Properties

Label 3696.2.a.bp.1.2
Level $3696$
Weight $2$
Character 3696.1
Self dual yes
Analytic conductor $29.513$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3696,2,Mod(1,3696)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3696, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3696.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3696 = 2^{4} \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3696.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: \(29.5127085871\)
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: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.52892\) of defining polynomial
Character \(\chi\) \(=\) 3696.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +0.133492 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +0.133492 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} +0.133492 q^{13} +0.133492 q^{15} -5.05784 q^{17} +0.924344 q^{19} +1.00000 q^{21} +7.05784 q^{23} -4.98218 q^{25} +1.00000 q^{27} +3.86651 q^{29} -2.79085 q^{31} -1.00000 q^{33} +0.133492 q^{35} +9.98218 q^{37} +0.133492 q^{39} +11.8487 q^{41} +3.05784 q^{43} +0.133492 q^{45} +3.07566 q^{47} +1.00000 q^{49} -5.05784 q^{51} -4.79085 q^{53} -0.133492 q^{55} +0.924344 q^{57} +12.6574 q^{59} +6.00000 q^{61} +1.00000 q^{63} +0.0178201 q^{65} -8.92434 q^{67} +7.05784 q^{69} +6.11567 q^{71} +7.86651 q^{73} -4.98218 q^{75} -1.00000 q^{77} -14.1157 q^{79} +1.00000 q^{81} +1.20915 q^{83} -0.675180 q^{85} +3.86651 q^{87} +15.5817 q^{89} +0.133492 q^{91} -2.79085 q^{93} +0.123392 q^{95} +12.7909 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9} - 3 q^{11} - 12 q^{19} + 3 q^{21} + 6 q^{23} + 15 q^{25} + 3 q^{27} + 12 q^{29} + 6 q^{31} - 3 q^{33} + 6 q^{41} - 6 q^{43} + 24 q^{47} + 3 q^{49} - 12 q^{57} + 24 q^{59} + 18 q^{61} + 3 q^{63} + 30 q^{65} - 12 q^{67} + 6 q^{69} - 12 q^{71} + 24 q^{73} + 15 q^{75} - 3 q^{77} - 12 q^{79} + 3 q^{81} + 18 q^{83} - 18 q^{85} + 12 q^{87} + 18 q^{89} + 6 q^{93} - 12 q^{95} + 24 q^{97} - 3 q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 0.133492 0.0596994 0.0298497 0.999554i \(-0.490497\pi\)
0.0298497 + 0.999554i \(0.490497\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 0.133492 0.0370240 0.0185120 0.999829i \(-0.494107\pi\)
0.0185120 + 0.999829i \(0.494107\pi\)
\(14\) 0 0
\(15\) 0.133492 0.0344675
\(16\) 0 0
\(17\) −5.05784 −1.22671 −0.613353 0.789809i \(-0.710180\pi\)
−0.613353 + 0.789809i \(0.710180\pi\)
\(18\) 0 0
\(19\) 0.924344 0.212059 0.106030 0.994363i \(-0.466186\pi\)
0.106030 + 0.994363i \(0.466186\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 7.05784 1.47166 0.735830 0.677166i \(-0.236792\pi\)
0.735830 + 0.677166i \(0.236792\pi\)
\(24\) 0 0
\(25\) −4.98218 −0.996436
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 3.86651 0.717993 0.358996 0.933339i \(-0.383119\pi\)
0.358996 + 0.933339i \(0.383119\pi\)
\(30\) 0 0
\(31\) −2.79085 −0.501252 −0.250626 0.968084i \(-0.580636\pi\)
−0.250626 + 0.968084i \(0.580636\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) 0.133492 0.0225643
\(36\) 0 0
\(37\) 9.98218 1.64106 0.820530 0.571603i \(-0.193678\pi\)
0.820530 + 0.571603i \(0.193678\pi\)
\(38\) 0 0
\(39\) 0.133492 0.0213758
\(40\) 0 0
\(41\) 11.8487 1.85045 0.925227 0.379414i \(-0.123874\pi\)
0.925227 + 0.379414i \(0.123874\pi\)
\(42\) 0 0
\(43\) 3.05784 0.466316 0.233158 0.972439i \(-0.425094\pi\)
0.233158 + 0.972439i \(0.425094\pi\)
\(44\) 0 0
\(45\) 0.133492 0.0198998
\(46\) 0 0
\(47\) 3.07566 0.448631 0.224315 0.974517i \(-0.427985\pi\)
0.224315 + 0.974517i \(0.427985\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −5.05784 −0.708239
\(52\) 0 0
\(53\) −4.79085 −0.658074 −0.329037 0.944317i \(-0.606724\pi\)
−0.329037 + 0.944317i \(0.606724\pi\)
\(54\) 0 0
\(55\) −0.133492 −0.0180000
\(56\) 0 0
\(57\) 0.924344 0.122432
\(58\) 0 0
\(59\) 12.6574 1.64785 0.823924 0.566700i \(-0.191780\pi\)
0.823924 + 0.566700i \(0.191780\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 0.0178201 0.00221031
\(66\) 0 0
\(67\) −8.92434 −1.09028 −0.545141 0.838344i \(-0.683524\pi\)
−0.545141 + 0.838344i \(0.683524\pi\)
\(68\) 0 0
\(69\) 7.05784 0.849664
\(70\) 0 0
\(71\) 6.11567 0.725797 0.362898 0.931829i \(-0.381787\pi\)
