Properties

Label 294.6.a.i.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 $1$

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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: no (minimal twist has level 42)
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)
 
\(f(q)\) \(=\) \(q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -24.0000 q^{5} -36.0000 q^{6} +64.0000 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -24.0000 q^{5} -36.0000 q^{6} +64.0000 q^{8} +81.0000 q^{9} -96.0000 q^{10} +66.0000 q^{11} -144.000 q^{12} -98.0000 q^{13} +216.000 q^{15} +256.000 q^{16} +216.000 q^{17} +324.000 q^{18} +340.000 q^{19} -384.000 q^{20} +264.000 q^{22} -1038.00 q^{23} -576.000 q^{24} -2549.00 q^{25} -392.000 q^{26} -729.000 q^{27} -2490.00 q^{29} +864.000 q^{30} +7048.00 q^{31} +1024.00 q^{32} -594.000 q^{33} +864.000 q^{34} +1296.00 q^{36} -12238.0 q^{37} +1360.00 q^{38} +882.000 q^{39} -1536.00 q^{40} -6468.00 q^{41} -15412.0 q^{43} +1056.00 q^{44} -1944.00 q^{45} -4152.00 q^{46} -20604.0 q^{47} -2304.00 q^{48} -10196.0 q^{50} -1944.00 q^{51} -1568.00 q^{52} +32490.0 q^{53} -2916.00 q^{54} -1584.00 q^{55} -3060.00 q^{57} -9960.00 q^{58} -34224.0 q^{59} +3456.00 q^{60} -35654.0 q^{61} +28192.0 q^{62} +4096.00 q^{64} +2352.00 q^{65} -2376.00 q^{66} +12680.0 q^{67} +3456.00 q^{68} +9342.00 q^{69} -42642.0 q^{71} +5184.00 q^{72} -33734.0 q^{73} -48952.0 q^{74} +22941.0 q^{75} +5440.00 q^{76} +3528.00 q^{78} -85108.0 q^{79} -6144.00 q^{80} +6561.00 q^{81} -25872.0 q^{82} +106764. q^{83} -5184.00 q^{85} -61648.0 q^{86} +22410.0 q^{87} +4224.00 q^{88} -34884.0 q^{89} -7776.00 q^{90} -16608.0 q^{92} -63432.0 q^{93} -82416.0 q^{94} -8160.00 q^{95} -9216.00 q^{96} -18662.0 q^{97} +5346.00 q^{99} +O(q^{100})\)

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
\(5\) −24.0000 −0.429325 −0.214663 0.976688i \(-0.568865\pi\)
−0.214663 + 0.976688i \(0.568865\pi\)
\(6\) −36.0000 −0.408248
\(7\) 0 0
\(8\) 64.0000 0.353553
\(9\) 81.0000 0.333333
\(10\) −96.0000 −0.303579
\(11\) 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 \(0.525625\pi\)
\(14\) 0 0
\(15\) 216.000 0.247871
\(16\) 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 \(-0.465544\pi\)
0.108035 + 0.994147i \(0.465544\pi\)
\(20\) −384.000 −0.214663
\(21\) 0 0
\(22\) 264.000 0.116291
\(23\) −1038.00 −0.409145 −0.204573 0.978851i \(-0.565580\pi\)
−0.204573 + 0.978851i \(0.565580\pi\)
\(24\) −576.000 −0.204124
\(25\) −2549.00 −0.815680
\(26\) −392.000 −0.113724
\(27\) −729.000 −0.192450
\(28\) 0 0
\(29\) −2490.00 −0.549800 −0.274900 0.961473i \(-0.588645\pi\)
−0.274900 + 0.961473i \(0.588645\pi\)
\(30\) 864.000 0.175271
\(31\) 7048.00 1.31723 0.658615 0.752480i \(-0.271143\pi\)
0.658615 + 0.752480i \(0.271143\pi\)
\(32\) 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 \(-0.762730\pi\)
−0.734812 + 0.678271i \(0.762730\pi\)
\(38\) 1360.00 0.152785
\(39\) 882.000 0.0928554
\(40\) −1536.00 −0.151789
\(41\) −6468.00 −0.600911 −0.300456 0.953796i \(-0.597139\pi\)
−0.300456 + 0.953796i \(0.597139\pi\)
\(42\) 0 0
\(43\) −15412.0 −1.27112 −0.635562 0.772050i \(-0.719232\pi\)
−0.635562 + 0.772050i \(0.719232\pi\)
\(44\) 1056.00 0.0822304
\(45\) −1944.00 −0.143108
\(46\) −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
\(49\) 0 0
\(50\) −10196.0 −0.576773
\(51\) −1944.00 −0.104658
\(52\) −1568.00 −0.0804151
\(53\) 32490.0 1.58877 0.794383 0.607417i \(-0.207794\pi\)
0.794383 + 0.607417i \(0.207794\pi\)
\(54\) −2916.00 −0.136083
\(55\) −1584.00 −0.0706071
\(56\) 0 0
\(57\) −3060.00 −0.124748
\(58\) −9960.00 −0.388767
\(59\) −34224.0 −1.27997 −0.639986 0.768386i \(-0.721060\pi\)
−0.639986 + 0.768386i \(0.721060\pi\)
\(60\) 3456.00 0.123935
\(61\) −35654.0 −1.22683 −0.613414 0.789762i \(-0.710204\pi\)
−0.613414 + 0.789762i \(0.710204\pi\)
\(62\) 28192.0 0.931422
\(63\) 0 0
\(64\) 4096.00 0.125000
\(65\) 2352.00 0.0690484
\(66\) −2376.00 −0.0671408
