Properties

Label 78.4.a.e.1.1
Level $78$
Weight $4$
Character 78.1
Self dual yes
Analytic conductor $4.602$
Analytic rank $0$
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] = [78,4,Mod(1,78)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(78, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("78.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 78 = 2 \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 78.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: \(4.60214898045\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 78.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +6.00000 q^{5} -6.00000 q^{6} +20.0000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +6.00000 q^{5} -6.00000 q^{6} +20.0000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +12.0000 q^{10} +24.0000 q^{11} -12.0000 q^{12} +13.0000 q^{13} +40.0000 q^{14} -18.0000 q^{15} +16.0000 q^{16} -30.0000 q^{17} +18.0000 q^{18} -16.0000 q^{19} +24.0000 q^{20} -60.0000 q^{21} +48.0000 q^{22} -72.0000 q^{23} -24.0000 q^{24} -89.0000 q^{25} +26.0000 q^{26} -27.0000 q^{27} +80.0000 q^{28} -282.000 q^{29} -36.0000 q^{30} +164.000 q^{31} +32.0000 q^{32} -72.0000 q^{33} -60.0000 q^{34} +120.000 q^{35} +36.0000 q^{36} +110.000 q^{37} -32.0000 q^{38} -39.0000 q^{39} +48.0000 q^{40} -126.000 q^{41} -120.000 q^{42} +164.000 q^{43} +96.0000 q^{44} +54.0000 q^{45} -144.000 q^{46} -204.000 q^{47} -48.0000 q^{48} +57.0000 q^{49} -178.000 q^{50} +90.0000 q^{51} +52.0000 q^{52} -738.000 q^{53} -54.0000 q^{54} +144.000 q^{55} +160.000 q^{56} +48.0000 q^{57} -564.000 q^{58} +120.000 q^{59} -72.0000 q^{60} +614.000 q^{61} +328.000 q^{62} +180.000 q^{63} +64.0000 q^{64} +78.0000 q^{65} -144.000 q^{66} +848.000 q^{67} -120.000 q^{68} +216.000 q^{69} +240.000 q^{70} +132.000 q^{71} +72.0000 q^{72} +218.000 q^{73} +220.000 q^{74} +267.000 q^{75} -64.0000 q^{76} +480.000 q^{77} -78.0000 q^{78} -1096.00 q^{79} +96.0000 q^{80} +81.0000 q^{81} -252.000 q^{82} +552.000 q^{83} -240.000 q^{84} -180.000 q^{85} +328.000 q^{86} +846.000 q^{87} +192.000 q^{88} +210.000 q^{89} +108.000 q^{90} +260.000 q^{91} -288.000 q^{92} -492.000 q^{93} -408.000 q^{94} -96.0000 q^{95} -96.0000 q^{96} -1726.00 q^{97} +114.000 q^{98} +216.000 q^{99} +O(q^{100})\)

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
\(5\) 6.00000 0.536656 0.268328 0.963328i \(-0.413529\pi\)
0.268328 + 0.963328i \(0.413529\pi\)
\(6\) −6.00000 −0.408248
\(7\) 20.0000 1.07990 0.539949 0.841698i \(-0.318443\pi\)
0.539949 + 0.841698i \(0.318443\pi\)
\(8\) 8.00000 0.353553
\(9\) 9.00000 0.333333
\(10\) 12.0000 0.379473
\(11\) 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
\(15\) −18.0000 −0.309839
\(16\) 16.0000 0.250000
\(17\) −30.0000 −0.428004 −0.214002 0.976833i \(-0.568650\pi\)
−0.214002 + 0.976833i \(0.568650\pi\)
\(18\) 18.0000 0.235702
\(19\) −16.0000 −0.193192 −0.0965961 0.995324i \(-0.530796\pi\)
−0.0965961 + 0.995324i \(0.530796\pi\)
\(20\) 24.0000 0.268328
\(21\) −60.0000 −0.623480
\(22\) 48.0000 0.465165
\(23\) −72.0000 −0.652741 −0.326370 0.945242i \(-0.605826\pi\)
−0.326370 + 0.945242i \(0.605826\pi\)
\(24\) −24.0000 −0.204124
\(25\) −89.0000 −0.712000
\(26\) 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 \(0.858539\pi\)
\(30\) −36.0000 −0.219089
\(31\) 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
\(35\) 120.000 0.579534
\(36\) 36.0000 0.166667
\(37\) 110.000 0.488754 0.244377 0.969680i \(-0.421417\pi\)
0.244377 + 0.969680i \(0.421417\pi\)
\(38\) −32.0000 −0.136608
\(39\) −39.0000 −0.160128
\(40\) 48.0000 0.189737
\(41\) −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 \(-0.406075\pi\)
0.290811 + 0.956780i \(0.406075\pi\)
\(44\) 96.0000 0.328921
\(45\) 54.0000 0.178885
\(46\) −144.000 −0.461557
\(47\) −204.000 −0.633116 −0.316558 0.948573i \(-0.602527\pi\)
−0.316558 + 0.948573i \(0.602527\pi\)
\(48\) −48.0000 −0.144338
\(49\) 57.0000 0.166181
\(50\) −178.000 −0.503460
\(51\) 90.0000 0.247108
\(52\) 52.0000 0.138675
\(53\) −738.000 −1.91268 −0.956341 0.292255i \(-0.905595\pi\)
−0.956341 + 0.292255i \(0.905595\pi\)
\(54\) −54.0000 −0.136083
\(55\) 144.000 0.353036
\(56\) 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 \(0.457733\pi\)
