Properties

Label 161.2.a.c.1.1
Level $161$
Weight $2$
Character 161.1
Self dual yes
Analytic conductor $1.286$
Analytic rank $0$
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] = [161,2,Mod(1,161)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(161, 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("161.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 161 = 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 161.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: \(1.28559147254\)
Analytic rank: \(0\)
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: \(\Z[a_1, a_2, a_3]\)
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 \(\chi\) \(=\) 161.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.70928 q^{2} -1.17009 q^{3} +5.34017 q^{4} -1.17009 q^{5} +3.17009 q^{6} -1.00000 q^{7} -9.04945 q^{8} -1.63090 q^{9} +3.17009 q^{10} +3.70928 q^{11} -6.24846 q^{12} +4.34017 q^{13} +2.70928 q^{14} +1.36910 q^{15} +13.8371 q^{16} +3.17009 q^{17} +4.41855 q^{18} +5.26180 q^{19} -6.24846 q^{20} +1.17009 q^{21} -10.0494 q^{22} +1.00000 q^{23} +10.5886 q^{24} -3.63090 q^{25} -11.7587 q^{26} +5.41855 q^{27} -5.34017 q^{28} +0.630898 q^{29} -3.70928 q^{30} +9.32684 q^{31} -19.3896 q^{32} -4.34017 q^{33} -8.58864 q^{34} +1.17009 q^{35} -8.70928 q^{36} -3.07838 q^{37} -14.2557 q^{38} -5.07838 q^{39} +10.5886 q^{40} +2.68035 q^{41} -3.17009 q^{42} -8.49693 q^{43} +19.8082 q^{44} +1.90829 q^{45} -2.70928 q^{46} +6.09171 q^{47} -16.1906 q^{48} +1.00000 q^{49} +9.83710 q^{50} -3.70928 q^{51} +23.1773 q^{52} +4.15676 q^{53} -14.6803 q^{54} -4.34017 q^{55} +9.04945 q^{56} -6.15676 q^{57} -1.70928 q^{58} -8.40522 q^{59} +7.31124 q^{60} -6.92881 q^{61} -25.2690 q^{62} +1.63090 q^{63} +24.8576 q^{64} -5.07838 q^{65} +11.7587 q^{66} +9.86603 q^{67} +16.9288 q^{68} -1.17009 q^{69} -3.17009 q^{70} +10.8371 q^{71} +14.7587 q^{72} -13.0205 q^{73} +8.34017 q^{74} +4.24846 q^{75} +28.0989 q^{76} -3.70928 q^{77} +13.7587 q^{78} -4.38962 q^{79} -16.1906 q^{80} -1.44748 q^{81} -7.26180 q^{82} +14.6537 q^{83} +6.24846 q^{84} -3.70928 q^{85} +23.0205 q^{86} -0.738205 q^{87} -33.5669 q^{88} -6.77205 q^{89} -5.17009 q^{90} -4.34017 q^{91} +5.34017 q^{92} -10.9132 q^{93} -16.5041 q^{94} -6.15676 q^{95} +22.6875 q^{96} +8.58864 q^{97} -2.70928 q^{98} -6.04945 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 2 q^{3} + 5 q^{4} + 2 q^{5} + 4 q^{6} - 3 q^{7} - 9 q^{8} - q^{9} + 4 q^{10} + 4 q^{11} - 10 q^{12} + 2 q^{13} + q^{14} + 8 q^{15} + 13 q^{16} + 4 q^{17} - q^{18} + 8 q^{19} - 10 q^{20}+ \cdots - q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.70928 −1.91575 −0.957873 0.287190i \(-0.907279\pi\)
−0.957873 + 0.287190i \(0.907279\pi\)
\(3\) −1.17009 −0.675550 −0.337775 0.941227i \(-0.609674\pi\)
−0.337775 + 0.941227i \(0.609674\pi\)
\(4\) 5.34017 2.67009
\(5\) −1.17009 −0.523279 −0.261639 0.965166i \(-0.584263\pi\)
−0.261639 + 0.965166i \(0.584263\pi\)
\(6\) 3.17009 1.29418
\(7\) −1.00000 −0.377964
\(8\) −9.04945 −3.19946
\(9\) −1.63090 −0.543633
\(10\) 3.17009 1.00247
\(11\) 3.70928 1.11839 0.559194 0.829037i \(-0.311111\pi\)
0.559194 + 0.829037i \(0.311111\pi\)
\(12\) −6.24846 −1.80378
\(13\) 4.34017 1.20375 0.601874 0.798591i \(-0.294421\pi\)
0.601874 + 0.798591i \(0.294421\pi\)
\(14\) 2.70928 0.724084
\(15\) 1.36910 0.353501
\(16\) 13.8371 3.45928
\(17\) 3.17009 0.768859 0.384429 0.923154i \(-0.374398\pi\)
0.384429 + 0.923154i \(0.374398\pi\)
\(18\) 4.41855 1.04146
\(19\) 5.26180 1.20714 0.603569 0.797311i \(-0.293745\pi\)
0.603569 + 0.797311i \(0.293745\pi\)
\(20\) −6.24846 −1.39720
\(21\) 1.17009 0.255334
\(22\) −10.0494 −2.14255
\(23\) 1.00000 0.208514
\(24\) 10.5886 2.16140
\(25\) −3.63090 −0.726180
\(26\) −11.7587 −2.30608
\(27\) 5.41855 1.04280
\(28\) −5.34017 −1.00920
\(29\) 0.630898 0.117155 0.0585774 0.998283i \(-0.481344\pi\)
0.0585774 + 0.998283i \(0.481344\pi\)
\(30\) −3.70928 −0.677218
\(31\) 9.32684 1.67515 0.837575 0.546322i \(-0.183973\pi\)
0.837575 + 0.546322i \(0.183973\pi\)
\(32\) −19.3896 −3.42763
\(33\) −4.34017 −0.755527
\(34\) −8.58864 −1.47294
\(35\) 1.17009 0.197781
\(36\) −8.70928 −1.45155
\(37\) −3.07838 −0.506082 −0.253041 0.967456i \(-0.581431\pi\)
−0.253041 + 0.967456i \(0.581431\pi\)
\(38\) −14.2557 −2.31257
\(39\) −5.07838 −0.813191
\(40\) 10.5886 1.67421
\(41\) 2.68035 0.418600 0.209300 0.977852i \(-0.432882\pi\)
0.209300 + 0.977852i \(0.432882\pi\)
\(42\) −3.17009 −0.489155
\(43\) −8.49693 −1.29577 −0.647885 0.761738i \(-0.724346\pi\)
−0.647885 + 0.761738i \(0.724346\pi\)
\(44\) 19.8082 2.98619
\(45\) 1.90829 0.284471
\(46\) −2.70928 −0.399461
\(47\) 6.09171 0.888567 0.444284 0.895886i \(-0.353458\pi\)
0.444284 + 0.895886i \(0.353458\pi\)
\(48\) −16.1906 −2.33691
\(49\) 1.00000 0.142857
\(50\) 9.83710 1.39118
\(51\) −3.70928 −0.519402
\(52\) 23.1773 3.21411
\(53\) 4.15676 0.570974 0.285487 0.958383i \(-0.407845\pi\)
0.285487 + 0.958383i \(0.407845\pi\)
\(54\) −14.6803 −1.99774
\(55\) −4.34017 −0.585229
\(56\) 9.04945 1.20928
\(57\) −6.15676 −0.815482
\(58\) −1.70928 −0.224439
\(59\) −8.40522 −1.09427 −0.547133 0.837046i \(-0.684281\pi\)
−0.547133 + 0.837046i \(0.684281\pi\)
