Properties

Label 4010.2.a.m.1.3
Level $4010$
Weight $2$
Character 4010.1
Self dual yes
Analytic conductor $32.020$
Analytic rank $0$
Dimension $20$
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] = [4010,2,Mod(1,4010)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4010, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4010.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4010 = 2 \cdot 5 \cdot 401 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4010.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: \(32.0200112105\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 4 x^{19} - 35 x^{18} + 154 x^{17} + 460 x^{16} - 2392 x^{15} - 2591 x^{14} + 19157 x^{13} + 2776 x^{12} - 83577 x^{11} + 34362 x^{10} + 190617 x^{9} - 150697 x^{8} + \cdots + 104 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.78165\) of defining polynomial
Character \(\chi\) \(=\) 4010.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.78165 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.78165 q^{6} -3.35211 q^{7} -1.00000 q^{8} +4.73759 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.78165 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.78165 q^{6} -3.35211 q^{7} -1.00000 q^{8} +4.73759 q^{9} -1.00000 q^{10} +3.90452 q^{11} -2.78165 q^{12} -4.76252 q^{13} +3.35211 q^{14} -2.78165 q^{15} +1.00000 q^{16} -4.89438 q^{17} -4.73759 q^{18} -2.47209 q^{19} +1.00000 q^{20} +9.32441 q^{21} -3.90452 q^{22} +6.29342 q^{23} +2.78165 q^{24} +1.00000 q^{25} +4.76252 q^{26} -4.83338 q^{27} -3.35211 q^{28} -2.05132 q^{29} +2.78165 q^{30} -5.06600 q^{31} -1.00000 q^{32} -10.8610 q^{33} +4.89438 q^{34} -3.35211 q^{35} +4.73759 q^{36} +1.59485 q^{37} +2.47209 q^{38} +13.2477 q^{39} -1.00000 q^{40} -4.70017 q^{41} -9.32441 q^{42} -4.61250 q^{43} +3.90452 q^{44} +4.73759 q^{45} -6.29342 q^{46} -1.83379 q^{47} -2.78165 q^{48} +4.23664 q^{49} -1.00000 q^{50} +13.6145 q^{51} -4.76252 q^{52} +9.36637 q^{53} +4.83338 q^{54} +3.90452 q^{55} +3.35211 q^{56} +6.87649 q^{57} +2.05132 q^{58} +14.0106 q^{59} -2.78165 q^{60} -14.8313 q^{61} +5.06600 q^{62} -15.8809 q^{63} +1.00000 q^{64} -4.76252 q^{65} +10.8610 q^{66} -8.21745 q^{67} -4.89438 q^{68} -17.5061 q^{69} +3.35211 q^{70} -1.00646 q^{71} -4.73759 q^{72} -4.30786 q^{73} -1.59485 q^{74} -2.78165 q^{75} -2.47209 q^{76} -13.0884 q^{77} -13.2477 q^{78} +4.93774 q^{79} +1.00000 q^{80} -0.768000 q^{81} +4.70017 q^{82} +3.19268 q^{83} +9.32441 q^{84} -4.89438 q^{85} +4.61250 q^{86} +5.70606 q^{87} -3.90452 q^{88} +12.5863 q^{89} -4.73759 q^{90} +15.9645 q^{91} +6.29342 q^{92} +14.0919 q^{93} +1.83379 q^{94} -2.47209 q^{95} +2.78165 q^{96} -0.763398 q^{97} -4.23664 q^{98} +18.4980 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 20 q^{2} + 4 q^{3} + 20 q^{4} + 20 q^{5} - 4 q^{6} + 11 q^{7} - 20 q^{8} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 20 q^{2} + 4 q^{3} + 20 q^{4} + 20 q^{5} - 4 q^{6} + 11 q^{7} - 20 q^{8} + 26 q^{9} - 20 q^{10} + 10 q^{11} + 4 q^{12} - 9 q^{13} - 11 q^{14} + 4 q^{15} + 20 q^{16} - 11 q^{17} - 26 q^{18} + 17 q^{19} + 20 q^{20} - 2 q^{21} - 10 q^{22} - 3 q^{23} - 4 q^{24} + 20 q^{25} + 9 q^{26} - 2 q^{27} + 11 q^{28} + 6 q^{29} - 4 q^{30} + 28 q^{31} - 20 q^{32} + 2 q^{33} + 11 q^{34} + 11 q^{35} + 26 q^{36} + 33 q^{37} - 17 q^{38} + 36 q^{39} - 20 q^{40} + 32 q^{41} + 2 q^{42} + 30 q^{43} + 10 q^{44} + 26 q^{45} + 3 q^{46} + 13 q^{47} + 4 q^{48} + 43 q^{49} - 20 q^{50} + 43 q^{51} - 9 q^{52} + 2 q^{54} + 10 q^{55} - 11 q^{56} + 19 q^{57} - 6 q^{58} + 52 q^{59} + 4 q^{60} + 25 q^{61} - 28 q^{62} + 16 q^{63} + 20 q^{64} - 9 q^{65} - 2 q^{66} + 40 q^{67} - 11 q^{68} + 39 q^{69} - 11 q^{70} + 25 q^{71} - 26 q^{72} - 5 q^{73} - 33 q^{74} + 4 q^{75} + 17 q^{76} - 9 q^{77} - 36 q^{78} + 40 q^{79} + 20 q^{80} + 48 q^{81} - 32 q^{82} + 10 q^{83} - 2 q^{84} - 11 q^{85} - 30 q^{86} + 10 q^{87} - 10 q^{88} + 17 q^{89} - 26 q^{90} + 88 q^{91} - 3 q^{92} - 4 q^{93} - 13 q^{94} + 17 q^{95} - 4 q^{96} - 2 q^{97} - 43 q^{98} + 50 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −2.78165 −1.60599 −0.802994 0.595987i \(-0.796761\pi\)
−0.802994 + 0.595987i \(0.796761\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 2.78165 1.13560
\(7\) −3.35211 −1.26698 −0.633489 0.773751i \(-0.718378\pi\)
−0.633489 + 0.773751i \(0.718378\pi\)
\(8\) −1.00000 −0.353553
\(9\) 4.73759 1.57920
\(10\) −1.00000 −0.316228
\(11\) 3.90452 1.17726 0.588628 0.808404i \(-0.299668\pi\)
0.588628 + 0.808404i \(0.299668\pi\)
\(12\) −2.78165 −0.802994
\(13\) −4.76252 −1.32089 −0.660443 0.750876i \(-0.729632\pi\)
−0.660443 + 0.750876i \(0.729632\pi\)
\(14\) 3.35211 0.895889
\(15\) −2.78165 −0.718220
\(16\) 1.00000 0.250000
\(17\) −4.89438 −1.18706 −0.593531 0.804811i \(-0.702267\pi\)
−0.593531 + 0.804811i \(0.702267\pi\)
\(18\) −4.73759 −1.11666
\(19\) −2.47209 −0.567136 −0.283568 0.958952i \(-0.591518\pi\)
−0.283568 + 0.958952i \(0.591518\pi\)
\(20\) 1.00000 0.223607
\(21\) 9.32441 2.03475
\(22\) −3.90452 −0.832446
\(23\) 6.29342 1.31227 0.656135 0.754644i \(-0.272190\pi\)
0.656135 + 0.754644i \(0.272190\pi\)
\(24\) 2.78165 0.567802
\(25\) 1.00000 0.200000
\(26\) 4.76252 0.934008
\(27\) −4.83338 −0.930184
\(28\) −3.35211 −0.633489
\(29\) −2.05132 −0.380921 −0.190460 0.981695i \(-0.560998\pi\)
−0.190460 + 0.981695i \(0.560998\pi\)
