Properties

Label 2013.2.a.e.1.6
Level $2013$
Weight $2$
Character 2013.1
Self dual yes
Analytic conductor $16.074$
Analytic rank $0$
Dimension $13$
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] = [2013,2,Mod(1,2013)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2013, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2013.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2013 = 3 \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2013.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: \(16.0738859269\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 2 x^{12} - 19 x^{11} + 35 x^{10} + 136 x^{9} - 220 x^{8} - 469 x^{7} + 610 x^{6} + 841 x^{5} - 760 x^{4} - 742 x^{3} + 366 x^{2} + 236 x - 47 \) 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 \(-0.805107\) of defining polynomial
Character \(\chi\) \(=\) 2013.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.805107 q^{2} -1.00000 q^{3} -1.35180 q^{4} +1.06503 q^{5} +0.805107 q^{6} -0.203035 q^{7} +2.69856 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.805107 q^{2} -1.00000 q^{3} -1.35180 q^{4} +1.06503 q^{5} +0.805107 q^{6} -0.203035 q^{7} +2.69856 q^{8} +1.00000 q^{9} -0.857467 q^{10} +1.00000 q^{11} +1.35180 q^{12} +0.801491 q^{13} +0.163465 q^{14} -1.06503 q^{15} +0.530979 q^{16} -0.376525 q^{17} -0.805107 q^{18} +2.62261 q^{19} -1.43972 q^{20} +0.203035 q^{21} -0.805107 q^{22} +2.93099 q^{23} -2.69856 q^{24} -3.86570 q^{25} -0.645286 q^{26} -1.00000 q^{27} +0.274464 q^{28} +0.907071 q^{29} +0.857467 q^{30} +2.07822 q^{31} -5.82461 q^{32} -1.00000 q^{33} +0.303143 q^{34} -0.216240 q^{35} -1.35180 q^{36} -2.14816 q^{37} -2.11148 q^{38} -0.801491 q^{39} +2.87406 q^{40} -1.83276 q^{41} -0.163465 q^{42} -2.25276 q^{43} -1.35180 q^{44} +1.06503 q^{45} -2.35976 q^{46} -0.685433 q^{47} -0.530979 q^{48} -6.95878 q^{49} +3.11230 q^{50} +0.376525 q^{51} -1.08346 q^{52} +14.0454 q^{53} +0.805107 q^{54} +1.06503 q^{55} -0.547903 q^{56} -2.62261 q^{57} -0.730289 q^{58} +7.73715 q^{59} +1.43972 q^{60} -1.00000 q^{61} -1.67319 q^{62} -0.203035 q^{63} +3.62748 q^{64} +0.853616 q^{65} +0.805107 q^{66} -7.59429 q^{67} +0.508987 q^{68} -2.93099 q^{69} +0.174096 q^{70} -9.23954 q^{71} +2.69856 q^{72} +7.49398 q^{73} +1.72950 q^{74} +3.86570 q^{75} -3.54525 q^{76} -0.203035 q^{77} +0.645286 q^{78} +7.41703 q^{79} +0.565511 q^{80} +1.00000 q^{81} +1.47557 q^{82} -8.55136 q^{83} -0.274464 q^{84} -0.401012 q^{85} +1.81371 q^{86} -0.907071 q^{87} +2.69856 q^{88} -2.97849 q^{89} -0.857467 q^{90} -0.162731 q^{91} -3.96212 q^{92} -2.07822 q^{93} +0.551846 q^{94} +2.79317 q^{95} +5.82461 q^{96} +17.8193 q^{97} +5.60256 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 2 q^{2} - 13 q^{3} + 16 q^{4} + 3 q^{5} - 2 q^{6} + 11 q^{7} + 9 q^{8} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q + 2 q^{2} - 13 q^{3} + 16 q^{4} + 3 q^{5} - 2 q^{6} + 11 q^{7} + 9 q^{8} + 13 q^{9} + 6 q^{10} + 13 q^{11} - 16 q^{12} + 13 q^{13} + q^{14} - 3 q^{15} + 18 q^{16} + 17 q^{17} + 2 q^{18} + 14 q^{19} - 7 q^{20} - 11 q^{21} + 2 q^{22} + 7 q^{23} - 9 q^{24} + 18 q^{25} - 10 q^{26} - 13 q^{27} + 19 q^{28} - 6 q^{29} - 6 q^{30} + 27 q^{31} + 5 q^{32} - 13 q^{33} + 6 q^{34} + 14 q^{35} + 16 q^{36} + 10 q^{37} + 2 q^{38} - 13 q^{39} + 8 q^{40} + 3 q^{41} - q^{42} + 29 q^{43} + 16 q^{44} + 3 q^{45} - 24 q^{46} + 8 q^{47} - 18 q^{48} + 8 q^{49} - 27 q^{50} - 17 q^{51} + 37 q^{52} - 24 q^{53} - 2 q^{54} + 3 q^{55} + 24 q^{56} - 14 q^{57} - 5 q^{58} + 13 q^{59} + 7 q^{60} - 13 q^{61} + 39 q^{62} + 11 q^{63} + 47 q^{64} - 11 q^{65} - 2 q^{66} + 44 q^{67} - 8 q^{68} - 7 q^{69} - 12 q^{70} + 3 q^{71} + 9 q^{72} + 48 q^{73} - 22 q^{74} - 18 q^{75} + 47 q^{76} + 11 q^{77} + 10 q^{78} - 17 q^{79} - 26 q^{80} + 13 q^{81} + 56 q^{82} + 50 q^{83} - 19 q^{84} + 8 q^{85} + 18 q^{86} + 6 q^{87} + 9 q^{88} - 15 q^{89} + 6 q^{90} + 47 q^{91} + 14 q^{92} - 27 q^{93} + 45 q^{94} - q^{95} - 5 q^{96} + 27 q^{97} + 47 q^{98} + 13 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.805107 −0.569296 −0.284648 0.958632i \(-0.591877\pi\)
−0.284648 + 0.958632i \(0.591877\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.35180 −0.675902
\(5\) 1.06503 0.476298 0.238149 0.971229i \(-0.423459\pi\)
0.238149 + 0.971229i \(0.423459\pi\)
\(6\) 0.805107 0.328683
\(7\) −0.203035 −0.0767401 −0.0383701 0.999264i \(-0.512217\pi\)
−0.0383701 + 0.999264i \(0.512217\pi\)
\(8\) 2.69856 0.954085
\(9\) 1.00000 0.333333
\(10\) −0.857467 −0.271155
\(11\) 1.00000 0.301511
\(12\) 1.35180 0.390232
\(13\) 0.801491 0.222294 0.111147 0.993804i \(-0.464548\pi\)
0.111147 + 0.993804i \(0.464548\pi\)
\(14\) 0.163465 0.0436879
\(15\) −1.06503 −0.274991
\(16\) 0.530979 0.132745
\(17\) −0.376525 −0.0913207 −0.0456603 0.998957i \(-0.514539\pi\)
−0.0456603 + 0.998957i \(0.514539\pi\)
\(18\) −0.805107 −0.189765
\(19\) 2.62261 0.601667 0.300833 0.953677i \(-0.402735\pi\)
0.300833 + 0.953677i \(0.402735\pi\)
\(20\) −1.43972 −0.321931
\(21\) 0.203035 0.0443059
\(22\) −0.805107 −0.171649
\(23\) 2.93099 0.611154 0.305577 0.952167i \(-0.401151\pi\)
0.305577 + 0.952167i \(0.401151\pi\)
\(24\) −2.69856 −0.550841
\(25\) −3.86570 −0.773140
\(26\) −0.645286 −0.126551
\(27\) −1.00000 −0.192450
\(28\) 0.274464 0.0518688
\(29\) 0.907071 0.168439 0.0842195 0.996447i \(-0.473160\pi\)
0.0842195 + 0.996447i \(0.473160\pi\)
\(30\) 0.857467 0.156551
\(31\) 2.07822 0.373260 0.186630 0.982430i \(-0.440243\pi\)
