Properties

Label 338.8.a.d.1.1
Level $338$
Weight $8$
Character 338.1
Self dual yes
Analytic conductor $105.586$
Analytic rank $1$
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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2)
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)
 
\(f(q)\) \(=\) \(q+8.00000 q^{2} +12.0000 q^{3} +64.0000 q^{4} +210.000 q^{5} +96.0000 q^{6} -1016.00 q^{7} +512.000 q^{8} -2043.00 q^{9} +O(q^{10})\) \(q+8.00000 q^{2} +12.0000 q^{3} +64.0000 q^{4} +210.000 q^{5} +96.0000 q^{6} -1016.00 q^{7} +512.000 q^{8} -2043.00 q^{9} +1680.00 q^{10} -1092.00 q^{11} +768.000 q^{12} -8128.00 q^{14} +2520.00 q^{15} +4096.00 q^{16} +14706.0 q^{17} -16344.0 q^{18} +39940.0 q^{19} +13440.0 q^{20} -12192.0 q^{21} -8736.00 q^{22} +68712.0 q^{23} +6144.00 q^{24} -34025.0 q^{25} -50760.0 q^{27} -65024.0 q^{28} -102570. q^{29} +20160.0 q^{30} -227552. q^{31} +32768.0 q^{32} -13104.0 q^{33} +117648. q^{34} -213360. q^{35} -130752. q^{36} -160526. q^{37} +319520. q^{38} +107520. q^{40} -10842.0 q^{41} -97536.0 q^{42} -630748. q^{43} -69888.0 q^{44} -429030. q^{45} +549696. q^{46} -472656. q^{47} +49152.0 q^{48} +208713. q^{49} -272200. q^{50} +176472. q^{51} -1.49402e6 q^{53} -406080. q^{54} -229320. q^{55} -520192. q^{56} +479280. q^{57} -820560. q^{58} -2.64066e6 q^{59} +161280. q^{60} +827702. q^{61} -1.82042e6 q^{62} +2.07569e6 q^{63} +262144. q^{64} -104832. q^{66} +126004. q^{67} +941184. q^{68} +824544. q^{69} -1.70688e6 q^{70} +1.41473e6 q^{71} -1.04602e6 q^{72} -980282. q^{73} -1.28421e6 q^{74} -408300. q^{75} +2.55616e6 q^{76} +1.10947e6 q^{77} -3.56680e6 q^{79} +860160. q^{80} +3.85892e6 q^{81} -86736.0 q^{82} -5.67289e6 q^{83} -780288. q^{84} +3.08826e6 q^{85} -5.04598e6 q^{86} -1.23084e6 q^{87} -559104. q^{88} +1.19512e7 q^{89} -3.43224e6 q^{90} +4.39757e6 q^{92} -2.73062e6 q^{93} -3.78125e6 q^{94} +8.38740e6 q^{95} +393216. q^{96} -8.68215e6 q^{97} +1.66970e6 q^{98} +2.23096e6 q^{99} +O(q^{100})\)

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
\(3\) 12.0000 0.256600 0.128300 0.991735i \(-0.459048\pi\)
0.128300 + 0.991735i \(0.459048\pi\)
\(4\) 64.0000 0.500000
\(5\) 210.000 0.751319 0.375659 0.926758i \(-0.377416\pi\)
0.375659 + 0.926758i \(0.377416\pi\)
\(6\) 96.0000 0.181444
\(7\) −1016.00 −1.11957 −0.559784 0.828638i \(-0.689116\pi\)
−0.559784 + 0.828638i \(0.689116\pi\)
\(8\) 512.000 0.353553
\(9\) −2043.00 −0.934156
\(10\) 1680.00 0.531263
\(11\) −1092.00 −0.247371 −0.123685 0.992321i \(-0.539471\pi\)
−0.123685 + 0.992321i \(0.539471\pi\)
\(12\) 768.000 0.128300
\(13\) 0 0
\(14\) −8128.00 −0.791654
\(15\) 2520.00 0.192789
\(16\) 4096.00 0.250000
\(17\) 14706.0 0.725978 0.362989 0.931793i \(-0.381756\pi\)
0.362989 + 0.931793i \(0.381756\pi\)
\(18\) −16344.0 −0.660548
\(19\) 39940.0 1.33589 0.667945 0.744211i \(-0.267174\pi\)
0.667945 + 0.744211i \(0.267174\pi\)
\(20\) 13440.0 0.375659
\(21\) −12192.0 −0.287281
\(22\) −8736.00 −0.174917
\(23\) 68712.0 1.17757 0.588783 0.808291i \(-0.299607\pi\)
0.588783 + 0.808291i \(0.299607\pi\)
\(24\) 6144.00 0.0907218
\(25\) −34025.0 −0.435520
\(26\) 0 0
\(27\) −50760.0 −0.496305
\(28\) −65024.0 −0.559784
\(29\) −102570. −0.780957 −0.390479 0.920612i \(-0.627690\pi\)
−0.390479 + 0.920612i \(0.627690\pi\)
\(30\) 20160.0 0.136322
\(31\) −227552. −1.37188 −0.685938 0.727660i \(-0.740608\pi\)
−0.685938 + 0.727660i \(0.740608\pi\)
\(32\) 32768.0 0.176777
\(33\) −13104.0 −0.0634753
\(34\) 117648. 0.513344
\(35\) −213360. −0.841153
\(36\) −130752. −0.467078
\(37\) −160526. −0.521002 −0.260501 0.965474i \(-0.583888\pi\)
−0.260501 + 0.965474i \(0.583888\pi\)
\(38\) 319520. 0.944616
\(39\) 0 0
\(40\) 107520. 0.265631
\(41\) −10842.0 −0.0245678 −0.0122839 0.999925i \(-0.503910\pi\)
−0.0122839 + 0.999925i \(0.503910\pi\)
\(42\) −97536.0 −0.203139
\(43\) −630748. −1.20981 −0.604904 0.796299i \(-0.706788\pi\)
−0.604904 + 0.796299i \(0.706788\pi\)
\(44\) −69888.0 −0.123685
\(45\) −429030. −0.701849
\(46\) 549696. 0.832665
\(47\) −472656. −0.664053 −0.332026 0.943270i \(-0.607732\pi\)
−0.332026 + 0.943270i \(0.607732\pi\)
\(48\) 49152.0 0.0641500
\(49\) 208713. 0.253433
\(50\) −272200. −0.307959
\(51\) 176472. 0.186286
\(52\) 0 0
\(53\) −1.49402e6 −1.37845 −0.689224 0.724548i \(-0.742048\pi\)
−0.689224 + 0.724548i \(0.742048\pi\)
\(54\) −406080. −0.350940
\(55\) −229320. −0.185854
\(56\) −520192. −0.395827
\(57\) 479280. 0.342789
\(58\) −820560. −0.552220
\(59\) −2.64066e6 −1.67390 −0.836952 0.547277i \(-0.815665\pi\)
−0.836952 + 0.547277i \(0.815665\pi\)
\(60\) 161280. 0.0963943
\(61\) 827702. 0.466895 0.233448 0.972369i \(-0.424999\pi\)
0.233448 + 0.972369i \(0.424999\pi\)
\(62\) −1.82042e6 −0.970063
\(63\) 2.07569e6 1.04585
\(64\) 262144. 0.125000
\(65\) 0 0
\(66\) −104832. −0.0448838
\(67\) 126004. 0.0511826 0.0255913 0.999672i \(-0.491853\pi\)
0.0255913 + 0.999672i \(0.491853\pi\)
\(68\) 941184. 0.362989
\(69\) 824544. 0.302164
\(70\) −1.70688e6 −0.594785
\(71\) 1.41473e6 0.469104 0.234552 0.972104i \(-0.424638\pi\)
0.234552 + 0.972104i \(0.424638\pi\)
\(72\) −1.04602e6 −0.330274
\(73\) −980282. −0.294931 −0.147466 0.989067i \(-0.547112\pi\)
−0.147466 + 0.989067i \(0.547112\pi\)
\(74\) −1.28421e6 −0.368404
\(75\) −408300. −0.111754
\(76\) 2.55616e6 0.667945
\(77\) 1.10947e6 0.276948
\(78\) 0 0
\(79\) −3.56680e6 −0.813924 −0.406962 0.913445i \(-0.633412\pi\)
−0.406962 + 0.913445i \(0.633412\pi\)
\(80\) 860160. 0.187830
