Properties

Label 10.38.a.a.1.2
Level $10$
Weight $38$
Character 10.1
Self dual yes
Analytic conductor $86.714$
Analytic rank $1$
Dimension $2$
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] = [10,38,Mod(1,10)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 38, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.1");
 
S:= CuspForms(chi, 38);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 38 \)
Character orbit: \([\chi]\) \(=\) 10.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: \(86.7140381246\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1222518952080 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3\cdot 5 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.10568e6\) of defining polynomial
Character \(\chi\) \(=\) 10.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+262144. q^{2} +5.04314e8 q^{3} +6.87195e10 q^{4} +3.81470e12 q^{5} +1.32203e14 q^{6} -3.94775e15 q^{7} +1.80144e16 q^{8} -1.95951e17 q^{9} +1.00000e18 q^{10} -2.18005e18 q^{11} +3.46562e19 q^{12} +1.81624e20 q^{13} -1.03488e21 q^{14} +1.92381e21 q^{15} +4.72237e21 q^{16} -2.91883e22 q^{17} -5.13673e22 q^{18} -4.63426e23 q^{19} +2.62144e23 q^{20} -1.99091e24 q^{21} -5.71488e23 q^{22} -5.39070e24 q^{23} +9.08492e24 q^{24} +1.45519e25 q^{25} +4.76117e25 q^{26} -3.25906e26 q^{27} -2.71287e26 q^{28} -1.38938e27 q^{29} +5.04314e26 q^{30} +1.08576e27 q^{31} +1.23794e27 q^{32} -1.09943e27 q^{33} -7.65153e27 q^{34} -1.50595e28 q^{35} -1.34656e28 q^{36} -8.17276e28 q^{37} -1.21484e29 q^{38} +9.15957e28 q^{39} +6.87195e28 q^{40} +4.55702e29 q^{41} -5.21904e29 q^{42} -6.17149e29 q^{43} -1.49812e29 q^{44} -7.47493e29 q^{45} -1.41314e30 q^{46} -1.66780e30 q^{47} +2.38156e30 q^{48} -2.97740e30 q^{49} +3.81470e30 q^{50} -1.47201e31 q^{51} +1.24811e31 q^{52} -5.11828e31 q^{53} -8.54342e31 q^{54} -8.31624e30 q^{55} -7.11163e31 q^{56} -2.33712e32 q^{57} -3.64219e32 q^{58} -4.79981e32 q^{59} +1.32203e32 q^{60} +1.48774e33 q^{61} +2.84626e32 q^{62} +7.73564e32 q^{63} +3.24519e32 q^{64} +6.92841e32 q^{65} -2.88210e32 q^{66} +1.20039e34 q^{67} -2.00580e33 q^{68} -2.71861e33 q^{69} -3.94775e33 q^{70} -1.19359e32 q^{71} -3.52994e33 q^{72} -4.41041e33 q^{73} -2.14244e34 q^{74} +7.33874e33 q^{75} -3.18464e34 q^{76} +8.60630e33 q^{77} +2.40113e34 q^{78} -1.73547e35 q^{79} +1.80144e34 q^{80} -7.61254e34 q^{81} +1.19460e35 q^{82} -1.57921e35 q^{83} -1.36814e35 q^{84} -1.11344e35 q^{85} -1.61782e35 q^{86} -7.00687e35 q^{87} -3.92723e34 q^{88} -9.73344e35 q^{89} -1.95951e35 q^{90} -7.17007e35 q^{91} -3.70446e35 q^{92} +5.47565e35 q^{93} -4.37205e35 q^{94} -1.76783e36 q^{95} +6.24311e35 q^{96} -8.76678e36 q^{97} -7.80509e35 q^{98} +4.27183e35 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 524288 q^{2} - 185500908 q^{3} + 137438953472 q^{4} + 7629394531250 q^{5} - 48627950026752 q^{6} - 651580251842156 q^{7} + 36\!\cdots\!68 q^{8} - 17\!\cdots\!94 q^{9} + 20\!\cdots\!00 q^{10} - 15\!\cdots\!96 q^{11}+ \cdots + 78\!\cdots\!12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 262144. 0.707107
\(3\) 5.04314e8 0.751551 0.375775 0.926711i \(-0.377376\pi\)
0.375775 + 0.926711i \(0.377376\pi\)
\(4\) 6.87195e10 0.500000
\(5\) 3.81470e12 0.447214
\(6\) 1.32203e14 0.531427
\(7\) −3.94775e15 −0.916296 −0.458148 0.888876i \(-0.651487\pi\)
−0.458148 + 0.888876i \(0.651487\pi\)
\(8\) 1.80144e16 0.353553
\(9\) −1.95951e17 −0.435172
\(10\) 1.00000e18 0.316228
\(11\) −2.18005e18 −0.118223 −0.0591115 0.998251i \(-0.518827\pi\)
−0.0591115 + 0.998251i \(0.518827\pi\)
\(12\) 3.46562e19 0.375775
\(13\) 1.81624e20 0.447942 0.223971 0.974596i \(-0.428098\pi\)
0.223971 + 0.974596i \(0.428098\pi\)
\(14\) −1.03488e21 −0.647919
\(15\) 1.92381e21 0.336104
\(16\) 4.72237e21 0.250000
\(17\) −2.91883e22 −0.503388 −0.251694 0.967807i \(-0.580988\pi\)
−0.251694 + 0.967807i \(0.580988\pi\)
\(18\) −5.13673e22 −0.307713
\(19\) −4.63426e23 −1.02103 −0.510515 0.859869i \(-0.670545\pi\)
−0.510515 + 0.859869i \(0.670545\pi\)
\(20\) 2.62144e23 0.223607
\(21\) −1.99091e24 −0.688643
\(22\) −5.71488e23 −0.0835964
\(23\) −5.39070e24 −0.346481 −0.173241 0.984880i \(-0.555424\pi\)
−0.173241 + 0.984880i \(0.555424\pi\)
\(24\) 9.08492e24 0.265713
\(25\) 1.45519e25 0.200000
\(26\) 4.76117e25 0.316743
\(27\) −3.25906e26 −1.07860
\(28\) −2.71287e26 −0.458148
\(29\) −1.38938e27 −1.22591 −0.612956 0.790117i \(-0.710020\pi\)
−0.612956 + 0.790117i \(0.710020\pi\)
\(30\) 5.04314e26 0.237661
\(31\) 1.08576e27 0.278961 0.139480 0.990225i \(-0.455457\pi\)
0.139480 + 0.990225i \(0.455457\pi\)
\(32\) 1.23794e27 0.176777
\(33\) −1.09943e27 −0.0888506
\(34\) −7.65153e27 −0.355949
\(35\) −1.50595e28 −0.409780
\(36\) −1.34656e28 −0.217586
\(37\) −8.17276e28 −0.795494 −0.397747 0.917495i \(-0.630208\pi\)
−0.397747 + 0.917495i \(0.630208\pi\)
\(38\) −1.21484e29 −0.721977
\(39\) 9.15957e28 0.336651
\(40\) 6.87195e28 0.158114
\(41\) 4.55702e29 0.664018 0.332009 0.943276i \(-0.392274\pi\)
0.332009 + 0.943276i \(0.392274\pi\)
\(42\) −5.21904e29 −0.486944
\(43\) −6.17149e29 −0.372584 −0.186292 0.982494i \(-0.559647\pi\)
−0.186292 + 0.982494i \(0.559647\pi\)
\(44\) −1.49812e29 −0.0591115
\(45\) −7.47493e29 −0.194615
\(46\) −1.41314e30 −0.244999
\(47\) −1.66780e30 −0.194238 −0.0971191 0.995273i \(-0.530963\pi\)
−0.0971191 + 0.995273i \(0.530963\pi\)
\(48\) 2.38156e30 0.187888
\(49\) −2.97740e30 −0.160402
\(50\) 3.81470e30 0.141421
\(51\) −1.47201e31 −0.378322
\(52\) 1.24811e31 0.223971
\(53\) −5.11828e31 −0.645684 −0.322842 0.946453i \(-0.604638\pi\)
−0.322842 + 0.946453i \(0.604638\pi\)
\(54\) −8.54342e31 −0.762688
\(55\) −8.31624e30 −0.0528710
