Properties

Label 69.8.a.c.1.6
Level $69$
Weight $8$
Character 69.1
Self dual yes
Analytic conductor $21.555$
Analytic rank $0$
Dimension $7$
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] = [69,8,Mod(1,69)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(69, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("69.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 69.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: \(21.5545667584\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 775x^{5} - 474x^{4} + 167184x^{3} - 33920x^{2} - 9348928x + 28965760 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(15.1764\) of defining polynomial
Character \(\chi\) \(=\) 69.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+15.1764 q^{2} -27.0000 q^{3} +102.322 q^{4} +149.395 q^{5} -409.762 q^{6} -45.6786 q^{7} -389.696 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+15.1764 q^{2} -27.0000 q^{3} +102.322 q^{4} +149.395 q^{5} -409.762 q^{6} -45.6786 q^{7} -389.696 q^{8} +729.000 q^{9} +2267.27 q^{10} +6854.49 q^{11} -2762.70 q^{12} +12280.7 q^{13} -693.235 q^{14} -4033.66 q^{15} -19011.4 q^{16} +7816.83 q^{17} +11063.6 q^{18} +17025.7 q^{19} +15286.4 q^{20} +1233.32 q^{21} +104026. q^{22} -12167.0 q^{23} +10521.8 q^{24} -55806.2 q^{25} +186376. q^{26} -19683.0 q^{27} -4673.93 q^{28} +232578. q^{29} -61216.3 q^{30} +317034. q^{31} -238643. q^{32} -185071. q^{33} +118631. q^{34} -6824.15 q^{35} +74592.9 q^{36} -180572. q^{37} +258388. q^{38} -331579. q^{39} -58218.5 q^{40} -313962. q^{41} +18717.3 q^{42} -659855. q^{43} +701366. q^{44} +108909. q^{45} -184651. q^{46} +553964. q^{47} +513308. q^{48} -821456. q^{49} -846935. q^{50} -211054. q^{51} +1.25659e6 q^{52} +839775. q^{53} -298716. q^{54} +1.02402e6 q^{55} +17800.8 q^{56} -459694. q^{57} +3.52970e6 q^{58} -166614. q^{59} -412733. q^{60} -1.79776e6 q^{61} +4.81143e6 q^{62} -33299.7 q^{63} -1.18828e6 q^{64} +1.83467e6 q^{65} -2.80871e6 q^{66} -4.71895e6 q^{67} +799835. q^{68} +328509. q^{69} -103566. q^{70} -1.33587e6 q^{71} -284088. q^{72} -2.18765e6 q^{73} -2.74043e6 q^{74} +1.50677e6 q^{75} +1.74211e6 q^{76} -313103. q^{77} -5.03216e6 q^{78} +8.31933e6 q^{79} -2.84021e6 q^{80} +531441. q^{81} -4.76480e6 q^{82} -5.82314e6 q^{83} +126196. q^{84} +1.16779e6 q^{85} -1.00142e7 q^{86} -6.27962e6 q^{87} -2.67116e6 q^{88} -4.66139e6 q^{89} +1.65284e6 q^{90} -560965. q^{91} -1.24495e6 q^{92} -8.55993e6 q^{93} +8.40716e6 q^{94} +2.54355e6 q^{95} +6.44336e6 q^{96} +9.60089e6 q^{97} -1.24667e7 q^{98} +4.99692e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 189 q^{3} + 654 q^{4} - 516 q^{5} + 1018 q^{7} + 1422 q^{8} + 5103 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 189 q^{3} + 654 q^{4} - 516 q^{5} + 1018 q^{7} + 1422 q^{8} + 5103 q^{9} - 15310 q^{10} + 9040 q^{11} - 17658 q^{12} + 3774 q^{13} + 4536 q^{14} + 13932 q^{15} + 52002 q^{16} - 40760 q^{17} + 81598 q^{19} - 88946 q^{20} - 27486 q^{21} + 245034 q^{22} - 85169 q^{23} - 38394 q^{24} + 321325 q^{25} + 412748 q^{26} - 137781 q^{27} + 965948 q^{28} + 154126 q^{29} + 413370 q^{30} + 243132 q^{31} + 1278286 q^{32} - 244080 q^{33} + 984836 q^{34} - 130296 q^{35} + 476766 q^{36} + 582114 q^{37} + 772558 q^{38} - 101898 q^{39} - 132618 q^{40} + 113062 q^{41} - 122472 q^{42} - 659778 q^{43} + 659390 q^{44} - 376164 q^{45} - 591032 q^{47} - 1404054 q^{48} + 3263235 q^{49} - 702684 q^{50} + 1100520 q^{51} + 1793280 q^{52} + 207128 q^{53} + 184664 q^{55} + 5390508 q^{56} - 2203146 q^{57} - 1142916 q^{58} + 447148 q^{59} + 2401542 q^{60} + 2248970 q^{61} - 5729060 q^{62} + 742122 q^{63} + 7212922 q^{64} - 827096 q^{65} - 6615918 q^{66} + 4467570 q^{67} - 5477620 q^{68} + 2299563 q^{69} - 12744284 q^{70} - 5154608 q^{71} + 1036638 q^{72} - 13239250 q^{73} - 2827426 q^{74} - 8675775 q^{75} - 527434 q^{76} - 18415912 q^{77} - 11144196 q^{78} + 9594446 q^{79} - 55932394 q^{80} + 3720087 q^{81} - 20889952 q^{82} - 573720 q^{83} - 26080596 q^{84} + 7477272 q^{85} - 28416910 q^{86} - 4161402 q^{87} + 26555702 q^{88} - 3810540 q^{89} - 11160990 q^{90} + 36092068 q^{91} - 7957218 q^{92} - 6564564 q^{93} + 33545768 q^{94} + 10497320 q^{95} - 34513722 q^{96} + 49497978 q^{97} - 1023376 q^{98} + 6590160 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 15.1764 1.34141 0.670707 0.741722i \(-0.265991\pi\)
0.670707 + 0.741722i \(0.265991\pi\)
\(3\) −27.0000 −0.577350
\(4\) 102.322 0.799392
\(5\) 149.395 0.534491 0.267246 0.963628i \(-0.413886\pi\)
0.267246 + 0.963628i \(0.413886\pi\)
\(6\) −409.762 −0.774466
\(7\) −45.6786 −0.0503349 −0.0251675 0.999683i \(-0.508012\pi\)
−0.0251675 + 0.999683i \(0.508012\pi\)
\(8\) −389.696 −0.269098
\(9\) 729.000 0.333333
\(10\) 2267.27 0.716974
\(11\) 6854.49 1.55275 0.776373 0.630274i \(-0.217057\pi\)
0.776373 + 0.630274i \(0.217057\pi\)
\(12\) −2762.70 −0.461529
\(13\) 12280.7 1.55032 0.775160 0.631765i \(-0.217669\pi\)
0.775160 + 0.631765i \(0.217669\pi\)
\(14\) −693.235 −0.0675200
\(15\) −4033.66 −0.308589
\(16\) −19011.4 −1.16036
\(17\) 7816.83 0.385886 0.192943 0.981210i \(-0.438197\pi\)
0.192943 + 0.981210i \(0.438197\pi\)
\(18\) 11063.6 0.447138
\(19\) 17025.7 0.569465 0.284733 0.958607i \(-0.408095\pi\)
0.284733 + 0.958607i \(0.408095\pi\)
\(20\) 15286.4 0.427268
\(21\) 1233.32 0.0290609
\(22\) 104026. 2.08288
\(23\) −12167.0 −0.208514
\(24\) 10521.8 0.155364
\(25\) −55806.2 −0.714319
\(26\) 186376. 2.07962
\(27\) −19683.0 −0.192450
\(28\) −4673.93 −0.0402374
\(29\) 232578. 1.77083 0.885414 0.464803i \(-0.153875\pi\)
0.885414 + 0.464803i \(0.153875\pi\)
\(30\) −61216.3 −0.413945
\(31\) 317034. 1.91135 0.955676 0.294422i \(-0.0951271\pi\)
