Properties

Label 1045.6.a.f.1.13
Level $1045$
Weight $6$
Character 1045.1
Self dual yes
Analytic conductor $167.601$
Analytic rank $0$
Dimension $38$
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] = [1045,6,Mod(1,1045)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1045, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1045.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1045.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: \(167.601091705\)
Analytic rank: \(0\)
Dimension: \(38\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Character \(\chi\) \(=\) 1045.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.27957 q^{2} +13.2431 q^{3} -13.6853 q^{4} +25.0000 q^{5} -56.6748 q^{6} +11.8777 q^{7} +195.513 q^{8} -67.6203 q^{9} +O(q^{10})\) \(q-4.27957 q^{2} +13.2431 q^{3} -13.6853 q^{4} +25.0000 q^{5} -56.6748 q^{6} +11.8777 q^{7} +195.513 q^{8} -67.6203 q^{9} -106.989 q^{10} -121.000 q^{11} -181.236 q^{12} -608.806 q^{13} -50.8313 q^{14} +331.078 q^{15} -398.784 q^{16} +2170.89 q^{17} +289.386 q^{18} -361.000 q^{19} -342.132 q^{20} +157.297 q^{21} +517.828 q^{22} +986.989 q^{23} +2589.20 q^{24} +625.000 q^{25} +2605.43 q^{26} -4113.58 q^{27} -162.549 q^{28} +1117.15 q^{29} -1416.87 q^{30} +145.291 q^{31} -4549.80 q^{32} -1602.42 q^{33} -9290.48 q^{34} +296.942 q^{35} +925.403 q^{36} -6997.18 q^{37} +1544.92 q^{38} -8062.48 q^{39} +4887.83 q^{40} -14203.9 q^{41} -673.164 q^{42} +12513.9 q^{43} +1655.92 q^{44} -1690.51 q^{45} -4223.89 q^{46} +4843.43 q^{47} -5281.14 q^{48} -16665.9 q^{49} -2674.73 q^{50} +28749.3 q^{51} +8331.68 q^{52} +19835.3 q^{53} +17604.3 q^{54} -3025.00 q^{55} +2322.24 q^{56} -4780.76 q^{57} -4780.91 q^{58} +20069.4 q^{59} -4530.89 q^{60} +39595.8 q^{61} -621.785 q^{62} -803.171 q^{63} +32232.3 q^{64} -15220.1 q^{65} +6857.65 q^{66} -12028.7 q^{67} -29709.3 q^{68} +13070.8 q^{69} -1270.78 q^{70} -3232.49 q^{71} -13220.7 q^{72} +2779.96 q^{73} +29944.9 q^{74} +8276.94 q^{75} +4940.39 q^{76} -1437.20 q^{77} +34503.9 q^{78} +22928.1 q^{79} -9969.60 q^{80} -38044.8 q^{81} +60786.8 q^{82} -103440. q^{83} -2152.66 q^{84} +54272.3 q^{85} -53554.3 q^{86} +14794.5 q^{87} -23657.1 q^{88} -61233.4 q^{89} +7234.64 q^{90} -7231.19 q^{91} -13507.2 q^{92} +1924.11 q^{93} -20727.8 q^{94} -9025.00 q^{95} -60253.5 q^{96} +155760. q^{97} +71323.0 q^{98} +8182.05 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 38 q + 8 q^{2} + 63 q^{3} + 616 q^{4} + 950 q^{5} + 149 q^{6} + 275 q^{7} + 264 q^{8} + 3029 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 38 q + 8 q^{2} + 63 q^{3} + 616 q^{4} + 950 q^{5} + 149 q^{6} + 275 q^{7} + 264 q^{8} + 3029 q^{9} + 200 q^{10} - 4598 q^{11} + 2312 q^{12} + 41 q^{13} + 23 q^{14} + 1575 q^{15} + 7196 q^{16} - 2431 q^{17} - 1689 q^{18} - 13718 q^{19} + 15400 q^{20} - 1577 q^{21} - 968 q^{22} + 9284 q^{23} + 7598 q^{24} + 23750 q^{25} + 13129 q^{26} + 9228 q^{27} - 1079 q^{28} - 559 q^{29} + 3725 q^{30} + 11147 q^{31} + 11051 q^{32} - 7623 q^{33} + 40895 q^{34} + 6875 q^{35} + 55887 q^{36} + 41579 q^{37} - 2888 q^{38} + 24982 q^{39} + 6600 q^{40} + 18597 q^{41} + 61360 q^{42} + 25353 q^{43} - 74536 q^{44} + 75725 q^{45} + 1611 q^{46} + 63516 q^{47} + 187737 q^{48} + 141609 q^{49} + 5000 q^{50} + 107546 q^{51} + 60018 q^{52} + 123045 q^{53} + 256696 q^{54} - 114950 q^{55} + 157335 q^{56} - 22743 q^{57} + 218938 q^{58} + 132925 q^{59} + 57800 q^{60} - 59107 q^{61} + 166982 q^{62} + 130582 q^{63} + 313126 q^{64} + 1025 q^{65} - 18029 q^{66} + 162534 q^{67} + 182980 q^{68} + 178552 q^{69} + 575 q^{70} + 157840 q^{71} + 98630 q^{72} - 63010 q^{73} + 122683 q^{74} + 39375 q^{75} - 222376 q^{76} - 33275 q^{77} + 277272 q^{78} - 16385 q^{79} + 179900 q^{80} + 290354 q^{81} + 362302 q^{82} + 138461 q^{83} + 446870 q^{84} - 60775 q^{85} + 643902 q^{86} + 291602 q^{87} - 31944 q^{88} + 224792 q^{89} - 42225 q^{90} + 498548 q^{91} + 581088 q^{92} + 134210 q^{93} + 35864 q^{94} - 342950 q^{95} + 377376 q^{96} + 292216 q^{97} - 58230 q^{98} - 366509 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.27957 −0.756528 −0.378264 0.925698i \(-0.623479\pi\)
−0.378264 + 0.925698i \(0.623479\pi\)
\(3\) 13.2431 0.849545 0.424773 0.905300i \(-0.360354\pi\)
0.424773 + 0.905300i \(0.360354\pi\)
\(4\) −13.6853 −0.427665
\(5\) 25.0000 0.447214
\(6\) −56.6748 −0.642705
\(7\) 11.8777 0.0916191 0.0458095 0.998950i \(-0.485413\pi\)
0.0458095 + 0.998950i \(0.485413\pi\)
\(8\) 195.513 1.08007
\(9\) −67.6203 −0.278273
\(10\) −106.989 −0.338330
\(11\) −121.000 −0.301511
\(12\) −181.236 −0.363321
\(13\) −608.806 −0.999126 −0.499563 0.866277i \(-0.666506\pi\)
−0.499563 + 0.866277i \(0.666506\pi\)
\(14\) −50.8313 −0.0693124
\(15\) 331.078 0.379928
\(16\) −398.784 −0.389437
\(17\) 2170.89 1.82186 0.910931 0.412558i \(-0.135365\pi\)
0.910931 + 0.412558i \(0.135365\pi\)
\(18\) 289.386 0.210521
\(19\) −361.000 −0.229416
\(20\) −342.132 −0.191258
\(21\) 157.297 0.0778346
\(22\) 517.828 0.228102
\(23\) 986.989 0.389039 0.194519 0.980899i \(-0.437685\pi\)
0.194519 + 0.980899i \(0.437685\pi\)
\(24\) 2589.20 0.917567
\(25\) 625.000 0.200000
\(26\) 2605.43 0.755867
\(27\) −4113.58 −1.08595
\(28\) −162.549 −0.0391823
\(29\) 1117.15 0.246669 0.123335 0.992365i \(-0.460641\pi\)
0.123335 + 0.992365i \(0.460641\pi\)
\(30\) −1416.87 −0.287426
\(31\) 145.291 0.0271541 0.0135771 0.999908i \(-0.495678\pi\)
0.0135771 + 0.999908i \(0.495678\pi\)
\(32\) −4549.80 −0.785448
\(33\) −1602.42 −0.256148
\(34\) −9290.48 −1.37829
