Properties

Label 2001.2.a.k.1.2
Level $2001$
Weight $2$
Character 2001.1
Self dual yes
Analytic conductor $15.978$
Analytic rank $0$
Dimension $10$
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] = [2001,2,Mod(1,2001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2001, 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("2001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2001 = 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2001.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: \(15.9780654445\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} - 17x^{8} + 23x^{7} + 69x^{6} - 88x^{5} - 106x^{4} + 101x^{3} + 60x^{2} - 23x - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-3.66000\) of defining polynomial
Character \(\chi\) \(=\) 2001.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.24431 q^{2} -1.00000 q^{3} +3.03693 q^{4} -1.31370 q^{5} +2.24431 q^{6} -2.96709 q^{7} -2.32720 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.24431 q^{2} -1.00000 q^{3} +3.03693 q^{4} -1.31370 q^{5} +2.24431 q^{6} -2.96709 q^{7} -2.32720 q^{8} +1.00000 q^{9} +2.94835 q^{10} +1.51682 q^{11} -3.03693 q^{12} +0.902202 q^{13} +6.65908 q^{14} +1.31370 q^{15} -0.850911 q^{16} +3.54263 q^{17} -2.24431 q^{18} +0.438527 q^{19} -3.98961 q^{20} +2.96709 q^{21} -3.40420 q^{22} -1.00000 q^{23} +2.32720 q^{24} -3.27420 q^{25} -2.02482 q^{26} -1.00000 q^{27} -9.01086 q^{28} +1.00000 q^{29} -2.94835 q^{30} -1.36633 q^{31} +6.56410 q^{32} -1.51682 q^{33} -7.95077 q^{34} +3.89786 q^{35} +3.03693 q^{36} -10.9378 q^{37} -0.984190 q^{38} -0.902202 q^{39} +3.05723 q^{40} +3.73744 q^{41} -6.65908 q^{42} -6.23701 q^{43} +4.60646 q^{44} -1.31370 q^{45} +2.24431 q^{46} +4.40935 q^{47} +0.850911 q^{48} +1.80364 q^{49} +7.34832 q^{50} -3.54263 q^{51} +2.73993 q^{52} -10.7318 q^{53} +2.24431 q^{54} -1.99264 q^{55} +6.90501 q^{56} -0.438527 q^{57} -2.24431 q^{58} +7.80539 q^{59} +3.98961 q^{60} +8.66624 q^{61} +3.06647 q^{62} -2.96709 q^{63} -13.0301 q^{64} -1.18522 q^{65} +3.40420 q^{66} +1.02600 q^{67} +10.7587 q^{68} +1.00000 q^{69} -8.74801 q^{70} -11.2518 q^{71} -2.32720 q^{72} -2.88266 q^{73} +24.5479 q^{74} +3.27420 q^{75} +1.33178 q^{76} -4.50053 q^{77} +2.02482 q^{78} +16.1535 q^{79} +1.11784 q^{80} +1.00000 q^{81} -8.38797 q^{82} +3.36143 q^{83} +9.01086 q^{84} -4.65395 q^{85} +13.9978 q^{86} -1.00000 q^{87} -3.52993 q^{88} -7.47996 q^{89} +2.94835 q^{90} -2.67692 q^{91} -3.03693 q^{92} +1.36633 q^{93} -9.89596 q^{94} -0.576092 q^{95} -6.56410 q^{96} -15.6206 q^{97} -4.04792 q^{98} +1.51682 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 3 q^{2} - 10 q^{3} + 17 q^{4} + 6 q^{5} - 3 q^{6} + 3 q^{7} - 6 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 3 q^{2} - 10 q^{3} + 17 q^{4} + 6 q^{5} - 3 q^{6} + 3 q^{7} - 6 q^{8} + 10 q^{9} - 4 q^{10} + 9 q^{11} - 17 q^{12} - 16 q^{13} + 16 q^{14} - 6 q^{15} + 27 q^{16} + 3 q^{18} + q^{19} + 21 q^{20} - 3 q^{21} + 17 q^{22} - 10 q^{23} + 6 q^{24} - 4 q^{25} + 28 q^{26} - 10 q^{27} - 14 q^{28} + 10 q^{29} + 4 q^{30} + 17 q^{31} + 21 q^{32} - 9 q^{33} - 3 q^{34} + 29 q^{35} + 17 q^{36} + q^{37} + 32 q^{38} + 16 q^{39} + 13 q^{40} - 16 q^{42} - 5 q^{43} + 33 q^{44} + 6 q^{45} - 3 q^{46} + 15 q^{47} - 27 q^{48} + 31 q^{49} - 22 q^{50} - 21 q^{52} + 35 q^{53} - 3 q^{54} - 20 q^{55} + 18 q^{56} - q^{57} + 3 q^{58} + 49 q^{59} - 21 q^{60} + 8 q^{61} + 15 q^{62} + 3 q^{63} + 12 q^{64} - 3 q^{65} - 17 q^{66} + 35 q^{67} - 18 q^{68} + 10 q^{69} - 16 q^{70} + 30 q^{71} - 6 q^{72} - 15 q^{73} + 23 q^{74} + 4 q^{75} + 10 q^{76} + 23 q^{77} - 28 q^{78} + 24 q^{79} + 23 q^{80} + 10 q^{81} - 5 q^{82} + q^{83} + 14 q^{84} - 10 q^{86} - 10 q^{87} + 18 q^{88} + 15 q^{89} - 4 q^{90} + 26 q^{91} - 17 q^{92} - 17 q^{93} + 3 q^{94} + 7 q^{95} - 21 q^{96} - 35 q^{97} + 3 q^{98} + 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.24431 −1.58697 −0.793484 0.608591i \(-0.791735\pi\)
−0.793484 + 0.608591i \(0.791735\pi\)
\(3\) −1.00000 −0.577350
\(4\) 3.03693 1.51847
\(5\) −1.31370 −0.587503 −0.293752 0.955882i \(-0.594904\pi\)
−0.293752 + 0.955882i \(0.594904\pi\)
\(6\) 2.24431 0.916236
\(7\) −2.96709 −1.12146 −0.560728 0.828000i \(-0.689479\pi\)
−0.560728 + 0.828000i \(0.689479\pi\)
\(8\) −2.32720 −0.822788
\(9\) 1.00000 0.333333
\(10\) 2.94835 0.932349
\(11\) 1.51682 0.457337 0.228668 0.973504i \(-0.426563\pi\)
0.228668 + 0.973504i \(0.426563\pi\)
\(12\) −3.03693 −0.876687
\(13\) 0.902202 0.250226 0.125113 0.992143i \(-0.460071\pi\)
0.125113 + 0.992143i \(0.460071\pi\)
\(14\) 6.65908 1.77971
\(15\) 1.31370 0.339195
\(16\) −0.850911 −0.212728
\(17\) 3.54263 0.859215 0.429607 0.903016i \(-0.358652\pi\)
0.429607 + 0.903016i \(0.358652\pi\)
\(18\) −2.24431 −0.528989
\(19\) 0.438527 0.100605 0.0503025 0.998734i \(-0.483981\pi\)
0.0503025 + 0.998734i \(0.483981\pi\)
\(20\) −3.98961 −0.892104
\(21\) 2.96709 0.647473
\(22\) −3.40420 −0.725779
\(23\) −1.00000 −0.208514
\(24\) 2.32720 0.475037
\(25\) −3.27420 −0.654840
\(26\) −2.02482 −0.397100
\(27\) −1.00000 −0.192450
\(28\) −9.01086 −1.70289
\(29\) 1.00000 0.185695
\(30\) −2.94835 −0.538292
\(31\) −1.36633 −0.245400 −0.122700 0.992444i \(-0.539155\pi\)
−0.122700 + 0.992444i \(0.539155\pi\)
\(32\) 6.56410 1.16038
\(33\) −1.51682 −0.264044
\(34\) −7.95077 −1.36355
\(35\) 3.89786 0.658859
