Properties

Label 1617.4.a.be.1.14
Level $1617$
Weight $4$
Character 1617.1
Self dual yes
Analytic conductor $95.406$
Analytic rank $1$
Dimension $16$
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] = [1617,4,Mod(1,1617)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1617, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1617.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1617.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: \(95.4060884793\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 92 x^{14} + 346 x^{13} + 3385 x^{12} - 11756 x^{11} - 63875 x^{10} + 199850 x^{9} + \cdots + 5479424 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4}\cdot 7^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.14
Root \(4.50124\) of defining polynomial
Character \(\chi\) \(=\) 1617.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.50124 q^{2} -3.00000 q^{3} +12.2612 q^{4} +3.51724 q^{5} -13.5037 q^{6} +19.1806 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+4.50124 q^{2} -3.00000 q^{3} +12.2612 q^{4} +3.51724 q^{5} -13.5037 q^{6} +19.1806 q^{8} +9.00000 q^{9} +15.8320 q^{10} +11.0000 q^{11} -36.7835 q^{12} -4.32358 q^{13} -10.5517 q^{15} -11.7531 q^{16} -87.2384 q^{17} +40.5112 q^{18} -70.0896 q^{19} +43.1255 q^{20} +49.5137 q^{22} +104.799 q^{23} -57.5417 q^{24} -112.629 q^{25} -19.4615 q^{26} -27.0000 q^{27} +8.09196 q^{29} -47.4959 q^{30} -79.0568 q^{31} -206.348 q^{32} -33.0000 q^{33} -392.681 q^{34} +110.351 q^{36} +24.4923 q^{37} -315.490 q^{38} +12.9708 q^{39} +67.4627 q^{40} -436.077 q^{41} +339.003 q^{43} +134.873 q^{44} +31.6552 q^{45} +471.724 q^{46} +164.630 q^{47} +35.2592 q^{48} -506.970 q^{50} +261.715 q^{51} -53.0122 q^{52} -289.755 q^{53} -121.534 q^{54} +38.6897 q^{55} +210.269 q^{57} +36.4239 q^{58} +49.4832 q^{59} -129.377 q^{60} -411.598 q^{61} -355.854 q^{62} -834.797 q^{64} -15.2071 q^{65} -148.541 q^{66} -189.946 q^{67} -1069.64 q^{68} -314.396 q^{69} -138.268 q^{71} +172.625 q^{72} -704.723 q^{73} +110.246 q^{74} +337.887 q^{75} -859.381 q^{76} +58.3845 q^{78} +988.061 q^{79} -41.3384 q^{80} +81.0000 q^{81} -1962.89 q^{82} -474.861 q^{83} -306.839 q^{85} +1525.93 q^{86} -24.2759 q^{87} +210.986 q^{88} +443.778 q^{89} +142.488 q^{90} +1284.95 q^{92} +237.170 q^{93} +741.040 q^{94} -246.522 q^{95} +619.043 q^{96} +147.704 q^{97} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{2} - 48 q^{3} + 72 q^{4} - 40 q^{5} - 12 q^{6} + 66 q^{8} + 144 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{2} - 48 q^{3} + 72 q^{4} - 40 q^{5} - 12 q^{6} + 66 q^{8} + 144 q^{9} - 178 q^{10} + 176 q^{11} - 216 q^{12} - 104 q^{13} + 120 q^{15} + 220 q^{16} - 180 q^{17} + 36 q^{18} - 152 q^{19} - 298 q^{20} + 44 q^{22} + 4 q^{23} - 198 q^{24} + 588 q^{25} - 406 q^{26} - 432 q^{27} + 412 q^{29} + 534 q^{30} - 628 q^{31} + 592 q^{32} - 528 q^{33} + 88 q^{34} + 648 q^{36} + 148 q^{37} - 446 q^{38} + 312 q^{39} - 1376 q^{40} - 596 q^{41} - 260 q^{43} + 792 q^{44} - 360 q^{45} - 148 q^{46} - 2220 q^{47} - 660 q^{48} + 82 q^{50} + 540 q^{51} - 1046 q^{52} + 168 q^{53} - 108 q^{54} - 440 q^{55} + 456 q^{57} + 538 q^{58} + 48 q^{59} + 894 q^{60} - 1504 q^{61} - 1276 q^{62} + 630 q^{64} + 1224 q^{65} - 132 q^{66} + 116 q^{67} - 356 q^{68} - 12 q^{69} + 320 q^{71} + 594 q^{72} - 652 q^{73} + 1062 q^{74} - 1764 q^{75} + 594 q^{76} + 1218 q^{78} + 1136 q^{79} - 1970 q^{80} + 1296 q^{81} + 416 q^{82} - 3300 q^{83} - 1148 q^{85} + 1864 q^{86} - 1236 q^{87} + 726 q^{88} - 2416 q^{89} - 1602 q^{90} - 1064 q^{92} + 1884 q^{93} - 914 q^{94} + 1120 q^{95} - 1776 q^{96} - 3616 q^{97} + 1584 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.50124 1.59143 0.795715 0.605672i \(-0.207096\pi\)
0.795715 + 0.605672i \(0.207096\pi\)
\(3\) −3.00000 −0.577350
\(4\) 12.2612 1.53265
\(5\) 3.51724 0.314592 0.157296 0.987552i \(-0.449722\pi\)
0.157296 + 0.987552i \(0.449722\pi\)
\(6\) −13.5037 −0.918812
\(7\) 0 0
\(8\) 19.1806 0.847669
\(9\) 9.00000 0.333333
\(10\) 15.8320 0.500651
\(11\) 11.0000 0.301511
\(12\) −36.7835 −0.884874
\(13\) −4.32358 −0.0922421 −0.0461210 0.998936i \(-0.514686\pi\)
−0.0461210 + 0.998936i \(0.514686\pi\)
\(14\) 0 0
\(15\) −10.5517 −0.181630
\(16\) −11.7531 −0.183642
\(17\) −87.2384 −1.24461 −0.622306 0.782774i \(-0.713804\pi\)
−0.622306 + 0.782774i \(0.713804\pi\)
\(18\) 40.5112 0.530476
\(19\) −70.0896 −0.846298 −0.423149 0.906060i \(-0.639075\pi\)
−0.423149 + 0.906060i \(0.639075\pi\)
\(20\) 43.1255 0.482158
\(21\) 0 0
\(22\) 49.5137 0.479834
\(23\) 104.799 0.950087 0.475044 0.879962i \(-0.342432\pi\)
0.475044 + 0.879962i \(0.342432\pi\)
\(24\) −57.5417 −0.489402
\(25\) −112.629 −0.901032
\(26\) −19.4615 −0.146797
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 8.09196 0.0518152 0.0259076 0.999664i \(-0.491752\pi\)
0.0259076 + 0.999664i \(0.491752\pi\)
\(30\) −47.4959 −0.289051
\(31\) −79.0568 −0.458033 −0.229016 0.973423i \(-0.573551\pi\)
−0.229016 + 0.973423i \(0.573551\pi\)
\(32\) −206.348 −1.13992
\(33\) −33.0000 −0.174078
\(34\) −392.681 −1.98071
