Properties

Label 2166.4.a.r.1.3
Level $2166$
Weight $4$
Character 2166.1
Self dual yes
Analytic conductor $127.798$
Analytic rank $0$
Dimension $3$
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] = [2166,4,Mod(1,2166)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2166, 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("2166.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2166 = 2 \cdot 3 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2166.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: \(127.798137072\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.373564.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 106x - 348 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-7.85460\) of defining polynomial
Character \(\chi\) \(=\) 2166.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +17.4112 q^{5} +6.00000 q^{6} -0.298020 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +17.4112 q^{5} +6.00000 q^{6} -0.298020 q^{7} -8.00000 q^{8} +9.00000 q^{9} -34.8224 q^{10} +53.7164 q^{11} -12.0000 q^{12} -45.9355 q^{13} +0.596040 q^{14} -52.2335 q^{15} +16.0000 q^{16} +87.9427 q^{17} -18.0000 q^{18} +69.6447 q^{20} +0.894059 q^{21} -107.433 q^{22} -65.1421 q^{23} +24.0000 q^{24} +178.149 q^{25} +91.8711 q^{26} -27.0000 q^{27} -1.19208 q^{28} -37.2750 q^{29} +104.467 q^{30} +176.191 q^{31} -32.0000 q^{32} -161.149 q^{33} -175.885 q^{34} -5.18888 q^{35} +36.0000 q^{36} -153.936 q^{37} +137.807 q^{39} -139.289 q^{40} +171.106 q^{41} -1.78812 q^{42} -248.680 q^{43} +214.866 q^{44} +156.701 q^{45} +130.284 q^{46} +372.119 q^{47} -48.0000 q^{48} -342.911 q^{49} -356.299 q^{50} -263.828 q^{51} -183.742 q^{52} +354.849 q^{53} +54.0000 q^{54} +935.266 q^{55} +2.38416 q^{56} +74.5500 q^{58} +734.163 q^{59} -208.934 q^{60} -172.166 q^{61} -352.381 q^{62} -2.68218 q^{63} +64.0000 q^{64} -799.792 q^{65} +322.299 q^{66} +277.911 q^{67} +351.771 q^{68} +195.426 q^{69} +10.3778 q^{70} +868.634 q^{71} -72.0000 q^{72} -844.039 q^{73} +307.871 q^{74} -534.448 q^{75} -16.0086 q^{77} -275.613 q^{78} -333.383 q^{79} +278.579 q^{80} +81.0000 q^{81} -342.213 q^{82} +1415.34 q^{83} +3.57624 q^{84} +1531.19 q^{85} +497.359 q^{86} +111.825 q^{87} -429.731 q^{88} +1329.02 q^{89} -313.401 q^{90} +13.6897 q^{91} -260.568 q^{92} -528.572 q^{93} -744.238 q^{94} +96.0000 q^{96} +1470.68 q^{97} +685.822 q^{98} +483.448 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 6 q^{2} - 9 q^{3} + 12 q^{4} - 5 q^{5} + 18 q^{6} - 11 q^{7} - 24 q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 6 q^{2} - 9 q^{3} + 12 q^{4} - 5 q^{5} + 18 q^{6} - 11 q^{7} - 24 q^{8} + 27 q^{9} + 10 q^{10} + 77 q^{11} - 36 q^{12} + 44 q^{13} + 22 q^{14} + 15 q^{15} + 48 q^{16} + 45 q^{17} - 54 q^{18} - 20 q^{20} + 33 q^{21} - 154 q^{22} + 20 q^{23} + 72 q^{24} + 282 q^{25} - 88 q^{26} - 81 q^{27} - 44 q^{28} + 272 q^{29} - 30 q^{30} + 64 q^{31} - 96 q^{32} - 231 q^{33} - 90 q^{34} + 439 q^{35} + 108 q^{36} - 280 q^{37} - 132 q^{39} + 40 q^{40} - 32 q^{41} - 66 q^{42} + 173 q^{43} + 308 q^{44} - 45 q^{45} - 40 q^{46} + 507 q^{47} - 144 q^{48} + 52 q^{49} - 564 q^{50} - 135 q^{51} + 176 q^{52} - 208 q^{53} + 162 q^{54} - 93 q^{55} + 88 q^{56} - 544 q^{58} + 344 q^{59} + 60 q^{60} + 847 q^{61} - 128 q^{62} - 99 q^{63} + 192 q^{64} - 2440 q^{65} + 462 q^{66} + 268 q^{67} + 180 q^{68} - 60 q^{69} - 878 q^{70} + 384 q^{71} - 216 q^{72} + 225 q^{73} + 560 q^{74} - 846 q^{75} - 2563 q^{77} + 264 q^{78} - 576 q^{79} - 80 q^{80} + 243 q^{81} + 64 q^{82} + 1492 q^{83} + 132 q^{84} + 2099 q^{85} - 346 q^{86} - 816 q^{87} - 616 q^{88} + 816 q^{89} + 90 q^{90} - 2464 q^{91} + 80 q^{92} - 192 q^{93} - 1014 q^{94} + 288 q^{96} + 4008 q^{97} - 104 q^{98} + 693 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) −3.00000 −0.577350
\(4\) 4.00000 0.500000
\(5\) 17.4112 1.55730 0.778652 0.627456i \(-0.215904\pi\)
0.778652 + 0.627456i \(0.215904\pi\)
\(6\) 6.00000 0.408248
\(7\) −0.298020 −0.0160916 −0.00804578 0.999968i \(-0.502561\pi\)
−0.00804578 + 0.999968i \(0.502561\pi\)
\(8\) −8.00000 −0.353553
\(9\) 9.00000 0.333333
\(10\) −34.8224 −1.10118
\(11\) 53.7164 1.47237 0.736187 0.676778i \(-0.236624\pi\)
0.736187 + 0.676778i \(0.236624\pi\)
\(12\) −12.0000 −0.288675
\(13\) −45.9355 −0.980017 −0.490009 0.871718i \(-0.663006\pi\)
−0.490009 + 0.871718i \(0.663006\pi\)
\(14\) 0.596040 0.0113784
\(15\) −52.2335 −0.899110
\(16\) 16.0000 0.250000
\(17\) 87.9427 1.25466 0.627331 0.778753i \(-0.284147\pi\)
0.627331 + 0.778753i \(0.284147\pi\)
\(18\) −18.0000 −0.235702
\(19\) 0 0
\(20\) 69.6447 0.778652
\(21\) 0.894059 0.00929046
\(22\) −107.433 −1.04113
\(23\) −65.1421 −0.590568 −0.295284 0.955410i \(-0.595414\pi\)
−0.295284 + 0.955410i \(0.595414\pi\)
\(24\) 24.0000 0.204124
\(25\) 178.149 1.42519
\(26\) 91.8711 0.692977
\(27\) −27.0000 −0.192450
\(28\) −1.19208 −0.00804578
\(29\) −37.2750 −0.238683 −0.119341 0.992853i \(-0.538078\pi\)
−0.119341 + 0.992853i \(0.538078\pi\)
\(30\) 104.467 0.635766
\(31\) 176.191 1.02080 0.510400 0.859937i \(-0.329497\pi\)
0.510400 + 0.859937i \(0.329497\pi\)
\(32\) −32.0000 −0.176777
\(33\) −161.149 −0.850075
\(34\) −175.885 −0.887180
\(35\) −5.18888 −0.0250594
\(36\) 36.0000 0.166667