\(56\) −7.11163e31 −0.323959
\(57\) −2.33712e32 −0.767355
\(58\) −3.64219e32 −0.866851
\(59\) −4.79981e32 −0.832649 −0.416325 0.909216i \(-0.636682\pi\)
−0.416325 + 0.909216i \(0.636682\pi\)
\(60\) 1.32203e32 0.168052
\(61\) 1.48774e33 1.39292 0.696460 0.717595i \(-0.254757\pi\)
0.696460 + 0.717595i \(0.254757\pi\)
\(62\) 2.84626e32 0.197255
\(63\) 7.73564e32 0.398746
\(64\) 3.24519e32 0.125000
\(65\) 6.92841e32 0.200326
\(66\) −2.88210e32 −0.0628269
\(67\) 1.20039e34 1.98125 0.990627 0.136596i \(-0.0436162\pi\)
0.990627 + 0.136596i \(0.0436162\pi\)
\(68\) −2.00580e33 −0.251694
\(69\) −2.71861e33 −0.260398
\(70\) −3.94775e33 −0.289758
\(71\) −1.19359e32 −0.00673869 −0.00336934 0.999994i \(-0.501072\pi\)
−0.00336934 + 0.999994i \(0.501072\pi\)
\(72\) −3.52994e33 −0.153856
\(73\) −4.41041e33 −0.148938 −0.0744691 0.997223i \(-0.523726\pi\)
−0.0744691 + 0.997223i \(0.523726\pi\)
\(74\) −2.14244e34 −0.562499
\(75\) 7.33874e33 0.150310
\(76\) −3.18464e34 −0.510515
\(77\) 8.60630e33 0.108327
\(78\) 2.40113e34 0.238048
\(79\) −1.73547e35 −1.35930 −0.679651 0.733536i \(-0.737869\pi\)
−0.679651 + 0.733536i \(0.737869\pi\)
\(80\) 1.80144e34 0.111803
\(81\) −7.61254e34 −0.375454
\(82\) 1.19460e35 0.469532
\(83\) −1.57921e35 −0.496015 −0.248008 0.968758i \(-0.579776\pi\)
−0.248008 + 0.968758i \(0.579776\pi\)
\(84\) −1.36814e35 −0.344321
\(85\) −1.11344e35 −0.225122
\(86\) −1.61782e35 −0.263456
\(87\) −7.00687e35 −0.921335
\(88\) −3.92723e34 −0.0417982
\(89\) −9.73344e35 −0.840526 −0.420263 0.907402i \(-0.638062\pi\)
−0.420263 + 0.907402i \(0.638062\pi\)
\(90\) −1.95951e35 −0.137613
\(91\) −7.17007e35 −0.410447
\(92\) −3.70446e35 −0.173241
\(93\) 5.47565e35 0.209653
\(94\) −4.37205e35 −0.137347
\(95\) −1.76783e36 −0.456618
\(96\) 6.24311e35 0.132857
\(97\) −8.76678e36 −1.54015 −0.770075 0.637954i \(-0.779781\pi\)
−0.770075 + 0.637954i \(0.779781\pi\)
\(98\) −7.80509e35 −0.113421
\(99\) 4.27183e35 0.0514473
\(100\) 1.00000e36 0.100000
\(101\) −9.95992e36 −0.828534 −0.414267 0.910155i \(-0.635962\pi\)
−0.414267 + 0.910155i \(0.635962\pi\)
\(102\) −3.85878e36 −0.267514
\(103\) −1.38183e37 −0.799773 −0.399887 0.916565i \(-0.630951\pi\)
−0.399887 + 0.916565i \(0.630951\pi\)
\(104\) 3.27185e36 0.158371
\(105\) −7.59470e36 −0.307970
\(106\) −1.34173e37 −0.456568
\(107\) 1.66394e37 0.475924 0.237962 0.971275i \(-0.423521\pi\)
0.237962 + 0.971275i \(0.423521\pi\)
\(108\) −2.23961e37 −0.539302
\(109\) −2.15440e37 −0.437458 −0.218729 0.975786i \(-0.570191\pi\)
−0.218729 + 0.975786i \(0.570191\pi\)
\(110\) −2.18005e36 −0.0373854
\(111\) −4.12164e37 −0.597854
\(112\) −1.86427e37 −0.229074
\(113\) −7.20846e37 −0.751435 −0.375717 0.926734i \(-0.622604\pi\)
−0.375717 + 0.926734i \(0.622604\pi\)
\(114\) −6.12663e37 −0.542602
\(115\) −2.05639e37 −0.154951
\(116\) −9.54778e37 −0.612956
\(117\) −3.55894e37 −0.194932
\(118\) −1.25824e38 −0.588772
\(119\) 1.15228e38 0.461253
\(120\) 3.46562e37 0.118831
\(121\) −3.35287e38 −0.986023
\(122\) 3.90003e38 0.984944
\(123\) 2.29817e38 0.499043
\(124\) 7.46129e37 0.139480
\(125\) 5.55112e37 0.0894427
\(126\) 2.02785e38 0.281956
\(127\) 5.59928e38 0.672610 0.336305 0.941753i \(-0.390823\pi\)
0.336305 + 0.941753i \(0.390823\pi\)
\(128\) 8.50706e37 0.0883883
\(129\) −3.11237e38 −0.280016
\(130\) 1.81624e38 0.141652
\(131\) 2.41935e39 1.63750 0.818748 0.574154i \(-0.194669\pi\)
0.818748 + 0.574154i \(0.194669\pi\)
\(132\) −7.55524e37 −0.0444253
\(133\) 1.82949e39 0.935565
\(134\) 3.14676e39 1.40096
\(135\) −1.24323e39 −0.482366
\(136\) −5.25809e38 −0.177975
\(137\) 4.22583e39 1.24905 0.624527 0.781003i \(-0.285292\pi\)
0.624527 + 0.781003i \(0.285292\pi\)
\(138\) −7.12666e38 −0.184129
\(139\) 1.36024e39 0.307497 0.153748 0.988110i \(-0.450866\pi\)
0.153748 + 0.988110i \(0.450866\pi\)
\(140\) −1.03488e39 −0.204890
\(141\) −8.41097e38 −0.145980
\(142\) −3.12892e37 −0.00476497
\(143\) −3.95950e38 −0.0529571
\(144\) −9.25352e38 −0.108793
\(145\) −5.30008e39 −0.548245
\(146\) −1.15616e39 −0.105315
\(147\) −1.50155e39 −0.120550
\(148\) −5.61628e39 −0.397747
\(149\) 2.43718e39 0.152385 0.0761924 0.997093i \(-0.475724\pi\)
0.0761924 + 0.997093i \(0.475724\pi\)
\(150\) 1.92381e39 0.106285
\(151\) 2.40615e40 1.17557 0.587786 0.809017i \(-0.300000\pi\)
0.587786 + 0.809017i \(0.300000\pi\)
\(152\) −8.34833e39 −0.360989
\(153\) 5.71946e39 0.219060
\(154\) 2.25609e39 0.0765990
\(155\) 4.14185e39 0.124755
\(156\) 6.29441e39 0.168326
\(157\) 3.62213e40 0.860638 0.430319 0.902677i \(-0.358401\pi\)
0.430319 + 0.902677i \(0.358401\pi\)
\(158\) −4.54943e40 −0.961172
\(159\) −2.58122e40 −0.485264
\(160\) 4.72237e39 0.0790569
\(161\) 2.12811e40 0.317479
\(162\) −1.99558e40 −0.265486
\(163\) −1.06162e41 −1.26037 −0.630186 0.776444i \(-0.717021\pi\)
−0.630186 + 0.776444i \(0.717021\pi\)
\(164\) 3.13156e40 0.332009
\(165\) −4.19400e39 −0.0397352
\(166\) −4.13980e40 −0.350736
\(167\) −3.16742e40 −0.240133 −0.120066 0.992766i \(-0.538311\pi\)
−0.120066 + 0.992766i \(0.538311\pi\)
\(168\) −3.58650e40 −0.243472
\(169\) −1.31413e41 −0.799348
\(170\) −2.91883e40 −0.159185
\(171\) 9.08086e40 0.444323
\(172\) −4.24102e40 −0.186292
\(173\) −3.69208e41 −1.45686 −0.728431 0.685119i \(-0.759750\pi\)
−0.728431 + 0.685119i \(0.759750\pi\)
\(174\) −1.83681e41 −0.651482
\(175\) −5.74473e40 −0.183259
\(176\) −1.02950e40 −0.0295558
\(177\) −2.42061e41 −0.625778
\(178\) −2.55156e41 −0.594342
\(179\) 4.82779e41 1.01384 0.506918 0.861994i \(-0.330785\pi\)
0.506918 + 0.861994i \(0.330785\pi\)
\(180\) −5.13673e40 −0.0973074
\(181\) 7.79879e41 1.33344 0.666720 0.745308i \(-0.267698\pi\)
0.666720 + 0.745308i \(0.267698\pi\)
