Properties

Label 1050.6.a.a.1.1
Level $1050$
Weight $6$
Character 1050.1
Self dual yes
Analytic conductor $168.403$
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] = [1050,6,Mod(1,1050)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1050, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1050.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1050.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: \(168.403010804\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1050.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} +36.0000 q^{6} -49.0000 q^{7} -64.0000 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} +36.0000 q^{6} -49.0000 q^{7} -64.0000 q^{8} +81.0000 q^{9} +66.0000 q^{11} -144.000 q^{12} -98.0000 q^{13} +196.000 q^{14} +256.000 q^{16} +216.000 q^{17} -324.000 q^{18} -340.000 q^{19} +441.000 q^{21} -264.000 q^{22} +1038.00 q^{23} +576.000 q^{24} +392.000 q^{26} -729.000 q^{27} -784.000 q^{28} -2490.00 q^{29} -7048.00 q^{31} -1024.00 q^{32} -594.000 q^{33} -864.000 q^{34} +1296.00 q^{36} +12238.0 q^{37} +1360.00 q^{38} +882.000 q^{39} +6468.00 q^{41} -1764.00 q^{42} +15412.0 q^{43} +1056.00 q^{44} -4152.00 q^{46} -20604.0 q^{47} -2304.00 q^{48} +2401.00 q^{49} -1944.00 q^{51} -1568.00 q^{52} -32490.0 q^{53} +2916.00 q^{54} +3136.00 q^{56} +3060.00 q^{57} +9960.00 q^{58} +34224.0 q^{59} +35654.0 q^{61} +28192.0 q^{62} -3969.00 q^{63} +4096.00 q^{64} +2376.00 q^{66} -12680.0 q^{67} +3456.00 q^{68} -9342.00 q^{69} -42642.0 q^{71} -5184.00 q^{72} -33734.0 q^{73} -48952.0 q^{74} -5440.00 q^{76} -3234.00 q^{77} -3528.00 q^{78} -85108.0 q^{79} +6561.00 q^{81} -25872.0 q^{82} +106764. q^{83} +7056.00 q^{84} -61648.0 q^{86} +22410.0 q^{87} -4224.00 q^{88} +34884.0 q^{89} +4802.00 q^{91} +16608.0 q^{92} +63432.0 q^{93} +82416.0 q^{94} +9216.00 q^{96} -18662.0 q^{97} -9604.00 q^{98} +5346.00 q^{99} +O(q^{100})\)

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
\(5\) 0 0
\(6\) 36.0000 0.408248
\(7\) −49.0000 −0.377964
\(8\) −64.0000 −0.353553
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 66.0000 0.164461 0.0822304 0.996613i \(-0.473796\pi\)
0.0822304 + 0.996613i \(0.473796\pi\)
\(12\) −144.000 −0.288675
\(13\) −98.0000 −0.160830 −0.0804151 0.996761i \(-0.525625\pi\)
−0.0804151 + 0.996761i \(0.525625\pi\)
\(14\) 196.000 0.267261
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) 216.000 0.181272 0.0906362 0.995884i \(-0.471110\pi\)
0.0906362 + 0.995884i \(0.471110\pi\)
\(18\) −324.000 −0.235702
\(19\) −340.000 −0.216070 −0.108035 0.994147i \(-0.534456\pi\)
−0.108035 + 0.994147i \(0.534456\pi\)
\(20\) 0 0
\(21\) 441.000 0.218218
\(22\) −264.000 −0.116291
\(23\) 1038.00 0.409145 0.204573 0.978851i \(-0.434420\pi\)
0.204573 + 0.978851i \(0.434420\pi\)
\(24\) 576.000 0.204124
\(25\) 0 0
\(26\) 392.000 0.113724
\(27\) −729.000 −0.192450
\(28\) −784.000 −0.188982
\(29\) −2490.00 −0.549800 −0.274900 0.961473i \(-0.588645\pi\)
−0.274900 + 0.961473i \(0.588645\pi\)
\(30\) 0 0
\(31\) −7048.00 −1.31723 −0.658615 0.752480i \(-0.728857\pi\)
−0.658615 + 0.752480i \(0.728857\pi\)
\(32\) −1024.00 −0.176777
\(33\) −594.000 −0.0949514
\(34\) −864.000 −0.128179
\(35\) 0 0
\(36\) 1296.00 0.166667
\(37\) 12238.0 1.46962 0.734812 0.678271i \(-0.237270\pi\)
0.734812 + 0.678271i \(0.237270\pi\)
\(38\) 1360.00 0.152785
\(39\) 882.000 0.0928554
\(40\) 0 0
\(41\) 6468.00 0.600911 0.300456 0.953796i \(-0.402861\pi\)
0.300456 + 0.953796i \(0.402861\pi\)
\(42\) −1764.00 −0.154303
\(43\) 15412.0 1.27112 0.635562 0.772050i \(-0.280768\pi\)
0.635562 + 0.772050i \(0.280768\pi\)
\(44\) 1056.00 0.0822304
\(45\) 0 0
\(46\) −4152.00 −0.289310
\(47\) −20604.0 −1.36053 −0.680263 0.732968i \(-0.738134\pi\)
−0.680263 + 0.732968i \(0.738134\pi\)
\(48\) −2304.00 −0.144338
\(49\) 2401.00 0.142857
\(50\) 0 0
\(51\) −1944.00 −0.104658
\(52\) −1568.00 −0.0804151
\(53\) −32490.0 −1.58877 −0.794383 0.607417i \(-0.792206\pi\)
−0.794383 + 0.607417i \(0.792206\pi\)
\(54\) 2916.00 0.136083
\(55\) 0 0
\(56\) 3136.00 0.133631
\(57\) 3060.00 0.124748
\(58\) 9960.00 0.388767
\(59\) 34224.0 1.27997 0.639986 0.768386i \(-0.278940\pi\)
0.639986 + 0.768386i \(0.278940\pi\)
\(60\) 0 0
\(61\) 35654.0 1.22683 0.613414 0.789762i \(-0.289796\pi\)
0.613414 + 0.789762i \(0.289796\pi\)
\(62\) 28192.0 0.931422
\(63\) −3969.00 −0.125988
\(64\) 4096.00 0.125000
\(65\) 0 0
\(66\) 2376.00 0.0671408
\(67\) −12680.0 −0.345090 −0.172545 0.985002i \(-0.555199\pi\)
−0.172545 + 0.985002i \(0.555199\pi\)
\(68\) 3456.00 0.0906362
\(69\) −9342.00 −0.236220
\(70\) 0 0
\(71\) −42642.0 −1.00390 −0.501951 0.864896i \(-0.667384\pi\)
−0.501951 + 0.864896i \(0.667384\pi\)
\(72\) −5184.00 −0.117851
\(73\) −33734.0 −0.740902 −0.370451 0.928852i \(-0.620797\pi\)
−0.370451 + 0.928852i \(0.620797\pi\)
\(74\) −48952.0 −1.03918
\(75\) 0 0
\(76\) −5440.00 −0.108035
\(77\) −3234.00 −0.0621603
\(78\) −3528.00 −0.0656587
