Properties

Label 441.6.a.k.1.1
Level $441$
Weight $6$
Character 441.1
Self dual yes
Analytic conductor $70.729$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [441,6,Mod(1,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 441.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: \(70.7292645375\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 441.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+10.0000 q^{2} +68.0000 q^{4} -56.0000 q^{5} +360.000 q^{8} +O(q^{10})\) \(q+10.0000 q^{2} +68.0000 q^{4} -56.0000 q^{5} +360.000 q^{8} -560.000 q^{10} -232.000 q^{11} +140.000 q^{13} +1424.00 q^{16} -1722.00 q^{17} +98.0000 q^{19} -3808.00 q^{20} -2320.00 q^{22} -1824.00 q^{23} +11.0000 q^{25} +1400.00 q^{26} -3418.00 q^{29} +7644.00 q^{31} +2720.00 q^{32} -17220.0 q^{34} -10398.0 q^{37} +980.000 q^{38} -20160.0 q^{40} -17962.0 q^{41} +10880.0 q^{43} -15776.0 q^{44} -18240.0 q^{46} +9324.00 q^{47} +110.000 q^{50} +9520.00 q^{52} -2262.00 q^{53} +12992.0 q^{55} -34180.0 q^{58} -2730.00 q^{59} -25648.0 q^{61} +76440.0 q^{62} -18368.0 q^{64} -7840.00 q^{65} -48404.0 q^{67} -117096. q^{68} +58560.0 q^{71} -68082.0 q^{73} -103980. q^{74} +6664.00 q^{76} +31784.0 q^{79} -79744.0 q^{80} -179620. q^{82} -20538.0 q^{83} +96432.0 q^{85} +108800. q^{86} -83520.0 q^{88} -50582.0 q^{89} -124032. q^{92} +93240.0 q^{94} -5488.00 q^{95} +58506.0 q^{97} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 10.0000 1.76777 0.883883 0.467707i \(-0.154920\pi\)
0.883883 + 0.467707i \(0.154920\pi\)
\(3\) 0 0
\(4\) 68.0000 2.12500
\(5\) −56.0000 −1.00176 −0.500879 0.865517i \(-0.666990\pi\)
−0.500879 + 0.865517i \(0.666990\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 360.000 1.98874
\(9\) 0 0
\(10\) −560.000 −1.77088
\(11\) −232.000 −0.578104 −0.289052 0.957313i \(-0.593340\pi\)
−0.289052 + 0.957313i \(0.593340\pi\)
\(12\) 0 0
\(13\) 140.000 0.229757 0.114879 0.993380i \(-0.463352\pi\)
0.114879 + 0.993380i \(0.463352\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1424.00 1.39062
\(17\) −1722.00 −1.44514 −0.722572 0.691296i \(-0.757040\pi\)
−0.722572 + 0.691296i \(0.757040\pi\)
\(18\) 0 0
\(19\) 98.0000 0.0622791 0.0311395 0.999515i \(-0.490086\pi\)
0.0311395 + 0.999515i \(0.490086\pi\)
\(20\) −3808.00 −2.12874
\(21\) 0 0
\(22\) −2320.00 −1.02195
\(23\) −1824.00 −0.718961 −0.359480 0.933153i \(-0.617046\pi\)
−0.359480 + 0.933153i \(0.617046\pi\)
\(24\) 0 0
\(25\) 11.0000 0.00352000
\(26\) 1400.00 0.406158
\(27\) 0 0
\(28\) 0 0
\(29\) −3418.00 −0.754705 −0.377352 0.926070i \(-0.623165\pi\)
−0.377352 + 0.926070i \(0.623165\pi\)
\(30\) 0 0
\(31\) 7644.00 1.42862 0.714310 0.699830i \(-0.246741\pi\)
0.714310 + 0.699830i \(0.246741\pi\)
\(32\) 2720.00 0.469563
\(33\) 0 0
\(34\) −17220.0 −2.55468
\(35\) 0 0
\(36\) 0 0
\(37\) −10398.0 −1.24866 −0.624332 0.781159i \(-0.714629\pi\)
−0.624332 + 0.781159i \(0.714629\pi\)
\(38\) 980.000 0.110095
\(39\) 0 0
\(40\) −20160.0 −1.99223
\(41\) −17962.0 −1.66876 −0.834382 0.551186i \(-0.814175\pi\)
−0.834382 + 0.551186i \(0.814175\pi\)
\(42\) 0 0
\(43\) 10880.0 0.897342 0.448671 0.893697i \(-0.351898\pi\)
0.448671 + 0.893697i \(0.351898\pi\)
\(44\) −15776.0 −1.22847
\(45\) 0 0
\(46\) −18240.0 −1.27096
\(47\) 9324.00 0.615684 0.307842 0.951438i \(-0.400393\pi\)
0.307842 + 0.951438i \(0.400393\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 110.000 0.00622254
\(51\) 0 0
\(52\) 9520.00 0.488235
\(53\) −2262.00 −0.110612 −0.0553061 0.998469i \(-0.517613\pi\)
−0.0553061 + 0.998469i \(0.517613\pi\)
\(54\) 0 0
\(55\) 12992.0 0.579121
\(56\) 0 0
\(57\) 0 0
\(58\) −34180.0 −1.33414
\(59\) −2730.00 −0.102102 −0.0510508 0.998696i \(-0.516257\pi\)
−0.0510508 + 0.998696i \(0.516257\pi\)
\(60\) 0 0
\(61\) −25648.0 −0.882529 −0.441264 0.897377i \(-0.645470\pi\)
−0.441264 + 0.897377i \(0.645470\pi\)
\(62\) 76440.0 2.52547
\(63\) 0 0
\(64\) −18368.0 −0.560547
\(65\) −7840.00 −0.230161
\(66\) 0 0
\(67\) −48404.0 −1.31733 −0.658664 0.752437i \(-0.728878\pi\)
−0.658664 + 0.752437i \(0.728878\pi\)
\(68\) −117096. −3.07093
\(69\) 0 0
\(70\) 0 0
\(71\) 58560.0 1.37865 0.689327 0.724450i \(-0.257906\pi\)
0.689327 + 0.724450i \(0.257906\pi\)
\(72\) 0 0
\(73\) −68082.0 −1.49529 −0.747645 0.664099i \(-0.768815\pi\)
−0.747645 + 0.664099i \(0.768815\pi\)
\(74\) −103980. −2.20735
\(75\) 0 0
\(76\) 6664.00 0.132343
\(77\) 0 0
\(78\) 0 0
\(79\) 31784.0 0.572982 0.286491 0.958083i \(-0.407511\pi\)
0.286491 + 0.958083i \(0.407511\pi\)
\(80\) −79744.0 −1.39307
\(81\) 0 0
\(82\) −179620. −2.94999
