Properties

Label 1339.2.a.c.1.1
Level $1339$
Weight $2$
Character 1339.1
Self dual yes
Analytic conductor $10.692$
Analytic rank $1$
Dimension $3$
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] = [1339,2,Mod(1,1339)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1339, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1339.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1339 = 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1339.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: \(10.6919688306\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.17009\) of defining polynomial
Character \(\chi\) \(=\) 1339.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.17009 q^{2} -2.70928 q^{3} +2.70928 q^{4} -4.17009 q^{5} +5.87936 q^{6} -1.70928 q^{7} -1.53919 q^{8} +4.34017 q^{9} +O(q^{10})\) \(q-2.17009 q^{2} -2.70928 q^{3} +2.70928 q^{4} -4.17009 q^{5} +5.87936 q^{6} -1.70928 q^{7} -1.53919 q^{8} +4.34017 q^{9} +9.04945 q^{10} -4.63090 q^{11} -7.34017 q^{12} +1.00000 q^{13} +3.70928 q^{14} +11.2979 q^{15} -2.07838 q^{16} -4.41855 q^{17} -9.41855 q^{18} -4.80098 q^{19} -11.2979 q^{20} +4.63090 q^{21} +10.0494 q^{22} +7.60197 q^{23} +4.17009 q^{24} +12.3896 q^{25} -2.17009 q^{26} -3.63090 q^{27} -4.63090 q^{28} -3.29072 q^{29} -24.5174 q^{30} +5.75872 q^{31} +7.58864 q^{32} +12.5464 q^{33} +9.58864 q^{34} +7.12783 q^{35} +11.7587 q^{36} +7.58864 q^{37} +10.4186 q^{38} -2.70928 q^{39} +6.41855 q^{40} -5.26180 q^{41} -10.0494 q^{42} +6.34017 q^{43} -12.5464 q^{44} -18.0989 q^{45} -16.4969 q^{46} +8.38962 q^{47} +5.63090 q^{48} -4.07838 q^{49} -26.8865 q^{50} +11.9711 q^{51} +2.70928 q^{52} +2.04945 q^{53} +7.87936 q^{54} +19.3112 q^{55} +2.63090 q^{56} +13.0072 q^{57} +7.14116 q^{58} -11.7721 q^{59} +30.6092 q^{60} +10.1278 q^{61} -12.4969 q^{62} -7.41855 q^{63} -12.3112 q^{64} -4.17009 q^{65} -27.2267 q^{66} -4.68035 q^{67} -11.9711 q^{68} -20.5958 q^{69} -15.4680 q^{70} +8.52359 q^{71} -6.68035 q^{72} -15.6875 q^{73} -16.4680 q^{74} -33.5669 q^{75} -13.0072 q^{76} +7.91548 q^{77} +5.87936 q^{78} +13.0205 q^{79} +8.66701 q^{80} -3.18342 q^{81} +11.4186 q^{82} +0.326842 q^{83} +12.5464 q^{84} +18.4257 q^{85} -13.7587 q^{86} +8.91548 q^{87} +7.12783 q^{88} +1.06505 q^{89} +39.2762 q^{90} -1.70928 q^{91} +20.5958 q^{92} -15.6020 q^{93} -18.2062 q^{94} +20.0205 q^{95} -20.5597 q^{96} +4.73820 q^{97} +8.85043 q^{98} -20.0989 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} - q^{3} + q^{4} - 7 q^{5} + 5 q^{6} + 2 q^{7} - 3 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} - q^{3} + q^{4} - 7 q^{5} + 5 q^{6} + 2 q^{7} - 3 q^{8} + 2 q^{9} + 9 q^{10} - 10 q^{11} - 11 q^{12} + 3 q^{13} + 4 q^{14} + 7 q^{15} - 3 q^{16} + q^{17} - 14 q^{18} - 5 q^{19} - 7 q^{20} + 10 q^{21} + 12 q^{22} + 4 q^{23} + 7 q^{24} + 8 q^{25} - q^{26} - 7 q^{27} - 10 q^{28} - 17 q^{29} - 23 q^{30} - 8 q^{31} + 3 q^{32} + 2 q^{33} + 9 q^{34} + 10 q^{36} + 3 q^{37} + 17 q^{38} - q^{39} + 5 q^{40} - 8 q^{41} - 12 q^{42} + 8 q^{43} - 2 q^{44} - 18 q^{45} - 32 q^{46} - 4 q^{47} + 13 q^{48} - 9 q^{49} - 34 q^{50} + 21 q^{51} + q^{52} - 12 q^{53} + 11 q^{54} + 32 q^{55} + 4 q^{56} + 5 q^{57} + q^{58} - 11 q^{59} + 39 q^{60} + 9 q^{61} - 20 q^{62} - 8 q^{63} - 11 q^{64} - 7 q^{65} - 24 q^{66} + 8 q^{67} - 21 q^{68} - 8 q^{69} - 14 q^{70} + 10 q^{71} + 2 q^{72} + 9 q^{73} - 17 q^{74} - 32 q^{75} - 5 q^{76} - 8 q^{77} + 5 q^{78} + 6 q^{79} + 3 q^{80} - 5 q^{81} + 20 q^{82} - 11 q^{83} + 2 q^{84} + 7 q^{85} - 16 q^{86} - 5 q^{87} - q^{89} + 42 q^{90} + 2 q^{91} + 8 q^{92} - 28 q^{93} - 30 q^{94} + 27 q^{95} - 27 q^{96} + 22 q^{97} - q^{98} - 24 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.17009 −1.53448 −0.767241 0.641358i \(-0.778371\pi\)
−0.767241 + 0.641358i \(0.778371\pi\)
\(3\) −2.70928 −1.56420 −0.782100 0.623152i \(-0.785852\pi\)
−0.782100 + 0.623152i \(0.785852\pi\)
\(4\) 2.70928 1.35464
\(5\) −4.17009 −1.86492 −0.932460 0.361274i \(-0.882342\pi\)
−0.932460 + 0.361274i \(0.882342\pi\)
\(6\) 5.87936 2.40024
\(7\) −1.70928 −0.646045 −0.323023 0.946391i \(-0.604699\pi\)
−0.323023 + 0.946391i \(0.604699\pi\)
\(8\) −1.53919 −0.544185
\(9\) 4.34017 1.44672
\(10\) 9.04945 2.86169
\(11\) −4.63090 −1.39627 −0.698134 0.715967i \(-0.745986\pi\)
−0.698134 + 0.715967i \(0.745986\pi\)
\(12\) −7.34017 −2.11893
\(13\) 1.00000 0.277350
\(14\) 3.70928 0.991346
\(15\) 11.2979 2.91711
\(16\) −2.07838 −0.519594
\(17\) −4.41855 −1.07166 −0.535828 0.844327i \(-0.680000\pi\)
−0.535828 + 0.844327i \(0.680000\pi\)
\(18\) −9.41855 −2.21997
\(19\) −4.80098 −1.10142 −0.550711 0.834696i \(-0.685643\pi\)
−0.550711 + 0.834696i \(0.685643\pi\)
\(20\) −11.2979 −2.52629
\(21\) 4.63090 1.01054
\(22\) 10.0494 2.14255
\(23\) 7.60197 1.58512 0.792560 0.609794i \(-0.208748\pi\)
0.792560 + 0.609794i \(0.208748\pi\)
\(24\) 4.17009 0.851215
\(25\) 12.3896 2.47792
\(26\) −2.17009 −0.425589
\(27\) −3.63090 −0.698767
\(28\) −4.63090 −0.875157
\(29\) −3.29072 −0.611072 −0.305536 0.952180i \(-0.598836\pi\)
−0.305536 + 0.952180i \(0.598836\pi\)
\(30\) −24.5174 −4.47625
\(31\) 5.75872 1.03430 0.517149 0.855896i \(-0.326993\pi\)
0.517149 + 0.855896i \(0.326993\pi\)
\(32\) 7.58864 1.34149
\(33\) 12.5464 2.18404
\(34\) 9.58864 1.64444
