Properties

Label 27.6.a.c.1.2
Level $27$
Weight $6$
Character 27.1
Self dual yes
Analytic conductor $4.330$
Analytic rank $0$
Dimension $2$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [27,6,Mod(1,27)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(27, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("27.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 27.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: \(4.33036313495\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.44949\) of defining polynomial
Character \(\chi\) \(=\) 27.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+7.34847 q^{2} +22.0000 q^{4} +58.7878 q^{5} +167.000 q^{7} -73.4847 q^{8} +432.000 q^{10} -764.241 q^{11} -235.000 q^{13} +1227.19 q^{14} -1244.00 q^{16} +176.363 q^{17} +1361.00 q^{19} +1293.33 q^{20} -5616.00 q^{22} -2410.30 q^{23} +331.000 q^{25} -1726.89 q^{26} +3674.00 q^{28} +470.302 q^{29} +3500.00 q^{31} -6789.99 q^{32} +1296.00 q^{34} +9817.55 q^{35} +13115.0 q^{37} +10001.3 q^{38} -4320.00 q^{40} +9406.04 q^{41} +104.000 q^{43} -16813.3 q^{44} -17712.0 q^{46} -20516.9 q^{47} +11082.0 q^{49} +2432.34 q^{50} -5170.00 q^{52} -1058.18 q^{53} -44928.0 q^{55} -12271.9 q^{56} +3456.00 q^{58} +30746.0 q^{59} -7393.00 q^{61} +25719.6 q^{62} -10088.0 q^{64} -13815.1 q^{65} +38861.0 q^{67} +3879.99 q^{68} +72144.0 q^{70} -2469.09 q^{71} +5465.00 q^{73} +96375.2 q^{74} +29942.0 q^{76} -127628. q^{77} -82903.0 q^{79} -73132.0 q^{80} +69120.0 q^{82} -13286.0 q^{83} +10368.0 q^{85} +764.241 q^{86} +56160.0 q^{88} +89768.9 q^{89} -39245.0 q^{91} -53026.6 q^{92} -150768. q^{94} +80010.1 q^{95} -49603.0 q^{97} +81435.7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 44 q^{4} + 334 q^{7} + 864 q^{10} - 470 q^{13} - 2488 q^{16} + 2722 q^{19} - 11232 q^{22} + 662 q^{25} + 7348 q^{28} + 7000 q^{31} + 2592 q^{34} + 26230 q^{37} - 8640 q^{40} + 208 q^{43} - 35424 q^{46}+ \cdots - 99206 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 7.34847 1.29904 0.649519 0.760345i \(-0.274970\pi\)
0.649519 + 0.760345i \(0.274970\pi\)
\(3\) 0 0
\(4\) 22.0000 0.687500
\(5\) 58.7878 1.05163 0.525814 0.850600i \(-0.323761\pi\)
0.525814 + 0.850600i \(0.323761\pi\)
\(6\) 0 0
\(7\) 167.000 1.28816 0.644082 0.764956i \(-0.277239\pi\)
0.644082 + 0.764956i \(0.277239\pi\)
\(8\) −73.4847 −0.405949
\(9\) 0 0
\(10\) 432.000 1.36610
\(11\) −764.241 −1.90436 −0.952179 0.305541i \(-0.901163\pi\)
−0.952179 + 0.305541i \(0.901163\pi\)
\(12\) 0 0
\(13\) −235.000 −0.385664 −0.192832 0.981232i \(-0.561767\pi\)
−0.192832 + 0.981232i \(0.561767\pi\)
\(14\) 1227.19 1.67337
\(15\) 0 0
\(16\) −1244.00 −1.21484
\(17\) 176.363 0.148008 0.0740041 0.997258i \(-0.476422\pi\)
0.0740041 + 0.997258i \(0.476422\pi\)
\(18\) 0 0
\(19\) 1361.00 0.864916 0.432458 0.901654i \(-0.357646\pi\)
0.432458 + 0.901654i \(0.357646\pi\)
\(20\) 1293.33 0.722994
\(21\) 0 0
\(22\) −5616.00 −2.47383
\(23\) −2410.30 −0.950060 −0.475030 0.879970i \(-0.657563\pi\)
−0.475030 + 0.879970i \(0.657563\pi\)
\(24\) 0 0
\(25\) 331.000 0.105920
\(26\) −1726.89 −0.500993
\(27\) 0 0
\(28\) 3674.00 0.885613
\(29\) 470.302 0.103844 0.0519221 0.998651i \(-0.483465\pi\)
0.0519221 + 0.998651i \(0.483465\pi\)
\(30\) 0 0
\(31\) 3500.00 0.654130 0.327065 0.945002i \(-0.393940\pi\)
0.327065 + 0.945002i \(0.393940\pi\)
\(32\) −6789.99 −1.17218
\(33\) 0 0
\(34\) 1296.00 0.192268
\(35\) 9817.55 1.35467
\(36\) 0 0
\(37\) 13115.0 1.57494 0.787470 0.616353i \(-0.211391\pi\)
0.787470 + 0.616353i \(0.211391\pi\)
\(38\) 10001.3 1.12356
\(39\) 0 0
\(40\) −4320.00 −0.426907
\(41\) 9406.04 0.873871 0.436935 0.899493i \(-0.356064\pi\)
0.436935 + 0.899493i \(0.356064\pi\)
\(42\) 0 0
\(43\) 104.000 0.00857753 0.00428876 0.999991i \(-0.498635\pi\)
0.00428876 + 0.999991i \(0.498635\pi\)
\(44\) −16813.3 −1.30925
\(45\) 0 0
\(46\) −17712.0 −1.23416
\(47\) −20516.9 −1.35478 −0.677388 0.735626i \(-0.736888\pi\)
−0.677388 + 0.735626i \(0.736888\pi\)
\(48\) 0 0
\(49\) 11082.0 0.659368
\(50\) 2432.34 0.137594
\(51\) 0 0
\(52\) −5170.00 −0.265144
\(53\) −1058.18 −0.0517452 −0.0258726 0.999665i \(-0.508236\pi\)
−0.0258726 + 0.999665i \(0.508236\pi\)
\(54\) 0 0
\(55\) −44928.0 −2.00267
\(56\) −12271.9 −0.522930
\(57\) 0 0
\(58\) 3456.00 0.134897
\(59\) 30746.0 1.14990 0.574948 0.818190i \(-0.305022\pi\)
0.574948 + 0.818190i \(0.305022\pi\)
\(60\) 0 0
\(61\) −7393.00 −0.254388 −0.127194 0.991878i \(-0.540597\pi\)
−0.127194 + 0.991878i \(0.540597\pi\)
\(62\) 25719.6 0.849739
\(63\) 0 0
\(64\) −10088.0 −0.307861
\(65\) −13815.1 −0.405575
\(66\) 0 0
\(67\) 38861.0 1.05761 0.528807 0.848742i \(-0.322640\pi\)
0.528807 + 0.848742i \(0.322640\pi\)
