Properties

Label 4010.2.a.m.1.20
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 1.20
Root \(3.32710\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +3.32710 q^{3} +1.00000 q^{4} +1.00000 q^{5} -3.32710 q^{6} -2.87743 q^{7} -1.00000 q^{8} +8.06959 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +3.32710 q^{3} +1.00000 q^{4} +1.00000 q^{5} -3.32710 q^{6} -2.87743 q^{7} -1.00000 q^{8} +8.06959 q^{9} -1.00000 q^{10} +3.79804 q^{11} +3.32710 q^{12} -2.47743 q^{13} +2.87743 q^{14} +3.32710 q^{15} +1.00000 q^{16} +2.94379 q^{17} -8.06959 q^{18} +0.783929 q^{19} +1.00000 q^{20} -9.57348 q^{21} -3.79804 q^{22} -0.0265160 q^{23} -3.32710 q^{24} +1.00000 q^{25} +2.47743 q^{26} +16.8670 q^{27} -2.87743 q^{28} -8.26058 q^{29} -3.32710 q^{30} +0.619229 q^{31} -1.00000 q^{32} +12.6365 q^{33} -2.94379 q^{34} -2.87743 q^{35} +8.06959 q^{36} +1.77151 q^{37} -0.783929 q^{38} -8.24264 q^{39} -1.00000 q^{40} +9.24986 q^{41} +9.57348 q^{42} +1.77538 q^{43} +3.79804 q^{44} +8.06959 q^{45} +0.0265160 q^{46} +10.6532 q^{47} +3.32710 q^{48} +1.27958 q^{49} -1.00000 q^{50} +9.79427 q^{51} -2.47743 q^{52} +5.35945 q^{53} -16.8670 q^{54} +3.79804 q^{55} +2.87743 q^{56} +2.60821 q^{57} +8.26058 q^{58} +4.82597 q^{59} +3.32710 q^{60} +0.650441 q^{61} -0.619229 q^{62} -23.2196 q^{63} +1.00000 q^{64} -2.47743 q^{65} -12.6365 q^{66} -10.3791 q^{67} +2.94379 q^{68} -0.0882214 q^{69} +2.87743 q^{70} +1.13986 q^{71} -8.06959 q^{72} -3.11153 q^{73} -1.77151 q^{74} +3.32710 q^{75} +0.783929 q^{76} -10.9286 q^{77} +8.24264 q^{78} +5.43079 q^{79} +1.00000 q^{80} +31.9095 q^{81} -9.24986 q^{82} -7.44281 q^{83} -9.57348 q^{84} +2.94379 q^{85} -1.77538 q^{86} -27.4838 q^{87} -3.79804 q^{88} +4.32672 q^{89} -8.06959 q^{90} +7.12861 q^{91} -0.0265160 q^{92} +2.06024 q^{93} -10.6532 q^{94} +0.783929 q^{95} -3.32710 q^{96} -6.14854 q^{97} -1.27958 q^{98} +30.6486 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 20 q^{2} + 4 q^{3} + 20 q^{4} + 20 q^{5} - 4 q^{6} + 11 q^{7} - 20 q^{8} + 26 q^{9}+O(q^{10}) \) 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
\(3\) 3.32710 1.92090 0.960451 0.278450i \(-0.0898207\pi\)
0.960451 + 0.278450i \(0.0898207\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −3.32710 −1.35828
\(7\) −2.87743 −1.08756 −0.543782 0.839226i \(-0.683008\pi\)
−0.543782 + 0.839226i \(0.683008\pi\)
\(8\) −1.00000 −0.353553
\(9\) 8.06959 2.68986
\(10\) −1.00000 −0.316228
\(11\) 3.79804 1.14515 0.572576 0.819852i \(-0.305944\pi\)
0.572576 + 0.819852i \(0.305944\pi\)
\(12\) 3.32710 0.960451
\(13\) −2.47743 −0.687115 −0.343557 0.939132i \(-0.611632\pi\)
−0.343557 + 0.939132i \(0.611632\pi\)
\(14\) 2.87743 0.769024
\(15\) 3.32710 0.859053
\(16\) 1.00000 0.250000
\(17\) 2.94379 0.713973 0.356986 0.934110i \(-0.383804\pi\)
0.356986 + 0.934110i \(0.383804\pi\)
\(18\) −8.06959 −1.90202
\(19\) 0.783929 0.179846 0.0899229 0.995949i \(-0.471338\pi\)
0.0899229 + 0.995949i \(0.471338\pi\)
\(20\) 1.00000 0.223607
\(21\) −9.57348 −2.08910
\(22\) −3.79804 −0.809745
\(23\) −0.0265160 −0.00552897 −0.00276449 0.999996i \(-0.500880\pi\)
−0.00276449 + 0.999996i \(0.500880\pi\)
\(24\) −3.32710 −0.679141
\(25\) 1.00000 0.200000
\(26\) 2.47743 0.485863
\(27\) 16.8670 3.24606
\(28\) −2.87743 −0.543782
\(29\) −8.26058 −1.53395 −0.766976 0.641676i \(-0.778239\pi\)
−0.766976 + 0.641676i \(0.778239\pi\)
\(30\) −3.32710 −0.607442
\(31\) 0.619229 0.111217 0.0556084 0.998453i \(-0.482290\pi\)
0.0556084 + 0.998453i \(0.482290\pi\)
\(32\) −1.00000 −0.176777
\(33\) 12.6365 2.19972
\(34\) −2.94379 −0.504855
\(35\) −2.87743 −0.486374
\(36\) 8.06959 1.34493
\(37\) 1.77151 0.291235 0.145617 0.989341i \(-0.453483\pi\)
0.145617 + 0.989341i \(0.453483\pi\)
\(38\) −0.783929 −0.127170
\(39\) −8.24264 −1.31988
\(40\) −1.00000 −0.158114
\(41\) 9.24986 1.44458 0.722292 0.691588i \(-0.243088\pi\)
0.722292 + 0.691588i \(0.243088\pi\)
\(42\) 9.57348 1.47722
\(43\) 1.77538 0.270742 0.135371 0.990795i \(-0.456777\pi\)
0.135371 + 0.990795i \(0.456777\pi\)
\(44\) 3.79804 0.572576
\(45\) 8.06959 1.20294
\(46\) 0.0265160 0.00390957
\(47\) 10.6532 1.55392 0.776962 0.629548i \(-0.216760\pi\)
0.776962 + 0.629548i \(0.216760\pi\)
\(48\) 3.32710 0.480225
\(49\) 1.27958 0.182797
\(50\) −1.00000 −0.141421
\(51\) 9.79427 1.37147
\(52\) −2.47743 −0.343557
\(53\) 5.35945 0.736177 0.368088 0.929791i \(-0.380012\pi\)
0.368088 + 0.929791i \(0.380012\pi\)
\(54\) −16.8670 −2.29531
\(55\) 3.79804 0.512128
\(56\) 2.87743 0.384512
\(57\) 2.60821 0.345466
\(58\) 8.26058 1.08467
\(59\) 4.82597 0.628287 0.314144 0.949375i \(-0.398283\pi\)
0.314144 + 0.949375i \(0.398283\pi\)
\(60\) 3.32710 0.429527
\(61\) 0.650441 0.0832805 0.0416402 0.999133i \(-0.486742\pi\)
0.0416402 + 0.999133i \(0.486742\pi\)
\(62\) −0.619229 −0.0786422
\(63\) −23.2196 −2.92540
\(64\) 1.00000 0.125000
\(65\) −2.47743 −0.307287
\(66\) −12.6365 −1.55544
\(67\) −10.3791 −1.26800 −0.634002 0.773332i \(-0.718589\pi\)
−0.634002 + 0.773332i \(0.718589\pi\)
\(68\) 2.94379 0.356986
\(69\) −0.0882214 −0.0106206
\(70\) 2.87743 0.343918
\(71\) 1.13986 0.135277 0.0676383 0.997710i \(-0.478454\pi\)