\(30\) 2.78165 0.507858
\(31\) −5.06600 −0.909881 −0.454941 0.890522i \(-0.650340\pi\)
−0.454941 + 0.890522i \(0.650340\pi\)
\(32\) −1.00000 −0.176777
\(33\) −10.8610 −1.89066
\(34\) 4.89438 0.839380
\(35\) −3.35211 −0.566610
\(36\) 4.73759 0.789599
\(37\) 1.59485 0.262191 0.131095 0.991370i \(-0.458151\pi\)
0.131095 + 0.991370i \(0.458151\pi\)
\(38\) 2.47209 0.401026
\(39\) 13.2477 2.12133
\(40\) −1.00000 −0.158114
\(41\) −4.70017 −0.734044 −0.367022 0.930212i \(-0.619623\pi\)
−0.367022 + 0.930212i \(0.619623\pi\)
\(42\) −9.32441 −1.43879
\(43\) −4.61250 −0.703400 −0.351700 0.936113i \(-0.614396\pi\)
−0.351700 + 0.936113i \(0.614396\pi\)
\(44\) 3.90452 0.588628
\(45\) 4.73759 0.706238
\(46\) −6.29342 −0.927915
\(47\) −1.83379 −0.267486 −0.133743 0.991016i \(-0.542700\pi\)
−0.133743 + 0.991016i \(0.542700\pi\)
\(48\) −2.78165 −0.401497
\(49\) 4.23664 0.605235
\(50\) −1.00000 −0.141421
\(51\) 13.6145 1.90641
\(52\) −4.76252 −0.660443
\(53\) 9.36637 1.28657 0.643285 0.765627i \(-0.277571\pi\)
0.643285 + 0.765627i \(0.277571\pi\)
\(54\) 4.83338 0.657739
\(55\) 3.90452 0.526485
\(56\) 3.35211 0.447945
\(57\) 6.87649 0.910814
\(58\) 2.05132 0.269352
\(59\) 14.0106 1.82402 0.912011 0.410166i \(-0.134529\pi\)
0.912011 + 0.410166i \(0.134529\pi\)
\(60\) −2.78165 −0.359110
\(61\) −14.8313 −1.89895 −0.949477 0.313837i \(-0.898385\pi\)
−0.949477 + 0.313837i \(0.898385\pi\)
\(62\) 5.06600 0.643383
\(63\) −15.8809 −2.00081
\(64\) 1.00000 0.125000
\(65\) −4.76252 −0.590718
\(66\) 10.8610 1.33690
\(67\) −8.21745 −1.00392 −0.501961 0.864890i \(-0.667388\pi\)
−0.501961 + 0.864890i \(0.667388\pi\)
\(68\) −4.89438 −0.593531
\(69\) −17.5061 −2.10749
\(70\) 3.35211 0.400654
\(71\) −1.00646 −0.119444 −0.0597221 0.998215i \(-0.519021\pi\)
−0.0597221 + 0.998215i \(0.519021\pi\)
\(72\) −4.73759 −0.558331
\(73\) −4.30786 −0.504197 −0.252099 0.967702i \(-0.581121\pi\)
−0.252099 + 0.967702i \(0.581121\pi\)
\(74\) −1.59485 −0.185397
\(75\) −2.78165 −0.321198
\(76\) −2.47209 −0.283568
\(77\) −13.0884 −1.49156
\(78\) −13.2477 −1.50001
\(79\) 4.93774 0.555539 0.277769 0.960648i \(-0.410405\pi\)
0.277769 + 0.960648i \(0.410405\pi\)
\(80\) 1.00000 0.111803
\(81\) −0.768000 −0.0853334
\(82\) 4.70017 0.519047
\(83\) 3.19268 0.350442 0.175221 0.984529i \(-0.443936\pi\)
0.175221 + 0.984529i \(0.443936\pi\)
\(84\) 9.32441 1.01738
\(85\) −4.89438 −0.530870
\(86\) 4.61250 0.497379
\(87\) 5.70606 0.611754
\(88\) −3.90452 −0.416223
\(89\) 12.5863 1.33415 0.667074 0.744991i \(-0.267546\pi\)
0.667074 + 0.744991i \(0.267546\pi\)
\(90\) −4.73759 −0.499386
\(91\) 15.9645 1.67353
\(92\) 6.29342 0.656135
\(93\) 14.0919 1.46126
\(94\) 1.83379 0.189141
\(95\) −2.47209 −0.253631
\(96\) 2.78165 0.283901
\(97\) −0.763398 −0.0775113 −0.0387557 0.999249i \(-0.512339\pi\)
−0.0387557 + 0.999249i \(0.512339\pi\)
\(98\) −4.23664 −0.427966
\(99\) 18.4980 1.85912
\(100\) 1.00000 0.100000
\(101\) −14.6484 −1.45757 −0.728784 0.684744i \(-0.759914\pi\)
−0.728784 + 0.684744i \(0.759914\pi\)
\(102\) −13.6145 −1.34803
\(103\) −12.3855 −1.22038 −0.610192 0.792254i \(-0.708908\pi\)
−0.610192 + 0.792254i \(0.708908\pi\)
\(104\) 4.76252 0.467004
\(105\) 9.32441 0.909969
\(106\) −9.36637 −0.909742
\(107\) −5.02509 −0.485794 −0.242897 0.970052i \(-0.578098\pi\)
−0.242897 + 0.970052i \(0.578098\pi\)
\(108\) −4.83338 −0.465092
\(109\) −13.2297 −1.26717 −0.633587 0.773671i \(-0.718418\pi\)
−0.633587 + 0.773671i \(0.718418\pi\)
\(110\) −3.90452 −0.372281
\(111\) −4.43631 −0.421076
\(112\) −3.35211 −0.316745
\(113\) −13.7021 −1.28899 −0.644495 0.764609i \(-0.722932\pi\)
−0.644495 + 0.764609i \(0.722932\pi\)
\(114\) −6.87649 −0.644043
\(115\) 6.29342 0.586865
\(116\) −2.05132 −0.190460
\(117\) −22.5629 −2.08594
\(118\) −14.0106 −1.28978
\(119\) 16.4065 1.50398
\(120\) 2.78165 0.253929
\(121\) 4.24525 0.385931
\(122\) 14.8313 1.34276
\(123\) 13.0742 1.17887
\(124\) −5.06600 −0.454941
\(125\) 1.00000 0.0894427
\(126\) 15.8809 1.41479
\(127\) 16.0966 1.42834 0.714170 0.699972i \(-0.246804\pi\)
0.714170 + 0.699972i \(0.246804\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 12.8304 1.12965
\(130\) 4.76252 0.417701
\(131\) 11.7477 1.02640 0.513199 0.858270i \(-0.328460\pi\)
0.513199 + 0.858270i \(0.328460\pi\)
\(132\) −10.8610 −0.945329
\(133\) 8.28671 0.718549
\(134\) 8.21745 0.709879
\(135\) −4.83338 −0.415991
\(136\) 4.89438 0.419690
\(137\) 4.15454 0.354946 0.177473 0.984126i \(-0.443208\pi\)
0.177473 + 0.984126i \(0.443208\pi\)
\(138\) 17.5061 1.49022
\(139\) −16.2327 −1.37684 −0.688421 0.725311i \(-0.741696\pi\)
−0.688421 + 0.725311i \(0.741696\pi\)
\(140\) −3.35211 −0.283305
\(141\) 5.10097 0.429579
\(142\) 1.00646 0.0844598
\(143\) −18.5954 −1.55502
\(144\) 4.73759 0.394799
\(145\) −2.05132 −0.170353
\(146\) 4.30786 0.356521
\(147\) −11.7849 −0.971999
\(148\) 1.59485 0.131095
\(149\) −20.1588 −1.65147 −0.825737 0.564056i \(-0.809240\pi\)
−0.825737 + 0.564056i \(0.809240\pi\)
\(150\) 2.78165 0.227121
\(151\) 15.2001 1.23696 0.618482 0.785799i \(-0.287748\pi\)
0.618482 + 0.785799i \(0.287748\pi\)
\(152\) 2.47209 0.200513
\(153\) −23.1876 −1.87461
\(154\) 13.0884 1.05469
\(155\) −5.06600 −0.406911
\(156\) 13.2477 1.06066
\(157\) 18.0499 1.44054 0.720268 0.693696i \(-0.244019\pi\)
0.720268 + 0.693696i \(0.244019\pi\)
\(158\) −4.93774 −0.392825
