Properties

Label 338.8.b.b.337.1
Level $338$
Weight $8$
Character 338.337
Analytic conductor $105.586$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [338,8,Mod(337,338)]
 
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([1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.337");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 338.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(105.586138614\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 338.337
Dual form 338.8.b.b.337.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.00000i q^{2} -39.0000 q^{3} -64.0000 q^{4} +385.000i q^{5} +312.000i q^{6} +293.000i q^{7} +512.000i q^{8} -666.000 q^{9} +O(q^{10})\) \(q-8.00000i q^{2} -39.0000 q^{3} -64.0000 q^{4} +385.000i q^{5} +312.000i q^{6} +293.000i q^{7} +512.000i q^{8} -666.000 q^{9} +3080.00 q^{10} +5402.00i q^{11} +2496.00 q^{12} +2344.00 q^{14} -15015.0i q^{15} +4096.00 q^{16} +21011.0 q^{17} +5328.00i q^{18} -27326.0i q^{19} -24640.0i q^{20} -11427.0i q^{21} +43216.0 q^{22} +63072.0 q^{23} -19968.0i q^{24} -70100.0 q^{25} +111267. q^{27} -18752.0i q^{28} +122238. q^{29} -120120. q^{30} -208396. i q^{31} -32768.0i q^{32} -210678. i q^{33} -168088. i q^{34} -112805. q^{35} +42624.0 q^{36} +442379. i q^{37} -218608. q^{38} -197120. q^{40} +58000.0i q^{41} -91416.0 q^{42} +202025. q^{43} -345728. i q^{44} -256410. i q^{45} -504576. i q^{46} -588511. i q^{47} -159744. q^{48} +737694. q^{49} +560800. i q^{50} -819429. q^{51} +1.68434e6 q^{53} -890136. i q^{54} -2.07977e6 q^{55} -150016. q^{56} +1.06571e6i q^{57} -977904. i q^{58} +442630. i q^{59} +960960. i q^{60} -1.08361e6 q^{61} -1.66717e6 q^{62} -195138. i q^{63} -262144. q^{64} -1.68542e6 q^{66} +3.44349e6i q^{67} -1.34470e6 q^{68} -2.45981e6 q^{69} +902440. i q^{70} +2.08470e6i q^{71} -340992. i q^{72} -5.93789e6i q^{73} +3.53903e6 q^{74} +2.73390e6 q^{75} +1.74886e6i q^{76} -1.58279e6 q^{77} -6.60926e6 q^{79} +1.57696e6i q^{80} -2.88287e6 q^{81} +464000. q^{82} -142740. i q^{83} +731328. i q^{84} +8.08924e6i q^{85} -1.61620e6i q^{86} -4.76728e6 q^{87} -2.76582e6 q^{88} +6.98529e6i q^{89} -2.05128e6 q^{90} -4.03661e6 q^{92} +8.12744e6i q^{93} -4.70809e6 q^{94} +1.05205e7 q^{95} +1.27795e6i q^{96} -200762. i q^{97} -5.90155e6i q^{98} -3.59773e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 78 q^{3} - 128 q^{4} - 1332 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 78 q^{3} - 128 q^{4} - 1332 q^{9} + 6160 q^{10} + 4992 q^{12} + 4688 q^{14} + 8192 q^{16} + 42022 q^{17} + 86432 q^{22} + 126144 q^{23} - 140200 q^{25} + 222534 q^{27} + 244476 q^{29} - 240240 q^{30} - 225610 q^{35} + 85248 q^{36} - 437216 q^{38} - 394240 q^{40} - 182832 q^{42} + 404050 q^{43} - 319488 q^{48} + 1475388 q^{49} - 1638858 q^{51} + 3368672 q^{53} - 4159540 q^{55} - 300032 q^{56} - 2167216 q^{61} - 3334336 q^{62} - 524288 q^{64} - 3370848 q^{66} - 2689408 q^{68} - 4919616 q^{69} + 7078064 q^{74} + 5467800 q^{75} - 3165572 q^{77} - 13218512 q^{79} - 5765742 q^{81} + 928000 q^{82} - 9534564 q^{87} - 5531648 q^{88} - 4102560 q^{90} - 8073216 q^{92} - 9416176 q^{94} + 21041020 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/338\mathbb{Z}\right)^\times\).

\(n\) \(171\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 8.00000i − 0.707107i
\(3\) −39.0000 −0.833950 −0.416975 0.908918i \(-0.636910\pi\)
−0.416975 + 0.908918i \(0.636910\pi\)
\(4\) −64.0000 −0.500000
\(5\) 385.000i 1.37742i 0.725038 + 0.688709i \(0.241822\pi\)
−0.725038 + 0.688709i \(0.758178\pi\)
\(6\) 312.000i 0.589692i
\(7\) 293.000i 0.322868i 0.986884 + 0.161434i \(0.0516118\pi\)
−0.986884 + 0.161434i \(0.948388\pi\)
\(8\) 512.000i 0.353553i
\(9\) −666.000 −0.304527
\(10\) 3080.00 0.973982
\(11\) 5402.00i 1.22371i 0.790968 + 0.611857i \(0.209577\pi\)
−0.790968 + 0.611857i \(0.790423\pi\)
\(12\) 2496.00 0.416975
\(13\) 0 0
\(14\) 2344.00 0.228302
\(15\) − 15015.0i − 1.14870i
\(16\) 4096.00 0.250000
\(17\) 21011.0 1.03723 0.518616 0.855008i \(-0.326448\pi\)
0.518616 + 0.855008i \(0.326448\pi\)
\(18\) 5328.00i 0.215333i
\(19\) − 27326.0i − 0.913984i −0.889471 0.456992i \(-0.848927\pi\)
0.889471 0.456992i \(-0.151073\pi\)
\(20\) − 24640.0i − 0.688709i
\(21\) − 11427.0i − 0.269256i
\(22\) 43216.0 0.865297
\(23\) 63072.0 1.08091 0.540455 0.841373i \(-0.318252\pi\)
0.540455 + 0.841373i \(0.318252\pi\)
\(24\) − 19968.0i − 0.294846i
\(25\) −70100.0 −0.897280
\(26\) 0 0
\(27\) 111267. 1.08791
\(28\) − 18752.0i − 0.161434i
\(29\) 122238. 0.930708 0.465354 0.885125i \(-0.345927\pi\)
0.465354 + 0.885125i \(0.345927\pi\)
\(30\) −120120. −0.812252
\(31\) − 208396.i − 1.25639i −0.778057 0.628194i \(-0.783795\pi\)
0.778057 0.628194i \(-0.216205\pi\)
\(32\) − 32768.0i − 0.176777i
\(33\) − 210678.i − 1.02052i
\(34\) − 168088.i − 0.733433i
\(35\) −112805. −0.444724
\(36\) 42624.0 0.152263
\(37\) 442379.i 1.43578i 0.696156 + 0.717891i \(0.254892\pi\)
−0.696156 + 0.717891i \(0.745108\pi\)
\(38\) −218608. −0.646284
\(39\) 0 0
\(40\) −197120. −0.486991
\(41\) 58000.0i 0.131427i 0.997839 + 0.0657135i \(0.0209323\pi\)
−0.997839 + 0.0657135i \(0.979068\pi\)
\(42\) −91416.0 −0.190392
\(43\) 202025. 0.387494 0.193747 0.981051i \(-0.437936\pi\)
0.193747 + 0.981051i \(0.437936\pi\)
\(44\) − 345728.i − 0.611857i
\(45\) − 256410.i − 0.419461i
\(46\) − 504576.i − 0.764318i
\(47\) − 588511.i − 0.826822i −0.910545 0.413411i \(-0.864337\pi\)
0.910545 0.413411i \(-0.135663\pi\)
\(48\) −159744. −0.208488
\(49\) 737694. 0.895757
\(50\) 560800.i 0.634473i
\(51\) −819429. −0.864999
\(52\) 0 0
