Properties

Label 6015.2.a.i.1.20
Level $6015$
Weight $2$
Character 6015.1
Self dual yes
Analytic conductor $48.030$
Analytic rank $0$
Dimension $43$
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] = [6015,2,Mod(1,6015)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6015, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6015.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6015 = 3 \cdot 5 \cdot 401 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6015.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: \(48.0300168158\)
Analytic rank: \(0\)
Dimension: \(43\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.20
Character \(\chi\) \(=\) 6015.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.211503 q^{2} +1.00000 q^{3} -1.95527 q^{4} +1.00000 q^{5} -0.211503 q^{6} +3.03461 q^{7} +0.836552 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.211503 q^{2} +1.00000 q^{3} -1.95527 q^{4} +1.00000 q^{5} -0.211503 q^{6} +3.03461 q^{7} +0.836552 q^{8} +1.00000 q^{9} -0.211503 q^{10} +2.42289 q^{11} -1.95527 q^{12} +5.84813 q^{13} -0.641830 q^{14} +1.00000 q^{15} +3.73360 q^{16} -0.920871 q^{17} -0.211503 q^{18} +2.43117 q^{19} -1.95527 q^{20} +3.03461 q^{21} -0.512449 q^{22} +0.265985 q^{23} +0.836552 q^{24} +1.00000 q^{25} -1.23690 q^{26} +1.00000 q^{27} -5.93347 q^{28} -0.369226 q^{29} -0.211503 q^{30} +6.48890 q^{31} -2.46277 q^{32} +2.42289 q^{33} +0.194767 q^{34} +3.03461 q^{35} -1.95527 q^{36} +8.25624 q^{37} -0.514202 q^{38} +5.84813 q^{39} +0.836552 q^{40} -5.57053 q^{41} -0.641830 q^{42} +6.44720 q^{43} -4.73739 q^{44} +1.00000 q^{45} -0.0562568 q^{46} -0.0721234 q^{47} +3.73360 q^{48} +2.20886 q^{49} -0.211503 q^{50} -0.920871 q^{51} -11.4346 q^{52} -10.5984 q^{53} -0.211503 q^{54} +2.42289 q^{55} +2.53861 q^{56} +2.43117 q^{57} +0.0780926 q^{58} +2.31999 q^{59} -1.95527 q^{60} -0.801821 q^{61} -1.37242 q^{62} +3.03461 q^{63} -6.94631 q^{64} +5.84813 q^{65} -0.512449 q^{66} +10.1489 q^{67} +1.80055 q^{68} +0.265985 q^{69} -0.641830 q^{70} +5.98942 q^{71} +0.836552 q^{72} -14.0198 q^{73} -1.74622 q^{74} +1.00000 q^{75} -4.75359 q^{76} +7.35252 q^{77} -1.23690 q^{78} -1.32234 q^{79} +3.73360 q^{80} +1.00000 q^{81} +1.17819 q^{82} -3.93580 q^{83} -5.93347 q^{84} -0.920871 q^{85} -1.36360 q^{86} -0.369226 q^{87} +2.02687 q^{88} -10.4647 q^{89} -0.211503 q^{90} +17.7468 q^{91} -0.520072 q^{92} +6.48890 q^{93} +0.0152543 q^{94} +2.43117 q^{95} -2.46277 q^{96} +0.184517 q^{97} -0.467182 q^{98} +2.42289 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 43 q + 3 q^{2} + 43 q^{3} + 61 q^{4} + 43 q^{5} + 3 q^{6} + 14 q^{7} + 18 q^{8} + 43 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 43 q + 3 q^{2} + 43 q^{3} + 61 q^{4} + 43 q^{5} + 3 q^{6} + 14 q^{7} + 18 q^{8} + 43 q^{9} + 3 q^{10} + 19 q^{11} + 61 q^{12} + 8 q^{13} + 6 q^{14} + 43 q^{15} + 85 q^{16} + 40 q^{17} + 3 q^{18} + 43 q^{19} + 61 q^{20} + 14 q^{21} + 19 q^{22} + 12 q^{23} + 18 q^{24} + 43 q^{25} + 43 q^{27} + 36 q^{28} + 41 q^{29} + 3 q^{30} + 33 q^{31} + 4 q^{32} + 19 q^{33} + 20 q^{34} + 14 q^{35} + 61 q^{36} + 12 q^{37} + 10 q^{38} + 8 q^{39} + 18 q^{40} + 47 q^{41} + 6 q^{42} + 73 q^{43} + 5 q^{44} + 43 q^{45} + 21 q^{46} + 32 q^{47} + 85 q^{48} + 87 q^{49} + 3 q^{50} + 40 q^{51} + 18 q^{52} + 17 q^{53} + 3 q^{54} + 19 q^{55} + 15 q^{56} + 43 q^{57} - 16 q^{58} + 21 q^{59} + 61 q^{60} + 77 q^{61} + 15 q^{62} + 14 q^{63} + 112 q^{64} + 8 q^{65} + 19 q^{66} + 26 q^{67} + 50 q^{68} + 12 q^{69} + 6 q^{70} + 2 q^{71} + 18 q^{72} + 49 q^{73} - 34 q^{74} + 43 q^{75} + 50 q^{76} - 2 q^{77} + 59 q^{79} + 85 q^{80} + 43 q^{81} - 45 q^{82} - 3 q^{83} + 36 q^{84} + 40 q^{85} - 35 q^{86} + 41 q^{87} - 13 q^{88} + 57 q^{89} + 3 q^{90} + 11 q^{91} - 9 q^{92} + 33 q^{93} + 52 q^{94} + 43 q^{95} + 4 q^{96} + 7 q^{97} - 32 q^{98} + 19 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.211503 −0.149556 −0.0747778 0.997200i \(-0.523825\pi\)
−0.0747778 + 0.997200i \(0.523825\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.95527 −0.977633
\(5\) 1.00000 0.447214
\(6\) −0.211503 −0.0863459
\(7\) 3.03461 1.14697 0.573487 0.819214i \(-0.305590\pi\)
0.573487 + 0.819214i \(0.305590\pi\)
\(8\) 0.836552 0.295766
\(9\) 1.00000 0.333333
\(10\) −0.211503 −0.0668833
\(11\) 2.42289 0.730528 0.365264 0.930904i \(-0.380979\pi\)
0.365264 + 0.930904i \(0.380979\pi\)
\(12\) −1.95527 −0.564437
\(13\) 5.84813 1.62198 0.810989 0.585061i \(-0.198929\pi\)
0.810989 + 0.585061i \(0.198929\pi\)
\(14\) −0.641830 −0.171536
\(15\) 1.00000 0.258199
\(16\) 3.73360 0.933400
\(17\) −0.920871 −0.223344 −0.111672 0.993745i \(-0.535621\pi\)
−0.111672 + 0.993745i \(0.535621\pi\)
\(18\) −0.211503 −0.0498518
\(19\) 2.43117 0.557750 0.278875 0.960327i \(-0.410039\pi\)
0.278875 + 0.960327i \(0.410039\pi\)
\(20\) −1.95527 −0.437211
\(21\) 3.03461 0.662206
\(22\) −0.512449 −0.109254
\(23\) 0.265985 0.0554618 0.0277309 0.999615i \(-0.491172\pi\)
0.0277309 + 0.999615i \(0.491172\pi\)
\(24\) 0.836552 0.170761
\(25\) 1.00000 0.200000
\(26\) −1.23690 −0.242576
\(27\) 1.00000 0.192450
\(28\) −5.93347 −1.12132
\(29\) −0.369226 −0.0685636 −0.0342818 0.999412i \(-0.510914\pi\)
−0.0342818 + 0.999412i \(0.510914\pi\)
\(30\) −0.211503 −0.0386151
\(31\) 6.48890 1.16544 0.582720 0.812673i \(-0.301988\pi\)
0.582720 + 0.812673i \(0.301988\pi\)
\(32\) −2.46277 −0.435361
\(33\) 2.42289 0.421770