0.955676 + 0.294422i \(0.0951271\pi\)
\(32\) −238643. −1.28743
\(33\) −185071. −0.896478
\(34\) 118631. 0.517633
\(35\) −6824.15 −0.0269036
\(36\) 74592.9 0.266464
\(37\) −180572. −0.586064 −0.293032 0.956103i \(-0.594664\pi\)
−0.293032 + 0.956103i \(0.594664\pi\)
\(38\) 258388. 0.763889
\(39\) −331579. −0.895078
\(40\) −58218.5 −0.143831
\(41\) −313962. −0.711432 −0.355716 0.934594i \(-0.615763\pi\)
−0.355716 + 0.934594i \(0.615763\pi\)
\(42\) 18717.3 0.0389827
\(43\) −659855. −1.26564 −0.632818 0.774301i \(-0.718102\pi\)
−0.632818 + 0.774301i \(0.718102\pi\)
\(44\) 701366. 1.24125
\(45\) 108909. 0.178164
\(46\) −184651. −0.279704
\(47\) 553964. 0.778286 0.389143 0.921177i \(-0.372771\pi\)
0.389143 + 0.921177i \(0.372771\pi\)
\(48\) 513308. 0.669937
\(49\) −821456. −0.997466
\(50\) −846935. −0.958198
\(51\) −211054. −0.222792
\(52\) 1.25659e6 1.23931
\(53\) 839775. 0.774814 0.387407 0.921909i \(-0.373371\pi\)
0.387407 + 0.921909i \(0.373371\pi\)
\(54\) −298716. −0.258155
\(55\) 1.02402e6 0.829929
\(56\) 17800.8 0.0135450
\(57\) −459694. −0.328781
\(58\) 3.52970e6 2.37541
\(59\) −166614. −0.105616 −0.0528081 0.998605i \(-0.516817\pi\)
−0.0528081 + 0.998605i \(0.516817\pi\)
\(60\) −412733. −0.246683
\(61\) −1.79776e6 −1.01409 −0.507045 0.861920i \(-0.669262\pi\)
−0.507045 + 0.861920i \(0.669262\pi\)
\(62\) 4.81143e6 2.56391
\(63\) −33299.7 −0.0167783
\(64\) −1.18828e6 −0.566614
\(65\) 1.83467e6 0.828633
\(66\) −2.80871e6 −1.20255
\(67\) −4.71895e6 −1.91683 −0.958415 0.285380i \(-0.907880\pi\)
−0.958415 + 0.285380i \(0.907880\pi\)
\(68\) 799835. 0.308474
\(69\) 328509. 0.120386
\(70\) −103566. −0.0360889
\(71\) −1.33587e6 −0.442954 −0.221477 0.975166i \(-0.571088\pi\)
−0.221477 + 0.975166i \(0.571088\pi\)
\(72\) −284088. −0.0896994
\(73\) −2.18765e6 −0.658186 −0.329093 0.944298i \(-0.606743\pi\)
−0.329093 + 0.944298i \(0.606743\pi\)
\(74\) −2.74043e6 −0.786155
\(75\) 1.50677e6 0.412412
\(76\) 1.74211e6 0.455226
\(77\) −313103. −0.0781574
\(78\) −5.03216e6 −1.20067
\(79\) 8.31933e6 1.89842 0.949212 0.314636i \(-0.101882\pi\)
0.949212 + 0.314636i \(0.101882\pi\)
\(80\) −2.84021e6 −0.620205
\(81\) 531441. 0.111111
\(82\) −4.76480e6 −0.954324
\(83\) −5.82314e6 −1.11785 −0.558926 0.829218i \(-0.688786\pi\)
−0.558926 + 0.829218i \(0.688786\pi\)
\(84\) 126196. 0.0232310
\(85\) 1.16779e6 0.206253
\(86\) −1.00142e7 −1.69774
\(87\) −6.27962e6 −1.02239
\(88\) −2.67116e6 −0.417841
\(89\) −4.66139e6 −0.700891 −0.350446 0.936583i \(-0.613970\pi\)
−0.350446 + 0.936583i \(0.613970\pi\)
\(90\) 1.65284e6 0.238991
\(91\) −560965. −0.0780353
\(92\) −1.24495e6 −0.166685
\(93\) −8.55993e6 −1.10352
\(94\) 8.40716e6 1.04400
\(95\) 2.54355e6 0.304374
\(96\) 6.44336e6 0.743299
\(97\) 9.60089e6 1.06810 0.534048 0.845454i \(-0.320670\pi\)
0.534048 + 0.845454i \(0.320670\pi\)
\(98\) −1.24667e7 −1.33802
\(99\) 4.99692e6 0.517582
\(100\) −5.71021e6 −0.571021
\(101\) −1.06823e7 −1.03167 −0.515834 0.856689i \(-0.672518\pi\)
−0.515834 + 0.856689i \(0.672518\pi\)
\(102\) −3.20304e6 −0.298856
\(103\) 1.20136e7 1.08329 0.541643 0.840608i \(-0.317802\pi\)
0.541643 + 0.840608i \(0.317802\pi\)
\(104\) −4.78574e6 −0.417188
\(105\) 184252. 0.0155328
\(106\) 1.27447e7 1.03935
\(107\) 5.52839e6 0.436270 0.218135 0.975919i \(-0.430003\pi\)
0.218135 + 0.975919i \(0.430003\pi\)
\(108\) −2.01401e6 −0.153843
\(109\) 1.75316e7 1.29667 0.648334 0.761356i \(-0.275466\pi\)
0.648334 + 0.761356i \(0.275466\pi\)
\(110\) 1.55410e7 1.11328
\(111\) 4.87545e6 0.338364
\(112\) 868414. 0.0584069
\(113\) −4.98398e6 −0.324939 −0.162469 0.986714i \(-0.551946\pi\)
−0.162469 + 0.986714i \(0.551946\pi\)
\(114\) −6.97648e6 −0.441031
\(115\) −1.81769e6 −0.111449
\(116\) 2.37979e7 1.41559
\(117\) 8.95263e6 0.516773
\(118\) −2.52860e6 −0.141675
\(119\) −357062. −0.0194236
\(120\) 1.57190e6 0.0830407
\(121\) 2.74968e7 1.41102
\(122\) −2.72834e7 −1.36031
\(123\) 8.47697e6 0.410745
\(124\) 3.24397e7 1.52792
\(125\) −2.00086e7 −0.916289
\(126\) −505368. −0.0225067
\(127\) 1.56305e7 0.677112 0.338556 0.940946i \(-0.390061\pi\)
0.338556 + 0.940946i \(0.390061\pi\)
\(128\) 1.25126e7 0.527367
\(129\) 1.78161e7 0.730715
\(130\) 2.78437e7 1.11154
\(131\) 4.19670e7 1.63102 0.815509 0.578744i \(-0.196457\pi\)
0.815509 + 0.578744i \(0.196457\pi\)
\(132\) −1.89369e7 −0.716638
\(133\) −777710. −0.0286640
\(134\) −7.16165e7 −2.57126
\(135\) −2.94054e6 −0.102863
\(136\) −3.04618e6 −0.103841
\(137\) −3.23047e7 −1.07336 −0.536678 0.843787i \(-0.680321\pi\)
−0.536678 + 0.843787i \(0.680321\pi\)
\(138\) 4.98557e6 0.161487
\(139\) −1.98327e7 −0.626369 −0.313184 0.949692i \(-0.601396\pi\)
−0.313184 + 0.949692i \(0.601396\pi\)
\(140\) −698262. −0.0215065
\(141\) −1.49570e7 −0.449344
\(142\) −2.02736e7 −0.594185
\(143\) 8.41779e7 2.40725
\(144\) −1.38593e7 −0.386788
\(145\) 3.47460e7 0.946492
\(146\) −3.32006e7 −0.882900
\(147\) 2.21793e7 0.575887
\(148\) −1.84766e7 −0.468495
\(149\) −3.65263e7 −0.904595 −0.452297 0.891867i \(-0.649395\pi\)
−0.452297 + 0.891867i \(0.649395\pi\)
\(150\) 2.28672e7 0.553216
\(151\) 2.05824e7 0.486492 0.243246 0.969965i \(-0.421788\pi\)
0.243246 + 0.969965i \(0.421788\pi\)
\(152\) −6.63484e6 −0.153242
\(153\) 5.69847e6 0.128629
\(154\) −4.75177e6 −0.104841
\(155\) 4.73633e7 1.02160
\(156\) −3.39279e7 −0.715518
\(157\) −5.88728e7 −1.21413 −0.607066 0.794652i \(-0.707654\pi\)
−0.607066 + 0.794652i \(0.707654\pi\)
\(158\) 1.26257e8 2.54657
\(159\) −2.26739e7 −0.447339
\(160\) −3.56521e7 −0.688121
\(161\) 555771. 0.0104956
\(162\) 8.06535e6 0.149046
\(163\) −2.85151e7 −0.515725 −0.257862 0.966182i \(-0.583018\pi\)
