Properties

Label 6006.2.a.cf.1.2
Level $6006$
Weight $2$
Character 6006.1
Self dual yes
Analytic conductor $47.958$
Analytic rank $0$
Dimension $6$
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] = [6006,2,Mod(1,6006)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6006, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6006.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6006 = 2 \cdot 3 \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6006.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: \(47.9581514540\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.72306708.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 11x^{4} - x^{3} + 10x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.916622\) of defining polynomial
Character \(\chi\) \(=\) 6006.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.18192 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.18192 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.18192 q^{10} +1.00000 q^{11} -1.00000 q^{12} -1.00000 q^{13} +1.00000 q^{14} +2.18192 q^{15} +1.00000 q^{16} -1.83324 q^{17} +1.00000 q^{18} -4.82525 q^{19} -2.18192 q^{20} -1.00000 q^{21} +1.00000 q^{22} -0.733562 q^{23} -1.00000 q^{24} -0.239205 q^{25} -1.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} +9.89770 q^{29} +2.18192 q^{30} +1.24900 q^{31} +1.00000 q^{32} -1.00000 q^{33} -1.83324 q^{34} -2.18192 q^{35} +1.00000 q^{36} +1.65132 q^{37} -4.82525 q^{38} +1.00000 q^{39} -2.18192 q^{40} -2.09169 q^{41} -1.00000 q^{42} +2.45161 q^{43} +1.00000 q^{44} -2.18192 q^{45} -0.733562 q^{46} +6.31961 q^{47} -1.00000 q^{48} +1.00000 q^{49} -0.239205 q^{50} +1.83324 q^{51} -1.00000 q^{52} -11.6233 q^{53} -1.00000 q^{54} -2.18192 q^{55} +1.00000 q^{56} +4.82525 q^{57} +9.89770 q^{58} +4.38488 q^{59} +2.18192 q^{60} +14.1289 q^{61} +1.24900 q^{62} +1.00000 q^{63} +1.00000 q^{64} +2.18192 q^{65} -1.00000 q^{66} -1.21488 q^{67} -1.83324 q^{68} +0.733562 q^{69} -2.18192 q^{70} +8.74873 q^{71} +1.00000 q^{72} -11.5316 q^{73} +1.65132 q^{74} +0.239205 q^{75} -4.82525 q^{76} +1.00000 q^{77} +1.00000 q^{78} +7.21993 q^{79} -2.18192 q^{80} +1.00000 q^{81} -2.09169 q^{82} -10.9625 q^{83} -1.00000 q^{84} +4.00000 q^{85} +2.45161 q^{86} -9.89770 q^{87} +1.00000 q^{88} +12.7595 q^{89} -2.18192 q^{90} -1.00000 q^{91} -0.733562 q^{92} -1.24900 q^{93} +6.31961 q^{94} +10.5283 q^{95} -1.00000 q^{96} +1.28161 q^{97} +1.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} - 6 q^{3} + 6 q^{4} - 6 q^{6} + 6 q^{7} + 6 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{2} - 6 q^{3} + 6 q^{4} - 6 q^{6} + 6 q^{7} + 6 q^{8} + 6 q^{9} + 6 q^{11} - 6 q^{12} - 6 q^{13} + 6 q^{14} + 6 q^{16} + 6 q^{18} + 2 q^{19} - 6 q^{21} + 6 q^{22} + 10 q^{23} - 6 q^{24} + 10 q^{25} - 6 q^{26} - 6 q^{27} + 6 q^{28} + 6 q^{29} + 2 q^{31} + 6 q^{32} - 6 q^{33} + 6 q^{36} + 12 q^{37} + 2 q^{38} + 6 q^{39} + 4 q^{41} - 6 q^{42} + 4 q^{43} + 6 q^{44} + 10 q^{46} + 4 q^{47} - 6 q^{48} + 6 q^{49} + 10 q^{50} - 6 q^{52} + 18 q^{53} - 6 q^{54} + 6 q^{56} - 2 q^{57} + 6 q^{58} + 14 q^{59} + 2 q^{62} + 6 q^{63} + 6 q^{64} - 6 q^{66} + 4 q^{67} - 10 q^{69} + 14 q^{71} + 6 q^{72} + 2 q^{73} + 12 q^{74} - 10 q^{75} + 2 q^{76} + 6 q^{77} + 6 q^{78} + 6 q^{79} + 6 q^{81} + 4 q^{82} + 24 q^{83} - 6 q^{84} + 24 q^{85} + 4 q^{86} - 6 q^{87} + 6 q^{88} - 14 q^{89} - 6 q^{91} + 10 q^{92} - 2 q^{93} + 4 q^{94} + 4 q^{95} - 6 q^{96} - 2 q^{97} + 6 q^{98} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −2.18192 −0.975786 −0.487893 0.872903i \(-0.662234\pi\)
−0.487893 + 0.872903i \(0.662234\pi\)
\(6\) −1.00000 −0.408248
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −2.18192 −0.689985
\(11\) 1.00000 0.301511
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350
\(14\) 1.00000 0.267261
\(15\) 2.18192 0.563371
\(16\) 1.00000 0.250000
\(17\) −1.83324 −0.444627 −0.222313 0.974975i \(-0.571361\pi\)
−0.222313 + 0.974975i \(0.571361\pi\)
\(18\) 1.00000 0.235702
\(19\) −4.82525 −1.10699 −0.553494 0.832853i \(-0.686706\pi\)
−0.553494 + 0.832853i \(0.686706\pi\)
\(20\) −2.18192 −0.487893
\(21\) −1.00000 −0.218218
\(22\) 1.00000 0.213201
\(23\) −0.733562 −0.152958 −0.0764791 0.997071i \(-0.524368\pi\)
−0.0764791 + 0.997071i \(0.524368\pi\)
\(24\) −1.00000 −0.204124
\(25\) −0.239205 −0.0478409
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) 9.89770 1.83796 0.918978 0.394308i \(-0.129016\pi\)
0.918978 + 0.394308i \(0.129016\pi\)
\(30\) 2.18192 0.398363
\(31\) 1.24900 0.224327 0.112163 0.993690i \(-0.464222\pi\)
0.112163 + 0.993690i \(0.464222\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.00000 −0.174078
\(34\) −1.83324 −0.314399
\(35\) −2.18192 −0.368813
\(36\) 1.00000 0.166667
\(37\) 1.65132 0.271475 0.135738 0.990745i \(-0.456660\pi\)
0.135738 + 0.990745i \(0.456660\pi\)
\(38\) −4.82525 −0.782759
\(39\) 1.00000 0.160128
\(40\) −2.18192 −0.344993
\(41\) −2.09169 −0.326667 −0.163333 0.986571i \(-0.552225\pi\)
−0.163333 + 0.986571i \(0.552225\pi\)
\(42\) −1.00000 −0.154303
\(43\) 2.45161 0.373867 0.186933 0.982373i \(-0.440145\pi\)
0.186933 + 0.982373i \(0.440145\pi\)
\(44\) 1.00000 0.150756
\(45\) −2.18192 −0.325262
\(46\) −0.733562 −0.108158
\(47\) 6.31961 0.921810 0.460905 0.887450i \(-0.347525\pi\)