0.186630 + 0.982430i \(0.440243\pi\)
\(32\) −5.82461 −1.02966
\(33\) −1.00000 −0.174078
\(34\) 0.303143 0.0519885
\(35\) −0.216240 −0.0365512
\(36\) −1.35180 −0.225301
\(37\) −2.14816 −0.353155 −0.176578 0.984287i \(-0.556503\pi\)
−0.176578 + 0.984287i \(0.556503\pi\)
\(38\) −2.11148 −0.342527
\(39\) −0.801491 −0.128341
\(40\) 2.87406 0.454429
\(41\) −1.83276 −0.286229 −0.143115 0.989706i \(-0.545712\pi\)
−0.143115 + 0.989706i \(0.545712\pi\)
\(42\) −0.163465 −0.0252232
\(43\) −2.25276 −0.343542 −0.171771 0.985137i \(-0.554949\pi\)
−0.171771 + 0.985137i \(0.554949\pi\)
\(44\) −1.35180 −0.203792
\(45\) 1.06503 0.158766
\(46\) −2.35976 −0.347928
\(47\) −0.685433 −0.0999806 −0.0499903 0.998750i \(-0.515919\pi\)
−0.0499903 + 0.998750i \(0.515919\pi\)
\(48\) −0.530979 −0.0766402
\(49\) −6.95878 −0.994111
\(50\) 3.11230 0.440146
\(51\) 0.376525 0.0527240
\(52\) −1.08346 −0.150249
\(53\) 14.0454 1.92928 0.964640 0.263570i \(-0.0849002\pi\)
0.964640 + 0.263570i \(0.0849002\pi\)
\(54\) 0.805107 0.109561
\(55\) 1.06503 0.143609
\(56\) −0.547903 −0.0732166
\(57\) −2.62261 −0.347373
\(58\) −0.730289 −0.0958917
\(59\) 7.73715 1.00729 0.503646 0.863910i \(-0.331992\pi\)
0.503646 + 0.863910i \(0.331992\pi\)
\(60\) 1.43972 0.185867
\(61\) −1.00000 −0.128037
\(62\) −1.67319 −0.212495
\(63\) −0.203035 −0.0255800
\(64\) 3.62748 0.453435
\(65\) 0.853616 0.105878
\(66\) 0.805107 0.0991018
\(67\) −7.59429 −0.927790 −0.463895 0.885890i \(-0.653549\pi\)
−0.463895 + 0.885890i \(0.653549\pi\)
\(68\) 0.508987 0.0617238
\(69\) −2.93099 −0.352850
\(70\) 0.174096 0.0208085
\(71\) −9.23954 −1.09653 −0.548266 0.836304i \(-0.684712\pi\)
−0.548266 + 0.836304i \(0.684712\pi\)
\(72\) 2.69856 0.318028
\(73\) 7.49398 0.877104 0.438552 0.898706i \(-0.355491\pi\)
0.438552 + 0.898706i \(0.355491\pi\)
\(74\) 1.72950 0.201050
\(75\) 3.86570 0.446373
\(76\) −3.54525 −0.406668
\(77\) −0.203035 −0.0231380
\(78\) 0.645286 0.0730642
\(79\) 7.41703 0.834481 0.417240 0.908796i \(-0.362997\pi\)
0.417240 + 0.908796i \(0.362997\pi\)
\(80\) 0.565511 0.0632260
\(81\) 1.00000 0.111111
\(82\) 1.47557 0.162949
\(83\) −8.55136 −0.938634 −0.469317 0.883030i \(-0.655500\pi\)
−0.469317 + 0.883030i \(0.655500\pi\)
\(84\) −0.274464 −0.0299465
\(85\) −0.401012 −0.0434959
\(86\) 1.81371 0.195577
\(87\) −0.907071 −0.0972483
\(88\) 2.69856 0.287667
\(89\) −2.97849 −0.315719 −0.157860 0.987462i \(-0.550459\pi\)
−0.157860 + 0.987462i \(0.550459\pi\)
\(90\) −0.857467 −0.0903849
\(91\) −0.162731 −0.0170588
\(92\) −3.96212 −0.413080
\(93\) −2.07822 −0.215502
\(94\) 0.551846 0.0569186
\(95\) 2.79317 0.286573
\(96\) 5.82461 0.594472
\(97\) 17.8193 1.80928 0.904638 0.426181i \(-0.140141\pi\)
0.904638 + 0.426181i \(0.140141\pi\)
\(98\) 5.60256 0.565944
\(99\) 1.00000 0.100504
\(100\) 5.22567 0.522567
\(101\) −4.50884 −0.448646 −0.224323 0.974515i \(-0.572017\pi\)
−0.224323 + 0.974515i \(0.572017\pi\)
\(102\) −0.303143 −0.0300156
\(103\) 19.8815 1.95898 0.979489 0.201499i \(-0.0645812\pi\)
0.979489 + 0.201499i \(0.0645812\pi\)
\(104\) 2.16287 0.212087
\(105\) 0.216240 0.0211028
\(106\) −11.3080 −1.09833
\(107\) 6.58861 0.636945 0.318472 0.947932i \(-0.396830\pi\)
0.318472 + 0.947932i \(0.396830\pi\)
\(108\) 1.35180 0.130077
\(109\) 19.1473 1.83398 0.916990 0.398910i \(-0.130612\pi\)
0.916990 + 0.398910i \(0.130612\pi\)
\(110\) −0.857467 −0.0817562
\(111\) 2.14816 0.203894
\(112\) −0.107807 −0.0101868
\(113\) −2.84072 −0.267232 −0.133616 0.991033i \(-0.542659\pi\)
−0.133616 + 0.991033i \(0.542659\pi\)
\(114\) 2.11148 0.197758
\(115\) 3.12161 0.291091
\(116\) −1.22618 −0.113848
\(117\) 0.801491 0.0740979
\(118\) −6.22923 −0.573447
\(119\) 0.0764478 0.00700796
\(120\) −2.87406 −0.262365
\(121\) 1.00000 0.0909091
\(122\) 0.805107 0.0728909
\(123\) 1.83276 0.165255
\(124\) −2.80935 −0.252287
\(125\) −9.44228 −0.844543
\(126\) 0.163465 0.0145626
\(127\) 5.00348 0.443987 0.221993 0.975048i \(-0.428744\pi\)
0.221993 + 0.975048i \(0.428744\pi\)
\(128\) 8.72872 0.771517
\(129\) 2.25276 0.198344
\(130\) −0.687252 −0.0602760
\(131\) −19.0550 −1.66485 −0.832423 0.554140i \(-0.813047\pi\)
−0.832423 + 0.554140i \(0.813047\pi\)
\(132\) 1.35180 0.117659
\(133\) −0.532481 −0.0461720
\(134\) 6.11422 0.528188
\(135\) −1.06503 −0.0916636
\(136\) −1.01607 −0.0871277
\(137\) 18.4521 1.57647 0.788237 0.615372i \(-0.210994\pi\)
0.788237 + 0.615372i \(0.210994\pi\)
\(138\) 2.35976 0.200876
\(139\) 14.6161 1.23972 0.619861 0.784711i \(-0.287189\pi\)
0.619861 + 0.784711i \(0.287189\pi\)
\(140\) 0.292313 0.0247050
\(141\) 0.685433 0.0577238
\(142\) 7.43882 0.624252
\(143\) 0.801491 0.0670241
\(144\) 0.530979 0.0442482
\(145\) 0.966063 0.0802271
\(146\) −6.03345 −0.499332
\(147\) 6.95878 0.573950
\(148\) 2.90389 0.238698
\(149\) −4.26078 −0.349057 −0.174528 0.984652i \(-0.555840\pi\)
−0.174528 + 0.984652i \(0.555840\pi\)
\(150\) −3.11230 −0.254118
\(151\) 19.9980 1.62741 0.813707 0.581276i \(-0.197446\pi\)
0.813707 + 0.581276i \(0.197446\pi\)
\(152\) 7.07726 0.574041
\(153\) −0.376525 −0.0304402
\(154\) 0.163465 0.0131724
\(155\) 2.21338 0.177783
\(156\) 1.08346 0.0867461
\(157\) 0.492192 0.0392812 0.0196406 0.999807i \(-0.493748\pi\)
0.0196406 + 0.999807i \(0.493748\pi\)
\(158\) −5.97150 −0.475067
\(159\) −14.0454 −1.11387
\(160\) −6.20342 −0.490423
\(161\) −0.595095 −0.0469000
\(162\) −0.805107 −0.0632552
\(163\) −1.93731 −0.151742 −0.0758708 0.997118i \(-0.524174\pi\)
