Properties

Label 2009.2.a.r.1.9
Level $2009$
Weight $2$
Character 2009.1
Self dual yes
Analytic conductor $16.042$
Analytic rank $0$
Dimension $17$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2009,2,Mod(1,2009)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2009, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2009.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2009 = 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2009.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: \(16.0419457661\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{17} - 3 x^{16} - 25 x^{15} + 77 x^{14} + 247 x^{13} - 790 x^{12} - 1231 x^{11} + 4173 x^{10} + 3251 x^{9} - 12183 x^{8} - 4259 x^{7} + 19567 x^{6} + 2029 x^{5} - 16136 x^{4} + \cdots - 464 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 287)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(0.381923\) of defining polynomial
Character \(\chi\) \(=\) 2009.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.381923 q^{2} +0.350486 q^{3} -1.85413 q^{4} +2.79025 q^{5} +0.133859 q^{6} -1.47198 q^{8} -2.87716 q^{9} +O(q^{10})\) \(q+0.381923 q^{2} +0.350486 q^{3} -1.85413 q^{4} +2.79025 q^{5} +0.133859 q^{6} -1.47198 q^{8} -2.87716 q^{9} +1.06566 q^{10} +2.56629 q^{11} -0.649848 q^{12} +2.81325 q^{13} +0.977941 q^{15} +3.14608 q^{16} +6.65325 q^{17} -1.09885 q^{18} -2.30819 q^{19} -5.17349 q^{20} +0.980128 q^{22} -4.36622 q^{23} -0.515909 q^{24} +2.78547 q^{25} +1.07445 q^{26} -2.05986 q^{27} -9.00821 q^{29} +0.373499 q^{30} +7.12027 q^{31} +4.14553 q^{32} +0.899449 q^{33} +2.54103 q^{34} +5.33464 q^{36} +11.1900 q^{37} -0.881551 q^{38} +0.986005 q^{39} -4.10720 q^{40} -1.00000 q^{41} +0.740251 q^{43} -4.75825 q^{44} -8.02798 q^{45} -1.66756 q^{46} -0.437347 q^{47} +1.10266 q^{48} +1.06384 q^{50} +2.33187 q^{51} -5.21615 q^{52} +9.64657 q^{53} -0.786709 q^{54} +7.16059 q^{55} -0.808987 q^{57} -3.44045 q^{58} +10.8160 q^{59} -1.81324 q^{60} -10.6452 q^{61} +2.71940 q^{62} -4.70889 q^{64} +7.84967 q^{65} +0.343521 q^{66} +13.5898 q^{67} -12.3360 q^{68} -1.53030 q^{69} +13.9653 q^{71} +4.23513 q^{72} -8.32207 q^{73} +4.27374 q^{74} +0.976269 q^{75} +4.27969 q^{76} +0.376578 q^{78} +5.09976 q^{79} +8.77835 q^{80} +7.90953 q^{81} -0.381923 q^{82} +5.83272 q^{83} +18.5642 q^{85} +0.282719 q^{86} -3.15725 q^{87} -3.77754 q^{88} -4.07202 q^{89} -3.06607 q^{90} +8.09556 q^{92} +2.49555 q^{93} -0.167033 q^{94} -6.44041 q^{95} +1.45295 q^{96} +9.55210 q^{97} -7.38364 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q + 3 q^{2} - q^{3} + 25 q^{4} + q^{5} - 2 q^{6} + 9 q^{8} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 17 q + 3 q^{2} - q^{3} + 25 q^{4} + q^{5} - 2 q^{6} + 9 q^{8} + 26 q^{9} + 2 q^{10} + 15 q^{11} - 4 q^{12} + 5 q^{13} + 24 q^{15} + 33 q^{16} - 4 q^{17} + 10 q^{18} - 5 q^{19} + 26 q^{20} + 16 q^{22} + 12 q^{23} - 16 q^{24} + 24 q^{25} - 31 q^{26} + 11 q^{27} + 14 q^{29} - 33 q^{30} + 3 q^{31} + 16 q^{32} - 4 q^{33} + 24 q^{34} + 57 q^{36} + 24 q^{37} + 45 q^{39} - 36 q^{40} - 17 q^{41} + 14 q^{43} - 9 q^{44} + 21 q^{45} + 44 q^{46} - 19 q^{47} + 60 q^{48} - 4 q^{50} + 2 q^{51} + 25 q^{52} + 4 q^{53} - 68 q^{54} - 9 q^{55} - 12 q^{57} - q^{58} + 27 q^{59} + 66 q^{60} + q^{61} + 23 q^{62} + 75 q^{64} + 22 q^{65} + 16 q^{66} + 49 q^{67} - 45 q^{68} - 12 q^{69} + 40 q^{71} - 23 q^{72} + 14 q^{73} + 33 q^{74} - 27 q^{75} + 9 q^{76} - 12 q^{78} + 61 q^{79} + 82 q^{80} + 53 q^{81} - 3 q^{82} + 18 q^{83} - 13 q^{85} - 4 q^{86} + 17 q^{87} + 74 q^{88} - 18 q^{89} + 20 q^{90} + 28 q^{92} - 36 q^{93} + 5 q^{94} + 20 q^{95} - 148 q^{96} - 26 q^{97} + 38 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.381923 0.270061 0.135030 0.990841i \(-0.456887\pi\)
0.135030 + 0.990841i \(0.456887\pi\)
\(3\) 0.350486 0.202353 0.101177 0.994868i \(-0.467739\pi\)
0.101177 + 0.994868i \(0.467739\pi\)
\(4\) −1.85413 −0.927067
\(5\) 2.79025 1.24784 0.623918 0.781490i \(-0.285540\pi\)
0.623918 + 0.781490i \(0.285540\pi\)
\(6\) 0.133859 0.0546476
\(7\) 0 0
\(8\) −1.47198 −0.520425
\(9\) −2.87716 −0.959053
\(10\) 1.06566 0.336991
\(11\) 2.56629 0.773767 0.386883 0.922129i \(-0.373552\pi\)
0.386883 + 0.922129i \(0.373552\pi\)
\(12\) −0.649848 −0.187595
\(13\) 2.81325 0.780256 0.390128 0.920761i \(-0.372431\pi\)
0.390128 + 0.920761i \(0.372431\pi\)
\(14\) 0 0
\(15\) 0.977941 0.252503
\(16\) 3.14608 0.786521
\(17\) 6.65325 1.61365 0.806825 0.590790i \(-0.201184\pi\)
0.806825 + 0.590790i \(0.201184\pi\)
\(18\) −1.09885 −0.259003
\(19\) −2.30819 −0.529535 −0.264767 0.964312i \(-0.585295\pi\)
−0.264767 + 0.964312i \(0.585295\pi\)
\(20\) −5.17349 −1.15683
\(21\) 0 0
\(22\) 0.980128 0.208964
\(23\) −4.36622 −0.910420 −0.455210 0.890384i \(-0.650436\pi\)
−0.455210 + 0.890384i \(0.650436\pi\)
\(24\) −0.515909 −0.105310
\(25\) 2.78547 0.557095
\(26\) 1.07445 0.210716
\(27\) −2.05986 −0.396420
\(28\) 0 0
\(29\) −9.00821 −1.67278 −0.836392 0.548132i \(-0.815339\pi\)
−0.836392 + 0.548132i \(0.815339\pi\)
\(30\) 0.373499 0.0681912
\(31\) 7.12027 1.27884 0.639419 0.768858i \(-0.279175\pi\)