0.460905 + 0.887450i \(0.347525\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) −0.239205 −0.0338286
\(51\) 1.83324 0.256705
\(52\) −1.00000 −0.138675
\(53\) −11.6233 −1.59658 −0.798289 0.602274i \(-0.794261\pi\)
−0.798289 + 0.602274i \(0.794261\pi\)
\(54\) −1.00000 −0.136083
\(55\) −2.18192 −0.294211
\(56\) 1.00000 0.133631
\(57\) 4.82525 0.639120
\(58\) 9.89770 1.29963
\(59\) 4.38488 0.570863 0.285431 0.958399i \(-0.407863\pi\)
0.285431 + 0.958399i \(0.407863\pi\)
\(60\) 2.18192 0.281685
\(61\) 14.1289 1.80902 0.904511 0.426451i \(-0.140236\pi\)
0.904511 + 0.426451i \(0.140236\pi\)
\(62\) 1.24900 0.158623
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 2.18192 0.270634
\(66\) −1.00000 −0.123091
\(67\) −1.21488 −0.148421 −0.0742106 0.997243i \(-0.523644\pi\)
−0.0742106 + 0.997243i \(0.523644\pi\)
\(68\) −1.83324 −0.222313
\(69\) 0.733562 0.0883105
\(70\) −2.18192 −0.260790
\(71\) 8.74873 1.03828 0.519142 0.854688i \(-0.326252\pi\)
0.519142 + 0.854688i \(0.326252\pi\)
\(72\) 1.00000 0.117851
\(73\) −11.5316 −1.34967 −0.674835 0.737969i \(-0.735785\pi\)
−0.674835 + 0.737969i \(0.735785\pi\)
\(74\) 1.65132 0.191962
\(75\) 0.239205 0.0276210
\(76\) −4.82525 −0.553494
\(77\) 1.00000 0.113961
\(78\) 1.00000 0.113228
\(79\) 7.21993 0.812305 0.406153 0.913805i \(-0.366870\pi\)
0.406153 + 0.913805i \(0.366870\pi\)
\(80\) −2.18192 −0.243947
\(81\) 1.00000 0.111111
\(82\) −2.09169 −0.230988
\(83\) −10.9625 −1.20329 −0.601646 0.798763i \(-0.705488\pi\)
−0.601646 + 0.798763i \(0.705488\pi\)
\(84\) −1.00000 −0.109109
\(85\) 4.00000 0.433861
\(86\) 2.45161 0.264364
\(87\) −9.89770 −1.06114
\(88\) 1.00000 0.106600
\(89\) 12.7595 1.35251 0.676255 0.736668i \(-0.263602\pi\)
0.676255 + 0.736668i \(0.263602\pi\)
\(90\) −2.18192 −0.229995
\(91\) −1.00000 −0.104828
\(92\) −0.733562 −0.0764791
\(93\) −1.24900 −0.129515
\(94\) 6.31961 0.651818
\(95\) 10.5283 1.08018
\(96\) −1.00000 −0.102062
\(97\) 1.28161 0.130127 0.0650637 0.997881i \(-0.479275\pi\)
0.0650637 + 0.997881i \(0.479275\pi\)
\(98\) 1.00000 0.101015
\(99\) 1.00000 0.100504
\(100\) −0.239205 −0.0239205
\(101\) −6.21018 −0.617936 −0.308968 0.951072i \(-0.599984\pi\)
−0.308968 + 0.951072i \(0.599984\pi\)
\(102\) 1.83324 0.181518
\(103\) −6.31961 −0.622690 −0.311345 0.950297i \(-0.600779\pi\)
−0.311345 + 0.950297i \(0.600779\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 2.18192 0.212934
\(106\) −11.6233 −1.12895
\(107\) −6.69799 −0.647519 −0.323759 0.946139i \(-0.604947\pi\)
−0.323759 + 0.946139i \(0.604947\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 4.49800 0.430830 0.215415 0.976523i \(-0.430890\pi\)
0.215415 + 0.976523i \(0.430890\pi\)
\(110\) −2.18192 −0.208038
\(111\) −1.65132 −0.156736
\(112\) 1.00000 0.0944911
\(113\) −8.75378 −0.823486 −0.411743 0.911300i \(-0.635080\pi\)
−0.411743 + 0.911300i \(0.635080\pi\)
\(114\) 4.82525 0.451926
\(115\) 1.60058 0.149255
\(116\) 9.89770 0.918978
\(117\) −1.00000 −0.0924500
\(118\) 4.38488 0.403661
\(119\) −1.83324 −0.168053
\(120\) 2.18192 0.199182
\(121\) 1.00000 0.0909091
\(122\) 14.1289 1.27917
\(123\) 2.09169 0.188601
\(124\) 1.24900 0.112163
\(125\) 11.4315 1.02247
\(126\) 1.00000 0.0890871
\(127\) 22.0622 1.95770 0.978851 0.204574i \(-0.0655808\pi\)
0.978851 + 0.204574i \(0.0655808\pi\)
\(128\) 1.00000 0.0883883
\(129\) −2.45161 −0.215852
\(130\) 2.18192 0.191367
\(131\) 0.503804 0.0440176 0.0220088 0.999758i \(-0.492994\pi\)
0.0220088 + 0.999758i \(0.492994\pi\)
\(132\) −1.00000 −0.0870388
\(133\) −4.82525 −0.418402
\(134\) −1.21488 −0.104950
\(135\) 2.18192 0.187790
\(136\) −1.83324 −0.157199
\(137\) 17.8767 1.52731 0.763653 0.645626i \(-0.223404\pi\)
0.763653 + 0.645626i \(0.223404\pi\)
\(138\) 0.733562 0.0624449
\(139\) −7.74258 −0.656717 −0.328358 0.944553i \(-0.606495\pi\)
−0.328358 + 0.944553i \(0.606495\pi\)
\(140\) −2.18192 −0.184406
\(141\) −6.31961 −0.532207
\(142\) 8.74873 0.734177
\(143\) −1.00000 −0.0836242
\(144\) 1.00000 0.0833333
\(145\) −21.5960 −1.79345
\(146\) −11.5316 −0.954360
\(147\) −1.00000 −0.0824786
\(148\) 1.65132 0.135738
\(149\) 18.3308 1.50171 0.750857 0.660465i \(-0.229641\pi\)
0.750857 + 0.660465i \(0.229641\pi\)
\(150\) 0.239205 0.0195310
\(151\) 7.62264 0.620322 0.310161 0.950684i \(-0.399617\pi\)
0.310161 + 0.950684i \(0.399617\pi\)
\(152\) −4.82525 −0.391380
\(153\) −1.83324 −0.148209
\(154\) 1.00000 0.0805823
\(155\) −2.72522 −0.218895
\(156\) 1.00000 0.0800641
\(157\) 8.68346 0.693015 0.346508 0.938047i \(-0.387367\pi\)
0.346508 + 0.938047i \(0.387367\pi\)
\(158\) 7.21993 0.574387
\(159\) 11.6233 0.921785
\(160\) −2.18192 −0.172496
\(161\) −0.733562 −0.0578128
\(162\) 1.00000 0.0785674
\(163\) −2.83505 −0.222058 −0.111029 0.993817i \(-0.535415\pi\)
−0.111029 + 0.993817i \(0.535415\pi\)
\(164\) −2.09169 −0.163333
\(165\) 2.18192 0.169863
\(166\) −10.9625 −0.850855
\(167\) −5.40369 −0.418150 −0.209075 0.977900i \(-0.567045\pi\)
−0.209075 + 0.977900i \(0.567045\pi\)
\(168\) −1.00000 −0.0771517
\(169\) 1.00000 0.0769231
\(170\) 4.00000 0.306786
\(171\) −4.82525 −0.368996
\(172\) 2.45161 0.186933
\(173\) 16.9444 1.28826 0.644128 0.764918i \(-0.277220\pi\)
0.644128 + 0.764918i \(0.277220\pi\)
\(174\) −9.89770 −0.750343
\(175\) −0.239205 −0.0180822
\(176\) 1.00000 0.0753778
\(177\) −4.38488 −0.329588
