Properties

Label 49.6.a.a.1.1
Level $49$
Weight $6$
Character 49.1
Self dual yes
Analytic conductor $7.859$
Analytic rank $0$
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] = [49,6,Mod(1,49)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(49, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("49.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 49.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: \(7.85880717084\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 49.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-10.0000 q^{2} +14.0000 q^{3} +68.0000 q^{4} +56.0000 q^{5} -140.000 q^{6} -360.000 q^{8} -47.0000 q^{9} +O(q^{10})\) \(q-10.0000 q^{2} +14.0000 q^{3} +68.0000 q^{4} +56.0000 q^{5} -140.000 q^{6} -360.000 q^{8} -47.0000 q^{9} -560.000 q^{10} +232.000 q^{11} +952.000 q^{12} +140.000 q^{13} +784.000 q^{15} +1424.00 q^{16} +1722.00 q^{17} +470.000 q^{18} +98.0000 q^{19} +3808.00 q^{20} -2320.00 q^{22} +1824.00 q^{23} -5040.00 q^{24} +11.0000 q^{25} -1400.00 q^{26} -4060.00 q^{27} +3418.00 q^{29} -7840.00 q^{30} +7644.00 q^{31} -2720.00 q^{32} +3248.00 q^{33} -17220.0 q^{34} -3196.00 q^{36} -10398.0 q^{37} -980.000 q^{38} +1960.00 q^{39} -20160.0 q^{40} +17962.0 q^{41} +10880.0 q^{43} +15776.0 q^{44} -2632.00 q^{45} -18240.0 q^{46} -9324.00 q^{47} +19936.0 q^{48} -110.000 q^{50} +24108.0 q^{51} +9520.00 q^{52} +2262.00 q^{53} +40600.0 q^{54} +12992.0 q^{55} +1372.00 q^{57} -34180.0 q^{58} +2730.00 q^{59} +53312.0 q^{60} -25648.0 q^{61} -76440.0 q^{62} -18368.0 q^{64} +7840.00 q^{65} -32480.0 q^{66} -48404.0 q^{67} +117096. q^{68} +25536.0 q^{69} -58560.0 q^{71} +16920.0 q^{72} -68082.0 q^{73} +103980. q^{74} +154.000 q^{75} +6664.00 q^{76} -19600.0 q^{78} +31784.0 q^{79} +79744.0 q^{80} -45419.0 q^{81} -179620. q^{82} +20538.0 q^{83} +96432.0 q^{85} -108800. q^{86} +47852.0 q^{87} -83520.0 q^{88} +50582.0 q^{89} +26320.0 q^{90} +124032. q^{92} +107016. q^{93} +93240.0 q^{94} +5488.00 q^{95} -38080.0 q^{96} +58506.0 q^{97} -10904.0 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −10.0000 −1.76777 −0.883883 0.467707i \(-0.845080\pi\)
−0.883883 + 0.467707i \(0.845080\pi\)
\(3\) 14.0000 0.898100 0.449050 0.893507i \(-0.351762\pi\)
0.449050 + 0.893507i \(0.351762\pi\)
\(4\) 68.0000 2.12500
\(5\) 56.0000 1.00176 0.500879 0.865517i \(-0.333010\pi\)
0.500879 + 0.865517i \(0.333010\pi\)
\(6\) −140.000 −1.58763
\(7\) 0 0
\(8\) −360.000 −1.98874
\(9\) −47.0000 −0.193416
\(10\) −560.000 −1.77088
\(11\) 232.000 0.578104 0.289052 0.957313i \(-0.406660\pi\)
0.289052 + 0.957313i \(0.406660\pi\)
\(12\) 952.000 1.90846
\(13\) 140.000 0.229757 0.114879 0.993380i \(-0.463352\pi\)
0.114879 + 0.993380i \(0.463352\pi\)
\(14\) 0 0
\(15\) 784.000 0.899680
\(16\) 1424.00 1.39062
\(17\) 1722.00 1.44514 0.722572 0.691296i \(-0.242960\pi\)
0.722572 + 0.691296i \(0.242960\pi\)
\(18\) 470.000 0.341914
\(19\) 98.0000 0.0622791 0.0311395 0.999515i \(-0.490086\pi\)
0.0311395 + 0.999515i \(0.490086\pi\)
\(20\) 3808.00 2.12874
\(21\) 0 0
\(22\) −2320.00 −1.02195
\(23\) 1824.00 0.718961 0.359480 0.933153i \(-0.382954\pi\)
0.359480 + 0.933153i \(0.382954\pi\)
\(24\) −5040.00 −1.78609
\(25\) 11.0000 0.00352000
\(26\) −1400.00 −0.406158
\(27\) −4060.00 −1.07181
\(28\) 0 0
\(29\) 3418.00 0.754705 0.377352 0.926070i \(-0.376835\pi\)
0.377352 + 0.926070i \(0.376835\pi\)
\(30\) −7840.00 −1.59042
\(31\) 7644.00 1.42862 0.714310 0.699830i \(-0.246741\pi\)
0.714310 + 0.699830i \(0.246741\pi\)
\(32\) −2720.00 −0.469563
\(33\) 3248.00 0.519196
\(34\) −17220.0 −2.55468
\(35\) 0 0
\(36\) −3196.00 −0.411008
\(37\) −10398.0 −1.24866 −0.624332 0.781159i \(-0.714629\pi\)
−0.624332 + 0.781159i \(0.714629\pi\)
\(38\) −980.000 −0.110095
\(39\) 1960.00 0.206345
\(40\) −20160.0 −1.99223
\(41\) 17962.0 1.66876 0.834382 0.551186i \(-0.185825\pi\)
0.834382 + 0.551186i \(0.185825\pi\)
\(42\) 0 0
\(43\) 10880.0 0.897342 0.448671 0.893697i \(-0.351898\pi\)
0.448671 + 0.893697i \(0.351898\pi\)
\(44\) 15776.0 1.22847
\(45\) −2632.00 −0.193756
\(46\) −18240.0 −1.27096
\(47\) −9324.00 −0.615684 −0.307842 0.951438i \(-0.599607\pi\)
−0.307842 + 0.951438i \(0.599607\pi\)
\(48\) 19936.0 1.24892
\(49\) 0 0
\(50\) −110.000 −0.00622254
\(51\) 24108.0 1.29788
\(52\) 9520.00 0.488235
\(53\) 2262.00 0.110612 0.0553061 0.998469i \(-0.482387\pi\)
0.0553061 + 0.998469i \(0.482387\pi\)
\(54\) 40600.0 1.89471
\(55\) 12992.0 0.579121
\(56\) 0 0
\(57\) 1372.00 0.0559329
\(58\) −34180.0 −1.33414
\(59\) 2730.00 0.102102 0.0510508 0.998696i \(-0.483743\pi\)
0.0510508 + 0.998696i \(0.483743\pi\)
\(60\) 53312.0 1.91182
\(61\) −25648.0 −0.882529 −0.441264 0.897377i \(-0.645470\pi\)
−0.441264 + 0.897377i \(0.645470\pi\)
\(62\) −76440.0 −2.52547
\(63\) 0 0
\(64\) −18368.0 −0.560547
\(65\) 7840.00 0.230161
\(66\) −32480.0 −0.917817
\(67\) −48404.0 −1.31733 −0.658664 0.752437i \(-0.728878\pi\)
−0.658664 + 0.752437i \(0.728878\pi\)
\(68\) 117096. 3.07093
\(69\) 25536.0 0.645699
\(70\) 0 0
\(71\) −58560.0 −1.37865 −0.689327 0.724450i \(-0.742094\pi\)
−0.689327 + 0.724450i \(0.742094\pi\)