\(182\) −1.87959e41 −0.290230
\(183\) 7.50291e41 1.04685
\(184\) −9.71102e40 −0.122500
\(185\) −3.11766e41 −0.355756
\(186\) 1.43541e41 0.148247
\(187\) 6.36320e40 0.0595121
\(188\) −1.14611e41 −0.0971191
\(189\) 1.28659e42 0.988320
\(190\) −4.63426e41 −0.322878
\(191\) 3.29328e41 0.208214 0.104107 0.994566i \(-0.466801\pi\)
0.104107 + 0.994566i \(0.466801\pi\)
\(192\) 1.63659e41 0.0939438
\(193\) −1.11821e41 −0.0583058 −0.0291529 0.999575i \(-0.509281\pi\)
−0.0291529 + 0.999575i \(0.509281\pi\)
\(194\) −2.29816e42 −1.08905
\(195\) 3.49410e41 0.150555
\(196\) −2.04606e41 −0.0802011
\(197\) 4.09822e42 1.46208 0.731038 0.682337i \(-0.239036\pi\)
0.731038 + 0.682337i \(0.239036\pi\)
\(198\) 1.11983e41 0.0363788
\(199\) 5.54584e42 1.64129 0.820646 0.571437i \(-0.193614\pi\)
0.820646 + 0.571437i \(0.193614\pi\)
\(200\) 2.62144e41 0.0707107
\(201\) 6.05376e42 1.48901
\(202\) −2.61093e42 −0.585862
\(203\) 5.48494e42 1.12330
\(204\) −1.01156e42 −0.189161
\(205\) 1.73837e42 0.296958
\(206\) −3.62239e42 −0.565525
\(207\) 1.05631e42 0.150779
\(208\) 8.57696e41 0.111985
\(209\) 1.01029e42 0.120709
\(210\) −1.99091e42 −0.217768
\(211\) −1.56109e43 −1.56388 −0.781939 0.623355i \(-0.785769\pi\)
−0.781939 + 0.623355i \(0.785769\pi\)
\(212\) −3.51725e42 −0.322842
\(213\) −6.01944e40 −0.00506447
\(214\) 4.36191e42 0.336529
\(215\) −2.35424e42 −0.166625
\(216\) −5.87099e42 −0.381344
\(217\) −4.28631e42 −0.255610
\(218\) −5.64764e42 −0.309330
\(219\) −2.22424e42 −0.111935
\(220\) −5.71488e41 −0.0264355
\(221\) −5.30130e42 −0.225489
\(222\) −1.08046e43 −0.422747
\(223\) 3.39504e42 0.122238 0.0611189 0.998130i \(-0.480533\pi\)
0.0611189 + 0.998130i \(0.480533\pi\)
\(224\) −4.88707e42 −0.161980
\(225\) −2.85146e42 −0.0870343
\(226\) −1.88965e43 −0.531345
\(227\) −5.49026e43 −1.42271 −0.711353 0.702835i \(-0.751917\pi\)
−0.711353 + 0.702835i \(0.751917\pi\)
\(228\) −1.60606e43 −0.383678
\(229\) 5.12775e43 1.12972 0.564860 0.825187i \(-0.308930\pi\)
0.564860 + 0.825187i \(0.308930\pi\)
\(230\) −5.39070e42 −0.109567
\(231\) 4.34028e42 0.0814135
\(232\) −2.50289e43 −0.433426
\(233\) 9.95511e43 1.59207 0.796034 0.605252i \(-0.206928\pi\)
0.796034 + 0.605252i \(0.206928\pi\)
\(234\) −9.32955e42 −0.137837
\(235\) −6.36217e42 −0.0868659
\(236\) −3.29841e43 −0.416325
\(237\) −8.75223e43 −1.02158
\(238\) 3.02063e43 0.326155
\(239\) −4.20147e43 −0.419798 −0.209899 0.977723i \(-0.567314\pi\)
−0.209899 + 0.977723i \(0.567314\pi\)
\(240\) 9.08492e42 0.0840259
\(241\) 9.20589e43 0.788407 0.394203 0.919023i \(-0.371021\pi\)
0.394203 + 0.919023i \(0.371021\pi\)
\(242\) −8.78934e43 −0.697224
\(243\) 1.08359e44 0.796432
\(244\) 1.02237e44 0.696460
\(245\) −1.13579e43 −0.0717341
\(246\) 6.02452e43 0.352877
\(247\) −8.41693e43 −0.457362
\(248\) 1.95593e43 0.0986275
\(249\) −7.96418e43 −0.372780
\(250\) 1.45519e43 0.0632456
\(251\) −8.81367e43 −0.355789 −0.177895 0.984050i \(-0.556929\pi\)
−0.177895 + 0.984050i \(0.556929\pi\)
\(252\) 5.31589e43 0.199373
\(253\) 1.17520e43 0.0409621
\(254\) 1.46782e44 0.475607
\(255\) −5.61526e43 −0.169191
\(256\) 2.23007e43 0.0625000
\(257\) −1.46131e44 −0.381049 −0.190524 0.981682i \(-0.561019\pi\)
−0.190524 + 0.981682i \(0.561019\pi\)
\(258\) −8.15890e43 −0.198001
\(259\) 3.22640e44 0.728908
\(260\) 4.76117e43 0.100163
\(261\) 2.72251e44 0.533482
\(262\) 6.34218e44 1.15788
\(263\) −2.73688e44 −0.465666 −0.232833 0.972517i \(-0.574800\pi\)
−0.232833 + 0.972517i \(0.574800\pi\)
\(264\) −1.98056e43 −0.0314134
\(265\) −1.95247e44 −0.288759
\(266\) 4.79589e44 0.661544
\(267\) −4.90872e44 −0.631698
\(268\) 8.24905e44 0.990627
\(269\) 1.32425e45 1.48440 0.742201 0.670177i \(-0.233782\pi\)
0.742201 + 0.670177i \(0.233782\pi\)
\(270\) −3.25906e44 −0.341085
\(271\) 5.21202e44 0.509418 0.254709 0.967018i \(-0.418020\pi\)
0.254709 + 0.967018i \(0.418020\pi\)
\(272\) −1.37838e44 −0.125847
\(273\) −3.61597e44 −0.308472
\(274\) 1.10777e45 0.883214
\(275\) −3.17239e43 −0.0236446
\(276\) −1.86821e44 −0.130199
\(277\) −1.63439e45 −1.06532 −0.532659 0.846330i \(-0.678807\pi\)
−0.532659 + 0.846330i \(0.678807\pi\)
\(278\) 3.56578e44 0.217433
\(279\) −2.12756e44 −0.121396
\(280\) −2.71287e44 −0.144879
\(281\) 1.81299e44 0.0906420 0.0453210 0.998972i \(-0.485569\pi\)
0.0453210 + 0.998972i \(0.485569\pi\)
\(282\) −2.20489e44 −0.103223
\(283\) −3.63001e45 −1.59169 −0.795845 0.605500i \(-0.792973\pi\)
−0.795845 + 0.605500i \(0.792973\pi\)
\(284\) −8.20228e42 −0.00336934
\(285\) −8.91541e44 −0.343172
\(286\) −1.03796e44 −0.0374463
\(287\) −1.79900e45 −0.608437
\(288\) −2.42575e44 −0.0769282
\(289\) −2.51014e45 −0.746600
\(290\) −1.38938e45 −0.387668
\(291\) −4.42121e45 −1.15750
\(292\) −3.03081e44 −0.0744691
\(293\) 1.89813e45 0.437800 0.218900 0.975747i \(-0.429753\pi\)
0.218900 + 0.975747i \(0.429753\pi\)
\(294\) −3.93622e44 −0.0852420
\(295\) −1.83098e45 −0.372372
\(296\) −1.47227e45 −0.281250
\(297\) 7.10491e44 0.127516
\(298\) 6.38892e44 0.107752
\(299\) −9.79081e44 −0.155204
\(300\) 5.04314e44 0.0751551
\(301\) 2.43635e45 0.341397
\(302\) 6.30757e45 0.831254
\(303\) −5.02293e45 −0.622686
\(304\) −2.18847e45 −0.255257
\(305\) 5.67529e45 0.622933
\(306\) 1.49932e45 0.154899
\(307\) −1.42121e45 −0.138229 −0.0691143 0.997609i \(-0.522017\pi\)
−0.0691143 + 0.997609i \(0.522017\pi\)
\(308\) 5.91420e44 0.0541637
\(309\) −6.96879e45 −0.601070
\(310\) 1.08576e45 0.0882151
\(311\) 1.48151e46 1.13406 0.567031 0.823696i \(-0.308092\pi\)
0.567031 + 0.823696i \(0.308092\pi\)
\(312\) 1.65004e45 0.119024
\(313\) 1.68494e46 1.14555 0.572775 0.819713i \(-0.305867\pi\)
0.572775 + 0.819713i \(0.305867\pi\)