\(81\) 3.85892e6 0.806805
\(82\) −86736.0 −0.0173720
\(83\) −5.67289e6 −1.08901 −0.544504 0.838758i \(-0.683282\pi\)
−0.544504 + 0.838758i \(0.683282\pi\)
\(84\) −780288. −0.143641
\(85\) 3.08826e6 0.545441
\(86\) −5.04598e6 −0.855463
\(87\) −1.23084e6 −0.200394
\(88\) −559104. −0.0874587
\(89\) 1.19512e7 1.79699 0.898496 0.438982i \(-0.144661\pi\)
0.898496 + 0.438982i \(0.144661\pi\)
\(90\) −3.43224e6 −0.496282
\(91\) 0 0
\(92\) 4.39757e6 0.588783
\(93\) −2.73062e6 −0.352023
\(94\) −3.78125e6 −0.469556
\(95\) 8.38740e6 1.00368
\(96\) 393216. 0.0453609
\(97\) −8.68215e6 −0.965886 −0.482943 0.875652i \(-0.660432\pi\)
−0.482943 + 0.875652i \(0.660432\pi\)
\(98\) 1.66970e6 0.179204
\(99\) 2.23096e6 0.231083
\(100\) −2.17760e6 −0.217760
\(101\) −1.00795e7 −0.973455 −0.486727 0.873554i \(-0.661810\pi\)
−0.486727 + 0.873554i \(0.661810\pi\)
\(102\) 1.41178e6 0.131724
\(103\) 3.74799e6 0.337962 0.168981 0.985619i \(-0.445952\pi\)
0.168981 + 0.985619i \(0.445952\pi\)
\(104\) 0 0
\(105\) −2.56032e6 −0.215840
\(106\) −1.19521e7 −0.974710
\(107\) −1.79856e7 −1.41932 −0.709661 0.704543i \(-0.751152\pi\)
−0.709661 + 0.704543i \(0.751152\pi\)
\(108\) −3.24864e6 −0.248152
\(109\) −1.22570e7 −0.906552 −0.453276 0.891370i \(-0.649745\pi\)
−0.453276 + 0.891370i \(0.649745\pi\)
\(110\) −1.83456e6 −0.131419
\(111\) −1.92631e6 −0.133689
\(112\) −4.16154e6 −0.279892
\(113\) 1.65950e7 1.08194 0.540968 0.841043i \(-0.318058\pi\)
0.540968 + 0.841043i \(0.318058\pi\)
\(114\) 3.83424e6 0.242389
\(115\) 1.44295e7 0.884727
\(116\) −6.56448e6 −0.390479
\(117\) 0 0
\(118\) −2.11253e7 −1.18363
\(119\) −1.49413e7 −0.812782
\(120\) 1.29024e6 0.0681610
\(121\) −1.82947e7 −0.938808
\(122\) 6.62162e6 0.330145
\(123\) −130104. −0.00630410
\(124\) −1.45633e7 −0.685938
\(125\) −2.35515e7 −1.07853
\(126\) 1.66055e7 0.739529
\(127\) 1.16826e6 0.0506087 0.0253043 0.999680i \(-0.491945\pi\)
0.0253043 + 0.999680i \(0.491945\pi\)
\(128\) 2.09715e6 0.0883883
\(129\) −7.56898e6 −0.310437
\(130\) 0 0
\(131\) −7.92383e6 −0.307954 −0.153977 0.988074i \(-0.549208\pi\)
−0.153977 + 0.988074i \(0.549208\pi\)
\(132\) −838656. −0.0317377
\(133\) −4.05790e7 −1.49562
\(134\) 1.00803e6 0.0361916
\(135\) −1.06596e7 −0.372883
\(136\) 7.52947e6 0.256672
\(137\) 315654. 0.0104879 0.00524396 0.999986i \(-0.498331\pi\)
0.00524396 + 0.999986i \(0.498331\pi\)
\(138\) 6.59635e6 0.213662
\(139\) 3.92038e7 1.23816 0.619079 0.785329i \(-0.287506\pi\)
0.619079 + 0.785329i \(0.287506\pi\)
\(140\) −1.36550e7 −0.420576
\(141\) −5.67187e6 −0.170396
\(142\) 1.13178e7 0.331706
\(143\) 0 0
\(144\) −8.36813e6 −0.233539
\(145\) −2.15397e7 −0.586748
\(146\) −7.84226e6 −0.208548
\(147\) 2.50456e6 0.0650309
\(148\) −1.02737e7 −0.260501
\(149\) 2.18860e7 0.542020 0.271010 0.962577i \(-0.412642\pi\)
0.271010 + 0.962577i \(0.412642\pi\)
\(150\) −3.26640e6 −0.0790224
\(151\) 2.94154e7 0.695274 0.347637 0.937629i \(-0.386984\pi\)
0.347637 + 0.937629i \(0.386984\pi\)
\(152\) 2.04493e7 0.472308
\(153\) −3.00444e7 −0.678177
\(154\) 8.87578e6 0.195832
\(155\) −4.77859e7 −1.03072
\(156\) 0 0
\(157\) 6.05550e7 1.24882 0.624412 0.781095i \(-0.285339\pi\)
0.624412 + 0.781095i \(0.285339\pi\)
\(158\) −2.85344e7 −0.575531
\(159\) −1.79282e7 −0.353710
\(160\) 6.88128e6 0.132816
\(161\) −6.98114e7 −1.31837
\(162\) 3.08714e7 0.570497
\(163\) −5.70853e7 −1.03245 −0.516223 0.856454i \(-0.672663\pi\)
−0.516223 + 0.856454i \(0.672663\pi\)
\(164\) −693888. −0.0122839
\(165\) −2.75184e6 −0.0476902
\(166\) −4.53831e7 −0.770045
\(167\) 8.77265e7 1.45755 0.728775 0.684754i \(-0.240090\pi\)
0.728775 + 0.684754i \(0.240090\pi\)
\(168\) −6.24230e6 −0.101569
\(169\) 0 0
\(170\) 2.47061e7 0.385685
\(171\) −8.15974e7 −1.24793
\(172\) −4.03679e7 −0.604904
\(173\) 8.56954e6 0.125833 0.0629167 0.998019i \(-0.479960\pi\)
0.0629167 + 0.998019i \(0.479960\pi\)
\(174\) −9.84672e6 −0.141700
\(175\) 3.45694e7 0.487594
\(176\) −4.47283e6 −0.0618427
\(177\) −3.16879e7 −0.429524
\(178\) 9.56095e7 1.27067
\(179\) 1.88041e7 0.245056 0.122528 0.992465i \(-0.460900\pi\)
0.122528 + 0.992465i \(0.460900\pi\)
\(180\) −2.74579e7 −0.350925
\(181\) −5.99625e7 −0.751631 −0.375816 0.926694i \(-0.622637\pi\)
−0.375816 + 0.926694i \(0.622637\pi\)
\(182\) 0 0
\(183\) 9.93242e6 0.119805
\(184\) 3.51805e7 0.416332
\(185\) −3.37105e7 −0.391439
\(186\) −2.18450e7 −0.248918
\(187\) −1.60590e7 −0.179586
\(188\) −3.02500e7 −0.332026
\(189\) 5.15722e7 0.555647
\(190\) 6.70992e7 0.709708
\(191\) 9.39861e7 0.975993 0.487997 0.872845i \(-0.337728\pi\)
0.487997 + 0.872845i \(0.337728\pi\)
\(192\) 3.14573e6 0.0320750
\(193\) 3.51946e7 0.352391 0.176196 0.984355i \(-0.443621\pi\)
0.176196 + 0.984355i \(0.443621\pi\)
\(194\) −6.94572e7 −0.682985
\(195\) 0 0
\(196\) 1.33576e7 0.126717
\(197\) −1.02985e8 −0.959718 −0.479859 0.877346i \(-0.659312\pi\)
−0.479859 + 0.877346i \(0.659312\pi\)
\(198\) 1.78476e7 0.163400
\(199\) 8.36376e7 0.752342 0.376171 0.926550i \(-0.377240\pi\)
0.376171 + 0.926550i \(0.377240\pi\)
\(200\) −1.74208e7 −0.153980
\(201\) 1.51205e6 0.0131335
\(202\) −8.06363e7 −0.688337
\(203\) 1.04211e8 0.874335
\(204\) 1.12942e7 0.0931430
\(205\) −2.27682e6 −0.0184582
\(206\) 2.99839e7 0.238975
\(207\) −1.40379e8 −1.10003
\(208\) 0 0
\(209\) −4.36145e7 −0.330460
\(210\) −2.04826e7 −0.152622
\(211\) −9.74010e7 −0.713797 −0.356899 0.934143i \(-0.616166\pi\)
−0.356899 + 0.934143i \(0.616166\pi\)
\(212\) −9.56172e7 −0.689224
\(213\) 1.69767e7 0.120372
