Properties

Label 8014.2.a.a.1.1
Level $8014$
Weight $2$
Character 8014.1
Self dual yes
Analytic conductor $63.992$
Analytic rank $0$
Dimension $2$
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] = [8014,2,Mod(1,8014)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8014, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8014.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8014 = 2 \cdot 4007 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8014.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: \(63.9921121799\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) 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 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 8014.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -3.23607 q^{3} +1.00000 q^{4} +1.61803 q^{5} -3.23607 q^{6} +2.85410 q^{7} +1.00000 q^{8} +7.47214 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -3.23607 q^{3} +1.00000 q^{4} +1.61803 q^{5} -3.23607 q^{6} +2.85410 q^{7} +1.00000 q^{8} +7.47214 q^{9} +1.61803 q^{10} +2.85410 q^{11} -3.23607 q^{12} +1.23607 q^{13} +2.85410 q^{14} -5.23607 q^{15} +1.00000 q^{16} -7.70820 q^{17} +7.47214 q^{18} -1.85410 q^{19} +1.61803 q^{20} -9.23607 q^{21} +2.85410 q^{22} +6.09017 q^{23} -3.23607 q^{24} -2.38197 q^{25} +1.23607 q^{26} -14.4721 q^{27} +2.85410 q^{28} +2.00000 q^{29} -5.23607 q^{30} +3.23607 q^{31} +1.00000 q^{32} -9.23607 q^{33} -7.70820 q^{34} +4.61803 q^{35} +7.47214 q^{36} +4.76393 q^{37} -1.85410 q^{38} -4.00000 q^{39} +1.61803 q^{40} +11.2361 q^{41} -9.23607 q^{42} -4.47214 q^{43} +2.85410 q^{44} +12.0902 q^{45} +6.09017 q^{46} +5.23607 q^{47} -3.23607 q^{48} +1.14590 q^{49} -2.38197 q^{50} +24.9443 q^{51} +1.23607 q^{52} +12.0902 q^{53} -14.4721 q^{54} +4.61803 q^{55} +2.85410 q^{56} +6.00000 q^{57} +2.00000 q^{58} +0.291796 q^{59} -5.23607 q^{60} +6.00000 q^{61} +3.23607 q^{62} +21.3262 q^{63} +1.00000 q^{64} +2.00000 q^{65} -9.23607 q^{66} -0.618034 q^{67} -7.70820 q^{68} -19.7082 q^{69} +4.61803 q^{70} -11.7984 q^{71} +7.47214 q^{72} -1.85410 q^{73} +4.76393 q^{74} +7.70820 q^{75} -1.85410 q^{76} +8.14590 q^{77} -4.00000 q^{78} +4.00000 q^{79} +1.61803 q^{80} +24.4164 q^{81} +11.2361 q^{82} -6.85410 q^{83} -9.23607 q^{84} -12.4721 q^{85} -4.47214 q^{86} -6.47214 q^{87} +2.85410 q^{88} -5.61803 q^{89} +12.0902 q^{90} +3.52786 q^{91} +6.09017 q^{92} -10.4721 q^{93} +5.23607 q^{94} -3.00000 q^{95} -3.23607 q^{96} -12.7639 q^{97} +1.14590 q^{98} +21.3262 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + q^{5} - 2 q^{6} - q^{7} + 2 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + q^{5} - 2 q^{6} - q^{7} + 2 q^{8} + 6 q^{9} + q^{10} - q^{11} - 2 q^{12} - 2 q^{13} - q^{14} - 6 q^{15} + 2 q^{16} - 2 q^{17} + 6 q^{18} + 3 q^{19} + q^{20} - 14 q^{21} - q^{22} + q^{23} - 2 q^{24} - 7 q^{25} - 2 q^{26} - 20 q^{27} - q^{28} + 4 q^{29} - 6 q^{30} + 2 q^{31} + 2 q^{32} - 14 q^{33} - 2 q^{34} + 7 q^{35} + 6 q^{36} + 14 q^{37} + 3 q^{38} - 8 q^{39} + q^{40} + 18 q^{41} - 14 q^{42} - q^{44} + 13 q^{45} + q^{46} + 6 q^{47} - 2 q^{48} + 9 q^{49} - 7 q^{50} + 32 q^{51} - 2 q^{52} + 13 q^{53} - 20 q^{54} + 7 q^{55} - q^{56} + 12 q^{57} + 4 q^{58} + 14 q^{59} - 6 q^{60} + 12 q^{61} + 2 q^{62} + 27 q^{63} + 2 q^{64} + 4 q^{65} - 14 q^{66} + q^{67} - 2 q^{68} - 26 q^{69} + 7 q^{70} + q^{71} + 6 q^{72} + 3 q^{73} + 14 q^{74} + 2 q^{75} + 3 q^{76} + 23 q^{77} - 8 q^{78} + 8 q^{79} + q^{80} + 22 q^{81} + 18 q^{82} - 7 q^{83} - 14 q^{84} - 16 q^{85} - 4 q^{87} - q^{88} - 9 q^{89} + 13 q^{90} + 16 q^{91} + q^{92} - 12 q^{93} + 6 q^{94} - 6 q^{95} - 2 q^{96} - 30 q^{97} + 9 q^{98} + 27 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −3.23607 −1.86834 −0.934172 0.356822i \(-0.883860\pi\)
−0.934172 + 0.356822i \(0.883860\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.61803 0.723607 0.361803 0.932254i \(-0.382161\pi\)
0.361803 + 0.932254i \(0.382161\pi\)
\(6\) −3.23607 −1.32112
\(7\) 2.85410 1.07875 0.539375 0.842066i \(-0.318661\pi\)
0.539375 + 0.842066i \(0.318661\pi\)
\(8\) 1.00000 0.353553
\(9\) 7.47214 2.49071
\(10\) 1.61803 0.511667
\(11\) 2.85410 0.860544 0.430272 0.902699i \(-0.358418\pi\)
0.430272 + 0.902699i \(0.358418\pi\)
\(12\) −3.23607 −0.934172
\(13\) 1.23607 0.342824 0.171412 0.985199i \(-0.445167\pi\)
0.171412 + 0.985199i \(0.445167\pi\)
\(14\) 2.85410 0.762791
\(15\) −5.23607 −1.35195
\(16\) 1.00000 0.250000
\(17\) −7.70820 −1.86951 −0.934757 0.355288i \(-0.884383\pi\)
−0.934757 + 0.355288i \(0.884383\pi\)
\(18\) 7.47214 1.76120
\(19\) −1.85410 −0.425360 −0.212680 0.977122i \(-0.568219\pi\)
−0.212680 + 0.977122i \(0.568219\pi\)
\(20\) 1.61803 0.361803
\(21\) −9.23607 −2.01548
\(22\) 2.85410 0.608497
\(23\) 6.09017 1.26989 0.634944 0.772558i \(-0.281023\pi\)
0.634944 + 0.772558i \(0.281023\pi\)
\(24\) −3.23607 −0.660560
\(25\) −2.38197 −0.476393
\(26\) 1.23607 0.242413
\(27\) −14.4721 −2.78516
\(28\) 2.85410 0.539375
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) −5.23607 −0.955971
\(31\) 3.23607 0.581215 0.290607 0.956842i \(-0.406143\pi\)
0.290607 + 0.956842i \(0.406143\pi\)
\(32\) 1.00000 0.176777
\(33\) −9.23607 −1.60779
\(34\) −7.70820 −1.32195
\(35\) 4.61803 0.780590
\(36\) 7.47214 1.24536