\(34\) 0.194767 0.0334023
\(35\) 3.03461 0.512943
\(36\) −1.95527 −0.325878
\(37\) 8.25624 1.35732 0.678659 0.734454i \(-0.262561\pi\)
0.678659 + 0.734454i \(0.262561\pi\)
\(38\) −0.514202 −0.0834145
\(39\) 5.84813 0.936450
\(40\) 0.836552 0.132271
\(41\) −5.57053 −0.869971 −0.434986 0.900437i \(-0.643247\pi\)
−0.434986 + 0.900437i \(0.643247\pi\)
\(42\) −0.641830 −0.0990366
\(43\) 6.44720 0.983189 0.491594 0.870824i \(-0.336414\pi\)
0.491594 + 0.870824i \(0.336414\pi\)
\(44\) −4.73739 −0.714188
\(45\) 1.00000 0.149071
\(46\) −0.0562568 −0.00829462
\(47\) −0.0721234 −0.0105203 −0.00526014 0.999986i \(-0.501674\pi\)
−0.00526014 + 0.999986i \(0.501674\pi\)
\(48\) 3.73360 0.538899
\(49\) 2.20886 0.315551
\(50\) −0.211503 −0.0299111
\(51\) −0.920871 −0.128948
\(52\) −11.4346 −1.58570
\(53\) −10.5984 −1.45580 −0.727899 0.685684i \(-0.759503\pi\)
−0.727899 + 0.685684i \(0.759503\pi\)
\(54\) −0.211503 −0.0287820
\(55\) 2.42289 0.326702
\(56\) 2.53861 0.339236
\(57\) 2.43117 0.322017
\(58\) 0.0780926 0.0102541
\(59\) 2.31999 0.302037 0.151018 0.988531i \(-0.451745\pi\)
0.151018 + 0.988531i \(0.451745\pi\)
\(60\) −1.95527 −0.252424
\(61\) −0.801821 −0.102663 −0.0513313 0.998682i \(-0.516346\pi\)
−0.0513313 + 0.998682i \(0.516346\pi\)
\(62\) −1.37242 −0.174298
\(63\) 3.03461 0.382325
\(64\) −6.94631 −0.868289
\(65\) 5.84813 0.725371
\(66\) −0.512449 −0.0630781
\(67\) 10.1489 1.23989 0.619945 0.784645i \(-0.287155\pi\)
0.619945 + 0.784645i \(0.287155\pi\)
\(68\) 1.80055 0.218348
\(69\) 0.265985 0.0320209
\(70\) −0.641830 −0.0767134
\(71\) 5.98942 0.710813 0.355406 0.934712i \(-0.384342\pi\)
0.355406 + 0.934712i \(0.384342\pi\)
\(72\) 0.836552 0.0985886
\(73\) −14.0198 −1.64089 −0.820446 0.571724i \(-0.806275\pi\)
−0.820446 + 0.571724i \(0.806275\pi\)
\(74\) −1.74622 −0.202994
\(75\) 1.00000 0.115470
\(76\) −4.75359 −0.545274
\(77\) 7.35252 0.837897
\(78\) −1.23690 −0.140051
\(79\) −1.32234 −0.148775 −0.0743874 0.997229i \(-0.523700\pi\)
−0.0743874 + 0.997229i \(0.523700\pi\)
\(80\) 3.73360 0.417429
\(81\) 1.00000 0.111111
\(82\) 1.17819 0.130109
\(83\) −3.93580 −0.432010 −0.216005 0.976392i \(-0.569303\pi\)
−0.216005 + 0.976392i \(0.569303\pi\)
\(84\) −5.93347 −0.647395
\(85\) −0.920871 −0.0998825
\(86\) −1.36360 −0.147041
\(87\) −0.369226 −0.0395852
\(88\) 2.02687 0.216065
\(89\) −10.4647 −1.10926 −0.554630 0.832097i \(-0.687140\pi\)
−0.554630 + 0.832097i \(0.687140\pi\)
\(90\) −0.211503 −0.0222944
\(91\) 17.7468 1.86037
\(92\) −0.520072 −0.0542213
\(93\) 6.48890 0.672867
\(94\) 0.0152543 0.00157337
\(95\) 2.43117 0.249433
\(96\) −2.46277 −0.251356
\(97\) 0.184517 0.0187349 0.00936745 0.999956i \(-0.497018\pi\)
0.00936745 + 0.999956i \(0.497018\pi\)
\(98\) −0.467182 −0.0471925
\(99\) 2.42289 0.243509
\(100\) −1.95527 −0.195527
\(101\) −4.35727 −0.433565 −0.216782 0.976220i \(-0.569556\pi\)
−0.216782 + 0.976220i \(0.569556\pi\)
\(102\) 0.194767 0.0192848
\(103\) −7.68993 −0.757712 −0.378856 0.925456i \(-0.623682\pi\)
−0.378856 + 0.925456i \(0.623682\pi\)
\(104\) 4.89226 0.479726
\(105\) 3.03461 0.296148
\(106\) 2.24159 0.217723
\(107\) −0.224002 −0.0216551 −0.0108276 0.999941i \(-0.503447\pi\)
−0.0108276 + 0.999941i \(0.503447\pi\)
\(108\) −1.95527 −0.188146
\(109\) −10.7052 −1.02538 −0.512688 0.858575i \(-0.671350\pi\)
−0.512688 + 0.858575i \(0.671350\pi\)
\(110\) −0.512449 −0.0488601
\(111\) 8.25624 0.783648
\(112\) 11.3300 1.07059
\(113\) −6.84989 −0.644383 −0.322192 0.946674i \(-0.604420\pi\)
−0.322192 + 0.946674i \(0.604420\pi\)
\(114\) −0.514202 −0.0481594
\(115\) 0.265985 0.0248033
\(116\) 0.721936 0.0670300
\(117\) 5.84813 0.540660
\(118\) −0.490685 −0.0451712
\(119\) −2.79448 −0.256170
\(120\) 0.836552 0.0763664
\(121\) −5.12962 −0.466329
\(122\) 0.169588 0.0153538
\(123\) −5.57053 −0.502278
\(124\) −12.6875 −1.13937
\(125\) 1.00000 0.0894427
\(126\) −0.641830 −0.0571788
\(127\) −5.47964 −0.486239 −0.243120 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(128\) 6.39472 0.565218
\(129\) 6.44720 0.567644
\(130\) −1.23690 −0.108483
\(131\) 17.0298 1.48790 0.743952 0.668233i \(-0.232949\pi\)
0.743952 + 0.668233i \(0.232949\pi\)
\(132\) −4.73739 −0.412337
\(133\) 7.37767 0.639725
\(134\) −2.14654 −0.185432
\(135\) 1.00000 0.0860663
\(136\) −0.770357 −0.0660575
\(137\) −18.1250 −1.54852 −0.774260 0.632867i \(-0.781878\pi\)
−0.774260 + 0.632867i \(0.781878\pi\)
\(138\) −0.0562568 −0.00478890
\(139\) −15.1610 −1.28594 −0.642971 0.765890i \(-0.722298\pi\)
−0.642971 + 0.765890i \(0.722298\pi\)
\(140\) −5.93347 −0.501470
\(141\) −0.0721234 −0.00607389
\(142\) −1.26678 −0.106306
\(143\) 14.1693 1.18490
\(144\) 3.73360 0.311133
\(145\) −0.369226 −0.0306626
\(146\) 2.96523 0.245404
\(147\) 2.20886 0.182184
\(148\) −16.1432 −1.32696
\(149\) −1.85053 −0.151601 −0.0758007 0.997123i \(-0.524151\pi\)
−0.0758007 + 0.997123i \(0.524151\pi\)
\(150\) −0.211503 −0.0172692
\(151\) 11.4334 0.930437 0.465219 0.885196i \(-0.345976\pi\)
0.465219 + 0.885196i \(0.345976\pi\)
\(152\) 2.03380 0.164963
\(153\) −0.920871 −0.0744480
\(154\) −1.55508 −0.125312
\(155\) 6.48890 0.521201
\(156\) −11.4346 −0.915504
\(157\) −3.26069 −0.260232 −0.130116 0.991499i \(-0.541535\pi\)
−0.130116 + 0.991499i \(0.541535\pi\)
\(158\) 0.279679 0.0222501
\(159\) −10.5984 −0.840505
\(160\) −2.46277 −0.194699
\(161\) 0.807162 0.0636133
\(162\) −0.211503 −0.0166173
