Properties

Label 294.6.a.e.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)
 
\(f(q)\) \(=\) \(q-4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} -86.0000 q^{5} -36.0000 q^{6} -64.0000 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} -86.0000 q^{5} -36.0000 q^{6} -64.0000 q^{8} +81.0000 q^{9} +344.000 q^{10} +34.0000 q^{11} +144.000 q^{12} +3.00000 q^{13} -774.000 q^{15} +256.000 q^{16} +1904.00 q^{17} -324.000 q^{18} +1489.00 q^{19} -1376.00 q^{20} -136.000 q^{22} -224.000 q^{23} -576.000 q^{24} +4271.00 q^{25} -12.0000 q^{26} +729.000 q^{27} -6508.00 q^{29} +3096.00 q^{30} -1731.00 q^{31} -1024.00 q^{32} +306.000 q^{33} -7616.00 q^{34} +1296.00 q^{36} -7633.00 q^{37} -5956.00 q^{38} +27.0000 q^{39} +5504.00 q^{40} -15414.0 q^{41} +18491.0 q^{43} +544.000 q^{44} -6966.00 q^{45} +896.000 q^{46} -18462.0 q^{47} +2304.00 q^{48} -17084.0 q^{50} +17136.0 q^{51} +48.0000 q^{52} -19956.0 q^{53} -2916.00 q^{54} -2924.00 q^{55} +13401.0 q^{57} +26032.0 q^{58} +31828.0 q^{59} -12384.0 q^{60} +57654.0 q^{61} +6924.00 q^{62} +4096.00 q^{64} -258.000 q^{65} -1224.00 q^{66} -60563.0 q^{67} +30464.0 q^{68} -2016.00 q^{69} -44834.0 q^{71} -5184.00 q^{72} -20821.0 q^{73} +30532.0 q^{74} +38439.0 q^{75} +23824.0 q^{76} -108.000 q^{78} -30531.0 q^{79} -22016.0 q^{80} +6561.00 q^{81} +61656.0 q^{82} -110602. q^{83} -163744. q^{85} -73964.0 q^{86} -58572.0 q^{87} -2176.00 q^{88} +58992.0 q^{89} +27864.0 q^{90} -3584.00 q^{92} -15579.0 q^{93} +73848.0 q^{94} -128054. q^{95} -9216.00 q^{96} +119846. q^{97} +2754.00 q^{99} +O(q^{100})\)

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
\(5\) −86.0000 −1.53841 −0.769207 0.638999i \(-0.779349\pi\)
−0.769207 + 0.638999i \(0.779349\pi\)
\(6\) −36.0000 −0.408248
\(7\) 0 0
\(8\) −64.0000 −0.353553
\(9\) 81.0000 0.333333
\(10\) 344.000 1.08782
\(11\) 34.0000 0.0847222 0.0423611 0.999102i \(-0.486512\pi\)
0.0423611 + 0.999102i \(0.486512\pi\)
\(12\) 144.000 0.288675
\(13\) 3.00000 0.00492337 0.00246169 0.999997i \(-0.499216\pi\)
0.00246169 + 0.999997i \(0.499216\pi\)
\(14\) 0 0
\(15\) −774.000 −0.888204
\(16\) 256.000 0.250000
\(17\) 1904.00 1.59788 0.798941 0.601410i \(-0.205394\pi\)
0.798941 + 0.601410i \(0.205394\pi\)
\(18\) −324.000 −0.235702
\(19\) 1489.00 0.946260 0.473130 0.880992i \(-0.343124\pi\)
0.473130 + 0.880992i \(0.343124\pi\)
\(20\) −1376.00 −0.769207
\(21\) 0 0
\(22\) −136.000 −0.0599076
\(23\) −224.000 −0.0882934 −0.0441467 0.999025i \(-0.514057\pi\)
−0.0441467 + 0.999025i \(0.514057\pi\)
\(24\) −576.000 −0.204124
\(25\) 4271.00 1.36672
\(26\) −12.0000 −0.00348135
\(27\) 729.000 0.192450
\(28\) 0 0
\(29\) −6508.00 −1.43699 −0.718493 0.695534i \(-0.755168\pi\)
−0.718493 + 0.695534i \(0.755168\pi\)
\(30\) 3096.00 0.628055
\(31\) −1731.00 −0.323514 −0.161757 0.986831i \(-0.551716\pi\)
−0.161757 + 0.986831i \(0.551716\pi\)
\(32\) −1024.00 −0.176777
\(33\) 306.000 0.0489144
\(34\) −7616.00 −1.12987
\(35\) 0 0
\(36\) 1296.00 0.166667
\(37\) −7633.00 −0.916623 −0.458312 0.888792i \(-0.651546\pi\)
−0.458312 + 0.888792i \(0.651546\pi\)
\(38\) −5956.00 −0.669107
\(39\) 27.0000 0.00284251
\(40\) 5504.00 0.543912
\(41\) −15414.0 −1.43204 −0.716021 0.698079i \(-0.754039\pi\)
−0.716021 + 0.698079i \(0.754039\pi\)
\(42\) 0 0
\(43\) 18491.0 1.52507 0.762534 0.646948i \(-0.223955\pi\)
0.762534 + 0.646948i \(0.223955\pi\)
\(44\) 544.000 0.0423611
\(45\) −6966.00 −0.512805
\(46\) 896.000 0.0624329
\(47\) −18462.0 −1.21909 −0.609543 0.792753i \(-0.708647\pi\)
−0.609543 + 0.792753i \(0.708647\pi\)
\(48\) 2304.00 0.144338
\(49\) 0 0
\(50\) −17084.0 −0.966417
\(51\) 17136.0 0.922537
\(52\) 48.0000 0.00246169
\(53\) −19956.0 −0.975852 −0.487926 0.872885i \(-0.662246\pi\)
−0.487926 + 0.872885i \(0.662246\pi\)
\(54\) −2916.00 −0.136083
\(55\) −2924.00 −0.130338
\(56\) 0 0
\(57\) 13401.0 0.546324
\(58\) 26032.0 1.01610
\(59\) 31828.0 1.19036 0.595181 0.803591i \(-0.297080\pi\)
0.595181 + 0.803591i \(0.297080\pi\)
\(60\) −12384.0 −0.444102
\(61\) 57654.0 1.98383 0.991916 0.126897i \(-0.0405017\pi\)
0.991916 + 0.126897i \(0.0405017\pi\)
\(62\) 6924.00 0.228759
\(63\) 0 0
\(64\) 4096.00 0.125000
\(65\) −258.000 −0.00757419
\(66\) −1224.00 −0.0345877
\(67\) −60563.0 −1.64824 −0.824120 0.566415i \(-0.808330\pi\)
−0.824120 + 0.566415i \(0.808330\pi\)
\(68\) 30464.0 0.798941
\(69\) −2016.00 −0.0509762
\(70\) 0 0
\(71\) −44834.0 −1.05551 −0.527754 0.849397i \(-0.676966\pi\)
−0.527754 + 0.849397i \(0.676966\pi\)
\(72\) −5184.00 −0.117851
\(73\) −20821.0 −0.457293 −0.228646 0.973510i \(-0.573430\pi\)
−0.228646 + 0.973510i \(0.573430\pi\)
\(74\) 30532.0 0.648151
\(75\) 38439.0 0.789076
\(76\) 23824.0 0.473130
\(77\) 0 0
\(78\) −108.000 −0.00200996
\(79\) −30531.0 −0.550394 −0.275197 0.961388i \(-0.588743\pi\)
−0.275197 + 0.961388i \(0.588743\pi\)
\(80\) −22016.0 −0.384604
\(81\) 6561.00 0.111111
\(82\) 61656.0 1.01261
\(83\) −110602. −1.76225 −0.881125 0.472883i \(-0.843213\pi\)
−0.881125 + 0.472883i \(0.843213\pi\)
\(84\) 0 0