\(37\) 4.76393 0.783186 0.391593 0.920139i \(-0.371924\pi\)
0.391593 + 0.920139i \(0.371924\pi\)
\(38\) −1.85410 −0.300775
\(39\) −4.00000 −0.640513
\(40\) 1.61803 0.255834
\(41\) 11.2361 1.75478 0.877390 0.479779i \(-0.159283\pi\)
0.877390 + 0.479779i \(0.159283\pi\)
\(42\) −9.23607 −1.42516
\(43\) −4.47214 −0.681994 −0.340997 0.940064i \(-0.610765\pi\)
−0.340997 + 0.940064i \(0.610765\pi\)
\(44\) 2.85410 0.430272
\(45\) 12.0902 1.80230
\(46\) 6.09017 0.897947
\(47\) 5.23607 0.763759 0.381880 0.924212i \(-0.375277\pi\)
0.381880 + 0.924212i \(0.375277\pi\)
\(48\) −3.23607 −0.467086
\(49\) 1.14590 0.163700
\(50\) −2.38197 −0.336861
\(51\) 24.9443 3.49290
\(52\) 1.23607 0.171412
\(53\) 12.0902 1.66071 0.830356 0.557233i \(-0.188137\pi\)
0.830356 + 0.557233i \(0.188137\pi\)
\(54\) −14.4721 −1.96941
\(55\) 4.61803 0.622696
\(56\) 2.85410 0.381395
\(57\) 6.00000 0.794719
\(58\) 2.00000 0.262613
\(59\) 0.291796 0.0379886 0.0189943 0.999820i \(-0.493954\pi\)
0.0189943 + 0.999820i \(0.493954\pi\)
\(60\) −5.23607 −0.675973
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 3.23607 0.410981
\(63\) 21.3262 2.68685
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) −9.23607 −1.13688
\(67\) −0.618034 −0.0755049 −0.0377524 0.999287i \(-0.512020\pi\)
−0.0377524 + 0.999287i \(0.512020\pi\)
\(68\) −7.70820 −0.934757
\(69\) −19.7082 −2.37259
\(70\) 4.61803 0.551961
\(71\) −11.7984 −1.40021 −0.700105 0.714040i \(-0.746863\pi\)
−0.700105 + 0.714040i \(0.746863\pi\)
\(72\) 7.47214 0.880600
\(73\) −1.85410 −0.217006 −0.108503 0.994096i \(-0.534606\pi\)
−0.108503 + 0.994096i \(0.534606\pi\)
\(74\) 4.76393 0.553796
\(75\) 7.70820 0.890067
\(76\) −1.85410 −0.212680
\(77\) 8.14590 0.928311
\(78\) −4.00000 −0.452911
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 1.61803 0.180902
\(81\) 24.4164 2.71293
\(82\) 11.2361 1.24082
\(83\) −6.85410 −0.752335 −0.376168 0.926552i \(-0.622758\pi\)
−0.376168 + 0.926552i \(0.622758\pi\)
\(84\) −9.23607 −1.00774
\(85\) −12.4721 −1.35279
\(86\) −4.47214 −0.482243
\(87\) −6.47214 −0.693886
\(88\) 2.85410 0.304248
\(89\) −5.61803 −0.595510 −0.297755 0.954642i \(-0.596238\pi\)
−0.297755 + 0.954642i \(0.596238\pi\)
\(90\) 12.0902 1.27442
\(91\) 3.52786 0.369821
\(92\) 6.09017 0.634944
\(93\) −10.4721 −1.08591
\(94\) 5.23607 0.540059
\(95\) −3.00000 −0.307794
\(96\) −3.23607 −0.330280
\(97\) −12.7639 −1.29598 −0.647990 0.761648i \(-0.724390\pi\)
−0.647990 + 0.761648i \(0.724390\pi\)
\(98\) 1.14590 0.115753
\(99\) 21.3262 2.14337
\(100\) −2.38197 −0.238197
\(101\) 0.763932 0.0760141 0.0380070 0.999277i \(-0.487899\pi\)
0.0380070 + 0.999277i \(0.487899\pi\)
\(102\) 24.9443 2.46985
\(103\) 14.7639 1.45473 0.727367 0.686249i \(-0.240744\pi\)
0.727367 + 0.686249i \(0.240744\pi\)
\(104\) 1.23607 0.121206
\(105\) −14.9443 −1.45841
\(106\) 12.0902 1.17430
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) −14.4721 −1.39258
\(109\) 10.4721 1.00305 0.501524 0.865144i \(-0.332773\pi\)
0.501524 + 0.865144i \(0.332773\pi\)
\(110\) 4.61803 0.440312
\(111\) −15.4164 −1.46326
\(112\) 2.85410 0.269687
\(113\) −2.38197 −0.224077 −0.112038 0.993704i \(-0.535738\pi\)
−0.112038 + 0.993704i \(0.535738\pi\)
\(114\) 6.00000 0.561951
\(115\) 9.85410 0.918900
\(116\) 2.00000 0.185695
\(117\) 9.23607 0.853875
\(118\) 0.291796 0.0268620
\(119\) −22.0000 −2.01674
\(120\) −5.23607 −0.477985
\(121\) −2.85410 −0.259464
\(122\) 6.00000 0.543214
\(123\) −36.3607 −3.27853
\(124\) 3.23607 0.290607
\(125\) −11.9443 −1.06833
\(126\) 21.3262 1.89989
\(127\) 11.2361 0.997040 0.498520 0.866878i \(-0.333877\pi\)
0.498520 + 0.866878i \(0.333877\pi\)
\(128\) 1.00000 0.0883883
\(129\) 14.4721 1.27420
\(130\) 2.00000 0.175412
\(131\) 1.56231 0.136499 0.0682497 0.997668i \(-0.478259\pi\)
0.0682497 + 0.997668i \(0.478259\pi\)
\(132\) −9.23607 −0.803897
\(133\) −5.29180 −0.458857
\(134\) −0.618034 −0.0533900
\(135\) −23.4164 −2.01536
\(136\) −7.70820 −0.660973
\(137\) −11.5279 −0.984892 −0.492446 0.870343i \(-0.663897\pi\)
−0.492446 + 0.870343i \(0.663897\pi\)
\(138\) −19.7082 −1.67767
\(139\) −2.47214 −0.209684 −0.104842 0.994489i \(-0.533434\pi\)
−0.104842 + 0.994489i \(0.533434\pi\)
\(140\) 4.61803 0.390295
\(141\) −16.9443 −1.42697
\(142\) −11.7984 −0.990098
\(143\) 3.52786 0.295015
\(144\) 7.47214 0.622678
\(145\) 3.23607 0.268741
\(146\) −1.85410 −0.153447
\(147\) −3.70820 −0.305848
\(148\) 4.76393 0.391593
\(149\) −7.23607 −0.592802 −0.296401 0.955064i \(-0.595786\pi\)
−0.296401 + 0.955064i \(0.595786\pi\)
\(150\) 7.70820 0.629372
\(151\) 7.70820 0.627285 0.313642 0.949541i \(-0.398451\pi\)
0.313642 + 0.949541i \(0.398451\pi\)
\(152\) −1.85410 −0.150388
\(153\) −57.5967 −4.65642
\(154\) 8.14590 0.656415
\(155\) 5.23607 0.420571
\(156\) −4.00000 −0.320256
\(157\) 20.6525 1.64825 0.824124 0.566410i \(-0.191668\pi\)
0.824124 + 0.566410i \(0.191668\pi\)
\(158\) 4.00000 0.318223
\(159\) −39.1246 −3.10278
\(160\) 1.61803 0.127917
\(161\) 17.3820 1.36989
\(162\) 24.4164 1.91833
\(163\) 9.32624 0.730487 0.365244 0.930912i \(-0.380986\pi\)
0.365244 + 0.930912i \(0.380986\pi\)
\(164\) 11.2361 0.877390
\(165\) −14.9443 −1.16341
\(166\) −6.85410 −0.531981
\(167\) −8.32624 −0.644304 −0.322152 0.946688i \(-0.604406\pi\)