\(53\) 1.68434e6 1.55404 0.777022 0.629474i \(-0.216729\pi\)
0.777022 + 0.629474i \(0.216729\pi\)
\(54\) − 890136.i − 0.769269i
\(55\) −2.07977e6 −1.68557
\(56\) −150016. −0.114151
\(57\) 1.06571e6i 0.762217i
\(58\) − 977904.i − 0.658110i
\(59\) 442630.i 0.280581i 0.990110 + 0.140291i \(0.0448037\pi\)
−0.990110 + 0.140291i \(0.955196\pi\)
\(60\) 960960.i 0.574349i
\(61\) −1.08361e6 −0.611248 −0.305624 0.952152i \(-0.598865\pi\)
−0.305624 + 0.952152i \(0.598865\pi\)
\(62\) −1.66717e6 −0.888400
\(63\) − 195138.i − 0.0983218i
\(64\) −262144. −0.125000
\(65\) 0 0
\(66\) −1.68542e6 −0.721615
\(67\) 3.44349e6i 1.39874i 0.714761 + 0.699369i \(0.246536\pi\)
−0.714761 + 0.699369i \(0.753464\pi\)
\(68\) −1.34470e6 −0.518616
\(69\) −2.45981e6 −0.901425
\(70\) 902440.i 0.314467i
\(71\) 2.08470e6i 0.691258i 0.938371 + 0.345629i \(0.112334\pi\)
−0.938371 + 0.345629i \(0.887666\pi\)
\(72\) − 340992.i − 0.107666i
\(73\) − 5.93789e6i − 1.78650i −0.449564 0.893248i \(-0.648421\pi\)
0.449564 0.893248i \(-0.351579\pi\)
\(74\) 3.53903e6 1.01525
\(75\) 2.73390e6 0.748287
\(76\) 1.74886e6i 0.456992i
\(77\) −1.58279e6 −0.395098
\(78\) 0 0
\(79\) −6.60926e6 −1.50820 −0.754098 0.656762i \(-0.771926\pi\)
−0.754098 + 0.656762i \(0.771926\pi\)
\(80\) 1.57696e6i 0.344354i
\(81\) −2.88287e6 −0.602737
\(82\) 464000. 0.0929329
\(83\) − 142740.i − 0.0274014i −0.999906 0.0137007i \(-0.995639\pi\)
0.999906 0.0137007i \(-0.00436120\pi\)
\(84\) 731328.i 0.134628i
\(85\) 8.08924e6i 1.42870i
\(86\) − 1.61620e6i − 0.274000i
\(87\) −4.76728e6 −0.776164
\(88\) −2.76582e6 −0.432648
\(89\) 6.98529e6i 1.05031i 0.851005 + 0.525157i \(0.175993\pi\)
−0.851005 + 0.525157i \(0.824007\pi\)
\(90\) −2.05128e6 −0.296603
\(91\) 0 0
\(92\) −4.03661e6 −0.540455
\(93\) 8.12744e6i 1.04776i
\(94\) −4.70809e6 −0.584652
\(95\) 1.05205e7 1.25894
\(96\) 1.27795e6i 0.147423i
\(97\) − 200762.i − 0.0223347i −0.999938 0.0111674i \(-0.996445\pi\)
0.999938 0.0111674i \(-0.00355475\pi\)
\(98\) − 5.90155e6i − 0.633395i
\(99\) − 3.59773e6i − 0.372654i
\(100\) 4.48640e6 0.448640
\(101\) 5.42144e6 0.523588 0.261794 0.965124i \(-0.415686\pi\)
0.261794 + 0.965124i \(0.415686\pi\)
\(102\) 6.55543e6i 0.611647i
\(103\) 1.71897e7 1.55002 0.775011 0.631948i \(-0.217745\pi\)
0.775011 + 0.631948i \(0.217745\pi\)
\(104\) 0 0
\(105\) 4.39940e6 0.370877
\(106\) − 1.34747e7i − 1.09887i
\(107\) 1.23582e7 0.975242 0.487621 0.873055i \(-0.337865\pi\)
0.487621 + 0.873055i \(0.337865\pi\)
\(108\) −7.12109e6 −0.543955
\(109\) 1.70569e7i 1.26156i 0.775964 + 0.630778i \(0.217264\pi\)
−0.775964 + 0.630778i \(0.782736\pi\)
\(110\) 1.66382e7i 1.19188i
\(111\) − 1.72528e7i − 1.19737i
\(112\) 1.20013e6i 0.0807169i
\(113\) 2.11250e7 1.37728 0.688639 0.725104i \(-0.258208\pi\)
0.688639 + 0.725104i \(0.258208\pi\)
\(114\) 8.52571e6 0.538969
\(115\) 2.42827e7i 1.48886i
\(116\) −7.82323e6 −0.465354
\(117\) 0 0
\(118\) 3.54104e6 0.198401
\(119\) 6.15622e6i 0.334888i
\(120\) 7.68768e6 0.406126
\(121\) −9.69443e6 −0.497478
\(122\) 8.66886e6i 0.432218i
\(123\) − 2.26200e6i − 0.109604i
\(124\) 1.33373e7i 0.628194i
\(125\) 3.08962e6i 0.141488i
\(126\) −1.56110e6 −0.0695240
\(127\) 3.24008e7 1.40360 0.701800 0.712374i \(-0.252380\pi\)
0.701800 + 0.712374i \(0.252380\pi\)
\(128\) 2.09715e6i 0.0883883i
\(129\) −7.87898e6 −0.323151
\(130\) 0 0
\(131\) −2.64669e7 −1.02862 −0.514308 0.857605i \(-0.671951\pi\)
−0.514308 + 0.857605i \(0.671951\pi\)
\(132\) 1.34834e7i 0.510259i
\(133\) 8.00652e6 0.295096
\(134\) 2.75479e7 0.989057
\(135\) 4.28378e7i 1.49851i
\(136\) 1.07576e7i 0.366717i
\(137\) − 5.36201e7i − 1.78158i −0.454413 0.890791i \(-0.650151\pi\)
0.454413 0.890791i \(-0.349849\pi\)
\(138\) 1.96785e7i 0.637403i
\(139\) 7.58784e6 0.239644 0.119822 0.992795i \(-0.461768\pi\)
0.119822 + 0.992795i \(0.461768\pi\)
\(140\) 7.21952e6 0.222362
\(141\) 2.29519e7i 0.689529i
\(142\) 1.66776e7 0.488793
\(143\) 0 0
\(144\) −2.72794e6 −0.0761317
\(145\) 4.70616e7i 1.28197i
\(146\) −4.75031e7 −1.26324
\(147\) −2.87701e7 −0.747016
\(148\) − 2.83123e7i − 0.717891i
\(149\) − 5.70297e7i − 1.41237i −0.708026 0.706187i \(-0.750414\pi\)
0.708026 0.706187i \(-0.249586\pi\)
\(150\) − 2.18712e7i − 0.529119i
\(151\) 2.00648e7i 0.474259i 0.971478 + 0.237130i \(0.0762066\pi\)
−0.971478 + 0.237130i \(0.923793\pi\)
\(152\) 1.39909e7 0.323142
\(153\) −1.39933e7 −0.315865
\(154\) 1.26623e7i 0.279376i
\(155\) 8.02325e7 1.73057
\(156\) 0 0
\(157\) −3.15314e7 −0.650272 −0.325136 0.945667i \(-0.605410\pi\)
−0.325136 + 0.945667i \(0.605410\pi\)
\(158\) 5.28740e7i 1.06646i
\(159\) −6.56891e7 −1.29600
\(160\) 1.26157e7 0.243495
\(161\) 1.84801e7i 0.348991i
\(162\) 2.30630e7i 0.426199i
\(163\) 3.13938e7i 0.567789i 0.958855 + 0.283895i \(0.0916266\pi\)
−0.958855 + 0.283895i \(0.908373\pi\)
\(164\) − 3.71200e6i − 0.0657135i
\(165\) 8.11110e7 1.40568
\(166\) −1.14192e6 −0.0193757
\(167\) − 9.22170e7i − 1.53216i −0.642747 0.766079i \(-0.722205\pi\)
0.642747 0.766079i \(-0.277795\pi\)
\(168\) 5.85062e6 0.0951962
\(169\) 0 0
\(170\) 6.47139e7 1.01024
\(171\) 1.81991e7i 0.278332i
\(172\) −1.29296e7 −0.193747
\(173\) 6.57015e7 0.964748 0.482374 0.875965i \(-0.339775\pi\)
0.482374 + 0.875965i \(0.339775\pi\)
\(174\) 3.81383e7i 0.548831i
\(175\) − 2.05393e7i − 0.289703i
\(176\) 2.21266e7i 0.305929i
\(177\) − 1.72626e7i − 0.233991i
\(178\) 5.58823e7 0.742684
\(179\) 3.20402e6 0.0417551 0.0208776 0.999782i \(-0.493354\pi\)
0.0208776 + 0.999782i \(0.493354\pi\)
\(180\) 1.64102e7i 0.209730i
\(181\) 4.45759e7 0.558760 0.279380 0.960181i \(-0.409871\pi\)
0.279380 + 0.960181i \(0.409871\pi\)