\(60\) 7.31124 0.943877
\(61\) −6.92881 −0.887143 −0.443572 0.896239i \(-0.646289\pi\)
−0.443572 + 0.896239i \(0.646289\pi\)
\(62\) −25.2690 −3.20916
\(63\) 1.63090 0.205474
\(64\) 24.8576 3.10720
\(65\) −5.07838 −0.629895
\(66\) 11.7587 1.44740
\(67\) 9.86603 1.20533 0.602664 0.797995i \(-0.294106\pi\)
0.602664 + 0.797995i \(0.294106\pi\)
\(68\) 16.9288 2.05292
\(69\) −1.17009 −0.140862
\(70\) −3.17009 −0.378898
\(71\) 10.8371 1.28613 0.643064 0.765813i \(-0.277663\pi\)
0.643064 + 0.765813i \(0.277663\pi\)
\(72\) 14.7587 1.73933
\(73\) −13.0205 −1.52394 −0.761968 0.647614i \(-0.775767\pi\)
−0.761968 + 0.647614i \(0.775767\pi\)
\(74\) 8.34017 0.969525
\(75\) 4.24846 0.490570
\(76\) 28.0989 3.22316
\(77\) −3.70928 −0.422711
\(78\) 13.7587 1.55787
\(79\) −4.38962 −0.493871 −0.246935 0.969032i \(-0.579424\pi\)
−0.246935 + 0.969032i \(0.579424\pi\)
\(80\) −16.1906 −1.81016
\(81\) −1.44748 −0.160831
\(82\) −7.26180 −0.801931
\(83\) 14.6537 1.60845 0.804225 0.594324i \(-0.202580\pi\)
0.804225 + 0.594324i \(0.202580\pi\)
\(84\) 6.24846 0.681763
\(85\) −3.70928 −0.402327
\(86\) 23.0205 2.48237
\(87\) −0.738205 −0.0791439
\(88\) −33.5669 −3.57824
\(89\) −6.77205 −0.717836 −0.358918 0.933369i \(-0.616854\pi\)
−0.358918 + 0.933369i \(0.616854\pi\)
\(90\) −5.17009 −0.544975
\(91\) −4.34017 −0.454974
\(92\) 5.34017 0.556752
\(93\) −10.9132 −1.13165
\(94\) −16.5041 −1.70227
\(95\) −6.15676 −0.631670
\(96\) 22.6875 2.31554
\(97\) 8.58864 0.872044 0.436022 0.899936i \(-0.356387\pi\)
0.436022 + 0.899936i \(0.356387\pi\)
\(98\) −2.70928 −0.273678
\(99\) −6.04945 −0.607992
\(100\) −19.3896 −1.93896
\(101\) −4.92162 −0.489720 −0.244860 0.969558i \(-0.578742\pi\)
−0.244860 + 0.969558i \(0.578742\pi\)
\(102\) 10.0494 0.995044
\(103\) 16.6803 1.64356 0.821782 0.569803i \(-0.192980\pi\)
0.821782 + 0.569803i \(0.192980\pi\)
\(104\) −39.2762 −3.85135
\(105\) −1.36910 −0.133611
\(106\) −11.2618 −1.09384
\(107\) −15.3340 −1.48240 −0.741198 0.671286i \(-0.765742\pi\)
−0.741198 + 0.671286i \(0.765742\pi\)
\(108\) 28.9360 2.78437
\(109\) −11.1773 −1.07059 −0.535294 0.844666i \(-0.679799\pi\)
−0.535294 + 0.844666i \(0.679799\pi\)
\(110\) 11.7587 1.12115
\(111\) 3.60197 0.341884
\(112\) −13.8371 −1.30748
\(113\) 13.5174 1.27161 0.635807 0.771848i \(-0.280667\pi\)
0.635807 + 0.771848i \(0.280667\pi\)
\(114\) 16.6803 1.56226
\(115\) −1.17009 −0.109111
\(116\) 3.36910 0.312813
\(117\) −7.07838 −0.654396
\(118\) 22.7721 2.09634
\(119\) −3.17009 −0.290601
\(120\) −12.3896 −1.13101
\(121\) 2.75872 0.250793
\(122\) 18.7721 1.69954
\(123\) −3.13624 −0.282785
\(124\) 49.8069 4.47280
\(125\) 10.0989 0.903273
\(126\) −4.41855 −0.393636
\(127\) 2.92162 0.259252 0.129626 0.991563i \(-0.458622\pi\)
0.129626 + 0.991563i \(0.458622\pi\)
\(128\) −28.5669 −2.52498
\(129\) 9.94214 0.875357
\(130\) 13.7587 1.20672
\(131\) −6.27513 −0.548260 −0.274130 0.961693i \(-0.588390\pi\)
−0.274130 + 0.961693i \(0.588390\pi\)
\(132\) −23.1773 −2.01732
\(133\) −5.26180 −0.456256
\(134\) −26.7298 −2.30910
\(135\) −6.34017 −0.545675
\(136\) −28.6875 −2.45994
\(137\) 7.75872 0.662873 0.331436 0.943478i \(-0.392467\pi\)
0.331436 + 0.943478i \(0.392467\pi\)
\(138\) 3.17009 0.269856
\(139\) −2.92881 −0.248418 −0.124209 0.992256i \(-0.539639\pi\)
−0.124209 + 0.992256i \(0.539639\pi\)
\(140\) 6.24846 0.528092
\(141\) −7.12783 −0.600271
\(142\) −29.3607 −2.46389
\(143\) 16.0989 1.34626
\(144\) −22.5669 −1.88057
\(145\) −0.738205 −0.0613046
\(146\) 35.2762 2.91948
\(147\) −1.17009 −0.0965071
\(148\) −16.4391 −1.35128
\(149\) 0.837101 0.0685780 0.0342890 0.999412i \(-0.489083\pi\)
0.0342890 + 0.999412i \(0.489083\pi\)
\(150\) −11.5103 −0.939809
\(151\) 15.9155 1.29518 0.647592 0.761988i \(-0.275776\pi\)
0.647592 + 0.761988i \(0.275776\pi\)
\(152\) −47.6163 −3.86220
\(153\) −5.17009 −0.417977
\(154\) 10.0494 0.809808
\(155\) −10.9132 −0.876570
\(156\) −27.1194 −2.17129
\(157\) −11.1701 −0.891470 −0.445735 0.895165i \(-0.647058\pi\)
−0.445735 + 0.895165i \(0.647058\pi\)
\(158\) 11.8927 0.946132
\(159\) −4.86376 −0.385722
\(160\) 22.6875 1.79361
\(161\) −1.00000 −0.0788110
\(162\) 3.92162 0.308112
\(163\) 14.6225 1.14532 0.572661 0.819792i \(-0.305911\pi\)
0.572661 + 0.819792i \(0.305911\pi\)
\(164\) 14.3135 1.11770
\(165\) 5.07838 0.395351
\(166\) −39.7009 −3.08138
\(167\) 6.00719 0.464850 0.232425 0.972614i \(-0.425334\pi\)
0.232425 + 0.972614i \(0.425334\pi\)
\(168\) −10.5886 −0.816931
\(169\) 5.83710 0.449008
\(170\) 10.0494 0.770758
\(171\) −8.58145 −0.656240
\(172\) −45.3751 −3.45982
\(173\) 14.0989 1.07192 0.535960 0.844244i \(-0.319950\pi\)
0.535960 + 0.844244i \(0.319950\pi\)
\(174\) 2.00000 0.151620
\(175\) 3.63090 0.274470
\(176\) 51.3256 3.86881
\(177\) 9.83483 0.739231
\(178\) 18.3474 1.37519
\(179\) −18.1256 −1.35477 −0.677384 0.735630i \(-0.736886\pi\)
−0.677384 + 0.735630i \(0.736886\pi\)
\(180\) 10.1906 0.759563
\(181\) 12.4319 0.924054 0.462027 0.886866i \(-0.347122\pi\)
0.462027 + 0.886866i \(0.347122\pi\)
\(182\) 11.7587 0.871615
\(183\) 8.10731 0.599309
\(184\) −9.04945 −0.667134
\(185\) 3.60197 0.264822
\(186\) 29.5669 2.16795
\(187\) 11.7587 0.859883
