Properties

Label 338.8.a.a.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)
 
\(f(q)\) \(=\) \(q-8.00000 q^{2} -87.0000 q^{3} +64.0000 q^{4} -321.000 q^{5} +696.000 q^{6} +181.000 q^{7} -512.000 q^{8} +5382.00 q^{9} +O(q^{10})\) \(q-8.00000 q^{2} -87.0000 q^{3} +64.0000 q^{4} -321.000 q^{5} +696.000 q^{6} +181.000 q^{7} -512.000 q^{8} +5382.00 q^{9} +2568.00 q^{10} -7782.00 q^{11} -5568.00 q^{12} -1448.00 q^{14} +27927.0 q^{15} +4096.00 q^{16} +9069.00 q^{17} -43056.0 q^{18} +37150.0 q^{19} -20544.0 q^{20} -15747.0 q^{21} +62256.0 q^{22} +19008.0 q^{23} +44544.0 q^{24} +24916.0 q^{25} -277965. q^{27} +11584.0 q^{28} +174750. q^{29} -223416. q^{30} -29012.0 q^{31} -32768.0 q^{32} +677034. q^{33} -72552.0 q^{34} -58101.0 q^{35} +344448. q^{36} -323669. q^{37} -297200. q^{38} +164352. q^{40} -795312. q^{41} +125976. q^{42} -314137. q^{43} -498048. q^{44} -1.72762e6 q^{45} -152064. q^{46} +447441. q^{47} -356352. q^{48} -790782. q^{49} -199328. q^{50} -789003. q^{51} -1.46923e6 q^{53} +2.22372e6 q^{54} +2.49802e6 q^{55} -92672.0 q^{56} -3.23205e6 q^{57} -1.39800e6 q^{58} -1.62777e6 q^{59} +1.78733e6 q^{60} -2.39961e6 q^{61} +232096. q^{62} +974142. q^{63} +262144. q^{64} -5.41627e6 q^{66} +64066.0 q^{67} +580416. q^{68} -1.65370e6 q^{69} +464808. q^{70} +322383. q^{71} -2.75558e6 q^{72} +4.45478e6 q^{73} +2.58935e6 q^{74} -2.16769e6 q^{75} +2.37760e6 q^{76} -1.40854e6 q^{77} +753560. q^{79} -1.31482e6 q^{80} +1.24125e7 q^{81} +6.36250e6 q^{82} +1.21909e6 q^{83} -1.00781e6 q^{84} -2.91115e6 q^{85} +2.51310e6 q^{86} -1.52032e7 q^{87} +3.98438e6 q^{88} -3.39033e6 q^{89} +1.38210e7 q^{90} +1.21651e6 q^{92} +2.52404e6 q^{93} -3.57953e6 q^{94} -1.19252e7 q^{95} +2.85082e6 q^{96} -1.62877e6 q^{97} +6.32626e6 q^{98} -4.18827e7 q^{99} +O(q^{100})\)

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
\(3\) −87.0000 −1.86035 −0.930175 0.367115i \(-0.880345\pi\)
−0.930175 + 0.367115i \(0.880345\pi\)
\(4\) 64.0000 0.500000
\(5\) −321.000 −1.14844 −0.574222 0.818699i \(-0.694695\pi\)
−0.574222 + 0.818699i \(0.694695\pi\)
\(6\) 696.000 1.31547
\(7\) 181.000 0.199451 0.0997253 0.995015i \(-0.468204\pi\)
0.0997253 + 0.995015i \(0.468204\pi\)
\(8\) −512.000 −0.353553
\(9\) 5382.00 2.46091
\(10\) 2568.00 0.812073
\(11\) −7782.00 −1.76286 −0.881428 0.472318i \(-0.843417\pi\)
−0.881428 + 0.472318i \(0.843417\pi\)
\(12\) −5568.00 −0.930175
\(13\) 0 0
\(14\) −1448.00 −0.141033
\(15\) 27927.0 2.13651
\(16\) 4096.00 0.250000
\(17\) 9069.00 0.447701 0.223851 0.974623i \(-0.428137\pi\)
0.223851 + 0.974623i \(0.428137\pi\)
\(18\) −43056.0 −1.74012
\(19\) 37150.0 1.24257 0.621286 0.783584i \(-0.286611\pi\)
0.621286 + 0.783584i \(0.286611\pi\)
\(20\) −20544.0 −0.574222
\(21\) −15747.0 −0.371048
\(22\) 62256.0 1.24653
\(23\) 19008.0 0.325753 0.162877 0.986646i \(-0.447923\pi\)
0.162877 + 0.986646i \(0.447923\pi\)
\(24\) 44544.0 0.657733
\(25\) 24916.0 0.318925
\(26\) 0 0
\(27\) −277965. −2.71780
\(28\) 11584.0 0.0997253
\(29\) 174750. 1.33053 0.665264 0.746608i \(-0.268319\pi\)
0.665264 + 0.746608i \(0.268319\pi\)
\(30\) −223416. −1.51074
\(31\) −29012.0 −0.174909 −0.0874544 0.996169i \(-0.527873\pi\)
−0.0874544 + 0.996169i \(0.527873\pi\)
\(32\) −32768.0 −0.176777
\(33\) 677034. 3.27953
\(34\) −72552.0 −0.316572
\(35\) −58101.0 −0.229058
\(36\) 344448. 1.23045
\(37\) −323669. −1.05050 −0.525249 0.850949i \(-0.676028\pi\)
−0.525249 + 0.850949i \(0.676028\pi\)
\(38\) −297200. −0.878630
\(39\) 0 0
\(40\) 164352. 0.406036
\(41\) −795312. −1.80216 −0.901081 0.433650i \(-0.857225\pi\)
−0.901081 + 0.433650i \(0.857225\pi\)
\(42\) 125976. 0.262371
\(43\) −314137. −0.602531 −0.301266 0.953540i \(-0.597409\pi\)
−0.301266 + 0.953540i \(0.597409\pi\)
\(44\) −498048. −0.881428
\(45\) −1.72762e6 −2.82621
\(46\) −152064. −0.230342
\(47\) 447441. 0.628627 0.314314 0.949319i \(-0.398226\pi\)
0.314314 + 0.949319i \(0.398226\pi\)
\(48\) −356352. −0.465088
\(49\) −790782. −0.960219
\(50\) −199328. −0.225514
\(51\) −789003. −0.832881
\(52\) 0 0
\(53\) −1.46923e6 −1.35558 −0.677790 0.735256i \(-0.737062\pi\)
−0.677790 + 0.735256i \(0.737062\pi\)
\(54\) 2.22372e6 1.92177
\(55\) 2.49802e6 2.02454
\(56\) −92672.0 −0.0705165
\(57\) −3.23205e6 −2.31162
\(58\) −1.39800e6 −0.940826
\(59\) −1.62777e6 −1.03184 −0.515918 0.856638i \(-0.672549\pi\)
−0.515918 + 0.856638i \(0.672549\pi\)
\(60\) 1.78733e6 1.06825
\(61\) −2.39961e6 −1.35359 −0.676793 0.736173i \(-0.736631\pi\)
−0.676793 + 0.736173i \(0.736631\pi\)
\(62\) 232096. 0.123679
\(63\) 974142. 0.490829
\(64\) 262144. 0.125000
\(65\) 0 0
\(66\) −5.41627e6 −2.31898
\(67\) 64066.0 0.0260235 0.0130118 0.999915i \(-0.495858\pi\)
0.0130118 + 0.999915i \(0.495858\pi\)
\(68\) 580416. 0.223851
\(69\) −1.65370e6 −0.606016
\(70\) 464808. 0.161968
\(71\) 322383. 0.106898 0.0534488 0.998571i \(-0.482979\pi\)
0.0534488 + 0.998571i \(0.482979\pi\)
\(72\) −2.75558e6 −0.870061
\(73\) 4.45478e6 1.34028 0.670141 0.742233i \(-0.266233\pi\)
0.670141 + 0.742233i \(0.266233\pi\)
\(74\) 2.58935e6 0.742814
\(75\) −2.16769e6 −0.593312
\(76\) 2.37760e6 0.621286
\(77\) −1.40854e6 −0.351603
\(78\) 0 0
\(79\) 753560. 0.171958 0.0859791 0.996297i \(-0.472598\pi\)
