Properties

Label 4015.2.a.i.1.15
Level $4015$
Weight $2$
Character 4015.1
Self dual yes
Analytic conductor $32.060$
Analytic rank $0$
Dimension $38$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4015,2,Mod(1,4015)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4015, 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("4015.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4015 = 5 \cdot 11 \cdot 73 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4015.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: \(32.0599364115\)
Analytic rank: \(0\)
Dimension: \(38\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.15
Character \(\chi\) \(=\) 4015.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.454453 q^{2} +2.93224 q^{3} -1.79347 q^{4} +1.00000 q^{5} -1.33257 q^{6} +4.01407 q^{7} +1.72395 q^{8} +5.59806 q^{9} +O(q^{10})\) \(q-0.454453 q^{2} +2.93224 q^{3} -1.79347 q^{4} +1.00000 q^{5} -1.33257 q^{6} +4.01407 q^{7} +1.72395 q^{8} +5.59806 q^{9} -0.454453 q^{10} -1.00000 q^{11} -5.25890 q^{12} -2.54805 q^{13} -1.82421 q^{14} +2.93224 q^{15} +2.80349 q^{16} -3.98319 q^{17} -2.54405 q^{18} -5.22272 q^{19} -1.79347 q^{20} +11.7702 q^{21} +0.454453 q^{22} +0.343009 q^{23} +5.05506 q^{24} +1.00000 q^{25} +1.15797 q^{26} +7.61814 q^{27} -7.19912 q^{28} +6.48025 q^{29} -1.33257 q^{30} +7.89077 q^{31} -4.72196 q^{32} -2.93224 q^{33} +1.81017 q^{34} +4.01407 q^{35} -10.0400 q^{36} +8.95232 q^{37} +2.37348 q^{38} -7.47151 q^{39} +1.72395 q^{40} +4.00932 q^{41} -5.34902 q^{42} -3.95029 q^{43} +1.79347 q^{44} +5.59806 q^{45} -0.155881 q^{46} +4.50057 q^{47} +8.22051 q^{48} +9.11274 q^{49} -0.454453 q^{50} -11.6797 q^{51} +4.56986 q^{52} +0.587202 q^{53} -3.46209 q^{54} -1.00000 q^{55} +6.92007 q^{56} -15.3143 q^{57} -2.94497 q^{58} +11.5878 q^{59} -5.25890 q^{60} +9.08260 q^{61} -3.58599 q^{62} +22.4710 q^{63} -3.46107 q^{64} -2.54805 q^{65} +1.33257 q^{66} -3.80200 q^{67} +7.14375 q^{68} +1.00579 q^{69} -1.82421 q^{70} -0.695920 q^{71} +9.65080 q^{72} -1.00000 q^{73} -4.06841 q^{74} +2.93224 q^{75} +9.36681 q^{76} -4.01407 q^{77} +3.39545 q^{78} +10.3893 q^{79} +2.80349 q^{80} +5.54408 q^{81} -1.82205 q^{82} -9.19674 q^{83} -21.1096 q^{84} -3.98319 q^{85} +1.79522 q^{86} +19.0017 q^{87} -1.72395 q^{88} +15.2413 q^{89} -2.54405 q^{90} -10.2280 q^{91} -0.615177 q^{92} +23.1377 q^{93} -2.04530 q^{94} -5.22272 q^{95} -13.8460 q^{96} -7.88949 q^{97} -4.14131 q^{98} -5.59806 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 38 q + 4 q^{2} + 5 q^{3} + 50 q^{4} + 38 q^{5} + 11 q^{6} + 15 q^{8} + 63 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 38 q + 4 q^{2} + 5 q^{3} + 50 q^{4} + 38 q^{5} + 11 q^{6} + 15 q^{8} + 63 q^{9} + 4 q^{10} - 38 q^{11} + 12 q^{12} - q^{13} + 23 q^{14} + 5 q^{15} + 74 q^{16} + 26 q^{17} + 16 q^{18} - 10 q^{19} + 50 q^{20} + 21 q^{21} - 4 q^{22} + 10 q^{23} + 41 q^{24} + 38 q^{25} + 25 q^{26} + 5 q^{27} + 2 q^{28} + 28 q^{29} + 11 q^{30} + 24 q^{31} + 39 q^{32} - 5 q^{33} + 38 q^{34} + 111 q^{36} + 12 q^{37} + 19 q^{38} - 18 q^{39} + 15 q^{40} + 62 q^{41} - 17 q^{42} - 32 q^{43} - 50 q^{44} + 63 q^{45} - 9 q^{46} + 31 q^{47} + 53 q^{48} + 88 q^{49} + 4 q^{50} - 3 q^{51} - 21 q^{52} + 30 q^{53} + 49 q^{54} - 38 q^{55} + 32 q^{56} + 49 q^{57} + 12 q^{58} + 31 q^{59} + 12 q^{60} + 25 q^{61} + 12 q^{62} + 15 q^{63} + 137 q^{64} - q^{65} - 11 q^{66} + 20 q^{67} + 75 q^{68} + 92 q^{69} + 23 q^{70} + 32 q^{71} + 6 q^{72} - 38 q^{73} + 55 q^{74} + 5 q^{75} - 57 q^{76} - 17 q^{78} - 2 q^{79} + 74 q^{80} + 118 q^{81} + 14 q^{82} + 4 q^{83} + 22 q^{84} + 26 q^{85} + 5 q^{86} + 24 q^{87} - 15 q^{88} + 143 q^{89} + 16 q^{90} + 66 q^{91} + 29 q^{92} - 8 q^{93} - 7 q^{94} - 10 q^{95} + 59 q^{96} + 41 q^{97} - 10 q^{98} - 63 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.454453 −0.321347 −0.160673 0.987008i \(-0.551367\pi\)
−0.160673 + 0.987008i \(0.551367\pi\)
\(3\) 2.93224 1.69293 0.846466 0.532443i \(-0.178726\pi\)
0.846466 + 0.532443i \(0.178726\pi\)
\(4\) −1.79347 −0.896736
\(5\) 1.00000 0.447214
\(6\) −1.33257 −0.544018
\(7\) 4.01407 1.51718 0.758588 0.651571i \(-0.225890\pi\)
0.758588 + 0.651571i \(0.225890\pi\)
\(8\) 1.72395 0.609510
\(9\) 5.59806 1.86602
\(10\) −0.454453 −0.143711
\(11\) −1.00000 −0.301511
\(12\) −5.25890 −1.51811
\(13\) −2.54805 −0.706702 −0.353351 0.935491i \(-0.614958\pi\)
−0.353351 + 0.935491i \(0.614958\pi\)
\(14\) −1.82421 −0.487539
\(15\) 2.93224 0.757102
\(16\) 2.80349 0.700872
\(17\) −3.98319 −0.966066 −0.483033 0.875602i \(-0.660465\pi\)
−0.483033 + 0.875602i \(0.660465\pi\)
\(18\) −2.54405 −0.599639
\(19\) −5.22272 −1.19817 −0.599087 0.800684i \(-0.704470\pi\)
−0.599087 + 0.800684i \(0.704470\pi\)
\(20\) −1.79347 −0.401033
\(21\) 11.7702 2.56847
\(22\) 0.454453 0.0968897
\(23\) 0.343009 0.0715223 0.0357612 0.999360i \(-0.488614\pi\)
0.0357612 + 0.999360i \(0.488614\pi\)
\(24\) 5.05506 1.03186
\(25\) 1.00000 0.200000
\(26\) 1.15797 0.227096
\(27\) 7.61814 1.46611
\(28\) −7.19912 −1.36051
\(29\) 6.48025 1.20335 0.601676 0.798740i \(-0.294500\pi\)
0.601676 + 0.798740i \(0.294500\pi\)
\(30\) −1.33257 −0.243292
\(31\) 7.89077 1.41722 0.708612 0.705598i \(-0.249322\pi\)
0.708612 + 0.705598i \(0.249322\pi\)
\(32\) −4.72196 −0.834733
\(33\) −2.93224 −0.510438
\(34\) 1.81017 0.310442