−0.322152 + 0.946688i \(0.604406\pi\)
\(168\) −9.23607 −0.712578
\(169\) −11.4721 −0.882472
\(170\) −12.4721 −0.956569
\(171\) −13.8541 −1.05945
\(172\) −4.47214 −0.340997
\(173\) −22.9443 −1.74442 −0.872210 0.489131i \(-0.837314\pi\)
−0.872210 + 0.489131i \(0.837314\pi\)
\(174\) −6.47214 −0.490651
\(175\) −6.79837 −0.513909
\(176\) 2.85410 0.215136
\(177\) −0.944272 −0.0709758
\(178\) −5.61803 −0.421089
\(179\) 10.0000 0.747435 0.373718 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 12.0902 0.901148
\(181\) 2.76393 0.205441 0.102721 0.994710i \(-0.467245\pi\)
0.102721 + 0.994710i \(0.467245\pi\)
\(182\) 3.52786 0.261503
\(183\) −19.4164 −1.43530
\(184\) 6.09017 0.448973
\(185\) 7.70820 0.566718
\(186\) −10.4721 −0.767854
\(187\) −22.0000 −1.60880
\(188\) 5.23607 0.381880
\(189\) −41.3050 −3.00449
\(190\) −3.00000 −0.217643
\(191\) 25.5623 1.84962 0.924812 0.380425i \(-0.124222\pi\)
0.924812 + 0.380425i \(0.124222\pi\)
\(192\) −3.23607 −0.233543
\(193\) −4.61803 −0.332413 −0.166207 0.986091i \(-0.553152\pi\)
−0.166207 + 0.986091i \(0.553152\pi\)
\(194\) −12.7639 −0.916397
\(195\) −6.47214 −0.463479
\(196\) 1.14590 0.0818499
\(197\) −10.0000 −0.712470 −0.356235 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) 21.3262 1.51559
\(199\) 12.6180 0.894469 0.447234 0.894417i \(-0.352409\pi\)
0.447234 + 0.894417i \(0.352409\pi\)
\(200\) −2.38197 −0.168430
\(201\) 2.00000 0.141069
\(202\) 0.763932 0.0537501
\(203\) 5.70820 0.400637
\(204\) 24.9443 1.74645
\(205\) 18.1803 1.26977
\(206\) 14.7639 1.02865
\(207\) 45.5066 3.16293
\(208\) 1.23607 0.0857059
\(209\) −5.29180 −0.366041
\(210\) −14.9443 −1.03125
\(211\) 4.32624 0.297831 0.148915 0.988850i \(-0.452422\pi\)
0.148915 + 0.988850i \(0.452422\pi\)
\(212\) 12.0902 0.830356
\(213\) 38.1803 2.61607
\(214\) 0 0
\(215\) −7.23607 −0.493496
\(216\) −14.4721 −0.984704
\(217\) 9.23607 0.626985
\(218\) 10.4721 0.709263
\(219\) 6.00000 0.405442
\(220\) 4.61803 0.311348
\(221\) −9.52786 −0.640913
\(222\) −15.4164 −1.03468
\(223\) −3.38197 −0.226473 −0.113237 0.993568i \(-0.536122\pi\)
−0.113237 + 0.993568i \(0.536122\pi\)
\(224\) 2.85410 0.190698
\(225\) −17.7984 −1.18656
\(226\) −2.38197 −0.158446
\(227\) −5.23607 −0.347530 −0.173765 0.984787i \(-0.555593\pi\)
−0.173765 + 0.984787i \(0.555593\pi\)
\(228\) 6.00000 0.397360
\(229\) −4.61803 −0.305168 −0.152584 0.988290i \(-0.548760\pi\)
−0.152584 + 0.988290i \(0.548760\pi\)
\(230\) 9.85410 0.649760
\(231\) −26.3607 −1.73441
\(232\) 2.00000 0.131306
\(233\) −8.09017 −0.530005 −0.265002 0.964248i \(-0.585373\pi\)
−0.265002 + 0.964248i \(0.585373\pi\)
\(234\) 9.23607 0.603781
\(235\) 8.47214 0.552661
\(236\) 0.291796 0.0189943
\(237\) −12.9443 −0.840821
\(238\) −22.0000 −1.42605
\(239\) 16.2918 1.05383 0.526914 0.849918i \(-0.323349\pi\)
0.526914 + 0.849918i \(0.323349\pi\)
\(240\) −5.23607 −0.337987
\(241\) 5.05573 0.325668 0.162834 0.986653i \(-0.447936\pi\)
0.162834 + 0.986653i \(0.447936\pi\)
\(242\) −2.85410 −0.183469
\(243\) −35.5967 −2.28353
\(244\) 6.00000 0.384111
\(245\) 1.85410 0.118454
\(246\) −36.3607 −2.31827
\(247\) −2.29180 −0.145823
\(248\) 3.23607 0.205491
\(249\) 22.1803 1.40562
\(250\) −11.9443 −0.755422
\(251\) 7.70820 0.486538 0.243269 0.969959i \(-0.421780\pi\)
0.243269 + 0.969959i \(0.421780\pi\)
\(252\) 21.3262 1.34343
\(253\) 17.3820 1.09279
\(254\) 11.2361 0.705014
\(255\) 40.3607 2.52748
\(256\) 1.00000 0.0625000
\(257\) 18.4721 1.15226 0.576130 0.817358i \(-0.304562\pi\)
0.576130 + 0.817358i \(0.304562\pi\)
\(258\) 14.4721 0.900996
\(259\) 13.5967 0.844861
\(260\) 2.00000 0.124035
\(261\) 14.9443 0.925027
\(262\) 1.56231 0.0965196
\(263\) 21.4164 1.32059 0.660296 0.751005i \(-0.270431\pi\)
0.660296 + 0.751005i \(0.270431\pi\)
\(264\) −9.23607 −0.568441
\(265\) 19.5623 1.20170
\(266\) −5.29180 −0.324461
\(267\) 18.1803 1.11262
\(268\) −0.618034 −0.0377524
\(269\) 7.09017 0.432295 0.216148 0.976361i \(-0.430651\pi\)
0.216148 + 0.976361i \(0.430651\pi\)
\(270\) −23.4164 −1.42508
\(271\) 4.76393 0.289388 0.144694 0.989476i \(-0.453780\pi\)
0.144694 + 0.989476i \(0.453780\pi\)
\(272\) −7.70820 −0.467379
\(273\) −11.4164 −0.690952
\(274\) −11.5279 −0.696424
\(275\) −6.79837 −0.409957
\(276\) −19.7082 −1.18629
\(277\) −29.4164 −1.76746 −0.883730 0.467996i \(-0.844976\pi\)
−0.883730 + 0.467996i \(0.844976\pi\)
\(278\) −2.47214 −0.148269
\(279\) 24.1803 1.44764
\(280\) 4.61803 0.275980
\(281\) −21.7082 −1.29500 −0.647501 0.762064i \(-0.724186\pi\)
−0.647501 + 0.762064i \(0.724186\pi\)
\(282\) −16.9443 −1.00902
\(283\) 7.05573 0.419419 0.209710 0.977764i \(-0.432748\pi\)
0.209710 + 0.977764i \(0.432748\pi\)
\(284\) −11.7984 −0.700105
\(285\) 9.70820 0.575064
\(286\) 3.52786 0.208607
\(287\) 32.0689 1.89297
\(288\) 7.47214 0.440300
\(289\) 42.4164 2.49508
\(290\) 3.23607 0.190028
\(291\) 41.3050 2.42134
\(292\) −1.85410 −0.108503
\(293\) −15.7082 −0.917683 −0.458842 0.888518i \(-0.651735\pi\)
−0.458842 + 0.888518i \(0.651735\pi\)
\(294\) −3.70820 −0.216267
\(295\) 0.472136 0.0274888
\(296\) 4.76393 0.276898
\(297\) −41.3050 −2.39676
\(298\) −7.23607 −0.419174
\(299\) 7.52786 0.435348
\(300\) 7.70820 0.445033
\(301\) −12.7639 −0.735701
\(302\) 7.70820 0.443557
