Properties

Label 538.2.a.b.1.2
Level $538$
Weight $2$
Character 538.1
Self dual yes
Analytic conductor $4.296$
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] = [538,2,Mod(1,538)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(538, 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("538.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 538 = 2 \cdot 269 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 538.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: \(4.29595162874\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 538.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.30278 q^{3} +1.00000 q^{4} -1.30278 q^{5} +2.30278 q^{6} -1.00000 q^{7} +1.00000 q^{8} +2.30278 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.30278 q^{3} +1.00000 q^{4} -1.30278 q^{5} +2.30278 q^{6} -1.00000 q^{7} +1.00000 q^{8} +2.30278 q^{9} -1.30278 q^{10} +3.00000 q^{11} +2.30278 q^{12} +5.00000 q^{13} -1.00000 q^{14} -3.00000 q^{15} +1.00000 q^{16} -5.60555 q^{17} +2.30278 q^{18} +2.00000 q^{19} -1.30278 q^{20} -2.30278 q^{21} +3.00000 q^{22} +8.21110 q^{23} +2.30278 q^{24} -3.30278 q^{25} +5.00000 q^{26} -1.60555 q^{27} -1.00000 q^{28} +4.69722 q^{29} -3.00000 q^{30} -9.60555 q^{31} +1.00000 q^{32} +6.90833 q^{33} -5.60555 q^{34} +1.30278 q^{35} +2.30278 q^{36} -3.60555 q^{37} +2.00000 q^{38} +11.5139 q^{39} -1.30278 q^{40} -8.60555 q^{41} -2.30278 q^{42} -12.2111 q^{43} +3.00000 q^{44} -3.00000 q^{45} +8.21110 q^{46} -9.90833 q^{47} +2.30278 q^{48} -6.00000 q^{49} -3.30278 q^{50} -12.9083 q^{51} +5.00000 q^{52} -1.60555 q^{54} -3.90833 q^{55} -1.00000 q^{56} +4.60555 q^{57} +4.69722 q^{58} +6.51388 q^{59} -3.00000 q^{60} -1.00000 q^{61} -9.60555 q^{62} -2.30278 q^{63} +1.00000 q^{64} -6.51388 q^{65} +6.90833 q^{66} +16.2111 q^{67} -5.60555 q^{68} +18.9083 q^{69} +1.30278 q^{70} +7.81665 q^{71} +2.30278 q^{72} -3.60555 q^{73} -3.60555 q^{74} -7.60555 q^{75} +2.00000 q^{76} -3.00000 q^{77} +11.5139 q^{78} -2.69722 q^{79} -1.30278 q^{80} -10.6056 q^{81} -8.60555 q^{82} +8.60555 q^{83} -2.30278 q^{84} +7.30278 q^{85} -12.2111 q^{86} +10.8167 q^{87} +3.00000 q^{88} -7.30278 q^{89} -3.00000 q^{90} -5.00000 q^{91} +8.21110 q^{92} -22.1194 q^{93} -9.90833 q^{94} -2.60555 q^{95} +2.30278 q^{96} +4.60555 q^{97} -6.00000 q^{98} +6.90833 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + q^{3} + 2 q^{4} + q^{5} + q^{6} - 2 q^{7} + 2 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + q^{3} + 2 q^{4} + q^{5} + q^{6} - 2 q^{7} + 2 q^{8} + q^{9} + q^{10} + 6 q^{11} + q^{12} + 10 q^{13} - 2 q^{14} - 6 q^{15} + 2 q^{16} - 4 q^{17} + q^{18} + 4 q^{19} + q^{20} - q^{21} + 6 q^{22} + 2 q^{23} + q^{24} - 3 q^{25} + 10 q^{26} + 4 q^{27} - 2 q^{28} + 13 q^{29} - 6 q^{30} - 12 q^{31} + 2 q^{32} + 3 q^{33} - 4 q^{34} - q^{35} + q^{36} + 4 q^{38} + 5 q^{39} + q^{40} - 10 q^{41} - q^{42} - 10 q^{43} + 6 q^{44} - 6 q^{45} + 2 q^{46} - 9 q^{47} + q^{48} - 12 q^{49} - 3 q^{50} - 15 q^{51} + 10 q^{52} + 4 q^{54} + 3 q^{55} - 2 q^{56} + 2 q^{57} + 13 q^{58} - 5 q^{59} - 6 q^{60} - 2 q^{61} - 12 q^{62} - q^{63} + 2 q^{64} + 5 q^{65} + 3 q^{66} + 18 q^{67} - 4 q^{68} + 27 q^{69} - q^{70} - 6 q^{71} + q^{72} - 8 q^{75} + 4 q^{76} - 6 q^{77} + 5 q^{78} - 9 q^{79} + q^{80} - 14 q^{81} - 10 q^{82} + 10 q^{83} - q^{84} + 11 q^{85} - 10 q^{86} + 6 q^{88} - 11 q^{89} - 6 q^{90} - 10 q^{91} + 2 q^{92} - 19 q^{93} - 9 q^{94} + 2 q^{95} + q^{96} + 2 q^{97} - 12 q^{98} + 3 q^{99}+O(q^{100}) \) 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
\(3\) 2.30278 1.32951 0.664754 0.747062i \(-0.268536\pi\)
0.664754 + 0.747062i \(0.268536\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.30278 −0.582619 −0.291309 0.956629i \(-0.594091\pi\)
−0.291309 + 0.956629i \(0.594091\pi\)
\(6\) 2.30278 0.940104
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 1.00000 0.353553
\(9\) 2.30278 0.767592
\(10\) −1.30278 −0.411974
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 2.30278 0.664754
\(13\) 5.00000 1.38675 0.693375 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) −1.00000 −0.267261
\(15\) −3.00000 −0.774597
\(16\) 1.00000 0.250000
\(17\) −5.60555 −1.35955 −0.679773 0.733423i \(-0.737922\pi\)
−0.679773 + 0.733423i \(0.737922\pi\)
\(18\) 2.30278 0.542769
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) −1.30278 −0.291309
\(21\) −2.30278 −0.502507
\(22\) 3.00000 0.639602
\(23\) 8.21110 1.71213 0.856067 0.516865i \(-0.172901\pi\)
0.856067 + 0.516865i \(0.172901\pi\)
\(24\) 2.30278 0.470052
\(25\) −3.30278 −0.660555
\(26\) 5.00000 0.980581
\(27\) −1.60555 −0.308988
\(28\) −1.00000 −0.188982
\(29\) 4.69722 0.872253 0.436126 0.899885i \(-0.356350\pi\)
0.436126 + 0.899885i \(0.356350\pi\)
\(30\) −3.00000 −0.547723
\(31\) −9.60555 −1.72521 −0.862604 0.505880i \(-0.831168\pi\)
−0.862604 + 0.505880i \(0.831168\pi\)
\(32\) 1.00000 0.176777
\(33\) 6.90833 1.20259
\(34\) −5.60555 −0.961344
\(35\) 1.30278 0.220209
\(36\) 2.30278 0.383796
\(37\) −3.60555 −0.592749 −0.296374 0.955072i \(-0.595778\pi\)
−0.296374 + 0.955072i \(0.595778\pi\)
\(38\) 2.00000 0.324443
\(39\) 11.5139 1.84370
\(40\) −1.30278 −0.205987
\(41\) −8.60555 −1.34396 −0.671981 0.740569i \(-0.734556\pi\)