\(178\) 12.7595 0.956368
\(179\) 19.7320 1.47484 0.737418 0.675436i \(-0.236045\pi\)
0.737418 + 0.675436i \(0.236045\pi\)
\(180\) −2.18192 −0.162631
\(181\) 2.27216 0.168888 0.0844442 0.996428i \(-0.473089\pi\)
0.0844442 + 0.996428i \(0.473089\pi\)
\(182\) −1.00000 −0.0741249
\(183\) −14.1289 −1.04444
\(184\) −0.733562 −0.0540789
\(185\) −3.60305 −0.264902
\(186\) −1.24900 −0.0915810
\(187\) −1.83324 −0.134060
\(188\) 6.31961 0.460905
\(189\) −1.00000 −0.0727393
\(190\) 10.5283 0.763806
\(191\) 23.0618 1.66870 0.834348 0.551238i \(-0.185844\pi\)
0.834348 + 0.551238i \(0.185844\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 1.75715 0.126483 0.0632414 0.997998i \(-0.479856\pi\)
0.0632414 + 0.997998i \(0.479856\pi\)
\(194\) 1.28161 0.0920140
\(195\) −2.18192 −0.156251
\(196\) 1.00000 0.0714286
\(197\) −20.3872 −1.45253 −0.726263 0.687417i \(-0.758744\pi\)
−0.726263 + 0.687417i \(0.758744\pi\)
\(198\) 1.00000 0.0710669
\(199\) −22.0426 −1.56256 −0.781280 0.624181i \(-0.785433\pi\)
−0.781280 + 0.624181i \(0.785433\pi\)
\(200\) −0.239205 −0.0169143
\(201\) 1.21488 0.0856910
\(202\) −6.21018 −0.436947
\(203\) 9.89770 0.694682
\(204\) 1.83324 0.128353
\(205\) 4.56391 0.318757
\(206\) −6.31961 −0.440308
\(207\) −0.733562 −0.0509861
\(208\) −1.00000 −0.0693375
\(209\) −4.82525 −0.333770
\(210\) 2.18192 0.150567
\(211\) −12.4660 −0.858192 −0.429096 0.903259i \(-0.641168\pi\)
−0.429096 + 0.903259i \(0.641168\pi\)
\(212\) −11.6233 −0.798289
\(213\) −8.74873 −0.599453
\(214\) −6.69799 −0.457865
\(215\) −5.34922 −0.364814
\(216\) −1.00000 −0.0680414
\(217\) 1.24900 0.0847876
\(218\) 4.49800 0.304643
\(219\) 11.5316 0.779232
\(220\) −2.18192 −0.147105
\(221\) 1.83324 0.123317
\(222\) −1.65132 −0.110829
\(223\) −0.272592 −0.0182541 −0.00912705 0.999958i \(-0.502905\pi\)
−0.00912705 + 0.999958i \(0.502905\pi\)
\(224\) 1.00000 0.0668153
\(225\) −0.239205 −0.0159470
\(226\) −8.75378 −0.582293
\(227\) 23.1263 1.53495 0.767473 0.641081i \(-0.221514\pi\)
0.767473 + 0.641081i \(0.221514\pi\)
\(228\) 4.82525 0.319560
\(229\) −4.19860 −0.277452 −0.138726 0.990331i \(-0.544301\pi\)
−0.138726 + 0.990331i \(0.544301\pi\)
\(230\) 1.60058 0.105539
\(231\) −1.00000 −0.0657952
\(232\) 9.89770 0.649816
\(233\) 6.44644 0.422320 0.211160 0.977451i \(-0.432276\pi\)
0.211160 + 0.977451i \(0.432276\pi\)
\(234\) −1.00000 −0.0653720
\(235\) −13.7889 −0.899489
\(236\) 4.38488 0.285431
\(237\) −7.21993 −0.468985
\(238\) −1.83324 −0.118832
\(239\) 14.7039 0.951113 0.475557 0.879685i \(-0.342247\pi\)
0.475557 + 0.879685i \(0.342247\pi\)
\(240\) 2.18192 0.140843
\(241\) 19.3315 1.24525 0.622626 0.782519i \(-0.286066\pi\)
0.622626 + 0.782519i \(0.286066\pi\)
\(242\) 1.00000 0.0642824
\(243\) −1.00000 −0.0641500
\(244\) 14.1289 0.904511
\(245\) −2.18192 −0.139398
\(246\) 2.09169 0.133361
\(247\) 4.82525 0.307023
\(248\) 1.24900 0.0793115
\(249\) 10.9625 0.694720
\(250\) 11.4315 0.722995
\(251\) 17.0067 1.07346 0.536728 0.843755i \(-0.319660\pi\)
0.536728 + 0.843755i \(0.319660\pi\)
\(252\) 1.00000 0.0629941
\(253\) −0.733562 −0.0461186
\(254\) 22.0622 1.38430
\(255\) −4.00000 −0.250490
\(256\) 1.00000 0.0625000
\(257\) −9.25108 −0.577066 −0.288533 0.957470i \(-0.593168\pi\)
−0.288533 + 0.957470i \(0.593168\pi\)
\(258\) −2.45161 −0.152630
\(259\) 1.65132 0.102608
\(260\) 2.18192 0.135317
\(261\) 9.89770 0.612652
\(262\) 0.503804 0.0311251
\(263\) 11.0663 0.682375 0.341187 0.939995i \(-0.389171\pi\)
0.341187 + 0.939995i \(0.389171\pi\)
\(264\) −1.00000 −0.0615457
\(265\) 25.3611 1.55792
\(266\) −4.82525 −0.295855
\(267\) −12.7595 −0.780872
\(268\) −1.21488 −0.0742106
\(269\) −10.6537 −0.649570 −0.324785 0.945788i \(-0.605292\pi\)
−0.324785 + 0.945788i \(0.605292\pi\)
\(270\) 2.18192 0.132788
\(271\) 28.4158 1.72614 0.863069 0.505086i \(-0.168539\pi\)
0.863069 + 0.505086i \(0.168539\pi\)
\(272\) −1.83324 −0.111157
\(273\) 1.00000 0.0605228
\(274\) 17.8767 1.07997
\(275\) −0.239205 −0.0144246
\(276\) 0.733562 0.0441552
\(277\) −15.7796 −0.948105 −0.474052 0.880497i \(-0.657209\pi\)
−0.474052 + 0.880497i \(0.657209\pi\)
\(278\) −7.74258 −0.464369
\(279\) 1.24900 0.0747756
\(280\) −2.18192 −0.130395
\(281\) 29.6619 1.76948 0.884742 0.466082i \(-0.154335\pi\)
0.884742 + 0.466082i \(0.154335\pi\)
\(282\) −6.31961 −0.376327
\(283\) −9.02923 −0.536732 −0.268366 0.963317i \(-0.586484\pi\)
−0.268366 + 0.963317i \(0.586484\pi\)
\(284\) 8.74873 0.519142
\(285\) −10.5283 −0.623645
\(286\) −1.00000 −0.0591312
\(287\) −2.09169 −0.123469
\(288\) 1.00000 0.0589256
\(289\) −13.6392 −0.802307
\(290\) −21.5960 −1.26816
\(291\) −1.28161 −0.0751291
\(292\) −11.5316 −0.674835
\(293\) 21.0756 1.23125 0.615624 0.788040i \(-0.288904\pi\)
0.615624 + 0.788040i \(0.288904\pi\)
\(294\) −1.00000 −0.0583212
\(295\) −9.56748 −0.557040
\(296\) 1.65132 0.0959810
\(297\) −1.00000 −0.0580259
\(298\) 18.3308 1.06187
\(299\) 0.733562 0.0424230
\(300\) 0.239205 0.0138105
\(301\) 2.45161 0.141308
\(302\) 7.62264 0.438634
\(303\) 6.21018 0.356766
\(304\) −4.82525 −0.276747
\(305\) −30.8282 −1.76522
\(306\) −1.83324 −0.104800
\(307\) −7.65854 −0.437096 −0.218548 0.975826i \(-0.570132\pi\)
−0.218548 + 0.975826i \(0.570132\pi\)
\(308\) 1.00000 0.0569803
\(309\) 6.31961 0.359510
\(310\) −2.72522 −0.154782