0.0676383 + 0.997710i \(0.478454\pi\)
\(72\) −8.06959 −0.951010
\(73\) −3.11153 −0.364177 −0.182089 0.983282i \(-0.558286\pi\)
−0.182089 + 0.983282i \(0.558286\pi\)
\(74\) −1.77151 −0.205934
\(75\) 3.32710 0.384180
\(76\) 0.783929 0.0899229
\(77\) −10.9286 −1.24543
\(78\) 8.24264 0.933296
\(79\) 5.43079 0.611012 0.305506 0.952190i \(-0.401174\pi\)
0.305506 + 0.952190i \(0.401174\pi\)
\(80\) 1.00000 0.111803
\(81\) 31.9095 3.54550
\(82\) −9.24986 −1.02148
\(83\) −7.44281 −0.816955 −0.408478 0.912768i \(-0.633940\pi\)
−0.408478 + 0.912768i \(0.633940\pi\)
\(84\) −9.57348 −1.04455
\(85\) 2.94379 0.319298
\(86\) −1.77538 −0.191444
\(87\) −27.4838 −2.94657
\(88\) −3.79804 −0.404872
\(89\) 4.32672 0.458631 0.229316 0.973352i \(-0.426351\pi\)
0.229316 + 0.973352i \(0.426351\pi\)
\(90\) −8.06959 −0.850609
\(91\) 7.12861 0.747282
\(92\) −0.0265160 −0.00276449
\(93\) 2.06024 0.213637
\(94\) −10.6532 −1.09879
\(95\) 0.783929 0.0804295
\(96\) −3.32710 −0.339571
\(97\) −6.14854 −0.624290 −0.312145 0.950034i \(-0.601047\pi\)
−0.312145 + 0.950034i \(0.601047\pi\)
\(98\) −1.27958 −0.129257
\(99\) 30.6486 3.08030
\(100\) 1.00000 0.100000
\(101\) 13.2042 1.31386 0.656931 0.753950i \(-0.271854\pi\)
0.656931 + 0.753950i \(0.271854\pi\)
\(102\) −9.79427 −0.969777
\(103\) −4.04644 −0.398708 −0.199354 0.979928i \(-0.563884\pi\)
−0.199354 + 0.979928i \(0.563884\pi\)
\(104\) 2.47743 0.242932
\(105\) −9.57348 −0.934276
\(106\) −5.35945 −0.520556
\(107\) −13.4285 −1.29818 −0.649092 0.760710i \(-0.724851\pi\)
−0.649092 + 0.760710i \(0.724851\pi\)
\(108\) 16.8670 1.62303
\(109\) −2.02380 −0.193845 −0.0969223 0.995292i \(-0.530900\pi\)
−0.0969223 + 0.995292i \(0.530900\pi\)
\(110\) −3.79804 −0.362129
\(111\) 5.89399 0.559433
\(112\) −2.87743 −0.271891
\(113\) 12.7031 1.19501 0.597506 0.801865i \(-0.296159\pi\)
0.597506 + 0.801865i \(0.296159\pi\)
\(114\) −2.60821 −0.244281
\(115\) −0.0265160 −0.00247263
\(116\) −8.26058 −0.766976
\(117\) −19.9918 −1.84824
\(118\) −4.82597 −0.444266
\(119\) −8.47053 −0.776492
\(120\) −3.32710 −0.303721
\(121\) 3.42510 0.311373
\(122\) −0.650441 −0.0588882
\(123\) 30.7752 2.77491
\(124\) 0.619229 0.0556084
\(125\) 1.00000 0.0894427
\(126\) 23.2196 2.06857
\(127\) 5.11512 0.453894 0.226947 0.973907i \(-0.427126\pi\)
0.226947 + 0.973907i \(0.427126\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 5.90686 0.520070
\(130\) 2.47743 0.217285
\(131\) 2.82921 0.247190 0.123595 0.992333i \(-0.460558\pi\)
0.123595 + 0.992333i \(0.460558\pi\)
\(132\) 12.6365 1.09986
\(133\) −2.25570 −0.195594
\(134\) 10.3791 0.896614
\(135\) 16.8670 1.45168
\(136\) −2.94379 −0.252428
\(137\) 8.47126 0.723749 0.361874 0.932227i \(-0.382137\pi\)
0.361874 + 0.932227i \(0.382137\pi\)
\(138\) 0.0882214 0.00750990
\(139\) −7.21459 −0.611934 −0.305967 0.952042i \(-0.598980\pi\)
−0.305967 + 0.952042i \(0.598980\pi\)
\(140\) −2.87743 −0.243187
\(141\) 35.4441 2.98493
\(142\) −1.13986 −0.0956550
\(143\) −9.40936 −0.786851
\(144\) 8.06959 0.672466
\(145\) −8.26058 −0.686004
\(146\) 3.11153 0.257512
\(147\) 4.25728 0.351135
\(148\) 1.77151 0.145617
\(149\) 4.04445 0.331334 0.165667 0.986182i \(-0.447022\pi\)
0.165667 + 0.986182i \(0.447022\pi\)
\(150\) −3.32710 −0.271656
\(151\) −4.20941 −0.342557 −0.171278 0.985223i \(-0.554790\pi\)
−0.171278 + 0.985223i \(0.554790\pi\)
\(152\) −0.783929 −0.0635851
\(153\) 23.7551 1.92049
\(154\) 10.9286 0.880650
\(155\) 0.619229 0.0497377
\(156\) −8.24264 −0.659940
\(157\) −22.1248 −1.76575 −0.882876 0.469606i \(-0.844396\pi\)
−0.882876 + 0.469606i \(0.844396\pi\)
\(158\) −5.43079 −0.432051
\(159\) 17.8314 1.41412
\(160\) −1.00000 −0.0790569
\(161\) 0.0762979 0.00601311
\(162\) −31.9095 −2.50705
\(163\) 13.6761 1.07120 0.535599 0.844472i \(-0.320086\pi\)
0.535599 + 0.844472i \(0.320086\pi\)
\(164\) 9.24986 0.722292
\(165\) 12.6365 0.983747
\(166\) 7.44281 0.577674
\(167\) −22.8840 −1.77082 −0.885409 0.464812i \(-0.846122\pi\)
−0.885409 + 0.464812i \(0.846122\pi\)
\(168\) 9.57348 0.738610
\(169\) −6.86236 −0.527874
\(170\) −2.94379 −0.225778
\(171\) 6.32599 0.483760
\(172\) 1.77538 0.135371
\(173\) −14.6738 −1.11563 −0.557815 0.829965i \(-0.688360\pi\)
−0.557815 + 0.829965i \(0.688360\pi\)
\(174\) 27.4838 2.08354
\(175\) −2.87743 −0.217513
\(176\) 3.79804 0.286288
\(177\) 16.0565 1.20688
\(178\) −4.32672 −0.324301
\(179\) 9.59560 0.717209 0.358604 0.933490i \(-0.383253\pi\)
0.358604 + 0.933490i \(0.383253\pi\)
\(180\) 8.06959 0.601472
\(181\) 10.2207 0.759700 0.379850 0.925048i \(-0.375976\pi\)
0.379850 + 0.925048i \(0.375976\pi\)
\(182\) −7.12861 −0.528408
\(183\) 2.16408 0.159974
\(184\) 0.0265160 0.00195479
\(185\) 1.77151 0.130244
\(186\) −2.06024 −0.151064
\(187\) 11.1806 0.817608
\(188\) 10.6532 0.776962
\(189\) −48.5336 −3.53030
\(190\) −0.783929 −0.0568722
\(191\) −2.14502 −0.155208 −0.0776040 0.996984i \(-0.524727\pi\)
−0.0776040 + 0.996984i \(0.524727\pi\)
\(192\) 3.32710 0.240113
\(193\) 14.3461 1.03265 0.516326 0.856392i \(-0.327299\pi\)
0.516326 + 0.856392i \(0.327299\pi\)
\(194\) 6.14854 0.441440
\(195\) −8.24264 −0.590268