−0.671981 + 0.740569i \(0.734556\pi\)
\(42\) −2.30278 −0.355326
\(43\) −12.2111 −1.86218 −0.931088 0.364795i \(-0.881139\pi\)
−0.931088 + 0.364795i \(0.881139\pi\)
\(44\) 3.00000 0.452267
\(45\) −3.00000 −0.447214
\(46\) 8.21110 1.21066
\(47\) −9.90833 −1.44528 −0.722639 0.691226i \(-0.757071\pi\)
−0.722639 + 0.691226i \(0.757071\pi\)
\(48\) 2.30278 0.332377
\(49\) −6.00000 −0.857143
\(50\) −3.30278 −0.467083
\(51\) −12.9083 −1.80753
\(52\) 5.00000 0.693375
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) −1.60555 −0.218488
\(55\) −3.90833 −0.526999
\(56\) −1.00000 −0.133631
\(57\) 4.60555 0.610020
\(58\) 4.69722 0.616776
\(59\) 6.51388 0.848035 0.424017 0.905654i \(-0.360620\pi\)
0.424017 + 0.905654i \(0.360620\pi\)
\(60\) −3.00000 −0.387298
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) −9.60555 −1.21991
\(63\) −2.30278 −0.290122
\(64\) 1.00000 0.125000
\(65\) −6.51388 −0.807947
\(66\) 6.90833 0.850356
\(67\) 16.2111 1.98050 0.990251 0.139297i \(-0.0444842\pi\)
0.990251 + 0.139297i \(0.0444842\pi\)
\(68\) −5.60555 −0.679773
\(69\) 18.9083 2.27630
\(70\) 1.30278 0.155711
\(71\) 7.81665 0.927666 0.463833 0.885923i \(-0.346474\pi\)
0.463833 + 0.885923i \(0.346474\pi\)
\(72\) 2.30278 0.271385
\(73\) −3.60555 −0.421998 −0.210999 0.977486i \(-0.567672\pi\)
−0.210999 + 0.977486i \(0.567672\pi\)
\(74\) −3.60555 −0.419137
\(75\) −7.60555 −0.878213
\(76\) 2.00000 0.229416
\(77\) −3.00000 −0.341882
\(78\) 11.5139 1.30369
\(79\) −2.69722 −0.303461 −0.151731 0.988422i \(-0.548485\pi\)
−0.151731 + 0.988422i \(0.548485\pi\)
\(80\) −1.30278 −0.145655
\(81\) −10.6056 −1.17839
\(82\) −8.60555 −0.950324
\(83\) 8.60555 0.944582 0.472291 0.881443i \(-0.343427\pi\)
0.472291 + 0.881443i \(0.343427\pi\)
\(84\) −2.30278 −0.251253
\(85\) 7.30278 0.792097
\(86\) −12.2111 −1.31676
\(87\) 10.8167 1.15967
\(88\) 3.00000 0.319801
\(89\) −7.30278 −0.774093 −0.387046 0.922060i \(-0.626505\pi\)
−0.387046 + 0.922060i \(0.626505\pi\)
\(90\) −3.00000 −0.316228
\(91\) −5.00000 −0.524142
\(92\) 8.21110 0.856067
\(93\) −22.1194 −2.29368
\(94\) −9.90833 −1.02197
\(95\) −2.60555 −0.267324
\(96\) 2.30278 0.235026
\(97\) 4.60555 0.467623 0.233811 0.972282i \(-0.424880\pi\)
0.233811 + 0.972282i \(0.424880\pi\)
\(98\) −6.00000 −0.606092
\(99\) 6.90833 0.694313
\(100\) −3.30278 −0.330278
\(101\) 6.90833 0.687404 0.343702 0.939079i \(-0.388319\pi\)
0.343702 + 0.939079i \(0.388319\pi\)
\(102\) −12.9083 −1.27811
\(103\) 11.1194 1.09563 0.547815 0.836600i \(-0.315460\pi\)
0.547815 + 0.836600i \(0.315460\pi\)
\(104\) 5.00000 0.490290
\(105\) 3.00000 0.292770
\(106\) 0 0
\(107\) 2.21110 0.213755 0.106878 0.994272i \(-0.465915\pi\)
0.106878 + 0.994272i \(0.465915\pi\)
\(108\) −1.60555 −0.154494
\(109\) 18.8167 1.80231 0.901154 0.433499i \(-0.142721\pi\)
0.901154 + 0.433499i \(0.142721\pi\)
\(110\) −3.90833 −0.372644
\(111\) −8.30278 −0.788065
\(112\) −1.00000 −0.0944911
\(113\) 0.908327 0.0854482 0.0427241 0.999087i \(-0.486396\pi\)
0.0427241 + 0.999087i \(0.486396\pi\)
\(114\) 4.60555 0.431349
\(115\) −10.6972 −0.997521
\(116\) 4.69722 0.436126
\(117\) 11.5139 1.06446
\(118\) 6.51388 0.599651
\(119\) 5.60555 0.513860
\(120\) −3.00000 −0.273861
\(121\) −2.00000 −0.181818
\(122\) −1.00000 −0.0905357
\(123\) −19.8167 −1.78681
\(124\) −9.60555 −0.862604
\(125\) 10.8167 0.967471
\(126\) −2.30278 −0.205148
\(127\) 0.302776 0.0268670 0.0134335 0.999910i \(-0.495724\pi\)
0.0134335 + 0.999910i \(0.495724\pi\)
\(128\) 1.00000 0.0883883
\(129\) −28.1194 −2.47578
\(130\) −6.51388 −0.571305
\(131\) 20.6056 1.80032 0.900158 0.435564i \(-0.143451\pi\)
0.900158 + 0.435564i \(0.143451\pi\)
\(132\) 6.90833 0.601293
\(133\) −2.00000 −0.173422
\(134\) 16.2111 1.40043
\(135\) 2.09167 0.180023
\(136\) −5.60555 −0.480672
\(137\) 8.09167 0.691318 0.345659 0.938360i \(-0.387655\pi\)
0.345659 + 0.938360i \(0.387655\pi\)
\(138\) 18.9083 1.60958
\(139\) −22.5139 −1.90960 −0.954801 0.297245i \(-0.903932\pi\)
−0.954801 + 0.297245i \(0.903932\pi\)
\(140\) 1.30278 0.110105
\(141\) −22.8167 −1.92151
\(142\) 7.81665 0.655959
\(143\) 15.0000 1.25436
\(144\) 2.30278 0.191898
\(145\) −6.11943 −0.508191
\(146\) −3.60555 −0.298398
\(147\) −13.8167 −1.13958
\(148\) −3.60555 −0.296374
\(149\) 7.69722 0.630581 0.315291 0.948995i \(-0.397898\pi\)
0.315291 + 0.948995i \(0.397898\pi\)
\(150\) −7.60555 −0.620991
\(151\) −15.6056 −1.26996 −0.634981 0.772528i \(-0.718992\pi\)
−0.634981 + 0.772528i \(0.718992\pi\)
\(152\) 2.00000 0.162221
\(153\) −12.9083 −1.04358
\(154\) −3.00000 −0.241747
\(155\) 12.5139 1.00514
\(156\) 11.5139 0.921848
\(157\) 10.6056 0.846415 0.423208 0.906033i \(-0.360904\pi\)
0.423208 + 0.906033i \(0.360904\pi\)
\(158\) −2.69722 −0.214580
\(159\) 0 0
\(160\) −1.30278 −0.102993
\(161\) −8.21110 −0.647126
\(162\) −10.6056 −0.833251
\(163\) −8.69722 −0.681219 −0.340610 0.940205i \(-0.610633\pi\)
−0.340610 + 0.940205i \(0.610633\pi\)
\(164\) −8.60555 −0.671981
\(165\) −9.00000 −0.700649
\(166\) 8.60555 0.667920
\(167\) −20.3305 −1.57322 −0.786612 0.617448i \(-0.788167\pi\)
−0.786612 + 0.617448i \(0.788167\pi\)
\(168\) −2.30278 −0.177663
\(169\) 12.0000 0.923077
\(170\) 7.30278 0.560097