0.0859791 + 0.996297i \(0.472598\pi\)
\(80\) −1.31482e6 −0.287111
\(81\) 1.24125e7 2.59515
\(82\) 6.36250e6 1.27432
\(83\) 1.21909e6 0.234025 0.117013 0.993130i \(-0.462668\pi\)
0.117013 + 0.993130i \(0.462668\pi\)
\(84\) −1.00781e6 −0.185524
\(85\) −2.91115e6 −0.514160
\(86\) 2.51310e6 0.426054
\(87\) −1.52032e7 −2.47525
\(88\) 3.98438e6 0.623264
\(89\) −3.39033e6 −0.509773 −0.254887 0.966971i \(-0.582038\pi\)
−0.254887 + 0.966971i \(0.582038\pi\)
\(90\) 1.38210e7 1.99843
\(91\) 0 0
\(92\) 1.21651e6 0.162877
\(93\) 2.52404e6 0.325392
\(94\) −3.57953e6 −0.444507
\(95\) −1.19252e7 −1.42702
\(96\) 2.85082e6 0.328867
\(97\) −1.62877e6 −0.181201 −0.0906003 0.995887i \(-0.528879\pi\)
−0.0906003 + 0.995887i \(0.528879\pi\)
\(98\) 6.32626e6 0.678978
\(99\) −4.18827e7 −4.33822
\(100\) 1.59462e6 0.159462
\(101\) −1.53503e7 −1.48249 −0.741244 0.671236i \(-0.765764\pi\)
−0.741244 + 0.671236i \(0.765764\pi\)
\(102\) 6.31202e6 0.588936
\(103\) 6.87643e6 0.620058 0.310029 0.950727i \(-0.399661\pi\)
0.310029 + 0.950727i \(0.399661\pi\)
\(104\) 0 0
\(105\) 5.05479e6 0.426128
\(106\) 1.17539e7 0.958539
\(107\) −1.52027e7 −1.19971 −0.599857 0.800107i \(-0.704776\pi\)
−0.599857 + 0.800107i \(0.704776\pi\)
\(108\) −1.77898e7 −1.35890
\(109\) −6.73260e6 −0.497955 −0.248978 0.968509i \(-0.580095\pi\)
−0.248978 + 0.968509i \(0.580095\pi\)
\(110\) −1.99842e7 −1.43157
\(111\) 2.81592e7 1.95429
\(112\) 741376. 0.0498627
\(113\) −1.15292e7 −0.751667 −0.375833 0.926687i \(-0.622644\pi\)
−0.375833 + 0.926687i \(0.622644\pi\)
\(114\) 2.58564e7 1.63456
\(115\) −6.10157e6 −0.374110
\(116\) 1.11840e7 0.665264
\(117\) 0 0
\(118\) 1.30222e7 0.729619
\(119\) 1.64149e6 0.0892943
\(120\) −1.42986e7 −0.755370
\(121\) 4.10724e7 2.10766
\(122\) 1.91969e7 0.957130
\(123\) 6.91921e7 3.35266
\(124\) −1.85677e6 −0.0874544
\(125\) 1.70801e7 0.782177
\(126\) −7.79314e6 −0.347069
\(127\) 2.06699e7 0.895418 0.447709 0.894179i \(-0.352240\pi\)
0.447709 + 0.894179i \(0.352240\pi\)
\(128\) −2.09715e6 −0.0883883
\(129\) 2.73299e7 1.12092
\(130\) 0 0
\(131\) −1.90949e7 −0.742107 −0.371054 0.928611i \(-0.621003\pi\)
−0.371054 + 0.928611i \(0.621003\pi\)
\(132\) 4.33302e7 1.63977
\(133\) 6.72415e6 0.247832
\(134\) −512528. −0.0184014
\(135\) 8.92268e7 3.12124
\(136\) −4.64333e6 −0.158286
\(137\) −2.96901e7 −0.986482 −0.493241 0.869893i \(-0.664188\pi\)
−0.493241 + 0.869893i \(0.664188\pi\)
\(138\) 1.32296e7 0.428518
\(139\) 1.55652e7 0.491591 0.245795 0.969322i \(-0.420951\pi\)
0.245795 + 0.969322i \(0.420951\pi\)
\(140\) −3.71846e6 −0.114529
\(141\) −3.89274e7 −1.16947
\(142\) −2.57906e6 −0.0755880
\(143\) 0 0
\(144\) 2.20447e7 0.615226
\(145\) −5.60948e7 −1.52804
\(146\) −3.56383e7 −0.947723
\(147\) 6.87980e7 1.78635
\(148\) −2.07148e7 −0.525249
\(149\) 2.49675e6 0.0618334 0.0309167 0.999522i \(-0.490157\pi\)
0.0309167 + 0.999522i \(0.490157\pi\)
\(150\) 1.73415e7 0.419535
\(151\) −2.39802e7 −0.566804 −0.283402 0.959001i \(-0.591463\pi\)
−0.283402 + 0.959001i \(0.591463\pi\)
\(152\) −1.90208e7 −0.439315
\(153\) 4.88094e7 1.10175
\(154\) 1.12683e7 0.248621
\(155\) 9.31285e6 0.200873
\(156\) 0 0
\(157\) 1.70550e7 0.351725 0.175863 0.984415i \(-0.443729\pi\)
0.175863 + 0.984415i \(0.443729\pi\)
\(158\) −6.02848e6 −0.121593
\(159\) 1.27823e8 2.52185
\(160\) 1.05185e7 0.203018
\(161\) 3.44045e6 0.0649717
\(162\) −9.93002e7 −1.83505
\(163\) 7.34586e7 1.32857 0.664287 0.747477i \(-0.268735\pi\)
0.664287 + 0.747477i \(0.268735\pi\)
\(164\) −5.09000e7 −0.901081
\(165\) −2.17328e8 −3.76636
\(166\) −9.75274e6 −0.165481
\(167\) −4.66860e7 −0.775674 −0.387837 0.921728i \(-0.626778\pi\)
−0.387837 + 0.921728i \(0.626778\pi\)
\(168\) 8.06246e6 0.131185
\(169\) 0 0
\(170\) 2.32892e7 0.363566
\(171\) 1.99941e8 3.05785
\(172\) −2.01048e7 −0.301266
\(173\) −7.80931e7 −1.14670 −0.573352 0.819309i \(-0.694357\pi\)
−0.573352 + 0.819309i \(0.694357\pi\)
\(174\) 1.21626e8 1.75027
\(175\) 4.50980e6 0.0636098
\(176\) −3.18751e7 −0.440714
\(177\) 1.41616e8 1.91958
\(178\) 2.71226e7 0.360464
\(179\) −5.56163e7 −0.724797 −0.362399 0.932023i \(-0.618042\pi\)
−0.362399 + 0.932023i \(0.618042\pi\)
\(180\) −1.10568e8 −1.41311
\(181\) −1.19435e8 −1.49711 −0.748557 0.663070i \(-0.769253\pi\)
−0.748557 + 0.663070i \(0.769253\pi\)
\(182\) 0 0
\(183\) 2.08766e8 2.51815
\(184\) −9.73210e6 −0.115171
\(185\) 1.03898e8 1.20644
\(186\) −2.01924e7 −0.230087
\(187\) −7.05750e7 −0.789233
\(188\) 2.86362e7 0.314314
\(189\) −5.03117e7 −0.542066
\(190\) 9.54012e7 1.00906
\(191\) 1.05485e8 1.09540 0.547700 0.836675i \(-0.315504\pi\)
0.547700 + 0.836675i \(0.315504\pi\)
\(192\) −2.28065e7 −0.232544
\(193\) −2.12059e7 −0.212327 −0.106164 0.994349i \(-0.533857\pi\)
−0.106164 + 0.994349i \(0.533857\pi\)
\(194\) 1.30302e7 0.128128
\(195\) 0 0
\(196\) −5.06100e7 −0.480110
\(197\) 1.66535e8 1.55194 0.775969 0.630771i \(-0.217261\pi\)
0.775969 + 0.630771i \(0.217261\pi\)
\(198\) 3.35062e8 3.06759
\(199\) −1.26351e8 −1.13656 −0.568279 0.822836i \(-0.692391\pi\)
−0.568279 + 0.822836i \(0.692391\pi\)
\(200\) −1.27570e7 −0.112757
\(201\) −5.57374e6 −0.0484129
\(202\) 1.22802e8 1.04828
\(203\) 3.16298e7 0.265375
\(204\) −5.04962e7 −0.416441
\(205\) 2.55295e8 2.06968