−0.257862 + 0.966182i \(0.583018\pi\)
\(164\) −3.21252e7 −0.568713
\(165\) −2.76487e7 −0.479160
\(166\) −8.83742e7 −1.49950
\(167\) −8.91994e7 −1.48202 −0.741010 0.671494i \(-0.765653\pi\)
−0.741010 + 0.671494i \(0.765653\pi\)
\(168\) −480620. −0.00782023
\(169\) 8.80671e7 1.40349
\(170\) 1.77229e7 0.276670
\(171\) 1.24117e7 0.189822
\(172\) −6.75178e7 −1.01174
\(173\) 2.98805e7 0.438759 0.219379 0.975640i \(-0.429597\pi\)
0.219379 + 0.975640i \(0.429597\pi\)
\(174\) −9.53018e7 −1.37145
\(175\) 2.54915e6 0.0359552
\(176\) −1.30313e8 −1.80175
\(177\) 4.49859e6 0.0609775
\(178\) −7.07430e7 −0.940186
\(179\) −4.75128e7 −0.619191 −0.309596 0.950868i \(-0.600194\pi\)
−0.309596 + 0.950868i \(0.600194\pi\)
\(180\) 1.11438e7 0.142423
\(181\) 7.79262e7 0.976806 0.488403 0.872618i \(-0.337580\pi\)
0.488403 + 0.872618i \(0.337580\pi\)
\(182\) −8.51341e6 −0.104678
\(183\) 4.85394e7 0.585485
\(184\) 4.74143e6 0.0561109
\(185\) −2.69766e7 −0.313246
\(186\) −1.29909e8 −1.48028
\(187\) 5.35803e7 0.599183
\(188\) 5.66828e7 0.622156
\(189\) 899092. 0.00968696
\(190\) 3.86019e7 0.408292
\(191\) 6.55466e7 0.680665 0.340332 0.940305i \(-0.389460\pi\)
0.340332 + 0.940305i \(0.389460\pi\)
\(192\) 3.20834e7 0.327135
\(193\) 3.05653e7 0.306040 0.153020 0.988223i \(-0.451100\pi\)
0.153020 + 0.988223i \(0.451100\pi\)
\(194\) 1.45707e8 1.43276
\(195\) −4.95362e7 −0.478411
\(196\) −8.40532e7 −0.797367
\(197\) −6.62379e7 −0.617269 −0.308635 0.951181i \(-0.599872\pi\)
−0.308635 + 0.951181i \(0.599872\pi\)
\(198\) 7.58351e7 0.694292
\(199\) 3.82627e7 0.344184 0.172092 0.985081i \(-0.444947\pi\)
0.172092 + 0.985081i \(0.444947\pi\)
\(200\) 2.17474e7 0.192222
\(201\) 1.27412e8 1.10668
\(202\) −1.62118e8 −1.38389
\(203\) −1.06239e7 −0.0891345
\(204\) −2.15955e7 −0.178098
\(205\) −4.69043e7 −0.380254
\(206\) 1.82323e8 1.45314
\(207\) −8.86974e6 −0.0695048
\(208\) −2.33473e8 −1.79894
\(209\) 1.16702e8 0.884235
\(210\) 2.79628e6 0.0208359
\(211\) −7.26236e7 −0.532218 −0.266109 0.963943i \(-0.585738\pi\)
−0.266109 + 0.963943i \(0.585738\pi\)
\(212\) 8.59276e7 0.619380
\(213\) 3.60684e7 0.255740
\(214\) 8.39009e7 0.585219
\(215\) −9.85789e7 −0.676471
\(216\) 7.67038e6 0.0517880
\(217\) −1.44817e7 −0.0962077
\(218\) 2.66066e8 1.73937
\(219\) 5.90666e7 0.380004
\(220\) 1.04780e8 0.663439
\(221\) 9.59961e7 0.598247
\(222\) 7.39917e7 0.453887
\(223\) 1.65402e8 0.998787 0.499393 0.866375i \(-0.333556\pi\)
0.499393 + 0.866375i \(0.333556\pi\)
\(224\) 1.09009e7 0.0648028
\(225\) −4.06827e7 −0.238106
\(226\) −7.56387e7 −0.435877
\(227\) −1.12210e8 −0.636708 −0.318354 0.947972i \(-0.603130\pi\)
−0.318354 + 0.947972i \(0.603130\pi\)
\(228\) −4.70369e7 −0.262825
\(229\) −1.99510e8 −1.09784 −0.548922 0.835873i \(-0.684962\pi\)
−0.548922 + 0.835873i \(0.684962\pi\)
\(230\) −2.75859e7 −0.149499
\(231\) 8.45379e6 0.0451242
\(232\) −9.06349e7 −0.476527
\(233\) −1.05157e8 −0.544618 −0.272309 0.962210i \(-0.587787\pi\)
−0.272309 + 0.962210i \(0.587787\pi\)
\(234\) 1.35868e8 0.693207
\(235\) 8.27594e7 0.415987
\(236\) −1.70484e7 −0.0844288
\(237\) −2.24622e8 −1.09606
\(238\) −5.41890e6 −0.0260550
\(239\) −2.54683e8 −1.20672 −0.603360 0.797469i \(-0.706172\pi\)
−0.603360 + 0.797469i \(0.706172\pi\)
\(240\) 7.66856e7 0.358075
\(241\) −2.61663e8 −1.20416 −0.602079 0.798437i \(-0.705661\pi\)
−0.602079 + 0.798437i \(0.705661\pi\)
\(242\) 4.17302e8 1.89276
\(243\) −1.43489e7 −0.0641500
\(244\) −1.83950e8 −0.810655
\(245\) −1.22721e8 −0.533137
\(246\) 1.28650e8 0.550979
\(247\) 2.09087e8 0.882854
\(248\) −1.23547e8 −0.514341
\(249\) 1.57225e8 0.645392
\(250\) −3.03658e8 −1.22912
\(251\) 1.15081e8 0.459351 0.229675 0.973267i \(-0.426234\pi\)
0.229675 + 0.973267i \(0.426234\pi\)
\(252\) −3.40730e6 −0.0134125
\(253\) −8.33985e7 −0.323770
\(254\) 2.37215e8 0.908288
\(255\) −3.15304e7 −0.119080
\(256\) 3.41995e8 1.27403
\(257\) −4.61638e8 −1.69643 −0.848215 0.529652i \(-0.822323\pi\)
−0.848215 + 0.529652i \(0.822323\pi\)
\(258\) 2.70383e8 0.980192
\(259\) 8.24829e6 0.0294995
\(260\) 1.87728e8 0.662402
\(261\) 1.69550e8 0.590276
\(262\) 6.36907e8 2.18787
\(263\) 2.86469e8 0.971030 0.485515 0.874228i \(-0.338632\pi\)
0.485515 + 0.874228i \(0.338632\pi\)
\(264\) 7.21214e7 0.241241
\(265\) 1.25458e8 0.414131
\(266\) −1.18028e7 −0.0384503
\(267\) 1.25858e8 0.404660
\(268\) −4.82853e8 −1.53230
\(269\) −2.73894e8 −0.857926 −0.428963 0.903322i \(-0.641121\pi\)
−0.428963 + 0.903322i \(0.641121\pi\)
\(270\) −4.46267e7 −0.137982
\(271\) −3.00122e8 −0.916020 −0.458010 0.888947i \(-0.651438\pi\)
−0.458010 + 0.888947i \(0.651438\pi\)
\(272\) −1.48609e8 −0.447769
\(273\) 1.51461e7 0.0450537
\(274\) −4.90268e8 −1.43981
\(275\) −3.82523e8 −1.10916
\(276\) 3.36138e7 0.0962355
\(277\) 4.92991e8 1.39367 0.696835 0.717231i \(-0.254591\pi\)
0.696835 + 0.717231i \(0.254591\pi\)
\(278\) −3.00988e8 −0.840220
\(279\) 2.31118e8 0.637117
\(280\) 2.65934e6 0.00723971
\(281\) 2.30293e8 0.619168 0.309584 0.950872i \(-0.399810\pi\)
0.309584 + 0.950872i \(0.399810\pi\)
\(282\) −2.26993e8 −0.602756
\(283\) 2.26945e8 0.595208 0.297604 0.954689i \(-0.403812\pi\)
0.297604 + 0.954689i \(0.403812\pi\)
\(284\) −1.36689e8 −0.354094
\(285\) −6.86759e7 −0.175731
\(286\) 1.27751e9 3.22912
\(287\) 1.43413e7 0.0358099
\(288\) −1.73971e8 −0.429144
\(289\) −3.49236e8 −0.851092
\(290\) 5.27319e8 1.26964
\(291\) −2.59224e8 −0.616665
\(292\) −2.23846e8 −0.526149
\(293\) 2.13906e7 0.0496805 0.0248403 0.999691i \(-0.492092\pi\)
0.0248403 + 0.999691i \(0.492092\pi\)
\(294\) 3.36602e8 0.772504
\(295\) −2.48913e7 −0.0564509
\(296\) 7.03683e7 0.157709
\(297\) −1.34917e8 −0.298826