\(303\) −2.47214 −0.142020
\(304\) −1.85410 −0.106340
\(305\) 9.70820 0.555890
\(306\) −57.5967 −3.29259
\(307\) −20.6180 −1.17673 −0.588367 0.808594i \(-0.700229\pi\)
−0.588367 + 0.808594i \(0.700229\pi\)
\(308\) 8.14590 0.464156
\(309\) −47.7771 −2.71794
\(310\) 5.23607 0.297389
\(311\) −10.5066 −0.595773 −0.297887 0.954601i \(-0.596282\pi\)
−0.297887 + 0.954601i \(0.596282\pi\)
\(312\) −4.00000 −0.226455
\(313\) 20.4721 1.15715 0.578577 0.815628i \(-0.303608\pi\)
0.578577 + 0.815628i \(0.303608\pi\)
\(314\) 20.6525 1.16549
\(315\) 34.5066 1.94423
\(316\) 4.00000 0.225018
\(317\) 30.5066 1.71342 0.856710 0.515798i \(-0.172505\pi\)
0.856710 + 0.515798i \(0.172505\pi\)
\(318\) −39.1246 −2.19400
\(319\) 5.70820 0.319598
\(320\) 1.61803 0.0904508
\(321\) 0 0
\(322\) 17.3820 0.968659
\(323\) 14.2918 0.795217
\(324\) 24.4164 1.35647
\(325\) −2.94427 −0.163319
\(326\) 9.32624 0.516533
\(327\) −33.8885 −1.87404
\(328\) 11.2361 0.620408
\(329\) 14.9443 0.823904
\(330\) −14.9443 −0.822655
\(331\) 26.8328 1.47486 0.737432 0.675421i \(-0.236038\pi\)
0.737432 + 0.675421i \(0.236038\pi\)
\(332\) −6.85410 −0.376168
\(333\) 35.5967 1.95069
\(334\) −8.32624 −0.455591
\(335\) −1.00000 −0.0546358
\(336\) −9.23607 −0.503869
\(337\) 34.5066 1.87969 0.939847 0.341597i \(-0.110968\pi\)
0.939847 + 0.341597i \(0.110968\pi\)
\(338\) −11.4721 −0.624002
\(339\) 7.70820 0.418652
\(340\) −12.4721 −0.676397
\(341\) 9.23607 0.500161
\(342\) −13.8541 −0.749144
\(343\) −16.7082 −0.902158
\(344\) −4.47214 −0.241121
\(345\) −31.8885 −1.71682
\(346\) −22.9443 −1.23349
\(347\) −14.9443 −0.802251 −0.401125 0.916023i \(-0.631381\pi\)
−0.401125 + 0.916023i \(0.631381\pi\)
\(348\) −6.47214 −0.346943
\(349\) −26.3607 −1.41105 −0.705527 0.708683i \(-0.749290\pi\)
−0.705527 + 0.708683i \(0.749290\pi\)
\(350\) −6.79837 −0.363388
\(351\) −17.8885 −0.954820
\(352\) 2.85410 0.152124
\(353\) −8.00000 −0.425797 −0.212899 0.977074i \(-0.568290\pi\)
−0.212899 + 0.977074i \(0.568290\pi\)
\(354\) −0.944272 −0.0501875
\(355\) −19.0902 −1.01320
\(356\) −5.61803 −0.297755
\(357\) 71.1935 3.76796
\(358\) 10.0000 0.528516
\(359\) 29.8541 1.57564 0.787820 0.615906i \(-0.211210\pi\)
0.787820 + 0.615906i \(0.211210\pi\)
\(360\) 12.0902 0.637208
\(361\) −15.5623 −0.819069
\(362\) 2.76393 0.145269
\(363\) 9.23607 0.484768
\(364\) 3.52786 0.184910
\(365\) −3.00000 −0.157027
\(366\) −19.4164 −1.01491
\(367\) −28.7426 −1.50035 −0.750177 0.661237i \(-0.770032\pi\)
−0.750177 + 0.661237i \(0.770032\pi\)
\(368\) 6.09017 0.317472
\(369\) 83.9574 4.37065
\(370\) 7.70820 0.400730
\(371\) 34.5066 1.79149
\(372\) −10.4721 −0.542955
\(373\) −8.03444 −0.416008 −0.208004 0.978128i \(-0.566697\pi\)
−0.208004 + 0.978128i \(0.566697\pi\)
\(374\) −22.0000 −1.13759
\(375\) 38.6525 1.99601
\(376\) 5.23607 0.270030
\(377\) 2.47214 0.127321
\(378\) −41.3050 −2.12450
\(379\) 7.61803 0.391312 0.195656 0.980673i \(-0.437316\pi\)
0.195656 + 0.980673i \(0.437316\pi\)
\(380\) −3.00000 −0.153897
\(381\) −36.3607 −1.86281
\(382\) 25.5623 1.30788
\(383\) 17.9098 0.915150 0.457575 0.889171i \(-0.348718\pi\)
0.457575 + 0.889171i \(0.348718\pi\)
\(384\) −3.23607 −0.165140
\(385\) 13.1803 0.671732
\(386\) −4.61803 −0.235052
\(387\) −33.4164 −1.69865
\(388\) −12.7639 −0.647990
\(389\) 32.3607 1.64075 0.820376 0.571825i \(-0.193764\pi\)
0.820376 + 0.571825i \(0.193764\pi\)
\(390\) −6.47214 −0.327729
\(391\) −46.9443 −2.37407
\(392\) 1.14590 0.0578766
\(393\) −5.05573 −0.255028
\(394\) −10.0000 −0.503793
\(395\) 6.47214 0.325649
\(396\) 21.3262 1.07168
\(397\) 11.5279 0.578567 0.289283 0.957243i \(-0.406583\pi\)
0.289283 + 0.957243i \(0.406583\pi\)
\(398\) 12.6180 0.632485
\(399\) 17.1246 0.857303
\(400\) −2.38197 −0.119098
\(401\) −12.2918 −0.613823 −0.306912 0.951738i \(-0.599296\pi\)
−0.306912 + 0.951738i \(0.599296\pi\)
\(402\) 2.00000 0.0997509
\(403\) 4.00000 0.199254
\(404\) 0.763932 0.0380070
\(405\) 39.5066 1.96310
\(406\) 5.70820 0.283293
\(407\) 13.5967 0.673966
\(408\) 24.9443 1.23493
\(409\) −27.3262 −1.35120 −0.675598 0.737270i \(-0.736114\pi\)
−0.675598 + 0.737270i \(0.736114\pi\)
\(410\) 18.1803 0.897863
\(411\) 37.3050 1.84012
\(412\) 14.7639 0.727367
\(413\) 0.832816 0.0409802
\(414\) 45.5066 2.23653
\(415\) −11.0902 −0.544395
\(416\) 1.23607 0.0606032
\(417\) 8.00000 0.391762
\(418\) −5.29180 −0.258830
\(419\) 26.8328 1.31087 0.655434 0.755252i \(-0.272486\pi\)
0.655434 + 0.755252i \(0.272486\pi\)
\(420\) −14.9443 −0.729206
\(421\) −11.0902 −0.540502 −0.270251 0.962790i \(-0.587107\pi\)
−0.270251 + 0.962790i \(0.587107\pi\)
\(422\) 4.32624 0.210598
\(423\) 39.1246 1.90230
\(424\) 12.0902 0.587151
\(425\) 18.3607 0.890624
\(426\) 38.1803 1.84984
\(427\) 17.1246 0.828718
\(428\) 0 0
\(429\) −11.4164 −0.551189
\(430\) −7.23607 −0.348954
\(431\) −24.3607 −1.17341 −0.586706 0.809800i \(-0.699576\pi\)
−0.586706 + 0.809800i \(0.699576\pi\)
\(432\) −14.4721 −0.696291
\(433\) −11.1246 −0.534615 −0.267307 0.963611i \(-0.586134\pi\)
−0.267307 + 0.963611i \(0.586134\pi\)
\(434\) 9.23607 0.443345
\(435\) −10.4721 −0.502100
\(436\) 10.4721 0.501524
\(437\) −11.2918 −0.540160
\(438\) 6.00000 0.286691
\(439\) −14.1803 −0.676791 −0.338395 0.941004i \(-0.609884\pi\)
