)
\(570\) 0 0
\(571\) 9.44554 0.395284 0.197642 0.980274i \(-0.436672\pi\)
0.197642 + 0.980274i \(0.436672\pi\)
\(572\) 0 0
\(573\) −6.48890 −0.271078
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 13.5646 0.564700 0.282350 0.959311i \(-0.408886\pi\)
0.282350 + 0.959311i \(0.408886\pi\)
\(578\) 0 0
\(579\) −6.93206 −0.288087
\(580\) 0 0
\(581\) 0.179232 0.00743578
\(582\) 0 0
\(583\) 10.7202 0.443987
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 35.2263 1.45394 0.726972 0.686667i \(-0.240927\pi\)
0.726972 + 0.686667i \(0.240927\pi\)
\(588\) 0 0
\(589\) 3.79085 0.156199
\(590\) 0 0
\(591\) −5.69975 −0.234457
\(592\) 0 0
\(593\) −39.7952 −1.63419 −0.817097 0.576500i \(-0.804418\pi\)
−0.817097 + 0.576500i \(0.804418\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −19.1284 −0.782875
\(598\) 0 0
\(599\) −19.1207 −0.781252 −0.390626 0.920550i \(-0.627741\pi\)
−0.390626 + 0.920550i \(0.627741\pi\)
\(600\) 0 0
\(601\) −22.1964 −0.905409 −0.452705 0.891661i \(-0.649541\pi\)
−0.452705 + 0.891661i \(0.649541\pi\)
\(602\) 0 0
\(603\) −1.50167 −0.0611528
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 12.9321 0.524896 0.262448 0.964946i \(-0.415470\pi\)
0.262448 + 0.964946i \(0.415470\pi\)
\(608\) 0 0
\(609\) 3.42335 0.138721
\(610\) 0 0
\(611\) −7.21687 −0.291963
\(612\) 0 0
\(613\) −18.4711 −0.746040 −0.373020 0.927823i \(-0.621678\pi\)
−0.373020 + 0.927823i \(0.621678\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 22.6096 0.910229 0.455115 0.890433i \(-0.349598\pi\)
0.455115 + 0.890433i \(0.349598\pi\)
\(618\) 0 0
\(619\) 37.1557 1.49341 0.746707 0.665154i \(-0.231634\pi\)
0.746707 + 0.665154i \(0.231634\pi\)
\(620\) 0 0
\(621\) −2.96169 −0.118848
\(622\) 0 0
\(623\) −5.76094 −0.230807
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 1.52892 0.0610591
\(628\) 0 0
\(629\) −29.1106 −1.16072
\(630\) 0 0
\(631\) 24.1506 0.961422 0.480711 0.876879i \(-0.340379\pi\)
0.480711 + 0.876879i \(0.340379\pi\)
\(632\) 0 0
\(633\) −35.2263 −1.40012
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −30.6089 −1.21277
\(638\) 0 0
\(639\) 6.65061 0.263094
\(640\) 0 0
\(641\) −34.3948 −1.35851 −0.679256 0.733902i \(-0.737697\pi\)
−0.679256 + 0.733902i \(0.737697\pi\)
\(642\) 0 0
\(643\) 44.4983 1.75484 0.877421 0.479720i \(-0.159262\pi\)
0.877421 + 0.479720i \(0.159262\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −40.2135 −1.58096 −0.790478 0.612490i \(-0.790168\pi\)
−0.790478 + 0.612490i \(0.790168\pi\)
\(648\) 0 0
\(649\) −13.1913 −0.517805
\(650\) 0 0
\(651\) −8.08775 −0.316984
\(652\) 0 0
\(653\) −31.3420 −1.22651 −0.613253 0.789887i \(-0.710139\pi\)
−0.613253 + 0.789887i \(0.710139\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −8.87326 −0.346179
\(658\) 0 0
\(659\) −32.3598 −1.26056 −0.630279 0.776369i \(-0.717060\pi\)
−0.630279 + 0.776369i \(0.717060\pi\)
\(660\) 0 0
\(661\) −22.8087 −0.887155 −0.443577 0.896236i \(-0.646291\pi\)
−0.443577 + 0.896236i \(0.646291\pi\)
\(662\) 0 0
\(663\) −30.2118 −1.17333
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −0.848688 −0.0328613
\(668\) 0 0
\(669\) 14.0528 0.543312
\(670\) 0 0
\(671\) −3.12844 −0.120772
\(672\) 0 0
\(673\) 27.7152 1.06834 0.534171 0.845376i \(-0.320624\pi\)
0.534171 + 0.845376i \(0.320624\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −16.0400 −0.616468 −0.308234 0.951311i \(-0.599738\pi\)
