Properties

Label 349.2.a.b.1.6
Level $349$
Weight $2$
Character 349.1
Self dual yes
Analytic conductor $2.787$
Analytic rank $0$
Dimension $17$
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] = [349,2,Mod(1,349)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(349, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("349.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 349 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 349.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: \(2.78677903054\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{17} - 5 x^{16} - 14 x^{15} + 102 x^{14} + 26 x^{13} - 792 x^{12} + 474 x^{11} + 2887 x^{10} - 3021 x^{9} - 4835 x^{8} + 6673 x^{7} + 2880 x^{6} - 5373 x^{5} - 164 x^{4} + 1075 x^{3} + \cdots - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-0.358260\) of defining polynomial
Character \(\chi\) \(=\) 349.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.358260 q^{2} -1.20564 q^{3} -1.87165 q^{4} -1.57855 q^{5} +0.431934 q^{6} +1.55949 q^{7} +1.38706 q^{8} -1.54642 q^{9} +O(q^{10})\) \(q-0.358260 q^{2} -1.20564 q^{3} -1.87165 q^{4} -1.57855 q^{5} +0.431934 q^{6} +1.55949 q^{7} +1.38706 q^{8} -1.54642 q^{9} +0.565531 q^{10} +0.789060 q^{11} +2.25654 q^{12} +2.87707 q^{13} -0.558704 q^{14} +1.90317 q^{15} +3.24637 q^{16} +4.64739 q^{17} +0.554022 q^{18} -2.65313 q^{19} +2.95449 q^{20} -1.88019 q^{21} -0.282688 q^{22} +6.97009 q^{23} -1.67230 q^{24} -2.50818 q^{25} -1.03074 q^{26} +5.48137 q^{27} -2.91882 q^{28} +9.67743 q^{29} -0.681829 q^{30} -1.20813 q^{31} -3.93716 q^{32} -0.951324 q^{33} -1.66497 q^{34} -2.46174 q^{35} +2.89437 q^{36} +7.25596 q^{37} +0.950510 q^{38} -3.46872 q^{39} -2.18954 q^{40} -6.83210 q^{41} +0.673597 q^{42} -5.06871 q^{43} -1.47684 q^{44} +2.44111 q^{45} -2.49711 q^{46} +3.70134 q^{47} -3.91397 q^{48} -4.56798 q^{49} +0.898580 q^{50} -5.60309 q^{51} -5.38487 q^{52} -10.8348 q^{53} -1.96375 q^{54} -1.24557 q^{55} +2.16310 q^{56} +3.19873 q^{57} -3.46704 q^{58} +3.23659 q^{59} -3.56206 q^{60} +14.5741 q^{61} +0.432824 q^{62} -2.41164 q^{63} -5.08222 q^{64} -4.54160 q^{65} +0.340821 q^{66} +9.43650 q^{67} -8.69829 q^{68} -8.40345 q^{69} +0.881942 q^{70} +13.6644 q^{71} -2.14498 q^{72} -6.93119 q^{73} -2.59952 q^{74} +3.02397 q^{75} +4.96573 q^{76} +1.23053 q^{77} +1.24270 q^{78} -0.331026 q^{79} -5.12456 q^{80} -1.96930 q^{81} +2.44767 q^{82} +8.48465 q^{83} +3.51906 q^{84} -7.33614 q^{85} +1.81592 q^{86} -11.6675 q^{87} +1.09447 q^{88} -1.23323 q^{89} -0.874552 q^{90} +4.48677 q^{91} -13.0456 q^{92} +1.45657 q^{93} -1.32604 q^{94} +4.18810 q^{95} +4.74681 q^{96} -3.08231 q^{97} +1.63653 q^{98} -1.22022 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q + 5 q^{2} + 6 q^{3} + 19 q^{4} + 5 q^{5} - 3 q^{6} + q^{7} + 9 q^{8} + 25 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 17 q + 5 q^{2} + 6 q^{3} + 19 q^{4} + 5 q^{5} - 3 q^{6} + q^{7} + 9 q^{8} + 25 q^{9} - 6 q^{10} + 39 q^{11} + 4 q^{12} - 6 q^{13} + 11 q^{14} + 12 q^{15} + 23 q^{16} + q^{17} + 7 q^{19} + 8 q^{20} - 3 q^{21} - q^{22} + 8 q^{23} - 21 q^{24} + 14 q^{25} + q^{26} + 9 q^{27} - 23 q^{28} + 9 q^{29} - 27 q^{30} - 2 q^{31} + 23 q^{32} - 5 q^{33} - 12 q^{34} + 16 q^{35} + 10 q^{36} - 7 q^{37} - 8 q^{38} - 39 q^{40} + 3 q^{41} - 30 q^{42} + q^{43} + 56 q^{44} - 21 q^{45} - q^{46} + 16 q^{47} - 31 q^{48} + 6 q^{49} - 5 q^{50} + 29 q^{51} - 37 q^{52} + 27 q^{53} - 36 q^{54} - 16 q^{55} + 14 q^{56} - 21 q^{57} - 44 q^{58} + 74 q^{59} - 27 q^{60} - 30 q^{61} - 18 q^{62} - 9 q^{63} - 17 q^{64} - 3 q^{65} - 66 q^{66} + 9 q^{67} - 51 q^{68} - 23 q^{69} - 77 q^{70} + 70 q^{71} - 55 q^{72} - 38 q^{73} + 26 q^{74} + 22 q^{75} - 50 q^{76} - 38 q^{77} - 75 q^{78} + 7 q^{79} - 28 q^{80} + 5 q^{81} - 56 q^{82} + 47 q^{83} - 69 q^{84} - 49 q^{85} + 17 q^{86} - 37 q^{87} - 30 q^{88} + 23 q^{89} - 64 q^{90} + 6 q^{91} + 17 q^{92} - 31 q^{93} - 40 q^{94} - 11 q^{95} - 49 q^{96} - 14 q^{97} + 5 q^{98} + 79 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.358260 −0.253328 −0.126664 0.991946i \(-0.540427\pi\)
−0.126664 + 0.991946i \(0.540427\pi\)
\(3\) −1.20564 −0.696078 −0.348039 0.937480i \(-0.613152\pi\)
−0.348039 + 0.937480i \(0.613152\pi\)
\(4\) −1.87165 −0.935825
\(5\) −1.57855 −0.705949 −0.352975 0.935633i \(-0.614830\pi\)
−0.352975 + 0.935633i \(0.614830\pi\)
\(6\) 0.431934 0.176336
\(7\) 1.55949 0.589433 0.294716 0.955585i \(-0.404775\pi\)
0.294716 + 0.955585i \(0.404775\pi\)
\(8\) 1.38706 0.490399
\(9\) −1.54642 −0.515475
\(10\) 0.565531 0.178837
\(11\) 0.789060 0.237910 0.118955 0.992900i \(-0.462046\pi\)
0.118955 + 0.992900i \(0.462046\pi\)
\(12\) 2.25654 0.651408
\(13\) 2.87707 0.797956 0.398978 0.916960i \(-0.369365\pi\)
0.398978 + 0.916960i \(0.369365\pi\)
\(14\) −0.558704 −0.149320
\(15\) 1.90317 0.491396
\(16\) 3.24637 0.811593
\(17\) 4.64739 1.12716 0.563579 0.826062i \(-0.309424\pi\)
0.563579 + 0.826062i \(0.309424\pi\)
\(18\) 0.554022 0.130584
\(19\) −2.65313 −0.608670 −0.304335 0.952565i \(-0.598434\pi\)
−0.304335 + 0.952565i \(0.598434\pi\)
\(20\) 2.95449 0.660645
\(21\) −1.88019 −0.410291
\(22\) −0.282688 −0.0602694
\(23\) 6.97009 1.45337 0.726683 0.686973i \(-0.241061\pi\)
0.726683 + 0.686973i \(0.241061\pi\)
\(24\) −1.67230 −0.341356
\(25\) −2.50818 −0.501636
\(26\) −1.03074 −0.202145
\(27\) 5.48137 1.05489
\(28\) −2.91882 −0.551606
