Properties

Label 338.8.a.b.1.1
Level $338$
Weight $8$
Character 338.1
Self dual yes
Analytic conductor $105.586$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [338,8,Mod(1,338)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(338, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 338.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: \(105.586138614\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 338.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +245.000 q^{5} +216.000 q^{6} +587.000 q^{7} -512.000 q^{8} -1458.00 q^{9} +O(q^{10})\) \(q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +245.000 q^{5} +216.000 q^{6} +587.000 q^{7} -512.000 q^{8} -1458.00 q^{9} -1960.00 q^{10} +3874.00 q^{11} -1728.00 q^{12} -4696.00 q^{14} -6615.00 q^{15} +4096.00 q^{16} +5229.00 q^{17} +11664.0 q^{18} +6522.00 q^{19} +15680.0 q^{20} -15849.0 q^{21} -30992.0 q^{22} -500.000 q^{23} +13824.0 q^{24} -18100.0 q^{25} +98415.0 q^{27} +37568.0 q^{28} +226954. q^{29} +52920.0 q^{30} -130156. q^{31} -32768.0 q^{32} -104598. q^{33} -41832.0 q^{34} +143815. q^{35} -93312.0 q^{36} +377769. q^{37} -52176.0 q^{38} -125440. q^{40} +539760. q^{41} +126792. q^{42} +13987.0 q^{43} +247936. q^{44} -357210. q^{45} +4000.00 q^{46} +526879. q^{47} -110592. q^{48} -478974. q^{49} +144800. q^{50} -141183. q^{51} -1.64994e6 q^{53} -787320. q^{54} +949130. q^{55} -300544. q^{56} -176094. q^{57} -1.81563e6 q^{58} +81194.0 q^{59} -423360. q^{60} -1.12695e6 q^{61} +1.04125e6 q^{62} -855846. q^{63} +262144. q^{64} +836784. q^{66} -478798. q^{67} +334656. q^{68} +13500.0 q^{69} -1.15052e6 q^{70} -940007. q^{71} +746496. q^{72} -1.67193e6 q^{73} -3.02215e6 q^{74} +488700. q^{75} +417408. q^{76} +2.27404e6 q^{77} -5.80119e6 q^{79} +1.00352e6 q^{80} +531441. q^{81} -4.31808e6 q^{82} -7.39882e6 q^{83} -1.01434e6 q^{84} +1.28110e6 q^{85} -111896. q^{86} -6.12776e6 q^{87} -1.98349e6 q^{88} +953754. q^{89} +2.85768e6 q^{90} -32000.0 q^{92} +3.51421e6 q^{93} -4.21503e6 q^{94} +1.59789e6 q^{95} +884736. q^{96} +1.03187e7 q^{97} +3.83179e6 q^{98} -5.64829e6 q^{99} +O(q^{100})\)

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 −0.288675 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 64.0000 0.500000
\(5\) 245.000 0.876539 0.438269 0.898844i \(-0.355592\pi\)
0.438269 + 0.898844i \(0.355592\pi\)
\(6\) 216.000 0.408248
\(7\) 587.000 0.646837 0.323419 0.946256i \(-0.395168\pi\)
0.323419 + 0.946256i \(0.395168\pi\)
\(8\) −512.000 −0.353553
\(9\) −1458.00 −0.666667
\(10\) −1960.00 −0.619806
\(11\) 3874.00 0.877577 0.438788 0.898590i \(-0.355408\pi\)
0.438788 + 0.898590i \(0.355408\pi\)
\(12\) −1728.00 −0.288675
\(13\) 0 0
\(14\) −4696.00 −0.457383
\(15\) −6615.00 −0.506070
\(16\) 4096.00 0.250000
\(17\) 5229.00 0.258135 0.129068 0.991636i \(-0.458802\pi\)
0.129068 + 0.991636i \(0.458802\pi\)
\(18\) 11664.0 0.471405
\(19\) 6522.00 0.218144 0.109072 0.994034i \(-0.465212\pi\)
0.109072 + 0.994034i \(0.465212\pi\)
\(20\) 15680.0 0.438269
\(21\) −15849.0 −0.373452
\(22\) −30992.0 −0.620541
\(23\) −500.000 −0.00856885 −0.00428443 0.999991i \(-0.501364\pi\)
−0.00428443 + 0.999991i \(0.501364\pi\)
\(24\) 13824.0 0.204124
\(25\) −18100.0 −0.231680
\(26\) 0 0
\(27\) 98415.0 0.962250
\(28\) 37568.0 0.323419
\(29\) 226954. 1.72800 0.864002 0.503488i \(-0.167950\pi\)
0.864002 + 0.503488i \(0.167950\pi\)
\(30\) 52920.0 0.357845
\(31\) −130156. −0.784690 −0.392345 0.919818i \(-0.628336\pi\)
−0.392345 + 0.919818i \(0.628336\pi\)
\(32\) −32768.0 −0.176777
\(33\) −104598. −0.506669
\(34\) −41832.0 −0.182529
\(35\) 143815. 0.566978
\(36\) −93312.0 −0.333333
\(37\) 377769. 1.22608 0.613042 0.790050i \(-0.289946\pi\)
0.613042 + 0.790050i \(0.289946\pi\)
\(38\) −52176.0 −0.154251
\(39\) 0 0
\(40\) −125440. −0.309903
\(41\) 539760. 1.22309 0.611543 0.791211i \(-0.290549\pi\)
0.611543 + 0.791211i \(0.290549\pi\)
\(42\) 126792. 0.264070
\(43\) 13987.0 0.0268278 0.0134139 0.999910i \(-0.495730\pi\)
0.0134139 + 0.999910i \(0.495730\pi\)
\(44\) 247936. 0.438788
\(45\) −357210. −0.584359
\(46\) 4000.00 0.00605909
\(47\) 526879. 0.740233 0.370116 0.928985i \(-0.379318\pi\)
0.370116 + 0.928985i \(0.379318\pi\)
\(48\) −110592. −0.144338
\(49\) −478974. −0.581602
\(50\) 144800. 0.163822
\(51\) −141183. −0.149034
\(52\) 0 0
\(53\) −1.64994e6 −1.52231 −0.761154 0.648571i \(-0.775367\pi\)
−0.761154 + 0.648571i \(0.775367\pi\)
\(54\) −787320. −0.680414
\(55\) 949130. 0.769230
\(56\) −300544. −0.228691
\(57\) −176094. −0.125945
\(58\) −1.81563e6 −1.22188
\(59\) 81194.0 0.0514685 0.0257343 0.999669i \(-0.491808\pi\)
0.0257343 + 0.999669i \(0.491808\pi\)
\(60\) −423360. −0.253035
\(61\) −1.12695e6 −0.635698 −0.317849 0.948141i \(-0.602961\pi\)
−0.317849 + 0.948141i \(0.602961\pi\)
\(62\) 1.04125e6 0.554860
\(63\) −855846. −0.431225
\(64\) 262144. 0.125000
\(65\) 0 0
\(66\) 836784. 0.358269
\(67\) −478798. −0.194487 −0.0972435 0.995261i \(-0.531003\pi\)
−0.0972435 + 0.995261i \(0.531003\pi\)
\(68\) 334656. 0.129068
\(69\) 13500.0 0.00494723
\(70\) −1.15052e6 −0.400914
\(71\) −940007. −0.311693 −0.155846 0.987781i \(-0.549810\pi\)
−0.155846 + 0.987781i \(0.549810\pi\)
\(72\) 746496. 0.235702
\(73\) −1.67193e6 −0.503022 −0.251511 0.967854i \(-0.580927\pi\)