\(298\) −5.54337e8 −1.21344
\(299\) −1.49419e8 −0.323264
\(300\) 1.54176e8 0.329679
\(301\) 3.01412e7 0.0637057
\(302\) 3.12366e8 0.652588
\(303\) 2.88422e8 0.595634
\(304\) −3.23682e8 −0.660787
\(305\) −2.68576e8 −0.542022
\(306\) 8.64820e7 0.172544
\(307\) −6.56081e8 −1.29412 −0.647058 0.762441i \(-0.724001\pi\)
−0.647058 + 0.762441i \(0.724001\pi\)
\(308\) −3.20374e7 −0.0624784
\(309\) −3.24368e8 −0.625436
\(310\) 7.18803e8 1.37039
\(311\) −2.04155e8 −0.384856 −0.192428 0.981311i \(-0.561636\pi\)
−0.192428 + 0.981311i \(0.561636\pi\)
\(312\) 1.29215e8 0.240864
\(313\) −3.33636e7 −0.0614989 −0.0307494 0.999527i \(-0.509789\pi\)
−0.0307494 + 0.999527i \(0.509789\pi\)
\(314\) −8.93475e8 −1.62865
\(315\) −4.97480e6 −0.00896786
\(316\) 8.51252e8 1.51759
\(317\) −7.39942e8 −1.30464 −0.652319 0.757945i \(-0.726204\pi\)
−0.652319 + 0.757945i \(0.726204\pi\)
\(318\) −3.44108e8 −0.600067
\(319\) 1.59421e9 2.74965
\(320\) −1.77522e8 −0.302850
\(321\) −1.49266e8 −0.251881
\(322\) 8.43459e6 0.0140789
\(323\) 1.33087e8 0.219749
\(324\) 5.43782e7 0.0888214
\(325\) −6.85339e8 −1.10742
\(326\) −4.32756e8 −0.691801
\(327\) −4.73353e8 −0.748631
\(328\) 1.22350e8 0.191445
\(329\) −2.53043e7 −0.0391750
\(330\) −4.19606e8 −0.642752
\(331\) 4.38506e8 0.664626 0.332313 0.943169i \(-0.392171\pi\)
0.332313 + 0.943169i \(0.392171\pi\)
\(332\) −5.95837e8 −0.893602
\(333\) −1.31637e8 −0.195355
\(334\) −1.35372e9 −1.98800
\(335\) −7.04987e8 −1.02453
\(336\) −2.34472e7 −0.0337212
\(337\) −5.92928e8 −0.843912 −0.421956 0.906616i \(-0.638656\pi\)
−0.421956 + 0.906616i \(0.638656\pi\)
\(338\) 1.33654e9 1.88267
\(339\) 1.34567e8 0.187603
\(340\) 1.19491e8 0.164877
\(341\) 2.17311e9 2.96784
\(342\) 1.88365e8 0.254630
\(343\) 7.51413e7 0.100542
\(344\) 2.57143e8 0.340580
\(345\) 4.90776e7 0.0643452
\(346\) 4.53477e8 0.588557
\(347\) −4.47661e8 −0.575170 −0.287585 0.957755i \(-0.592852\pi\)
−0.287585 + 0.957755i \(0.592852\pi\)
\(348\) −6.42544e8 −0.817289
\(349\) −2.47824e8 −0.312071 −0.156036 0.987751i \(-0.549871\pi\)
−0.156036 + 0.987751i \(0.549871\pi\)
\(350\) 3.86868e7 0.0482308
\(351\) −2.41721e8 −0.298359
\(352\) −1.63578e9 −1.99905
\(353\) −2.71331e8 −0.328313 −0.164156 0.986434i \(-0.552490\pi\)
−0.164156 + 0.986434i \(0.552490\pi\)
\(354\) 6.82723e7 0.0817962
\(355\) −1.99571e8 −0.236755
\(356\) −4.76964e8 −0.560287
\(357\) 9.64066e6 0.0112142
\(358\) −7.21071e8 −0.830592
\(359\) −1.44169e9 −1.64453 −0.822266 0.569103i \(-0.807291\pi\)
−0.822266 + 0.569103i \(0.807291\pi\)
\(360\) −4.24413e7 −0.0479435
\(361\) −6.03997e8 −0.675709
\(362\) 1.18264e9 1.31030
\(363\) −7.42414e8 −0.814653
\(364\) −5.73992e7 −0.0623808
\(365\) −3.26824e8 −0.351795
\(366\) 7.36652e8 0.785378
\(367\) −1.37060e8 −0.144737 −0.0723687 0.997378i \(-0.523056\pi\)
−0.0723687 + 0.997378i \(0.523056\pi\)
\(368\) 2.31312e8 0.241953
\(369\) −2.28878e8 −0.237144
\(370\) −4.09406e8 −0.420193
\(371\) −3.83597e7 −0.0390002
\(372\) −8.75871e8 −0.882144
\(373\) 1.58651e9 1.58293 0.791466 0.611214i \(-0.209318\pi\)
0.791466 + 0.611214i \(0.209318\pi\)
\(374\) 8.13155e8 0.803753
\(375\) 5.40233e8 0.529019
\(376\) −2.15877e8 −0.209435
\(377\) 2.85623e9 2.74535
\(378\) 1.36449e7 0.0129942
\(379\) 1.33727e9 1.26177 0.630886 0.775876i \(-0.282692\pi\)
0.630886 + 0.775876i \(0.282692\pi\)
\(380\) 2.60262e8 0.243314
\(381\) −4.22024e8 −0.390931
\(382\) 9.94759e8 0.913054
\(383\) 3.13397e8 0.285036 0.142518 0.989792i \(-0.454480\pi\)
0.142518 + 0.989792i \(0.454480\pi\)
\(384\) −3.37840e8 −0.304475
\(385\) −4.67760e7 −0.0417744
\(386\) 4.63870e8 0.410526
\(387\) −4.81034e8 −0.421879
\(388\) 9.82384e8 0.853828
\(389\) −1.89508e9 −1.63231 −0.816156 0.577832i \(-0.803899\pi\)
−0.816156 + 0.577832i \(0.803899\pi\)
\(390\) −7.51779e8 −0.641748
\(391\) −9.51073e7 −0.0804628
\(392\) 3.20118e8 0.268416
\(393\) −1.13311e9 −0.941669
\(394\) −1.00525e9 −0.828014
\(395\) 1.24286e9 1.01469
\(396\) 5.11296e8 0.413751
\(397\) 1.22322e9 0.981159 0.490580 0.871396i \(-0.336785\pi\)
0.490580 + 0.871396i \(0.336785\pi\)
\(398\) 5.80689e8 0.461693
\(399\) 2.09982e7 0.0165492
\(400\) 1.06095e9 0.828870
\(401\) 1.76123e9 1.36399 0.681994 0.731358i \(-0.261113\pi\)
0.681994 + 0.731358i \(0.261113\pi\)
\(402\) 1.93365e9 1.48452
\(403\) 3.89340e9 2.96321
\(404\) −1.09304e9 −0.824707
\(405\) 7.93946e7 0.0593879
\(406\) −1.61232e8 −0.119566
\(407\) −1.23773e9 −0.910009
\(408\) 8.22470e7 0.0599528
\(409\) 1.17048e7 0.00845928 0.00422964 0.999991i \(-0.498654\pi\)
0.00422964 + 0.999991i \(0.498654\pi\)
\(410\) −7.11836e8 −0.510078
\(411\) 8.72227e8 0.619702
\(412\) 1.22926e9 0.865971
\(413\) 7.61071e6 0.00531619
\(414\) −1.34610e8 −0.0932347
\(415\) −8.69947e8 −0.597482
\(416\) −2.93070e9 −1.99593
\(417\) 5.35483e8 0.361634
\(418\) 1.77112e9 1.18613
\(419\) 2.29028e9 1.52103 0.760517 0.649318i \(-0.224946\pi\)
0.760517 + 0.649318i \(0.224946\pi\)
\(420\) 1.88531e7 0.0124168
\(421\) 1.64294e9 1.07308 0.536542 0.843874i \(-0.319730\pi\)
0.536542 + 0.843874i \(0.319730\pi\)
\(422\) −1.10216e9 −0.713924
\(423\) 4.03840e8 0.259429
\(424\) −3.27257e8 −0.208501
\(425\) −4.36227e8 −0.275646
\(426\) 5.47387e8 0.343053
\(427\) 8.21190e7 0.0510441
\(428\) 5.65677e8 0.348751
\(429\) −2.27280e9 −1.38983
\(430\) −1.49607e9 −0.907428
\(431\) 1.72851e9 1.03992 0.519962 0.854189i \(-0.325946\pi\)
0.519962 + 0.854189i \(0.325946\pi\)
\(432\) 3.74202e8 0.223312
\(433\) 6.64910e8 0.393600 0.196800 0.980444i \(-0.436945\pi\)
0.196800 + 0.980444i \(0.436945\pi\)
\(434\) −2.19779e8 −0.129054
\(435\) −9.38143e8 −0.546458
\(436\) 1.79387e9 1.03655
\(437\) −2.07152e8 −0.118742
\(438\) 8.96417e8 0.509742
