Properties

Label 77.6.a.a.1.1
Level $77$
Weight $6$
Character 77.1
Self dual yes
Analytic conductor $12.350$
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] = [77,6,Mod(1,77)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(77, 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("77.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 77 = 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 77.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: \(12.3495541256\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 77.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} -6.00000 q^{3} -28.0000 q^{4} -74.0000 q^{5} +12.0000 q^{6} -49.0000 q^{7} +120.000 q^{8} -207.000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} -6.00000 q^{3} -28.0000 q^{4} -74.0000 q^{5} +12.0000 q^{6} -49.0000 q^{7} +120.000 q^{8} -207.000 q^{9} +148.000 q^{10} +121.000 q^{11} +168.000 q^{12} +364.000 q^{13} +98.0000 q^{14} +444.000 q^{15} +656.000 q^{16} +148.000 q^{17} +414.000 q^{18} -1320.00 q^{19} +2072.00 q^{20} +294.000 q^{21} -242.000 q^{22} -436.000 q^{23} -720.000 q^{24} +2351.00 q^{25} -728.000 q^{26} +2700.00 q^{27} +1372.00 q^{28} +2970.00 q^{29} -888.000 q^{30} +8842.00 q^{31} -5152.00 q^{32} -726.000 q^{33} -296.000 q^{34} +3626.00 q^{35} +5796.00 q^{36} +138.000 q^{37} +2640.00 q^{38} -2184.00 q^{39} -8880.00 q^{40} +532.000 q^{41} -588.000 q^{42} -20676.0 q^{43} -3388.00 q^{44} +15318.0 q^{45} +872.000 q^{46} -11722.0 q^{47} -3936.00 q^{48} +2401.00 q^{49} -4702.00 q^{50} -888.000 q^{51} -10192.0 q^{52} +5274.00 q^{53} -5400.00 q^{54} -8954.00 q^{55} -5880.00 q^{56} +7920.00 q^{57} -5940.00 q^{58} -27670.0 q^{59} -12432.0 q^{60} +19512.0 q^{61} -17684.0 q^{62} +10143.0 q^{63} -10688.0 q^{64} -26936.0 q^{65} +1452.00 q^{66} +64088.0 q^{67} -4144.00 q^{68} +2616.00 q^{69} -7252.00 q^{70} -3708.00 q^{71} -24840.0 q^{72} -24296.0 q^{73} -276.000 q^{74} -14106.0 q^{75} +36960.0 q^{76} -5929.00 q^{77} +4368.00 q^{78} -2200.00 q^{79} -48544.0 q^{80} +34101.0 q^{81} -1064.00 q^{82} +74424.0 q^{83} -8232.00 q^{84} -10952.0 q^{85} +41352.0 q^{86} -17820.0 q^{87} +14520.0 q^{88} +34170.0 q^{89} -30636.0 q^{90} -17836.0 q^{91} +12208.0 q^{92} -53052.0 q^{93} +23444.0 q^{94} +97680.0 q^{95} +30912.0 q^{96} +151718. q^{97} -4802.00 q^{98} -25047.0 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.353553 −0.176777 0.984251i \(-0.556567\pi\)
−0.176777 + 0.984251i \(0.556567\pi\)
\(3\) −6.00000 −0.384900 −0.192450 0.981307i \(-0.561643\pi\)
−0.192450 + 0.981307i \(0.561643\pi\)
\(4\) −28.0000 −0.875000
\(5\) −74.0000 −1.32375 −0.661876 0.749613i \(-0.730240\pi\)
−0.661876 + 0.749613i \(0.730240\pi\)
\(6\) 12.0000 0.136083
\(7\) −49.0000 −0.377964
\(8\) 120.000 0.662913
\(9\) −207.000 −0.851852
\(10\) 148.000 0.468017
\(11\) 121.000 0.301511
\(12\) 168.000 0.336788
\(13\) 364.000 0.597369 0.298685 0.954352i \(-0.403452\pi\)
0.298685 + 0.954352i \(0.403452\pi\)
\(14\) 98.0000 0.133631
\(15\) 444.000 0.509512
\(16\) 656.000 0.640625
\(17\) 148.000 0.124205 0.0621025 0.998070i \(-0.480219\pi\)
0.0621025 + 0.998070i \(0.480219\pi\)
\(18\) 414.000 0.301175
\(19\) −1320.00 −0.838861 −0.419430 0.907787i \(-0.637770\pi\)
−0.419430 + 0.907787i \(0.637770\pi\)
\(20\) 2072.00 1.15828
\(21\) 294.000 0.145479
\(22\) −242.000 −0.106600
\(23\) −436.000 −0.171857 −0.0859284 0.996301i \(-0.527386\pi\)
−0.0859284 + 0.996301i \(0.527386\pi\)
\(24\) −720.000 −0.255155
\(25\) 2351.00 0.752320
\(26\) −728.000 −0.211202
\(27\) 2700.00 0.712778
\(28\) 1372.00 0.330719
\(29\) 2970.00 0.655785 0.327892 0.944715i \(-0.393662\pi\)
0.327892 + 0.944715i \(0.393662\pi\)
\(30\) −888.000 −0.180140
\(31\) 8842.00 1.65252 0.826259 0.563290i \(-0.190465\pi\)
0.826259 + 0.563290i \(0.190465\pi\)
\(32\) −5152.00 −0.889408
\(33\) −726.000 −0.116052
\(34\) −296.000 −0.0439131
\(35\) 3626.00 0.500331
\(36\) 5796.00 0.745370
\(37\) 138.000 0.0165720 0.00828600 0.999966i \(-0.497362\pi\)
0.00828600 + 0.999966i \(0.497362\pi\)
\(38\) 2640.00 0.296582
\(39\) −2184.00 −0.229928
\(40\) −8880.00 −0.877532
\(41\) 532.000 0.0494256 0.0247128 0.999695i \(-0.492133\pi\)
0.0247128 + 0.999695i \(0.492133\pi\)
\(42\) −588.000 −0.0514344
\(43\) −20676.0 −1.70528 −0.852639 0.522500i \(-0.825000\pi\)
−0.852639 + 0.522500i \(0.825000\pi\)
\(44\) −3388.00 −0.263822
\(45\) 15318.0 1.12764
\(46\) 872.000 0.0607606
\(47\) −11722.0 −0.774029 −0.387014 0.922074i \(-0.626494\pi\)
−0.387014 + 0.922074i \(0.626494\pi\)
\(48\) −3936.00 −0.246577
\(49\) 2401.00 0.142857
\(50\) −4702.00 −0.265985
\(51\) −888.000 −0.0478066
\(52\) −10192.0 −0.522698
\(53\) 5274.00 0.257899 0.128950 0.991651i \(-0.458839\pi\)
0.128950 + 0.991651i \(0.458839\pi\)
\(54\) −5400.00 −0.252005
\(55\) −8954.00 −0.399126
\(56\) −5880.00 −0.250557
\(57\) 7920.00 0.322878
\(58\) −5940.00 −0.231855
\(59\) −27670.0 −1.03485 −0.517427 0.855727i \(-0.673110\pi\)
−0.517427 + 0.855727i \(0.673110\pi\)
\(60\) −12432.0 −0.445823
\(61\) 19512.0 0.671394 0.335697 0.941970i \(-0.391028\pi\)
0.335697 + 0.941970i \(0.391028\pi\)
\(62\) −17684.0 −0.584253
\(63\) 10143.0 0.321970
\(64\) −10688.0 −0.326172
\(65\) −26936.0 −0.790769
\(66\) 1452.00 0.0410305
\(67\) 64088.0 1.74417 0.872087 0.489351i \(-0.162766\pi\)
0.872087 + 0.489351i \(0.162766\pi\)