\(37\) −153.936 −0.683969 −0.341984 0.939706i \(-0.611099\pi\)
−0.341984 + 0.939706i \(0.611099\pi\)
\(38\) 0 0
\(39\) 137.807 0.565813
\(40\) −139.289 −0.550590
\(41\) 171.106 0.651764 0.325882 0.945410i \(-0.394339\pi\)
0.325882 + 0.945410i \(0.394339\pi\)
\(42\) −1.78812 −0.00656935
\(43\) −248.680 −0.881936 −0.440968 0.897523i \(-0.645365\pi\)
−0.440968 + 0.897523i \(0.645365\pi\)
\(44\) 214.866 0.736187
\(45\) 156.701 0.519101
\(46\) 130.284 0.417594
\(47\) 372.119 1.15488 0.577438 0.816435i \(-0.304053\pi\)
0.577438 + 0.816435i \(0.304053\pi\)
\(48\) −48.0000 −0.144338
\(49\) −342.911 −0.999741
\(50\) −356.299 −1.00776
\(51\) −263.828 −0.724379
\(52\) −183.742 −0.490009
\(53\) 354.849 0.919664 0.459832 0.888006i \(-0.347910\pi\)
0.459832 + 0.888006i \(0.347910\pi\)
\(54\) 54.0000 0.136083
\(55\) 935.266 2.29293
\(56\) 2.38416 0.00568922
\(57\) 0 0
\(58\) 74.5500 0.168774
\(59\) 734.163 1.62000 0.809999 0.586431i \(-0.199467\pi\)
0.809999 + 0.586431i \(0.199467\pi\)
\(60\) −208.934 −0.449555
\(61\) −172.166 −0.361371 −0.180686 0.983541i \(-0.557832\pi\)
−0.180686 + 0.983541i \(0.557832\pi\)
\(62\) −352.381 −0.721814
\(63\) −2.68218 −0.00536385
\(64\) 64.0000 0.125000
\(65\) −799.792 −1.52618
\(66\) 322.299 0.601094
\(67\) 277.911 0.506749 0.253375 0.967368i \(-0.418459\pi\)
0.253375 + 0.967368i \(0.418459\pi\)
\(68\) 351.771 0.627331
\(69\) 195.426 0.340964
\(70\) 10.3778 0.0177197
\(71\) 868.634 1.45194 0.725971 0.687725i \(-0.241391\pi\)
0.725971 + 0.687725i \(0.241391\pi\)
\(72\) −72.0000 −0.117851
\(73\) −844.039 −1.35325 −0.676625 0.736328i \(-0.736558\pi\)
−0.676625 + 0.736328i \(0.736558\pi\)
\(74\) 307.871 0.483639
\(75\) −534.448 −0.822836
\(76\) 0 0
\(77\) −16.0086 −0.0236928
\(78\) −275.613 −0.400090
\(79\) −333.383 −0.474791 −0.237396 0.971413i \(-0.576294\pi\)
−0.237396 + 0.971413i \(0.576294\pi\)
\(80\) 278.579 0.389326
\(81\) 81.0000 0.111111
\(82\) −342.213 −0.460867
\(83\) 1415.34 1.87173 0.935865 0.352359i \(-0.114621\pi\)
0.935865 + 0.352359i \(0.114621\pi\)
\(84\) 3.57624 0.00464523
\(85\) 1531.19 1.95389
\(86\) 497.359 0.623623
\(87\) 111.825 0.137803
\(88\) −429.731 −0.520563
\(89\) 1329.02 1.58287 0.791437 0.611251i \(-0.209333\pi\)
0.791437 + 0.611251i \(0.209333\pi\)
\(90\) −313.401 −0.367060
\(91\) 13.6897 0.0157700
\(92\) −260.568 −0.295284
\(93\) −528.572 −0.589359
\(94\) −744.238 −0.816620
\(95\) 0 0
\(96\) 96.0000 0.102062
\(97\) 1470.68 1.53943 0.769716 0.638387i \(-0.220398\pi\)
0.769716 + 0.638387i \(0.220398\pi\)
\(98\) 685.822 0.706924
\(99\) 483.448 0.490791
\(100\) 712.597 0.712597
\(101\) −677.796 −0.667755 −0.333877 0.942617i \(-0.608357\pi\)
−0.333877 + 0.942617i \(0.608357\pi\)
\(102\) 527.656 0.512214
\(103\) −1646.65 −1.57523 −0.787617 0.616165i \(-0.788685\pi\)
−0.787617 + 0.616165i \(0.788685\pi\)
\(104\) 367.484 0.346488
\(105\) 15.5666 0.0144681
\(106\) −709.697 −0.650301
\(107\) −1960.88 −1.77164 −0.885819 0.464030i \(-0.846403\pi\)
−0.885819 + 0.464030i \(0.846403\pi\)
\(108\) −108.000 −0.0962250
\(109\) −352.283 −0.309565 −0.154782 0.987949i \(-0.549468\pi\)
−0.154782 + 0.987949i \(0.549468\pi\)
\(110\) −1870.53 −1.62135
\(111\) 461.807 0.394890
\(112\) −4.76832 −0.00402289
\(113\) 2216.11 1.84490 0.922452 0.386111i \(-0.126182\pi\)
0.922452 + 0.386111i \(0.126182\pi\)
\(114\) 0 0
\(115\) −1134.20 −0.919693
\(116\) −149.100 −0.119341
\(117\) −413.420 −0.326672
\(118\) −1468.33 −1.14551
\(119\) −26.2087 −0.0201895
\(120\) 417.868 0.317883
\(121\) 1554.45 1.16788
\(122\) 344.333 0.255528
\(123\) −513.319 −0.376296
\(124\) 704.763 0.510400
\(125\) 925.392 0.662156
\(126\) 5.36436 0.00379282
\(127\) −2438.78 −1.70399 −0.851996 0.523549i \(-0.824608\pi\)
−0.851996 + 0.523549i \(0.824608\pi\)
\(128\) −128.000 −0.0883883
\(129\) 746.039 0.509186
\(130\) 1599.58 1.07918
\(131\) −1533.40 −1.02270 −0.511351 0.859372i \(-0.670855\pi\)
−0.511351 + 0.859372i \(0.670855\pi\)
\(132\) −644.597 −0.425038
\(133\) 0 0
\(134\) −555.821 −0.358326
\(135\) −470.102 −0.299703
\(136\) −703.542 −0.443590
\(137\) −1087.94 −0.678461 −0.339230 0.940703i \(-0.610167\pi\)
−0.339230 + 0.940703i \(0.610167\pi\)
\(138\) −390.852 −0.241098
\(139\) −49.3719 −0.0301271 −0.0150635 0.999887i \(-0.504795\pi\)
−0.0150635 + 0.999887i \(0.504795\pi\)
\(140\) −20.7555 −0.0125297
\(141\) −1116.36 −0.666768
\(142\) −1737.27 −1.02668
\(143\) −2467.49 −1.44295
\(144\) 144.000 0.0833333
\(145\) −649.002 −0.371701
\(146\) 1688.08 0.956892
\(147\) 1028.73 0.577201
\(148\) −615.742 −0.341984
\(149\) 2565.82 1.41074 0.705369 0.708840i \(-0.250781\pi\)
0.705369 + 0.708840i \(0.250781\pi\)
\(150\) 1068.90 0.581833
\(151\) 46.8825 0.0252665 0.0126333 0.999920i \(-0.495979\pi\)
0.0126333 + 0.999920i \(0.495979\pi\)
\(152\) 0 0
\(153\) 791.485 0.418221
\(154\) 32.0171 0.0167533
\(155\) 3067.69 1.58969
\(156\) 551.226 0.282907
\(157\) 2888.34 1.46824 0.734122 0.679017i \(-0.237594\pi\)
0.734122 + 0.679017i \(0.237594\pi\)
\(158\) 666.766 0.335728
\(159\) −1064.55 −0.530968
\(160\) −557.158 −0.275295
\(161\) 19.4136 0.00950315
\(162\) −162.000 −0.0785674
\(163\) −183.569 −0.0882099 −0.0441050 0.999027i \(-0.514044\pi\)
−0.0441050 + 0.999027i \(0.514044\pi\)
\(164\) 684.426 0.325882
