Properties

Label 507.4.a.r.1.5
Level $507$
Weight $4$
Character 507.1
Self dual yes
Analytic conductor $29.914$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [507,4,Mod(1,507)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(507, 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("507.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 507.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: \(29.9139683729\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 70x^{8} + 1645x^{6} - 14700x^{4} + 44100x^{2} - 27648 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-0.917374\) of defining polynomial
Character \(\chi\) \(=\) 507.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.917374 q^{2} +3.00000 q^{3} -7.15843 q^{4} +15.4704 q^{5} -2.75212 q^{6} +20.5833 q^{7} +13.9059 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-0.917374 q^{2} +3.00000 q^{3} -7.15843 q^{4} +15.4704 q^{5} -2.75212 q^{6} +20.5833 q^{7} +13.9059 q^{8} +9.00000 q^{9} -14.1922 q^{10} +65.8420 q^{11} -21.4753 q^{12} -18.8826 q^{14} +46.4112 q^{15} +44.5105 q^{16} +44.2956 q^{17} -8.25636 q^{18} -147.053 q^{19} -110.744 q^{20} +61.7500 q^{21} -60.4017 q^{22} +53.1586 q^{23} +41.7178 q^{24} +114.334 q^{25} +27.0000 q^{27} -147.344 q^{28} -38.6257 q^{29} -42.5765 q^{30} -88.3894 q^{31} -152.080 q^{32} +197.526 q^{33} -40.6357 q^{34} +318.433 q^{35} -64.4258 q^{36} -78.9587 q^{37} +134.903 q^{38} +215.131 q^{40} +354.966 q^{41} -56.6478 q^{42} -407.846 q^{43} -471.325 q^{44} +139.234 q^{45} -48.7663 q^{46} +67.9674 q^{47} +133.531 q^{48} +80.6738 q^{49} -104.887 q^{50} +132.887 q^{51} +226.572 q^{53} -24.7691 q^{54} +1018.60 q^{55} +286.231 q^{56} -441.160 q^{57} +35.4342 q^{58} -142.031 q^{59} -332.231 q^{60} +266.831 q^{61} +81.0862 q^{62} +185.250 q^{63} -216.569 q^{64} -181.205 q^{66} +411.187 q^{67} -317.087 q^{68} +159.476 q^{69} -292.122 q^{70} +91.5052 q^{71} +125.153 q^{72} -63.1328 q^{73} +72.4347 q^{74} +343.001 q^{75} +1052.67 q^{76} +1355.25 q^{77} -287.115 q^{79} +688.595 q^{80} +81.0000 q^{81} -325.637 q^{82} +373.812 q^{83} -442.033 q^{84} +685.272 q^{85} +374.147 q^{86} -115.877 q^{87} +915.595 q^{88} -119.403 q^{89} -127.729 q^{90} -380.532 q^{92} -265.168 q^{93} -62.3515 q^{94} -2274.97 q^{95} -456.241 q^{96} -554.650 q^{97} -74.0080 q^{98} +592.578 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 30 q^{3} + 60 q^{4} + 90 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 30 q^{3} + 60 q^{4} + 90 q^{9} + 80 q^{10} + 180 q^{12} - 60 q^{14} + 500 q^{16} + 210 q^{17} + 580 q^{22} - 120 q^{23} + 960 q^{25} + 270 q^{27} + 990 q^{29} + 240 q^{30} - 120 q^{35} + 540 q^{36} + 1380 q^{38} + 2000 q^{40} - 180 q^{42} - 740 q^{43} + 1500 q^{48} + 1550 q^{49} + 630 q^{51} + 330 q^{53} + 520 q^{55} - 5340 q^{56} + 2750 q^{61} - 1560 q^{62} + 3140 q^{64} + 1740 q^{66} + 1200 q^{68} - 360 q^{69} - 4380 q^{74} + 2880 q^{75} + 4320 q^{77} + 1100 q^{79} + 810 q^{81} - 4780 q^{82} + 2970 q^{87} + 6340 q^{88} + 720 q^{90} - 1740 q^{92} + 6460 q^{94} - 2760 q^{95}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.917374 −0.324341 −0.162170 0.986763i \(-0.551849\pi\)
−0.162170 + 0.986763i \(0.551849\pi\)
\(3\) 3.00000 0.577350
\(4\) −7.15843 −0.894803
\(5\) 15.4704 1.38372 0.691858 0.722034i \(-0.256792\pi\)
0.691858 + 0.722034i \(0.256792\pi\)
\(6\) −2.75212 −0.187258
\(7\) 20.5833 1.11140 0.555698 0.831384i \(-0.312451\pi\)
0.555698 + 0.831384i \(0.312451\pi\)
\(8\) 13.9059 0.614562
\(9\) 9.00000 0.333333
\(10\) −14.1922 −0.448795
\(11\) 65.8420 1.80474 0.902369 0.430964i \(-0.141827\pi\)
0.902369 + 0.430964i \(0.141827\pi\)
\(12\) −21.4753 −0.516615
\(13\) 0 0
\(14\) −18.8826 −0.360471
\(15\) 46.4112 0.798889
\(16\) 44.5105 0.695476
\(17\) 44.2956 0.631957 0.315979 0.948766i \(-0.397667\pi\)
0.315979 + 0.948766i \(0.397667\pi\)
\(18\) −8.25636 −0.108114
\(19\) −147.053 −1.77560 −0.887798 0.460234i \(-0.847766\pi\)
−0.887798 + 0.460234i \(0.847766\pi\)
\(20\) −110.744 −1.23815
\(21\) 61.7500 0.641665
\(22\) −60.4017 −0.585350
\(23\) 53.1586 0.481928 0.240964 0.970534i \(-0.422536\pi\)
0.240964 + 0.970534i \(0.422536\pi\)
\(24\) 41.7178 0.354817
\(25\) 114.334 0.914669
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) −147.344 −0.994480
\(29\) −38.6257 −0.247331 −0.123666 0.992324i \(-0.539465\pi\)
−0.123666 + 0.992324i \(0.539465\pi\)
\(30\) −42.5765 −0.259112
\(31\) −88.3894 −0.512104 −0.256052 0.966663i \(-0.582422\pi\)
−0.256052 + 0.966663i \(0.582422\pi\)
\(32\) −152.080 −0.840133
\(33\) 197.526 1.04197
\(34\) −40.6357 −0.204969
\(35\) 318.433 1.53786
\(36\) −64.4258 −0.298268
\(37\) −78.9587 −0.350831 −0.175415 0.984495i \(-0.556127\pi\)
−0.175415 + 0.984495i \(0.556127\pi\)
\(38\) 134.903 0.575898
\(39\) 0 0
\(40\) 215.131 0.850379
\(41\) 354.966 1.35211 0.676054 0.736852i \(-0.263689\pi\)
0.676054 + 0.736852i \(0.263689\pi\)
\(42\) −56.6478 −0.208118
\(43\) −407.846 −1.44642 −0.723208 0.690630i \(-0.757333\pi\)
−0.723208 + 0.690630i \(0.757333\pi\)
\(44\) −471.325 −1.61489
\(45\) 139.234 0.461239
\(46\) −48.7663 −0.156309
\(47\) 67.9674 0.210938 0.105469 0.994423i \(-0.466366\pi\)
0.105469 + 0.994423i \(0.466366\pi\)