\(60\) −72.0000 −0.154919
\(61\) 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
\(65\) 78.0000 0.148842
\(66\) −144.000 −0.268563
\(67\) 848.000 1.54626 0.773132 0.634245i \(-0.218689\pi\)
0.773132 + 0.634245i \(0.218689\pi\)
\(68\) −120.000 −0.214002
\(69\) 216.000 0.376860
\(70\) 240.000 0.409793
\(71\) 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 \(-0.444085\pi\)
0.174760 + 0.984611i \(0.444085\pi\)
\(74\) 220.000 0.345601
\(75\) 267.000 0.411073
\(76\) −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 \(0.785005\pi\)
\(80\) 96.0000 0.134164
\(81\) 81.0000 0.111111
\(82\) −252.000 −0.339375
\(83\) 552.000 0.729998 0.364999 0.931008i \(-0.381069\pi\)
0.364999 + 0.931008i \(0.381069\pi\)
\(84\) −240.000 −0.311740
\(85\) −180.000 −0.229691
\(86\) 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 \(0.460089\pi\)
\(90\) 108.000 0.126491
\(91\) 260.000 0.299510
\(92\) −288.000 −0.326370
\(93\) −492.000 −0.548581
\(94\) −408.000 −0.447681
\(95\) −96.0000 −0.103678
\(96\) −96.0000 −0.102062
\(97\) −1726.00 −1.80669 −0.903344 0.428917i \(-0.858895\pi\)
−0.903344 + 0.428917i \(0.858895\pi\)
\(98\) 114.000 0.117508
\(99\) 216.000 0.219281
\(100\) −356.000 −0.356000
\(101\) 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 \(-0.580011\pi\)
−0.248724 + 0.968574i \(0.580011\pi\)
\(104\) 104.000 0.0980581
\(105\) −360.000 −0.334594
\(106\) −1476.00 −1.35247
\(107\) 12.0000 0.0108419 0.00542095 0.999985i \(-0.498274\pi\)
0.00542095 + 0.999985i \(0.498274\pi\)
\(108\) −108.000 −0.0962250
\(109\) −1834.00 −1.61161 −0.805804 0.592182i \(-0.798267\pi\)
−0.805804 + 0.592182i \(0.798267\pi\)
\(110\) 288.000 0.249634
\(111\) −330.000 −0.282182
\(112\) 320.000 0.269975
\(113\) −366.000 −0.304694 −0.152347 0.988327i \(-0.548683\pi\)
−0.152347 + 0.988327i \(0.548683\pi\)
\(114\) 96.0000 0.0788704
\(115\) −432.000 −0.350297
\(116\) −1128.00 −0.902864
\(117\) 117.000 0.0924500
\(118\) 240.000 0.187236
\(119\) −600.000 −0.462201
\(120\) −144.000 −0.109545
\(121\) −755.000 −0.567243
\(122\) 1228.00 0.911294
\(123\) 378.000 0.277098
\(124\) 656.000 0.475085
\(125\) −1284.00 −0.918756
\(126\) 360.000 0.254535
\(127\) 2144.00 1.49803 0.749013 0.662556i \(-0.230528\pi\)
0.749013 + 0.662556i \(0.230528\pi\)
\(128\) 128.000 0.0883883
\(129\) −492.000 −0.335800
\(130\) 156.000 0.105247
\(131\) −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
\(135\) −162.000 −0.103280
\(136\) −240.000 −0.151322
\(137\) 2754.00 1.71745 0.858723 0.512440i \(-0.171258\pi\)
0.858723 + 0.512440i \(0.171258\pi\)
\(138\) 432.000 0.266480
\(139\) 2252.00 1.37419 0.687094 0.726568i \(-0.258886\pi\)
0.687094 + 0.726568i \(0.258886\pi\)
\(140\) 480.000 0.289767
\(141\) 612.000 0.365530
\(142\) 264.000 0.156017
\(143\) 312.000 0.182453
\(144\) 144.000 0.0833333
\(145\) −1692.00 −0.969055
\(146\) 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 \(0.661760\pi\)
\(150\) 534.000 0.290673
\(151\) −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
\(155\) 984.000 0.509915
\(156\) −156.000 −0.0800641
\(157\) 326.000 0.165717 0.0828587 0.996561i \(-0.473595\pi\)
0.0828587 + 0.996561i \(0.473595\pi\)
\(158\) −2192.00 −1.10371
\(159\) 2214.00 1.10429
\(160\) 192.000 0.0948683
\(161\) −1440.00 −0.704894
\(162\) 162.000 0.0785674
\(163\) 1496.00 0.718870 0.359435 0.933170i \(-0.382969\pi\)
0.359435 + 0.933170i \(0.382969\pi\)
\(164\) −504.000 −0.239974
\(165\) −432.000 −0.203825
\(166\) 1104.00 0.516187
\(167\) 1116.00 0.517118 0.258559 0.965995i \(-0.416752\pi\)
0.258559 + 0.965995i \(0.416752\pi\)
\(168\) −480.000 −0.220433
\(169\) 169.000 0.0769231
\(170\) −360.000 −0.162416
\(171\) −144.000 −0.0643974
\(172\) 656.000 0.290811
\(173\) 4374.00 1.92225 0.961124 0.276116i \(-0.0890472\pi\)
0.961124 + 0.276116i \(0.0890472\pi\)
\(174\) 1692.00 0.737185
\(175\) −1780.00 −0.768888
\(176\) 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 \(0.499203\pi\)
\(180\) 216.000 0.0894427
\(181\) 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
\(185\) 660.000 0.262293
\(186\) −984.000 −0.387905
\(187\) −720.000 −0.281559
\(188\) −816.000 −0.316558
\(189\) −540.000 −0.207827
\(190\) −192.000 −0.0733113
\(191\) −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 \(-0.711754\pi\)