\(311\) 24.4586 1.38692 0.693461 0.720494i \(-0.256085\pi\)
0.693461 + 0.720494i \(0.256085\pi\)
\(312\) 1.00000 0.0566139
\(313\) 26.9065 1.52085 0.760424 0.649427i \(-0.224991\pi\)
0.760424 + 0.649427i \(0.224991\pi\)
\(314\) 8.68346 0.490036
\(315\) −2.18192 −0.122938
\(316\) 7.21993 0.406153
\(317\) 26.1731 1.47003 0.735013 0.678053i \(-0.237176\pi\)
0.735013 + 0.678053i \(0.237176\pi\)
\(318\) 11.6233 0.651801
\(319\) 9.89770 0.554165
\(320\) −2.18192 −0.121973
\(321\) 6.69799 0.373845
\(322\) −0.733562 −0.0408798
\(323\) 8.84586 0.492197
\(324\) 1.00000 0.0555556
\(325\) 0.239205 0.0132687
\(326\) −2.83505 −0.157019
\(327\) −4.49800 −0.248740
\(328\) −2.09169 −0.115494
\(329\) 6.31961 0.348411
\(330\) 2.18192 0.120111
\(331\) 9.76211 0.536574 0.268287 0.963339i \(-0.413542\pi\)
0.268287 + 0.963339i \(0.413542\pi\)
\(332\) −10.9625 −0.601646
\(333\) 1.65132 0.0904917
\(334\) −5.40369 −0.295677
\(335\) 2.65078 0.144827
\(336\) −1.00000 −0.0545545
\(337\) 18.7182 1.01965 0.509823 0.860279i \(-0.329711\pi\)
0.509823 + 0.860279i \(0.329711\pi\)
\(338\) 1.00000 0.0543928
\(339\) 8.75378 0.475440
\(340\) 4.00000 0.216930
\(341\) 1.24900 0.0676371
\(342\) −4.82525 −0.260920
\(343\) 1.00000 0.0539949
\(344\) 2.45161 0.132182
\(345\) −1.60058 −0.0861722
\(346\) 16.9444 0.910935
\(347\) 15.5758 0.836154 0.418077 0.908412i \(-0.362704\pi\)
0.418077 + 0.908412i \(0.362704\pi\)
\(348\) −9.89770 −0.530572
\(349\) 14.2771 0.764238 0.382119 0.924113i \(-0.375194\pi\)
0.382119 + 0.924113i \(0.375194\pi\)
\(350\) −0.239205 −0.0127860
\(351\) 1.00000 0.0533761
\(352\) 1.00000 0.0533002
\(353\) 2.45665 0.130754 0.0653772 0.997861i \(-0.479175\pi\)
0.0653772 + 0.997861i \(0.479175\pi\)
\(354\) −4.38488 −0.233054
\(355\) −19.0891 −1.01314
\(356\) 12.7595 0.676255
\(357\) 1.83324 0.0970256
\(358\) 19.7320 1.04287
\(359\) −25.5915 −1.35067 −0.675334 0.737512i \(-0.736000\pi\)
−0.675334 + 0.737512i \(0.736000\pi\)
\(360\) −2.18192 −0.114998
\(361\) 4.28305 0.225424
\(362\) 2.27216 0.119422
\(363\) −1.00000 −0.0524864
\(364\) −1.00000 −0.0524142
\(365\) 25.1610 1.31699
\(366\) −14.1289 −0.738530
\(367\) −13.4271 −0.700891 −0.350446 0.936583i \(-0.613970\pi\)
−0.350446 + 0.936583i \(0.613970\pi\)
\(368\) −0.733562 −0.0382396
\(369\) −2.09169 −0.108889
\(370\) −3.60305 −0.187314
\(371\) −11.6233 −0.603450
\(372\) −1.24900 −0.0647576
\(373\) −2.15944 −0.111812 −0.0559058 0.998436i \(-0.517805\pi\)
−0.0559058 + 0.998436i \(0.517805\pi\)
\(374\) −1.83324 −0.0947948
\(375\) −11.4315 −0.590323
\(376\) 6.31961 0.325909
\(377\) −9.89770 −0.509757
\(378\) −1.00000 −0.0514344
\(379\) −37.4146 −1.92186 −0.960928 0.276797i \(-0.910727\pi\)
−0.960928 + 0.276797i \(0.910727\pi\)
\(380\) 10.5283 0.540092
\(381\) −22.0622 −1.13028
\(382\) 23.0618 1.17995
\(383\) 20.5340 1.04924 0.524620 0.851337i \(-0.324208\pi\)
0.524620 + 0.851337i \(0.324208\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −2.18192 −0.111201
\(386\) 1.75715 0.0894368
\(387\) 2.45161 0.124622
\(388\) 1.28161 0.0650637
\(389\) −19.4714 −0.987237 −0.493618 0.869679i \(-0.664326\pi\)
−0.493618 + 0.869679i \(0.664326\pi\)
\(390\) −2.18192 −0.110486
\(391\) 1.34480 0.0680093
\(392\) 1.00000 0.0505076
\(393\) −0.503804 −0.0254135
\(394\) −20.3872 −1.02709
\(395\) −15.7533 −0.792636
\(396\) 1.00000 0.0502519
\(397\) 4.14996 0.208280 0.104140 0.994563i \(-0.466791\pi\)
0.104140 + 0.994563i \(0.466791\pi\)
\(398\) −22.0426 −1.10490
\(399\) 4.82525 0.241565
\(400\) −0.239205 −0.0119602
\(401\) −4.94437 −0.246910 −0.123455 0.992350i \(-0.539397\pi\)
−0.123455 + 0.992350i \(0.539397\pi\)
\(402\) 1.21488 0.0605927
\(403\) −1.24900 −0.0622171
\(404\) −6.21018 −0.308968
\(405\) −2.18192 −0.108421
\(406\) 9.89770 0.491215
\(407\) 1.65132 0.0818529
\(408\) 1.83324 0.0907591
\(409\) −27.7874 −1.37400 −0.686998 0.726659i \(-0.741072\pi\)
−0.686998 + 0.726659i \(0.741072\pi\)
\(410\) 4.56391 0.225395
\(411\) −17.8767 −0.881791
\(412\) −6.31961 −0.311345
\(413\) 4.38488 0.215766
\(414\) −0.733562 −0.0360526
\(415\) 23.9194 1.17416
\(416\) −1.00000 −0.0490290
\(417\) 7.74258 0.379156
\(418\) −4.82525 −0.236011
\(419\) 27.7533 1.35584 0.677920 0.735136i \(-0.262882\pi\)
0.677920 + 0.735136i \(0.262882\pi\)
\(420\) 2.18192 0.106467
\(421\) −22.3811 −1.09079 −0.545393 0.838180i \(-0.683620\pi\)
−0.545393 + 0.838180i \(0.683620\pi\)
\(422\) −12.4660 −0.606833
\(423\) 6.31961 0.307270
\(424\) −11.6233 −0.564476
\(425\) 0.438520 0.0212714
\(426\) −8.74873 −0.423877
\(427\) 14.1289 0.683746
\(428\) −6.69799 −0.323759
\(429\) 1.00000 0.0482805
\(430\) −5.34922 −0.257962
\(431\) −25.3366 −1.22042 −0.610210 0.792240i \(-0.708915\pi\)
−0.610210 + 0.792240i \(0.708915\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −34.2673 −1.64678 −0.823391 0.567474i \(-0.807921\pi\)
−0.823391 + 0.567474i \(0.807921\pi\)
\(434\) 1.24900 0.0599539
\(435\) 21.5960 1.03545
\(436\) 4.49800 0.215415
\(437\) 3.53962 0.169323
\(438\) 11.5316 0.551000
\(439\) 2.09725 0.100096 0.0500482 0.998747i \(-0.484062\pi\)
0.0500482 + 0.998747i \(0.484062\pi\)
\(440\) −2.18192 −0.104019
\(441\) 1.00000 0.0476190
\(442\) 1.83324 0.0871985
\(443\) −15.1828 −0.721359 −0.360679 0.932690i \(-0.617455\pi\)
−0.360679 + 0.932690i \(0.617455\pi\)
\(444\) −1.65132 −0.0783681