\(48\) 133.531 0.401533
\(49\) 80.6738 0.235201
\(50\) −104.887 −0.296664
\(51\) 132.887 0.364861
\(52\) 0 0
\(53\) 226.572 0.587209 0.293604 0.955927i \(-0.405145\pi\)
0.293604 + 0.955927i \(0.405145\pi\)
\(54\) −24.7691 −0.0624194
\(55\) 1018.60 2.49724
\(56\) 286.231 0.683021
\(57\) −441.160 −1.02514
\(58\) 35.4342 0.0802196
\(59\) −142.031 −0.313404 −0.156702 0.987646i \(-0.550086\pi\)
−0.156702 + 0.987646i \(0.550086\pi\)
\(60\) −332.231 −0.714848
\(61\) 266.831 0.560069 0.280035 0.959990i \(-0.409654\pi\)
0.280035 + 0.959990i \(0.409654\pi\)
\(62\) 81.0862 0.166096
\(63\) 185.250 0.370465
\(64\) −216.569 −0.422987
\(65\) 0 0
\(66\) −181.205 −0.337952
\(67\) 411.187 0.749768 0.374884 0.927072i \(-0.377683\pi\)
0.374884 + 0.927072i \(0.377683\pi\)
\(68\) −317.087 −0.565477
\(69\) 159.476 0.278241
\(70\) −292.122 −0.498789
\(71\) 91.5052 0.152953 0.0764765 0.997071i \(-0.475633\pi\)
0.0764765 + 0.997071i \(0.475633\pi\)
\(72\) 125.153 0.204854
\(73\) −63.1328 −0.101221 −0.0506105 0.998718i \(-0.516117\pi\)
−0.0506105 + 0.998718i \(0.516117\pi\)
\(74\) 72.4347 0.113789
\(75\) 343.001 0.528084
\(76\) 1052.67 1.58881
\(77\) 1355.25 2.00578
\(78\) 0 0
\(79\) −287.115 −0.408899 −0.204449 0.978877i \(-0.565540\pi\)
−0.204449 + 0.978877i \(0.565540\pi\)
\(80\) 688.595 0.962341
\(81\) 81.0000 0.111111
\(82\) −325.637 −0.438543
\(83\) 373.812 0.494352 0.247176 0.968971i \(-0.420497\pi\)
0.247176 + 0.968971i \(0.420497\pi\)
\(84\) −442.033 −0.574164
\(85\) 685.272 0.874449
\(86\) 374.147 0.469132
\(87\) −115.877 −0.142797
\(88\) 915.595 1.10912
\(89\) −119.403 −0.142209 −0.0711047 0.997469i \(-0.522652\pi\)
−0.0711047 + 0.997469i \(0.522652\pi\)
\(90\) −127.729 −0.149598
\(91\) 0 0
\(92\) −380.532 −0.431231
\(93\) −265.168 −0.295663
\(94\) −62.3515 −0.0684157
\(95\) −2274.97 −2.45692
\(96\) −456.241 −0.485051
\(97\) −554.650 −0.580579 −0.290290 0.956939i \(-0.593752\pi\)
−0.290290 + 0.956939i \(0.593752\pi\)
\(98\) −74.0080 −0.0762851
\(99\) 592.578 0.601579
\(100\) −818.449 −0.818449
\(101\) −462.565 −0.455712 −0.227856 0.973695i \(-0.573172\pi\)
−0.227856 + 0.973695i \(0.573172\pi\)
\(102\) −121.907 −0.118339
\(103\) 1122.07 1.07341 0.536704 0.843771i \(-0.319669\pi\)
0.536704 + 0.843771i \(0.319669\pi\)
\(104\) 0 0
\(105\) 955.298 0.887881
\(106\) −207.851 −0.190456
\(107\) 602.756 0.544585 0.272293 0.962214i \(-0.412218\pi\)
0.272293 + 0.962214i \(0.412218\pi\)
\(108\) −193.277 −0.172205
\(109\) −1421.89 −1.24947 −0.624735 0.780837i \(-0.714793\pi\)
−0.624735 + 0.780837i \(0.714793\pi\)
\(110\) −934.440 −0.809958
\(111\) −236.876 −0.202552
\(112\) 916.174 0.772949
\(113\) −396.719 −0.330267 −0.165134 0.986271i \(-0.552806\pi\)
−0.165134 + 0.986271i \(0.552806\pi\)
\(114\) 404.708 0.332495
\(115\) 822.386 0.666851
\(116\) 276.499 0.221313
\(117\) 0 0
\(118\) 130.295 0.101650
\(119\) 911.752 0.702354
\(120\) 645.392 0.490966
\(121\) 3004.17 2.25708
\(122\) −244.784 −0.181653
\(123\) 1064.90 0.780639
\(124\) 632.729 0.458232
\(125\) −165.013 −0.118074
\(126\) −169.944 −0.120157
\(127\) 437.868 0.305941 0.152970 0.988231i \(-0.451116\pi\)
0.152970 + 0.988231i \(0.451116\pi\)
\(128\) 1415.32 0.977324
\(129\) −1223.54 −0.835089
\(130\) 0 0
\(131\) 1657.44 1.10543 0.552715 0.833370i \(-0.313592\pi\)
0.552715 + 0.833370i \(0.313592\pi\)
\(132\) −1413.98 −0.932354
\(133\) −3026.85 −1.97339
\(134\) −377.212 −0.243180
\(135\) 417.701 0.266296
\(136\) 615.973 0.388377
\(137\) −1967.42 −1.22692 −0.613460 0.789725i \(-0.710223\pi\)
−0.613460 + 0.789725i \(0.710223\pi\)
\(138\) −146.299 −0.0902449
\(139\) 2825.13 1.72392 0.861960 0.506977i \(-0.169237\pi\)
0.861960 + 0.506977i \(0.169237\pi\)
\(140\) −2279.48 −1.37608
\(141\) 203.902 0.121785
\(142\) −83.9445 −0.0496089
\(143\) 0 0
\(144\) 400.594 0.231825
\(145\) −597.555 −0.342236
\(146\) 57.9164 0.0328301
\(147\) 242.021 0.135793
\(148\) 565.220 0.313924
\(149\) −797.658 −0.438569 −0.219284 0.975661i \(-0.570372\pi\)
−0.219284 + 0.975661i \(0.570372\pi\)
\(150\) −314.660 −0.171279
\(151\) 161.987 0.0873003 0.0436501 0.999047i \(-0.486101\pi\)
0.0436501 + 0.999047i \(0.486101\pi\)
\(152\) −2044.91 −1.09121
\(153\) 398.661 0.210652
\(154\) −1243.27 −0.650555
\(155\) −1367.42 −0.708606
\(156\) 0 0
\(157\) −342.000 −0.173851 −0.0869255 0.996215i \(-0.527704\pi\)
−0.0869255 + 0.996215i \(0.527704\pi\)
\(158\) 263.392 0.132622
\(159\) 679.716 0.339025
\(160\) −2352.74 −1.16250
\(161\) 1094.18 0.535613
\(162\) −74.3073 −0.0360378
\(163\) 556.597 0.267460 0.133730 0.991018i \(-0.457304\pi\)
0.133730 + 0.991018i \(0.457304\pi\)
\(164\) −2541.00 −1.20987
\(165\) 3055.81 1.44178
\(166\) −342.926 −0.160339
\(167\) −3136.42 −1.45332 −0.726658 0.686999i \(-0.758928\pi\)
−0.726658 + 0.686999i \(0.758928\pi\)
\(168\) 858.692 0.394342
\(169\) 0 0
\(170\) −628.650 −0.283619
\(171\) −1323.48 −0.591865
\(172\) 2919.53 1.29426
\(173\) 3324.19 1.46089 0.730444 0.682972i \(-0.239313\pi\)
0.730444 + 0.682972i \(0.239313\pi\)
\(174\) 106.303 0.0463148
\(175\) 2353.37 1.01656
\(176\) 2930.66 1.25515
\(177\) −426.092 −0.180944
\(178\) 109.537 0.0461243
\(179\) −3313.25 −1.38349 −0.691743 0.722144i \(-0.743157\pi\)
