Properties

Label 1617.4.a.be.1.13
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 1.13
Root \(4.13448\) 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)
 
\(f(q)\) \(=\) \(q+4.13448 q^{2} -3.00000 q^{3} +9.09391 q^{4} +10.9590 q^{5} -12.4034 q^{6} +4.52277 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+4.13448 q^{2} -3.00000 q^{3} +9.09391 q^{4} +10.9590 q^{5} -12.4034 q^{6} +4.52277 q^{8} +9.00000 q^{9} +45.3099 q^{10} +11.0000 q^{11} -27.2817 q^{12} -55.7486 q^{13} -32.8771 q^{15} -54.0520 q^{16} +55.2997 q^{17} +37.2103 q^{18} -116.629 q^{19} +99.6605 q^{20} +45.4793 q^{22} -74.0341 q^{23} -13.5683 q^{24} -4.89972 q^{25} -230.491 q^{26} -27.0000 q^{27} +189.891 q^{29} -135.930 q^{30} -142.372 q^{31} -259.659 q^{32} -33.0000 q^{33} +228.635 q^{34} +81.8452 q^{36} +241.223 q^{37} -482.199 q^{38} +167.246 q^{39} +49.5651 q^{40} -215.447 q^{41} -252.189 q^{43} +100.033 q^{44} +98.6312 q^{45} -306.093 q^{46} -557.633 q^{47} +162.156 q^{48} -20.2578 q^{50} -165.899 q^{51} -506.973 q^{52} +170.675 q^{53} -111.631 q^{54} +120.549 q^{55} +349.886 q^{57} +785.102 q^{58} -791.190 q^{59} -298.981 q^{60} +257.636 q^{61} -588.633 q^{62} -641.139 q^{64} -610.950 q^{65} -136.438 q^{66} +219.083 q^{67} +502.890 q^{68} +222.102 q^{69} -12.4721 q^{71} +40.7049 q^{72} +107.351 q^{73} +997.330 q^{74} +14.6992 q^{75} -1060.61 q^{76} +691.474 q^{78} +11.9028 q^{79} -592.358 q^{80} +81.0000 q^{81} -890.763 q^{82} +509.214 q^{83} +606.030 q^{85} -1042.67 q^{86} -569.674 q^{87} +49.7504 q^{88} -1357.83 q^{89} +407.789 q^{90} -673.260 q^{92} +427.115 q^{93} -2305.52 q^{94} -1278.14 q^{95} +778.977 q^{96} +405.813 q^{97} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{2} - 48 q^{3} + 72 q^{4} - 40 q^{5} - 12 q^{6} + 66 q^{8} + 144 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.13448 1.46176 0.730879 0.682507i \(-0.239110\pi\)
0.730879 + 0.682507i \(0.239110\pi\)
\(3\) −3.00000 −0.577350
\(4\) 9.09391 1.13674
\(5\) 10.9590 0.980205 0.490103 0.871665i \(-0.336959\pi\)
0.490103 + 0.871665i \(0.336959\pi\)
\(6\) −12.4034 −0.843947
\(7\) 0 0
\(8\) 4.52277 0.199880
\(9\) 9.00000 0.333333
\(10\) 45.3099 1.43282
\(11\) 11.0000 0.301511
\(12\) −27.2817 −0.656297
\(13\) −55.7486 −1.18937 −0.594687 0.803957i \(-0.702724\pi\)
−0.594687 + 0.803957i \(0.702724\pi\)
\(14\) 0 0
\(15\) −32.8771 −0.565922
\(16\) −54.0520 −0.844563
\(17\) 55.2997 0.788949 0.394475 0.918907i \(-0.370927\pi\)
0.394475 + 0.918907i \(0.370927\pi\)
\(18\) 37.2103 0.487253
\(19\) −116.629 −1.40824 −0.704118 0.710083i \(-0.748657\pi\)
−0.704118 + 0.710083i \(0.748657\pi\)
\(20\) 99.6605 1.11424
\(21\) 0 0
\(22\) 45.4793 0.440737
\(23\) −74.0341 −0.671182 −0.335591 0.942008i \(-0.608936\pi\)
−0.335591 + 0.942008i \(0.608936\pi\)
\(24\) −13.5683 −0.115401
\(25\) −4.89972 −0.0391978
\(26\) −230.491 −1.73858
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 189.891 1.21593 0.607965 0.793964i \(-0.291986\pi\)
0.607965 + 0.793964i \(0.291986\pi\)
\(30\) −135.930 −0.827241
\(31\) −142.372 −0.824862 −0.412431 0.910989i \(-0.635320\pi\)
−0.412431 + 0.910989i \(0.635320\pi\)
\(32\) −259.659 −1.43443
\(33\) −33.0000 −0.174078
\(34\) 228.635 1.15325
\(35\) 0 0
\(36\) 81.8452 0.378913
\(37\) 241.223 1.07180 0.535902 0.844280i \(-0.319972\pi\)
0.535902 + 0.844280i \(0.319972\pi\)
\(38\) −482.199 −2.05850
\(39\) 167.246 0.686686
\(40\) 49.5651 0.195923
\(41\) −215.447 −0.820664 −0.410332 0.911936i \(-0.634587\pi\)
−0.410332 + 0.911936i \(0.634587\pi\)
\(42\) 0 0
\(43\) −252.189 −0.894384 −0.447192 0.894438i \(-0.647576\pi\)
−0.447192 + 0.894438i \(0.647576\pi\)
\(44\) 100.033 0.342740
\(45\) 98.6312 0.326735
\(46\) −306.093 −0.981106
\(47\) −557.633 −1.73062 −0.865311 0.501236i \(-0.832879\pi\)
−0.865311 + 0.501236i \(0.832879\pi\)
\(48\) 162.156 0.487609
\(49\) 0 0
\(50\) −20.2578 −0.0572977
\(51\) −165.899 −0.455500
\(52\) −506.973 −1.35201
\(53\) 170.675 0.442339 0.221169 0.975235i \(-0.429013\pi\)
0.221169 + 0.975235i \(0.429013\pi\)
\(54\) −111.631 −0.281316
\(55\) 120.549 0.295543
\(56\) 0 0
\(57\) 349.886 0.813045
\(58\) 785.102 1.77740
\(59\) −791.190 −1.74583 −0.872917 0.487869i \(-0.837774\pi\)
−0.872917 + 0.487869i \(0.837774\pi\)
\(60\) −298.981 −0.643305
\(61\) 257.636 0.540769 0.270384 0.962752i \(-0.412849\pi\)
0.270384 + 0.962752i \(0.412849\pi\)
\(62\) −588.633 −1.20575
\(63\) 0 0
\(64\) −641.139 −1.25222
\(65\) −610.950 −1.16583
\(66\) −136.438 −0.254460
\(67\) 219.083 0.399482 0.199741 0.979849i \(-0.435990\pi\)
0.199741 + 0.979849i \(0.435990\pi\)
\(68\) 502.890 0.896830
\(69\) 222.102 0.387507
\(70\) 0 0
\(71\) −12.4721 −0.0208475 −0.0104237 0.999946i \(-0.503318\pi\)
−0.0104237 + 0.999946i \(0.503318\pi\)
\(72\) 40.7049 0.0666266
\(73\) 107.351 0.172116 0.0860578 0.996290i \(-0.472573\pi\)