−0.617251 + 0.786766i \(0.711754\pi\)
\(194\) −3452.00 −1.27752
\(195\) −234.000 −0.0859338
\(196\) 228.000 0.0830904
\(197\) 3126.00 1.13055 0.565275 0.824903i \(-0.308770\pi\)
0.565275 + 0.824903i \(0.308770\pi\)
\(198\) 432.000 0.155055
\(199\) 4664.00 1.66142 0.830709 0.556707i \(-0.187935\pi\)
0.830709 + 0.556707i \(0.187935\pi\)
\(200\) −712.000 −0.251730
\(201\) −2544.00 −0.892736
\(202\) 1596.00 0.555912
\(203\) −5640.00 −1.95000
\(204\) 360.000 0.123554
\(205\) −756.000 −0.257567
\(206\) −1040.00 −0.351749
\(207\) −648.000 −0.217580
\(208\) 208.000 0.0693375
\(209\) −384.000 −0.127090
\(210\) −720.000 −0.236594
\(211\) −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
\(215\) 984.000 0.312131
\(216\) −216.000 −0.0680414
\(217\) 3280.00 1.02609
\(218\) −3668.00 −1.13958
\(219\) −654.000 −0.201796
\(220\) 576.000 0.176518
\(221\) −390.000 −0.118707
\(222\) −660.000 −0.199533
\(223\) −268.000 −0.0804781 −0.0402390 0.999190i \(-0.512812\pi\)
−0.0402390 + 0.999190i \(0.512812\pi\)
\(224\) 640.000 0.190901
\(225\) −801.000 −0.237333
\(226\) −732.000 −0.215451
\(227\) 1800.00 0.526300 0.263150 0.964755i \(-0.415239\pi\)
0.263150 + 0.964755i \(0.415239\pi\)
\(228\) 192.000 0.0557698
\(229\) 2990.00 0.862816 0.431408 0.902157i \(-0.358017\pi\)
0.431408 + 0.902157i \(0.358017\pi\)
\(230\) −864.000 −0.247698
\(231\) −1440.00 −0.410152
\(232\) −2256.00 −0.638421
\(233\) 2826.00 0.794581 0.397291 0.917693i \(-0.369951\pi\)
0.397291 + 0.917693i \(0.369951\pi\)
\(234\) 234.000 0.0653720
\(235\) −1224.00 −0.339766
\(236\) 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 \(0.578856\pi\)
\(240\) −288.000 −0.0774597
\(241\) −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
\(245\) 342.000 0.0891820
\(246\) 756.000 0.195938
\(247\) −208.000 −0.0535819
\(248\) 1312.00 0.335936
\(249\) −1656.00 −0.421465
\(250\) −2568.00 −0.649658
\(251\) 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
\(255\) 540.000 0.132612
\(256\) 256.000 0.0625000
\(257\) −558.000 −0.135436 −0.0677181 0.997704i \(-0.521572\pi\)
−0.0677181 + 0.997704i \(0.521572\pi\)
\(258\) −984.000 −0.237446
\(259\) 2200.00 0.527804
\(260\) 312.000 0.0744208
\(261\) −2538.00 −0.601909
\(262\) −5496.00 −1.29597
\(263\) 2112.00 0.495177 0.247588 0.968865i \(-0.420362\pi\)
0.247588 + 0.968865i \(0.420362\pi\)
\(264\) −576.000 −0.134282
\(265\) −4428.00 −1.02645
\(266\) −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 \(0.306223\pi\)
\(270\) −324.000 −0.0730297
\(271\) −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
\(275\) −2136.00 −0.468384
\(276\) 864.000 0.188430
\(277\) 5582.00 1.21079 0.605397 0.795924i \(-0.293014\pi\)
0.605397 + 0.795924i \(0.293014\pi\)
\(278\) 4504.00 0.971698
\(279\) 1476.00 0.316723
\(280\) 960.000 0.204896
\(281\) −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 \(-0.665556\pi\)
−0.496976 + 0.867765i \(0.665556\pi\)
\(284\) 528.000 0.110321
\(285\) 288.000 0.0598584
\(286\) 624.000 0.129014
\(287\) −2520.00 −0.518296
\(288\) 288.000 0.0589256
\(289\) −4013.00 −0.816813
\(290\) −3384.00 −0.685225
\(291\) 5178.00 1.04309
\(292\) 872.000 0.174760
\(293\) 4998.00 0.996540 0.498270 0.867022i \(-0.333969\pi\)
0.498270 + 0.867022i \(0.333969\pi\)
\(294\) −342.000 −0.0678430
\(295\) 720.000 0.142102
\(296\) 880.000 0.172801
\(297\) −648.000 −0.126602
\(298\) −3540.00 −0.688143
\(299\) −936.000 −0.181038
\(300\) 1068.00 0.205537
\(301\) 3280.00 0.628093
\(302\) −1976.00 −0.376510
\(303\) −2394.00 −0.453900
\(304\) −256.000 −0.0482980
\(305\) 3684.00 0.691624
\(306\) −540.000 −0.100882
\(307\) 6824.00 1.26862 0.634310 0.773079i \(-0.281284\pi\)
0.634310 + 0.773079i \(0.281284\pi\)
\(308\) 1920.00 0.355202
\(309\) 1560.00 0.287202
\(310\) 1968.00 0.360564
\(311\) −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 \(-0.383547\pi\)
0.357740 + 0.933821i \(0.383547\pi\)
\(314\) 652.000 0.117180
\(315\) 1080.00 0.193178
\(316\) −4384.00 −0.780441
\(317\) 7086.00 1.25549 0.627744 0.778420i \(-0.283979\pi\)
0.627744 + 0.778420i \(0.283979\pi\)
\(318\) 4428.00 0.780849
\(319\) −6768.00 −1.18788
\(320\) 384.000 0.0670820
\(321\) −36.0000 −0.00625958
\(322\) −2880.00 −0.498435