\(79\) −85108.0 −1.53427 −0.767137 0.641484i \(-0.778319\pi\)
−0.767137 + 0.641484i \(0.778319\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) −25872.0 −0.424908
\(83\) 106764. 1.70110 0.850550 0.525895i \(-0.176270\pi\)
0.850550 + 0.525895i \(0.176270\pi\)
\(84\) 7056.00 0.109109
\(85\) 0 0
\(86\) −61648.0 −0.898820
\(87\) 22410.0 0.317427
\(88\) −4224.00 −0.0581456
\(89\) 34884.0 0.466822 0.233411 0.972378i \(-0.425011\pi\)
0.233411 + 0.972378i \(0.425011\pi\)
\(90\) 0 0
\(91\) 4802.00 0.0607881
\(92\) 16608.0 0.204573
\(93\) 63432.0 0.760503
\(94\) 82416.0 0.962037
\(95\) 0 0
\(96\) 9216.00 0.102062
\(97\) −18662.0 −0.201386 −0.100693 0.994918i \(-0.532106\pi\)
−0.100693 + 0.994918i \(0.532106\pi\)
\(98\) −9604.00 −0.101015
\(99\) 5346.00 0.0548202
\(100\) 0 0
\(101\) 153084. 1.49323 0.746614 0.665257i \(-0.231678\pi\)
0.746614 + 0.665257i \(0.231678\pi\)
\(102\) 7776.00 0.0740041
\(103\) −35864.0 −0.333093 −0.166547 0.986034i \(-0.553262\pi\)
−0.166547 + 0.986034i \(0.553262\pi\)
\(104\) 6272.00 0.0568621
\(105\) 0 0
\(106\) 129960. 1.12343
\(107\) 95454.0 0.805999 0.403000 0.915200i \(-0.367968\pi\)
0.403000 + 0.915200i \(0.367968\pi\)
\(108\) −11664.0 −0.0962250
\(109\) 212222. 1.71090 0.855449 0.517887i \(-0.173281\pi\)
0.855449 + 0.517887i \(0.173281\pi\)
\(110\) 0 0
\(111\) −110142. −0.848488
\(112\) −12544.0 −0.0944911
\(113\) −62106.0 −0.457549 −0.228774 0.973479i \(-0.573472\pi\)
−0.228774 + 0.973479i \(0.573472\pi\)
\(114\) −12240.0 −0.0882103
\(115\) 0 0
\(116\) −39840.0 −0.274900
\(117\) −7938.00 −0.0536101
\(118\) −136896. −0.905077
\(119\) −10584.0 −0.0685145
\(120\) 0 0
\(121\) −156695. −0.972953
\(122\) −142616. −0.867498
\(123\) −58212.0 −0.346936
\(124\) −112768. −0.658615
\(125\) 0 0
\(126\) 15876.0 0.0890871
\(127\) 53044.0 0.291828 0.145914 0.989297i \(-0.453388\pi\)
0.145914 + 0.989297i \(0.453388\pi\)
\(128\) −16384.0 −0.0883883
\(129\) −138708. −0.733884
\(130\) 0 0
\(131\) 69324.0 0.352944 0.176472 0.984306i \(-0.443532\pi\)
0.176472 + 0.984306i \(0.443532\pi\)
\(132\) −9504.00 −0.0474757
\(133\) 16660.0 0.0816669
\(134\) 50720.0 0.244015
\(135\) 0 0
\(136\) −13824.0 −0.0640894
\(137\) −129846. −0.591054 −0.295527 0.955334i \(-0.595495\pi\)
−0.295527 + 0.955334i \(0.595495\pi\)
\(138\) 37368.0 0.167033
\(139\) −104356. −0.458121 −0.229061 0.973412i \(-0.573565\pi\)
−0.229061 + 0.973412i \(0.573565\pi\)
\(140\) 0 0
\(141\) 185436. 0.785500
\(142\) 170568. 0.709867
\(143\) −6468.00 −0.0264503
\(144\) 20736.0 0.0833333
\(145\) 0 0
\(146\) 134936. 0.523897
\(147\) −21609.0 −0.0824786
\(148\) 195808. 0.734812
\(149\) 217194. 0.801461 0.400730 0.916196i \(-0.368756\pi\)
0.400730 + 0.916196i \(0.368756\pi\)
\(150\) 0 0
\(151\) 221000. 0.788769 0.394385 0.918945i \(-0.370958\pi\)
0.394385 + 0.918945i \(0.370958\pi\)
\(152\) 21760.0 0.0763924
\(153\) 17496.0 0.0604241
\(154\) 12936.0 0.0439540
\(155\) 0 0
\(156\) 14112.0 0.0464277
\(157\) 378370. 1.22509 0.612544 0.790436i \(-0.290146\pi\)
0.612544 + 0.790436i \(0.290146\pi\)
\(158\) 340432. 1.08489
\(159\) 292410. 0.917275
\(160\) 0 0
\(161\) −50862.0 −0.154642
\(162\) −26244.0 −0.0785674
\(163\) −104816. −0.309000 −0.154500 0.987993i \(-0.549377\pi\)
−0.154500 + 0.987993i \(0.549377\pi\)
\(164\) 103488. 0.300456
\(165\) 0 0
\(166\) −427056. −1.20286
\(167\) 426972. 1.18470 0.592350 0.805681i \(-0.298200\pi\)
0.592350 + 0.805681i \(0.298200\pi\)
\(168\) −28224.0 −0.0771517
\(169\) −361689. −0.974134
\(170\) 0 0
\(171\) −27540.0 −0.0720234
\(172\) 246592. 0.635562
\(173\) −331068. −0.841012 −0.420506 0.907290i \(-0.638147\pi\)
−0.420506 + 0.907290i \(0.638147\pi\)
\(174\) −89640.0 −0.224455
\(175\) 0 0
\(176\) 16896.0 0.0411152
\(177\) −308016. −0.738993
\(178\) −139536. −0.330093
\(179\) −400194. −0.933551 −0.466775 0.884376i \(-0.654584\pi\)
−0.466775 + 0.884376i \(0.654584\pi\)
\(180\) 0 0
\(181\) 588098. 1.33430 0.667150 0.744924i \(-0.267514\pi\)
0.667150 + 0.744924i \(0.267514\pi\)
\(182\) −19208.0 −0.0429837
\(183\) −320886. −0.708309
\(184\) −66432.0 −0.144655
\(185\) 0 0
\(186\) −253728. −0.537757
\(187\) 14256.0 0.0298122
\(188\) −329664. −0.680263
\(189\) 35721.0 0.0727393
\(190\) 0 0
\(191\) 939342. 1.86312 0.931559 0.363590i \(-0.118449\pi\)
0.931559 + 0.363590i \(0.118449\pi\)
\(192\) −36864.0 −0.0721688
\(193\) −338390. −0.653919 −0.326960 0.945038i \(-0.606024\pi\)
−0.326960 + 0.945038i \(0.606024\pi\)
\(194\) 74648.0 0.142401
\(195\) 0 0
\(196\) 38416.0 0.0714286
\(197\) 237942. 0.436823 0.218412 0.975857i \(-0.429912\pi\)
0.218412 + 0.975857i \(0.429912\pi\)
\(198\) −21384.0 −0.0387638
\(199\) 204464. 0.366003 0.183001 0.983113i \(-0.441419\pi\)
0.183001 + 0.983113i \(0.441419\pi\)
\(200\) 0 0
\(201\) 114120. 0.199238
\(202\) −612336. −1.05587
\(203\) 122010. 0.207805
\(204\) −31104.0 −0.0523288
\(205\) 0 0
\(206\) 143456. 0.235532
\(207\) 84078.0 0.136382
\(208\) −25088.0 −0.0402076