0.0860578 + 0.996290i \(0.472573\pi\)
\(74\) 997.330 1.56672
\(75\) 14.6992 0.0226308
\(76\) −1060.61 −1.60080
\(77\) 0 0
\(78\) 691.474 1.00377
\(79\) 11.9028 0.0169515 0.00847575 0.999964i \(-0.497302\pi\)
0.00847575 + 0.999964i \(0.497302\pi\)
\(80\) −592.358 −0.827845
\(81\) 81.0000 0.111111
\(82\) −890.763 −1.19961
\(83\) 509.214 0.673415 0.336708 0.941609i \(-0.390687\pi\)
0.336708 + 0.941609i \(0.390687\pi\)
\(84\) 0 0
\(85\) 606.030 0.773332
\(86\) −1042.67 −1.30737
\(87\) −569.674 −0.702017
\(88\) 49.7504 0.0602661
\(89\) −1357.83 −1.61719 −0.808597 0.588363i \(-0.799773\pi\)
−0.808597 + 0.588363i \(0.799773\pi\)
\(90\) 407.789 0.477608
\(91\) 0 0
\(92\) −673.260 −0.762959
\(93\) 427.115 0.476234
\(94\) −2305.52 −2.52975
\(95\) −1278.14 −1.38036
\(96\) 778.977 0.828167
\(97\) 405.813 0.424785 0.212392 0.977184i \(-0.431875\pi\)
0.212392 + 0.977184i \(0.431875\pi\)
\(98\) 0 0
\(99\) 99.0000 0.100504
\(100\) −44.5576 −0.0445576
\(101\) 618.367 0.609206 0.304603 0.952479i \(-0.401476\pi\)
0.304603 + 0.952479i \(0.401476\pi\)
\(102\) −685.906 −0.665831
\(103\) 1220.77 1.16783 0.583914 0.811815i \(-0.301520\pi\)
0.583914 + 0.811815i \(0.301520\pi\)
\(104\) −252.138 −0.237732
\(105\) 0 0
\(106\) 705.651 0.646593
\(107\) 1047.79 0.946667 0.473334 0.880883i \(-0.343051\pi\)
0.473334 + 0.880883i \(0.343051\pi\)
\(108\) −245.536 −0.218766
\(109\) −519.147 −0.456195 −0.228098 0.973638i \(-0.573251\pi\)
−0.228098 + 0.973638i \(0.573251\pi\)
\(110\) 498.409 0.432013
\(111\) −723.668 −0.618806
\(112\) 0 0
\(113\) 2341.62 1.94939 0.974694 0.223542i \(-0.0717621\pi\)
0.974694 + 0.223542i \(0.0717621\pi\)
\(114\) 1446.60 1.18848
\(115\) −811.342 −0.657896
\(116\) 1726.86 1.38219
\(117\) −501.737 −0.396458
\(118\) −3271.16 −2.55199
\(119\) 0 0
\(120\) −148.695 −0.113116
\(121\) 121.000 0.0909091
\(122\) 1065.19 0.790473
\(123\) 646.342 0.473811
\(124\) −1294.72 −0.937653
\(125\) −1423.57 −1.01863
\(126\) 0 0
\(127\) −1468.50 −1.02605 −0.513025 0.858374i \(-0.671475\pi\)
−0.513025 + 0.858374i \(0.671475\pi\)
\(128\) −573.502 −0.396022
\(129\) 756.568 0.516373
\(130\) −2525.96 −1.70416
\(131\) −2128.17 −1.41938 −0.709692 0.704512i \(-0.751166\pi\)
−0.709692 + 0.704512i \(0.751166\pi\)
\(132\) −300.099 −0.197881
\(133\) 0 0
\(134\) 905.795 0.583946
\(135\) −295.894 −0.188641
\(136\) 250.107 0.157695
\(137\) −1654.03 −1.03149 −0.515743 0.856743i \(-0.672484\pi\)
−0.515743 + 0.856743i \(0.672484\pi\)
\(138\) 918.278 0.566442
\(139\) −2311.64 −1.41058 −0.705290 0.708919i \(-0.749183\pi\)
−0.705290 + 0.708919i \(0.749183\pi\)
\(140\) 0 0
\(141\) 1672.90 0.999175
\(142\) −51.5657 −0.0304740
\(143\) −613.234 −0.358610
\(144\) −486.468 −0.281521
\(145\) 2081.03 1.19186
\(146\) 443.839 0.251592
\(147\) 0 0
\(148\) 2193.66 1.21836
\(149\) −1496.00 −0.822528 −0.411264 0.911516i \(-0.634913\pi\)
−0.411264 + 0.911516i \(0.634913\pi\)
\(150\) 60.7734 0.0330808
\(151\) 37.2530 0.0200769 0.0100384 0.999950i \(-0.496805\pi\)
0.0100384 + 0.999950i \(0.496805\pi\)
\(152\) −527.485 −0.281478
\(153\) 497.697 0.262983
\(154\) 0 0
\(155\) −1560.26 −0.808534
\(156\) 1520.92 0.780583
\(157\) −539.971 −0.274486 −0.137243 0.990537i \(-0.543824\pi\)
−0.137243 + 0.990537i \(0.543824\pi\)
\(158\) 49.2118 0.0247790
\(159\) −512.024 −0.255384
\(160\) −2845.61 −1.40603
\(161\) 0 0
\(162\) 334.893 0.162418
\(163\) −2002.98 −0.962487 −0.481244 0.876587i \(-0.659815\pi\)
−0.481244 + 0.876587i \(0.659815\pi\)
\(164\) −1959.26 −0.932881
\(165\) −361.648 −0.170632
\(166\) 2105.33 0.984371
\(167\) −3346.12 −1.55048 −0.775241 0.631666i \(-0.782371\pi\)
−0.775241 + 0.631666i \(0.782371\pi\)
\(168\) 0 0
\(169\) 910.901 0.414611
\(170\) 2505.62 1.13043
\(171\) −1049.66 −0.469412
\(172\) −2293.39 −1.01668
\(173\) −2538.50 −1.11560 −0.557799 0.829976i \(-0.688354\pi\)
−0.557799 + 0.829976i \(0.688354\pi\)
\(174\) −2355.31 −1.02618
\(175\) 0 0
\(176\) −594.572 −0.254645
\(177\) 2373.57 1.00796
\(178\) −5613.94 −2.36395
\(179\) −4575.15 −1.91041 −0.955203 0.295951i \(-0.904363\pi\)
−0.955203 + 0.295951i \(0.904363\pi\)
\(180\) 896.944 0.371413
\(181\) 495.478 0.203473 0.101736 0.994811i \(-0.467560\pi\)
0.101736 + 0.994811i \(0.467560\pi\)
\(182\) 0 0
\(183\) −772.907 −0.312213
\(184\) −334.839 −0.134156
\(185\) 2643.57 1.05059
\(186\) 1765.90 0.696140
\(187\) 608.296 0.237877
\(188\) −5071.07 −1.96726
\(189\) 0 0
\(190\) −5284.43 −2.01775
\(191\) 1787.89 0.677314 0.338657 0.940910i \(-0.390027\pi\)
0.338657 + 0.940910i \(0.390027\pi\)
\(192\) 1923.42 0.722972
\(193\) −4947.29 −1.84515 −0.922575 0.385818i \(-0.873919\pi\)
−0.922575 + 0.385818i \(0.873919\pi\)
\(194\) 1677.83 0.620933
\(195\) 1832.85 0.673093
\(196\) 0 0
\(197\) 2792.79 1.01004 0.505020 0.863108i \(-0.331485\pi\)
0.505020 + 0.863108i \(0.331485\pi\)