\(85\) −163744. −2.45820
\(86\) −73964.0 −1.07839
\(87\) −58572.0 −0.829644
\(88\) −2176.00 −0.0299538
\(89\) 58992.0 0.789438 0.394719 0.918802i \(-0.370842\pi\)
0.394719 + 0.918802i \(0.370842\pi\)
\(90\) 27864.0 0.362608
\(91\) 0 0
\(92\) −3584.00 −0.0441467
\(93\) −15579.0 −0.186781
\(94\) 73848.0 0.862023
\(95\) −128054. −1.45574
\(96\) −9216.00 −0.102062
\(97\) 119846. 1.29328 0.646642 0.762793i \(-0.276173\pi\)
0.646642 + 0.762793i \(0.276173\pi\)
\(98\) 0 0
\(99\) 2754.00 0.0282407
\(100\) 68336.0 0.683360
\(101\) −100010. −0.975529 −0.487764 0.872975i \(-0.662187\pi\)
−0.487764 + 0.872975i \(0.662187\pi\)
\(102\) −68544.0 −0.652332
\(103\) −121691. −1.13023 −0.565113 0.825013i \(-0.691167\pi\)
−0.565113 + 0.825013i \(0.691167\pi\)
\(104\) −192.000 −0.00174068
\(105\) 0 0
\(106\) 79824.0 0.690031
\(107\) −48648.0 −0.410776 −0.205388 0.978681i \(-0.565846\pi\)
−0.205388 + 0.978681i \(0.565846\pi\)
\(108\) 11664.0 0.0962250
\(109\) −152075. −1.22600 −0.613002 0.790082i \(-0.710038\pi\)
−0.613002 + 0.790082i \(0.710038\pi\)
\(110\) 11696.0 0.0921628
\(111\) −68697.0 −0.529213
\(112\) 0 0
\(113\) −60886.0 −0.448561 −0.224280 0.974525i \(-0.572003\pi\)
−0.224280 + 0.974525i \(0.572003\pi\)
\(114\) −53604.0 −0.386309
\(115\) 19264.0 0.135832
\(116\) −104128. −0.718493
\(117\) 243.000 0.00164112
\(118\) −127312. −0.841714
\(119\) 0 0
\(120\) 49536.0 0.314028
\(121\) −159895. −0.992822
\(122\) −230616. −1.40278
\(123\) −138726. −0.826790
\(124\) −27696.0 −0.161757
\(125\) −98556.0 −0.564167
\(126\) 0 0
\(127\) −151965. −0.836054 −0.418027 0.908435i \(-0.637278\pi\)
−0.418027 + 0.908435i \(0.637278\pi\)
\(128\) −16384.0 −0.0883883
\(129\) 166419. 0.880499
\(130\) 1032.00 0.00535576
\(131\) −235502. −1.19899 −0.599496 0.800378i \(-0.704632\pi\)
−0.599496 + 0.800378i \(0.704632\pi\)
\(132\) 4896.00 0.0244572
\(133\) 0 0
\(134\) 242252. 1.16548
\(135\) −62694.0 −0.296068
\(136\) −121856. −0.564937
\(137\) 325508. 1.48170 0.740850 0.671671i \(-0.234423\pi\)
0.740850 + 0.671671i \(0.234423\pi\)
\(138\) 8064.00 0.0360456
\(139\) −3211.00 −0.0140962 −0.00704812 0.999975i \(-0.502244\pi\)
−0.00704812 + 0.999975i \(0.502244\pi\)
\(140\) 0 0
\(141\) −166158. −0.703839
\(142\) 179336. 0.746357
\(143\) 102.000 0.000417119 0
\(144\) 20736.0 0.0833333
\(145\) 559688. 2.21068
\(146\) 83284.0 0.323355
\(147\) 0 0
\(148\) −122128. −0.458312
\(149\) −151884. −0.560462 −0.280231 0.959933i \(-0.590411\pi\)
−0.280231 + 0.959933i \(0.590411\pi\)
\(150\) −153756. −0.557961
\(151\) 76648.0 0.273564 0.136782 0.990601i \(-0.456324\pi\)
0.136782 + 0.990601i \(0.456324\pi\)
\(152\) −95296.0 −0.334554
\(153\) 154224. 0.532627
\(154\) 0 0
\(155\) 148866. 0.497698
\(156\) 432.000 0.00142126
\(157\) 389710. 1.26181 0.630903 0.775862i \(-0.282685\pi\)
0.630903 + 0.775862i \(0.282685\pi\)
\(158\) 122124. 0.389187
\(159\) −179604. −0.563408
\(160\) 88064.0 0.271956
\(161\) 0 0
\(162\) −26244.0 −0.0785674
\(163\) −112372. −0.331275 −0.165638 0.986187i \(-0.552968\pi\)
−0.165638 + 0.986187i \(0.552968\pi\)
\(164\) −246624. −0.716021
\(165\) −26316.0 −0.0752506
\(166\) 442408. 1.24610
\(167\) −52550.0 −0.145808 −0.0729040 0.997339i \(-0.523227\pi\)
−0.0729040 + 0.997339i \(0.523227\pi\)
\(168\) 0 0
\(169\) −371284. −0.999976
\(170\) 654976. 1.73821
\(171\) 120609. 0.315420
\(172\) 295856. 0.762534
\(173\) 135256. 0.343591 0.171795 0.985133i \(-0.445043\pi\)
0.171795 + 0.985133i \(0.445043\pi\)
\(174\) 234288. 0.586647
\(175\) 0 0
\(176\) 8704.00 0.0211805
\(177\) 286452. 0.687256
\(178\) −235968. −0.558217
\(179\) 250638. 0.584675 0.292337 0.956315i \(-0.405567\pi\)
0.292337 + 0.956315i \(0.405567\pi\)
\(180\) −111456. −0.256402
\(181\) −199233. −0.452027 −0.226014 0.974124i \(-0.572569\pi\)
−0.226014 + 0.974124i \(0.572569\pi\)
\(182\) 0 0
\(183\) 518886. 1.14537
\(184\) 14336.0 0.0312164
\(185\) 656438. 1.41015
\(186\) 62316.0 0.132074
\(187\) 64736.0 0.135376
\(188\) −295392. −0.609543
\(189\) 0 0
\(190\) 512216. 1.02936
\(191\) −238770. −0.473583 −0.236792 0.971560i \(-0.576096\pi\)
−0.236792 + 0.971560i \(0.576096\pi\)
\(192\) 36864.0 0.0721688
\(193\) 85691.0 0.165593 0.0827965 0.996566i \(-0.473615\pi\)
0.0827965 + 0.996566i \(0.473615\pi\)
\(194\) −479384. −0.914491
\(195\) −2322.00 −0.00437296
\(196\) 0 0
\(197\) −71408.0 −0.131094 −0.0655468 0.997849i \(-0.520879\pi\)
−0.0655468 + 0.997849i \(0.520879\pi\)
\(198\) −11016.0 −0.0199692
\(199\) 711352. 1.27336 0.636681 0.771127i \(-0.280307\pi\)
0.636681 + 0.771127i \(0.280307\pi\)
\(200\) −273344. −0.483208
\(201\) −545067. −0.951612
\(202\) 400040. 0.689803
\(203\) 0 0
\(204\) 274176. 0.461269
\(205\) 1.32560e6 2.20307
\(206\) 486764. 0.799191
\(207\) −18144.0 −0.0294311
\(208\) 768.000 0.00123084
\(209\) 50626.0 0.0801693
\(210\) 0 0
\(211\) −260260. −0.402440 −0.201220 0.979546i \(-0.564491\pi\)
−0.201220 + 0.979546i \(0.564491\pi\)
\(212\) −319296. −0.487926
\(213\) −403506. −0.609398
\(214\) 194592. 0.290463
\(215\) −1.59023e6 −2.34619