\(36\) 3.03693 0.506155
\(37\) −10.9378 −1.79817 −0.899084 0.437776i \(-0.855766\pi\)
−0.899084 + 0.437776i \(0.855766\pi\)
\(38\) −0.984190 −0.159657
\(39\) −0.902202 −0.144468
\(40\) 3.05723 0.483391
\(41\) 3.73744 0.583690 0.291845 0.956466i \(-0.405731\pi\)
0.291845 + 0.956466i \(0.405731\pi\)
\(42\) −6.65908 −1.02752
\(43\) −6.23701 −0.951135 −0.475568 0.879679i \(-0.657757\pi\)
−0.475568 + 0.879679i \(0.657757\pi\)
\(44\) 4.60646 0.694450
\(45\) −1.31370 −0.195834
\(46\) 2.24431 0.330906
\(47\) 4.40935 0.643170 0.321585 0.946881i \(-0.395784\pi\)
0.321585 + 0.946881i \(0.395784\pi\)
\(48\) 0.850911 0.122818
\(49\) 1.80364 0.257663
\(50\) 7.34832 1.03921
\(51\) −3.54263 −0.496068
\(52\) 2.73993 0.379959
\(53\) −10.7318 −1.47413 −0.737065 0.675822i \(-0.763789\pi\)
−0.737065 + 0.675822i \(0.763789\pi\)
\(54\) 2.24431 0.305412
\(55\) −1.99264 −0.268687
\(56\) 6.90501 0.922720
\(57\) −0.438527 −0.0580843
\(58\) −2.24431 −0.294692
\(59\) 7.80539 1.01618 0.508088 0.861305i \(-0.330353\pi\)
0.508088 + 0.861305i \(0.330353\pi\)
\(60\) 3.98961 0.515056
\(61\) 8.66624 1.10960 0.554799 0.831984i \(-0.312795\pi\)
0.554799 + 0.831984i \(0.312795\pi\)
\(62\) 3.06647 0.389442
\(63\) −2.96709 −0.373819
\(64\) −13.0301 −1.62876
\(65\) −1.18522 −0.147009
\(66\) 3.40420 0.419029
\(67\) 1.02600 0.125346 0.0626729 0.998034i \(-0.480038\pi\)
0.0626729 + 0.998034i \(0.480038\pi\)
\(68\) 10.7587 1.30469
\(69\) 1.00000 0.120386
\(70\) −8.74801 −1.04559
\(71\) −11.2518 −1.33535 −0.667674 0.744454i \(-0.732710\pi\)
−0.667674 + 0.744454i \(0.732710\pi\)
\(72\) −2.32720 −0.274263
\(73\) −2.88266 −0.337390 −0.168695 0.985668i \(-0.553955\pi\)
−0.168695 + 0.985668i \(0.553955\pi\)
\(74\) 24.5479 2.85363
\(75\) 3.27420 0.378072
\(76\) 1.33178 0.152765
\(77\) −4.50053 −0.512883
\(78\) 2.02482 0.229266
\(79\) 16.1535 1.81741 0.908703 0.417443i \(-0.137074\pi\)
0.908703 + 0.417443i \(0.137074\pi\)
\(80\) 1.11784 0.124978
\(81\) 1.00000 0.111111
\(82\) −8.38797 −0.926296
\(83\) 3.36143 0.368965 0.184483 0.982836i \(-0.440939\pi\)
0.184483 + 0.982836i \(0.440939\pi\)
\(84\) 9.01086 0.983165
\(85\) −4.65395 −0.504792
\(86\) 13.9978 1.50942
\(87\) −1.00000 −0.107211
\(88\) −3.52993 −0.376291
\(89\) −7.47996 −0.792874 −0.396437 0.918062i \(-0.629753\pi\)
−0.396437 + 0.918062i \(0.629753\pi\)
\(90\) 2.94835 0.310783
\(91\) −2.67692 −0.280617
\(92\) −3.03693 −0.316622
\(93\) 1.36633 0.141682
\(94\) −9.89596 −1.02069
\(95\) −0.576092 −0.0591057
\(96\) −6.56410 −0.669946
\(97\) −15.6206 −1.58603 −0.793014 0.609203i \(-0.791489\pi\)
−0.793014 + 0.609203i \(0.791489\pi\)
\(98\) −4.04792 −0.408902
\(99\) 1.51682 0.152446
\(100\) −9.94352 −0.994352
\(101\) −3.63336 −0.361533 −0.180767 0.983526i \(-0.557858\pi\)
−0.180767 + 0.983526i \(0.557858\pi\)
\(102\) 7.95077 0.787243
\(103\) 10.2810 1.01302 0.506510 0.862234i \(-0.330935\pi\)
0.506510 + 0.862234i \(0.330935\pi\)
\(104\) −2.09960 −0.205883
\(105\) −3.89786 −0.380392
\(106\) 24.0856 2.33940
\(107\) 7.31175 0.706854 0.353427 0.935462i \(-0.385016\pi\)
0.353427 + 0.935462i \(0.385016\pi\)
\(108\) −3.03693 −0.292229
\(109\) −3.37760 −0.323515 −0.161758 0.986831i \(-0.551716\pi\)
−0.161758 + 0.986831i \(0.551716\pi\)
\(110\) 4.47210 0.426398
\(111\) 10.9378 1.03817
\(112\) 2.52473 0.238565
\(113\) −2.02102 −0.190122 −0.0950610 0.995471i \(-0.530305\pi\)
−0.0950610 + 0.995471i \(0.530305\pi\)
\(114\) 0.984190 0.0921779
\(115\) 1.31370 0.122503
\(116\) 3.03693 0.281972
\(117\) 0.902202 0.0834086
\(118\) −17.5177 −1.61264
\(119\) −10.5113 −0.963571
\(120\) −3.05723 −0.279086
\(121\) −8.69927 −0.790843
\(122\) −19.4497 −1.76090
\(123\) −3.73744 −0.336993
\(124\) −4.14945 −0.372632
\(125\) 10.8698 0.972224
\(126\) 6.65908 0.593238
\(127\) 2.41766 0.214533 0.107266 0.994230i \(-0.465790\pi\)
0.107266 + 0.994230i \(0.465790\pi\)
\(128\) 16.1153 1.42441
\(129\) 6.23701 0.549138
\(130\) 2.66000 0.233298
\(131\) 3.99026 0.348630 0.174315 0.984690i \(-0.444229\pi\)
0.174315 + 0.984690i \(0.444229\pi\)
\(132\) −4.60646 −0.400941
\(133\) −1.30115 −0.112824
\(134\) −2.30266 −0.198920
\(135\) 1.31370 0.113065
\(136\) −8.24440 −0.706951
\(137\) 6.48948 0.554434 0.277217 0.960807i \(-0.410588\pi\)
0.277217 + 0.960807i \(0.410588\pi\)
\(138\) −2.24431 −0.191048
\(139\) −18.9540 −1.60766 −0.803830 0.594859i \(-0.797208\pi\)
−0.803830 + 0.594859i \(0.797208\pi\)
\(140\) 11.8375 1.00045
\(141\) −4.40935 −0.371335
\(142\) 25.2526 2.11915
\(143\) 1.36847 0.114438
\(144\) −0.850911 −0.0709093
\(145\) −1.31370 −0.109097
\(146\) 6.46958 0.535427
\(147\) −1.80364 −0.148762
\(148\) −33.2174 −2.73046
\(149\) 22.8616 1.87290 0.936448 0.350807i \(-0.114093\pi\)
0.936448 + 0.350807i \(0.114093\pi\)
\(150\) −7.34832 −0.599988
\(151\) 16.8032 1.36743 0.683714 0.729750i \(-0.260364\pi\)
0.683714 + 0.729750i \(0.260364\pi\)
\(152\) −1.02054 −0.0827765
\(153\) 3.54263 0.286405
\(154\) 10.1006 0.813929
\(155\) 1.79495 0.144174
\(156\) −2.73993 −0.219370
\(157\) 4.66486 0.372296 0.186148 0.982522i \(-0.440400\pi\)
0.186148 + 0.982522i \(0.440400\pi\)
\(158\) −36.2534 −2.88416
\(159\) 10.7318 0.851089
\(160\) −8.62324 −0.681727
\(161\) 2.96709 0.233840
\(162\) −2.24431 −0.176330
\(163\) −6.11817 −0.479212 −0.239606 0.970870i \(-0.577018\pi\)
−0.239606 + 0.970870i \(0.577018\pi\)
