Properties

Label 6.8.a.a.1.1
Level 6
Weight 8
Character 6.1
Self dual yes
Analytic conductor 1.874
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6 = 2 \cdot 3 \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 6.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.87431015290\)
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
Root \(0\)
Character \(\chi\) = 6.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -114.000 q^{5} +216.000 q^{6} -1576.00 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -114.000 q^{5} +216.000 q^{6} -1576.00 q^{7} +512.000 q^{8} +729.000 q^{9} -912.000 q^{10} +7332.00 q^{11} +1728.00 q^{12} -3802.00 q^{13} -12608.0 q^{14} -3078.00 q^{15} +4096.00 q^{16} -6606.00 q^{17} +5832.00 q^{18} +24860.0 q^{19} -7296.00 q^{20} -42552.0 q^{21} +58656.0 q^{22} +41448.0 q^{23} +13824.0 q^{24} -65129.0 q^{25} -30416.0 q^{26} +19683.0 q^{27} -100864. q^{28} -41610.0 q^{29} -24624.0 q^{30} +33152.0 q^{31} +32768.0 q^{32} +197964. q^{33} -52848.0 q^{34} +179664. q^{35} +46656.0 q^{36} -36466.0 q^{37} +198880. q^{38} -102654. q^{39} -58368.0 q^{40} -639078. q^{41} -340416. q^{42} -156412. q^{43} +469248. q^{44} -83106.0 q^{45} +331584. q^{46} -433776. q^{47} +110592. q^{48} +1.66023e6 q^{49} -521032. q^{50} -178362. q^{51} -243328. q^{52} +786078. q^{53} +157464. q^{54} -835848. q^{55} -806912. q^{56} +671220. q^{57} -332880. q^{58} +745140. q^{59} -196992. q^{60} -1.66062e6 q^{61} +265216. q^{62} -1.14890e6 q^{63} +262144. q^{64} +433428. q^{65} +1.58371e6 q^{66} -3.29084e6 q^{67} -422784. q^{68} +1.11910e6 q^{69} +1.43731e6 q^{70} +5.71615e6 q^{71} +373248. q^{72} +2.65990e6 q^{73} -291728. q^{74} -1.75848e6 q^{75} +1.59104e6 q^{76} -1.15552e7 q^{77} -821232. q^{78} +3.80744e6 q^{79} -466944. q^{80} +531441. q^{81} -5.11262e6 q^{82} +2.22947e6 q^{83} -2.72333e6 q^{84} +753084. q^{85} -1.25130e6 q^{86} -1.12347e6 q^{87} +3.75398e6 q^{88} +5.99121e6 q^{89} -664848. q^{90} +5.99195e6 q^{91} +2.65267e6 q^{92} +895104. q^{93} -3.47021e6 q^{94} -2.83404e6 q^{95} +884736. q^{96} -4.06013e6 q^{97} +1.32819e7 q^{98} +5.34503e6 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.00000 0.707107
\(3\) 27.0000 0.577350
\(4\) 64.0000 0.500000
\(5\) −114.000 −0.407859 −0.203929 0.978986i \(-0.565371\pi\)
−0.203929 + 0.978986i \(0.565371\pi\)
\(6\) 216.000 0.408248
\(7\) −1576.00 −1.73665 −0.868327 0.495993i \(-0.834804\pi\)
−0.868327 + 0.495993i \(0.834804\pi\)
\(8\) 512.000 0.353553
\(9\) 729.000 0.333333
\(10\) −912.000 −0.288400
\(11\) 7332.00 1.66092 0.830459 0.557080i \(-0.188078\pi\)
0.830459 + 0.557080i \(0.188078\pi\)
\(12\) 1728.00 0.288675
\(13\) −3802.00 −0.479966 −0.239983 0.970777i \(-0.577142\pi\)
−0.239983 + 0.970777i \(0.577142\pi\)
\(14\) −12608.0 −1.22800
\(15\) −3078.00 −0.235477
\(16\) 4096.00 0.250000
\(17\) −6606.00 −0.326112 −0.163056 0.986617i \(-0.552135\pi\)
−0.163056 + 0.986617i \(0.552135\pi\)
\(18\) 5832.00 0.235702
\(19\) 24860.0 0.831502 0.415751 0.909478i \(-0.363519\pi\)
0.415751 + 0.909478i \(0.363519\pi\)
\(20\) −7296.00 −0.203929
\(21\) −42552.0 −1.00266
\(22\) 58656.0 1.17445
\(23\) 41448.0 0.710323 0.355162 0.934805i \(-0.384426\pi\)
0.355162 + 0.934805i \(0.384426\pi\)
\(24\) 13824.0 0.204124
\(25\) −65129.0 −0.833651
\(26\) −30416.0 −0.339387
\(27\) 19683.0 0.192450
\(28\) −100864. −0.868327
\(29\) −41610.0 −0.316814 −0.158407 0.987374i \(-0.550636\pi\)
−0.158407 + 0.987374i \(0.550636\pi\)
\(30\) −24624.0 −0.166508
\(31\) 33152.0 0.199868 0.0999341 0.994994i \(-0.468137\pi\)
0.0999341 + 0.994994i \(0.468137\pi\)
\(32\) 32768.0 0.176777
\(33\) 197964. 0.958931
\(34\) −52848.0 −0.230596
\(35\) 179664. 0.708309
\(36\) 46656.0 0.166667
\(37\) −36466.0 −0.118354 −0.0591769 0.998248i \(-0.518848\pi\)
−0.0591769 + 0.998248i \(0.518848\pi\)
\(38\) 198880. 0.587961
\(39\) −102654. −0.277108
\(40\) −58368.0 −0.144200
\(41\) −639078. −1.44814 −0.724070 0.689727i \(-0.757731\pi\)
−0.724070 + 0.689727i \(0.757731\pi\)
\(42\) −340416. −0.708986
\(43\) −156412. −0.300006 −0.150003 0.988686i \(-0.547928\pi\)
−0.150003 + 0.988686i \(0.547928\pi\)
\(44\) 469248. 0.830459
\(45\) −83106.0 −0.135953
\(46\) 331584. 0.502275
\(47\) −433776. −0.609429 −0.304714 0.952444i \(-0.598561\pi\)
−0.304714 + 0.952444i \(0.598561\pi\)
\(48\) 110592. 0.144338
\(49\) 1.66023e6 2.01596
\(50\) −521032. −0.589480
\(51\) −178362. −0.188281
\(52\) −243328. −0.239983
\(53\) 786078. 0.725271 0.362635 0.931931i \(-0.381877\pi\)
0.362635 + 0.931931i \(0.381877\pi\)
\(54\) 157464. 0.136083
\(55\) −835848. −0.677420
\(56\) −806912. −0.614000
\(57\) 671220. 0.480068
\(58\) −332880. −0.224022
\(59\) 745140. 0.472341 0.236171 0.971712i \(-0.424108\pi\)
0.236171 + 0.971712i \(0.424108\pi\)
\(60\) −196992. −0.117739
\(61\) −1.66062e6 −0.936732 −0.468366 0.883535i \(-0.655157\pi\)
−0.468366 + 0.883535i \(0.655157\pi\)
\(62\) 265216. 0.141328
\(63\) −1.14890e6 −0.578884
\(64\) 262144. 0.125000
\(65\) 433428. 0.195758
\(66\) 1.58371e6 0.678067
\(67\) −3.29084e6 −1.33673 −0.668366 0.743832i \(-0.733006\pi\)
−0.668366 + 0.743832i \(0.733006\pi\)
\(68\) −422784. −0.163056
\(69\) 1.11910e6 0.410105
\(70\) 1.43731e6 0.500850
\(71\) 5.71615e6 1.89539 0.947697 0.319171i \(-0.103404\pi\)
0.947697 + 0.319171i \(0.103404\pi\)
\(72\) 373248. 0.117851
\(73\) 2.65990e6 0.800267 0.400134 0.916457i \(-0.368964\pi\)
0.400134 + 0.916457i \(0.368964\pi\)
\(74\) −291728. −0.0836888
\(75\) −1.75848e6 −0.481309
\(76\) 1.59104e6 0.415751
\(77\) −1.15552e7 −2.88444
