Properties

Label 1617.4.a.be.1.2
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 1.2
Root \(-4.48532\) 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)
 
\(f(q)\) \(=\) \(q-4.48532 q^{2} -3.00000 q^{3} +12.1181 q^{4} +17.6644 q^{5} +13.4559 q^{6} -18.4708 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-4.48532 q^{2} -3.00000 q^{3} +12.1181 q^{4} +17.6644 q^{5} +13.4559 q^{6} -18.4708 q^{8} +9.00000 q^{9} -79.2305 q^{10} +11.0000 q^{11} -36.3542 q^{12} -12.6777 q^{13} -52.9933 q^{15} -14.0972 q^{16} -64.0201 q^{17} -40.3678 q^{18} +164.419 q^{19} +214.058 q^{20} -49.3385 q^{22} -69.8858 q^{23} +55.4123 q^{24} +187.032 q^{25} +56.8635 q^{26} -27.0000 q^{27} +193.871 q^{29} +237.692 q^{30} -23.9089 q^{31} +210.997 q^{32} -33.0000 q^{33} +287.150 q^{34} +109.062 q^{36} -135.778 q^{37} -737.472 q^{38} +38.0331 q^{39} -326.276 q^{40} -400.174 q^{41} -557.868 q^{43} +133.299 q^{44} +158.980 q^{45} +313.460 q^{46} -385.997 q^{47} +42.2916 q^{48} -838.898 q^{50} +192.060 q^{51} -153.629 q^{52} -401.985 q^{53} +121.104 q^{54} +194.309 q^{55} -493.258 q^{57} -869.571 q^{58} +518.682 q^{59} -642.175 q^{60} -172.267 q^{61} +107.239 q^{62} -833.608 q^{64} -223.945 q^{65} +148.015 q^{66} -30.6570 q^{67} -775.799 q^{68} +209.657 q^{69} -300.247 q^{71} -166.237 q^{72} +90.7014 q^{73} +609.007 q^{74} -561.096 q^{75} +1992.44 q^{76} -170.591 q^{78} -500.357 q^{79} -249.019 q^{80} +81.0000 q^{81} +1794.91 q^{82} -935.293 q^{83} -1130.88 q^{85} +2502.21 q^{86} -581.612 q^{87} -203.178 q^{88} -143.060 q^{89} -713.075 q^{90} -846.880 q^{92} +71.7266 q^{93} +1731.32 q^{94} +2904.37 q^{95} -632.990 q^{96} -217.510 q^{97} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{2} - 48 q^{3} + 72 q^{4} - 40 q^{5} - 12 q^{6} + 66 q^{8} + 144 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.48532 −1.58580 −0.792899 0.609353i \(-0.791429\pi\)
−0.792899 + 0.609353i \(0.791429\pi\)
\(3\) −3.00000 −0.577350
\(4\) 12.1181 1.51476
\(5\) 17.6644 1.57995 0.789977 0.613136i \(-0.210092\pi\)
0.789977 + 0.613136i \(0.210092\pi\)
\(6\) 13.4559 0.915561
\(7\) 0 0
\(8\) −18.4708 −0.816300
\(9\) 9.00000 0.333333
\(10\) −79.2305 −2.50549
\(11\) 11.0000 0.301511
\(12\) −36.3542 −0.874545
\(13\) −12.6777 −0.270474 −0.135237 0.990813i \(-0.543180\pi\)
−0.135237 + 0.990813i \(0.543180\pi\)
\(14\) 0 0
\(15\) −52.9933 −0.912187
\(16\) −14.0972 −0.220269
\(17\) −64.0201 −0.913362 −0.456681 0.889631i \(-0.650962\pi\)
−0.456681 + 0.889631i \(0.650962\pi\)
\(18\) −40.3678 −0.528599
\(19\) 164.419 1.98528 0.992641 0.121092i \(-0.0386395\pi\)
0.992641 + 0.121092i \(0.0386395\pi\)
\(20\) 214.058 2.39325
\(21\) 0 0
\(22\) −49.3385 −0.478136
\(23\) −69.8858 −0.633574 −0.316787 0.948497i \(-0.602604\pi\)
−0.316787 + 0.948497i \(0.602604\pi\)
\(24\) 55.4123 0.471291
\(25\) 187.032 1.49626
\(26\) 56.8635 0.428918
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 193.871 1.24141 0.620705 0.784044i \(-0.286847\pi\)
0.620705 + 0.784044i \(0.286847\pi\)
\(30\) 237.692 1.44654
\(31\) −23.9089 −0.138521 −0.0692607 0.997599i \(-0.522064\pi\)
−0.0692607 + 0.997599i \(0.522064\pi\)
\(32\) 210.997 1.16560
\(33\) −33.0000 −0.174078
\(34\) 287.150 1.44841
\(35\) 0 0
\(36\) 109.062 0.504919
\(37\) −135.778 −0.603291 −0.301645 0.953420i \(-0.597536\pi\)
−0.301645 + 0.953420i \(0.597536\pi\)
\(38\) −737.472 −3.14826
\(39\) 38.0331 0.156158
\(40\) −326.276 −1.28972
\(41\) −400.174 −1.52431 −0.762155 0.647394i \(-0.775859\pi\)
−0.762155 + 0.647394i \(0.775859\pi\)
\(42\) 0 0
\(43\) −557.868 −1.97847 −0.989234 0.146344i \(-0.953249\pi\)
−0.989234 + 0.146344i \(0.953249\pi\)
\(44\) 133.299 0.456716
\(45\) 158.980 0.526652
\(46\) 313.460 1.00472
\(47\) −385.997 −1.19795 −0.598973 0.800769i \(-0.704424\pi\)
−0.598973 + 0.800769i \(0.704424\pi\)
\(48\) 42.2916 0.127172
\(49\) 0 0
\(50\) −838.898 −2.37276
\(51\) 192.060 0.527330
\(52\) −153.629 −0.409703
\(53\) −401.985 −1.04183 −0.520914 0.853609i \(-0.674409\pi\)
−0.520914 + 0.853609i \(0.674409\pi\)
\(54\) 121.104 0.305187
\(55\) 194.309 0.476374
\(56\) 0 0
\(57\) −493.258 −1.14620
\(58\) −869.571 −1.96863
\(59\) 518.682 1.14452 0.572260 0.820072i \(-0.306067\pi\)
0.572260 + 0.820072i \(0.306067\pi\)
\(60\) −642.175 −1.38174
\(61\) −172.267 −0.361582 −0.180791 0.983522i \(-0.557866\pi\)
−0.180791 + 0.983522i \(0.557866\pi\)
\(62\) 107.239 0.219667
\(63\) 0 0
\(64\) −833.608 −1.62814
\(65\) −223.945 −0.427337
\(66\) 148.015 0.276052
\(67\) −30.6570 −0.0559007 −0.0279503 0.999609i \(-0.508898\pi\)
−0.0279503 + 0.999609i \(0.508898\pi\)
\(68\) −775.799 −1.38352
\(69\) 209.657 0.365794
\(70\) 0 0
\(71\) −300.247 −0.501870 −0.250935 0.968004i \(-0.580738\pi\)
−0.250935 + 0.968004i \(0.580738\pi\)
\(72\) −166.237 −0.272100
\(73\) 90.7014 0.145422 0.0727109 0.997353i \(-0.476835\pi\)
