Properties

Label 10.34.a.a.1.1
Level $10$
Weight $34$
Character 10.1
Self dual yes
Analytic conductor $68.983$
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] = [10,34,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, 34, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.1");
 
S:= CuspForms(chi, 34);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 34 \)
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: \(68.9828288810\)
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: \( x^{2} - x - 3937184160 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(62747.5\) 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)
 
\(f(q)\) \(=\) \(q+65536.0 q^{2} -7.12076e7 q^{3} +4.29497e9 q^{4} +1.52588e11 q^{5} -4.66666e12 q^{6} -7.62676e13 q^{7} +2.81475e14 q^{8} -4.88541e14 q^{9} +1.00000e16 q^{10} +1.14348e17 q^{11} -3.05834e17 q^{12} +3.91852e18 q^{13} -4.99828e18 q^{14} -1.08654e19 q^{15} +1.84467e19 q^{16} -1.94707e20 q^{17} -3.20170e19 q^{18} -1.21873e21 q^{19} +6.55360e20 q^{20} +5.43083e21 q^{21} +7.49389e21 q^{22} -4.03724e21 q^{23} -2.00432e22 q^{24} +2.32831e22 q^{25} +2.56804e23 q^{26} +4.30635e23 q^{27} -3.27567e23 q^{28} +1.32420e23 q^{29} -7.12076e23 q^{30} +4.55893e24 q^{31} +1.20893e24 q^{32} -8.14242e24 q^{33} -1.27603e25 q^{34} -1.16375e25 q^{35} -2.09827e24 q^{36} -4.62500e25 q^{37} -7.98704e25 q^{38} -2.79028e26 q^{39} +4.29497e25 q^{40} -4.25054e26 q^{41} +3.55915e26 q^{42} +5.95880e26 q^{43} +4.91120e26 q^{44} -7.45455e25 q^{45} -2.64585e26 q^{46} -3.08054e27 q^{47} -1.31355e27 q^{48} -1.91424e27 q^{49} +1.52588e27 q^{50} +1.38646e28 q^{51} +1.68299e28 q^{52} -5.55613e28 q^{53} +2.82221e28 q^{54} +1.74481e28 q^{55} -2.14674e28 q^{56} +8.67825e28 q^{57} +8.67826e27 q^{58} -1.85966e29 q^{59} -4.66666e28 q^{60} -1.55295e29 q^{61} +2.98774e29 q^{62} +3.72599e28 q^{63} +7.92282e28 q^{64} +5.97918e29 q^{65} -5.33622e29 q^{66} -2.16431e30 q^{67} -8.36261e29 q^{68} +2.87482e29 q^{69} -7.62676e29 q^{70} -3.10911e30 q^{71} -1.37512e29 q^{72} -7.37969e30 q^{73} -3.03104e30 q^{74} -1.65793e30 q^{75} -5.23439e30 q^{76} -8.72103e30 q^{77} -1.82864e31 q^{78} +6.89383e30 q^{79} +2.81475e30 q^{80} -2.79487e31 q^{81} -2.78563e31 q^{82} +5.09132e31 q^{83} +2.33253e31 q^{84} -2.97099e31 q^{85} +3.90516e31 q^{86} -9.42929e30 q^{87} +3.21860e31 q^{88} -2.16513e32 q^{89} -4.88541e30 q^{90} -2.98856e32 q^{91} -1.73398e31 q^{92} -3.24631e32 q^{93} -2.01886e32 q^{94} -1.85963e32 q^{95} -8.60847e31 q^{96} +1.46729e32 q^{97} -1.25452e32 q^{98} -5.58636e31 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 131072 q^{2} - 74648412 q^{3} + 8589934592 q^{4} + 305175781250 q^{5} - 4892158328832 q^{6} + 20327789415556 q^{7} + 562949953421312 q^{8} - 60\!\cdots\!74 q^{9} + 20\!\cdots\!00 q^{10} - 11\!\cdots\!76 q^{11}+ \cdots + 64\!\cdots\!12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 65536.0 0.707107