\(165\) −2805.80 −1.32383
\(166\) −2830.68 −1.32351
\(167\) −3522.58 −1.63225 −0.816125 0.577876i \(-0.803882\pi\)
−0.816125 + 0.577876i \(0.803882\pi\)
\(168\) −7.15247 −0.00328468
\(169\) −86.9273 −0.0395664
\(170\) −3062.37 −1.38161
\(171\) 0 0
\(172\) −994.718 −0.440968
\(173\) 3142.83 1.38118 0.690592 0.723245i \(-0.257350\pi\)
0.690592 + 0.723245i \(0.257350\pi\)
\(174\) −223.650 −0.0974418
\(175\) −53.0920 −0.0229336
\(176\) 859.463 0.368093
\(177\) −2202.49 −0.935307
\(178\) −2658.04 −1.11926
\(179\) −325.374 −0.135864 −0.0679318 0.997690i \(-0.521640\pi\)
−0.0679318 + 0.997690i \(0.521640\pi\)
\(180\) 626.803 0.259551
\(181\) 2639.37 1.08388 0.541942 0.840416i \(-0.317689\pi\)
0.541942 + 0.840416i \(0.317689\pi\)
\(182\) −27.3794 −0.0111511
\(183\) 516.499 0.208638
\(184\) 521.136 0.208797
\(185\) −2680.20 −1.06515
\(186\) 1057.14 0.416740
\(187\) 4723.97 1.84733
\(188\) 1488.48 0.577438
\(189\) 8.04653 0.00309682
\(190\) 0 0
\(191\) −3801.94 −1.44031 −0.720154 0.693814i \(-0.755929\pi\)
−0.720154 + 0.693814i \(0.755929\pi\)
\(192\) −192.000 −0.0721688
\(193\) −2534.43 −0.945245 −0.472623 0.881265i \(-0.656693\pi\)
−0.472623 + 0.881265i \(0.656693\pi\)
\(194\) −2941.36 −1.08854
\(195\) 2399.38 0.881143
\(196\) −1371.64 −0.499871
\(197\) −4115.93 −1.48857 −0.744284 0.667864i \(-0.767209\pi\)
−0.744284 + 0.667864i \(0.767209\pi\)
\(198\) −966.896 −0.347042
\(199\) 1531.33 0.545493 0.272747 0.962086i \(-0.412068\pi\)
0.272747 + 0.962086i \(0.412068\pi\)
\(200\) −1425.19 −0.503882
\(201\) −833.732 −0.292572
\(202\) 1355.59 0.472174
\(203\) 11.1087 0.00384077
\(204\) −1055.31 −0.362190
\(205\) 2979.17 1.01499
\(206\) 3293.30 1.11386
\(207\) −586.278 −0.196856
\(208\) −734.968 −0.245004
\(209\) 0 0
\(210\) −31.1333 −0.0102305
\(211\) 1431.90 0.467185 0.233593 0.972335i \(-0.424952\pi\)
0.233593 + 0.972335i \(0.424952\pi\)
\(212\) 1419.39 0.459832
\(213\) −2605.90 −0.838279
\(214\) 3921.76 1.25274
\(215\) −4329.80 −1.37344
\(216\) 216.000 0.0680414
\(217\) −52.5083 −0.0164263
\(218\) 704.566 0.218895
\(219\) 2532.12 0.781299
\(220\) 3741.07 1.14647
\(221\) −4039.70 −1.22959
\(222\) −923.613 −0.279229
\(223\) 3138.32 0.942409 0.471205 0.882024i \(-0.343819\pi\)
0.471205 + 0.882024i \(0.343819\pi\)
\(224\) 9.53663 0.00284461
\(225\) 1603.34 0.475065
\(226\) −4432.22 −1.30454
\(227\) −3887.66 −1.13671 −0.568355 0.822784i \(-0.692420\pi\)
−0.568355 + 0.822784i \(0.692420\pi\)
\(228\) 0 0
\(229\) −458.623 −0.132344 −0.0661718 0.997808i \(-0.521079\pi\)
−0.0661718 + 0.997808i \(0.521079\pi\)
\(230\) 2268.40 0.650321
\(231\) 48.0257 0.0136790
\(232\) 298.200 0.0843871
\(233\) 4351.86 1.22360 0.611802 0.791011i \(-0.290445\pi\)
0.611802 + 0.791011i \(0.290445\pi\)
\(234\) 826.839 0.230992
\(235\) 6479.03 1.79849
\(236\) 2936.65 0.809999
\(237\) 1000.15 0.274121
\(238\) 52.4174 0.0142761
\(239\) −68.9788 −0.0186689 −0.00933445 0.999956i \(-0.502971\pi\)
−0.00933445 + 0.999956i \(0.502971\pi\)
\(240\) −835.737 −0.224777
\(241\) −39.6721 −0.0106038 −0.00530188 0.999986i \(-0.501688\pi\)
−0.00530188 + 0.999986i \(0.501688\pi\)
\(242\) −3108.91 −0.825819
\(243\) −243.000 −0.0641500
\(244\) −688.666 −0.180686
\(245\) −5970.49 −1.55690
\(246\) 1026.64 0.266082
\(247\) 0 0
\(248\) −1409.53 −0.360907
\(249\) −4246.01 −1.08064
\(250\) −1850.78 −0.468215
\(251\) −5883.34 −1.47950 −0.739748 0.672884i \(-0.765055\pi\)
−0.739748 + 0.672884i \(0.765055\pi\)
\(252\) −10.7287 −0.00268193
\(253\) −3499.20 −0.869536
\(254\) 4877.56 1.20490
\(255\) −4593.56 −1.12808
\(256\) 256.000 0.0625000
\(257\) −3076.70 −0.746767 −0.373384 0.927677i \(-0.621803\pi\)
−0.373384 + 0.927677i \(0.621803\pi\)
\(258\) −1492.08 −0.360049
\(259\) 45.8758 0.0110061
\(260\) −3199.17 −0.763092
\(261\) −335.475 −0.0795609
\(262\) 3066.81 0.723160
\(263\) −1610.93 −0.377698 −0.188849 0.982006i \(-0.560476\pi\)
−0.188849 + 0.982006i \(0.560476\pi\)
\(264\) 1289.19 0.300547
\(265\) 6178.33 1.43220
\(266\) 0 0
\(267\) −3987.06 −0.913873
\(268\) 1111.64 0.253375
\(269\) 5892.43 1.33557 0.667784 0.744355i \(-0.267243\pi\)
0.667784 + 0.744355i \(0.267243\pi\)
\(270\) 940.204 0.211922
\(271\) 2430.30 0.544761 0.272380 0.962190i \(-0.412189\pi\)
0.272380 + 0.962190i \(0.412189\pi\)
\(272\) 1407.08 0.313665
\(273\) −41.0691 −0.00910481
\(274\) 2175.88 0.479744
\(275\) 9569.54 2.09842
\(276\) 781.705 0.170482
\(277\) 1573.19 0.341241 0.170621 0.985337i \(-0.445423\pi\)
0.170621 + 0.985337i \(0.445423\pi\)
\(278\) 98.7437 0.0213031
\(279\) 1585.72 0.340267
\(280\) 41.5110 0.00885985
\(281\) 3250.35 0.690035 0.345017 0.938596i \(-0.387873\pi\)
0.345017 + 0.938596i \(0.387873\pi\)
\(282\) 2232.71 0.471476
\(283\) 1283.72 0.269643 0.134822 0.990870i \(-0.456954\pi\)
0.134822 + 0.990870i \(0.456954\pi\)
\(284\) 3474.54 0.725971
\(285\) 0 0
\(286\) 4934.98 1.02032
\(287\) −50.9931 −0.0104879
\(288\) −288.000 −0.0589256
\(289\) 2820.93 0.574176
\(290\) 1298.00 0.262833
\(291\) −4412.04 −0.888791
\(292\) −3376.15 −0.676625
\(293\) 5208.20 1.03845 0.519226 0.854637i \(-0.326220\pi\)
0.519226 + 0.854637i \(0.326220\pi\)
\(294\) −2057.47 −0.408143
\(295\) 12782.6 2.52283
\(296\) 1231.48 0.241819
\(297\) −1450.34 −0.283358
\(298\) −5131.63 −0.997542