\(196\) 1.27958 0.0913985
\(197\) 12.6963 0.904575 0.452287 0.891872i \(-0.350608\pi\)
0.452287 + 0.891872i \(0.350608\pi\)
\(198\) −30.6486 −2.17810
\(199\) −12.5782 −0.891645 −0.445823 0.895121i \(-0.647089\pi\)
−0.445823 + 0.895121i \(0.647089\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −34.5321 −2.43571
\(202\) −13.2042 −0.929041
\(203\) 23.7692 1.66827
\(204\) 9.79427 0.685736
\(205\) 9.24986 0.646038
\(206\) 4.04644 0.281929
\(207\) −0.213973 −0.0148722
\(208\) −2.47743 −0.171779
\(209\) 2.97739 0.205951
\(210\) 9.57348 0.660633
\(211\) −6.56554 −0.451991 −0.225995 0.974128i \(-0.572563\pi\)
−0.225995 + 0.974128i \(0.572563\pi\)
\(212\) 5.35945 0.368088
\(213\) 3.79243 0.259853
\(214\) 13.4285 0.917954
\(215\) 1.77538 0.121080
\(216\) −16.8670 −1.14766
\(217\) −1.78179 −0.120956
\(218\) 2.02380 0.137069
\(219\) −10.3524 −0.699548
\(220\) 3.79804 0.256064
\(221\) −7.29301 −0.490581
\(222\) −5.89399 −0.395579
\(223\) 22.0602 1.47726 0.738631 0.674110i \(-0.235473\pi\)
0.738631 + 0.674110i \(0.235473\pi\)
\(224\) 2.87743 0.192256
\(225\) 8.06959 0.537972
\(226\) −12.7031 −0.845000
\(227\) 12.4121 0.823822 0.411911 0.911224i \(-0.364861\pi\)
0.411911 + 0.911224i \(0.364861\pi\)
\(228\) 2.60821 0.172733
\(229\) 8.44156 0.557834 0.278917 0.960315i \(-0.410025\pi\)
0.278917 + 0.960315i \(0.410025\pi\)
\(230\) 0.0265160 0.00174841
\(231\) −36.3605 −2.39234
\(232\) 8.26058 0.542334
\(233\) −21.0833 −1.38122 −0.690608 0.723230i \(-0.742657\pi\)
−0.690608 + 0.723230i \(0.742657\pi\)
\(234\) 19.9918 1.30691
\(235\) 10.6532 0.694936
\(236\) 4.82597 0.314144
\(237\) 18.0688 1.17369
\(238\) 8.47053 0.549063
\(239\) 23.5046 1.52039 0.760194 0.649696i \(-0.225104\pi\)
0.760194 + 0.649696i \(0.225104\pi\)
\(240\) 3.32710 0.214763
\(241\) 27.0218 1.74063 0.870314 0.492497i \(-0.163916\pi\)
0.870314 + 0.492497i \(0.163916\pi\)
\(242\) −3.42510 −0.220174
\(243\) 55.5649 3.56449
\(244\) 0.650441 0.0416402
\(245\) 1.27958 0.0817493
\(246\) −30.7752 −1.96215
\(247\) −1.94213 −0.123575
\(248\) −0.619229 −0.0393211
\(249\) −24.7630 −1.56929
\(250\) −1.00000 −0.0632456
\(251\) 11.5473 0.728858 0.364429 0.931231i \(-0.381264\pi\)
0.364429 + 0.931231i \(0.381264\pi\)
\(252\) −23.2196 −1.46270
\(253\) −0.100709 −0.00633151
\(254\) −5.11512 −0.320951
\(255\) 9.79427 0.613341
\(256\) 1.00000 0.0625000
\(257\) −11.9453 −0.745128 −0.372564 0.928006i \(-0.621521\pi\)
−0.372564 + 0.928006i \(0.621521\pi\)
\(258\) −5.90686 −0.367745
\(259\) −5.09739 −0.316736
\(260\) −2.47743 −0.153643
\(261\) −66.6595 −4.12612
\(262\) −2.82921 −0.174789
\(263\) −21.0381 −1.29727 −0.648633 0.761102i \(-0.724659\pi\)
−0.648633 + 0.761102i \(0.724659\pi\)
\(264\) −12.6365 −0.777720
\(265\) 5.35945 0.329228
\(266\) 2.25570 0.138306
\(267\) 14.3954 0.880985
\(268\) −10.3791 −0.634002
\(269\) −8.55608 −0.521673 −0.260837 0.965383i \(-0.583998\pi\)
−0.260837 + 0.965383i \(0.583998\pi\)
\(270\) −16.8670 −1.02649
\(271\) −23.1493 −1.40622 −0.703110 0.711081i \(-0.748206\pi\)
−0.703110 + 0.711081i \(0.748206\pi\)
\(272\) 2.94379 0.178493
\(273\) 23.7176 1.43545
\(274\) −8.47126 −0.511768
\(275\) 3.79804 0.229030
\(276\) −0.0882214 −0.00531030
\(277\) −23.3367 −1.40217 −0.701083 0.713079i \(-0.747300\pi\)
−0.701083 + 0.713079i \(0.747300\pi\)
\(278\) 7.21459 0.432703
\(279\) 4.99692 0.299158
\(280\) 2.87743 0.171959
\(281\) −20.0167 −1.19410 −0.597048 0.802206i \(-0.703660\pi\)
−0.597048 + 0.802206i \(0.703660\pi\)
\(282\) −35.4441 −2.11067
\(283\) 18.7895 1.11692 0.558460 0.829531i \(-0.311392\pi\)
0.558460 + 0.829531i \(0.311392\pi\)
\(284\) 1.13986 0.0676383
\(285\) 2.60821 0.154497
\(286\) 9.40936 0.556387
\(287\) −26.6158 −1.57108
\(288\) −8.06959 −0.475505
\(289\) −8.33412 −0.490243
\(290\) 8.26058 0.485078
\(291\) −20.4568 −1.19920
\(292\) −3.11153 −0.182089
\(293\) −15.9251 −0.930353 −0.465176 0.885218i \(-0.654009\pi\)
−0.465176 + 0.885218i \(0.654009\pi\)
\(294\) −4.25728 −0.248290
\(295\) 4.82597 0.280979
\(296\) −1.77151 −0.102967
\(297\) 64.0616 3.71723
\(298\) −4.04445 −0.234289
\(299\) 0.0656915 0.00379904
\(300\) 3.32710 0.192090
\(301\) −5.10852 −0.294450
\(302\) 4.20941 0.242224
\(303\) 43.9315 2.52380
\(304\) 0.783929 0.0449614
\(305\) 0.650441 0.0372442
\(306\) −23.7551 −1.35799
\(307\) −13.0738 −0.746159 −0.373080 0.927799i \(-0.621698\pi\)
−0.373080 + 0.927799i \(0.621698\pi\)
\(308\) −10.9286 −0.622713
\(309\) −13.4629 −0.765879
\(310\) −0.619229 −0.0351699
\(311\) 4.57265 0.259291 0.129646 0.991560i \(-0.458616\pi\)
0.129646 + 0.991560i \(0.458616\pi\)
\(312\) 8.24264 0.466648
\(313\) −22.5845 −1.27655 −0.638275 0.769808i \(-0.720352\pi\)
−0.638275 + 0.769808i \(0.720352\pi\)
\(314\) 22.1248 1.24858
\(315\) −23.2196 −1.30828
\(316\) 5.43079 0.305506
\(317\) −15.0700 −0.846418 −0.423209 0.906032i \(-0.639096\pi\)
−0.423209 + 0.906032i \(0.639096\pi\)
\(318\) −17.8314 −0.999936
\(319\) −31.3740 −1.75661
\(320\) 1.00000 0.0559017
\(321\) −44.6780 −2.49368
\(322\) −0.0762979 −0.00425191
\(323\) 2.30772 0.128405
\(324\) 31.9095 1.77275
\(325\) −2.47743 −0.137423