\(323\) 480.000 0.0826870
\(324\) 324.000 0.0555556
\(325\) −1157.00 −0.197473
\(326\) 2992.00 0.508318
\(327\) 5502.00 0.930463
\(328\) −1008.00 −0.169687
\(329\) −4080.00 −0.683701
\(330\) −864.000 −0.144126
\(331\) −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
\(335\) 5088.00 0.829812
\(336\) −960.000 −0.155870
\(337\) 2306.00 0.372747 0.186374 0.982479i \(-0.440327\pi\)
0.186374 + 0.982479i \(0.440327\pi\)
\(338\) 338.000 0.0543928
\(339\) 1098.00 0.175915
\(340\) −720.000 −0.114846
\(341\) 3936.00 0.625063
\(342\) −288.000 −0.0455358
\(343\) −5720.00 −0.900440
\(344\) 1312.00 0.205635
\(345\) 1296.00 0.202244
\(346\) 8748.00 1.35924
\(347\) −11076.0 −1.71352 −0.856759 0.515717i \(-0.827526\pi\)
−0.856759 + 0.515717i \(0.827526\pi\)
\(348\) 3384.00 0.521269
\(349\) 2342.00 0.359210 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(350\) −3560.00 −0.543686
\(351\) −351.000 −0.0533761
\(352\) 768.000 0.116291
\(353\) 4650.00 0.701118 0.350559 0.936541i \(-0.385992\pi\)
0.350559 + 0.936541i \(0.385992\pi\)
\(354\) −720.000 −0.108100
\(355\) 792.000 0.118408
\(356\) 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 \(0.810678\pi\)
\(360\) 432.000 0.0632456
\(361\) −6603.00 −0.962677
\(362\) 9436.00 1.37001
\(363\) 2265.00 0.327498
\(364\) 1040.00 0.149755
\(365\) 1308.00 0.187572
\(366\) −3684.00 −0.526136
\(367\) −7288.00 −1.03660 −0.518298 0.855200i \(-0.673434\pi\)
−0.518298 + 0.855200i \(0.673434\pi\)
\(368\) −1152.00 −0.163185
\(369\) −1134.00 −0.159983
\(370\) 1320.00 0.185469
\(371\) −14760.0 −2.06550
\(372\) −1968.00 −0.274290
\(373\) −9970.00 −1.38399 −0.691993 0.721904i \(-0.743267\pi\)
−0.691993 + 0.721904i \(0.743267\pi\)
\(374\) −1440.00 −0.199093
\(375\) 3852.00 0.530444
\(376\) −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 \(0.135068\pi\)
\(380\) −384.000 −0.0518389
\(381\) −6432.00 −0.864885
\(382\) −2736.00 −0.366455
\(383\) 11820.0 1.57696 0.788478 0.615064i \(-0.210870\pi\)
0.788478 + 0.615064i \(0.210870\pi\)
\(384\) −384.000 −0.0510310
\(385\) 2880.00 0.381243
\(386\) −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 \(0.496390\pi\)
\(390\) −468.000 −0.0607644
\(391\) 2160.00 0.279376
\(392\) 456.000 0.0587538
\(393\) 8244.00 1.05815
\(394\) 6252.00 0.799419
\(395\) −6576.00 −0.837657
\(396\) 864.000 0.109640
\(397\) −2986.00 −0.377489 −0.188744 0.982026i \(-0.560442\pi\)
−0.188744 + 0.982026i \(0.560442\pi\)
\(398\) 9328.00 1.17480
\(399\) 960.000 0.120451
\(400\) −1424.00 −0.178000
\(401\) −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
\(405\) 486.000 0.0596285
\(406\) −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 \(0.644830\pi\)
\(410\) −1512.00 −0.182128
\(411\) −8262.00 −0.991568
\(412\) −2080.00 −0.248724
\(413\) 2400.00 0.285947
\(414\) −1296.00 −0.153852
\(415\) 3312.00 0.391758
\(416\) 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 \(0.640091\pi\)
\(420\) −1440.00 −0.167297
\(421\) −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
\(425\) 2670.00 0.304739
\(426\) −792.000 −0.0900764
\(427\) 12280.0 1.39174
\(428\) 48.0000 0.00542095
\(429\) −936.000 −0.105339
\(430\) 1968.00 0.220710
\(431\) 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 \(-0.512686\pi\)
−0.0398440 + 0.999206i \(0.512686\pi\)
\(434\) 6560.00 0.725553
\(435\) 5076.00 0.559484
\(436\) −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 \(0.337594\pi\)
\(440\) 1152.00 0.124817
\(441\) 513.000 0.0553936
\(442\) −780.000 −0.0839385
\(443\) 2604.00 0.279277 0.139639 0.990203i \(-0.455406\pi\)
0.139639 + 0.990203i \(0.455406\pi\)
\(444\) −1320.00 −0.141091
\(445\) 1260.00 0.134224
\(446\) −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 \(0.744162\pi\)
\(450\) −1602.00 −0.167820
\(451\) −3024.00 −0.315731
\(452\) −1464.00 −0.152347
\(453\) 2964.00 0.307419
\(454\) 3600.00 0.372151
\(455\) 1560.00 0.160734
\(456\) 384.000 0.0394352
\(457\) 8426.00 0.862476 0.431238 0.902238i \(-0.358077\pi\)
0.431238 + 0.902238i \(0.358077\pi\)
\(458\) 5980.00 0.610103
\(459\) 810.000 0.0823694
\(460\) −1728.00 −0.175149
\(461\) 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 \(-0.205049\pi\)
0.799593 + 0.600543i \(0.205049\pi\)
\(464\) −4512.00 −0.451432