\(164\) 11.3503 0.886312
\(165\) 1.99264 0.155127
\(166\) −7.54410 −0.585536
\(167\) 5.11459 0.395779 0.197889 0.980224i \(-0.436591\pi\)
0.197889 + 0.980224i \(0.436591\pi\)
\(168\) −6.90501 −0.532733
\(169\) −12.1860 −0.937387
\(170\) 10.4449 0.801088
\(171\) 0.438527 0.0335350
\(172\) −18.9414 −1.44427
\(173\) 10.5173 0.799614 0.399807 0.916599i \(-0.369077\pi\)
0.399807 + 0.916599i \(0.369077\pi\)
\(174\) 2.24431 0.170141
\(175\) 9.71485 0.734374
\(176\) −1.29068 −0.0972883
\(177\) −7.80539 −0.586689
\(178\) 16.7873 1.25826
\(179\) 10.1812 0.760981 0.380491 0.924785i \(-0.375755\pi\)
0.380491 + 0.924785i \(0.375755\pi\)
\(180\) −3.98961 −0.297368
\(181\) 21.6445 1.60882 0.804412 0.594072i \(-0.202481\pi\)
0.804412 + 0.594072i \(0.202481\pi\)
\(182\) 6.00784 0.445330
\(183\) −8.66624 −0.640627
\(184\) 2.32720 0.171563
\(185\) 14.3690 1.05643
\(186\) −3.06647 −0.224845
\(187\) 5.37352 0.392951
\(188\) 13.3909 0.976632
\(189\) 2.96709 0.215824
\(190\) 1.29293 0.0937989
\(191\) 17.2119 1.24541 0.622705 0.782457i \(-0.286034\pi\)
0.622705 + 0.782457i \(0.286034\pi\)
\(192\) 13.0301 0.940364
\(193\) −7.97678 −0.574181 −0.287090 0.957903i \(-0.592688\pi\)
−0.287090 + 0.957903i \(0.592688\pi\)
\(194\) 35.0574 2.51698
\(195\) 1.18522 0.0848754
\(196\) 5.47752 0.391252
\(197\) 5.13962 0.366183 0.183092 0.983096i \(-0.441390\pi\)
0.183092 + 0.983096i \(0.441390\pi\)
\(198\) −3.40420 −0.241926
\(199\) 21.9501 1.55600 0.777999 0.628265i \(-0.216235\pi\)
0.777999 + 0.628265i \(0.216235\pi\)
\(200\) 7.61970 0.538794
\(201\) −1.02600 −0.0723684
\(202\) 8.15440 0.573741
\(203\) −2.96709 −0.208249
\(204\) −10.7587 −0.753262
\(205\) −4.90986 −0.342920
\(206\) −23.0738 −1.60763
\(207\) −1.00000 −0.0695048
\(208\) −0.767694 −0.0532300
\(209\) 0.665164 0.0460104
\(210\) 8.74801 0.603670
\(211\) 7.00040 0.481927 0.240964 0.970534i \(-0.422537\pi\)
0.240964 + 0.970534i \(0.422537\pi\)
\(212\) −32.5918 −2.23841
\(213\) 11.2518 0.770963
\(214\) −16.4098 −1.12175
\(215\) 8.19355 0.558795
\(216\) 2.32720 0.158346
\(217\) 4.05403 0.275206
\(218\) 7.58038 0.513408
\(219\) 2.88266 0.194792
\(220\) −6.05150 −0.407992
\(221\) 3.19617 0.214998
\(222\) −24.5479 −1.64755
\(223\) 19.3205 1.29380 0.646899 0.762575i \(-0.276065\pi\)
0.646899 + 0.762575i \(0.276065\pi\)
\(224\) −19.4763 −1.30131
\(225\) −3.27420 −0.218280
\(226\) 4.53581 0.301717
\(227\) −4.35027 −0.288738 −0.144369 0.989524i \(-0.546115\pi\)
−0.144369 + 0.989524i \(0.546115\pi\)
\(228\) −1.33178 −0.0881990
\(229\) 14.2691 0.942927 0.471464 0.881886i \(-0.343726\pi\)
0.471464 + 0.881886i \(0.343726\pi\)
\(230\) −2.94835 −0.194408
\(231\) 4.50053 0.296113
\(232\) −2.32720 −0.152788
\(233\) 1.10104 0.0721315 0.0360657 0.999349i \(-0.488517\pi\)
0.0360657 + 0.999349i \(0.488517\pi\)
\(234\) −2.02482 −0.132367
\(235\) −5.79256 −0.377865
\(236\) 23.7044 1.54303
\(237\) −16.1535 −1.04928
\(238\) 23.5907 1.52916
\(239\) 3.57602 0.231314 0.115657 0.993289i \(-0.463103\pi\)
0.115657 + 0.993289i \(0.463103\pi\)
\(240\) −1.11784 −0.0721563
\(241\) 23.8181 1.53426 0.767130 0.641492i \(-0.221684\pi\)
0.767130 + 0.641492i \(0.221684\pi\)
\(242\) 19.5239 1.25504
\(243\) −1.00000 −0.0641500
\(244\) 26.3188 1.68489
\(245\) −2.36943 −0.151378
\(246\) 8.38797 0.534797
\(247\) 0.395640 0.0251740
\(248\) 3.17972 0.201912
\(249\) −3.36143 −0.213022
\(250\) −24.3952 −1.54289
\(251\) −11.0028 −0.694489 −0.347245 0.937775i \(-0.612883\pi\)
−0.347245 + 0.937775i \(0.612883\pi\)
\(252\) −9.01086 −0.567631
\(253\) −1.51682 −0.0953614
\(254\) −5.42599 −0.340457
\(255\) 4.65395 0.291442
\(256\) −10.1076 −0.631727
\(257\) 22.5544 1.40691 0.703453 0.710742i \(-0.251640\pi\)
0.703453 + 0.710742i \(0.251640\pi\)
\(258\) −13.9978 −0.871465
\(259\) 32.4536 2.01657
\(260\) −3.59943 −0.223227
\(261\) 1.00000 0.0618984
\(262\) −8.95538 −0.553265
\(263\) 10.0124 0.617390 0.308695 0.951161i \(-0.400108\pi\)
0.308695 + 0.951161i \(0.400108\pi\)
\(264\) 3.52993 0.217252
\(265\) 14.0984 0.866056
\(266\) 2.92018 0.179048
\(267\) 7.47996 0.457766
\(268\) 3.11589 0.190333
\(269\) −12.9882 −0.791902 −0.395951 0.918272i \(-0.629585\pi\)
−0.395951 + 0.918272i \(0.629585\pi\)
\(270\) −2.94835 −0.179431
\(271\) 15.5385 0.943894 0.471947 0.881627i \(-0.343551\pi\)
0.471947 + 0.881627i \(0.343551\pi\)
\(272\) −3.01447 −0.182779
\(273\) 2.67692 0.162014
\(274\) −14.5644 −0.879868
\(275\) −4.96635 −0.299482
\(276\) 3.03693 0.182802
\(277\) 17.5683 1.05557 0.527787 0.849377i \(-0.323022\pi\)
0.527787 + 0.849377i \(0.323022\pi\)
\(278\) 42.5387 2.55130
\(279\) −1.36633 −0.0818001
\(280\) −9.07109 −0.542101
\(281\) −13.1371 −0.783693 −0.391846 0.920031i \(-0.628164\pi\)
−0.391846 + 0.920031i \(0.628164\pi\)
\(282\) 9.89596 0.589296
\(283\) −7.30957 −0.434509 −0.217254 0.976115i \(-0.569710\pi\)
−0.217254 + 0.976115i \(0.569710\pi\)
\(284\) −34.1710 −2.02768
\(285\) 0.576092 0.0341247
\(286\) −3.07128 −0.181609
\(287\) −11.0893 −0.654582
\(288\) 6.56410 0.386793
\(289\) −4.44975 −0.261750
\(290\) 2.94835 0.173133
\(291\) 15.6206 0.915694
\(292\) −8.75444 −0.512315
\(293\) −8.00780 −0.467821 −0.233910 0.972258i \(-0.575152\pi\)
−0.233910 + 0.972258i \(0.575152\pi\)
\(294\) 4.04792 0.236080
\(295\) −10.2539 −0.597007
\(296\) 25.4545 1.47951
\(297\) −1.51682 −0.0880145