\(198\) 409.313 0.146912
\(199\) 2512.22 0.894907 0.447454 0.894307i \(-0.352331\pi\)
0.447454 + 0.894307i \(0.352331\pi\)
\(200\) −22.1603 −0.00783484
\(201\) −657.249 −0.230641
\(202\) 2556.62 0.890512
\(203\) 0 0
\(204\) −1508.67 −0.517785
\(205\) −2361.09 −0.804419
\(206\) 5047.26 1.70708
\(207\) −666.307 −0.223727
\(208\) 3013.32 1.00450
\(209\) −1282.92 −0.424599
\(210\) 0 0
\(211\) 3069.47 1.00147 0.500736 0.865600i \(-0.333063\pi\)
0.500736 + 0.865600i \(0.333063\pi\)
\(212\) 1552.10 0.502824
\(213\) 37.4164 0.0120363
\(214\) 4332.05 1.38380
\(215\) −2763.75 −0.876680
\(216\) −122.115 −0.0384669
\(217\) 0 0
\(218\) −2146.40 −0.666847
\(219\) −322.052 −0.0993710
\(220\) 1096.27 0.335955
\(221\) −3082.88 −0.938356
\(222\) −2991.99 −0.904546
\(223\) 6545.98 1.96570 0.982850 0.184407i \(-0.0590363\pi\)
0.982850 + 0.184407i \(0.0590363\pi\)
\(224\) 0 0
\(225\) −44.0975 −0.0130659
\(226\) 9681.37 2.84954
\(227\) 431.720 0.126230 0.0631151 0.998006i \(-0.479896\pi\)
0.0631151 + 0.998006i \(0.479896\pi\)
\(228\) 3181.84 0.924220
\(229\) 1091.44 0.314953 0.157477 0.987523i \(-0.449664\pi\)
0.157477 + 0.987523i \(0.449664\pi\)
\(230\) −3354.48 −0.961685
\(231\) 0 0
\(232\) 858.834 0.243040
\(233\) 722.874 0.203249 0.101625 0.994823i \(-0.467596\pi\)
0.101625 + 0.994823i \(0.467596\pi\)
\(234\) −2074.42 −0.579526
\(235\) −6111.12 −1.69636
\(236\) −7195.02 −1.98456
\(237\) −35.7084 −0.00978695
\(238\) 0 0
\(239\) 3029.83 0.820015 0.410008 0.912082i \(-0.365526\pi\)
0.410008 + 0.912082i \(0.365526\pi\)
\(240\) 1777.07 0.477957
\(241\) 3604.71 0.963484 0.481742 0.876313i \(-0.340004\pi\)
0.481742 + 0.876313i \(0.340004\pi\)
\(242\) 500.272 0.132887
\(243\) −243.000 −0.0641500
\(244\) 2342.92 0.614713
\(245\) 0 0
\(246\) 2672.29 0.692597
\(247\) 6501.89 1.67492
\(248\) −643.914 −0.164873
\(249\) −1527.64 −0.388796
\(250\) −5885.74 −1.48899
\(251\) 1623.57 0.408281 0.204141 0.978942i \(-0.434560\pi\)
0.204141 + 0.978942i \(0.434560\pi\)
\(252\) 0 0
\(253\) −814.375 −0.202369
\(254\) −6071.48 −1.49984
\(255\) −1818.09 −0.446483
\(256\) 2757.98 0.673335
\(257\) −2385.67 −0.579043 −0.289522 0.957171i \(-0.593496\pi\)
−0.289522 + 0.957171i \(0.593496\pi\)
\(258\) 3128.02 0.754813
\(259\) 0 0
\(260\) −5555.93 −1.32525
\(261\) 1709.02 0.405310
\(262\) −8798.88 −2.07480
\(263\) 3806.19 0.892393 0.446197 0.894935i \(-0.352778\pi\)
0.446197 + 0.894935i \(0.352778\pi\)
\(264\) −149.251 −0.0347946
\(265\) 1870.43 0.433583
\(266\) 0 0
\(267\) 4073.50 0.933687
\(268\) 1992.32 0.454106
\(269\) 920.445 0.208626 0.104313 0.994544i \(-0.466736\pi\)
0.104313 + 0.994544i \(0.466736\pi\)
\(270\) −1223.37 −0.275747
\(271\) 4574.65 1.02542 0.512712 0.858560i \(-0.328641\pi\)
0.512712 + 0.858560i \(0.328641\pi\)
\(272\) −2989.06 −0.666317
\(273\) 0 0
\(274\) −6838.56 −1.50778
\(275\) −53.8969 −0.0118186
\(276\) 2019.78 0.440494
\(277\) 1236.78 0.268271 0.134136 0.990963i \(-0.457174\pi\)
0.134136 + 0.990963i \(0.457174\pi\)
\(278\) −9557.41 −2.06193
\(279\) −1281.35 −0.274954
\(280\) 0 0
\(281\) −4590.07 −0.974450 −0.487225 0.873277i \(-0.661991\pi\)
−0.487225 + 0.873277i \(0.661991\pi\)
\(282\) 6916.57 1.46055
\(283\) −4826.37 −1.01377 −0.506887 0.862012i \(-0.669204\pi\)
−0.506887 + 0.862012i \(0.669204\pi\)
\(284\) −113.420 −0.0236981
\(285\) 3834.41 0.796951
\(286\) −2535.40 −0.524201
\(287\) 0 0
\(288\) −2336.93 −0.478143
\(289\) −1854.95 −0.377559
\(290\) 8603.96 1.74221
\(291\) −1217.44 −0.245250
\(292\) 976.238 0.195651
\(293\) 1517.59 0.302590 0.151295 0.988489i \(-0.451656\pi\)
0.151295 + 0.988489i \(0.451656\pi\)
\(294\) 0 0
\(295\) −8670.68 −1.71128
\(296\) 1090.99 0.214232
\(297\) −297.000 −0.0580259
\(298\) −6185.16 −1.20234
\(299\) 4127.30 0.798287
\(300\) 133.673 0.0257254
\(301\) 0 0
\(302\) 154.022 0.0293475
\(303\) −1855.10 −0.351725
\(304\) 6304.02 1.18934
\(305\) 2823.44 0.530064
\(306\) 2057.72 0.384418
\(307\) −2860.08 −0.531704 −0.265852 0.964014i \(-0.585653\pi\)
−0.265852 + 0.964014i \(0.585653\pi\)
\(308\) 0 0
\(309\) −3662.32 −0.674246
\(310\) −6450.85 −1.18188
\(311\) 10151.0 1.85084 0.925420 0.378942i \(-0.123712\pi\)
0.925420 + 0.378942i \(0.123712\pi\)
\(312\) 756.413 0.137255
\(313\) 2693.16 0.486347 0.243174 0.969983i \(-0.421812\pi\)
0.243174 + 0.969983i \(0.421812\pi\)
\(314\) −2232.50 −0.401233
\(315\) 0 0
\(316\) 108.243 0.0192694
\(317\) 805.888 0.142786 0.0713930 0.997448i \(-0.477256\pi\)
0.0713930 + 0.997448i \(0.477256\pi\)
\(318\) −2116.95 −0.373311
\(319\) 2088.81 0.366617
\(320\) −7026.26 −1.22744
\(321\) −3143.36 −0.546558
\(322\) 0 0
\(323\) −6449.53 −1.11103
\(324\) 736.607 0.126304
\(325\) 273.152 0.0466208
\(326\) −8281.28 −1.40692
\(327\) 1557.44 0.263384
\(328\) −974.418 −0.164034
\(329\) 0 0
\(330\) −1495.23 −0.249423