\(182\) 0 0
\(183\) 4.22607e7 0.509751
\(184\) 3.22929e7i 0.382159i
\(185\) −1.70316e8 −1.97767
\(186\) 6.50196e7 0.740881
\(187\) 1.13501e8i 1.26927i
\(188\) 3.76647e7i 0.413411i
\(189\) 3.26012e7i 0.351251i
\(190\) − 8.41641e7i − 0.890203i
\(191\) 1.86394e8 1.93559 0.967797 0.251733i \(-0.0810004\pi\)
0.967797 + 0.251733i \(0.0810004\pi\)
\(192\) 1.02236e7 0.104244
\(193\) 1.52927e8i 1.53120i 0.643314 + 0.765602i \(0.277559\pi\)
−0.643314 + 0.765602i \(0.722441\pi\)
\(194\) −1.60610e6 −0.0157930
\(195\) 0 0
\(196\) −4.72124e7 −0.447878
\(197\) 9.51837e7i 0.887015i 0.896271 + 0.443507i \(0.146266\pi\)
−0.896271 + 0.443507i \(0.853734\pi\)
\(198\) −2.87819e7 −0.263506
\(199\) −1.78585e8 −1.60642 −0.803212 0.595693i \(-0.796878\pi\)
−0.803212 + 0.595693i \(0.796878\pi\)
\(200\) − 3.58912e7i − 0.317236i
\(201\) − 1.34296e8i − 1.16648i
\(202\) − 4.33715e7i − 0.370232i
\(203\) 3.58157e7i 0.300495i
\(204\) 5.24435e7 0.432500
\(205\) −2.23300e7 −0.181030
\(206\) − 1.37517e8i − 1.09603i
\(207\) −4.20060e7 −0.329166
\(208\) 0 0
\(209\) 1.47615e8 1.11846
\(210\) − 3.51952e7i − 0.262250i
\(211\) −1.33235e8 −0.976406 −0.488203 0.872730i \(-0.662348\pi\)
−0.488203 + 0.872730i \(0.662348\pi\)
\(212\) −1.07798e8 −0.777022
\(213\) − 8.13035e7i − 0.576475i
\(214\) − 9.88657e7i − 0.689600i
\(215\) 7.77796e7i 0.533742i
\(216\) 5.69687e7i 0.384634i
\(217\) 6.10600e7 0.405647
\(218\) 1.36455e8 0.892054
\(219\) 2.31578e8i 1.48985i
\(220\) 1.33105e8 0.842783
\(221\) 0 0
\(222\) −1.38022e8 −0.846669
\(223\) 1.19394e8i 0.720969i 0.932765 + 0.360484i \(0.117389\pi\)
−0.932765 + 0.360484i \(0.882611\pi\)
\(224\) 9.60102e6 0.0570755
\(225\) 4.66866e7 0.273246
\(226\) − 1.69000e8i − 0.973883i
\(227\) 1.13656e7i 0.0644911i 0.999480 + 0.0322456i \(0.0102659\pi\)
−0.999480 + 0.0322456i \(0.989734\pi\)
\(228\) − 6.82057e7i − 0.381109i
\(229\) 1.46559e7i 0.0806470i 0.999187 + 0.0403235i \(0.0128389\pi\)
−0.999187 + 0.0403235i \(0.987161\pi\)
\(230\) 1.94262e8 1.05279
\(231\) 6.17287e7 0.329492
\(232\) 6.25859e7i 0.329055i
\(233\) 2.46924e8 1.27885 0.639423 0.768855i \(-0.279173\pi\)
0.639423 + 0.768855i \(0.279173\pi\)
\(234\) 0 0
\(235\) 2.26577e8 1.13888
\(236\) − 2.83283e7i − 0.140291i
\(237\) 2.57761e8 1.25776
\(238\) 4.92498e7 0.236802
\(239\) − 1.61239e7i − 0.0763971i −0.999270 0.0381985i \(-0.987838\pi\)
0.999270 0.0381985i \(-0.0121619\pi\)
\(240\) − 6.15014e7i − 0.287175i
\(241\) − 1.14256e8i − 0.525798i −0.964823 0.262899i \(-0.915321\pi\)
0.964823 0.262899i \(-0.0846787\pi\)
\(242\) 7.75555e7i 0.351770i
\(243\) −1.30909e8 −0.585258
\(244\) 6.93509e7 0.305624
\(245\) 2.84012e8i 1.23383i
\(246\) −1.80960e7 −0.0775014
\(247\) 0 0
\(248\) 1.06699e8 0.444200
\(249\) 5.56686e6i 0.0228514i
\(250\) 2.47170e7 0.100047
\(251\) −2.22704e8 −0.888935 −0.444467 0.895795i \(-0.646607\pi\)
−0.444467 + 0.895795i \(0.646607\pi\)
\(252\) 1.24888e7i 0.0491609i
\(253\) 3.40715e8i 1.32272i
\(254\) − 2.59207e8i − 0.992494i
\(255\) − 3.15480e8i − 1.19147i
\(256\) 1.67772e7 0.0625000
\(257\) −2.82302e8 −1.03741 −0.518703 0.854955i \(-0.673585\pi\)
−0.518703 + 0.854955i \(0.673585\pi\)
\(258\) 6.30318e7i 0.228502i
\(259\) −1.29617e8 −0.463567
\(260\) 0 0
\(261\) −8.14105e7 −0.283425
\(262\) 2.11735e8i 0.727342i
\(263\) −2.36490e8 −0.801619 −0.400809 0.916162i \(-0.631271\pi\)
−0.400809 + 0.916162i \(0.631271\pi\)
\(264\) 1.07867e8 0.360807
\(265\) 6.48469e8i 2.14057i
\(266\) − 6.40521e7i − 0.208664i
\(267\) − 2.72426e8i − 0.875910i
\(268\) − 2.20383e8i − 0.699369i
\(269\) −4.82172e8 −1.51032 −0.755160 0.655541i \(-0.772441\pi\)
−0.755160 + 0.655541i \(0.772441\pi\)
\(270\) 3.42702e8 1.05960
\(271\) − 4.66372e8i − 1.42344i −0.702462 0.711721i \(-0.747916\pi\)
0.702462 0.711721i \(-0.252084\pi\)
\(272\) 8.60611e7 0.259308
\(273\) 0 0
\(274\) −4.28961e8 −1.25977
\(275\) − 3.78680e8i − 1.09801i
\(276\) 1.57428e8 0.450712
\(277\) 1.88709e8 0.533475 0.266738 0.963769i \(-0.414054\pi\)
0.266738 + 0.963769i \(0.414054\pi\)
\(278\) − 6.07027e7i − 0.169454i
\(279\) 1.38792e8i 0.382603i
\(280\) − 5.77562e7i − 0.157234i
\(281\) 7.15402e8i 1.92344i 0.274040 + 0.961718i \(0.411640\pi\)
−0.274040 + 0.961718i \(0.588360\pi\)
\(282\) 1.83615e8 0.487570
\(283\) 4.04602e8 1.06115 0.530573 0.847639i \(-0.321977\pi\)
0.530573 + 0.847639i \(0.321977\pi\)
\(284\) − 1.33421e8i − 0.345629i
\(285\) −4.10300e8 −1.04989
\(286\) 0 0
\(287\) −1.69940e7 −0.0424335
\(288\) 2.18235e7i 0.0538332i
\(289\) 3.11234e7 0.0758482
\(290\) 3.76493e8 0.906492
\(291\) 7.82972e6i 0.0186260i
\(292\) 3.80025e8i 0.893248i
\(293\) 8.11321e8i 1.88433i 0.335156 + 0.942163i \(0.391211\pi\)
−0.335156 + 0.942163i \(0.608789\pi\)
\(294\) 2.30161e8i 0.528220i
\(295\) −1.70413e8 −0.386478
\(296\) −2.26498e8 −0.507626
\(297\) 6.01064e8i 1.33129i
\(298\) −4.56238e8 −0.998699
\(299\) 0 0
\(300\) −1.74970e8 −0.374144
\(301\) 5.91933e7i 0.125109i
\(302\) 1.60519e8 0.335352
\(303\) −2.11436e8 −0.436646
\(304\) − 1.11927e8i − 0.228496i
\(305\) − 4.17189e8i − 0.841945i
\(306\) 1.11947e8i 0.223350i
\(307\) − 4.60958e8i − 0.909237i −0.890686 0.454618i \(-0.849776\pi\)
0.890686 0.454618i \(-0.150224\pi\)
\(308\) 1.01298e8 0.197549
\(309\) −6.70398e8 −1.29264
\(310\) − 6.41860e8i − 1.22370i
\(311\) 2.87718e8 0.542383 0.271192 0.962525i \(-0.412582\pi\)
0.271192 + 0.962525i \(0.412582\pi\)
\(312\) 0 0
\(313\) −9.56179e8 −1.76252 −0.881260 0.472632i \(-0.843304\pi\)
−0.881260 + 0.472632i \(0.843304\pi\)
\(314\) 2.52252e8i 0.459812i
\(315\) 7.51281e7 0.135430
\(316\) 4.22992e8 0.754098
