Properties

Label 1911.2.a.x.1.7
Level $1911$
Weight $2$
Character 1911.1
Self dual yes
Analytic conductor $15.259$
Analytic rank $0$
Dimension $10$
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] = [1911,2,Mod(1,1911)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1911, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1911.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1911 = 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1911.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: \(15.2594118263\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 4x^{9} - 10x^{8} + 52x^{7} + 16x^{6} - 212x^{5} + 64x^{4} + 300x^{3} - 159x^{2} - 80x + 46 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(1.67947\) of defining polynomial
Character \(\chi\) \(=\) 1911.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.67947 q^{2} -1.00000 q^{3} +0.820610 q^{4} -1.01562 q^{5} -1.67947 q^{6} -1.98075 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.67947 q^{2} -1.00000 q^{3} +0.820610 q^{4} -1.01562 q^{5} -1.67947 q^{6} -1.98075 q^{8} +1.00000 q^{9} -1.70571 q^{10} +4.05316 q^{11} -0.820610 q^{12} +1.00000 q^{13} +1.01562 q^{15} -4.96782 q^{16} -7.26007 q^{17} +1.67947 q^{18} +2.90686 q^{19} -0.833432 q^{20} +6.80714 q^{22} +9.53118 q^{23} +1.98075 q^{24} -3.96851 q^{25} +1.67947 q^{26} -1.00000 q^{27} +6.53200 q^{29} +1.70571 q^{30} +9.86851 q^{31} -4.38180 q^{32} -4.05316 q^{33} -12.1930 q^{34} +0.820610 q^{36} +4.24217 q^{37} +4.88198 q^{38} -1.00000 q^{39} +2.01169 q^{40} -8.83058 q^{41} +0.566950 q^{43} +3.32606 q^{44} -1.01562 q^{45} +16.0073 q^{46} +12.1032 q^{47} +4.96782 q^{48} -6.66498 q^{50} +7.26007 q^{51} +0.820610 q^{52} +8.89147 q^{53} -1.67947 q^{54} -4.11648 q^{55} -2.90686 q^{57} +10.9703 q^{58} +5.44378 q^{59} +0.833432 q^{60} -5.51703 q^{61} +16.5738 q^{62} +2.57655 q^{64} -1.01562 q^{65} -6.80714 q^{66} -3.03034 q^{67} -5.95769 q^{68} -9.53118 q^{69} -6.18315 q^{71} -1.98075 q^{72} +3.99200 q^{73} +7.12458 q^{74} +3.96851 q^{75} +2.38540 q^{76} -1.67947 q^{78} +1.06343 q^{79} +5.04544 q^{80} +1.00000 q^{81} -14.8307 q^{82} -8.02803 q^{83} +7.37350 q^{85} +0.952174 q^{86} -6.53200 q^{87} -8.02827 q^{88} +6.60402 q^{89} -1.70571 q^{90} +7.82138 q^{92} -9.86851 q^{93} +20.3269 q^{94} -2.95228 q^{95} +4.38180 q^{96} +1.63178 q^{97} +4.05316 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 4 q^{2} - 10 q^{3} + 16 q^{4} + 6 q^{5} - 4 q^{6} + 12 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 4 q^{2} - 10 q^{3} + 16 q^{4} + 6 q^{5} - 4 q^{6} + 12 q^{8} + 10 q^{9} - 8 q^{10} + 12 q^{11} - 16 q^{12} + 10 q^{13} - 6 q^{15} + 24 q^{16} + 4 q^{18} - 10 q^{19} + 16 q^{20} + 8 q^{22} + 14 q^{23} - 12 q^{24} + 32 q^{25} + 4 q^{26} - 10 q^{27} + 18 q^{29} + 8 q^{30} - 14 q^{31} + 28 q^{32} - 12 q^{33} - 4 q^{34} + 16 q^{36} + 24 q^{37} + 4 q^{38} - 10 q^{39} - 16 q^{40} + 24 q^{41} + 2 q^{43} + 48 q^{44} + 6 q^{45} + 20 q^{46} + 18 q^{47} - 24 q^{48} - 28 q^{50} + 16 q^{52} + 10 q^{53} - 4 q^{54} - 12 q^{55} + 10 q^{57} + 12 q^{58} + 12 q^{59} - 16 q^{60} + 4 q^{61} - 4 q^{62} + 32 q^{64} + 6 q^{65} - 8 q^{66} - 12 q^{67} + 40 q^{68} - 14 q^{69} + 32 q^{71} + 12 q^{72} + 18 q^{73} + 24 q^{74} - 32 q^{75} - 32 q^{76} - 4 q^{78} + 34 q^{79} + 32 q^{80} + 10 q^{81} - 48 q^{82} + 30 q^{83} + 40 q^{86} - 18 q^{87} + 32 q^{88} + 10 q^{89} - 8 q^{90} - 40 q^{92} + 14 q^{93} + 24 q^{94} - 30 q^{95} - 28 q^{96} + 2 q^{97} + 12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.67947 1.18756 0.593781 0.804626i \(-0.297634\pi\)
0.593781 + 0.804626i \(0.297634\pi\)
\(3\) −1.00000 −0.577350
\(4\) 0.820610 0.410305
\(5\) −1.01562 −0.454201 −0.227100 0.973871i \(-0.572925\pi\)
−0.227100 + 0.973871i \(0.572925\pi\)
\(6\) −1.67947 −0.685640
\(7\) 0 0
\(8\) −1.98075 −0.700300
\(9\) 1.00000 0.333333
\(10\) −1.70571 −0.539392
\(11\) 4.05316 1.22207 0.611036 0.791603i \(-0.290753\pi\)
0.611036 + 0.791603i \(0.290753\pi\)
\(12\) −0.820610 −0.236890
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 1.01562 0.262233
\(16\) −4.96782 −1.24195
\(17\) −7.26007 −1.76082 −0.880412 0.474209i \(-0.842734\pi\)
−0.880412 + 0.474209i \(0.842734\pi\)
\(18\) 1.67947 0.395854
\(19\) 2.90686 0.666879 0.333440 0.942771i \(-0.391791\pi\)
0.333440 + 0.942771i \(0.391791\pi\)
\(20\) −0.833432 −0.186361
\(21\) 0 0
\(22\) 6.80714 1.45129
\(23\) 9.53118 1.98739 0.993694 0.112127i \(-0.0357664\pi\)
0.993694 + 0.112127i \(0.0357664\pi\)
\(24\) 1.98075 0.404318
\(25\) −3.96851 −0.793702
\(26\) 1.67947 0.329371
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 6.53200 1.21296 0.606481 0.795098i \(-0.292581\pi\)
0.606481 + 0.795098i \(0.292581\pi\)
\(30\) 1.70571 0.311418
\(31\) 9.86851 1.77244 0.886218 0.463268i \(-0.153323\pi\)
0.886218 + 0.463268i \(0.153323\pi\)
\(32\) −4.38180 −0.774600
\(33\) −4.05316 −0.705564
\(34\) −12.1930 −2.09109
\(35\) 0 0
\(36\) 0.820610 0.136768
\(37\) 4.24217 0.697408 0.348704 0.937233i \(-0.386622\pi\)
0.348704 + 0.937233i \(0.386622\pi\)
\(38\) 4.88198 0.791961