\(171\) 4.60555 0.352195
\(172\) −12.2111 −0.931088
\(173\) 7.42221 0.564300 0.282150 0.959370i \(-0.408952\pi\)
0.282150 + 0.959370i \(0.408952\pi\)
\(174\) 10.8167 0.820008
\(175\) 3.30278 0.249666
\(176\) 3.00000 0.226134
\(177\) 15.0000 1.12747
\(178\) −7.30278 −0.547366
\(179\) −24.6333 −1.84118 −0.920590 0.390531i \(-0.872292\pi\)
−0.920590 + 0.390531i \(0.872292\pi\)
\(180\) −3.00000 −0.223607
\(181\) 13.2111 0.981974 0.490987 0.871167i \(-0.336636\pi\)
0.490987 + 0.871167i \(0.336636\pi\)
\(182\) −5.00000 −0.370625
\(183\) −2.30278 −0.170226
\(184\) 8.21110 0.605331
\(185\) 4.69722 0.345347
\(186\) −22.1194 −1.62188
\(187\) −16.8167 −1.22976
\(188\) −9.90833 −0.722639
\(189\) 1.60555 0.116787
\(190\) −2.60555 −0.189027
\(191\) 0.788897 0.0570826 0.0285413 0.999593i \(-0.490914\pi\)
0.0285413 + 0.999593i \(0.490914\pi\)
\(192\) 2.30278 0.166189
\(193\) −14.0278 −1.00974 −0.504870 0.863195i \(-0.668460\pi\)
−0.504870 + 0.863195i \(0.668460\pi\)
\(194\) 4.60555 0.330659
\(195\) −15.0000 −1.07417
\(196\) −6.00000 −0.428571
\(197\) 14.0917 1.00399 0.501995 0.864870i \(-0.332600\pi\)
0.501995 + 0.864870i \(0.332600\pi\)
\(198\) 6.90833 0.490953
\(199\) −3.09167 −0.219163 −0.109581 0.993978i \(-0.534951\pi\)
−0.109581 + 0.993978i \(0.534951\pi\)
\(200\) −3.30278 −0.233542
\(201\) 37.3305 2.63309
\(202\) 6.90833 0.486068
\(203\) −4.69722 −0.329681
\(204\) −12.9083 −0.903764
\(205\) 11.2111 0.783017
\(206\) 11.1194 0.774727
\(207\) 18.9083 1.31422
\(208\) 5.00000 0.346688
\(209\) 6.00000 0.415029
\(210\) 3.00000 0.207020
\(211\) −1.11943 −0.0770647 −0.0385324 0.999257i \(-0.512268\pi\)
−0.0385324 + 0.999257i \(0.512268\pi\)
\(212\) 0 0
\(213\) 18.0000 1.23334
\(214\) 2.21110 0.151148
\(215\) 15.9083 1.08494
\(216\) −1.60555 −0.109244
\(217\) 9.60555 0.652067
\(218\) 18.8167 1.27442
\(219\) −8.30278 −0.561050
\(220\) −3.90833 −0.263499
\(221\) −28.0278 −1.88535
\(222\) −8.30278 −0.557246
\(223\) −13.9083 −0.931370 −0.465685 0.884950i \(-0.654192\pi\)
−0.465685 + 0.884950i \(0.654192\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −7.60555 −0.507037
\(226\) 0.908327 0.0604210
\(227\) −8.72498 −0.579097 −0.289549 0.957163i \(-0.593505\pi\)
−0.289549 + 0.957163i \(0.593505\pi\)
\(228\) 4.60555 0.305010
\(229\) 6.02776 0.398325 0.199163 0.979966i \(-0.436178\pi\)
0.199163 + 0.979966i \(0.436178\pi\)
\(230\) −10.6972 −0.705354
\(231\) −6.90833 −0.454535
\(232\) 4.69722 0.308388
\(233\) 16.8167 1.10170 0.550848 0.834606i \(-0.314304\pi\)
0.550848 + 0.834606i \(0.314304\pi\)
\(234\) 11.5139 0.752686
\(235\) 12.9083 0.842046
\(236\) 6.51388 0.424017
\(237\) −6.21110 −0.403454
\(238\) 5.60555 0.363354
\(239\) 0.908327 0.0587548 0.0293774 0.999568i \(-0.490648\pi\)
0.0293774 + 0.999568i \(0.490648\pi\)
\(240\) −3.00000 −0.193649
\(241\) 18.0278 1.16127 0.580635 0.814164i \(-0.302804\pi\)
0.580635 + 0.814164i \(0.302804\pi\)
\(242\) −2.00000 −0.128565
\(243\) −19.6056 −1.25770
\(244\) −1.00000 −0.0640184
\(245\) 7.81665 0.499388
\(246\) −19.8167 −1.26346
\(247\) 10.0000 0.636285
\(248\) −9.60555 −0.609953
\(249\) 19.8167 1.25583
\(250\) 10.8167 0.684105
\(251\) −11.7250 −0.740074 −0.370037 0.929017i \(-0.620655\pi\)
−0.370037 + 0.929017i \(0.620655\pi\)
\(252\) −2.30278 −0.145061
\(253\) 24.6333 1.54868
\(254\) 0.302776 0.0189978
\(255\) 16.8167 1.05310
\(256\) 1.00000 0.0625000
\(257\) −3.00000 −0.187135 −0.0935674 0.995613i \(-0.529827\pi\)
−0.0935674 + 0.995613i \(0.529827\pi\)
\(258\) −28.1194 −1.75064
\(259\) 3.60555 0.224038
\(260\) −6.51388 −0.403974
\(261\) 10.8167 0.669534
\(262\) 20.6056 1.27302
\(263\) 21.5139 1.32660 0.663301 0.748352i \(-0.269155\pi\)
0.663301 + 0.748352i \(0.269155\pi\)
\(264\) 6.90833 0.425178
\(265\) 0 0
\(266\) −2.00000 −0.122628
\(267\) −16.8167 −1.02916
\(268\) 16.2111 0.990251
\(269\) −1.00000 −0.0609711
\(270\) 2.09167 0.127295
\(271\) 11.0000 0.668202 0.334101 0.942537i \(-0.391567\pi\)
0.334101 + 0.942537i \(0.391567\pi\)
\(272\) −5.60555 −0.339886
\(273\) −11.5139 −0.696852
\(274\) 8.09167 0.488836
\(275\) −9.90833 −0.597495
\(276\) 18.9083 1.13815
\(277\) −16.1194 −0.968523 −0.484261 0.874923i \(-0.660912\pi\)
−0.484261 + 0.874923i \(0.660912\pi\)
\(278\) −22.5139 −1.35029
\(279\) −22.1194 −1.32426
\(280\) 1.30278 0.0778557
\(281\) 10.8167 0.645267 0.322634 0.946524i \(-0.395432\pi\)
0.322634 + 0.946524i \(0.395432\pi\)
\(282\) −22.8167 −1.35871
\(283\) −13.0000 −0.772770 −0.386385 0.922338i \(-0.626276\pi\)
−0.386385 + 0.922338i \(0.626276\pi\)
\(284\) 7.81665 0.463833
\(285\) −6.00000 −0.355409
\(286\) 15.0000 0.886969
\(287\) 8.60555 0.507970
\(288\) 2.30278 0.135692
\(289\) 14.4222 0.848365
\(290\) −6.11943 −0.359345
\(291\) 10.6056 0.621708
\(292\) −3.60555 −0.210999
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) −13.8167 −0.805804
\(295\) −8.48612 −0.494081
\(296\) −3.60555 −0.209568
\(297\) −4.81665 −0.279491
\(298\) 7.69722 0.445888
\(299\) 41.0555 2.37430
\(300\) −7.60555 −0.439107
\(301\) 12.2111 0.703836
\(302\) −15.6056 −0.897999
\(303\) 15.9083 0.913910
\(304\) 2.00000 0.114708
\(305\) 1.30278 0.0745967
\(306\) −12.9083 −0.737920
\(307\) 15.4222 0.880192 0.440096 0.897951i \(-0.354944\pi\)