0.639419 + 0.768858i \(0.279175\pi\)
\(32\) 4.14553 0.732833
\(33\) 0.899449 0.156574
\(34\) 2.54103 0.435783
\(35\) 0 0
\(36\) 5.33464 0.889107
\(37\) 11.1900 1.83963 0.919816 0.392349i \(-0.128338\pi\)
0.919816 + 0.392349i \(0.128338\pi\)
\(38\) −0.881551 −0.143006
\(39\) 0.986005 0.157887
\(40\) −4.10720 −0.649405
\(41\) −1.00000 −0.156174
\(42\) 0 0
\(43\) 0.740251 0.112887 0.0564436 0.998406i \(-0.482024\pi\)
0.0564436 + 0.998406i \(0.482024\pi\)
\(44\) −4.75825 −0.717334
\(45\) −8.02798 −1.19674
\(46\) −1.66756 −0.245869
\(47\) −0.437347 −0.0637936 −0.0318968 0.999491i \(-0.510155\pi\)
−0.0318968 + 0.999491i \(0.510155\pi\)
\(48\) 1.10266 0.159155
\(49\) 0 0
\(50\) 1.06384 0.150449
\(51\) 2.33187 0.326527
\(52\) −5.21615 −0.723350
\(53\) 9.64657 1.32506 0.662529 0.749036i \(-0.269483\pi\)
0.662529 + 0.749036i \(0.269483\pi\)
\(54\) −0.786709 −0.107058
\(55\) 7.16059 0.965534
\(56\) 0 0
\(57\) −0.808987 −0.107153
\(58\) −3.44045 −0.451753
\(59\) 10.8160 1.40812 0.704062 0.710139i \(-0.251368\pi\)
0.704062 + 0.710139i \(0.251368\pi\)
\(60\) −1.81324 −0.234088
\(61\) −10.6452 −1.36298 −0.681491 0.731827i \(-0.738668\pi\)
−0.681491 + 0.731827i \(0.738668\pi\)
\(62\) 2.71940 0.345364
\(63\) 0 0
\(64\) −4.70889 −0.588612
\(65\) 7.84967 0.973631
\(66\) 0.343521 0.0422845
\(67\) 13.5898 1.66026 0.830130 0.557569i \(-0.188266\pi\)
0.830130 + 0.557569i \(0.188266\pi\)
\(68\) −12.3360 −1.49596
\(69\) −1.53030 −0.184226
\(70\) 0 0
\(71\) 13.9653 1.65738 0.828691 0.559707i \(-0.189086\pi\)
0.828691 + 0.559707i \(0.189086\pi\)
\(72\) 4.23513 0.499115
\(73\) −8.32207 −0.974025 −0.487012 0.873395i \(-0.661913\pi\)
−0.487012 + 0.873395i \(0.661913\pi\)
\(74\) 4.27374 0.496812
\(75\) 0.976269 0.112730
\(76\) 4.27969 0.490914
\(77\) 0 0
\(78\) 0.376578 0.0426391
\(79\) 5.09976 0.573768 0.286884 0.957965i \(-0.407381\pi\)
0.286884 + 0.957965i \(0.407381\pi\)
\(80\) 8.77835 0.981449
\(81\) 7.90953 0.878836
\(82\) −0.381923 −0.0421764
\(83\) 5.83272 0.640224 0.320112 0.947380i \(-0.396279\pi\)
0.320112 + 0.947380i \(0.396279\pi\)
\(84\) 0 0
\(85\) 18.5642 2.01357
\(86\) 0.282719 0.0304864
\(87\) −3.15725 −0.338493
\(88\) −3.77754 −0.402688
\(89\) −4.07202 −0.431633 −0.215817 0.976434i \(-0.569241\pi\)
−0.215817 + 0.976434i \(0.569241\pi\)
\(90\) −3.06607 −0.323193
\(91\) 0 0
\(92\) 8.09556 0.844021
\(93\) 2.49555 0.258777
\(94\) −0.167033 −0.0172281
\(95\) −6.44041 −0.660772
\(96\) 1.45295 0.148291
\(97\) 9.55210 0.969869 0.484935 0.874550i \(-0.338843\pi\)
0.484935 + 0.874550i \(0.338843\pi\)
\(98\) 0 0
\(99\) −7.38364 −0.742084
\(100\) −5.16464 −0.516464
\(101\) −9.23725 −0.919141 −0.459571 0.888141i \(-0.651997\pi\)
−0.459571 + 0.888141i \(0.651997\pi\)
\(102\) 0.890596 0.0881821
\(103\) 10.8162 1.06576 0.532878 0.846192i \(-0.321111\pi\)
0.532878 + 0.846192i \(0.321111\pi\)
\(104\) −4.14106 −0.406065
\(105\) 0 0
\(106\) 3.68425 0.357846
\(107\) −10.2115 −0.987185 −0.493592 0.869693i \(-0.664316\pi\)
−0.493592 + 0.869693i \(0.664316\pi\)
\(108\) 3.81926 0.367508
\(109\) −15.4796 −1.48267 −0.741336 0.671134i \(-0.765808\pi\)
−0.741336 + 0.671134i \(0.765808\pi\)
\(110\) 2.73480 0.260753
\(111\) 3.92195 0.372255
\(112\) 0 0
\(113\) 5.39660 0.507670 0.253835 0.967248i \(-0.418308\pi\)
0.253835 + 0.967248i \(0.418308\pi\)
\(114\) −0.308971 −0.0289378
\(115\) −12.1828 −1.13605
\(116\) 16.7024 1.55078
\(117\) −8.09418 −0.748307
\(118\) 4.13088 0.380279
\(119\) 0 0
\(120\) −1.43951 −0.131409
\(121\) −4.41414 −0.401285
\(122\) −4.06566 −0.368087
\(123\) −0.350486 −0.0316022
\(124\) −13.2019 −1.18557
\(125\) −6.17907 −0.552673
\(126\) 0 0
\(127\) 4.53912 0.402782 0.201391 0.979511i \(-0.435454\pi\)
0.201391 + 0.979511i \(0.435454\pi\)
\(128\) −10.0895 −0.891794
\(129\) 0.259447 0.0228431
\(130\) 2.99797 0.262939
\(131\) 12.1667 1.06301 0.531503 0.847057i \(-0.321628\pi\)
0.531503 + 0.847057i \(0.321628\pi\)
\(132\) −1.66770 −0.145155
\(133\) 0 0
\(134\) 5.19027 0.448371
\(135\) −5.74752 −0.494668
\(136\) −9.79348 −0.839784
\(137\) 1.28517 0.109800 0.0548999 0.998492i \(-0.482516\pi\)
0.0548999 + 0.998492i \(0.482516\pi\)
\(138\) −0.584457 −0.0497523
\(139\) −12.1515 −1.03068 −0.515339 0.856986i \(-0.672334\pi\)
−0.515339 + 0.856986i \(0.672334\pi\)
\(140\) 0 0
\(141\) −0.153284 −0.0129088
\(142\) 5.33369 0.447593
\(143\) 7.21963 0.603736
\(144\) −9.05179 −0.754316
\(145\) −25.1351 −2.08736
\(146\) −3.17839 −0.263046
\(147\) 0 0
\(148\) −20.7479 −1.70546
\(149\) 15.2610 1.25023 0.625113 0.780534i \(-0.285053\pi\)
0.625113 + 0.780534i \(0.285053\pi\)
\(150\) 0.372860 0.0304439
\(151\) −4.51478 −0.367407 −0.183704 0.982982i \(-0.558809\pi\)
−0.183704 + 0.982982i \(0.558809\pi\)
\(152\) 3.39762 0.275583
\(153\) −19.1425 −1.54758
\(154\) 0 0
\(155\) 19.8673 1.59578
\(156\) −1.82819 −0.146372
\(157\) 15.5519 1.24118 0.620589 0.784136i \(-0.286893\pi\)
0.620589 + 0.784136i \(0.286893\pi\)
\(158\) 1.94772 0.154952
\(159\) 3.38099 0.268130
\(160\) 11.5671 0.914456
\(161\) 0 0
\(162\) 3.02083 0.237339
\(163\) 1.61931 0.126834 0.0634172 0.997987i \(-0.479800\pi\)
0.0634172 + 0.997987i \(0.479800\pi\)