\(83\) −20538.0 −0.327237 −0.163619 0.986524i \(-0.552317\pi\)
−0.163619 + 0.986524i \(0.552317\pi\)
\(84\) 0 0
\(85\) 96432.0 1.44768
\(86\) 108800. 1.58629
\(87\) 0 0
\(88\) −83520.0 −1.14970
\(89\) −50582.0 −0.676894 −0.338447 0.940985i \(-0.609902\pi\)
−0.338447 + 0.940985i \(0.609902\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −124032. −1.52779
\(93\) 0 0
\(94\) 93240.0 1.08839
\(95\) −5488.00 −0.0623886
\(96\) 0 0
\(97\) 58506.0 0.631351 0.315676 0.948867i \(-0.397769\pi\)
0.315676 + 0.948867i \(0.397769\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 748.000 0.00748000
\(101\) 38696.0 0.377453 0.188726 0.982030i \(-0.439564\pi\)
0.188726 + 0.982030i \(0.439564\pi\)
\(102\) 0 0
\(103\) −53060.0 −0.492804 −0.246402 0.969168i \(-0.579248\pi\)
−0.246402 + 0.969168i \(0.579248\pi\)
\(104\) 50400.0 0.456927
\(105\) 0 0
\(106\) −22620.0 −0.195537
\(107\) 146324. 1.23554 0.617769 0.786360i \(-0.288037\pi\)
0.617769 + 0.786360i \(0.288037\pi\)
\(108\) 0 0
\(109\) 92898.0 0.748928 0.374464 0.927241i \(-0.377827\pi\)
0.374464 + 0.927241i \(0.377827\pi\)
\(110\) 129920. 1.02375
\(111\) 0 0
\(112\) 0 0
\(113\) 83354.0 0.614088 0.307044 0.951695i \(-0.400660\pi\)
0.307044 + 0.951695i \(0.400660\pi\)
\(114\) 0 0
\(115\) 102144. 0.720225
\(116\) −232424. −1.60375
\(117\) 0 0
\(118\) −27300.0 −0.180492
\(119\) 0 0
\(120\) 0 0
\(121\) −107227. −0.665795
\(122\) −256480. −1.56011
\(123\) 0 0
\(124\) 519792. 3.03582
\(125\) 174384. 0.998232
\(126\) 0 0
\(127\) 60384.0 0.332210 0.166105 0.986108i \(-0.446881\pi\)
0.166105 + 0.986108i \(0.446881\pi\)
\(128\) −270720. −1.46048
\(129\) 0 0
\(130\) −78400.0 −0.406872
\(131\) −61586.0 −0.313548 −0.156774 0.987635i \(-0.550109\pi\)
−0.156774 + 0.987635i \(0.550109\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −484040. −2.32873
\(135\) 0 0
\(136\) −619920. −2.87401
\(137\) 204462. 0.930703 0.465352 0.885126i \(-0.345928\pi\)
0.465352 + 0.885126i \(0.345928\pi\)
\(138\) 0 0
\(139\) 35406.0 0.155432 0.0777159 0.996976i \(-0.475237\pi\)
0.0777159 + 0.996976i \(0.475237\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 585600. 2.43714
\(143\) −32480.0 −0.132824
\(144\) 0 0
\(145\) 191408. 0.756032
\(146\) −680820. −2.64332
\(147\) 0 0
\(148\) −707064. −2.65341
\(149\) 20226.0 0.0746353 0.0373177 0.999303i \(-0.488119\pi\)
0.0373177 + 0.999303i \(0.488119\pi\)
\(150\) 0 0
\(151\) 70904.0 0.253063 0.126531 0.991963i \(-0.459616\pi\)
0.126531 + 0.991963i \(0.459616\pi\)
\(152\) 35280.0 0.123857
\(153\) 0 0
\(154\) 0 0
\(155\) −428064. −1.43113
\(156\) 0 0
\(157\) −293524. −0.950374 −0.475187 0.879885i \(-0.657620\pi\)
−0.475187 + 0.879885i \(0.657620\pi\)
\(158\) 317840. 1.01290
\(159\) 0 0
\(160\) −152320. −0.470389
\(161\) 0 0
\(162\) 0 0
\(163\) 13192.0 0.0388903 0.0194452 0.999811i \(-0.493810\pi\)
0.0194452 + 0.999811i \(0.493810\pi\)
\(164\) −1.22142e6 −3.54612
\(165\) 0 0
\(166\) −205380. −0.578479
\(167\) 493612. 1.36960 0.684801 0.728730i \(-0.259889\pi\)
0.684801 + 0.728730i \(0.259889\pi\)
\(168\) 0 0
\(169\) −351693. −0.947212
\(170\) 964320. 2.55917
\(171\) 0 0
\(172\) 739840. 1.90685
\(173\) 240716. 0.611490 0.305745 0.952113i \(-0.401094\pi\)
0.305745 + 0.952113i \(0.401094\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −330368. −0.803926
\(177\) 0 0
\(178\) −505820. −1.19659
\(179\) −294932. −0.688001 −0.344001 0.938969i \(-0.611782\pi\)
−0.344001 + 0.938969i \(0.611782\pi\)
\(180\) 0 0
\(181\) 336980. 0.764553 0.382277 0.924048i \(-0.375140\pi\)
0.382277 + 0.924048i \(0.375140\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −656640. −1.42982
\(185\) 582288. 1.25086
\(186\) 0 0
\(187\) 399504. 0.835444
\(188\) 634032. 1.30833
\(189\) 0 0
\(190\) −54880.0 −0.110288
\(191\) −358264. −0.710591 −0.355296 0.934754i \(-0.615620\pi\)
−0.355296 + 0.934754i \(0.615620\pi\)
\(192\) 0 0
\(193\) −989554. −1.91226 −0.956128 0.292948i \(-0.905364\pi\)
−0.956128 + 0.292948i \(0.905364\pi\)
\(194\) 585060. 1.11608
\(195\) 0 0
\(196\) 0 0
\(197\) 990050. 1.81757 0.908786 0.417263i \(-0.137011\pi\)
0.908786 + 0.417263i \(0.137011\pi\)
\(198\) 0 0
\(199\) 840756. 1.50500 0.752501 0.658591i \(-0.228847\pi\)
0.752501 + 0.658591i \(0.228847\pi\)
\(200\) 3960.00 0.00700036
\(201\) 0 0
\(202\) 386960. 0.667249
\(203\) 0 0
\(204\) 0 0
\(205\) 1.00587e6 1.67170
\(206\) −530600. −0.871163
\(207\) 0 0
\(208\) 199360. 0.319506
\(209\) −22736.0 −0.0360038
\(210\) 0 0
\(211\) 1.15073e6 1.77938 0.889689 0.456568i \(-0.150921\pi\)