\(35\) 4.01407 0.678501
\(36\) −10.0400 −1.67333
\(37\) 8.95232 1.47175 0.735876 0.677116i \(-0.236770\pi\)
0.735876 + 0.677116i \(0.236770\pi\)
\(38\) 2.37348 0.385030
\(39\) −7.47151 −1.19640
\(40\) 1.72395 0.272581
\(41\) 4.00932 0.626151 0.313076 0.949728i \(-0.398641\pi\)
0.313076 + 0.949728i \(0.398641\pi\)
\(42\) −5.34902 −0.825371
\(43\) −3.95029 −0.602413 −0.301207 0.953559i \(-0.597389\pi\)
−0.301207 + 0.953559i \(0.597389\pi\)
\(44\) 1.79347 0.270376
\(45\) 5.59806 0.834509
\(46\) −0.155881 −0.0229835
\(47\) 4.50057 0.656476 0.328238 0.944595i \(-0.393545\pi\)
0.328238 + 0.944595i \(0.393545\pi\)
\(48\) 8.22051 1.18653
\(49\) 9.11274 1.30182
\(50\) −0.454453 −0.0642694
\(51\) −11.6797 −1.63549
\(52\) 4.56986 0.633725
\(53\) 0.587202 0.0806584 0.0403292 0.999186i \(-0.487159\pi\)
0.0403292 + 0.999186i \(0.487159\pi\)
\(54\) −3.46209 −0.471130
\(55\) −1.00000 −0.134840
\(56\) 6.92007 0.924733
\(57\) −15.3143 −2.02843
\(58\) −2.94497 −0.386694
\(59\) 11.5878 1.50860 0.754298 0.656532i \(-0.227977\pi\)
0.754298 + 0.656532i \(0.227977\pi\)
\(60\) −5.25890 −0.678921
\(61\) 9.08260 1.16291 0.581454 0.813580i \(-0.302484\pi\)
0.581454 + 0.813580i \(0.302484\pi\)
\(62\) −3.58599 −0.455421
\(63\) 22.4710 2.83108
\(64\) −3.46107 −0.432633
\(65\) −2.54805 −0.316047
\(66\) 1.33257 0.164028
\(67\) −3.80200 −0.464489 −0.232244 0.972657i \(-0.574607\pi\)
−0.232244 + 0.972657i \(0.574607\pi\)
\(68\) 7.14375 0.866307
\(69\) 1.00579 0.121082
\(70\) −1.82421 −0.218034
\(71\) −0.695920 −0.0825905 −0.0412952 0.999147i \(-0.513148\pi\)
−0.0412952 + 0.999147i \(0.513148\pi\)
\(72\) 9.65080 1.13736
\(73\) −1.00000 −0.117041
\(74\) −4.06841 −0.472943
\(75\) 2.93224 0.338586
\(76\) 9.36681 1.07445
\(77\) −4.01407 −0.457445
\(78\) 3.39545 0.384459
\(79\) 10.3893 1.16889 0.584444 0.811434i \(-0.301313\pi\)
0.584444 + 0.811434i \(0.301313\pi\)
\(80\) 2.80349 0.313440
\(81\) 5.54408 0.616009
\(82\) −1.82205 −0.201212
\(83\) −9.19674 −1.00947 −0.504737 0.863273i \(-0.668410\pi\)
−0.504737 + 0.863273i \(0.668410\pi\)
\(84\) −21.1096 −2.30324
\(85\) −3.98319 −0.432038
\(86\) 1.79522 0.193584
\(87\) 19.0017 2.03719
\(88\) −1.72395 −0.183774
\(89\) 15.2413 1.61558 0.807789 0.589472i \(-0.200664\pi\)
0.807789 + 0.589472i \(0.200664\pi\)
\(90\) −2.54405 −0.268167
\(91\) −10.2280 −1.07219
\(92\) −0.615177 −0.0641367
\(93\) 23.1377 2.39927
\(94\) −2.04530 −0.210956
\(95\) −5.22272 −0.535840
\(96\) −13.8460 −1.41315
\(97\) −7.88949 −0.801056 −0.400528 0.916284i \(-0.631173\pi\)
−0.400528 + 0.916284i \(0.631173\pi\)
\(98\) −4.14131 −0.418336
\(99\) −5.59806 −0.562626
\(100\) −1.79347 −0.179347
\(101\) 17.7507 1.76626 0.883131 0.469126i \(-0.155431\pi\)
0.883131 + 0.469126i \(0.155431\pi\)
\(102\) 5.30787 0.525558
\(103\) −14.1523 −1.39446 −0.697232 0.716846i \(-0.745585\pi\)
−0.697232 + 0.716846i \(0.745585\pi\)
\(104\) −4.39272 −0.430742
\(105\) 11.7702 1.14866
\(106\) −0.266856 −0.0259193
\(107\) 3.16194 0.305676 0.152838 0.988251i \(-0.451159\pi\)
0.152838 + 0.988251i \(0.451159\pi\)
\(108\) −13.6629 −1.31472
\(109\) 2.34608 0.224714 0.112357 0.993668i \(-0.464160\pi\)
0.112357 + 0.993668i \(0.464160\pi\)
\(110\) 0.454453 0.0433304
\(111\) 26.2504 2.49158
\(112\) 11.2534 1.06335
\(113\) −7.71763 −0.726013 −0.363007 0.931787i \(-0.618250\pi\)
−0.363007 + 0.931787i \(0.618250\pi\)
\(114\) 6.95963 0.651829
\(115\) 0.343009 0.0319858
\(116\) −11.6222 −1.07909
\(117\) −14.2641 −1.31872
\(118\) −5.26609 −0.484783
\(119\) −15.9888 −1.46569
\(120\) 5.05506 0.461461
\(121\) 1.00000 0.0909091
\(122\) −4.12761 −0.373697
\(123\) 11.7563 1.06003
\(124\) −14.1519 −1.27088
\(125\) 1.00000 0.0894427
\(126\) −10.2120 −0.909758
\(127\) 9.28633 0.824028 0.412014 0.911177i \(-0.364825\pi\)
0.412014 + 0.911177i \(0.364825\pi\)
\(128\) 11.0168 0.973758
\(129\) −11.5832 −1.01984
\(130\) 1.15797 0.101561
\(131\) −19.0112 −1.66102 −0.830509 0.557006i \(-0.811950\pi\)
−0.830509 + 0.557006i \(0.811950\pi\)
\(132\) 5.25890 0.457729
\(133\) −20.9644 −1.81784
\(134\) 1.72783 0.149262
\(135\) 7.61814 0.655665
\(136\) −6.86685 −0.588827
\(137\) −1.59211 −0.136023 −0.0680115 0.997685i \(-0.521665\pi\)
−0.0680115 + 0.997685i \(0.521665\pi\)
\(138\) −0.457083 −0.0389095
\(139\) −14.9012 −1.26390 −0.631950 0.775009i \(-0.717745\pi\)
−0.631950 + 0.775009i \(0.717745\pi\)
\(140\) −7.19912 −0.608437
\(141\) 13.1968 1.11137
\(142\) 0.316263 0.0265402
\(143\) 2.54805 0.213079
\(144\) 15.6941 1.30784
\(145\) 6.48025 0.538156
\(146\) 0.454453 0.0376108
\(147\) 26.7208 2.20389
\(148\) −16.0557 −1.31977
\(149\) −20.0774 −1.64480 −0.822401 0.568909i \(-0.807366\pi\)
−0.822401 + 0.568909i \(0.807366\pi\)
\(150\) −1.33257 −0.108804
\(151\) −15.8960 −1.29360 −0.646799 0.762661i \(-0.723893\pi\)
−0.646799 + 0.762661i \(0.723893\pi\)
\(152\) −9.00374 −0.730300
\(153\) −22.2982 −1.80270
\(154\) 1.82421 0.146999
\(155\) 7.89077 0.633802
\(156\) 13.3999 1.07285
\(157\) 19.3992 1.54822 0.774111 0.633050i \(-0.218197\pi\)
0.774111 + 0.633050i \(0.218197\pi\)
\(158\) −4.72145 −0.375619
\(159\) 1.72182 0.136549
\(160\) −4.72196 −0.373304
\(161\) 1.37686 0.108512
\(162\) −2.51953 −0.197953
\(163\) −3.88963 −0.304659 −0.152330 0.988330i \(-0.548678\pi\)
−0.152330 + 0.988330i \(0.548678\pi\)
\(164\) −7.19061 −0.561492