\(465\) −2952.00 −0.294399
\(466\) 5652.00 0.561854
\(467\) 18540.0 1.83711 0.918553 0.395297i \(-0.129358\pi\)
0.918553 + 0.395297i \(0.129358\pi\)
\(468\) 468.000 0.0462250
\(469\) 16960.0 1.66981
\(470\) −2448.00 −0.240251
\(471\) −978.000 −0.0956770
\(472\) 960.000 0.0936178
\(473\) 3936.00 0.382616
\(474\) 6576.00 0.637227
\(475\) 1424.00 0.137553
\(476\) −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 \(0.404763\pi\)
\(480\) −576.000 −0.0547723
\(481\) 1430.00 0.135556
\(482\) −3164.00 −0.298996
\(483\) 4320.00 0.406971
\(484\) −3020.00 −0.283621
\(485\) −10356.0 −0.969571
\(486\) −486.000 −0.0453609
\(487\) 11756.0 1.09387 0.546936 0.837175i \(-0.315794\pi\)
0.546936 + 0.837175i \(0.315794\pi\)
\(488\) 4912.00 0.455647
\(489\) −4488.00 −0.415040
\(490\) 684.000 0.0630612
\(491\) 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
\(495\) 1296.00 0.117679
\(496\) 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 \(0.631404\pi\)
\(500\) −5136.00 −0.459378
\(501\) −3348.00 −0.298558
\(502\) 4296.00 0.381952
\(503\) −6528.00 −0.578666 −0.289333 0.957228i \(-0.593434\pi\)
−0.289333 + 0.957228i \(0.593434\pi\)
\(504\) 1440.00 0.127267
\(505\) 4788.00 0.421907
\(506\) −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 \(0.676852\pi\)
\(510\) 1080.00 0.0937710
\(511\) 4360.00 0.377446
\(512\) 512.000 0.0441942
\(513\) 432.000 0.0371799
\(514\) −1116.00 −0.0957678
\(515\) −3120.00 −0.266958
\(516\) −1968.00 −0.167900
\(517\) −4896.00 −0.416491
\(518\) 4400.00 0.373214
\(519\) −13122.0 −1.10981
\(520\) 624.000 0.0526235
\(521\) −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 \(-0.778095\pi\)
−0.766685 + 0.642024i \(0.778095\pi\)
\(524\) −10992.0 −0.916389
\(525\) 5340.00 0.443918
\(526\) 4224.00 0.350143
\(527\) −4920.00 −0.406677
\(528\) −1152.00 −0.0949514
\(529\) −6983.00 −0.573929
\(530\) −8856.00 −0.725811
\(531\) 1080.00 0.0882637
\(532\) −1280.00 −0.104314
\(533\) −1638.00 −0.133114
\(534\) −1260.00 −0.102108
\(535\) 72.0000 0.00581838
\(536\) 6784.00 0.546687
\(537\) −36.0000 −0.00289295
\(538\) 10092.0 0.808731
\(539\) 1368.00 0.109321
\(540\) −648.000 −0.0516398
\(541\) 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
\(545\) −11004.0 −0.864880
\(546\) −1560.00 −0.122274
\(547\) 17444.0 1.36353 0.681766 0.731571i \(-0.261212\pi\)
0.681766 + 0.731571i \(0.261212\pi\)
\(548\) 11016.0 0.858723
\(549\) 5526.00 0.429588
\(550\) −4272.00 −0.331198
\(551\) 4512.00 0.348852
\(552\) 1728.00 0.133240
\(553\) −21920.0 −1.68559
\(554\) 11164.0 0.856160
\(555\) −1980.00 −0.151435
\(556\) 9008.00 0.687094
\(557\) −3714.00 −0.282526 −0.141263 0.989972i \(-0.545116\pi\)
−0.141263 + 0.989972i \(0.545116\pi\)
\(558\) 2952.00 0.223957
\(559\) 2132.00 0.161313
\(560\) 1920.00 0.144884
\(561\) 2160.00 0.162558
\(562\) −3900.00 −0.292725
\(563\) −13812.0 −1.03394 −0.516968 0.856004i \(-0.672940\pi\)
−0.516968 + 0.856004i \(0.672940\pi\)
\(564\) 2448.00 0.182765
\(565\) −2196.00 −0.163516
\(566\) −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 \(0.699801\pi\)
\(570\) 576.000 0.0423263
\(571\) 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
\(575\) 6408.00 0.464751
\(576\) 576.000 0.0416667
\(577\) −10654.0 −0.768686 −0.384343 0.923190i \(-0.625572\pi\)
−0.384343 + 0.923190i \(0.625572\pi\)
\(578\) −8026.00 −0.577574
\(579\) 9930.00 0.712740
\(580\) −6768.00 −0.484527
\(581\) 11040.0 0.788324
\(582\) 10356.0 0.737577
\(583\) −17712.0 −1.25824
\(584\) 1744.00 0.123574
\(585\) 702.000 0.0496139
\(586\) 9996.00 0.704660
\(587\) −9984.00 −0.702017 −0.351008 0.936372i \(-0.614161\pi\)
−0.351008 + 0.936372i \(0.614161\pi\)
\(588\) −684.000 −0.0479723
\(589\) −2624.00 −0.183565
\(590\) 1440.00 0.100481
\(591\) −9378.00 −0.652723
\(592\) 1760.00 0.122188
\(593\) 12618.0 0.873793 0.436896 0.899512i \(-0.356078\pi\)
0.436896 + 0.899512i \(0.356078\pi\)
\(594\) −1296.00 −0.0895211
\(595\) −3600.00 −0.248043
\(596\) −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 \(0.375428\pi\)
\(600\) 2136.00 0.145336
\(601\) 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
\(605\) −4530.00 −0.304414
\(606\) −4788.00 −0.320956
\(607\) 1064.00 0.0711473 0.0355737 0.999367i \(-0.488674\pi\)