−0.691743 + 0.722144i \(0.743157\pi\)
\(180\) −996.694 −0.412718
\(181\) 76.0118 0.0312150 0.0156075 0.999878i \(-0.495032\pi\)
0.0156075 + 0.999878i \(0.495032\pi\)
\(182\) 0 0
\(183\) 800.494 0.323356
\(184\) 739.221 0.296174
\(185\) −1221.52 −0.485450
\(186\) 243.258 0.0958956
\(187\) 2916.51 1.14052
\(188\) −486.540 −0.188748
\(189\) 555.750 0.213888
\(190\) 2087.00 0.796879
\(191\) −4073.39 −1.54314 −0.771572 0.636142i \(-0.780529\pi\)
−0.771572 + 0.636142i \(0.780529\pi\)
\(192\) −649.708 −0.244211
\(193\) −867.581 −0.323574 −0.161787 0.986826i \(-0.551726\pi\)
−0.161787 + 0.986826i \(0.551726\pi\)
\(194\) 508.821 0.188305
\(195\) 0 0
\(196\) −577.497 −0.210458
\(197\) 1887.69 0.682703 0.341352 0.939936i \(-0.389115\pi\)
0.341352 + 0.939936i \(0.389115\pi\)
\(198\) −543.616 −0.195117
\(199\) 2786.22 0.992511 0.496256 0.868176i \(-0.334708\pi\)
0.496256 + 0.868176i \(0.334708\pi\)
\(200\) 1589.92 0.562120
\(201\) 1233.56 0.432879
\(202\) 424.345 0.147806
\(203\) −795.045 −0.274883
\(204\) −951.261 −0.326478
\(205\) 5491.47 1.87093
\(206\) −1029.36 −0.348150
\(207\) 478.428 0.160643
\(208\) 0 0
\(209\) −9682.28 −3.20448
\(210\) −876.365 −0.287976
\(211\) 708.789 0.231256 0.115628 0.993293i \(-0.463112\pi\)
0.115628 + 0.993293i \(0.463112\pi\)
\(212\) −1621.90 −0.525436
\(213\) 274.516 0.0883075
\(214\) −552.953 −0.176631
\(215\) −6309.54 −2.00143
\(216\) 375.460 0.118272
\(217\) −1819.35 −0.569150
\(218\) 1304.40 0.405254
\(219\) −189.398 −0.0584400
\(220\) −7291.59 −2.23454
\(221\) 0 0
\(222\) 217.304 0.0656959
\(223\) −5396.64 −1.62056 −0.810281 0.586041i \(-0.800686\pi\)
−0.810281 + 0.586041i \(0.800686\pi\)
\(224\) −3130.32 −0.933720
\(225\) 1029.00 0.304890
\(226\) 363.940 0.107119
\(227\) −3181.79 −0.930321 −0.465160 0.885226i \(-0.654003\pi\)
−0.465160 + 0.885226i \(0.654003\pi\)
\(228\) 3158.01 0.917299
\(229\) −2034.00 −0.586945 −0.293473 0.955967i \(-0.594811\pi\)
−0.293473 + 0.955967i \(0.594811\pi\)
\(230\) −754.435 −0.216287
\(231\) 4065.75 1.15804
\(232\) −537.126 −0.152000
\(233\) −2794.22 −0.785645 −0.392823 0.919614i \(-0.628501\pi\)
−0.392823 + 0.919614i \(0.628501\pi\)
\(234\) 0 0
\(235\) 1051.48 0.291878
\(236\) 1016.72 0.280435
\(237\) −861.346 −0.236078
\(238\) −836.417 −0.227802
\(239\) 5493.81 1.48688 0.743442 0.668800i \(-0.233192\pi\)
0.743442 + 0.668800i \(0.233192\pi\)
\(240\) 2065.79 0.555608
\(241\) −4061.27 −1.08552 −0.542758 0.839889i \(-0.682620\pi\)
−0.542758 + 0.839889i \(0.682620\pi\)
\(242\) −2755.95 −0.732062
\(243\) 243.000 0.0641500
\(244\) −1910.09 −0.501152
\(245\) 1248.06 0.325451
\(246\) −976.910 −0.253193
\(247\) 0 0
\(248\) −1229.14 −0.314719
\(249\) 1121.44 0.285414
\(250\) 151.379 0.0382961
\(251\) −3570.88 −0.897977 −0.448988 0.893538i \(-0.648216\pi\)
−0.448988 + 0.893538i \(0.648216\pi\)
\(252\) −1326.10 −0.331493
\(253\) 3500.07 0.869753
\(254\) −401.688 −0.0992290
\(255\) 2055.82 0.504863
\(256\) 434.179 0.106001
\(257\) −7518.26 −1.82481 −0.912405 0.409288i \(-0.865777\pi\)
−0.912405 + 0.409288i \(0.865777\pi\)
\(258\) 1122.44 0.270853
\(259\) −1625.23 −0.389912
\(260\) 0 0
\(261\) −347.631 −0.0824437
\(262\) −1520.49 −0.358536
\(263\) −4660.18 −1.09262 −0.546310 0.837583i \(-0.683968\pi\)
−0.546310 + 0.837583i \(0.683968\pi\)
\(264\) 2746.79 0.640352
\(265\) 3505.16 0.812530
\(266\) 2776.75 0.640050
\(267\) −358.208 −0.0821047
\(268\) −2943.45 −0.670895
\(269\) 5347.44 1.21204 0.606021 0.795449i \(-0.292765\pi\)
0.606021 + 0.795449i \(0.292765\pi\)
\(270\) −383.188 −0.0863707
\(271\) −2973.08 −0.666427 −0.333214 0.942851i \(-0.608133\pi\)
−0.333214 + 0.942851i \(0.608133\pi\)
\(272\) 1971.62 0.439511
\(273\) 0 0
\(274\) 1804.86 0.397940
\(275\) 7527.96 1.65074
\(276\) −1141.60 −0.248971
\(277\) −764.153 −0.165753 −0.0828764 0.996560i \(-0.526411\pi\)
−0.0828764 + 0.996560i \(0.526411\pi\)
\(278\) −2591.70 −0.559137
\(279\) −795.505 −0.170701
\(280\) 4428.11 0.945107
\(281\) 7040.34 1.49463 0.747316 0.664469i \(-0.231342\pi\)
0.747316 + 0.664469i \(0.231342\pi\)
\(282\) −187.055 −0.0394998
\(283\) −9035.17 −1.89783 −0.948913 0.315537i \(-0.897815\pi\)
−0.948913 + 0.315537i \(0.897815\pi\)
\(284\) −655.033 −0.136863
\(285\) −6824.92 −1.41850
\(286\) 0 0
\(287\) 7306.39 1.50273
\(288\) −1368.72 −0.280044
\(289\) −2950.90 −0.600630
\(290\) 548.181 0.111001
\(291\) −1663.95 −0.335197
\(292\) 451.931 0.0905729
\(293\) −1785.23 −0.355954 −0.177977 0.984035i \(-0.556955\pi\)
−0.177977 + 0.984035i \(0.556955\pi\)
\(294\) −222.024 −0.0440432
\(295\) −2197.27 −0.433662
\(296\) −1098.00 −0.215607
\(297\) 1777.73 0.347322
\(298\) 731.751 0.142246
\(299\) 0 0
\(300\) −2455.35 −0.472532
\(301\) −8394.83 −1.60754
\(302\) −148.603 −0.0283150
\(303\) −1387.69 −0.263105
\(304\) −6545.40 −1.23488
\(305\) 4127.99 0.774977
\(306\) −365.721 −0.0683231
\(307\) −5323.13 −0.989600 −0.494800 0.869007i \(-0.664759\pi\)
−0.494800 + 0.869007i \(0.664759\pi\)
\(308\) −9701.45 −1.79478
\(309\) 3366.22 0.619733
\(310\) 1254.44 0.229830
\(311\) 6265.64 1.14242 0.571209 0.820805i \(-0.306475\pi\)
0.571209 + 0.820805i \(0.306475\pi\)
\(312\) 0 0
\(313\) 7193.77 1.29909 0.649547 0.760322i \(-0.274959\pi\)