\(165\) −2.93224 −0.228275
\(166\) 4.17949 0.324391
\(167\) 7.95012 0.615199 0.307599 0.951516i \(-0.400474\pi\)
0.307599 + 0.951516i \(0.400474\pi\)
\(168\) 20.2913 1.56551
\(169\) −6.50744 −0.500572
\(170\) 1.81017 0.138834
\(171\) −29.2371 −2.23582
\(172\) 7.08473 0.540206
\(173\) −19.6741 −1.49579 −0.747896 0.663816i \(-0.768936\pi\)
−0.747896 + 0.663816i \(0.768936\pi\)
\(174\) −8.63537 −0.654646
\(175\) 4.01407 0.303435
\(176\) −2.80349 −0.211321
\(177\) 33.9781 2.55395
\(178\) −6.92647 −0.519161
\(179\) −15.8383 −1.18381 −0.591905 0.806008i \(-0.701624\pi\)
−0.591905 + 0.806008i \(0.701624\pi\)
\(180\) −10.0400 −0.748335
\(181\) −8.72608 −0.648604 −0.324302 0.945954i \(-0.605129\pi\)
−0.324302 + 0.945954i \(0.605129\pi\)
\(182\) 4.64817 0.344545
\(183\) 26.6324 1.96872
\(184\) 0.591332 0.0435936
\(185\) 8.95232 0.658188
\(186\) −10.5150 −0.770996
\(187\) 3.98319 0.291280
\(188\) −8.07165 −0.588686
\(189\) 30.5797 2.22435
\(190\) 2.37348 0.172190
\(191\) 11.0526 0.799738 0.399869 0.916572i \(-0.369056\pi\)
0.399869 + 0.916572i \(0.369056\pi\)
\(192\) −10.1487 −0.732419
\(193\) 1.62235 0.116779 0.0583895 0.998294i \(-0.481403\pi\)
0.0583895 + 0.998294i \(0.481403\pi\)
\(194\) 3.58540 0.257417
\(195\) −7.47151 −0.535046
\(196\) −16.3435 −1.16739
\(197\) 16.2364 1.15680 0.578399 0.815754i \(-0.303678\pi\)
0.578399 + 0.815754i \(0.303678\pi\)
\(198\) 2.54405 0.180798
\(199\) −20.4639 −1.45065 −0.725325 0.688407i \(-0.758310\pi\)
−0.725325 + 0.688407i \(0.758310\pi\)
\(200\) 1.72395 0.121902
\(201\) −11.1484 −0.786348
\(202\) −8.06687 −0.567583
\(203\) 26.0122 1.82570
\(204\) 20.9472 1.46660
\(205\) 4.00932 0.280023
\(206\) 6.43154 0.448106
\(207\) 1.92018 0.133462
\(208\) −7.14343 −0.495308
\(209\) 5.22272 0.361263
\(210\) −5.34902 −0.369117
\(211\) 16.9237 1.16508 0.582539 0.812803i \(-0.302059\pi\)
0.582539 + 0.812803i \(0.302059\pi\)
\(212\) −1.05313 −0.0723294
\(213\) −2.04061 −0.139820
\(214\) −1.43695 −0.0982280
\(215\) −3.95029 −0.269407
\(216\) 13.1333 0.893610
\(217\) 31.6741 2.15018
\(218\) −1.06618 −0.0722111
\(219\) −2.93224 −0.198143
\(220\) 1.79347 0.120916
\(221\) 10.1494 0.682721
\(222\) −11.9296 −0.800660
\(223\) −10.2499 −0.686387 −0.343193 0.939265i \(-0.611509\pi\)
−0.343193 + 0.939265i \(0.611509\pi\)
\(224\) −18.9543 −1.26644
\(225\) 5.59806 0.373204
\(226\) 3.50730 0.233302
\(227\) −20.5357 −1.36300 −0.681502 0.731816i \(-0.738673\pi\)
−0.681502 + 0.731816i \(0.738673\pi\)
\(228\) 27.4658 1.81897
\(229\) 8.20194 0.541999 0.271000 0.962579i \(-0.412646\pi\)
0.271000 + 0.962579i \(0.412646\pi\)
\(230\) −0.155881 −0.0102785
\(231\) −11.7702 −0.774424
\(232\) 11.1717 0.733456
\(233\) 17.8722 1.17085 0.585423 0.810728i \(-0.300929\pi\)
0.585423 + 0.810728i \(0.300929\pi\)
\(234\) 6.48238 0.423766
\(235\) 4.50057 0.293585
\(236\) −20.7823 −1.35281
\(237\) 30.4640 1.97885
\(238\) 7.26616 0.470995
\(239\) 1.70551 0.110320 0.0551601 0.998478i \(-0.482433\pi\)
0.0551601 + 0.998478i \(0.482433\pi\)
\(240\) 8.22051 0.530632
\(241\) −14.1559 −0.911863 −0.455932 0.890015i \(-0.650694\pi\)
−0.455932 + 0.890015i \(0.650694\pi\)
\(242\) −0.454453 −0.0292133
\(243\) −6.59782 −0.423250
\(244\) −16.2894 −1.04282
\(245\) 9.11274 0.582192
\(246\) −5.34269 −0.340638
\(247\) 13.3078 0.846753
\(248\) 13.6033 0.863813
\(249\) −26.9671 −1.70897
\(250\) −0.454453 −0.0287421
\(251\) 6.81720 0.430298 0.215149 0.976581i \(-0.430976\pi\)
0.215149 + 0.976581i \(0.430976\pi\)
\(252\) −40.3011 −2.53873
\(253\) −0.343009 −0.0215648
\(254\) −4.22020 −0.264799
\(255\) −11.6797 −0.731411
\(256\) 1.91551 0.119719
\(257\) 14.1277 0.881264 0.440632 0.897688i \(-0.354754\pi\)
0.440632 + 0.897688i \(0.354754\pi\)
\(258\) 5.26402 0.327724
\(259\) 35.9352 2.23291
\(260\) 4.56986 0.283411
\(261\) 36.2768 2.24548
\(262\) 8.63970 0.533763
\(263\) 10.9108 0.672787 0.336393 0.941722i \(-0.390793\pi\)
0.336393 + 0.941722i \(0.390793\pi\)
\(264\) −5.05506 −0.311117
\(265\) 0.587202 0.0360716
\(266\) 9.52732 0.584157
\(267\) 44.6913 2.73506
\(268\) 6.81879 0.416524
\(269\) 28.5175 1.73874 0.869372 0.494158i \(-0.164524\pi\)
0.869372 + 0.494158i \(0.164524\pi\)
\(270\) −3.46209 −0.210696
\(271\) 22.8996 1.39105 0.695526 0.718501i \(-0.255172\pi\)
0.695526 + 0.718501i \(0.255172\pi\)
\(272\) −11.1668 −0.677089
\(273\) −29.9911 −1.81515
\(274\) 0.723538 0.0437105
\(275\) −1.00000 −0.0603023
\(276\) −1.80385 −0.108579
\(277\) 11.9686 0.719125 0.359562 0.933121i \(-0.382926\pi\)
0.359562 + 0.933121i \(0.382926\pi\)
\(278\) 6.77188 0.406150
\(279\) 44.1730 2.64457
\(280\) 6.92007 0.413553
\(281\) −13.0516 −0.778595 −0.389298 0.921112i \(-0.627282\pi\)
−0.389298 + 0.921112i \(0.627282\pi\)
\(282\) −5.99731 −0.357135
\(283\) 19.1469 1.13816 0.569081 0.822281i \(-0.307299\pi\)
0.569081 + 0.822281i \(0.307299\pi\)
\(284\) 1.24811 0.0740619
\(285\) −15.3143 −0.907141
\(286\) −1.15797 −0.0684721
\(287\) 16.0937 0.949981
\(288\) −26.4338 −1.55763
\(289\) −1.13417 −0.0667156
\(290\) −2.94497 −0.172935
\(291\) −23.1339 −1.35613
\(292\) 1.79347 0.104955
\(293\) −26.5060 −1.54850 −0.774249 0.632882i \(-0.781872\pi\)
−0.774249 + 0.632882i \(0.781872\pi\)
\(294\) −12.1433 −0.708214
\(295\) 11.5878 0.674665
\(296\) 15.4334 0.897048
\(297\) −7.61814 −0.442050
\(298\) 9.12421 0.528552
\(299\) −0.874004 −0.0505450
