Properties

Label 2009.2.a.u.1.9
Level $2009$
Weight $2$
Character 2009.1
Self dual yes
Analytic conductor $16.042$
Analytic rank $0$
Dimension $20$
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] = [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: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 2 x^{19} - 27 x^{18} + 52 x^{17} + 302 x^{16} - 552 x^{15} - 1820 x^{14} + 3090 x^{13} + 6449 x^{12} - 9852 x^{11} - 13797 x^{10} + 18080 x^{9} + 17721 x^{8} - 18446 x^{7} + \cdots + 49 \) 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 1.9
Root \(0.845808\) 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)
 
\(f(q)\) \(=\) \(q-0.845808 q^{2} +0.0216736 q^{3} -1.28461 q^{4} -2.12752 q^{5} -0.0183317 q^{6} +2.77815 q^{8} -2.99953 q^{9} +O(q^{10})\) \(q-0.845808 q^{2} +0.0216736 q^{3} -1.28461 q^{4} -2.12752 q^{5} -0.0183317 q^{6} +2.77815 q^{8} -2.99953 q^{9} +1.79947 q^{10} -5.19192 q^{11} -0.0278421 q^{12} -3.20762 q^{13} -0.0461110 q^{15} +0.219438 q^{16} -6.26875 q^{17} +2.53703 q^{18} +1.56448 q^{19} +2.73303 q^{20} +4.39137 q^{22} -4.17962 q^{23} +0.0602125 q^{24} -0.473663 q^{25} +2.71303 q^{26} -0.130031 q^{27} +3.92688 q^{29} +0.0390010 q^{30} +3.32925 q^{31} -5.74190 q^{32} -0.112528 q^{33} +5.30216 q^{34} +3.85322 q^{36} +0.150653 q^{37} -1.32325 q^{38} -0.0695207 q^{39} -5.91056 q^{40} +1.00000 q^{41} -4.19682 q^{43} +6.66959 q^{44} +6.38156 q^{45} +3.53515 q^{46} -0.154774 q^{47} +0.00475602 q^{48} +0.400628 q^{50} -0.135866 q^{51} +4.12054 q^{52} -6.90674 q^{53} +0.109982 q^{54} +11.0459 q^{55} +0.0339078 q^{57} -3.32139 q^{58} +0.905932 q^{59} +0.0592346 q^{60} -7.22479 q^{61} -2.81590 q^{62} +4.41767 q^{64} +6.82428 q^{65} +0.0951768 q^{66} -1.72181 q^{67} +8.05290 q^{68} -0.0905874 q^{69} -12.9514 q^{71} -8.33314 q^{72} +11.0549 q^{73} -0.127424 q^{74} -0.0102660 q^{75} -2.00974 q^{76} +0.0588012 q^{78} +0.331420 q^{79} -0.466859 q^{80} +8.99577 q^{81} -0.845808 q^{82} -6.16966 q^{83} +13.3369 q^{85} +3.54971 q^{86} +0.0851097 q^{87} -14.4239 q^{88} -1.82708 q^{89} -5.39757 q^{90} +5.36918 q^{92} +0.0721568 q^{93} +0.130909 q^{94} -3.32845 q^{95} -0.124448 q^{96} +7.73595 q^{97} +15.5733 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 2 q^{2} + 8 q^{3} + 18 q^{4} + 8 q^{5} + 12 q^{6} - 6 q^{8} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 2 q^{2} + 8 q^{3} + 18 q^{4} + 8 q^{5} + 12 q^{6} - 6 q^{8} + 16 q^{9} + 16 q^{10} + 6 q^{12} + 12 q^{13} + 14 q^{16} + 8 q^{17} - 18 q^{18} + 36 q^{19} + 24 q^{20} - 8 q^{22} - 12 q^{23} + 36 q^{24} + 20 q^{25} - 22 q^{26} + 32 q^{27} + 4 q^{29} + 28 q^{30} + 80 q^{31} + 6 q^{32} - 12 q^{33} + 48 q^{34} + 26 q^{36} + 4 q^{37} + 12 q^{38} - 28 q^{39} - 4 q^{40} + 20 q^{41} - 20 q^{44} + 40 q^{45} + 8 q^{46} + 32 q^{47} + 16 q^{48} + 6 q^{50} - 20 q^{51} + 36 q^{52} + 4 q^{53} - 50 q^{54} + 64 q^{55} - 4 q^{57} + 32 q^{59} + 20 q^{60} + 44 q^{61} - 8 q^{62} - 30 q^{64} - 8 q^{65} + 32 q^{66} - 4 q^{67} - 48 q^{68} - 24 q^{69} + 8 q^{71} - 8 q^{72} + 48 q^{73} - 38 q^{74} + 24 q^{75} + 84 q^{76} + 30 q^{78} - 4 q^{79} + 56 q^{80} - 2 q^{82} + 8 q^{83} - 12 q^{85} - 24 q^{86} + 40 q^{87} - 48 q^{88} + 20 q^{89} + 48 q^{90} - 50 q^{92} + 48 q^{93} + 26 q^{94} + 20 q^{95} + 70 q^{96} + 8 q^{97} - 8 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.845808 −0.598076 −0.299038 0.954241i \(-0.596666\pi\)
−0.299038 + 0.954241i \(0.596666\pi\)
\(3\) 0.0216736 0.0125133 0.00625663 0.999980i \(-0.498008\pi\)
0.00625663 + 0.999980i \(0.498008\pi\)
\(4\) −1.28461 −0.642305
\(5\) −2.12752 −0.951455 −0.475728 0.879593i \(-0.657815\pi\)
−0.475728 + 0.879593i \(0.657815\pi\)
\(6\) −0.0183317 −0.00748388
\(7\) 0 0
\(8\) 2.77815 0.982224
\(9\) −2.99953 −0.999843
\(10\) 1.79947 0.569043
\(11\) −5.19192 −1.56542 −0.782712 0.622384i \(-0.786164\pi\)
−0.782712 + 0.622384i \(0.786164\pi\)
\(12\) −0.0278421 −0.00803732
\(13\) −3.20762 −0.889635 −0.444817 0.895621i \(-0.646731\pi\)
−0.444817 + 0.895621i \(0.646731\pi\)
\(14\) 0 0
\(15\) −0.0461110 −0.0119058
\(16\) 0.219438 0.0548596
\(17\) −6.26875 −1.52040 −0.760198 0.649691i \(-0.774898\pi\)
−0.760198 + 0.649691i \(0.774898\pi\)
\(18\) 2.53703 0.597983
\(19\) 1.56448 0.358916 0.179458 0.983766i \(-0.442566\pi\)
0.179458 + 0.983766i \(0.442566\pi\)
\(20\) 2.73303 0.611124
\(21\) 0 0
\(22\) 4.39137 0.936243
\(23\) −4.17962 −0.871511 −0.435755 0.900065i \(-0.643519\pi\)
−0.435755 + 0.900065i \(0.643519\pi\)
\(24\) 0.0602125 0.0122908
\(25\) −0.473663 −0.0947326
\(26\) 2.71303 0.532070
\(27\) −0.130031 −0.0250246
\(28\) 0 0
\(29\) 3.92688 0.729204 0.364602 0.931163i \(-0.381205\pi\)
0.364602 + 0.931163i \(0.381205\pi\)
\(30\) 0.0390010 0.00712058
\(31\) 3.32925 0.597951 0.298975 0.954261i \(-0.403355\pi\)
0.298975 + 0.954261i \(0.403355\pi\)
\(32\) −5.74190 −1.01503
\(33\) −0.112528 −0.0195885
\(34\) 5.30216 0.909313
\(35\) 0 0
\(36\) 3.85322 0.642204
\(37\) 0.150653 0.0247672 0.0123836 0.999923i \(-0.496058\pi\)
0.0123836 + 0.999923i \(0.496058\pi\)
\(38\) −1.32325 −0.214659
\(39\) −0.0695207 −0.0111322
\(40\) −5.91056 −0.934542
\(41\) 1.00000 0.156174