\(35\) 296.942 0.0409733
\(36\) 925.403 0.119008
\(37\) −6997.18 −0.840270 −0.420135 0.907462i \(-0.638017\pi\)
−0.420135 + 0.907462i \(0.638017\pi\)
\(38\) 1544.92 0.173559
\(39\) −8062.48 −0.848803
\(40\) 4887.83 0.483021
\(41\) −14203.9 −1.31962 −0.659811 0.751432i \(-0.729364\pi\)
−0.659811 + 0.751432i \(0.729364\pi\)
\(42\) −673.164 −0.0588840
\(43\) 12513.9 1.03210 0.516052 0.856557i \(-0.327401\pi\)
0.516052 + 0.856557i \(0.327401\pi\)
\(44\) 1655.92 0.128946
\(45\) −1690.51 −0.124447
\(46\) −4223.89 −0.294319
\(47\) 4843.43 0.319822 0.159911 0.987131i \(-0.448879\pi\)
0.159911 + 0.987131i \(0.448879\pi\)
\(48\) −5281.14 −0.330845
\(49\) −16665.9 −0.991606
\(50\) −2674.73 −0.151306
\(51\) 28749.3 1.54775
\(52\) 8331.68 0.427292
\(53\) 19835.3 0.969948 0.484974 0.874529i \(-0.338829\pi\)
0.484974 + 0.874529i \(0.338829\pi\)
\(54\) 17604.3 0.821552
\(55\) −3025.00 −0.134840
\(56\) 2322.24 0.0989549
\(57\) −4780.76 −0.194899
\(58\) −4780.91 −0.186612
\(59\) 20069.4 0.750592 0.375296 0.926905i \(-0.377541\pi\)
0.375296 + 0.926905i \(0.377541\pi\)
\(60\) −4530.89 −0.162482
\(61\) 39595.8 1.36246 0.681231 0.732068i \(-0.261445\pi\)
0.681231 + 0.732068i \(0.261445\pi\)
\(62\) −621.785 −0.0205429
\(63\) −803.171 −0.0254951
\(64\) 32232.3 0.983651
\(65\) −15220.1 −0.446823
\(66\) 6857.65 0.193783
\(67\) −12028.7 −0.327365 −0.163683 0.986513i \(-0.552337\pi\)
−0.163683 + 0.986513i \(0.552337\pi\)
\(68\) −29709.3 −0.779147
\(69\) 13070.8 0.330506
\(70\) −1270.78 −0.0309975
\(71\) −3232.49 −0.0761011 −0.0380505 0.999276i \(-0.512115\pi\)
−0.0380505 + 0.999276i \(0.512115\pi\)
\(72\) −13220.7 −0.300554
\(73\) 2779.96 0.0610565 0.0305282 0.999534i \(-0.490281\pi\)
0.0305282 + 0.999534i \(0.490281\pi\)
\(74\) 29944.9 0.635688
\(75\) 8276.94 0.169909
\(76\) 4940.39 0.0981131
\(77\) −1437.20 −0.0276242
\(78\) 34503.9 0.642144
\(79\) 22928.1 0.413332 0.206666 0.978412i \(-0.433739\pi\)
0.206666 + 0.978412i \(0.433739\pi\)
\(80\) −9969.60 −0.174162
\(81\) −38044.8 −0.644292
\(82\) 60786.8 0.998331
\(83\) −103440. −1.64814 −0.824070 0.566487i \(-0.808302\pi\)
−0.824070 + 0.566487i \(0.808302\pi\)
\(84\) −2152.66 −0.0332871
\(85\) 54272.3 0.814762
\(86\) −53554.3 −0.780815
\(87\) 14794.5 0.209557
\(88\) −23657.1 −0.325653
\(89\) −61233.4 −0.819433 −0.409716 0.912213i \(-0.634372\pi\)
−0.409716 + 0.912213i \(0.634372\pi\)
\(90\) 7234.64 0.0941479
\(91\) −7231.19 −0.0915390
\(92\) −13507.2 −0.166378
\(93\) 1924.11 0.0230687
\(94\) −20727.8 −0.241954
\(95\) −9025.00 −0.102598
\(96\) −60253.5 −0.667274
\(97\) 155760. 1.68084 0.840420 0.541935i \(-0.182308\pi\)
0.840420 + 0.541935i \(0.182308\pi\)
\(98\) 71323.0 0.750178
\(99\) 8182.05 0.0839024
\(100\) −8553.30 −0.0855330
\(101\) 116411. 1.13551 0.567754 0.823198i \(-0.307813\pi\)
0.567754 + 0.823198i \(0.307813\pi\)
\(102\) −123035. −1.17092
\(103\) −77913.2 −0.723632 −0.361816 0.932249i \(-0.617843\pi\)
−0.361816 + 0.932249i \(0.617843\pi\)
\(104\) −119030. −1.07913
\(105\) 3932.43 0.0348087
\(106\) −84886.4 −0.733793
\(107\) 81538.6 0.688500 0.344250 0.938878i \(-0.388133\pi\)
0.344250 + 0.938878i \(0.388133\pi\)
\(108\) 56295.5 0.464423
\(109\) 55981.5 0.451313 0.225657 0.974207i \(-0.427547\pi\)
0.225657 + 0.974207i \(0.427547\pi\)
\(110\) 12945.7 0.102010
\(111\) −92664.4 −0.713847
\(112\) −4736.62 −0.0356799
\(113\) −48284.2 −0.355720 −0.177860 0.984056i \(-0.556917\pi\)
−0.177860 + 0.984056i \(0.556917\pi\)
\(114\) 20459.6 0.147447
\(115\) 24674.7 0.173983
\(116\) −15288.5 −0.105492
\(117\) 41167.6 0.278030
\(118\) −85888.3 −0.567844
\(119\) 25785.1 0.166917
\(120\) 64730.1 0.410349
\(121\) 14641.0 0.0909091
\(122\) −169453. −1.03074
\(123\) −188104. −1.12108
\(124\) −1988.35 −0.0116129
\(125\) 15625.0 0.0894427
\(126\) 3437.23 0.0192878
\(127\) 137758. 0.757892 0.378946 0.925419i \(-0.376287\pi\)
0.378946 + 0.925419i \(0.376287\pi\)
\(128\) 7653.40 0.0412885
\(129\) 165723. 0.876819
\(130\) 65135.7 0.338034
\(131\) −183015. −0.931769 −0.465884 0.884846i \(-0.654264\pi\)
−0.465884 + 0.884846i \(0.654264\pi\)
\(132\) 21929.5 0.109545
\(133\) −4287.84 −0.0210189
\(134\) 51477.8 0.247661
\(135\) −102839. −0.485652
\(136\) 424438. 1.96774
\(137\) −284891. −1.29681 −0.648406 0.761295i \(-0.724564\pi\)
−0.648406 + 0.761295i \(0.724564\pi\)
\(138\) −55937.4 −0.250037
\(139\) 26645.6 0.116974 0.0584868 0.998288i \(-0.481372\pi\)
0.0584868 + 0.998288i \(0.481372\pi\)
\(140\) −4063.73 −0.0175228
\(141\) 64142.0 0.271703
\(142\) 13833.7 0.0575726
\(143\) 73665.5 0.301248
\(144\) 26965.9 0.108370
\(145\) 27928.7 0.110314
\(146\) −11897.0 −0.0461909
\(147\) −220708. −0.842414
\(148\) 95758.4 0.359354
\(149\) 303506. 1.11996 0.559978 0.828507i \(-0.310810\pi\)
0.559978 + 0.828507i \(0.310810\pi\)
\(150\) −35421.7 −0.128541
\(151\) 91029.9 0.324894 0.162447 0.986717i \(-0.448061\pi\)
0.162447 + 0.986717i \(0.448061\pi\)
\(152\) −70580.3 −0.247785
\(153\) −146796. −0.506975
\(154\) 6150.59 0.0208985
\(155\) 3632.29 0.0121437
\(156\) 110337. 0.363004
\(157\) −294041. −0.952049 −0.476024 0.879432i \(-0.657923\pi\)
−0.476024 + 0.879432i \(0.657923\pi\)
\(158\) −98122.2 −0.312698
\(159\) 262680. 0.824014
\(160\) −113745. −0.351263
\(161\) 11723.1 0.0356433
\(162\) 162815. 0.487425
\(163\) −216759. −0.639010 −0.319505 0.947585i \(-0.603517\pi\)
−0.319505 + 0.947585i \(0.603517\pi\)
\(164\) 194385. 0.564356
\(165\) −40060.4 −0.114553
\(166\) 442680. 1.24686
\(167\) 115741. 0.321141 0.160570 0.987024i \(-0.448667\pi\)