\(216\) −46656.0 −0.0680414
\(217\) 0 0
\(218\) 608300. 0.866915
\(219\) −187389. −0.264018
\(220\) −46784.0 −0.0651689
\(221\) 5712.00 0.00786697
\(222\) 274788. 0.374210
\(223\) −105656. −0.142276 −0.0711381 0.997466i \(-0.522663\pi\)
−0.0711381 + 0.997466i \(0.522663\pi\)
\(224\) 0 0
\(225\) 345951. 0.455573
\(226\) 243544. 0.317180
\(227\) −654750. −0.843356 −0.421678 0.906746i \(-0.638559\pi\)
−0.421678 + 0.906746i \(0.638559\pi\)
\(228\) 214416. 0.273162
\(229\) −557713. −0.702784 −0.351392 0.936228i \(-0.614292\pi\)
−0.351392 + 0.936228i \(0.614292\pi\)
\(230\) −77056.0 −0.0960477
\(231\) 0 0
\(232\) 416512. 0.508051
\(233\) −1.24759e6 −1.50551 −0.752755 0.658301i \(-0.771275\pi\)
−0.752755 + 0.658301i \(0.771275\pi\)
\(234\) −972.000 −0.00116045
\(235\) 1.58773e6 1.87546
\(236\) 509248. 0.595181
\(237\) −274779. −0.317770
\(238\) 0 0
\(239\) −496926. −0.562726 −0.281363 0.959601i \(-0.590786\pi\)
−0.281363 + 0.959601i \(0.590786\pi\)
\(240\) −198144. −0.222051
\(241\) 277618. 0.307897 0.153948 0.988079i \(-0.450801\pi\)
0.153948 + 0.988079i \(0.450801\pi\)
\(242\) 639580. 0.702031
\(243\) 59049.0 0.0641500
\(244\) 922464. 0.991916
\(245\) 0 0
\(246\) 554904. 0.584629
\(247\) 4467.00 0.00465879
\(248\) 110784. 0.114379
\(249\) −995418. −1.01744
\(250\) 394224. 0.398927
\(251\) 308328. 0.308908 0.154454 0.988000i \(-0.450638\pi\)
0.154454 + 0.988000i \(0.450638\pi\)
\(252\) 0 0
\(253\) −7616.00 −0.00748041
\(254\) 607860. 0.591179
\(255\) −1.47370e6 −1.41925
\(256\) 65536.0 0.0625000
\(257\) −408762. −0.386045 −0.193022 0.981194i \(-0.561829\pi\)
−0.193022 + 0.981194i \(0.561829\pi\)
\(258\) −665676. −0.622606
\(259\) 0 0
\(260\) −4128.00 −0.00378710
\(261\) −527148. −0.478995
\(262\) 942008. 0.847816
\(263\) 1.08812e6 0.970039 0.485019 0.874503i \(-0.338813\pi\)
0.485019 + 0.874503i \(0.338813\pi\)
\(264\) −19584.0 −0.0172938
\(265\) 1.71622e6 1.50126
\(266\) 0 0
\(267\) 530928. 0.455782
\(268\) −969008. −0.824120
\(269\) 668290. 0.563098 0.281549 0.959547i \(-0.409152\pi\)
0.281549 + 0.959547i \(0.409152\pi\)
\(270\) 250776. 0.209352
\(271\) 830664. 0.687072 0.343536 0.939140i \(-0.388375\pi\)
0.343536 + 0.939140i \(0.388375\pi\)
\(272\) 487424. 0.399470
\(273\) 0 0
\(274\) −1.30203e6 −1.04772
\(275\) 145214. 0.115792
\(276\) −32256.0 −0.0254881
\(277\) 925073. 0.724397 0.362198 0.932101i \(-0.382026\pi\)
0.362198 + 0.932101i \(0.382026\pi\)
\(278\) 12844.0 0.00996755
\(279\) −140211. −0.107838
\(280\) 0 0
\(281\) 1.33635e6 1.00961 0.504805 0.863233i \(-0.331564\pi\)
0.504805 + 0.863233i \(0.331564\pi\)
\(282\) 664632. 0.497689
\(283\) 992957. 0.736995 0.368497 0.929629i \(-0.379872\pi\)
0.368497 + 0.929629i \(0.379872\pi\)
\(284\) −717344. −0.527754
\(285\) −1.15249e6 −0.840473
\(286\) −408.000 −0.000294948 0
\(287\) 0 0
\(288\) −82944.0 −0.0589256
\(289\) 2.20536e6 1.55323
\(290\) −2.23875e6 −1.56319
\(291\) 1.07861e6 0.746678
\(292\) −333136. −0.228646
\(293\) −563544. −0.383494 −0.191747 0.981444i \(-0.561415\pi\)
−0.191747 + 0.981444i \(0.561415\pi\)
\(294\) 0 0
\(295\) −2.73721e6 −1.83127
\(296\) 488512. 0.324075
\(297\) 24786.0 0.0163048
\(298\) 607536. 0.396307
\(299\) −672.000 −0.000434702 0
\(300\) 615024. 0.394538
\(301\) 0 0
\(302\) −306592. −0.193439
\(303\) −900090. −0.563222
\(304\) 381184. 0.236565
\(305\) −4.95824e6 −3.05196
\(306\) −616896. −0.376624
\(307\) −2.82703e6 −1.71193 −0.855963 0.517037i \(-0.827035\pi\)
−0.855963 + 0.517037i \(0.827035\pi\)
\(308\) 0 0
\(309\) −1.09522e6 −0.652536
\(310\) −595464. −0.351926
\(311\) 1.12731e6 0.660912 0.330456 0.943821i \(-0.392797\pi\)
0.330456 + 0.943821i \(0.392797\pi\)
\(312\) −1728.00 −0.00100498
\(313\) 2.36013e6 1.36168 0.680840 0.732432i \(-0.261615\pi\)
0.680840 + 0.732432i \(0.261615\pi\)
\(314\) −1.55884e6 −0.892231
\(315\) 0 0
\(316\) −488496. −0.275197
\(317\) −2.22420e6 −1.24316 −0.621578 0.783352i \(-0.713508\pi\)
−0.621578 + 0.783352i \(0.713508\pi\)
\(318\) 718416. 0.398390
\(319\) −221272. −0.121745
\(320\) −352256. −0.192302
\(321\) −437832. −0.237162
\(322\) 0 0
\(323\) 2.83506e6 1.51201
\(324\) 104976. 0.0555556
\(325\) 12813.0 0.00672887
\(326\) 449488. 0.234247
\(327\) −1.36868e6 −0.707833
\(328\) 986496. 0.506303
\(329\) 0 0
\(330\) 105264. 0.0532102
\(331\) −3.70304e6 −1.85775 −0.928877 0.370388i \(-0.879225\pi\)
−0.928877 + 0.370388i \(0.879225\pi\)
\(332\) −1.76963e6 −0.881125
\(333\) −618273. −0.305541
\(334\) 210200. 0.103102
\(335\) 5.20842e6 2.53568
\(336\) 0 0
\(337\) 1.21432e6 0.582452 0.291226 0.956654i \(-0.405937\pi\)
0.291226 + 0.956654i \(0.405937\pi\)
\(338\) 1.48514e6 0.707090
\(339\) −547974. −0.258977
\(340\) −2.61990e6 −1.22910
\(341\) −58854.0 −0.0274088
\(342\) −482436. −0.223036
\(343\) 0 0
\(344\) −1.18342e6 −0.539193
\(345\) 173376. 0.0784226
\(346\) −541024. −0.242955
\(347\) −977904. −0.435986 −0.217993 0.975950i \(-0.569951\pi\)
−0.217993 + 0.975950i \(0.569951\pi\)
\(348\) −937152. −0.414822