\(35\) 7.12783 1.20482
\(36\) 11.7587 1.95979
\(37\) 7.58864 1.24756 0.623782 0.781598i \(-0.285595\pi\)
0.623782 + 0.781598i \(0.285595\pi\)
\(38\) 10.4186 1.69011
\(39\) −2.70928 −0.433831
\(40\) 6.41855 1.01486
\(41\) −5.26180 −0.821754 −0.410877 0.911691i \(-0.634777\pi\)
−0.410877 + 0.911691i \(0.634777\pi\)
\(42\) −10.0494 −1.55066
\(43\) 6.34017 0.966867 0.483434 0.875381i \(-0.339389\pi\)
0.483434 + 0.875381i \(0.339389\pi\)
\(44\) −12.5464 −1.89144
\(45\) −18.0989 −2.69802
\(46\) −16.4969 −2.43234
\(47\) 8.38962 1.22375 0.611876 0.790954i \(-0.290415\pi\)
0.611876 + 0.790954i \(0.290415\pi\)
\(48\) 5.63090 0.812750
\(49\) −4.07838 −0.582625
\(50\) −26.8865 −3.80233
\(51\) 11.9711 1.67629
\(52\) 2.70928 0.375709
\(53\) 2.04945 0.281513 0.140757 0.990044i \(-0.455046\pi\)
0.140757 + 0.990044i \(0.455046\pi\)
\(54\) 7.87936 1.07225
\(55\) 19.3112 2.60393
\(56\) 2.63090 0.351568
\(57\) 13.0072 1.72284
\(58\) 7.14116 0.937680
\(59\) −11.7721 −1.53259 −0.766295 0.642488i \(-0.777902\pi\)
−0.766295 + 0.642488i \(0.777902\pi\)
\(60\) 30.6092 3.95163
\(61\) 10.1278 1.29674 0.648368 0.761327i \(-0.275452\pi\)
0.648368 + 0.761327i \(0.275452\pi\)
\(62\) −12.4969 −1.58711
\(63\) −7.41855 −0.934650
\(64\) −12.3112 −1.53891
\(65\) −4.17009 −0.517236
\(66\) −27.2267 −3.35138
\(67\) −4.68035 −0.571795 −0.285898 0.958260i \(-0.592292\pi\)
−0.285898 + 0.958260i \(0.592292\pi\)
\(68\) −11.9711 −1.45171
\(69\) −20.5958 −2.47945
\(70\) −15.4680 −1.84878
\(71\) 8.52359 1.01156 0.505782 0.862661i \(-0.331204\pi\)
0.505782 + 0.862661i \(0.331204\pi\)
\(72\) −6.68035 −0.787286
\(73\) −15.6875 −1.83609 −0.918043 0.396480i \(-0.870232\pi\)
−0.918043 + 0.396480i \(0.870232\pi\)
\(74\) −16.4680 −1.91437
\(75\) −33.5669 −3.87597
\(76\) −13.0072 −1.49203
\(77\) 7.91548 0.902053
\(78\) 5.87936 0.665707
\(79\) 13.0205 1.46492 0.732461 0.680809i \(-0.238371\pi\)
0.732461 + 0.680809i \(0.238371\pi\)
\(80\) 8.66701 0.969002
\(81\) −3.18342 −0.353713
\(82\) 11.4186 1.26097
\(83\) 0.326842 0.0358756 0.0179378 0.999839i \(-0.494290\pi\)
0.0179378 + 0.999839i \(0.494290\pi\)
\(84\) 12.5464 1.36892
\(85\) 18.4257 1.99855
\(86\) −13.7587 −1.48364
\(87\) 8.91548 0.955840
\(88\) 7.12783 0.759829
\(89\) 1.06505 0.112895 0.0564474 0.998406i \(-0.482023\pi\)
0.0564474 + 0.998406i \(0.482023\pi\)
\(90\) 39.2762 4.14007
\(91\) −1.70928 −0.179181
\(92\) 20.5958 2.14726
\(93\) −15.6020 −1.61785
\(94\) −18.2062 −1.87783
\(95\) 20.0205 2.05406
\(96\) −20.5597 −2.09837
\(97\) 4.73820 0.481092 0.240546 0.970638i \(-0.422674\pi\)
0.240546 + 0.970638i \(0.422674\pi\)
\(98\) 8.85043 0.894029
\(99\) −20.0989 −2.02002
\(100\) 33.5669 3.35669
\(101\) −14.4969 −1.44250 −0.721249 0.692676i \(-0.756432\pi\)
−0.721249 + 0.692676i \(0.756432\pi\)
\(102\) −25.9783 −2.57223
\(103\) 1.00000 0.0985329
\(104\) −1.53919 −0.150930
\(105\) −19.3112 −1.88458
\(106\) −4.44748 −0.431977
\(107\) 7.07838 0.684293 0.342146 0.939647i \(-0.388846\pi\)
0.342146 + 0.939647i \(0.388846\pi\)
\(108\) −9.83710 −0.946576
\(109\) 3.21235 0.307687 0.153844 0.988095i \(-0.450835\pi\)
0.153844 + 0.988095i \(0.450835\pi\)
\(110\) −41.9071 −3.99568
\(111\) −20.5597 −1.95144
\(112\) 3.55252 0.335682
\(113\) 0.340173 0.0320008 0.0160004 0.999872i \(-0.494907\pi\)
0.0160004 + 0.999872i \(0.494907\pi\)
\(114\) −28.2267 −2.64367
\(115\) −31.7009 −2.95612
\(116\) −8.91548 −0.827781
\(117\) 4.34017 0.401249
\(118\) 25.5464 2.35173
\(119\) 7.55252 0.692338
\(120\) −17.3896 −1.58745
\(121\) 10.4452 0.949565
\(122\) −21.9783 −1.98982
\(123\) 14.2557 1.28539
\(124\) 15.6020 1.40110
\(125\) −30.8154 −2.75621
\(126\) 16.0989 1.43420
\(127\) −15.0989 −1.33981 −0.669905 0.742447i \(-0.733665\pi\)
−0.669905 + 0.742447i \(0.733665\pi\)
\(128\) 11.5392 1.01993
\(129\) −17.1773 −1.51237
\(130\) 9.04945 0.793689
\(131\) −0.183417 −0.0160253 −0.00801263 0.999968i \(-0.502551\pi\)
−0.00801263 + 0.999968i \(0.502551\pi\)
\(132\) 33.9916 2.95859
\(133\) 8.20620 0.711568
\(134\) 10.1568 0.877410
\(135\) 15.1412 1.30314
\(136\) 6.80098 0.583180
\(137\) −7.95055 −0.679261 −0.339631 0.940559i \(-0.610302\pi\)
−0.339631 + 0.940559i \(0.610302\pi\)
\(138\) 44.6947 3.80467
\(139\) 9.75872 0.827724 0.413862 0.910340i \(-0.364180\pi\)
0.413862 + 0.910340i \(0.364180\pi\)
\(140\) 19.3112 1.63210
\(141\) −22.7298 −1.91419
\(142\) −18.4969 −1.55223
\(143\) −4.63090 −0.387255
\(144\) −9.02052 −0.751710
\(145\) 13.7226 1.13960
\(146\) 34.0433 2.81744
\(147\) 11.0494 0.911343
\(148\) 20.5597 1.69000
\(149\) −19.8082 −1.62275 −0.811374 0.584527i \(-0.801280\pi\)
−0.811374 + 0.584527i \(0.801280\pi\)
\(150\) 72.8431 5.94761
\(151\) 20.5958 1.67606 0.838032 0.545621i \(-0.183706\pi\)
0.838032 + 0.545621i \(0.183706\pi\)
\(152\) 7.38962 0.599377
\(153\) −19.1773 −1.55039
\(154\) −17.1773 −1.38418
\(155\) −24.0144 −1.92888
\(156\) −7.34017 −0.587684
\(157\) 0.133969 0.0106919 0.00534595 0.999986i \(-0.498298\pi\)
0.00534595 + 0.999986i \(0.498298\pi\)
\(158\) −28.2557 −2.24790
\(159\) −5.55252 −0.440343
\(160\) −31.6453 −2.50178
\(161\) −12.9939 −1.02406
\(162\) 6.90829 0.542767
\(163\) −18.3268 −1.43547 −0.717735 0.696317i \(-0.754821\pi\)
−0.717735 + 0.696317i \(0.754821\pi\)
\(164\) −14.2557 −1.11318
\(165\) −52.3195 −4.07307