\(317\) 4.92761e8i 0.868818i 0.900716 + 0.434409i \(0.143043\pi\)
−0.900716 + 0.434409i \(0.856957\pi\)
\(318\) 5.25513e8i 0.916407i
\(319\) 6.60330e8i 1.13892i
\(320\) − 1.00925e8i − 0.172177i
\(321\) −4.81970e8 −0.813303
\(322\) 1.47841e8 0.246774
\(323\) − 5.74147e8i − 0.948012i
\(324\) 1.84504e8 0.301368
\(325\) 0 0
\(326\) 2.51150e8 0.401488
\(327\) − 6.65218e8i − 1.05207i
\(328\) −2.96960e7 −0.0464665
\(329\) 1.72434e8 0.266954
\(330\) − 6.48888e8i − 0.993965i
\(331\) − 4.83358e8i − 0.732607i −0.930495 0.366304i \(-0.880623\pi\)
0.930495 0.366304i \(-0.119377\pi\)
\(332\) 9.13536e6i 0.0137007i
\(333\) − 2.94624e8i − 0.437234i
\(334\) −7.37736e8 −1.08340
\(335\) −1.32574e9 −1.92665
\(336\) − 4.68050e7i − 0.0673139i
\(337\) −1.30823e9 −1.86200 −0.930998 0.365025i \(-0.881060\pi\)
−0.930998 + 0.365025i \(0.881060\pi\)
\(338\) 0 0
\(339\) −8.23874e8 −1.14858
\(340\) − 5.17711e8i − 0.714350i
\(341\) 1.12576e9 1.53746
\(342\) 1.45593e8 0.196811
\(343\) 4.57442e8i 0.612078i
\(344\) 1.03437e8i 0.137000i
\(345\) − 9.47026e8i − 1.24164i
\(346\) − 5.25612e8i − 0.682180i
\(347\) −8.94842e8 −1.14972 −0.574861 0.818251i \(-0.694944\pi\)
−0.574861 + 0.818251i \(0.694944\pi\)
\(348\) 3.05106e8 0.388082
\(349\) 5.41626e8i 0.682041i 0.940056 + 0.341020i \(0.110772\pi\)
−0.940056 + 0.341020i \(0.889228\pi\)
\(350\) −1.64314e8 −0.204851
\(351\) 0 0
\(352\) 1.77013e8 0.216324
\(353\) 2.25334e8i 0.272656i 0.990664 + 0.136328i \(0.0435301\pi\)
−0.990664 + 0.136328i \(0.956470\pi\)
\(354\) −1.38101e8 −0.165457
\(355\) −8.02611e8 −0.952152
\(356\) − 4.47058e8i − 0.525157i
\(357\) − 2.40093e8i − 0.279280i
\(358\) − 2.56322e7i − 0.0295253i
\(359\) 4.38763e8i 0.500495i 0.968182 + 0.250247i \(0.0805120\pi\)
−0.968182 + 0.250247i \(0.919488\pi\)
\(360\) 1.31282e8 0.148302
\(361\) 1.47161e8 0.164634
\(362\) − 3.56607e8i − 0.395103i
\(363\) 3.78083e8 0.414872
\(364\) 0 0
\(365\) 2.28609e9 2.46075
\(366\) − 3.38086e8i − 0.360448i
\(367\) 8.08568e8 0.853857 0.426929 0.904285i \(-0.359596\pi\)
0.426929 + 0.904285i \(0.359596\pi\)
\(368\) 2.58343e8 0.270227
\(369\) − 3.86280e7i − 0.0400230i
\(370\) 1.36253e9i 1.39843i
\(371\) 4.93510e8i 0.501750i
\(372\) − 5.20156e8i − 0.523882i
\(373\) −1.17884e9 −1.17618 −0.588092 0.808794i \(-0.700121\pi\)
−0.588092 + 0.808794i \(0.700121\pi\)
\(374\) 9.08011e8 0.897513
\(375\) − 1.20495e8i − 0.117994i
\(376\) 3.01318e8 0.292326
\(377\) 0 0
\(378\) 2.60810e8 0.248372
\(379\) − 1.79168e9i − 1.69053i −0.534345 0.845266i \(-0.679442\pi\)
0.534345 0.845266i \(-0.320558\pi\)
\(380\) −6.73313e8 −0.629469
\(381\) −1.26363e9 −1.17053
\(382\) − 1.49115e9i − 1.36867i
\(383\) − 1.19775e9i − 1.08936i −0.838644 0.544680i \(-0.816651\pi\)
0.838644 0.544680i \(-0.183349\pi\)
\(384\) − 8.17889e7i − 0.0737115i
\(385\) − 6.09373e8i − 0.544215i
\(386\) 1.22341e9 1.08273
\(387\) −1.34549e8 −0.118002
\(388\) 1.28488e7i 0.0111674i
\(389\) 1.43672e8 0.123751 0.0618754 0.998084i \(-0.480292\pi\)
0.0618754 + 0.998084i \(0.480292\pi\)
\(390\) 0 0
\(391\) 1.32521e9 1.12115
\(392\) 3.77699e8i 0.316698i
\(393\) 1.03221e9 0.857815
\(394\) 7.61470e8 0.627214
\(395\) − 2.54456e9i − 2.07742i
\(396\) 2.30255e8i 0.186327i
\(397\) 6.17334e8i 0.495169i 0.968866 + 0.247584i \(0.0796367\pi\)
−0.968866 + 0.247584i \(0.920363\pi\)
\(398\) 1.42868e9i 1.13591i
\(399\) −3.12254e8 −0.246095
\(400\) −2.87130e8 −0.224320
\(401\) − 1.13305e9i − 0.877491i −0.898611 0.438746i \(-0.855423\pi\)
0.898611 0.438746i \(-0.144577\pi\)
\(402\) −1.07437e9 −0.824825
\(403\) 0 0
\(404\) −3.46972e8 −0.261794
\(405\) − 1.10991e9i − 0.830220i
\(406\) 2.86526e8 0.212482
\(407\) −2.38973e9 −1.75699
\(408\) − 4.19548e8i − 0.305823i
\(409\) − 1.04283e9i − 0.753670i −0.926280 0.376835i \(-0.877012\pi\)
0.926280 0.376835i \(-0.122988\pi\)
\(410\) 1.78640e8i 0.128007i
\(411\) 2.09119e9i 1.48575i
\(412\) −1.10014e9 −0.775011
\(413\) −1.29691e8 −0.0905906
\(414\) 3.36048e8i 0.232755i
\(415\) 5.49549e7 0.0377431
\(416\) 0 0
\(417\) −2.95926e8 −0.199851
\(418\) − 1.18092e9i − 0.790867i
\(419\) 7.09302e8 0.471066 0.235533 0.971866i \(-0.424316\pi\)
0.235533 + 0.971866i \(0.424316\pi\)
\(420\) −2.81561e8 −0.185439
\(421\) − 1.19877e9i − 0.782974i −0.920184 0.391487i \(-0.871961\pi\)
0.920184 0.391487i \(-0.128039\pi\)
\(422\) 1.06588e9i 0.690424i
\(423\) 3.91948e8i 0.251789i
\(424\) 8.62380e8i 0.549437i
\(425\) −1.47287e9 −0.930687
\(426\) −6.50428e8 −0.407629
\(427\) − 3.17497e8i − 0.197352i
\(428\) −7.90926e8 −0.487621
\(429\) 0 0
\(430\) 6.22237e8 0.377412
\(431\) 9.54153e8i 0.574047i 0.957924 + 0.287024i \(0.0926658\pi\)
−0.957924 + 0.287024i \(0.907334\pi\)
\(432\) 4.55750e8 0.271978
\(433\) 3.81628e8 0.225908 0.112954 0.993600i \(-0.463969\pi\)
0.112954 + 0.993600i \(0.463969\pi\)
\(434\) − 4.88480e8i − 0.286836i
\(435\) − 1.83540e9i − 1.06910i
\(436\) − 1.09164e9i − 0.630778i
\(437\) − 1.72351e9i − 0.987933i
\(438\) 1.85262e9 1.05348
\(439\) −1.11683e8 −0.0630031 −0.0315015 0.999504i \(-0.510029\pi\)
−0.0315015 + 0.999504i \(0.510029\pi\)
\(440\) − 1.06484e9i − 0.595938i
\(441\) −4.91304e8 −0.272782
\(442\) 0 0
\(443\) 1.45991e9 0.797837 0.398919 0.916986i \(-0.369386\pi\)
0.398919 + 0.916986i \(0.369386\pi\)
\(444\) 1.10418e9i 0.598685i
\(445\) −2.68934e9 −1.44672
\(446\) 9.55154e8 0.509802
\(447\) 2.22416e9i 1.17785i
\(448\) − 7.68082e7i − 0.0403585i
\(449\) 6.34009e8i 0.330547i 0.986248 + 0.165273i \(0.0528507\pi\)
−0.986248 + 0.165273i \(0.947149\pi\)
\(450\) − 3.73493e8i − 0.193214i
\(451\) −3.13316e8 −0.160829
\(452\) −1.35200e9 −0.688639
\(453\) − 7.82528e8i − 0.395509i