\(67\) 12680.0 0.345090 0.172545 0.985002i \(-0.444801\pi\)
0.172545 + 0.985002i \(0.444801\pi\)
\(68\) 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
\(75\) 22941.0 0.470933
\(76\) 5440.00 0.108035
\(77\) 0 0
\(78\) 3528.00 0.0656587
\(79\) −85108.0 −1.53427 −0.767137 0.641484i \(-0.778319\pi\)
−0.767137 + 0.641484i \(0.778319\pi\)
\(80\) −6144.00 −0.107331
\(81\) 6561.00 0.111111
\(82\) −25872.0 −0.424908
\(83\) 106764. 1.70110 0.850550 0.525895i \(-0.176270\pi\)
0.850550 + 0.525895i \(0.176270\pi\)
\(84\) 0 0
\(85\) −5184.00 −0.0778247
\(86\) −61648.0 −0.898820
\(87\) 22410.0 0.317427
\(88\) 4224.00 0.0581456
\(89\) −34884.0 −0.466822 −0.233411 0.972378i \(-0.574989\pi\)
−0.233411 + 0.972378i \(0.574989\pi\)
\(90\) −7776.00 −0.101193
\(91\) 0 0
\(92\) −16608.0 −0.204573
\(93\) −63432.0 −0.760503
\(94\) −82416.0 −0.962037
\(95\) −8160.00 −0.0927644
\(96\) −9216.00 −0.102062
\(97\) −18662.0 −0.201386 −0.100693 0.994918i \(-0.532106\pi\)
−0.100693 + 0.994918i \(0.532106\pi\)
\(98\) 0 0
\(99\) 5346.00 0.0548202
\(100\) −40784.0 −0.407840
\(101\) −153084. −1.49323 −0.746614 0.665257i \(-0.768322\pi\)
−0.746614 + 0.665257i \(0.768322\pi\)
\(102\) −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 \(-0.632032\pi\)
−0.403000 + 0.915200i \(0.632032\pi\)
\(108\) −11664.0 −0.0962250
\(109\) 212222. 1.71090 0.855449 0.517887i \(-0.173281\pi\)
0.855449 + 0.517887i \(0.173281\pi\)
\(110\) −6336.00 −0.0499268
\(111\) 110142. 0.848488
\(112\) 0 0
\(113\) 62106.0 0.457549 0.228774 0.973479i \(-0.426528\pi\)
0.228774 + 0.973479i \(0.426528\pi\)
\(114\) −12240.0 −0.0882103
\(115\) 24912.0 0.175656
\(116\) −39840.0 −0.274900
\(117\) −7938.00 −0.0536101
\(118\) −136896. −0.905077
\(119\) 0 0
\(120\) 13824.0 0.0876356
\(121\) −156695. −0.972953
\(122\) −142616. −0.867498
\(123\) 58212.0 0.346936
\(124\) 112768. 0.658615
\(125\) 136176. 0.779517
\(126\) 0 0
\(127\) −53044.0 −0.291828 −0.145914 0.989297i \(-0.546612\pi\)
−0.145914 + 0.989297i \(0.546612\pi\)
\(128\) 16384.0 0.0883883
\(129\) 138708. 0.733884
\(130\) 9408.00 0.0488246
\(131\) −69324.0 −0.352944 −0.176472 0.984306i \(-0.556468\pi\)
−0.176472 + 0.984306i \(0.556468\pi\)
\(132\) −9504.00 −0.0474757
\(133\) 0 0
\(134\) 50720.0 0.244015
\(135\) 17496.0 0.0826236
\(136\) 13824.0 0.0640894
\(137\) 129846. 0.591054 0.295527 0.955334i \(-0.404505\pi\)
0.295527 + 0.955334i \(0.404505\pi\)
\(138\) 37368.0 0.167033
\(139\) 104356. 0.458121 0.229061 0.973412i \(-0.426435\pi\)
0.229061 + 0.973412i \(0.426435\pi\)
\(140\) 0 0
\(141\) 185436. 0.785500
\(142\) −170568. −0.709867
\(143\) −6468.00 −0.0264503
\(144\) 20736.0 0.0833333
\(145\) 59760.0 0.236043
\(146\) −134936. −0.523897
\(147\) 0 0
\(148\) −195808. −0.734812
\(149\) 217194. 0.801461 0.400730 0.916196i \(-0.368756\pi\)
0.400730 + 0.916196i \(0.368756\pi\)
\(150\) 91764.0 0.333000
\(151\) 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
\(154\) 0 0
\(155\) −169152. −0.565520
\(156\) 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
\(160\) −24576.0 −0.0758947
\(161\) 0 0
\(162\) 26244.0 0.0785674
\(163\) 104816. 0.309000 0.154500 0.987993i \(-0.450623\pi\)
0.154500 + 0.987993i \(0.450623\pi\)
\(164\) −103488. −0.300456
\(165\) 14256.0 0.0407650
\(166\) 427056. 1.20286
\(167\) 426972. 1.18470 0.592350 0.805681i \(-0.298200\pi\)
0.592350 + 0.805681i \(0.298200\pi\)
\(168\) 0 0
\(169\) −361689. −0.974134
\(170\) −20736.0 −0.0550304
\(171\) 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 \(0.654584\pi\)
\(180\) −31104.0 −0.0715542
\(181\) −588098. −1.33430 −0.667150 0.744924i \(-0.732486\pi\)
−0.667150 + 0.744924i \(0.732486\pi\)
\(182\) 0 0
\(183\) 320886. 0.708309
\(184\) −66432.0 −0.144655
\(185\) 293712. 0.630946
\(186\) −253728. −0.537757
\(187\) 14256.0 0.0298122
\(188\) −329664. −0.680263
\(189\) 0 0
\(190\) −32640.0 −0.0655943
\(191\) 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 \(-0.393976\pi\)