0.440096 + 0.897951i \(0.354944\pi\)
\(308\) −3.00000 −0.170941
\(309\) 25.6056 1.45665
\(310\) 12.5139 0.710741
\(311\) 13.8167 0.783471 0.391735 0.920078i \(-0.371875\pi\)
0.391735 + 0.920078i \(0.371875\pi\)
\(312\) 11.5139 0.651845
\(313\) −17.3028 −0.978011 −0.489005 0.872281i \(-0.662640\pi\)
−0.489005 + 0.872281i \(0.662640\pi\)
\(314\) 10.6056 0.598506
\(315\) 3.00000 0.169031
\(316\) −2.69722 −0.151731
\(317\) 24.3944 1.37013 0.685064 0.728483i \(-0.259774\pi\)
0.685064 + 0.728483i \(0.259774\pi\)
\(318\) 0 0
\(319\) 14.0917 0.788982
\(320\) −1.30278 −0.0728274
\(321\) 5.09167 0.284189
\(322\) −8.21110 −0.457587
\(323\) −11.2111 −0.623802
\(324\) −10.6056 −0.589197
\(325\) −16.5139 −0.916025
\(326\) −8.69722 −0.481695
\(327\) 43.3305 2.39618
\(328\) −8.60555 −0.475162
\(329\) 9.90833 0.546264
\(330\) −9.00000 −0.495434
\(331\) −1.00000 −0.0549650 −0.0274825 0.999622i \(-0.508749\pi\)
−0.0274825 + 0.999622i \(0.508749\pi\)
\(332\) 8.60555 0.472291
\(333\) −8.30278 −0.454989
\(334\) −20.3305 −1.11244
\(335\) −21.1194 −1.15388
\(336\) −2.30278 −0.125627
\(337\) −23.5416 −1.28239 −0.641197 0.767376i \(-0.721562\pi\)
−0.641197 + 0.767376i \(0.721562\pi\)
\(338\) 12.0000 0.652714
\(339\) 2.09167 0.113604
\(340\) 7.30278 0.396049
\(341\) −28.8167 −1.56051
\(342\) 4.60555 0.249040
\(343\) 13.0000 0.701934
\(344\) −12.2111 −0.658379
\(345\) −24.6333 −1.32621
\(346\) 7.42221 0.399020
\(347\) 7.18335 0.385622 0.192811 0.981236i \(-0.438240\pi\)
0.192811 + 0.981236i \(0.438240\pi\)
\(348\) 10.8167 0.579834
\(349\) −8.30278 −0.444437 −0.222219 0.974997i \(-0.571330\pi\)
−0.222219 + 0.974997i \(0.571330\pi\)
\(350\) 3.30278 0.176541
\(351\) −8.02776 −0.428490
\(352\) 3.00000 0.159901
\(353\) 11.3305 0.603063 0.301532 0.953456i \(-0.402502\pi\)
0.301532 + 0.953456i \(0.402502\pi\)
\(354\) 15.0000 0.797241
\(355\) −10.1833 −0.540476
\(356\) −7.30278 −0.387046
\(357\) 12.9083 0.683181
\(358\) −24.6333 −1.30191
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) −3.00000 −0.158114
\(361\) −15.0000 −0.789474
\(362\) 13.2111 0.694360
\(363\) −4.60555 −0.241729
\(364\) −5.00000 −0.262071
\(365\) 4.69722 0.245864
\(366\) −2.30278 −0.120368
\(367\) −22.9083 −1.19580 −0.597902 0.801569i \(-0.703999\pi\)
−0.597902 + 0.801569i \(0.703999\pi\)
\(368\) 8.21110 0.428033
\(369\) −19.8167 −1.03161
\(370\) 4.69722 0.244197
\(371\) 0 0
\(372\) −22.1194 −1.14684
\(373\) 20.7889 1.07641 0.538204 0.842815i \(-0.319103\pi\)
0.538204 + 0.842815i \(0.319103\pi\)
\(374\) −16.8167 −0.869568
\(375\) 24.9083 1.28626
\(376\) −9.90833 −0.510983
\(377\) 23.4861 1.20960
\(378\) 1.60555 0.0825806
\(379\) 4.72498 0.242706 0.121353 0.992609i \(-0.461277\pi\)
0.121353 + 0.992609i \(0.461277\pi\)
\(380\) −2.60555 −0.133662
\(381\) 0.697224 0.0357199
\(382\) 0.788897 0.0403635
\(383\) −21.5139 −1.09931 −0.549654 0.835392i \(-0.685240\pi\)
−0.549654 + 0.835392i \(0.685240\pi\)
\(384\) 2.30278 0.117513
\(385\) 3.90833 0.199187
\(386\) −14.0278 −0.713994
\(387\) −28.1194 −1.42939
\(388\) 4.60555 0.233811
\(389\) −17.2111 −0.872638 −0.436319 0.899792i \(-0.643718\pi\)
−0.436319 + 0.899792i \(0.643718\pi\)
\(390\) −15.0000 −0.759555
\(391\) −46.0278 −2.32772
\(392\) −6.00000 −0.303046
\(393\) 47.4500 2.39353
\(394\) 14.0917 0.709928
\(395\) 3.51388 0.176802
\(396\) 6.90833 0.347156
\(397\) 2.51388 0.126168 0.0630840 0.998008i \(-0.479906\pi\)
0.0630840 + 0.998008i \(0.479906\pi\)
\(398\) −3.09167 −0.154972
\(399\) −4.60555 −0.230566
\(400\) −3.30278 −0.165139
\(401\) 39.1194 1.95353 0.976766 0.214311i \(-0.0687505\pi\)
0.976766 + 0.214311i \(0.0687505\pi\)
\(402\) 37.3305 1.86188
\(403\) −48.0278 −2.39243
\(404\) 6.90833 0.343702
\(405\) 13.8167 0.686555
\(406\) −4.69722 −0.233119
\(407\) −10.8167 −0.536162
\(408\) −12.9083 −0.639057
\(409\) −27.6056 −1.36501 −0.682503 0.730882i \(-0.739109\pi\)
−0.682503 + 0.730882i \(0.739109\pi\)
\(410\) 11.2111 0.553677
\(411\) 18.6333 0.919113
\(412\) 11.1194 0.547815
\(413\) −6.51388 −0.320527
\(414\) 18.9083 0.929294
\(415\) −11.2111 −0.550331
\(416\) 5.00000 0.245145
\(417\) −51.8444 −2.53883
\(418\) 6.00000 0.293470
\(419\) 27.9083 1.36341 0.681705 0.731627i \(-0.261239\pi\)
0.681705 + 0.731627i \(0.261239\pi\)
\(420\) 3.00000 0.146385
\(421\) 21.4222 1.04405 0.522027 0.852929i \(-0.325176\pi\)
0.522027 + 0.852929i \(0.325176\pi\)
\(422\) −1.11943 −0.0544930
\(423\) −22.8167 −1.10938
\(424\) 0 0
\(425\) 18.5139 0.898055
\(426\) 18.0000 0.872103
\(427\) 1.00000 0.0483934
\(428\) 2.21110 0.106878
\(429\) 34.5416 1.66769
\(430\) 15.9083 0.767168
\(431\) −23.6056 −1.13704 −0.568520 0.822670i \(-0.692484\pi\)
−0.568520 + 0.822670i \(0.692484\pi\)
\(432\) −1.60555 −0.0772471
\(433\) 23.9083 1.14896 0.574480 0.818518i \(-0.305204\pi\)
0.574480 + 0.818518i \(0.305204\pi\)
\(434\) 9.60555 0.461081
\(435\) −14.0917 −0.675644
\(436\) 18.8167 0.901154
\(437\) 16.4222 0.785581
\(438\) −8.30278 −0.396722
\(439\) −18.3305 −0.874869 −0.437434 0.899250i \(-0.644113\pi\)
−0.437434 + 0.899250i \(0.644113\pi\)
\(440\) −3.90833 −0.186322
\(441\) −13.8167 −0.657936
\(442\) −28.0278 −1.33314
\(443\) 16.4222 0.780243 0.390121 0.920763i \(-0.372433\pi\)
0.390121 + 0.920763i \(0.372433\pi\)