\(35\) 0 0
\(36\) 110.351 0.510882
\(37\) 24.4923 0.108825 0.0544123 0.998519i \(-0.482671\pi\)
0.0544123 + 0.998519i \(0.482671\pi\)
\(38\) −315.490 −1.34682
\(39\) 12.9708 0.0532560
\(40\) 67.4627 0.266670
\(41\) −436.077 −1.66107 −0.830533 0.556969i \(-0.811964\pi\)
−0.830533 + 0.556969i \(0.811964\pi\)
\(42\) 0 0
\(43\) 339.003 1.20227 0.601134 0.799148i \(-0.294716\pi\)
0.601134 + 0.799148i \(0.294716\pi\)
\(44\) 134.873 0.462110
\(45\) 31.6552 0.104864
\(46\) 471.724 1.51200
\(47\) 164.630 0.510931 0.255466 0.966818i \(-0.417771\pi\)
0.255466 + 0.966818i \(0.417771\pi\)
\(48\) 35.2592 0.106026
\(49\) 0 0
\(50\) −506.970 −1.43393
\(51\) 261.715 0.718578
\(52\) −53.0122 −0.141374
\(53\) −289.755 −0.750961 −0.375480 0.926830i \(-0.622522\pi\)
−0.375480 + 0.926830i \(0.622522\pi\)
\(54\) −121.534 −0.306271
\(55\) 38.6897 0.0948530
\(56\) 0 0
\(57\) 210.269 0.488610
\(58\) 36.4239 0.0824602
\(59\) 49.4832 0.109189 0.0545947 0.998509i \(-0.482613\pi\)
0.0545947 + 0.998509i \(0.482613\pi\)
\(60\) −129.377 −0.278374
\(61\) −411.598 −0.863930 −0.431965 0.901890i \(-0.642180\pi\)
−0.431965 + 0.901890i \(0.642180\pi\)
\(62\) −355.854 −0.728927
\(63\) 0 0
\(64\) −834.797 −1.63046
\(65\) −15.2071 −0.0290186
\(66\) −148.541 −0.277032
\(67\) −189.946 −0.346352 −0.173176 0.984891i \(-0.555403\pi\)
−0.173176 + 0.984891i \(0.555403\pi\)
\(68\) −1069.64 −1.90755
\(69\) −314.396 −0.548533
\(70\) 0 0
\(71\) −138.268 −0.231118 −0.115559 0.993301i \(-0.536866\pi\)
−0.115559 + 0.993301i \(0.536866\pi\)
\(72\) 172.625 0.282556
\(73\) −704.723 −1.12989 −0.564943 0.825130i \(-0.691102\pi\)
−0.564943 + 0.825130i \(0.691102\pi\)
\(74\) 110.246 0.173187
\(75\) 337.887 0.520211
\(76\) −859.381 −1.29708
\(77\) 0 0
\(78\) 58.3845 0.0847531
\(79\) 988.061 1.40716 0.703579 0.710617i \(-0.251584\pi\)
0.703579 + 0.710617i \(0.251584\pi\)
\(80\) −41.3384 −0.0577722
\(81\) 81.0000 0.111111
\(82\) −1962.89 −2.64347
\(83\) −474.861 −0.627985 −0.313992 0.949425i \(-0.601667\pi\)
−0.313992 + 0.949425i \(0.601667\pi\)
\(84\) 0 0
\(85\) −306.839 −0.391545
\(86\) 1525.93 1.91332
\(87\) −24.2759 −0.0299155
\(88\) 210.986 0.255582
\(89\) 443.778 0.528543 0.264272 0.964448i \(-0.414868\pi\)
0.264272 + 0.964448i \(0.414868\pi\)
\(90\) 142.488 0.166884
\(91\) 0 0
\(92\) 1284.95 1.45615
\(93\) 237.170 0.264445
\(94\) 741.040 0.813111
\(95\) −246.522 −0.266239
\(96\) 619.043 0.658134
\(97\) 147.704 0.154609 0.0773046 0.997008i \(-0.475369\pi\)
0.0773046 + 0.997008i \(0.475369\pi\)
\(98\) 0 0
\(99\) 99.0000 0.100504
\(100\) −1380.96 −1.38096
\(101\) −775.300 −0.763814 −0.381907 0.924201i \(-0.624733\pi\)
−0.381907 + 0.924201i \(0.624733\pi\)
\(102\) 1178.04 1.14357
\(103\) −838.423 −0.802061 −0.401031 0.916065i \(-0.631348\pi\)
−0.401031 + 0.916065i \(0.631348\pi\)
\(104\) −82.9287 −0.0781907
\(105\) 0 0
\(106\) −1304.26 −1.19510
\(107\) −345.140 −0.311832 −0.155916 0.987770i \(-0.549833\pi\)
−0.155916 + 0.987770i \(0.549833\pi\)
\(108\) −331.052 −0.294958
\(109\) 772.585 0.678901 0.339451 0.940624i \(-0.389759\pi\)
0.339451 + 0.940624i \(0.389759\pi\)
\(110\) 174.152 0.150952
\(111\) −73.4769 −0.0628299
\(112\) 0 0
\(113\) −1630.41 −1.35731 −0.678655 0.734458i \(-0.737437\pi\)
−0.678655 + 0.734458i \(0.737437\pi\)
\(114\) 946.471 0.777589
\(115\) 368.602 0.298890
\(116\) 99.2169 0.0794143
\(117\) −38.9123 −0.0307474
\(118\) 222.736 0.173767
\(119\) 0 0
\(120\) −202.388 −0.153962
\(121\) 121.000 0.0909091
\(122\) −1852.70 −1.37488
\(123\) 1308.23 0.959017
\(124\) −969.329 −0.702002
\(125\) −835.799 −0.598049
\(126\) 0 0
\(127\) −1694.58 −1.18402 −0.592008 0.805932i \(-0.701665\pi\)
−0.592008 + 0.805932i \(0.701665\pi\)
\(128\) −2106.84 −1.45484
\(129\) −1017.01 −0.694129
\(130\) −68.4508 −0.0461810
\(131\) 519.573 0.346529 0.173265 0.984875i \(-0.444568\pi\)
0.173265 + 0.984875i \(0.444568\pi\)
\(132\) −404.619 −0.266799
\(133\) 0 0
\(134\) −854.992 −0.551195
\(135\) −94.9656 −0.0605432
\(136\) −1673.28 −1.05502
\(137\) 956.954 0.596774 0.298387 0.954445i \(-0.403551\pi\)
0.298387 + 0.954445i \(0.403551\pi\)
\(138\) −1415.17 −0.872952
\(139\) 822.763 0.502056 0.251028 0.967980i \(-0.419231\pi\)
0.251028 + 0.967980i \(0.419231\pi\)
\(140\) 0 0
\(141\) −493.890 −0.294986
\(142\) −622.376 −0.367807
\(143\) −47.5594 −0.0278120
\(144\) −105.778 −0.0612139
\(145\) 28.4614 0.0163006
\(146\) −3172.13 −1.79813
\(147\) 0 0
\(148\) 300.304 0.166790
\(149\) −2126.90 −1.16941 −0.584705 0.811246i \(-0.698790\pi\)
−0.584705 + 0.811246i \(0.698790\pi\)
\(150\) 1520.91 0.827879
\(151\) 935.949 0.504413 0.252207 0.967673i \(-0.418844\pi\)
0.252207 + 0.967673i \(0.418844\pi\)
\(152\) −1344.36 −0.717380
\(153\) −785.146 −0.414871
\(154\) 0 0
\(155\) −278.062 −0.144093
\(156\) 159.037 0.0816226
\(157\) −451.308 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(158\) 4447.50 2.23939
\(159\) 869.265 0.433567
\(160\) −725.776 −0.358610
\(161\) 0 0
\(162\) 364.601 0.176825
\(163\) −2849.10 −1.36907 −0.684535 0.728980i \(-0.739995\pi\)
−0.684535 + 0.728980i \(0.739995\pi\)