\(209\) −22440.0 −0.0355351
\(210\) 0 0
\(211\) −348724. −0.539232 −0.269616 0.962968i \(-0.586897\pi\)
−0.269616 + 0.962968i \(0.586897\pi\)
\(212\) −519840. −0.794383
\(213\) 383778. 0.579604
\(214\) −381816. −0.569928
\(215\) 0 0
\(216\) 46656.0 0.0680414
\(217\) 345352. 0.497866
\(218\) −848888. −1.20979
\(219\) 303606. 0.427760
\(220\) 0 0
\(221\) −21168.0 −0.0291541
\(222\) 440568. 0.599971
\(223\) −1.47006e6 −1.97957 −0.989787 0.142554i \(-0.954468\pi\)
−0.989787 + 0.142554i \(0.954468\pi\)
\(224\) 50176.0 0.0668153
\(225\) 0 0
\(226\) 248424. 0.323536
\(227\) 589560. 0.759387 0.379694 0.925112i \(-0.376029\pi\)
0.379694 + 0.925112i \(0.376029\pi\)
\(228\) 48960.0 0.0623741
\(229\) −1.04534e6 −1.31725 −0.658627 0.752469i \(-0.728863\pi\)
−0.658627 + 0.752469i \(0.728863\pi\)
\(230\) 0 0
\(231\) 29106.0 0.0358883
\(232\) 159360. 0.194383
\(233\) −651222. −0.785849 −0.392925 0.919571i \(-0.628537\pi\)
−0.392925 + 0.919571i \(0.628537\pi\)
\(234\) 31752.0 0.0379080
\(235\) 0 0
\(236\) 547584. 0.639986
\(237\) 765972. 0.885813
\(238\) 42336.0 0.0484471
\(239\) −513462. −0.581452 −0.290726 0.956806i \(-0.593897\pi\)
−0.290726 + 0.956806i \(0.593897\pi\)
\(240\) 0 0
\(241\) −694714. −0.770484 −0.385242 0.922816i \(-0.625882\pi\)
−0.385242 + 0.922816i \(0.625882\pi\)
\(242\) 626780. 0.687981
\(243\) −59049.0 −0.0641500
\(244\) 570464. 0.613414
\(245\) 0 0
\(246\) 232848. 0.245321
\(247\) 33320.0 0.0347506
\(248\) 451072. 0.465711
\(249\) −960876. −0.982130
\(250\) 0 0
\(251\) −1.39608e6 −1.39870 −0.699352 0.714777i \(-0.746528\pi\)
−0.699352 + 0.714777i \(0.746528\pi\)
\(252\) −63504.0 −0.0629941
\(253\) 68508.0 0.0672884
\(254\) −212176. −0.206354
\(255\) 0 0
\(256\) 65536.0 0.0625000
\(257\) 1.00520e6 0.949339 0.474670 0.880164i \(-0.342568\pi\)
0.474670 + 0.880164i \(0.342568\pi\)
\(258\) 554832. 0.518934
\(259\) −599662. −0.555466
\(260\) 0 0
\(261\) −201690. −0.183267
\(262\) −277296. −0.249569
\(263\) −1.25301e6 −1.11703 −0.558515 0.829494i \(-0.688629\pi\)
−0.558515 + 0.829494i \(0.688629\pi\)
\(264\) 38016.0 0.0335704
\(265\) 0 0
\(266\) −66640.0 −0.0577472
\(267\) −313956. −0.269520
\(268\) −202880. −0.172545
\(269\) −1.76069e6 −1.48355 −0.741774 0.670650i \(-0.766015\pi\)
−0.741774 + 0.670650i \(0.766015\pi\)
\(270\) 0 0
\(271\) 770528. 0.637331 0.318666 0.947867i \(-0.396765\pi\)
0.318666 + 0.947867i \(0.396765\pi\)
\(272\) 55296.0 0.0453181
\(273\) −43218.0 −0.0350960
\(274\) 519384. 0.417938
\(275\) 0 0
\(276\) −149472. −0.118110
\(277\) −707738. −0.554208 −0.277104 0.960840i \(-0.589375\pi\)
−0.277104 + 0.960840i \(0.589375\pi\)
\(278\) 417424. 0.323941
\(279\) −570888. −0.439077
\(280\) 0 0
\(281\) 2.30432e6 1.74091 0.870456 0.492247i \(-0.163824\pi\)
0.870456 + 0.492247i \(0.163824\pi\)
\(282\) −741744. −0.555432
\(283\) −1.60903e6 −1.19426 −0.597128 0.802146i \(-0.703692\pi\)
−0.597128 + 0.802146i \(0.703692\pi\)
\(284\) −682272. −0.501951
\(285\) 0 0
\(286\) 25872.0 0.0187032
\(287\) −316932. −0.227123
\(288\) −82944.0 −0.0589256
\(289\) −1.37320e6 −0.967140
\(290\) 0 0
\(291\) 167958. 0.116270
\(292\) −539744. −0.370451
\(293\) −517020. −0.351834 −0.175917 0.984405i \(-0.556289\pi\)
−0.175917 + 0.984405i \(0.556289\pi\)
\(294\) 86436.0 0.0583212
\(295\) 0 0
\(296\) −783232. −0.519590
\(297\) −48114.0 −0.0316505
\(298\) −868776. −0.566718
\(299\) −101724. −0.0658030
\(300\) 0 0
\(301\) −755188. −0.480440
\(302\) −884000. −0.557744
\(303\) −1.37776e6 −0.862116
\(304\) −87040.0 −0.0540176
\(305\) 0 0
\(306\) −69984.0 −0.0427263
\(307\) −1.35002e6 −0.817512 −0.408756 0.912644i \(-0.634037\pi\)
−0.408756 + 0.912644i \(0.634037\pi\)
\(308\) −51744.0 −0.0310802
\(309\) 322776. 0.192311
\(310\) 0 0
\(311\) 1.34538e6 0.788758 0.394379 0.918948i \(-0.370960\pi\)
0.394379 + 0.918948i \(0.370960\pi\)
\(312\) −56448.0 −0.0328293
\(313\) −256154. −0.147788 −0.0738942 0.997266i \(-0.523543\pi\)
−0.0738942 + 0.997266i \(0.523543\pi\)
\(314\) −1.51348e6 −0.866269
\(315\) 0 0
\(316\) −1.36173e6 −0.767137
\(317\) −1.84629e6 −1.03193 −0.515967 0.856609i \(-0.672567\pi\)
−0.515967 + 0.856609i \(0.672567\pi\)
\(318\) −1.16964e6 −0.648611
\(319\) −164340. −0.0904204
\(320\) 0 0
\(321\) −859086. −0.465344
\(322\) 203448. 0.109349
\(323\) −73440.0 −0.0391675
\(324\) 104976. 0.0555556
\(325\) 0 0
\(326\) 419264. 0.218496
\(327\) −1.91000e6 −0.987788
\(328\) −413952. −0.212454
\(329\) 1.00960e6 0.514231
\(330\) 0 0
\(331\) −3.33238e6 −1.67180 −0.835900 0.548881i \(-0.815054\pi\)
−0.835900 + 0.548881i \(0.815054\pi\)
\(332\) 1.70822e6 0.850550
\(333\) 991278. 0.489875
\(334\) −1.70789e6 −0.837709
\(335\) 0 0
\(336\) 112896. 0.0545545
\(337\) 1.63481e6 0.784136 0.392068 0.919936i \(-0.371760\pi\)
0.392068 + 0.919936i \(0.371760\pi\)
\(338\) 1.44676e6 0.688816
\(339\) 558954. 0.264166
\(340\) 0 0
\(341\) −465168. −0.216633
\(342\) 110160. 0.0509282
\(343\) −117649. −0.0539949