\(331\) 634.320 0.105333 0.0526667 0.998612i \(-0.483228\pi\)
0.0526667 + 0.998612i \(0.483228\pi\)
\(332\) 4630.74 0.765497
\(333\) 2171.00 0.357268
\(334\) −13834.5 −2.26643
\(335\) 2400.94 0.391574
\(336\) 0 0
\(337\) −1070.63 −0.173060 −0.0865299 0.996249i \(-0.527578\pi\)
−0.0865299 + 0.996249i \(0.527578\pi\)
\(338\) 3766.10 0.606062
\(339\) −7024.85 −1.12548
\(340\) 5511.19 0.879077
\(341\) −1566.09 −0.248705
\(342\) −4339.79 −0.686167
\(343\) 0 0
\(344\) −1140.59 −0.178769
\(345\) 2434.03 0.379836
\(346\) −10495.4 −1.63073
\(347\) −4767.97 −0.737631 −0.368816 0.929503i \(-0.620237\pi\)
−0.368816 + 0.929503i \(0.620237\pi\)
\(348\) −5180.57 −0.798011
\(349\) 5046.40 0.774005 0.387002 0.922079i \(-0.373510\pi\)
0.387002 + 0.922079i \(0.373510\pi\)
\(350\) 0 0
\(351\) 1505.21 0.228895
\(352\) −2856.25 −0.432496
\(353\) 3080.35 0.464449 0.232224 0.972662i \(-0.425400\pi\)
0.232224 + 0.972662i \(0.425400\pi\)
\(354\) 9813.48 1.47339
\(355\) −136.682 −0.0204348
\(356\) −12348.0 −1.83833
\(357\) 0 0
\(358\) −18915.9 −2.79255
\(359\) −6997.76 −1.02877 −0.514384 0.857560i \(-0.671979\pi\)
−0.514384 + 0.857560i \(0.671979\pi\)
\(360\) 446.086 0.0653078
\(361\) 6743.27 0.983128
\(362\) 2048.54 0.297428
\(363\) −363.000 −0.0524864
\(364\) 0 0
\(365\) 1176.46 0.168709
\(366\) −3195.57 −0.456380
\(367\) 2459.20 0.349780 0.174890 0.984588i \(-0.444043\pi\)
0.174890 + 0.984588i \(0.444043\pi\)
\(368\) 4001.70 0.566855
\(369\) −1939.03 −0.273555
\(370\) 10929.8 1.53571
\(371\) 0 0
\(372\) 3884.15 0.541354
\(373\) 9454.51 1.31243 0.656214 0.754575i \(-0.272157\pi\)
0.656214 + 0.754575i \(0.272157\pi\)
\(374\) 2514.99 0.347719
\(375\) 4270.72 0.588105
\(376\) −2522.05 −0.345916
\(377\) −10586.2 −1.44620
\(378\) 0 0
\(379\) −11712.9 −1.58747 −0.793736 0.608262i \(-0.791867\pi\)
−0.793736 + 0.608262i \(0.791867\pi\)
\(380\) −11623.3 −1.56911
\(381\) 4405.50 0.592390
\(382\) 7391.98 0.990069
\(383\) 11835.3 1.57900 0.789498 0.613753i \(-0.210341\pi\)
0.789498 + 0.613753i \(0.210341\pi\)
\(384\) 1720.51 0.228644
\(385\) 0 0
\(386\) −20454.5 −2.69716
\(387\) −2269.70 −0.298128
\(388\) 3690.43 0.482869
\(389\) −6793.97 −0.885522 −0.442761 0.896640i \(-0.646001\pi\)
−0.442761 + 0.896640i \(0.646001\pi\)
\(390\) 7577.88 0.983899
\(391\) −4094.06 −0.529528
\(392\) 0 0
\(393\) 6384.52 0.819482
\(394\) 11546.7 1.47644
\(395\) 130.443 0.0166160
\(396\) 900.298 0.114247
\(397\) −9567.26 −1.20949 −0.604745 0.796420i \(-0.706725\pi\)
−0.604745 + 0.796420i \(0.706725\pi\)
\(398\) 10386.7 1.30814
\(399\) 0 0
\(400\) 264.840 0.0331050
\(401\) −5521.26 −0.687577 −0.343789 0.939047i \(-0.611710\pi\)
−0.343789 + 0.939047i \(0.611710\pi\)
\(402\) −2717.38 −0.337141
\(403\) 7937.02 0.981070
\(404\) 5623.38 0.692508
\(405\) 887.681 0.108912
\(406\) 0 0
\(407\) 2653.45 0.323161
\(408\) −750.322 −0.0910453
\(409\) −2847.96 −0.344310 −0.172155 0.985070i \(-0.555073\pi\)
−0.172155 + 0.985070i \(0.555073\pi\)
\(410\) −9761.90 −1.17587
\(411\) 4962.10 0.595529
\(412\) 11101.6 1.32752
\(413\) 0 0
\(414\) −2754.83 −0.327035
\(415\) 5580.49 0.660085
\(416\) 14475.6 1.70607
\(417\) 6934.91 0.814398
\(418\) −5304.19 −0.620661
\(419\) −886.838 −0.103401 −0.0517003 0.998663i \(-0.516464\pi\)
−0.0517003 + 0.998663i \(0.516464\pi\)
\(420\) 0 0
\(421\) 10167.4 1.17702 0.588511 0.808489i \(-0.299714\pi\)
0.588511 + 0.808489i \(0.299714\pi\)
\(422\) 12690.6 1.46391
\(423\) −5018.70 −0.576874
\(424\) 771.921 0.0884146
\(425\) −270.953 −0.0309250
\(426\) 154.697 0.0175941
\(427\) 0 0
\(428\) 9528.48 1.07611
\(429\) 1839.70 0.207044
\(430\) −11426.7 −1.28150
\(431\) −5702.23 −0.637278 −0.318639 0.947876i \(-0.603226\pi\)
−0.318639 + 0.947876i \(0.603226\pi\)
\(432\) 1459.40 0.162536
\(433\) −3981.78 −0.441922 −0.220961 0.975283i \(-0.570919\pi\)
−0.220961 + 0.975283i \(0.570919\pi\)
\(434\) 0 0
\(435\) −6243.08 −0.688121
\(436\) −4721.08 −0.518575
\(437\) 8634.51 0.945182
\(438\) −1331.52 −0.145256
\(439\) 6369.61 0.692495 0.346247 0.938143i \(-0.387456\pi\)
0.346247 + 0.938143i \(0.387456\pi\)
\(440\) 545.216 0.0590731
\(441\) 0 0
\(442\) −12746.1 −1.37165
\(443\) 9283.06 0.995602 0.497801 0.867291i \(-0.334141\pi\)
0.497801 + 0.867291i \(0.334141\pi\)
\(444\) −6580.97 −0.703422
\(445\) −14880.6 −1.58518
\(446\) 27064.2 2.87338
\(447\) 4487.99 0.474887
\(448\) 0 0
\(449\) 12119.9 1.27389 0.636943 0.770911i \(-0.280199\pi\)
0.636943 + 0.770911i \(0.280199\pi\)
\(450\) −182.320 −0.0190992
\(451\) −2369.92 −0.247440
\(452\) 21294.5 2.21595
\(453\) −111.759 −0.0115914
\(454\) 1784.94 0.184518
\(455\) 0 0
\(456\) 1582.45 0.162511
\(457\) 9956.75 1.01916 0.509581 0.860423i \(-0.329800\pi\)
0.509581 + 0.860423i \(0.329800\pi\)
\(458\) 4512.53 0.460386
\(459\) −1493.09 −0.151833
\(460\) −7378.28 −0.747856
