Properties

Label 1881.2.a.k.1.1
Level $1881$
Weight $2$
Character 1881.1
Self dual yes
Analytic conductor $15.020$
Analytic rank $0$
Dimension $5$
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] = [1881,2,Mod(1,1881)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1881, 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("1881.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1881 = 3^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1881.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: \(15.0198606202\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.246832.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 209)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.71457\) of defining polynomial
Character \(\chi\) \(=\) 1881.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.65432 q^{2} +5.04540 q^{4} +1.06025 q^{5} -3.36889 q^{7} -8.08346 q^{8} -2.81425 q^{10} -1.00000 q^{11} -2.15993 q^{13} +8.94209 q^{14} +11.3653 q^{16} -3.67288 q^{17} -1.00000 q^{19} +5.34940 q^{20} +2.65432 q^{22} -3.15468 q^{23} -3.87587 q^{25} +5.73313 q^{26} -16.9974 q^{28} -7.17849 q^{29} +4.65295 q^{31} -14.0001 q^{32} +9.74900 q^{34} -3.57187 q^{35} +2.27446 q^{37} +2.65432 q^{38} -8.57050 q^{40} +11.3852 q^{41} +9.38838 q^{43} -5.04540 q^{44} +8.37352 q^{46} +5.77094 q^{47} +4.34940 q^{49} +10.2878 q^{50} -10.8977 q^{52} +5.65820 q^{53} -1.06025 q^{55} +27.2322 q^{56} +19.0540 q^{58} +13.7944 q^{59} +6.98152 q^{61} -12.3504 q^{62} +14.4301 q^{64} -2.29007 q^{65} -4.81332 q^{67} -18.5312 q^{68} +9.48087 q^{70} -15.2629 q^{71} -8.08806 q^{73} -6.03713 q^{74} -5.04540 q^{76} +3.36889 q^{77} +13.4291 q^{79} +12.0500 q^{80} -30.2199 q^{82} -9.96666 q^{83} -3.89418 q^{85} -24.9197 q^{86} +8.08346 q^{88} +4.61626 q^{89} +7.27655 q^{91} -15.9166 q^{92} -15.3179 q^{94} -1.06025 q^{95} -4.09907 q^{97} -11.5447 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{2} + 6 q^{4} + 5 q^{5} + 6 q^{7} - 6 q^{8} + 12 q^{10} - 5 q^{11} + 4 q^{13} + 14 q^{14} + 8 q^{16} + 4 q^{17} - 5 q^{19} + 8 q^{20} + 2 q^{22} - 3 q^{23} + 6 q^{25} + 6 q^{26} - 10 q^{28} - 10 q^{29}+ \cdots + 10 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.65432 −1.87689 −0.938443 0.345435i \(-0.887732\pi\)
−0.938443 + 0.345435i \(0.887732\pi\)
\(3\) 0 0
\(4\) 5.04540 2.52270
\(5\) 1.06025 0.474159 0.237080 0.971490i \(-0.423810\pi\)
0.237080 + 0.971490i \(0.423810\pi\)
\(6\) 0 0
\(7\) −3.36889 −1.27332 −0.636660 0.771145i \(-0.719684\pi\)
−0.636660 + 0.771145i \(0.719684\pi\)
\(8\) −8.08346 −2.85793
\(9\) 0 0
\(10\) −2.81425 −0.889943
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −2.15993 −0.599056 −0.299528 0.954087i \(-0.596829\pi\)
−0.299528 + 0.954087i \(0.596829\pi\)
\(14\) 8.94209 2.38987
\(15\) 0 0
\(16\) 11.3653 2.84131
\(17\) −3.67288 −0.890805 −0.445402 0.895330i \(-0.646939\pi\)
−0.445402 + 0.895330i \(0.646939\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 5.34940 1.19616
\(21\) 0 0
\(22\) 2.65432 0.565902
\(23\) −3.15468 −0.657797 −0.328898 0.944365i \(-0.606677\pi\)
−0.328898 + 0.944365i \(0.606677\pi\)
\(24\) 0 0
\(25\) −3.87587 −0.775173
\(26\) 5.73313 1.12436
\(27\) 0 0
\(28\) −16.9974 −3.21220
\(29\) −7.17849 −1.33301 −0.666506 0.745499i \(-0.732211\pi\)
−0.666506 + 0.745499i \(0.732211\pi\)
\(30\) 0 0
\(31\) 4.65295 0.835694 0.417847 0.908517i \(-0.362785\pi\)
0.417847 + 0.908517i \(0.362785\pi\)
\(32\) −14.0001 −2.47489
\(33\) 0 0
\(34\) 9.74900 1.67194
\(35\) −3.57187 −0.603756
\(36\) 0 0
\(37\) 2.27446 0.373918 0.186959 0.982368i \(-0.440137\pi\)
0.186959 + 0.982368i \(0.440137\pi\)
\(38\) 2.65432 0.430587
\(39\) 0 0
\(40\) −8.57050 −1.35512
\(41\) 11.3852 1.77807 0.889034 0.457841i \(-0.151377\pi\)
0.889034 + 0.457841i \(0.151377\pi\)
\(42\) 0 0
\(43\) 9.38838 1.43171 0.715857 0.698247i \(-0.246036\pi\)
0.715857 + 0.698247i \(0.246036\pi\)
\(44\) −5.04540 −0.760623
\(45\) 0 0
\(46\) 8.37352 1.23461
\(47\) 5.77094 0.841778 0.420889 0.907112i \(-0.361718\pi\)
0.420889 + 0.907112i \(0.361718\pi\)
\(48\) 0 0
\(49\) 4.34940 0.621342
\(50\) 10.2878 1.45491
\(51\) 0 0
\(52\) −10.8977 −1.51124
\(53\) 5.65820 0.777213 0.388607 0.921404i \(-0.372957\pi\)
0.388607 + 0.921404i \(0.372957\pi\)
\(54\) 0 0
\(55\) −1.06025 −0.142964
\(56\) 27.2322 3.63906
\(57\) 0 0
\(58\) 19.0540 2.50191
\(59\) 13.7944 1.79588 0.897939 0.440121i \(-0.145064\pi\)
0.897939 + 0.440121i \(0.145064\pi\)
\(60\) 0 0
\(61\) 6.98152 0.893892 0.446946 0.894561i \(-0.352512\pi\)
0.446946 + 0.894561i \(0.352512\pi\)