\(344\) −986368. −0.449410
\(345\) 0 0
\(346\) 1.32427e6 0.594685
\(347\) 841530. 0.375185 0.187593 0.982247i \(-0.439932\pi\)
0.187593 + 0.982247i \(0.439932\pi\)
\(348\) 358560. 0.158713
\(349\) −977242. −0.429476 −0.214738 0.976672i \(-0.568890\pi\)
−0.214738 + 0.976672i \(0.568890\pi\)
\(350\) 0 0
\(351\) 71442.0 0.0309518
\(352\) −67584.0 −0.0290728
\(353\) −3.45857e6 −1.47727 −0.738634 0.674106i \(-0.764529\pi\)
−0.738634 + 0.674106i \(0.764529\pi\)
\(354\) 1.23206e6 0.522547
\(355\) 0 0
\(356\) 558144. 0.233411
\(357\) 95256.0 0.0395569
\(358\) 1.60078e6 0.660120
\(359\) −3.47301e6 −1.42223 −0.711115 0.703076i \(-0.751810\pi\)
−0.711115 + 0.703076i \(0.751810\pi\)
\(360\) 0 0
\(361\) −2.36050e6 −0.953314
\(362\) −2.35239e6 −0.943492
\(363\) 1.41026e6 0.561734
\(364\) 76832.0 0.0303941
\(365\) 0 0
\(366\) 1.28354e6 0.500850
\(367\) −3.11994e6 −1.20915 −0.604575 0.796548i \(-0.706657\pi\)
−0.604575 + 0.796548i \(0.706657\pi\)
\(368\) 265728. 0.102286
\(369\) 523908. 0.200304
\(370\) 0 0
\(371\) 1.59201e6 0.600497
\(372\) 1.01491e6 0.380252
\(373\) 2.01673e6 0.750543 0.375272 0.926915i \(-0.377549\pi\)
0.375272 + 0.926915i \(0.377549\pi\)
\(374\) −57024.0 −0.0210804
\(375\) 0 0
\(376\) 1.31866e6 0.481019
\(377\) 244020. 0.0884244
\(378\) −142884. −0.0514344
\(379\) −5.38083e6 −1.92420 −0.962102 0.272690i \(-0.912087\pi\)
−0.962102 + 0.272690i \(0.912087\pi\)
\(380\) 0 0
\(381\) −477396. −0.168487
\(382\) −3.75737e6 −1.31742
\(383\) −807432. −0.281261 −0.140630 0.990062i \(-0.544913\pi\)
−0.140630 + 0.990062i \(0.544913\pi\)
\(384\) 147456. 0.0510310
\(385\) 0 0
\(386\) 1.35356e6 0.462391
\(387\) 1.24837e6 0.423708
\(388\) −298592. −0.100693
\(389\) 891390. 0.298671 0.149336 0.988787i \(-0.452286\pi\)
0.149336 + 0.988787i \(0.452286\pi\)
\(390\) 0 0
\(391\) 224208. 0.0741667
\(392\) −153664. −0.0505076
\(393\) −623916. −0.203772
\(394\) −951768. −0.308881
\(395\) 0 0
\(396\) 85536.0 0.0274101
\(397\) −1.12345e6 −0.357749 −0.178875 0.983872i \(-0.557246\pi\)
−0.178875 + 0.983872i \(0.557246\pi\)
\(398\) −817856. −0.258803
\(399\) −149940. −0.0471504
\(400\) 0 0
\(401\) 1.72037e6 0.534271 0.267136 0.963659i \(-0.413923\pi\)
0.267136 + 0.963659i \(0.413923\pi\)
\(402\) −456480. −0.140882
\(403\) 690704. 0.211850
\(404\) 2.44934e6 0.746614
\(405\) 0 0
\(406\) −488040. −0.146940
\(407\) 807708. 0.241695
\(408\) 124416. 0.0370021
\(409\) 77246.0 0.0228332 0.0114166 0.999935i \(-0.496366\pi\)
0.0114166 + 0.999935i \(0.496366\pi\)
\(410\) 0 0
\(411\) 1.16861e6 0.341245
\(412\) −573824. −0.166547
\(413\) −1.67698e6 −0.483784
\(414\) −336312. −0.0964365
\(415\) 0 0
\(416\) 100352. 0.0284310
\(417\) 939204. 0.264496
\(418\) 89760.0 0.0251271
\(419\) −5.20615e6 −1.44871 −0.724356 0.689427i \(-0.757863\pi\)
−0.724356 + 0.689427i \(0.757863\pi\)
\(420\) 0 0
\(421\) 1.71847e6 0.472539 0.236270 0.971688i \(-0.424075\pi\)
0.236270 + 0.971688i \(0.424075\pi\)
\(422\) 1.39490e6 0.381295
\(423\) −1.66892e6 −0.453509
\(424\) 2.07936e6 0.561714
\(425\) 0 0
\(426\) −1.53511e6 −0.409842
\(427\) −1.74705e6 −0.463697
\(428\) 1.52726e6 0.403000
\(429\) 58212.0 0.0152711
\(430\) 0 0
\(431\) −580626. −0.150558 −0.0752789 0.997163i \(-0.523985\pi\)
−0.0752789 + 0.997163i \(0.523985\pi\)
\(432\) −186624. −0.0481125
\(433\) −4.15087e6 −1.06395 −0.531973 0.846761i \(-0.678549\pi\)
−0.531973 + 0.846761i \(0.678549\pi\)
\(434\) −1.38141e6 −0.352045
\(435\) 0 0
\(436\) 3.39555e6 0.855449
\(437\) −352920. −0.0884042
\(438\) −1.21442e6 −0.302472
\(439\) 3.88407e6 0.961891 0.480946 0.876750i \(-0.340293\pi\)
0.480946 + 0.876750i \(0.340293\pi\)
\(440\) 0 0
\(441\) 194481. 0.0476190
\(442\) 84672.0 0.0206150
\(443\) 2.31499e6 0.560453 0.280226 0.959934i \(-0.409590\pi\)
0.280226 + 0.959934i \(0.409590\pi\)
\(444\) −1.76227e6 −0.424244
\(445\) 0 0
\(446\) 5.88022e6 1.39977
\(447\) −1.95475e6 −0.462723
\(448\) −200704. −0.0472456
\(449\) −1.92281e6 −0.450113 −0.225056 0.974346i \(-0.572257\pi\)
−0.225056 + 0.974346i \(0.572257\pi\)
\(450\) 0 0
\(451\) 426888. 0.0988263
\(452\) −993696. −0.228774
\(453\) −1.98900e6 −0.455396
\(454\) −2.35824e6 −0.536968
\(455\) 0 0
\(456\) −195840. −0.0441051
\(457\) −6.86215e6 −1.53699 −0.768493 0.639858i \(-0.778993\pi\)
−0.768493 + 0.639858i \(0.778993\pi\)
\(458\) 4.18137e6 0.931440
\(459\) −157464. −0.0348859
\(460\) 0 0
\(461\) 2.97167e6 0.651250 0.325625 0.945499i \(-0.394425\pi\)
0.325625 + 0.945499i \(0.394425\pi\)
\(462\) −116424. −0.0253768
\(463\) −4.87423e6 −1.05670 −0.528352 0.849025i \(-0.677190\pi\)
−0.528352 + 0.849025i \(0.677190\pi\)
\(464\) −637440. −0.137450
\(465\) 0 0
\(466\) 2.60489e6 0.555679
\(467\) 8.17301e6 1.73416 0.867081 0.498167i \(-0.165993\pi\)
0.867081 + 0.498167i \(0.165993\pi\)
\(468\) −127008. −0.0268050
\(469\) 621320. 0.130432
\(470\) 0 0
\(471\) −3.40533e6 −0.707305
\(472\) −2.19034e6 −0.452539
\(473\) 1.01719e6 0.209050
\(474\) −3.06389e6 −0.626364