\(68\) 3879.99 0.101756
\(69\) 0 0
\(70\) 72144.0 1.75977
\(71\) −2469.09 −0.0581287 −0.0290643 0.999578i \(-0.509253\pi\)
−0.0290643 + 0.999578i \(0.509253\pi\)
\(72\) 0 0
\(73\) 5465.00 0.120028 0.0600141 0.998198i \(-0.480885\pi\)
0.0600141 + 0.998198i \(0.480885\pi\)
\(74\) 96375.2 2.04591
\(75\) 0 0
\(76\) 29942.0 0.594630
\(77\) −127628. −2.45313
\(78\) 0 0
\(79\) −82903.0 −1.49452 −0.747261 0.664530i \(-0.768632\pi\)
−0.747261 + 0.664530i \(0.768632\pi\)
\(80\) −73132.0 −1.27756
\(81\) 0 0
\(82\) 69120.0 1.13519
\(83\) −13286.0 −0.211690 −0.105845 0.994383i \(-0.533755\pi\)
−0.105845 + 0.994383i \(0.533755\pi\)
\(84\) 0 0
\(85\) 10368.0 0.155649
\(86\) 764.241 0.0111425
\(87\) 0 0
\(88\) 56160.0 0.773073
\(89\) 89768.9 1.20130 0.600649 0.799513i \(-0.294909\pi\)
0.600649 + 0.799513i \(0.294909\pi\)
\(90\) 0 0
\(91\) −39245.0 −0.496799
\(92\) −53026.6 −0.653166
\(93\) 0 0
\(94\) −150768. −1.75991
\(95\) 80010.1 0.909570
\(96\) 0 0
\(97\) −49603.0 −0.535277 −0.267639 0.963519i \(-0.586243\pi\)
−0.267639 + 0.963519i \(0.586243\pi\)
\(98\) 81435.7 0.856544
\(99\) 0 0
\(100\) 7282.00 0.0728200
\(101\) 138739. 1.35330 0.676652 0.736303i \(-0.263430\pi\)
0.676652 + 0.736303i \(0.263430\pi\)
\(102\) 0 0
\(103\) −94933.0 −0.881707 −0.440853 0.897579i \(-0.645324\pi\)
−0.440853 + 0.897579i \(0.645324\pi\)
\(104\) 17268.9 0.156560
\(105\) 0 0
\(106\) −7776.00 −0.0672189
\(107\) −168780. −1.42515 −0.712575 0.701596i \(-0.752471\pi\)
−0.712575 + 0.701596i \(0.752471\pi\)
\(108\) 0 0
\(109\) 124850. 1.00652 0.503260 0.864135i \(-0.332134\pi\)
0.503260 + 0.864135i \(0.332134\pi\)
\(110\) −330152. −2.60155
\(111\) 0 0
\(112\) −207748. −1.56492
\(113\) 172542. 1.27116 0.635578 0.772037i \(-0.280762\pi\)
0.635578 + 0.772037i \(0.280762\pi\)
\(114\) 0 0
\(115\) −141696. −0.999109
\(116\) 10346.6 0.0713928
\(117\) 0 0
\(118\) 225936. 1.49376
\(119\) 29452.7 0.190659
\(120\) 0 0
\(121\) 423013. 2.62658
\(122\) −54327.2 −0.330459
\(123\) 0 0
\(124\) 77000.0 0.449714
\(125\) −164253. −0.940239
\(126\) 0 0
\(127\) −70108.0 −0.385708 −0.192854 0.981227i \(-0.561774\pi\)
−0.192854 + 0.981227i \(0.561774\pi\)
\(128\) 143148. 0.772255
\(129\) 0 0
\(130\) −101520. −0.526858
\(131\) −139445. −0.709943 −0.354971 0.934877i \(-0.615509\pi\)
−0.354971 + 0.934877i \(0.615509\pi\)
\(132\) 0 0
\(133\) 227287. 1.11415
\(134\) 285569. 1.37388
\(135\) 0 0
\(136\) −12960.0 −0.0600838
\(137\) 9464.83 0.0430835 0.0215418 0.999768i \(-0.493143\pi\)
0.0215418 + 0.999768i \(0.493143\pi\)
\(138\) 0 0
\(139\) −150913. −0.662506 −0.331253 0.943542i \(-0.607471\pi\)
−0.331253 + 0.943542i \(0.607471\pi\)
\(140\) 215986. 0.931335
\(141\) 0 0
\(142\) −18144.0 −0.0755113
\(143\) 179597. 0.734443
\(144\) 0 0
\(145\) 27648.0 0.109205
\(146\) 40159.4 0.155921
\(147\) 0 0
\(148\) 288530. 1.08277
\(149\) −230801. −0.851670 −0.425835 0.904801i \(-0.640020\pi\)
−0.425835 + 0.904801i \(0.640020\pi\)
\(150\) 0 0
\(151\) 129461. 0.462058 0.231029 0.972947i \(-0.425791\pi\)
0.231029 + 0.972947i \(0.425791\pi\)
\(152\) −100013. −0.351112
\(153\) 0 0
\(154\) −937872. −3.18670
\(155\) 205757. 0.687901
\(156\) 0 0
\(157\) 257822. 0.834778 0.417389 0.908728i \(-0.362945\pi\)
0.417389 + 0.908728i \(0.362945\pi\)
\(158\) −609210. −1.94144
\(159\) 0 0
\(160\) −399168. −1.23270
\(161\) −402520. −1.22383
\(162\) 0 0
\(163\) −131569. −0.387869 −0.193934 0.981015i \(-0.562125\pi\)
−0.193934 + 0.981015i \(0.562125\pi\)
\(164\) 206933. 0.600786
\(165\) 0 0
\(166\) −97632.0 −0.274993
\(167\) −257079. −0.713305 −0.356652 0.934237i \(-0.616082\pi\)
−0.356652 + 0.934237i \(0.616082\pi\)
\(168\) 0 0
\(169\) −316068. −0.851263
\(170\) 76188.9 0.202195
\(171\) 0 0
\(172\) 2288.00 0.00589705
\(173\) −167428. −0.425316 −0.212658 0.977127i \(-0.568212\pi\)
−0.212658 + 0.977127i \(0.568212\pi\)
\(174\) 0 0
\(175\) 55277.0 0.136442
\(176\) 950716. 2.31350
\(177\) 0 0
\(178\) 659664. 1.56053
\(179\) 389057. 0.907572 0.453786 0.891111i \(-0.350073\pi\)
0.453786 + 0.891111i \(0.350073\pi\)
\(180\) 0 0
\(181\) 165305. 0.375050 0.187525 0.982260i \(-0.439953\pi\)
0.187525 + 0.982260i \(0.439953\pi\)
\(182\) −288391. −0.645361
\(183\) 0 0
\(184\) 177120. 0.385676
\(185\) 771001. 1.65625
\(186\) 0 0
\(187\) −134784. −0.281861
\(188\) −451372. −0.931409
\(189\) 0 0
\(190\) 587952. 1.18157
\(191\) −343967. −0.682234 −0.341117 0.940021i \(-0.610805\pi\)
−0.341117 + 0.940021i \(0.610805\pi\)
\(192\) 0 0
\(193\) −251785. −0.486560 −0.243280 0.969956i \(-0.578223\pi\)
−0.243280 + 0.969956i \(0.578223\pi\)
\(194\) −364506. −0.695345
\(195\) 0 0
\(196\) 243804. 0.453316
\(197\) 820971. 1.50717 0.753585 0.657350i \(-0.228323\pi\)