\(159\) −26.0540 −2.06622
\(160\) −1.00000 −0.0790569
\(161\) −21.0962 −1.66262
\(162\) 0.768000 0.0603398
\(163\) −15.5264 −1.21612 −0.608061 0.793890i \(-0.708052\pi\)
−0.608061 + 0.793890i \(0.708052\pi\)
\(164\) −4.70017 −0.367022
\(165\) −10.8610 −0.845528
\(166\) −3.19268 −0.247800
\(167\) 8.16505 0.631830 0.315915 0.948787i \(-0.397688\pi\)
0.315915 + 0.948787i \(0.397688\pi\)
\(168\) −9.32441 −0.719394
\(169\) 9.68163 0.744741
\(170\) 4.89438 0.375382
\(171\) −11.7117 −0.895620
\(172\) −4.61250 −0.351700
\(173\) 7.06357 0.537033 0.268517 0.963275i \(-0.413467\pi\)
0.268517 + 0.963275i \(0.413467\pi\)
\(174\) −5.70606 −0.432576
\(175\) −3.35211 −0.253396
\(176\) 3.90452 0.294314
\(177\) −38.9726 −2.92936
\(178\) −12.5863 −0.943386
\(179\) 6.59782 0.493144 0.246572 0.969124i \(-0.420696\pi\)
0.246572 + 0.969124i \(0.420696\pi\)
\(180\) 4.73759 0.353119
\(181\) −13.7708 −1.02357 −0.511786 0.859113i \(-0.671016\pi\)
−0.511786 + 0.859113i \(0.671016\pi\)
\(182\) −15.9645 −1.18337
\(183\) 41.2555 3.04970
\(184\) −6.29342 −0.463957
\(185\) 1.59485 0.117255
\(186\) −14.0919 −1.03327
\(187\) −19.1102 −1.39748
\(188\) −1.83379 −0.133743
\(189\) 16.2020 1.17852
\(190\) 2.47209 0.179344
\(191\) −25.8062 −1.86727 −0.933635 0.358224i \(-0.883382\pi\)
−0.933635 + 0.358224i \(0.883382\pi\)
\(192\) −2.78165 −0.200748
\(193\) 6.45048 0.464316 0.232158 0.972678i \(-0.425421\pi\)
0.232158 + 0.972678i \(0.425421\pi\)
\(194\) 0.763398 0.0548088
\(195\) 13.2477 0.948687
\(196\) 4.23664 0.302617
\(197\) 20.5238 1.46226 0.731130 0.682239i \(-0.238993\pi\)
0.731130 + 0.682239i \(0.238993\pi\)
\(198\) −18.4980 −1.31460
\(199\) 2.93538 0.208084 0.104042 0.994573i \(-0.466822\pi\)
0.104042 + 0.994573i \(0.466822\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 22.8581 1.61229
\(202\) 14.6484 1.03066
\(203\) 6.87626 0.482619
\(204\) 13.6145 0.953204
\(205\) −4.70017 −0.328274
\(206\) 12.3855 0.862942
\(207\) 29.8157 2.07233
\(208\) −4.76252 −0.330222
\(209\) −9.65231 −0.667664
\(210\) −9.32441 −0.643445
\(211\) 24.7638 1.70481 0.852404 0.522883i \(-0.175144\pi\)
0.852404 + 0.522883i \(0.175144\pi\)
\(212\) 9.36637 0.643285
\(213\) 2.79961 0.191826
\(214\) 5.02509 0.343508
\(215\) −4.61250 −0.314570
\(216\) 4.83338 0.328870
\(217\) 16.9818 1.15280
\(218\) 13.2297 0.896027
\(219\) 11.9830 0.809735
\(220\) 3.90452 0.263242
\(221\) 23.3096 1.56797
\(222\) 4.43631 0.297745
\(223\) −2.06642 −0.138378 −0.0691888 0.997604i \(-0.522041\pi\)
−0.0691888 + 0.997604i \(0.522041\pi\)
\(224\) 3.35211 0.223972
\(225\) 4.73759 0.315839
\(226\) 13.7021 0.911453
\(227\) 15.1770 1.00733 0.503666 0.863898i \(-0.331984\pi\)
0.503666 + 0.863898i \(0.331984\pi\)
\(228\) 6.87649 0.455407
\(229\) 0.871005 0.0575576 0.0287788 0.999586i \(-0.490838\pi\)
0.0287788 + 0.999586i \(0.490838\pi\)
\(230\) −6.29342 −0.414976
\(231\) 36.4073 2.39542
\(232\) 2.05132 0.134676
\(233\) 1.34295 0.0879798 0.0439899 0.999032i \(-0.485993\pi\)
0.0439899 + 0.999032i \(0.485993\pi\)
\(234\) 22.5629 1.47498
\(235\) −1.83379 −0.119623
\(236\) 14.0106 0.912011
\(237\) −13.7351 −0.892188
\(238\) −16.4065 −1.06348
\(239\) 26.2745 1.69956 0.849779 0.527139i \(-0.176735\pi\)
0.849779 + 0.527139i \(0.176735\pi\)
\(240\) −2.78165 −0.179555
\(241\) −5.07033 −0.326609 −0.163304 0.986576i \(-0.552215\pi\)
−0.163304 + 0.986576i \(0.552215\pi\)
\(242\) −4.24525 −0.272895
\(243\) 16.6364 1.06723
\(244\) −14.8313 −0.949477
\(245\) 4.23664 0.270669
\(246\) −13.0742 −0.833584
\(247\) 11.7734 0.749122
\(248\) 5.06600 0.321692
\(249\) −8.88091 −0.562805
\(250\) −1.00000 −0.0632456
\(251\) 13.9205 0.878653 0.439327 0.898327i \(-0.355217\pi\)
0.439327 + 0.898327i \(0.355217\pi\)
\(252\) −15.8809 −1.00040
\(253\) 24.5728 1.54488
\(254\) −16.0966 −1.00999
\(255\) 13.6145 0.852572
\(256\) 1.00000 0.0625000
\(257\) 19.2699 1.20202 0.601011 0.799241i \(-0.294765\pi\)
0.601011 + 0.799241i \(0.294765\pi\)
\(258\) −12.8304 −0.798785
\(259\) −5.34610 −0.332190
\(260\) −4.76252 −0.295359
\(261\) −9.71833 −0.601549
\(262\) −11.7477 −0.725772
\(263\) 21.1485 1.30407 0.652036 0.758188i \(-0.273915\pi\)
0.652036 + 0.758188i \(0.273915\pi\)
\(264\) 10.8610 0.668449
\(265\) 9.36637 0.575371
\(266\) −8.28671 −0.508091
\(267\) −35.0108 −2.14263
\(268\) −8.21745 −0.501961
\(269\) 21.3660 1.30271 0.651355 0.758773i \(-0.274201\pi\)
0.651355 + 0.758773i \(0.274201\pi\)
\(270\) 4.83338 0.294150
\(271\) 1.49664 0.0909145 0.0454572 0.998966i \(-0.485526\pi\)
0.0454572 + 0.998966i \(0.485526\pi\)
\(272\) −4.89438 −0.296766
\(273\) −44.4077 −2.68768
\(274\) −4.15454 −0.250985
\(275\) 3.90452 0.235451
\(276\) −17.5061 −1.05374
\(277\) 19.6599 1.18125 0.590625 0.806946i \(-0.298881\pi\)
0.590625 + 0.806946i \(0.298881\pi\)
\(278\) 16.2327 0.973574
\(279\) −24.0007 −1.43688
\(280\) 3.35211 0.200327
\(281\) −21.3948 −1.27631 −0.638153 0.769909i \(-0.720301\pi\)
−0.638153 + 0.769909i \(0.720301\pi\)
\(282\) −5.10097 −0.303758
\(283\) −14.3052 −0.850354 −0.425177 0.905110i \(-0.639788\pi\)
−0.425177 + 0.905110i \(0.639788\pi\)
\(284\) −1.00646 −0.0597221
\(285\) 6.87649 0.407328
\(286\) 18.5954 1.09957
\(287\) 15.7555 0.930017
\(288\) −4.73759 −0.279165
\(289\) 6.95500 0.409117
\(290\) 2.05132 0.120458
\(291\) 2.12351 0.124482
\(292\) −4.30786 −0.252099