0.889689 + 0.456568i \(0.150921\pi\)
\(212\) −153816. −0.235051
\(213\) 0 0
\(214\) 1.46324e6 2.18414
\(215\) −609280. −0.898919
\(216\) 0 0
\(217\) 0 0
\(218\) 928980. 1.32393
\(219\) 0 0
\(220\) 883456. 1.23063
\(221\) −241080. −0.332032
\(222\) 0 0
\(223\) 824264. 1.10995 0.554976 0.831866i \(-0.312727\pi\)
0.554976 + 0.831866i \(0.312727\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 833540. 1.08556
\(227\) 74382.0 0.0958083 0.0479042 0.998852i \(-0.484746\pi\)
0.0479042 + 0.998852i \(0.484746\pi\)
\(228\) 0 0
\(229\) −1.13196e6 −1.42640 −0.713199 0.700961i \(-0.752755\pi\)
−0.713199 + 0.700961i \(0.752755\pi\)
\(230\) 1.02144e6 1.27319
\(231\) 0 0
\(232\) −1.23048e6 −1.50091
\(233\) 198726. 0.239809 0.119904 0.992785i \(-0.461741\pi\)
0.119904 + 0.992785i \(0.461741\pi\)
\(234\) 0 0
\(235\) −522144. −0.616766
\(236\) −185640. −0.216966
\(237\) 0 0
\(238\) 0 0
\(239\) −482904. −0.546847 −0.273424 0.961894i \(-0.588156\pi\)
−0.273424 + 0.961894i \(0.588156\pi\)
\(240\) 0 0
\(241\) −805910. −0.893807 −0.446904 0.894582i \(-0.647473\pi\)
−0.446904 + 0.894582i \(0.647473\pi\)
\(242\) −1.07227e6 −1.17697
\(243\) 0 0
\(244\) −1.74406e6 −1.87537
\(245\) 0 0
\(246\) 0 0
\(247\) 13720.0 0.0143091
\(248\) 2.75184e6 2.84115
\(249\) 0 0
\(250\) 1.74384e6 1.76464
\(251\) 430738. 0.431548 0.215774 0.976443i \(-0.430773\pi\)
0.215774 + 0.976443i \(0.430773\pi\)
\(252\) 0 0
\(253\) 423168. 0.415634
\(254\) 603840. 0.587270
\(255\) 0 0
\(256\) −2.11942e6 −2.02124
\(257\) −1.17691e6 −1.11150 −0.555751 0.831349i \(-0.687569\pi\)
−0.555751 + 0.831349i \(0.687569\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −533120. −0.489093
\(261\) 0 0
\(262\) −615860. −0.554279
\(263\) −1.29098e6 −1.15088 −0.575438 0.817845i \(-0.695169\pi\)
−0.575438 + 0.817845i \(0.695169\pi\)
\(264\) 0 0
\(265\) 126672. 0.110807
\(266\) 0 0
\(267\) 0 0
\(268\) −3.29147e6 −2.79932
\(269\) −1.27756e6 −1.07646 −0.538232 0.842797i \(-0.680907\pi\)
−0.538232 + 0.842797i \(0.680907\pi\)
\(270\) 0 0
\(271\) −1.65054e6 −1.36522 −0.682612 0.730781i \(-0.739156\pi\)
−0.682612 + 0.730781i \(0.739156\pi\)
\(272\) −2.45213e6 −2.00965
\(273\) 0 0
\(274\) 2.04462e6 1.64527
\(275\) −2552.00 −0.00203493
\(276\) 0 0
\(277\) −1.06409e6 −0.833257 −0.416628 0.909077i \(-0.636788\pi\)
−0.416628 + 0.909077i \(0.636788\pi\)
\(278\) 354060. 0.274767
\(279\) 0 0
\(280\) 0 0
\(281\) 22342.0 0.0168794 0.00843969 0.999964i \(-0.497314\pi\)
0.00843969 + 0.999964i \(0.497314\pi\)
\(282\) 0 0
\(283\) 2.49574e6 1.85239 0.926196 0.377042i \(-0.123059\pi\)
0.926196 + 0.377042i \(0.123059\pi\)
\(284\) 3.98208e6 2.92964
\(285\) 0 0
\(286\) −324800. −0.234802
\(287\) 0 0
\(288\) 0 0
\(289\) 1.54543e6 1.08844
\(290\) 1.91408e6 1.33649
\(291\) 0 0
\(292\) −4.62958e6 −3.17749
\(293\) −1.93178e6 −1.31458 −0.657291 0.753637i \(-0.728298\pi\)
−0.657291 + 0.753637i \(0.728298\pi\)
\(294\) 0 0
\(295\) 152880. 0.102281
\(296\) −3.74328e6 −2.48326
\(297\) 0 0
\(298\) 202260. 0.131938
\(299\) −255360. −0.165187
\(300\) 0 0
\(301\) 0 0
\(302\) 709040. 0.447356
\(303\) 0 0
\(304\) 139552. 0.0866068
\(305\) 1.43629e6 0.884081
\(306\) 0 0
\(307\) 459074. 0.277995 0.138997 0.990293i \(-0.455612\pi\)
0.138997 + 0.990293i \(0.455612\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −4.28064e6 −2.52991
\(311\) 667128. 0.391118 0.195559 0.980692i \(-0.437348\pi\)
0.195559 + 0.980692i \(0.437348\pi\)
\(312\) 0 0
\(313\) 111034. 0.0640612 0.0320306 0.999487i \(-0.489803\pi\)
0.0320306 + 0.999487i \(0.489803\pi\)
\(314\) −2.93524e6 −1.68004
\(315\) 0 0
\(316\) 2.16131e6 1.21759
\(317\) 68778.0 0.0384416 0.0192208 0.999815i \(-0.493881\pi\)
0.0192208 + 0.999815i \(0.493881\pi\)
\(318\) 0 0
\(319\) 792976. 0.436298
\(320\) 1.02861e6 0.561533
\(321\) 0 0
\(322\) 0 0
\(323\) −168756. −0.0900022
\(324\) 0 0
\(325\) 1540.00 0.000808746 0
\(326\) 131920. 0.0687490
\(327\) 0 0
\(328\) −6.46632e6 −3.31874
\(329\) 0 0
\(330\) 0 0
\(331\) −564448. −0.283174 −0.141587 0.989926i \(-0.545221\pi\)
−0.141587 + 0.989926i \(0.545221\pi\)
\(332\) −1.39658e6 −0.695379
\(333\) 0 0
\(334\) 4.93612e6 2.42114
\(335\) 2.71062e6 1.31965
\(336\) 0 0
\(337\) 2.07729e6 0.996376 0.498188 0.867069i \(-0.333999\pi\)
0.498188 + 0.867069i \(0.333999\pi\)
\(338\) −3.51693e6 −1.67445
\(339\) 0 0
\(340\) 6.55738e6 3.07633
\(341\) −1.77341e6 −0.825891
\(342\) 0 0
\(343\) 0 0
\(344\) 3.91680e6 1.78458
\(345\) 0 0
\(346\) 2.40716e6 1.08097
\(347\) 53248.0 0.0237399 0.0118700 0.999930i \(-0.496222\pi\)