\(444\) −8.30278 −0.394032
\(445\) 9.51388 0.451001
\(446\) −13.9083 −0.658578
\(447\) 17.7250 0.838363
\(448\) −1.00000 −0.0472456
\(449\) 5.09167 0.240291 0.120145 0.992756i \(-0.461664\pi\)
0.120145 + 0.992756i \(0.461664\pi\)
\(450\) −7.60555 −0.358529
\(451\) −25.8167 −1.21566
\(452\) 0.908327 0.0427241
\(453\) −35.9361 −1.68842
\(454\) −8.72498 −0.409484
\(455\) 6.51388 0.305375
\(456\) 4.60555 0.215675
\(457\) 10.7250 0.501693 0.250847 0.968027i \(-0.419291\pi\)
0.250847 + 0.968027i \(0.419291\pi\)
\(458\) 6.02776 0.281659
\(459\) 9.00000 0.420084
\(460\) −10.6972 −0.498761
\(461\) 13.0278 0.606763 0.303382 0.952869i \(-0.401884\pi\)
0.303382 + 0.952869i \(0.401884\pi\)
\(462\) −6.90833 −0.321404
\(463\) 35.9083 1.66880 0.834401 0.551158i \(-0.185814\pi\)
0.834401 + 0.551158i \(0.185814\pi\)
\(464\) 4.69722 0.218063
\(465\) 28.8167 1.33634
\(466\) 16.8167 0.779016
\(467\) 4.42221 0.204635 0.102318 0.994752i \(-0.467374\pi\)
0.102318 + 0.994752i \(0.467374\pi\)
\(468\) 11.5139 0.532229
\(469\) −16.2111 −0.748559
\(470\) 12.9083 0.595417
\(471\) 24.4222 1.12532
\(472\) 6.51388 0.299826
\(473\) −36.6333 −1.68440
\(474\) −6.21110 −0.285285
\(475\) −6.60555 −0.303083
\(476\) 5.60555 0.256930
\(477\) 0 0
\(478\) 0.908327 0.0415459
\(479\) 9.78890 0.447266 0.223633 0.974673i \(-0.428208\pi\)
0.223633 + 0.974673i \(0.428208\pi\)
\(480\) −3.00000 −0.136931
\(481\) −18.0278 −0.821995
\(482\) 18.0278 0.821142
\(483\) −18.9083 −0.860359
\(484\) −2.00000 −0.0909091
\(485\) −6.00000 −0.272446
\(486\) −19.6056 −0.889326
\(487\) −38.4222 −1.74108 −0.870538 0.492101i \(-0.836229\pi\)
−0.870538 + 0.492101i \(0.836229\pi\)
\(488\) −1.00000 −0.0452679
\(489\) −20.0278 −0.905686
\(490\) 7.81665 0.353120
\(491\) −37.8167 −1.70664 −0.853321 0.521386i \(-0.825415\pi\)
−0.853321 + 0.521386i \(0.825415\pi\)
\(492\) −19.8167 −0.893404
\(493\) −26.3305 −1.18587
\(494\) 10.0000 0.449921
\(495\) −9.00000 −0.404520
\(496\) −9.60555 −0.431302
\(497\) −7.81665 −0.350625
\(498\) 19.8167 0.888005
\(499\) 35.2389 1.57751 0.788754 0.614709i \(-0.210727\pi\)
0.788754 + 0.614709i \(0.210727\pi\)
\(500\) 10.8167 0.483735
\(501\) −46.8167 −2.09161
\(502\) −11.7250 −0.523312
\(503\) −33.6333 −1.49964 −0.749818 0.661645i \(-0.769859\pi\)
−0.749818 + 0.661645i \(0.769859\pi\)
\(504\) −2.30278 −0.102574
\(505\) −9.00000 −0.400495
\(506\) 24.6333 1.09508
\(507\) 27.6333 1.22724
\(508\) 0.302776 0.0134335
\(509\) 28.9361 1.28257 0.641285 0.767303i \(-0.278402\pi\)
0.641285 + 0.767303i \(0.278402\pi\)
\(510\) 16.8167 0.744654
\(511\) 3.60555 0.159500
\(512\) 1.00000 0.0441942
\(513\) −3.21110 −0.141774
\(514\) −3.00000 −0.132324
\(515\) −14.4861 −0.638335
\(516\) −28.1194 −1.23789
\(517\) −29.7250 −1.30730
\(518\) 3.60555 0.158419
\(519\) 17.0917 0.750241
\(520\) −6.51388 −0.285652
\(521\) 6.63331 0.290610 0.145305 0.989387i \(-0.453584\pi\)
0.145305 + 0.989387i \(0.453584\pi\)
\(522\) 10.8167 0.473432
\(523\) −23.3028 −1.01896 −0.509480 0.860483i \(-0.670162\pi\)
−0.509480 + 0.860483i \(0.670162\pi\)
\(524\) 20.6056 0.900158
\(525\) 7.60555 0.331933
\(526\) 21.5139 0.938050
\(527\) 53.8444 2.34550
\(528\) 6.90833 0.300646
\(529\) 44.4222 1.93140
\(530\) 0 0
\(531\) 15.0000 0.650945
\(532\) −2.00000 −0.0867110
\(533\) −43.0278 −1.86374
\(534\) −16.8167 −0.727728
\(535\) −2.88057 −0.124538
\(536\) 16.2111 0.700213
\(537\) −56.7250 −2.44786
\(538\) −1.00000 −0.0431131
\(539\) −18.0000 −0.775315
\(540\) 2.09167 0.0900113
\(541\) −20.4222 −0.878019 −0.439010 0.898482i \(-0.644671\pi\)
−0.439010 + 0.898482i \(0.644671\pi\)
\(542\) 11.0000 0.472490
\(543\) 30.4222 1.30554
\(544\) −5.60555 −0.240336
\(545\) −24.5139 −1.05006
\(546\) −11.5139 −0.492748
\(547\) 39.9361 1.70754 0.853772 0.520647i \(-0.174309\pi\)
0.853772 + 0.520647i \(0.174309\pi\)
\(548\) 8.09167 0.345659
\(549\) −2.30278 −0.0982801
\(550\) −9.90833 −0.422492
\(551\) 9.39445 0.400217
\(552\) 18.9083 0.804792
\(553\) 2.69722 0.114698
\(554\) −16.1194 −0.684849
\(555\) 10.8167 0.459141
\(556\) −22.5139 −0.954801
\(557\) −12.2389 −0.518577 −0.259289 0.965800i \(-0.583488\pi\)
−0.259289 + 0.965800i \(0.583488\pi\)
\(558\) −22.1194 −0.936390
\(559\) −61.0555 −2.58237
\(560\) 1.30278 0.0550523
\(561\) −38.7250 −1.63497
\(562\) 10.8167 0.456273
\(563\) −0.633308 −0.0266907 −0.0133454 0.999911i \(-0.504248\pi\)
−0.0133454 + 0.999911i \(0.504248\pi\)
\(564\) −22.8167 −0.960754
\(565\) −1.18335 −0.0497837
\(566\) −13.0000 −0.546431
\(567\) 10.6056 0.445391
\(568\) 7.81665 0.327980
\(569\) −20.0917 −0.842287 −0.421143 0.906994i \(-0.638371\pi\)
−0.421143 + 0.906994i \(0.638371\pi\)
\(570\) −6.00000 −0.251312
\(571\) −30.8444 −1.29080 −0.645399 0.763845i \(-0.723309\pi\)
−0.645399 + 0.763845i \(0.723309\pi\)
\(572\) 15.0000 0.627182
\(573\) 1.81665 0.0758918
\(574\) 8.60555 0.359189
\(575\) −27.1194 −1.13096
\(576\) 2.30278 0.0959490
\(577\) −40.6333 −1.69159 −0.845793 0.533511i \(-0.820872\pi\)
−0.845793 + 0.533511i \(0.820872\pi\)
\(578\) 14.4222 0.599885
\(579\) −32.3028 −1.34246
\(580\) −6.11943 −0.254095
\(581\) −8.60555 −0.357018
\(582\) 10.6056 0.439614
\(583\) 0 0
\(584\) −3.60555 −0.149199
\(585\) −15.0000 −0.620174
