Properties

Label 294.6.a.g.1.1
Level $294$
Weight $6$
Character 294.1
Self dual yes
Analytic conductor $47.153$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [294,6,Mod(1,294)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(294, 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("294.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 294.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: \(47.1528430250\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 294.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} +26.0000 q^{5} -36.0000 q^{6} -64.0000 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} +26.0000 q^{5} -36.0000 q^{6} -64.0000 q^{8} +81.0000 q^{9} -104.000 q^{10} -358.000 q^{11} +144.000 q^{12} +332.000 q^{13} +234.000 q^{15} +256.000 q^{16} +126.000 q^{17} -324.000 q^{18} -2200.00 q^{19} +416.000 q^{20} +1432.00 q^{22} -2142.00 q^{23} -576.000 q^{24} -2449.00 q^{25} -1328.00 q^{26} +729.000 q^{27} -3610.00 q^{29} -936.000 q^{30} +5668.00 q^{31} -1024.00 q^{32} -3222.00 q^{33} -504.000 q^{34} +1296.00 q^{36} -2922.00 q^{37} +8800.00 q^{38} +2988.00 q^{39} -1664.00 q^{40} -2142.00 q^{41} +6388.00 q^{43} -5728.00 q^{44} +2106.00 q^{45} +8568.00 q^{46} -6520.00 q^{47} +2304.00 q^{48} +9796.00 q^{50} +1134.00 q^{51} +5312.00 q^{52} -10702.0 q^{53} -2916.00 q^{54} -9308.00 q^{55} -19800.0 q^{57} +14440.0 q^{58} +42524.0 q^{59} +3744.00 q^{60} -44840.0 q^{61} -22672.0 q^{62} +4096.00 q^{64} +8632.00 q^{65} +12888.0 q^{66} -1448.00 q^{67} +2016.00 q^{68} -19278.0 q^{69} -4402.00 q^{71} -5184.00 q^{72} +20500.0 q^{73} +11688.0 q^{74} -22041.0 q^{75} -35200.0 q^{76} -11952.0 q^{78} +65236.0 q^{79} +6656.00 q^{80} +6561.00 q^{81} +8568.00 q^{82} -102804. q^{83} +3276.00 q^{85} -25552.0 q^{86} -32490.0 q^{87} +22912.0 q^{88} -128006. q^{89} -8424.00 q^{90} -34272.0 q^{92} +51012.0 q^{93} +26080.0 q^{94} -57200.0 q^{95} -9216.00 q^{96} -113324. q^{97} -28998.0 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.00000 −0.707107
\(3\) 9.00000 0.577350
\(4\) 16.0000 0.500000
\(5\) 26.0000 0.465102 0.232551 0.972584i \(-0.425293\pi\)
0.232551 + 0.972584i \(0.425293\pi\)
\(6\) −36.0000 −0.408248
\(7\) 0 0
\(8\) −64.0000 −0.353553
\(9\) 81.0000 0.333333
\(10\) −104.000 −0.328877
\(11\) −358.000 −0.892075 −0.446037 0.895014i \(-0.647165\pi\)
−0.446037 + 0.895014i \(0.647165\pi\)
\(12\) 144.000 0.288675
\(13\) 332.000 0.544853 0.272427 0.962177i \(-0.412174\pi\)
0.272427 + 0.962177i \(0.412174\pi\)
\(14\) 0 0
\(15\) 234.000 0.268527
\(16\) 256.000 0.250000
\(17\) 126.000 0.105742 0.0528711 0.998601i \(-0.483163\pi\)
0.0528711 + 0.998601i \(0.483163\pi\)
\(18\) −324.000 −0.235702
\(19\) −2200.00 −1.39810 −0.699051 0.715072i \(-0.746394\pi\)
−0.699051 + 0.715072i \(0.746394\pi\)
\(20\) 416.000 0.232551
\(21\) 0 0
\(22\) 1432.00 0.630792
\(23\) −2142.00 −0.844306 −0.422153 0.906525i \(-0.638726\pi\)
−0.422153 + 0.906525i \(0.638726\pi\)
\(24\) −576.000 −0.204124
\(25\) −2449.00 −0.783680
\(26\) −1328.00 −0.385270
\(27\) 729.000 0.192450
\(28\) 0 0
\(29\) −3610.00 −0.797099 −0.398549 0.917147i \(-0.630486\pi\)
−0.398549 + 0.917147i \(0.630486\pi\)
\(30\) −936.000 −0.189877
\(31\) 5668.00 1.05932 0.529658 0.848211i \(-0.322320\pi\)
0.529658 + 0.848211i \(0.322320\pi\)
\(32\) −1024.00 −0.176777
\(33\) −3222.00 −0.515040
\(34\) −504.000 −0.0747710
\(35\) 0 0
\(36\) 1296.00 0.166667
\(37\) −2922.00 −0.350894 −0.175447 0.984489i \(-0.556137\pi\)
−0.175447 + 0.984489i \(0.556137\pi\)
\(38\) 8800.00 0.988607
\(39\) 2988.00 0.314571
\(40\) −1664.00 −0.164438
\(41\) −2142.00 −0.199003 −0.0995015 0.995037i \(-0.531725\pi\)
−0.0995015 + 0.995037i \(0.531725\pi\)
\(42\) 0 0
\(43\) 6388.00 0.526858 0.263429 0.964679i \(-0.415146\pi\)
0.263429 + 0.964679i \(0.415146\pi\)
\(44\) −5728.00 −0.446037
\(45\) 2106.00 0.155034
\(46\) 8568.00 0.597014
\(47\) −6520.00 −0.430530 −0.215265 0.976556i \(-0.569061\pi\)
−0.215265 + 0.976556i \(0.569061\pi\)
\(48\) 2304.00 0.144338
\(49\) 0 0
\(50\) 9796.00 0.554145
\(51\) 1134.00 0.0610503
\(52\) 5312.00 0.272427
\(53\) −10702.0 −0.523330 −0.261665 0.965159i \(-0.584271\pi\)
−0.261665 + 0.965159i \(0.584271\pi\)
\(54\) −2916.00 −0.136083
\(55\) −9308.00 −0.414906
\(56\) 0 0
\(57\) −19800.0 −0.807194
\(58\) 14440.0 0.563634
\(59\) 42524.0 1.59039 0.795196 0.606353i \(-0.207368\pi\)
0.795196 + 0.606353i \(0.207368\pi\)
\(60\) 3744.00 0.134263
\(61\) −44840.0 −1.54291 −0.771456 0.636283i \(-0.780471\pi\)
−0.771456 + 0.636283i \(0.780471\pi\)
\(62\) −22672.0 −0.749050
\(63\) 0 0
\(64\) 4096.00 0.125000
\(65\) 8632.00 0.253413
\(66\) 12888.0 0.364188
\(67\) −1448.00 −0.0394077 −0.0197039 0.999806i \(-0.506272\pi\)
−0.0197039 + 0.999806i \(0.506272\pi\)
\(68\) 2016.00 0.0528711
\(69\) −19278.0 −0.487460
\(70\) 0 0
\(71\) −4402.00 −0.103634 −0.0518172 0.998657i \(-0.516501\pi\)
−0.0518172 + 0.998657i \(0.516501\pi\)
\(72\) −5184.00 −0.117851
\(73\) 20500.0 0.450243 0.225121 0.974331i \(-0.427722\pi\)
0.225121 + 0.974331i \(0.427722\pi\)