\(72\) 16920.0 0.384653
\(73\) −68082.0 −1.49529 −0.747645 0.664099i \(-0.768815\pi\)
−0.747645 + 0.664099i \(0.768815\pi\)
\(74\) 103980. 2.20735
\(75\) 154.000 0.00316131
\(76\) 6664.00 0.132343
\(77\) 0 0
\(78\) −19600.0 −0.364770
\(79\) 31784.0 0.572982 0.286491 0.958083i \(-0.407511\pi\)
0.286491 + 0.958083i \(0.407511\pi\)
\(80\) 79744.0 1.39307
\(81\) −45419.0 −0.769175
\(82\) −179620. −2.94999
\(83\) 20538.0 0.327237 0.163619 0.986524i \(-0.447683\pi\)
0.163619 + 0.986524i \(0.447683\pi\)
\(84\) 0 0
\(85\) 96432.0 1.44768
\(86\) −108800. −1.58629
\(87\) 47852.0 0.677801
\(88\) −83520.0 −1.14970
\(89\) 50582.0 0.676894 0.338447 0.940985i \(-0.390098\pi\)
0.338447 + 0.940985i \(0.390098\pi\)
\(90\) 26320.0 0.342515
\(91\) 0 0
\(92\) 124032. 1.52779
\(93\) 107016. 1.28304
\(94\) 93240.0 1.08839
\(95\) 5488.00 0.0623886
\(96\) −38080.0 −0.421715
\(97\) 58506.0 0.631351 0.315676 0.948867i \(-0.397769\pi\)
0.315676 + 0.948867i \(0.397769\pi\)
\(98\) 0 0
\(99\) −10904.0 −0.111814
\(100\) 748.000 0.00748000
\(101\) −38696.0 −0.377453 −0.188726 0.982030i \(-0.560436\pi\)
−0.188726 + 0.982030i \(0.560436\pi\)
\(102\) −241080. −2.29436
\(103\) −53060.0 −0.492804 −0.246402 0.969168i \(-0.579248\pi\)
−0.246402 + 0.969168i \(0.579248\pi\)
\(104\) −50400.0 −0.456927
\(105\) 0 0
\(106\) −22620.0 −0.195537
\(107\) −146324. −1.23554 −0.617769 0.786360i \(-0.711963\pi\)
−0.617769 + 0.786360i \(0.711963\pi\)
\(108\) −276080. −2.27759
\(109\) 92898.0 0.748928 0.374464 0.927241i \(-0.377827\pi\)
0.374464 + 0.927241i \(0.377827\pi\)
\(110\) −129920. −1.02375
\(111\) −145572. −1.12143
\(112\) 0 0
\(113\) −83354.0 −0.614088 −0.307044 0.951695i \(-0.599340\pi\)
−0.307044 + 0.951695i \(0.599340\pi\)
\(114\) −13720.0 −0.0988762
\(115\) 102144. 0.720225
\(116\) 232424. 1.60375
\(117\) −6580.00 −0.0444387
\(118\) −27300.0 −0.180492
\(119\) 0 0
\(120\) −282240. −1.78923
\(121\) −107227. −0.665795
\(122\) 256480. 1.56011
\(123\) 251468. 1.49872
\(124\) 519792. 3.03582
\(125\) −174384. −0.998232
\(126\) 0 0
\(127\) 60384.0 0.332210 0.166105 0.986108i \(-0.446881\pi\)
0.166105 + 0.986108i \(0.446881\pi\)
\(128\) 270720. 1.46048
\(129\) 152320. 0.805903
\(130\) −78400.0 −0.406872
\(131\) 61586.0 0.313548 0.156774 0.987635i \(-0.449891\pi\)
0.156774 + 0.987635i \(0.449891\pi\)
\(132\) 220864. 1.10329
\(133\) 0 0
\(134\) 484040. 2.32873
\(135\) −227360. −1.07369
\(136\) −619920. −2.87401
\(137\) −204462. −0.930703 −0.465352 0.885126i \(-0.654072\pi\)
−0.465352 + 0.885126i \(0.654072\pi\)
\(138\) −255360. −1.14145
\(139\) 35406.0 0.155432 0.0777159 0.996976i \(-0.475237\pi\)
0.0777159 + 0.996976i \(0.475237\pi\)
\(140\) 0 0
\(141\) −130536. −0.552946
\(142\) 585600. 2.43714
\(143\) 32480.0 0.132824
\(144\) −66928.0 −0.268969
\(145\) 191408. 0.756032
\(146\) 680820. 2.64332
\(147\) 0 0
\(148\) −707064. −2.65341
\(149\) −20226.0 −0.0746353 −0.0373177 0.999303i \(-0.511881\pi\)
−0.0373177 + 0.999303i \(0.511881\pi\)
\(150\) −1540.00 −0.00558847
\(151\) 70904.0 0.253063 0.126531 0.991963i \(-0.459616\pi\)
0.126531 + 0.991963i \(0.459616\pi\)
\(152\) −35280.0 −0.123857
\(153\) −80934.0 −0.279513
\(154\) 0 0
\(155\) 428064. 1.43113
\(156\) 133280. 0.438484
\(157\) −293524. −0.950374 −0.475187 0.879885i \(-0.657620\pi\)
−0.475187 + 0.879885i \(0.657620\pi\)
\(158\) −317840. −1.01290
\(159\) 31668.0 0.0993408
\(160\) −152320. −0.470389
\(161\) 0 0
\(162\) 454190. 1.35972
\(163\) 13192.0 0.0388903 0.0194452 0.999811i \(-0.493810\pi\)
0.0194452 + 0.999811i \(0.493810\pi\)
\(164\) 1.22142e6 3.54612
\(165\) 181888. 0.520109
\(166\) −205380. −0.578479
\(167\) −493612. −1.36960 −0.684801 0.728730i \(-0.740111\pi\)
−0.684801 + 0.728730i \(0.740111\pi\)
\(168\) 0 0
\(169\) −351693. −0.947212
\(170\) −964320. −2.55917
\(171\) −4606.00 −0.0120457
\(172\) 739840. 1.90685
\(173\) −240716. −0.611490 −0.305745 0.952113i \(-0.598906\pi\)
−0.305745 + 0.952113i \(0.598906\pi\)
\(174\) −478520. −1.19819
\(175\) 0 0
\(176\) 330368. 0.803926
\(177\) 38220.0 0.0916975
\(178\) −505820. −1.19659
\(179\) 294932. 0.688001 0.344001 0.938969i \(-0.388218\pi\)
0.344001 + 0.938969i \(0.388218\pi\)
\(180\) −178976. −0.411731
\(181\) 336980. 0.764553 0.382277 0.924048i \(-0.375140\pi\)
0.382277 + 0.924048i \(0.375140\pi\)
\(182\) 0 0
\(183\) −359072. −0.792600
\(184\) −656640. −1.42982
\(185\) −582288. −1.25086
\(186\) −1.07016e6 −2.26812
\(187\) 399504. 0.835444
\(188\) −634032. −1.30833
\(189\) 0 0
\(190\) −54880.0 −0.110288
\(191\) 358264. 0.710591 0.355296 0.934754i \(-0.384380\pi\)
0.355296 + 0.934754i \(0.384380\pi\)
\(192\) −257152. −0.503427
\(193\) −989554. −1.91226 −0.956128 0.292948i \(-0.905364\pi\)
−0.956128 + 0.292948i \(0.905364\pi\)
\(194\) −585060. −1.11608
\(195\) 109760. 0.206708
\(196\) 0 0
\(197\) −990050. −1.81757 −0.908786 0.417263i \(-0.862989\pi\)
−0.908786 + 0.417263i \(0.862989\pi\)
\(198\) 109040. 0.197662
\(199\) 840756. 1.50500 0.752501 0.658591i \(-0.228847\pi\)
0.752501 + 0.658591i \(0.228847\pi\)
\(200\) −3960.00 −0.00700036
\(201\) −677656. −1.18309
\(202\) 386960. 0.667249