\(164\) −5346.81 −2.54583
\(165\) −116.069 −0.0547634
\(166\) −2137.46 −0.999393
\(167\) −1595.42 −0.739267 −0.369633 0.929178i \(-0.620517\pi\)
−0.369633 + 0.929178i \(0.620517\pi\)
\(168\) 0 0
\(169\) −2178.31 −0.991491
\(170\) −1381.16 −0.623116
\(171\) −630.807 −0.282099
\(172\) 4156.58 1.84265
\(173\) 2663.40 1.17049 0.585245 0.810857i \(-0.300998\pi\)
0.585245 + 0.810857i \(0.300998\pi\)
\(174\) −109.272 −0.0476084
\(175\) 0 0
\(176\) −129.284 −0.0553701
\(177\) −148.450 −0.0630405
\(178\) 1997.55 0.841139
\(179\) −300.187 −0.125347 −0.0626733 0.998034i \(-0.519963\pi\)
−0.0626733 + 0.998034i \(0.519963\pi\)
\(180\) 388.130 0.160719
\(181\) −3066.86 −1.25944 −0.629718 0.776824i \(-0.716829\pi\)
−0.629718 + 0.776824i \(0.716829\pi\)
\(182\) 0 0
\(183\) 1234.79 0.498790
\(184\) 2010.09 0.805359
\(185\) 86.1454 0.0342353
\(186\) 1067.56 0.420846
\(187\) −959.622 −0.375265
\(188\) 2018.56 0.783077
\(189\) 0 0
\(190\) −1109.66 −0.423700
\(191\) −3519.94 −1.33348 −0.666738 0.745293i \(-0.732310\pi\)
−0.666738 + 0.745293i \(0.732310\pi\)
\(192\) 2504.39 0.941348
\(193\) 1603.62 0.598087 0.299043 0.954240i \(-0.403332\pi\)
0.299043 + 0.954240i \(0.403332\pi\)
\(194\) 664.852 0.246049
\(195\) 45.6213 0.0167539
\(196\) 0 0
\(197\) 2599.80 0.940245 0.470123 0.882601i \(-0.344210\pi\)
0.470123 + 0.882601i \(0.344210\pi\)
\(198\) 445.623 0.159945
\(199\) 4317.59 1.53802 0.769010 0.639236i \(-0.220749\pi\)
0.769010 + 0.639236i \(0.220749\pi\)
\(200\) −2160.29 −0.763777
\(201\) 569.838 0.199966
\(202\) −3489.81 −1.21556
\(203\) 0 0
\(204\) 3208.93 1.10133
\(205\) −1533.79 −0.522558
\(206\) −3773.94 −1.27642
\(207\) 943.187 0.316696
\(208\) 50.8154 0.0169395
\(209\) −770.986 −0.255168
\(210\) 0 0
\(211\) −5041.22 −1.64480 −0.822399 0.568912i \(-0.807365\pi\)
−0.822399 + 0.568912i \(0.807365\pi\)
\(212\) −3552.74 −1.15096
\(213\) 414.803 0.133436
\(214\) −1553.56 −0.496258
\(215\) 1192.36 0.378224
\(216\) −517.875 −0.163134
\(217\) 0 0
\(218\) 3477.59 1.08042
\(219\) 2114.17 0.652340
\(220\) 474.381 0.145376
\(221\) 377.183 0.114806
\(222\) −330.737 −0.0999893
\(223\) 257.569 0.0773456 0.0386728 0.999252i \(-0.487687\pi\)
0.0386728 + 0.999252i \(0.487687\pi\)
\(224\) 0 0
\(225\) −1013.66 −0.300344
\(226\) −7338.86 −2.16006
\(227\) −4107.22 −1.20091 −0.600453 0.799660i \(-0.705013\pi\)
−0.600453 + 0.799660i \(0.705013\pi\)
\(228\) 2578.14 0.748867
\(229\) 3428.49 0.989350 0.494675 0.869078i \(-0.335287\pi\)
0.494675 + 0.869078i \(0.335287\pi\)
\(230\) 1659.17 0.475662
\(231\) 0 0
\(232\) 155.208 0.0439221
\(233\) 5035.51 1.41583 0.707913 0.706300i \(-0.249637\pi\)
0.707913 + 0.706300i \(0.249637\pi\)
\(234\) −175.153 −0.0489322
\(235\) 579.044 0.160735
\(236\) 606.722 0.167349
\(237\) −2964.18 −0.812423
\(238\) 0 0
\(239\) 2911.58 0.788010 0.394005 0.919108i \(-0.371089\pi\)
0.394005 + 0.919108i \(0.371089\pi\)
\(240\) 124.015 0.0333548
\(241\) 5004.15 1.33753 0.668767 0.743472i \(-0.266822\pi\)
0.668767 + 0.743472i \(0.266822\pi\)
\(242\) 544.650 0.144675
\(243\) −243.000 −0.0641500
\(244\) −5046.67 −1.32410
\(245\) 0 0
\(246\) 5888.66 1.52621
\(247\) 303.038 0.0780643
\(248\) −1516.35 −0.388260
\(249\) 1424.58 0.362567
\(250\) −3762.13 −0.951753
\(251\) 4251.88 1.06923 0.534614 0.845096i \(-0.320457\pi\)
0.534614 + 0.845096i \(0.320457\pi\)
\(252\) 0 0
\(253\) 1152.78 0.286462
\(254\) −7627.73 −1.88428
\(255\) 920.516 0.226059
\(256\) −2805.01 −0.684818
\(257\) 5094.47 1.23651 0.618257 0.785976i \(-0.287839\pi\)
0.618257 + 0.785976i \(0.287839\pi\)
\(258\) −4577.80 −1.10466
\(259\) 0 0
\(260\) −186.457 −0.0444752
\(261\) 72.8277 0.0172717
\(262\) 2338.73 0.551477
\(263\) 277.845 0.0651432 0.0325716 0.999469i \(-0.489630\pi\)
0.0325716 + 0.999469i \(0.489630\pi\)
\(264\) −632.958 −0.147560
\(265\) −1019.14 −0.236246
\(266\) 0 0
\(267\) −1331.33 −0.305155
\(268\) −2328.96 −0.530835
\(269\) −363.012 −0.0822798 −0.0411399 0.999153i \(-0.513099\pi\)
−0.0411399 + 0.999153i \(0.513099\pi\)
\(270\) −427.463 −0.0963503
\(271\) 1593.33 0.357152 0.178576 0.983926i \(-0.442851\pi\)
0.178576 + 0.983926i \(0.442851\pi\)
\(272\) 1025.32 0.228563
\(273\) 0 0
\(274\) 4307.48 0.949724
\(275\) −1238.92 −0.271671
\(276\) −3854.86 −0.840707
\(277\) 1596.25 0.346243 0.173122 0.984900i \(-0.444615\pi\)
0.173122 + 0.984900i \(0.444615\pi\)
\(278\) 3703.45 0.798987
\(279\) −711.511 −0.152678
\(280\) 0 0
\(281\) −927.809 −0.196970 −0.0984848 0.995139i \(-0.531400\pi\)
−0.0984848 + 0.995139i \(0.531400\pi\)
\(282\) −2223.12 −0.469450
\(283\) −1807.57 −0.379679 −0.189839 0.981815i \(-0.560797\pi\)
−0.189839 + 0.981815i \(0.560797\pi\)
\(284\) −1695.32 −0.354222
\(285\) 739.567 0.153713
\(286\) −214.076 −0.0442609
\(287\) 0 0
\(288\) −1857.13 −0.379974
\(289\) 2697.54 0.549061
\(290\) 128.112 0.0259413
\(291\) −443.113 −0.0892636
\(292\) −8640.73 −1.73171
\(293\) 4933.72 0.983724 0.491862 0.870673i \(-0.336316\pi\)
0.491862 + 0.870673i \(0.336316\pi\)
\(294\) 0 0
\(295\) 174.045 0.0343501
\(296\) 469.776 0.0922472
\(297\) −297.000 −0.0580259