−0.338395 + 0.941004i \(0.609884\pi\)
\(440\) 4.61803 0.220156
\(441\) 8.56231 0.407729
\(442\) −9.52786 −0.453194
\(443\) −0.111456 −0.00529544 −0.00264772 0.999996i \(-0.500843\pi\)
−0.00264772 + 0.999996i \(0.500843\pi\)
\(444\) −15.4164 −0.731630
\(445\) −9.09017 −0.430915
\(446\) −3.38197 −0.160141
\(447\) 23.4164 1.10756
\(448\) 2.85410 0.134844
\(449\) 4.94427 0.233335 0.116667 0.993171i \(-0.462779\pi\)
0.116667 + 0.993171i \(0.462779\pi\)
\(450\) −17.7984 −0.839023
\(451\) 32.0689 1.51006
\(452\) −2.38197 −0.112038
\(453\) −24.9443 −1.17198
\(454\) −5.23607 −0.245741
\(455\) 5.70820 0.267605
\(456\) 6.00000 0.280976
\(457\) −11.8197 −0.552900 −0.276450 0.961028i \(-0.589158\pi\)
−0.276450 + 0.961028i \(0.589158\pi\)
\(458\) −4.61803 −0.215787
\(459\) 111.554 5.20690
\(460\) 9.85410 0.459450
\(461\) 8.58359 0.399778 0.199889 0.979819i \(-0.435942\pi\)
0.199889 + 0.979819i \(0.435942\pi\)
\(462\) −26.3607 −1.22641
\(463\) −0.583592 −0.0271218 −0.0135609 0.999908i \(-0.504317\pi\)
−0.0135609 + 0.999908i \(0.504317\pi\)
\(464\) 2.00000 0.0928477
\(465\) −16.9443 −0.785772
\(466\) −8.09017 −0.374770
\(467\) 6.47214 0.299495 0.149747 0.988724i \(-0.452154\pi\)
0.149747 + 0.988724i \(0.452154\pi\)
\(468\) 9.23607 0.426937
\(469\) −1.76393 −0.0814508
\(470\) 8.47214 0.390790
\(471\) −66.8328 −3.07949
\(472\) 0.291796 0.0134310
\(473\) −12.7639 −0.586886
\(474\) −12.9443 −0.594550
\(475\) 4.41641 0.202639
\(476\) −22.0000 −1.00837
\(477\) 90.3394 4.13636
\(478\) 16.2918 0.745169
\(479\) −27.1246 −1.23936 −0.619678 0.784856i \(-0.712737\pi\)
−0.619678 + 0.784856i \(0.712737\pi\)
\(480\) −5.23607 −0.238993
\(481\) 5.88854 0.268494
\(482\) 5.05573 0.230282
\(483\) −56.2492 −2.55943
\(484\) −2.85410 −0.129732
\(485\) −20.6525 −0.937781
\(486\) −35.5967 −1.61470
\(487\) 3.41641 0.154812 0.0774061 0.997000i \(-0.475336\pi\)
0.0774061 + 0.997000i \(0.475336\pi\)
\(488\) 6.00000 0.271607
\(489\) −30.1803 −1.36480
\(490\) 1.85410 0.0837598
\(491\) 18.4377 0.832081 0.416041 0.909346i \(-0.363417\pi\)
0.416041 + 0.909346i \(0.363417\pi\)
\(492\) −36.3607 −1.63927
\(493\) −15.4164 −0.694320
\(494\) −2.29180 −0.103113
\(495\) 34.5066 1.55096
\(496\) 3.23607 0.145304
\(497\) −33.6738 −1.51047
\(498\) 22.1803 0.993925
\(499\) −14.4721 −0.647862 −0.323931 0.946081i \(-0.605005\pi\)
−0.323931 + 0.946081i \(0.605005\pi\)
\(500\) −11.9443 −0.534164
\(501\) 26.9443 1.20378
\(502\) 7.70820 0.344034
\(503\) 32.9443 1.46891 0.734456 0.678656i \(-0.237437\pi\)
0.734456 + 0.678656i \(0.237437\pi\)
\(504\) 21.3262 0.949946
\(505\) 1.23607 0.0550043
\(506\) 17.3820 0.772723
\(507\) 37.1246 1.64876
\(508\) 11.2361 0.498520
\(509\) 8.47214 0.375521 0.187760 0.982215i \(-0.439877\pi\)
0.187760 + 0.982215i \(0.439877\pi\)
\(510\) 40.3607 1.78720
\(511\) −5.29180 −0.234095
\(512\) 1.00000 0.0441942
\(513\) 26.8328 1.18470
\(514\) 18.4721 0.814771
\(515\) 23.8885 1.05266
\(516\) 14.4721 0.637100
\(517\) 14.9443 0.657248
\(518\) 13.5967 0.597407
\(519\) 74.2492 3.25918
\(520\) 2.00000 0.0877058
\(521\) −33.7984 −1.48073 −0.740367 0.672203i \(-0.765348\pi\)
−0.740367 + 0.672203i \(0.765348\pi\)
\(522\) 14.9443 0.654093
\(523\) −10.1803 −0.445155 −0.222578 0.974915i \(-0.571447\pi\)
−0.222578 + 0.974915i \(0.571447\pi\)
\(524\) 1.56231 0.0682497
\(525\) 22.0000 0.960159
\(526\) 21.4164 0.933800
\(527\) −24.9443 −1.08659
\(528\) −9.23607 −0.401948
\(529\) 14.0902 0.612616
\(530\) 19.5623 0.849732
\(531\) 2.18034 0.0946187
\(532\) −5.29180 −0.229428
\(533\) 13.8885 0.601580
\(534\) 18.1803 0.786740
\(535\) 0 0
\(536\) −0.618034 −0.0266950
\(537\) −32.3607 −1.39647
\(538\) 7.09017 0.305679
\(539\) 3.27051 0.140871
\(540\) −23.4164 −1.00768
\(541\) 20.9098 0.898984 0.449492 0.893284i \(-0.351605\pi\)
0.449492 + 0.893284i \(0.351605\pi\)
\(542\) 4.76393 0.204628
\(543\) −8.94427 −0.383835
\(544\) −7.70820 −0.330487
\(545\) 16.9443 0.725813
\(546\) −11.4164 −0.488577
\(547\) 15.6738 0.670162 0.335081 0.942189i \(-0.391236\pi\)
0.335081 + 0.942189i \(0.391236\pi\)
\(548\) −11.5279 −0.492446
\(549\) 44.8328 1.91342
\(550\) −6.79837 −0.289884
\(551\) −3.70820 −0.157975
\(552\) −19.7082 −0.838837
\(553\) 11.4164 0.485475
\(554\) −29.4164 −1.24978
\(555\) −24.9443 −1.05883
\(556\) −2.47214 −0.104842
\(557\) −28.3262 −1.20022 −0.600111 0.799917i \(-0.704877\pi\)
−0.600111 + 0.799917i \(0.704877\pi\)
\(558\) 24.1803 1.02364
\(559\) −5.52786 −0.233804
\(560\) 4.61803 0.195148
\(561\) 71.1935 3.00579
\(562\) −21.7082 −0.915705
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) −16.9443 −0.713483
\(565\) −3.85410 −0.162143
\(566\) 7.05573 0.296574
\(567\) 69.6869 2.92658
\(568\) −11.7984 −0.495049
\(569\) 16.2918 0.682988 0.341494 0.939884i \(-0.389067\pi\)
0.341494 + 0.939884i \(0.389067\pi\)
\(570\) 9.70820 0.406632
\(571\) 40.4721 1.69371 0.846853 0.531827i \(-0.178494\pi\)
0.846853 + 0.531827i \(0.178494\pi\)
\(572\) 3.52786 0.147507
\(573\) −82.7214 −3.45573
\(574\) 32.0689 1.33853
\(575\) −14.5066 −0.604966
\(576\) 7.47214 0.311339
\(577\) −13.0344 −0.542631 −0.271315 0.962490i \(-0.587459\pi\)
−0.271315 + 0.962490i \(0.587459\pi\)
\(578\) 42.4164 1.76429
\(579\) 14.9443 0.621063
\(580\) 3.23607 0.134370
\(581\) −19.5623 −0.811581