\(188\) 32.5308 2.37255
\(189\) −5.41855 −0.394142
\(190\) 16.6803 1.21012
\(191\) −4.86376 −0.351930 −0.175965 0.984396i \(-0.556304\pi\)
−0.175965 + 0.984396i \(0.556304\pi\)
\(192\) −29.0856 −2.09907
\(193\) −17.2351 −1.24061 −0.620306 0.784360i \(-0.712992\pi\)
−0.620306 + 0.784360i \(0.712992\pi\)
\(194\) −23.2690 −1.67062
\(195\) 5.94214 0.425526
\(196\) 5.34017 0.381441
\(197\) −12.5236 −0.892269 −0.446134 0.894966i \(-0.647200\pi\)
−0.446134 + 0.894966i \(0.647200\pi\)
\(198\) 16.3896 1.16476
\(199\) 1.16290 0.0824357 0.0412178 0.999150i \(-0.486876\pi\)
0.0412178 + 0.999150i \(0.486876\pi\)
\(200\) 32.8576 2.32338
\(201\) −11.5441 −0.814259
\(202\) 13.3340 0.938179
\(203\) −0.630898 −0.0442803
\(204\) −19.8082 −1.38685
\(205\) −3.13624 −0.219044
\(206\) −45.1917 −3.14865
\(207\) −1.63090 −0.113355
\(208\) 60.0554 4.16409
\(209\) 19.5174 1.35005
\(210\) 3.70928 0.255964
\(211\) 4.76487 0.328027 0.164013 0.986458i \(-0.447556\pi\)
0.164013 + 0.986458i \(0.447556\pi\)
\(212\) 22.1978 1.52455
\(213\) −12.6803 −0.868843
\(214\) 41.5441 2.83990
\(215\) 9.94214 0.678048
\(216\) −49.0349 −3.33640
\(217\) −9.32684 −0.633147
\(218\) 30.2823 2.05098
\(219\) 15.2351 1.02949
\(220\) −23.1773 −1.56261
\(221\) 13.7587 0.925512
\(222\) −9.75872 −0.654963
\(223\) −21.3679 −1.43090 −0.715450 0.698664i \(-0.753778\pi\)
−0.715450 + 0.698664i \(0.753778\pi\)
\(224\) 19.3896 1.29552
\(225\) 5.92162 0.394775
\(226\) −36.6225 −2.43609
\(227\) −5.84324 −0.387830 −0.193915 0.981018i \(-0.562119\pi\)
−0.193915 + 0.981018i \(0.562119\pi\)
\(228\) −32.8781 −2.17741
\(229\) 2.85658 0.188768 0.0943839 0.995536i \(-0.469912\pi\)
0.0943839 + 0.995536i \(0.469912\pi\)
\(230\) 3.17009 0.209029
\(231\) 4.34017 0.285562
\(232\) −5.70928 −0.374832
\(233\) −13.5259 −0.886108 −0.443054 0.896495i \(-0.646105\pi\)
−0.443054 + 0.896495i \(0.646105\pi\)
\(234\) 19.1773 1.25366
\(235\) −7.12783 −0.464968
\(236\) −44.8853 −2.92179
\(237\) 5.13624 0.333634
\(238\) 8.58864 0.556719
\(239\) 4.76487 0.308214 0.154107 0.988054i \(-0.450750\pi\)
0.154107 + 0.988054i \(0.450750\pi\)
\(240\) 18.9444 1.22286
\(241\) 24.4775 1.57673 0.788366 0.615207i \(-0.210928\pi\)
0.788366 + 0.615207i \(0.210928\pi\)
\(242\) −7.47414 −0.480456
\(243\) −14.5620 −0.934151
\(244\) −37.0010 −2.36875
\(245\) −1.17009 −0.0747541
\(246\) 8.49693 0.541744
\(247\) 22.8371 1.45309
\(248\) −84.4028 −5.35958
\(249\) −17.1461 −1.08659
\(250\) −27.3607 −1.73044
\(251\) −26.2557 −1.65724 −0.828621 0.559810i \(-0.810874\pi\)
−0.828621 + 0.559810i \(0.810874\pi\)
\(252\) 8.70928 0.548633
\(253\) 3.70928 0.233200
\(254\) −7.91548 −0.496661
\(255\) 4.34017 0.271792
\(256\) 27.6803 1.73002
\(257\) 7.84324 0.489248 0.244624 0.969618i \(-0.421335\pi\)
0.244624 + 0.969618i \(0.421335\pi\)
\(258\) −26.9360 −1.67696
\(259\) 3.07838 0.191281
\(260\) −27.1194 −1.68187
\(261\) −1.02893 −0.0636891
\(262\) 17.0010 1.05033
\(263\) 10.9132 0.672937 0.336469 0.941695i \(-0.390767\pi\)
0.336469 + 0.941695i \(0.390767\pi\)
\(264\) 39.2762 2.41728
\(265\) −4.86376 −0.298779
\(266\) 14.2557 0.874070
\(267\) 7.92389 0.484934
\(268\) 52.6863 3.21833
\(269\) −3.57531 −0.217990 −0.108995 0.994042i \(-0.534763\pi\)
−0.108995 + 0.994042i \(0.534763\pi\)
\(270\) 17.1773 1.04538
\(271\) −5.98053 −0.363291 −0.181646 0.983364i \(-0.558142\pi\)
−0.181646 + 0.983364i \(0.558142\pi\)
\(272\) 43.8648 2.65969
\(273\) 5.07838 0.307357
\(274\) −21.0205 −1.26990
\(275\) −13.4680 −0.812151
\(276\) −6.24846 −0.376113
\(277\) −13.3112 −0.799795 −0.399898 0.916560i \(-0.630954\pi\)
−0.399898 + 0.916560i \(0.630954\pi\)
\(278\) 7.93495 0.475907
\(279\) −15.2111 −0.910666
\(280\) −10.5886 −0.632792
\(281\) −2.13009 −0.127071 −0.0635354 0.997980i \(-0.520238\pi\)
−0.0635354 + 0.997980i \(0.520238\pi\)
\(282\) 19.3112 1.14997
\(283\) 13.6742 0.812847 0.406423 0.913685i \(-0.366776\pi\)
0.406423 + 0.913685i \(0.366776\pi\)
\(284\) 57.8720 3.43407
\(285\) 7.20394 0.426724
\(286\) −43.6163 −2.57909
\(287\) −2.68035 −0.158216
\(288\) 31.6225 1.86337
\(289\) −6.95055 −0.408856
\(290\) 2.00000 0.117444
\(291\) −10.0494 −0.589109
\(292\) −69.5318 −4.06904
\(293\) −13.2690 −0.775182 −0.387591 0.921831i \(-0.626693\pi\)
−0.387591 + 0.921831i \(0.626693\pi\)
\(294\) 3.17009 0.184883
\(295\) 9.83483 0.572606
\(296\) 27.8576 1.61919
\(297\) 20.0989 1.16626
\(298\) −2.26794 −0.131378
\(299\) 4.34017 0.250999
\(300\) 22.6875 1.30987
\(301\) 8.49693 0.489755
\(302\) −43.1194 −2.48124
\(303\) 5.75872 0.330830
\(304\) 72.8080 4.17582
\(305\) 8.10731 0.464223
\(306\) 14.0072 0.800738
\(307\) −17.5103 −0.999363 −0.499682 0.866209i \(-0.666550\pi\)
−0.499682 + 0.866209i \(0.666550\pi\)
\(308\) −19.8082 −1.12868
\(309\) −19.5174 −1.11031
\(310\) 29.5669 1.67929
\(311\) 2.03385 0.115329 0.0576645 0.998336i \(-0.481635\pi\)
0.0576645 + 0.998336i \(0.481635\pi\)
\(312\) 45.9565 2.60178
\(313\) 12.3786 0.699677 0.349839 0.936810i \(-0.386236\pi\)
0.349839 + 0.936810i \(0.386236\pi\)
\(314\) 30.2628 1.70783
\(315\) −1.90829 −0.107520
\(316\) −23.4413 −1.31868
\(317\) −20.1483 −1.13164 −0.565822 0.824527i \(-0.691441\pi\)
−0.565822 + 0.824527i \(0.691441\pi\)