0.0118700 + 0.999930i \(0.496222\pi\)
\(348\) 0 0
\(349\) 2.27200e6 0.998494 0.499247 0.866460i \(-0.333610\pi\)
0.499247 + 0.866460i \(0.333610\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −631040. −0.271456
\(353\) 4.00645e6 1.71129 0.855644 0.517565i \(-0.173162\pi\)
0.855644 + 0.517565i \(0.173162\pi\)
\(354\) 0 0
\(355\) −3.27936e6 −1.38108
\(356\) −3.43958e6 −1.43840
\(357\) 0 0
\(358\) −2.94932e6 −1.21623
\(359\) −73784.0 −0.0302152 −0.0151076 0.999886i \(-0.504809\pi\)
−0.0151076 + 0.999886i \(0.504809\pi\)
\(360\) 0 0
\(361\) −2.46650e6 −0.996121
\(362\) 3.36980e6 1.35155
\(363\) 0 0
\(364\) 0 0
\(365\) 3.81259e6 1.49792
\(366\) 0 0
\(367\) −1.40431e6 −0.544250 −0.272125 0.962262i \(-0.587726\pi\)
−0.272125 + 0.962262i \(0.587726\pi\)
\(368\) −2.59738e6 −0.999805
\(369\) 0 0
\(370\) 5.82288e6 2.21123
\(371\) 0 0
\(372\) 0 0
\(373\) −1.60323e6 −0.596657 −0.298329 0.954463i \(-0.596429\pi\)
−0.298329 + 0.954463i \(0.596429\pi\)
\(374\) 3.99504e6 1.47687
\(375\) 0 0
\(376\) 3.35664e6 1.22443
\(377\) −478520. −0.173399
\(378\) 0 0
\(379\) −4.77012e6 −1.70581 −0.852906 0.522064i \(-0.825162\pi\)
−0.852906 + 0.522064i \(0.825162\pi\)
\(380\) −373184. −0.132576
\(381\) 0 0
\(382\) −3.58264e6 −1.25616
\(383\) −2.23079e6 −0.777072 −0.388536 0.921434i \(-0.627019\pi\)
−0.388536 + 0.921434i \(0.627019\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −9.89554e6 −3.38042
\(387\) 0 0
\(388\) 3.97841e6 1.34162
\(389\) −4.84024e6 −1.62178 −0.810892 0.585196i \(-0.801018\pi\)
−0.810892 + 0.585196i \(0.801018\pi\)
\(390\) 0 0
\(391\) 3.14093e6 1.03900
\(392\) 0 0
\(393\) 0 0
\(394\) 9.90050e6 3.21304
\(395\) −1.77990e6 −0.573989
\(396\) 0 0
\(397\) −995820. −0.317106 −0.158553 0.987350i \(-0.550683\pi\)
−0.158553 + 0.987350i \(0.550683\pi\)
\(398\) 8.40756e6 2.66049
\(399\) 0 0
\(400\) 15664.0 0.00489500
\(401\) 3.31605e6 1.02982 0.514909 0.857245i \(-0.327826\pi\)
0.514909 + 0.857245i \(0.327826\pi\)
\(402\) 0 0
\(403\) 1.07016e6 0.328236
\(404\) 2.63133e6 0.802087
\(405\) 0 0
\(406\) 0 0
\(407\) 2.41234e6 0.721858
\(408\) 0 0
\(409\) −3.07273e6 −0.908274 −0.454137 0.890932i \(-0.650052\pi\)
−0.454137 + 0.890932i \(0.650052\pi\)
\(410\) 1.00587e7 2.95517
\(411\) 0 0
\(412\) −3.60808e6 −1.04721
\(413\) 0 0
\(414\) 0 0
\(415\) 1.15013e6 0.327813
\(416\) 380800. 0.107886
\(417\) 0 0
\(418\) −227360. −0.0636463
\(419\) 2.81438e6 0.783154 0.391577 0.920145i \(-0.371930\pi\)
0.391577 + 0.920145i \(0.371930\pi\)
\(420\) 0 0
\(421\) 3.05802e6 0.840883 0.420441 0.907320i \(-0.361875\pi\)
0.420441 + 0.907320i \(0.361875\pi\)
\(422\) 1.15073e7 3.14552
\(423\) 0 0
\(424\) −814320. −0.219979
\(425\) −18942.0 −0.00508690
\(426\) 0 0
\(427\) 0 0
\(428\) 9.95003e6 2.62552
\(429\) 0 0
\(430\) −6.09280e6 −1.58908
\(431\) −1.93750e6 −0.502398 −0.251199 0.967936i \(-0.580825\pi\)
−0.251199 + 0.967936i \(0.580825\pi\)
\(432\) 0 0
\(433\) −3.94790e6 −1.01192 −0.505961 0.862557i \(-0.668862\pi\)
−0.505961 + 0.862557i \(0.668862\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 6.31706e6 1.59147
\(437\) −178752. −0.0447762
\(438\) 0 0
\(439\) 7.41770e6 1.83700 0.918498 0.395426i \(-0.129403\pi\)
0.918498 + 0.395426i \(0.129403\pi\)
\(440\) 4.67712e6 1.15172
\(441\) 0 0
\(442\) −2.41080e6 −0.586956
\(443\) −1.40269e6 −0.339589 −0.169794 0.985480i \(-0.554310\pi\)
−0.169794 + 0.985480i \(0.554310\pi\)
\(444\) 0 0
\(445\) 2.83259e6 0.678085
\(446\) 8.24264e6 1.96214
\(447\) 0 0
\(448\) 0 0
\(449\) 590574. 0.138248 0.0691239 0.997608i \(-0.477980\pi\)
0.0691239 + 0.997608i \(0.477980\pi\)
\(450\) 0 0
\(451\) 4.16718e6 0.964720
\(452\) 5.66807e6 1.30494
\(453\) 0 0
\(454\) 743820. 0.169367
\(455\) 0 0
\(456\) 0 0
\(457\) −2.90484e6 −0.650627 −0.325313 0.945606i \(-0.605470\pi\)
−0.325313 + 0.945606i \(0.605470\pi\)
\(458\) −1.13196e7 −2.52154
\(459\) 0 0
\(460\) 6.94579e6 1.53048
\(461\) −922684. −0.202209 −0.101105 0.994876i \(-0.532238\pi\)
−0.101105 + 0.994876i \(0.532238\pi\)
\(462\) 0 0
\(463\) 7.18235e6 1.55709 0.778546 0.627588i \(-0.215958\pi\)
0.778546 + 0.627588i \(0.215958\pi\)
\(464\) −4.86723e6 −1.04951
\(465\) 0 0
\(466\) 1.98726e6 0.423926
\(467\) −612570. −0.129976 −0.0649881 0.997886i \(-0.520701\pi\)
−0.0649881 + 0.997886i \(0.520701\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −5.22144e6 −1.09030
\(471\) 0 0
\(472\) −982800. −0.203053
\(473\) −2.52416e6 −0.518757
\(474\) 0 0
\(475\) 1078.00 0.000219222 0
\(476\) 0 0
\(477\) 0 0
\(478\) −4.82904e6 −0.966699
\(479\) 2.60330e6 0.518424 0.259212 0.965820i \(-0.416537\pi\)
