Properties

Label 1950.4.a.c.1.1
Level $1950$
Weight $4$
Character 1950.1
Self dual yes
Analytic conductor $115.054$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1950,4,Mod(1,1950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1950, 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("1950.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1950.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: \(115.053724511\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1950.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -6.00000 q^{6} -20.0000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -6.00000 q^{6} -20.0000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +24.0000 q^{11} +12.0000 q^{12} -13.0000 q^{13} +40.0000 q^{14} +16.0000 q^{16} +30.0000 q^{17} -18.0000 q^{18} -16.0000 q^{19} -60.0000 q^{21} -48.0000 q^{22} +72.0000 q^{23} -24.0000 q^{24} +26.0000 q^{26} +27.0000 q^{27} -80.0000 q^{28} -282.000 q^{29} +164.000 q^{31} -32.0000 q^{32} +72.0000 q^{33} -60.0000 q^{34} +36.0000 q^{36} -110.000 q^{37} +32.0000 q^{38} -39.0000 q^{39} -126.000 q^{41} +120.000 q^{42} -164.000 q^{43} +96.0000 q^{44} -144.000 q^{46} +204.000 q^{47} +48.0000 q^{48} +57.0000 q^{49} +90.0000 q^{51} -52.0000 q^{52} +738.000 q^{53} -54.0000 q^{54} +160.000 q^{56} -48.0000 q^{57} +564.000 q^{58} +120.000 q^{59} +614.000 q^{61} -328.000 q^{62} -180.000 q^{63} +64.0000 q^{64} -144.000 q^{66} -848.000 q^{67} +120.000 q^{68} +216.000 q^{69} +132.000 q^{71} -72.0000 q^{72} -218.000 q^{73} +220.000 q^{74} -64.0000 q^{76} -480.000 q^{77} +78.0000 q^{78} -1096.00 q^{79} +81.0000 q^{81} +252.000 q^{82} -552.000 q^{83} -240.000 q^{84} +328.000 q^{86} -846.000 q^{87} -192.000 q^{88} +210.000 q^{89} +260.000 q^{91} +288.000 q^{92} +492.000 q^{93} -408.000 q^{94} -96.0000 q^{96} +1726.00 q^{97} -114.000 q^{98} +216.000 q^{99} +O(q^{100})\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) 3.00000 0.577350
\(4\) 4.00000 0.500000
\(5\) 0 0
\(6\) −6.00000 −0.408248
\(7\) −20.0000 −1.07990 −0.539949 0.841698i \(-0.681557\pi\)
−0.539949 + 0.841698i \(0.681557\pi\)
\(8\) −8.00000 −0.353553
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 24.0000 0.657843 0.328921 0.944357i \(-0.393315\pi\)
0.328921 + 0.944357i \(0.393315\pi\)
\(12\) 12.0000 0.288675
\(13\) −13.0000 −0.277350
\(14\) 40.0000 0.763604
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 30.0000 0.428004 0.214002 0.976833i \(-0.431350\pi\)
0.214002 + 0.976833i \(0.431350\pi\)
\(18\) −18.0000 −0.235702
\(19\) −16.0000 −0.193192 −0.0965961 0.995324i \(-0.530796\pi\)
−0.0965961 + 0.995324i \(0.530796\pi\)
\(20\) 0 0
\(21\) −60.0000 −0.623480
\(22\) −48.0000 −0.465165
\(23\) 72.0000 0.652741 0.326370 0.945242i \(-0.394174\pi\)
0.326370 + 0.945242i \(0.394174\pi\)
\(24\) −24.0000 −0.204124
\(25\) 0 0
\(26\) 26.0000 0.196116
\(27\) 27.0000 0.192450
\(28\) −80.0000 −0.539949
\(29\) −282.000 −1.80573 −0.902864 0.429927i \(-0.858539\pi\)
−0.902864 + 0.429927i \(0.858539\pi\)
\(30\) 0 0
\(31\) 164.000 0.950170 0.475085 0.879940i \(-0.342417\pi\)
0.475085 + 0.879940i \(0.342417\pi\)
\(32\) −32.0000 −0.176777
\(33\) 72.0000 0.379806
\(34\) −60.0000 −0.302645
\(35\) 0 0
\(36\) 36.0000 0.166667
\(37\) −110.000 −0.488754 −0.244377 0.969680i \(-0.578583\pi\)
−0.244377 + 0.969680i \(0.578583\pi\)
\(38\) 32.0000 0.136608
\(39\) −39.0000 −0.160128
\(40\) 0 0
\(41\) −126.000 −0.479949 −0.239974 0.970779i \(-0.577139\pi\)
−0.239974 + 0.970779i \(0.577139\pi\)
\(42\) 120.000 0.440867
\(43\) −164.000 −0.581622 −0.290811 0.956780i \(-0.593925\pi\)
−0.290811 + 0.956780i \(0.593925\pi\)
\(44\) 96.0000 0.328921
\(45\) 0 0
\(46\) −144.000 −0.461557
\(47\) 204.000 0.633116 0.316558 0.948573i \(-0.397473\pi\)
0.316558 + 0.948573i \(0.397473\pi\)
\(48\) 48.0000 0.144338
\(49\) 57.0000 0.166181
\(50\) 0 0
\(51\) 90.0000 0.247108
\(52\) −52.0000 −0.138675
\(53\) 738.000 1.91268 0.956341 0.292255i \(-0.0944055\pi\)
0.956341 + 0.292255i \(0.0944055\pi\)
\(54\) −54.0000 −0.136083
\(55\) 0 0
\(56\) 160.000 0.381802
\(57\) −48.0000 −0.111540
\(58\) 564.000 1.27684
\(59\) 120.000 0.264791 0.132396 0.991197i \(-0.457733\pi\)
0.132396 + 0.991197i \(0.457733\pi\)
\(60\) 0 0
\(61\) 614.000 1.28876 0.644382 0.764703i \(-0.277115\pi\)
0.644382 + 0.764703i \(0.277115\pi\)
\(62\) −328.000 −0.671872
\(63\) −180.000 −0.359966
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) −144.000 −0.268563
\(67\) −848.000 −1.54626 −0.773132 0.634245i \(-0.781311\pi\)
−0.773132 + 0.634245i \(0.781311\pi\)
\(68\) 120.000 0.214002
\(69\) 216.000 0.376860
\(70\) 0 0
\(71\) 132.000 0.220641 0.110321 0.993896i \(-0.464812\pi\)
0.110321 + 0.993896i \(0.464812\pi\)
\(72\) −72.0000 −0.117851
\(73\) −218.000 −0.349520 −0.174760 0.984611i \(-0.555915\pi\)