\(78\) −821232. −0.195945
\(79\) 3.80744e6 0.868837 0.434418 0.900711i \(-0.356954\pi\)
0.434418 + 0.900711i \(0.356954\pi\)
\(80\) −466944. −0.101965
\(81\) 531441. 0.111111
\(82\) −5.11262e6 −1.02399
\(83\) 2.22947e6 0.427984 0.213992 0.976835i \(-0.431353\pi\)
0.213992 + 0.976835i \(0.431353\pi\)
\(84\) −2.72333e6 −0.501329
\(85\) 753084. 0.133008
\(86\) −1.25130e6 −0.212137
\(87\) −1.12347e6 −0.182913
\(88\) 3.75398e6 0.587223
\(89\) 5.99121e6 0.900844 0.450422 0.892816i \(-0.351274\pi\)
0.450422 + 0.892816i \(0.351274\pi\)
\(90\) −664848. −0.0961332
\(91\) 5.99195e6 0.833534
\(92\) 2.65267e6 0.355162
\(93\) 895104. 0.115394
\(94\) −3.47021e6 −0.430931
\(95\) −2.83404e6 −0.339136
\(96\) 884736. 0.102062
\(97\) −4.06013e6 −0.451688 −0.225844 0.974163i \(-0.572514\pi\)
−0.225844 + 0.974163i \(0.572514\pi\)
\(98\) 1.32819e7 1.42550
\(99\) 5.34503e6 0.553639
\(100\) −4.16826e6 −0.416826
\(101\) −1.72819e7 −1.66904 −0.834522 0.550975i \(-0.814256\pi\)
−0.834522 + 0.550975i \(0.814256\pi\)
\(102\) −1.42690e6 −0.133135
\(103\) −1.43623e7 −1.29507 −0.647536 0.762035i \(-0.724201\pi\)
−0.647536 + 0.762035i \(0.724201\pi\)
\(104\) −1.94662e6 −0.169694
\(105\) 4.85093e6 0.408943
\(106\) 6.28862e6 0.512844
\(107\) 6.45440e6 0.509346 0.254673 0.967027i \(-0.418032\pi\)
0.254673 + 0.967027i \(0.418032\pi\)
\(108\) 1.25971e6 0.0962250
\(109\) −884410. −0.0654125 −0.0327063 0.999465i \(-0.510413\pi\)
−0.0327063 + 0.999465i \(0.510413\pi\)
\(110\) −6.68678e6 −0.479008
\(111\) −984582. −0.0683316
\(112\) −6.45530e6 −0.434163
\(113\) 1.21325e7 0.790999 0.395499 0.918466i \(-0.370572\pi\)
0.395499 + 0.918466i \(0.370572\pi\)
\(114\) 5.36976e6 0.339459
\(115\) −4.72507e6 −0.289712
\(116\) −2.66304e6 −0.158407
\(117\) −2.77166e6 −0.159989
\(118\) 5.96112e6 0.333996
\(119\) 1.04111e7 0.566344
\(120\) −1.57594e6 −0.0832538
\(121\) 3.42711e7 1.75865
\(122\) −1.32849e7 −0.662369
\(123\) −1.72551e7 −0.836084
\(124\) 2.12173e6 0.0999341
\(125\) 1.63310e7 0.747871
\(126\) −9.19123e6 −0.409333
\(127\) 6.86806e6 0.297524 0.148762 0.988873i \(-0.452471\pi\)
0.148762 + 0.988873i \(0.452471\pi\)
\(128\) 2.09715e6 0.0883883
\(129\) −4.22312e6 −0.173209
\(130\) 3.46742e6 0.138422
\(131\) −3.95208e7 −1.53595 −0.767973 0.640482i \(-0.778735\pi\)
−0.767973 + 0.640482i \(0.778735\pi\)
\(132\) 1.26697e7 0.479466
\(133\) −3.91794e7 −1.44403
\(134\) −2.63267e7 −0.945212
\(135\) −2.24386e6 −0.0784925
\(136\) −3.38227e6 −0.115298
\(137\) 1.91741e7 0.637078 0.318539 0.947910i \(-0.396808\pi\)
0.318539 + 0.947910i \(0.396808\pi\)
\(138\) 8.95277e6 0.289988
\(139\) 1.32449e7 0.418309 0.209154 0.977883i \(-0.432929\pi\)
0.209154 + 0.977883i \(0.432929\pi\)
\(140\) 1.14985e7 0.354155
\(141\) −1.17120e7 −0.351854
\(142\) 4.57292e7 1.34025
\(143\) −2.78763e7 −0.797184
\(144\) 2.98598e6 0.0833333
\(145\) 4.74354e6 0.129215
\(146\) 2.12792e7 0.565874
\(147\) 4.48263e7 1.16392
\(148\) −2.33382e6 −0.0591769
\(149\) 5.73624e7 1.42061 0.710306 0.703893i \(-0.248556\pi\)
0.710306 + 0.703893i \(0.248556\pi\)
\(150\) −1.40679e7 −0.340337
\(151\) −3.10873e7 −0.734790 −0.367395 0.930065i \(-0.619750\pi\)
−0.367395 + 0.930065i \(0.619750\pi\)
\(152\) 1.27283e7 0.293981
\(153\) −4.81577e6 −0.108704
\(154\) −9.24419e7 −2.03961
\(155\) −3.77933e6 −0.0815180
\(156\) −6.56986e6 −0.138554
\(157\) −3.37835e7 −0.696715 −0.348358 0.937362i \(-0.613261\pi\)
−0.348358 + 0.937362i \(0.613261\pi\)
\(158\) 3.04595e7 0.614360
\(159\) 2.12241e7 0.418735
\(160\) −3.73555e6 −0.0720999
\(161\) −6.53220e7 −1.23359
\(162\) 4.25153e6 0.0785674
\(163\) 6.26659e7 1.13338 0.566689 0.823932i \(-0.308224\pi\)
0.566689 + 0.823932i \(0.308224\pi\)
\(164\) −4.09010e7 −0.724070
\(165\) −2.25679e7 −0.391109
\(166\) 1.78357e7 0.302631
\(167\) 6.27072e7 1.04186 0.520931 0.853599i \(-0.325585\pi\)
0.520931 + 0.853599i \(0.325585\pi\)
\(168\) −2.17866e7 −0.354493
\(169\) −4.82933e7 −0.769633
\(170\) 6.02467e6 0.0940507
\(171\) 1.81229e7 0.277167
\(172\) −1.00104e7 −0.150003
\(173\) −2.70521e7 −0.397228 −0.198614 0.980078i \(-0.563644\pi\)
−0.198614 + 0.980078i \(0.563644\pi\)
\(174\) −8.98776e6 −0.129339
\(175\) 1.02643e8 1.44776
\(176\) 3.00319e7 0.415229
\(177\) 2.01188e7 0.272706
\(178\) 4.79297e7 0.636993
\(179\) −1.34281e8 −1.74996 −0.874981 0.484157i \(-0.839126\pi\)
−0.874981 + 0.484157i \(0.839126\pi\)
\(180\) −5.31878e6 −0.0679765
\(181\) 1.14661e8 1.43727 0.718636 0.695386i \(-0.244767\pi\)
0.718636 + 0.695386i \(0.244767\pi\)
\(182\) 4.79356e7 0.589398
\(183\) −4.48367e7 −0.540822
\(184\) 2.12214e7 0.251137
\(185\) 4.15712e6 0.0482716
\(186\) 7.16083e6 0.0815959
\(187\) −4.84352e7 −0.541646
\(188\) −2.77617e7 −0.304714
\(189\) −3.10204e7 −0.334219
\(190\) −2.26723e7 −0.239805
\(191\) 1.63605e7 0.169895 0.0849474 0.996385i \(-0.472928\pi\)
0.0849474 + 0.996385i \(0.472928\pi\)
\(192\) 7.07789e6 0.0721688
\(193\) −1.54198e8 −1.54394 −0.771968 0.635661i \(-0.780728\pi\)
−0.771968 + 0.635661i \(0.780728\pi\)
\(194\) −3.24810e7 −0.319392
\(195\) 1.17026e7 0.113021
\(196\) 1.06255e8 1.00798
\(197\) 8.32288e7 0.775607 0.387804 0.921742i \(-0.373234\pi\)
0.387804 + 0.921742i \(0.373234\pi\)
\(198\) 4.27602e7 0.391482
\(199\) −7.61722e7 −0.685190 −0.342595 0.939483i \(-0.611306\pi\)
−0.342595 + 0.939483i \(0.611306\pi\)
\(200\) −3.33460e7 −0.294740
\(201\) −8.88526e7 −0.771763
\(202\) −1.38256e8 −1.18019
\(203\) 6.55774e7 0.550196
\(204\) −1.14152e7 −0.0941405
\(205\) 7.28549e7 0.590636
\(206\) −1.14898e8 −0.915755
\(207\) 3.02156e7 0.236774