\(163\) 10.8738 0.851699 0.425850 0.904794i \(-0.359975\pi\)
0.425850 + 0.904794i \(0.359975\pi\)
\(164\) 10.8919 0.850513
\(165\) 2.42289 0.188621
\(166\) 0.832435 0.0646095
\(167\) −10.0862 −0.780494 −0.390247 0.920710i \(-0.627610\pi\)
−0.390247 + 0.920710i \(0.627610\pi\)
\(168\) 2.53861 0.195858
\(169\) 21.2006 1.63081
\(170\) 0.194767 0.0149380
\(171\) 2.43117 0.185917
\(172\) −12.6060 −0.961198
\(173\) 16.8481 1.28093 0.640467 0.767985i \(-0.278741\pi\)
0.640467 + 0.767985i \(0.278741\pi\)
\(174\) 0.0780926 0.00592019
\(175\) 3.03461 0.229395
\(176\) 9.04609 0.681874
\(177\) 2.31999 0.174381
\(178\) 2.21333 0.165896
\(179\) −22.9185 −1.71301 −0.856504 0.516140i \(-0.827368\pi\)
−0.856504 + 0.516140i \(0.827368\pi\)
\(180\) −1.95527 −0.145737
\(181\) −11.1785 −0.830888 −0.415444 0.909619i \(-0.636374\pi\)
−0.415444 + 0.909619i \(0.636374\pi\)
\(182\) −3.75351 −0.278228
\(183\) −0.801821 −0.0592723
\(184\) 0.222511 0.0164037
\(185\) 8.25624 0.607011
\(186\) −1.37242 −0.100631
\(187\) −2.23117 −0.163159
\(188\) 0.141020 0.0102850
\(189\) 3.03461 0.220735
\(190\) −0.514202 −0.0373041
\(191\) −16.3225 −1.18105 −0.590526 0.807018i \(-0.701080\pi\)
−0.590526 + 0.807018i \(0.701080\pi\)
\(192\) −6.94631 −0.501307
\(193\) 13.9218 1.00211 0.501057 0.865414i \(-0.332945\pi\)
0.501057 + 0.865414i \(0.332945\pi\)
\(194\) −0.0390260 −0.00280191
\(195\) 5.84813 0.418793
\(196\) −4.31891 −0.308494
\(197\) 9.62146 0.685501 0.342750 0.939427i \(-0.388641\pi\)
0.342750 + 0.939427i \(0.388641\pi\)
\(198\) −0.512449 −0.0364181
\(199\) −21.0696 −1.49359 −0.746794 0.665056i \(-0.768408\pi\)
−0.746794 + 0.665056i \(0.768408\pi\)
\(200\) 0.836552 0.0591532
\(201\) 10.1489 0.715851
\(202\) 0.921578 0.0648420
\(203\) −1.12046 −0.0786407
\(204\) 1.80055 0.126064
\(205\) −5.57053 −0.389063
\(206\) 1.62645 0.113320
\(207\) 0.265985 0.0184873
\(208\) 21.8346 1.51395
\(209\) 5.89046 0.407452
\(210\) −0.641830 −0.0442905
\(211\) 10.1585 0.699338 0.349669 0.936873i \(-0.386294\pi\)
0.349669 + 0.936873i \(0.386294\pi\)
\(212\) 20.7226 1.42324
\(213\) 5.98942 0.410388
\(214\) 0.0473773 0.00323864
\(215\) 6.44720 0.439695
\(216\) 0.836552 0.0569202
\(217\) 19.6913 1.33673
\(218\) 2.26419 0.153351
\(219\) −14.0198 −0.947369
\(220\) −4.73739 −0.319395
\(221\) −5.38537 −0.362259
\(222\) −1.74622 −0.117199
\(223\) 12.6691 0.848382 0.424191 0.905573i \(-0.360558\pi\)
0.424191 + 0.905573i \(0.360558\pi\)
\(224\) −7.47356 −0.499348
\(225\) 1.00000 0.0666667
\(226\) 1.44877 0.0963711
\(227\) 15.5838 1.03433 0.517166 0.855885i \(-0.326987\pi\)
0.517166 + 0.855885i \(0.326987\pi\)
\(228\) −4.75359 −0.314814
\(229\) 6.10570 0.403476 0.201738 0.979440i \(-0.435341\pi\)
0.201738 + 0.979440i \(0.435341\pi\)
\(230\) −0.0562568 −0.00370947
\(231\) 7.35252 0.483760
\(232\) −0.308877 −0.0202788
\(233\) −7.81283 −0.511835 −0.255918 0.966699i \(-0.582378\pi\)
−0.255918 + 0.966699i \(0.582378\pi\)
\(234\) −1.23690 −0.0808586
\(235\) −0.0721234 −0.00470481
\(236\) −4.53619 −0.295281
\(237\) −1.32234 −0.0858952
\(238\) 0.591043 0.0383116
\(239\) −18.3863 −1.18931 −0.594656 0.803980i \(-0.702712\pi\)
−0.594656 + 0.803980i \(0.702712\pi\)
\(240\) 3.73360 0.241003
\(241\) −17.4321 −1.12290 −0.561449 0.827511i \(-0.689756\pi\)
−0.561449 + 0.827511i \(0.689756\pi\)
\(242\) 1.08493 0.0697421
\(243\) 1.00000 0.0641500
\(244\) 1.56777 0.100366
\(245\) 2.20886 0.141119
\(246\) 1.17819 0.0751185
\(247\) 14.2178 0.904658
\(248\) 5.42830 0.344697
\(249\) −3.93580 −0.249421
\(250\) −0.211503 −0.0133767
\(251\) 19.7790 1.24844 0.624219 0.781250i \(-0.285417\pi\)
0.624219 + 0.781250i \(0.285417\pi\)
\(252\) −5.93347 −0.373774
\(253\) 0.644452 0.0405164
\(254\) 1.15896 0.0727198
\(255\) −0.920871 −0.0576672
\(256\) 12.5401 0.783758
\(257\) 25.4638 1.58839 0.794194 0.607664i \(-0.207893\pi\)
0.794194 + 0.607664i \(0.207893\pi\)
\(258\) −1.36360 −0.0848943
\(259\) 25.0545 1.55681
\(260\) −11.4346 −0.709147
\(261\) −0.369226 −0.0228545
\(262\) −3.60187 −0.222524
\(263\) −4.94984 −0.305220 −0.152610 0.988286i \(-0.548768\pi\)
−0.152610 + 0.988286i \(0.548768\pi\)
\(264\) 2.02687 0.124745
\(265\) −10.5984 −0.651053
\(266\) −1.56040 −0.0956744
\(267\) −10.4647 −0.640432
\(268\) −19.8439 −1.21216
\(269\) 29.9145 1.82392 0.911960 0.410279i \(-0.134569\pi\)
0.911960 + 0.410279i \(0.134569\pi\)
\(270\) −0.211503 −0.0128717
\(271\) 16.8727 1.02494 0.512471 0.858704i \(-0.328730\pi\)
0.512471 + 0.858704i \(0.328730\pi\)
\(272\) −3.43816 −0.208469
\(273\) 17.7468 1.07408
\(274\) 3.83349 0.231590
\(275\) 2.42289 0.146106
\(276\) −0.520072 −0.0313047
\(277\) 29.6902 1.78391 0.891956 0.452122i \(-0.149333\pi\)
0.891956 + 0.452122i \(0.149333\pi\)
\(278\) 3.20661 0.192320
\(279\) 6.48890 0.388480
\(280\) 2.53861 0.151711
\(281\) −9.05519 −0.540187 −0.270094 0.962834i \(-0.587055\pi\)
−0.270094 + 0.962834i \(0.587055\pi\)
\(282\) 0.0152543 0.000908383 0
\(283\) −2.42399 −0.144091 −0.0720457 0.997401i \(-0.522953\pi\)
−0.0720457 + 0.997401i \(0.522953\pi\)
\(284\) −11.7109 −0.694914
\(285\) 2.43117 0.144010
\(286\) −2.99687 −0.177208
\(287\) −16.9044 −0.997835
\(288\) −2.46277 −0.145120
\(289\) −16.1520 −0.950117
\(290\) 0.0780926 0.00458576
\(291\) 0.184517 0.0108166
\(292\) 27.4124 1.60419
\(293\) 11.7098 0.684092 0.342046 0.939683i \(-0.388880\pi\)
0.342046 + 0.939683i \(0.388880\pi\)
\(294\) −0.467182 −0.0272466
\(295\) 2.31999 0.135075