\(68\) −4144.00 −0.108679
\(69\) 2616.00 0.0661477
\(70\) −7252.00 −0.176894
\(71\) −3708.00 −0.0872959 −0.0436480 0.999047i \(-0.513898\pi\)
−0.0436480 + 0.999047i \(0.513898\pi\)
\(72\) −24840.0 −0.564703
\(73\) −24296.0 −0.533615 −0.266807 0.963750i \(-0.585969\pi\)
−0.266807 + 0.963750i \(0.585969\pi\)
\(74\) −276.000 −0.00585908
\(75\) −14106.0 −0.289568
\(76\) 36960.0 0.734003
\(77\) −5929.00 −0.113961
\(78\) 4368.00 0.0812917
\(79\) −2200.00 −0.0396602 −0.0198301 0.999803i \(-0.506313\pi\)
−0.0198301 + 0.999803i \(0.506313\pi\)
\(80\) −48544.0 −0.848029
\(81\) 34101.0 0.577503
\(82\) −1064.00 −0.0174746
\(83\) 74424.0 1.18582 0.592909 0.805270i \(-0.297980\pi\)
0.592909 + 0.805270i \(0.297980\pi\)
\(84\) −8232.00 −0.127294
\(85\) −10952.0 −0.164417
\(86\) 41352.0 0.602907
\(87\) −17820.0 −0.252412
\(88\) 14520.0 0.199876
\(89\) 34170.0 0.457267 0.228634 0.973513i \(-0.426574\pi\)
0.228634 + 0.973513i \(0.426574\pi\)
\(90\) −30636.0 −0.398681
\(91\) −17836.0 −0.225784
\(92\) 12208.0 0.150375
\(93\) −53052.0 −0.636055
\(94\) 23444.0 0.273660
\(95\) 97680.0 1.11044
\(96\) 30912.0 0.342333
\(97\) 151718. 1.63722 0.818611 0.574348i \(-0.194744\pi\)
0.818611 + 0.574348i \(0.194744\pi\)
\(98\) −4802.00 −0.0505076
\(99\) −25047.0 −0.256843
\(100\) −65828.0 −0.658280
\(101\) 116852. 1.13981 0.569905 0.821710i \(-0.306980\pi\)
0.569905 + 0.821710i \(0.306980\pi\)
\(102\) 1776.00 0.0169022
\(103\) 103694. 0.963076 0.481538 0.876425i \(-0.340078\pi\)
0.481538 + 0.876425i \(0.340078\pi\)
\(104\) 43680.0 0.396004
\(105\) −21756.0 −0.192578
\(106\) −10548.0 −0.0911812
\(107\) −97092.0 −0.819830 −0.409915 0.912124i \(-0.634442\pi\)
−0.409915 + 0.912124i \(0.634442\pi\)
\(108\) −75600.0 −0.623681
\(109\) 52930.0 0.426713 0.213356 0.976974i \(-0.431560\pi\)
0.213356 + 0.976974i \(0.431560\pi\)
\(110\) 17908.0 0.141112
\(111\) −828.000 −0.00637856
\(112\) −32144.0 −0.242133
\(113\) −80526.0 −0.593253 −0.296627 0.954994i \(-0.595862\pi\)
−0.296627 + 0.954994i \(0.595862\pi\)
\(114\) −15840.0 −0.114155
\(115\) 32264.0 0.227496
\(116\) −83160.0 −0.573812
\(117\) −75348.0 −0.508870
\(118\) 55340.0 0.365876
\(119\) −7252.00 −0.0469451
\(120\) 53280.0 0.337762
\(121\) 14641.0 0.0909091
\(122\) −39024.0 −0.237373
\(123\) −3192.00 −0.0190239
\(124\) −247576. −1.44595
\(125\) 57276.0 0.327867
\(126\) −20286.0 −0.113833
\(127\) 221048. 1.21612 0.608061 0.793890i \(-0.291948\pi\)
0.608061 + 0.793890i \(0.291948\pi\)
\(128\) 186240. 1.00473
\(129\) 124056. 0.656362
\(130\) 53872.0 0.279579
\(131\) 37572.0 0.191287 0.0956436 0.995416i \(-0.469509\pi\)
0.0956436 + 0.995416i \(0.469509\pi\)
\(132\) 20328.0 0.101545
\(133\) 64680.0 0.317060
\(134\) −128176. −0.616658
\(135\) −199800. −0.943542
\(136\) 17760.0 0.0823371
\(137\) −290602. −1.32281 −0.661405 0.750029i \(-0.730039\pi\)
−0.661405 + 0.750029i \(0.730039\pi\)
\(138\) −5232.00 −0.0233868
\(139\) 367360. 1.61270 0.806352 0.591435i \(-0.201439\pi\)
0.806352 + 0.591435i \(0.201439\pi\)
\(140\) −101528. −0.437790
\(141\) 70332.0 0.297924
\(142\) 7416.00 0.0308638
\(143\) 44044.0 0.180114
\(144\) −135792. −0.545718
\(145\) −219780. −0.868097
\(146\) 48592.0 0.188661
\(147\) −14406.0 −0.0549857
\(148\) −3864.00 −0.0145005
\(149\) −462730. −1.70751 −0.853753 0.520679i \(-0.825679\pi\)
−0.853753 + 0.520679i \(0.825679\pi\)
\(150\) 28212.0 0.102378
\(151\) −7648.00 −0.0272964 −0.0136482 0.999907i \(-0.504344\pi\)
−0.0136482 + 0.999907i \(0.504344\pi\)
\(152\) −158400. −0.556091
\(153\) −30636.0 −0.105804
\(154\) 11858.0 0.0402911
\(155\) −654308. −2.18752
\(156\) 61152.0 0.201187
\(157\) −161482. −0.522847 −0.261424 0.965224i \(-0.584192\pi\)
−0.261424 + 0.965224i \(0.584192\pi\)
\(158\) 4400.00 0.0140220
\(159\) −31644.0 −0.0992656
\(160\) 381248. 1.17736
\(161\) 21364.0 0.0649558
\(162\) −68202.0 −0.204178
\(163\) 179464. 0.529064 0.264532 0.964377i \(-0.414782\pi\)
0.264532 + 0.964377i \(0.414782\pi\)
\(164\) −14896.0 −0.0432474
\(165\) 53724.0 0.153624
\(166\) −148848. −0.419250
\(167\) 316848. 0.879144 0.439572 0.898207i \(-0.355130\pi\)
0.439572 + 0.898207i \(0.355130\pi\)
\(168\) 35280.0 0.0964396
\(169\) −238797. −0.643150
\(170\) 21904.0 0.0581301
\(171\) 273240. 0.714585
\(172\) 578928. 1.49212
\(173\) −175116. −0.444847 −0.222423 0.974950i \(-0.571397\pi\)
−0.222423 + 0.974950i \(0.571397\pi\)
\(174\) 35640.0 0.0892410
\(175\) −115199. −0.284350
\(176\) 79376.0 0.193156
\(177\) 166020. 0.398316
\(178\) −68340.0 −0.161668
\(179\) −69780.0 −0.162779 −0.0813895 0.996682i \(-0.525936\pi\)
−0.0813895 + 0.996682i \(0.525936\pi\)
\(180\) −428904. −0.986686
\(181\) −78638.0 −0.178417 −0.0892085 0.996013i \(-0.528434\pi\)
−0.0892085 + 0.996013i \(0.528434\pi\)
\(182\) 35672.0 0.0798268
\(183\) −117072. −0.258420
\(184\) −52320.0 −0.113926
\(185\) −10212.0 −0.0219372
\(186\) 106104. 0.224879
\(187\) 17908.0 0.0374492
\(188\) 328216. 0.677275
\(189\) −132300. −0.269405
\(190\) −195360. −0.392601
\(191\) −927208. −1.83905 −0.919525 0.393030i \(-0.871427\pi\)
−0.919525 + 0.393030i \(0.871427\pi\)
\(192\) 64128.0 0.125544
\(193\) 877474. 1.69567 0.847834 0.530261i \(-0.177906\pi\)
0.847834 + 0.530261i \(0.177906\pi\)
\(194\) −303436. −0.578846
\(195\) 161616. 0.304367
\(196\) −67228.0 −0.125000
\(197\) −744602. −1.36697 −0.683484 0.729965i \(-0.739536\pi\)