\(300\) −5.25890 −0.303623
\(301\) −15.8567 −0.913966
\(302\) 7.22398 0.415693
\(303\) 52.0494 2.99016
\(304\) −14.6418 −0.839767
\(305\) 9.08260 0.520068
\(306\) 10.1335 0.579291
\(307\) −18.8383 −1.07516 −0.537581 0.843212i \(-0.680662\pi\)
−0.537581 + 0.843212i \(0.680662\pi\)
\(308\) 7.19912 0.410208
\(309\) −41.4979 −2.36073
\(310\) −3.58599 −0.203670
\(311\) −4.77177 −0.270582 −0.135291 0.990806i \(-0.543197\pi\)
−0.135291 + 0.990806i \(0.543197\pi\)
\(312\) −12.8805 −0.729217
\(313\) −22.1675 −1.25298 −0.626490 0.779429i \(-0.715509\pi\)
−0.626490 + 0.779429i \(0.715509\pi\)
\(314\) −8.81601 −0.497516
\(315\) 22.4710 1.26610
\(316\) −18.6329 −1.04818
\(317\) −16.6837 −0.937052 −0.468526 0.883450i \(-0.655215\pi\)
−0.468526 + 0.883450i \(0.655215\pi\)
\(318\) −0.782487 −0.0438797
\(319\) −6.48025 −0.362825
\(320\) −3.46107 −0.193480
\(321\) 9.27157 0.517489
\(322\) −0.625719 −0.0348699
\(323\) 20.8031 1.15752
\(324\) −9.94316 −0.552398
\(325\) −2.54805 −0.141340
\(326\) 1.76765 0.0979012
\(327\) 6.87928 0.380425
\(328\) 6.91189 0.381645
\(329\) 18.0656 0.995989
\(330\) 1.33257 0.0733554
\(331\) −28.6451 −1.57448 −0.787239 0.616648i \(-0.788490\pi\)
−0.787239 + 0.616648i \(0.788490\pi\)
\(332\) 16.4941 0.905231
\(333\) 50.1156 2.74632
\(334\) −3.61296 −0.197692
\(335\) −3.80200 −0.207726
\(336\) 32.9977 1.80017
\(337\) −16.8036 −0.915350 −0.457675 0.889120i \(-0.651318\pi\)
−0.457675 + 0.889120i \(0.651318\pi\)
\(338\) 2.95733 0.160857
\(339\) −22.6300 −1.22909
\(340\) 7.14375 0.387424
\(341\) −7.89077 −0.427309
\(342\) 13.2869 0.718473
\(343\) 8.48069 0.457914
\(344\) −6.81012 −0.367177
\(345\) 1.00579 0.0541497
\(346\) 8.94094 0.480668
\(347\) −19.7364 −1.05951 −0.529753 0.848152i \(-0.677715\pi\)
−0.529753 + 0.848152i \(0.677715\pi\)
\(348\) −34.0790 −1.82683
\(349\) −8.95453 −0.479325 −0.239663 0.970856i \(-0.577037\pi\)
−0.239663 + 0.970856i \(0.577037\pi\)
\(350\) −1.82421 −0.0975079
\(351\) −19.4114 −1.03610
\(352\) 4.72196 0.251681
\(353\) −14.6499 −0.779735 −0.389867 0.920871i \(-0.627479\pi\)
−0.389867 + 0.920871i \(0.627479\pi\)
\(354\) −15.4415 −0.820704
\(355\) −0.695920 −0.0369356
\(356\) −27.3349 −1.44875
\(357\) −46.8831 −2.48132
\(358\) 7.19776 0.380413
\(359\) −28.2818 −1.49265 −0.746327 0.665579i \(-0.768185\pi\)
−0.746327 + 0.665579i \(0.768185\pi\)
\(360\) 9.65080 0.508642
\(361\) 8.27684 0.435623
\(362\) 3.96559 0.208427
\(363\) 2.93224 0.153903
\(364\) 18.3437 0.961472
\(365\) −1.00000 −0.0523424
\(366\) −12.1032 −0.632643
\(367\) −6.95945 −0.363280 −0.181640 0.983365i \(-0.558141\pi\)
−0.181640 + 0.983365i \(0.558141\pi\)
\(368\) 0.961622 0.0501280
\(369\) 22.4444 1.16841
\(370\) −4.06841 −0.211507
\(371\) 2.35707 0.122373
\(372\) −41.4968 −2.15151
\(373\) −14.9021 −0.771600 −0.385800 0.922583i \(-0.626074\pi\)
−0.385800 + 0.922583i \(0.626074\pi\)
\(374\) −1.81017 −0.0936019
\(375\) 2.93224 0.151420
\(376\) 7.75878 0.400129
\(377\) −16.5120 −0.850412
\(378\) −13.8971 −0.714787
\(379\) 30.8831 1.58636 0.793180 0.608987i \(-0.208424\pi\)
0.793180 + 0.608987i \(0.208424\pi\)
\(380\) 9.36681 0.480507
\(381\) 27.2298 1.39502
\(382\) −5.02289 −0.256993
\(383\) 30.5292 1.55997 0.779985 0.625798i \(-0.215226\pi\)
0.779985 + 0.625798i \(0.215226\pi\)
\(384\) 32.3040 1.64851
\(385\) −4.01407 −0.204576
\(386\) −0.737280 −0.0375266
\(387\) −22.1139 −1.12411
\(388\) 14.1496 0.718336
\(389\) 21.1418 1.07193 0.535965 0.844240i \(-0.319948\pi\)
0.535965 + 0.844240i \(0.319948\pi\)
\(390\) 3.39545 0.171935
\(391\) −1.36627 −0.0690953
\(392\) 15.7100 0.793473
\(393\) −55.7455 −2.81199
\(394\) −7.37870 −0.371734
\(395\) 10.3893 0.522743
\(396\) 10.0400 0.504527
\(397\) 9.35012 0.469269 0.234634 0.972084i \(-0.424611\pi\)
0.234634 + 0.972084i \(0.424611\pi\)
\(398\) 9.29989 0.466161
\(399\) −61.4726 −3.07748
\(400\) 2.80349 0.140174
\(401\) 27.1940 1.35800 0.679002 0.734137i \(-0.262413\pi\)
0.679002 + 0.734137i \(0.262413\pi\)
\(402\) 5.06642 0.252690
\(403\) −20.1061 −1.00156
\(404\) −31.8354 −1.58387
\(405\) 5.54408 0.275488
\(406\) −11.8213 −0.586682
\(407\) −8.95232 −0.443750
\(408\) −20.1353 −0.996845
\(409\) −16.9629 −0.838762 −0.419381 0.907810i \(-0.637753\pi\)
−0.419381 + 0.907810i \(0.637753\pi\)
\(410\) −1.82205 −0.0899846
\(411\) −4.66845 −0.230278
\(412\) 25.3817 1.25047
\(413\) 46.5140 2.28881
\(414\) −0.872634 −0.0428876
\(415\) −9.19674 −0.451450
\(416\) 12.0318 0.589908
\(417\) −43.6938 −2.13970
\(418\) −2.37348 −0.116091
\(419\) 5.54865 0.271069 0.135535 0.990773i \(-0.456725\pi\)
0.135535 + 0.990773i \(0.456725\pi\)
\(420\) −21.1096 −1.03004
\(421\) 28.6696 1.39727 0.698637 0.715477i \(-0.253790\pi\)
0.698637 + 0.715477i \(0.253790\pi\)
\(422\) −7.69104 −0.374394
\(423\) 25.1945 1.22500
\(424\) 1.01231 0.0491621
\(425\) −3.98319 −0.193213
\(426\) 0.927360 0.0449307
\(427\) 36.4582 1.76433
\(428\) −5.67085 −0.274111
\(429\) 7.47151 0.360728
\(430\) 1.79522 0.0865732
\(431\) 1.92806 0.0928715 0.0464357 0.998921i \(-0.485214\pi\)
0.0464357 + 0.998921i \(0.485214\pi\)
\(432\) 21.3574 1.02756
\(433\) −0.0632902 −0.00304153 −0.00152077 0.999999i \(-0.500484\pi\)
−0.00152077 + 0.999999i \(0.500484\pi\)
\(434\) −14.3944 −0.690953
\(435\) 19.0017 0.911061
\(436\) −4.20763 −0.201509
\(437\) −1.79144 −0.0856962