−0.0758708 + 0.997118i \(0.524174\pi\)
\(164\) 2.47753 0.193463
\(165\) −1.06503 −0.0829129
\(166\) 6.88476 0.534361
\(167\) 14.8338 1.14788 0.573938 0.818899i \(-0.305415\pi\)
0.573938 + 0.818899i \(0.305415\pi\)
\(168\) 0.547903 0.0422716
\(169\) −12.3576 −0.950586
\(170\) 0.322857 0.0247620
\(171\) 2.62261 0.200556
\(172\) 3.04529 0.232201
\(173\) −10.3859 −0.789623 −0.394812 0.918762i \(-0.629190\pi\)
−0.394812 + 0.918762i \(0.629190\pi\)
\(174\) 0.730289 0.0553631
\(175\) 0.784874 0.0593309
\(176\) 0.530979 0.0400240
\(177\) −7.73715 −0.581560
\(178\) 2.39800 0.179738
\(179\) 16.9049 1.26353 0.631765 0.775160i \(-0.282331\pi\)
0.631765 + 0.775160i \(0.282331\pi\)
\(180\) −1.43972 −0.107310
\(181\) 2.91712 0.216828 0.108414 0.994106i \(-0.465423\pi\)
0.108414 + 0.994106i \(0.465423\pi\)
\(182\) 0.131016 0.00971154
\(183\) 1.00000 0.0739221
\(184\) 7.90945 0.583093
\(185\) −2.28787 −0.168207
\(186\) 1.67319 0.122684
\(187\) −0.376525 −0.0275342
\(188\) 0.926570 0.0675771
\(189\) 0.203035 0.0147686
\(190\) −2.24880 −0.163145
\(191\) 10.8546 0.785410 0.392705 0.919664i \(-0.371539\pi\)
0.392705 + 0.919664i \(0.371539\pi\)
\(192\) −3.62748 −0.261791
\(193\) 9.85294 0.709230 0.354615 0.935012i \(-0.384612\pi\)
0.354615 + 0.935012i \(0.384612\pi\)
\(194\) −14.3464 −1.03001
\(195\) −0.853616 −0.0611287
\(196\) 9.40690 0.671921
\(197\) 1.58403 0.112857 0.0564286 0.998407i \(-0.482029\pi\)
0.0564286 + 0.998407i \(0.482029\pi\)
\(198\) −0.805107 −0.0572164
\(199\) 25.5545 1.81151 0.905756 0.423799i \(-0.139304\pi\)
0.905756 + 0.423799i \(0.139304\pi\)
\(200\) −10.4318 −0.737641
\(201\) 7.59429 0.535660
\(202\) 3.63010 0.255413
\(203\) −0.184168 −0.0129260
\(204\) −0.508987 −0.0356363
\(205\) −1.95196 −0.136330
\(206\) −16.0067 −1.11524
\(207\) 2.93099 0.203718
\(208\) 0.425575 0.0295083
\(209\) 2.62261 0.181409
\(210\) −0.174096 −0.0120138
\(211\) 16.1904 1.11460 0.557298 0.830313i \(-0.311838\pi\)
0.557298 + 0.830313i \(0.311838\pi\)
\(212\) −18.9866 −1.30400
\(213\) 9.23954 0.633083
\(214\) −5.30453 −0.362610
\(215\) −2.39927 −0.163629
\(216\) −2.69856 −0.183614
\(217\) −0.421952 −0.0286440
\(218\) −15.4156 −1.04408
\(219\) −7.49398 −0.506396
\(220\) −1.43972 −0.0970657
\(221\) −0.301781 −0.0203000
\(222\) −1.72950 −0.116076
\(223\) −4.93169 −0.330250 −0.165125 0.986273i \(-0.552803\pi\)
−0.165125 + 0.986273i \(0.552803\pi\)
\(224\) 1.18260 0.0790159
\(225\) −3.86570 −0.257713
\(226\) 2.28708 0.152134
\(227\) −2.59452 −0.172204 −0.0861022 0.996286i \(-0.527441\pi\)
−0.0861022 + 0.996286i \(0.527441\pi\)
\(228\) 3.54525 0.234790
\(229\) −6.75414 −0.446326 −0.223163 0.974781i \(-0.571638\pi\)
−0.223163 + 0.974781i \(0.571638\pi\)
\(230\) −2.51323 −0.165717
\(231\) 0.203035 0.0133587
\(232\) 2.44779 0.160705
\(233\) −27.8481 −1.82439 −0.912195 0.409757i \(-0.865614\pi\)
−0.912195 + 0.409757i \(0.865614\pi\)
\(234\) −0.645286 −0.0421837
\(235\) −0.730010 −0.0476206
\(236\) −10.4591 −0.680830
\(237\) −7.41703 −0.481788
\(238\) −0.0615487 −0.00398961
\(239\) 16.9202 1.09448 0.547240 0.836976i \(-0.315679\pi\)
0.547240 + 0.836976i \(0.315679\pi\)
\(240\) −0.565511 −0.0365036
\(241\) −1.06455 −0.0685739 −0.0342869 0.999412i \(-0.510916\pi\)
−0.0342869 + 0.999412i \(0.510916\pi\)
\(242\) −0.805107 −0.0517542
\(243\) −1.00000 −0.0641500
\(244\) 1.35180 0.0865403
\(245\) −7.41134 −0.473493
\(246\) −1.47557 −0.0940788
\(247\) 2.10199 0.133747
\(248\) 5.60821 0.356121
\(249\) 8.55136 0.541920
\(250\) 7.60204 0.480795
\(251\) −3.19791 −0.201850 −0.100925 0.994894i \(-0.532180\pi\)
−0.100925 + 0.994894i \(0.532180\pi\)
\(252\) 0.274464 0.0172896
\(253\) 2.93099 0.184270
\(254\) −4.02833 −0.252760
\(255\) 0.401012 0.0251124
\(256\) −14.2825 −0.892657
\(257\) −12.0296 −0.750388 −0.375194 0.926946i \(-0.622424\pi\)
−0.375194 + 0.926946i \(0.622424\pi\)
\(258\) −1.81371 −0.112917
\(259\) 0.436152 0.0271012
\(260\) −1.15392 −0.0715631
\(261\) 0.907071 0.0561463
\(262\) 15.3413 0.947791
\(263\) −3.98169 −0.245521 −0.122761 0.992436i \(-0.539175\pi\)
−0.122761 + 0.992436i \(0.539175\pi\)
\(264\) −2.69856 −0.166085
\(265\) 14.9588 0.918912
\(266\) 0.428704 0.0262855
\(267\) 2.97849 0.182281
\(268\) 10.2660 0.627095
\(269\) 2.20633 0.134522 0.0672611 0.997735i \(-0.478574\pi\)
0.0672611 + 0.997735i \(0.478574\pi\)
\(270\) 0.857467 0.0521838
\(271\) 10.9268 0.663754 0.331877 0.943323i \(-0.392318\pi\)
0.331877 + 0.943323i \(0.392318\pi\)
\(272\) −0.199927 −0.0121223
\(273\) 0.162731 0.00984893
\(274\) −14.8559 −0.897480
\(275\) −3.86570 −0.233111
\(276\) 3.96212 0.238492
\(277\) 7.40923 0.445177 0.222589 0.974912i \(-0.428549\pi\)
0.222589 + 0.974912i \(0.428549\pi\)
\(278\) −11.7675 −0.705770
\(279\) 2.07822 0.124420
\(280\) −0.583535 −0.0348729
\(281\) −0.155459 −0.00927391 −0.00463696 0.999989i \(-0.501476\pi\)
−0.00463696 + 0.999989i \(0.501476\pi\)
\(282\) −0.551846 −0.0328620
\(283\) −11.9494 −0.710316 −0.355158 0.934806i \(-0.615573\pi\)
−0.355158 + 0.934806i \(0.615573\pi\)
\(284\) 12.4900 0.741148
\(285\) −2.79317 −0.165453
\(286\) −0.645286 −0.0381565
\(287\) 0.372115 0.0219653
\(288\) −5.82461 −0.343219
\(289\) −16.8582 −0.991661
\(290\) −0.777784 −0.0456730
\(291\) −17.8193 −1.04459
\(292\) −10.1304 −0.592836
\(293\) 16.9701 0.991404 0.495702 0.868493i \(-0.334911\pi\)
0.495702 + 0.868493i \(0.334911\pi\)
\(294\) −5.60256 −0.326748
\(295\) 8.24034 0.479771