\(299\) 2992.33 0.578766
\(300\) −2137.79 −0.411418
\(301\) 74.1114 0.0141917
\(302\) −93.7651 −0.0178661
\(303\) 2033.39 0.385528
\(304\) 0 0
\(305\) −2997.62 −0.562765
\(306\) −1582.97 −0.295727
\(307\) 479.565 0.0891538 0.0445769 0.999006i \(-0.485806\pi\)
0.0445769 + 0.999006i \(0.485806\pi\)
\(308\) −64.0342 −0.0118464
\(309\) 4939.95 0.909462
\(310\) −6135.38 −1.12408
\(311\) 1642.23 0.299428 0.149714 0.988729i \(-0.452165\pi\)
0.149714 + 0.988729i \(0.452165\pi\)
\(312\) −1102.45 −0.200045
\(313\) 8018.68 1.44806 0.724029 0.689769i \(-0.242288\pi\)
0.724029 + 0.689769i \(0.242288\pi\)
\(314\) −5776.67 −1.03821
\(315\) −46.6999 −0.00835315
\(316\) −1333.53 −0.237396
\(317\) 3071.10 0.544132 0.272066 0.962279i \(-0.412293\pi\)
0.272066 + 0.962279i \(0.412293\pi\)
\(318\) 2129.09 0.375451
\(319\) −2002.28 −0.351430
\(320\) 1114.32 0.194663
\(321\) 5882.64 1.02286
\(322\) −38.8272 −0.00671974
\(323\) 0 0
\(324\) 324.000 0.0555556
\(325\) −8183.38 −1.39671
\(326\) 367.138 0.0623738
\(327\) 1056.85 0.178727
\(328\) −1368.85 −0.230434
\(329\) −110.899 −0.0185837
\(330\) 5611.60 0.936086
\(331\) −2172.26 −0.360719 −0.180360 0.983601i \(-0.557726\pi\)
−0.180360 + 0.983601i \(0.557726\pi\)
\(332\) 5661.35 0.935865
\(333\) −1385.42 −0.227990
\(334\) 7045.17 1.15417
\(335\) 4838.75 0.789162
\(336\) 14.3049 0.00232262
\(337\) −9700.80 −1.56806 −0.784030 0.620722i \(-0.786839\pi\)
−0.784030 + 0.620722i \(0.786839\pi\)
\(338\) 173.855 0.0279777
\(339\) −6648.33 −1.06516
\(340\) 6124.75 0.976945
\(341\) 9464.34 1.50300
\(342\) 0 0
\(343\) 204.415 0.0321789
\(344\) 1989.44 0.311812
\(345\) 3402.60 0.530985
\(346\) −6285.65 −0.976644
\(347\) −483.729 −0.0748356 −0.0374178 0.999300i \(-0.511913\pi\)
−0.0374178 + 0.999300i \(0.511913\pi\)
\(348\) 447.300 0.0689017
\(349\) 11947.6 1.83250 0.916250 0.400607i \(-0.131201\pi\)
0.916250 + 0.400607i \(0.131201\pi\)
\(350\) 106.184 0.0162165
\(351\) 1240.26 0.188604
\(352\) −1718.93 −0.260281
\(353\) 7542.02 1.13717 0.568585 0.822624i \(-0.307491\pi\)
0.568585 + 0.822624i \(0.307491\pi\)
\(354\) 4404.98 0.661362
\(355\) 15123.9 2.26111
\(356\) 5316.08 0.791437
\(357\) 78.6260 0.0116564
\(358\) 650.748 0.0960701
\(359\) 8295.22 1.21951 0.609756 0.792589i \(-0.291267\pi\)
0.609756 + 0.792589i \(0.291267\pi\)
\(360\) −1253.61 −0.183530
\(361\) 0 0
\(362\) −5278.75 −0.766422
\(363\) −4663.36 −0.674278
\(364\) 54.7588 0.00788500
\(365\) −14695.7 −2.10742
\(366\) −1033.00 −0.147529
\(367\) −2998.67 −0.426510 −0.213255 0.976997i \(-0.568407\pi\)
−0.213255 + 0.976997i \(0.568407\pi\)
\(368\) −1042.27 −0.147642
\(369\) 1539.96 0.217255
\(370\) 5360.40 0.753173
\(371\) −105.752 −0.0147988
\(372\) −2114.29 −0.294679
\(373\) −6597.74 −0.915866 −0.457933 0.888987i \(-0.651410\pi\)
−0.457933 + 0.888987i \(0.651410\pi\)
\(374\) −9447.94 −1.30626
\(375\) −2776.18 −0.382296
\(376\) −2976.95 −0.408310
\(377\) 1712.25 0.233913
\(378\) −16.0931 −0.00218978
\(379\) −6283.53 −0.851618 −0.425809 0.904813i \(-0.640010\pi\)
−0.425809 + 0.904813i \(0.640010\pi\)
\(380\) 0 0
\(381\) 7316.35 0.983800
\(382\) 7603.88 1.01845
\(383\) −3098.49 −0.413383 −0.206691 0.978406i \(-0.566270\pi\)
−0.206691 + 0.978406i \(0.566270\pi\)
\(384\) 384.000 0.0510310
\(385\) −278.728 −0.0368969
\(386\) 5068.86 0.668389
\(387\) −2238.12 −0.293979
\(388\) 5882.72 0.769716
\(389\) 10304.6 1.34309 0.671546 0.740963i \(-0.265631\pi\)
0.671546 + 0.740963i \(0.265631\pi\)
\(390\) −4798.75 −0.623062
\(391\) −5728.77 −0.740963
\(392\) 2743.29 0.353462
\(393\) 4600.21 0.590458
\(394\) 8231.86 1.05258
\(395\) −5804.59 −0.739394
\(396\) 1933.79 0.245396
\(397\) 10609.4 1.34124 0.670620 0.741801i \(-0.266028\pi\)
0.670620 + 0.741801i \(0.266028\pi\)
\(398\) −3062.66 −0.385722
\(399\) 0 0
\(400\) 2850.39 0.356299
\(401\) −13842.3 −1.72382 −0.861908 0.507065i \(-0.830730\pi\)
−0.861908 + 0.507065i \(0.830730\pi\)
\(402\) 1667.46 0.206879
\(403\) −8093.41 −1.00040
\(404\) −2711.18 −0.333877
\(405\) 1410.31 0.173034
\(406\) −22.2174 −0.00271584
\(407\) −8268.87 −1.00706
\(408\) 2110.63 0.256107
\(409\) 3161.28 0.382189 0.191095 0.981572i \(-0.438796\pi\)
0.191095 + 0.981572i \(0.438796\pi\)
\(410\) −5958.33 −0.717710
\(411\) 3263.83 0.391710
\(412\) −6586.60 −0.787617
\(413\) −218.795 −0.0260683
\(414\) 1172.56 0.139198
\(415\) 24642.7 2.91485
\(416\) 1469.94 0.173244
\(417\) 148.116 0.0173939
\(418\) 0 0
\(419\) 7703.02 0.898132 0.449066 0.893499i \(-0.351757\pi\)
0.449066 + 0.893499i \(0.351757\pi\)
\(420\) 62.2665 0.00723404
\(421\) 4734.35 0.548071 0.274036 0.961720i \(-0.411641\pi\)
0.274036 + 0.961720i \(0.411641\pi\)
\(422\) −2863.80 −0.330350
\(423\) 3349.07 0.384958
\(424\) −2838.79 −0.325150
\(425\) 15666.9 1.78814
\(426\) 5211.80 0.592753
\(427\) 51.3090 0.00581503
\(428\) −7843.52 −0.885819
\(429\) 7402.48 0.833088
\(430\) 8659.61 0.971171
\(431\) 9552.77 1.06761 0.533806 0.845607i \(-0.320761\pi\)
0.533806 + 0.845607i \(0.320761\pi\)
\(432\) −432.000 −0.0481125
\(433\) 15829.3 1.75684 0.878418 0.477894i \(-0.158600\pi\)
0.878418 + 0.477894i \(0.158600\pi\)
\(434\) 105.017 0.0116151
\(435\) 1947.01 0.214602
\(436\) −1409.13 −0.154782
\(437\) 0 0
\(438\) −5064.23 −0.552462
\(439\) 12163.0 1.32234 0.661169 0.750237i \(-0.270061\pi\)