\(314\) 9.49520e45 0.608563
\(315\) 2.95091e45 0.178325
\(316\) −1.19261e46 −0.679651
\(317\) 9.92215e45 0.533347 0.266673 0.963787i \(-0.414076\pi\)
0.266673 + 0.963787i \(0.414076\pi\)
\(318\) −6.76651e45 −0.343134
\(319\) 3.02893e45 0.144931
\(320\) 1.23794e45 0.0559017
\(321\) 8.39148e45 0.357681
\(322\) 5.57872e45 0.224492
\(323\) 1.35266e46 0.513975
\(324\) −5.23130e45 −0.187727
\(325\) 2.64298e45 0.0895884
\(326\) −2.78297e46 −0.891217
\(327\) −1.08650e46 −0.328772
\(328\) 8.20920e45 0.234766
\(329\) 6.58407e45 0.177980
\(330\) −1.09943e45 −0.0280970
\(331\) 4.31846e46 1.04354 0.521772 0.853085i \(-0.325271\pi\)
0.521772 + 0.853085i \(0.325271\pi\)
\(332\) −1.08522e46 −0.248008
\(333\) 1.60146e46 0.346177
\(334\) −8.30321e45 −0.169800
\(335\) 4.57914e46 0.886044
\(336\) −9.40179e45 −0.172161
\(337\) 5.65717e45 0.0980497 0.0490248 0.998798i \(-0.484389\pi\)
0.0490248 + 0.998798i \(0.484389\pi\)
\(338\) −3.44493e46 −0.565224
\(339\) −3.63533e46 −0.564741
\(340\) −7.65153e45 −0.112561
\(341\) −2.36702e45 −0.0329796
\(342\) 2.38049e46 0.314184
\(343\) 8.50326e46 1.06327
\(344\) −1.11176e46 −0.131728
\(345\) −1.03707e46 −0.116454
\(346\) −9.67857e46 −1.03016
\(347\) 5.80965e46 0.586211 0.293106 0.956080i \(-0.405311\pi\)
0.293106 + 0.956080i \(0.405311\pi\)
\(348\) −4.81508e46 −0.460668
\(349\) −1.34423e47 −1.21956 −0.609780 0.792571i \(-0.708742\pi\)
−0.609780 + 0.792571i \(0.708742\pi\)
\(350\) −1.50595e46 −0.129584
\(351\) −5.91923e46 −0.483152
\(352\) −2.69877e45 −0.0208991
\(353\) −1.24862e47 −0.917484 −0.458742 0.888570i \(-0.651700\pi\)
−0.458742 + 0.888570i \(0.651700\pi\)
\(354\) −6.34550e46 −0.442492
\(355\) −4.55318e44 −0.00301363
\(356\) −6.68877e46 −0.420263
\(357\) 5.81111e46 0.346655
\(358\) 1.26558e47 0.716890
\(359\) −9.59735e46 −0.516302 −0.258151 0.966105i \(-0.583113\pi\)
−0.258151 + 0.966105i \(0.583113\pi\)
\(360\) −1.34656e46 −0.0688067
\(361\) 8.75566e45 0.0425016
\(362\) 2.04441e47 0.942884
\(363\) −1.69090e47 −0.741046
\(364\) −4.92723e46 −0.205224
\(365\) −1.68244e46 −0.0666072
\(366\) 1.96684e47 0.740235
\(367\) 1.16622e47 0.417310 0.208655 0.977989i \(-0.433091\pi\)
0.208655 + 0.977989i \(0.433091\pi\)
\(368\) −2.54568e46 −0.0866204
\(369\) −8.92952e46 −0.288962
\(370\) −8.17276e46 −0.251557
\(371\) 2.02057e47 0.591638
\(372\) 3.76284e46 0.104827
\(373\) −3.24328e47 −0.859749 −0.429874 0.902889i \(-0.641442\pi\)
−0.429874 + 0.902889i \(0.641442\pi\)
\(374\) 1.66807e46 0.0420814
\(375\) 2.79951e46 0.0672207
\(376\) −3.00445e46 −0.0686735
\(377\) −2.52346e47 −0.549138
\(378\) 3.37273e47 0.698848
\(379\) −8.85067e46 −0.174643 −0.0873214 0.996180i \(-0.527831\pi\)
−0.0873214 + 0.996180i \(0.527831\pi\)
\(380\) −1.21484e47 −0.228309
\(381\) 2.82380e47 0.505501
\(382\) 8.63313e46 0.147230
\(383\) −1.64510e46 −0.0267310 −0.0133655 0.999911i \(-0.504255\pi\)
−0.0133655 + 0.999911i \(0.504255\pi\)
\(384\) 4.29023e46 0.0664283
\(385\) 3.28304e46 0.0484454
\(386\) −2.93131e46 −0.0412284
\(387\) 1.20931e47 0.162138
\(388\) −6.02448e47 −0.770075
\(389\) 2.33292e47 0.284336 0.142168 0.989843i \(-0.454593\pi\)
0.142168 + 0.989843i \(0.454593\pi\)
\(390\) 9.15957e46 0.106458
\(391\) 1.57345e47 0.174415
\(392\) −5.36362e46 −0.0567107
\(393\) 1.22011e48 1.23066
\(394\) 1.07432e48 1.03384
\(395\) −6.62029e47 −0.607898
\(396\) 2.93558e46 0.0257237
\(397\) −1.77175e48 −1.48176 −0.740879 0.671639i \(-0.765591\pi\)
−0.740879 + 0.671639i \(0.765591\pi\)
\(398\) 1.45381e48 1.16057
\(399\) 9.22637e47 0.703125
\(400\) 6.87195e46 0.0500000
\(401\) 9.35091e47 0.649656 0.324828 0.945773i \(-0.394694\pi\)
0.324828 + 0.945773i \(0.394694\pi\)
\(402\) 1.58696e48 1.05289
\(403\) 1.97200e47 0.124958
\(404\) −6.84441e47 −0.414267
\(405\) −2.90395e47 −0.167908
\(406\) 1.43784e48 0.794292
\(407\) 1.78171e47 0.0940458
\(408\) −2.65173e47 −0.133757
\(409\) −2.24904e48 −1.08422 −0.542108 0.840309i \(-0.682374\pi\)
−0.542108 + 0.840309i \(0.682374\pi\)
\(410\) 4.55702e47 0.209981
\(411\) 2.13115e48 0.938727
\(412\) −9.49589e47 −0.399887
\(413\) 1.89484e48 0.762953
\(414\) 2.76906e47 0.106617
\(415\) −6.02420e47 −0.221825
\(416\) 2.24840e47 0.0791857
\(417\) 6.85987e47 0.231099
\(418\) 2.64842e47 0.0853544
\(419\) −5.89430e48 −1.81749 −0.908747 0.417348i \(-0.862960\pi\)
−0.908747 + 0.417348i \(0.862960\pi\)
\(420\) −5.21904e47 −0.153985
\(421\) −6.89314e48 −1.94625 −0.973125 0.230278i \(-0.926037\pi\)
−0.973125 + 0.230278i \(0.926037\pi\)
\(422\) −4.09231e48 −1.10583
\(423\) 3.26807e47 0.0845269
\(424\) −9.22027e47 −0.228284
\(425\) −4.24745e47 −0.100678
\(426\) −1.57796e46 −0.00358112
\(427\) −5.87324e48 −1.27633
\(428\) 1.14345e48 0.237962
\(429\) −1.99684e47 −0.0397999
\(430\) −6.17149e47 −0.117821
\(431\) −3.31632e47 −0.0606495 −0.0303247 0.999540i \(-0.509654\pi\)
−0.0303247 + 0.999540i \(0.509654\pi\)
\(432\) −1.53905e48 −0.269651
\(433\) 7.05114e48 1.18368 0.591838 0.806057i \(-0.298402\pi\)
0.591838 + 0.806057i \(0.298402\pi\)
\(434\) −1.12363e48 −0.180744
\(435\) −2.67291e48 −0.412034
\(436\) −1.48049e48 −0.218729
\(437\) 2.49819e48 0.353768
\(438\) −5.83070e47 −0.0791498
\(439\) −4.37776e48 −0.569716 −0.284858 0.958570i \(-0.591946\pi\)
−0.284858 + 0.958570i \(0.591946\pi\)
\(440\) −1.49812e47 −0.0186927
\(441\) 5.83425e47 0.0698025
\(442\) −1.38970e48 −0.159445
\(443\) 1.23141e49 1.35498 0.677491 0.735531i \(-0.263067\pi\)
0.677491 + 0.735531i \(0.263067\pi\)
\(444\) −2.83237e48 −0.298927
\(445\) −3.71301e48 −0.375895
\(446\) 8.89990e47 0.0864351
\(447\) 1.22910e48 0.114525
\(448\) −1.28112e48 −0.114537
\(449\) 1.51088e49 1.29621 0.648103 0.761553i \(-0.275563\pi\)