\(206\) −5.50114e7 −0.438448
\(207\) 1.02301e8 0.801648
\(208\) 0 0
\(209\) −2.89101e8 −2.19047
\(210\) −4.04383e7 −0.301318
\(211\) 1.08571e8 0.795655 0.397828 0.917460i \(-0.369764\pi\)
0.397828 + 0.917460i \(0.369764\pi\)
\(212\) −9.40308e7 −0.677790
\(213\) −2.80473e7 −0.198867
\(214\) 1.21622e8 0.848326
\(215\) 1.00838e8 0.691974
\(216\) 1.42318e8 0.960886
\(217\) −5.25117e6 −0.0348857
\(218\) 5.38608e7 0.352108
\(219\) −3.87566e8 −2.49340
\(220\) 1.59873e8 1.01227
\(221\) 0 0
\(222\) −2.25274e8 −1.38189
\(223\) 1.25603e8 0.758459 0.379229 0.925303i \(-0.376189\pi\)
0.379229 + 0.925303i \(0.376189\pi\)
\(224\) −5.93101e6 −0.0352582
\(225\) 1.34098e8 0.784844
\(226\) 9.22338e7 0.531509
\(227\) −1.90774e8 −1.08250 −0.541252 0.840861i \(-0.682049\pi\)
−0.541252 + 0.840861i \(0.682049\pi\)
\(228\) −2.06851e8 −1.15581
\(229\) 5.28911e7 0.291044 0.145522 0.989355i \(-0.453514\pi\)
0.145522 + 0.989355i \(0.453514\pi\)
\(230\) 4.88125e7 0.264536
\(231\) 1.22543e8 0.654104
\(232\) −8.94720e7 −0.470413
\(233\) 1.51254e8 0.783359 0.391680 0.920102i \(-0.371894\pi\)
0.391680 + 0.920102i \(0.371894\pi\)
\(234\) 0 0
\(235\) −1.43629e8 −0.721944
\(236\) −1.04177e8 −0.515918
\(237\) −6.55597e7 −0.319903
\(238\) −1.31319e7 −0.0631406
\(239\) −2.61917e8 −1.24100 −0.620498 0.784208i \(-0.713070\pi\)
−0.620498 + 0.784208i \(0.713070\pi\)
\(240\) 1.14389e8 0.534127
\(241\) 1.31752e8 0.606312 0.303156 0.952941i \(-0.401960\pi\)
0.303156 + 0.952941i \(0.401960\pi\)
\(242\) −3.28579e8 −1.49034
\(243\) −4.71980e8 −2.11009
\(244\) −1.53575e8 −0.676793
\(245\) 2.53841e8 1.10276
\(246\) −5.53537e8 −2.37069
\(247\) 0 0
\(248\) 1.48541e7 0.0618396
\(249\) −1.06061e8 −0.435370
\(250\) −1.36641e8 −0.553083
\(251\) 2.47061e8 0.986159 0.493080 0.869984i \(-0.335871\pi\)
0.493080 + 0.869984i \(0.335871\pi\)
\(252\) 6.23451e7 0.245415
\(253\) −1.47920e8 −0.574256
\(254\) −1.65359e8 −0.633156
\(255\) 2.53270e8 0.956518
\(256\) 1.67772e7 0.0625000
\(257\) 2.27286e8 0.835231 0.417616 0.908624i \(-0.362866\pi\)
0.417616 + 0.908624i \(0.362866\pi\)
\(258\) −2.18639e8 −0.792610
\(259\) −5.85841e7 −0.209522
\(260\) 0 0
\(261\) 9.40504e8 3.27430
\(262\) 1.52759e8 0.524749
\(263\) −4.25872e8 −1.44356 −0.721779 0.692124i \(-0.756675\pi\)
−0.721779 + 0.692124i \(0.756675\pi\)
\(264\) −3.46641e8 −1.15949
\(265\) 4.71623e8 1.55681
\(266\) −5.37932e7 −0.175243
\(267\) 2.94959e8 0.948357
\(268\) 4.10022e6 0.0130118
\(269\) −5.14154e8 −1.61050 −0.805250 0.592936i \(-0.797969\pi\)
−0.805250 + 0.592936i \(0.797969\pi\)
\(270\) −7.13814e8 −2.20705
\(271\) −4.57096e7 −0.139513 −0.0697565 0.997564i \(-0.522222\pi\)
−0.0697565 + 0.997564i \(0.522222\pi\)
\(272\) 3.71466e7 0.111925
\(273\) 0 0
\(274\) 2.37521e8 0.697548
\(275\) −1.93896e8 −0.562218
\(276\) −1.05837e8 −0.303008
\(277\) −2.73964e8 −0.774487 −0.387244 0.921977i \(-0.626573\pi\)
−0.387244 + 0.921977i \(0.626573\pi\)
\(278\) −1.24522e8 −0.347607
\(279\) −1.56143e8 −0.430434
\(280\) 2.97477e7 0.0809842
\(281\) −4.21707e8 −1.13381 −0.566903 0.823784i \(-0.691859\pi\)
−0.566903 + 0.823784i \(0.691859\pi\)
\(282\) 3.11419e8 0.826938
\(283\) 3.81957e8 1.00176 0.500878 0.865518i \(-0.333010\pi\)
0.500878 + 0.865518i \(0.333010\pi\)
\(284\) 2.06325e7 0.0534488
\(285\) 1.03749e9 2.65477
\(286\) 0 0
\(287\) −1.43951e8 −0.359443
\(288\) −1.76357e8 −0.435031
\(289\) −3.28092e8 −0.799564
\(290\) 4.48758e8 1.08049
\(291\) 1.41703e8 0.337097
\(292\) 2.85106e8 0.670141
\(293\) 4.04833e8 0.940240 0.470120 0.882602i \(-0.344211\pi\)
0.470120 + 0.882602i \(0.344211\pi\)
\(294\) −5.50384e8 −1.26314
\(295\) 5.22514e8 1.18501
\(296\) 1.65719e8 0.371407
\(297\) 2.16312e9 4.79108
\(298\) −1.99740e7 −0.0437228
\(299\) 0 0
\(300\) −1.38732e8 −0.296656
\(301\) −5.68588e7 −0.120175
\(302\) 1.91841e8 0.400791
\(303\) 1.33547e9 2.75795
\(304\) 1.52166e8 0.310643
\(305\) 7.70274e8 1.55452
\(306\) −3.90475e8 −0.779055
\(307\) 4.75520e7 0.0937960 0.0468980 0.998900i \(-0.485066\pi\)
0.0468980 + 0.998900i \(0.485066\pi\)
\(308\) −9.01467e7 −0.175801
\(309\) −5.98249e8 −1.15353
\(310\) −7.45028e7 −0.142039
\(311\) −3.02841e8 −0.570892 −0.285446 0.958395i \(-0.592142\pi\)
−0.285446 + 0.958395i \(0.592142\pi\)
\(312\) 0 0
\(313\) −6.31685e8 −1.16438 −0.582191 0.813052i \(-0.697804\pi\)
−0.582191 + 0.813052i \(0.697804\pi\)
\(314\) −1.36440e8 −0.248707
\(315\) −3.12700e8 −0.563690
\(316\) 4.82278e7 0.0859791
\(317\) −7.93332e8 −1.39877 −0.699387 0.714743i \(-0.746544\pi\)
−0.699387 + 0.714743i \(0.746544\pi\)
\(318\) −1.02259e9 −1.78322
\(319\) −1.35990e9 −2.34553
\(320\) −8.41482e7 −0.143556
\(321\) 1.32264e9 2.23189
\(322\) −2.75236e7 −0.0459420
\(323\) 3.36913e8 0.556300
\(324\) 7.94401e8 1.29757
\(325\) 0 0
\(326\) −5.87669e8 −0.939444
\(327\) 5.85737e8 0.926372
\(328\) 4.07200e8 0.637161
\(329\) 8.09868e7 0.125380
\(330\) 1.73862e9 2.66322
\(331\) 1.21628e9 1.84346 0.921731 0.387829i \(-0.126775\pi\)
0.921731 + 0.387829i \(0.126775\pi\)
\(332\) 7.80219e7 0.117013
\(333\) −1.74199e9 −2.58518
\(334\) 3.73488e8 0.548484
\(335\) −2.05652e7 −0.0298866
\(336\) −6.44997e7 −0.0927620
\(337\) −1.51221e8 −0.215232 −0.107616 0.994193i \(-0.534322\pi\)