\(439\) −2.39908e9 −1.35338 −0.676688 0.736270i \(-0.736585\pi\)
−0.676688 + 0.736270i \(0.736585\pi\)
\(440\) −3.99058e8 −0.223332
\(441\) −5.98842e8 −0.332489
\(442\) 1.45687e9 0.802497
\(443\) 1.19429e9 0.652675 0.326338 0.945253i \(-0.394185\pi\)
0.326338 + 0.945253i \(0.394185\pi\)
\(444\) 4.98867e8 0.270486
\(445\) −6.96388e8 −0.374620
\(446\) 2.51020e9 1.33979
\(447\) 9.86211e8 0.522268
\(448\) 5.42788e7 0.0285205
\(449\) −3.02443e9 −1.57681 −0.788407 0.615154i \(-0.789094\pi\)
−0.788407 + 0.615154i \(0.789094\pi\)
\(450\) −6.17416e8 −0.319399
\(451\) −2.15205e9 −1.10467
\(452\) −5.09971e8 −0.259753
\(453\) −5.55724e8 −0.280877
\(454\) −1.70294e9 −0.854089
\(455\) −8.38053e7 −0.0417092
\(456\) 1.79141e8 0.0884744
\(457\) −2.48857e9 −1.21967 −0.609836 0.792528i \(-0.708765\pi\)
−0.609836 + 0.792528i \(0.708765\pi\)
\(458\) −3.02784e9 −1.47266
\(459\) −1.53859e8 −0.0742638
\(460\) −1.85990e8 −0.0890916
\(461\) −2.04628e9 −0.972776 −0.486388 0.873743i \(-0.661686\pi\)
−0.486388 + 0.873743i \(0.661686\pi\)
\(462\) 1.28298e8 0.0605302
\(463\) 1.21358e9 0.568246 0.284123 0.958788i \(-0.408298\pi\)
0.284123 + 0.958788i \(0.408298\pi\)
\(464\) −4.42164e9 −2.05481
\(465\) −1.27881e9 −0.589821
\(466\) −1.59590e9 −0.730558
\(467\) −1.28395e9 −0.583364 −0.291682 0.956515i \(-0.594215\pi\)
−0.291682 + 0.956515i \(0.594215\pi\)
\(468\) 9.16053e8 0.413105
\(469\) 2.15555e8 0.0964835
\(470\) 1.25599e9 0.558011
\(471\) 1.58956e9 0.700979
\(472\) 6.49289e7 0.0284211
\(473\) −4.52297e9 −1.96521
\(474\) −3.40894e9 −1.47027
\(475\) −9.50139e8 −0.406780
\(476\) −3.65353e7 −0.0155270
\(477\) 6.12196e8 0.258271
\(478\) −3.86516e9 −1.61871
\(479\) 2.62181e9 1.09000 0.545001 0.838435i \(-0.316529\pi\)
0.545001 + 0.838435i \(0.316529\pi\)
\(480\) 9.62605e8 0.397287
\(481\) −2.21755e9 −0.908587
\(482\) −3.97110e9 −1.61527
\(483\) −1.50058e7 −0.00605961
\(484\) 2.81353e9 1.12796
\(485\) 1.43432e9 0.570888
\(486\) −2.17764e8 −0.0860518
\(487\) 4.30189e9 1.68775 0.843875 0.536540i \(-0.180269\pi\)
0.843875 + 0.536540i \(0.180269\pi\)
\(488\) 7.00578e8 0.272890
\(489\) 7.69907e8 0.297754
\(490\) −1.86246e9 −0.715158
\(491\) 2.28666e8 0.0871798 0.0435899 0.999050i \(-0.486121\pi\)
0.0435899 + 0.999050i \(0.486121\pi\)
\(492\) 8.67382e8 0.328347
\(493\) 1.81803e9 0.683338
\(494\) 3.17319e9 1.18427
\(495\) 7.46514e8 0.276643
\(496\) −6.02727e9 −2.21786
\(497\) 6.10205e7 0.0222961
\(498\) 2.38610e9 0.865738
\(499\) 1.73024e9 0.623382 0.311691 0.950184i \(-0.399105\pi\)
0.311691 + 0.950184i \(0.399105\pi\)
\(500\) −2.04733e9 −0.732474
\(501\) 2.40838e9 0.855645
\(502\) 1.74651e9 0.616180
\(503\) 8.27541e8 0.289936 0.144968 0.989436i \(-0.453692\pi\)
0.144968 + 0.989436i \(0.453692\pi\)
\(504\) 1.29768e7 0.00451501
\(505\) −1.59588e9 −0.551417
\(506\) −1.26569e9 −0.434310
\(507\) −2.37781e9 −0.810307
\(508\) 1.59935e9 0.541278
\(509\) −1.63228e9 −0.548632 −0.274316 0.961640i \(-0.588451\pi\)
−0.274316 + 0.961640i \(0.588451\pi\)
\(510\) −4.78518e8 −0.159736
\(511\) 9.99289e7 0.0331297
\(512\) 3.58863e9 1.18164
\(513\) −3.35117e8 −0.109594
\(514\) −7.00600e9 −2.27562
\(515\) 1.79477e9 0.579007
\(516\) 1.82298e9 0.584128
\(517\) 3.79714e9 1.20848
\(518\) 1.25179e8 0.0395710
\(519\) −8.06772e8 −0.253318
\(520\) −7.14965e8 −0.222984
\(521\) 1.95096e7 0.00604389 0.00302195 0.999995i \(-0.499038\pi\)
0.00302195 + 0.999995i \(0.499038\pi\)
\(522\) 2.57315e9 0.791805
\(523\) −2.79054e9 −0.852966 −0.426483 0.904496i \(-0.640248\pi\)
−0.426483 + 0.904496i \(0.640248\pi\)
\(524\) 4.29416e9 1.30382
\(525\) −6.88270e7 −0.0207587
\(526\) 4.34756e9 1.30255
\(527\) 2.47820e9 0.737564
\(528\) 3.51846e9 1.04024
\(529\) 1.48036e8 0.0434783
\(530\) 1.90400e9 0.555521
\(531\) −1.21462e8 −0.0352054
\(532\) −7.95770e7 −0.0229138
\(533\) −3.85567e9 −1.10295
\(534\) 1.91006e9 0.542816
\(535\) 8.25913e8 0.233183
\(536\) 1.83895e9 0.515815
\(537\) 1.28284e9 0.357490
\(538\) −4.15672e9 −1.15083
\(539\) −5.63066e9 −1.54881
\(540\) −3.00882e8 −0.0822278
\(541\) 2.92896e9 0.795285 0.397643 0.917540i \(-0.369828\pi\)
0.397643 + 0.917540i \(0.369828\pi\)
\(542\) −4.55476e9 −1.22876
\(543\) −2.10401e9 −0.563959
\(544\) −1.86543e9 −0.496802
\(545\) 2.61913e9 0.693058
\(546\) 2.29862e8 0.0604357
\(547\) 3.11875e9 0.814751 0.407376 0.913261i \(-0.366444\pi\)
0.407376 + 0.913261i \(0.366444\pi\)
\(548\) −3.30549e9 −0.858032
\(549\) −1.31056e9 −0.338030
\(550\) −5.80531e9 −1.48784
\(551\) 3.95981e9 1.00843
\(552\) −1.28019e8 −0.0323956
\(553\) −3.80015e8 −0.0955571
\(554\) 7.48182e9 1.86949
\(555\) 7.28367e8 0.180853
\(556\) −2.02933e9 −0.500714
\(557\) 4.07990e9 1.00036 0.500180 0.865921i \(-0.333267\pi\)
0.500180 + 0.865921i \(0.333267\pi\)
\(558\) 3.50753e9 0.854638
\(559\) −8.10348e9 −1.96214
\(560\) 1.29737e8 0.0312180
\(561\) −1.44667e9 −0.345939
\(562\) 3.49501e9 0.830561
\(563\) 6.90942e9 1.63178 0.815891 0.578205i \(-0.196247\pi\)
0.815891 + 0.578205i \(0.196247\pi\)
\(564\) −1.53044e9 −0.359202
\(565\) −7.44581e8 −0.173677
\(566\) 3.44421e9 0.798421
\(567\) −2.42755e7 −0.00559277
\(568\) 5.20581e8 0.119198
\(569\) −1.15725e9 −0.263350 −0.131675 0.991293i \(-0.542036\pi\)
−0.131675 + 0.991293i \(0.542036\pi\)
\(570\) −1.04225e9 −0.235727
\(571\) −4.86086e9 −1.09266 −0.546332 0.837569i \(-0.683976\pi\)
−0.546332 + 0.837569i \(0.683976\pi\)
\(572\) 8.61327e9 1.92434
\(573\) −1.76976e9 −0.392982
\(574\) 2.17649e8 0.0480359
\(575\) 6.78994e8 0.148946
\(576\) −8.66253e8 −0.188871
\(577\) −5.87427e9 −1.27303 −0.636514 0.771265i \(-0.719624\pi\)
−0.636514 + 0.771265i \(0.719624\pi\)
\(578\) −5.30013e9 −1.14167
\(579\) −8.25263e8 −0.176692