0.648103 + 0.761553i \(0.275563\pi\)
\(450\) −7.47493e47 −0.0615426
\(451\) −9.93455e47 −0.0785023
\(452\) −4.95362e48 −0.375717
\(453\) 1.21346e49 0.883501
\(454\) −1.43924e49 −1.00600
\(455\) −2.73516e48 −0.183558
\(456\) −4.21019e48 −0.271301
\(457\) 2.42148e49 1.49841 0.749207 0.662336i \(-0.230435\pi\)
0.749207 + 0.662336i \(0.230435\pi\)
\(458\) 1.34421e49 0.798832
\(459\) 9.51262e48 0.542957
\(460\) −1.41314e48 −0.0774756
\(461\) 1.88635e49 0.993471 0.496736 0.867902i \(-0.334532\pi\)
0.496736 + 0.867902i \(0.334532\pi\)
\(462\) 1.13778e48 0.0575680
\(463\) 5.15519e48 0.250609 0.125304 0.992118i \(-0.460009\pi\)
0.125304 + 0.992118i \(0.460009\pi\)
\(464\) −6.56118e48 −0.306478
\(465\) 2.08879e48 0.0937597
\(466\) 2.60967e49 1.12576
\(467\) 1.96969e49 0.816649 0.408324 0.912837i \(-0.366113\pi\)
0.408324 + 0.912837i \(0.366113\pi\)
\(468\) −2.44569e48 −0.0974658
\(469\) −4.73885e49 −1.81541
\(470\) −1.66780e48 −0.0614235
\(471\) 1.82669e49 0.646813
\(472\) −8.64657e48 −0.294386
\(473\) 1.34542e48 0.0440480
\(474\) −2.29434e49 −0.722369
\(475\) −6.74373e48 −0.204206
\(476\) 7.91840e48 0.230626
\(477\) 1.00293e49 0.280983
\(478\) −1.10139e49 −0.296842
\(479\) −6.22680e49 −1.61458 −0.807288 0.590158i \(-0.799065\pi\)
−0.807288 + 0.590158i \(0.799065\pi\)
\(480\) 2.38156e48 0.0594153
\(481\) −1.48437e49 −0.356335
\(482\) 2.41327e49 0.557488
\(483\) 1.07324e49 0.238602
\(484\) −2.30407e49 −0.493012
\(485\) −3.34426e49 −0.688776
\(486\) 2.84056e49 0.563162
\(487\) 4.56184e49 0.870671 0.435335 0.900268i \(-0.356630\pi\)
0.435335 + 0.900268i \(0.356630\pi\)
\(488\) 2.68008e49 0.492472
\(489\) −5.35390e49 −0.947233
\(490\) −2.97740e48 −0.0507236
\(491\) 7.50460e49 1.23117 0.615587 0.788069i \(-0.288919\pi\)
0.615587 + 0.788069i \(0.288919\pi\)
\(492\) 1.57929e49 0.249522
\(493\) 4.05537e49 0.617110
\(494\) −2.20645e49 −0.323404
\(495\) 1.62957e48 0.0230080
\(496\) 5.12736e48 0.0697402
\(497\) 4.71199e47 0.00617463
\(498\) −2.08776e49 −0.263596
\(499\) −1.22644e50 −1.49206 −0.746030 0.665912i \(-0.768043\pi\)
−0.746030 + 0.665912i \(0.768043\pi\)
\(500\) 3.81470e48 0.0447214
\(501\) −1.59738e49 −0.180472
\(502\) −2.31045e49 −0.251581
\(503\) −6.93196e48 −0.0727526 −0.0363763 0.999338i \(-0.511581\pi\)
−0.0363763 + 0.999338i \(0.511581\pi\)
\(504\) 1.39353e49 0.140978
\(505\) −3.79941e49 −0.370532
\(506\) 3.08072e48 0.0289646
\(507\) −6.62737e49 −0.600750
\(508\) 3.84780e49 0.336305
\(509\) −1.92494e50 −1.62233 −0.811165 0.584817i \(-0.801166\pi\)
−0.811165 + 0.584817i \(0.801166\pi\)
\(510\) −1.47201e49 −0.119636
\(511\) 1.74112e49 0.136472
\(512\) 5.84601e48 0.0441942
\(513\) 1.51033e50 1.10129
\(514\) −3.83075e49 −0.269442
\(515\) −5.27128e49 −0.357670
\(516\) −2.13881e49 −0.140008
\(517\) 3.63590e48 0.0229634
\(518\) 8.45781e49 0.515416
\(519\) −1.86197e50 −1.09491
\(520\) 1.24811e49 0.0708258
\(521\) −7.74303e49 −0.424046 −0.212023 0.977265i \(-0.568005\pi\)
−0.212023 + 0.977265i \(0.568005\pi\)
\(522\) 7.13690e49 0.377229
\(523\) 1.36453e50 0.696147 0.348074 0.937467i \(-0.386836\pi\)
0.348074 + 0.937467i \(0.386836\pi\)
\(524\) 1.66257e50 0.818748
\(525\) −2.89715e49 −0.137729
\(526\) −7.17458e49 −0.329276
\(527\) −3.16915e49 −0.140426
\(528\) −5.19192e48 −0.0222127
\(529\) −2.13004e50 −0.879951
\(530\) −5.11828e49 −0.204183
\(531\) 9.40527e49 0.362345
\(532\) 1.25721e50 0.467783
\(533\) 8.27666e49 0.297442
\(534\) −1.28679e50 −0.446678
\(535\) 6.34742e49 0.212840
\(536\) 2.16244e50 0.700479
\(537\) 2.43473e50 0.761949
\(538\) 3.47143e50 1.04963
\(539\) 6.49090e48 0.0189632
\(540\) −8.54342e49 −0.241183
\(541\) −3.69295e49 −0.100745 −0.0503726 0.998730i \(-0.516041\pi\)
−0.0503726 + 0.998730i \(0.516041\pi\)
\(542\) 1.36630e50 0.360213
\(543\) 3.93304e50 1.00215
\(544\) −3.61333e49 −0.0889873
\(545\) −8.21840e49 −0.195637
\(546\) −9.47904e49 −0.218123
\(547\) −3.43870e50 −0.764942 −0.382471 0.923968i \(-0.624927\pi\)
−0.382471 + 0.923968i \(0.624927\pi\)
\(548\) 2.90397e50 0.624527
\(549\) −2.91525e50 −0.606160
\(550\) −8.31624e48 −0.0167193
\(551\) 6.43876e50 1.25169
\(552\) −4.89741e49 −0.0920647
\(553\) 6.85120e50 1.24552
\(554\) −4.28445e50 −0.753294
\(555\) −1.57228e50 −0.267368
\(556\) 9.34747e49 0.153748
\(557\) −1.65111e50 −0.262697 −0.131348 0.991336i \(-0.541931\pi\)
−0.131348 + 0.991336i \(0.541931\pi\)
\(558\) −5.57726e49 −0.0858398
\(559\) −1.12089e50 −0.166896
\(560\) −7.11163e49 −0.102445
\(561\) 3.20905e49 0.0447264
\(562\) 4.75265e49 0.0640935
\(563\) 8.28221e50 1.08079 0.540395 0.841412i \(-0.318275\pi\)
0.540395 + 0.841412i \(0.318275\pi\)
\(564\) −5.77998e49 −0.0729899
\(565\) −2.74981e50 −0.336052
\(566\) −9.51585e50 −1.12550
\(567\) 3.00524e50 0.344027
\(568\) −2.15018e48 −0.00238249
\(569\) 5.31805e50 0.570394 0.285197 0.958469i \(-0.407941\pi\)
0.285197 + 0.958469i \(0.407941\pi\)
\(570\) −2.33712e50 −0.242659
\(571\) 7.10776e50 0.714438 0.357219 0.934021i \(-0.383725\pi\)
0.357219 + 0.934021i \(0.383725\pi\)
\(572\) −2.72095e49 −0.0264785
\(573\) 1.66085e50 0.156484
\(574\) −4.71596e50 −0.430230
\(575\) −7.84450e49 −0.0692963
\(576\) −6.35897e49 −0.0543965
\(577\) −1.53160e51 −1.26880 −0.634399 0.773006i \(-0.718752\pi\)
−0.634399 + 0.773006i \(0.718752\pi\)
\(578\) −6.58018e50 −0.527926
\(579\) −5.63928e49 −0.0438198
\(580\) −3.64219e50 −0.274122
\(581\) 6.23432e50 0.454496
\(582\) −1.15899e51 −0.818476
\(583\) 1.11581e50 0.0763348
\(584\) −7.94509e49 −0.0526576
\(585\) −1.35763e50 −0.0871761
\(586\) 4.97584e50 0.309571
\(587\) −3.05746e51 −1.84313 −0.921565 0.388223i \(-0.873089\pi\)
−0.921565 + 0.388223i \(0.873089\pi\)