0.326960 + 0.945038i \(0.393976\pi\)
\(194\) −74648.0 −0.142401
\(195\) −21168.0 −0.0398651
\(196\) 0 0
\(197\) −237942. −0.436823 −0.218412 0.975857i \(-0.570088\pi\)
−0.218412 + 0.975857i \(0.570088\pi\)
\(198\) 21384.0 0.0387638
\(199\) −204464. −0.366003 −0.183001 0.983113i \(-0.558581\pi\)
−0.183001 + 0.983113i \(0.558581\pi\)
\(200\) −163136. −0.288386
\(201\) −114120. −0.199238
\(202\) −612336. −1.05587
\(203\) 0 0
\(204\) −31104.0 −0.0523288
\(205\) 155232. 0.257986
\(206\) −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
\(215\) 369888. 0.545725
\(216\) −46656.0 −0.0680414
\(217\) 0 0
\(218\) 848888. 1.20979
\(219\) 303606. 0.427760
\(220\) −25344.0 −0.0353036
\(221\) −21168.0 −0.0291541
\(222\) 440568. 0.599971
\(223\) −1.47006e6 −1.97957 −0.989787 0.142554i \(-0.954468\pi\)
−0.989787 + 0.142554i \(0.954468\pi\)
\(224\) 0 0
\(225\) −206469. −0.271893
\(226\) 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 \(-0.271137\pi\)
0.658627 + 0.752469i \(0.271137\pi\)
\(230\) 99648.0 0.124208
\(231\) 0 0
\(232\) −159360. −0.194383
\(233\) 651222. 0.785849 0.392925 0.919571i \(-0.371463\pi\)
0.392925 + 0.919571i \(0.371463\pi\)
\(234\) −31752.0 −0.0379080
\(235\) 494496. 0.584108
\(236\) −547584. −0.639986
\(237\) 765972. 0.885813
\(238\) 0 0
\(239\) −513462. −0.581452 −0.290726 0.956806i \(-0.593897\pi\)
−0.290726 + 0.956806i \(0.593897\pi\)
\(240\) 55296.0 0.0619677
\(241\) 694714. 0.770484 0.385242 0.922816i \(-0.374118\pi\)
0.385242 + 0.922816i \(0.374118\pi\)
\(242\) −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
\(250\) 544704. 0.551202
\(251\) 1.39608e6 1.39870 0.699352 0.714777i \(-0.253472\pi\)
0.699352 + 0.714777i \(0.253472\pi\)
\(252\) 0 0
\(253\) −68508.0 −0.0672884
\(254\) −212176. −0.206354
\(255\) 46656.0 0.0449321
\(256\) 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
\(259\) 0 0
\(260\) 37632.0 0.0345242
\(261\) −201690. −0.183267
\(262\) −277296. −0.249569
\(263\) 1.25301e6 1.11703 0.558515 0.829494i \(-0.311371\pi\)
0.558515 + 0.829494i \(0.311371\pi\)
\(264\) −38016.0 −0.0335704
\(265\) −779760. −0.682097
\(266\) 0 0
\(267\) 313956. 0.269520
\(268\) 202880. 0.172545
\(269\) 1.76069e6 1.48355 0.741774 0.670650i \(-0.233985\pi\)
0.741774 + 0.670650i \(0.233985\pi\)
\(270\) 69984.0 0.0584237
\(271\) −770528. −0.637331 −0.318666 0.947867i \(-0.603235\pi\)
−0.318666 + 0.947867i \(0.603235\pi\)
\(272\) 55296.0 0.0453181
\(273\) 0 0
\(274\) 519384. 0.417938
\(275\) −168234. −0.134147
\(276\) 149472. 0.118110
\(277\) 707738. 0.554208 0.277104 0.960840i \(-0.410625\pi\)
0.277104 + 0.960840i \(0.410625\pi\)
\(278\) 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
\(285\) 73440.0 0.0535575
\(286\) −25872.0 −0.0187032
\(287\) 0 0
\(288\) 82944.0 0.0589256
\(289\) −1.37320e6 −0.967140
\(290\) 239040. 0.166907
\(291\) 167958. 0.116270
\(292\) −539744. −0.370451
\(293\) −517020. −0.351834 −0.175917 0.984405i \(-0.556289\pi\)
−0.175917 + 0.984405i \(0.556289\pi\)
\(294\) 0 0
\(295\) 821376. 0.549524
\(296\) −783232. −0.519590
\(297\) −48114.0 −0.0316505
\(298\) 868776. 0.566718
\(299\) 101724. 0.0658030
\(300\) 367056. 0.235467
\(301\) 0 0
\(302\) 884000. 0.557744
\(303\) 1.37776e6 0.862116
\(304\) 87040.0 0.0540176
\(305\) 855696. 0.526708
\(306\) 69984.0 0.0427263
\(307\) −1.35002e6 −0.817512 −0.408756 0.912644i \(-0.634037\pi\)
−0.408756 + 0.912644i \(0.634037\pi\)
\(308\) 0 0
\(309\) 322776. 0.192311
\(310\) −676608. −0.399883
\(311\) −1.34538e6 −0.788758 −0.394379 0.918948i \(-0.629040\pi\)
−0.394379 + 0.918948i \(0.629040\pi\)
\(312\) 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 \(-0.327433\pi\)
0.515967 + 0.856609i \(0.327433\pi\)
\(318\) −1.16964e6 −0.648611
\(319\) −164340. −0.0904204
\(320\) −98304.0 −0.0536656
\(321\) 859086. 0.465344
\(322\) 0 0
\(323\) 73440.0 0.0391675
\(324\) 104976. 0.0555556
\(325\) 249802. 0.131186
\(326\) 419264. 0.218496
\(327\) −1.91000e6 −0.987788