0.753585 + 0.657350i \(0.228323\pi\)
\(198\) 0 0
\(199\) −336157. −0.601741 −0.300870 0.953665i \(-0.597277\pi\)
−0.300870 + 0.953665i \(0.597277\pi\)
\(200\) −24323.4 −0.0429982
\(201\) 0 0
\(202\) 1.01952e6 1.75799
\(203\) 78540.4 0.133768
\(204\) 0 0
\(205\) 552960. 0.918986
\(206\) −697612. −1.14537
\(207\) 0 0
\(208\) 292340. 0.468522
\(209\) −1.04013e6 −1.64711
\(210\) 0 0
\(211\) −821557. −1.27037 −0.635187 0.772358i \(-0.719077\pi\)
−0.635187 + 0.772358i \(0.719077\pi\)
\(212\) −23280.0 −0.0355748
\(213\) 0 0
\(214\) −1.24027e6 −1.85132
\(215\) 6113.93 0.00902036
\(216\) 0 0
\(217\) 584500. 0.842627
\(218\) 917456. 1.30751
\(219\) 0 0
\(220\) −988416. −1.37684
\(221\) −41445.4 −0.0570815
\(222\) 0 0
\(223\) 670388. 0.902743 0.451371 0.892336i \(-0.350935\pi\)
0.451371 + 0.892336i \(0.350935\pi\)
\(224\) −1.13393e6 −1.50996
\(225\) 0 0
\(226\) 1.26792e6 1.65128
\(227\) 860065. 1.10781 0.553907 0.832579i \(-0.313136\pi\)
0.553907 + 0.832579i \(0.313136\pi\)
\(228\) 0 0
\(229\) −1.43277e6 −1.80546 −0.902732 0.430203i \(-0.858442\pi\)
−0.902732 + 0.430203i \(0.858442\pi\)
\(230\) −1.04125e6 −1.29788
\(231\) 0 0
\(232\) −34560.0 −0.0421555
\(233\) 407046. 0.491195 0.245598 0.969372i \(-0.421016\pi\)
0.245598 + 0.969372i \(0.421016\pi\)
\(234\) 0 0
\(235\) −1.20614e6 −1.42472
\(236\) 676412. 0.790553
\(237\) 0 0
\(238\) 216432. 0.247673
\(239\) 115577. 0.130881 0.0654404 0.997856i \(-0.479155\pi\)
0.0654404 + 0.997856i \(0.479155\pi\)
\(240\) 0 0
\(241\) −1.55192e6 −1.72118 −0.860590 0.509298i \(-0.829905\pi\)
−0.860590 + 0.509298i \(0.829905\pi\)
\(242\) 3.10850e6 3.41202
\(243\) 0 0
\(244\) −162646. −0.174892
\(245\) 651486. 0.693410
\(246\) 0 0
\(247\) −319835. −0.333567
\(248\) −257196. −0.265544
\(249\) 0 0
\(250\) −1.20701e6 −1.22141
\(251\) −1.70226e6 −1.70546 −0.852729 0.522353i \(-0.825054\pi\)
−0.852729 + 0.522353i \(0.825054\pi\)
\(252\) 0 0
\(253\) 1.84205e6 1.80925
\(254\) −515186. −0.501049
\(255\) 0 0
\(256\) 1.37474e6 1.31105
\(257\) −1.38210e6 −1.30529 −0.652645 0.757664i \(-0.726340\pi\)
−0.652645 + 0.757664i \(0.726340\pi\)
\(258\) 0 0
\(259\) 2.19020e6 2.02878
\(260\) −303933. −0.278833
\(261\) 0 0
\(262\) −1.02470e6 −0.922243
\(263\) −1.61678e6 −1.44132 −0.720662 0.693286i \(-0.756162\pi\)
−0.720662 + 0.693286i \(0.756162\pi\)
\(264\) 0 0
\(265\) −62208.0 −0.0544166
\(266\) 1.67021e6 1.44733
\(267\) 0 0
\(268\) 854942. 0.727109
\(269\) −406870. −0.342827 −0.171413 0.985199i \(-0.554833\pi\)
−0.171413 + 0.985199i \(0.554833\pi\)
\(270\) 0 0
\(271\) 246053. 0.203519 0.101760 0.994809i \(-0.467553\pi\)
0.101760 + 0.994809i \(0.467553\pi\)
\(272\) −219396. −0.179807
\(273\) 0 0
\(274\) 69552.0 0.0559672
\(275\) −252964. −0.201710
\(276\) 0 0
\(277\) −347350. −0.271999 −0.136000 0.990709i \(-0.543425\pi\)
−0.136000 + 0.990709i \(0.543425\pi\)
\(278\) −1.10898e6 −0.860620
\(279\) 0 0
\(280\) −721440. −0.549927
\(281\) 611393. 0.461907 0.230953 0.972965i \(-0.425815\pi\)
0.230953 + 0.972965i \(0.425815\pi\)
\(282\) 0 0
\(283\) 2.05827e6 1.52770 0.763848 0.645397i \(-0.223308\pi\)
0.763848 + 0.645397i \(0.223308\pi\)
\(284\) −54319.9 −0.0399635
\(285\) 0 0
\(286\) 1.31976e6 0.954069
\(287\) 1.57081e6 1.12569
\(288\) 0 0
\(289\) −1.38875e6 −0.978094
\(290\) 203170. 0.141862
\(291\) 0 0
\(292\) 120230. 0.0825193
\(293\) 2.26844e6 1.54369 0.771843 0.635813i \(-0.219335\pi\)
0.771843 + 0.635813i \(0.219335\pi\)
\(294\) 0 0
\(295\) 1.80749e6 1.20926
\(296\) −963752. −0.639346
\(297\) 0 0
\(298\) −1.69603e6 −1.10635
\(299\) 566420. 0.366404
\(300\) 0 0
\(301\) 17368.0 0.0110493
\(302\) 951340. 0.600231
\(303\) 0 0
\(304\) −1.69308e6 −1.05074
\(305\) −434618. −0.267521
\(306\) 0 0
\(307\) 902576. 0.546560 0.273280 0.961935i \(-0.411891\pi\)
0.273280 + 0.961935i \(0.411891\pi\)
\(308\) −2.80782e6 −1.68652
\(309\) 0 0
\(310\) 1.51200e6 0.893609
\(311\) 1.45964e6 0.855746 0.427873 0.903839i \(-0.359263\pi\)
0.427873 + 0.903839i \(0.359263\pi\)
\(312\) 0 0
\(313\) −2.75992e6 −1.59234 −0.796169 0.605075i \(-0.793143\pi\)
−0.796169 + 0.605075i \(0.793143\pi\)
\(314\) 1.89460e6 1.08441
\(315\) 0 0
\(316\) −1.82387e6 −1.02748
\(317\) 17048.4 0.00952877 0.00476438 0.999989i \(-0.498483\pi\)
0.00476438 + 0.999989i \(0.498483\pi\)
\(318\) 0 0
\(319\) −359424. −0.197756
\(320\) −593051. −0.323755
\(321\) 0 0
\(322\) −2.95790e6 −1.58981
\(323\) 240030. 0.128015
\(324\) 0 0
\(325\) −77785.0 −0.0408496
\(326\) −966831. −0.503856
\(327\) 0 0
\(328\) −691200. −0.354747
\(329\) −3.42633e6 −1.74518
\(330\) 0 0
\(331\) 3.37236e6 1.69186 0.845929 0.533296i \(-0.179047\pi\)
0.845929 + 0.533296i \(0.179047\pi\)
\(332\) −292293. −0.145537