\(62\) −12.3504 −1.56850
\(63\) 0 0
\(64\) 14.4301 1.80377
\(65\) −2.29007 −0.284048
\(66\) 0 0
\(67\) −4.81332 −0.588041 −0.294020 0.955799i \(-0.594993\pi\)
−0.294020 + 0.955799i \(0.594993\pi\)
\(68\) −18.5312 −2.24723
\(69\) 0 0
\(70\) 9.48087 1.13318
\(71\) −15.2629 −1.81137 −0.905685 0.423951i \(-0.860643\pi\)
−0.905685 + 0.423951i \(0.860643\pi\)
\(72\) 0 0
\(73\) −8.08806 −0.946636 −0.473318 0.880892i \(-0.656944\pi\)
−0.473318 + 0.880892i \(0.656944\pi\)
\(74\) −6.03713 −0.701802
\(75\) 0 0
\(76\) −5.04540 −0.578747
\(77\) 3.36889 0.383920
\(78\) 0 0
\(79\) 13.4291 1.51090 0.755448 0.655209i \(-0.227419\pi\)
0.755448 + 0.655209i \(0.227419\pi\)
\(80\) 12.0500 1.34724
\(81\) 0 0
\(82\) −30.2199 −3.33723
\(83\) −9.96666 −1.09398 −0.546992 0.837138i \(-0.684227\pi\)
−0.546992 + 0.837138i \(0.684227\pi\)
\(84\) 0 0
\(85\) −3.89418 −0.422383
\(86\) −24.9197 −2.68716
\(87\) 0 0
\(88\) 8.08346 0.861699
\(89\) 4.61626 0.489323 0.244661 0.969609i \(-0.421323\pi\)
0.244661 + 0.969609i \(0.421323\pi\)
\(90\) 0 0
\(91\) 7.27655 0.762790
\(92\) −15.9166 −1.65942
\(93\) 0 0
\(94\) −15.3179 −1.57992
\(95\) −1.06025 −0.108780
\(96\) 0 0
\(97\) −4.09907 −0.416197 −0.208099 0.978108i \(-0.566727\pi\)
−0.208099 + 0.978108i \(0.566727\pi\)
\(98\) −11.5447 −1.16619
\(99\) 0 0
\(100\) −19.5553 −1.95553
\(101\) 10.8730 1.08190 0.540950 0.841055i \(-0.318065\pi\)
0.540950 + 0.841055i \(0.318065\pi\)
\(102\) 0 0
\(103\) 8.85864 0.872867 0.436434 0.899736i \(-0.356241\pi\)
0.436434 + 0.899736i \(0.356241\pi\)
\(104\) 17.4597 1.71206
\(105\) 0 0
\(106\) −15.0186 −1.45874
\(107\) 18.1669 1.75626 0.878131 0.478421i \(-0.158791\pi\)
0.878131 + 0.478421i \(0.158791\pi\)
\(108\) 0 0
\(109\) −0.211303 −0.0202392 −0.0101196 0.999949i \(-0.503221\pi\)
−0.0101196 + 0.999949i \(0.503221\pi\)
\(110\) 2.81425 0.268328
\(111\) 0 0
\(112\) −38.2883 −3.61790
\(113\) 0.198345 0.0186587 0.00932935 0.999956i \(-0.497030\pi\)
0.00932935 + 0.999956i \(0.497030\pi\)
\(114\) 0 0
\(115\) −3.34476 −0.311900
\(116\) −36.2184 −3.36279
\(117\) 0 0
\(118\) −36.6147 −3.37066
\(119\) 12.3735 1.13428
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −18.5312 −1.67773
\(123\) 0 0
\(124\) 23.4760 2.10821
\(125\) −9.41066 −0.841715
\(126\) 0 0
\(127\) −3.60331 −0.319742 −0.159871 0.987138i \(-0.551108\pi\)
−0.159871 + 0.987138i \(0.551108\pi\)
\(128\) −10.3020 −0.910578
\(129\) 0 0
\(130\) 6.07857 0.533126
\(131\) −16.8741 −1.47430 −0.737150 0.675729i \(-0.763829\pi\)
−0.737150 + 0.675729i \(0.763829\pi\)
\(132\) 0 0
\(133\) 3.36889 0.292119
\(134\) 12.7761 1.10369
\(135\) 0 0
\(136\) 29.6896 2.54586
\(137\) 6.10839 0.521875 0.260937 0.965356i \(-0.415968\pi\)
0.260937 + 0.965356i \(0.415968\pi\)
\(138\) 0 0
\(139\) 11.3347 0.961398 0.480699 0.876886i \(-0.340383\pi\)
0.480699 + 0.876886i \(0.340383\pi\)
\(140\) −18.0215 −1.52310
\(141\) 0 0
\(142\) 40.5125 3.39973
\(143\) 2.15993 0.180622
\(144\) 0 0
\(145\) −7.61101 −0.632060
\(146\) 21.4683 1.77673
\(147\) 0 0
\(148\) 11.4755 0.943284
\(149\) −6.75698 −0.553554 −0.276777 0.960934i \(-0.589266\pi\)
−0.276777 + 0.960934i \(0.589266\pi\)
\(150\) 0 0
\(151\) 6.46231 0.525895 0.262948 0.964810i \(-0.415305\pi\)
0.262948 + 0.964810i \(0.415305\pi\)
\(152\) 8.08346 0.655655
\(153\) 0 0
\(154\) −8.94209 −0.720574
\(155\) 4.93330 0.396252
\(156\) 0 0
\(157\) 0.248382 0.0198230 0.00991152 0.999951i \(-0.496845\pi\)
0.00991152 + 0.999951i \(0.496845\pi\)
\(158\) −35.6452 −2.83578
\(159\) 0 0
\(160\) −14.8436 −1.17349
\(161\) 10.6278 0.837585
\(162\) 0 0
\(163\) −16.9710 −1.32927 −0.664637 0.747167i \(-0.731414\pi\)
−0.664637 + 0.747167i \(0.731414\pi\)
\(164\) 57.4428 4.48553
\(165\) 0 0
\(166\) 26.4547 2.05328
\(167\) 0.865538 0.0669773 0.0334887 0.999439i \(-0.489338\pi\)
0.0334887 + 0.999439i \(0.489338\pi\)
\(168\) 0 0
\(169\) −8.33471 −0.641132
\(170\) 10.3364 0.792765
\(171\) 0 0
\(172\) 47.3681 3.61178
\(173\) 4.14483 0.315125 0.157563 0.987509i \(-0.449636\pi\)
0.157563 + 0.987509i \(0.449636\pi\)
\(174\) 0 0
\(175\) 13.0574 0.987043
\(176\) −11.3653 −0.856688
\(177\) 0 0
\(178\) −12.2530 −0.918403
\(179\) −2.04584 −0.152913 −0.0764567 0.997073i \(-0.524361\pi\)
−0.0764567 + 0.997073i \(0.524361\pi\)
\(180\) 0 0
\(181\) 5.27519 0.392101 0.196051 0.980594i \(-0.437188\pi\)
0.196051 + 0.980594i \(0.437188\pi\)
\(182\) −19.3143 −1.43167
\(183\) 0 0
\(184\) 25.5007 1.87994
\(185\) 2.41150 0.177297
\(186\) 0 0
\(187\) 3.67288 0.268588
\(188\) 29.1167 2.12355
\(189\) 0 0
\(190\) 2.81425 0.204167