\(326\) −13.6761 −0.757452
\(327\) −6.73337 −0.372356
\(328\) −9.24986 −0.510738
\(329\) −30.6537 −1.68999
\(330\) −12.6365 −0.695614
\(331\) −0.290068 −0.0159436 −0.00797178 0.999968i \(-0.502538\pi\)
−0.00797178 + 0.999968i \(0.502538\pi\)
\(332\) −7.44281 −0.408478
\(333\) 14.2954 0.783381
\(334\) 22.8840 1.25216
\(335\) −10.3791 −0.567068
\(336\) −9.57348 −0.522276
\(337\) −14.2582 −0.776696 −0.388348 0.921513i \(-0.626954\pi\)
−0.388348 + 0.921513i \(0.626954\pi\)
\(338\) 6.86236 0.373263
\(339\) 42.2646 2.29550
\(340\) 2.94379 0.159649
\(341\) 2.35186 0.127360
\(342\) −6.32599 −0.342070
\(343\) 16.4601 0.888761
\(344\) −1.77538 −0.0957219
\(345\) −0.0882214 −0.00474968
\(346\) 14.6738 0.788870
\(347\) −19.9345 −1.07014 −0.535069 0.844808i \(-0.679714\pi\)
−0.535069 + 0.844808i \(0.679714\pi\)
\(348\) −27.4838 −1.47329
\(349\) −5.22658 −0.279773 −0.139886 0.990168i \(-0.544674\pi\)
−0.139886 + 0.990168i \(0.544674\pi\)
\(350\) 2.87743 0.153805
\(351\) −41.7868 −2.23041
\(352\) −3.79804 −0.202436
\(353\) −20.6145 −1.09720 −0.548599 0.836085i \(-0.684839\pi\)
−0.548599 + 0.836085i \(0.684839\pi\)
\(354\) −16.0565 −0.853392
\(355\) 1.13986 0.0604975
\(356\) 4.32672 0.229316
\(357\) −28.1823 −1.49156
\(358\) −9.59560 −0.507143
\(359\) −17.7824 −0.938517 −0.469259 0.883061i \(-0.655479\pi\)
−0.469259 + 0.883061i \(0.655479\pi\)
\(360\) −8.06959 −0.425305
\(361\) −18.3855 −0.967656
\(362\) −10.2207 −0.537189
\(363\) 11.3957 0.598117
\(364\) 7.12861 0.373641
\(365\) −3.11153 −0.162865
\(366\) −2.16408 −0.113118
\(367\) −3.36265 −0.175529 −0.0877643 0.996141i \(-0.527972\pi\)
−0.0877643 + 0.996141i \(0.527972\pi\)
\(368\) −0.0265160 −0.00138224
\(369\) 74.6425 3.88573
\(370\) −1.77151 −0.0920965
\(371\) −15.4214 −0.800640
\(372\) 2.06024 0.106818
\(373\) 6.13110 0.317457 0.158728 0.987322i \(-0.449261\pi\)
0.158728 + 0.987322i \(0.449261\pi\)
\(374\) −11.1806 −0.578136
\(375\) 3.32710 0.171811
\(376\) −10.6532 −0.549395
\(377\) 20.4650 1.05400
\(378\) 48.5336 2.49630
\(379\) 29.4156 1.51098 0.755488 0.655163i \(-0.227400\pi\)
0.755488 + 0.655163i \(0.227400\pi\)
\(380\) 0.783929 0.0402147
\(381\) 17.0185 0.871885
\(382\) 2.14502 0.109749
\(383\) −15.2014 −0.776756 −0.388378 0.921500i \(-0.626964\pi\)
−0.388378 + 0.921500i \(0.626964\pi\)
\(384\) −3.32710 −0.169785
\(385\) −10.9286 −0.556972
\(386\) −14.3461 −0.730195
\(387\) 14.3266 0.728260
\(388\) −6.14854 −0.312145
\(389\) 13.5439 0.686704 0.343352 0.939207i \(-0.388438\pi\)
0.343352 + 0.939207i \(0.388438\pi\)
\(390\) 8.24264 0.417382
\(391\) −0.0780575 −0.00394754
\(392\) −1.27958 −0.0646285
\(393\) 9.41307 0.474827
\(394\) −12.6963 −0.639631
\(395\) 5.43079 0.273253
\(396\) 30.6486 1.54015
\(397\) −20.7482 −1.04132 −0.520661 0.853763i \(-0.674314\pi\)
−0.520661 + 0.853763i \(0.674314\pi\)
\(398\) 12.5782 0.630488
\(399\) −7.50493 −0.375717
\(400\) 1.00000 0.0500000
\(401\) −1.00000 −0.0499376
\(402\) 34.5321 1.72231
\(403\) −1.53410 −0.0764187
\(404\) 13.2042 0.656931
\(405\) 31.9095 1.58559
\(406\) −23.7692 −1.17965
\(407\) 6.72827 0.333508
\(408\) −9.79427 −0.484888
\(409\) 32.3855 1.60136 0.800679 0.599093i \(-0.204472\pi\)
0.800679 + 0.599093i \(0.204472\pi\)
\(410\) −9.24986 −0.456818
\(411\) 28.1847 1.39025
\(412\) −4.04644 −0.199354
\(413\) −13.8864 −0.683303
\(414\) 0.213973 0.0105162
\(415\) −7.44281 −0.365353
\(416\) 2.47743 0.121466
\(417\) −24.0037 −1.17546
\(418\) −2.97739 −0.145629
\(419\) −18.4374 −0.900725 −0.450362 0.892846i \(-0.648705\pi\)
−0.450362 + 0.892846i \(0.648705\pi\)
\(420\) −9.57348 −0.467138
\(421\) 15.3683 0.749007 0.374503 0.927226i \(-0.377813\pi\)
0.374503 + 0.927226i \(0.377813\pi\)
\(422\) 6.56554 0.319606
\(423\) 85.9666 4.17984
\(424\) −5.35945 −0.260278
\(425\) 2.94379 0.142795
\(426\) −3.79243 −0.183744
\(427\) −1.87160 −0.0905729
\(428\) −13.4285 −0.649092
\(429\) −31.3059 −1.51146
\(430\) −1.77538 −0.0856163
\(431\) 37.0475 1.78452 0.892258 0.451525i \(-0.149120\pi\)
0.892258 + 0.451525i \(0.149120\pi\)
\(432\) 16.8670 0.811515
\(433\) 30.8682 1.48343 0.741716 0.670714i \(-0.234012\pi\)
0.741716 + 0.670714i \(0.234012\pi\)
\(434\) 1.78179 0.0855285
\(435\) −27.4838 −1.31775
\(436\) −2.02380 −0.0969223
\(437\) −0.0207867 −0.000994362 0
\(438\) 10.3524 0.494655
\(439\) −38.2997 −1.82795 −0.913973 0.405774i \(-0.867002\pi\)
−0.913973 + 0.405774i \(0.867002\pi\)
\(440\) −3.79804 −0.181064
\(441\) 10.3257 0.491699
\(442\) 7.29301 0.346893
\(443\) −9.53641 −0.453089 −0.226544 0.974001i \(-0.572743\pi\)
−0.226544 + 0.974001i \(0.572743\pi\)
\(444\) 5.89399 0.279716
\(445\) 4.32672 0.205106
\(446\) −22.0602 −1.04458
\(447\) 13.4563 0.636460
\(448\) −2.87743 −0.135946
\(449\) −36.7230 −1.73306 −0.866532 0.499121i \(-0.833656\pi\)
−0.866532 + 0.499121i \(0.833656\pi\)
\(450\) −8.06959 −0.380404
\(451\) 35.1313 1.65427
\(452\) 12.7031 0.597506
\(453\) −14.0051 −0.658018
\(454\) −12.4121 −0.582530
\(455\) 7.12861 0.334194
\(456\) −2.60821 −0.122141
\(457\) 34.4625 1.61209 0.806044 0.591856i \(-0.201604\pi\)
0.806044 + 0.591856i \(0.201604\pi\)
\(458\) −8.44156 −0.394448
