Properties

Label 39.4.a.b.1.2
Level $39$
Weight $4$
Character 39.1
Self dual yes
Analytic conductor $2.301$
Analytic rank $0$
Dimension $2$
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] = [39,4,Mod(1,39)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(39, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("39.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 39 = 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 39.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(2.30107449022\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{14}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.74166\) of defining polynomial
Character \(\chi\) \(=\) 39.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.74166 q^{2} -3.00000 q^{3} +14.4833 q^{4} +4.51669 q^{5} -14.2250 q^{6} -7.48331 q^{7} +30.7417 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+4.74166 q^{2} -3.00000 q^{3} +14.4833 q^{4} +4.51669 q^{5} -14.2250 q^{6} -7.48331 q^{7} +30.7417 q^{8} +9.00000 q^{9} +21.4166 q^{10} -66.8999 q^{11} -43.4499 q^{12} -13.0000 q^{13} -35.4833 q^{14} -13.5501 q^{15} +29.8999 q^{16} +96.9666 q^{17} +42.6749 q^{18} +31.4833 q^{19} +65.4166 q^{20} +22.4499 q^{21} -317.216 q^{22} +183.600 q^{23} -92.2250 q^{24} -104.600 q^{25} -61.6415 q^{26} -27.0000 q^{27} -108.383 q^{28} +112.200 q^{29} -64.2497 q^{30} -77.2831 q^{31} -104.158 q^{32} +200.700 q^{33} +459.783 q^{34} -33.7998 q^{35} +130.350 q^{36} +54.7664 q^{37} +149.283 q^{38} +39.0000 q^{39} +138.850 q^{40} +451.716 q^{41} +106.450 q^{42} -113.434 q^{43} -968.932 q^{44} +40.6502 q^{45} +870.566 q^{46} -42.2670 q^{47} -89.6997 q^{48} -287.000 q^{49} -495.975 q^{50} -290.900 q^{51} -188.283 q^{52} -530.999 q^{53} -128.025 q^{54} -302.166 q^{55} -230.049 q^{56} -94.4499 q^{57} +532.015 q^{58} +219.666 q^{59} -196.250 q^{60} +822.865 q^{61} -366.450 q^{62} -67.3498 q^{63} -733.082 q^{64} -58.7169 q^{65} +951.649 q^{66} -872.082 q^{67} +1404.40 q^{68} -550.799 q^{69} -160.267 q^{70} -100.299 q^{71} +276.675 q^{72} -165.634 q^{73} +259.684 q^{74} +313.799 q^{75} +455.983 q^{76} +500.633 q^{77} +184.925 q^{78} -545.266 q^{79} +135.048 q^{80} +81.0000 q^{81} +2141.88 q^{82} -454.534 q^{83} +325.150 q^{84} +437.968 q^{85} -537.864 q^{86} -336.601 q^{87} -2056.61 q^{88} -230.915 q^{89} +192.749 q^{90} +97.2831 q^{91} +2659.13 q^{92} +231.849 q^{93} -200.415 q^{94} +142.200 q^{95} +312.475 q^{96} -1089.16 q^{97} -1360.86 q^{98} -602.099 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 6 q^{3} + 14 q^{4} + 24 q^{5} - 6 q^{6} + 54 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 6 q^{3} + 14 q^{4} + 24 q^{5} - 6 q^{6} + 54 q^{8} + 18 q^{9} - 32 q^{10} - 44 q^{11} - 42 q^{12} - 26 q^{13} - 56 q^{14} - 72 q^{15} - 30 q^{16} + 164 q^{17} + 18 q^{18} + 48 q^{19} + 56 q^{20} - 380 q^{22} + 8 q^{23} - 162 q^{24} + 150 q^{25} - 26 q^{26} - 54 q^{27} - 112 q^{28} + 404 q^{29} + 96 q^{30} + 40 q^{31} - 126 q^{32} + 132 q^{33} + 276 q^{34} + 112 q^{35} + 126 q^{36} - 100 q^{37} + 104 q^{38} + 78 q^{39} + 592 q^{40} + 200 q^{41} + 168 q^{42} - 616 q^{43} - 980 q^{44} + 216 q^{45} + 1352 q^{46} - 324 q^{47} + 90 q^{48} - 574 q^{49} - 1194 q^{50} - 492 q^{51} - 182 q^{52} - 164 q^{53} - 54 q^{54} + 144 q^{55} - 56 q^{56} - 144 q^{57} - 268 q^{58} + 140 q^{59} - 168 q^{60} + 628 q^{61} - 688 q^{62} - 194 q^{64} - 312 q^{65} + 1140 q^{66} - 472 q^{67} + 1372 q^{68} - 24 q^{69} - 560 q^{70} + 428 q^{71} + 486 q^{72} - 900 q^{73} + 684 q^{74} - 450 q^{75} + 448 q^{76} + 672 q^{77} + 78 q^{78} - 432 q^{79} - 1032 q^{80} + 162 q^{81} + 2832 q^{82} - 1388 q^{83} + 336 q^{84} + 1744 q^{85} + 840 q^{86} - 1212 q^{87} - 1524 q^{88} + 960 q^{89} - 288 q^{90} + 2744 q^{92} - 120 q^{93} + 572 q^{94} + 464 q^{95} + 378 q^{96} - 532 q^{97} - 574 q^{98} - 396 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.74166 1.67643 0.838215 0.545341i \(-0.183600\pi\)
0.838215 + 0.545341i \(0.183600\pi\)
\(3\) −3.00000 −0.577350
\(4\) 14.4833 1.81041
\(5\) 4.51669 0.403985 0.201992 0.979387i \(-0.435258\pi\)
0.201992 + 0.979387i \(0.435258\pi\)
\(6\) −14.2250 −0.967887
\(7\) −7.48331 −0.404061 −0.202031 0.979379i \(-0.564754\pi\)
−0.202031 + 0.979379i \(0.564754\pi\)
\(8\) 30.7417 1.35860
\(9\) 9.00000 0.333333
\(10\) 21.4166 0.677252
\(11\) −66.8999 −1.83373 −0.916867 0.399193i \(-0.869290\pi\)
−0.916867 + 0.399193i \(0.869290\pi\)
\(12\) −43.4499 −1.04524
\(13\) −13.0000 −0.277350
\(14\) −35.4833 −0.677380
\(15\) −13.5501 −0.233241
\(16\) 29.8999 0.467186
\(17\) 96.9666 1.38340 0.691702 0.722183i \(-0.256861\pi\)
0.691702 + 0.722183i \(0.256861\pi\)
\(18\) 42.6749 0.558810
\(19\) 31.4833 0.380146 0.190073 0.981770i \(-0.439128\pi\)
0.190073 + 0.981770i \(0.439128\pi\)
\(20\) 65.4166 0.731380
\(21\) 22.4499 0.233285
\(22\) −317.216 −3.07413
\(23\) 183.600 1.66448 0.832242 0.554412i \(-0.187057\pi\)
0.832242 + 0.554412i \(0.187057\pi\)
\(24\) −92.2250 −0.784389
\(25\) −104.600 −0.836796
\(26\) −61.6415 −0.464958
\(27\) −27.0000 −0.192450
\(28\) −108.383 −0.731518
\(29\) 112.200 0.718450 0.359225 0.933251i \(-0.383041\pi\)
0.359225 + 0.933251i \(0.383041\pi\)
\(30\) −64.2497 −0.391011
\(31\) −77.2831 −0.447757 −0.223878 0.974617i \(-0.571872\pi\)
−0.223878 + 0.974617i \(0.571872\pi\)
\(32\) −104.158 −0.575398
\(33\) 200.700 1.05871
\(34\) 459.783 2.31918
\(35\) −33.7998 −0.163234
\(36\) 130.350 0.603471