\(445\) −27.8404 −1.31976
\(446\) −0.272592 −0.0129076
\(447\) −18.3308 −0.867015
\(448\) 1.00000 0.0472456
\(449\) 34.3812 1.62255 0.811274 0.584667i \(-0.198775\pi\)
0.811274 + 0.584667i \(0.198775\pi\)
\(450\) −0.239205 −0.0112762
\(451\) −2.09169 −0.0984938
\(452\) −8.75378 −0.411743
\(453\) −7.62264 −0.358143
\(454\) 23.1263 1.08537
\(455\) 2.18192 0.102290
\(456\) 4.82525 0.225963
\(457\) 23.0258 1.07710 0.538550 0.842594i \(-0.318972\pi\)
0.538550 + 0.842594i \(0.318972\pi\)
\(458\) −4.19860 −0.196188
\(459\) 1.83324 0.0855685
\(460\) 1.60058 0.0746273
\(461\) −5.79994 −0.270130 −0.135065 0.990837i \(-0.543124\pi\)
−0.135065 + 0.990837i \(0.543124\pi\)
\(462\) −1.00000 −0.0465242
\(463\) −20.9602 −0.974105 −0.487052 0.873373i \(-0.661928\pi\)
−0.487052 + 0.873373i \(0.661928\pi\)
\(464\) 9.89770 0.459489
\(465\) 2.72522 0.126379
\(466\) 6.44644 0.298626
\(467\) 27.1046 1.25425 0.627126 0.778918i \(-0.284231\pi\)
0.627126 + 0.778918i \(0.284231\pi\)
\(468\) −1.00000 −0.0462250
\(469\) −1.21488 −0.0560980
\(470\) −13.7889 −0.636035
\(471\) −8.68346 −0.400113
\(472\) 4.38488 0.201831
\(473\) 2.45161 0.112725
\(474\) −7.21993 −0.331622
\(475\) 1.15422 0.0529593
\(476\) −1.83324 −0.0840266
\(477\) −11.6233 −0.532193
\(478\) 14.7039 0.672538
\(479\) −13.2625 −0.605980 −0.302990 0.952994i \(-0.597985\pi\)
−0.302990 + 0.952994i \(0.597985\pi\)
\(480\) 2.18192 0.0995908
\(481\) −1.65132 −0.0752937
\(482\) 19.3315 0.880527
\(483\) 0.733562 0.0333782
\(484\) 1.00000 0.0454545
\(485\) −2.79637 −0.126977
\(486\) −1.00000 −0.0453609
\(487\) 14.4688 0.655645 0.327822 0.944739i \(-0.393685\pi\)
0.327822 + 0.944739i \(0.393685\pi\)
\(488\) 14.1289 0.639586
\(489\) 2.83505 0.128205
\(490\) −2.18192 −0.0985693
\(491\) −13.6098 −0.614200 −0.307100 0.951677i \(-0.599359\pi\)
−0.307100 + 0.951677i \(0.599359\pi\)
\(492\) 2.09169 0.0943006
\(493\) −18.1449 −0.817205
\(494\) 4.82525 0.217098
\(495\) −2.18192 −0.0980702
\(496\) 1.24900 0.0560817
\(497\) 8.74873 0.392434
\(498\) 10.9625 0.491242
\(499\) −41.2145 −1.84501 −0.922507 0.385980i \(-0.873863\pi\)
−0.922507 + 0.385980i \(0.873863\pi\)
\(500\) 11.4315 0.511234
\(501\) 5.40369 0.241419
\(502\) 17.0067 0.759048
\(503\) 10.3949 0.463485 0.231742 0.972777i \(-0.425557\pi\)
0.231742 + 0.972777i \(0.425557\pi\)
\(504\) 1.00000 0.0445435
\(505\) 13.5501 0.602974
\(506\) −0.733562 −0.0326108
\(507\) −1.00000 −0.0444116
\(508\) 22.0622 0.978851
\(509\) 22.8036 1.01075 0.505376 0.862899i \(-0.331354\pi\)
0.505376 + 0.862899i \(0.331354\pi\)
\(510\) −4.00000 −0.177123
\(511\) −11.5316 −0.510127
\(512\) 1.00000 0.0441942
\(513\) 4.82525 0.213040
\(514\) −9.25108 −0.408048
\(515\) 13.7889 0.607612
\(516\) −2.45161 −0.107926
\(517\) 6.31961 0.277936
\(518\) 1.65132 0.0725548
\(519\) −16.9444 −0.743775
\(520\) 2.18192 0.0956837
\(521\) −5.95503 −0.260895 −0.130447 0.991455i \(-0.541641\pi\)
−0.130447 + 0.991455i \(0.541641\pi\)
\(522\) 9.89770 0.433211
\(523\) −6.73788 −0.294627 −0.147313 0.989090i \(-0.547063\pi\)
−0.147313 + 0.989090i \(0.547063\pi\)
\(524\) 0.503804 0.0220088
\(525\) 0.239205 0.0104397
\(526\) 11.0663 0.482512
\(527\) −2.28972 −0.0997418
\(528\) −1.00000 −0.0435194
\(529\) −22.4619 −0.976604
\(530\) 25.3611 1.10162
\(531\) 4.38488 0.190288
\(532\) −4.82525 −0.209201
\(533\) 2.09169 0.0906011
\(534\) −12.7595 −0.552160
\(535\) 14.6145 0.631840
\(536\) −1.21488 −0.0524748
\(537\) −19.7320 −0.851497
\(538\) −10.6537 −0.459316
\(539\) 1.00000 0.0430730
\(540\) 2.18192 0.0938951
\(541\) −26.8197 −1.15307 −0.576534 0.817073i \(-0.695595\pi\)
−0.576534 + 0.817073i \(0.695595\pi\)
\(542\) 28.4158 1.22056
\(543\) −2.27216 −0.0975077
\(544\) −1.83324 −0.0785997
\(545\) −9.81429 −0.420398
\(546\) 1.00000 0.0427960
\(547\) −7.24096 −0.309601 −0.154801 0.987946i \(-0.549473\pi\)
−0.154801 + 0.987946i \(0.549473\pi\)
\(548\) 17.8767 0.763653
\(549\) 14.1289 0.603007
\(550\) −0.239205 −0.0101997
\(551\) −47.7589 −2.03460
\(552\) 0.733562 0.0312225
\(553\) 7.21993 0.307023
\(554\) −15.7796 −0.670411
\(555\) 3.60305 0.152941
\(556\) −7.74258 −0.328358
\(557\) 13.5118 0.572514 0.286257 0.958153i \(-0.407589\pi\)
0.286257 + 0.958153i \(0.407589\pi\)
\(558\) 1.24900 0.0528743
\(559\) −2.45161 −0.103692
\(560\) −2.18192 −0.0922031
\(561\) 1.83324 0.0773996
\(562\) 29.6619 1.25121
\(563\) −31.8630 −1.34286 −0.671432 0.741066i \(-0.734320\pi\)
−0.671432 + 0.741066i \(0.734320\pi\)
\(564\) −6.31961 −0.266104
\(565\) 19.1001 0.803546
\(566\) −9.02923 −0.379527
\(567\) 1.00000 0.0419961
\(568\) 8.74873 0.367089
\(569\) 24.8849 1.04323 0.521615 0.853181i \(-0.325330\pi\)
0.521615 + 0.853181i \(0.325330\pi\)
\(570\) −10.5283 −0.440983
\(571\) −14.1701 −0.592999 −0.296499 0.955033i \(-0.595819\pi\)
−0.296499 + 0.955033i \(0.595819\pi\)
\(572\) −1.00000 −0.0418121
\(573\) −23.0618 −0.963422
\(574\) −2.09169 −0.0873054
\(575\) 0.175471 0.00731766
\(576\) 1.00000 0.0416667
\(577\) −19.1126 −0.795667 −0.397834 0.917458i \(-0.630238\pi\)
−0.397834 + 0.917458i \(0.630238\pi\)
\(578\) −13.6392 −0.567317
\(579\) −1.75715 −0.0730249
\(580\) −21.5960 −0.896727
\(581\) −10.9625 −0.454801
\(582\) −1.28161 −0.0531243
\(583\) −11.6233 −0.481387
\(584\) −11.5316 −0.477180
\(585\) 2.18192 0.0902115
\(586\) 21.0756 0.870623