\(166\) −0.709275 −0.0550504
\(167\) 14.5597 1.12666 0.563332 0.826231i \(-0.309519\pi\)
0.563332 + 0.826231i \(0.309519\pi\)
\(168\) −7.12783 −0.549924
\(169\) 1.00000 0.0769231
\(170\) −39.9854 −3.06674
\(171\) −20.8371 −1.59345
\(172\) 17.1773 1.30975
\(173\) 17.1773 1.30596 0.652982 0.757373i \(-0.273518\pi\)
0.652982 + 0.757373i \(0.273518\pi\)
\(174\) −19.3474 −1.46672
\(175\) −21.1773 −1.60085
\(176\) 9.62475 0.725493
\(177\) 31.8937 2.39728
\(178\) −2.31124 −0.173235
\(179\) −12.8865 −0.963186 −0.481593 0.876395i \(-0.659942\pi\)
−0.481593 + 0.876395i \(0.659942\pi\)
\(180\) −49.0349 −3.65485
\(181\) −1.21235 −0.0901130 −0.0450565 0.998984i \(-0.514347\pi\)
−0.0450565 + 0.998984i \(0.514347\pi\)
\(182\) 3.70928 0.274950
\(183\) −27.4391 −2.02835
\(184\) −11.7009 −0.862599
\(185\) −31.6453 −2.32661
\(186\) 33.8576 2.48256
\(187\) 20.4619 1.49632
\(188\) 22.7298 1.65774
\(189\) 6.20620 0.451435
\(190\) −43.4463 −3.15192
\(191\) 6.89269 0.498738 0.249369 0.968409i \(-0.419777\pi\)
0.249369 + 0.968409i \(0.419777\pi\)
\(192\) 33.3545 2.40716
\(193\) 3.22568 0.232189 0.116095 0.993238i \(-0.462962\pi\)
0.116095 + 0.993238i \(0.462962\pi\)
\(194\) −10.2823 −0.738227
\(195\) 11.2979 0.809060
\(196\) −11.0494 −0.789246
\(197\) 8.77432 0.625145 0.312572 0.949894i \(-0.398809\pi\)
0.312572 + 0.949894i \(0.398809\pi\)
\(198\) 43.6163 3.09968
\(199\) −20.6020 −1.46043 −0.730217 0.683215i \(-0.760581\pi\)
−0.730217 + 0.683215i \(0.760581\pi\)
\(200\) −19.0700 −1.34845
\(201\) 12.6803 0.894403
\(202\) 31.4596 2.21349
\(203\) 5.62475 0.394780
\(204\) 32.4329 2.27076
\(205\) 21.9421 1.53251
\(206\) −2.17009 −0.151197
\(207\) 32.9939 2.29323
\(208\) −2.07838 −0.144110
\(209\) 22.2329 1.53788
\(210\) 41.9071 2.89186
\(211\) −4.85762 −0.334412 −0.167206 0.985922i \(-0.553475\pi\)
−0.167206 + 0.985922i \(0.553475\pi\)
\(212\) 5.55252 0.381349
\(213\) −23.0928 −1.58229
\(214\) −15.3607 −1.05004
\(215\) −26.4391 −1.80313
\(216\) 5.58864 0.380259
\(217\) −9.84324 −0.668203
\(218\) −6.97107 −0.472140
\(219\) 42.5018 2.87201
\(220\) 52.3195 3.52738
\(221\) −4.41855 −0.297224
\(222\) 44.6163 2.99445
\(223\) −5.05664 −0.338617 −0.169309 0.985563i \(-0.554153\pi\)
−0.169309 + 0.985563i \(0.554153\pi\)
\(224\) −12.9711 −0.866666
\(225\) 53.7731 3.58487
\(226\) −0.738205 −0.0491047
\(227\) 16.9360 1.12408 0.562041 0.827109i \(-0.310016\pi\)
0.562041 + 0.827109i \(0.310016\pi\)
\(228\) 35.2401 2.33383
\(229\) −1.75872 −0.116220 −0.0581099 0.998310i \(-0.518507\pi\)
−0.0581099 + 0.998310i \(0.518507\pi\)
\(230\) 68.7936 4.53612
\(231\) −21.4452 −1.41099
\(232\) 5.06505 0.332537
\(233\) 23.9916 1.57174 0.785871 0.618391i \(-0.212215\pi\)
0.785871 + 0.618391i \(0.212215\pi\)
\(234\) −9.41855 −0.615710
\(235\) −34.9854 −2.28220
\(236\) −31.8937 −2.07611
\(237\) −35.2762 −2.29143
\(238\) −16.3896 −1.06238
\(239\) −12.3535 −0.799082 −0.399541 0.916715i \(-0.630830\pi\)
−0.399541 + 0.916715i \(0.630830\pi\)
\(240\) −23.4813 −1.51571
\(241\) 3.00719 0.193710 0.0968550 0.995299i \(-0.469122\pi\)
0.0968550 + 0.995299i \(0.469122\pi\)
\(242\) −22.6670 −1.45709
\(243\) 19.5174 1.25204
\(244\) 27.4391 1.75661
\(245\) 17.0072 1.08655
\(246\) −30.9360 −1.97241
\(247\) −4.80098 −0.305479
\(248\) −8.86376 −0.562850
\(249\) −0.885505 −0.0561166
\(250\) 66.8720 4.22936
\(251\) 16.6781 1.05271 0.526355 0.850265i \(-0.323558\pi\)
0.526355 + 0.850265i \(0.323558\pi\)
\(252\) −20.0989 −1.26611
\(253\) −35.2039 −2.21325
\(254\) 32.7659 2.05592
\(255\) −49.9204 −3.12614
\(256\) −0.418551 −0.0261594
\(257\) −16.1217 −1.00564 −0.502821 0.864390i \(-0.667705\pi\)
−0.502821 + 0.864390i \(0.667705\pi\)
\(258\) 37.2762 2.32071
\(259\) −12.9711 −0.805983
\(260\) −11.2979 −0.700667
\(261\) −14.2823 −0.884053
\(262\) 0.398032 0.0245905
\(263\) 5.30510 0.327127 0.163563 0.986533i \(-0.447701\pi\)
0.163563 + 0.986533i \(0.447701\pi\)
\(264\) −19.3112 −1.18852
\(265\) −8.54638 −0.525000
\(266\) −17.8082 −1.09189
\(267\) −2.88550 −0.176590
\(268\) −12.6803 −0.774575
\(269\) 25.3340 1.54464 0.772322 0.635232i \(-0.219095\pi\)
0.772322 + 0.635232i \(0.219095\pi\)
\(270\) −32.8576 −1.99965
\(271\) 13.4947 0.819742 0.409871 0.912143i \(-0.365574\pi\)
0.409871 + 0.912143i \(0.365574\pi\)
\(272\) 9.18342 0.556826
\(273\) 4.63090 0.280275
\(274\) 17.2534 1.04232
\(275\) −57.3751 −3.45985
\(276\) −55.7998 −3.35875
\(277\) 3.07838 0.184962 0.0924809 0.995714i \(-0.470520\pi\)
0.0924809 + 0.995714i \(0.470520\pi\)
\(278\) −21.1773 −1.27013
\(279\) 24.9939 1.49634
\(280\) −10.9711 −0.655647
\(281\) 8.64423 0.515671 0.257836 0.966189i \(-0.416991\pi\)
0.257836 + 0.966189i \(0.416991\pi\)
\(282\) 49.3256 2.93730
\(283\) −1.04945 −0.0623832 −0.0311916 0.999513i \(-0.509930\pi\)
−0.0311916 + 0.999513i \(0.509930\pi\)
\(284\) 23.0928 1.37030
\(285\) −54.2411 −3.21297
\(286\) 10.0494 0.594236
\(287\) 8.99386 0.530891
\(288\) 32.9360 1.94077
\(289\) 2.52359 0.148446
\(290\) −29.7792 −1.74870
\(291\) −12.8371 −0.752524
\(292\) −42.5018 −2.48723
\(293\) 11.2123 0.655032 0.327516 0.944846i \(-0.393788\pi\)
0.327516 + 0.944846i \(0.393788\pi\)
\(294\) −23.9783 −1.39844
\(295\) 49.0905 2.85816
\(296\) −11.6803 −0.678906
\(297\) 16.8143 0.975666
\(298\) 42.9854 2.49008