0.362898 + 0.931829i \(0.381787\pi\)
\(72\) 0 0
\(73\) 7.86651 0.920705 0.460353 0.887736i \(-0.347723\pi\)
0.460353 + 0.887736i \(0.347723\pi\)
\(74\) 0 0
\(75\) −4.98218 −0.575293
\(76\) 0 0
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) −14.1157 −1.58814 −0.794069 0.607828i \(-0.792041\pi\)
−0.794069 + 0.607828i \(0.792041\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 1.20915 0.132721 0.0663606 0.997796i \(-0.478861\pi\)
0.0663606 + 0.997796i \(0.478861\pi\)
\(84\) 0 0
\(85\) −0.675180 −0.0732336
\(86\) 0 0
\(87\) 3.86651 0.414533
\(88\) 0 0
\(89\) 15.5817 1.65166 0.825829 0.563921i \(-0.190708\pi\)
0.825829 + 0.563921i \(0.190708\pi\)
\(90\) 0 0
\(91\) 0.133492 0.0139938
\(92\) 0 0
\(93\) −2.79085 −0.289398
\(94\) 0 0
\(95\) 0.123392 0.0126598
\(96\) 0 0
\(97\) 12.7909 1.29871 0.649357 0.760484i \(-0.275038\pi\)
0.649357 + 0.760484i \(0.275038\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −9.59180 −0.954420 −0.477210 0.878789i \(-0.658352\pi\)
−0.477210 + 0.878789i \(0.658352\pi\)
\(102\) 0 0
\(103\) −9.84869 −0.970420 −0.485210 0.874398i \(-0.661257\pi\)
−0.485210 + 0.874398i \(0.661257\pi\)
\(104\) 0 0
\(105\) 0.133492 0.0130275
\(106\) 0 0
\(107\) −0.924344 −0.0893597 −0.0446799 0.999001i \(-0.514227\pi\)
−0.0446799 + 0.999001i \(0.514227\pi\)
\(108\) 0 0
\(109\) −8.52387 −0.816439 −0.408219 0.912884i \(-0.633850\pi\)
−0.408219 + 0.912884i \(0.633850\pi\)
\(110\) 0 0
\(111\) 9.98218 0.947467
\(112\) 0 0
\(113\) −12.1157 −1.13975 −0.569873 0.821733i \(-0.693008\pi\)
−0.569873 + 0.821733i \(0.693008\pi\)
\(114\) 0 0
\(115\) 0.942164 0.0878573
\(116\) 0 0
\(117\) 0.133492 0.0123413
\(118\) 0 0
\(119\) −5.05784 −0.463651
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 11.8487 1.06836
\(124\) 0 0
\(125\) −1.33254 −0.119186
\(126\) 0 0
\(127\) 8.90652 0.790326 0.395163 0.918611i \(-0.370688\pi\)
0.395163 + 0.918611i \(0.370688\pi\)
\(128\) 0 0
\(129\) 3.05784 0.269227
\(130\) 0 0
\(131\) 15.6974 1.37149 0.685743 0.727844i \(-0.259477\pi\)
0.685743 + 0.727844i \(0.259477\pi\)
\(132\) 0 0
\(133\) 0.924344 0.0801508
\(134\) 0 0
\(135\) 0.133492 0.0114892
\(136\) 0 0
\(137\) 14.6395 1.25074 0.625370 0.780328i \(-0.284948\pi\)
0.625370 + 0.780328i \(0.284948\pi\)
\(138\) 0 0
\(139\) −18.6496 −1.58184 −0.790921 0.611918i \(-0.790398\pi\)
−0.790921 + 0.611918i \(0.790398\pi\)
\(140\) 0 0
\(141\) 3.07566 0.259017
\(142\) 0 0
\(143\) −0.133492 −0.0111632
\(144\) 0 0
\(145\) 0.516148 0.0428637
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 11.8665 0.972142 0.486071 0.873919i \(-0.338430\pi\)
0.486071 + 0.873919i \(0.338430\pi\)
\(150\) 0 0
\(151\) 11.3248 0.921601 0.460800 0.887504i \(-0.347562\pi\)
0.460800 + 0.887504i \(0.347562\pi\)
\(152\) 0 0
\(153\) −5.05784 −0.408902
\(154\) 0 0
\(155\) −0.372556 −0.0299244
\(156\) 0 0
\(157\) −20.3827 −1.62671 −0.813357 0.581766i \(-0.802362\pi\)
−0.813357 + 0.581766i \(0.802362\pi\)
\(158\) 0 0
\(159\) −4.79085 −0.379939
\(160\) 0 0
\(161\) 7.05784 0.556235
\(162\) 0 0
\(163\) 19.3070 1.51224 0.756120 0.654432i \(-0.227092\pi\)
0.756120 + 0.654432i \(0.227092\pi\)
\(164\) 0 0
\(165\) −0.133492 −0.0103923
\(166\) 0 0
\(167\) −12.6395 −0.978077 −0.489038 0.872262i \(-0.662652\pi\)
−0.489038 + 0.872262i \(0.662652\pi\)
\(168\) 0 0
\(169\) −12.9822 −0.998629
\(170\) 0 0
\(171\) 0.924344 0.0706864
\(172\) 0 0
\(173\) −19.8487 −1.50907 −0.754534 0.656261i \(-0.772137\pi\)
−0.754534 + 0.656261i \(0.772137\pi\)
\(174\) 0 0
\(175\) −4.98218 −0.376617
\(176\) 0 0
\(177\) 12.6574 0.951385
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 2.67518 0.198845 0.0994223 0.995045i \(-0.468301\pi\)
0.0994223 + 0.995045i \(0.468301\pi\)
\(182\) 0 0
\(183\) 6.00000 0.443533
\(184\) 0 0
\(185\) 1.33254 0.0979703
\(186\) 0 0
\(187\) 5.05784 0.369866
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −3.32482 −0.240576 −0.120288 0.992739i \(-0.538382\pi\)
−0.120288 + 0.992739i \(0.538382\pi\)
\(192\) 0 0
\(193\) 18.3726 1.32249 0.661243 0.750172i \(-0.270029\pi\)
0.661243 + 0.750172i \(0.270029\pi\)
\(194\) 0 0
\(195\) 0.0178201 0.00127612
\(196\) 0 0
\(197\) −8.11567 −0.578218 −0.289109 0.957296i \(-0.593359\pi\)