\(29\) 9.67743 1.79705 0.898527 0.438919i \(-0.144638\pi\)
0.898527 + 0.438919i \(0.144638\pi\)
\(30\) −0.681829 −0.124484
\(31\) −1.20813 −0.216986 −0.108493 0.994097i \(-0.534603\pi\)
−0.108493 + 0.994097i \(0.534603\pi\)
\(32\) −3.93716 −0.695998
\(33\) −0.951324 −0.165604
\(34\) −1.66497 −0.285541
\(35\) −2.46174 −0.416109
\(36\) 2.89437 0.482394
\(37\) 7.25596 1.19287 0.596436 0.802661i \(-0.296583\pi\)
0.596436 + 0.802661i \(0.296583\pi\)
\(38\) 0.950510 0.154193
\(39\) −3.46872 −0.555440
\(40\) −2.18954 −0.346197
\(41\) −6.83210 −1.06699 −0.533497 0.845802i \(-0.679123\pi\)
−0.533497 + 0.845802i \(0.679123\pi\)
\(42\) 0.673597 0.103938
\(43\) −5.06871 −0.772971 −0.386485 0.922295i \(-0.626311\pi\)
−0.386485 + 0.922295i \(0.626311\pi\)
\(44\) −1.47684 −0.222642
\(45\) 2.44111 0.363899
\(46\) −2.49711 −0.368178
\(47\) 3.70134 0.539896 0.269948 0.962875i \(-0.412993\pi\)
0.269948 + 0.962875i \(0.412993\pi\)
\(48\) −3.91397 −0.564932
\(49\) −4.56798 −0.652569
\(50\) 0.898580 0.127078
\(51\) −5.60309 −0.784590
\(52\) −5.38487 −0.746747
\(53\) −10.8348 −1.48828 −0.744138 0.668026i \(-0.767139\pi\)
−0.744138 + 0.668026i \(0.767139\pi\)
\(54\) −1.96375 −0.267233
\(55\) −1.24557 −0.167953
\(56\) 2.16310 0.289057
\(57\) 3.19873 0.423682
\(58\) −3.46704 −0.455244
\(59\) 3.23659 0.421368 0.210684 0.977554i \(-0.432431\pi\)
0.210684 + 0.977554i \(0.432431\pi\)
\(60\) −3.56206 −0.459861
\(61\) 14.5741 1.86603 0.933014 0.359840i \(-0.117169\pi\)
0.933014 + 0.359840i \(0.117169\pi\)
\(62\) 0.432824 0.0549686
\(63\) −2.41164 −0.303838
\(64\) −5.08222 −0.635277
\(65\) −4.54160 −0.563316
\(66\) 0.340821 0.0419522
\(67\) 9.43650 1.15285 0.576426 0.817149i \(-0.304447\pi\)
0.576426 + 0.817149i \(0.304447\pi\)
\(68\) −8.69829 −1.05482
\(69\) −8.40345 −1.01166
\(70\) 0.881942 0.105412
\(71\) 13.6644 1.62167 0.810834 0.585276i \(-0.199014\pi\)
0.810834 + 0.585276i \(0.199014\pi\)
\(72\) −2.14498 −0.252788
\(73\) −6.93119 −0.811235 −0.405617 0.914043i \(-0.632943\pi\)
−0.405617 + 0.914043i \(0.632943\pi\)
\(74\) −2.59952 −0.302188
\(75\) 3.02397 0.349178
\(76\) 4.96573 0.569608
\(77\) 1.23053 0.140232
\(78\) 1.24270 0.140709
\(79\) −0.331026 −0.0372433 −0.0186216 0.999827i \(-0.505928\pi\)
−0.0186216 + 0.999827i \(0.505928\pi\)
\(80\) −5.12456 −0.572944
\(81\) −1.96930 −0.218811
\(82\) 2.44767 0.270300
\(83\) 8.48465 0.931311 0.465656 0.884966i \(-0.345819\pi\)
0.465656 + 0.884966i \(0.345819\pi\)
\(84\) 3.51906 0.383961
\(85\) −7.33614 −0.795716
\(86\) 1.81592 0.195815
\(87\) −11.6675 −1.25089
\(88\) 1.09447 0.116671
\(89\) −1.23323 −0.130722 −0.0653609 0.997862i \(-0.520820\pi\)
−0.0653609 + 0.997862i \(0.520820\pi\)
\(90\) −0.874552 −0.0921858
\(91\) 4.48677 0.470341
\(92\) −13.0456 −1.36010
\(93\) 1.45657 0.151039
\(94\) −1.32604 −0.136771
\(95\) 4.18810 0.429690
\(96\) 4.74681 0.484469
\(97\) −3.08231 −0.312961 −0.156481 0.987681i \(-0.550015\pi\)
−0.156481 + 0.987681i \(0.550015\pi\)
\(98\) 1.63653 0.165314
\(99\) −1.22022 −0.122637
\(100\) 4.69443 0.469443
\(101\) 1.86510 0.185585 0.0927923 0.995685i \(-0.470421\pi\)
0.0927923 + 0.995685i \(0.470421\pi\)
\(102\) 2.00736 0.198759
\(103\) −14.2911 −1.40815 −0.704074 0.710127i \(-0.748638\pi\)
−0.704074 + 0.710127i \(0.748638\pi\)
\(104\) 3.99066 0.391317
\(105\) 2.96798 0.289645
\(106\) 3.88168 0.377022
\(107\) 7.90169 0.763885 0.381943 0.924186i \(-0.375255\pi\)
0.381943 + 0.924186i \(0.375255\pi\)
\(108\) −10.2592 −0.987192
\(109\) −9.71011 −0.930060 −0.465030 0.885295i \(-0.653956\pi\)
−0.465030 + 0.885295i \(0.653956\pi\)
\(110\) 0.446238 0.0425471
\(111\) −8.74809 −0.830332
\(112\) 5.06269 0.478380
\(113\) 11.1633 1.05015 0.525076 0.851055i \(-0.324037\pi\)
0.525076 + 0.851055i \(0.324037\pi\)
\(114\) −1.14598 −0.107331
\(115\) −11.0026 −1.02600
\(116\) −18.1128 −1.68173
\(117\) −4.44917 −0.411326
\(118\) −1.15954 −0.106744
\(119\) 7.24757 0.664383
\(120\) 2.63980 0.240980
\(121\) −10.3774 −0.943399
\(122\) −5.22133 −0.472717
\(123\) 8.23708 0.742712
\(124\) 2.26119 0.203061
\(125\) 11.8520 1.06008
\(126\) 0.863993 0.0769706
\(127\) −11.4985 −1.02032 −0.510162 0.860078i \(-0.670415\pi\)
−0.510162 + 0.860078i \(0.670415\pi\)
\(128\) 9.69507 0.856932
\(129\) 6.11106 0.538048
\(130\) 1.62707 0.142704
\(131\) 15.0200 1.31231 0.656154 0.754627i \(-0.272182\pi\)
0.656154 + 0.754627i \(0.272182\pi\)
\(132\) 1.78055 0.154977
\(133\) −4.13754 −0.358770
\(134\) −3.38072 −0.292050
\(135\) −8.65261 −0.744698
\(136\) 6.44619 0.552757
\(137\) 5.18646 0.443110 0.221555 0.975148i \(-0.428887\pi\)
0.221555 + 0.975148i \(0.428887\pi\)
\(138\) 3.01062 0.256281
\(139\) 16.8266 1.42721 0.713605 0.700548i \(-0.247061\pi\)
0.713605 + 0.700548i \(0.247061\pi\)
\(140\) 4.60751 0.389406
\(141\) −4.46250 −0.375810
\(142\) −4.89542 −0.410814
\(143\) 2.27018 0.189842
\(144\) −5.02027 −0.418356
\(145\) −15.2763 −1.26863
\(146\) 2.48317 0.205508
\(147\) 5.50736 0.454239
\(148\) −13.5806 −1.11632
\(149\) 6.55102 0.536681 0.268340 0.963324i \(-0.413525\pi\)
0.268340 + 0.963324i \(0.413525\pi\)
\(150\) −1.08337 −0.0884565
\(151\) 20.9181 1.70229 0.851144 0.524932i \(-0.175909\pi\)
0.851144 + 0.524932i \(0.175909\pi\)
\(152\) −3.68004 −0.298491
\(153\) −7.18684 −0.581021
\(154\) −0.440850 −0.0355247
\(155\) 1.90709 0.153181
\(156\) 6.49223 0.519795
\(157\) −2.27007 −0.181172 −0.0905858 0.995889i \(-0.528874\pi\)
−0.0905858 + 0.995889i \(0.528874\pi\)