\(299\) 7.60197 0.439633
\(300\) −90.9420 −5.25054
\(301\) −10.8371 −0.624640
\(302\) −44.6947 −2.57189
\(303\) 39.2762 2.25636
\(304\) 9.97826 0.572292
\(305\) −42.2339 −2.41831
\(306\) 41.6163 2.37905
\(307\) −2.21008 −0.126136 −0.0630679 0.998009i \(-0.520088\pi\)
−0.0630679 + 0.998009i \(0.520088\pi\)
\(308\) 21.4452 1.22195
\(309\) −2.70928 −0.154125
\(310\) 52.1133 2.95983
\(311\) −5.65142 −0.320462 −0.160231 0.987080i \(-0.551224\pi\)
−0.160231 + 0.987080i \(0.551224\pi\)
\(312\) 4.17009 0.236085
\(313\) −29.3318 −1.65793 −0.828965 0.559301i \(-0.811070\pi\)
−0.828965 + 0.559301i \(0.811070\pi\)
\(314\) −0.290725 −0.0164065
\(315\) 30.9360 1.74305
\(316\) 35.2762 1.98444
\(317\) −1.40417 −0.0788663 −0.0394332 0.999222i \(-0.512555\pi\)
−0.0394332 + 0.999222i \(0.512555\pi\)
\(318\) 12.0494 0.675700
\(319\) 15.2390 0.853221
\(320\) 51.3390 2.86993
\(321\) −19.1773 −1.07037
\(322\) 28.1978 1.57140
\(323\) 21.2134 1.18034
\(324\) −8.62475 −0.479153
\(325\) 12.3896 0.687253
\(326\) 39.7708 2.20270
\(327\) −8.70313 −0.481284
\(328\) 8.09890 0.447187
\(329\) −14.3402 −0.790599
\(330\) 113.538 6.25005
\(331\) −3.39576 −0.186648 −0.0933240 0.995636i \(-0.529749\pi\)
−0.0933240 + 0.995636i \(0.529749\pi\)
\(332\) 0.885505 0.0485984
\(333\) 32.9360 1.80488
\(334\) −31.5958 −1.72885
\(335\) 19.5174 1.06635
\(336\) −9.62475 −0.525073
\(337\) 29.9132 1.62948 0.814738 0.579829i \(-0.196881\pi\)
0.814738 + 0.579829i \(0.196881\pi\)
\(338\) −2.17009 −0.118037
\(339\) −0.921622 −0.0500556
\(340\) 49.9204 2.70731
\(341\) −26.6681 −1.44416
\(342\) 45.2183 2.44513
\(343\) 18.9360 1.02245
\(344\) −9.75872 −0.526155
\(345\) 85.8864 4.62397
\(346\) −37.2762 −2.00398
\(347\) −6.05786 −0.325203 −0.162601 0.986692i \(-0.551988\pi\)
−0.162601 + 0.986692i \(0.551988\pi\)
\(348\) 24.1545 1.29482
\(349\) 14.3451 0.767875 0.383938 0.923359i \(-0.374568\pi\)
0.383938 + 0.923359i \(0.374568\pi\)
\(350\) 45.9565 2.45648
\(351\) −3.63090 −0.193803
\(352\) −35.1422 −1.87309
\(353\) −16.2918 −0.867123 −0.433562 0.901124i \(-0.642743\pi\)
−0.433562 + 0.901124i \(0.642743\pi\)
\(354\) −69.2122 −3.67859
\(355\) −35.5441 −1.88649
\(356\) 2.88550 0.152931
\(357\) −20.4619 −1.08296
\(358\) 27.9649 1.47799
\(359\) 10.8722 0.573811 0.286906 0.957959i \(-0.407373\pi\)
0.286906 + 0.957959i \(0.407373\pi\)
\(360\) 27.8576 1.46823
\(361\) 4.04945 0.213129
\(362\) 2.63090 0.138277
\(363\) −28.2990 −1.48531
\(364\) −4.63090 −0.242725
\(365\) 65.4184 3.42415
\(366\) 59.5452 3.11248
\(367\) 17.7093 0.924417 0.462208 0.886771i \(-0.347057\pi\)
0.462208 + 0.886771i \(0.347057\pi\)
\(368\) −15.7998 −0.823620
\(369\) −22.8371 −1.18885
\(370\) 68.6730 3.57014
\(371\) −3.50307 −0.181870
\(372\) −42.2700 −2.19160
\(373\) −21.8166 −1.12962 −0.564810 0.825221i \(-0.691051\pi\)
−0.564810 + 0.825221i \(0.691051\pi\)
\(374\) −44.4040 −2.29608
\(375\) 83.4873 4.31127
\(376\) −12.9132 −0.665948
\(377\) −3.29072 −0.169481
\(378\) −13.4680 −0.692719
\(379\) 34.5464 1.77453 0.887264 0.461262i \(-0.152603\pi\)
0.887264 + 0.461262i \(0.152603\pi\)
\(380\) 54.2411 2.78251
\(381\) 40.9071 2.09573
\(382\) −14.9577 −0.765305
\(383\) 21.1506 1.08075 0.540373 0.841426i \(-0.318283\pi\)
0.540373 + 0.841426i \(0.318283\pi\)
\(384\) −31.2628 −1.59538
\(385\) −33.0082 −1.68226
\(386\) −7.00000 −0.356291
\(387\) 27.5174 1.39879
\(388\) 12.8371 0.651705
\(389\) −25.0700 −1.27110 −0.635549 0.772061i \(-0.719226\pi\)
−0.635549 + 0.772061i \(0.719226\pi\)
\(390\) −24.5174 −1.24149
\(391\) −33.5897 −1.69870
\(392\) 6.27739 0.317056
\(393\) 0.496928 0.0250667
\(394\) −19.0410 −0.959274
\(395\) −54.2967 −2.73196
\(396\) −54.4534 −2.73639
\(397\) −36.0010 −1.80684 −0.903420 0.428756i \(-0.858952\pi\)
−0.903420 + 0.428756i \(0.858952\pi\)
\(398\) 44.7081 2.24101
\(399\) −22.2329 −1.11304
\(400\) −25.7503 −1.28752
\(401\) −7.17727 −0.358416 −0.179208 0.983811i \(-0.557354\pi\)
−0.179208 + 0.983811i \(0.557354\pi\)
\(402\) −27.5174 −1.37245
\(403\) 5.75872 0.286862
\(404\) −39.2762 −1.95406
\(405\) 13.2751 0.659646
\(406\) −12.2062 −0.605784
\(407\) −35.1422 −1.74193
\(408\) −18.4257 −0.912210
\(409\) 19.5669 0.967521 0.483760 0.875200i \(-0.339271\pi\)
0.483760 + 0.875200i \(0.339271\pi\)
\(410\) −47.6163 −2.35160
\(411\) 21.5402 1.06250
\(412\) 2.70928 0.133476
\(413\) 20.1217 0.990123
\(414\) −71.5995 −3.51892
\(415\) −1.36296 −0.0669050
\(416\) 7.58864 0.372064
\(417\) −26.4391 −1.29473
\(418\) −48.2472 −2.35985
\(419\) −16.7877 −0.820131 −0.410065 0.912056i \(-0.634494\pi\)
−0.410065 + 0.912056i \(0.634494\pi\)
\(420\) −52.3195 −2.55293
\(421\) −6.89496 −0.336040 −0.168020 0.985784i \(-0.553737\pi\)
−0.168020 + 0.985784i \(0.553737\pi\)
\(422\) 10.5415 0.513150
\(423\) 36.4124 1.77043
\(424\) −3.15449 −0.153195
\(425\) −54.7442 −2.65548
\(426\) 50.1133 2.42800
\(427\) −17.3112 −0.837750
\(428\) 19.1773 0.926969
\(429\) 12.5464 0.605745
\(430\) 57.3751 2.76687
\(431\) 20.6765 0.995951 0.497975 0.867191i \(-0.334077\pi\)
0.497975 + 0.867191i \(0.334077\pi\)
\(432\) 7.54638 0.363075
\(433\) −13.3919 −0.643573 −0.321787 0.946812i \(-0.604283\pi\)
−0.321787 + 0.946812i \(0.604283\pi\)
\(434\) 21.3607 1.02535
\(435\) −37.1783 −1.78256
\(436\) 8.70313 0.416804
\(437\) −36.4969 −1.74588