\(3\) −7.12076e7 −0.955049 −0.477524 0.878619i \(-0.658466\pi\)
−0.477524 + 0.878619i \(0.658466\pi\)
\(4\) 4.29497e9 0.500000
\(5\) 1.52588e11 0.447214
\(6\) −4.66666e12 −0.675321
\(7\) −7.62676e13 −0.867406 −0.433703 0.901056i \(-0.642793\pi\)
−0.433703 + 0.901056i \(0.642793\pi\)
\(8\) 2.81475e14 0.353553
\(9\) −4.88541e14 −0.0878820
\(10\) 1.00000e16 0.316228
\(11\) 1.14348e17 0.750323 0.375161 0.926959i \(-0.377587\pi\)
0.375161 + 0.926959i \(0.377587\pi\)
\(12\) −3.05834e17 −0.477524
\(13\) 3.91852e18 1.63326 0.816632 0.577159i \(-0.195839\pi\)
0.816632 + 0.577159i \(0.195839\pi\)
\(14\) −4.99828e18 −0.613349
\(15\) −1.08654e19 −0.427111
\(16\) 1.84467e19 0.250000
\(17\) −1.94707e20 −0.970453 −0.485227 0.874388i \(-0.661263\pi\)
−0.485227 + 0.874388i \(0.661263\pi\)
\(18\) −3.20170e19 −0.0621420
\(19\) −1.21873e21 −0.969330 −0.484665 0.874700i \(-0.661058\pi\)
−0.484665 + 0.874700i \(0.661058\pi\)
\(20\) 6.55360e20 0.223607
\(21\) 5.43083e21 0.828415
\(22\) 7.49389e21 0.530558
\(23\) −4.03724e21 −0.137270 −0.0686350 0.997642i \(-0.521864\pi\)
−0.0686350 + 0.997642i \(0.521864\pi\)
\(24\) −2.00432e22 −0.337661
\(25\) 2.32831e22 0.200000
\(26\) 2.56804e23 1.15489
\(27\) 4.30635e23 1.03898
\(28\) −3.27567e23 −0.433703
\(29\) 1.32420e23 0.0982621 0.0491310 0.998792i \(-0.484355\pi\)
0.0491310 + 0.998792i \(0.484355\pi\)
\(30\) −7.12076e23 −0.302013
\(31\) 4.55893e24 1.12563 0.562814 0.826583i \(-0.309719\pi\)
0.562814 + 0.826583i \(0.309719\pi\)
\(32\) 1.20893e24 0.176777
\(33\) −8.14242e24 −0.716595
\(34\) −1.27603e25 −0.686214
\(35\) −1.16375e25 −0.387916
\(36\) −2.09827e24 −0.0439410
\(37\) −4.62500e25 −0.616288 −0.308144 0.951340i \(-0.599708\pi\)
−0.308144 + 0.951340i \(0.599708\pi\)
\(38\) −7.98704e25 −0.685420
\(39\) −2.79028e26 −1.55985
\(40\) 4.29497e25 0.158114
\(41\) −4.25054e26 −1.04114 −0.520571 0.853818i \(-0.674281\pi\)
−0.520571 + 0.853818i \(0.674281\pi\)
\(42\) 3.55915e26 0.585778
\(43\) 5.95880e26 0.665165 0.332582 0.943074i \(-0.392080\pi\)
0.332582 + 0.943074i \(0.392080\pi\)
\(44\) 4.91120e26 0.375161
\(45\) −7.45455e25 −0.0393020
\(46\) −2.64585e26 −0.0970645
\(47\) −3.08054e27 −0.792524 −0.396262 0.918138i \(-0.629693\pi\)
−0.396262 + 0.918138i \(0.629693\pi\)
\(48\) −1.31355e27 −0.238762
\(49\) −1.91424e27 −0.247606
\(50\) 1.52588e27 0.141421
\(51\) 1.38646e28 0.926830
\(52\) 1.68299e28 0.816632
\(53\) −5.55613e28 −1.96889 −0.984443 0.175706i \(-0.943779\pi\)
−0.984443 + 0.175706i \(0.943779\pi\)
\(54\) 2.82221e28 0.734670
\(55\) 1.74481e28 0.335555
\(56\) −2.14674e28 −0.306674
\(57\) 8.67825e28 0.925757
\(58\) 8.67826e27 0.0694818
\(59\) −1.85966e29 −1.12299 −0.561494 0.827481i \(-0.689773\pi\)
−0.561494 + 0.827481i \(0.689773\pi\)
\(60\) −4.66666e28 −0.213555