\(191\) 16.1899 1.17146 0.585729 0.810507i \(-0.300808\pi\)
0.585729 + 0.810507i \(0.300808\pi\)
\(192\) 0 0
\(193\) −20.0620 −1.44409 −0.722045 0.691846i \(-0.756798\pi\)
−0.722045 + 0.691846i \(0.756798\pi\)
\(194\) 10.8802 0.781155
\(195\) 0 0
\(196\) 21.9444 1.56746
\(197\) 23.8398 1.69851 0.849257 0.527979i \(-0.177050\pi\)
0.849257 + 0.527979i \(0.177050\pi\)
\(198\) 0 0
\(199\) −1.90194 −0.134825 −0.0674125 0.997725i \(-0.521474\pi\)
−0.0674125 + 0.997725i \(0.521474\pi\)
\(200\) 31.3304 2.21539
\(201\) 0 0
\(202\) −28.8603 −2.03060
\(203\) 24.1835 1.69735
\(204\) 0 0
\(205\) 12.0712 0.843087
\(206\) −23.5136 −1.63827
\(207\) 0 0
\(208\) −24.5481 −1.70211
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) 12.3048 0.847100 0.423550 0.905873i \(-0.360784\pi\)
0.423550 + 0.905873i \(0.360784\pi\)
\(212\) 28.5479 1.96068
\(213\) 0 0
\(214\) −48.2207 −3.29630
\(215\) 9.95405 0.678860
\(216\) 0 0
\(217\) −15.6753 −1.06411
\(218\) 0.560865 0.0379866
\(219\) 0 0
\(220\) −5.34940 −0.360656
\(221\) 7.93316 0.533642
\(222\) 0 0
\(223\) 22.8473 1.52997 0.764985 0.644048i \(-0.222746\pi\)
0.764985 + 0.644048i \(0.222746\pi\)
\(224\) 47.1647 3.15132
\(225\) 0 0
\(226\) −0.526470 −0.0350202
\(227\) −4.00654 −0.265923 −0.132962 0.991121i \(-0.542449\pi\)
−0.132962 + 0.991121i \(0.542449\pi\)
\(228\) 0 0
\(229\) −24.6024 −1.62577 −0.812887 0.582422i \(-0.802105\pi\)
−0.812887 + 0.582422i \(0.802105\pi\)
\(230\) 8.87805 0.585401
\(231\) 0 0
\(232\) 58.0270 3.80966
\(233\) −16.2994 −1.06781 −0.533903 0.845545i \(-0.679275\pi\)
−0.533903 + 0.845545i \(0.679275\pi\)
\(234\) 0 0
\(235\) 6.11865 0.399137
\(236\) 69.5982 4.53046
\(237\) 0 0
\(238\) −32.8433 −2.12891
\(239\) 15.4182 0.997324 0.498662 0.866797i \(-0.333825\pi\)
0.498662 + 0.866797i \(0.333825\pi\)
\(240\) 0 0
\(241\) 24.1441 1.55526 0.777629 0.628723i \(-0.216422\pi\)
0.777629 + 0.628723i \(0.216422\pi\)
\(242\) −2.65432 −0.170626
\(243\) 0 0
\(244\) 35.2245 2.25502
\(245\) 4.61146 0.294615
\(246\) 0 0
\(247\) 2.15993 0.137433
\(248\) −37.6119 −2.38836
\(249\) 0 0
\(250\) 24.9789 1.57980
\(251\) 19.9099 1.25670 0.628350 0.777931i \(-0.283731\pi\)
0.628350 + 0.777931i \(0.283731\pi\)
\(252\) 0 0
\(253\) 3.15468 0.198333
\(254\) 9.56433 0.600119
\(255\) 0 0
\(256\) −1.51547 −0.0947166
\(257\) 22.3126 1.39182 0.695911 0.718128i \(-0.255001\pi\)
0.695911 + 0.718128i \(0.255001\pi\)
\(258\) 0 0
\(259\) −7.66239 −0.476118
\(260\) −11.5543 −0.716568
\(261\) 0 0
\(262\) 44.7893 2.76709
\(263\) −13.7963 −0.850716 −0.425358 0.905025i \(-0.639852\pi\)
−0.425358 + 0.905025i \(0.639852\pi\)
\(264\) 0 0
\(265\) 5.99912 0.368523
\(266\) −8.94209 −0.548275
\(267\) 0 0
\(268\) −24.2851 −1.48345
\(269\) 11.2028 0.683048 0.341524 0.939873i \(-0.389057\pi\)
0.341524 + 0.939873i \(0.389057\pi\)
\(270\) 0 0
\(271\) 17.1234 1.04017 0.520085 0.854115i \(-0.325900\pi\)
0.520085 + 0.854115i \(0.325900\pi\)
\(272\) −41.7433 −2.53106
\(273\) 0 0
\(274\) −16.2136 −0.979499
\(275\) 3.87587 0.233723
\(276\) 0 0
\(277\) −13.7127 −0.823919 −0.411960 0.911202i \(-0.635156\pi\)
−0.411960 + 0.911202i \(0.635156\pi\)
\(278\) −30.0859 −1.80443
\(279\) 0 0
\(280\) 28.8730 1.72549
\(281\) −25.8599 −1.54267 −0.771337 0.636427i \(-0.780412\pi\)
−0.771337 + 0.636427i \(0.780412\pi\)
\(282\) 0 0
\(283\) 28.8711 1.71621 0.858103 0.513477i \(-0.171643\pi\)
0.858103 + 0.513477i \(0.171643\pi\)
\(284\) −77.0073 −4.56954
\(285\) 0 0
\(286\) −5.73313 −0.339007
\(287\) −38.3554 −2.26405
\(288\) 0 0
\(289\) −3.50993 −0.206467
\(290\) 20.2020 1.18630
\(291\) 0 0
\(292\) −40.8075 −2.38808
\(293\) −6.37971 −0.372707 −0.186353 0.982483i \(-0.559667\pi\)
−0.186353 + 0.982483i \(0.559667\pi\)
\(294\) 0 0
\(295\) 14.6255 0.851532
\(296\) −18.3855 −1.06863
\(297\) 0 0
\(298\) 17.9352 1.03896
\(299\) 6.81389 0.394057
\(300\) 0 0
\(301\) −31.6284 −1.82303
\(302\) −17.1530 −0.987045
\(303\) 0 0
\(304\) −11.3653 −0.651842
\(305\) 7.40217 0.423847
\(306\) 0 0
\(307\) 10.4299 0.595264 0.297632 0.954681i \(-0.403803\pi\)
0.297632 + 0.954681i \(0.403803\pi\)
\(308\) 16.9974 0.968515
\(309\) 0 0
\(310\) −13.0945 −0.743720
\(311\) 8.26224 0.468508 0.234254 0.972175i \(-0.424735\pi\)
0.234254 + 0.972175i \(0.424735\pi\)
\(312\) 0 0
\(313\) −24.1167 −1.36315 −0.681577 0.731746i \(-0.738706\pi\)
−0.681577 + 0.731746i \(0.738706\pi\)
\(314\) −0.659285 −0.0372056
\(315\) 0 0
\(316\) 67.7554 3.81154
\(317\) 11.7330 0.658989 0.329495 0.944157i \(-0.393122\pi\)
0.329495 + 0.944157i \(0.393122\pi\)
\(318\) 0 0
\(319\) 7.17849 0.401918