−0.251511 + 0.967854i \(0.580927\pi\)
\(74\) −3.02215e6 −0.866972
\(75\) 488700. 0.133761
\(76\) 417408. 0.109072
\(77\) 2.27404e6 0.567649
\(78\) 0 0
\(79\) −5.80119e6 −1.32380 −0.661900 0.749592i \(-0.730249\pi\)
−0.661900 + 0.749592i \(0.730249\pi\)
\(80\) 1.00352e6 0.219135
\(81\) 531441. 0.111111
\(82\) −4.31808e6 −0.864853
\(83\) −7.39882e6 −1.42033 −0.710164 0.704036i \(-0.751379\pi\)
−0.710164 + 0.704036i \(0.751379\pi\)
\(84\) −1.01434e6 −0.186726
\(85\) 1.28110e6 0.226266
\(86\) −111896. −0.0189701
\(87\) −6.12776e6 −0.997664
\(88\) −1.98349e6 −0.310270
\(89\) 953754. 0.143407 0.0717037 0.997426i \(-0.477156\pi\)
0.0717037 + 0.997426i \(0.477156\pi\)
\(90\) 2.85768e6 0.413204
\(91\) 0 0
\(92\) −32000.0 −0.00428443
\(93\) 3.51421e6 0.453041
\(94\) −4.21503e6 −0.523424
\(95\) 1.59789e6 0.191212
\(96\) 884736. 0.102062
\(97\) 1.03187e7 1.14795 0.573976 0.818872i \(-0.305400\pi\)
0.573976 + 0.818872i \(0.305400\pi\)
\(98\) 3.83179e6 0.411254
\(99\) −5.64829e6 −0.585051
\(100\) −1.15840e6 −0.115840
\(101\) 4.73503e6 0.457296 0.228648 0.973509i \(-0.426569\pi\)
0.228648 + 0.973509i \(0.426569\pi\)
\(102\) 1.12946e6 0.105383
\(103\) 1.60974e7 1.45153 0.725763 0.687945i \(-0.241487\pi\)
0.725763 + 0.687945i \(0.241487\pi\)
\(104\) 0 0
\(105\) −3.88301e6 −0.327345
\(106\) 1.31995e7 1.07643
\(107\) 1.77823e7 1.40328 0.701642 0.712530i \(-0.252451\pi\)
0.701642 + 0.712530i \(0.252451\pi\)
\(108\) 6.29856e6 0.481125
\(109\) 2.20506e7 1.63090 0.815450 0.578827i \(-0.196489\pi\)
0.815450 + 0.578827i \(0.196489\pi\)
\(110\) −7.59304e6 −0.543928
\(111\) −1.01998e7 −0.707880
\(112\) 2.40435e6 0.161709
\(113\) −1.80880e7 −1.17928 −0.589639 0.807667i \(-0.700730\pi\)
−0.589639 + 0.807667i \(0.700730\pi\)
\(114\) 1.40875e6 0.0890569
\(115\) −122500. −0.00751093
\(116\) 1.45251e7 0.864002
\(117\) 0 0
\(118\) −649552. −0.0363938
\(119\) 3.06942e6 0.166972
\(120\) 3.38688e6 0.178923
\(121\) −4.47930e6 −0.229859
\(122\) 9.01562e6 0.449507
\(123\) −1.45735e7 −0.706149
\(124\) −8.32998e6 −0.392345
\(125\) −2.35751e7 −1.07962
\(126\) 6.84677e6 0.304922
\(127\) −2.52728e6 −0.109481 −0.0547407 0.998501i \(-0.517433\pi\)
−0.0547407 + 0.998501i \(0.517433\pi\)
\(128\) −2.09715e6 −0.0883883
\(129\) −377649. −0.0154890
\(130\) 0 0
\(131\) 4.19410e7 1.63001 0.815004 0.579456i \(-0.196735\pi\)
0.815004 + 0.579456i \(0.196735\pi\)
\(132\) −6.69427e6 −0.253335
\(133\) 3.82841e6 0.141104
\(134\) 3.83038e6 0.137523
\(135\) 2.41117e7 0.843450
\(136\) −2.67725e6 −0.0912646
\(137\) 2.45947e7 0.817182 0.408591 0.912718i \(-0.366020\pi\)
0.408591 + 0.912718i \(0.366020\pi\)
\(138\) −108000. −0.00349822
\(139\) −5.79349e7 −1.82974 −0.914869 0.403752i \(-0.867706\pi\)
−0.914869 + 0.403752i \(0.867706\pi\)
\(140\) 9.20416e6 0.283489
\(141\) −1.42257e7 −0.427374
\(142\) 7.52006e6 0.220400
\(143\) 0 0
\(144\) −5.97197e6 −0.166667
\(145\) 5.56037e7 1.51466
\(146\) 1.33754e7 0.355690
\(147\) 1.29323e7 0.335788
\(148\) 2.41772e7 0.613042
\(149\) 2.90993e7 0.720660 0.360330 0.932825i \(-0.382664\pi\)
0.360330 + 0.932825i \(0.382664\pi\)
\(150\) −3.90960e6 −0.0945830
\(151\) 4.13849e7 0.978187 0.489094 0.872231i \(-0.337328\pi\)
0.489094 + 0.872231i \(0.337328\pi\)
\(152\) −3.33926e6 −0.0771255
\(153\) −7.62388e6 −0.172090
\(154\) −1.81923e7 −0.401389
\(155\) −3.18882e7 −0.687811
\(156\) 0 0
\(157\) 8.60728e6 0.177508 0.0887538 0.996054i \(-0.471712\pi\)
0.0887538 + 0.996054i \(0.471712\pi\)
\(158\) 4.64095e7 0.936067
\(159\) 4.45484e7 0.878905
\(160\) −8.02816e6 −0.154952
\(161\) −293500. −0.00554265
\(162\) −4.25153e6 −0.0785674
\(163\) 8.32173e7 1.50507 0.752535 0.658552i \(-0.228831\pi\)
0.752535 + 0.658552i \(0.228831\pi\)
\(164\) 3.45446e7 0.611543
\(165\) −2.56265e7 −0.444115
\(166\) 5.91905e7 1.00432
\(167\) −1.15768e8 −1.92346 −0.961729 0.274004i \(-0.911652\pi\)
−0.961729 + 0.274004i \(0.911652\pi\)
\(168\) 8.11469e6 0.132035
\(169\) 0 0
\(170\) −1.02488e7 −0.159994
\(171\) −9.50908e6 −0.145429
\(172\) 895168. 0.0134139
\(173\) 9.10083e7 1.33635 0.668174 0.744005i \(-0.267076\pi\)
0.668174 + 0.744005i \(0.267076\pi\)
\(174\) 4.90221e7 0.705455
\(175\) −1.06247e7 −0.149859
\(176\) 1.58679e7 0.219394
\(177\) −2.19224e6 −0.0297154
\(178\) −7.63003e6 −0.101404
\(179\) 1.38177e8 1.80074 0.900370 0.435124i \(-0.143296\pi\)
0.900370 + 0.435124i \(0.143296\pi\)
\(180\) −2.28614e7 −0.292180
\(181\) 1.05517e8 1.32266 0.661332 0.750094i \(-0.269992\pi\)
0.661332 + 0.750094i \(0.269992\pi\)
\(182\) 0 0
\(183\) 3.04277e7 0.367021
\(184\) 256000. 0.00302955
\(185\) 9.25534e7 1.07471
\(186\) −2.81137e7 −0.320348
\(187\) 2.02571e7 0.226534
\(188\) 3.37203e7 0.370116
\(189\) 5.77696e7 0.622419
\(190\) −1.27831e7 −0.135207
\(191\) −9.36485e7 −0.972487 −0.486244 0.873823i \(-0.661633\pi\)
−0.486244 + 0.873823i \(0.661633\pi\)
\(192\) −7.07789e6 −0.0721688
\(193\) −1.31216e8 −1.31382 −0.656910 0.753969i \(-0.728137\pi\)
−0.656910 + 0.753969i \(0.728137\pi\)
\(194\) −8.25495e7 −0.811724
\(195\) 0 0
\(196\) −3.06543e7 −0.290801
\(197\) 1.35325e8 1.26109 0.630546 0.776152i \(-0.282831\pi\)
0.630546 + 0.776152i \(0.282831\pi\)
\(198\) 4.51863e7 0.413694
\(199\) −1.24505e8 −1.11996 −0.559980 0.828506i \(-0.689191\pi\)
−0.559980 + 0.828506i \(0.689191\pi\)
