Properties

Label 539.6.a.e.1.2
Level $539$
Weight $6$
Character 539.1
Self dual yes
Analytic conductor $86.447$
Analytic rank $1$
Dimension $3$
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] = [539,6,Mod(1,539)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(539, 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("539.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 539.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: \(86.4468788792\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.54492.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 52x - 38 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(8.04796\) of defining polynomial
Character \(\chi\) \(=\) 539.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.20859 q^{2} -16.8394 q^{3} -27.1221 q^{4} -75.2230 q^{5} -37.1913 q^{6} -130.577 q^{8} +40.5643 q^{9} +O(q^{10})\) \(q+2.20859 q^{2} -16.8394 q^{3} -27.1221 q^{4} -75.2230 q^{5} -37.1913 q^{6} -130.577 q^{8} +40.5643 q^{9} -166.137 q^{10} +121.000 q^{11} +456.719 q^{12} -455.465 q^{13} +1266.71 q^{15} +579.518 q^{16} -190.657 q^{17} +89.5900 q^{18} +135.393 q^{19} +2040.21 q^{20} +267.240 q^{22} +2796.65 q^{23} +2198.83 q^{24} +2533.51 q^{25} -1005.94 q^{26} +3408.89 q^{27} -2608.58 q^{29} +2797.64 q^{30} +1056.76 q^{31} +5458.37 q^{32} -2037.56 q^{33} -421.082 q^{34} -1100.19 q^{36} +12536.8 q^{37} +299.028 q^{38} +7669.74 q^{39} +9822.37 q^{40} -1130.09 q^{41} -14671.0 q^{43} -3281.78 q^{44} -3051.37 q^{45} +6176.65 q^{46} +16882.2 q^{47} -9758.71 q^{48} +5595.48 q^{50} +3210.54 q^{51} +12353.2 q^{52} +3313.02 q^{53} +7528.84 q^{54} -9101.99 q^{55} -2279.93 q^{57} -5761.29 q^{58} -11454.0 q^{59} -34355.8 q^{60} +28227.5 q^{61} +2333.95 q^{62} -6489.25 q^{64} +34261.4 q^{65} -4500.15 q^{66} -51431.0 q^{67} +5171.01 q^{68} -47093.8 q^{69} -16218.0 q^{71} -5296.75 q^{72} +10168.8 q^{73} +27688.7 q^{74} -42662.6 q^{75} -3672.15 q^{76} +16939.3 q^{78} +60841.2 q^{79} -43593.1 q^{80} -67260.7 q^{81} -2495.90 q^{82} -45770.6 q^{83} +14341.8 q^{85} -32402.3 q^{86} +43926.9 q^{87} -15799.8 q^{88} +82267.9 q^{89} -6739.23 q^{90} -75851.0 q^{92} -17795.1 q^{93} +37285.8 q^{94} -10184.7 q^{95} -91915.5 q^{96} -53097.0 q^{97} +4908.28 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 34 q^{3} + 84 q^{4} - 24 q^{5} + 206 q^{6} - 564 q^{8} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 34 q^{3} + 84 q^{4} - 24 q^{5} + 206 q^{6} - 564 q^{8} - 7 q^{9} + 414 q^{10} + 363 q^{11} - 992 q^{12} - 486 q^{13} + 1654 q^{15} + 1992 q^{16} - 1086 q^{17} - 3706 q^{18} - 1380 q^{19} + 3480 q^{20} - 3066 q^{23} + 11748 q^{24} - 57 q^{25} - 12132 q^{26} + 2990 q^{27} - 3426 q^{29} + 2650 q^{30} + 4098 q^{31} - 12408 q^{32} - 4114 q^{33} - 25320 q^{34} + 4756 q^{36} + 17724 q^{37} + 9240 q^{38} - 6560 q^{39} + 15276 q^{40} - 5994 q^{41} - 26208 q^{43} + 10164 q^{44} - 18458 q^{45} - 16806 q^{46} + 17232 q^{47} - 61064 q^{48} + 41070 q^{50} - 22724 q^{51} + 35304 q^{52} + 50586 q^{53} - 18814 q^{54} - 2904 q^{55} + 20160 q^{57} - 29172 q^{58} + 3738 q^{59} - 13456 q^{60} - 18486 q^{61} + 19974 q^{62} - 20352 q^{64} - 7668 q^{65} + 24926 q^{66} - 47754 q^{67} + 12600 q^{68} - 35042 q^{69} + 39282 q^{71} - 95040 q^{72} - 15426 q^{73} + 153294 q^{74} + 21916 q^{75} - 103920 q^{76} + 124984 q^{78} + 125148 q^{79} - 118680 q^{80} - 86917 q^{81} + 255372 q^{82} + 143928 q^{83} - 104040 q^{85} + 243060 q^{86} + 19368 q^{87} - 68244 q^{88} + 106824 q^{89} - 103424 q^{90} - 336528 q^{92} - 16622 q^{93} + 74928 q^{94} - 22200 q^{95} + 76456 q^{96} - 9684 q^{97} - 847 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.20859 0.390428 0.195214 0.980761i \(-0.437460\pi\)
0.195214 + 0.980761i \(0.437460\pi\)
\(3\) −16.8394 −1.08025 −0.540123 0.841586i \(-0.681622\pi\)
−0.540123 + 0.841586i \(0.681622\pi\)
\(4\) −27.1221 −0.847566
\(5\) −75.2230 −1.34563 −0.672815 0.739810i \(-0.734915\pi\)
−0.672815 + 0.739810i \(0.734915\pi\)
\(6\) −37.1913 −0.421758
\(7\) 0 0
\(8\) −130.577 −0.721341
\(9\) 40.5643 0.166931
\(10\) −166.137 −0.525371
\(11\) 121.000 0.301511
\(12\) 456.719 0.915580
\(13\) −455.465 −0.747474 −0.373737 0.927535i \(-0.621924\pi\)
−0.373737 + 0.927535i \(0.621924\pi\)
\(14\) 0 0
\(15\) 1266.71 1.45361
\(16\) 579.518 0.565935
\(17\) −190.657 −0.160003 −0.0800017 0.996795i \(-0.525493\pi\)
−0.0800017 + 0.996795i \(0.525493\pi\)
\(18\) 89.5900 0.0651746
\(19\) 135.393 0.0860424 0.0430212 0.999074i \(-0.486302\pi\)
0.0430212 + 0.999074i \(0.486302\pi\)
\(20\) 2040.21 1.14051
\(21\) 0 0
\(22\) 267.240 0.117718
\(23\) 2796.65 1.10235 0.551173 0.834391i \(-0.314180\pi\)
0.551173 + 0.834391i \(0.314180\pi\)
\(24\) 2198.83 0.779225
\(25\) 2533.51 0.810722
\(26\) −1005.94 −0.291835
\(27\) 3408.89 0.899919
\(28\) 0 0
\(29\) −2608.58 −0.575983 −0.287991 0.957633i \(-0.592987\pi\)
−0.287991 + 0.957633i \(0.592987\pi\)
\(30\) 2797.64 0.567530
\(31\) 1056.76 0.197502 0.0987510 0.995112i \(-0.468515\pi\)
0.0987510 + 0.995112i \(0.468515\pi\)
\(32\) 5458.37 0.942297
\(33\) −2037.56 −0.325706
\(34\) −421.082 −0.0624697
\(35\) 0 0
\(36\) −1100.19 −0.141485
\(37\) 12536.8 1.50550 0.752752 0.658304i \(-0.228726\pi\)
0.752752 + 0.658304i \(0.228726\pi\)
\(38\) 299.028 0.0335933