\(164\) 1.85413 0.144784
\(165\) 2.50969 0.195379
\(166\) 2.22765 0.172899
\(167\) 20.8888 1.61643 0.808214 0.588890i \(-0.200435\pi\)
0.808214 + 0.588890i \(0.200435\pi\)
\(168\) 0 0
\(169\) −5.08561 −0.391201
\(170\) 7.09011 0.543786
\(171\) 6.64103 0.507852
\(172\) −1.37252 −0.104654
\(173\) −7.19393 −0.546944 −0.273472 0.961880i \(-0.588172\pi\)
−0.273472 + 0.961880i \(0.588172\pi\)
\(174\) −1.20583 −0.0914136
\(175\) 0 0
\(176\) 8.07378 0.608584
\(177\) 3.79085 0.284938
\(178\) −1.55520 −0.116567
\(179\) 1.36999 0.102398 0.0511991 0.998688i \(-0.483696\pi\)
0.0511991 + 0.998688i \(0.483696\pi\)
\(180\) 14.8850 1.10946
\(181\) −8.57407 −0.637306 −0.318653 0.947872i \(-0.603230\pi\)
−0.318653 + 0.947872i \(0.603230\pi\)
\(182\) 0 0
\(183\) −3.73100 −0.275803
\(184\) 6.42701 0.473805
\(185\) 31.2230 2.29556
\(186\) 0.953110 0.0698854
\(187\) 17.0742 1.24859
\(188\) 0.810900 0.0591410
\(189\) 0 0
\(190\) −2.45974 −0.178449
\(191\) 2.48778 0.180009 0.0900046 0.995941i \(-0.471312\pi\)
0.0900046 + 0.995941i \(0.471312\pi\)
\(192\) −1.65040 −0.119107
\(193\) 4.18984 0.301592 0.150796 0.988565i \(-0.451816\pi\)
0.150796 + 0.988565i \(0.451816\pi\)
\(194\) 3.64817 0.261923
\(195\) 2.75120 0.197017
\(196\) 0 0
\(197\) −8.70173 −0.619972 −0.309986 0.950741i \(-0.600324\pi\)
−0.309986 + 0.950741i \(0.600324\pi\)
\(198\) −2.81998 −0.200408
\(199\) −3.29100 −0.233293 −0.116646 0.993174i \(-0.537214\pi\)
−0.116646 + 0.993174i \(0.537214\pi\)
\(200\) −4.10017 −0.289926
\(201\) 4.76304 0.335959
\(202\) −3.52792 −0.248224
\(203\) 0 0
\(204\) −4.32360 −0.302713
\(205\) −2.79025 −0.194879
\(206\) 4.13097 0.287818
\(207\) 12.5623 0.873141
\(208\) 8.85073 0.613688
\(209\) −5.92349 −0.409736
\(210\) 0 0
\(211\) −13.2898 −0.914905 −0.457452 0.889234i \(-0.651238\pi\)
−0.457452 + 0.889234i \(0.651238\pi\)
\(212\) −17.8860 −1.22842
\(213\) 4.89465 0.335376
\(214\) −3.90002 −0.266600
\(215\) 2.06548 0.140865
\(216\) 3.03208 0.206307
\(217\) 0 0
\(218\) −5.91200 −0.400411
\(219\) −2.91677 −0.197097
\(220\) −13.2767 −0.895115
\(221\) 18.7173 1.25906
\(222\) 1.49789 0.100531
\(223\) −0.947240 −0.0634319 −0.0317159 0.999497i \(-0.510097\pi\)
−0.0317159 + 0.999497i \(0.510097\pi\)
\(224\) 0 0
\(225\) −8.01425 −0.534284
\(226\) 2.06109 0.137102
\(227\) −0.0180577 −0.00119853 −0.000599267 1.00000i \(-0.500191\pi\)
−0.000599267 1.00000i \(0.500191\pi\)
\(228\) 1.49997 0.0993380
\(229\) −3.88646 −0.256824 −0.128412 0.991721i \(-0.540988\pi\)
−0.128412 + 0.991721i \(0.540988\pi\)
\(230\) −4.65291 −0.306804
\(231\) 0 0
\(232\) 13.2599 0.870558
\(233\) −14.5343 −0.952171 −0.476085 0.879399i \(-0.657945\pi\)
−0.476085 + 0.879399i \(0.657945\pi\)
\(234\) −3.09135 −0.202088
\(235\) −1.22031 −0.0796040
\(236\) −20.0543 −1.30542
\(237\) 1.78739 0.116104
\(238\) 0 0
\(239\) −12.0608 −0.780145 −0.390073 0.920784i \(-0.627550\pi\)
−0.390073 + 0.920784i \(0.627550\pi\)
\(240\) 3.07669 0.198599
\(241\) −2.65986 −0.171336 −0.0856682 0.996324i \(-0.527302\pi\)
−0.0856682 + 0.996324i \(0.527302\pi\)
\(242\) −1.68586 −0.108371
\(243\) 8.95176 0.574256
\(244\) 19.7377 1.26358
\(245\) 0 0
\(246\) −0.133859 −0.00853452
\(247\) −6.49352 −0.413172
\(248\) −10.4809 −0.665540
\(249\) 2.04429 0.129551
\(250\) −2.35993 −0.149255
\(251\) −12.4762 −0.787489 −0.393744 0.919220i \(-0.628820\pi\)
−0.393744 + 0.919220i \(0.628820\pi\)
\(252\) 0 0
\(253\) −11.2050 −0.704453
\(254\) 1.73360 0.108775
\(255\) 6.50649 0.407452
\(256\) 5.56437 0.347773
\(257\) −2.73873 −0.170838 −0.0854188 0.996345i \(-0.527223\pi\)
−0.0854188 + 0.996345i \(0.527223\pi\)
\(258\) 0.0990890 0.00616901
\(259\) 0 0
\(260\) −14.5543 −0.902622
\(261\) 25.9181 1.60429
\(262\) 4.64673 0.287076
\(263\) −0.230482 −0.0142121 −0.00710607 0.999975i \(-0.502262\pi\)
−0.00710607 + 0.999975i \(0.502262\pi\)
\(264\) −1.32398 −0.0814850
\(265\) 26.9163 1.65346
\(266\) 0 0
\(267\) −1.42719 −0.0873423
\(268\) −25.1974 −1.53917
\(269\) 8.89980 0.542630 0.271315 0.962491i \(-0.412541\pi\)
0.271315 + 0.962491i \(0.412541\pi\)
\(270\) −2.19511 −0.133590
\(271\) −15.7432 −0.956329 −0.478165 0.878270i \(-0.658698\pi\)
−0.478165 + 0.878270i \(0.658698\pi\)
\(272\) 20.9317 1.26917
\(273\) 0 0
\(274\) 0.490838 0.0296526
\(275\) 7.14835 0.431061
\(276\) 2.83738 0.170790
\(277\) 14.8431 0.891833 0.445917 0.895074i \(-0.352878\pi\)
0.445917 + 0.895074i \(0.352878\pi\)
\(278\) −4.64095 −0.278346
\(279\) −20.4862 −1.22647
\(280\) 0 0
\(281\) −7.52162 −0.448702 −0.224351 0.974508i \(-0.572026\pi\)
−0.224351 + 0.974508i \(0.572026\pi\)
\(282\) −0.0585427 −0.00348617
\(283\) −0.386790 −0.0229923 −0.0114961 0.999934i \(-0.503659\pi\)
−0.0114961 + 0.999934i \(0.503659\pi\)
\(284\) −25.8936 −1.53650
\(285\) −2.25727 −0.133709
\(286\) 2.75735 0.163045
\(287\) 0 0
\(288\) −11.9274 −0.702826
\(289\) 27.2658 1.60387
\(290\) −9.59970 −0.563714
\(291\) 3.34788 0.196256
\(292\) 15.4302 0.902986
\(293\) 0.553673 0.0323459 0.0161729 0.999869i \(-0.494852\pi\)
0.0161729 + 0.999869i \(0.494852\pi\)
\(294\) 0 0
\(295\) 30.1793 1.75711
\(296\) −16.4716 −0.957391
\(297\) −5.28621 −0.306737