\(349\) −511282. −0.224697 −0.112348 0.993669i \(-0.535837\pi\)
−0.112348 + 0.993669i \(0.535837\pi\)
\(350\) 0 0
\(351\) 2187.00 0.000947504 0
\(352\) −34816.0 −0.0149769
\(353\) −3.02752e6 −1.29315 −0.646577 0.762848i \(-0.723800\pi\)
−0.646577 + 0.762848i \(0.723800\pi\)
\(354\) −1.14581e6 −0.485964
\(355\) 3.85572e6 1.62381
\(356\) 943872. 0.394719
\(357\) 0 0
\(358\) −1.00255e6 −0.413427
\(359\) 4.59456e6 1.88151 0.940757 0.339082i \(-0.110116\pi\)
0.940757 + 0.339082i \(0.110116\pi\)
\(360\) 445824. 0.181304
\(361\) −258978. −0.104591
\(362\) 796932. 0.319632
\(363\) −1.43906e6 −0.573206
\(364\) 0 0
\(365\) 1.79061e6 0.703506
\(366\) −2.07554e6 −0.809896
\(367\) 1.11273e6 0.431246 0.215623 0.976477i \(-0.430822\pi\)
0.215623 + 0.976477i \(0.430822\pi\)
\(368\) −57344.0 −0.0220734
\(369\) −1.24853e6 −0.477347
\(370\) −2.62575e6 −0.997125
\(371\) 0 0
\(372\) −249264. −0.0933904
\(373\) −2.45895e6 −0.915119 −0.457559 0.889179i \(-0.651276\pi\)
−0.457559 + 0.889179i \(0.651276\pi\)
\(374\) −258944. −0.0957253
\(375\) −887004. −0.325722
\(376\) 1.18157e6 0.431012
\(377\) −19524.0 −0.00707482
\(378\) 0 0
\(379\) −4.10130e6 −1.46664 −0.733320 0.679884i \(-0.762030\pi\)
−0.733320 + 0.679884i \(0.762030\pi\)
\(380\) −2.04886e6 −0.727871
\(381\) −1.36768e6 −0.482696
\(382\) 955080. 0.334874
\(383\) 2.59413e6 0.903639 0.451820 0.892109i \(-0.350775\pi\)
0.451820 + 0.892109i \(0.350775\pi\)
\(384\) −147456. −0.0510310
\(385\) 0 0
\(386\) −342764. −0.117092
\(387\) 1.49777e6 0.508356
\(388\) 1.91754e6 0.646642
\(389\) 2.23407e6 0.748552 0.374276 0.927317i \(-0.377891\pi\)
0.374276 + 0.927317i \(0.377891\pi\)
\(390\) 9288.00 0.00309215
\(391\) −426496. −0.141082
\(392\) 0 0
\(393\) −2.11952e6 −0.692238
\(394\) 285632. 0.0926971
\(395\) 2.62567e6 0.846733
\(396\) 44064.0 0.0141204
\(397\) 2.06400e6 0.657253 0.328627 0.944460i \(-0.393414\pi\)
0.328627 + 0.944460i \(0.393414\pi\)
\(398\) −2.84541e6 −0.900403
\(399\) 0 0
\(400\) 1.09338e6 0.341680
\(401\) 1.15283e6 0.358017 0.179008 0.983848i \(-0.442711\pi\)
0.179008 + 0.983848i \(0.442711\pi\)
\(402\) 2.18027e6 0.672891
\(403\) −5193.00 −0.00159278
\(404\) −1.60016e6 −0.487764
\(405\) −564246. −0.170935
\(406\) 0 0
\(407\) −259522. −0.0776583
\(408\) −1.09670e6 −0.326166
\(409\) −5.93412e6 −1.75408 −0.877038 0.480421i \(-0.840484\pi\)
−0.877038 + 0.480421i \(0.840484\pi\)
\(410\) −5.30242e6 −1.55781
\(411\) 2.92957e6 0.855460
\(412\) −1.94706e6 −0.565113
\(413\) 0 0
\(414\) 72576.0 0.0208110
\(415\) 9.51177e6 2.71107
\(416\) −3072.00 −0.000870338 0
\(417\) −28899.0 −0.00813847
\(418\) −202504. −0.0566882
\(419\) −771666. −0.214731 −0.107365 0.994220i \(-0.534241\pi\)
−0.107365 + 0.994220i \(0.534241\pi\)
\(420\) 0 0
\(421\) −2.87542e6 −0.790671 −0.395336 0.918537i \(-0.629372\pi\)
−0.395336 + 0.918537i \(0.629372\pi\)
\(422\) 1.04104e6 0.284568
\(423\) −1.49542e6 −0.406362
\(424\) 1.27718e6 0.345016
\(425\) 8.13198e6 2.18386
\(426\) 1.61402e6 0.430909
\(427\) 0 0
\(428\) −778368. −0.205388
\(429\) 918.000 0.000240824 0
\(430\) 6.36090e6 1.65901
\(431\) −137862. −0.0357480 −0.0178740 0.999840i \(-0.505690\pi\)
−0.0178740 + 0.999840i \(0.505690\pi\)
\(432\) 186624. 0.0481125
\(433\) −1.56526e6 −0.401204 −0.200602 0.979673i \(-0.564290\pi\)
−0.200602 + 0.979673i \(0.564290\pi\)
\(434\) 0 0
\(435\) 5.03719e6 1.27634
\(436\) −2.43320e6 −0.613002
\(437\) −333536. −0.0835486
\(438\) 749556. 0.186689
\(439\) −4.88158e6 −1.20892 −0.604462 0.796634i \(-0.706612\pi\)
−0.604462 + 0.796634i \(0.706612\pi\)
\(440\) 187136. 0.0460814
\(441\) 0 0
\(442\) −22848.0 −0.00556279
\(443\) −1.30152e6 −0.315094 −0.157547 0.987511i \(-0.550359\pi\)
−0.157547 + 0.987511i \(0.550359\pi\)
\(444\) −1.09915e6 −0.264606
\(445\) −5.07331e6 −1.21448
\(446\) 422624. 0.100604
\(447\) −1.36696e6 −0.323583
\(448\) 0 0
\(449\) −3.13141e6 −0.733034 −0.366517 0.930411i \(-0.619450\pi\)
−0.366517 + 0.930411i \(0.619450\pi\)
\(450\) −1.38380e6 −0.322139
\(451\) −524076. −0.121326
\(452\) −974176. −0.224280
\(453\) 689832. 0.157942
\(454\) 2.61900e6 0.596343
\(455\) 0 0
\(456\) −857664. −0.193155
\(457\) 6.49268e6 1.45423 0.727116 0.686514i \(-0.240860\pi\)
0.727116 + 0.686514i \(0.240860\pi\)
\(458\) 2.23085e6 0.496944
\(459\) 1.38802e6 0.307512
\(460\) 308224. 0.0679160
\(461\) −5.34717e6 −1.17185 −0.585925 0.810365i \(-0.699269\pi\)
−0.585925 + 0.810365i \(0.699269\pi\)
\(462\) 0 0
\(463\) 3.37285e6 0.731215 0.365607 0.930769i \(-0.380861\pi\)
0.365607 + 0.930769i \(0.380861\pi\)
\(464\) −1.66605e6 −0.359247
\(465\) 1.33979e6 0.287346
\(466\) 4.99038e6 1.06456
\(467\) −2.23452e6 −0.474125 −0.237062 0.971494i \(-0.576185\pi\)
−0.237062 + 0.971494i \(0.576185\pi\)
\(468\) 3888.00 0.000820562 0
\(469\) 0 0
\(470\) −6.35093e6 −1.32615
\(471\) 3.50739e6 0.728504
\(472\) −2.03699e6 −0.420857
\(473\) 628694. 0.129207
\(474\) 1.09912e6 0.224697
\(475\) 6.35952e6 1.29327
\(476\) 0 0
\(477\) −1.61644e6 −0.325284
\(478\) 1.98770e6 0.397907
\(479\) 2.52136e6 0.502108 0.251054 0.967973i \(-0.419223\pi\)