\(39\) −1.00000 −0.160128
\(40\) 2.01169 0.318077
\(41\) −8.83058 −1.37911 −0.689553 0.724236i \(-0.742193\pi\)
−0.689553 + 0.724236i \(0.742193\pi\)
\(42\) 0 0
\(43\) 0.566950 0.0864590 0.0432295 0.999065i \(-0.486235\pi\)
0.0432295 + 0.999065i \(0.486235\pi\)
\(44\) 3.32606 0.501423
\(45\) −1.01562 −0.151400
\(46\) 16.0073 2.36015
\(47\) 12.1032 1.76543 0.882717 0.469905i \(-0.155712\pi\)
0.882717 + 0.469905i \(0.155712\pi\)
\(48\) 4.96782 0.717043
\(49\) 0 0
\(50\) −6.66498 −0.942570
\(51\) 7.26007 1.01661
\(52\) 0.820610 0.113798
\(53\) 8.89147 1.22134 0.610669 0.791886i \(-0.290901\pi\)
0.610669 + 0.791886i \(0.290901\pi\)
\(54\) −1.67947 −0.228547
\(55\) −4.11648 −0.555066
\(56\) 0 0
\(57\) −2.90686 −0.385023
\(58\) 10.9703 1.44047
\(59\) 5.44378 0.708720 0.354360 0.935109i \(-0.384699\pi\)
0.354360 + 0.935109i \(0.384699\pi\)
\(60\) 0.833432 0.107596
\(61\) −5.51703 −0.706384 −0.353192 0.935551i \(-0.614904\pi\)
−0.353192 + 0.935551i \(0.614904\pi\)
\(62\) 16.5738 2.10488
\(63\) 0 0
\(64\) 2.57655 0.322069
\(65\) −1.01562 −0.125973
\(66\) −6.80714 −0.837901
\(67\) −3.03034 −0.370214 −0.185107 0.982718i \(-0.559263\pi\)
−0.185107 + 0.982718i \(0.559263\pi\)
\(68\) −5.95769 −0.722476
\(69\) −9.53118 −1.14742
\(70\) 0 0
\(71\) −6.18315 −0.733805 −0.366902 0.930259i \(-0.619582\pi\)
−0.366902 + 0.930259i \(0.619582\pi\)
\(72\) −1.98075 −0.233433
\(73\) 3.99200 0.467229 0.233614 0.972329i \(-0.424945\pi\)
0.233614 + 0.972329i \(0.424945\pi\)
\(74\) 7.12458 0.828216
\(75\) 3.96851 0.458244
\(76\) 2.38540 0.273624
\(77\) 0 0
\(78\) −1.67947 −0.190162
\(79\) 1.06343 0.119645 0.0598227 0.998209i \(-0.480946\pi\)
0.0598227 + 0.998209i \(0.480946\pi\)
\(80\) 5.04544 0.564097
\(81\) 1.00000 0.111111
\(82\) −14.8307 −1.63777
\(83\) −8.02803 −0.881191 −0.440596 0.897706i \(-0.645233\pi\)
−0.440596 + 0.897706i \(0.645233\pi\)
\(84\) 0 0
\(85\) 7.37350 0.799768
\(86\) 0.952174 0.102676
\(87\) −6.53200 −0.700304
\(88\) −8.02827 −0.855817
\(89\) 6.60402 0.700025 0.350013 0.936745i \(-0.386177\pi\)
0.350013 + 0.936745i \(0.386177\pi\)
\(90\) −1.70571 −0.179797
\(91\) 0 0
\(92\) 7.82138 0.815436
\(93\) −9.86851 −1.02332
\(94\) 20.3269 2.09656
\(95\) −2.95228 −0.302897
\(96\) 4.38180 0.447215
\(97\) 1.63178 0.165682 0.0828410 0.996563i \(-0.473601\pi\)
0.0828410 + 0.996563i \(0.473601\pi\)
\(98\) 0 0
\(99\) 4.05316 0.407357
\(100\) −3.25660 −0.325660
\(101\) 15.5527 1.54755 0.773774 0.633461i \(-0.218366\pi\)
0.773774 + 0.633461i \(0.218366\pi\)
\(102\) 12.1930 1.20729
\(103\) 14.5504 1.43369 0.716847 0.697230i \(-0.245584\pi\)
0.716847 + 0.697230i \(0.245584\pi\)
\(104\) −1.98075 −0.194228
\(105\) 0 0
\(106\) 14.9329 1.45041
\(107\) 18.7040 1.80818 0.904092 0.427338i \(-0.140548\pi\)
0.904092 + 0.427338i \(0.140548\pi\)
\(108\) −0.820610 −0.0789633
\(109\) −10.8252 −1.03687 −0.518433 0.855118i \(-0.673484\pi\)
−0.518433 + 0.855118i \(0.673484\pi\)
\(110\) −6.91350 −0.659176
\(111\) −4.24217 −0.402649
\(112\) 0 0
\(113\) −9.52791 −0.896311 −0.448155 0.893956i \(-0.647919\pi\)
−0.448155 + 0.893956i \(0.647919\pi\)
\(114\) −4.88198 −0.457239
\(115\) −9.68009 −0.902673
\(116\) 5.36023 0.497685
\(117\) 1.00000 0.0924500
\(118\) 9.14265 0.841649
\(119\) 0 0
\(120\) −2.01169 −0.183642
\(121\) 5.42807 0.493461
\(122\) −9.26568 −0.838875
\(123\) 8.83058 0.796227
\(124\) 8.09820 0.727240
\(125\) 9.10863 0.814701
\(126\) 0 0
\(127\) 10.8621 0.963854 0.481927 0.876211i \(-0.339937\pi\)
0.481927 + 0.876211i \(0.339937\pi\)
\(128\) 13.0908 1.15708
\(129\) −0.566950 −0.0499172
\(130\) −1.70571 −0.149600
\(131\) 3.62137 0.316401 0.158200 0.987407i \(-0.449431\pi\)
0.158200 + 0.987407i \(0.449431\pi\)
\(132\) −3.32606 −0.289497
\(133\) 0 0
\(134\) −5.08935 −0.439653
\(135\) 1.01562 0.0874110
\(136\) 14.3804 1.23310
\(137\) −14.6162 −1.24874 −0.624372 0.781127i \(-0.714645\pi\)
−0.624372 + 0.781127i \(0.714645\pi\)
\(138\) −16.0073 −1.36263
\(139\) −11.1222 −0.943370 −0.471685 0.881767i \(-0.656354\pi\)
−0.471685 + 0.881767i \(0.656354\pi\)
\(140\) 0 0
\(141\) −12.1032 −1.01927
\(142\) −10.3844 −0.871439
\(143\) 4.05316 0.338942
\(144\) −4.96782 −0.413985
\(145\) −6.63406 −0.550928
\(146\) 6.70444 0.554863
\(147\) 0 0
\(148\) 3.48117 0.286150
\(149\) 5.23259 0.428671 0.214335 0.976760i \(-0.431241\pi\)
0.214335 + 0.976760i \(0.431241\pi\)
\(150\) 6.66498 0.544193
\(151\) −15.0293 −1.22307 −0.611535 0.791217i \(-0.709448\pi\)
−0.611535 + 0.791217i \(0.709448\pi\)
\(152\) −5.75775 −0.467015
\(153\) −7.26007 −0.586942
\(154\) 0 0
\(155\) −10.0227 −0.805042
\(156\) −0.820610 −0.0657014
\(157\) 1.33422 0.106482 0.0532410 0.998582i \(-0.483045\pi\)
0.0532410 + 0.998582i \(0.483045\pi\)
\(158\) 1.78600 0.142086
\(159\) −8.89147 −0.705139
\(160\) 4.45026 0.351824
\(161\) 0 0
\(162\) 1.67947 0.131951
\(163\) −23.6878 −1.85537 −0.927686 0.373361i \(-0.878205\pi\)
−0.927686 + 0.373361i \(0.878205\pi\)
\(164\) −7.24647 −0.565854
\(165\) 4.11648 0.320468
\(166\) −13.4828 −1.04647
\(167\) −9.08825 −0.703270 −0.351635 0.936137i \(-0.614374\pi\)