\(296\) −5.79694 −0.336940
\(297\) −1.00000 −0.0580259
\(298\) 3.43038 0.198717
\(299\) 2.34916 0.135856
\(300\) −5.22567 −0.301704
\(301\) 0.457390 0.0263635
\(302\) −16.1005 −0.926480
\(303\) 4.50884 0.259026
\(304\) 1.39255 0.0798681
\(305\) −1.06503 −0.0609837
\(306\) 0.303143 0.0173295
\(307\) 13.3520 0.762041 0.381021 0.924567i \(-0.375573\pi\)
0.381021 + 0.924567i \(0.375573\pi\)
\(308\) 0.274464 0.0156390
\(309\) −19.8815 −1.13102
\(310\) −1.78201 −0.101211
\(311\) −12.6506 −0.717352 −0.358676 0.933462i \(-0.616772\pi\)
−0.358676 + 0.933462i \(0.616772\pi\)
\(312\) −2.16287 −0.122448
\(313\) 8.15405 0.460894 0.230447 0.973085i \(-0.425981\pi\)
0.230447 + 0.973085i \(0.425981\pi\)
\(314\) −0.396267 −0.0223627
\(315\) −0.216240 −0.0121837
\(316\) −10.0264 −0.564027
\(317\) −22.7512 −1.27784 −0.638918 0.769274i \(-0.720618\pi\)
−0.638918 + 0.769274i \(0.720618\pi\)
\(318\) 11.3080 0.634122
\(319\) 0.907071 0.0507863
\(320\) 3.86339 0.215970
\(321\) −6.58861 −0.367740
\(322\) 0.479115 0.0267000
\(323\) −0.987476 −0.0549446
\(324\) −1.35180 −0.0751002
\(325\) −3.09832 −0.171864
\(326\) 1.55974 0.0863860
\(327\) −19.1473 −1.05885
\(328\) −4.94582 −0.273087
\(329\) 0.139167 0.00767253
\(330\) 0.857467 0.0472020
\(331\) 2.47607 0.136097 0.0680486 0.997682i \(-0.478323\pi\)
0.0680486 + 0.997682i \(0.478323\pi\)
\(332\) 11.5598 0.634424
\(333\) −2.14816 −0.117718
\(334\) −11.9428 −0.653481
\(335\) −8.08819 −0.441905
\(336\) 0.107807 0.00588138
\(337\) −11.8667 −0.646419 −0.323210 0.946327i \(-0.604762\pi\)
−0.323210 + 0.946327i \(0.604762\pi\)
\(338\) 9.94920 0.541165
\(339\) 2.84072 0.154287
\(340\) 0.542089 0.0293989
\(341\) 2.07822 0.112542
\(342\) −2.11148 −0.114176
\(343\) 2.83412 0.153028
\(344\) −6.07920 −0.327769
\(345\) −3.12161 −0.168062
\(346\) 8.36174 0.449530
\(347\) 33.4479 1.79558 0.897789 0.440426i \(-0.145173\pi\)
0.897789 + 0.440426i \(0.145173\pi\)
\(348\) 1.22618 0.0657303
\(349\) −19.7785 −1.05872 −0.529359 0.848398i \(-0.677567\pi\)
−0.529359 + 0.848398i \(0.677567\pi\)
\(350\) −0.631907 −0.0337768
\(351\) −0.801491 −0.0427804
\(352\) −5.82461 −0.310453
\(353\) −15.2833 −0.813448 −0.406724 0.913551i \(-0.633329\pi\)
−0.406724 + 0.913551i \(0.633329\pi\)
\(354\) 6.22923 0.331080
\(355\) −9.84043 −0.522276
\(356\) 4.02633 0.213395
\(357\) −0.0764478 −0.00404605
\(358\) −13.6102 −0.719324
\(359\) 10.2459 0.540759 0.270380 0.962754i \(-0.412851\pi\)
0.270380 + 0.962754i \(0.412851\pi\)
\(360\) 2.87406 0.151476
\(361\) −12.1219 −0.637997
\(362\) −2.34859 −0.123439
\(363\) −1.00000 −0.0524864
\(364\) 0.219980 0.0115301
\(365\) 7.98135 0.417763
\(366\) −0.805107 −0.0420836
\(367\) 11.5481 0.602807 0.301403 0.953497i \(-0.402545\pi\)
0.301403 + 0.953497i \(0.402545\pi\)
\(368\) 1.55629 0.0811274
\(369\) −1.83276 −0.0954098
\(370\) 1.84198 0.0957598
\(371\) −2.85171 −0.148053
\(372\) 2.80935 0.145658
\(373\) −26.2319 −1.35824 −0.679119 0.734028i \(-0.737638\pi\)
−0.679119 + 0.734028i \(0.737638\pi\)
\(374\) 0.303143 0.0156751
\(375\) 9.44228 0.487597
\(376\) −1.84968 −0.0953900
\(377\) 0.727010 0.0374429
\(378\) −0.163465 −0.00840774
\(379\) 11.1404 0.572245 0.286123 0.958193i \(-0.407634\pi\)
0.286123 + 0.958193i \(0.407634\pi\)
\(380\) −3.77581 −0.193695
\(381\) −5.00348 −0.256336
\(382\) −8.73910 −0.447131
\(383\) 37.6696 1.92483 0.962413 0.271590i \(-0.0875494\pi\)
0.962413 + 0.271590i \(0.0875494\pi\)
\(384\) −8.72872 −0.445436
\(385\) −0.216240 −0.0110206
\(386\) −7.93266 −0.403762
\(387\) −2.25276 −0.114514
\(388\) −24.0882 −1.22289
\(389\) −10.9370 −0.554527 −0.277263 0.960794i \(-0.589427\pi\)
−0.277263 + 0.960794i \(0.589427\pi\)
\(390\) 0.687252 0.0348004
\(391\) −1.10359 −0.0558110
\(392\) −18.7787 −0.948466
\(393\) 19.0550 0.961200
\(394\) −1.27531 −0.0642492
\(395\) 7.89939 0.397461
\(396\) −1.35180 −0.0679307
\(397\) 33.2094 1.66673 0.833365 0.552722i \(-0.186411\pi\)
0.833365 + 0.552722i \(0.186411\pi\)
\(398\) −20.5741 −1.03129
\(399\) 0.532481 0.0266574
\(400\) −2.05260 −0.102630
\(401\) −18.7562 −0.936640 −0.468320 0.883559i \(-0.655141\pi\)
−0.468320 + 0.883559i \(0.655141\pi\)
\(402\) −6.11422 −0.304949
\(403\) 1.66568 0.0829733
\(404\) 6.09506 0.303241
\(405\) 1.06503 0.0529220
\(406\) 0.148274 0.00735874
\(407\) −2.14816 −0.106480
\(408\) 1.01607 0.0503032
\(409\) 5.97380 0.295385 0.147693 0.989033i \(-0.452815\pi\)
0.147693 + 0.989033i \(0.452815\pi\)
\(410\) 1.57153 0.0776125
\(411\) −18.4521 −0.910177
\(412\) −26.8758 −1.32408
\(413\) −1.57091 −0.0772997
\(414\) −2.35976 −0.115976
\(415\) −9.10750 −0.447069
\(416\) −4.66838 −0.228886
\(417\) −14.6161 −0.715754
\(418\) −2.11148 −0.103276
\(419\) −21.7376 −1.06195 −0.530976 0.847387i \(-0.678174\pi\)
−0.530976 + 0.847387i \(0.678174\pi\)
\(420\) −0.292313 −0.0142634
\(421\) −16.2177 −0.790402 −0.395201 0.918595i \(-0.629325\pi\)
−0.395201 + 0.918595i \(0.629325\pi\)
\(422\) −13.0350 −0.634535
\(423\) −0.685433 −0.0333269
\(424\) 37.9023 1.84070
\(425\) 1.45553 0.0706037
\(426\) −7.43882 −0.360412
\(427\) 0.203035 0.00982557
\(428\) −8.90650 −0.430512
\(429\) −0.801491 −0.0386964
\(430\) 1.93167 0.0931532
\(431\) −31.0756 −1.49686 −0.748430 0.663214i \(-0.769192\pi\)
−0.748430 + 0.663214i \(0.769192\pi\)
\(432\) −0.530979 −0.0255467
\(433\) 37.4627 1.80034 0.900171 0.435536i \(-0.143441\pi\)
0.900171 + 0.435536i \(0.143441\pi\)
\(434\) 0.339717 0.0163069