0.649547 + 0.760322i \(0.274959\pi\)
\(314\) 313.742 0.0563869
\(315\) 2865.89 0.512619
\(316\) 2055.29 0.365884
\(317\) 9576.87 1.69682 0.848408 0.529343i \(-0.177561\pi\)
0.848408 + 0.529343i \(0.177561\pi\)
\(318\) −623.554 −0.109960
\(319\) −2543.19 −0.446368
\(320\) −3350.41 −0.585293
\(321\) 1808.27 0.314417
\(322\) −1003.77 −0.173721
\(323\) −6513.81 −1.12210
\(324\) −579.832 −0.0994226
\(325\) 0 0
\(326\) −510.608 −0.0867483
\(327\) −4265.67 −0.721382
\(328\) 4936.14 0.830953
\(329\) 1399.00 0.234435
\(330\) −2803.32 −0.467629
\(331\) −5773.13 −0.958670 −0.479335 0.877632i \(-0.659122\pi\)
−0.479335 + 0.877632i \(0.659122\pi\)
\(332\) −2675.91 −0.442348
\(333\) −710.629 −0.116944
\(334\) 2877.27 0.471370
\(335\) 6361.23 1.03747
\(336\) 2748.52 0.446262
\(337\) 1238.09 0.200127 0.100063 0.994981i \(-0.468095\pi\)
0.100063 + 0.994981i \(0.468095\pi\)
\(338\) 0 0
\(339\) −1190.16 −0.190680
\(340\) −4905.47 −0.782460
\(341\) −5819.74 −0.924213
\(342\) 1214.12 0.191966
\(343\) −5399.55 −0.849995
\(344\) −5671.48 −0.888912
\(345\) 2467.16 0.385007
\(346\) −3049.53 −0.473826
\(347\) 5449.97 0.843140 0.421570 0.906796i \(-0.361479\pi\)
0.421570 + 0.906796i \(0.361479\pi\)
\(348\) 829.497 0.127775
\(349\) 1374.03 0.210746 0.105373 0.994433i \(-0.466396\pi\)
0.105373 + 0.994433i \(0.466396\pi\)
\(350\) −2158.92 −0.329711
\(351\) 0 0
\(352\) −10013.3 −1.51622
\(353\) 5970.44 0.900211 0.450106 0.892975i \(-0.351386\pi\)
0.450106 + 0.892975i \(0.351386\pi\)
\(354\) 390.886 0.0586874
\(355\) 1415.62 0.211644
\(356\) 854.734 0.127249
\(357\) 2735.26 0.405504
\(358\) 3039.49 0.448720
\(359\) 7813.71 1.14872 0.574362 0.818602i \(-0.305250\pi\)
0.574362 + 0.818602i \(0.305250\pi\)
\(360\) 1936.18 0.283460
\(361\) 14765.6 2.15274
\(362\) −69.7313 −0.0101243
\(363\) 9012.52 1.30312
\(364\) 0 0
\(365\) −976.690 −0.140061
\(366\) −734.352 −0.104878
\(367\) 1688.39 0.240145 0.120073 0.992765i \(-0.461687\pi\)
0.120073 + 0.992765i \(0.461687\pi\)
\(368\) 2366.11 0.335169
\(369\) 3194.69 0.450702
\(370\) 1120.59 0.157451
\(371\) 4663.61 0.652622
\(372\) 1898.19 0.264560
\(373\) 1870.61 0.259670 0.129835 0.991536i \(-0.458555\pi\)
0.129835 + 0.991536i \(0.458555\pi\)
\(374\) −2675.53 −0.369916
\(375\) −495.039 −0.0681699
\(376\) 945.151 0.129634
\(377\) 0 0
\(378\) −509.831 −0.0693726
\(379\) 11667.0 1.58125 0.790627 0.612298i \(-0.209755\pi\)
0.790627 + 0.612298i \(0.209755\pi\)
\(380\) 16285.2 2.19846
\(381\) 1313.60 0.176635
\(382\) 3736.82 0.500504
\(383\) −6676.60 −0.890753 −0.445376 0.895343i \(-0.646930\pi\)
−0.445376 + 0.895343i \(0.646930\pi\)
\(384\) 4245.95 0.564259
\(385\) 20966.3 2.77543
\(386\) 795.896 0.104948
\(387\) −3670.61 −0.482139
\(388\) 3970.42 0.519504
\(389\) 14285.3 1.86194 0.930969 0.365099i \(-0.118965\pi\)
0.930969 + 0.365099i \(0.118965\pi\)
\(390\) 0 0
\(391\) 2354.69 0.304558
\(392\) 1121.85 0.144545
\(393\) 4972.32 0.638220
\(394\) −1731.72 −0.221428
\(395\) −4441.79 −0.565800
\(396\) −4241.93 −0.538295
\(397\) 3569.28 0.451227 0.225614 0.974217i \(-0.427561\pi\)
0.225614 + 0.974217i \(0.427561\pi\)
\(398\) −2556.00 −0.321912
\(399\) −9080.54 −1.13934
\(400\) 5089.04 0.636130
\(401\) 499.348 0.0621852 0.0310926 0.999517i \(-0.490101\pi\)
0.0310926 + 0.999517i \(0.490101\pi\)
\(402\) −1131.64 −0.140400
\(403\) 0 0
\(404\) 3311.23 0.407772
\(405\) 1253.10 0.153746
\(406\) 729.354 0.0891557
\(407\) −5198.80 −0.633158
\(408\) 1847.92 0.224229
\(409\) 10457.5 1.26428 0.632140 0.774855i \(-0.282177\pi\)
0.632140 + 0.774855i \(0.282177\pi\)
\(410\) −5037.73 −0.606819
\(411\) −5902.26 −0.708363
\(412\) −8032.27 −0.960489
\(413\) −2923.47 −0.348316
\(414\) −438.897 −0.0521029
\(415\) 5783.03 0.684043
\(416\) 0 0
\(417\) 8475.40 0.995305
\(418\) 8882.27 1.03934
\(419\) −1705.42 −0.198843 −0.0994215 0.995045i \(-0.531699\pi\)
−0.0994215 + 0.995045i \(0.531699\pi\)
\(420\) −6838.43 −0.794479
\(421\) −8765.57 −1.01475 −0.507373 0.861727i \(-0.669383\pi\)
−0.507373 + 0.861727i \(0.669383\pi\)
\(422\) −650.225 −0.0750058
\(423\) 611.707 0.0703126
\(424\) 3150.70 0.360876
\(425\) 5064.48 0.578032
\(426\) −251.833 −0.0286417
\(427\) 5492.28 0.622459
\(428\) −4314.79 −0.487297
\(429\) 0 0
\(430\) 5788.21 0.649145
\(431\) −8080.37 −0.903057 −0.451529 0.892257i \(-0.649121\pi\)
−0.451529 + 0.892257i \(0.649121\pi\)
\(432\) 1201.78 0.133844
\(433\) −4124.48 −0.457760 −0.228880 0.973455i \(-0.573506\pi\)
−0.228880 + 0.973455i \(0.573506\pi\)
\(434\) 1669.02 0.184598
\(435\) −1792.66 −0.197590
\(436\) 10178.5 1.11803
\(437\) −7817.14 −0.855709
\(438\) 173.749 0.0189545
\(439\) −6115.52 −0.664870 −0.332435 0.943126i \(-0.607870\pi\)
−0.332435 + 0.943126i \(0.607870\pi\)
\(440\) 14164.6 1.53471
\(441\) 726.064 0.0784002
\(442\) 0 0
\(443\) −11058.8 −1.18605 −0.593025 0.805184i \(-0.702067\pi\)
−0.593025 + 0.805184i \(0.702067\pi\)
\(444\) 1695.66 0.181244
\(445\) −1847.21 −0.196777
\(446\) 4950.73 0.525614
\(447\) −2392.97 −0.253208
\(448\) −4457.72 −0.470106
\(449\) 242.012 0.0254371 0.0127185 0.999919i \(-0.495951\pi\)
0.0127185 + 0.999919i \(0.495951\pi\)
\(450\) −943.980 −0.0988881
\(451\) 23371.7 2.44020
\(452\) 2839.88 0.295524
\(453\) 485.962 0.0504028