\(298\) −9573.67 −1.86103
\(299\) −453.105 −0.0876380
\(300\) 4142.89 0.797300
\(301\) 0 0
\(302\) 4212.93 0.802738
\(303\) 2325.90 0.440988
\(304\) 823.768 0.155416
\(305\) −1447.69 −0.271785
\(306\) −3534.13 −0.660238
\(307\) 9665.11 1.79680 0.898399 0.439181i \(-0.144731\pi\)
0.898399 + 0.439181i \(0.144731\pi\)
\(308\) 0 0
\(309\) 2515.27 0.463070
\(310\) −1251.62 −0.229314
\(311\) −8084.15 −1.47399 −0.736994 0.675899i \(-0.763755\pi\)
−0.736994 + 0.675899i \(0.763755\pi\)
\(312\) 248.786 0.0451434
\(313\) 1389.42 0.250909 0.125454 0.992099i \(-0.459961\pi\)
0.125454 + 0.992099i \(0.459961\pi\)
\(314\) −2031.45 −0.365099
\(315\) 0 0
\(316\) 12114.8 2.15668
\(317\) −5478.74 −0.970715 −0.485357 0.874316i \(-0.661311\pi\)
−0.485357 + 0.874316i \(0.661311\pi\)
\(318\) 3912.77 0.689992
\(319\) 89.0116 0.0156229
\(320\) −2936.18 −0.512930
\(321\) 1035.42 0.180036
\(322\) 0 0
\(323\) 6114.51 1.05331
\(324\) 993.155 0.170294
\(325\) 486.961 0.0831130
\(326\) −12824.5 −2.17878
\(327\) −2317.76 −0.391964
\(328\) −8364.19 −1.40803
\(329\) 0 0
\(330\) −522.455 −0.0871521
\(331\) −3774.34 −0.626756 −0.313378 0.949628i \(-0.601461\pi\)
−0.313378 + 0.949628i \(0.601461\pi\)
\(332\) −5822.35 −0.962479
\(333\) 220.431 0.0362749
\(334\) −7181.38 −1.17649
\(335\) −668.086 −0.108960
\(336\) 0 0
\(337\) 537.073 0.0868138 0.0434069 0.999057i \(-0.486179\pi\)
0.0434069 + 0.999057i \(0.486179\pi\)
\(338\) −9805.08 −1.57789
\(339\) 4891.23 0.783643
\(340\) −3762.20 −0.600100
\(341\) −869.625 −0.138102
\(342\) −2839.41 −0.448941
\(343\) 0 0
\(344\) 6502.27 1.01912
\(345\) −1105.81 −0.172564
\(346\) 11988.6 1.86275
\(347\) −868.090 −0.134298 −0.0671492 0.997743i \(-0.521390\pi\)
−0.0671492 + 0.997743i \(0.521390\pi\)
\(348\) −297.651 −0.0458499
\(349\) −3069.02 −0.470719 −0.235359 0.971908i \(-0.575627\pi\)
−0.235359 + 0.971908i \(0.575627\pi\)
\(350\) 0 0
\(351\) 116.737 0.0177520
\(352\) −2269.83 −0.343699
\(353\) 2961.35 0.446507 0.223253 0.974760i \(-0.428332\pi\)
0.223253 + 0.974760i \(0.428332\pi\)
\(354\) −668.208 −0.100324
\(355\) −486.321 −0.0727078
\(356\) 5441.24 0.810070
\(357\) 0 0
\(358\) −1351.22 −0.199480
\(359\) 780.106 0.114686 0.0573432 0.998355i \(-0.481737\pi\)
0.0573432 + 0.998355i \(0.481737\pi\)
\(360\) 607.164 0.0888899
\(361\) −1946.44 −0.283780
\(362\) −13804.7 −2.00430
\(363\) −363.000 −0.0524864
\(364\) 0 0
\(365\) −2478.68 −0.355453
\(366\) 5558.10 0.793789
\(367\) 6007.56 0.854474 0.427237 0.904140i \(-0.359487\pi\)
0.427237 + 0.904140i \(0.359487\pi\)
\(368\) −1231.70 −0.174476
\(369\) −3924.69 −0.553689
\(370\) 387.761 0.0544831
\(371\) 0 0
\(372\) 2907.99 0.405301
\(373\) 4262.21 0.591659 0.295829 0.955241i \(-0.404404\pi\)
0.295829 + 0.955241i \(0.404404\pi\)
\(374\) −4319.49 −0.597207
\(375\) 2507.40 0.345284
\(376\) 3157.70 0.433100
\(377\) −34.9863 −0.00477954
\(378\) 0 0
\(379\) 1746.46 0.236701 0.118350 0.992972i \(-0.462239\pi\)
0.118350 + 0.992972i \(0.462239\pi\)
\(380\) −3022.65 −0.408049
\(381\) 5083.75 0.683592
\(382\) −15844.1 −2.12213
\(383\) 5243.49 0.699556 0.349778 0.936833i \(-0.386257\pi\)
0.349778 + 0.936833i \(0.386257\pi\)
\(384\) 6320.52 0.839954
\(385\) 0 0
\(386\) 7218.26 0.951813
\(387\) 3051.03 0.400756
\(388\) 1811.03 0.236961
\(389\) 11247.9 1.46604 0.733022 0.680205i \(-0.238109\pi\)
0.733022 + 0.680205i \(0.238109\pi\)
\(390\) 205.353 0.0266626
\(391\) −9142.46 −1.18249
\(392\) 0 0
\(393\) −1558.72 −0.200069
\(394\) 11702.3 1.49633
\(395\) 3475.25 0.442681
\(396\) 1213.86 0.154037
\(397\) 1763.94 0.222996 0.111498 0.993765i \(-0.464435\pi\)
0.111498 + 0.993765i \(0.464435\pi\)
\(398\) 19434.5 2.44765
\(399\) 0 0
\(400\) 1323.74 0.165467
\(401\) −9191.59 −1.14465 −0.572327 0.820025i \(-0.693959\pi\)
−0.572327 + 0.820025i \(0.693959\pi\)
\(402\) 2564.98 0.318232
\(403\) 341.809 0.0422499
\(404\) −9506.08 −1.17066
\(405\) 284.897 0.0349547
\(406\) 0 0
\(407\) 269.415 0.0328119
\(408\) 5019.84 0.609116
\(409\) 1535.09 0.185587 0.0927936 0.995685i \(-0.470420\pi\)
0.0927936 + 0.995685i \(0.470420\pi\)
\(410\) −6903.95 −0.831614
\(411\) −2870.86 −0.344548
\(412\) −10280.0 −1.22928
\(413\) 0 0
\(414\) 4245.51 0.503999
\(415\) −1670.20 −0.197559
\(416\) 892.162 0.105149
\(417\) −2468.29 −0.289862
\(418\) −3470.39 −0.406083
\(419\) −16887.5 −1.96899 −0.984496 0.175406i \(-0.943876\pi\)
−0.984496 + 0.175406i \(0.943876\pi\)
\(420\) 0 0
\(421\) −203.001 −0.0235004 −0.0117502 0.999931i \(-0.503740\pi\)
−0.0117502 + 0.999931i \(0.503740\pi\)
\(422\) −22691.8 −2.61758
\(423\) 1481.67 0.170310
\(424\) −5557.66 −0.636566
\(425\) 9825.57 1.12144
\(426\) 1867.13 0.212354
\(427\) 0 0
\(428\) −4231.82 −0.477927
\(429\) 142.678 0.0160573
\(430\) 5367.09 0.601916
\(431\) 4326.32 0.483506 0.241753 0.970338i \(-0.422278\pi\)
0.241753 + 0.970338i \(0.422278\pi\)
\(432\) 317.333 0.0353419
\(433\) −9703.19 −1.07692 −0.538459 0.842652i \(-0.680993\pi\)
−0.538459 + 0.842652i \(0.680993\pi\)
\(434\) 0 0
\(435\) −85.3842 −0.00941117
\(436\) 9472.80 1.04052
\(437\) −7345.29 −0.804057