\(208\) −1.55730e7 −0.119991
\(209\) 1.82274e8 1.38106
\(210\) 3.88074e7 0.289166
\(211\) 3.52446e7 0.258288 0.129144 0.991626i \(-0.458777\pi\)
0.129144 + 0.991626i \(0.458777\pi\)
\(212\) 5.03090e7 0.362635
\(213\) 1.54336e8 1.09431
\(214\) 5.16352e7 0.360162
\(215\) 1.78310e7 0.122360
\(216\) 1.00777e7 0.0680414
\(217\) −5.22476e7 −0.347102
\(218\) −7.07528e6 −0.0462536
\(219\) 7.18172e7 0.462034
\(220\) −5.34943e7 −0.338710
\(221\) 2.51160e7 0.156523
\(222\) −7.87666e6 −0.0483177
\(223\) −1.89131e8 −1.14208 −0.571040 0.820922i \(-0.693460\pi\)
−0.571040 + 0.820922i \(0.693460\pi\)
\(224\) −5.16424e7 −0.307000
\(225\) −4.74790e7 −0.277884
\(226\) 9.70600e7 0.559320
\(227\) −1.76100e8 −0.999239 −0.499620 0.866245i \(-0.666527\pi\)
−0.499620 + 0.866245i \(0.666527\pi\)
\(228\) 4.29581e7 0.240034
\(229\) 6.50396e7 0.357894 0.178947 0.983859i \(-0.442731\pi\)
0.178947 + 0.983859i \(0.442731\pi\)
\(230\) −3.78006e7 −0.204857
\(231\) −3.11991e8 −1.66533
\(232\) −2.13043e7 −0.112011
\(233\) −2.51319e8 −1.30160 −0.650802 0.759248i \(-0.725567\pi\)
−0.650802 + 0.759248i \(0.725567\pi\)
\(234\) −2.21733e7 −0.113129
\(235\) 4.94505e7 0.248561
\(236\) 4.76890e7 0.236171
\(237\) 1.02801e8 0.501623
\(238\) 8.32884e7 0.400466
\(239\) 2.13079e8 1.00960 0.504799 0.863237i \(-0.331566\pi\)
0.504799 + 0.863237i \(0.331566\pi\)
\(240\) −1.26075e7 −0.0588693
\(241\) 2.57284e8 1.18400 0.592001 0.805937i \(-0.298338\pi\)
0.592001 + 0.805937i \(0.298338\pi\)
\(242\) 2.74168e8 1.24355
\(243\) 1.43489e7 0.0641500
\(244\) −1.06280e8 −0.468366
\(245\) −1.89267e8 −0.822229
\(246\) −1.38041e8 −0.591200
\(247\) −9.45177e7 −0.399093
\(248\) 1.69738e7 0.0706641
\(249\) 6.01956e7 0.247097
\(250\) 1.30648e8 0.528824
\(251\) 1.23058e8 0.491193 0.245596 0.969372i \(-0.421016\pi\)
0.245596 + 0.969372i \(0.421016\pi\)
\(252\) −7.35299e7 −0.289442
\(253\) 3.03897e8 1.17979
\(254\) 5.49445e7 0.210381
\(255\) 2.03333e7 0.0767921
\(256\) 1.67772e7 0.0625000
\(257\) −4.43334e8 −1.62916 −0.814582 0.580048i \(-0.803034\pi\)
−0.814582 + 0.580048i \(0.803034\pi\)
\(258\) −3.37850e7 −0.122477
\(259\) 5.74704e7 0.205539
\(260\) 2.77394e7 0.0978792
\(261\) −3.03337e7 −0.105605
\(262\) −3.16166e8 −1.08608
\(263\) 2.98925e8 1.01325 0.506625 0.862166i \(-0.330893\pi\)
0.506625 + 0.862166i \(0.330893\pi\)
\(264\) 1.01358e8 0.339033
\(265\) −8.96129e7 −0.295808
\(266\) −3.13435e8 −1.02108
\(267\) 1.61763e8 0.520102
\(268\) −2.10614e8 −0.668366
\(269\) 2.08908e8 0.654368 0.327184 0.944961i \(-0.393900\pi\)
0.327184 + 0.944961i \(0.393900\pi\)
\(270\) −1.79509e7 −0.0555026
\(271\) −1.12749e7 −0.0344129 −0.0172064 0.999852i \(-0.505477\pi\)
−0.0172064 + 0.999852i \(0.505477\pi\)
\(272\) −2.70582e7 −0.0815281
\(273\) 1.61783e8 0.481241
\(274\) 1.53393e8 0.450482
\(275\) −4.77526e8 −1.38463
\(276\) 7.16221e7 0.205053
\(277\) −6.58964e8 −1.86287 −0.931435 0.363907i \(-0.881443\pi\)
−0.931435 + 0.363907i \(0.881443\pi\)
\(278\) 1.05959e8 0.295789
\(279\) 2.41678e7 0.0666227
\(280\) 9.19880e7 0.250425
\(281\) −1.05123e8 −0.282634 −0.141317 0.989964i \(-0.545134\pi\)
−0.141317 + 0.989964i \(0.545134\pi\)
\(282\) −9.36956e7 −0.248798
\(283\) 3.30161e8 0.865911 0.432956 0.901415i \(-0.357471\pi\)
0.432956 + 0.901415i \(0.357471\pi\)
\(284\) 3.65834e8 0.947697
\(285\) −7.65191e7 −0.195800
\(286\) −2.23010e8 −0.563694
\(287\) 1.00719e9 2.51492
\(288\) 2.38879e7 0.0589256
\(289\) −3.66699e8 −0.893651
\(290\) 3.79483e7 0.0913691
\(291\) −1.09623e8 −0.260782
\(292\) 1.70233e8 0.400134
\(293\) −8.71002e7 −0.202294 −0.101147 0.994871i \(-0.532251\pi\)
−0.101147 + 0.994871i \(0.532251\pi\)
\(294\) 3.58610e8 0.823014
\(295\) −8.49460e7 −0.192649
\(296\) −1.86706e7 −0.0418444
\(297\) 1.44316e8 0.319644
\(298\) 4.58899e8 1.00452
\(299\) −1.57585e8 −0.340931
\(300\) −1.12543e8 −0.240654
\(301\) 2.46505e8 0.521007
\(302\) −2.48698e8 −0.519575
\(303\) −4.66612e8 −0.963623
\(304\) 1.01827e8 0.207876
\(305\) 1.89310e8 0.382054
\(306\) −3.85262e7 −0.0768654
\(307\) −3.91709e8 −0.772644 −0.386322 0.922364i \(-0.626255\pi\)
−0.386322 + 0.922364i \(0.626255\pi\)
\(308\) −7.39535e8 −1.44222
\(309\) −3.87782e8 −0.747710
\(310\) −3.02346e7 −0.0576419
\(311\) −2.04936e8 −0.386328 −0.193164 0.981166i \(-0.561875\pi\)
−0.193164 + 0.981166i \(0.561875\pi\)
\(312\) −5.25588e7 −0.0979726
\(313\) 8.77202e8 1.61694 0.808471 0.588536i \(-0.200295\pi\)
0.808471 + 0.588536i \(0.200295\pi\)
\(314\) −2.70268e8 −0.492652
\(315\) 1.30975e8 0.236103
\(316\) 2.43676e8 0.434418
\(317\) −4.40831e8 −0.777256 −0.388628 0.921395i \(-0.627051\pi\)
−0.388628 + 0.921395i \(0.627051\pi\)
\(318\) 1.69793e8 0.296090
\(319\) −3.05085e8 −0.526202
\(320\) −2.98844e7 −0.0509823
\(321\) 1.74269e8 0.294071
\(322\) −5.22576e8 −0.872277
\(323\) −1.64225e8 −0.271163
\(324\) 3.40122e7 0.0555556
\(325\) 2.47620e8 0.400124
\(326\) 5.01327e8 0.801419
\(327\) −2.38791e7 −0.0377659
\(328\) −3.27208e8 −0.511995
\(329\) 6.83631e8 1.05837
\(330\) −1.80543e8 −0.276555
\(331\) 1.11223e9 1.68576 0.842882 0.538099i \(-0.180857\pi\)
0.842882 + 0.538099i \(0.180857\pi\)
\(332\) 1.42686e8 0.213992
\(333\) −2.65837e7 −0.0394513
\(334\) 5.01658e8 0.736707
\(335\) 3.75155e8 0.545198
\(336\) −1.74293e8 −0.250664
\(337\) 2.88198e8 0.410191 0.205096 0.978742i \(-0.434249\pi\)
0.205096 + 0.978742i \(0.434249\pi\)
\(338\) −3.86347e8 −0.544213
\(339\) 3.27577e8 0.456683
\(340\) 4.81974e7 0.0665039
\(341\) 2.43070e8 0.331965
\(342\) 1.44984e8 0.195987
\(343\) −1.31862e9 −1.76438
\(344\) −8.00829e7 −0.106068