\(200\) 9.26720e6 0.0819112
\(201\) 1.29275e7 0.112287
\(202\) −3.78802e7 −0.323357
\(203\) 1.33222e8 1.11774
\(204\) −9.03571e6 −0.0745172
\(205\) 1.32241e8 1.07208
\(206\) −1.28779e8 −1.02638
\(207\) 729000. 0.00571257
\(208\) 0 0
\(209\) 2.52662e7 0.191438
\(210\) 3.10640e7 0.231468
\(211\) 8.06435e7 0.590991 0.295496 0.955344i \(-0.404515\pi\)
0.295496 + 0.955344i \(0.404515\pi\)
\(212\) −1.05596e8 −0.761154
\(213\) 2.53802e7 0.179956
\(214\) −1.42259e8 −0.992271
\(215\) 3.42682e6 0.0235156
\(216\) −5.03885e7 −0.340207
\(217\) −7.64016e7 −0.507567
\(218\) −1.76405e8 −1.15322
\(219\) 4.51420e7 0.290420
\(220\) 6.07443e7 0.384615
\(221\) 0 0
\(222\) 8.15981e7 0.500547
\(223\) 1.30612e8 0.788706 0.394353 0.918959i \(-0.370969\pi\)
0.394353 + 0.918959i \(0.370969\pi\)
\(224\) −1.92348e7 −0.114346
\(225\) 2.63898e7 0.154453
\(226\) 1.44704e8 0.833875
\(227\) 1.44302e8 0.818809 0.409405 0.912353i \(-0.365736\pi\)
0.409405 + 0.912353i \(0.365736\pi\)
\(228\) −1.12700e7 −0.0629727
\(229\) 1.03229e8 0.568039 0.284019 0.958819i \(-0.408332\pi\)
0.284019 + 0.958819i \(0.408332\pi\)
\(230\) 980000. 0.00531103
\(231\) −6.13990e7 −0.327733
\(232\) −1.16200e8 −0.610942
\(233\) −3.19575e7 −0.165511 −0.0827556 0.996570i \(-0.526372\pi\)
−0.0827556 + 0.996570i \(0.526372\pi\)
\(234\) 0 0
\(235\) 1.29085e8 0.648843
\(236\) 5.19642e6 0.0257343
\(237\) 1.56632e8 0.764296
\(238\) −2.45554e7 −0.118067
\(239\) 1.50183e8 0.711588 0.355794 0.934564i \(-0.384210\pi\)
0.355794 + 0.934564i \(0.384210\pi\)
\(240\) −2.70950e7 −0.126517
\(241\) −1.36186e8 −0.626718 −0.313359 0.949635i \(-0.601454\pi\)
−0.313359 + 0.949635i \(0.601454\pi\)
\(242\) 3.58344e7 0.162535
\(243\) −2.29583e8 −1.02640
\(244\) −7.21249e7 −0.317849
\(245\) −1.17349e8 −0.509796
\(246\) 1.16588e8 0.499323
\(247\) 0 0
\(248\) 6.66399e7 0.277430
\(249\) 1.99768e8 0.820027
\(250\) 1.88601e8 0.763403
\(251\) 1.89898e8 0.757989 0.378995 0.925399i \(-0.376270\pi\)
0.378995 + 0.925399i \(0.376270\pi\)
\(252\) −5.47741e7 −0.215612
\(253\) −1.93700e6 −0.00751983
\(254\) 2.02182e7 0.0774150
\(255\) −3.45898e7 −0.130634
\(256\) 1.67772e7 0.0625000
\(257\) 1.72850e8 0.635190 0.317595 0.948227i \(-0.397125\pi\)
0.317595 + 0.948227i \(0.397125\pi\)
\(258\) 3.02119e6 0.0109524
\(259\) 2.21750e8 0.793077
\(260\) 0 0
\(261\) −3.30899e8 −1.15200
\(262\) −3.35528e8 −1.15259
\(263\) −1.54338e8 −0.523151 −0.261576 0.965183i \(-0.584242\pi\)
−0.261576 + 0.965183i \(0.584242\pi\)
\(264\) 5.35542e7 0.179135
\(265\) −4.04235e8 −1.33436
\(266\) −3.06273e7 −0.0997753
\(267\) −2.57514e7 −0.0827963
\(268\) −3.06431e7 −0.0972435
\(269\) −1.34962e7 −0.0422745 −0.0211372 0.999777i \(-0.506729\pi\)
−0.0211372 + 0.999777i \(0.506729\pi\)
\(270\) −1.92893e8 −0.596409
\(271\) 6.48939e8 1.98067 0.990333 0.138707i \(-0.0442947\pi\)
0.990333 + 0.138707i \(0.0442947\pi\)
\(272\) 2.14180e7 0.0645338
\(273\) 0 0
\(274\) −1.96757e8 −0.577835
\(275\) −7.01194e7 −0.203317
\(276\) 864000. 0.00247361
\(277\) 5.55195e8 1.56952 0.784760 0.619800i \(-0.212786\pi\)
0.784760 + 0.619800i \(0.212786\pi\)
\(278\) 4.63479e8 1.29382
\(279\) 1.89767e8 0.523127
\(280\) −7.36333e7 −0.200457
\(281\) −3.34956e8 −0.900565 −0.450283 0.892886i \(-0.648677\pi\)
−0.450283 + 0.892886i \(0.648677\pi\)
\(282\) 1.13806e8 0.302199
\(283\) 5.76535e8 1.51207 0.756037 0.654529i \(-0.227133\pi\)
0.756037 + 0.654529i \(0.227133\pi\)
\(284\) −6.01604e7 −0.155846
\(285\) −4.31430e7 −0.110396
\(286\) 0 0
\(287\) 3.16839e8 0.791138
\(288\) 4.77757e7 0.117851
\(289\) −3.82996e8 −0.933366
\(290\) −4.44830e8 −1.07103
\(291\) −2.78605e8 −0.662770
\(292\) −1.07003e8 −0.251511
\(293\) 1.77399e8 0.412016 0.206008 0.978550i \(-0.433953\pi\)
0.206008 + 0.978550i \(0.433953\pi\)
\(294\) −1.03458e8 −0.237438
\(295\) 1.98925e7 0.0451142
\(296\) −1.93418e8 −0.433486
\(297\) 3.81260e8 0.844449
\(298\) −2.32794e8 −0.509583
\(299\) 0 0
\(300\) 3.12768e7 0.0668803
\(301\) 8.21037e6 0.0173532
\(302\) −3.31079e8 −0.691683
\(303\) −1.27846e8 −0.264020
\(304\) 2.67141e7 0.0545360
\(305\) −2.76103e8 −0.557214
\(306\) 6.09911e7 0.121686
\(307\) −2.94122e8 −0.580155 −0.290077 0.957003i \(-0.593681\pi\)
−0.290077 + 0.957003i \(0.593681\pi\)
\(308\) 1.45538e8 0.283825
\(309\) −4.34629e8 −0.838038
\(310\) 2.55106e8 0.486356
\(311\) −2.86005e8 −0.539154 −0.269577 0.962979i \(-0.586884\pi\)
−0.269577 + 0.962979i \(0.586884\pi\)
\(312\) 0 0
\(313\) −2.76110e8 −0.508951 −0.254476 0.967079i \(-0.581903\pi\)
−0.254476 + 0.967079i \(0.581903\pi\)
\(314\) −6.88582e7 −0.125517
\(315\) −2.09682e8 −0.377985
\(316\) −3.71276e8 −0.661900
\(317\) −5.45989e8 −0.962667 −0.481334 0.876537i \(-0.659847\pi\)
−0.481334 + 0.876537i \(0.659847\pi\)
\(318\) −3.56387e8 −0.621480
\(319\) 8.79220e8 1.51646
\(320\) 6.42253e7 0.109567
\(321\) −4.80123e8 −0.810186
\(322\) 2.34800e6 0.00391925
\(323\) 3.41035e7 0.0563107
\(324\) 3.40122e7 0.0555556
\(325\) 0 0
\(326\) −6.65738e8 −1.06425
\(327\) −5.95366e8 −0.941601
\(328\) −2.76357e8 −0.432426
\(329\) 3.09278e8 0.478810
\(330\) 2.05012e8 0.314037
\(331\) 3.60921e8 0.547034 0.273517 0.961867i \(-0.411813\pi\)
0.273517 + 0.961867i \(0.411813\pi\)
\(332\) −4.73524e8 −0.710164
\(333\) −5.50787e8 −0.817389
\(334\) 9.26148e8 1.36009