−0.308234 + 0.951311i \(0.599738\pi\)
\(678\) 0 0
\(679\) 15.9573 0.612385
\(680\) 0 0
\(681\) −22.9872 −0.880873
\(682\) 0 0
\(683\) 35.8208 1.37064 0.685322 0.728240i \(-0.259662\pi\)
0.685322 + 0.728240i \(0.259662\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 7.01277 0.267554
\(688\) 0 0
\(689\) 64.9415 2.47407
\(690\) 0 0
\(691\) −38.5333 −1.46588 −0.732938 0.680296i \(-0.761851\pi\)
−0.732938 + 0.680296i \(0.761851\pi\)
\(692\) 0 0
\(693\) 0.924344 0.0351129
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 6.08843 0.230616
\(698\) 0 0
\(699\) −0.516148 −0.0195225
\(700\) 0 0
\(701\) −52.7596 −1.99270 −0.996351 0.0853497i \(-0.972799\pi\)
−0.996351 + 0.0853497i \(0.972799\pi\)
\(702\) 0 0
\(703\) 8.92434 0.336588
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −19.9015 −0.748472
\(708\) 0 0
\(709\) 5.04774 0.189572 0.0947859 0.995498i \(-0.469783\pi\)
0.0947859 + 0.995498i \(0.469783\pi\)
\(710\) 0 0
\(711\) −1.71349 −0.0642609
\(712\) 0 0
\(713\) 2.00505 0.0750897
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −39.0044 −1.45665
\(718\) 0 0
\(719\) −16.2848 −0.607321 −0.303660 0.952780i \(-0.598209\pi\)
−0.303660 + 0.952780i \(0.598209\pi\)
\(720\) 0 0
\(721\) −25.6197 −0.954128
\(722\) 0 0
\(723\) 2.96169 0.110146
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 9.68461 0.359182 0.179591 0.983741i \(-0.442523\pi\)
0.179591 + 0.983741i \(0.442523\pi\)
\(728\) 0 0
\(729\) 30.0383 1.11253
\(730\) 0 0
\(731\) −2.82649 −0.104542
\(732\) 0 0
\(733\) 13.8964 0.513276 0.256638 0.966508i \(-0.417385\pi\)
0.256638 + 0.966508i \(0.417385\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2.26698 0.0835054
\(738\) 0 0
\(739\) 17.9092 0.658800 0.329400 0.944190i \(-0.393153\pi\)
0.329400 + 0.944190i \(0.393153\pi\)
\(740\) 0 0
\(741\) 9.26193 0.340246
\(742\) 0 0
\(743\) −9.77808 −0.358723 −0.179362 0.983783i \(-0.557403\pi\)
−0.179362 + 0.983783i \(0.557403\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0.0850814 0.00311296
\(748\) 0 0
\(749\) −7.13854 −0.260837
\(750\) 0 0
\(751\) −24.5212 −0.894791 −0.447396 0.894336i \(-0.647648\pi\)
−0.447396 + 0.894336i \(0.647648\pi\)
\(752\) 0 0
\(753\) 26.5689 0.968226
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −43.9822 −1.59856 −0.799280 0.600959i \(-0.794785\pi\)
−0.799280 + 0.600959i \(0.794785\pi\)
\(758\) 0 0
\(759\) 0.808672 0.0293529
\(760\) 0 0
\(761\) 24.2313 0.878386 0.439193 0.898393i \(-0.355265\pi\)
0.439193 + 0.898393i \(0.355265\pi\)
\(762\) 0 0
\(763\) −11.5041 −0.416475
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −79.9109 −2.88542
\(768\) 0 0
\(769\) −23.5282 −0.848450 −0.424225 0.905557i \(-0.639453\pi\)
−0.424225 + 0.905557i \(0.639453\pi\)
\(770\) 0 0
\(771\) 4.55951 0.164207
\(772\) 0 0
\(773\) −2.74579 −0.0987591 −0.0493795 0.998780i \(-0.515724\pi\)
−0.0493795 + 0.998780i \(0.515724\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −19.0400 −0.683057
\(778\) 0 0
\(779\) −1.86651 −0.0668746
\(780\) 0 0
\(781\) −10.0400 −0.359260
\(782\) 0 0
\(783\) 8.98485 0.321092
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 31.4482 1.12101 0.560504 0.828152i \(-0.310608\pi\)
0.560504 + 0.828152i \(0.310608\pi\)
\(788\) 0 0
\(789\) −34.8181 −1.23956
\(790\) 0 0
\(791\) −15.2441 −0.542018
\(792\) 0 0