\(298\) 5.82852 0.337637
\(299\) −12.2833 −0.710360
\(300\) −1.81013 −0.104508
\(301\) 0 0
\(302\) −1.72430 −0.0992222
\(303\) −3.23753 −0.185991
\(304\) −7.26175 −0.416490
\(305\) −29.7028 −1.70078
\(306\) −7.31096 −0.417940
\(307\) −31.1079 −1.77542 −0.887710 0.460403i \(-0.847705\pi\)
−0.887710 + 0.460403i \(0.847705\pi\)
\(308\) 0 0
\(309\) 3.79094 0.215659
\(310\) 7.58779 0.430958
\(311\) −22.5749 −1.28011 −0.640053 0.768331i \(-0.721087\pi\)
−0.640053 + 0.768331i \(0.721087\pi\)
\(312\) −1.45138 −0.0821684
\(313\) 20.1533 1.13913 0.569565 0.821946i \(-0.307112\pi\)
0.569565 + 0.821946i \(0.307112\pi\)
\(314\) 5.93964 0.335193
\(315\) 0 0
\(316\) −9.45564 −0.531921
\(317\) 1.47763 0.0829920 0.0414960 0.999139i \(-0.486788\pi\)
0.0414960 + 0.999139i \(0.486788\pi\)
\(318\) 1.29128 0.0724113
\(319\) −23.1177 −1.29434
\(320\) −13.1390 −0.734491
\(321\) −3.57899 −0.199760
\(322\) 0 0
\(323\) −15.3570 −0.854484
\(324\) −14.6653 −0.814740
\(325\) 7.83624 0.434676
\(326\) 0.618454 0.0342530
\(327\) −5.42536 −0.300023
\(328\) 1.47198 0.0812767
\(329\) 0 0
\(330\) 0.958507 0.0527641
\(331\) −9.90120 −0.544219 −0.272110 0.962266i \(-0.587721\pi\)
−0.272110 + 0.962266i \(0.587721\pi\)
\(332\) −10.8146 −0.593531
\(333\) −32.1956 −1.76431
\(334\) 7.97794 0.436533
\(335\) 37.9189 2.07173
\(336\) 0 0
\(337\) −5.08380 −0.276932 −0.138466 0.990367i \(-0.544217\pi\)
−0.138466 + 0.990367i \(0.544217\pi\)
\(338\) −1.94231 −0.105648
\(339\) 1.89143 0.102729
\(340\) −34.4205 −1.86672
\(341\) 18.2727 0.989523
\(342\) 2.53636 0.137151
\(343\) 0 0
\(344\) −1.08964 −0.0587493
\(345\) −4.26991 −0.229884
\(346\) −2.74753 −0.147708
\(347\) 1.43251 0.0769011 0.0384506 0.999261i \(-0.487758\pi\)
0.0384506 + 0.999261i \(0.487758\pi\)
\(348\) 5.85397 0.313806
\(349\) −30.5990 −1.63792 −0.818962 0.573847i \(-0.805450\pi\)
−0.818962 + 0.573847i \(0.805450\pi\)
\(350\) 0 0
\(351\) −5.79491 −0.309309
\(352\) 10.6387 0.567042
\(353\) −11.6733 −0.621307 −0.310653 0.950523i \(-0.600548\pi\)
−0.310653 + 0.950523i \(0.600548\pi\)
\(354\) 1.44782 0.0769505
\(355\) 38.9667 2.06814
\(356\) 7.55008 0.400153
\(357\) 0 0
\(358\) 0.523233 0.0276537
\(359\) −14.4087 −0.760462 −0.380231 0.924891i \(-0.624156\pi\)
−0.380231 + 0.924891i \(0.624156\pi\)
\(360\) 11.8171 0.622814
\(361\) −13.6723 −0.719593
\(362\) −3.27464 −0.172111
\(363\) −1.54709 −0.0812012
\(364\) 0 0
\(365\) −23.2206 −1.21542
\(366\) −1.42496 −0.0744836
\(367\) −33.8514 −1.76703 −0.883513 0.468406i \(-0.844828\pi\)
−0.883513 + 0.468406i \(0.844828\pi\)
\(368\) −13.7365 −0.716064
\(369\) 2.87716 0.149779
\(370\) 11.9248 0.619940
\(371\) 0 0
\(372\) −4.62709 −0.239904
\(373\) −14.1001 −0.730077 −0.365039 0.930992i \(-0.618944\pi\)
−0.365039 + 0.930992i \(0.618944\pi\)
\(374\) 6.52104 0.337195
\(375\) −2.16568 −0.111835
\(376\) 0.643768 0.0331998
\(377\) −25.3424 −1.30520
\(378\) 0 0
\(379\) 33.0376 1.69703 0.848513 0.529175i \(-0.177499\pi\)
0.848513 + 0.529175i \(0.177499\pi\)
\(380\) 11.9414 0.612581
\(381\) 1.59090 0.0815041
\(382\) 0.950140 0.0486134
\(383\) −6.88197 −0.351652 −0.175826 0.984421i \(-0.556260\pi\)
−0.175826 + 0.984421i \(0.556260\pi\)
\(384\) −3.53623 −0.180457
\(385\) 0 0
\(386\) 1.60020 0.0814480
\(387\) −2.12982 −0.108265
\(388\) −17.7109 −0.899134
\(389\) 27.6030 1.39953 0.699765 0.714373i \(-0.253288\pi\)
0.699765 + 0.714373i \(0.253288\pi\)
\(390\) 1.05075 0.0532066
\(391\) −29.0496 −1.46910
\(392\) 0 0
\(393\) 4.26424 0.215102
\(394\) −3.32339 −0.167430
\(395\) 14.2296 0.715968
\(396\) 13.6903 0.687961
\(397\) −25.8854 −1.29915 −0.649576 0.760297i \(-0.725054\pi\)
−0.649576 + 0.760297i \(0.725054\pi\)
\(398\) −1.25691 −0.0630032
\(399\) 0 0
\(400\) 8.76334 0.438167
\(401\) 3.79311 0.189419 0.0947093 0.995505i \(-0.469808\pi\)
0.0947093 + 0.995505i \(0.469808\pi\)
\(402\) 1.81912 0.0907292
\(403\) 20.0311 0.997821
\(404\) 17.1271 0.852106
\(405\) 22.0695 1.09664
\(406\) 0 0
\(407\) 28.7170 1.42345
\(408\) −3.43247 −0.169933
\(409\) 25.0191 1.23712 0.618558 0.785739i \(-0.287717\pi\)
0.618558 + 0.785739i \(0.287717\pi\)
\(410\) −1.06566 −0.0526292
\(411\) 0.450435 0.0222183
\(412\) −20.0548 −0.988027
\(413\) 0 0
\(414\) 4.79784 0.235801
\(415\) 16.2747 0.798895
\(416\) 11.6624 0.571797
\(417\) −4.25894 −0.208561
\(418\) −2.26232 −0.110654
\(419\) −23.1834 −1.13258 −0.566291 0.824205i \(-0.691622\pi\)
−0.566291 + 0.824205i \(0.691622\pi\)
\(420\) 0 0
\(421\) −7.00007 −0.341163 −0.170581 0.985344i \(-0.554565\pi\)
−0.170581 + 0.985344i \(0.554565\pi\)
\(422\) −5.07567 −0.247080
\(423\) 1.25832 0.0611815
\(424\) −14.1996 −0.689594
\(425\) 18.5325 0.898956
\(426\) 1.86938 0.0905719
\(427\) 0 0
\(428\) 18.9335 0.915186
\(429\) 2.53038 0.122168
\(430\) 0.788856 0.0380420
\(431\) −18.9909 −0.914759 −0.457379 0.889272i \(-0.651212\pi\)
−0.457379 + 0.889272i \(0.651212\pi\)
\(432\) −6.48049 −0.311793
\(433\) −1.94264 −0.0933571 −0.0466786 0.998910i \(-0.514864\pi\)
−0.0466786 + 0.998910i \(0.514864\pi\)
\(434\) 0 0
\(435\) −8.80951 −0.422383
\(436\) 28.7012 1.37454
\(437\) 10.0781 0.482099