\(335\) −1.17306e8 −0.170475
\(336\) −6.49175e7 −0.0933629
\(337\) 4.27270e8 0.608132 0.304066 0.952651i \(-0.401656\pi\)
0.304066 + 0.952651i \(0.401656\pi\)
\(338\) 0 0
\(339\) 4.88376e8 0.680856
\(340\) 8.19907e7 0.113133
\(341\) −5.04224e8 −0.688626
\(342\) 7.60726e7 0.102834
\(343\) −7.64577e8 −1.02304
\(344\) −7.16134e6 −0.00948506
\(345\) 3.30750e6 0.00433644
\(346\) −7.28067e8 −0.944941
\(347\) 1.31637e9 1.69131 0.845656 0.533728i \(-0.179209\pi\)
0.845656 + 0.533728i \(0.179209\pi\)
\(348\) −3.92177e8 −0.498832
\(349\) 1.26585e9 1.59402 0.797012 0.603963i \(-0.206413\pi\)
0.797012 + 0.603963i \(0.206413\pi\)
\(350\) 8.49976e7 0.105966
\(351\) 0 0
\(352\) −1.26943e8 −0.155135
\(353\) −1.10618e9 −1.33848 −0.669241 0.743046i \(-0.733381\pi\)
−0.669241 + 0.743046i \(0.733381\pi\)
\(354\) 1.75379e7 0.0210119
\(355\) −2.30302e8 −0.273211
\(356\) 6.10403e7 0.0717037
\(357\) −8.28744e7 −0.0964010
\(358\) −1.10542e9 −1.27332
\(359\) 1.02959e9 1.17445 0.587223 0.809425i \(-0.300221\pi\)
0.587223 + 0.809425i \(0.300221\pi\)
\(360\) 1.82892e8 0.206602
\(361\) −8.51335e8 −0.952413
\(362\) −8.44140e8 −0.935264
\(363\) 1.20941e8 0.132709
\(364\) 0 0
\(365\) −4.09622e8 −0.440918
\(366\) −2.43422e8 −0.259523
\(367\) 7.84516e8 0.828458 0.414229 0.910173i \(-0.364051\pi\)
0.414229 + 0.910173i \(0.364051\pi\)
\(368\) −2.04800e6 −0.00214221
\(369\) −7.86970e8 −0.815391
\(370\) −7.40427e8 −0.759935
\(371\) −9.68515e8 −0.984686
\(372\) 2.24910e8 0.226521
\(373\) −5.52719e8 −0.551472 −0.275736 0.961233i \(-0.588922\pi\)
−0.275736 + 0.961233i \(0.588922\pi\)
\(374\) −1.62057e8 −0.160183
\(375\) 6.36528e8 0.623316
\(376\) −2.69762e8 −0.261712
\(377\) 0 0
\(378\) −4.62157e8 −0.440117
\(379\) −6.93760e8 −0.654594 −0.327297 0.944921i \(-0.606138\pi\)
−0.327297 + 0.944921i \(0.606138\pi\)
\(380\) 1.02265e8 0.0956058
\(381\) 6.82366e7 0.0632091
\(382\) 7.49188e8 0.687652
\(383\) 1.56765e9 1.42578 0.712890 0.701276i \(-0.247386\pi\)
0.712890 + 0.701276i \(0.247386\pi\)
\(384\) 5.66231e7 0.0510310
\(385\) 5.57139e8 0.497567
\(386\) 1.04973e9 0.929010
\(387\) −2.03930e7 −0.0178852
\(388\) 6.60396e8 0.573976
\(389\) −2.11817e9 −1.82447 −0.912235 0.409668i \(-0.865645\pi\)
−0.912235 + 0.409668i \(0.865645\pi\)
\(390\) 0 0
\(391\) −2.61450e6 −0.00221192
\(392\) 2.45235e8 0.205627
\(393\) −1.13241e9 −0.941085
\(394\) −1.08260e9 −0.891727
\(395\) −1.42129e9 −1.16036
\(396\) −3.61491e8 −0.292526
\(397\) −3.65973e8 −0.293550 −0.146775 0.989170i \(-0.546889\pi\)
−0.146775 + 0.989170i \(0.546889\pi\)
\(398\) 9.96043e8 0.791931
\(399\) −1.03367e8 −0.0814662
\(400\) −7.41376e7 −0.0579200
\(401\) 5.66697e8 0.438880 0.219440 0.975626i \(-0.429577\pi\)
0.219440 + 0.975626i \(0.429577\pi\)
\(402\) −1.03420e8 −0.0793990
\(403\) 0 0
\(404\) 3.03042e8 0.228648
\(405\) 1.30203e8 0.0973932
\(406\) −1.06578e9 −0.790360
\(407\) 1.46348e9 1.07598
\(408\) 7.22857e7 0.0526916
\(409\) 1.64659e9 1.19002 0.595011 0.803717i \(-0.297148\pi\)
0.595011 + 0.803717i \(0.297148\pi\)
\(410\) −1.05793e9 −0.758077
\(411\) −6.64056e8 −0.471800
\(412\) 1.03023e9 0.725763
\(413\) 4.76609e7 0.0332918
\(414\) −5.83200e6 −0.00403939
\(415\) −1.81271e9 −1.24497
\(416\) 0 0
\(417\) 1.56424e9 1.05640
\(418\) −2.02130e8 −0.135367
\(419\) −2.07538e8 −0.137831 −0.0689157 0.997622i \(-0.521954\pi\)
−0.0689157 + 0.997622i \(0.521954\pi\)
\(420\) −2.48512e8 −0.163672
\(421\) −1.28086e9 −0.836590 −0.418295 0.908311i \(-0.637372\pi\)
−0.418295 + 0.908311i \(0.637372\pi\)
\(422\) −6.45148e8 −0.417894
\(423\) −7.68190e8 −0.493489
\(424\) 8.44769e8 0.538217
\(425\) −9.46449e7 −0.0598048
\(426\) −2.03042e8 −0.127248
\(427\) −6.61521e8 −0.411193
\(428\) 1.13807e9 0.701642
\(429\) 0 0
\(430\) −2.74145e7 −0.0166280
\(431\) 1.30893e8 0.0787494 0.0393747 0.999225i \(-0.487463\pi\)
0.0393747 + 0.999225i \(0.487463\pi\)
\(432\) 4.03108e8 0.240563
\(433\) −3.01643e8 −0.178560 −0.0892802 0.996007i \(-0.528457\pi\)
−0.0892802 + 0.996007i \(0.528457\pi\)
\(434\) 6.11213e8 0.358904
\(435\) −1.50130e9 −0.874491
\(436\) 1.41124e9 0.815450
\(437\) −3.26100e6 −0.00186924
\(438\) −3.61136e8 −0.205358
\(439\) 8.03991e8 0.453550 0.226775 0.973947i \(-0.427182\pi\)
0.226775 + 0.973947i \(0.427182\pi\)
\(440\) −4.85955e8 −0.271964
\(441\) 6.98344e8 0.387734
\(442\) 0 0
\(443\) 1.78013e8 0.0972832 0.0486416 0.998816i \(-0.484511\pi\)
0.0486416 + 0.998816i \(0.484511\pi\)
\(444\) −6.52785e8 −0.353940
\(445\) 2.33670e8 0.125702
\(446\) −1.04489e9 −0.557699
\(447\) −7.85680e8 −0.416073
\(448\) 1.53879e8 0.0808546
\(449\) 1.61622e9 0.842634 0.421317 0.906914i \(-0.361568\pi\)
0.421317 + 0.906914i \(0.361568\pi\)
\(450\) −2.11118e8 −0.109215
\(451\) 2.09103e9 1.07335
\(452\) −1.15763e9 −0.589639
\(453\) −1.11739e9 −0.564757
\(454\) −1.15442e9 −0.578986
\(455\) 0 0
\(456\) 9.01601e7 0.0445284
\(457\) −1.23751e9 −0.606517 −0.303258 0.952908i \(-0.598074\pi\)
−0.303258 + 0.952908i \(0.598074\pi\)
\(458\) −8.25833e8 −0.401664
\(459\) 5.14612e8 0.248391
\(460\) −7.84000e6 −0.00375546
\(461\) −1.25678e8 −0.0597457 −0.0298729 0.999554i \(-0.509510\pi\)
−0.0298729 + 0.999554i \(0.509510\pi\)
\(462\) 4.91192e8 0.231742
\(463\) 1.01176e9 0.473743 0.236872 0.971541i \(-0.423878\pi\)