0.0355737 + 0.999367i \(0.488674\pi\)
\(608\) −512.000 −0.0341519
\(609\) 16920.0 1.12583
\(610\) 7368.00 0.489052
\(611\) −2652.00 −0.175595
\(612\) −1080.00 −0.0713340
\(613\) −20914.0 −1.37799 −0.688996 0.724766i \(-0.741948\pi\)
−0.688996 + 0.724766i \(0.741948\pi\)
\(614\) 13648.0 0.897050
\(615\) 2268.00 0.148707
\(616\) 3840.00 0.251166
\(617\) 9714.00 0.633826 0.316913 0.948455i \(-0.397354\pi\)
0.316913 + 0.948455i \(0.397354\pi\)
\(618\) 3120.00 0.203082
\(619\) −14848.0 −0.964122 −0.482061 0.876138i \(-0.660112\pi\)
−0.482061 + 0.876138i \(0.660112\pi\)
\(620\) 3936.00 0.254957
\(621\) 1944.00 0.125620
\(622\) −17520.0 −1.12940
\(623\) 4200.00 0.270095
\(624\) −624.000 −0.0400320
\(625\) 3421.00 0.218944
\(626\) 7924.00 0.505921
\(627\) 1152.00 0.0733755
\(628\) 1304.00 0.0828587
\(629\) −3300.00 −0.209189
\(630\) 2160.00 0.136598
\(631\) 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
\(635\) 12864.0 0.803925
\(636\) 8856.00 0.552143
\(637\) 741.000 0.0460902
\(638\) −13536.0 −0.839961
\(639\) 1188.00 0.0735470
\(640\) 768.000 0.0474342
\(641\) −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 \(-0.658422\pi\)
−0.477404 + 0.878684i \(0.658422\pi\)
\(644\) −5760.00 −0.352447
\(645\) −2952.00 −0.180209
\(646\) 960.000 0.0584686
\(647\) 1128.00 0.0685414 0.0342707 0.999413i \(-0.489089\pi\)
0.0342707 + 0.999413i \(0.489089\pi\)
\(648\) 648.000 0.0392837
\(649\) 2880.00 0.174191
\(650\) −2314.00 −0.139635
\(651\) −9840.00 −0.592412
\(652\) 5984.00 0.359435
\(653\) 8118.00 0.486496 0.243248 0.969964i \(-0.421787\pi\)
0.243248 + 0.969964i \(0.421787\pi\)
\(654\) 11004.0 0.657936
\(655\) −16488.0 −0.983572
\(656\) −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 \(0.368617\pi\)
\(660\) −1728.00 −0.101913
\(661\) −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
\(665\) −1920.00 −0.111962
\(666\) 1980.00 0.115200
\(667\) 20304.0 1.17867
\(668\) 4464.00 0.258559
\(669\) 804.000 0.0464640
\(670\) 10176.0 0.586766
\(671\) 14736.0 0.847805
\(672\) −1920.00 −0.110217
\(673\) −718.000 −0.0411246 −0.0205623 0.999789i \(-0.506546\pi\)
−0.0205623 + 0.999789i \(0.506546\pi\)
\(674\) 4612.00 0.263572
\(675\) 2403.00 0.137024
\(676\) 676.000 0.0384615
\(677\) −2994.00 −0.169969 −0.0849843 0.996382i \(-0.527084\pi\)
−0.0849843 + 0.996382i \(0.527084\pi\)
\(678\) 2196.00 0.124391
\(679\) −34520.0 −1.95104
\(680\) −1440.00 −0.0812081
\(681\) −5400.00 −0.303860
\(682\) 7872.00 0.441986
\(683\) 27384.0 1.53414 0.767071 0.641562i \(-0.221713\pi\)
0.767071 + 0.641562i \(0.221713\pi\)
\(684\) −576.000 −0.0321987
\(685\) 16524.0 0.921678
\(686\) −11440.0 −0.636707
\(687\) −8970.00 −0.498147
\(688\) 2624.00 0.145406
\(689\) −9594.00 −0.530482
\(690\) 2592.00 0.143008
\(691\) 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
\(695\) 13512.0 0.737467
\(696\) 6768.00 0.368592
\(697\) 3780.00 0.205420
\(698\) 4684.00 0.254000
\(699\) −8478.00 −0.458752
\(700\) −7120.00 −0.384444
\(701\) 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
\(705\) 3672.00 0.196164
\(706\) 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 \(0.467452\pi\)
\(710\) 1584.00 0.0837274
\(711\) −9864.00 −0.520294
\(712\) 1680.00 0.0884279
\(713\) −11808.0 −0.620215
\(714\) 3600.00 0.188693
\(715\) 1872.00 0.0979144
\(716\) 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 \(0.316909\pi\)
\(720\) 864.000 0.0447214
\(721\) −10400.0 −0.537193
\(722\) −13206.0 −0.680715
\(723\) 4746.00 0.244130
\(724\) 18872.0 0.968746
\(725\) 25098.0 1.28568
\(726\) 4530.00 0.231576
\(727\) −29464.0 −1.50311 −0.751554 0.659672i \(-0.770695\pi\)
−0.751554 + 0.659672i \(0.770695\pi\)
\(728\) 2080.00 0.105893
\(729\) 729.000 0.0370370
\(730\) 2616.00 0.132634
\(731\) −4920.00 −0.248937
\(732\) −7368.00 −0.372034
\(733\) −2698.00 −0.135952 −0.0679761 0.997687i \(-0.521654\pi\)
−0.0679761 + 0.997687i \(0.521654\pi\)
\(734\) −14576.0 −0.732984
\(735\) −1026.00 −0.0514892
\(736\) −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 \(0.494993\pi\)
\(740\) 2640.00 0.131146
\(741\) 624.000 0.0309355
\(742\) −29520.0 −1.46053
\(743\) −20844.0 −1.02920 −0.514598 0.857432i \(-0.672059\pi\)
−0.514598 + 0.857432i \(0.672059\pi\)
\(744\) −3936.00 −0.193953
\(745\) −10620.0 −0.522264