\(328\) −413952. −0.212454
\(329\) 0 0
\(330\) 57024.0 0.0288252
\(331\) −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
\(335\) −304320. −0.148156
\(336\) 0 0
\(337\) −1.63481e6 −0.784136 −0.392068 0.919936i \(-0.628240\pi\)
−0.392068 + 0.919936i \(0.628240\pi\)
\(338\) −1.44676e6 −0.688816
\(339\) −558954. −0.264166
\(340\) −82944.0 −0.0389124
\(341\) 465168. 0.216633
\(342\) 110160. 0.0509282
\(343\) 0 0
\(344\) −986368. −0.449410
\(345\) −224208. −0.101415
\(346\) −1.32427e6 −0.594685
\(347\) −841530. −0.375185 −0.187593 0.982247i \(-0.560068\pi\)
−0.187593 + 0.982247i \(0.560068\pi\)
\(348\) 358560. 0.158713
\(349\) 977242. 0.429476 0.214738 0.976672i \(-0.431110\pi\)
0.214738 + 0.976672i \(0.431110\pi\)
\(350\) 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
\(355\) 1.02341e6 0.431001
\(356\) −558144. −0.233411
\(357\) 0 0
\(358\) −1.60078e6 −0.660120
\(359\) −3.47301e6 −1.42223 −0.711115 0.703076i \(-0.751810\pi\)
−0.711115 + 0.703076i \(0.751810\pi\)
\(360\) −124416. −0.0505964
\(361\) −2.36050e6 −0.953314
\(362\) −2.35239e6 −0.943492
\(363\) 1.41026e6 0.561734
\(364\) 0 0
\(365\) 809616. 0.318088
\(366\) 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
\(370\) 1.17485e6 0.446146
\(371\) 0 0
\(372\) −1.01491e6 −0.380252
\(373\) −2.01673e6 −0.750543 −0.375272 0.926915i \(-0.622451\pi\)
−0.375272 + 0.926915i \(0.622451\pi\)
\(374\) 57024.0 0.0210804
\(375\) −1.22558e6 −0.450054
\(376\) −1.31866e6 −0.481019
\(377\) 244020. 0.0884244
\(378\) 0 0
\(379\) −5.38083e6 −1.92420 −0.962102 0.272690i \(-0.912087\pi\)
−0.962102 + 0.272690i \(0.912087\pi\)
\(380\) −130560. −0.0463822
\(381\) 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 \(0.452286\pi\)
\(390\) −84672.0 −0.0281889
\(391\) −224208. −0.0741667
\(392\) 0 0
\(393\) 623916. 0.203772
\(394\) −951768. −0.308881
\(395\) 2.04259e6 0.658702
\(396\) 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
\(399\) 0 0
\(400\) −652544. −0.203920
\(401\) 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
\(405\) −157464. −0.0477028
\(406\) 0 0
\(407\) −807708. −0.241695
\(408\) −124416. −0.0370021
\(409\) −77246.0 −0.0228332 −0.0114166 0.999935i \(-0.503634\pi\)
−0.0114166 + 0.999935i \(0.503634\pi\)
\(410\) 620928. 0.182424
\(411\) −1.16861e6 −0.341245
\(412\) −573824. −0.166547
\(413\) 0 0
\(414\) −336312. −0.0964365
\(415\) −2.56234e6 −0.730324
\(416\) −100352. −0.0284310
\(417\) −939204. −0.264496
\(418\) 89760.0 0.0251271
\(419\) 5.20615e6 1.44871 0.724356 0.689427i \(-0.242137\pi\)
0.724356 + 0.689427i \(0.242137\pi\)
\(420\) 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
\(425\) −550584. −0.147860
\(426\) 1.53511e6 0.409842
\(427\) 0 0
\(428\) −1.52726e6 −0.403000
\(429\) 58212.0 0.0152711
\(430\) 1.47955e6 0.385886
\(431\) −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 \(0.678549\pi\)
\(434\) 0 0
\(435\) −537840. −0.136279
\(436\) 3.39555e6 0.855449
\(437\) −352920. −0.0884042
\(438\) 1.21442e6 0.302472
\(439\) −3.88407e6 −0.961891 −0.480946 0.876750i \(-0.659707\pi\)
−0.480946 + 0.876750i \(0.659707\pi\)
\(440\) −101376. −0.0249634
\(441\) 0 0
\(442\) −84672.0 −0.0206150
\(443\) −2.31499e6 −0.560453 −0.280226 0.959934i \(-0.590410\pi\)
−0.280226 + 0.959934i \(0.590410\pi\)
\(444\) 1.76227e6 0.424244
\(445\) 837216. 0.200418
\(446\) −5.88022e6 −1.39977
\(447\) −1.95475e6 −0.462723
\(448\) 0 0
\(449\) −1.92281e6 −0.450113 −0.225056 0.974346i \(-0.572257\pi\)
−0.225056 + 0.974346i \(0.572257\pi\)
\(450\) −825876. −0.192258
\(451\) −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 \(-0.221007\pi\)
0.768493 + 0.639858i \(0.221007\pi\)
\(458\) 4.18137e6 0.931440
\(459\) −157464. −0.0348859
\(460\) 398592. 0.0878282
\(461\) −2.97167e6 −0.651250 −0.325625 0.945499i \(-0.605575\pi\)
−0.325625 + 0.945499i \(0.605575\pi\)
\(462\) 0 0
\(463\) 4.87423e6 1.05670 0.528352 0.849025i \(-0.322810\pi\)
0.528352 + 0.849025i \(0.322810\pi\)
\(464\) −637440. −0.137450