\(454\) 2918.89 0.301741
\(455\) 0 0
\(456\) −6134.74 −0.630012
\(457\) 11052.2 1.13129 0.565644 0.824650i \(-0.308628\pi\)
0.565644 + 0.824650i \(0.308628\pi\)
\(458\) 1865.94 0.190370
\(459\) 1195.98 0.121620
\(460\) −5886.99 −0.596700
\(461\) −273.763 −0.0276582 −0.0138291 0.999904i \(-0.504402\pi\)
−0.0138291 + 0.999904i \(0.504402\pi\)
\(462\) −3729.81 −0.375598
\(463\) −11579.2 −1.16227 −0.581134 0.813808i \(-0.697391\pi\)
−0.581134 + 0.813808i \(0.697391\pi\)
\(464\) −1719.25 −0.172013
\(465\) −4102.26 −0.409114
\(466\) 2563.34 0.254817
\(467\) 902.915 0.0894688 0.0447344 0.998999i \(-0.485756\pi\)
0.0447344 + 0.998999i \(0.485756\pi\)
\(468\) 0 0
\(469\) 8463.59 0.833289
\(470\) −964.604 −0.0946678
\(471\) −1026.00 −0.100373
\(472\) −1975.07 −0.192606
\(473\) −26853.4 −2.61040
\(474\) 790.176 0.0765696
\(475\) −16813.1 −1.62408
\(476\) −6526.71 −0.628469
\(477\) 2039.15 0.195736
\(478\) −5039.88 −0.482257
\(479\) 12068.8 1.15122 0.575611 0.817723i \(-0.304764\pi\)
0.575611 + 0.817723i \(0.304764\pi\)
\(480\) −7058.23 −0.671172
\(481\) 0 0
\(482\) 3725.70 0.352077
\(483\) 3282.55 0.309236
\(484\) −21505.1 −2.01964
\(485\) −8580.66 −0.803356
\(486\) −222.922 −0.0208065
\(487\) −11492.5 −1.06936 −0.534678 0.845056i \(-0.679567\pi\)
−0.534678 + 0.845056i \(0.679567\pi\)
\(488\) 3710.54 0.344197
\(489\) 1669.79 0.154418
\(490\) −1144.93 −0.105557
\(491\) 15704.2 1.44342 0.721710 0.692196i \(-0.243357\pi\)
0.721710 + 0.692196i \(0.243357\pi\)
\(492\) −7622.99 −0.698519
\(493\) −1710.95 −0.156303
\(494\) 0 0
\(495\) 9167.43 0.832415
\(496\) −3934.25 −0.356156
\(497\) 1883.48 0.169991
\(498\) −1028.78 −0.0925715
\(499\) −9019.80 −0.809181 −0.404591 0.914498i \(-0.632586\pi\)
−0.404591 + 0.914498i \(0.632586\pi\)
\(500\) 1181.23 0.105653
\(501\) −9409.27 −0.839073
\(502\) 3275.83 0.291250
\(503\) −6033.84 −0.534862 −0.267431 0.963577i \(-0.586175\pi\)
−0.267431 + 0.963577i \(0.586175\pi\)
\(504\) 2576.08 0.227674
\(505\) −7156.06 −0.630576
\(506\) −3210.87 −0.282096
\(507\) 0 0
\(508\) −3134.44 −0.273757
\(509\) −22451.5 −1.95510 −0.977551 0.210699i \(-0.932426\pi\)
−0.977551 + 0.210699i \(0.932426\pi\)
\(510\) −1885.95 −0.163748
\(511\) −1299.48 −0.112497
\(512\) −11720.8 −1.01170
\(513\) −3970.44 −0.341714
\(514\) 6897.06 0.591860
\(515\) 17358.9 1.48529
\(516\) 8758.60 0.747240
\(517\) 4475.11 0.380687
\(518\) 1490.95 0.126464
\(519\) 9972.58 0.843445
\(520\) 0 0
\(521\) 15674.7 1.31808 0.659040 0.752108i \(-0.270963\pi\)
0.659040 + 0.752108i \(0.270963\pi\)
\(522\) 318.908 0.0267399
\(523\) 13510.8 1.12961 0.564804 0.825225i \(-0.308952\pi\)
0.564804 + 0.825225i \(0.308952\pi\)
\(524\) −11864.7 −0.989142
\(525\) 7060.10 0.586911
\(526\) 4275.13 0.354381
\(527\) −3915.27 −0.323627
\(528\) 8791.97 0.724662
\(529\) −9341.16 −0.767746
\(530\) −3215.55 −0.263537
\(531\) −1278.28 −0.104468
\(532\) 21667.4 1.76580
\(533\) 0 0
\(534\) 328.610 0.0266299
\(535\) 9324.89 0.753551
\(536\) 5717.94 0.460778
\(537\) −9939.75 −0.798756
\(538\) −4905.60 −0.393114
\(539\) 5311.73 0.424475
\(540\) −2990.08 −0.238283
\(541\) −12103.6 −0.961875 −0.480937 0.876755i \(-0.659704\pi\)
−0.480937 + 0.876755i \(0.659704\pi\)
\(542\) 2727.43 0.216149
\(543\) 228.036 0.0180220
\(544\) −6736.49 −0.530928
\(545\) −21997.2 −1.72891
\(546\) 0 0
\(547\) −15228.6 −1.19036 −0.595181 0.803592i \(-0.702920\pi\)
−0.595181 + 0.803592i \(0.702920\pi\)
\(548\) 14083.6 1.09785
\(549\) 2401.48 0.186690
\(550\) −6905.95 −0.535401
\(551\) 5680.03 0.439160
\(552\) 2217.66 0.170996
\(553\) −5909.79 −0.454448
\(554\) 701.014 0.0537603
\(555\) −3664.57 −0.280275
\(556\) −20223.5 −1.54257
\(557\) −23596.9 −1.79503 −0.897516 0.440982i \(-0.854630\pi\)
−0.897516 + 0.440982i \(0.854630\pi\)
\(558\) 729.775 0.0553653
\(559\) 0 0
\(560\) 14173.6 1.06954
\(561\) 8749.54 0.658478
\(562\) −6458.63 −0.484770
\(563\) 7941.62 0.594493 0.297246 0.954801i \(-0.403932\pi\)
0.297246 + 0.954801i \(0.403932\pi\)
\(564\) −1459.62 −0.108974
\(565\) −6137.40 −0.456996
\(566\) 8288.63 0.615542
\(567\) 1667.25 0.123488
\(568\) 1272.47 0.0939991
\(569\) 2274.65 0.167590 0.0837948 0.996483i \(-0.473296\pi\)
0.0837948 + 0.996483i \(0.473296\pi\)
\(570\) 6261.00 0.460078
\(571\) −4499.84 −0.329794 −0.164897 0.986311i \(-0.552729\pi\)
−0.164897 + 0.986311i \(0.552729\pi\)
\(572\) 0 0
\(573\) −12220.2 −0.890934
\(574\) −6702.69 −0.487395
\(575\) 6077.82 0.440804
\(576\) −1949.12 −0.140996
\(577\) 25253.3 1.82202 0.911011 0.412381i \(-0.135303\pi\)
0.911011 + 0.412381i \(0.135303\pi\)
\(578\) 2707.08 0.194809
\(579\) −2602.74 −0.186816
\(580\) 4277.55 0.306234
\(581\) 7694.31 0.549421
\(582\) 1526.46 0.108718
\(583\) 14918.0 1.05976
\(584\) −877.921 −0.0622066
\(585\) 0 0
\(586\) 1637.73 0.115450
\(587\) 11285.2 0.793508 0.396754 0.917925i \(-0.370137\pi\)
0.396754 + 0.917925i \(0.370137\pi\)
\(588\) −1732.49 −0.121508
\(589\) 12997.9 0.909289
\(590\) 2015.72 0.140654
\(591\) 5663.08 0.394159
\(592\) −3514.49 −0.243994
\(593\) −12824.5 −0.888090 −0.444045 0.896005i \(-0.646457\pi\)
−0.444045 + 0.896005i \(0.646457\pi\)
\(594\) −1630.85 −0.112651
\(595\) 14105.2 0.971859
\(596\) 5709.98 0.392433