\(318\) 13.1773 0.738945
\(319\) 2.34017 0.131025
\(320\) −29.0856 −1.62593
\(321\) 17.9421 1.00143
\(322\) 2.70928 0.150982
\(323\) 16.6803 0.928119
\(324\) −7.72979 −0.429433
\(325\) −15.7587 −0.874137
\(326\) −39.6163 −2.19415
\(327\) 13.0784 0.723236
\(328\) −24.2557 −1.33929
\(329\) −6.09171 −0.335847
\(330\) −13.7587 −0.757393
\(331\) −6.70701 −0.368650 −0.184325 0.982865i \(-0.559010\pi\)
−0.184325 + 0.982865i \(0.559010\pi\)
\(332\) 78.2532 4.29470
\(333\) 5.02052 0.275123
\(334\) −16.2751 −0.890535
\(335\) −11.5441 −0.630722
\(336\) 16.1906 0.883270
\(337\) −29.0205 −1.58085 −0.790424 0.612560i \(-0.790140\pi\)
−0.790424 + 0.612560i \(0.790140\pi\)
\(338\) −15.8143 −0.860185
\(339\) −15.8166 −0.859039
\(340\) −19.8082 −1.07425
\(341\) 34.5958 1.87347
\(342\) 23.2495 1.25719
\(343\) −1.00000 −0.0539949
\(344\) 76.8925 4.14577
\(345\) 1.36910 0.0737100
\(346\) −38.1978 −2.05353
\(347\) −18.2413 −0.979243 −0.489622 0.871935i \(-0.662865\pi\)
−0.489622 + 0.871935i \(0.662865\pi\)
\(348\) −3.94214 −0.211321
\(349\) −8.35455 −0.447209 −0.223604 0.974680i \(-0.571782\pi\)
−0.223604 + 0.974680i \(0.571782\pi\)
\(350\) −9.83710 −0.525815
\(351\) 23.5174 1.25527
\(352\) −71.9214 −3.83343
\(353\) −31.5897 −1.68135 −0.840675 0.541541i \(-0.817841\pi\)
−0.840675 + 0.541541i \(0.817841\pi\)
\(354\) −26.6453 −1.41618
\(355\) −12.6803 −0.673003
\(356\) −36.1639 −1.91669
\(357\) 3.70928 0.196316
\(358\) 49.1071 2.59539
\(359\) −4.48852 −0.236895 −0.118447 0.992960i \(-0.537792\pi\)
−0.118447 + 0.992960i \(0.537792\pi\)
\(360\) −17.2690 −0.910155
\(361\) 8.68649 0.457184
\(362\) −33.6814 −1.77025
\(363\) −3.22795 −0.169423
\(364\) −23.1773 −1.21482
\(365\) 15.2351 0.797443
\(366\) −21.9649 −1.14813
\(367\) −18.7526 −0.978877 −0.489438 0.872038i \(-0.662798\pi\)
−0.489438 + 0.872038i \(0.662798\pi\)
\(368\) 13.8371 0.721309
\(369\) −4.37137 −0.227564
\(370\) −9.75872 −0.507332
\(371\) −4.15676 −0.215808
\(372\) −58.2784 −3.02160
\(373\) 22.5113 1.16559 0.582796 0.812619i \(-0.301959\pi\)
0.582796 + 0.812619i \(0.301959\pi\)
\(374\) −31.8576 −1.64732
\(375\) −11.8166 −0.610206
\(376\) −55.1266 −2.84294
\(377\) 2.73820 0.141025
\(378\) 14.6803 0.755076
\(379\) −22.3318 −1.14711 −0.573553 0.819169i \(-0.694435\pi\)
−0.573553 + 0.819169i \(0.694435\pi\)
\(380\) −32.8781 −1.68661
\(381\) −3.41855 −0.175138
\(382\) 13.1773 0.674208
\(383\) 15.2351 0.778479 0.389239 0.921137i \(-0.372738\pi\)
0.389239 + 0.921137i \(0.372738\pi\)
\(384\) 33.4257 1.70575
\(385\) 4.34017 0.221196
\(386\) 46.6947 2.37670
\(387\) 13.8576 0.704422
\(388\) 45.8648 2.32843
\(389\) 21.3874 1.08438 0.542191 0.840255i \(-0.317595\pi\)
0.542191 + 0.840255i \(0.317595\pi\)
\(390\) −16.0989 −0.815199
\(391\) 3.17009 0.160318
\(392\) −9.04945 −0.457066
\(393\) 7.34244 0.370377
\(394\) 33.9299 1.70936
\(395\) 5.13624 0.258432
\(396\) −32.3051 −1.62339
\(397\) 20.6537 1.03658 0.518289 0.855205i \(-0.326569\pi\)
0.518289 + 0.855205i \(0.326569\pi\)
\(398\) −3.15061 −0.157926
\(399\) 6.15676 0.308223
\(400\) −50.2411 −2.51205
\(401\) 24.3402 1.21549 0.607745 0.794132i \(-0.292074\pi\)
0.607745 + 0.794132i \(0.292074\pi\)
\(402\) 31.2762 1.55991
\(403\) 40.4801 2.01646
\(404\) −26.2823 −1.30759
\(405\) 1.69368 0.0841595
\(406\) 1.70928 0.0848299
\(407\) −11.4186 −0.565997
\(408\) 33.5669 1.66181
\(409\) 17.2885 0.854859 0.427430 0.904049i \(-0.359419\pi\)
0.427430 + 0.904049i \(0.359419\pi\)
\(410\) 8.49693 0.419633
\(411\) −9.07838 −0.447803
\(412\) 89.0759 4.38846
\(413\) 8.40522 0.413594
\(414\) 4.41855 0.217160
\(415\) −17.1461 −0.841668
\(416\) −84.1543 −4.12600
\(417\) 3.42696 0.167819
\(418\) −52.8781 −2.58635
\(419\) −3.05172 −0.149086 −0.0745430 0.997218i \(-0.523750\pi\)
−0.0745430 + 0.997218i \(0.523750\pi\)
\(420\) −7.31124 −0.356752
\(421\) 13.8843 0.676679 0.338339 0.941024i \(-0.390135\pi\)
0.338339 + 0.941024i \(0.390135\pi\)
\(422\) −12.9093 −0.628417
\(423\) −9.93495 −0.483054
\(424\) −37.6163 −1.82681
\(425\) −11.5103 −0.558330
\(426\) 34.3545 1.66448
\(427\) 6.92881 0.335309
\(428\) −81.8864 −3.95813
\(429\) −18.8371 −0.909464
\(430\) −26.9360 −1.29897
\(431\) 15.2039 0.732348 0.366174 0.930546i \(-0.380668\pi\)
0.366174 + 0.930546i \(0.380668\pi\)
\(432\) 74.9770 3.60733
\(433\) 12.8566 0.617848 0.308924 0.951087i \(-0.400031\pi\)
0.308924 + 0.951087i \(0.400031\pi\)
\(434\) 25.2690 1.21295
\(435\) 0.863763 0.0414143
\(436\) −59.6886 −2.85856
\(437\) 5.26180 0.251706
\(438\) −41.2762 −1.97225
\(439\) 14.5620 0.695005 0.347503 0.937679i \(-0.387030\pi\)
0.347503 + 0.937679i \(0.387030\pi\)
\(440\) 39.2762 1.87242
\(441\) −1.63090 −0.0776618
\(442\) −37.2762 −1.77305
\(443\) −30.2557 −1.43749 −0.718745 0.695274i \(-0.755283\pi\)
−0.718745 + 0.695274i \(0.755283\pi\)
\(444\) 19.2351 0.912859
\(445\) 7.92389 0.375628
\(446\) 57.8915 2.74124
\(447\) −0.979481 −0.0463279
\(448\) −24.8576 −1.17441
\(449\) −4.68422 −0.221062 −0.110531 0.993873i \(-0.535255\pi\)
−0.110531 + 0.993873i \(0.535255\pi\)
\(450\) −16.0433 −0.756289
\(451\) 9.94214 0.468157
\(452\) 72.1855 3.39532
\(453\) −18.6225 −0.874961
\(454\) 15.8310 0.742984
\(455\) 5.07838 0.238078