\(461\) 2985.32 0.301606 0.150803 0.988564i \(-0.451814\pi\)
0.150803 + 0.988564i \(0.451814\pi\)
\(462\) 0 0
\(463\) 4409.74 0.442630 0.221315 0.975202i \(-0.428965\pi\)
0.221315 + 0.975202i \(0.428965\pi\)
\(464\) −10264.0 −1.02693
\(465\) 4680.77 0.466807
\(466\) 2988.71 0.297101
\(467\) −2233.71 −0.221335 −0.110668 0.993857i \(-0.535299\pi\)
−0.110668 + 0.993857i \(0.535299\pi\)
\(468\) −4562.75 −0.450670
\(469\) 0 0
\(470\) −25266.3 −2.47967
\(471\) 1619.91 0.158475
\(472\) −3578.37 −0.348957
\(473\) −2774.08 −0.269667
\(474\) −147.636 −0.0143062
\(475\) 571.448 0.0551997
\(476\) 0 0
\(477\) 1536.07 0.147446
\(478\) 12526.8 1.19866
\(479\) −4715.63 −0.449818 −0.224909 0.974380i \(-0.572208\pi\)
−0.224909 + 0.974380i \(0.572208\pi\)
\(480\) 8536.83 0.811774
\(481\) −13447.8 −1.27478
\(482\) 14903.6 1.40838
\(483\) 0 0
\(484\) 1100.36 0.103340
\(485\) 4447.32 0.416376
\(486\) −1004.68 −0.0937719
\(487\) −3311.08 −0.308089 −0.154045 0.988064i \(-0.549230\pi\)
−0.154045 + 0.988064i \(0.549230\pi\)
\(488\) 1165.23 0.108089
\(489\) 6008.94 0.555692
\(490\) 0 0
\(491\) 18909.3 1.73801 0.869007 0.494800i \(-0.164759\pi\)
0.869007 + 0.494800i \(0.164759\pi\)
\(492\) 5877.78 0.538599
\(493\) 10500.9 0.959307
\(494\) 26881.9 2.44833
\(495\) 1084.94 0.0985143
\(496\) 7695.49 0.696648
\(497\) 0 0
\(498\) −6316.00 −0.568327
\(499\) 19151.7 1.71813 0.859067 0.511863i \(-0.171044\pi\)
0.859067 + 0.511863i \(0.171044\pi\)
\(500\) −12945.9 −1.15791
\(501\) 10038.4 0.895171
\(502\) 6712.60 0.596809
\(503\) −5085.07 −0.450759 −0.225380 0.974271i \(-0.572362\pi\)
−0.225380 + 0.974271i \(0.572362\pi\)
\(504\) 0 0
\(505\) 6776.70 0.597147
\(506\) −3367.02 −0.295815
\(507\) −2732.70 −0.239376
\(508\) −13354.4 −1.16635
\(509\) 5714.36 0.497612 0.248806 0.968553i \(-0.419962\pi\)
0.248806 + 0.968553i \(0.419962\pi\)
\(510\) −7516.86 −0.652651
\(511\) 0 0
\(512\) 15990.8 1.38028
\(513\) 3148.98 0.271015
\(514\) −9863.51 −0.846422
\(515\) 13378.5 1.14471
\(516\) 6880.17 0.586982
\(517\) −6133.97 −0.521802
\(518\) 0 0
\(519\) 7615.49 0.644091
\(520\) −2763.18 −0.233026
\(521\) −4605.81 −0.387301 −0.193651 0.981071i \(-0.562033\pi\)
−0.193651 + 0.981071i \(0.562033\pi\)
\(522\) 7065.92 0.592465
\(523\) 1660.38 0.138821 0.0694103 0.997588i \(-0.477888\pi\)
0.0694103 + 0.997588i \(0.477888\pi\)
\(524\) −19353.4 −1.61347
\(525\) 0 0
\(526\) 15736.6 1.30446
\(527\) −7873.11 −0.650774
\(528\) 1783.72 0.147020
\(529\) −6685.95 −0.549515
\(530\) 7733.24 0.633794
\(531\) −7120.71 −0.581945
\(532\) 0 0
\(533\) 12010.9 0.976077
\(534\) 16841.8 1.36483
\(535\) 11482.7 0.927928
\(536\) 990.862 0.0798483
\(537\) 13725.4 1.10297
\(538\) 3805.56 0.304962
\(539\) 0 0
\(540\) −2690.83 −0.214435
\(541\) −13229.9 −1.05138 −0.525690 0.850676i \(-0.676193\pi\)
−0.525690 + 0.850676i \(0.676193\pi\)
\(542\) 18913.8 1.49892
\(543\) −1486.43 −0.117475
\(544\) −14359.1 −1.13169
\(545\) −5689.35 −0.447165
\(546\) 0 0
\(547\) 13908.5 1.08718 0.543588 0.839352i \(-0.317065\pi\)
0.543588 + 0.839352i \(0.317065\pi\)
\(548\) −15041.6 −1.17253
\(549\) 2318.72 0.180256
\(550\) −222.836 −0.0172759
\(551\) −22146.8 −1.71232
\(552\) 1004.52 0.0774549
\(553\) 0 0
\(554\) 5113.45 0.392148
\(555\) −7930.70 −0.606557
\(556\) −21021.8 −1.60346
\(557\) −15886.9 −1.20853 −0.604264 0.796784i \(-0.706533\pi\)
−0.604264 + 0.796784i \(0.706533\pi\)
\(558\) −5297.70 −0.401917
\(559\) 14059.2 1.06376
\(560\) 0 0
\(561\) −1824.89 −0.137338
\(562\) −18977.5 −1.42441
\(563\) 18993.4 1.42180 0.710901 0.703292i \(-0.248287\pi\)
0.710901 + 0.703292i \(0.248287\pi\)
\(564\) 15213.2 1.13580
\(565\) 25661.9 1.91080
\(566\) −19954.5 −1.48189
\(567\) 0 0
\(568\) −56.4085 −0.00416699
\(569\) 5613.98 0.413621 0.206810 0.978381i \(-0.433692\pi\)
0.206810 + 0.978381i \(0.433692\pi\)
\(570\) 15853.3 1.16495
\(571\) −14949.8 −1.09567 −0.547835 0.836586i \(-0.684548\pi\)
−0.547835 + 0.836586i \(0.684548\pi\)
\(572\) −5576.70 −0.407646
\(573\) −5363.66 −0.391047
\(574\) 0 0
\(575\) 362.747 0.0263088
\(576\) −5770.25 −0.417408
\(577\) −892.541 −0.0643969 −0.0321984 0.999481i \(-0.510251\pi\)
−0.0321984 + 0.999481i \(0.510251\pi\)
\(578\) −7669.24 −0.551901
\(579\) 14841.9 1.06530
\(580\) 18924.7 1.35483
\(581\) 0 0
\(582\) −5033.48 −0.358496
\(583\) 1877.42 0.133370
\(584\) 485.522 0.0344025
\(585\) −5498.55 −0.388610
\(586\) 6274.46 0.442313
\(587\) −19851.1 −1.39582 −0.697908 0.716187i \(-0.745886\pi\)
−0.697908 + 0.716187i \(0.745886\pi\)
\(588\) 0 0
\(589\) 16604.6 1.16160
\(590\) −35848.7 −2.50147
\(591\) −8378.36 −0.583147
\(592\) −13038.6 −0.905206
\(593\) 6639.01 0.459749 0.229875 0.973220i \(-0.426168\pi\)
0.229875 + 0.973220i \(0.426168\pi\)
\(594\) −1227.94 −0.0848199
\(595\) 0 0
\(596\) −13604.5 −0.935000
\(597\) −7536.66 −0.516675
\(598\) 17064.2 1.16690