−0.174760 + 0.984611i \(0.555915\pi\)
\(74\) 220.000 0.345601
\(75\) 0 0
\(76\) −64.0000 −0.0965961
\(77\) −480.000 −0.710404
\(78\) 78.0000 0.113228
\(79\) −1096.00 −1.56088 −0.780441 0.625230i \(-0.785005\pi\)
−0.780441 + 0.625230i \(0.785005\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 252.000 0.339375
\(83\) −552.000 −0.729998 −0.364999 0.931008i \(-0.618931\pi\)
−0.364999 + 0.931008i \(0.618931\pi\)
\(84\) −240.000 −0.311740
\(85\) 0 0
\(86\) 328.000 0.411269
\(87\) −846.000 −1.04254
\(88\) −192.000 −0.232583
\(89\) 210.000 0.250112 0.125056 0.992150i \(-0.460089\pi\)
0.125056 + 0.992150i \(0.460089\pi\)
\(90\) 0 0
\(91\) 260.000 0.299510
\(92\) 288.000 0.326370
\(93\) 492.000 0.548581
\(94\) −408.000 −0.447681
\(95\) 0 0
\(96\) −96.0000 −0.102062
\(97\) 1726.00 1.80669 0.903344 0.428917i \(-0.141105\pi\)
0.903344 + 0.428917i \(0.141105\pi\)
\(98\) −114.000 −0.117508
\(99\) 216.000 0.219281
\(100\) 0 0
\(101\) 798.000 0.786178 0.393089 0.919500i \(-0.371406\pi\)
0.393089 + 0.919500i \(0.371406\pi\)
\(102\) −180.000 −0.174732
\(103\) 520.000 0.497448 0.248724 0.968574i \(-0.419989\pi\)
0.248724 + 0.968574i \(0.419989\pi\)
\(104\) 104.000 0.0980581
\(105\) 0 0
\(106\) −1476.00 −1.35247
\(107\) −12.0000 −0.0108419 −0.00542095 0.999985i \(-0.501726\pi\)
−0.00542095 + 0.999985i \(0.501726\pi\)
\(108\) 108.000 0.0962250
\(109\) −1834.00 −1.61161 −0.805804 0.592182i \(-0.798267\pi\)
−0.805804 + 0.592182i \(0.798267\pi\)
\(110\) 0 0
\(111\) −330.000 −0.282182
\(112\) −320.000 −0.269975
\(113\) 366.000 0.304694 0.152347 0.988327i \(-0.451317\pi\)
0.152347 + 0.988327i \(0.451317\pi\)
\(114\) 96.0000 0.0788704
\(115\) 0 0
\(116\) −1128.00 −0.902864
\(117\) −117.000 −0.0924500
\(118\) −240.000 −0.187236
\(119\) −600.000 −0.462201
\(120\) 0 0
\(121\) −755.000 −0.567243
\(122\) −1228.00 −0.911294
\(123\) −378.000 −0.277098
\(124\) 656.000 0.475085
\(125\) 0 0
\(126\) 360.000 0.254535
\(127\) −2144.00 −1.49803 −0.749013 0.662556i \(-0.769472\pi\)
−0.749013 + 0.662556i \(0.769472\pi\)
\(128\) −128.000 −0.0883883
\(129\) −492.000 −0.335800
\(130\) 0 0
\(131\) −2748.00 −1.83278 −0.916389 0.400289i \(-0.868910\pi\)
−0.916389 + 0.400289i \(0.868910\pi\)
\(132\) 288.000 0.189903
\(133\) 320.000 0.208628
\(134\) 1696.00 1.09337
\(135\) 0 0
\(136\) −240.000 −0.151322
\(137\) −2754.00 −1.71745 −0.858723 0.512440i \(-0.828742\pi\)
−0.858723 + 0.512440i \(0.828742\pi\)
\(138\) −432.000 −0.266480
\(139\) 2252.00 1.37419 0.687094 0.726568i \(-0.258886\pi\)
0.687094 + 0.726568i \(0.258886\pi\)
\(140\) 0 0
\(141\) 612.000 0.365530
\(142\) −264.000 −0.156017
\(143\) −312.000 −0.182453
\(144\) 144.000 0.0833333
\(145\) 0 0
\(146\) 436.000 0.247148
\(147\) 171.000 0.0959445
\(148\) −440.000 −0.244377
\(149\) −1770.00 −0.973182 −0.486591 0.873630i \(-0.661760\pi\)
−0.486591 + 0.873630i \(0.661760\pi\)
\(150\) 0 0
\(151\) −988.000 −0.532466 −0.266233 0.963909i \(-0.585779\pi\)
−0.266233 + 0.963909i \(0.585779\pi\)
\(152\) 128.000 0.0683038
\(153\) 270.000 0.142668
\(154\) 960.000 0.502331
\(155\) 0 0
\(156\) −156.000 −0.0800641
\(157\) −326.000 −0.165717 −0.0828587 0.996561i \(-0.526405\pi\)
−0.0828587 + 0.996561i \(0.526405\pi\)
\(158\) 2192.00 1.10371
\(159\) 2214.00 1.10429
\(160\) 0 0
\(161\) −1440.00 −0.704894
\(162\) −162.000 −0.0785674
\(163\) −1496.00 −0.718870 −0.359435 0.933170i \(-0.617031\pi\)
−0.359435 + 0.933170i \(0.617031\pi\)
\(164\) −504.000 −0.239974
\(165\) 0 0
\(166\) 1104.00 0.516187
\(167\) −1116.00 −0.517118 −0.258559 0.965995i \(-0.583248\pi\)
−0.258559 + 0.965995i \(0.583248\pi\)
\(168\) 480.000 0.220433
\(169\) 169.000 0.0769231
\(170\) 0 0
\(171\) −144.000 −0.0643974
\(172\) −656.000 −0.290811
\(173\) −4374.00 −1.92225 −0.961124 0.276116i \(-0.910953\pi\)
−0.961124 + 0.276116i \(0.910953\pi\)
\(174\) 1692.00 0.737185
\(175\) 0 0
\(176\) 384.000 0.164461
\(177\) 360.000 0.152877
\(178\) −420.000 −0.176856
\(179\) 12.0000 0.00501074 0.00250537 0.999997i \(-0.499203\pi\)
0.00250537 + 0.999997i \(0.499203\pi\)
\(180\) 0 0
\(181\) 4718.00 1.93749 0.968746 0.248053i \(-0.0797909\pi\)
0.968746 + 0.248053i \(0.0797909\pi\)
\(182\) −520.000 −0.211786
\(183\) 1842.00 0.744069
\(184\) −576.000 −0.230779
\(185\) 0 0
\(186\) −984.000 −0.387905
\(187\) 720.000 0.281559
\(188\) 816.000 0.316558
\(189\) −540.000 −0.207827
\(190\) 0 0
\(191\) −1368.00 −0.518246 −0.259123 0.965844i \(-0.583434\pi\)
−0.259123 + 0.965844i \(0.583434\pi\)
\(192\) 192.000 0.0721688
\(193\) 3310.00 1.23450 0.617251 0.786766i \(-0.288246\pi\)
0.617251 + 0.786766i \(0.288246\pi\)
\(194\) −3452.00 −1.27752
\(195\) 0 0
\(196\) 228.000 0.0830904
\(197\) −3126.00 −1.13055 −0.565275 0.824903i \(-0.691230\pi\)
−0.565275 + 0.824903i \(0.691230\pi\)
\(198\) −432.000 −0.155055