\(465\) 1.52237e6 0.326503
\(466\) 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
\(469\) 0 0
\(470\) 1.97798e6 0.413027
\(471\) −3.40533e6 −0.707305
\(472\) −2.19034e6 −0.452539
\(473\) −1.01719e6 −0.209050
\(474\) 3.06389e6 0.626364
\(475\) −866660. −0.176244
\(476\) 0 0
\(477\) 2.63169e6 0.529589
\(478\) −2.05385e6 −0.411148
\(479\) −2.34397e6 −0.466782 −0.233391 0.972383i \(-0.574982\pi\)
−0.233391 + 0.972383i \(0.574982\pi\)
\(480\) 221184. 0.0438178
\(481\) 1.19932e6 0.236360
\(482\) 2.77886e6 0.544814
\(483\) 0 0
\(484\) −2.50712e6 −0.486476
\(485\) 447888. 0.0864600
\(486\) −236196. −0.0453609
\(487\) 316928. 0.0605534 0.0302767 0.999542i \(-0.490361\pi\)
0.0302767 + 0.999542i \(0.490361\pi\)
\(488\) −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
\(495\) −128304. −0.0235357
\(496\) 1.80429e6 0.329308
\(497\) 0 0
\(498\) −3.84350e6 −0.694471
\(499\) −4.86773e6 −0.875135 −0.437568 0.899185i \(-0.644160\pi\)
−0.437568 + 0.899185i \(0.644160\pi\)
\(500\) 2.17882e6 0.389758
\(501\) −3.84275e6 −0.683987
\(502\) 5.58432e6 0.989034
\(503\) −426888. −0.0752305 −0.0376153 0.999292i \(-0.511976\pi\)
−0.0376153 + 0.999292i \(0.511976\pi\)
\(504\) 0 0
\(505\) 3.67402e6 0.641081
\(506\) −274032. −0.0475801
\(507\) 3.25520e6 0.562416
\(508\) −848704. −0.145914
\(509\) 9.41621e6 1.61095 0.805474 0.592631i \(-0.201911\pi\)
0.805474 + 0.592631i \(0.201911\pi\)
\(510\) 186624. 0.0317718
\(511\) 0 0
\(512\) 262144. 0.0441942
\(513\) −247860. −0.0415827
\(514\) 4.02082e6 0.671284
\(515\) 860736. 0.143005
\(516\) 2.21933e6 0.366942
\(517\) −1.35986e6 −0.223753
\(518\) 0 0
\(519\) 2.97961e6 0.485558
\(520\) 150528. 0.0244123
\(521\) −1.84039e6 −0.297041 −0.148520 0.988909i \(-0.547451\pi\)
−0.148520 + 0.988909i \(0.547451\pi\)
\(522\) −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
\(530\) −3.11904e6 −0.482316
\(531\) −2.77214e6 −0.426658
\(532\) 0 0
\(533\) 633864. 0.0966447
\(534\) 1.25582e6 0.190579
\(535\) 2.29090e6 0.346036
\(536\) 811520. 0.122008
\(537\) 3.60175e6 0.538986
\(538\) 7.04275e6 1.04903
\(539\) 0 0
\(540\) 279936. 0.0413118
\(541\) 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
\(545\) −5.09333e6 −0.734531
\(546\) 0 0
\(547\) 8.73025e6 1.24755 0.623775 0.781604i \(-0.285598\pi\)
0.623775 + 0.781604i \(0.285598\pi\)
\(548\) 2.07754e6 0.295527
\(549\) −2.88797e6 −0.408943
\(550\) −672936. −0.0948565
\(551\) −846600. −0.118795
\(552\) 597888. 0.0835165
\(553\) 0 0
\(554\) 2.83095e6 0.391885
\(555\) −2.64341e6 −0.364277
\(556\) 1.66970e6 0.229061
\(557\) −3.01066e6 −0.411172 −0.205586 0.978639i \(-0.565910\pi\)
−0.205586 + 0.978639i \(0.565910\pi\)
\(558\) 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
\(565\) −1.49054e6 −0.196437
\(566\) −6.43611e6 −0.844467
\(567\) 0 0
\(568\) −2.72909e6 −0.354933
\(569\) 1.31578e7 1.70374 0.851870 0.523754i \(-0.175469\pi\)
0.851870 + 0.523754i \(0.175469\pi\)
\(570\) 293760. 0.0378709
\(571\) −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
\(574\) 0 0
\(575\) 2.64586e6 0.333732
\(576\) 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
\(580\) 956160. 0.118021
\(581\) 0 0
\(582\) 671832. 0.0822154
\(583\) 2.14434e6 0.261290
\(584\) −2.15898e6 −0.261948
\(585\) 190512. 0.0230161
\(586\) −2.06808e6 −0.248784
\(587\) 554568. 0.0664293 0.0332146 0.999448i \(-0.489426\pi\)
0.0332146 + 0.999448i \(0.489426\pi\)
\(588\) 0 0
\(589\) 2.39632e6 0.284614
\(590\) 3.28550e6 0.388572
\(591\) 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 \(0.338303\pi\)
\(600\) 1.46822e6 0.166500
\(601\) 9.61555e6 1.08590 0.542948 0.839767i \(-0.317308\pi\)
0.542948 + 0.839767i \(0.317308\pi\)
\(602\) 0 0
\(603\) 1.02708e6 0.115030
\(604\) 3.53600e6 0.394385
\(605\) 3.76068e6 0.417713
\(606\) 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
\(609\) 0 0
\(610\) 3.42278e6 0.372439
\(611\) 2.01919e6 0.218814
\(612\) 279936. 0.0302121