−0.289109 + 0.957296i \(0.593359\pi\)
\(198\) 0 0
\(199\) 6.52387 0.462465 0.231232 0.972899i \(-0.425724\pi\)
0.231232 + 0.972899i \(0.425724\pi\)
\(200\) 0 0
\(201\) −8.92434 −0.629475
\(202\) 0 0
\(203\) 3.86651 0.271376
\(204\) 0 0
\(205\) 1.58170 0.110471
\(206\) 0 0
\(207\) 7.05784 0.490554
\(208\) 0 0
\(209\) −0.924344 −0.0639382
\(210\) 0 0
\(211\) −13.8487 −0.953383 −0.476691 0.879071i \(-0.658164\pi\)
−0.476691 + 0.879071i \(0.658164\pi\)
\(212\) 0 0
\(213\) 6.11567 0.419039
\(214\) 0 0
\(215\) 0.408196 0.0278388
\(216\) 0 0
\(217\) −2.79085 −0.189455
\(218\) 0 0
\(219\) 7.86651 0.531569
\(220\) 0 0
\(221\) −0.675180 −0.0454175
\(222\) 0 0
\(223\) 24.4983 1.64053 0.820265 0.571984i \(-0.193826\pi\)
0.820265 + 0.571984i \(0.193826\pi\)
\(224\) 0 0
\(225\) −4.98218 −0.332145
\(226\) 0 0
\(227\) 7.73302 0.513258 0.256629 0.966510i \(-0.417388\pi\)
0.256629 + 0.966510i \(0.417388\pi\)
\(228\) 0 0
\(229\) −4.79085 −0.316588 −0.158294 0.987392i \(-0.550599\pi\)
−0.158294 + 0.987392i \(0.550599\pi\)
\(230\) 0 0
\(231\) −1.00000 −0.0657952
\(232\) 0 0
\(233\) −9.69738 −0.635296 −0.317648 0.948209i \(-0.602893\pi\)
−0.317648 + 0.948209i \(0.602893\pi\)
\(234\) 0 0
\(235\) 0.410575 0.0267830
\(236\) 0 0
\(237\) −14.1157 −0.916911
\(238\) 0 0
\(239\) −23.3070 −1.50760 −0.753802 0.657101i \(-0.771782\pi\)
−0.753802 + 0.657101i \(0.771782\pi\)
\(240\) 0 0
\(241\) 9.71520 0.625811 0.312905 0.949784i \(-0.398698\pi\)
0.312905 + 0.949784i \(0.398698\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0.133492 0.00852849
\(246\) 0 0
\(247\) 0.123392 0.00785127
\(248\) 0 0
\(249\) 1.20915 0.0766266
\(250\) 0 0
\(251\) 15.0400 0.949317 0.474659 0.880170i \(-0.342572\pi\)
0.474659 + 0.880170i \(0.342572\pi\)
\(252\) 0 0
\(253\) −7.05784 −0.443722
\(254\) 0 0
\(255\) −0.675180 −0.0422814
\(256\) 0 0
\(257\) 15.8309 0.987502 0.493751 0.869603i \(-0.335625\pi\)
0.493751 + 0.869603i \(0.335625\pi\)
\(258\) 0 0
\(259\) 9.98218 0.620262
\(260\) 0 0
\(261\) 3.86651 0.239331
\(262\) 0 0
\(263\) 3.07566 0.189653 0.0948265 0.995494i \(-0.469770\pi\)
0.0948265 + 0.995494i \(0.469770\pi\)
\(264\) 0 0
\(265\) −0.639540 −0.0392866
\(266\) 0 0
\(267\) 15.5817 0.953585
\(268\) 0 0
\(269\) −10.5340 −0.642267 −0.321134 0.947034i \(-0.604064\pi\)
−0.321134 + 0.947034i \(0.604064\pi\)
\(270\) 0 0
\(271\) 4.92434 0.299133 0.149566 0.988752i \(-0.452212\pi\)
0.149566 + 0.988752i \(0.452212\pi\)
\(272\) 0 0
\(273\) 0.133492 0.00807930
\(274\) 0 0
\(275\) 4.98218 0.300437
\(276\) 0 0
\(277\) −0.151312 −0.00909146 −0.00454573 0.999990i \(-0.501447\pi\)
−0.00454573 + 0.999990i \(0.501447\pi\)
\(278\) 0 0
\(279\) −2.79085 −0.167084
\(280\) 0 0
\(281\) 6.51615 0.388721 0.194360 0.980930i \(-0.437737\pi\)
0.194360 + 0.980930i \(0.437737\pi\)
\(282\) 0 0
\(283\) 0.390376 0.0232055 0.0116027 0.999933i \(-0.496307\pi\)
0.0116027 + 0.999933i \(0.496307\pi\)
\(284\) 0 0
\(285\) 0.123392 0.00730914
\(286\) 0 0
\(287\) 11.8487 0.699406
\(288\) 0 0
\(289\) 8.58170 0.504806
\(290\) 0 0
\(291\) 12.7909 0.749813
\(292\) 0 0
\(293\) 29.4304 1.71934 0.859671 0.510848i \(-0.170669\pi\)
0.859671 + 0.510848i \(0.170669\pi\)
\(294\) 0 0
\(295\) 1.68966 0.0983755
\(296\) 0 0
\(297\) −1.00000 −0.0580259
\(298\) 0 0
\(299\) 0.942164 0.0544868
\(300\) 0 0
\(301\) 3.05784 0.176251
\(302\) 0 0
\(303\) −9.59180 −0.551035
\(304\) 0 0
\(305\) 0.800952 0.0458624
\(306\) 0 0
\(307\) −23.6974 −1.35248 −0.676240 0.736681i \(-0.736392\pi\)
−0.676240 + 0.736681i \(0.736392\pi\)
\(308\) 0 0
\(309\) −9.84869 −0.560272
\(310\) 0 0
\(311\) 12.2313 0.693576 0.346788 0.937944i \(-0.387272\pi\)
0.346788 + 0.937944i \(0.387272\pi\)
\(312\) 0 0
\(313\) −19.1735 −1.08375 −0.541875 0.840459i \(-0.682286\pi\)
−0.541875 + 0.840459i \(0.682286\pi\)
\(314\) 0 0
\(315\) 0.133492 0.00752142
\(316\) 0 0
\(317\) 29.5562 1.66004 0.830020 0.557734i \(-0.188329\pi\)
0.830020 + 0.557734i \(0.188329\pi\)
\(318\) 0 0
\(319\) −3.86651 −0.216483
\(320\) 0 0
\(321\) −0.924344 −0.0515919
\(322\) 0 0
\(323\) −4.67518 −0.260134
\(324\) 0 0
\(325\) −0.665081 −0.0368920
\(326\) 0 0
\(327\) −8.52387 −0.471371
\(328\) 0 0
\(329\) 3.07566 0.169566
\(330\) 0 0
\(331\) −18.6496 −1.02508 −0.512538 0.858664i \(-0.671295\pi\)