\(438\) 9516.39 1.03815
\(439\) −761.339 −0.0827715 −0.0413858 0.999143i \(-0.513177\pi\)
−0.0413858 + 0.999143i \(0.513177\pi\)
\(440\) 742.090 0.0804039
\(441\) 0 0
\(442\) 1697.79 0.182705
\(443\) −8110.91 −0.869889 −0.434945 0.900457i \(-0.643232\pi\)
−0.434945 + 0.900457i \(0.643232\pi\)
\(444\) −900.913 −0.0962960
\(445\) 1560.88 0.166275
\(446\) 1159.38 0.123090
\(447\) 6380.69 0.675159
\(448\) 0 0
\(449\) −1306.72 −0.137345 −0.0686727 0.997639i \(-0.521876\pi\)
−0.0686727 + 0.997639i \(0.521876\pi\)
\(450\) −4562.73 −0.477976
\(451\) −4796.84 −0.500830
\(452\) −19990.7 −2.08028
\(453\) −2807.85 −0.291223
\(454\) −18487.6 −1.91116
\(455\) 0 0
\(456\) 4033.07 0.414180
\(457\) −10236.0 −1.04775 −0.523873 0.851796i \(-0.675513\pi\)
−0.523873 + 0.851796i \(0.675513\pi\)
\(458\) 15432.5 1.57448
\(459\) 2355.44 0.239526
\(460\) 4519.49 0.458092
\(461\) 18227.8 1.84155 0.920773 0.390100i \(-0.127559\pi\)
0.920773 + 0.390100i \(0.127559\pi\)
\(462\) 0 0
\(463\) 18797.6 1.88682 0.943412 0.331624i \(-0.107597\pi\)
0.943412 + 0.331624i \(0.107597\pi\)
\(464\) −95.1054 −0.00951542
\(465\) 834.186 0.0831924
\(466\) 22666.1 2.25319
\(467\) 3847.41 0.381235 0.190617 0.981664i \(-0.438951\pi\)
0.190617 + 0.981664i \(0.438951\pi\)
\(468\) −477.110 −0.0471248
\(469\) 0 0
\(470\) 2606.42 0.255798
\(471\) 1353.92 0.132453
\(472\) 949.116 0.0925564
\(473\) 3729.03 0.362497
\(474\) −13342.5 −1.29291
\(475\) 7894.12 0.762542
\(476\) 0 0
\(477\) −2607.80 −0.250320
\(478\) 13105.7 1.25406
\(479\) 3718.08 0.354663 0.177331 0.984151i \(-0.443254\pi\)
0.177331 + 0.984151i \(0.443254\pi\)
\(480\) 2177.33 0.207044
\(481\) −105.895 −0.0100382
\(482\) 22524.9 2.12859
\(483\) 0 0
\(484\) 1483.60 0.139331
\(485\) 519.512 0.0486388
\(486\) −1093.80 −0.102090
\(487\) 15470.2 1.43947 0.719736 0.694248i \(-0.244263\pi\)
0.719736 + 0.694248i \(0.244263\pi\)
\(488\) −7894.68 −0.732326
\(489\) 8547.29 0.790433
\(490\) 0 0
\(491\) −18899.6 −1.73713 −0.868563 0.495579i \(-0.834956\pi\)
−0.868563 + 0.495579i \(0.834956\pi\)
\(492\) 16040.4 1.46983
\(493\) −705.930 −0.0644898
\(494\) 1364.05 0.124234
\(495\) 348.207 0.0316177
\(496\) 929.160 0.0841139
\(497\) 0 0
\(498\) 6412.39 0.577000
\(499\) −9179.85 −0.823541 −0.411770 0.911288i \(-0.635089\pi\)
−0.411770 + 0.911288i \(0.635089\pi\)
\(500\) −10247.9 −0.916598
\(501\) 4786.27 0.426816
\(502\) 19138.7 1.70160
\(503\) −6929.02 −0.614214 −0.307107 0.951675i \(-0.599361\pi\)
−0.307107 + 0.951675i \(0.599361\pi\)
\(504\) 0 0
\(505\) −2726.92 −0.240290
\(506\) 5188.96 0.455884
\(507\) 6534.92 0.572438
\(508\) −20777.6 −1.81468
\(509\) 14105.0 1.22828 0.614139 0.789198i \(-0.289503\pi\)
0.614139 + 0.789198i \(0.289503\pi\)
\(510\) 4143.47 0.359756
\(511\) 0 0
\(512\) 4228.66 0.365004
\(513\) 1892.42 0.162870
\(514\) 22931.4 1.96782
\(515\) −2948.94 −0.252322
\(516\) −12469.7 −1.06385
\(517\) 1810.93 0.154052
\(518\) 0 0
\(519\) −7990.21 −0.675782
\(520\) −291.681 −0.0245982
\(521\) −12971.8 −1.09079 −0.545396 0.838178i \(-0.683621\pi\)
−0.545396 + 0.838178i \(0.683621\pi\)
\(522\) 327.815 0.0274867
\(523\) 8337.84 0.697109 0.348555 0.937288i \(-0.386673\pi\)
0.348555 + 0.937288i \(0.386673\pi\)
\(524\) 6370.58 0.531107
\(525\) 0 0
\(526\) 1250.65 0.103671
\(527\) 6896.79 0.570074
\(528\) 387.851 0.0319679
\(529\) −1184.26 −0.0973338
\(530\) −4587.39 −0.375969
\(531\) 445.349 0.0363964
\(532\) 0 0
\(533\) 1885.41 0.153220
\(534\) −5992.65 −0.485632
\(535\) −1213.94 −0.0980997
\(536\) −3643.27 −0.293592
\(537\) 900.562 0.0723689
\(538\) −1634.01 −0.130942
\(539\) 0 0
\(540\) −1164.39 −0.0927914
\(541\) 22647.9 1.79983 0.899915 0.436065i \(-0.143628\pi\)
0.899915 + 0.436065i \(0.143628\pi\)
\(542\) 7171.98 0.568382
\(543\) 9200.58 0.727136
\(544\) 18001.5 1.41876
\(545\) 2717.37 0.213577
\(546\) 0 0
\(547\) −1363.33 −0.106566 −0.0532831 0.998579i \(-0.516969\pi\)
−0.0532831 + 0.998579i \(0.516969\pi\)
\(548\) 11733.4 0.914644
\(549\) −3704.38 −0.287977
\(550\) −5576.67 −0.432346
\(551\) −567.163 −0.0438511
\(552\) −6030.28 −0.464974
\(553\) 0 0
\(554\) 7185.11 0.551022
\(555\) −258.436 −0.0197658
\(556\) 10088.0 0.769475
\(557\) 21547.2 1.63911 0.819555 0.573001i \(-0.194221\pi\)
0.819555 + 0.573001i \(0.194221\pi\)
\(558\) −3202.68 −0.242976
\(559\) −1465.71 −0.110900
\(560\) 0 0
\(561\) 2878.87 0.216659
\(562\) −4176.29 −0.313463
\(563\) −9865.47 −0.738508 −0.369254 0.929329i \(-0.620387\pi\)
−0.369254 + 0.929329i \(0.620387\pi\)
\(564\) −6055.67 −0.452110
\(565\) −5734.55 −0.426998
\(566\) −8136.32 −0.604232
\(567\) 0 0
\(568\) −2652.05 −0.195911
\(569\) 22438.4 1.65320 0.826598 0.562793i \(-0.190273\pi\)
0.826598 + 0.562793i \(0.190273\pi\)
\(570\) 3328.97 0.244623
\(571\) 14815.0 1.08579 0.542897 0.839799i \(-0.317327\pi\)
0.542897 + 0.839799i \(0.317327\pi\)
\(572\) −583.134 −0.0426260
\(573\) 10559.8 0.769882
\(574\) 0 0
\(575\) −11803.4 −0.856059
\(576\) −7513.17 −0.543487
\(577\) −1859.54 −0.134165 −0.0670827 0.997747i \(-0.521369\pi\)
−0.0670827 + 0.997747i \(0.521369\pi\)