\(454\) 9.09244e7 0.0456021
\(455\) 0 0
\(456\) −5.45646e8 −0.269484
\(457\) 6.04376e8i 0.296211i 0.988972 + 0.148105i \(0.0473175\pi\)
−0.988972 + 0.148105i \(0.952683\pi\)
\(458\) 1.17247e8 0.0570261
\(459\) 2.33783e9 1.12841
\(460\) − 1.55409e9i − 0.744432i
\(461\) 2.20565e9i 1.04853i 0.851554 + 0.524267i \(0.175661\pi\)
−0.851554 + 0.524267i \(0.824339\pi\)
\(462\) − 4.93829e8i − 0.232986i
\(463\) − 1.04925e9i − 0.491299i −0.969359 0.245650i \(-0.920999\pi\)
0.969359 0.245650i \(-0.0790012\pi\)
\(464\) 5.00687e8 0.232677
\(465\) −3.12907e9 −1.44321
\(466\) − 1.97539e9i − 0.904281i
\(467\) 2.01461e9 0.915337 0.457668 0.889123i \(-0.348685\pi\)
0.457668 + 0.889123i \(0.348685\pi\)
\(468\) 0 0
\(469\) −1.00894e9 −0.451607
\(470\) − 1.81261e9i − 0.805309i
\(471\) 1.22973e9 0.542295
\(472\) −2.26627e8 −0.0992005
\(473\) 1.09134e9i 0.474183i
\(474\) − 2.06209e9i − 0.889371i
\(475\) 1.91555e9i 0.820099i
\(476\) − 3.93998e8i − 0.167444i
\(477\) −1.12177e9 −0.473248
\(478\) −1.28991e8 −0.0540209
\(479\) − 3.67842e9i − 1.52928i −0.644458 0.764639i \(-0.722917\pi\)
0.644458 0.764639i \(-0.277083\pi\)
\(480\) −4.92012e8 −0.203063
\(481\) 0 0
\(482\) −9.14048e8 −0.371796
\(483\) − 7.20724e8i − 0.291041i
\(484\) 6.20444e8 0.248739
\(485\) 7.72934e7 0.0307642
\(486\) 1.04727e9i 0.413840i
\(487\) − 1.91497e8i − 0.0751294i −0.999294 0.0375647i \(-0.988040\pi\)
0.999294 0.0375647i \(-0.0119600\pi\)
\(488\) − 5.54807e8i − 0.216109i
\(489\) − 1.22436e9i − 0.473508i
\(490\) 2.27210e9 0.872450
\(491\) 3.22321e8 0.122886 0.0614431 0.998111i \(-0.480430\pi\)
0.0614431 + 0.998111i \(0.480430\pi\)
\(492\) 1.44768e8i 0.0548018i
\(493\) 2.56834e9 0.965359
\(494\) 0 0
\(495\) 1.38513e9 0.513300
\(496\) − 8.53590e8i − 0.314097i
\(497\) −6.10819e8 −0.223185
\(498\) 4.45349e7 0.0161584
\(499\) 3.86695e9i 1.39321i 0.717455 + 0.696604i \(0.245307\pi\)
−0.717455 + 0.696604i \(0.754693\pi\)
\(500\) − 1.97736e8i − 0.0707442i
\(501\) 3.59646e9i 1.27774i
\(502\) 1.78163e9i 0.628572i
\(503\) −3.43814e8 −0.120458 −0.0602290 0.998185i \(-0.519183\pi\)
−0.0602290 + 0.998185i \(0.519183\pi\)
\(504\) 9.99107e7 0.0347620
\(505\) 2.08725e9i 0.721199i
\(506\) 2.72572e9 0.935307
\(507\) 0 0
\(508\) −2.07365e9 −0.701800
\(509\) 2.11533e9i 0.710993i 0.934678 + 0.355497i \(0.115688\pi\)
−0.934678 + 0.355497i \(0.884312\pi\)
\(510\) −2.52384e9 −0.842493
\(511\) 1.73980e9 0.576802
\(512\) − 1.34218e8i − 0.0441942i
\(513\) − 3.04048e9i − 0.994333i
\(514\) 2.25842e9i 0.733556i
\(515\) 6.61803e9i 2.13503i
\(516\) 5.04254e8 0.161576
\(517\) 3.17914e9 1.01179
\(518\) 1.03694e9i 0.327792i
\(519\) −2.56236e9 −0.804552
\(520\) 0 0
\(521\) −1.40622e9 −0.435634 −0.217817 0.975990i \(-0.569894\pi\)
−0.217817 + 0.975990i \(0.569894\pi\)
\(522\) 6.51284e8i 0.200412i
\(523\) 2.18120e9 0.666712 0.333356 0.942801i \(-0.391819\pi\)
0.333356 + 0.942801i \(0.391819\pi\)
\(524\) 1.69388e9 0.514308
\(525\) 8.01033e8i 0.241598i
\(526\) 1.89192e9i 0.566830i
\(527\) − 4.37861e9i − 1.30316i
\(528\) − 8.62937e8i − 0.255129i
\(529\) 5.73252e8 0.168365
\(530\) 5.18775e9 1.51361
\(531\) − 2.94792e8i − 0.0854445i
\(532\) −5.12417e8 −0.147548
\(533\) 0 0
\(534\) −2.17941e9 −0.619362
\(535\) 4.75791e9i 1.34332i
\(536\) −1.76306e9 −0.494529
\(537\) −1.24957e8 −0.0348217
\(538\) 3.85737e9i 1.06796i
\(539\) 3.98502e9i 1.09615i
\(540\) − 2.74162e9i − 0.749254i
\(541\) − 2.54634e8i − 0.0691395i −0.999402 0.0345698i \(-0.988994\pi\)
0.999402 0.0345698i \(-0.0110061\pi\)
\(542\) −3.73097e9 −1.00653
\(543\) −1.73846e9 −0.465978
\(544\) − 6.88488e8i − 0.183358i
\(545\) −6.56689e9 −1.73769
\(546\) 0 0
\(547\) 2.15158e9 0.562085 0.281043 0.959695i \(-0.409320\pi\)
0.281043 + 0.959695i \(0.409320\pi\)
\(548\) 3.43169e9i 0.890791i
\(549\) 7.21683e8 0.186142
\(550\) −3.02944e9 −0.776414
\(551\) − 3.34028e9i − 0.850652i
\(552\) − 1.25942e9i − 0.318702i
\(553\) − 1.93651e9i − 0.486948i
\(554\) − 1.50967e9i − 0.377224i
\(555\) 6.64232e9 1.64928
\(556\) −4.85622e8 −0.119822
\(557\) 7.71518e9i 1.89170i 0.324599 + 0.945852i \(0.394771\pi\)
−0.324599 + 0.945852i \(0.605229\pi\)
\(558\) 1.11033e9 0.270542
\(559\) 0 0
\(560\) −4.62049e8 −0.111181
\(561\) − 4.42656e9i − 1.05851i
\(562\) 5.72321e9 1.36008
\(563\) 8.12996e7 0.0192003 0.00960017 0.999954i \(-0.496944\pi\)
0.00960017 + 0.999954i \(0.496944\pi\)
\(564\) − 1.46892e9i − 0.344764i
\(565\) 8.13312e9i 1.89709i
\(566\) − 3.23681e9i − 0.750343i
\(567\) − 8.44681e8i − 0.194604i
\(568\) −1.06737e9 −0.244397
\(569\) 5.08814e9 1.15789 0.578944 0.815367i \(-0.303465\pi\)
0.578944 + 0.815367i \(0.303465\pi\)
\(570\) 3.28240e9i 0.742385i
\(571\) 5.61762e9 1.26277 0.631387 0.775468i \(-0.282486\pi\)
0.631387 + 0.775468i \(0.282486\pi\)
\(572\) 0 0
\(573\) −7.26935e9 −1.61419
\(574\) 1.35952e8i 0.0300050i
\(575\) −4.42135e9 −0.969878
\(576\) 1.74588e8 0.0380658
\(577\) 4.12728e9i 0.894435i 0.894425 + 0.447218i \(0.147585\pi\)
−0.894425 + 0.447218i \(0.852415\pi\)
\(578\) − 2.48988e8i − 0.0536328i
\(579\) − 5.96415e9i − 1.27695i
\(580\) − 3.01194e9i − 0.640987i
\(581\) 4.18228e7 0.00884702
\(582\) 6.26377e7 0.0131706
\(583\) 9.09878e9i 1.90171i
\(584\) 3.04020e9 0.631622
\(585\) 0 0
\(586\) 6.49057e9 1.33242
\(587\) 1.86734e9i 0.381056i 0.981682 + 0.190528i \(0.0610201\pi\)
−0.981682 + 0.190528i \(0.938980\pi\)
\(588\) 1.84128e9 0.373508
\(589\) −5.69463e9 −1.14832
\(590\) 1.36330e9i 0.273281i
\(591\) − 3.71216e9i − 0.739726i
\(592\) 1.81198e9i 0.358945i
\(593\) − 3.31544e9i − 0.652905i −0.945214 0.326453i \(-0.894147\pi\)
0.945214 0.326453i \(-0.105853\pi\)
