Properties

Label 2009.2.a.u.1.12
Level $2009$
Weight $2$
Character 2009.1
Self dual yes
Analytic conductor $16.042$
Analytic rank $0$
Dimension $20$
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: \(20\)
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 1.12
Root \(-0.463326\) 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)
 
\(f(q)\) \(=\) \(q+0.463326 q^{2} +0.981674 q^{3} -1.78533 q^{4} -3.31637 q^{5} +0.454835 q^{6} -1.75384 q^{8} -2.03632 q^{9} +O(q^{10})\) \(q+0.463326 q^{2} +0.981674 q^{3} -1.78533 q^{4} -3.31637 q^{5} +0.454835 q^{6} -1.75384 q^{8} -2.03632 q^{9} -1.53656 q^{10} -2.89859 q^{11} -1.75261 q^{12} -6.95519 q^{13} -3.25559 q^{15} +2.75806 q^{16} +7.66901 q^{17} -0.943479 q^{18} +1.90411 q^{19} +5.92081 q^{20} -1.34299 q^{22} +3.76800 q^{23} -1.72170 q^{24} +5.99830 q^{25} -3.22252 q^{26} -4.94402 q^{27} -3.58717 q^{29} -1.50840 q^{30} +8.42330 q^{31} +4.78557 q^{32} -2.84547 q^{33} +3.55325 q^{34} +3.63549 q^{36} -3.65525 q^{37} +0.882224 q^{38} -6.82773 q^{39} +5.81639 q^{40} +1.00000 q^{41} -1.24584 q^{43} +5.17494 q^{44} +6.75318 q^{45} +1.74581 q^{46} +8.86125 q^{47} +2.70751 q^{48} +2.77917 q^{50} +7.52846 q^{51} +12.4173 q^{52} +4.12816 q^{53} -2.29070 q^{54} +9.61280 q^{55} +1.86922 q^{57} -1.66203 q^{58} -10.5998 q^{59} +5.81230 q^{60} +8.56444 q^{61} +3.90274 q^{62} -3.29883 q^{64} +23.0660 q^{65} -1.31838 q^{66} -4.56055 q^{67} -13.6917 q^{68} +3.69895 q^{69} +2.16682 q^{71} +3.57138 q^{72} +3.80196 q^{73} -1.69358 q^{74} +5.88838 q^{75} -3.39946 q^{76} -3.16347 q^{78} -11.0070 q^{79} -9.14673 q^{80} +1.25553 q^{81} +0.463326 q^{82} +12.5558 q^{83} -25.4333 q^{85} -0.577230 q^{86} -3.52143 q^{87} +5.08367 q^{88} -7.65506 q^{89} +3.12892 q^{90} -6.72712 q^{92} +8.26893 q^{93} +4.10565 q^{94} -6.31473 q^{95} +4.69786 q^{96} -0.742199 q^{97} +5.90245 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 2 q^{2} + 8 q^{3} + 18 q^{4} + 8 q^{5} + 12 q^{6} - 6 q^{8} + 16 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.463326 0.327621 0.163811 0.986492i \(-0.447621\pi\)
0.163811 + 0.986492i \(0.447621\pi\)
\(3\) 0.981674 0.566770 0.283385 0.959006i \(-0.408543\pi\)
0.283385 + 0.959006i \(0.408543\pi\)
\(4\) −1.78533 −0.892664
\(5\) −3.31637 −1.48313 −0.741563 0.670884i \(-0.765915\pi\)
−0.741563 + 0.670884i \(0.765915\pi\)
\(6\) 0.454835 0.185686
\(7\) 0 0
\(8\) −1.75384 −0.620077
\(9\) −2.03632 −0.678772
\(10\) −1.53656 −0.485903
\(11\) −2.89859 −0.873958 −0.436979 0.899472i \(-0.643952\pi\)
−0.436979 + 0.899472i \(0.643952\pi\)
\(12\) −1.75261 −0.505935
\(13\) −6.95519 −1.92902 −0.964512 0.264040i \(-0.914945\pi\)
−0.964512 + 0.264040i \(0.914945\pi\)
\(14\) 0 0
\(15\) −3.25559 −0.840590
\(16\) 2.75806 0.689514
\(17\) 7.66901 1.86001 0.930004 0.367550i \(-0.119803\pi\)
0.930004 + 0.367550i \(0.119803\pi\)
\(18\) −0.943479 −0.222380
\(19\) 1.90411 0.436833 0.218416 0.975856i \(-0.429911\pi\)
0.218416 + 0.975856i \(0.429911\pi\)
\(20\) 5.92081 1.32393
\(21\) 0 0
\(22\) −1.34299 −0.286327
\(23\) 3.76800 0.785682 0.392841 0.919606i \(-0.371492\pi\)
0.392841 + 0.919606i \(0.371492\pi\)
\(24\) −1.72170 −0.351441
\(25\) 5.99830 1.19966
\(26\) −3.22252 −0.631989
\(27\) −4.94402 −0.951477
\(28\) 0 0
\(29\) −3.58717 −0.666120 −0.333060 0.942906i \(-0.608081\pi\)
−0.333060 + 0.942906i \(0.608081\pi\)
\(30\) −1.50840 −0.275395
\(31\) 8.42330 1.51287 0.756435 0.654069i \(-0.226940\pi\)
0.756435 + 0.654069i \(0.226940\pi\)
\(32\) 4.78557 0.845976
\(33\) −2.84547 −0.495333
\(34\) 3.55325 0.609378
\(35\) 0 0
\(36\) 3.63549 0.605916
\(37\) −3.65525 −0.600920 −0.300460 0.953794i \(-0.597140\pi\)
−0.300460 + 0.953794i \(0.597140\pi\)
\(38\) 0.882224 0.143116
\(39\) −6.82773 −1.09331
\(40\) 5.81639 0.919652
\(41\) 1.00000 0.156174
\(42\) 0 0
\(43\) −1.24584 −0.189989 −0.0949944 0.995478i \(-0.530283\pi\)
−0.0949944 + 0.995478i \(0.530283\pi\)
\(44\) 5.17494 0.780151
\(45\) 6.75318 1.00670
\(46\) 1.74581 0.257406
\(47\) 8.86125 1.29255 0.646273 0.763106i \(-0.276327\pi\)
0.646273 + 0.763106i \(0.276327\pi\)
\(48\) 2.70751 0.390796
\(49\) 0 0
\(50\) 2.77917 0.393034
\(51\) 7.52846 1.05420
\(52\) 12.4173 1.72197
\(53\) 4.12816 0.567047 0.283523 0.958965i \(-0.408497\pi\)
0.283523 + 0.958965i \(0.408497\pi\)
\(54\) −2.29070 −0.311724
\(55\) 9.61280 1.29619
\(56\) 0 0
\(57\) 1.86922 0.247584
\(58\) −1.66203 −0.218235
\(59\) −10.5998 −1.37997 −0.689987 0.723822i \(-0.742384\pi\)
−0.689987 + 0.723822i \(0.742384\pi\)
\(60\) 5.81230 0.750365
\(61\) 8.56444 1.09656 0.548282 0.836293i \(-0.315282\pi\)
0.548282 + 0.836293i \(0.315282\pi\)
\(62\) 3.90274 0.495648
\(63\) 0 0
\(64\) −3.29883 −0.412354
\(65\) 23.0660 2.86098
\(66\) −1.31838 −0.162282
\(67\) −4.56055 −0.557160 −0.278580 0.960413i \(-0.589864\pi\)
−0.278580 + 0.960413i \(0.589864\pi\)
\(68\) −13.6917 −1.66036