\(586\) 6.00000 0.247858
\(587\) 19.3028 0.796711 0.398355 0.917231i \(-0.369581\pi\)
0.398355 + 0.917231i \(0.369581\pi\)
\(588\) −13.8167 −0.569789
\(589\) −19.2111 −0.791580
\(590\) −8.48612 −0.349368
\(591\) 32.4500 1.33481
\(592\) −3.60555 −0.148187
\(593\) −15.6333 −0.641983 −0.320991 0.947082i \(-0.604016\pi\)
−0.320991 + 0.947082i \(0.604016\pi\)
\(594\) −4.81665 −0.197630
\(595\) −7.30278 −0.299385
\(596\) 7.69722 0.315291
\(597\) −7.11943 −0.291379
\(598\) 41.0555 1.67888
\(599\) −19.8167 −0.809687 −0.404843 0.914386i \(-0.632674\pi\)
−0.404843 + 0.914386i \(0.632674\pi\)
\(600\) −7.60555 −0.310495
\(601\) 34.3305 1.40037 0.700186 0.713961i \(-0.253100\pi\)
0.700186 + 0.713961i \(0.253100\pi\)
\(602\) 12.2111 0.497687
\(603\) 37.3305 1.52022
\(604\) −15.6056 −0.634981
\(605\) 2.60555 0.105931
\(606\) 15.9083 0.646232
\(607\) 33.5416 1.36141 0.680706 0.732556i \(-0.261673\pi\)
0.680706 + 0.732556i \(0.261673\pi\)
\(608\) 2.00000 0.0811107
\(609\) −10.8167 −0.438313
\(610\) 1.30278 0.0527478
\(611\) −49.5416 −2.00424
\(612\) −12.9083 −0.521788
\(613\) −36.2111 −1.46255 −0.731276 0.682081i \(-0.761075\pi\)
−0.731276 + 0.682081i \(0.761075\pi\)
\(614\) 15.4222 0.622390
\(615\) 25.8167 1.04103
\(616\) −3.00000 −0.120873
\(617\) 28.9361 1.16492 0.582461 0.812858i \(-0.302090\pi\)
0.582461 + 0.812858i \(0.302090\pi\)
\(618\) 25.6056 1.03001
\(619\) 19.2111 0.772159 0.386080 0.922465i \(-0.373829\pi\)
0.386080 + 0.922465i \(0.373829\pi\)
\(620\) 12.5139 0.502569
\(621\) −13.1833 −0.529029
\(622\) 13.8167 0.553997
\(623\) 7.30278 0.292580
\(624\) 11.5139 0.460924
\(625\) 2.42221 0.0968882
\(626\) −17.3028 −0.691558
\(627\) 13.8167 0.551784
\(628\) 10.6056 0.423208
\(629\) 20.2111 0.805869
\(630\) 3.00000 0.119523
\(631\) 8.11943 0.323229 0.161615 0.986854i \(-0.448330\pi\)
0.161615 + 0.986854i \(0.448330\pi\)
\(632\) −2.69722 −0.107290
\(633\) −2.57779 −0.102458
\(634\) 24.3944 0.968827
\(635\) −0.394449 −0.0156532
\(636\) 0 0
\(637\) −30.0000 −1.18864
\(638\) 14.0917 0.557895
\(639\) 18.0000 0.712069
\(640\) −1.30278 −0.0514967
\(641\) 33.2389 1.31286 0.656428 0.754389i \(-0.272067\pi\)
0.656428 + 0.754389i \(0.272067\pi\)
\(642\) 5.09167 0.200952
\(643\) 24.5416 0.967828 0.483914 0.875116i \(-0.339215\pi\)
0.483914 + 0.875116i \(0.339215\pi\)
\(644\) −8.21110 −0.323563
\(645\) 36.6333 1.44244
\(646\) −11.2111 −0.441095
\(647\) −6.63331 −0.260782 −0.130391 0.991463i \(-0.541623\pi\)
−0.130391 + 0.991463i \(0.541623\pi\)
\(648\) −10.6056 −0.416625
\(649\) 19.5416 0.767076
\(650\) −16.5139 −0.647728
\(651\) 22.1194 0.866929
\(652\) −8.69722 −0.340610
\(653\) −15.2389 −0.596343 −0.298171 0.954512i \(-0.596377\pi\)
−0.298171 + 0.954512i \(0.596377\pi\)
\(654\) 43.3305 1.69436
\(655\) −26.8444 −1.04890
\(656\) −8.60555 −0.335990
\(657\) −8.30278 −0.323922
\(658\) 9.90833 0.386267
\(659\) −28.9361 −1.12719 −0.563595 0.826051i \(-0.690582\pi\)
−0.563595 + 0.826051i \(0.690582\pi\)
\(660\) −9.00000 −0.350325
\(661\) 25.7250 1.00059 0.500293 0.865856i \(-0.333226\pi\)
0.500293 + 0.865856i \(0.333226\pi\)
\(662\) −1.00000 −0.0388661
\(663\) −64.5416 −2.50659
\(664\) 8.60555 0.333960
\(665\) 2.60555 0.101039
\(666\) −8.30278 −0.321726
\(667\) 38.5694 1.49341
\(668\) −20.3305 −0.786612
\(669\) −32.0278 −1.23826
\(670\) −21.1194 −0.815915
\(671\) −3.00000 −0.115814
\(672\) −2.30278 −0.0888315
\(673\) 24.8167 0.956612 0.478306 0.878193i \(-0.341251\pi\)
0.478306 + 0.878193i \(0.341251\pi\)
\(674\) −23.5416 −0.906790
\(675\) 5.30278 0.204104
\(676\) 12.0000 0.461538
\(677\) 19.3028 0.741866 0.370933 0.928660i \(-0.379038\pi\)
0.370933 + 0.928660i \(0.379038\pi\)
\(678\) 2.09167 0.0803302
\(679\) −4.60555 −0.176745
\(680\) 7.30278 0.280049
\(681\) −20.0917 −0.769915
\(682\) −28.8167 −1.10345
\(683\) 3.39445 0.129885 0.0649425 0.997889i \(-0.479314\pi\)
0.0649425 + 0.997889i \(0.479314\pi\)
\(684\) 4.60555 0.176098
\(685\) −10.5416 −0.402775
\(686\) 13.0000 0.496342
\(687\) 13.8806 0.529577
\(688\) −12.2111 −0.465544
\(689\) 0 0
\(690\) −24.6333 −0.937774
\(691\) −33.4500 −1.27250 −0.636248 0.771484i \(-0.719515\pi\)
−0.636248 + 0.771484i \(0.719515\pi\)
\(692\) 7.42221 0.282150
\(693\) −6.90833 −0.262426
\(694\) 7.18335 0.272676
\(695\) 29.3305 1.11257
\(696\) 10.8167 0.410004
\(697\) 48.2389 1.82718
\(698\) −8.30278 −0.314265
\(699\) 38.7250 1.46471
\(700\) 3.30278 0.124833
\(701\) −5.09167 −0.192310 −0.0961549 0.995366i \(-0.530654\pi\)
−0.0961549 + 0.995366i \(0.530654\pi\)
\(702\) −8.02776 −0.302988
\(703\) −7.21110 −0.271972
\(704\) 3.00000 0.113067
\(705\) 29.7250 1.11951
\(706\) 11.3305 0.426430
\(707\) −6.90833 −0.259814
\(708\) 15.0000 0.563735
\(709\) 32.6333 1.22557 0.612785 0.790250i \(-0.290049\pi\)
0.612785 + 0.790250i \(0.290049\pi\)
\(710\) −10.1833 −0.382174
\(711\) −6.21110 −0.232935
\(712\) −7.30278 −0.273683
\(713\) −78.8722 −2.95379
\(714\) 12.9083 0.483082
\(715\) −19.5416 −0.730816
\(716\) −24.6333 −0.920590
\(717\) 2.09167 0.0781150
\(718\) 0 0
\(719\) 10.1833 0.379775 0.189887 0.981806i \(-0.439188\pi\)
0.189887 + 0.981806i \(0.439188\pi\)
\(720\) −3.00000 −0.111803
\(721\) −11.1194 −0.414109
\(722\) −15.0000 −0.558242