\(333\) 0 0
\(334\) −1.88914e6 −0.926610
\(335\) 2.28455e6 1.11222
\(336\) 0 0
\(337\) −360523. −0.172925 −0.0864626 0.996255i \(-0.527556\pi\)
−0.0864626 + 0.996255i \(0.527556\pi\)
\(338\) −2.32262e6 −1.10582
\(339\) 0 0
\(340\) 228096. 0.107009
\(341\) −2.67484e6 −1.24570
\(342\) 0 0
\(343\) −956075. −0.438790
\(344\) −7642.41 −0.00348204
\(345\) 0 0
\(346\) −1.23034e6 −0.552502
\(347\) −1.44206e6 −0.642926 −0.321463 0.946922i \(-0.604175\pi\)
−0.321463 + 0.946922i \(0.604175\pi\)
\(348\) 0 0
\(349\) 677579. 0.297781 0.148890 0.988854i \(-0.452430\pi\)
0.148890 + 0.988854i \(0.452430\pi\)
\(350\) 406201. 0.177244
\(351\) 0 0
\(352\) 5.18918e6 2.23225
\(353\) 3.84484e6 1.64226 0.821129 0.570743i \(-0.193345\pi\)
0.821129 + 0.570743i \(0.193345\pi\)
\(354\) 0 0
\(355\) −145152. −0.0611297
\(356\) 1.97492e6 0.825893
\(357\) 0 0
\(358\) 2.85898e6 1.17897
\(359\) 1.36699e6 0.559796 0.279898 0.960030i \(-0.409699\pi\)
0.279898 + 0.960030i \(0.409699\pi\)
\(360\) 0 0
\(361\) −623778. −0.251920
\(362\) 1.21474e6 0.487205
\(363\) 0 0
\(364\) −863390. −0.341549
\(365\) 321275. 0.126225
\(366\) 0 0
\(367\) −1.16951e6 −0.453251 −0.226625 0.973982i \(-0.572769\pi\)
−0.226625 + 0.973982i \(0.572769\pi\)
\(368\) 2.99841e6 1.15417
\(369\) 0 0
\(370\) 5.66568e6 2.15153
\(371\) −176716. −0.0666563
\(372\) 0 0
\(373\) 2.52666e6 0.940318 0.470159 0.882582i \(-0.344197\pi\)
0.470159 + 0.882582i \(0.344197\pi\)
\(374\) −990456. −0.366148
\(375\) 0 0
\(376\) 1.50768e6 0.549971
\(377\) −110521. −0.0400490
\(378\) 0 0
\(379\) 219269. 0.0784114 0.0392057 0.999231i \(-0.487517\pi\)
0.0392057 + 0.999231i \(0.487517\pi\)
\(380\) 1.76022e6 0.625329
\(381\) 0 0
\(382\) −2.52763e6 −0.886248
\(383\) −59728.4 −0.0208058 −0.0104029 0.999946i \(-0.503311\pi\)
−0.0104029 + 0.999946i \(0.503311\pi\)
\(384\) 0 0
\(385\) −7.50298e6 −2.57977
\(386\) −1.85023e6 −0.632060
\(387\) 0 0
\(388\) −1.09127e6 −0.368003
\(389\) −2.70653e6 −0.906857 −0.453428 0.891293i \(-0.649799\pi\)
−0.453428 + 0.891293i \(0.649799\pi\)
\(390\) 0 0
\(391\) −425088. −0.140617
\(392\) −814357. −0.267670
\(393\) 0 0
\(394\) 6.03288e6 1.95787
\(395\) −4.87368e6 −1.57168
\(396\) 0 0
\(397\) 4.43128e6 1.41108 0.705542 0.708668i \(-0.250704\pi\)
0.705542 + 0.708668i \(0.250704\pi\)
\(398\) −2.47024e6 −0.781684
\(399\) 0 0
\(400\) −411764. −0.128676
\(401\) 2.37585e6 0.737832 0.368916 0.929463i \(-0.379729\pi\)
0.368916 + 0.929463i \(0.379729\pi\)
\(402\) 0 0
\(403\) −822500. −0.252274
\(404\) 3.05226e6 0.930397
\(405\) 0 0
\(406\) 577152. 0.173770
\(407\) −1.00230e7 −2.99925
\(408\) 0 0
\(409\) −1.71805e6 −0.507840 −0.253920 0.967225i \(-0.581720\pi\)
−0.253920 + 0.967225i \(0.581720\pi\)
\(410\) 4.06341e6 1.19380
\(411\) 0 0
\(412\) −2.08853e6 −0.606173
\(413\) 5.13458e6 1.48126
\(414\) 0 0
\(415\) −781056. −0.222619
\(416\) 1.59565e6 0.452068
\(417\) 0 0
\(418\) −7.64338e6 −2.13966
\(419\) 5.34633e6 1.48772 0.743860 0.668336i \(-0.232993\pi\)
0.743860 + 0.668336i \(0.232993\pi\)
\(420\) 0 0
\(421\) −5.68202e6 −1.56242 −0.781209 0.624269i \(-0.785397\pi\)
−0.781209 + 0.624269i \(0.785397\pi\)
\(422\) −6.03719e6 −1.65026
\(423\) 0 0
\(424\) 77760.0 0.0210059
\(425\) 58376.2 0.0156770
\(426\) 0 0
\(427\) −1.23463e6 −0.327693
\(428\) −3.71315e6 −0.979791
\(429\) 0 0
\(430\) 44928.0 0.0117178
\(431\) −2.29890e6 −0.596109 −0.298055 0.954549i \(-0.596338\pi\)
−0.298055 + 0.954549i \(0.596338\pi\)
\(432\) 0 0
\(433\) −4.72608e6 −1.21138 −0.605691 0.795700i \(-0.707103\pi\)
−0.605691 + 0.795700i \(0.707103\pi\)
\(434\) 4.29518e6 1.09460
\(435\) 0 0
\(436\) 2.74670e6 0.691982
\(437\) −3.28042e6 −0.821723
\(438\) 0 0
\(439\) 6.12223e6 1.51617 0.758086 0.652155i \(-0.226135\pi\)
0.758086 + 0.652155i \(0.226135\pi\)
\(440\) 3.30152e6 0.812985
\(441\) 0 0
\(442\) −304560. −0.0741510
\(443\) 6.68429e6 1.61825 0.809125 0.587636i \(-0.199941\pi\)
0.809125 + 0.587636i \(0.199941\pi\)
\(444\) 0 0
\(445\) 5.27731e6 1.26332
\(446\) 4.92633e6 1.17270
\(447\) 0 0
\(448\) −1.68470e6 −0.396576
\(449\) 5.01701e6 1.17443 0.587217 0.809429i \(-0.300223\pi\)
0.587217 + 0.809429i \(0.300223\pi\)
\(450\) 0 0
\(451\) −7.18848e6 −1.66416
\(452\) 3.79593e6 0.873920
\(453\) 0 0
\(454\) 6.32016e6 1.43909
\(455\) −2.30713e6 −0.522448
\(456\) 0 0
\(457\) 683222. 0.153028 0.0765141 0.997069i \(-0.475621\pi\)
0.0765141 + 0.997069i \(0.475621\pi\)
\(458\) −1.05287e7 −2.34537
\(459\) 0 0
\(460\) −3.11731e6 −0.686888
\(461\) −5.04123e6 −1.10480 −0.552400 0.833579i \(-0.686288\pi\)
−0.552400 + 0.833579i \(0.686288\pi\)
\(462\) 0 0
\(463\) 5.04086e6 1.09283 0.546415 0.837515i \(-0.315992\pi\)
0.546415 + 0.837515i \(0.315992\pi\)