\(39\) 7669.74 0.807456
\(40\) 9822.37 0.970658
\(41\) −1130.09 −0.104991 −0.0524954 0.998621i \(-0.516718\pi\)
−0.0524954 + 0.998621i \(0.516718\pi\)
\(42\) 0 0
\(43\) −14671.0 −1.21001 −0.605005 0.796222i \(-0.706829\pi\)
−0.605005 + 0.796222i \(0.706829\pi\)
\(44\) −3281.78 −0.255551
\(45\) −3051.37 −0.224628
\(46\) 6176.65 0.430386
\(47\) 16882.2 1.11477 0.557383 0.830256i \(-0.311806\pi\)
0.557383 + 0.830256i \(0.311806\pi\)
\(48\) −9758.71 −0.611349
\(49\) 0 0
\(50\) 5595.48 0.316528
\(51\) 3210.54 0.172843
\(52\) 12353.2 0.633534
\(53\) 3313.02 0.162007 0.0810035 0.996714i \(-0.474187\pi\)
0.0810035 + 0.996714i \(0.474187\pi\)
\(54\) 7528.84 0.351353
\(55\) −9101.99 −0.405723
\(56\) 0 0
\(57\) −2279.93 −0.0929469
\(58\) −5761.29 −0.224880
\(59\) −11454.0 −0.428378 −0.214189 0.976792i \(-0.568711\pi\)
−0.214189 + 0.976792i \(0.568711\pi\)
\(60\) −34355.8 −1.23203
\(61\) 28227.5 0.971286 0.485643 0.874157i \(-0.338585\pi\)
0.485643 + 0.874157i \(0.338585\pi\)
\(62\) 2333.95 0.0771102
\(63\) 0 0
\(64\) −6489.25 −0.198036
\(65\) 34261.4 1.00582
\(66\) −4500.15 −0.127165
\(67\) −51431.0 −1.39971 −0.699855 0.714285i \(-0.746752\pi\)
−0.699855 + 0.714285i \(0.746752\pi\)
\(68\) 5171.01 0.135614
\(69\) −47093.8 −1.19081
\(70\) 0 0
\(71\) −16218.0 −0.381814 −0.190907 0.981608i \(-0.561143\pi\)
−0.190907 + 0.981608i \(0.561143\pi\)
\(72\) −5296.75 −0.120414
\(73\) 10168.8 0.223337 0.111669 0.993745i \(-0.464380\pi\)
0.111669 + 0.993745i \(0.464380\pi\)
\(74\) 27688.7 0.587791
\(75\) −42662.6 −0.875779
\(76\) −3672.15 −0.0729266
\(77\) 0 0
\(78\) 16939.3 0.315253
\(79\) 60841.2 1.09681 0.548404 0.836214i \(-0.315236\pi\)
0.548404 + 0.836214i \(0.315236\pi\)
\(80\) −43593.1 −0.761540
\(81\) −67260.7 −1.13907
\(82\) −2495.90 −0.0409913
\(83\) −45770.6 −0.729275 −0.364638 0.931150i \(-0.618807\pi\)
−0.364638 + 0.931150i \(0.618807\pi\)
\(84\) 0 0
\(85\) 14341.8 0.215306
\(86\) −32402.3 −0.472421
\(87\) 43926.9 0.622203
\(88\) −15799.8 −0.217492
\(89\) 82267.9 1.10092 0.550460 0.834862i \(-0.314452\pi\)
0.550460 + 0.834862i \(0.314452\pi\)
\(90\) −6739.23 −0.0877009
\(91\) 0 0
\(92\) −75851.0 −0.934312
\(93\) −17795.1 −0.213351
\(94\) 37285.8 0.435235
\(95\) −10184.7 −0.115781
\(96\) −91915.5 −1.01791
\(97\) −53097.0 −0.572981 −0.286491 0.958083i \(-0.592489\pi\)
−0.286491 + 0.958083i \(0.592489\pi\)
\(98\) 0 0
\(99\) 4908.28 0.0503317
\(100\) −68714.1 −0.687141
\(101\) −186821. −1.82231 −0.911153 0.412069i \(-0.864806\pi\)
−0.911153 + 0.412069i \(0.864806\pi\)
\(102\) 7090.76 0.0674827
\(103\) −34290.5 −0.318479 −0.159240 0.987240i \(-0.550904\pi\)
−0.159240 + 0.987240i \(0.550904\pi\)
\(104\) 59473.0 0.539184
\(105\) 0 0
\(106\) 7317.10 0.0632520
\(107\) −224117. −1.89241 −0.946206 0.323565i \(-0.895119\pi\)
−0.946206 + 0.323565i \(0.895119\pi\)
\(108\) −92456.3 −0.762741
\(109\) 162229. 1.30786 0.653931 0.756554i \(-0.273118\pi\)
0.653931 + 0.756554i \(0.273118\pi\)
\(110\) −20102.6 −0.158405
\(111\) −211112. −1.62632
\(112\) 0 0
\(113\) 92225.0 0.679442 0.339721 0.940526i \(-0.389667\pi\)
0.339721 + 0.940526i \(0.389667\pi\)
\(114\) −5035.44 −0.0362890
\(115\) −210372. −1.48335
\(116\) 70750.3 0.488184
\(117\) −18475.6 −0.124777
\(118\) −25297.2 −0.167251
\(119\) 0 0
\(120\) −165403. −1.04855
\(121\) 14641.0 0.0909091
\(122\) 62342.9 0.379217
\(123\) 19029.9 0.113416
\(124\) −28661.5 −0.167396
\(125\) 44493.9 0.254698
\(126\) 0 0
\(127\) 138299. 0.760868 0.380434 0.924808i \(-0.375775\pi\)
0.380434 + 0.924808i \(0.375775\pi\)
\(128\) −189000. −1.01962
\(129\) 247051. 1.30711
\(130\) 75669.5 0.392702
\(131\) 54420.4 0.277066 0.138533 0.990358i \(-0.455761\pi\)
0.138533 + 0.990358i \(0.455761\pi\)
\(132\) 55263.0 0.276058
\(133\) 0 0
\(134\) −113590. −0.546485
\(135\) −256427. −1.21096
\(136\) 24895.3 0.115417
\(137\) 40555.1 0.184605 0.0923025 0.995731i \(-0.470577\pi\)
0.0923025 + 0.995731i \(0.470577\pi\)
\(138\) −104011. −0.464923
\(139\) −140537. −0.616955 −0.308477 0.951232i \(-0.599819\pi\)
−0.308477 + 0.951232i \(0.599819\pi\)
\(140\) 0 0
\(141\) −284285. −1.20422
\(142\) −35818.9 −0.149071
\(143\) −55111.2 −0.225372
\(144\) 23507.7 0.0944723
\(145\) 196225. 0.775060
\(146\) 22458.7 0.0871971
\(147\) 0 0
\(148\) −340024. −1.27602
\(149\) 176073. 0.649722 0.324861 0.945762i \(-0.394683\pi\)
0.324861 + 0.945762i \(0.394683\pi\)
\(150\) −94224.4 −0.341928
\(151\) 409241. 1.46062 0.730309 0.683117i \(-0.239376\pi\)
0.730309 + 0.683117i \(0.239376\pi\)
\(152\) −17679.2 −0.0620659
\(153\) −7733.85 −0.0267096
\(154\) 0 0
\(155\) −79492.6 −0.265765
\(156\) −208020. −0.684373
\(157\) −14294.5 −0.0462829 −0.0231414 0.999732i \(-0.507367\pi\)
−0.0231414 + 0.999732i \(0.507367\pi\)
\(158\) 134373. 0.428224
\(159\) −55789.1 −0.175007
\(160\) −410595. −1.26798
\(161\) 0 0
\(162\) −148551. −0.444722
\(163\) −418474. −1.23367 −0.616836 0.787091i \(-0.711586\pi\)
−0.616836 + 0.787091i \(0.711586\pi\)
\(164\) 30650.3 0.0889868
\(165\) 153272. 0.438281
\(166\) −101089. −0.284729
\(167\) 139747. 0.387749 0.193875 0.981026i \(-0.437894\pi\)
0.193875 + 0.981026i \(0.437894\pi\)
\(168\) 0 0
\(169\) −163845. −0.441282
\(170\) 31675.1 0.0840612
\(171\) 5492.13 0.0143632
\(172\) 397909. 1.02556