\(580\) 3.55529e9 0.756619
\(581\) 2.65993e8 0.0562670
\(582\) −3.93408e9 −0.827204
\(583\) 5.75622e9 1.20309
\(584\) 8.52519e8 0.177117
\(585\) 1.33748e9 0.276211
\(586\) 3.24632e8 0.0666422
\(587\) −4.42899e9 −0.903797 −0.451899 0.892069i \(-0.649253\pi\)
−0.451899 + 0.892069i \(0.649253\pi\)
\(588\) 2.26944e9 0.460360
\(589\) 5.39773e9 1.08845
\(590\) −3.77760e8 −0.0757241
\(591\) 1.78842e9 0.356381
\(592\) 3.43293e9 0.680048
\(593\) −4.19380e9 −0.825878 −0.412939 0.910759i \(-0.635498\pi\)
−0.412939 + 0.910759i \(0.635498\pi\)
\(594\) −2.04755e9 −0.400850
\(595\) −5.33432e7 −0.0103817
\(596\) −3.73745e9 −0.723126
\(597\) −1.03309e9 −0.198714
\(598\) −2.26764e9 −0.433631
\(599\) 8.36071e9 1.58946 0.794729 0.606964i \(-0.207613\pi\)
0.794729 + 0.606964i \(0.207613\pi\)
\(600\) −5.87181e8 −0.110979
\(601\) −6.82190e8 −0.128187 −0.0640936 0.997944i \(-0.520416\pi\)
−0.0640936 + 0.997944i \(0.520416\pi\)
\(602\) 4.57435e8 0.0854557
\(603\) −3.44011e9 −0.638943
\(604\) 2.10603e9 0.388898
\(605\) 4.10788e9 0.754178
\(606\) 4.37720e9 0.798991
\(607\) 7.75495e9 1.40740 0.703702 0.710495i \(-0.251529\pi\)
0.703702 + 0.710495i \(0.251529\pi\)
\(608\) −4.06306e9 −0.733147
\(609\) 2.86844e8 0.0514619
\(610\) −4.07600e9 −0.727076
\(611\) 6.80307e9 1.20659
\(612\) 5.83080e8 0.102825
\(613\) −2.87474e9 −0.504065 −0.252032 0.967719i \(-0.581099\pi\)
−0.252032 + 0.967719i \(0.581099\pi\)
\(614\) −9.95693e9 −1.73595
\(615\) 1.26642e9 0.219540
\(616\) 1.22015e8 0.0210320
\(617\) 1.19191e9 0.204290 0.102145 0.994770i \(-0.467429\pi\)
0.102145 + 0.994770i \(0.467429\pi\)
\(618\) −4.92272e9 −0.838968
\(619\) −1.27744e9 −0.216483 −0.108242 0.994125i \(-0.534522\pi\)
−0.108242 + 0.994125i \(0.534522\pi\)
\(620\) 4.84632e9 0.816659
\(621\) 2.39483e8 0.0401286
\(622\) −3.09833e9 −0.516251
\(623\) 2.12926e8 0.0352793
\(624\) 6.30378e9 1.03862
\(625\) 1.37067e9 0.224571
\(626\) −5.06338e8 −0.0824955
\(627\) −3.15096e9 −0.510513
\(628\) −6.02399e9 −0.970567
\(629\) −1.41150e9 −0.226154
\(630\) −7.54994e7 −0.0120296
\(631\) 1.09977e10 1.74261 0.871306 0.490739i \(-0.163273\pi\)
0.871306 + 0.490739i \(0.163273\pi\)
\(632\) −3.24201e9 −0.510863
\(633\) 1.96084e9 0.307276
\(634\) −1.12296e10 −1.75006
\(635\) 2.33512e9 0.361911
\(636\) −2.32005e9 −0.357599
\(637\) −1.00881e10 −1.54639
\(638\) 2.41943e10 3.68842
\(639\) −9.73846e8 −0.147651
\(640\) 1.86932e9 0.281873
\(641\) 2.50921e9 0.376300 0.188150 0.982140i \(-0.439751\pi\)
0.188150 + 0.982140i \(0.439751\pi\)
\(642\) −2.26532e9 −0.337876
\(643\) 4.00837e9 0.594606 0.297303 0.954783i \(-0.403913\pi\)
0.297303 + 0.954783i \(0.403913\pi\)
\(644\) 5.68678e7 0.00839007
\(645\) 2.66163e9 0.390561
\(646\) 2.01978e9 0.294774
\(647\) −1.09131e10 −1.58410 −0.792052 0.610454i \(-0.790987\pi\)
−0.792052 + 0.610454i \(0.790987\pi\)
\(648\) −2.07100e8 −0.0298998
\(649\) −1.14206e9 −0.163995
\(650\) −1.04010e10 −1.48551
\(651\) 3.91005e8 0.0555456
\(652\) −2.91773e9 −0.412267
\(653\) −6.66520e9 −0.936735 −0.468368 0.883534i \(-0.655158\pi\)
−0.468368 + 0.883534i \(0.655158\pi\)
\(654\) −7.18378e9 −1.00422
\(655\) 6.26966e9 0.871765
\(656\) 5.96885e9 0.825520
\(657\) −1.59480e9 −0.219395
\(658\) −3.84027e8 −0.0525499
\(659\) 2.60764e9 0.354935 0.177468 0.984127i \(-0.443209\pi\)
0.177468 + 0.984127i \(0.443209\pi\)
\(660\) −2.82907e9 −0.383037
\(661\) 4.72170e9 0.635906 0.317953 0.948106i \(-0.397005\pi\)
0.317953 + 0.948106i \(0.397005\pi\)
\(662\) 6.65492e9 0.891538
\(663\) −2.59190e9 −0.345398
\(664\) 2.26925e9 0.300812
\(665\) −1.16186e8 −0.0153207
\(666\) −1.99777e9 −0.262052
\(667\) −2.82978e9 −0.369243
\(668\) −9.12708e9 −1.18472
\(669\) −4.46584e9 −0.576650
\(670\) −1.06991e10 −1.37432
\(671\) −1.23227e10 −1.57462
\(672\) −2.94324e8 −0.0374139
\(673\) 4.41241e8 0.0557986 0.0278993 0.999611i \(-0.491118\pi\)
0.0278993 + 0.999611i \(0.491118\pi\)
\(674\) −8.99849e9 −1.13204
\(675\) 1.09843e9 0.137471
\(676\) 9.01122e9 1.12194
\(677\) −7.16818e8 −0.0887868 −0.0443934 0.999014i \(-0.514136\pi\)
−0.0443934 + 0.999014i \(0.514136\pi\)
\(678\) 2.04224e9 0.251654
\(679\) −4.38555e8 −0.0537625
\(680\) −4.55084e8 −0.0555023
\(681\) 3.02966e9 0.367603
\(682\) 3.29799e10 3.98111
\(683\) 1.46026e10 1.75370 0.876852 0.480761i \(-0.159639\pi\)
0.876852 + 0.480761i \(0.159639\pi\)
\(684\) 1.27000e9 0.151742
\(685\) −4.82616e9 −0.573699
\(686\) 1.14037e9 0.134869
\(687\) 5.38677e9 0.633841
\(688\) 1.25448e10 1.46860
\(689\) 1.03130e10 1.20121
\(690\) 7.44819e8 0.0863136
\(691\) −1.63420e10 −1.88422 −0.942111 0.335301i \(-0.891162\pi\)
−0.942111 + 0.335301i \(0.891162\pi\)
\(692\) 3.05743e9 0.350740
\(693\) −2.28252e8 −0.0260525
\(694\) −6.79388e9 −0.771542
\(695\) −2.96290e9 −0.334789
\(696\) 2.44714e9 0.275123
\(697\) −2.45418e9 −0.274532
\(698\) −3.76106e9 −0.418617
\(699\) 2.83923e9 0.314435
\(700\) 2.60834e8 0.0287423
\(701\) −1.33002e10 −1.45829 −0.729147 0.684357i \(-0.760083\pi\)
−0.729147 + 0.684357i \(0.760083\pi\)
\(702\) −3.66845e9 −0.400223
\(703\) −3.07437e9 −0.333743
\(704\) −8.14502e9 −0.879808
\(705\) −2.23450e9 −0.240170
\(706\) −4.11782e9 −0.440404
\(707\) 4.87952e8 0.0519289
\(708\) 4.60306e8 0.0487450
\(709\) −5.40175e9 −0.569211 −0.284605 0.958645i \(-0.591863\pi\)
−0.284605 + 0.958645i \(0.591863\pi\)
\(710\) −3.02877e9 −0.317587
\(711\) 6.06479e9 0.632808
\(712\) 1.81653e9 0.188609
\(713\) −3.85736e9 −0.398544
\(714\) 1.46310e8 0.0150429
\(715\) 1.25757e10 1.28666
\(716\) −4.86161e9 −0.494976
\(717\) 6.87643e9 0.696700
\(718\) −2.18797e10 −2.20600
\(719\) 8.57269e9 0.860134 0.430067 0.902797i \(-0.358490\pi\)
0.430067 + 0.902797i \(0.358490\pi\)