\(214\) −1.43885e8 −1.00361
\(215\) −1.32457e8 −0.908951
\(216\) −2.59891e7 −0.175470
\(217\) 2.31193e8 1.53591
\(218\) −9.80562e7 −0.641029
\(219\) −1.17634e7 −0.0756794
\(220\) −1.46765e7 −0.0929271
\(221\) 0 0
\(222\) −1.54105e7 −0.0945325
\(223\) 1.46457e7 0.0884390 0.0442195 0.999022i \(-0.485920\pi\)
0.0442195 + 0.999022i \(0.485920\pi\)
\(224\) −3.32923e7 −0.197914
\(225\) 6.95131e7 0.406844
\(226\) 1.32760e8 0.765045
\(227\) 1.84541e8 1.04713 0.523567 0.851985i \(-0.324601\pi\)
0.523567 + 0.851985i \(0.324601\pi\)
\(228\) 3.06739e7 0.171395
\(229\) 8.75461e6 0.0481740 0.0240870 0.999710i \(-0.492332\pi\)
0.0240870 + 0.999710i \(0.492332\pi\)
\(230\) 1.15436e8 0.625597
\(231\) 1.33137e7 0.0710650
\(232\) −5.25158e7 −0.276110
\(233\) −1.19556e8 −0.619193 −0.309597 0.950868i \(-0.600194\pi\)
−0.309597 + 0.950868i \(0.600194\pi\)
\(234\) 0 0
\(235\) −9.92578e7 −0.498915
\(236\) −1.69002e8 −0.836952
\(237\) −4.28016e7 −0.208853
\(238\) −1.19530e8 −0.574723
\(239\) −3.96209e8 −1.87729 −0.938646 0.344883i \(-0.887919\pi\)
−0.938646 + 0.344883i \(0.887919\pi\)
\(240\) 1.03219e7 0.0481971
\(241\) 2.56606e8 1.18089 0.590443 0.807080i \(-0.298953\pi\)
0.590443 + 0.807080i \(0.298953\pi\)
\(242\) −1.46358e8 −0.663837
\(243\) 1.57319e8 0.703331
\(244\) 5.29729e7 0.233448
\(245\) 4.38297e7 0.190409
\(246\) −1.04083e6 −0.00445767
\(247\) 0 0
\(248\) −1.16507e8 −0.485031
\(249\) −6.80747e7 −0.279440
\(250\) −1.88412e8 −0.762638
\(251\) −7.34775e7 −0.293290 −0.146645 0.989189i \(-0.546847\pi\)
−0.146645 + 0.989189i \(0.546847\pi\)
\(252\) 1.32844e8 0.522926
\(253\) −7.50335e7 −0.291295
\(254\) 9.34605e6 0.0357857
\(255\) 3.70591e7 0.139960
\(256\) 1.67772e7 0.0625000
\(257\) −2.02701e8 −0.744886 −0.372443 0.928055i \(-0.621480\pi\)
−0.372443 + 0.928055i \(0.621480\pi\)
\(258\) −6.05518e7 −0.219512
\(259\) 1.63094e8 0.583297
\(260\) 0 0
\(261\) 2.09551e8 0.729536
\(262\) −6.33906e7 −0.217756
\(263\) 1.54254e8 0.522867 0.261434 0.965221i \(-0.415805\pi\)
0.261434 + 0.965221i \(0.415805\pi\)
\(264\) −6.70925e6 −0.0224419
\(265\) −3.13744e8 −1.03565
\(266\) −3.24632e8 −1.05756
\(267\) 1.43414e8 0.461108
\(268\) 8.06426e6 0.0255913
\(269\) −6.24018e8 −1.95463 −0.977315 0.211793i \(-0.932070\pi\)
−0.977315 + 0.211793i \(0.932070\pi\)
\(270\) −8.52768e7 −0.263668
\(271\) 3.87983e8 1.18419 0.592094 0.805869i \(-0.298302\pi\)
0.592094 + 0.805869i \(0.298302\pi\)
\(272\) 6.02358e7 0.181494
\(273\) 0 0
\(274\) 2.52523e6 0.00741608
\(275\) 3.71553e7 0.107735
\(276\) 5.27708e7 0.151082
\(277\) 4.53952e8 1.28331 0.641654 0.766994i \(-0.278248\pi\)
0.641654 + 0.766994i \(0.278248\pi\)
\(278\) 3.13630e8 0.875510
\(279\) 4.64889e8 1.28155
\(280\) −1.09240e8 −0.297392
\(281\) −3.33770e8 −0.897377 −0.448689 0.893688i \(-0.648109\pi\)
−0.448689 + 0.893688i \(0.648109\pi\)
\(282\) −4.53750e7 −0.120488
\(283\) 5.37695e8 1.41021 0.705104 0.709104i \(-0.250900\pi\)
0.705104 + 0.709104i \(0.250900\pi\)
\(284\) 9.05426e7 0.234552
\(285\) 1.00649e8 0.257544
\(286\) 0 0
\(287\) 1.10155e7 0.0275053
\(288\) −6.69450e7 −0.165137
\(289\) −1.94072e8 −0.472956
\(290\) −1.72318e8 −0.414894
\(291\) −1.04186e8 −0.247847
\(292\) −6.27380e7 −0.147466
\(293\) −3.35600e8 −0.779445 −0.389722 0.920932i \(-0.627429\pi\)
−0.389722 + 0.920932i \(0.627429\pi\)
\(294\) 2.00364e7 0.0459838
\(295\) −5.54539e8 −1.25764
\(296\) −8.21893e7 −0.184202
\(297\) 5.54299e7 0.122771
\(298\) 1.75088e8 0.383266
\(299\) 0 0
\(300\) −2.61312e7 −0.0558772
\(301\) 6.40840e8 1.35446
\(302\) 2.35324e8 0.491633
\(303\) −1.20954e8 −0.249789
\(304\) 1.63594e8 0.333972
\(305\) 1.73817e8 0.350787
\(306\) −2.40355e8 −0.479543
\(307\) −2.15029e8 −0.424143 −0.212072 0.977254i \(-0.568021\pi\)
−0.212072 + 0.977254i \(0.568021\pi\)
\(308\) 7.10062e7 0.138474
\(309\) 4.49759e7 0.0867212
\(310\) −3.82287e8 −0.728826
\(311\) 7.92062e8 1.49313 0.746565 0.665313i \(-0.231702\pi\)
0.746565 + 0.665313i \(0.231702\pi\)
\(312\) 0 0
\(313\) −1.18457e8 −0.218352 −0.109176 0.994022i \(-0.534821\pi\)
−0.109176 + 0.994022i \(0.534821\pi\)
\(314\) 4.84440e8 0.883051
\(315\) 4.35894e8 0.785768
\(316\) −2.28275e8 −0.406962
\(317\) 5.07310e7 0.0894470 0.0447235 0.998999i \(-0.485759\pi\)
0.0447235 + 0.998999i \(0.485759\pi\)
\(318\) −1.43426e8 −0.250111
\(319\) 1.12006e8 0.193186
\(320\) 5.50502e7 0.0939149
\(321\) −2.15827e8 −0.364198
\(322\) −5.58491e8 −0.932225
\(323\) 5.87358e8 0.969826
\(324\) 2.46971e8 0.403402
\(325\) 0 0
\(326\) −4.56682e8 −0.730050
\(327\) −1.47084e8 −0.232621
\(328\) −5.55110e6 −0.00868602
\(329\) 4.80218e8 0.743453
\(330\) −2.20147e7 −0.0337221
\(331\) −2.73757e8 −0.414923 −0.207461 0.978243i \(-0.566520\pi\)
−0.207461 + 0.978243i \(0.566520\pi\)
\(332\) −3.63065e8 −0.544504
\(333\) 3.27955e8 0.486697
\(334\) 7.01812e8 1.03064
\(335\) 2.64608e7 0.0384545
\(336\) −4.99384e7 −0.0718203
\(337\) −9.18512e7 −0.130732 −0.0653658 0.997861i \(-0.520821\pi\)
−0.0653658 + 0.997861i \(0.520821\pi\)
\(338\) 0 0
\(339\) 1.99140e8 0.277625
\(340\) 1.97649e8 0.272720
\(341\) 2.48487e8 0.339362
\(342\) −6.52779e8 −0.882419
\(343\) 6.24667e8 0.835833
\(344\) −3.22943e8 −0.427732
\(345\) 1.73154e8 0.227021
\(346\) 6.85563e7 0.0889777
\(347\) −1.36700e9 −1.75637 −0.878187 0.478318i \(-0.841247\pi\)
−0.878187 + 0.478318i \(0.841247\pi\)
\(348\) −7.87738e7 −0.100197
\(349\) −1.13143e9 −1.42475 −0.712377 0.701797i \(-0.752381\pi\)
−0.712377 + 0.701797i \(0.752381\pi\)
\(350\) 2.76555e8 0.344781