\(438\) 1.33257 0.0636725
\(439\) −39.8008 −1.89959 −0.949794 0.312876i \(-0.898708\pi\)
−0.949794 + 0.312876i \(0.898708\pi\)
\(440\) −1.72395 −0.0821863
\(441\) 51.0137 2.42922
\(442\) −4.61242 −0.219390
\(443\) −33.6711 −1.59976 −0.799881 0.600159i \(-0.795104\pi\)
−0.799881 + 0.600159i \(0.795104\pi\)
\(444\) −47.0794 −2.23429
\(445\) 15.2413 0.722508
\(446\) 4.65812 0.220568
\(447\) −58.8717 −2.78454
\(448\) −13.8930 −0.656381
\(449\) −35.7073 −1.68513 −0.842565 0.538595i \(-0.818955\pi\)
−0.842565 + 0.538595i \(0.818955\pi\)
\(450\) −2.54405 −0.119928
\(451\) −4.00932 −0.188792
\(452\) 13.8414 0.651043
\(453\) −46.6109 −2.18997
\(454\) 9.33252 0.437997
\(455\) −10.2280 −0.479498
\(456\) −26.4012 −1.23635
\(457\) −17.1178 −0.800737 −0.400369 0.916354i \(-0.631118\pi\)
−0.400369 + 0.916354i \(0.631118\pi\)
\(458\) −3.72740 −0.174170
\(459\) −30.3445 −1.41636
\(460\) −0.615177 −0.0286828
\(461\) 25.1811 1.17280 0.586399 0.810022i \(-0.300545\pi\)
0.586399 + 0.810022i \(0.300545\pi\)
\(462\) 5.34902 0.248859
\(463\) −9.26234 −0.430458 −0.215229 0.976564i \(-0.569050\pi\)
−0.215229 + 0.976564i \(0.569050\pi\)
\(464\) 18.1673 0.843397
\(465\) 23.1377 1.07298
\(466\) −8.12207 −0.376247
\(467\) 14.4145 0.667024 0.333512 0.942746i \(-0.391766\pi\)
0.333512 + 0.942746i \(0.391766\pi\)
\(468\) 25.5823 1.18254
\(469\) −15.2615 −0.704710
\(470\) −2.04530 −0.0943426
\(471\) 56.8831 2.62104
\(472\) 19.9768 0.919505
\(473\) 3.95029 0.181634
\(474\) −13.8445 −0.635897
\(475\) −5.22272 −0.239635
\(476\) 28.6755 1.31434
\(477\) 3.28719 0.150510
\(478\) −0.775074 −0.0354511
\(479\) 36.0074 1.64522 0.822609 0.568607i \(-0.192517\pi\)
0.822609 + 0.568607i \(0.192517\pi\)
\(480\) −13.8460 −0.631978
\(481\) −22.8110 −1.04009
\(482\) 6.43320 0.293024
\(483\) 4.03729 0.183703
\(484\) −1.79347 −0.0815215
\(485\) −7.88949 −0.358243
\(486\) 2.99840 0.136010
\(487\) 43.4785 1.97020 0.985100 0.171983i \(-0.0550174\pi\)
0.985100 + 0.171983i \(0.0550174\pi\)
\(488\) 15.6580 0.708804
\(489\) −11.4053 −0.515767
\(490\) −4.14131 −0.187085
\(491\) −15.4476 −0.697141 −0.348570 0.937283i \(-0.613333\pi\)
−0.348570 + 0.937283i \(0.613333\pi\)
\(492\) −21.0846 −0.950569
\(493\) −25.8121 −1.16252
\(494\) −6.04775 −0.272101
\(495\) −5.59806 −0.251614
\(496\) 22.1217 0.993293
\(497\) −2.79347 −0.125304
\(498\) 12.2553 0.549172
\(499\) −11.4349 −0.511899 −0.255949 0.966690i \(-0.582388\pi\)
−0.255949 + 0.966690i \(0.582388\pi\)
\(500\) −1.79347 −0.0802065
\(501\) 23.3117 1.04149
\(502\) −3.09810 −0.138275
\(503\) 6.07764 0.270989 0.135494 0.990778i \(-0.456738\pi\)
0.135494 + 0.990778i \(0.456738\pi\)
\(504\) 38.7390 1.72557
\(505\) 17.7507 0.789897
\(506\) 0.155881 0.00692978
\(507\) −19.0814 −0.847435
\(508\) −16.6548 −0.738936
\(509\) −9.74848 −0.432094 −0.216047 0.976383i \(-0.569316\pi\)
−0.216047 + 0.976383i \(0.569316\pi\)
\(510\) 5.30787 0.235037
\(511\) −4.01407 −0.177572
\(512\) −22.9041 −1.01223
\(513\) −39.7875 −1.75666
\(514\) −6.42039 −0.283191
\(515\) −14.1523 −0.623623
\(516\) 20.7742 0.914532
\(517\) −4.50057 −0.197935
\(518\) −16.3309 −0.717537
\(519\) −57.6892 −2.53227
\(520\) −4.39272 −0.192634
\(521\) 5.39978 0.236569 0.118284 0.992980i \(-0.462261\pi\)
0.118284 + 0.992980i \(0.462261\pi\)
\(522\) −16.4861 −0.721578
\(523\) −35.7850 −1.56477 −0.782383 0.622797i \(-0.785996\pi\)
−0.782383 + 0.622797i \(0.785996\pi\)
\(524\) 34.0961 1.48949
\(525\) 11.7702 0.513695
\(526\) −4.95843 −0.216198
\(527\) −31.4305 −1.36913
\(528\) −8.22051 −0.357752
\(529\) −22.8823 −0.994885
\(530\) −0.266856 −0.0115915
\(531\) 64.8689 2.81507
\(532\) 37.5990 1.63012
\(533\) −10.2160 −0.442502
\(534\) −20.3101 −0.878904
\(535\) 3.16194 0.136702
\(536\) −6.55448 −0.283110
\(537\) −46.4417 −2.00411
\(538\) −12.9599 −0.558740
\(539\) −9.11274 −0.392514
\(540\) −13.6629 −0.587959
\(541\) −2.55972 −0.110051 −0.0550256 0.998485i \(-0.517524\pi\)
−0.0550256 + 0.998485i \(0.517524\pi\)
\(542\) −10.4068 −0.447010
\(543\) −25.5870 −1.09804
\(544\) 18.8085 0.806408
\(545\) 2.34608 0.100495
\(546\) 13.6296 0.583291
\(547\) −1.16722 −0.0499068 −0.0249534 0.999689i \(-0.507944\pi\)
−0.0249534 + 0.999689i \(0.507944\pi\)
\(548\) 2.85540 0.121977
\(549\) 50.8449 2.17001
\(550\) 0.454453 0.0193779
\(551\) −33.8446 −1.44183
\(552\) 1.73393 0.0738010
\(553\) 41.7034 1.77341
\(554\) −5.43918 −0.231088
\(555\) 26.2504 1.11427
\(556\) 26.7248 1.13338
\(557\) −36.7799 −1.55841 −0.779207 0.626766i \(-0.784378\pi\)
−0.779207 + 0.626766i \(0.784378\pi\)
\(558\) −20.0746 −0.849824
\(559\) 10.0655 0.425727
\(560\) 11.2534 0.475543
\(561\) 11.6797 0.493117
\(562\) 5.93135 0.250199
\(563\) 6.11013 0.257511 0.128756 0.991676i \(-0.458902\pi\)
0.128756 + 0.991676i \(0.458902\pi\)
\(564\) −23.6681 −0.996605
\(565\) −7.71763 −0.324683
\(566\) −8.70135 −0.365745
\(567\) 22.2543 0.934594
\(568\) −1.19973 −0.0503397
\(569\) −7.31923 −0.306838 −0.153419 0.988161i \(-0.549028\pi\)
−0.153419 + 0.988161i \(0.549028\pi\)
\(570\) 6.95963 0.291507
\(571\) 16.3404 0.683824 0.341912 0.939732i \(-0.388926\pi\)
0.341912 + 0.939732i \(0.388926\pi\)
\(572\) −4.56986 −0.191075
\(573\) 32.4089 1.35390
\(574\) −7.31383 −0.305273
\(575\) 0.343009 0.0143045
\(576\) −19.3753 −0.807302
\(577\) 19.9481 0.830450 0.415225 0.909719i \(-0.363703\pi\)