\(296\) 6.90678 0.401448
\(297\) 2.42289 0.140590
\(298\) 0.391393 0.0226728
\(299\) 1.55552 0.0899579
\(300\) −1.95527 −0.112887
\(301\) 19.5647 1.12769
\(302\) −2.41820 −0.139152
\(303\) −4.35727 −0.250319
\(304\) 9.07703 0.520603
\(305\) −0.801821 −0.0459121
\(306\) 0.194767 0.0111341
\(307\) −4.24442 −0.242242 −0.121121 0.992638i \(-0.538649\pi\)
−0.121121 + 0.992638i \(0.538649\pi\)
\(308\) −14.3761 −0.819156
\(309\) −7.68993 −0.437465
\(310\) −1.37242 −0.0779484
\(311\) −12.3533 −0.700493 −0.350246 0.936658i \(-0.613902\pi\)
−0.350246 + 0.936658i \(0.613902\pi\)
\(312\) 4.89226 0.276970
\(313\) 7.57303 0.428053 0.214026 0.976828i \(-0.431342\pi\)
0.214026 + 0.976828i \(0.431342\pi\)
\(314\) 0.689648 0.0389191
\(315\) 3.03461 0.170981
\(316\) 2.58553 0.145447
\(317\) −8.60857 −0.483505 −0.241753 0.970338i \(-0.577722\pi\)
−0.241753 + 0.970338i \(0.577722\pi\)
\(318\) 2.24159 0.125702
\(319\) −0.894593 −0.0500876
\(320\) −6.94631 −0.388311
\(321\) −0.224002 −0.0125026
\(322\) −0.170718 −0.00951372
\(323\) −2.23880 −0.124570
\(324\) −1.95527 −0.108626
\(325\) 5.84813 0.324396
\(326\) −2.29984 −0.127376
\(327\) −10.7052 −0.592001
\(328\) −4.66004 −0.257308
\(329\) −0.218866 −0.0120665
\(330\) −0.512449 −0.0282094
\(331\) −23.1407 −1.27193 −0.635965 0.771718i \(-0.719398\pi\)
−0.635965 + 0.771718i \(0.719398\pi\)
\(332\) 7.69553 0.422347
\(333\) 8.25624 0.452439
\(334\) 2.13327 0.116727
\(335\) 10.1489 0.554496
\(336\) 11.3300 0.618103
\(337\) −9.24214 −0.503451 −0.251726 0.967799i \(-0.580998\pi\)
−0.251726 + 0.967799i \(0.580998\pi\)
\(338\) −4.48400 −0.243897
\(339\) −6.84989 −0.372035
\(340\) 1.80055 0.0976484
\(341\) 15.7219 0.851386
\(342\) −0.514202 −0.0278048
\(343\) −14.5392 −0.785045
\(344\) 5.39342 0.290794
\(345\) 0.265985 0.0143202
\(346\) −3.56342 −0.191571
\(347\) −5.09499 −0.273513 −0.136757 0.990605i \(-0.543668\pi\)
−0.136757 + 0.990605i \(0.543668\pi\)
\(348\) 0.721936 0.0386998
\(349\) −28.7025 −1.53641 −0.768205 0.640204i \(-0.778850\pi\)
−0.768205 + 0.640204i \(0.778850\pi\)
\(350\) −0.641830 −0.0343073
\(351\) 5.84813 0.312150
\(352\) −5.96702 −0.318043
\(353\) 9.05416 0.481904 0.240952 0.970537i \(-0.422540\pi\)
0.240952 + 0.970537i \(0.422540\pi\)
\(354\) −0.490685 −0.0260796
\(355\) 5.98942 0.317885
\(356\) 20.4613 1.08445
\(357\) −2.79448 −0.147900
\(358\) 4.84734 0.256190
\(359\) 25.1956 1.32977 0.664886 0.746945i \(-0.268480\pi\)
0.664886 + 0.746945i \(0.268480\pi\)
\(360\) 0.836552 0.0440902
\(361\) −13.0894 −0.688915
\(362\) 2.36428 0.124264
\(363\) −5.12962 −0.269235
\(364\) −34.6997 −1.81876
\(365\) −14.0198 −0.733829
\(366\) 0.169588 0.00886450
\(367\) 11.8528 0.618712 0.309356 0.950946i \(-0.399887\pi\)
0.309356 + 0.950946i \(0.399887\pi\)
\(368\) 0.993083 0.0517680
\(369\) −5.57053 −0.289990
\(370\) −1.74622 −0.0907818
\(371\) −32.1619 −1.66976
\(372\) −12.6875 −0.657817
\(373\) −1.34905 −0.0698512 −0.0349256 0.999390i \(-0.511119\pi\)
−0.0349256 + 0.999390i \(0.511119\pi\)
\(374\) 0.471899 0.0244013
\(375\) 1.00000 0.0516398
\(376\) −0.0603350 −0.00311154
\(377\) −2.15928 −0.111209
\(378\) −0.641830 −0.0330122
\(379\) 1.90003 0.0975979 0.0487990 0.998809i \(-0.484461\pi\)
0.0487990 + 0.998809i \(0.484461\pi\)
\(380\) −4.75359 −0.243854
\(381\) −5.47964 −0.280730
\(382\) 3.45226 0.176633
\(383\) 14.2734 0.729336 0.364668 0.931138i \(-0.381182\pi\)
0.364668 + 0.931138i \(0.381182\pi\)
\(384\) 6.39472 0.326329
\(385\) 7.35252 0.374719
\(386\) −2.94451 −0.149872
\(387\) 6.44720 0.327730
\(388\) −0.360780 −0.0183159
\(389\) 30.4225 1.54248 0.771241 0.636544i \(-0.219636\pi\)
0.771241 + 0.636544i \(0.219636\pi\)
\(390\) −1.23690 −0.0626328
\(391\) −0.244938 −0.0123871
\(392\) 1.84783 0.0933294
\(393\) 17.0298 0.859041
\(394\) −2.03497 −0.102520
\(395\) −1.32234 −0.0665341
\(396\) −4.73739 −0.238063
\(397\) 1.15575 0.0580055 0.0290028 0.999579i \(-0.490767\pi\)
0.0290028 + 0.999579i \(0.490767\pi\)
\(398\) 4.45630 0.223374
\(399\) 7.37767 0.369345
\(400\) 3.73360 0.186680
\(401\) 1.00000 0.0499376
\(402\) −2.14654 −0.107059
\(403\) 37.9479 1.89032
\(404\) 8.51963 0.423867
\(405\) 1.00000 0.0496904
\(406\) 0.236981 0.0117612
\(407\) 20.0039 0.991558
\(408\) −0.770357 −0.0381383
\(409\) −16.7064 −0.826077 −0.413038 0.910714i \(-0.635532\pi\)
−0.413038 + 0.910714i \(0.635532\pi\)
\(410\) 1.17819 0.0581865
\(411\) −18.1250 −0.894039
\(412\) 15.0359 0.740764
\(413\) 7.04026 0.346429
\(414\) −0.0562568 −0.00276487
\(415\) −3.93580 −0.193201
\(416\) −14.4026 −0.706146
\(417\) −15.1610 −0.742439
\(418\) −1.24585 −0.0609366
\(419\) 24.3068 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(420\) −5.93347 −0.289524
\(421\) 17.4587 0.850886 0.425443 0.904985i \(-0.360118\pi\)
0.425443 + 0.904985i \(0.360118\pi\)
\(422\) −2.14855 −0.104590
\(423\) −0.0721234 −0.00350676
\(424\) −8.86609 −0.430575
\(425\) −0.920871 −0.0446688
\(426\) −1.26678 −0.0613758
\(427\) −2.43322 −0.117752
\(428\) 0.437984 0.0211708
\(429\) 14.1693 0.684103
\(430\) −1.36360 −0.0657589
\(431\) 22.1901 1.06886 0.534431 0.845212i \(-0.320526\pi\)
0.534431 + 0.845212i \(0.320526\pi\)
\(432\) 3.73360 0.179633
\(433\) 14.4121 0.692603 0.346301 0.938123i \(-0.387438\pi\)
0.346301 + 0.938123i \(0.387438\pi\)
\(434\) −4.16477 −0.199915
\(435\) −0.369226 −0.0177030
\(436\) 20.9316 1.00244
\(437\) 0.646657 0.0309338
\(438\) 2.96523 0.141684