\(173\) 687104. 1.74545 0.872725 0.488213i \(-0.162351\pi\)
0.872725 + 0.488213i \(0.162351\pi\)
\(174\) 97016.5 0.242925
\(175\) 0 0
\(176\) 70121.6 0.170636
\(177\) 192878. 0.462754
\(178\) 181696. 0.429829
\(179\) 35496.4 0.0828042 0.0414021 0.999143i \(-0.486818\pi\)
0.0414021 + 0.999143i \(0.486818\pi\)
\(180\) 82759.7 0.190387
\(181\) −260469. −0.590963 −0.295481 0.955349i \(-0.595480\pi\)
−0.295481 + 0.955349i \(0.595480\pi\)
\(182\) 0 0
\(183\) −475333. −1.04923
\(184\) −365177. −0.795167
\(185\) −943056. −2.02585
\(186\) −39302.2 −0.0832980
\(187\) −23069.4 −0.0482429
\(188\) −457880. −0.944837
\(189\) 0 0
\(190\) −22493.8 −0.0452042
\(191\) 392051. 0.777605 0.388803 0.921321i \(-0.372889\pi\)
0.388803 + 0.921321i \(0.372889\pi\)
\(192\) 109275. 0.213928
\(193\) 15776.8 0.0304878 0.0152439 0.999884i \(-0.495148\pi\)
0.0152439 + 0.999884i \(0.495148\pi\)
\(194\) −117270. −0.223708
\(195\) −576941. −1.08654
\(196\) 0 0
\(197\) 545551. 1.00154 0.500771 0.865580i \(-0.333050\pi\)
0.500771 + 0.865580i \(0.333050\pi\)
\(198\) 10840.4 0.0196509
\(199\) 546514. 0.978293 0.489146 0.872202i \(-0.337308\pi\)
0.489146 + 0.872202i \(0.337308\pi\)
\(200\) −330817. −0.584807
\(201\) 866065. 1.51203
\(202\) −412610. −0.711478
\(203\) 0 0
\(204\) −87076.5 −0.146496
\(205\) 85008.5 0.141279
\(206\) −75733.7 −0.124343
\(207\) 113444. 0.184016
\(208\) −263950. −0.423022
\(209\) 16382.6 0.0259428
\(210\) 0 0
\(211\) 537150. 0.830596 0.415298 0.909686i \(-0.363677\pi\)
0.415298 + 0.909686i \(0.363677\pi\)
\(212\) −89856.0 −0.137312
\(213\) 273101. 0.412453
\(214\) −494983. −0.738850
\(215\) 1.10360e6 1.62823
\(216\) −445121. −0.649148
\(217\) 0 0
\(218\) 358298. 0.510626
\(219\) −171236. −0.241259
\(220\) 246865. 0.343877
\(221\) 86837.3 0.119598
\(222\) −466259. −0.634958
\(223\) 189640. 0.255368 0.127684 0.991815i \(-0.459246\pi\)
0.127684 + 0.991815i \(0.459246\pi\)
\(224\) 0 0
\(225\) 102770. 0.135335
\(226\) 203687. 0.265273
\(227\) −363428. −0.468116 −0.234058 0.972223i \(-0.575201\pi\)
−0.234058 + 0.972223i \(0.575201\pi\)
\(228\) 61836.6 0.0787787
\(229\) −504331. −0.635516 −0.317758 0.948172i \(-0.602930\pi\)
−0.317758 + 0.948172i \(0.602930\pi\)
\(230\) −464627. −0.579141
\(231\) 0 0
\(232\) 340620. 0.415480
\(233\) −1.20159e6 −1.45000 −0.724999 0.688750i \(-0.758160\pi\)
−0.724999 + 0.688750i \(0.758160\pi\)
\(234\) −40805.1 −0.0487163
\(235\) −1.26993e6 −1.50006
\(236\) 310657. 0.363079
\(237\) −1.02453e6 −1.18482
\(238\) 0 0
\(239\) −185929. −0.210549 −0.105275 0.994443i \(-0.533572\pi\)
−0.105275 + 0.994443i \(0.533572\pi\)
\(240\) 734080. 0.822650
\(241\) −174842. −0.193911 −0.0969556 0.995289i \(-0.530910\pi\)
−0.0969556 + 0.995289i \(0.530910\pi\)
\(242\) 32336.0 0.0354934
\(243\) 304267. 0.330552
\(244\) −765589. −0.823229
\(245\) 0 0
\(246\) 42029.3 0.0442807
\(247\) −61666.8 −0.0643145
\(248\) −137988. −0.142466
\(249\) 770748. 0.787797
\(250\) 98269.0 0.0994412
\(251\) −447906. −0.448748 −0.224374 0.974503i \(-0.572034\pi\)
−0.224374 + 0.974503i \(0.572034\pi\)
\(252\) 0 0
\(253\) 338394. 0.332370
\(254\) 305446. 0.297064
\(255\) −241506. −0.232583
\(256\) −209768. −0.200050
\(257\) 1.14572e6 1.08204 0.541022 0.841009i \(-0.318038\pi\)
0.541022 + 0.841009i \(0.318038\pi\)
\(258\) 545634. 0.510331
\(259\) 0 0
\(260\) −929243. −0.852503
\(261\) −105815. −0.0961496
\(262\) 120192. 0.108174
\(263\) 443228. 0.395128 0.197564 0.980290i \(-0.436697\pi\)
0.197564 + 0.980290i \(0.436697\pi\)
\(264\) 266058. 0.234945
\(265\) −249215. −0.218002
\(266\) 0 0
\(267\) −1.38534e6 −1.18926
\(268\) 1.39492e6 1.18635
\(269\) 1.88722e6 1.59016 0.795082 0.606502i \(-0.207428\pi\)
0.795082 + 0.606502i \(0.207428\pi\)
\(270\) −566343. −0.472792
\(271\) −2.24203e6 −1.85446 −0.927230 0.374491i \(-0.877817\pi\)
−0.927230 + 0.374491i \(0.877817\pi\)
\(272\) −110489. −0.0905516
\(273\) 0 0
\(274\) 89569.6 0.0720749
\(275\) 306554. 0.244442
\(276\) 1.27728e6 1.00929
\(277\) 1.27824e6 1.00095 0.500474 0.865751i \(-0.333159\pi\)
0.500474 + 0.865751i \(0.333159\pi\)
\(278\) −310389. −0.240876
\(279\) 42866.7 0.0329693
\(280\) 0 0
\(281\) 549325. 0.415015 0.207508 0.978233i \(-0.433465\pi\)
0.207508 + 0.978233i \(0.433465\pi\)
\(282\) −627869. −0.470161
\(283\) 135813. 0.100803 0.0504016 0.998729i \(-0.483950\pi\)
0.0504016 + 0.998729i \(0.483950\pi\)
\(284\) 439867. 0.323612
\(285\) 171504. 0.125072
\(286\) −121718. −0.0879914
\(287\) 0 0
\(288\) 221415. 0.157299
\(289\) −1.38351e6 −0.974399
\(290\) 433382. 0.302605
\(291\) 894120. 0.618961
\(292\) −275799. −0.189293
\(293\) 1.76403e6 1.20043 0.600215 0.799839i \(-0.295082\pi\)
0.600215 + 0.799839i \(0.295082\pi\)
\(294\) 0 0
\(295\) 861606. 0.576439
\(296\) −1.63701e6 −1.08598
\(297\) 412476. 0.271336
\(298\) 388874. 0.253669
\(299\) −1.27377e6 −0.823976
\(300\) 1.15710e6 0.742281
\(301\) 0 0
\(302\) 903846. 0.570266
\(303\) 3.14594e6 1.96854
\(304\) 78462.7 0.0486944
\(305\) −2.12336e6 −1.30699
\(306\) −17080.9 −0.0104282
\(307\) 1.93533e6 1.17195 0.585975 0.810329i \(-0.300712\pi\)
0.585975 + 0.810329i \(0.300712\pi\)
\(308\) 0 0
\(309\) 577431. 0.344036
\(310\) −175567. −0.103762
\(311\) −2.98327e6 −1.74901 −0.874504 0.485019i \(-0.838813\pi\)
−0.874504 + 0.485019i \(0.838813\pi\)