\(199\) 4664.00 1.66142 0.830709 0.556707i \(-0.187935\pi\)
0.830709 + 0.556707i \(0.187935\pi\)
\(200\) 0 0
\(201\) −2544.00 −0.892736
\(202\) −1596.00 −0.555912
\(203\) 5640.00 1.95000
\(204\) 360.000 0.123554
\(205\) 0 0
\(206\) −1040.00 −0.351749
\(207\) 648.000 0.217580
\(208\) −208.000 −0.0693375
\(209\) −384.000 −0.127090
\(210\) 0 0
\(211\) −556.000 −0.181406 −0.0907029 0.995878i \(-0.528911\pi\)
−0.0907029 + 0.995878i \(0.528911\pi\)
\(212\) 2952.00 0.956341
\(213\) 396.000 0.127387
\(214\) 24.0000 0.00766638
\(215\) 0 0
\(216\) −216.000 −0.0680414
\(217\) −3280.00 −1.02609
\(218\) 3668.00 1.13958
\(219\) −654.000 −0.201796
\(220\) 0 0
\(221\) −390.000 −0.118707
\(222\) 660.000 0.199533
\(223\) 268.000 0.0804781 0.0402390 0.999190i \(-0.487188\pi\)
0.0402390 + 0.999190i \(0.487188\pi\)
\(224\) 640.000 0.190901
\(225\) 0 0
\(226\) −732.000 −0.215451
\(227\) −1800.00 −0.526300 −0.263150 0.964755i \(-0.584761\pi\)
−0.263150 + 0.964755i \(0.584761\pi\)
\(228\) −192.000 −0.0557698
\(229\) 2990.00 0.862816 0.431408 0.902157i \(-0.358017\pi\)
0.431408 + 0.902157i \(0.358017\pi\)
\(230\) 0 0
\(231\) −1440.00 −0.410152
\(232\) 2256.00 0.638421
\(233\) −2826.00 −0.794581 −0.397291 0.917693i \(-0.630049\pi\)
−0.397291 + 0.917693i \(0.630049\pi\)
\(234\) 234.000 0.0653720
\(235\) 0 0
\(236\) 480.000 0.132396
\(237\) −3288.00 −0.901175
\(238\) 1200.00 0.326825
\(239\) −1812.00 −0.490412 −0.245206 0.969471i \(-0.578856\pi\)
−0.245206 + 0.969471i \(0.578856\pi\)
\(240\) 0 0
\(241\) −1582.00 −0.422845 −0.211422 0.977395i \(-0.567810\pi\)
−0.211422 + 0.977395i \(0.567810\pi\)
\(242\) 1510.00 0.401101
\(243\) 243.000 0.0641500
\(244\) 2456.00 0.644382
\(245\) 0 0
\(246\) 756.000 0.195938
\(247\) 208.000 0.0535819
\(248\) −1312.00 −0.335936
\(249\) −1656.00 −0.421465
\(250\) 0 0
\(251\) 2148.00 0.540162 0.270081 0.962838i \(-0.412950\pi\)
0.270081 + 0.962838i \(0.412950\pi\)
\(252\) −720.000 −0.179983
\(253\) 1728.00 0.429401
\(254\) 4288.00 1.05926
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) 558.000 0.135436 0.0677181 0.997704i \(-0.478428\pi\)
0.0677181 + 0.997704i \(0.478428\pi\)
\(258\) 984.000 0.237446
\(259\) 2200.00 0.527804
\(260\) 0 0
\(261\) −2538.00 −0.601909
\(262\) 5496.00 1.29597
\(263\) −2112.00 −0.495177 −0.247588 0.968865i \(-0.579638\pi\)
−0.247588 + 0.968865i \(0.579638\pi\)
\(264\) −576.000 −0.134282
\(265\) 0 0
\(266\) −640.000 −0.147522
\(267\) 630.000 0.144402
\(268\) −3392.00 −0.773132
\(269\) 5046.00 1.14372 0.571859 0.820352i \(-0.306223\pi\)
0.571859 + 0.820352i \(0.306223\pi\)
\(270\) 0 0
\(271\) −3796.00 −0.850888 −0.425444 0.904985i \(-0.639882\pi\)
−0.425444 + 0.904985i \(0.639882\pi\)
\(272\) 480.000 0.107001
\(273\) 780.000 0.172922
\(274\) 5508.00 1.21442
\(275\) 0 0
\(276\) 864.000 0.188430
\(277\) −5582.00 −1.21079 −0.605397 0.795924i \(-0.706986\pi\)
−0.605397 + 0.795924i \(0.706986\pi\)
\(278\) −4504.00 −0.971698
\(279\) 1476.00 0.316723
\(280\) 0 0
\(281\) −1950.00 −0.413976 −0.206988 0.978343i \(-0.566366\pi\)
−0.206988 + 0.978343i \(0.566366\pi\)
\(282\) −1224.00 −0.258469
\(283\) 4732.00 0.993951 0.496976 0.867765i \(-0.334444\pi\)
0.496976 + 0.867765i \(0.334444\pi\)
\(284\) 528.000 0.110321
\(285\) 0 0
\(286\) 624.000 0.129014
\(287\) 2520.00 0.518296
\(288\) −288.000 −0.0589256
\(289\) −4013.00 −0.816813
\(290\) 0 0
\(291\) 5178.00 1.04309
\(292\) −872.000 −0.174760
\(293\) −4998.00 −0.996540 −0.498270 0.867022i \(-0.666031\pi\)
−0.498270 + 0.867022i \(0.666031\pi\)
\(294\) −342.000 −0.0678430
\(295\) 0 0
\(296\) 880.000 0.172801
\(297\) 648.000 0.126602
\(298\) 3540.00 0.688143
\(299\) −936.000 −0.181038
\(300\) 0 0
\(301\) 3280.00 0.628093
\(302\) 1976.00 0.376510
\(303\) 2394.00 0.453900
\(304\) −256.000 −0.0482980
\(305\) 0 0
\(306\) −540.000 −0.100882
\(307\) −6824.00 −1.26862 −0.634310 0.773079i \(-0.718716\pi\)
−0.634310 + 0.773079i \(0.718716\pi\)
\(308\) −1920.00 −0.355202
\(309\) 1560.00 0.287202
\(310\) 0 0
\(311\) −8760.00 −1.59722 −0.798608 0.601852i \(-0.794430\pi\)
−0.798608 + 0.601852i \(0.794430\pi\)
\(312\) 312.000 0.0566139
\(313\) −3962.00 −0.715481 −0.357740 0.933821i \(-0.616453\pi\)
−0.357740 + 0.933821i \(0.616453\pi\)
\(314\) 652.000 0.117180
\(315\) 0 0
\(316\) −4384.00 −0.780441
\(317\) −7086.00 −1.25549 −0.627744 0.778420i \(-0.716021\pi\)
−0.627744 + 0.778420i \(0.716021\pi\)
\(318\) −4428.00 −0.780849
\(319\) −6768.00 −1.18788
\(320\) 0 0
\(321\) −36.0000 −0.00625958
\(322\) 2880.00 0.498435
\(323\) −480.000 −0.0826870
\(324\) 324.000 0.0555556
\(325\) 0 0
\(326\) 2992.00 0.508318
\(327\) −5502.00 −0.930463
\(328\) 1008.00 0.169687
\(329\) −4080.00 −0.683701
\(330\) 0 0
\(331\) −9016.00 −1.49717 −0.748586 0.663037i \(-0.769267\pi\)
−0.748586 + 0.663037i \(0.769267\pi\)
\(332\) −2208.00 −0.364999