\(37\) 54.7664 0.243339 0.121669 0.992571i \(-0.461175\pi\)
0.121669 + 0.992571i \(0.461175\pi\)
\(38\) 149.283 0.637287
\(39\) 39.0000 0.160128
\(40\) 138.850 0.548854
\(41\) 451.716 1.72064 0.860319 0.509756i \(-0.170264\pi\)
0.860319 + 0.509756i \(0.170264\pi\)
\(42\) 106.450 0.391085
\(43\) −113.434 −0.402291 −0.201145 0.979561i \(-0.564466\pi\)
−0.201145 + 0.979561i \(0.564466\pi\)
\(44\) −968.932 −3.31982
\(45\) 40.6502 0.134662
\(46\) 870.566 2.79039
\(47\) −42.2670 −0.131176 −0.0655880 0.997847i \(-0.520892\pi\)
−0.0655880 + 0.997847i \(0.520892\pi\)
\(48\) −89.6997 −0.269730
\(49\) −287.000 −0.836735
\(50\) −495.975 −1.40283
\(51\) −290.900 −0.798708
\(52\) −188.283 −0.502119
\(53\) −530.999 −1.37619 −0.688097 0.725618i \(-0.741554\pi\)
−0.688097 + 0.725618i \(0.741554\pi\)
\(54\) −128.025 −0.322629
\(55\) −302.166 −0.740800
\(56\) −230.049 −0.548958
\(57\) −94.4499 −0.219477
\(58\) 532.015 1.20443
\(59\) 219.666 0.484714 0.242357 0.970187i \(-0.422080\pi\)
0.242357 + 0.970187i \(0.422080\pi\)
\(60\) −196.250 −0.422262
\(61\) 822.865 1.72717 0.863583 0.504207i \(-0.168215\pi\)
0.863583 + 0.504207i \(0.168215\pi\)
\(62\) −366.450 −0.750632
\(63\) −67.3498 −0.134687
\(64\) −733.082 −1.43180
\(65\) −58.7169 −0.112045
\(66\) 951.649 1.77485
\(67\) −872.082 −1.59018 −0.795088 0.606495i \(-0.792575\pi\)
−0.795088 + 0.606495i \(0.792575\pi\)
\(68\) 1404.40 2.50453
\(69\) −550.799 −0.960991
\(70\) −160.267 −0.273651
\(71\) −100.299 −0.167653 −0.0838263 0.996480i \(-0.526714\pi\)
−0.0838263 + 0.996480i \(0.526714\pi\)
\(72\) 276.675 0.452867
\(73\) −165.634 −0.265562 −0.132781 0.991145i \(-0.542391\pi\)
−0.132781 + 0.991145i \(0.542391\pi\)
\(74\) 259.684 0.407941
\(75\) 313.799 0.483125
\(76\) 455.983 0.688221
\(77\) 500.633 0.740940
\(78\) 184.925 0.268443
\(79\) −545.266 −0.776547 −0.388273 0.921544i \(-0.626928\pi\)
−0.388273 + 0.921544i \(0.626928\pi\)
\(80\) 135.048 0.188736
\(81\) 81.0000 0.111111
\(82\) 2141.88 2.88453
\(83\) −454.534 −0.601103 −0.300552 0.953766i \(-0.597171\pi\)
−0.300552 + 0.953766i \(0.597171\pi\)
\(84\) 325.150 0.422342
\(85\) 437.968 0.558874
\(86\) −537.864 −0.674412
\(87\) −336.601 −0.414797
\(88\) −2056.61 −2.49132
\(89\) −230.915 −0.275022 −0.137511 0.990500i \(-0.543910\pi\)
−0.137511 + 0.990500i \(0.543910\pi\)
\(90\) 192.749 0.225751
\(91\) 97.2831 0.112066
\(92\) 2659.13 3.01341
\(93\) 231.849 0.258512
\(94\) −200.415 −0.219907
\(95\) 142.200 0.153573
\(96\) 312.475 0.332206
\(97\) −1089.16 −1.14008 −0.570041 0.821616i \(-0.693073\pi\)
−0.570041 + 0.821616i \(0.693073\pi\)
\(98\) −1360.86 −1.40273
\(99\) −602.099 −0.611245
\(100\) −1514.95 −1.51495
\(101\) −77.2336 −0.0760894 −0.0380447 0.999276i \(-0.512113\pi\)
−0.0380447 + 0.999276i \(0.512113\pi\)
\(102\) −1379.35 −1.33898
\(103\) 1351.36 1.29275 0.646377 0.763018i \(-0.276283\pi\)
0.646377 + 0.763018i \(0.276283\pi\)
\(104\) −399.642 −0.376808
\(105\) 101.399 0.0942434
\(106\) −2517.81 −2.30709
\(107\) 1133.67 1.02426 0.512129 0.858908i \(-0.328857\pi\)
0.512129 + 0.858908i \(0.328857\pi\)
\(108\) −391.049 −0.348414
\(109\) 1017.66 0.894262 0.447131 0.894469i \(-0.352446\pi\)
0.447131 + 0.894469i \(0.352446\pi\)
\(110\) −1432.77 −1.24190
\(111\) −164.299 −0.140492
\(112\) −223.750 −0.188772
\(113\) 1570.06 1.30707 0.653536 0.756895i \(-0.273285\pi\)
0.653536 + 0.756895i \(0.273285\pi\)
\(114\) −447.849 −0.367938
\(115\) 829.261 0.672426
\(116\) 1625.03 1.30069
\(117\) −117.000 −0.0924500
\(118\) 1041.58 0.812588
\(119\) −725.632 −0.558979
\(120\) −416.551 −0.316881
\(121\) 3144.60 2.36258
\(122\) 3901.75 2.89547
\(123\) −1355.15 −0.993411
\(124\) −1119.32 −0.810625
\(125\) −1037.03 −0.742037
\(126\) −319.350 −0.225793
\(127\) −1248.16 −0.872099 −0.436050 0.899923i \(-0.643623\pi\)
−0.436050 + 0.899923i \(0.643623\pi\)
\(128\) −2642.76 −1.82491
\(129\) 340.301 0.232263
\(130\) −278.415 −0.187836
\(131\) −1274.80 −0.850227 −0.425113 0.905140i \(-0.639766\pi\)
−0.425113 + 0.905140i \(0.639766\pi\)
\(132\) 2906.80 1.91670
\(133\) −235.600 −0.153602
\(134\) −4135.11 −2.66582
\(135\) −121.951 −0.0777469
\(136\) 2980.91 1.87950
\(137\) 874.915 0.545613 0.272807 0.962069i \(-0.412048\pi\)
0.272807 + 0.962069i \(0.412048\pi\)
\(138\) −2611.70 −1.61103
\(139\) 310.334 0.189368 0.0946840 0.995507i \(-0.469816\pi\)
0.0946840 + 0.995507i \(0.469816\pi\)
\(140\) −489.533 −0.295522
\(141\) 126.801 0.0757345
\(142\) −475.585 −0.281058
\(143\) 869.699 0.508586
\(144\) 269.099 0.155729
\(145\) 506.773 0.290243
\(146\) −785.380 −0.445195
\(147\) 861.000 0.483089
\(148\) 793.199 0.440544
\(149\) 5.08064 0.00279344 0.00139672 0.999999i \(-0.499555\pi\)
0.00139672 + 0.999999i \(0.499555\pi\)
\(150\) 1487.93 0.809924
\(151\) 6.54894 0.00352944 0.00176472 0.999998i \(-0.499438\pi\)
0.00176472 + 0.999998i \(0.499438\pi\)
\(152\) 967.849 0.516467
\(153\) 872.700 0.461135
\(154\) 2373.83 1.24213
\(155\) −349.063 −0.180887
\(156\) 564.849 0.289898
\(157\) −2297.60 −1.16795 −0.583975 0.811772i \(-0.698503\pi\)
−0.583975 + 0.811772i \(0.698503\pi\)
\(158\) −2585.46 −1.30183
\(159\) 1593.00 0.794546
\(160\) −470.450 −0.232452
\(161\) −1373.93 −0.672553
\(162\) 384.074 0.186270
\(163\) −1085.49 −0.521606 −0.260803 0.965392i \(-0.583987\pi\)
−0.260803 + 0.965392i \(0.583987\pi\)
\(164\) 6542.34 3.11507
\(165\) 906.497 0.427701
\(166\) −2155.24 −1.00771