0.661169 + 0.750237i \(0.270061\pi\)
\(440\) −7482.13 −0.810674
\(441\) −3086.20 −0.333247
\(442\) 8079.39 0.869451
\(443\) 10738.1 1.15165 0.575826 0.817573i \(-0.304681\pi\)
0.575826 + 0.817573i \(0.304681\pi\)
\(444\) 1847.23 0.197445
\(445\) 23139.8 2.46501
\(446\) −6276.64 −0.666384
\(447\) −7697.45 −0.814490
\(448\) −19.0733 −0.00201144
\(449\) 11902.2 1.25101 0.625503 0.780222i \(-0.284894\pi\)
0.625503 + 0.780222i \(0.284894\pi\)
\(450\) −3206.69 −0.335921
\(451\) 9191.23 0.959641
\(452\) 8864.45 0.922452
\(453\) −140.648 −0.0145876
\(454\) 7775.32 0.803775
\(455\) 238.354 0.0245587
\(456\) 0 0
\(457\) −14980.8 −1.53342 −0.766709 0.641995i \(-0.778107\pi\)
−0.766709 + 0.641995i \(0.778107\pi\)
\(458\) 917.247 0.0935811
\(459\) −2374.45 −0.241460
\(460\) −4536.80 −0.459847
\(461\) 1153.32 0.116520 0.0582598 0.998301i \(-0.481445\pi\)
0.0582598 + 0.998301i \(0.481445\pi\)
\(462\) −96.0513 −0.00967254
\(463\) 9780.17 0.981691 0.490846 0.871247i \(-0.336688\pi\)
0.490846 + 0.871247i \(0.336688\pi\)
\(464\) −596.400 −0.0596707
\(465\) −9203.07 −0.917811
\(466\) −8703.71 −0.865218
\(467\) 14300.1 1.41698 0.708492 0.705719i \(-0.249376\pi\)
0.708492 + 0.705719i \(0.249376\pi\)
\(468\) −1653.68 −0.163336
\(469\) −82.8229 −0.00815438
\(470\) −12958.1 −1.27173
\(471\) −8665.01 −0.847691
\(472\) −5873.31 −0.572756
\(473\) −13358.2 −1.29854
\(474\) −2000.30 −0.193833
\(475\) 0 0
\(476\) −104.835 −0.0100947
\(477\) 3193.64 0.306555
\(478\) 137.958 0.0132009
\(479\) −1760.50 −0.167932 −0.0839658 0.996469i \(-0.526759\pi\)
−0.0839658 + 0.996469i \(0.526759\pi\)
\(480\) 1671.47 0.158942
\(481\) 7071.11 0.670301
\(482\) 79.3442 0.00749799
\(483\) −58.2409 −0.00548665
\(484\) 6217.82 0.583942
\(485\) 25606.3 2.39736
\(486\) 486.000 0.0453609
\(487\) −8313.61 −0.773564 −0.386782 0.922171i \(-0.626413\pi\)
−0.386782 + 0.922171i \(0.626413\pi\)
\(488\) 1377.33 0.127764
\(489\) 550.706 0.0509280
\(490\) 11941.0 1.10089
\(491\) 5072.82 0.466259 0.233130 0.972446i \(-0.425103\pi\)
0.233130 + 0.972446i \(0.425103\pi\)
\(492\) −2053.28 −0.188148
\(493\) −3278.07 −0.299466
\(494\) 0 0
\(495\) 8417.40 0.764311
\(496\) 2819.05 0.255200
\(497\) −258.870 −0.0233640
\(498\) 8492.03 0.764130
\(499\) 7893.74 0.708161 0.354080 0.935215i \(-0.384794\pi\)
0.354080 + 0.935215i \(0.384794\pi\)
\(500\) 3701.57 0.331078
\(501\) 10567.8 0.942380
\(502\) 11766.7 1.04616
\(503\) 14575.6 1.29203 0.646016 0.763323i \(-0.276434\pi\)
0.646016 + 0.763323i \(0.276434\pi\)
\(504\) 21.4574 0.00189641
\(505\) −11801.2 −1.03990
\(506\) 6998.40 0.614855
\(507\) 260.782 0.0228437
\(508\) −9755.13 −0.851996
\(509\) −4331.64 −0.377204 −0.188602 0.982054i \(-0.560396\pi\)
−0.188602 + 0.982054i \(0.560396\pi\)
\(510\) 9187.12 0.797672
\(511\) 251.540 0.0217759
\(512\) −512.000 −0.0441942
\(513\) 0 0
\(514\) 6153.40 0.528044
\(515\) −28670.1 −2.45312
\(516\) 2984.15 0.254593
\(517\) 19988.9 1.70041
\(518\) −91.7517 −0.00778250
\(519\) −9428.48 −0.797427
\(520\) 6398.33 0.539588
\(521\) −13878.8 −1.16707 −0.583535 0.812088i \(-0.698331\pi\)
−0.583535 + 0.812088i \(0.698331\pi\)
\(522\) 670.950 0.0562580
\(523\) −4580.75 −0.382987 −0.191494 0.981494i \(-0.561333\pi\)
−0.191494 + 0.981494i \(0.561333\pi\)
\(524\) −6133.61 −0.511351
\(525\) 159.276 0.0132407
\(526\) 3221.87 0.267073
\(527\) 15494.7 1.28076
\(528\) −2578.39 −0.212519
\(529\) −7923.51 −0.651230
\(530\) −12356.7 −1.01272
\(531\) 6607.47 0.539999
\(532\) 0 0
\(533\) −7859.87 −0.638740
\(534\) 7974.12 0.646205
\(535\) −34141.2 −2.75898
\(536\) −2223.29 −0.179163
\(537\) 976.122 0.0784409
\(538\) −11784.9 −0.944389
\(539\) −18420.0 −1.47199
\(540\) −1880.41 −0.149852
\(541\) −4696.29 −0.373215 −0.186608 0.982435i \(-0.559749\pi\)
−0.186608 + 0.982435i \(0.559749\pi\)
\(542\) −4860.60 −0.385204
\(543\) −7918.12 −0.625781
\(544\) −2814.17 −0.221795
\(545\) −6133.66 −0.482086
\(546\) 82.1382 0.00643808
\(547\) −8293.13 −0.648243 −0.324121 0.946016i \(-0.605069\pi\)
−0.324121 + 0.946016i \(0.605069\pi\)
\(548\) −4351.77 −0.339230
\(549\) −1549.50 −0.120457
\(550\) −19139.1 −1.48381
\(551\) 0 0
\(552\) −1563.41 −0.120549
\(553\) 99.3547 0.00764013
\(554\) −3146.38 −0.241294
\(555\) 8040.60 0.614963
\(556\) −197.487 −0.0150635
\(557\) −11787.1 −0.896654 −0.448327 0.893870i \(-0.647980\pi\)
−0.448327 + 0.893870i \(0.647980\pi\)
\(558\) −3171.43 −0.240605
\(559\) 11423.2 0.864313
\(560\) −83.0220 −0.00626486
\(561\) −14171.9 −1.06656
\(562\) −6500.71 −0.487928
\(563\) −9868.41 −0.738728 −0.369364 0.929285i \(-0.620424\pi\)
−0.369364 + 0.929285i \(0.620424\pi\)
\(564\) −4465.43 −0.333384
\(565\) 38585.1 2.87308
\(566\) −2567.43 −0.190667
\(567\) −24.1396 −0.00178795
\(568\) −6949.07 −0.513339
\(569\) 16579.6 1.22154 0.610768 0.791810i \(-0.290861\pi\)
0.610768 + 0.791810i \(0.290861\pi\)
\(570\) 0 0
\(571\) −9453.06 −0.692817 −0.346408 0.938084i \(-0.612599\pi\)
−0.346408 + 0.938084i \(0.612599\pi\)
\(572\) −9869.97 −0.721476
\(573\) 11405.8 0.831562
\(574\) 101.986 0.00741607
\(575\) −11605.0 −0.841674
\(576\) 576.000 0.0416667
\(577\) −23284.1 −1.67995 −0.839974 0.542627i \(-0.817430\pi\)
−0.839974 + 0.542627i \(0.817430\pi\)
\(578\) −5641.85 −0.406004