\(333\) −990.000 −0.162918
\(334\) 2232.00 0.365658
\(335\) 0 0
\(336\) −960.000 −0.155870
\(337\) −2306.00 −0.372747 −0.186374 0.982479i \(-0.559673\pi\)
−0.186374 + 0.982479i \(0.559673\pi\)
\(338\) −338.000 −0.0543928
\(339\) 1098.00 0.175915
\(340\) 0 0
\(341\) 3936.00 0.625063
\(342\) 288.000 0.0455358
\(343\) 5720.00 0.900440
\(344\) 1312.00 0.205635
\(345\) 0 0
\(346\) 8748.00 1.35924
\(347\) 11076.0 1.71352 0.856759 0.515717i \(-0.172474\pi\)
0.856759 + 0.515717i \(0.172474\pi\)
\(348\) −3384.00 −0.521269
\(349\) 2342.00 0.359210 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(350\) 0 0
\(351\) −351.000 −0.0533761
\(352\) −768.000 −0.116291
\(353\) −4650.00 −0.701118 −0.350559 0.936541i \(-0.614008\pi\)
−0.350559 + 0.936541i \(0.614008\pi\)
\(354\) −720.000 −0.108100
\(355\) 0 0
\(356\) 840.000 0.125056
\(357\) −1800.00 −0.266852
\(358\) −24.0000 −0.00354313
\(359\) −11268.0 −1.65655 −0.828276 0.560320i \(-0.810678\pi\)
−0.828276 + 0.560320i \(0.810678\pi\)
\(360\) 0 0
\(361\) −6603.00 −0.962677
\(362\) −9436.00 −1.37001
\(363\) −2265.00 −0.327498
\(364\) 1040.00 0.149755
\(365\) 0 0
\(366\) −3684.00 −0.526136
\(367\) 7288.00 1.03660 0.518298 0.855200i \(-0.326566\pi\)
0.518298 + 0.855200i \(0.326566\pi\)
\(368\) 1152.00 0.163185
\(369\) −1134.00 −0.159983
\(370\) 0 0
\(371\) −14760.0 −2.06550
\(372\) 1968.00 0.274290
\(373\) 9970.00 1.38399 0.691993 0.721904i \(-0.256733\pi\)
0.691993 + 0.721904i \(0.256733\pi\)
\(374\) −1440.00 −0.199093
\(375\) 0 0
\(376\) −1632.00 −0.223840
\(377\) 3666.00 0.500819
\(378\) 1080.00 0.146956
\(379\) 13448.0 1.82263 0.911316 0.411708i \(-0.135068\pi\)
0.911316 + 0.411708i \(0.135068\pi\)
\(380\) 0 0
\(381\) −6432.00 −0.864885
\(382\) 2736.00 0.366455
\(383\) −11820.0 −1.57696 −0.788478 0.615064i \(-0.789130\pi\)
−0.788478 + 0.615064i \(0.789130\pi\)
\(384\) −384.000 −0.0510310
\(385\) 0 0
\(386\) −6620.00 −0.872925
\(387\) −1476.00 −0.193874
\(388\) 6904.00 0.903344
\(389\) 174.000 0.0226790 0.0113395 0.999936i \(-0.496390\pi\)
0.0113395 + 0.999936i \(0.496390\pi\)
\(390\) 0 0
\(391\) 2160.00 0.279376
\(392\) −456.000 −0.0587538
\(393\) −8244.00 −1.05815
\(394\) 6252.00 0.799419
\(395\) 0 0
\(396\) 864.000 0.109640
\(397\) 2986.00 0.377489 0.188744 0.982026i \(-0.439558\pi\)
0.188744 + 0.982026i \(0.439558\pi\)
\(398\) −9328.00 −1.17480
\(399\) 960.000 0.120451
\(400\) 0 0
\(401\) −10566.0 −1.31581 −0.657906 0.753100i \(-0.728558\pi\)
−0.657906 + 0.753100i \(0.728558\pi\)
\(402\) 5088.00 0.631260
\(403\) −2132.00 −0.263530
\(404\) 3192.00 0.393089
\(405\) 0 0
\(406\) −11280.0 −1.37886
\(407\) −2640.00 −0.321523
\(408\) −720.000 −0.0873660
\(409\) −7270.00 −0.878920 −0.439460 0.898262i \(-0.644830\pi\)
−0.439460 + 0.898262i \(0.644830\pi\)
\(410\) 0 0
\(411\) −8262.00 −0.991568
\(412\) 2080.00 0.248724
\(413\) −2400.00 −0.285947
\(414\) −1296.00 −0.153852
\(415\) 0 0
\(416\) 416.000 0.0490290
\(417\) 6756.00 0.793388
\(418\) 768.000 0.0898663
\(419\) −7308.00 −0.852074 −0.426037 0.904706i \(-0.640091\pi\)
−0.426037 + 0.904706i \(0.640091\pi\)
\(420\) 0 0
\(421\) −5938.00 −0.687412 −0.343706 0.939077i \(-0.611682\pi\)
−0.343706 + 0.939077i \(0.611682\pi\)
\(422\) 1112.00 0.128273
\(423\) 1836.00 0.211039
\(424\) −5904.00 −0.676235
\(425\) 0 0
\(426\) −792.000 −0.0900764
\(427\) −12280.0 −1.39174
\(428\) −48.0000 −0.00542095
\(429\) −936.000 −0.105339
\(430\) 0 0
\(431\) 11532.0 1.28881 0.644405 0.764685i \(-0.277105\pi\)
0.644405 + 0.764685i \(0.277105\pi\)
\(432\) 432.000 0.0481125
\(433\) 718.000 0.0796879 0.0398440 0.999206i \(-0.487314\pi\)
0.0398440 + 0.999206i \(0.487314\pi\)
\(434\) 6560.00 0.725553
\(435\) 0 0
\(436\) −7336.00 −0.805804
\(437\) −1152.00 −0.126104
\(438\) 1308.00 0.142691
\(439\) 8984.00 0.976726 0.488363 0.872640i \(-0.337594\pi\)
0.488363 + 0.872640i \(0.337594\pi\)
\(440\) 0 0
\(441\) 513.000 0.0553936
\(442\) 780.000 0.0839385
\(443\) −2604.00 −0.279277 −0.139639 0.990203i \(-0.544594\pi\)
−0.139639 + 0.990203i \(0.544594\pi\)
\(444\) −1320.00 −0.141091
\(445\) 0 0
\(446\) −536.000 −0.0569066
\(447\) −5310.00 −0.561867
\(448\) −1280.00 −0.134987
\(449\) −13206.0 −1.38804 −0.694020 0.719956i \(-0.744162\pi\)
−0.694020 + 0.719956i \(0.744162\pi\)
\(450\) 0 0
\(451\) −3024.00 −0.315731
\(452\) 1464.00 0.152347
\(453\) −2964.00 −0.307419
\(454\) 3600.00 0.372151
\(455\) 0 0
\(456\) 384.000 0.0394352
\(457\) −8426.00 −0.862476 −0.431238 0.902238i \(-0.641923\pi\)
−0.431238 + 0.902238i \(0.641923\pi\)
\(458\) −5980.00 −0.610103
\(459\) 810.000 0.0823694
\(460\) 0 0
\(461\) 16686.0 1.68578 0.842890 0.538086i \(-0.180852\pi\)
0.842890 + 0.538086i \(0.180852\pi\)
\(462\) 2880.00 0.290021
\(463\) −15932.0 −1.59919 −0.799593 0.600543i \(-0.794951\pi\)
−0.799593 + 0.600543i \(0.794951\pi\)