\(69\) 3.69895 0.445301
\(70\) 0 0
\(71\) 2.16682 0.257154 0.128577 0.991700i \(-0.458959\pi\)
0.128577 + 0.991700i \(0.458959\pi\)
\(72\) 3.57138 0.420891
\(73\) 3.80196 0.444985 0.222493 0.974934i \(-0.428581\pi\)
0.222493 + 0.974934i \(0.428581\pi\)
\(74\) −1.69358 −0.196874
\(75\) 5.88838 0.679931
\(76\) −3.39946 −0.389945
\(77\) 0 0
\(78\) −3.16347 −0.358192
\(79\) −11.0070 −1.23838 −0.619191 0.785240i \(-0.712539\pi\)
−0.619191 + 0.785240i \(0.712539\pi\)
\(80\) −9.14673 −1.02264
\(81\) 1.25553 0.139504
\(82\) 0.463326 0.0511658
\(83\) 12.5558 1.37818 0.689089 0.724677i \(-0.258011\pi\)
0.689089 + 0.724677i \(0.258011\pi\)
\(84\) 0 0
\(85\) −25.4333 −2.75862
\(86\) −0.577230 −0.0622443
\(87\) −3.52143 −0.377537
\(88\) 5.08367 0.541921
\(89\) −7.65506 −0.811435 −0.405717 0.913999i \(-0.632978\pi\)
−0.405717 + 0.913999i \(0.632978\pi\)
\(90\) 3.12892 0.329818
\(91\) 0 0
\(92\) −6.72712 −0.701350
\(93\) 8.26893 0.857449
\(94\) 4.10565 0.423465
\(95\) −6.31473 −0.647878
\(96\) 4.69786 0.479474
\(97\) −0.742199 −0.0753589 −0.0376795 0.999290i \(-0.511997\pi\)
−0.0376795 + 0.999290i \(0.511997\pi\)
\(98\) 0 0
\(99\) 5.90245 0.593218
\(100\) −10.7089 −1.07089
\(101\) −3.22970 −0.321367 −0.160683 0.987006i \(-0.551370\pi\)
−0.160683 + 0.987006i \(0.551370\pi\)
\(102\) 3.48814 0.345377
\(103\) 9.82813 0.968394 0.484197 0.874959i \(-0.339112\pi\)
0.484197 + 0.874959i \(0.339112\pi\)
\(104\) 12.1983 1.19614
\(105\) 0 0
\(106\) 1.91269 0.185777
\(107\) −18.3990 −1.77870 −0.889350 0.457226i \(-0.848843\pi\)
−0.889350 + 0.457226i \(0.848843\pi\)
\(108\) 8.82670 0.849350
\(109\) 6.43698 0.616551 0.308276 0.951297i \(-0.400248\pi\)
0.308276 + 0.951297i \(0.400248\pi\)
\(110\) 4.45386 0.424659
\(111\) −3.58827 −0.340583
\(112\) 0 0
\(113\) −13.3856 −1.25921 −0.629605 0.776915i \(-0.716783\pi\)
−0.629605 + 0.776915i \(0.716783\pi\)
\(114\) 0.866057 0.0811137
\(115\) −12.4961 −1.16527
\(116\) 6.40427 0.594622
\(117\) 14.1630 1.30937
\(118\) −4.91116 −0.452109
\(119\) 0 0
\(120\) 5.70980 0.521231
\(121\) −2.59817 −0.236198
\(122\) 3.96813 0.359258
\(123\) 0.981674 0.0885146
\(124\) −15.0384 −1.35048
\(125\) −3.31074 −0.296121
\(126\) 0 0
\(127\) −6.62684 −0.588037 −0.294018 0.955800i \(-0.594993\pi\)
−0.294018 + 0.955800i \(0.594993\pi\)
\(128\) −11.0996 −0.981072
\(129\) −1.22301 −0.107680
\(130\) 10.6871 0.937319
\(131\) 15.7977 1.38025 0.690124 0.723692i \(-0.257556\pi\)
0.690124 + 0.723692i \(0.257556\pi\)
\(132\) 5.08010 0.442166
\(133\) 0 0
\(134\) −2.11302 −0.182537
\(135\) 16.3962 1.41116
\(136\) −13.4502 −1.15335
\(137\) 4.59477 0.392558 0.196279 0.980548i \(-0.437114\pi\)
0.196279 + 0.980548i \(0.437114\pi\)
\(138\) 1.71382 0.145890
\(139\) −2.72020 −0.230724 −0.115362 0.993324i \(-0.536803\pi\)
−0.115362 + 0.993324i \(0.536803\pi\)
\(140\) 0 0
\(141\) 8.69886 0.732576
\(142\) 1.00394 0.0842490
\(143\) 20.1603 1.68589
\(144\) −5.61627 −0.468023
\(145\) 11.8964 0.987939
\(146\) 1.76155 0.145787
\(147\) 0 0
\(148\) 6.52583 0.536420
\(149\) 8.74660 0.716549 0.358275 0.933616i \(-0.383365\pi\)
0.358275 + 0.933616i \(0.383365\pi\)
\(150\) 2.72824 0.222760
\(151\) 19.9371 1.62246 0.811228 0.584730i \(-0.198799\pi\)
0.811228 + 0.584730i \(0.198799\pi\)
\(152\) −3.33951 −0.270870
\(153\) −15.6165 −1.26252
\(154\) 0 0
\(155\) −27.9348 −2.24377
\(156\) 12.1897 0.975961
\(157\) 18.4568 1.47301 0.736505 0.676432i \(-0.236475\pi\)
0.736505 + 0.676432i \(0.236475\pi\)
\(158\) −5.09982 −0.405720
\(159\) 4.05251 0.321385
\(160\) −15.8707 −1.25469
\(161\) 0 0
\(162\) 0.581721 0.0457043
\(163\) −7.80147 −0.611058 −0.305529 0.952183i \(-0.598833\pi\)
−0.305529 + 0.952183i \(0.598833\pi\)
\(164\) −1.78533 −0.139411
\(165\) 9.43663 0.734641
\(166\) 5.81743 0.451520
\(167\) −5.52402 −0.427461 −0.213731 0.976893i \(-0.568561\pi\)
−0.213731 + 0.976893i \(0.568561\pi\)
\(168\) 0 0
\(169\) 35.3747 2.72113
\(170\) −11.7839 −0.903784
\(171\) −3.87737 −0.296510
\(172\) 2.22423 0.169596
\(173\) −8.52314 −0.648002 −0.324001 0.946057i \(-0.605028\pi\)
−0.324001 + 0.946057i \(0.605028\pi\)
\(174\) −1.63157 −0.123689
\(175\) 0 0
\(176\) −7.99447 −0.602606
\(177\) −10.4055 −0.782127
\(178\) −3.54679 −0.265843
\(179\) −11.1040 −0.829950 −0.414975 0.909833i \(-0.636210\pi\)
−0.414975 + 0.909833i \(0.636210\pi\)
\(180\) −12.0566 −0.898649
\(181\) 25.7288 1.91241 0.956203 0.292704i \(-0.0945552\pi\)
0.956203 + 0.292704i \(0.0945552\pi\)
\(182\) 0 0
\(183\) 8.40749 0.621499
\(184\) −6.60848 −0.487183
\(185\) 12.1222 0.891239
\(186\) 3.83122 0.280918
\(187\) −22.2293 −1.62557
\(188\) −15.8202 −1.15381
\(189\) 0 0
\(190\) −2.92578 −0.212259
\(191\) 11.4290 0.826975 0.413487 0.910510i \(-0.364311\pi\)
0.413487 + 0.910510i \(0.364311\pi\)
\(192\) −3.23838 −0.233710
\(193\) 14.0055 1.00814 0.504068 0.863664i \(-0.331836\pi\)
0.504068 + 0.863664i \(0.331836\pi\)