\(203\) 0 0
\(204\) 1.63934e6 2.75800
\(205\) 1.00587e6 1.67170
\(206\) 530600. 0.871163
\(207\) −85728.0 −0.139058
\(208\) 199360. 0.319506
\(209\) 22736.0 0.0360038
\(210\) 0 0
\(211\) 1.15073e6 1.77938 0.889689 0.456568i \(-0.150921\pi\)
0.889689 + 0.456568i \(0.150921\pi\)
\(212\) 153816. 0.235051
\(213\) −819840. −1.23817
\(214\) 1.46324e6 2.18414
\(215\) 609280. 0.898919
\(216\) 1.46160e6 2.13154
\(217\) 0 0
\(218\) −928980. −1.32393
\(219\) −953148. −1.34292
\(220\) 883456. 1.23063
\(221\) 241080. 0.332032
\(222\) 1.45572e6 1.98242
\(223\) 824264. 1.10995 0.554976 0.831866i \(-0.312727\pi\)
0.554976 + 0.831866i \(0.312727\pi\)
\(224\) 0 0
\(225\) −517.000 −0.000680823 0
\(226\) 833540. 1.08556
\(227\) −74382.0 −0.0958083 −0.0479042 0.998852i \(-0.515254\pi\)
−0.0479042 + 0.998852i \(0.515254\pi\)
\(228\) 93296.0 0.118857
\(229\) −1.13196e6 −1.42640 −0.713199 0.700961i \(-0.752755\pi\)
−0.713199 + 0.700961i \(0.752755\pi\)
\(230\) −1.02144e6 −1.27319
\(231\) 0 0
\(232\) −1.23048e6 −1.50091
\(233\) −198726. −0.239809 −0.119904 0.992785i \(-0.538259\pi\)
−0.119904 + 0.992785i \(0.538259\pi\)
\(234\) 65800.0 0.0785572
\(235\) −522144. −0.616766
\(236\) 185640. 0.216966
\(237\) 444976. 0.514595
\(238\) 0 0
\(239\) 482904. 0.546847 0.273424 0.961894i \(-0.411844\pi\)
0.273424 + 0.961894i \(0.411844\pi\)
\(240\) 1.11642e6 1.25112
\(241\) −805910. −0.893807 −0.446904 0.894582i \(-0.647473\pi\)
−0.446904 + 0.894582i \(0.647473\pi\)
\(242\) 1.07227e6 1.17697
\(243\) 350714. 0.381011
\(244\) −1.74406e6 −1.87537
\(245\) 0 0
\(246\) −2.51468e6 −2.64938
\(247\) 13720.0 0.0143091
\(248\) −2.75184e6 −2.84115
\(249\) 287532. 0.293892
\(250\) 1.74384e6 1.76464
\(251\) −430738. −0.431548 −0.215774 0.976443i \(-0.569227\pi\)
−0.215774 + 0.976443i \(0.569227\pi\)
\(252\) 0 0
\(253\) 423168. 0.415634
\(254\) −603840. −0.587270
\(255\) 1.35005e6 1.30017
\(256\) −2.11942e6 −2.02124
\(257\) 1.17691e6 1.11150 0.555751 0.831349i \(-0.312431\pi\)
0.555751 + 0.831349i \(0.312431\pi\)
\(258\) −1.52320e6 −1.42465
\(259\) 0 0
\(260\) 533120. 0.489093
\(261\) −160646. −0.145972
\(262\) −615860. −0.554279
\(263\) 1.29098e6 1.15088 0.575438 0.817845i \(-0.304831\pi\)
0.575438 + 0.817845i \(0.304831\pi\)
\(264\) −1.16928e6 −1.03254
\(265\) 126672. 0.110807
\(266\) 0 0
\(267\) 708148. 0.607919
\(268\) −3.29147e6 −2.79932
\(269\) 1.27756e6 1.07646 0.538232 0.842797i \(-0.319093\pi\)
0.538232 + 0.842797i \(0.319093\pi\)
\(270\) 2.27360e6 1.89804
\(271\) −1.65054e6 −1.36522 −0.682612 0.730781i \(-0.739156\pi\)
−0.682612 + 0.730781i \(0.739156\pi\)
\(272\) 2.45213e6 2.00965
\(273\) 0 0
\(274\) 2.04462e6 1.64527
\(275\) 2552.00 0.00203493
\(276\) 1.73645e6 1.37211
\(277\) −1.06409e6 −0.833257 −0.416628 0.909077i \(-0.636788\pi\)
−0.416628 + 0.909077i \(0.636788\pi\)
\(278\) −354060. −0.274767
\(279\) −359268. −0.276317
\(280\) 0 0
\(281\) −22342.0 −0.0168794 −0.00843969 0.999964i \(-0.502686\pi\)
−0.00843969 + 0.999964i \(0.502686\pi\)
\(282\) 1.30536e6 0.977479
\(283\) 2.49574e6 1.85239 0.926196 0.377042i \(-0.123059\pi\)
0.926196 + 0.377042i \(0.123059\pi\)
\(284\) −3.98208e6 −2.92964
\(285\) 76832.0 0.0560312
\(286\) −324800. −0.234802
\(287\) 0 0
\(288\) 127840. 0.0908208
\(289\) 1.54543e6 1.08844
\(290\) −1.91408e6 −1.33649
\(291\) 819084. 0.567017
\(292\) −4.62958e6 −3.17749
\(293\) 1.93178e6 1.31458 0.657291 0.753637i \(-0.271702\pi\)
0.657291 + 0.753637i \(0.271702\pi\)
\(294\) 0 0
\(295\) 152880. 0.102281
\(296\) 3.74328e6 2.48326
\(297\) −941920. −0.619616
\(298\) 202260. 0.131938
\(299\) 255360. 0.165187
\(300\) 10472.0 0.00671779
\(301\) 0 0
\(302\) −709040. −0.447356
\(303\) −541744. −0.338991
\(304\) 139552. 0.0866068
\(305\) −1.43629e6 −0.884081
\(306\) 809340. 0.494114
\(307\) 459074. 0.277995 0.138997 0.990293i \(-0.455612\pi\)
0.138997 + 0.990293i \(0.455612\pi\)
\(308\) 0 0
\(309\) −742840. −0.442587
\(310\) −4.28064e6 −2.52991
\(311\) −667128. −0.391118 −0.195559 0.980692i \(-0.562652\pi\)
−0.195559 + 0.980692i \(0.562652\pi\)
\(312\) −705600. −0.410367
\(313\) 111034. 0.0640612 0.0320306 0.999487i \(-0.489803\pi\)
0.0320306 + 0.999487i \(0.489803\pi\)
\(314\) 2.93524e6 1.68004
\(315\) 0 0
\(316\) 2.16131e6 1.21759
\(317\) −68778.0 −0.0384416 −0.0192208 0.999815i \(-0.506119\pi\)
−0.0192208 + 0.999815i \(0.506119\pi\)
\(318\) −316680. −0.175611
\(319\) 792976. 0.436298
\(320\) −1.02861e6 −0.561533
\(321\) −2.04854e6 −1.10964
\(322\) 0 0
\(323\) 168756. 0.0900022
\(324\) −3.08849e6 −1.63450
\(325\) 1540.00 0.000808746 0
\(326\) −131920. −0.0687490
\(327\) 1.30057e6 0.672613
\(328\) −6.46632e6 −3.31874
\(329\) 0 0
\(330\) −1.81888e6 −0.919431
\(331\) −564448. −0.283174 −0.141587 0.989926i \(-0.545221\pi\)
−0.141587 + 0.989926i \(0.545221\pi\)
\(332\) 1.39658e6 0.695379
\(333\) 488706. 0.241511
\(334\) 4.93612e6 2.42114
\(335\) −2.71062e6 −1.31965
\(336\) 0 0
\(337\) 2.07729e6 0.996376 0.498188 0.867069i \(-0.333999\pi\)
0.498188 + 0.867069i \(0.333999\pi\)
\(338\) 3.51693e6 1.67445
\(339\) −1.16696e6 −0.551512
\(340\) 6.55738e6 3.07633