\(351\) 0 0
\(352\) −3.57827e7 −0.0437294
\(353\) 4.48395e7 0.0542562 0.0271281 0.999632i \(-0.491364\pi\)
0.0271281 + 0.999632i \(0.491364\pi\)
\(354\) −2.53503e8 −0.303719
\(355\) 2.97093e8 0.352446
\(356\) 7.64876e8 0.898496
\(357\) −1.79296e8 −0.208560
\(358\) 1.50432e8 0.173281
\(359\) −3.98281e8 −0.454317 −0.227158 0.973858i \(-0.572943\pi\)
−0.227158 + 0.973858i \(0.572943\pi\)
\(360\) −2.19663e8 −0.248141
\(361\) 7.01332e8 0.784600
\(362\) −4.79700e8 −0.531483
\(363\) −2.19536e8 −0.240898
\(364\) 0 0
\(365\) −2.05859e8 −0.221588
\(366\) 7.94594e7 0.0847152
\(367\) 1.63472e9 1.72628 0.863140 0.504964i \(-0.168494\pi\)
0.863140 + 0.504964i \(0.168494\pi\)
\(368\) 2.81444e8 0.294391
\(369\) 2.21502e7 0.0229501
\(370\) −2.69684e8 −0.276789
\(371\) 1.51792e9 1.54327
\(372\) −1.74760e8 −0.176012
\(373\) −1.54633e9 −1.54284 −0.771421 0.636325i \(-0.780454\pi\)
−0.771421 + 0.636325i \(0.780454\pi\)
\(374\) −1.28472e8 −0.126986
\(375\) −2.82618e8 −0.276752
\(376\) −2.42000e8 −0.234778
\(377\) 0 0
\(378\) 4.12577e8 0.392902
\(379\) 1.05688e9 0.997216 0.498608 0.866828i \(-0.333845\pi\)
0.498608 + 0.866828i \(0.333845\pi\)
\(380\) 5.36794e8 0.501839
\(381\) 1.40191e7 0.0129862
\(382\) 7.51889e8 0.690132
\(383\) −2.24910e8 −0.204556 −0.102278 0.994756i \(-0.532613\pi\)
−0.102278 + 0.994756i \(0.532613\pi\)
\(384\) 2.51658e7 0.0226805
\(385\) 2.32989e8 0.208077
\(386\) 2.81556e8 0.249178
\(387\) 1.28862e9 1.13015
\(388\) −5.55657e8 −0.482943
\(389\) 1.01788e9 0.876746 0.438373 0.898793i \(-0.355555\pi\)
0.438373 + 0.898793i \(0.355555\pi\)
\(390\) 0 0
\(391\) 1.01048e9 0.854887
\(392\) 1.06861e8 0.0896021
\(393\) −9.50859e7 −0.0790210
\(394\) −8.23883e8 −0.678623
\(395\) −7.49028e8 −0.611517
\(396\) 1.42781e8 0.115541
\(397\) 1.47565e9 1.18363 0.591817 0.806072i \(-0.298411\pi\)
0.591817 + 0.806072i \(0.298411\pi\)
\(398\) 6.69100e8 0.531986
\(399\) −4.86948e8 −0.383776
\(400\) −1.39366e8 −0.108880
\(401\) −2.74912e8 −0.212906 −0.106453 0.994318i \(-0.533949\pi\)
−0.106453 + 0.994318i \(0.533949\pi\)
\(402\) 1.20964e7 0.00928676
\(403\) 0 0
\(404\) −6.45090e8 −0.486727
\(405\) 8.10373e8 0.606167
\(406\) 8.33689e8 0.618248
\(407\) 1.75294e8 0.128881
\(408\) 9.03537e7 0.0658620
\(409\) 1.63427e9 1.18112 0.590558 0.806995i \(-0.298908\pi\)
0.590558 + 0.806995i \(0.298908\pi\)
\(410\) −1.82146e7 −0.0130519
\(411\) 3.78785e6 0.00269120
\(412\) 2.39871e8 0.168981
\(413\) 2.68291e9 1.87405
\(414\) −1.12303e9 −0.777839
\(415\) −1.19131e9 −0.818192
\(416\) 0 0
\(417\) 4.70445e8 0.317712
\(418\) −3.48916e8 −0.233670
\(419\) −1.11280e9 −0.739039 −0.369519 0.929223i \(-0.620478\pi\)
−0.369519 + 0.929223i \(0.620478\pi\)
\(420\) −1.63860e8 −0.107920
\(421\) −9.22528e8 −0.602549 −0.301274 0.953537i \(-0.597412\pi\)
−0.301274 + 0.953537i \(0.597412\pi\)
\(422\) −7.79208e8 −0.504731
\(423\) 9.65636e8 0.620329
\(424\) −7.64937e8 −0.487355
\(425\) −5.00372e8 −0.316178
\(426\) 1.35814e8 0.0851159
\(427\) −8.40945e8 −0.522721
\(428\) −1.15108e9 −0.709661
\(429\) 0 0
\(430\) −1.05966e9 −0.642726
\(431\) 9.81508e8 0.590505 0.295252 0.955419i \(-0.404596\pi\)
0.295252 + 0.955419i \(0.404596\pi\)
\(432\) −2.07913e8 −0.124076
\(433\) 2.84998e9 1.68707 0.843537 0.537071i \(-0.180469\pi\)
0.843537 + 0.537071i \(0.180469\pi\)
\(434\) 1.84954e9 1.08605
\(435\) −2.58476e8 −0.150560
\(436\) −7.84450e8 −0.453276
\(437\) 2.74436e9 1.57310
\(438\) −9.41071e7 −0.0535134
\(439\) −1.05622e9 −0.595838 −0.297919 0.954591i \(-0.596292\pi\)
−0.297919 + 0.954591i \(0.596292\pi\)
\(440\) −1.17412e8 −0.0657094
\(441\) −4.26401e8 −0.236746
\(442\) 0 0
\(443\) 1.82325e9 0.996401 0.498201 0.867062i \(-0.333994\pi\)
0.498201 + 0.867062i \(0.333994\pi\)
\(444\) −1.23284e8 −0.0668446
\(445\) 2.50975e9 1.35011
\(446\) 1.17166e8 0.0625358
\(447\) 2.62633e8 0.139082
\(448\) −2.66338e8 −0.139946
\(449\) −1.84846e9 −0.963713 −0.481856 0.876250i \(-0.660037\pi\)
−0.481856 + 0.876250i \(0.660037\pi\)
\(450\) 5.56105e8 0.287682
\(451\) 1.18395e7 0.00607735
\(452\) 1.06208e9 0.540968
\(453\) 3.52985e8 0.178407
\(454\) 1.47633e9 0.740435
\(455\) 0 0
\(456\) 2.45391e8 0.121194
\(457\) 2.98066e9 1.46085 0.730425 0.682993i \(-0.239322\pi\)
0.730425 + 0.682993i \(0.239322\pi\)
\(458\) 7.00369e7 0.0340642
\(459\) −7.46477e8 −0.360306
\(460\) 9.23489e8 0.442364
\(461\) 2.52781e9 1.20169 0.600843 0.799367i \(-0.294832\pi\)
0.600843 + 0.799367i \(0.294832\pi\)
\(462\) 1.06509e8 0.0502505
\(463\) 8.90291e8 0.416868 0.208434 0.978036i \(-0.433163\pi\)
0.208434 + 0.978036i \(0.433163\pi\)
\(464\) −4.20127e8 −0.195239
\(465\) −5.73431e8 −0.264482
\(466\) −9.56450e8 −0.437836
\(467\) 2.65667e9 1.20706 0.603529 0.797341i \(-0.293761\pi\)
0.603529 + 0.797341i \(0.293761\pi\)
\(468\) 0 0
\(469\) −1.28020e8 −0.0573024
\(470\) −7.94062e8 −0.352786
\(471\) 7.26660e8 0.320448
\(472\) −1.35202e9 −0.591814
\(473\) 6.88777e8 0.299271
\(474\) −3.42413e8 −0.147681
\(475\) −1.35896e9 −0.581806
\(476\) −9.56243e8 −0.406391
\(477\) 3.05228e9 1.28769
\(478\) −3.16967e9 −1.32745
\(479\) −1.30093e9 −0.540855 −0.270428 0.962740i \(-0.587165\pi\)
−0.270428 + 0.962740i \(0.587165\pi\)
\(480\) 8.25754e7 0.0340805
\(481\) 0 0
\(482\) 2.05285e9 0.835012
\(483\) −8.37737e8 −0.338293
\(484\) −1.17086e9 −0.469404
\(485\) −1.82325e9 −0.725689
\(486\) 1.25855e9 0.497330
\(487\) 1.07447e9 0.421542 0.210771 0.977535i \(-0.432402\pi\)
0.210771 + 0.977535i \(0.432402\pi\)
\(488\) 4.23783e8 0.165072