−0.512538 + 0.858664i \(0.671295\pi\)
\(332\) 0 0
\(333\) 9.98218 0.547020
\(334\) 0 0
\(335\) −1.19133 −0.0650892
\(336\) 0 0
\(337\) −21.0222 −1.14515 −0.572576 0.819852i \(-0.694056\pi\)
−0.572576 + 0.819852i \(0.694056\pi\)
\(338\) 0 0
\(339\) −12.1157 −0.658033
\(340\) 0 0
\(341\) 2.79085 0.151133
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0.942164 0.0507244
\(346\) 0 0
\(347\) −23.1634 −1.24348 −0.621738 0.783225i \(-0.713573\pi\)
−0.621738 + 0.783225i \(0.713573\pi\)
\(348\) 0 0
\(349\) −28.0979 −1.50404 −0.752022 0.659138i \(-0.770921\pi\)
−0.752022 + 0.659138i \(0.770921\pi\)
\(350\) 0 0
\(351\) 0.133492 0.00712527
\(352\) 0 0
\(353\) −33.4126 −1.77837 −0.889186 0.457546i \(-0.848728\pi\)
−0.889186 + 0.457546i \(0.848728\pi\)
\(354\) 0 0
\(355\) 0.816393 0.0433296
\(356\) 0 0
\(357\) −5.05784 −0.267689
\(358\) 0 0
\(359\) −13.5817 −0.716815 −0.358407 0.933565i \(-0.616680\pi\)
−0.358407 + 0.933565i \(0.616680\pi\)
\(360\) 0 0
\(361\) −18.1456 −0.955031
\(362\) 0 0
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) 1.05012 0.0549655
\(366\) 0 0
\(367\) 3.73302 0.194862 0.0974309 0.995242i \(-0.468937\pi\)
0.0974309 + 0.995242i \(0.468937\pi\)
\(368\) 0 0
\(369\) 11.8487 0.616818
\(370\) 0 0
\(371\) −4.79085 −0.248729
\(372\) 0 0
\(373\) 6.94216 0.359452 0.179726 0.983717i \(-0.442479\pi\)
0.179726 + 0.983717i \(0.442479\pi\)
\(374\) 0 0
\(375\) −1.33254 −0.0688121
\(376\) 0 0
\(377\) 0.516148 0.0265830
\(378\) 0 0
\(379\) 0.390376 0.0200523 0.0100261 0.999950i \(-0.496809\pi\)
0.0100261 + 0.999950i \(0.496809\pi\)
\(380\) 0 0
\(381\) 8.90652 0.456295
\(382\) 0 0
\(383\) 0.533968 0.0272845 0.0136422 0.999907i \(-0.495657\pi\)
0.0136422 + 0.999907i \(0.495657\pi\)
\(384\) 0 0
\(385\) −0.133492 −0.00680338
\(386\) 0 0
\(387\) 3.05784 0.155439
\(388\) 0 0
\(389\) −10.4983 −0.532286 −0.266143 0.963934i \(-0.585749\pi\)
−0.266143 + 0.963934i \(0.585749\pi\)
\(390\) 0 0
\(391\) −35.6974 −1.80529
\(392\) 0 0
\(393\) 15.6974 0.791828
\(394\) 0 0
\(395\) −1.88433 −0.0948108
\(396\) 0 0
\(397\) −20.7552 −1.04167 −0.520837 0.853656i \(-0.674380\pi\)
−0.520837 + 0.853656i \(0.674380\pi\)
\(398\) 0 0
\(399\) 0.924344 0.0462751
\(400\) 0 0
\(401\) 21.3248 1.06491 0.532455 0.846458i \(-0.321269\pi\)
0.532455 + 0.846458i \(0.321269\pi\)
\(402\) 0 0
\(403\) −0.372556 −0.0185583
\(404\) 0 0
\(405\) 0.133492 0.00663327
\(406\) 0 0
\(407\) −9.98218 −0.494798
\(408\) 0 0
\(409\) 33.9287 1.67767 0.838834 0.544388i \(-0.183238\pi\)
0.838834 + 0.544388i \(0.183238\pi\)
\(410\) 0 0
\(411\) 14.6395 0.722115
\(412\) 0 0
\(413\) 12.6574 0.622828
\(414\) 0 0
\(415\) 0.161411 0.00792338
\(416\) 0 0
\(417\) −18.6496 −0.913277
\(418\) 0 0
\(419\) 9.99228 0.488155 0.244077 0.969756i \(-0.421515\pi\)
0.244077 + 0.969756i \(0.421515\pi\)
\(420\) 0 0
\(421\) −29.9822 −1.46124 −0.730621 0.682783i \(-0.760769\pi\)
−0.730621 + 0.682783i \(0.760769\pi\)
\(422\) 0 0
\(423\) 3.07566 0.149544
\(424\) 0 0
\(425\) 25.1990 1.22233
\(426\) 0 0
\(427\) 6.00000 0.290360
\(428\) 0 0
\(429\) −0.133492 −0.00644505
\(430\) 0 0
\(431\) 2.54169 0.122429 0.0612144 0.998125i \(-0.480503\pi\)
0.0612144 + 0.998125i \(0.480503\pi\)
\(432\) 0 0
\(433\) 20.3827 0.979528 0.489764 0.871855i \(-0.337083\pi\)
0.489764 + 0.871855i \(0.337083\pi\)
\(434\) 0 0
\(435\) 0.516148 0.0247474
\(436\) 0 0
\(437\) 6.52387 0.312079
\(438\) 0 0
\(439\) 5.45831 0.260511 0.130256 0.991480i \(-0.458420\pi\)
0.130256 + 0.991480i \(0.458420\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 10.6496 0.505980 0.252990 0.967469i \(-0.418586\pi\)
0.252990 + 0.967469i \(0.418586\pi\)
\(444\) 0 0
\(445\) 2.08003 0.0986030
\(446\) 0 0
\(447\) 11.8665 0.561267
\(448\) 0 0
\(449\) 27.0323 1.27573 0.637866 0.770147i \(-0.279817\pi\)
0.637866 + 0.770147i \(0.279817\pi\)
\(450\) 0 0
\(451\) −11.8487 −0.557933
\(452\) 0 0
\(453\) 11.3248 0.532086
\(454\) 0 0
\(455\) 0.0178201 0.000835419 0
\(456\) 0 0
\(457\) 4.79085 0.224107 0.112053 0.993702i \(-0.464257\pi\)
0.112053 + 0.993702i \(0.464257\pi\)
\(458\) 0 0
\(459\) −5.05784 −0.236080
\(460\) 0 0
\(461\) 22.4983 1.04785 0.523926 0.851764i \(-0.324467\pi\)
0.523926 + 0.851764i \(0.324467\pi\)