\(578\) 12142.3 0.873792
\(579\) −4810.85 −0.345306
\(580\) 348.970 0.0249831
\(581\) 0 0
\(582\) −1994.56 −0.142057
\(583\) −3187.31 −0.226423
\(584\) −13517.0 −0.957769
\(585\) −136.864 −0.00967287
\(586\) 22207.9 1.56553
\(587\) −10171.6 −0.715211 −0.357605 0.933873i \(-0.616407\pi\)
−0.357605 + 0.933873i \(0.616407\pi\)
\(588\) 0 0
\(589\) 5541.06 0.387632
\(590\) 783.417 0.0546657
\(591\) −7799.41 −0.542851
\(592\) −287.860 −0.0199847
\(593\) 13732.9 0.950997 0.475498 0.879717i \(-0.342268\pi\)
0.475498 + 0.879717i \(0.342268\pi\)
\(594\) −1336.87 −0.0923441
\(595\) 0 0
\(596\) −26078.2 −1.79229
\(597\) −12952.8 −0.887977
\(598\) −2039.54 −0.139470
\(599\) 13180.5 0.899066 0.449533 0.893264i \(-0.351590\pi\)
0.449533 + 0.893264i \(0.351590\pi\)
\(600\) 6480.86 0.440967
\(601\) 14739.8 1.00041 0.500206 0.865907i \(-0.333258\pi\)
0.500206 + 0.865907i \(0.333258\pi\)
\(602\) 0 0
\(603\) −1709.51 −0.115451
\(604\) 11475.8 0.773087
\(605\) 425.587 0.0285993
\(606\) 10469.4 0.701802
\(607\) 25367.7 1.69628 0.848139 0.529773i \(-0.177723\pi\)
0.848139 + 0.529773i \(0.177723\pi\)
\(608\) 14462.8 0.964713
\(609\) 0 0
\(610\) −6516.40 −0.432527
\(611\) −711.792 −0.0471293
\(612\) −9626.80 −0.635850
\(613\) 1656.51 0.109145 0.0545725 0.998510i \(-0.482620\pi\)
0.0545725 + 0.998510i \(0.482620\pi\)
\(614\) 43505.0 2.85948
\(615\) 4601.36 0.301699
\(616\) 0 0
\(617\) −16741.5 −1.09236 −0.546181 0.837667i \(-0.683919\pi\)
−0.546181 + 0.837667i \(0.683919\pi\)
\(618\) 11321.8 0.736943
\(619\) 14817.1 0.962114 0.481057 0.876689i \(-0.340253\pi\)
0.481057 + 0.876689i \(0.340253\pi\)
\(620\) −3409.37 −0.220844
\(621\) −2829.56 −0.182844
\(622\) −36388.7 −2.34575
\(623\) 0 0
\(624\) −152.446 −0.00978002
\(625\) 11138.9 0.712891
\(626\) 6254.09 0.399303
\(627\) 2312.96 0.147322
\(628\) −5533.57 −0.351614
\(629\) −2136.67 −0.135445
\(630\) 0 0
\(631\) −27736.3 −1.74987 −0.874934 0.484242i \(-0.839095\pi\)
−0.874934 + 0.484242i \(0.839095\pi\)
\(632\) 18951.5 1.19280
\(633\) 15123.7 0.949624
\(634\) −24661.1 −1.54482
\(635\) −5960.27 −0.372482
\(636\) 10658.2 0.664506
\(637\) 0 0
\(638\) 400.663 0.0248627
\(639\) −1244.41 −0.0770392
\(640\) −7410.27 −0.457682
\(641\) 10887.3 0.670861 0.335431 0.942065i \(-0.391118\pi\)
0.335431 + 0.942065i \(0.391118\pi\)
\(642\) 4660.68 0.286515
\(643\) −12370.5 −0.758703 −0.379352 0.925253i \(-0.623853\pi\)
−0.379352 + 0.925253i \(0.623853\pi\)
\(644\) 0 0
\(645\) −3577.07 −0.218367
\(646\) 27522.9 1.67627
\(647\) 19992.7 1.21483 0.607414 0.794386i \(-0.292207\pi\)
0.607414 + 0.794386i \(0.292207\pi\)
\(648\) 1553.62 0.0941854
\(649\) 544.316 0.0329218
\(650\) 2191.93 0.132268
\(651\) 0 0
\(652\) −34933.3 −2.09830
\(653\) −25232.8 −1.51215 −0.756075 0.654485i \(-0.772886\pi\)
−0.756075 + 0.654485i \(0.772886\pi\)
\(654\) −10432.8 −0.623783
\(655\) 1827.47 0.109015
\(656\) 5125.24 0.305041
\(657\) −6342.51 −0.376628
\(658\) 0 0
\(659\) −12349.7 −0.730006 −0.365003 0.931006i \(-0.618932\pi\)
−0.365003 + 0.931006i \(0.618932\pi\)
\(660\) −1423.14 −0.0839329
\(661\) −25788.0 −1.51746 −0.758728 0.651407i \(-0.774179\pi\)
−0.758728 + 0.651407i \(0.774179\pi\)
\(662\) −16989.2 −0.997438
\(663\) −1131.55 −0.0662831
\(664\) −9108.09 −0.532323
\(665\) 0 0
\(666\) 992.212 0.0577289
\(667\) 848.026 0.0492289
\(668\) −19561.7 −1.13303
\(669\) −772.706 −0.0446555
\(670\) −3007.22 −0.173401
\(671\) −4527.58 −0.260485
\(672\) 0 0
\(673\) −26052.3 −1.49219 −0.746093 0.665841i \(-0.768073\pi\)
−0.746093 + 0.665841i \(0.768073\pi\)
\(674\) 2417.50 0.138158
\(675\) 3040.98 0.173404
\(676\) −26708.6 −1.51961
\(677\) −21618.7 −1.22729 −0.613645 0.789582i \(-0.710297\pi\)
−0.613645 + 0.789582i \(0.710297\pi\)
\(678\) 22016.6 1.24711
\(679\) 0 0
\(680\) −5885.34 −0.331901
\(681\) 12321.7 0.693343
\(682\) −3914.39 −0.219780
\(683\) −23085.3 −1.29331 −0.646657 0.762781i \(-0.723834\pi\)
−0.646657 + 0.762781i \(0.723834\pi\)
\(684\) −7734.43 −0.432359
\(685\) 3365.84 0.187740
\(686\) 0 0
\(687\) −10285.5 −0.571201
\(688\) −3984.33 −0.220786
\(689\) 1252.78 0.0692702
\(690\) −4977.50 −0.274624
\(691\) 19261.9 1.06043 0.530215 0.847863i \(-0.322111\pi\)
0.530215 + 0.847863i \(0.322111\pi\)
\(692\) 32656.4 1.79395
\(693\) 0 0
\(694\) −3907.48 −0.213726
\(695\) 2893.86 0.157943
\(696\) −465.625 −0.0253584
\(697\) 38042.6 2.06738
\(698\) −13814.4 −0.749116
\(699\) −15106.5 −0.817427
\(700\) 0 0
\(701\) 15422.5 0.830957 0.415478 0.909603i \(-0.363614\pi\)
0.415478 + 0.909603i \(0.363614\pi\)
\(702\) 525.460 0.0282510
\(703\) −1716.66 −0.0920981
\(704\) −9182.76 −0.491603
\(705\) −1737.13 −0.0928003
\(706\) 13329.8 0.710584
\(707\) 0 0
\(708\) −1820.17 −0.0966187
\(709\) −11970.7 −0.634091 −0.317046 0.948410i \(-0.602691\pi\)
−0.317046 + 0.948410i \(0.602691\pi\)
\(710\) −2189.05 −0.115709
\(711\) 8892.54 0.469053
\(712\) 8511.91 0.448030
\(713\) −8285.04 −0.435171
\(714\) 0 0
\(715\) −167.278 −0.00874944
\(716\) −3680.65 −0.192112
\(717\) −8734.73 −0.454958
\(718\) 3511.45 0.182515
\(719\) −19840.4 −1.02910 −0.514550 0.857460i \(-0.672041\pi\)