\(439\) −36.6471 −1.74907 −0.874536 0.484961i \(-0.838834\pi\)
−0.874536 + 0.484961i \(0.838834\pi\)
\(440\) 2.02687 0.0966273
\(441\) 2.20886 0.105184
\(442\) 1.13902 0.0541779
\(443\) −13.4046 −0.636870 −0.318435 0.947945i \(-0.603157\pi\)
−0.318435 + 0.947945i \(0.603157\pi\)
\(444\) −16.1432 −0.766120
\(445\) −10.4647 −0.496076
\(446\) −2.67955 −0.126880
\(447\) −1.85053 −0.0875271
\(448\) −21.0794 −0.995906
\(449\) −32.1891 −1.51910 −0.759550 0.650449i \(-0.774581\pi\)
−0.759550 + 0.650449i \(0.774581\pi\)
\(450\) −0.211503 −0.00997037
\(451\) −13.4968 −0.635538
\(452\) 13.3934 0.629971
\(453\) 11.4334 0.537188
\(454\) −3.29602 −0.154690
\(455\) 17.7468 0.831982
\(456\) 2.03380 0.0952416
\(457\) 3.86165 0.180640 0.0903201 0.995913i \(-0.471211\pi\)
0.0903201 + 0.995913i \(0.471211\pi\)
\(458\) −1.29138 −0.0603421
\(459\) −0.920871 −0.0429826
\(460\) −0.520072 −0.0242485
\(461\) 0.462987 0.0215634 0.0107817 0.999942i \(-0.496568\pi\)
0.0107817 + 0.999942i \(0.496568\pi\)
\(462\) −1.55508 −0.0723490
\(463\) 30.6724 1.42547 0.712734 0.701434i \(-0.247457\pi\)
0.712734 + 0.701434i \(0.247457\pi\)
\(464\) −1.37854 −0.0639972
\(465\) 6.48890 0.300915
\(466\) 1.65244 0.0765478
\(467\) −35.5085 −1.64314 −0.821569 0.570109i \(-0.806901\pi\)
−0.821569 + 0.570109i \(0.806901\pi\)
\(468\) −11.4346 −0.528567
\(469\) 30.7981 1.42212
\(470\) 0.0152543 0.000703631 0
\(471\) −3.26069 −0.150245
\(472\) 1.94079 0.0893322
\(473\) 15.6208 0.718247
\(474\) 0.279679 0.0128461
\(475\) 2.43117 0.111550
\(476\) 5.46396 0.250440
\(477\) −10.5984 −0.485266
\(478\) 3.88877 0.177868
\(479\) 18.5970 0.849717 0.424859 0.905260i \(-0.360324\pi\)
0.424859 + 0.905260i \(0.360324\pi\)
\(480\) −2.46277 −0.112410
\(481\) 48.2836 2.20154
\(482\) 3.68694 0.167936
\(483\) 0.807162 0.0367272
\(484\) 10.0298 0.455899
\(485\) 0.184517 0.00837850
\(486\) −0.211503 −0.00959399
\(487\) 18.8938 0.856158 0.428079 0.903741i \(-0.359190\pi\)
0.428079 + 0.903741i \(0.359190\pi\)
\(488\) −0.670765 −0.0303641
\(489\) 10.8738 0.491729
\(490\) −0.467182 −0.0211051
\(491\) 16.5401 0.746445 0.373222 0.927742i \(-0.378253\pi\)
0.373222 + 0.927742i \(0.378253\pi\)
\(492\) 10.8919 0.491044
\(493\) 0.340010 0.0153133
\(494\) −3.00712 −0.135297
\(495\) 2.42289 0.108901
\(496\) 24.2269 1.08782
\(497\) 18.1755 0.815285
\(498\) 0.832435 0.0373023
\(499\) 35.4985 1.58913 0.794565 0.607179i \(-0.207699\pi\)
0.794565 + 0.607179i \(0.207699\pi\)
\(500\) −1.95527 −0.0874422
\(501\) −10.0862 −0.450619
\(502\) −4.18332 −0.186711
\(503\) 7.75457 0.345759 0.172880 0.984943i \(-0.444693\pi\)
0.172880 + 0.984943i \(0.444693\pi\)
\(504\) 2.53861 0.113079
\(505\) −4.35727 −0.193896
\(506\) −0.136304 −0.00605945
\(507\) 21.2006 0.941551
\(508\) 10.7142 0.475364
\(509\) −9.24652 −0.409845 −0.204922 0.978778i \(-0.565694\pi\)
−0.204922 + 0.978778i \(0.565694\pi\)
\(510\) 0.194767 0.00862444
\(511\) −42.5446 −1.88206
\(512\) −15.4417 −0.682434
\(513\) 2.43117 0.107339
\(514\) −5.38568 −0.237552
\(515\) −7.68993 −0.338859
\(516\) −12.6060 −0.554948
\(517\) −0.174747 −0.00768536
\(518\) −5.29911 −0.232829
\(519\) 16.8481 0.739548
\(520\) 4.89226 0.214540
\(521\) −21.1535 −0.926753 −0.463376 0.886162i \(-0.653362\pi\)
−0.463376 + 0.886162i \(0.653362\pi\)
\(522\) 0.0780926 0.00341802
\(523\) 19.6153 0.857718 0.428859 0.903371i \(-0.358916\pi\)
0.428859 + 0.903371i \(0.358916\pi\)
\(524\) −33.2979 −1.45462
\(525\) 3.03461 0.132441
\(526\) 1.04691 0.0456473
\(527\) −5.97544 −0.260294
\(528\) 9.04609 0.393680
\(529\) −22.9293 −0.996924
\(530\) 2.24159 0.0973685
\(531\) 2.31999 0.100679
\(532\) −14.4253 −0.625416
\(533\) −32.5772 −1.41107
\(534\) 2.21333 0.0957801
\(535\) −0.224002 −0.00968447
\(536\) 8.49012 0.366717
\(537\) −22.9185 −0.989006
\(538\) −6.32702 −0.272777
\(539\) 5.35182 0.230519
\(540\) −1.95527 −0.0841413
\(541\) 35.5697 1.52926 0.764631 0.644469i \(-0.222921\pi\)
0.764631 + 0.644469i \(0.222921\pi\)
\(542\) −3.56863 −0.153286
\(543\) −11.1785 −0.479713
\(544\) 2.26790 0.0972353
\(545\) −10.7052 −0.458562
\(546\) −3.75351 −0.160635
\(547\) −20.8798 −0.892755 −0.446377 0.894845i \(-0.647286\pi\)
−0.446377 + 0.894845i \(0.647286\pi\)
\(548\) 35.4391 1.51389
\(549\) −0.801821 −0.0342209
\(550\) −0.512449 −0.0218509
\(551\) −0.897653 −0.0382413
\(552\) 0.222511 0.00947069
\(553\) −4.01278 −0.170641
\(554\) −6.27958 −0.266794
\(555\) 8.25624 0.350458
\(556\) 29.6439 1.25718
\(557\) −20.6420 −0.874629 −0.437315 0.899309i \(-0.644070\pi\)
−0.437315 + 0.899309i \(0.644070\pi\)
\(558\) −1.37242 −0.0580993
\(559\) 37.7040 1.59471
\(560\) 11.3300 0.478781
\(561\) −2.23117 −0.0941999
\(562\) 1.91520 0.0807880
\(563\) −23.3531 −0.984218 −0.492109 0.870534i \(-0.663774\pi\)
−0.492109 + 0.870534i \(0.663774\pi\)
\(564\) 0.141020 0.00593803
\(565\) −6.84989 −0.288177
\(566\) 0.512682 0.0215497
\(567\) 3.03461 0.127442
\(568\) 5.01046 0.210234
\(569\) 4.22051 0.176933 0.0884664 0.996079i \(-0.471803\pi\)
0.0884664 + 0.996079i \(0.471803\pi\)
\(570\) −0.514202 −0.0215375
\(571\) −27.5757 −1.15401 −0.577004 0.816742i \(-0.695778\pi\)
−0.577004 + 0.816742i \(0.695778\pi\)
\(572\) −27.7048 −1.15840
\(573\) −16.3225 −0.681881
\(574\) 3.57534 0.149232
\(575\) 0.265985 0.0110924
\(576\) −6.94631 −0.289430
\(577\) −8.44884 −0.351730 −0.175865 0.984414i \(-0.556272\pi\)
−0.175865 + 0.984414i \(0.556272\pi\)