\(462\) 0 0
\(463\) −25.6897 −1.19390 −0.596950 0.802279i \(-0.703621\pi\)
−0.596950 + 0.802279i \(0.703621\pi\)
\(464\) 0 0
\(465\) −0.372556 −0.0172769
\(466\) 0 0
\(467\) −4.62172 −0.213868 −0.106934 0.994266i \(-0.534103\pi\)
−0.106934 + 0.994266i \(0.534103\pi\)
\(468\) 0 0
\(469\) −8.92434 −0.412088
\(470\) 0 0
\(471\) −20.3827 −0.939183
\(472\) 0 0
\(473\) −3.05784 −0.140599
\(474\) 0 0
\(475\) −4.60525 −0.211303
\(476\) 0 0
\(477\) −4.79085 −0.219358
\(478\) 0 0
\(479\) 0.372556 0.0170225 0.00851126 0.999964i \(-0.497291\pi\)
0.00851126 + 0.999964i \(0.497291\pi\)
\(480\) 0 0
\(481\) 1.33254 0.0607586
\(482\) 0 0
\(483\) 7.05784 0.321143
\(484\) 0 0
\(485\) 1.70748 0.0775325
\(486\) 0 0
\(487\) −4.30262 −0.194971 −0.0974853 0.995237i \(-0.531080\pi\)
−0.0974853 + 0.995237i \(0.531080\pi\)
\(488\) 0 0
\(489\) 19.3070 0.873093
\(490\) 0 0
\(491\) −25.4583 −1.14892 −0.574459 0.818534i \(-0.694787\pi\)
−0.574459 + 0.818534i \(0.694787\pi\)
\(492\) 0 0
\(493\) −19.5562 −0.880765
\(494\) 0 0
\(495\) −0.133492 −0.00600002
\(496\) 0 0
\(497\) 6.11567 0.274325
\(498\) 0 0
\(499\) 8.39038 0.375605 0.187802 0.982207i \(-0.439864\pi\)
0.187802 + 0.982207i \(0.439864\pi\)
\(500\) 0 0
\(501\) −12.6395 −0.564693
\(502\) 0 0
\(503\) 26.7552 1.19296 0.596478 0.802629i \(-0.296566\pi\)
0.596478 + 0.802629i \(0.296566\pi\)
\(504\) 0 0
\(505\) −1.28043 −0.0569783
\(506\) 0 0
\(507\) −12.9822 −0.576559
\(508\) 0 0
\(509\) −18.0000 −0.797836 −0.398918 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 0 0
\(511\) 7.86651 0.347994
\(512\) 0 0
\(513\) 0.924344 0.0408108
\(514\) 0 0
\(515\) −1.31472 −0.0579335
\(516\) 0 0
\(517\) −3.07566 −0.135267
\(518\) 0 0
\(519\) −19.8487 −0.871261
\(520\) 0 0
\(521\) −1.44821 −0.0634473 −0.0317237 0.999497i \(-0.510100\pi\)
−0.0317237 + 0.999497i \(0.510100\pi\)
\(522\) 0 0
\(523\) −23.5740 −1.03082 −0.515409 0.856944i \(-0.672360\pi\)
−0.515409 + 0.856944i \(0.672360\pi\)
\(524\) 0 0
\(525\) −4.98218 −0.217440
\(526\) 0 0
\(527\) 14.1157 0.614888
\(528\) 0 0
\(529\) 26.8130 1.16578
\(530\) 0 0
\(531\) 12.6574 0.549283
\(532\) 0 0
\(533\) 1.58170 0.0685112
\(534\) 0 0
\(535\) −0.123392 −0.00533472
\(536\) 0 0
\(537\) 12.0000 0.517838
\(538\) 0 0
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −18.4983 −0.795305 −0.397653 0.917536i \(-0.630175\pi\)
−0.397653 + 0.917536i \(0.630175\pi\)
\(542\) 0 0
\(543\) 2.67518 0.114803
\(544\) 0 0
\(545\) −1.13787 −0.0487409
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 0 0
\(549\) 6.00000 0.256074
\(550\) 0 0
\(551\) 3.57398 0.152257
\(552\) 0 0
\(553\) −14.1157 −0.600259
\(554\) 0 0
\(555\) 1.33254 0.0565632
\(556\) 0 0
\(557\) 1.98218 0.0839877 0.0419938 0.999118i \(-0.486629\pi\)
0.0419938 + 0.999118i \(0.486629\pi\)
\(558\) 0 0
\(559\) 0.408196 0.0172649
\(560\) 0 0
\(561\) 5.05784 0.213542
\(562\) 0 0
\(563\) −1.98990 −0.0838643 −0.0419322 0.999120i \(-0.513351\pi\)
−0.0419322 + 0.999120i \(0.513351\pi\)
\(564\) 0 0
\(565\) −1.61734 −0.0680422
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −14.5340 −0.609296 −0.304648 0.952465i \(-0.598539\pi\)
−0.304648 + 0.952465i \(0.598539\pi\)
\(570\) 0 0
\(571\) 29.2791 1.22529 0.612646 0.790358i \(-0.290105\pi\)
0.612646 + 0.790358i \(0.290105\pi\)
\(572\) 0 0
\(573\) −3.32482 −0.138896
\(574\) 0 0
\(575\) −35.1634 −1.46642
\(576\) 0 0
\(577\) −5.59180 −0.232790 −0.116395 0.993203i \(-0.537134\pi\)
−0.116395 + 0.993203i \(0.537134\pi\)
\(578\) 0 0
\(579\) 18.3726 0.763537
\(580\) 0 0
\(581\) 1.20915 0.0501639
\(582\) 0 0
\(583\) 4.79085 0.198417
\(584\) 0 0
\(585\) 0.0178201 0.000736770 0
\(586\) 0 0
\(587\) 2.80867 0.115926 0.0579632 0.998319i \(-0.481539\pi\)
0.0579632 + 0.998319i \(0.481539\pi\)
\(588\) 0 0
\(589\) −2.57971 −0.106295
\(590\) 0 0
\(591\) −8.11567 −0.333834
\(592\) 0 0
\(593\) −42.1957 −1.73277 −0.866385 0.499377i \(-0.833562\pi\)
−0.866385 + 0.499377i \(0.833562\pi\)
\(594\) 0 0
\(595\) −0.675180 −0.0276797
\(596\) 0 0
\(597\) 6.52387 0.267004
\(598\) 0 0
\(599\) −34.3470 −1.40338 −0.701691 0.712482i \(-0.747571\pi\)
−0.701691 + 0.712482i \(0.747571\pi\)
\(600\) 0 0
\(601\) 5.98218 0.244018 0.122009 0.992529i \(-0.461066\pi\)
0.122009 + 0.992529i \(0.461066\pi\)