\(594\) 4.80851e9 0.941366
\(595\) −2.37015e9 −0.461281
\(596\) 3.64990e9i 0.706187i
\(597\) 6.96483e9 1.33968
\(598\) 0 0
\(599\) 1.93367e9 0.367610 0.183805 0.982963i \(-0.441158\pi\)
0.183805 + 0.982963i \(0.441158\pi\)
\(600\) 1.39976e9i 0.264559i
\(601\) −5.88820e9 −1.10643 −0.553213 0.833040i \(-0.686598\pi\)
−0.553213 + 0.833040i \(0.686598\pi\)
\(602\) 4.73547e8 0.0884657
\(603\) − 2.29336e9i − 0.425953i
\(604\) − 1.28415e9i − 0.237130i
\(605\) − 3.73236e9i − 0.685235i
\(606\) 1.69149e9i 0.308756i
\(607\) 7.94197e9 1.44135 0.720673 0.693276i \(-0.243833\pi\)
0.720673 + 0.693276i \(0.243833\pi\)
\(608\) −8.95418e8 −0.161571
\(609\) − 1.39681e9i − 0.250598i
\(610\) −3.33751e9 −0.595345
\(611\) 0 0
\(612\) 8.95573e8 0.157932
\(613\) − 2.36146e8i − 0.0414065i −0.999786 0.0207033i \(-0.993409\pi\)
0.999786 0.0207033i \(-0.00659052\pi\)
\(614\) −3.68766e9 −0.642927
\(615\) 8.70870e8 0.150970
\(616\) − 8.10386e8i − 0.139688i
\(617\) 1.27029e9i 0.217723i 0.994057 + 0.108862i \(0.0347206\pi\)
−0.994057 + 0.108862i \(0.965279\pi\)
\(618\) 5.36318e9i 0.914035i
\(619\) − 1.63555e9i − 0.277170i −0.990351 0.138585i \(-0.955745\pi\)
0.990351 0.138585i \(-0.0442554\pi\)
\(620\) −5.13488e9 −0.865285
\(621\) 7.01783e9 1.17593
\(622\) − 2.30175e9i − 0.383523i
\(623\) −2.04669e9 −0.339112
\(624\) 0 0
\(625\) −6.66607e9 −1.09217
\(626\) 7.64943e9i 1.24629i
\(627\) −5.75699e9 −0.932736
\(628\) 2.01801e9 0.325136
\(629\) 9.29483e9i 1.48924i
\(630\) − 6.01025e8i − 0.0957636i
\(631\) − 1.68242e9i − 0.266582i −0.991077 0.133291i \(-0.957445\pi\)
0.991077 0.133291i \(-0.0425546\pi\)
\(632\) − 3.38394e9i − 0.533228i
\(633\) 5.19618e9 0.814275
\(634\) 3.94209e9 0.614347
\(635\) 1.24743e10i 1.93334i
\(636\) 4.20410e9 0.647998
\(637\) 0 0
\(638\) 5.28264e9 0.805338
\(639\) − 1.38841e9i − 0.210507i
\(640\) −8.07404e8 −0.121748
\(641\) 1.70575e9 0.255807 0.127903 0.991787i \(-0.459175\pi\)
0.127903 + 0.991787i \(0.459175\pi\)
\(642\) 3.85576e9i 0.575092i
\(643\) − 1.45635e9i − 0.216036i −0.994149 0.108018i \(-0.965550\pi\)
0.994149 0.108018i \(-0.0344505\pi\)
\(644\) − 1.18273e9i − 0.174495i
\(645\) − 3.03341e9i − 0.445114i
\(646\) −4.59317e9 −0.670346
\(647\) 3.56464e9 0.517430 0.258715 0.965954i \(-0.416701\pi\)
0.258715 + 0.965954i \(0.416701\pi\)
\(648\) − 1.47603e9i − 0.213100i
\(649\) −2.39109e9 −0.343352
\(650\) 0 0
\(651\) −2.38134e9 −0.338289
\(652\) − 2.00920e9i − 0.283895i
\(653\) −5.86806e9 −0.824705 −0.412352 0.911024i \(-0.635293\pi\)
−0.412352 + 0.911024i \(0.635293\pi\)
\(654\) −5.32174e9 −0.743929
\(655\) − 1.01898e10i − 1.41683i
\(656\) 2.37568e8i 0.0328567i
\(657\) 3.95463e9i 0.544036i
\(658\) − 1.37947e9i − 0.188765i
\(659\) −2.73239e9 −0.371915 −0.185958 0.982558i \(-0.559539\pi\)
−0.185958 + 0.982558i \(0.559539\pi\)
\(660\) −5.19111e9 −0.702839
\(661\) − 8.50066e9i − 1.14485i −0.819958 0.572424i \(-0.806003\pi\)
0.819958 0.572424i \(-0.193997\pi\)
\(662\) −3.86687e9 −0.518032
\(663\) 0 0
\(664\) 7.30829e7 0.00968785
\(665\) 3.08251e9i 0.406470i
\(666\) −2.35700e9 −0.309171
\(667\) 7.70980e9 1.00601
\(668\) 5.90189e9i 0.766079i
\(669\) − 4.65638e9i − 0.601252i
\(670\) 1.06059e10i 1.36235i
\(671\) − 5.85365e9i − 0.747994i
\(672\) −3.74440e8 −0.0475981
\(673\) 3.85727e7 0.00487784 0.00243892 0.999997i \(-0.499224\pi\)
0.00243892 + 0.999997i \(0.499224\pi\)
\(674\) 1.04658e10i 1.31663i
\(675\) −7.79982e9 −0.976160
\(676\) 0 0
\(677\) −7.34428e9 −0.909681 −0.454840 0.890573i \(-0.650304\pi\)
−0.454840 + 0.890573i \(0.650304\pi\)
\(678\) 6.59100e9i 0.812170i
\(679\) 5.88233e7 0.00721116
\(680\) −4.14169e9 −0.505122
\(681\) − 4.43257e8i − 0.0537824i
\(682\) − 9.00604e9i − 1.08715i
\(683\) 7.49577e9i 0.900210i 0.892976 + 0.450105i \(0.148613\pi\)
−0.892976 + 0.450105i \(0.851387\pi\)
\(684\) − 1.16474e9i − 0.139166i
\(685\) 2.06438e10 2.45398
\(686\) 3.65954e9 0.432805
\(687\) − 5.71580e8i − 0.0672556i
\(688\) 8.27494e8 0.0968736
\(689\) 0 0
\(690\) −7.57621e9 −0.877971
\(691\) 1.66382e10i 1.91838i 0.282769 + 0.959188i \(0.408747\pi\)
−0.282769 + 0.959188i \(0.591253\pi\)
\(692\) −4.20490e9 −0.482374
\(693\) 1.05414e9 0.120318
\(694\) 7.15873e9i 0.812976i
\(695\) 2.92132e9i 0.330090i
\(696\) − 2.44085e9i − 0.274415i
\(697\) 1.21864e9i 0.136320i
\(698\) 4.33300e9 0.482275
\(699\) −9.63005e9 −1.06649
\(700\) 1.31452e9i 0.144851i
\(701\) −3.26804e9 −0.358323 −0.179161 0.983820i \(-0.557338\pi\)
−0.179161 + 0.983820i \(0.557338\pi\)
\(702\) 0 0
\(703\) 1.20884e10 1.31228
\(704\) − 1.41610e9i − 0.152964i
\(705\) −8.83649e9 −0.949769
\(706\) 1.80267e9 0.192797
\(707\) 1.58848e9i 0.169050i
\(708\) 1.10480e9i 0.116995i
\(709\) − 4.48613e9i − 0.472727i −0.971665 0.236363i \(-0.924044\pi\)
0.971665 0.236363i \(-0.0759555\pi\)
\(710\) 6.42089e9i 0.673273i
\(711\) 4.40176e9 0.459286
\(712\) −3.57647e9 −0.371342
\(713\) − 1.31440e10i − 1.35804i
\(714\) −1.92074e9 −0.197481
\(715\) 0 0
\(716\) −2.05057e8 −0.0208776
\(717\) 6.28831e8i 0.0637114i
\(718\) 3.51011e9 0.353903
\(719\) −5.42385e9 −0.544198 −0.272099 0.962269i \(-0.587718\pi\)
−0.272099 + 0.962269i \(0.587718\pi\)
\(720\) − 1.05026e9i − 0.104865i
\(721\) 5.03658e9i 0.500452i
\(722\) − 1.17729e9i − 0.116414i
\(723\) 4.45598e9i 0.438490i
\(724\) −2.85286e9 −0.279380
\(725\) −8.56888e9 −0.835105
\(726\) − 3.02466e9i − 0.293359i
\(727\) 1.50827e10 1.45582 0.727911 0.685672i \(-0.240492\pi\)
0.727911 + 0.685672i \(0.240492\pi\)
\(728\) 0 0
\(729\) 1.14103e10 1.09081
\(730\) − 1.82887e10i − 1.74001i
\(731\) 4.24475e9 0.401921
\(732\) −2.70469e9 −0.254875