\(312\) −1.00149e6 −0.582451
\(313\) 10701.5 0.00617426 0.00308713 0.999995i \(-0.499017\pi\)
0.00308713 + 0.999995i \(0.499017\pi\)
\(314\) −31570.8 −0.0180701
\(315\) 0 0
\(316\) −1.65014e6 −0.929617
\(317\) −2.43658e6 −1.36186 −0.680929 0.732349i \(-0.738424\pi\)
−0.680929 + 0.732349i \(0.738424\pi\)
\(318\) −123215. −0.0683277
\(319\) −315638. −0.173665
\(320\) 488141. 0.266484
\(321\) 3.77399e6 2.04427
\(322\) 0 0
\(323\) −25813.6 −0.0137671
\(324\) 1.82425e6 0.965433
\(325\) −1.15392e6 −0.605994
\(326\) −924239. −0.481660
\(327\) −2.73183e6 −1.41281
\(328\) 147563. 0.0757342
\(329\) 0 0
\(330\) 338515. 0.171117
\(331\) 119576. 0.0599894 0.0299947 0.999550i \(-0.490451\pi\)
0.0299947 + 0.999550i \(0.490451\pi\)
\(332\) 1.24140e6 0.618109
\(333\) 508546. 0.251316
\(334\) 308644. 0.151388
\(335\) 3.86880e6 1.88349
\(336\) 0 0
\(337\) −2.02195e6 −0.969830 −0.484915 0.874561i \(-0.661149\pi\)
−0.484915 + 0.874561i \(0.661149\pi\)
\(338\) −361867. −0.172289
\(339\) −1.55301e6 −0.733965
\(340\) −388979. −0.182486
\(341\) 127868. 0.0595491
\(342\) 12129.9 0.00560778
\(343\) 0 0
\(344\) 1.91569e6 0.872829
\(345\) 3.54254e6 1.60238
\(346\) 1.51753e6 0.681471
\(347\) 3.01864e6 1.34582 0.672912 0.739723i \(-0.265043\pi\)
0.672912 + 0.739723i \(0.265043\pi\)
\(348\) −1.19139e6 −0.527358
\(349\) −2.40399e6 −1.05650 −0.528250 0.849089i \(-0.677152\pi\)
−0.528250 + 0.849089i \(0.677152\pi\)
\(350\) 0 0
\(351\) −1.55263e6 −0.672666
\(352\) 660463. 0.284113
\(353\) −3.62981e6 −1.55041 −0.775206 0.631709i \(-0.782354\pi\)
−0.775206 + 0.631709i \(0.782354\pi\)
\(354\) 425989. 0.180672
\(355\) 1.21997e6 0.513780
\(356\) −2.23128e6 −0.933102
\(357\) 0 0
\(358\) 78397.1 0.0323290
\(359\) 939181. 0.384603 0.192302 0.981336i \(-0.438405\pi\)
0.192302 + 0.981336i \(0.438405\pi\)
\(360\) 398438. 0.162033
\(361\) −2.45777e6 −0.992597
\(362\) −575270. −0.230728
\(363\) −246545. −0.0982042
\(364\) 0 0
\(365\) −764926. −0.300530
\(366\) −1.04982e6 −0.409647
\(367\) 2.26697e6 0.878577 0.439288 0.898346i \(-0.355231\pi\)
0.439288 + 0.898346i \(0.355231\pi\)
\(368\) 1.62071e6 0.623857
\(369\) −45841.1 −0.0175263
\(370\) −2.08283e6 −0.790949
\(371\) 0 0
\(372\) 482642. 0.180829
\(373\) −4.55029e6 −1.69343 −0.846714 0.532048i \(-0.821423\pi\)
−0.846714 + 0.532048i \(0.821423\pi\)
\(374\) −50951.0 −0.0188353
\(375\) −749250. −0.275137
\(376\) −2.20442e6 −0.804125
\(377\) 1.18812e6 0.430532
\(378\) 0 0
\(379\) 618788. 0.221281 0.110641 0.993860i \(-0.464710\pi\)
0.110641 + 0.993860i \(0.464710\pi\)
\(380\) 276230. 0.0981323
\(381\) −2.32886e6 −0.821924
\(382\) 865881. 0.303599
\(383\) −2.23829e6 −0.779686 −0.389843 0.920881i \(-0.627471\pi\)
−0.389843 + 0.920881i \(0.627471\pi\)
\(384\) 3.18264e6 1.10144
\(385\) 0 0
\(386\) 34844.5 0.0119033
\(387\) −595120. −0.201989
\(388\) 1.44010e6 0.485640
\(389\) −4.60206e6 −1.54198 −0.770989 0.636848i \(-0.780238\pi\)
−0.770989 + 0.636848i \(0.780238\pi\)
\(390\) −1.27423e6 −0.424214
\(391\) −533199. −0.176379
\(392\) 0 0
\(393\) −916405. −0.299299
\(394\) 1.20490e6 0.391030
\(395\) −4.57666e6 −1.47590
\(396\) −133123. −0.0426595
\(397\) 4.35532e6 1.38690 0.693448 0.720506i \(-0.256091\pi\)
0.693448 + 0.720506i \(0.256091\pi\)
\(398\) 1.20703e6 0.381952
\(399\) 0 0
\(400\) 1.46821e6 0.458816
\(401\) 3.62515e6 1.12581 0.562905 0.826522i \(-0.309684\pi\)
0.562905 + 0.826522i \(0.309684\pi\)
\(402\) 1.91278e6 0.590338
\(403\) −481316. −0.147628
\(404\) 5.06697e6 1.54453
\(405\) 5.05955e6 1.53276
\(406\) 0 0
\(407\) 1.51695e6 0.453927
\(408\) −419221. −0.124679
\(409\) −4.13585e6 −1.22252 −0.611260 0.791430i \(-0.709337\pi\)
−0.611260 + 0.791430i \(0.709337\pi\)
\(410\) 187749. 0.0551592
\(411\) −682922. −0.199419
\(412\) 930032. 0.269932
\(413\) 0 0
\(414\) 250552. 0.0718450
\(415\) 3.44300e6 0.981335
\(416\) −2.48609e6 −0.704343
\(417\) 2.36655e6 0.666463
\(418\) 36182.4 0.0101288
\(419\) 2.46691e6 0.686464 0.343232 0.939251i \(-0.388478\pi\)
0.343232 + 0.939251i \(0.388478\pi\)
\(420\) 0 0
\(421\) 3.45258e6 0.949376 0.474688 0.880154i \(-0.342561\pi\)
0.474688 + 0.880154i \(0.342561\pi\)
\(422\) 1.18635e6 0.324287
\(423\) 684813. 0.186089
\(424\) −432602. −0.116862
\(425\) −483029. −0.129718
\(426\) 603168. 0.161033
\(427\) 0 0
\(428\) 6.07853e6 1.60394
\(429\) 928038. 0.243457
\(430\) 2.43740e6 0.635704
\(431\) −3.65893e6 −0.948770 −0.474385 0.880318i \(-0.657329\pi\)
−0.474385 + 0.880318i \(0.657329\pi\)
\(432\) 1.97551e6 0.509296
\(433\) 1.59716e6 0.409381 0.204690 0.978827i \(-0.434381\pi\)
0.204690 + 0.978827i \(0.434381\pi\)
\(434\) 0 0
\(435\) −3.30431e6 −0.837256
\(436\) −4.39999e6 −1.10850
\(437\) 378647. 0.0948485
\(438\) −378190. −0.0941943
\(439\) 1.58464e6 0.392437 0.196219 0.980560i \(-0.437134\pi\)
0.196219 + 0.980560i \(0.437134\pi\)
\(440\) 1.18851e6 0.292664
\(441\) 0 0
\(442\) 191788. 0.0466945
\(443\) 2.29633e6 0.555936 0.277968 0.960590i \(-0.410339\pi\)
0.277968 + 0.960590i \(0.410339\pi\)
\(444\) 5.72580e6 1.37841
\(445\) −6.18844e6 −1.48143
\(446\) 418837. 0.0997029
\(447\) −2.96496e6 −0.701859
\(448\) 0 0
\(449\) 3.31569e6 0.776171 0.388086 0.921623i \(-0.373136\pi\)
0.388086 + 0.921623i \(0.373136\pi\)
\(450\) 226977. 0.0528385