\(579\) 7603.29 0.545738
\(580\) −2596.01 −0.185851
\(581\) −421.799 −0.0301190
\(582\) 8824.07 0.628470
\(583\) 19061.2 1.35409
\(584\) 6752.31 0.478446
\(585\) −7198.13 −0.508728
\(586\) −10416.4 −0.734296
\(587\) −18491.4 −1.30021 −0.650103 0.759846i \(-0.725274\pi\)
−0.650103 + 0.759846i \(0.725274\pi\)
\(588\) 4114.93 0.288600
\(589\) 0 0
\(590\) −25565.3 −1.78391
\(591\) 12347.8 0.859425
\(592\) −2462.97 −0.170992
\(593\) −4642.52 −0.321493 −0.160747 0.986996i \(-0.551390\pi\)
−0.160747 + 0.986996i \(0.551390\pi\)
\(594\) 2900.69 0.200365
\(595\) −456.324 −0.0314411
\(596\) 10263.3 0.705369
\(597\) −4593.99 −0.314941
\(598\) −5984.67 −0.409250
\(599\) 13658.6 0.931675 0.465838 0.884870i \(-0.345753\pi\)
0.465838 + 0.884870i \(0.345753\pi\)
\(600\) 4275.58 0.290917
\(601\) −9070.65 −0.615640 −0.307820 0.951445i \(-0.599599\pi\)
−0.307820 + 0.951445i \(0.599599\pi\)
\(602\) −148.223 −0.0100351
\(603\) 2501.20 0.168916
\(604\) 187.530 0.0126333
\(605\) 27064.9 1.81875
\(606\) −4066.78 −0.272610
\(607\) 9500.94 0.635307 0.317653 0.948207i \(-0.397105\pi\)
0.317653 + 0.948207i \(0.397105\pi\)
\(608\) 0 0
\(609\) −33.3261 −0.00221747
\(610\) 5995.24 0.397935
\(611\) −17093.5 −1.13180
\(612\) 3165.94 0.209110
\(613\) 12685.1 0.835802 0.417901 0.908493i \(-0.362766\pi\)
0.417901 + 0.908493i \(0.362766\pi\)
\(614\) −959.130 −0.0630413
\(615\) −8937.50 −0.586008
\(616\) 128.068 0.00837666
\(617\) −17506.1 −1.14225 −0.571125 0.820863i \(-0.693493\pi\)
−0.571125 + 0.820863i \(0.693493\pi\)
\(618\) −9879.89 −0.643087
\(619\) −11070.4 −0.718834 −0.359417 0.933177i \(-0.617024\pi\)
−0.359417 + 0.933177i \(0.617024\pi\)
\(620\) 12270.8 0.794847
\(621\) 1758.84 0.113655
\(622\) −3284.45 −0.211727
\(623\) −396.074 −0.0254709
\(624\) 2204.91 0.141453
\(625\) −6156.50 −0.394016
\(626\) −16037.4 −1.02393
\(627\) 0 0
\(628\) 11553.3 0.734122
\(629\) −13537.5 −0.858149
\(630\) 93.3998 0.00590657
\(631\) −14996.3 −0.946108 −0.473054 0.881033i \(-0.656848\pi\)
−0.473054 + 0.881033i \(0.656848\pi\)
\(632\) 2667.06 0.167864
\(633\) −4295.70 −0.269730
\(634\) −6142.19 −0.384760
\(635\) −42462.1 −2.65363
\(636\) −4258.18 −0.265484
\(637\) 15751.8 0.979763
\(638\) 4004.56 0.248499
\(639\) 7817.71 0.483981
\(640\) −2228.63 −0.137647
\(641\) 5497.70 0.338762 0.169381 0.985551i \(-0.445823\pi\)
0.169381 + 0.985551i \(0.445823\pi\)
\(642\) −11765.3 −0.723268
\(643\) 4295.23 0.263433 0.131716 0.991287i \(-0.457951\pi\)
0.131716 + 0.991287i \(0.457951\pi\)
\(644\) 77.6545 0.00475158
\(645\) 12989.4 0.792958
\(646\) 0 0
\(647\) −8674.61 −0.527101 −0.263550 0.964646i \(-0.584893\pi\)
−0.263550 + 0.964646i \(0.584893\pi\)
\(648\) −648.000 −0.0392837
\(649\) 39436.6 2.38524
\(650\) 16366.8 0.987626
\(651\) 157.525 0.00948370
\(652\) −734.275 −0.0441050
\(653\) 4186.04 0.250861 0.125430 0.992102i \(-0.459969\pi\)
0.125430 + 0.992102i \(0.459969\pi\)
\(654\) −2113.70 −0.126379
\(655\) −26698.4 −1.59266
\(656\) 2737.70 0.162941
\(657\) −7596.35 −0.451083
\(658\) 221.798 0.0131407
\(659\) −701.546 −0.0414694 −0.0207347 0.999785i \(-0.506601\pi\)
−0.0207347 + 0.999785i \(0.506601\pi\)
\(660\) −11223.2 −0.661913
\(661\) 30247.4 1.77986 0.889931 0.456095i \(-0.150752\pi\)
0.889931 + 0.456095i \(0.150752\pi\)
\(662\) 4344.52 0.255067
\(663\) 12119.1 0.709904
\(664\) −11322.7 −0.661756
\(665\) 0 0
\(666\) 2770.84 0.161213
\(667\) 2428.17 0.140958
\(668\) −14090.3 −0.816125
\(669\) −9414.95 −0.544100
\(670\) −9677.51 −0.558022
\(671\) −9248.16 −0.532074
\(672\) −28.6099 −0.00164234
\(673\) −12697.3 −0.727259 −0.363629 0.931544i \(-0.618463\pi\)
−0.363629 + 0.931544i \(0.618463\pi\)
\(674\) 19401.6 1.10879
\(675\) −4810.03 −0.274279
\(676\) −347.709 −0.0197832
\(677\) 17211.1 0.977071 0.488535 0.872544i \(-0.337531\pi\)
0.488535 + 0.872544i \(0.337531\pi\)
\(678\) 13296.7 0.753179
\(679\) −438.291 −0.0247718
\(680\) −12249.5 −0.690804
\(681\) 11663.0 0.656279
\(682\) −18928.7 −1.06278
\(683\) −3009.45 −0.168599 −0.0842996 0.996440i \(-0.526865\pi\)
−0.0842996 + 0.996440i \(0.526865\pi\)
\(684\) 0 0
\(685\) −18942.4 −1.05657
\(686\) −408.830 −0.0227539
\(687\) 1375.87 0.0764086
\(688\) −3978.87 −0.220484
\(689\) −16300.2 −0.901287
\(690\) −6805.20 −0.375463
\(691\) −15027.3 −0.827303 −0.413652 0.910435i \(-0.635747\pi\)
−0.413652 + 0.910435i \(0.635747\pi\)
\(692\) 12571.3 0.690592
\(693\) −144.077 −0.00789759
\(694\) 967.458 0.0529167
\(695\) −859.623 −0.0469170
\(696\) −894.600 −0.0487209
\(697\) 15047.6 0.817744
\(698\) −23895.3 −1.29577
\(699\) −13055.6 −0.706448
\(700\) −212.368 −0.0114668
\(701\) 6827.82 0.367879 0.183940 0.982938i \(-0.441115\pi\)
0.183940 + 0.982938i \(0.441115\pi\)
\(702\) −2480.52 −0.133363
\(703\) 0 0
\(704\) 3437.85 0.184047
\(705\) −19437.1 −1.03836
\(706\) −15084.0 −0.804101
\(707\) 201.997 0.0107452
\(708\) −8809.96 −0.467653
\(709\) 19077.9 1.01056 0.505280 0.862956i \(-0.331389\pi\)
0.505280 + 0.862956i \(0.331389\pi\)
\(710\) −30247.9 −1.59885
\(711\) −3000.45 −0.158264
\(712\) −10632.2 −0.559630
\(713\) −11477.4 −0.602851
\(714\) −157.252 −0.00824231
\(715\) −42962.0 −2.24711
\(716\) −1301.50 −0.0679318
\(717\) 206.936 0.0107785
\(718\) −16590.4 −0.862326
\(719\) −26690.8 −1.38442 −0.692210 0.721696i \(-0.743363\pi\)