\(320\) 15.2996 0.855273
\(321\) 0 0
\(322\) −28.2095 −1.57205
\(323\) 3.67288 0.204365
\(324\) 0 0
\(325\) 8.37159 0.464372
\(326\) 45.0465 2.49489
\(327\) 0 0
\(328\) −92.0317 −5.08160
\(329\) −19.4416 −1.07185
\(330\) 0 0
\(331\) −6.95223 −0.382129 −0.191065 0.981577i \(-0.561194\pi\)
−0.191065 + 0.981577i \(0.561194\pi\)
\(332\) −50.2858 −2.75979
\(333\) 0 0
\(334\) −2.29741 −0.125709
\(335\) −5.10333 −0.278825
\(336\) 0 0
\(337\) 29.4416 1.60379 0.801893 0.597468i \(-0.203826\pi\)
0.801893 + 0.597468i \(0.203826\pi\)
\(338\) 22.1230 1.20333
\(339\) 0 0
\(340\) −19.6477 −1.06555
\(341\) −4.65295 −0.251971
\(342\) 0 0
\(343\) 8.92959 0.482152
\(344\) −75.8905 −4.09174
\(345\) 0 0
\(346\) −11.0017 −0.591454
\(347\) 17.2180 0.924311 0.462155 0.886799i \(-0.347076\pi\)
0.462155 + 0.886799i \(0.347076\pi\)
\(348\) 0 0
\(349\) 4.32405 0.231461 0.115731 0.993281i \(-0.463079\pi\)
0.115731 + 0.993281i \(0.463079\pi\)
\(350\) −34.6583 −1.85257
\(351\) 0 0
\(352\) 14.0001 0.746207
\(353\) −23.4421 −1.24770 −0.623848 0.781545i \(-0.714432\pi\)
−0.623848 + 0.781545i \(0.714432\pi\)
\(354\) 0 0
\(355\) −16.1825 −0.858878
\(356\) 23.2909 1.23441
\(357\) 0 0
\(358\) 5.43031 0.287001
\(359\) −8.20544 −0.433066 −0.216533 0.976275i \(-0.569475\pi\)
−0.216533 + 0.976275i \(0.569475\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −14.0020 −0.735930
\(363\) 0 0
\(364\) 36.7131 1.92429
\(365\) −8.57539 −0.448856
\(366\) 0 0
\(367\) 33.1494 1.73039 0.865193 0.501439i \(-0.167196\pi\)
0.865193 + 0.501439i \(0.167196\pi\)
\(368\) −35.8538 −1.86901
\(369\) 0 0
\(370\) −6.40088 −0.332766
\(371\) −19.0618 −0.989640
\(372\) 0 0
\(373\) −16.5840 −0.858685 −0.429343 0.903142i \(-0.641255\pi\)
−0.429343 + 0.903142i \(0.641255\pi\)
\(374\) −9.74900 −0.504109
\(375\) 0 0
\(376\) −46.6492 −2.40575
\(377\) 15.5050 0.798550
\(378\) 0 0
\(379\) 5.79467 0.297652 0.148826 0.988863i \(-0.452451\pi\)
0.148826 + 0.988863i \(0.452451\pi\)
\(380\) −5.34940 −0.274418
\(381\) 0 0
\(382\) −42.9730 −2.19869
\(383\) −5.89016 −0.300973 −0.150487 0.988612i \(-0.548084\pi\)
−0.150487 + 0.988612i \(0.548084\pi\)
\(384\) 0 0
\(385\) 3.57187 0.182039
\(386\) 53.2508 2.71039
\(387\) 0 0
\(388\) −20.6814 −1.04994
\(389\) −13.8055 −0.699964 −0.349982 0.936756i \(-0.613812\pi\)
−0.349982 + 0.936756i \(0.613812\pi\)
\(390\) 0 0
\(391\) 11.5868 0.585968
\(392\) −35.1581 −1.77575
\(393\) 0 0
\(394\) −63.2784 −3.18792
\(395\) 14.2383 0.716405
\(396\) 0 0
\(397\) 6.48471 0.325458 0.162729 0.986671i \(-0.447970\pi\)
0.162729 + 0.986671i \(0.447970\pi\)
\(398\) 5.04835 0.253051
\(399\) 0 0
\(400\) −44.0502 −2.20251
\(401\) −3.70806 −0.185172 −0.0925858 0.995705i \(-0.529513\pi\)
−0.0925858 + 0.995705i \(0.529513\pi\)
\(402\) 0 0
\(403\) −10.0500 −0.500628
\(404\) 54.8584 2.72931
\(405\) 0 0
\(406\) −64.1908 −3.18573
\(407\) −2.27446 −0.112741
\(408\) 0 0
\(409\) 31.0569 1.53567 0.767833 0.640650i \(-0.221335\pi\)
0.767833 + 0.640650i \(0.221335\pi\)
\(410\) −32.0407 −1.58238
\(411\) 0 0
\(412\) 44.6954 2.20198
\(413\) −46.4717 −2.28673
\(414\) 0 0
\(415\) −10.5672 −0.518722
\(416\) 30.2392 1.48260
\(417\) 0 0
\(418\) −2.65432 −0.129827
\(419\) 11.2871 0.551410 0.275705 0.961242i \(-0.411089\pi\)
0.275705 + 0.961242i \(0.411089\pi\)
\(420\) 0 0
\(421\) 34.5194 1.68237 0.841186 0.540746i \(-0.181858\pi\)
0.841186 + 0.540746i \(0.181858\pi\)
\(422\) −32.6609 −1.58991
\(423\) 0 0
\(424\) −45.7378 −2.22122
\(425\) 14.2356 0.690528
\(426\) 0 0
\(427\) −23.5199 −1.13821
\(428\) 91.6593 4.43052
\(429\) 0 0
\(430\) −26.4212 −1.27414
\(431\) −18.1654 −0.874996 −0.437498 0.899219i \(-0.644135\pi\)
−0.437498 + 0.899219i \(0.644135\pi\)
\(432\) 0 0
\(433\) 10.7804 0.518075 0.259037 0.965867i \(-0.416595\pi\)
0.259037 + 0.965867i \(0.416595\pi\)
\(434\) 41.6071 1.99720
\(435\) 0 0
\(436\) −1.06611 −0.0510573
\(437\) 3.15468 0.150909
\(438\) 0 0
\(439\) −1.38742 −0.0662180 −0.0331090 0.999452i \(-0.510541\pi\)
−0.0331090 + 0.999452i \(0.510541\pi\)
\(440\) 8.57050 0.408583
\(441\) 0 0
\(442\) −21.0571 −1.00159
\(443\) 7.26792 0.345310 0.172655 0.984982i \(-0.444766\pi\)
0.172655 + 0.984982i \(0.444766\pi\)
\(444\) 0 0
\(445\) 4.89440 0.232017
\(446\) −60.6441 −2.87158
\(447\) 0 0
\(448\) −48.6135 −2.29677
\(449\) −21.4298 −1.01134 −0.505668 0.862728i \(-0.668754\pi\)
−0.505668 + 0.862728i \(0.668754\pi\)
\(450\) 0 0
\(451\) −11.3852 −0.536108
\(452\) 1.00073 0.0470703
\(453\) 0 0
\(454\) 10.6346 0.499108
\(455\) 7.71498 0.361684
\(456\) 0 0