\(464\) −585056. −0.126154
\(465\) 0 0
\(466\) 2.99117e6 0.638081
\(467\) 8.12347e6 1.72365 0.861825 0.507205i \(-0.169321\pi\)
0.861825 + 0.507205i \(0.169321\pi\)
\(468\) 0 0
\(469\) 6.48979e6 1.36238
\(470\) −8.86331e6 −1.85077
\(471\) 0 0
\(472\) −2.25936e6 −0.466800
\(473\) −79481.0 −0.0163347
\(474\) 0 0
\(475\) 450491. 0.0916119
\(476\) 647959. 0.131078
\(477\) 0 0
\(478\) 849312. 0.170019
\(479\) 6.60739e6 1.31580 0.657902 0.753104i \(-0.271444\pi\)
0.657902 + 0.753104i \(0.271444\pi\)
\(480\) 0 0
\(481\) −3.08202e6 −0.607398
\(482\) −1.14042e7 −2.23588
\(483\) 0 0
\(484\) 9.30629e6 1.80577
\(485\) −2.91605e6 −0.562912
\(486\) 0 0
\(487\) −2.49806e6 −0.477289 −0.238644 0.971107i \(-0.576703\pi\)
−0.238644 + 0.971107i \(0.576703\pi\)
\(488\) 543272. 0.103269
\(489\) 0 0
\(490\) 4.78742e6 0.900765
\(491\) −5.70141e6 −1.06728 −0.533640 0.845711i \(-0.679176\pi\)
−0.533640 + 0.845711i \(0.679176\pi\)
\(492\) 0 0
\(493\) 82944.0 0.0153698
\(494\) −2.35030e6 −0.433317
\(495\) 0 0
\(496\) −4.35400e6 −0.794665
\(497\) −412337. −0.0748793
\(498\) 0 0
\(499\) 4.19754e6 0.754646 0.377323 0.926082i \(-0.376845\pi\)
0.377323 + 0.926082i \(0.376845\pi\)
\(500\) −3.61357e6 −0.646414
\(501\) 0 0
\(502\) −1.25090e7 −2.21546
\(503\) 5.37784e6 0.947738 0.473869 0.880595i \(-0.342857\pi\)
0.473869 + 0.880595i \(0.342857\pi\)
\(504\) 0 0
\(505\) 8.15616e6 1.42317
\(506\) 1.35362e7 2.35029
\(507\) 0 0
\(508\) −1.54238e6 −0.265174
\(509\) 4.35623e6 0.745275 0.372637 0.927977i \(-0.378454\pi\)
0.372637 + 0.927977i \(0.378454\pi\)
\(510\) 0 0
\(511\) 912655. 0.154616
\(512\) 5.52146e6 0.930849
\(513\) 0 0
\(514\) −1.01563e7 −1.69562
\(515\) −5.58090e6 −0.927227
\(516\) 0 0
\(517\) 1.56799e7 2.57998
\(518\) 1.60947e7 2.63546
\(519\) 0 0
\(520\) 1.01520e6 0.164643
\(521\) −4.29639e6 −0.693440 −0.346720 0.937969i \(-0.612705\pi\)
−0.346720 + 0.937969i \(0.612705\pi\)
\(522\) 0 0
\(523\) −1.07625e7 −1.72052 −0.860261 0.509854i \(-0.829699\pi\)
−0.860261 + 0.509854i \(0.829699\pi\)
\(524\) −3.06778e6 −0.488086
\(525\) 0 0
\(526\) −1.18809e7 −1.87234
\(527\) 617271. 0.0968166
\(528\) 0 0
\(529\) −626807. −0.0973856
\(530\) −457134. −0.0706893
\(531\) 0 0
\(532\) 5.00031e6 0.765981
\(533\) −2.21042e6 −0.337021
\(534\) 0 0
\(535\) −9.92218e6 −1.49873
\(536\) −2.85569e6 −0.429338
\(537\) 0 0
\(538\) −2.98987e6 −0.445345
\(539\) −8.46932e6 −1.25567
\(540\) 0 0
\(541\) 7.49825e6 1.10146 0.550728 0.834685i \(-0.314350\pi\)
0.550728 + 0.834685i \(0.314350\pi\)
\(542\) 1.80811e6 0.264379
\(543\) 0 0
\(544\) −1.19750e6 −0.173492
\(545\) 7.33965e6 1.05848
\(546\) 0 0
\(547\) −3.63295e6 −0.519148 −0.259574 0.965723i \(-0.583582\pi\)
−0.259574 + 0.965723i \(0.583582\pi\)
\(548\) 208226. 0.0296199
\(549\) 0 0
\(550\) −1.85890e6 −0.262028
\(551\) 640081. 0.0898165
\(552\) 0 0
\(553\) −1.38448e7 −1.92519
\(554\) −2.55249e6 −0.353338
\(555\) 0 0
\(556\) −3.32009e6 −0.455473
\(557\) −7.12137e6 −0.972581 −0.486290 0.873797i \(-0.661650\pi\)
−0.486290 + 0.873797i \(0.661650\pi\)
\(558\) 0 0
\(559\) −24440.0 −0.00330805
\(560\) −1.22130e7 −1.64571
\(561\) 0 0
\(562\) 4.49280e6 0.600035
\(563\) −3.41768e6 −0.454424 −0.227212 0.973845i \(-0.572961\pi\)
−0.227212 + 0.973845i \(0.572961\pi\)
\(564\) 0 0
\(565\) 1.01434e7 1.33678
\(566\) 1.51251e7 1.98453
\(567\) 0 0
\(568\) 181440. 0.0235973
\(569\) −5.43699e6 −0.704008 −0.352004 0.935999i \(-0.614500\pi\)
−0.352004 + 0.935999i \(0.614500\pi\)
\(570\) 0 0
\(571\) −3.92190e6 −0.503391 −0.251696 0.967806i \(-0.580988\pi\)
−0.251696 + 0.967806i \(0.580988\pi\)
\(572\) 3.95112e6 0.504929
\(573\) 0 0
\(574\) 1.15430e7 1.46231
\(575\) −797809. −0.100630
\(576\) 0 0
\(577\) 1.07034e7 1.33839 0.669193 0.743088i \(-0.266640\pi\)
0.669193 + 0.743088i \(0.266640\pi\)
\(578\) −1.02052e7 −1.27058
\(579\) 0 0
\(580\) 608256. 0.0750786
\(581\) −2.21877e6 −0.272691
\(582\) 0 0
\(583\) 808704. 0.0985413
\(584\) −401594. −0.0487253
\(585\) 0 0
\(586\) 1.66696e7 2.00531
\(587\) 2.99424e6 0.358667 0.179333 0.983788i \(-0.442606\pi\)
0.179333 + 0.983788i \(0.442606\pi\)
\(588\) 0 0
\(589\) 4.76350e6 0.565767
\(590\) 1.32823e7 1.57088
\(591\) 0 0
\(592\) −1.63151e7 −1.91331
\(593\) −5.85279e6 −0.683481 −0.341740 0.939794i \(-0.611016\pi\)
−0.341740 + 0.939794i \(0.611016\pi\)
\(594\) 0 0
\(595\) 1.73146e6 0.200502
\(596\) −5.07762e6 −0.585523
\(597\) 0 0
\(598\) 4.16232e6 0.475973
\(599\) −1.93059e6 −0.219848 −0.109924 0.993940i \(-0.535061\pi\)
−0.109924 + 0.993940i \(0.535061\pi\)
\(600\) 0 0
\(601\) −6.64461e6 −0.750384 −0.375192 0.926947i \(-0.622423\pi\)
−0.375192 + 0.926947i \(0.622423\pi\)
\(602\) 127628. 0.0143534
\(603\) 0 0
\(604\) 2.84814e6 0.317665