\(167\) −109.066 −0.0505374 −0.0252687 0.999681i \(-0.508044\pi\)
−0.0252687 + 0.999681i \(0.508044\pi\)
\(168\) 690.148 0.316941
\(169\) 169.000 0.0769231
\(170\) 2076.69 0.936912
\(171\) 283.350 0.126715
\(172\) −1642.90 −0.728313
\(173\) −1889.17 −0.830236 −0.415118 0.909768i \(-0.636260\pi\)
−0.415118 + 0.909768i \(0.636260\pi\)
\(174\) −1596.05 −0.695379
\(175\) 782.751 0.338117
\(176\) −2000.30 −0.856694
\(177\) −658.999 −0.279850
\(178\) −1094.92 −0.461054
\(179\) 3427.86 1.43134 0.715672 0.698437i \(-0.246121\pi\)
0.715672 + 0.698437i \(0.246121\pi\)
\(180\) 588.749 0.243793
\(181\) 208.403 0.0855826 0.0427913 0.999084i \(-0.486375\pi\)
0.0427913 + 0.999084i \(0.486375\pi\)
\(182\) 461.283 0.187871
\(183\) −2468.60 −0.997180
\(184\) 5644.15 2.26137
\(185\) 247.363 0.0983052
\(186\) 1099.35 0.433378
\(187\) −6487.06 −2.53679
\(188\) −612.166 −0.237483
\(189\) 202.049 0.0777616
\(190\) 674.265 0.257454
\(191\) 957.735 0.362824 0.181412 0.983407i \(-0.441933\pi\)
0.181412 + 0.983407i \(0.441933\pi\)
\(192\) 2199.25 0.826650
\(193\) −512.730 −0.191228 −0.0956142 0.995418i \(-0.530482\pi\)
−0.0956142 + 0.995418i \(0.530482\pi\)
\(194\) −5164.45 −1.91127
\(195\) 176.151 0.0646893
\(196\) −4156.71 −1.51484
\(197\) −3870.35 −1.39975 −0.699876 0.714265i \(-0.746761\pi\)
−0.699876 + 0.714265i \(0.746761\pi\)
\(198\) −2854.95 −1.02471
\(199\) 2305.83 0.821388 0.410694 0.911773i \(-0.365286\pi\)
0.410694 + 0.911773i \(0.365286\pi\)
\(200\) −3215.56 −1.13687
\(201\) 2616.25 0.918088
\(202\) −366.215 −0.127558
\(203\) −839.630 −0.290298
\(204\) −4213.19 −1.44599
\(205\) 2040.26 0.695111
\(206\) 6407.70 2.16721
\(207\) 1652.40 0.554828
\(208\) −388.699 −0.129574
\(209\) −2106.23 −0.697086
\(210\) 480.801 0.157992
\(211\) −3672.40 −1.19819 −0.599096 0.800677i \(-0.704473\pi\)
−0.599096 + 0.800677i \(0.704473\pi\)
\(212\) −7690.62 −2.49148
\(213\) 300.898 0.0967942
\(214\) 5375.46 1.71710
\(215\) −512.345 −0.162519
\(216\) −830.025 −0.261463
\(217\) 578.334 0.180921
\(218\) 4825.41 1.49917
\(219\) 496.902 0.153322
\(220\) −4376.36 −1.34116
\(221\) −1260.57 −0.383687
\(222\) −779.051 −0.235525
\(223\) 5087.05 1.52760 0.763798 0.645455i \(-0.223332\pi\)
0.763798 + 0.645455i \(0.223332\pi\)
\(224\) 779.449 0.232496
\(225\) −941.396 −0.278932
\(226\) 7444.71 2.19122
\(227\) −2625.83 −0.767763 −0.383882 0.923382i \(-0.625413\pi\)
−0.383882 + 0.923382i \(0.625413\pi\)
\(228\) −1367.95 −0.397345
\(229\) 1678.73 0.484425 0.242213 0.970223i \(-0.422127\pi\)
0.242213 + 0.970223i \(0.422127\pi\)
\(230\) 3932.07 1.12727
\(231\) −1501.90 −0.427782
\(232\) 3449.22 0.976088
\(233\) 648.506 0.182339 0.0911696 0.995835i \(-0.470939\pi\)
0.0911696 + 0.995835i \(0.470939\pi\)
\(234\) −554.774 −0.154986
\(235\) −190.907 −0.0529931
\(236\) 3181.50 0.877533
\(237\) 1635.80 0.448340
\(238\) −3440.70 −0.937089
\(239\) 5219.69 1.41269 0.706347 0.707866i \(-0.250342\pi\)
0.706347 + 0.707866i \(0.250342\pi\)
\(240\) −405.145 −0.108967
\(241\) 6103.56 1.63139 0.815695 0.578483i \(-0.196355\pi\)
0.815695 + 0.578483i \(0.196355\pi\)
\(242\) 14910.6 3.96070
\(243\) −243.000 −0.0641500
\(244\) 11917.8 3.12689
\(245\) −1296.29 −0.338028
\(246\) −6425.64 −1.66538
\(247\) −409.283 −0.105433
\(248\) −2375.81 −0.608323
\(249\) 1363.60 0.347047
\(250\) −4917.24 −1.24397
\(251\) 6423.40 1.61530 0.807652 0.589660i \(-0.200738\pi\)
0.807652 + 0.589660i \(0.200738\pi\)
\(252\) −975.449 −0.243839
\(253\) −12282.8 −3.05222
\(254\) −5918.36 −1.46201
\(255\) −1313.90 −0.322666
\(256\) −6666.39 −1.62754
\(257\) −1230.23 −0.298597 −0.149299 0.988792i \(-0.547702\pi\)
−0.149299 + 0.988792i \(0.547702\pi\)
\(258\) 1613.59 0.389372
\(259\) −409.834 −0.0983238
\(260\) −850.415 −0.202848
\(261\) 1009.80 0.239483
\(262\) −6044.66 −1.42534
\(263\) −514.992 −0.120744 −0.0603722 0.998176i \(-0.519229\pi\)
−0.0603722 + 0.998176i \(0.519229\pi\)
\(264\) 6169.84 1.43836
\(265\) −2398.35 −0.555961
\(266\) −1117.13 −0.257503
\(267\) 692.745 0.158784
\(268\) −12630.6 −2.87888
\(269\) −5132.60 −1.16335 −0.581673 0.813423i \(-0.697602\pi\)
−0.581673 + 0.813423i \(0.697602\pi\)
\(270\) −578.247 −0.130337
\(271\) −4300.00 −0.963862 −0.481931 0.876209i \(-0.660064\pi\)
−0.481931 + 0.876209i \(0.660064\pi\)
\(272\) 2899.29 0.646306
\(273\) −291.849 −0.0647015
\(274\) 4148.55 0.914682
\(275\) 6997.70 1.53446
\(276\) −7977.39 −1.73979
\(277\) −1812.80 −0.393215 −0.196607 0.980482i \(-0.562992\pi\)
−0.196607 + 0.980482i \(0.562992\pi\)
\(278\) 1471.50 0.317462
\(279\) −695.548 −0.149252
\(280\) −1039.06 −0.221771
\(281\) −4073.08 −0.864696 −0.432348 0.901707i \(-0.642315\pi\)
−0.432348 + 0.901707i \(0.642315\pi\)
\(282\) 601.246 0.126963
\(283\) 6346.29 1.33303 0.666516 0.745491i \(-0.267785\pi\)
0.666516 + 0.745491i \(0.267785\pi\)
\(284\) −1452.67 −0.303520
\(285\) −426.601 −0.0886654
\(286\) 4123.81 0.852609
\(287\) −3380.33 −0.695243
\(288\) −937.424 −0.191799
\(289\) 4489.53 0.913806
\(290\) 2402.94 0.486572
\(291\) 3267.49 0.658226
\(292\) −2398.93 −0.480777
\(293\) −8390.97 −1.67306 −0.836529 0.547923i \(-0.815419\pi\)
−0.836529 + 0.547923i \(0.815419\pi\)
\(294\) 4082.57 0.809864
\(295\) 992.164 0.195817
\(296\) 1683.61 0.330601
\(297\) 1806.30 0.352902
\(298\) 24.0907 0.00468300
\(299\) −2386.79 −0.461645
\(300\) 4544.84 0.874656
\(301\) 848.861 0.162550
\(302\) 31.0528 0.00591685
\(303\) 231.701 0.0439302
\(304\) 941.348 0.177599