\(435\) −0.966063 −0.0463192
\(436\) −25.8834 −1.23959
\(437\) 7.68683 0.367711
\(438\) 6.03345 0.288290
\(439\) 15.0607 0.718809 0.359405 0.933182i \(-0.382980\pi\)
0.359405 + 0.933182i \(0.382980\pi\)
\(440\) 2.87406 0.137015
\(441\) −6.95878 −0.331370
\(442\) 0.242966 0.0115567
\(443\) 7.50403 0.356527 0.178264 0.983983i \(-0.442952\pi\)
0.178264 + 0.983983i \(0.442952\pi\)
\(444\) −2.90389 −0.137813
\(445\) −3.17220 −0.150376
\(446\) 3.97054 0.188010
\(447\) 4.26078 0.201528
\(448\) −0.736506 −0.0347966
\(449\) 13.5424 0.639103 0.319552 0.947569i \(-0.396468\pi\)
0.319552 + 0.947569i \(0.396468\pi\)
\(450\) 3.11230 0.146715
\(451\) −1.83276 −0.0863014
\(452\) 3.84009 0.180623
\(453\) −19.9980 −0.939587
\(454\) 2.08887 0.0980353
\(455\) −0.173314 −0.00812509
\(456\) −7.07726 −0.331423
\(457\) 10.1840 0.476388 0.238194 0.971218i \(-0.423445\pi\)
0.238194 + 0.971218i \(0.423445\pi\)
\(458\) 5.43780 0.254092
\(459\) 0.376525 0.0175747
\(460\) −4.21980 −0.196749
\(461\) −14.0649 −0.655067 −0.327533 0.944840i \(-0.606217\pi\)
−0.327533 + 0.944840i \(0.606217\pi\)
\(462\) −0.163465 −0.00760508
\(463\) −27.3090 −1.26916 −0.634578 0.772859i \(-0.718826\pi\)
−0.634578 + 0.772859i \(0.718826\pi\)
\(464\) 0.481636 0.0223594
\(465\) −2.21338 −0.102643
\(466\) 22.4207 1.03862
\(467\) 16.1833 0.748875 0.374437 0.927252i \(-0.377836\pi\)
0.374437 + 0.927252i \(0.377836\pi\)
\(468\) −1.08346 −0.0500829
\(469\) 1.54191 0.0711988
\(470\) 0.587736 0.0271102
\(471\) −0.492192 −0.0226790
\(472\) 20.8792 0.961041
\(473\) −2.25276 −0.103582
\(474\) 5.97150 0.274280
\(475\) −10.1382 −0.465173
\(476\) −0.103342 −0.00473669
\(477\) 14.0454 0.643093
\(478\) −13.6226 −0.623083
\(479\) −19.5798 −0.894623 −0.447311 0.894378i \(-0.647618\pi\)
−0.447311 + 0.894378i \(0.647618\pi\)
\(480\) 6.20342 0.283146
\(481\) −1.72173 −0.0785042
\(482\) 0.857078 0.0390389
\(483\) 0.595095 0.0270777
\(484\) −1.35180 −0.0614456
\(485\) 18.9782 0.861754
\(486\) 0.805107 0.0365204
\(487\) −25.6857 −1.16393 −0.581966 0.813213i \(-0.697716\pi\)
−0.581966 + 0.813213i \(0.697716\pi\)
\(488\) −2.69856 −0.122158
\(489\) 1.93731 0.0876081
\(490\) 5.96692 0.269558
\(491\) 30.0246 1.35499 0.677497 0.735526i \(-0.263065\pi\)
0.677497 + 0.735526i \(0.263065\pi\)
\(492\) −2.47753 −0.111696
\(493\) −0.341535 −0.0153820
\(494\) −1.69233 −0.0761415
\(495\) 1.06503 0.0478698
\(496\) 1.10349 0.0495482
\(497\) 1.87595 0.0841480
\(498\) −6.88476 −0.308513
\(499\) 16.6668 0.746109 0.373054 0.927810i \(-0.378311\pi\)
0.373054 + 0.927810i \(0.378311\pi\)
\(500\) 12.7641 0.570828
\(501\) −14.8338 −0.662726
\(502\) 2.57466 0.114912
\(503\) −27.2800 −1.21635 −0.608176 0.793802i \(-0.708099\pi\)
−0.608176 + 0.793802i \(0.708099\pi\)
\(504\) −0.547903 −0.0244055
\(505\) −4.80207 −0.213689
\(506\) −2.35976 −0.104904
\(507\) 12.3576 0.548821
\(508\) −6.76372 −0.300092
\(509\) −21.2805 −0.943243 −0.471622 0.881801i \(-0.656331\pi\)
−0.471622 + 0.881801i \(0.656331\pi\)
\(510\) −0.322857 −0.0142964
\(511\) −1.52154 −0.0673091
\(512\) −5.95850 −0.263331
\(513\) −2.62261 −0.115791
\(514\) 9.68514 0.427193
\(515\) 21.1744 0.933057
\(516\) −3.04529 −0.134061
\(517\) −0.685433 −0.0301453
\(518\) −0.351149 −0.0154286
\(519\) 10.3859 0.455889
\(520\) 2.30353 0.101017
\(521\) −25.1670 −1.10258 −0.551292 0.834312i \(-0.685865\pi\)
−0.551292 + 0.834312i \(0.685865\pi\)
\(522\) −0.730289 −0.0319639
\(523\) 20.0188 0.875362 0.437681 0.899130i \(-0.355800\pi\)
0.437681 + 0.899130i \(0.355800\pi\)
\(524\) 25.7587 1.12527
\(525\) −0.784874 −0.0342547
\(526\) 3.20568 0.139774
\(527\) −0.782502 −0.0340863
\(528\) −0.530979 −0.0231079
\(529\) −14.4093 −0.626491
\(530\) −12.0434 −0.523134
\(531\) 7.73715 0.335764
\(532\) 0.719810 0.0312077
\(533\) −1.46894 −0.0636270
\(534\) −2.39800 −0.103772
\(535\) 7.01710 0.303376
\(536\) −20.4936 −0.885191
\(537\) −16.9049 −0.729500
\(538\) −1.77633 −0.0765830
\(539\) −6.95878 −0.299736
\(540\) 1.43972 0.0619556
\(541\) 33.9981 1.46169 0.730847 0.682541i \(-0.239125\pi\)
0.730847 + 0.682541i \(0.239125\pi\)
\(542\) −8.79722 −0.377873
\(543\) −2.91712 −0.125186
\(544\) 2.19311 0.0940289
\(545\) 20.3926 0.873521
\(546\) −0.131016 −0.00560696
\(547\) −17.6728 −0.755636 −0.377818 0.925880i \(-0.623325\pi\)
−0.377818 + 0.925880i \(0.623325\pi\)
\(548\) −24.9437 −1.06554
\(549\) −1.00000 −0.0426790
\(550\) 3.11230 0.132709
\(551\) 2.37889 0.101344
\(552\) −7.90945 −0.336649
\(553\) −1.50592 −0.0640381
\(554\) −5.96522 −0.253438
\(555\) 2.28787 0.0971145
\(556\) −19.7581 −0.837931
\(557\) 40.6753 1.72347 0.861734 0.507360i \(-0.169379\pi\)
0.861734 + 0.507360i \(0.169379\pi\)
\(558\) −1.67319 −0.0708318
\(559\) −1.80557 −0.0763673
\(560\) −0.114819 −0.00485197
\(561\) 0.376525 0.0158969
\(562\) 0.125161 0.00527961
\(563\) 43.3214 1.82578 0.912890 0.408207i \(-0.133846\pi\)
0.912890 + 0.408207i \(0.133846\pi\)
\(564\) −0.926570 −0.0390156
\(565\) −3.02546 −0.127282
\(566\) 9.62051 0.404380
\(567\) −0.203035 −0.00852668
\(568\) −24.9335 −1.04618
\(569\) −26.7783 −1.12260 −0.561302 0.827611i \(-0.689699\pi\)
−0.561302 + 0.827611i \(0.689699\pi\)
\(570\) 2.24880 0.0941917
\(571\) 29.9130 1.25182 0.625910 0.779896i \(-0.284728\pi\)
0.625910 + 0.779896i \(0.284728\pi\)
\(572\) −1.08346 −0.0453017
\(573\) −10.8546 −0.453457
\(574\) −0.299593 −0.0125048
\(575\) −11.3303 −0.472508
\(576\) 3.62748 0.151145
\(577\) −19.3583 −0.805897 −0.402948 0.915223i \(-0.632015\pi\)