−0.107616 + 0.994193i \(0.534322\pi\)
\(338\) 0 0
\(339\) 1.00304e9 1.39836
\(340\) −1.86314e8 −0.257080
\(341\) 2.25771e8 0.308339
\(342\) −1.59953e9 −2.16223
\(343\) −2.92193e8 −0.390967
\(344\) 1.60838e8 0.213027
\(345\) 5.30836e8 0.695975
\(346\) 6.24745e8 0.810842
\(347\) −5.97234e8 −0.767347 −0.383673 0.923469i \(-0.625341\pi\)
−0.383673 + 0.923469i \(0.625341\pi\)
\(348\) −9.73008e8 −1.23762
\(349\) −1.19600e8 −0.150606 −0.0753029 0.997161i \(-0.523992\pi\)
−0.0753029 + 0.997161i \(0.523992\pi\)
\(350\) −3.60784e7 −0.0449789
\(351\) 0 0
\(352\) 2.55001e8 0.311632
\(353\) 4.66414e8 0.564366 0.282183 0.959361i \(-0.408942\pi\)
0.282183 + 0.959361i \(0.408942\pi\)
\(354\) −1.13293e9 −1.35735
\(355\) −1.03485e8 −0.122766
\(356\) −2.16981e8 −0.254887
\(357\) −1.42810e8 −0.166119
\(358\) 4.44931e8 0.512509
\(359\) 7.70102e8 0.878451 0.439225 0.898377i \(-0.355253\pi\)
0.439225 + 0.898377i \(0.355253\pi\)
\(360\) 8.84542e8 0.999217
\(361\) 4.86251e8 0.543983
\(362\) 9.55477e8 1.05862
\(363\) −3.57329e9 −3.92099
\(364\) 0 0
\(365\) −1.42999e9 −1.53924
\(366\) −1.67013e9 −1.78060
\(367\) −8.55319e8 −0.903227 −0.451613 0.892214i \(-0.649151\pi\)
−0.451613 + 0.892214i \(0.649151\pi\)
\(368\) 7.78568e7 0.0814384
\(369\) −4.28037e9 −4.43495
\(370\) −8.31182e8 −0.853081
\(371\) −2.65931e8 −0.270371
\(372\) 1.61539e8 0.162696
\(373\) −5.29609e8 −0.528414 −0.264207 0.964466i \(-0.585110\pi\)
−0.264207 + 0.964466i \(0.585110\pi\)
\(374\) 5.64600e8 0.558072
\(375\) −1.48597e9 −1.45512
\(376\) −2.29090e8 −0.222253
\(377\) 0 0
\(378\) 4.02493e8 0.383299
\(379\) −1.98358e9 −1.87159 −0.935797 0.352540i \(-0.885318\pi\)
−0.935797 + 0.352540i \(0.885318\pi\)
\(380\) −7.63210e8 −0.713512
\(381\) −1.79828e9 −1.66579
\(382\) −8.43877e8 −0.774564
\(383\) −8.98756e8 −0.817422 −0.408711 0.912664i \(-0.634022\pi\)
−0.408711 + 0.912664i \(0.634022\pi\)
\(384\) 1.82452e8 0.164433
\(385\) 4.52142e8 0.403796
\(386\) 1.69647e8 0.150138
\(387\) −1.69069e9 −1.48277
\(388\) −1.04242e8 −0.0906003
\(389\) 1.82475e9 1.57174 0.785868 0.618395i \(-0.212217\pi\)
0.785868 + 0.618395i \(0.212217\pi\)
\(390\) 0 0
\(391\) 1.72384e8 0.145840
\(392\) 4.04880e8 0.339489
\(393\) 1.66125e9 1.38058
\(394\) −1.33228e9 −1.09739
\(395\) −2.41893e8 −0.197485
\(396\) −2.68049e9 −2.16911
\(397\) −4.93083e8 −0.395506 −0.197753 0.980252i \(-0.563364\pi\)
−0.197753 + 0.980252i \(0.563364\pi\)
\(398\) 1.01081e9 0.803668
\(399\) −5.85001e8 −0.461054
\(400\) 1.02056e8 0.0797312
\(401\) 5.68280e8 0.440105 0.220053 0.975488i \(-0.429377\pi\)
0.220053 + 0.975488i \(0.429377\pi\)
\(402\) 4.45899e7 0.0342331
\(403\) 0 0
\(404\) −9.82417e8 −0.741244
\(405\) −3.98442e9 −2.98039
\(406\) −2.53038e8 −0.187648
\(407\) 2.51879e9 1.85188
\(408\) 4.03970e8 0.294468
\(409\) 1.28472e9 0.928489 0.464245 0.885707i \(-0.346326\pi\)
0.464245 + 0.885707i \(0.346326\pi\)
\(410\) −2.04236e9 −1.46349
\(411\) 2.58304e9 1.83520
\(412\) 4.40091e8 0.310029
\(413\) −2.94626e8 −0.205801
\(414\) −8.18408e8 −0.566851
\(415\) −3.91329e8 −0.268765
\(416\) 0 0
\(417\) −1.35418e9 −0.914532
\(418\) 2.31281e9 1.54890
\(419\) 2.74847e8 0.182533 0.0912667 0.995826i \(-0.470908\pi\)
0.0912667 + 0.995826i \(0.470908\pi\)
\(420\) 3.23506e8 0.213064
\(421\) −7.51368e8 −0.490756 −0.245378 0.969428i \(-0.578912\pi\)
−0.245378 + 0.969428i \(0.578912\pi\)
\(422\) −8.68568e8 −0.562613
\(423\) 2.40813e9 1.54699
\(424\) 7.52247e8 0.479270
\(425\) 2.25963e8 0.142783
\(426\) 2.24379e8 0.140620
\(427\) −4.34329e8 −0.269974
\(428\) −9.72974e8 −0.599857
\(429\) 0 0
\(430\) −8.06704e8 −0.489299
\(431\) 1.30756e8 0.0786668 0.0393334 0.999226i \(-0.487477\pi\)
0.0393334 + 0.999226i \(0.487477\pi\)
\(432\) −1.13854e9 −0.679449
\(433\) 1.66736e9 0.987010 0.493505 0.869743i \(-0.335716\pi\)
0.493505 + 0.869743i \(0.335716\pi\)
\(434\) 4.20094e7 0.0246679
\(435\) 4.88024e9 2.84269
\(436\) −4.30887e8 −0.248978
\(437\) 7.06147e8 0.404772
\(438\) 3.10053e9 1.76310
\(439\) 2.31478e9 1.30582 0.652910 0.757436i \(-0.273548\pi\)
0.652910 + 0.757436i \(0.273548\pi\)
\(440\) −1.27899e9 −0.715784
\(441\) −4.25599e9 −2.36301
\(442\) 0 0
\(443\) −6.90047e8 −0.377108 −0.188554 0.982063i \(-0.560380\pi\)
−0.188554 + 0.982063i \(0.560380\pi\)
\(444\) 1.80219e9 0.977147
\(445\) 1.08830e9 0.585446
\(446\) −1.00482e9 −0.536311
\(447\) −2.17217e8 −0.115032
\(448\) 4.74481e7 0.0249313
\(449\) −2.63806e9 −1.37538 −0.687690 0.726004i \(-0.741375\pi\)
−0.687690 + 0.726004i \(0.741375\pi\)
\(450\) −1.07278e9 −0.554968
\(451\) 6.18912e9 3.17695
\(452\) −7.37870e8 −0.375833
\(453\) 2.08627e9 1.05445
\(454\) 1.52619e9 0.765445
\(455\) 0 0
\(456\) 1.65481e9 0.817280
\(457\) 6.16222e8 0.302016 0.151008 0.988533i \(-0.451748\pi\)
0.151008 + 0.988533i \(0.451748\pi\)
\(458\) −4.23129e8 −0.205799
\(459\) −2.52086e9 −1.21676
\(460\) −3.90500e8 −0.187055
\(461\) −1.23621e9 −0.587679 −0.293839 0.955855i \(-0.594933\pi\)
−0.293839 + 0.955855i \(0.594933\pi\)
\(462\) −9.80345e8 −0.462522
\(463\) −6.78469e7 −0.0317685 −0.0158843 0.999874i \(-0.505056\pi\)
−0.0158843 + 0.999874i \(0.505056\pi\)
\(464\) 7.15776e8 0.332632
\(465\) −8.10218e8 −0.373694
\(466\) −1.21003e9 −0.553919