\(438\) −1.11398 −0.0532281
\(439\) −20.0057 −0.954818 −0.477409 0.878681i \(-0.658424\pi\)
−0.477409 + 0.878681i \(0.658424\pi\)
\(440\) −10.5403 −0.502488
\(441\) 0 0
\(442\) 7.14856 0.340023
\(443\) −2.61996 −0.124478 −0.0622390 0.998061i \(-0.519824\pi\)
−0.0622390 + 0.998061i \(0.519824\pi\)
\(444\) −7.27183 −0.345106
\(445\) −11.3619 −0.538608
\(446\) −0.361773 −0.0171304
\(447\) 5.34875 0.252987
\(448\) 0 0
\(449\) −14.8631 −0.701434 −0.350717 0.936482i \(-0.614062\pi\)
−0.350717 + 0.936482i \(0.614062\pi\)
\(450\) −3.06083 −0.144289
\(451\) −2.56629 −0.120842
\(452\) −10.0060 −0.470644
\(453\) −1.58236 −0.0743460
\(454\) −0.00689667 −0.000323677 0
\(455\) 0 0
\(456\) 1.19082 0.0557651
\(457\) −35.4909 −1.66020 −0.830098 0.557618i \(-0.811715\pi\)
−0.830098 + 0.557618i \(0.811715\pi\)
\(458\) −1.48433 −0.0693582
\(459\) −13.7048 −0.639684
\(460\) 22.5886 1.05320
\(461\) −26.2318 −1.22174 −0.610868 0.791733i \(-0.709179\pi\)
−0.610868 + 0.791733i \(0.709179\pi\)
\(462\) 0 0
\(463\) 32.8514 1.52673 0.763366 0.645966i \(-0.223545\pi\)
0.763366 + 0.645966i \(0.223545\pi\)
\(464\) −28.3406 −1.31568
\(465\) 6.96321 0.322911
\(466\) −5.55097 −0.257144
\(467\) 3.27993 0.151777 0.0758886 0.997116i \(-0.475821\pi\)
0.0758886 + 0.997116i \(0.475821\pi\)
\(468\) 15.0077 0.693731
\(469\) 0 0
\(470\) −0.466063 −0.0214979
\(471\) 5.45073 0.251156
\(472\) −15.9210 −0.732822
\(473\) 1.89970 0.0873483
\(474\) 0.682647 0.0313550
\(475\) −6.42940 −0.295001
\(476\) 0 0
\(477\) −27.7547 −1.27080
\(478\) −4.60628 −0.210686
\(479\) 6.31527 0.288552 0.144276 0.989537i \(-0.453915\pi\)
0.144276 + 0.989537i \(0.453915\pi\)
\(480\) 4.05409 0.185043
\(481\) 31.4804 1.43538
\(482\) −1.01586 −0.0462712
\(483\) 0 0
\(484\) 8.18440 0.372018
\(485\) 26.6527 1.21024
\(486\) 3.41889 0.155084
\(487\) 13.9699 0.633039 0.316519 0.948586i \(-0.397486\pi\)
0.316519 + 0.948586i \(0.397486\pi\)
\(488\) 15.6696 0.709329
\(489\) 0.567546 0.0256653
\(490\) 0 0
\(491\) −27.9855 −1.26297 −0.631484 0.775389i \(-0.717554\pi\)
−0.631484 + 0.775389i \(0.717554\pi\)
\(492\) 0.649848 0.0292974
\(493\) −59.9339 −2.69929
\(494\) −2.48003 −0.111582
\(495\) −20.6022 −0.925999
\(496\) 22.4010 1.00583
\(497\) 0 0
\(498\) 0.780760 0.0349867
\(499\) −6.61178 −0.295984 −0.147992 0.988989i \(-0.547281\pi\)
−0.147992 + 0.988989i \(0.547281\pi\)
\(500\) 11.4568 0.512365
\(501\) 7.32124 0.327089
\(502\) −4.76494 −0.212670
\(503\) 18.2722 0.814717 0.407358 0.913268i \(-0.366450\pi\)
0.407358 + 0.913268i \(0.366450\pi\)
\(504\) 0 0
\(505\) −25.7742 −1.14694
\(506\) −4.27945 −0.190245
\(507\) −1.78243 −0.0791607
\(508\) −8.41614 −0.373406
\(509\) −0.735146 −0.0325848 −0.0162924 0.999867i \(-0.505186\pi\)
−0.0162924 + 0.999867i \(0.505186\pi\)
\(510\) 2.48498 0.110037
\(511\) 0 0
\(512\) 22.3042 0.985714
\(513\) 4.75455 0.209918
\(514\) −1.04599 −0.0461365
\(515\) 30.1800 1.32989
\(516\) −0.481050 −0.0211771
\(517\) −1.12236 −0.0493614
\(518\) 0 0
\(519\) −2.52137 −0.110676
\(520\) −11.5546 −0.506702
\(521\) −39.6253 −1.73602 −0.868008 0.496550i \(-0.834600\pi\)
−0.868008 + 0.496550i \(0.834600\pi\)
\(522\) 9.89872 0.433255
\(523\) 30.1005 1.31620 0.658101 0.752930i \(-0.271360\pi\)
0.658101 + 0.752930i \(0.271360\pi\)
\(524\) −22.5586 −0.985477
\(525\) 0 0
\(526\) −0.0880265 −0.00383814
\(527\) 47.3730 2.06360
\(528\) 2.82974 0.123149
\(529\) −3.93611 −0.171135
\(530\) 10.2800 0.446533
\(531\) −31.1194 −1.35047
\(532\) 0 0
\(533\) −2.81325 −0.121855
\(534\) −0.545076 −0.0235877
\(535\) −28.4926 −1.23184
\(536\) −20.0040 −0.864041
\(537\) 0.480163 0.0207206
\(538\) 3.39904 0.146543
\(539\) 0 0
\(540\) 10.6567 0.458590
\(541\) 16.2139 0.697088 0.348544 0.937292i \(-0.386676\pi\)
0.348544 + 0.937292i \(0.386676\pi\)
\(542\) −6.01268 −0.258267
\(543\) −3.00509 −0.128961
\(544\) 27.5813 1.18254
\(545\) −43.1918 −1.85013
\(546\) 0 0
\(547\) 40.2961 1.72294 0.861468 0.507812i \(-0.169546\pi\)
0.861468 + 0.507812i \(0.169546\pi\)
\(548\) −2.38289 −0.101792
\(549\) 30.6280 1.30717
\(550\) 2.73012 0.116413
\(551\) 20.7927 0.885797
\(552\) 2.25257 0.0958759
\(553\) 0 0
\(554\) 5.66891 0.240849
\(555\) 10.9432 0.464513
\(556\) 22.5306 0.955509
\(557\) −17.3769 −0.736283 −0.368141 0.929770i \(-0.620006\pi\)
−0.368141 + 0.929770i \(0.620006\pi\)
\(558\) −7.82414 −0.331222
\(559\) 2.08251 0.0880809
\(560\) 0 0
\(561\) 5.98426 0.252656
\(562\) −2.87268 −0.121177
\(563\) 25.7302 1.08440 0.542199 0.840250i \(-0.317592\pi\)
0.542199 + 0.840250i \(0.317592\pi\)
\(564\) 0.284209 0.0119674
\(565\) 15.0579 0.633489
\(566\) −0.147724 −0.00620931
\(567\) 0 0
\(568\) −20.5568 −0.862543
\(569\) 19.1948 0.804686 0.402343 0.915489i \(-0.368196\pi\)
0.402343 + 0.915489i \(0.368196\pi\)
\(570\) −0.862105 −0.0361096
\(571\) 3.03484 0.127004 0.0635020 0.997982i \(-0.479773\pi\)
0.0635020 + 0.997982i \(0.479773\pi\)
\(572\) −13.3862 −0.559704
\(573\) 0.871931 0.0364254
\(574\) 0 0
\(575\) −12.1620 −0.507190
\(576\) 13.5482 0.564510
\(577\) 19.4877 0.811282 0.405641 0.914033i \(-0.367048\pi\)
0.405641 + 0.914033i \(0.367048\pi\)
\(578\) 10.4134 0.433142