0.259212 + 0.965820i \(0.416537\pi\)
\(480\) 0 0
\(481\) −1.45572e6 −0.286890
\(482\) −8.05910e6 −1.58004
\(483\) 0 0
\(484\) −7.29144e6 −1.41482
\(485\) −3.27634e6 −0.632461
\(486\) 0 0
\(487\) 5.46309e6 1.04380 0.521898 0.853008i \(-0.325224\pi\)
0.521898 + 0.853008i \(0.325224\pi\)
\(488\) −9.23328e6 −1.75512
\(489\) 0 0
\(490\) 0 0
\(491\) −1.64090e6 −0.307170 −0.153585 0.988135i \(-0.549082\pi\)
−0.153585 + 0.988135i \(0.549082\pi\)
\(492\) 0 0
\(493\) 5.88580e6 1.09066
\(494\) 137200. 0.0252951
\(495\) 0 0
\(496\) 1.08851e7 1.98667
\(497\) 0 0
\(498\) 0 0
\(499\) 2.99796e6 0.538983 0.269491 0.963003i \(-0.413144\pi\)
0.269491 + 0.963003i \(0.413144\pi\)
\(500\) 1.18581e7 2.12124
\(501\) 0 0
\(502\) 4.30738e6 0.762876
\(503\) −6.89405e6 −1.21494 −0.607469 0.794343i \(-0.707815\pi\)
−0.607469 + 0.794343i \(0.707815\pi\)
\(504\) 0 0
\(505\) −2.16698e6 −0.378117
\(506\) 4.23168e6 0.734745
\(507\) 0 0
\(508\) 4.10611e6 0.705946
\(509\) 2.30476e6 0.394305 0.197152 0.980373i \(-0.436831\pi\)
0.197152 + 0.980373i \(0.436831\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.25312e7 −2.11260
\(513\) 0 0
\(514\) −1.17691e7 −1.96488
\(515\) 2.97136e6 0.493671
\(516\) 0 0
\(517\) −2.16317e6 −0.355929
\(518\) 0 0
\(519\) 0 0
\(520\) −2.82240e6 −0.457731
\(521\) −1.20960e7 −1.95231 −0.976155 0.217073i \(-0.930349\pi\)
−0.976155 + 0.217073i \(0.930349\pi\)
\(522\) 0 0
\(523\) −5.48443e6 −0.876753 −0.438377 0.898791i \(-0.644446\pi\)
−0.438377 + 0.898791i \(0.644446\pi\)
\(524\) −4.18785e6 −0.666289
\(525\) 0 0
\(526\) −1.29098e7 −2.03448
\(527\) −1.31630e7 −2.06456
\(528\) 0 0
\(529\) −3.10937e6 −0.483095
\(530\) 1.26672e6 0.195880
\(531\) 0 0
\(532\) 0 0
\(533\) −2.51468e6 −0.383411
\(534\) 0 0
\(535\) −8.19414e6 −1.23771
\(536\) −1.74254e7 −2.61982
\(537\) 0 0
\(538\) −1.27756e7 −1.90294
\(539\) 0 0
\(540\) 0 0
\(541\) −6.71799e6 −0.986839 −0.493420 0.869791i \(-0.664253\pi\)
−0.493420 + 0.869791i \(0.664253\pi\)
\(542\) −1.65054e7 −2.41340
\(543\) 0 0
\(544\) −4.68384e6 −0.678586
\(545\) −5.20229e6 −0.750245
\(546\) 0 0
\(547\) −5.00235e6 −0.714835 −0.357418 0.933945i \(-0.616343\pi\)
−0.357418 + 0.933945i \(0.616343\pi\)
\(548\) 1.39034e7 1.97774
\(549\) 0 0
\(550\) −25520.0 −0.00359728
\(551\) −334964. −0.0470023
\(552\) 0 0
\(553\) 0 0
\(554\) −1.06409e7 −1.47300
\(555\) 0 0
\(556\) 2.40761e6 0.330293
\(557\) −9.01961e6 −1.23183 −0.615913 0.787814i \(-0.711213\pi\)
−0.615913 + 0.787814i \(0.711213\pi\)
\(558\) 0 0
\(559\) 1.52320e6 0.206171
\(560\) 0 0
\(561\) 0 0
\(562\) 223420. 0.0298388
\(563\) 1.24051e7 1.64941 0.824707 0.565561i \(-0.191340\pi\)
0.824707 + 0.565561i \(0.191340\pi\)
\(564\) 0 0
\(565\) −4.66782e6 −0.615167
\(566\) 2.49574e7 3.27460
\(567\) 0 0
\(568\) 2.10816e7 2.74178
\(569\) −6.48804e6 −0.840103 −0.420052 0.907500i \(-0.637988\pi\)
−0.420052 + 0.907500i \(0.637988\pi\)
\(570\) 0 0
\(571\) −1.02285e7 −1.31287 −0.656435 0.754382i \(-0.727936\pi\)
−0.656435 + 0.754382i \(0.727936\pi\)
\(572\) −2.20864e6 −0.282251
\(573\) 0 0
\(574\) 0 0
\(575\) −20064.0 −0.00253074
\(576\) 0 0
\(577\) −2.65338e6 −0.331787 −0.165894 0.986144i \(-0.553051\pi\)
−0.165894 + 0.986144i \(0.553051\pi\)
\(578\) 1.54543e7 1.92411
\(579\) 0 0
\(580\) 1.30157e7 1.60657
\(581\) 0 0
\(582\) 0 0
\(583\) 524784. 0.0639454
\(584\) −2.45095e7 −2.97374
\(585\) 0 0
\(586\) −1.93178e7 −2.32387
\(587\) −1.43044e7 −1.71346 −0.856729 0.515766i \(-0.827507\pi\)
−0.856729 + 0.515766i \(0.827507\pi\)
\(588\) 0 0
\(589\) 749112. 0.0889731
\(590\) 1.52880e6 0.180809
\(591\) 0 0
\(592\) −1.48068e7 −1.73642
\(593\) −1.00265e7 −1.17088 −0.585442 0.810714i \(-0.699079\pi\)
−0.585442 + 0.810714i \(0.699079\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 1.37537e6 0.158600
\(597\) 0 0
\(598\) −2.55360e6 −0.292011
\(599\) 7.52292e6 0.856681 0.428341 0.903617i \(-0.359098\pi\)
0.428341 + 0.903617i \(0.359098\pi\)
\(600\) 0 0
\(601\) −3.38625e6 −0.382413 −0.191207 0.981550i \(-0.561240\pi\)
−0.191207 + 0.981550i \(0.561240\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 4.82147e6 0.537759
\(605\) 6.00471e6 0.666966
\(606\) 0 0
\(607\) 6.90861e6 0.761060 0.380530 0.924769i \(-0.375742\pi\)
0.380530 + 0.924769i \(0.375742\pi\)
\(608\) 266560. 0.0292439
\(609\) 0 0
\(610\) 1.43629e7 1.56285
\(611\) 1.30536e6 0.141458
\(612\) 0 0
\(613\) −9.68896e6 −1.04142 −0.520710 0.853734i \(-0.674333\pi\)
−0.520710 + 0.853734i \(0.674333\pi\)
\(614\) 4.59074e6 0.491430
\(615\) 0 0
\(616\) 0 0