\(746\) −19940.0 −0.978626
\(747\) 4968.00 0.243333
\(748\) −2880.00 −0.140780
\(749\) 240.000 0.0117082
\(750\) 7704.00 0.375080
\(751\) 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
\(755\) −5928.00 −0.285751
\(756\) −2160.00 −0.103913
\(757\) 37550.0 1.80288 0.901439 0.432907i \(-0.142512\pi\)
0.901439 + 0.432907i \(0.142512\pi\)
\(758\) 26896.0 1.28880
\(759\) 5184.00 0.247915
\(760\) −768.000 −0.0366556
\(761\) 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
\(765\) −1620.00 −0.0765637
\(766\) 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 \(0.617639\pi\)
\(770\) 5760.00 0.269579
\(771\) 1674.00 0.0781941
\(772\) −13240.0 −0.617251
\(773\) −29514.0 −1.37328 −0.686640 0.726998i \(-0.740915\pi\)
−0.686640 + 0.726998i \(0.740915\pi\)
\(774\) 2952.00 0.137090
\(775\) −14596.0 −0.676521
\(776\) −13808.0 −0.638761
\(777\) −6600.00 −0.304728
\(778\) 348.000 0.0160365
\(779\) 2016.00 0.0927223
\(780\) −936.000 −0.0429669
\(781\) 3168.00 0.145147
\(782\) 4320.00 0.197548
\(783\) 7614.00 0.347512
\(784\) 912.000 0.0415452
\(785\) 1956.00 0.0889333
\(786\) 16488.0 0.748228
\(787\) 33176.0 1.50266 0.751332 0.659924i \(-0.229412\pi\)
0.751332 + 0.659924i \(0.229412\pi\)
\(788\) 12504.0 0.565275
\(789\) −6336.00 −0.285890
\(790\) −13152.0 −0.592313
\(791\) −7320.00 −0.329038
\(792\) 1728.00 0.0775275
\(793\) 7982.00 0.357439
\(794\) −5972.00 −0.266925
\(795\) 13284.0 0.592623
\(796\) 18656.0 0.830709
\(797\) −16746.0 −0.744258 −0.372129 0.928181i \(-0.621372\pi\)
−0.372129 + 0.928181i \(0.621372\pi\)
\(798\) 1920.00 0.0851720
\(799\) 6120.00 0.270976
\(800\) −2848.00 −0.125865
\(801\) 1890.00 0.0833706
\(802\) −21132.0 −0.930420
\(803\) 5232.00 0.229929
\(804\) −10176.0 −0.446368
\(805\) −8640.00 −0.378286
\(806\) 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 \(0.611892\pi\)
\(810\) 972.000 0.0421637
\(811\) 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
\(815\) 8976.00 0.385786
\(816\) 1440.00 0.0617771
\(817\) −2624.00 −0.112365
\(818\) −14540.0 −0.621490
\(819\) 2340.00 0.0998367
\(820\) −3024.00 −0.128784
\(821\) −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 \(-0.565112\pi\)
−0.203133 + 0.979151i \(0.565112\pi\)
\(824\) −4160.00 −0.175874
\(825\) 6408.00 0.270422
\(826\) 4800.00 0.202195
\(827\) −39960.0 −1.68022 −0.840112 0.542413i \(-0.817511\pi\)
−0.840112 + 0.542413i \(0.817511\pi\)
\(828\) −2592.00 −0.108790
\(829\) −3706.00 −0.155265 −0.0776325 0.996982i \(-0.524736\pi\)
−0.0776325 + 0.996982i \(0.524736\pi\)
\(830\) 6624.00 0.277015
\(831\) −16746.0 −0.699052
\(832\) 832.000 0.0346688
\(833\) −1710.00 −0.0711260
\(834\) −13512.0 −0.561010
\(835\) 6696.00 0.277515
\(836\) −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 \(0.435671\pi\)
\(840\) −2880.00 −0.118297
\(841\) 55135.0 2.26065
\(842\) −11876.0 −0.486074
\(843\) 5850.00 0.239009
\(844\) −2224.00 −0.0907029
\(845\) 1014.00 0.0412813
\(846\) −3672.00 −0.149227
\(847\) −15100.0 −0.612565
\(848\) −11808.0 −0.478170
\(849\) 14196.0 0.573858
\(850\) 5340.00 0.215483
\(851\) −7920.00 −0.319029
\(852\) −1584.00 −0.0636936
\(853\) 11342.0 0.455267 0.227633 0.973747i \(-0.426901\pi\)
0.227633 + 0.973747i \(0.426901\pi\)
\(854\) 24560.0 0.984105
\(855\) −864.000 −0.0345593
\(856\) 96.0000 0.00383319
\(857\) −16134.0 −0.643089 −0.321544 0.946895i \(-0.604202\pi\)
−0.321544 + 0.946895i \(0.604202\pi\)
\(858\) −1872.00 −0.0744860
\(859\) −20932.0 −0.831421 −0.415710 0.909497i \(-0.636467\pi\)
−0.415710 + 0.909497i \(0.636467\pi\)
\(860\) 3936.00 0.156066
\(861\) 7560.00 0.299238
\(862\) 23064.0 0.911326
\(863\) 10044.0 0.396178 0.198089 0.980184i \(-0.436526\pi\)
0.198089 + 0.980184i \(0.436526\pi\)
\(864\) −864.000 −0.0340207
\(865\) 26244.0 1.03159
\(866\) −1436.00 −0.0563479
\(867\) 12039.0 0.471587
\(868\) 13120.0 0.513044
\(869\) −26304.0 −1.02681
\(870\) 10152.0 0.395615
\(871\) 11024.0 0.428856
\(872\) −14672.0 −0.569790
\(873\) −15534.0 −0.602229
\(874\) 2304.00 0.0891693
\(875\) −25680.0 −0.992163
\(876\) −2616.00 −0.100898
\(877\) −26314.0 −1.01318 −0.506591 0.862186i \(-0.669095\pi\)
−0.506591 + 0.862186i \(0.669095\pi\)
\(878\) 17968.0 0.690650
\(879\) −14994.0 −0.575353
\(880\) 2304.00 0.0882589