\(587\) 2.27032 0.0937062 0.0468531 0.998902i \(-0.485081\pi\)
0.0468531 + 0.998902i \(0.485081\pi\)
\(588\) −1.00000 −0.0412393
\(589\) −6.02673 −0.248327
\(590\) −9.56748 −0.393887
\(591\) 20.3872 0.838616
\(592\) 1.65132 0.0678688
\(593\) 1.62636 0.0667864 0.0333932 0.999442i \(-0.489369\pi\)
0.0333932 + 0.999442i \(0.489369\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 4.00000 0.163984
\(596\) 18.3308 0.750857
\(597\) 22.0426 0.902144
\(598\) 0.733562 0.0299976
\(599\) −14.3907 −0.587986 −0.293993 0.955808i \(-0.594984\pi\)
−0.293993 + 0.955808i \(0.594984\pi\)
\(600\) 0.239205 0.00976549
\(601\) −13.6310 −0.556019 −0.278009 0.960578i \(-0.589675\pi\)
−0.278009 + 0.960578i \(0.589675\pi\)
\(602\) 2.45161 0.0999201
\(603\) −1.21488 −0.0494737
\(604\) 7.62264 0.310161
\(605\) −2.18192 −0.0887079
\(606\) 6.21018 0.252271
\(607\) −10.5698 −0.429014 −0.214507 0.976722i \(-0.568814\pi\)
−0.214507 + 0.976722i \(0.568814\pi\)
\(608\) −4.82525 −0.195690
\(609\) −9.89770 −0.401075
\(610\) −30.8282 −1.24820
\(611\) −6.31961 −0.255664
\(612\) −1.83324 −0.0741045
\(613\) −11.4717 −0.463336 −0.231668 0.972795i \(-0.574418\pi\)
−0.231668 + 0.972795i \(0.574418\pi\)
\(614\) −7.65854 −0.309074
\(615\) −4.56391 −0.184035
\(616\) 1.00000 0.0402911
\(617\) −45.5174 −1.83246 −0.916231 0.400651i \(-0.868784\pi\)
−0.916231 + 0.400651i \(0.868784\pi\)
\(618\) 6.31961 0.254212
\(619\) 24.4404 0.982342 0.491171 0.871063i \(-0.336569\pi\)
0.491171 + 0.871063i \(0.336569\pi\)
\(620\) −2.72522 −0.109448
\(621\) 0.733562 0.0294368
\(622\) 24.4586 0.980702
\(623\) 12.7595 0.511200
\(624\) 1.00000 0.0400320
\(625\) −23.7468 −0.949870
\(626\) 26.9065 1.07540
\(627\) 4.82525 0.192702
\(628\) 8.68346 0.346508
\(629\) −3.02727 −0.120705
\(630\) −2.18192 −0.0869300
\(631\) 8.54937 0.340345 0.170172 0.985414i \(-0.445568\pi\)
0.170172 + 0.985414i \(0.445568\pi\)
\(632\) 7.21993 0.287193
\(633\) 12.4660 0.495477
\(634\) 26.1731 1.03947
\(635\) −48.1380 −1.91030
\(636\) 11.6233 0.460893
\(637\) −1.00000 −0.0396214
\(638\) 9.89770 0.391854
\(639\) 8.74873 0.346094
\(640\) −2.18192 −0.0862481
\(641\) −11.9856 −0.473405 −0.236702 0.971582i \(-0.576067\pi\)
−0.236702 + 0.971582i \(0.576067\pi\)
\(642\) 6.69799 0.264349
\(643\) 23.3792 0.921985 0.460992 0.887404i \(-0.347494\pi\)
0.460992 + 0.887404i \(0.347494\pi\)
\(644\) −0.733562 −0.0289064
\(645\) 5.34922 0.210625
\(646\) 8.84586 0.348036
\(647\) −8.04119 −0.316132 −0.158066 0.987429i \(-0.550526\pi\)
−0.158066 + 0.987429i \(0.550526\pi\)
\(648\) 1.00000 0.0392837
\(649\) 4.38488 0.172122
\(650\) 0.239205 0.00938238
\(651\) −1.24900 −0.0489521
\(652\) −2.83505 −0.111029
\(653\) 1.36918 0.0535800 0.0267900 0.999641i \(-0.491471\pi\)
0.0267900 + 0.999641i \(0.491471\pi\)
\(654\) −4.49800 −0.175886
\(655\) −1.09926 −0.0429517
\(656\) −2.09169 −0.0816667
\(657\) −11.5316 −0.449890
\(658\) 6.31961 0.246364
\(659\) 2.32339 0.0905064 0.0452532 0.998976i \(-0.485591\pi\)
0.0452532 + 0.998976i \(0.485591\pi\)
\(660\) 2.18192 0.0849313
\(661\) −6.65249 −0.258752 −0.129376 0.991596i \(-0.541297\pi\)
−0.129376 + 0.991596i \(0.541297\pi\)
\(662\) 9.76211 0.379415
\(663\) −1.83324 −0.0711973
\(664\) −10.9625 −0.425428
\(665\) 10.5283 0.408271
\(666\) 1.65132 0.0639873
\(667\) −7.26058 −0.281131
\(668\) −5.40369 −0.209075
\(669\) 0.272592 0.0105390
\(670\) 2.65078 0.102408
\(671\) 14.1289 0.545441
\(672\) −1.00000 −0.0385758
\(673\) 29.6793 1.14405 0.572026 0.820235i \(-0.306158\pi\)
0.572026 + 0.820235i \(0.306158\pi\)
\(674\) 18.7182 0.720998
\(675\) 0.239205 0.00920699
\(676\) 1.00000 0.0384615
\(677\) −26.2614 −1.00931 −0.504653 0.863322i \(-0.668380\pi\)
−0.504653 + 0.863322i \(0.668380\pi\)
\(678\) 8.75378 0.336187
\(679\) 1.28161 0.0491835
\(680\) 4.00000 0.153393
\(681\) −23.1263 −0.886201
\(682\) 1.24900 0.0478266
\(683\) 19.6923 0.753504 0.376752 0.926314i \(-0.377041\pi\)
0.376752 + 0.926314i \(0.377041\pi\)
\(684\) −4.82525 −0.184498
\(685\) −39.0055 −1.49033
\(686\) 1.00000 0.0381802
\(687\) 4.19860 0.160187
\(688\) 2.45161 0.0934667
\(689\) 11.6233 0.442811
\(690\) −1.60058 −0.0609329
\(691\) 30.9151 1.17607 0.588033 0.808837i \(-0.299902\pi\)
0.588033 + 0.808837i \(0.299902\pi\)
\(692\) 16.9444 0.644128
\(693\) 1.00000 0.0379869
\(694\) 15.5758 0.591250
\(695\) 16.8937 0.640815
\(696\) −9.89770 −0.375171
\(697\) 3.83458 0.145245
\(698\) 14.2771 0.540398
\(699\) −6.44644 −0.243827
\(700\) −0.239205 −0.00904108
\(701\) −16.1855 −0.611320 −0.305660 0.952141i \(-0.598877\pi\)
−0.305660 + 0.952141i \(0.598877\pi\)
\(702\) 1.00000 0.0377426
\(703\) −7.96803 −0.300520
\(704\) 1.00000 0.0376889
\(705\) 13.7889 0.519320
\(706\) 2.45665 0.0924574
\(707\) −6.21018 −0.233558
\(708\) −4.38488 −0.164794
\(709\) 17.4467 0.655225 0.327613 0.944812i \(-0.393756\pi\)
0.327613 + 0.944812i \(0.393756\pi\)
\(710\) −19.0891 −0.716400
\(711\) 7.21993 0.270768
\(712\) 12.7595 0.478184
\(713\) −0.916218 −0.0343126
\(714\) 1.83324 0.0686074
\(715\) 2.18192 0.0815994
\(716\) 19.7320 0.737418
\(717\) −14.7039 −0.549125
\(718\) −25.5915 −0.955066
\(719\) −18.0170 −0.671921 −0.335960 0.941876i \(-0.609061\pi\)
−0.335960 + 0.941876i \(0.609061\pi\)
\(720\) −2.18192 −0.0813155
\(721\) −6.31961 −0.235355
\(722\) 4.28305 0.159399