0.236872 + 0.971541i \(0.423878\pi\)
\(464\) 9.29604e8 0.432001
\(465\) 8.60982e8 0.397108
\(466\) 2.55660e8 0.117034
\(467\) −1.61515e9 −0.733846 −0.366923 0.930251i \(-0.619589\pi\)
−0.366923 + 0.930251i \(0.619589\pi\)
\(468\) 0 0
\(469\) −2.81054e8 −0.125801
\(470\) −1.03268e9 −0.458801
\(471\) −2.32397e8 −0.102484
\(472\) −4.15713e7 −0.0181969
\(473\) 5.41856e7 0.0235435
\(474\) −1.25306e9 −0.540439
\(475\) −1.18048e8 −0.0505396
\(476\) 1.96443e8 0.0834858
\(477\) 2.40561e9 1.01487
\(478\) −1.20147e9 −0.503169
\(479\) −4.98553e8 −0.207270 −0.103635 0.994615i \(-0.533047\pi\)
−0.103635 + 0.994615i \(0.533047\pi\)
\(480\) 2.16760e8 0.0894614
\(481\) 0 0
\(482\) 1.08949e9 0.443157
\(483\) 7.92450e6 0.00320005
\(484\) −2.86675e8 −0.114929
\(485\) 2.52808e9 1.00622
\(486\) 1.83666e9 0.725775
\(487\) 4.81147e9 1.88767 0.943836 0.330413i \(-0.107188\pi\)
0.943836 + 0.330413i \(0.107188\pi\)
\(488\) 5.76999e8 0.224753
\(489\) −2.24687e9 −0.868953
\(490\) 9.38789e8 0.360480
\(491\) 1.50724e9 0.574643 0.287322 0.957834i \(-0.407235\pi\)
0.287322 + 0.957834i \(0.407235\pi\)
\(492\) −9.32705e8 −0.353075
\(493\) 1.18674e9 0.446059
\(494\) 0 0
\(495\) −1.38383e9 −0.512820
\(496\) −5.33119e8 −0.196173
\(497\) −5.51784e8 −0.201615
\(498\) −1.59814e9 −0.579847
\(499\) 3.47961e9 1.25366 0.626829 0.779157i \(-0.284353\pi\)
0.626829 + 0.779157i \(0.284353\pi\)
\(500\) −1.50881e9 −0.539808
\(501\) 3.12575e9 1.11051
\(502\) −1.51919e9 −0.535979
\(503\) −1.65804e9 −0.580907 −0.290454 0.956889i \(-0.593806\pi\)
−0.290454 + 0.956889i \(0.593806\pi\)
\(504\) 4.38193e8 0.152461
\(505\) 1.16008e9 0.400838
\(506\) 1.54960e7 0.00531732
\(507\) 0 0
\(508\) −1.61746e8 −0.0547407
\(509\) 8.62799e8 0.289999 0.145000 0.989432i \(-0.453682\pi\)
0.145000 + 0.989432i \(0.453682\pi\)
\(510\) 2.76719e8 0.0923725
\(511\) −9.81421e8 −0.325373
\(512\) −1.34218e8 −0.0441942
\(513\) 6.41863e8 0.209909
\(514\) −1.38280e9 −0.449147
\(515\) 3.94386e9 1.27232
\(516\) −2.41695e7 −0.00774452
\(517\) 2.04113e9 0.649611
\(518\) −1.77400e9 −0.560790
\(519\) −2.45723e9 −0.771541
\(520\) 0 0
\(521\) −1.33343e9 −0.413085 −0.206542 0.978438i \(-0.566221\pi\)
−0.206542 + 0.978438i \(0.566221\pi\)
\(522\) 2.64719e9 0.814589
\(523\) 942900. 0.000288210 0 0.000144105 1.00000i \(-0.499954\pi\)
0.000144105 1.00000i \(0.499954\pi\)
\(524\) 2.68423e9 0.815004
\(525\) 2.86867e8 0.0865213
\(526\) 1.23470e9 0.369924
\(527\) −6.80586e8 −0.202556
\(528\) −4.28433e8 −0.126667
\(529\) −3.40458e9 −0.999927
\(530\) 3.23388e9 0.943536
\(531\) −1.18381e8 −0.0343124
\(532\) 2.45018e8 0.0705518
\(533\) 0 0
\(534\) 2.06011e8 0.0585458
\(535\) 4.35667e9 1.23003
\(536\) 2.45145e8 0.0687615
\(537\) −3.73079e9 −1.03966
\(538\) 1.07969e8 0.0298926
\(539\) −1.85555e9 −0.510400
\(540\) 1.54315e9 0.421725
\(541\) −3.42111e9 −0.928916 −0.464458 0.885595i \(-0.653751\pi\)
−0.464458 + 0.885595i \(0.653751\pi\)
\(542\) −5.19151e9 −1.40054
\(543\) −2.84897e9 −0.763640
\(544\) −1.71344e8 −0.0456323
\(545\) 5.40240e9 1.42955
\(546\) 0 0
\(547\) 4.07726e9 1.06515 0.532577 0.846381i \(-0.321224\pi\)
0.532577 + 0.846381i \(0.321224\pi\)
\(548\) 1.57406e9 0.408591
\(549\) 1.64310e9 0.423799
\(550\) 5.60955e8 0.143767
\(551\) 1.48019e9 0.376954
\(552\) −6.91200e6 −0.00174911
\(553\) −3.40530e9 −0.856283
\(554\) −4.44156e9 −1.10982
\(555\) −2.49894e9 −0.620484
\(556\) −3.70784e9 −0.914869
\(557\) 5.23587e9 1.28379 0.641897 0.766791i \(-0.278148\pi\)
0.641897 + 0.766791i \(0.278148\pi\)
\(558\) −1.51814e9 −0.369907
\(559\) 0 0
\(560\) 5.89066e8 0.141744
\(561\) −5.46943e8 −0.130789
\(562\) 2.67965e9 0.636796
\(563\) 7.04236e9 1.66318 0.831589 0.555392i \(-0.187432\pi\)
0.831589 + 0.555392i \(0.187432\pi\)
\(564\) −9.10447e8 −0.213687
\(565\) −4.43156e9 −1.03368
\(566\) −4.61228e9 −1.06920
\(567\) 3.11956e8 0.0718708
\(568\) 4.81284e8 0.110200
\(569\) 8.80025e8 0.200264 0.100132 0.994974i \(-0.468074\pi\)
0.100132 + 0.994974i \(0.468074\pi\)
\(570\) 3.45144e8 0.0780618
\(571\) 8.46430e9 1.90267 0.951337 0.308151i \(-0.0997102\pi\)
0.951337 + 0.308151i \(0.0997102\pi\)
\(572\) 0 0
\(573\) 2.52851e9 0.561466
\(574\) −2.53471e9 −0.559419
\(575\) 9.05000e6 0.00198523
\(576\) −3.82206e8 −0.0833333
\(577\) 2.74152e8 0.0594122 0.0297061 0.999559i \(-0.490543\pi\)
0.0297061 + 0.999559i \(0.490543\pi\)
\(578\) 3.06397e9 0.659990
\(579\) 3.54283e9 0.758534
\(580\) 3.55864e9 0.757331
\(581\) −4.34310e9 −0.918721
\(582\) 2.22884e9 0.468649
\(583\) −6.39187e9 −1.33594
\(584\) 8.56026e8 0.177845
\(585\) 0 0
\(586\) −1.41919e9 −0.291339
\(587\) 4.71162e9 0.961472 0.480736 0.876865i \(-0.340369\pi\)
0.480736 + 0.876865i \(0.340369\pi\)
\(588\) 8.27667e8 0.167894
\(589\) −8.48877e8 −0.171175
\(590\) −1.59140e8 −0.0319005
\(591\) −3.65378e9 −0.728092
\(592\) 1.54734e9 0.306521
\(593\) 4.59365e9 0.904620 0.452310 0.891861i \(-0.350600\pi\)
0.452310 + 0.891861i \(0.350600\pi\)
\(594\) −3.05008e9 −0.597115
\(595\) 7.52009e8 0.146357
\(596\) 1.86235e9 0.360330
\(597\) 3.36164e9 0.646609
\(598\) 0 0
\(599\) −2.06624e9 −0.392815 −0.196407 0.980522i \(-0.562927\pi\)
−0.196407 + 0.980522i \(0.562927\pi\)
\(600\) −2.50214e8 −0.0472915
\(601\) 8.75128e9 1.64441 0.822206 0.569190i \(-0.192743\pi\)
0.822206 + 0.569190i \(0.192743\pi\)