0.160570 + 0.987024i \(0.448667\pi\)
\(168\) 30753.7 0.0840667
\(169\) −648.409 −0.00174635
\(170\) −232262. −0.616390
\(171\) 24410.9 0.0638402
\(172\) −171257. −0.441395
\(173\) −217742. −0.553130 −0.276565 0.960995i \(-0.589196\pi\)
−0.276565 + 0.960995i \(0.589196\pi\)
\(174\) −63314.0 −0.158536
\(175\) 7423.54 0.0183238
\(176\) 48252.9 0.117420
\(177\) 265781. 0.637662
\(178\) 262053. 0.619924
\(179\) 132007. 0.307940 0.153970 0.988076i \(-0.450794\pi\)
0.153970 + 0.988076i \(0.450794\pi\)
\(180\) 23135.1 0.0532218
\(181\) 677944. 1.53815 0.769073 0.639161i \(-0.220718\pi\)
0.769073 + 0.639161i \(0.220718\pi\)
\(182\) 30946.4 0.0692519
\(183\) 524371. 1.15747
\(184\) 192969. 0.420188
\(185\) −174929. −0.375780
\(186\) −8234.36 −0.0174521
\(187\) −262678. −0.549312
\(188\) −66283.7 −0.136777
\(189\) −48859.7 −0.0994938
\(190\) 38623.1 0.0776182
\(191\) 756704. 1.50087 0.750434 0.660946i \(-0.229845\pi\)
0.750434 + 0.660946i \(0.229845\pi\)
\(192\) 426855. 0.835656
\(193\) 221018. 0.427105 0.213552 0.976932i \(-0.431497\pi\)
0.213552 + 0.976932i \(0.431497\pi\)
\(194\) −666586. −1.27160
\(195\) −201562. −0.379596
\(196\) 228078. 0.424075
\(197\) 480123. 0.881428 0.440714 0.897647i \(-0.354725\pi\)
0.440714 + 0.897647i \(0.354725\pi\)
\(198\) −35015.7 −0.0634745
\(199\) 43013.5 0.0769967 0.0384983 0.999259i \(-0.487743\pi\)
0.0384983 + 0.999259i \(0.487743\pi\)
\(200\) 122196. 0.216014
\(201\) −159298. −0.278112
\(202\) −498188. −0.859043
\(203\) 13269.1 0.0225996
\(204\) −393443. −0.661921
\(205\) −355099. −0.590153
\(206\) 333435. 0.547448
\(207\) −66740.5 −0.108259
\(208\) 242782. 0.389097
\(209\) 43681.0 0.0691714
\(210\) −16829.1 −0.0263337
\(211\) 393510. 0.608484 0.304242 0.952595i \(-0.401597\pi\)
0.304242 + 0.952595i \(0.401597\pi\)
\(212\) −271451. −0.414813
\(213\) −42808.1 −0.0646513
\(214\) −348950. −0.520870
\(215\) 312849. 0.461571
\(216\) −804259. −1.17290
\(217\) 1725.72 0.00248784
\(218\) −239577. −0.341431
\(219\) 36815.3 0.0518702
\(220\) 41398.0 0.0576664
\(221\) −1.32165e6 −1.82027
\(222\) 396564. 0.540045
\(223\) 1.30521e6 1.75760 0.878798 0.477195i \(-0.158346\pi\)
0.878798 + 0.477195i \(0.158346\pi\)
\(224\) −54041.0 −0.0719621
\(225\) −42262.7 −0.0556546
\(226\) 206635. 0.269112
\(227\) −1.31152e6 −1.68931 −0.844655 0.535311i \(-0.820195\pi\)
−0.844655 + 0.535311i \(0.820195\pi\)
\(228\) 65426.0 0.0833515
\(229\) −495209. −0.624022 −0.312011 0.950078i \(-0.601003\pi\)
−0.312011 + 0.950078i \(0.601003\pi\)
\(230\) −105597. −0.131623
\(231\) −19032.9 −0.0234680
\(232\) 218417. 0.266420
\(233\) −407725. −0.492015 −0.246007 0.969268i \(-0.579119\pi\)
−0.246007 + 0.969268i \(0.579119\pi\)
\(234\) −176180. −0.210337
\(235\) 121086. 0.143029
\(236\) −274655. −0.321002
\(237\) 303638. 0.351145
\(238\) −110349. −0.126278
\(239\) −959369. −1.08640 −0.543201 0.839603i \(-0.682788\pi\)
−0.543201 + 0.839603i \(0.682788\pi\)
\(240\) −132028. −0.147958
\(241\) 1.40519e6 1.55845 0.779223 0.626747i \(-0.215614\pi\)
0.779223 + 0.626747i \(0.215614\pi\)
\(242\) −62657.2 −0.0687753
\(243\) 495768. 0.538596
\(244\) −541880. −0.582678
\(245\) −416648. −0.443460
\(246\) 805005. 0.848127
\(247\) 219779. 0.229215
\(248\) 28406.4 0.0293283
\(249\) −1.36987e6 −1.40017
\(250\) −66868.3 −0.0676659
\(251\) 1.16660e6 1.16879 0.584397 0.811468i \(-0.301331\pi\)
0.584397 + 0.811468i \(0.301331\pi\)
\(252\) 10991.6 0.0109034
\(253\) −119426. −0.117300
\(254\) −589544. −0.573366
\(255\) 718733. 0.692177
\(256\) −1.06419e6 −1.01489
\(257\) −1.04400e6 −0.985976 −0.492988 0.870036i \(-0.664095\pi\)
−0.492988 + 0.870036i \(0.664095\pi\)
\(258\) −709225. −0.663338
\(259\) −83110.1 −0.0769847
\(260\) 208292. 0.191091
\(261\) −75541.8 −0.0686414
\(262\) 783225. 0.704909
\(263\) 1.52744e6 1.36168 0.680839 0.732433i \(-0.261615\pi\)
0.680839 + 0.732433i \(0.261615\pi\)
\(264\) −313294. −0.276657
\(265\) 495882. 0.433774
\(266\) 18350.1 0.0159014
\(267\) −810920. −0.696145
\(268\) 164616. 0.140003
\(269\) −757695. −0.638431 −0.319215 0.947682i \(-0.603419\pi\)
−0.319215 + 0.947682i \(0.603419\pi\)
\(270\) 440108. 0.367409
\(271\) −488209. −0.403815 −0.201908 0.979405i \(-0.564714\pi\)
−0.201908 + 0.979405i \(0.564714\pi\)
\(272\) −865716. −0.709501
\(273\) −95763.4 −0.0777666
\(274\) 1.21921e6 0.981075
\(275\) −75625.0 −0.0603023
\(276\) −178877. −0.141346
\(277\) −876260. −0.686173 −0.343086 0.939304i \(-0.611472\pi\)
−0.343086 + 0.939304i \(0.611472\pi\)
\(278\) −114032. −0.0884939
\(279\) −9824.65 −0.00755625
\(280\) 58056.0 0.0442540
\(281\) 829031. 0.626333 0.313166 0.949698i \(-0.398610\pi\)
0.313166 + 0.949698i \(0.398610\pi\)
\(282\) −274500. −0.205551
\(283\) −197711. −0.146745 −0.0733726 0.997305i \(-0.523376\pi\)
−0.0733726 + 0.997305i \(0.523376\pi\)
\(284\) 44237.5 0.0325458
\(285\) −119519. −0.0871615
\(286\) −315257. −0.227903
\(287\) −168710. −0.120902
\(288\) 307659. 0.218569
\(289\) 3.29291e6 2.31918
\(290\) −119523. −0.0834556
\(291\) 2.06275e6 1.42795
\(292\) −38044.6 −0.0261117
\(293\) 1.83759e6 1.25049 0.625244 0.780429i \(-0.284999\pi\)
0.625244 + 0.780429i \(0.284999\pi\)
\(294\) 944537. 0.637310
\(295\) 501734. 0.335675
\(296\) −1.36804e6 −0.907549
\(297\) 497743. 0.327426
\(298\) −1.29887e6 −0.847278
\(299\) −600885. −0.388699
\(300\) −113272. −0.0726642
\(301\) 148636. 0.0945604
\(302\) −389569. −0.245791
\(303\) 1.54164e6 0.964665
\(304\) 143961. 0.0893431
\(305\) 989895. 0.609312
\(306\) 628225. 0.383541