\(158\) 0.118593 0.00943477
\(159\) 13.0629 1.03596
\(160\) 6.21500 0.491339
\(161\) 10.8698 0.856661
\(162\) 0.705521 0.0554309
\(163\) 9.57334 0.749842 0.374921 0.927057i \(-0.377670\pi\)
0.374921 + 0.927057i \(0.377670\pi\)
\(164\) 12.7873 0.998520
\(165\) 1.50171 0.116908
\(166\) −3.03971 −0.235927
\(167\) 18.4356 1.42659 0.713297 0.700862i \(-0.247201\pi\)
0.713297 + 0.700862i \(0.247201\pi\)
\(168\) −2.60793 −0.201206
\(169\) −4.72246 −0.363266
\(170\) 2.62824 0.201577
\(171\) 4.10287 0.313754
\(172\) 9.48685 0.723366
\(173\) −7.03539 −0.534890 −0.267445 0.963573i \(-0.586179\pi\)
−0.267445 + 0.963573i \(0.586179\pi\)
\(174\) 4.18001 0.316886
\(175\) −3.91149 −0.295680
\(176\) 2.56158 0.193086
\(177\) −3.90217 −0.293305
\(178\) 0.441816 0.0331155
\(179\) −18.8686 −1.41031 −0.705153 0.709055i \(-0.749122\pi\)
−0.705153 + 0.709055i \(0.749122\pi\)
\(180\) −4.56890 −0.340546
\(181\) −2.10777 −0.156669 −0.0783345 0.996927i \(-0.524960\pi\)
−0.0783345 + 0.996927i \(0.524960\pi\)
\(182\) −1.60743 −0.119151
\(183\) −17.5712 −1.29890
\(184\) 9.66792 0.712728
\(185\) −11.4539 −0.842107
\(186\) −0.521831 −0.0382625
\(187\) 3.66707 0.268162
\(188\) −6.92761 −0.505248
\(189\) 8.54815 0.621786
\(190\) −1.50043 −0.108853
\(191\) −6.25482 −0.452583 −0.226292 0.974060i \(-0.572660\pi\)
−0.226292 + 0.974060i \(0.572660\pi\)
\(192\) 6.12734 0.442203
\(193\) −12.1063 −0.871431 −0.435716 0.900084i \(-0.643505\pi\)
−0.435716 + 0.900084i \(0.643505\pi\)
\(194\) 1.10427 0.0792819
\(195\) 5.47555 0.392112
\(196\) 8.54967 0.610690
\(197\) −12.3250 −0.878121 −0.439061 0.898457i \(-0.644689\pi\)
−0.439061 + 0.898457i \(0.644689\pi\)
\(198\) 0.437156 0.0310673
\(199\) 0.0989600 0.00701509 0.00350754 0.999994i \(-0.498884\pi\)
0.00350754 + 0.999994i \(0.498884\pi\)
\(200\) −3.47899 −0.246002
\(201\) −11.3771 −0.802476
\(202\) −0.668191 −0.0470138
\(203\) 15.0919 1.05924
\(204\) 10.4870 0.734239
\(205\) 10.7848 0.753244
\(206\) 5.11994 0.356723
\(207\) −10.7787 −0.749173
\(208\) 9.34005 0.647616
\(209\) −2.09348 −0.144809
\(210\) −1.06331 −0.0733752
\(211\) 22.9635 1.58087 0.790437 0.612543i \(-0.209854\pi\)
0.790437 + 0.612543i \(0.209854\pi\)
\(212\) 20.2790 1.39277
\(213\) −16.4744 −1.12881
\(214\) −2.83086 −0.193514
\(215\) 8.00121 0.545678
\(216\) 7.60297 0.517316
\(217\) −1.88406 −0.127899
\(218\) 3.47874 0.235610
\(219\) 8.35654 0.564683
\(220\) 2.33127 0.157174
\(221\) 13.3709 0.899422
\(222\) 3.13409 0.210346
\(223\) 11.5656 0.774488 0.387244 0.921977i \(-0.373427\pi\)
0.387244 + 0.921977i \(0.373427\pi\)
\(224\) −6.13997 −0.410244
\(225\) 3.87871 0.258581
\(226\) −3.99935 −0.266033
\(227\) −16.5148 −1.09612 −0.548062 0.836438i \(-0.684634\pi\)
−0.548062 + 0.836438i \(0.684634\pi\)
\(228\) −5.98690 −0.396492
\(229\) −24.0669 −1.59039 −0.795194 0.606355i \(-0.792631\pi\)
−0.795194 + 0.606355i \(0.792631\pi\)
\(230\) 3.94181 0.259915
\(231\) −1.48358 −0.0976126
\(232\) 13.4231 0.881273
\(233\) −0.181911 −0.0119174 −0.00595868 0.999982i \(-0.501897\pi\)
−0.00595868 + 0.999982i \(0.501897\pi\)
\(234\) 1.59396 0.104200
\(235\) −5.84275 −0.381139
\(236\) −6.05776 −0.394327
\(237\) 0.399099 0.0259243
\(238\) −2.59651 −0.168307
\(239\) 18.2061 1.17766 0.588828 0.808258i \(-0.299590\pi\)
0.588828 + 0.808258i \(0.299590\pi\)
\(240\) 6.17839 0.398814
\(241\) −11.2575 −0.725162 −0.362581 0.931952i \(-0.618104\pi\)
−0.362581 + 0.931952i \(0.618104\pi\)
\(242\) 3.71780 0.238989
\(243\) −14.0698 −0.902580
\(244\) −27.2777 −1.74628
\(245\) 7.21079 0.460681
\(246\) −2.95101 −0.188150
\(247\) −7.63325 −0.485692
\(248\) −1.67574 −0.106410
\(249\) −10.2295 −0.648266
\(250\) −4.24611 −0.268548
\(251\) 9.20365 0.580929 0.290465 0.956886i \(-0.406190\pi\)
0.290465 + 0.956886i \(0.406190\pi\)
\(252\) 4.51374 0.284339
\(253\) 5.49982 0.345771
\(254\) 4.11944 0.258477
\(255\) 8.84476 0.553881
\(256\) 6.69108 0.418193
\(257\) −29.8651 −1.86293 −0.931466 0.363827i \(-0.881470\pi\)
−0.931466 + 0.363827i \(0.881470\pi\)
\(258\) −2.18935 −0.136303
\(259\) 11.3156 0.703118
\(260\) 8.50029 0.527166
\(261\) −14.9654 −0.926336
\(262\) −5.38108 −0.332444
\(263\) 7.09679 0.437607 0.218803 0.975769i \(-0.429785\pi\)
0.218803 + 0.975769i \(0.429785\pi\)
\(264\) −1.31954 −0.0812121
\(265\) 17.1033 1.05065
\(266\) 1.48231 0.0908865
\(267\) 1.48683 0.0909926
\(268\) −17.6618 −1.07887
\(269\) 0.833960 0.0508474 0.0254237 0.999677i \(-0.491907\pi\)
0.0254237 + 0.999677i \(0.491907\pi\)
\(270\) 3.09988 0.188653
\(271\) −21.9704 −1.33461 −0.667304 0.744786i \(-0.732552\pi\)
−0.667304 + 0.744786i \(0.732552\pi\)
\(272\) 15.0872 0.914793
\(273\) −5.40944 −0.327394
\(274\) −1.85810 −0.112252
\(275\) −1.97910 −0.119344
\(276\) 15.7283 0.946733
\(277\) 9.95992 0.598434 0.299217 0.954185i \(-0.403275\pi\)
0.299217 + 0.954185i \(0.403275\pi\)
\(278\) −6.02828 −0.361552
\(279\) 1.86828 0.111851
\(280\) −3.41457 −0.204060
\(281\) 13.2894 0.792781 0.396391 0.918082i \(-0.370263\pi\)
0.396391 + 0.918082i \(0.370263\pi\)
\(282\) 1.59873 0.0952032
\(283\) −16.2832 −0.967937 −0.483969 0.875085i \(-0.660805\pi\)
−0.483969 + 0.875085i \(0.660805\pi\)
\(284\) −25.5750 −1.51760
\(285\) −5.04935 −0.299098
\(286\) −0.813315 −0.0480923
\(287\) −10.6546 −0.628922
\(288\) 6.08852 0.358769
\(289\) 4.59823 0.270484
\(290\) 5.47289 0.321379
\(291\) 3.71617 0.217846
\(292\) 12.9728 0.759173