\(613\) −7.96215e6 −0.855814 −0.427907 0.903823i \(-0.640749\pi\)
−0.427907 + 0.903823i \(0.640749\pi\)
\(614\) −5.40008e6 −0.578068
\(615\) −1.39709e6 −0.148948
\(616\) 0 0
\(617\) −1.37397e7 −1.45299 −0.726497 0.687170i \(-0.758853\pi\)
−0.726497 + 0.687170i \(0.758853\pi\)
\(618\) 1.29110e6 0.135985
\(619\) 8.70113e6 0.912744 0.456372 0.889789i \(-0.349149\pi\)
0.456372 + 0.889789i \(0.349149\pi\)
\(620\) −2.70643e6 −0.282760
\(621\) 756702. 0.0787401
\(622\) −5.38152e6 −0.557736
\(623\) 0 0
\(624\) 225792. 0.0232138
\(625\) 4.69740e6 0.481014
\(626\) −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
\(635\) 1.27306e6 0.125289
\(636\) −4.67856e6 −0.458637
\(637\) 0 0
\(638\) −657360. −0.0639369
\(639\) −3.45400e6 −0.334634
\(640\) −393216. −0.0379473
\(641\) −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 \(0.227419\pi\)
\(644\) 0 0
\(645\) −3.32899e6 −0.315075
\(646\) 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
\(650\) 999208. 0.0927625
\(651\) 0 0
\(652\) 1.67706e6 0.154500
\(653\) 7.34149e6 0.673753 0.336877 0.941549i \(-0.390629\pi\)
0.336877 + 0.941549i \(0.390629\pi\)
\(654\) −7.63999e6 −0.698471
\(655\) 1.66378e6 0.151528
\(656\) −1.65581e6 −0.150228
\(657\) −2.73245e6 −0.246967
\(658\) 0 0
\(659\) −6.18934e6 −0.555176 −0.277588 0.960700i \(-0.589535\pi\)
−0.277588 + 0.960700i \(0.589535\pi\)
\(660\) 228096. 0.0203825
\(661\) 1.96690e7 1.75097 0.875484 0.483248i \(-0.160543\pi\)
0.875484 + 0.483248i \(0.160543\pi\)
\(662\) −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
\(670\) −1.21728e6 −0.104762
\(671\) −2.35316e6 −0.201765
\(672\) 0 0
\(673\) 7.18259e6 0.611285 0.305642 0.952146i \(-0.401129\pi\)
0.305642 + 0.952146i \(0.401129\pi\)
\(674\) −6.53922e6 −0.554468
\(675\) 1.85822e6 0.156978
\(676\) −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
\(679\) 0 0
\(680\) −331776. −0.0275152
\(681\) −5.30604e6 −0.438432
\(682\) 1.86067e6 0.153182
\(683\) 2.12204e7 1.74061 0.870306 0.492512i \(-0.163921\pi\)
0.870306 + 0.492512i \(0.163921\pi\)
\(684\) 440640. 0.0360117
\(685\) −3.11630e6 −0.253754
\(686\) 0 0
\(687\) −9.40808e6 −0.760517
\(688\) −3.94547e6 −0.317781
\(689\) −3.18402e6 −0.255522
\(690\) −896832. −0.0717114
\(691\) −1.63276e7 −1.30085 −0.650424 0.759571i \(-0.725409\pi\)
−0.650424 + 0.759571i \(0.725409\pi\)
\(692\) −5.29709e6 −0.420506
\(693\) 0 0
\(694\) −3.36612e6 −0.265296
\(695\) −2.50454e6 −0.196683
\(696\) 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
\(705\) −4.45046e6 −0.337235
\(706\) −1.38343e7 −1.04459
\(707\) 0 0
\(708\) 4.92826e6 0.369496
\(709\) 2.21195e7 1.65257 0.826284 0.563253i \(-0.190450\pi\)
0.826284 + 0.563253i \(0.190450\pi\)
\(710\) 4.09363e6 0.304763
\(711\) −6.89375e6 −0.511424
\(712\) −2.23258e6 −0.165046
\(713\) −7.31582e6 −0.538939
\(714\) 0 0
\(715\) 155232. 0.0113558
\(716\) −6.40310e6 −0.466775
\(717\) 4.62116e6 0.335701
\(718\) −1.38920e7 −1.00567
\(719\) −2.55819e7 −1.84548 −0.922742 0.385418i \(-0.874057\pi\)
−0.922742 + 0.385418i \(0.874057\pi\)
\(720\) −497664. −0.0357771
\(721\) 0 0
\(722\) −9.44200e6 −0.674095
\(723\) −6.25243e6 −0.444839
\(724\) −9.40957e6 −0.667150
\(725\) 6.34701e6 0.448460
\(726\) 5.64102e6 0.397206
\(727\) 9.29438e6 0.652205 0.326103 0.945334i \(-0.394265\pi\)
0.326103 + 0.945334i \(0.394265\pi\)
\(728\) 0 0
\(729\) 531441. 0.0370370
\(730\) 3.23846e6 0.224922
\(731\) −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 \(0.237343\pi\)
\(740\) 4.69939e6 0.315473
\(741\) 299880. 0.0200633
\(742\) 0 0
\(743\) 3.79246e6 0.252028 0.126014 0.992028i \(-0.459782\pi\)
0.126014 + 0.992028i \(0.459782\pi\)
\(744\) −4.05965e6 −0.268878
\(745\) −5.21266e6 −0.344087
\(746\) −8.06692e6 −0.530714
\(747\) 8.64788e6 0.567033
\(748\) 228096. 0.0149061
\(749\) 0 0
\(750\) −4.90234e6 −0.318236
\(751\) −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
\(755\) −5.30400e6 −0.338638
\(756\) 0 0
\(757\) 1.18427e7 0.751126 0.375563 0.926797i \(-0.377449\pi\)