0.0727109 + 0.997353i \(0.476835\pi\)
\(74\) 609.007 0.956698
\(75\) −561.096 −0.863864
\(76\) 1992.44 3.00722
\(77\) 0 0
\(78\) −170.591 −0.247636
\(79\) −500.357 −0.712590 −0.356295 0.934374i \(-0.615960\pi\)
−0.356295 + 0.934374i \(0.615960\pi\)
\(80\) −249.019 −0.348015
\(81\) 81.0000 0.111111
\(82\) 1794.91 2.41725
\(83\) −935.293 −1.23689 −0.618444 0.785829i \(-0.712237\pi\)
−0.618444 + 0.785829i \(0.712237\pi\)
\(84\) 0 0
\(85\) −1130.88 −1.44307
\(86\) 2502.21 3.13745
\(87\) −581.612 −0.716728
\(88\) −203.178 −0.246124
\(89\) −143.060 −0.170386 −0.0851931 0.996364i \(-0.527151\pi\)
−0.0851931 + 0.996364i \(0.527151\pi\)
\(90\) −713.075 −0.835163
\(91\) 0 0
\(92\) −846.880 −0.959710
\(93\) 71.7266 0.0799753
\(94\) 1731.32 1.89970
\(95\) 2904.37 3.13666
\(96\) −632.990 −0.672961
\(97\) −217.510 −0.227679 −0.113839 0.993499i \(-0.536315\pi\)
−0.113839 + 0.993499i \(0.536315\pi\)
\(98\) 0 0
\(99\) 99.0000 0.100504
\(100\) 2266.46 2.26646
\(101\) 1510.62 1.48824 0.744122 0.668044i \(-0.232868\pi\)
0.744122 + 0.668044i \(0.232868\pi\)
\(102\) −861.451 −0.836238
\(103\) −466.405 −0.446177 −0.223088 0.974798i \(-0.571614\pi\)
−0.223088 + 0.974798i \(0.571614\pi\)
\(104\) 234.167 0.220788
\(105\) 0 0
\(106\) 1803.03 1.65213
\(107\) −610.806 −0.551859 −0.275929 0.961178i \(-0.588986\pi\)
−0.275929 + 0.961178i \(0.588986\pi\)
\(108\) −327.187 −0.291515
\(109\) −1514.80 −1.33111 −0.665556 0.746348i \(-0.731805\pi\)
−0.665556 + 0.746348i \(0.731805\pi\)
\(110\) −871.536 −0.755433
\(111\) 407.334 0.348310
\(112\) 0 0
\(113\) −1971.57 −1.64132 −0.820661 0.571415i \(-0.806395\pi\)
−0.820661 + 0.571415i \(0.806395\pi\)
\(114\) 2212.42 1.81765
\(115\) −1234.49 −1.00102
\(116\) 2349.34 1.88043
\(117\) −114.099 −0.0901581
\(118\) −2326.45 −1.81498
\(119\) 0 0
\(120\) 978.827 0.744619
\(121\) 121.000 0.0909091
\(122\) 772.672 0.573397
\(123\) 1200.52 0.880061
\(124\) −289.729 −0.209826
\(125\) 1095.76 0.784062
\(126\) 0 0
\(127\) 489.124 0.341754 0.170877 0.985292i \(-0.445340\pi\)
0.170877 + 0.985292i \(0.445340\pi\)
\(128\) 2051.02 1.41630
\(129\) 1673.60 1.14227
\(130\) 1004.46 0.677670
\(131\) 40.7008 0.0271454 0.0135727 0.999908i \(-0.495680\pi\)
0.0135727 + 0.999908i \(0.495680\pi\)
\(132\) −399.896 −0.263685
\(133\) 0 0
\(134\) 137.506 0.0886472
\(135\) −476.940 −0.304062
\(136\) 1182.50 0.745577
\(137\) 2051.83 1.27956 0.639780 0.768558i \(-0.279025\pi\)
0.639780 + 0.768558i \(0.279025\pi\)
\(138\) −940.380 −0.580076
\(139\) −2495.57 −1.52281 −0.761407 0.648274i \(-0.775491\pi\)
−0.761407 + 0.648274i \(0.775491\pi\)
\(140\) 0 0
\(141\) 1157.99 0.691634
\(142\) 1346.70 0.795864
\(143\) −139.455 −0.0815510
\(144\) −126.875 −0.0734230
\(145\) 3424.61 1.96137
\(146\) −406.824 −0.230610
\(147\) 0 0
\(148\) −1645.36 −0.913839
\(149\) 2450.34 1.34725 0.673623 0.739075i \(-0.264737\pi\)
0.673623 + 0.739075i \(0.264737\pi\)
\(150\) 2516.69 1.36991
\(151\) 935.344 0.504087 0.252044 0.967716i \(-0.418897\pi\)
0.252044 + 0.967716i \(0.418897\pi\)
\(152\) −3036.95 −1.62059
\(153\) −576.181 −0.304454
\(154\) 0 0
\(155\) −422.337 −0.218857
\(156\) 460.887 0.236542
\(157\) 2169.85 1.10301 0.551506 0.834171i \(-0.314053\pi\)
0.551506 + 0.834171i \(0.314053\pi\)
\(158\) 2244.26 1.13002
\(159\) 1205.96 0.601500
\(160\) 3727.13 1.84160
\(161\) 0 0
\(162\) −363.311 −0.176200
\(163\) 1092.44 0.524948 0.262474 0.964939i \(-0.415462\pi\)
0.262474 + 0.964939i \(0.415462\pi\)
\(164\) −4849.33 −2.30896
\(165\) −582.926 −0.275035
\(166\) 4195.08 1.96146
\(167\) 1112.55 0.515521 0.257761 0.966209i \(-0.417015\pi\)
0.257761 + 0.966209i \(0.417015\pi\)
\(168\) 0 0
\(169\) −2036.28 −0.926844
\(170\) 5072.34 2.28842
\(171\) 1479.77 0.661761
\(172\) −6760.28 −2.99690
\(173\) 4463.99 1.96180 0.980899 0.194518i \(-0.0623143\pi\)
0.980899 + 0.194518i \(0.0623143\pi\)
\(174\) 2608.71 1.13659
\(175\) 0 0
\(176\) −155.069 −0.0664136
\(177\) −1556.05 −0.660789
\(178\) 641.671 0.270198
\(179\) −2435.80 −1.01710 −0.508548 0.861033i \(-0.669818\pi\)
−0.508548 + 0.861033i \(0.669818\pi\)
\(180\) 1926.53 0.797749
\(181\) −1943.67 −0.798188 −0.399094 0.916910i \(-0.630675\pi\)
−0.399094 + 0.916910i \(0.630675\pi\)
\(182\) 0 0
\(183\) 516.801 0.208760
\(184\) 1290.84 0.517186
\(185\) −2398.44 −0.953172
\(186\) −321.716 −0.126825
\(187\) −704.221 −0.275389
\(188\) −4677.53 −1.81460
\(189\) 0 0
\(190\) −13027.0 −4.97410
\(191\) 1841.30 0.697550 0.348775 0.937206i \(-0.386598\pi\)
0.348775 + 0.937206i \(0.386598\pi\)
\(192\) 2500.83 0.940008
\(193\) −3214.92 −1.19904 −0.599522 0.800359i \(-0.704642\pi\)
−0.599522 + 0.800359i \(0.704642\pi\)
\(194\) 975.602 0.361052
\(195\) 671.834 0.246723
\(196\) 0 0
\(197\) 3560.56 1.28771 0.643856 0.765147i \(-0.277334\pi\)
0.643856 + 0.765147i \(0.277334\pi\)