\(293\) −23.1953 −1.35508 −0.677542 0.735484i \(-0.736955\pi\)
−0.677542 + 0.735484i \(0.736955\pi\)
\(294\) −1.97307 −0.115072
\(295\) −5.10912 −0.297464
\(296\) 10.0644 0.584983
\(297\) 4.32512 0.250969
\(298\) −2.34697 −0.135956
\(299\) 20.0535 1.15972
\(300\) −5.65981 −0.326769
\(301\) −7.90461 −0.455614
\(302\) −7.49411 −0.431237
\(303\) −2.24865 −0.129181
\(304\) −8.61305 −0.493992
\(305\) −23.0060 −1.31732
\(306\) 2.57476 0.147189
\(307\) −12.3670 −0.705825 −0.352912 0.935656i \(-0.614809\pi\)
−0.352912 + 0.935656i \(0.614809\pi\)
\(308\) −2.30313 −0.131233
\(309\) 17.2300 0.980181
\(310\) −0.683234 −0.0388051
\(311\) −19.4193 −1.10117 −0.550583 0.834781i \(-0.685594\pi\)
−0.550583 + 0.834781i \(0.685594\pi\)
\(312\) −4.81131 −0.272387
\(313\) 11.2697 0.636999 0.318499 0.947923i \(-0.396821\pi\)
0.318499 + 0.947923i \(0.396821\pi\)
\(314\) 0.813276 0.0458958
\(315\) 3.80689 0.214494
\(316\) 0.619564 0.0348532
\(317\) 9.30611 0.522683 0.261342 0.965246i \(-0.415835\pi\)
0.261342 + 0.965246i \(0.415835\pi\)
\(318\) −4.67992 −0.262437
\(319\) 7.63607 0.427538
\(320\) 8.02254 0.448474
\(321\) −9.52662 −0.531724
\(322\) −3.89422 −0.217016
\(323\) −12.3301 −0.686067
\(324\) 3.68584 0.204769
\(325\) −7.21621 −0.400283
\(326\) −3.42974 −0.189956
\(327\) 11.7069 0.647394
\(328\) −9.47651 −0.523253
\(329\) 5.77221 0.318232
\(330\) −0.538004 −0.0296161
\(331\) −4.26779 −0.234579 −0.117290 0.993098i \(-0.537421\pi\)
−0.117290 + 0.993098i \(0.537421\pi\)
\(332\) −15.8803 −0.871544
\(333\) −11.2208 −0.614895
\(334\) −6.60475 −0.361396
\(335\) −14.8960 −0.813855
\(336\) −6.10380 −0.332990
\(337\) 25.2589 1.37594 0.687971 0.725738i \(-0.258502\pi\)
0.687971 + 0.725738i \(0.258502\pi\)
\(338\) 1.69187 0.0920255
\(339\) −13.4589 −0.730988
\(340\) 13.7307 0.744651
\(341\) −0.953284 −0.0516232
\(342\) −1.46989 −0.0794827
\(343\) −18.0402 −0.974078
\(344\) −7.03059 −0.379064
\(345\) 13.2653 0.714178
\(346\) 2.52050 0.135503
\(347\) 0.724410 0.0388884 0.0194442 0.999811i \(-0.493810\pi\)
0.0194442 + 0.999811i \(0.493810\pi\)
\(348\) 21.8375 1.17061
\(349\) 1.00000 0.0535288
\(350\) 1.40133 0.0749042
\(351\) 15.7703 0.841755
\(352\) −3.10665 −0.165585
\(353\) −21.4860 −1.14359 −0.571793 0.820398i \(-0.693752\pi\)
−0.571793 + 0.820398i \(0.693752\pi\)
\(354\) 1.39799 0.0743024
\(355\) −21.5700 −1.14482
\(356\) 2.30817 0.122333
\(357\) −8.73798 −0.462463
\(358\) 6.75987 0.357270
\(359\) −12.4632 −0.657780 −0.328890 0.944368i \(-0.606675\pi\)
−0.328890 + 0.944368i \(0.606675\pi\)
\(360\) 3.38596 0.178456
\(361\) −11.9609 −0.629521
\(362\) 0.755128 0.0396886
\(363\) 12.5114 0.656679
\(364\) −8.39766 −0.440157
\(365\) 10.9412 0.572690
\(366\) 6.29506 0.329048
\(367\) −27.7310 −1.44755 −0.723773 0.690038i \(-0.757594\pi\)
−0.723773 + 0.690038i \(0.757594\pi\)
\(368\) 22.6275 1.17954
\(369\) 10.5653 0.550009
\(370\) 4.10347 0.213329
\(371\) −16.8968 −0.877238
\(372\) −2.72619 −0.141346
\(373\) 19.0340 0.985542 0.492771 0.870159i \(-0.335984\pi\)
0.492771 + 0.870159i \(0.335984\pi\)
\(374\) −1.31376 −0.0679331
\(375\) −14.2893 −0.737898
\(376\) 5.13397 0.264764
\(377\) 27.8427 1.43397
\(378\) −3.06246 −0.157516
\(379\) −18.0869 −0.929062 −0.464531 0.885557i \(-0.653777\pi\)
−0.464531 + 0.885557i \(0.653777\pi\)
\(380\) −7.83866 −0.402115
\(381\) 13.8631 0.710226
\(382\) 2.24085 0.114652
\(383\) −24.1015 −1.23153 −0.615765 0.787930i \(-0.711153\pi\)
−0.615765 + 0.787930i \(0.711153\pi\)
\(384\) −11.6888 −0.596492
\(385\) −1.94246 −0.0989968
\(386\) 4.33721 0.220758
\(387\) 7.83838 0.398447
\(388\) 5.76901 0.292877
\(389\) 8.92229 0.452378 0.226189 0.974083i \(-0.427373\pi\)
0.226189 + 0.974083i \(0.427373\pi\)
\(390\) −1.96167 −0.0993331
\(391\) 32.3927 1.63817
\(392\) −6.33605 −0.320019
\(393\) −18.1088 −0.913469
\(394\) 4.41556 0.222453
\(395\) 0.522541 0.0262919
\(396\) 2.28383 0.114767
\(397\) −32.1747 −1.61480 −0.807401 0.590004i \(-0.799126\pi\)
−0.807401 + 0.590004i \(0.799126\pi\)
\(398\) −0.0354534 −0.00177712
\(399\) 4.98839 0.249732
\(400\) −8.14248 −0.407124
\(401\) 11.1234 0.555474 0.277737 0.960657i \(-0.410416\pi\)
0.277737 + 0.960657i \(0.410416\pi\)
\(402\) 4.07594 0.203290
\(403\) −3.47587 −0.173145
\(404\) −3.49082 −0.173675
\(405\) 3.10864 0.154469
\(406\) −5.40681 −0.268336
\(407\) 5.72538 0.283797
\(408\) −7.77181 −0.384762
\(409\) 26.0533 1.28825 0.644126 0.764920i \(-0.277221\pi\)
0.644126 + 0.764920i \(0.277221\pi\)
\(410\) −3.86377 −0.190818
\(411\) −6.25303 −0.308439
\(412\) 26.7480 1.31778
\(413\) 5.04743 0.248368
\(414\) 3.86159 0.189787
\(415\) −13.3934 −0.657458
\(416\) −11.3275 −0.555376
\(417\) −20.2868 −0.993450
\(418\) 0.750009 0.0366842
\(419\) 4.22720 0.206512 0.103256 0.994655i \(-0.467074\pi\)
0.103256 + 0.994655i \(0.467074\pi\)
\(420\) −5.55501 −0.271057
\(421\) 36.0653 1.75772 0.878859 0.477082i \(-0.158306\pi\)
0.878859 + 0.477082i \(0.158306\pi\)
\(422\) −8.22691 −0.400480
\(423\) −5.72384 −0.278303
\(424\) −15.0285 −0.729848
\(425\) −11.6565 −0.565422
\(426\) 5.90213 0.285959
\(427\) 22.7283 1.09990
\(428\) −14.7892 −0.714863
\(429\) −2.73703 −0.132145
\(430\) −2.86651 −0.138236
\(431\) 11.3277 0.545635 0.272818 0.962066i \(-0.412044\pi\)
0.272818 + 0.962066i \(0.412044\pi\)
\(432\) 17.7946 0.856141
\(433\) −12.8261 −0.616382 −0.308191 0.951325i \(-0.599724\pi\)
−0.308191 + 0.951325i \(0.599724\pi\)
\(434\) 0.674985 0.0324003