\(345\) −1.27577e8 −0.167265
\(346\) −2.16417e8 −0.280883
\(347\) −1.10601e9 −1.42103 −0.710517 0.703680i \(-0.751539\pi\)
−0.710517 + 0.703680i \(0.751539\pi\)
\(348\) −7.19021e7 −0.0914564
\(349\) −1.32184e9 −1.66453 −0.832264 0.554379i \(-0.812956\pi\)
−0.832264 + 0.554379i \(0.812956\pi\)
\(350\) 8.21146e8 1.02372
\(351\) −7.48348e7 −0.0923695
\(352\) 2.40255e8 0.293612
\(353\) 1.20395e9 1.45679 0.728396 0.685157i \(-0.240266\pi\)
0.728396 + 0.685157i \(0.240266\pi\)
\(354\) 1.60950e8 0.192832
\(355\) −6.51641e8 −0.773053
\(356\) 3.83437e8 0.450422
\(357\) 2.81099e8 0.326979
\(358\) −1.07425e9 −1.23741
\(359\) −1.32057e9 −1.50637 −0.753185 0.657809i \(-0.771484\pi\)
−0.753185 + 0.657809i \(0.771484\pi\)
\(360\) −4.25503e7 −0.0480666
\(361\) −2.75852e8 −0.308604
\(362\) 9.17284e8 1.01630
\(363\) 9.25318e8 1.01536
\(364\) 3.83485e8 0.416767
\(365\) −3.03228e8 −0.326396
\(366\) −3.58693e8 −0.382419
\(367\) 1.75107e9 1.84915 0.924575 0.381000i \(-0.124420\pi\)
0.924575 + 0.381000i \(0.124420\pi\)
\(368\) 1.69771e8 0.177581
\(369\) −4.65888e8 −0.482713
\(370\) 3.32570e7 0.0341332
\(371\) −1.23886e9 −1.25954
\(372\) 5.72867e7 0.0576970
\(373\) −4.87945e8 −0.486844 −0.243422 0.969920i \(-0.578270\pi\)
−0.243422 + 0.969920i \(0.578270\pi\)
\(374\) −3.87482e8 −0.383001
\(375\) 4.40936e8 0.431783
\(376\) −2.22093e8 −0.215466
\(377\) 1.58201e8 0.152060
\(378\) −2.48163e8 −0.236329
\(379\) 1.11007e9 1.04740 0.523700 0.851903i \(-0.324551\pi\)
0.523700 + 0.851903i \(0.324551\pi\)
\(380\) −1.81379e8 −0.169568
\(381\) 1.85438e8 0.171775
\(382\) 1.30884e8 0.120134
\(383\) 1.86912e9 1.69997 0.849983 0.526810i \(-0.176612\pi\)
0.849983 + 0.526810i \(0.176612\pi\)
\(384\) 5.66231e7 0.0510310
\(385\) 1.31730e9 1.17644
\(386\) −1.23359e9 −1.09173
\(387\) −1.14024e8 −0.100002
\(388\) −2.59848e8 −0.225844
\(389\) −2.73895e8 −0.235918 −0.117959 0.993018i \(-0.537635\pi\)
−0.117959 + 0.993018i \(0.537635\pi\)
\(390\) 9.36204e7 0.0799180
\(391\) −2.73805e8 −0.231645
\(392\) 8.50039e8 0.712751
\(393\) −1.06706e9 −0.886779
\(394\) 6.65831e8 0.548437
\(395\) −4.34048e8 −0.354363
\(396\) 3.42082e8 0.276820
\(397\) 6.24552e8 0.500958 0.250479 0.968122i \(-0.419412\pi\)
0.250479 + 0.968122i \(0.419412\pi\)
\(398\) −6.09378e8 −0.484502
\(399\) −1.05784e9 −0.833712
\(400\) −2.66768e8 −0.208413
\(401\) 5.55500e8 0.430208 0.215104 0.976591i \(-0.430991\pi\)
0.215104 + 0.976591i \(0.430991\pi\)
\(402\) −7.10821e8 −0.545719
\(403\) −1.26044e8 −0.0959299
\(404\) −1.10604e9 −0.834522
\(405\) −6.05843e7 −0.0453176
\(406\) 5.24619e8 0.389048
\(407\) −2.67369e8 −0.196576
\(408\) −9.13213e7 −0.0665674
\(409\) −2.15770e9 −1.55941 −0.779704 0.626149i \(-0.784630\pi\)
−0.779704 + 0.626149i \(0.784630\pi\)
\(410\) 5.82839e8 0.417643
\(411\) 5.17700e8 0.367817
\(412\) −9.19188e8 −0.647536
\(413\) −1.17434e9 −0.820293
\(414\) 2.41725e8 0.167425
\(415\) −2.54159e8 −0.174557
\(416\) −1.24584e8 −0.0848468
\(417\) 3.57612e8 0.241511
\(418\) 1.45819e9 0.976555
\(419\) 1.67797e9 1.11438 0.557191 0.830384i \(-0.311879\pi\)
0.557191 + 0.830384i \(0.311879\pi\)
\(420\) 3.10459e8 0.204471
\(421\) −5.25233e8 −0.343056 −0.171528 0.985179i \(-0.554870\pi\)
−0.171528 + 0.985179i \(0.554870\pi\)
\(422\) 2.81957e8 0.182637
\(423\) −3.16223e8 −0.203143
\(424\) 4.02472e8 0.256422
\(425\) 4.30242e8 0.271864
\(426\) 1.23469e9 0.773791
\(427\) 2.61713e9 1.62678
\(428\) 4.13082e8 0.254673
\(429\) −7.52659e8 −0.460254
\(430\) 1.42648e8 0.0865218
\(431\) 1.70593e8 0.102634 0.0513169 0.998682i \(-0.483658\pi\)
0.0513169 + 0.998682i \(0.483658\pi\)
\(432\) 8.06216e7 0.0481125
\(433\) −1.68797e9 −0.999210 −0.499605 0.866253i \(-0.666522\pi\)
−0.499605 + 0.866253i \(0.666522\pi\)
\(434\) −4.17980e8 −0.245438
\(435\) 1.28076e8 0.0746026
\(436\) −5.66022e7 −0.0327063
\(437\) 1.03040e9 0.590636
\(438\) 5.74538e8 0.326708
\(439\) −1.17850e9 −0.664817 −0.332409 0.943135i \(-0.607861\pi\)
−0.332409 + 0.943135i \(0.607861\pi\)
\(440\) −4.27954e8 −0.239504
\(441\) 1.21031e9 0.671988
\(442\) 2.00928e8 0.110678
\(443\) 7.15755e8 0.391157 0.195579 0.980688i \(-0.437342\pi\)
0.195579 + 0.980688i \(0.437342\pi\)
\(444\) −6.30132e7 −0.0341658
\(445\) −6.82998e8 −0.367417
\(446\) −1.51305e9 −0.807573
\(447\) 1.54879e9 0.820191
\(448\) −4.13139e8 −0.217082
\(449\) −1.37358e9 −0.716132 −0.358066 0.933696i \(-0.616564\pi\)
−0.358066 + 0.933696i \(0.616564\pi\)
\(450\) −3.79832e8 −0.196493
\(451\) −4.68572e9 −2.40524
\(452\) 7.76480e8 0.395499
\(453\) −8.39357e8 −0.424231
\(454\) −1.40880e9 −0.706569
\(455\) −6.83083e8 −0.339964
\(456\) 3.43665e8 0.169730
\(457\) 1.84752e9 0.905488 0.452744 0.891641i \(-0.350445\pi\)
0.452744 + 0.891641i \(0.350445\pi\)
\(458\) 5.20317e8 0.253069
\(459\) −1.30026e8 −0.0627604
\(460\) −3.02405e8 −0.144856
\(461\) 3.09414e9 1.47091 0.735455 0.677573i \(-0.236968\pi\)
0.735455 + 0.677573i \(0.236968\pi\)
\(462\) −2.49593e9 −1.17757
\(463\) 3.00451e9 1.40682 0.703412 0.710782i \(-0.251659\pi\)
0.703412 + 0.710782i \(0.251659\pi\)
\(464\) −1.70435e8 −0.0792036
\(465\) −1.02042e8 −0.0470645
\(466\) −2.01055e9 −0.920373
\(467\) −2.99252e9 −1.35965 −0.679825 0.733374i \(-0.737944\pi\)
−0.679825 + 0.733374i \(0.737944\pi\)
\(468\) −1.77386e8 −0.0799943
\(469\) 5.18636e9 2.32144
\(470\) 3.95604e8 0.175759
\(471\) −9.12154e8 −0.402249
\(472\) 3.81512e8 0.166998
\(473\) −1.14681e9 −0.498286
\(474\) 8.22407e8 0.354701
\(475\) −1.61911e9 −0.693183
\(476\) 6.66308e8 0.283172
\(477\) 5.73051e8 0.241757
\(478\) 1.70464e9 0.713894