\(451\) −136740. −0.0316559
\(452\) −2.50134e6 −0.575872
\(453\) −6.89136e6 −1.57783
\(454\) −802664. −0.182765
\(455\) 0 0
\(456\) 297706. 0.0670464
\(457\) 2.20892e6 0.494754 0.247377 0.968919i \(-0.420431\pi\)
0.247377 + 0.968919i \(0.420431\pi\)
\(458\) −1.11386e6 −0.248123
\(459\) −649927. −0.143990
\(460\) 5.70574e6 1.25724
\(461\) 1.86064e6 0.407764 0.203882 0.978995i \(-0.434644\pi\)
0.203882 + 0.978995i \(0.434644\pi\)
\(462\) 0 0
\(463\) 1.20592e6 0.261437 0.130718 0.991420i \(-0.458272\pi\)
0.130718 + 0.991420i \(0.458272\pi\)
\(464\) −1.51172e6 −0.325969
\(465\) 1.33861e6 0.287091
\(466\) −2.65383e6 −0.566119
\(467\) −2.29388e6 −0.486719 −0.243360 0.969936i \(-0.578250\pi\)
−0.243360 + 0.969936i \(0.578250\pi\)
\(468\) 501098. 0.105757
\(469\) 0 0
\(470\) −2.80475e6 −0.585666
\(471\) 240711. 0.0499969
\(472\) 1.49563e6 0.309007
\(473\) −1.77519e6 −0.364832
\(474\) −2.26276e6 −0.462587
\(475\) 343019. 0.0697565
\(476\) 0 0
\(477\) 134390. 0.0270441
\(478\) −410642. −0.0822041
\(479\) 7.90892e6 1.57499 0.787496 0.616320i \(-0.211377\pi\)
0.787496 + 0.616320i \(0.211377\pi\)
\(480\) 6.91416e6 1.36973
\(481\) −5.71007e6 −1.12533
\(482\) −386154. −0.0757083
\(483\) 0 0
\(484\) −397095. −0.0770515
\(485\) 3.99412e6 0.771021
\(486\) 672002. 0.129056
\(487\) 3.48410e6 0.665684 0.332842 0.942983i \(-0.391992\pi\)
0.332842 + 0.942983i \(0.391992\pi\)
\(488\) −3.68585e6 −0.700628
\(489\) 7.04684e6 1.33267
\(490\) 0 0
\(491\) −8.98096e6 −1.68120 −0.840599 0.541658i \(-0.817797\pi\)
−0.840599 + 0.541658i \(0.817797\pi\)
\(492\) −516132. −0.0961276
\(493\) 497343. 0.0921592
\(494\) −136197. −0.0251101
\(495\) −369216. −0.0677279
\(496\) 612410. 0.111773
\(497\) 0 0
\(498\) 1.70227e6 0.307577
\(499\) 6.67736e6 1.20048 0.600238 0.799821i \(-0.295072\pi\)
0.600238 + 0.799821i \(0.295072\pi\)
\(500\) −1.20677e6 −0.215874
\(501\) −2.35325e6 −0.418865
\(502\) −989242. −0.175204
\(503\) −7.58428e6 −1.33658 −0.668289 0.743902i \(-0.732973\pi\)
−0.668289 + 0.743902i \(0.732973\pi\)
\(504\) 0 0
\(505\) 1.40532e7 2.45215
\(506\) 747375. 0.129766
\(507\) 2.75905e6 0.476693
\(508\) −3.75096e6 −0.644886
\(509\) 6.02580e6 1.03091 0.515454 0.856917i \(-0.327623\pi\)
0.515454 + 0.856917i \(0.327623\pi\)
\(510\) −533389. −0.0908068
\(511\) 0 0
\(512\) 5.58471e6 0.941511
\(513\) 461540. 0.0774312
\(514\) 2.53042e6 0.422460
\(515\) 2.57944e6 0.428555
\(516\) −6.70054e6 −1.10786
\(517\) 2.04274e6 0.336114
\(518\) 0 0
\(519\) −1.15704e7 −1.88551
\(520\) −4.47374e6 −0.725542
\(521\) 4.58541e6 0.740088 0.370044 0.929014i \(-0.379343\pi\)
0.370044 + 0.929014i \(0.379343\pi\)
\(522\) −233703. −0.0375394
\(523\) 4.88145e6 0.780359 0.390179 0.920739i \(-0.372413\pi\)
0.390179 + 0.920739i \(0.372413\pi\)
\(524\) −1.47600e6 −0.234832
\(525\) 0 0
\(526\) 978910. 0.154269
\(527\) −201478. −0.0316010
\(528\) −1.18080e6 −0.184329
\(529\) 1.38489e6 0.215168
\(530\) −550414. −0.0851138
\(531\) −464624. −0.0715098
\(532\) 0 0
\(533\) 514714. 0.0784780
\(534\) −3.05965e6 −0.464321
\(535\) 1.68588e7 2.54649
\(536\) 6.71568e6 1.00967
\(537\) −597738. −0.0894489
\(538\) 4.16810e6 0.620844
\(539\) 0 0
\(540\) 6.95484e6 1.02637
\(541\) 6.21940e6 0.913598 0.456799 0.889570i \(-0.348996\pi\)
0.456799 + 0.889570i \(0.348996\pi\)
\(542\) −4.95172e6 −0.724033
\(543\) 4.38614e6 0.638385
\(544\) −1.04067e6 −0.150771
\(545\) −1.22034e7 −1.75990
\(546\) 0 0
\(547\) −9.49047e6 −1.35619 −0.678093 0.734976i \(-0.737193\pi\)
−0.678093 + 0.734976i \(0.737193\pi\)
\(548\) −1.09994e6 −0.156465
\(549\) 1.14503e6 0.162138
\(550\) 677053. 0.0954368
\(551\) −353184. −0.0495589
\(552\) 6.14935e6 0.858976
\(553\) 0 0
\(554\) 2.82310e6 0.390798
\(555\) 1.58805e7 2.18842
\(556\) 3.81166e6 0.522910
\(557\) −2.92907e6 −0.400029 −0.200014 0.979793i \(-0.564099\pi\)
−0.200014 + 0.979793i \(0.564099\pi\)
\(558\) 94675.0 0.0128721
\(559\) 6.68213e6 0.904451
\(560\) 0 0
\(561\) 388475. 0.0521141
\(562\) 1.21324e6 0.162033
\(563\) 455079. 0.0605084 0.0302542 0.999542i \(-0.490368\pi\)
0.0302542 + 0.999542i \(0.490368\pi\)
\(564\) 7.71041e6 1.02066
\(565\) −6.93744e6 −0.914278
\(566\) 299955. 0.0393563
\(567\) 0 0
\(568\) 2.11769e6 0.275418
\(569\) −6.27664e6 −0.812730 −0.406365 0.913711i \(-0.633204\pi\)
−0.406365 + 0.913711i \(0.633204\pi\)
\(570\) 378781. 0.0488316
\(571\) −621794. −0.0798098 −0.0399049 0.999203i \(-0.512706\pi\)
−0.0399049 + 0.999203i \(0.512706\pi\)
\(572\) 1.49473e6 0.191018
\(573\) −6.60189e6 −0.840005
\(574\) 0 0
\(575\) 7.08532e6 0.893697
\(576\) −263232. −0.0330585
\(577\) −1.28776e7 −1.61026 −0.805130 0.593098i \(-0.797905\pi\)
−0.805130 + 0.593098i \(0.797905\pi\)
\(578\) −3.05560e6 −0.380432
\(579\) −265671. −0.0329343
\(580\) −5.32205e6 −0.656915
\(581\) 0 0
\(582\) 1.97475e6 0.241659
\(583\) 400875. 0.0488470
\(584\) −1.32780e6 −0.161102
\(585\) 1.38979e6 0.167904
\(586\) 3.89602e6 0.468681
\(587\) −1.08775e7 −1.30296 −0.651482 0.758664i \(-0.725852\pi\)
−0.651482 + 0.758664i \(0.725852\pi\)
\(588\) 0 0
\(589\) 143078. 0.0169935
\(590\) 1.90293e6 0.225058
\(591\) −9.18673e6 −1.08191
\(592\) 7.26529e6 0.852018
\(593\) 7.50449e6 0.876364 0.438182 0.898886i \(-0.355622\pi\)
0.438182 + 0.898886i \(0.355622\pi\)
\(594\) 910990. 0.105937
\(595\) 0 0