\(194\) −0.343881 −0.0246892
\(195\) 22.6433 1.62152
\(196\) 0 0
\(197\) 15.0830 1.07462 0.537310 0.843385i \(-0.319440\pi\)
0.537310 + 0.843385i \(0.319440\pi\)
\(198\) 2.73476 0.194351
\(199\) 18.4669 1.30909 0.654543 0.756025i \(-0.272861\pi\)
0.654543 + 0.756025i \(0.272861\pi\)
\(200\) −10.5201 −0.743882
\(201\) −4.47697 −0.315781
\(202\) −1.49640 −0.105287
\(203\) 0 0
\(204\) −13.4408 −0.941043
\(205\) −3.31637 −0.231625
\(206\) 4.55363 0.317266
\(207\) −7.67284 −0.533299
\(208\) −19.1828 −1.33009
\(209\) −5.51924 −0.381774
\(210\) 0 0
\(211\) −13.8693 −0.954805 −0.477402 0.878685i \(-0.658422\pi\)
−0.477402 + 0.878685i \(0.658422\pi\)
\(212\) −7.37013 −0.506182
\(213\) 2.12711 0.145747
\(214\) −8.52475 −0.582740
\(215\) 4.13166 0.281777
\(216\) 8.67103 0.589989
\(217\) 0 0
\(218\) 2.98242 0.201995
\(219\) 3.73228 0.252204
\(220\) −17.1620 −1.15706
\(221\) −53.3394 −3.58800
\(222\) −1.66254 −0.111582
\(223\) 25.2567 1.69132 0.845658 0.533725i \(-0.179208\pi\)
0.845658 + 0.533725i \(0.179208\pi\)
\(224\) 0 0
\(225\) −12.2144 −0.814296
\(226\) −6.20190 −0.412544
\(227\) 6.76044 0.448706 0.224353 0.974508i \(-0.427973\pi\)
0.224353 + 0.974508i \(0.427973\pi\)
\(228\) −3.33716 −0.221009
\(229\) −13.6311 −0.900771 −0.450385 0.892834i \(-0.648713\pi\)
−0.450385 + 0.892834i \(0.648713\pi\)
\(230\) −5.78976 −0.381766
\(231\) 0 0
\(232\) 6.29132 0.413046
\(233\) 0.294047 0.0192637 0.00963183 0.999954i \(-0.496934\pi\)
0.00963183 + 0.999954i \(0.496934\pi\)
\(234\) 6.56208 0.428977
\(235\) −29.3872 −1.91701
\(236\) 18.9241 1.23185
\(237\) −10.8053 −0.701877
\(238\) 0 0
\(239\) −8.90226 −0.575839 −0.287920 0.957655i \(-0.592964\pi\)
−0.287920 + 0.957655i \(0.592964\pi\)
\(240\) −8.97911 −0.579599
\(241\) −9.51807 −0.613113 −0.306556 0.951853i \(-0.599177\pi\)
−0.306556 + 0.951853i \(0.599177\pi\)
\(242\) −1.20380 −0.0773833
\(243\) 16.0646 1.03054
\(244\) −15.2903 −0.978864
\(245\) 0 0
\(246\) 0.454835 0.0289992
\(247\) −13.2435 −0.842661
\(248\) −14.7731 −0.938095
\(249\) 12.3257 0.781109
\(250\) −1.53395 −0.0970156
\(251\) −28.4152 −1.79355 −0.896774 0.442488i \(-0.854096\pi\)
−0.896774 + 0.442488i \(0.854096\pi\)
\(252\) 0 0
\(253\) −10.9219 −0.686653
\(254\) −3.07039 −0.192653
\(255\) −24.9672 −1.56350
\(256\) 1.45494 0.0909340
\(257\) −14.2422 −0.888403 −0.444202 0.895927i \(-0.646513\pi\)
−0.444202 + 0.895927i \(0.646513\pi\)
\(258\) −0.566652 −0.0352782
\(259\) 0 0
\(260\) −41.1804 −2.55390
\(261\) 7.30460 0.452144
\(262\) 7.31947 0.452198
\(263\) 25.1201 1.54897 0.774487 0.632590i \(-0.218008\pi\)
0.774487 + 0.632590i \(0.218008\pi\)
\(264\) 4.99051 0.307145
\(265\) −13.6905 −0.841001
\(266\) 0 0
\(267\) −7.51478 −0.459897
\(268\) 8.14208 0.497357
\(269\) 8.45011 0.515212 0.257606 0.966250i \(-0.417066\pi\)
0.257606 + 0.966250i \(0.417066\pi\)
\(270\) 7.59679 0.462326
\(271\) −2.52722 −0.153518 −0.0767588 0.997050i \(-0.524457\pi\)
−0.0767588 + 0.997050i \(0.524457\pi\)
\(272\) 21.1515 1.28250
\(273\) 0 0
\(274\) 2.12888 0.128610
\(275\) −17.3866 −1.04845
\(276\) −6.60384 −0.397504
\(277\) 19.9949 1.20138 0.600689 0.799482i \(-0.294893\pi\)
0.600689 + 0.799482i \(0.294893\pi\)
\(278\) −1.26034 −0.0755901
\(279\) −17.1525 −1.02689
\(280\) 0 0
\(281\) 6.76965 0.403843 0.201922 0.979402i \(-0.435281\pi\)
0.201922 + 0.979402i \(0.435281\pi\)
\(282\) 4.03041 0.240007
\(283\) 6.32699 0.376101 0.188050 0.982159i \(-0.439783\pi\)
0.188050 + 0.982159i \(0.439783\pi\)
\(284\) −3.86848 −0.229552
\(285\) −6.19901 −0.367198
\(286\) 9.34078 0.552332
\(287\) 0 0
\(288\) −9.74492 −0.574225
\(289\) 41.8137 2.45963
\(290\) 5.51190 0.323670
\(291\) −0.728598 −0.0427112
\(292\) −6.78774 −0.397223
\(293\) −27.0292 −1.57906 −0.789530 0.613712i \(-0.789676\pi\)
−0.789530 + 0.613712i \(0.789676\pi\)
\(294\) 0 0
\(295\) 35.1528 2.04667
\(296\) 6.41074 0.372617
\(297\) 14.3307 0.831551
\(298\) 4.05253 0.234757
\(299\) −26.2072 −1.51560
\(300\) −10.5127 −0.606950
\(301\) 0 0
\(302\) 9.23737 0.531551
\(303\) −3.17051 −0.182141
\(304\) 5.25164 0.301202
\(305\) −28.4028 −1.62634
\(306\) −7.23555 −0.413629
\(307\) 5.48187 0.312867 0.156433 0.987689i \(-0.450000\pi\)
0.156433 + 0.987689i \(0.450000\pi\)
\(308\) 0 0
\(309\) 9.64802 0.548856
\(310\) −12.9429 −0.735108
\(311\) 7.22580 0.409737 0.204869 0.978789i \(-0.434323\pi\)
0.204869 + 0.978789i \(0.434323\pi\)
\(312\) 11.9748 0.677938
\(313\) 18.5794 1.05017 0.525086 0.851049i \(-0.324033\pi\)
0.525086 + 0.851049i \(0.324033\pi\)
\(314\) 8.55151 0.482589
\(315\) 0 0
\(316\) 19.6511 1.10546
\(317\) 29.0493 1.63157 0.815786 0.578353i \(-0.196305\pi\)
0.815786 + 0.578353i \(0.196305\pi\)
\(318\) 1.87763 0.105293
\(319\) 10.3977 0.582161
\(320\) 10.9401 0.611573
\(321\) −18.0618 −1.00811
\(322\) 0 0
\(323\) 14.6026 0.812512
\(324\) −2.24154 −0.124530