\(599\) −25797.7 −1.75971 −0.879856 0.475241i \(-0.842361\pi\)
−0.879856 + 0.475241i \(0.842361\pi\)
\(600\) 66.4809 0.00452345
\(601\) 26416.1 1.79291 0.896453 0.443138i \(-0.146135\pi\)
0.896453 + 0.443138i \(0.146135\pi\)
\(602\) 0 0
\(603\) 1971.75 0.133161
\(604\) 338.776 0.0228222
\(605\) 1326.04 0.0891096
\(606\) −7669.87 −0.514138
\(607\) 4301.46 0.287629 0.143815 0.989605i \(-0.454063\pi\)
0.143815 + 0.989605i \(0.454063\pi\)
\(608\) 30283.7 2.02001
\(609\) 0 0
\(610\) 11673.4 0.774826
\(611\) 31087.3 2.05836
\(612\) 4526.01 0.298943
\(613\) 19926.9 1.31295 0.656477 0.754346i \(-0.272046\pi\)
0.656477 + 0.754346i \(0.272046\pi\)
\(614\) −11824.9 −0.777224
\(615\) 7083.28 0.464432
\(616\) 0 0
\(617\) −20225.1 −1.31967 −0.659833 0.751412i \(-0.729373\pi\)
−0.659833 + 0.751412i \(0.729373\pi\)
\(618\) −15141.8 −0.985585
\(619\) −12966.4 −0.841947 −0.420974 0.907073i \(-0.638311\pi\)
−0.420974 + 0.907073i \(0.638311\pi\)
\(620\) −14188.8 −0.919093
\(621\) 1998.92 0.129169
\(622\) 41969.2 2.70548
\(623\) 0 0
\(624\) −9039.97 −0.579949
\(625\) −14988.5 −0.959266
\(626\) 11134.8 0.710922
\(627\) 3848.75 0.245142
\(628\) −4910.45 −0.312019
\(629\) 13339.5 0.845599
\(630\) 0 0
\(631\) 18440.1 1.16337 0.581687 0.813413i \(-0.302393\pi\)
0.581687 + 0.813413i \(0.302393\pi\)
\(632\) 53.8335 0.00338826
\(633\) −9208.40 −0.578201
\(634\) 3331.92 0.208719
\(635\) −16093.3 −1.00574
\(636\) −4656.30 −0.290306
\(637\) 0 0
\(638\) 8636.12 0.535905
\(639\) −112.249 −0.00694915
\(640\) −6285.02 −0.388183
\(641\) −11567.3 −0.712764 −0.356382 0.934340i \(-0.615990\pi\)
−0.356382 + 0.934340i \(0.615990\pi\)
\(642\) −12996.2 −0.798937
\(643\) 16921.6 1.03782 0.518912 0.854827i \(-0.326337\pi\)
0.518912 + 0.854827i \(0.326337\pi\)
\(644\) 0 0
\(645\) 8291.25 0.506152
\(646\) −26665.4 −1.62405
\(647\) 1962.25 0.119234 0.0596168 0.998221i \(-0.481012\pi\)
0.0596168 + 0.998221i \(0.481012\pi\)
\(648\) 366.344 0.0222089
\(649\) −8703.09 −0.526389
\(650\) 1129.34 0.0681484
\(651\) 0 0
\(652\) −18214.9 −1.09410
\(653\) 19466.5 1.16659 0.583293 0.812261i \(-0.301764\pi\)
0.583293 + 0.812261i \(0.301764\pi\)
\(654\) 6439.21 0.385005
\(655\) −23322.7 −1.39129
\(656\) 11645.4 0.693103
\(657\) 966.156 0.0573719
\(658\) 0 0
\(659\) 9995.31 0.590838 0.295419 0.955368i \(-0.404541\pi\)
0.295419 + 0.955368i \(0.404541\pi\)
\(660\) −3288.80 −0.193964
\(661\) −15473.1 −0.910492 −0.455246 0.890366i \(-0.650449\pi\)
−0.455246 + 0.890366i \(0.650449\pi\)
\(662\) 2622.58 0.153972
\(663\) 9248.63 0.541760
\(664\) 2303.05 0.134602
\(665\) 0 0
\(666\) 8975.97 0.522240
\(667\) −14058.4 −0.816110
\(668\) −30429.3 −1.76249
\(669\) −19637.9 −1.13490
\(670\) 9926.63 0.572387
\(671\) 2833.99 0.163048
\(672\) 0 0
\(673\) 22075.3 1.26440 0.632200 0.774805i \(-0.282152\pi\)
0.632200 + 0.774805i \(0.282152\pi\)
\(674\) −4426.51 −0.252972
\(675\) 132.292 0.00754361
\(676\) 8283.66 0.471305
\(677\) −10736.1 −0.609489 −0.304744 0.952434i \(-0.598571\pi\)
−0.304744 + 0.952434i \(0.598571\pi\)
\(678\) −29044.1 −1.64518
\(679\) 0 0
\(680\) 2740.93 0.154574
\(681\) −1295.16 −0.0728791
\(682\) −6474.96 −0.363547
\(683\) −18453.5 −1.03383 −0.516913 0.856038i \(-0.672919\pi\)
−0.516913 + 0.856038i \(0.672919\pi\)
\(684\) −9545.51 −0.533599
\(685\) −18126.6 −1.01107
\(686\) 0 0
\(687\) −3274.32 −0.181838
\(688\) 13631.4 0.755364
\(689\) −9514.86 −0.526107
\(690\) 10063.4 0.555229
\(691\) −9601.59 −0.528599 −0.264299 0.964441i \(-0.585141\pi\)
−0.264299 + 0.964441i \(0.585141\pi\)
\(692\) −23084.9 −1.26814
\(693\) 0 0
\(694\) −19713.1 −1.07824
\(695\) −25333.3 −1.38266
\(696\) −2576.50 −0.140319
\(697\) −11914.2 −0.647462
\(698\) 20864.2 1.13141
\(699\) −2168.62 −0.117346
\(700\) 0 0
\(701\) 14973.3 0.806753 0.403377 0.915034i \(-0.367836\pi\)
0.403377 + 0.915034i \(0.367836\pi\)
\(702\) 6223.26 0.334590
\(703\) −28133.5 −1.50935
\(704\) −7052.53 −0.377560
\(705\) 18333.4 0.979396
\(706\) 12735.6 0.678912
\(707\) 0 0
\(708\) 21585.1 1.14579
\(709\) −17611.7 −0.932895 −0.466447 0.884549i \(-0.654466\pi\)
−0.466447 + 0.884549i \(0.654466\pi\)
\(710\) −565.110 −0.0298707
\(711\) 107.125 0.00565050
\(712\) −6141.17 −0.323244
\(713\) 10540.4 0.553633
\(714\) 0 0
\(715\) −6720.45 −0.351511
\(716\) −41606.0 −2.17163
\(717\) −9089.50 −0.473436
\(718\) −28932.1 −1.50381
\(719\) 36062.1 1.87050 0.935249 0.353990i \(-0.115175\pi\)
0.935249 + 0.353990i \(0.115175\pi\)
\(720\) −5331.22 −0.275948
\(721\) 0 0
\(722\) 27879.9 1.43710
\(723\) −10814.1 −0.556268
\(724\) 4505.83 0.231296
\(725\) −930.415 −0.0476617
\(726\) −1500.82 −0.0767224
\(727\) −38125.3 −1.94496 −0.972482 0.232978i \(-0.925153\pi\)
−0.972482 + 0.232978i \(0.925153\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 4864.04 0.246611
\(731\) −13946.0 −0.705624