\(42\) 0 0
\(43\) −4.19682 −0.640009 −0.320005 0.947416i \(-0.603685\pi\)
−0.320005 + 0.947416i \(0.603685\pi\)
\(44\) 6.66959 1.00548
\(45\) 6.38156 0.951306
\(46\) 3.53515 0.521230
\(47\) −0.154774 −0.0225761 −0.0112881 0.999936i \(-0.503593\pi\)
−0.0112881 + 0.999936i \(0.503593\pi\)
\(48\) 0.00475602 0.000686472 0
\(49\) 0 0
\(50\) 0.400628 0.0566574
\(51\) −0.135866 −0.0190251
\(52\) 4.12054 0.571417
\(53\) −6.90674 −0.948714 −0.474357 0.880333i \(-0.657319\pi\)
−0.474357 + 0.880333i \(0.657319\pi\)
\(54\) 0.109982 0.0149666
\(55\) 11.0459 1.48943
\(56\) 0 0
\(57\) 0.0339078 0.00449120
\(58\) −3.32139 −0.436120
\(59\) 0.905932 0.117942 0.0589712 0.998260i \(-0.481218\pi\)
0.0589712 + 0.998260i \(0.481218\pi\)
\(60\) 0.0592346 0.00764715
\(61\) −7.22479 −0.925040 −0.462520 0.886609i \(-0.653055\pi\)
−0.462520 + 0.886609i \(0.653055\pi\)
\(62\) −2.81590 −0.357620
\(63\) 0 0
\(64\) 4.41767 0.552208
\(65\) 6.82428 0.846448
\(66\) 0.0951768 0.0117154
\(67\) −1.72181 −0.210353 −0.105176 0.994454i \(-0.533541\pi\)
−0.105176 + 0.994454i \(0.533541\pi\)
\(68\) 8.05290 0.976557
\(69\) −0.0905874 −0.0109054
\(70\) 0 0
\(71\) −12.9514 −1.53705 −0.768527 0.639818i \(-0.779010\pi\)
−0.768527 + 0.639818i \(0.779010\pi\)
\(72\) −8.33314 −0.982070
\(73\) 11.0549 1.29387 0.646937 0.762543i \(-0.276050\pi\)
0.646937 + 0.762543i \(0.276050\pi\)
\(74\) −0.127424 −0.0148127
\(75\) −0.0102660 −0.00118541
\(76\) −2.00974 −0.230533
\(77\) 0 0
\(78\) 0.0588012 0.00665792
\(79\) 0.331420 0.0372877 0.0186438 0.999826i \(-0.494065\pi\)
0.0186438 + 0.999826i \(0.494065\pi\)
\(80\) −0.466859 −0.0521964
\(81\) 8.99577 0.999530
\(82\) −0.845808 −0.0934039
\(83\) −6.16966 −0.677208 −0.338604 0.940929i \(-0.609955\pi\)
−0.338604 + 0.940929i \(0.609955\pi\)
\(84\) 0 0
\(85\) 13.3369 1.44659
\(86\) 3.54971 0.382775
\(87\) 0.0851097 0.00912472
\(88\) −14.4239 −1.53760
\(89\) −1.82708 −0.193670 −0.0968349 0.995300i \(-0.530872\pi\)
−0.0968349 + 0.995300i \(0.530872\pi\)
\(90\) −5.39757 −0.568954
\(91\) 0 0
\(92\) 5.36918 0.559775
\(93\) 0.0721568 0.00748231
\(94\) 0.130909 0.0135022
\(95\) −3.32845 −0.341492
\(96\) −0.124448 −0.0127014
\(97\) 7.73595 0.785467 0.392733 0.919652i \(-0.371529\pi\)
0.392733 + 0.919652i \(0.371529\pi\)
\(98\) 0 0
\(99\) 15.5733 1.56518
\(100\) 0.608472 0.0608472
\(101\) 7.56840 0.753084 0.376542 0.926400i \(-0.377113\pi\)
0.376542 + 0.926400i \(0.377113\pi\)
\(102\) 0.114917 0.0113785
\(103\) 1.75418 0.172844 0.0864220 0.996259i \(-0.472457\pi\)
0.0864220 + 0.996259i \(0.472457\pi\)
\(104\) −8.91126 −0.873821
\(105\) 0 0
\(106\) 5.84178 0.567403
\(107\) −14.3693 −1.38914 −0.694568 0.719427i \(-0.744404\pi\)
−0.694568 + 0.719427i \(0.744404\pi\)
\(108\) 0.167039 0.0160734
\(109\) 2.69508 0.258142 0.129071 0.991635i \(-0.458801\pi\)
0.129071 + 0.991635i \(0.458801\pi\)
\(110\) −9.34272 −0.890794
\(111\) 0.00326519 0.000309919 0
\(112\) 0 0
\(113\) 18.2788 1.71952 0.859761 0.510697i \(-0.170613\pi\)
0.859761 + 0.510697i \(0.170613\pi\)
\(114\) −0.0286795 −0.00268608
\(115\) 8.89222 0.829204
\(116\) −5.04451 −0.468371
\(117\) 9.62137 0.889496
\(118\) −0.766245 −0.0705385
\(119\) 0 0
\(120\) −0.128103 −0.0116942
\(121\) 15.9561 1.45055
\(122\) 6.11079 0.553245
\(123\) 0.0216736 0.00195424
\(124\) −4.27678 −0.384066
\(125\) 11.6453 1.04159
\(126\) 0 0
\(127\) −1.85675 −0.164760 −0.0823798 0.996601i \(-0.526252\pi\)
−0.0823798 + 0.996601i \(0.526252\pi\)
\(128\) 7.74730 0.684771
\(129\) −0.0909602 −0.00800860
\(130\) −5.77203 −0.506241
\(131\) 17.1914 1.50202 0.751011 0.660290i \(-0.229567\pi\)
0.751011 + 0.660290i \(0.229567\pi\)
\(132\) 0.144554 0.0125818
\(133\) 0 0
\(134\) 1.45632 0.125807
\(135\) 0.276644 0.0238097
\(136\) −17.4155 −1.49337
\(137\) −8.53668 −0.729338 −0.364669 0.931137i \(-0.618818\pi\)
−0.364669 + 0.931137i \(0.618818\pi\)
\(138\) 0.0766195 0.00652228
\(139\) 15.0184 1.27385 0.636923 0.770928i \(-0.280207\pi\)
0.636923 + 0.770928i \(0.280207\pi\)
\(140\) 0 0
\(141\) −0.00335451 −0.000282501 0
\(142\) 10.9544 0.919275
\(143\) 16.6537 1.39266
\(144\) −0.658212 −0.0548510
\(145\) −8.35452 −0.693805
\(146\) −9.35029 −0.773836
\(147\) 0 0
\(148\) −0.193530 −0.0159081
\(149\) −13.5199 −1.10760 −0.553798 0.832651i \(-0.686822\pi\)
−0.553798 + 0.832651i \(0.686822\pi\)
\(150\) 0.00868305 0.000708968 0
\(151\) 2.98665 0.243050 0.121525 0.992588i \(-0.461222\pi\)
0.121525 + 0.992588i \(0.461222\pi\)
\(152\) 4.34635 0.352535
\(153\) 18.8033 1.52016
\(154\) 0 0
\(155\) −7.08304 −0.568923
\(156\) 0.0893070 0.00715028
\(157\) 11.7665 0.939065 0.469533 0.882915i \(-0.344422\pi\)
0.469533 + 0.882915i \(0.344422\pi\)
\(158\) −0.280318 −0.0223009
\(159\) −0.149694 −0.0118715
\(160\) 12.2160 0.965760
\(161\) 0 0
\(162\) −7.60870 −0.597796
\(163\) −11.1721 −0.875069 −0.437535 0.899202i \(-0.644148\pi\)
−0.437535 + 0.899202i \(0.644148\pi\)
\(164\) −1.28461 −0.100311
\(165\) 0.239405 0.0186376
\(166\) 5.21834 0.405022
\(167\) 24.8646 1.92408 0.962041 0.272904i \(-0.0879842\pi\)
0.962041 + 0.272904i \(0.0879842\pi\)
\(168\) 0 0
\(169\) −2.71115 −0.208550