\(733\) − 6.75596e9i − 0.633612i −0.948490 0.316806i \(-0.897390\pi\)
0.948490 0.316806i \(-0.102610\pi\)
\(734\) − 6.46854e9i − 0.603768i
\(735\) − 1.10765e10i − 1.02895i
\(736\) − 2.06674e9i − 0.191080i
\(737\) −1.86017e10 −1.71166
\(738\) −3.09024e8 −0.0283006
\(739\) − 1.08154e10i − 0.985797i −0.870087 0.492899i \(-0.835937\pi\)
0.870087 0.492899i \(-0.164063\pi\)
\(740\) 1.09002e10 0.988836
\(741\) 0 0
\(742\) 3.94808e9 0.354791
\(743\) − 3.71897e9i − 0.332630i −0.986073 0.166315i \(-0.946813\pi\)
0.986073 0.166315i \(-0.0531869\pi\)
\(744\) −4.16125e9 −0.370441
\(745\) 2.19565e10 1.94543
\(746\) 9.43075e9i 0.831687i
\(747\) 9.50648e7i 0.00834445i
\(748\) − 7.26409e9i − 0.634637i
\(749\) 3.62096e9i 0.314874i
\(750\) −9.63963e8 −0.0834346
\(751\) 2.15786e10 1.85902 0.929510 0.368797i \(-0.120230\pi\)
0.929510 + 0.368797i \(0.120230\pi\)
\(752\) − 2.41054e9i − 0.206706i
\(753\) 8.68546e9 0.741328
\(754\) 0 0
\(755\) −7.72496e9 −0.653253
\(756\) − 2.08648e9i − 0.175626i
\(757\) 7.42446e9 0.622056 0.311028 0.950401i \(-0.399327\pi\)
0.311028 + 0.950401i \(0.399327\pi\)
\(758\) −1.43335e10 −1.19539
\(759\) − 1.32879e10i − 1.10309i
\(760\) 5.38650e9i 0.445102i
\(761\) − 8.57002e9i − 0.704913i −0.935828 0.352457i \(-0.885346\pi\)
0.935828 0.352457i \(-0.114654\pi\)
\(762\) 1.01091e10i 0.827691i
\(763\) −4.99766e9 −0.407315
\(764\) −1.19292e10 −0.967797
\(765\) − 5.38743e9i − 0.435078i
\(766\) −9.58202e9 −0.770294
\(767\) 0 0
\(768\) −6.54311e8 −0.0521219
\(769\) 7.81741e9i 0.619899i 0.950753 + 0.309949i \(0.100312\pi\)
−0.950753 + 0.309949i \(0.899688\pi\)
\(770\) −4.87498e9 −0.384818
\(771\) 1.10098e10 0.865144
\(772\) − 9.78732e9i − 0.765602i
\(773\) 1.30864e10i 1.01904i 0.860457 + 0.509522i \(0.170178\pi\)
−0.860457 + 0.509522i \(0.829822\pi\)
\(774\) 1.07639e9i 0.0834403i
\(775\) 1.46086e10i 1.12733i
\(776\) 1.02790e8 0.00789651
\(777\) 5.05506e9 0.386592
\(778\) − 1.14937e9i − 0.0875051i
\(779\) 1.58491e9 0.120122
\(780\) 0 0
\(781\) −1.12616e10 −0.845903
\(782\) − 1.06016e10i − 0.792775i
\(783\) 1.36011e10 1.01253
\(784\) 3.02159e9 0.223939
\(785\) − 1.21396e10i − 0.895696i
\(786\) − 8.25767e9i − 0.606567i
\(787\) − 5.35561e9i − 0.391649i −0.980639 0.195825i \(-0.937262\pi\)
0.980639 0.195825i \(-0.0627383\pi\)
\(788\) − 6.09176e9i − 0.443507i
\(789\) 9.22311e9 0.668510
\(790\) −2.03565e10 −1.46895
\(791\) 6.18962e9i 0.444679i
\(792\) 1.84204e9 0.131753
\(793\) 0 0
\(794\) 4.93867e9 0.350137
\(795\) − 2.52903e10i − 1.78513i
\(796\) 1.14295e10 0.803212
\(797\) −1.21863e10 −0.852641 −0.426320 0.904572i \(-0.640190\pi\)
−0.426320 + 0.904572i \(0.640190\pi\)
\(798\) 2.49803e9i 0.174016i
\(799\) − 1.23652e10i − 0.857606i
\(800\) 2.29704e9i 0.158618i
\(801\) − 4.65220e9i − 0.319849i
\(802\) −9.06438e9 −0.620480
\(803\) 3.20765e10 2.18616
\(804\) 8.59494e9i 0.583239i
\(805\) −7.11484e9 −0.480706
\(806\) 0 0
\(807\) 1.88047e10 1.25953
\(808\) 2.77578e9i 0.185116i
\(809\) 1.32472e10 0.879636 0.439818 0.898087i \(-0.355043\pi\)
0.439818 + 0.898087i \(0.355043\pi\)
\(810\) −8.87924e9 −0.587054
\(811\) 1.45473e10i 0.957658i 0.877908 + 0.478829i \(0.158939\pi\)
−0.877908 + 0.478829i \(0.841061\pi\)
\(812\) − 2.29221e9i − 0.150248i
\(813\) 1.81885e10i 1.18708i
\(814\) 1.91179e10i 1.24238i
\(815\) −1.20866e10 −0.782083
\(816\) −3.35638e9 −0.216250
\(817\) − 5.52054e9i − 0.354164i
\(818\) −8.34262e9 −0.532925
\(819\) 0 0
\(820\) 1.42912e9 0.0905149
\(821\) 6.51876e9i 0.411115i 0.978645 + 0.205558i \(0.0659008\pi\)
−0.978645 + 0.205558i \(0.934099\pi\)
\(822\) 1.67295e10 1.05059
\(823\) 6.77944e9 0.423930 0.211965 0.977277i \(-0.432014\pi\)
0.211965 + 0.977277i \(0.432014\pi\)
\(824\) 8.80112e9i 0.548015i
\(825\) 1.47685e10i 0.915690i
\(826\) 1.03752e9i 0.0640572i
\(827\) 7.96808e9i 0.489874i 0.969539 + 0.244937i \(0.0787673\pi\)
−0.969539 + 0.244937i \(0.921233\pi\)
\(828\) 2.68838e9 0.164583
\(829\) −3.74439e9 −0.228265 −0.114133 0.993466i \(-0.536409\pi\)
−0.114133 + 0.993466i \(0.536409\pi\)
\(830\) − 4.39639e8i − 0.0266884i
\(831\) −7.35966e9 −0.444892
\(832\) 0 0
\(833\) 1.54997e10 0.929106
\(834\) 2.36741e9i 0.141316i
\(835\) 3.55035e10 2.11042
\(836\) −9.44736e9 −0.559228
\(837\) − 2.31876e10i − 1.36684i
\(838\) − 5.67442e9i − 0.333094i
\(839\) 8.06205e9i 0.471280i 0.971840 + 0.235640i \(0.0757186\pi\)
−0.971840 + 0.235640i \(0.924281\pi\)
\(840\) 2.25249e9i 0.131125i
\(841\) −2.30775e9 −0.133783
\(842\) −9.59013e9 −0.553646
\(843\) − 2.79007e10i − 1.60405i
\(844\) 8.52706e9 0.488203
\(845\) 0 0
\(846\) 3.13559e9 0.178042
\(847\) − 2.84047e9i − 0.160619i
\(848\) 6.89904e9 0.388511
\(849\) −1.57795e10 −0.884943
\(850\) 1.17830e10i 0.658095i
\(851\) 2.79017e10i 1.55195i
\(852\) 5.20342e9i 0.288238i
\(853\) 3.31854e9i 0.183074i 0.995802 + 0.0915368i \(0.0291779\pi\)
−0.995802 + 0.0915368i \(0.970822\pi\)
\(854\) −2.53998e9 −0.139549
\(855\) −7.00666e9 −0.383380
\(856\) 6.32740e9i 0.344800i
\(857\) 1.81939e10 0.987398 0.493699 0.869633i \(-0.335644\pi\)
0.493699 + 0.869633i \(0.335644\pi\)
\(858\) 0 0
\(859\) −1.91859e10 −1.03278 −0.516388 0.856355i \(-0.672724\pi\)
−0.516388 + 0.856355i \(0.672724\pi\)
\(860\) − 4.97790e9i − 0.266871i
\(861\) 6.62766e8 0.0353874
\(862\) 7.63322e9 0.405913
\(863\) − 2.77943e10i − 1.47203i −0.676963 0.736017i \(-0.736704\pi\)
0.676963 0.736017i \(-0.263296\pi\)
\(864\) − 3.64600e9i − 0.192317i
\(865\) 2.52951e10i 1.32886i
\(866\) − 3.05302e9i − 0.159741i
\(867\) −1.21381e9 −0.0632536
\(868\) −3.90784e9 −0.202823
\(869\) − 3.57032e10i − 1.84560i
\(870\) −1.46832e10 −0.755969
\(871\) 0 0