\(456\) 55.7152 2.60911
\(457\) 23.4596 1.09739 0.548697 0.836022i \(-0.315124\pi\)
0.548697 + 0.836022i \(0.315124\pi\)
\(458\) −7.73925 −0.361631
\(459\) 17.1773 0.801767
\(460\) −6.24846 −0.291336
\(461\) 6.58145 0.306529 0.153264 0.988185i \(-0.451021\pi\)
0.153264 + 0.988185i \(0.451021\pi\)
\(462\) −11.7587 −0.547065
\(463\) 20.0144 0.930147 0.465073 0.885272i \(-0.346028\pi\)
0.465073 + 0.885272i \(0.346028\pi\)
\(464\) 8.72979 0.405271
\(465\) 12.7694 0.592167
\(466\) 36.6453 1.69756
\(467\) 21.4596 0.993031 0.496516 0.868028i \(-0.334613\pi\)
0.496516 + 0.868028i \(0.334613\pi\)
\(468\) −37.7998 −1.74729
\(469\) −9.86603 −0.455571
\(470\) 19.3112 0.890761
\(471\) 13.0700 0.602232
\(472\) 76.0626 3.50107
\(473\) −31.5174 −1.44917
\(474\) −13.9155 −0.639159
\(475\) −19.1050 −0.876599
\(476\) −16.9288 −0.775931
\(477\) −6.77924 −0.310400
\(478\) −12.9093 −0.590459
\(479\) −14.3090 −0.653794 −0.326897 0.945060i \(-0.606003\pi\)
−0.326897 + 0.945060i \(0.606003\pi\)
\(480\) −26.5464 −1.21167
\(481\) −13.3607 −0.609195
\(482\) −66.3162 −3.02062
\(483\) 1.17009 0.0532408
\(484\) 14.7321 0.669639
\(485\) −10.0494 −0.456322
\(486\) 39.4524 1.78960
\(487\) −5.36069 −0.242916 −0.121458 0.992597i \(-0.538757\pi\)
−0.121458 + 0.992597i \(0.538757\pi\)
\(488\) 62.7019 2.83838
\(489\) −17.1096 −0.773722
\(490\) 3.17009 0.143210
\(491\) 8.99386 0.405887 0.202944 0.979190i \(-0.434949\pi\)
0.202944 + 0.979190i \(0.434949\pi\)
\(492\) −16.7480 −0.755060
\(493\) 2.00000 0.0900755
\(494\) −61.8720 −2.78375
\(495\) 7.07838 0.318149
\(496\) 129.056 5.79481
\(497\) −10.8371 −0.486110
\(498\) 46.4534 2.08163
\(499\) 2.02666 0.0907259 0.0453629 0.998971i \(-0.485556\pi\)
0.0453629 + 0.998971i \(0.485556\pi\)
\(500\) 53.9299 2.41182
\(501\) −7.02893 −0.314029
\(502\) 71.1338 3.17486
\(503\) −32.4124 −1.44520 −0.722599 0.691268i \(-0.757053\pi\)
−0.722599 + 0.691268i \(0.757053\pi\)
\(504\) −14.7587 −0.657406
\(505\) 5.75872 0.256260
\(506\) −10.0494 −0.446752
\(507\) −6.82991 −0.303327
\(508\) 15.6020 0.692225
\(509\) −10.4826 −0.464631 −0.232315 0.972640i \(-0.574630\pi\)
−0.232315 + 0.972640i \(0.574630\pi\)
\(510\) −11.7587 −0.520685
\(511\) 13.0205 0.575994
\(512\) −17.8599 −0.789303
\(513\) 28.5113 1.25880
\(514\) −21.2495 −0.937276
\(515\) −19.5174 −0.860041
\(516\) 53.0928 2.33728
\(517\) 22.5958 0.993763
\(518\) −8.34017 −0.366446
\(519\) −16.4969 −0.724135
\(520\) 45.9565 2.01533
\(521\) −24.6030 −1.07788 −0.538939 0.842345i \(-0.681175\pi\)
−0.538939 + 0.842345i \(0.681175\pi\)
\(522\) 2.78765 0.122012
\(523\) 12.1978 0.533372 0.266686 0.963783i \(-0.414071\pi\)
0.266686 + 0.963783i \(0.414071\pi\)
\(524\) −33.5103 −1.46390
\(525\) −4.24846 −0.185418
\(526\) −29.5669 −1.28918
\(527\) 29.5669 1.28795
\(528\) −60.0554 −2.61358
\(529\) 1.00000 0.0434783
\(530\) 13.1773 0.572384
\(531\) 13.7081 0.594879
\(532\) −28.0989 −1.21824
\(533\) 11.6332 0.503888
\(534\) −21.4680 −0.929011
\(535\) 17.9421 0.775706
\(536\) −89.2821 −3.85640
\(537\) 21.2085 0.915213
\(538\) 9.68649 0.417614
\(539\) 3.70928 0.159770
\(540\) −33.8576 −1.45700
\(541\) 17.5259 0.753495 0.376748 0.926316i \(-0.377042\pi\)
0.376748 + 0.926316i \(0.377042\pi\)
\(542\) 16.2029 0.695974
\(543\) −14.5464 −0.624245
\(544\) −61.4668 −2.63537
\(545\) 13.0784 0.560216
\(546\) −13.7587 −0.588819
\(547\) 12.5958 0.538559 0.269279 0.963062i \(-0.413215\pi\)
0.269279 + 0.963062i \(0.413215\pi\)
\(548\) 41.4329 1.76993
\(549\) 11.3002 0.482280
\(550\) 36.4885 1.55588
\(551\) 3.31965 0.141422
\(552\) 10.5886 0.450682
\(553\) 4.38962 0.186666
\(554\) 36.0638 1.53221
\(555\) −4.21461 −0.178900
\(556\) −15.6404 −0.663299
\(557\) −26.5113 −1.12332 −0.561660 0.827368i \(-0.689837\pi\)
−0.561660 + 0.827368i \(0.689837\pi\)
\(558\) 41.2111 1.74461
\(559\) −36.8781 −1.55978
\(560\) 16.1906 0.684178
\(561\) −13.7587 −0.580894
\(562\) 5.77101 0.243435
\(563\) −17.0472 −0.718453 −0.359227 0.933250i \(-0.616959\pi\)
−0.359227 + 0.933250i \(0.616959\pi\)
\(564\) −38.0638 −1.60278
\(565\) −15.8166 −0.665409
\(566\) −37.0472 −1.55721
\(567\) 1.44748 0.0607885
\(568\) −98.0698 −4.11492
\(569\) −25.1194 −1.05306 −0.526530 0.850156i \(-0.676507\pi\)
−0.526530 + 0.850156i \(0.676507\pi\)
\(570\) −19.5174 −0.817496
\(571\) −12.0144 −0.502786 −0.251393 0.967885i \(-0.580889\pi\)
−0.251393 + 0.967885i \(0.580889\pi\)
\(572\) 85.9709 3.59462
\(573\) 5.69102 0.237746
\(574\) 7.26180 0.303101
\(575\) −3.63090 −0.151419
\(576\) −40.5402 −1.68918
\(577\) −10.1133 −0.421021 −0.210511 0.977592i \(-0.567513\pi\)
−0.210511 + 0.977592i \(0.567513\pi\)
\(578\) 18.8310 0.783265
\(579\) 20.1666 0.838095
\(580\) −3.94214 −0.163689
\(581\) −14.6537 −0.607937
\(582\) 27.2267 1.12858
\(583\) 15.4186 0.638571
\(584\) 117.829 4.87578
\(585\) 8.28231 0.342432
\(586\) 35.9493 1.48505
\(587\) −14.2485 −0.588097 −0.294049 0.955790i \(-0.595003\pi\)
−0.294049 + 0.955790i \(0.595003\pi\)
\(588\) −6.24846 −0.257682
\(589\) 49.0759 2.02214
\(590\) −26.6453 −1.09697
\(591\) 14.6537 0.602772
\(592\) −42.5958 −1.75068
\(593\) −20.2557 −0.831800 −0.415900 0.909410i \(-0.636533\pi\)
−0.415900 + 0.909410i \(0.636533\pi\)