\(198\) −444.046 −0.159379
\(199\) −1379.18 −0.491295 −0.245648 0.969359i \(-0.579001\pi\)
−0.245648 + 0.969359i \(0.579001\pi\)
\(200\) −3454.62 −1.22139
\(201\) 91.9709 0.0322743
\(202\) −6775.62 −2.36005
\(203\) 0 0
\(204\) 2327.40 0.798776
\(205\) −7068.85 −2.40834
\(206\) 2091.97 0.707547
\(207\) −628.972 −0.211191
\(208\) 178.720 0.0595771
\(209\) 1808.61 0.598585
\(210\) 0 0
\(211\) 2395.25 0.781498 0.390749 0.920497i \(-0.372216\pi\)
0.390749 + 0.920497i \(0.372216\pi\)
\(212\) −4871.28 −1.57812
\(213\) 900.741 0.289755
\(214\) 2739.66 0.875137
\(215\) −9854.42 −3.12589
\(216\) 498.711 0.157097
\(217\) 0 0
\(218\) 6794.34 2.11087
\(219\) −272.104 −0.0839593
\(220\) 2354.64 0.721591
\(221\) 811.628 0.247041
\(222\) −1827.02 −0.552350
\(223\) −3425.31 −1.02859 −0.514295 0.857614i \(-0.671946\pi\)
−0.514295 + 0.857614i \(0.671946\pi\)
\(224\) 0 0
\(225\) 1683.29 0.498752
\(226\) 8843.10 2.60281
\(227\) 4888.33 1.42929 0.714647 0.699485i \(-0.246587\pi\)
0.714647 + 0.699485i \(0.246587\pi\)
\(228\) −5977.32 −1.73622
\(229\) −5076.02 −1.46477 −0.732386 0.680890i \(-0.761593\pi\)
−0.732386 + 0.680890i \(0.761593\pi\)
\(230\) 5537.09 1.58741
\(231\) 0 0
\(232\) −3580.94 −1.01336
\(233\) 2337.03 0.657099 0.328550 0.944487i \(-0.393440\pi\)
0.328550 + 0.944487i \(0.393440\pi\)
\(234\) 511.772 0.142973
\(235\) −6818.42 −1.89270
\(236\) 6285.42 1.73367
\(237\) 1501.07 0.411414
\(238\) 0 0
\(239\) −945.904 −0.256006 −0.128003 0.991774i \(-0.540857\pi\)
−0.128003 + 0.991774i \(0.540857\pi\)
\(240\) 747.058 0.200927
\(241\) 1633.69 0.436661 0.218331 0.975875i \(-0.429939\pi\)
0.218331 + 0.975875i \(0.429939\pi\)
\(242\) −542.723 −0.144163
\(243\) −243.000 −0.0641500
\(244\) −2087.54 −0.547709
\(245\) 0 0
\(246\) −5384.72 −1.39560
\(247\) −2084.46 −0.536968
\(248\) 441.615 0.113075
\(249\) 2805.88 0.714118
\(250\) −4914.83 −1.24336
\(251\) −274.771 −0.0690973 −0.0345486 0.999403i \(-0.510999\pi\)
−0.0345486 + 0.999403i \(0.510999\pi\)
\(252\) 0 0
\(253\) −768.744 −0.191030
\(254\) −2193.88 −0.541953
\(255\) 3392.63 0.833157
\(256\) −2530.62 −0.617828
\(257\) −2957.52 −0.717841 −0.358920 0.933368i \(-0.616855\pi\)
−0.358920 + 0.933368i \(0.616855\pi\)
\(258\) −7506.64 −1.81141
\(259\) 0 0
\(260\) −2713.77 −0.647311
\(261\) 1744.84 0.413803
\(262\) −182.556 −0.0430471
\(263\) 5364.89 1.25785 0.628923 0.777468i \(-0.283496\pi\)
0.628923 + 0.777468i \(0.283496\pi\)
\(264\) 609.535 0.142100
\(265\) −7100.84 −1.64604
\(266\) 0 0
\(267\) 429.181 0.0983725
\(268\) −371.503 −0.0846759
\(269\) −5294.56 −1.20006 −0.600028 0.799979i \(-0.704844\pi\)
−0.600028 + 0.799979i \(0.704844\pi\)
\(270\) 2139.22 0.482182
\(271\) −7630.90 −1.71050 −0.855248 0.518219i \(-0.826595\pi\)
−0.855248 + 0.518219i \(0.826595\pi\)
\(272\) 902.505 0.201185
\(273\) 0 0
\(274\) −9203.11 −2.02912
\(275\) 2057.35 0.451138
\(276\) 2540.64 0.554089
\(277\) −1269.76 −0.275425 −0.137713 0.990472i \(-0.543975\pi\)
−0.137713 + 0.990472i \(0.543975\pi\)
\(278\) 11193.4 2.41488
\(279\) −215.180 −0.0461738
\(280\) 0 0
\(281\) −524.744 −0.111401 −0.0557004 0.998448i \(-0.517739\pi\)
−0.0557004 + 0.998448i \(0.517739\pi\)
\(282\) −5193.95 −1.09679
\(283\) −4784.16 −1.00491 −0.502454 0.864604i \(-0.667569\pi\)
−0.502454 + 0.864604i \(0.667569\pi\)
\(284\) −3638.41 −0.760211
\(285\) −8713.12 −1.81095
\(286\) 625.499 0.129323
\(287\) 0 0
\(288\) 1898.97 0.388534
\(289\) −814.431 −0.165771
\(290\) −15360.5 −3.11034
\(291\) 652.531 0.131450
\(292\) 1099.12 0.220279
\(293\) 3941.60 0.785908 0.392954 0.919558i \(-0.371453\pi\)
0.392954 + 0.919558i \(0.371453\pi\)
\(294\) 0 0
\(295\) 9162.22 1.80829
\(296\) 2507.92 0.492466
\(297\) −297.000 −0.0580259
\(298\) −10990.5 −2.13646
\(299\) 885.992 0.171365
\(300\) −6799.39 −1.30854
\(301\) 0 0
\(302\) −4195.31 −0.799381
\(303\) −4531.87 −0.859238
\(304\) −2317.85 −0.437296
\(305\) −3043.00 −0.571284
\(306\) 2584.35 0.482802
\(307\) −6307.31 −1.17256 −0.586282 0.810107i \(-0.699409\pi\)
−0.586282 + 0.810107i \(0.699409\pi\)
\(308\) 0 0
\(309\) 1399.21 0.257600
\(310\) 1894.31 0.347064
\(311\) −1209.51 −0.220530 −0.110265 0.993902i \(-0.535170\pi\)
−0.110265 + 0.993902i \(0.535170\pi\)
\(312\) −702.501 −0.127472
\(313\) −3408.92 −0.615602 −0.307801 0.951451i \(-0.599593\pi\)
−0.307801 + 0.951451i \(0.599593\pi\)
\(314\) −9732.47 −1.74916
\(315\) 0 0
\(316\) −6063.36 −1.07940
\(317\) −3425.65 −0.606951 −0.303476 0.952839i \(-0.598147\pi\)
−0.303476 + 0.952839i \(0.598147\pi\)
\(318\) −5409.09 −0.953857
\(319\) 2132.58 0.374299
\(320\) −14725.2 −2.57239
\(321\) 1832.42 0.318616
\(322\) 0 0
\(323\) −10526.1 −1.81328
\(324\) 981.562 0.168306
\(325\) −2371.14 −0.404699
\(326\) −4899.94 −0.832462
\(327\) 4544.39 0.768517
\(328\) 7391.53 1.24430
\(329\) 0 0
\(330\) 2614.61 0.436150