\(74\) 11688.0 0.248120
\(75\) −22041.0 −0.452458
\(76\) −35200.0 −0.699051
\(77\) 0 0
\(78\) −11952.0 −0.222435
\(79\) 65236.0 1.17603 0.588017 0.808849i \(-0.299909\pi\)
0.588017 + 0.808849i \(0.299909\pi\)
\(80\) 6656.00 0.116276
\(81\) 6561.00 0.111111
\(82\) 8568.00 0.140716
\(83\) −102804. −1.63800 −0.819002 0.573791i \(-0.805472\pi\)
−0.819002 + 0.573791i \(0.805472\pi\)
\(84\) 0 0
\(85\) 3276.00 0.0491809
\(86\) −25552.0 −0.372545
\(87\) −32490.0 −0.460205
\(88\) 22912.0 0.315396
\(89\) −128006. −1.71299 −0.856496 0.516154i \(-0.827363\pi\)
−0.856496 + 0.516154i \(0.827363\pi\)
\(90\) −8424.00 −0.109626
\(91\) 0 0
\(92\) −34272.0 −0.422153
\(93\) 51012.0 0.611596
\(94\) 26080.0 0.304430
\(95\) −57200.0 −0.650260
\(96\) −9216.00 −0.102062
\(97\) −113324. −1.22290 −0.611452 0.791281i \(-0.709414\pi\)
−0.611452 + 0.791281i \(0.709414\pi\)
\(98\) 0 0
\(99\) −28998.0 −0.297358
\(100\) −39184.0 −0.391840
\(101\) −139714. −1.36281 −0.681407 0.731905i \(-0.738632\pi\)
−0.681407 + 0.731905i \(0.738632\pi\)
\(102\) −4536.00 −0.0431691
\(103\) −142180. −1.32052 −0.660261 0.751036i \(-0.729554\pi\)
−0.660261 + 0.751036i \(0.729554\pi\)
\(104\) −21248.0 −0.192635
\(105\) 0 0
\(106\) 42808.0 0.370050
\(107\) −198518. −1.67626 −0.838128 0.545473i \(-0.816350\pi\)
−0.838128 + 0.545473i \(0.816350\pi\)
\(108\) 11664.0 0.0962250
\(109\) 132538. 1.06850 0.534250 0.845327i \(-0.320594\pi\)
0.534250 + 0.845327i \(0.320594\pi\)
\(110\) 37232.0 0.293383
\(111\) −26298.0 −0.202589
\(112\) 0 0
\(113\) 47026.0 0.346451 0.173226 0.984882i \(-0.444581\pi\)
0.173226 + 0.984882i \(0.444581\pi\)
\(114\) 79200.0 0.570773
\(115\) −55692.0 −0.392689
\(116\) −57760.0 −0.398549
\(117\) 26892.0 0.181618
\(118\) −170096. −1.12458
\(119\) 0 0
\(120\) −14976.0 −0.0949386
\(121\) −32887.0 −0.204202
\(122\) 179360. 1.09100
\(123\) −19278.0 −0.114894
\(124\) 90688.0 0.529658
\(125\) −144924. −0.829593
\(126\) 0 0
\(127\) 165548. 0.910782 0.455391 0.890291i \(-0.349499\pi\)
0.455391 + 0.890291i \(0.349499\pi\)
\(128\) −16384.0 −0.0883883
\(129\) 57492.0 0.304182
\(130\) −34528.0 −0.179190
\(131\) −139308. −0.709248 −0.354624 0.935009i \(-0.615391\pi\)
−0.354624 + 0.935009i \(0.615391\pi\)
\(132\) −51552.0 −0.257520
\(133\) 0 0
\(134\) 5792.00 0.0278655
\(135\) 18954.0 0.0895089
\(136\) −8064.00 −0.0373855
\(137\) −332842. −1.51508 −0.757542 0.652786i \(-0.773600\pi\)
−0.757542 + 0.652786i \(0.773600\pi\)
\(138\) 77112.0 0.344686
\(139\) 8556.00 0.0375607 0.0187804 0.999824i \(-0.494022\pi\)
0.0187804 + 0.999824i \(0.494022\pi\)
\(140\) 0 0
\(141\) −58680.0 −0.248566
\(142\) 17608.0 0.0732806
\(143\) −118856. −0.486050
\(144\) 20736.0 0.0833333
\(145\) −93860.0 −0.370732
\(146\) −82000.0 −0.318370
\(147\) 0 0
\(148\) −46752.0 −0.175447
\(149\) 69554.0 0.256659 0.128329 0.991732i \(-0.459038\pi\)
0.128329 + 0.991732i \(0.459038\pi\)
\(150\) 88164.0 0.319936
\(151\) 529240. 1.88891 0.944453 0.328647i \(-0.106593\pi\)
0.944453 + 0.328647i \(0.106593\pi\)
\(152\) 140800. 0.494303
\(153\) 10206.0 0.0352474
\(154\) 0 0
\(155\) 147368. 0.492690
\(156\) 47808.0 0.157286
\(157\) 13040.0 0.0422210 0.0211105 0.999777i \(-0.493280\pi\)
0.0211105 + 0.999777i \(0.493280\pi\)
\(158\) −260944. −0.831581
\(159\) −96318.0 −0.302144
\(160\) −26624.0 −0.0822192
\(161\) 0 0
\(162\) −26244.0 −0.0785674
\(163\) −351240. −1.03546 −0.517732 0.855543i \(-0.673224\pi\)
−0.517732 + 0.855543i \(0.673224\pi\)
\(164\) −34272.0 −0.0995015
\(165\) −83772.0 −0.239546
\(166\) 411216. 1.15824
\(167\) 626128. 1.73729 0.868644 0.495436i \(-0.164992\pi\)
0.868644 + 0.495436i \(0.164992\pi\)
\(168\) 0 0
\(169\) −261069. −0.703135
\(170\) −13104.0 −0.0347762
\(171\) −178200. −0.466034
\(172\) 102208. 0.263429
\(173\) −184826. −0.469513 −0.234757 0.972054i \(-0.575429\pi\)
−0.234757 + 0.972054i \(0.575429\pi\)
\(174\) 129960. 0.325414
\(175\) 0 0
\(176\) −91648.0 −0.223019
\(177\) 382716. 0.918213
\(178\) 512024. 1.21127
\(179\) −357522. −0.834008 −0.417004 0.908905i \(-0.636920\pi\)
−0.417004 + 0.908905i \(0.636920\pi\)
\(180\) 33696.0 0.0775170
\(181\) 696508. 1.58026 0.790132 0.612937i \(-0.210012\pi\)
0.790132 + 0.612937i \(0.210012\pi\)
\(182\) 0 0
\(183\) −403560. −0.890800
\(184\) 137088. 0.298507
\(185\) −75972.0 −0.163202
\(186\) −204048. −0.432464
\(187\) −45108.0 −0.0943299
\(188\) −104320. −0.215265
\(189\) 0 0
\(190\) 228800. 0.459803
\(191\) 68670.0 0.136202 0.0681010 0.997678i \(-0.478306\pi\)
0.0681010 + 0.997678i \(0.478306\pi\)
\(192\) 36864.0 0.0721688
\(193\) 827222. 1.59856 0.799280 0.600959i \(-0.205215\pi\)
0.799280 + 0.600959i \(0.205215\pi\)
\(194\) 453296. 0.864724
\(195\) 77688.0 0.146308
\(196\) 0 0
\(197\) −143382. −0.263226 −0.131613 0.991301i \(-0.542016\pi\)
−0.131613 + 0.991301i \(0.542016\pi\)
\(198\) 115992. 0.210264
\(199\) −542600. −0.971286 −0.485643 0.874157i \(-0.661414\pi\)
−0.485643 + 0.874157i \(0.661414\pi\)
\(200\) 156736. 0.277073
\(201\) −13032.0 −0.0227521
\(202\) 558856. 0.963655