\(594\) −54.4534 −2.23425
\(595\) 3.70928 0.152065
\(596\) 4.47027 0.183109
\(597\) −1.36069 −0.0556894
\(598\) −11.7587 −0.480850
\(599\) −19.2351 −0.785926 −0.392963 0.919554i \(-0.628550\pi\)
−0.392963 + 0.919554i \(0.628550\pi\)
\(600\) −38.4463 −1.56956
\(601\) −21.4329 −0.874267 −0.437134 0.899397i \(-0.644006\pi\)
−0.437134 + 0.899397i \(0.644006\pi\)
\(602\) −23.0205 −0.938246
\(603\) −16.0905 −0.655255
\(604\) 84.9914 3.45825
\(605\) −3.22795 −0.131235
\(606\) −15.6020 −0.633787
\(607\) −37.7926 −1.53395 −0.766977 0.641675i \(-0.778240\pi\)
−0.766977 + 0.641675i \(0.778240\pi\)
\(608\) −102.024 −4.13763
\(609\) 0.738205 0.0299136
\(610\) −21.9649 −0.889334
\(611\) 26.4391 1.06961
\(612\) −27.6092 −1.11603
\(613\) 14.6947 0.593514 0.296757 0.954953i \(-0.404095\pi\)
0.296757 + 0.954953i \(0.404095\pi\)
\(614\) 47.4401 1.91453
\(615\) 3.66967 0.147975
\(616\) 33.5669 1.35245
\(617\) 12.1256 0.488157 0.244078 0.969756i \(-0.421515\pi\)
0.244078 + 0.969756i \(0.421515\pi\)
\(618\) 52.8781 2.12707
\(619\) −12.0144 −0.482899 −0.241449 0.970413i \(-0.577623\pi\)
−0.241449 + 0.970413i \(0.577623\pi\)
\(620\) −58.2784 −2.34052
\(621\) 5.41855 0.217439
\(622\) −5.51026 −0.220941
\(623\) 6.77205 0.271317
\(624\) −70.2700 −2.81305
\(625\) 6.33791 0.253516
\(626\) −33.5369 −1.34040
\(627\) −22.8371 −0.912026
\(628\) −59.6502 −2.38030
\(629\) −9.75872 −0.389106
\(630\) 5.17009 0.205981
\(631\) 10.9300 0.435118 0.217559 0.976047i \(-0.430191\pi\)
0.217559 + 0.976047i \(0.430191\pi\)
\(632\) 39.7237 1.58012
\(633\) −5.57531 −0.221599
\(634\) 54.5874 2.16794
\(635\) −3.41855 −0.135661
\(636\) −25.9733 −1.02991
\(637\) 4.34017 0.171964
\(638\) −6.34017 −0.251010
\(639\) −17.6742 −0.699181
\(640\) 33.4257 1.32127
\(641\) −33.1194 −1.30814 −0.654069 0.756435i \(-0.726939\pi\)
−0.654069 + 0.756435i \(0.726939\pi\)
\(642\) −48.6102 −1.91849
\(643\) −17.5297 −0.691305 −0.345653 0.938363i \(-0.612342\pi\)
−0.345653 + 0.938363i \(0.612342\pi\)
\(644\) −5.34017 −0.210432
\(645\) −11.6332 −0.458055
\(646\) −45.1917 −1.77804
\(647\) −6.88777 −0.270786 −0.135393 0.990792i \(-0.543230\pi\)
−0.135393 + 0.990792i \(0.543230\pi\)
\(648\) 13.0989 0.514573
\(649\) −31.1773 −1.22382
\(650\) 42.6947 1.67462
\(651\) 10.9132 0.427722
\(652\) 78.0866 3.05811
\(653\) −36.3545 −1.42266 −0.711332 0.702856i \(-0.751908\pi\)
−0.711332 + 0.702856i \(0.751908\pi\)
\(654\) −35.4329 −1.38554
\(655\) 7.34244 0.286893
\(656\) 37.0882 1.44805
\(657\) 21.2351 0.828461
\(658\) 16.5041 0.643397
\(659\) −2.63931 −0.102813 −0.0514064 0.998678i \(-0.516370\pi\)
−0.0514064 + 0.998678i \(0.516370\pi\)
\(660\) 27.1194 1.05562
\(661\) 21.0010 0.816846 0.408423 0.912793i \(-0.366079\pi\)
0.408423 + 0.912793i \(0.366079\pi\)
\(662\) 18.1711 0.706241
\(663\) −16.0989 −0.625229
\(664\) −132.608 −5.14618
\(665\) 6.15676 0.238749
\(666\) −13.6020 −0.527066
\(667\) 0.630898 0.0244285
\(668\) 32.0794 1.24119
\(669\) 25.0023 0.966644
\(670\) 31.2762 1.20830
\(671\) −25.7009 −0.992171
\(672\) −22.6875 −0.875191
\(673\) −51.2534 −1.97567 −0.987836 0.155497i \(-0.950302\pi\)
−0.987836 + 0.155497i \(0.950302\pi\)
\(674\) 78.6246 3.02851
\(675\) −19.6742 −0.757260
\(676\) 31.1711 1.19889
\(677\) 41.4668 1.59370 0.796849 0.604179i \(-0.206499\pi\)
0.796849 + 0.604179i \(0.206499\pi\)
\(678\) 42.8515 1.64570
\(679\) −8.58864 −0.329602
\(680\) 33.5669 1.28723
\(681\) 6.83710 0.261998
\(682\) −93.7296 −3.58909
\(683\) −26.2557 −1.00464 −0.502322 0.864680i \(-0.667521\pi\)
−0.502322 + 0.864680i \(0.667521\pi\)
\(684\) −45.8264 −1.75222
\(685\) −9.07838 −0.346867
\(686\) 2.70928 0.103441
\(687\) −3.34244 −0.127522
\(688\) −117.573 −4.48242
\(689\) 18.0410 0.687309
\(690\) −3.70928 −0.141210
\(691\) −12.0338 −0.457789 −0.228895 0.973451i \(-0.573511\pi\)
−0.228895 + 0.973451i \(0.573511\pi\)
\(692\) 75.2905 2.86212
\(693\) 6.04945 0.229800
\(694\) 49.4206 1.87598
\(695\) 3.42696 0.129992
\(696\) 6.68035 0.253218
\(697\) 8.49693 0.321844
\(698\) 22.6348 0.856739
\(699\) 15.8264 0.598610
\(700\) 19.3896 0.732859
\(701\) 34.6491 1.30868 0.654340 0.756200i \(-0.272946\pi\)
0.654340 + 0.756200i \(0.272946\pi\)
\(702\) −63.7152 −2.40478
\(703\) −16.1978 −0.610911
\(704\) 92.2038 3.47506
\(705\) 8.34017 0.314109
\(706\) 85.5851 3.22104
\(707\) 4.92162 0.185097
\(708\) 52.5197 1.97381
\(709\) 7.65983 0.287671 0.143835 0.989602i \(-0.454056\pi\)
0.143835 + 0.989602i \(0.454056\pi\)
\(710\) 34.3545 1.28930
\(711\) 7.15902 0.268484
\(712\) 61.2834 2.29669
\(713\) 9.32684 0.349293
\(714\) −10.0494 −0.376091
\(715\) −18.8371 −0.704468
\(716\) −96.7936 −3.61735
\(717\) −5.57531 −0.208214
\(718\) 12.1606 0.453831
\(719\) 22.4196 0.836110 0.418055 0.908422i \(-0.362712\pi\)
0.418055 + 0.908422i \(0.362712\pi\)
\(720\) 26.4052 0.984064
\(721\) −16.6803 −0.621209
\(722\) −23.5341 −0.875848
\(723\) −28.6407 −1.06516
\(724\) 66.3884 2.46731
\(725\) −2.29072 −0.0850754
\(726\) 8.74539 0.324572
\(727\) 34.8371 1.29204 0.646018 0.763322i \(-0.276433\pi\)
0.646018 + 0.763322i \(0.276433\pi\)
\(728\) 39.2762 1.45567
\(729\) 21.3812 0.791897
\(730\) −41.2762 −1.52770
\(731\) −26.9360 −0.996264