\(438\) −92.2327 −4.40705
\(439\) 5.44521 0.259886 0.129943 0.991521i \(-0.458521\pi\)
0.129943 + 0.991521i \(0.458521\pi\)
\(440\) −29.7237 −1.41702
\(441\) −17.7009 −0.842898
\(442\) 9.58864 0.456085
\(443\) 37.3051 1.77242 0.886209 0.463285i \(-0.153329\pi\)
0.886209 + 0.463285i \(0.153329\pi\)
\(444\) −55.7019 −2.64350
\(445\) −4.44134 −0.210540
\(446\) 10.9733 0.519603
\(447\) 53.6658 2.53830
\(448\) 21.0433 0.994203
\(449\) 18.9977 0.896558 0.448279 0.893894i \(-0.352037\pi\)
0.448279 + 0.893894i \(0.352037\pi\)
\(450\) −116.692 −5.50093
\(451\) 24.3668 1.14739
\(452\) 0.921622 0.0433495
\(453\) −55.7998 −2.62170
\(454\) −36.7526 −1.72488
\(455\) 7.12783 0.334158
\(456\) −20.0205 −0.937547
\(457\) −26.6309 −1.24574 −0.622870 0.782325i \(-0.714034\pi\)
−0.622870 + 0.782325i \(0.714034\pi\)
\(458\) 3.81658 0.178337
\(459\) 16.0433 0.748837
\(460\) −85.8864 −4.00447
\(461\) 14.9444 0.696030 0.348015 0.937489i \(-0.386856\pi\)
0.348015 + 0.937489i \(0.386856\pi\)
\(462\) 46.5380 2.16514
\(463\) 7.18115 0.333736 0.166868 0.985979i \(-0.446635\pi\)
0.166868 + 0.985979i \(0.446635\pi\)
\(464\) 6.83937 0.317510
\(465\) 65.0616 3.01716
\(466\) −52.0638 −2.41181
\(467\) −20.3896 −0.943519 −0.471760 0.881727i \(-0.656381\pi\)
−0.471760 + 0.881727i \(0.656381\pi\)
\(468\) 11.7587 0.543547
\(469\) 8.00000 0.369406
\(470\) 75.9214 3.50200
\(471\) −0.362959 −0.0167243
\(472\) 18.1194 0.834014
\(473\) −29.3607 −1.35001
\(474\) 76.5523 3.51617
\(475\) −59.4824 −2.72924
\(476\) 20.4619 0.937868
\(477\) 8.89496 0.407272
\(478\) 26.8082 1.22618
\(479\) 10.6947 0.488654 0.244327 0.969693i \(-0.421433\pi\)
0.244327 + 0.969693i \(0.421433\pi\)
\(480\) 85.7358 3.91328
\(481\) 7.58864 0.346012
\(482\) −6.52586 −0.297245
\(483\) 35.2039 1.60183
\(484\) 28.2990 1.28632
\(485\) −19.7587 −0.897197
\(486\) −42.3545 −1.92124
\(487\) −39.3295 −1.78219 −0.891095 0.453817i \(-0.850062\pi\)
−0.891095 + 0.453817i \(0.850062\pi\)
\(488\) −15.5886 −0.705664
\(489\) 49.6525 2.24536
\(490\) −36.9071 −1.66729
\(491\) −17.3074 −0.781071 −0.390535 0.920588i \(-0.627710\pi\)
−0.390535 + 0.920588i \(0.627710\pi\)
\(492\) 38.6225 1.74124
\(493\) 14.5402 0.654859
\(494\) 10.4186 0.468753
\(495\) 83.8141 3.76717
\(496\) −11.9688 −0.537415
\(497\) −14.5692 −0.653516
\(498\) 1.92162 0.0861100
\(499\) −22.5380 −1.00894 −0.504469 0.863430i \(-0.668312\pi\)
−0.504469 + 0.863430i \(0.668312\pi\)
\(500\) −83.4873 −3.73367
\(501\) −39.4463 −1.76233
\(502\) −36.1929 −1.61537
\(503\) −21.8394 −0.973769 −0.486885 0.873466i \(-0.661867\pi\)
−0.486885 + 0.873466i \(0.661867\pi\)
\(504\) 11.4186 0.508623
\(505\) 60.4534 2.69014
\(506\) 76.3956 3.39620
\(507\) −2.70928 −0.120323
\(508\) −40.9071 −1.81496
\(509\) −43.7770 −1.94038 −0.970190 0.242345i \(-0.922083\pi\)
−0.970190 + 0.242345i \(0.922083\pi\)
\(510\) 108.332 4.79700
\(511\) 26.8143 1.18620
\(512\) −22.1701 −0.979789
\(513\) 17.4319 0.769636
\(514\) 34.9854 1.54314
\(515\) −4.17009 −0.183756
\(516\) −46.5380 −2.04872
\(517\) −38.8515 −1.70869
\(518\) 28.1483 1.23677
\(519\) −46.5380 −2.04279
\(520\) 6.41855 0.281472
\(521\) −8.39803 −0.367924 −0.183962 0.982933i \(-0.558892\pi\)
−0.183962 + 0.982933i \(0.558892\pi\)
\(522\) 30.9939 1.35656
\(523\) 31.6781 1.38519 0.692593 0.721329i \(-0.256468\pi\)
0.692593 + 0.721329i \(0.256468\pi\)
\(524\) −0.496928 −0.0217084
\(525\) 57.3751 2.50405
\(526\) −11.5125 −0.501970
\(527\) −25.4452 −1.10841
\(528\) −26.0761 −1.13482
\(529\) 34.7899 1.51261
\(530\) 18.5464 0.805603
\(531\) −51.0928 −2.21724
\(532\) 22.2329 0.963917
\(533\) −5.26180 −0.227914
\(534\) 6.26180 0.270974
\(535\) −29.5174 −1.27615
\(536\) 7.20394 0.311163
\(537\) 34.9132 1.50662
\(538\) −54.9770 −2.37023
\(539\) 18.8865 0.813501
\(540\) 41.0216 1.76529
\(541\) −2.48255 −0.106733 −0.0533666 0.998575i \(-0.516995\pi\)
−0.0533666 + 0.998575i \(0.516995\pi\)
\(542\) −29.2846 −1.25788
\(543\) 3.28458 0.140955
\(544\) −33.5308 −1.43762
\(545\) −13.3958 −0.573812
\(546\) −10.0494 −0.430077
\(547\) −21.5525 −0.921519 −0.460760 0.887525i \(-0.652423\pi\)
−0.460760 + 0.887525i \(0.652423\pi\)
\(548\) −21.5402 −0.920153
\(549\) 43.9565 1.87602
\(550\) 124.509 5.30908
\(551\) 15.7987 0.673048
\(552\) 31.7009 1.34928
\(553\) −22.2557 −0.946407
\(554\) −6.68035 −0.283821
\(555\) 85.7358 3.63928
\(556\) 26.4391 1.12127
\(557\) −4.58986 −0.194479 −0.0972393 0.995261i \(-0.531001\pi\)
−0.0972393 + 0.995261i \(0.531001\pi\)
\(558\) −54.2388 −2.29611
\(559\) 6.34017 0.268161
\(560\) −14.8143 −0.626019
\(561\) −55.4368 −2.34054
\(562\) −18.7587 −0.791289
\(563\) 7.97107 0.335941 0.167970 0.985792i \(-0.446279\pi\)
0.167970 + 0.985792i \(0.446279\pi\)
\(564\) −61.5813 −2.59304
\(565\) −1.41855 −0.0596789
\(566\) 2.27739 0.0957260
\(567\) 5.44134 0.228515
\(568\) −13.1194 −0.550478
\(569\) −21.9688 −0.920980 −0.460490 0.887665i \(-0.652326\pi\)
−0.460490 + 0.887665i \(0.652326\pi\)
\(570\) 117.708 4.93024
\(571\) 3.85166 0.161187 0.0805934 0.996747i \(-0.474318\pi\)
0.0805934 + 0.996747i \(0.474318\pi\)
\(572\) −12.5464 −0.524590
\(573\) −18.6742 −0.780126
\(574\) −19.5174 −0.814643
\(575\) 94.1855 3.92781
\(576\) −53.4329 −2.22637
\(577\) −5.48133 −0.228191 −0.114095 0.993470i \(-0.536397\pi\)