−0.351635 + 0.936137i \(0.614374\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 12.3836 0.949775
\(171\) 2.90686 0.222293
\(172\) 0.465245 0.0354746
\(173\) −6.51688 −0.495469 −0.247735 0.968828i \(-0.579686\pi\)
−0.247735 + 0.968828i \(0.579686\pi\)
\(174\) −10.9703 −0.831655
\(175\) 0 0
\(176\) −20.1353 −1.51776
\(177\) −5.44378 −0.409180
\(178\) 11.0912 0.831324
\(179\) −0.541948 −0.0405071 −0.0202536 0.999795i \(-0.506447\pi\)
−0.0202536 + 0.999795i \(0.506447\pi\)
\(180\) −0.833432 −0.0621203
\(181\) 17.9887 1.33709 0.668543 0.743674i \(-0.266918\pi\)
0.668543 + 0.743674i \(0.266918\pi\)
\(182\) 0 0
\(183\) 5.51703 0.407831
\(184\) −18.8788 −1.39177
\(185\) −4.30845 −0.316764
\(186\) −16.5738 −1.21525
\(187\) −29.4262 −2.15186
\(188\) 9.93202 0.724367
\(189\) 0 0
\(190\) −4.95825 −0.359709
\(191\) −15.9819 −1.15641 −0.578204 0.815892i \(-0.696246\pi\)
−0.578204 + 0.815892i \(0.696246\pi\)
\(192\) −2.57655 −0.185947
\(193\) −8.62801 −0.621058 −0.310529 0.950564i \(-0.600506\pi\)
−0.310529 + 0.950564i \(0.600506\pi\)
\(194\) 2.74052 0.196758
\(195\) 1.01562 0.0727304
\(196\) 0 0
\(197\) −0.179690 −0.0128024 −0.00640118 0.999980i \(-0.502038\pi\)
−0.00640118 + 0.999980i \(0.502038\pi\)
\(198\) 6.80714 0.483763
\(199\) 0.414150 0.0293583 0.0146791 0.999892i \(-0.495327\pi\)
0.0146791 + 0.999892i \(0.495327\pi\)
\(200\) 7.86061 0.555829
\(201\) 3.03034 0.213743
\(202\) 26.1202 1.83781
\(203\) 0 0
\(204\) 5.95769 0.417122
\(205\) 8.96855 0.626391
\(206\) 24.4369 1.70260
\(207\) 9.53118 0.662463
\(208\) −4.96782 −0.344456
\(209\) 11.7820 0.814975
\(210\) 0 0
\(211\) −8.50902 −0.585785 −0.292893 0.956145i \(-0.594618\pi\)
−0.292893 + 0.956145i \(0.594618\pi\)
\(212\) 7.29643 0.501121
\(213\) 6.18315 0.423662
\(214\) 31.4128 2.14733
\(215\) −0.575808 −0.0392698
\(216\) 1.98075 0.134773
\(217\) 0 0
\(218\) −18.1806 −1.23134
\(219\) −3.99200 −0.269755
\(220\) −3.37803 −0.227747
\(221\) −7.26007 −0.488365
\(222\) −7.12458 −0.478171
\(223\) −22.6174 −1.51457 −0.757287 0.653083i \(-0.773475\pi\)
−0.757287 + 0.653083i \(0.773475\pi\)
\(224\) 0 0
\(225\) −3.96851 −0.264567
\(226\) −16.0018 −1.06443
\(227\) −2.15420 −0.142980 −0.0714898 0.997441i \(-0.522775\pi\)
−0.0714898 + 0.997441i \(0.522775\pi\)
\(228\) −2.38540 −0.157977
\(229\) 5.06718 0.334849 0.167424 0.985885i \(-0.446455\pi\)
0.167424 + 0.985885i \(0.446455\pi\)
\(230\) −16.2574 −1.07198
\(231\) 0 0
\(232\) −12.9382 −0.849437
\(233\) −8.97668 −0.588082 −0.294041 0.955793i \(-0.595000\pi\)
−0.294041 + 0.955793i \(0.595000\pi\)
\(234\) 1.67947 0.109790
\(235\) −12.2923 −0.801862
\(236\) 4.46722 0.290791
\(237\) −1.06343 −0.0690773
\(238\) 0 0
\(239\) 27.3970 1.77216 0.886081 0.463530i \(-0.153417\pi\)
0.886081 + 0.463530i \(0.153417\pi\)
\(240\) −5.04544 −0.325682
\(241\) 1.42092 0.0915295 0.0457648 0.998952i \(-0.485428\pi\)
0.0457648 + 0.998952i \(0.485428\pi\)
\(242\) 9.11627 0.586016
\(243\) −1.00000 −0.0641500
\(244\) −4.52734 −0.289833
\(245\) 0 0
\(246\) 14.8307 0.945569
\(247\) 2.90686 0.184959
\(248\) −19.5470 −1.24124
\(249\) 8.02803 0.508756
\(250\) 15.2977 0.967508
\(251\) 12.7519 0.804895 0.402447 0.915443i \(-0.368160\pi\)
0.402447 + 0.915443i \(0.368160\pi\)
\(252\) 0 0
\(253\) 38.6313 2.42873
\(254\) 18.2425 1.14464
\(255\) −7.37350 −0.461746
\(256\) 16.8325 1.05203
\(257\) −12.7298 −0.794064 −0.397032 0.917805i \(-0.629960\pi\)
−0.397032 + 0.917805i \(0.629960\pi\)
\(258\) −0.952174 −0.0592797
\(259\) 0 0
\(260\) −0.833432 −0.0516872
\(261\) 6.53200 0.404321
\(262\) 6.08198 0.375746
\(263\) −28.9952 −1.78792 −0.893961 0.448144i \(-0.852085\pi\)
−0.893961 + 0.448144i \(0.852085\pi\)
\(264\) 8.02827 0.494106
\(265\) −9.03039 −0.554733
\(266\) 0 0
\(267\) −6.60402 −0.404160
\(268\) −2.48673 −0.151901
\(269\) −1.34213 −0.0818314 −0.0409157 0.999163i \(-0.513028\pi\)
−0.0409157 + 0.999163i \(0.513028\pi\)
\(270\) 1.70571 0.103806
\(271\) 11.3410 0.688916 0.344458 0.938802i \(-0.388063\pi\)
0.344458 + 0.938802i \(0.388063\pi\)
\(272\) 36.0667 2.18687
\(273\) 0 0
\(274\) −24.5474 −1.48296
\(275\) −16.0850 −0.969961
\(276\) −7.82138 −0.470792
\(277\) 13.0962 0.786875 0.393437 0.919351i \(-0.371286\pi\)
0.393437 + 0.919351i \(0.371286\pi\)
\(278\) −18.6793 −1.12031
\(279\) 9.86851 0.590812
\(280\) 0 0
\(281\) 2.93491 0.175082 0.0875410 0.996161i \(-0.472099\pi\)
0.0875410 + 0.996161i \(0.472099\pi\)
\(282\) −20.3269 −1.21045
\(283\) −21.2779 −1.26484 −0.632420 0.774626i \(-0.717938\pi\)
−0.632420 + 0.774626i \(0.717938\pi\)
\(284\) −5.07396 −0.301084
\(285\) 2.95228 0.174878
\(286\) 6.80714 0.402515
\(287\) 0 0
\(288\) −4.38180 −0.258200
\(289\) 35.7086 2.10050
\(290\) −11.1417 −0.654262
\(291\) −1.63178 −0.0956566
\(292\) 3.27588 0.191706
\(293\) −7.81571 −0.456599 −0.228299 0.973591i \(-0.573317\pi\)
−0.228299 + 0.973591i \(0.573317\pi\)
\(294\) 0 0
\(295\) −5.52883 −0.321901
\(296\) −8.40266 −0.488395
\(297\) −4.05316 −0.235188
\(298\) 8.78797 0.509073
\(299\) 9.53118 0.551202
\(300\) 3.25660 0.188020
\(301\) 0 0
\(302\) −25.2413 −1.45247