\(617\) 7.84742e6 0.829877 0.414939 0.909849i \(-0.363803\pi\)
0.414939 + 0.909849i \(0.363803\pi\)
\(618\) 0 0
\(619\) 1.01972e7 1.06968 0.534840 0.844953i \(-0.320372\pi\)
0.534840 + 0.844953i \(0.320372\pi\)
\(620\) −2.91084e7 −3.04115
\(621\) 0 0
\(622\) 6.67128e6 0.691406
\(623\) 0 0
\(624\) 0 0
\(625\) −9.79988e6 −1.00351
\(626\) 1.11034e6 0.113245
\(627\) 0 0
\(628\) −1.99596e7 −2.01954
\(629\) 1.79054e7 1.80450
\(630\) 0 0
\(631\) −8.36258e6 −0.836116 −0.418058 0.908420i \(-0.637289\pi\)
−0.418058 + 0.908420i \(0.637289\pi\)
\(632\) 1.14422e7 1.13951
\(633\) 0 0
\(634\) 687780. 0.0679558
\(635\) −3.38150e6 −0.332794
\(636\) 0 0
\(637\) 0 0
\(638\) 7.92976e6 0.771273
\(639\) 0 0
\(640\) 1.51603e7 1.46305
\(641\) −1.10283e6 −0.106014 −0.0530070 0.998594i \(-0.516881\pi\)
−0.0530070 + 0.998594i \(0.516881\pi\)
\(642\) 0 0
\(643\) −1.71354e7 −1.63443 −0.817217 0.576330i \(-0.804484\pi\)
−0.817217 + 0.576330i \(0.804484\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −1.68756e6 −0.159103
\(647\) −54964.0 −0.00516200 −0.00258100 0.999997i \(-0.500822\pi\)
−0.00258100 + 0.999997i \(0.500822\pi\)
\(648\) 0 0
\(649\) 633360. 0.0590254
\(650\) 15400.0 0.00142968
\(651\) 0 0
\(652\) 897056. 0.0826420
\(653\) 485166. 0.0445254 0.0222627 0.999752i \(-0.492913\pi\)
0.0222627 + 0.999752i \(0.492913\pi\)
\(654\) 0 0
\(655\) 3.44882e6 0.314099
\(656\) −2.55779e7 −2.32063
\(657\) 0 0
\(658\) 0 0
\(659\) 2.72136e6 0.244103 0.122051 0.992524i \(-0.461053\pi\)
0.122051 + 0.992524i \(0.461053\pi\)
\(660\) 0 0
\(661\) 2.14525e6 0.190974 0.0954869 0.995431i \(-0.469559\pi\)
0.0954869 + 0.995431i \(0.469559\pi\)
\(662\) −5.64448e6 −0.500586
\(663\) 0 0
\(664\) −7.39368e6 −0.650789
\(665\) 0 0
\(666\) 0 0
\(667\) 6.23443e6 0.542603
\(668\) 3.35656e7 2.91041
\(669\) 0 0
\(670\) 2.71062e7 2.33283
\(671\) 5.95034e6 0.510194
\(672\) 0 0
\(673\) 2.92796e6 0.249188 0.124594 0.992208i \(-0.460237\pi\)
0.124594 + 0.992208i \(0.460237\pi\)
\(674\) 2.07729e7 1.76136
\(675\) 0 0
\(676\) −2.39151e7 −2.01282
\(677\) −1.34992e7 −1.13198 −0.565988 0.824414i \(-0.691505\pi\)
−0.565988 + 0.824414i \(0.691505\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 3.47155e7 2.87906
\(681\) 0 0
\(682\) −1.77341e7 −1.45998
\(683\) 5.42972e6 0.445375 0.222688 0.974890i \(-0.428517\pi\)
0.222688 + 0.974890i \(0.428517\pi\)
\(684\) 0 0
\(685\) −1.14499e7 −0.932340
\(686\) 0 0
\(687\) 0 0
\(688\) 1.54931e7 1.24787
\(689\) −316680. −0.0254140
\(690\) 0 0
\(691\) −2.08280e7 −1.65940 −0.829702 0.558207i \(-0.811490\pi\)
−0.829702 + 0.558207i \(0.811490\pi\)
\(692\) 1.63687e7 1.29942
\(693\) 0 0
\(694\) 532480. 0.0419667
\(695\) −1.98274e6 −0.155705
\(696\) 0 0
\(697\) 3.09306e7 2.41160
\(698\) 2.27200e7 1.76510
\(699\) 0 0
\(700\) 0 0
\(701\) −2.35141e7 −1.80731 −0.903655 0.428261i \(-0.859126\pi\)
−0.903655 + 0.428261i \(0.859126\pi\)
\(702\) 0 0
\(703\) −1.01900e6 −0.0777656
\(704\) 4.26138e6 0.324055
\(705\) 0 0
\(706\) 4.00645e7 3.02516
\(707\) 0 0
\(708\) 0 0
\(709\) −1.95747e7 −1.46244 −0.731221 0.682140i \(-0.761049\pi\)
−0.731221 + 0.682140i \(0.761049\pi\)
\(710\) −3.27936e7 −2.44142
\(711\) 0 0
\(712\) −1.82095e7 −1.34617
\(713\) −1.39427e7 −1.02712
\(714\) 0 0
\(715\) 1.81888e6 0.133057
\(716\) −2.00554e7 −1.46200
\(717\) 0 0
\(718\) −737840. −0.0534135
\(719\) −2.61152e7 −1.88396 −0.941978 0.335674i \(-0.891036\pi\)
−0.941978 + 0.335674i \(0.891036\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −2.46650e7 −1.76091
\(723\) 0 0
\(724\) 2.29146e7 1.62468
\(725\) −37598.0 −0.00265656
\(726\) 0 0
\(727\) −1.54126e7 −1.08154 −0.540768 0.841172i \(-0.681866\pi\)
−0.540768 + 0.841172i \(0.681866\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 3.81259e7 2.64797
\(731\) −1.87354e7 −1.29679
\(732\) 0 0
\(733\) 1.69868e7 1.16776 0.583878 0.811841i \(-0.301535\pi\)
0.583878 + 0.811841i \(0.301535\pi\)
\(734\) −1.40431e7 −0.962107
\(735\) 0 0
\(736\) −4.96128e6 −0.337597
\(737\) 1.12297e7 0.761554
\(738\) 0 0
\(739\) 2.01511e6 0.135734 0.0678669 0.997694i \(-0.478381\pi\)
0.0678669 + 0.997694i \(0.478381\pi\)
\(740\) 3.95956e7 2.65808
\(741\) 0 0
\(742\) 0 0
\(743\) 1.51381e7 1.00600 0.503001 0.864286i \(-0.332229\pi\)
0.503001 + 0.864286i \(0.332229\pi\)
\(744\) 0 0
\(745\) −1.13266e6 −0.0747666
\(746\) −1.60323e7 −1.05475
\(747\) 0 0
\(748\) 2.71663e7 1.77532
\(749\) 0 0
\(750\) 0 0
\(751\) 7.21401e6 0.466742 0.233371 0.972388i \(-0.425024\pi\)
0.233371 + 0.972388i \(0.425024\pi\)
\(752\) 1.32774e7 0.856185
\(753\) 0 0