\(298\) −51.3085 −2.97222
\(299\) −0.902202 −0.0521757
\(300\) 9.94352 0.574089
\(301\) 18.5058 1.06666
\(302\) −37.7117 −2.17006
\(303\) 3.63336 0.208731
\(304\) −0.373147 −0.0214015
\(305\) −11.3848 −0.651893
\(306\) −7.95077 −0.454515
\(307\) −30.5296 −1.74242 −0.871208 0.490915i \(-0.836663\pi\)
−0.871208 + 0.490915i \(0.836663\pi\)
\(308\) −13.6678 −0.778795
\(309\) −10.2810 −0.584867
\(310\) −4.02842 −0.228799
\(311\) 26.8315 1.52148 0.760738 0.649060i \(-0.224837\pi\)
0.760738 + 0.649060i \(0.224837\pi\)
\(312\) 2.09960 0.118867
\(313\) 7.67441 0.433783 0.216892 0.976196i \(-0.430408\pi\)
0.216892 + 0.976196i \(0.430408\pi\)
\(314\) −10.4694 −0.590822
\(315\) 3.89786 0.219620
\(316\) 49.0569 2.75967
\(317\) −11.3137 −0.635441 −0.317721 0.948184i \(-0.602917\pi\)
−0.317721 + 0.948184i \(0.602917\pi\)
\(318\) −24.0856 −1.35065
\(319\) 1.51682 0.0849253
\(320\) 17.1176 0.956901
\(321\) −7.31175 −0.408102
\(322\) −6.65908 −0.371096
\(323\) 1.55354 0.0864412
\(324\) 3.03693 0.168718
\(325\) −2.95399 −0.163858
\(326\) 13.7311 0.760493
\(327\) 3.37760 0.186782
\(328\) −8.69775 −0.480253
\(329\) −13.0830 −0.721287
\(330\) −4.47210 −0.246181
\(331\) 34.5401 1.89849 0.949247 0.314531i \(-0.101847\pi\)
0.949247 + 0.314531i \(0.101847\pi\)
\(332\) 10.2084 0.560261
\(333\) −10.9378 −0.599389
\(334\) −11.4787 −0.628088
\(335\) −1.34785 −0.0736411
\(336\) −2.52473 −0.137735
\(337\) 22.5432 1.22801 0.614004 0.789303i \(-0.289558\pi\)
0.614004 + 0.789303i \(0.289558\pi\)
\(338\) 27.3492 1.48760
\(339\) 2.02102 0.109767
\(340\) −14.1337 −0.766509
\(341\) −2.07247 −0.112231
\(342\) −0.984190 −0.0532189
\(343\) 15.4181 0.832498
\(344\) 14.5147 0.782583
\(345\) −1.31370 −0.0707271
\(346\) −23.6041 −1.26896
\(347\) 15.7661 0.846367 0.423184 0.906044i \(-0.360913\pi\)
0.423184 + 0.906044i \(0.360913\pi\)
\(348\) −3.03693 −0.162797
\(349\) −18.6754 −0.999673 −0.499836 0.866120i \(-0.666607\pi\)
−0.499836 + 0.866120i \(0.666607\pi\)
\(350\) −21.8031 −1.16543
\(351\) −0.902202 −0.0481560
\(352\) 9.95653 0.530685
\(353\) −22.1152 −1.17708 −0.588538 0.808470i \(-0.700296\pi\)
−0.588538 + 0.808470i \(0.700296\pi\)
\(354\) 17.5177 0.931057
\(355\) 14.7815 0.784521
\(356\) −22.7161 −1.20395
\(357\) 10.5113 0.556318
\(358\) −22.8499 −1.20765
\(359\) −31.7410 −1.67523 −0.837613 0.546264i \(-0.816049\pi\)
−0.837613 + 0.546264i \(0.816049\pi\)
\(360\) 3.05723 0.161130
\(361\) −18.8077 −0.989879
\(362\) −48.5770 −2.55315
\(363\) 8.69927 0.456593
\(364\) −8.12961 −0.426108
\(365\) 3.78694 0.198218
\(366\) 19.4497 1.01665
\(367\) 7.88341 0.411511 0.205755 0.978603i \(-0.434035\pi\)
0.205755 + 0.978603i \(0.434035\pi\)
\(368\) 0.850911 0.0443568
\(369\) 3.73744 0.194563
\(370\) −32.2485 −1.67652
\(371\) 31.8423 1.65317
\(372\) 4.14945 0.215139
\(373\) −19.3349 −1.00112 −0.500561 0.865701i \(-0.666873\pi\)
−0.500561 + 0.865701i \(0.666873\pi\)
\(374\) −12.0598 −0.623600
\(375\) −10.8698 −0.561314
\(376\) −10.2614 −0.529193
\(377\) 0.902202 0.0464658
\(378\) −6.65908 −0.342506
\(379\) −13.7522 −0.706405 −0.353202 0.935547i \(-0.614907\pi\)
−0.353202 + 0.935547i \(0.614907\pi\)
\(380\) −1.74955 −0.0897500
\(381\) −2.41766 −0.123861
\(382\) −38.6289 −1.97642
\(383\) 9.78378 0.499928 0.249964 0.968255i \(-0.419581\pi\)
0.249964 + 0.968255i \(0.419581\pi\)
\(384\) −16.1153 −0.822381
\(385\) 5.91234 0.301321
\(386\) 17.9024 0.911206
\(387\) −6.23701 −0.317045
\(388\) −47.4386 −2.40833
\(389\) 20.3246 1.03050 0.515250 0.857040i \(-0.327699\pi\)
0.515250 + 0.857040i \(0.327699\pi\)
\(390\) −2.66000 −0.134695
\(391\) −3.54263 −0.179159
\(392\) −4.19742 −0.212002
\(393\) −3.99026 −0.201282
\(394\) −11.5349 −0.581120
\(395\) −21.2208 −1.06773
\(396\) 4.60646 0.231483
\(397\) −15.0836 −0.757024 −0.378512 0.925596i \(-0.623564\pi\)
−0.378512 + 0.925596i \(0.623564\pi\)
\(398\) −49.2628 −2.46932
\(399\) 1.30115 0.0651390
\(400\) 2.78605 0.139303
\(401\) 31.7667 1.58635 0.793176 0.608993i \(-0.208426\pi\)
0.793176 + 0.608993i \(0.208426\pi\)
\(402\) 2.30266 0.114846
\(403\) −1.23271 −0.0614055
\(404\) −11.0343 −0.548976
\(405\) −1.31370 −0.0652782
\(406\) 6.65908 0.330484
\(407\) −16.5907 −0.822369
\(408\) 8.24440 0.408159
\(409\) −2.67330 −0.132186 −0.0660931 0.997813i \(-0.521053\pi\)
−0.0660931 + 0.997813i \(0.521053\pi\)
\(410\) 11.0193 0.544202
\(411\) −6.48948 −0.320102
\(412\) 31.2228 1.53824
\(413\) −23.1593 −1.13960
\(414\) 2.24431 0.110302
\(415\) −4.41591 −0.216768
\(416\) 5.92215 0.290357
\(417\) 18.9540 0.928183
\(418\) −1.49283 −0.0730169
\(419\) 28.1776 1.37657 0.688284 0.725442i \(-0.258364\pi\)
0.688284 + 0.725442i \(0.258364\pi\)
\(420\) −11.8375 −0.577613
\(421\) 27.5851 1.34441 0.672207 0.740363i \(-0.265346\pi\)
0.672207 + 0.740363i \(0.265346\pi\)
\(422\) −15.7111 −0.764803
\(423\) 4.40935 0.214390
\(424\) 24.9751 1.21290
\(425\) −11.5993 −0.562648
\(426\) −25.2526 −1.22349
\(427\) −25.7135 −1.24437
\(428\) 22.2053 1.07333
\(429\) −1.36847 −0.0660706
\(430\) −18.3889 −0.886790
\(431\) 5.24056 0.252429 0.126215 0.992003i \(-0.459717\pi\)
0.126215 + 0.992003i \(0.459717\pi\)
\(432\) 0.850911 0.0409395
\(433\) −21.8854 −1.05174 −0.525872 0.850564i \(-0.676261\pi\)
−0.525872 + 0.850564i \(0.676261\pi\)
\(434\) −9.09851 −0.436742
\(435\) 1.31370 0.0629870
\(436\) −10.2575 −0.491247