\(303\) −15.5527 −0.893478
\(304\) −14.4408 −0.828234
\(305\) 5.60323 0.320840
\(306\) −12.1930 −0.697030
\(307\) 1.63049 0.0930568 0.0465284 0.998917i \(-0.485184\pi\)
0.0465284 + 0.998917i \(0.485184\pi\)
\(308\) 0 0
\(309\) −14.5504 −0.827744
\(310\) −16.8328 −0.956038
\(311\) 11.3479 0.643483 0.321741 0.946828i \(-0.395732\pi\)
0.321741 + 0.946828i \(0.395732\pi\)
\(312\) 1.98075 0.112138
\(313\) 5.83580 0.329859 0.164930 0.986305i \(-0.447260\pi\)
0.164930 + 0.986305i \(0.447260\pi\)
\(314\) 2.24077 0.126454
\(315\) 0 0
\(316\) 0.872664 0.0490912
\(317\) 23.6802 1.33001 0.665006 0.746838i \(-0.268429\pi\)
0.665006 + 0.746838i \(0.268429\pi\)
\(318\) −14.9329 −0.837397
\(319\) 26.4752 1.48233
\(320\) −2.61681 −0.146284
\(321\) −18.7040 −1.04396
\(322\) 0 0
\(323\) −21.1040 −1.17426
\(324\) 0.820610 0.0455895
\(325\) −3.96851 −0.220133
\(326\) −39.7829 −2.20337
\(327\) 10.8252 0.598634
\(328\) 17.4911 0.965787
\(329\) 0 0
\(330\) 6.91350 0.380576
\(331\) −19.2273 −1.05683 −0.528413 0.848987i \(-0.677213\pi\)
−0.528413 + 0.848987i \(0.677213\pi\)
\(332\) −6.58789 −0.361557
\(333\) 4.24217 0.232469
\(334\) −15.2634 −0.835177
\(335\) 3.07768 0.168152
\(336\) 0 0
\(337\) −19.8950 −1.08375 −0.541875 0.840459i \(-0.682285\pi\)
−0.541875 + 0.840459i \(0.682285\pi\)
\(338\) 1.67947 0.0913510
\(339\) 9.52791 0.517485
\(340\) 6.05077 0.328149
\(341\) 39.9986 2.16605
\(342\) 4.88198 0.263987
\(343\) 0 0
\(344\) −1.12298 −0.0605472
\(345\) 9.68009 0.521159
\(346\) −10.9449 −0.588401
\(347\) 16.6933 0.896145 0.448072 0.893997i \(-0.352111\pi\)
0.448072 + 0.893997i \(0.352111\pi\)
\(348\) −5.36023 −0.287338
\(349\) 11.9128 0.637676 0.318838 0.947809i \(-0.396707\pi\)
0.318838 + 0.947809i \(0.396707\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) −17.7601 −0.946617
\(353\) −11.9997 −0.638681 −0.319340 0.947640i \(-0.603461\pi\)
−0.319340 + 0.947640i \(0.603461\pi\)
\(354\) −9.14265 −0.485926
\(355\) 6.27976 0.333295
\(356\) 5.41933 0.287224
\(357\) 0 0
\(358\) −0.910184 −0.0481047
\(359\) −6.50164 −0.343143 −0.171572 0.985172i \(-0.554884\pi\)
−0.171572 + 0.985172i \(0.554884\pi\)
\(360\) 2.01169 0.106026
\(361\) −10.5502 −0.555272
\(362\) 30.2113 1.58787
\(363\) −5.42807 −0.284900
\(364\) 0 0
\(365\) −4.05437 −0.212216
\(366\) 9.26568 0.484325
\(367\) 11.2963 0.589660 0.294830 0.955550i \(-0.404737\pi\)
0.294830 + 0.955550i \(0.404737\pi\)
\(368\) −47.3492 −2.46825
\(369\) −8.83058 −0.459702
\(370\) −7.23590 −0.376177
\(371\) 0 0
\(372\) −8.09820 −0.419872
\(373\) −30.2400 −1.56577 −0.782884 0.622167i \(-0.786252\pi\)
−0.782884 + 0.622167i \(0.786252\pi\)
\(374\) −49.4203 −2.55546
\(375\) −9.10863 −0.470368
\(376\) −23.9734 −1.23633
\(377\) 6.53200 0.336415
\(378\) 0 0
\(379\) −2.33348 −0.119863 −0.0599313 0.998203i \(-0.519088\pi\)
−0.0599313 + 0.998203i \(0.519088\pi\)
\(380\) −2.42267 −0.124280
\(381\) −10.8621 −0.556481
\(382\) −26.8411 −1.37331
\(383\) 25.4474 1.30030 0.650151 0.759805i \(-0.274705\pi\)
0.650151 + 0.759805i \(0.274705\pi\)
\(384\) −13.0908 −0.668039
\(385\) 0 0
\(386\) −14.4905 −0.737545
\(387\) 0.566950 0.0288197
\(388\) 1.33905 0.0679802
\(389\) −7.31939 −0.371108 −0.185554 0.982634i \(-0.559408\pi\)
−0.185554 + 0.982634i \(0.559408\pi\)
\(390\) 1.70571 0.0863719
\(391\) −69.1970 −3.49944
\(392\) 0 0
\(393\) −3.62137 −0.182674
\(394\) −0.301783 −0.0152036
\(395\) −1.08005 −0.0543431
\(396\) 3.32606 0.167141
\(397\) 9.01082 0.452240 0.226120 0.974099i \(-0.427396\pi\)
0.226120 + 0.974099i \(0.427396\pi\)
\(398\) 0.695551 0.0348648
\(399\) 0 0
\(400\) 19.7148 0.985741
\(401\) 34.9597 1.74580 0.872902 0.487895i \(-0.162235\pi\)
0.872902 + 0.487895i \(0.162235\pi\)
\(402\) 5.08935 0.253834
\(403\) 9.86851 0.491585
\(404\) 12.7627 0.634967
\(405\) −1.01562 −0.0504668
\(406\) 0 0
\(407\) 17.1942 0.852283
\(408\) −14.3804 −0.711933
\(409\) −4.05390 −0.200452 −0.100226 0.994965i \(-0.531957\pi\)
−0.100226 + 0.994965i \(0.531957\pi\)
\(410\) 15.0624 0.743879
\(411\) 14.6162 0.720963
\(412\) 11.9402 0.588252
\(413\) 0 0
\(414\) 16.0073 0.786716
\(415\) 8.15346 0.400238
\(416\) −4.38180 −0.214835
\(417\) 11.1222 0.544655
\(418\) 19.7874 0.967834
\(419\) 20.7873 1.01552 0.507762 0.861497i \(-0.330473\pi\)
0.507762 + 0.861497i \(0.330473\pi\)
\(420\) 0 0
\(421\) 19.7027 0.960249 0.480125 0.877200i \(-0.340591\pi\)
0.480125 + 0.877200i \(0.340591\pi\)
\(422\) −14.2906 −0.695657
\(423\) 12.1032 0.588478
\(424\) −17.6117 −0.855302
\(425\) 28.8116 1.39757
\(426\) 10.3844 0.503126
\(427\) 0 0
\(428\) 15.3487 0.741907
\(429\) −4.05316 −0.195688
\(430\) −0.967051 −0.0466353
\(431\) 3.54528 0.170770 0.0853850 0.996348i \(-0.472788\pi\)
0.0853850 + 0.996348i \(0.472788\pi\)
\(432\) 4.96782 0.239014
\(433\) −10.2828 −0.494159 −0.247079 0.968995i \(-0.579471\pi\)
−0.247079 + 0.968995i \(0.579471\pi\)
\(434\) 0 0
\(435\) 6.63406 0.318079
\(436\) −8.88327 −0.425431
\(437\) 27.7058 1.32535
\(438\) −6.70444 −0.320350
\(439\) 13.7824 0.657800 0.328900 0.944365i \(-0.393322\pi\)
0.328900 + 0.944365i \(0.393322\pi\)