\(723\) 41.5139 1.54392
\(724\) 13.2111 0.490987
\(725\) −15.5139 −0.576171
\(726\) −4.60555 −0.170928
\(727\) −5.42221 −0.201098 −0.100549 0.994932i \(-0.532060\pi\)
−0.100549 + 0.994932i \(0.532060\pi\)
\(728\) −5.00000 −0.185312
\(729\) −13.3305 −0.493723
\(730\) 4.69722 0.173852
\(731\) 68.4500 2.53171
\(732\) −2.30278 −0.0851130
\(733\) 5.00000 0.184679 0.0923396 0.995728i \(-0.470565\pi\)
0.0923396 + 0.995728i \(0.470565\pi\)
\(734\) −22.9083 −0.845562
\(735\) 18.0000 0.663940
\(736\) 8.21110 0.302665
\(737\) 48.6333 1.79143
\(738\) −19.8167 −0.729461
\(739\) −23.3028 −0.857206 −0.428603 0.903493i \(-0.640994\pi\)
−0.428603 + 0.903493i \(0.640994\pi\)
\(740\) 4.69722 0.172673
\(741\) 23.0278 0.845946
\(742\) 0 0
\(743\) −22.5416 −0.826972 −0.413486 0.910510i \(-0.635689\pi\)
−0.413486 + 0.910510i \(0.635689\pi\)
\(744\) −22.1194 −0.810938
\(745\) −10.0278 −0.367389
\(746\) 20.7889 0.761136
\(747\) 19.8167 0.725053
\(748\) −16.8167 −0.614878
\(749\) −2.21110 −0.0807919
\(750\) 24.9083 0.909524
\(751\) −23.5416 −0.859046 −0.429523 0.903056i \(-0.641318\pi\)
−0.429523 + 0.903056i \(0.641318\pi\)
\(752\) −9.90833 −0.361320
\(753\) −27.0000 −0.983935
\(754\) 23.4861 0.855314
\(755\) 20.3305 0.739904
\(756\) 1.60555 0.0583933
\(757\) −6.84441 −0.248764 −0.124382 0.992234i \(-0.539695\pi\)
−0.124382 + 0.992234i \(0.539695\pi\)
\(758\) 4.72498 0.171619
\(759\) 56.7250 2.05899
\(760\) −2.60555 −0.0945133
\(761\) −30.0000 −1.08750 −0.543750 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 0.697224 0.0252578
\(763\) −18.8167 −0.681209
\(764\) 0.788897 0.0285413
\(765\) 16.8167 0.608007
\(766\) −21.5139 −0.777328
\(767\) 32.5694 1.17601
\(768\) 2.30278 0.0830943
\(769\) −38.1472 −1.37562 −0.687811 0.725889i \(-0.741428\pi\)
−0.687811 + 0.725889i \(0.741428\pi\)
\(770\) 3.90833 0.140846
\(771\) −6.90833 −0.248797
\(772\) −14.0278 −0.504870
\(773\) 23.4500 0.843436 0.421718 0.906727i \(-0.361427\pi\)
0.421718 + 0.906727i \(0.361427\pi\)
\(774\) −28.1194 −1.01073
\(775\) 31.7250 1.13959
\(776\) 4.60555 0.165330
\(777\) 8.30278 0.297860
\(778\) −17.2111 −0.617048
\(779\) −17.2111 −0.616652
\(780\) −15.0000 −0.537086
\(781\) 23.4500 0.839106
\(782\) −46.0278 −1.64595
\(783\) −7.54163 −0.269516
\(784\) −6.00000 −0.214286
\(785\) −13.8167 −0.493138
\(786\) 47.4500 1.69248
\(787\) −3.84441 −0.137038 −0.0685192 0.997650i \(-0.521827\pi\)
−0.0685192 + 0.997650i \(0.521827\pi\)
\(788\) 14.0917 0.501995
\(789\) 49.5416 1.76373
\(790\) 3.51388 0.125018
\(791\) −0.908327 −0.0322964
\(792\) 6.90833 0.245477
\(793\) −5.00000 −0.177555
\(794\) 2.51388 0.0892142
\(795\) 0 0
\(796\) −3.09167 −0.109581
\(797\) −4.54163 −0.160873 −0.0804365 0.996760i \(-0.525631\pi\)
−0.0804365 + 0.996760i \(0.525631\pi\)
\(798\) −4.60555 −0.163035
\(799\) 55.5416 1.96492
\(800\) −3.30278 −0.116771
\(801\) −16.8167 −0.594187
\(802\) 39.1194 1.38136
\(803\) −10.8167 −0.381711
\(804\) 37.3305 1.31655
\(805\) 10.6972 0.377028
\(806\) −48.0278 −1.69171
\(807\) −2.30278 −0.0810615
\(808\) 6.90833 0.243034
\(809\) 26.8444 0.943799 0.471900 0.881652i \(-0.343568\pi\)
0.471900 + 0.881652i \(0.343568\pi\)
\(810\) 13.8167 0.485468
\(811\) −8.18335 −0.287356 −0.143678 0.989624i \(-0.545893\pi\)
−0.143678 + 0.989624i \(0.545893\pi\)
\(812\) −4.69722 −0.164840
\(813\) 25.3305 0.888381
\(814\) −10.8167 −0.379124
\(815\) 11.3305 0.396891
\(816\) −12.9083 −0.451882
\(817\) −24.4222 −0.854425
\(818\) −27.6056 −0.965205
\(819\) −11.5139 −0.402327
\(820\) 11.2111 0.391509
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) 18.6333 0.649911
\(823\) 1.60555 0.0559660 0.0279830 0.999608i \(-0.491092\pi\)
0.0279830 + 0.999608i \(0.491092\pi\)
\(824\) 11.1194 0.387364
\(825\) −22.8167 −0.794374
\(826\) −6.51388 −0.226647
\(827\) −12.9083 −0.448867 −0.224433 0.974489i \(-0.572053\pi\)
−0.224433 + 0.974489i \(0.572053\pi\)
\(828\) 18.9083 0.657110
\(829\) −20.9361 −0.727140 −0.363570 0.931567i \(-0.618442\pi\)
−0.363570 + 0.931567i \(0.618442\pi\)
\(830\) −11.2111 −0.389143
\(831\) −37.1194 −1.28766
\(832\) 5.00000 0.173344
\(833\) 33.6333 1.16533
\(834\) −51.8444 −1.79523
\(835\) 26.4861 0.916590
\(836\) 6.00000 0.207514
\(837\) 15.4222 0.533069
\(838\) 27.9083 0.964077
\(839\) 5.72498 0.197648 0.0988241 0.995105i \(-0.468492\pi\)
0.0988241 + 0.995105i \(0.468492\pi\)
\(840\) 3.00000 0.103510
\(841\) −6.93608 −0.239175
\(842\) 21.4222 0.738258
\(843\) 24.9083 0.857888
\(844\) −1.11943 −0.0385324
\(845\) −15.6333 −0.537802
\(846\) −22.8167 −0.784453
\(847\) 2.00000 0.0687208
\(848\) 0 0
\(849\) −29.9361 −1.02740
\(850\) 18.5139 0.635021
\(851\) −29.6056 −1.01487
\(852\) 18.0000 0.616670
\(853\) −11.6972 −0.400505 −0.200253 0.979744i \(-0.564176\pi\)
−0.200253 + 0.979744i \(0.564176\pi\)
\(854\) 1.00000 0.0342193
\(855\) −6.00000 −0.205196
\(856\) 2.21110 0.0755739
\(857\) −11.4500 −0.391123 −0.195562 0.980691i \(-0.562653\pi\)
−0.195562 + 0.980691i \(0.562653\pi\)
\(858\) 34.5416 1.17923
\(859\) −11.3028 −0.385646 −0.192823 0.981234i \(-0.561764\pi\)
−0.192823 + 0.981234i \(0.561764\pi\)
\(860\) 15.9083 0.542469
\(861\) 19.8167 0.675350
\(862\) −23.6056 −0.804008
\(863\) 30.7527 1.04684 0.523418 0.852076i \(-0.324657\pi\)