\(305\) 3716.62 0.697748
\(306\) 4138.04 0.773059
\(307\) 4005.27 0.744603 0.372301 0.928112i \(-0.378569\pi\)
0.372301 + 0.928112i \(0.378569\pi\)
\(308\) 7250.82 1.34141
\(309\) −4054.09 −0.746372
\(310\) −1655.14 −0.303244
\(311\) 5836.53 1.06418 0.532088 0.846689i \(-0.321407\pi\)
0.532088 + 0.846689i \(0.321407\pi\)
\(312\) 1198.92 0.217550
\(313\) 1763.19 0.318407 0.159204 0.987246i \(-0.449107\pi\)
0.159204 + 0.987246i \(0.449107\pi\)
\(314\) −10894.4 −1.95798
\(315\) −304.198 −0.0544115
\(316\) −7897.26 −1.40587
\(317\) −6106.35 −1.08191 −0.540957 0.841050i \(-0.681938\pi\)
−0.540957 + 0.841050i \(0.681938\pi\)
\(318\) 7553.44 1.33200
\(319\) −7506.18 −1.31745
\(320\) −3311.10 −0.578425
\(321\) −3401.00 −0.591356
\(322\) −6514.72 −1.12749
\(323\) 3052.83 0.525895
\(324\) 1173.15 0.201157
\(325\) 1359.79 0.232086
\(326\) −5147.00 −0.874436
\(327\) −3052.99 −0.516302
\(328\) 13886.5 2.33766
\(329\) 316.297 0.0530031
\(330\) 4298.30 0.717011
\(331\) 7490.38 1.24383 0.621916 0.783084i \(-0.286354\pi\)
0.621916 + 0.783084i \(0.286354\pi\)
\(332\) −6583.16 −1.08825
\(333\) 492.898 0.0811130
\(334\) −517.152 −0.0847224
\(335\) −3938.92 −0.642406
\(336\) 671.251 0.108987
\(337\) 9462.46 1.52953 0.764767 0.644307i \(-0.222854\pi\)
0.764767 + 0.644307i \(0.222854\pi\)
\(338\) 801.340 0.128956
\(339\) −4710.19 −0.754639
\(340\) 6343.22 1.01179
\(341\) 5170.23 0.821066
\(342\) 1343.55 0.212429
\(343\) 4714.49 0.742153
\(344\) −3487.14 −0.546553
\(345\) −2487.78 −0.388226
\(346\) −8957.79 −1.39183
\(347\) 11460.3 1.77297 0.886487 0.462753i \(-0.153138\pi\)
0.886487 + 0.462753i \(0.153138\pi\)
\(348\) −4875.09 −0.750955
\(349\) −3673.96 −0.563503 −0.281751 0.959487i \(-0.590915\pi\)
−0.281751 + 0.959487i \(0.590915\pi\)
\(350\) 3711.54 0.566829
\(351\) 351.000 0.0533761
\(352\) 6968.17 1.05513
\(353\) 3388.65 0.510935 0.255467 0.966818i \(-0.417771\pi\)
0.255467 + 0.966818i \(0.417771\pi\)
\(354\) −3124.75 −0.469148
\(355\) −453.020 −0.0677290
\(356\) −3344.41 −0.497903
\(357\) 2176.90 0.322727
\(358\) 16253.7 2.39955
\(359\) −9673.98 −1.42221 −0.711105 0.703086i \(-0.751805\pi\)
−0.711105 + 0.703086i \(0.751805\pi\)
\(360\) 1249.65 0.182951
\(361\) −5867.80 −0.855489
\(362\) 988.174 0.143473
\(363\) −9433.79 −1.36404
\(364\) 1408.98 0.202887
\(365\) −748.117 −0.107283
\(366\) −11705.2 −1.67170
\(367\) −8715.98 −1.23970 −0.619851 0.784720i \(-0.712807\pi\)
−0.619851 + 0.784720i \(0.712807\pi\)
\(368\) 5489.61 0.777624
\(369\) 4065.44 0.573546
\(370\) 1172.91 0.164802
\(371\) 3973.63 0.556067
\(372\) 3357.95 0.468014
\(373\) 4667.99 0.647987 0.323994 0.946059i \(-0.394974\pi\)
0.323994 + 0.946059i \(0.394974\pi\)
\(374\) −30759.4 −4.25276
\(375\) 3111.09 0.428416
\(376\) −1299.36 −0.178216
\(377\) −1458.60 −0.199262
\(378\) 958.049 0.130362
\(379\) 10862.7 1.47225 0.736123 0.676848i \(-0.236655\pi\)
0.736123 + 0.676848i \(0.236655\pi\)
\(380\) 2059.53 0.278031
\(381\) 3744.49 0.503507
\(382\) 4541.25 0.608248
\(383\) 10054.2 1.34137 0.670686 0.741742i \(-0.266000\pi\)
0.670686 + 0.741742i \(0.266000\pi\)
\(384\) 7928.27 1.05361
\(385\) 2261.20 0.299329
\(386\) −2431.19 −0.320581
\(387\) −1020.90 −0.134097
\(388\) −15774.7 −2.06402
\(389\) 6418.50 0.836584 0.418292 0.908313i \(-0.362629\pi\)
0.418292 + 0.908313i \(0.362629\pi\)
\(390\) 835.246 0.108447
\(391\) 17803.0 2.30265
\(392\) −8822.86 −1.13679
\(393\) 3824.40 0.490879
\(394\) −18351.9 −2.34658
\(395\) −2462.79 −0.313713
\(396\) −8720.39 −1.10661
\(397\) −12019.9 −1.51955 −0.759775 0.650186i \(-0.774691\pi\)
−0.759775 + 0.650186i \(0.774691\pi\)
\(398\) 10933.5 1.37700
\(399\) 706.799 0.0886822
\(400\) −3127.52 −0.390939
\(401\) 3599.80 0.448293 0.224147 0.974555i \(-0.428041\pi\)
0.224147 + 0.974555i \(0.428041\pi\)
\(402\) 12405.3 1.53911
\(403\) 1004.68 0.124185
\(404\) −1118.60 −0.137753
\(405\) 365.852 0.0448872
\(406\) −3981.24 −0.486664
\(407\) −3663.87 −0.446219
\(408\) −8942.74 −1.08513
\(409\) −48.0968 −0.00581475 −0.00290737 0.999996i \(-0.500925\pi\)
−0.00290737 + 0.999996i \(0.500925\pi\)
\(410\) 9674.20 1.16530
\(411\) −2624.74 −0.315010
\(412\) 19572.2 2.34042
\(413\) −1643.83 −0.195854
\(414\) 7835.10 0.930130
\(415\) −2052.99 −0.242837
\(416\) 1354.06 0.159587
\(417\) −931.001 −0.109332
\(418\) −9987.02 −1.16862
\(419\) 723.462 0.0843518 0.0421759 0.999110i \(-0.486571\pi\)
0.0421759 + 0.999110i \(0.486571\pi\)
\(420\) 1468.60 0.170620
\(421\) −14845.5 −1.71859 −0.859295 0.511481i \(-0.829097\pi\)
−0.859295 + 0.511481i \(0.829097\pi\)
\(422\) −17413.3 −2.00868
\(423\) −380.403 −0.0437253
\(424\) −16323.8 −1.86970
\(425\) −10142.7 −1.15763
\(426\) 1426.75 0.162269
\(427\) −6157.76 −0.697880
\(428\) 16419.2 1.85433
\(429\) −2609.10 −0.293632
\(430\) −2429.36 −0.272452
\(431\) 1103.11 0.123283 0.0616417 0.998098i \(-0.480366\pi\)
0.0616417 + 0.998098i \(0.480366\pi\)
\(432\) −807.297 −0.0899099
\(433\) 8893.53 0.987057 0.493528 0.869730i \(-0.335707\pi\)
0.493528 + 0.869730i \(0.335707\pi\)
\(434\) 2742.26 0.303301
\(435\) −1520.32 −0.167572
\(436\) 14739.1 1.61898
\(437\) 5780.32 0.632747
\(438\) 2356.14 0.257034
\(439\) −10901.7 −1.18521 −0.592607 0.805492i \(-0.701901\pi\)
−0.592607 + 0.805492i \(0.701901\pi\)
\(440\) −9289.08 −1.00645
\(441\) −2583.00 −0.278912
\(442\) −5977.17 −0.643224
\(443\) −3781.37 −0.405550 −0.202775 0.979225i \(-0.564996\pi\)