\(588\) −1.03186e50 −0.0602752
\(589\) −5.03169e50 −0.284827
\(590\) −4.79981e50 −0.263307
\(591\) 2.06679e51 1.09882
\(592\) −3.85948e50 −0.198874
\(593\) −3.18854e51 −1.59250 −0.796252 0.604966i \(-0.793187\pi\)
−0.796252 + 0.604966i \(0.793187\pi\)
\(594\) 1.86251e50 0.0901674
\(595\) 4.39560e50 0.206278
\(596\) 1.67482e50 0.0761924
\(597\) 2.79685e51 1.23351
\(598\) −2.56660e50 −0.109745
\(599\) −2.67703e51 −1.10983 −0.554916 0.831906i \(-0.687250\pi\)
−0.554916 + 0.831906i \(0.687250\pi\)
\(600\) 1.32203e50 0.0531427
\(601\) −1.40525e51 −0.547743 −0.273872 0.961766i \(-0.588304\pi\)
−0.273872 + 0.961766i \(0.588304\pi\)
\(602\) 6.38675e50 0.241404
\(603\) −2.35218e51 −0.862186
\(604\) 1.65349e51 0.587786
\(605\) −1.27902e51 −0.440963
\(606\) −1.31673e51 −0.440305
\(607\) 2.84184e51 0.921741 0.460870 0.887467i \(-0.347537\pi\)
0.460870 + 0.887467i \(0.347537\pi\)
\(608\) −5.73693e50 −0.180494
\(609\) 2.76613e51 0.844216
\(610\) 1.48774e51 0.440480
\(611\) −3.02913e50 −0.0870074
\(612\) 3.93039e50 0.109530
\(613\) 5.74303e51 1.55282 0.776412 0.630226i \(-0.217038\pi\)
0.776412 + 0.630226i \(0.217038\pi\)
\(614\) −3.72561e50 −0.0977424
\(615\) 8.76683e50 0.223179
\(616\) 1.55037e50 0.0382995
\(617\) 5.29231e51 1.26873 0.634366 0.773033i \(-0.281261\pi\)
0.634366 + 0.773033i \(0.281261\pi\)
\(618\) −1.82683e51 −0.425021
\(619\) 1.88485e50 0.0425599 0.0212799 0.999774i \(-0.493226\pi\)
0.0212799 + 0.999774i \(0.493226\pi\)
\(620\) 2.84626e50 0.0623775
\(621\) 1.75686e51 0.373716
\(622\) 3.88369e51 0.801903
\(623\) 3.84252e51 0.770171
\(624\) 4.32549e50 0.0841628
\(625\) 2.11758e50 0.0400000
\(626\) 4.41697e51 0.810026
\(627\) 5.09505e50 0.0907191
\(628\) 2.48911e51 0.430319
\(629\) 2.38549e51 0.400442
\(630\) 7.73564e50 0.126095
\(631\) −5.48789e51 −0.868685 −0.434343 0.900748i \(-0.643019\pi\)
−0.434343 + 0.900748i \(0.643019\pi\)
\(632\) −3.12635e51 −0.480586
\(633\) −7.87281e51 −1.17533
\(634\) 2.60103e51 0.377133
\(635\) 2.13596e51 0.300800
\(636\) −1.77380e51 −0.242632
\(637\) −5.40769e50 −0.0718509
\(638\) 7.94016e50 0.102482
\(639\) 2.33885e49 0.00293249
\(640\) 3.24519e50 0.0395285
\(641\) −1.26934e52 −1.50213 −0.751063 0.660231i \(-0.770458\pi\)
−0.751063 + 0.660231i \(0.770458\pi\)
\(642\) 2.19978e51 0.252918
\(643\) −2.67003e50 −0.0298273 −0.0149136 0.999889i \(-0.504747\pi\)
−0.0149136 + 0.999889i \(0.504747\pi\)
\(644\) 1.46243e51 0.158740
\(645\) −1.18728e51 −0.125227
\(646\) 3.54591e51 0.363435
\(647\) −7.38492e51 −0.735557 −0.367778 0.929913i \(-0.619882\pi\)
−0.367778 + 0.929913i \(0.619882\pi\)
\(648\) −1.37135e51 −0.132743
\(649\) 1.04638e51 0.0984384
\(650\) 6.92841e50 0.0633486
\(651\) −2.16165e51 −0.192104
\(652\) −7.29540e51 −0.630186
\(653\) −1.43531e52 −1.20518 −0.602592 0.798050i \(-0.705865\pi\)
−0.602592 + 0.798050i \(0.705865\pi\)
\(654\) −2.84819e51 −0.232477
\(655\) 9.22909e51 0.732310
\(656\) 2.15199e51 0.166004
\(657\) 8.64224e50 0.0648137
\(658\) 1.72597e51 0.125851
\(659\) 1.21039e52 0.858112 0.429056 0.903278i \(-0.358846\pi\)
0.429056 + 0.903278i \(0.358846\pi\)
\(660\) −2.88210e50 −0.0198676
\(661\) −1.90863e52 −1.27937 −0.639683 0.768638i \(-0.720935\pi\)
−0.639683 + 0.768638i \(0.720935\pi\)
\(662\) 1.13206e52 0.737897
\(663\) −2.67352e51 −0.169466
\(664\) −2.84485e51 −0.175368
\(665\) 6.97894e51 0.418397
\(666\) 4.19813e51 0.244784
\(667\) 7.48975e51 0.424756
\(668\) −2.17664e51 −0.120066
\(669\) 1.71217e51 0.0918678
\(670\) 1.20039e52 0.626527
\(671\) −3.24336e51 −0.164675
\(672\) −2.46462e51 −0.121736
\(673\) −1.67548e52 −0.805120 −0.402560 0.915394i \(-0.631879\pi\)
−0.402560 + 0.915394i \(0.631879\pi\)
\(674\) 1.48299e51 0.0693316
\(675\) −4.74255e51 −0.215721
\(676\) −9.03067e51 −0.399674
\(677\) −2.49877e52 −1.07606 −0.538029 0.842926i \(-0.680831\pi\)
−0.538029 + 0.842926i \(0.680831\pi\)
\(678\) −9.52980e51 −0.399332
\(679\) 3.46090e52 1.41123
\(680\) −2.00580e51 −0.0795927
\(681\) −2.76882e52 −1.06923
\(682\) −6.20499e50 −0.0233201
\(683\) −4.73250e52 −1.73104 −0.865520 0.500874i \(-0.833012\pi\)
−0.865520 + 0.500874i \(0.833012\pi\)
\(684\) 6.24032e51 0.222162
\(685\) 1.61202e52 0.558594
\(686\) 2.22908e52 0.751847
\(687\) 2.58600e52 0.849041
\(688\) −2.91441e51 −0.0931459
\(689\) −9.29603e51 −0.289229
\(690\) −2.71861e51 −0.0823452
\(691\) 1.73179e52 0.510683 0.255341 0.966851i \(-0.417812\pi\)
0.255341 + 0.966851i \(0.417812\pi\)
\(692\) −2.53718e52 −0.728431
\(693\) −1.68641e51 −0.0471410
\(694\) 1.52296e52 0.414514
\(695\) 5.18889e51 0.137517
\(696\) −1.26225e52 −0.325741
\(697\) −1.33012e52 −0.334259
\(698\) −3.52382e52 −0.862359
\(699\) 5.02051e52 1.19652
\(700\) −3.94775e51 −0.0916296
\(701\) 8.62925e51 0.195070 0.0975348 0.995232i \(-0.468904\pi\)
0.0975348 + 0.995232i \(0.468904\pi\)
\(702\) −1.55169e52 −0.341640
\(703\) 3.78747e52 0.812223
\(704\) −7.07468e50 −0.0147779
\(705\) −3.20853e51 −0.0652841
\(706\) −3.27318e52 −0.648759
\(707\) 3.93193e52 0.759183
\(708\) −1.66343e52 −0.312889
\(709\) 2.82932e52 0.518473 0.259236 0.965814i \(-0.416529\pi\)
0.259236 + 0.965814i \(0.416529\pi\)
\(710\) −1.19359e50 −0.00213096
\(711\) 3.40067e52 0.591530
\(712\) −1.75342e52 −0.297171
\(713\) −5.85301e51 −0.0966547
\(714\) 1.52335e52 0.245122
\(715\) −1.51043e51 −0.0236831
\(716\) 3.31763e52 0.506918
\(717\) −2.11886e52 −0.315500
\(718\) −2.51589e52 −0.365081
\(719\) 1.27280e53 1.80001 0.900007 0.435875i \(-0.143561\pi\)
0.900007 + 0.435875i \(0.143561\pi\)
\(720\) −3.52994e51 −0.0486537
\(721\) 5.45513e52 0.732829
\(722\) 2.29524e51 0.0300532
\(723\) 4.64266e52 0.592528
\(724\) 5.35929e52 0.666720