\(602\) −6.56830e7 −0.0122706
\(603\) 6.98087e8 0.129658
\(604\) 2.64863e9 0.489094
\(605\) −1.09743e9 −0.201480
\(606\) 1.02277e9 0.186690
\(607\) −5.99457e9 −1.08792 −0.543961 0.839111i \(-0.683076\pi\)
−0.543961 + 0.839111i \(0.683076\pi\)
\(608\) −2.13713e8 −0.0385628
\(609\) −3.59699e9 −0.645326
\(610\) 2.20883e9 0.394010
\(611\) 0 0
\(612\) −4.87928e8 −0.0860451
\(613\) 3.28777e9 0.576487 0.288244 0.957557i \(-0.406929\pi\)
0.288244 + 0.957557i \(0.406929\pi\)
\(614\) 2.35298e9 0.410231
\(615\) −3.57051e9 −0.618967
\(616\) −1.16431e9 −0.200694
\(617\) −2.90178e9 −0.497355 −0.248677 0.968586i \(-0.579996\pi\)
−0.248677 + 0.968586i \(0.579996\pi\)
\(618\) 3.47703e9 0.592583
\(619\) −6.02255e9 −1.02062 −0.510309 0.859991i \(-0.670469\pi\)
−0.510309 + 0.859991i \(0.670469\pi\)
\(620\) −2.04085e9 −0.343906
\(621\) −4.92075e7 −0.00824538
\(622\) 2.28804e9 0.381239
\(623\) 5.59854e8 0.0927612
\(624\) 0 0
\(625\) −4.36184e9 −0.714644
\(626\) 2.20888e9 0.359883
\(627\) −6.82188e8 −0.110527
\(628\) 5.50866e8 0.0887538
\(629\) 1.97535e9 0.316496
\(630\) 1.67746e9 0.267276
\(631\) 2.50727e8 0.0397282 0.0198641 0.999803i \(-0.493677\pi\)
0.0198641 + 0.999803i \(0.493677\pi\)
\(632\) 2.97021e9 0.468034
\(633\) −2.17738e9 −0.341209
\(634\) 4.36791e9 0.680709
\(635\) −6.19184e8 −0.0959647
\(636\) 2.85110e9 0.439453
\(637\) 0 0
\(638\) −7.03376e9 −1.07230
\(639\) 1.37053e9 0.207795
\(640\) −5.13802e8 −0.0774758
\(641\) −1.25739e10 −1.88568 −0.942839 0.333248i \(-0.891856\pi\)
−0.942839 + 0.333248i \(0.891856\pi\)
\(642\) 3.84098e9 0.572888
\(643\) −1.95354e9 −0.289790 −0.144895 0.989447i \(-0.546284\pi\)
−0.144895 + 0.989447i \(0.546284\pi\)
\(644\) −1.87840e7 −0.00277133
\(645\) −9.25240e7 −0.0135767
\(646\) −2.72828e8 −0.0398176
\(647\) −7.96393e9 −1.15601 −0.578006 0.816032i \(-0.696169\pi\)
−0.578006 + 0.816032i \(0.696169\pi\)
\(648\) −2.72098e8 −0.0392837
\(649\) 3.14546e8 0.0451676
\(650\) 0 0
\(651\) 2.06284e9 0.293044
\(652\) 5.32591e9 0.752535
\(653\) 7.42841e8 0.104400 0.0521999 0.998637i \(-0.483377\pi\)
0.0521999 + 0.998637i \(0.483377\pi\)
\(654\) 4.76293e9 0.665812
\(655\) 1.02756e10 1.42876
\(656\) 2.21086e9 0.305772
\(657\) 2.43767e9 0.335348
\(658\) −2.47422e9 −0.338570
\(659\) −4.38274e9 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(660\) −1.64010e9 −0.222058
\(661\) −1.01201e10 −1.36296 −0.681478 0.731839i \(-0.738662\pi\)
−0.681478 + 0.731839i \(0.738662\pi\)
\(662\) −2.88737e9 −0.386812
\(663\) 0 0
\(664\) 3.78819e9 0.502162
\(665\) 9.37961e8 0.123683
\(666\) 4.40630e9 0.577982
\(667\) −1.13477e8 −0.0148070
\(668\) −7.40918e9 −0.961729
\(669\) −3.52651e9 −0.455359
\(670\) 9.38444e8 0.120544
\(671\) −4.36581e9 −0.557874
\(672\) 5.19340e8 0.0660175
\(673\) 1.41044e10 1.78362 0.891808 0.452415i \(-0.149437\pi\)
0.891808 + 0.452415i \(0.149437\pi\)
\(674\) −3.41816e9 −0.430014
\(675\) −1.78131e9 −0.222934
\(676\) 0 0
\(677\) −1.31550e10 −1.62942 −0.814708 0.579872i \(-0.803103\pi\)
−0.814708 + 0.579872i \(0.803103\pi\)
\(678\) −3.90701e9 −0.481438
\(679\) 6.05707e9 0.742538
\(680\) −6.55926e8 −0.0799970
\(681\) −3.89616e9 −0.472740
\(682\) 4.03379e9 0.486932
\(683\) −8.18437e9 −0.982907 −0.491454 0.870904i \(-0.663534\pi\)
−0.491454 + 0.870904i \(0.663534\pi\)
\(684\) −6.08581e8 −0.0727147
\(685\) 6.02569e9 0.716292
\(686\) 6.11662e9 0.723398
\(687\) −2.78718e9 −0.327957
\(688\) 5.72908e7 0.00670695
\(689\) 0 0
\(690\) −2.64600e7 −0.00306632
\(691\) 6.59230e9 0.760088 0.380044 0.924968i \(-0.375909\pi\)
0.380044 + 0.924968i \(0.375909\pi\)
\(692\) 5.82453e9 0.668174
\(693\) −3.31555e9 −0.378433
\(694\) −1.05309e10 −1.19594
\(695\) −1.41941e10 −1.60384
\(696\) 3.13741e9 0.352727
\(697\) 2.82241e9 0.315722
\(698\) −1.01268e10 −1.12715
\(699\) 8.62853e8 0.0955580
\(700\) −6.79981e8 −0.0749296
\(701\) −7.42154e9 −0.813731 −0.406866 0.913488i \(-0.633378\pi\)
−0.406866 + 0.913488i \(0.633378\pi\)
\(702\) 0 0
\(703\) 2.46381e9 0.267463
\(704\) 1.01555e9 0.109697
\(705\) −3.48530e9 −0.374610
\(706\) 8.84940e9 0.946449
\(707\) 2.77946e9 0.295796
\(708\) −1.40303e8 −0.0148577
\(709\) 1.28256e10 1.35150 0.675748 0.737133i \(-0.263821\pi\)
0.675748 + 0.737133i \(0.263821\pi\)
\(710\) 1.84241e9 0.193189
\(711\) 8.45813e9 0.882533
\(712\) −4.88322e8 −0.0507021
\(713\) 6.50780e7 0.00672389
\(714\) 6.62995e8 0.0681658
\(715\) 0 0
\(716\) 8.84335e9 0.900370
\(717\) −4.05495e9 −0.410836
\(718\) −8.23671e9 −0.830459
\(719\) −1.34066e8 −0.0134514 −0.00672568 0.999977i \(-0.502141\pi\)
−0.00672568 + 0.999977i \(0.502141\pi\)
\(720\) −1.46313e9 −0.146090
\(721\) 9.44916e9 0.938900
\(722\) 6.81068e9 0.673458
\(723\) 3.67702e9 0.361836
\(724\) 6.75312e9 0.661332
\(725\) −4.10787e9 −0.400344
\(726\) −9.67528e8 −0.0938394
\(727\) 8.04352e8 0.0776383 0.0388191 0.999246i \(-0.487640\pi\)
0.0388191 + 0.999246i \(0.487640\pi\)
\(728\) 0 0
\(729\) 5.03647e9 0.481481
\(730\) 3.27697e9 0.311776
\(731\) 7.31380e7 0.00692520
\(732\) 1.94737e9 0.183510
\(733\) 1.64250e10 1.54043 0.770213 0.637786i \(-0.220150\pi\)
0.770213 + 0.637786i \(0.220150\pi\)
\(734\) −6.27613e9 −0.585808
\(735\) 3.16841e9 0.294331
\(736\) 1.63840e7 0.00151477
\(737\) −1.85486e9 −0.170677
\(738\) 6.29576e9 0.576569