\(170\) −11.2804 −0.865171
\(171\) −4.69270 −0.358859
\(172\) 5.39128 0.411081
\(173\) −9.36733 −0.712185 −0.356092 0.934451i \(-0.615891\pi\)
−0.356092 + 0.934451i \(0.615891\pi\)
\(174\) −0.0719865 −0.00545728
\(175\) 0 0
\(176\) −1.13931 −0.0858785
\(177\) 0.0196348 0.00147584
\(178\) 1.54536 0.115829
\(179\) 11.0566 0.826405 0.413203 0.910639i \(-0.364410\pi\)
0.413203 + 0.910639i \(0.364410\pi\)
\(180\) −8.19781 −0.611028
\(181\) −7.84517 −0.583127 −0.291564 0.956551i \(-0.594175\pi\)
−0.291564 + 0.956551i \(0.594175\pi\)
\(182\) 0 0
\(183\) −0.156587 −0.0115753
\(184\) −11.6116 −0.856018
\(185\) −0.320517 −0.0235649
\(186\) −0.0610308 −0.00447499
\(187\) 32.5469 2.38006
\(188\) 0.198824 0.0145007
\(189\) 0 0
\(190\) 2.81523 0.204238
\(191\) 17.6747 1.27890 0.639448 0.768835i \(-0.279163\pi\)
0.639448 + 0.768835i \(0.279163\pi\)
\(192\) 0.0957467 0.00690992
\(193\) −13.7455 −0.989422 −0.494711 0.869057i \(-0.664726\pi\)
−0.494711 + 0.869057i \(0.664726\pi\)
\(194\) −6.54313 −0.469769
\(195\) 0.147907 0.0105918
\(196\) 0 0
\(197\) −16.6689 −1.18761 −0.593807 0.804608i \(-0.702376\pi\)
−0.593807 + 0.804608i \(0.702376\pi\)
\(198\) −13.1720 −0.936097
\(199\) 0.732788 0.0519460 0.0259730 0.999663i \(-0.491732\pi\)
0.0259730 + 0.999663i \(0.491732\pi\)
\(200\) −1.31591 −0.0930486
\(201\) −0.0373178 −0.00263220
\(202\) −6.40141 −0.450402
\(203\) 0 0
\(204\) 0.174535 0.0122199
\(205\) −2.12752 −0.148592
\(206\) −1.48370 −0.103374
\(207\) 12.5369 0.871374
\(208\) −0.703876 −0.0488050
\(209\) −8.12264 −0.561855
\(210\) 0 0
\(211\) −25.1457 −1.73110 −0.865551 0.500821i \(-0.833031\pi\)
−0.865551 + 0.500821i \(0.833031\pi\)
\(212\) 8.87246 0.609363
\(213\) −0.280704 −0.0192335
\(214\) 12.1537 0.830810
\(215\) 8.92882 0.608940
\(216\) −0.361246 −0.0245797
\(217\) 0 0
\(218\) −2.27952 −0.154389
\(219\) 0.239599 0.0161906
\(220\) −14.1897 −0.956668
\(221\) 20.1078 1.35260
\(222\) −0.00276173 −0.000185355 0
\(223\) −27.2560 −1.82519 −0.912597 0.408859i \(-0.865927\pi\)
−0.912597 + 0.408859i \(0.865927\pi\)
\(224\) 0 0
\(225\) 1.42077 0.0947178
\(226\) −15.4603 −1.02841
\(227\) 13.9987 0.929127 0.464564 0.885540i \(-0.346211\pi\)
0.464564 + 0.885540i \(0.346211\pi\)
\(228\) −0.0435583 −0.00288472
\(229\) 25.7545 1.70190 0.850951 0.525244i \(-0.176026\pi\)
0.850951 + 0.525244i \(0.176026\pi\)
\(230\) −7.52111 −0.495927
\(231\) 0 0
\(232\) 10.9095 0.716242
\(233\) −20.9873 −1.37492 −0.687462 0.726220i \(-0.741275\pi\)
−0.687462 + 0.726220i \(0.741275\pi\)
\(234\) −8.13783 −0.531986
\(235\) 0.329285 0.0214802
\(236\) −1.16377 −0.0757549
\(237\) 0.00718306 0.000466590 0
\(238\) 0 0
\(239\) −25.0254 −1.61876 −0.809380 0.587285i \(-0.800197\pi\)
−0.809380 + 0.587285i \(0.800197\pi\)
\(240\) −0.0101185 −0.000653147 0
\(241\) −8.06559 −0.519551 −0.259775 0.965669i \(-0.583648\pi\)
−0.259775 + 0.965669i \(0.583648\pi\)
\(242\) −13.4958 −0.867541
\(243\) 0.585065 0.0375319
\(244\) 9.28104 0.594157
\(245\) 0 0
\(246\) −0.0183317 −0.00116879
\(247\) −5.01825 −0.319304
\(248\) 9.24915 0.587321
\(249\) −0.133719 −0.00847407
\(250\) −9.84971 −0.622950
\(251\) −27.2270 −1.71856 −0.859278 0.511509i \(-0.829087\pi\)
−0.859278 + 0.511509i \(0.829087\pi\)
\(252\) 0 0
\(253\) 21.7003 1.36428
\(254\) 1.57045 0.0985388
\(255\) 0.289058 0.0181015
\(256\) −15.3881 −0.961754
\(257\) 14.0475 0.876261 0.438130 0.898911i \(-0.355641\pi\)
0.438130 + 0.898911i \(0.355641\pi\)
\(258\) 0.0769349 0.00478976
\(259\) 0 0
\(260\) −8.76653 −0.543677
\(261\) −11.7788 −0.729090
\(262\) −14.5406 −0.898324
\(263\) 8.55511 0.527531 0.263765 0.964587i \(-0.415036\pi\)
0.263765 + 0.964587i \(0.415036\pi\)
\(264\) −0.312618 −0.0192403
\(265\) 14.6942 0.902659
\(266\) 0 0
\(267\) −0.0395993 −0.00242344
\(268\) 2.21185 0.135110
\(269\) −30.6009 −1.86577 −0.932883 0.360178i \(-0.882716\pi\)
−0.932883 + 0.360178i \(0.882716\pi\)
\(270\) −0.233988 −0.0142400
\(271\) −28.4723 −1.72957 −0.864786 0.502141i \(-0.832546\pi\)
−0.864786 + 0.502141i \(0.832546\pi\)
\(272\) −1.37560 −0.0834083
\(273\) 0 0
\(274\) 7.22040 0.436200
\(275\) 2.45922 0.148297
\(276\) 0.116369 0.00700461
\(277\) 21.4231 1.28719 0.643595 0.765366i \(-0.277442\pi\)
0.643595 + 0.765366i \(0.277442\pi\)
\(278\) −12.7027 −0.761857
\(279\) −9.98618 −0.597857
\(280\) 0 0
\(281\) 16.0553 0.957778 0.478889 0.877875i \(-0.341040\pi\)
0.478889 + 0.877875i \(0.341040\pi\)
\(282\) 0.00283727 0.000168957 0
\(283\) −11.3955 −0.677390 −0.338695 0.940896i \(-0.609985\pi\)
−0.338695 + 0.940896i \(0.609985\pi\)
\(284\) 16.6375 0.987256
\(285\) −0.0721396 −0.00427318
\(286\) −14.0859 −0.832915
\(287\) 0 0
\(288\) 17.2230 1.01488
\(289\) 22.2973 1.31160
\(290\) 7.06632 0.414949
\(291\) 0.167666 0.00982875
\(292\) −14.2012 −0.831061
\(293\) −15.7980 −0.922926 −0.461463 0.887159i \(-0.652675\pi\)
−0.461463 + 0.887159i \(0.652675\pi\)
\(294\) 0 0
\(295\) −1.92739 −0.112217
\(296\) 0.418537 0.0243269
\(297\) 0.675113 0.0391740
\(298\) 11.4353 0.662427
\(299\) 13.4066 0.775326
\(300\) 0.0131878 0.000761396 0
\(301\) 0 0
\(302\) −2.52613 −0.145362
\(303\) 0.164034 0.00942353