\(475\) 0 0
\(476\) −169344. −0.0342572
\(477\) −2.63169e6 −0.529589
\(478\) 2.05385e6 0.411148
\(479\) 2.34397e6 0.466782 0.233391 0.972383i \(-0.425018\pi\)
0.233391 + 0.972383i \(0.425018\pi\)
\(480\) 0 0
\(481\) −1.19932e6 −0.236360
\(482\) 2.77886e6 0.544814
\(483\) 457758. 0.0892829
\(484\) −2.50712e6 −0.486476
\(485\) 0 0
\(486\) 236196. 0.0453609
\(487\) −316928. −0.0605534 −0.0302767 0.999542i \(-0.509639\pi\)
−0.0302767 + 0.999542i \(0.509639\pi\)
\(488\) −2.28186e6 −0.433749
\(489\) 943344. 0.178401
\(490\) 0 0
\(491\) −5.20041e6 −0.973495 −0.486748 0.873543i \(-0.661817\pi\)
−0.486748 + 0.873543i \(0.661817\pi\)
\(492\) −931392. −0.173468
\(493\) −537840. −0.0996634
\(494\) −133280. −0.0245724
\(495\) 0 0
\(496\) −1.80429e6 −0.329308
\(497\) 2.08946e6 0.379440
\(498\) 3.84350e6 0.694471
\(499\) −4.86773e6 −0.875135 −0.437568 0.899185i \(-0.644160\pi\)
−0.437568 + 0.899185i \(0.644160\pi\)
\(500\) 0 0
\(501\) −3.84275e6 −0.683987
\(502\) 5.58432e6 0.989034
\(503\) −426888. −0.0752305 −0.0376153 0.999292i \(-0.511976\pi\)
−0.0376153 + 0.999292i \(0.511976\pi\)
\(504\) 254016. 0.0445435
\(505\) 0 0
\(506\) −274032. −0.0475801
\(507\) 3.25520e6 0.562416
\(508\) 848704. 0.145914
\(509\) −9.41621e6 −1.61095 −0.805474 0.592631i \(-0.798089\pi\)
−0.805474 + 0.592631i \(0.798089\pi\)
\(510\) 0 0
\(511\) 1.65297e6 0.280035
\(512\) −262144. −0.0441942
\(513\) 247860. 0.0415827
\(514\) −4.02082e6 −0.671284
\(515\) 0 0
\(516\) −2.21933e6 −0.366942
\(517\) −1.35986e6 −0.223753
\(518\) 2.39865e6 0.392773
\(519\) 2.97961e6 0.485558
\(520\) 0 0
\(521\) 1.84039e6 0.297041 0.148520 0.988909i \(-0.452549\pi\)
0.148520 + 0.988909i \(0.452549\pi\)
\(522\) 806760. 0.129589
\(523\) 979108. 0.156522 0.0782612 0.996933i \(-0.475063\pi\)
0.0782612 + 0.996933i \(0.475063\pi\)
\(524\) 1.10918e6 0.176472
\(525\) 0 0
\(526\) 5.01204e6 0.789860
\(527\) −1.52237e6 −0.238777
\(528\) −152064. −0.0237379
\(529\) −5.35890e6 −0.832600
\(530\) 0 0
\(531\) 2.77214e6 0.426658
\(532\) 266560. 0.0408334
\(533\) −633864. −0.0966447
\(534\) 1.25582e6 0.190579
\(535\) 0 0
\(536\) 811520. 0.122008
\(537\) 3.60175e6 0.538986
\(538\) 7.04275e6 1.04903
\(539\) 158466. 0.0234944
\(540\) 0 0
\(541\) 5.96117e6 0.875666 0.437833 0.899056i \(-0.355746\pi\)
0.437833 + 0.899056i \(0.355746\pi\)
\(542\) −3.08211e6 −0.450661
\(543\) −5.29288e6 −0.770358
\(544\) −221184. −0.0320447
\(545\) 0 0
\(546\) 172872. 0.0248166
\(547\) −8.73025e6 −1.24755 −0.623775 0.781604i \(-0.714402\pi\)
−0.623775 + 0.781604i \(0.714402\pi\)
\(548\) −2.07754e6 −0.295527
\(549\) 2.88797e6 0.408943
\(550\) 0 0
\(551\) 846600. 0.118795
\(552\) 597888. 0.0835165
\(553\) 4.17029e6 0.579901
\(554\) 2.83095e6 0.391885
\(555\) 0 0
\(556\) −1.66970e6 −0.229061
\(557\) 3.01066e6 0.411172 0.205586 0.978639i \(-0.434090\pi\)
0.205586 + 0.978639i \(0.434090\pi\)
\(558\) 2.28355e6 0.310474
\(559\) −1.51038e6 −0.204435
\(560\) 0 0
\(561\) −128304. −0.0172121
\(562\) −9.21727e6 −1.23101
\(563\) −1.17573e7 −1.56327 −0.781637 0.623733i \(-0.785615\pi\)
−0.781637 + 0.623733i \(0.785615\pi\)
\(564\) 2.96698e6 0.392750
\(565\) 0 0
\(566\) 6.43611e6 0.844467
\(567\) −321489. −0.0419961
\(568\) 2.72909e6 0.354933
\(569\) 1.31578e7 1.70374 0.851870 0.523754i \(-0.175469\pi\)
0.851870 + 0.523754i \(0.175469\pi\)
\(570\) 0 0
\(571\) −1.03344e7 −1.32647 −0.663234 0.748412i \(-0.730817\pi\)
−0.663234 + 0.748412i \(0.730817\pi\)
\(572\) −103488. −0.0132251
\(573\) −8.45408e6 −1.07567
\(574\) 1.26773e6 0.160600
\(575\) 0 0
\(576\) 331776. 0.0416667
\(577\) 7.88133e6 0.985508 0.492754 0.870169i \(-0.335990\pi\)
0.492754 + 0.870169i \(0.335990\pi\)
\(578\) 5.49280e6 0.683872
\(579\) 3.04551e6 0.377541
\(580\) 0 0
\(581\) −5.23144e6 −0.642955
\(582\) −671832. −0.0822154
\(583\) −2.14434e6 −0.261290
\(584\) 2.15898e6 0.261948
\(585\) 0 0
\(586\) 2.06808e6 0.248784
\(587\) 554568. 0.0664293 0.0332146 0.999448i \(-0.489426\pi\)
0.0332146 + 0.999448i \(0.489426\pi\)
\(588\) −345744. −0.0412393
\(589\) 2.39632e6 0.284614
\(590\) 0 0
\(591\) −2.14148e6 −0.252200
\(592\) 3.13293e6 0.367406
\(593\) 9.20369e6 1.07479 0.537397 0.843329i \(-0.319408\pi\)
0.537397 + 0.843329i \(0.319408\pi\)
\(594\) 192456. 0.0223803
\(595\) 0 0
\(596\) 3.47510e6 0.400730
\(597\) −1.84018e6 −0.211312
\(598\) 406896. 0.0465297
\(599\) 8.54295e6 0.972839 0.486419 0.873725i \(-0.338303\pi\)
0.486419 + 0.873725i \(0.338303\pi\)
\(600\) 0 0
\(601\) −9.61555e6 −1.08590 −0.542948 0.839767i \(-0.682692\pi\)
−0.542948 + 0.839767i \(0.682692\pi\)
\(602\) 3.02075e6 0.339722
\(603\) −1.02708e6 −0.115030
\(604\) 3.53600e6 0.394385
\(605\) 0 0
\(606\) 5.51102e6 0.609608
\(607\) −2.21264e6 −0.243747 −0.121873 0.992546i \(-0.538890\pi\)
−0.121873 + 0.992546i \(0.538890\pi\)
\(608\) 348160. 0.0381962
\(609\) −1.09809e6 −0.119976
\(610\) 0 0
\(611\) 2.01919e6 0.218814
\(612\) 279936. 0.0302121