\(467\) −1.17502e9 −0.533869 −0.266934 0.963715i \(-0.586011\pi\)
−0.266934 + 0.963715i \(0.586011\pi\)
\(468\) 0 0
\(469\) 1.15959e7 0.00519040
\(470\) 1.14903e9 0.510491
\(471\) −1.48379e9 −0.654332
\(472\) 8.33418e8 0.364809
\(473\) 2.44461e9 1.06218
\(474\) 5.24478e8 0.226205
\(475\) 9.25629e8 0.396287
\(476\) 1.05055e8 0.0446471
\(477\) −7.90741e9 −3.33595
\(478\) 2.09533e9 0.877517
\(479\) −3.96154e8 −0.164699 −0.0823494 0.996604i \(-0.526242\pi\)
−0.0823494 + 0.996604i \(0.526242\pi\)
\(480\) −9.15112e8 −0.377685
\(481\) 0 0
\(482\) −1.05401e9 −0.428728
\(483\) −2.99319e8 −0.120870
\(484\) 2.62863e9 1.05383
\(485\) 5.22836e8 0.208099
\(486\) 3.77584e9 1.49206
\(487\) 3.03665e9 1.19136 0.595680 0.803222i \(-0.296883\pi\)
0.595680 + 0.803222i \(0.296883\pi\)
\(488\) 1.22860e9 0.478565
\(489\) −6.39090e9 −2.47162
\(490\) −2.03073e9 −0.779768
\(491\) −2.91974e9 −1.11316 −0.556582 0.830793i \(-0.687887\pi\)
−0.556582 + 0.830793i \(0.687887\pi\)
\(492\) 4.42830e9 1.67633
\(493\) 1.58481e9 0.595679
\(494\) 0 0
\(495\) 1.34444e10 4.98221
\(496\) −1.18833e8 −0.0437272
\(497\) 5.83513e7 0.0213208
\(498\) 8.48488e8 0.307853
\(499\) 1.62343e9 0.584898 0.292449 0.956281i \(-0.405530\pi\)
0.292449 + 0.956281i \(0.405530\pi\)
\(500\) 1.09313e9 0.391089
\(501\) 4.06168e9 1.44303
\(502\) −1.97649e9 −0.697320
\(503\) 4.75888e9 1.66731 0.833655 0.552285i \(-0.186244\pi\)
0.833655 + 0.552285i \(0.186244\pi\)
\(504\) −4.98761e8 −0.173534
\(505\) 4.92744e9 1.70256
\(506\) 1.18336e9 0.406061
\(507\) 0 0
\(508\) 1.32288e9 0.447709
\(509\) 9.19375e8 0.309016 0.154508 0.987992i \(-0.450621\pi\)
0.154508 + 0.987992i \(0.450621\pi\)
\(510\) −2.02616e9 −0.676360
\(511\) 8.06316e8 0.267320
\(512\) −1.34218e8 −0.0441942
\(513\) −1.03264e10 −3.37706
\(514\) −1.81829e9 −0.590598
\(515\) −2.20733e9 −0.712103
\(516\) 1.74911e9 0.560460
\(517\) −3.48199e9 −1.10818
\(518\) 4.68673e8 0.148155
\(519\) 6.79410e9 2.13327
\(520\) 0 0
\(521\) −1.46089e9 −0.452569 −0.226284 0.974061i \(-0.572658\pi\)
−0.226284 + 0.974061i \(0.572658\pi\)
\(522\) −7.52404e9 −2.31528
\(523\) 2.12856e9 0.650624 0.325312 0.945607i \(-0.394531\pi\)
0.325312 + 0.945607i \(0.394531\pi\)
\(524\) −1.22207e9 −0.371054
\(525\) −3.92352e8 −0.118336
\(526\) 3.40698e9 1.02075
\(527\) −2.63110e8 −0.0783069
\(528\) 2.77313e9 0.819883
\(529\) −3.04352e9 −0.893885
\(530\) −3.77299e9 −1.10083
\(531\) −8.76066e9 −2.53925
\(532\) 4.30346e8 0.123916
\(533\) 0 0
\(534\) −2.35967e9 −0.670590
\(535\) 4.88007e9 1.37781
\(536\) −3.28018e7 −0.00920070
\(537\) 4.83862e9 1.34838
\(538\) 4.11323e9 1.13879
\(539\) 6.15387e9 1.69273
\(540\) 5.71051e9 1.56062
\(541\) 1.72479e7 0.00468324 0.00234162 0.999997i \(-0.499255\pi\)
0.00234162 + 0.999997i \(0.499255\pi\)
\(542\) 3.65677e8 0.0986507
\(543\) 1.03908e10 2.78516
\(544\) −2.97173e8 −0.0791431
\(545\) 2.16117e9 0.571874
\(546\) 0 0
\(547\) 7.51154e8 0.196234 0.0981168 0.995175i \(-0.468718\pi\)
0.0981168 + 0.995175i \(0.468718\pi\)
\(548\) −1.90016e9 −0.493241
\(549\) −1.29147e10 −3.33105
\(550\) 1.55117e9 0.397549
\(551\) 6.49196e9 1.65328
\(552\) 8.46692e8 0.214259
\(553\) 1.36394e8 0.0342972
\(554\) 2.19171e9 0.547645
\(555\) −9.03910e9 −2.24440
\(556\) 9.96175e8 0.245795
\(557\) −3.00701e9 −0.737295 −0.368647 0.929569i \(-0.620179\pi\)
−0.368647 + 0.929569i \(0.620179\pi\)
\(558\) 1.24914e9 0.304363
\(559\) 0 0
\(560\) −2.37982e8 −0.0572645
\(561\) 6.14002e9 1.46825
\(562\) 3.37366e9 0.801722
\(563\) 2.82880e9 0.668070 0.334035 0.942561i \(-0.391590\pi\)
0.334035 + 0.942561i \(0.391590\pi\)
\(564\) −2.49135e9 −0.584734
\(565\) 3.70088e9 0.863248
\(566\) −3.05566e9 −0.708349
\(567\) 2.24667e9 0.517604
\(568\) −1.65060e8 −0.0377940
\(569\) 7.67290e9 1.74609 0.873045 0.487639i \(-0.162142\pi\)
0.873045 + 0.487639i \(0.162142\pi\)
\(570\) −8.29990e9 −1.87720
\(571\) −3.09363e9 −0.695411 −0.347706 0.937604i \(-0.613039\pi\)
−0.347706 + 0.937604i \(0.613039\pi\)
\(572\) 0 0
\(573\) −9.17717e9 −2.03783
\(574\) 1.15161e9 0.254164
\(575\) 4.73603e8 0.103891
\(576\) 1.41086e9 0.307613
\(577\) −3.71815e9 −0.805770 −0.402885 0.915251i \(-0.631993\pi\)
−0.402885 + 0.915251i \(0.631993\pi\)
\(578\) 2.62474e9 0.565377
\(579\) 1.84491e9 0.395003
\(580\) −3.59006e9 −0.764019
\(581\) 2.20656e8 0.0466765
\(582\) −1.13363e9 −0.238363
\(583\) 1.14336e10 2.38969
\(584\) −2.28085e9 −0.473862
\(585\) 0 0
\(586\) −3.23866e9 −0.664850
\(587\) 2.74853e9 0.560876 0.280438 0.959872i \(-0.409520\pi\)
0.280438 + 0.959872i \(0.409520\pi\)
\(588\) 4.40307e9 0.893173
\(589\) −1.07780e9 −0.217337
\(590\) −4.18011e9 −0.837927
\(591\) −1.44886e10 −2.88715
\(592\) −1.32575e9 −0.262624
\(593\) 9.11262e9 1.79453 0.897267 0.441488i \(-0.145549\pi\)
0.897267 + 0.441488i \(0.145549\pi\)
\(594\) −1.73050e10 −3.38781
\(595\) −5.26918e8 −0.102550
\(596\) 1.59792e8 0.0309167
\(597\) 1.09925e10 2.11440
\(598\) 0 0
\(599\) −5.52493e9 −1.05035 −0.525174 0.850995i \(-0.676000\pi\)
−0.525174 + 0.850995i \(0.676000\pi\)
\(600\) 1.10986e9 0.209767
\(601\) −1.78219e9 −0.334883 −0.167441 0.985882i \(-0.553551\pi\)
−0.167441 + 0.985882i \(0.553551\pi\)
\(602\) 4.54870e8 0.0849767
\(603\) 3.44803e8 0.0640414
\(604\) −1.53473e9 −0.283402