−0.692210 + 0.721696i \(0.743363\pi\)
\(720\) 2507.21 0.129775
\(721\) 490.734 0.0253480
\(722\) 0 0
\(723\) 119.016 0.00612208
\(724\) 10557.5 0.541942
\(725\) −6640.52 −0.340169
\(726\) 9326.73 0.476787
\(727\) 24306.5 1.24000 0.619998 0.784604i \(-0.287134\pi\)
0.619998 + 0.784604i \(0.287134\pi\)
\(728\) −109.518 −0.00557554
\(729\) 729.000 0.0370370
\(730\) 29391.4 1.49017
\(731\) −21869.6 −1.10653
\(732\) 2066.00 0.104319
\(733\) 27102.2 1.36568 0.682841 0.730567i \(-0.260744\pi\)
0.682841 + 0.730567i \(0.260744\pi\)
\(734\) 5997.34 0.301588
\(735\) 17911.5 0.898877
\(736\) 2084.55 0.104399
\(737\) 14928.4 0.746124
\(738\) −3079.92 −0.153622
\(739\) 14638.0 0.728643 0.364321 0.931273i \(-0.381301\pi\)
0.364321 + 0.931273i \(0.381301\pi\)
\(740\) −10720.8 −0.532573
\(741\) 0 0
\(742\) 211.504 0.0104644
\(743\) 19681.2 0.971782 0.485891 0.874020i \(-0.338495\pi\)
0.485891 + 0.874020i \(0.338495\pi\)
\(744\) 4228.58 0.208370
\(745\) 44673.9 2.19695
\(746\) 13195.5 0.647615
\(747\) 12738.0 0.623910
\(748\) 18895.9 0.923666
\(749\) 584.381 0.0285084
\(750\) 5552.35 0.270324
\(751\) 7288.23 0.354129 0.177065 0.984199i \(-0.443340\pi\)
0.177065 + 0.984199i \(0.443340\pi\)
\(752\) 5953.90 0.288719
\(753\) 17650.0 0.854187
\(754\) −3424.49 −0.165402
\(755\) 816.280 0.0393477
\(756\) 32.1861 0.00154841
\(757\) −25114.6 −1.20582 −0.602911 0.797808i \(-0.705993\pi\)
−0.602911 + 0.797808i \(0.705993\pi\)
\(758\) 12567.1 0.602185
\(759\) 10497.6 0.502027
\(760\) 0 0
\(761\) 36270.2 1.72772 0.863859 0.503734i \(-0.168041\pi\)
0.863859 + 0.503734i \(0.168041\pi\)
\(762\) −14632.7 −0.695652
\(763\) 104.987 0.00498138
\(764\) −15207.8 −0.720154
\(765\) 13780.7 0.651296
\(766\) 6196.99 0.292306
\(767\) −33724.2 −1.58763
\(768\) −768.000 −0.0360844
\(769\) −11999.4 −0.562690 −0.281345 0.959607i \(-0.590781\pi\)
−0.281345 + 0.959607i \(0.590781\pi\)
\(770\) 557.456 0.0260900
\(771\) 9230.10 0.431146
\(772\) −10137.7 −0.472623
\(773\) −27176.5 −1.26451 −0.632257 0.774759i \(-0.717871\pi\)
−0.632257 + 0.774759i \(0.717871\pi\)
\(774\) 4476.23 0.207874
\(775\) 31388.3 1.45484
\(776\) −11765.4 −0.544271
\(777\) −137.627 −0.00635439
\(778\) −20609.1 −0.949709
\(779\) 0 0
\(780\) 9597.50 0.440571
\(781\) 46659.9 2.13780
\(782\) 11457.5 0.523940
\(783\) 1006.43 0.0459345
\(784\) −5486.58 −0.249935
\(785\) 50289.4 2.28650
\(786\) −9200.42 −0.417517
\(787\) 549.906 0.0249073 0.0124536 0.999922i \(-0.496036\pi\)
0.0124536 + 0.999922i \(0.496036\pi\)
\(788\) −16463.7 −0.744284
\(789\) 4832.80 0.218064
\(790\) 11609.2 0.522830
\(791\) −660.445 −0.0296874
\(792\) −3867.58 −0.173521
\(793\) 7908.55 0.354150
\(794\) −21218.9 −0.948399
\(795\) −18535.0 −0.826879
\(796\) 6125.32 0.272747
\(797\) 30526.9 1.35674 0.678368 0.734722i \(-0.262687\pi\)
0.678368 + 0.734722i \(0.262687\pi\)
\(798\) 0 0
\(799\) 32725.2 1.44898
\(800\) −5700.78 −0.251941
\(801\) 11961.2 0.527625
\(802\) 27684.5 1.21892
\(803\) −45338.7 −1.99249
\(804\) −3334.93 −0.146286
\(805\) 338.014 0.0147993
\(806\) 16186.8 0.707390
\(807\) −17677.3 −0.771091
\(808\) 5422.37 0.236087
\(809\) 37891.7 1.64673 0.823364 0.567514i \(-0.192095\pi\)
0.823364 + 0.567514i \(0.192095\pi\)
\(810\) −2820.61 −0.122353
\(811\) −2925.25 −0.126658 −0.0633289 0.997993i \(-0.520172\pi\)
−0.0633289 + 0.997993i \(0.520172\pi\)
\(812\) 44.4348 0.00192039
\(813\) −7290.90 −0.314518
\(814\) 16537.7 0.712097
\(815\) −3196.15 −0.137370
\(816\) −4221.25 −0.181095
\(817\) 0 0
\(818\) −6322.57 −0.270249
\(819\) 123.207 0.00525667
\(820\) 11916.7 0.507497
\(821\) −5014.51 −0.213164 −0.106582 0.994304i \(-0.533991\pi\)
−0.106582 + 0.994304i \(0.533991\pi\)
\(822\) −6527.65 −0.276980
\(823\) −6409.06 −0.271453 −0.135726 0.990746i \(-0.543337\pi\)
−0.135726 + 0.990746i \(0.543337\pi\)
\(824\) 13173.2 0.556929
\(825\) −28708.6 −1.21152
\(826\) 437.590 0.0184331
\(827\) −41427.8 −1.74194 −0.870970 0.491336i \(-0.836509\pi\)
−0.870970 + 0.491336i \(0.836509\pi\)
\(828\) −2345.11 −0.0984279
\(829\) −35010.4 −1.46678 −0.733390 0.679808i \(-0.762063\pi\)
−0.733390 + 0.679808i \(0.762063\pi\)
\(830\) −49285.4 −2.06111
\(831\) −4719.57 −0.197016
\(832\) −2939.87 −0.122502
\(833\) −30156.6 −1.25434
\(834\) −296.231 −0.0122993
\(835\) −61332.3 −2.54191
\(836\) 0 0
\(837\) −4757.15 −0.196453
\(838\) −15406.0 −0.635075
\(839\) 2489.62 0.102445 0.0512223 0.998687i \(-0.483688\pi\)
0.0512223 + 0.998687i \(0.483688\pi\)
\(840\) −124.533 −0.00511524
\(841\) −22999.6 −0.943031
\(842\) −9468.69 −0.387545
\(843\) −9751.06 −0.398392
\(844\) 5727.60 0.233593
\(845\) −1513.51 −0.0616169
\(846\) −6698.14 −0.272207
\(847\) −463.258 −0.0187931
\(848\) 5677.58 0.229916
\(849\) −3851.15 −0.155679
\(850\) −31333.9 −1.26440
\(851\) 10027.7 0.403930
\(852\) −10423.6 −0.419140
\(853\) 619.844 0.0248805 0.0124402 0.999923i \(-0.496040\pi\)
0.0124402 + 0.999923i \(0.496040\pi\)
\(854\) −102.618 −0.00411184
\(855\) 0 0
\(856\) 15687.0 0.626369
\(857\) 40352.1 1.60840 0.804201 0.594357i \(-0.202594\pi\)
0.804201 + 0.594357i \(0.202594\pi\)
\(858\) −14805.0 −0.589082
\(859\) −32744.2 −1.30060 −0.650302 0.759676i \(-0.725358\pi\)
−0.650302 + 0.759676i \(0.725358\pi\)
\(860\) −17319.2 −0.686721