\(579\) 1.46848 0.0610280
\(580\) 46.6039 1.93512
\(581\) 0 0
\(582\) 1.27863 0.0530010
\(583\) 24.7559 1.02529
\(584\) 12.2500 0.506907
\(585\) −22.5847 −0.933764
\(586\) 0.211460 0.00873535
\(587\) 26.7821 1.10541 0.552707 0.833375i \(-0.313595\pi\)
0.552707 + 0.833375i \(0.313595\pi\)
\(588\) 0 0
\(589\) −16.4349 −0.677189
\(590\) 11.5262 0.474525
\(591\) −3.04983 −0.125453
\(592\) 35.2048 1.44691
\(593\) −6.46300 −0.265403 −0.132702 0.991156i \(-0.542365\pi\)
−0.132702 + 0.991156i \(0.542365\pi\)
\(594\) −2.01893 −0.0828376
\(595\) 0 0
\(596\) −28.2959 −1.15904
\(597\) −1.15345 −0.0472075
\(598\) −4.69127 −0.191840
\(599\) 12.0599 0.492753 0.246376 0.969174i \(-0.420760\pi\)
0.246376 + 0.969174i \(0.420760\pi\)
\(600\) −1.43705 −0.0586674
\(601\) 1.73965 0.0709620 0.0354810 0.999370i \(-0.488704\pi\)
0.0354810 + 0.999370i \(0.488704\pi\)
\(602\) 0 0
\(603\) −39.1001 −1.59228
\(604\) 8.37100 0.340611
\(605\) −12.3165 −0.500738
\(606\) −1.23649 −0.0502288
\(607\) −31.4098 −1.27488 −0.637442 0.770498i \(-0.720008\pi\)
−0.637442 + 0.770498i \(0.720008\pi\)
\(608\) −9.56867 −0.388061
\(609\) 0 0
\(610\) −11.3442 −0.459313
\(611\) −1.23037 −0.0497753
\(612\) 35.4927 1.43471
\(613\) 11.1482 0.450271 0.225135 0.974327i \(-0.427718\pi\)
0.225135 + 0.974327i \(0.427718\pi\)
\(614\) −11.8808 −0.479471
\(615\) −0.977941 −0.0394344
\(616\) 0 0
\(617\) −13.4284 −0.540608 −0.270304 0.962775i \(-0.587124\pi\)
−0.270304 + 0.962775i \(0.587124\pi\)
\(618\) 1.44785 0.0582409
\(619\) 15.1941 0.610700 0.305350 0.952240i \(-0.401226\pi\)
0.305350 + 0.952240i \(0.401226\pi\)
\(620\) −36.8367 −1.47940
\(621\) 8.99381 0.360909
\(622\) −8.62188 −0.345706
\(623\) 0 0
\(624\) 3.10205 0.124182
\(625\) −31.1685 −1.24674
\(626\) 7.69701 0.307634
\(627\) −2.07610 −0.0829114
\(628\) −28.8354 −1.15066
\(629\) 74.4502 2.96852
\(630\) 0 0
\(631\) 31.1970 1.24193 0.620966 0.783837i \(-0.286740\pi\)
0.620966 + 0.783837i \(0.286740\pi\)
\(632\) −7.50676 −0.298603
\(633\) −4.65787 −0.185134
\(634\) 0.564342 0.0224129
\(635\) 12.6653 0.502606
\(636\) −6.26880 −0.248574
\(637\) 0 0
\(638\) −8.82920 −0.349551
\(639\) −40.1805 −1.58952
\(640\) −28.1522 −1.11281
\(641\) 36.0536 1.42403 0.712016 0.702163i \(-0.247782\pi\)
0.712016 + 0.702163i \(0.247782\pi\)
\(642\) −1.36690 −0.0539472
\(643\) 33.5678 1.32378 0.661892 0.749600i \(-0.269754\pi\)
0.661892 + 0.749600i \(0.269754\pi\)
\(644\) 0 0
\(645\) 0.723922 0.0285044
\(646\) −5.86518 −0.230762
\(647\) 0.274010 0.0107724 0.00538621 0.999985i \(-0.498286\pi\)
0.00538621 + 0.999985i \(0.498286\pi\)
\(648\) −11.6427 −0.457368
\(649\) 27.7570 1.08956
\(650\) 2.99284 0.117389
\(651\) 0 0
\(652\) −3.00242 −0.117584
\(653\) −0.861511 −0.0337135 −0.0168568 0.999858i \(-0.505366\pi\)
−0.0168568 + 0.999858i \(0.505366\pi\)
\(654\) −2.07207 −0.0810244
\(655\) 33.9480 1.32646
\(656\) −3.14608 −0.122834
\(657\) 23.9439 0.934141
\(658\) 0 0
\(659\) −16.9842 −0.661609 −0.330805 0.943699i \(-0.607320\pi\)
−0.330805 + 0.943699i \(0.607320\pi\)
\(660\) −4.65329 −0.181129
\(661\) −19.9732 −0.776868 −0.388434 0.921477i \(-0.626984\pi\)
−0.388434 + 0.921477i \(0.626984\pi\)
\(662\) −3.78150 −0.146972
\(663\) 6.56014 0.254775
\(664\) −8.58567 −0.333189
\(665\) 0 0
\(666\) −12.2962 −0.476469
\(667\) 39.3319 1.52294
\(668\) −38.7307 −1.49854
\(669\) −0.331994 −0.0128356
\(670\) 14.4821 0.559494
\(671\) −27.3188 −1.05463
\(672\) 0 0
\(673\) 14.4496 0.556992 0.278496 0.960437i \(-0.410164\pi\)
0.278496 + 0.960437i \(0.410164\pi\)
\(674\) −1.94162 −0.0747885
\(675\) −5.73769 −0.220844
\(676\) 9.42941 0.362670
\(677\) 17.6835 0.679631 0.339815 0.940492i \(-0.389635\pi\)
0.339815 + 0.940492i \(0.389635\pi\)
\(678\) 0.722382 0.0277429
\(679\) 0 0
\(680\) −27.3262 −1.04791
\(681\) −0.00632897 −0.000242527 0
\(682\) 6.97878 0.267231
\(683\) 1.11202 0.0425504 0.0212752 0.999774i \(-0.493227\pi\)
0.0212752 + 0.999774i \(0.493227\pi\)
\(684\) −12.3134 −0.470813
\(685\) 3.58595 0.137012
\(686\) 0 0
\(687\) −1.36215 −0.0519692
\(688\) 2.32889 0.0887881
\(689\) 27.1382 1.03388
\(690\) −1.63078 −0.0620827
\(691\) −15.2935 −0.581794 −0.290897 0.956754i \(-0.593954\pi\)
−0.290897 + 0.956754i \(0.593954\pi\)
\(692\) 13.3385 0.507054
\(693\) 0 0
\(694\) 0.547109 0.0207680
\(695\) −33.9057 −1.28612
\(696\) 4.64742 0.176160
\(697\) −6.65325 −0.252010
\(698\) −11.6865 −0.442339
\(699\) −5.09405 −0.192675
\(700\) 0 0
\(701\) −37.9453 −1.43317 −0.716586 0.697498i \(-0.754296\pi\)
−0.716586 + 0.697498i \(0.754296\pi\)
\(702\) −2.21321 −0.0835322
\(703\) −25.8287 −0.974149
\(704\) −12.0844 −0.455448
\(705\) −0.427700 −0.0161081
\(706\) −4.45830 −0.167791
\(707\) 0 0
\(708\) −7.02875 −0.264157
\(709\) −29.2638 −1.09903 −0.549513 0.835485i \(-0.685187\pi\)
−0.549513 + 0.835485i \(0.685187\pi\)
\(710\) 14.8823 0.558523
\(711\) −14.6728 −0.550274
\(712\) 5.99395 0.224633
\(713\) −31.0887 −1.16428
\(714\) 0 0
\(715\) 20.1446 0.753363
\(716\) −2.54015 −0.0949299
\(717\) −4.22712 −0.157865
\(718\) −5.50302 −0.205371
\(719\) 24.4954 0.913524 0.456762 0.889589i \(-0.349009\pi\)