0.415225 + 0.909719i \(0.363703\pi\)
\(578\) 0.515425 0.0214388
\(579\) 4.75712 0.197699
\(580\) −11.6222 −0.482584
\(581\) −36.9163 −1.53155
\(582\) 10.5133 0.435789
\(583\) −0.587202 −0.0243194
\(584\) −1.72395 −0.0713378
\(585\) −14.2641 −0.589749
\(586\) 12.0457 0.497605
\(587\) −8.39196 −0.346373 −0.173187 0.984889i \(-0.555406\pi\)
−0.173187 + 0.984889i \(0.555406\pi\)
\(588\) −47.9230 −1.97631
\(589\) −41.2113 −1.69808
\(590\) −5.26609 −0.216801
\(591\) 47.6092 1.95838
\(592\) 25.0977 1.03151
\(593\) −37.2001 −1.52763 −0.763813 0.645438i \(-0.776675\pi\)
−0.763813 + 0.645438i \(0.776675\pi\)
\(594\) 3.46209 0.142051
\(595\) −15.9888 −0.655477
\(596\) 36.0082 1.47495
\(597\) −60.0052 −2.45585
\(598\) 0.397194 0.0162425
\(599\) −6.57215 −0.268531 −0.134265 0.990945i \(-0.542867\pi\)
−0.134265 + 0.990945i \(0.542867\pi\)
\(600\) 5.05506 0.206372
\(601\) −4.81474 −0.196397 −0.0981987 0.995167i \(-0.531308\pi\)
−0.0981987 + 0.995167i \(0.531308\pi\)
\(602\) 7.20613 0.293700
\(603\) −21.2838 −0.866745
\(604\) 28.5090 1.16002
\(605\) 1.00000 0.0406558
\(606\) −23.6540 −0.960879
\(607\) 10.0570 0.408202 0.204101 0.978950i \(-0.434573\pi\)
0.204101 + 0.978950i \(0.434573\pi\)
\(608\) 24.6615 1.00016
\(609\) 76.2741 3.09078
\(610\) −4.12761 −0.167122
\(611\) −11.4677 −0.463933
\(612\) 39.9911 1.61655
\(613\) −31.3737 −1.26717 −0.633587 0.773672i \(-0.718418\pi\)
−0.633587 + 0.773672i \(0.718418\pi\)
\(614\) 8.56114 0.345500
\(615\) 11.7563 0.474061
\(616\) −6.92007 −0.278818
\(617\) −13.9568 −0.561878 −0.280939 0.959726i \(-0.590646\pi\)
−0.280939 + 0.959726i \(0.590646\pi\)
\(618\) 18.8588 0.758614
\(619\) −25.5234 −1.02587 −0.512936 0.858427i \(-0.671442\pi\)
−0.512936 + 0.858427i \(0.671442\pi\)
\(620\) −14.1519 −0.568353
\(621\) 2.61309 0.104860
\(622\) 2.16855 0.0869508
\(623\) 61.1797 2.45111
\(624\) −20.9463 −0.838522
\(625\) 1.00000 0.0400000
\(626\) 10.0741 0.402641
\(627\) 15.3143 0.611594
\(628\) −34.7919 −1.38835
\(629\) −35.6588 −1.42181
\(630\) −10.2120 −0.406856
\(631\) 20.6445 0.821846 0.410923 0.911670i \(-0.365206\pi\)
0.410923 + 0.911670i \(0.365206\pi\)
\(632\) 17.9107 0.712449
\(633\) 49.6245 1.97240
\(634\) 7.58197 0.301119
\(635\) 9.28633 0.368517
\(636\) −3.08804 −0.122449
\(637\) −23.2197 −0.919999
\(638\) 2.94497 0.116592
\(639\) −3.89580 −0.154115
\(640\) 11.0168 0.435478
\(641\) 45.3629 1.79173 0.895863 0.444330i \(-0.146558\pi\)
0.895863 + 0.444330i \(0.146558\pi\)
\(642\) −4.21349 −0.166293
\(643\) 45.4885 1.79389 0.896946 0.442140i \(-0.145781\pi\)
0.896946 + 0.442140i \(0.145781\pi\)
\(644\) −2.46936 −0.0973065
\(645\) −11.5832 −0.456088
\(646\) −9.45404 −0.371964
\(647\) −8.30027 −0.326317 −0.163159 0.986600i \(-0.552168\pi\)
−0.163159 + 0.986600i \(0.552168\pi\)
\(648\) 9.55775 0.375464
\(649\) −11.5878 −0.454859
\(650\) 1.15797 0.0454193
\(651\) 92.8762 3.64011
\(652\) 6.97594 0.273199
\(653\) 21.0375 0.823263 0.411631 0.911350i \(-0.364959\pi\)
0.411631 + 0.911350i \(0.364959\pi\)
\(654\) −3.12631 −0.122248
\(655\) −19.0112 −0.742829
\(656\) 11.2401 0.438852
\(657\) −5.59806 −0.218401
\(658\) −8.20997 −0.320058
\(659\) −20.9096 −0.814522 −0.407261 0.913312i \(-0.633516\pi\)
−0.407261 + 0.913312i \(0.633516\pi\)
\(660\) 5.25890 0.204702
\(661\) 8.89183 0.345852 0.172926 0.984935i \(-0.444678\pi\)
0.172926 + 0.984935i \(0.444678\pi\)
\(662\) 13.0179 0.505953
\(663\) 29.7605 1.15580
\(664\) −15.8548 −0.615284
\(665\) −20.9644 −0.812963
\(666\) −22.7752 −0.882521
\(667\) 2.22279 0.0860666
\(668\) −14.2583 −0.551671
\(669\) −30.0553 −1.16201
\(670\) 1.72783 0.0667519
\(671\) −9.08260 −0.350630
\(672\) −55.5786 −2.14399
\(673\) 36.7588 1.41695 0.708474 0.705737i \(-0.249384\pi\)
0.708474 + 0.705737i \(0.249384\pi\)
\(674\) 7.63644 0.294145
\(675\) 7.61814 0.293222
\(676\) 11.6709 0.448881
\(677\) 4.83714 0.185906 0.0929532 0.995670i \(-0.470369\pi\)
0.0929532 + 0.995670i \(0.470369\pi\)
\(678\) 10.2843 0.394965
\(679\) −31.6689 −1.21534
\(680\) −6.86685 −0.263332
\(681\) −60.2158 −2.30747
\(682\) 3.58599 0.137314
\(683\) 11.8370 0.452929 0.226464 0.974019i \(-0.427283\pi\)
0.226464 + 0.974019i \(0.427283\pi\)
\(684\) 52.4360 2.00494
\(685\) −1.59211 −0.0608313
\(686\) −3.85407 −0.147149
\(687\) 24.0501 0.917568
\(688\) −11.0746 −0.422215
\(689\) −1.49622 −0.0570015
\(690\) −0.457083 −0.0174008
\(691\) 9.08395 0.345570 0.172785 0.984960i \(-0.444723\pi\)
0.172785 + 0.984960i \(0.444723\pi\)
\(692\) 35.2849 1.34133
\(693\) −22.4710 −0.853602
\(694\) 8.96927 0.340469
\(695\) −14.9012 −0.565233
\(696\) 32.7581 1.24169
\(697\) −15.9699 −0.604904
\(698\) 4.06941 0.154030
\(699\) 52.4056 1.98216
\(700\) −7.19912 −0.272101
\(701\) 0.954313 0.0360439 0.0180219 0.999838i \(-0.494263\pi\)
0.0180219 + 0.999838i \(0.494263\pi\)
\(702\) 8.82157 0.332949
\(703\) −46.7555 −1.76342
\(704\) 3.46107 0.130444
\(705\) 13.1968 0.497019
\(706\) 6.65769 0.250565
\(707\) 71.2526 2.67973
\(708\) −60.9388 −2.29022
\(709\) −31.6264 −1.18776 −0.593878 0.804555i \(-0.702404\pi\)
−0.593878 + 0.804555i \(0.702404\pi\)
\(710\) 0.316263 0.0118691
\(711\) 58.1600 2.18117
\(712\) 26.2754 0.984711
\(713\) 2.70661 0.101363
\(714\) 21.3062 0.797363
\(715\) 2.54805 0.0952917
\(716\) 28.4055 1.06156
\(717\) 5.00097 0.186765
\(718\) 12.8527 0.479660