\(596\) −4.77548e6 −0.550682
\(597\) −9.20295e6 −1.05680
\(598\) −2.81325e6 −0.321703
\(599\) −7.69438e6 −0.876207 −0.438104 0.898925i \(-0.644350\pi\)
−0.438104 + 0.898925i \(0.644350\pi\)
\(600\) 5.57074e6 0.631735
\(601\) −3.14770e6 −0.355473 −0.177737 0.984078i \(-0.556877\pi\)
−0.177737 + 0.984078i \(0.556877\pi\)
\(602\) 0 0
\(603\) −2.08626e6 −0.233655
\(604\) −1.10995e7 −1.23797
\(605\) −1.10134e6 −0.122330
\(606\) 6.94810e6 0.768572
\(607\) 4.57397e6 0.503874 0.251937 0.967744i \(-0.418932\pi\)
0.251937 + 0.967744i \(0.418932\pi\)
\(608\) 739025. 0.0810775
\(609\) 0 0
\(610\) −4.68962e6 −0.510286
\(611\) −7.68923e6 −0.833258
\(612\) 209758. 0.0226381
\(613\) −1.56075e7 −1.67758 −0.838790 0.544455i \(-0.816736\pi\)
−0.838790 + 0.544455i \(0.816736\pi\)
\(614\) 4.27436e6 0.457562
\(615\) −1.43149e6 −0.152616
\(616\) 0 0
\(617\) −1.18602e7 −1.25424 −0.627119 0.778924i \(-0.715766\pi\)
−0.627119 + 0.778924i \(0.715766\pi\)
\(618\) 1.27531e6 0.134321
\(619\) −7.91821e6 −0.830616 −0.415308 0.909681i \(-0.636326\pi\)
−0.415308 + 0.909681i \(0.636326\pi\)
\(620\) 2.15601e6 0.225253
\(621\) 9.53346e6 0.992023
\(622\) −6.58883e6 −0.682861
\(623\) 0 0
\(624\) 4.44475e6 0.456968
\(625\) −1.12642e7 −1.15345
\(626\) 23635.3 0.00241060
\(627\) −275872. −0.0280246
\(628\) 387698. 0.0392278
\(629\) −2.39022e6 −0.240886
\(630\) 0 0
\(631\) −1.11561e7 −1.11542 −0.557709 0.830037i \(-0.688319\pi\)
−0.557709 + 0.830037i \(0.688319\pi\)
\(632\) −7.94444e6 −0.791172
\(633\) −9.04527e6 −0.897248
\(634\) −5.38140e6 −0.531707
\(635\) −1.04033e7 −1.02385
\(636\) 1.51312e6 0.148330
\(637\) 0 0
\(638\) −697116. −0.0678037
\(639\) −657872. −0.0637367
\(640\) 1.42172e7 1.37203
\(641\) 7.17389e6 0.689620 0.344810 0.938672i \(-0.387943\pi\)
0.344810 + 0.938672i \(0.387943\pi\)
\(642\) 8.33521e6 0.798139
\(643\) −7.14025e6 −0.681061 −0.340531 0.940233i \(-0.610607\pi\)
−0.340531 + 0.940233i \(0.610607\pi\)
\(644\) 0 0
\(645\) −1.85839e7 −1.75889
\(646\) −57011.6 −0.00537505
\(647\) −1.56897e7 −1.47351 −0.736756 0.676159i \(-0.763643\pi\)
−0.736756 + 0.676159i \(0.763643\pi\)
\(648\) 8.78267e6 0.821654
\(649\) −1.38594e6 −0.129161
\(650\) −2.54854e6 −0.236597
\(651\) 0 0
\(652\) 1.13499e7 1.04562
\(653\) −5.04236e6 −0.462755 −0.231378 0.972864i \(-0.574323\pi\)
−0.231378 + 0.972864i \(0.574323\pi\)
\(654\) −6.03350e6 −0.551601
\(655\) −4.09367e6 −0.372829
\(656\) −654904. −0.0594180
\(657\) 412489. 0.0372820
\(658\) 0 0
\(659\) −9.10902e6 −0.817068 −0.408534 0.912743i \(-0.633960\pi\)
−0.408534 + 0.912743i \(0.633960\pi\)
\(660\) −4.15705e6 −0.371472
\(661\) −1.31308e7 −1.16893 −0.584464 0.811420i \(-0.698695\pi\)
−0.584464 + 0.811420i \(0.698695\pi\)
\(662\) 264095. 0.0234215
\(663\) −1.46229e6 −0.129196
\(664\) 5.97657e6 0.526056
\(665\) 0 0
\(666\) 1.12317e6 0.0981207
\(667\) −7.29528e6 −0.634933
\(668\) −3.79023e6 −0.328643
\(669\) −3.19341e6 −0.275861
\(670\) 8.54459e6 0.735367
\(671\) 3.41552e6 0.292854
\(672\) 0 0
\(673\) 1.55171e7 1.32061 0.660303 0.750999i \(-0.270428\pi\)
0.660303 + 0.750999i \(0.270428\pi\)
\(674\) −4.46566e6 −0.378648
\(675\) 8.63644e6 0.729584
\(676\) 4.44382e6 0.374016
\(677\) 1.40356e7 1.17695 0.588476 0.808515i \(-0.299728\pi\)
0.588476 + 0.808515i \(0.299728\pi\)
\(678\) −3.42997e6 −0.286560
\(679\) 0 0
\(680\) −1.87270e6 −0.155309
\(681\) 6.11990e6 0.505681
\(682\) 282408. 0.0232496
\(683\) 5.34969e6 0.438810 0.219405 0.975634i \(-0.429588\pi\)
0.219405 + 0.975634i \(0.429588\pi\)
\(684\) −148958. −0.0121737
\(685\) −3.05068e6 −0.248410
\(686\) 0 0
\(687\) 8.49261e6 0.686514
\(688\) −8.50211e6 −0.684787
\(689\) −1.50896e6 −0.121096
\(690\) 7.82402e6 0.625615
\(691\) −1.31390e7 −1.04681 −0.523404 0.852084i \(-0.675338\pi\)
−0.523404 + 0.852084i \(0.675338\pi\)
\(692\) −1.86357e7 −1.47938
\(693\) 0 0
\(694\) 6.66695e6 0.525447
\(695\) 1.05716e7 0.830194
\(696\) −5.73582e6 −0.448820
\(697\) 215458. 0.0167989
\(698\) −5.30944e6 −0.412487
\(699\) 2.02341e7 1.56636
\(700\) 0 0
\(701\) −2.49888e7 −1.92066 −0.960330 0.278865i \(-0.910042\pi\)
−0.960330 + 0.278865i \(0.910042\pi\)
\(702\) −3.42912e6 −0.262627
\(703\) 1.69740e6 0.129537
\(704\) −785200. −0.0597102
\(705\) 2.13848e7 1.62044
\(706\) −8.01676e6 −0.605323
\(707\) 0 0
\(708\) −5.23127e6 −0.392215
\(709\) −8.86200e6 −0.662089 −0.331044 0.943615i \(-0.607401\pi\)
−0.331044 + 0.943615i \(0.607401\pi\)
\(710\) 2.69441e6 0.200594
\(711\) 2.46798e6 0.183092
\(712\) −1.07423e7 −0.794138
\(713\) 2.95538e6 0.217716
\(714\) 0 0
\(715\) 4.14563e6 0.303268
\(716\) −962739. −0.0701820
\(717\) 3.13093e6 0.227445
\(718\) 2.07427e6 0.150160
\(719\) −2.58635e7 −1.86580 −0.932901 0.360132i \(-0.882732\pi\)
−0.932901 + 0.360132i \(0.882732\pi\)
\(720\) −1.76832e6 −0.127125
\(721\) 0 0
\(722\) −5.42820e6 −0.387537
\(723\) 2.94423e6 0.209472
\(724\) 7.06448e6 0.500880
\(725\) −6.60886e6 −0.466962
\(726\) −544518. −0.0383416
\(727\) 1.71871e7 1.20605 0.603026 0.797721i \(-0.293961\pi\)
0.603026 + 0.797721i \(0.293961\pi\)
\(728\) 0 0
\(729\) 1.12207e7 0.781988
\(730\) −1.68941e6 −0.117335
\(731\) 2.79712e6 0.193606
\(732\) 1.28920e7 0.889290
\(733\) −1.85650e7 −1.27625 −0.638125 0.769932i \(-0.720290\pi\)
−0.638125 + 0.769932i \(0.720290\pi\)