\(325\) −41.7194 −2.31417
\(326\) −3.61463 −0.200196
\(327\) 6.31902 0.349443
\(328\) −1.75384 −0.0968398
\(329\) 0 0
\(330\) 4.37224 0.240684
\(331\) 4.58876 0.252221 0.126111 0.992016i \(-0.459751\pi\)
0.126111 + 0.992016i \(0.459751\pi\)
\(332\) −22.4162 −1.23025
\(333\) 7.44325 0.407888
\(334\) −2.55942 −0.140045
\(335\) 15.1245 0.826338
\(336\) 0 0
\(337\) 32.2998 1.75948 0.879742 0.475451i \(-0.157715\pi\)
0.879742 + 0.475451i \(0.157715\pi\)
\(338\) 16.3900 0.891501
\(339\) −13.1403 −0.713683
\(340\) 45.4067 2.46253
\(341\) −24.4157 −1.32218
\(342\) −1.79649 −0.0971429
\(343\) 0 0
\(344\) 2.18501 0.117808
\(345\) −12.2671 −0.660437
\(346\) −3.94899 −0.212299
\(347\) −13.6395 −0.732207 −0.366103 0.930574i \(-0.619308\pi\)
−0.366103 + 0.930574i \(0.619308\pi\)
\(348\) 6.28691 0.337014
\(349\) −15.2222 −0.814824 −0.407412 0.913244i \(-0.633569\pi\)
−0.407412 + 0.913244i \(0.633569\pi\)
\(350\) 0 0
\(351\) 34.3866 1.83542
\(352\) −13.8714 −0.739348
\(353\) −10.9932 −0.585106 −0.292553 0.956249i \(-0.594505\pi\)
−0.292553 + 0.956249i \(0.594505\pi\)
\(354\) −4.82115 −0.256241
\(355\) −7.18596 −0.381391
\(356\) 13.6668 0.724339
\(357\) 0 0
\(358\) −5.14476 −0.271909
\(359\) −2.33360 −0.123163 −0.0615814 0.998102i \(-0.519614\pi\)
−0.0615814 + 0.998102i \(0.519614\pi\)
\(360\) −11.8440 −0.624234
\(361\) −15.3744 −0.809177
\(362\) 11.9208 0.626545
\(363\) −2.55056 −0.133870
\(364\) 0 0
\(365\) −12.6087 −0.659969
\(366\) 3.89541 0.203616
\(367\) −27.5423 −1.43770 −0.718849 0.695166i \(-0.755331\pi\)
−0.718849 + 0.695166i \(0.755331\pi\)
\(368\) 10.3924 0.541739
\(369\) −2.03632 −0.106006
\(370\) 5.61652 0.291989
\(371\) 0 0
\(372\) −14.7628 −0.765414
\(373\) −0.841692 −0.0435811 −0.0217906 0.999763i \(-0.506937\pi\)
−0.0217906 + 0.999763i \(0.506937\pi\)
\(374\) −10.2994 −0.532571
\(375\) −3.25006 −0.167833
\(376\) −15.5412 −0.801478
\(377\) 24.9494 1.28496
\(378\) 0 0
\(379\) 12.8659 0.660875 0.330437 0.943828i \(-0.392804\pi\)
0.330437 + 0.943828i \(0.392804\pi\)
\(380\) 11.2739 0.578337
\(381\) −6.50540 −0.333282
\(382\) 5.29537 0.270935
\(383\) −23.4028 −1.19583 −0.597914 0.801560i \(-0.704004\pi\)
−0.597914 + 0.801560i \(0.704004\pi\)
\(384\) −10.8962 −0.556042
\(385\) 0 0
\(386\) 6.48910 0.330287
\(387\) 2.53692 0.128959
\(388\) 1.32507 0.0672702
\(389\) 4.88275 0.247565 0.123783 0.992309i \(-0.460497\pi\)
0.123783 + 0.992309i \(0.460497\pi\)
\(390\) 10.4912 0.531244
\(391\) 28.8968 1.46137
\(392\) 0 0
\(393\) 15.5081 0.782282
\(394\) 6.98836 0.352068
\(395\) 36.5032 1.83668
\(396\) −10.5378 −0.529545
\(397\) 25.0538 1.25741 0.628707 0.777642i \(-0.283585\pi\)
0.628707 + 0.777642i \(0.283585\pi\)
\(398\) 8.55621 0.428884
\(399\) 0 0
\(400\) 16.5437 0.827183
\(401\) −9.10250 −0.454557 −0.227279 0.973830i \(-0.572983\pi\)
−0.227279 + 0.973830i \(0.572983\pi\)
\(402\) −2.07430 −0.103457
\(403\) −58.5857 −2.91836
\(404\) 5.76607 0.286873
\(405\) −4.16381 −0.206901
\(406\) 0 0
\(407\) 10.5951 0.525179
\(408\) −13.2037 −0.653683
\(409\) 4.42310 0.218708 0.109354 0.994003i \(-0.465122\pi\)
0.109354 + 0.994003i \(0.465122\pi\)
\(410\) −1.53656 −0.0758853
\(411\) 4.51057 0.222490
\(412\) −17.5464 −0.864451
\(413\) 0 0
\(414\) −3.55503 −0.174720
\(415\) −41.6396 −2.04401
\(416\) −33.2845 −1.63191
\(417\) −2.67035 −0.130767
\(418\) −2.55721 −0.125077
\(419\) −1.58417 −0.0773919 −0.0386959 0.999251i \(-0.512320\pi\)
−0.0386959 + 0.999251i \(0.512320\pi\)
\(420\) 0 0
\(421\) 21.6911 1.05716 0.528579 0.848884i \(-0.322725\pi\)
0.528579 + 0.848884i \(0.322725\pi\)
\(422\) −6.42603 −0.312814
\(423\) −18.0443 −0.877344
\(424\) −7.24015 −0.351613
\(425\) 46.0010 2.23138
\(426\) 0.985544 0.0477498
\(427\) 0 0
\(428\) 32.8483 1.58778
\(429\) 19.7908 0.955509
\(430\) 1.91431 0.0923162
\(431\) −4.06913 −0.196003 −0.0980015 0.995186i \(-0.531245\pi\)
−0.0980015 + 0.995186i \(0.531245\pi\)
\(432\) −13.6359 −0.656057
\(433\) 24.6813 1.18611 0.593055 0.805162i \(-0.297922\pi\)
0.593055 + 0.805162i \(0.297922\pi\)
\(434\) 0 0
\(435\) 11.6784 0.559934
\(436\) −11.4921 −0.550373
\(437\) 7.17469 0.343212
\(438\) 1.72926 0.0826275
\(439\) −24.9536 −1.19097 −0.595484 0.803367i \(-0.703040\pi\)
−0.595484 + 0.803367i \(0.703040\pi\)
\(440\) −16.8593 −0.803737
\(441\) 0 0
\(442\) −24.7136 −1.17550
\(443\) −15.1895 −0.721676 −0.360838 0.932629i \(-0.617509\pi\)
−0.360838 + 0.932629i \(0.617509\pi\)
\(444\) 6.40624 0.304026
\(445\) 25.3870 1.20346
\(446\) 11.7021 0.554111
\(447\) 8.58631 0.406118
\(448\) 0 0
\(449\) −1.42027 −0.0670267 −0.0335134 0.999438i \(-0.510670\pi\)
−0.0335134 + 0.999438i \(0.510670\pi\)
\(450\) −5.65927 −0.266781
\(451\) −2.89859 −0.136489
\(452\) 23.8977 1.12405
\(453\) 19.5717 0.919559
\(454\) 3.13229 0.147006
\(455\) 0 0
\(456\) −3.27831 −0.153521
\(457\) 13.0274 0.609396 0.304698 0.952449i \(-0.401445\pi\)