\(293\) −26.8485 −1.56851 −0.784254 0.620440i \(-0.786954\pi\)
−0.784254 + 0.620440i \(0.786954\pi\)
\(294\) 11.7849 0.687307
\(295\) 14.0106 0.815727
\(296\) −1.59485 −0.0926985
\(297\) −18.8720 −1.09506
\(298\) 20.1588 1.16777
\(299\) −29.9726 −1.73336
\(300\) −2.78165 −0.160599
\(301\) 15.4616 0.891193
\(302\) −15.2001 −0.874666
\(303\) 40.7467 2.34084
\(304\) −2.47209 −0.141784
\(305\) −14.8313 −0.849238
\(306\) 23.1876 1.32555
\(307\) 10.3721 0.591970 0.295985 0.955193i \(-0.404352\pi\)
0.295985 + 0.955193i \(0.404352\pi\)
\(308\) −13.0884 −0.745779
\(309\) 34.4523 1.95992
\(310\) 5.06600 0.287730
\(311\) 9.19579 0.521445 0.260723 0.965414i \(-0.416039\pi\)
0.260723 + 0.965414i \(0.416039\pi\)
\(312\) −13.2477 −0.750003
\(313\) −1.64229 −0.0928279 −0.0464139 0.998922i \(-0.514779\pi\)
−0.0464139 + 0.998922i \(0.514779\pi\)
\(314\) −18.0499 −1.01861
\(315\) −15.8809 −0.894789
\(316\) 4.93774 0.277769
\(317\) 11.8898 0.667798 0.333899 0.942609i \(-0.391635\pi\)
0.333899 + 0.942609i \(0.391635\pi\)
\(318\) 26.0540 1.46103
\(319\) −8.00942 −0.448441
\(320\) 1.00000 0.0559017
\(321\) 13.9781 0.780179
\(322\) 21.0962 1.17565
\(323\) 12.0994 0.673226
\(324\) −0.768000 −0.0426667
\(325\) −4.76252 −0.264177
\(326\) 15.5264 0.859928
\(327\) 36.8004 2.03507
\(328\) 4.70017 0.259524
\(329\) 6.14707 0.338899
\(330\) 10.8610 0.597879
\(331\) 27.8492 1.53073 0.765364 0.643598i \(-0.222559\pi\)
0.765364 + 0.643598i \(0.222559\pi\)
\(332\) 3.19268 0.175221
\(333\) 7.55573 0.414051
\(334\) −8.16505 −0.446772
\(335\) −8.21745 −0.448967
\(336\) 9.32441 0.508688
\(337\) 2.30475 0.125548 0.0627739 0.998028i \(-0.480005\pi\)
0.0627739 + 0.998028i \(0.480005\pi\)
\(338\) −9.68163 −0.526611
\(339\) 38.1146 2.07010
\(340\) −4.89438 −0.265435
\(341\) −19.7803 −1.07116
\(342\) 11.7117 0.633299
\(343\) 9.26308 0.500159
\(344\) 4.61250 0.248690
\(345\) −17.5061 −0.942498
\(346\) −7.06357 −0.379740
\(347\) −12.0815 −0.648568 −0.324284 0.945960i \(-0.605123\pi\)
−0.324284 + 0.945960i \(0.605123\pi\)
\(348\) 5.70606 0.305877
\(349\) −16.6586 −0.891716 −0.445858 0.895104i \(-0.647101\pi\)
−0.445858 + 0.895104i \(0.647101\pi\)
\(350\) 3.35211 0.179178
\(351\) 23.0191 1.22867
\(352\) −3.90452 −0.208111
\(353\) 1.81409 0.0965543 0.0482772 0.998834i \(-0.484627\pi\)
0.0482772 + 0.998834i \(0.484627\pi\)
\(354\) 38.9726 2.07137
\(355\) −1.00646 −0.0534171
\(356\) 12.5863 0.667074
\(357\) −45.6372 −2.41538
\(358\) −6.59782 −0.348706
\(359\) 6.16066 0.325147 0.162574 0.986696i \(-0.448020\pi\)
0.162574 + 0.986696i \(0.448020\pi\)
\(360\) −4.73759 −0.249693
\(361\) −12.8888 −0.678357
\(362\) 13.7708 0.723775
\(363\) −11.8088 −0.619801
\(364\) 15.9645 0.836767
\(365\) −4.30786 −0.225484
\(366\) −41.2555 −2.15646
\(367\) 26.0479 1.35969 0.679844 0.733357i \(-0.262047\pi\)
0.679844 + 0.733357i \(0.262047\pi\)
\(368\) 6.29342 0.328067
\(369\) −22.2675 −1.15920
\(370\) −1.59485 −0.0829121
\(371\) −31.3971 −1.63006
\(372\) 14.0919 0.730629
\(373\) 6.44018 0.333460 0.166730 0.986003i \(-0.446679\pi\)
0.166730 + 0.986003i \(0.446679\pi\)
\(374\) 19.1102 0.988165
\(375\) −2.78165 −0.143644
\(376\) 1.83379 0.0945705
\(377\) 9.76947 0.503153
\(378\) −16.2020 −0.833341
\(379\) −7.87991 −0.404764 −0.202382 0.979307i \(-0.564868\pi\)
−0.202382 + 0.979307i \(0.564868\pi\)
\(380\) −2.47209 −0.126815
\(381\) −44.7751 −2.29390
\(382\) 25.8062 1.32036
\(383\) −20.5899 −1.05209 −0.526047 0.850455i \(-0.676326\pi\)
−0.526047 + 0.850455i \(0.676326\pi\)
\(384\) 2.78165 0.141951
\(385\) −13.0884 −0.667045
\(386\) −6.45048 −0.328321
\(387\) −21.8522 −1.11081
\(388\) −0.763398 −0.0387557
\(389\) 34.7031 1.75952 0.879758 0.475423i \(-0.157705\pi\)
0.879758 + 0.475423i \(0.157705\pi\)
\(390\) −13.2477 −0.670823
\(391\) −30.8024 −1.55775
\(392\) −4.23664 −0.213983
\(393\) −32.6779 −1.64838
\(394\) −20.5238 −1.03397
\(395\) 4.93774 0.248444
\(396\) 18.4980 0.929560
\(397\) 0.294316 0.0147713 0.00738566 0.999973i \(-0.497649\pi\)
0.00738566 + 0.999973i \(0.497649\pi\)
\(398\) −2.93538 −0.147137
\(399\) −23.0508 −1.15398
\(400\) 1.00000 0.0500000
\(401\) −1.00000 −0.0499376
\(402\) −22.8581 −1.14006
\(403\) 24.1270 1.20185
\(404\) −14.6484 −0.728784
\(405\) −0.768000 −0.0381623
\(406\) −6.87626 −0.341263
\(407\) 6.22710 0.308666
\(408\) −13.6145 −0.674017
\(409\) 8.24295 0.407588 0.203794 0.979014i \(-0.434673\pi\)
0.203794 + 0.979014i \(0.434673\pi\)
\(410\) 4.70017 0.232125
\(411\) −11.5565 −0.570039
\(412\) −12.3855 −0.610192
\(413\) −46.9650 −2.31100
\(414\) −29.8157 −1.46536
\(415\) 3.19268 0.156722
\(416\) 4.76252 0.233502
\(417\) 45.1538 2.21119
\(418\) 9.65231 0.472110
\(419\) 27.0969 1.32377 0.661885 0.749605i \(-0.269757\pi\)
0.661885 + 0.749605i \(0.269757\pi\)
\(420\) 9.32441 0.454984
\(421\) 30.6725 1.49488 0.747442 0.664327i \(-0.231282\pi\)
0.747442 + 0.664327i \(0.231282\pi\)
\(422\) −24.7638 −1.20548
\(423\) −8.68775 −0.422413
\(424\) −9.36637 −0.454871
\(425\) −4.89438 −0.237413
\(426\) −2.79961 −0.135641
\(427\) 49.7162 2.40593
\(428\) −5.02509 −0.242897
\(429\) 51.7258 2.49735
\(430\) 4.61250 0.222435
\(431\) −27.4086 −1.32022 −0.660112 0.751167i \(-0.729491\pi\)
−0.660112 + 0.751167i \(0.729491\pi\)
\(432\) −4.83338 −0.232546
\(433\) 19.8145 0.952223 0.476112 0.879385i \(-0.342046\pi\)
0.476112 + 0.879385i \(0.342046\pi\)