\(613\) 7.96215e6 0.855814 0.427907 0.903823i \(-0.359251\pi\)
0.427907 + 0.903823i \(0.359251\pi\)
\(614\) 5.40008e6 0.578068
\(615\) 0 0
\(616\) 206976. 0.0219770
\(617\) 1.37397e7 1.45299 0.726497 0.687170i \(-0.241147\pi\)
0.726497 + 0.687170i \(0.241147\pi\)
\(618\) −1.29110e6 −0.135985
\(619\) −8.70113e6 −0.912744 −0.456372 0.889789i \(-0.650851\pi\)
−0.456372 + 0.889789i \(0.650851\pi\)
\(620\) 0 0
\(621\) −756702. −0.0787401
\(622\) −5.38152e6 −0.557736
\(623\) −1.70932e6 −0.176442
\(624\) 225792. 0.0232138
\(625\) 0 0
\(626\) 1.02462e6 0.104502
\(627\) 201960. 0.0205162
\(628\) 6.05392e6 0.612544
\(629\) 2.64341e6 0.266402
\(630\) 0 0
\(631\) 445412. 0.0445337 0.0222668 0.999752i \(-0.492912\pi\)
0.0222668 + 0.999752i \(0.492912\pi\)
\(632\) 5.44691e6 0.542447
\(633\) 3.13852e6 0.311326
\(634\) 7.38516e6 0.729687
\(635\) 0 0
\(636\) 4.67856e6 0.458637
\(637\) −235298. −0.0229757
\(638\) 657360. 0.0639369
\(639\) −3.45400e6 −0.334634
\(640\) 0 0
\(641\) −8.00119e6 −0.769147 −0.384573 0.923094i \(-0.625651\pi\)
−0.384573 + 0.923094i \(0.625651\pi\)
\(642\) 3.43634e6 0.329048
\(643\) 1.58402e7 1.51090 0.755448 0.655209i \(-0.227419\pi\)
0.755448 + 0.655209i \(0.227419\pi\)
\(644\) −813792. −0.0773212
\(645\) 0 0
\(646\) 293760. 0.0276956
\(647\) −1.30187e6 −0.122266 −0.0611331 0.998130i \(-0.519471\pi\)
−0.0611331 + 0.998130i \(0.519471\pi\)
\(648\) −419904. −0.0392837
\(649\) 2.25878e6 0.210505
\(650\) 0 0
\(651\) −3.10817e6 −0.287443
\(652\) −1.67706e6 −0.154500
\(653\) −7.34149e6 −0.673753 −0.336877 0.941549i \(-0.609371\pi\)
−0.336877 + 0.941549i \(0.609371\pi\)
\(654\) 7.63999e6 0.698471
\(655\) 0 0
\(656\) 1.65581e6 0.150228
\(657\) −2.73245e6 −0.246967
\(658\) −4.03838e6 −0.363616
\(659\) −6.18934e6 −0.555176 −0.277588 0.960700i \(-0.589535\pi\)
−0.277588 + 0.960700i \(0.589535\pi\)
\(660\) 0 0
\(661\) −1.96690e7 −1.75097 −0.875484 0.483248i \(-0.839457\pi\)
−0.875484 + 0.483248i \(0.839457\pi\)
\(662\) 1.33295e7 1.18214
\(663\) 190512. 0.0168321
\(664\) −6.83290e6 −0.601429
\(665\) 0 0
\(666\) −3.96511e6 −0.346394
\(667\) −2.58462e6 −0.224948
\(668\) 6.83155e6 0.592350
\(669\) 1.32305e7 1.14291
\(670\) 0 0
\(671\) 2.35316e6 0.201765
\(672\) −451584. −0.0385758
\(673\) −7.18259e6 −0.611285 −0.305642 0.952146i \(-0.598871\pi\)
−0.305642 + 0.952146i \(0.598871\pi\)
\(674\) −6.53922e6 −0.554468
\(675\) 0 0
\(676\) −5.78702e6 −0.487067
\(677\) 1.89192e7 1.58647 0.793234 0.608917i \(-0.208396\pi\)
0.793234 + 0.608917i \(0.208396\pi\)
\(678\) −2.23582e6 −0.186794
\(679\) 914438. 0.0761167
\(680\) 0 0
\(681\) −5.30604e6 −0.438432
\(682\) 1.86067e6 0.153182
\(683\) −2.12204e7 −1.74061 −0.870306 0.492512i \(-0.836079\pi\)
−0.870306 + 0.492512i \(0.836079\pi\)
\(684\) −440640. −0.0360117
\(685\) 0 0
\(686\) 470596. 0.0381802
\(687\) 9.40808e6 0.760517
\(688\) 3.94547e6 0.317781
\(689\) 3.18402e6 0.255522
\(690\) 0 0
\(691\) 1.63276e7 1.30085 0.650424 0.759571i \(-0.274591\pi\)
0.650424 + 0.759571i \(0.274591\pi\)
\(692\) −5.29709e6 −0.420506
\(693\) −261954. −0.0207201
\(694\) −3.36612e6 −0.265296
\(695\) 0 0
\(696\) −1.43424e6 −0.112227
\(697\) 1.39709e6 0.108929
\(698\) 3.90897e6 0.303685
\(699\) 5.86100e6 0.453710
\(700\) 0 0
\(701\) −5.40470e6 −0.415409 −0.207705 0.978192i \(-0.566599\pi\)
−0.207705 + 0.978192i \(0.566599\pi\)
\(702\) −285768. −0.0218862
\(703\) −4.16092e6 −0.317542
\(704\) 270336. 0.0205576
\(705\) 0 0
\(706\) 1.38343e7 1.04459
\(707\) −7.50112e6 −0.564387
\(708\) −4.92826e6 −0.369496
\(709\) 2.21195e7 1.65257 0.826284 0.563253i \(-0.190450\pi\)
0.826284 + 0.563253i \(0.190450\pi\)
\(710\) 0 0
\(711\) −6.89375e6 −0.511424
\(712\) −2.23258e6 −0.165046
\(713\) −7.31582e6 −0.538939
\(714\) −381024. −0.0279709
\(715\) 0 0
\(716\) −6.40310e6 −0.466775
\(717\) 4.62116e6 0.335701
\(718\) 1.38920e7 1.00567
\(719\) 2.55819e7 1.84548 0.922742 0.385418i \(-0.125943\pi\)
0.922742 + 0.385418i \(0.125943\pi\)
\(720\) 0 0
\(721\) 1.75734e6 0.125897
\(722\) 9.44200e6 0.674095
\(723\) 6.25243e6 0.444839
\(724\) 9.40957e6 0.667150
\(725\) 0 0
\(726\) −5.64102e6 −0.397206
\(727\) 9.29438e6 0.652205 0.326103 0.945334i \(-0.394265\pi\)
0.326103 + 0.945334i \(0.394265\pi\)
\(728\) −307328. −0.0214918
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) 3.32899e6 0.230420
\(732\) −5.13418e6 −0.354155
\(733\) −3.40699e6 −0.234213 −0.117107 0.993119i \(-0.537362\pi\)
−0.117107 + 0.993119i \(0.537362\pi\)
\(734\) 1.24797e7 0.854999
\(735\) 0 0
\(736\) −1.06291e6 −0.0723274
\(737\) −836880. −0.0567537
\(738\) −2.09563e6 −0.141636
\(739\) 2.18135e7 1.46932 0.734658 0.678438i \(-0.237343\pi\)
0.734658 + 0.678438i \(0.237343\pi\)
\(740\) 0 0
\(741\) −299880. −0.0200633
\(742\) −6.36804e6 −0.424616
\(743\) −3.79246e6 −0.252028 −0.126014 0.992028i \(-0.540218\pi\)
−0.126014 + 0.992028i \(0.540218\pi\)
\(744\) −4.05965e6 −0.268878
\(745\) 0 0
\(746\) −8.06692e6 −0.530714
\(747\) 8.64788e6 0.567033