\(732\) −7028.75 −0.354905
\(733\) −10828.6 −0.545654 −0.272827 0.962063i \(-0.587959\pi\)
−0.272827 + 0.962063i \(0.587959\pi\)
\(734\) 10167.5 0.511293
\(735\) 0 0
\(736\) 19223.6 0.962762
\(737\) 2409.91 0.120448
\(738\) −8016.87 −0.399871
\(739\) 31316.5 1.55886 0.779429 0.626491i \(-0.215509\pi\)
0.779429 + 0.626491i \(0.215509\pi\)
\(740\) 24040.4 1.19424
\(741\) −19505.7 −0.967015
\(742\) 0 0
\(743\) −3528.10 −0.174204 −0.0871019 0.996199i \(-0.527761\pi\)
−0.0871019 + 0.996199i \(0.527761\pi\)
\(744\) 1931.74 0.0951897
\(745\) −16394.7 −0.806246
\(746\) 39089.5 1.91845
\(747\) 4582.92 0.224472
\(748\) 5531.79 0.270404
\(749\) 0 0
\(750\) 17657.2 0.859667
\(751\) −16211.7 −0.787715 −0.393858 0.919171i \(-0.628860\pi\)
−0.393858 + 0.919171i \(0.628860\pi\)
\(752\) 30141.2 1.46162
\(753\) −4870.70 −0.235721
\(754\) −43768.3 −2.11399
\(755\) 408.257 0.0196795
\(756\) 0 0
\(757\) 37954.3 1.82229 0.911144 0.412088i \(-0.135200\pi\)
0.911144 + 0.412088i \(0.135200\pi\)
\(758\) −48426.8 −2.32050
\(759\) 2443.13 0.116838
\(760\) −5780.72 −0.275906
\(761\) −109.775 −0.00522907 −0.00261454 0.999997i \(-0.500832\pi\)
−0.00261454 + 0.999997i \(0.500832\pi\)
\(762\) 18214.4 0.865931
\(763\) 0 0
\(764\) 16258.9 0.769929
\(765\) 5454.27 0.257777
\(766\) 48932.8 2.30811
\(767\) 44107.7 2.07645
\(768\) −8273.94 −0.388750
\(769\) 36591.1 1.71587 0.857937 0.513755i \(-0.171746\pi\)
0.857937 + 0.513755i \(0.171746\pi\)
\(770\) 0 0
\(771\) 7157.02 0.334311
\(772\) −44990.3 −2.09745
\(773\) −13202.5 −0.614310 −0.307155 0.951659i \(-0.599377\pi\)
−0.307155 + 0.951659i \(0.599377\pi\)
\(774\) −9384.05 −0.435791
\(775\) 697.582 0.0323328
\(776\) 1835.40 0.0849059
\(777\) 0 0
\(778\) −28089.5 −1.29442
\(779\) 25127.4 1.15569
\(780\) 16667.8 0.765131
\(781\) −137.193 −0.00628574
\(782\) −16926.8 −0.774043
\(783\) −5127.07 −0.234006
\(784\) 0 0
\(785\) −5917.55 −0.269053
\(786\) 26396.6 1.19788
\(787\) −8288.79 −0.375430 −0.187715 0.982224i \(-0.560108\pi\)
−0.187715 + 0.982224i \(0.560108\pi\)
\(788\) 25397.4 1.14815
\(789\) −11418.6 −0.515224
\(790\) 539.314 0.0242885
\(791\) 0 0
\(792\) 447.754 0.0200887
\(793\) −14362.8 −0.643176
\(794\) −39555.6 −1.76798
\(795\) −5611.28 −0.250329
\(796\) 22845.9 1.01728
\(797\) −42473.0 −1.88767 −0.943834 0.330419i \(-0.892810\pi\)
−0.943834 + 0.330419i \(0.892810\pi\)
\(798\) 0 0
\(799\) −30836.9 −1.36537
\(800\) 1272.26 0.0562263
\(801\) −12220.5 −0.539064
\(802\) −22827.5 −1.00507
\(803\) 1180.86 0.0518948
\(804\) −5976.97 −0.262178
\(805\) 0 0
\(806\) 32815.4 1.43409
\(807\) −2761.33 −0.120451
\(808\) 2796.73 0.121768
\(809\) 23398.9 1.01689 0.508444 0.861095i \(-0.330221\pi\)
0.508444 + 0.861095i \(0.330221\pi\)
\(810\) 3670.10 0.159203
\(811\) 40985.0 1.77457 0.887286 0.461220i \(-0.152588\pi\)
0.887286 + 0.461220i \(0.152588\pi\)
\(812\) 0 0
\(813\) −13723.9 −0.592029
\(814\) 10970.6 0.472384
\(815\) −21950.7 −0.943435
\(816\) 8967.18 0.384699
\(817\) 29412.5 1.25950
\(818\) −11774.8 −0.503298
\(819\) 0 0
\(820\) −21471.6 −0.914415
\(821\) 21819.6 0.927538 0.463769 0.885956i \(-0.346497\pi\)
0.463769 + 0.885956i \(0.346497\pi\)
\(822\) 20515.7 0.870519
\(823\) −30412.0 −1.28809 −0.644043 0.764989i \(-0.722744\pi\)
−0.644043 + 0.764989i \(0.722744\pi\)
\(824\) 5521.27 0.233425
\(825\) 161.691 0.00682345
\(826\) 0 0
\(827\) 27535.2 1.15779 0.578896 0.815401i \(-0.303484\pi\)
0.578896 + 0.815401i \(0.303484\pi\)
\(828\) −6059.34 −0.254320
\(829\) −36911.1 −1.54641 −0.773206 0.634155i \(-0.781348\pi\)
−0.773206 + 0.634155i \(0.781348\pi\)
\(830\) 23072.4 0.964885
\(831\) −3710.35 −0.154886
\(832\) 35742.6 1.48936
\(833\) 0 0
\(834\) 28672.2 1.19045
\(835\) −36670.2 −1.51979
\(836\) −11666.7 −0.482658
\(837\) 3844.04 0.158745
\(838\) −3666.61 −0.151147
\(839\) 9468.89 0.389634 0.194817 0.980840i \(-0.437589\pi\)
0.194817 + 0.980840i \(0.437589\pi\)
\(840\) 0 0
\(841\) 11669.8 0.478485
\(842\) 42036.7 1.72052
\(843\) 13770.2 0.562599
\(844\) 27913.5 1.13841
\(845\) 9982.59 0.406404
\(846\) −20749.7 −0.843250
\(847\) 0 0
\(848\) −9225.31 −0.373583
\(849\) 14479.1 0.585303
\(850\) −1120.25 −0.0452050
\(851\) −17858.7 −0.719376
\(852\) 340.261 0.0136821
\(853\) −47631.0 −1.91191 −0.955953 0.293520i \(-0.905173\pi\)
−0.955953 + 0.293520i \(0.905173\pi\)
\(854\) 0 0
\(855\) −11503.2 −0.460120
\(856\) 4738.89 0.189220
\(857\) −30786.4 −1.22712 −0.613561 0.789647i \(-0.710264\pi\)
−0.613561 + 0.789647i \(0.710264\pi\)
\(858\) 7606.21 0.302648
\(859\) −45755.7 −1.81742 −0.908710 0.417428i \(-0.862932\pi\)
−0.908710 + 0.417428i \(0.862932\pi\)
\(860\) −25133.3 −0.996557
\(861\) 0 0
\(862\) −23575.7 −0.931546
\(863\) −40160.7 −1.58411 −0.792054 0.610451i \(-0.790988\pi\)
−0.792054 + 0.610451i \(0.790988\pi\)
\(864\) 7010.80 0.276056
\(865\) −27819.5 −1.09351