\(582\) 41.3050 1.71215
\(583\) 34.5066 1.42912
\(584\) −1.85410 −0.0767233
\(585\) 14.9443 0.617870
\(586\) −15.7082 −0.648900
\(587\) −8.74265 −0.360848 −0.180424 0.983589i \(-0.557747\pi\)
−0.180424 + 0.983589i \(0.557747\pi\)
\(588\) −3.70820 −0.152924
\(589\) −6.00000 −0.247226
\(590\) 0.472136 0.0194375
\(591\) 32.3607 1.33114
\(592\) 4.76393 0.195796
\(593\) −3.90983 −0.160557 −0.0802787 0.996772i \(-0.525581\pi\)
−0.0802787 + 0.996772i \(0.525581\pi\)
\(594\) −41.3050 −1.69476
\(595\) −35.5967 −1.45932
\(596\) −7.23607 −0.296401
\(597\) −40.8328 −1.67118
\(598\) 7.52786 0.307837
\(599\) −6.00000 −0.245153 −0.122577 0.992459i \(-0.539116\pi\)
−0.122577 + 0.992459i \(0.539116\pi\)
\(600\) 7.70820 0.314686
\(601\) −34.5623 −1.40983 −0.704913 0.709294i \(-0.749014\pi\)
−0.704913 + 0.709294i \(0.749014\pi\)
\(602\) −12.7639 −0.520219
\(603\) −4.61803 −0.188061
\(604\) 7.70820 0.313642
\(605\) −4.61803 −0.187750
\(606\) −2.47214 −0.100424
\(607\) −27.7984 −1.12830 −0.564151 0.825672i \(-0.690796\pi\)
−0.564151 + 0.825672i \(0.690796\pi\)
\(608\) −1.85410 −0.0751938
\(609\) −18.4721 −0.748529
\(610\) 9.70820 0.393074
\(611\) 6.47214 0.261835
\(612\) −57.5967 −2.32821
\(613\) 11.7082 0.472890 0.236445 0.971645i \(-0.424018\pi\)
0.236445 + 0.971645i \(0.424018\pi\)
\(614\) −20.6180 −0.832076
\(615\) −58.8328 −2.37237
\(616\) 8.14590 0.328208
\(617\) 0.909830 0.0366284 0.0183142 0.999832i \(-0.494170\pi\)
0.0183142 + 0.999832i \(0.494170\pi\)
\(618\) −47.7771 −1.92188
\(619\) 29.3262 1.17872 0.589361 0.807870i \(-0.299380\pi\)
0.589361 + 0.807870i \(0.299380\pi\)
\(620\) 5.23607 0.210286
\(621\) −88.1378 −3.53685
\(622\) −10.5066 −0.421275
\(623\) −16.0344 −0.642406
\(624\) −4.00000 −0.160128
\(625\) −7.41641 −0.296656
\(626\) 20.4721 0.818231
\(627\) 17.1246 0.683891
\(628\) 20.6525 0.824124
\(629\) −36.7214 −1.46418
\(630\) 34.5066 1.37477
\(631\) 3.09017 0.123018 0.0615089 0.998107i \(-0.480409\pi\)
0.0615089 + 0.998107i \(0.480409\pi\)
\(632\) 4.00000 0.159111
\(633\) −14.0000 −0.556450
\(634\) 30.5066 1.21157
\(635\) 18.1803 0.721465
\(636\) −39.1246 −1.55139
\(637\) 1.41641 0.0561201
\(638\) 5.70820 0.225990
\(639\) −88.1591 −3.48752
\(640\) 1.61803 0.0639584
\(641\) 18.5836 0.734008 0.367004 0.930219i \(-0.380384\pi\)
0.367004 + 0.930219i \(0.380384\pi\)
\(642\) 0 0
\(643\) 48.3262 1.90580 0.952900 0.303283i \(-0.0980829\pi\)
0.952900 + 0.303283i \(0.0980829\pi\)
\(644\) 17.3820 0.684945
\(645\) 23.4164 0.922020
\(646\) 14.2918 0.562303
\(647\) −1.85410 −0.0728923 −0.0364461 0.999336i \(-0.511604\pi\)
−0.0364461 + 0.999336i \(0.511604\pi\)
\(648\) 24.4164 0.959167
\(649\) 0.832816 0.0326909
\(650\) −2.94427 −0.115484
\(651\) −29.8885 −1.17142
\(652\) 9.32624 0.365244
\(653\) 26.5066 1.03728 0.518641 0.854992i \(-0.326438\pi\)
0.518641 + 0.854992i \(0.326438\pi\)
\(654\) −33.8885 −1.32515
\(655\) 2.52786 0.0987718
\(656\) 11.2361 0.438695
\(657\) −13.8541 −0.540500
\(658\) 14.9443 0.582588
\(659\) 25.3050 0.985741 0.492870 0.870103i \(-0.335948\pi\)
0.492870 + 0.870103i \(0.335948\pi\)
\(660\) −14.9443 −0.581705
\(661\) 8.76393 0.340877 0.170439 0.985368i \(-0.445482\pi\)
0.170439 + 0.985368i \(0.445482\pi\)
\(662\) 26.8328 1.04289
\(663\) 30.8328 1.19745
\(664\) −6.85410 −0.265991
\(665\) −8.56231 −0.332032
\(666\) 35.5967 1.37935
\(667\) 12.1803 0.471625
\(668\) −8.32624 −0.322152
\(669\) 10.9443 0.423130
\(670\) −1.00000 −0.0386334
\(671\) 17.1246 0.661088
\(672\) −9.23607 −0.356289
\(673\) −50.2492 −1.93697 −0.968483 0.249081i \(-0.919871\pi\)
−0.968483 + 0.249081i \(0.919871\pi\)
\(674\) 34.5066 1.32914
\(675\) 34.4721 1.32683
\(676\) −11.4721 −0.441236
\(677\) 7.88854 0.303181 0.151591 0.988443i \(-0.451560\pi\)
0.151591 + 0.988443i \(0.451560\pi\)
\(678\) 7.70820 0.296032
\(679\) −36.4296 −1.39804
\(680\) −12.4721 −0.478285
\(681\) 16.9443 0.649306
\(682\) 9.23607 0.353667
\(683\) −12.5836 −0.481498 −0.240749 0.970587i \(-0.577393\pi\)
−0.240749 + 0.970587i \(0.577393\pi\)
\(684\) −13.8541 −0.529725
\(685\) −18.6525 −0.712674
\(686\) −16.7082 −0.637922
\(687\) 14.9443 0.570160
\(688\) −4.47214 −0.170499
\(689\) 14.9443 0.569331
\(690\) −31.8885 −1.21398
\(691\) 30.0689 1.14387 0.571937 0.820297i \(-0.306192\pi\)
0.571937 + 0.820297i \(0.306192\pi\)
\(692\) −22.9443 −0.872210
\(693\) 60.8673 2.31216
\(694\) −14.9443 −0.567277
\(695\) −4.00000 −0.151729
\(696\) −6.47214 −0.245326
\(697\) −86.6099 −3.28058
\(698\) −26.3607 −0.997766
\(699\) 26.1803 0.990231
\(700\) −6.79837 −0.256954
\(701\) 40.1803 1.51759 0.758795 0.651329i \(-0.225788\pi\)
0.758795 + 0.651329i \(0.225788\pi\)
\(702\) −17.8885 −0.675160
\(703\) −8.83282 −0.333136
\(704\) 2.85410 0.107568
\(705\) −27.4164 −1.03256
\(706\) −8.00000 −0.301084
\(707\) 2.18034 0.0820001
\(708\) −0.944272 −0.0354879
\(709\) 37.7771 1.41875 0.709374 0.704832i \(-0.248978\pi\)
0.709374 + 0.704832i \(0.248978\pi\)
\(710\) −19.0902 −0.716441
\(711\) 29.8885 1.12091
\(712\) −5.61803 −0.210545
\(713\) 19.7082 0.738078
\(714\) 71.1935 2.66435
\(715\) 5.70820 0.213475
\(716\) 10.0000 0.373718
\(717\) −52.7214 −1.96892
\(718\) 29.8541 1.11415
\(719\) 38.2918 1.42804 0.714022 0.700124i \(-0.246872\pi\)