\(720\) −2.07051e9 −0.206735
\(721\) −5.48765e8 −0.0545272
\(722\) −9.16649e9 −0.906406
\(723\) 7.06491e9 0.695221
\(724\) 7.97358e9 0.780851
\(725\) −1.29793e10 −1.26494
\(726\) −1.12671e10 −1.09279
\(727\) 1.61791e10 1.56165 0.780827 0.624747i \(-0.214798\pi\)
0.780827 + 0.624747i \(0.214798\pi\)
\(728\) 2.18606e8 0.0209992
\(729\) 3.87420e8 0.0370370
\(730\) −4.96000e9 −0.471902
\(731\) −5.15797e9 −0.488392
\(732\) 4.96666e9 0.468032
\(733\) −1.43824e10 −1.34886 −0.674431 0.738338i \(-0.735611\pi\)
−0.674431 + 0.738338i \(0.735611\pi\)
\(734\) −2.08008e9 −0.194153
\(735\) 3.31348e9 0.307807
\(736\) 2.90357e9 0.268448
\(737\) −3.23460e10 −2.97635
\(738\) −3.47354e9 −0.318108
\(739\) 1.15738e10 1.05492 0.527461 0.849579i \(-0.323144\pi\)
0.527461 + 0.849579i \(0.323144\pi\)
\(740\) −2.76030e9 −0.250407
\(741\) −5.64536e9 −0.509716
\(742\) −5.82161e8 −0.0523154
\(743\) −1.91780e10 −1.71531 −0.857656 0.514223i \(-0.828080\pi\)
−0.857656 + 0.514223i \(0.828080\pi\)
\(744\) 3.33577e9 0.296955
\(745\) −5.45685e9 −0.483498
\(746\) 2.40775e10 2.12337
\(747\) −4.24507e9 −0.372617
\(748\) 5.48246e9 0.478983
\(749\) −2.52529e8 −0.0219596
\(750\) 8.19878e9 0.709634
\(751\) 1.46703e10 1.26386 0.631929 0.775026i \(-0.282263\pi\)
0.631929 + 0.775026i \(0.282263\pi\)
\(752\) −1.05316e10 −0.903095
\(753\) −3.10718e9 −0.265206
\(754\) 4.33471e10 3.68265
\(755\) 3.07490e9 0.260026
\(756\) 9.19970e7 0.00774368
\(757\) 4.15177e9 0.347855 0.173927 0.984758i \(-0.444354\pi\)
0.173927 + 0.984758i \(0.444354\pi\)
\(758\) 2.02949e10 1.69256
\(759\) 2.25176e9 0.186929
\(760\) −9.91211e8 −0.0819066
\(761\) −1.08914e10 −0.895857 −0.447929 0.894069i \(-0.647838\pi\)
−0.447929 + 0.894069i \(0.647838\pi\)
\(762\) −6.40480e9 −0.524400
\(763\) −8.00819e8 −0.0652677
\(764\) 6.70687e9 0.544118
\(765\) 8.51322e8 0.0687509
\(766\) 4.75623e9 0.382351
\(767\) −2.04614e9 −0.163739
\(768\) −9.23387e9 −0.735562
\(769\) 1.94850e10 1.54511 0.772554 0.634949i \(-0.218979\pi\)
0.772554 + 0.634949i \(0.218979\pi\)
\(770\) −7.09890e8 −0.0560368
\(771\) 1.24642e10 0.979435
\(772\) 3.12751e9 0.244646
\(773\) −1.02798e10 −0.800492 −0.400246 0.916408i \(-0.631075\pi\)
−0.400246 + 0.916408i \(0.631075\pi\)
\(774\) −7.30035e9 −0.565914
\(775\) −1.76925e10 −1.36531
\(776\) −3.74143e9 −0.287423
\(777\) −2.22704e8 −0.0170315
\(778\) −2.87604e10 −2.18961
\(779\) −5.34542e9 −0.405136
\(780\) −5.06865e9 −0.382438
\(781\) −9.15667e9 −0.687795
\(782\) −1.44338e9 −0.107934
\(783\) −4.57784e9 −0.340796
\(784\) 1.56170e10 1.15742
\(785\) −8.79529e9 −0.648943
\(786\) −1.71965e10 −1.26317
\(787\) −8.67563e9 −0.634438 −0.317219 0.948352i \(-0.602749\pi\)
−0.317219 + 0.948352i \(0.602749\pi\)
\(788\) −6.77761e9 −0.493440
\(789\) −7.73467e9 −0.560625
\(790\) 1.88622e10 1.36112
\(791\) 2.27661e8 0.0163558
\(792\) −1.94728e9 −0.139280
\(793\) −2.20777e10 −1.57216
\(794\) 1.85641e10 1.31614
\(795\) −3.38737e9 −0.239099
\(796\) 3.91513e9 0.275138
\(797\) 6.59704e9 0.461578 0.230789 0.973004i \(-0.425869\pi\)
0.230789 + 0.973004i \(0.425869\pi\)
\(798\) 3.18676e8 0.0221993
\(799\) 4.33024e9 0.300330
\(800\) 1.33178e10 0.919637
\(801\) −3.39816e9 −0.233630
\(802\) 2.67291e10 1.82967
\(803\) −1.49952e10 −1.02200
\(804\) 1.30370e10 0.884673
\(805\) 8.30294e7 0.00560979
\(806\) 5.90877e10 3.97489
\(807\) 7.39514e9 0.495324
\(808\) 4.16285e9 0.277620
\(809\) −1.17222e10 −0.778374 −0.389187 0.921159i \(-0.627244\pi\)
−0.389187 + 0.921159i \(0.627244\pi\)
\(810\) 1.20492e9 0.0796638
\(811\) 3.14796e9 0.207232 0.103616 0.994617i \(-0.466959\pi\)
0.103616 + 0.994617i \(0.466959\pi\)
\(812\) −1.08706e9 −0.0712535
\(813\) 8.10329e9 0.528865
\(814\) −1.87843e10 −1.22070
\(815\) −4.26001e9 −0.275650
\(816\) 4.01244e9 0.258519
\(817\) −1.12345e10 −0.720736
\(818\) 1.77637e8 0.0113474
\(819\) −4.08944e8 −0.0260118
\(820\) −4.79935e9 −0.303972
\(821\) 1.04194e10 0.657113 0.328556 0.944484i \(-0.393438\pi\)
0.328556 + 0.944484i \(0.393438\pi\)
\(822\) 1.32372e10 0.831277
\(823\) −1.05440e10 −0.659334 −0.329667 0.944097i \(-0.606936\pi\)
−0.329667 + 0.944097i \(0.606936\pi\)
\(824\) −4.68165e9 −0.291510
\(825\) 1.03281e10 0.640372
\(826\) 1.15503e8 0.00713121
\(827\) 2.77244e10 1.70448 0.852241 0.523149i \(-0.175243\pi\)
0.852241 + 0.523149i \(0.175243\pi\)
\(828\) −9.07572e8 −0.0555616
\(829\) 3.00153e10 1.82979 0.914896 0.403691i \(-0.132273\pi\)
0.914896 + 0.403691i \(0.132273\pi\)
\(830\) −1.32026e10 −0.801471
\(831\) −1.33108e10 −0.804636
\(832\) −1.45929e10 −0.878433
\(833\) −6.42118e9 −0.384909
\(834\) 8.12669e9 0.485101
\(835\) −1.33259e10 −0.792127
\(836\) 1.19412e10 0.706851
\(837\) −6.24019e9 −0.367840
\(838\) 3.47581e10 2.04034
\(839\) −1.61747e10 −0.945516 −0.472758 0.881192i \(-0.656741\pi\)
−0.472758 + 0.881192i \(0.656741\pi\)
\(840\) −7.18022e7 −0.00417985
\(841\) 3.68429e10 2.13583
\(842\) 2.49338e10 1.43945
\(843\) −6.21791e9 −0.357477
\(844\) −7.43101e9 −0.425451
\(845\) 1.31568e10 0.750155
\(846\) 6.12882e9 0.348001
\(847\) −1.25602e9 −0.0710236
\(848\) −1.59653e10 −0.899066
\(849\) −6.12752e9 −0.343644
\(850\) −6.62035e9 −0.369755
\(851\) 2.19702e9 0.122203
\(852\) 3.69060e9 0.204436
\(853\) 1.01140e10 0.557958 0.278979 0.960297i \(-0.410004\pi\)
0.278979 + 0.960297i \(0.410004\pi\)
\(854\) 1.24627e9 0.0684713
\(855\) 1.85425e9 0.101458
\(856\) −2.15439e9 −0.117399
\(857\) 2.45418e10 1.33190 0.665952 0.745995i \(-0.268026\pi\)
0.665952 + 0.745995i \(0.268026\pi\)
\(858\) −3.44929e10 −1.86434
\(859\) −7.47642e9 −0.402455 −0.201228 0.979545i \(-0.564493\pi\)
−0.201228 + 0.979545i \(0.564493\pi\)
\(860\) −1.00868e10 −0.540766