\(341\) 1.77341e6 0.825891
\(342\) 46060.0 0.0212941
\(343\) 0 0
\(344\) −3.91680e6 −1.78458
\(345\) 1.43002e6 0.646834
\(346\) 2.40716e6 1.08097
\(347\) −53248.0 −0.0237399 −0.0118700 0.999930i \(-0.503778\pi\)
−0.0118700 + 0.999930i \(0.503778\pi\)
\(348\) 3.25394e6 1.44033
\(349\) 2.27200e6 0.998494 0.499247 0.866460i \(-0.333610\pi\)
0.499247 + 0.866460i \(0.333610\pi\)
\(350\) 0 0
\(351\) −568400. −0.246256
\(352\) −631040. −0.271456
\(353\) −4.00645e6 −1.71129 −0.855644 0.517565i \(-0.826838\pi\)
−0.855644 + 0.517565i \(0.826838\pi\)
\(354\) −382200. −0.162100
\(355\) −3.27936e6 −1.38108
\(356\) 3.43958e6 1.43840
\(357\) 0 0
\(358\) −2.94932e6 −1.21623
\(359\) 73784.0 0.0302152 0.0151076 0.999886i \(-0.495191\pi\)
0.0151076 + 0.999886i \(0.495191\pi\)
\(360\) 947520. 0.385329
\(361\) −2.46650e6 −0.996121
\(362\) −3.36980e6 −1.35155
\(363\) −1.50118e6 −0.597951
\(364\) 0 0
\(365\) −3.81259e6 −1.49792
\(366\) 3.59072e6 1.40113
\(367\) −1.40431e6 −0.544250 −0.272125 0.962262i \(-0.587726\pi\)
−0.272125 + 0.962262i \(0.587726\pi\)
\(368\) 2.59738e6 0.999805
\(369\) −844214. −0.322765
\(370\) 5.82288e6 2.21123
\(371\) 0 0
\(372\) 7.27709e6 2.72647
\(373\) −1.60323e6 −0.596657 −0.298329 0.954463i \(-0.596429\pi\)
−0.298329 + 0.954463i \(0.596429\pi\)
\(374\) −3.99504e6 −1.47687
\(375\) −2.44138e6 −0.896513
\(376\) 3.35664e6 1.22443
\(377\) 478520. 0.173399
\(378\) 0 0
\(379\) −4.77012e6 −1.70581 −0.852906 0.522064i \(-0.825162\pi\)
−0.852906 + 0.522064i \(0.825162\pi\)
\(380\) 373184. 0.132576
\(381\) 845376. 0.298358
\(382\) −3.58264e6 −1.25616
\(383\) 2.23079e6 0.777072 0.388536 0.921434i \(-0.372981\pi\)
0.388536 + 0.921434i \(0.372981\pi\)
\(384\) 3.79008e6 1.31166
\(385\) 0 0
\(386\) 9.89554e6 3.38042
\(387\) −511360. −0.173560
\(388\) 3.97841e6 1.34162
\(389\) 4.84024e6 1.62178 0.810892 0.585196i \(-0.198982\pi\)
0.810892 + 0.585196i \(0.198982\pi\)
\(390\) −1.09760e6 −0.365412
\(391\) 3.14093e6 1.03900
\(392\) 0 0
\(393\) 862204. 0.281597
\(394\) 9.90050e6 3.21304
\(395\) 1.77990e6 0.573989
\(396\) −741472. −0.237606
\(397\) −995820. −0.317106 −0.158553 0.987350i \(-0.550683\pi\)
−0.158553 + 0.987350i \(0.550683\pi\)
\(398\) −8.40756e6 −2.66049
\(399\) 0 0
\(400\) 15664.0 0.00489500
\(401\) −3.31605e6 −1.02982 −0.514909 0.857245i \(-0.672174\pi\)
−0.514909 + 0.857245i \(0.672174\pi\)
\(402\) 6.77656e6 2.09143
\(403\) 1.07016e6 0.328236
\(404\) −2.63133e6 −0.802087
\(405\) −2.54346e6 −0.770527
\(406\) 0 0
\(407\) −2.41234e6 −0.721858
\(408\) −8.67888e6 −2.58115
\(409\) −3.07273e6 −0.908274 −0.454137 0.890932i \(-0.650052\pi\)
−0.454137 + 0.890932i \(0.650052\pi\)
\(410\) −1.00587e7 −2.95517
\(411\) −2.86247e6 −0.835865
\(412\) −3.60808e6 −1.04721
\(413\) 0 0
\(414\) 857280. 0.245823
\(415\) 1.15013e6 0.327813
\(416\) −380800. −0.107886
\(417\) 495684. 0.139593
\(418\) −227360. −0.0636463
\(419\) −2.81438e6 −0.783154 −0.391577 0.920145i \(-0.628070\pi\)
−0.391577 + 0.920145i \(0.628070\pi\)
\(420\) 0 0
\(421\) 3.05802e6 0.840883 0.420441 0.907320i \(-0.361875\pi\)
0.420441 + 0.907320i \(0.361875\pi\)
\(422\) −1.15073e7 −3.14552
\(423\) 438228. 0.119083
\(424\) −814320. −0.219979
\(425\) 18942.0 0.00508690
\(426\) 8.19840e6 2.18880
\(427\) 0 0
\(428\) −9.95003e6 −2.62552
\(429\) 454720. 0.119289
\(430\) −6.09280e6 −1.58908
\(431\) 1.93750e6 0.502398 0.251199 0.967936i \(-0.419175\pi\)
0.251199 + 0.967936i \(0.419175\pi\)
\(432\) −5.78144e6 −1.49048
\(433\) −3.94790e6 −1.01192 −0.505961 0.862557i \(-0.668862\pi\)
−0.505961 + 0.862557i \(0.668862\pi\)
\(434\) 0 0
\(435\) 2.67971e6 0.678993
\(436\) 6.31706e6 1.59147
\(437\) 178752. 0.0447762
\(438\) 9.53148e6 2.37397
\(439\) 7.41770e6 1.83700 0.918498 0.395426i \(-0.129403\pi\)
0.918498 + 0.395426i \(0.129403\pi\)
\(440\) −4.67712e6 −1.15172
\(441\) 0 0
\(442\) −2.41080e6 −0.586956
\(443\) 1.40269e6 0.339589 0.169794 0.985480i \(-0.445690\pi\)
0.169794 + 0.985480i \(0.445690\pi\)
\(444\) −9.89890e6 −2.38303
\(445\) 2.83259e6 0.678085
\(446\) −8.24264e6 −1.96214
\(447\) −283164. −0.0670300
\(448\) 0 0
\(449\) −590574. −0.138248 −0.0691239 0.997608i \(-0.522020\pi\)
−0.0691239 + 0.997608i \(0.522020\pi\)
\(450\) 5170.00 0.00120354
\(451\) 4.16718e6 0.964720
\(452\) −5.66807e6 −1.30494
\(453\) 992656. 0.227276
\(454\) 743820. 0.169367
\(455\) 0 0
\(456\) −493920. −0.111236
\(457\) −2.90484e6 −0.650627 −0.325313 0.945606i \(-0.605470\pi\)
−0.325313 + 0.945606i \(0.605470\pi\)
\(458\) 1.13196e7 2.52154
\(459\) −6.99132e6 −1.54891
\(460\) 6.94579e6 1.53048
\(461\) 922684. 0.202209 0.101105 0.994876i \(-0.467762\pi\)
0.101105 + 0.994876i \(0.467762\pi\)
\(462\) 0 0
\(463\) 7.18235e6 1.55709 0.778546 0.627588i \(-0.215958\pi\)
0.778546 + 0.627588i \(0.215958\pi\)
\(464\) 4.86723e6 1.04951
\(465\) 5.99290e6 1.28530
\(466\) 1.98726e6 0.423926
\(467\) 612570. 0.129976 0.0649881 0.997886i \(-0.479299\pi\)
0.0649881 + 0.997886i \(0.479299\pi\)
\(468\) −447440. −0.0944322
\(469\) 0 0
\(470\) 5.22144e6 1.09030
\(471\) −4.10934e6 −0.853531
\(472\) −982800. −0.203053
\(473\) 2.52416e6 0.518757