\(331\) −2057.54 −0.341670 −0.170835 0.985300i \(-0.554646\pi\)
−0.170835 + 0.985300i \(0.554646\pi\)
\(332\) −11333.9 −1.87359
\(333\) −1222.00 −0.201097
\(334\) −4990.16 −0.817513
\(335\) −541.538 −0.0883205
\(336\) 0 0
\(337\) 3000.29 0.484975 0.242487 0.970155i \(-0.422037\pi\)
0.242487 + 0.970155i \(0.422037\pi\)
\(338\) 9133.34 1.46979
\(339\) 5914.70 0.947618
\(340\) −13704.0 −2.18590
\(341\) −262.998 −0.0417657
\(342\) −6637.25 −1.04942
\(343\) 0 0
\(344\) 10304.3 1.61502
\(345\) 3703.48 0.577938
\(346\) −20022.4 −3.11102
\(347\) 7851.02 1.21460 0.607298 0.794474i \(-0.292253\pi\)
0.607298 + 0.794474i \(0.292253\pi\)
\(348\) −7048.01 −1.08567
\(349\) 7743.52 1.18768 0.593841 0.804582i \(-0.297611\pi\)
0.593841 + 0.804582i \(0.297611\pi\)
\(350\) 0 0
\(351\) 342.298 0.0520528
\(352\) 2320.96 0.351442
\(353\) −6241.13 −0.941026 −0.470513 0.882393i \(-0.655931\pi\)
−0.470513 + 0.882393i \(0.655931\pi\)
\(354\) 6979.36 1.04788
\(355\) −5303.69 −0.792932
\(356\) −1733.61 −0.258094
\(357\) 0 0
\(358\) 10925.3 1.61291
\(359\) 11449.7 1.68326 0.841631 0.540052i \(-0.181596\pi\)
0.841631 + 0.540052i \(0.181596\pi\)
\(360\) −2936.48 −0.429906
\(361\) 20174.7 2.94135
\(362\) 8717.98 1.26577
\(363\) −363.000 −0.0524864
\(364\) 0 0
\(365\) 1602.19 0.229760
\(366\) −2318.01 −0.331051
\(367\) −7354.04 −1.04599 −0.522994 0.852336i \(-0.675185\pi\)
−0.522994 + 0.852336i \(0.675185\pi\)
\(368\) 985.195 0.139557
\(369\) −3601.57 −0.508104
\(370\) 10757.8 1.51154
\(371\) 0 0
\(372\) 869.187 0.121143
\(373\) −9294.68 −1.29024 −0.645121 0.764080i \(-0.723193\pi\)
−0.645121 + 0.764080i \(0.723193\pi\)
\(374\) 3158.65 0.436711
\(375\) −3287.28 −0.452679
\(376\) 7129.66 0.977883
\(377\) −2457.84 −0.335769
\(378\) 0 0
\(379\) 7965.09 1.07952 0.539762 0.841818i \(-0.318514\pi\)
0.539762 + 0.841818i \(0.318514\pi\)
\(380\) 35195.3 4.75127
\(381\) −1467.37 −0.197312
\(382\) −8258.83 −1.10617
\(383\) −9222.45 −1.23041 −0.615203 0.788369i \(-0.710926\pi\)
−0.615203 + 0.788369i \(0.710926\pi\)
\(384\) −6153.07 −0.817702
\(385\) 0 0
\(386\) 14420.0 1.90144
\(387\) −5020.81 −0.659489
\(388\) −2635.80 −0.344878
\(389\) −9155.98 −1.19338 −0.596692 0.802470i \(-0.703519\pi\)
−0.596692 + 0.802470i \(0.703519\pi\)
\(390\) −3013.39 −0.391253
\(391\) 4474.09 0.578682
\(392\) 0 0
\(393\) −122.102 −0.0156724
\(394\) −15970.2 −2.04205
\(395\) −8838.53 −1.12586
\(396\) 1199.69 0.152239
\(397\) −11300.7 −1.42863 −0.714314 0.699825i \(-0.753261\pi\)
−0.714314 + 0.699825i \(0.753261\pi\)
\(398\) 6186.07 0.779095
\(399\) 0 0
\(400\) −2636.63 −0.329579
\(401\) −1590.16 −0.198026 −0.0990132 0.995086i \(-0.531569\pi\)
−0.0990132 + 0.995086i \(0.531569\pi\)
\(402\) −412.518 −0.0511805
\(403\) 303.110 0.0374664
\(404\) 18305.8 2.25433
\(405\) 1430.82 0.175551
\(406\) 0 0
\(407\) −1493.56 −0.181899
\(408\) −3547.50 −0.430459
\(409\) 8973.22 1.08483 0.542417 0.840109i \(-0.317509\pi\)
0.542417 + 0.840109i \(0.317509\pi\)
\(410\) 31706.0 3.81914
\(411\) −6155.49 −0.738754
\(412\) −5651.92 −0.675850
\(413\) 0 0
\(414\) 2821.14 0.334907
\(415\) −16521.4 −1.95423
\(416\) −2674.95 −0.315265
\(417\) 7486.70 0.879197
\(418\) −8112.20 −0.949236
\(419\) −13859.3 −1.61593 −0.807963 0.589233i \(-0.799430\pi\)
−0.807963 + 0.589233i \(0.799430\pi\)
\(420\) 0 0
\(421\) 6966.55 0.806482 0.403241 0.915094i \(-0.367884\pi\)
0.403241 + 0.915094i \(0.367884\pi\)
\(422\) −10743.5 −1.23930
\(423\) −3473.97 −0.399315
\(424\) 7424.97 0.850445
\(425\) −11973.8 −1.36662
\(426\) −4040.11 −0.459493
\(427\) 0 0
\(428\) −7401.78 −0.835932
\(429\) 418.364 0.0470835
\(430\) 44200.2 4.95703
\(431\) 2609.03 0.291584 0.145792 0.989315i \(-0.453427\pi\)
0.145792 + 0.989315i \(0.453427\pi\)
\(432\) 380.625 0.0423908
\(433\) 2263.17 0.251180 0.125590 0.992082i \(-0.459918\pi\)
0.125590 + 0.992082i \(0.459918\pi\)
\(434\) 0 0
\(435\) −10273.8 −1.13240
\(436\) −18356.4 −2.01631
\(437\) −11490.6 −1.25782
\(438\) 1220.47 0.133143
\(439\) 15196.2 1.65211 0.826056 0.563589i \(-0.190580\pi\)
0.826056 + 0.563589i \(0.190580\pi\)
\(440\) −3589.03 −0.388864
\(441\) 0 0
\(442\) −3640.41 −0.391757
\(443\) −866.219 −0.0929013 −0.0464507 0.998921i \(-0.514791\pi\)
−0.0464507 + 0.998921i \(0.514791\pi\)
\(444\) 4936.09 0.527605
\(445\) −2527.08 −0.269202
\(446\) 15363.6 1.63114
\(447\) −7351.02 −0.777832
\(448\) 0 0
\(449\) 17643.6 1.85446 0.927231 0.374489i \(-0.122182\pi\)
0.927231 + 0.374489i \(0.122182\pi\)
\(450\) −7550.08 −0.790920
\(451\) −4401.92 −0.459597
\(452\) −23891.6 −2.48620
\(453\) −2806.03 −0.291035
\(454\) −21925.7 −2.26657
\(455\) 0 0
\(456\) 9110.85 0.935646
\(457\) 11593.9 1.18674 0.593372 0.804929i \(-0.297797\pi\)
0.593372 + 0.804929i \(0.297797\pi\)
\(458\) 22767.5 2.32283
\(459\) 1728.54 0.175777
\(460\) −14959.7 −1.51630