\(739\) 3.94761e9 0.359815 0.179907 0.983684i \(-0.442420\pi\)
0.179907 + 0.983684i \(0.442420\pi\)
\(740\) 5.92342e9 0.537355
\(741\) 0 0
\(742\) 7.74812e9 0.696278
\(743\) −1.81094e10 −1.61974 −0.809868 0.586612i \(-0.800461\pi\)
−0.809868 + 0.586612i \(0.800461\pi\)
\(744\) −1.79928e9 −0.160174
\(745\) 7.12932e9 0.631686
\(746\) 4.42175e9 0.389950
\(747\) 1.07875e10 0.946886
\(748\) 1.29646e9 0.113267
\(749\) 1.04382e10 0.907696
\(750\) −5.09223e9 −0.440751
\(751\) −7.45449e9 −0.642211 −0.321106 0.947043i \(-0.604054\pi\)
−0.321106 + 0.947043i \(0.604054\pi\)
\(752\) 2.15810e9 0.185058
\(753\) −5.12726e9 −0.437625
\(754\) 0 0
\(755\) 1.01393e10 0.857419
\(756\) 3.69725e9 0.311210
\(757\) −1.20015e10 −1.00554 −0.502770 0.864420i \(-0.667686\pi\)
−0.502770 + 0.864420i \(0.667686\pi\)
\(758\) 5.55008e9 0.462868
\(759\) 5.22990e7 0.00434157
\(760\) −8.18120e8 −0.0676035
\(761\) −1.21973e10 −1.00327 −0.501635 0.865080i \(-0.667268\pi\)
−0.501635 + 0.865080i \(0.667268\pi\)
\(762\) −5.45892e8 −0.0446956
\(763\) 1.29437e10 1.05493
\(764\) −5.99350e9 −0.486244
\(765\) −1.86785e9 −0.150844
\(766\) −1.25412e10 −1.00818
\(767\) 0 0
\(768\) −4.52985e8 −0.0360844
\(769\) 7.65065e8 0.0606675 0.0303338 0.999540i \(-0.490343\pi\)
0.0303338 + 0.999540i \(0.490343\pi\)
\(770\) −4.45711e9 −0.351833
\(771\) −4.66695e9 −0.366727
\(772\) −8.39781e9 −0.656910
\(773\) 5.75818e8 0.0448391 0.0224195 0.999749i \(-0.492863\pi\)
0.0224195 + 0.999749i \(0.492863\pi\)
\(774\) 1.63144e8 0.0126467
\(775\) 2.35582e9 0.181797
\(776\) −5.28317e9 −0.405862
\(777\) −5.98726e9 −0.457883
\(778\) 1.69453e10 1.29009
\(779\) 3.52031e9 0.266809
\(780\) 0 0
\(781\) −3.64159e9 −0.273534
\(782\) 2.09160e7 0.00156407
\(783\) 2.23357e10 1.66277
\(784\) −1.96188e9 −0.145400
\(785\) 2.10878e9 0.155592
\(786\) 9.05926e9 0.665448
\(787\) −5.85027e9 −0.427823 −0.213912 0.976853i \(-0.568620\pi\)
−0.213912 + 0.976853i \(0.568620\pi\)
\(788\) 8.66081e9 0.630546
\(789\) 4.16712e9 0.302041
\(790\) 1.13703e10 0.820499
\(791\) −1.06177e10 −0.762800
\(792\) 2.89193e9 0.206847
\(793\) 0 0
\(794\) 2.92779e9 0.207571
\(795\) 1.09144e10 0.770394
\(796\) −7.96834e9 −0.559980
\(797\) 7.90313e9 0.552962 0.276481 0.961019i \(-0.410832\pi\)
0.276481 + 0.961019i \(0.410832\pi\)
\(798\) 8.26937e8 0.0576053
\(799\) 2.75505e9 0.191080
\(800\) 5.93101e8 0.0409556
\(801\) −1.39057e9 −0.0956049
\(802\) −4.53358e9 −0.310335
\(803\) −6.47704e9 −0.441441
\(804\) 8.27363e8 0.0561436
\(805\) −7.19075e7 −0.00485835
\(806\) 0 0
\(807\) 3.64397e8 0.0244072
\(808\) −2.42433e9 −0.161679
\(809\) −1.36831e10 −0.908582 −0.454291 0.890853i \(-0.650107\pi\)
−0.454291 + 0.890853i \(0.650107\pi\)
\(810\) −1.04162e9 −0.0688674
\(811\) 2.02619e10 1.33385 0.666924 0.745126i \(-0.267611\pi\)
0.666924 + 0.745126i \(0.267611\pi\)
\(812\) 8.52621e9 0.558869
\(813\) −1.75214e10 −1.14354
\(814\) −1.17078e10 −0.760835
\(815\) 2.03882e10 1.31925
\(816\) −5.78286e8 −0.0372586
\(817\) 9.12232e7 0.00585232
\(818\) −1.31728e10 −0.841473
\(819\) 0 0
\(820\) 8.46344e9 0.536041
\(821\) −1.27444e10 −0.803742 −0.401871 0.915696i \(-0.631640\pi\)
−0.401871 + 0.915696i \(0.631640\pi\)
\(822\) 5.31245e9 0.333613
\(823\) 1.30392e9 0.0815366 0.0407683 0.999169i \(-0.487019\pi\)
0.0407683 + 0.999169i \(0.487019\pi\)
\(824\) −8.24185e9 −0.513192
\(825\) 1.89322e9 0.117385
\(826\) −3.81287e8 −0.0235408
\(827\) −6.58079e9 −0.404584 −0.202292 0.979325i \(-0.564839\pi\)
−0.202292 + 0.979325i \(0.564839\pi\)
\(828\) 4.66560e7 0.00285628
\(829\) −1.84124e10 −1.12246 −0.561229 0.827661i \(-0.689671\pi\)
−0.561229 + 0.827661i \(0.689671\pi\)
\(830\) 1.45017e10 0.880329
\(831\) −1.49903e10 −0.906162
\(832\) 0 0
\(833\) −2.50456e9 −0.150132
\(834\) −1.25139e10 −0.746987
\(835\) −2.83633e10 −1.68598
\(836\) 1.61704e9 0.0957191
\(837\) −1.28093e10 −0.755069
\(838\) 1.66030e9 0.0974616
\(839\) −3.25291e10 −1.90154 −0.950769 0.309901i \(-0.899704\pi\)
−0.950769 + 0.309901i \(0.899704\pi\)
\(840\) 1.98810e9 0.115734
\(841\) 3.42582e10 1.98600
\(842\) 1.02468e10 0.591559
\(843\) 9.04380e9 0.519942
\(844\) 5.16119e9 0.295496
\(845\) 0 0
\(846\) 6.14552e9 0.348949
\(847\) −2.62935e9 −0.148681
\(848\) −6.75815e9 −0.380577
\(849\) −1.55664e10 −0.872996
\(850\) 7.57159e8 0.0422884
\(851\) −1.88884e8 −0.0105061
\(852\) 1.62433e9 0.0899780
\(853\) −2.08665e10 −1.15114 −0.575569 0.817753i \(-0.695220\pi\)
−0.575569 + 0.817753i \(0.695220\pi\)
\(854\) 5.29217e9 0.290758
\(855\) −2.32972e9 −0.127474
\(856\) −9.10455e9 −0.496136
\(857\) −4.27453e9 −0.231983 −0.115991 0.993250i \(-0.537004\pi\)
−0.115991 + 0.993250i \(0.537004\pi\)
\(858\) 0 0
\(859\) −3.16915e8 −0.0170595 −0.00852976 0.999964i \(-0.502715\pi\)
−0.00852976 + 0.999964i \(0.502715\pi\)
\(860\) 2.19316e8 0.0117578
\(861\) −8.55466e9 −0.456764
\(862\) −1.04715e9 −0.0556842
\(863\) −2.64396e10 −1.40029 −0.700144 0.714002i \(-0.746881\pi\)
−0.700144 + 0.714002i \(0.746881\pi\)
\(864\) −3.22486e9 −0.170103
\(865\) 2.22970e10 1.17136
\(866\) 2.41314e9 0.126261
\(867\) 1.03409e10 0.538879
\(868\) −4.88970e9 −0.253783
\(869\) −2.24738e10 −1.16174
\(870\) 1.20104e10 0.618358
\(871\) 0 0
\(872\) −1.12899e10 −0.576610
\(873\) −1.50447e10 −0.765301
\(874\) 2.60880e7 0.00132175