\(474\) −4.44976e6 −0.909684
\(475\) 1078.00 0.000219222 0
\(476\) 0 0
\(477\) −106314. −0.0213941
\(478\) −4.82904e6 −0.966699
\(479\) −2.60330e6 −0.518424 −0.259212 0.965820i \(-0.583463\pi\)
−0.259212 + 0.965820i \(0.583463\pi\)
\(480\) −2.13248e6 −0.422456
\(481\) −1.45572e6 −0.286890
\(482\) 8.05910e6 1.58004
\(483\) 0 0
\(484\) −7.29144e6 −1.41482
\(485\) 3.27634e6 0.632461
\(486\) −3.50714e6 −0.673539
\(487\) 5.46309e6 1.04380 0.521898 0.853008i \(-0.325224\pi\)
0.521898 + 0.853008i \(0.325224\pi\)
\(488\) 9.23328e6 1.75512
\(489\) 184688. 0.0349274
\(490\) 0 0
\(491\) 1.64090e6 0.307170 0.153585 0.988135i \(-0.450918\pi\)
0.153585 + 0.988135i \(0.450918\pi\)
\(492\) 1.70998e7 3.18478
\(493\) 5.88580e6 1.09066
\(494\) −137200. −0.0252951
\(495\) −610624. −0.112011
\(496\) 1.08851e7 1.98667
\(497\) 0 0
\(498\) −2.87532e6 −0.519533
\(499\) 2.99796e6 0.538983 0.269491 0.963003i \(-0.413144\pi\)
0.269491 + 0.963003i \(0.413144\pi\)
\(500\) −1.18581e7 −2.12124
\(501\) −6.91057e6 −1.23004
\(502\) 4.30738e6 0.762876
\(503\) 6.89405e6 1.21494 0.607469 0.794343i \(-0.292185\pi\)
0.607469 + 0.794343i \(0.292185\pi\)
\(504\) 0 0
\(505\) −2.16698e6 −0.378117
\(506\) −4.23168e6 −0.734745
\(507\) −4.92370e6 −0.850691
\(508\) 4.10611e6 0.705946
\(509\) −2.30476e6 −0.394305 −0.197152 0.980373i \(-0.563169\pi\)
−0.197152 + 0.980373i \(0.563169\pi\)
\(510\) −1.35005e7 −2.29839
\(511\) 0 0
\(512\) 1.25312e7 2.11260
\(513\) −397880. −0.0667511
\(514\) −1.17691e7 −1.96488
\(515\) −2.97136e6 −0.493671
\(516\) 1.03578e7 1.71254
\(517\) −2.16317e6 −0.355929
\(518\) 0 0
\(519\) −3.37002e6 −0.549180
\(520\) −2.82240e6 −0.457731
\(521\) 1.20960e7 1.95231 0.976155 0.217073i \(-0.0696509\pi\)
0.976155 + 0.217073i \(0.0696509\pi\)
\(522\) 1.60646e6 0.258044
\(523\) −5.48443e6 −0.876753 −0.438377 0.898791i \(-0.644446\pi\)
−0.438377 + 0.898791i \(0.644446\pi\)
\(524\) 4.18785e6 0.666289
\(525\) 0 0
\(526\) −1.29098e7 −2.03448
\(527\) 1.31630e7 2.06456
\(528\) 4.62515e6 0.722007
\(529\) −3.10937e6 −0.483095
\(530\) −1.26672e6 −0.195880
\(531\) −128310. −0.0197480
\(532\) 0 0
\(533\) 2.51468e6 0.383411
\(534\) −7.08148e6 −1.07466
\(535\) −8.19414e6 −1.23771
\(536\) 1.74254e7 2.61982
\(537\) 4.12905e6 0.617894
\(538\) −1.27756e7 −1.90294
\(539\) 0 0
\(540\) −1.54605e7 −2.28160
\(541\) −6.71799e6 −0.986839 −0.493420 0.869791i \(-0.664253\pi\)
−0.493420 + 0.869791i \(0.664253\pi\)
\(542\) 1.65054e7 2.41340
\(543\) 4.71772e6 0.686646
\(544\) −4.68384e6 −0.678586
\(545\) 5.20229e6 0.750245
\(546\) 0 0
\(547\) −5.00235e6 −0.714835 −0.357418 0.933945i \(-0.616343\pi\)
−0.357418 + 0.933945i \(0.616343\pi\)
\(548\) −1.39034e7 −1.97774
\(549\) 1.20546e6 0.170695
\(550\) −25520.0 −0.00359728
\(551\) 334964. 0.0470023
\(552\) −9.19296e6 −1.28413
\(553\) 0 0
\(554\) 1.06409e7 1.47300
\(555\) −8.15203e6 −1.12340
\(556\) 2.40761e6 0.330293
\(557\) 9.01961e6 1.23183 0.615913 0.787814i \(-0.288787\pi\)
0.615913 + 0.787814i \(0.288787\pi\)
\(558\) 3.59268e6 0.488465
\(559\) 1.52320e6 0.206171
\(560\) 0 0
\(561\) 5.59306e6 0.750312
\(562\) 223420. 0.0298388
\(563\) −1.24051e7 −1.64941 −0.824707 0.565561i \(-0.808660\pi\)
−0.824707 + 0.565561i \(0.808660\pi\)
\(564\) −8.87645e6 −1.17501
\(565\) −4.66782e6 −0.615167
\(566\) −2.49574e7 −3.27460
\(567\) 0 0
\(568\) 2.10816e7 2.74178
\(569\) 6.48804e6 0.840103 0.420052 0.907500i \(-0.362012\pi\)
0.420052 + 0.907500i \(0.362012\pi\)
\(570\) −768320. −0.0990501
\(571\) −1.02285e7 −1.31287 −0.656435 0.754382i \(-0.727936\pi\)
−0.656435 + 0.754382i \(0.727936\pi\)
\(572\) 2.20864e6 0.282251
\(573\) 5.01570e6 0.638182
\(574\) 0 0
\(575\) 20064.0 0.00253074
\(576\) 863296. 0.108419
\(577\) −2.65338e6 −0.331787 −0.165894 0.986144i \(-0.553051\pi\)
−0.165894 + 0.986144i \(0.553051\pi\)
\(578\) −1.54543e7 −1.92411
\(579\) −1.38538e7 −1.71740
\(580\) 1.30157e7 1.60657
\(581\) 0 0
\(582\) −8.19084e6 −1.00235
\(583\) 524784. 0.0639454
\(584\) 2.45095e7 2.97374
\(585\) −368480. −0.0445168
\(586\) −1.93178e7 −2.32387
\(587\) 1.43044e7 1.71346 0.856729 0.515766i \(-0.172493\pi\)
0.856729 + 0.515766i \(0.172493\pi\)
\(588\) 0 0
\(589\) 749112. 0.0889731
\(590\) −1.52880e6 −0.180809
\(591\) −1.38607e7 −1.63236
\(592\) −1.48068e7 −1.73642
\(593\) 1.00265e7 1.17088 0.585442 0.810714i \(-0.300921\pi\)
0.585442 + 0.810714i \(0.300921\pi\)
\(594\) 9.41920e6 1.09534
\(595\) 0 0
\(596\) −1.37537e6 −0.158600
\(597\) 1.17706e7 1.35164
\(598\) −2.55360e6 −0.292011
\(599\) −7.52292e6 −0.856681 −0.428341 0.903617i \(-0.640902\pi\)
−0.428341 + 0.903617i \(0.640902\pi\)
\(600\) −55440.0 −0.00628702
\(601\) −3.38625e6 −0.382413 −0.191207 0.981550i \(-0.561240\pi\)
−0.191207 + 0.981550i \(0.561240\pi\)
\(602\) 0 0
\(603\) 2.27499e6 0.254792
\(604\) 4.82147e6 0.537759
\(605\) −6.00471e6 −0.666966
\(606\) 5.41744e6 0.599256
\(607\) 6.90861e6 0.761060 0.380530 0.924769i \(-0.375742\pi\)
0.380530 + 0.924769i \(0.375742\pi\)
\(608\) −266560. −0.0292439
\(609\) 0 0
\(610\) 1.43629e7 1.56285
\(611\) −1.30536e6 −0.141458
\(612\) −5.50351e6 −0.593966