\(457\) −17.6312 −0.824754 −0.412377 0.911013i \(-0.635301\pi\)
−0.412377 + 0.911013i \(0.635301\pi\)
\(458\) 65.3026 3.05139
\(459\) 0 0
\(460\) −16.8756 −0.786831
\(461\) −34.2073 −1.59320 −0.796598 0.604510i \(-0.793369\pi\)
−0.796598 + 0.604510i \(0.793369\pi\)
\(462\) 0 0
\(463\) −11.0009 −0.511253 −0.255626 0.966776i \(-0.582282\pi\)
−0.255626 + 0.966776i \(0.582282\pi\)
\(464\) −81.5854 −3.78751
\(465\) 0 0
\(466\) 43.2637 2.00415
\(467\) −4.38453 −0.202892 −0.101446 0.994841i \(-0.532347\pi\)
−0.101446 + 0.994841i \(0.532347\pi\)
\(468\) 0 0
\(469\) 16.2155 0.748764
\(470\) −16.2408 −0.749134
\(471\) 0 0
\(472\) −111.506 −5.13250
\(473\) −9.38838 −0.431678
\(474\) 0 0
\(475\) 3.87587 0.177837
\(476\) 62.4294 2.86145
\(477\) 0 0
\(478\) −40.9249 −1.87186
\(479\) 14.3683 0.656505 0.328253 0.944590i \(-0.393540\pi\)
0.328253 + 0.944590i \(0.393540\pi\)
\(480\) 0 0
\(481\) −4.91266 −0.223998
\(482\) −64.0861 −2.91904
\(483\) 0 0
\(484\) 5.04540 0.229336
\(485\) −4.34604 −0.197344
\(486\) 0 0
\(487\) 1.01712 0.0460899 0.0230449 0.999734i \(-0.492664\pi\)
0.0230449 + 0.999734i \(0.492664\pi\)
\(488\) −56.4348 −2.55468
\(489\) 0 0
\(490\) −12.2403 −0.552959
\(491\) 2.41045 0.108782 0.0543910 0.998520i \(-0.482678\pi\)
0.0543910 + 0.998520i \(0.482678\pi\)
\(492\) 0 0
\(493\) 26.3658 1.18745
\(494\) −5.73313 −0.257946
\(495\) 0 0
\(496\) 52.8820 2.37447
\(497\) 51.4189 2.30645
\(498\) 0 0
\(499\) 4.07426 0.182389 0.0911945 0.995833i \(-0.470931\pi\)
0.0911945 + 0.995833i \(0.470931\pi\)
\(500\) −47.4805 −2.12339
\(501\) 0 0
\(502\) −52.8471 −2.35868
\(503\) 19.9553 0.889762 0.444881 0.895590i \(-0.353246\pi\)
0.444881 + 0.895590i \(0.353246\pi\)
\(504\) 0 0
\(505\) 11.5281 0.512993
\(506\) −8.37352 −0.372249
\(507\) 0 0
\(508\) −18.1801 −0.806613
\(509\) 8.79239 0.389716 0.194858 0.980831i \(-0.437575\pi\)
0.194858 + 0.980831i \(0.437575\pi\)
\(510\) 0 0
\(511\) 27.2478 1.20537
\(512\) 24.6266 1.08835
\(513\) 0 0
\(514\) −59.2247 −2.61229
\(515\) 9.39239 0.413878
\(516\) 0 0
\(517\) −5.77094 −0.253806
\(518\) 20.3384 0.893618
\(519\) 0 0
\(520\) 18.5117 0.811790
\(521\) −27.3483 −1.19815 −0.599075 0.800693i \(-0.704465\pi\)
−0.599075 + 0.800693i \(0.704465\pi\)
\(522\) 0 0
\(523\) −41.7043 −1.82360 −0.911800 0.410634i \(-0.865307\pi\)
−0.911800 + 0.410634i \(0.865307\pi\)
\(524\) −85.1368 −3.71922
\(525\) 0 0
\(526\) 36.6197 1.59670
\(527\) −17.0897 −0.744441
\(528\) 0 0
\(529\) −13.0480 −0.567304
\(530\) −15.9236 −0.691675
\(531\) 0 0
\(532\) 16.9974 0.736930
\(533\) −24.5912 −1.06516
\(534\) 0 0
\(535\) 19.2615 0.832748
\(536\) 38.9083 1.68058
\(537\) 0 0
\(538\) −29.7358 −1.28200
\(539\) −4.34940 −0.187342
\(540\) 0 0
\(541\) −20.7584 −0.892472 −0.446236 0.894915i \(-0.647236\pi\)
−0.446236 + 0.894915i \(0.647236\pi\)
\(542\) −45.4508 −1.95228
\(543\) 0 0
\(544\) 51.4207 2.20464
\(545\) −0.224035 −0.00959659
\(546\) 0 0
\(547\) 5.39396 0.230629 0.115315 0.993329i \(-0.463212\pi\)
0.115315 + 0.993329i \(0.463212\pi\)
\(548\) 30.8193 1.31653
\(549\) 0 0
\(550\) −10.2878 −0.438672
\(551\) 7.17849 0.305814
\(552\) 0 0
\(553\) −45.2412 −1.92385
\(554\) 36.3980 1.54640
\(555\) 0 0
\(556\) 57.1881 2.42532
\(557\) 38.0076 1.61043 0.805217 0.592981i \(-0.202049\pi\)
0.805217 + 0.592981i \(0.202049\pi\)
\(558\) 0 0
\(559\) −20.2782 −0.857677
\(560\) −40.5952 −1.71546
\(561\) 0 0
\(562\) 68.6404 2.89542
\(563\) 26.4313 1.11395 0.556973 0.830530i \(-0.311963\pi\)
0.556973 + 0.830530i \(0.311963\pi\)
\(564\) 0 0
\(565\) 0.210295 0.00884719
\(566\) −76.6330 −3.22112
\(567\) 0 0
\(568\) 123.377 5.17677
\(569\) −40.6105 −1.70248 −0.851239 0.524778i \(-0.824148\pi\)
−0.851239 + 0.524778i \(0.824148\pi\)
\(570\) 0 0
\(571\) 8.44299 0.353328 0.176664 0.984271i \(-0.443469\pi\)
0.176664 + 0.984271i \(0.443469\pi\)
\(572\) 10.8977 0.455656
\(573\) 0 0
\(574\) 101.807 4.24936
\(575\) 12.2271 0.509906
\(576\) 0 0
\(577\) 20.9771 0.873288 0.436644 0.899634i \(-0.356167\pi\)
0.436644 + 0.899634i \(0.356167\pi\)
\(578\) 9.31648 0.387514
\(579\) 0 0
\(580\) −38.4006 −1.59450
\(581\) 33.5766 1.39299
\(582\) 0 0
\(583\) −5.65820 −0.234339
\(584\) 65.3795 2.70542
\(585\) 0 0
\(586\) 16.9338 0.699528
\(587\) 21.1452 0.872756 0.436378 0.899764i \(-0.356261\pi\)
0.436378 + 0.899764i \(0.356261\pi\)
\(588\) 0 0
\(589\) −4.65295 −0.191721
\(590\) −38.8208 −1.59823
\(591\) 0 0
\(592\) 25.8498 1.06242
\(593\) −27.7731 −1.14051 −0.570253 0.821469i \(-0.693155\pi\)
−0.570253 + 0.821469i \(0.693155\pi\)
\(594\) 0 0
\(595\) 13.1191 0.537829