\(61\) −1.55295e29 −0.541023 −0.270511 0.962717i \(-0.587193\pi\)
−0.270511 + 0.962717i \(0.587193\pi\)
\(62\) 2.98774e29 0.795940
\(63\) 3.72599e28 0.0762294
\(64\) 7.92282e28 0.125000
\(65\) 5.97918e29 0.730418
\(66\) −5.33622e29 −0.506709
\(67\) −2.16431e30 −1.60356 −0.801778 0.597622i \(-0.796112\pi\)
−0.801778 + 0.597622i \(0.796112\pi\)
\(68\) −8.36261e29 −0.485227
\(69\) 2.87482e29 0.131099
\(70\) −7.62676e29 −0.274298
\(71\) −3.10911e30 −0.884856 −0.442428 0.896804i \(-0.645883\pi\)
−0.442428 + 0.896804i \(0.645883\pi\)
\(72\) −1.37512e29 −0.0310710
\(73\) −7.37969e30 −1.32804 −0.664020 0.747715i \(-0.731151\pi\)
−0.664020 + 0.747715i \(0.731151\pi\)
\(74\) −3.03104e30 −0.435781
\(75\) −1.65793e30 −0.191010
\(76\) −5.23439e30 −0.484665
\(77\) −8.72103e30 −0.650835
\(78\) −1.82864e31 −1.10298
\(79\) 6.89383e30 0.336988 0.168494 0.985703i \(-0.446110\pi\)
0.168494 + 0.985703i \(0.446110\pi\)
\(80\) 2.81475e30 0.111803
\(81\) −2.79487e31 −0.904395
\(82\) −2.78563e31 −0.736199
\(83\) 5.09132e31 1.10165 0.550823 0.834622i \(-0.314314\pi\)
0.550823 + 0.834622i \(0.314314\pi\)
\(84\) 2.33253e31 0.414208
\(85\) −2.97099e31 −0.434000
\(86\) 3.90516e31 0.470342
\(87\) −9.42929e30 −0.0938450
\(88\) 3.21860e31 0.265279
\(89\) −2.16513e32 −1.48098 −0.740489 0.672069i \(-0.765406\pi\)
−0.740489 + 0.672069i \(0.765406\pi\)
\(90\) −4.88541e30 −0.0277907
\(91\) −2.98856e32 −1.41670
\(92\) −1.73398e31 −0.0686350
\(93\) −3.24631e32 −1.07503
\(94\) −2.01886e32 −0.560399
\(95\) −1.85963e32 −0.433498
\(96\) −8.60847e31 −0.168830
\(97\) 1.46729e32 0.242539 0.121270 0.992620i \(-0.461303\pi\)
0.121270 + 0.992620i \(0.461303\pi\)
\(98\) −1.25452e32 −0.175084
\(99\) −5.58636e31 −0.0659399
\(100\) 1.00000e32 0.100000
\(101\) 1.73454e32 0.147191 0.0735957 0.997288i \(-0.476553\pi\)
0.0735957 + 0.997288i \(0.476553\pi\)
\(102\) 9.08632e32 0.655368
\(103\) 2.04881e33 1.25802 0.629010 0.777397i \(-0.283460\pi\)
0.629010 + 0.777397i \(0.283460\pi\)
\(104\) 1.10296e33 0.577446
\(105\) 8.28679e32 0.370479
\(106\) −3.64126e33 −1.39221
\(107\) 2.99134e33 0.979564 0.489782 0.871845i \(-0.337076\pi\)
0.489782 + 0.871845i \(0.337076\pi\)
\(108\) 1.84956e33 0.519490
\(109\) 1.01197e33 0.244136 0.122068 0.992522i \(-0.461047\pi\)
0.122068 + 0.992522i \(0.461047\pi\)
\(110\) 1.14348e33 0.237273
\(111\) 3.29335e33 0.588585
\(112\) −1.40689e33 −0.216852
\(113\) 1.11805e34 1.48822 0.744112 0.668055i \(-0.232873\pi\)
0.744112 + 0.668055i \(0.232873\pi\)
\(114\) 5.68738e33 0.654609
\(115\) −6.16034e32 −0.0613890
\(116\) 5.68739e32 0.0491310
\(117\) −1.91436e33 −0.143535
\(118\) −1.21875e34 −0.794073
\(119\) 1.48498e34 0.841777
\(120\) −3.05834e33 −0.151006
\(121\) −1.01498e34 −0.437015
\(122\) −1.01774e34 −0.382561
\(123\) 3.02670e34 0.994342
\(124\) 1.95805e34 0.562814