0.523418 + 0.852076i \(0.324657\pi\)
\(864\) −1.60555 −0.0546220
\(865\) −9.66947 −0.328772
\(866\) 23.9083 0.812438
\(867\) 33.2111 1.12791
\(868\) 9.60555 0.326034
\(869\) −8.09167 −0.274491
\(870\) −14.0917 −0.477752
\(871\) 81.0555 2.74646
\(872\) 18.8167 0.637212
\(873\) 10.6056 0.358944
\(874\) 16.4222 0.555489
\(875\) −10.8167 −0.365670
\(876\) −8.30278 −0.280525
\(877\) 24.5416 0.828712 0.414356 0.910115i \(-0.364007\pi\)
0.414356 + 0.910115i \(0.364007\pi\)
\(878\) −18.3305 −0.618625
\(879\) 13.8167 0.466024
\(880\) −3.90833 −0.131750
\(881\) −39.7527 −1.33930 −0.669652 0.742675i \(-0.733557\pi\)
−0.669652 + 0.742675i \(0.733557\pi\)
\(882\) −13.8167 −0.465231
\(883\) 41.2389 1.38780 0.693899 0.720072i \(-0.255891\pi\)
0.693899 + 0.720072i \(0.255891\pi\)
\(884\) −28.0278 −0.942675
\(885\) −19.5416 −0.656885
\(886\) 16.4222 0.551715
\(887\) 56.4500 1.89540 0.947702 0.319156i \(-0.103399\pi\)
0.947702 + 0.319156i \(0.103399\pi\)
\(888\) −8.30278 −0.278623
\(889\) −0.302776 −0.0101548
\(890\) 9.51388 0.318906
\(891\) −31.8167 −1.06590
\(892\) −13.9083 −0.465685
\(893\) −19.8167 −0.663139
\(894\) 17.7250 0.592812
\(895\) 32.0917 1.07271
\(896\) −1.00000 −0.0334077
\(897\) 94.5416 3.15665
\(898\) 5.09167 0.169911
\(899\) −45.1194 −1.50482
\(900\) −7.60555 −0.253518
\(901\) 0 0
\(902\) −25.8167 −0.859601
\(903\) 28.1194 0.935756
\(904\) 0.908327 0.0302105
\(905\) −17.2111 −0.572116
\(906\) −35.9361 −1.19390
\(907\) 10.2111 0.339054 0.169527 0.985526i \(-0.445776\pi\)
0.169527 + 0.985526i \(0.445776\pi\)
\(908\) −8.72498 −0.289549
\(909\) 15.9083 0.527646
\(910\) 6.51388 0.215933
\(911\) −42.3583 −1.40339 −0.701696 0.712476i \(-0.747574\pi\)
−0.701696 + 0.712476i \(0.747574\pi\)
\(912\) 4.60555 0.152505
\(913\) 25.8167 0.854407
\(914\) 10.7250 0.354751
\(915\) 3.00000 0.0991769
\(916\) 6.02776 0.199163
\(917\) −20.6056 −0.680455
\(918\) 9.00000 0.297044
\(919\) 12.1833 0.401892 0.200946 0.979602i \(-0.435599\pi\)
0.200946 + 0.979602i \(0.435599\pi\)
\(920\) −10.6972 −0.352677
\(921\) 35.5139 1.17022
\(922\) 13.0278 0.429046
\(923\) 39.0833 1.28644
\(924\) −6.90833 −0.227267
\(925\) 11.9083 0.391543
\(926\) 35.9083 1.18002
\(927\) 25.6056 0.840997
\(928\) 4.69722 0.154194
\(929\) 31.0278 1.01799 0.508994 0.860770i \(-0.330018\pi\)
0.508994 + 0.860770i \(0.330018\pi\)
\(930\) 28.8167 0.944935
\(931\) −12.0000 −0.393284
\(932\) 16.8167 0.550848
\(933\) 31.8167 1.04163
\(934\) 4.42221 0.144699
\(935\) 21.9083 0.716479
\(936\) 11.5139 0.376343
\(937\) −2.06392 −0.0674252 −0.0337126 0.999432i \(-0.510733\pi\)
−0.0337126 + 0.999432i \(0.510733\pi\)
\(938\) −16.2111 −0.529311
\(939\) −39.8444 −1.30027
\(940\) 12.9083 0.421023
\(941\) −48.3583 −1.57644 −0.788218 0.615397i \(-0.788996\pi\)
−0.788218 + 0.615397i \(0.788996\pi\)
\(942\) 24.4222 0.795718
\(943\) −70.6611 −2.30104
\(944\) 6.51388 0.212009
\(945\) −2.09167 −0.0680421
\(946\) −36.6333 −1.19105
\(947\) 23.8806 0.776014 0.388007 0.921656i \(-0.373164\pi\)
0.388007 + 0.921656i \(0.373164\pi\)
\(948\) −6.21110 −0.201727
\(949\) −18.0278 −0.585206
\(950\) −6.60555 −0.214312
\(951\) 56.1749 1.82160
\(952\) 5.60555 0.181677
\(953\) 22.4222 0.726326 0.363163 0.931726i \(-0.381697\pi\)
0.363163 + 0.931726i \(0.381697\pi\)
\(954\) 0 0
\(955\) −1.02776 −0.0332574
\(956\) 0.908327 0.0293774
\(957\) 32.4500 1.04896
\(958\) 9.78890 0.316265
\(959\) −8.09167 −0.261294
\(960\) −3.00000 −0.0968246
\(961\) 61.2666 1.97634
\(962\) −18.0278 −0.581238
\(963\) 5.09167 0.164077
\(964\) 18.0278 0.580635
\(965\) 18.2750 0.588294
\(966\) −18.9083 −0.608365
\(967\) 45.9361 1.47720 0.738602 0.674141i \(-0.235486\pi\)
0.738602 + 0.674141i \(0.235486\pi\)
\(968\) −2.00000 −0.0642824
\(969\) −25.8167 −0.829350
\(970\) −6.00000 −0.192648
\(971\) −10.1833 −0.326799 −0.163400 0.986560i \(-0.552246\pi\)
−0.163400 + 0.986560i \(0.552246\pi\)
\(972\) −19.6056 −0.628848
\(973\) 22.5139 0.721762
\(974\) −38.4222 −1.23113
\(975\) −38.0278 −1.21786
\(976\) −1.00000 −0.0320092
\(977\) 25.5416 0.817149 0.408575 0.912725i \(-0.366026\pi\)
0.408575 + 0.912725i \(0.366026\pi\)
\(978\) −20.0278 −0.640417
\(979\) −21.9083 −0.700193
\(980\) 7.81665 0.249694
\(981\) 43.3305 1.38344
\(982\) −37.8167 −1.20678
\(983\) 34.9361 1.11429 0.557144 0.830416i \(-0.311897\pi\)
0.557144 + 0.830416i \(0.311897\pi\)
\(984\) −19.8167 −0.631732
\(985\) −18.3583 −0.584944
\(986\) −26.3305 −0.838535
\(987\) 22.8167 0.726262
\(988\) 10.0000 0.318142
\(989\) −100.267 −3.18829
\(990\) −9.00000 −0.286039
\(991\) −17.5416 −0.557228 −0.278614 0.960403i \(-0.589875\pi\)
−0.278614 + 0.960403i \(0.589875\pi\)
\(992\) −9.60555 −0.304977
\(993\) −2.30278 −0.0730764
\(994\) −7.81665 −0.247929
\(995\) 4.02776 0.127688
\(996\) 19.8167 0.627915
\(997\) −25.7889 −0.816743 −0.408371 0.912816i \(-0.633903\pi\)
−0.408371 + 0.912816i \(0.633903\pi\)
\(998\) 35.2389 1.11547
\(999\) 5.78890 0.183153
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 538.2.a.b.1.2 2
3.2 odd 2 4842.2.a.d.1.2 2
4.3 odd 2 4304.2.a.c.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
538.2.a.b.1.2 2 1.1 even 1 trivial
4304.2.a.c.1.1 2 4.3 odd 2
4842.2.a.d.1.2 2 3.2 odd 2