−0.402948 + 0.915223i \(0.632015\pi\)
\(578\) 13.5727 0.564549
\(579\) −9.85294 −0.409474
\(580\) −1.30593 −0.0542257
\(581\) 1.73623 0.0720309
\(582\) 14.3464 0.594679
\(583\) 14.0454 0.581700
\(584\) 20.2230 0.836832
\(585\) 0.853616 0.0352927
\(586\) −13.6627 −0.564402
\(587\) −33.1146 −1.36679 −0.683394 0.730050i \(-0.739497\pi\)
−0.683394 + 0.730050i \(0.739497\pi\)
\(588\) −9.40690 −0.387934
\(589\) 5.45036 0.224578
\(590\) −6.63435 −0.273132
\(591\) −1.58403 −0.0651581
\(592\) −1.14063 −0.0468795
\(593\) −11.3857 −0.467555 −0.233778 0.972290i \(-0.575109\pi\)
−0.233778 + 0.972290i \(0.575109\pi\)
\(594\) 0.805107 0.0330339
\(595\) 0.0814196 0.00333788
\(596\) 5.75974 0.235928
\(597\) −25.5545 −1.04588
\(598\) −1.89133 −0.0773421
\(599\) 14.0221 0.572927 0.286463 0.958091i \(-0.407520\pi\)
0.286463 + 0.958091i \(0.407520\pi\)
\(600\) 10.4318 0.425877
\(601\) 1.14648 0.0467660 0.0233830 0.999727i \(-0.492556\pi\)
0.0233830 + 0.999727i \(0.492556\pi\)
\(602\) −0.368247 −0.0150086
\(603\) −7.59429 −0.309263
\(604\) −27.0333 −1.09997
\(605\) 1.06503 0.0432998
\(606\) −3.63010 −0.147463
\(607\) −15.6771 −0.636315 −0.318157 0.948038i \(-0.603064\pi\)
−0.318157 + 0.948038i \(0.603064\pi\)
\(608\) −15.2757 −0.619510
\(609\) 0.184168 0.00746284
\(610\) 0.857467 0.0347178
\(611\) −0.549368 −0.0222251
\(612\) 0.508987 0.0205746
\(613\) 18.6826 0.754584 0.377292 0.926094i \(-0.376855\pi\)
0.377292 + 0.926094i \(0.376855\pi\)
\(614\) −10.7498 −0.433827
\(615\) 1.95196 0.0787104
\(616\) −0.547903 −0.0220756
\(617\) 3.28234 0.132142 0.0660709 0.997815i \(-0.478954\pi\)
0.0660709 + 0.997815i \(0.478954\pi\)
\(618\) 16.0067 0.643883
\(619\) −20.4789 −0.823116 −0.411558 0.911383i \(-0.635015\pi\)
−0.411558 + 0.911383i \(0.635015\pi\)
\(620\) −2.99205 −0.120164
\(621\) −2.93099 −0.117617
\(622\) 10.1851 0.408386
\(623\) 0.604738 0.0242283
\(624\) −0.425575 −0.0170366
\(625\) 9.27215 0.370886
\(626\) −6.56488 −0.262385
\(627\) −2.62261 −0.104737
\(628\) −0.665347 −0.0265502
\(629\) 0.808836 0.0322504
\(630\) 0.174096 0.00693615
\(631\) −21.1120 −0.840454 −0.420227 0.907419i \(-0.638050\pi\)
−0.420227 + 0.907419i \(0.638050\pi\)
\(632\) 20.0153 0.796165
\(633\) −16.1904 −0.643512
\(634\) 18.3172 0.727468
\(635\) 5.32888 0.211470
\(636\) 18.9866 0.752867
\(637\) −5.57740 −0.220985
\(638\) −0.730289 −0.0289124
\(639\) −9.23954 −0.365511
\(640\) 9.29639 0.367472
\(641\) −28.2912 −1.11743 −0.558717 0.829358i \(-0.688706\pi\)
−0.558717 + 0.829358i \(0.688706\pi\)
\(642\) 5.30453 0.209353
\(643\) −20.9767 −0.827238 −0.413619 0.910450i \(-0.635736\pi\)
−0.413619 + 0.910450i \(0.635736\pi\)
\(644\) 0.804451 0.0316998
\(645\) 2.39927 0.0944710
\(646\) 0.795024 0.0312798
\(647\) 20.5640 0.808456 0.404228 0.914658i \(-0.367540\pi\)
0.404228 + 0.914658i \(0.367540\pi\)
\(648\) 2.69856 0.106009
\(649\) 7.73715 0.303710
\(650\) 2.49448 0.0978416
\(651\) 0.421952 0.0165376
\(652\) 2.61886 0.102562
\(653\) 3.95699 0.154849 0.0774246 0.996998i \(-0.475330\pi\)
0.0774246 + 0.996998i \(0.475330\pi\)
\(654\) 15.4156 0.602799
\(655\) −20.2943 −0.792963
\(656\) −0.973158 −0.0379954
\(657\) 7.49398 0.292368
\(658\) −0.112044 −0.00436794
\(659\) −13.5419 −0.527519 −0.263759 0.964589i \(-0.584962\pi\)
−0.263759 + 0.964589i \(0.584962\pi\)
\(660\) 1.43972 0.0560409
\(661\) −37.6849 −1.46577 −0.732886 0.680351i \(-0.761827\pi\)
−0.732886 + 0.680351i \(0.761827\pi\)
\(662\) −1.99350 −0.0774797
\(663\) 0.301781 0.0117202
\(664\) −23.0764 −0.895536
\(665\) −0.567111 −0.0219916
\(666\) 1.72950 0.0670167
\(667\) 2.65862 0.102942
\(668\) −20.0524 −0.775851
\(669\) 4.93169 0.190670
\(670\) 6.51185 0.251575
\(671\) −1.00000 −0.0386046
\(672\) −1.18260 −0.0456199
\(673\) 46.2878 1.78427 0.892133 0.451774i \(-0.149209\pi\)
0.892133 + 0.451774i \(0.149209\pi\)
\(674\) 9.55394 0.368004
\(675\) 3.86570 0.148791
\(676\) 16.7051 0.642502
\(677\) 26.6561 1.02448 0.512239 0.858843i \(-0.328816\pi\)
0.512239 + 0.858843i \(0.328816\pi\)
\(678\) −2.28708 −0.0878348
\(679\) −3.61795 −0.138844
\(680\) −1.08215 −0.0414987
\(681\) 2.59452 0.0994222
\(682\) −1.67319 −0.0640698
\(683\) 27.7294 1.06104 0.530519 0.847673i \(-0.321997\pi\)
0.530519 + 0.847673i \(0.321997\pi\)
\(684\) −3.54525 −0.135556
\(685\) 19.6522 0.750871
\(686\) −2.28177 −0.0871185
\(687\) 6.75414 0.257687
\(688\) −1.19617 −0.0456034
\(689\) 11.2572 0.428867
\(690\) 2.51323 0.0956769
\(691\) −14.4148 −0.548365 −0.274183 0.961678i \(-0.588407\pi\)
−0.274183 + 0.961678i \(0.588407\pi\)
\(692\) 14.0397 0.533708
\(693\) −0.203035 −0.00771267
\(694\) −26.9291 −1.02222
\(695\) 15.5667 0.590477
\(696\) −2.44779 −0.0927831
\(697\) 0.690080 0.0261387
\(698\) 15.9238 0.602724
\(699\) 27.8481 1.05331
\(700\) −1.06099 −0.0401018
\(701\) −21.9567 −0.829295 −0.414647 0.909982i \(-0.636095\pi\)
−0.414647 + 0.909982i \(0.636095\pi\)
\(702\) 0.645286 0.0243547
\(703\) −5.63378 −0.212482
\(704\) 3.62748 0.136716
\(705\) 0.730010 0.0274938
\(706\) 12.3047 0.463093
\(707\) 0.915453 0.0344292
\(708\) 10.4591 0.393077
\(709\) −5.41410 −0.203331 −0.101665 0.994819i \(-0.532417\pi\)
−0.101665 + 0.994819i \(0.532417\pi\)
\(710\) 7.92260 0.297330
\(711\) 7.41703 0.278160
\(712\) −8.03763 −0.301223
\(713\) 6.09125 0.228119
\(714\) 0.0615487 0.00230340
\(715\) 0.853616 0.0319234
\(716\) −22.8521 −0.854023
\(717\) −16.9202 −0.631898
\(718\) −8.24906 −0.307852