\(866\) −16462.6 −0.645984
\(867\) 5564.84 0.217984
\(868\) 0 0
\(869\) 130.931 0.00511107
\(870\) −25811.9 −1.00587
\(871\) −12213.6 −0.475133
\(872\) −2347.98 −0.0911842
\(873\) 3652.32 0.141595
\(874\) 35699.2 1.38163
\(875\) 0 0
\(876\) −2928.71 −0.112959
\(877\) −12085.7 −0.465341 −0.232671 0.972556i \(-0.574746\pi\)
−0.232671 + 0.972556i \(0.574746\pi\)
\(878\) 26335.0 1.01226
\(879\) −4552.78 −0.174700
\(880\) −6515.94 −0.249605
\(881\) 43533.0 1.66477 0.832387 0.554196i \(-0.186974\pi\)
0.832387 + 0.554196i \(0.186974\pi\)
\(882\) 0 0
\(883\) −31480.4 −1.19977 −0.599887 0.800084i \(-0.704788\pi\)
−0.599887 + 0.800084i \(0.704788\pi\)
\(884\) −28035.4 −1.06667
\(885\) 26012.0 0.988005
\(886\) 38380.6 1.45533
\(887\) −22002.2 −0.832875 −0.416437 0.909164i \(-0.636722\pi\)
−0.416437 + 0.909164i \(0.636722\pi\)
\(888\) −3272.98 −0.123687
\(889\) 0 0
\(890\) −61523.3 −2.31715
\(891\) 891.000 0.0335013
\(892\) 59528.6 2.23449
\(893\) 65036.1 2.43712
\(894\) 18555.5 0.694170
\(895\) −50139.2 −1.87259
\(896\) 0 0
\(897\) −12381.9 −0.460891
\(898\) 50109.6 1.86211
\(899\) −27035.2 −1.00297
\(900\) −401.019 −0.0148525
\(901\) 9438.25 0.348983
\(902\) −9798.39 −0.361697
\(903\) 0 0
\(904\) 10590.6 0.389644
\(905\) 5429.95 0.199445
\(906\) −462.065 −0.0169438
\(907\) −48048.1 −1.75900 −0.879500 0.475900i \(-0.842123\pi\)
−0.879500 + 0.475900i \(0.842123\pi\)
\(908\) 3926.03 0.143491
\(909\) 5565.30 0.203069
\(910\) 0 0
\(911\) 32477.6 1.18115 0.590577 0.806981i \(-0.298900\pi\)
0.590577 + 0.806981i \(0.298900\pi\)
\(912\) −18912.1 −0.686668
\(913\) 5601.35 0.203042
\(914\) 41166.0 1.48977
\(915\) −8470.31 −0.306033
\(916\) 9925.45 0.358020
\(917\) 0 0
\(918\) −6173.15 −0.221944
\(919\) −48705.4 −1.74825 −0.874127 0.485698i \(-0.838565\pi\)
−0.874127 + 0.485698i \(0.838565\pi\)
\(920\) −3669.51 −0.131500
\(921\) 8580.23 0.306980
\(922\) 12342.8 0.440876
\(923\) 695.303 0.0247954
\(924\) 0 0
\(925\) −1181.92 −0.0420123
\(926\) 18232.0 0.647019
\(927\) 10987.0 0.389276
\(928\) −49307.0 −1.74416
\(929\) −28313.6 −0.999933 −0.499967 0.866045i \(-0.666654\pi\)
−0.499967 + 0.866045i \(0.666654\pi\)
\(930\) 19352.5 0.682360
\(931\) 0 0
\(932\) 6573.75 0.231041
\(933\) −30453.1 −1.06858
\(934\) −9235.22 −0.323539
\(935\) 6666.33 0.233168
\(936\) −2269.24 −0.0792440
\(937\) 3815.98 0.133044 0.0665222 0.997785i \(-0.478810\pi\)
0.0665222 + 0.997785i \(0.478810\pi\)
\(938\) 0 0
\(939\) −8079.49 −0.280793
\(940\) −55574.0 −1.92832
\(941\) 48453.2 1.67856 0.839282 0.543696i \(-0.182976\pi\)
0.839282 + 0.543696i \(0.182976\pi\)
\(942\) 6697.49 0.231652
\(943\) 15950.5 0.550815
\(944\) 42765.4 1.47447
\(945\) 0 0
\(946\) −11469.4 −0.394188
\(947\) −15831.8 −0.543256 −0.271628 0.962402i \(-0.587562\pi\)
−0.271628 + 0.962402i \(0.587562\pi\)
\(948\) −324.729 −0.0111252
\(949\) −5984.64 −0.204710
\(950\) 2362.64 0.0806886
\(951\) −2417.66 −0.0824375
\(952\) 0 0
\(953\) 26284.5 0.893430 0.446715 0.894676i \(-0.352594\pi\)
0.446715 + 0.894676i \(0.352594\pi\)
\(954\) 6350.85 0.215531
\(955\) 19593.5 0.663906
\(956\) 27553.0 0.932143
\(957\) −6266.42 −0.211666
\(958\) −19496.7 −0.657526
\(959\) 0 0
\(960\) 21078.8 0.708661
\(961\) −9521.27 −0.319602
\(962\) −55599.7 −1.86342
\(963\) 9430.08 0.315556
\(964\) 32780.9 1.09523
\(965\) −54217.5 −1.80863
\(966\) 0 0
\(967\) −36542.0 −1.21521 −0.607607 0.794238i \(-0.707870\pi\)
−0.607607 + 0.794238i \(0.707870\pi\)
\(968\) 547.255 0.0181709
\(969\) 19348.6 0.641451
\(970\) 18387.4 0.608641
\(971\) 24753.7 0.818109 0.409055 0.912510i \(-0.365859\pi\)
0.409055 + 0.912510i \(0.365859\pi\)
\(972\) −2209.82 −0.0729219
\(973\) 0 0
\(974\) −13689.6 −0.450352
\(975\) −819.457 −0.0269165
\(976\) −13925.7 −0.456713
\(977\) −47421.6 −1.55287 −0.776434 0.630198i \(-0.782974\pi\)
−0.776434 + 0.630198i \(0.782974\pi\)
\(978\) 24843.8 0.812288
\(979\) −14936.2 −0.487602
\(980\) 0 0
\(981\) −4672.32 −0.152065
\(982\) 78180.1 2.54056
\(983\) 9479.42 0.307575 0.153788 0.988104i \(-0.450853\pi\)
0.153788 + 0.988104i \(0.450853\pi\)
\(984\) 2923.25 0.0947052
\(985\) 30606.2 0.990047
\(986\) 43415.9 1.40228
\(987\) 0 0
\(988\) 59127.6 1.90395
\(989\) 18670.6 0.600295
\(990\) 4485.68 0.144004
\(991\) 2290.12 0.0734086 0.0367043 0.999326i \(-0.488314\pi\)
0.0367043 + 0.999326i \(0.488314\pi\)
\(992\) 36968.1 1.18321
\(993\) −1902.96 −0.0608143
\(994\) 0 0
\(995\) 27531.5 0.877193
\(996\) −13892.2 −0.441960
\(997\) 14477.0 0.459871 0.229935 0.973206i \(-0.426148\pi\)
0.229935 + 0.973206i \(0.426148\pi\)
\(998\) 79182.4 2.51150
\(999\) −6513.01 −0.206269
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.1.13 16
7.6 odd 2 1617.4.a.bf.1.13 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1617.4.a.be.1.13 16 1.1 even 1 trivial
1617.4.a.bf.1.13 yes 16 7.6 odd 2