−0.683484 + 0.729965i \(0.739536\pi\)
\(198\) 50094.0 0.0908077
\(199\) 1.07931e6 1.93203 0.966014 0.258489i \(-0.0832246\pi\)
0.966014 + 0.258489i \(0.0832246\pi\)
\(200\) 282120. 0.498722
\(201\) −384528. −0.671333
\(202\) −233704. −0.402984
\(203\) −145530. −0.247863
\(204\) 24864.0 0.0418307
\(205\) −39368.0 −0.0654273
\(206\) −207388. −0.340499
\(207\) 90252.0 0.146397
\(208\) 238784. 0.382690
\(209\) −159720. −0.252926
\(210\) 43512.0 0.0680865
\(211\) 728772. 1.12690 0.563450 0.826150i \(-0.309474\pi\)
0.563450 + 0.826150i \(0.309474\pi\)
\(212\) −147672. −0.225662
\(213\) 22248.0 0.0336002
\(214\) 194184. 0.289854
\(215\) 1.53002e6 2.25737
\(216\) 324000. 0.472510
\(217\) −433258. −0.624593
\(218\) −105860. −0.150866
\(219\) 145776. 0.205388
\(220\) 250712. 0.349236
\(221\) 53872.0 0.0741963
\(222\) 1656.00 0.00225516
\(223\) 38374.0 0.0516743 0.0258372 0.999666i \(-0.491775\pi\)
0.0258372 + 0.999666i \(0.491775\pi\)
\(224\) 252448. 0.336165
\(225\) −486657. −0.640865
\(226\) 161052. 0.209747
\(227\) 323268. 0.416388 0.208194 0.978088i \(-0.433241\pi\)
0.208194 + 0.978088i \(0.433241\pi\)
\(228\) −221760. −0.282518
\(229\) 813690. 1.02535 0.512673 0.858584i \(-0.328655\pi\)
0.512673 + 0.858584i \(0.328655\pi\)
\(230\) −64528.0 −0.0804320
\(231\) 35574.0 0.0438634
\(232\) 356400. 0.434728
\(233\) 1.10801e6 1.33707 0.668537 0.743679i \(-0.266921\pi\)
0.668537 + 0.743679i \(0.266921\pi\)
\(234\) 150696. 0.179913
\(235\) 867428. 1.02462
\(236\) 774760. 0.905497
\(237\) 13200.0 0.0152652
\(238\) 14504.0 0.0165976
\(239\) −1.31352e6 −1.48745 −0.743724 0.668487i \(-0.766942\pi\)
−0.743724 + 0.668487i \(0.766942\pi\)
\(240\) 291264. 0.326406
\(241\) −1.05607e6 −1.17125 −0.585625 0.810582i \(-0.699151\pi\)
−0.585625 + 0.810582i \(0.699151\pi\)
\(242\) −29282.0 −0.0321412
\(243\) −860706. −0.935059
\(244\) −546336. −0.587469
\(245\) −177674. −0.189107
\(246\) 6384.00 0.00672597
\(247\) −480480. −0.501110
\(248\) 1.06104e6 1.09548
\(249\) −446544. −0.456421
\(250\) −114552. −0.115918
\(251\) 1.68914e6 1.69232 0.846159 0.532931i \(-0.178909\pi\)
0.846159 + 0.532931i \(0.178909\pi\)
\(252\) −284004. −0.281724
\(253\) −52756.0 −0.0518168
\(254\) −442096. −0.429964
\(255\) 65712.0 0.0632840
\(256\) −30464.0 −0.0290527
\(257\) 641938. 0.606262 0.303131 0.952949i \(-0.401968\pi\)
0.303131 + 0.952949i \(0.401968\pi\)
\(258\) −248112. −0.232059
\(259\) −6762.00 −0.00626363
\(260\) 754208. 0.691923
\(261\) −614790. −0.558632
\(262\) −75144.0 −0.0676303
\(263\) −1.10150e6 −0.981959 −0.490980 0.871171i \(-0.663361\pi\)
−0.490980 + 0.871171i \(0.663361\pi\)
\(264\) −87120.0 −0.0769322
\(265\) −390276. −0.341395
\(266\) −129360. −0.112097
\(267\) −205020. −0.176002
\(268\) −1.79446e6 −1.52615
\(269\) −2.15147e6 −1.81282 −0.906410 0.422399i \(-0.861188\pi\)
−0.906410 + 0.422399i \(0.861188\pi\)
\(270\) 399600. 0.333592
\(271\) −1.08327e6 −0.896010 −0.448005 0.894031i \(-0.647865\pi\)
−0.448005 + 0.894031i \(0.647865\pi\)
\(272\) 97088.0 0.0795689
\(273\) 107016. 0.0869045
\(274\) 581204. 0.467684
\(275\) 284471. 0.226833
\(276\) −73248.0 −0.0578793
\(277\) −2.22372e6 −1.74133 −0.870665 0.491877i \(-0.836311\pi\)
−0.870665 + 0.491877i \(0.836311\pi\)
\(278\) −734720. −0.570177
\(279\) −1.83029e6 −1.40770
\(280\) 435120. 0.331676
\(281\) −153018. −0.115605 −0.0578025 0.998328i \(-0.518409\pi\)
−0.0578025 + 0.998328i \(0.518409\pi\)
\(282\) −140664. −0.105332
\(283\) 715324. 0.530929 0.265465 0.964121i \(-0.414475\pi\)
0.265465 + 0.964121i \(0.414475\pi\)
\(284\) 103824. 0.0763839
\(285\) −586080. −0.427410
\(286\) −88088.0 −0.0636798
\(287\) −26068.0 −0.0186811
\(288\) 1.06646e6 0.757644
\(289\) −1.39795e6 −0.984573
\(290\) 439560. 0.306919
\(291\) −910308. −0.630167
\(292\) 680288. 0.466913
\(293\) 347424. 0.236424 0.118212 0.992988i \(-0.462284\pi\)
0.118212 + 0.992988i \(0.462284\pi\)
\(294\) 28812.0 0.0194404
\(295\) 2.04758e6 1.36989
\(296\) 16560.0 0.0109858
\(297\) 326700. 0.214911
\(298\) 925460. 0.603694
\(299\) −158704. −0.102662
\(300\) 394968. 0.253372
\(301\) 1.01312e6 0.644535
\(302\) 15296.0 0.00965074
\(303\) −701112. −0.438713
\(304\) −865920. −0.537395
\(305\) −1.44389e6 −0.888759
\(306\) 61272.0 0.0374075
\(307\) 2.64043e6 1.59893 0.799463 0.600715i \(-0.205117\pi\)
0.799463 + 0.600715i \(0.205117\pi\)
\(308\) 166012. 0.0997155
\(309\) −622164. −0.370688
\(310\) 1.30862e6 0.773407
\(311\) −947778. −0.555656 −0.277828 0.960631i \(-0.589614\pi\)
−0.277828 + 0.960631i \(0.589614\pi\)
\(312\) −262080. −0.152422
\(313\) −248686. −0.143480 −0.0717399 0.997423i \(-0.522855\pi\)
−0.0717399 + 0.997423i \(0.522855\pi\)
\(314\) 322964. 0.184854
\(315\) −750582. −0.426208
\(316\) 61600.0 0.0347027
\(317\) 2.60904e6 1.45825 0.729125 0.684380i \(-0.239927\pi\)
0.729125 + 0.684380i \(0.239927\pi\)
\(318\) 63288.0 0.0350957
\(319\) 359370. 0.197727
\(320\) 790912. 0.431771
\(321\) 582552. 0.315553
\(322\) −42728.0 −0.0229653
\(323\) −195360. −0.104191
\(324\) −954828. −0.505316
\(325\) 855764. 0.449413
\(326\) −358928. −0.187052
\(327\) −317580. −0.164242
\(328\) 63840.0 0.0327649
\(329\) 574378. 0.292555
\(330\) −107448. −0.0543142
\(331\) 152332. 0.0764225 0.0382112 0.999270i \(-0.487834\pi\)
0.0382112 + 0.999270i \(0.487834\pi\)