0.251054 + 0.967973i \(0.419223\pi\)
\(480\) 792576. 0.157014
\(481\) −22899.0 −0.00451288
\(482\) −1.11047e6 −0.217716
\(483\) 0 0
\(484\) −2.55832e6 −0.496411
\(485\) −1.03068e7 −1.98961
\(486\) −236196. −0.0453609
\(487\) 1.91672e6 0.366215 0.183107 0.983093i \(-0.441384\pi\)
0.183107 + 0.983093i \(0.441384\pi\)
\(488\) −3.68986e6 −0.701390
\(489\) −1.01135e6 −0.191262
\(490\) 0 0
\(491\) 5.82875e6 1.09112 0.545559 0.838073i \(-0.316317\pi\)
0.545559 + 0.838073i \(0.316317\pi\)
\(492\) −2.21962e6 −0.413395
\(493\) −1.23912e7 −2.29613
\(494\) −17868.0 −0.00329427
\(495\) −236844. −0.0434460
\(496\) −443136. −0.0808785
\(497\) 0 0
\(498\) 3.98167e6 0.719436
\(499\) −1.00049e7 −1.79870 −0.899352 0.437225i \(-0.855961\pi\)
−0.899352 + 0.437225i \(0.855961\pi\)
\(500\) −1.57690e6 −0.282084
\(501\) −472950. −0.0841823
\(502\) −1.23331e6 −0.218431
\(503\) −1.13666e6 −0.200313 −0.100157 0.994972i \(-0.531934\pi\)
−0.100157 + 0.994972i \(0.531934\pi\)
\(504\) 0 0
\(505\) 8.60086e6 1.50077
\(506\) 30464.0 0.00528945
\(507\) −3.34156e6 −0.577336
\(508\) −2.43144e6 −0.418027
\(509\) −4.01937e6 −0.687644 −0.343822 0.939035i \(-0.611722\pi\)
−0.343822 + 0.939035i \(0.611722\pi\)
\(510\) 5.89478e6 1.00356
\(511\) 0 0
\(512\) −262144. −0.0441942
\(513\) 1.08548e6 0.182108
\(514\) 1.63505e6 0.272975
\(515\) 1.04654e7 1.73876
\(516\) 2.66270e6 0.440249
\(517\) −627708. −0.103284
\(518\) 0 0
\(519\) 1.21730e6 0.198372
\(520\) 16512.0 0.00267788
\(521\) 5.52028e6 0.890977 0.445488 0.895288i \(-0.353030\pi\)
0.445488 + 0.895288i \(0.353030\pi\)
\(522\) 2.10859e6 0.338701
\(523\) −8.94472e6 −1.42992 −0.714961 0.699164i \(-0.753556\pi\)
−0.714961 + 0.699164i \(0.753556\pi\)
\(524\) −3.76803e6 −0.599496
\(525\) 0 0
\(526\) −4.35250e6 −0.685921
\(527\) −3.29582e6 −0.516937
\(528\) 78336.0 0.0122286
\(529\) −6.38617e6 −0.992204
\(530\) −6.86486e6 −1.06155
\(531\) 2.57807e6 0.396788
\(532\) 0 0
\(533\) −46242.0 −0.00705048
\(534\) −2.12371e6 −0.322287
\(535\) 4.18373e6 0.631945
\(536\) 3.87603e6 0.582741
\(537\) 2.25574e6 0.337562
\(538\) −2.67316e6 −0.398171
\(539\) 0 0
\(540\) −1.00310e6 −0.148034
\(541\) 849057. 0.124722 0.0623611 0.998054i \(-0.480137\pi\)
0.0623611 + 0.998054i \(0.480137\pi\)
\(542\) −3.32266e6 −0.485833
\(543\) −1.79310e6 −0.260978
\(544\) −1.94970e6 −0.282468
\(545\) 1.30784e7 1.88610
\(546\) 0 0
\(547\) 8.61340e6 1.23085 0.615426 0.788194i \(-0.288984\pi\)
0.615426 + 0.788194i \(0.288984\pi\)
\(548\) 5.20813e6 0.740850
\(549\) 4.66997e6 0.661277
\(550\) −580856. −0.0818770
\(551\) −9.69041e6 −1.35976
\(552\) 129024. 0.0180228
\(553\) 0 0
\(554\) −3.70029e6 −0.512226
\(555\) 5.90794e6 0.814149
\(556\) −51376.0 −0.00704812
\(557\) 7.79879e6 1.06510 0.532549 0.846399i \(-0.321234\pi\)
0.532549 + 0.846399i \(0.321234\pi\)
\(558\) 560844. 0.0762529
\(559\) 55473.0 0.00750848
\(560\) 0 0
\(561\) 582624. 0.0781594
\(562\) −5.34539e6 −0.713902
\(563\) −1.07271e6 −0.142631 −0.0713153 0.997454i \(-0.522720\pi\)
−0.0713153 + 0.997454i \(0.522720\pi\)
\(564\) −2.65853e6 −0.351920
\(565\) 5.23620e6 0.690073
\(566\) −3.97183e6 −0.521134
\(567\) 0 0
\(568\) 2.86938e6 0.373179
\(569\) −1.01524e6 −0.131459 −0.0657293 0.997837i \(-0.520937\pi\)
−0.0657293 + 0.997837i \(0.520937\pi\)
\(570\) 4.60994e6 0.594304
\(571\) −6.78093e6 −0.870361 −0.435180 0.900343i \(-0.643315\pi\)
−0.435180 + 0.900343i \(0.643315\pi\)
\(572\) 1632.00 0.000208560 0
\(573\) −2.14893e6 −0.273423
\(574\) 0 0
\(575\) −956704. −0.120672
\(576\) 331776. 0.0416667
\(577\) 3.30537e6 0.413314 0.206657 0.978413i \(-0.433742\pi\)
0.206657 + 0.978413i \(0.433742\pi\)
\(578\) −8.82144e6 −1.09830
\(579\) 771219. 0.0956052
\(580\) 8.95501e6 1.10534
\(581\) 0 0
\(582\) −4.31446e6 −0.527981
\(583\) −678504. −0.0826763
\(584\) 1.33254e6 0.161677
\(585\) −20898.0 −0.00252473
\(586\) 2.25418e6 0.271171
\(587\) 1.19833e7 1.43543 0.717715 0.696337i \(-0.245188\pi\)
0.717715 + 0.696337i \(0.245188\pi\)
\(588\) 0 0
\(589\) −2.57746e6 −0.306128
\(590\) 1.09488e7 1.29490
\(591\) −642672. −0.0756869
\(592\) −1.95405e6 −0.229156
\(593\) 5.31020e6 0.620117 0.310059 0.950717i \(-0.399651\pi\)
0.310059 + 0.950717i \(0.399651\pi\)
\(594\) −99144.0 −0.0115292
\(595\) 0 0
\(596\) −2.43014e6 −0.280231
\(597\) 6.40217e6 0.735176
\(598\) 2688.00 0.000307380 0
\(599\) 2.42236e6 0.275849 0.137924 0.990443i \(-0.455957\pi\)
0.137924 + 0.990443i \(0.455957\pi\)
\(600\) −2.46010e6 −0.278981
\(601\) 7.10659e6 0.802556 0.401278 0.915956i \(-0.368566\pi\)
0.401278 + 0.915956i \(0.368566\pi\)
\(602\) 0 0
\(603\) −4.90560e6 −0.549413
\(604\) 1.22637e6 0.136782
\(605\) 1.37510e7 1.52737
\(606\) 3.60036e6 0.398258
\(607\) −1.79194e7 −1.97402 −0.987011 0.160655i \(-0.948640\pi\)
−0.987011 + 0.160655i \(0.948640\pi\)
\(608\) −1.52474e6 −0.167277
\(609\) 0 0
\(610\) 1.98330e7 2.15806
\(611\) −55386.0 −0.00600201
\(612\) 2.46758e6 0.266314
\(613\) 1.39790e7 1.50254 0.751269 0.659997i \(-0.229442\pi\)
0.751269 + 0.659997i \(0.229442\pi\)