\(125\) 3.55271e33 0.0894427
\(126\) 2.44186e33 0.0539023
\(127\) −3.93900e33 −0.0763176 −0.0381588 0.999272i \(-0.512149\pi\)
−0.0381588 + 0.999272i \(0.512149\pi\)
\(128\) 5.19230e33 0.0883883
\(129\) −4.24312e34 −0.635265
\(130\) 3.91852e34 0.516484
\(131\) −1.47962e35 −1.71860 −0.859298 0.511475i \(-0.829099\pi\)
−0.859298 + 0.511475i \(0.829099\pi\)
\(132\) −3.49714e34 −0.358297
\(133\) 9.29493e34 0.840803
\(134\) −1.41840e35 −1.13389
\(135\) 6.57097e34 0.464646
\(136\) −5.48052e34 −0.343107
\(137\) −1.94112e35 −1.07687 −0.538433 0.842668i \(-0.680984\pi\)
−0.538433 + 0.842668i \(0.680984\pi\)
\(138\) 1.88404e34 0.0927013
\(139\) −2.46485e35 −1.07658 −0.538290 0.842760i \(-0.680929\pi\)
−0.538290 + 0.842760i \(0.680929\pi\)
\(140\) −4.99828e34 −0.193958
\(141\) 2.19358e35 0.756899
\(142\) −2.03758e35 −0.625688
\(143\) 4.48074e35 1.22548
\(144\) −9.01200e33 −0.0219705
\(145\) 2.02057e34 0.0439441
\(146\) −4.83636e35 −0.939066
\(147\) 1.36309e35 0.236476
\(148\) −1.98642e35 −0.308144
\(149\) 3.11387e35 0.432243 0.216121 0.976367i \(-0.430659\pi\)
0.216121 + 0.976367i \(0.430659\pi\)
\(150\) −1.08654e35 −0.135064
\(151\) 9.07358e34 0.101079 0.0505393 0.998722i \(-0.483906\pi\)
0.0505393 + 0.998722i \(0.483906\pi\)
\(152\) −3.43041e35 −0.342710
\(153\) 9.51225e34 0.0852854
\(154\) −5.71541e35 −0.460210
\(155\) 6.95638e35 0.503397
\(156\) −1.19842e36 −0.779923
\(157\) −1.67723e36 −0.982311 −0.491156 0.871072i \(-0.663425\pi\)
−0.491156 + 0.871072i \(0.663425\pi\)
\(158\) 4.51794e35 0.238286
\(159\) 3.95638e36 1.88038
\(160\) 1.84467e35 0.0790569
\(161\) 3.07911e35 0.119069
\(162\) −1.83164e36 −0.639504
\(163\) 1.61172e36 0.508386 0.254193 0.967154i \(-0.418190\pi\)
0.254193 + 0.967154i \(0.418190\pi\)
\(164\) −1.82559e36 −0.520571
\(165\) −1.24244e36 −0.320471
\(166\) 3.33665e36 0.778981
\(167\) 4.78201e36 1.01109 0.505544 0.862801i \(-0.331292\pi\)
0.505544 + 0.862801i \(0.331292\pi\)
\(168\) 1.52864e36 0.292889
\(169\) 9.59865e36 1.66755
\(170\) −1.94707e36 −0.306884
\(171\) 5.95398e35 0.0851867
\(172\) 2.55928e36 0.332582
\(173\) 3.21818e36 0.380057 0.190029 0.981779i \(-0.439142\pi\)
0.190029 + 0.981779i \(0.439142\pi\)
\(174\) −6.17958e35 −0.0663585
\(175\) −1.77574e36 −0.173481
\(176\) 2.10934e36 0.187581
\(177\) 1.32422e37 1.07251
\(178\) −1.41894e37 −1.04721
\(179\) 7.43648e35 0.0500372 0.0250186 0.999687i \(-0.492036\pi\)
0.0250186 + 0.999687i \(0.492036\pi\)
\(180\) −3.20170e35 −0.0196510
\(181\) 1.48729e37 0.833103 0.416551 0.909112i \(-0.363239\pi\)
0.416551 + 0.909112i \(0.363239\pi\)
\(182\) −1.95858e37 −1.00176
\(183\) 1.10582e37 0.516703
\(184\) −1.13638e36 −0.0485323
\(185\) −7.05719e36 −0.275612
\(186\) −2.12750e37 −0.760161
\(187\) −2.22643e37 −0.728153
\(188\) −1.32308e37 −0.396262
\(189\) −3.28435e37 −0.901218
\(190\) −1.21873e37 −0.306529