\(332\) −2.08387e6 −1.03759
\(333\) −28566.0 −0.0141169
\(334\) −633696. −0.310824
\(335\) −4.74251e6 −2.30885
\(336\) 192864. 0.0931972
\(337\) 206558. 0.0990757 0.0495379 0.998772i \(-0.484225\pi\)
0.0495379 + 0.998772i \(0.484225\pi\)
\(338\) 477594. 0.227388
\(339\) 483156. 0.228343
\(340\) 306656. 0.143865
\(341\) 1.06988e6 0.498253
\(342\) −546480. −0.252644
\(343\) −117649. −0.0539949
\(344\) −2.48112e6 −1.13045
\(345\) −193584. −0.0875632
\(346\) 350232. 0.157277
\(347\) 2.01807e6 0.899730 0.449865 0.893097i \(-0.351472\pi\)
0.449865 + 0.893097i \(0.351472\pi\)
\(348\) 498960. 0.220860
\(349\) −580440. −0.255090 −0.127545 0.991833i \(-0.540710\pi\)
−0.127545 + 0.991833i \(0.540710\pi\)
\(350\) 230398. 0.100533
\(351\) 982800. 0.425792
\(352\) −623392. −0.268167
\(353\) 572034. 0.244335 0.122167 0.992510i \(-0.461016\pi\)
0.122167 + 0.992510i \(0.461016\pi\)
\(354\) −332040. −0.140826
\(355\) 274392. 0.115558
\(356\) −956760. −0.400109
\(357\) 43512.0 0.0180692
\(358\) 139560. 0.0575511
\(359\) 4.56544e6 1.86959 0.934795 0.355187i \(-0.115583\pi\)
0.934795 + 0.355187i \(0.115583\pi\)
\(360\) 1.83816e6 0.747527
\(361\) −733699. −0.296312
\(362\) 157276. 0.0630799
\(363\) −87846.0 −0.0349909
\(364\) 499408. 0.197561
\(365\) 1.79790e6 0.706373
\(366\) 234144. 0.0913651
\(367\) 924358. 0.358241 0.179120 0.983827i \(-0.442675\pi\)
0.179120 + 0.983827i \(0.442675\pi\)
\(368\) −286016. −0.110096
\(369\) −110124. −0.0421033
\(370\) 20424.0 0.00775598
\(371\) −258426. −0.0974768
\(372\) 1.48546e6 0.556548
\(373\) 4.92021e6 1.83110 0.915550 0.402205i \(-0.131756\pi\)
0.915550 + 0.402205i \(0.131756\pi\)
\(374\) −35816.0 −0.0132403
\(375\) −343656. −0.126196
\(376\) −1.40664e6 −0.513113
\(377\) 1.08108e6 0.391746
\(378\) 264600. 0.0952490
\(379\) 3.97540e6 1.42162 0.710809 0.703385i \(-0.248329\pi\)
0.710809 + 0.703385i \(0.248329\pi\)
\(380\) −2.73504e6 −0.971638
\(381\) −1.32629e6 −0.468086
\(382\) 1.85442e6 0.650203
\(383\) −982846. −0.342364 −0.171182 0.985239i \(-0.554759\pi\)
−0.171182 + 0.985239i \(0.554759\pi\)
\(384\) −1.11744e6 −0.386720
\(385\) 438746. 0.150856
\(386\) −1.75495e6 −0.599509
\(387\) 4.27993e6 1.45264
\(388\) −4.24810e6 −1.43257
\(389\) −744090. −0.249317 −0.124658 0.992200i \(-0.539783\pi\)
−0.124658 + 0.992200i \(0.539783\pi\)
\(390\) −323232. −0.107610
\(391\) −64528.0 −0.0213455
\(392\) 288120. 0.0947018
\(393\) −225432. −0.0736265
\(394\) 1.48920e6 0.483296
\(395\) 162800. 0.0525003
\(396\) 701316. 0.224738
\(397\) 5.73024e6 1.82472 0.912360 0.409388i \(-0.134258\pi\)
0.912360 + 0.409388i \(0.134258\pi\)
\(398\) −2.15862e6 −0.683075
\(399\) −388080. −0.122036
\(400\) 1.54226e6 0.481955
\(401\) 4.26756e6 1.32531 0.662657 0.748923i \(-0.269429\pi\)
0.662657 + 0.748923i \(0.269429\pi\)
\(402\) 769056. 0.237352
\(403\) 3.21849e6 0.987164
\(404\) −3.27186e6 −0.997334
\(405\) −2.52347e6 −0.764471
\(406\) 291060. 0.0876330
\(407\) 16698.0 0.00499664
\(408\) −106560. −0.0316916
\(409\) −5.81772e6 −1.71967 −0.859834 0.510574i \(-0.829433\pi\)
−0.859834 + 0.510574i \(0.829433\pi\)
\(410\) 78736.0 0.0231320
\(411\) 1.74361e6 0.509149
\(412\) −2.90343e6 −0.842692
\(413\) 1.35583e6 0.391138
\(414\) −180504. −0.0517590
\(415\) −5.50738e6 −1.56973
\(416\) −1.87533e6 −0.531305
\(417\) −2.20416e6 −0.620730
\(418\) 319440. 0.0894229
\(419\) −4.38823e6 −1.22111 −0.610554 0.791974i \(-0.709053\pi\)
−0.610554 + 0.791974i \(0.709053\pi\)
\(420\) 609168. 0.168505
\(421\) −1.48456e6 −0.408218 −0.204109 0.978948i \(-0.565430\pi\)
−0.204109 + 0.978948i \(0.565430\pi\)
\(422\) −1.45754e6 −0.398419
\(423\) 2.42645e6 0.659358
\(424\) 632880. 0.170965
\(425\) 347948. 0.0934420
\(426\) −44496.0 −0.0118795
\(427\) −956088. −0.253763
\(428\) 2.71858e6 0.717352
\(429\) −264264. −0.0693258
\(430\) −3.06005e6 −0.798100
\(431\) −206448. −0.0535325 −0.0267662 0.999642i \(-0.508521\pi\)
−0.0267662 + 0.999642i \(0.508521\pi\)
\(432\) 1.77120e6 0.456623
\(433\) −5.67867e6 −1.45555 −0.727774 0.685817i \(-0.759445\pi\)
−0.727774 + 0.685817i \(0.759445\pi\)
\(434\) 866516. 0.220827
\(435\) 1.31868e6 0.334131
\(436\) −1.48204e6 −0.373374
\(437\) 575520. 0.144164
\(438\) −291552. −0.0726157
\(439\) −4.43666e6 −1.09874 −0.549370 0.835579i \(-0.685132\pi\)
−0.549370 + 0.835579i \(0.685132\pi\)
\(440\) −1.07448e6 −0.264586
\(441\) −497007. −0.121693
\(442\) −107744. −0.0262324
\(443\) 6.17328e6 1.49454 0.747269 0.664522i \(-0.231365\pi\)
0.747269 + 0.664522i \(0.231365\pi\)
\(444\) 23184.0 0.00558124
\(445\) −2.52858e6 −0.605308
\(446\) −76748.0 −0.0182696
\(447\) 2.77638e6 0.657219
\(448\) 523712. 0.123281
\(449\) −4.93105e6 −1.15431 −0.577156 0.816634i \(-0.695838\pi\)
−0.577156 + 0.816634i \(0.695838\pi\)
\(450\) 973314. 0.226580
\(451\) 64372.0 0.0149024
\(452\) 2.25473e6 0.519096
\(453\) 45888.0 0.0105064
\(454\) −646536. −0.147215
\(455\) 1.31986e6 0.298883
\(456\) 950400. 0.214040
\(457\) −4.15030e6 −0.929585 −0.464793 0.885420i \(-0.653871\pi\)
−0.464793 + 0.885420i \(0.653871\pi\)
\(458\) −1.62738e6 −0.362514
\(459\) 399600. 0.0885307
\(460\) −903392. −0.199059
\(461\) −4.40345e6 −0.965029 −0.482515 0.875888i \(-0.660277\pi\)
−0.482515 + 0.875888i \(0.660277\pi\)
\(462\) −71148.0 −0.0155081
\(463\) 6.82728e6 1.48012 0.740058 0.672544i \(-0.234798\pi\)