\(723\) −19.3315 −0.718947
\(724\) 2.27216 0.0844442
\(725\) −2.36758 −0.0879295
\(726\) −1.00000 −0.0371135
\(727\) 21.5257 0.798345 0.399172 0.916876i \(-0.369298\pi\)
0.399172 + 0.916876i \(0.369298\pi\)
\(728\) −1.00000 −0.0370625
\(729\) 1.00000 0.0370370
\(730\) 25.1610 0.931252
\(731\) −4.49439 −0.166231
\(732\) −14.1289 −0.522220
\(733\) −12.8172 −0.473414 −0.236707 0.971581i \(-0.576068\pi\)
−0.236707 + 0.971581i \(0.576068\pi\)
\(734\) −13.4271 −0.495605
\(735\) 2.18192 0.0804815
\(736\) −0.733562 −0.0270394
\(737\) −1.21488 −0.0447507
\(738\) −2.09169 −0.0769961
\(739\) −0.757692 −0.0278721 −0.0139361 0.999903i \(-0.504436\pi\)
−0.0139361 + 0.999903i \(0.504436\pi\)
\(740\) −3.60305 −0.132451
\(741\) −4.82525 −0.177260
\(742\) −11.6233 −0.426704
\(743\) 34.8980 1.28028 0.640141 0.768257i \(-0.278876\pi\)
0.640141 + 0.768257i \(0.278876\pi\)
\(744\) −1.24900 −0.0457905
\(745\) −39.9963 −1.46535
\(746\) −2.15944 −0.0790627
\(747\) −10.9625 −0.401097
\(748\) −1.83324 −0.0670300
\(749\) −6.69799 −0.244739
\(750\) −11.4315 −0.417421
\(751\) 10.2778 0.375044 0.187522 0.982260i \(-0.439954\pi\)
0.187522 + 0.982260i \(0.439954\pi\)
\(752\) 6.31961 0.230452
\(753\) −17.0067 −0.619760
\(754\) −9.89770 −0.360453
\(755\) −16.6320 −0.605302
\(756\) −1.00000 −0.0363696
\(757\) 25.7302 0.935179 0.467590 0.883946i \(-0.345122\pi\)
0.467590 + 0.883946i \(0.345122\pi\)
\(758\) −37.4146 −1.35896
\(759\) 0.733562 0.0266266
\(760\) 10.5283 0.381903
\(761\) −49.2245 −1.78439 −0.892193 0.451654i \(-0.850834\pi\)
−0.892193 + 0.451654i \(0.850834\pi\)
\(762\) −22.0622 −0.799229
\(763\) 4.49800 0.162838
\(764\) 23.0618 0.834348
\(765\) 4.00000 0.144620
\(766\) 20.5340 0.741924
\(767\) −4.38488 −0.158329
\(768\) −1.00000 −0.0360844
\(769\) −35.1595 −1.26788 −0.633942 0.773380i \(-0.718564\pi\)
−0.633942 + 0.773380i \(0.718564\pi\)
\(770\) −2.18192 −0.0786311
\(771\) 9.25108 0.333169
\(772\) 1.75715 0.0632414
\(773\) −18.5154 −0.665954 −0.332977 0.942935i \(-0.608053\pi\)
−0.332977 + 0.942935i \(0.608053\pi\)
\(774\) 2.45161 0.0881212
\(775\) −0.298766 −0.0107320
\(776\) 1.28161 0.0460070
\(777\) −1.65132 −0.0592407
\(778\) −19.4714 −0.698082
\(779\) 10.0929 0.361617
\(780\) −2.18192 −0.0781254
\(781\) 8.74873 0.313054
\(782\) 1.34480 0.0480899
\(783\) −9.89770 −0.353715
\(784\) 1.00000 0.0357143
\(785\) −18.9467 −0.676235
\(786\) −0.503804 −0.0179701
\(787\) −22.9820 −0.819221 −0.409610 0.912261i \(-0.634335\pi\)
−0.409610 + 0.912261i \(0.634335\pi\)
\(788\) −20.3872 −0.726263
\(789\) −11.0663 −0.393969
\(790\) −15.7533 −0.560479
\(791\) −8.75378 −0.311248
\(792\) 1.00000 0.0355335
\(793\) −14.1289 −0.501732
\(794\) 4.14996 0.147276
\(795\) −25.3611 −0.899465
\(796\) −22.0426 −0.781280
\(797\) −43.6908 −1.54761 −0.773804 0.633425i \(-0.781649\pi\)
−0.773804 + 0.633425i \(0.781649\pi\)
\(798\) 4.82525 0.170812
\(799\) −11.5854 −0.409861
\(800\) −0.239205 −0.00845716
\(801\) 12.7595 0.450836
\(802\) −4.94437 −0.174592
\(803\) −11.5316 −0.406941
\(804\) 1.21488 0.0428455
\(805\) 1.60058 0.0564129
\(806\) −1.24900 −0.0439941
\(807\) 10.6537 0.375030
\(808\) −6.21018 −0.218473
\(809\) 33.1790 1.16651 0.583256 0.812289i \(-0.301779\pi\)
0.583256 + 0.812289i \(0.301779\pi\)
\(810\) −2.18192 −0.0766650
\(811\) −14.0825 −0.494503 −0.247251 0.968951i \(-0.579527\pi\)
−0.247251 + 0.968951i \(0.579527\pi\)
\(812\) 9.89770 0.347341
\(813\) −28.4158 −0.996586
\(814\) 1.65132 0.0578787
\(815\) 6.18586 0.216681
\(816\) 1.83324 0.0641764
\(817\) −11.8296 −0.413866
\(818\) −27.7874 −0.971562
\(819\) −1.00000 −0.0349428
\(820\) 4.56391 0.159379
\(821\) −42.8899 −1.49687 −0.748433 0.663210i \(-0.769194\pi\)
−0.748433 + 0.663210i \(0.769194\pi\)
\(822\) −17.8767 −0.623520
\(823\) −8.74892 −0.304968 −0.152484 0.988306i \(-0.548727\pi\)
−0.152484 + 0.988306i \(0.548727\pi\)
\(824\) −6.31961 −0.220154
\(825\) 0.239205 0.00832803
\(826\) 4.38488 0.152570
\(827\) −29.4594 −1.02440 −0.512202 0.858865i \(-0.671170\pi\)
−0.512202 + 0.858865i \(0.671170\pi\)
\(828\) −0.733562 −0.0254930
\(829\) −10.2105 −0.354626 −0.177313 0.984155i \(-0.556740\pi\)
−0.177313 + 0.984155i \(0.556740\pi\)
\(830\) 23.9194 0.830253
\(831\) 15.7796 0.547388
\(832\) −1.00000 −0.0346688
\(833\) −1.83324 −0.0635181
\(834\) 7.74258 0.268104
\(835\) 11.7904 0.408025
\(836\) −4.82525 −0.166885
\(837\) −1.24900 −0.0431717
\(838\) 27.7533 0.958723
\(839\) 1.59559 0.0550858 0.0275429 0.999621i \(-0.491232\pi\)
0.0275429 + 0.999621i \(0.491232\pi\)
\(840\) 2.18192 0.0752836
\(841\) 68.9645 2.37808
\(842\) −22.3811 −0.771302
\(843\) −29.6619 −1.02161
\(844\) −12.4660 −0.429096
\(845\) −2.18192 −0.0750605
\(846\) 6.31961 0.217273
\(847\) 1.00000 0.0343604
\(848\) −11.6233 −0.399145
\(849\) 9.02923 0.309882
\(850\) 0.438520 0.0150411
\(851\) −1.21134 −0.0415244
\(852\) −8.74873 −0.299727
\(853\) −11.5068 −0.393986 −0.196993 0.980405i \(-0.563118\pi\)
−0.196993 + 0.980405i \(0.563118\pi\)
\(854\) 14.1289 0.483481
\(855\) 10.5283 0.360061
\(856\) −6.69799 −0.228933
\(857\) −35.5980 −1.21601 −0.608003 0.793935i \(-0.708029\pi\)
−0.608003 + 0.793935i \(0.708029\pi\)
\(858\) 1.00000 0.0341394
\(859\) 10.4220 0.355596 0.177798 0.984067i \(-0.443103\pi\)
0.177798 + 0.984067i \(0.443103\pi\)
\(860\) −5.34922 −0.182407