\(459\) 49.6529 2.31760
\(460\) −0.0265160 −0.00123632
\(461\) 32.8634 1.53060 0.765300 0.643674i \(-0.222591\pi\)
0.765300 + 0.643674i \(0.222591\pi\)
\(462\) 36.3605 1.69164
\(463\) −3.98341 −0.185125 −0.0925624 0.995707i \(-0.529506\pi\)
−0.0925624 + 0.995707i \(0.529506\pi\)
\(464\) −8.26058 −0.383488
\(465\) 2.06024 0.0955412
\(466\) 21.0833 0.976667
\(467\) 1.64490 0.0761171 0.0380586 0.999276i \(-0.487883\pi\)
0.0380586 + 0.999276i \(0.487883\pi\)
\(468\) −19.9918 −0.924122
\(469\) 29.8650 1.37904
\(470\) −10.6532 −0.491394
\(471\) −73.6114 −3.39184
\(472\) −4.82597 −0.222133
\(473\) 6.74295 0.310041
\(474\) −18.0688 −0.829927
\(475\) 0.783929 0.0359691
\(476\) −8.47053 −0.388246
\(477\) 43.2485 1.98021
\(478\) −23.5046 −1.07508
\(479\) 0.132786 0.00606715 0.00303357 0.999995i \(-0.499034\pi\)
0.00303357 + 0.999995i \(0.499034\pi\)
\(480\) −3.32710 −0.151861
\(481\) −4.38879 −0.200112
\(482\) −27.0218 −1.23081
\(483\) 0.253851 0.0115506
\(484\) 3.42510 0.155687
\(485\) −6.14854 −0.279191
\(486\) −55.5649 −2.52048
\(487\) −14.6785 −0.665148 −0.332574 0.943077i \(-0.607917\pi\)
−0.332574 + 0.943077i \(0.607917\pi\)
\(488\) −0.650441 −0.0294441
\(489\) 45.5019 2.05767
\(490\) −1.27958 −0.0578055
\(491\) 6.99801 0.315816 0.157908 0.987454i \(-0.449525\pi\)
0.157908 + 0.987454i \(0.449525\pi\)
\(492\) 30.7752 1.38745
\(493\) −24.3174 −1.09520
\(494\) 1.94213 0.0873805
\(495\) 30.6486 1.37755
\(496\) 0.619229 0.0278042
\(497\) −3.27986 −0.147122
\(498\) 24.7630 1.10966
\(499\) −0.837087 −0.0374732 −0.0187366 0.999824i \(-0.505964\pi\)
−0.0187366 + 0.999824i \(0.505964\pi\)
\(500\) 1.00000 0.0447214
\(501\) −76.1374 −3.40157
\(502\) −11.5473 −0.515380
\(503\) 22.7655 1.01507 0.507533 0.861633i \(-0.330558\pi\)
0.507533 + 0.861633i \(0.330558\pi\)
\(504\) 23.2196 1.03428
\(505\) 13.2042 0.587577
\(506\) 0.100709 0.00447706
\(507\) −22.8317 −1.01399
\(508\) 5.11512 0.226947
\(509\) 5.32735 0.236131 0.118065 0.993006i \(-0.462331\pi\)
0.118065 + 0.993006i \(0.462331\pi\)
\(510\) −9.79427 −0.433697
\(511\) 8.95320 0.396066
\(512\) −1.00000 −0.0441942
\(513\) 13.2226 0.583790
\(514\) 11.9453 0.526885
\(515\) −4.04644 −0.178308
\(516\) 5.90686 0.260035
\(517\) 40.4611 1.77948
\(518\) 5.09739 0.223966
\(519\) −48.8213 −2.14302
\(520\) 2.47743 0.108642
\(521\) −0.814586 −0.0356877 −0.0178438 0.999841i \(-0.505680\pi\)
−0.0178438 + 0.999841i \(0.505680\pi\)
\(522\) 66.6595 2.91761
\(523\) 34.2010 1.49551 0.747753 0.663977i \(-0.231133\pi\)
0.747753 + 0.663977i \(0.231133\pi\)
\(524\) 2.82921 0.123595
\(525\) −9.57348 −0.417821
\(526\) 21.0381 0.917305
\(527\) 1.82288 0.0794058
\(528\) 12.6365 0.549931
\(529\) −22.9993 −0.999969
\(530\) −5.35945 −0.232800
\(531\) 38.9436 1.69001
\(532\) −2.25570 −0.0977969
\(533\) −22.9158 −0.992595
\(534\) −14.3954 −0.622951
\(535\) −13.4285 −0.580565
\(536\) 10.3791 0.448307
\(537\) 31.9255 1.37769
\(538\) 8.55608 0.368879
\(539\) 4.85989 0.209330
\(540\) 16.8670 0.725841
\(541\) 40.9278 1.75962 0.879812 0.475321i \(-0.157668\pi\)
0.879812 + 0.475321i \(0.157668\pi\)
\(542\) 23.1493 0.994348
\(543\) 34.0053 1.45931
\(544\) −2.94379 −0.126214
\(545\) −2.02380 −0.0866899
\(546\) −23.7176 −1.01502
\(547\) 25.3583 1.08424 0.542121 0.840301i \(-0.317622\pi\)
0.542121 + 0.840301i \(0.317622\pi\)
\(548\) 8.47126 0.361874
\(549\) 5.24879 0.224013
\(550\) −3.79804 −0.161949
\(551\) −6.47571 −0.275875
\(552\) 0.0882214 0.00375495
\(553\) −15.6267 −0.664515
\(554\) 23.3367 0.991481
\(555\) 5.89399 0.250186
\(556\) −7.21459 −0.305967
\(557\) −34.5302 −1.46309 −0.731546 0.681792i \(-0.761201\pi\)
−0.731546 + 0.681792i \(0.761201\pi\)
\(558\) −4.99692 −0.211537
\(559\) −4.39837 −0.186031
\(560\) −2.87743 −0.121593
\(561\) 37.1990 1.57054
\(562\) 20.0167 0.844353
\(563\) −8.33455 −0.351259 −0.175630 0.984456i \(-0.556196\pi\)
−0.175630 + 0.984456i \(0.556196\pi\)
\(564\) 35.4441 1.49247
\(565\) 12.7031 0.534425
\(566\) −18.7895 −0.789782
\(567\) −91.8171 −3.85596
\(568\) −1.13986 −0.0478275
\(569\) −27.0660 −1.13467 −0.567333 0.823489i \(-0.692025\pi\)
−0.567333 + 0.823489i \(0.692025\pi\)
\(570\) −2.60821 −0.109246
\(571\) −19.9083 −0.833137 −0.416569 0.909104i \(-0.636767\pi\)
−0.416569 + 0.909104i \(0.636767\pi\)
\(572\) −9.40936 −0.393425
\(573\) −7.13668 −0.298139
\(574\) 26.6158 1.11092
\(575\) −0.0265160 −0.00110579
\(576\) 8.06959 0.336233
\(577\) −30.6239 −1.27489 −0.637444 0.770496i \(-0.720008\pi\)
−0.637444 + 0.770496i \(0.720008\pi\)
\(578\) 8.33412 0.346654
\(579\) 47.7307 1.98362
\(580\) −8.26058 −0.343002
\(581\) 21.4161 0.888491
\(582\) 20.4568 0.847962
\(583\) 20.3554 0.843035
\(584\) 3.11153 0.128756
\(585\) −19.9918 −0.826560
\(586\) 15.9251 0.657859
\(587\) 7.03383 0.290317 0.145159 0.989408i \(-0.453631\pi\)
0.145159 + 0.989408i \(0.453631\pi\)
\(588\) 4.25728 0.175567
\(589\) 0.485432 0.0200019
\(590\) −4.82597 −0.198682
\(591\) 42.2419 1.73760
\(592\) 1.77151 0.0728086
\(593\) −38.6111 −1.58557 −0.792784 0.609503i \(-0.791369\pi\)
−0.792784 + 0.609503i \(0.791369\pi\)
\(594\) −64.0616 −2.62848
\(595\) −8.47053 −0.347258