\(203\) 0 0
\(204\) 18144.0 0.0305251
\(205\) −55692.0 −0.0925568
\(206\) 568720. 0.933750
\(207\) −173502. −0.281435
\(208\) 84992.0 0.136213
\(209\) 787600. 1.24721
\(210\) 0 0
\(211\) 1.12776e6 1.74385 0.871925 0.489640i \(-0.162872\pi\)
0.871925 + 0.489640i \(0.162872\pi\)
\(212\) −171232. −0.261665
\(213\) −39618.0 −0.0598334
\(214\) 794072. 1.18529
\(215\) 166088. 0.245043
\(216\) −46656.0 −0.0680414
\(217\) 0 0
\(218\) −530152. −0.755543
\(219\) 184500. 0.259948
\(220\) −148928. −0.207453
\(221\) 41832.0 0.0576140
\(222\) 105192. 0.143252
\(223\) 897976. 1.20921 0.604606 0.796525i \(-0.293331\pi\)
0.604606 + 0.796525i \(0.293331\pi\)
\(224\) 0 0
\(225\) −198369. −0.261227
\(226\) −188104. −0.244978
\(227\) −467612. −0.602311 −0.301156 0.953575i \(-0.597372\pi\)
−0.301156 + 0.953575i \(0.597372\pi\)
\(228\) −316800. −0.403597
\(229\) −446140. −0.562189 −0.281095 0.959680i \(-0.590697\pi\)
−0.281095 + 0.959680i \(0.590697\pi\)
\(230\) 222768. 0.277673
\(231\) 0 0
\(232\) 231040. 0.281817
\(233\) 701486. 0.846504 0.423252 0.906012i \(-0.360888\pi\)
0.423252 + 0.906012i \(0.360888\pi\)
\(234\) −107568. −0.128423
\(235\) −169520. −0.200240
\(236\) 680384. 0.795196
\(237\) 587124. 0.678983
\(238\) 0 0
\(239\) −384198. −0.435071 −0.217536 0.976052i \(-0.569802\pi\)
−0.217536 + 0.976052i \(0.569802\pi\)
\(240\) 59904.0 0.0671317
\(241\) −953780. −1.05780 −0.528902 0.848683i \(-0.677396\pi\)
−0.528902 + 0.848683i \(0.677396\pi\)
\(242\) 131548. 0.144393
\(243\) 59049.0 0.0641500
\(244\) −717440. −0.771456
\(245\) 0 0
\(246\) 77112.0 0.0812427
\(247\) −730400. −0.761760
\(248\) −362752. −0.374525
\(249\) −925236. −0.945702
\(250\) 579696. 0.586611
\(251\) 569540. 0.570611 0.285305 0.958437i \(-0.407905\pi\)
0.285305 + 0.958437i \(0.407905\pi\)
\(252\) 0 0
\(253\) 766836. 0.753184
\(254\) −662192. −0.644020
\(255\) 29484.0 0.0283946
\(256\) 65536.0 0.0625000
\(257\) 1.06664e6 1.00736 0.503681 0.863890i \(-0.331979\pi\)
0.503681 + 0.863890i \(0.331979\pi\)
\(258\) −229968. −0.215089
\(259\) 0 0
\(260\) 138112. 0.126706
\(261\) −292410. −0.265700
\(262\) 557232. 0.501514
\(263\) −1.48243e6 −1.32155 −0.660777 0.750582i \(-0.729773\pi\)
−0.660777 + 0.750582i \(0.729773\pi\)
\(264\) 206208. 0.182094
\(265\) −278252. −0.243402
\(266\) 0 0
\(267\) −1.15205e6 −0.988996
\(268\) −23168.0 −0.0197039
\(269\) −215110. −0.181251 −0.0906254 0.995885i \(-0.528887\pi\)
−0.0906254 + 0.995885i \(0.528887\pi\)
\(270\) −75816.0 −0.0632924
\(271\) 1.93104e6 1.59723 0.798614 0.601843i \(-0.205567\pi\)
0.798614 + 0.601843i \(0.205567\pi\)
\(272\) 32256.0 0.0264355
\(273\) 0 0
\(274\) 1.33137e6 1.07133
\(275\) 876742. 0.699101
\(276\) −308448. −0.243730
\(277\) 2.03756e6 1.59555 0.797777 0.602953i \(-0.206009\pi\)
0.797777 + 0.602953i \(0.206009\pi\)
\(278\) −34224.0 −0.0265594
\(279\) 459108. 0.353105
\(280\) 0 0
\(281\) −639066. −0.482814 −0.241407 0.970424i \(-0.577609\pi\)
−0.241407 + 0.970424i \(0.577609\pi\)
\(282\) 234720. 0.175763
\(283\) 37744.0 0.0280144 0.0140072 0.999902i \(-0.495541\pi\)
0.0140072 + 0.999902i \(0.495541\pi\)
\(284\) −70432.0 −0.0518172
\(285\) −514800. −0.375428
\(286\) 475424. 0.343689
\(287\) 0 0
\(288\) −82944.0 −0.0589256
\(289\) −1.40398e6 −0.988819
\(290\) 375440. 0.262147
\(291\) −1.01992e6 −0.706044
\(292\) 328000. 0.225121
\(293\) −1.83921e6 −1.25159 −0.625795 0.779987i \(-0.715225\pi\)
−0.625795 + 0.779987i \(0.715225\pi\)
\(294\) 0 0
\(295\) 1.10562e6 0.739695
\(296\) 187008. 0.124060
\(297\) −260982. −0.171680
\(298\) −278216. −0.181485
\(299\) −711144. −0.460023
\(300\) −352656. −0.226229
\(301\) 0 0
\(302\) −2.11696e6 −1.33566
\(303\) −1.25743e6 −0.786821
\(304\) −563200. −0.349525
\(305\) −1.16584e6 −0.717611
\(306\) −40824.0 −0.0249237
\(307\) −1.06472e6 −0.644747 −0.322374 0.946613i \(-0.604481\pi\)
−0.322374 + 0.946613i \(0.604481\pi\)
\(308\) 0 0
\(309\) −1.27962e6 −0.762403
\(310\) −589472. −0.348385
\(311\) −1.00952e6 −0.591853 −0.295927 0.955211i \(-0.595628\pi\)
−0.295927 + 0.955211i \(0.595628\pi\)
\(312\) −191232. −0.111218
\(313\) −1.44910e6 −0.836058 −0.418029 0.908434i \(-0.637279\pi\)
−0.418029 + 0.908434i \(0.637279\pi\)
\(314\) −52160.0 −0.0298548
\(315\) 0 0
\(316\) 1.04378e6 0.588017
\(317\) 2.72311e6 1.52201 0.761003 0.648748i \(-0.224707\pi\)
0.761003 + 0.648748i \(0.224707\pi\)
\(318\) 385272. 0.213648
\(319\) 1.29238e6 0.711072
\(320\) 106496. 0.0581378
\(321\) −1.78666e6 −0.967787
\(322\) 0 0
\(323\) −277200. −0.147838
\(324\) 104976. 0.0555556
\(325\) −813068. −0.426991
\(326\) 1.40496e6 0.732184
\(327\) 1.19284e6 0.616898
\(328\) 137088. 0.0703582
\(329\) 0 0
\(330\) 335088. 0.169385
\(331\) −1.10040e6 −0.552055 −0.276027 0.961150i \(-0.589018\pi\)
−0.276027 + 0.961150i \(0.589018\pi\)
\(332\) −1.64486e6 −0.819002
\(333\) −236682. −0.116965
\(334\) −2.50451e6 −1.22845
\(335\) −37648.0 −0.0183286
\(336\) 0 0
\(337\) 1.73512e6 0.832251 0.416125 0.909307i \(-0.363388\pi\)
0.416125 + 0.909307i \(0.363388\pi\)
\(338\) 1.04428e6 0.497191