0.740058 + 0.672544i \(0.234798\pi\)
\(464\) 1.94832e6 0.420112
\(465\) 3.92585e6 0.841979
\(466\) −2.21603e6 −0.472727
\(467\) 6.88244e6 1.46033 0.730163 0.683272i \(-0.239444\pi\)
0.730163 + 0.683272i \(0.239444\pi\)
\(468\) 2.10974e6 0.445261
\(469\) −3.14031e6 −0.659236
\(470\) −1.73486e6 −0.362259
\(471\) 968892. 0.201244
\(472\) −3.32040e6 −0.686018
\(473\) −2.50180e6 −0.514161
\(474\) −26400.0 −0.00539707
\(475\) −3.10332e6 −0.631092
\(476\) 203056. 0.0410770
\(477\) −1.09172e6 −0.219692
\(478\) 2.62704e6 0.525892
\(479\) 5.02430e6 1.00055 0.500273 0.865868i \(-0.333233\pi\)
0.500273 + 0.865868i \(0.333233\pi\)
\(480\) −2.28749e6 −0.453164
\(481\) 50232.0 0.00989960
\(482\) 2.11214e6 0.414099
\(483\) −128184. −0.0250015
\(484\) −409948. −0.0795455
\(485\) −1.12271e7 −2.16728
\(486\) 1.72141e6 0.330593
\(487\) −4.51601e6 −0.862845 −0.431422 0.902150i \(-0.641988\pi\)
−0.431422 + 0.902150i \(0.641988\pi\)
\(488\) 2.34144e6 0.445075
\(489\) −1.07678e6 −0.203637
\(490\) 355348. 0.0668596
\(491\) 5.55737e6 1.04032 0.520159 0.854070i \(-0.325873\pi\)
0.520159 + 0.854070i \(0.325873\pi\)
\(492\) 89376.0 0.0166459
\(493\) 439560. 0.0814518
\(494\) 960960. 0.177169
\(495\) 1.85348e6 0.339996
\(496\) 5.80035e6 1.05864
\(497\) 181692. 0.0329947
\(498\) 893088. 0.161369
\(499\) −3.49744e6 −0.628780 −0.314390 0.949294i \(-0.601800\pi\)
−0.314390 + 0.949294i \(0.601800\pi\)
\(500\) −1.60373e6 −0.286884
\(501\) −1.90109e6 −0.338383
\(502\) −3.37828e6 −0.598325
\(503\) 4.61280e6 0.812915 0.406457 0.913670i \(-0.366764\pi\)
0.406457 + 0.913670i \(0.366764\pi\)
\(504\) 1.21716e6 0.213438
\(505\) −8.64705e6 −1.50883
\(506\) 105512. 0.0183200
\(507\) 1.43278e6 0.247548
\(508\) −6.18934e6 −1.06411
\(509\) 7.41609e6 1.26876 0.634382 0.773020i \(-0.281255\pi\)
0.634382 + 0.773020i \(0.281255\pi\)
\(510\) −131424. −0.0223743
\(511\) 1.19050e6 0.201687
\(512\) −5.89875e6 −0.994455
\(513\) −3.56400e6 −0.597922
\(514\) −1.28388e6 −0.214346
\(515\) −7.67336e6 −1.27487
\(516\) −3.47357e6 −0.574317
\(517\) −1.41836e6 −0.233378
\(518\) 13524.0 0.00221453
\(519\) 1.05070e6 0.171222
\(520\) −3.23232e6 −0.524211
\(521\) 9.75970e6 1.57522 0.787612 0.616172i \(-0.211317\pi\)
0.787612 + 0.616172i \(0.211317\pi\)
\(522\) 1.22958e6 0.197506
\(523\) 1.66084e6 0.265506 0.132753 0.991149i \(-0.457618\pi\)
0.132753 + 0.991149i \(0.457618\pi\)
\(524\) −1.05202e6 −0.167376
\(525\) 691194. 0.109446
\(526\) 2.20299e6 0.347175
\(527\) 1.30862e6 0.205251
\(528\) −476256. −0.0743457
\(529\) −6.24625e6 −0.970465
\(530\) 780552. 0.120701
\(531\) 5.72769e6 0.881542
\(532\) −1.81104e6 −0.277427
\(533\) 193648. 0.0295253
\(534\) 410040. 0.0622262
\(535\) 7.18481e6 1.08525
\(536\) 7.69056e6 1.15623
\(537\) 418680. 0.0626537
\(538\) 4.30294e6 0.640929
\(539\) 290521. 0.0430730
\(540\) 5.59440e6 0.825599
\(541\) 6.97250e6 1.02423 0.512113 0.858918i \(-0.328863\pi\)
0.512113 + 0.858918i \(0.328863\pi\)
\(542\) 2.16654e6 0.316787
\(543\) 471828. 0.0686727
\(544\) −762496. −0.110469
\(545\) −3.91682e6 −0.564862
\(546\) −214032. −0.0307254
\(547\) 3.08503e6 0.440850 0.220425 0.975404i \(-0.429256\pi\)
0.220425 + 0.975404i \(0.429256\pi\)
\(548\) 8.13686e6 1.15746
\(549\) −4.03898e6 −0.571928
\(550\) −568942. −0.0801976
\(551\) −3.92040e6 −0.550112
\(552\) 313920. 0.0438502
\(553\) 107800. 0.0149901
\(554\) 4.44744e6 0.615653
\(555\) 61272.0 0.00844364
\(556\) −1.02861e7 −1.41112
\(557\) −3.29052e6 −0.449394 −0.224697 0.974429i \(-0.572139\pi\)
−0.224697 + 0.974429i \(0.572139\pi\)
\(558\) 3.66059e6 0.497697
\(559\) −7.52606e6 −1.01868
\(560\) 2.37866e6 0.320525
\(561\) −107448. −0.0144142
\(562\) 306036. 0.0408725
\(563\) 5.45754e6 0.725648 0.362824 0.931858i \(-0.381813\pi\)
0.362824 + 0.931858i \(0.381813\pi\)
\(564\) −1.96930e6 −0.260683
\(565\) 5.95892e6 0.785320
\(566\) −1.43065e6 −0.187712
\(567\) −1.67095e6 −0.218276
\(568\) −444960. −0.0578696
\(569\) 7.28571e6 0.943390 0.471695 0.881762i \(-0.343642\pi\)
0.471695 + 0.881762i \(0.343642\pi\)
\(570\) 1.17216e6 0.151112
\(571\) 1.07129e7 1.37504 0.687519 0.726166i \(-0.258700\pi\)
0.687519 + 0.726166i \(0.258700\pi\)
\(572\) −1.23323e6 −0.157599
\(573\) 5.56325e6 0.707851
\(574\) 52136.0 0.00660477
\(575\) −1.02504e6 −0.129291
\(576\) 2.21242e6 0.277850
\(577\) 4.22024e6 0.527713 0.263856 0.964562i \(-0.415006\pi\)
0.263856 + 0.964562i \(0.415006\pi\)
\(578\) 2.79591e6 0.348099
\(579\) −5.26484e6 −0.652663
\(580\) 6.15384e6 0.759585
\(581\) −3.64678e6 −0.448197
\(582\) 1.82062e6 0.222798
\(583\) 638154. 0.0777596
\(584\) −2.91552e6 −0.353740
\(585\) 5.57575e6 0.673618
\(586\) −694848. −0.0835884
\(587\) −6.24180e6 −0.747678 −0.373839 0.927494i \(-0.621959\pi\)
−0.373839 + 0.927494i \(0.621959\pi\)
\(588\) 403368. 0.0481125
\(589\) −1.16714e7 −1.38623
\(590\) −4.09516e6 −0.484329
\(591\) 4.46761e6 0.526147
\(592\) 90528.0 0.0106164
\(593\) 493664. 0.0576494 0.0288247 0.999584i \(-0.490824\pi\)
0.0288247 + 0.999584i \(0.490824\pi\)
\(594\) −653400. −0.0759824
\(595\) 536648. 0.0621437
\(596\) 1.29564e7 1.49407
\(597\) −6.47586e6 −0.743638
\(598\) 317408. 0.0362965
\(599\) −8.28890e6 −0.943908 −0.471954 0.881623i \(-0.656451\pi\)
−0.471954 + 0.881623i \(0.656451\pi\)
\(600\) −1.69272e6 −0.191958
\(601\) 80612.0 0.00910361 0.00455180 0.999990i \(-0.498551\pi\)