\(719\) 34.7613 1.29638 0.648188 0.761480i \(-0.275527\pi\)
0.648188 + 0.761480i \(0.275527\pi\)
\(720\) 0.565511 0.0210753
\(721\) −4.03664 −0.150332
\(722\) 9.75946 0.363209
\(723\) 1.06455 0.0395911
\(724\) −3.94337 −0.146554
\(725\) −3.50647 −0.130227
\(726\) 0.805107 0.0298803
\(727\) 34.1852 1.26786 0.633930 0.773390i \(-0.281441\pi\)
0.633930 + 0.773390i \(0.281441\pi\)
\(728\) −0.439139 −0.0162756
\(729\) 1.00000 0.0370370
\(730\) −6.42584 −0.237831
\(731\) 0.848220 0.0313725
\(732\) −1.35180 −0.0499641
\(733\) −21.4554 −0.792474 −0.396237 0.918148i \(-0.629684\pi\)
−0.396237 + 0.918148i \(0.629684\pi\)
\(734\) −9.29746 −0.343176
\(735\) 7.41134 0.273371
\(736\) −17.0719 −0.629278
\(737\) −7.59429 −0.279739
\(738\) 1.47557 0.0543164
\(739\) 33.6003 1.23601 0.618004 0.786175i \(-0.287942\pi\)
0.618004 + 0.786175i \(0.287942\pi\)
\(740\) 3.09274 0.113692
\(741\) −2.10199 −0.0772187
\(742\) 2.29593 0.0842862
\(743\) 3.15148 0.115617 0.0578083 0.998328i \(-0.481589\pi\)
0.0578083 + 0.998328i \(0.481589\pi\)
\(744\) −5.60821 −0.205607
\(745\) −4.53788 −0.166255
\(746\) 21.1195 0.773240
\(747\) −8.55136 −0.312878
\(748\) 0.508987 0.0186104
\(749\) −1.33772 −0.0488792
\(750\) −7.60204 −0.277587
\(751\) −21.1165 −0.770551 −0.385276 0.922802i \(-0.625894\pi\)
−0.385276 + 0.922802i \(0.625894\pi\)
\(752\) −0.363950 −0.0132719
\(753\) 3.19791 0.116538
\(754\) −0.585320 −0.0213161
\(755\) 21.2986 0.775134
\(756\) −0.274464 −0.00998215
\(757\) 25.3065 0.919779 0.459890 0.887976i \(-0.347889\pi\)
0.459890 + 0.887976i \(0.347889\pi\)
\(758\) −8.96923 −0.325777
\(759\) −2.93099 −0.106388
\(760\) 7.53752 0.273415
\(761\) −26.3449 −0.955000 −0.477500 0.878632i \(-0.658457\pi\)
−0.477500 + 0.878632i \(0.658457\pi\)
\(762\) 4.02833 0.145931
\(763\) −3.88758 −0.140740
\(764\) −14.6733 −0.530860
\(765\) −0.401012 −0.0144986
\(766\) −30.3280 −1.09580
\(767\) 6.20126 0.223914
\(768\) 14.2825 0.515375
\(769\) −0.854669 −0.0308202 −0.0154101 0.999881i \(-0.504905\pi\)
−0.0154101 + 0.999881i \(0.504905\pi\)
\(770\) 0.174096 0.00627398
\(771\) 12.0296 0.433237
\(772\) −13.3192 −0.479370
\(773\) 14.3339 0.515555 0.257778 0.966204i \(-0.417010\pi\)
0.257778 + 0.966204i \(0.417010\pi\)
\(774\) 1.81371 0.0651925
\(775\) −8.03378 −0.288582
\(776\) 48.0864 1.72620
\(777\) −0.436152 −0.0156469
\(778\) 8.80543 0.315690
\(779\) −4.80661 −0.172215
\(780\) 1.15392 0.0413170
\(781\) −9.23954 −0.330617
\(782\) 0.888508 0.0317730
\(783\) −0.907071 −0.0324161
\(784\) −3.69496 −0.131963
\(785\) 0.524202 0.0187096
\(786\) −15.3413 −0.547208
\(787\) 46.4807 1.65686 0.828430 0.560093i \(-0.189235\pi\)
0.828430 + 0.560093i \(0.189235\pi\)
\(788\) −2.14129 −0.0762804
\(789\) 3.98169 0.141752
\(790\) −6.35985 −0.226273
\(791\) 0.576766 0.0205074
\(792\) 2.69856 0.0958891
\(793\) −0.801491 −0.0284618
\(794\) −26.7371 −0.948864
\(795\) −14.9588 −0.530534
\(796\) −34.5447 −1.22440
\(797\) −13.5176 −0.478817 −0.239409 0.970919i \(-0.576954\pi\)
−0.239409 + 0.970919i \(0.576954\pi\)
\(798\) −0.428704 −0.0151760
\(799\) 0.258082 0.00913030
\(800\) 22.5162 0.796068
\(801\) −2.97849 −0.105240
\(802\) 15.1007 0.533226
\(803\) 7.49398 0.264457
\(804\) −10.2660 −0.362054
\(805\) −0.633797 −0.0223384
\(806\) −1.34105 −0.0472364
\(807\) −2.20633 −0.0776665
\(808\) −12.1674 −0.428046
\(809\) 44.3814 1.56037 0.780183 0.625551i \(-0.215126\pi\)
0.780183 + 0.625551i \(0.215126\pi\)
\(810\) −0.857467 −0.0301283
\(811\) −2.02464 −0.0710946 −0.0355473 0.999368i \(-0.511317\pi\)
−0.0355473 + 0.999368i \(0.511317\pi\)
\(812\) 0.248958 0.00873672
\(813\) −10.9268 −0.383219
\(814\) 1.72950 0.0606189
\(815\) −2.06330 −0.0722743
\(816\) 0.199927 0.00699883
\(817\) −5.90810 −0.206698
\(818\) −4.80954 −0.168162
\(819\) −0.162731 −0.00568628
\(820\) 2.63866 0.0921460
\(821\) 26.4603 0.923472 0.461736 0.887017i \(-0.347227\pi\)
0.461736 + 0.887017i \(0.347227\pi\)
\(822\) 14.8559 0.518161
\(823\) −40.6265 −1.41615 −0.708075 0.706137i \(-0.750436\pi\)
−0.708075 + 0.706137i \(0.750436\pi\)
\(824\) 53.6513 1.86903
\(825\) 3.86570 0.134586
\(826\) 1.26475 0.0440064
\(827\) −39.7573 −1.38250 −0.691248 0.722617i \(-0.742939\pi\)
−0.691248 + 0.722617i \(0.742939\pi\)
\(828\) −3.96212 −0.137693
\(829\) −8.76395 −0.304385 −0.152192 0.988351i \(-0.548633\pi\)
−0.152192 + 0.988351i \(0.548633\pi\)
\(830\) 7.33251 0.254515
\(831\) −7.40923 −0.257023
\(832\) 2.90739 0.100796
\(833\) 2.62015 0.0907829
\(834\) 11.7675 0.407476
\(835\) 15.7985 0.546731
\(836\) −3.54525 −0.122615
\(837\) −2.07822 −0.0718339
\(838\) 17.5011 0.604565
\(839\) −0.750852 −0.0259223 −0.0129611 0.999916i \(-0.504126\pi\)
−0.0129611 + 0.999916i \(0.504126\pi\)
\(840\) 0.583535 0.0201339
\(841\) −28.1772 −0.971628
\(842\) 13.0570 0.449973
\(843\) 0.155459 0.00535430
\(844\) −21.8863 −0.753357
\(845\) −13.1613 −0.452762
\(846\) 0.551846 0.0189729
\(847\) −0.203035 −0.00697637
\(848\) 7.45779 0.256102
\(849\) 11.9494 0.410101
\(850\) −1.17186 −0.0401944
\(851\) −6.29624 −0.215832
\(852\) −12.4900 −0.427902
\(853\) 27.3127 0.935168 0.467584 0.883949i \(-0.345124\pi\)
0.467584 + 0.883949i \(0.345124\pi\)
\(854\) −0.163465 −0.00559366
\(855\) 2.79317 0.0955243
\(856\) 17.7798 0.607699
\(857\) −8.31434 −0.284012 −0.142006 0.989866i \(-0.545355\pi\)
−0.142006 + 0.989866i \(0.545355\pi\)
\(858\) 0.645286 0.0220297
\(859\) 27.0139 0.921703 0.460852 0.887477i \(-0.347544\pi\)