\(605\) −1.31842e10 −2.42053
\(606\) −1.06838e10 −1.95016
\(607\) 9.53705e9 1.73083 0.865414 0.501058i \(-0.167056\pi\)
0.865414 + 0.501058i \(0.167056\pi\)
\(608\) −1.21733e9 −0.219658
\(609\) −2.75179e9 −0.493690
\(610\) −6.16219e9 −1.09921
\(611\) 0 0
\(612\) 3.12380e9 0.550875
\(613\) 1.18627e9 0.208004 0.104002 0.994577i \(-0.466835\pi\)
0.104002 + 0.994577i \(0.466835\pi\)
\(614\) −3.80416e8 −0.0663238
\(615\) −2.22107e10 −3.85034
\(616\) 7.21174e8 0.124310
\(617\) −1.32256e9 −0.226682 −0.113341 0.993556i \(-0.536155\pi\)
−0.113341 + 0.993556i \(0.536155\pi\)
\(618\) 4.78599e9 0.815666
\(619\) 3.59450e9 0.609147 0.304573 0.952489i \(-0.401486\pi\)
0.304573 + 0.952489i \(0.401486\pi\)
\(620\) 5.96023e8 0.100437
\(621\) −5.28356e9 −0.885332
\(622\) 2.42273e9 0.403681
\(623\) −6.13650e8 −0.101675
\(624\) 0 0
\(625\) −7.42927e9 −1.21721
\(626\) 5.05348e9 0.823343
\(627\) 2.51518e10 4.07505
\(628\) 1.09152e9 0.175863
\(629\) −2.93535e9 −0.470309
\(630\) 2.50160e9 0.398589
\(631\) −7.49102e6 −0.00118697 −0.000593483 1.00000i \(-0.500189\pi\)
−0.000593483 1.00000i \(0.500189\pi\)
\(632\) −3.85823e8 −0.0607964
\(633\) −9.44567e9 −1.48020
\(634\) 6.34666e9 0.989083
\(635\) −6.63505e9 −1.02834
\(636\) 8.18068e9 1.26093
\(637\) 0 0
\(638\) 1.08792e10 1.65854
\(639\) 1.73507e9 0.263065
\(640\) 6.73186e8 0.101509
\(641\) 4.06396e9 0.609462 0.304731 0.952438i \(-0.401433\pi\)
0.304731 + 0.952438i \(0.401433\pi\)
\(642\) −1.05811e10 −1.57818
\(643\) −1.56544e6 −0.000232219 0 −0.000116109 1.00000i \(-0.500037\pi\)
−0.000116109 1.00000i \(0.500037\pi\)
\(644\) 2.20189e8 0.0324859
\(645\) −8.77290e9 −1.28731
\(646\) −2.69531e9 −0.393364
\(647\) 1.31025e10 1.90191 0.950956 0.309325i \(-0.100103\pi\)
0.950956 + 0.309325i \(0.100103\pi\)
\(648\) −6.35521e9 −0.917524
\(649\) 1.26673e10 1.81898
\(650\) 0 0
\(651\) 4.56852e8 0.0648996
\(652\) 4.70135e9 0.664287
\(653\) 7.63326e9 1.07279 0.536394 0.843968i \(-0.319786\pi\)
0.536394 + 0.843968i \(0.319786\pi\)
\(654\) −4.68589e9 −0.655044
\(655\) 6.12945e9 0.852269
\(656\) −3.25760e9 −0.450541
\(657\) 2.39756e10 3.29831
\(658\) −6.47895e8 −0.0886571
\(659\) −9.25900e9 −1.26027 −0.630137 0.776484i \(-0.717001\pi\)
−0.630137 + 0.776484i \(0.717001\pi\)
\(660\) −1.39090e10 −1.88318
\(661\) −4.79962e9 −0.646401 −0.323201 0.946330i \(-0.604759\pi\)
−0.323201 + 0.946330i \(0.604759\pi\)
\(662\) −9.73021e9 −1.30352
\(663\) 0 0
\(664\) −6.24175e8 −0.0827405
\(665\) −2.15845e9 −0.284621
\(666\) 1.39359e10 1.82799
\(667\) 3.32165e9 0.433424
\(668\) −2.98791e9 −0.387837
\(669\) −1.09274e10 −1.41100
\(670\) 1.64521e8 0.0211330
\(671\) 1.86737e10 2.38618
\(672\) 5.15998e8 0.0655927
\(673\) −1.08997e10 −1.37836 −0.689182 0.724589i \(-0.742030\pi\)
−0.689182 + 0.724589i \(0.742030\pi\)
\(674\) 1.20977e9 0.152192
\(675\) −6.92578e9 −0.866773
\(676\) 0 0
\(677\) 3.44099e9 0.426210 0.213105 0.977029i \(-0.431642\pi\)
0.213105 + 0.977029i \(0.431642\pi\)
\(678\) −8.02434e9 −0.988793
\(679\) −2.94808e8 −0.0361406
\(680\) 1.49051e9 0.181783
\(681\) 1.65974e10 2.01384
\(682\) −1.80617e9 −0.218029
\(683\) −5.53553e9 −0.664794 −0.332397 0.943140i \(-0.607857\pi\)
−0.332397 + 0.943140i \(0.607857\pi\)
\(684\) 1.27962e10 1.52892
\(685\) 9.53051e9 1.13292
\(686\) 2.33754e9 0.276455
\(687\) −4.60153e9 −0.541444
\(688\) −1.28671e9 −0.150633
\(689\) 0 0
\(690\) −4.24669e9 −0.492129
\(691\) −4.21595e8 −0.0486097 −0.0243048 0.999705i \(-0.507737\pi\)
−0.0243048 + 0.999705i \(0.507737\pi\)
\(692\) −4.99796e9 −0.573352
\(693\) −7.58077e9 −0.865261
\(694\) 4.77788e9 0.542596
\(695\) −4.99644e9 −0.564565
\(696\) 7.78406e9 0.875133
\(697\) −7.21268e9 −0.806830
\(698\) 9.56799e8 0.106494
\(699\) −1.31591e10 −1.45732
\(700\) 2.88627e8 0.0318049
\(701\) −5.14995e9 −0.564663 −0.282332 0.959317i \(-0.591108\pi\)
−0.282332 + 0.959317i \(0.591108\pi\)
\(702\) 0 0
\(703\) −1.20243e10 −1.30532
\(704\) −2.04000e9 −0.220357
\(705\) 1.24957e10 1.34307
\(706\) −3.73132e9 −0.399067
\(707\) −2.77840e9 −0.295683
\(708\) 9.06342e9 0.959789
\(709\) −1.05683e10 −1.11363 −0.556817 0.830635i \(-0.687978\pi\)
−0.556817 + 0.830635i \(0.687978\pi\)
\(710\) 8.27880e8 0.0868086
\(711\) 4.05566e9 0.423173
\(712\) 1.73585e9 0.180232
\(713\) −5.51460e8 −0.0569772
\(714\) 1.14248e9 0.117464
\(715\) 0 0
\(716\) −3.55944e9 −0.362399
\(717\) 2.27868e10 2.30869
\(718\) −6.16081e9 −0.621159
\(719\) 1.53690e10 1.54204 0.771020 0.636811i \(-0.219747\pi\)
0.771020 + 0.636811i \(0.219747\pi\)
\(720\) −7.07634e9 −0.706553
\(721\) 1.24463e9 0.123671
\(722\) −3.89001e9 −0.384654
\(723\) −1.14624e10 −1.12795
\(724\) −7.64381e9 −0.748557
\(725\) 4.35407e9 0.424339
\(726\) 2.85864e10 2.77256
\(727\) 4.88599e9 0.471609 0.235804 0.971801i \(-0.424228\pi\)
0.235804 + 0.971801i \(0.424228\pi\)
\(728\) 0 0
\(729\) 1.39161e10 1.33036
\(730\) 1.14399e10 1.08841
\(731\) −2.84891e9 −0.269754
\(732\) 1.33610e10 1.25907
\(733\) −3.59889e9 −0.337524 −0.168762 0.985657i \(-0.553977\pi\)
−0.168762 + 0.985657i \(0.553977\pi\)
\(734\) 6.84255e9 0.638678
\(735\) −2.20842e10 −2.05152
\(736\) −6.22854e8 −0.0575856
\(737\) −4.98562e8 −0.0458757
\(738\) 3.42430e10 3.13598
\(739\) −2.78886e9 −0.254198 −0.127099 0.991890i \(-0.540567\pi\)