\(461\) −4932.41 −0.498319 −0.249160 0.968462i \(-0.580154\pi\)
−0.249160 + 0.968462i \(0.580154\pi\)
\(462\) 0 0
\(463\) 7375.40 0.740311 0.370155 0.928970i \(-0.379304\pi\)
0.370155 + 0.928970i \(0.379304\pi\)
\(464\) −2733.04 −0.273444
\(465\) 1267.01 0.126357
\(466\) −10482.3 −1.04203
\(467\) 16097.9 1.59512 0.797560 0.603239i \(-0.206124\pi\)
0.797560 + 0.603239i \(0.206124\pi\)
\(468\) −1382.66 −0.136568
\(469\) 0 0
\(470\) 30582.7 3.00144
\(471\) −6509.55 −0.636825
\(472\) −9580.46 −0.934272
\(473\) −6136.55 −0.596530
\(474\) −6732.78 −0.652419
\(475\) 30751.7 2.97049
\(476\) 0 0
\(477\) −3617.87 −0.347276
\(478\) 4242.68 0.405974
\(479\) −16191.7 −1.54451 −0.772254 0.635314i \(-0.780870\pi\)
−0.772254 + 0.635314i \(0.780870\pi\)
\(480\) −11181.4 −1.06325
\(481\) 1721.35 0.163175
\(482\) −7327.63 −0.692457
\(483\) 0 0
\(484\) 1466.28 0.137705
\(485\) −3842.19 −0.359722
\(486\) 1089.93 0.101729
\(487\) 14337.6 1.33408 0.667041 0.745021i \(-0.267560\pi\)
0.667041 + 0.745021i \(0.267560\pi\)
\(488\) 3181.90 0.295160
\(489\) −3277.32 −0.303079
\(490\) 0 0
\(491\) −12343.6 −1.13454 −0.567270 0.823532i \(-0.692001\pi\)
−0.567270 + 0.823532i \(0.692001\pi\)
\(492\) 14548.0 1.33308
\(493\) −12411.6 −1.13386
\(494\) 9349.46 0.851523
\(495\) 1748.78 0.158791
\(496\) 337.049 0.0305119
\(497\) 0 0
\(498\) −12585.3 −1.13245
\(499\) 6780.54 0.608294 0.304147 0.952625i \(-0.401629\pi\)
0.304147 + 0.952625i \(0.401629\pi\)
\(500\) 13278.5 1.18766
\(501\) −3337.66 −0.297636
\(502\) 1232.44 0.109574
\(503\) 93.9106 0.00832458 0.00416229 0.999991i \(-0.498675\pi\)
0.00416229 + 0.999991i \(0.498675\pi\)
\(504\) 0 0
\(505\) 26684.3 2.35136
\(506\) 3448.06 0.302935
\(507\) 6108.83 0.535113
\(508\) 5927.23 0.517674
\(509\) 3411.80 0.297103 0.148552 0.988905i \(-0.452539\pi\)
0.148552 + 0.988905i \(0.452539\pi\)
\(510\) −15217.0 −1.32122
\(511\) 0 0
\(512\) −5057.56 −0.436552
\(513\) −4439.32 −0.382068
\(514\) 13265.4 1.13835
\(515\) −8238.77 −0.704939
\(516\) 20280.8 1.73026
\(517\) −4245.97 −0.361194
\(518\) 0 0
\(519\) −13392.0 −1.13264
\(520\) 4136.43 0.348835
\(521\) −14418.7 −1.21247 −0.606234 0.795286i \(-0.707321\pi\)
−0.606234 + 0.795286i \(0.707321\pi\)
\(522\) −7826.14 −0.656209
\(523\) −3093.54 −0.258644 −0.129322 0.991603i \(-0.541280\pi\)
−0.129322 + 0.991603i \(0.541280\pi\)
\(524\) 493.215 0.0411187
\(525\) 0 0
\(526\) −24063.2 −1.99469
\(527\) 1530.65 0.126520
\(528\) 465.208 0.0383439
\(529\) −7282.97 −0.598584
\(530\) 31849.5 2.61029
\(531\) 4668.14 0.381507
\(532\) 0 0
\(533\) 5073.29 0.412287
\(534\) −1925.01 −0.155999
\(535\) −10789.5 −0.871912
\(536\) 566.258 0.0456317
\(537\) 7307.40 0.587221
\(538\) 23747.8 1.90305
\(539\) 0 0
\(540\) −5779.58 −0.460581
\(541\) −8016.37 −0.637062 −0.318531 0.947912i \(-0.603190\pi\)
−0.318531 + 0.947912i \(0.603190\pi\)
\(542\) 34227.0 2.71250
\(543\) 5831.02 0.460834
\(544\) −13508.0 −1.06462
\(545\) −26758.0 −2.10310
\(546\) 0 0
\(547\) −21742.1 −1.69950 −0.849749 0.527188i \(-0.823246\pi\)
−0.849749 + 0.527188i \(0.823246\pi\)
\(548\) 24864.2 1.93822
\(549\) −1550.40 −0.120527
\(550\) −9227.87 −0.715414
\(551\) 31876.1 2.46455
\(552\) −3872.53 −0.298598
\(553\) 0 0
\(554\) 5695.29 0.436769
\(555\) 7195.32 0.550314
\(556\) −30241.4 −2.30669
\(557\) 14452.8 1.09943 0.549715 0.835352i \(-0.314736\pi\)
0.549715 + 0.835352i \(0.314736\pi\)
\(558\) 965.149 0.0732223
\(559\) 7072.49 0.535124
\(560\) 0 0
\(561\) 2112.66 0.158996
\(562\) 2353.64 0.176659
\(563\) −20969.6 −1.56974 −0.784869 0.619661i \(-0.787270\pi\)
−0.784869 + 0.619661i \(0.787270\pi\)
\(564\) 14032.6 1.04766
\(565\) −34826.6 −2.59322
\(566\) 21458.5 1.59358
\(567\) 0 0
\(568\) 5545.79 0.409676
\(569\) 19786.0 1.45777 0.728887 0.684634i \(-0.240038\pi\)
0.728887 + 0.684634i \(0.240038\pi\)
\(570\) 39081.1 2.87180
\(571\) −23309.9 −1.70839 −0.854193 0.519956i \(-0.825948\pi\)
−0.854193 + 0.519956i \(0.825948\pi\)
\(572\) −1689.92 −0.123530
\(573\) −5523.91 −0.402731
\(574\) 0 0
\(575\) −13070.9 −0.947989
\(576\) −7502.48 −0.542714
\(577\) 23703.7 1.71023 0.855113 0.518442i \(-0.173488\pi\)
0.855113 + 0.518442i \(0.173488\pi\)
\(578\) 3652.98 0.262879
\(579\) 9644.77 0.692268
\(580\) 41499.7 2.97100
\(581\) 0 0
\(582\) −2926.81 −0.208454
\(583\) −4421.84 −0.314123
\(584\) −1675.32 −0.118708
\(585\) −2015.50 −0.142446
\(586\) −17679.3 −1.24629
\(587\) −24778.3 −1.74227 −0.871134 0.491046i \(-0.836615\pi\)
−0.871134 + 0.491046i \(0.836615\pi\)
\(588\) 0 0
\(589\) −3931.08 −0.275004
\(590\) −41095.5 −2.86758
\(591\) −10681.7 −0.743461
\(592\) 1914.09 0.132886
\(593\) 5983.47 0.414353 0.207177 0.978304i \(-0.433573\pi\)
0.207177 + 0.978304i \(0.433573\pi\)
\(594\) 1332.14 0.0920174
\(595\) 0 0
\(596\) 29693.3 2.04075
\(597\) 4137.55 0.283649