−0.114095 + 0.993470i \(0.536397\pi\)
\(578\) −5.47641 −0.227789
\(579\) −8.73925 −0.363191
\(580\) 37.1783 1.54375
\(581\) −0.558663 −0.0231772
\(582\) 27.8576 1.15474
\(583\) −9.49079 −0.393068
\(584\) 24.1461 0.999172
\(585\) −18.0989 −0.748297
\(586\) −24.3318 −1.00514
\(587\) 18.7877 0.775449 0.387725 0.921775i \(-0.373261\pi\)
0.387725 + 0.921775i \(0.373261\pi\)
\(588\) 29.9360 1.23454
\(589\) −27.6475 −1.13920
\(590\) −106.531 −4.38580
\(591\) −23.7721 −0.977852
\(592\) −15.7721 −0.648227
\(593\) 31.7370 1.30328 0.651641 0.758528i \(-0.274081\pi\)
0.651641 + 0.758528i \(0.274081\pi\)
\(594\) −36.4885 −1.49714
\(595\) −31.4947 −1.29116
\(596\) −53.6658 −2.19824
\(597\) 55.8164 2.28441
\(598\) −16.4969 −0.674610
\(599\) 37.3857 1.52754 0.763770 0.645488i \(-0.223346\pi\)
0.763770 + 0.645488i \(0.223346\pi\)
\(600\) 51.6658 2.10925
\(601\) −25.0928 −1.02355 −0.511777 0.859118i \(-0.671013\pi\)
−0.511777 + 0.859118i \(0.671013\pi\)
\(602\) 23.5174 0.958500
\(603\) −20.3135 −0.827230
\(604\) 55.7998 2.27046
\(605\) −43.5574 −1.77086
\(606\) −85.2327 −3.46234
\(607\) −37.6514 −1.52822 −0.764112 0.645084i \(-0.776822\pi\)
−0.764112 + 0.645084i \(0.776822\pi\)
\(608\) −36.4329 −1.47755
\(609\) −15.2390 −0.617516
\(610\) 91.6512 3.71085
\(611\) 8.38962 0.339408
\(612\) −51.9565 −2.10022
\(613\) 11.3691 0.459194 0.229597 0.973286i \(-0.426259\pi\)
0.229597 + 0.973286i \(0.426259\pi\)
\(614\) 4.79606 0.193553
\(615\) −59.4473 −2.39715
\(616\) −12.1834 −0.490884
\(617\) −22.1701 −0.892534 −0.446267 0.894900i \(-0.647247\pi\)
−0.446267 + 0.894900i \(0.647247\pi\)
\(618\) 5.87936 0.236503
\(619\) −11.5525 −0.464335 −0.232167 0.972676i \(-0.574582\pi\)
−0.232167 + 0.972676i \(0.574582\pi\)
\(620\) −65.0616 −2.61293
\(621\) −27.6020 −1.10763
\(622\) 12.2641 0.491744
\(623\) −1.82046 −0.0729351
\(624\) 5.63090 0.225416
\(625\) 66.5546 2.66218
\(626\) 63.6525 2.54406
\(627\) −60.2350 −2.40555
\(628\) 0.362959 0.0144837
\(629\) −33.5308 −1.33696
\(630\) −67.1338 −2.67467
\(631\) −15.5981 −0.620950 −0.310475 0.950582i \(-0.600488\pi\)
−0.310475 + 0.950582i \(0.600488\pi\)
\(632\) −20.0410 −0.797190
\(633\) 13.1606 0.523088
\(634\) 3.04718 0.121019
\(635\) 62.9637 2.49864
\(636\) −15.0433 −0.596506
\(637\) −4.07838 −0.161591
\(638\) −33.0700 −1.30925
\(639\) 36.9939 1.46345
\(640\) −48.1194 −1.90209
\(641\) 13.3028 0.525430 0.262715 0.964873i \(-0.415382\pi\)
0.262715 + 0.964873i \(0.415382\pi\)
\(642\) 41.6163 1.64247
\(643\) 6.88163 0.271385 0.135692 0.990751i \(-0.456674\pi\)
0.135692 + 0.990751i \(0.456674\pi\)
\(644\) −35.2039 −1.38723
\(645\) 71.6307 2.82046
\(646\) −46.0349 −1.81122
\(647\) 3.29299 0.129461 0.0647304 0.997903i \(-0.479381\pi\)
0.0647304 + 0.997903i \(0.479381\pi\)
\(648\) 4.89988 0.192485
\(649\) 54.5152 2.13991
\(650\) −26.8865 −1.05458
\(651\) 26.6681 1.04520
\(652\) −49.6525 −1.94454
\(653\) 24.9216 0.975258 0.487629 0.873051i \(-0.337862\pi\)
0.487629 + 0.873051i \(0.337862\pi\)
\(654\) 18.8865 0.738523
\(655\) 0.764867 0.0298858
\(656\) 10.9360 0.426979
\(657\) −68.0866 −2.65631
\(658\) 31.1194 1.21316
\(659\) 10.2062 0.397577 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(660\) −141.748 −5.51753
\(661\) −26.4163 −1.02747 −0.513737 0.857948i \(-0.671739\pi\)
−0.513737 + 0.857948i \(0.671739\pi\)
\(662\) 7.36910 0.286408
\(663\) 11.9711 0.464918
\(664\) −0.503072 −0.0195230
\(665\) −34.2206 −1.32702
\(666\) −71.4740 −2.76956
\(667\) −25.0160 −0.968623
\(668\) 39.4463 1.52622
\(669\) 13.6998 0.529666
\(670\) −42.3545 −1.63630
\(671\) −46.9009 −1.81059
\(672\) 35.1422 1.35564
\(673\) 14.1857 0.546818 0.273409 0.961898i \(-0.411849\pi\)
0.273409 + 0.961898i \(0.411849\pi\)
\(674\) −64.9143 −2.50040
\(675\) −44.9854 −1.73149
\(676\) 2.70928 0.104203
\(677\) 6.45959 0.248262 0.124131 0.992266i \(-0.460386\pi\)
0.124131 + 0.992266i \(0.460386\pi\)
\(678\) 2.00000 0.0768095
\(679\) −8.09890 −0.310807
\(680\) −28.3607 −1.08758
\(681\) −45.8843 −1.75829
\(682\) 57.8720 2.21603
\(683\) 0.366835 0.0140365 0.00701827 0.999975i \(-0.497766\pi\)
0.00701827 + 0.999975i \(0.497766\pi\)
\(684\) −56.4534 −2.15855
\(685\) 33.1545 1.26677
\(686\) −41.0928 −1.56893
\(687\) 4.76487 0.181791
\(688\) −13.1773 −0.502379
\(689\) 2.04945 0.0780778
\(690\) −186.381 −7.09540
\(691\) 35.6430 1.35592 0.677962 0.735097i \(-0.262863\pi\)
0.677962 + 0.735097i \(0.262863\pi\)
\(692\) 46.5380 1.76911
\(693\) 34.3545 1.30502
\(694\) 13.1461 0.499018
\(695\) −40.6947 −1.54364
\(696\) −13.7226 −0.520154
\(697\) 23.2495 0.880638
\(698\) −31.1301 −1.17829
\(699\) −64.9998 −2.45852
\(700\) −57.3751 −2.16857
\(701\) 40.9709 1.54745 0.773725 0.633522i \(-0.218391\pi\)
0.773725 + 0.633522i \(0.218391\pi\)
\(702\) 7.87936 0.297387
\(703\) −36.4329 −1.37409
\(704\) 57.0121 2.14872
\(705\) 94.7852 3.56982
\(706\) 35.3545 1.33059
\(707\) 24.7792 0.931919
\(708\) 86.4089 3.24745
\(709\) −19.7321 −0.741053 −0.370527 0.928822i \(-0.620823\pi\)
−0.370527 + 0.928822i \(0.620823\pi\)
\(710\) 77.1338 2.89478
\(711\) 56.5113 2.11934
\(712\) −1.63931 −0.0614357
\(713\) 43.7776 1.63949
\(714\) 44.4040 1.66178
\(715\) 19.3112 0.722200
\(716\) −34.9132 −1.30477
\(717\) 33.4690 1.24992
\(718\) −23.5936 −0.880504