\(437\) −0.438527 −0.0209776
\(438\) −6.46958 −0.309129
\(439\) −13.9271 −0.664702 −0.332351 0.943156i \(-0.607842\pi\)
−0.332351 + 0.943156i \(0.607842\pi\)
\(440\) 4.63726 0.221072
\(441\) 1.80364 0.0858875
\(442\) −7.17320 −0.341194
\(443\) −27.5227 −1.30764 −0.653821 0.756649i \(-0.726835\pi\)
−0.653821 + 0.756649i \(0.726835\pi\)
\(444\) 33.2174 1.57643
\(445\) 9.82640 0.465816
\(446\) −43.3613 −2.05322
\(447\) −22.8616 −1.08132
\(448\) 38.6614 1.82658
\(449\) 35.1874 1.66060 0.830298 0.557320i \(-0.188170\pi\)
0.830298 + 0.557320i \(0.188170\pi\)
\(450\) 7.34832 0.346403
\(451\) 5.66900 0.266943
\(452\) −6.13771 −0.288694
\(453\) −16.8032 −0.789484
\(454\) 9.76336 0.458217
\(455\) 3.51666 0.164864
\(456\) 1.02054 0.0477911
\(457\) −18.3507 −0.858409 −0.429205 0.903207i \(-0.641206\pi\)
−0.429205 + 0.903207i \(0.641206\pi\)
\(458\) −32.0242 −1.49639
\(459\) −3.54263 −0.165356
\(460\) 3.98961 0.186016
\(461\) −24.7369 −1.15211 −0.576056 0.817410i \(-0.695409\pi\)
−0.576056 + 0.817410i \(0.695409\pi\)
\(462\) −10.1006 −0.469922
\(463\) 12.6364 0.587264 0.293632 0.955919i \(-0.405136\pi\)
0.293632 + 0.955919i \(0.405136\pi\)
\(464\) −0.850911 −0.0395026
\(465\) −1.79495 −0.0832386
\(466\) −2.47107 −0.114470
\(467\) −20.5427 −0.950603 −0.475301 0.879823i \(-0.657661\pi\)
−0.475301 + 0.879823i \(0.657661\pi\)
\(468\) 2.73993 0.126653
\(469\) −3.04424 −0.140570
\(470\) 13.0003 0.599659
\(471\) −4.66486 −0.214945
\(472\) −18.1647 −0.836097
\(473\) −9.46039 −0.434989
\(474\) 36.2534 1.66517
\(475\) −1.43582 −0.0658801
\(476\) −31.9222 −1.46315
\(477\) −10.7318 −0.491376
\(478\) −8.02570 −0.367087
\(479\) −14.3319 −0.654842 −0.327421 0.944879i \(-0.606179\pi\)
−0.327421 + 0.944879i \(0.606179\pi\)
\(480\) 8.62324 0.393595
\(481\) −9.86814 −0.449948
\(482\) −53.4553 −2.43482
\(483\) −2.96709 −0.135007
\(484\) −26.4191 −1.20087
\(485\) 20.5207 0.931797
\(486\) 2.24431 0.101804
\(487\) −6.62111 −0.300031 −0.150016 0.988684i \(-0.547932\pi\)
−0.150016 + 0.988684i \(0.547932\pi\)
\(488\) −20.1680 −0.912964
\(489\) 6.11817 0.276673
\(490\) 5.31775 0.240231
\(491\) −16.2298 −0.732440 −0.366220 0.930528i \(-0.619348\pi\)
−0.366220 + 0.930528i \(0.619348\pi\)
\(492\) −11.3503 −0.511713
\(493\) 3.54263 0.159552
\(494\) −0.887939 −0.0399503
\(495\) −1.99264 −0.0895624
\(496\) 1.16263 0.0522035
\(497\) 33.3852 1.49753
\(498\) 7.54410 0.338059
\(499\) −11.1031 −0.497043 −0.248521 0.968626i \(-0.579945\pi\)
−0.248521 + 0.968626i \(0.579945\pi\)
\(500\) 33.0108 1.47629
\(501\) −5.11459 −0.228503
\(502\) 24.6937 1.10213
\(503\) 7.73894 0.345062 0.172531 0.985004i \(-0.444805\pi\)
0.172531 + 0.985004i \(0.444805\pi\)
\(504\) 6.90501 0.307573
\(505\) 4.77314 0.212402
\(506\) 3.40420 0.151335
\(507\) 12.1860 0.541201
\(508\) 7.34227 0.325761
\(509\) 12.1628 0.539105 0.269553 0.962986i \(-0.413124\pi\)
0.269553 + 0.962986i \(0.413124\pi\)
\(510\) −10.4449 −0.462508
\(511\) 8.55312 0.378368
\(512\) −9.54594 −0.421875
\(513\) −0.438527 −0.0193614
\(514\) −50.6191 −2.23271
\(515\) −13.5062 −0.595152
\(516\) 18.9414 0.833848
\(517\) 6.68817 0.294146
\(518\) −72.8359 −3.20022
\(519\) −10.5173 −0.461658
\(520\) 2.75824 0.120957
\(521\) −19.3594 −0.848152 −0.424076 0.905627i \(-0.639401\pi\)
−0.424076 + 0.905627i \(0.639401\pi\)
\(522\) −2.24431 −0.0982308
\(523\) 4.25739 0.186163 0.0930814 0.995659i \(-0.470328\pi\)
0.0930814 + 0.995659i \(0.470328\pi\)
\(524\) 12.1181 0.529383
\(525\) −9.71485 −0.423991
\(526\) −22.4709 −0.979777
\(527\) −4.84041 −0.210852
\(528\) 1.29068 0.0561694
\(529\) 1.00000 0.0434783
\(530\) −31.6411 −1.37440
\(531\) 7.80539 0.338725
\(532\) −3.95150 −0.171319
\(533\) 3.37192 0.146054
\(534\) −16.7873 −0.726459
\(535\) −9.60543 −0.415279
\(536\) −2.38770 −0.103133
\(537\) −10.1812 −0.439353
\(538\) 29.1495 1.25672
\(539\) 2.73579 0.117839
\(540\) 3.98961 0.171685
\(541\) 16.3992 0.705057 0.352528 0.935801i \(-0.385322\pi\)
0.352528 + 0.935801i \(0.385322\pi\)
\(542\) −34.8731 −1.49793
\(543\) −21.6445 −0.928855
\(544\) 23.2542 0.997016
\(545\) 4.43714 0.190066
\(546\) −6.00784 −0.257112
\(547\) −16.7707 −0.717065 −0.358533 0.933517i \(-0.616723\pi\)
−0.358533 + 0.933517i \(0.616723\pi\)
\(548\) 19.7081 0.841889
\(549\) 8.66624 0.369866
\(550\) 11.1460 0.475269
\(551\) 0.438527 0.0186819
\(552\) −2.32720 −0.0990520
\(553\) −47.9288 −2.03814
\(554\) −39.4286 −1.67516
\(555\) −14.3690 −0.609930
\(556\) −57.5621 −2.44118
\(557\) 36.1688 1.53252 0.766261 0.642530i \(-0.222115\pi\)
0.766261 + 0.642530i \(0.222115\pi\)
\(558\) 3.06647 0.129814
\(559\) −5.62705 −0.237999
\(560\) −3.31674 −0.140158
\(561\) −5.37352 −0.226870
\(562\) 29.4837 1.24370
\(563\) 15.4439 0.650883 0.325441 0.945562i \(-0.394487\pi\)
0.325441 + 0.945562i \(0.394487\pi\)
\(564\) −13.3909 −0.563859
\(565\) 2.65501 0.111697
\(566\) 16.4049 0.689551
\(567\) −2.96709 −0.124606
\(568\) 26.1852 1.09871
\(569\) −18.1051 −0.759007 −0.379503 0.925190i \(-0.623905\pi\)
−0.379503 + 0.925190i \(0.623905\pi\)
\(570\) −1.29293 −0.0541548
\(571\) 40.4246 1.69171 0.845857 0.533409i \(-0.179089\pi\)
0.845857 + 0.533409i \(0.179089\pi\)
\(572\) 4.15596 0.173769
\(573\) −17.2119 −0.719038
\(574\) 24.8879 1.03880
\(575\) 3.27420 0.136544
\(576\) −13.0301 −0.542919
\(577\) −5.89094 −0.245243 −0.122621 0.992454i \(-0.539130\pi\)