\(339\) 423234. 0.200024
\(340\) 52416.0 0.0245905
\(341\) −2.02914e6 −0.944989
\(342\) 712800. 0.329536
\(343\) 0 0
\(344\) −408832. −0.186273
\(345\) −501228. −0.226719
\(346\) 739304. 0.331996
\(347\) 1.59145e6 0.709526 0.354763 0.934956i \(-0.384562\pi\)
0.354763 + 0.934956i \(0.384562\pi\)
\(348\) −519840. −0.230103
\(349\) −2.33376e6 −1.02563 −0.512817 0.858498i \(-0.671398\pi\)
−0.512817 + 0.858498i \(0.671398\pi\)
\(350\) 0 0
\(351\) 242028. 0.104857
\(352\) 366592. 0.157698
\(353\) 2.81081e6 1.20059 0.600296 0.799778i \(-0.295049\pi\)
0.600296 + 0.799778i \(0.295049\pi\)
\(354\) −1.53086e6 −0.649275
\(355\) −114452. −0.0482006
\(356\) −2.04810e6 −0.856496
\(357\) 0 0
\(358\) 1.43009e6 0.589733
\(359\) 939310. 0.384656 0.192328 0.981331i \(-0.438396\pi\)
0.192328 + 0.981331i \(0.438396\pi\)
\(360\) −134784. −0.0548128
\(361\) 2.36390e6 0.954688
\(362\) −2.78603e6 −1.11742
\(363\) −295983. −0.117896
\(364\) 0 0
\(365\) 533000. 0.209409
\(366\) 1.61424e6 0.629891
\(367\) 3.09851e6 1.20085 0.600424 0.799682i \(-0.294999\pi\)
0.600424 + 0.799682i \(0.294999\pi\)
\(368\) −548352. −0.211077
\(369\) −173502. −0.0663344
\(370\) 303888. 0.115401
\(371\) 0 0
\(372\) 816192. 0.305798
\(373\) −228266. −0.0849511 −0.0424756 0.999098i \(-0.513524\pi\)
−0.0424756 + 0.999098i \(0.513524\pi\)
\(374\) 180432. 0.0667013
\(375\) −1.30432e6 −0.478966
\(376\) 417280. 0.152215
\(377\) −1.19852e6 −0.434302
\(378\) 0 0
\(379\) −1.03669e6 −0.370725 −0.185362 0.982670i \(-0.559346\pi\)
−0.185362 + 0.982670i \(0.559346\pi\)
\(380\) −915200. −0.325130
\(381\) 1.48993e6 0.525840
\(382\) −274680. −0.0963094
\(383\) −211776. −0.0737700 −0.0368850 0.999320i \(-0.511744\pi\)
−0.0368850 + 0.999320i \(0.511744\pi\)
\(384\) −147456. −0.0510310
\(385\) 0 0
\(386\) −3.30889e6 −1.13035
\(387\) 517428. 0.175619
\(388\) −1.81318e6 −0.611452
\(389\) 1.41325e6 0.473526 0.236763 0.971567i \(-0.423914\pi\)
0.236763 + 0.971567i \(0.423914\pi\)
\(390\) −310752. −0.103455
\(391\) −269892. −0.0892788
\(392\) 0 0
\(393\) −1.25377e6 −0.409484
\(394\) 573528. 0.186129
\(395\) 1.69614e6 0.546976
\(396\) −463968. −0.148679
\(397\) −1.09034e6 −0.347203 −0.173602 0.984816i \(-0.555541\pi\)
−0.173602 + 0.984816i \(0.555541\pi\)
\(398\) 2.17040e6 0.686803
\(399\) 0 0
\(400\) −626944. −0.195920
\(401\) 2.64253e6 0.820651 0.410325 0.911939i \(-0.365415\pi\)
0.410325 + 0.911939i \(0.365415\pi\)
\(402\) 52128.0 0.0160881
\(403\) 1.88178e6 0.577172
\(404\) −2.23542e6 −0.681407
\(405\) 170586. 0.0516780
\(406\) 0 0
\(407\) 1.04608e6 0.313024
\(408\) −72576.0 −0.0215845
\(409\) 6.25427e6 1.84871 0.924354 0.381536i \(-0.124605\pi\)
0.924354 + 0.381536i \(0.124605\pi\)
\(410\) 222768. 0.0654475
\(411\) −2.99558e6 −0.874734
\(412\) −2.27488e6 −0.660261
\(413\) 0 0
\(414\) 694008. 0.199005
\(415\) −2.67290e6 −0.761839
\(416\) −339968. −0.0963174
\(417\) 77004.0 0.0216857
\(418\) −3.15040e6 −0.881911
\(419\) −973924. −0.271013 −0.135506 0.990776i \(-0.543266\pi\)
−0.135506 + 0.990776i \(0.543266\pi\)
\(420\) 0 0
\(421\) 864618. 0.237749 0.118875 0.992909i \(-0.462071\pi\)
0.118875 + 0.992909i \(0.462071\pi\)
\(422\) −4.51102e6 −1.23309
\(423\) −528120. −0.143510
\(424\) 684928. 0.185025
\(425\) −308574. −0.0828680
\(426\) 158472. 0.0423086
\(427\) 0 0
\(428\) −3.17629e6 −0.838128
\(429\) −1.06970e6 −0.280621
\(430\) −664352. −0.173271
\(431\) −3.66046e6 −0.949166 −0.474583 0.880211i \(-0.657401\pi\)
−0.474583 + 0.880211i \(0.657401\pi\)
\(432\) 186624. 0.0481125
\(433\) −4.93667e6 −1.26536 −0.632681 0.774413i \(-0.718045\pi\)
−0.632681 + 0.774413i \(0.718045\pi\)
\(434\) 0 0
\(435\) −844740. −0.214042
\(436\) 2.12061e6 0.534250
\(437\) 4.71240e6 1.18043
\(438\) −738000. −0.183811
\(439\) 731304. 0.181108 0.0905538 0.995892i \(-0.471136\pi\)
0.0905538 + 0.995892i \(0.471136\pi\)
\(440\) 595712. 0.146691
\(441\) 0 0
\(442\) −167328. −0.0407392
\(443\) 4.86620e6 1.17810 0.589048 0.808098i \(-0.299503\pi\)
0.589048 + 0.808098i \(0.299503\pi\)
\(444\) −420768. −0.101294
\(445\) −3.32816e6 −0.796716
\(446\) −3.59190e6 −0.855042
\(447\) 625986. 0.148182
\(448\) 0 0
\(449\) 5.71987e6 1.33897 0.669484 0.742827i \(-0.266515\pi\)
0.669484 + 0.742827i \(0.266515\pi\)
\(450\) 793476. 0.184715
\(451\) 766836. 0.177526
\(452\) 752416. 0.173226
\(453\) 4.76316e6 1.09056
\(454\) 1.87045e6 0.425898
\(455\) 0 0
\(456\) 1.26720e6 0.285386
\(457\) −6.82034e6 −1.52762 −0.763811 0.645440i \(-0.776674\pi\)
−0.763811 + 0.645440i \(0.776674\pi\)
\(458\) 1.78456e6 0.397528
\(459\) 91854.0 0.0203501
\(460\) −891072. −0.196344
\(461\) −7.45934e6 −1.63474 −0.817369 0.576115i \(-0.804568\pi\)
−0.817369 + 0.576115i \(0.804568\pi\)
\(462\) 0 0
\(463\) −5.23848e6 −1.13567 −0.567836 0.823142i \(-0.692219\pi\)
−0.567836 + 0.823142i \(0.692219\pi\)
\(464\) −924160. −0.199275
\(465\) 1.32631e6 0.284455
\(466\) −2.80594e6 −0.598569
\(467\) 8.95995e6 1.90114 0.950568 0.310516i \(-0.100502\pi\)
0.950568 + 0.310516i \(0.100502\pi\)
\(468\) 430272. 0.0908089