\(435\) 18.4178 0.883065
\(436\) 18.1739 0.870373
\(437\) −18.4926 −0.884620
\(438\) −2.99381 −0.143050
\(439\) 11.5063 0.549166 0.274583 0.961563i \(-0.411460\pi\)
0.274583 + 0.961563i \(0.411460\pi\)
\(440\) −1.72768 −0.0823638
\(441\) 7.06404 0.336383
\(442\) −4.79025 −0.227849
\(443\) 11.3816 0.540755 0.270378 0.962754i \(-0.412851\pi\)
0.270378 + 0.962754i \(0.412851\pi\)
\(444\) 16.3734 0.777046
\(445\) 1.94671 0.0922829
\(446\) −4.14348 −0.196200
\(447\) −7.89819 −0.373572
\(448\) −7.92568 −0.374453
\(449\) −18.6872 −0.881904 −0.440952 0.897531i \(-0.645359\pi\)
−0.440952 + 0.897531i \(0.645359\pi\)
\(450\) −1.38959 −0.0655057
\(451\) −5.39093 −0.253849
\(452\) −20.8937 −0.982758
\(453\) −25.2197 −1.18493
\(454\) 5.91658 0.277679
\(455\) −7.08259 −0.332037
\(456\) 4.43682 0.207773
\(457\) −11.2023 −0.524023 −0.262012 0.965065i \(-0.584386\pi\)
−0.262012 + 0.965065i \(0.584386\pi\)
\(458\) 8.62222 0.402890
\(459\) 25.4740 1.18903
\(460\) 20.5931 0.960158
\(461\) −9.04204 −0.421130 −0.210565 0.977580i \(-0.567530\pi\)
−0.210565 + 0.977580i \(0.567530\pi\)
\(462\) 0.531508 0.0247280
\(463\) −0.803861 −0.0373586 −0.0186793 0.999826i \(-0.505946\pi\)
−0.0186793 + 0.999826i \(0.505946\pi\)
\(464\) 31.4165 1.45848
\(465\) −2.29927 −0.106626
\(466\) 0.0651713 0.00301900
\(467\) 8.77838 0.406215 0.203108 0.979156i \(-0.434896\pi\)
0.203108 + 0.979156i \(0.434896\pi\)
\(468\) 8.32729 0.384929
\(469\) 14.7162 0.679529
\(470\) 2.09322 0.0965532
\(471\) 2.73690 0.126110
\(472\) 4.48933 0.206638
\(473\) −3.99951 −0.183898
\(474\) −0.142981 −0.00656734
\(475\) 6.65453 0.305331
\(476\) −13.5649 −0.621747
\(477\) 16.7552 0.767169
\(478\) −6.52252 −0.298333
\(479\) 32.9704 1.50645 0.753227 0.657761i \(-0.228496\pi\)
0.753227 + 0.657761i \(0.228496\pi\)
\(480\) −7.49308 −0.342011
\(481\) 20.8759 0.951859
\(482\) 4.03313 0.183704
\(483\) −13.1051 −0.596303
\(484\) 19.4228 0.882856
\(485\) 4.86559 0.220935
\(486\) 5.04066 0.228649
\(487\) −10.6257 −0.481496 −0.240748 0.970588i \(-0.577393\pi\)
−0.240748 + 0.970588i \(0.577393\pi\)
\(488\) 20.2152 0.915098
\(489\) −11.5420 −0.521949
\(490\) −2.58334 −0.116703
\(491\) −21.9301 −0.989692 −0.494846 0.868981i \(-0.664776\pi\)
−0.494846 + 0.868981i \(0.664776\pi\)
\(492\) −15.4169 −0.695049
\(493\) 44.9748 2.02556
\(494\) 2.73469 0.123039
\(495\) 1.92618 0.0865754
\(496\) −3.92203 −0.176104
\(497\) 21.3096 0.955865
\(498\) 3.66481 0.164224
\(499\) 43.5451 1.94935 0.974673 0.223635i \(-0.0717922\pi\)
0.974673 + 0.223635i \(0.0717922\pi\)
\(500\) −22.1829 −0.992048
\(501\) −22.2268 −0.993021
\(502\) −3.29730 −0.147166
\(503\) 8.73420 0.389439 0.194719 0.980859i \(-0.437620\pi\)
0.194719 + 0.980859i \(0.437620\pi\)
\(504\) −3.34508 −0.149002
\(505\) −2.94416 −0.131013
\(506\) −1.97037 −0.0875934
\(507\) 5.69360 0.252862
\(508\) 21.5211 0.954845
\(509\) −15.6844 −0.695200 −0.347600 0.937643i \(-0.613003\pi\)
−0.347600 + 0.937643i \(0.613003\pi\)
\(510\) −3.16872 −0.140313
\(511\) −10.8091 −0.478168
\(512\) −21.7873 −0.962871
\(513\) −14.5428 −0.642079
\(514\) 10.6995 0.471933
\(515\) 22.5593 0.994081
\(516\) −11.4378 −0.503519
\(517\) 2.92058 0.128447
\(518\) −4.05393 −0.178119
\(519\) 8.48217 0.372326
\(520\) −6.29946 −0.276250
\(521\) −24.9842 −1.09458 −0.547288 0.836944i \(-0.684340\pi\)
−0.547288 + 0.836944i \(0.684340\pi\)
\(522\) 5.36151 0.234667
\(523\) 9.50069 0.415436 0.207718 0.978189i \(-0.433396\pi\)
0.207718 + 0.978189i \(0.433396\pi\)
\(524\) −28.1123 −1.22809
\(525\) 4.71586 0.205817
\(526\) −2.54249 −0.110858
\(527\) −5.61464 −0.244577
\(528\) −3.08835 −0.134403
\(529\) 25.5822 1.11227
\(530\) −6.12743 −0.266158
\(531\) −5.00514 −0.217205
\(532\) 7.74402 0.335746
\(533\) −19.6564 −0.851415
\(534\) −0.532672 −0.0230510
\(535\) −12.4732 −0.539264
\(536\) 13.0890 0.565357
\(537\) 22.7488 0.981683
\(538\) −0.298774 −0.0128811
\(539\) −3.60441 −0.155253
\(540\) 16.1947 0.696907
\(541\) 15.3139 0.658395 0.329197 0.944261i \(-0.393222\pi\)
0.329197 + 0.944261i \(0.393222\pi\)
\(542\) 7.87112 0.338093
\(543\) 2.54121 0.109054
\(544\) −18.2975 −0.784499
\(545\) 15.3279 0.656575
\(546\) 1.93799 0.0829382
\(547\) 27.4217 1.17247 0.586235 0.810141i \(-0.300610\pi\)
0.586235 + 0.810141i \(0.300610\pi\)
\(548\) −9.70725 −0.414673
\(549\) −22.5378 −0.961891
\(550\) 0.709033 0.0302333
\(551\) −25.6755 −1.09381
\(552\) −11.6561 −0.496115
\(553\) −0.516232 −0.0219524
\(554\) −3.56824 −0.151600
\(555\) 13.8093 0.586172
\(556\) −31.4934 −1.33562
\(557\) 1.30097 0.0551239 0.0275619 0.999620i \(-0.491226\pi\)
0.0275619 + 0.999620i \(0.491226\pi\)
\(558\) −0.669329 −0.0283350
\(559\) −14.5830 −0.616797
\(560\) −7.99172 −0.337712
\(561\) −4.42117 −0.186662
\(562\) −4.76107 −0.200834
\(563\) 42.9705 1.81099 0.905494 0.424358i \(-0.139500\pi\)
0.905494 + 0.424358i \(0.139500\pi\)
\(564\) 8.35223 0.351692
\(565\) −17.6218 −0.741354
\(566\) 5.83363 0.245206
\(567\) −3.07110 −0.128974
\(568\) 18.9533 0.795264
\(569\) −4.22327 −0.177049 −0.0885243 0.996074i \(-0.528215\pi\)
−0.0885243 + 0.996074i \(0.528215\pi\)
\(570\) 1.80898 0.0757699
\(571\) −4.12901 −0.172794 −0.0863968 0.996261i \(-0.527535\pi\)
−0.0863968 + 0.996261i \(0.527535\pi\)
\(572\) −4.24898 −0.177659
\(573\) 7.54109 0.315033
\(574\) 3.81712 0.159323
\(575\) −17.4822 −0.729060
\(576\) 7.85927 0.327469
\(577\) 35.5668 1.48067 0.740333 0.672241i \(-0.234668\pi\)