\(861\) −3.87216e8 −0.0206748
\(862\) 2.62325e10 1.39497
\(863\) 1.37132e10 0.726276 0.363138 0.931735i \(-0.381705\pi\)
0.363138 + 0.931735i \(0.381705\pi\)
\(864\) 4.69721e9 0.247766
\(865\) 4.46399e9 0.234513
\(866\) 1.00909e10 0.527981
\(867\) 9.42937e9 0.491378
\(868\) −1.48180e9 −0.0769077
\(869\) 5.70247e10 2.94777
\(870\) −1.42376e10 −0.733026
\(871\) −5.79520e10 −2.97170
\(872\) −6.83199e9 −0.348931
\(873\) 6.99905e9 0.356032
\(874\) −3.14381e9 −0.159282
\(875\) 9.13966e8 0.0461213
\(876\) 6.04383e9 0.303772
\(877\) −1.47072e10 −0.736260 −0.368130 0.929774i \(-0.620002\pi\)
−0.368130 + 0.929774i \(0.620002\pi\)
\(878\) −3.64093e10 −1.81544
\(879\) −5.77546e8 −0.0286831
\(880\) −1.94682e10 −0.963020
\(881\) −9.76304e9 −0.481027 −0.240513 0.970646i \(-0.577316\pi\)
−0.240513 + 0.970646i \(0.577316\pi\)
\(882\) −9.08824e9 −0.446005
\(883\) 2.10864e10 1.03072 0.515358 0.856975i \(-0.327659\pi\)
0.515358 + 0.856975i \(0.327659\pi\)
\(884\) 9.82253e9 0.478234
\(885\) 6.72066e8 0.0325920
\(886\) 1.81250e10 0.875508
\(887\) −1.84126e10 −0.885897 −0.442948 0.896547i \(-0.646068\pi\)
−0.442948 + 0.896547i \(0.646068\pi\)
\(888\) −1.89994e9 −0.0910532
\(889\) −7.13981e8 −0.0340824
\(890\) −1.05686e10 −0.502521
\(891\) 3.64275e9 0.172527
\(892\) 1.69243e10 0.798422
\(893\) 9.43162e9 0.443207
\(894\) 1.49671e10 0.700578
\(895\) −7.09816e9 −0.330952
\(896\) −5.71558e8 −0.0265450
\(897\) 4.03432e9 0.186637
\(898\) −4.58998e10 −2.11516
\(899\) 7.37354e10 3.38467
\(900\) −4.16274e9 −0.190340
\(901\) 6.56437e9 0.298990
\(902\) −3.26602e10 −1.48182
\(903\) −8.13813e8 −0.0367805
\(904\) 1.94224e9 0.0874404
\(905\) 1.16418e10 0.522094
\(906\) −8.43387e9 −0.376772
\(907\) −2.49645e10 −1.11096 −0.555480 0.831530i \(-0.687465\pi\)
−0.555480 + 0.831530i \(0.687465\pi\)
\(908\) −1.14815e10 −0.508979
\(909\) −7.78739e9 −0.343889
\(910\) −1.27186e9 −0.0559493
\(911\) 2.05743e10 0.901594 0.450797 0.892627i \(-0.351140\pi\)
0.450797 + 0.892627i \(0.351140\pi\)
\(912\) 8.73943e9 0.381506
\(913\) −3.99146e10 −1.73574
\(914\) −3.77674e10 −1.63608
\(915\) 7.25154e9 0.312937
\(916\) −2.04143e10 −0.877608
\(917\) −1.91699e9 −0.0820972
\(918\) −2.33502e9 −0.0996186
\(919\) −2.63487e10 −1.11984 −0.559918 0.828548i \(-0.689167\pi\)
−0.559918 + 0.828548i \(0.689167\pi\)
\(920\) 7.08345e8 0.0299908
\(921\) 1.77142e10 0.747158
\(922\) −3.10552e10 −1.30489
\(923\) −1.64054e10 −0.686720
\(924\) 8.65010e8 0.0360719
\(925\) 1.00770e10 0.418637
\(926\) 1.84178e10 0.762254
\(927\) 8.75792e9 0.361095
\(928\) −5.55032e10 −2.27982
\(929\) −4.62120e10 −1.89104 −0.945518 0.325571i \(-0.894443\pi\)
−0.945518 + 0.325571i \(0.894443\pi\)
\(930\) −1.94077e10 −0.791195
\(931\) −1.39859e10 −0.568022
\(932\) −1.07599e10 −0.435363
\(933\) 5.51217e9 0.222196
\(934\) −1.94857e10 −0.782533
\(935\) 8.00463e9 0.320258
\(936\) −3.48880e9 −0.139063
\(937\) −6.52016e9 −0.258922 −0.129461 0.991584i \(-0.541325\pi\)
−0.129461 + 0.991584i \(0.541325\pi\)
\(938\) 3.27134e9 0.129424
\(939\) 9.00816e8 0.0355064
\(940\) 8.46812e9 0.332537
\(941\) 6.12084e9 0.239468 0.119734 0.992806i \(-0.461796\pi\)
0.119734 + 0.992806i \(0.461796\pi\)
\(942\) 2.41238e10 0.940303
\(943\) 3.81997e9 0.148344
\(944\) 3.16758e9 0.122553
\(945\) 1.34320e8 0.00517760
\(946\) −6.86422e10 −2.63616
\(947\) 6.84254e9 0.261814 0.130907 0.991395i \(-0.458211\pi\)
0.130907 + 0.991395i \(0.458211\pi\)
\(948\) −2.29838e10 −0.876179
\(949\) −2.68659e10 −1.02040
\(950\) −1.44197e10 −0.545660
\(951\) 1.99784e10 0.753233
\(952\) 1.39145e8 0.00522684
\(953\) 3.05467e10 1.14324 0.571622 0.820517i \(-0.306314\pi\)
0.571622 + 0.820517i \(0.306314\pi\)
\(954\) 9.29091e9 0.346449
\(955\) 9.79232e9 0.363809
\(956\) −2.60597e10 −0.964643
\(957\) −4.30436e10 −1.58751
\(958\) 3.97896e10 1.46214
\(959\) 1.47563e9 0.0540273
\(960\) 4.79310e9 0.174851
\(961\) 7.29982e10 2.65326
\(962\) −3.36544e10 −1.21879
\(963\) 4.03020e9 0.145423
\(964\) −2.67740e10 −0.962594
\(965\) 4.56630e9 0.163576
\(966\) −2.27734e8 −0.00812845
\(967\) −1.72169e10 −0.612296 −0.306148 0.951984i \(-0.599040\pi\)
−0.306148 + 0.951984i \(0.599040\pi\)
\(968\) −1.07154e10 −0.379703
\(969\) −3.59335e9 −0.126872
\(970\) 2.17678e10 0.765797
\(971\) −1.63286e10 −0.572377 −0.286188 0.958173i \(-0.592388\pi\)
−0.286188 + 0.958173i \(0.592388\pi\)
\(972\) −1.46821e9 −0.0512810
\(973\) 9.05930e8 0.0315282
\(974\) 6.52871e10 2.26397
\(975\) 1.85042e10 0.639371
\(976\) 3.41779e10 1.17671
\(977\) 3.33959e9 0.114568 0.0572838 0.998358i \(-0.481756\pi\)
0.0572838 + 0.998358i \(0.481756\pi\)
\(978\) 1.16844e10 0.399411
\(979\) −3.19514e10 −1.08831
\(980\) −1.25571e10 −0.426186
\(981\) 1.27805e10 0.432223
\(982\) 3.47032e9 0.116944
\(983\) −1.18002e10 −0.396233 −0.198116 0.980178i \(-0.563482\pi\)
−0.198116 + 0.980178i \(0.563482\pi\)
\(984\) −3.30344e9 −0.110531
\(985\) −9.89560e9 −0.329925
\(986\) 2.75910e10 0.916640
\(987\) 6.83216e8 0.0226177
\(988\) 2.13943e10 0.705746
\(989\) 8.02845e9 0.263903
\(990\) 1.13294e10 0.371093
\(991\) 2.60974e10 0.851803 0.425901 0.904770i \(-0.359957\pi\)
0.425901 + 0.904770i \(0.359957\pi\)
\(992\) −7.56581e10 −2.46073
\(993\) −1.18396e10 −0.383722
\(994\) 9.26069e8 0.0299083
\(995\) 5.71625e9 0.183963
\(996\) 1.60876e10 0.515921
\(997\) 1.94933e10 0.622949 0.311474 0.950255i \(-0.399177\pi\)
0.311474 + 0.950255i \(0.399177\pi\)
\(998\) 2.62587e10 0.836213
\(999\) 3.55420e9 0.112788
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 69.8.a.c.1.6 7
3.2 odd 2 207.8.a.d.1.2 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
69.8.a.c.1.6 7 1.1 even 1 trivial
207.8.a.d.1.2 7 3.2 odd 2