\(602\) 0 0
\(603\) −8.92434 −0.363427
\(604\) 0 0
\(605\) 0.133492 0.00542722
\(606\) 0 0
\(607\) −8.12339 −0.329718 −0.164859 0.986317i \(-0.552717\pi\)
−0.164859 + 0.986317i \(0.552717\pi\)
\(608\) 0 0
\(609\) 3.86651 0.156679
\(610\) 0 0
\(611\) 0.410575 0.0166101
\(612\) 0 0
\(613\) −16.5239 −0.667393 −0.333696 0.942681i \(-0.608296\pi\)
−0.333696 + 0.942681i \(0.608296\pi\)
\(614\) 0 0
\(615\) 1.58170 0.0637805
\(616\) 0 0
\(617\) 11.8843 0.478445 0.239223 0.970965i \(-0.423107\pi\)
0.239223 + 0.970965i \(0.423107\pi\)
\(618\) 0 0
\(619\) −43.0222 −1.72921 −0.864604 0.502454i \(-0.832431\pi\)
−0.864604 + 0.502454i \(0.832431\pi\)
\(620\) 0 0
\(621\) 7.05784 0.283221
\(622\) 0 0
\(623\) 15.5817 0.624268
\(624\) 0 0
\(625\) 24.7330 0.989321
\(626\) 0 0
\(627\) −0.924344 −0.0369147
\(628\) 0 0
\(629\) −50.4882 −2.01310
\(630\) 0 0
\(631\) −13.5817 −0.540679 −0.270340 0.962765i \(-0.587136\pi\)
−0.270340 + 0.962765i \(0.587136\pi\)
\(632\) 0 0
\(633\) −13.8487 −0.550436
\(634\) 0 0
\(635\) 1.18895 0.0471820
\(636\) 0 0
\(637\) 0.133492 0.00528914
\(638\) 0 0
\(639\) 6.11567 0.241932
\(640\) 0 0
\(641\) 36.7552 1.45174 0.725872 0.687830i \(-0.241437\pi\)
0.725872 + 0.687830i \(0.241437\pi\)
\(642\) 0 0
\(643\) 44.3369 1.74848 0.874239 0.485496i \(-0.161361\pi\)
0.874239 + 0.485496i \(0.161361\pi\)
\(644\) 0 0
\(645\) 0.408196 0.0160727
\(646\) 0 0
\(647\) 17.9721 0.706555 0.353278 0.935519i \(-0.385067\pi\)
0.353278 + 0.935519i \(0.385067\pi\)
\(648\) 0 0
\(649\) −12.6574 −0.496845
\(650\) 0 0
\(651\) −2.79085 −0.109382
\(652\) 0 0
\(653\) −22.1957 −0.868585 −0.434292 0.900772i \(-0.643002\pi\)
−0.434292 + 0.900772i \(0.643002\pi\)
\(654\) 0 0
\(655\) 2.09547 0.0818769
\(656\) 0 0
\(657\) 7.86651 0.306902
\(658\) 0 0
\(659\) −9.99228 −0.389244 −0.194622 0.980878i \(-0.562348\pi\)
−0.194622 + 0.980878i \(0.562348\pi\)
\(660\) 0 0
\(661\) 27.9543 1.08729 0.543647 0.839314i \(-0.317043\pi\)
0.543647 + 0.839314i \(0.317043\pi\)
\(662\) 0 0
\(663\) −0.675180 −0.0262218
\(664\) 0 0
\(665\) 0.123392 0.00478495
\(666\) 0 0
\(667\) 27.2892 1.05664
\(668\) 0 0
\(669\) 24.4983 0.947160
\(670\) 0 0
\(671\) −6.00000 −0.231627
\(672\) 0 0
\(673\) −43.4203 −1.67373 −0.836865 0.547410i \(-0.815614\pi\)
−0.836865 + 0.547410i \(0.815614\pi\)
\(674\) 0 0
\(675\) −4.98218 −0.191764
\(676\) 0 0
\(677\) −12.3470 −0.474534 −0.237267 0.971444i \(-0.576252\pi\)
−0.237267 + 0.971444i \(0.576252\pi\)
\(678\) 0 0
\(679\) 12.7909 0.490868
\(680\) 0 0
\(681\) 7.73302 0.296330
\(682\) 0 0
\(683\) 27.8386 1.06521 0.532607 0.846363i \(-0.321212\pi\)
0.532607 + 0.846363i \(0.321212\pi\)
\(684\) 0 0
\(685\) 1.95426 0.0746684
\(686\) 0 0
\(687\) −4.79085 −0.182782
\(688\) 0 0
\(689\) −0.639540 −0.0243645
\(690\) 0 0
\(691\) 16.8010 0.639138 0.319569 0.947563i \(-0.396462\pi\)
0.319569 + 0.947563i \(0.396462\pi\)
\(692\) 0 0
\(693\) −1.00000 −0.0379869
\(694\) 0 0
\(695\) −2.48958 −0.0944350
\(696\) 0 0
\(697\) −59.9287 −2.26996
\(698\) 0 0
\(699\) −9.69738 −0.366788
\(700\) 0 0
\(701\) 9.16341 0.346097 0.173049 0.984913i \(-0.444638\pi\)
0.173049 + 0.984913i \(0.444638\pi\)
\(702\) 0 0
\(703\) 9.22697 0.348002
\(704\) 0 0
\(705\) 0.410575 0.0154632
\(706\) 0 0
\(707\) −9.59180 −0.360737
\(708\) 0 0
\(709\) −34.7475 −1.30497 −0.652485 0.757802i \(-0.726273\pi\)
−0.652485 + 0.757802i \(0.726273\pi\)
\(710\) 0 0
\(711\) −14.1157 −0.529379
\(712\) 0 0
\(713\) −19.6974 −0.737673
\(714\) 0 0
\(715\) −0.0178201 −0.000666434 0
\(716\) 0 0
\(717\) −23.3070 −0.870416
\(718\) 0 0
\(719\) 25.1557 0.938149 0.469074 0.883159i \(-0.344588\pi\)
0.469074 + 0.883159i \(0.344588\pi\)
\(720\) 0 0
\(721\) −9.84869 −0.366784
\(722\) 0 0
\(723\) 9.71520 0.361312
\(724\) 0 0
\(725\) −19.2636 −0.715434
\(726\) 0 0
\(727\) 23.5918 0.874972 0.437486 0.899225i \(-0.355869\pi\)
0.437486 + 0.899225i \(0.355869\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −15.4660 −0.572032
\(732\) 0 0
\(733\) 21.9287 0.809956 0.404978 0.914326i \(-0.367279\pi\)
0.404978 + 0.914326i \(0.367279\pi\)
\(734\) 0 0
\(735\) 0.133492 0.00492392
\(736\) 0 0
\(737\) 8.92434 0.328732
\(738\) 0 0
\(739\) −35.0578 −1.28962 −0.644812 0.764341i \(-0.723064\pi\)