0.740333 + 0.672241i \(0.234668\pi\)
\(578\) −1.64736 −0.0685212
\(579\) 14.5959 0.606584
\(580\) 28.5919 1.18721
\(581\) 13.2317 0.548945
\(582\) −1.33135 −0.0551864
\(583\) −8.54931 −0.354076
\(584\) −9.61396 −0.397828
\(585\) 7.02324 0.290375
\(586\) 8.30994 0.343281
\(587\) −17.3760 −0.717185 −0.358592 0.933494i \(-0.616743\pi\)
−0.358592 + 0.933494i \(0.616743\pi\)
\(588\) −10.3078 −0.425088
\(589\) 3.20532 0.132073
\(590\) 1.83039 0.0753561
\(591\) 14.8596 0.611241
\(592\) 23.5555 0.968127
\(593\) 19.0958 0.784172 0.392086 0.919929i \(-0.371754\pi\)
0.392086 + 0.919929i \(0.371754\pi\)
\(594\) −1.54952 −0.0635775
\(595\) −11.4406 −0.469021
\(596\) −12.2612 −0.502239
\(597\) −0.119310 −0.00488305
\(598\) −7.18435 −0.293790
\(599\) −9.20830 −0.376241 −0.188120 0.982146i \(-0.560240\pi\)
−0.188120 + 0.982146i \(0.560240\pi\)
\(600\) 4.19442 0.171236
\(601\) −8.39228 −0.342328 −0.171164 0.985243i \(-0.554753\pi\)
−0.171164 + 0.985243i \(0.554753\pi\)
\(602\) 2.83191 0.115420
\(603\) −14.5928 −0.594266
\(604\) −39.1513 −1.59304
\(605\) 16.3812 0.665991
\(606\) 0.805600 0.0327253
\(607\) −23.7898 −0.965599 −0.482800 0.875731i \(-0.660380\pi\)
−0.482800 + 0.875731i \(0.660380\pi\)
\(608\) 10.4458 0.423633
\(609\) −18.1954 −0.737316
\(610\) 8.24214 0.333714
\(611\) 10.6490 0.430813
\(612\) 13.4512 0.543734
\(613\) 15.9458 0.644043 0.322021 0.946732i \(-0.395638\pi\)
0.322021 + 0.946732i \(0.395638\pi\)
\(614\) 4.43062 0.178805
\(615\) −13.0026 −0.524317
\(616\) 1.70682 0.0687697
\(617\) −14.0750 −0.566638 −0.283319 0.959026i \(-0.591436\pi\)
−0.283319 + 0.959026i \(0.591436\pi\)
\(618\) −6.17282 −0.248307
\(619\) 6.01473 0.241752 0.120876 0.992668i \(-0.461430\pi\)
0.120876 + 0.992668i \(0.461430\pi\)
\(620\) −3.56940 −0.143351
\(621\) 38.2056 1.53314
\(622\) 6.95714 0.278956
\(623\) −1.92321 −0.0770517
\(624\) −11.2608 −0.450791
\(625\) −6.16815 −0.246726
\(626\) −4.03747 −0.161370
\(627\) 2.52399 0.100798
\(628\) 4.24878 0.169545
\(629\) 33.7213 1.34455
\(630\) −1.36386 −0.0543373
\(631\) −2.12252 −0.0844960 −0.0422480 0.999107i \(-0.513452\pi\)
−0.0422480 + 0.999107i \(0.513452\pi\)
\(632\) −0.459152 −0.0182641
\(633\) −27.6858 −1.10041
\(634\) −3.33401 −0.132410
\(635\) 18.1509 0.720298
\(636\) −24.4492 −0.969474
\(637\) −13.1424 −0.520721
\(638\) −2.73570 −0.108307
\(639\) −21.1310 −0.835929
\(640\) −15.3042 −0.604950
\(641\) 6.77267 0.267504 0.133752 0.991015i \(-0.457297\pi\)
0.133752 + 0.991015i \(0.457297\pi\)
\(642\) 3.41301 0.134701
\(643\) −11.2524 −0.443751 −0.221875 0.975075i \(-0.571218\pi\)
−0.221875 + 0.975075i \(0.571218\pi\)
\(644\) −20.3445 −0.801685
\(645\) −9.64661 −0.379835
\(646\) 4.41739 0.173800
\(647\) 34.2906 1.34810 0.674051 0.738685i \(-0.264553\pi\)
0.674051 + 0.738685i \(0.264553\pi\)
\(648\) −2.73153 −0.107305
\(649\) 2.55386 0.100248
\(650\) 2.58528 0.101403
\(651\) 2.27151 0.0890275
\(652\) −17.9179 −0.701721
\(653\) 25.1564 0.984447 0.492224 0.870469i \(-0.336184\pi\)
0.492224 + 0.870469i \(0.336184\pi\)
\(654\) −4.19412 −0.164003
\(655\) −23.7099 −0.926422
\(656\) −22.1795 −0.865966
\(657\) 10.7186 0.418171
\(658\) −2.06795 −0.0806172
\(659\) −5.93357 −0.231139 −0.115570 0.993299i \(-0.536869\pi\)
−0.115570 + 0.993299i \(0.536869\pi\)
\(660\) −2.81068 −0.109406
\(661\) 23.0235 0.895511 0.447756 0.894156i \(-0.352223\pi\)
0.447756 + 0.894156i \(0.352223\pi\)
\(662\) 1.52898 0.0594255
\(663\) −16.1205 −0.626068
\(664\) 11.7687 0.456714
\(665\) 6.53131 0.253273
\(666\) 4.01996 0.155770
\(667\) 67.4526 2.61178
\(668\) −34.5051 −1.33504
\(669\) −13.9440 −0.539105
\(670\) 5.33664 0.206172
\(671\) 11.4999 0.443948
\(672\) 7.40261 0.285562
\(673\) −33.7956 −1.30272 −0.651362 0.758767i \(-0.725802\pi\)
−0.651362 + 0.758767i \(0.725802\pi\)
\(674\) −9.04927 −0.348565
\(675\) −13.7482 −0.529170
\(676\) 8.83879 0.339954
\(677\) −10.0119 −0.384788 −0.192394 0.981318i \(-0.561625\pi\)
−0.192394 + 0.981318i \(0.561625\pi\)
\(678\) 4.82179 0.185180
\(679\) −4.80684 −0.184470
\(680\) −10.1756 −0.390218
\(681\) 19.9109 0.762989
\(682\) 0.341524 0.0130776
\(683\) 28.5554 1.09264 0.546321 0.837576i \(-0.316028\pi\)
0.546321 + 0.837576i \(0.316028\pi\)
\(684\) −7.67913 −0.293619
\(685\) −8.18710 −0.312813
\(686\) 6.46307 0.246761
\(687\) 29.0161 1.10704
\(688\) −16.4549 −0.627338
\(689\) −31.1725 −1.18758
\(690\) −4.75241 −0.180921
\(691\) −28.4025 −1.08048 −0.540240 0.841511i \(-0.681667\pi\)
−0.540240 + 0.841511i \(0.681667\pi\)
\(692\) 13.1678 0.500564
\(693\) −1.90293 −0.0722861
\(694\) −0.259527 −0.00985152
\(695\) −26.5616 −1.00754
\(696\) −16.1835 −0.613435
\(697\) −31.7514 −1.20267
\(698\) −0.358260 −0.0135603
\(699\) 0.219319 0.00829541
\(700\) 7.32093 0.276705
\(701\) 10.9741 0.414488 0.207244 0.978289i \(-0.433551\pi\)
0.207244 + 0.978289i \(0.433551\pi\)
\(702\) −5.64986 −0.213240
\(703\) −19.2510 −0.726065
\(704\) −4.01017 −0.151139
\(705\) 7.04428 0.265303
\(706\) 7.69758 0.289702
\(707\) 2.90861 0.109390
\(708\) 7.30350 0.274482
\(709\) 50.6627 1.90268 0.951338 0.308150i \(-0.0997097\pi\)
0.951338 + 0.308150i \(0.0997097\pi\)
\(710\) 7.72766 0.290014
\(711\) 0.511906 0.0191980
\(712\) −1.71056 −0.0641058
\(713\) −8.42076 −0.315360
\(714\) 3.13047 0.117155
\(715\) −3.58359 −0.134019
\(716\) 35.3154 1.31980
\(717\) −21.9501 −0.819741
\(718\) 4.46505 0.166634
\(719\) −25.5125 −0.951456 −0.475728 0.879592i \(-0.657815\pi\)