0.304698 + 0.952449i \(0.401445\pi\)
\(458\) −6.31566 −0.295112
\(459\) −37.9157 −1.76975
\(460\) 22.3096 1.04019
\(461\) 32.4046 1.50923 0.754617 0.656165i \(-0.227823\pi\)
0.754617 + 0.656165i \(0.227823\pi\)
\(462\) 0 0
\(463\) 3.32911 0.154717 0.0773584 0.997003i \(-0.475351\pi\)
0.0773584 + 0.997003i \(0.475351\pi\)
\(464\) −9.89360 −0.459299
\(465\) −27.4228 −1.27170
\(466\) 0.136240 0.00631118
\(467\) 11.3449 0.524979 0.262490 0.964935i \(-0.415456\pi\)
0.262490 + 0.964935i \(0.415456\pi\)
\(468\) −25.2856 −1.16883
\(469\) 0 0
\(470\) −13.6158 −0.628052
\(471\) 18.1185 0.834858
\(472\) 18.5903 0.855690
\(473\) 3.61118 0.166042
\(474\) −5.00636 −0.229950
\(475\) 11.4214 0.524051
\(476\) 0 0
\(477\) −8.40625 −0.384896
\(478\) −4.12465 −0.188657
\(479\) −17.9451 −0.819931 −0.409965 0.912101i \(-0.634459\pi\)
−0.409965 + 0.912101i \(0.634459\pi\)
\(480\) −15.5799 −0.711120
\(481\) 25.4230 1.15919
\(482\) −4.40997 −0.200869
\(483\) 0 0
\(484\) 4.63859 0.210845
\(485\) 2.46141 0.111767
\(486\) 7.44315 0.337628
\(487\) 5.37906 0.243748 0.121874 0.992546i \(-0.461110\pi\)
0.121874 + 0.992546i \(0.461110\pi\)
\(488\) −15.0207 −0.679954
\(489\) −7.65850 −0.346329
\(490\) 0 0
\(491\) 17.1637 0.774586 0.387293 0.921957i \(-0.373410\pi\)
0.387293 + 0.921957i \(0.373410\pi\)
\(492\) −1.75261 −0.0790138
\(493\) −27.5100 −1.23899
\(494\) −6.13604 −0.276074
\(495\) −19.5747 −0.879817
\(496\) 23.2319 1.04314
\(497\) 0 0
\(498\) 5.71082 0.255908
\(499\) −24.0541 −1.07681 −0.538405 0.842686i \(-0.680973\pi\)
−0.538405 + 0.842686i \(0.680973\pi\)
\(500\) 5.91075 0.264337
\(501\) −5.42278 −0.242272
\(502\) −13.1655 −0.587605
\(503\) −34.1088 −1.52084 −0.760418 0.649434i \(-0.775006\pi\)
−0.760418 + 0.649434i \(0.775006\pi\)
\(504\) 0 0
\(505\) 10.7109 0.476627
\(506\) −5.06040 −0.224962
\(507\) 34.7264 1.54226
\(508\) 11.8311 0.524920
\(509\) −21.1292 −0.936534 −0.468267 0.883587i \(-0.655122\pi\)
−0.468267 + 0.883587i \(0.655122\pi\)
\(510\) −11.5679 −0.512237
\(511\) 0 0
\(512\) 22.8732 1.01086
\(513\) −9.41396 −0.415636
\(514\) −6.59878 −0.291060
\(515\) −32.5937 −1.43625
\(516\) 2.18347 0.0961220
\(517\) −25.6851 −1.12963
\(518\) 0 0
\(519\) −8.36694 −0.367268
\(520\) −40.4541 −1.77403
\(521\) 21.1323 0.925822 0.462911 0.886405i \(-0.346805\pi\)
0.462911 + 0.886405i \(0.346805\pi\)
\(522\) 3.38442 0.148132
\(523\) −20.8085 −0.909890 −0.454945 0.890519i \(-0.650341\pi\)
−0.454945 + 0.890519i \(0.650341\pi\)
\(524\) −28.2040 −1.23210
\(525\) 0 0
\(526\) 11.6388 0.507477
\(527\) 64.5983 2.81395
\(528\) −7.84797 −0.341539
\(529\) −8.80218 −0.382704
\(530\) −6.34318 −0.275530
\(531\) 21.5845 0.936687
\(532\) 0 0
\(533\) −6.95519 −0.301263
\(534\) −3.48179 −0.150672
\(535\) 61.0179 2.63804
\(536\) 7.99849 0.345482
\(537\) −10.9005 −0.470390
\(538\) 3.91516 0.168794
\(539\) 0 0
\(540\) −29.2726 −1.25969
\(541\) 6.38057 0.274322 0.137161 0.990549i \(-0.456202\pi\)
0.137161 + 0.990549i \(0.456202\pi\)
\(542\) −1.17093 −0.0502956
\(543\) 25.2573 1.08389
\(544\) 36.7005 1.57352
\(545\) −21.3474 −0.914422
\(546\) 0 0
\(547\) −34.3335 −1.46799 −0.733997 0.679152i \(-0.762347\pi\)
−0.733997 + 0.679152i \(0.762347\pi\)
\(548\) −8.20318 −0.350423
\(549\) −17.4399 −0.744317
\(550\) −8.05568 −0.343495
\(551\) −6.83036 −0.290983
\(552\) −6.48737 −0.276121
\(553\) 0 0
\(554\) 9.26418 0.393597
\(555\) 11.9000 0.505128
\(556\) 4.85645 0.205959
\(557\) 3.13283 0.132742 0.0663712 0.997795i \(-0.478858\pi\)
0.0663712 + 0.997795i \(0.478858\pi\)
\(558\) −7.94721 −0.336432
\(559\) 8.66506 0.366493
\(560\) 0 0
\(561\) −21.8219 −0.921323
\(562\) 3.13656 0.132308
\(563\) 4.81748 0.203032 0.101516 0.994834i \(-0.467631\pi\)
0.101516 + 0.994834i \(0.467631\pi\)
\(564\) −15.5303 −0.653944
\(565\) 44.3916 1.86757
\(566\) 2.93146 0.123219
\(567\) 0 0
\(568\) −3.80025 −0.159455
\(569\) −17.7546 −0.744310 −0.372155 0.928171i \(-0.621381\pi\)
−0.372155 + 0.928171i \(0.621381\pi\)
\(570\) −2.87216 −0.120302
\(571\) −27.2787 −1.14158 −0.570788 0.821097i \(-0.693362\pi\)
−0.570788 + 0.821097i \(0.693362\pi\)
\(572\) −35.9927 −1.50493
\(573\) 11.2196 0.468704
\(574\) 0 0
\(575\) 22.6016 0.942552
\(576\) 6.71747 0.279894
\(577\) −23.2141 −0.966415 −0.483207 0.875506i \(-0.660528\pi\)
−0.483207 + 0.875506i \(0.660528\pi\)
\(578\) 19.3734 0.805826
\(579\) 13.7488 0.571381
\(580\) −21.2389 −0.881898
\(581\) 0 0
\(582\) −0.337579 −0.0139931
\(583\) −11.9659 −0.495575
\(584\) −6.66803 −0.275925
\(585\) −46.9696 −1.94196
\(586\) −12.5233 −0.517334
\(587\) 11.8476 0.489001 0.244501 0.969649i \(-0.421376\pi\)
0.244501 + 0.969649i \(0.421376\pi\)
\(588\) 0 0
\(589\) 16.0389 0.660871
\(590\) 16.2872 0.670534
\(591\) 14.8066 0.609062
\(592\) −10.0814 −0.414343
\(593\) 42.6426 1.75112 0.875562 0.483106i \(-0.160492\pi\)
0.875562 + 0.483106i \(0.160492\pi\)