\(861\) 2.09169 0.0712846
\(862\) −25.3366 −0.862967
\(863\) 49.1051 1.67156 0.835779 0.549066i \(-0.185016\pi\)
0.835779 + 0.549066i \(0.185016\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −36.9713 −1.25706
\(866\) −34.2673 −1.16445
\(867\) 13.6392 0.463212
\(868\) 1.24900 0.0423938
\(869\) 7.21993 0.244919
\(870\) 21.5960 0.732174
\(871\) 1.21488 0.0411646
\(872\) 4.49800 0.152321
\(873\) 1.28161 0.0433758
\(874\) 3.53962 0.119729
\(875\) 11.4315 0.386457
\(876\) 11.5316 0.389616
\(877\) −9.22238 −0.311417 −0.155709 0.987803i \(-0.549766\pi\)
−0.155709 + 0.987803i \(0.549766\pi\)
\(878\) 2.09725 0.0707789
\(879\) −21.0756 −0.710861
\(880\) −2.18192 −0.0735527
\(881\) 7.33367 0.247078 0.123539 0.992340i \(-0.460576\pi\)
0.123539 + 0.992340i \(0.460576\pi\)
\(882\) 1.00000 0.0336718
\(883\) 9.08080 0.305593 0.152797 0.988258i \(-0.451172\pi\)
0.152797 + 0.988258i \(0.451172\pi\)
\(884\) 1.83324 0.0616587
\(885\) 9.56748 0.321607
\(886\) −15.1828 −0.510078
\(887\) 48.2562 1.62028 0.810142 0.586233i \(-0.199390\pi\)
0.810142 + 0.586233i \(0.199390\pi\)
\(888\) −1.65132 −0.0554146
\(889\) 22.0622 0.739942
\(890\) −27.8404 −0.933211
\(891\) 1.00000 0.0335013
\(892\) −0.272592 −0.00912705
\(893\) −30.4937 −1.02043
\(894\) −18.3308 −0.613072
\(895\) −43.0537 −1.43913
\(896\) 1.00000 0.0334077
\(897\) −0.733562 −0.0244929
\(898\) 34.3812 1.14731
\(899\) 12.3622 0.412303
\(900\) −0.239205 −0.00797349
\(901\) 21.3083 0.709882
\(902\) −2.09169 −0.0696456
\(903\) −2.45161 −0.0815844
\(904\) −8.75378 −0.291146
\(905\) −4.95768 −0.164799
\(906\) −7.62264 −0.253245
\(907\) 18.2513 0.606026 0.303013 0.952986i \(-0.402008\pi\)
0.303013 + 0.952986i \(0.402008\pi\)
\(908\) 23.1263 0.767473
\(909\) −6.21018 −0.205979
\(910\) 2.18192 0.0723301
\(911\) 13.1556 0.435863 0.217932 0.975964i \(-0.430069\pi\)
0.217932 + 0.975964i \(0.430069\pi\)
\(912\) 4.82525 0.159780
\(913\) −10.9625 −0.362806
\(914\) 23.0258 0.761624
\(915\) 30.8282 1.01915
\(916\) −4.19860 −0.138726
\(917\) 0.503804 0.0166371
\(918\) 1.83324 0.0605061
\(919\) −10.4863 −0.345911 −0.172955 0.984930i \(-0.555332\pi\)
−0.172955 + 0.984930i \(0.555332\pi\)
\(920\) 1.60058 0.0527695
\(921\) 7.65854 0.252358
\(922\) −5.79994 −0.191011
\(923\) −8.74873 −0.287968
\(924\) −1.00000 −0.0328976
\(925\) −0.395003 −0.0129876
\(926\) −20.9602 −0.688796
\(927\) −6.31961 −0.207563
\(928\) 9.89770 0.324908
\(929\) −35.9632 −1.17992 −0.589958 0.807434i \(-0.700856\pi\)
−0.589958 + 0.807434i \(0.700856\pi\)
\(930\) 2.72522 0.0893635
\(931\) −4.82525 −0.158141
\(932\) 6.44644 0.211160
\(933\) −24.4586 −0.800740
\(934\) 27.1046 0.886890
\(935\) 4.00000 0.130814
\(936\) −1.00000 −0.0326860
\(937\) −40.1360 −1.31119 −0.655594 0.755114i \(-0.727582\pi\)
−0.655594 + 0.755114i \(0.727582\pi\)
\(938\) −1.21488 −0.0396672
\(939\) −26.9065 −0.878062
\(940\) −13.7889 −0.449745
\(941\) −9.89193 −0.322468 −0.161234 0.986916i \(-0.551547\pi\)
−0.161234 + 0.986916i \(0.551547\pi\)
\(942\) −8.68346 −0.282922
\(943\) 1.53438 0.0499664
\(944\) 4.38488 0.142716
\(945\) 2.18192 0.0709780
\(946\) 2.45161 0.0797086
\(947\) −2.93793 −0.0954697 −0.0477349 0.998860i \(-0.515200\pi\)
−0.0477349 + 0.998860i \(0.515200\pi\)
\(948\) −7.21993 −0.234492
\(949\) 11.5316 0.374331
\(950\) 1.15422 0.0374479
\(951\) −26.1731 −0.848720
\(952\) −1.83324 −0.0594158
\(953\) −56.7108 −1.83704 −0.918521 0.395373i \(-0.870615\pi\)
−0.918521 + 0.395373i \(0.870615\pi\)
\(954\) −11.6233 −0.376317
\(955\) −50.3192 −1.62829
\(956\) 14.7039 0.475557
\(957\) −9.89770 −0.319947
\(958\) −13.2625 −0.428493
\(959\) 17.8767 0.577268
\(960\) 2.18192 0.0704213
\(961\) −29.4400 −0.949677
\(962\) −1.65132 −0.0532407
\(963\) −6.69799 −0.215840
\(964\) 19.3315 0.622626
\(965\) −3.83398 −0.123420
\(966\) 0.733562 0.0236020
\(967\) 41.0517 1.32013 0.660067 0.751207i \(-0.270528\pi\)
0.660067 + 0.751207i \(0.270528\pi\)
\(968\) 1.00000 0.0321412
\(969\) −8.84586 −0.284170
\(970\) −2.79637 −0.0897860
\(971\) 41.9125 1.34504 0.672518 0.740081i \(-0.265213\pi\)
0.672518 + 0.740081i \(0.265213\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −7.74258 −0.248216
\(974\) 14.4688 0.463611
\(975\) −0.239205 −0.00766068
\(976\) 14.1289 0.452255
\(977\) 36.8188 1.17794 0.588969 0.808156i \(-0.299534\pi\)
0.588969 + 0.808156i \(0.299534\pi\)
\(978\) 2.83505 0.0906548
\(979\) 12.7595 0.407797
\(980\) −2.18192 −0.0696990
\(981\) 4.49800 0.143610
\(982\) −13.6098 −0.434305
\(983\) 8.15638 0.260148 0.130074 0.991504i \(-0.458479\pi\)
0.130074 + 0.991504i \(0.458479\pi\)
\(984\) 2.09169 0.0666806
\(985\) 44.4833 1.41735
\(986\) −18.1449 −0.577851
\(987\) −6.31961 −0.201155
\(988\) 4.82525 0.153512
\(989\) −1.79841 −0.0571860
\(990\) −2.18192 −0.0693461
\(991\) 38.3123 1.21703 0.608516 0.793542i \(-0.291765\pi\)
0.608516 + 0.793542i \(0.291765\pi\)
\(992\) 1.24900 0.0396558
\(993\) −9.76211 −0.309791
\(994\) 8.74873 0.277493
\(995\) 48.0953 1.52472
\(996\) 10.9625 0.347360
\(997\) −32.1201 −1.01725 −0.508627 0.860987i \(-0.669847\pi\)
−0.508627 + 0.860987i \(0.669847\pi\)
\(998\) −41.2145 −1.30462
\(999\) −1.65132 −0.0522454
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 6006.2.a.cf.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6006.2.a.cf.1.2 6 1.1 even 1 trivial