\(861\) 152.979 0.00605519
\(862\) −19105.5 −0.754916
\(863\) −14658.3 −0.578186 −0.289093 0.957301i \(-0.593354\pi\)
−0.289093 + 0.957301i \(0.593354\pi\)
\(864\) 864.000 0.0340207
\(865\) 54720.3 2.15092
\(866\) −31658.7 −1.24227
\(867\) −8462.78 −0.331501
\(868\) −210.033 −0.00821313
\(869\) −17908.1 −0.699070
\(870\) −3894.01 −0.151746
\(871\) −12766.0 −0.496623
\(872\) 2818.26 0.109448
\(873\) 13236.1 0.513144
\(874\) 0 0
\(875\) −275.785 −0.0106551
\(876\) 10128.5 0.390650
\(877\) −13235.2 −0.509604 −0.254802 0.966993i \(-0.582010\pi\)
−0.254802 + 0.966993i \(0.582010\pi\)
\(878\) −24325.9 −0.935033
\(879\) −15624.6 −0.599550
\(880\) 14964.3 0.573233
\(881\) 4433.78 0.169555 0.0847775 0.996400i \(-0.472982\pi\)
0.0847775 + 0.996400i \(0.472982\pi\)
\(882\) 6172.40 0.235641
\(883\) 41092.0 1.56609 0.783044 0.621967i \(-0.213666\pi\)
0.783044 + 0.621967i \(0.213666\pi\)
\(884\) −16158.8 −0.614795
\(885\) −38347.9 −1.45656
\(886\) −21476.2 −0.814340
\(887\) 19295.4 0.730414 0.365207 0.930926i \(-0.380998\pi\)
0.365207 + 0.930926i \(0.380998\pi\)
\(888\) −3694.45 −0.139615
\(889\) 726.805 0.0274199
\(890\) −46279.6 −1.74303
\(891\) 4351.03 0.163597
\(892\) 12553.3 0.471205
\(893\) 0 0
\(894\) 15394.9 0.575931
\(895\) −5665.15 −0.211581
\(896\) 38.1465 0.00142231
\(897\) −8977.00 −0.334151
\(898\) −23804.5 −0.884594
\(899\) −6567.51 −0.243647
\(900\) 6413.37 0.237532
\(901\) 31206.4 1.15387
\(902\) −18382.5 −0.678569
\(903\) −222.334 −0.00819360
\(904\) −17728.9 −0.652272
\(905\) 45954.6 1.68794
\(906\) 281.295 0.0103150
\(907\) 32079.8 1.17441 0.587207 0.809437i \(-0.300227\pi\)
0.587207 + 0.809437i \(0.300227\pi\)
\(908\) −15550.6 −0.568355
\(909\) −6100.16 −0.222585
\(910\) −476.708 −0.0173656
\(911\) 2449.21 0.0890735 0.0445367 0.999008i \(-0.485819\pi\)
0.0445367 + 0.999008i \(0.485819\pi\)
\(912\) 0 0
\(913\) 76026.9 2.75589
\(914\) 29961.6 1.08429
\(915\) 8992.86 0.324912
\(916\) −1834.49 −0.0661718
\(917\) 456.985 0.0164569
\(918\) 4748.91 0.170738
\(919\) 26639.3 0.956201 0.478100 0.878305i \(-0.341326\pi\)
0.478100 + 0.878305i \(0.341326\pi\)
\(920\) 9073.60 0.325161
\(921\) −1438.69 −0.0514730
\(922\) −2306.64 −0.0823918
\(923\) −39901.2 −1.42293
\(924\) 192.103 0.00683952
\(925\) −27423.5 −0.974788
\(926\) −19560.3 −0.694161
\(927\) −14819.8 −0.525078
\(928\) 1192.80 0.0421935
\(929\) 4232.31 0.149470 0.0747350 0.997203i \(-0.476189\pi\)
0.0747350 + 0.997203i \(0.476189\pi\)
\(930\) 18406.1 0.648990
\(931\) 0 0
\(932\) 17407.4 0.611802
\(933\) −4926.68 −0.172875
\(934\) −28600.3 −1.00196
\(935\) 82249.9 2.87686
\(936\) 3307.36 0.115496
\(937\) −3428.88 −0.119548 −0.0597740 0.998212i \(-0.519038\pi\)
−0.0597740 + 0.998212i \(0.519038\pi\)
\(938\) 165.646 0.00576602
\(939\) −24056.0 −0.836037
\(940\) 25916.1 0.899246
\(941\) −22357.8 −0.774541 −0.387270 0.921966i \(-0.626582\pi\)
−0.387270 + 0.921966i \(0.626582\pi\)
\(942\) 17330.0 0.599408
\(943\) −11146.2 −0.384911
\(944\) 11746.6 0.405000
\(945\) 140.100 0.00482269
\(946\) 26716.3 0.918207
\(947\) 18852.7 0.646918 0.323459 0.946242i \(-0.395154\pi\)
0.323459 + 0.946242i \(0.395154\pi\)
\(948\) 4000.59 0.137060
\(949\) 38771.4 1.32621
\(950\) 0 0
\(951\) −9213.29 −0.314155
\(952\) 209.669 0.00713805
\(953\) 49446.0 1.68071 0.840353 0.542040i \(-0.182348\pi\)
0.840353 + 0.542040i \(0.182348\pi\)
\(954\) −6387.27 −0.216767
\(955\) −66196.3 −2.24300
\(956\) −275.915 −0.00933445
\(957\) 6006.84 0.202898
\(958\) 3521.00 0.118746
\(959\) 324.228 0.0109175
\(960\) −3342.95 −0.112389
\(961\) 1252.18 0.0420321
\(962\) −14142.2 −0.473974
\(963\) −17647.9 −0.590546
\(964\) −158.688 −0.00530188
\(965\) −44127.4 −1.47203
\(966\) 116.482 0.00387965
\(967\) 10689.7 0.355488 0.177744 0.984077i \(-0.443120\pi\)
0.177744 + 0.984077i \(0.443120\pi\)
\(968\) −12435.6 −0.412910
\(969\) 0 0
\(970\) −51212.5 −1.69519
\(971\) −43553.1 −1.43943 −0.719715 0.694270i \(-0.755727\pi\)
−0.719715 + 0.694270i \(0.755727\pi\)
\(972\) −972.000 −0.0320750
\(973\) 14.7138 0.000484792 0
\(974\) 16627.2 0.546992
\(975\) 24550.1 0.806394
\(976\) −2754.66 −0.0903428
\(977\) −27972.4 −0.915983 −0.457991 0.888957i \(-0.651431\pi\)
−0.457991 + 0.888957i \(0.651431\pi\)
\(978\) −1101.41 −0.0360115
\(979\) 71390.2 2.33058
\(980\) −23882.0 −0.778450
\(981\) −3170.54 −0.103188
\(982\) −10145.6 −0.329695
\(983\) −47151.0 −1.52989 −0.764945 0.644096i \(-0.777234\pi\)
−0.764945 + 0.644096i \(0.777234\pi\)
\(984\) 4106.56 0.133041
\(985\) −71663.2 −2.31815
\(986\) 6556.13 0.211754
\(987\) 332.697 0.0107293
\(988\) 0 0
\(989\) 16199.5 0.520843
\(990\) −16834.8 −0.540449
\(991\) 23895.1 0.765945 0.382972 0.923760i \(-0.374900\pi\)
0.382972 + 0.923760i \(0.374900\pi\)
\(992\) −5638.10 −0.180454
\(993\) 6516.77 0.208261
\(994\) 517.740 0.0165208
\(995\) 26662.3 0.849499
\(996\) −16984.1 −0.540322
\(997\) −38584.2 −1.22565 −0.612826 0.790218i \(-0.709967\pi\)
−0.612826 + 0.790218i \(0.709967\pi\)
\(998\) −15787.5 −0.500745
\(999\) 4156.26 0.131630
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 2166.4.a.r.1.3 3
19.18 odd 2 2166.4.a.v.1.3 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2166.4.a.r.1.3 3 1.1 even 1 trivial
2166.4.a.v.1.3 yes 3 19.18 odd 2