\(598\) −3973.95 −0.271751
\(599\) 19091.5 1.30227 0.651134 0.758963i \(-0.274294\pi\)
0.651134 + 0.758963i \(0.274294\pi\)
\(600\) 10363.9 0.705172
\(601\) 8064.08 0.547322 0.273661 0.961826i \(-0.411765\pi\)
0.273661 + 0.961826i \(0.411765\pi\)
\(602\) 0 0
\(603\) −275.913 −0.0186336
\(604\) 11334.5 0.763570
\(605\) 2137.40 0.143632
\(606\) 20326.9 1.36258
\(607\) −772.307 −0.0516425 −0.0258212 0.999667i \(-0.508220\pi\)
−0.0258212 + 0.999667i \(0.508220\pi\)
\(608\) 34691.9 2.31405
\(609\) 0 0
\(610\) 13648.8 0.905941
\(611\) 4893.56 0.324013
\(612\) −6982.19 −0.461174
\(613\) −3462.58 −0.228144 −0.114072 0.993472i \(-0.536389\pi\)
−0.114072 + 0.993472i \(0.536389\pi\)
\(614\) 28290.3 1.85945
\(615\) 21206.6 1.39046
\(616\) 0 0
\(617\) −9668.33 −0.630847 −0.315423 0.948951i \(-0.602147\pi\)
−0.315423 + 0.948951i \(0.602147\pi\)
\(618\) −6275.92 −0.408502
\(619\) −11678.4 −0.758312 −0.379156 0.925333i \(-0.623786\pi\)
−0.379156 + 0.925333i \(0.623786\pi\)
\(620\) −5117.90 −0.331516
\(621\) 1886.92 0.121931
\(622\) 5425.02 0.349717
\(623\) 0 0
\(624\) −536.161 −0.0343968
\(625\) −4023.02 −0.257473
\(626\) 15290.1 0.976220
\(627\) −5425.84 −0.345593
\(628\) 26294.4 1.67080
\(629\) 8692.52 0.551023
\(630\) 0 0
\(631\) −25953.7 −1.63740 −0.818702 0.574218i \(-0.805306\pi\)
−0.818702 + 0.574218i \(0.805306\pi\)
\(632\) 9241.98 0.581687
\(633\) −7185.76 −0.451198
\(634\) 15365.1 0.962502
\(635\) 8640.10 0.539956
\(636\) 14613.8 0.911126
\(637\) 0 0
\(638\) −9565.28 −0.593563
\(639\) −2702.22 −0.167290
\(640\) 36230.2 2.23769
\(641\) −5837.77 −0.359716 −0.179858 0.983693i \(-0.557564\pi\)
−0.179858 + 0.983693i \(0.557564\pi\)
\(642\) −8218.98 −0.505260
\(643\) −21648.0 −1.32771 −0.663853 0.747863i \(-0.731080\pi\)
−0.663853 + 0.747863i \(0.731080\pi\)
\(644\) 0 0
\(645\) 29563.3 1.80473
\(646\) 47213.0 2.87550
\(647\) −12635.4 −0.767775 −0.383888 0.923380i \(-0.625415\pi\)
−0.383888 + 0.923380i \(0.625415\pi\)
\(648\) −1496.13 −0.0907000
\(649\) 5705.50 0.345086
\(650\) 10635.3 0.641771
\(651\) 0 0
\(652\) 13238.2 0.795168
\(653\) 21730.9 1.30229 0.651144 0.758954i \(-0.274289\pi\)
0.651144 + 0.758954i \(0.274289\pi\)
\(654\) −20383.0 −1.21871
\(655\) 718.957 0.0428885
\(656\) 5641.34 0.335758
\(657\) 816.312 0.0484739
\(658\) 0 0
\(659\) −21795.9 −1.28839 −0.644194 0.764862i \(-0.722807\pi\)
−0.644194 + 0.764862i \(0.722807\pi\)
\(660\) −7063.93 −0.416611
\(661\) 2134.94 0.125627 0.0628137 0.998025i \(-0.479993\pi\)
0.0628137 + 0.998025i \(0.479993\pi\)
\(662\) 9228.72 0.541820
\(663\) −2434.88 −0.142629
\(664\) 17275.6 1.00967
\(665\) 0 0
\(666\) 5481.06 0.318899
\(667\) −13548.8 −0.786525
\(668\) 13482.0 0.780889
\(669\) 10275.9 0.593856
\(670\) 2428.97 0.140059
\(671\) −1894.94 −0.109021
\(672\) 0 0
\(673\) −5744.89 −0.329048 −0.164524 0.986373i \(-0.552609\pi\)
−0.164524 + 0.986373i \(0.552609\pi\)
\(674\) −13457.3 −0.769072
\(675\) −5049.86 −0.287955
\(676\) −24675.7 −1.40394
\(677\) 1725.42 0.0979518 0.0489759 0.998800i \(-0.484404\pi\)
0.0489759 + 0.998800i \(0.484404\pi\)
\(678\) −26529.3 −1.50273
\(679\) 0 0
\(680\) 20888.2 1.17798
\(681\) −14665.0 −0.825204
\(682\) 1179.63 0.0662321
\(683\) −10687.6 −0.598754 −0.299377 0.954135i \(-0.596779\pi\)
−0.299377 + 0.954135i \(0.596779\pi\)
\(684\) 17932.0 1.00241
\(685\) 36244.4 2.02165
\(686\) 0 0
\(687\) 15228.1 0.845686
\(688\) 7864.39 0.435795
\(689\) 5096.25 0.281788
\(690\) −16611.3 −0.916493
\(691\) 1711.25 0.0942099 0.0471050 0.998890i \(-0.485000\pi\)
0.0471050 + 0.998890i \(0.485000\pi\)
\(692\) 54094.9 2.97165
\(693\) 0 0
\(694\) −35214.3 −1.92610
\(695\) −44082.8 −2.40598
\(696\) 10742.8 0.585065
\(697\) 25619.2 1.39225
\(698\) −34732.1 −1.88343
\(699\) −7011.10 −0.379376
\(700\) 0 0
\(701\) −6517.71 −0.351170 −0.175585 0.984464i \(-0.556182\pi\)
−0.175585 + 0.984464i \(0.556182\pi\)
\(702\) −1535.32 −0.0825452
\(703\) −22324.5 −1.19770
\(704\) −9169.69 −0.490903
\(705\) 20455.2 1.09275
\(706\) 27993.5 1.49228
\(707\) 0 0
\(708\) −18856.3 −1.00093
\(709\) 21389.4 1.13300 0.566499 0.824062i \(-0.308297\pi\)
0.566499 + 0.824062i \(0.308297\pi\)
\(710\) 23788.7 1.25743
\(711\) −4503.22 −0.237530
\(712\) 2642.43 0.139086
\(713\) 1670.89 0.0877635
\(714\) 0 0
\(715\) −2463.39 −0.128847
\(716\) −29517.2 −1.54065
\(717\) 2837.71 0.147805
\(718\) −51355.5 −2.66932
\(719\) −28850.8 −1.49646 −0.748230 0.663440i \(-0.769096\pi\)
−0.748230 + 0.663440i \(0.769096\pi\)
\(720\) −2241.17 −0.116005
\(721\) 0 0
\(722\) −90489.9 −4.66438
\(723\) −4901.08 −0.252107
\(724\) −23553.5 −1.20906
\(725\) 36260.0 1.85747
\(726\) 1628.17 0.0832328
\(727\) −27958.6 −1.42631 −0.713156 0.701006i \(-0.752735\pi\)
−0.713156 + 0.701006i \(0.752735\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) −7186.32 −0.364353