−0.202775 + 0.979225i \(0.564996\pi\)
\(444\) −2379.60 −0.254348
\(445\) −1042.97 −0.111105
\(446\) 24121.0 2.56091
\(447\) −15.2419 −0.00161279
\(448\) 5485.88 0.578535
\(449\) 106.834 0.0112289 0.00561447 0.999984i \(-0.498213\pi\)
0.00561447 + 0.999984i \(0.498213\pi\)
\(450\) −4463.78 −0.467610
\(451\) −30219.7 −3.15519
\(452\) 22739.7 2.36634
\(453\) −19.6468 −0.00203772
\(454\) −12450.8 −1.28710
\(455\) 439.397 0.0452731
\(456\) −2903.55 −0.298182
\(457\) −1237.64 −0.126684 −0.0633419 0.997992i \(-0.520176\pi\)
−0.0633419 + 0.997992i \(0.520176\pi\)
\(458\) 7959.95 0.812105
\(459\) −2618.10 −0.266236
\(460\) 12010.5 1.21737
\(461\) 8790.90 0.888141 0.444071 0.895992i \(-0.353534\pi\)
0.444071 + 0.895992i \(0.353534\pi\)
\(462\) −7121.49 −0.717146
\(463\) −3861.55 −0.387606 −0.193803 0.981040i \(-0.562082\pi\)
−0.193803 + 0.981040i \(0.562082\pi\)
\(464\) 3354.77 0.335650
\(465\) 1047.19 0.104435
\(466\) 3074.99 0.305679
\(467\) −8991.38 −0.890945 −0.445473 0.895296i \(-0.646964\pi\)
−0.445473 + 0.895296i \(0.646964\pi\)
\(468\) −1694.55 −0.167373
\(469\) 6526.06 0.642528
\(470\) −905.214 −0.0888391
\(471\) 6892.79 0.674316
\(472\) 6752.91 0.658533
\(473\) 7588.71 0.737694
\(474\) 7756.39 0.751609
\(475\) −3293.14 −0.318105
\(476\) −10509.6 −1.01198
\(477\) −4778.99 −0.458731
\(478\) 24750.0 2.36828
\(479\) 4179.82 0.398707 0.199354 0.979928i \(-0.436116\pi\)
0.199354 + 0.979928i \(0.436116\pi\)
\(480\) 1411.35 0.134206
\(481\) −711.963 −0.0674901
\(482\) 28941.0 2.73491
\(483\) 4121.80 0.388299
\(484\) 45544.2 4.27725
\(485\) −4919.41 −0.460575
\(486\) −1152.22 −0.107543
\(487\) −18443.8 −1.71616 −0.858078 0.513519i \(-0.828342\pi\)
−0.858078 + 0.513519i \(0.828342\pi\)
\(488\) 25296.2 2.34653
\(489\) 3256.46 0.301149
\(490\) −6146.56 −0.566680
\(491\) 8093.26 0.743877 0.371939 0.928257i \(-0.378693\pi\)
0.371939 + 0.928257i \(0.378693\pi\)
\(492\) −19627.0 −1.79849
\(493\) 10879.7 0.993907
\(494\) −1940.68 −0.176752
\(495\) −2719.49 −0.246933
\(496\) −2310.76 −0.209185
\(497\) 750.571 0.0677418
\(498\) 6465.73 0.581800
\(499\) −10941.6 −0.981591 −0.490796 0.871275i \(-0.663294\pi\)
−0.490796 + 0.871275i \(0.663294\pi\)
\(500\) −15019.6 −1.34340
\(501\) 327.197 0.0291778
\(502\) 30457.5 2.70794
\(503\) −9260.11 −0.820851 −0.410425 0.911894i \(-0.634620\pi\)
−0.410425 + 0.911894i \(0.634620\pi\)
\(504\) −2070.45 −0.182986
\(505\) −348.840 −0.0307389
\(506\) −58240.8 −5.11684
\(507\) −507.000 −0.0444116
\(508\) −18077.5 −1.57886
\(509\) −9996.40 −0.870497 −0.435248 0.900310i \(-0.643339\pi\)
−0.435248 + 0.900310i \(0.643339\pi\)
\(510\) −6230.08 −0.540927
\(511\) 1239.49 0.107303
\(512\) −10467.7 −0.903538
\(513\) −850.049 −0.0731591
\(514\) −5833.32 −0.500577
\(515\) 6103.68 0.522253
\(516\) 4928.69 0.420491
\(517\) 2827.66 0.240542
\(518\) −1943.29 −0.164833
\(519\) 5667.51 0.479337
\(520\) −1805.06 −0.152225
\(521\) 11427.9 0.960973 0.480486 0.877002i \(-0.340460\pi\)
0.480486 + 0.877002i \(0.340460\pi\)
\(522\) 4788.14 0.401477
\(523\) −4810.47 −0.402193 −0.201097 0.979571i \(-0.564451\pi\)
−0.201097 + 0.979571i \(0.564451\pi\)
\(524\) −18463.3 −1.53926
\(525\) −2348.25 −0.195212
\(526\) −2441.92 −0.202419
\(527\) −7493.88 −0.619428
\(528\) 6000.90 0.494613
\(529\) 21541.8 1.77051
\(530\) −11372.2 −0.932030
\(531\) 1977.00 0.161571
\(532\) −3412.26 −0.278083
\(533\) −5872.31 −0.477219
\(534\) 3284.76 0.266190
\(535\) 5120.41 0.413785
\(536\) −26809.2 −2.16042
\(537\) −10283.6 −0.826386
\(538\) −24337.0 −1.95027
\(539\) 19200.3 1.53435
\(540\) −1766.25 −0.140754
\(541\) 2411.99 0.191681 0.0958406 0.995397i \(-0.469446\pi\)
0.0958406 + 0.995397i \(0.469446\pi\)
\(542\) −20389.1 −1.61585
\(543\) −625.208 −0.0494111
\(544\) −10099.9 −0.796008
\(545\) 4596.47 0.361268
\(546\) −1383.85 −0.108468
\(547\) −4396.34 −0.343646 −0.171823 0.985128i \(-0.554966\pi\)
−0.171823 + 0.985128i \(0.554966\pi\)
\(548\) 12671.7 0.987786
\(549\) 7405.79 0.575722
\(550\) 33180.7 2.57242
\(551\) 3532.43 0.273116
\(552\) −16932.5 −1.30560
\(553\) 4080.40 0.313772
\(554\) −8595.67 −0.659197
\(555\) −742.088 −0.0567565
\(556\) 4494.66 0.342835
\(557\) −17488.0 −1.33032 −0.665160 0.746701i \(-0.731637\pi\)
−0.665160 + 0.746701i \(0.731637\pi\)
\(558\) −3298.05 −0.250211
\(559\) 1474.64 0.111575
\(560\) −1010.61 −0.0762608
\(561\) 19461.2 1.46462
\(562\) −19313.2 −1.44960
\(563\) −6881.77 −0.515154 −0.257577 0.966258i \(-0.582924\pi\)
−0.257577 + 0.966258i \(0.582924\pi\)
\(564\) 1836.50 0.137111
\(565\) 7091.49 0.528037
\(566\) 30092.0 2.23473
\(567\) −606.148 −0.0448957
\(568\) −3083.36 −0.227773
\(569\) 14733.5 1.08552 0.542758 0.839889i \(-0.317380\pi\)
0.542758 + 0.839889i \(0.317380\pi\)
\(570\) −2022.79 −0.148641
\(571\) 4488.51 0.328964 0.164482 0.986380i \(-0.447405\pi\)
0.164482 + 0.986380i \(0.447405\pi\)
\(572\) 12596.1 0.920752
\(573\) −2873.21 −0.209476
\(574\) −16028.4 −1.16553
\(575\) −19204.4 −1.39284
\(576\) −6597.74 −0.477267
\(577\) −10552.2 −0.761338 −0.380669 0.924711i \(-0.624306\pi\)
−0.380669 + 0.924711i \(0.624306\pi\)
\(578\) 21287.8 1.53193
\(579\) 1538.19 0.110406
\(580\) 7339.75 0.525460
\(581\) 3401.42 0.242882
\(582\) 15493.3 1.10347
\(583\) 35523.8 2.52357
\(584\) −5091.86 −0.360793
\(585\) −528.452 −0.0373484
\(586\) −39787.1 −2.80476
\(587\) −1637.20 −0.115118 −0.0575591 0.998342i \(-0.518332\pi\)
−0.0575591 + 0.998342i \(0.518332\pi\)