\(464\) −4512.00 −0.451432
\(465\) 0 0
\(466\) 5652.00 0.561854
\(467\) −18540.0 −1.83711 −0.918553 0.395297i \(-0.870642\pi\)
−0.918553 + 0.395297i \(0.870642\pi\)
\(468\) −468.000 −0.0462250
\(469\) 16960.0 1.66981
\(470\) 0 0
\(471\) −978.000 −0.0956770
\(472\) −960.000 −0.0936178
\(473\) −3936.00 −0.382616
\(474\) 6576.00 0.637227
\(475\) 0 0
\(476\) −2400.00 −0.231100
\(477\) 6642.00 0.637560
\(478\) 3624.00 0.346774
\(479\) 6180.00 0.589502 0.294751 0.955574i \(-0.404763\pi\)
0.294751 + 0.955574i \(0.404763\pi\)
\(480\) 0 0
\(481\) 1430.00 0.135556
\(482\) 3164.00 0.298996
\(483\) −4320.00 −0.406971
\(484\) −3020.00 −0.283621
\(485\) 0 0
\(486\) −486.000 −0.0453609
\(487\) −11756.0 −1.09387 −0.546936 0.837175i \(-0.684206\pi\)
−0.546936 + 0.837175i \(0.684206\pi\)
\(488\) −4912.00 −0.455647
\(489\) −4488.00 −0.415040
\(490\) 0 0
\(491\) 1908.00 0.175370 0.0876852 0.996148i \(-0.472053\pi\)
0.0876852 + 0.996148i \(0.472053\pi\)
\(492\) −1512.00 −0.138549
\(493\) −8460.00 −0.772858
\(494\) −416.000 −0.0378881
\(495\) 0 0
\(496\) 2624.00 0.237542
\(497\) −2640.00 −0.238270
\(498\) 3312.00 0.298021
\(499\) −8944.00 −0.802382 −0.401191 0.915995i \(-0.631404\pi\)
−0.401191 + 0.915995i \(0.631404\pi\)
\(500\) 0 0
\(501\) −3348.00 −0.298558
\(502\) −4296.00 −0.381952
\(503\) 6528.00 0.578666 0.289333 0.957228i \(-0.406566\pi\)
0.289333 + 0.957228i \(0.406566\pi\)
\(504\) 1440.00 0.127267
\(505\) 0 0
\(506\) −3456.00 −0.303632
\(507\) 507.000 0.0444116
\(508\) −8576.00 −0.749013
\(509\) −12114.0 −1.05490 −0.527450 0.849586i \(-0.676852\pi\)
−0.527450 + 0.849586i \(0.676852\pi\)
\(510\) 0 0
\(511\) 4360.00 0.377446
\(512\) −512.000 −0.0441942
\(513\) −432.000 −0.0371799
\(514\) −1116.00 −0.0957678
\(515\) 0 0
\(516\) −1968.00 −0.167900
\(517\) 4896.00 0.416491
\(518\) −4400.00 −0.373214
\(519\) −13122.0 −1.10981
\(520\) 0 0
\(521\) −14310.0 −1.20333 −0.601663 0.798750i \(-0.705495\pi\)
−0.601663 + 0.798750i \(0.705495\pi\)
\(522\) 5076.00 0.425614
\(523\) 18340.0 1.53337 0.766685 0.642024i \(-0.221905\pi\)
0.766685 + 0.642024i \(0.221905\pi\)
\(524\) −10992.0 −0.916389
\(525\) 0 0
\(526\) 4224.00 0.350143
\(527\) 4920.00 0.406677
\(528\) 1152.00 0.0949514
\(529\) −6983.00 −0.573929
\(530\) 0 0
\(531\) 1080.00 0.0882637
\(532\) 1280.00 0.104314
\(533\) 1638.00 0.133114
\(534\) −1260.00 −0.102108
\(535\) 0 0
\(536\) 6784.00 0.546687
\(537\) 36.0000 0.00289295
\(538\) −10092.0 −0.808731
\(539\) 1368.00 0.109321
\(540\) 0 0
\(541\) 9254.00 0.735417 0.367708 0.929941i \(-0.380142\pi\)
0.367708 + 0.929941i \(0.380142\pi\)
\(542\) 7592.00 0.601668
\(543\) 14154.0 1.11861
\(544\) −960.000 −0.0756611
\(545\) 0 0
\(546\) −1560.00 −0.122274
\(547\) −17444.0 −1.36353 −0.681766 0.731571i \(-0.738788\pi\)
−0.681766 + 0.731571i \(0.738788\pi\)
\(548\) −11016.0 −0.858723
\(549\) 5526.00 0.429588
\(550\) 0 0
\(551\) 4512.00 0.348852
\(552\) −1728.00 −0.133240
\(553\) 21920.0 1.68559
\(554\) 11164.0 0.856160
\(555\) 0 0
\(556\) 9008.00 0.687094
\(557\) 3714.00 0.282526 0.141263 0.989972i \(-0.454884\pi\)
0.141263 + 0.989972i \(0.454884\pi\)
\(558\) −2952.00 −0.223957
\(559\) 2132.00 0.161313
\(560\) 0 0
\(561\) 2160.00 0.162558
\(562\) 3900.00 0.292725
\(563\) 13812.0 1.03394 0.516968 0.856004i \(-0.327060\pi\)
0.516968 + 0.856004i \(0.327060\pi\)
\(564\) 2448.00 0.182765
\(565\) 0 0
\(566\) −9464.00 −0.702830
\(567\) −1620.00 −0.119989
\(568\) −1056.00 −0.0780084
\(569\) −15942.0 −1.17456 −0.587279 0.809385i \(-0.699801\pi\)
−0.587279 + 0.809385i \(0.699801\pi\)
\(570\) 0 0
\(571\) 1604.00 0.117557 0.0587787 0.998271i \(-0.481279\pi\)
0.0587787 + 0.998271i \(0.481279\pi\)
\(572\) −1248.00 −0.0912264
\(573\) −4104.00 −0.299210
\(574\) −5040.00 −0.366490
\(575\) 0 0
\(576\) 576.000 0.0416667
\(577\) 10654.0 0.768686 0.384343 0.923190i \(-0.374428\pi\)
0.384343 + 0.923190i \(0.374428\pi\)
\(578\) 8026.00 0.577574
\(579\) 9930.00 0.712740
\(580\) 0 0
\(581\) 11040.0 0.788324
\(582\) −10356.0 −0.737577
\(583\) 17712.0 1.25824
\(584\) 1744.00 0.123574
\(585\) 0 0
\(586\) 9996.00 0.704660
\(587\) 9984.00 0.702017 0.351008 0.936372i \(-0.385839\pi\)
0.351008 + 0.936372i \(0.385839\pi\)
\(588\) 684.000 0.0479723
\(589\) −2624.00 −0.183565
\(590\) 0 0
\(591\) −9378.00 −0.652723
\(592\) −1760.00 −0.122188
\(593\) −12618.0 −0.873793 −0.436896 0.899512i \(-0.643922\pi\)
−0.436896 + 0.899512i \(0.643922\pi\)
\(594\) −1296.00 −0.0895211
\(595\) 0 0
\(596\) −7080.00 −0.486591
\(597\) 13992.0 0.959220
\(598\) 1872.00 0.128013
\(599\) 11184.0 0.762881 0.381441 0.924393i \(-0.375428\pi\)
0.381441 + 0.924393i \(0.375428\pi\)
\(600\) 0 0
\(601\) 2810.00 0.190719 0.0953596 0.995443i \(-0.469600\pi\)
0.0953596 + 0.995443i \(0.469600\pi\)
\(602\) −6560.00 −0.444129
\(603\) −7632.00 −0.515421