−0.475728 + 0.879592i \(0.657815\pi\)
\(720\) 7.92475 0.295338
\(721\) −22.2869 −0.830008
\(722\) 4.28511 0.159475
\(723\) 13.5726 0.504770
\(724\) 3.94500 0.146615
\(725\) −24.2727 −0.901466
\(726\) −4.48234 −0.166355
\(727\) −9.11030 −0.337882 −0.168941 0.985626i \(-0.554035\pi\)
−0.168941 + 0.985626i \(0.554035\pi\)
\(728\) 6.22341 0.230655
\(729\) 22.8711 0.847077
\(730\) −3.91981 −0.145079
\(731\) −23.5563 −0.871260
\(732\) 32.8872 1.21554
\(733\) 4.30094 0.158859 0.0794294 0.996840i \(-0.474690\pi\)
0.0794294 + 0.996840i \(0.474690\pi\)
\(734\) 9.93491 0.366704
\(735\) −8.69364 −0.320670
\(736\) −27.4424 −1.01154
\(737\) 7.44596 0.274276
\(738\) −3.78513 −0.139333
\(739\) 9.26480 0.340811 0.170406 0.985374i \(-0.445492\pi\)
0.170406 + 0.985374i \(0.445492\pi\)
\(740\) 21.4377 0.788065
\(741\) 9.20297 0.338080
\(742\) 6.05345 0.222229
\(743\) −44.1214 −1.61866 −0.809328 0.587357i \(-0.800169\pi\)
−0.809328 + 0.587357i \(0.800169\pi\)
\(744\) 2.02035 0.0740695
\(745\) −10.3411 −0.378869
\(746\) −6.81912 −0.249666
\(747\) −13.1209 −0.480067
\(748\) −6.86347 −0.250953
\(749\) 12.3226 0.450259
\(750\) 5.11929 0.186930
\(751\) −23.6421 −0.862711 −0.431355 0.902182i \(-0.641964\pi\)
−0.431355 + 0.902182i \(0.641964\pi\)
\(752\) 12.0159 0.438176
\(753\) −11.0963 −0.404372
\(754\) −9.97491 −0.363265
\(755\) −33.0202 −1.20173
\(756\) −15.9991 −0.581883
\(757\) −51.5916 −1.87513 −0.937565 0.347810i \(-0.886925\pi\)
−0.937565 + 0.347810i \(0.886925\pi\)
\(758\) 6.47981 0.235357
\(759\) −6.63082 −0.240684
\(760\) 5.80913 0.210719
\(761\) −31.7407 −1.15060 −0.575300 0.817942i \(-0.695115\pi\)
−0.575300 + 0.817942i \(0.695115\pi\)
\(762\) −4.96658 −0.179920
\(763\) −15.1428 −0.548207
\(764\) 11.7068 0.423539
\(765\) 11.3448 0.410172
\(766\) 8.63460 0.311981
\(767\) 9.31190 0.336233
\(768\) −8.06706 −0.291095
\(769\) 48.7532 1.75808 0.879042 0.476744i \(-0.158183\pi\)
0.879042 + 0.476744i \(0.158183\pi\)
\(770\) 0.695905 0.0250787
\(771\) 36.0066 1.29675
\(772\) 22.6588 0.815507
\(773\) −15.4314 −0.555028 −0.277514 0.960722i \(-0.589510\pi\)
−0.277514 + 0.960722i \(0.589510\pi\)
\(774\) −2.80818 −0.100938
\(775\) 3.03020 0.108848
\(776\) −4.27534 −0.153476
\(777\) −13.6426 −0.489425
\(778\) −3.19650 −0.114600
\(779\) 18.1265 0.649448
\(780\) −10.2483 −0.366949
\(781\) 10.7820 0.385812
\(782\) −11.6050 −0.414995
\(783\) 53.0455 1.89569
\(784\) −14.8294 −0.529621
\(785\) 3.58342 0.127898
\(786\) 6.48766 0.231407
\(787\) 29.7489 1.06043 0.530216 0.847862i \(-0.322111\pi\)
0.530216 + 0.847862i \(0.322111\pi\)
\(788\) 23.0681 0.821768
\(789\) −8.55619 −0.304609
\(790\) −0.187205 −0.00666047
\(791\) 17.4090 0.618994
\(792\) −1.69252 −0.0601409
\(793\) 41.9309 1.48901
\(794\) 11.5269 0.409074
\(795\) −20.6205 −0.731333
\(796\) −0.185218 −0.00656489
\(797\) 5.55839 0.196888 0.0984442 0.995143i \(-0.468613\pi\)
0.0984442 + 0.995143i \(0.468613\pi\)
\(798\) −1.78714 −0.0632641
\(799\) 17.2016 0.608548
\(800\) 9.87510 0.349137
\(801\) 1.90709 0.0673838
\(802\) −3.98506 −0.140717
\(803\) −5.46912 −0.193001
\(804\) 21.2939 0.750977
\(805\) −17.1585 −0.604759
\(806\) 1.24526 0.0438626
\(807\) −1.00546 −0.0353938
\(808\) 2.58700 0.0910104
\(809\) 8.22890 0.289313 0.144656 0.989482i \(-0.453792\pi\)
0.144656 + 0.989482i \(0.453792\pi\)
\(810\) −1.11370 −0.0391314
\(811\) −4.19373 −0.147262 −0.0736310 0.997286i \(-0.523459\pi\)
−0.0736310 + 0.997286i \(0.523459\pi\)
\(812\) −28.2467 −0.991265
\(813\) 26.4885 0.928991
\(814\) −2.05117 −0.0718936
\(815\) −15.1120 −0.529350
\(816\) −18.1897 −0.636768
\(817\) 13.4479 0.470484
\(818\) −9.33384 −0.326350
\(819\) −6.93845 −0.242449
\(820\) −20.1854 −0.704905
\(821\) −12.6670 −0.442081 −0.221041 0.975265i \(-0.570945\pi\)
−0.221041 + 0.975265i \(0.570945\pi\)
\(822\) 2.24021 0.0781363
\(823\) −24.6674 −0.859853 −0.429926 0.902864i \(-0.641460\pi\)
−0.429926 + 0.902864i \(0.641460\pi\)
\(824\) −19.8226 −0.690554
\(825\) 2.38609 0.0830730
\(826\) −1.80829 −0.0629186
\(827\) −39.0120 −1.35658 −0.678290 0.734795i \(-0.737278\pi\)
−0.678290 + 0.734795i \(0.737278\pi\)
\(828\) 20.1740 0.701095
\(829\) −54.1581 −1.88099 −0.940494 0.339811i \(-0.889637\pi\)
−0.940494 + 0.339811i \(0.889637\pi\)
\(830\) 4.79833 0.166553
\(831\) −12.0081 −0.416557
\(832\) −14.6219 −0.506923
\(833\) −21.2292 −0.735548
\(834\) 7.26796 0.251669
\(835\) −29.1016 −1.00710
\(836\) 3.91826 0.135516
\(837\) −6.62219 −0.228896
\(838\) −1.51444 −0.0523154
\(839\) −34.3418 −1.18561 −0.592804 0.805346i \(-0.701979\pi\)
−0.592804 + 0.805346i \(0.701979\pi\)
\(840\) 4.11675 0.142041
\(841\) 64.6526 2.22940
\(842\) −12.9208 −0.445279
\(843\) −16.0223 −0.551838
\(844\) −42.9797 −1.47942
\(845\) 7.45464 0.256447
\(846\) 2.05062 0.0705019
\(847\) −16.1835 −0.556070
\(848\) −35.1738 −1.20787
\(849\) 19.6318 0.673760
\(850\) 4.17605 0.143237
\(851\) 50.5747 1.73368
\(852\) 30.8343 1.05637
\(853\) 16.9647 0.580862 0.290431 0.956896i \(-0.406201\pi\)
0.290431 + 0.956896i \(0.406201\pi\)
\(854\) −8.14263 −0.278635
\(855\) −6.47658 −0.221494
\(856\) 10.9601 0.374608
\(857\) −35.7135 −1.21995 −0.609974 0.792421i \(-0.708820\pi\)
−0.609974 + 0.792421i \(0.708820\pi\)
\(858\) 0.980567 0.0334760
\(859\) −41.2222 −1.40648 −0.703241 0.710951i \(-0.748265\pi\)
−0.703241 + 0.710951i \(0.748265\pi\)
\(860\) −14.9755 −0.510659
\(861\) 12.8457 0.437779