\(307\) 2.58579e6 1.56584 0.782921 0.622121i \(-0.213729\pi\)
0.782921 + 0.622121i \(0.213729\pi\)
\(308\) 19668.5 0.0118139
\(309\) −1.03181e6 −0.614758
\(310\) −15544.6 −0.00918705
\(311\) 3.12725e6 1.83342 0.916710 0.399552i \(-0.130834\pi\)
0.916710 + 0.399552i \(0.130834\pi\)
\(312\) −1.57632e6 −0.916766
\(313\) −1.79490e6 −1.03557 −0.517784 0.855511i \(-0.673243\pi\)
−0.517784 + 0.855511i \(0.673243\pi\)
\(314\) 1.25837e6 0.720252
\(315\) −20079.3 −0.0114018
\(316\) −313777. −0.176768
\(317\) 713831. 0.398977 0.199488 0.979900i \(-0.436072\pi\)
0.199488 + 0.979900i \(0.436072\pi\)
\(318\) −1.12416e6 −0.623390
\(319\) −135175. −0.0743736
\(320\) 805807. 0.439902
\(321\) 1.07982e6 0.584912
\(322\) −50169.9 −0.0269652
\(323\) −783691. −0.417964
\(324\) 520653. 0.275541
\(325\) −380504. −0.199825
\(326\) 927634. 0.483429
\(327\) 741368. 0.383411
\(328\) −2.77706e6 −1.42528
\(329\) 57528.6 0.0293018
\(330\) 171441. 0.0866623
\(331\) 2.96536e6 1.48767 0.743836 0.668362i \(-0.233004\pi\)
0.743836 + 0.668362i \(0.233004\pi\)
\(332\) 1.41561e6 0.704852
\(333\) 473151. 0.233824
\(334\) −495321. −0.242952
\(335\) −300718. −0.146402
\(336\) −62727.5 −0.0303117
\(337\) 2.33783e6 1.12134 0.560670 0.828039i \(-0.310543\pi\)
0.560670 + 0.828039i \(0.310543\pi\)
\(338\) 2774.91 0.00132117
\(339\) −639432. −0.302200
\(340\) −742731. −0.348445
\(341\) −17580.3 −0.00818728
\(342\) −104468. −0.0482969
\(343\) −397580. −0.182469
\(344\) 2.44664e6 1.11474
\(345\) 326770. 0.147807
\(346\) 931843. 0.418459
\(347\) −2.55615e6 −1.13963 −0.569814 0.821774i \(-0.692985\pi\)
−0.569814 + 0.821774i \(0.692985\pi\)
\(348\) −202467. −0.0896201
\(349\) 97407.5 0.0428084 0.0214042 0.999771i \(-0.493186\pi\)
0.0214042 + 0.999771i \(0.493186\pi\)
\(350\) −31769.6 −0.0138625
\(351\) 2.50437e6 1.08500
\(352\) 550526. 0.236822
\(353\) 4.35548e6 1.86037 0.930184 0.367094i \(-0.119647\pi\)
0.930184 + 0.367094i \(0.119647\pi\)
\(354\) −1.13743e6 −0.482409
\(355\) −80812.2 −0.0340334
\(356\) 837996. 0.350443
\(357\) 341475. 0.141804
\(358\) −564935. −0.232965
\(359\) 4.02320e6 1.64754 0.823769 0.566926i \(-0.191867\pi\)
0.823769 + 0.566926i \(0.191867\pi\)
\(360\) −330517. −0.134412
\(361\) 130321. 0.0526316
\(362\) −2.90131e6 −1.16365
\(363\) 193892. 0.0772314
\(364\) 98960.9 0.0391481
\(365\) 69499.0 0.0273053
\(366\) −2.24408e6 −0.875662
\(367\) −427192. −0.165561 −0.0827805 0.996568i \(-0.526380\pi\)
−0.0827805 + 0.996568i \(0.526380\pi\)
\(368\) −393595. −0.151506
\(369\) 960475. 0.367215
\(370\) 748623. 0.284288
\(371\) 235597. 0.0888657
\(372\) −26332.0 −0.00986566
\(373\) 3.08228e6 1.14710 0.573548 0.819172i \(-0.305567\pi\)
0.573548 + 0.819172i \(0.305567\pi\)
\(374\) 1.12415e6 0.415570
\(375\) 206923. 0.0759856
\(376\) 946955. 0.345430
\(377\) −680126. −0.246454
\(378\) 209098. 0.0752699
\(379\) 3.00884e6 1.07597 0.537987 0.842953i \(-0.319185\pi\)
0.537987 + 0.842953i \(0.319185\pi\)
\(380\) 123510. 0.0438775
\(381\) 1.82434e6 0.643863
\(382\) −3.23837e6 −1.13545
\(383\) −155731. −0.0542472 −0.0271236 0.999632i \(-0.508635\pi\)
−0.0271236 + 0.999632i \(0.508635\pi\)
\(384\) 101355. 0.0350765
\(385\) −35929.9 −0.0123539
\(386\) −945862. −0.323117
\(387\) −846197. −0.287206
\(388\) −2.13162e6 −0.718837
\(389\) −3.16932e6 −1.06192 −0.530960 0.847397i \(-0.678169\pi\)
−0.530960 + 0.847397i \(0.678169\pi\)
\(390\) 862598. 0.287175
\(391\) 2.14264e6 0.708775
\(392\) −3.25841e6 −1.07100
\(393\) −2.42368e6 −0.791580
\(394\) −2.05472e6 −0.666825
\(395\) 573201. 0.184848
\(396\) −111974. −0.0358821
\(397\) 4.88674e6 1.55612 0.778060 0.628191i \(-0.216204\pi\)
0.778060 + 0.628191i \(0.216204\pi\)
\(398\) −184079. −0.0582502
\(399\) −56784.2 −0.0178565
\(400\) −249240. −0.0778875
\(401\) 227783. 0.0707391 0.0353696 0.999374i \(-0.488739\pi\)
0.0353696 + 0.999374i \(0.488739\pi\)
\(402\) 681725. 0.210399
\(403\) −88454.3 −0.0271304
\(404\) −1.59312e6 −0.485617
\(405\) −951119. −0.288136
\(406\) −56786.0 −0.0170973
\(407\) 846659. 0.253351
\(408\) 5.62088e6 1.67168
\(409\) 1.30452e6 0.385605 0.192803 0.981238i \(-0.438242\pi\)
0.192803 + 0.981238i \(0.438242\pi\)
\(410\) 1.51967e6 0.446467
\(411\) −3.77284e6 −1.10170
\(412\) 1.06626e6 0.309472
\(413\) 238377. 0.0687685
\(414\) 285620. 0.0819009
\(415\) −2.58601e6 −0.737071
\(416\) 2.76995e6 0.784762
\(417\) 352870. 0.0993745
\(418\) −186936. −0.0523301
\(419\) 5.68370e6 1.58160 0.790799 0.612076i \(-0.209665\pi\)
0.790799 + 0.612076i \(0.209665\pi\)
\(420\) −53816.4 −0.0148865
\(421\) 2.49111e6 0.684996 0.342498 0.939519i \(-0.388727\pi\)
0.342498 + 0.939519i \(0.388727\pi\)
\(422\) −1.68405e6 −0.460335
\(423\) −327514. −0.0889978
\(424\) 3.87806e6 1.04761
\(425\) 1.35681e6 0.364373
\(426\) 183200. 0.0489105
\(427\) 470306. 0.124828
\(428\) −1.11588e6 −0.294447
\(429\) 975560. 0.255924
\(430\) −1.33886e6 −0.349191
\(431\) 1.62364e6 0.421015 0.210508 0.977592i \(-0.432488\pi\)
0.210508 + 0.977592i \(0.432488\pi\)
\(432\) 1.64043e6 0.422910
\(433\) 4.11880e6 1.05573 0.527863 0.849330i \(-0.322994\pi\)
0.527863 + 0.849330i \(0.322994\pi\)
\(434\) −7385.35 −0.00188212
\(435\) 369862. 0.0937167
\(436\) −766122. −0.193011
\(437\) −356303. −0.0892516
\(438\) −157554. −0.0392413
\(439\) −498345. −0.123415 −0.0617076 0.998094i \(-0.519655\pi\)
−0.0617076 + 0.998094i \(0.519655\pi\)
\(440\) −591428. −0.145636
\(441\) 1.12695e6 0.275937
\(442\) 5.65610e6 1.37709
\(443\) −497639. −0.120477 −0.0602386 0.998184i \(-0.519186\pi\)