\(578\) 3.41620 0.142095
\(579\) 13.9218 0.578571
\(580\) 0.721936 0.0299767
\(581\) −11.9436 −0.495505
\(582\) −0.0390260 −0.00161768
\(583\) −25.6786 −1.06350
\(584\) −11.7283 −0.485320
\(585\) 5.84813 0.241790
\(586\) −2.47666 −0.102310
\(587\) −23.5838 −0.973409 −0.486705 0.873567i \(-0.661801\pi\)
−0.486705 + 0.873567i \(0.661801\pi\)
\(588\) −4.31891 −0.178109
\(589\) 15.7756 0.650024
\(590\) −0.490685 −0.0202012
\(591\) 9.62146 0.395774
\(592\) 30.8255 1.26692
\(593\) −16.4635 −0.676073 −0.338037 0.941133i \(-0.609763\pi\)
−0.338037 + 0.941133i \(0.609763\pi\)
\(594\) −0.512449 −0.0210260
\(595\) −2.79448 −0.114563
\(596\) 3.61828 0.148211
\(597\) −21.0696 −0.862323
\(598\) −0.328997 −0.0134537
\(599\) −42.1650 −1.72281 −0.861407 0.507915i \(-0.830416\pi\)
−0.861407 + 0.507915i \(0.830416\pi\)
\(600\) 0.836552 0.0341521
\(601\) −7.03460 −0.286947 −0.143474 0.989654i \(-0.545827\pi\)
−0.143474 + 0.989654i \(0.545827\pi\)
\(602\) −4.13801 −0.168653
\(603\) 10.1489 0.413297
\(604\) −22.3553 −0.909626
\(605\) −5.12962 −0.208549
\(606\) 0.921578 0.0374366
\(607\) 26.6821 1.08299 0.541496 0.840703i \(-0.317858\pi\)
0.541496 + 0.840703i \(0.317858\pi\)
\(608\) −5.98743 −0.242822
\(609\) −1.12046 −0.0454032
\(610\) 0.169588 0.00686641
\(611\) −0.421787 −0.0170637
\(612\) 1.80055 0.0727828
\(613\) 24.5956 0.993406 0.496703 0.867921i \(-0.334544\pi\)
0.496703 + 0.867921i \(0.334544\pi\)
\(614\) 0.897710 0.0362286
\(615\) −5.57053 −0.224626
\(616\) 6.15076 0.247821
\(617\) −18.7309 −0.754078 −0.377039 0.926197i \(-0.623058\pi\)
−0.377039 + 0.926197i \(0.623058\pi\)
\(618\) 1.62645 0.0654253
\(619\) −15.1561 −0.609175 −0.304588 0.952484i \(-0.598519\pi\)
−0.304588 + 0.952484i \(0.598519\pi\)
\(620\) −12.6875 −0.509543
\(621\) 0.265985 0.0106736
\(622\) 2.61277 0.104763
\(623\) −31.7564 −1.27229
\(624\) 21.8346 0.874082
\(625\) 1.00000 0.0400000
\(626\) −1.60172 −0.0640177
\(627\) 5.89046 0.235242
\(628\) 6.37552 0.254411
\(629\) −7.60293 −0.303149
\(630\) −0.641830 −0.0255711
\(631\) 21.4986 0.855847 0.427924 0.903815i \(-0.359245\pi\)
0.427924 + 0.903815i \(0.359245\pi\)
\(632\) −1.10621 −0.0440025
\(633\) 10.1585 0.403763
\(634\) 1.82074 0.0723109
\(635\) −5.47964 −0.217453
\(636\) 20.7226 0.821706
\(637\) 12.9177 0.511818
\(638\) 0.189210 0.00749088
\(639\) 5.98942 0.236938
\(640\) 6.39472 0.252773
\(641\) 22.4259 0.885768 0.442884 0.896579i \(-0.353955\pi\)
0.442884 + 0.896579i \(0.353955\pi\)
\(642\) 0.0473773 0.00186983
\(643\) 20.1049 0.792859 0.396429 0.918065i \(-0.370249\pi\)
0.396429 + 0.918065i \(0.370249\pi\)
\(644\) −1.57822 −0.0621905
\(645\) 6.44720 0.253858
\(646\) 0.473513 0.0186301
\(647\) −23.7482 −0.933637 −0.466819 0.884353i \(-0.654600\pi\)
−0.466819 + 0.884353i \(0.654600\pi\)
\(648\) 0.836552 0.0328629
\(649\) 5.62107 0.220646
\(650\) −1.23690 −0.0485152
\(651\) 19.6913 0.771762
\(652\) −21.2611 −0.832650
\(653\) 21.3475 0.835391 0.417695 0.908587i \(-0.362838\pi\)
0.417695 + 0.908587i \(0.362838\pi\)
\(654\) 2.26419 0.0885370
\(655\) 17.0298 0.665411
\(656\) −20.7981 −0.812031
\(657\) −14.0198 −0.546964
\(658\) 0.0462910 0.00180461
\(659\) 7.69749 0.299852 0.149926 0.988697i \(-0.452097\pi\)
0.149926 + 0.988697i \(0.452097\pi\)
\(660\) −4.73739 −0.184403
\(661\) 43.2616 1.68268 0.841341 0.540505i \(-0.181767\pi\)
0.841341 + 0.540505i \(0.181767\pi\)
\(662\) 4.89434 0.190224
\(663\) −5.38537 −0.209150
\(664\) −3.29250 −0.127774
\(665\) 7.37767 0.286094
\(666\) −1.74622 −0.0676648
\(667\) −0.0982088 −0.00380266
\(668\) 19.7212 0.763037
\(669\) 12.6691 0.489814
\(670\) −2.14654 −0.0829279
\(671\) −1.94272 −0.0749979
\(672\) −7.47356 −0.288299
\(673\) −28.4016 −1.09480 −0.547400 0.836871i \(-0.684382\pi\)
−0.547400 + 0.836871i \(0.684382\pi\)
\(674\) 1.95474 0.0752939
\(675\) 1.00000 0.0384900
\(676\) −41.4528 −1.59434
\(677\) −27.8235 −1.06935 −0.534673 0.845059i \(-0.679565\pi\)
−0.534673 + 0.845059i \(0.679565\pi\)
\(678\) 1.44877 0.0556399
\(679\) 0.559938 0.0214885
\(680\) −0.770357 −0.0295418
\(681\) 15.5838 0.597172
\(682\) −3.32523 −0.127330
\(683\) −9.73555 −0.372521 −0.186260 0.982500i \(-0.559637\pi\)
−0.186260 + 0.982500i \(0.559637\pi\)
\(684\) −4.75359 −0.181758
\(685\) −18.1250 −0.692520
\(686\) 3.07510 0.117408
\(687\) 6.10570 0.232947
\(688\) 24.0713 0.917708
\(689\) −61.9806 −2.36127
\(690\) −0.0562568 −0.00214166
\(691\) −4.52388 −0.172096 −0.0860482 0.996291i \(-0.527424\pi\)
−0.0860482 + 0.996291i \(0.527424\pi\)
\(692\) −32.9425 −1.25228
\(693\) 7.35252 0.279299
\(694\) 1.07761 0.0409054
\(695\) −15.1610 −0.575091
\(696\) −0.308877 −0.0117080
\(697\) 5.12974 0.194303
\(698\) 6.07068 0.229779
\(699\) −7.81283 −0.295508
\(700\) −5.93347 −0.224264
\(701\) 46.2086 1.74527 0.872636 0.488370i \(-0.162408\pi\)
0.872636 + 0.488370i \(0.162408\pi\)
\(702\) −1.23690 −0.0466837
\(703\) 20.0724 0.757044
\(704\) −16.8301 −0.634309
\(705\) −0.0721234 −0.00271633
\(706\) −1.91499 −0.0720715
\(707\) −13.2226 −0.497288
\(708\) −4.53619 −0.170481
\(709\) −28.8628 −1.08396 −0.541982 0.840390i \(-0.682326\pi\)
−0.541982 + 0.840390i \(0.682326\pi\)
\(710\) −1.26678 −0.0475415
\(711\) −1.32234 −0.0495916
\(712\) −8.75430 −0.328081
\(713\) 1.72595 0.0646374
\(714\) 0.591043 0.0221192
\(715\) 14.1693 0.529904
\(716\) 44.8117 1.67469
\(717\) −18.3863 −0.686650
\(718\) −5.32895 −0.198875
\(719\) 21.4317 0.799267 0.399633 0.916675i \(-0.369137\pi\)