\(489\) −6.85024e8 −0.264926
\(490\) 3.50638e8 0.134640
\(491\) −7.83344e8 −0.298653 −0.149327 0.988788i \(-0.547711\pi\)
−0.149327 + 0.988788i \(0.547711\pi\)
\(492\) −8.32666e6 −0.00315205
\(493\) −1.50839e9 −0.566958
\(494\) 0 0
\(495\) 4.68501e8 0.173617
\(496\) −9.32053e8 −0.342969
\(497\) −1.43736e9 −0.525193
\(498\) −5.44598e8 −0.197594
\(499\) 6.23188e8 0.224526 0.112263 0.993679i \(-0.464190\pi\)
0.112263 + 0.993679i \(0.464190\pi\)
\(500\) −1.50730e9 −0.539267
\(501\) 1.05272e9 0.374007
\(502\) −5.87820e8 −0.207387
\(503\) −2.70927e9 −0.949215 −0.474607 0.880198i \(-0.657410\pi\)
−0.474607 + 0.880198i \(0.657410\pi\)
\(504\) 1.06275e9 0.369764
\(505\) −2.11670e9 −0.731375
\(506\) −6.00268e8 −0.205977
\(507\) 0 0
\(508\) 7.47684e7 0.0253043
\(509\) −3.49943e9 −1.17621 −0.588106 0.808784i \(-0.700126\pi\)
−0.588106 + 0.808784i \(0.700126\pi\)
\(510\) 2.96473e8 0.0989668
\(511\) 9.95967e8 0.330196
\(512\) 1.34218e8 0.0441942
\(513\) −2.02735e9 −0.663008
\(514\) −1.62161e9 −0.526714
\(515\) 7.87078e8 0.253918
\(516\) −4.84414e8 −0.155218
\(517\) 5.16140e8 0.164267
\(518\) 1.30476e9 0.412453
\(519\) 1.02835e8 0.0322889
\(520\) 0 0
\(521\) −1.37683e9 −0.426530 −0.213265 0.976994i \(-0.568410\pi\)
−0.213265 + 0.976994i \(0.568410\pi\)
\(522\) 1.67640e9 0.515860
\(523\) −2.86154e9 −0.874669 −0.437334 0.899299i \(-0.644077\pi\)
−0.437334 + 0.899299i \(0.644077\pi\)
\(524\) −5.07125e8 −0.153977
\(525\) 4.14833e8 0.125117
\(526\) 1.23403e9 0.369723
\(527\) −3.34638e9 −0.995951
\(528\) −5.36740e7 −0.0158688
\(529\) 1.31651e9 0.386661
\(530\) −2.50995e9 −0.732318
\(531\) 5.39487e9 1.56369
\(532\) −2.59706e9 −0.747810
\(533\) 0 0
\(534\) 1.14731e9 0.326053
\(535\) −3.77697e9 −1.06636
\(536\) 6.45140e7 0.0180958
\(537\) 2.25649e8 0.0628815
\(538\) −4.99215e9 −1.38213
\(539\) −2.27915e8 −0.0626919
\(540\) −6.82214e8 −0.186442
\(541\) −5.34467e9 −1.45121 −0.725605 0.688111i \(-0.758440\pi\)
−0.725605 + 0.688111i \(0.758440\pi\)
\(542\) 3.10387e9 0.837347
\(543\) −7.19550e8 −0.192869
\(544\) 4.81886e8 0.128336
\(545\) −2.57398e9 −0.681109
\(546\) 0 0
\(547\) −3.37135e9 −0.880740 −0.440370 0.897816i \(-0.645153\pi\)
−0.440370 + 0.897816i \(0.645153\pi\)
\(548\) 2.02019e7 0.00524396
\(549\) −1.69100e9 −0.436153
\(550\) 2.97242e8 0.0761801
\(551\) −4.09665e9 −1.04327
\(552\) 4.22167e8 0.106831
\(553\) 3.62387e9 0.911244
\(554\) 3.63162e9 0.907436
\(555\) −4.04526e8 −0.100443
\(556\) 2.50904e9 0.619079
\(557\) 5.61106e9 1.37579 0.687894 0.725811i \(-0.258535\pi\)
0.687894 + 0.725811i \(0.258535\pi\)
\(558\) 3.71911e9 0.906190
\(559\) 0 0
\(560\) −8.73923e8 −0.210288
\(561\) −1.92707e8 −0.0460817
\(562\) −2.67016e9 −0.634542
\(563\) 6.69690e9 1.58159 0.790795 0.612081i \(-0.209667\pi\)
0.790795 + 0.612081i \(0.209667\pi\)
\(564\) −3.63000e8 −0.0851980
\(565\) 3.48494e9 0.812879
\(566\) 4.30156e9 0.997168
\(567\) −3.92066e9 −0.903273
\(568\) 7.24341e8 0.165853
\(569\) 1.96850e9 0.447964 0.223982 0.974593i \(-0.428094\pi\)
0.223982 + 0.974593i \(0.428094\pi\)
\(570\) 8.05190e8 0.182111
\(571\) 1.02926e9 0.231365 0.115682 0.993286i \(-0.463094\pi\)
0.115682 + 0.993286i \(0.463094\pi\)
\(572\) 0 0
\(573\) 1.12783e9 0.250440
\(574\) 8.81238e7 0.0194492
\(575\) −2.33793e9 −0.512853
\(576\) −5.35560e8 −0.116770
\(577\) −3.31179e9 −0.717708 −0.358854 0.933394i \(-0.616832\pi\)
−0.358854 + 0.933394i \(0.616832\pi\)
\(578\) −1.55258e9 −0.334431
\(579\) 4.22335e8 0.0904236
\(580\) −1.37854e9 −0.293374
\(581\) 5.76366e9 1.21922
\(582\) −8.33486e8 −0.175254
\(583\) 1.63147e9 0.340988
\(584\) −5.01904e8 −0.104274
\(585\) 0 0
\(586\) −2.68480e9 −0.551151
\(587\) 5.59411e8 0.114156 0.0570778 0.998370i \(-0.481822\pi\)
0.0570778 + 0.998370i \(0.481822\pi\)
\(588\) 1.60292e8 0.0325155
\(589\) −9.08843e9 −1.83267
\(590\) −4.43631e9 −0.889282
\(591\) −1.23582e9 −0.246264
\(592\) −6.57514e8 −0.130250
\(593\) 3.02459e9 0.595628 0.297814 0.954624i \(-0.403742\pi\)
0.297814 + 0.954624i \(0.403742\pi\)
\(594\) 4.43439e8 0.0868124
\(595\) −3.13767e9 −0.610658
\(596\) 1.40071e9 0.271010
\(597\) 1.00365e9 0.193051
\(598\) 0 0
\(599\) −5.63246e9 −1.07079 −0.535395 0.844602i \(-0.679837\pi\)
−0.535395 + 0.844602i \(0.679837\pi\)
\(600\) −2.09050e8 −0.0395112
\(601\) 3.40792e8 0.0640366 0.0320183 0.999487i \(-0.489807\pi\)
0.0320183 + 0.999487i \(0.489807\pi\)
\(602\) 5.12672e9 0.957749
\(603\) −2.57426e8 −0.0478126
\(604\) 1.88259e9 0.347637
\(605\) −3.84189e9 −0.705344
\(606\) −9.67636e8 −0.176627
\(607\) 3.85420e9 0.699477 0.349739 0.936847i \(-0.386270\pi\)
0.349739 + 0.936847i \(0.386270\pi\)
\(608\) 1.30875e9 0.236154
\(609\) 1.25053e9 0.224355
\(610\) 1.39054e9 0.248044
\(611\) 0 0
\(612\) −1.92284e9 −0.339088
\(613\) −9.22245e9 −1.61709 −0.808545 0.588434i \(-0.799745\pi\)
−0.808545 + 0.588434i \(0.799745\pi\)
\(614\) −1.72023e9 −0.299915
\(615\) −2.73218e7 −0.00473639
\(616\) 5.68050e8 0.0979160
\(617\) −6.53611e9 −1.12027 −0.560133 0.828402i \(-0.689250\pi\)
−0.560133 + 0.828402i \(0.689250\pi\)
\(618\) 3.59807e8 0.0613211
\(619\) −1.36559e9 −0.231420 −0.115710 0.993283i \(-0.536914\pi\)
−0.115710 + 0.993283i \(0.536914\pi\)
\(620\) −3.05830e9 −0.515358
\(621\) −3.48782e9 −0.584431
\(622\) 6.33649e9 1.05580
\(623\) −1.21424e10 −2.01186
\(624\) 0 0
\(625\) −2.28761e9 −0.374802
\(626\) −9.47659e8 −0.154398
\(627\) −5.23374e8 −0.0847960
\(628\) 3.87552e9 0.624412
\(629\) −2.36070e9 −0.378236
\(630\) 3.48716e9 0.555622