\(440\) 8.15371 0.388713
\(441\) 0 0
\(442\) −12.1930 −0.579964
\(443\) 16.2294 0.771082 0.385541 0.922691i \(-0.374015\pi\)
0.385541 + 0.922691i \(0.374015\pi\)
\(444\) −3.48117 −0.165209
\(445\) −6.70721 −0.317952
\(446\) −37.9852 −1.79865
\(447\) −5.23259 −0.247493
\(448\) 0 0
\(449\) 39.8213 1.87928 0.939642 0.342159i \(-0.111158\pi\)
0.939642 + 0.342159i \(0.111158\pi\)
\(450\) −6.66498 −0.314190
\(451\) −35.7917 −1.68537
\(452\) −7.81871 −0.367761
\(453\) 15.0293 0.706140
\(454\) −3.61792 −0.169797
\(455\) 0 0
\(456\) 5.75775 0.269631
\(457\) −3.04965 −0.142656 −0.0713282 0.997453i \(-0.522724\pi\)
−0.0713282 + 0.997453i \(0.522724\pi\)
\(458\) 8.51016 0.397654
\(459\) 7.26007 0.338871
\(460\) −7.94359 −0.370372
\(461\) 14.0689 0.655252 0.327626 0.944808i \(-0.393751\pi\)
0.327626 + 0.944808i \(0.393751\pi\)
\(462\) 0 0
\(463\) 26.2561 1.22022 0.610112 0.792315i \(-0.291125\pi\)
0.610112 + 0.792315i \(0.291125\pi\)
\(464\) −32.4498 −1.50644
\(465\) 10.0227 0.464791
\(466\) −15.0760 −0.698384
\(467\) 0.631924 0.0292419 0.0146210 0.999893i \(-0.495346\pi\)
0.0146210 + 0.999893i \(0.495346\pi\)
\(468\) 0.820610 0.0379327
\(469\) 0 0
\(470\) −20.6445 −0.952261
\(471\) −1.33422 −0.0614774
\(472\) −10.7827 −0.496316
\(473\) 2.29794 0.105659
\(474\) −1.78600 −0.0820337
\(475\) −11.5359 −0.529303
\(476\) 0 0
\(477\) 8.89147 0.407112
\(478\) 46.0123 2.10455
\(479\) 17.2080 0.786253 0.393127 0.919484i \(-0.371393\pi\)
0.393127 + 0.919484i \(0.371393\pi\)
\(480\) −4.45026 −0.203126
\(481\) 4.24217 0.193426
\(482\) 2.38639 0.108697
\(483\) 0 0
\(484\) 4.45433 0.202470
\(485\) −1.65727 −0.0752529
\(486\) −1.67947 −0.0761822
\(487\) −30.9293 −1.40154 −0.700770 0.713387i \(-0.747160\pi\)
−0.700770 + 0.713387i \(0.747160\pi\)
\(488\) 10.9278 0.494680
\(489\) 23.6878 1.07120
\(490\) 0 0
\(491\) 23.0434 1.03993 0.519966 0.854187i \(-0.325944\pi\)
0.519966 + 0.854187i \(0.325944\pi\)
\(492\) 7.24647 0.326696
\(493\) −47.4228 −2.13581
\(494\) 4.88198 0.219650
\(495\) −4.11648 −0.185022
\(496\) −49.0250 −2.20129
\(497\) 0 0
\(498\) 13.4828 0.604180
\(499\) −12.7385 −0.570252 −0.285126 0.958490i \(-0.592035\pi\)
−0.285126 + 0.958490i \(0.592035\pi\)
\(500\) 7.47464 0.334276
\(501\) 9.08825 0.406033
\(502\) 21.4164 0.955863
\(503\) −12.7285 −0.567536 −0.283768 0.958893i \(-0.591584\pi\)
−0.283768 + 0.958893i \(0.591584\pi\)
\(504\) 0 0
\(505\) −15.7957 −0.702898
\(506\) 64.8801 2.88427
\(507\) −1.00000 −0.0444116
\(508\) 8.91354 0.395474
\(509\) −24.8747 −1.10255 −0.551276 0.834323i \(-0.685859\pi\)
−0.551276 + 0.834323i \(0.685859\pi\)
\(510\) −12.3836 −0.548353
\(511\) 0 0
\(512\) 2.08800 0.0922776
\(513\) −2.90686 −0.128341
\(514\) −21.3793 −0.943001
\(515\) −14.7777 −0.651185
\(516\) −0.465245 −0.0204813
\(517\) 49.0562 2.15749
\(518\) 0 0
\(519\) 6.51688 0.286059
\(520\) 2.01169 0.0882186
\(521\) −36.7391 −1.60957 −0.804784 0.593568i \(-0.797719\pi\)
−0.804784 + 0.593568i \(0.797719\pi\)
\(522\) 10.9703 0.480156
\(523\) −19.5499 −0.854858 −0.427429 0.904049i \(-0.640581\pi\)
−0.427429 + 0.904049i \(0.640581\pi\)
\(524\) 2.97174 0.129821
\(525\) 0 0
\(526\) −48.6965 −2.12327
\(527\) −71.6460 −3.12095
\(528\) 20.1353 0.876278
\(529\) 67.8433 2.94971
\(530\) −15.1662 −0.658780
\(531\) 5.44378 0.236240
\(532\) 0 0
\(533\) −8.83058 −0.382495
\(534\) −11.0912 −0.479965
\(535\) −18.9962 −0.821279
\(536\) 6.00233 0.259261
\(537\) 0.541948 0.0233868
\(538\) −2.25407 −0.0971799
\(539\) 0 0
\(540\) 0.833432 0.0358652
\(541\) −13.1112 −0.563695 −0.281847 0.959459i \(-0.590947\pi\)
−0.281847 + 0.959459i \(0.590947\pi\)
\(542\) 19.0468 0.818131
\(543\) −17.9887 −0.771967
\(544\) 31.8121 1.36393
\(545\) 10.9943 0.470945
\(546\) 0 0
\(547\) −28.5723 −1.22166 −0.610831 0.791761i \(-0.709165\pi\)
−0.610831 + 0.791761i \(0.709165\pi\)
\(548\) −11.9942 −0.512366
\(549\) −5.51703 −0.235461
\(550\) −27.0142 −1.15189
\(551\) 18.9876 0.808899
\(552\) 18.8788 0.803537
\(553\) 0 0
\(554\) 21.9946 0.934463
\(555\) 4.30845 0.182884
\(556\) −9.12696 −0.387069
\(557\) 8.09164 0.342854 0.171427 0.985197i \(-0.445162\pi\)
0.171427 + 0.985197i \(0.445162\pi\)
\(558\) 16.5738 0.701626
\(559\) 0.566950 0.0239794
\(560\) 0 0
\(561\) 29.4262 1.24237
\(562\) 4.92909 0.207921
\(563\) 12.0169 0.506450 0.253225 0.967407i \(-0.418509\pi\)
0.253225 + 0.967407i \(0.418509\pi\)
\(564\) −9.93202 −0.418213
\(565\) 9.67678 0.407105
\(566\) −35.7355 −1.50208
\(567\) 0 0
\(568\) 12.2472 0.513883
\(569\) 33.6891 1.41232 0.706161 0.708051i \(-0.250426\pi\)
0.706161 + 0.708051i \(0.250426\pi\)
\(570\) 4.95825 0.207678
\(571\) −21.4169 −0.896269 −0.448135 0.893966i \(-0.647912\pi\)
−0.448135 + 0.893966i \(0.647912\pi\)
\(572\) 3.32606 0.139070
\(573\) 15.9819 0.667653
\(574\) 0 0
\(575\) −37.8245 −1.57739
\(576\) 2.57655 0.107356
\(577\) −46.5340 −1.93723 −0.968617 0.248557i \(-0.920044\pi\)
−0.968617 + 0.248557i \(0.920044\pi\)
\(578\) 59.9714 2.49448
\(579\) 8.62801 0.358568
\(580\) −5.44398 −0.226049
\(581\) 0 0
\(582\) −2.74052 −0.113598