\(719\) 4.03832 0.150604 0.0753020 0.997161i \(-0.476008\pi\)
0.0753020 + 0.997161i \(0.476008\pi\)
\(720\) 15.6941 0.584884
\(721\) −56.8081 −2.11565
\(722\) −3.76143 −0.139986
\(723\) −41.5086 −1.54372
\(724\) 15.6500 0.581627
\(725\) 6.48025 0.240671
\(726\) −1.33257 −0.0494562
\(727\) −15.1500 −0.561882 −0.280941 0.959725i \(-0.590646\pi\)
−0.280941 + 0.959725i \(0.590646\pi\)
\(728\) −17.6327 −0.653511
\(729\) −35.9787 −1.33254
\(730\) 0.454453 0.0168201
\(731\) 15.7348 0.581971
\(732\) −47.7645 −1.76543
\(733\) 32.9810 1.21818 0.609091 0.793101i \(-0.291535\pi\)
0.609091 + 0.793101i \(0.291535\pi\)
\(734\) 3.16274 0.116739
\(735\) 26.7208 0.985611
\(736\) −1.61968 −0.0597020
\(737\) 3.80200 0.140049
\(738\) −10.1999 −0.375465
\(739\) 33.4284 1.22968 0.614842 0.788650i \(-0.289220\pi\)
0.614842 + 0.788650i \(0.289220\pi\)
\(740\) −16.0557 −0.590221
\(741\) 39.0216 1.43349
\(742\) −1.07118 −0.0393242
\(743\) −11.0010 −0.403589 −0.201794 0.979428i \(-0.564677\pi\)
−0.201794 + 0.979428i \(0.564677\pi\)
\(744\) 39.8883 1.46238
\(745\) −20.0774 −0.735577
\(746\) 6.77229 0.247951
\(747\) −51.4839 −1.88370
\(748\) −7.14375 −0.261201
\(749\) 12.6922 0.463764
\(750\) −1.33257 −0.0486585
\(751\) −9.50545 −0.346859 −0.173429 0.984846i \(-0.555485\pi\)
−0.173429 + 0.984846i \(0.555485\pi\)
\(752\) 12.6173 0.460106
\(753\) 19.9897 0.728465
\(754\) 7.50393 0.273277
\(755\) −15.8960 −0.578514
\(756\) −54.8439 −1.99465
\(757\) −45.7176 −1.66163 −0.830817 0.556546i \(-0.812126\pi\)
−0.830817 + 0.556546i \(0.812126\pi\)
\(758\) −14.0349 −0.509772
\(759\) −1.00579 −0.0365077
\(760\) −9.00374 −0.326600
\(761\) −44.6478 −1.61848 −0.809241 0.587477i \(-0.800121\pi\)
−0.809241 + 0.587477i \(0.800121\pi\)
\(762\) −12.3747 −0.448287
\(763\) 9.41733 0.340930
\(764\) −19.8225 −0.717154
\(765\) −22.2982 −0.806191
\(766\) −13.8741 −0.501292
\(767\) −29.5262 −1.06613
\(768\) 5.61674 0.202677
\(769\) 26.5064 0.955844 0.477922 0.878402i \(-0.341390\pi\)
0.477922 + 0.878402i \(0.341390\pi\)
\(770\) 1.82421 0.0657398
\(771\) 41.4260 1.49192
\(772\) −2.90963 −0.104720
\(773\) 23.9999 0.863217 0.431609 0.902061i \(-0.357946\pi\)
0.431609 + 0.902061i \(0.357946\pi\)
\(774\) 10.0497 0.361231
\(775\) 7.89077 0.283445
\(776\) −13.6011 −0.488252
\(777\) 105.371 3.78016
\(778\) −9.60794 −0.344461
\(779\) −20.9396 −0.750239
\(780\) 13.3999 0.479795
\(781\) 0.695920 0.0249020
\(782\) 0.620906 0.0222036
\(783\) 49.3675 1.76425
\(784\) 25.5475 0.912410
\(785\) 19.3992 0.692386
\(786\) 25.3337 0.903624
\(787\) −44.6725 −1.59240 −0.796202 0.605031i \(-0.793161\pi\)
−0.796202 + 0.605031i \(0.793161\pi\)
\(788\) −29.1196 −1.03734
\(789\) 31.9930 1.13898
\(790\) −4.72145 −0.167982
\(791\) −30.9791 −1.10149
\(792\) −9.65080 −0.342926
\(793\) −23.1429 −0.821829
\(794\) −4.24919 −0.150798
\(795\) 1.72182 0.0610667
\(796\) 36.7015 1.30085
\(797\) 19.2437 0.681646 0.340823 0.940128i \(-0.389294\pi\)
0.340823 + 0.940128i \(0.389294\pi\)
\(798\) 27.9364 0.988939
\(799\) −17.9266 −0.634199
\(800\) −4.72196 −0.166947
\(801\) 85.3219 3.01470
\(802\) −12.3584 −0.436390
\(803\) 1.00000 0.0352892
\(804\) 19.9944 0.705146
\(805\) 1.37686 0.0485280
\(806\) 9.13727 0.321847
\(807\) 83.6204 2.94358
\(808\) 30.6014 1.07655
\(809\) 47.5771 1.67272 0.836361 0.548180i \(-0.184679\pi\)
0.836361 + 0.548180i \(0.184679\pi\)
\(810\) −2.51953 −0.0885271
\(811\) −18.4707 −0.648593 −0.324296 0.945956i \(-0.605128\pi\)
−0.324296 + 0.945956i \(0.605128\pi\)
\(812\) −46.6521 −1.63717
\(813\) 67.1472 2.35496
\(814\) 4.06841 0.142598
\(815\) −3.88963 −0.136248
\(816\) −32.7439 −1.14627
\(817\) 20.6313 0.721796
\(818\) 7.70884 0.269533
\(819\) −57.2572 −2.00073
\(820\) −7.19061 −0.251107
\(821\) 9.00762 0.314368 0.157184 0.987569i \(-0.449758\pi\)
0.157184 + 0.987569i \(0.449758\pi\)
\(822\) 2.12159 0.0739990
\(823\) −38.4563 −1.34050 −0.670251 0.742135i \(-0.733813\pi\)
−0.670251 + 0.742135i \(0.733813\pi\)
\(824\) −24.3979 −0.849940
\(825\) −2.93224 −0.102088
\(826\) −21.1384 −0.735500
\(827\) 18.5542 0.645192 0.322596 0.946537i \(-0.395444\pi\)
0.322596 + 0.946537i \(0.395444\pi\)
\(828\) −3.44380 −0.119680
\(829\) −25.4625 −0.884349 −0.442174 0.896929i \(-0.645793\pi\)
−0.442174 + 0.896929i \(0.645793\pi\)
\(830\) 4.17949 0.145072
\(831\) 35.0949 1.21743
\(832\) 8.81897 0.305743
\(833\) −36.2978 −1.25764
\(834\) 19.8568 0.687585
\(835\) 7.95012 0.275125
\(836\) −9.36681 −0.323958
\(837\) 60.1130 2.07781
\(838\) −2.52160 −0.0871072
\(839\) −23.2802 −0.803722 −0.401861 0.915701i \(-0.631637\pi\)
−0.401861 + 0.915701i \(0.631637\pi\)
\(840\) 20.2913 0.700118
\(841\) 12.9937 0.448058
\(842\) −13.0290 −0.449009
\(843\) −38.2706 −1.31811
\(844\) −30.3522 −1.04477
\(845\) −6.50744 −0.223863
\(846\) −11.4497 −0.393649
\(847\) 4.01407 0.137925
\(848\) 1.64622 0.0565313
\(849\) 56.1433 1.92683
\(850\) 1.81017 0.0620885
\(851\) 3.07073 0.105263
\(852\) 3.65977 0.125382
\(853\) −6.18516 −0.211776 −0.105888 0.994378i \(-0.533768\pi\)
−0.105888 + 0.994378i \(0.533768\pi\)
\(854\) −16.5685 −0.566963
\(855\) −29.2371 −0.999888
\(856\) 5.45104 0.186313
\(857\) −19.3546 −0.661139 −0.330570 0.943782i \(-0.607241\pi\)
−0.330570 + 0.943782i \(0.607241\pi\)
\(858\) −3.39545 −0.115919
\(859\) −6.42562 −0.219239 −0.109620 0.993974i \(-0.534963\pi\)