\(597\) 8358.65 0.573027
\(598\) 0 0
\(599\) −26180.3 −1.78581 −0.892905 0.450245i \(-0.851336\pi\)
−0.892905 + 0.450245i \(0.851336\pi\)
\(600\) 4769.75 0.324540
\(601\) 16012.6 1.08680 0.543399 0.839474i \(-0.317137\pi\)
0.543399 + 0.839474i \(0.317137\pi\)
\(602\) 7701.20 0.521391
\(603\) 3700.68 0.249923
\(604\) −1159.57 −0.0781166
\(605\) 46475.8 3.12316
\(606\) 1273.03 0.0853357
\(607\) −9531.48 −0.637349 −0.318674 0.947864i \(-0.603238\pi\)
−0.318674 + 0.947864i \(0.603238\pi\)
\(608\) 22363.9 1.49174
\(609\) −2385.14 −0.158704
\(610\) −3786.91 −0.251356
\(611\) 0 0
\(612\) −2853.78 −0.188492
\(613\) −10187.1 −0.671211 −0.335605 0.942003i \(-0.608941\pi\)
−0.335605 + 0.942003i \(0.608941\pi\)
\(614\) 4883.30 0.320967
\(615\) 16474.4 1.08018
\(616\) 18846.0 1.23267
\(617\) −8012.62 −0.522813 −0.261407 0.965229i \(-0.584186\pi\)
−0.261407 + 0.965229i \(0.584186\pi\)
\(618\) −3088.08 −0.201004
\(619\) −1886.59 −0.122501 −0.0612506 0.998122i \(-0.519509\pi\)
−0.0612506 + 0.998122i \(0.519509\pi\)
\(620\) 9788.58 0.634063
\(621\) 1435.28 0.0927470
\(622\) −5747.94 −0.370533
\(623\) −2457.70 −0.158051
\(624\) 0 0
\(625\) −16844.5 −1.07805
\(626\) −6599.38 −0.421349
\(627\) −29046.8 −1.85011
\(628\) 2448.18 0.155562
\(629\) −3497.53 −0.221710
\(630\) −2629.10 −0.166263
\(631\) −14956.1 −0.943569 −0.471784 0.881714i \(-0.656390\pi\)
−0.471784 + 0.881714i \(0.656390\pi\)
\(632\) −3992.61 −0.251293
\(633\) 2126.37 0.133516
\(634\) −8785.57 −0.550346
\(635\) 6773.99 0.423335
\(636\) −4865.70 −0.303361
\(637\) 0 0
\(638\) 2333.06 0.144775
\(639\) 823.547 0.0509844
\(640\) 21895.5 1.35234
\(641\) 23691.7 1.45985 0.729927 0.683525i \(-0.239554\pi\)
0.729927 + 0.683525i \(0.239554\pi\)
\(642\) −1658.86 −0.101978
\(643\) −13651.3 −0.837254 −0.418627 0.908158i \(-0.637489\pi\)
−0.418627 + 0.908158i \(0.637489\pi\)
\(644\) −7832.62 −0.479268
\(645\) −18928.6 −1.15553
\(646\) 5975.60 0.363943
\(647\) −18132.3 −1.10178 −0.550892 0.834577i \(-0.685712\pi\)
−0.550892 + 0.834577i \(0.685712\pi\)
\(648\) 1126.38 0.0682846
\(649\) −9351.59 −0.565612
\(650\) 0 0
\(651\) −5458.05 −0.328599
\(652\) −3984.36 −0.239324
\(653\) 6363.47 0.381351 0.190675 0.981653i \(-0.438932\pi\)
0.190675 + 0.981653i \(0.438932\pi\)
\(654\) 3913.21 0.233973
\(655\) 25641.3 1.52960
\(656\) 15799.7 0.940358
\(657\) −568.195 −0.0337403
\(658\) −1283.40 −0.0760369
\(659\) −1051.43 −0.0621518 −0.0310759 0.999517i \(-0.509893\pi\)
−0.0310759 + 0.999517i \(0.509893\pi\)
\(660\) −21874.8 −1.29011
\(661\) −8119.75 −0.477794 −0.238897 0.971045i \(-0.576786\pi\)
−0.238897 + 0.971045i \(0.576786\pi\)
\(662\) 5296.12 0.310936
\(663\) 0 0
\(664\) 5198.21 0.303810
\(665\) −46826.5 −2.73061
\(666\) 651.912 0.0379296
\(667\) −2053.29 −0.119196
\(668\) 22451.9 1.30043
\(669\) −16189.9 −0.935632
\(670\) −5835.62 −0.336492
\(671\) 17568.7 1.01078
\(672\) −9390.96 −0.539083
\(673\) −190.264 −0.0108977 −0.00544885 0.999985i \(-0.501734\pi\)
−0.00544885 + 0.999985i \(0.501734\pi\)
\(674\) −1135.79 −0.0649093
\(675\) 3087.01 0.176028
\(676\) 0 0
\(677\) 4861.93 0.276010 0.138005 0.990431i \(-0.455931\pi\)
0.138005 + 0.990431i \(0.455931\pi\)
\(678\) 1091.82 0.0618452
\(679\) −11416.5 −0.645253
\(680\) 9529.35 0.537403
\(681\) −9545.37 −0.537121
\(682\) 5338.88 0.299760
\(683\) 14539.6 0.814556 0.407278 0.913304i \(-0.366478\pi\)
0.407278 + 0.913304i \(0.366478\pi\)
\(684\) 9474.02 0.529603
\(685\) −30436.8 −1.69771
\(686\) 4953.40 0.275688
\(687\) −6102.00 −0.338873
\(688\) −18153.4 −1.00595
\(689\) 0 0
\(690\) −2263.31 −0.124873
\(691\) 22106.5 1.21703 0.608517 0.793541i \(-0.291765\pi\)
0.608517 + 0.793541i \(0.291765\pi\)
\(692\) −23796.0 −1.30721
\(693\) 12197.2 0.668593
\(694\) −4999.66 −0.273465
\(695\) 43706.0 2.38541
\(696\) −1611.38 −0.0877574
\(697\) 15723.4 0.854474
\(698\) −1260.50 −0.0683534
\(699\) −8382.66 −0.453593
\(700\) −16846.4 −0.909620
\(701\) 229.971 0.0123907 0.00619535 0.999981i \(-0.498028\pi\)
0.00619535 + 0.999981i \(0.498028\pi\)
\(702\) 0 0
\(703\) 11611.1 0.622934
\(704\) −14259.4 −0.763380
\(705\) 3154.45 0.168516
\(706\) −5477.13 −0.291975
\(707\) −9521.12 −0.506476
\(708\) 3050.15 0.161909
\(709\) 4802.22 0.254374 0.127187 0.991879i \(-0.459405\pi\)
0.127187 + 0.991879i \(0.459405\pi\)
\(710\) −1298.66 −0.0686446
\(711\) −2584.04 −0.136300
\(712\) −1660.40 −0.0873965
\(713\) −4698.66 −0.246797
\(714\) −2509.25 −0.131522
\(715\) 0 0
\(716\) 23717.6 1.23795
\(717\) 16481.4 0.858453
\(718\) −7168.09 −0.372578
\(719\) −11016.9 −0.571435 −0.285718 0.958314i \(-0.592232\pi\)
−0.285718 + 0.958314i \(0.592232\pi\)
\(720\) 6197.36 0.320780
\(721\) 23096.0 1.19298
\(722\) −13545.6 −0.698221
\(723\) −12183.8 −0.626723
\(724\) −544.125 −0.0279313
\(725\) −4416.21 −0.226226
\(726\) −8267.85 −0.422656
\(727\) −13498.2 −0.688612 −0.344306 0.938857i \(-0.611886\pi\)
−0.344306 + 0.938857i \(0.611886\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 895.990 0.0454275
\(731\) −18065.8 −0.914073
\(732\) −5730.27 −0.289340
\(733\) 17014.7 0.857368 0.428684 0.903455i \(-0.358977\pi\)
0.428684 + 0.903455i \(0.358977\pi\)
\(734\) −1548.88 −0.0778888
\(735\) 3744.17 0.187899