\(793\) −18.9516 −0.672991
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −12.1685 −0.431029 −0.215514 0.976501i \(-0.569143\pi\)
−0.215514 + 0.976501i \(0.569143\pi\)
\(798\) 0 0
\(799\) −3.88603 −0.137478
\(800\) 0 0
\(801\) −2.73472 −0.0966266
\(802\) 0 0
\(803\) 13.3954 0.472714
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 38.9771 1.37206
\(808\) 0 0
\(809\) 49.1480 1.72795 0.863975 0.503534i \(-0.167967\pi\)
0.863975 + 0.503534i \(0.167967\pi\)
\(810\) 0 0
\(811\) −29.7253 −1.04380 −0.521898 0.853008i \(-0.674776\pi\)
−0.521898 + 0.853008i \(0.674776\pi\)
\(812\) 0 0
\(813\) 14.0373 0.492311
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0.866508 0.0303153
\(818\) 0 0
\(819\) 5.59952 0.195663
\(820\) 0 0
\(821\) 49.3648 1.72284 0.861422 0.507889i \(-0.169574\pi\)
0.861422 + 0.507889i \(0.169574\pi\)
\(822\) 0 0
\(823\) −37.5817 −1.31002 −0.655008 0.755622i \(-0.727335\pi\)
−0.655008 + 0.755622i \(0.727335\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −35.3749 −1.23011 −0.615054 0.788485i \(-0.710866\pi\)
−0.615054 + 0.788485i \(0.710866\pi\)
\(828\) 0 0
\(829\) 44.1786 1.53438 0.767192 0.641417i \(-0.221653\pi\)
0.767192 + 0.641417i \(0.221653\pi\)
\(830\) 0 0
\(831\) −36.8436 −1.27809
\(832\) 0 0
\(833\) −16.4819 −0.571062
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −21.2270 −0.733711
\(838\) 0 0
\(839\) −47.7424 −1.64825 −0.824126 0.566406i \(-0.808333\pi\)
−0.824126 + 0.566406i \(0.808333\pi\)
\(840\) 0 0
\(841\) −26.4253 −0.911219
\(842\) 0 0
\(843\) −41.9116 −1.44351
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −1.39543 −0.0479474
\(848\) 0 0
\(849\) −26.9771 −0.925853
\(850\) 0 0
\(851\) 4.72025 0.161808
\(852\) 0 0
\(853\) −39.7502 −1.36102 −0.680510 0.732739i \(-0.738242\pi\)
−0.680510 + 0.732739i \(0.738242\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 12.1080 0.413600 0.206800 0.978383i \(-0.433695\pi\)
0.206800 + 0.978383i \(0.433695\pi\)
\(858\) 0 0
\(859\) 55.4049 1.89039 0.945195 0.326508i \(-0.105872\pi\)
0.945195 + 0.326508i \(0.105872\pi\)
\(860\) 0 0
\(861\) 3.98218 0.135712
\(862\) 0 0
\(863\) −10.4660 −0.356268 −0.178134 0.984006i \(-0.557006\pi\)
−0.178134 + 0.984006i \(0.557006\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 9.72359 0.330230
\(868\) 0 0
\(869\) 2.58675 0.0877496
\(870\) 0 0
\(871\) 13.7330 0.465326
\(872\) 0 0
\(873\) 7.57495 0.256373
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 21.4959 0.725867 0.362933 0.931815i \(-0.381775\pi\)
0.362933 + 0.931815i \(0.381775\pi\)
\(878\) 0 0
\(879\) 38.4728 1.29765
\(880\) 0 0
\(881\) −38.3521 −1.29211 −0.646057 0.763289i \(-0.723583\pi\)
−0.646057 + 0.763289i \(0.723583\pi\)
\(882\) 0 0
\(883\) −25.8665 −0.870477 −0.435239 0.900315i \(-0.643336\pi\)
−0.435239 + 0.900315i \(0.643336\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 7.19133 0.241461 0.120731 0.992685i \(-0.461476\pi\)
0.120731 + 0.992685i \(0.461476\pi\)
\(888\) 0 0
\(889\) −11.8208 −0.396456
\(890\) 0 0
\(891\) −6.57398 −0.220237
\(892\) 0 0
\(893\) 1.19133 0.0398663
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 4.89880 0.163566
\(898\) 0 0
\(899\) −6.08270 −0.202869
\(900\) 0 0
\(901\) 34.9687 1.16498
\(902\) 0 0
\(903\) −1.84869 −0.0615205
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 14.0356 0.466046 0.233023 0.972471i \(-0.425138\pi\)