\(719\) −19.3652 −0.722201 −0.361101 0.932527i \(-0.617599\pi\)
−0.361101 + 0.932527i \(0.617599\pi\)
\(720\) 37.6163 1.40188
\(721\) −1.70928 −0.0636567
\(722\) −8.78765 −0.327043
\(723\) −8.14730 −0.303001
\(724\) −3.28458 −0.122071
\(725\) −40.7708 −1.51419
\(726\) 61.4112 2.27918
\(727\) 16.6332 0.616890 0.308445 0.951242i \(-0.400191\pi\)
0.308445 + 0.951242i \(0.400191\pi\)
\(728\) 2.63090 0.0975076
\(729\) −43.3279 −1.60474
\(730\) −141.964 −5.25431
\(731\) −28.0144 −1.03615
\(732\) −74.3400 −2.74769
\(733\) 15.6248 0.577113 0.288557 0.957463i \(-0.406825\pi\)
0.288557 + 0.957463i \(0.406825\pi\)
\(734\) −38.4307 −1.41850
\(735\) −46.0772 −1.69958
\(736\) 57.6886 2.12643
\(737\) 21.6742 0.798380
\(738\) 49.5585 1.82427
\(739\) −1.69982 −0.0625289 −0.0312644 0.999511i \(-0.509953\pi\)
−0.0312644 + 0.999511i \(0.509953\pi\)
\(740\) −85.7358 −3.15171
\(741\) 13.0072 0.477831
\(742\) 7.60197 0.279077
\(743\) −32.3812 −1.18795 −0.593976 0.804483i \(-0.702442\pi\)
−0.593976 + 0.804483i \(0.702442\pi\)
\(744\) 24.0144 0.880410
\(745\) 82.6018 3.02630
\(746\) 47.3439 1.73338
\(747\) 1.41855 0.0519021
\(748\) 55.4368 2.02697
\(749\) −12.0989 −0.442084
\(750\) −181.175 −6.61556
\(751\) −17.3607 −0.633501 −0.316750 0.948509i \(-0.602592\pi\)
−0.316750 + 0.948509i \(0.602592\pi\)
\(752\) −17.4368 −0.635855
\(753\) −45.1855 −1.64665
\(754\) 7.14116 0.260066
\(755\) −85.8864 −3.12573
\(756\) 16.8143 0.611531
\(757\) 24.4596 0.888999 0.444499 0.895779i \(-0.353382\pi\)
0.444499 + 0.895779i \(0.353382\pi\)
\(758\) −74.9686 −2.72298
\(759\) 95.3772 3.46197
\(760\) −30.8154 −1.11779
\(761\) 39.2411 1.42249 0.711244 0.702945i \(-0.248132\pi\)
0.711244 + 0.702945i \(0.248132\pi\)
\(762\) −88.7719 −3.21587
\(763\) −5.49079 −0.198780
\(764\) 18.6742 0.675609
\(765\) 79.9709 2.89135
\(766\) −45.8987 −1.65839
\(767\) −11.7721 −0.425064
\(768\) 1.13397 0.0409186
\(769\) −29.9204 −1.07896 −0.539479 0.841999i \(-0.681379\pi\)
−0.539479 + 0.841999i \(0.681379\pi\)
\(770\) 71.6307 2.58139
\(771\) 43.6781 1.57303
\(772\) 8.73925 0.314532
\(773\) 1.94602 0.0699934 0.0349967 0.999387i \(-0.488858\pi\)
0.0349967 + 0.999387i \(0.488858\pi\)
\(774\) −59.7152 −2.14642
\(775\) 71.3484 2.56291
\(776\) −7.29299 −0.261803
\(777\) 35.1422 1.26072
\(778\) 54.4040 1.95048
\(779\) 25.2618 0.905098
\(780\) 30.6092 1.09598
\(781\) −39.4719 −1.41241
\(782\) 72.8925 2.60663
\(783\) 11.9483 0.426997
\(784\) 8.47641 0.302729
\(785\) −0.558663 −0.0199395
\(786\) −1.07838 −0.0384645
\(787\) 47.8443 1.70546 0.852732 0.522348i \(-0.174944\pi\)
0.852732 + 0.522348i \(0.174944\pi\)
\(788\) 23.7721 0.846844
\(789\) −14.3730 −0.511692
\(790\) 117.829 4.19215
\(791\) −0.581449 −0.0206740
\(792\) 30.9360 1.09926
\(793\) 10.1278 0.359650
\(794\) 78.1254 2.77257
\(795\) 23.1545 0.821205
\(796\) −55.8164 −1.97836
\(797\) −30.5936 −1.08368 −0.541840 0.840482i \(-0.682272\pi\)
−0.541840 + 0.840482i \(0.682272\pi\)
\(798\) 48.2472 1.70793
\(799\) −37.0700 −1.31144
\(800\) 94.0203 3.32412
\(801\) 4.62249 0.163328
\(802\) 15.5753 0.549983
\(803\) 72.6474 2.56367
\(804\) 34.3545 1.21159
\(805\) 54.1855 1.90979
\(806\) −12.4969 −0.440186
\(807\) −68.6369 −2.41613
\(808\) 22.3135 0.784987
\(809\) −51.0265 −1.79400 −0.896998 0.442035i \(-0.854257\pi\)
−0.896998 + 0.442035i \(0.854257\pi\)
\(810\) −28.8082 −1.01222
\(811\) −20.0410 −0.703736 −0.351868 0.936050i \(-0.614453\pi\)
−0.351868 + 0.936050i \(0.614453\pi\)
\(812\) 15.2390 0.534784
\(813\) −36.5608 −1.28224
\(814\) 76.2616 2.67297
\(815\) 76.4245 2.67703
\(816\) −24.8804 −0.870988
\(817\) −30.4391 −1.06493
\(818\) −42.4619 −1.48464
\(819\) −7.41855 −0.259225
\(820\) 59.4473 2.07599
\(821\) −36.9093 −1.28814 −0.644072 0.764965i \(-0.722756\pi\)
−0.644072 + 0.764965i \(0.722756\pi\)
\(822\) −46.7442 −1.63039
\(823\) 22.9672 0.800586 0.400293 0.916387i \(-0.368908\pi\)
0.400293 + 0.916387i \(0.368908\pi\)
\(824\) −1.53919 −0.0536202
\(825\) 155.445 5.41190
\(826\) −43.6658 −1.51933
\(827\) −5.90707 −0.205409 −0.102704 0.994712i \(-0.532750\pi\)
−0.102704 + 0.994712i \(0.532750\pi\)
\(828\) 89.3894 3.10650
\(829\) 46.9360 1.63015 0.815077 0.579352i \(-0.196694\pi\)
0.815077 + 0.579352i \(0.196694\pi\)
\(830\) 2.95774 0.102665
\(831\) −8.34017 −0.289317
\(832\) −12.3112 −0.426816
\(833\) 18.0205 0.624374
\(834\) 57.3751 1.98674
\(835\) −60.7152 −2.10114
\(836\) 60.2350 2.08327
\(837\) −20.9093 −0.722732
\(838\) 36.4307 1.25848
\(839\) 13.6525 0.471335 0.235668 0.971834i \(-0.424272\pi\)
0.235668 + 0.971834i \(0.424272\pi\)
\(840\) 29.7237 1.02556
\(841\) −18.1711 −0.626591
\(842\) 14.9627 0.515647
\(843\) −23.4196 −0.806614
\(844\) −13.1606 −0.453008
\(845\) −4.17009 −0.143455
\(846\) −79.0181 −2.71670
\(847\) −17.8537 −0.613462
\(848\) −4.25953 −0.146273
\(849\) 2.84324 0.0975799
\(850\) 118.800 4.07479
\(851\) 57.6886 1.97754
\(852\) −62.5646 −2.14343
\(853\) −25.9506 −0.888530 −0.444265 0.895895i \(-0.646535\pi\)
−0.444265 + 0.895895i \(0.646535\pi\)
\(854\) 37.5669 1.28551
\(855\) 86.8925 2.97166
\(856\) −10.8950 −0.372382
\(857\) 44.8636 1.53251 0.766255 0.642536i \(-0.222118\pi\)
0.766255 + 0.642536i \(0.222118\pi\)
\(858\) −27.2267 −0.929505
\(859\) −46.0722 −1.57196 −0.785982 0.618249i \(-0.787842\pi\)