−0.109620 + 0.993974i \(0.534963\pi\)
\(860\) 7.08473 0.241587
\(861\) 47.1907 1.60825
\(862\) −0.876213 −0.0298439
\(863\) 14.3184 0.487404 0.243702 0.969850i \(-0.421638\pi\)
0.243702 + 0.969850i \(0.421638\pi\)
\(864\) −35.9726 −1.22381
\(865\) −19.6741 −0.668939
\(866\) 0.0287624 0.000977387 0
\(867\) −3.32565 −0.112945
\(868\) −56.8066 −1.92814
\(869\) −10.3893 −0.352433
\(870\) −8.63537 −0.292767
\(871\) 9.68769 0.328255
\(872\) 4.04454 0.136965
\(873\) −44.1658 −1.49479
\(874\) 0.814126 0.0275382
\(875\) 4.01407 0.135700
\(876\) 5.25890 0.177682
\(877\) 20.4040 0.688995 0.344497 0.938787i \(-0.388049\pi\)
0.344497 + 0.938787i \(0.388049\pi\)
\(878\) 18.0876 0.610427
\(879\) −77.7221 −2.62150
\(880\) −2.80349 −0.0945056
\(881\) 2.90247 0.0977867 0.0488934 0.998804i \(-0.484431\pi\)
0.0488934 + 0.998804i \(0.484431\pi\)
\(882\) −23.1833 −0.780623
\(883\) 49.7621 1.67463 0.837314 0.546722i \(-0.184125\pi\)
0.837314 + 0.546722i \(0.184125\pi\)
\(884\) −18.2026 −0.612221
\(885\) 33.9781 1.14216
\(886\) 15.3019 0.514078
\(887\) 32.3733 1.08699 0.543495 0.839413i \(-0.317101\pi\)
0.543495 + 0.839413i \(0.317101\pi\)
\(888\) 45.2545 1.51864
\(889\) 37.2760 1.25020
\(890\) −6.92647 −0.232176
\(891\) −5.54408 −0.185734
\(892\) 18.3830 0.615508
\(893\) −23.5052 −0.786573
\(894\) 26.7544 0.894802
\(895\) −15.8383 −0.529416
\(896\) 44.2223 1.47736
\(897\) −2.56279 −0.0855692
\(898\) 16.2273 0.541511
\(899\) 51.1342 1.70542
\(900\) −10.0400 −0.334665
\(901\) −2.33894 −0.0779214
\(902\) 1.82205 0.0606676
\(903\) −46.4958 −1.54728
\(904\) −13.3048 −0.442513
\(905\) −8.72608 −0.290065
\(906\) 21.1825 0.703741
\(907\) 6.24920 0.207501 0.103751 0.994603i \(-0.466916\pi\)
0.103751 + 0.994603i \(0.466916\pi\)
\(908\) 36.8302 1.22225
\(909\) 99.3696 3.29588
\(910\) 4.64817 0.154085
\(911\) 18.8447 0.624351 0.312176 0.950024i \(-0.398942\pi\)
0.312176 + 0.950024i \(0.398942\pi\)
\(912\) −42.9335 −1.42167
\(913\) 9.19674 0.304368
\(914\) 7.77924 0.257314
\(915\) 26.6324 0.880440
\(916\) −14.7100 −0.486031
\(917\) −76.3123 −2.52005
\(918\) 13.7902 0.455143
\(919\) 33.0120 1.08897 0.544483 0.838772i \(-0.316726\pi\)
0.544483 + 0.838772i \(0.316726\pi\)
\(920\) 0.591332 0.0194956
\(921\) −55.2386 −1.82017
\(922\) −11.4436 −0.376875
\(923\) 1.77324 0.0583668
\(924\) 21.1096 0.694454
\(925\) 8.95232 0.294351
\(926\) 4.20930 0.138326
\(927\) −79.2252 −2.60210
\(928\) −30.5995 −1.00448
\(929\) 1.26418 0.0414764 0.0207382 0.999785i \(-0.493398\pi\)
0.0207382 + 0.999785i \(0.493398\pi\)
\(930\) −10.5150 −0.344800
\(931\) −47.5933 −1.55981
\(932\) −32.0533 −1.04994
\(933\) −13.9920 −0.458078
\(934\) −6.55072 −0.214346
\(935\) 3.98319 0.130264
\(936\) −24.5907 −0.803773
\(937\) 22.1573 0.723848 0.361924 0.932208i \(-0.382120\pi\)
0.361924 + 0.932208i \(0.382120\pi\)
\(938\) 6.93563 0.226456
\(939\) −65.0005 −2.12121
\(940\) −8.07165 −0.263268
\(941\) 2.99531 0.0976443 0.0488221 0.998807i \(-0.484453\pi\)
0.0488221 + 0.998807i \(0.484453\pi\)
\(942\) −25.8507 −0.842261
\(943\) 1.37523 0.0447838
\(944\) 32.4861 1.05733
\(945\) 30.5797 0.994759
\(946\) −1.79522 −0.0583676
\(947\) −30.1720 −0.980459 −0.490230 0.871593i \(-0.663087\pi\)
−0.490230 + 0.871593i \(0.663087\pi\)
\(948\) −54.6363 −1.77451
\(949\) 2.54805 0.0827132
\(950\) 2.37348 0.0770059
\(951\) −48.9208 −1.58637
\(952\) −27.5640 −0.893354
\(953\) −30.0505 −0.973432 −0.486716 0.873560i \(-0.661805\pi\)
−0.486716 + 0.873560i \(0.661805\pi\)
\(954\) −1.49387 −0.0483660
\(955\) 11.0526 0.357654
\(956\) −3.05879 −0.0989282
\(957\) −19.0017 −0.614237
\(958\) −16.3637 −0.528686
\(959\) −6.39083 −0.206371
\(960\) −10.1487 −0.327548
\(961\) 31.2643 1.00853
\(962\) 10.3665 0.334230
\(963\) 17.7007 0.570397
\(964\) 25.3883 0.817701
\(965\) 1.62235 0.0522252
\(966\) −1.83476 −0.0590324
\(967\) −38.6126 −1.24170 −0.620848 0.783931i \(-0.713212\pi\)
−0.620848 + 0.783931i \(0.713212\pi\)
\(968\) 1.72395 0.0554100
\(969\) 60.9998 1.95960
\(970\) 3.58540 0.115120
\(971\) −15.8100 −0.507369 −0.253684 0.967287i \(-0.581642\pi\)
−0.253684 + 0.967287i \(0.581642\pi\)
\(972\) 11.8330 0.379544
\(973\) −59.8143 −1.91756
\(974\) −19.7589 −0.633117
\(975\) −7.47151 −0.239280
\(976\) 25.4630 0.815049
\(977\) 18.9126 0.605066 0.302533 0.953139i \(-0.402168\pi\)
0.302533 + 0.953139i \(0.402168\pi\)
\(978\) 5.18319 0.165740
\(979\) −15.2413 −0.487115
\(980\) −16.3435 −0.522072
\(981\) 13.1335 0.419320
\(982\) 7.02021 0.224024
\(983\) −50.9819 −1.62607 −0.813035 0.582215i \(-0.802186\pi\)
−0.813035 + 0.582215i \(0.802186\pi\)
\(984\) 20.2674 0.646100
\(985\) 16.2364 0.517336
\(986\) 11.7304 0.373572
\(987\) 52.9728 1.68614
\(988\) −23.8671 −0.759314
\(989\) −1.35498 −0.0430860
\(990\) 2.54405 0.0808554
\(991\) −27.7207 −0.880578 −0.440289 0.897856i \(-0.645124\pi\)
−0.440289 + 0.897856i \(0.645124\pi\)
\(992\) −37.2599 −1.18300
\(993\) −83.9945 −2.66548
\(994\) 1.26950 0.0402661
\(995\) −20.4639 −0.648750
\(996\) 48.3647 1.53250
\(997\) −12.8864 −0.408116 −0.204058 0.978959i \(-0.565413\pi\)
−0.204058 + 0.978959i \(0.565413\pi\)
\(998\) 5.19665 0.164497
\(999\) 68.2001 2.15776
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 4015.2.a.i.1.15 38
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4015.2.a.i.1.15 38 1.1 even 1 trivial