\(469\) 0 0
\(470\) 678080. 0.141591
\(471\) 117360. 0.0243763
\(472\) −2.72154e6 −0.562288
\(473\) −2.28690e6 −0.469997
\(474\) −2.34850e6 −0.480114
\(475\) 5.38780e6 1.09566
\(476\) 0 0
\(477\) −866862. −0.174443
\(478\) 1.53679e6 0.307642
\(479\) −1.75354e6 −0.349201 −0.174601 0.984639i \(-0.555863\pi\)
−0.174601 + 0.984639i \(0.555863\pi\)
\(480\) −239616. −0.0474693
\(481\) −970104. −0.191186
\(482\) 3.81512e6 0.747981
\(483\) 0 0
\(484\) −526192. −0.102101
\(485\) −2.94642e6 −0.568776
\(486\) −236196. −0.0453609
\(487\) 927568. 0.177224 0.0886122 0.996066i \(-0.471757\pi\)
0.0886122 + 0.996066i \(0.471757\pi\)
\(488\) 2.86976e6 0.545502
\(489\) −3.16116e6 −0.597825
\(490\) 0 0
\(491\) 8.43733e6 1.57943 0.789716 0.613472i \(-0.210228\pi\)
0.789716 + 0.613472i \(0.210228\pi\)
\(492\) −308448. −0.0574472
\(493\) −454860. −0.0842870
\(494\) 2.92160e6 0.538646
\(495\) −753948. −0.138302
\(496\) 1.45101e6 0.264829
\(497\) 0 0
\(498\) 3.70094e6 0.668712
\(499\) 1.33278e6 0.239611 0.119806 0.992797i \(-0.461773\pi\)
0.119806 + 0.992797i \(0.461773\pi\)
\(500\) −2.31878e6 −0.414797
\(501\) 5.63515e6 1.00302
\(502\) −2.27816e6 −0.403483
\(503\) 3.64494e6 0.642349 0.321174 0.947020i \(-0.395922\pi\)
0.321174 + 0.947020i \(0.395922\pi\)
\(504\) 0 0
\(505\) −3.63256e6 −0.633848
\(506\) −3.06734e6 −0.532582
\(507\) −2.34962e6 −0.405955
\(508\) 2.64877e6 0.455391
\(509\) −3.26166e6 −0.558013 −0.279007 0.960289i \(-0.590005\pi\)
−0.279007 + 0.960289i \(0.590005\pi\)
\(510\) −117936. −0.0200780
\(511\) 0 0
\(512\) −262144. −0.0441942
\(513\) −1.60380e6 −0.269065
\(514\) −4.26657e6 −0.712313
\(515\) −3.69668e6 −0.614177
\(516\) 919872. 0.152091
\(517\) 2.33416e6 0.384065
\(518\) 0 0
\(519\) −1.66343e6 −0.271074
\(520\) −552448. −0.0895948
\(521\) −2.18741e6 −0.353050 −0.176525 0.984296i \(-0.556486\pi\)
−0.176525 + 0.984296i \(0.556486\pi\)
\(522\) 1.16964e6 0.187878
\(523\) 1.03890e7 1.66081 0.830406 0.557159i \(-0.188109\pi\)
0.830406 + 0.557159i \(0.188109\pi\)
\(524\) −2.22893e6 −0.354624
\(525\) 0 0
\(526\) 5.92972e6 0.934480
\(527\) 714168. 0.112014
\(528\) −824832. −0.128760
\(529\) −1.84818e6 −0.287147
\(530\) 1.11301e6 0.172111
\(531\) 3.44444e6 0.530131
\(532\) 0 0
\(533\) −711144. −0.108428
\(534\) 4.60822e6 0.699326
\(535\) −5.16147e6 −0.779630
\(536\) 92672.0 0.0139327
\(537\) −3.21770e6 −0.481515
\(538\) 860440. 0.128164
\(539\) 0 0
\(540\) 303264. 0.0447545
\(541\) 1.27724e7 1.87620 0.938101 0.346363i \(-0.112583\pi\)
0.938101 + 0.346363i \(0.112583\pi\)
\(542\) −7.72414e6 −1.12941
\(543\) 6.26857e6 0.912366
\(544\) −129024. −0.0186928
\(545\) 3.44599e6 0.496961
\(546\) 0 0
\(547\) −5.22238e6 −0.746278 −0.373139 0.927776i \(-0.621719\pi\)
−0.373139 + 0.927776i \(0.621719\pi\)
\(548\) −5.32547e6 −0.757542
\(549\) −3.63204e6 −0.514304
\(550\) −3.50697e6 −0.494339
\(551\) 7.94200e6 1.11443
\(552\) 1.23379e6 0.172343
\(553\) 0 0
\(554\) −8.15025e6 −1.12823
\(555\) −683748. −0.0942245
\(556\) 136896. 0.0187804
\(557\) −5.74047e6 −0.783988 −0.391994 0.919968i \(-0.628215\pi\)
−0.391994 + 0.919968i \(0.628215\pi\)
\(558\) −1.83643e6 −0.249683
\(559\) 2.12082e6 0.287061
\(560\) 0 0
\(561\) −405972. −0.0544614
\(562\) 2.55626e6 0.341401
\(563\) −2.30448e6 −0.306409 −0.153204 0.988195i \(-0.548959\pi\)
−0.153204 + 0.988195i \(0.548959\pi\)
\(564\) −938880. −0.124283
\(565\) 1.22268e6 0.161135
\(566\) −150976. −0.0198092
\(567\) 0 0
\(568\) 281728. 0.0366403
\(569\) −5.12150e6 −0.663157 −0.331578 0.943428i \(-0.607581\pi\)
−0.331578 + 0.943428i \(0.607581\pi\)
\(570\) 2.05920e6 0.265468
\(571\) −2.38637e6 −0.306300 −0.153150 0.988203i \(-0.548942\pi\)
−0.153150 + 0.988203i \(0.548942\pi\)
\(572\) −1.90170e6 −0.243025
\(573\) 618030. 0.0786363
\(574\) 0 0
\(575\) 5.24576e6 0.661666
\(576\) 331776. 0.0416667
\(577\) −5.24151e6 −0.655416 −0.327708 0.944779i \(-0.606276\pi\)
−0.327708 + 0.944779i \(0.606276\pi\)
\(578\) 5.61592e6 0.699200
\(579\) 7.44500e6 0.922929
\(580\) −1.50176e6 −0.185366
\(581\) 0 0
\(582\) 4.07966e6 0.499249
\(583\) 3.83132e6 0.466849
\(584\) −1.31200e6 −0.159185
\(585\) 699192. 0.0844708
\(586\) 7.35684e6 0.885008
\(587\) 9.11548e6 1.09190 0.545952 0.837816i \(-0.316168\pi\)
0.545952 + 0.837816i \(0.316168\pi\)
\(588\) 0 0
\(589\) −1.24696e7 −1.48103
\(590\) −4.42250e6 −0.523043
\(591\) −1.29044e6 −0.151974
\(592\) −748032. −0.0877235
\(593\) 3.05043e6 0.356225 0.178112 0.984010i \(-0.443001\pi\)
0.178112 + 0.984010i \(0.443001\pi\)
\(594\) 1.04393e6 0.121396
\(595\) 0 0
\(596\) 1.11286e6 0.128329
\(597\) −4.88340e6 −0.560772
\(598\) 2.84458e6 0.325285
\(599\) 1.43408e7 1.63308 0.816539 0.577290i \(-0.195890\pi\)
0.816539 + 0.577290i \(0.195890\pi\)
\(600\) 1.41062e6 0.159968
\(601\) −3.12662e6 −0.353092 −0.176546 0.984292i \(-0.556492\pi\)
−0.176546 + 0.984292i \(0.556492\pi\)
\(602\) 0 0
\(603\) −117288. −0.0131359
\(604\) 8.46784e6 0.944453
\(605\) −855062. −0.0949750
\(606\) 5.02970e6 0.556366
\(607\) 1.15098e7 1.26794 0.633969 0.773359i \(-0.281425\pi\)