−0.127099 + 0.991890i \(0.540567\pi\)
\(740\) 6.64946e9 0.603219
\(741\) 0 0
\(742\) 2.12745e9 0.191181
\(743\) 3.08130e9 0.275597 0.137798 0.990460i \(-0.455997\pi\)
0.137798 + 0.990460i \(0.455997\pi\)
\(744\) −1.29231e9 −0.115043
\(745\) −8.01457e8 −0.0710122
\(746\) 4.23687e9 0.373645
\(747\) 6.56115e9 0.575915
\(748\) −4.51680e9 −0.394616
\(749\) −2.75169e9 −0.239284
\(750\) 1.18877e10 1.02893
\(751\) −6.41281e8 −0.0552470 −0.0276235 0.999618i \(-0.508794\pi\)
−0.0276235 + 0.999618i \(0.508794\pi\)
\(752\) 1.83272e9 0.157157
\(753\) −2.14943e10 −1.83460
\(754\) 0 0
\(755\) 7.69763e9 0.650943
\(756\) −3.21995e9 −0.271033
\(757\) −1.60219e10 −1.34239 −0.671195 0.741280i \(-0.734219\pi\)
−0.671195 + 0.741280i \(0.734219\pi\)
\(758\) 1.58686e10 1.32342
\(759\) 1.28691e10 1.06832
\(760\) 6.10568e9 0.504529
\(761\) 5.73623e9 0.471824 0.235912 0.971774i \(-0.424192\pi\)
0.235912 + 0.971774i \(0.424192\pi\)
\(762\) 1.43863e10 1.17789
\(763\) −1.21860e9 −0.0993175
\(764\) 6.75102e9 0.547700
\(765\) −1.56678e10 −1.26530
\(766\) 7.19005e9 0.578005
\(767\) 0 0
\(768\) −1.45962e9 −0.116272
\(769\) −2.45874e10 −1.94971 −0.974857 0.222832i \(-0.928470\pi\)
−0.974857 + 0.222832i \(0.928470\pi\)
\(770\) −3.61714e9 −0.285527
\(771\) −1.97739e10 −1.55382
\(772\) −1.35718e9 −0.106164
\(773\) 1.31517e10 1.02413 0.512065 0.858947i \(-0.328881\pi\)
0.512065 + 0.858947i \(0.328881\pi\)
\(774\) 1.35255e10 1.04848
\(775\) −7.22863e8 −0.0557828
\(776\) 8.33932e8 0.0640641
\(777\) 5.09682e9 0.389785
\(778\) −1.45980e10 −1.11138
\(779\) −2.95458e10 −2.23932
\(780\) 0 0
\(781\) −2.50878e9 −0.188445
\(782\) −1.37907e9 −0.103125
\(783\) −4.85744e10 −3.61611
\(784\) −3.23904e9 −0.240055
\(785\) −5.47466e9 −0.403937
\(786\) −1.32900e10 −0.976218
\(787\) 7.38863e9 0.540322 0.270161 0.962815i \(-0.412923\pi\)
0.270161 + 0.962815i \(0.412923\pi\)
\(788\) 1.06583e10 0.775969
\(789\) 3.70509e10 2.68552
\(790\) 1.93514e9 0.139643
\(791\) −2.08679e9 −0.149920
\(792\) 2.14440e10 1.53379
\(793\) 0 0
\(794\) 3.94467e9 0.279665
\(795\) −4.10312e10 −2.89621
\(796\) −8.08645e9 −0.568279
\(797\) −5.22399e9 −0.365509 −0.182754 0.983159i \(-0.558501\pi\)
−0.182754 + 0.983159i \(0.558501\pi\)
\(798\) 4.68001e9 0.326014
\(799\) 4.05784e9 0.281437
\(800\) −8.16447e8 −0.0563785
\(801\) −1.82468e10 −1.25450
\(802\) −4.54624e9 −0.311202
\(803\) −3.46671e10 −2.36273
\(804\) −3.56719e8 −0.0242064
\(805\) −1.10438e9 −0.0746164
\(806\) 0 0
\(807\) 4.47314e10 2.99609
\(808\) 7.85934e9 0.524139
\(809\) −7.92102e9 −0.525970 −0.262985 0.964800i \(-0.584707\pi\)
−0.262985 + 0.964800i \(0.584707\pi\)
\(810\) 3.18754e10 2.10745
\(811\) 8.16607e9 0.537576 0.268788 0.963199i \(-0.413377\pi\)
0.268788 + 0.963199i \(0.413377\pi\)
\(812\) 2.02430e9 0.132687
\(813\) 3.97674e9 0.259543
\(814\) −2.01503e10 −1.30947
\(815\) −2.35802e10 −1.52579
\(816\) −3.23176e9 −0.208220
\(817\) −1.16702e10 −0.748688
\(818\) −1.02778e10 −0.656541
\(819\) 0 0
\(820\) 1.63389e10 1.03484
\(821\) −2.63749e10 −1.66338 −0.831688 0.555244i \(-0.812625\pi\)
−0.831688 + 0.555244i \(0.812625\pi\)
\(822\) −2.06643e10 −1.29768
\(823\) 2.04085e10 1.27618 0.638090 0.769962i \(-0.279725\pi\)
0.638090 + 0.769962i \(0.279725\pi\)
\(824\) −3.52073e9 −0.219224
\(825\) 1.68690e10 1.04592
\(826\) 2.35701e9 0.145523
\(827\) 2.55307e10 1.56962 0.784809 0.619738i \(-0.212761\pi\)
0.784809 + 0.619738i \(0.212761\pi\)
\(828\) 6.54727e9 0.400824
\(829\) 8.48208e9 0.517085 0.258542 0.966000i \(-0.416758\pi\)
0.258542 + 0.966000i \(0.416758\pi\)
\(830\) 3.13063e9 0.190046
\(831\) 2.38349e10 1.44082
\(832\) 0 0
\(833\) −7.17160e9 −0.429891
\(834\) 1.08334e10 0.646672
\(835\) 1.49862e10 0.890819
\(836\) −1.85025e10 −1.09524
\(837\) 8.06432e9 0.475367
\(838\) −2.19878e9 −0.129071
\(839\) 2.29323e10 1.34055 0.670273 0.742115i \(-0.266177\pi\)
0.670273 + 0.742115i \(0.266177\pi\)
\(840\) −2.58805e9 −0.150659
\(841\) 1.32877e10 0.770306
\(842\) 6.01094e9 0.347017
\(843\) 3.66885e10 2.10928
\(844\) 6.94854e9 0.397828
\(845\) 0 0
\(846\) −1.92650e10 −1.09389
\(847\) 7.43410e9 0.420374
\(848\) −6.01797e9 −0.338895
\(849\) −3.32303e10 −1.86362
\(850\) −1.80771e9 −0.100963
\(851\) −6.15230e9 −0.342203
\(852\) −1.79503e9 −0.0994335
\(853\) −2.47175e10 −1.36358 −0.681792 0.731546i \(-0.738799\pi\)
−0.681792 + 0.731546i \(0.738799\pi\)
\(854\) 3.47463e9 0.190900
\(855\) −6.41812e10 −3.51177
\(856\) 7.78379e9 0.424163
\(857\) 1.19081e10 0.646265 0.323133 0.946354i \(-0.395264\pi\)
0.323133 + 0.946354i \(0.395264\pi\)
\(858\) 0 0
\(859\) −4.94214e9 −0.266035 −0.133018 0.991114i \(-0.542467\pi\)
−0.133018 + 0.991114i \(0.542467\pi\)
\(860\) 6.45363e9 0.345987
\(861\) 1.25238e10 0.668689
\(862\) −1.04605e9 −0.0556259
\(863\) 2.05387e10 1.08776 0.543881 0.839162i \(-0.316954\pi\)
0.543881 + 0.839162i \(0.316954\pi\)
\(864\) 9.10836e9 0.480443
\(865\) 2.50679e10 1.31693
\(866\) −1.33389e10 −0.697921
\(867\) 2.85440e10 1.48747
\(868\) −3.36075e8 −0.0174428
\(869\) −5.86420e9 −0.303138
\(870\) −3.90419e10 −2.01008
\(871\) 0 0
\(872\) 3.44709e9 0.176054
\(873\) −8.76606e9 −0.445918
\(874\) −5.64918e9 −0.286217
\(875\) 3.09150e9 0.156006