\(594\) 6.63979 0.272434
\(595\) 0 0
\(596\) −15.6155 −0.639638
\(597\) 18.1285 0.741950
\(598\) −12.1425 −0.496543
\(599\) 41.4555 1.69383 0.846913 0.531732i \(-0.178459\pi\)
0.846913 + 0.531732i \(0.178459\pi\)
\(600\) −10.3273 −0.421610
\(601\) −18.5531 −0.756799 −0.378399 0.925642i \(-0.623525\pi\)
−0.378399 + 0.925642i \(0.623525\pi\)
\(602\) 0 0
\(603\) 9.28672 0.378185
\(604\) −35.5942 −1.44831
\(605\) 8.61650 0.350311
\(606\) −1.46898 −0.0596732
\(607\) 17.5428 0.712039 0.356019 0.934479i \(-0.384134\pi\)
0.356019 + 0.934479i \(0.384134\pi\)
\(608\) 9.11224 0.369550
\(609\) 0 0
\(610\) −13.1598 −0.532824
\(611\) −61.6317 −2.49335
\(612\) 27.8806 1.12701
\(613\) 5.18107 0.209262 0.104631 0.994511i \(-0.466634\pi\)
0.104631 + 0.994511i \(0.466634\pi\)
\(614\) 2.53990 0.102502
\(615\) −3.25559 −0.131278
\(616\) 0 0
\(617\) 16.7278 0.673434 0.336717 0.941606i \(-0.390683\pi\)
0.336717 + 0.941606i \(0.390683\pi\)
\(618\) 4.47018 0.179817
\(619\) 18.0170 0.724164 0.362082 0.932146i \(-0.382066\pi\)
0.362082 + 0.932146i \(0.382066\pi\)
\(620\) 49.8727 2.00294
\(621\) −18.6291 −0.747559
\(622\) 3.34790 0.134239
\(623\) 0 0
\(624\) −18.8313 −0.753854
\(625\) −19.0119 −0.760475
\(626\) 8.60835 0.344059
\(627\) −5.41809 −0.216378
\(628\) −32.9514 −1.31490
\(629\) −28.0322 −1.11772
\(630\) 0 0
\(631\) −0.604726 −0.0240738 −0.0120369 0.999928i \(-0.503832\pi\)
−0.0120369 + 0.999928i \(0.503832\pi\)
\(632\) 19.3045 0.767892
\(633\) −13.6152 −0.541154
\(634\) 13.4593 0.534538
\(635\) 21.9770 0.872132
\(636\) −7.23506 −0.286889
\(637\) 0 0
\(638\) 4.81754 0.190728
\(639\) −4.41232 −0.174549
\(640\) 36.8103 1.45505
\(641\) −28.2472 −1.11570 −0.557848 0.829943i \(-0.688373\pi\)
−0.557848 + 0.829943i \(0.688373\pi\)
\(642\) −8.36853 −0.330279
\(643\) 22.1196 0.872312 0.436156 0.899871i \(-0.356340\pi\)
0.436156 + 0.899871i \(0.356340\pi\)
\(644\) 0 0
\(645\) 4.05595 0.159703
\(646\) 6.76579 0.266196
\(647\) −12.8921 −0.506843 −0.253421 0.967356i \(-0.581556\pi\)
−0.253421 + 0.967356i \(0.581556\pi\)
\(648\) −2.20201 −0.0865030
\(649\) 30.7244 1.20604
\(650\) −19.3297 −0.758172
\(651\) 0 0
\(652\) 13.9282 0.545470
\(653\) −9.61112 −0.376112 −0.188056 0.982158i \(-0.560219\pi\)
−0.188056 + 0.982158i \(0.560219\pi\)
\(654\) 2.92777 0.114485
\(655\) −52.3908 −2.04708
\(656\) 2.75806 0.107684
\(657\) −7.74199 −0.302044
\(658\) 0 0
\(659\) 39.0986 1.52307 0.761533 0.648126i \(-0.224447\pi\)
0.761533 + 0.648126i \(0.224447\pi\)
\(660\) −16.8475 −0.655788
\(661\) 14.0488 0.546436 0.273218 0.961952i \(-0.411912\pi\)
0.273218 + 0.961952i \(0.411912\pi\)
\(662\) 2.12610 0.0826331
\(663\) −52.3619 −2.03357
\(664\) −22.0209 −0.854576
\(665\) 0 0
\(666\) 3.44865 0.133633
\(667\) −13.5164 −0.523359
\(668\) 9.86219 0.381580
\(669\) 24.7939 0.958587
\(670\) 7.00757 0.270726
\(671\) −24.8248 −0.958351
\(672\) 0 0
\(673\) −31.9310 −1.23085 −0.615426 0.788195i \(-0.711016\pi\)
−0.615426 + 0.788195i \(0.711016\pi\)
\(674\) 14.9654 0.576444
\(675\) −29.6557 −1.14145
\(676\) −63.1555 −2.42906
\(677\) 29.5898 1.13723 0.568614 0.822604i \(-0.307480\pi\)
0.568614 + 0.822604i \(0.307480\pi\)
\(678\) −6.08824 −0.233818
\(679\) 0 0
\(680\) 44.6059 1.71056
\(681\) 6.63655 0.254313
\(682\) −11.3124 −0.433176
\(683\) 4.62301 0.176895 0.0884473 0.996081i \(-0.471810\pi\)
0.0884473 + 0.996081i \(0.471810\pi\)
\(684\) 6.92238 0.264684
\(685\) −15.2380 −0.582213
\(686\) 0 0
\(687\) −13.3813 −0.510530
\(688\) −3.43610 −0.131000
\(689\) −28.7122 −1.09385
\(690\) −5.68366 −0.216373
\(691\) 19.5317 0.743021 0.371511 0.928429i \(-0.378840\pi\)
0.371511 + 0.928429i \(0.378840\pi\)
\(692\) 15.2166 0.578448
\(693\) 0 0
\(694\) −6.31954 −0.239886
\(695\) 9.02118 0.342193
\(696\) 6.17603 0.234102
\(697\) 7.66901 0.290484
\(698\) −7.05284 −0.266954
\(699\) 0.288658 0.0109181
\(700\) 0 0
\(701\) −21.2862 −0.803968 −0.401984 0.915647i \(-0.631679\pi\)
−0.401984 + 0.915647i \(0.631679\pi\)
\(702\) 15.9322 0.601323
\(703\) −6.96000 −0.262502
\(704\) 9.56197 0.360380
\(705\) −28.8486 −1.08650
\(706\) −5.09342 −0.191693
\(707\) 0 0
\(708\) 18.5773 0.698177
\(709\) 19.7977 0.743520 0.371760 0.928329i \(-0.378754\pi\)
0.371760 + 0.928329i \(0.378754\pi\)
\(710\) −3.32944 −0.124952
\(711\) 22.4137 0.840579
\(712\) 13.4258 0.503152
\(713\) 31.7390 1.18863
\(714\) 0 0
\(715\) −66.8589 −2.50038
\(716\) 19.8242 0.740866
\(717\) −8.73911 −0.326368
\(718\) −1.08122 −0.0403508
\(719\) 9.67617 0.360860 0.180430 0.983588i \(-0.442251\pi\)
0.180430 + 0.983588i \(0.442251\pi\)
\(720\) 18.6256 0.694136
\(721\) 0 0
\(722\) −7.12335 −0.265104
\(723\) −9.34364 −0.347494
\(724\) −45.9343 −1.70714
\(725\) −21.5169 −0.799118
\(726\) −1.18174 −0.0438585
\(727\) 28.6421 1.06228 0.531138 0.847285i \(-0.321764\pi\)
0.531138 + 0.847285i \(0.321764\pi\)
\(728\) 0 0
\(729\) 12.0036 0.444577