−0.514550 + 0.857460i \(0.672041\pi\)
\(720\) −372.046 −0.0192574
\(721\) 0 0
\(722\) −8761.41 −0.451615
\(723\) −15012.5 −0.772226
\(724\) −37603.3 −1.93027
\(725\) −911.390 −0.0466871
\(726\) −1633.95 −0.0835284
\(727\) −27682.1 −1.41220 −0.706102 0.708110i \(-0.749548\pi\)
−0.706102 + 0.708110i \(0.749548\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) −11157.2 −0.565678
\(731\) −29574.1 −1.49636
\(732\) 15140.0 0.764469
\(733\) 1687.23 0.0850192 0.0425096 0.999096i \(-0.486465\pi\)
0.0425096 + 0.999096i \(0.486465\pi\)
\(734\) 27041.5 1.35983
\(735\) 0 0
\(736\) −21625.0 −1.08303
\(737\) −2089.41 −0.104429
\(738\) −17666.0 −0.881156
\(739\) −36046.7 −1.79432 −0.897158 0.441711i \(-0.854372\pi\)
−0.897158 + 0.441711i \(0.854372\pi\)
\(740\) 1056.24 0.0524707
\(741\) −909.115 −0.0450704
\(742\) 0 0
\(743\) 13448.3 0.664027 0.332013 0.943275i \(-0.392272\pi\)
0.332013 + 0.943275i \(0.392272\pi\)
\(744\) 4549.06 0.224162
\(745\) −7480.81 −0.367887
\(746\) 19185.2 0.941583
\(747\) −4273.75 −0.209328
\(748\) −11766.1 −0.575148
\(749\) 0 0
\(750\) 11286.4 0.549495
\(751\) 30941.5 1.50342 0.751711 0.659492i \(-0.229229\pi\)
0.751711 + 0.659492i \(0.229229\pi\)
\(752\) −1934.91 −0.0938283
\(753\) −12755.6 −0.617319
\(754\) −157.482 −0.00760629
\(755\) 3291.96 0.158684
\(756\) 0 0
\(757\) −24433.7 −1.17313 −0.586565 0.809902i \(-0.699520\pi\)
−0.586565 + 0.809902i \(0.699520\pi\)
\(758\) 7861.23 0.376692
\(759\) −3458.35 −0.165389
\(760\) −4728.43 −0.225682
\(761\) 18941.4 0.902265 0.451133 0.892457i \(-0.351020\pi\)
0.451133 + 0.892457i \(0.351020\pi\)
\(762\) 22883.2 1.08789
\(763\) 0 0
\(764\) −43158.6 −2.04375
\(765\) −2761.55 −0.130515
\(766\) 23602.2 1.11329
\(767\) −213.945 −0.0100718
\(768\) 8415.04 0.395380
\(769\) −19551.0 −0.916810 −0.458405 0.888743i \(-0.651579\pi\)
−0.458405 + 0.888743i \(0.651579\pi\)
\(770\) 0 0
\(771\) −15283.4 −0.713902
\(772\) 19662.2 0.916656
\(773\) 16899.5 0.786331 0.393165 0.919468i \(-0.371380\pi\)
0.393165 + 0.919468i \(0.371380\pi\)
\(774\) 13733.4 0.637774
\(775\) 8904.09 0.412702
\(776\) 2833.05 0.131057
\(777\) 0 0
\(778\) 50629.5 2.33310
\(779\) 30564.4 1.40576
\(780\) 559.371 0.0256778
\(781\) −1520.94 −0.0696846
\(782\) −41152.4 −1.88185
\(783\) −218.483 −0.00997183
\(784\) 0 0
\(785\) −1587.36 −0.0721724
\(786\) −7016.18 −0.318395
\(787\) −19764.5 −0.895207 −0.447604 0.894232i \(-0.647722\pi\)
−0.447604 + 0.894232i \(0.647722\pi\)
\(788\) 31876.6 1.44106
\(789\) −833.536 −0.0376105
\(790\) 15642.9 0.704495
\(791\) 0 0
\(792\) 1898.87 0.0851939
\(793\) 1779.58 0.0796907
\(794\) 7939.90 0.354882
\(795\) 3057.42 0.136397
\(796\) 52938.8 2.35724
\(797\) −7851.98 −0.348973 −0.174486 0.984660i \(-0.555826\pi\)
−0.174486 + 0.984660i \(0.555826\pi\)
\(798\) 0 0
\(799\) −14362.1 −0.635912
\(800\) 23240.7 1.02711
\(801\) 3994.00 0.176181
\(802\) −41373.6 −1.82164
\(803\) −7751.96 −0.340673
\(804\) 6986.88 0.306478
\(805\) 0 0
\(806\) 1538.56 0.0672377
\(807\) 1089.04 0.0475043
\(808\) −14870.7 −0.647461
\(809\) 38484.1 1.67247 0.836235 0.548371i \(-0.184752\pi\)
0.836235 + 0.548371i \(0.184752\pi\)
\(810\) 1282.39 0.0556278
\(811\) −3444.50 −0.149140 −0.0745702 0.997216i \(-0.523758\pi\)
−0.0745702 + 0.997216i \(0.523758\pi\)
\(812\) 0 0
\(813\) −4780.00 −0.206202
\(814\) 1212.70 0.0522177
\(815\) −10021.0 −0.430699
\(816\) −3075.96 −0.131961
\(817\) −23760.6 −1.01748
\(818\) 6909.80 0.295349
\(819\) 0 0
\(820\) −18806.0 −0.800897
\(821\) 23322.5 0.991426 0.495713 0.868486i \(-0.334907\pi\)
0.495713 + 0.868486i \(0.334907\pi\)
\(822\) −12922.4 −0.548324
\(823\) −5127.50 −0.217173 −0.108587 0.994087i \(-0.534632\pi\)
−0.108587 + 0.994087i \(0.534632\pi\)
\(824\) −16081.4 −0.679882
\(825\) 3716.76 0.156850
\(826\) 0 0
\(827\) 20030.5 0.842235 0.421117 0.907006i \(-0.361638\pi\)
0.421117 + 0.907006i \(0.361638\pi\)
\(828\) 11564.6 0.485383
\(829\) 17841.2 0.747467 0.373733 0.927536i \(-0.378077\pi\)
0.373733 + 0.927536i \(0.378077\pi\)
\(830\) −7517.98 −0.314401
\(831\) −4788.75 −0.199904
\(832\) 3609.31 0.150397
\(833\) 0 0
\(834\) −11110.4 −0.461295
\(835\) −5611.49 −0.232567
\(836\) −9453.19 −0.391083
\(837\) 2134.53 0.0881485
\(838\) −76014.7 −3.13351
\(839\) −78.6997 −0.00323840 −0.00161920 0.999999i \(-0.500515\pi\)
−0.00161920 + 0.999999i \(0.500515\pi\)
\(840\) 0 0
\(841\) −24323.5 −0.997315
\(842\) −913.756 −0.0373992
\(843\) 2783.43 0.113720
\(844\) −61811.3 −2.52089
\(845\) −7661.64 −0.311915
\(846\) 6669.36 0.271037
\(847\) 0 0
\(848\) 3405.51 0.137908
\(849\) 5422.72 0.219208
\(850\) 44227.3 1.78469
\(851\) 2566.76 0.103393
\(852\) 5085.97 0.204510
\(853\) 21540.4 0.864630 0.432315 0.901723i \(-0.357697\pi\)
0.432315 + 0.901723i \(0.357697\pi\)
\(854\) 0 0
\(855\) −2218.70 −0.0887462
\(856\) −6619.98 −0.264330
\(857\) −47343.8 −1.88709 −0.943543 0.331251i \(-0.892529\pi\)
−0.943543 + 0.331251i \(0.892529\pi\)
\(858\) 642.229 0.0255540
\(859\) −41886.8 −1.66375 −0.831873 0.554966i \(-0.812731\pi\)
−0.831873 + 0.554966i \(0.812731\pi\)