\(605\) 2.48680e7 2.76218
\(606\) 0 0
\(607\) −6.72545e6 −0.740883 −0.370441 0.928856i \(-0.620794\pi\)
−0.370441 + 0.928856i \(0.620794\pi\)
\(608\) −9.24117e6 −1.01384
\(609\) 0 0
\(610\) −3.19378e6 −0.347520
\(611\) 4.82148e6 0.522489
\(612\) 0 0
\(613\) 3.03643e6 0.326372 0.163186 0.986595i \(-0.447823\pi\)
0.163186 + 0.986595i \(0.447823\pi\)
\(614\) 6.63255e6 0.710002
\(615\) 0 0
\(616\) 9.37872e6 0.995845
\(617\) 5.93504e6 0.627640 0.313820 0.949483i \(-0.398391\pi\)
0.313820 + 0.949483i \(0.398391\pi\)
\(618\) 0 0
\(619\) 1.21004e7 1.26932 0.634661 0.772791i \(-0.281140\pi\)
0.634661 + 0.772791i \(0.281140\pi\)
\(620\) 4.52666e6 0.472932
\(621\) 0 0
\(622\) 1.07261e7 1.11165
\(623\) 1.49914e7 1.54747
\(624\) 0 0
\(625\) −1.06904e7 −1.09470
\(626\) −2.02812e7 −2.06851
\(627\) 0 0
\(628\) 5.67208e6 0.573910
\(629\) 2.31300e6 0.233104
\(630\) 0 0
\(631\) −6.17037e6 −0.616933 −0.308466 0.951235i \(-0.599816\pi\)
−0.308466 + 0.951235i \(0.599816\pi\)
\(632\) 6.09210e6 0.606701
\(633\) 0 0
\(634\) 125280. 0.0123782
\(635\) −4.12149e6 −0.405621
\(636\) 0 0
\(637\) −2.60427e6 −0.254295
\(638\) −2.64122e6 −0.256893
\(639\) 0 0
\(640\) 8.41536e6 0.812125
\(641\) −1.13478e7 −1.09085 −0.545427 0.838158i \(-0.683632\pi\)
−0.545427 + 0.838158i \(0.683632\pi\)
\(642\) 0 0
\(643\) −1.00778e7 −0.961257 −0.480629 0.876924i \(-0.659592\pi\)
−0.480629 + 0.876924i \(0.659592\pi\)
\(644\) −8.85543e6 −0.841386
\(645\) 0 0
\(646\) 1.76386e6 0.166296
\(647\) −7.07287e6 −0.664255 −0.332128 0.943234i \(-0.607767\pi\)
−0.332128 + 0.943234i \(0.607767\pi\)
\(648\) 0 0
\(649\) −2.34973e7 −2.18981
\(650\) −571601. −0.0530651
\(651\) 0 0
\(652\) −2.89452e6 −0.266660
\(653\) −1.84773e7 −1.69573 −0.847865 0.530213i \(-0.822112\pi\)
−0.847865 + 0.530213i \(0.822112\pi\)
\(654\) 0 0
\(655\) −8.19763e6 −0.746595
\(656\) −1.17011e7 −1.06162
\(657\) 0 0
\(658\) −2.51783e7 −2.26705
\(659\) −1.53893e7 −1.38040 −0.690202 0.723616i \(-0.742479\pi\)
−0.690202 + 0.723616i \(0.742479\pi\)
\(660\) 0 0
\(661\) 5.21093e6 0.463886 0.231943 0.972729i \(-0.425492\pi\)
0.231943 + 0.972729i \(0.425492\pi\)
\(662\) 2.47817e7 2.19779
\(663\) 0 0
\(664\) 976320. 0.0859354
\(665\) 1.33617e7 1.17168
\(666\) 0 0
\(667\) −1.13357e6 −0.0986582
\(668\) −5.65573e6 −0.490397
\(669\) 0 0
\(670\) 1.67880e7 1.44481
\(671\) 5.65003e6 0.484445
\(672\) 0 0
\(673\) 1.60404e7 1.36514 0.682569 0.730821i \(-0.260863\pi\)
0.682569 + 0.730821i \(0.260863\pi\)
\(674\) −2.64929e6 −0.224636
\(675\) 0 0
\(676\) −6.95350e6 −0.585243
\(677\) 1.48811e7 1.24785 0.623927 0.781482i \(-0.285536\pi\)
0.623927 + 0.781482i \(0.285536\pi\)
\(678\) 0 0
\(679\) −8.28370e6 −0.689525
\(680\) −761889. −0.0631858
\(681\) 0 0
\(682\) −1.96560e7 −1.61821
\(683\) 5.62881e6 0.461705 0.230853 0.972989i \(-0.425848\pi\)
0.230853 + 0.972989i \(0.425848\pi\)
\(684\) 0 0
\(685\) 556416. 0.0453078
\(686\) −7.02569e6 −0.570005
\(687\) 0 0
\(688\) −129376. −0.0104204
\(689\) 248672. 0.0199563
\(690\) 0 0
\(691\) −1.01916e7 −0.811987 −0.405993 0.913876i \(-0.633074\pi\)
−0.405993 + 0.913876i \(0.633074\pi\)
\(692\) −3.68341e6 −0.292405
\(693\) 0 0
\(694\) −1.05970e7 −0.835185
\(695\) −8.87184e6 −0.696709
\(696\) 0 0
\(697\) 1.65888e6 0.129340
\(698\) 4.97917e6 0.386828
\(699\) 0 0
\(700\) 1.21609e6 0.0938041
\(701\) 2.39524e7 1.84100 0.920501 0.390739i \(-0.127781\pi\)
0.920501 + 0.390739i \(0.127781\pi\)
\(702\) 0 0
\(703\) 1.78495e7 1.36219
\(704\) 7.70966e6 0.586278
\(705\) 0 0
\(706\) 2.82537e7 2.13335
\(707\) 2.31694e7 1.74328
\(708\) 0 0
\(709\) 1.49472e7 1.11672 0.558360 0.829599i \(-0.311431\pi\)
0.558360 + 0.829599i \(0.311431\pi\)
\(710\) −1.06665e6 −0.0794098
\(711\) 0 0
\(712\) −6.59664e6 −0.487666
\(713\) −8.43604e6 −0.621463
\(714\) 0 0
\(715\) 1.05581e7 0.772360
\(716\) 8.55926e6 0.623956
\(717\) 0 0
\(718\) 1.00453e7 0.727196
\(719\) −2.61670e7 −1.88770 −0.943848 0.330380i \(-0.892823\pi\)
−0.943848 + 0.330380i \(0.892823\pi\)
\(720\) 0 0
\(721\) −1.58538e7 −1.13578
\(722\) −4.58381e6 −0.327253
\(723\) 0 0
\(724\) 3.63671e6 0.257847
\(725\) 155670. 0.0109992
\(726\) 0 0
\(727\) −2.91140e6 −0.204299 −0.102149 0.994769i \(-0.532572\pi\)
−0.102149 + 0.994769i \(0.532572\pi\)
\(728\) 2.88391e6 0.201675
\(729\) 0 0
\(730\) 2.36088e6 0.163971
\(731\) 18341.8 0.00126954
\(732\) 0 0
\(733\) 6.78250e6 0.466262 0.233131 0.972445i \(-0.425103\pi\)
0.233131 + 0.972445i \(0.425103\pi\)
\(734\) −8.59410e6 −0.588790
\(735\) 0 0
\(736\) 1.63659e7 1.11364
\(737\) −2.96992e7 −2.01407
\(738\) 0 0
\(739\) 1.52588e7 1.02780 0.513901 0.857850i \(-0.328200\pi\)
0.513901 + 0.857850i \(0.328200\pi\)
\(740\) 1.69620e7 1.13867
\(741\) 0 0