\(631\) −1.54079e9 −0.244141 −0.122070 0.992521i \(-0.538953\pi\)
−0.122070 + 0.992521i \(0.538953\pi\)
\(632\) −1.82620e9 −0.287766
\(633\) −1.16881e9 −0.183160
\(634\) 4.05848e8 0.0632486
\(635\) 2.45334e8 0.0380233
\(636\) −1.14741e9 −0.176855
\(637\) 0 0
\(638\) 8.96052e8 0.136603
\(639\) −2.89029e9 −0.438216
\(640\) 4.40402e8 0.0664078
\(641\) −4.54018e9 −0.680879 −0.340440 0.940266i \(-0.610576\pi\)
−0.340440 + 0.940266i \(0.610576\pi\)
\(642\) −1.72661e9 −0.257527
\(643\) −1.14054e10 −1.69189 −0.845944 0.533272i \(-0.820962\pi\)
−0.845944 + 0.533272i \(0.820962\pi\)
\(644\) −4.46793e9 −0.659183
\(645\) −1.58948e9 −0.233237
\(646\) 4.69886e9 0.685770
\(647\) −1.26393e10 −1.83468 −0.917338 0.398109i \(-0.869666\pi\)
−0.917338 + 0.398109i \(0.869666\pi\)
\(648\) 1.97577e9 0.285248
\(649\) 2.88360e9 0.414075
\(650\) 0 0
\(651\) 2.77431e9 0.394114
\(652\) −3.65346e9 −0.516223
\(653\) −1.05004e10 −1.47575 −0.737873 0.674940i \(-0.764170\pi\)
−0.737873 + 0.674940i \(0.764170\pi\)
\(654\) −1.17667e9 −0.164488
\(655\) −1.66400e9 −0.231371
\(656\) −4.44088e7 −0.00614194
\(657\) 2.00272e9 0.275512
\(658\) 3.84175e9 0.525700
\(659\) 9.64818e9 1.31325 0.656624 0.754219i \(-0.271984\pi\)
0.656624 + 0.754219i \(0.271984\pi\)
\(660\) −1.76118e8 −0.0238451
\(661\) 6.58299e9 0.886580 0.443290 0.896378i \(-0.353811\pi\)
0.443290 + 0.896378i \(0.353811\pi\)
\(662\) −2.19006e9 −0.293395
\(663\) 0 0
\(664\) −2.90452e9 −0.385023
\(665\) −8.52160e9 −1.12369
\(666\) 2.62364e9 0.344147
\(667\) −7.04779e9 −0.919629
\(668\) 5.61450e9 0.728775
\(669\) 1.75749e8 0.0226935
\(670\) 2.11687e8 0.0271914
\(671\) −9.03851e8 −0.115496
\(672\) −3.99507e8 −0.0507846
\(673\) −8.54649e9 −1.08077 −0.540387 0.841416i \(-0.681722\pi\)
−0.540387 + 0.841416i \(0.681722\pi\)
\(674\) −7.34810e8 −0.0924411
\(675\) 1.72711e9 0.216151
\(676\) 0 0
\(677\) 8.71305e9 1.07922 0.539610 0.841915i \(-0.318572\pi\)
0.539610 + 0.841915i \(0.318572\pi\)
\(678\) 1.59312e9 0.196311
\(679\) 8.82106e9 1.08138
\(680\) 1.58119e9 0.192842
\(681\) 2.21449e9 0.268695
\(682\) 1.98789e9 0.239965
\(683\) −1.46109e10 −1.75470 −0.877351 0.479849i \(-0.840692\pi\)
−0.877351 + 0.479849i \(0.840692\pi\)
\(684\) −5.22223e9 −0.623965
\(685\) 6.62873e7 0.00787977
\(686\) 4.99734e9 0.591023
\(687\) 1.05055e8 0.0123615
\(688\) −2.58354e9 −0.302452
\(689\) 0 0
\(690\) 1.38523e9 0.160528
\(691\) 1.47348e10 1.69891 0.849454 0.527662i \(-0.176931\pi\)
0.849454 + 0.527662i \(0.176931\pi\)
\(692\) 5.48451e8 0.0629167
\(693\) −2.26665e9 −0.258713
\(694\) −1.09360e10 −1.24194
\(695\) 8.23279e9 0.930252
\(696\) −6.30190e8 −0.0708499
\(697\) −1.59442e8 −0.0178357
\(698\) −9.05146e9 −1.00745
\(699\) −1.43467e9 −0.158885
\(700\) 2.21244e9 0.243797
\(701\) 1.31502e9 0.144185 0.0720923 0.997398i \(-0.477032\pi\)
0.0720923 + 0.997398i \(0.477032\pi\)
\(702\) 0 0
\(703\) −6.41141e9 −0.696001
\(704\) −2.86261e8 −0.0309213
\(705\) −1.19109e9 −0.128022
\(706\) 3.58716e8 0.0383649
\(707\) 1.02408e10 1.08985
\(708\) −2.02803e9 −0.214762
\(709\) −6.64028e8 −0.0699721 −0.0349860 0.999388i \(-0.511139\pi\)
−0.0349860 + 0.999388i \(0.511139\pi\)
\(710\) 2.37674e9 0.249217
\(711\) 7.28697e9 0.760332
\(712\) 6.11901e9 0.635333
\(713\) −1.56356e10 −1.61547
\(714\) −1.43436e9 −0.147474
\(715\) 0 0
\(716\) 1.20346e9 0.122528
\(717\) −4.75451e9 −0.481713
\(718\) −3.18624e9 −0.321250
\(719\) 4.95034e9 0.496689 0.248344 0.968672i \(-0.420114\pi\)
0.248344 + 0.968672i \(0.420114\pi\)
\(720\) −1.75731e9 −0.175462
\(721\) −3.80796e9 −0.378372
\(722\) 5.61065e9 0.554796
\(723\) 3.07928e9 0.303015
\(724\) −3.83760e9 −0.375816
\(725\) 3.48994e9 0.340123
\(726\) −1.75629e9 −0.170341
\(727\) 8.81101e9 0.850463 0.425231 0.905085i \(-0.360193\pi\)
0.425231 + 0.905085i \(0.360193\pi\)
\(728\) 0 0
\(729\) −6.55163e9 −0.626330
\(730\) −1.64687e9 −0.156686
\(731\) −9.27578e9 −0.878293
\(732\) 6.35675e8 0.0599027
\(733\) 1.49414e8 0.0140129 0.00700643 0.999975i \(-0.497770\pi\)
0.00700643 + 0.999975i \(0.497770\pi\)
\(734\) 1.30777e10 1.22066
\(735\) 5.25957e8 0.0488590
\(736\) 2.25155e9 0.208166
\(737\) −1.37596e8 −0.0126611
\(738\) 1.77202e8 0.0162282
\(739\) 4.70806e9 0.429127 0.214564 0.976710i \(-0.431167\pi\)
0.214564 + 0.976710i \(0.431167\pi\)
\(740\) −2.15747e9 −0.195719
\(741\) 0 0
\(742\) 1.21434e10 1.09125
\(743\) −1.69676e9 −0.151761 −0.0758805 0.997117i \(-0.524177\pi\)
−0.0758805 + 0.997117i \(0.524177\pi\)
\(744\) −1.39808e9 −0.124459
\(745\) 4.59607e9 0.407230
\(746\) −1.23707e10 −1.09095
\(747\) 1.15897e10 1.01730
\(748\) −1.02777e9 −0.0897928
\(749\) 1.82733e10 1.58903
\(750\) −2.26094e9 −0.195693
\(751\) 1.06650e10 0.918800 0.459400 0.888229i \(-0.348064\pi\)
0.459400 + 0.888229i \(0.348064\pi\)
\(752\) −1.93600e9 −0.166013
\(753\) −8.81731e8 −0.0752581
\(754\) 0 0
\(755\) 6.17724e9 0.522373
\(756\) 3.30062e9 0.277824
\(757\) 6.22876e9 0.521874 0.260937 0.965356i \(-0.415968\pi\)
0.260937 + 0.965356i \(0.415968\pi\)
\(758\) 8.45506e9 0.705138
\(759\) −9.00402e8 −0.0747464
\(760\) 4.29435e9 0.354854
\(761\) 8.38334e9 0.689558 0.344779 0.938684i \(-0.387954\pi\)
0.344779 + 0.938684i \(0.387954\pi\)
\(762\) 1.12153e8 0.00918263
\(763\) 1.24531e10 1.01495
\(764\) 6.01511e9 0.487997
\(765\) −6.30932e9 −0.509527
\(766\) −1.79928e9 −0.144643
\(767\) 0 0
\(768\) 2.01327e8 0.0160375
\(769\) 1.18649e10 0.940852 0.470426 0.882439i \(-0.344100\pi\)
0.470426 + 0.882439i \(0.344100\pi\)
\(770\) 1.86391e9 0.147132