\(588\) 12470.1 0.874591
\(589\) −2433.13 −0.170213
\(590\) 4704.50 0.328273
\(591\) 11611.1 0.808147
\(592\) 1637.51 0.113685
\(593\) 14024.0 0.971155 0.485577 0.874194i \(-0.338609\pi\)
0.485577 + 0.874194i \(0.338609\pi\)
\(594\) 8564.84 0.591616
\(595\) −3277.45 −0.225819
\(596\) 73.5845 0.00505728
\(597\) −6917.50 −0.474229
\(598\) −11317.4 −0.773915
\(599\) 365.860 0.0249560 0.0124780 0.999922i \(-0.496028\pi\)
0.0124780 + 0.999922i \(0.496028\pi\)
\(600\) 9646.69 0.656374
\(601\) −10128.9 −0.687465 −0.343733 0.939068i \(-0.611691\pi\)
−0.343733 + 0.939068i \(0.611691\pi\)
\(602\) 4025.01 0.272503
\(603\) −7848.74 −0.530058
\(604\) 94.8504 0.00638975
\(605\) 14203.1 0.954446
\(606\) 1098.65 0.0736459
\(607\) 2324.97 0.155465 0.0777327 0.996974i \(-0.475232\pi\)
0.0777327 + 0.996974i \(0.475232\pi\)
\(608\) −3279.25 −0.218735
\(609\) 2518.89 0.167603
\(610\) 17623.0 1.16973
\(611\) 549.471 0.0363817
\(612\) 12639.6 0.834845
\(613\) 633.133 0.0417162 0.0208581 0.999782i \(-0.493360\pi\)
0.0208581 + 0.999782i \(0.493360\pi\)
\(614\) 18991.6 1.24827
\(615\) −6120.77 −0.401323
\(616\) 15390.3 1.00664
\(617\) −2981.85 −0.194562 −0.0972810 0.995257i \(-0.531015\pi\)
−0.0972810 + 0.995257i \(0.531015\pi\)
\(618\) −19223.1 −1.25124
\(619\) 15158.6 0.984292 0.492146 0.870513i \(-0.336213\pi\)
0.492146 + 0.870513i \(0.336213\pi\)
\(620\) −5055.60 −0.327480
\(621\) −4957.19 −0.320330
\(622\) 27674.8 1.78402
\(623\) 1728.01 0.111126
\(624\) 1166.10 0.0748096
\(625\) 8391.01 0.537025
\(626\) 8360.45 0.533787
\(627\) 6318.69 0.402463
\(628\) −33276.8 −2.11447
\(629\) 5310.51 0.336636
\(630\) −1442.40 −0.0912170
\(631\) 5562.30 0.350922 0.175461 0.984486i \(-0.443858\pi\)
0.175461 + 0.984486i \(0.443858\pi\)
\(632\) −16762.4 −1.05502
\(633\) 11017.2 0.691776
\(634\) −28954.2 −1.81375
\(635\) −5637.56 −0.352315
\(636\) 23071.9 1.43846
\(637\) 3731.00 0.232068
\(638\) −35591.7 −2.20861
\(639\) −902.693 −0.0558842
\(640\) −11936.5 −0.737237
\(641\) −24140.7 −1.48752 −0.743761 0.668446i \(-0.766960\pi\)
−0.743761 + 0.668446i \(0.766960\pi\)
\(642\) −16126.4 −0.991366
\(643\) −1749.69 −0.107311 −0.0536555 0.998560i \(-0.517087\pi\)
−0.0536555 + 0.998560i \(0.517087\pi\)
\(644\) −19899.1 −1.21760
\(645\) 1537.03 0.0938305
\(646\) 14475.5 0.881626
\(647\) 1489.51 0.0905083 0.0452542 0.998976i \(-0.485590\pi\)
0.0452542 + 0.998976i \(0.485590\pi\)
\(648\) 2490.07 0.150956
\(649\) −14695.7 −0.888836
\(650\) 6447.68 0.389075
\(651\) −1735.00 −0.104455
\(652\) −15721.4 −0.944323
\(653\) 10668.2 0.639327 0.319663 0.947531i \(-0.396430\pi\)
0.319663 + 0.947531i \(0.396430\pi\)
\(654\) −14476.2 −0.865544
\(655\) −5757.86 −0.343478
\(656\) 13506.3 0.803858
\(657\) −1490.71 −0.0885205
\(658\) 1499.77 0.0888559
\(659\) −13933.1 −0.823608 −0.411804 0.911272i \(-0.635101\pi\)
−0.411804 + 0.911272i \(0.635101\pi\)
\(660\) 13129.1 0.774317
\(661\) −2349.69 −0.138264 −0.0691320 0.997608i \(-0.522023\pi\)
−0.0691320 + 0.997608i \(0.522023\pi\)
\(662\) 35516.8 2.08520
\(663\) 3781.70 0.221522
\(664\) −13973.1 −0.816660
\(665\) −1064.13 −0.0620529
\(666\) 2337.15 0.135980
\(667\) 20599.9 1.19585
\(668\) −1579.63 −0.0914937
\(669\) −15261.1 −0.881958
\(670\) −18677.0 −1.07695
\(671\) −55049.6 −3.16716
\(672\) −2338.35 −0.134232
\(673\) −32421.0 −1.85697 −0.928483 0.371374i \(-0.878887\pi\)
−0.928483 + 0.371374i \(0.878887\pi\)
\(674\) 44867.7 2.56415
\(675\) 2824.19 0.161042
\(676\) 2447.68 0.139263
\(677\) −1071.74 −0.0608421 −0.0304211 0.999537i \(-0.509685\pi\)
−0.0304211 + 0.999537i \(0.509685\pi\)
\(678\) −22334.1 −1.26510
\(679\) 8150.56 0.460663
\(680\) 13463.9 0.759287
\(681\) 7877.48 0.443268
\(682\) 24515.5 1.37646
\(683\) −305.487 −0.0171144 −0.00855721 0.999963i \(-0.502724\pi\)
−0.00855721 + 0.999963i \(0.502724\pi\)
\(684\) 4103.84 0.229407
\(685\) 3951.72 0.220419
\(686\) 22354.5 1.24417
\(687\) −5036.18 −0.279683
\(688\) −3391.66 −0.187944
\(689\) 6902.99 0.381688
\(690\) −11796.2 −0.650833
\(691\) 2180.81 0.120061 0.0600303 0.998197i \(-0.480880\pi\)
0.0600303 + 0.998197i \(0.480880\pi\)
\(692\) −27361.4 −1.50307
\(693\) 4505.70 0.246980
\(694\) 54340.9 2.97227
\(695\) 1401.68 0.0765018
\(696\) −10347.7 −0.563545
\(697\) 43801.4 2.38034
\(698\) −17420.6 −0.944672
\(699\) −1945.52 −0.105274
\(700\) 11336.8 0.612132
\(701\) −15168.3 −0.817259 −0.408629 0.912700i \(-0.633993\pi\)
−0.408629 + 0.912700i \(0.633993\pi\)
\(702\) 1664.32 0.0894812
\(703\) 1724.23 0.0925043
\(704\) 49043.1 2.62554
\(705\) 572.720 0.0305956
\(706\) 16067.8 0.856545
\(707\) 577.963 0.0307448
\(708\) −9544.49 −0.506644
\(709\) −8988.17 −0.476104 −0.238052 0.971252i \(-0.576509\pi\)
−0.238052 + 0.971252i \(0.576509\pi\)
\(710\) −2148.07 −0.113543
\(711\) −4907.39 −0.258849
\(712\) −7098.71 −0.373645
\(713\) −14189.1 −0.745284
\(714\) 10322.1 0.541029
\(715\) 3928.15 0.205461
\(716\) 49646.8 2.59132
\(717\) −15659.1 −0.815619
\(718\) −45870.7 −2.38423
\(719\) 8448.18 0.438198 0.219099 0.975703i \(-0.429688\pi\)
0.219099 + 0.975703i \(0.429688\pi\)
\(720\) 1215.44 0.0629120
\(721\) −10112.7 −0.522352
\(722\) −27823.1 −1.43417
\(723\) −18310.7 −0.941883
\(724\) 3018.36 0.154940
\(725\) −11736.1 −0.601197
\(726\) −44731.8 −2.28671
\(727\) −8624.18 −0.439963 −0.219982 0.975504i \(-0.570600\pi\)
−0.219982 + 0.975504i \(0.570600\pi\)
\(728\) 2990.64 0.152254
\(729\) 729.000 0.0370370