\(736\) −8084.38 −0.404883
\(737\) 27073.4 1.35313
\(738\) −2930.73 −0.146181
\(739\) −13392.6 −0.666651 −0.333326 0.942812i \(-0.608171\pi\)
−0.333326 + 0.942812i \(0.608171\pi\)
\(740\) 8744.19 0.434382
\(741\) 0 0
\(742\) −4278.27 −0.211672
\(743\) −11796.5 −0.582467 −0.291233 0.956652i \(-0.594066\pi\)
−0.291233 + 0.956652i \(0.594066\pi\)
\(744\) −3687.42 −0.181703
\(745\) −12340.1 −0.606854
\(746\) −1716.05 −0.0842214
\(747\) 3364.31 0.164784
\(748\) −20877.6 −1.02054
\(749\) 12406.7 0.605250
\(750\) 454.136 0.0221103
\(751\) −12450.5 −0.604961 −0.302480 0.953156i \(-0.597815\pi\)
−0.302480 + 0.953156i \(0.597815\pi\)
\(752\) 3025.26 0.146702
\(753\) −10712.6 −0.518447
\(754\) 0 0
\(755\) 2506.01 0.120799
\(756\) −3978.30 −0.191388
\(757\) −7029.84 −0.337522 −0.168761 0.985657i \(-0.553977\pi\)
−0.168761 + 0.985657i \(0.553977\pi\)
\(758\) −10703.0 −0.512865
\(759\) 10500.2 0.502152
\(760\) −31635.6 −1.50993
\(761\) −17740.6 −0.845067 −0.422533 0.906347i \(-0.638859\pi\)
−0.422533 + 0.906347i \(0.638859\pi\)
\(762\) −1205.06 −0.0572899
\(763\) −29267.2 −1.38866
\(764\) 29159.1 1.38081
\(765\) 6167.45 0.291483
\(766\) 6124.94 0.288907
\(767\) 0 0
\(768\) 1302.54 0.0611995
\(769\) −31976.5 −1.49948 −0.749741 0.661732i \(-0.769822\pi\)
−0.749741 + 0.661732i \(0.769822\pi\)
\(770\) −19233.9 −0.900184
\(771\) −22554.8 −1.05355
\(772\) 6210.51 0.289535
\(773\) 2135.11 0.0993463 0.0496732 0.998766i \(-0.484182\pi\)
0.0496732 + 0.998766i \(0.484182\pi\)
\(774\) 3367.32 0.156377
\(775\) −10105.9 −0.468405
\(776\) −7712.93 −0.356802
\(777\) −4875.70 −0.225116
\(778\) −13105.0 −0.603902
\(779\) −52198.9 −2.40080
\(780\) 0 0
\(781\) 6024.89 0.276040
\(782\) −2160.14 −0.0987804
\(783\) −1042.89 −0.0475989
\(784\) 3590.83 0.163576
\(785\) −5290.89 −0.240560
\(786\) −4561.48 −0.207001
\(787\) 11704.4 0.530134 0.265067 0.964230i \(-0.414606\pi\)
0.265067 + 0.964230i \(0.414606\pi\)
\(788\) −13512.9 −0.610885
\(789\) −13980.6 −0.630825
\(790\) 4074.78 0.183512
\(791\) −8165.80 −0.367057
\(792\) 8240.36 0.369708
\(793\) 0 0
\(794\) −3274.37 −0.146351
\(795\) 10515.5 0.469115
\(796\) −19944.9 −0.888102
\(797\) −149.733 −0.00665474 −0.00332737 0.999994i \(-0.501059\pi\)
−0.00332737 + 0.999994i \(0.501059\pi\)
\(798\) 8330.25 0.369533
\(799\) 3010.66 0.133304
\(800\) −17387.9 −0.768443
\(801\) −1074.62 −0.0474032
\(802\) −458.089 −0.0201692
\(803\) −4156.79 −0.182677
\(804\) −8830.35 −0.387341
\(805\) 16927.4 0.741135
\(806\) 0 0
\(807\) 16042.3 0.699772
\(808\) −6432.40 −0.280063
\(809\) 23520.8 1.02219 0.511093 0.859526i \(-0.329241\pi\)
0.511093 + 0.859526i \(0.329241\pi\)
\(810\) −1149.56 −0.0498661
\(811\) −29604.8 −1.28183 −0.640915 0.767612i \(-0.721445\pi\)
−0.640915 + 0.767612i \(0.721445\pi\)
\(812\) 5691.27 0.245966
\(813\) −8919.24 −0.384762
\(814\) 4769.25 0.205359
\(815\) 8610.79 0.370089
\(816\) 5914.86 0.253752
\(817\) 59975.0 2.56825
\(818\) −9593.44 −0.410057
\(819\) 0 0
\(820\) −39310.3 −1.67412
\(821\) −43034.7 −1.82938 −0.914691 0.404154i \(-0.867566\pi\)
−0.914691 + 0.404154i \(0.867566\pi\)
\(822\) 5414.58 0.229751
\(823\) 5584.95 0.236548 0.118274 0.992981i \(-0.462264\pi\)
0.118274 + 0.992981i \(0.462264\pi\)
\(824\) 15603.5 0.659675
\(825\) 22583.9 0.953054
\(826\) 2681.91 0.112973
\(827\) 4788.13 0.201330 0.100665 0.994920i \(-0.467903\pi\)
0.100665 + 0.994920i \(0.467903\pi\)
\(828\) −3424.79 −0.143744
\(829\) −32392.2 −1.35709 −0.678546 0.734558i \(-0.737390\pi\)
−0.678546 + 0.734558i \(0.737390\pi\)
\(830\) −5305.20 −0.221863
\(831\) −2292.46 −0.0956974
\(832\) 0 0
\(833\) 3573.50 0.148637
\(834\) −7775.11 −0.322818
\(835\) −48521.8 −2.01098
\(836\) 69309.9 2.86738
\(837\) −2386.51 −0.0985544
\(838\) 1564.51 0.0644929
\(839\) −13598.9 −0.559577 −0.279788 0.960062i \(-0.590264\pi\)
−0.279788 + 0.960062i \(0.590264\pi\)
\(840\) 13284.3 0.545658
\(841\) −22897.1 −0.938827
\(842\) 8041.31 0.329123
\(843\) 21121.0 0.862926
\(844\) −5073.82 −0.206929
\(845\) 0 0
\(846\) −561.164 −0.0228052
\(847\) 61835.9 2.50851
\(848\) 10084.8 0.408390
\(849\) −27105.5 −1.09571
\(850\) −4646.02 −0.187479
\(851\) −4197.34 −0.169075
\(852\) −1965.10 −0.0790178
\(853\) −41037.0 −1.64722 −0.823611 0.567155i \(-0.808044\pi\)
−0.823611 + 0.567155i \(0.808044\pi\)
\(854\) −5038.47 −0.201889
\(855\) −20474.8 −0.818973
\(856\) 8381.89 0.334681
\(857\) −39959.3 −1.59275 −0.796374 0.604804i \(-0.793251\pi\)
−0.796374 + 0.604804i \(0.793251\pi\)
\(858\) 0 0
\(859\) −32570.5 −1.29371 −0.646853 0.762615i \(-0.723915\pi\)
−0.646853 + 0.762615i \(0.723915\pi\)
\(860\) 45166.4 1.79088
\(861\) 21919.2 0.867599
\(862\) 7412.72 0.292898
\(863\) 16951.8 0.668652 0.334326 0.942457i \(-0.391491\pi\)
0.334326 + 0.942457i \(0.391491\pi\)
\(864\) −4106.17 −0.161684
\(865\) 51426.6 2.02145
\(866\) 3783.69 0.148470
\(867\) −8852.69 −0.346774
\(868\) 13023.7 0.509277
\(869\) −18904.3 −0.737955
\(870\) 1644.54 0.0640865
\(871\) 0 0
\(872\) −19772.7 −0.767876
\(873\) −4991.85 −0.193526
\(874\) 7171.24 0.277541
\(875\) −3396.52 −0.131227
\(876\) 1355.79 0.0522923
\(877\) 45939.2 1.76882 0.884411 0.466709i \(-0.154561\pi\)
0.884411 + 0.466709i \(0.154561\pi\)
\(878\) 5610.22 0.215644