−0.0602386 + 0.998184i \(0.519186\pi\)
\(444\) 1.26814e6 0.305288
\(445\) −1.53083e6 −0.366461
\(446\) −5.58575e6 −1.32967
\(447\) 4.01935e6 0.951453
\(448\) 382844. 0.0901212
\(449\) 7.53957e6 1.76494 0.882471 0.470367i \(-0.155878\pi\)
0.882471 + 0.470367i \(0.155878\pi\)
\(450\) 180866. 0.0421042
\(451\) 1.71868e6 0.397881
\(452\) 660782. 0.152129
\(453\) 1.20552e6 0.276012
\(454\) 5.61273e6 1.27801
\(455\) −180780. −0.0409375
\(456\) −934702. −0.210504
\(457\) −4.56957e6 −1.02349 −0.511747 0.859136i \(-0.671001\pi\)
−0.511747 + 0.859136i \(0.671001\pi\)
\(458\) 2.11928e6 0.472091
\(459\) −8.93012e6 −1.97845
\(460\) −337681. −0.0744066
\(461\) −2.01890e6 −0.442448 −0.221224 0.975223i \(-0.571005\pi\)
−0.221224 + 0.975223i \(0.571005\pi\)
\(462\) 81452.8 0.0177542
\(463\) 4.61836e6 1.00123 0.500617 0.865669i \(-0.333107\pi\)
0.500617 + 0.865669i \(0.333107\pi\)
\(464\) −445500. −0.0960623
\(465\) 48102.7 0.0103166
\(466\) 1.74489e6 0.372223
\(467\) 4.66977e6 0.990839 0.495420 0.868654i \(-0.335014\pi\)
0.495420 + 0.868654i \(0.335014\pi\)
\(468\) −563391. −0.118904
\(469\) −142873. −0.0299929
\(470\) −518195. −0.108205
\(471\) −3.89402e6 −0.808809
\(472\) 3.92383e6 0.810691
\(473\) −1.51419e6 −0.311191
\(474\) −1.29944e6 −0.265651
\(475\) −225625. −0.0458831
\(476\) −352876. −0.0713847
\(477\) −1.34127e6 −0.269910
\(478\) 4.10568e6 0.821894
\(479\) 3.46392e6 0.689810 0.344905 0.938638i \(-0.387911\pi\)
0.344905 + 0.938638i \(0.387911\pi\)
\(480\) −1.50634e6 −0.298414
\(481\) 4.25992e6 0.839536
\(482\) −6.01360e6 −1.17901
\(483\) 155250. 0.0302806
\(484\) −200366. −0.0388786
\(485\) 3.89400e6 0.751695
\(486\) −2.12167e6 −0.407463
\(487\) −6.76913e6 −1.29333 −0.646667 0.762773i \(-0.723838\pi\)
−0.646667 + 0.762773i \(0.723838\pi\)
\(488\) 7.74151e6 1.47155
\(489\) −2.87056e6 −0.542868
\(490\) 1.78307e6 0.335490
\(491\) −184736. −0.0345819 −0.0172910 0.999851i \(-0.505504\pi\)
−0.0172910 + 0.999851i \(0.505504\pi\)
\(492\) 2.57426e6 0.479446
\(493\) 2.42520e6 0.449398
\(494\) −940559. −0.173408
\(495\) 204551. 0.0375223
\(496\) −57939.9 −0.0105748
\(497\) −38394.4 −0.00697231
\(498\) 5.86245e6 1.05927
\(499\) −5.96534e6 −1.07247 −0.536233 0.844070i \(-0.680153\pi\)
−0.536233 + 0.844070i \(0.680153\pi\)
\(500\) −213833. −0.0382515
\(501\) 1.53277e6 0.272824
\(502\) −4.99255e6 −0.884225
\(503\) −1.10592e7 −1.94896 −0.974478 0.224483i \(-0.927931\pi\)
−0.974478 + 0.224483i \(0.927931\pi\)
\(504\) −157031. −0.0275365
\(505\) 2.91027e6 0.507814
\(506\) 511090. 0.0887404
\(507\) −8586.94 −0.00148361
\(508\) −1.88526e6 −0.324124
\(509\) −9.46412e6 −1.61915 −0.809573 0.587019i \(-0.800301\pi\)
−0.809573 + 0.587019i \(0.800301\pi\)
\(510\) −3.07587e6 −0.523651
\(511\) 33019.4 0.00559394
\(512\) 4.30935e6 0.726502
\(513\) 1.48500e6 0.249134
\(514\) 4.46786e6 0.745919
\(515\) −1.94783e6 −0.323618
\(516\) −2.26797e6 −0.374985
\(517\) −586055. −0.0964300
\(518\) 355676. 0.0582411
\(519\) −2.88358e6 −0.469909
\(520\) −2.97574e6 −0.482600
\(521\) 4.23835e6 0.684073 0.342036 0.939687i \(-0.388883\pi\)
0.342036 + 0.939687i \(0.388883\pi\)
\(522\) 323286. 0.0519291
\(523\) −6.57389e6 −1.05092 −0.525458 0.850820i \(-0.676106\pi\)
−0.525458 + 0.850820i \(0.676106\pi\)
\(524\) 2.50461e6 0.398485
\(525\) 98310.7 0.0155669
\(526\) −6.53678e6 −1.03015
\(527\) 315412. 0.0494711
\(528\) 639017. 0.0997534
\(529\) −5.46220e6 −0.848649
\(530\) −2.12216e6 −0.328162
\(531\) −1.35710e6 −0.208869
\(532\) 58680.3 0.00898903
\(533\) 8.64744e6 1.31847
\(534\) 3.47039e6 0.526653
\(535\) 2.03847e6 0.307907
\(536\) −2.35178e6 −0.353577
\(537\) 1.74819e6 0.261609
\(538\) 3.24261e6 0.482991
\(539\) 2.01658e6 0.298980
\(540\) 1.40739e6 0.207696
\(541\) 7.89424e6 1.15962 0.579812 0.814750i \(-0.303126\pi\)
0.579812 + 0.814750i \(0.303126\pi\)
\(542\) 2.08933e6 0.305498
\(543\) 8.97808e6 1.30672
\(544\) −9.87713e6 −1.43098
\(545\) 1.39954e6 0.201833
\(546\) 409826. 0.0588326
\(547\) −3.40867e6 −0.487099 −0.243549 0.969888i \(-0.578312\pi\)
−0.243549 + 0.969888i \(0.578312\pi\)
\(548\) 3.89881e6 0.554601
\(549\) −2.67748e6 −0.379136
\(550\) 323642. 0.0456204
\(551\) −403290. −0.0565898
\(552\) 2.55551e6 0.356969
\(553\) 272332. 0.0378691
\(554\) 3.75002e6 0.519109
\(555\) −2.31661e6 −0.319242
\(556\) −364652. −0.0500256
\(557\) 8.69872e6 1.18800 0.594001 0.804464i \(-0.297547\pi\)
0.594001 + 0.804464i \(0.297547\pi\)
\(558\) 42045.3 0.00571652
\(559\) −7.61856e6 −1.03120
\(560\) −118416. −0.0159565
\(561\) −3.47867e6 −0.466666
\(562\) −3.54790e6 −0.473838
\(563\) −3.05136e6 −0.405717 −0.202858 0.979208i \(-0.565023\pi\)
−0.202858 + 0.979208i \(0.565023\pi\)
\(564\) −877802. −0.116198
\(565\) −1.20710e6 −0.159083
\(566\) 846116. 0.111017
\(567\) −451883. −0.0590294
\(568\) −631994. −0.0821944
\(569\) −8.67457e6 −1.12323 −0.561613 0.827400i \(-0.689819\pi\)
−0.561613 + 0.827400i \(0.689819\pi\)
\(570\) 511490. 0.0659401
\(571\) 963909. 0.123722 0.0618608 0.998085i \(-0.480297\pi\)
0.0618608 + 0.998085i \(0.480297\pi\)
\(572\) −1.00813e6 −0.128833
\(573\) 1.00211e7 1.27506
\(574\) 722005. 0.0914661
\(575\) 616868. 0.0778077
\(576\) −2.17956e6 −0.273723
\(577\) −6.94281e6 −0.868152 −0.434076 0.900876i \(-0.642925\pi\)
−0.434076 + 0.900876i \(0.642925\pi\)
\(578\) −1.40922e7 −1.75453
\(579\) 2.92696e6 0.362845
\(580\) −382212. −0.0471774
\(581\) −1.22863e6 −0.151001
\(582\) −8.82766e6 −1.08028
\(583\) −2.40007e6 −0.292450
\(584\) 543520. 0.0659452
\(585\) 1.02919e6 0.124339