−0.644812 + 0.764341i \(0.723064\pi\)
\(740\) 0 0
\(741\) 0.123392 0.00453294
\(742\) 0 0
\(743\) 31.3070 1.14854 0.574271 0.818665i \(-0.305285\pi\)
0.574271 + 0.818665i \(0.305285\pi\)
\(744\) 0 0
\(745\) 1.58408 0.0580363
\(746\) 0 0
\(747\) 1.20915 0.0442404
\(748\) 0 0
\(749\) −0.924344 −0.0337748
\(750\) 0 0
\(751\) 4.42602 0.161508 0.0807538 0.996734i \(-0.474267\pi\)
0.0807538 + 0.996734i \(0.474267\pi\)
\(752\) 0 0
\(753\) 15.0400 0.548089
\(754\) 0 0
\(755\) 1.51177 0.0550190
\(756\) 0 0
\(757\) 2.51615 0.0914509 0.0457255 0.998954i \(-0.485440\pi\)
0.0457255 + 0.998954i \(0.485440\pi\)
\(758\) 0 0
\(759\) −7.05784 −0.256183
\(760\) 0 0
\(761\) 13.7330 0.497821 0.248911 0.968526i \(-0.419927\pi\)
0.248911 + 0.968526i \(0.419927\pi\)
\(762\) 0 0
\(763\) −8.52387 −0.308585
\(764\) 0 0
\(765\) −0.675180 −0.0244112
\(766\) 0 0
\(767\) 1.68966 0.0610099
\(768\) 0 0
\(769\) 46.4805 1.67613 0.838065 0.545570i \(-0.183687\pi\)
0.838065 + 0.545570i \(0.183687\pi\)
\(770\) 0 0
\(771\) 15.8309 0.570135
\(772\) 0 0
\(773\) −45.9822 −1.65386 −0.826932 0.562302i \(-0.809916\pi\)
−0.826932 + 0.562302i \(0.809916\pi\)
\(774\) 0 0
\(775\) 13.9045 0.499465
\(776\) 0 0
\(777\) 9.98218 0.358109
\(778\) 0 0
\(779\) 10.9523 0.392406
\(780\) 0 0
\(781\) −6.11567 −0.218836
\(782\) 0 0
\(783\) 3.86651 0.138178
\(784\) 0 0
\(785\) −2.72092 −0.0971138
\(786\) 0 0
\(787\) −40.9243 −1.45880 −0.729398 0.684090i \(-0.760200\pi\)
−0.729398 + 0.684090i \(0.760200\pi\)
\(788\) 0 0
\(789\) 3.07566 0.109496
\(790\) 0 0
\(791\) −12.1157 −0.430784
\(792\) 0 0
\(793\) 0.800952 0.0284426
\(794\) 0 0
\(795\) −0.639540 −0.0226821
\(796\) 0 0
\(797\) −28.8786 −1.02293 −0.511466 0.859303i \(-0.670898\pi\)
−0.511466 + 0.859303i \(0.670898\pi\)
\(798\) 0 0
\(799\) −15.5562 −0.550338
\(800\) 0 0
\(801\) 15.5817 0.550552
\(802\) 0 0
\(803\) −7.86651 −0.277603
\(804\) 0 0
\(805\) 0.942164 0.0332069
\(806\) 0 0
\(807\) −10.5340 −0.370813
\(808\) 0 0
\(809\) 6.51615 0.229096 0.114548 0.993418i \(-0.463458\pi\)
0.114548 + 0.993418i \(0.463458\pi\)
\(810\) 0 0
\(811\) −38.5060 −1.35213 −0.676065 0.736842i \(-0.736316\pi\)
−0.676065 + 0.736842i \(0.736316\pi\)
\(812\) 0 0
\(813\) 4.92434 0.172704
\(814\) 0 0
\(815\) 2.57733 0.0902799
\(816\) 0 0
\(817\) 2.82649 0.0988864
\(818\) 0 0
\(819\) 0.133492 0.00466459
\(820\) 0 0
\(821\) −37.3769 −1.30446 −0.652232 0.758019i \(-0.726167\pi\)
−0.652232 + 0.758019i \(0.726167\pi\)
\(822\) 0 0
\(823\) −43.0400 −1.50028 −0.750140 0.661279i \(-0.770014\pi\)
−0.750140 + 0.661279i \(0.770014\pi\)
\(824\) 0 0
\(825\) 4.98218 0.173457
\(826\) 0 0
\(827\) −11.3426 −0.394422 −0.197211 0.980361i \(-0.563188\pi\)
−0.197211 + 0.980361i \(0.563188\pi\)
\(828\) 0 0
\(829\) 14.4983 0.503548 0.251774 0.967786i \(-0.418986\pi\)
0.251774 + 0.967786i \(0.418986\pi\)
\(830\) 0 0
\(831\) −0.151312 −0.00524896
\(832\) 0 0
\(833\) −5.05784 −0.175244
\(834\) 0 0
\(835\) −1.68728 −0.0583906
\(836\) 0 0
\(837\) −2.79085 −0.0964660
\(838\) 0 0
\(839\) 3.60962 0.124618 0.0623090 0.998057i \(-0.480154\pi\)
0.0623090 + 0.998057i \(0.480154\pi\)
\(840\) 0 0
\(841\) −14.0501 −0.484487
\(842\) 0 0
\(843\) 6.51615 0.224428
\(844\) 0 0
\(845\) −1.73302 −0.0596176
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 0 0
\(849\) 0.390376 0.0133977
\(850\) 0 0
\(851\) 70.4526 2.41508
\(852\) 0 0
\(853\) −24.6496 −0.843988 −0.421994 0.906599i \(-0.638670\pi\)
−0.421994 + 0.906599i \(0.638670\pi\)
\(854\) 0 0
\(855\) 0.123392 0.00421993
\(856\) 0 0
\(857\) 17.9287 0.612433 0.306217 0.951962i \(-0.400937\pi\)
0.306217 + 0.951962i \(0.400937\pi\)
\(858\) 0 0
\(859\) 15.6974 0.535588 0.267794 0.963476i \(-0.413705\pi\)
0.267794 + 0.963476i \(0.413705\pi\)
\(860\) 0 0
\(861\) 11.8487 0.403802
\(862\) 0 0
\(863\) 15.0578 0.512575 0.256287 0.966601i \(-0.417501\pi\)
0.256287 + 0.966601i \(0.417501\pi\)
\(864\) 0 0
\(865\) −2.64964 −0.0900904
\(866\) 0 0
\(867\) 8.58170 0.291450
\(868\) 0 0
\(869\) 14.1157 0.478841
\(870\) 0 0
\(871\) −1.19133 −0.0403666
\(872\) 0 0
\(873\) 12.7909 0.432905
\(874\) 0 0
\(875\) −1.33254 −0.0450481
\(876\) 0 0
\(877\) −43.8487 −1.48066 −0.740332 0.672241i \(-0.765332\pi\)
−0.740332 + 0.672241i \(0.765332\pi\)
\(878\) 0 0