\(771\) −2.43241e9 −0.191138
\(772\) 2.25245e9 0.176196
\(773\) −5.56680e9 −0.433488 −0.216744 0.976228i \(-0.569544\pi\)
−0.216744 + 0.976228i \(0.569544\pi\)
\(774\) 1.03089e10 0.799136
\(775\) 7.74246e9 0.597479
\(776\) −4.44526e9 −0.341492
\(777\) 1.95713e9 0.149674
\(778\) 8.14306e9 0.619953
\(779\) −4.33029e8 −0.0328198
\(780\) 0 0
\(781\) −1.54488e9 −0.116042
\(782\) 8.08383e9 0.604496
\(783\) 5.20645e9 0.387593
\(784\) 8.54888e8 0.0633583
\(785\) 1.27165e10 0.938264
\(786\) −7.60687e8 −0.0558763
\(787\) −1.34611e8 −0.00984395 −0.00492198 0.999988i \(-0.501567\pi\)
−0.00492198 + 0.999988i \(0.501567\pi\)
\(788\) −6.59106e9 −0.479859
\(789\) 1.85105e9 0.134168
\(790\) −5.99222e9 −0.432408
\(791\) −1.68605e10 −1.21130
\(792\) 1.14225e9 0.0817001
\(793\) 0 0
\(794\) 1.18052e10 0.836955
\(795\) −3.76493e9 −0.265749
\(796\) 5.35280e9 0.376171
\(797\) −7.41548e9 −0.518842 −0.259421 0.965764i \(-0.583532\pi\)
−0.259421 + 0.965764i \(0.583532\pi\)
\(798\) −3.89559e9 −0.271371
\(799\) −6.95088e9 −0.482088
\(800\) −1.11493e9 −0.0769898
\(801\) −2.44163e10 −1.67867
\(802\) −2.19930e9 −0.150548
\(803\) 1.07047e9 0.0729574
\(804\) 9.67711e7 0.00656673
\(805\) −1.46604e10 −0.990513
\(806\) 0 0
\(807\) −7.48822e9 −0.501558
\(808\) −5.16072e9 −0.344168
\(809\) −1.41542e10 −0.939863 −0.469932 0.882703i \(-0.655721\pi\)
−0.469932 + 0.882703i \(0.655721\pi\)
\(810\) 6.48299e9 0.428625
\(811\) 2.63708e10 1.73600 0.868001 0.496563i \(-0.165405\pi\)
0.868001 + 0.496563i \(0.165405\pi\)
\(812\) 6.66951e9 0.437168
\(813\) 4.65580e9 0.303863
\(814\) 1.40236e9 0.0911324
\(815\) −1.19879e10 −0.775697
\(816\) 7.22829e8 0.0465715
\(817\) −2.51921e10 −1.61617
\(818\) 1.30742e10 0.835176
\(819\) 0 0
\(820\) −1.45716e8 −0.00922912
\(821\) −8.06264e9 −0.508483 −0.254241 0.967141i \(-0.581826\pi\)
−0.254241 + 0.967141i \(0.581826\pi\)
\(822\) 3.03028e7 0.00190297
\(823\) −2.34202e10 −1.46451 −0.732253 0.681033i \(-0.761531\pi\)
−0.732253 + 0.681033i \(0.761531\pi\)
\(824\) 1.91897e9 0.119488
\(825\) 4.45864e8 0.0276448
\(826\) 2.14633e10 1.32515
\(827\) −5.55722e9 −0.341655 −0.170828 0.985301i \(-0.554644\pi\)
−0.170828 + 0.985301i \(0.554644\pi\)
\(828\) −8.98423e9 −0.550015
\(829\) 2.84256e10 1.73288 0.866440 0.499281i \(-0.166403\pi\)
0.866440 + 0.499281i \(0.166403\pi\)
\(830\) −9.53046e9 −0.578549
\(831\) 5.44743e9 0.329297
\(832\) 0 0
\(833\) 3.06933e9 0.183987
\(834\) 3.76356e9 0.224656
\(835\) 1.84226e10 1.09508
\(836\) −2.79133e9 −0.165230
\(837\) 1.15505e10 0.680868
\(838\) −8.90238e9 −0.522579
\(839\) −1.04036e10 −0.608156 −0.304078 0.952647i \(-0.598348\pi\)
−0.304078 + 0.952647i \(0.598348\pi\)
\(840\) −1.31088e9 −0.0763109
\(841\) −6.72927e9 −0.390105
\(842\) −7.38023e9 −0.426066
\(843\) −4.00524e9 −0.230267
\(844\) −6.23367e9 −0.356899
\(845\) 0 0
\(846\) 7.72509e9 0.438639
\(847\) 1.85874e10 1.05106
\(848\) −6.11950e9 −0.344612
\(849\) 6.45234e9 0.361860
\(850\) −4.00297e9 −0.223572
\(851\) −1.10301e10 −0.613514
\(852\) 1.08651e9 0.0601860
\(853\) 1.80580e10 0.996205 0.498102 0.867118i \(-0.334030\pi\)
0.498102 + 0.867118i \(0.334030\pi\)
\(854\) −6.72756e9 −0.369620
\(855\) −1.71355e10 −0.937593
\(856\) −9.20861e9 −0.501806
\(857\) −6.34034e9 −0.344096 −0.172048 0.985089i \(-0.555038\pi\)
−0.172048 + 0.985089i \(0.555038\pi\)
\(858\) 0 0
\(859\) 1.21489e10 0.653973 0.326987 0.945029i \(-0.393967\pi\)
0.326987 + 0.945029i \(0.393967\pi\)
\(860\) −8.47725e9 −0.454476
\(861\) 1.32186e8 0.00705786
\(862\) 7.85206e9 0.417550
\(863\) 2.87111e10 1.52059 0.760295 0.649578i \(-0.225054\pi\)
0.760295 + 0.649578i \(0.225054\pi\)
\(864\) −1.66330e9 −0.0877351
\(865\) 1.79960e9 0.0945411
\(866\) 2.27998e10 1.19294
\(867\) −2.32887e9 −0.121361
\(868\) 1.47963e10 0.767954
\(869\) 3.89495e9 0.201341
\(870\) −2.06781e9 −0.106462
\(871\) 0 0
\(872\) −6.27560e9 −0.320514
\(873\) 1.77376e10 0.902289
\(874\) 2.19549e10 1.11235
\(875\) 2.39283e10 1.20749
\(876\) −7.52857e8 −0.0378397
\(877\) −2.46021e10 −1.23161 −0.615806 0.787898i \(-0.711169\pi\)
−0.615806 + 0.787898i \(0.711169\pi\)
\(878\) −8.44975e9 −0.421321
\(879\) −4.02720e9 −0.200006
\(880\) −9.39295e8 −0.0464636
\(881\) −1.25378e10 −0.617738 −0.308869 0.951105i \(-0.599951\pi\)
−0.308869 + 0.951105i \(0.599951\pi\)
\(882\) −3.41121e9 −0.167405
\(883\) 1.93097e10 0.943873 0.471937 0.881633i \(-0.343555\pi\)
0.471937 + 0.881633i \(0.343555\pi\)
\(884\) 0 0
\(885\) −6.65446e9 −0.322709
\(886\) 1.45860e10 0.704562
\(887\) 3.20268e10 1.54092 0.770462 0.637486i \(-0.220026\pi\)
0.770462 + 0.637486i \(0.220026\pi\)
\(888\) −9.86272e8 −0.0472663
\(889\) −1.18695e9 −0.0566599
\(890\) 2.00780e10 0.954675
\(891\) −4.21394e9 −0.199580
\(892\) 9.37327e8 0.0442195
\(893\) −1.88779e10 −0.887101
\(894\) 2.10106e9 0.0983461
\(895\) 3.94885e9 0.184115
\(896\) −2.13071e9 −0.0989568
\(897\) 0 0
\(898\) −1.47877e10 −0.681448
\(899\) 2.33400e10 1.07138
\(900\) 4.44884e9 0.203422
\(901\) −2.19710e10 −1.00072
\(902\) 9.47157e7 0.00429733
\(903\) 7.69008e9 0.347555
\(904\) 8.49662e9 0.382522
\(905\) −1.25921e10 −0.564715
\(906\) 2.82388e9 0.126153
\(907\) 2.33703e9 0.104002 0.0520008 0.998647i \(-0.483440\pi\)
0.0520008 + 0.998647i \(0.483440\pi\)
\(908\) 1.18106e10 0.523567
\(909\) 2.05925e10 0.909359
\(910\) 0 0
\(911\) 2.20343e10 0.965573 0.482786 0.875738i \(-0.339625\pi\)
0.482786 + 0.875738i \(0.339625\pi\)
\(912\) 1.96313e9 0.0856973
\(913\) 6.19480e9 0.269389
\(914\) 2.38453e10 1.03298