\(586\) −7.86409e6 −0.946029
\(587\) 2.88979e6 0.346155 0.173077 0.984908i \(-0.444629\pi\)
0.173077 + 0.984908i \(0.444629\pi\)
\(588\) 3.02046e6 0.360271
\(589\) −52450.2 −0.00622958
\(590\) −2.14721e6 −0.253947
\(591\) 6.35832e6 0.748813
\(592\) 2.79036e6 0.327232
\(593\) 4.76414e6 0.556349 0.278175 0.960531i \(-0.410271\pi\)
0.278175 + 0.960531i \(0.410271\pi\)
\(594\) −2.13012e6 −0.247707
\(595\) 644628. 0.0746477
\(596\) −4.15356e6 −0.478966
\(597\) 569632. 0.0654122
\(598\) 2.57153e6 0.294061
\(599\) −7.22809e6 −0.823108 −0.411554 0.911385i \(-0.635014\pi\)
−0.411554 + 0.911385i \(0.635014\pi\)
\(600\) 1.61825e6 0.183513
\(601\) 3.81027e6 0.430299 0.215149 0.976581i \(-0.430976\pi\)
0.215149 + 0.976581i \(0.430976\pi\)
\(602\) −636100. −0.0715376
\(603\) 813386. 0.0910968
\(604\) −1.24577e6 −0.138946
\(605\) 366025. 0.0406558
\(606\) −6.59756e6 −0.729796
\(607\) 1.64080e7 1.80752 0.903762 0.428034i \(-0.140794\pi\)
0.903762 + 0.428034i \(0.140794\pi\)
\(608\) 1.64248e6 0.180194
\(609\) 175724. 0.0191994
\(610\) −4.23633e6 −0.460962
\(611\) −2.94871e6 −0.319543
\(612\) 2.00895e6 0.216815
\(613\) 2.02821e6 0.218002 0.109001 0.994042i \(-0.465235\pi\)
0.109001 + 0.994042i \(0.465235\pi\)
\(614\) −1.10661e7 −1.18460
\(615\) −4.70261e6 −0.501361
\(616\) −280991. −0.0298360
\(617\) 9.37403e6 0.991320 0.495660 0.868517i \(-0.334926\pi\)
0.495660 + 0.868517i \(0.334926\pi\)
\(618\) 4.41571e6 0.465082
\(619\) 7.59697e6 0.796918 0.398459 0.917186i \(-0.369545\pi\)
0.398459 + 0.917186i \(0.369545\pi\)
\(620\) −49708.9 −0.00519343
\(621\) −4.06005e6 −0.422477
\(622\) −1.33833e7 −1.38703
\(623\) −727309. −0.0750757
\(624\) 3.21519e6 0.330556
\(625\) 390625. 0.0400000
\(626\) 7.68139e6 0.783437
\(627\) 578472. 0.0587643
\(628\) 4.02404e6 0.407158
\(629\) −1.51901e7 −1.53086
\(630\) 85930.6 0.00862575
\(631\) 1.50154e7 1.50128 0.750642 0.660709i \(-0.229744\pi\)
0.750642 + 0.660709i \(0.229744\pi\)
\(632\) 4.48274e6 0.446427
\(633\) 5.21129e6 0.516935
\(634\) −3.05489e6 −0.301837
\(635\) 3.44395e6 0.338939
\(636\) −3.59486e6 −0.352402
\(637\) 1.01463e7 0.990740
\(638\) 578490. 0.0562657
\(639\) 218582. 0.0211769
\(640\) 191335. 0.0184648
\(641\) 1.96960e7 1.89336 0.946679 0.322180i \(-0.104415\pi\)
0.946679 + 0.322180i \(0.104415\pi\)
\(642\) −4.62118e6 −0.442502
\(643\) 1.52750e7 1.45698 0.728492 0.685055i \(-0.240222\pi\)
0.728492 + 0.685055i \(0.240222\pi\)
\(644\) −160434. −0.0152434
\(645\) 4.14309e6 0.392125
\(646\) 3.35386e6 0.316201
\(647\) −1.37109e7 −1.28767 −0.643836 0.765164i \(-0.722658\pi\)
−0.643836 + 0.765164i \(0.722658\pi\)
\(648\) −7.43826e6 −0.695879
\(649\) −2.42839e6 −0.226312
\(650\) 1.62839e6 0.151173
\(651\) 22853.9 0.00211353
\(652\) 2.96640e6 0.273282
\(653\) −3.26845e6 −0.299957 −0.149979 0.988689i \(-0.547920\pi\)
−0.149979 + 0.988689i \(0.547920\pi\)
\(654\) −3.17274e6 −0.290061
\(655\) −4.57537e6 −0.416700
\(656\) 5.66430e6 0.513910
\(657\) −187982. −0.0169904
\(658\) −246198. −0.0221676
\(659\) −2.79749e6 −0.250931 −0.125466 0.992098i \(-0.540042\pi\)
−0.125466 + 0.992098i \(0.540042\pi\)
\(660\) 548238. 0.0489902
\(661\) 8.71937e6 0.776214 0.388107 0.921614i \(-0.373129\pi\)
0.388107 + 0.921614i \(0.373129\pi\)
\(662\) −1.26905e7 −1.12547
\(663\) −1.75028e7 −1.54640
\(664\) −2.02239e7 −1.78011
\(665\) −107196. −0.00939992
\(666\) −2.02488e6 −0.176895
\(667\) 1.10261e6 0.0959639
\(668\) −1.58395e6 −0.137341
\(669\) 1.72851e7 1.49316
\(670\) 1.28694e6 0.110757
\(671\) −4.79109e6 −0.410798
\(672\) −715671. −0.0611350
\(673\) 9.68767e6 0.824483 0.412242 0.911075i \(-0.364746\pi\)
0.412242 + 0.911075i \(0.364746\pi\)
\(674\) −1.00049e7 −0.848326
\(675\) −2.57098e6 −0.217190
\(676\) 8873.66 0.000746855 0
\(677\) −1.68290e7 −1.41119 −0.705596 0.708614i \(-0.749321\pi\)
−0.705596 + 0.708614i \(0.749321\pi\)
\(678\) 2.73649e6 0.228623
\(679\) 1.85006e6 0.153997
\(680\) 1.06110e7 0.879999
\(681\) −1.73686e7 −1.43515
\(682\) 75236.0 0.00619391
\(683\) 1.26965e7 1.04143 0.520717 0.853729i \(-0.325665\pi\)
0.520717 + 0.853729i \(0.325665\pi\)
\(684\) −334070. −0.0273022
\(685\) −7.12227e6 −0.579952
\(686\) 1.70147e6 0.138043
\(687\) −6.55811e6 −0.530135
\(688\) −4.99036e6 −0.401940
\(689\) −1.20758e7 −0.969100
\(690\) −1.39843e6 −0.111820
\(691\) −1.72598e7 −1.37512 −0.687560 0.726127i \(-0.741318\pi\)
−0.687560 + 0.726127i \(0.741318\pi\)
\(692\) 2.97986e6 0.236554
\(693\) 97183.7 0.00768706
\(694\) 1.09392e7 0.862160
\(695\) 666140. 0.0523122
\(696\) 2.89252e6 0.226336
\(697\) −3.08352e7 −2.40417
\(698\) −416862. −0.0323857
\(699\) −5.39955e6 −0.417989
\(700\) −101593. −0.00783646
\(701\) −1.78321e7 −1.37059 −0.685294 0.728266i \(-0.740326\pi\)
−0.685294 + 0.728266i \(0.740326\pi\)
\(702\) −1.07176e7 −0.820835
\(703\) 2.52598e6 0.192771
\(704\) −3.90011e6 −0.296582
\(705\) 1.60355e6 0.121509
\(706\) −1.86396e7 −1.40742
\(707\) 1.38269e6 0.104034
\(708\) −3.63728e6 −0.272706
\(709\) −2.03015e7 −1.51675 −0.758373 0.651821i \(-0.774005\pi\)
−0.758373 + 0.651821i \(0.774005\pi\)
\(710\) 345841. 0.0257473
\(711\) −1.55040e6 −0.115019
\(712\) −1.19719e7 −0.885044
\(713\) 143401. 0.0105640
\(714\) −1.46136e6 −0.107279
\(715\) 1.84164e6 0.134722
\(716\) −1.80656e6 −0.131695
\(717\) −1.27050e7 −0.922948
\(718\) −1.72176e7 −1.24641
\(719\) −4.58061e6 −0.330446 −0.165223 0.986256i \(-0.552834\pi\)
−0.165223 + 0.986256i \(0.552834\pi\)
\(720\) 674147. 0.0484645
\(721\) −925427. −0.0662985
\(722\) −557718. −0.0398173