0.399633 + 0.916675i \(0.369137\pi\)
\(720\) 3.73360 0.139143
\(721\) −23.3360 −0.869076
\(722\) 2.76845 0.103031
\(723\) −17.4321 −0.648305
\(724\) 21.8569 0.812304
\(725\) −0.369226 −0.0137127
\(726\) 1.08493 0.0402656
\(727\) −25.2662 −0.937071 −0.468536 0.883445i \(-0.655218\pi\)
−0.468536 + 0.883445i \(0.655218\pi\)
\(728\) 14.8461 0.550234
\(729\) 1.00000 0.0370370
\(730\) 2.96523 0.109748
\(731\) −5.93704 −0.219589
\(732\) 1.56777 0.0579466
\(733\) −11.6821 −0.431489 −0.215745 0.976450i \(-0.569218\pi\)
−0.215745 + 0.976450i \(0.569218\pi\)
\(734\) −2.50691 −0.0925318
\(735\) 2.20886 0.0814750
\(736\) −0.655062 −0.0241459
\(737\) 24.5897 0.905775
\(738\) 1.17819 0.0433697
\(739\) −31.8904 −1.17311 −0.586554 0.809910i \(-0.699516\pi\)
−0.586554 + 0.809910i \(0.699516\pi\)
\(740\) −16.1432 −0.593434
\(741\) 14.2178 0.522304
\(742\) 6.80236 0.249722
\(743\) −40.3014 −1.47851 −0.739257 0.673424i \(-0.764823\pi\)
−0.739257 + 0.673424i \(0.764823\pi\)
\(744\) 5.42830 0.199011
\(745\) −1.85053 −0.0677982
\(746\) 0.285329 0.0104466
\(747\) −3.93580 −0.144003
\(748\) 4.36252 0.159510
\(749\) −0.679760 −0.0248379
\(750\) −0.211503 −0.00772301
\(751\) −27.7590 −1.01294 −0.506470 0.862258i \(-0.669050\pi\)
−0.506470 + 0.862258i \(0.669050\pi\)
\(752\) −0.269280 −0.00981963
\(753\) 19.7790 0.720786
\(754\) 0.456696 0.0166319
\(755\) 11.4334 0.416104
\(756\) −5.93347 −0.215798
\(757\) 26.2546 0.954240 0.477120 0.878838i \(-0.341681\pi\)
0.477120 + 0.878838i \(0.341681\pi\)
\(758\) −0.401862 −0.0145963
\(759\) 0.644452 0.0233921
\(760\) 2.03380 0.0737738
\(761\) −11.0992 −0.402347 −0.201173 0.979556i \(-0.564475\pi\)
−0.201173 + 0.979556i \(0.564475\pi\)
\(762\) 1.15896 0.0419848
\(763\) −32.4862 −1.17608
\(764\) 31.9148 1.15464
\(765\) −0.920871 −0.0332942
\(766\) −3.01887 −0.109076
\(767\) 13.5676 0.489897
\(768\) 12.5401 0.452503
\(769\) 10.0998 0.364209 0.182105 0.983279i \(-0.441709\pi\)
0.182105 + 0.983279i \(0.441709\pi\)
\(770\) −1.55508 −0.0560413
\(771\) 25.4638 0.917056
\(772\) −27.2209 −0.979701
\(773\) −26.6875 −0.959881 −0.479941 0.877301i \(-0.659342\pi\)
−0.479941 + 0.877301i \(0.659342\pi\)
\(774\) −1.36360 −0.0490138
\(775\) 6.48890 0.233088
\(776\) 0.154358 0.00554114
\(777\) 25.0545 0.898825
\(778\) −6.43446 −0.230687
\(779\) −13.5429 −0.485226
\(780\) −11.4346 −0.409426
\(781\) 14.5117 0.519268
\(782\) 0.0518053 0.00185255
\(783\) −0.369226 −0.0131951
\(784\) 8.24700 0.294536
\(785\) −3.26069 −0.116379
\(786\) −3.60187 −0.128474
\(787\) 43.5061 1.55083 0.775413 0.631454i \(-0.217542\pi\)
0.775413 + 0.631454i \(0.217542\pi\)
\(788\) −18.8125 −0.670168
\(789\) −4.94984 −0.176219
\(790\) 0.279679 0.00995054
\(791\) −20.7867 −0.739092
\(792\) 2.02687 0.0720217
\(793\) −4.68915 −0.166517
\(794\) −0.244445 −0.00867504
\(795\) −10.5984 −0.375885
\(796\) 41.1968 1.46018
\(797\) −50.3062 −1.78194 −0.890969 0.454064i \(-0.849974\pi\)
−0.890969 + 0.454064i \(0.849974\pi\)
\(798\) −1.56040 −0.0552376
\(799\) 0.0664164 0.00234964
\(800\) −2.46277 −0.0870722
\(801\) −10.4647 −0.369753
\(802\) −0.211503 −0.00746845
\(803\) −33.9683 −1.19872
\(804\) −19.8439 −0.699840
\(805\) 0.807162 0.0284487
\(806\) −8.02611 −0.282708
\(807\) 29.9145 1.05304
\(808\) −3.64509 −0.128234
\(809\) 1.54720 0.0543967 0.0271983 0.999630i \(-0.491341\pi\)
0.0271983 + 0.999630i \(0.491341\pi\)
\(810\) −0.211503 −0.00743147
\(811\) 24.6350 0.865052 0.432526 0.901622i \(-0.357622\pi\)
0.432526 + 0.901622i \(0.357622\pi\)
\(812\) 2.19079 0.0768818
\(813\) 16.8727 0.591751
\(814\) −4.23090 −0.148293
\(815\) 10.8738 0.380892
\(816\) −3.43816 −0.120360
\(817\) 15.6743 0.548373
\(818\) 3.53345 0.123544
\(819\) 17.7468 0.620123
\(820\) 10.8919 0.380361
\(821\) 2.59844 0.0906860 0.0453430 0.998971i \(-0.485562\pi\)
0.0453430 + 0.998971i \(0.485562\pi\)
\(822\) 3.83349 0.133708
\(823\) −41.2340 −1.43733 −0.718664 0.695358i \(-0.755246\pi\)
−0.718664 + 0.695358i \(0.755246\pi\)
\(824\) −6.43303 −0.224105
\(825\) 2.42289 0.0843541
\(826\) −1.48904 −0.0518103
\(827\) −20.5095 −0.713186 −0.356593 0.934260i \(-0.616062\pi\)
−0.356593 + 0.934260i \(0.616062\pi\)
\(828\) −0.520072 −0.0180738
\(829\) −5.45389 −0.189421 −0.0947107 0.995505i \(-0.530193\pi\)
−0.0947107 + 0.995505i \(0.530193\pi\)
\(830\) 0.832435 0.0288942
\(831\) 29.6902 1.02994
\(832\) −40.6229 −1.40835
\(833\) −2.03408 −0.0704765
\(834\) 3.20661 0.111036
\(835\) −10.0862 −0.349048
\(836\) −11.5174 −0.398338
\(837\) 6.48890 0.224289
\(838\) −5.14097 −0.177592
\(839\) 20.3373 0.702123 0.351061 0.936352i \(-0.385821\pi\)
0.351061 + 0.936352i \(0.385821\pi\)
\(840\) 2.53861 0.0875904
\(841\) −28.8637 −0.995299
\(842\) −3.69258 −0.127255
\(843\) −9.05519 −0.311877
\(844\) −19.8625 −0.683696
\(845\) 21.2006 0.729322
\(846\) 0.0152543 0.000524455 0
\(847\) −15.5664 −0.534868
\(848\) −39.5701 −1.35884
\(849\) −2.42399 −0.0831912
\(850\) 0.194767 0.00668046
\(851\) 2.19604 0.0752793
\(852\) −11.7109 −0.401209
\(853\) −9.03803 −0.309456 −0.154728 0.987957i \(-0.549450\pi\)
−0.154728 + 0.987957i \(0.549450\pi\)
\(854\) 0.514633 0.0176104
\(855\) 2.43117 0.0831444
\(856\) −0.187390 −0.00640485
\(857\) 1.87095 0.0639105 0.0319553 0.999489i \(-0.489827\pi\)
0.0319553 + 0.999489i \(0.489827\pi\)
\(858\) −2.99687 −0.102311
\(859\) 9.11687 0.311064 0.155532 0.987831i \(-0.450291\pi\)