\(596\) −34.0917 −1.39645
\(597\) 0 0
\(598\) −18.0862 −0.739600
\(599\) −23.0129 −0.940281 −0.470140 0.882592i \(-0.655797\pi\)
−0.470140 + 0.882592i \(0.655797\pi\)
\(600\) 0 0
\(601\) 23.3322 0.951738 0.475869 0.879516i \(-0.342134\pi\)
0.475869 + 0.879516i \(0.342134\pi\)
\(602\) 83.9517 3.42162
\(603\) 0 0
\(604\) 32.6049 1.32668
\(605\) 1.06025 0.0431054
\(606\) 0 0
\(607\) −3.22857 −0.131044 −0.0655218 0.997851i \(-0.520871\pi\)
−0.0655218 + 0.997851i \(0.520871\pi\)
\(608\) 14.0001 0.567778
\(609\) 0 0
\(610\) −19.6477 −0.795512
\(611\) −12.4648 −0.504273
\(612\) 0 0
\(613\) −17.2202 −0.695519 −0.347759 0.937584i \(-0.613057\pi\)
−0.347759 + 0.937584i \(0.613057\pi\)
\(614\) −27.6842 −1.11724
\(615\) 0 0
\(616\) −27.2322 −1.09722
\(617\) 27.4776 1.10621 0.553103 0.833113i \(-0.313443\pi\)
0.553103 + 0.833113i \(0.313443\pi\)
\(618\) 0 0
\(619\) −9.53870 −0.383393 −0.191696 0.981454i \(-0.561399\pi\)
−0.191696 + 0.981454i \(0.561399\pi\)
\(620\) 24.8905 0.999625
\(621\) 0 0
\(622\) −21.9306 −0.879337
\(623\) −15.5517 −0.623064
\(624\) 0 0
\(625\) 9.40166 0.376066
\(626\) 64.0133 2.55848
\(627\) 0 0
\(628\) 1.25319 0.0500076
\(629\) −8.35381 −0.333088
\(630\) 0 0
\(631\) 1.16647 0.0464363 0.0232182 0.999730i \(-0.492609\pi\)
0.0232182 + 0.999730i \(0.492609\pi\)
\(632\) −108.554 −4.31804
\(633\) 0 0
\(634\) −31.1430 −1.23685
\(635\) −3.82042 −0.151609
\(636\) 0 0
\(637\) −9.39438 −0.372219
\(638\) −19.0540 −0.754355
\(639\) 0 0
\(640\) −10.9227 −0.431759
\(641\) −10.0249 −0.395960 −0.197980 0.980206i \(-0.563438\pi\)
−0.197980 + 0.980206i \(0.563438\pi\)
\(642\) 0 0
\(643\) −40.7197 −1.60583 −0.802915 0.596094i \(-0.796719\pi\)
−0.802915 + 0.596094i \(0.796719\pi\)
\(644\) 53.6213 2.11298
\(645\) 0 0
\(646\) −9.74900 −0.383569
\(647\) 16.6460 0.654423 0.327211 0.944951i \(-0.393891\pi\)
0.327211 + 0.944951i \(0.393891\pi\)
\(648\) 0 0
\(649\) −13.7944 −0.541477
\(650\) −22.2209 −0.871574
\(651\) 0 0
\(652\) −85.6256 −3.35336
\(653\) 34.7416 1.35954 0.679771 0.733424i \(-0.262079\pi\)
0.679771 + 0.733424i \(0.262079\pi\)
\(654\) 0 0
\(655\) −17.8908 −0.699053
\(656\) 129.396 5.05205
\(657\) 0 0
\(658\) 51.6043 2.01175
\(659\) 22.9537 0.894148 0.447074 0.894497i \(-0.352466\pi\)
0.447074 + 0.894497i \(0.352466\pi\)
\(660\) 0 0
\(661\) −22.7141 −0.883476 −0.441738 0.897144i \(-0.645638\pi\)
−0.441738 + 0.897144i \(0.645638\pi\)
\(662\) 18.4534 0.717213
\(663\) 0 0
\(664\) 80.5651 3.12653
\(665\) 3.57187 0.138511
\(666\) 0 0
\(667\) 22.6459 0.876851
\(668\) 4.36698 0.168964
\(669\) 0 0
\(670\) 13.5459 0.523322
\(671\) −6.98152 −0.269518
\(672\) 0 0
\(673\) 20.9448 0.807364 0.403682 0.914899i \(-0.367730\pi\)
0.403682 + 0.914899i \(0.367730\pi\)
\(674\) −78.1473 −3.01012
\(675\) 0 0
\(676\) −42.0519 −1.61738
\(677\) −11.4153 −0.438726 −0.219363 0.975643i \(-0.570398\pi\)
−0.219363 + 0.975643i \(0.570398\pi\)
\(678\) 0 0
\(679\) 13.8093 0.529952
\(680\) 31.4784 1.20714
\(681\) 0 0
\(682\) 12.3504 0.472921
\(683\) 34.9550 1.33752 0.668759 0.743479i \(-0.266826\pi\)
0.668759 + 0.743479i \(0.266826\pi\)
\(684\) 0 0
\(685\) 6.47643 0.247452
\(686\) −23.7020 −0.904945
\(687\) 0 0
\(688\) 106.701 4.06795
\(689\) −12.2213 −0.465594
\(690\) 0 0
\(691\) −15.8717 −0.603789 −0.301894 0.953341i \(-0.597619\pi\)
−0.301894 + 0.953341i \(0.597619\pi\)
\(692\) 20.9123 0.794967
\(693\) 0 0
\(694\) −45.7020 −1.73483
\(695\) 12.0177 0.455855
\(696\) 0 0
\(697\) −41.8165 −1.58391
\(698\) −11.4774 −0.434426
\(699\) 0 0
\(700\) 65.8795 2.49001
\(701\) 18.5830 0.701869 0.350935 0.936400i \(-0.385864\pi\)
0.350935 + 0.936400i \(0.385864\pi\)
\(702\) 0 0
\(703\) −2.27446 −0.0857828
\(704\) −14.4301 −0.543857
\(705\) 0 0
\(706\) 62.2228 2.34178
\(707\) −36.6298 −1.37760
\(708\) 0 0
\(709\) 23.8063 0.894064 0.447032 0.894518i \(-0.352481\pi\)
0.447032 + 0.894518i \(0.352481\pi\)
\(710\) 42.9535 1.61202
\(711\) 0 0
\(712\) −37.3153 −1.39845
\(713\) −14.6786 −0.549717
\(714\) 0 0
\(715\) 2.29007 0.0856437
\(716\) −10.3221 −0.385755
\(717\) 0 0
\(718\) 21.7798 0.812816
\(719\) −37.0144 −1.38040 −0.690202 0.723616i \(-0.742478\pi\)
−0.690202 + 0.723616i \(0.742478\pi\)
\(720\) 0 0
\(721\) −29.8437 −1.11144
\(722\) −2.65432 −0.0987835
\(723\) 0 0
\(724\) 26.6154 0.989154
\(725\) 27.8229 1.03332
\(726\) 0 0
\(727\) −22.0744 −0.818694 −0.409347 0.912379i \(-0.634243\pi\)
−0.409347 + 0.912379i \(0.634243\pi\)
\(728\) −58.8197 −2.18000
\(729\) 0 0
\(730\) 22.7618 0.842452
\(731\) −34.4824 −1.27538
\(732\) 0 0