\(881\) 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 \(-0.538844\pi\)
−0.121729 + 0.992563i \(0.538844\pi\)
\(884\) −1560.00 −0.0593535
\(885\) −2160.00 −0.0820425
\(886\) 5208.00 0.197479
\(887\) −5472.00 −0.207138 −0.103569 0.994622i \(-0.533026\pi\)
−0.103569 + 0.994622i \(0.533026\pi\)
\(888\) −2640.00 −0.0997664
\(889\) 42880.0 1.61772
\(890\) 2520.00 0.0949108
\(891\) 1944.00 0.0730937
\(892\) −1072.00 −0.0402390
\(893\) 3264.00 0.122313
\(894\) 10620.0 0.397300
\(895\) 72.0000 0.00268904
\(896\) 2560.00 0.0954504
\(897\) 2808.00 0.104522
\(898\) −26412.0 −0.981492
\(899\) −46248.0 −1.71575
\(900\) −3204.00 −0.118667
\(901\) 22140.0 0.818635
\(902\) −6048.00 −0.223255
\(903\) −9840.00 −0.362630
\(904\) −2928.00 −0.107725
\(905\) 28308.0 1.03977
\(906\) 5928.00 0.217378
\(907\) −7180.00 −0.262853 −0.131427 0.991326i \(-0.541956\pi\)
−0.131427 + 0.991326i \(0.541956\pi\)
\(908\) 7200.00 0.263150
\(909\) 7182.00 0.262059
\(910\) 3120.00 0.113656
\(911\) 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
\(915\) −11052.0 −0.399309
\(916\) 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 \(0.682716\pi\)
\(920\) −3456.00 −0.123849
\(921\) −20472.0 −0.732438
\(922\) 33372.0 1.19203
\(923\) 1716.00 0.0611948
\(924\) −5760.00 −0.205076
\(925\) −9790.00 −0.347993
\(926\) 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 \(0.510828\pi\)
\(930\) −5904.00 −0.208172
\(931\) −912.000 −0.0321048
\(932\) 11304.0 0.397291
\(933\) 26280.0 0.922153
\(934\) 37080.0 1.29903
\(935\) −4320.00 −0.151101
\(936\) 936.000 0.0326860
\(937\) 3962.00 0.138135 0.0690677 0.997612i \(-0.477998\pi\)
0.0690677 + 0.997612i \(0.477998\pi\)
\(938\) 33920.0 1.18073
\(939\) −11886.0 −0.413083
\(940\) −4896.00 −0.169883
\(941\) −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
\(945\) −3240.00 −0.111531
\(946\) 7872.00 0.270551
\(947\) 4848.00 0.166356 0.0831778 0.996535i \(-0.473493\pi\)
0.0831778 + 0.996535i \(0.473493\pi\)
\(948\) 13152.0 0.450588
\(949\) 2834.00 0.0969394
\(950\) 2848.00 0.0972645
\(951\) −21258.0 −0.724856
\(952\) −4800.00 −0.163413
\(953\) 762.000 0.0259009 0.0129505 0.999916i \(-0.495878\pi\)
0.0129505 + 0.999916i \(0.495878\pi\)
\(954\) −13284.0 −0.450823
\(955\) −8208.00 −0.278120
\(956\) −7248.00 −0.245206
\(957\) 20304.0 0.685826
\(958\) 12360.0 0.416841
\(959\) 55080.0 1.85467
\(960\) −1152.00 −0.0387298
\(961\) −2895.00 −0.0971770
\(962\) 2860.00 0.0958525
\(963\) 108.000 0.00361397
\(964\) −6328.00 −0.211422
\(965\) −19860.0 −0.662504
\(966\) 8640.00 0.287772
\(967\) 35804.0 1.19067 0.595336 0.803477i \(-0.297019\pi\)
0.595336 + 0.803477i \(0.297019\pi\)
\(968\) −6040.00 −0.200551
\(969\) −1440.00 −0.0477394
\(970\) −20712.0 −0.685590
\(971\) −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
\(975\) 3471.00 0.114011
\(976\) 9824.00 0.322191
\(977\) −28710.0 −0.940137 −0.470069 0.882630i \(-0.655771\pi\)
−0.470069 + 0.882630i \(0.655771\pi\)
\(978\) −8976.00 −0.293477
\(979\) 5040.00 0.164534
\(980\) 1368.00 0.0445910
\(981\) −16506.0 −0.537203
\(982\) 3816.00 0.124006
\(983\) −49524.0 −1.60689 −0.803444 0.595381i \(-0.797001\pi\)
−0.803444 + 0.595381i \(0.797001\pi\)
\(984\) 3024.00 0.0979691
\(985\) 18756.0 0.606717
\(986\) 16920.0 0.546493
\(987\) 12240.0 0.394735
\(988\) −832.000 −0.0267909
\(989\) −11808.0 −0.379649
\(990\) 2592.00 0.0832113
\(991\) 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
\(995\) 27984.0 0.891610
\(996\) −6624.00 −0.210732
\(997\) 18398.0 0.584424 0.292212 0.956354i \(-0.405609\pi\)
0.292212 + 0.956354i \(0.405609\pi\)
\(998\) −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 78.4.a.e.1.1 1
3.2 odd 2 234.4.a.b.1.1 1
4.3 odd 2 624.4.a.i.1.1 1
5.4 even 2 1950.4.a.c.1.1 1
8.3 odd 2 2496.4.a.b.1.1 1
8.5 even 2 2496.4.a.k.1.1 1
12.11 even 2 1872.4.a.e.1.1 1
13.5 odd 4 1014.4.b.c.337.1 2
13.8 odd 4 1014.4.b.c.337.2 2
13.12 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 1.1 even 1 trivial
234.4.a.b.1.1 1 3.2 odd 2
624.4.a.i.1.1 1 4.3 odd 2
1014.4.a.b.1.1 1 13.12 even 2
1014.4.b.c.337.1 2 13.5 odd 4
1014.4.b.c.337.2 2 13.8 odd 4
1872.4.a.e.1.1 1 12.11 even 2
1950.4.a.c.1.1 1 5.4 even 2
2496.4.a.b.1.1 1 8.3 odd 2
2496.4.a.k.1.1 1 8.5 even 2