\(583\) 36.0385 1.49256
\(584\) −7.90714 −0.327200
\(585\) −1.01562 −0.0419909
\(586\) −13.1262 −0.542240
\(587\) −1.76869 −0.0730016 −0.0365008 0.999334i \(-0.511621\pi\)
−0.0365008 + 0.999334i \(0.511621\pi\)
\(588\) 0 0
\(589\) 28.6864 1.18200
\(590\) −9.28550 −0.382278
\(591\) 0.179690 0.00739144
\(592\) −21.0743 −0.866150
\(593\) 3.65783 0.150209 0.0751045 0.997176i \(-0.476071\pi\)
0.0751045 + 0.997176i \(0.476071\pi\)
\(594\) −6.80714 −0.279300
\(595\) 0 0
\(596\) 4.29392 0.175886
\(597\) −0.414150 −0.0169500
\(598\) 16.0073 0.654587
\(599\) 29.2941 1.19692 0.598461 0.801152i \(-0.295779\pi\)
0.598461 + 0.801152i \(0.295779\pi\)
\(600\) −7.86061 −0.320908
\(601\) −17.0717 −0.696368 −0.348184 0.937426i \(-0.613202\pi\)
−0.348184 + 0.937426i \(0.613202\pi\)
\(602\) 0 0
\(603\) −3.03034 −0.123405
\(604\) −12.3332 −0.501832
\(605\) −5.51288 −0.224130
\(606\) −26.1202 −1.06106
\(607\) −26.4068 −1.07182 −0.535909 0.844276i \(-0.680031\pi\)
−0.535909 + 0.844276i \(0.680031\pi\)
\(608\) −12.7373 −0.516565
\(609\) 0 0
\(610\) 9.41045 0.381018
\(611\) 12.1032 0.489643
\(612\) −5.95769 −0.240825
\(613\) −27.9308 −1.12812 −0.564058 0.825735i \(-0.690760\pi\)
−0.564058 + 0.825735i \(0.690760\pi\)
\(614\) 2.73835 0.110511
\(615\) −8.96855 −0.361647
\(616\) 0 0
\(617\) 32.4751 1.30740 0.653699 0.756755i \(-0.273216\pi\)
0.653699 + 0.756755i \(0.273216\pi\)
\(618\) −24.4369 −0.982998
\(619\) 7.57894 0.304624 0.152312 0.988332i \(-0.451328\pi\)
0.152312 + 0.988332i \(0.451328\pi\)
\(620\) −8.22473 −0.330313
\(621\) −9.53118 −0.382473
\(622\) 19.0585 0.764176
\(623\) 0 0
\(624\) 4.96782 0.198872
\(625\) 10.5916 0.423664
\(626\) 9.80104 0.391728
\(627\) −11.7820 −0.470526
\(628\) 1.09487 0.0436901
\(629\) −30.7984 −1.22801
\(630\) 0 0
\(631\) −16.5735 −0.659780 −0.329890 0.944019i \(-0.607012\pi\)
−0.329890 + 0.944019i \(0.607012\pi\)
\(632\) −2.10639 −0.0837877
\(633\) 8.50902 0.338203
\(634\) 39.7701 1.57947
\(635\) −11.0318 −0.437783
\(636\) −7.29643 −0.289322
\(637\) 0 0
\(638\) 44.4642 1.76036
\(639\) −6.18315 −0.244602
\(640\) −13.2954 −0.525545
\(641\) −19.8649 −0.784616 −0.392308 0.919834i \(-0.628323\pi\)
−0.392308 + 0.919834i \(0.628323\pi\)
\(642\) −31.4128 −1.23976
\(643\) −9.68565 −0.381965 −0.190982 0.981593i \(-0.561167\pi\)
−0.190982 + 0.981593i \(0.561167\pi\)
\(644\) 0 0
\(645\) 0.575808 0.0226724
\(646\) −35.4435 −1.39450
\(647\) 34.2542 1.34667 0.673337 0.739336i \(-0.264860\pi\)
0.673337 + 0.739336i \(0.264860\pi\)
\(648\) −1.98075 −0.0778111
\(649\) 22.0645 0.866107
\(650\) −6.66498 −0.261422
\(651\) 0 0
\(652\) −19.4385 −0.761269
\(653\) −37.5001 −1.46749 −0.733746 0.679424i \(-0.762230\pi\)
−0.733746 + 0.679424i \(0.762230\pi\)
\(654\) 18.1806 0.710916
\(655\) −3.67795 −0.143710
\(656\) 43.8687 1.71279
\(657\) 3.99200 0.155743
\(658\) 0 0
\(659\) 40.1575 1.56432 0.782158 0.623081i \(-0.214119\pi\)
0.782158 + 0.623081i \(0.214119\pi\)
\(660\) 3.37803 0.131490
\(661\) 30.8232 1.19888 0.599441 0.800419i \(-0.295390\pi\)
0.599441 + 0.800419i \(0.295390\pi\)
\(662\) −32.2916 −1.25505
\(663\) 7.26007 0.281958
\(664\) 15.9015 0.617098
\(665\) 0 0
\(666\) 7.12458 0.276072
\(667\) 62.2576 2.41063
\(668\) −7.45792 −0.288555
\(669\) 22.6174 0.874439
\(670\) 5.16887 0.199691
\(671\) −22.3614 −0.863252
\(672\) 0 0
\(673\) −27.2528 −1.05052 −0.525260 0.850942i \(-0.676032\pi\)
−0.525260 + 0.850942i \(0.676032\pi\)
\(674\) −33.4130 −1.28702
\(675\) 3.96851 0.152748
\(676\) 0.820610 0.0315619
\(677\) −0.689513 −0.0265001 −0.0132501 0.999912i \(-0.504218\pi\)
−0.0132501 + 0.999912i \(0.504218\pi\)
\(678\) 16.0018 0.614546
\(679\) 0 0
\(680\) −14.6050 −0.560077
\(681\) 2.15420 0.0825493
\(682\) 67.1763 2.57231
\(683\) −0.746528 −0.0285651 −0.0142825 0.999898i \(-0.504546\pi\)
−0.0142825 + 0.999898i \(0.504546\pi\)
\(684\) 2.38540 0.0912080
\(685\) 14.8445 0.567181
\(686\) 0 0
\(687\) −5.06718 −0.193325
\(688\) −2.81650 −0.107378
\(689\) 8.89147 0.338738
\(690\) 16.2574 0.618909
\(691\) 36.8456 1.40167 0.700837 0.713321i \(-0.252810\pi\)
0.700837 + 0.713321i \(0.252810\pi\)
\(692\) −5.34782 −0.203294
\(693\) 0 0
\(694\) 28.0359 1.06423
\(695\) 11.2959 0.428479
\(696\) 12.9382 0.490422
\(697\) 64.1106 2.42836
\(698\) 20.0071 0.757280
\(699\) 8.97668 0.339529
\(700\) 0 0
\(701\) −19.8455 −0.749556 −0.374778 0.927115i \(-0.622281\pi\)
−0.374778 + 0.927115i \(0.622281\pi\)
\(702\) −1.67947 −0.0633874
\(703\) 12.3314 0.465087
\(704\) 10.4432 0.393592
\(705\) 12.2923 0.462955
\(706\) −20.1531 −0.758473
\(707\) 0 0
\(708\) −4.46722 −0.167889
\(709\) 6.77166 0.254315 0.127157 0.991883i \(-0.459415\pi\)
0.127157 + 0.991883i \(0.459415\pi\)
\(710\) 10.5466 0.395808
\(711\) 1.06343 0.0398818
\(712\) −13.0809 −0.490227
\(713\) 94.0585 3.52252
\(714\) 0 0
\(715\) −4.11648 −0.153948
\(716\) −0.444728 −0.0166203
\(717\) −27.3970 −1.02316
\(718\) −10.9193 −0.407504
\(719\) 23.7828 0.886949 0.443475 0.896287i \(-0.353746\pi\)
0.443475 + 0.896287i \(0.353746\pi\)
\(720\) 5.04544 0.188032
\(721\) 0 0
\(722\) −17.7187 −0.659420