\(734\) 5.00680e6 0.343021
\(735\) 0 0
\(736\) 1.52651e7 1.03874
\(737\) −6.22315e6 −0.422028
\(738\) −101244. −0.00684274
\(739\) 5.94724e6 0.400594 0.200297 0.979735i \(-0.435809\pi\)
0.200297 + 0.979735i \(0.435809\pi\)
\(740\) 2.55777e7 1.71705
\(741\) 1.03843e6 0.0694755
\(742\) 0 0
\(743\) −2.72654e7 −1.81193 −0.905963 0.423357i \(-0.860852\pi\)
−0.905963 + 0.423357i \(0.860852\pi\)
\(744\) 2.32363e6 0.153899
\(745\) −1.32448e7 −0.874285
\(746\) −1.00497e7 −0.661161
\(747\) −1.85665e6 −0.121739
\(748\) 625692. 0.0408890
\(749\) 0 0
\(750\) −1.65479e6 −0.107421
\(751\) 1.30069e7 0.841541 0.420770 0.907167i \(-0.361760\pi\)
0.420770 + 0.907167i \(0.361760\pi\)
\(752\) 9.78351e6 0.630885
\(753\) 7.54245e6 0.484758
\(754\) 2.62407e6 0.168092
\(755\) −3.07844e7 −1.96545
\(756\) 0 0
\(757\) −9.22009e6 −0.584784 −0.292392 0.956299i \(-0.594451\pi\)
−0.292392 + 0.956299i \(0.594451\pi\)
\(758\) 1.36665e6 0.0863942
\(759\) −5.69835e6 −0.359041
\(760\) 1.32988e6 0.0835177
\(761\) −328083. −0.0205363 −0.0102682 0.999947i \(-0.503269\pi\)
−0.0102682 + 0.999947i \(0.503269\pi\)
\(762\) −5.14351e6 −0.320902
\(763\) 0 0
\(764\) −1.06333e7 −0.659072
\(765\) 581764. 0.0359412
\(766\) −4.94347e6 −0.304411
\(767\) 5.21690e6 0.320202
\(768\) 3.53235e6 0.216103
\(769\) −2.19214e6 −0.133676 −0.0668380 0.997764i \(-0.521291\pi\)
−0.0668380 + 0.997764i \(0.521291\pi\)
\(770\) 0 0
\(771\) −1.92932e7 −1.16887
\(772\) −427900. −0.0258404
\(773\) −2.18539e7 −1.31547 −0.657735 0.753249i \(-0.728485\pi\)
−0.657735 + 0.753249i \(0.728485\pi\)
\(774\) −1.31438e6 −0.0788619
\(775\) 2.67730e6 0.160119
\(776\) 6.93323e6 0.413315
\(777\) 0 0
\(778\) −1.01641e7 −0.602031
\(779\) −153006. −0.00903367
\(780\) 1.56479e7 0.920913
\(781\) −1.96238e6 −0.115121
\(782\) −1.17762e6 −0.0688633
\(783\) −8.89237e6 −0.518338
\(784\) 0 0
\(785\) 1.07528e6 0.0622797
\(786\) −2.02396e6 −0.116855
\(787\) −2.61010e7 −1.50217 −0.751087 0.660203i \(-0.770470\pi\)
−0.751087 + 0.660203i \(0.770470\pi\)
\(788\) −1.47965e7 −0.848874
\(789\) −7.46368e6 −0.426835
\(790\) −1.01080e7 −0.576231
\(791\) 0 0
\(792\) −640907. −0.0363063
\(793\) −1.28566e7 −0.726011
\(794\) 9.61913e6 0.541483
\(795\) 4.19663e6 0.235495
\(796\) −1.48226e7 −0.829168
\(797\) −1.39846e7 −0.779840 −0.389920 0.920849i \(-0.627497\pi\)
−0.389920 + 0.920849i \(0.627497\pi\)
\(798\) 0 0
\(799\) −3.21869e6 −0.178366
\(800\) 1.38288e7 0.763941
\(801\) 3.33714e6 0.183778
\(802\) 8.00647e6 0.439547
\(803\) 1.23042e6 0.0673388
\(804\) −2.34895e7 −1.28155
\(805\) 0 0
\(806\) −1.06303e6 −0.0576379
\(807\) −3.17796e7 −1.71777
\(808\) 2.43944e7 1.31450
\(809\) 2.70989e7 1.45573 0.727865 0.685721i \(-0.240513\pi\)
0.727865 + 0.685721i \(0.240513\pi\)
\(810\) 1.11745e7 0.598432
\(811\) −1.99644e7 −1.06587 −0.532936 0.846156i \(-0.678911\pi\)
−0.532936 + 0.846156i \(0.678911\pi\)
\(812\) 0 0
\(813\) 3.77543e7 2.00327
\(814\) 3.35033e6 0.177226
\(815\) 3.14789e7 1.66007
\(816\) 1.86056e6 0.0978180
\(817\) −1.98635e6 −0.104112
\(818\) −9.13439e6 −0.477306
\(819\) 0 0
\(820\) −2.30561e6 −0.119743
\(821\) 3.18829e7 1.65082 0.825410 0.564533i \(-0.190944\pi\)
0.825410 + 0.564533i \(0.190944\pi\)
\(822\) −1.50829e6 −0.0778586
\(823\) −1.34203e7 −0.690655 −0.345328 0.938482i \(-0.612232\pi\)
−0.345328 + 0.938482i \(0.612232\pi\)
\(824\) 4.47754e6 0.229732
\(825\) −5.16218e6 −0.264057
\(826\) 0 0
\(827\) 1.19386e7 0.607002 0.303501 0.952831i \(-0.401844\pi\)
0.303501 + 0.952831i \(0.401844\pi\)
\(828\) −3.07684e6 −0.155966
\(829\) −2.59274e7 −1.31031 −0.655153 0.755497i \(-0.727396\pi\)
−0.655153 + 0.755497i \(0.727396\pi\)
\(830\) 7.60419e6 0.383140
\(831\) −2.15247e7 −1.08127
\(832\) 2.95563e6 0.148027
\(833\) 0 0
\(834\) 5.22675e6 0.260206
\(835\) −1.05122e7 −0.521768
\(836\) −444330. −0.0219882
\(837\) 3.60237e6 0.177736
\(838\) 5.44839e6 0.268015
\(839\) 3.21482e7 1.57671 0.788354 0.615222i \(-0.210933\pi\)
0.788354 + 0.615222i \(0.210933\pi\)
\(840\) 0 0
\(841\) −1.37064e7 −0.668244
\(842\) 7.62533e6 0.370662
\(843\) −9.25029e6 −0.448318
\(844\) −1.45687e7 −0.703985
\(845\) 1.23249e7 0.593803
\(846\) 1.51247e6 0.0726544
\(847\) 0 0
\(848\) 1.91995e6 0.0916855
\(849\) −2.28700e6 −0.108892
\(850\) −1.06681e6 −0.0506456
\(851\) 3.50610e7 1.65959
\(852\) −7.40708e6 −0.349581
\(853\) 5.64308e6 0.265548 0.132774 0.991146i \(-0.457612\pi\)
0.132774 + 0.991146i \(0.457612\pi\)
\(854\) 0 0
\(855\) −413135. −0.0193275
\(856\) 2.92645e7 1.36507
\(857\) 1.77067e7 0.823543 0.411772 0.911287i \(-0.364910\pi\)
0.411772 + 0.911287i \(0.364910\pi\)
\(858\) 2.04966e6 0.0950524
\(859\) −1.57119e7 −0.726515 −0.363258 0.931689i \(-0.618336\pi\)
−0.363258 + 0.931689i \(0.618336\pi\)
\(860\) −2.99319e7 −1.38003
\(861\) 0 0
\(862\) −8.08108e6 −0.370426
\(863\) 245263. 0.0112100 0.00560500 0.999984i \(-0.498216\pi\)
0.00560500 + 0.999984i \(0.498216\pi\)
\(864\) 1.86070e7 0.847991
\(865\) −5.16861e7 −2.34873
\(866\) 3.52747e6 0.159834
\(867\) 2.32974e7 1.05259
\(868\) 0 0
\(869\) 7.36179e6 0.330700
\(870\) −7.29788e6 −0.326888
\(871\) 2.34250e7 1.04625
\(872\) −2.11833e7 −0.943415
\(873\) −2.15384e6 −0.0956486
\(874\) 836276. 0.0370315
\(875\) 0 0
\(876\) 4.64428e6 0.204483
\(877\) 1.04352e7 0.458142 0.229071 0.973410i \(-0.426431\pi\)