\(479\) 1.84041e9 0.765141 0.382570 0.923926i \(-0.375039\pi\)
0.382570 + 0.923926i \(0.375039\pi\)
\(480\) −1.00860e8 −0.0416269
\(481\) 1.38644e8 0.0568058
\(482\) 2.05827e9 0.837216
\(483\) −1.76370e9 −0.712211
\(484\) 2.19335e9 0.879323
\(485\) 4.62854e8 0.184225
\(486\) 1.14791e8 0.0453609
\(487\) −4.26676e8 −0.167397 −0.0836983 0.996491i \(-0.526673\pi\)
−0.0836983 + 0.996491i \(0.526673\pi\)
\(488\) −8.50236e8 −0.331185
\(489\) 1.69198e9 0.654356
\(490\) −1.51413e9 −0.581403
\(491\) 6.07547e7 0.0231630 0.0115815 0.999933i \(-0.496313\pi\)
0.0115815 + 0.999933i \(0.496313\pi\)
\(492\) −1.10433e9 −0.418042
\(493\) 2.74876e8 0.103317
\(494\) −7.56142e8 −0.282201
\(495\) −6.09333e8 −0.225807
\(496\) 1.35791e8 0.0499671
\(497\) −9.00866e9 −3.29164
\(498\) 4.81565e8 0.174724
\(499\) 3.24588e9 1.16945 0.584723 0.811233i \(-0.301203\pi\)
0.584723 + 0.811233i \(0.301203\pi\)
\(500\) 1.04518e9 0.373935
\(501\) 1.69310e9 0.601519
\(502\) 9.84464e8 0.347326
\(503\) −7.44381e8 −0.260800 −0.130400 0.991461i \(-0.541626\pi\)
−0.130400 + 0.991461i \(0.541626\pi\)
\(504\) −5.88239e8 −0.204667
\(505\) 1.97014e9 0.680734
\(506\) 2.43117e9 0.834237
\(507\) −1.30392e9 −0.444348
\(508\) 4.39556e8 0.148762
\(509\) −4.44155e8 −0.149287 −0.0746436 0.997210i \(-0.523782\pi\)
−0.0746436 + 0.997210i \(0.523782\pi\)
\(510\) 1.62666e8 0.0543002
\(511\) −4.19200e9 −1.38979
\(512\) 1.34218e8 0.0441942
\(513\) 4.89319e8 0.160023
\(514\) −3.54667e9 −1.15199
\(515\) 1.63730e9 0.528207
\(516\) −2.70280e8 −0.0866044
\(517\) −3.18045e9 −1.01221
\(518\) 4.59763e8 0.145338
\(519\) −7.30407e8 −0.229340
\(520\) 2.21915e8 0.0692110
\(521\) 3.04963e9 0.944745 0.472372 0.881399i \(-0.343398\pi\)
0.472372 + 0.881399i \(0.343398\pi\)
\(522\) −2.42670e8 −0.0746738
\(523\) −1.40306e9 −0.428866 −0.214433 0.976739i \(-0.568790\pi\)
−0.214433 + 0.976739i \(0.568790\pi\)
\(524\) −2.52933e9 −0.767973
\(525\) 2.77137e9 0.835866
\(526\) 2.39140e9 0.716476
\(527\) −2.19002e8 −0.0651795
\(528\) 8.10861e8 0.239733
\(529\) −1.68689e9 −0.495441
\(530\) −7.16903e8 −0.209168
\(531\) 5.43207e8 0.157447
\(532\) −2.50748e9 −0.722016
\(533\) 2.42977e9 0.695058
\(534\) 1.29410e9 0.367768
\(535\) −7.35802e8 −0.207741
\(536\) −1.68491e9 −0.472606
\(537\) −3.62558e9 −1.01034
\(538\) 1.67126e9 0.462708
\(539\) 1.21728e10 3.34835
\(540\) −1.43607e8 −0.0392462
\(541\) 4.21106e9 1.14341 0.571704 0.820460i \(-0.306283\pi\)
0.571704 + 0.820460i \(0.306283\pi\)
\(542\) −9.01994e7 −0.0243336
\(543\) 3.09583e9 0.829809
\(544\) −2.16465e8 −0.0576491
\(545\) 1.00823e8 0.0266791
\(546\) 1.29426e9 0.340289
\(547\) 1.99956e9 0.522371 0.261185 0.965289i \(-0.415887\pi\)
0.261185 + 0.965289i \(0.415887\pi\)
\(548\) 1.22714e9 0.318539
\(549\) −1.21059e9 −0.312244
\(550\) −3.82021e9 −0.979078
\(551\) −1.03442e9 −0.263432
\(552\) 5.72977e8 0.144994
\(553\) −6.00053e9 −1.50887
\(554\) −5.27172e9 −1.31725
\(555\) 1.12242e8 0.0278696
\(556\) 8.47674e8 0.209154
\(557\) −3.37403e9 −0.827287 −0.413643 0.910439i \(-0.635744\pi\)
−0.413643 + 0.910439i \(0.635744\pi\)
\(558\) 1.93342e8 0.0471094
\(559\) 5.94678e8 0.143993
\(560\) 7.35904e8 0.177077
\(561\) −1.30775e9 −0.312719
\(562\) −8.40983e8 −0.199853
\(563\) −5.58021e9 −1.31787 −0.658933 0.752201i \(-0.728992\pi\)
−0.658933 + 0.752201i \(0.728992\pi\)
\(564\) −7.49565e8 −0.175927
\(565\) −1.38310e9 −0.322616
\(566\) 2.64129e9 0.612292
\(567\) −8.37551e8 −0.192961
\(568\) 2.92667e9 0.670123
\(569\) −8.88310e8 −0.202149 −0.101074 0.994879i \(-0.532228\pi\)
−0.101074 + 0.994879i \(0.532228\pi\)
\(570\) −6.12153e8 −0.138452
\(571\) −1.79171e9 −0.402755 −0.201377 0.979514i \(-0.564542\pi\)
−0.201377 + 0.979514i \(0.564542\pi\)
\(572\) −1.78408e9 −0.398592
\(573\) 4.41734e8 0.0980888
\(574\) 8.05750e9 1.77831
\(575\) −2.69947e9 −0.592162
\(576\) 1.91103e8 0.0416667
\(577\) 3.82103e9 0.828066 0.414033 0.910262i \(-0.364120\pi\)
0.414033 + 0.910262i \(0.364120\pi\)
\(578\) −2.93360e9 −0.631906
\(579\) −4.16336e9 −0.891392
\(580\) 3.03587e8 0.0646077
\(581\) −3.51364e9 −0.743260
\(582\) −8.76987e8 −0.184401
\(583\) 5.76352e9 1.20461
\(584\) 1.36187e9 0.282937
\(585\) 3.15969e8 0.0652528
\(586\) −6.96802e8 −0.143043
\(587\) 4.36219e9 0.890166 0.445083 0.895489i \(-0.353174\pi\)
0.445083 + 0.895489i \(0.353174\pi\)
\(588\) 2.86888e9 0.581959
\(589\) 8.24159e8 0.166191
\(590\) −6.79568e8 −0.136223
\(591\) 2.24718e9 0.447797
\(592\) −1.49365e8 −0.0295884
\(593\) 6.38531e9 1.25745 0.628724 0.777628i \(-0.283577\pi\)
0.628724 + 0.777628i \(0.283577\pi\)
\(594\) 1.15453e9 0.226022
\(595\) −1.18686e9 −0.230988
\(596\) 3.67120e9 0.710306
\(597\) −2.05665e9 −0.395594
\(598\) −1.26068e9 −0.241075
\(599\) 8.04297e8 0.152905 0.0764527 0.997073i \(-0.475641\pi\)
0.0764527 + 0.997073i \(0.475641\pi\)
\(600\) −9.00343e8 −0.170168
\(601\) −4.87162e9 −0.915403 −0.457702 0.889106i \(-0.651327\pi\)
−0.457702 + 0.889106i \(0.651327\pi\)
\(602\) 1.97204e9 0.368408
\(603\) −2.39902e9 −0.445577
\(604\) −1.98959e9 −0.367395
\(605\) −3.90690e9 −0.717280
\(606\) −3.73290e9 −0.681384
\(607\) 7.17517e9 1.30218 0.651091 0.759000i \(-0.274312\pi\)
0.651091 + 0.759000i \(0.274312\pi\)
\(608\) 8.14612e8 0.146990
\(609\) 1.77059e9 0.317656
\(610\) 1.51448e9 0.270153
\(611\) 1.64922e9 0.292505
\(612\) −3.08210e8 −0.0543521
\(613\) 3.47891e9 0.610002 0.305001 0.952352i \(-0.401343\pi\)
0.305001 + 0.952352i \(0.401343\pi\)
\(614\) −3.13367e9 −0.546342
\(615\) 1.96708e9 0.341004
\(616\) −5.91628e9 −1.01980