\(725\) −2.02182e52 −0.245183
\(726\) −4.43259e52 −0.523999
\(727\) −5.42190e52 −0.624834 −0.312417 0.949945i \(-0.601139\pi\)
−0.312417 + 0.949945i \(0.601139\pi\)
\(728\) −1.29164e52 −0.145115
\(729\) 8.89250e52 0.974012
\(730\) −4.41041e51 −0.0470984
\(731\) 1.80135e52 0.187554
\(732\) 5.15596e52 0.523425
\(733\) −4.67036e52 −0.462304 −0.231152 0.972918i \(-0.574249\pi\)
−0.231152 + 0.972918i \(0.574249\pi\)
\(734\) 3.05718e52 0.295083
\(735\) −5.72795e51 −0.0539118
\(736\) −6.67336e51 −0.0612498
\(737\) −2.61692e52 −0.234230
\(738\) −2.34082e52 −0.204327
\(739\) −3.22372e52 −0.274432 −0.137216 0.990541i \(-0.543815\pi\)
−0.137216 + 0.990541i \(0.543815\pi\)
\(740\) −2.14244e52 −0.177878
\(741\) −4.24478e52 −0.343731
\(742\) 5.29679e52 0.418351
\(743\) −5.43932e52 −0.419036 −0.209518 0.977805i \(-0.567190\pi\)
−0.209518 + 0.977805i \(0.567190\pi\)
\(744\) 9.86405e51 0.0741235
\(745\) 9.29710e51 0.0681485
\(746\) −8.50207e52 −0.607934
\(747\) 3.09447e52 0.215852
\(748\) 4.37275e51 0.0297561
\(749\) −6.56881e52 −0.436087
\(750\) 7.33874e51 0.0475322
\(751\) −1.15095e53 −0.727304 −0.363652 0.931535i \(-0.618470\pi\)
−0.363652 + 0.931535i \(0.618470\pi\)
\(752\) −7.87598e51 −0.0485595
\(753\) −4.44486e52 −0.267394
\(754\) −6.61510e52 −0.388299
\(755\) 9.17873e52 0.525731
\(756\) 8.84140e52 0.494160
\(757\) −3.51043e53 −1.91464 −0.957318 0.289038i \(-0.906665\pi\)
−0.957318 + 0.289038i \(0.906665\pi\)
\(758\) −2.32015e52 −0.123491
\(759\) 5.92671e51 0.0307851
\(760\) −3.18464e52 −0.161439
\(761\) −1.73091e53 −0.856363 −0.428182 0.903693i \(-0.640846\pi\)
−0.428182 + 0.903693i \(0.640846\pi\)
\(762\) 7.40242e52 0.357443
\(763\) 8.50504e52 0.400841
\(764\) 2.26312e52 0.104107
\(765\) 2.18180e52 0.0979668
\(766\) −4.31254e51 −0.0189017
\(767\) −8.71762e52 −0.372978
\(768\) 1.12466e52 0.0469719
\(769\) −3.14722e53 −1.28319 −0.641594 0.767045i \(-0.721727\pi\)
−0.641594 + 0.767045i \(0.721727\pi\)
\(770\) 8.60630e51 0.0342561
\(771\) −7.36961e52 −0.286377
\(772\) −7.68426e51 −0.0291529
\(773\) −1.31709e53 −0.487861 −0.243931 0.969793i \(-0.578437\pi\)
−0.243931 + 0.969793i \(0.578437\pi\)
\(774\) 3.17013e52 0.114649
\(775\) 1.57999e52 0.0557921
\(776\) −1.57928e53 −0.544525
\(777\) 1.62712e53 0.547811
\(778\) 6.11560e52 0.201056
\(779\) −2.11184e53 −0.677982
\(780\) 2.40113e52 0.0752775
\(781\) 2.60209e50 0.000796669 0
\(782\) 4.12471e52 0.123330
\(783\) 4.52808e53 1.32227
\(784\) −1.40604e52 −0.0401006
\(785\) 1.38173e53 0.384889
\(786\) 3.19846e53 0.870208
\(787\) 2.46211e53 0.654296 0.327148 0.944973i \(-0.393912\pi\)
0.327148 + 0.944973i \(0.393912\pi\)
\(788\) 2.81628e53 0.731038
\(789\) −1.38025e53 −0.349972
\(790\) −1.73547e53 −0.429849
\(791\) 2.84572e53 0.688536
\(792\) 7.69545e51 0.0181894
\(793\) 2.70210e53 0.623948
\(794\) −4.64453e53 −1.04776
\(795\) −9.84658e52 −0.217017
\(796\) 3.81107e53 0.820646
\(797\) −6.55944e53 −1.38003 −0.690013 0.723797i \(-0.742395\pi\)
−0.690013 + 0.723797i \(0.742395\pi\)
\(798\) 2.41864e53 0.497184
\(799\) 4.86803e52 0.0977772
\(800\) 1.80144e52 0.0353553
\(801\) 1.90728e53 0.365773
\(802\) 2.45129e53 0.459376
\(803\) 9.61493e51 0.0176079
\(804\) 4.16011e53 0.744506
\(805\) 8.11810e52 0.141981
\(806\) 5.16949e52 0.0883588
\(807\) 6.67837e53 1.11560
\(808\) −1.79422e53 −0.292931
\(809\) −1.13014e54 −1.80338 −0.901688 0.432387i \(-0.857671\pi\)
−0.901688 + 0.432387i \(0.857671\pi\)
\(810\) −7.61254e52 −0.118729
\(811\) 4.98428e53 0.759831 0.379915 0.925021i \(-0.375953\pi\)
0.379915 + 0.925021i \(0.375953\pi\)
\(812\) 3.76922e53 0.561649
\(813\) 2.62850e53 0.382853
\(814\) 4.67063e52 0.0665004
\(815\) −4.04976e53 −0.563655
\(816\) −6.95135e52 −0.0945805
\(817\) 2.86003e53 0.380419
\(818\) −5.89572e53 −0.766656
\(819\) 1.40498e53 0.178615
\(820\) 1.19460e53 0.148479
\(821\) −6.32222e53 −0.768283 −0.384142 0.923274i \(-0.625503\pi\)
−0.384142 + 0.923274i \(0.625503\pi\)
\(822\) 5.58667e53 0.663780
\(823\) 1.53088e54 1.77846 0.889228 0.457465i \(-0.151243\pi\)
0.889228 + 0.457465i \(0.151243\pi\)
\(824\) −2.48929e53 −0.282763
\(825\) −1.59988e52 −0.0177701
\(826\) 4.96722e53 0.539489
\(827\) 1.35864e54 1.44295 0.721477 0.692439i \(-0.243464\pi\)
0.721477 + 0.692439i \(0.243464\pi\)
\(828\) 7.25892e52 0.0753895
\(829\) −5.63827e53 −0.572648 −0.286324 0.958133i \(-0.592433\pi\)
−0.286324 + 0.958133i \(0.592433\pi\)
\(830\) −1.57921e53 −0.156854
\(831\) −8.24245e53 −0.800640
\(832\) 5.89404e52 0.0559927
\(833\) 8.69053e52 0.0807446
\(834\) 1.79827e53 0.163412
\(835\) −1.20828e53 −0.107391
\(836\) 6.94268e52 0.0603546
\(837\) −3.53856e53 −0.300888
\(838\) −1.54516e54 −1.28516
\(839\) −2.01357e54 −1.63821 −0.819106 0.573642i \(-0.805530\pi\)
−0.819106 + 0.573642i \(0.805530\pi\)
\(840\) −1.36814e53 −0.108884
\(841\) 6.45914e53 0.502862
\(842\) −1.80700e54 −1.37621
\(843\) 9.14319e52 0.0681220
\(844\) −1.07277e54 −0.781939
\(845\) −5.01303e53 −0.357479
\(846\) 8.56706e52 0.0597696
\(847\) 1.32363e54 0.903489
\(848\) −2.41704e53 −0.161421
\(849\) −1.83067e54 −1.19624
\(850\) −1.11344e53 −0.0711899
\(851\) 4.40569e53 0.275624
\(852\) −4.13653e51 −0.00253223
\(853\) −5.77259e53 −0.345791 −0.172895 0.984940i \(-0.555312\pi\)
−0.172895 + 0.984940i \(0.555312\pi\)
\(854\) −1.53963e54 −0.902500
\(855\) 3.46407e53 0.198707
\(856\) 2.99748e53 0.168264
\(857\) −1.17381e54 −0.644842 −0.322421 0.946596i \(-0.604497\pi\)
−0.322421 + 0.946596i \(0.604497\pi\)
\(858\) −5.23458e52 −0.0281428
\(859\) 3.05143e54 1.60557 0.802786 0.596267i \(-0.203350\pi\)
0.802786 + 0.596267i \(0.203350\pi\)
\(860\) −1.61782e53 −0.0833123
\(861\) −9.07261e53 −0.457271