\(604\) −3952.00 −0.266233
\(605\) 0 0
\(606\) −4788.00 −0.320956
\(607\) −1064.00 −0.0711473 −0.0355737 0.999367i \(-0.511326\pi\)
−0.0355737 + 0.999367i \(0.511326\pi\)
\(608\) 512.000 0.0341519
\(609\) 16920.0 1.12583
\(610\) 0 0
\(611\) −2652.00 −0.175595
\(612\) 1080.00 0.0713340
\(613\) 20914.0 1.37799 0.688996 0.724766i \(-0.258052\pi\)
0.688996 + 0.724766i \(0.258052\pi\)
\(614\) 13648.0 0.897050
\(615\) 0 0
\(616\) 3840.00 0.251166
\(617\) −9714.00 −0.633826 −0.316913 0.948455i \(-0.602646\pi\)
−0.316913 + 0.948455i \(0.602646\pi\)
\(618\) −3120.00 −0.203082
\(619\) −14848.0 −0.964122 −0.482061 0.876138i \(-0.660112\pi\)
−0.482061 + 0.876138i \(0.660112\pi\)
\(620\) 0 0
\(621\) 1944.00 0.125620
\(622\) 17520.0 1.12940
\(623\) −4200.00 −0.270095
\(624\) −624.000 −0.0400320
\(625\) 0 0
\(626\) 7924.00 0.505921
\(627\) −1152.00 −0.0733755
\(628\) −1304.00 −0.0828587
\(629\) −3300.00 −0.209189
\(630\) 0 0
\(631\) 19172.0 1.20955 0.604774 0.796397i \(-0.293263\pi\)
0.604774 + 0.796397i \(0.293263\pi\)
\(632\) 8768.00 0.551855
\(633\) −1668.00 −0.104735
\(634\) 14172.0 0.887763
\(635\) 0 0
\(636\) 8856.00 0.552143
\(637\) −741.000 −0.0460902
\(638\) 13536.0 0.839961
\(639\) 1188.00 0.0735470
\(640\) 0 0
\(641\) −11502.0 −0.708739 −0.354369 0.935105i \(-0.615304\pi\)
−0.354369 + 0.935105i \(0.615304\pi\)
\(642\) 72.0000 0.00442619
\(643\) 15568.0 0.954809 0.477404 0.878684i \(-0.341578\pi\)
0.477404 + 0.878684i \(0.341578\pi\)
\(644\) −5760.00 −0.352447
\(645\) 0 0
\(646\) 960.000 0.0584686
\(647\) −1128.00 −0.0685414 −0.0342707 0.999413i \(-0.510911\pi\)
−0.0342707 + 0.999413i \(0.510911\pi\)
\(648\) −648.000 −0.0392837
\(649\) 2880.00 0.174191
\(650\) 0 0
\(651\) −9840.00 −0.592412
\(652\) −5984.00 −0.359435
\(653\) −8118.00 −0.486496 −0.243248 0.969964i \(-0.578213\pi\)
−0.243248 + 0.969964i \(0.578213\pi\)
\(654\) 11004.0 0.657936
\(655\) 0 0
\(656\) −2016.00 −0.119987
\(657\) −1962.00 −0.116507
\(658\) 8160.00 0.483450
\(659\) 13572.0 0.802261 0.401131 0.916021i \(-0.368617\pi\)
0.401131 + 0.916021i \(0.368617\pi\)
\(660\) 0 0
\(661\) −13138.0 −0.773085 −0.386542 0.922272i \(-0.626331\pi\)
−0.386542 + 0.922272i \(0.626331\pi\)
\(662\) 18032.0 1.05866
\(663\) −1170.00 −0.0685355
\(664\) 4416.00 0.258093
\(665\) 0 0
\(666\) 1980.00 0.115200
\(667\) −20304.0 −1.17867
\(668\) −4464.00 −0.258559
\(669\) 804.000 0.0464640
\(670\) 0 0
\(671\) 14736.0 0.847805
\(672\) 1920.00 0.110217
\(673\) 718.000 0.0411246 0.0205623 0.999789i \(-0.493454\pi\)
0.0205623 + 0.999789i \(0.493454\pi\)
\(674\) 4612.00 0.263572
\(675\) 0 0
\(676\) 676.000 0.0384615
\(677\) 2994.00 0.169969 0.0849843 0.996382i \(-0.472916\pi\)
0.0849843 + 0.996382i \(0.472916\pi\)
\(678\) −2196.00 −0.124391
\(679\) −34520.0 −1.95104
\(680\) 0 0
\(681\) −5400.00 −0.303860
\(682\) −7872.00 −0.441986
\(683\) −27384.0 −1.53414 −0.767071 0.641562i \(-0.778287\pi\)
−0.767071 + 0.641562i \(0.778287\pi\)
\(684\) −576.000 −0.0321987
\(685\) 0 0
\(686\) −11440.0 −0.636707
\(687\) 8970.00 0.498147
\(688\) −2624.00 −0.145406
\(689\) −9594.00 −0.530482
\(690\) 0 0
\(691\) 27632.0 1.52123 0.760616 0.649202i \(-0.224897\pi\)
0.760616 + 0.649202i \(0.224897\pi\)
\(692\) −17496.0 −0.961124
\(693\) −4320.00 −0.236801
\(694\) −22152.0 −1.21164
\(695\) 0 0
\(696\) 6768.00 0.368592
\(697\) −3780.00 −0.205420
\(698\) −4684.00 −0.254000
\(699\) −8478.00 −0.458752
\(700\) 0 0
\(701\) 19062.0 1.02705 0.513525 0.858075i \(-0.328339\pi\)
0.513525 + 0.858075i \(0.328339\pi\)
\(702\) 702.000 0.0377426
\(703\) 1760.00 0.0944234
\(704\) 1536.00 0.0822304
\(705\) 0 0
\(706\) 9300.00 0.495765
\(707\) −15960.0 −0.848992
\(708\) 1440.00 0.0764386
\(709\) 3854.00 0.204147 0.102073 0.994777i \(-0.467452\pi\)
0.102073 + 0.994777i \(0.467452\pi\)
\(710\) 0 0
\(711\) −9864.00 −0.520294
\(712\) −1680.00 −0.0884279
\(713\) 11808.0 0.620215
\(714\) 3600.00 0.188693
\(715\) 0 0
\(716\) 48.0000 0.00250537
\(717\) −5436.00 −0.283140
\(718\) 22536.0 1.17136
\(719\) 20976.0 1.08800 0.544001 0.839085i \(-0.316909\pi\)
0.544001 + 0.839085i \(0.316909\pi\)
\(720\) 0 0
\(721\) −10400.0 −0.537193
\(722\) 13206.0 0.680715
\(723\) −4746.00 −0.244130
\(724\) 18872.0 0.968746
\(725\) 0 0
\(726\) 4530.00 0.231576
\(727\) 29464.0 1.50311 0.751554 0.659672i \(-0.229305\pi\)
0.751554 + 0.659672i \(0.229305\pi\)
\(728\) −2080.00 −0.105893
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −4920.00 −0.248937
\(732\) 7368.00 0.372034
\(733\) 2698.00 0.135952 0.0679761 0.997687i \(-0.478346\pi\)
0.0679761 + 0.997687i \(0.478346\pi\)
\(734\) −14576.0 −0.732984
\(735\) 0 0
\(736\) −2304.00 −0.115389
\(737\) −20352.0 −1.01720
\(738\) 2268.00 0.113125
\(739\) 632.000 0.0314594 0.0157297 0.999876i \(-0.494993\pi\)
0.0157297 + 0.999876i \(0.494993\pi\)