0.456762 + 0.889589i \(0.349009\pi\)
\(720\) −25.2567 −0.941262
\(721\) 0 0
\(722\) −5.22176 −0.194334
\(723\) −0.932241 −0.0346704
\(724\) 15.8975 0.590825
\(725\) −25.0921 −0.931899
\(726\) −0.590870 −0.0219293
\(727\) −13.8973 −0.515422 −0.257711 0.966222i \(-0.582968\pi\)
−0.257711 + 0.966222i \(0.582968\pi\)
\(728\) 0 0
\(729\) −20.5911 −0.762634
\(730\) −8.86850 −0.328238
\(731\) 4.92507 0.182160
\(732\) 6.91777 0.255688
\(733\) 7.28624 0.269123 0.134562 0.990905i \(-0.457037\pi\)
0.134562 + 0.990905i \(0.457037\pi\)
\(734\) −12.9286 −0.477204
\(735\) 0 0
\(736\) −18.1003 −0.667186
\(737\) 34.8755 1.28465
\(738\) 1.09885 0.0404494
\(739\) −3.20790 −0.118005 −0.0590023 0.998258i \(-0.518792\pi\)
−0.0590023 + 0.998258i \(0.518792\pi\)
\(740\) −57.8916 −2.12814
\(741\) −2.27588 −0.0836067
\(742\) 0 0
\(743\) 1.42278 0.0521967 0.0260984 0.999659i \(-0.491692\pi\)
0.0260984 + 0.999659i \(0.491692\pi\)
\(744\) −3.67342 −0.134674
\(745\) 42.5818 1.56008
\(746\) −5.38517 −0.197165
\(747\) −16.7817 −0.614009
\(748\) −31.6579 −1.15753
\(749\) 0 0
\(750\) −0.827123 −0.0302022
\(751\) −28.9997 −1.05821 −0.529107 0.848555i \(-0.677473\pi\)
−0.529107 + 0.848555i \(0.677473\pi\)
\(752\) −1.37593 −0.0501750
\(753\) −4.37272 −0.159351
\(754\) −9.67885 −0.352483
\(755\) −12.5973 −0.458464
\(756\) 0 0
\(757\) 25.8833 0.940744 0.470372 0.882468i \(-0.344120\pi\)
0.470372 + 0.882468i \(0.344120\pi\)
\(758\) 12.6178 0.458300
\(759\) −3.92720 −0.142548
\(760\) 9.48019 0.343882
\(761\) −36.0482 −1.30675 −0.653374 0.757036i \(-0.726647\pi\)
−0.653374 + 0.757036i \(0.726647\pi\)
\(762\) 0.607601 0.0220110
\(763\) 0 0
\(764\) −4.61267 −0.166881
\(765\) −53.4122 −1.93112
\(766\) −2.62838 −0.0949674
\(767\) 30.4281 1.09870
\(768\) 1.95023 0.0703729
\(769\) −35.4285 −1.27758 −0.638791 0.769380i \(-0.720565\pi\)
−0.638791 + 0.769380i \(0.720565\pi\)
\(770\) 0 0
\(771\) −0.959887 −0.0345695
\(772\) −7.76854 −0.279596
\(773\) −37.8547 −1.36154 −0.680769 0.732498i \(-0.738354\pi\)
−0.680769 + 0.732498i \(0.738354\pi\)
\(774\) −0.813428 −0.0292381
\(775\) 19.8333 0.712434
\(776\) −14.0605 −0.504744
\(777\) 0 0
\(778\) 10.5422 0.377958
\(779\) 2.30819 0.0826994
\(780\) −5.10109 −0.182648
\(781\) 35.8392 1.28243
\(782\) −11.0947 −0.396746
\(783\) 18.5557 0.663125
\(784\) 0 0
\(785\) 43.3937 1.54879
\(786\) 1.62861 0.0580907
\(787\) 43.7967 1.56118 0.780592 0.625041i \(-0.214918\pi\)
0.780592 + 0.625041i \(0.214918\pi\)
\(788\) 16.1342 0.574756
\(789\) −0.0807807 −0.00287587
\(790\) 5.43461 0.193355
\(791\) 0 0
\(792\) 10.8686 0.386199
\(793\) −29.9477 −1.06347
\(794\) −9.88625 −0.350850
\(795\) 9.43378 0.334582
\(796\) 6.10196 0.216278
\(797\) 5.00505 0.177288 0.0886439 0.996063i \(-0.471747\pi\)
0.0886439 + 0.996063i \(0.471747\pi\)
\(798\) 0 0
\(799\) −2.90978 −0.102941
\(800\) 11.5473 0.408258
\(801\) 11.7159 0.413959
\(802\) 1.44868 0.0511545
\(803\) −21.3569 −0.753668
\(804\) −8.83131 −0.311456
\(805\) 0 0
\(806\) 7.65035 0.269472
\(807\) 3.11925 0.109803
\(808\) 13.5971 0.478344
\(809\) −15.6219 −0.549236 −0.274618 0.961553i \(-0.588551\pi\)
−0.274618 + 0.961553i \(0.588551\pi\)
\(810\) 8.42887 0.296160
\(811\) 30.9054 1.08523 0.542617 0.839980i \(-0.317433\pi\)
0.542617 + 0.839980i \(0.317433\pi\)
\(812\) 0 0
\(813\) −5.51775 −0.193516
\(814\) 10.9677 0.384417
\(815\) 4.51828 0.158269
\(816\) 7.33626 0.256820
\(817\) −1.70864 −0.0597777
\(818\) 9.55539 0.334096
\(819\) 0 0
\(820\) 5.17349 0.180666
\(821\) −8.03281 −0.280347 −0.140173 0.990127i \(-0.544766\pi\)
−0.140173 + 0.990127i \(0.544766\pi\)
\(822\) 0.172032 0.00600030
\(823\) −52.6132 −1.83398 −0.916991 0.398908i \(-0.869389\pi\)
−0.916991 + 0.398908i \(0.869389\pi\)
\(824\) −15.9213 −0.554646
\(825\) 2.50539 0.0872266
\(826\) 0 0
\(827\) −12.2906 −0.427385 −0.213693 0.976901i \(-0.568549\pi\)
−0.213693 + 0.976901i \(0.568549\pi\)
\(828\) −23.2922 −0.809461
\(829\) −21.6800 −0.752978 −0.376489 0.926421i \(-0.622869\pi\)
−0.376489 + 0.926421i \(0.622869\pi\)
\(830\) 6.21570 0.215750
\(831\) 5.20228 0.180465
\(832\) −13.2473 −0.459268
\(833\) 0 0
\(834\) −1.62659 −0.0563241
\(835\) 58.2850 2.01704
\(836\) 10.9829 0.379853
\(837\) −14.6668 −0.506958
\(838\) −8.85428 −0.305866
\(839\) 26.8087 0.925540 0.462770 0.886478i \(-0.346856\pi\)
0.462770 + 0.886478i \(0.346856\pi\)
\(840\) 0 0
\(841\) 52.1479 1.79820
\(842\) −2.67349 −0.0921346
\(843\) −2.63622 −0.0907963
\(844\) 24.6410 0.848178
\(845\) −14.1901 −0.488155
\(846\) 0.480581 0.0165227
\(847\) 0 0
\(848\) 30.3489 1.04219
\(849\) −0.135564 −0.00465255
\(850\) 7.07798 0.242773
\(851\) −48.8582 −1.67484
\(852\) −9.07534 −0.310916
\(853\) 36.5753 1.25231 0.626157 0.779697i \(-0.284627\pi\)
0.626157 + 0.779697i \(0.284627\pi\)
\(854\) 0 0
\(855\) 18.5301 0.633716
\(856\) 15.0312 0.513755
\(857\) 14.1015 0.481700 0.240850 0.970562i \(-0.422574\pi\)
0.240850 + 0.970562i \(0.422574\pi\)
\(858\) 0.966410 0.0329927
\(859\) −19.3798 −0.661232 −0.330616 0.943765i \(-0.607256\pi\)
−0.330616 + 0.943765i \(0.607256\pi\)
\(860\) −3.82968 −0.130591
\(861\) 0 0
\(862\) −7.25306 −0.247040