\(754\) −4.78520e6 −0.306529
\(755\) −3.97062e6 −0.253508
\(756\) 0 0
\(757\) −1.09697e7 −0.695755 −0.347877 0.937540i \(-0.613097\pi\)
−0.347877 + 0.937540i \(0.613097\pi\)
\(758\) −4.77012e7 −3.01548
\(759\) 0 0
\(760\) −1.97568e6 −0.124075
\(761\) 1.92442e7 1.20459 0.602293 0.798275i \(-0.294254\pi\)
0.602293 + 0.798275i \(0.294254\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −2.43620e7 −1.51001
\(765\) 0 0
\(766\) −2.23079e7 −1.37368
\(767\) −382200. −0.0234586
\(768\) 0 0
\(769\) −8.21185e6 −0.500755 −0.250378 0.968148i \(-0.580555\pi\)
−0.250378 + 0.968148i \(0.580555\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −6.72897e7 −4.06355
\(773\) 1.86187e7 1.12073 0.560363 0.828247i \(-0.310662\pi\)
0.560363 + 0.828247i \(0.310662\pi\)
\(774\) 0 0
\(775\) 84084.0 0.00502874
\(776\) 2.10622e7 1.25559
\(777\) 0 0
\(778\) −4.84024e7 −2.86694
\(779\) −1.76028e6 −0.103929
\(780\) 0 0
\(781\) −1.35859e7 −0.797006
\(782\) 3.14093e7 1.83671
\(783\) 0 0
\(784\) 0 0
\(785\) 1.64373e7 0.952045
\(786\) 0 0
\(787\) −2.62501e7 −1.51075 −0.755377 0.655291i \(-0.772546\pi\)
−0.755377 + 0.655291i \(0.772546\pi\)
\(788\) 6.73234e7 3.86234
\(789\) 0 0
\(790\) −1.77990e7 −1.01468
\(791\) 0 0
\(792\) 0 0
\(793\) −3.59072e6 −0.202768
\(794\) −9.95820e6 −0.560570
\(795\) 0 0
\(796\) 5.71714e7 3.19813
\(797\) −1.00373e7 −0.559720 −0.279860 0.960041i \(-0.590288\pi\)
−0.279860 + 0.960041i \(0.590288\pi\)
\(798\) 0 0
\(799\) −1.60559e7 −0.889751
\(800\) 29920.0 0.00165286
\(801\) 0 0
\(802\) 3.31605e7 1.82048
\(803\) 1.57950e7 0.864433
\(804\) 0 0
\(805\) 0 0
\(806\) 1.07016e7 0.580245
\(807\) 0 0
\(808\) 1.39306e7 0.750655
\(809\) −1.40884e7 −0.756816 −0.378408 0.925639i \(-0.623528\pi\)
−0.378408 + 0.925639i \(0.623528\pi\)
\(810\) 0 0
\(811\) −1.81433e7 −0.968646 −0.484323 0.874889i \(-0.660934\pi\)
−0.484323 + 0.874889i \(0.660934\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 2.41234e7 1.27608
\(815\) −738752. −0.0389587
\(816\) 0 0
\(817\) 1.06624e6 0.0558856
\(818\) −3.07273e7 −1.60562
\(819\) 0 0
\(820\) 6.83993e7 3.55236
\(821\) 2.13669e7 1.10633 0.553164 0.833072i \(-0.313420\pi\)
0.553164 + 0.833072i \(0.313420\pi\)
\(822\) 0 0
\(823\) 1.78017e7 0.916142 0.458071 0.888916i \(-0.348541\pi\)
0.458071 + 0.888916i \(0.348541\pi\)
\(824\) −1.91016e7 −0.980058
\(825\) 0 0
\(826\) 0 0
\(827\) −1.62921e7 −0.828350 −0.414175 0.910197i \(-0.635930\pi\)
−0.414175 + 0.910197i \(0.635930\pi\)
\(828\) 0 0
\(829\) 2.08499e6 0.105370 0.0526851 0.998611i \(-0.483222\pi\)
0.0526851 + 0.998611i \(0.483222\pi\)
\(830\) 1.15013e7 0.579497
\(831\) 0 0
\(832\) −2.57152e6 −0.128790
\(833\) 0 0
\(834\) 0 0
\(835\) −2.76423e7 −1.37201
\(836\) −1.54605e6 −0.0765081
\(837\) 0 0
\(838\) 2.81438e7 1.38443
\(839\) −2.27850e7 −1.11749 −0.558745 0.829340i \(-0.688717\pi\)
−0.558745 + 0.829340i \(0.688717\pi\)
\(840\) 0 0
\(841\) −8.82842e6 −0.430421
\(842\) 3.05802e7 1.48648
\(843\) 0 0
\(844\) 7.82498e7 3.78118
\(845\) 1.96948e7 0.948877
\(846\) 0 0
\(847\) 0 0
\(848\) −3.22109e6 −0.153820
\(849\) 0 0
\(850\) −189420. −0.00899246
\(851\) 1.89660e7 0.897740
\(852\) 0 0
\(853\) 2.26975e7 1.06808 0.534042 0.845458i \(-0.320672\pi\)
0.534042 + 0.845458i \(0.320672\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 5.26766e7 2.45716
\(857\) 2.52900e7 1.17624 0.588120 0.808774i \(-0.299868\pi\)
0.588120 + 0.808774i \(0.299868\pi\)
\(858\) 0 0
\(859\) 1.03947e7 0.480652 0.240326 0.970692i \(-0.422746\pi\)
0.240326 + 0.970692i \(0.422746\pi\)
\(860\) −4.14310e7 −1.91020
\(861\) 0 0
\(862\) −1.93750e7 −0.888122
\(863\) −4.33399e7 −1.98089 −0.990447 0.137892i \(-0.955967\pi\)
−0.990447 + 0.137892i \(0.955967\pi\)
\(864\) 0 0
\(865\) −1.34801e7 −0.612566
\(866\) −3.94790e7 −1.78884
\(867\) 0 0
\(868\) 0 0
\(869\) −7.37389e6 −0.331243
\(870\) 0 0
\(871\) −6.77656e6 −0.302666
\(872\) 3.34433e7 1.48942
\(873\) 0 0
\(874\) −1.78752e6 −0.0791539
\(875\) 0 0
\(876\) 0 0
\(877\) 3.71659e7 1.63172 0.815861 0.578248i \(-0.196264\pi\)
0.815861 + 0.578248i \(0.196264\pi\)
\(878\) 7.41770e7 3.24738
\(879\) 0 0
\(880\) 1.85006e7 0.805340
\(881\) 9.04785e6 0.392740 0.196370 0.980530i \(-0.437085\pi\)
0.196370 + 0.980530i \(0.437085\pi\)
\(882\) 0 0
\(883\) 3.29679e7 1.42295 0.711474 0.702712i \(-0.248028\pi\)
0.711474 + 0.702712i \(0.248028\pi\)
\(884\) −1.63934e7 −0.705569
\(885\) 0 0
\(886\) −1.40269e7 −0.600313
\(887\) −1.61099e7 −0.687517 −0.343758 0.939058i \(-0.611700\pi\)
−0.343758 + 0.939058i \(0.611700\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 2.83259e7 1.19870