\(732\) 43.2944 1.60021
\(733\) −38.6752 −1.42850 −0.714251 0.699889i \(-0.753233\pi\)
−0.714251 + 0.699889i \(0.753233\pi\)
\(734\) 50.8059 1.87528
\(735\) 1.36910 0.0505001
\(736\) −19.3896 −0.714711
\(737\) 36.5958 1.34802
\(738\) 11.8432 0.435956
\(739\) −51.3484 −1.88888 −0.944441 0.328682i \(-0.893396\pi\)
−0.944441 + 0.328682i \(0.893396\pi\)
\(740\) 19.2351 0.707098
\(741\) −26.7214 −0.981635
\(742\) 11.2618 0.413434
\(743\) 8.00597 0.293710 0.146855 0.989158i \(-0.453085\pi\)
0.146855 + 0.989158i \(0.453085\pi\)
\(744\) 98.7585 3.62066
\(745\) −0.979481 −0.0358854
\(746\) −60.9893 −2.23298
\(747\) −23.8987 −0.874406
\(748\) 62.7936 2.29596
\(749\) 15.3340 0.560293
\(750\) 32.0144 1.16900
\(751\) 23.3958 0.853724 0.426862 0.904317i \(-0.359619\pi\)
0.426862 + 0.904317i \(0.359619\pi\)
\(752\) 84.2916 3.07380
\(753\) 30.7214 1.11955
\(754\) −7.41855 −0.270168
\(755\) −18.6225 −0.677742
\(756\) −28.9360 −1.05239
\(757\) −29.1194 −1.05836 −0.529182 0.848509i \(-0.677501\pi\)
−0.529182 + 0.848509i \(0.677501\pi\)
\(758\) 60.5029 2.19756
\(759\) −4.34017 −0.157538
\(760\) 55.7152 2.02100
\(761\) −10.1834 −0.369149 −0.184574 0.982819i \(-0.559091\pi\)
−0.184574 + 0.982819i \(0.559091\pi\)
\(762\) 9.26180 0.335519
\(763\) 11.1773 0.404645
\(764\) −25.9733 −0.939682
\(765\) 6.04945 0.218718
\(766\) −41.2762 −1.49137
\(767\) −36.4801 −1.31722
\(768\) −32.3884 −1.16872
\(769\) 0.431882 0.0155741 0.00778703 0.999970i \(-0.497521\pi\)
0.00778703 + 0.999970i \(0.497521\pi\)
\(770\) −11.7587 −0.423755
\(771\) −9.17727 −0.330511
\(772\) −92.0386 −3.31254
\(773\) −0.0650468 −0.00233957 −0.00116978 0.999999i \(-0.500372\pi\)
−0.00116978 + 0.999999i \(0.500372\pi\)
\(774\) −37.5441 −1.34950
\(775\) −33.8648 −1.21646
\(776\) −77.7224 −2.79007
\(777\) −3.60197 −0.129220
\(778\) −57.9442 −2.07740
\(779\) 14.1034 0.505308
\(780\) 31.7321 1.13619
\(781\) 40.1978 1.43839
\(782\) −8.58864 −0.307129
\(783\) 3.41855 0.122169
\(784\) 13.8371 0.494182
\(785\) 13.0700 0.466487
\(786\) −19.8927 −0.709549
\(787\) 31.5486 1.12459 0.562294 0.826937i \(-0.309919\pi\)
0.562294 + 0.826937i \(0.309919\pi\)
\(788\) −66.8781 −2.38244
\(789\) −12.7694 −0.454603
\(790\) −13.9155 −0.495091
\(791\) −13.5174 −0.480625
\(792\) 54.7442 1.94525
\(793\) −30.0722 −1.06790
\(794\) −55.9565 −1.98582
\(795\) 5.69102 0.201840
\(796\) 6.21008 0.220110
\(797\) 51.8336 1.83604 0.918020 0.396533i \(-0.129787\pi\)
0.918020 + 0.396533i \(0.129787\pi\)
\(798\) −16.6803 −0.590478
\(799\) 19.3112 0.683183
\(800\) 70.4017 2.48908
\(801\) 11.0445 0.390239
\(802\) −65.9442 −2.32857
\(803\) −48.2967 −1.70435
\(804\) −61.6475 −2.17414
\(805\) 1.17009 0.0412401
\(806\) −109.672 −3.86302
\(807\) 4.18342 0.147263
\(808\) 44.5380 1.56684
\(809\) 44.4391 1.56239 0.781197 0.624284i \(-0.214609\pi\)
0.781197 + 0.624284i \(0.214609\pi\)
\(810\) −4.58864 −0.161228
\(811\) −23.6358 −0.829966 −0.414983 0.909829i \(-0.636212\pi\)
−0.414983 + 0.909829i \(0.636212\pi\)
\(812\) −3.36910 −0.118232
\(813\) 6.99773 0.245421
\(814\) 30.9360 1.08431
\(815\) −17.1096 −0.599322
\(816\) −51.3256 −1.79676
\(817\) −44.7091 −1.56417
\(818\) −46.8392 −1.63769
\(819\) 7.07838 0.247339
\(820\) −16.7480 −0.584867
\(821\) −14.9399 −0.521405 −0.260703 0.965419i \(-0.583954\pi\)
−0.260703 + 0.965419i \(0.583954\pi\)
\(822\) 24.5958 0.857878
\(823\) 30.7670 1.07247 0.536234 0.844069i \(-0.319846\pi\)
0.536234 + 0.844069i \(0.319846\pi\)
\(824\) −150.948 −5.25852
\(825\) 15.7587 0.548648
\(826\) −22.7721 −0.792341
\(827\) −24.5113 −0.852342 −0.426171 0.904643i \(-0.640138\pi\)
−0.426171 + 0.904643i \(0.640138\pi\)
\(828\) −8.70928 −0.302668
\(829\) 43.8720 1.52374 0.761869 0.647731i \(-0.224282\pi\)
0.761869 + 0.647731i \(0.224282\pi\)
\(830\) 46.4534 1.61242
\(831\) 15.5753 0.540301
\(832\) 107.886 3.74029
\(833\) 3.17009 0.109837
\(834\) −9.28458 −0.321499
\(835\) −7.02893 −0.243246
\(836\) 104.227 3.60475
\(837\) 50.5380 1.74685
\(838\) 8.26794 0.285611
\(839\) −5.39189 −0.186149 −0.0930743 0.995659i \(-0.529669\pi\)
−0.0930743 + 0.995659i \(0.529669\pi\)
\(840\) 12.3896 0.427483
\(841\) −28.6020 −0.986275
\(842\) −37.6163 −1.29634
\(843\) 2.49239 0.0858426
\(844\) 25.4452 0.875860
\(845\) −6.82991 −0.234956
\(846\) 26.9165 0.925409
\(847\) −2.75872 −0.0947909
\(848\) 57.5174 1.97516
\(849\) −16.0000 −0.549119
\(850\) 31.1845 1.06962
\(851\) −3.07838 −0.105525
\(852\) −67.7152 −2.31989
\(853\) −1.33403 −0.0456763 −0.0228382 0.999739i \(-0.507270\pi\)
−0.0228382 + 0.999739i \(0.507270\pi\)
\(854\) −18.7721 −0.642366
\(855\) 10.0410 0.343396
\(856\) 138.765 4.74287
\(857\) 12.1034 0.413445 0.206723 0.978400i \(-0.433720\pi\)
0.206723 + 0.978400i \(0.433720\pi\)
\(858\) 51.0349 1.74230
\(859\) 40.5718 1.38429 0.692146 0.721757i \(-0.256665\pi\)
0.692146 + 0.721757i \(0.256665\pi\)
\(860\) 53.0928 1.81045
\(861\) 3.13624 0.106883
\(862\) −41.1917 −1.40299
\(863\) 34.0944 1.16059 0.580293 0.814408i \(-0.302938\pi\)
0.580293 + 0.814408i \(0.302938\pi\)
\(864\) −105.064 −3.57434
\(865\) −16.4969 −0.560912
\(866\) −34.8320 −1.18364
\(867\) 8.13275 0.276203
\(868\) −49.8069 −1.69056
\(869\) −16.2823 −0.552340