0.633969 + 0.773359i \(0.281425\pi\)
\(608\) 2.25280e6 0.247152
\(609\) 0 0
\(610\) 4.66336e6 0.507428
\(611\) −2.16464e6 −0.234576
\(612\) 163296. 0.0176237
\(613\) −1.21782e7 −1.30898 −0.654488 0.756072i \(-0.727116\pi\)
−0.654488 + 0.756072i \(0.727116\pi\)
\(614\) 4.25888e6 0.455905
\(615\) −501228. −0.0534377
\(616\) 0 0
\(617\) 1.77629e6 0.187845 0.0939226 0.995580i \(-0.470059\pi\)
0.0939226 + 0.995580i \(0.470059\pi\)
\(618\) 5.11848e6 0.539101
\(619\) −5.95516e6 −0.624694 −0.312347 0.949968i \(-0.601115\pi\)
−0.312347 + 0.949968i \(0.601115\pi\)
\(620\) 2.35789e6 0.246345
\(621\) −1.56152e6 −0.162487
\(622\) 4.03808e6 0.418503
\(623\) 0 0
\(624\) 764928. 0.0786428
\(625\) 3.88510e6 0.397834
\(626\) 5.79638e6 0.591182
\(627\) 7.08840e6 0.720078
\(628\) 208640. 0.0211105
\(629\) −368172. −0.0371043
\(630\) 0 0
\(631\) −1.45351e7 −1.45327 −0.726633 0.687026i \(-0.758916\pi\)
−0.726633 + 0.687026i \(0.758916\pi\)
\(632\) −4.17510e6 −0.415791
\(633\) 1.01498e7 1.00681
\(634\) −1.08924e7 −1.07622
\(635\) 4.30425e6 0.423607
\(636\) −1.54109e6 −0.151072
\(637\) 0 0
\(638\) −5.16952e6 −0.502804
\(639\) −356562. −0.0345448
\(640\) −425984. −0.0411096
\(641\) −1.07349e7 −1.03194 −0.515970 0.856607i \(-0.672568\pi\)
−0.515970 + 0.856607i \(0.672568\pi\)
\(642\) 7.14665e6 0.684329
\(643\) 1.62815e7 1.55298 0.776492 0.630127i \(-0.216997\pi\)
0.776492 + 0.630127i \(0.216997\pi\)
\(644\) 0 0
\(645\) 1.49479e6 0.141476
\(646\) 1.10880e6 0.104537
\(647\) −7.91947e6 −0.743765 −0.371882 0.928280i \(-0.621288\pi\)
−0.371882 + 0.928280i \(0.621288\pi\)
\(648\) −419904. −0.0392837
\(649\) −1.52236e7 −1.41875
\(650\) 3.25227e6 0.301928
\(651\) 0 0
\(652\) −5.61984e6 −0.517732
\(653\) 1.34478e6 0.123415 0.0617076 0.998094i \(-0.480345\pi\)
0.0617076 + 0.998094i \(0.480345\pi\)
\(654\) −4.77137e6 −0.436213
\(655\) −3.62201e6 −0.329873
\(656\) −548352. −0.0497508
\(657\) 1.66050e6 0.150081
\(658\) 0 0
\(659\) 2.02235e7 1.81402 0.907010 0.421109i \(-0.138359\pi\)
0.907010 + 0.421109i \(0.138359\pi\)
\(660\) −1.34035e6 −0.119773
\(661\) 7.17802e6 0.639001 0.319500 0.947586i \(-0.396485\pi\)
0.319500 + 0.947586i \(0.396485\pi\)
\(662\) 4.40162e6 0.390362
\(663\) 376488. 0.0332635
\(664\) 6.57946e6 0.579122
\(665\) 0 0
\(666\) 946728. 0.0827065
\(667\) 7.73262e6 0.672995
\(668\) 1.00180e7 0.868644
\(669\) 8.08178e6 0.698139
\(670\) 150592. 0.0129603
\(671\) 1.60527e7 1.37639
\(672\) 0 0
\(673\) 9.61217e6 0.818057 0.409029 0.912522i \(-0.365868\pi\)
0.409029 + 0.912522i \(0.365868\pi\)
\(674\) −6.94047e6 −0.588490
\(675\) −1.78532e6 −0.150819
\(676\) −4.17710e6 −0.351567
\(677\) −9.66815e6 −0.810722 −0.405361 0.914157i \(-0.632854\pi\)
−0.405361 + 0.914157i \(0.632854\pi\)
\(678\) −1.69294e6 −0.141438
\(679\) 0 0
\(680\) −209664. −0.0173881
\(681\) −4.20851e6 −0.347745
\(682\) 8.11658e6 0.668208
\(683\) 389854. 0.0319779 0.0159890 0.999872i \(-0.494910\pi\)
0.0159890 + 0.999872i \(0.494910\pi\)
\(684\) −2.85120e6 −0.233017
\(685\) −8.65389e6 −0.704669
\(686\) 0 0
\(687\) −4.01526e6 −0.324580
\(688\) 1.63533e6 0.131715
\(689\) −3.55306e6 −0.285138
\(690\) 2.00491e6 0.160314
\(691\) 3.73985e6 0.297961 0.148980 0.988840i \(-0.452401\pi\)
0.148980 + 0.988840i \(0.452401\pi\)
\(692\) −2.95722e6 −0.234757
\(693\) 0 0
\(694\) −6.36578e6 −0.501711
\(695\) 222456. 0.0174696
\(696\) 2.07936e6 0.162707
\(697\) −269892. −0.0210430
\(698\) 9.33504e6 0.725233
\(699\) 6.31337e6 0.488730
\(700\) 0 0
\(701\) 2.49886e7 1.92064 0.960322 0.278893i \(-0.0899674\pi\)
0.960322 + 0.278893i \(0.0899674\pi\)
\(702\) −968112. −0.0741452
\(703\) 6.42840e6 0.490585
\(704\) −1.46637e6 −0.111509
\(705\) −1.52568e6 −0.115609
\(706\) −1.12433e7 −0.848947
\(707\) 0 0
\(708\) 6.12346e6 0.459107
\(709\) −9.83584e6 −0.734845 −0.367423 0.930054i \(-0.619760\pi\)
−0.367423 + 0.930054i \(0.619760\pi\)
\(710\) 457808. 0.0340830
\(711\) 5.28412e6 0.392011
\(712\) 8.19238e6 0.605634
\(713\) −1.21409e7 −0.894387
\(714\) 0 0
\(715\) −3.09026e6 −0.226063
\(716\) −5.72035e6 −0.417004
\(717\) −3.45778e6 −0.251188
\(718\) −3.75724e6 −0.271993
\(719\) −2.13624e7 −1.54109 −0.770546 0.637384i \(-0.780017\pi\)
−0.770546 + 0.637384i \(0.780017\pi\)
\(720\) 539136. 0.0387585
\(721\) 0 0
\(722\) −9.45560e6 −0.675066
\(723\) −8.58402e6 −0.610724
\(724\) 1.11441e7 0.790132
\(725\) 8.84089e6 0.624670
\(726\) 1.18393e6 0.0833653
\(727\) 6.53025e6 0.458241 0.229120 0.973398i \(-0.426415\pi\)
0.229120 + 0.973398i \(0.426415\pi\)
\(728\) 0 0
\(729\) 531441. 0.0370370
\(730\) −2.13200e6 −0.148074
\(731\) 804888. 0.0557111
\(732\) −6.45696e6 −0.445400
\(733\) −1.66571e7 −1.14509 −0.572545 0.819873i \(-0.694044\pi\)
−0.572545 + 0.819873i \(0.694044\pi\)
\(734\) −1.23940e7 −0.849128
\(735\) 0 0
\(736\) 2.19341e6 0.149254
\(737\) 518384. 0.0351547
\(738\) 694008. 0.0469055
\(739\) −2.39536e7 −1.61347 −0.806733 0.590917i \(-0.798766\pi\)
−0.806733 + 0.590917i \(0.798766\pi\)
\(740\) −1.21555e6 −0.0816008
\(741\) −6.57360e6 −0.439803
\(742\) 0 0