\(304\) 0.343306 0.0196900
\(305\) 15.3709 0.880134
\(306\) −15.9040 −0.909171
\(307\) 9.37252 0.534918 0.267459 0.963569i \(-0.413816\pi\)
0.267459 + 0.963569i \(0.413816\pi\)
\(308\) 0 0
\(309\) 0.0380193 0.00216284
\(310\) 5.99089 0.340260
\(311\) −0.975361 −0.0553076 −0.0276538 0.999618i \(-0.508804\pi\)
−0.0276538 + 0.999618i \(0.508804\pi\)
\(312\) −0.193139 −0.0109343
\(313\) 32.2917 1.82523 0.912616 0.408817i \(-0.134059\pi\)
0.912616 + 0.408817i \(0.134059\pi\)
\(314\) −9.95216 −0.561633
\(315\) 0 0
\(316\) −0.425745 −0.0239500
\(317\) −20.2286 −1.13615 −0.568075 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(318\) 0.126612 0.00710006
\(319\) −20.3881 −1.14151
\(320\) −9.39867 −0.525402
\(321\) −0.311435 −0.0173826
\(322\) 0 0
\(323\) −9.80732 −0.545694
\(324\) −11.5561 −0.642003
\(325\) 1.51933 0.0842774
\(326\) 9.44948 0.523358
\(327\) 0.0584121 0.00323020
\(328\) 2.77815 0.153398
\(329\) 0 0
\(330\) −0.202490 −0.0111467
\(331\) 30.2925 1.66503 0.832514 0.554004i \(-0.186901\pi\)
0.832514 + 0.554004i \(0.186901\pi\)
\(332\) 7.92560 0.434974
\(333\) −0.451889 −0.0247633
\(334\) −21.0307 −1.15075
\(335\) 3.66318 0.200141
\(336\) 0 0
\(337\) −13.2607 −0.722359 −0.361179 0.932496i \(-0.617626\pi\)
−0.361179 + 0.932496i \(0.617626\pi\)
\(338\) 2.29311 0.124729
\(339\) 0.396167 0.0215168
\(340\) −17.1327 −0.929151
\(341\) −17.2852 −0.936046
\(342\) 3.96912 0.214625
\(343\) 0 0
\(344\) −11.6594 −0.628632
\(345\) 0.192726 0.0103760
\(346\) 7.92296 0.425941
\(347\) 25.1449 1.34985 0.674924 0.737887i \(-0.264176\pi\)
0.674924 + 0.737887i \(0.264176\pi\)
\(348\) −0.109333 −0.00586085
\(349\) 15.9360 0.853034 0.426517 0.904480i \(-0.359740\pi\)
0.426517 + 0.904480i \(0.359740\pi\)
\(350\) 0 0
\(351\) 0.417092 0.0222627
\(352\) 29.8115 1.58896
\(353\) −12.5243 −0.666602 −0.333301 0.942820i \(-0.608163\pi\)
−0.333301 + 0.942820i \(0.608163\pi\)
\(354\) −0.0166073 −0.000882667 0
\(355\) 27.5544 1.46244
\(356\) 2.34708 0.124395
\(357\) 0 0
\(358\) −9.35172 −0.494254
\(359\) 5.54354 0.292577 0.146288 0.989242i \(-0.453267\pi\)
0.146288 + 0.989242i \(0.453267\pi\)
\(360\) 17.7289 0.934396
\(361\) −16.5524 −0.871180
\(362\) 6.63551 0.348755
\(363\) 0.345825 0.0181511
\(364\) 0 0
\(365\) −23.5194 −1.23106
\(366\) 0.132443 0.00692289
\(367\) −2.95894 −0.154455 −0.0772277 0.997013i \(-0.524607\pi\)
−0.0772277 + 0.997013i \(0.524607\pi\)
\(368\) −0.917168 −0.0478107
\(369\) −2.99953 −0.156149
\(370\) 0.271096 0.0140936
\(371\) 0 0
\(372\) −0.0926933 −0.00480592
\(373\) 11.1625 0.577971 0.288986 0.957333i \(-0.406682\pi\)
0.288986 + 0.957333i \(0.406682\pi\)
\(374\) −27.5284 −1.42346
\(375\) 0.252396 0.0130337
\(376\) −0.429985 −0.0221748
\(377\) −12.5960 −0.648725
\(378\) 0 0
\(379\) −13.4510 −0.690933 −0.345467 0.938431i \(-0.612279\pi\)
−0.345467 + 0.938431i \(0.612279\pi\)
\(380\) 4.27576 0.219342
\(381\) −0.0402424 −0.00206168
\(382\) −14.9494 −0.764877
\(383\) −33.5431 −1.71397 −0.856987 0.515338i \(-0.827666\pi\)
−0.856987 + 0.515338i \(0.827666\pi\)
\(384\) 0.167912 0.00856872
\(385\) 0 0
\(386\) 11.6260 0.591750
\(387\) 12.5885 0.639909
\(388\) −9.93767 −0.504509
\(389\) 36.6850 1.86000 0.930001 0.367558i \(-0.119806\pi\)
0.930001 + 0.367558i \(0.119806\pi\)
\(390\) −0.125101 −0.00633472
\(391\) 26.2010 1.32504
\(392\) 0 0
\(393\) 0.372600 0.0187952
\(394\) 14.0987 0.710283
\(395\) −0.705102 −0.0354776
\(396\) −20.0056 −1.00532
\(397\) −20.8915 −1.04851 −0.524256 0.851560i \(-0.675657\pi\)
−0.524256 + 0.851560i \(0.675657\pi\)
\(398\) −0.619798 −0.0310677
\(399\) 0 0
\(400\) −0.103940 −0.00519699
\(401\) −18.5820 −0.927939 −0.463969 0.885851i \(-0.653575\pi\)
−0.463969 + 0.885851i \(0.653575\pi\)
\(402\) 0.0315637 0.00157425
\(403\) −10.6790 −0.531958
\(404\) −9.72243 −0.483709
\(405\) −19.1387 −0.951008
\(406\) 0 0
\(407\) −0.782179 −0.0387712
\(408\) −0.377457 −0.0186869
\(409\) 28.9182 1.42992 0.714958 0.699168i \(-0.246446\pi\)
0.714958 + 0.699168i \(0.246446\pi\)
\(410\) 1.79947 0.0888696
\(411\) −0.185021 −0.00912640
\(412\) −2.25343 −0.111018
\(413\) 0 0
\(414\) −10.6038 −0.521148
\(415\) 13.1261 0.644333
\(416\) 18.4179 0.903010
\(417\) 0.325503 0.0159399
\(418\) 6.87020 0.336032
\(419\) −30.7335 −1.50143 −0.750715 0.660626i \(-0.770291\pi\)
−0.750715 + 0.660626i \(0.770291\pi\)
\(420\) 0 0
\(421\) −18.2914 −0.891470 −0.445735 0.895165i \(-0.647058\pi\)
−0.445735 + 0.895165i \(0.647058\pi\)
\(422\) 21.2684 1.03533
\(423\) 0.464249 0.0225726
\(424\) −19.1880 −0.931849
\(425\) 2.96928 0.144031
\(426\) 0.237422 0.0115031
\(427\) 0 0
\(428\) 18.4590 0.892248
\(429\) 0.360946 0.0174267
\(430\) −7.55207 −0.364193
\(431\) 6.26079 0.301572 0.150786 0.988566i \(-0.451820\pi\)
0.150786 + 0.988566i \(0.451820\pi\)
\(432\) −0.0285339 −0.00137284
\(433\) 17.7835 0.854622 0.427311 0.904105i \(-0.359461\pi\)
0.427311 + 0.904105i \(0.359461\pi\)
\(434\) 0 0
\(435\) −0.181073 −0.00868176
\(436\) −3.46212 −0.165806
\(437\) −6.53892 −0.312799
\(438\) −0.202654 −0.00968320
\(439\) 21.0267 1.00355 0.501774 0.864999i \(-0.332681\pi\)
0.501774 + 0.864999i \(0.332681\pi\)