\(870\) −2.34017 −0.0793393
\(871\) 42.8203 1.45091
\(872\) 101.148 3.42531
\(873\) −14.0072 −0.474071
\(874\) −14.2557 −0.482205
\(875\) −10.0989 −0.341405
\(876\) 81.3582 2.74884
\(877\) −20.4163 −0.689409 −0.344704 0.938711i \(-0.612021\pi\)
−0.344704 + 0.938711i \(0.612021\pi\)
\(878\) −39.4524 −1.33145
\(879\) 15.5259 0.523674
\(880\) −60.0554 −2.02447
\(881\) −35.4668 −1.19491 −0.597453 0.801904i \(-0.703821\pi\)
−0.597453 + 0.801904i \(0.703821\pi\)
\(882\) 4.41855 0.148780
\(883\) 7.91548 0.266377 0.133189 0.991091i \(-0.457478\pi\)
0.133189 + 0.991091i \(0.457478\pi\)
\(884\) 73.4740 2.47120
\(885\) −11.5076 −0.386824
\(886\) 81.9709 2.75387
\(887\) −8.10608 −0.272176 −0.136088 0.990697i \(-0.543453\pi\)
−0.136088 + 0.990697i \(0.543453\pi\)
\(888\) −32.5958 −1.09384
\(889\) −2.92162 −0.0979881
\(890\) −21.4680 −0.719609
\(891\) −5.36910 −0.179872
\(892\) −114.108 −3.82062
\(893\) 32.0533 1.07262
\(894\) 2.65368 0.0887525
\(895\) 21.2085 0.708921
\(896\) 28.5669 0.954353
\(897\) −5.07838 −0.169562
\(898\) 12.6908 0.423499
\(899\) 5.88428 0.196252
\(900\) 31.6225 1.05408
\(901\) 13.1773 0.438999
\(902\) −26.9360 −0.896871
\(903\) −9.94214 −0.330854
\(904\) −122.325 −4.06848
\(905\) −14.5464 −0.483538
\(906\) 50.4534 1.67620
\(907\) −18.8371 −0.625476 −0.312738 0.949839i \(-0.601246\pi\)
−0.312738 + 0.949839i \(0.601246\pi\)
\(908\) −31.2039 −1.03554
\(909\) 8.02666 0.266228
\(910\) −13.7587 −0.456097
\(911\) 33.6514 1.11492 0.557461 0.830203i \(-0.311776\pi\)
0.557461 + 0.830203i \(0.311776\pi\)
\(912\) −85.1917 −2.82098
\(913\) 54.3545 1.79887
\(914\) −63.5585 −2.10233
\(915\) −9.48625 −0.313606
\(916\) 15.2546 0.504026
\(917\) 6.27513 0.207223
\(918\) −46.5380 −1.53598
\(919\) 42.5464 1.40348 0.701738 0.712435i \(-0.252408\pi\)
0.701738 + 0.712435i \(0.252408\pi\)
\(920\) 10.5886 0.349097
\(921\) 20.4885 0.675120
\(922\) −17.8310 −0.587231
\(923\) 47.0349 1.54817
\(924\) 23.1773 0.762476
\(925\) 11.1773 0.367507
\(926\) −54.2245 −1.78193
\(927\) −27.2039 −0.893495
\(928\) −12.2329 −0.401563
\(929\) 36.3012 1.19100 0.595502 0.803354i \(-0.296953\pi\)
0.595502 + 0.803354i \(0.296953\pi\)
\(930\) −34.5958 −1.13444
\(931\) 5.26180 0.172448
\(932\) −72.2304 −2.36599
\(933\) −2.37978 −0.0779105
\(934\) −58.1399 −1.90240
\(935\) −13.7587 −0.449958
\(936\) 64.0554 2.09372
\(937\) 6.50412 0.212480 0.106240 0.994341i \(-0.466119\pi\)
0.106240 + 0.994341i \(0.466119\pi\)
\(938\) 26.7298 0.872759
\(939\) −14.4840 −0.472667
\(940\) −38.0638 −1.24151
\(941\) −2.35965 −0.0769223 −0.0384611 0.999260i \(-0.512246\pi\)
−0.0384611 + 0.999260i \(0.512246\pi\)
\(942\) −35.4101 −1.15372
\(943\) 2.68035 0.0872841
\(944\) −116.304 −3.78537
\(945\) 6.34017 0.206246
\(946\) 85.3894 2.77625
\(947\) 1.17727 0.0382563 0.0191281 0.999817i \(-0.493911\pi\)
0.0191281 + 0.999817i \(0.493911\pi\)
\(948\) 27.4284 0.890833
\(949\) −56.5113 −1.83443
\(950\) 51.7608 1.67934
\(951\) 23.5753 0.764482
\(952\) 28.6875 0.929768
\(953\) −31.8264 −1.03096 −0.515479 0.856902i \(-0.672386\pi\)
−0.515479 + 0.856902i \(0.672386\pi\)
\(954\) 18.3668 0.594648
\(955\) 5.69102 0.184157
\(956\) 25.4452 0.822957
\(957\) −2.73820 −0.0885136
\(958\) 38.7670 1.25250
\(959\) −7.75872 −0.250542
\(960\) 34.0326 1.09840
\(961\) 55.9900 1.80613
\(962\) 36.1978 1.16706
\(963\) 25.0082 0.805879
\(964\) 130.714 4.21001
\(965\) 20.1666 0.649186
\(966\) −3.17009 −0.101996
\(967\) 42.6225 1.37065 0.685323 0.728239i \(-0.259661\pi\)
0.685323 + 0.728239i \(0.259661\pi\)
\(968\) −24.9649 −0.802403
\(969\) −19.5174 −0.626991
\(970\) 27.2267 0.874197
\(971\) 55.7998 1.79070 0.895350 0.445364i \(-0.146926\pi\)
0.895350 + 0.445364i \(0.146926\pi\)
\(972\) −77.7635 −2.49426
\(973\) 2.92881 0.0938933
\(974\) 14.5236 0.465366
\(975\) 18.4391 0.590523
\(976\) −95.8746 −3.06887
\(977\) 52.2245 1.67081 0.835404 0.549636i \(-0.185234\pi\)
0.835404 + 0.549636i \(0.185234\pi\)
\(978\) 46.3545 1.48226
\(979\) −25.1194 −0.802820
\(980\) −6.24846 −0.199600
\(981\) 18.2290 0.582007
\(982\) −24.3668 −0.777577
\(983\) −16.7649 −0.534716 −0.267358 0.963597i \(-0.586151\pi\)
−0.267358 + 0.963597i \(0.586151\pi\)
\(984\) 28.3812 0.904760
\(985\) 14.6537 0.466905
\(986\) −5.41855 −0.172562
\(987\) 7.12783 0.226881
\(988\) 121.954 3.87988
\(989\) −8.49693 −0.270187
\(990\) −19.1773 −0.609494
\(991\) −39.0349 −1.23998 −0.619992 0.784608i \(-0.712864\pi\)
−0.619992 + 0.784608i \(0.712864\pi\)
\(992\) −180.844 −5.74180
\(993\) 7.84778 0.249042
\(994\) 29.3607 0.931265
\(995\) −1.36069 −0.0431368
\(996\) −91.5630 −2.90129
\(997\) −11.2618 −0.356665 −0.178332 0.983970i \(-0.557070\pi\)
−0.178332 + 0.983970i \(0.557070\pi\)
\(998\) −5.49079 −0.173808
\(999\) −16.6803 −0.527743
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 161.2.a.c.1.1 3
3.2 odd 2 1449.2.a.m.1.3 3
4.3 odd 2 2576.2.a.v.1.3 3
5.4 even 2 4025.2.a.j.1.3 3
7.6 odd 2 1127.2.a.f.1.1 3
23.22 odd 2 3703.2.a.c.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
161.2.a.c.1.1 3 1.1 even 1 trivial
1127.2.a.f.1.1 3 7.6 odd 2
1449.2.a.m.1.3 3 3.2 odd 2
2576.2.a.v.1.3 3 4.3 odd 2
3703.2.a.c.1.1 3 23.22 odd 2
4025.2.a.j.1.3 3 5.4 even 2