\(617\) −2.39378e8 −0.0410286 −0.0205143 0.999790i \(-0.506530\pi\)
−0.0205143 + 0.999790i \(0.506530\pi\)
\(618\) −3.10226e9 −0.528711
\(619\) −5.52959e9 −0.937078 −0.468539 0.883443i \(-0.655219\pi\)
−0.468539 + 0.883443i \(0.655219\pi\)
\(620\) −2.41877e8 −0.0407590
\(621\) 8.15821e8 0.136702
\(622\) −1.63949e9 −0.273175
\(623\) −9.44215e9 −1.56445
\(624\) −4.20471e8 −0.0692771
\(625\) 3.22647e9 0.528626
\(626\) 7.01762e9 1.14335
\(627\) 4.92139e9 0.797354
\(628\) −2.16214e9 −0.348358
\(629\) 2.40894e8 0.0385966
\(630\) 1.04780e9 0.166950
\(631\) −6.13683e9 −0.972392 −0.486196 0.873850i \(-0.661616\pi\)
−0.486196 + 0.873850i \(0.661616\pi\)
\(632\) 1.94941e9 0.307180
\(633\) 9.51603e8 0.149122
\(634\) −3.52664e9 −0.549603
\(635\) −7.82959e8 −0.121348
\(636\) 1.35834e9 0.209368
\(637\) −6.31221e9 −0.967594
\(638\) −2.44068e9 −0.372081
\(639\) 4.16707e9 0.631798
\(640\) −2.39075e8 −0.0360500
\(641\) 1.07038e10 1.60522 0.802611 0.596503i \(-0.203444\pi\)
0.802611 + 0.596503i \(0.203444\pi\)
\(642\) 1.39415e9 0.207940
\(643\) −1.39803e9 −0.207385 −0.103692 0.994609i \(-0.533066\pi\)
−0.103692 + 0.994609i \(0.533066\pi\)
\(644\) −4.18061e9 −0.616793
\(645\) 4.81436e8 0.0706447
\(646\) −1.31380e9 −0.191741
\(647\) −5.31605e9 −0.771656 −0.385828 0.922571i \(-0.626084\pi\)
−0.385828 + 0.922571i \(0.626084\pi\)
\(648\) 2.72098e8 0.0392837
\(649\) 5.46337e9 0.784520
\(650\) 1.98096e9 0.282930
\(651\) −1.41068e9 −0.200399
\(652\) 4.01062e9 0.566689
\(653\) 3.24403e9 0.455921 0.227960 0.973670i \(-0.426794\pi\)
0.227960 + 0.973670i \(0.426794\pi\)
\(654\) −1.91033e8 −0.0267046
\(655\) 4.50537e9 0.626449
\(656\) −2.61766e9 −0.362035
\(657\) 1.93907e9 0.266756
\(658\) 5.46905e9 0.748378
\(659\) −5.16506e9 −0.703034 −0.351517 0.936181i \(-0.614334\pi\)
−0.351517 + 0.936181i \(0.614334\pi\)
\(660\) −1.44435e9 −0.195554
\(661\) −3.22515e9 −0.434356 −0.217178 0.976132i \(-0.569685\pi\)
−0.217178 + 0.976132i \(0.569685\pi\)
\(662\) 8.89784e9 1.19201
\(663\) 6.78132e8 0.0903685
\(664\) 1.14149e9 0.151315
\(665\) 4.46645e9 0.588961
\(666\) −2.12670e8 −0.0278963
\(667\) −1.72465e9 −0.225041
\(668\) 4.01326e9 0.520931
\(669\) −5.10655e9 −0.659380
\(670\) 3.00124e9 0.385513
\(671\) −1.21757e10 −1.55583
\(672\) −1.39434e9 −0.177246
\(673\) −2.00633e9 −0.253718 −0.126859 0.991921i \(-0.540490\pi\)
−0.126859 + 0.991921i \(0.540490\pi\)
\(674\) 2.30559e9 0.290049
\(675\) −1.28193e9 −0.160436
\(676\) −3.09077e9 −0.384816
\(677\) 1.00211e10 1.24124 0.620619 0.784112i \(-0.286881\pi\)
0.620619 + 0.784112i \(0.286881\pi\)
\(678\) 2.62062e9 0.322924
\(679\) 6.39876e9 0.784425
\(680\) 3.85579e8 0.0470254
\(681\) −4.75471e9 −0.576911
\(682\) 1.94456e9 0.234734
\(683\) −5.84861e9 −0.702393 −0.351196 0.936302i \(-0.614225\pi\)
−0.351196 + 0.936302i \(0.614225\pi\)
\(684\) 1.15987e9 0.138584
\(685\) −2.18584e9 −0.259838
\(686\) −1.05490e10 −1.24760
\(687\) 1.75607e9 0.206630
\(688\) −6.40664e8 −0.0750016
\(689\) −2.98867e9 −0.348105
\(690\) −1.02062e9 −0.118274
\(691\) 2.58686e9 0.298263 0.149131 0.988817i \(-0.452352\pi\)
0.149131 + 0.988817i \(0.452352\pi\)
\(692\) −1.73134e9 −0.198614
\(693\) −8.42376e9 −0.961479
\(694\) −8.84805e9 −1.00482
\(695\) −1.50992e9 −0.170611
\(696\) −5.75217e8 −0.0646694
\(697\) 4.22175e9 0.472256
\(698\) −1.05747e10 −1.17700
\(699\) −6.78560e9 −0.751481
\(700\) 6.56917e9 0.723881
\(701\) −1.74460e9 −0.191286 −0.0956429 0.995416i \(-0.530491\pi\)
−0.0956429 + 0.995416i \(0.530491\pi\)
\(702\) −5.98678e8 −0.0653151
\(703\) −9.06545e8 −0.0984114
\(704\) 1.92204e9 0.207615
\(705\) 1.33516e9 0.143507
\(706\) 9.63161e9 1.03011
\(707\) 2.72363e10 2.89855
\(708\) 1.28760e9 0.136353
\(709\) −1.12051e10 −1.18074 −0.590368 0.807134i \(-0.701017\pi\)
−0.590368 + 0.807134i \(0.701017\pi\)
\(710\) −5.21313e9 −0.546631
\(711\) 2.77562e9 0.289612
\(712\) 3.06750e9 0.318496
\(713\) 1.37408e9 0.141971
\(714\) 2.24879e9 0.231209
\(715\) 3.17789e9 0.325138
\(716\) −8.59398e9 −0.874981
\(717\) 5.75314e9 0.582892
\(718\) −1.05646e10 −1.06516
\(719\) −9.36568e8 −0.0939698 −0.0469849 0.998896i \(-0.514961\pi\)
−0.0469849 + 0.998896i \(0.514961\pi\)
\(720\) −3.40402e8 −0.0339882
\(721\) 2.26350e10 2.24909
\(722\) −2.20682e9 −0.218216
\(723\) 6.94666e9 0.683584
\(724\) 7.33827e9 0.718636
\(725\) 2.71002e9 0.264113
\(726\) 7.40255e9 0.717965
\(727\) 4.20445e9 0.405825 0.202913 0.979197i \(-0.434959\pi\)
0.202913 + 0.979197i \(0.434959\pi\)
\(728\) 3.06788e9 0.294699
\(729\) 3.87420e8 0.0370370
\(730\) −2.42583e9 −0.230797
\(731\) 1.03326e9 0.0978358
\(732\) −2.86955e9 −0.270411
\(733\) −1.15491e10 −1.08314 −0.541571 0.840655i \(-0.682170\pi\)
−0.541571 + 0.840655i \(0.682170\pi\)
\(734\) 1.40086e10 1.30755
\(735\) −5.11020e9 −0.474714
\(736\) 1.35817e9 0.125569
\(737\) −2.41284e10 −2.22020
\(738\) −3.72710e9 −0.341330
\(739\) −1.39655e10 −1.27292 −0.636460 0.771310i \(-0.719602\pi\)
−0.636460 + 0.771310i \(0.719602\pi\)
\(740\) 2.66056e8 0.0241358
\(741\) −2.55198e9 −0.230416
\(742\) −9.91087e9 −0.890632
\(743\) 1.43832e10 1.28646 0.643230 0.765673i \(-0.277594\pi\)
0.643230 + 0.765673i \(0.277594\pi\)
\(744\) 4.58293e8 0.0407979
\(745\) −6.53932e9 −0.579409
\(746\) −3.90356e9 −0.344251
\(747\) 1.62528e9 0.142661
\(748\) −3.09985e9 −0.270823
\(749\) −1.01721e10 −0.884557
\(750\) 3.52749e9 0.305317
\(751\) −6.70841e8 −0.0577936 −0.0288968 0.999582i \(-0.509199\pi\)
−0.0288968 + 0.999582i \(0.509199\pi\)
\(752\) −1.77675e9 −0.152357
\(753\) 3.32257e9 0.283590
\(754\) 1.26561e9 0.107523
\(755\) 3.54395e9 0.299691