\(613\) −9.68896e6 −1.04142 −0.520710 0.853734i \(-0.674333\pi\)
−0.520710 + 0.853734i \(0.674333\pi\)
\(614\) −4.59074e6 −0.491430
\(615\) 1.40822e7 1.50135
\(616\) 0 0
\(617\) −7.84742e6 −0.829877 −0.414939 0.909849i \(-0.636197\pi\)
−0.414939 + 0.909849i \(0.636197\pi\)
\(618\) 7.42840e6 0.782391
\(619\) 1.01972e7 1.06968 0.534840 0.844953i \(-0.320372\pi\)
0.534840 + 0.844953i \(0.320372\pi\)
\(620\) 2.91084e7 3.04115
\(621\) −7.40544e6 −0.770587
\(622\) 6.67128e6 0.691406
\(623\) 0 0
\(624\) 2.79104e6 0.286949
\(625\) −9.79988e6 −1.00351
\(626\) −1.11034e6 −0.113245
\(627\) 318304. 0.0323350
\(628\) −1.99596e7 −2.01954
\(629\) −1.79054e7 −1.80450
\(630\) 0 0
\(631\) −8.36258e6 −0.836116 −0.418058 0.908420i \(-0.637289\pi\)
−0.418058 + 0.908420i \(0.637289\pi\)
\(632\) −1.14422e7 −1.13951
\(633\) 1.61102e7 1.59806
\(634\) 687780. 0.0679558
\(635\) 3.38150e6 0.332794
\(636\) 2.15342e6 0.211099
\(637\) 0 0
\(638\) −7.92976e6 −0.771273
\(639\) 2.75232e6 0.266653
\(640\) 1.51603e7 1.46305
\(641\) 1.10283e6 0.106014 0.0530070 0.998594i \(-0.483119\pi\)
0.0530070 + 0.998594i \(0.483119\pi\)
\(642\) 2.04854e7 1.96158
\(643\) −1.71354e7 −1.63443 −0.817217 0.576330i \(-0.804484\pi\)
−0.817217 + 0.576330i \(0.804484\pi\)
\(644\) 0 0
\(645\) 8.52992e6 0.807320
\(646\) −1.68756e6 −0.159103
\(647\) 54964.0 0.00516200 0.00258100 0.999997i \(-0.499178\pi\)
0.00258100 + 0.999997i \(0.499178\pi\)
\(648\) 1.63508e7 1.52969
\(649\) 633360. 0.0590254
\(650\) −15400.0 −0.00142968
\(651\) 0 0
\(652\) 897056. 0.0826420
\(653\) −485166. −0.0445254 −0.0222627 0.999752i \(-0.507087\pi\)
−0.0222627 + 0.999752i \(0.507087\pi\)
\(654\) −1.30057e7 −1.18902
\(655\) 3.44882e6 0.314099
\(656\) 2.55779e7 2.32063
\(657\) 3.19985e6 0.289212
\(658\) 0 0
\(659\) −2.72136e6 −0.244103 −0.122051 0.992524i \(-0.538947\pi\)
−0.122051 + 0.992524i \(0.538947\pi\)
\(660\) 1.23684e7 1.10523
\(661\) 2.14525e6 0.190974 0.0954869 0.995431i \(-0.469559\pi\)
0.0954869 + 0.995431i \(0.469559\pi\)
\(662\) 5.64448e6 0.500586
\(663\) 3.37512e6 0.298198
\(664\) −7.39368e6 −0.650789
\(665\) 0 0
\(666\) −4.88706e6 −0.426935
\(667\) 6.23443e6 0.542603
\(668\) −3.35656e7 −2.91041
\(669\) 1.15397e7 0.996848
\(670\) 2.71062e7 2.33283
\(671\) −5.95034e6 −0.510194
\(672\) 0 0
\(673\) 2.92796e6 0.249188 0.124594 0.992208i \(-0.460237\pi\)
0.124594 + 0.992208i \(0.460237\pi\)
\(674\) −2.07729e7 −1.76136
\(675\) −44660.0 −0.00377276
\(676\) −2.39151e7 −2.01282
\(677\) 1.34992e7 1.13198 0.565988 0.824414i \(-0.308495\pi\)
0.565988 + 0.824414i \(0.308495\pi\)
\(678\) 1.16696e7 0.974945
\(679\) 0 0
\(680\) −3.47155e7 −2.87906
\(681\) −1.04135e6 −0.0860455
\(682\) −1.77341e7 −1.45998
\(683\) −5.42972e6 −0.445375 −0.222688 0.974890i \(-0.571483\pi\)
−0.222688 + 0.974890i \(0.571483\pi\)
\(684\) −313208. −0.0255972
\(685\) −1.14499e7 −0.932340
\(686\) 0 0
\(687\) −1.58474e7 −1.28105
\(688\) 1.54931e7 1.24787
\(689\) 316680. 0.0254140
\(690\) −1.43002e7 −1.14345
\(691\) −2.08280e7 −1.65940 −0.829702 0.558207i \(-0.811490\pi\)
−0.829702 + 0.558207i \(0.811490\pi\)
\(692\) −1.63687e7 −1.29942
\(693\) 0 0
\(694\) 532480. 0.0419667
\(695\) 1.98274e6 0.155705
\(696\) −1.72267e7 −1.34797
\(697\) 3.09306e7 2.41160
\(698\) −2.27200e7 −1.76510
\(699\) −2.78216e6 −0.215372
\(700\) 0 0
\(701\) 2.35141e7 1.80731 0.903655 0.428261i \(-0.140874\pi\)
0.903655 + 0.428261i \(0.140874\pi\)
\(702\) 5.68400e6 0.435323
\(703\) −1.01900e6 −0.0777656
\(704\) −4.26138e6 −0.324055
\(705\) −7.31002e6 −0.553918
\(706\) 4.00645e7 3.02516
\(707\) 0 0
\(708\) 2.59896e6 0.194857
\(709\) −1.95747e7 −1.46244 −0.731221 0.682140i \(-0.761049\pi\)
−0.731221 + 0.682140i \(0.761049\pi\)
\(710\) 3.27936e7 2.44142
\(711\) −1.49385e6 −0.110824
\(712\) −1.82095e7 −1.34617
\(713\) 1.39427e7 1.02712
\(714\) 0 0
\(715\) 1.81888e6 0.133057
\(716\) 2.00554e7 1.46200
\(717\) 6.76066e6 0.491124
\(718\) −737840. −0.0534135
\(719\) 2.61152e7 1.88396 0.941978 0.335674i \(-0.108964\pi\)
0.941978 + 0.335674i \(0.108964\pi\)
\(720\) −3.74797e6 −0.269442
\(721\) 0 0
\(722\) 2.46650e7 1.76091
\(723\) −1.12827e7 −0.802729
\(724\) 2.29146e7 1.62468
\(725\) 37598.0 0.00265656
\(726\) 1.50118e7 1.05704
\(727\) −1.54126e7 −1.08154 −0.540768 0.841172i \(-0.681866\pi\)
−0.540768 + 0.841172i \(0.681866\pi\)
\(728\) 0 0
\(729\) 1.59468e7 1.11136
\(730\) 3.81259e7 2.64797
\(731\) 1.87354e7 1.29679
\(732\) −2.44169e7 −1.68427
\(733\) 1.69868e7 1.16776 0.583878 0.811841i \(-0.301535\pi\)
0.583878 + 0.811841i \(0.301535\pi\)
\(734\) 1.40431e7 0.962107
\(735\) 0 0
\(736\) −4.96128e6 −0.337597
\(737\) −1.12297e7 −0.761554
\(738\) 8.44214e6 0.570574
\(739\) 2.01511e6 0.135734 0.0678669 0.997694i \(-0.478381\pi\)
0.0678669 + 0.997694i \(0.478381\pi\)
\(740\) −3.95956e7 −2.65808
\(741\) 192080. 0.0128510
\(742\) 0 0
\(743\) −1.51381e7 −1.00600 −0.503001 0.864286i \(-0.667771\pi\)
−0.503001 + 0.864286i \(0.667771\pi\)
\(744\) −3.85258e7 −2.55164
\(745\) −1.13266e6 −0.0747666
\(746\) 1.60323e7 1.05475
\(747\) −965286. −0.0632928
\(748\) 2.71663e7 1.77532
\(749\) 0 0
\(750\) 2.44138e7 1.58483