0.00455180 + 0.999990i \(0.498551\pi\)
\(602\) −2.02625e6 −0.227877
\(603\) −1.32662e7 −1.48578
\(604\) 214144. 0.0238844
\(605\) −1.08343e6 −0.120341
\(606\) 1.40222e6 0.155109
\(607\) 914168. 0.100706 0.0503529 0.998731i \(-0.483965\pi\)
0.0503529 + 0.998731i \(0.483965\pi\)
\(608\) 6.80064e6 0.746089
\(609\) 873180. 0.0954027
\(610\) 2.88778e6 0.314224
\(611\) −4.26681e6 −0.462381
\(612\) 857808. 0.0925788
\(613\) 9.97327e6 1.07198 0.535990 0.844224i \(-0.319939\pi\)
0.535990 + 0.844224i \(0.319939\pi\)
\(614\) −5.28086e6 −0.565306
\(615\) 236208. 0.0251830
\(616\) −711480. −0.0755459
\(617\) −4.60636e6 −0.487130 −0.243565 0.969885i \(-0.578317\pi\)
−0.243565 + 0.969885i \(0.578317\pi\)
\(618\) 1.24433e6 0.131058
\(619\) 2.51711e6 0.264044 0.132022 0.991247i \(-0.457853\pi\)
0.132022 + 0.991247i \(0.457853\pi\)
\(620\) 1.83206e7 1.91408
\(621\) −1.17720e6 −0.122496
\(622\) 1.89556e6 0.196454
\(623\) −1.67433e6 −0.172831
\(624\) −1.43270e6 −0.147297
\(625\) −1.15853e7 −1.18633
\(626\) 497372. 0.0507277
\(627\) 958320. 0.0973513
\(628\) 4.52150e6 0.457492
\(629\) 20424.0 0.00205833
\(630\) 1.50116e6 0.150687
\(631\) −2.02153e6 −0.202119 −0.101059 0.994880i \(-0.532223\pi\)
−0.101059 + 0.994880i \(0.532223\pi\)
\(632\) −264000. −0.0262912
\(633\) −4.37263e6 −0.433744
\(634\) −5.21808e6 −0.515570
\(635\) −1.63576e7 −1.60984
\(636\) 886032. 0.0868574
\(637\) 873964. 0.0853385
\(638\) −718740. −0.0699069
\(639\) 767556. 0.0743632
\(640\) −1.37818e7 −1.33001
\(641\) 1.55870e7 1.49837 0.749183 0.662363i \(-0.230446\pi\)
0.749183 + 0.662363i \(0.230446\pi\)
\(642\) −1.16510e6 −0.111565
\(643\) −5.88755e6 −0.561574 −0.280787 0.959770i \(-0.590595\pi\)
−0.280787 + 0.959770i \(0.590595\pi\)
\(644\) −598192. −0.0568363
\(645\) −9.18014e6 −0.868861
\(646\) 390720. 0.0368370
\(647\) 5.58958e6 0.524950 0.262475 0.964939i \(-0.415461\pi\)
0.262475 + 0.964939i \(0.415461\pi\)
\(648\) 4.09212e6 0.382834
\(649\) −3.34807e6 −0.312020
\(650\) −1.71153e6 −0.158891
\(651\) 2.59955e6 0.240406
\(652\) −5.02499e6 −0.462931
\(653\) 1.76227e7 1.61730 0.808648 0.588293i \(-0.200200\pi\)
0.808648 + 0.588293i \(0.200200\pi\)
\(654\) 635160. 0.0580683
\(655\) −2.78033e6 −0.253217
\(656\) 348992. 0.0316633
\(657\) 5.02927e6 0.454561
\(658\) −1.14876e6 −0.103434
\(659\) −9.75566e6 −0.875071 −0.437535 0.899201i \(-0.644149\pi\)
−0.437535 + 0.899201i \(0.644149\pi\)
\(660\) −1.50427e6 −0.134421
\(661\) −9.79522e6 −0.871988 −0.435994 0.899950i \(-0.643603\pi\)
−0.435994 + 0.899950i \(0.643603\pi\)
\(662\) −304664. −0.0270194
\(663\) −323232. −0.0285582
\(664\) 8.93088e6 0.786093
\(665\) −4.78632e6 −0.419708
\(666\) 57132.0 0.00499107
\(667\) −1.29492e6 −0.112701
\(668\) −8.87174e6 −0.769251
\(669\) −230244. −0.0198895
\(670\) 9.48502e6 0.816303
\(671\) 2.36095e6 0.202433
\(672\) −1.51469e6 −0.129390
\(673\) −4.72727e6 −0.402321 −0.201160 0.979558i \(-0.564471\pi\)
−0.201160 + 0.979558i \(0.564471\pi\)
\(674\) −413116. −0.0350286
\(675\) 6.34770e6 0.536237
\(676\) 6.68632e6 0.562756
\(677\) −1.93989e7 −1.62669 −0.813344 0.581783i \(-0.802355\pi\)
−0.813344 + 0.581783i \(0.802355\pi\)
\(678\) −966312. −0.0807315
\(679\) −7.43418e6 −0.618812
\(680\) −1.31424e6 −0.108994
\(681\) −1.93961e6 −0.160268
\(682\) −2.13976e6 −0.176159
\(683\) −6.22004e6 −0.510201 −0.255100 0.966915i \(-0.582109\pi\)
−0.255100 + 0.966915i \(0.582109\pi\)
\(684\) −7.65072e6 −0.625262
\(685\) 2.15045e7 1.75107
\(686\) 235298. 0.0190901
\(687\) −4.88214e6 −0.394656
\(688\) −1.35635e7 −1.09244
\(689\) 1.91974e6 0.154061
\(690\) 387168. 0.0309583
\(691\) 6.04140e6 0.481330 0.240665 0.970608i \(-0.422635\pi\)
0.240665 + 0.970608i \(0.422635\pi\)
\(692\) 4.90325e6 0.389241
\(693\) 1.22730e6 0.0970775
\(694\) −4.03614e6 −0.318103
\(695\) −2.71846e7 −2.13482
\(696\) −2.13840e6 −0.167327
\(697\) 78736.0 0.00613891
\(698\) 1.16088e6 0.0901880
\(699\) −6.64808e6 −0.514640
\(700\) 3.22557e6 0.248806
\(701\) 3.97068e6 0.305190 0.152595 0.988289i \(-0.451237\pi\)
0.152595 + 0.988289i \(0.451237\pi\)
\(702\) −1.96560e6 −0.150540
\(703\) −182160. −0.0139016
\(704\) −1.29325e6 −0.0983445
\(705\) −5.20457e6 −0.394377
\(706\) −1.14407e6 −0.0863853
\(707\) −5.72575e6 −0.430808
\(708\) −4.64856e6 −0.348526
\(709\) 5.15939e6 0.385463 0.192732 0.981252i \(-0.438265\pi\)
0.192732 + 0.981252i \(0.438265\pi\)
\(710\) −548784. −0.0408560
\(711\) 455400. 0.0337846
\(712\) 4.10040e6 0.303128
\(713\) −3.85511e6 −0.283997
\(714\) −87024.0 −0.00638842
\(715\) −3.25926e6 −0.238426
\(716\) 1.95384e6 0.142432
\(717\) 7.88112e6 0.572519
\(718\) −9.13088e6 −0.661000
\(719\) −1.26734e7 −0.914259 −0.457129 0.889400i \(-0.651122\pi\)
−0.457129 + 0.889400i \(0.651122\pi\)
\(720\) 1.00486e7 0.722395
\(721\) −5.08101e6 −0.364009
\(722\) 1.46740e6 0.104762
\(723\) 6.33641e6 0.450814
\(724\) 2.20186e6 0.156115
\(725\) 6.98247e6 0.493360
\(726\) 175692. 0.0123712
\(727\) 1.32783e7 0.931762 0.465881 0.884847i \(-0.345737\pi\)
0.465881 + 0.884847i \(0.345737\pi\)
\(728\) −2.14032e6 −0.149675
\(729\) −3.12231e6 −0.217599
\(730\) −3.59581e6 −0.249741
\(731\) −3.06005e6 −0.211804
\(732\) 3.27802e6 0.226117
\(733\) 3.84050e6 0.264015 0.132007 0.991249i \(-0.457858\pi\)
0.132007 + 0.991249i \(0.457858\pi\)
\(734\) −1.84872e6 −0.126657