\(596\) 4.04445 0.165667
\(597\) −41.8489 −1.71276
\(598\) −0.0656915 −0.00268632
\(599\) −13.7963 −0.563703 −0.281852 0.959458i \(-0.590949\pi\)
−0.281852 + 0.959458i \(0.590949\pi\)
\(600\) −3.32710 −0.135828
\(601\) 37.2149 1.51803 0.759014 0.651075i \(-0.225682\pi\)
0.759014 + 0.651075i \(0.225682\pi\)
\(602\) 5.10852 0.208208
\(603\) −83.7547 −3.41075
\(604\) −4.20941 −0.171278
\(605\) 3.42510 0.139250
\(606\) −43.9315 −1.78460
\(607\) −31.3955 −1.27430 −0.637152 0.770738i \(-0.719888\pi\)
−0.637152 + 0.770738i \(0.719888\pi\)
\(608\) −0.783929 −0.0317925
\(609\) 79.0825 3.20459
\(610\) −0.650441 −0.0263356
\(611\) −26.3924 −1.06772
\(612\) 23.7551 0.960245
\(613\) 44.8344 1.81084 0.905422 0.424513i \(-0.139555\pi\)
0.905422 + 0.424513i \(0.139555\pi\)
\(614\) 13.0738 0.527614
\(615\) 30.7752 1.24098
\(616\) 10.9286 0.440325
\(617\) −35.8872 −1.44476 −0.722382 0.691494i \(-0.756953\pi\)
−0.722382 + 0.691494i \(0.756953\pi\)
\(618\) 13.4629 0.541558
\(619\) 5.64832 0.227025 0.113513 0.993537i \(-0.463790\pi\)
0.113513 + 0.993537i \(0.463790\pi\)
\(620\) 0.619229 0.0248688
\(621\) −0.447246 −0.0179474
\(622\) −4.57265 −0.183347
\(623\) −12.4498 −0.498791
\(624\) −8.24264 −0.329970
\(625\) 1.00000 0.0400000
\(626\) 22.5845 0.902657
\(627\) 9.90609 0.395611
\(628\) −22.1248 −0.882876
\(629\) 5.21495 0.207934
\(630\) 23.2196 0.925092
\(631\) −35.7341 −1.42255 −0.711277 0.702912i \(-0.751883\pi\)
−0.711277 + 0.702912i \(0.751883\pi\)
\(632\) −5.43079 −0.216025
\(633\) −21.8442 −0.868230
\(634\) 15.0700 0.598508
\(635\) 5.11512 0.202987
\(636\) 17.8314 0.707062
\(637\) −3.17006 −0.125602
\(638\) 31.3740 1.24211
\(639\) 9.19820 0.363875
\(640\) −1.00000 −0.0395285
\(641\) −11.8594 −0.468418 −0.234209 0.972186i \(-0.575250\pi\)
−0.234209 + 0.972186i \(0.575250\pi\)
\(642\) 44.6780 1.76330
\(643\) −13.4394 −0.530000 −0.265000 0.964248i \(-0.585372\pi\)
−0.265000 + 0.964248i \(0.585372\pi\)
\(644\) 0.0762979 0.00300656
\(645\) 5.90686 0.232582
\(646\) −2.30772 −0.0907960
\(647\) −11.8536 −0.466013 −0.233007 0.972475i \(-0.574856\pi\)
−0.233007 + 0.972475i \(0.574856\pi\)
\(648\) −31.9095 −1.25352
\(649\) 18.3292 0.719485
\(650\) 2.47743 0.0971727
\(651\) −5.92818 −0.232344
\(652\) 13.6761 0.535599
\(653\) −25.5777 −1.00093 −0.500466 0.865756i \(-0.666838\pi\)
−0.500466 + 0.865756i \(0.666838\pi\)
\(654\) 6.73337 0.263296
\(655\) 2.82921 0.110547
\(656\) 9.24986 0.361146
\(657\) −25.1088 −0.979586
\(658\) 30.6537 1.19501
\(659\) −25.7646 −1.00365 −0.501823 0.864970i \(-0.667337\pi\)
−0.501823 + 0.864970i \(0.667337\pi\)
\(660\) 12.6365 0.491873
\(661\) −32.9276 −1.28074 −0.640368 0.768068i \(-0.721218\pi\)
−0.640368 + 0.768068i \(0.721218\pi\)
\(662\) 0.290068 0.0112738
\(663\) −24.2646 −0.942358
\(664\) 7.44281 0.288837
\(665\) −2.25570 −0.0874722
\(666\) −14.2954 −0.553934
\(667\) 0.219038 0.00848118
\(668\) −22.8840 −0.885409
\(669\) 73.3965 2.83767
\(670\) 10.3791 0.400978
\(671\) 2.47040 0.0953688
\(672\) 9.57348 0.369305
\(673\) 21.7958 0.840166 0.420083 0.907486i \(-0.362001\pi\)
0.420083 + 0.907486i \(0.362001\pi\)
\(674\) 14.2582 0.549207
\(675\) 16.8670 0.649212
\(676\) −6.86236 −0.263937
\(677\) 5.89169 0.226436 0.113218 0.993570i \(-0.463884\pi\)
0.113218 + 0.993570i \(0.463884\pi\)
\(678\) −42.2646 −1.62316
\(679\) 17.6920 0.678956
\(680\) −2.94379 −0.112889
\(681\) 41.2964 1.58248
\(682\) −2.35186 −0.0900573
\(683\) 16.0949 0.615854 0.307927 0.951410i \(-0.400365\pi\)
0.307927 + 0.951410i \(0.400365\pi\)
\(684\) 6.32599 0.241880
\(685\) 8.47126 0.323670
\(686\) −16.4601 −0.628449
\(687\) 28.0859 1.07154
\(688\) 1.77538 0.0676856
\(689\) −13.2776 −0.505838
\(690\) 0.0882214 0.00335853
\(691\) 30.5236 1.16117 0.580586 0.814199i \(-0.302823\pi\)
0.580586 + 0.814199i \(0.302823\pi\)
\(692\) −14.6738 −0.557815
\(693\) −88.1891 −3.35003
\(694\) 19.9345 0.756702
\(695\) −7.21459 −0.273665
\(696\) 27.4838 1.04177
\(697\) 27.2296 1.03139
\(698\) 5.22658 0.197829
\(699\) −70.1463 −2.65318
\(700\) −2.87743 −0.108756
\(701\) −34.5197 −1.30379 −0.651895 0.758309i \(-0.726026\pi\)
−0.651895 + 0.758309i \(0.726026\pi\)
\(702\) 41.7868 1.57714
\(703\) 1.38874 0.0523773
\(704\) 3.79804 0.143144
\(705\) 35.4441 1.33490
\(706\) 20.6145 0.775837
\(707\) −37.9940 −1.42891
\(708\) 16.0565 0.603439
\(709\) 6.79341 0.255132 0.127566 0.991830i \(-0.459284\pi\)
0.127566 + 0.991830i \(0.459284\pi\)
\(710\) −1.13986 −0.0427782
\(711\) 43.8242 1.64354
\(712\) −4.32672 −0.162151
\(713\) −0.0164195 −0.000614915 0
\(714\) 28.1823 1.05470
\(715\) −9.40936 −0.351890
\(716\) 9.59560 0.358604
\(717\) 78.2022 2.92052
\(718\) 17.7824 0.663632
\(719\) −1.31472 −0.0490307 −0.0245153 0.999699i \(-0.507804\pi\)
−0.0245153 + 0.999699i \(0.507804\pi\)
\(720\) 8.06959 0.300736
\(721\) 11.6433 0.433621
\(722\) 18.3855 0.684236
\(723\) 89.9043 3.34358
\(724\) 10.2207 0.379850
\(725\) −8.26058 −0.306790
\(726\) −11.3957 −0.422933
\(727\) 27.9140 1.03527 0.517636 0.855601i \(-0.326812\pi\)
0.517636 + 0.855601i \(0.326812\pi\)
\(728\) −7.12861 −0.264204
\(729\) 89.1416 3.30154
\(730\) 3.11153 0.115163