\(733\) −18.7090 −0.691031 −0.345515 0.938413i \(-0.612296\pi\)
−0.345515 + 0.938413i \(0.612296\pi\)
\(734\) −87.9891 −3.24774
\(735\) 0 0
\(736\) 44.1658 1.62797
\(737\) 4.81332 0.177301
\(738\) 0 0
\(739\) −5.91630 −0.217635 −0.108817 0.994062i \(-0.534706\pi\)
−0.108817 + 0.994062i \(0.534706\pi\)
\(740\) 12.1670 0.447267
\(741\) 0 0
\(742\) 50.5961 1.85744
\(743\) −14.0422 −0.515159 −0.257579 0.966257i \(-0.582925\pi\)
−0.257579 + 0.966257i \(0.582925\pi\)
\(744\) 0 0
\(745\) −7.16411 −0.262473
\(746\) 44.0191 1.61165
\(747\) 0 0
\(748\) 18.5312 0.677566
\(749\) −61.2023 −2.23628
\(750\) 0 0
\(751\) 4.91166 0.179229 0.0896145 0.995977i \(-0.471437\pi\)
0.0896145 + 0.995977i \(0.471437\pi\)
\(752\) 65.5882 2.39176
\(753\) 0 0
\(754\) −41.1553 −1.49879
\(755\) 6.85168 0.249358
\(756\) 0 0
\(757\) 49.2829 1.79122 0.895609 0.444843i \(-0.146741\pi\)
0.895609 + 0.444843i \(0.146741\pi\)
\(758\) −15.3809 −0.558659
\(759\) 0 0
\(760\) 8.57050 0.310885
\(761\) 7.25922 0.263146 0.131573 0.991306i \(-0.457997\pi\)
0.131573 + 0.991306i \(0.457997\pi\)
\(762\) 0 0
\(763\) 0.711856 0.0257709
\(764\) 81.6843 2.95524
\(765\) 0 0
\(766\) 15.6344 0.564892
\(767\) −29.7949 −1.07583
\(768\) 0 0
\(769\) −21.5123 −0.775755 −0.387877 0.921711i \(-0.626792\pi\)
−0.387877 + 0.921711i \(0.626792\pi\)
\(770\) −9.48087 −0.341667
\(771\) 0 0
\(772\) −101.221 −3.64301
\(773\) −4.54038 −0.163306 −0.0816530 0.996661i \(-0.526020\pi\)
−0.0816530 + 0.996661i \(0.526020\pi\)
\(774\) 0 0
\(775\) −18.0342 −0.647808
\(776\) 33.1346 1.18946
\(777\) 0 0
\(778\) 36.6440 1.31375
\(779\) −11.3852 −0.407917
\(780\) 0 0
\(781\) 15.2629 0.546149
\(782\) −30.7550 −1.09980
\(783\) 0 0
\(784\) 49.4320 1.76543
\(785\) 0.263348 0.00939928
\(786\) 0 0
\(787\) −7.25028 −0.258444 −0.129222 0.991616i \(-0.541248\pi\)
−0.129222 + 0.991616i \(0.541248\pi\)
\(788\) 120.281 4.28484
\(789\) 0 0
\(790\) −37.7929 −1.34461
\(791\) −0.668201 −0.0237585
\(792\) 0 0
\(793\) −15.0796 −0.535491
\(794\) −17.2125 −0.610848
\(795\) 0 0
\(796\) −9.59605 −0.340123
\(797\) −18.8674 −0.668319 −0.334159 0.942517i \(-0.608452\pi\)
−0.334159 + 0.942517i \(0.608452\pi\)
\(798\) 0 0
\(799\) −21.1960 −0.749860
\(800\) 54.2624 1.91847
\(801\) 0 0
\(802\) 9.84236 0.347546
\(803\) 8.08806 0.285422
\(804\) 0 0
\(805\) 11.2681 0.397149
\(806\) 26.6760 0.939622
\(807\) 0 0
\(808\) −87.8911 −3.09200
\(809\) 19.4948 0.685399 0.342700 0.939445i \(-0.388659\pi\)
0.342700 + 0.939445i \(0.388659\pi\)
\(810\) 0 0
\(811\) 4.28635 0.150514 0.0752570 0.997164i \(-0.476022\pi\)
0.0752570 + 0.997164i \(0.476022\pi\)
\(812\) 122.016 4.28191
\(813\) 0 0
\(814\) 6.03713 0.211601
\(815\) −17.9936 −0.630287
\(816\) 0 0
\(817\) −9.38838 −0.328458
\(818\) −82.4350 −2.88227
\(819\) 0 0
\(820\) 60.9039 2.12686
\(821\) −5.37419 −0.187561 −0.0937803 0.995593i \(-0.529895\pi\)
−0.0937803 + 0.995593i \(0.529895\pi\)
\(822\) 0 0
\(823\) 26.3196 0.917445 0.458723 0.888580i \(-0.348307\pi\)
0.458723 + 0.888580i \(0.348307\pi\)
\(824\) −71.6084 −2.49460
\(825\) 0 0
\(826\) 123.351 4.29192
\(827\) 8.11186 0.282077 0.141039 0.990004i \(-0.454956\pi\)
0.141039 + 0.990004i \(0.454956\pi\)
\(828\) 0 0
\(829\) −21.5948 −0.750017 −0.375009 0.927021i \(-0.622360\pi\)
−0.375009 + 0.927021i \(0.622360\pi\)
\(830\) 28.0486 0.973582
\(831\) 0 0
\(832\) −31.1681 −1.08056
\(833\) −15.9748 −0.553495
\(834\) 0 0
\(835\) 0.917688 0.0317579
\(836\) 5.04540 0.174499
\(837\) 0 0
\(838\) −29.9595 −1.03493
\(839\) 4.35109 0.150216 0.0751082 0.997175i \(-0.476070\pi\)
0.0751082 + 0.997175i \(0.476070\pi\)
\(840\) 0 0
\(841\) 22.5308 0.776923
\(842\) −91.6254 −3.15762
\(843\) 0 0
\(844\) 62.0828 2.13698
\(845\) −8.83689 −0.303998
\(846\) 0 0
\(847\) −3.36889 −0.115756
\(848\) 64.3069 2.20831
\(849\) 0 0
\(850\) −37.7858 −1.29604
\(851\) −7.17519 −0.245962
\(852\) 0 0
\(853\) 19.5022 0.667744 0.333872 0.942618i \(-0.391645\pi\)
0.333872 + 0.942618i \(0.391645\pi\)
\(854\) 62.4294 2.13629
\(855\) 0 0
\(856\) −146.851 −5.01928
\(857\) −13.3462 −0.455898 −0.227949 0.973673i \(-0.573202\pi\)
−0.227949 + 0.973673i \(0.573202\pi\)
\(858\) 0 0
\(859\) 21.6016 0.737036 0.368518 0.929621i \(-0.379865\pi\)
0.368518 + 0.929621i \(0.379865\pi\)
\(860\) 50.2221 1.71256
\(861\) 0 0
\(862\) 48.2167 1.64227
\(863\) 2.72425 0.0927345 0.0463673 0.998924i \(-0.485236\pi\)
0.0463673 + 0.998924i \(0.485236\pi\)
\(864\) 0 0
\(865\) 4.39456 0.149420
\(866\) −28.6147 −0.972367
\(867\) 0 0
\(868\) −79.0879 −2.68442
\(869\) −13.4291 −0.455552
\(870\) 0 0
\(871\) 10.3964 0.352269