\(879\) 29.4304 0.992662
\(880\) 0 0
\(881\) 11.0656 0.372808 0.186404 0.982473i \(-0.440317\pi\)
0.186404 + 0.982473i \(0.440317\pi\)
\(882\) 0 0
\(883\) −5.72530 −0.192672 −0.0963358 0.995349i \(-0.530712\pi\)
−0.0963358 + 0.995349i \(0.530712\pi\)
\(884\) 0 0
\(885\) 1.68966 0.0567971
\(886\) 0 0
\(887\) −15.0935 −0.506789 −0.253395 0.967363i \(-0.581547\pi\)
−0.253395 + 0.967363i \(0.581547\pi\)
\(888\) 0 0
\(889\) 8.90652 0.298715
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 0 0
\(893\) 2.84296 0.0951362
\(894\) 0 0
\(895\) 1.60190 0.0535457
\(896\) 0 0
\(897\) 0.942164 0.0314579
\(898\) 0 0
\(899\) −10.7909 −0.359895
\(900\) 0 0
\(901\) 24.2313 0.807263
\(902\) 0 0
\(903\) 3.05784 0.101758
\(904\) 0 0
\(905\) 0.357115 0.0118709
\(906\) 0 0
\(907\) −41.2993 −1.37132 −0.685660 0.727922i \(-0.740486\pi\)
−0.685660 + 0.727922i \(0.740486\pi\)
\(908\) 0 0
\(909\) −9.59180 −0.318140
\(910\) 0 0
\(911\) −3.45059 −0.114323 −0.0571616 0.998365i \(-0.518205\pi\)
−0.0571616 + 0.998365i \(0.518205\pi\)
\(912\) 0 0
\(913\) −1.20915 −0.0400170
\(914\) 0 0
\(915\) 0.800952 0.0264786
\(916\) 0 0
\(917\) 15.6974 0.518373
\(918\) 0 0
\(919\) 10.8265 0.357133 0.178567 0.983928i \(-0.442854\pi\)
0.178567 + 0.983928i \(0.442854\pi\)
\(920\) 0 0
\(921\) −23.6974 −0.780855
\(922\) 0 0
\(923\) 0.816393 0.0268719
\(924\) 0 0
\(925\) −49.7330 −1.63521
\(926\) 0 0
\(927\) −9.84869 −0.323473
\(928\) 0 0
\(929\) 17.4684 0.573120 0.286560 0.958062i \(-0.407488\pi\)
0.286560 + 0.958062i \(0.407488\pi\)
\(930\) 0 0
\(931\) 0.924344 0.0302942
\(932\) 0 0
\(933\) 12.2313 0.400436
\(934\) 0 0
\(935\) 0.675180 0.0220808
\(936\) 0 0
\(937\) 10.5340 0.344130 0.172065 0.985086i \(-0.444956\pi\)
0.172065 + 0.985086i \(0.444956\pi\)
\(938\) 0 0
\(939\) −19.1735 −0.625704
\(940\) 0 0
\(941\) −16.1513 −0.526518 −0.263259 0.964725i \(-0.584797\pi\)
−0.263259 + 0.964725i \(0.584797\pi\)
\(942\) 0 0
\(943\) 83.6261 2.72324
\(944\) 0 0
\(945\) 0.133492 0.00434249
\(946\) 0 0
\(947\) 6.91662 0.224760 0.112380 0.993665i \(-0.464153\pi\)
0.112380 + 0.993665i \(0.464153\pi\)
\(948\) 0 0
\(949\) 1.05012 0.0340882
\(950\) 0 0
\(951\) 29.5562 0.958424
\(952\) 0 0
\(953\) −16.1335 −0.522615 −0.261308 0.965256i \(-0.584154\pi\)
−0.261308 + 0.965256i \(0.584154\pi\)
\(954\) 0 0
\(955\) −0.443837 −0.0143622
\(956\) 0 0
\(957\) −3.86651 −0.124986
\(958\) 0 0
\(959\) 14.6395 0.472735
\(960\) 0 0
\(961\) −23.2111 −0.748747
\(962\) 0 0
\(963\) −0.924344 −0.0297866
\(964\) 0 0
\(965\) 2.45259 0.0789516
\(966\) 0 0
\(967\) −17.8487 −0.573975 −0.286988 0.957934i \(-0.592654\pi\)
−0.286988 + 0.957934i \(0.592654\pi\)
\(968\) 0 0
\(969\) −4.67518 −0.150188
\(970\) 0 0
\(971\) −33.9566 −1.08972 −0.544860 0.838527i \(-0.683417\pi\)
−0.544860 + 0.838527i \(0.683417\pi\)
\(972\) 0 0
\(973\) −18.6496 −0.597880
\(974\) 0 0
\(975\) −0.665081 −0.0212996
\(976\) 0 0
\(977\) 41.0222 1.31242 0.656208 0.754580i \(-0.272159\pi\)
0.656208 + 0.754580i \(0.272159\pi\)
\(978\) 0 0
\(979\) −15.5817 −0.497993
\(980\) 0 0
\(981\) −8.52387 −0.272146
\(982\) 0 0
\(983\) −42.3470 −1.35066 −0.675330 0.737516i \(-0.735999\pi\)
−0.675330 + 0.737516i \(0.735999\pi\)
\(984\) 0 0
\(985\) −1.08338 −0.0345192
\(986\) 0 0
\(987\) 3.07566 0.0978992
\(988\) 0 0
\(989\) 21.5817 0.686258
\(990\) 0 0
\(991\) −50.9687 −1.61908 −0.809538 0.587068i \(-0.800282\pi\)
−0.809538 + 0.587068i \(0.800282\pi\)
\(992\) 0 0
\(993\) −18.6496 −0.591828
\(994\) 0 0
\(995\) 0.870884 0.0276089
\(996\) 0 0
\(997\) −32.8608 −1.04071 −0.520356 0.853950i \(-0.674201\pi\)
−0.520356 + 0.853950i \(0.674201\pi\)
\(998\) 0 0
\(999\) 9.98218 0.315822
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 3696.2.a.bp.1.2 3
4.3 odd 2 231.2.a.d.1.3 3
12.11 even 2 693.2.a.m.1.1 3
20.19 odd 2 5775.2.a.bw.1.1 3
28.27 even 2 1617.2.a.s.1.3 3
44.43 even 2 2541.2.a.bi.1.1 3
84.83 odd 2 4851.2.a.bp.1.1 3
132.131 odd 2 7623.2.a.cb.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.d.1.3 3 4.3 odd 2
693.2.a.m.1.1 3 12.11 even 2
1617.2.a.s.1.3 3 28.27 even 2
2541.2.a.bi.1.1 3 44.43 even 2
3696.2.a.bp.1.2 3 1.1 even 1 trivial
4851.2.a.bp.1.1 3 84.83 odd 2
5775.2.a.bw.1.1 3 20.19 odd 2
7623.2.a.cb.1.3 3 132.131 odd 2