\(730\) −5.84194 −0.216220
\(731\) −9.55435 −0.353380
\(732\) −15.0101 −0.554790
\(733\) −42.4156 −1.56666 −0.783328 0.621608i \(-0.786480\pi\)
−0.783328 + 0.621608i \(0.786480\pi\)
\(734\) −12.7611 −0.471021
\(735\) 0 0
\(736\) 18.0320 0.664669
\(737\) 13.2192 0.486934
\(738\) −0.943479 −0.0347299
\(739\) 47.5757 1.75010 0.875051 0.484031i \(-0.160828\pi\)
0.875051 + 0.484031i \(0.160828\pi\)
\(740\) −21.6421 −0.795578
\(741\) −13.0008 −0.477595
\(742\) 0 0
\(743\) −44.7704 −1.64247 −0.821234 0.570591i \(-0.806714\pi\)
−0.821234 + 0.570591i \(0.806714\pi\)
\(744\) −14.5024 −0.531684
\(745\) −29.0069 −1.06273
\(746\) −0.389978 −0.0142781
\(747\) −25.5676 −0.935468
\(748\) 39.6866 1.45109
\(749\) 0 0
\(750\) −1.50584 −0.0549855
\(751\) −17.6077 −0.642513 −0.321256 0.946992i \(-0.604105\pi\)
−0.321256 + 0.946992i \(0.604105\pi\)
\(752\) 24.4398 0.891228
\(753\) −27.8944 −1.01653
\(754\) 11.5597 0.420981
\(755\) −66.1187 −2.40631
\(756\) 0 0
\(757\) −24.6213 −0.894876 −0.447438 0.894315i \(-0.647663\pi\)
−0.447438 + 0.894315i \(0.647663\pi\)
\(758\) 5.96109 0.216517
\(759\) −10.7217 −0.389174
\(760\) 11.0750 0.401734
\(761\) 5.91079 0.214266 0.107133 0.994245i \(-0.465833\pi\)
0.107133 + 0.994245i \(0.465833\pi\)
\(762\) −3.01412 −0.109190
\(763\) 0 0
\(764\) −20.4046 −0.738211
\(765\) 51.7902 1.87248
\(766\) −10.8431 −0.391779
\(767\) 73.7235 2.66200
\(768\) 1.42828 0.0515386
\(769\) −10.0471 −0.362308 −0.181154 0.983455i \(-0.557983\pi\)
−0.181154 + 0.983455i \(0.557983\pi\)
\(770\) 0 0
\(771\) −13.9812 −0.503520
\(772\) −25.0044 −0.899927
\(773\) −23.6963 −0.852297 −0.426148 0.904653i \(-0.640130\pi\)
−0.426148 + 0.904653i \(0.640130\pi\)
\(774\) 1.17542 0.0422497
\(775\) 50.5255 1.81493
\(776\) 1.30170 0.0467283
\(777\) 0 0
\(778\) 2.26231 0.0811076
\(779\) 1.90411 0.0682218
\(780\) −40.4257 −1.44747
\(781\) −6.28071 −0.224741
\(782\) 13.3887 0.478777
\(783\) 17.7350 0.633798
\(784\) 0 0
\(785\) −61.2094 −2.18466
\(786\) 7.18533 0.256292
\(787\) 29.2904 1.04409 0.522045 0.852918i \(-0.325169\pi\)
0.522045 + 0.852918i \(0.325169\pi\)
\(788\) −26.9281 −0.959275
\(789\) 24.6598 0.877912
\(790\) 16.9129 0.601734
\(791\) 0 0
\(792\) −10.3520 −0.367841
\(793\) −59.5673 −2.11530
\(794\) 11.6081 0.411955
\(795\) −13.4396 −0.476654
\(796\) −32.9695 −1.16857
\(797\) 16.9100 0.598983 0.299491 0.954099i \(-0.403183\pi\)
0.299491 + 0.954099i \(0.403183\pi\)
\(798\) 0 0
\(799\) 67.9570 2.40415
\(800\) 28.7053 1.01488
\(801\) 15.5881 0.550779
\(802\) −4.21743 −0.148923
\(803\) −11.0203 −0.388899
\(804\) 7.99287 0.281887
\(805\) 0 0
\(806\) −27.1443 −0.956117
\(807\) 8.29525 0.292007
\(808\) 5.66438 0.199272
\(809\) 4.00253 0.140721 0.0703607 0.997522i \(-0.477585\pi\)
0.0703607 + 0.997522i \(0.477585\pi\)
\(810\) −1.92920 −0.0677852
\(811\) 18.9202 0.664380 0.332190 0.943213i \(-0.392213\pi\)
0.332190 + 0.943213i \(0.392213\pi\)
\(812\) 0 0
\(813\) −2.48091 −0.0870092
\(814\) 4.90898 0.172060
\(815\) 25.8725 0.906276
\(816\) 20.7639 0.726883
\(817\) −2.37222 −0.0829933
\(818\) 2.04934 0.0716535
\(819\) 0 0
\(820\) 5.92081 0.206764
\(821\) −1.63520 −0.0570688 −0.0285344 0.999593i \(-0.509084\pi\)
−0.0285344 + 0.999593i \(0.509084\pi\)
\(822\) 2.08987 0.0728925
\(823\) 33.2336 1.15845 0.579225 0.815168i \(-0.303355\pi\)
0.579225 + 0.815168i \(0.303355\pi\)
\(824\) −17.2370 −0.600479
\(825\) −17.0680 −0.594231
\(826\) 0 0
\(827\) −0.0333819 −0.00116080 −0.000580400 1.00000i \(-0.500185\pi\)
−0.000580400 1.00000i \(0.500185\pi\)
\(828\) 13.6985 0.476057
\(829\) 52.9231 1.83810 0.919048 0.394147i \(-0.128960\pi\)
0.919048 + 0.394147i \(0.128960\pi\)
\(830\) −19.2927 −0.669661
\(831\) 19.6285 0.680905
\(832\) 22.9440 0.795441
\(833\) 0 0
\(834\) −1.23724 −0.0428422
\(835\) 18.3197 0.633979
\(836\) 9.85365 0.340796
\(837\) −41.6450 −1.43946
\(838\) −0.733989 −0.0253552
\(839\) 10.1155 0.349227 0.174613 0.984637i \(-0.444132\pi\)
0.174613 + 0.984637i \(0.444132\pi\)
\(840\) 0 0
\(841\) −16.1322 −0.556284
\(842\) 10.0500 0.346347
\(843\) 6.64558 0.228886
\(844\) 24.7613 0.852320
\(845\) −117.316 −4.03578
\(846\) −8.36040 −0.287437
\(847\) 0 0
\(848\) 11.3857 0.390987
\(849\) 6.21104 0.213162
\(850\) 21.3135 0.731047
\(851\) −13.7730 −0.472132
\(852\) −3.79758 −0.130103
\(853\) 33.0103 1.13025 0.565126 0.825005i \(-0.308828\pi\)
0.565126 + 0.825005i \(0.308828\pi\)
\(854\) 0 0
\(855\) 12.8588 0.439761
\(856\) 32.2690 1.10293
\(857\) 24.4904 0.836576 0.418288 0.908315i \(-0.362630\pi\)
0.418288 + 0.908315i \(0.362630\pi\)
\(858\) 9.16960 0.313045
\(859\) 36.0524 1.23009 0.615045 0.788492i \(-0.289138\pi\)
0.615045 + 0.788492i \(0.289138\pi\)
\(860\) −7.37638 −0.251532
\(861\) 0 0
\(862\) −1.88533 −0.0642147
\(863\) 12.4612 0.424183 0.212091 0.977250i \(-0.431973\pi\)
0.212091 + 0.977250i \(0.431973\pi\)