0.714022 + 0.700124i \(0.246872\pi\)
\(720\) 12.0902 0.450574
\(721\) 42.1378 1.56929
\(722\) −15.5623 −0.579169
\(723\) −16.3607 −0.608460
\(724\) 2.76393 0.102721
\(725\) −4.76393 −0.176928
\(726\) 9.23607 0.342783
\(727\) −16.3607 −0.606784 −0.303392 0.952866i \(-0.598119\pi\)
−0.303392 + 0.952866i \(0.598119\pi\)
\(728\) 3.52786 0.130751
\(729\) 41.9443 1.55349
\(730\) −3.00000 −0.111035
\(731\) 34.4721 1.27500
\(732\) −19.4164 −0.717651
\(733\) −17.6180 −0.650737 −0.325368 0.945587i \(-0.605488\pi\)
−0.325368 + 0.945587i \(0.605488\pi\)
\(734\) −28.7426 −1.06091
\(735\) −6.00000 −0.221313
\(736\) 6.09017 0.224487
\(737\) −1.76393 −0.0649753
\(738\) 83.9574 3.09052
\(739\) 20.4721 0.753080 0.376540 0.926400i \(-0.377114\pi\)
0.376540 + 0.926400i \(0.377114\pi\)
\(740\) 7.70820 0.283359
\(741\) 7.41641 0.272449
\(742\) 34.5066 1.26678
\(743\) 17.2016 0.631066 0.315533 0.948915i \(-0.397817\pi\)
0.315533 + 0.948915i \(0.397817\pi\)
\(744\) −10.4721 −0.383927
\(745\) −11.7082 −0.428955
\(746\) −8.03444 −0.294162
\(747\) −51.2148 −1.87385
\(748\) −22.0000 −0.804400
\(749\) 0 0
\(750\) 38.6525 1.41139
\(751\) −42.6312 −1.55563 −0.777817 0.628491i \(-0.783673\pi\)
−0.777817 + 0.628491i \(0.783673\pi\)
\(752\) 5.23607 0.190940
\(753\) −24.9443 −0.909020
\(754\) 2.47214 0.0900299
\(755\) 12.4721 0.453908
\(756\) −41.3050 −1.50225
\(757\) 1.81966 0.0661367 0.0330683 0.999453i \(-0.489472\pi\)
0.0330683 + 0.999453i \(0.489472\pi\)
\(758\) 7.61803 0.276699
\(759\) −56.2492 −2.04172
\(760\) −3.00000 −0.108821
\(761\) −45.2705 −1.64105 −0.820527 0.571607i \(-0.806320\pi\)
−0.820527 + 0.571607i \(0.806320\pi\)
\(762\) −36.3607 −1.31721
\(763\) 29.8885 1.08204
\(764\) 25.5623 0.924812
\(765\) −93.1935 −3.36942
\(766\) 17.9098 0.647108
\(767\) 0.360680 0.0130234
\(768\) −3.23607 −0.116772
\(769\) 25.5623 0.921800 0.460900 0.887452i \(-0.347527\pi\)
0.460900 + 0.887452i \(0.347527\pi\)
\(770\) 13.1803 0.474986
\(771\) −59.7771 −2.15282
\(772\) −4.61803 −0.166207
\(773\) −40.8328 −1.46865 −0.734327 0.678796i \(-0.762502\pi\)
−0.734327 + 0.678796i \(0.762502\pi\)
\(774\) −33.4164 −1.20113
\(775\) −7.70820 −0.276887
\(776\) −12.7639 −0.458198
\(777\) −44.0000 −1.57849
\(778\) 32.3607 1.16019
\(779\) −20.8328 −0.746413
\(780\) −6.47214 −0.231740
\(781\) −33.6738 −1.20494
\(782\) −46.9443 −1.67872
\(783\) −28.9443 −1.03438
\(784\) 1.14590 0.0409249
\(785\) 33.4164 1.19268
\(786\) −5.05573 −0.180332
\(787\) −8.18034 −0.291598 −0.145799 0.989314i \(-0.546575\pi\)
−0.145799 + 0.989314i \(0.546575\pi\)
\(788\) −10.0000 −0.356235
\(789\) −69.3050 −2.46732
\(790\) 6.47214 0.230268
\(791\) −6.79837 −0.241722
\(792\) 21.3262 0.757795
\(793\) 7.41641 0.263364
\(794\) 11.5279 0.409109
\(795\) −63.3050 −2.24520
\(796\) 12.6180 0.447234
\(797\) −13.2016 −0.467626 −0.233813 0.972282i \(-0.575120\pi\)
−0.233813 + 0.972282i \(0.575120\pi\)
\(798\) 17.1246 0.606205
\(799\) −40.3607 −1.42786
\(800\) −2.38197 −0.0842152
\(801\) −41.9787 −1.48324
\(802\) −12.2918 −0.434038
\(803\) −5.29180 −0.186743
\(804\) 2.00000 0.0705346
\(805\) 28.1246 0.991262
\(806\) 4.00000 0.140894
\(807\) −22.9443 −0.807677
\(808\) 0.763932 0.0268750
\(809\) −34.3607 −1.20806 −0.604029 0.796963i \(-0.706439\pi\)
−0.604029 + 0.796963i \(0.706439\pi\)
\(810\) 39.5066 1.38812
\(811\) 38.8541 1.36435 0.682176 0.731188i \(-0.261034\pi\)
0.682176 + 0.731188i \(0.261034\pi\)
\(812\) 5.70820 0.200319
\(813\) −15.4164 −0.540677
\(814\) 13.5967 0.476566
\(815\) 15.0902 0.528586
\(816\) 24.9443 0.873224
\(817\) 8.29180 0.290093
\(818\) −27.3262 −0.955440
\(819\) 26.3607 0.921117
\(820\) 18.1803 0.634885
\(821\) 31.9098 1.11366 0.556830 0.830626i \(-0.312017\pi\)
0.556830 + 0.830626i \(0.312017\pi\)
\(822\) 37.3050 1.30116
\(823\) −7.81966 −0.272576 −0.136288 0.990669i \(-0.543517\pi\)
−0.136288 + 0.990669i \(0.543517\pi\)
\(824\) 14.7639 0.514326
\(825\) 22.0000 0.765942
\(826\) 0.832816 0.0289774
\(827\) 40.7426 1.41676 0.708380 0.705831i \(-0.249426\pi\)
0.708380 + 0.705831i \(0.249426\pi\)
\(828\) 45.5066 1.58146
\(829\) 40.6312 1.41118 0.705590 0.708621i \(-0.250682\pi\)
0.705590 + 0.708621i \(0.250682\pi\)
\(830\) −11.0902 −0.384945
\(831\) 95.1935 3.30223
\(832\) 1.23607 0.0428529
\(833\) −8.83282 −0.306039
\(834\) 8.00000 0.277017
\(835\) −13.4721 −0.466222
\(836\) −5.29180 −0.183021
\(837\) −46.8328 −1.61878
\(838\) 26.8328 0.926924
\(839\) 14.4721 0.499634 0.249817 0.968293i \(-0.419630\pi\)
0.249817 + 0.968293i \(0.419630\pi\)
\(840\) −14.9443 −0.515626
\(841\) −25.0000 −0.862069
\(842\) −11.0902 −0.382192
\(843\) 70.2492 2.41951
\(844\) 4.32624 0.148915
\(845\) −18.5623 −0.638563
\(846\) 39.1246 1.34513
\(847\) −8.14590 −0.279896
\(848\) 12.0902 0.415178
\(849\) −22.8328 −0.783620
\(850\) 18.3607 0.629766
\(851\) 29.0132 0.994558
\(852\) 38.1803 1.30804
\(853\) −38.9230 −1.33270 −0.666349 0.745640i \(-0.732144\pi\)
−0.666349 + 0.745640i \(0.732144\pi\)
\(854\) 17.1246 0.585992
\(855\) −22.4164 −0.766625
\(856\) 0 0
\(857\) −9.12461 −0.311691 −0.155845 0.987781i \(-0.549810\pi\)
−0.155845 + 0.987781i \(0.549810\pi\)
\(858\) −11.4164 −0.389750
\(859\) 2.67376 0.0912276 0.0456138 0.998959i \(-0.485476\pi\)
0.0456138 + 0.998959i \(0.485476\pi\)