\(731\) 35714.8 1.80706
\(732\) 6262.62 0.316220
\(733\) 17330.6 0.873286 0.436643 0.899635i \(-0.356167\pi\)
0.436643 + 0.899635i \(0.356167\pi\)
\(734\) 32985.2 1.65873
\(735\) 0 0
\(736\) −14745.7 −0.738495
\(737\) −337.227 −0.0168547
\(738\) 16154.2 0.805750
\(739\) −25893.2 −1.28890 −0.644451 0.764646i \(-0.722914\pi\)
−0.644451 + 0.764646i \(0.722914\pi\)
\(740\) −29064.4 −1.44382
\(741\) 6253.38 0.310018
\(742\) 0 0
\(743\) −16921.8 −0.835533 −0.417766 0.908554i \(-0.637187\pi\)
−0.417766 + 0.908554i \(0.637187\pi\)
\(744\) −1324.85 −0.0652839
\(745\) 43283.8 2.12859
\(746\) 41689.6 2.04606
\(747\) −8417.64 −0.412296
\(748\) −8533.78 −0.417147
\(749\) 0 0
\(750\) 14744.5 0.717857
\(751\) −22370.2 −1.08695 −0.543477 0.839424i \(-0.682892\pi\)
−0.543477 + 0.839424i \(0.682892\pi\)
\(752\) 5441.48 0.263870
\(753\) 824.314 0.0398933
\(754\) 11024.2 0.532462
\(755\) 16522.3 0.796435
\(756\) 0 0
\(757\) −11839.7 −0.568454 −0.284227 0.958757i \(-0.591737\pi\)
−0.284227 + 0.958757i \(0.591737\pi\)
\(758\) −35725.9 −1.71191
\(759\) 2306.23 0.110291
\(760\) −53646.0 −2.56045
\(761\) −5016.00 −0.238935 −0.119468 0.992838i \(-0.538119\pi\)
−0.119468 + 0.992838i \(0.538119\pi\)
\(762\) 6581.63 0.312897
\(763\) 0 0
\(764\) 22313.0 1.05662
\(765\) −10177.9 −0.481023
\(766\) 41365.6 1.95117
\(767\) −6575.70 −0.309563
\(768\) 7591.87 0.356703
\(769\) 3621.32 0.169815 0.0849077 0.996389i \(-0.472940\pi\)
0.0849077 + 0.996389i \(0.472940\pi\)
\(770\) 0 0
\(771\) 8872.56 0.414446
\(772\) −38958.6 −1.81626
\(773\) −40624.8 −1.89026 −0.945131 0.326690i \(-0.894067\pi\)
−0.945131 + 0.326690i \(0.894067\pi\)
\(774\) 22519.9 1.04582
\(775\) −4471.73 −0.207263
\(776\) 4017.58 0.185854
\(777\) 0 0
\(778\) 41067.5 1.89247
\(779\) −65796.4 −3.02619
\(780\) 8141.31 0.373725
\(781\) −3302.72 −0.151319
\(782\) −20067.7 −0.917673
\(783\) −5234.51 −0.238909
\(784\) 0 0
\(785\) 38329.2 1.74271
\(786\) 547.668 0.0248533
\(787\) −34921.0 −1.58170 −0.790851 0.612009i \(-0.790362\pi\)
−0.790851 + 0.612009i \(0.790362\pi\)
\(788\) 43147.0 1.95057
\(789\) −16094.7 −0.726218
\(790\) 39643.6 1.78539
\(791\) 0 0
\(792\) −1828.61 −0.0820413
\(793\) 2183.95 0.0977987
\(794\) 50687.1 2.26552
\(795\) 21302.5 0.950342
\(796\) −16713.0 −0.744193
\(797\) −5587.68 −0.248339 −0.124169 0.992261i \(-0.539627\pi\)
−0.124169 + 0.992261i \(0.539627\pi\)
\(798\) 0 0
\(799\) 24711.6 1.09416
\(800\) 39463.1 1.74404
\(801\) −1287.54 −0.0567954
\(802\) 7132.35 0.314030
\(803\) 997.715 0.0438463
\(804\) 1114.51 0.0488877
\(805\) 0 0
\(806\) −1359.54 −0.0594142
\(807\) 15883.7 0.692853
\(808\) −27902.4 −1.21485
\(809\) 5916.64 0.257130 0.128565 0.991701i \(-0.458963\pi\)
0.128565 + 0.991701i \(0.458963\pi\)
\(810\) −6417.67 −0.278388
\(811\) 14652.2 0.634414 0.317207 0.948356i \(-0.397255\pi\)
0.317207 + 0.948356i \(0.397255\pi\)
\(812\) 0 0
\(813\) 22892.7 0.987555
\(814\) 6699.08 0.288455
\(815\) 19297.3 0.829394
\(816\) −2707.51 −0.116154
\(817\) −91724.3 −3.92782
\(818\) −40247.7 −1.72033
\(819\) 0 0
\(820\) −85660.7 −3.64805
\(821\) −6358.86 −0.270311 −0.135156 0.990824i \(-0.543153\pi\)
−0.135156 + 0.990824i \(0.543153\pi\)
\(822\) 27609.3 1.17152
\(823\) 3082.98 0.130578 0.0652892 0.997866i \(-0.479203\pi\)
0.0652892 + 0.997866i \(0.479203\pi\)
\(824\) 8614.85 0.364214
\(825\) −6172.06 −0.260465
\(826\) 0 0
\(827\) 31270.9 1.31487 0.657433 0.753513i \(-0.271642\pi\)
0.657433 + 0.753513i \(0.271642\pi\)
\(828\) −7621.92 −0.319903
\(829\) 32518.5 1.36238 0.681192 0.732105i \(-0.261462\pi\)
0.681192 + 0.732105i \(0.261462\pi\)
\(830\) 74103.8 3.09901
\(831\) 3809.29 0.159017
\(832\) 10568.2 0.440370
\(833\) 0 0
\(834\) −33580.2 −1.39423
\(835\) 19652.6 0.814500
\(836\) 21916.9 0.906711
\(837\) 645.540 0.0266584
\(838\) 62163.5 2.56253
\(839\) 14469.5 0.595404 0.297702 0.954659i \(-0.403780\pi\)
0.297702 + 0.954659i \(0.403780\pi\)
\(840\) 0 0
\(841\) 13196.8 0.541098
\(842\) −31247.2 −1.27892
\(843\) 1574.23 0.0643172
\(844\) 29025.8 1.18378
\(845\) −35969.6 −1.46437
\(846\) 15581.9 0.633233
\(847\) 0 0
\(848\) 5666.87 0.229482
\(849\) 14352.5 0.580184
\(850\) 53706.3 2.16719
\(851\) 9488.95 0.382229
\(852\) 10915.2 0.438908
\(853\) 10391.1 0.417100 0.208550 0.978012i \(-0.433126\pi\)
0.208550 + 0.978012i \(0.433126\pi\)
\(854\) 0 0
\(855\) 26139.4 1.04555
\(856\) 11282.1 0.450482
\(857\) 29846.9 1.18968 0.594838 0.803846i \(-0.297216\pi\)
0.594838 + 0.803846i \(0.297216\pi\)
\(858\) −1876.50 −0.0746650
\(859\) −3583.98 −0.142356 −0.0711780 0.997464i \(-0.522676\pi\)
−0.0711780 + 0.997464i \(0.522676\pi\)
\(860\) −119416. −4.73496
\(861\) 0 0
\(862\) −11702.3 −0.462393
\(863\) 25228.5 0.995120 0.497560 0.867430i \(-0.334229\pi\)
0.497560 + 0.867430i \(0.334229\pi\)
\(864\) −5696.91 −0.224320
\(865\) 78853.9 3.09955