\(860\) 14619.7 0.579683
\(861\) 0 0
\(862\) 19473.8 0.769466
\(863\) 6778.32 0.267366 0.133683 0.991024i \(-0.457320\pi\)
0.133683 + 0.991024i \(0.457320\pi\)
\(864\) 5571.39 0.219378
\(865\) 9367.83 0.368227
\(866\) −43676.4 −1.71384
\(867\) −8092.61 −0.317001
\(868\) 0 0
\(869\) 10868.7 0.424274
\(870\) −384.335 −0.0149772
\(871\) 821.247 0.0319482
\(872\) 14818.6 0.575483
\(873\) 1329.34 0.0515364
\(874\) −33062.9 −1.27960
\(875\) 0 0
\(876\) 25922.2 0.999806
\(877\) 16658.1 0.641396 0.320698 0.947182i \(-0.396083\pi\)
0.320698 + 0.947182i \(0.396083\pi\)
\(878\) −3426.97 −0.131725
\(879\) −14801.2 −0.567953
\(880\) −454.723 −0.0174190
\(881\) −18477.7 −0.706618 −0.353309 0.935507i \(-0.614944\pi\)
−0.353309 + 0.935507i \(0.614944\pi\)
\(882\) 0 0
\(883\) −17383.7 −0.662523 −0.331261 0.943539i \(-0.607474\pi\)
−0.331261 + 0.943539i \(0.607474\pi\)
\(884\) 4624.70 0.175956
\(885\) −522.134 −0.0198320
\(886\) −36509.2 −1.38437
\(887\) 37232.6 1.40941 0.704706 0.709500i \(-0.251079\pi\)
0.704706 + 0.709500i \(0.251079\pi\)
\(888\) −1409.33 −0.0532590
\(889\) 0 0
\(890\) 7025.88 0.264616
\(891\) 891.000 0.0335013
\(892\) 3158.09 0.118543
\(893\) −11538.9 −0.432400
\(894\) 28721.0 1.07447
\(895\) −1055.83 −0.0394330
\(896\) 0 0
\(897\) 1359.32 0.0505978
\(898\) −5881.88 −0.218576
\(899\) −639.725 −0.0237330
\(900\) −12428.7 −0.460321
\(901\) 25277.8 0.934656
\(902\) −21591.7 −0.797036
\(903\) 0 0
\(904\) −31272.1 −1.15055
\(905\) −10786.9 −0.396208
\(906\) −12638.8 −0.463461
\(907\) −2483.45 −0.0909168 −0.0454584 0.998966i \(-0.514475\pi\)
−0.0454584 + 0.998966i \(0.514475\pi\)
\(908\) −50359.3 −1.84056
\(909\) −6977.70 −0.254605
\(910\) 0 0
\(911\) −17734.9 −0.644987 −0.322494 0.946572i \(-0.604521\pi\)
−0.322494 + 0.946572i \(0.604521\pi\)
\(912\) −2471.30 −0.0897292
\(913\) −5223.47 −0.189345
\(914\) −46074.7 −1.66741
\(915\) 4343.07 0.156915
\(916\) 42037.3 1.51632
\(917\) 0 0
\(918\) 10602.4 0.381188
\(919\) −54311.4 −1.94947 −0.974737 0.223356i \(-0.928299\pi\)
−0.974737 + 0.223356i \(0.928299\pi\)
\(920\) 7069.99 0.253360
\(921\) −28995.3 −1.03738
\(922\) 82047.6 2.93069
\(923\) 597.812 0.0213188
\(924\) 0 0
\(925\) −2758.54 −0.0980544
\(926\) 84612.6 3.00275
\(927\) −7545.81 −0.267354
\(928\) −1669.76 −0.0590652
\(929\) −25708.7 −0.907939 −0.453970 0.891017i \(-0.649993\pi\)
−0.453970 + 0.891017i \(0.649993\pi\)
\(930\) 3754.87 0.132395
\(931\) 0 0
\(932\) 61741.3 2.16996
\(933\) 24252.5 0.851007
\(934\) 17318.1 0.606708
\(935\) −3375.23 −0.118055
\(936\) −746.359 −0.0260636
\(937\) 38383.7 1.33825 0.669125 0.743150i \(-0.266669\pi\)
0.669125 + 0.743150i \(0.266669\pi\)
\(938\) 0 0
\(939\) −4168.25 −0.144862
\(940\) 7099.76 0.246350
\(941\) −5989.47 −0.207493 −0.103747 0.994604i \(-0.533083\pi\)
−0.103747 + 0.994604i \(0.533083\pi\)
\(942\) 6094.34 0.210790
\(943\) −45700.2 −1.57816
\(944\) −581.580 −0.0200517
\(945\) 0 0
\(946\) 16785.3 0.576889
\(947\) 4018.91 0.137906 0.0689530 0.997620i \(-0.478034\pi\)
0.0689530 + 0.997620i \(0.478034\pi\)
\(948\) −36344.3 −1.24516
\(949\) 3046.93 0.104223
\(950\) 35533.4 1.21353
\(951\) 16436.2 0.560442
\(952\) 0 0
\(953\) 1591.38 0.0540920 0.0270460 0.999634i \(-0.491390\pi\)
0.0270460 + 0.999634i \(0.491390\pi\)
\(954\) −11738.3 −0.398367
\(955\) −12380.5 −0.419500
\(956\) 35699.3 1.20774
\(957\) −267.035 −0.00901986
\(958\) 16736.0 0.564420
\(959\) 0 0
\(960\) 8808.55 0.296140
\(961\) −23541.0 −0.790206
\(962\) −476.657 −0.0159751
\(963\) −3106.26 −0.103944
\(964\) 61356.8 2.04997
\(965\) 5640.31 0.188153
\(966\) 0 0
\(967\) −51810.1 −1.72296 −0.861478 0.507794i \(-0.830461\pi\)
−0.861478 + 0.507794i \(0.830461\pi\)
\(968\) 2320.85 0.0770608
\(969\) −18343.5 −0.608131
\(970\) 2338.45 0.0774052
\(971\) 39288.4 1.29848 0.649240 0.760584i \(-0.275087\pi\)
0.649240 + 0.760584i \(0.275087\pi\)
\(972\) −2979.46 −0.0983193
\(973\) 0 0
\(974\) 69635.2 2.29082
\(975\) −1460.88 −0.0479853
\(976\) 4837.54 0.158654
\(977\) 54140.1 1.77287 0.886436 0.462852i \(-0.153174\pi\)
0.886436 + 0.462852i \(0.153174\pi\)
\(978\) 38473.4 1.25792
\(979\) 4881.56 0.159362
\(980\) 0 0
\(981\) 6953.27 0.226300
\(982\) −85071.8 −2.76451
\(983\) 18394.4 0.596835 0.298417 0.954435i \(-0.403541\pi\)
0.298417 + 0.954435i \(0.403541\pi\)
\(984\) 25092.6 0.812929
\(985\) 9144.14 0.295794
\(986\) −3177.56 −0.102631
\(987\) 0 0
\(988\) 3715.61 0.119645
\(989\) 35527.0 1.14226
\(990\) 1567.36 0.0503173
\(991\) −38764.2 −1.24257 −0.621283 0.783586i \(-0.713388\pi\)
−0.621283 + 0.783586i \(0.713388\pi\)
\(992\) 16313.2 0.522121
\(993\) 11323.0 0.361858
\(994\) 0 0
\(995\) 15186.0 0.483849
\(996\) 17467.0 0.555687
\(997\) −34556.3 −1.09770 −0.548851 0.835920i \(-0.684935\pi\)
−0.548851 + 0.835920i \(0.684935\pi\)
\(998\) −41320.7 −1.31061
\(999\) −661.292 −0.0209433
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 1617.4.a.be.1.14 16
7.6 odd 2 1617.4.a.bf.1.14 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1617.4.a.be.1.14 16 1.1 even 1 trivial
1617.4.a.bf.1.14 yes 16 7.6 odd 2