\(915\) 2.08581e9 0.0900121
\(916\) 5.60295e8 0.0240870
\(917\) 8.05061e9 0.344775
\(918\) −5.97181e9 −0.254775
\(919\) −1.43277e10 −0.608938 −0.304469 0.952522i \(-0.598479\pi\)
−0.304469 + 0.952522i \(0.598479\pi\)
\(920\) 7.38791e9 0.312798
\(921\) −2.58035e9 −0.108835
\(922\) 2.02225e10 0.849720
\(923\) 0 0
\(924\) 8.52074e8 0.0355325
\(925\) 5.46190e9 0.226907
\(926\) 7.12233e9 0.294770
\(927\) −7.65715e9 −0.315710
\(928\) −3.36101e9 −0.138055
\(929\) −1.31280e10 −0.537208 −0.268604 0.963251i \(-0.586562\pi\)
−0.268604 + 0.963251i \(0.586562\pi\)
\(930\) −4.58745e9 −0.187017
\(931\) 8.33600e9 0.338558
\(932\) −7.65160e9 −0.309597
\(933\) 9.50474e9 0.383137
\(934\) 2.12533e10 0.853519
\(935\) −3.37238e9 −0.134926
\(936\) 0 0
\(937\) −3.87626e10 −1.53930 −0.769652 0.638463i \(-0.779571\pi\)
−0.769652 + 0.638463i \(0.779571\pi\)
\(938\) −1.02416e9 −0.0405189
\(939\) −1.42149e9 −0.0560291
\(940\) −6.35250e9 −0.249458
\(941\) −2.06279e10 −0.807035 −0.403517 0.914972i \(-0.632212\pi\)
−0.403517 + 0.914972i \(0.632212\pi\)
\(942\) 5.81328e9 0.226591
\(943\) −7.44976e8 −0.0289302
\(944\) −1.08161e10 −0.418476
\(945\) 1.08302e10 0.417468
\(946\) 5.51021e9 0.211617
\(947\) 2.11705e10 0.810040 0.405020 0.914308i \(-0.367264\pi\)
0.405020 + 0.914308i \(0.367264\pi\)
\(948\) −2.73930e9 −0.104427
\(949\) 0 0
\(950\) −1.08717e10 −0.411399
\(951\) 6.08771e8 0.0229521
\(952\) −7.64994e9 −0.287362
\(953\) 2.14876e10 0.804196 0.402098 0.915597i \(-0.368281\pi\)
0.402098 + 0.915597i \(0.368281\pi\)
\(954\) 2.44182e10 0.910531
\(955\) 1.97371e10 0.733282
\(956\) −2.53574e10 −0.938646
\(957\) 1.34408e9 0.0495715
\(958\) −1.04075e10 −0.382442
\(959\) −3.20704e8 −0.0117419
\(960\) 6.60603e8 0.0240986
\(961\) 2.42673e10 0.882043
\(962\) 0 0
\(963\) 3.67445e10 1.32587
\(964\) 1.64228e10 0.590443
\(965\) 7.39086e9 0.264758
\(966\) −6.70189e9 −0.239209
\(967\) −3.92625e10 −1.39632 −0.698161 0.715941i \(-0.745998\pi\)
−0.698161 + 0.715941i \(0.745998\pi\)
\(968\) −9.36689e9 −0.331919
\(969\) 7.04829e9 0.248857
\(970\) −1.45860e10 −0.513139
\(971\) −5.62647e10 −1.97228 −0.986140 0.165917i \(-0.946941\pi\)
−0.986140 + 0.165917i \(0.946941\pi\)
\(972\) 1.00684e10 0.351665
\(973\) −3.98310e10 −1.38620
\(974\) 8.59573e9 0.298076
\(975\) 0 0
\(976\) 3.39027e9 0.116724
\(977\) 8.43437e9 0.289349 0.144674 0.989479i \(-0.453787\pi\)
0.144674 + 0.989479i \(0.453787\pi\)
\(978\) −5.48019e9 −0.187331
\(979\) −1.30507e10 −0.444523
\(980\) 2.80510e9 0.0952045
\(981\) 2.50411e10 0.846861
\(982\) −6.26675e9 −0.211180
\(983\) 2.24230e10 0.752932 0.376466 0.926430i \(-0.377139\pi\)
0.376466 + 0.926430i \(0.377139\pi\)
\(984\) −6.66132e7 −0.00222883
\(985\) −2.16269e10 −0.721054
\(986\) −1.20672e10 −0.400900
\(987\) 5.76262e9 0.190770
\(988\) 0 0
\(989\) −4.33400e10 −1.42463
\(990\) 3.74801e9 0.122766
\(991\) 3.46728e10 1.13170 0.565849 0.824509i \(-0.308548\pi\)
0.565849 + 0.824509i \(0.308548\pi\)
\(992\) −7.45642e9 −0.242516
\(993\) −3.28508e9 −0.106469
\(994\) −1.14989e10 −0.371368
\(995\) 1.75639e10 0.565249
\(996\) −4.35678e9 −0.139720
\(997\) −2.96474e10 −0.947444 −0.473722 0.880674i \(-0.657090\pi\)
−0.473722 + 0.880674i \(0.657090\pi\)
\(998\) 4.98550e9 0.158764
\(999\) 8.14830e9 0.258576
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.d.1.1 1
13.5 odd 4 338.8.b.d.337.1 2
13.8 odd 4 338.8.b.d.337.2 2
13.12 even 2 2.8.a.a.1.1 1
39.38 odd 2 18.8.a.b.1.1 1
52.51 odd 2 16.8.a.b.1.1 1
65.12 odd 4 50.8.b.c.49.1 2
65.38 odd 4 50.8.b.c.49.2 2
65.64 even 2 50.8.a.g.1.1 1
91.12 odd 6 98.8.c.e.67.1 2
91.25 even 6 98.8.c.d.79.1 2
91.38 odd 6 98.8.c.e.79.1 2
91.51 even 6 98.8.c.d.67.1 2
91.90 odd 2 98.8.a.a.1.1 1
104.51 odd 2 64.8.a.e.1.1 1
104.77 even 2 64.8.a.c.1.1 1
117.25 even 6 162.8.c.l.109.1 2
117.38 odd 6 162.8.c.a.109.1 2
117.77 odd 6 162.8.c.a.55.1 2
117.103 even 6 162.8.c.l.55.1 2
143.142 odd 2 242.8.a.e.1.1 1
156.155 even 2 144.8.a.i.1.1 1
195.38 even 4 450.8.c.g.199.1 2
195.77 even 4 450.8.c.g.199.2 2
195.194 odd 2 450.8.a.c.1.1 1
208.51 odd 4 256.8.b.f.129.1 2
208.77 even 4 256.8.b.b.129.2 2
208.155 odd 4 256.8.b.f.129.2 2
208.181 even 4 256.8.b.b.129.1 2
221.220 even 2 578.8.a.b.1.1 1
260.103 even 4 400.8.c.j.49.1 2
260.207 even 4 400.8.c.j.49.2 2
260.259 odd 2 400.8.a.l.1.1 1
312.77 odd 2 576.8.a.g.1.1 1
312.155 even 2 576.8.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.8.a.a.1.1 1 13.12 even 2
16.8.a.b.1.1 1 52.51 odd 2
18.8.a.b.1.1 1 39.38 odd 2
50.8.a.g.1.1 1 65.64 even 2
50.8.b.c.49.1 2 65.12 odd 4
50.8.b.c.49.2 2 65.38 odd 4
64.8.a.c.1.1 1 104.77 even 2
64.8.a.e.1.1 1 104.51 odd 2
98.8.a.a.1.1 1 91.90 odd 2
98.8.c.d.67.1 2 91.51 even 6
98.8.c.d.79.1 2 91.25 even 6
98.8.c.e.67.1 2 91.12 odd 6
98.8.c.e.79.1 2 91.38 odd 6
144.8.a.i.1.1 1 156.155 even 2
162.8.c.a.55.1 2 117.77 odd 6
162.8.c.a.109.1 2 117.38 odd 6
162.8.c.l.55.1 2 117.103 even 6
162.8.c.l.109.1 2 117.25 even 6
242.8.a.e.1.1 1 143.142 odd 2
256.8.b.b.129.1 2 208.181 even 4
256.8.b.b.129.2 2 208.77 even 4
256.8.b.f.129.1 2 208.51 odd 4
256.8.b.f.129.2 2 208.155 odd 4
338.8.a.d.1.1 1 1.1 even 1 trivial
338.8.b.d.337.1 2 13.5 odd 4
338.8.b.d.337.2 2 13.8 odd 4
400.8.a.l.1.1 1 260.259 odd 2
400.8.c.j.49.1 2 260.103 even 4
400.8.c.j.49.2 2 260.207 even 4
450.8.a.c.1.1 1 195.194 odd 2
450.8.c.g.199.1 2 195.38 even 4
450.8.c.g.199.2 2 195.77 even 4
576.8.a.f.1.1 1 312.155 even 2
576.8.a.g.1.1 1 312.77 odd 2
578.8.a.b.1.1 1 221.220 even 2