\(614\) 1.13081e7 1.21051
\(615\) 1.19304e7 1.27195
\(616\) 0 0
\(617\) −5.25594e6 −0.555824 −0.277912 0.960606i \(-0.589642\pi\)
−0.277912 + 0.960606i \(0.589642\pi\)
\(618\) 4.38088e6 0.461413
\(619\) −1.44301e6 −0.151371 −0.0756857 0.997132i \(-0.524115\pi\)
−0.0756857 + 0.997132i \(0.524115\pi\)
\(620\) 2.38186e6 0.248849
\(621\) −163296. −0.0169921
\(622\) −4.50926e6 −0.467336
\(623\) 0 0
\(624\) 6912.00 0.000710628 0
\(625\) −4.87106e6 −0.498796
\(626\) −9.44052e6 −0.962853
\(627\) 455634. 0.0462857
\(628\) 6.23536e6 0.630903
\(629\) −1.45332e7 −1.46466
\(630\) 0 0
\(631\) 1.51723e7 1.51697 0.758487 0.651688i \(-0.225939\pi\)
0.758487 + 0.651688i \(0.225939\pi\)
\(632\) 1.95398e6 0.194593
\(633\) −2.34234e6 −0.232349
\(634\) 8.89680e6 0.879044
\(635\) 1.30690e7 1.28620
\(636\) −2.87366e6 −0.281704
\(637\) 0 0
\(638\) 885088. 0.0860864
\(639\) −3.63155e6 −0.351836
\(640\) 1.40902e6 0.135978
\(641\) −144848. −0.0139241 −0.00696205 0.999976i \(-0.502216\pi\)
−0.00696205 + 0.999976i \(0.502216\pi\)
\(642\) 1.75133e6 0.167699
\(643\) 27469.0 0.00262009 0.00131004 0.999999i \(-0.499583\pi\)
0.00131004 + 0.999999i \(0.499583\pi\)
\(644\) 0 0
\(645\) −1.43120e7 −1.35457
\(646\) −1.13402e7 −1.06915
\(647\) −8.55783e6 −0.803717 −0.401858 0.915702i \(-0.631636\pi\)
−0.401858 + 0.915702i \(0.631636\pi\)
\(648\) −419904. −0.0392837
\(649\) 1.08215e6 0.100850
\(650\) −51252.0 −0.00475803
\(651\) 0 0
\(652\) −1.79795e6 −0.165638
\(653\) 1.25030e7 1.14745 0.573723 0.819049i \(-0.305498\pi\)
0.573723 + 0.819049i \(0.305498\pi\)
\(654\) 5.47470e6 0.500514
\(655\) 2.02532e7 1.84455
\(656\) −3.94598e6 −0.358010
\(657\) −1.68650e6 −0.152431
\(658\) 0 0
\(659\) 1.80471e7 1.61880 0.809400 0.587258i \(-0.199793\pi\)
0.809400 + 0.587258i \(0.199793\pi\)
\(660\) −421056. −0.0376253
\(661\) −3.34144e6 −0.297461 −0.148731 0.988878i \(-0.547519\pi\)
−0.148731 + 0.988878i \(0.547519\pi\)
\(662\) 1.48122e7 1.31363
\(663\) 51408.0 0.00454200
\(664\) 7.07853e6 0.623050
\(665\) 0 0
\(666\) 2.47309e6 0.216050
\(667\) 1.45779e6 0.126876
\(668\) −840800. −0.0729040
\(669\) −950904. −0.0821432
\(670\) −2.08337e7 −1.79299
\(671\) 1.96024e6 0.168075
\(672\) 0 0
\(673\) −8.47066e6 −0.720907 −0.360454 0.932777i \(-0.617378\pi\)
−0.360454 + 0.932777i \(0.617378\pi\)
\(674\) −4.85730e6 −0.411856
\(675\) 3.11356e6 0.263025
\(676\) −5.94054e6 −0.499988
\(677\) −1.46553e7 −1.22892 −0.614458 0.788949i \(-0.710625\pi\)
−0.614458 + 0.788949i \(0.710625\pi\)
\(678\) 2.19190e6 0.183124
\(679\) 0 0
\(680\) 1.04796e7 0.869107
\(681\) −5.89275e6 −0.486912
\(682\) 235416. 0.0193809
\(683\) 1.97616e7 1.62095 0.810477 0.585771i \(-0.199208\pi\)
0.810477 + 0.585771i \(0.199208\pi\)
\(684\) 1.92974e6 0.157710
\(685\) −2.79937e7 −2.27947
\(686\) 0 0
\(687\) −5.01942e6 −0.405753
\(688\) 4.73370e6 0.381267
\(689\) −59868.0 −0.00480448
\(690\) −693504. −0.0554532
\(691\) −1.35832e7 −1.08220 −0.541101 0.840958i \(-0.681992\pi\)
−0.541101 + 0.840958i \(0.681992\pi\)
\(692\) 2.16410e6 0.171795
\(693\) 0 0
\(694\) 3.91162e6 0.308289
\(695\) 276146. 0.0216859
\(696\) 3.74861e6 0.293324
\(697\) −2.93483e7 −2.28823
\(698\) 2.04513e6 0.158885
\(699\) −1.12283e7 −0.869206
\(700\) 0 0
\(701\) −1.29915e7 −0.998538 −0.499269 0.866447i \(-0.666398\pi\)
−0.499269 + 0.866447i \(0.666398\pi\)
\(702\) −8748.00 −0.000669986 0
\(703\) −1.13655e7 −0.867365
\(704\) 139264. 0.0105903
\(705\) 1.42896e7 1.08280
\(706\) 1.21101e7 0.914399
\(707\) 0 0
\(708\) 4.58323e6 0.343628
\(709\) 1.90873e6 0.142603 0.0713017 0.997455i \(-0.477285\pi\)
0.0713017 + 0.997455i \(0.477285\pi\)
\(710\) −1.54229e7 −1.14821
\(711\) −2.47301e6 −0.183465
\(712\) −3.77549e6 −0.279109
\(713\) 387744. 0.0285641
\(714\) 0 0
\(715\) −8772.00 −0.000641702 0
\(716\) 4.01021e6 0.292337
\(717\) −4.47233e6 −0.324890
\(718\) −1.83782e7 −1.33043
\(719\) 1.11200e7 0.802198 0.401099 0.916035i \(-0.368628\pi\)
0.401099 + 0.916035i \(0.368628\pi\)
\(720\) −1.78330e6 −0.128201
\(721\) 0 0
\(722\) 1.03591e6 0.0739571
\(723\) 2.49856e6 0.177764
\(724\) −3.18773e6 −0.226014
\(725\) −2.77957e7 −1.96396
\(726\) 5.75622e6 0.405318
\(727\) −8.37406e6 −0.587624 −0.293812 0.955863i \(-0.594924\pi\)
−0.293812 + 0.955863i \(0.594924\pi\)
\(728\) 0 0
\(729\) 531441. 0.0370370
\(730\) −7.16242e6 −0.497454
\(731\) 3.52069e7 2.43688
\(732\) 8.30218e6 0.572683
\(733\) −4.64448e6 −0.319284 −0.159642 0.987175i \(-0.551034\pi\)
−0.159642 + 0.987175i \(0.551034\pi\)
\(734\) −4.45092e6 −0.304937
\(735\) 0 0
\(736\) 229376. 0.0156082
\(737\) −2.05914e6 −0.139642
\(738\) 4.99414e6 0.337536
\(739\) −1.10623e7 −0.745136 −0.372568 0.928005i \(-0.621523\pi\)
−0.372568 + 0.928005i \(0.621523\pi\)
\(740\) 1.05030e7 0.705074
\(741\) 40203.0 0.00268976
\(742\) 0 0
\(743\) 1.97245e6 0.131079 0.0655395 0.997850i \(-0.479123\pi\)
0.0655395 + 0.997850i \(0.479123\pi\)
\(744\) 997056. 0.0660370
\(745\) 1.30620e7 0.862223
\(746\) 9.83580e6 0.647087
\(747\) −8.95876e6 −0.587417
\(748\) 1.03578e6 0.0676880