\(723\) 1.86090e7 1.32397
\(724\) −9.27786e6 −0.657811
\(725\) 698217. 0.0493339
\(726\) −829775. −0.0584277
\(727\) −3.12165e6 −0.219053 −0.109526 0.993984i \(-0.534933\pi\)
−0.109526 + 0.993984i \(0.534933\pi\)
\(728\) −1.41379e6 −0.0988685
\(729\) 1.58104e7 1.10185
\(730\) −297426. −0.0206572
\(731\) 2.71664e7 1.88035
\(732\) −7.17617e6 −0.495011
\(733\) 1.64680e7 1.13209 0.566046 0.824374i \(-0.308473\pi\)
0.566046 + 0.824374i \(0.308473\pi\)
\(734\) 1.82820e6 0.125252
\(735\) −5.51771e6 −0.376739
\(736\) −4.49060e6 −0.305570
\(737\) 1.45548e6 0.0987043
\(738\) −4.11042e6 −0.277808
\(739\) −2.24853e7 −1.51456 −0.757281 0.653089i \(-0.773473\pi\)
−0.757281 + 0.653089i \(0.773473\pi\)
\(740\) 2.39396e6 0.160708
\(741\) 2.91055e6 0.194729
\(742\) −1.00825e6 −0.0672294
\(743\) −2.11668e7 −1.40664 −0.703321 0.710873i \(-0.748300\pi\)
−0.703321 + 0.710873i \(0.748300\pi\)
\(744\) 376189. 0.0249157
\(745\) 7.58764e6 0.500860
\(746\) −1.31908e7 −0.867810
\(747\) 6.99466e6 0.458633
\(748\) 3.59482e6 0.234922
\(749\) 968488. 0.0630797
\(750\) −885543. −0.0574853
\(751\) −6.33885e6 −0.410119 −0.205060 0.978749i \(-0.565739\pi\)
−0.205060 + 0.978749i \(0.565739\pi\)
\(752\) −1.93148e6 −0.124551
\(753\) 1.54494e7 0.992943
\(754\) 2.91064e6 0.186449
\(755\) 2.27575e6 0.145297
\(756\) 668658. 0.0425500
\(757\) −2.14477e7 −1.36032 −0.680160 0.733064i \(-0.738090\pi\)
−0.680160 + 0.733064i \(0.738090\pi\)
\(758\) −1.28766e7 −0.814004
\(759\) −1.58157e6 −0.0996513
\(760\) −1.76451e6 −0.110813
\(761\) −1.27796e6 −0.0799939 −0.0399970 0.999200i \(-0.512735\pi\)
−0.0399970 + 0.999200i \(0.512735\pi\)
\(762\) −7.80739e6 −0.487101
\(763\) 664929. 0.0413489
\(764\) −1.03557e7 −0.641869
\(765\) −3.66991e6 −0.226726
\(766\) 666461. 0.0410395
\(767\) −1.22183e7 −0.749936
\(768\) −1.40931e7 −0.862193
\(769\) −1.25278e7 −0.763937 −0.381969 0.924175i \(-0.624754\pi\)
−0.381969 + 0.924175i \(0.624754\pi\)
\(770\) 153765. 0.00934608
\(771\) −1.38258e7 −0.837631
\(772\) −3.02469e6 −0.182658
\(773\) 1.37232e7 0.826053 0.413027 0.910719i \(-0.364472\pi\)
0.413027 + 0.910719i \(0.364472\pi\)
\(774\) 3.62136e6 0.217280
\(775\) 90807.1 0.00543082
\(776\) 3.04532e7 1.81542
\(777\) −1.10064e6 −0.0654020
\(778\) 1.35633e7 0.803373
\(779\) 5.12762e6 0.302742
\(780\) 2.75843e6 0.162340
\(781\) 391131. 0.0229453
\(782\) −9.16960e6 −0.536208
\(783\) −4.59547e6 −0.267871
\(784\) 6.64610e6 0.386168
\(785\) −7.35103e6 −0.425769
\(786\) 1.03723e7 0.598853
\(787\) 4.20499e6 0.242007 0.121004 0.992652i \(-0.461389\pi\)
0.121004 + 0.992652i \(0.461389\pi\)
\(788\) −6.57062e6 −0.376956
\(789\) 2.02280e7 1.15681
\(790\) −2.45305e6 −0.139843
\(791\) −573503. −0.0325908
\(792\) 1.59970e6 0.0906204
\(793\) −2.41062e7 −1.36127
\(794\) −2.09131e7 −1.17725
\(795\) 6.56701e6 0.368510
\(796\) −588652. −0.0329288
\(797\) −2.00669e7 −1.11901 −0.559507 0.828826i \(-0.689009\pi\)
−0.559507 + 0.828826i \(0.689009\pi\)
\(798\) 243012. 0.0135089
\(799\) 1.05146e7 0.582672
\(800\) −2.84363e6 −0.157090
\(801\) 4.14062e6 0.228026
\(802\) −974812. −0.0535161
\(803\) −336375. −0.0184092
\(804\) 2.18003e6 0.118939
\(805\) 293078. 0.0159402
\(806\) 378546. 0.0205249
\(807\) −1.00342e7 −0.542376
\(808\) 2.27599e7 1.22643
\(809\) −5.43542e6 −0.291986 −0.145993 0.989286i \(-0.546638\pi\)
−0.145993 + 0.989286i \(0.546638\pi\)
\(810\) 4.07038e6 0.217983
\(811\) −1.68868e7 −0.901563 −0.450782 0.892634i \(-0.648855\pi\)
−0.450782 + 0.892634i \(0.648855\pi\)
\(812\) −181591. −0.00966507
\(813\) −6.46541e6 −0.343059
\(814\) −3.62333e6 −0.191667
\(815\) −5.41897e6 −0.285774
\(816\) −1.14648e7 −0.602754
\(817\) −4.51753e6 −0.236781
\(818\) −5.58279e6 −0.291721
\(819\) 488975. 0.0254728
\(820\) 4.85962e6 0.252388
\(821\) −1.99548e7 −1.03321 −0.516606 0.856223i \(-0.672805\pi\)
−0.516606 + 0.856223i \(0.672805\pi\)
\(822\) 1.61461e7 0.833467
\(823\) 3.06729e7 1.57854 0.789270 0.614046i \(-0.210459\pi\)
0.789270 + 0.614046i \(0.210459\pi\)
\(824\) −1.52331e7 −0.781573
\(825\) −1.00151e6 −0.0512295
\(826\) −1.02015e6 −0.0520253
\(827\) 647257. 0.0329088 0.0164544 0.999865i \(-0.494762\pi\)
0.0164544 + 0.999865i \(0.494762\pi\)
\(828\) 913362. 0.0462985
\(829\) 5.31655e6 0.268685 0.134343 0.990935i \(-0.457108\pi\)
0.134343 + 0.990935i \(0.457108\pi\)
\(830\) 1.10670e7 0.557615
\(831\) −1.16044e7 −0.582935
\(832\) −1.96232e7 −0.982792
\(833\) −3.61799e7 −1.80657
\(834\) −1.51013e6 −0.0751796
\(835\) 2.89352e6 0.143619
\(836\) −597787. −0.0295822
\(837\) −597667. −0.0294880
\(838\) −2.43238e7 −1.19652
\(839\) −2.26426e7 −1.11051 −0.555254 0.831681i \(-0.687379\pi\)
−0.555254 + 0.831681i \(0.687379\pi\)
\(840\) 768842. 0.0375958
\(841\) −1.92631e7 −0.939154
\(842\) −1.06609e7 −0.518219
\(843\) 1.09789e7 0.532098
\(844\) −5.38529e6 −0.260227
\(845\) −16210.2 −0.000780993 0
\(846\) 1.40162e6 0.0673293
\(847\) 173901. 0.00832901
\(848\) −7.90998e6 −0.377734
\(849\) −2.61830e6 −0.124667
\(850\) −5.80655e6 −0.275658
\(851\) −6.90614e6 −0.326897
\(852\) 585842. 0.0276491
\(853\) 1.55183e7 0.730251 0.365125 0.930958i \(-0.381026\pi\)
0.365125 + 0.930958i \(0.381026\pi\)
\(854\) −2.01271e6 −0.0944356
\(855\) 610273. 0.0285502
\(856\) 1.59419e7 0.743628
\(857\) −2.74647e7 −1.27739 −0.638695 0.769460i \(-0.720525\pi\)
−0.638695 + 0.769460i \(0.720525\pi\)
\(858\) −4.17498e6 −0.193614
\(859\) −2.81821e7 −1.30314 −0.651569 0.758589i \(-0.725889\pi\)
−0.651569 + 0.758589i \(0.725889\pi\)
\(860\) −4.28142e6 −0.197398