\(862\) −8.69353e52 −0.0428856
\(863\) 3.37124e54 1.62776 0.813879 0.581034i \(-0.197352\pi\)
0.813879 + 0.581034i \(0.197352\pi\)
\(864\) −4.03451e53 −0.190672
\(865\) −1.40842e54 −0.651528
\(866\) 1.84841e54 0.836986
\(867\) −1.26590e54 −0.561108
\(868\) −2.94553e53 −0.127805
\(869\) 3.78342e53 0.160701
\(870\) −7.00687e53 −0.291352
\(871\) 2.18021e54 0.887487
\(872\) −3.88103e53 −0.154665
\(873\) 1.71786e54 0.670229
\(874\) 6.54885e53 0.250152
\(875\) −2.19144e53 −0.0819560
\(876\) −1.52848e53 −0.0559673
\(877\) 4.59456e52 0.0164722 0.00823610 0.999966i \(-0.497378\pi\)
0.00823610 + 0.999966i \(0.497378\pi\)
\(878\) −1.14760e54 −0.402850
\(879\) 9.57256e53 0.329029
\(880\) −3.92723e52 −0.0132177
\(881\) −4.37096e54 −1.44053 −0.720266 0.693698i \(-0.755980\pi\)
−0.720266 + 0.693698i \(0.755980\pi\)
\(882\) 1.52941e53 0.0493578
\(883\) 5.60045e54 1.76991 0.884953 0.465680i \(-0.154190\pi\)
0.884953 + 0.465680i \(0.154190\pi\)
\(884\) −3.64302e53 −0.112744
\(885\) −9.23391e53 −0.279856
\(886\) 3.22806e54 0.958116
\(887\) −3.86725e54 −1.12413 −0.562064 0.827094i \(-0.689992\pi\)
−0.562064 + 0.827094i \(0.689992\pi\)
\(888\) −7.42489e53 −0.211373
\(889\) −2.21045e54 −0.616310
\(890\) −9.73344e53 −0.265798
\(891\) 1.65957e53 0.0443873
\(892\) 2.33306e53 0.0611189
\(893\) 7.72903e53 0.198323
\(894\) 3.22202e53 0.0809813
\(895\) 1.84166e54 0.453401
\(896\) −3.35837e53 −0.0809899
\(897\) −4.93765e53 −0.116643
\(898\) 3.96069e54 0.916556
\(899\) −1.50854e54 −0.341981
\(900\) −1.95951e53 −0.0435172
\(901\) 1.49394e54 0.325030
\(902\) −2.60428e53 −0.0555095
\(903\) 1.22869e54 0.256577
\(904\) −1.29856e54 −0.265672
\(905\) 2.97500e54 0.596332
\(906\) 3.18100e54 0.624730
\(907\) 2.47231e54 0.475739 0.237869 0.971297i \(-0.423551\pi\)
0.237869 + 0.971297i \(0.423551\pi\)
\(908\) −3.77288e54 −0.711353
\(909\) 1.95166e54 0.360555
\(910\) −7.17007e53 −0.129795
\(911\) −7.75991e54 −1.37647 −0.688235 0.725488i \(-0.741614\pi\)
−0.688235 + 0.725488i \(0.741614\pi\)
\(912\) −1.10367e54 −0.191839
\(913\) 3.44276e53 0.0586404
\(914\) 6.34777e54 1.05954
\(915\) 2.86213e54 0.468166
\(916\) 3.52376e54 0.564860
\(917\) −9.55099e54 −1.50043
\(918\) 2.49368e54 0.383928
\(919\) −3.05017e53 −0.0460243 −0.0230121 0.999735i \(-0.507326\pi\)
−0.0230121 + 0.999735i \(0.507326\pi\)
\(920\) −3.70446e53 −0.0547835
\(921\) −7.16736e53 −0.103886
\(922\) 4.94496e54 0.702490
\(923\) −2.16785e52 −0.00301854
\(924\) 2.98262e53 0.0407067
\(925\) −1.18929e54 −0.159099
\(926\) 1.35140e54 0.177207
\(927\) 2.70771e54 0.348039
\(928\) −1.71997e54 −0.216713
\(929\) 4.09018e54 0.505187 0.252593 0.967573i \(-0.418717\pi\)
0.252593 + 0.967573i \(0.418717\pi\)
\(930\) 5.47565e53 0.0662981
\(931\) 1.37981e54 0.163775
\(932\) 6.84110e54 0.796034
\(933\) 7.47146e54 0.852305
\(934\) 5.16342e54 0.577458
\(935\) 2.42737e53 0.0266146
\(936\) −6.41122e53 −0.0689187
\(937\) 6.61227e54 0.696897 0.348448 0.937328i \(-0.386709\pi\)
0.348448 + 0.937328i \(0.386709\pi\)
\(938\) −1.24226e55 −1.28369
\(939\) 8.49739e54 0.860938
\(940\) −4.37205e53 −0.0434330
\(941\) −1.99049e55 −1.93889 −0.969444 0.245314i \(-0.921109\pi\)
−0.969444 + 0.245314i \(0.921109\pi\)
\(942\) 4.78857e54 0.457366
\(943\) −2.45655e54 −0.230070
\(944\) −2.26665e54 −0.208162
\(945\) 4.90796e54 0.441990
\(946\) 3.52693e53 0.0311466
\(947\) −2.25303e55 −1.95116 −0.975580 0.219644i \(-0.929510\pi\)
−0.975580 + 0.219644i \(0.929510\pi\)
\(948\) −6.01448e54 −0.510792
\(949\) −8.01038e53 −0.0667157
\(950\) −1.76783e54 −0.144395
\(951\) 5.00389e54 0.400837
\(952\) 2.07576e54 0.163077
\(953\) −1.62950e55 −1.25555 −0.627777 0.778393i \(-0.716035\pi\)
−0.627777 + 0.778393i \(0.716035\pi\)
\(954\) 2.62912e54 0.198685
\(955\) 1.25629e54 0.0931163
\(956\) −2.88723e54 −0.209899
\(957\) 1.52753e54 0.108923
\(958\) −1.63232e55 −1.14168
\(959\) −1.66825e55 −1.14450
\(960\) 6.24311e53 0.0420130
\(961\) −1.39701e55 −0.922181
\(962\) −3.89119e54 −0.251967
\(963\) −3.26050e54 −0.207109
\(964\) 6.32624e54 0.394203
\(965\) −4.26562e53 −0.0260751
\(966\) 2.81343e54 0.168717
\(967\) −4.18975e54 −0.246489 −0.123245 0.992376i \(-0.539330\pi\)
−0.123245 + 0.992376i \(0.539330\pi\)
\(968\) −6.03999e54 −0.348612
\(969\) 6.82165e54 0.386278
\(970\) −8.76678e54 −0.487038
\(971\) −9.28589e54 −0.506137 −0.253068 0.967448i \(-0.581440\pi\)
−0.253068 + 0.967448i \(0.581440\pi\)
\(972\) 7.44637e54 0.398216
\(973\) −5.36987e54 −0.281758
\(974\) 1.19586e55 0.615657
\(975\) 1.33289e54 0.0673302
\(976\) 7.02567e54 0.348230
\(977\) 1.35583e55 0.659409 0.329705 0.944084i \(-0.393051\pi\)
0.329705 + 0.944084i \(0.393051\pi\)
\(978\) −1.40349e55 −0.669795
\(979\) 2.12194e54 0.0993696
\(980\) −7.80509e53 −0.0358670
\(981\) 4.22157e54 0.190370
\(982\) 1.96728e55 0.870572
\(983\) −1.34843e55 −0.585583 −0.292791 0.956176i \(-0.594584\pi\)
−0.292791 + 0.956176i \(0.594584\pi\)
\(984\) 4.14002e54 0.176438
\(985\) 1.56335e55 0.653860
\(986\) 1.06309e55 0.436363
\(987\) 3.32044e54 0.133761
\(988\) −5.78407e54 −0.228681
\(989\) 3.32687e54 0.129093
\(990\) 4.27183e53 0.0162691
\(991\) 1.84939e55 0.691298 0.345649 0.938364i \(-0.387659\pi\)
0.345649 + 0.938364i \(0.387659\pi\)
\(992\) 1.34411e54 0.0493137
\(993\) 2.17786e55 0.784276
\(994\) 1.23522e53 0.00436612
\(995\) 2.11557e55 0.734008
\(996\) −5.47294e54 −0.186390
\(997\) −2.97900e54 −0.0995885 −0.0497943 0.998759i \(-0.515857\pi\)
−0.0497943 + 0.998759i \(0.515857\pi\)
\(998\) −3.21504e55 −1.05505
\(999\) 2.66355e55 0.858023
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 10.38.a.a.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.38.a.a.1.2 2 1.1 even 1 trivial