\(440\) 30.6872 1.46295
\(441\) 0 0
\(442\) −17.0073 −0.808957
\(443\) 14.2736 0.678158 0.339079 0.940758i \(-0.389885\pi\)
0.339079 + 0.940758i \(0.389885\pi\)
\(444\) −0.00419450 −0.000199062 0
\(445\) 3.88714 0.184268
\(446\) 23.0533 1.09161
\(447\) −0.293025 −0.0138596
\(448\) 0 0
\(449\) −23.6951 −1.11824 −0.559122 0.829086i \(-0.688862\pi\)
−0.559122 + 0.829086i \(0.688862\pi\)
\(450\) −1.20170 −0.0566485
\(451\) −5.19192 −0.244478
\(452\) −23.4811 −1.10446
\(453\) 0.0647314 0.00304135
\(454\) −11.8402 −0.555689
\(455\) 0 0
\(456\) 0.0942010 0.00441137
\(457\) −2.48464 −0.116226 −0.0581132 0.998310i \(-0.518508\pi\)
−0.0581132 + 0.998310i \(0.518508\pi\)
\(458\) −21.7833 −1.01787
\(459\) 0.815135 0.0380472
\(460\) −11.4230 −0.532601
\(461\) −25.8203 −1.20257 −0.601285 0.799035i \(-0.705344\pi\)
−0.601285 + 0.799035i \(0.705344\pi\)
\(462\) 0 0
\(463\) 25.2725 1.17451 0.587256 0.809401i \(-0.300208\pi\)
0.587256 + 0.809401i \(0.300208\pi\)
\(464\) 0.861709 0.0400038
\(465\) −0.153515 −0.00711908
\(466\) 17.7512 0.822310
\(467\) 33.2250 1.53747 0.768734 0.639569i \(-0.220887\pi\)
0.768734 + 0.639569i \(0.220887\pi\)
\(468\) −12.3597 −0.571327
\(469\) 0 0
\(470\) −0.278512 −0.0128468
\(471\) 0.255021 0.0117508
\(472\) 2.51681 0.115846
\(473\) 21.7896 1.00189
\(474\) −0.00607549 −0.000279057 0
\(475\) −0.741035 −0.0340010
\(476\) 0 0
\(477\) 20.7170 0.948565
\(478\) 21.1667 0.968142
\(479\) −18.0657 −0.825442 −0.412721 0.910857i \(-0.635422\pi\)
−0.412721 + 0.910857i \(0.635422\pi\)
\(480\) 0.264765 0.0120848
\(481\) −0.483239 −0.0220338
\(482\) 6.82194 0.310731
\(483\) 0 0
\(484\) −20.4973 −0.931696
\(485\) −16.4584 −0.747337
\(486\) −0.494852 −0.0224470
\(487\) 10.3071 0.467060 0.233530 0.972350i \(-0.424972\pi\)
0.233530 + 0.972350i \(0.424972\pi\)
\(488\) −20.0716 −0.908596
\(489\) −0.242140 −0.0109500
\(490\) 0 0
\(491\) −18.2743 −0.824709 −0.412354 0.911023i \(-0.635293\pi\)
−0.412354 + 0.911023i \(0.635293\pi\)
\(492\) −0.0278421 −0.00125522
\(493\) −24.6167 −1.10868
\(494\) 4.24448 0.190968
\(495\) −33.1326 −1.48920
\(496\) 0.730565 0.0328033
\(497\) 0 0
\(498\) 0.113100 0.00506814
\(499\) 18.8781 0.845101 0.422550 0.906339i \(-0.361135\pi\)
0.422550 + 0.906339i \(0.361135\pi\)
\(500\) −14.9597 −0.669017
\(501\) 0.538906 0.0240765
\(502\) 23.0288 1.02783
\(503\) −32.4514 −1.44694 −0.723469 0.690357i \(-0.757454\pi\)
−0.723469 + 0.690357i \(0.757454\pi\)
\(504\) 0 0
\(505\) −16.1019 −0.716526
\(506\) −18.3542 −0.815946
\(507\) −0.0587603 −0.00260964
\(508\) 2.38519 0.105826
\(509\) 3.87039 0.171552 0.0857760 0.996314i \(-0.472663\pi\)
0.0857760 + 0.996314i \(0.472663\pi\)
\(510\) −0.244488 −0.0108261
\(511\) 0 0
\(512\) −2.47926 −0.109569
\(513\) −0.203431 −0.00898170
\(514\) −11.8815 −0.524071
\(515\) −3.73204 −0.164453
\(516\) 0.116848 0.00514396
\(517\) 0.803575 0.0353412
\(518\) 0 0
\(519\) −0.203024 −0.00891175
\(520\) 18.9589 0.831401
\(521\) −43.4935 −1.90549 −0.952743 0.303777i \(-0.901752\pi\)
−0.952743 + 0.303777i \(0.901752\pi\)
\(522\) 9.96261 0.436052
\(523\) −22.7651 −0.995450 −0.497725 0.867335i \(-0.665831\pi\)
−0.497725 + 0.867335i \(0.665831\pi\)
\(524\) −22.0843 −0.964755
\(525\) 0 0
\(526\) −7.23598 −0.315504
\(527\) −20.8702 −0.909122
\(528\) −0.0246929 −0.00107462
\(529\) −5.53079 −0.240469
\(530\) −12.4285 −0.539859
\(531\) −2.71737 −0.117924
\(532\) 0 0
\(533\) −3.20762 −0.138938
\(534\) 0.0334934 0.00144940
\(535\) 30.5710 1.32170
\(536\) −4.78344 −0.206613
\(537\) 0.239635 0.0103410
\(538\) 25.8824 1.11587
\(539\) 0 0
\(540\) −0.355380 −0.0152931
\(541\) −15.7172 −0.675736 −0.337868 0.941193i \(-0.609706\pi\)
−0.337868 + 0.941193i \(0.609706\pi\)
\(542\) 24.0821 1.03442
\(543\) −0.170033 −0.00729682
\(544\) 35.9946 1.54325
\(545\) −5.73384 −0.245611
\(546\) 0 0
\(547\) −0.590413 −0.0252442 −0.0126221 0.999920i \(-0.504018\pi\)
−0.0126221 + 0.999920i \(0.504018\pi\)
\(548\) 10.9663 0.468457
\(549\) 21.6710 0.924895
\(550\) −2.08003 −0.0886928
\(551\) 6.14352 0.261723
\(552\) −0.251665 −0.0107116
\(553\) 0 0
\(554\) −18.1198 −0.769838
\(555\) −0.00694676 −0.000294874 0
\(556\) −19.2928 −0.818196
\(557\) 33.4498 1.41732 0.708658 0.705553i \(-0.249301\pi\)
0.708658 + 0.705553i \(0.249301\pi\)
\(558\) 8.44639 0.357564
\(559\) 13.4618 0.569375
\(560\) 0 0
\(561\) 0.705408 0.0297823
\(562\) −13.5797 −0.572824
\(563\) −40.5522 −1.70907 −0.854536 0.519392i \(-0.826158\pi\)
−0.854536 + 0.519392i \(0.826158\pi\)
\(564\) 0.00430923 0.000181451 0
\(565\) −38.8884 −1.63605
\(566\) 9.63837 0.405131
\(567\) 0 0
\(568\) −35.9810 −1.50973
\(569\) −2.35745 −0.0988296 −0.0494148 0.998778i \(-0.515736\pi\)
−0.0494148 + 0.998778i \(0.515736\pi\)
\(570\) 0.0610162 0.00255569
\(571\) 6.02445 0.252115 0.126058 0.992023i \(-0.459768\pi\)
0.126058 + 0.992023i \(0.459768\pi\)
\(572\) −21.3935 −0.894509
\(573\) 0.383074 0.0160031
\(574\) 0 0
\(575\) 1.97973 0.0825605
\(576\) −13.2509 −0.552122
\(577\) −14.4100 −0.599895 −0.299947 0.953956i \(-0.596969\pi\)
−0.299947 + 0.953956i \(0.596969\pi\)
\(578\) −18.8592 −0.784440
\(579\) −0.297914 −0.0123809
\(580\) 10.7323 0.445634