\(756\) −1.98531e9 −0.167110
\(757\) −1.91569e10 −1.60506 −0.802529 0.596613i \(-0.796513\pi\)
−0.802529 + 0.596613i \(0.796513\pi\)
\(758\) 8.88055e9 0.740624
\(759\) 8.20521e9 0.681151
\(760\) −1.45103e9 −0.119903
\(761\) 1.79120e10 1.47332 0.736660 0.676263i \(-0.236402\pi\)
0.736660 + 0.676263i \(0.236402\pi\)
\(762\) 1.48350e9 0.121463
\(763\) 1.39383e9 0.113599
\(764\) 1.04707e9 0.0849474
\(765\) 5.48998e8 0.0443359
\(766\) 1.49529e10 1.20206
\(767\) −2.83302e9 −0.226708
\(768\) 4.52985e8 0.0360844
\(769\) −2.14072e10 −1.69753 −0.848765 0.528771i \(-0.822653\pi\)
−0.848765 + 0.528771i \(0.822653\pi\)
\(770\) 1.05384e10 0.831871
\(771\) −1.19700e10 −0.940599
\(772\) −9.86870e9 −0.771968
\(773\) −7.55163e8 −0.0588047 −0.0294024 0.999568i \(-0.509360\pi\)
−0.0294024 + 0.999568i \(0.509360\pi\)
\(774\) −9.12195e8 −0.0707122
\(775\) −2.15916e9 −0.166620
\(776\) −2.07878e9 −0.159696
\(777\) 1.55170e9 0.118668
\(778\) −2.19116e9 −0.166819
\(779\) −1.58875e10 −1.20413
\(780\) 7.48964e8 0.0565106
\(781\) 4.19108e10 3.14809
\(782\) −2.19044e9 −0.163798
\(783\) −8.19010e8 −0.0609709
\(784\) 6.80031e9 0.503991
\(785\) 3.85132e9 0.284162
\(786\) −8.53649e9 −0.627047
\(787\) 2.04665e10 1.49669 0.748347 0.663308i \(-0.230848\pi\)
0.748347 + 0.663308i \(0.230848\pi\)
\(788\) 5.32664e9 0.387804
\(789\) 8.07097e9 0.585000
\(790\) −3.47239e9 −0.250572
\(791\) −1.91208e10 −1.37369
\(792\) 2.73665e9 0.195741
\(793\) 6.31367e9 0.449599
\(794\) 4.99641e9 0.354231
\(795\) −2.41955e9 −0.170785
\(796\) −4.87502e9 −0.342595
\(797\) −1.03098e10 −0.721348 −0.360674 0.932692i \(-0.617453\pi\)
−0.360674 + 0.932692i \(0.617453\pi\)
\(798\) −8.46274e9 −0.589523
\(799\) 2.86552e9 0.198742
\(800\) −2.13415e9 −0.147370
\(801\) 4.36759e9 0.300281
\(802\) 4.44400e9 0.304203
\(803\) 1.95024e10 1.32918
\(804\) −5.68656e9 −0.385881
\(805\) 7.44671e9 0.503129
\(806\) −1.00835e9 −0.0678327
\(807\) 5.64051e9 0.377799
\(808\) −8.84835e9 −0.590096
\(809\) −4.16428e8 −0.0276516 −0.0138258 0.999904i \(-0.504401\pi\)
−0.0138258 + 0.999904i \(0.504401\pi\)
\(810\) −4.84674e8 −0.0320444
\(811\) −5.82687e9 −0.383586 −0.191793 0.981435i \(-0.561430\pi\)
−0.191793 + 0.981435i \(0.561430\pi\)
\(812\) 4.19695e9 0.275098
\(813\) −3.04423e8 −0.0198683
\(814\) −2.13895e9 −0.139000
\(815\) −7.14391e9 −0.462258
\(816\) −7.30571e8 −0.0470703
\(817\) −3.88840e9 −0.249456
\(818\) −1.72616e10 −1.10267
\(819\) 4.36813e9 0.277845
\(820\) 4.66271e9 0.295318
\(821\) −2.08333e10 −1.31388 −0.656941 0.753942i \(-0.728150\pi\)
−0.656941 + 0.753942i \(0.728150\pi\)
\(822\) 4.14160e9 0.260086
\(823\) 4.23403e9 0.264761 0.132381 0.991199i \(-0.457738\pi\)
0.132381 + 0.991199i \(0.457738\pi\)
\(824\) −7.35350e9 −0.457877
\(825\) −1.28932e10 −0.799414
\(826\) −9.39473e9 −0.580035
\(827\) 5.70597e9 0.350800 0.175400 0.984497i \(-0.443878\pi\)
0.175400 + 0.984497i \(0.443878\pi\)
\(828\) 1.93380e9 0.118387
\(829\) 2.51612e10 1.53388 0.766940 0.641719i \(-0.221779\pi\)
0.766940 + 0.641719i \(0.221779\pi\)
\(830\) −2.03327e9 −0.123431
\(831\) −1.77920e10 −1.07553
\(832\) −9.96671e8 −0.0599957
\(833\) −1.09675e10 −0.657431
\(834\) 2.86090e9 0.170774
\(835\) −7.14862e9 −0.424932
\(836\) 1.16655e10 0.690528
\(837\) 6.52531e8 0.0384647
\(838\) 1.34237e10 0.787988
\(839\) 2.27048e10 1.32724 0.663622 0.748068i \(-0.269018\pi\)
0.663622 + 0.748068i \(0.269018\pi\)
\(840\) 2.48368e9 0.144583
\(841\) −1.55185e10 −0.899629
\(842\) −4.20187e9 −0.242577
\(843\) −2.83832e9 −0.163179
\(844\) 2.25565e9 0.129144
\(845\) 5.50544e9 0.313901
\(846\) −2.52978e9 −0.143644
\(847\) −5.40112e10 −3.05416
\(848\) 3.21978e9 0.181318
\(849\) 8.91435e9 0.499934
\(850\) 3.44194e9 0.192237
\(851\) −1.51144e9 −0.0840695
\(852\) 9.87751e9 0.547153
\(853\) 2.86872e10 1.58258 0.791292 0.611439i \(-0.209409\pi\)
0.791292 + 0.611439i \(0.209409\pi\)
\(854\) 2.09371e10 1.15031
\(855\) −2.06602e9 −0.113045
\(856\) 3.30465e9 0.180081
\(857\) 5.34950e9 0.290322 0.145161 0.989408i \(-0.453630\pi\)
0.145161 + 0.989408i \(0.453630\pi\)
\(858\) −6.02127e9 −0.325449
\(859\) −1.40330e10 −0.755394 −0.377697 0.925929i \(-0.623284\pi\)
−0.377697 + 0.925929i \(0.623284\pi\)
\(860\) 1.14118e9 0.0611801
\(861\) 2.71940e10 1.45199
\(862\) 1.36474e9 0.0725731
\(863\) 3.24994e10 1.72122 0.860612 0.509261i \(-0.170081\pi\)
0.860612 + 0.509261i \(0.170081\pi\)
\(864\) 6.44973e8 0.0340207
\(865\) 3.08394e9 0.162013
\(866\) −1.35038e10 −0.706549
\(867\) −9.90088e9 −0.515949
\(868\) −3.34384e9 −0.173551
\(869\) 2.79162e10 1.44307
\(870\) 1.02460e9 0.0527520
\(871\) 1.25118e10 0.641586
\(872\) −4.52818e8 −0.0231268
\(873\) −2.95983e9 −0.150563
\(874\) 8.24318e9 0.417642
\(875\) −2.57376e10 −1.29879
\(876\) 4.59630e9 0.231017
\(877\) −3.38694e10 −1.69554 −0.847772 0.530361i \(-0.822056\pi\)
−0.847772 + 0.530361i \(0.822056\pi\)
\(878\) −9.42797e9 −0.470097
\(879\) −2.35171e9 −0.116794
\(880\) −3.42363e9 −0.169355
\(881\) −1.52708e10 −0.752397 −0.376198 0.926539i \(-0.622769\pi\)
−0.376198 + 0.926539i \(0.622769\pi\)
\(882\) 9.68248e9 0.475167
\(883\) −1.12045e10 −0.547685 −0.273842 0.961775i \(-0.588295\pi\)
−0.273842 + 0.961775i \(0.588295\pi\)
\(884\) 1.60742e9 0.0782614
\(885\) −2.29354e9 −0.111226
\(886\) 5.72604e9 0.276590
\(887\) 6.97232e9 0.335463 0.167732 0.985833i \(-0.446356\pi\)
0.167732 + 0.985833i \(0.446356\pi\)
\(888\) −5.04106e8 −0.0241589
\(889\) −1.08241e10 −0.516695
\(890\) −5.46398e9 −0.259803
\(891\) 3.89653e9 0.184546
\(892\) −1.21044e10 −0.571040