\(862\) −4.05826 −0.138225
\(863\) 25.7153 0.875359 0.437680 0.899131i \(-0.355800\pi\)
0.437680 + 0.899131i \(0.355800\pi\)
\(864\) −21.5810 −0.734201
\(865\) 11.1057 0.377606
\(866\) 4.59507 0.156147
\(867\) −5.54382 −0.188278
\(868\) 3.52631 0.119691
\(869\) −0.261199 −0.00886057
\(870\) −6.59835 −0.223705
\(871\) 27.1495 0.919926
\(872\) −13.4685 −0.456100
\(873\) 4.76656 0.161324
\(874\) 6.62515 0.224099
\(875\) 18.4832 0.624845
\(876\) −15.6405 −0.528444
\(877\) −33.6362 −1.13581 −0.567907 0.823093i \(-0.692247\pi\)
−0.567907 + 0.823093i \(0.692247\pi\)
\(878\) −4.12225 −0.139119
\(879\) 27.9652 0.943244
\(880\) −4.04359 −0.136309
\(881\) 17.0869 0.575671 0.287835 0.957680i \(-0.407064\pi\)
0.287835 + 0.957680i \(0.407064\pi\)
\(882\) −2.53076 −0.0852152
\(883\) −27.7133 −0.932628 −0.466314 0.884619i \(-0.654418\pi\)
−0.466314 + 0.884619i \(0.654418\pi\)
\(884\) −25.0256 −0.841702
\(885\) 6.15977 0.207059
\(886\) −4.07757 −0.136988
\(887\) 50.1913 1.68526 0.842630 0.538493i \(-0.181006\pi\)
0.842630 + 0.538493i \(0.181006\pi\)
\(888\) −12.1341 −0.407194
\(889\) −17.9318 −0.601413
\(890\) −0.697428 −0.0233778
\(891\) −1.55389 −0.0520574
\(892\) −21.6467 −0.724785
\(893\) −9.82014 −0.328618
\(894\) 2.82961 0.0946362
\(895\) 29.7850 0.995604
\(896\) 15.1194 0.505103
\(897\) −24.1773 −0.807257
\(898\) 6.69488 0.223411
\(899\) −11.6916 −0.389936
\(900\) −7.25959 −0.241986
\(901\) −50.3536 −1.67752
\(902\) 1.93136 0.0643071
\(903\) 9.53014 0.317143
\(904\) 15.4841 0.514993
\(905\) 3.32721 0.110600
\(906\) 9.03522 0.300175
\(907\) 42.8683 1.42342 0.711709 0.702475i \(-0.247922\pi\)
0.711709 + 0.702475i \(0.247922\pi\)
\(908\) 30.9099 1.02578
\(909\) −2.88424 −0.0956642
\(910\) 2.53741 0.0841143
\(911\) −46.7726 −1.54964 −0.774822 0.632179i \(-0.782161\pi\)
−0.774822 + 0.632179i \(0.782161\pi\)
\(912\) 10.3843 0.343857
\(913\) 6.69489 0.221569
\(914\) 4.01335 0.132750
\(915\) 27.7371 0.916959
\(916\) 45.0449 1.48833
\(917\) 23.4236 0.773517
\(918\) −9.12633 −0.301214
\(919\) 10.6493 0.351288 0.175644 0.984454i \(-0.443799\pi\)
0.175644 + 0.984454i \(0.443799\pi\)
\(920\) −15.2613 −0.503150
\(921\) 14.9102 0.491309
\(922\) 3.23940 0.106684
\(923\) 39.3135 1.29402
\(924\) 2.77675 0.0913483
\(925\) −18.1992 −0.598387
\(926\) 0.287991 0.00946397
\(927\) 22.1002 0.725865
\(928\) −38.1016 −1.25075
\(929\) 59.9077 1.96551 0.982754 0.184918i \(-0.0592019\pi\)
0.982754 + 0.184918i \(0.0592019\pi\)
\(930\) 0.823736 0.0270114
\(931\) 12.1195 0.397199
\(932\) 0.340473 0.0111526
\(933\) 23.4127 0.766497
\(934\) −3.14494 −0.102906
\(935\) −5.78865 −0.189309
\(936\) −6.17126 −0.201714
\(937\) −48.4932 −1.58420 −0.792101 0.610390i \(-0.791013\pi\)
−0.792101 + 0.610390i \(0.791013\pi\)
\(938\) −5.27221 −0.172144
\(939\) −13.5872 −0.443401
\(940\) 10.9356 0.356680
\(941\) −31.7973 −1.03656 −0.518282 0.855210i \(-0.673428\pi\)
−0.518282 + 0.855210i \(0.673428\pi\)
\(942\) −0.980521 −0.0319471
\(943\) −47.6204 −1.55073
\(944\) 10.5072 0.341979
\(945\) −13.4937 −0.438949
\(946\) 1.43287 0.0465865
\(947\) −31.0555 −1.00917 −0.504583 0.863363i \(-0.668354\pi\)
−0.504583 + 0.863363i \(0.668354\pi\)
\(948\) −0.746973 −0.0242606
\(949\) −19.9415 −0.647329
\(950\) −2.38405 −0.0773488
\(951\) −11.2199 −0.363829
\(952\) 10.0528 0.325813
\(953\) −8.61722 −0.279139 −0.139570 0.990212i \(-0.544572\pi\)
−0.139570 + 0.990212i \(0.544572\pi\)
\(954\) −6.00272 −0.194345
\(955\) 9.87355 0.319501
\(956\) −34.0755 −1.10208
\(957\) −9.20637 −0.297600
\(958\) −11.8120 −0.381627
\(959\) 8.08825 0.261183
\(960\) −9.67232 −0.312173
\(961\) −29.5404 −0.952917
\(962\) −7.47900 −0.241133
\(963\) −12.2194 −0.393764
\(964\) 21.0702 0.678625
\(965\) 19.1104 0.615186
\(966\) 4.69504 0.151060
\(967\) 44.5830 1.43369 0.716846 0.697232i \(-0.245585\pi\)
0.716846 + 0.697232i \(0.245585\pi\)
\(968\) −14.3940 −0.462641
\(969\) 14.8657 0.477556
\(970\) −1.74314 −0.0559690
\(971\) −10.0866 −0.323695 −0.161848 0.986816i \(-0.551745\pi\)
−0.161848 + 0.986816i \(0.551745\pi\)
\(972\) 26.3338 0.844657
\(973\) 26.2409 0.841244
\(974\) 3.80676 0.121976
\(975\) 8.70017 0.278629
\(976\) 47.3131 1.51446
\(977\) −42.6221 −1.36360 −0.681801 0.731538i \(-0.738803\pi\)
−0.681801 + 0.731538i \(0.738803\pi\)
\(978\) 4.13505 0.132224
\(979\) −0.973089 −0.0311001
\(980\) −13.4961 −0.431116
\(981\) 15.0159 0.479422
\(982\) 7.85668 0.250717
\(983\) 8.80618 0.280874 0.140437 0.990090i \(-0.455149\pi\)
0.140437 + 0.990090i \(0.455149\pi\)
\(984\) 11.4253 0.364225
\(985\) 19.4557 0.619909
\(986\) −16.1127 −0.513132
\(987\) −6.95923 −0.221515
\(988\) 14.2868 0.454522
\(989\) −35.3294 −1.12341
\(990\) −0.690073 −0.0219320
\(991\) 60.1419 1.91047 0.955236 0.295846i \(-0.0956017\pi\)
0.955236 + 0.295846i \(0.0956017\pi\)
\(992\) 4.75659 0.151022
\(993\) 5.14543 0.163285
\(994\) −7.63436 −0.242147
\(995\) −0.156213 −0.00495229
\(996\) 19.1460 0.606663
\(997\) 18.0567 0.571861 0.285930 0.958250i \(-0.407697\pi\)
0.285930 + 0.958250i \(0.407697\pi\)
\(998\) −15.6005 −0.493824
\(999\) 39.7725 1.25835
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 349.2.a.b.1.6 17
3.2 odd 2 3141.2.a.e.1.12 17
4.3 odd 2 5584.2.a.m.1.12 17
5.4 even 2 8725.2.a.m.1.12 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
349.2.a.b.1.6 17 1.1 even 1 trivial
3141.2.a.e.1.12 17 3.2 odd 2
5584.2.a.m.1.12 17 4.3 odd 2
8725.2.a.m.1.12 17 5.4 even 2