\(581\) 0 0
\(582\) −0.141813 −0.00587834
\(583\) 35.8593 1.48514
\(584\) 30.7121 1.27087
\(585\) −20.4696 −0.846315
\(586\) 13.3620 0.551981
\(587\) 0.234221 0.00966735 0.00483367 0.999988i \(-0.498461\pi\)
0.00483367 + 0.999988i \(0.498461\pi\)
\(588\) 0 0
\(589\) 5.20853 0.214614
\(590\) 1.63020 0.0671143
\(591\) −0.361276 −0.0148609
\(592\) 0.0330591 0.00135872
\(593\) −28.6508 −1.17655 −0.588274 0.808661i \(-0.700193\pi\)
−0.588274 + 0.808661i \(0.700193\pi\)
\(594\) −0.571016 −0.0234291
\(595\) 0 0
\(596\) 17.3678 0.711413
\(597\) 0.0158822 0.000650013 0
\(598\) −11.3394 −0.463704
\(599\) −25.7299 −1.05129 −0.525647 0.850703i \(-0.676177\pi\)
−0.525647 + 0.850703i \(0.676177\pi\)
\(600\) −0.0285204 −0.00116434
\(601\) −25.8320 −1.05371 −0.526855 0.849955i \(-0.676629\pi\)
−0.526855 + 0.849955i \(0.676629\pi\)
\(602\) 0 0
\(603\) 5.16462 0.210320
\(604\) −3.83668 −0.156112
\(605\) −33.9468 −1.38013
\(606\) −0.138742 −0.00563599
\(607\) 14.6431 0.594346 0.297173 0.954824i \(-0.403956\pi\)
0.297173 + 0.954824i \(0.403956\pi\)
\(608\) −8.98307 −0.364312
\(609\) 0 0
\(610\) −13.0008 −0.526388
\(611\) 0.496457 0.0200845
\(612\) −24.1549 −0.976404
\(613\) −30.1823 −1.21905 −0.609526 0.792766i \(-0.708640\pi\)
−0.609526 + 0.792766i \(0.708640\pi\)
\(614\) −7.92735 −0.319922
\(615\) −0.0461110 −0.00185937
\(616\) 0 0
\(617\) −13.5169 −0.544170 −0.272085 0.962273i \(-0.587713\pi\)
−0.272085 + 0.962273i \(0.587713\pi\)
\(618\) −0.0321570 −0.00129354
\(619\) −14.2452 −0.572565 −0.286282 0.958145i \(-0.592420\pi\)
−0.286282 + 0.958145i \(0.592420\pi\)
\(620\) 9.09894 0.365422
\(621\) 0.543482 0.0218092
\(622\) 0.824968 0.0330782
\(623\) 0 0
\(624\) −0.0152555 −0.000610709 0
\(625\) −22.4073 −0.896293
\(626\) −27.3125 −1.09163
\(627\) −0.176047 −0.00703064
\(628\) −15.1153 −0.603166
\(629\) −0.944407 −0.0376560
\(630\) 0 0
\(631\) 10.1182 0.402798 0.201399 0.979509i \(-0.435451\pi\)
0.201399 + 0.979509i \(0.435451\pi\)
\(632\) 0.920734 0.0366248
\(633\) −0.544998 −0.0216617
\(634\) 17.1095 0.679504
\(635\) 3.95026 0.156761
\(636\) 0.192298 0.00762512
\(637\) 0 0
\(638\) 17.2444 0.682712
\(639\) 38.8482 1.53681
\(640\) −16.4825 −0.651529
\(641\) 48.2183 1.90451 0.952254 0.305307i \(-0.0987592\pi\)
0.952254 + 0.305307i \(0.0987592\pi\)
\(642\) 0.263414 0.0103961
\(643\) 20.1040 0.792823 0.396412 0.918073i \(-0.370255\pi\)
0.396412 + 0.918073i \(0.370255\pi\)
\(644\) 0 0
\(645\) 0.193520 0.00761983
\(646\) 8.29511 0.326367
\(647\) 12.0746 0.474701 0.237350 0.971424i \(-0.423721\pi\)
0.237350 + 0.971424i \(0.423721\pi\)
\(648\) 24.9916 0.981762
\(649\) −4.70353 −0.184630
\(650\) −1.28506 −0.0504044
\(651\) 0 0
\(652\) 14.3518 0.562061
\(653\) −22.6661 −0.886991 −0.443496 0.896277i \(-0.646262\pi\)
−0.443496 + 0.896277i \(0.646262\pi\)
\(654\) −0.0494054 −0.00193190
\(655\) −36.5751 −1.42911
\(656\) 0.219438 0.00856763
\(657\) −33.1594 −1.29367
\(658\) 0 0
\(659\) 25.4712 0.992217 0.496109 0.868260i \(-0.334762\pi\)
0.496109 + 0.868260i \(0.334762\pi\)
\(660\) −0.307541 −0.0119710
\(661\) 47.3837 1.84301 0.921507 0.388362i \(-0.126959\pi\)
0.921507 + 0.388362i \(0.126959\pi\)
\(662\) −25.6217 −0.995814
\(663\) 0.435808 0.0169254
\(664\) −17.1402 −0.665170
\(665\) 0 0
\(666\) 0.382211 0.0148104
\(667\) −16.4129 −0.635509
\(668\) −31.9413 −1.23585
\(669\) −0.590735 −0.0228391
\(670\) −3.09835 −0.119700
\(671\) 37.5106 1.44808
\(672\) 0 0
\(673\) −3.32956 −0.128345 −0.0641726 0.997939i \(-0.520441\pi\)
−0.0641726 + 0.997939i \(0.520441\pi\)
\(674\) 11.2160 0.432026
\(675\) 0.0615911 0.00237064
\(676\) 3.48276 0.133952
\(677\) 4.42275 0.169980 0.0849901 0.996382i \(-0.472914\pi\)
0.0849901 + 0.996382i \(0.472914\pi\)
\(678\) −0.335081 −0.0128687
\(679\) 0 0
\(680\) 37.0519 1.42087
\(681\) 0.303402 0.0116264
\(682\) 14.6200 0.559827
\(683\) −35.3517 −1.35270 −0.676348 0.736583i \(-0.736438\pi\)
−0.676348 + 0.736583i \(0.736438\pi\)
\(684\) 6.02828 0.230497
\(685\) 18.1620 0.693933
\(686\) 0 0
\(687\) 0.558192 0.0212963
\(688\) −0.920944 −0.0351106
\(689\) 22.1542 0.844009
\(690\) −0.163009 −0.00620566
\(691\) 27.4126 1.04282 0.521412 0.853305i \(-0.325405\pi\)
0.521412 + 0.853305i \(0.325405\pi\)
\(692\) 12.0334 0.457439
\(693\) 0 0
\(694\) −21.2677 −0.807312
\(695\) −31.9520 −1.21201
\(696\) 0.236447 0.00896251
\(697\) −6.26875 −0.237446
\(698\) −13.4788 −0.510180
\(699\) −0.454870 −0.0172048
\(700\) 0 0
\(701\) 45.3878 1.71427 0.857137 0.515088i \(-0.172241\pi\)
0.857137 + 0.515088i \(0.172241\pi\)
\(702\) −0.352780 −0.0133148
\(703\) 0.235693 0.00888934
\(704\) −22.9362 −0.864440
\(705\) 0.00713678 0.000268787 0
\(706\) 10.5932 0.398679
\(707\) 0 0
\(708\) −0.0252230 −0.000947940 0
\(709\) −8.87595 −0.333343 −0.166672 0.986012i \(-0.553302\pi\)
−0.166672 + 0.986012i \(0.553302\pi\)
\(710\) −23.3058 −0.874650
\(711\) −0.994105 −0.0372818
\(712\) −5.07589 −0.190227
\(713\) −13.9150 −0.521120
\(714\) 0 0
\(715\) −35.4311 −1.32505
\(716\) −14.2033 −0.530804
\(717\) −0.542391 −0.0202560
\(718\) −4.68877 −0.174983
\(719\) 21.2366 0.791993 0.395997 0.918252i \(-0.370399\pi\)