\(731\) 5.22633 0.193303
\(732\) 2.16408 0.0799868
\(733\) −46.0733 −1.70176 −0.850878 0.525363i \(-0.823930\pi\)
−0.850878 + 0.525363i \(0.823930\pi\)
\(734\) 3.36265 0.124117
\(735\) 4.25728 0.157032
\(736\) 0.0265160 0.000977393 0
\(737\) −39.4201 −1.45206
\(738\) −74.6425 −2.74763
\(739\) 7.96261 0.292909 0.146455 0.989217i \(-0.453214\pi\)
0.146455 + 0.989217i \(0.453214\pi\)
\(740\) 1.77151 0.0651220
\(741\) −6.46165 −0.237375
\(742\) 15.4214 0.566138
\(743\) −2.89731 −0.106292 −0.0531461 0.998587i \(-0.516925\pi\)
−0.0531461 + 0.998587i \(0.516925\pi\)
\(744\) −2.06024 −0.0755320
\(745\) 4.04445 0.148177
\(746\) −6.13110 −0.224476
\(747\) −60.0604 −2.19750
\(748\) 11.1806 0.408804
\(749\) 38.6395 1.41186
\(750\) −3.32710 −0.121488
\(751\) −24.0517 −0.877658 −0.438829 0.898570i \(-0.644607\pi\)
−0.438829 + 0.898570i \(0.644607\pi\)
\(752\) 10.6532 0.388481
\(753\) 38.4189 1.40006
\(754\) −20.4650 −0.745291
\(755\) −4.20941 −0.153196
\(756\) −48.5336 −1.76515
\(757\) 50.6602 1.84128 0.920638 0.390418i \(-0.127670\pi\)
0.920638 + 0.390418i \(0.127670\pi\)
\(758\) −29.4156 −1.06842
\(759\) −0.335068 −0.0121622
\(760\) −0.783929 −0.0284361
\(761\) 9.81709 0.355869 0.177935 0.984042i \(-0.443058\pi\)
0.177935 + 0.984042i \(0.443058\pi\)
\(762\) −17.0185 −0.616516
\(763\) 5.82332 0.210819
\(764\) −2.14502 −0.0776040
\(765\) 23.7551 0.858869
\(766\) 15.2014 0.549249
\(767\) −11.9560 −0.431705
\(768\) 3.32710 0.120056
\(769\) 41.6658 1.50251 0.751254 0.660014i \(-0.229450\pi\)
0.751254 + 0.660014i \(0.229450\pi\)
\(770\) 10.9286 0.393839
\(771\) −39.7432 −1.43132
\(772\) 14.3461 0.516326
\(773\) −0.937508 −0.0337198 −0.0168599 0.999858i \(-0.505367\pi\)
−0.0168599 + 0.999858i \(0.505367\pi\)
\(774\) −14.3266 −0.514958
\(775\) 0.619229 0.0222434
\(776\) 6.14854 0.220720
\(777\) −16.9595 −0.608419
\(778\) −13.5439 −0.485573
\(779\) 7.25123 0.259802
\(780\) −8.24264 −0.295134
\(781\) 4.32924 0.154912
\(782\) 0.0780575 0.00279133
\(783\) −139.331 −4.97930
\(784\) 1.27958 0.0456992
\(785\) −22.1248 −0.789668
\(786\) −9.41307 −0.335753
\(787\) 9.10469 0.324547 0.162274 0.986746i \(-0.448117\pi\)
0.162274 + 0.986746i \(0.448117\pi\)
\(788\) 12.6963 0.452287
\(789\) −69.9959 −2.49192
\(790\) −5.43079 −0.193219
\(791\) −36.5523 −1.29965
\(792\) −30.6486 −1.08905
\(793\) −1.61142 −0.0572232
\(794\) 20.7482 0.736326
\(795\) 17.8314 0.632415
\(796\) −12.5782 −0.445823
\(797\) 13.9593 0.494464 0.247232 0.968956i \(-0.420479\pi\)
0.247232 + 0.968956i \(0.420479\pi\)
\(798\) 7.50493 0.265672
\(799\) 31.3606 1.10946
\(800\) −1.00000 −0.0353553
\(801\) 34.9148 1.23365
\(802\) 1.00000 0.0353112
\(803\) −11.8177 −0.417038
\(804\) −34.5321 −1.21785
\(805\) 0.0762979 0.00268915
\(806\) 1.53410 0.0540362
\(807\) −28.4669 −1.00208
\(808\) −13.2042 −0.464521
\(809\) 29.2557 1.02857 0.514287 0.857618i \(-0.328057\pi\)
0.514287 + 0.857618i \(0.328057\pi\)
\(810\) −31.9095 −1.12118
\(811\) −23.9151 −0.839772 −0.419886 0.907577i \(-0.637930\pi\)
−0.419886 + 0.907577i \(0.637930\pi\)
\(812\) 23.7692 0.834136
\(813\) −77.0200 −2.70121
\(814\) −6.72827 −0.235826
\(815\) 13.6761 0.479055
\(816\) 9.79427 0.342868
\(817\) 1.39177 0.0486919
\(818\) −32.3855 −1.13233
\(819\) 57.5249 2.01008
\(820\) 9.24986 0.323019
\(821\) −47.7783 −1.66747 −0.833737 0.552161i \(-0.813803\pi\)
−0.833737 + 0.552161i \(0.813803\pi\)
\(822\) −28.1847 −0.983056
\(823\) −7.84950 −0.273616 −0.136808 0.990598i \(-0.543684\pi\)
−0.136808 + 0.990598i \(0.543684\pi\)
\(824\) 4.04644 0.140965
\(825\) 12.6365 0.439945
\(826\) 13.8864 0.483168
\(827\) −31.7419 −1.10378 −0.551888 0.833918i \(-0.686092\pi\)
−0.551888 + 0.833918i \(0.686092\pi\)
\(828\) −0.213973 −0.00743609
\(829\) 0.0702186 0.00243879 0.00121940 0.999999i \(-0.499612\pi\)
0.00121940 + 0.999999i \(0.499612\pi\)
\(830\) 7.44281 0.258344
\(831\) −77.6435 −2.69342
\(832\) −2.47743 −0.0858893
\(833\) 3.76681 0.130512
\(834\) 24.0037 0.831179
\(835\) −22.8840 −0.791934
\(836\) 2.97739 0.102975
\(837\) 10.4446 0.361017
\(838\) 18.4374 0.636909
\(839\) −41.0691 −1.41786 −0.708932 0.705277i \(-0.750823\pi\)
−0.708932 + 0.705277i \(0.750823\pi\)
\(840\) 9.57348 0.330316
\(841\) 39.2372 1.35301
\(842\) −15.3683 −0.529628
\(843\) −66.5975 −2.29374
\(844\) −6.56554 −0.225995
\(845\) −6.86236 −0.236072
\(846\) −85.9666 −2.95559
\(847\) −9.85548 −0.338638
\(848\) 5.35945 0.184044
\(849\) 62.5145 2.14549
\(850\) −2.94379 −0.100971
\(851\) −0.0469734 −0.00161023
\(852\) 3.79243 0.129926
\(853\) 37.9699 1.30006 0.650032 0.759907i \(-0.274755\pi\)
0.650032 + 0.759907i \(0.274755\pi\)
\(854\) 1.87160 0.0640447
\(855\) 6.32599 0.216344
\(856\) 13.4285 0.458977
\(857\) 23.2158 0.793038 0.396519 0.918026i \(-0.370218\pi\)
0.396519 + 0.918026i \(0.370218\pi\)
\(858\) 31.3059 1.06877
\(859\) −4.72303 −0.161148 −0.0805739 0.996749i \(-0.525675\pi\)
−0.0805739 + 0.996749i \(0.525675\pi\)
\(860\) 1.77538 0.0605399
\(861\) −88.5533 −3.01789
\(862\) −37.0475 −1.26184
\(863\) 24.1263 0.821270 0.410635 0.911800i \(-0.365307\pi\)
0.410635 + 0.911800i \(0.365307\pi\)