\(434\) −16.9818 −0.815153
\(435\) 5.70606 0.273585
\(436\) −13.2297 −0.633587
\(437\) −15.5579 −0.744235
\(438\) −11.9830 −0.572569
\(439\) 13.3948 0.639298 0.319649 0.947536i \(-0.396435\pi\)
0.319649 + 0.947536i \(0.396435\pi\)
\(440\) −3.90452 −0.186140
\(441\) 20.0715 0.955785
\(442\) −23.3096 −1.10873
\(443\) 20.8818 0.992124 0.496062 0.868287i \(-0.334779\pi\)
0.496062 + 0.868287i \(0.334779\pi\)
\(444\) −4.43631 −0.210538
\(445\) 12.5863 0.596650
\(446\) 2.06642 0.0978478
\(447\) 56.0748 2.65225
\(448\) −3.35211 −0.158372
\(449\) 35.9911 1.69852 0.849262 0.527972i \(-0.177048\pi\)
0.849262 + 0.527972i \(0.177048\pi\)
\(450\) −4.73759 −0.223332
\(451\) −18.3519 −0.864157
\(452\) −13.7021 −0.644495
\(453\) −42.2813 −1.98655
\(454\) −15.1770 −0.712291
\(455\) 15.9645 0.748427
\(456\) −6.87649 −0.322021
\(457\) 34.2136 1.60045 0.800223 0.599703i \(-0.204715\pi\)
0.800223 + 0.599703i \(0.204715\pi\)
\(458\) −0.871005 −0.0406994
\(459\) 23.6564 1.10419
\(460\) 6.29342 0.293432
\(461\) −15.9895 −0.744707 −0.372354 0.928091i \(-0.621449\pi\)
−0.372354 + 0.928091i \(0.621449\pi\)
\(462\) −36.4073 −1.69382
\(463\) −17.7535 −0.825074 −0.412537 0.910941i \(-0.635357\pi\)
−0.412537 + 0.910941i \(0.635357\pi\)
\(464\) −2.05132 −0.0952302
\(465\) 14.0919 0.653494
\(466\) −1.34295 −0.0622111
\(467\) 20.1913 0.934343 0.467172 0.884167i \(-0.345273\pi\)
0.467172 + 0.884167i \(0.345273\pi\)
\(468\) −22.5629 −1.04297
\(469\) 27.5458 1.27195
\(470\) 1.83379 0.0845865
\(471\) −50.2084 −2.31348
\(472\) −14.0106 −0.644889
\(473\) −18.0096 −0.828082
\(474\) 13.7351 0.630872
\(475\) −2.47209 −0.113427
\(476\) 16.4065 0.751991
\(477\) 44.3740 2.03175
\(478\) −26.2745 −1.20177
\(479\) 14.8374 0.677939 0.338969 0.940797i \(-0.389922\pi\)
0.338969 + 0.940797i \(0.389922\pi\)
\(480\) 2.78165 0.126964
\(481\) −7.59549 −0.346325
\(482\) 5.07033 0.230947
\(483\) 58.6824 2.67014
\(484\) 4.24525 0.192966
\(485\) −0.763398 −0.0346641
\(486\) −16.6364 −0.754644
\(487\) 13.9455 0.631933 0.315967 0.948770i \(-0.397671\pi\)
0.315967 + 0.948770i \(0.397671\pi\)
\(488\) 14.8313 0.671381
\(489\) 43.1891 1.95308
\(490\) −4.23664 −0.191392
\(491\) 25.8866 1.16825 0.584124 0.811664i \(-0.301438\pi\)
0.584124 + 0.811664i \(0.301438\pi\)
\(492\) 13.0742 0.589433
\(493\) 10.0400 0.452177
\(494\) −11.7734 −0.529710
\(495\) 18.4980 0.831423
\(496\) −5.06600 −0.227470
\(497\) 3.37375 0.151333
\(498\) 8.88091 0.397963
\(499\) 0.374418 0.0167612 0.00838062 0.999965i \(-0.497332\pi\)
0.00838062 + 0.999965i \(0.497332\pi\)
\(500\) 1.00000 0.0447214
\(501\) −22.7123 −1.01471
\(502\) −13.9205 −0.621302
\(503\) −19.3432 −0.862471 −0.431235 0.902239i \(-0.641922\pi\)
−0.431235 + 0.902239i \(0.641922\pi\)
\(504\) 15.8809 0.707393
\(505\) −14.6484 −0.651844
\(506\) −24.5728 −1.09239
\(507\) −26.9309 −1.19605
\(508\) 16.0966 0.714170
\(509\) −2.61635 −0.115968 −0.0579838 0.998318i \(-0.518467\pi\)
−0.0579838 + 0.998318i \(0.518467\pi\)
\(510\) −13.6145 −0.602859
\(511\) 14.4404 0.638807
\(512\) −1.00000 −0.0441942
\(513\) 11.9485 0.527541
\(514\) −19.2699 −0.849958
\(515\) −12.3855 −0.545772
\(516\) 12.8304 0.564826
\(517\) −7.16007 −0.314899
\(518\) 5.34610 0.234894
\(519\) −19.6484 −0.862469
\(520\) 4.76252 0.208850
\(521\) −44.6210 −1.95488 −0.977440 0.211213i \(-0.932259\pi\)
−0.977440 + 0.211213i \(0.932259\pi\)
\(522\) 9.71833 0.425360
\(523\) 28.1019 1.22881 0.614404 0.788992i \(-0.289396\pi\)
0.614404 + 0.788992i \(0.289396\pi\)
\(524\) 11.7477 0.513199
\(525\) 9.32441 0.406950
\(526\) −21.1485 −0.922118
\(527\) 24.7950 1.08009
\(528\) −10.8610 −0.472665
\(529\) 16.6072 0.722051
\(530\) −9.36637 −0.406849
\(531\) 66.3764 2.88049
\(532\) 8.28671 0.359275
\(533\) 22.3847 0.969588
\(534\) 35.0108 1.51507
\(535\) −5.02509 −0.217254
\(536\) 8.21745 0.354940
\(537\) −18.3528 −0.791984
\(538\) −21.3660 −0.921155
\(539\) 16.5420 0.712516
\(540\) −4.83338 −0.207995
\(541\) 30.2716 1.30148 0.650739 0.759302i \(-0.274459\pi\)
0.650739 + 0.759302i \(0.274459\pi\)
\(542\) −1.49664 −0.0642862
\(543\) 38.3055 1.64385
\(544\) 4.89438 0.209845
\(545\) −13.2297 −0.566697
\(546\) 44.4077 1.90047
\(547\) 4.71926 0.201781 0.100890 0.994898i \(-0.467831\pi\)
0.100890 + 0.994898i \(0.467831\pi\)
\(548\) 4.15454 0.177473
\(549\) −70.2647 −2.99882
\(550\) −3.90452 −0.166489
\(551\) 5.07105 0.216034
\(552\) 17.5061 0.745110
\(553\) −16.5518 −0.703856
\(554\) −19.6599 −0.835270
\(555\) −4.43631 −0.188311
\(556\) −16.2327 −0.688421
\(557\) −33.3579 −1.41342 −0.706709 0.707504i \(-0.749821\pi\)
−0.706709 + 0.707504i \(0.749821\pi\)
\(558\) 24.0007 1.01603
\(559\) 21.9672 0.929112
\(560\) −3.35211 −0.141653
\(561\) 53.1579 2.24433
\(562\) 21.3948 0.902485
\(563\) 33.1694 1.39792 0.698962 0.715158i \(-0.253645\pi\)
0.698962 + 0.715158i \(0.253645\pi\)
\(564\) 5.10097 0.214790
\(565\) −13.7021 −0.576454
\(566\) 14.3052 0.601291
\(567\) 2.57442 0.108116
\(568\) 1.00646 0.0422299
\(569\) −10.4438 −0.437826 −0.218913 0.975744i \(-0.570251\pi\)
−0.218913 + 0.975744i \(0.570251\pi\)
\(570\) −6.87649 −0.288025
\(571\) 20.4106 0.854158 0.427079 0.904214i \(-0.359543\pi\)
0.427079 + 0.904214i \(0.359543\pi\)
\(572\) −18.5954 −0.777511
\(573\) 71.7839 2.99881
\(574\) −15.7555 −0.657622
\(575\) 6.29342 0.262454
\(576\) 4.73759 0.197400