0.233023 + 0.972471i \(0.425138\pi\)
\(908\) 0 0
\(909\) −9.44725 −0.313345
\(910\) 0 0
\(911\) −18.9166 −0.626736 −0.313368 0.949632i \(-0.601457\pi\)
−0.313368 + 0.949632i \(0.601457\pi\)
\(912\) 0 0
\(913\) −0.128442 −0.00425082
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −15.9573 −0.526957
\(918\) 0 0
\(919\) −32.7451 −1.08016 −0.540081 0.841613i \(-0.681606\pi\)
−0.540081 + 0.841613i \(0.681606\pi\)
\(920\) 0 0
\(921\) −45.2697 −1.49169
\(922\) 0 0
\(923\) −60.8208 −2.00194
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −12.1617 −0.399443
\(928\) 0 0
\(929\) −9.37256 −0.307504 −0.153752 0.988110i \(-0.549136\pi\)
−0.153752 + 0.988110i \(0.549136\pi\)
\(930\) 0 0
\(931\) 5.05279 0.165598
\(932\) 0 0
\(933\) 16.4099 0.537236
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 49.6617 1.62238 0.811189 0.584784i \(-0.198821\pi\)
0.811189 + 0.584784i \(0.198821\pi\)
\(938\) 0 0
\(939\) −50.8964 −1.66094
\(940\) 0 0
\(941\) 18.0178 0.587364 0.293682 0.955903i \(-0.405119\pi\)
0.293682 + 0.955903i \(0.405119\pi\)
\(942\) 0 0
\(943\) −0.987230 −0.0321486
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −29.6096 −0.962183 −0.481092 0.876670i \(-0.659760\pi\)
−0.481092 + 0.876670i \(0.659760\pi\)
\(948\) 0 0
\(949\) 81.1473 2.63415
\(950\) 0 0
\(951\) 28.3631 0.919738
\(952\) 0 0
\(953\) 0.649640 0.0210439 0.0105219 0.999945i \(-0.496651\pi\)
0.0105219 + 0.999945i \(0.496651\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −2.45326 −0.0793027
\(958\) 0 0
\(959\) −9.29690 −0.300213
\(960\) 0 0
\(961\) −16.6294 −0.536434
\(962\) 0 0
\(963\) −3.38867 −0.109198
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 30.7381 0.988470 0.494235 0.869328i \(-0.335448\pi\)
0.494235 + 0.869328i \(0.335448\pi\)
\(968\) 0 0
\(969\) 4.98723 0.160213
\(970\) 0 0
\(971\) −24.4634 −0.785067 −0.392533 0.919738i \(-0.628401\pi\)
−0.392533 + 0.919738i \(0.628401\pi\)
\(972\) 0 0
\(973\) 17.9294 0.574790
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −2.14359 −0.0685796 −0.0342898 0.999412i \(-0.510917\pi\)
−0.0342898 + 0.999412i \(0.510917\pi\)
\(978\) 0 0
\(979\) 4.12844 0.131946
\(980\) 0 0
\(981\) −5.46098 −0.174356
\(982\) 0 0
\(983\) −6.56388 −0.209355 −0.104678 0.994506i \(-0.533381\pi\)
−0.104678 + 0.994506i \(0.533381\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −2.54169 −0.0809028
\(988\) 0 0
\(989\) 0.458312 0.0145735
\(990\) 0 0
\(991\) −27.6345 −0.877839 −0.438919 0.898527i \(-0.644639\pi\)
−0.438919 + 0.898527i \(0.644639\pi\)
\(992\) 0 0
\(993\) −8.96941 −0.284636
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 19.2008 0.608094 0.304047 0.952657i \(-0.401662\pi\)
0.304047 + 0.952657i \(0.401662\pi\)
\(998\) 0 0
\(999\) −49.9721 −1.58105
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.ca.1.1 3
4.3 odd 2 2200.2.a.t.1.3 3
5.2 odd 4 4400.2.b.bc.4049.5 6
5.3 odd 4 4400.2.b.bc.4049.2 6
5.4 even 2 4400.2.a.bx.1.3 3
20.3 even 4 2200.2.b.l.1849.5 6
20.7 even 4 2200.2.b.l.1849.2 6
20.19 odd 2 2200.2.a.w.1.1 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2200.2.a.t.1.3 3 4.3 odd 2
2200.2.a.w.1.1 yes 3 20.19 odd 2
2200.2.b.l.1849.2 6 20.7 even 4
2200.2.b.l.1849.5 6 20.3 even 4
4400.2.a.bx.1.3 3 5.4 even 2
4400.2.a.ca.1.1 3 1.1 even 1 trivial
4400.2.b.bc.4049.2 6 5.3 odd 4
4400.2.b.bc.4049.5 6 5.2 odd 4