\(879\) −5355.70 −0.205510
\(880\) 45338.5 1.73677
\(881\) 37960.5 1.45167 0.725836 0.687868i \(-0.241453\pi\)
0.725836 + 0.687868i \(0.241453\pi\)
\(882\) −666.072 −0.0254284
\(883\) 43172.9 1.64539 0.822697 0.568480i \(-0.192468\pi\)
0.822697 + 0.568480i \(0.192468\pi\)
\(884\) 0 0
\(885\) −6591.82 −0.250375
\(886\) 10145.1 0.384684
\(887\) 28706.3 1.08665 0.543327 0.839521i \(-0.317164\pi\)
0.543327 + 0.839521i \(0.317164\pi\)
\(888\) −3293.99 −0.124481
\(889\) 9012.78 0.340021
\(890\) 1694.58 0.0638229
\(891\) 5333.20 0.200526
\(892\) 38631.4 1.45008
\(893\) −9994.83 −0.374540
\(894\) 2195.25 0.0821255
\(895\) −51257.3 −1.91435
\(896\) 29131.9 1.08619
\(897\) 0 0
\(898\) −222.015 −0.00825027
\(899\) 3414.10 0.126659
\(900\) −7366.04 −0.272816
\(901\) 10036.2 0.371091
\(902\) −21440.6 −0.791456
\(903\) −25184.5 −0.928114
\(904\) −5516.75 −0.202969
\(905\) 1175.93 0.0431927
\(906\) −445.809 −0.0163477
\(907\) −22356.1 −0.818435 −0.409218 0.912437i \(-0.634198\pi\)
−0.409218 + 0.912437i \(0.634198\pi\)
\(908\) 22776.6 0.832454
\(909\) −4163.08 −0.151904
\(910\) 0 0
\(911\) 6953.80 0.252897 0.126449 0.991973i \(-0.459642\pi\)
0.126449 + 0.991973i \(0.459642\pi\)
\(912\) −19636.2 −0.712960
\(913\) 24612.6 0.892176
\(914\) −10139.0 −0.366923
\(915\) 12384.0 0.447433
\(916\) 14560.2 0.525200
\(917\) 34115.7 1.22857
\(918\) −1097.16 −0.0394464
\(919\) 40625.2 1.45822 0.729108 0.684399i \(-0.239935\pi\)
0.729108 + 0.684399i \(0.239935\pi\)
\(920\) 11436.0 0.409821
\(921\) −15969.4 −0.571346
\(922\) 251.143 0.00897068
\(923\) 0 0
\(924\) −29104.3 −1.03621
\(925\) −9027.64 −0.320894
\(926\) 10622.4 0.376971
\(927\) 10098.6 0.357803
\(928\) 5874.20 0.207791
\(929\) −42813.8 −1.51203 −0.756014 0.654555i \(-0.772856\pi\)
−0.756014 + 0.654555i \(0.772856\pi\)
\(930\) 3763.31 0.132692
\(931\) −11863.3 −0.417621
\(932\) 20002.2 0.702998
\(933\) 18796.9 0.659575
\(934\) −828.311 −0.0290184
\(935\) 45119.7 1.57815
\(936\) 0 0
\(937\) 43484.1 1.51608 0.758038 0.652210i \(-0.226158\pi\)
0.758038 + 0.652210i \(0.226158\pi\)
\(938\) −7764.28 −0.270269
\(939\) 21581.3 0.750032
\(940\) −7526.97 −0.261173
\(941\) 7108.58 0.246263 0.123131 0.992390i \(-0.460706\pi\)
0.123131 + 0.992390i \(0.460706\pi\)
\(942\) 941.227 0.0325550
\(943\) 18869.5 0.651618
\(944\) −6321.85 −0.217965
\(945\) 8597.68 0.295960
\(946\) 24634.6 0.846660
\(947\) −1938.99 −0.0665351 −0.0332676 0.999446i \(-0.510591\pi\)
−0.0332676 + 0.999446i \(0.510591\pi\)
\(948\) 6165.88 0.211243
\(949\) 0 0
\(950\) 15423.9 0.526756
\(951\) 28730.6 0.979657
\(952\) 12678.8 0.431640
\(953\) −47806.7 −1.62498 −0.812492 0.582972i \(-0.801890\pi\)
−0.812492 + 0.582972i \(0.801890\pi\)
\(954\) −1870.66 −0.0634852
\(955\) −63017.1 −2.13527
\(956\) −39327.1 −1.33047
\(957\) −7629.58 −0.257711
\(958\) −11071.6 −0.373388
\(959\) −40496.1 −1.36359
\(960\) −10051.2 −0.337919
\(961\) −21978.3 −0.737750
\(962\) 0 0
\(963\) 5424.81 0.181528
\(964\) 29072.3 0.971323
\(965\) −13421.8 −0.447735
\(966\) −3011.32 −0.100298
\(967\) 2832.71 0.0942025 0.0471013 0.998890i \(-0.485002\pi\)
0.0471013 + 0.998890i \(0.485002\pi\)
\(968\) 41775.8 1.38711
\(969\) −19541.4 −0.647845
\(970\) 7871.68 0.260561
\(971\) −41276.3 −1.36418 −0.682090 0.731268i \(-0.738929\pi\)
−0.682090 + 0.731268i \(0.738929\pi\)
\(972\) −1739.50 −0.0574016
\(973\) 58150.7 1.91596
\(974\) 10542.9 0.346835
\(975\) 0 0
\(976\) 11876.8 0.389515
\(977\) −4278.89 −0.140117 −0.0700583 0.997543i \(-0.522319\pi\)
−0.0700583 + 0.997543i \(0.522319\pi\)
\(978\) −1531.82 −0.0500842
\(979\) −7861.70 −0.256651
\(980\) −8934.12 −0.291214
\(981\) −12797.0 −0.416490
\(982\) −14406.6 −0.468159
\(983\) −22652.9 −0.735011 −0.367505 0.930021i \(-0.619788\pi\)
−0.367505 + 0.930021i \(0.619788\pi\)
\(984\) 14808.4 0.479751
\(985\) 29203.4 0.944667
\(986\) 1569.58 0.0506953
\(987\) 4196.99 0.135351
\(988\) 0 0
\(989\) −21680.5 −0.697068
\(990\) −8409.96 −0.269986
\(991\) 22060.2 0.707131 0.353565 0.935410i \(-0.384969\pi\)
0.353565 + 0.935410i \(0.384969\pi\)
\(992\) 13442.3 0.430235
\(993\) −17319.4 −0.553488
\(994\) −1727.86 −0.0551351
\(995\) 43103.9 1.37335
\(996\) −8027.72 −0.255390
\(997\) 35635.5 1.13198 0.565992 0.824411i \(-0.308493\pi\)
0.565992 + 0.824411i \(0.308493\pi\)
\(998\) 8274.52 0.262450
\(999\) −2131.89 −0.0675174
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 507.4.a.r.1.5 10
3.2 odd 2 1521.4.a.bk.1.6 10
13.2 odd 12 39.4.j.c.4.3 10
13.5 odd 4 507.4.b.i.337.6 10
13.7 odd 12 39.4.j.c.10.3 yes 10
13.8 odd 4 507.4.b.i.337.5 10
13.12 even 2 inner 507.4.a.r.1.6 10
39.2 even 12 117.4.q.e.82.3 10
39.20 even 12 117.4.q.e.10.3 10
39.38 odd 2 1521.4.a.bk.1.5 10
52.7 even 12 624.4.bv.h.49.4 10
52.15 even 12 624.4.bv.h.433.2 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.c.4.3 10 13.2 odd 12
39.4.j.c.10.3 yes 10 13.7 odd 12
117.4.q.e.10.3 10 39.20 even 12
117.4.q.e.82.3 10 39.2 even 12
507.4.a.r.1.5 10 1.1 even 1 trivial
507.4.a.r.1.6 10 13.12 even 2 inner
507.4.b.i.337.5 10 13.8 odd 4
507.4.b.i.337.6 10 13.5 odd 4
624.4.bv.h.49.4 10 52.7 even 12
624.4.bv.h.433.2 10 52.15 even 12
1521.4.a.bk.1.5 10 39.38 odd 2
1521.4.a.bk.1.6 10 3.2 odd 2