\(864\) −23.6599 −0.804927
\(865\) 28.2659 0.961068
\(866\) 11.4355 0.388595
\(867\) 41.0474 1.39404
\(868\) 0 0
\(869\) 31.9047 1.08229
\(870\) 5.41089 0.183446
\(871\) 31.7195 1.07477
\(872\) −11.2895 −0.382309
\(873\) 1.51135 0.0511515
\(874\) 3.32422 0.112443
\(875\) 0 0
\(876\) −6.66335 −0.225134
\(877\) 36.5137 1.23298 0.616490 0.787363i \(-0.288554\pi\)
0.616490 + 0.787363i \(0.288554\pi\)
\(878\) −11.5616 −0.390187
\(879\) −26.5338 −0.894963
\(880\) 26.5126 0.893740
\(881\) −9.53691 −0.321307 −0.160653 0.987011i \(-0.551360\pi\)
−0.160653 + 0.987011i \(0.551360\pi\)
\(882\) 0 0
\(883\) 51.2983 1.72633 0.863163 0.504925i \(-0.168480\pi\)
0.863163 + 0.504925i \(0.168480\pi\)
\(884\) 95.2284 3.20288
\(885\) 34.5086 1.15999
\(886\) −7.03770 −0.236436
\(887\) 50.8458 1.70723 0.853617 0.520901i \(-0.174404\pi\)
0.853617 + 0.520901i \(0.174404\pi\)
\(888\) 6.29326 0.211188
\(889\) 0 0
\(890\) 11.7625 0.394279
\(891\) −3.63927 −0.121920
\(892\) −45.0916 −1.50978
\(893\) 16.8728 0.564626
\(894\) 3.97826 0.133053
\(895\) 36.8249 1.23092
\(896\) 0 0
\(897\) −25.7269 −0.858996
\(898\) −0.658049 −0.0219594
\(899\) −30.2158 −1.00775
\(900\) 21.8068 0.726893
\(901\) 31.6589 1.05471
\(902\) −1.34299 −0.0447168
\(903\) 0 0
\(904\) 23.4762 0.780808
\(905\) −85.3262 −2.83634
\(906\) 9.06809 0.301267
\(907\) 45.7791 1.52007 0.760034 0.649883i \(-0.225182\pi\)
0.760034 + 0.649883i \(0.225182\pi\)
\(908\) −12.0696 −0.400544
\(909\) 6.57668 0.218135
\(910\) 0 0
\(911\) −47.0642 −1.55931 −0.779653 0.626212i \(-0.784605\pi\)
−0.779653 + 0.626212i \(0.784605\pi\)
\(912\) 5.15540 0.170712
\(913\) −36.3941 −1.20447
\(914\) 6.03593 0.199651
\(915\) −27.8823 −0.921761
\(916\) 24.3361 0.804086
\(917\) 0 0
\(918\) −17.5674 −0.579809
\(919\) 39.0485 1.28809 0.644045 0.764987i \(-0.277255\pi\)
0.644045 + 0.764987i \(0.277255\pi\)
\(920\) 21.9161 0.722554
\(921\) 5.38141 0.177323
\(922\) 15.0139 0.494457
\(923\) −15.0706 −0.496055
\(924\) 0 0
\(925\) −21.9253 −0.720900
\(926\) 1.54246 0.0506885
\(927\) −20.0132 −0.657319
\(928\) −17.1666 −0.563522
\(929\) −23.8926 −0.783891 −0.391945 0.919988i \(-0.628198\pi\)
−0.391945 + 0.919988i \(0.628198\pi\)
\(930\) −12.7057 −0.416637
\(931\) 0 0
\(932\) −0.524971 −0.0171960
\(933\) 7.09338 0.232227
\(934\) 5.25639 0.171994
\(935\) 73.7206 2.41092
\(936\) −24.8396 −0.811909
\(937\) −36.1258 −1.18018 −0.590090 0.807338i \(-0.700908\pi\)
−0.590090 + 0.807338i \(0.700908\pi\)
\(938\) 0 0
\(939\) 18.2390 0.595206
\(940\) 52.4658 1.71124
\(941\) 22.9141 0.746980 0.373490 0.927634i \(-0.378161\pi\)
0.373490 + 0.927634i \(0.378161\pi\)
\(942\) 8.39479 0.273517
\(943\) 3.76800 0.122703
\(944\) −29.2348 −0.951511
\(945\) 0 0
\(946\) 1.67315 0.0543989
\(947\) 17.4074 0.565664 0.282832 0.959170i \(-0.408726\pi\)
0.282832 + 0.959170i \(0.408726\pi\)
\(948\) 19.2909 0.626541
\(949\) −26.4434 −0.858387
\(950\) 5.29185 0.171690
\(951\) 28.5170 0.924726
\(952\) 0 0
\(953\) −20.6396 −0.668583 −0.334292 0.942470i \(-0.608497\pi\)
−0.334292 + 0.942470i \(0.608497\pi\)
\(954\) −3.89484 −0.126100
\(955\) −37.9028 −1.22651
\(956\) 15.8935 0.514031
\(957\) 10.2072 0.329951
\(958\) −8.31442 −0.268627
\(959\) 0 0
\(960\) 10.7397 0.346621
\(961\) 39.9520 1.28877
\(962\) 11.7791 0.379775
\(963\) 37.4662 1.20733
\(964\) 16.9929 0.547304
\(965\) −46.4473 −1.49519
\(966\) 0 0
\(967\) −2.77753 −0.0893193 −0.0446597 0.999002i \(-0.514220\pi\)
−0.0446597 + 0.999002i \(0.514220\pi\)
\(968\) 4.55679 0.146461
\(969\) 14.3350 0.460507
\(970\) 1.14043 0.0366172
\(971\) −18.3051 −0.587438 −0.293719 0.955892i \(-0.594893\pi\)
−0.293719 + 0.955892i \(0.594893\pi\)
\(972\) −28.6806 −0.919930
\(973\) 0 0
\(974\) 2.49226 0.0798571
\(975\) −40.9548 −1.31160
\(976\) 23.6212 0.756096
\(977\) 3.18355 0.101851 0.0509253 0.998702i \(-0.483783\pi\)
0.0509253 + 0.998702i \(0.483783\pi\)
\(978\) −3.54838 −0.113465
\(979\) 22.1889 0.709160
\(980\) 0 0
\(981\) −13.1077 −0.418498
\(982\) 7.95239 0.253771
\(983\) −19.8284 −0.632426 −0.316213 0.948688i \(-0.602412\pi\)
−0.316213 + 0.948688i \(0.602412\pi\)
\(984\) −1.72170 −0.0548858
\(985\) −50.0208 −1.59380
\(986\) −12.7461 −0.405919
\(987\) 0 0
\(988\) 23.6439 0.752213
\(989\) −4.69432 −0.149271
\(990\) −9.06947 −0.288247
\(991\) −29.6097 −0.940583 −0.470292 0.882511i \(-0.655851\pi\)
−0.470292 + 0.882511i \(0.655851\pi\)
\(992\) 40.3103 1.27985
\(993\) 4.50467 0.142951
\(994\) 0 0
\(995\) −61.2431 −1.94154
\(996\) −22.0054 −0.697268
\(997\) 34.8793 1.10464 0.552319 0.833633i \(-0.313743\pi\)
0.552319 + 0.833633i \(0.313743\pi\)
\(998\) −11.1449 −0.352786
\(999\) 18.0716 0.571762
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.1.12 yes 20
7.6 odd 2 2009.2.a.t.1.12 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2009.2.a.t.1.12 20 7.6 odd 2
2009.2.a.u.1.12 yes 20 1.1 even 1 trivial