\(866\) −10151.0 −0.398321
\(867\) 2443.29 0.0957077
\(868\) 0 0
\(869\) −5503.93 −0.214854
\(870\) 46081.4 1.79575
\(871\) 388.660 0.0151197
\(872\) 27979.4 1.08659
\(873\) −1957.59 −0.0758929
\(874\) 51538.9 1.99465
\(875\) 0 0
\(876\) −3297.37 −0.127178
\(877\) 26369.4 1.01531 0.507657 0.861559i \(-0.330512\pi\)
0.507657 + 0.861559i \(0.330512\pi\)
\(878\) −68159.9 −2.61992
\(879\) −11824.8 −0.453744
\(880\) −2739.21 −0.104930
\(881\) −9559.77 −0.365581 −0.182791 0.983152i \(-0.558513\pi\)
−0.182791 + 0.983152i \(0.558513\pi\)
\(882\) 0 0
\(883\) 7436.71 0.283426 0.141713 0.989908i \(-0.454739\pi\)
0.141713 + 0.989908i \(0.454739\pi\)
\(884\) 9835.35 0.374207
\(885\) −27486.7 −1.04402
\(886\) 3885.26 0.147323
\(887\) −13874.1 −0.525193 −0.262596 0.964906i \(-0.584579\pi\)
−0.262596 + 0.964906i \(0.584579\pi\)
\(888\) −7523.77 −0.284326
\(889\) 0 0
\(890\) 11334.8 0.426901
\(891\) 891.000 0.0335013
\(892\) −41508.0 −1.55806
\(893\) −63465.3 −2.37826
\(894\) 32971.6 1.23349
\(895\) −43027.0 −1.60697
\(896\) 0 0
\(897\) −2657.98 −0.0989378
\(898\) −79137.2 −2.94080
\(899\) −4635.23 −0.171962
\(900\) 20398.2 0.755488
\(901\) 25735.1 0.951566
\(902\) 19744.0 0.728828
\(903\) 0 0
\(904\) 36416.4 1.33981
\(905\) −34333.9 −1.26110
\(906\) 12585.9 0.461523
\(907\) 16.8917 0.000618390 0 0.000309195 1.00000i \(-0.499902\pi\)
0.000309195 1.00000i \(0.499902\pi\)
\(908\) 59237.0 2.16503
\(909\) 13595.6 0.496081
\(910\) 0 0
\(911\) −7523.49 −0.273616 −0.136808 0.990598i \(-0.543684\pi\)
−0.136808 + 0.990598i \(0.543684\pi\)
\(912\) 6953.56 0.252473
\(913\) −10288.2 −0.372936
\(914\) −52002.5 −1.88194
\(915\) 9128.99 0.329831
\(916\) −61511.5 −2.21877
\(917\) 0 0
\(918\) −7753.06 −0.278746
\(919\) −2001.97 −0.0718594 −0.0359297 0.999354i \(-0.511439\pi\)
−0.0359297 + 0.999354i \(0.511439\pi\)
\(920\) 22802.0 0.817131
\(921\) 18921.9 0.676980
\(922\) 22123.4 0.790234
\(923\) 3806.44 0.135743
\(924\) 0 0
\(925\) −25394.8 −0.902678
\(926\) −33081.0 −1.17398
\(927\) −4197.64 −0.148726
\(928\) 40906.1 1.44699
\(929\) 44165.6 1.55977 0.779886 0.625922i \(-0.215277\pi\)
0.779886 + 0.625922i \(0.215277\pi\)
\(930\) −5682.94 −0.200377
\(931\) 0 0
\(932\) 28320.3 0.995345
\(933\) 3628.52 0.127323
\(934\) −72204.1 −2.52954
\(935\) −12439.7 −0.435102
\(936\) 2107.50 0.0735960
\(937\) 24035.4 0.837996 0.418998 0.907987i \(-0.362381\pi\)
0.418998 + 0.907987i \(0.362381\pi\)
\(938\) 0 0
\(939\) 10226.8 0.355418
\(940\) −82625.9 −2.86698
\(941\) 9556.56 0.331068 0.165534 0.986204i \(-0.447065\pi\)
0.165534 + 0.986204i \(0.447065\pi\)
\(942\) 29197.4 1.00988
\(943\) 27966.5 0.965763
\(944\) −7311.98 −0.252102
\(945\) 0 0
\(946\) 27524.4 0.945977
\(947\) 52400.2 1.79808 0.899038 0.437870i \(-0.144267\pi\)
0.899038 + 0.437870i \(0.144267\pi\)
\(948\) 18190.1 0.623192
\(949\) −1149.89 −0.0393328
\(950\) −137931. −4.71060
\(951\) 10276.9 0.350423
\(952\) 0 0
\(953\) −14168.6 −0.481600 −0.240800 0.970575i \(-0.577410\pi\)
−0.240800 + 0.970575i \(0.577410\pi\)
\(954\) 16227.3 0.550710
\(955\) 32525.6 1.10210
\(956\) −11462.5 −0.387787
\(957\) −6397.73 −0.216102
\(958\) 72625.0 2.44928
\(959\) 0 0
\(960\) 44175.7 1.48517
\(961\) −29219.4 −0.980812
\(962\) −7720.81 −0.258762
\(963\) −5497.26 −0.183953
\(964\) 19797.2 0.661436
\(965\) −56789.8 −1.89443
\(966\) 0 0
\(967\) −39570.4 −1.31593 −0.657963 0.753051i \(-0.728582\pi\)
−0.657963 + 0.753051i \(0.728582\pi\)
\(968\) −2234.96 −0.0742091
\(969\) 31578.4 1.04690
\(970\) 17233.5 0.570446
\(971\) −8374.48 −0.276776 −0.138388 0.990378i \(-0.544192\pi\)
−0.138388 + 0.990378i \(0.544192\pi\)
\(972\) −2944.69 −0.0971717
\(973\) 0 0
\(974\) −64308.6 −2.11559
\(975\) 7113.41 0.233653
\(976\) 2428.48 0.0796454
\(977\) 16934.0 0.554519 0.277260 0.960795i \(-0.410574\pi\)
0.277260 + 0.960795i \(0.410574\pi\)
\(978\) 14699.8 0.480622
\(979\) −1573.66 −0.0513734
\(980\) 0 0
\(981\) −13633.2 −0.443704
\(982\) 55365.0 1.79915
\(983\) −49440.9 −1.60419 −0.802096 0.597195i \(-0.796282\pi\)
−0.802096 + 0.597195i \(0.796282\pi\)
\(984\) −22174.6 −0.718394
\(985\) 62895.2 2.03453
\(986\) 55670.0 1.79807
\(987\) 0 0
\(988\) −25259.6 −0.813375
\(989\) 38987.1 1.25351
\(990\) −7843.82 −0.251811
\(991\) 52151.6 1.67169 0.835847 0.548962i \(-0.184977\pi\)
0.835847 + 0.548962i \(0.184977\pi\)
\(992\) −5044.69 −0.161461
\(993\) 6172.63 0.197263
\(994\) 0 0
\(995\) −24362.5 −0.776224
\(996\) 34001.8 1.08171
\(997\) −55040.8 −1.74840 −0.874202 0.485562i \(-0.838615\pi\)
−0.874202 + 0.485562i \(0.838615\pi\)
\(998\) −30412.9 −0.964632
\(999\) 3666.01 0.116103
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.1.2 16
7.6 odd 2 1617.4.a.bf.1.2 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1617.4.a.be.1.2 16 1.1 even 1 trivial
1617.4.a.bf.1.2 yes 16 7.6 odd 2