0.375563 + 0.926797i \(0.377449\pi\)
\(758\) −2.15233e7 −1.36062
\(759\) 616572. 0.0388490
\(760\) −522240. −0.0327972
\(761\) −2.97791e6 −0.186402 −0.0932008 0.995647i \(-0.529710\pi\)
−0.0932008 + 0.995647i \(0.529710\pi\)
\(762\) 1.90958e6 0.119138
\(763\) 0 0
\(764\) 1.50295e7 0.931559
\(765\) −419904. −0.0259416
\(766\) −3.22973e6 −0.198881
\(767\) 3.35395e6 0.205858
\(768\) −589824. −0.0360844
\(769\) 2.02441e7 1.23447 0.617237 0.786777i \(-0.288252\pi\)
0.617237 + 0.786777i \(0.288252\pi\)
\(770\) 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
\(775\) −1.79654e7 −1.07444
\(776\) −1.19437e6 −0.0712006
\(777\) 0 0
\(778\) 3.56556e6 0.211193
\(779\) −2.19912e6 −0.129839
\(780\) −338688. −0.0199326
\(781\) −2.81437e6 −0.165103
\(782\) −896832. −0.0524438
\(783\) 1.81521e6 0.105809
\(784\) 0 0
\(785\) −9.08088e6 −0.525961
\(786\) 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
\(790\) 8.17037e6 0.465773
\(791\) 0 0
\(792\) 342144. 0.0193819
\(793\) 3.49409e6 0.197311
\(794\) −4.49382e6 −0.252967
\(795\) 7.01784e6 0.393809
\(796\) −3.27142e6 −0.183001
\(797\) 1.49548e7 0.833938 0.416969 0.908921i \(-0.363092\pi\)
0.416969 + 0.908921i \(0.363092\pi\)
\(798\) 0 0
\(799\) −4.45046e6 −0.246626
\(800\) −2.61018e6 −0.144193
\(801\) −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 \(0.219497\pi\)
\(810\) −629856. −0.0337310
\(811\) 1.52265e7 0.812922 0.406461 0.913668i \(-0.366763\pi\)
0.406461 + 0.913668i \(0.366763\pi\)
\(812\) 0 0
\(813\) 6.93475e6 0.367963
\(814\) −3.23083e6 −0.170904
\(815\) −2.51558e6 −0.132661
\(816\) −497664. −0.0261644
\(817\) −5.24008e6 −0.274652
\(818\) −308984. −0.0161455
\(819\) 0 0
\(820\) 2.48371e6 0.128993
\(821\) −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 \(-0.613728\pi\)
−0.349735 + 0.936849i \(0.613728\pi\)
\(824\) −2.29530e6 −0.117766
\(825\) 1.51411e6 0.0774500
\(826\) 0 0
\(827\) 3.13936e6 0.159616 0.0798082 0.996810i \(-0.474569\pi\)
0.0798082 + 0.996810i \(0.474569\pi\)
\(828\) −1.34525e6 −0.0681909
\(829\) −1.27081e7 −0.642234 −0.321117 0.947040i \(-0.604058\pi\)
−0.321117 + 0.947040i \(0.604058\pi\)
\(830\) −1.02493e7 −0.516417
\(831\) −6.36964e6 −0.319972
\(832\) −401408. −0.0201038
\(833\) 0 0
\(834\) −3.75682e6 −0.187027
\(835\) −1.02473e7 −0.508621
\(836\) 359040. 0.0177675
\(837\) −5.13799e6 −0.253501
\(838\) 2.08246e7 1.02439
\(839\) 2.98312e7 1.46307 0.731536 0.681803i \(-0.238804\pi\)
0.731536 + 0.681803i \(0.238804\pi\)
\(840\) 0 0
\(841\) −1.43110e7 −0.697720
\(842\) 6.87390e6 0.334136
\(843\) −2.07389e7 −1.00512
\(844\) −5.57958e6 −0.269616
\(845\) 8.68054e6 0.418220
\(846\) −6.67570e6 −0.320679
\(847\) 0 0
\(848\) 8.31744e6 0.397192
\(849\) 1.44813e7 0.689504
\(850\) −2.20234e6 −0.104553
\(851\) 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 \(0.350619\pi\)
\(854\) 0 0
\(855\) −660960. −0.0309215
\(856\) −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 \(-0.432011\pi\)
0.211974 + 0.977275i \(0.432011\pi\)
\(860\) 5.91821e6 0.272863
\(861\) 0 0
\(862\) −2.32250e6 −0.106460
\(863\) −2.92196e7 −1.33551 −0.667755 0.744381i \(-0.732745\pi\)
−0.667755 + 0.744381i \(0.732745\pi\)
\(864\) −746496. −0.0340207
\(865\) 7.94563e6 0.361067
\(866\) −1.66035e7 −0.752324
\(867\) 1.23588e7 0.558379
\(868\) 0 0
\(869\) −5.61713e6 −0.252328
\(870\) −2.15136e6 −0.0963640
\(871\) −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 \(-0.431606\pi\)
0.213215 + 0.977005i \(0.431606\pi\)
\(878\) −1.55363e7 −0.680160
\(879\) 4.65318e6 0.203132
\(880\) −405504. −0.0176518
\(881\) −1.65372e7 −0.717833 −0.358917 0.933370i \(-0.616854\pi\)
−0.358917 + 0.933370i \(0.616854\pi\)
\(882\) 0 0
\(883\) −2.39487e7 −1.03367 −0.516833 0.856086i \(-0.672889\pi\)
−0.516833 + 0.856086i \(0.672889\pi\)
\(884\) −338688. −0.0145770
\(885\) −7.39238e6 −0.317268
\(886\) −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
\(889\) 0 0
\(890\) 3.34886e6 0.141717