\(891\) 0 0
\(892\) 5.60500e7 2.35865
\(893\) 913752. 0.0383442
\(894\) 0 0
\(895\) 1.65162e7 0.689211
\(896\) 0 0
\(897\) 0 0
\(898\) 5.90574e6 0.244390
\(899\) −2.61272e7 −1.07819
\(900\) 0 0
\(901\) 3.89516e6 0.159850
\(902\) 4.16718e7 1.70540
\(903\) 0 0
\(904\) 3.00074e7 1.22126
\(905\) −1.88709e7 −0.765898
\(906\) 0 0
\(907\) −4.47286e7 −1.80537 −0.902686 0.430300i \(-0.858408\pi\)
−0.902686 + 0.430300i \(0.858408\pi\)
\(908\) 5.05798e6 0.203593
\(909\) 0 0
\(910\) 0 0
\(911\) 6.60518e6 0.263687 0.131844 0.991271i \(-0.457910\pi\)
0.131844 + 0.991271i \(0.457910\pi\)
\(912\) 0 0
\(913\) 4.76482e6 0.189177
\(914\) −2.90484e7 −1.15016
\(915\) 0 0
\(916\) −7.69730e7 −3.03110
\(917\) 0 0
\(918\) 0 0
\(919\) −3.08930e7 −1.20662 −0.603311 0.797506i \(-0.706152\pi\)
−0.603311 + 0.797506i \(0.706152\pi\)
\(920\) 3.67718e7 1.43234
\(921\) 0 0
\(922\) −9.22684e6 −0.357459
\(923\) 8.19840e6 0.316756
\(924\) 0 0
\(925\) −114378. −0.00439530
\(926\) 7.18235e7 2.75258
\(927\) 0 0
\(928\) −9.29696e6 −0.354381
\(929\) −4.87215e6 −0.185217 −0.0926087 0.995703i \(-0.529521\pi\)
−0.0926087 + 0.995703i \(0.529521\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 1.35134e7 0.509593
\(933\) 0 0
\(934\) −6.12570e6 −0.229767
\(935\) −2.23722e7 −0.836913
\(936\) 0 0
\(937\) 3.25004e7 1.20932 0.604658 0.796485i \(-0.293310\pi\)
0.604658 + 0.796485i \(0.293310\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −3.55058e7 −1.31063
\(941\) −2.64040e6 −0.0972066 −0.0486033 0.998818i \(-0.515477\pi\)
−0.0486033 + 0.998818i \(0.515477\pi\)
\(942\) 0 0
\(943\) 3.27627e7 1.19978
\(944\) −3.88752e6 −0.141985
\(945\) 0 0
\(946\) −2.52416e7 −0.917042
\(947\) 4.08179e7 1.47903 0.739513 0.673142i \(-0.235056\pi\)
0.739513 + 0.673142i \(0.235056\pi\)
\(948\) 0 0
\(949\) −9.53148e6 −0.343554
\(950\) 10780.0 0.000387534 0
\(951\) 0 0
\(952\) 0 0
\(953\) 6.71983e6 0.239677 0.119838 0.992793i \(-0.461762\pi\)
0.119838 + 0.992793i \(0.461762\pi\)
\(954\) 0 0
\(955\) 2.00628e7 0.711841
\(956\) −3.28375e7 −1.16205
\(957\) 0 0
\(958\) 2.60330e7 0.916454
\(959\) 0 0
\(960\) 0 0
\(961\) 2.98016e7 1.04095
\(962\) −1.45572e7 −0.507154
\(963\) 0 0
\(964\) −5.48019e7 −1.89934
\(965\) 5.54150e7 1.91562
\(966\) 0 0
\(967\) −2.78979e6 −0.0959413 −0.0479707 0.998849i \(-0.515275\pi\)
−0.0479707 + 0.998849i \(0.515275\pi\)
\(968\) −3.86017e7 −1.32409
\(969\) 0 0
\(970\) −3.27634e7 −1.11804
\(971\) 3.33594e7 1.13545 0.567727 0.823217i \(-0.307823\pi\)
0.567727 + 0.823217i \(0.307823\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 5.46309e7 1.84519
\(975\) 0 0
\(976\) −3.65228e7 −1.22727
\(977\) 7.60033e6 0.254739 0.127370 0.991855i \(-0.459347\pi\)
0.127370 + 0.991855i \(0.459347\pi\)
\(978\) 0 0
\(979\) 1.17350e7 0.391316
\(980\) 0 0
\(981\) 0 0
\(982\) −1.64090e7 −0.543004
\(983\) −5.79760e6 −0.191366 −0.0956829 0.995412i \(-0.530503\pi\)
−0.0956829 + 0.995412i \(0.530503\pi\)
\(984\) 0 0
\(985\) −5.54428e7 −1.82077
\(986\) 5.88580e7 1.92803
\(987\) 0 0
\(988\) 932960. 0.0304068
\(989\) −1.98451e7 −0.645153
\(990\) 0 0
\(991\) 1.26825e7 0.410224 0.205112 0.978739i \(-0.434244\pi\)
0.205112 + 0.978739i \(0.434244\pi\)
\(992\) 2.07917e7 0.670827
\(993\) 0 0
\(994\) 0 0
\(995\) −4.70823e7 −1.50765
\(996\) 0 0
\(997\) −1.44400e7 −0.460077 −0.230039 0.973182i \(-0.573885\pi\)
−0.230039 + 0.973182i \(0.573885\pi\)
\(998\) 2.99796e7 0.952796
\(999\) 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 441.6.a.k.1.1 1
3.2 odd 2 49.6.a.a.1.1 1
7.6 odd 2 63.6.a.e.1.1 1
12.11 even 2 784.6.a.c.1.1 1
21.2 odd 6 49.6.c.b.18.1 2
21.5 even 6 49.6.c.c.18.1 2
21.11 odd 6 49.6.c.b.30.1 2
21.17 even 6 49.6.c.c.30.1 2
21.20 even 2 7.6.a.a.1.1 1
28.27 even 2 1008.6.a.y.1.1 1
84.83 odd 2 112.6.a.g.1.1 1
105.62 odd 4 175.6.b.a.99.1 2
105.83 odd 4 175.6.b.a.99.2 2
105.104 even 2 175.6.a.b.1.1 1
168.83 odd 2 448.6.a.c.1.1 1
168.125 even 2 448.6.a.m.1.1 1
231.230 odd 2 847.6.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.6.a.a.1.1 1 21.20 even 2
49.6.a.a.1.1 1 3.2 odd 2
49.6.c.b.18.1 2 21.2 odd 6
49.6.c.b.30.1 2 21.11 odd 6
49.6.c.c.18.1 2 21.5 even 6
49.6.c.c.30.1 2 21.17 even 6
63.6.a.e.1.1 1 7.6 odd 2
112.6.a.g.1.1 1 84.83 odd 2
175.6.a.b.1.1 1 105.104 even 2
175.6.b.a.99.1 2 105.62 odd 4
175.6.b.a.99.2 2 105.83 odd 4
441.6.a.k.1.1 1 1.1 even 1 trivial
448.6.a.c.1.1 1 168.83 odd 2
448.6.a.m.1.1 1 168.125 even 2
784.6.a.c.1.1 1 12.11 even 2
847.6.a.b.1.1 1 231.230 odd 2
1008.6.a.y.1.1 1 28.27 even 2