\(723\) −1.42092 −0.0528446
\(724\) 14.7617 0.548613
\(725\) −25.9223 −0.962730
\(726\) −9.11627 −0.338336
\(727\) 32.3712 1.20058 0.600290 0.799783i \(-0.295052\pi\)
0.600290 + 0.799783i \(0.295052\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −6.80919 −0.252019
\(731\) −4.11609 −0.152239
\(732\) 4.52734 0.167335
\(733\) −43.2848 −1.59876 −0.799379 0.600827i \(-0.794838\pi\)
−0.799379 + 0.600827i \(0.794838\pi\)
\(734\) 18.9717 0.700258
\(735\) 0 0
\(736\) −41.7637 −1.53943
\(737\) −12.2824 −0.452429
\(738\) −14.8307 −0.545925
\(739\) 23.2717 0.856063 0.428032 0.903764i \(-0.359207\pi\)
0.428032 + 0.903764i \(0.359207\pi\)
\(740\) −3.53556 −0.129970
\(741\) −2.90686 −0.106786
\(742\) 0 0
\(743\) 30.9402 1.13509 0.567543 0.823344i \(-0.307894\pi\)
0.567543 + 0.823344i \(0.307894\pi\)
\(744\) 19.5470 0.716628
\(745\) −5.31435 −0.194703
\(746\) −50.7871 −1.85945
\(747\) −8.02803 −0.293730
\(748\) −24.1474 −0.882918
\(749\) 0 0
\(750\) −15.2977 −0.558591
\(751\) 13.4036 0.489106 0.244553 0.969636i \(-0.421359\pi\)
0.244553 + 0.969636i \(0.421359\pi\)
\(752\) −60.1265 −2.19259
\(753\) −12.7519 −0.464706
\(754\) 10.9703 0.399514
\(755\) 15.2642 0.555520
\(756\) 0 0
\(757\) 31.8271 1.15677 0.578387 0.815762i \(-0.303682\pi\)
0.578387 + 0.815762i \(0.303682\pi\)
\(758\) −3.91900 −0.142344
\(759\) −38.6313 −1.40223
\(760\) 5.84771 0.212119
\(761\) 3.91844 0.142043 0.0710216 0.997475i \(-0.477374\pi\)
0.0710216 + 0.997475i \(0.477374\pi\)
\(762\) −18.2425 −0.660857
\(763\) 0 0
\(764\) −13.1149 −0.474481
\(765\) 7.37350 0.266589
\(766\) 42.7381 1.54419
\(767\) 5.44378 0.196564
\(768\) −16.8325 −0.607391
\(769\) −10.6889 −0.385453 −0.192727 0.981252i \(-0.561733\pi\)
−0.192727 + 0.981252i \(0.561733\pi\)
\(770\) 0 0
\(771\) 12.7298 0.458453
\(772\) −7.08023 −0.254823
\(773\) −11.7012 −0.420862 −0.210431 0.977609i \(-0.567487\pi\)
−0.210431 + 0.977609i \(0.567487\pi\)
\(774\) 0.952174 0.0342252
\(775\) −39.1632 −1.40679
\(776\) −3.23214 −0.116027
\(777\) 0 0
\(778\) −12.2927 −0.440714
\(779\) −25.6693 −0.919697
\(780\) 0.833432 0.0298416
\(781\) −25.0613 −0.896763
\(782\) −116.214 −4.15581
\(783\) −6.53200 −0.233435
\(784\) 0 0
\(785\) −1.35506 −0.0483642
\(786\) −6.08198 −0.216937
\(787\) 12.8318 0.457403 0.228702 0.973497i \(-0.426552\pi\)
0.228702 + 0.973497i \(0.426552\pi\)
\(788\) −0.147455 −0.00525287
\(789\) 28.9952 1.03226
\(790\) −1.81390 −0.0645358
\(791\) 0 0
\(792\) −8.02827 −0.285272
\(793\) −5.51703 −0.195916
\(794\) 15.1334 0.537063
\(795\) 9.03039 0.320275
\(796\) 0.339856 0.0120459
\(797\) −1.12352 −0.0397970 −0.0198985 0.999802i \(-0.506334\pi\)
−0.0198985 + 0.999802i \(0.506334\pi\)
\(798\) 0 0
\(799\) −87.8701 −3.10862
\(800\) 17.3892 0.614801
\(801\) 6.60402 0.233342
\(802\) 58.7137 2.07325
\(803\) 16.1802 0.570987
\(804\) 2.48673 0.0877000
\(805\) 0 0
\(806\) 16.5738 0.583788
\(807\) 1.34213 0.0472454
\(808\) −30.8059 −1.08375
\(809\) 7.84953 0.275975 0.137987 0.990434i \(-0.455937\pi\)
0.137987 + 0.990434i \(0.455937\pi\)
\(810\) −1.70571 −0.0599325
\(811\) −7.38224 −0.259225 −0.129613 0.991565i \(-0.541373\pi\)
−0.129613 + 0.991565i \(0.541373\pi\)
\(812\) 0 0
\(813\) −11.3410 −0.397746
\(814\) 28.8771 1.01214
\(815\) 24.0579 0.842712
\(816\) −36.0667 −1.26259
\(817\) 1.64804 0.0576577
\(818\) −6.80839 −0.238050
\(819\) 0 0
\(820\) 7.35969 0.257011
\(821\) −27.7269 −0.967674 −0.483837 0.875158i \(-0.660757\pi\)
−0.483837 + 0.875158i \(0.660757\pi\)
\(822\) 24.5474 0.856188
\(823\) −8.41214 −0.293229 −0.146614 0.989194i \(-0.546838\pi\)
−0.146614 + 0.989194i \(0.546838\pi\)
\(824\) −28.8207 −1.00402
\(825\) 16.0850 0.560007
\(826\) 0 0
\(827\) −6.80644 −0.236683 −0.118342 0.992973i \(-0.537758\pi\)
−0.118342 + 0.992973i \(0.537758\pi\)
\(828\) 7.82138 0.271812
\(829\) −50.0143 −1.73707 −0.868534 0.495629i \(-0.834938\pi\)
−0.868534 + 0.495629i \(0.834938\pi\)
\(830\) 13.6935 0.475308
\(831\) −13.0962 −0.454302
\(832\) 2.57655 0.0893259
\(833\) 0 0
\(834\) 18.6793 0.646812
\(835\) 9.23025 0.319426
\(836\) 9.66840 0.334388
\(837\) −9.86851 −0.341106
\(838\) 34.9115 1.20600
\(839\) 18.8435 0.650552 0.325276 0.945619i \(-0.394543\pi\)
0.325276 + 0.945619i \(0.394543\pi\)
\(840\) 0 0
\(841\) 13.6670 0.471276
\(842\) 33.0900 1.14036
\(843\) −2.93491 −0.101084
\(844\) −6.98259 −0.240351
\(845\) −1.01562 −0.0349385
\(846\) 20.3269 0.698855
\(847\) 0 0
\(848\) −44.1712 −1.51685
\(849\) 21.2779 0.730256
\(850\) 48.3882 1.65970
\(851\) 40.4329 1.38602
\(852\) 5.07396 0.173831
\(853\) −41.3669 −1.41638 −0.708189 0.706023i \(-0.750487\pi\)
−0.708189 + 0.706023i \(0.750487\pi\)
\(854\) 0 0
\(855\) −2.95228 −0.100966
\(856\) −37.0479 −1.26627
\(857\) 29.5295 1.00871 0.504355 0.863497i \(-0.331730\pi\)
0.504355 + 0.863497i \(0.331730\pi\)
\(858\) −6.80714 −0.232392
\(859\) −36.0619 −1.23041 −0.615207 0.788365i \(-0.710928\pi\)
−0.615207 + 0.788365i \(0.710928\pi\)
\(860\) −0.472514 −0.0161126
\(861\) 0 0
\(862\) 5.95418 0.202800
\(863\) −2.68662 −0.0914537 −0.0457269 0.998954i \(-0.514560\pi\)