\(893\) −1.07837e10 −0.506742
\(894\) 1.23903e10 0.579963
\(895\) 1.53080e10 0.713737
\(896\) −3.30511e9 −0.153500
\(897\) −4.25480e9 −0.196837
\(898\) −1.09887e10 −0.506382
\(899\) −1.37945e9 −0.0633211
\(900\) −3.03866e9 −0.138942
\(901\) −5.19283e9 −0.236520
\(902\) −3.74858e10 −1.70076
\(903\) 6.65564e9 0.300804
\(904\) 6.21184e9 0.279660
\(905\) −1.30713e10 −0.586204
\(906\) −6.71485e9 −0.299977
\(907\) 1.18095e9 0.0525539 0.0262770 0.999655i \(-0.491635\pi\)
0.0262770 + 0.999655i \(0.491635\pi\)
\(908\) −1.12704e10 −0.499620
\(909\) −1.25985e10 −0.556348
\(910\) −5.46466e9 −0.240391
\(911\) 1.27915e10 0.560541 0.280271 0.959921i \(-0.409576\pi\)
0.280271 + 0.959921i \(0.409576\pi\)
\(912\) 2.74932e9 0.120017
\(913\) 1.63465e10 0.710847
\(914\) 1.47802e10 0.640276
\(915\) 5.11138e9 0.220579
\(916\) 4.16254e9 0.178947
\(917\) 6.22848e10 2.66741
\(918\) −1.04021e9 −0.0443783
\(919\) 3.52353e10 1.49752 0.748761 0.662840i \(-0.230649\pi\)
0.748761 + 0.662840i \(0.230649\pi\)
\(920\) −2.41924e9 −0.102429
\(921\) −1.05761e10 −0.446086
\(922\) 2.47531e10 1.04009
\(923\) −2.17328e10 −0.909725
\(924\) −1.99674e10 −0.832665
\(925\) 2.37499e9 0.0986658
\(926\) 2.40361e10 0.994775
\(927\) −1.04701e10 −0.431691
\(928\) −1.36348e9 −0.0560054
\(929\) −2.17764e10 −0.891111 −0.445556 0.895254i \(-0.646994\pi\)
−0.445556 + 0.895254i \(0.646994\pi\)
\(930\) −8.16335e8 −0.0332796
\(931\) 4.12734e10 1.67628
\(932\) −1.60844e10 −0.650802
\(933\) −5.53327e9 −0.223047
\(934\) −2.39401e10 −0.961418
\(935\) 5.52161e9 0.220915
\(936\) −1.41909e9 −0.0565645
\(937\) 1.15795e10 0.459833 0.229916 0.973210i \(-0.426155\pi\)
0.229916 + 0.973210i \(0.426155\pi\)
\(938\) 4.14909e10 1.64151
\(939\) 2.36845e10 0.933542
\(940\) 3.16483e9 0.124280
\(941\) −3.83930e10 −1.50207 −0.751033 0.660265i \(-0.770444\pi\)
−0.751033 + 0.660265i \(0.770444\pi\)
\(942\) −7.29723e9 −0.284433
\(943\) −2.64885e10 −1.02865
\(944\) 3.05209e9 0.118085
\(945\) 3.53633e9 0.136314
\(946\) −9.17450e9 −0.352341
\(947\) 1.11841e10 0.427933 0.213966 0.976841i \(-0.431362\pi\)
0.213966 + 0.976841i \(0.431362\pi\)
\(948\) 6.57926e9 0.250812
\(949\) −1.01129e10 −0.384101
\(950\) −1.29529e10 −0.490154
\(951\) −1.19024e10 −0.448749
\(952\) 5.33046e9 0.200233
\(953\) −5.19835e9 −0.194554 −0.0972770 0.995257i \(-0.531013\pi\)
−0.0972770 + 0.995257i \(0.531013\pi\)
\(954\) 4.58441e9 0.170948
\(955\) −1.86510e9 −0.0692931
\(956\) 1.36371e10 0.504799
\(957\) −8.23728e9 −0.303803
\(958\) 1.47233e10 0.541036
\(959\) −3.02183e10 −1.10638
\(960\) −8.06879e8 −0.0294347
\(961\) −2.64136e10 −0.960053
\(962\) 1.10915e9 0.0401677
\(963\) 4.70526e9 0.169782
\(964\) 1.64662e10 0.592001
\(965\) 1.75786e10 0.629708
\(966\) −1.41096e10 −0.503609
\(967\) −2.69243e8 −0.00957529 −0.00478765 0.999989i \(-0.501524\pi\)
−0.00478765 + 0.999989i \(0.501524\pi\)
\(968\) 1.75468e10 0.621776
\(969\) −4.43408e9 −0.156556
\(970\) 3.70283e9 0.130267
\(971\) 4.37283e9 0.153284 0.0766418 0.997059i \(-0.475580\pi\)
0.0766418 + 0.997059i \(0.475580\pi\)
\(972\) 9.18330e8 0.0320750
\(973\) −2.08740e10 −0.726457
\(974\) −3.41341e9 −0.118367
\(975\) 6.68575e9 0.231012
\(976\) −6.80189e9 −0.234183
\(977\) −3.74991e10 −1.28644 −0.643220 0.765681i \(-0.722402\pi\)
−0.643220 + 0.765681i \(0.722402\pi\)
\(978\) 1.35358e10 0.462699
\(979\) 4.39276e10 1.49623
\(980\) −1.21131e10 −0.411114
\(981\) −6.44735e8 −0.0218042
\(982\) 4.86038e8 0.0163787
\(983\) −3.06190e10 −1.02814 −0.514071 0.857748i \(-0.671863\pi\)
−0.514071 + 0.857748i \(0.671863\pi\)
\(984\) −8.83461e9 −0.295600
\(985\) −9.48808e9 −0.316338
\(986\) 2.19901e9 0.0730562
\(987\) 1.84580e10 0.611048
\(988\) −6.04913e9 −0.199546
\(989\) −6.48296e9 −0.213102
\(990\) −4.87467e9 −0.159669
\(991\) −1.72703e10 −0.563693 −0.281847 0.959459i \(-0.590947\pi\)
−0.281847 + 0.959459i \(0.590947\pi\)
\(992\) 1.08632e9 0.0353320
\(993\) 3.00302e10 0.973276
\(994\) −7.20692e10 −2.32754
\(995\) 8.68363e9 0.279461
\(996\) 3.85252e9 0.123548
\(997\) −3.75077e9 −0.119864 −0.0599319 0.998202i \(-0.519088\pi\)
−0.0599319 + 0.998202i \(0.519088\pi\)
\(998\) 2.59670e10 0.826923
\(999\) −7.17760e8 −0.0227772
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 6.8.a.a.1.1 1
3.2 odd 2 18.8.a.a.1.1 1
4.3 odd 2 48.8.a.b.1.1 1
5.2 odd 4 150.8.c.k.49.2 2
5.3 odd 4 150.8.c.k.49.1 2
5.4 even 2 150.8.a.e.1.1 1
7.2 even 3 294.8.e.c.67.1 2
7.3 odd 6 294.8.e.d.79.1 2
7.4 even 3 294.8.e.c.79.1 2
7.5 odd 6 294.8.e.d.67.1 2
7.6 odd 2 294.8.a.l.1.1 1
8.3 odd 2 192.8.a.n.1.1 1
8.5 even 2 192.8.a.f.1.1 1
9.2 odd 6 162.8.c.i.109.1 2
9.4 even 3 162.8.c.d.55.1 2
9.5 odd 6 162.8.c.i.55.1 2
9.7 even 3 162.8.c.d.109.1 2
12.11 even 2 144.8.a.h.1.1 1
15.2 even 4 450.8.c.a.199.1 2
15.8 even 4 450.8.c.a.199.2 2
15.14 odd 2 450.8.a.ba.1.1 1
24.5 odd 2 576.8.a.h.1.1 1
24.11 even 2 576.8.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6.8.a.a.1.1 1 1.1 even 1 trivial
18.8.a.a.1.1 1 3.2 odd 2
48.8.a.b.1.1 1 4.3 odd 2
144.8.a.h.1.1 1 12.11 even 2
150.8.a.e.1.1 1 5.4 even 2
150.8.c.k.49.1 2 5.3 odd 4
150.8.c.k.49.2 2 5.2 odd 4
162.8.c.d.55.1 2 9.4 even 3
162.8.c.d.109.1 2 9.7 even 3
162.8.c.i.55.1 2 9.5 odd 6
162.8.c.i.109.1 2 9.2 odd 6
192.8.a.f.1.1 1 8.5 even 2
192.8.a.n.1.1 1 8.3 odd 2
294.8.a.l.1.1 1 7.6 odd 2
294.8.e.c.67.1 2 7.2 even 3
294.8.e.c.79.1 2 7.4 even 3
294.8.e.d.67.1 2 7.5 odd 6
294.8.e.d.79.1 2 7.3 odd 6
450.8.a.ba.1.1 1 15.14 odd 2
450.8.c.a.199.1 2 15.2 even 4
450.8.c.a.199.2 2 15.8 even 4
576.8.a.h.1.1 1 24.5 odd 2
576.8.a.i.1.1 1 24.11 even 2