\(577\) 8.73757 0.363750 0.181875 0.983322i \(-0.441783\pi\)
0.181875 + 0.983322i \(0.441783\pi\)
\(578\) −6.95500 −0.289290
\(579\) −17.9430 −0.745686
\(580\) −2.05132 −0.0851765
\(581\) −10.7022 −0.444002
\(582\) −2.12351 −0.0880223
\(583\) 36.5711 1.51462
\(584\) 4.30786 0.178261
\(585\) −22.5629 −0.932861
\(586\) 26.8485 1.10910
\(587\) −32.8822 −1.35719 −0.678597 0.734511i \(-0.737412\pi\)
−0.678597 + 0.734511i \(0.737412\pi\)
\(588\) −11.7849 −0.486000
\(589\) 12.5236 0.516026
\(590\) −14.0106 −0.576806
\(591\) −57.0901 −2.34837
\(592\) 1.59485 0.0655477
\(593\) −31.9516 −1.31210 −0.656048 0.754719i \(-0.727773\pi\)
−0.656048 + 0.754719i \(0.727773\pi\)
\(594\) 18.8720 0.774327
\(595\) 16.4065 0.672602
\(596\) −20.1588 −0.825737
\(597\) −8.16521 −0.334180
\(598\) 29.9726 1.22567
\(599\) −27.2584 −1.11375 −0.556874 0.830597i \(-0.688000\pi\)
−0.556874 + 0.830597i \(0.688000\pi\)
\(600\) 2.78165 0.113560
\(601\) 17.4698 0.712610 0.356305 0.934370i \(-0.384036\pi\)
0.356305 + 0.934370i \(0.384036\pi\)
\(602\) −15.4616 −0.630169
\(603\) −38.9309 −1.58539
\(604\) 15.2001 0.618482
\(605\) 4.24525 0.172594
\(606\) −40.7467 −1.65522
\(607\) 10.0867 0.409405 0.204703 0.978824i \(-0.434377\pi\)
0.204703 + 0.978824i \(0.434377\pi\)
\(608\) 2.47209 0.100256
\(609\) −19.1274 −0.775080
\(610\) 14.8313 0.600502
\(611\) 8.73347 0.353318
\(612\) −23.1876 −0.937303
\(613\) −33.2200 −1.34174 −0.670871 0.741574i \(-0.734080\pi\)
−0.670871 + 0.741574i \(0.734080\pi\)
\(614\) −10.3721 −0.418586
\(615\) 13.0742 0.527205
\(616\) 13.0884 0.527345
\(617\) −16.6887 −0.671863 −0.335932 0.941886i \(-0.609051\pi\)
−0.335932 + 0.941886i \(0.609051\pi\)
\(618\) −34.4523 −1.38587
\(619\) 27.9923 1.12511 0.562553 0.826761i \(-0.309819\pi\)
0.562553 + 0.826761i \(0.309819\pi\)
\(620\) −5.06600 −0.203456
\(621\) −30.4185 −1.22065
\(622\) −9.19579 −0.368718
\(623\) −42.1908 −1.69034
\(624\) 13.2477 0.530332
\(625\) 1.00000 0.0400000
\(626\) 1.64229 0.0656392
\(627\) 26.8494 1.07226
\(628\) 18.0499 0.720268
\(629\) −7.80579 −0.311237
\(630\) 15.8809 0.632711
\(631\) 6.41953 0.255558 0.127779 0.991803i \(-0.459215\pi\)
0.127779 + 0.991803i \(0.459215\pi\)
\(632\) −4.93774 −0.196413
\(633\) −68.8842 −2.73790
\(634\) −11.8898 −0.472205
\(635\) 16.0966 0.638773
\(636\) −26.0540 −1.03311
\(637\) −20.1771 −0.799446
\(638\) 8.00942 0.317096
\(639\) −4.76817 −0.188626
\(640\) −1.00000 −0.0395285
\(641\) 14.5422 0.574380 0.287190 0.957874i \(-0.407279\pi\)
0.287190 + 0.957874i \(0.407279\pi\)
\(642\) −13.9781 −0.551670
\(643\) −15.9655 −0.629619 −0.314809 0.949155i \(-0.601941\pi\)
−0.314809 + 0.949155i \(0.601941\pi\)
\(644\) −21.0962 −0.831309
\(645\) 12.8304 0.505196
\(646\) −12.0994 −0.476043
\(647\) −0.286663 −0.0112699 −0.00563494 0.999984i \(-0.501794\pi\)
−0.00563494 + 0.999984i \(0.501794\pi\)
\(648\) 0.768000 0.0301699
\(649\) 54.7045 2.14734
\(650\) 4.76252 0.186802
\(651\) −47.2375 −1.85138
\(652\) −15.5264 −0.608061
\(653\) −32.0906 −1.25580 −0.627902 0.778293i \(-0.716086\pi\)
−0.627902 + 0.778293i \(0.716086\pi\)
\(654\) −36.8004 −1.43901
\(655\) 11.7477 0.459019
\(656\) −4.70017 −0.183511
\(657\) −20.4089 −0.796227
\(658\) −6.14707 −0.239638
\(659\) −13.7101 −0.534071 −0.267035 0.963687i \(-0.586044\pi\)
−0.267035 + 0.963687i \(0.586044\pi\)
\(660\) −10.8610 −0.422764
\(661\) −9.37600 −0.364684 −0.182342 0.983235i \(-0.558368\pi\)
−0.182342 + 0.983235i \(0.558368\pi\)
\(662\) −27.8492 −1.08239
\(663\) −64.8393 −2.51815
\(664\) −3.19268 −0.123900
\(665\) 8.28671 0.321345
\(666\) −7.55573 −0.292778
\(667\) −12.9098 −0.499871
\(668\) 8.16505 0.315915
\(669\) 5.74806 0.222233
\(670\) 8.21745 0.317468
\(671\) −57.9091 −2.23555
\(672\) −9.32441 −0.359697
\(673\) 14.4602 0.557401 0.278701 0.960378i \(-0.410096\pi\)
0.278701 + 0.960378i \(0.410096\pi\)
\(674\) −2.30475 −0.0887757
\(675\) −4.83338 −0.186037
\(676\) 9.68163 0.372370
\(677\) 18.0780 0.694795 0.347398 0.937718i \(-0.387065\pi\)
0.347398 + 0.937718i \(0.387065\pi\)
\(678\) −38.1146 −1.46378
\(679\) 2.55899 0.0982052
\(680\) 4.89438 0.187691
\(681\) −42.2171 −1.61776
\(682\) 19.7803 0.757427
\(683\) 23.8751 0.913556 0.456778 0.889581i \(-0.349003\pi\)
0.456778 + 0.889581i \(0.349003\pi\)
\(684\) −11.7117 −0.447810
\(685\) 4.15454 0.158737
\(686\) −9.26308 −0.353666
\(687\) −2.42283 −0.0924368
\(688\) −4.61250 −0.175850
\(689\) −44.6076 −1.69941
\(690\) 17.5061 0.666447
\(691\) −45.1913 −1.71916 −0.859580 0.511002i \(-0.829275\pi\)
−0.859580 + 0.511002i \(0.829275\pi\)
\(692\) 7.06357 0.268517
\(693\) −62.0073 −2.35546
\(694\) 12.0815 0.458607
\(695\) −16.2327 −0.615742
\(696\) −5.70606 −0.216288
\(697\) 23.0044 0.871356
\(698\) 16.6586 0.630538
\(699\) −3.73563 −0.141295
\(700\) −3.35211 −0.126698
\(701\) 44.8665 1.69458 0.847292 0.531128i \(-0.178232\pi\)
0.847292 + 0.531128i \(0.178232\pi\)
\(702\) −23.0191 −0.868799
\(703\) −3.94260 −0.148698
\(704\) 3.90452 0.147157
\(705\) 5.10097 0.192114
\(706\) −1.81409 −0.0682742
\(707\) 49.1030 1.84671
\(708\) −38.9726 −1.46468
\(709\) 31.6879 1.19006 0.595031 0.803703i \(-0.297140\pi\)
0.595031 + 0.803703i \(0.297140\pi\)
\(710\) 1.00646 0.0377716
\(711\) 23.3930 0.877305
\(712\) −12.5863 −0.471693
\(713\) −31.8825 −1.19401
\(714\) 45.6372 1.70793
\(715\) −18.5954 −0.695427
\(716\) 6.59782 0.246572