−0.0457269 + 0.998954i \(0.514560\pi\)
\(864\) 4.38180 0.149072
\(865\) 6.61870 0.225043
\(866\) −17.2696 −0.586844
\(867\) −35.7086 −1.21273
\(868\) 0 0
\(869\) 4.31026 0.146215
\(870\) 11.1417 0.377738
\(871\) −3.03034 −0.102679
\(872\) 21.4420 0.726116
\(873\) 1.63178 0.0552273
\(874\) 46.5310 1.57393
\(875\) 0 0
\(876\) −3.27588 −0.110682
\(877\) −6.40742 −0.216363 −0.108182 0.994131i \(-0.534503\pi\)
−0.108182 + 0.994131i \(0.534503\pi\)
\(878\) 23.1472 0.781179
\(879\) 7.81571 0.263618
\(880\) 20.4499 0.689367
\(881\) 13.3454 0.449617 0.224808 0.974403i \(-0.427824\pi\)
0.224808 + 0.974403i \(0.427824\pi\)
\(882\) 0 0
\(883\) −14.8329 −0.499167 −0.249584 0.968353i \(-0.580294\pi\)
−0.249584 + 0.968353i \(0.580294\pi\)
\(884\) −5.95769 −0.200379
\(885\) 5.52883 0.185850
\(886\) 27.2568 0.915709
\(887\) −32.8467 −1.10288 −0.551442 0.834213i \(-0.685922\pi\)
−0.551442 + 0.834213i \(0.685922\pi\)
\(888\) 8.40266 0.281975
\(889\) 0 0
\(890\) −11.2645 −0.377588
\(891\) 4.05316 0.135786
\(892\) −18.5601 −0.621437
\(893\) 35.1823 1.17733
\(894\) −8.78797 −0.293914
\(895\) 0.550416 0.0183984
\(896\) 0 0
\(897\) −9.53118 −0.318237
\(898\) 66.8786 2.23177
\(899\) 64.4611 2.14990
\(900\) −3.25660 −0.108553
\(901\) −64.5527 −2.15056
\(902\) −60.1110 −2.00148
\(903\) 0 0
\(904\) 18.8724 0.627686
\(905\) −18.2697 −0.607306
\(906\) 25.2413 0.838586
\(907\) −6.78020 −0.225133 −0.112566 0.993644i \(-0.535907\pi\)
−0.112566 + 0.993644i \(0.535907\pi\)
\(908\) −1.76776 −0.0586653
\(909\) 15.5527 0.515850
\(910\) 0 0
\(911\) 19.2486 0.637736 0.318868 0.947799i \(-0.396697\pi\)
0.318868 + 0.947799i \(0.396697\pi\)
\(912\) 14.4408 0.478181
\(913\) −32.5389 −1.07688
\(914\) −5.12178 −0.169413
\(915\) −5.60323 −0.185237
\(916\) 4.15818 0.137390
\(917\) 0 0
\(918\) 12.1930 0.402430
\(919\) 23.6687 0.780757 0.390379 0.920654i \(-0.372344\pi\)
0.390379 + 0.920654i \(0.372344\pi\)
\(920\) 19.1738 0.632142
\(921\) −1.63049 −0.0537264
\(922\) 23.6282 0.778152
\(923\) −6.18315 −0.203521
\(924\) 0 0
\(925\) −16.8351 −0.553534
\(926\) 44.0962 1.44909
\(927\) 14.5504 0.477898
\(928\) −28.6219 −0.939560
\(929\) −18.2611 −0.599126 −0.299563 0.954076i \(-0.596841\pi\)
−0.299563 + 0.954076i \(0.596841\pi\)
\(930\) 16.8328 0.551969
\(931\) 0 0
\(932\) −7.36635 −0.241293
\(933\) −11.3479 −0.371515
\(934\) 1.06129 0.0347266
\(935\) 29.8859 0.977375
\(936\) −1.98075 −0.0647427
\(937\) 2.02078 0.0660159 0.0330079 0.999455i \(-0.489491\pi\)
0.0330079 + 0.999455i \(0.489491\pi\)
\(938\) 0 0
\(939\) −5.83580 −0.190444
\(940\) −10.0872 −0.329008
\(941\) −26.6332 −0.868217 −0.434108 0.900861i \(-0.642937\pi\)
−0.434108 + 0.900861i \(0.642937\pi\)
\(942\) −2.24077 −0.0730083
\(943\) −84.1658 −2.74082
\(944\) −27.0437 −0.880198
\(945\) 0 0
\(946\) 3.85931 0.125477
\(947\) 3.57351 0.116124 0.0580618 0.998313i \(-0.481508\pi\)
0.0580618 + 0.998313i \(0.481508\pi\)
\(948\) −0.872664 −0.0283428
\(949\) 3.99200 0.129586
\(950\) −19.3742 −0.628581
\(951\) −23.6802 −0.767883
\(952\) 0 0
\(953\) −5.18119 −0.167835 −0.0839176 0.996473i \(-0.526743\pi\)
−0.0839176 + 0.996473i \(0.526743\pi\)
\(954\) 14.9329 0.483472
\(955\) 16.2316 0.525242
\(956\) 22.4822 0.727128
\(957\) −26.4752 −0.855822
\(958\) 28.9003 0.933725
\(959\) 0 0
\(960\) 2.61681 0.0844571
\(961\) 66.3874 2.14153
\(962\) 7.12458 0.229706
\(963\) 18.7040 0.602728
\(964\) 1.16602 0.0375551
\(965\) 8.76281 0.282085
\(966\) 0 0
\(967\) −33.3558 −1.07265 −0.536325 0.844012i \(-0.680188\pi\)
−0.536325 + 0.844012i \(0.680188\pi\)
\(968\) −10.7516 −0.345570
\(969\) 21.1040 0.677958
\(970\) −2.78334 −0.0893676
\(971\) −11.0917 −0.355949 −0.177975 0.984035i \(-0.556954\pi\)
−0.177975 + 0.984035i \(0.556954\pi\)
\(972\) −0.820610 −0.0263211
\(973\) 0 0
\(974\) −51.9447 −1.66442
\(975\) 3.96851 0.127094
\(976\) 27.4076 0.877297
\(977\) 31.3247 1.00217 0.501083 0.865399i \(-0.332935\pi\)
0.501083 + 0.865399i \(0.332935\pi\)
\(978\) 39.7829 1.27212
\(979\) 26.7671 0.855481
\(980\) 0 0
\(981\) −10.8252 −0.345622
\(982\) 38.7006 1.23499
\(983\) −19.7083 −0.628596 −0.314298 0.949324i \(-0.601769\pi\)
−0.314298 + 0.949324i \(0.601769\pi\)
\(984\) −17.4911 −0.557597
\(985\) 0.182497 0.00581484
\(986\) −79.6450 −2.53641
\(987\) 0 0
\(988\) 2.38540 0.0758897
\(989\) 5.40370 0.171828
\(990\) −6.91350 −0.219725
\(991\) −36.9147 −1.17263 −0.586316 0.810082i \(-0.699422\pi\)
−0.586316 + 0.810082i \(0.699422\pi\)
\(992\) −43.2418 −1.37293
\(993\) 19.2273 0.610159
\(994\) 0 0
\(995\) −0.420620 −0.0133346
\(996\) 6.58789 0.208745
\(997\) 1.04532 0.0331057 0.0165528 0.999863i \(-0.494731\pi\)
0.0165528 + 0.999863i \(0.494731\pi\)
\(998\) −21.3938 −0.677210
\(999\) −4.24217 −0.134216
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 1911.2.a.x.1.7 10
3.2 odd 2 5733.2.a.bw.1.4 10
7.6 odd 2 1911.2.a.y.1.7 yes 10
21.20 even 2 5733.2.a.bx.1.4 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1911.2.a.x.1.7 10 1.1 even 1 trivial
1911.2.a.y.1.7 yes 10 7.6 odd 2
5733.2.a.bw.1.4 10 3.2 odd 2
5733.2.a.bx.1.4 10 21.20 even 2