\(876\) −2.48042e10 −1.24670
\(877\) 1.42584e10 0.713791 0.356895 0.934144i \(-0.383835\pi\)
0.356895 + 0.934144i \(0.383835\pi\)
\(878\) −1.85182e10 −0.923354
\(879\) −3.52204e10 −1.74918
\(880\) 1.02319e10 0.506136
\(881\) 1.78398e10 0.878971 0.439486 0.898250i \(-0.355161\pi\)
0.439486 + 0.898250i \(0.355161\pi\)
\(882\) 3.40479e10 1.67090
\(883\) 3.79954e10 1.85724 0.928622 0.371028i \(-0.120995\pi\)
0.928622 + 0.371028i \(0.120995\pi\)
\(884\) 0 0
\(885\) −4.54587e10 −2.20453
\(886\) 5.52038e9 0.266656
\(887\) 8.45195e7 0.00406653 0.00203327 0.999998i \(-0.499353\pi\)
0.00203327 + 0.999998i \(0.499353\pi\)
\(888\) −1.44175e10 −0.690947
\(889\) 3.74126e9 0.178592
\(890\) −8.70637e9 −0.413973
\(891\) −9.65942e10 −4.57488
\(892\) 8.03857e9 0.379229
\(893\) 1.66224e10 0.781114
\(894\) 1.73774e9 0.0813398
\(895\) 1.78528e10 0.832390
\(896\) −3.79585e8 −0.0176291
\(897\) 0 0
\(898\) 2.11045e10 0.972541
\(899\) −5.06985e9 −0.232721
\(900\) 8.58227e9 0.392422
\(901\) −1.33245e10 −0.606894
\(902\) −4.95129e10 −2.24645
\(903\) 4.94672e9 0.223568
\(904\) 5.90296e9 0.265754
\(905\) 3.83385e10 1.71935
\(906\) −1.66902e10 −0.745611
\(907\) 1.82024e10 0.810033 0.405017 0.914309i \(-0.367266\pi\)
0.405017 + 0.914309i \(0.367266\pi\)
\(908\) −1.22096e10 −0.541252
\(909\) −8.26151e10 −3.64826
\(910\) 0 0
\(911\) −3.66963e10 −1.60808 −0.804040 0.594575i \(-0.797320\pi\)
−0.804040 + 0.594575i \(0.797320\pi\)
\(912\) −1.32385e10 −0.577905
\(913\) −9.48697e9 −0.412553
\(914\) −4.92978e9 −0.213558
\(915\) −6.70139e10 −2.89195
\(916\) 3.38503e9 0.145522
\(917\) −3.45617e9 −0.148014
\(918\) 2.01669e10 0.860380
\(919\) 1.33474e10 0.567275 0.283638 0.958932i \(-0.408459\pi\)
0.283638 + 0.958932i \(0.408459\pi\)
\(920\) 3.12400e9 0.132268
\(921\) −4.13702e9 −0.174493
\(922\) 9.88970e9 0.415552
\(923\) 0 0
\(924\) 7.84276e9 0.327052
\(925\) −8.06454e9 −0.335030
\(926\) 5.42775e8 0.0224637
\(927\) 3.70089e10 1.52591
\(928\) −5.72621e9 −0.235206
\(929\) −2.71771e10 −1.11211 −0.556055 0.831146i \(-0.687686\pi\)
−0.556055 + 0.831146i \(0.687686\pi\)
\(930\) 6.48174e9 0.264242
\(931\) −2.93776e10 −1.19314
\(932\) 9.68025e9 0.391680
\(933\) 2.63472e10 1.06206
\(934\) 9.40012e9 0.377502
\(935\) 2.26546e10 0.906390
\(936\) 0 0
\(937\) 4.04333e10 1.60565 0.802825 0.596214i \(-0.203329\pi\)
0.802825 + 0.596214i \(0.203329\pi\)
\(938\) −9.27676e7 −0.00367017
\(939\) 5.49566e10 2.16616
\(940\) −9.19223e9 −0.360972
\(941\) 8.49843e9 0.332487 0.166244 0.986085i \(-0.446836\pi\)
0.166244 + 0.986085i \(0.446836\pi\)
\(942\) 1.18703e10 0.462683
\(943\) −1.51173e10 −0.587061
\(944\) −6.66735e9 −0.257959
\(945\) 1.61500e10 0.622533
\(946\) −1.95569e10 −0.751072
\(947\) 4.40082e9 0.168387 0.0841935 0.996449i \(-0.473169\pi\)
0.0841935 + 0.996449i \(0.473169\pi\)
\(948\) −4.19582e9 −0.159951
\(949\) 0 0
\(950\) −7.40504e9 −0.280217
\(951\) 6.90199e10 2.60221
\(952\) −8.40442e8 −0.0315703
\(953\) −1.73133e10 −0.647970 −0.323985 0.946062i \(-0.605023\pi\)
−0.323985 + 0.946062i \(0.605023\pi\)
\(954\) 6.32593e10 2.35887
\(955\) −3.38606e10 −1.25801
\(956\) −1.67627e10 −0.620498
\(957\) 1.18312e11 4.36351
\(958\) 3.16924e9 0.116460
\(959\) −5.37390e9 −0.196754
\(960\) 7.32090e9 0.267064
\(961\) −2.66709e10 −0.969407
\(962\) 0 0
\(963\) −8.18210e10 −2.95238
\(964\) 8.43211e9 0.303156
\(965\) 6.80708e9 0.243846
\(966\) 2.39455e9 0.0854681
\(967\) −1.40918e10 −0.501158 −0.250579 0.968096i \(-0.580621\pi\)
−0.250579 + 0.968096i \(0.580621\pi\)
\(968\) −2.10290e10 −0.745171
\(969\) −2.93115e10 −1.03491
\(970\) −4.18269e9 −0.147148
\(971\) −7.27843e9 −0.255135 −0.127568 0.991830i \(-0.540717\pi\)
−0.127568 + 0.991830i \(0.540717\pi\)
\(972\) −3.02067e10 −1.05505
\(973\) 2.81731e9 0.0980481
\(974\) −2.42932e10 −0.842419
\(975\) 0 0
\(976\) −9.82879e9 −0.338397
\(977\) −2.43791e10 −0.836348 −0.418174 0.908367i \(-0.637330\pi\)
−0.418174 + 0.908367i \(0.637330\pi\)
\(978\) 5.11272e10 1.74770
\(979\) 2.63835e10 0.898657
\(980\) 1.62458e10 0.551379
\(981\) −3.62349e10 −1.22542
\(982\) 2.33579e10 0.787126
\(983\) −4.06556e10 −1.36516 −0.682579 0.730811i \(-0.739142\pi\)
−0.682579 + 0.730811i \(0.739142\pi\)
\(984\) −3.54264e10 −1.18534
\(985\) −5.34578e10 −1.78231
\(986\) −1.26785e10 −0.421209
\(987\) −7.04585e9 −0.233251
\(988\) 0 0
\(989\) −5.97112e9 −0.196277
\(990\) −1.07555e11 −3.52295
\(991\) −4.86636e10 −1.58835 −0.794175 0.607689i \(-0.792097\pi\)
−0.794175 + 0.607689i \(0.792097\pi\)
\(992\) 9.50665e8 0.0309198
\(993\) −1.05816e11 −3.42949
\(994\) −4.66811e8 −0.0150761
\(995\) 4.05586e10 1.30527
\(996\) −6.78790e9 −0.217685
\(997\) −1.76682e10 −0.564622 −0.282311 0.959323i \(-0.591101\pi\)
−0.282311 + 0.959323i \(0.591101\pi\)
\(998\) −1.29874e10 −0.413586
\(999\) 8.99687e10 2.85504
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.a.1.1 1
13.5 odd 4 338.8.b.a.337.2 2
13.8 odd 4 338.8.b.a.337.1 2
13.12 even 2 26.8.a.b.1.1 1
39.38 odd 2 234.8.a.a.1.1 1
52.51 odd 2 208.8.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
26.8.a.b.1.1 1 13.12 even 2
208.8.a.e.1.1 1 52.51 odd 2
234.8.a.a.1.1 1 39.38 odd 2
338.8.a.a.1.1 1 1.1 even 1 trivial
338.8.b.a.337.1 2 13.8 odd 4
338.8.b.a.337.2 2 13.5 odd 4