0.395997 + 0.918252i \(0.370399\pi\)
\(720\) 1.40036 0.0521883
\(721\) 0 0
\(722\) 14.0002 0.521032
\(723\) −0.174810 −0.00650127
\(724\) 10.0780 0.374545
\(725\) −1.86002 −0.0690794
\(726\) −0.292502 −0.0108558
\(727\) 8.30528 0.308026 0.154013 0.988069i \(-0.450780\pi\)
0.154013 + 0.988069i \(0.450780\pi\)
\(728\) 0 0
\(729\) −26.9746 −0.999061
\(730\) 19.8929 0.736270
\(731\) 26.3088 0.973068
\(732\) 0.201153 0.00743484
\(733\) 5.01851 0.185363 0.0926814 0.995696i \(-0.470456\pi\)
0.0926814 + 0.995696i \(0.470456\pi\)
\(734\) 2.50269 0.0923761
\(735\) 0 0
\(736\) 23.9989 0.884613
\(737\) 8.93950 0.329291
\(738\) 2.53703 0.0933892
\(739\) −30.8086 −1.13331 −0.566656 0.823955i \(-0.691763\pi\)
−0.566656 + 0.823955i \(0.691763\pi\)
\(740\) 0.411739 0.0151358
\(741\) −0.108764 −0.00399553
\(742\) 0 0
\(743\) 35.4367 1.30005 0.650023 0.759915i \(-0.274759\pi\)
0.650023 + 0.759915i \(0.274759\pi\)
\(744\) 0.200462 0.00734930
\(745\) 28.7639 1.05383
\(746\) −9.44131 −0.345671
\(747\) 18.5061 0.677102
\(748\) −41.8100 −1.52873
\(749\) 0 0
\(750\) −0.213479 −0.00779513
\(751\) 7.27823 0.265586 0.132793 0.991144i \(-0.457605\pi\)
0.132793 + 0.991144i \(0.457605\pi\)
\(752\) −0.0339634 −0.00123852
\(753\) −0.590108 −0.0215047
\(754\) 10.6538 0.387987
\(755\) −6.35415 −0.231251
\(756\) 0 0
\(757\) 25.0166 0.909243 0.454621 0.890685i \(-0.349775\pi\)
0.454621 + 0.890685i \(0.349775\pi\)
\(758\) 11.3770 0.413231
\(759\) 0.470323 0.0170716
\(760\) −9.24694 −0.335422
\(761\) 23.2193 0.841701 0.420850 0.907130i \(-0.361732\pi\)
0.420850 + 0.907130i \(0.361732\pi\)
\(762\) 0.0340373 0.00123304
\(763\) 0 0
\(764\) −22.7051 −0.821440
\(765\) −40.0044 −1.44636
\(766\) 28.3710 1.02509
\(767\) −2.90589 −0.104926
\(768\) −0.333515 −0.0120347
\(769\) 40.2927 1.45299 0.726496 0.687170i \(-0.241147\pi\)
0.726496 + 0.687170i \(0.241147\pi\)
\(770\) 0 0
\(771\) 0.304461 0.0109649
\(772\) 17.6576 0.635510
\(773\) −11.5294 −0.414685 −0.207343 0.978268i \(-0.566481\pi\)
−0.207343 + 0.978268i \(0.566481\pi\)
\(774\) −10.6474 −0.382715
\(775\) −1.57694 −0.0566454
\(776\) 21.4916 0.771504
\(777\) 0 0
\(778\) −31.0284 −1.11242
\(779\) 1.56448 0.0560532
\(780\) −0.190002 −0.00680317
\(781\) 67.2429 2.40614
\(782\) −22.1610 −0.792476
\(783\) −0.510618 −0.0182480
\(784\) 0 0
\(785\) −25.0334 −0.893479
\(786\) −0.315148 −0.0112410
\(787\) 52.0845 1.85661 0.928306 0.371818i \(-0.121266\pi\)
0.928306 + 0.371818i \(0.121266\pi\)
\(788\) 21.4131 0.762809
\(789\) 0.185420 0.00660113
\(790\) 0.596381 0.0212183
\(791\) 0 0
\(792\) 43.2650 1.53736
\(793\) 23.1744 0.822948
\(794\) 17.6702 0.627091
\(795\) 0.318477 0.0112952
\(796\) −0.941347 −0.0333651
\(797\) −19.7674 −0.700197 −0.350098 0.936713i \(-0.613852\pi\)
−0.350098 + 0.936713i \(0.613852\pi\)
\(798\) 0 0
\(799\) 0.970240 0.0343246
\(800\) 2.71973 0.0961568
\(801\) 5.48038 0.193640
\(802\) 15.7168 0.554978
\(803\) −57.3960 −2.02546
\(804\) 0.0479388 0.00169067
\(805\) 0 0
\(806\) 9.03236 0.318151
\(807\) −0.663230 −0.0233468
\(808\) 21.0261 0.739697
\(809\) 34.1602 1.20101 0.600504 0.799621i \(-0.294967\pi\)
0.600504 + 0.799621i \(0.294967\pi\)
\(810\) 16.1876 0.568776
\(811\) 48.6449 1.70815 0.854077 0.520147i \(-0.174123\pi\)
0.854077 + 0.520147i \(0.174123\pi\)
\(812\) 0 0
\(813\) −0.617098 −0.0216426
\(814\) 0.661573 0.0231881
\(815\) 23.7689 0.832589
\(816\) −0.0298143 −0.00104371
\(817\) −6.56583 −0.229709
\(818\) −24.4593 −0.855199
\(819\) 0 0
\(820\) 2.73303 0.0954415
\(821\) 18.1850 0.634659 0.317330 0.948315i \(-0.397214\pi\)
0.317330 + 0.948315i \(0.397214\pi\)
\(822\) 0.156492 0.00545828
\(823\) 11.6225 0.405135 0.202567 0.979268i \(-0.435072\pi\)
0.202567 + 0.979268i \(0.435072\pi\)
\(824\) 4.87336 0.169772
\(825\) 0.0533002 0.00185567
\(826\) 0 0
\(827\) −7.09639 −0.246766 −0.123383 0.992359i \(-0.539374\pi\)
−0.123383 + 0.992359i \(0.539374\pi\)
\(828\) −16.1050 −0.559688
\(829\) 8.50362 0.295343 0.147672 0.989036i \(-0.452822\pi\)
0.147672 + 0.989036i \(0.452822\pi\)
\(830\) −11.1021 −0.385360
\(831\) 0.464316 0.0161069
\(832\) −14.1702 −0.491264
\(833\) 0 0
\(834\) −0.275313 −0.00953331
\(835\) −52.9000 −1.83068
\(836\) 10.4344 0.360882
\(837\) −0.432907 −0.0149634
\(838\) 25.9947 0.897970
\(839\) 39.3940 1.36003 0.680015 0.733198i \(-0.261973\pi\)
0.680015 + 0.733198i \(0.261973\pi\)
\(840\) 0 0
\(841\) −13.5796 −0.468261
\(842\) 15.4710 0.533167
\(843\) 0.347976 0.0119849
\(844\) 32.3024 1.11189
\(845\) 5.76801 0.198426
\(846\) −0.392666 −0.0135001
\(847\) 0 0
\(848\) −1.51560 −0.0520460
\(849\) −0.246981 −0.00847635
\(850\) −2.51144 −0.0861416
\(851\) −0.629672 −0.0215849
\(852\) 0.360595 0.0123538
\(853\) −50.8194 −1.74002 −0.870012 0.493031i \(-0.835889\pi\)
−0.870012 + 0.493031i \(0.835889\pi\)
\(854\) 0 0
\(855\) 9.98380 0.341439
\(856\) −39.9201 −1.36444
\(857\) −17.4259 −0.595257 −0.297628 0.954682i \(-0.596196\pi\)
−0.297628 + 0.954682i \(0.596196\pi\)
\(858\) −0.305291 −0.0104225
\(859\) 14.7578 0.503530 0.251765 0.967788i \(-0.418989\pi\)
0.251765 + 0.967788i \(0.418989\pi\)
\(860\) −11.4700 −0.391125
\(861\) 0 0