\(872\) 1.70806 0.0578422
\(873\) 0 0
\(874\) −8.37352 −0.283239
\(875\) 31.7034 1.07177
\(876\) 0 0
\(877\) 1.29130 0.0436041 0.0218021 0.999762i \(-0.493060\pi\)
0.0218021 + 0.999762i \(0.493060\pi\)
\(878\) 3.68266 0.124284
\(879\) 0 0
\(880\) −12.0500 −0.406207
\(881\) −24.9873 −0.841844 −0.420922 0.907097i \(-0.638293\pi\)
−0.420922 + 0.907097i \(0.638293\pi\)
\(882\) 0 0
\(883\) 15.5046 0.521772 0.260886 0.965370i \(-0.415985\pi\)
0.260886 + 0.965370i \(0.415985\pi\)
\(884\) 40.0260 1.34622
\(885\) 0 0
\(886\) −19.2914 −0.648107
\(887\) 9.98029 0.335105 0.167553 0.985863i \(-0.446414\pi\)
0.167553 + 0.985863i \(0.446414\pi\)
\(888\) 0 0
\(889\) 12.1391 0.407134
\(890\) −12.9913 −0.435469
\(891\) 0 0
\(892\) 115.274 3.85966
\(893\) −5.77094 −0.193117
\(894\) 0 0
\(895\) −2.16911 −0.0725053
\(896\) 34.7063 1.15946
\(897\) 0 0
\(898\) 56.8815 1.89816
\(899\) −33.4012 −1.11399
\(900\) 0 0
\(901\) −20.7819 −0.692345
\(902\) 30.2199 1.00621
\(903\) 0 0
\(904\) −1.60331 −0.0533253
\(905\) 5.59303 0.185919
\(906\) 0 0
\(907\) 11.8887 0.394758 0.197379 0.980327i \(-0.436757\pi\)
0.197379 + 0.980327i \(0.436757\pi\)
\(908\) −20.2146 −0.670845
\(909\) 0 0
\(910\) −20.4780 −0.678839
\(911\) 23.7166 0.785767 0.392883 0.919588i \(-0.371478\pi\)
0.392883 + 0.919588i \(0.371478\pi\)
\(912\) 0 0
\(913\) 9.96666 0.329848
\(914\) 46.7989 1.54797
\(915\) 0 0
\(916\) −124.129 −4.10134
\(917\) 56.8471 1.87726
\(918\) 0 0
\(919\) 18.1426 0.598470 0.299235 0.954179i \(-0.403269\pi\)
0.299235 + 0.954179i \(0.403269\pi\)
\(920\) 27.0372 0.891390
\(921\) 0 0
\(922\) 90.7972 2.99024
\(923\) 32.9667 1.08511
\(924\) 0 0
\(925\) −8.81549 −0.289852
\(926\) 29.1997 0.959563
\(927\) 0 0
\(928\) 100.500 3.29906
\(929\) 50.5019 1.65691 0.828457 0.560053i \(-0.189219\pi\)
0.828457 + 0.560053i \(0.189219\pi\)
\(930\) 0 0
\(931\) −4.34940 −0.142546
\(932\) −82.2368 −2.69376
\(933\) 0 0
\(934\) 11.6379 0.380805
\(935\) 3.89418 0.127353
\(936\) 0 0
\(937\) −39.6587 −1.29559 −0.647797 0.761813i \(-0.724309\pi\)
−0.647797 + 0.761813i \(0.724309\pi\)
\(938\) −43.0412 −1.40534
\(939\) 0 0
\(940\) 30.8711 1.00690
\(941\) 34.8809 1.13709 0.568543 0.822653i \(-0.307507\pi\)
0.568543 + 0.822653i \(0.307507\pi\)
\(942\) 0 0
\(943\) −35.9166 −1.16961
\(944\) 156.777 5.10265
\(945\) 0 0
\(946\) 24.9197 0.810210
\(947\) 30.3067 0.984835 0.492417 0.870359i \(-0.336113\pi\)
0.492417 + 0.870359i \(0.336113\pi\)
\(948\) 0 0
\(949\) 17.4696 0.567088
\(950\) −10.2878 −0.333780
\(951\) 0 0
\(952\) −100.021 −3.24169
\(953\) 6.65804 0.215675 0.107838 0.994169i \(-0.465607\pi\)
0.107838 + 0.994169i \(0.465607\pi\)
\(954\) 0 0
\(955\) 17.1653 0.555457
\(956\) 77.7912 2.51595
\(957\) 0 0
\(958\) −38.1381 −1.23219
\(959\) −20.5785 −0.664513
\(960\) 0 0
\(961\) −9.35006 −0.301615
\(962\) 13.0398 0.420419
\(963\) 0 0
\(964\) 121.817 3.92345
\(965\) −21.2707 −0.684729
\(966\) 0 0
\(967\) −20.0020 −0.643221 −0.321611 0.946872i \(-0.604224\pi\)
−0.321611 + 0.946872i \(0.604224\pi\)
\(968\) −8.08346 −0.259812
\(969\) 0 0
\(970\) 11.5358 0.370392
\(971\) 0.491941 0.0157871 0.00789357 0.999969i \(-0.497487\pi\)
0.00789357 + 0.999969i \(0.497487\pi\)
\(972\) 0 0
\(973\) −38.1853 −1.22417
\(974\) −2.69975 −0.0865055
\(975\) 0 0
\(976\) 79.3467 2.53983
\(977\) −23.6515 −0.756679 −0.378339 0.925667i \(-0.623505\pi\)
−0.378339 + 0.925667i \(0.623505\pi\)
\(978\) 0 0
\(979\) −4.61626 −0.147536
\(980\) 23.2666 0.743225
\(981\) 0 0
\(982\) −6.39809 −0.204171
\(983\) −39.5609 −1.26180 −0.630899 0.775865i \(-0.717314\pi\)
−0.630899 + 0.775865i \(0.717314\pi\)
\(984\) 0 0
\(985\) 25.2762 0.805366
\(986\) −69.9831 −2.22872
\(987\) 0 0
\(988\) 10.8977 0.346702
\(989\) −29.6173 −0.941777
\(990\) 0 0
\(991\) −13.6096 −0.432322 −0.216161 0.976358i \(-0.569354\pi\)
−0.216161 + 0.976358i \(0.569354\pi\)
\(992\) −65.1417 −2.06825
\(993\) 0 0
\(994\) −136.482 −4.32895
\(995\) −2.01654 −0.0639285
\(996\) 0 0
\(997\) 41.1821 1.30425 0.652125 0.758111i \(-0.273878\pi\)
0.652125 + 0.758111i \(0.273878\pi\)
\(998\) −10.8144 −0.342323
\(999\) 0 0
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 1881.2.a.k.1.1 5
3.2 odd 2 209.2.a.c.1.5 5
12.11 even 2 3344.2.a.t.1.3 5
15.14 odd 2 5225.2.a.h.1.1 5
33.32 even 2 2299.2.a.n.1.1 5
57.56 even 2 3971.2.a.h.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.a.c.1.5 5 3.2 odd 2
1881.2.a.k.1.1 5 1.1 even 1 trivial
2299.2.a.n.1.1 5 33.32 even 2
3344.2.a.t.1.3 5 12.11 even 2
3971.2.a.h.1.1 5 57.56 even 2
5225.2.a.h.1.1 5 15.14 odd 2