\(861\) −2.23424e6 −0.102712
\(862\) −6.94850e6 −0.318510
\(863\) 1.36345e7 0.623180 0.311590 0.950217i \(-0.399139\pi\)
0.311590 + 0.950217i \(0.399139\pi\)
\(864\) 1.87160e7 0.852958
\(865\) −5.44355e6 −0.247367
\(866\) −1.76267e7 −0.798686
\(867\) 4.36083e7 1.97025
\(868\) −23617.0 −0.00106396
\(869\) −2.77429e6 −0.124624
\(870\) −1.58285e6 −0.0708993
\(871\) 7.32316e6 0.327079
\(872\) 1.09451e7 0.487449
\(873\) −1.05325e7 −0.467732
\(874\) 1.52482e6 0.0675213
\(875\) 185588. 0.00819466
\(876\) −503828. −0.0221831
\(877\) 2.04955e7 0.899826 0.449913 0.893072i \(-0.351455\pi\)
0.449913 + 0.893072i \(0.351455\pi\)
\(878\) 2.13270e6 0.0933671
\(879\) 2.43354e7 1.06235
\(880\) 1.20632e6 0.0525117
\(881\) 2.63579e7 1.14412 0.572060 0.820212i \(-0.306145\pi\)
0.572060 + 0.820212i \(0.306145\pi\)
\(882\) −4.82288e6 −0.208754
\(883\) −3.38025e6 −0.145897 −0.0729486 0.997336i \(-0.523241\pi\)
−0.0729486 + 0.997336i \(0.523241\pi\)
\(884\) 1.80872e7 0.778466
\(885\) 6.64452e6 0.285171
\(886\) 2.12968e6 0.0911444
\(887\) −1.56679e7 −0.668655 −0.334327 0.942457i \(-0.608509\pi\)
−0.334327 + 0.942457i \(0.608509\pi\)
\(888\) −1.81171e7 −0.771004
\(889\) 1.63624e6 0.0694373
\(890\) 6.55131e6 0.277238
\(891\) 4.60342e6 0.194261
\(892\) −1.78622e7 −0.751662
\(893\) −1.74848e6 −0.0733722
\(894\) −1.72011e7 −0.719801
\(895\) 3.30018e6 0.137715
\(896\) 90904.5 0.00378282
\(897\) −7.95757e6 −0.330217
\(898\) −3.22661e7 −1.33523
\(899\) 162312. 0.00669809
\(900\) 578377. 0.0238015
\(901\) 4.30602e7 1.76711
\(902\) −7.35520e6 −0.301008
\(903\) 1.96841e6 0.0803333
\(904\) −9.44020e6 −0.384202
\(905\) 1.69486e7 0.687880
\(906\) −5.15910e6 −0.208811
\(907\) 3.97386e7 1.60396 0.801981 0.597349i \(-0.203779\pi\)
0.801981 + 0.597349i \(0.203779\pi\)
\(908\) 1.79485e7 0.722459
\(909\) −7.87173e6 −0.315981
\(910\) 773660. 0.0309704
\(911\) −8.60111e6 −0.343367 −0.171684 0.985152i \(-0.554921\pi\)
−0.171684 + 0.985152i \(0.554921\pi\)
\(912\) 1.90649e6 0.0759010
\(913\) 1.25163e7 0.496933
\(914\) 1.95558e7 0.774302
\(915\) 1.31093e7 0.517638
\(916\) 6.77708e6 0.266873
\(917\) −2.17379e6 −0.0853678
\(918\) 3.82171e7 1.49676
\(919\) −2.53760e7 −0.991140 −0.495570 0.868568i \(-0.665041\pi\)
−0.495570 + 0.868568i \(0.665041\pi\)
\(920\) 4.82424e6 0.187914
\(921\) 3.42439e7 1.33025
\(922\) 8.64001e6 0.334724
\(923\) 1.96796e6 0.0760346
\(924\) 260471. 0.0100364
\(925\) −4.37324e6 −0.168054
\(926\) −1.97646e7 −0.757462
\(927\) 5.26851e6 0.201367
\(928\) −5.08280e6 −0.193746
\(929\) 1.42066e7 0.540073 0.270036 0.962850i \(-0.412964\pi\)
0.270036 + 0.962850i \(0.412964\pi\)
\(930\) −205859. −0.00780481
\(931\) 6.01640e6 0.227490
\(932\) 5.57984e6 0.210417
\(933\) 4.14145e7 1.55757
\(934\) −1.99846e7 −0.749598
\(935\) −6.56694e6 −0.245660
\(936\) 8.04882e6 0.300291
\(937\) 5.58399e6 0.207776 0.103888 0.994589i \(-0.466872\pi\)
0.103888 + 0.994589i \(0.466872\pi\)
\(938\) 611435. 0.0226905
\(939\) −2.37700e7 −0.879763
\(940\) −1.65709e6 −0.0611684
\(941\) 2.63319e7 0.969411 0.484706 0.874677i \(-0.338927\pi\)
0.484706 + 0.874677i \(0.338927\pi\)
\(942\) 1.66647e7 0.611887
\(943\) −1.40191e7 −0.513383
\(944\) −8.00334e6 −0.292308
\(945\) −1.22149e6 −0.0444950
\(946\) 6.48007e6 0.235425
\(947\) 5.13035e7 1.85897 0.929484 0.368863i \(-0.120253\pi\)
0.929484 + 0.368863i \(0.120253\pi\)
\(948\) −4.15538e6 −0.150172
\(949\) −1.69246e6 −0.0610031
\(950\) 965578. 0.0347119
\(951\) 9.45334e6 0.338949
\(952\) 5.04133e6 0.180282
\(953\) 1.28686e7 0.458986 0.229493 0.973310i \(-0.426293\pi\)
0.229493 + 0.973310i \(0.426293\pi\)
\(954\) 5.74004e6 0.204195
\(955\) 1.89176e7 0.671208
\(956\) 1.31292e7 0.464616
\(957\) −1.79013e6 −0.0631838
\(958\) −1.48241e7 −0.521861
\(959\) −3.38384e6 −0.118813
\(960\) 1.06714e7 0.373717
\(961\) −2.86080e7 −0.999263
\(962\) −1.82306e7 −0.635132
\(963\) −5.51367e6 −0.191591
\(964\) −1.92304e7 −0.666493
\(965\) 5.52545e6 0.191007
\(966\) −664405. −0.0229082
\(967\) −1.21851e6 −0.0419047 −0.0209524 0.999780i \(-0.506670\pi\)
−0.0209524 + 0.999780i \(0.506670\pi\)
\(968\) 2.86251e6 0.0981881
\(969\) −1.03785e7 −0.355079
\(970\) −1.66646e7 −0.568678
\(971\) −1.46249e7 −0.497789 −0.248895 0.968531i \(-0.580067\pi\)
−0.248895 + 0.968531i \(0.580067\pi\)
\(972\) −6.78473e6 −0.230339
\(973\) 316487. 0.0107170
\(974\) 2.89690e7 0.978443
\(975\) −5.03905e6 −0.169761
\(976\) −1.57902e7 −0.530594
\(977\) −3.17616e7 −1.06455 −0.532274 0.846572i \(-0.678662\pi\)
−0.532274 + 0.846572i \(0.678662\pi\)
\(978\) 1.22848e7 0.410695
\(979\) 7.40924e6 0.247068
\(980\) 5.70195e6 0.189652
\(981\) −3.78548e6 −0.125588
\(982\) 790593. 0.0261622
\(983\) 2.31472e7 0.764039 0.382019 0.924154i \(-0.375229\pi\)
0.382019 + 0.924154i \(0.375229\pi\)
\(984\) −3.67769e7 −1.21084
\(985\) 1.20031e7 0.394187
\(986\) −1.03788e7 −0.339982
\(987\) 761857. 0.0248932
\(988\) −3.00774e6 −0.0980274
\(989\) 1.23511e7 0.401528
\(990\) −875392. −0.0283867
\(991\) −3.16204e7 −1.02278 −0.511392 0.859348i \(-0.670870\pi\)
−0.511392 + 0.859348i \(0.670870\pi\)
\(992\) −661048. −0.0213282
\(993\) 3.92705e7 1.26384
\(994\) 164311. 0.00527475
\(995\) 1.07534e6 0.0344340
\(996\) 1.87471e7 0.598804
\(997\) 4.01493e7 1.27921 0.639603 0.768706i \(-0.279099\pi\)
0.639603 + 0.768706i \(0.279099\pi\)
\(998\) 2.55291e7 0.811351
\(999\) 2.87834e7 0.912491
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 1045.6.a.f.1.13 38
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.6.a.f.1.13 38 1.1 even 1 trivial