\(860\) −7.23607 −0.246748
\(861\) −103.777 −3.53671
\(862\) −24.3607 −0.829728
\(863\) 8.74265 0.297603 0.148802 0.988867i \(-0.452458\pi\)
0.148802 + 0.988867i \(0.452458\pi\)
\(864\) −14.4721 −0.492352
\(865\) −37.1246 −1.26227
\(866\) −11.1246 −0.378030
\(867\) −137.262 −4.66167
\(868\) 9.23607 0.313493
\(869\) 11.4164 0.387275
\(870\) −10.4721 −0.355039
\(871\) −0.763932 −0.0258848
\(872\) 10.4721 0.354631
\(873\) −95.3738 −3.22792
\(874\) −11.2918 −0.381951
\(875\) −34.0902 −1.15246
\(876\) 6.00000 0.202721
\(877\) −46.8673 −1.58259 −0.791297 0.611431i \(-0.790594\pi\)
−0.791297 + 0.611431i \(0.790594\pi\)
\(878\) −14.1803 −0.478563
\(879\) 50.8328 1.71455
\(880\) 4.61803 0.155674
\(881\) 10.6525 0.358891 0.179446 0.983768i \(-0.442570\pi\)
0.179446 + 0.983768i \(0.442570\pi\)
\(882\) 8.56231 0.288308
\(883\) 31.4853 1.05956 0.529782 0.848134i \(-0.322274\pi\)
0.529782 + 0.848134i \(0.322274\pi\)
\(884\) −9.52786 −0.320457
\(885\) −1.52786 −0.0513586
\(886\) −0.111456 −0.00374444
\(887\) −40.3262 −1.35402 −0.677011 0.735973i \(-0.736725\pi\)
−0.677011 + 0.735973i \(0.736725\pi\)
\(888\) −15.4164 −0.517341
\(889\) 32.0689 1.07556
\(890\) −9.09017 −0.304703
\(891\) 69.6869 2.33460
\(892\) −3.38197 −0.113237
\(893\) −9.70820 −0.324873
\(894\) 23.4164 0.783162
\(895\) 16.1803 0.540849
\(896\) 2.85410 0.0953489
\(897\) −24.3607 −0.813379
\(898\) 4.94427 0.164992
\(899\) 6.47214 0.215858
\(900\) −17.7984 −0.593279
\(901\) −93.1935 −3.10473
\(902\) 32.0689 1.06778
\(903\) 41.3050 1.37454
\(904\) −2.38197 −0.0792230
\(905\) 4.47214 0.148659
\(906\) −24.9443 −0.828718
\(907\) −54.1803 −1.79903 −0.899514 0.436891i \(-0.856079\pi\)
−0.899514 + 0.436891i \(0.856079\pi\)
\(908\) −5.23607 −0.173765
\(909\) 5.70820 0.189329
\(910\) 5.70820 0.189225
\(911\) −54.3607 −1.80105 −0.900525 0.434805i \(-0.856817\pi\)
−0.900525 + 0.434805i \(0.856817\pi\)
\(912\) 6.00000 0.198680
\(913\) −19.5623 −0.647418
\(914\) −11.8197 −0.390960
\(915\) −31.4164 −1.03859
\(916\) −4.61803 −0.152584
\(917\) 4.45898 0.147249
\(918\) 111.554 3.68184
\(919\) 19.7082 0.650114 0.325057 0.945694i \(-0.394617\pi\)
0.325057 + 0.945694i \(0.394617\pi\)
\(920\) 9.85410 0.324880
\(921\) 66.7214 2.19854
\(922\) 8.58359 0.282686
\(923\) −14.5836 −0.480025
\(924\) −26.3607 −0.867203
\(925\) −11.3475 −0.373104
\(926\) −0.583592 −0.0191780
\(927\) 110.318 3.62332
\(928\) 2.00000 0.0656532
\(929\) −24.4508 −0.802206 −0.401103 0.916033i \(-0.631373\pi\)
−0.401103 + 0.916033i \(0.631373\pi\)
\(930\) −16.9443 −0.555625
\(931\) −2.12461 −0.0696313
\(932\) −8.09017 −0.265002
\(933\) 34.0000 1.11311
\(934\) 6.47214 0.211775
\(935\) −35.5967 −1.16414
\(936\) 9.23607 0.301890
\(937\) −48.0689 −1.57034 −0.785171 0.619279i \(-0.787425\pi\)
−0.785171 + 0.619279i \(0.787425\pi\)
\(938\) −1.76393 −0.0575944
\(939\) −66.2492 −2.16196
\(940\) 8.47214 0.276331
\(941\) 3.88854 0.126763 0.0633815 0.997989i \(-0.479812\pi\)
0.0633815 + 0.997989i \(0.479812\pi\)
\(942\) −66.8328 −2.17753
\(943\) 68.4296 2.22837
\(944\) 0.291796 0.00949715
\(945\) −66.8328 −2.17407
\(946\) −12.7639 −0.414991
\(947\) −24.6869 −0.802217 −0.401109 0.916031i \(-0.631375\pi\)
−0.401109 + 0.916031i \(0.631375\pi\)
\(948\) −12.9443 −0.420410
\(949\) −2.29180 −0.0743948
\(950\) 4.41641 0.143287
\(951\) −98.7214 −3.20126
\(952\) −22.0000 −0.713024
\(953\) 28.3951 0.919808 0.459904 0.887969i \(-0.347884\pi\)
0.459904 + 0.887969i \(0.347884\pi\)
\(954\) 90.3394 2.92485
\(955\) 41.3607 1.33840
\(956\) 16.2918 0.526914
\(957\) −18.4721 −0.597119
\(958\) −27.1246 −0.876356
\(959\) −32.9017 −1.06245
\(960\) −5.23607 −0.168993
\(961\) −20.5279 −0.662189
\(962\) 5.88854 0.189854
\(963\) 0 0
\(964\) 5.05573 0.162834
\(965\) −7.47214 −0.240537
\(966\) −56.2492 −1.80979
\(967\) 34.5410 1.11076 0.555382 0.831595i \(-0.312572\pi\)
0.555382 + 0.831595i \(0.312572\pi\)
\(968\) −2.85410 −0.0917343
\(969\) −46.2492 −1.48574
\(970\) −20.6525 −0.663111
\(971\) 20.0000 0.641831 0.320915 0.947108i \(-0.396010\pi\)
0.320915 + 0.947108i \(0.396010\pi\)
\(972\) −35.5967 −1.14177
\(973\) −7.05573 −0.226196
\(974\) 3.41641 0.109469
\(975\) 9.52786 0.305136
\(976\) 6.00000 0.192055
\(977\) 32.4721 1.03888 0.519438 0.854508i \(-0.326141\pi\)
0.519438 + 0.854508i \(0.326141\pi\)
\(978\) −30.1803 −0.965061
\(979\) −16.0344 −0.512463
\(980\) 1.85410 0.0592271
\(981\) 78.2492 2.49831
\(982\) 18.4377 0.588370
\(983\) −11.7082 −0.373434 −0.186717 0.982414i \(-0.559785\pi\)
−0.186717 + 0.982414i \(0.559785\pi\)
\(984\) −36.3607 −1.15914
\(985\) −16.1803 −0.515548
\(986\) −15.4164 −0.490958
\(987\) −48.3607 −1.53934
\(988\) −2.29180 −0.0729117
\(989\) −27.2361 −0.866057
\(990\) 34.5066 1.09669
\(991\) 13.5279 0.429727 0.214863 0.976644i \(-0.431069\pi\)
0.214863 + 0.976644i \(0.431069\pi\)
\(992\) 3.23607 0.102745
\(993\) −86.8328 −2.75556
\(994\) −33.6738 −1.06807
\(995\) 20.4164 0.647244
\(996\) 22.1803 0.702811
\(997\) 16.9443 0.536630 0.268315 0.963331i \(-0.413533\pi\)
0.268315 + 0.963331i \(0.413533\pi\)
\(998\) −14.4721 −0.458107
\(999\) −68.9443 −2.18130
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 8014.2.a.a.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8014.2.a.a.1.1 2 1.1 even 1 trivial