\(875\) −1.38386e10 −0.698335
\(876\) 2.88909e9 0.145210
\(877\) 1.09380e9 0.0547568 0.0273784 0.999625i \(-0.491284\pi\)
0.0273784 + 0.999625i \(0.491284\pi\)
\(878\) −6.43193e9 −0.320708
\(879\) −4.78977e9 −0.237878
\(880\) 3.88764e9 0.192308
\(881\) −5.62114e8 −0.0276955 −0.0138477 0.999904i \(-0.504408\pi\)
−0.0138477 + 0.999904i \(0.504408\pi\)
\(882\) −5.58675e9 −0.274170
\(883\) 6.92318e8 0.0338410 0.0169205 0.999857i \(-0.494614\pi\)
0.0169205 + 0.999857i \(0.494614\pi\)
\(884\) 0 0
\(885\) −5.37098e8 −0.0260467
\(886\) −1.42410e9 −0.0687896
\(887\) 9.12313e9 0.438946 0.219473 0.975619i \(-0.429566\pi\)
0.219473 + 0.975619i \(0.429566\pi\)
\(888\) 5.22228e9 0.250273
\(889\) −1.48351e9 −0.0708166
\(890\) −1.86936e9 −0.0888848
\(891\) 2.05880e9 0.0975086
\(892\) 8.35915e9 0.394353
\(893\) 3.43630e9 0.161477
\(894\) 6.28544e9 0.294208
\(895\) 3.38534e10 1.57842
\(896\) −1.23103e9 −0.0571729
\(897\) 0 0
\(898\) −1.29298e10 −0.595832
\(899\) −2.95394e10 −1.35595
\(900\) 1.68895e9 0.0772267
\(901\) −8.62754e9 −0.392962
\(902\) −1.67282e10 −0.758975
\(903\) −2.21680e8 −0.0100189
\(904\) 9.26105e9 0.416937
\(905\) 2.58518e10 1.15937
\(906\) 8.93913e9 0.399343
\(907\) −3.52627e10 −1.56924 −0.784622 0.619975i \(-0.787143\pi\)
−0.784622 + 0.619975i \(0.787143\pi\)
\(908\) 9.23535e9 0.409405
\(909\) −6.90367e9 −0.304864
\(910\) 0 0
\(911\) −2.42397e10 −1.06222 −0.531108 0.847304i \(-0.678224\pi\)
−0.531108 + 0.847304i \(0.678224\pi\)
\(912\) −7.21281e8 −0.0314864
\(913\) −2.86630e10 −1.24645
\(914\) 9.90009e9 0.428872
\(915\) 7.45479e9 0.321708
\(916\) 6.60666e9 0.284019
\(917\) 2.46194e10 1.05435
\(918\) −4.11690e9 −0.175639
\(919\) −2.87487e10 −1.22184 −0.610920 0.791692i \(-0.709200\pi\)
−0.610920 + 0.791692i \(0.709200\pi\)
\(920\) 6.27200e7 0.00265551
\(921\) 7.94131e9 0.334952
\(922\) 1.00543e9 0.0422466
\(923\) 0 0
\(924\) −3.92954e9 −0.163866
\(925\) −6.83762e9 −0.284059
\(926\) −8.09406e9 −0.334987
\(927\) −2.34700e10 −0.967683
\(928\) −7.43683e9 −0.305471
\(929\) −1.67658e10 −0.686072 −0.343036 0.939322i \(-0.611455\pi\)
−0.343036 + 0.939322i \(0.611455\pi\)
\(930\) −6.88786e9 −0.280798
\(931\) −3.12387e9 −0.126873
\(932\) −2.04528e9 −0.0827556
\(933\) 7.72214e9 0.311281
\(934\) 1.29212e10 0.518907
\(935\) 4.96300e9 0.198565
\(936\) 0 0
\(937\) 4.14519e9 0.164610 0.0823049 0.996607i \(-0.473772\pi\)
0.0823049 + 0.996607i \(0.473772\pi\)
\(938\) 2.24844e9 0.0889550
\(939\) 7.45496e9 0.293843
\(940\) 8.26146e9 0.324421
\(941\) −2.40106e10 −0.939376 −0.469688 0.882833i \(-0.655634\pi\)
−0.469688 + 0.882833i \(0.655634\pi\)
\(942\) 1.85917e9 0.0724672
\(943\) −2.69880e8 −0.0104804
\(944\) 3.32571e8 0.0128671
\(945\) 1.41536e10 0.545575
\(946\) −4.33485e8 −0.0166477
\(947\) 5.10040e10 1.95155 0.975774 0.218783i \(-0.0702087\pi\)
0.975774 + 0.218783i \(0.0702087\pi\)
\(948\) 1.00245e10 0.382148
\(949\) 0 0
\(950\) 9.44386e8 0.0357369
\(951\) 1.47417e10 0.555796
\(952\) −1.57154e9 −0.0590333
\(953\) 1.53610e10 0.574902 0.287451 0.957795i \(-0.407192\pi\)
0.287451 + 0.957795i \(0.407192\pi\)
\(954\) −1.92449e10 −0.717623
\(955\) −2.29439e10 −0.852423
\(956\) 9.61172e9 0.355794
\(957\) −2.37389e10 −0.875527
\(958\) 3.98842e9 0.146562
\(959\) 1.44371e10 0.528584
\(960\) −1.73408e9 −0.0632587
\(961\) −1.05720e10 −0.384261
\(962\) 0 0
\(963\) −2.59266e10 −0.935522
\(964\) −8.71589e9 −0.313359
\(965\) −3.21479e10 −1.15161
\(966\) −6.33960e7 −0.00226278
\(967\) −3.46233e10 −1.23133 −0.615666 0.788007i \(-0.711113\pi\)
−0.615666 + 0.788007i \(0.711113\pi\)
\(968\) 2.29340e9 0.0812673
\(969\) −9.20796e8 −0.0325110
\(970\) −2.02246e10 −0.711508
\(971\) −2.01887e10 −0.707688 −0.353844 0.935304i \(-0.615126\pi\)
−0.353844 + 0.935304i \(0.615126\pi\)
\(972\) −1.46933e10 −0.513200
\(973\) −3.40078e10 −1.18354
\(974\) −3.84918e10 −1.33479
\(975\) 0 0
\(976\) −4.61600e9 −0.158925
\(977\) −3.10610e10 −1.06558 −0.532788 0.846249i \(-0.678856\pi\)
−0.532788 + 0.846249i \(0.678856\pi\)
\(978\) 1.79749e10 0.614442
\(979\) 3.69484e9 0.125851
\(980\) −7.51031e9 −0.254898
\(981\) −3.21498e10 −1.08727
\(982\) −1.20580e10 −0.406334
\(983\) 3.95227e10 1.32712 0.663559 0.748124i \(-0.269045\pi\)
0.663559 + 0.748124i \(0.269045\pi\)
\(984\) 7.46164e9 0.249662
\(985\) 3.31547e10 1.10540
\(986\) −9.49394e9 −0.315411
\(987\) −8.35051e9 −0.276441
\(988\) 0 0
\(989\) −6.99350e6 −0.000229883 0
\(990\) 1.10707e10 0.362619
\(991\) 3.60450e10 1.17649 0.588244 0.808684i \(-0.299820\pi\)
0.588244 + 0.808684i \(0.299820\pi\)
\(992\) 4.26495e9 0.138715
\(993\) −9.74488e9 −0.315830
\(994\) 4.41427e9 0.142563
\(995\) −3.05038e10 −0.981688
\(996\) 1.27852e10 0.410014
\(997\) 1.64165e10 0.524623 0.262311 0.964983i \(-0.415515\pi\)
0.262311 + 0.964983i \(0.415515\pi\)
\(998\) −2.78369e10 −0.886470
\(999\) 3.71781e10 1.17980
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 338.8.a.b.1.1 1
13.5 odd 4 338.8.b.c.337.2 2
13.8 odd 4 338.8.b.c.337.1 2
13.12 even 2 26.8.a.c.1.1 1
39.38 odd 2 234.8.a.b.1.1 1
52.51 odd 2 208.8.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
26.8.a.c.1.1 1 13.12 even 2
208.8.a.a.1.1 1 52.51 odd 2
234.8.a.b.1.1 1 39.38 odd 2
338.8.a.b.1.1 1 1.1 even 1 trivial
338.8.b.c.337.1 2 13.8 odd 4
338.8.b.c.337.2 2 13.5 odd 4