\(862\) −5.29543 −0.180363
\(863\) −10.2274 −0.348143 −0.174072 0.984733i \(-0.555692\pi\)
−0.174072 + 0.984733i \(0.555692\pi\)
\(864\) 0.746627 0.0254008
\(865\) 19.9292 0.677612
\(866\) −15.0415 −0.511129
\(867\) 0.483262 0.0164124
\(868\) 0 0
\(869\) −1.72071 −0.0583710
\(870\) 0.153153 0.00519236
\(871\) 5.52292 0.187137
\(872\) 7.48733 0.253553
\(873\) −23.2042 −0.785344
\(874\) 5.53067 0.187078
\(875\) 0 0
\(876\) −0.307791 −0.0103993
\(877\) −42.7787 −1.44453 −0.722266 0.691615i \(-0.756900\pi\)
−0.722266 + 0.691615i \(0.756900\pi\)
\(878\) −17.7845 −0.600199
\(879\) −0.342399 −0.0115488
\(880\) 2.42390 0.0817095
\(881\) 0.101554 0.00342143 0.00171072 0.999999i \(-0.499455\pi\)
0.00171072 + 0.999999i \(0.499455\pi\)
\(882\) 0 0
\(883\) −17.0311 −0.573142 −0.286571 0.958059i \(-0.592515\pi\)
−0.286571 + 0.958059i \(0.592515\pi\)
\(884\) −25.8307 −0.868779
\(885\) −0.0417734 −0.00140420
\(886\) −12.0727 −0.405590
\(887\) −44.0812 −1.48010 −0.740051 0.672550i \(-0.765199\pi\)
−0.740051 + 0.672550i \(0.765199\pi\)
\(888\) 0.00907119 0.000304409 0
\(889\) 0 0
\(890\) −3.28778 −0.110207
\(891\) −46.7054 −1.56469
\(892\) 35.0133 1.17233
\(893\) −0.242140 −0.00810292
\(894\) 0.247843 0.00828911
\(895\) −23.5230 −0.786288
\(896\) 0 0
\(897\) 0.290570 0.00970186
\(898\) 20.0415 0.668795
\(899\) 13.0736 0.436028
\(900\) −1.82513 −0.0608377
\(901\) 43.2967 1.44242
\(902\) 4.39137 0.146217
\(903\) 0 0
\(904\) 50.7811 1.68895
\(905\) 16.6908 0.554819
\(906\) −0.0547503 −0.00181896
\(907\) 6.16615 0.204744 0.102372 0.994746i \(-0.467357\pi\)
0.102372 + 0.994746i \(0.467357\pi\)
\(908\) −17.9829 −0.596783
\(909\) −22.7016 −0.752966
\(910\) 0 0
\(911\) 38.0446 1.26048 0.630238 0.776402i \(-0.282957\pi\)
0.630238 + 0.776402i \(0.282957\pi\)
\(912\) 0.00744068 0.000246385 0
\(913\) 32.0324 1.06012
\(914\) 2.10153 0.0695123
\(915\) 0.333142 0.0110133
\(916\) −33.0844 −1.09314
\(917\) 0 0
\(918\) −0.689447 −0.0227552
\(919\) 47.2627 1.55905 0.779526 0.626370i \(-0.215460\pi\)
0.779526 + 0.626370i \(0.215460\pi\)
\(920\) 24.7039 0.814463
\(921\) 0.203136 0.00669357
\(922\) 21.8390 0.719229
\(923\) 41.5434 1.36742
\(924\) 0 0
\(925\) −0.0713588 −0.00234626
\(926\) −21.3757 −0.702448
\(927\) −5.26170 −0.172817
\(928\) −22.5478 −0.740167
\(929\) −23.4189 −0.768349 −0.384174 0.923261i \(-0.625514\pi\)
−0.384174 + 0.923261i \(0.625514\pi\)
\(930\) 0.129844 0.00425776
\(931\) 0 0
\(932\) 26.9605 0.883120
\(933\) −0.0211396 −0.000692078 0
\(934\) −28.1019 −0.919523
\(935\) −69.2441 −2.26452
\(936\) 26.7296 0.873684
\(937\) 38.4910 1.25745 0.628723 0.777629i \(-0.283578\pi\)
0.628723 + 0.777629i \(0.283578\pi\)
\(938\) 0 0
\(939\) 0.699876 0.0228396
\(940\) −0.423002 −0.0137968
\(941\) 22.3251 0.727777 0.363889 0.931442i \(-0.381449\pi\)
0.363889 + 0.931442i \(0.381449\pi\)
\(942\) −0.215699 −0.00702786
\(943\) −4.17962 −0.136107
\(944\) 0.198796 0.00647027
\(945\) 0 0
\(946\) −18.4298 −0.599204
\(947\) −12.6943 −0.412510 −0.206255 0.978498i \(-0.566128\pi\)
−0.206255 + 0.978498i \(0.566128\pi\)
\(948\) −0.00922743 −0.000299693 0
\(949\) −35.4599 −1.15108
\(950\) 0.626773 0.0203352
\(951\) −0.438426 −0.0142169
\(952\) 0 0
\(953\) −6.70948 −0.217341 −0.108671 0.994078i \(-0.534659\pi\)
−0.108671 + 0.994078i \(0.534659\pi\)
\(954\) −17.5226 −0.567315
\(955\) −37.6032 −1.21681
\(956\) 32.1479 1.03974
\(957\) −0.441883 −0.0142840
\(958\) 15.2801 0.493678
\(959\) 0 0
\(960\) −0.203703 −0.00657448
\(961\) −19.9161 −0.642455
\(962\) 0.408727 0.0131779
\(963\) 43.1013 1.38892
\(964\) 10.3611 0.333710
\(965\) 29.2438 0.941391
\(966\) 0 0
\(967\) 21.1576 0.680382 0.340191 0.940356i \(-0.389508\pi\)
0.340191 + 0.940356i \(0.389508\pi\)
\(968\) 44.3283 1.42477
\(969\) −0.212560 −0.00682841
\(970\) 13.9206 0.446965
\(971\) 15.8074 0.507283 0.253642 0.967298i \(-0.418372\pi\)
0.253642 + 0.967298i \(0.418372\pi\)
\(972\) −0.751580 −0.0241069
\(973\) 0 0
\(974\) −8.71784 −0.279338
\(975\) 0.0329294 0.00105459
\(976\) −1.58540 −0.0507473
\(977\) 44.0859 1.41043 0.705216 0.708993i \(-0.250850\pi\)
0.705216 + 0.708993i \(0.250850\pi\)
\(978\) 0.204804 0.00654892
\(979\) 9.48605 0.303175
\(980\) 0 0
\(981\) −8.08398 −0.258101
\(982\) 15.4566 0.493239
\(983\) 17.4650 0.557047 0.278523 0.960429i \(-0.410155\pi\)
0.278523 + 0.960429i \(0.410155\pi\)
\(984\) 0.0602125 0.00191950
\(985\) 35.4635 1.12996
\(986\) 20.8210 0.663075
\(987\) 0 0
\(988\) 6.44650 0.205090
\(989\) 17.5411 0.557775
\(990\) 28.0238 0.890654
\(991\) 23.4368 0.744495 0.372248 0.928133i \(-0.378587\pi\)
0.372248 + 0.928133i \(0.378587\pi\)
\(992\) −19.1162 −0.606940
\(993\) 0.656548 0.0208349
\(994\) 0 0
\(995\) −1.55902 −0.0494243
\(996\) 0.171776 0.00544294
\(997\) 14.6814 0.464963 0.232482 0.972601i \(-0.425315\pi\)
0.232482 + 0.972601i \(0.425315\pi\)
\(998\) −15.9673 −0.505435
\(999\) −0.0195896 −0.000619789 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 2009.2.a.u.1.9 yes 20
7.6 odd 2 2009.2.a.t.1.9 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2009.2.a.t.1.9 20 7.6 odd 2
2009.2.a.u.1.9 yes 20 1.1 even 1 trivial