\(743\) −7.48982e6 −0.497736 −0.248868 0.968537i \(-0.580059\pi\)
−0.248868 + 0.968537i \(0.580059\pi\)
\(744\) −3.26477e6 −0.216232
\(745\) 1.80840e6 0.119373
\(746\) 913064. 0.0600695
\(747\) −8.32712e6 −0.546001
\(748\) −721728. −0.0471650
\(749\) 0 0
\(750\) 5.21726e6 0.338680
\(751\) −4.71845e6 −0.305281 −0.152640 0.988282i \(-0.548778\pi\)
−0.152640 + 0.988282i \(0.548778\pi\)
\(752\) −1.66912e6 −0.107632
\(753\) 5.12586e6 0.329442
\(754\) 4.79408e6 0.307098
\(755\) 1.37602e7 0.878534
\(756\) 0 0
\(757\) −2.67397e7 −1.69597 −0.847983 0.530024i \(-0.822183\pi\)
−0.847983 + 0.530024i \(0.822183\pi\)
\(758\) 4.14677e6 0.262142
\(759\) 6.90152e6 0.434851
\(760\) 3.66080e6 0.229902
\(761\) −1.44331e7 −0.903435 −0.451718 0.892161i \(-0.649189\pi\)
−0.451718 + 0.892161i \(0.649189\pi\)
\(762\) −5.95973e6 −0.371825
\(763\) 0 0
\(764\) 1.09872e6 0.0681010
\(765\) 265356. 0.0163936
\(766\) 847104. 0.0521633
\(767\) 1.41180e7 0.866530
\(768\) 589824. 0.0360844
\(769\) −8.55510e6 −0.521686 −0.260843 0.965381i \(-0.584001\pi\)
−0.260843 + 0.965381i \(0.584001\pi\)
\(770\) 0 0
\(771\) 9.59978e6 0.581601
\(772\) 1.32356e7 0.799280
\(773\) 1.92272e7 1.15735 0.578677 0.815557i \(-0.303569\pi\)
0.578677 + 0.815557i \(0.303569\pi\)
\(774\) −2.06971e6 −0.124182
\(775\) −1.38809e7 −0.830165
\(776\) 7.25274e6 0.432362
\(777\) 0 0
\(778\) −5.65298e6 −0.334833
\(779\) 4.71240e6 0.278227
\(780\) 1.24301e6 0.0731539
\(781\) 1.57592e6 0.0924497
\(782\) 1.07957e6 0.0631296
\(783\) −2.63169e6 −0.153402
\(784\) 0 0
\(785\) 339040. 0.0196371
\(786\) 5.01509e6 0.289549
\(787\) −2.53316e7 −1.45789 −0.728947 0.684570i \(-0.759990\pi\)
−0.728947 + 0.684570i \(0.759990\pi\)
\(788\) −2.29411e6 −0.131613
\(789\) −1.33419e7 −0.762999
\(790\) −6.78454e6 −0.386770
\(791\) 0 0
\(792\) 1.85587e6 0.105132
\(793\) −1.48869e7 −0.840661
\(794\) 4.36134e6 0.245510
\(795\) −2.50427e6 −0.140528
\(796\) −8.68160e6 −0.485643
\(797\) −3.13162e7 −1.74632 −0.873158 0.487437i \(-0.837932\pi\)
−0.873158 + 0.487437i \(0.837932\pi\)
\(798\) 0 0
\(799\) −821520. −0.0455251
\(800\) 2.50778e6 0.138536
\(801\) −1.03685e7 −0.570997
\(802\) −1.05701e7 −0.580288
\(803\) −7.33900e6 −0.401650
\(804\) −208512. −0.0113760
\(805\) 0 0
\(806\) −7.52710e6 −0.408122
\(807\) −1.93599e6 −0.104645
\(808\) 8.94170e6 0.481827
\(809\) 484890. 0.0260479 0.0130239 0.999915i \(-0.495854\pi\)
0.0130239 + 0.999915i \(0.495854\pi\)
\(810\) −682344. −0.0365419
\(811\) −5.32623e6 −0.284359 −0.142180 0.989841i \(-0.545411\pi\)
−0.142180 + 0.989841i \(0.545411\pi\)
\(812\) 0 0
\(813\) 1.73793e7 0.922161
\(814\) −4.18430e6 −0.221341
\(815\) −9.13224e6 −0.481596
\(816\) 290304. 0.0152626
\(817\) −1.40536e7 −0.736601
\(818\) −2.50171e7 −1.30723
\(819\) 0 0
\(820\) −891072. −0.0462784
\(821\) 3.21777e7 1.66609 0.833043 0.553209i \(-0.186597\pi\)
0.833043 + 0.553209i \(0.186597\pi\)
\(822\) 1.19823e7 0.618530
\(823\) −8.07408e6 −0.415521 −0.207761 0.978180i \(-0.566618\pi\)
−0.207761 + 0.978180i \(0.566618\pi\)
\(824\) 9.09952e6 0.466875
\(825\) 7.89068e6 0.403626
\(826\) 0 0
\(827\) −8.04922e6 −0.409251 −0.204626 0.978840i \(-0.565598\pi\)
−0.204626 + 0.978840i \(0.565598\pi\)
\(828\) −2.77603e6 −0.140718
\(829\) −1.35889e7 −0.686751 −0.343375 0.939198i \(-0.611570\pi\)
−0.343375 + 0.939198i \(0.611570\pi\)
\(830\) 1.06916e7 0.538701
\(831\) 1.83381e7 0.921193
\(832\) 1.35987e6 0.0681067
\(833\) 0 0
\(834\) −308016. −0.0153341
\(835\) 1.62793e7 0.808017
\(836\) 1.26016e7 0.623606
\(837\) 4.13197e6 0.203865
\(838\) 3.89570e6 0.191635
\(839\) −3.67721e6 −0.180349 −0.0901744 0.995926i \(-0.528742\pi\)
−0.0901744 + 0.995926i \(0.528742\pi\)
\(840\) 0 0
\(841\) −7.47905e6 −0.364633
\(842\) −3.45847e6 −0.168114
\(843\) −5.75159e6 −0.278753
\(844\) 1.80441e7 0.871925
\(845\) −6.78779e6 −0.327029
\(846\) 2.11248e6 0.101477
\(847\) 0 0
\(848\) −2.73971e6 −0.130832
\(849\) 339696. 0.0161741
\(850\) 1.23430e6 0.0585965
\(851\) 6.25892e6 0.296262
\(852\) −633888. −0.0299167
\(853\) −3.25379e7 −1.53115 −0.765573 0.643349i \(-0.777544\pi\)
−0.765573 + 0.643349i \(0.777544\pi\)
\(854\) 0 0
\(855\) −4.63320e6 −0.216753
\(856\) 1.27052e7 0.592646
\(857\) 1.12723e7 0.524278 0.262139 0.965030i \(-0.415572\pi\)
0.262139 + 0.965030i \(0.415572\pi\)
\(858\) 4.27882e6 0.198429
\(859\) −7.69694e6 −0.355906 −0.177953 0.984039i \(-0.556948\pi\)
−0.177953 + 0.984039i \(0.556948\pi\)
\(860\) 2.65741e6 0.122521
\(861\) 0 0
\(862\) 1.46418e7 0.671162
\(863\) −4.58785e6 −0.209692 −0.104846 0.994488i \(-0.533435\pi\)
−0.104846 + 0.994488i \(0.533435\pi\)
\(864\) −746496. −0.0340207
\(865\) −4.80548e6 −0.218372
\(866\) 1.97467e7 0.894746
\(867\) −1.26358e7 −0.570895
\(868\) 0 0
\(869\) −2.33545e7 −1.04911
\(870\) 3.37896e6 0.151351
\(871\) −480736. −0.0214714
\(872\) −8.48243e6 −0.377771
\(873\) −9.17924e6 −0.407635
\(874\) −1.88496e7 −0.834687
\(875\) 0 0
\(876\) 2.95200e6 0.129974
\(877\) −1.14666e7 −0.503424 −0.251712 0.967802i \(-0.580994\pi\)
−0.251712 + 0.967802i \(0.580994\pi\)