0.155532 + 0.987831i \(0.450291\pi\)
\(860\) −12.6060 −0.429861
\(861\) −16.9044 −0.576100
\(862\) −4.69329 −0.159854
\(863\) −45.5201 −1.54952 −0.774761 0.632255i \(-0.782130\pi\)
−0.774761 + 0.632255i \(0.782130\pi\)
\(864\) −2.46277 −0.0837853
\(865\) 16.8481 0.572852
\(866\) −3.04821 −0.103583
\(867\) −16.1520 −0.548551
\(868\) −38.5017 −1.30683
\(869\) −3.20388 −0.108684
\(870\) 0.0780926 0.00264759
\(871\) 59.3523 2.01108
\(872\) −8.95549 −0.303271
\(873\) 0.184517 0.00624497
\(874\) −0.136770 −0.00462632
\(875\) 3.03461 0.102589
\(876\) 27.4124 0.926180
\(877\) 12.9307 0.436640 0.218320 0.975877i \(-0.429942\pi\)
0.218320 + 0.975877i \(0.429942\pi\)
\(878\) 7.75099 0.261583
\(879\) 11.7098 0.394961
\(880\) 9.04609 0.304943
\(881\) −39.7481 −1.33915 −0.669574 0.742745i \(-0.733523\pi\)
−0.669574 + 0.742745i \(0.733523\pi\)
\(882\) −0.467182 −0.0157308
\(883\) 20.4010 0.686548 0.343274 0.939235i \(-0.388464\pi\)
0.343274 + 0.939235i \(0.388464\pi\)
\(884\) 10.5298 0.354157
\(885\) 2.31999 0.0779855
\(886\) 2.83511 0.0952475
\(887\) −21.7993 −0.731948 −0.365974 0.930625i \(-0.619264\pi\)
−0.365974 + 0.930625i \(0.619264\pi\)
\(888\) 6.90678 0.231776
\(889\) −16.6286 −0.557704
\(890\) 2.21333 0.0741909
\(891\) 2.42289 0.0811697
\(892\) −24.7714 −0.829407
\(893\) −0.175345 −0.00586768
\(894\) 0.391393 0.0130902
\(895\) −22.9185 −0.766081
\(896\) 19.4055 0.648291
\(897\) 1.55552 0.0519372
\(898\) 6.80811 0.227190
\(899\) −2.39587 −0.0799068
\(900\) −1.95527 −0.0651755
\(901\) 9.75973 0.325144
\(902\) 2.85461 0.0950482
\(903\) 19.5647 0.651074
\(904\) −5.73029 −0.190587
\(905\) −11.1785 −0.371584
\(906\) −2.41820 −0.0803394
\(907\) −13.0315 −0.432702 −0.216351 0.976316i \(-0.569416\pi\)
−0.216351 + 0.976316i \(0.569416\pi\)
\(908\) −30.4704 −1.01120
\(909\) −4.35727 −0.144522
\(910\) −3.75351 −0.124428
\(911\) −7.43704 −0.246400 −0.123200 0.992382i \(-0.539316\pi\)
−0.123200 + 0.992382i \(0.539316\pi\)
\(912\) 9.07703 0.300570
\(913\) −9.53599 −0.315595
\(914\) −0.816752 −0.0270157
\(915\) −0.801821 −0.0265074
\(916\) −11.9383 −0.394452
\(917\) 51.6789 1.70659
\(918\) 0.194767 0.00642828
\(919\) 27.3370 0.901765 0.450882 0.892583i \(-0.351109\pi\)
0.450882 + 0.892583i \(0.351109\pi\)
\(920\) 0.222511 0.00733596
\(921\) −4.24442 −0.139858
\(922\) −0.0979233 −0.00322493
\(923\) 35.0269 1.15292
\(924\) −14.3761 −0.472940
\(925\) 8.25624 0.271464
\(926\) −6.48732 −0.213187
\(927\) −7.68993 −0.252571
\(928\) 0.909321 0.0298499
\(929\) 35.0498 1.14995 0.574973 0.818172i \(-0.305013\pi\)
0.574973 + 0.818172i \(0.305013\pi\)
\(930\) −1.37242 −0.0450035
\(931\) 5.37012 0.175999
\(932\) 15.2762 0.500387
\(933\) −12.3533 −0.404430
\(934\) 7.51017 0.245740
\(935\) −2.23117 −0.0729669
\(936\) 4.89226 0.159909
\(937\) 34.9984 1.14335 0.571674 0.820481i \(-0.306294\pi\)
0.571674 + 0.820481i \(0.306294\pi\)
\(938\) −6.51390 −0.212686
\(939\) 7.57303 0.247136
\(940\) 0.141020 0.00459958
\(941\) 55.7197 1.81641 0.908205 0.418526i \(-0.137453\pi\)
0.908205 + 0.418526i \(0.137453\pi\)
\(942\) 0.689648 0.0224699
\(943\) −1.48168 −0.0482502
\(944\) 8.66190 0.281921
\(945\) 3.03461 0.0987159
\(946\) −3.30386 −0.107418
\(947\) 5.51743 0.179292 0.0896461 0.995974i \(-0.471426\pi\)
0.0896461 + 0.995974i \(0.471426\pi\)
\(948\) 2.58553 0.0839740
\(949\) −81.9895 −2.66149
\(950\) −0.514202 −0.0166829
\(951\) −8.60857 −0.279152
\(952\) −2.33773 −0.0757663
\(953\) −55.7412 −1.80563 −0.902817 0.430026i \(-0.858504\pi\)
−0.902817 + 0.430026i \(0.858504\pi\)
\(954\) 2.24159 0.0725742
\(955\) −16.3225 −0.528183
\(956\) 35.9502 1.16271
\(957\) −0.894593 −0.0289181
\(958\) −3.93332 −0.127080
\(959\) −55.0022 −1.77611
\(960\) −6.94631 −0.224191
\(961\) 11.1058 0.358251
\(962\) −10.2121 −0.329253
\(963\) −0.224002 −0.00721838
\(964\) 34.0843 1.09778
\(965\) 13.9218 0.448159
\(966\) −0.170718 −0.00549275
\(967\) 22.3789 0.719658 0.359829 0.933018i \(-0.382835\pi\)
0.359829 + 0.933018i \(0.382835\pi\)
\(968\) −4.29120 −0.137924
\(969\) −2.23880 −0.0719205
\(970\) −0.0390260 −0.00125305
\(971\) 46.0083 1.47648 0.738239 0.674539i \(-0.235658\pi\)
0.738239 + 0.674539i \(0.235658\pi\)
\(972\) −1.95527 −0.0627152
\(973\) −46.0078 −1.47494
\(974\) −3.99610 −0.128043
\(975\) 5.84813 0.187290
\(976\) −2.99368 −0.0958253
\(977\) 0.229801 0.00735199 0.00367600 0.999993i \(-0.498830\pi\)
0.00367600 + 0.999993i \(0.498830\pi\)
\(978\) −2.29984 −0.0735408
\(979\) −25.3549 −0.810345
\(980\) −4.31891 −0.137963
\(981\) −10.7052 −0.341792
\(982\) −3.49829 −0.111635
\(983\) −19.6189 −0.625747 −0.312873 0.949795i \(-0.601292\pi\)
−0.312873 + 0.949795i \(0.601292\pi\)
\(984\) −4.66004 −0.148557
\(985\) 9.62146 0.306565
\(986\) −0.0719132 −0.00229018
\(987\) −0.218866 −0.00696660
\(988\) −27.7996 −0.884424
\(989\) 1.71486 0.0545294
\(990\) −0.512449 −0.0162867
\(991\) 25.8041 0.819695 0.409848 0.912154i \(-0.365582\pi\)
0.409848 + 0.912154i \(0.365582\pi\)
\(992\) −15.9807 −0.507387
\(993\) −23.1407 −0.734349
\(994\) −3.84419 −0.121930
\(995\) −21.0696 −0.667952
\(996\) 7.69553 0.243842
\(997\) −6.05796 −0.191857 −0.0959287 0.995388i \(-0.530582\pi\)
−0.0959287 + 0.995388i \(0.530582\pi\)
\(998\) −7.50805 −0.237663
\(999\) 8.25624 0.261216
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 6015.2.a.i.1.20 43
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6015.2.a.i.1.20 43 1.1 even 1 trivial