\(749\) 0 0
\(750\) 3.54802e6 0.230320
\(751\) 1.50246e7 0.972085 0.486042 0.873935i \(-0.338440\pi\)
0.486042 + 0.873935i \(0.338440\pi\)
\(752\) −4.72627e6 −0.304771
\(753\) 2.77495e6 0.178348
\(754\) 78096.0 0.00500265
\(755\) −6.59173e6 −0.420854
\(756\) 0 0
\(757\) −4.72426e6 −0.299636 −0.149818 0.988714i \(-0.547869\pi\)
−0.149818 + 0.988714i \(0.547869\pi\)
\(758\) 1.64052e7 1.03707
\(759\) −68544.0 −0.00431882
\(760\) 8.19546e6 0.514682
\(761\) 8.57835e6 0.536960 0.268480 0.963285i \(-0.413479\pi\)
0.268480 + 0.963285i \(0.413479\pi\)
\(762\) 5.47074e6 0.341318
\(763\) 0 0
\(764\) −3.82032e6 −0.236792
\(765\) −1.32633e7 −0.819402
\(766\) −1.03765e7 −0.638970
\(767\) 95484.0 0.00586060
\(768\) 589824. 0.0360844
\(769\) −1.76168e7 −1.07426 −0.537131 0.843499i \(-0.680492\pi\)
−0.537131 + 0.843499i \(0.680492\pi\)
\(770\) 0 0
\(771\) −3.67886e6 −0.222883
\(772\) 1.37106e6 0.0827965
\(773\) 1.25420e6 0.0754951 0.0377475 0.999287i \(-0.487982\pi\)
0.0377475 + 0.999287i \(0.487982\pi\)
\(774\) −5.99108e6 −0.359462
\(775\) −7.39310e6 −0.442153
\(776\) −7.67014e6 −0.457245
\(777\) 0 0
\(778\) −8.93626e6 −0.529306
\(779\) −2.29514e7 −1.35508
\(780\) −37152.0 −0.00218648
\(781\) −1.52436e6 −0.0894250
\(782\) 1.70598e6 0.0997604
\(783\) −4.74433e6 −0.276548
\(784\) 0 0
\(785\) −3.35151e7 −1.94118
\(786\) 8.47807e6 0.489487
\(787\) −7.09121e6 −0.408116 −0.204058 0.978959i \(-0.565413\pi\)
−0.204058 + 0.978959i \(0.565413\pi\)
\(788\) −1.14253e6 −0.0655468
\(789\) 9.79312e6 0.560052
\(790\) −1.05027e7 −0.598731
\(791\) 0 0
\(792\) −176256. −0.00998461
\(793\) 172962. 0.00976715
\(794\) −8.25599e6 −0.464748
\(795\) 1.54459e7 0.866756
\(796\) 1.13816e7 0.636681
\(797\) 2.71630e6 0.151472 0.0757358 0.997128i \(-0.475869\pi\)
0.0757358 + 0.997128i \(0.475869\pi\)
\(798\) 0 0
\(799\) −3.51516e7 −1.94795
\(800\) −4.37350e6 −0.241604
\(801\) 4.77835e6 0.263146
\(802\) −4.61131e6 −0.253156
\(803\) −707914. −0.0387429
\(804\) −8.72107e6 −0.475806
\(805\) 0 0
\(806\) 20772.0 0.00112627
\(807\) 6.01461e6 0.325105
\(808\) 6.40064e6 0.344901
\(809\) −2.18739e7 −1.17505 −0.587524 0.809207i \(-0.699897\pi\)
−0.587524 + 0.809207i \(0.699897\pi\)
\(810\) 2.25698e6 0.120869
\(811\) 1.72352e7 0.920164 0.460082 0.887876i \(-0.347820\pi\)
0.460082 + 0.887876i \(0.347820\pi\)
\(812\) 0 0
\(813\) 7.47598e6 0.396681
\(814\) 1.03809e6 0.0549127
\(815\) 9.66399e6 0.509639
\(816\) 4.38682e6 0.230634
\(817\) 2.75331e7 1.44311
\(818\) 2.37365e7 1.24032
\(819\) 0 0
\(820\) 2.12097e7 1.10154
\(821\) −2.26191e7 −1.17116 −0.585582 0.810613i \(-0.699134\pi\)
−0.585582 + 0.810613i \(0.699134\pi\)
\(822\) −1.17183e7 −0.604901
\(823\) 1.62633e6 0.0836967 0.0418484 0.999124i \(-0.486675\pi\)
0.0418484 + 0.999124i \(0.486675\pi\)
\(824\) 7.78822e6 0.399595
\(825\) 1.30693e6 0.0668523
\(826\) 0 0
\(827\) −2.28304e7 −1.16078 −0.580391 0.814338i \(-0.697100\pi\)
−0.580391 + 0.814338i \(0.697100\pi\)
\(828\) −290304. −0.0147156
\(829\) −2.58593e7 −1.30686 −0.653432 0.756985i \(-0.726672\pi\)
−0.653432 + 0.756985i \(0.726672\pi\)
\(830\) −3.80471e7 −1.91702
\(831\) 8.32566e6 0.418231
\(832\) 12288.0 0.000615422 0
\(833\) 0 0
\(834\) 115596. 0.00575477
\(835\) 4.51930e6 0.224313
\(836\) 810016. 0.0400846
\(837\) −1.26190e6 −0.0622603
\(838\) 3.08666e6 0.151838
\(839\) 1.23061e7 0.603554 0.301777 0.953379i \(-0.402420\pi\)
0.301777 + 0.953379i \(0.402420\pi\)
\(840\) 0 0
\(841\) 2.18429e7 1.06493
\(842\) 1.15017e7 0.559089
\(843\) 1.20271e7 0.582899
\(844\) −4.16416e6 −0.201220
\(845\) 3.19304e7 1.53838
\(846\) 5.98169e6 0.287341
\(847\) 0 0
\(848\) −5.10874e6 −0.243963
\(849\) 8.93661e6 0.425504
\(850\) −3.25279e7 −1.54422
\(851\) 1.70979e6 0.0809318
\(852\) −6.45610e6 −0.304699
\(853\) 1.91416e7 0.900753 0.450377 0.892839i \(-0.351290\pi\)
0.450377 + 0.892839i \(0.351290\pi\)
\(854\) 0 0
\(855\) −1.03724e7 −0.485247
\(856\) 3.11347e6 0.145231
\(857\) 4.75507e6 0.221159 0.110580 0.993867i \(-0.464729\pi\)
0.110580 + 0.993867i \(0.464729\pi\)
\(858\) −3672.00 −0.000170288 0
\(859\) 9.19876e6 0.425350 0.212675 0.977123i \(-0.431782\pi\)
0.212675 + 0.977123i \(0.431782\pi\)
\(860\) −2.54436e7 −1.17309
\(861\) 0 0
\(862\) 551448. 0.0252776
\(863\) −1.73532e7 −0.793144 −0.396572 0.918004i \(-0.629800\pi\)
−0.396572 + 0.918004i \(0.629800\pi\)
\(864\) −746496. −0.0340207
\(865\) −1.16320e7 −0.528585
\(866\) 6.26102e6 0.283694
\(867\) 1.98482e7 0.896756
\(868\) 0 0
\(869\) −1.03805e6 −0.0466305
\(870\) −2.01488e7 −0.902507
\(871\) −181689. −0.00811490
\(872\) 9.73280e6 0.433458
\(873\) 9.70753e6 0.431095
\(874\) 1.33414e6 0.0590778
\(875\) 0 0
\(876\) −2.99822e6 −0.132009
\(877\) −3.55622e7 −1.56131 −0.780655 0.624962i \(-0.785115\pi\)
−0.780655 + 0.624962i \(0.785115\pi\)
\(878\) 1.95263e7 0.854838
\(879\) −5.07190e6 −0.221410
\(880\) −748544. −0.0325845
\(881\) 2.63056e7 1.14185 0.570923 0.821003i \(-0.306585\pi\)
0.570923 + 0.821003i \(0.306585\pi\)
\(882\) 0 0
\(883\) 1.20394e7 0.519642 0.259821 0.965657i \(-0.416336\pi\)