\(878\) −2.92522e6 −0.128062
\(879\) −1.65529e7 −0.722606
\(880\) −2.38285e6 −0.103726
\(881\) −3.02550e7 −1.31328 −0.656640 0.754204i \(-0.728023\pi\)
−0.656640 + 0.754204i \(0.728023\pi\)
\(882\) 0 0
\(883\) 9.83052e6 0.424302 0.212151 0.977237i \(-0.431953\pi\)
0.212151 + 0.977237i \(0.431953\pi\)
\(884\) 669312. 0.0288070
\(885\) 9.95062e6 0.427063
\(886\) −1.94648e7 −0.833039
\(887\) 2.32272e7 0.991263 0.495631 0.868533i \(-0.334937\pi\)
0.495631 + 0.868533i \(0.334937\pi\)
\(888\) 1.68307e6 0.0716259
\(889\) 0 0
\(890\) 1.33126e7 0.563363
\(891\) −2.34884e6 −0.0991194
\(892\) 1.43676e7 0.604606
\(893\) 1.43440e7 0.601924
\(894\) −2.50394e6 −0.104781
\(895\) −9.29557e6 −0.387899
\(896\) 0 0
\(897\) −6.40030e6 −0.265594
\(898\) −2.28795e7 −0.946793
\(899\) −2.04615e7 −0.844380
\(900\) −3.17390e6 −0.130613
\(901\) −1.34845e6 −0.0553380
\(902\) −3.06734e6 −0.125530
\(903\) 0 0
\(904\) −3.00966e6 −0.122489
\(905\) 1.81092e7 0.734984
\(906\) −1.90526e7 −0.771143
\(907\) 3.05501e7 1.23309 0.616544 0.787321i \(-0.288532\pi\)
0.616544 + 0.787321i \(0.288532\pi\)
\(908\) −7.48179e6 −0.301156
\(909\) −1.13168e7 −0.454271
\(910\) 0 0
\(911\) 2.21502e7 0.884265 0.442133 0.896950i \(-0.354222\pi\)
0.442133 + 0.896950i \(0.354222\pi\)
\(912\) −5.06880e6 −0.201799
\(913\) 3.68038e7 1.46122
\(914\) 2.72814e7 1.08019
\(915\) −1.04926e7 −0.414313
\(916\) −7.13824e6 −0.281095
\(917\) 0 0
\(918\) −367416. −0.0143897
\(919\) −1.26723e7 −0.494955 −0.247477 0.968894i \(-0.579602\pi\)
−0.247477 + 0.968894i \(0.579602\pi\)
\(920\) 3.56429e6 0.138836
\(921\) −9.58248e6 −0.372245
\(922\) 2.98374e7 1.15593
\(923\) −1.46146e6 −0.0564656
\(924\) 0 0
\(925\) 7.15598e6 0.274989
\(926\) 2.09539e7 0.803042
\(927\) −1.15166e7 −0.440174
\(928\) 3.69664e6 0.140909
\(929\) −4.02840e7 −1.53142 −0.765709 0.643187i \(-0.777612\pi\)
−0.765709 + 0.643187i \(0.777612\pi\)
\(930\) −5.30525e6 −0.201140
\(931\) 0 0
\(932\) 1.12238e7 0.423252
\(933\) −9.08568e6 −0.341707
\(934\) −3.58398e7 −1.34431
\(935\) −1.17281e6 −0.0438731
\(936\) −1.72109e6 −0.0642116
\(937\) 1.34104e7 0.498992 0.249496 0.968376i \(-0.419735\pi\)
0.249496 + 0.968376i \(0.419735\pi\)
\(938\) 0 0
\(939\) −1.30419e7 −0.482698
\(940\) −2.71232e6 −0.100120
\(941\) −2.73213e7 −1.00584 −0.502918 0.864334i \(-0.667740\pi\)
−0.502918 + 0.864334i \(0.667740\pi\)
\(942\) −469440. −0.0172366
\(943\) 4.58816e6 0.168020
\(944\) 1.08861e7 0.397598
\(945\) 0 0
\(946\) 9.14762e6 0.332338
\(947\) 8.71745e6 0.315874 0.157937 0.987449i \(-0.449516\pi\)
0.157937 + 0.987449i \(0.449516\pi\)
\(948\) 9.39398e6 0.339492
\(949\) 6.80600e6 0.245316
\(950\) −2.15512e7 −0.774752
\(951\) 2.45080e7 0.878731
\(952\) 0 0
\(953\) −1.62984e7 −0.581315 −0.290658 0.956827i \(-0.593874\pi\)
−0.290658 + 0.956827i \(0.593874\pi\)
\(954\) 3.46745e6 0.123350
\(955\) 1.78542e6 0.0633479
\(956\) −6.14717e6 −0.217536
\(957\) 1.16314e7 0.410538
\(958\) 7.01414e6 0.246923
\(959\) 0 0
\(960\) 958464. 0.0335659
\(961\) 3.49707e6 0.122151
\(962\) 3.88042e6 0.135189
\(963\) −1.60800e7 −0.558752
\(964\) −1.52605e7 −0.528902
\(965\) 2.15078e7 0.743493
\(966\) 0 0
\(967\) −5.49067e6 −0.188825 −0.0944124 0.995533i \(-0.530097\pi\)
−0.0944124 + 0.995533i \(0.530097\pi\)
\(968\) 2.10477e6 0.0721964
\(969\) −2.49480e6 −0.0853545
\(970\) 1.17857e7 0.402185
\(971\) 4.51675e7 1.53737 0.768685 0.639628i \(-0.220911\pi\)
0.768685 + 0.639628i \(0.220911\pi\)
\(972\) 944784. 0.0320750
\(973\) 0 0
\(974\) −3.71027e6 −0.125317
\(975\) −7.31761e6 −0.246523
\(976\) −1.14790e7 −0.385728
\(977\) −2.38010e7 −0.797737 −0.398868 0.917008i \(-0.630597\pi\)
−0.398868 + 0.917008i \(0.630597\pi\)
\(978\) 1.26446e7 0.422726
\(979\) 4.58261e7 1.52812
\(980\) 0 0
\(981\) 1.07356e7 0.356166
\(982\) −3.37493e7 −1.11683
\(983\) 9.36478e6 0.309111 0.154555 0.987984i \(-0.450606\pi\)
0.154555 + 0.987984i \(0.450606\pi\)
\(984\) 1.23379e6 0.0406213
\(985\) −3.72793e6 −0.122427
\(986\) 1.81944e6 0.0595999
\(987\) 0 0
\(988\) −1.16864e7 −0.380880
\(989\) −1.36831e7 −0.444830
\(990\) 3.01579e6 0.0977943
\(991\) −4.33916e7 −1.40353 −0.701764 0.712409i \(-0.747604\pi\)
−0.701764 + 0.712409i \(0.747604\pi\)
\(992\) −5.80403e6 −0.187262
\(993\) −9.90364e6 −0.318729
\(994\) 0 0
\(995\) −1.41076e7 −0.451747
\(996\) −1.48038e7 −0.472851
\(997\) 4.35294e6 0.138690 0.0693449 0.997593i \(-0.477909\pi\)
0.0693449 + 0.997593i \(0.477909\pi\)
\(998\) −5.33112e6 −0.169431
\(999\) −2.13014e6 −0.0675296
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 294.6.a.g.1.1 yes 1
3.2 odd 2 882.6.a.p.1.1 1
7.2 even 3 294.6.e.j.67.1 2
7.3 odd 6 294.6.e.q.79.1 2
7.4 even 3 294.6.e.j.79.1 2
7.5 odd 6 294.6.e.q.67.1 2
7.6 odd 2 294.6.a.a.1.1 1
21.20 even 2 882.6.a.t.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
294.6.a.a.1.1 1 7.6 odd 2
294.6.a.g.1.1 yes 1 1.1 even 1 trivial
294.6.e.j.67.1 2 7.2 even 3
294.6.e.j.79.1 2 7.4 even 3
294.6.e.q.67.1 2 7.5 odd 6
294.6.e.q.79.1 2 7.3 odd 6
882.6.a.p.1.1 1 3.2 odd 2
882.6.a.t.1.1 1 21.20 even 2