\(751\) 7.21401e6 0.466742 0.233371 0.972388i \(-0.425024\pi\)
0.233371 + 0.972388i \(0.425024\pi\)
\(752\) −1.32774e7 −0.856185
\(753\) −6.03033e6 −0.387573
\(754\) −4.78520e6 −0.306529
\(755\) 3.97062e6 0.253508
\(756\) 0 0
\(757\) −1.09697e7 −0.695755 −0.347877 0.937540i \(-0.613097\pi\)
−0.347877 + 0.937540i \(0.613097\pi\)
\(758\) 4.77012e7 3.01548
\(759\) 5.92435e6 0.373281
\(760\) −1.97568e6 −0.124075
\(761\) −1.92442e7 −1.20459 −0.602293 0.798275i \(-0.705746\pi\)
−0.602293 + 0.798275i \(0.705746\pi\)
\(762\) −8.45376e6 −0.527427
\(763\) 0 0
\(764\) 2.43620e7 1.51001
\(765\) −4.53230e6 −0.280005
\(766\) −2.23079e7 −1.37368
\(767\) 382200. 0.0234586
\(768\) −2.96719e7 −1.81528
\(769\) −8.21185e6 −0.500755 −0.250378 0.968148i \(-0.580555\pi\)
−0.250378 + 0.968148i \(0.580555\pi\)
\(770\) 0 0
\(771\) 1.64767e7 0.998241
\(772\) −6.72897e7 −4.06355
\(773\) −1.86187e7 −1.12073 −0.560363 0.828247i \(-0.689338\pi\)
−0.560363 + 0.828247i \(0.689338\pi\)
\(774\) 5.11360e6 0.306813
\(775\) 84084.0 0.00502874
\(776\) −2.10622e7 −1.25559
\(777\) 0 0
\(778\) −4.84024e7 −2.86694
\(779\) 1.76028e6 0.103929
\(780\) 7.46368e6 0.439255
\(781\) −1.35859e7 −0.797006
\(782\) −3.14093e7 −1.83671
\(783\) −1.38771e7 −0.808898
\(784\) 0 0
\(785\) −1.64373e7 −0.952045
\(786\) −8.62204e6 −0.497799
\(787\) −2.62501e7 −1.51075 −0.755377 0.655291i \(-0.772546\pi\)
−0.755377 + 0.655291i \(0.772546\pi\)
\(788\) −6.73234e7 −3.86234
\(789\) 1.80737e7 1.03360
\(790\) −1.77990e7 −1.01468
\(791\) 0 0
\(792\) 3.92544e6 0.222370
\(793\) −3.59072e6 −0.202768
\(794\) 9.95820e6 0.560570
\(795\) 1.77341e6 0.0995155
\(796\) 5.71714e7 3.19813
\(797\) 1.00373e7 0.559720 0.279860 0.960041i \(-0.409712\pi\)
0.279860 + 0.960041i \(0.409712\pi\)
\(798\) 0 0
\(799\) −1.60559e7 −0.889751
\(800\) −29920.0 −0.00165286
\(801\) −2.37735e6 −0.130922
\(802\) 3.31605e7 1.82048
\(803\) −1.57950e7 −0.864433
\(804\) −4.60806e7 −2.51407
\(805\) 0 0
\(806\) −1.07016e7 −0.580245
\(807\) 1.78858e7 0.966772
\(808\) 1.39306e7 0.750655
\(809\) 1.40884e7 0.756816 0.378408 0.925639i \(-0.376472\pi\)
0.378408 + 0.925639i \(0.376472\pi\)
\(810\) 2.54346e7 1.36211
\(811\) −1.81433e7 −0.968646 −0.484323 0.874889i \(-0.660934\pi\)
−0.484323 + 0.874889i \(0.660934\pi\)
\(812\) 0 0
\(813\) −2.31076e7 −1.22611
\(814\) 2.41234e7 1.27608
\(815\) 738752. 0.0389587
\(816\) 3.43298e7 1.80487
\(817\) 1.06624e6 0.0558856
\(818\) 3.07273e7 1.60562
\(819\) 0 0
\(820\) 6.83993e7 3.55236
\(821\) −2.13669e7 −1.10633 −0.553164 0.833072i \(-0.686580\pi\)
−0.553164 + 0.833072i \(0.686580\pi\)
\(822\) 2.86247e7 1.47761
\(823\) 1.78017e7 0.916142 0.458071 0.888916i \(-0.348541\pi\)
0.458071 + 0.888916i \(0.348541\pi\)
\(824\) 1.91016e7 0.980058
\(825\) 35728.0 0.00182757
\(826\) 0 0
\(827\) 1.62921e7 0.828350 0.414175 0.910197i \(-0.364070\pi\)
0.414175 + 0.910197i \(0.364070\pi\)
\(828\) −5.82950e6 −0.295499
\(829\) 2.08499e6 0.105370 0.0526851 0.998611i \(-0.483222\pi\)
0.0526851 + 0.998611i \(0.483222\pi\)
\(830\) −1.15013e7 −0.579497
\(831\) −1.48973e7 −0.748348
\(832\) −2.57152e6 −0.128790
\(833\) 0 0
\(834\) −4.95684e6 −0.246769
\(835\) −2.76423e7 −1.37201
\(836\) 1.54605e6 0.0765081
\(837\) −3.10346e7 −1.53120
\(838\) 2.81438e7 1.38443
\(839\) 2.27850e7 1.11749 0.558745 0.829340i \(-0.311283\pi\)
0.558745 + 0.829340i \(0.311283\pi\)
\(840\) 0 0
\(841\) −8.82842e6 −0.430421
\(842\) −3.05802e7 −1.48648
\(843\) −312788. −0.0151594
\(844\) 7.82498e7 3.78118
\(845\) −1.96948e7 −0.948877
\(846\) −4.38228e6 −0.210511
\(847\) 0 0
\(848\) 3.22109e6 0.153820
\(849\) 3.49403e7 1.66363
\(850\) −189420. −0.00899246
\(851\) −1.89660e7 −0.897740
\(852\) −5.57491e7 −2.63111
\(853\) 2.26975e7 1.06808 0.534042 0.845458i \(-0.320672\pi\)
0.534042 + 0.845458i \(0.320672\pi\)
\(854\) 0 0
\(855\) −257936. −0.0120669
\(856\) 5.26766e7 2.45716
\(857\) −2.52900e7 −1.17624 −0.588120 0.808774i \(-0.700132\pi\)
−0.588120 + 0.808774i \(0.700132\pi\)
\(858\) −4.54720e6 −0.210875
\(859\) 1.03947e7 0.480652 0.240326 0.970692i \(-0.422746\pi\)
0.240326 + 0.970692i \(0.422746\pi\)
\(860\) 4.14310e7 1.91020
\(861\) 0 0
\(862\) −1.93750e7 −0.888122
\(863\) 4.33399e7 1.98089 0.990447 0.137892i \(-0.0440327\pi\)
0.990447 + 0.137892i \(0.0440327\pi\)
\(864\) 1.10432e7 0.503281
\(865\) −1.34801e7 −0.612566
\(866\) 3.94790e7 1.78884
\(867\) 2.16360e7 0.977527
\(868\) 0 0
\(869\) 7.37389e6 0.331243
\(870\) −2.67971e7 −1.20030
\(871\) −6.77656e6 −0.302666
\(872\) −3.34433e7 −1.48942
\(873\) −2.74978e6 −0.122113
\(874\) −1.78752e6 −0.0791539
\(875\) 0 0
\(876\) −6.48141e7 −2.85370
\(877\) 3.71659e7 1.63172 0.815861 0.578248i \(-0.196264\pi\)
0.815861 + 0.578248i \(0.196264\pi\)
\(878\) −7.41770e7 −3.24738
\(879\) 2.70449e7 1.18063
\(880\) 1.85006e7 0.805340
\(881\) −9.04785e6 −0.392740 −0.196370 0.980530i \(-0.562915\pi\)
−0.196370 + 0.980530i \(0.562915\pi\)
\(882\) 0 0
\(883\) 3.29679e7 1.42295 0.711474 0.702712i \(-0.248028\pi\)
0.711474 + 0.702712i \(0.248028\pi\)
\(884\) 1.63934e7 0.705569
\(885\) 2.14032e6 0.0918588
\(886\) −1.40269e7 −0.600313