0.259821 + 0.965657i \(0.416336\pi\)
\(884\) 91392.0 0.00393349
\(885\) −2.46349e7 −1.05729
\(886\) 5.20606e6 0.222805
\(887\) 133590. 0.00570118 0.00285059 0.999996i \(-0.499093\pi\)
0.00285059 + 0.999996i \(0.499093\pi\)
\(888\) 4.39661e6 0.187105
\(889\) 0 0
\(890\) 2.02932e7 0.858769
\(891\) 223074. 0.00941358
\(892\) −1.69050e6 −0.0711381
\(893\) −2.74899e7 −1.15357
\(894\) 5.46782e6 0.228808
\(895\) −2.15549e7 −0.899472
\(896\) 0 0
\(897\) −6048.00 −0.000250975 0
\(898\) 1.25256e7 0.518333
\(899\) 1.12653e7 0.464885
\(900\) 5.53522e6 0.227787
\(901\) −3.79962e7 −1.55930
\(902\) 2.09630e6 0.0857902
\(903\) 0 0
\(904\) 3.89670e6 0.158590
\(905\) 1.71340e7 0.695406
\(906\) −2.75933e6 −0.111682
\(907\) −1.27476e7 −0.514529 −0.257264 0.966341i \(-0.582821\pi\)
−0.257264 + 0.966341i \(0.582821\pi\)
\(908\) −1.04760e7 −0.421678
\(909\) −8.10081e6 −0.325176
\(910\) 0 0
\(911\) 2.43253e7 0.971095 0.485548 0.874210i \(-0.338620\pi\)
0.485548 + 0.874210i \(0.338620\pi\)
\(912\) 3.43066e6 0.136581
\(913\) −3.76047e6 −0.149302
\(914\) −2.59707e7 −1.02830
\(915\) −4.46242e7 −1.76205
\(916\) −8.92341e6 −0.351392
\(917\) 0 0
\(918\) −5.55206e6 −0.217444
\(919\) −3.84805e7 −1.50297 −0.751487 0.659748i \(-0.770663\pi\)
−0.751487 + 0.659748i \(0.770663\pi\)
\(920\) −1.23290e6 −0.0480238
\(921\) −2.54433e7 −0.988381
\(922\) 2.13887e7 0.828623
\(923\) −134502. −0.00519666
\(924\) 0 0
\(925\) −3.26005e7 −1.25277
\(926\) −1.34914e7 −0.517047
\(927\) −9.85697e6 −0.376742
\(928\) 6.66419e6 0.254026
\(929\) −3.96678e7 −1.50799 −0.753996 0.656879i \(-0.771876\pi\)
−0.753996 + 0.656879i \(0.771876\pi\)
\(930\) −5.35918e6 −0.203185
\(931\) 0 0
\(932\) −1.99615e7 −0.752755
\(933\) 1.01458e7 0.381578
\(934\) 8.93809e6 0.335257
\(935\) −5.56730e6 −0.208265
\(936\) −15552.0 −0.000580225 0
\(937\) −1.48428e7 −0.552288 −0.276144 0.961116i \(-0.589057\pi\)
−0.276144 + 0.961116i \(0.589057\pi\)
\(938\) 0 0
\(939\) 2.12412e7 0.786166
\(940\) 2.54037e7 0.937729
\(941\) −4.67063e7 −1.71950 −0.859748 0.510718i \(-0.829380\pi\)
−0.859748 + 0.510718i \(0.829380\pi\)
\(942\) −1.40296e7 −0.515130
\(943\) 3.45274e6 0.126440
\(944\) 8.14797e6 0.297591
\(945\) 0 0
\(946\) −2.51478e6 −0.0913632
\(947\) 4.02379e7 1.45801 0.729006 0.684508i \(-0.239983\pi\)
0.729006 + 0.684508i \(0.239983\pi\)
\(948\) −4.39646e6 −0.158885
\(949\) −62463.0 −0.00225142
\(950\) −2.54381e7 −0.914482
\(951\) −2.00178e7 −0.717737
\(952\) 0 0
\(953\) 2.45579e7 0.875908 0.437954 0.898997i \(-0.355703\pi\)
0.437954 + 0.898997i \(0.355703\pi\)
\(954\) 6.46574e6 0.230010
\(955\) 2.05342e7 0.728567
\(956\) −7.95082e6 −0.281363
\(957\) −1.99145e6 −0.0702893
\(958\) −1.00855e7 −0.355044
\(959\) 0 0
\(960\) −3.17030e6 −0.111026
\(961\) −2.56328e7 −0.895339
\(962\) 91596.0 0.00319109
\(963\) −3.94049e6 −0.136925
\(964\) 4.44189e6 0.153948
\(965\) −7.36943e6 −0.254751
\(966\) 0 0
\(967\) 5.34313e7 1.83751 0.918754 0.394830i \(-0.129197\pi\)
0.918754 + 0.394830i \(0.129197\pi\)
\(968\) 1.02333e7 0.351016
\(969\) 2.55155e7 0.872961
\(970\) 4.12270e7 1.40687
\(971\) 2.81485e6 0.0958093 0.0479046 0.998852i \(-0.484746\pi\)
0.0479046 + 0.998852i \(0.484746\pi\)
\(972\) 944784. 0.0320750
\(973\) 0 0
\(974\) −7.66687e6 −0.258953
\(975\) 115317. 0.00388492
\(976\) 1.47594e7 0.495958
\(977\) 2.54713e7 0.853718 0.426859 0.904318i \(-0.359620\pi\)
0.426859 + 0.904318i \(0.359620\pi\)
\(978\) 4.04539e6 0.135243
\(979\) 2.00573e6 0.0668829
\(980\) 0 0
\(981\) −1.23181e7 −0.408668
\(982\) −2.33150e7 −0.771537
\(983\) 2.90135e6 0.0957670 0.0478835 0.998853i \(-0.484752\pi\)
0.0478835 + 0.998853i \(0.484752\pi\)
\(984\) 8.87846e6 0.292314
\(985\) 6.14109e6 0.201676
\(986\) 4.95649e7 1.62361
\(987\) 0 0
\(988\) 71472.0 0.00232940
\(989\) −4.14198e6 −0.134654
\(990\) 947376. 0.0307209
\(991\) 2.73784e6 0.0885570 0.0442785 0.999019i \(-0.485901\pi\)
0.0442785 + 0.999019i \(0.485901\pi\)
\(992\) 1.77254e6 0.0571897
\(993\) −3.33274e7 −1.07258
\(994\) 0 0
\(995\) −6.11763e7 −1.95896
\(996\) −1.59267e7 −0.508718
\(997\) 2.74079e7 0.873249 0.436624 0.899644i \(-0.356174\pi\)
0.436624 + 0.899644i \(0.356174\pi\)
\(998\) 4.00194e7 1.27188
\(999\) −5.56446e6 −0.176404
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.e.1.1 1
3.2 odd 2 882.6.a.y.1.1 1
7.2 even 3 294.6.e.m.67.1 2
7.3 odd 6 42.6.e.b.37.1 yes 2
7.4 even 3 294.6.e.m.79.1 2
7.5 odd 6 42.6.e.b.25.1 2
7.6 odd 2 294.6.a.d.1.1 1
21.5 even 6 126.6.g.b.109.1 2
21.17 even 6 126.6.g.b.37.1 2
21.20 even 2 882.6.a.m.1.1 1
28.3 even 6 336.6.q.a.289.1 2
28.19 even 6 336.6.q.a.193.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.6.e.b.25.1 2 7.5 odd 6
42.6.e.b.37.1 yes 2 7.3 odd 6
126.6.g.b.37.1 2 21.17 even 6
126.6.g.b.109.1 2 21.5 even 6
294.6.a.d.1.1 1 7.6 odd 2
294.6.a.e.1.1 1 1.1 even 1 trivial
294.6.e.m.67.1 2 7.2 even 3
294.6.e.m.79.1 2 7.4 even 3
336.6.q.a.193.1 2 28.19 even 6
336.6.q.a.289.1 2 28.3 even 6
882.6.a.m.1.1 1 21.20 even 2
882.6.a.y.1.1 1 3.2 odd 2