\(742\) −1.29859e6 −0.0865890
\(743\) 1.96863e6 0.130825 0.0654125 0.997858i \(-0.479164\pi\)
0.0654125 + 0.997858i \(0.479164\pi\)
\(744\) 0 0
\(745\) −1.35683e7 −0.895640
\(746\) 1.85671e7 1.22151
\(747\) 0 0
\(748\) −2.96525e6 −0.193779
\(749\) −2.81862e7 −1.83583
\(750\) 0 0
\(751\) −1.62379e7 −1.05058 −0.525290 0.850924i \(-0.676043\pi\)
−0.525290 + 0.850924i \(0.676043\pi\)
\(752\) 2.55231e7 1.64584
\(753\) 0 0
\(754\) −812160. −0.0520251
\(755\) 7.61072e6 0.485913
\(756\) 0 0
\(757\) 9.17096e6 0.581668 0.290834 0.956774i \(-0.406067\pi\)
0.290834 + 0.956774i \(0.406067\pi\)
\(758\) 1.61129e6 0.101859
\(759\) 0 0
\(760\) −5.87952e6 −0.369239
\(761\) −2.31222e7 −1.44733 −0.723666 0.690151i \(-0.757544\pi\)
−0.723666 + 0.690151i \(0.757544\pi\)
\(762\) 0 0
\(763\) 2.08500e7 1.29656
\(764\) −7.56728e6 −0.469036
\(765\) 0 0
\(766\) −438912. −0.0270275
\(767\) −7.22531e6 −0.443474
\(768\) 0 0
\(769\) 9.38318e6 0.572182 0.286091 0.958202i \(-0.407644\pi\)
0.286091 + 0.958202i \(0.407644\pi\)
\(770\) −5.51354e7 −3.35123
\(771\) 0 0
\(772\) −5.53927e6 −0.334510
\(773\) −89239.8 −0.00537168 −0.00268584 0.999996i \(-0.500855\pi\)
−0.00268584 + 0.999996i \(0.500855\pi\)
\(774\) 0 0
\(775\) 1.15850e6 0.0692854
\(776\) 3.64506e6 0.217295
\(777\) 0 0
\(778\) −1.98888e7 −1.17804
\(779\) 1.28016e7 0.755825
\(780\) 0 0
\(781\) 1.88698e6 0.110698
\(782\) −3.12375e6 −0.182666
\(783\) 0 0
\(784\) −1.37860e7 −0.801029
\(785\) 1.51568e7 0.877875
\(786\) 0 0
\(787\) 3.04100e7 1.75017 0.875084 0.483971i \(-0.160806\pi\)
0.875084 + 0.483971i \(0.160806\pi\)
\(788\) 1.80614e7 1.03618
\(789\) 0 0
\(790\) −3.58141e7 −2.04167
\(791\) 2.88145e7 1.63746
\(792\) 0 0
\(793\) 1.73736e6 0.0981083
\(794\) 3.25631e7 1.83305
\(795\) 0 0
\(796\) −7.39545e6 −0.413697
\(797\) 2.44534e6 0.136362 0.0681809 0.997673i \(-0.478281\pi\)
0.0681809 + 0.997673i \(0.478281\pi\)
\(798\) 0 0
\(799\) −3.61843e6 −0.200518
\(800\) −2.24749e6 −0.124157
\(801\) 0 0
\(802\) 1.74588e7 0.958473
\(803\) −4.17658e6 −0.228576
\(804\) 0 0
\(805\) −2.36632e7 −1.28702
\(806\) −6.04412e6 −0.327714
\(807\) 0 0
\(808\) −1.01952e7 −0.549373
\(809\) 2.11075e7 1.13388 0.566938 0.823760i \(-0.308128\pi\)
0.566938 + 0.823760i \(0.308128\pi\)
\(810\) 0 0
\(811\) −1.56012e7 −0.832925 −0.416462 0.909153i \(-0.636730\pi\)
−0.416462 + 0.909153i \(0.636730\pi\)
\(812\) 1.72789e6 0.0919657
\(813\) 0 0
\(814\) −7.36538e7 −3.89614
\(815\) −7.73465e6 −0.407893
\(816\) 0 0
\(817\) 141544. 0.00741885
\(818\) −1.26250e7 −0.659703
\(819\) 0 0
\(820\) 1.21651e7 0.631803
\(821\) 4.04677e6 0.209532 0.104766 0.994497i \(-0.466591\pi\)
0.104766 + 0.994497i \(0.466591\pi\)
\(822\) 0 0
\(823\) 2.21529e7 1.14007 0.570035 0.821620i \(-0.306929\pi\)
0.570035 + 0.821620i \(0.306929\pi\)
\(824\) 6.97612e6 0.357928
\(825\) 0 0
\(826\) 3.77313e7 1.92421
\(827\) 1.72252e7 0.875792 0.437896 0.899026i \(-0.355724\pi\)
0.437896 + 0.899026i \(0.355724\pi\)
\(828\) 0 0
\(829\) −2.31834e7 −1.17163 −0.585816 0.810444i \(-0.699226\pi\)
−0.585816 + 0.810444i \(0.699226\pi\)
\(830\) −5.73957e6 −0.289190
\(831\) 0 0
\(832\) 2.37068e6 0.118731
\(833\) 1.95446e6 0.0975919
\(834\) 0 0
\(835\) −1.51131e7 −0.750131
\(836\) −2.28829e7 −1.13239
\(837\) 0 0
\(838\) 3.92874e7 1.93260
\(839\) 1.07272e7 0.526118 0.263059 0.964780i \(-0.415269\pi\)
0.263059 + 0.964780i \(0.415269\pi\)
\(840\) 0 0
\(841\) −2.02900e7 −0.989216
\(842\) −4.17541e7 −2.02964
\(843\) 0 0
\(844\) −1.80743e7 −0.873382
\(845\) −1.85809e7 −0.895211
\(846\) 0 0
\(847\) 7.06432e7 3.38346
\(848\) 1.31638e6 0.0628623
\(849\) 0 0
\(850\) 428976. 0.0203651
\(851\) −3.16111e7 −1.49629
\(852\) 0 0
\(853\) −421069. −0.0198144 −0.00990719 0.999951i \(-0.503154\pi\)
−0.00990719 + 0.999951i \(0.503154\pi\)
\(854\) −9.07265e6 −0.425686
\(855\) 0 0
\(856\) 1.24027e7 0.578539
\(857\) 2.71932e7 1.26476 0.632381 0.774658i \(-0.282078\pi\)
0.632381 + 0.774658i \(0.282078\pi\)
\(858\) 0 0
\(859\) 2.01529e7 0.931869 0.465934 0.884819i \(-0.345718\pi\)
0.465934 + 0.884819i \(0.345718\pi\)
\(860\) 134506. 0.00620150
\(861\) 0 0
\(862\) −1.68934e7 −0.774369
\(863\) −1.88086e7 −0.859666 −0.429833 0.902908i \(-0.641428\pi\)
−0.429833 + 0.902908i \(0.641428\pi\)
\(864\) 0 0
\(865\) −9.84269e6 −0.447274
\(866\) −3.47294e7 −1.57363
\(867\) 0 0
\(868\) 1.28590e7 0.579306
\(869\) 6.33579e7 2.84611
\(870\) 0 0
\(871\) −9.13234e6 −0.407884
\(872\) −9.17456e6 −0.408596
\(873\) 0 0
\(874\) −2.41060e7 −1.06745
\(875\) −2.74302e7 −1.21118
\(876\) 0 0
\(877\) −2.21143e7 −0.970898 −0.485449 0.874265i \(-0.661344\pi\)
−0.485449 + 0.874265i \(0.661344\pi\)
\(878\) 4.49890e7 1.96956
\(879\) 0 0
\(880\) 5.58904e7 2.43294