\(740\) 0 0
\(741\) 624.000 0.0309355
\(742\) 29520.0 1.46053
\(743\) 20844.0 1.02920 0.514598 0.857432i \(-0.327941\pi\)
0.514598 + 0.857432i \(0.327941\pi\)
\(744\) −3936.00 −0.193953
\(745\) 0 0
\(746\) −19940.0 −0.978626
\(747\) −4968.00 −0.243333
\(748\) 2880.00 0.140780
\(749\) 240.000 0.0117082
\(750\) 0 0
\(751\) 272.000 0.0132163 0.00660814 0.999978i \(-0.497897\pi\)
0.00660814 + 0.999978i \(0.497897\pi\)
\(752\) 3264.00 0.158279
\(753\) 6444.00 0.311862
\(754\) −7332.00 −0.354132
\(755\) 0 0
\(756\) −2160.00 −0.103913
\(757\) −37550.0 −1.80288 −0.901439 0.432907i \(-0.857488\pi\)
−0.901439 + 0.432907i \(0.857488\pi\)
\(758\) −26896.0 −1.28880
\(759\) 5184.00 0.247915
\(760\) 0 0
\(761\) 33330.0 1.58766 0.793832 0.608138i \(-0.208083\pi\)
0.793832 + 0.608138i \(0.208083\pi\)
\(762\) 12864.0 0.611566
\(763\) 36680.0 1.74037
\(764\) −5472.00 −0.259123
\(765\) 0 0
\(766\) 23640.0 1.11508
\(767\) −1560.00 −0.0734398
\(768\) 768.000 0.0360844
\(769\) −15406.0 −0.722438 −0.361219 0.932481i \(-0.617639\pi\)
−0.361219 + 0.932481i \(0.617639\pi\)
\(770\) 0 0
\(771\) 1674.00 0.0781941
\(772\) 13240.0 0.617251
\(773\) 29514.0 1.37328 0.686640 0.726998i \(-0.259085\pi\)
0.686640 + 0.726998i \(0.259085\pi\)
\(774\) 2952.00 0.137090
\(775\) 0 0
\(776\) −13808.0 −0.638761
\(777\) 6600.00 0.304728
\(778\) −348.000 −0.0160365
\(779\) 2016.00 0.0927223
\(780\) 0 0
\(781\) 3168.00 0.145147
\(782\) −4320.00 −0.197548
\(783\) −7614.00 −0.347512
\(784\) 912.000 0.0415452
\(785\) 0 0
\(786\) 16488.0 0.748228
\(787\) −33176.0 −1.50266 −0.751332 0.659924i \(-0.770588\pi\)
−0.751332 + 0.659924i \(0.770588\pi\)
\(788\) −12504.0 −0.565275
\(789\) −6336.00 −0.285890
\(790\) 0 0
\(791\) −7320.00 −0.329038
\(792\) −1728.00 −0.0775275
\(793\) −7982.00 −0.357439
\(794\) −5972.00 −0.266925
\(795\) 0 0
\(796\) 18656.0 0.830709
\(797\) 16746.0 0.744258 0.372129 0.928181i \(-0.378628\pi\)
0.372129 + 0.928181i \(0.378628\pi\)
\(798\) −1920.00 −0.0851720
\(799\) 6120.00 0.270976
\(800\) 0 0
\(801\) 1890.00 0.0833706
\(802\) 21132.0 0.930420
\(803\) −5232.00 −0.229929
\(804\) −10176.0 −0.446368
\(805\) 0 0
\(806\) 4264.00 0.186344
\(807\) 15138.0 0.660326
\(808\) −6384.00 −0.277956
\(809\) −15846.0 −0.688647 −0.344324 0.938851i \(-0.611892\pi\)
−0.344324 + 0.938851i \(0.611892\pi\)
\(810\) 0 0
\(811\) 22952.0 0.993778 0.496889 0.867814i \(-0.334476\pi\)
0.496889 + 0.867814i \(0.334476\pi\)
\(812\) 22560.0 0.975001
\(813\) −11388.0 −0.491260
\(814\) 5280.00 0.227351
\(815\) 0 0
\(816\) 1440.00 0.0617771
\(817\) 2624.00 0.112365
\(818\) 14540.0 0.621490
\(819\) 2340.00 0.0998367
\(820\) 0 0
\(821\) −37146.0 −1.57906 −0.789528 0.613715i \(-0.789674\pi\)
−0.789528 + 0.613715i \(0.789674\pi\)
\(822\) 16524.0 0.701144
\(823\) 9592.00 0.406265 0.203133 0.979151i \(-0.434888\pi\)
0.203133 + 0.979151i \(0.434888\pi\)
\(824\) −4160.00 −0.175874
\(825\) 0 0
\(826\) 4800.00 0.202195
\(827\) 39960.0 1.68022 0.840112 0.542413i \(-0.182489\pi\)
0.840112 + 0.542413i \(0.182489\pi\)
\(828\) 2592.00 0.108790
\(829\) −3706.00 −0.155265 −0.0776325 0.996982i \(-0.524736\pi\)
−0.0776325 + 0.996982i \(0.524736\pi\)
\(830\) 0 0
\(831\) −16746.0 −0.699052
\(832\) −832.000 −0.0346688
\(833\) 1710.00 0.0711260
\(834\) −13512.0 −0.561010
\(835\) 0 0
\(836\) −1536.00 −0.0635451
\(837\) 4428.00 0.182860
\(838\) 14616.0 0.602508
\(839\) 9756.00 0.401448 0.200724 0.979648i \(-0.435671\pi\)
0.200724 + 0.979648i \(0.435671\pi\)
\(840\) 0 0
\(841\) 55135.0 2.26065
\(842\) 11876.0 0.486074
\(843\) −5850.00 −0.239009
\(844\) −2224.00 −0.0907029
\(845\) 0 0
\(846\) −3672.00 −0.149227
\(847\) 15100.0 0.612565
\(848\) 11808.0 0.478170
\(849\) 14196.0 0.573858
\(850\) 0 0
\(851\) −7920.00 −0.319029
\(852\) 1584.00 0.0636936
\(853\) −11342.0 −0.455267 −0.227633 0.973747i \(-0.573099\pi\)
−0.227633 + 0.973747i \(0.573099\pi\)
\(854\) 24560.0 0.984105
\(855\) 0 0
\(856\) 96.0000 0.00383319
\(857\) 16134.0 0.643089 0.321544 0.946895i \(-0.395798\pi\)
0.321544 + 0.946895i \(0.395798\pi\)
\(858\) 1872.00 0.0744860
\(859\) −20932.0 −0.831421 −0.415710 0.909497i \(-0.636467\pi\)
−0.415710 + 0.909497i \(0.636467\pi\)
\(860\) 0 0
\(861\) 7560.00 0.299238
\(862\) −23064.0 −0.911326
\(863\) −10044.0 −0.396178 −0.198089 0.980184i \(-0.563474\pi\)
−0.198089 + 0.980184i \(0.563474\pi\)
\(864\) −864.000 −0.0340207
\(865\) 0 0
\(866\) −1436.00 −0.0563479
\(867\) −12039.0 −0.471587
\(868\) −13120.0 −0.513044
\(869\) −26304.0 −1.02681
\(870\) 0 0
\(871\) 11024.0 0.428856
\(872\) 14672.0 0.569790
\(873\) 15534.0 0.602229
\(874\) 2304.00 0.0891693
\(875\) 0 0
\(876\) −2616.00 −0.100898
\(877\) 26314.0 1.01318 0.506591 0.862186i \(-0.330905\pi\)
0.506591 + 0.862186i \(0.330905\pi\)
\(878\) −17968.0 −0.690650
\(879\) −14994.0 −0.575353