−0.122621 + 0.992454i \(0.539130\pi\)
\(578\) 9.98663 0.415389
\(579\) 7.97678 0.331504
\(580\) −3.98961 −0.165660
\(581\) −9.97368 −0.413778
\(582\) −35.0574 −1.45318
\(583\) −16.2782 −0.674174
\(584\) 6.70851 0.277600
\(585\) −1.18522 −0.0490029
\(586\) 17.9720 0.742416
\(587\) −13.3990 −0.553036 −0.276518 0.961009i \(-0.589181\pi\)
−0.276518 + 0.961009i \(0.589181\pi\)
\(588\) −5.47752 −0.225889
\(589\) −0.599173 −0.0246885
\(590\) 23.0130 0.947430
\(591\) −5.13962 −0.211416
\(592\) 9.30713 0.382521
\(593\) 7.11166 0.292041 0.146020 0.989282i \(-0.453353\pi\)
0.146020 + 0.989282i \(0.453353\pi\)
\(594\) 3.40420 0.139676
\(595\) 13.8087 0.566101
\(596\) 69.4291 2.84393
\(597\) −21.9501 −0.898356
\(598\) 2.02482 0.0828011
\(599\) 0.867674 0.0354522 0.0177261 0.999843i \(-0.494357\pi\)
0.0177261 + 0.999843i \(0.494357\pi\)
\(600\) −7.61970 −0.311073
\(601\) 2.04317 0.0833424 0.0416712 0.999131i \(-0.486732\pi\)
0.0416712 + 0.999131i \(0.486732\pi\)
\(602\) −41.5327 −1.69275
\(603\) 1.02600 0.0417819
\(604\) 51.0302 2.07639
\(605\) 11.4282 0.464623
\(606\) −8.15440 −0.331250
\(607\) 45.7094 1.85529 0.927644 0.373465i \(-0.121830\pi\)
0.927644 + 0.373465i \(0.121830\pi\)
\(608\) 2.87853 0.116740
\(609\) 2.96709 0.120233
\(610\) 25.5511 1.03453
\(611\) 3.97813 0.160938
\(612\) 10.7587 0.434896
\(613\) 21.0161 0.848831 0.424416 0.905467i \(-0.360480\pi\)
0.424416 + 0.905467i \(0.360480\pi\)
\(614\) 68.5179 2.76516
\(615\) 4.90986 0.197985
\(616\) 10.4736 0.421994
\(617\) −16.5056 −0.664489 −0.332244 0.943193i \(-0.607806\pi\)
−0.332244 + 0.943193i \(0.607806\pi\)
\(618\) 23.0738 0.928165
\(619\) 30.3608 1.22030 0.610152 0.792284i \(-0.291108\pi\)
0.610152 + 0.792284i \(0.291108\pi\)
\(620\) 5.45113 0.218923
\(621\) 1.00000 0.0401286
\(622\) −60.2182 −2.41453
\(623\) 22.1937 0.889173
\(624\) 0.767694 0.0307324
\(625\) 2.09137 0.0836548
\(626\) −17.2238 −0.688400
\(627\) −0.665164 −0.0265641
\(628\) 14.1669 0.565319
\(629\) −38.7487 −1.54501
\(630\) −8.74801 −0.348529
\(631\) 0.182379 0.00726040 0.00363020 0.999993i \(-0.498844\pi\)
0.00363020 + 0.999993i \(0.498844\pi\)
\(632\) −37.5923 −1.49534
\(633\) −7.00040 −0.278241
\(634\) 25.3915 1.00842
\(635\) −3.17608 −0.126039
\(636\) 32.5918 1.29235
\(637\) 1.62725 0.0644738
\(638\) −3.40420 −0.134774
\(639\) −11.2518 −0.445116
\(640\) −21.1706 −0.836843
\(641\) −7.86588 −0.310684 −0.155342 0.987861i \(-0.549648\pi\)
−0.155342 + 0.987861i \(0.549648\pi\)
\(642\) 16.4098 0.647645
\(643\) 11.3455 0.447423 0.223711 0.974655i \(-0.428183\pi\)
0.223711 + 0.974655i \(0.428183\pi\)
\(644\) 9.01086 0.355077
\(645\) −8.19355 −0.322621
\(646\) −3.48663 −0.137179
\(647\) 22.3414 0.878330 0.439165 0.898406i \(-0.355274\pi\)
0.439165 + 0.898406i \(0.355274\pi\)
\(648\) −2.32720 −0.0914209
\(649\) 11.8393 0.464735
\(650\) 6.62967 0.260037
\(651\) −4.05403 −0.158890
\(652\) −18.5804 −0.727667
\(653\) 39.1961 1.53386 0.766931 0.641729i \(-0.221783\pi\)
0.766931 + 0.641729i \(0.221783\pi\)
\(654\) −7.58038 −0.296416
\(655\) −5.24199 −0.204821
\(656\) −3.18023 −0.124167
\(657\) −2.88266 −0.112463
\(658\) 29.3622 1.14466
\(659\) −15.0318 −0.585557 −0.292778 0.956180i \(-0.594580\pi\)
−0.292778 + 0.956180i \(0.594580\pi\)
\(660\) 6.05150 0.235554
\(661\) −39.2171 −1.52537 −0.762685 0.646770i \(-0.776119\pi\)
−0.762685 + 0.646770i \(0.776119\pi\)
\(662\) −77.5187 −3.01285
\(663\) −3.19617 −0.124129
\(664\) −7.82271 −0.303580
\(665\) 1.70932 0.0662845
\(666\) 24.5479 0.951212
\(667\) −1.00000 −0.0387202
\(668\) 15.5327 0.600976
\(669\) −19.3205 −0.746975
\(670\) 3.02500 0.116866
\(671\) 13.1451 0.507460
\(672\) 19.4763 0.751314
\(673\) 11.9758 0.461635 0.230818 0.972997i \(-0.425860\pi\)
0.230818 + 0.972997i \(0.425860\pi\)
\(674\) −50.5941 −1.94881
\(675\) 3.27420 0.126024
\(676\) −37.0081 −1.42339
\(677\) 22.1732 0.852185 0.426093 0.904680i \(-0.359890\pi\)
0.426093 + 0.904680i \(0.359890\pi\)
\(678\) −4.53581 −0.174197
\(679\) 46.3477 1.77866
\(680\) 10.8306 0.415336
\(681\) 4.35027 0.166703
\(682\) 4.65127 0.178106
\(683\) −3.66223 −0.140131 −0.0700657 0.997542i \(-0.522321\pi\)
−0.0700657 + 0.997542i \(0.522321\pi\)
\(684\) 1.33178 0.0509217
\(685\) −8.52522 −0.325732
\(686\) −34.6030 −1.32115
\(687\) −14.2691 −0.544399
\(688\) 5.30715 0.202333
\(689\) −9.68228 −0.368865
\(690\) 2.94835 0.112242
\(691\) 14.7825 0.562354 0.281177 0.959656i \(-0.409275\pi\)
0.281177 + 0.959656i \(0.409275\pi\)
\(692\) 31.9403 1.21419
\(693\) −4.50053 −0.170961
\(694\) −35.3840 −1.34316
\(695\) 24.8999 0.944505
\(696\) 2.32720 0.0882121
\(697\) 13.2404 0.501515
\(698\) 41.9135 1.58645
\(699\) −1.10104 −0.0416451
\(700\) 29.5033 1.11512
\(701\) 19.8395 0.749328 0.374664 0.927161i \(-0.377758\pi\)
0.374664 + 0.927161i \(0.377758\pi\)
\(702\) 2.02482 0.0764220
\(703\) −4.79653 −0.180905
\(704\) −19.7642 −0.744891
\(705\) 5.79256 0.218160
\(706\) 49.6335 1.86798
\(707\) 10.7805 0.405443
\(708\) −23.7044 −0.890867
\(709\) 17.7266 0.665735 0.332868 0.942974i \(-0.391984\pi\)
0.332868 + 0.942974i \(0.391984\pi\)
\(710\) −33.1743 −1.24501
\(711\) 16.1535 0.605802
\(712\) 17.4073 0.652367
\(713\) 1.36633 0.0511695
\(714\) −23.5907 −0.882859
\(715\) −1.79776 −0.0672325
\(716\) 30.9197 1.15552
\(717\) −3.57602 −0.133549
\(718\) 71.2367 2.65853