\(717\) −73.0866 −2.72947
\(718\) −6.16066 −0.229914
\(719\) 3.34223 0.124644 0.0623220 0.998056i \(-0.480149\pi\)
0.0623220 + 0.998056i \(0.480149\pi\)
\(720\) 4.73759 0.176560
\(721\) 41.5177 1.54620
\(722\) 12.8888 0.479671
\(723\) 14.1039 0.524529
\(724\) −13.7708 −0.511786
\(725\) −2.05132 −0.0761842
\(726\) 11.8088 0.438266
\(727\) −31.9893 −1.18642 −0.593208 0.805049i \(-0.702139\pi\)
−0.593208 + 0.805049i \(0.702139\pi\)
\(728\) −15.9645 −0.591684
\(729\) −43.9728 −1.62862
\(730\) 4.30786 0.159441
\(731\) 22.5754 0.834980
\(732\) 41.2555 1.52485
\(733\) 7.68058 0.283689 0.141844 0.989889i \(-0.454697\pi\)
0.141844 + 0.989889i \(0.454697\pi\)
\(734\) −26.0479 −0.961445
\(735\) −11.7849 −0.434691
\(736\) −6.29342 −0.231979
\(737\) −32.0852 −1.18187
\(738\) 22.2675 0.819678
\(739\) 4.71296 0.173369 0.0866846 0.996236i \(-0.472373\pi\)
0.0866846 + 0.996236i \(0.472373\pi\)
\(740\) 1.59485 0.0586277
\(741\) −32.7495 −1.20308
\(742\) 31.3971 1.15262
\(743\) 32.3832 1.18802 0.594012 0.804456i \(-0.297543\pi\)
0.594012 + 0.804456i \(0.297543\pi\)
\(744\) −14.0919 −0.516633
\(745\) −20.1588 −0.738561
\(746\) −6.44018 −0.235792
\(747\) 15.1256 0.553417
\(748\) −19.1102 −0.698738
\(749\) 16.8447 0.615491
\(750\) 2.78165 0.101572
\(751\) −15.4349 −0.563229 −0.281614 0.959528i \(-0.590870\pi\)
−0.281614 + 0.959528i \(0.590870\pi\)
\(752\) −1.83379 −0.0668715
\(753\) −38.7220 −1.41111
\(754\) −9.76947 −0.355783
\(755\) 15.2001 0.553187
\(756\) 16.2020 0.589261
\(757\) −45.5686 −1.65622 −0.828109 0.560567i \(-0.810583\pi\)
−0.828109 + 0.560567i \(0.810583\pi\)
\(758\) 7.87991 0.286211
\(759\) −68.3529 −2.48105
\(760\) 2.47209 0.0896721
\(761\) 34.3933 1.24676 0.623379 0.781920i \(-0.285759\pi\)
0.623379 + 0.781920i \(0.285759\pi\)
\(762\) 44.7751 1.62203
\(763\) 44.3474 1.60548
\(764\) −25.8062 −0.933635
\(765\) −23.1876 −0.838349
\(766\) 20.5899 0.743943
\(767\) −66.7257 −2.40933
\(768\) −2.78165 −0.100374
\(769\) 18.9543 0.683511 0.341756 0.939789i \(-0.388978\pi\)
0.341756 + 0.939789i \(0.388978\pi\)
\(770\) 13.0884 0.471672
\(771\) −53.6021 −1.93043
\(772\) 6.45048 0.232158
\(773\) −18.7400 −0.674031 −0.337016 0.941499i \(-0.609417\pi\)
−0.337016 + 0.941499i \(0.609417\pi\)
\(774\) 21.8522 0.785460
\(775\) −5.06600 −0.181976
\(776\) 0.763398 0.0274044
\(777\) 14.8710 0.533494
\(778\) −34.7031 −1.24417
\(779\) 11.6192 0.416303
\(780\) 13.2477 0.474343
\(781\) −3.92972 −0.140616
\(782\) 30.8024 1.10149
\(783\) 9.91481 0.354326
\(784\) 4.23664 0.151309
\(785\) 18.0499 0.644227
\(786\) 32.6779 1.16558
\(787\) −31.1940 −1.11195 −0.555973 0.831200i \(-0.687654\pi\)
−0.555973 + 0.831200i \(0.687654\pi\)
\(788\) 20.5238 0.731130
\(789\) −58.8278 −2.09432
\(790\) −4.93774 −0.175677
\(791\) 45.9311 1.63312
\(792\) −18.4980 −0.657298
\(793\) 70.6344 2.50830
\(794\) −0.294316 −0.0104449
\(795\) −26.0540 −0.924040
\(796\) 2.93538 0.104042
\(797\) −22.3821 −0.792816 −0.396408 0.918075i \(-0.629743\pi\)
−0.396408 + 0.918075i \(0.629743\pi\)
\(798\) 23.0508 0.815988
\(799\) 8.97528 0.317522
\(800\) −1.00000 −0.0353553
\(801\) 59.6289 2.10688
\(802\) 1.00000 0.0353112
\(803\) −16.8201 −0.593569
\(804\) 22.8581 0.806143
\(805\) −21.0962 −0.743545
\(806\) −24.1270 −0.849836
\(807\) −59.4329 −2.09214
\(808\) 14.6484 0.515328
\(809\) −38.8534 −1.36601 −0.683007 0.730412i \(-0.739328\pi\)
−0.683007 + 0.730412i \(0.739328\pi\)
\(810\) 0.768000 0.0269848
\(811\) 1.91575 0.0672712 0.0336356 0.999434i \(-0.489291\pi\)
0.0336356 + 0.999434i \(0.489291\pi\)
\(812\) 6.87626 0.241309
\(813\) −4.16314 −0.146008
\(814\) −6.22710 −0.218260
\(815\) −15.5264 −0.543866
\(816\) 13.6145 0.476602
\(817\) 11.4025 0.398924
\(818\) −8.24295 −0.288208
\(819\) 75.6333 2.64284
\(820\) −4.70017 −0.164137
\(821\) −48.9423 −1.70810 −0.854049 0.520192i \(-0.825860\pi\)
−0.854049 + 0.520192i \(0.825860\pi\)
\(822\) 11.5565 0.403079
\(823\) −51.3985 −1.79164 −0.895820 0.444418i \(-0.853411\pi\)
−0.895820 + 0.444418i \(0.853411\pi\)
\(824\) 12.3855 0.431471
\(825\) −10.8610 −0.378132
\(826\) 46.9650 1.63412
\(827\) 17.0730 0.593688 0.296844 0.954926i \(-0.404066\pi\)
0.296844 + 0.954926i \(0.404066\pi\)
\(828\) 29.8157 1.03617
\(829\) −7.12097 −0.247322 −0.123661 0.992325i \(-0.539463\pi\)
−0.123661 + 0.992325i \(0.539463\pi\)
\(830\) −3.19268 −0.110819
\(831\) −54.6871 −1.89707
\(832\) −4.76252 −0.165111
\(833\) −20.7358 −0.718451
\(834\) −45.1538 −1.56355
\(835\) 8.16505 0.282563
\(836\) −9.65231 −0.333832
\(837\) 24.4859 0.846357
\(838\) −27.0969 −0.936047
\(839\) 7.18510 0.248057 0.124029 0.992279i \(-0.460419\pi\)
0.124029 + 0.992279i \(0.460419\pi\)
\(840\) −9.32441 −0.321723
\(841\) −24.7921 −0.854899
\(842\) −30.6725 −1.05704
\(843\) 59.5129 2.04973
\(844\) 24.7638 0.852404
\(845\) 9.68163 0.333058
\(846\) 8.68775 0.298691
\(847\) −14.2305 −0.488967
\(848\) 9.36637 0.321642
\(849\) 39.7920 1.36566
\(850\) 4.89438 0.167876
\(851\) 10.0370 0.344065
\(852\) 2.79961 0.0959130
\(853\) 16.4768 0.564156 0.282078 0.959391i \(-0.408976\pi\)
0.282078 + 0.959391i \(0.408976\pi\)
\(854\) −49.7162 −1.70125
\(855\) −11.7117 −0.400533
\(856\) 5.02509 0.171754
\(857\) −11.1040 −0.379306 −0.189653 0.981851i \(-0.560736\pi\)
−0.189653 + 0.981851i \(0.560736\pi\)
\(858\) −51.7258 −1.76589
\(859\) −9.03637 −0.308317 −0.154158 0.988046i \(-0.549267\pi\)