\(880\) 0 0
\(881\) 37506.0 1.43429 0.717145 0.696924i \(-0.245449\pi\)
0.717145 + 0.696924i \(0.245449\pi\)
\(882\) −1026.00 −0.0391692
\(883\) 6388.00 0.243458 0.121729 0.992563i \(-0.461156\pi\)
0.121729 + 0.992563i \(0.461156\pi\)
\(884\) −1560.00 −0.0593535
\(885\) 0 0
\(886\) 5208.00 0.197479
\(887\) 5472.00 0.207138 0.103569 0.994622i \(-0.466974\pi\)
0.103569 + 0.994622i \(0.466974\pi\)
\(888\) 2640.00 0.0997664
\(889\) 42880.0 1.61772
\(890\) 0 0
\(891\) 1944.00 0.0730937
\(892\) 1072.00 0.0402390
\(893\) −3264.00 −0.122313
\(894\) 10620.0 0.397300
\(895\) 0 0
\(896\) 2560.00 0.0954504
\(897\) −2808.00 −0.104522
\(898\) 26412.0 0.981492
\(899\) −46248.0 −1.71575
\(900\) 0 0
\(901\) 22140.0 0.818635
\(902\) 6048.00 0.223255
\(903\) 9840.00 0.362630
\(904\) −2928.00 −0.107725
\(905\) 0 0
\(906\) 5928.00 0.217378
\(907\) 7180.00 0.262853 0.131427 0.991326i \(-0.458044\pi\)
0.131427 + 0.991326i \(0.458044\pi\)
\(908\) −7200.00 −0.263150
\(909\) 7182.00 0.262059
\(910\) 0 0
\(911\) 27624.0 1.00464 0.502318 0.864683i \(-0.332481\pi\)
0.502318 + 0.864683i \(0.332481\pi\)
\(912\) −768.000 −0.0278849
\(913\) −13248.0 −0.480224
\(914\) 16852.0 0.609863
\(915\) 0 0
\(916\) 11960.0 0.431408
\(917\) 54960.0 1.97921
\(918\) −1620.00 −0.0582440
\(919\) −30256.0 −1.08602 −0.543011 0.839726i \(-0.682716\pi\)
−0.543011 + 0.839726i \(0.682716\pi\)
\(920\) 0 0
\(921\) −20472.0 −0.732438
\(922\) −33372.0 −1.19203
\(923\) −1716.00 −0.0611948
\(924\) −5760.00 −0.205076
\(925\) 0 0
\(926\) 31864.0 1.13079
\(927\) 4680.00 0.165816
\(928\) 9024.00 0.319210
\(929\) −1926.00 −0.0680194 −0.0340097 0.999422i \(-0.510828\pi\)
−0.0340097 + 0.999422i \(0.510828\pi\)
\(930\) 0 0
\(931\) −912.000 −0.0321048
\(932\) −11304.0 −0.397291
\(933\) −26280.0 −0.922153
\(934\) 37080.0 1.29903
\(935\) 0 0
\(936\) 936.000 0.0326860
\(937\) −3962.00 −0.138135 −0.0690677 0.997612i \(-0.522002\pi\)
−0.0690677 + 0.997612i \(0.522002\pi\)
\(938\) −33920.0 −1.18073
\(939\) −11886.0 −0.413083
\(940\) 0 0
\(941\) −1074.00 −0.0372066 −0.0186033 0.999827i \(-0.505922\pi\)
−0.0186033 + 0.999827i \(0.505922\pi\)
\(942\) 1956.00 0.0676538
\(943\) −9072.00 −0.313282
\(944\) 1920.00 0.0661978
\(945\) 0 0
\(946\) 7872.00 0.270551
\(947\) −4848.00 −0.166356 −0.0831778 0.996535i \(-0.526507\pi\)
−0.0831778 + 0.996535i \(0.526507\pi\)
\(948\) −13152.0 −0.450588
\(949\) 2834.00 0.0969394
\(950\) 0 0
\(951\) −21258.0 −0.724856
\(952\) 4800.00 0.163413
\(953\) −762.000 −0.0259009 −0.0129505 0.999916i \(-0.504122\pi\)
−0.0129505 + 0.999916i \(0.504122\pi\)
\(954\) −13284.0 −0.450823
\(955\) 0 0
\(956\) −7248.00 −0.245206
\(957\) −20304.0 −0.685826
\(958\) −12360.0 −0.416841
\(959\) 55080.0 1.85467
\(960\) 0 0
\(961\) −2895.00 −0.0971770
\(962\) −2860.00 −0.0958525
\(963\) −108.000 −0.00361397
\(964\) −6328.00 −0.211422
\(965\) 0 0
\(966\) 8640.00 0.287772
\(967\) −35804.0 −1.19067 −0.595336 0.803477i \(-0.702981\pi\)
−0.595336 + 0.803477i \(0.702981\pi\)
\(968\) 6040.00 0.200551
\(969\) −1440.00 −0.0477394
\(970\) 0 0
\(971\) −4260.00 −0.140793 −0.0703964 0.997519i \(-0.522426\pi\)
−0.0703964 + 0.997519i \(0.522426\pi\)
\(972\) 972.000 0.0320750
\(973\) −45040.0 −1.48398
\(974\) 23512.0 0.773484
\(975\) 0 0
\(976\) 9824.00 0.322191
\(977\) 28710.0 0.940137 0.470069 0.882630i \(-0.344229\pi\)
0.470069 + 0.882630i \(0.344229\pi\)
\(978\) 8976.00 0.293477
\(979\) 5040.00 0.164534
\(980\) 0 0
\(981\) −16506.0 −0.537203
\(982\) −3816.00 −0.124006
\(983\) 49524.0 1.60689 0.803444 0.595381i \(-0.202999\pi\)
0.803444 + 0.595381i \(0.202999\pi\)
\(984\) 3024.00 0.0979691
\(985\) 0 0
\(986\) 16920.0 0.546493
\(987\) −12240.0 −0.394735
\(988\) 832.000 0.0267909
\(989\) −11808.0 −0.379649
\(990\) 0 0
\(991\) 44408.0 1.42348 0.711739 0.702444i \(-0.247908\pi\)
0.711739 + 0.702444i \(0.247908\pi\)
\(992\) −5248.00 −0.167968
\(993\) −27048.0 −0.864393
\(994\) 5280.00 0.168482
\(995\) 0 0
\(996\) −6624.00 −0.210732
\(997\) −18398.0 −0.584424 −0.292212 0.956354i \(-0.594391\pi\)
−0.292212 + 0.956354i \(0.594391\pi\)
\(998\) 17888.0 0.567369
\(999\) −2970.00 −0.0940607
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 1950.4.a.c.1.1 1
5.4 even 2 78.4.a.e.1.1 1
15.14 odd 2 234.4.a.b.1.1 1
20.19 odd 2 624.4.a.i.1.1 1
40.19 odd 2 2496.4.a.b.1.1 1
40.29 even 2 2496.4.a.k.1.1 1
60.59 even 2 1872.4.a.e.1.1 1
65.34 odd 4 1014.4.b.c.337.2 2
65.44 odd 4 1014.4.b.c.337.1 2
65.64 even 2 1014.4.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.4.a.e.1.1 1 5.4 even 2
234.4.a.b.1.1 1 15.14 odd 2
624.4.a.i.1.1 1 20.19 odd 2
1014.4.a.b.1.1 1 65.64 even 2
1014.4.b.c.337.1 2 65.44 odd 4
1014.4.b.c.337.2 2 65.34 odd 4
1872.4.a.e.1.1 1 60.59 even 2
1950.4.a.c.1.1 1 1.1 even 1 trivial
2496.4.a.b.1.1 1 40.19 odd 2
2496.4.a.k.1.1 1 40.29 even 2