\(735\) 1.06604e6 0.0727875
\(736\) 2.24627e6 0.152851
\(737\) 7.75465e6 0.525888
\(738\) 220248. 0.0148858
\(739\) 1.19394e7 0.804215 0.402107 0.915592i \(-0.368278\pi\)
0.402107 + 0.915592i \(0.368278\pi\)
\(740\) 285936. 0.0191951
\(741\) 2.88288e6 0.192877
\(742\) 516852. 0.0344633
\(743\) 2.53631e7 1.68551 0.842754 0.538298i \(-0.180933\pi\)
0.842754 + 0.538298i \(0.180933\pi\)
\(744\) −6.36624e6 −0.421649
\(745\) 3.42420e7 2.26031
\(746\) −9.84043e6 −0.647391
\(747\) −1.54058e7 −1.01014
\(748\) −501424. −0.0327681
\(749\) 4.75751e6 0.309867
\(750\) 687312. 0.0446170
\(751\) 1.43903e7 0.931046 0.465523 0.885036i \(-0.345866\pi\)
0.465523 + 0.885036i \(0.345866\pi\)
\(752\) −7.68963e6 −0.495862
\(753\) −1.01349e7 −0.651373
\(754\) −2.16216e6 −0.138503
\(755\) 565952. 0.0361337
\(756\) 3.70440e6 0.235729
\(757\) −1.85431e7 −1.17609 −0.588047 0.808827i \(-0.700103\pi\)
−0.588047 + 0.808827i \(0.700103\pi\)
\(758\) −7.95080e6 −0.502618
\(759\) 316536. 0.0199443
\(760\) 1.17216e7 0.736127
\(761\) 6.43123e6 0.402562 0.201281 0.979534i \(-0.435490\pi\)
0.201281 + 0.979534i \(0.435490\pi\)
\(762\) 2.65258e6 0.165493
\(763\) −2.59357e6 −0.161282
\(764\) 2.59618e7 1.60917
\(765\) 2.26706e6 0.140059
\(766\) 1.96569e6 0.121044
\(767\) −1.00719e7 −0.618190
\(768\) 182784. 0.0111824
\(769\) 1.48249e7 0.904014 0.452007 0.892014i \(-0.350708\pi\)
0.452007 + 0.892014i \(0.350708\pi\)
\(770\) −877492. −0.0533355
\(771\) −3.85163e6 −0.233350
\(772\) −2.45693e7 −1.48371
\(773\) 1.96325e7 1.18175 0.590876 0.806762i \(-0.298782\pi\)
0.590876 + 0.806762i \(0.298782\pi\)
\(774\) −8.55986e6 −0.513588
\(775\) 2.07875e7 1.24322
\(776\) 1.82062e7 1.08534
\(777\) 40572.0 0.00241087
\(778\) 1.48818e6 0.0881468
\(779\) −702240. −0.0414612
\(780\) −4.52525e6 −0.266321
\(781\) −448668. −0.0263207
\(782\) 129056. 0.00754677
\(783\) 8.01900e6 0.467429
\(784\) 1.57506e6 0.0915179
\(785\) 1.19497e7 0.692120
\(786\) 450864. 0.0260309
\(787\) −9.48639e6 −0.545964 −0.272982 0.962019i \(-0.588010\pi\)
−0.272982 + 0.962019i \(0.588010\pi\)
\(788\) 2.08489e7 1.19610
\(789\) 6.60898e6 0.377956
\(790\) −325600. −0.0185617
\(791\) 3.94577e6 0.224229
\(792\) −3.00564e6 −0.170264
\(793\) 7.10237e6 0.401070
\(794\) −1.14605e7 −0.645136
\(795\) 2.34166e6 0.131403
\(796\) −3.02207e7 −1.69052
\(797\) 3.32326e7 1.85318 0.926592 0.376068i \(-0.122724\pi\)
0.926592 + 0.376068i \(0.122724\pi\)
\(798\) 776160. 0.0431463
\(799\) −1.73486e6 −0.0961383
\(800\) −1.21124e7 −0.669119
\(801\) −7.07319e6 −0.389524
\(802\) −8.53512e6 −0.468569
\(803\) −2.93982e6 −0.160891
\(804\) 1.07668e7 0.587416
\(805\) −1.58094e6 −0.0859854
\(806\) −6.43698e6 −0.349015
\(807\) 1.29088e7 0.697755
\(808\) 1.40222e7 0.755595
\(809\) −2.75792e7 −1.48153 −0.740764 0.671766i \(-0.765536\pi\)
−0.740764 + 0.671766i \(0.765536\pi\)
\(810\) 5.04695e6 0.270281
\(811\) −7.76107e6 −0.414352 −0.207176 0.978304i \(-0.566427\pi\)
−0.207176 + 0.978304i \(0.566427\pi\)
\(812\) 4.07484e6 0.216880
\(813\) 6.49961e6 0.344874
\(814\) −33396.0 −0.00176658
\(815\) −1.32803e7 −0.700350
\(816\) −582528. −0.0306261
\(817\) 2.72923e7 1.43049
\(818\) 1.16354e7 0.607994
\(819\) 3.69205e6 0.192335
\(820\) 1.10230e6 0.0572488
\(821\) −1.60578e7 −0.831434 −0.415717 0.909494i \(-0.636469\pi\)
−0.415717 + 0.909494i \(0.636469\pi\)
\(822\) −3.48722e6 −0.180012
\(823\) 1.04666e6 0.0538651 0.0269326 0.999637i \(-0.491426\pi\)
0.0269326 + 0.999637i \(0.491426\pi\)
\(824\) 1.24433e7 0.638435
\(825\) −1.70683e6 −0.0873081
\(826\) −2.71166e6 −0.138288
\(827\) 8.91799e6 0.453423 0.226711 0.973962i \(-0.427203\pi\)
0.226711 + 0.973962i \(0.427203\pi\)
\(828\) −2.52706e6 −0.128097
\(829\) −2.53821e7 −1.28275 −0.641374 0.767229i \(-0.721635\pi\)
−0.641374 + 0.767229i \(0.721635\pi\)
\(830\) 1.10148e7 0.554983
\(831\) 1.33423e7 0.670238
\(832\) −3.89043e6 −0.194845
\(833\) 355348. 0.0177436
\(834\) 4.40832e6 0.219461
\(835\) −2.34468e7 −1.16377
\(836\) 4.47216e6 0.221310
\(837\) 2.38734e7 1.17788
\(838\) 8.77646e6 0.431727
\(839\) −3.10636e7 −1.52351 −0.761757 0.647863i \(-0.775663\pi\)
−0.761757 + 0.647863i \(0.775663\pi\)
\(840\) −2.61072e6 −0.127662
\(841\) −1.16902e7 −0.569946
\(842\) 2.96912e6 0.144327
\(843\) 918108. 0.0444964
\(844\) −2.04056e7 −0.986038
\(845\) 1.76710e7 0.851371
\(846\) −4.85291e6 −0.233118
\(847\) −717409. −0.0343604
\(848\) 3.45974e6 0.165217
\(849\) −4.29194e6 −0.204355
\(850\) −695896. −0.0330367
\(851\) −60168.0 −0.00284801
\(852\) −622944. −0.0294002
\(853\) 2.24337e7 1.05567 0.527835 0.849347i \(-0.323004\pi\)
0.527835 + 0.849347i \(0.323004\pi\)
\(854\) 1.91218e6 0.0897187
\(855\) −2.02198e7 −0.945934
\(856\) −1.16510e7 −0.543476
\(857\) 4.76449e6 0.221597 0.110799 0.993843i \(-0.464659\pi\)
0.110799 + 0.993843i \(0.464659\pi\)
\(858\) 528528. 0.0245104
\(859\) 468030. 0.0216417 0.0108208 0.999941i \(-0.496556\pi\)
0.0108208 + 0.999941i \(0.496556\pi\)
\(860\) −4.28407e7 −1.97520
\(861\) 156408. 0.00719037
\(862\) 412896. 0.0189266
\(863\) −1.20487e7 −0.550697 −0.275349 0.961344i \(-0.588793\pi\)
−0.275349 + 0.961344i \(0.588793\pi\)
\(864\) −1.39104e7 −0.633950
\(865\) 1.29586e7 0.588867
\(866\) 1.13573e7 0.514614
\(867\) 8.38772e6 0.378962
\(868\) 1.21312e7 0.546519
\(869\) −266200. −0.0119580
\(870\) −2.63736e6 −0.118133