−0.154158 + 0.988046i \(0.549267\pi\)
\(860\) −4.61250 −0.157285
\(861\) −43.8263 −1.49360
\(862\) 27.4086 0.933539
\(863\) −21.3777 −0.727704 −0.363852 0.931457i \(-0.618539\pi\)
−0.363852 + 0.931457i \(0.618539\pi\)
\(864\) 4.83338 0.164435
\(865\) 7.06357 0.240169
\(866\) −19.8145 −0.673324
\(867\) −19.3464 −0.657038
\(868\) 16.9818 0.576400
\(869\) 19.2795 0.654011
\(870\) −5.70606 −0.193454
\(871\) 39.1358 1.32607
\(872\) 13.2297 0.448014
\(873\) −3.61667 −0.122406
\(874\) 15.5579 0.526254
\(875\) −3.35211 −0.113322
\(876\) 11.9830 0.404867
\(877\) −7.38760 −0.249462 −0.124731 0.992191i \(-0.539807\pi\)
−0.124731 + 0.992191i \(0.539807\pi\)
\(878\) −13.3948 −0.452052
\(879\) 74.6833 2.51900
\(880\) 3.90452 0.131621
\(881\) −11.6556 −0.392688 −0.196344 0.980535i \(-0.562907\pi\)
−0.196344 + 0.980535i \(0.562907\pi\)
\(882\) −20.0715 −0.675842
\(883\) −26.0091 −0.875275 −0.437638 0.899151i \(-0.644185\pi\)
−0.437638 + 0.899151i \(0.644185\pi\)
\(884\) 23.3096 0.783987
\(885\) −38.9726 −1.31005
\(886\) −20.8818 −0.701537
\(887\) −51.8479 −1.74088 −0.870441 0.492273i \(-0.836166\pi\)
−0.870441 + 0.492273i \(0.836166\pi\)
\(888\) 4.43631 0.148873
\(889\) −53.9575 −1.80968
\(890\) −12.5863 −0.421895
\(891\) −2.99867 −0.100459
\(892\) −2.06642 −0.0691888
\(893\) 4.53329 0.151701
\(894\) −56.0748 −1.87542
\(895\) 6.59782 0.220541
\(896\) 3.35211 0.111986
\(897\) 83.3733 2.78375
\(898\) −35.9911 −1.20104
\(899\) 10.3920 0.346593
\(900\) 4.73759 0.157920
\(901\) −45.8426 −1.52724
\(902\) 18.3519 0.611051
\(903\) −43.0089 −1.43125
\(904\) 13.7021 0.455727
\(905\) −13.7708 −0.457756
\(906\) 42.2813 1.40470
\(907\) 15.8793 0.527263 0.263631 0.964624i \(-0.415080\pi\)
0.263631 + 0.964624i \(0.415080\pi\)
\(908\) 15.1770 0.503666
\(909\) −69.3980 −2.30179
\(910\) −15.9645 −0.529218
\(911\) 31.1585 1.03233 0.516163 0.856490i \(-0.327360\pi\)
0.516163 + 0.856490i \(0.327360\pi\)
\(912\) 6.87649 0.227703
\(913\) 12.4659 0.412560
\(914\) −34.2136 −1.13169
\(915\) 41.2555 1.36387
\(916\) 0.871005 0.0287788
\(917\) −39.3794 −1.30042
\(918\) −23.6564 −0.780778
\(919\) −37.4097 −1.23403 −0.617017 0.786950i \(-0.711659\pi\)
−0.617017 + 0.786950i \(0.711659\pi\)
\(920\) −6.29342 −0.207488
\(921\) −28.8517 −0.950696
\(922\) 15.9895 0.526587
\(923\) 4.79327 0.157772
\(924\) 36.4073 1.19771
\(925\) 1.59485 0.0524382
\(926\) 17.7535 0.583415
\(927\) −58.6776 −1.92723
\(928\) 2.05132 0.0673379
\(929\) −16.1968 −0.531399 −0.265699 0.964056i \(-0.585603\pi\)
−0.265699 + 0.964056i \(0.585603\pi\)
\(930\) −14.0919 −0.462090
\(931\) −10.4734 −0.343250
\(932\) 1.34295 0.0439899
\(933\) −25.5795 −0.837435
\(934\) −20.1913 −0.660680
\(935\) −19.1102 −0.624970
\(936\) 22.5629 0.737491
\(937\) −25.0407 −0.818043 −0.409022 0.912525i \(-0.634130\pi\)
−0.409022 + 0.912525i \(0.634130\pi\)
\(938\) −27.5458 −0.899402
\(939\) 4.56829 0.149080
\(940\) −1.83379 −0.0598117
\(941\) 55.6029 1.81260 0.906301 0.422633i \(-0.138894\pi\)
0.906301 + 0.422633i \(0.138894\pi\)
\(942\) 50.2084 1.63588
\(943\) −29.5802 −0.963263
\(944\) 14.0106 0.456005
\(945\) 16.2020 0.527051
\(946\) 18.0096 0.585543
\(947\) 6.24011 0.202776 0.101388 0.994847i \(-0.467672\pi\)
0.101388 + 0.994847i \(0.467672\pi\)
\(948\) −13.7351 −0.446094
\(949\) 20.5163 0.665987
\(950\) 2.47209 0.0802052
\(951\) −33.0733 −1.07248
\(952\) −16.4065 −0.531738
\(953\) 37.0942 1.20160 0.600799 0.799400i \(-0.294849\pi\)
0.600799 + 0.799400i \(0.294849\pi\)
\(954\) −44.3740 −1.43666
\(955\) −25.8062 −0.835069
\(956\) 26.2745 0.849779
\(957\) 22.2794 0.720191
\(958\) −14.8374 −0.479375
\(959\) −13.9265 −0.449709
\(960\) −2.78165 −0.0897775
\(961\) −5.33561 −0.172116
\(962\) 7.59549 0.244888
\(963\) −23.8068 −0.767165
\(964\) −5.07033 −0.163304
\(965\) 6.45048 0.207648
\(966\) −58.6824 −1.88808
\(967\) 27.4648 0.883210 0.441605 0.897210i \(-0.354409\pi\)
0.441605 + 0.897210i \(0.354409\pi\)
\(968\) −4.24525 −0.136447
\(969\) −33.6562 −1.08119
\(970\) 0.763398 0.0245112
\(971\) 6.78986 0.217897 0.108948 0.994047i \(-0.465252\pi\)
0.108948 + 0.994047i \(0.465252\pi\)
\(972\) 16.6364 0.533614
\(973\) 54.4139 1.74443
\(974\) −13.9455 −0.446844
\(975\) 13.2477 0.424266
\(976\) −14.8313 −0.474738
\(977\) 26.4271 0.845477 0.422738 0.906252i \(-0.361069\pi\)
0.422738 + 0.906252i \(0.361069\pi\)
\(978\) −43.1891 −1.38103
\(979\) 49.1435 1.57063
\(980\) 4.23664 0.135335
\(981\) −62.6768 −2.00112
\(982\) −25.8866 −0.826076
\(983\) −42.2133 −1.34639 −0.673197 0.739463i \(-0.735080\pi\)
−0.673197 + 0.739463i \(0.735080\pi\)
\(984\) −13.0742 −0.416792
\(985\) 20.5238 0.653942
\(986\) −10.0400 −0.319737
\(987\) −17.0990 −0.544267
\(988\) 11.7734 0.374561
\(989\) −29.0284 −0.923051
\(990\) −18.4980 −0.587905
\(991\) −42.5555 −1.35182 −0.675911 0.736984i \(-0.736249\pi\)
−0.675911 + 0.736984i \(0.736249\pi\)
\(992\) 5.06600 0.160846
\(993\) −77.4667 −2.45833
\(994\) −3.37375 −0.107009
\(995\) 2.93538 0.0930578
\(996\) −8.88091 −0.281403
\(997\) 23.1029 0.731677 0.365839 0.930678i \(-0.380782\pi\)
0.365839 + 0.930678i \(0.380782\pi\)
\(998\) −0.374418 −0.0118520
\(999\) −7.70849 −0.243886
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 4010.2.a.m.1.3 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4010.2.a.m.1.3 20 1.1 even 1 trivial