\(891\) 433026. 0.0182734
\(892\) −2.35209e7 −0.989787
\(893\) −7.00536e6 −0.293969
\(894\) −7.81898e6 −0.327195
\(895\) 9.60466e6 0.400797
\(896\) 0 0
\(897\) −915516. −0.0379914
\(898\) −7.69126e6 −0.318278
\(899\) −1.75495e7 −0.724212
\(900\) −3.30350e6 −0.135947
\(901\) 7.01784e6 0.287999
\(902\) −1.70755e6 −0.0698808
\(903\) 0 0
\(904\) 3.97478e6 0.161768
\(905\) 1.41144e7 0.572848
\(906\) −7.95600e6 −0.322014
\(907\) 2.06126e7 0.831983 0.415991 0.909369i \(-0.363435\pi\)
0.415991 + 0.909369i \(0.363435\pi\)
\(908\) 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
\(915\) −7.70126e6 −0.304095
\(916\) 1.67255e7 0.658627
\(917\) 0 0
\(918\) −629856. −0.0246680
\(919\) −3.61227e7 −1.41088 −0.705442 0.708767i \(-0.749252\pi\)
−0.705442 + 0.708767i \(0.749252\pi\)
\(920\) 1.59437e6 0.0621039
\(921\) 1.21502e7 0.471991
\(922\) −1.18867e7 −0.460504
\(923\) 4.17892e6 0.161458
\(924\) 0 0
\(925\) 3.11947e7 1.19874
\(926\) 1.94969e7 0.747203
\(927\) −2.90498e6 −0.111031
\(928\) −2.54976e6 −0.0971917
\(929\) −1.29366e7 −0.491792 −0.245896 0.969296i \(-0.579082\pi\)
−0.245896 + 0.969296i \(0.579082\pi\)
\(930\) 6.08947e6 0.230873
\(931\) 0 0
\(932\) 1.04196e7 0.392925
\(933\) 1.21084e7 0.455390
\(934\) 3.26920e7 1.22624
\(935\) −342144. −0.0127991
\(936\) −508032. −0.0189540
\(937\) −5.01394e7 −1.86565 −0.932824 0.360332i \(-0.882664\pi\)
−0.932824 + 0.360332i \(0.882664\pi\)
\(938\) 0 0
\(939\) 2.30539e6 0.0853257
\(940\) 7.91194e6 0.292054
\(941\) 1.05568e7 0.388651 0.194325 0.980937i \(-0.437748\pi\)
0.194325 + 0.980937i \(0.437748\pi\)
\(942\) −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 \(-0.518157\pi\)
−0.0570124 + 0.998373i \(0.518157\pi\)
\(948\) 1.22556e7 0.442906
\(949\) 3.30593e6 0.119159
\(950\) −3.46664e6 −0.124623
\(951\) −1.66166e7 −0.595787
\(952\) 0 0
\(953\) 5.22829e7 1.86478 0.932389 0.361455i \(-0.117720\pi\)
0.932389 + 0.361455i \(0.117720\pi\)
\(954\) 1.05268e7 0.374476
\(955\) −2.25442e7 −0.799883
\(956\) −8.21539e6 −0.290726
\(957\) 1.47906e6 0.0522043
\(958\) −9.37589e6 −0.330064
\(959\) 0 0
\(960\) 884736. 0.0309839
\(961\) 2.10452e7 0.735095
\(962\) 4.79730e6 0.167132
\(963\) −7.73177e6 −0.268666
\(964\) 1.11154e7 0.385242
\(965\) −8.12136e6 −0.280744
\(966\) 0 0
\(967\) −2.48235e7 −0.853682 −0.426841 0.904327i \(-0.640374\pi\)
−0.426841 + 0.904327i \(0.640374\pi\)
\(968\) −1.00285e7 −0.343991
\(969\) −660960. −0.0226134
\(970\) 1.79155e6 0.0611364
\(971\) −1.33077e7 −0.452956 −0.226478 0.974016i \(-0.572721\pi\)
−0.226478 + 0.974016i \(0.572721\pi\)
\(972\) −944784. −0.0320750
\(973\) 0 0
\(974\) 1.26771e6 0.0428177
\(975\) −2.24822e6 −0.0757403
\(976\) −9.12742e6 −0.306707
\(977\) 8.17705e6 0.274069 0.137035 0.990566i \(-0.456243\pi\)
0.137035 + 0.990566i \(0.456243\pi\)
\(978\) −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
\(985\) 5.71061e6 0.187539
\(986\) −2.15136e6 −0.0704727
\(987\) 0 0
\(988\) −533120. −0.0173753
\(989\) 1.59977e7 0.520075
\(990\) −513216. −0.0166423
\(991\) −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
\(994\) 0 0
\(995\) 4.90714e6 0.157134
\(996\) −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 294.6.a.i.1.1 1
3.2 odd 2 882.6.a.i.1.1 1
7.2 even 3 294.6.e.f.67.1 2
7.3 odd 6 294.6.e.b.79.1 2
7.4 even 3 294.6.e.f.79.1 2
7.5 odd 6 294.6.e.b.67.1 2
7.6 odd 2 42.6.a.f.1.1 1
21.20 even 2 126.6.a.b.1.1 1
28.27 even 2 336.6.a.g.1.1 1
35.13 even 4 1050.6.g.m.799.1 2
35.27 even 4 1050.6.g.m.799.2 2
35.34 odd 2 1050.6.a.a.1.1 1
84.83 odd 2 1008.6.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.6.a.f.1.1 1 7.6 odd 2
126.6.a.b.1.1 1 21.20 even 2
294.6.a.i.1.1 1 1.1 even 1 trivial
294.6.e.b.67.1 2 7.5 odd 6
294.6.e.b.79.1 2 7.3 odd 6
294.6.e.f.67.1 2 7.2 even 3
294.6.e.f.79.1 2 7.4 even 3
336.6.a.g.1.1 1 28.27 even 2
882.6.a.i.1.1 1 3.2 odd 2
1008.6.a.k.1.1 1 84.83 odd 2
1050.6.a.a.1.1 1 35.34 odd 2
1050.6.g.m.799.1 2 35.13 even 4
1050.6.g.m.799.2 2 35.27 even 4