\(719\) 18.0624 0.673612 0.336806 0.941574i \(-0.390653\pi\)
0.336806 + 0.941574i \(0.390653\pi\)
\(720\) 1.11784 0.0416595
\(721\) −30.5048 −1.13606
\(722\) 42.2103 1.57091
\(723\) −23.8181 −0.885805
\(724\) 65.7329 2.44294
\(725\) −3.27420 −0.121601
\(726\) −19.5239 −0.724599
\(727\) −20.3044 −0.753048 −0.376524 0.926407i \(-0.622881\pi\)
−0.376524 + 0.926407i \(0.622881\pi\)
\(728\) 6.22971 0.230888
\(729\) 1.00000 0.0370370
\(730\) −8.49908 −0.314565
\(731\) −22.0954 −0.817229
\(732\) −26.3188 −0.972770
\(733\) −6.63182 −0.244952 −0.122476 0.992471i \(-0.539083\pi\)
−0.122476 + 0.992471i \(0.539083\pi\)
\(734\) −17.6928 −0.653054
\(735\) 2.36943 0.0873979
\(736\) −6.56410 −0.241956
\(737\) 1.55625 0.0573253
\(738\) −8.38797 −0.308765
\(739\) 26.4937 0.974588 0.487294 0.873238i \(-0.337984\pi\)
0.487294 + 0.873238i \(0.337984\pi\)
\(740\) 43.6377 1.60415
\(741\) −0.395640 −0.0145342
\(742\) −71.4641 −2.62353
\(743\) −41.8653 −1.53589 −0.767945 0.640516i \(-0.778721\pi\)
−0.767945 + 0.640516i \(0.778721\pi\)
\(744\) −3.17972 −0.116574
\(745\) −30.0332 −1.10033
\(746\) 43.3935 1.58875
\(747\) 3.36143 0.122988
\(748\) 16.3190 0.596682
\(749\) −21.6946 −0.792705
\(750\) 24.3952 0.890787
\(751\) −13.9711 −0.509811 −0.254905 0.966966i \(-0.582044\pi\)
−0.254905 + 0.966966i \(0.582044\pi\)
\(752\) −3.75197 −0.136820
\(753\) 11.0028 0.400964
\(754\) −2.02482 −0.0737397
\(755\) −22.0744 −0.803368
\(756\) 9.01086 0.327722
\(757\) 1.79857 0.0653703 0.0326852 0.999466i \(-0.489594\pi\)
0.0326852 + 0.999466i \(0.489594\pi\)
\(758\) 30.8643 1.12104
\(759\) 1.51682 0.0550569
\(760\) 1.34068 0.0486315
\(761\) −22.2173 −0.805378 −0.402689 0.915337i \(-0.631924\pi\)
−0.402689 + 0.915337i \(0.631924\pi\)
\(762\) 5.42599 0.196563
\(763\) 10.0216 0.362808
\(764\) 52.2714 1.89111
\(765\) −4.65395 −0.168264
\(766\) −21.9578 −0.793369
\(767\) 7.04204 0.254273
\(768\) 10.1076 0.364728
\(769\) −41.3008 −1.48934 −0.744672 0.667431i \(-0.767394\pi\)
−0.744672 + 0.667431i \(0.767394\pi\)
\(770\) −13.2691 −0.478186
\(771\) −22.5544 −0.812278
\(772\) −24.2249 −0.871874
\(773\) −19.5525 −0.703254 −0.351627 0.936140i \(-0.614371\pi\)
−0.351627 + 0.936140i \(0.614371\pi\)
\(774\) 13.9978 0.503140
\(775\) 4.47364 0.160698
\(776\) 36.3521 1.30497
\(777\) −32.4536 −1.16426
\(778\) −45.6148 −1.63537
\(779\) 1.63897 0.0587221
\(780\) 3.59943 0.128880
\(781\) −17.0670 −0.610704
\(782\) 7.95077 0.284319
\(783\) −1.00000 −0.0357371
\(784\) −1.53474 −0.0548120
\(785\) −6.12821 −0.218725
\(786\) 8.95538 0.319428
\(787\) 13.6255 0.485695 0.242848 0.970064i \(-0.421919\pi\)
0.242848 + 0.970064i \(0.421919\pi\)
\(788\) 15.6087 0.556036
\(789\) −10.0124 −0.356450
\(790\) 47.6260 1.69446
\(791\) 5.99656 0.213213
\(792\) −3.52993 −0.125430
\(793\) 7.81870 0.277650
\(794\) 33.8523 1.20137
\(795\) −14.0984 −0.500018
\(796\) 66.6608 2.36273
\(797\) 14.3368 0.507836 0.253918 0.967226i \(-0.418281\pi\)
0.253918 + 0.967226i \(0.418281\pi\)
\(798\) −2.92018 −0.103373
\(799\) 15.6207 0.552621
\(800\) −21.4922 −0.759863
\(801\) −7.47996 −0.264291
\(802\) −71.2943 −2.51749
\(803\) −4.37246 −0.154301
\(804\) −3.11589 −0.109889
\(805\) −3.89786 −0.137382
\(806\) 2.76658 0.0974486
\(807\) 12.9882 0.457205
\(808\) 8.45555 0.297465
\(809\) −45.8588 −1.61231 −0.806155 0.591705i \(-0.798455\pi\)
−0.806155 + 0.591705i \(0.798455\pi\)
\(810\) 2.94835 0.103594
\(811\) 13.4624 0.472728 0.236364 0.971665i \(-0.424044\pi\)
0.236364 + 0.971665i \(0.424044\pi\)
\(812\) −9.01086 −0.316219
\(813\) −15.5385 −0.544958
\(814\) 37.2346 1.30507
\(815\) 8.03742 0.281539
\(816\) 3.01447 0.105527
\(817\) −2.73510 −0.0956889
\(818\) 5.99972 0.209775
\(819\) −2.67692 −0.0935391
\(820\) −14.9109 −0.520712
\(821\) 48.9229 1.70742 0.853711 0.520748i \(-0.174347\pi\)
0.853711 + 0.520748i \(0.174347\pi\)
\(822\) 14.5644 0.507992
\(823\) 6.18455 0.215580 0.107790 0.994174i \(-0.465623\pi\)
0.107790 + 0.994174i \(0.465623\pi\)
\(824\) −23.9260 −0.833500
\(825\) 4.96635 0.172906
\(826\) 51.9767 1.80850
\(827\) −21.6665 −0.753417 −0.376709 0.926332i \(-0.622944\pi\)
−0.376709 + 0.926332i \(0.622944\pi\)
\(828\) −3.03693 −0.105541
\(829\) −51.9818 −1.80540 −0.902702 0.430267i \(-0.858420\pi\)
−0.902702 + 0.430267i \(0.858420\pi\)
\(830\) 9.91066 0.344004
\(831\) −17.5683 −0.609436
\(832\) −11.7558 −0.407557
\(833\) 6.38963 0.221387
\(834\) −42.5387 −1.47300
\(835\) −6.71902 −0.232521
\(836\) 2.02006 0.0698652
\(837\) 1.36633 0.0472273
\(838\) −63.2394 −2.18457
\(839\) −29.1361 −1.00589 −0.502946 0.864318i \(-0.667750\pi\)
−0.502946 + 0.864318i \(0.667750\pi\)
\(840\) 9.07109 0.312982
\(841\) 1.00000 0.0344828
\(842\) −61.9095 −2.13354
\(843\) 13.1371 0.452465
\(844\) 21.2597 0.731790
\(845\) 16.0088 0.550718
\(846\) −9.89596 −0.340230
\(847\) 25.8115 0.886895
\(848\) 9.13183 0.313588
\(849\) 7.30957 0.250864
\(850\) 26.0324 0.892904
\(851\) 10.9378 0.374944
\(852\) 34.1710 1.17068
\(853\) −6.26515 −0.214515 −0.107257 0.994231i \(-0.534207\pi\)
−0.107257 + 0.994231i \(0.534207\pi\)
\(854\) 57.7092 1.97477
\(855\) −0.576092 −0.0197019
\(856\) −17.0159 −0.581591
\(857\) 32.0046 1.09326 0.546628 0.837376i \(-0.315911\pi\)
0.546628 + 0.837376i \(0.315911\pi\)
\(858\) 3.07128 0.104852
\(859\) −2.27931 −0.0777690 −0.0388845 0.999244i \(-0.512380\pi\)