\(881\) 1.00370e7 0.435677 0.217838 0.975985i \(-0.430099\pi\)
0.217838 + 0.975985i \(0.430099\pi\)
\(882\) 0 0
\(883\) −2.12042e7 −0.915207 −0.457604 0.889156i \(-0.651292\pi\)
−0.457604 + 0.889156i \(0.651292\pi\)
\(884\) −911798. −0.0392435
\(885\) 0 0
\(886\) 4.91193e7 2.10217
\(887\) −1.73507e7 −0.740472 −0.370236 0.928938i \(-0.620723\pi\)
−0.370236 + 0.928938i \(0.620723\pi\)
\(888\) 0 0
\(889\) −1.17080e7 −0.496855
\(890\) 3.87802e7 1.64110
\(891\) 0 0
\(892\) 1.47485e7 0.620636
\(893\) −2.79235e7 −1.17177
\(894\) 0 0
\(895\) 2.28718e7 0.954427
\(896\) 2.39057e7 0.994792
\(897\) 0 0
\(898\) 3.68673e7 1.52563
\(899\) 1.64606e6 0.0679275
\(900\) 0 0
\(901\) −186624. −0.00765871
\(902\) −5.28243e7 −2.16181
\(903\) 0 0
\(904\) −1.26792e7 −0.516025
\(905\) 9.71791e6 0.394413
\(906\) 0 0
\(907\) −1.02757e7 −0.414756 −0.207378 0.978261i \(-0.566493\pi\)
−0.207378 + 0.978261i \(0.566493\pi\)
\(908\) 1.89214e7 0.761622
\(909\) 0 0
\(910\) −1.69538e7 −0.678679
\(911\) 3.94713e7 1.57574 0.787871 0.615840i \(-0.211183\pi\)
0.787871 + 0.615840i \(0.211183\pi\)
\(912\) 0 0
\(913\) 1.01537e7 0.403133
\(914\) 5.02064e6 0.198789
\(915\) 0 0
\(916\) −3.15210e7 −1.24126
\(917\) −2.32872e7 −0.914523
\(918\) 0 0
\(919\) −1.65850e6 −0.0647779 −0.0323889 0.999475i \(-0.510312\pi\)
−0.0323889 + 0.999475i \(0.510312\pi\)
\(920\) 1.04125e7 0.405588
\(921\) 0 0
\(922\) −3.70453e7 −1.43518
\(923\) 580235. 0.0224181
\(924\) 0 0
\(925\) 4.34106e6 0.166818
\(926\) 3.70426e7 1.41963
\(927\) 0 0
\(928\) −3.19334e6 −0.121724
\(929\) −4.65517e6 −0.176969 −0.0884843 0.996078i \(-0.528202\pi\)
−0.0884843 + 0.996078i \(0.528202\pi\)
\(930\) 0 0
\(931\) 1.50826e7 0.570298
\(932\) 8.95502e6 0.337697
\(933\) 0 0
\(934\) 5.96951e7 2.23909
\(935\) −7.92365e6 −0.296412
\(936\) 0 0
\(937\) −3.86106e7 −1.43667 −0.718337 0.695696i \(-0.755096\pi\)
−0.718337 + 0.695696i \(0.755096\pi\)
\(938\) 4.76900e7 1.76978
\(939\) 0 0
\(940\) −2.65352e7 −0.979495
\(941\) −3.14452e7 −1.15766 −0.578828 0.815449i \(-0.696490\pi\)
−0.578828 + 0.815449i \(0.696490\pi\)
\(942\) 0 0
\(943\) −2.26714e7 −0.830230
\(944\) −3.82480e7 −1.39694
\(945\) 0 0
\(946\) −584064. −0.0212194
\(947\) −9.34273e6 −0.338531 −0.169266 0.985570i \(-0.554140\pi\)
−0.169266 + 0.985570i \(0.554140\pi\)
\(948\) 0 0
\(949\) −1.28428e6 −0.0462906
\(950\) 3.31042e6 0.119007
\(951\) 0 0
\(952\) −2.16432e6 −0.0773979
\(953\) −4.89849e6 −0.174715 −0.0873574 0.996177i \(-0.527842\pi\)
−0.0873574 + 0.996177i \(0.527842\pi\)
\(954\) 0 0
\(955\) −2.02211e7 −0.717456
\(956\) 2.54269e6 0.0899805
\(957\) 0 0
\(958\) 4.85542e7 1.70928
\(959\) 1.58063e6 0.0554987
\(960\) 0 0
\(961\) −1.63792e7 −0.572114
\(962\) −2.26482e7 −0.789033
\(963\) 0 0
\(964\) −3.41422e7 −1.18331
\(965\) −1.48019e7 −0.511680
\(966\) 0 0
\(967\) −1.13925e7 −0.391791 −0.195896 0.980625i \(-0.562761\pi\)
−0.195896 + 0.980625i \(0.562761\pi\)
\(968\) −3.10850e7 −1.06626
\(969\) 0 0
\(970\) −2.14285e7 −0.731244
\(971\) −1.48394e7 −0.505089 −0.252544 0.967585i \(-0.581267\pi\)
−0.252544 + 0.967585i \(0.581267\pi\)
\(972\) 0 0
\(973\) −2.52025e7 −0.853416
\(974\) −1.83570e7 −0.620016
\(975\) 0 0
\(976\) 9.19689e6 0.309041
\(977\) −2.53280e7 −0.848915 −0.424458 0.905448i \(-0.639535\pi\)
−0.424458 + 0.905448i \(0.639535\pi\)
\(978\) 0 0
\(979\) −6.86051e7 −2.28770
\(980\) 1.43327e7 0.476719
\(981\) 0 0
\(982\) −4.18967e7 −1.38644
\(983\) 3.48906e7 1.15166 0.575830 0.817569i \(-0.304679\pi\)
0.575830 + 0.817569i \(0.304679\pi\)
\(984\) 0 0
\(985\) 4.82630e7 1.58498
\(986\) 609511. 0.0199659
\(987\) 0 0
\(988\) −7.03637e6 −0.229328
\(989\) −250671. −0.00814917
\(990\) 0 0
\(991\) 2.38192e7 0.770447 0.385224 0.922823i \(-0.374124\pi\)
0.385224 + 0.922823i \(0.374124\pi\)
\(992\) −2.37649e7 −0.766757
\(993\) 0 0
\(994\) −3.03005e6 −0.0972710
\(995\) −1.97619e7 −0.632807
\(996\) 0 0
\(997\) 3.78624e7 1.20634 0.603171 0.797612i \(-0.293904\pi\)
0.603171 + 0.797612i \(0.293904\pi\)
\(998\) 3.08455e7 0.980313
\(999\) 0 0
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 27.6.a.c.1.2 yes 2
3.2 odd 2 inner 27.6.a.c.1.1 2
4.3 odd 2 432.6.a.o.1.2 2
5.4 even 2 675.6.a.j.1.1 2
9.2 odd 6 81.6.c.f.28.2 4
9.4 even 3 81.6.c.f.55.1 4
9.5 odd 6 81.6.c.f.55.2 4
9.7 even 3 81.6.c.f.28.1 4
12.11 even 2 432.6.a.o.1.1 2
15.14 odd 2 675.6.a.j.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.6.a.c.1.1 2 3.2 odd 2 inner
27.6.a.c.1.2 yes 2 1.1 even 1 trivial
81.6.c.f.28.1 4 9.7 even 3
81.6.c.f.28.2 4 9.2 odd 6
81.6.c.f.55.1 4 9.4 even 3
81.6.c.f.55.2 4 9.5 odd 6
432.6.a.o.1.1 2 12.11 even 2
432.6.a.o.1.2 2 4.3 odd 2
675.6.a.j.1.1 2 5.4 even 2
675.6.a.j.1.2 2 15.14 odd 2