\(748\) 228096. 0.0149061
\(749\) −4.67725e6 −0.304639
\(750\) 0 0
\(751\) −2.01483e7 −1.30358 −0.651790 0.758400i \(-0.725982\pi\)
−0.651790 + 0.758400i \(0.725982\pi\)
\(752\) −5.27462e6 −0.340132
\(753\) 1.25647e7 0.807542
\(754\) −976080. −0.0625255
\(755\) 0 0
\(756\) 571536. 0.0363696
\(757\) −1.18427e7 −0.751126 −0.375563 0.926797i \(-0.622551\pi\)
−0.375563 + 0.926797i \(0.622551\pi\)
\(758\) 2.15233e7 1.36062
\(759\) −616572. −0.0388490
\(760\) 0 0
\(761\) 2.97791e6 0.186402 0.0932008 0.995647i \(-0.470290\pi\)
0.0932008 + 0.995647i \(0.470290\pi\)
\(762\) 1.90958e6 0.119138
\(763\) −1.03989e7 −0.646659
\(764\) 1.50295e7 0.931559
\(765\) 0 0
\(766\) 3.22973e6 0.198881
\(767\) −3.35395e6 −0.205858
\(768\) −589824. −0.0360844
\(769\) −2.02441e7 −1.23447 −0.617237 0.786777i \(-0.711748\pi\)
−0.617237 + 0.786777i \(0.711748\pi\)
\(770\) 0 0
\(771\) −9.04684e6 −0.548101
\(772\) −5.41424e6 −0.326960
\(773\) 7.37953e6 0.444202 0.222101 0.975024i \(-0.428709\pi\)
0.222101 + 0.975024i \(0.428709\pi\)
\(774\) −4.99349e6 −0.299607
\(775\) 0 0
\(776\) 1.19437e6 0.0712006
\(777\) 5.39696e6 0.320698
\(778\) −3.56556e6 −0.211193
\(779\) −2.19912e6 −0.129839
\(780\) 0 0
\(781\) −2.81437e6 −0.165103
\(782\) −896832. −0.0524438
\(783\) 1.81521e6 0.105809
\(784\) 614656. 0.0357143
\(785\) 0 0
\(786\) 2.49566e6 0.144089
\(787\) −1.36289e7 −0.784377 −0.392188 0.919885i \(-0.628282\pi\)
−0.392188 + 0.919885i \(0.628282\pi\)
\(788\) 3.80707e6 0.218412
\(789\) 1.12771e7 0.644918
\(790\) 0 0
\(791\) 3.04319e6 0.172937
\(792\) −342144. −0.0193819
\(793\) −3.49409e6 −0.197311
\(794\) 4.49382e6 0.252967
\(795\) 0 0
\(796\) 3.27142e6 0.183001
\(797\) 1.49548e7 0.833938 0.416969 0.908921i \(-0.363092\pi\)
0.416969 + 0.908921i \(0.363092\pi\)
\(798\) 599760. 0.0333404
\(799\) −4.45046e6 −0.246626
\(800\) 0 0
\(801\) 2.82560e6 0.155607
\(802\) −6.88150e6 −0.377787
\(803\) −2.22644e6 −0.121849
\(804\) 1.82592e6 0.0996189
\(805\) 0 0
\(806\) −2.76282e6 −0.149801
\(807\) 1.58462e7 0.856527
\(808\) −9.79738e6 −0.527936
\(809\) 2.87242e7 1.54304 0.771519 0.636206i \(-0.219497\pi\)
0.771519 + 0.636206i \(0.219497\pi\)
\(810\) 0 0
\(811\) −1.52265e7 −0.812922 −0.406461 0.913668i \(-0.633237\pi\)
−0.406461 + 0.913668i \(0.633237\pi\)
\(812\) 1.95216e6 0.103902
\(813\) −6.93475e6 −0.367963
\(814\) −3.23083e6 −0.170904
\(815\) 0 0
\(816\) −497664. −0.0261644
\(817\) −5.24008e6 −0.274652
\(818\) −308984. −0.0161455
\(819\) 388962. 0.0202627
\(820\) 0 0
\(821\) −3.31001e7 −1.71384 −0.856921 0.515447i \(-0.827626\pi\)
−0.856921 + 0.515447i \(0.827626\pi\)
\(822\) −4.67446e6 −0.241297
\(823\) 1.35915e7 0.699470 0.349735 0.936849i \(-0.386272\pi\)
0.349735 + 0.936849i \(0.386272\pi\)
\(824\) 2.29530e6 0.117766
\(825\) 0 0
\(826\) 6.70790e6 0.342087
\(827\) −3.13936e6 −0.159616 −0.0798082 0.996810i \(-0.525431\pi\)
−0.0798082 + 0.996810i \(0.525431\pi\)
\(828\) 1.34525e6 0.0681909
\(829\) 1.27081e7 0.642234 0.321117 0.947040i \(-0.395942\pi\)
0.321117 + 0.947040i \(0.395942\pi\)
\(830\) 0 0
\(831\) 6.36964e6 0.319972
\(832\) −401408. −0.0201038
\(833\) 518616. 0.0258960
\(834\) −3.75682e6 −0.187027
\(835\) 0 0
\(836\) −359040. −0.0177675
\(837\) 5.13799e6 0.253501
\(838\) 2.08246e7 1.02439
\(839\) −2.98312e7 −1.46307 −0.731536 0.681803i \(-0.761196\pi\)
−0.731536 + 0.681803i \(0.761196\pi\)
\(840\) 0 0
\(841\) −1.43110e7 −0.697720
\(842\) −6.87390e6 −0.334136
\(843\) −2.07389e7 −1.00512
\(844\) −5.57958e6 −0.269616
\(845\) 0 0
\(846\) 6.67570e6 0.320679
\(847\) 7.67806e6 0.367742
\(848\) −8.31744e6 −0.397192
\(849\) 1.44813e7 0.689504
\(850\) 0 0
\(851\) 1.27030e7 0.601290
\(852\) 6.14045e6 0.289802
\(853\) 1.92215e7 0.904515 0.452257 0.891888i \(-0.350619\pi\)
0.452257 + 0.891888i \(0.350619\pi\)
\(854\) 6.98818e6 0.327884
\(855\) 0 0
\(856\) −6.10906e6 −0.284964
\(857\) 2.65655e7 1.23556 0.617782 0.786349i \(-0.288031\pi\)
0.617782 + 0.786349i \(0.288031\pi\)
\(858\) −232848. −0.0107983
\(859\) −9.16844e6 −0.423948 −0.211974 0.977275i \(-0.567989\pi\)
−0.211974 + 0.977275i \(0.567989\pi\)
\(860\) 0 0
\(861\) 2.85239e6 0.131130
\(862\) 2.32250e6 0.106460
\(863\) 2.92196e7 1.33551 0.667755 0.744381i \(-0.267255\pi\)
0.667755 + 0.744381i \(0.267255\pi\)
\(864\) 746496. 0.0340207
\(865\) 0 0
\(866\) 1.66035e7 0.752324
\(867\) 1.23588e7 0.558379
\(868\) 5.52563e6 0.248933
\(869\) −5.61713e6 −0.252328
\(870\) 0 0
\(871\) 1.24264e6 0.0555009
\(872\) −1.35822e7 −0.604894
\(873\) −1.51162e6 −0.0671286
\(874\) 1.41168e6 0.0625112
\(875\) 0 0
\(876\) 4.85770e6 0.213880
\(877\) −9.71286e6 −0.426430 −0.213215 0.977005i \(-0.568394\pi\)
−0.213215 + 0.977005i \(0.568394\pi\)
\(878\) −1.55363e7 −0.680160
\(879\) 4.65318e6 0.203132
\(880\) 0 0
\(881\) 1.65372e7 0.717833 0.358917 0.933370i \(-0.383146\pi\)
0.358917 + 0.933370i \(0.383146\pi\)
\(882\) −777924. −0.0336718
\(883\) 2.39487e7 1.03367 0.516833 0.856086i \(-0.327111\pi\)