0.229071 + 0.973410i \(0.426431\pi\)
\(878\) 3.49983e6 0.153218
\(879\) −2.97051e7 −1.29676
\(880\) −5.27476e6 −0.229613
\(881\) 1.10430e7 0.479344 0.239672 0.970854i \(-0.422960\pi\)
0.239672 + 0.970854i \(0.422960\pi\)
\(882\) 0 0
\(883\) 5.41498e6 0.233720 0.116860 0.993148i \(-0.462717\pi\)
0.116860 + 0.993148i \(0.462717\pi\)
\(884\) −2.35521e6 −0.101368
\(885\) −1.45089e7 −0.622696
\(886\) 5.07166e6 0.217053
\(887\) 1.52663e7 0.651515 0.325757 0.945453i \(-0.394381\pi\)
0.325757 + 0.945453i \(0.394381\pi\)
\(888\) 2.75663e7 1.17313
\(889\) 0 0
\(890\) −1.36677e7 −0.578391
\(891\) −8.13854e6 −0.343441
\(892\) −5.14343e6 −0.216442
\(893\) 2.28573e6 0.0959170
\(894\) −6.54838e6 −0.274025
\(895\) −2.67015e6 −0.111424
\(896\) 0 0
\(897\) 2.14496e7 0.890096
\(898\) 7.32300e6 0.303039
\(899\) −2.75664e6 −0.113758
\(900\) −2.78734e6 −0.114705
\(901\) −631648. −0.0259217
\(902\) −302004. −0.0123594
\(903\) 0 0
\(904\) −1.20424e7 −0.490109
\(905\) 1.95933e7 0.795218
\(906\) −1.52202e7 −0.616027
\(907\) 2.02437e7 0.817094 0.408547 0.912737i \(-0.366036\pi\)
0.408547 + 0.912737i \(0.366036\pi\)
\(908\) 9.85694e6 0.396759
\(909\) −7.57825e6 −0.304200
\(910\) 0 0
\(911\) −1.17158e7 −0.467708 −0.233854 0.972272i \(-0.575134\pi\)
−0.233854 + 0.972272i \(0.575134\pi\)
\(912\) −1.32126e6 −0.0526019
\(913\) −5.53824e6 −0.219885
\(914\) 4.87860e6 0.193166
\(915\) 3.57560e7 1.41187
\(916\) 1.36785e7 0.538642
\(917\) 0 0
\(918\) −1.43542e6 −0.0562177
\(919\) −1.95296e7 −0.762788 −0.381394 0.924413i \(-0.624556\pi\)
−0.381394 + 0.924413i \(0.624556\pi\)
\(920\) 2.74697e7 1.07000
\(921\) −3.25898e7 −1.26599
\(922\) 4.10939e6 0.159202
\(923\) 7.38673e6 0.285396
\(924\) 0 0
\(925\) 3.17620e7 1.22055
\(926\) 2.66339e6 0.102072
\(927\) −1.39097e6 −0.0531641
\(928\) −1.42386e7 −0.542747
\(929\) 4.29425e7 1.63248 0.816240 0.577713i \(-0.196055\pi\)
0.816240 + 0.577713i \(0.196055\pi\)
\(930\) 2.95643e6 0.112088
\(931\) 0 0
\(932\) 3.25898e7 1.22897
\(933\) 5.02364e7 1.88936
\(934\) −5.06625e6 −0.190029
\(935\) 1.73535e6 0.0649171
\(936\) 2.41248e6 0.0900067
\(937\) −2.27191e7 −0.845361 −0.422681 0.906279i \(-0.638911\pi\)
−0.422681 + 0.906279i \(0.638911\pi\)
\(938\) 0 0
\(939\) −180207. −0.00666972
\(940\) 3.44431e7 1.27140
\(941\) −3.98095e6 −0.146559 −0.0732795 0.997311i \(-0.523347\pi\)
−0.0732795 + 0.997311i \(0.523347\pi\)
\(942\) 531632. 0.0195202
\(943\) −3.16045e6 −0.115736
\(944\) −6.63780e6 −0.242434
\(945\) 0 0
\(946\) −3.92067e6 −0.142440
\(947\) −2.43639e7 −0.882818 −0.441409 0.897306i \(-0.645521\pi\)
−0.441409 + 0.897306i \(0.645521\pi\)
\(948\) 2.77874e7 1.00422
\(949\) −4.63152e6 −0.166939
\(950\) 757589. 0.0272348
\(951\) 4.10304e7 1.47114
\(952\) 0 0
\(953\) 1.39017e7 0.495833 0.247916 0.968781i \(-0.420254\pi\)
0.247916 + 0.968781i \(0.420254\pi\)
\(954\) 296813. 0.0105587
\(955\) −2.94913e7 −1.04637
\(956\) 5.04280e6 0.178454
\(957\) 5.31515e6 0.187601
\(958\) 1.74676e7 0.614920
\(959\) 0 0
\(960\) −8.21999e6 −0.287868
\(961\) −2.75124e7 −0.960993
\(962\) −1.26112e7 −0.439358
\(963\) −9.09116e6 −0.315903
\(964\) 4.74208e6 0.164353
\(965\) −1.18678e6 −0.0410253
\(966\) 0 0
\(967\) 5.16682e7 1.77688 0.888438 0.458998i \(-0.151791\pi\)
0.888438 + 0.458998i \(0.151791\pi\)
\(968\) −1.91177e6 −0.0655764
\(969\) 434684. 0.0148718
\(970\) 8.82137e6 0.301028
\(971\) 1.45794e7 0.496240 0.248120 0.968729i \(-0.420187\pi\)
0.248120 + 0.968729i \(0.420187\pi\)
\(972\) −8.25237e6 −0.280164
\(973\) 0 0
\(974\) 7.69495e6 0.259901
\(975\) 1.94313e7 0.654623
\(976\) 1.63583e7 0.549685
\(977\) −3.09921e6 −0.103876 −0.0519379 0.998650i \(-0.516540\pi\)
−0.0519379 + 0.998650i \(0.516540\pi\)
\(978\) 1.55636e7 0.520311
\(979\) 9.95442e6 0.331940
\(980\) 0 0
\(981\) 6.58071e6 0.218323
\(982\) −1.98353e7 −0.656386
\(983\) −1.53445e7 −0.506489 −0.253245 0.967402i \(-0.581498\pi\)
−0.253245 + 0.967402i \(0.581498\pi\)
\(984\) −2.48486e6 −0.0818116
\(985\) −4.10380e7 −1.34771
\(986\) 1.09843e6 0.0359815
\(987\) 0 0
\(988\) 1.67253e6 0.0545108
\(989\) −4.10296e7 −1.33385
\(990\) −815447. −0.0264428
\(991\) 1.57747e7 0.510244 0.255122 0.966909i \(-0.417884\pi\)
0.255122 + 0.966909i \(0.417884\pi\)
\(992\) 5.76818e6 0.186106
\(993\) −2.01359e6 −0.0648033
\(994\) 0 0
\(995\) −4.11105e7 −1.31642
\(996\) −2.09043e7 −0.667710
\(997\) −1.85577e6 −0.0591270 −0.0295635 0.999563i \(-0.509412\pi\)
−0.0295635 + 0.999563i \(0.509412\pi\)
\(998\) 1.47476e7 0.468699
\(999\) 4.27365e7 1.35483
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 539.6.a.e.1.2 3
7.6 odd 2 11.6.a.b.1.2 3
21.20 even 2 99.6.a.g.1.2 3
28.27 even 2 176.6.a.i.1.2 3
35.13 even 4 275.6.b.b.199.3 6
35.27 even 4 275.6.b.b.199.4 6
35.34 odd 2 275.6.a.b.1.2 3
56.13 odd 2 704.6.a.q.1.2 3
56.27 even 2 704.6.a.t.1.2 3
77.76 even 2 121.6.a.d.1.2 3
231.230 odd 2 1089.6.a.r.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.6.a.b.1.2 3 7.6 odd 2
99.6.a.g.1.2 3 21.20 even 2
121.6.a.d.1.2 3 77.76 even 2
176.6.a.i.1.2 3 28.27 even 2
275.6.a.b.1.2 3 35.34 odd 2
275.6.b.b.199.3 6 35.13 even 4
275.6.b.b.199.4 6 35.27 even 4
539.6.a.e.1.2 3 1.1 even 1 trivial
704.6.a.q.1.2 3 56.13 odd 2
704.6.a.t.1.2 3 56.27 even 2
1089.6.a.r.1.2 3 231.230 odd 2