\(887\) 1.61099e7 0.687517 0.343758 0.939058i \(-0.388300\pi\)
0.343758 + 0.939058i \(0.388300\pi\)
\(888\) 5.24059e7 2.23022
\(889\) 0 0
\(890\) −2.83259e7 −1.19870
\(891\) −1.05372e7 −0.444663
\(892\) 5.60500e7 2.35865
\(893\) −913752. −0.0383442
\(894\) 2.83164e6 0.118493
\(895\) 1.65162e7 0.689211
\(896\) 0 0
\(897\) 3.57504e6 0.148354
\(898\) 5.90574e6 0.244390
\(899\) 2.61272e7 1.07819
\(900\) −35156.0 −0.00144675
\(901\) 3.89516e6 0.159850
\(902\) −4.16718e7 −1.70540
\(903\) 0 0
\(904\) 3.00074e7 1.22126
\(905\) 1.88709e7 0.765898
\(906\) −9.92656e6 −0.401771
\(907\) −4.47286e7 −1.80537 −0.902686 0.430300i \(-0.858408\pi\)
−0.902686 + 0.430300i \(0.858408\pi\)
\(908\) −5.05798e6 −0.203593
\(909\) 1.81871e6 0.0730053
\(910\) 0 0
\(911\) −6.60518e6 −0.263687 −0.131844 0.991271i \(-0.542090\pi\)
−0.131844 + 0.991271i \(0.542090\pi\)
\(912\) 1.95373e6 0.0777816
\(913\) 4.76482e6 0.189177
\(914\) 2.90484e7 1.15016
\(915\) −2.01080e7 −0.793993
\(916\) −7.69730e7 −3.03110
\(917\) 0 0
\(918\) 6.99132e7 2.73812
\(919\) −3.08930e7 −1.20662 −0.603311 0.797506i \(-0.706152\pi\)
−0.603311 + 0.797506i \(0.706152\pi\)
\(920\) −3.67718e7 −1.43234
\(921\) 6.42704e6 0.249667
\(922\) −9.22684e6 −0.357459
\(923\) −8.19840e6 −0.316756
\(924\) 0 0
\(925\) −114378. −0.00439530
\(926\) −7.18235e7 −2.75258
\(927\) 2.49382e6 0.0953160
\(928\) −9.29696e6 −0.354381
\(929\) 4.87215e6 0.185217 0.0926087 0.995703i \(-0.470479\pi\)
0.0926087 + 0.995703i \(0.470479\pi\)
\(930\) −5.99290e7 −2.27211
\(931\) 0 0
\(932\) −1.35134e7 −0.509593
\(933\) −9.33979e6 −0.351264
\(934\) −6.12570e6 −0.229767
\(935\) 2.23722e7 0.836913
\(936\) 2.36880e6 0.0883769
\(937\) 3.25004e7 1.20932 0.604658 0.796485i \(-0.293310\pi\)
0.604658 + 0.796485i \(0.293310\pi\)
\(938\) 0 0
\(939\) 1.55448e6 0.0575334
\(940\) −3.55058e7 −1.31063
\(941\) 2.64040e6 0.0972066 0.0486033 0.998818i \(-0.484523\pi\)
0.0486033 + 0.998818i \(0.484523\pi\)
\(942\) 4.10934e7 1.50884
\(943\) 3.27627e7 1.19978
\(944\) 3.88752e6 0.141985
\(945\) 0 0
\(946\) −2.52416e7 −0.917042
\(947\) −4.08179e7 −1.47903 −0.739513 0.673142i \(-0.764944\pi\)
−0.739513 + 0.673142i \(0.764944\pi\)
\(948\) 3.02584e7 1.09351
\(949\) −9.53148e6 −0.343554
\(950\) −10780.0 −0.000387534 0
\(951\) −962892. −0.0345244
\(952\) 0 0
\(953\) −6.71983e6 −0.239677 −0.119838 0.992793i \(-0.538238\pi\)
−0.119838 + 0.992793i \(0.538238\pi\)
\(954\) 1.06314e6 0.0378198
\(955\) 2.00628e7 0.711841
\(956\) 3.28375e7 1.16205
\(957\) 1.11017e7 0.391840
\(958\) 2.60330e7 0.916454
\(959\) 0 0
\(960\) −1.44005e7 −0.504313
\(961\) 2.98016e7 1.04095
\(962\) 1.45572e7 0.507154
\(963\) 6.87723e6 0.238972
\(964\) −5.48019e7 −1.89934
\(965\) −5.54150e7 −1.91562
\(966\) 0 0
\(967\) −2.78979e6 −0.0959413 −0.0479707 0.998849i \(-0.515275\pi\)
−0.0479707 + 0.998849i \(0.515275\pi\)
\(968\) 3.86017e7 1.32409
\(969\) 2.36258e6 0.0808310
\(970\) −3.27634e7 −1.11804
\(971\) −3.33594e7 −1.13545 −0.567727 0.823217i \(-0.692177\pi\)
−0.567727 + 0.823217i \(0.692177\pi\)
\(972\) 2.38486e7 0.809648
\(973\) 0 0
\(974\) −5.46309e7 −1.84519
\(975\) 21560.0 0.000726335 0
\(976\) −3.65228e7 −1.22727
\(977\) −7.60033e6 −0.254739 −0.127370 0.991855i \(-0.540653\pi\)
−0.127370 + 0.991855i \(0.540653\pi\)
\(978\) −1.84688e6 −0.0617435
\(979\) 1.17350e7 0.391316
\(980\) 0 0
\(981\) −4.36621e6 −0.144854
\(982\) −1.64090e7 −0.543004
\(983\) 5.79760e6 0.191366 0.0956829 0.995412i \(-0.469497\pi\)
0.0956829 + 0.995412i \(0.469497\pi\)
\(984\) −9.05285e7 −2.98056
\(985\) −5.54428e7 −1.82077
\(986\) −5.88580e7 −1.92803
\(987\) 0 0
\(988\) 932960. 0.0304068
\(989\) 1.98451e7 0.645153
\(990\) 6.10624e6 0.198009
\(991\) 1.26825e7 0.410224 0.205112 0.978739i \(-0.434244\pi\)
0.205112 + 0.978739i \(0.434244\pi\)
\(992\) −2.07917e7 −0.670827
\(993\) −7.90227e6 −0.254319
\(994\) 0 0
\(995\) 4.70823e7 1.50765
\(996\) 1.95522e7 0.624521
\(997\) −1.44400e7 −0.460077 −0.230039 0.973182i \(-0.573885\pi\)
−0.230039 + 0.973182i \(0.573885\pi\)
\(998\) −2.99796e7 −0.952796
\(999\) 4.22159e7 1.33833
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 49.6.a.a.1.1 1
3.2 odd 2 441.6.a.k.1.1 1
4.3 odd 2 784.6.a.c.1.1 1
7.2 even 3 49.6.c.b.18.1 2
7.3 odd 6 49.6.c.c.30.1 2
7.4 even 3 49.6.c.b.30.1 2
7.5 odd 6 49.6.c.c.18.1 2
7.6 odd 2 7.6.a.a.1.1 1
21.20 even 2 63.6.a.e.1.1 1
28.27 even 2 112.6.a.g.1.1 1
35.13 even 4 175.6.b.a.99.2 2
35.27 even 4 175.6.b.a.99.1 2
35.34 odd 2 175.6.a.b.1.1 1
56.13 odd 2 448.6.a.m.1.1 1
56.27 even 2 448.6.a.c.1.1 1
77.76 even 2 847.6.a.b.1.1 1
84.83 odd 2 1008.6.a.y.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.6.a.a.1.1 1 7.6 odd 2
49.6.a.a.1.1 1 1.1 even 1 trivial
49.6.c.b.18.1 2 7.2 even 3
49.6.c.b.30.1 2 7.4 even 3
49.6.c.c.18.1 2 7.5 odd 6
49.6.c.c.30.1 2 7.3 odd 6
63.6.a.e.1.1 1 21.20 even 2
112.6.a.g.1.1 1 28.27 even 2
175.6.a.b.1.1 1 35.34 odd 2
175.6.b.a.99.1 2 35.27 even 4
175.6.b.a.99.2 2 35.13 even 4
441.6.a.k.1.1 1 3.2 odd 2
448.6.a.c.1.1 1 56.27 even 2
448.6.a.m.1.1 1 56.13 odd 2
784.6.a.c.1.1 1 4.3 odd 2
847.6.a.b.1.1 1 77.76 even 2
1008.6.a.y.1.1 1 84.83 odd 2