\(730\) −3547.31 −0.179852
\(731\) −10999.3 −0.556530
\(732\) −35753.5 −1.80531
\(733\) 31124.2 1.56835 0.784174 0.620541i \(-0.213087\pi\)
0.784174 + 0.620541i \(0.213087\pi\)
\(734\) −41328.2 −2.07827
\(735\) 3888.87 0.195161
\(736\) −19123.4 −0.957742
\(737\) 58342.2 2.91596
\(738\) 19276.9 0.961509
\(739\) −17671.1 −0.879626 −0.439813 0.898089i \(-0.644955\pi\)
−0.439813 + 0.898089i \(0.644955\pi\)
\(740\) 3582.63 0.177973
\(741\) 1227.85 0.0608720
\(742\) 18841.6 0.932206
\(743\) −21331.1 −1.05325 −0.526623 0.850099i \(-0.676542\pi\)
−0.526623 + 0.850099i \(0.676542\pi\)
\(744\) 7127.43 0.351215
\(745\) 22.9477 0.00112851
\(746\) 22134.0 1.08630
\(747\) −4090.81 −0.200368
\(748\) −93954.1 −4.59265
\(749\) −8483.58 −0.413863
\(750\) 14751.7 0.718208
\(751\) 11712.9 0.569122 0.284561 0.958658i \(-0.408152\pi\)
0.284561 + 0.958658i \(0.408152\pi\)
\(752\) −1263.78 −0.0612835
\(753\) −19270.2 −0.932596
\(754\) −6916.20 −0.334049
\(755\) 29.5795 0.00142584
\(756\) 2926.35 0.140781
\(757\) −16610.9 −0.797537 −0.398768 0.917052i \(-0.630562\pi\)
−0.398768 + 0.917052i \(0.630562\pi\)
\(758\) 51507.4 2.46812
\(759\) 36848.4 1.76220
\(760\) 4371.47 0.208645
\(761\) −29365.5 −1.39882 −0.699408 0.714723i \(-0.746553\pi\)
−0.699408 + 0.714723i \(0.746553\pi\)
\(762\) 17755.1 0.844093
\(763\) −7615.50 −0.361336
\(764\) 13871.2 0.656861
\(765\) 3941.71 0.186291
\(766\) 47673.5 2.24871
\(767\) −2855.66 −0.134435
\(768\) 19999.2 0.939659
\(769\) −28599.9 −1.34114 −0.670572 0.741844i \(-0.733951\pi\)
−0.670572 + 0.741844i \(0.733951\pi\)
\(770\) 10721.8 0.501803
\(771\) 3690.68 0.172395
\(772\) −7426.03 −0.346203
\(773\) 13491.8 0.627772 0.313886 0.949461i \(-0.398369\pi\)
0.313886 + 0.949461i \(0.398369\pi\)
\(774\) −4840.78 −0.224804
\(775\) 8083.78 0.374681
\(776\) −33482.7 −1.54892
\(777\) 1229.50 0.0567673
\(778\) 30434.3 1.40247
\(779\) 14221.5 0.654093
\(780\) 2551.25 0.117114
\(781\) 6710.01 0.307430
\(782\) 84415.9 3.86024
\(783\) −3029.41 −0.138266
\(784\) −8581.27 −0.390911
\(785\) −10377.5 −0.471834
\(786\) 18134.0 0.822923
\(787\) 25876.7 1.17205 0.586025 0.810293i \(-0.300692\pi\)
0.586025 + 0.810293i \(0.300692\pi\)
\(788\) −56055.5 −2.53413
\(789\) 1544.98 0.0697118
\(790\) −11677.7 −0.525918
\(791\) −11749.3 −0.528137
\(792\) −18509.5 −0.830438
\(793\) −10697.3 −0.479030
\(794\) −56994.2 −2.54742
\(795\) 7195.06 0.320984
\(796\) 33396.1 1.48705
\(797\) 21936.4 0.974938 0.487469 0.873140i \(-0.337920\pi\)
0.487469 + 0.873140i \(0.337920\pi\)
\(798\) 3351.40 0.148669
\(799\) −4098.49 −0.181469
\(800\) 10894.9 0.481491
\(801\) −2078.23 −0.0916739
\(802\) 17069.0 0.751531
\(803\) 11080.9 0.486969
\(804\) 37891.9 1.66212
\(805\) −6205.62 −0.271701
\(806\) 4763.85 0.208188
\(807\) 15397.8 0.671658
\(808\) −2374.29 −0.103375
\(809\) 5583.23 0.242640 0.121320 0.992613i \(-0.461287\pi\)
0.121320 + 0.992613i \(0.461287\pi\)
\(810\) 1734.74 0.0752502
\(811\) 12925.4 0.559647 0.279823 0.960052i \(-0.409724\pi\)
0.279823 + 0.960052i \(0.409724\pi\)
\(812\) −12160.6 −0.525559
\(813\) 12900.0 0.556486
\(814\) −17372.8 −0.748054
\(815\) −4902.80 −0.210721
\(816\) −8697.87 −0.373145
\(817\) −3571.27 −0.152929
\(818\) −228.058 −0.00974801
\(819\) 875.548 0.0373555
\(820\) 29549.7 1.25844
\(821\) −10153.2 −0.431608 −0.215804 0.976437i \(-0.569237\pi\)
−0.215804 + 0.976437i \(0.569237\pi\)
\(822\) −12445.6 −0.528092
\(823\) 3282.93 0.139047 0.0695235 0.997580i \(-0.477852\pi\)
0.0695235 + 0.997580i \(0.477852\pi\)
\(824\) 41543.1 1.75634
\(825\) −20993.1 −0.885922
\(826\) −7794.49 −0.328335
\(827\) −17689.3 −0.743795 −0.371897 0.928274i \(-0.621293\pi\)
−0.371897 + 0.928274i \(0.621293\pi\)
\(828\) 23932.2 1.00447
\(829\) 38181.5 1.59964 0.799818 0.600243i \(-0.204930\pi\)
0.799818 + 0.600243i \(0.204930\pi\)
\(830\) −9734.56 −0.407098
\(831\) 5438.40 0.227023
\(832\) 9530.06 0.397110
\(833\) −27829.4 −1.15754
\(834\) −4414.49 −0.183287
\(835\) −492.615 −0.0204163
\(836\) −30505.2 −1.26201
\(837\) 2086.64 0.0861708
\(838\) 3430.41 0.141410
\(839\) 43895.2 1.80623 0.903117 0.429395i \(-0.141273\pi\)
0.903117 + 0.429395i \(0.141273\pi\)
\(840\) 3117.18 0.128039
\(841\) −11800.1 −0.483829
\(842\) −70392.4 −2.88109
\(843\) 12219.2 0.499233
\(844\) −53188.5 −2.16922
\(845\) 763.320 0.0310757
\(846\) −1803.74 −0.0733024
\(847\) −23532.0 −0.954627
\(848\) −15876.8 −0.642938
\(849\) −19038.9 −0.769626
\(850\) −48093.0 −1.94068
\(851\) 10055.1 0.405034
\(852\) 4358.00 0.175238
\(853\) −19955.2 −0.800998 −0.400499 0.916297i \(-0.631163\pi\)
−0.400499 + 0.916297i \(0.631163\pi\)
\(854\) −29198.0 −1.16995
\(855\) 1279.80 0.0511910
\(856\) 34850.8 1.39156
\(857\) 26030.4 1.03755 0.518776 0.854910i \(-0.326388\pi\)
0.518776 + 0.854910i \(0.326388\pi\)
\(858\) −12371.4 −0.492254
\(859\) −45617.4 −1.81193 −0.905964 0.423354i \(-0.860853\pi\)
−0.905964 + 0.423354i \(0.860853\pi\)
\(860\) −7420.45 −0.294227
\(861\) 10141.0 0.401399
\(862\) 5230.59 0.206676
\(863\) −2010.93 −0.0793195 −0.0396597 0.999213i \(-0.512627\pi\)
−0.0396597 + 0.999213i \(0.512627\pi\)
\(864\) 2812.27 0.110735
\(865\) −8532.78 −0.335403
\(866\) 42170.1 1.65473
\(867\) −13468.6 −0.527586
\(868\) 8376.19 0.327542
\(869\) 36478.2 1.42398
\(870\) −7208.83 −0.280922
\(871\) 11337.1 0.441035
\(872\) 31284.7 1.21495
\(873\) −9802.48 −0.380027
\(874\) 27408.3 1.06076
\(875\) 7760.41 0.299828