\(871\) 2.33280e7 1.04192
\(872\) 6.35160e6 0.282873
\(873\) −3.14056e7 −1.39467
\(874\) −1.15104e6 −0.0509697
\(875\) −2.80652e6 −0.123922
\(876\) −4.08173e6 −0.179715
\(877\) 3.61718e6 0.158807 0.0794037 0.996843i \(-0.474698\pi\)
0.0794037 + 0.996843i \(0.474698\pi\)
\(878\) 8.87332e6 0.388463
\(879\) −2.08454e6 −0.0909995
\(880\) −5.87382e6 −0.255690
\(881\) −2.27025e7 −0.985448 −0.492724 0.870186i \(-0.663999\pi\)
−0.492724 + 0.870186i \(0.663999\pi\)
\(882\) 994014. 0.0430250
\(883\) 2.41926e7 1.04419 0.522097 0.852886i \(-0.325150\pi\)
0.522097 + 0.852886i \(0.325150\pi\)
\(884\) −1.50842e6 −0.0649218
\(885\) −1.22855e7 −0.527271
\(886\) −1.23466e7 −0.528399
\(887\) −4.06125e7 −1.73321 −0.866604 0.498997i \(-0.833702\pi\)
−0.866604 + 0.498997i \(0.833702\pi\)
\(888\) −99360.0 −0.00422843
\(889\) −1.08314e7 −0.459651
\(890\) 5.05716e6 0.214009
\(891\) 4.12622e6 0.174124
\(892\) −1.07447e6 −0.0452150
\(893\) 1.54730e7 0.649302
\(894\) −5.55276e6 −0.232362
\(895\) 5.16372e6 0.215479
\(896\) −9.12576e6 −0.379751
\(897\) 952224. 0.0395146
\(898\) 9.86210e6 0.408111
\(899\) 2.62607e7 1.08370
\(900\) 1.36264e7 0.560757
\(901\) 780552. 0.0320324
\(902\) −128744. −0.00526879
\(903\) −6.07874e6 −0.248082
\(904\) −9.66312e6 −0.393275
\(905\) 5.81921e6 0.236180
\(906\) −91776.0 −0.00371457
\(907\) −1.98235e7 −0.800132 −0.400066 0.916486i \(-0.631013\pi\)
−0.400066 + 0.916486i \(0.631013\pi\)
\(908\) −9.05150e6 −0.364339
\(909\) −2.41884e7 −0.970950
\(910\) −2.63973e6 −0.105671
\(911\) −2.21209e7 −0.883094 −0.441547 0.897238i \(-0.645570\pi\)
−0.441547 + 0.897238i \(0.645570\pi\)
\(912\) 5.19552e6 0.206844
\(913\) 9.00530e6 0.357537
\(914\) 8.30060e6 0.328658
\(915\) 8.66333e6 0.342083
\(916\) −2.27833e7 −0.897177
\(917\) −1.84103e6 −0.0722998
\(918\) −799200. −0.0313003
\(919\) 4.42949e7 1.73008 0.865038 0.501706i \(-0.167294\pi\)
0.865038 + 0.501706i \(0.167294\pi\)
\(920\) 3.87168e6 0.150810
\(921\) −1.58426e7 −0.615427
\(922\) 8.80690e6 0.341189
\(923\) −1.34971e6 −0.0521479
\(924\) −996072. −0.0383805
\(925\) 324438. 0.0124674
\(926\) −1.36546e7 −0.523300
\(927\) −2.14647e7 −0.820398
\(928\) −1.53014e7 −0.583260
\(929\) 1.13166e7 0.430207 0.215104 0.976591i \(-0.430991\pi\)
0.215104 + 0.976591i \(0.430991\pi\)
\(930\) −7.85170e6 −0.297684
\(931\) −3.16932e6 −0.119837
\(932\) −3.10244e7 −1.16994
\(933\) 5.68667e6 0.213872
\(934\) −1.37649e7 −0.516304
\(935\) −1.32519e6 −0.0495735
\(936\) −9.04176e6 −0.337337
\(937\) 3.37578e7 1.25610 0.628052 0.778172i \(-0.283853\pi\)
0.628052 + 0.778172i \(0.283853\pi\)
\(938\) 6.28062e6 0.233075
\(939\) 1.49212e6 0.0552254
\(940\) −2.42880e7 −0.896544
\(941\) −3.30036e7 −1.21503 −0.607516 0.794307i \(-0.707834\pi\)
−0.607516 + 0.794307i \(0.707834\pi\)
\(942\) −1.93778e6 −0.0711505
\(943\) −231952. −0.00849413
\(944\) −1.81515e7 −0.662953
\(945\) 9.79020e6 0.356625
\(946\) 5.00359e6 0.181783
\(947\) 1.92599e7 0.697876 0.348938 0.937146i \(-0.386542\pi\)
0.348938 + 0.937146i \(0.386542\pi\)
\(948\) −369600. −0.0133571
\(949\) −8.84374e6 −0.318765
\(950\) 6.20664e6 0.223125
\(951\) −1.56542e7 −0.561281
\(952\) −870240. −0.0311205
\(953\) 1.18503e7 0.422665 0.211332 0.977414i \(-0.432220\pi\)
0.211332 + 0.977414i \(0.432220\pi\)
\(954\) 2.18344e6 0.0776729
\(955\) 6.86134e7 2.43445
\(956\) 3.67786e7 1.30152
\(957\) −2.15622e6 −0.0761050
\(958\) −1.00486e7 −0.353746
\(959\) 1.42395e7 0.499975
\(960\) −4.74547e6 −0.166189
\(961\) 4.95518e7 1.73082
\(962\) −100464. −0.00350004
\(963\) 2.00980e7 0.698374
\(964\) 2.95699e7 1.02484
\(965\) −6.49331e7 −2.24465
\(966\) 256368. 0.00883936
\(967\) −1.44196e7 −0.495892 −0.247946 0.968774i \(-0.579755\pi\)
−0.247946 + 0.968774i \(0.579755\pi\)
\(968\) 1.75692e6 0.0602648
\(969\) 1.17216e6 0.0401031
\(970\) 2.24543e7 0.766248
\(971\) −1.37494e7 −0.467990 −0.233995 0.972238i \(-0.575180\pi\)
−0.233995 + 0.972238i \(0.575180\pi\)
\(972\) 2.40998e7 0.818177
\(973\) −1.80006e7 −0.609545
\(974\) 9.03202e6 0.305062
\(975\) −5.13458e6 −0.172979
\(976\) 1.27999e7 0.430112
\(977\) −9.18802e6 −0.307954 −0.153977 0.988074i \(-0.549208\pi\)
−0.153977 + 0.988074i \(0.549208\pi\)
\(978\) 2.15357e6 0.0719965
\(979\) 4.13457e6 0.137871
\(980\) 4.97487e6 0.165469
\(981\) −1.09565e7 −0.363496
\(982\) −1.11147e7 −0.367808
\(983\) −3.43732e7 −1.13458 −0.567292 0.823517i \(-0.692009\pi\)
−0.567292 + 0.823517i \(0.692009\pi\)
\(984\) −383040. −0.0126112
\(985\) 5.51005e7 1.80953
\(986\) −879120. −0.0287976
\(987\) −3.44627e6 −0.112605
\(988\) 1.34534e7 0.438471
\(989\) 9.01474e6 0.293064
\(990\) −3.70696e6 −0.120207
\(991\) −4.32179e7 −1.39791 −0.698956 0.715164i \(-0.746352\pi\)
−0.698956 + 0.715164i \(0.746352\pi\)
\(992\) −4.55540e7 −1.46976
\(993\) −913992. −0.0294150
\(994\) −363384. −0.0116654
\(995\) −7.98689e7 −2.55753
\(996\) 1.25032e7 0.399369
\(997\) 2.54793e7 0.811801 0.405901 0.913917i \(-0.366958\pi\)
0.405901 + 0.913917i \(0.366958\pi\)
\(998\) 6.99488e6 0.222307
\(999\) 372600. 0.0118122
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 77.6.a.a.1.1 1
3.2 odd 2 693.6.a.a.1.1 1
7.6 odd 2 539.6.a.d.1.1 1
11.10 odd 2 847.6.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.6.a.a.1.1 1 1.1 even 1 trivial
539.6.a.d.1.1 1 7.6 odd 2
693.6.a.a.1.1 1 3.2 odd 2
847.6.a.a.1.1 1 11.10 odd 2