\(872\) −8.73311e9 −0.446027
\(873\) 1.33707e8i 0.00680152i
\(874\) −1.37880e10 −0.698574
\(875\) −9.05260e8 −0.0456820
\(876\) − 1.48210e10i − 0.744925i
\(877\) − 3.40401e10i − 1.70409i −0.523471 0.852043i \(-0.675363\pi\)
0.523471 0.852043i \(-0.324637\pi\)
\(878\) 8.93465e8i 0.0445499i
\(879\) − 3.16415e10i − 1.57143i
\(880\) −8.51874e9 −0.421392
\(881\) −3.24476e10 −1.59870 −0.799350 0.600866i \(-0.794823\pi\)
−0.799350 + 0.600866i \(0.794823\pi\)
\(882\) 3.93043e9i 0.192886i
\(883\) −1.15866e10 −0.566362 −0.283181 0.959066i \(-0.591390\pi\)
−0.283181 + 0.959066i \(0.591390\pi\)
\(884\) 0 0
\(885\) 6.64609e9 0.322303
\(886\) − 1.16793e10i − 0.564156i
\(887\) −2.86160e10 −1.37682 −0.688408 0.725323i \(-0.741690\pi\)
−0.688408 + 0.725323i \(0.741690\pi\)
\(888\) 8.83342e9 0.423335
\(889\) 9.49344e9i 0.453177i
\(890\) 2.15147e10i 1.02299i
\(891\) − 1.55733e10i − 0.737578i
\(892\) − 7.64123e9i − 0.360484i
\(893\) −1.60817e10 −0.755702
\(894\) 1.77933e10 0.832865
\(895\) 1.23355e9i 0.0575142i
\(896\) −6.14466e8 −0.0285377
\(897\) 0 0
\(898\) 5.07207e9 0.233732
\(899\) − 2.54739e10i − 1.16933i
\(900\) −2.98794e9 −0.136623
\(901\) 3.53896e10 1.61190
\(902\) 2.50653e9i 0.113723i
\(903\) − 2.30854e9i − 0.104335i
\(904\) 1.08160e10i 0.486942i
\(905\) 1.71617e10i 0.769646i
\(906\) −6.26022e9 −0.279667
\(907\) −2.28278e10 −1.01587 −0.507936 0.861395i \(-0.669591\pi\)
−0.507936 + 0.861395i \(0.669591\pi\)
\(908\) − 7.27395e8i − 0.0322456i
\(909\) −3.61068e9 −0.159446
\(910\) 0 0
\(911\) −1.76175e10 −0.772024 −0.386012 0.922494i \(-0.626148\pi\)
−0.386012 + 0.922494i \(0.626148\pi\)
\(912\) 4.36516e9i 0.190554i
\(913\) 7.71081e8 0.0335315
\(914\) 4.83501e9 0.209453
\(915\) 1.62704e10i 0.702140i
\(916\) − 9.37978e8i − 0.0403235i
\(917\) − 7.75480e9i − 0.332107i
\(918\) − 1.87026e10i − 0.797910i
\(919\) −4.93202e9 −0.209614 −0.104807 0.994493i \(-0.533423\pi\)
−0.104807 + 0.994493i \(0.533423\pi\)
\(920\) −1.24328e10 −0.526393
\(921\) 1.79774e10i 0.758258i
\(922\) 1.76452e10 0.741425
\(923\) 0 0
\(924\) −3.95063e9 −0.164746
\(925\) − 3.10108e10i − 1.28830i
\(926\) −8.39401e9 −0.347401
\(927\) −1.14483e10 −0.472023
\(928\) − 4.00549e9i − 0.164527i
\(929\) − 2.68352e10i − 1.09812i −0.835783 0.549061i \(-0.814986\pi\)
0.835783 0.549061i \(-0.185014\pi\)
\(930\) 2.50325e10i 1.02050i
\(931\) − 2.01582e10i − 0.818707i
\(932\) −1.58032e10 −0.639423
\(933\) −1.12210e10 −0.452321
\(934\) − 1.61168e10i − 0.647241i
\(935\) −4.36980e10 −1.74832
\(936\) 0 0
\(937\) −2.08650e10 −0.828570 −0.414285 0.910147i \(-0.635968\pi\)
−0.414285 + 0.910147i \(0.635968\pi\)
\(938\) 8.07153e9i 0.319335i
\(939\) 3.72910e10 1.46985
\(940\) −1.45009e10 −0.569440
\(941\) − 3.07099e10i − 1.20147i −0.799447 0.600737i \(-0.794874\pi\)
0.799447 0.600737i \(-0.205126\pi\)
\(942\) − 9.83781e9i − 0.383460i
\(943\) 3.65818e9i 0.142061i
\(944\) 1.81301e9i 0.0701453i
\(945\) −1.25515e10 −0.483820
\(946\) 8.73071e9 0.335298
\(947\) 1.03377e9i 0.0395548i 0.999804 + 0.0197774i \(0.00629575\pi\)
−0.999804 + 0.0197774i \(0.993704\pi\)
\(948\) −1.64967e10 −0.628880
\(949\) 0 0
\(950\) 1.53244e10 0.579898
\(951\) − 1.92177e10i − 0.724552i
\(952\) −3.15199e9 −0.118401
\(953\) 1.78629e9 0.0668540 0.0334270 0.999441i \(-0.489358\pi\)
0.0334270 + 0.999441i \(0.489358\pi\)
\(954\) 8.97414e9i 0.334637i
\(955\) 7.17615e10i 2.66612i
\(956\) 1.03193e9i 0.0381985i
\(957\) − 2.57529e10i − 0.949803i
\(958\) −2.94273e10 −1.08136
\(959\) 1.57107e10 0.575215
\(960\) 3.93609e9i 0.143587i
\(961\) −1.59163e10 −0.578508
\(962\) 0 0
\(963\) −8.23057e9 −0.296987
\(964\) 7.31238e9i 0.262899i
\(965\) −5.88768e10 −2.10911
\(966\) −5.76579e9 −0.205797
\(967\) − 1.53418e10i − 0.545613i −0.962069 0.272806i \(-0.912048\pi\)
0.962069 0.272806i \(-0.0879518\pi\)
\(968\) − 4.96355e9i − 0.175885i
\(969\) 2.23917e10i 0.790595i
\(970\) − 6.18347e8i − 0.0217536i
\(971\) −5.22182e8 −0.0183043 −0.00915217 0.999958i \(-0.502913\pi\)
−0.00915217 + 0.999958i \(0.502913\pi\)
\(972\) 8.37817e9 0.292629
\(973\) 2.22324e9i 0.0773733i
\(974\) −1.53197e9 −0.0531245
\(975\) 0 0
\(976\) −4.43846e9 −0.152812
\(977\) 5.68857e10i 1.95152i 0.218851 + 0.975758i \(0.429769\pi\)
−0.218851 + 0.975758i \(0.570231\pi\)
\(978\) −9.79486e9 −0.334821
\(979\) −3.77345e10 −1.28528
\(980\) − 1.81768e10i − 0.616916i
\(981\) − 1.13599e10i − 0.384177i
\(982\) − 2.57857e9i − 0.0868936i
\(983\) 1.15736e10i 0.388624i 0.980940 + 0.194312i \(0.0622474\pi\)
−0.980940 + 0.194312i \(0.937753\pi\)
\(984\) 1.15814e9 0.0387507
\(985\) −3.66457e10 −1.22179
\(986\) − 2.05467e10i − 0.682612i
\(987\) −6.72492e9 −0.222626
\(988\) 0 0
\(989\) 1.27421e10 0.418846
\(990\) − 1.10810e10i − 0.362958i
\(991\) 1.38509e10 0.452086 0.226043 0.974117i \(-0.427421\pi\)
0.226043 + 0.974117i \(0.427421\pi\)
\(992\) −6.82872e9 −0.222100
\(993\) 1.88510e10i 0.610958i
\(994\) 4.88655e9i 0.157816i
\(995\) − 6.87554e10i − 2.21272i
\(996\) − 3.56279e8i − 0.0114257i
\(997\) 5.45316e9 0.174267 0.0871334 0.996197i \(-0.472229\pi\)
0.0871334 + 0.996197i \(0.472229\pi\)
\(998\) 3.09356e10 0.985147
\(999\) 4.92222e10i 1.56200i
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.b.b.337.1 2
13.5 odd 4 26.8.a.a.1.1 1
13.8 odd 4 338.8.a.c.1.1 1
13.12 even 2 inner 338.8.b.b.337.2 2
39.5 even 4 234.8.a.d.1.1 1
52.31 even 4 208.8.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
26.8.a.a.1.1 1 13.5 odd 4
208.8.a.c.1.1 1 52.31 even 4
234.8.a.d.1.1 1 39.5 even 4
338.8.a.c.1.1 1 13.8 odd 4
338.8.b.b.337.1 2 1.1 even 1 trivial
338.8.b.b.337.2 2 13.12 even 2 inner