0.460852 + 0.887477i \(0.347544\pi\)
\(860\) 3.24334 0.110597
\(861\) −0.372115 −0.0126817
\(862\) 25.0192 0.852157
\(863\) −7.54619 −0.256875 −0.128438 0.991718i \(-0.540996\pi\)
−0.128438 + 0.991718i \(0.540996\pi\)
\(864\) 5.82461 0.198157
\(865\) −11.0613 −0.376096
\(866\) −30.1615 −1.02493
\(867\) 16.8582 0.572535
\(868\) 0.570397 0.0193605
\(869\) 7.41703 0.251605
\(870\) 0.777784 0.0263693
\(871\) −6.08676 −0.206242
\(872\) 51.6702 1.74977
\(873\) 17.8193 0.603092
\(874\) −6.18872 −0.209337
\(875\) 1.91712 0.0648104
\(876\) 10.1304 0.342274
\(877\) −40.4243 −1.36503 −0.682515 0.730871i \(-0.739114\pi\)
−0.682515 + 0.730871i \(0.739114\pi\)
\(878\) −12.1255 −0.409215
\(879\) −16.9701 −0.572387
\(880\) 0.565511 0.0190634
\(881\) −24.8846 −0.838383 −0.419191 0.907898i \(-0.637686\pi\)
−0.419191 + 0.907898i \(0.637686\pi\)
\(882\) 5.60256 0.188648
\(883\) 43.2827 1.45658 0.728289 0.685270i \(-0.240316\pi\)
0.728289 + 0.685270i \(0.240316\pi\)
\(884\) 0.407949 0.0137208
\(885\) −8.24034 −0.276996
\(886\) −6.04154 −0.202970
\(887\) 10.2272 0.343396 0.171698 0.985150i \(-0.445075\pi\)
0.171698 + 0.985150i \(0.445075\pi\)
\(888\) 5.79694 0.194533
\(889\) −1.01588 −0.0340716
\(890\) 2.55396 0.0856088
\(891\) 1.00000 0.0335013
\(892\) 6.66667 0.223217
\(893\) −1.79762 −0.0601550
\(894\) −3.43038 −0.114729
\(895\) 18.0043 0.601817
\(896\) −1.77224 −0.0592063
\(897\) −2.34916 −0.0784363
\(898\) −10.9030 −0.363839
\(899\) 1.88510 0.0628715
\(900\) 5.22567 0.174189
\(901\) −5.28843 −0.176183
\(902\) 1.47557 0.0491311
\(903\) −0.457390 −0.0152210
\(904\) −7.66584 −0.254962
\(905\) 3.10683 0.103275
\(906\) 16.1005 0.534904
\(907\) −34.3774 −1.14148 −0.570741 0.821130i \(-0.693344\pi\)
−0.570741 + 0.821130i \(0.693344\pi\)
\(908\) 3.50728 0.116393
\(909\) −4.50884 −0.149549
\(910\) 0.139536 0.00462559
\(911\) −20.2548 −0.671072 −0.335536 0.942027i \(-0.608917\pi\)
−0.335536 + 0.942027i \(0.608917\pi\)
\(912\) −1.39255 −0.0461118
\(913\) −8.55136 −0.283009
\(914\) −8.19921 −0.271206
\(915\) 1.06503 0.0352090
\(916\) 9.13027 0.301673
\(917\) 3.86885 0.127761
\(918\) −0.303143 −0.0100052
\(919\) −26.9266 −0.888227 −0.444114 0.895970i \(-0.646481\pi\)
−0.444114 + 0.895970i \(0.646481\pi\)
\(920\) 8.42384 0.277726
\(921\) −13.3520 −0.439965
\(922\) 11.3237 0.372927
\(923\) −7.40541 −0.243752
\(924\) −0.274464 −0.00902919
\(925\) 8.30415 0.273039
\(926\) 21.9866 0.722526
\(927\) 19.8815 0.652993
\(928\) −5.28334 −0.173434
\(929\) 13.1495 0.431421 0.215711 0.976457i \(-0.430793\pi\)
0.215711 + 0.976457i \(0.430793\pi\)
\(930\) 1.78201 0.0584343
\(931\) −18.2501 −0.598124
\(932\) 37.6452 1.23311
\(933\) 12.6506 0.414163
\(934\) −13.0293 −0.426332
\(935\) −0.401012 −0.0131145
\(936\) 2.16287 0.0706957
\(937\) 19.9881 0.652981 0.326491 0.945200i \(-0.394134\pi\)
0.326491 + 0.945200i \(0.394134\pi\)
\(938\) −1.24140 −0.0405332
\(939\) −8.15405 −0.266097
\(940\) 0.986829 0.0321868
\(941\) 24.3479 0.793720 0.396860 0.917879i \(-0.370100\pi\)
0.396860 + 0.917879i \(0.370100\pi\)
\(942\) 0.396267 0.0129111
\(943\) −5.37181 −0.174930
\(944\) 4.10826 0.133713
\(945\) 0.216240 0.00703428
\(946\) 1.81371 0.0589688
\(947\) −33.4192 −1.08598 −0.542989 0.839740i \(-0.682707\pi\)
−0.542989 + 0.839740i \(0.682707\pi\)
\(948\) 10.0264 0.325641
\(949\) 6.00636 0.194975
\(950\) 8.16234 0.264821
\(951\) 22.7512 0.737760
\(952\) 0.206299 0.00668619
\(953\) −1.05530 −0.0341847 −0.0170923 0.999854i \(-0.505441\pi\)
−0.0170923 + 0.999854i \(0.505441\pi\)
\(954\) −11.3080 −0.366111
\(955\) 11.5605 0.374089
\(956\) −22.8728 −0.739761
\(957\) −0.907071 −0.0293215
\(958\) 15.7638 0.509305
\(959\) −3.74644 −0.120979
\(960\) −3.86339 −0.124690
\(961\) −26.6810 −0.860677
\(962\) 1.38618 0.0446922
\(963\) 6.58861 0.212315
\(964\) 1.43907 0.0463492
\(965\) 10.4937 0.337805
\(966\) −0.479115 −0.0154153
\(967\) 15.5418 0.499791 0.249895 0.968273i \(-0.419604\pi\)
0.249895 + 0.968273i \(0.419604\pi\)
\(968\) 2.69856 0.0867350
\(969\) 0.987476 0.0317223
\(970\) −15.2795 −0.490594
\(971\) −27.9120 −0.895738 −0.447869 0.894099i \(-0.647817\pi\)
−0.447869 + 0.894099i \(0.647817\pi\)
\(972\) 1.35180 0.0433591
\(973\) −2.96759 −0.0951365
\(974\) 20.6797 0.662622
\(975\) 3.09832 0.0992258
\(976\) −0.530979 −0.0169962
\(977\) 10.2968 0.329423 0.164712 0.986342i \(-0.447331\pi\)
0.164712 + 0.986342i \(0.447331\pi\)
\(978\) −1.55974 −0.0498750
\(979\) −2.97849 −0.0951929
\(980\) 10.0187 0.320035
\(981\) 19.1473 0.611327
\(982\) −24.1730 −0.771393
\(983\) −53.4728 −1.70552 −0.852759 0.522304i \(-0.825073\pi\)
−0.852759 + 0.522304i \(0.825073\pi\)
\(984\) 4.94582 0.157667
\(985\) 1.68704 0.0537537
\(986\) 0.274972 0.00875689
\(987\) −0.139167 −0.00442973
\(988\) −2.84148 −0.0903996
\(989\) −6.60282 −0.209957
\(990\) −0.857467 −0.0272521
\(991\) −22.7926 −0.724031 −0.362015 0.932172i \(-0.617911\pi\)
−0.362015 + 0.932172i \(0.617911\pi\)
\(992\) −12.1048 −0.384329
\(993\) −2.47607 −0.0785758
\(994\) −1.51034 −0.0479051
\(995\) 27.2165 0.862820
\(996\) −11.5598 −0.366285
\(997\) 53.2933 1.68782 0.843908 0.536488i \(-0.180249\pi\)
0.843908 + 0.536488i \(0.180249\pi\)
\(998\) −13.4186 −0.424757
\(999\) 2.14816 0.0679648
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 2013.2.a.e.1.6 13
3.2 odd 2 6039.2.a.i.1.8 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2013.2.a.e.1.6 13 1.1 even 1 trivial
6039.2.a.i.1.8 13 3.2 odd 2