−0.785982 + 0.618249i \(0.787842\pi\)
\(860\) −71.6307 −2.44259
\(861\) −24.3668 −0.830420
\(862\) −44.8697 −1.52827
\(863\) 19.4224 0.661147 0.330574 0.943780i \(-0.392758\pi\)
0.330574 + 0.943780i \(0.392758\pi\)
\(864\) −27.5536 −0.937391
\(865\) −71.6307 −2.43552
\(866\) 29.0616 0.987552
\(867\) −6.83710 −0.232200
\(868\) −26.6681 −0.905173
\(869\) −60.2967 −2.04543
\(870\) 80.6802 2.73531
\(871\) −4.68035 −0.158587
\(872\) −4.94441 −0.167439
\(873\) 20.5646 0.696007
\(874\) 79.2015 2.67903
\(875\) 52.6719 1.78064
\(876\) 115.149 3.89053
\(877\) −15.9821 −0.539678 −0.269839 0.962905i \(-0.586971\pi\)
−0.269839 + 0.962905i \(0.586971\pi\)
\(878\) −11.8166 −0.398790
\(879\) −30.3773 −1.02460
\(880\) −40.1361 −1.35299
\(881\) 0.805905 0.0271516 0.0135758 0.999908i \(-0.495679\pi\)
0.0135758 + 0.999908i \(0.495679\pi\)
\(882\) 38.4124 1.29341
\(883\) −33.6209 −1.13143 −0.565716 0.824600i \(-0.691400\pi\)
−0.565716 + 0.824600i \(0.691400\pi\)
\(884\) −11.9711 −0.402631
\(885\) −133.000 −4.47073
\(886\) −80.9553 −2.71975
\(887\) 28.9216 0.971093 0.485547 0.874211i \(-0.338621\pi\)
0.485547 + 0.874211i \(0.338621\pi\)
\(888\) 31.6453 1.06195
\(889\) 25.8082 0.865578
\(890\) 9.63809 0.323069
\(891\) 14.7421 0.493878
\(892\) −13.6998 −0.458704
\(893\) −40.2784 −1.34787
\(894\) −116.459 −3.89499
\(895\) 53.7380 1.79626
\(896\) −19.7237 −0.658921
\(897\) −20.5958 −0.687675
\(898\) −41.2267 −1.37575
\(899\) −18.9504 −0.632030
\(900\) 145.686 4.85620
\(901\) −9.05559 −0.301685
\(902\) −52.8781 −1.76065
\(903\) 29.3607 0.977063
\(904\) −0.523590 −0.0174144
\(905\) 5.05559 0.168054
\(906\) 121.090 4.02296
\(907\) 39.5813 1.31427 0.657137 0.753771i \(-0.271767\pi\)
0.657137 + 0.753771i \(0.271767\pi\)
\(908\) 45.8843 1.52272
\(909\) −62.9192 −2.08690
\(910\) −15.4680 −0.512759
\(911\) −48.7114 −1.61388 −0.806940 0.590634i \(-0.798878\pi\)
−0.806940 + 0.590634i \(0.798878\pi\)
\(912\) −27.0338 −0.895180
\(913\) −1.51357 −0.0500919
\(914\) 57.7914 1.91157
\(915\) 114.423 3.78272
\(916\) −4.76487 −0.157436
\(917\) 0.313511 0.0103530
\(918\) −34.8154 −1.14908
\(919\) 1.11118 0.0366545 0.0183273 0.999832i \(-0.494166\pi\)
0.0183273 + 0.999832i \(0.494166\pi\)
\(920\) 48.7936 1.60868
\(921\) 5.98771 0.197302
\(922\) −32.4307 −1.06805
\(923\) 8.52359 0.280557
\(924\) −58.1010 −1.91138
\(925\) 94.0203 3.09137
\(926\) −15.5837 −0.512113
\(927\) 4.34017 0.142550
\(928\) −24.9721 −0.819750
\(929\) −51.1650 −1.67867 −0.839334 0.543615i \(-0.817055\pi\)
−0.839334 + 0.543615i \(0.817055\pi\)
\(930\) −141.189 −4.62978
\(931\) 19.5802 0.641716
\(932\) 64.9998 2.12914
\(933\) 15.3112 0.501268
\(934\) 44.2472 1.44781
\(935\) −85.3277 −2.79051
\(936\) −6.68035 −0.218354
\(937\) 2.51971 0.0823155 0.0411577 0.999153i \(-0.486895\pi\)
0.0411577 + 0.999153i \(0.486895\pi\)
\(938\) −17.3607 −0.566847
\(939\) 79.4678 2.59333
\(940\) −94.7852 −3.09155
\(941\) −23.0784 −0.752334 −0.376167 0.926552i \(-0.622758\pi\)
−0.376167 + 0.926552i \(0.622758\pi\)
\(942\) 0.787653 0.0256631
\(943\) −40.0000 −1.30258
\(944\) 24.4668 0.796326
\(945\) −25.8804 −0.841890
\(946\) 63.7152 2.07156
\(947\) −11.5174 −0.374267 −0.187133 0.982335i \(-0.559920\pi\)
−0.187133 + 0.982335i \(0.559920\pi\)
\(948\) −95.5729 −3.10406
\(949\) −15.6875 −0.509239
\(950\) 129.082 4.18797
\(951\) 3.80430 0.123363
\(952\) −11.6248 −0.376760
\(953\) 22.2967 0.722261 0.361130 0.932515i \(-0.382391\pi\)
0.361130 + 0.932515i \(0.382391\pi\)
\(954\) −19.3028 −0.624952
\(955\) −28.7431 −0.930106
\(956\) −33.4690 −1.08247
\(957\) −41.2867 −1.33461
\(958\) −23.2085 −0.749832
\(959\) 13.5897 0.438834
\(960\) −139.091 −4.48915
\(961\) 2.16290 0.0697709
\(962\) −16.4680 −0.530950
\(963\) 30.7214 0.989983
\(964\) 8.14730 0.262407
\(965\) −13.4514 −0.433014
\(966\) −76.3956 −2.45799
\(967\) 21.6787 0.697141 0.348571 0.937283i \(-0.386667\pi\)
0.348571 + 0.937283i \(0.386667\pi\)
\(968\) −16.0772 −0.516739
\(969\) −57.4729 −1.84630
\(970\) 42.8781 1.37673
\(971\) 41.7419 1.33956 0.669781 0.742559i \(-0.266388\pi\)
0.669781 + 0.742559i \(0.266388\pi\)
\(972\) 52.8781 1.69607
\(973\) −16.6803 −0.534747
\(974\) 85.3484 2.73474
\(975\) −33.5669 −1.07500
\(976\) −21.0494 −0.673776
\(977\) −27.3835 −0.876075 −0.438038 0.898957i \(-0.644326\pi\)
−0.438038 + 0.898957i \(0.644326\pi\)
\(978\) −107.750 −3.44547
\(979\) −4.93212 −0.157631
\(980\) 46.0772 1.47188
\(981\) 13.9421 0.445138
\(982\) 37.5585 1.19854
\(983\) −24.1617 −0.770638 −0.385319 0.922783i \(-0.625909\pi\)
−0.385319 + 0.922783i \(0.625909\pi\)
\(984\) −21.9421 −0.699490
\(985\) −36.5897 −1.16584
\(986\) −31.5536 −1.00487
\(987\) 38.8515 1.23666
\(988\) −13.0072 −0.413814
\(989\) 48.1978 1.53260
\(990\) −181.884 −5.78065
\(991\) −6.39350 −0.203096 −0.101548 0.994831i \(-0.532380\pi\)
−0.101548 + 0.994831i \(0.532380\pi\)
\(992\) 43.7009 1.38750
\(993\) 9.20006 0.291955
\(994\) 31.6163 1.00281
\(995\) 85.9120 2.72359
\(996\) −2.39908 −0.0760177
\(997\) 42.5074 1.34622 0.673112 0.739541i \(-0.264957\pi\)
0.673112 + 0.739541i \(0.264957\pi\)
\(998\) 48.9093 1.54820
\(999\) −27.5536 −0.871756
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 1339.2.a.c.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1339.2.a.c.1.1 3 1.1 even 1 trivial