\(876\) 7196.79 0.277576
\(877\) −36767.3 −1.41567 −0.707836 0.706376i \(-0.750329\pi\)
−0.707836 + 0.706376i \(0.750329\pi\)
\(878\) −51692.0 −1.98693
\(879\) 25172.9 0.965940
\(880\) −9034.72 −0.346091
\(881\) 35401.2 1.35380 0.676899 0.736076i \(-0.263323\pi\)
0.676899 + 0.736076i \(0.263323\pi\)
\(882\) −12247.7 −0.467575
\(883\) −11928.0 −0.454596 −0.227298 0.973825i \(-0.572989\pi\)
−0.227298 + 0.973825i \(0.572989\pi\)
\(884\) −18257.2 −0.694633
\(885\) −2976.49 −0.113055
\(886\) −17930.0 −0.679875
\(887\) −32939.3 −1.24689 −0.623447 0.781866i \(-0.714268\pi\)
−0.623447 + 0.781866i \(0.714268\pi\)
\(888\) −5050.83 −0.190872
\(889\) 9340.40 0.352381
\(890\) −4945.41 −0.186259
\(891\) −5418.89 −0.203748
\(892\) 73677.3 2.76558
\(893\) −1330.70 −0.0498660
\(894\) −72.2720 −0.00270373
\(895\) 15482.6 0.578241
\(896\) 19776.6 0.737376
\(897\) 7160.38 0.266531
\(898\) 506.569 0.0188245
\(899\) −8671.18 −0.321691
\(900\) −13634.5 −0.504983
\(901\) −51489.2 −1.90383
\(902\) −143292. −5.28946
\(903\) −2546.58 −0.0938483
\(904\) 48266.4 1.77579
\(905\) 941.289 0.0345740
\(906\) −93.1585 −0.00341610
\(907\) 1500.89 0.0549461 0.0274731 0.999623i \(-0.491254\pi\)
0.0274731 + 0.999623i \(0.491254\pi\)
\(908\) −38030.7 −1.38997
\(909\) −695.102 −0.0253631
\(910\) 2083.47 0.0758971
\(911\) 19728.2 0.717481 0.358740 0.933437i \(-0.383206\pi\)
0.358740 + 0.933437i \(0.383206\pi\)
\(912\) −2824.04 −0.102537
\(913\) 30408.3 1.10226
\(914\) −5868.48 −0.212376
\(915\) −11149.9 −0.402845
\(916\) 24313.5 0.877011
\(917\) 9539.72 0.343543
\(918\) −12414.1 −0.446326
\(919\) 19992.0 0.717602 0.358801 0.933414i \(-0.383186\pi\)
0.358801 + 0.933414i \(0.383186\pi\)
\(920\) 25492.9 0.913560
\(921\) −12015.8 −0.429897
\(922\) 41683.4 1.48891
\(923\) 1303.89 0.0464984
\(924\) −21752.5 −0.774463
\(925\) −5728.54 −0.203625
\(926\) −18310.2 −0.649794
\(927\) 12162.3 0.430918
\(928\) −11686.6 −0.413395
\(929\) −1923.62 −0.0679354 −0.0339677 0.999423i \(-0.510814\pi\)
−0.0339677 + 0.999423i \(0.510814\pi\)
\(930\) 4965.42 0.175078
\(931\) −9035.71 −0.318081
\(932\) 9392.52 0.330110
\(933\) −17509.6 −0.614403
\(934\) −42634.0 −1.49361
\(935\) −29300.0 −1.02483
\(936\) −3596.77 −0.125603
\(937\) −10252.9 −0.357468 −0.178734 0.983897i \(-0.557200\pi\)
−0.178734 + 0.983897i \(0.557200\pi\)
\(938\) 30944.4 1.07715
\(939\) −5289.58 −0.183833
\(940\) −2764.96 −0.0959394
\(941\) 11043.0 0.382563 0.191282 0.981535i \(-0.438736\pi\)
0.191282 + 0.981535i \(0.438736\pi\)
\(942\) 32683.2 1.13044
\(943\) 82934.8 2.86398
\(944\) 6568.00 0.226451
\(945\) 912.594 0.0314145
\(946\) 35983.1 1.23669
\(947\) 32105.2 1.10167 0.550833 0.834615i \(-0.314310\pi\)
0.550833 + 0.834615i \(0.314310\pi\)
\(948\) 23691.8 0.811680
\(949\) 2153.24 0.0736535
\(950\) −15614.9 −0.533280
\(951\) 18319.0 0.624643
\(952\) −22307.1 −0.759431
\(953\) −9473.37 −0.322007 −0.161003 0.986954i \(-0.551473\pi\)
−0.161003 + 0.986954i \(0.551473\pi\)
\(954\) −22660.3 −0.769031
\(955\) 4325.79 0.146575
\(956\) 75598.4 2.55756
\(957\) 22518.5 0.760628
\(958\) 19819.3 0.668404
\(959\) −6547.26 −0.220461
\(960\) 9933.30 0.333954
\(961\) −23818.3 −0.799514
\(962\) −3375.89 −0.113142
\(963\) 10203.0 0.341419
\(964\) 88399.8 2.95349
\(965\) −2315.84 −0.0772534
\(966\) 19544.2 0.650956
\(967\) −5310.75 −0.176610 −0.0883052 0.996093i \(-0.528145\pi\)
−0.0883052 + 0.996093i \(0.528145\pi\)
\(968\) 96670.1 3.20981
\(969\) −9158.49 −0.303626
\(970\) −23326.2 −0.772122
\(971\) 24271.2 0.802164 0.401082 0.916042i \(-0.368634\pi\)
0.401082 + 0.916042i \(0.368634\pi\)
\(972\) −3519.45 −0.116138
\(973\) −2322.32 −0.0765163
\(974\) −87454.2 −2.87702
\(975\) −4079.38 −0.133995
\(976\) 24603.6 0.806907
\(977\) −49602.5 −1.62428 −0.812142 0.583460i \(-0.801699\pi\)
−0.812142 + 0.583460i \(0.801699\pi\)
\(978\) 15441.0 0.504856
\(979\) 15448.2 0.504317
\(980\) −18774.6 −0.611971
\(981\) 9158.98 0.298087
\(982\) 38375.5 1.24706
\(983\) 47385.7 1.53751 0.768753 0.639545i \(-0.220877\pi\)
0.768753 + 0.639545i \(0.220877\pi\)
\(984\) −41659.5 −1.34965
\(985\) −17481.2 −0.565478
\(986\) 51587.7 1.66621
\(987\) −948.891 −0.0306014
\(988\) −5927.78 −0.190878
\(989\) −20826.4 −0.669607
\(990\) −12894.9 −0.413966
\(991\) −8947.33 −0.286802 −0.143401 0.989665i \(-0.545804\pi\)
−0.143401 + 0.989665i \(0.545804\pi\)
\(992\) 8049.67 0.257638
\(993\) −22471.1 −0.718127
\(994\) 3558.95 0.113564
\(995\) 10414.7 0.331828
\(996\) 19749.5 0.628299
\(997\) −14908.6 −0.473582 −0.236791 0.971561i \(-0.576096\pi\)
−0.236791 + 0.971561i \(0.576096\pi\)
\(998\) −51881.4 −1.64557
\(999\) −1478.69 −0.0468306
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 39.4.a.b.1.2 2
3.2 odd 2 117.4.a.c.1.1 2
4.3 odd 2 624.4.a.r.1.1 2
5.4 even 2 975.4.a.j.1.1 2
7.6 odd 2 1911.4.a.h.1.2 2
8.3 odd 2 2496.4.a.s.1.2 2
8.5 even 2 2496.4.a.bc.1.2 2
12.11 even 2 1872.4.a.t.1.2 2
13.5 odd 4 507.4.b.f.337.1 4
13.8 odd 4 507.4.b.f.337.4 4
13.12 even 2 507.4.a.f.1.1 2
39.38 odd 2 1521.4.a.s.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.a.b.1.2 2 1.1 even 1 trivial
117.4.a.c.1.1 2 3.2 odd 2
507.4.a.f.1.1 2 13.12 even 2
507.4.b.f.337.1 4 13.5 odd 4
507.4.b.f.337.4 4 13.8 odd 4
624.4.a.r.1.1 2 4.3 odd 2
975.4.a.j.1.1 2 5.4 even 2
1521.4.a.s.1.2 2 39.38 odd 2
1872.4.a.t.1.2 2 12.11 even 2
1911.4.a.h.1.2 2 7.6 odd 2
2496.4.a.s.1.2 2 8.3 odd 2
2496.4.a.bc.1.2 2 8.5 even 2