0.516833 + 0.856086i \(0.327111\pi\)
\(884\) −338688. −0.0145770
\(885\) 0 0
\(886\) −9.25994e6 −0.396300
\(887\) 4.62846e6 0.197527 0.0987637 0.995111i \(-0.468511\pi\)
0.0987637 + 0.995111i \(0.468511\pi\)
\(888\) 7.04909e6 0.299986
\(889\) −2.59916e6 −0.110301
\(890\) 0 0
\(891\) 433026. 0.0182734
\(892\) −2.35209e7 −0.989787
\(893\) 7.00536e6 0.293969
\(894\) 7.81898e6 0.327195
\(895\) 0 0
\(896\) 802816. 0.0334077
\(897\) 915516. 0.0379914
\(898\) 7.69126e6 0.318278
\(899\) 1.75495e7 0.724212
\(900\) 0 0
\(901\) −7.01784e6 −0.287999
\(902\) −1.70755e6 −0.0698808
\(903\) 6.79669e6 0.277382
\(904\) 3.97478e6 0.161768
\(905\) 0 0
\(906\) 7.95600e6 0.322014
\(907\) −2.06126e7 −0.831983 −0.415991 0.909369i \(-0.636565\pi\)
−0.415991 + 0.909369i \(0.636565\pi\)
\(908\) 9.43296e6 0.379694
\(909\) 1.23998e7 0.497743
\(910\) 0 0
\(911\) −3.46749e6 −0.138427 −0.0692133 0.997602i \(-0.522049\pi\)
−0.0692133 + 0.997602i \(0.522049\pi\)
\(912\) 783360. 0.0311870
\(913\) 7.04642e6 0.279764
\(914\) 2.74486e7 1.08681
\(915\) 0 0
\(916\) −1.67255e7 −0.658627
\(917\) −3.39688e6 −0.133400
\(918\) 629856. 0.0246680
\(919\) −3.61227e7 −1.41088 −0.705442 0.708767i \(-0.749252\pi\)
−0.705442 + 0.708767i \(0.749252\pi\)
\(920\) 0 0
\(921\) 1.21502e7 0.471991
\(922\) −1.18867e7 −0.460504
\(923\) 4.17892e6 0.161458
\(924\) 465696. 0.0179441
\(925\) 0 0
\(926\) 1.94969e7 0.747203
\(927\) −2.90498e6 −0.111031
\(928\) 2.54976e6 0.0971917
\(929\) 1.29366e7 0.491792 0.245896 0.969296i \(-0.420918\pi\)
0.245896 + 0.969296i \(0.420918\pi\)
\(930\) 0 0
\(931\) −816340. −0.0308672
\(932\) −1.04196e7 −0.392925
\(933\) −1.21084e7 −0.455390
\(934\) −3.26920e7 −1.22624
\(935\) 0 0
\(936\) 508032. 0.0189540
\(937\) −5.01394e7 −1.86565 −0.932824 0.360332i \(-0.882664\pi\)
−0.932824 + 0.360332i \(0.882664\pi\)
\(938\) −2.48528e6 −0.0922292
\(939\) 2.30539e6 0.0853257
\(940\) 0 0
\(941\) −1.05568e7 −0.388651 −0.194325 0.980937i \(-0.562252\pi\)
−0.194325 + 0.980937i \(0.562252\pi\)
\(942\) 1.36213e7 0.500140
\(943\) 6.71378e6 0.245860
\(944\) 8.76134e6 0.319993
\(945\) 0 0
\(946\) −4.06877e6 −0.147821
\(947\) 3.14684e6 0.114025 0.0570124 0.998373i \(-0.481843\pi\)
0.0570124 + 0.998373i \(0.481843\pi\)
\(948\) 1.22556e7 0.442906
\(949\) 3.30593e6 0.119159
\(950\) 0 0
\(951\) 1.66166e7 0.595787
\(952\) 677376. 0.0242235
\(953\) −5.22829e7 −1.86478 −0.932389 0.361455i \(-0.882280\pi\)
−0.932389 + 0.361455i \(0.882280\pi\)
\(954\) 1.05268e7 0.374476
\(955\) 0 0
\(956\) −8.21539e6 −0.290726
\(957\) 1.47906e6 0.0522043
\(958\) −9.37589e6 −0.330064
\(959\) 6.36245e6 0.223397
\(960\) 0 0
\(961\) 2.10452e7 0.735095
\(962\) 4.79730e6 0.167132
\(963\) 7.73177e6 0.268666
\(964\) −1.11154e7 −0.385242
\(965\) 0 0
\(966\) −1.83103e6 −0.0631325
\(967\) 2.48235e7 0.853682 0.426841 0.904327i \(-0.359626\pi\)
0.426841 + 0.904327i \(0.359626\pi\)
\(968\) 1.00285e7 0.343991
\(969\) 660960. 0.0226134
\(970\) 0 0
\(971\) 1.33077e7 0.452956 0.226478 0.974016i \(-0.427279\pi\)
0.226478 + 0.974016i \(0.427279\pi\)
\(972\) −944784. −0.0320750
\(973\) 5.11344e6 0.173154
\(974\) 1.26771e6 0.0428177
\(975\) 0 0
\(976\) 9.12742e6 0.306707
\(977\) −8.17705e6 −0.274069 −0.137035 0.990566i \(-0.543757\pi\)
−0.137035 + 0.990566i \(0.543757\pi\)
\(978\) −3.77338e6 −0.126149
\(979\) 2.30234e6 0.0767739
\(980\) 0 0
\(981\) 1.71900e7 0.570299
\(982\) 2.08016e7 0.688365
\(983\) 1.32465e7 0.437238 0.218619 0.975810i \(-0.429845\pi\)
0.218619 + 0.975810i \(0.429845\pi\)
\(984\) 3.72557e6 0.122661
\(985\) 0 0
\(986\) 2.15136e6 0.0704727
\(987\) −9.08636e6 −0.296891
\(988\) 533120. 0.0173753
\(989\) 1.59977e7 0.520075
\(990\) 0 0
\(991\) −1.48550e7 −0.480494 −0.240247 0.970712i \(-0.577228\pi\)
−0.240247 + 0.970712i \(0.577228\pi\)
\(992\) 7.21715e6 0.232856
\(993\) 2.99914e7 0.965215
\(994\) −8.35783e6 −0.268304
\(995\) 0 0
\(996\) −1.53740e7 −0.491065
\(997\) 3.33769e6 0.106343 0.0531714 0.998585i \(-0.483067\pi\)
0.0531714 + 0.998585i \(0.483067\pi\)
\(998\) 1.94709e7 0.618814
\(999\) −8.92150e6 −0.282829
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 1050.6.a.a.1.1 1
5.2 odd 4 1050.6.g.m.799.1 2
5.3 odd 4 1050.6.g.m.799.2 2
5.4 even 2 42.6.a.f.1.1 1
15.14 odd 2 126.6.a.b.1.1 1
20.19 odd 2 336.6.a.g.1.1 1
35.4 even 6 294.6.e.b.79.1 2
35.9 even 6 294.6.e.b.67.1 2
35.19 odd 6 294.6.e.f.67.1 2
35.24 odd 6 294.6.e.f.79.1 2
35.34 odd 2 294.6.a.i.1.1 1
60.59 even 2 1008.6.a.k.1.1 1
105.104 even 2 882.6.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.6.a.f.1.1 1 5.4 even 2
126.6.a.b.1.1 1 15.14 odd 2
294.6.a.i.1.1 1 35.34 odd 2
294.6.e.b.67.1 2 35.9 even 6
294.6.e.b.79.1 2 35.4 even 6
294.6.e.f.67.1 2 35.19 odd 6
294.6.e.f.79.1 2 35.24 odd 6
336.6.a.g.1.1 1 20.19 odd 2
882.6.a.i.1.1 1 105.104 even 2
1008.6.a.k.1.1 1 60.59 even 2
1050.6.a.a.1.1 1 1.1 even 1 trivial
1050.6.g.m.799.1 2 5.2 odd 4
1050.6.g.m.799.2 2 5.3 odd 4