\(864\) −16.8670 −0.573828
\(865\) −14.6738 −0.498925
\(866\) −30.8682 −1.04895
\(867\) −27.7285 −0.941708
\(868\) −1.78179 −0.0604778
\(869\) 20.6264 0.699701
\(870\) 27.4838 0.931787
\(871\) 25.7133 0.871263
\(872\) 2.02380 0.0685344
\(873\) −49.6162 −1.67925
\(874\) 0.0207867 0.000703120 0
\(875\) −2.87743 −0.0972747
\(876\) −10.3524 −0.349774
\(877\) 1.52197 0.0513933 0.0256966 0.999670i \(-0.491820\pi\)
0.0256966 + 0.999670i \(0.491820\pi\)
\(878\) 38.2997 1.29255
\(879\) −52.9843 −1.78712
\(880\) 3.79804 0.128032
\(881\) 8.08799 0.272491 0.136246 0.990675i \(-0.456496\pi\)
0.136246 + 0.990675i \(0.456496\pi\)
\(882\) −10.3257 −0.347683
\(883\) −11.8485 −0.398735 −0.199367 0.979925i \(-0.563889\pi\)
−0.199367 + 0.979925i \(0.563889\pi\)
\(884\) −7.29301 −0.245291
\(885\) 16.0565 0.539732
\(886\) 9.53641 0.320382
\(887\) −9.32235 −0.313014 −0.156507 0.987677i \(-0.550023\pi\)
−0.156507 + 0.987677i \(0.550023\pi\)
\(888\) −5.89399 −0.197789
\(889\) −14.7184 −0.493639
\(890\) −4.32672 −0.145032
\(891\) 121.193 4.06013
\(892\) 22.0602 0.738631
\(893\) 8.35133 0.279467
\(894\) −13.4563 −0.450045
\(895\) 9.59560 0.320745
\(896\) 2.87743 0.0961280
\(897\) 0.218562 0.00729757
\(898\) 36.7230 1.22546
\(899\) −5.11520 −0.170601
\(900\) 8.06959 0.268986
\(901\) 15.7771 0.525610
\(902\) −35.1313 −1.16975
\(903\) −16.9965 −0.565609
\(904\) −12.7031 −0.422500
\(905\) 10.2207 0.339748
\(906\) 14.0051 0.465289
\(907\) −55.2371 −1.83412 −0.917058 0.398753i \(-0.869443\pi\)
−0.917058 + 0.398753i \(0.869443\pi\)
\(908\) 12.4121 0.411911
\(909\) 106.552 3.53411
\(910\) −7.12861 −0.236311
\(911\) 39.1120 1.29584 0.647919 0.761709i \(-0.275639\pi\)
0.647919 + 0.761709i \(0.275639\pi\)
\(912\) 2.60821 0.0863665
\(913\) −28.2681 −0.935538
\(914\) −34.4625 −1.13992
\(915\) 2.16408 0.0715424
\(916\) 8.44156 0.278917
\(917\) −8.14085 −0.268835
\(918\) −49.6529 −1.63879
\(919\) −54.4431 −1.79591 −0.897957 0.440084i \(-0.854949\pi\)
−0.897957 + 0.440084i \(0.854949\pi\)
\(920\) 0.0265160 0.000874207 0
\(921\) −43.4977 −1.43330
\(922\) −32.8634 −1.08230
\(923\) −2.82392 −0.0929505
\(924\) −36.3605 −1.19617
\(925\) 1.77151 0.0582469
\(926\) 3.98341 0.130903
\(927\) −32.6531 −1.07247
\(928\) 8.26058 0.271167
\(929\) −1.74749 −0.0573334 −0.0286667 0.999589i \(-0.509126\pi\)
−0.0286667 + 0.999589i \(0.509126\pi\)
\(930\) −2.06024 −0.0675578
\(931\) 1.00310 0.0328752
\(932\) −21.0833 −0.690608
\(933\) 15.2137 0.498073
\(934\) −1.64490 −0.0538229
\(935\) 11.1806 0.365645
\(936\) 19.9918 0.653453
\(937\) −51.7617 −1.69098 −0.845491 0.533989i \(-0.820692\pi\)
−0.845491 + 0.533989i \(0.820692\pi\)
\(938\) −29.8650 −0.975125
\(939\) −75.1408 −2.45213
\(940\) 10.6532 0.347468
\(941\) −15.7490 −0.513404 −0.256702 0.966491i \(-0.582636\pi\)
−0.256702 + 0.966491i \(0.582636\pi\)
\(942\) 73.6114 2.39839
\(943\) −0.245269 −0.00798707
\(944\) 4.82597 0.157072
\(945\) −48.5336 −1.57880
\(946\) −6.74295 −0.219232
\(947\) −30.5208 −0.991793 −0.495896 0.868382i \(-0.665160\pi\)
−0.495896 + 0.868382i \(0.665160\pi\)
\(948\) 18.0688 0.586847
\(949\) 7.70859 0.250231
\(950\) −0.783929 −0.0254340
\(951\) −50.1395 −1.62589
\(952\) 8.47053 0.274531
\(953\) −50.3980 −1.63255 −0.816275 0.577663i \(-0.803965\pi\)
−0.816275 + 0.577663i \(0.803965\pi\)
\(954\) −43.2485 −1.40022
\(955\) −2.14502 −0.0694111
\(956\) 23.5046 0.760194
\(957\) −104.384 −3.37427
\(958\) −0.132786 −0.00429012
\(959\) −24.3754 −0.787124
\(960\) 3.32710 0.107382
\(961\) −30.6166 −0.987631
\(962\) 4.38879 0.141500
\(963\) −108.363 −3.49193
\(964\) 27.0218 0.870314
\(965\) 14.3461 0.461816
\(966\) −0.253851 −0.00816751
\(967\) 8.83722 0.284186 0.142093 0.989853i \(-0.454617\pi\)
0.142093 + 0.989853i \(0.454617\pi\)
\(968\) −3.42510 −0.110087
\(969\) 7.67801 0.246653
\(970\) 6.14854 0.197418
\(971\) 44.5253 1.42889 0.714443 0.699693i \(-0.246680\pi\)
0.714443 + 0.699693i \(0.246680\pi\)
\(972\) 55.5649 1.78225
\(973\) 20.7595 0.665518
\(974\) 14.6785 0.470331
\(975\) −8.24264 −0.263976
\(976\) 0.650441 0.0208201
\(977\) −33.0350 −1.05688 −0.528441 0.848970i \(-0.677223\pi\)
−0.528441 + 0.848970i \(0.677223\pi\)
\(978\) −45.5019 −1.45499
\(979\) 16.4330 0.525202
\(980\) 1.27958 0.0408746
\(981\) −16.3312 −0.521415
\(982\) −6.99801 −0.223315
\(983\) 39.3209 1.25414 0.627071 0.778962i \(-0.284254\pi\)
0.627071 + 0.778962i \(0.284254\pi\)
\(984\) −30.7752 −0.981077
\(985\) 12.6963 0.404538
\(986\) 24.3174 0.774424
\(987\) −101.988 −3.24631
\(988\) −1.94213 −0.0617873
\(989\) −0.0470759 −0.00149693
\(990\) −30.6486 −0.974077
\(991\) 29.9847 0.952496 0.476248 0.879311i \(-0.341996\pi\)
0.476248 + 0.879311i \(0.341996\pi\)
\(992\) −0.619229 −0.0196606
\(993\) −0.965084 −0.0306260
\(994\) 3.27986 0.104031
\(995\) −12.5782 −0.398756
\(996\) −24.7630 −0.784645
\(997\) 23.5075 0.744490 0.372245 0.928135i \(-0.378588\pi\)
0.372245 + 0.928135i \(0.378588\pi\)
\(998\) 0.837087 0.0264975
\(999\) 29.8801 0.945365
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.1.20 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4010.2.a.m.1.20 20 1.1 even 1 trivial