−0.0388845 + 0.999244i \(0.512380\pi\)
\(860\) 24.8832 0.848511
\(861\) 11.0893 0.377923
\(862\) −11.7615 −0.400597
\(863\) 23.3854 0.796048 0.398024 0.917375i \(-0.369696\pi\)
0.398024 + 0.917375i \(0.369696\pi\)
\(864\) −6.56410 −0.223315
\(865\) −13.8165 −0.469776
\(866\) 49.1176 1.66908
\(867\) 4.44975 0.151122
\(868\) 12.3118 0.417890
\(869\) 24.5018 0.831167
\(870\) −2.94835 −0.0999583
\(871\) 0.925659 0.0313648
\(872\) 7.86033 0.266185
\(873\) −15.6206 −0.528676
\(874\) 0.984190 0.0332907
\(875\) −32.2517 −1.09031
\(876\) 8.75444 0.295785
\(877\) −37.0607 −1.25145 −0.625725 0.780044i \(-0.715197\pi\)
−0.625725 + 0.780044i \(0.715197\pi\)
\(878\) 31.2566 1.05486
\(879\) 8.00780 0.270096
\(880\) 1.69556 0.0571572
\(881\) 20.0136 0.674274 0.337137 0.941456i \(-0.390541\pi\)
0.337137 + 0.941456i \(0.390541\pi\)
\(882\) −4.04792 −0.136301
\(883\) 32.4866 1.09326 0.546631 0.837374i \(-0.315910\pi\)
0.546631 + 0.837374i \(0.315910\pi\)
\(884\) 9.70655 0.326467
\(885\) 10.2539 0.344682
\(886\) 61.7695 2.07519
\(887\) 19.3042 0.648170 0.324085 0.946028i \(-0.394944\pi\)
0.324085 + 0.946028i \(0.394944\pi\)
\(888\) −25.4545 −0.854196
\(889\) −7.17343 −0.240589
\(890\) −22.0535 −0.739235
\(891\) 1.51682 0.0508152
\(892\) 58.6751 1.96459
\(893\) 1.93362 0.0647061
\(894\) 51.3085 1.71601
\(895\) −13.3751 −0.447079
\(896\) −47.8156 −1.59741
\(897\) 0.902202 0.0301237
\(898\) −78.9715 −2.63531
\(899\) −1.36633 −0.0455697
\(900\) −9.94352 −0.331451
\(901\) −38.0189 −1.26659
\(902\) −12.7230 −0.423630
\(903\) −18.5058 −0.615834
\(904\) 4.70332 0.156430
\(905\) −28.4343 −0.945189
\(906\) 37.7117 1.25289
\(907\) −10.6991 −0.355259 −0.177629 0.984097i \(-0.556843\pi\)
−0.177629 + 0.984097i \(0.556843\pi\)
\(908\) −13.2115 −0.438438
\(909\) −3.63336 −0.120511
\(910\) −7.89248 −0.261633
\(911\) 30.1038 0.997384 0.498692 0.866779i \(-0.333814\pi\)
0.498692 + 0.866779i \(0.333814\pi\)
\(912\) 0.373147 0.0123561
\(913\) 5.09867 0.168741
\(914\) 41.1847 1.36227
\(915\) 11.3848 0.376370
\(916\) 43.3342 1.43180
\(917\) −11.8395 −0.390973
\(918\) 7.95077 0.262414
\(919\) 7.97055 0.262924 0.131462 0.991321i \(-0.458033\pi\)
0.131462 + 0.991321i \(0.458033\pi\)
\(920\) −3.05723 −0.100794
\(921\) 30.5296 1.00598
\(922\) 55.5173 1.82837
\(923\) −10.1514 −0.334138
\(924\) 13.6678 0.449638
\(925\) 35.8126 1.17751
\(926\) −28.3600 −0.931969
\(927\) 10.2810 0.337673
\(928\) 6.56410 0.215477
\(929\) 45.5813 1.49548 0.747738 0.663994i \(-0.231140\pi\)
0.747738 + 0.663994i \(0.231140\pi\)
\(930\) 4.02842 0.132097
\(931\) 0.790944 0.0259221
\(932\) 3.34378 0.109529
\(933\) −26.8315 −0.878424
\(934\) 46.1042 1.50858
\(935\) −7.05918 −0.230860
\(936\) −2.09960 −0.0686276
\(937\) 42.9955 1.40460 0.702301 0.711880i \(-0.252156\pi\)
0.702301 + 0.711880i \(0.252156\pi\)
\(938\) 6.83221 0.223080
\(939\) −7.67441 −0.250445
\(940\) −17.5916 −0.573775
\(941\) −22.3893 −0.729870 −0.364935 0.931033i \(-0.618909\pi\)
−0.364935 + 0.931033i \(0.618909\pi\)
\(942\) 10.4694 0.341111
\(943\) −3.73744 −0.121708
\(944\) −6.64170 −0.216169
\(945\) −3.89786 −0.126797
\(946\) 21.2321 0.690314
\(947\) −14.3161 −0.465211 −0.232605 0.972571i \(-0.574725\pi\)
−0.232605 + 0.972571i \(0.574725\pi\)
\(948\) −49.0569 −1.59330
\(949\) −2.60074 −0.0844237
\(950\) 3.22243 0.104550
\(951\) 11.3137 0.366872
\(952\) 24.4619 0.792815
\(953\) 46.4935 1.50607 0.753036 0.657979i \(-0.228589\pi\)
0.753036 + 0.657979i \(0.228589\pi\)
\(954\) 24.0856 0.779798
\(955\) −22.6112 −0.731682
\(956\) 10.8601 0.351242
\(957\) −1.51682 −0.0490317
\(958\) 32.1653 1.03921
\(959\) −19.2549 −0.621773
\(960\) −17.1176 −0.552467
\(961\) −29.1331 −0.939779
\(962\) 22.1472 0.714053
\(963\) 7.31175 0.235618
\(964\) 72.3340 2.32972
\(965\) 10.4791 0.337333
\(966\) 6.65908 0.214252
\(967\) −6.58785 −0.211851 −0.105926 0.994374i \(-0.533781\pi\)
−0.105926 + 0.994374i \(0.533781\pi\)
\(968\) 20.2449 0.650696
\(969\) −1.55354 −0.0499069
\(970\) −46.0548 −1.47873
\(971\) 7.43166 0.238493 0.119247 0.992865i \(-0.461952\pi\)
0.119247 + 0.992865i \(0.461952\pi\)
\(972\) −3.03693 −0.0974096
\(973\) 56.2383 1.80292
\(974\) 14.8598 0.476140
\(975\) 2.95399 0.0946034
\(976\) −7.37420 −0.236042
\(977\) 36.1933 1.15793 0.578963 0.815354i \(-0.303458\pi\)
0.578963 + 0.815354i \(0.303458\pi\)
\(978\) −13.7311 −0.439071
\(979\) −11.3457 −0.362610
\(980\) −7.19581 −0.229862
\(981\) −3.37760 −0.107838
\(982\) 36.4247 1.16236
\(983\) 25.3261 0.807779 0.403889 0.914808i \(-0.367658\pi\)
0.403889 + 0.914808i \(0.367658\pi\)
\(984\) 8.69775 0.277274
\(985\) −6.75191 −0.215134
\(986\) −7.95077 −0.253204
\(987\) 13.0830 0.416435
\(988\) 1.20153 0.0382258
\(989\) 6.23701 0.198325
\(990\) 4.47210 0.142133
\(991\) −23.1884 −0.736603 −0.368302 0.929706i \(-0.620061\pi\)
−0.368302 + 0.929706i \(0.620061\pi\)
\(992\) −8.96874 −0.284758
\(993\) −34.5401 −1.09610
\(994\) −74.9268 −2.37654
\(995\) −28.8357 −0.914154
\(996\) −10.2084 −0.323467
\(997\) 7.28351 0.230671 0.115336 0.993327i \(-0.463206\pi\)
0.115336 + 0.993327i \(0.463206\pi\)
\(998\) 24.9188 0.788791
\(999\) 10.9378 0.346058
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 2001.2.a.k.1.2 10
3.2 odd 2 6003.2.a.k.1.9 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2001.2.a.k.1.2 10 1.1 even 1 trivial
6003.2.a.k.1.9 10 3.2 odd 2