\(863\) 36.9877 1.25908 0.629538 0.776970i \(-0.283244\pi\)
0.629538 + 0.776970i \(0.283244\pi\)
\(864\) −8.53922 −0.290510
\(865\) −20.0728 −0.682497
\(866\) −0.741938 −0.0252121
\(867\) 9.55626 0.324548
\(868\) 0 0
\(869\) 13.0875 0.443962
\(870\) −3.36456 −0.114069
\(871\) 38.2316 1.29543
\(872\) 22.7857 0.771620
\(873\) −27.4829 −0.930156
\(874\) 3.84905 0.130196
\(875\) 0 0
\(876\) 5.40808 0.182722
\(877\) −8.16131 −0.275588 −0.137794 0.990461i \(-0.544001\pi\)
−0.137794 + 0.990461i \(0.544001\pi\)
\(878\) −7.64063 −0.257859
\(879\) 0.194054 0.00654529
\(880\) 22.5278 0.759413
\(881\) −54.1517 −1.82442 −0.912209 0.409725i \(-0.865625\pi\)
−0.912209 + 0.409725i \(0.865625\pi\)
\(882\) 0 0
\(883\) 45.6230 1.53533 0.767667 0.640848i \(-0.221417\pi\)
0.767667 + 0.640848i \(0.221417\pi\)
\(884\) −34.7043 −1.16723
\(885\) 10.5774 0.355556
\(886\) −1.00062 −0.0336166
\(887\) −17.8641 −0.599817 −0.299908 0.953968i \(-0.596956\pi\)
−0.299908 + 0.953968i \(0.596956\pi\)
\(888\) −5.77305 −0.193731
\(889\) 0 0
\(890\) −4.33939 −0.145457
\(891\) 20.2982 0.680014
\(892\) 1.75631 0.0588056
\(893\) 1.00948 0.0337809
\(894\) 2.04281 0.0683219
\(895\) 3.82262 0.127776
\(896\) 0 0
\(897\) −4.30511 −0.143744
\(898\) −5.67657 −0.189430
\(899\) −64.1409 −2.13922
\(900\) 14.8595 0.495317
\(901\) 64.1811 2.13818
\(902\) −0.980128 −0.0326347
\(903\) 0 0
\(904\) −7.94372 −0.264204
\(905\) −23.9238 −0.795253
\(906\) −0.604342 −0.0200779
\(907\) 30.9726 1.02843 0.514214 0.857662i \(-0.328084\pi\)
0.514214 + 0.857662i \(0.328084\pi\)
\(908\) 0.0334814 0.00111112
\(909\) 26.5771 0.881505
\(910\) 0 0
\(911\) 38.5068 1.27579 0.637893 0.770125i \(-0.279806\pi\)
0.637893 + 0.770125i \(0.279806\pi\)
\(912\) −2.54514 −0.0842780
\(913\) 14.9685 0.495384
\(914\) −13.5548 −0.448353
\(915\) −10.4104 −0.344157
\(916\) 7.20602 0.238094
\(917\) 0 0
\(918\) −5.23417 −0.172753
\(919\) 53.4161 1.76203 0.881017 0.473085i \(-0.156860\pi\)
0.881017 + 0.473085i \(0.156860\pi\)
\(920\) 17.9329 0.591231
\(921\) −10.9029 −0.359262
\(922\) −10.0185 −0.329943
\(923\) 39.2880 1.29318
\(924\) 0 0
\(925\) 31.1696 1.02485
\(926\) 12.5467 0.412310
\(927\) −31.1200 −1.02212
\(928\) −37.3438 −1.22587
\(929\) −32.7404 −1.07418 −0.537089 0.843525i \(-0.680476\pi\)
−0.537089 + 0.843525i \(0.680476\pi\)
\(930\) 2.65941 0.0872056
\(931\) 0 0
\(932\) 26.9485 0.882726
\(933\) −7.91218 −0.259033
\(934\) 1.25268 0.0409890
\(935\) 47.6412 1.55803
\(936\) 11.9145 0.389438
\(937\) −28.0682 −0.916948 −0.458474 0.888708i \(-0.651604\pi\)
−0.458474 + 0.888708i \(0.651604\pi\)
\(938\) 0 0
\(939\) 7.06343 0.230506
\(940\) 2.26261 0.0737983
\(941\) 56.3824 1.83801 0.919006 0.394243i \(-0.128993\pi\)
0.919006 + 0.394243i \(0.128993\pi\)
\(942\) 2.08176 0.0678274
\(943\) 4.36622 0.142184
\(944\) 34.0280 1.10752
\(945\) 0 0
\(946\) 0.725540 0.0235893
\(947\) −24.5587 −0.798052 −0.399026 0.916940i \(-0.630652\pi\)
−0.399026 + 0.916940i \(0.630652\pi\)
\(948\) −3.31407 −0.107636
\(949\) −23.4121 −0.759988
\(950\) −2.45554 −0.0796682
\(951\) 0.517888 0.0167937
\(952\) 0 0
\(953\) 11.4811 0.371910 0.185955 0.982558i \(-0.440462\pi\)
0.185955 + 0.982558i \(0.440462\pi\)
\(954\) −10.6002 −0.343194
\(955\) 6.94151 0.224622
\(956\) 22.3623 0.723247
\(957\) −8.10243 −0.261914
\(958\) 2.41195 0.0779266
\(959\) 0 0
\(960\) −4.60502 −0.148626
\(961\) 19.6983 0.635428
\(962\) 12.0231 0.387641
\(963\) 29.3802 0.946763
\(964\) 4.93173 0.158840
\(965\) 11.6907 0.376337
\(966\) 0 0
\(967\) −55.7786 −1.79372 −0.896859 0.442316i \(-0.854157\pi\)
−0.896859 + 0.442316i \(0.854157\pi\)
\(968\) 6.49754 0.208839
\(969\) −5.38239 −0.172907
\(970\) 10.1793 0.326838
\(971\) 6.99246 0.224399 0.112199 0.993686i \(-0.464210\pi\)
0.112199 + 0.993686i \(0.464210\pi\)
\(972\) −16.5978 −0.532374
\(973\) 0 0
\(974\) 5.33545 0.170959
\(975\) 2.74649 0.0879581
\(976\) −33.4908 −1.07201
\(977\) −19.6472 −0.628571 −0.314285 0.949329i \(-0.601765\pi\)
−0.314285 + 0.949329i \(0.601765\pi\)
\(978\) 0.216759 0.00693119
\(979\) −10.4500 −0.333984
\(980\) 0 0
\(981\) 44.5371 1.42196
\(982\) −10.6883 −0.341078
\(983\) −28.0238 −0.893820 −0.446910 0.894579i \(-0.647476\pi\)
−0.446910 + 0.894579i \(0.647476\pi\)
\(984\) 0.515909 0.0164466
\(985\) −24.2800 −0.773624
\(986\) −22.8902 −0.728971
\(987\) 0 0
\(988\) 12.0399 0.383039
\(989\) −3.23210 −0.102775
\(990\) −7.86845 −0.250076
\(991\) 35.0606 1.11374 0.556869 0.830601i \(-0.312003\pi\)
0.556869 + 0.830601i \(0.312003\pi\)
\(992\) 29.5173 0.937175
\(993\) −3.47023 −0.110124
\(994\) 0 0
\(995\) −9.18270 −0.291111
\(996\) −3.79038 −0.120103
\(997\) −41.7816 −1.32324 −0.661619 0.749841i \(-0.730130\pi\)
−0.661619 + 0.749841i \(0.730130\pi\)
\(998\) −2.52519 −0.0799336
\(999\) −23.0499 −0.729268
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 2009.2.a.r.1.9 17
7.3 odd 6 287.2.e.d.247.9 yes 34
7.5 odd 6 287.2.e.d.165.9 34
7.6 odd 2 2009.2.a.s.1.9 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
287.2.e.d.165.9 34 7.5 odd 6
287.2.e.d.247.9 yes 34 7.3 odd 6
2009.2.a.r.1.9 17 1.1 even 1 trivial
2009.2.a.s.1.9 17 7.6 odd 2