\(191\) −4.87779e36 −0.112505 −0.0562526 0.998417i \(-0.517915\pi\)
−0.0562526 + 0.998417i \(0.517915\pi\)
\(192\) −5.64165e36 −0.119381
\(193\) −7.11316e37 −1.38155 −0.690776 0.723069i \(-0.742731\pi\)
−0.690776 + 0.723069i \(0.742731\pi\)
\(194\) 9.61602e36 0.171501
\(195\) −4.25763e37 −0.697585
\(196\) −8.22161e36 −0.123803
\(197\) −1.21522e38 −1.68252 −0.841261 0.540630i \(-0.818186\pi\)
−0.841261 + 0.540630i \(0.818186\pi\)
\(198\) −3.66108e36 −0.0466265
\(199\) 1.59347e38 1.86754 0.933768 0.357878i \(-0.116500\pi\)
0.933768 + 0.357878i \(0.116500\pi\)
\(200\) 6.55360e36 0.0707107
\(201\) 1.54115e38 1.53147
\(202\) 1.13675e37 0.104080
\(203\) −1.00993e37 −0.0852331
\(204\) 5.95481e37 0.463415
\(205\) −6.48580e37 −0.465613
\(206\) 1.34271e38 0.889555
\(207\) 1.97236e36 0.0120636
\(208\) 7.22839e37 0.408316
\(209\) −1.39359e38 −0.727311
\(210\) 5.43083e37 0.261968
\(211\) −1.86059e38 −0.829832 −0.414916 0.909860i \(-0.636189\pi\)
−0.414916 + 0.909860i \(0.636189\pi\)
\(212\) −2.38634e38 −0.984443
\(213\) 2.21392e38 0.845081
\(214\) 1.96040e38 0.692656
\(215\) 9.09241e37 0.297471
\(216\) 1.21213e38 0.367335
\(217\) −3.47699e38 −0.976378
\(218\) 6.63207e37 0.172630
\(219\) 5.25490e38 1.26834
\(220\) 7.49389e37 0.167777
\(221\) −7.62963e38 −1.58501
\(222\) 2.15833e38 0.416192
\(223\) 2.35939e38 0.422444 0.211222 0.977438i \(-0.432256\pi\)
0.211222 + 0.977438i \(0.432256\pi\)
\(224\) −9.22019e37 −0.153337
\(225\) −1.13747e37 −0.0175764
\(226\) 7.32727e38 1.05233
\(227\) 3.89426e38 0.519994 0.259997 0.965609i \(-0.416278\pi\)
0.259997 + 0.965609i \(0.416278\pi\)
\(228\) 3.72728e38 0.462879
\(229\) −6.42546e38 −0.742368 −0.371184 0.928559i \(-0.621048\pi\)
−0.371184 + 0.928559i \(0.621048\pi\)
\(230\) −4.03724e37 −0.0434086
\(231\) 6.21003e38 0.621579
\(232\) 3.72728e37 0.0347409
\(233\) 9.96582e38 0.865248 0.432624 0.901574i \(-0.357588\pi\)
0.432624 + 0.901574i \(0.357588\pi\)
\(234\) −1.25459e38 −0.101494
\(235\) −4.70053e38 −0.354427
\(236\) −7.98718e38 −0.561494
\(237\) −4.90893e38 −0.321839
\(238\) 9.73200e38 0.595226
\(239\) 2.06735e39 1.17991 0.589956 0.807435i \(-0.299145\pi\)
0.589956 + 0.807435i \(0.299145\pi\)
\(240\) −2.00432e38 −0.106778
\(241\) −2.94996e39 −1.46735 −0.733677 0.679498i \(-0.762197\pi\)
−0.733677 + 0.679498i \(0.762197\pi\)
\(242\) −6.65174e38 −0.309017
\(243\) −4.03771e38 −0.175239
\(244\) −6.66987e38 −0.270511
\(245\) −2.92090e38 −0.110733
\(246\) 1.98358e39 0.703106
\(247\) −4.77560e39 −1.58317
\(248\) 1.28323e39 0.397970
\(249\) −3.62541e39 −1.05213
\(250\) 2.32831e38 0.0632456
\(251\) −5.46260e39 −1.38926 −0.694630 0.719368i \(-0.744432\pi\)
−0.694630 + 0.719368i \(0.744432\pi\)
\(252\) 1.60030e38 0.0381147
\(253\) −4.61649e38 −0.102997
\(254\) −2.58146e38 −0.0539647
\(255\) 2.11557e39 0.414491
\(256\) 3.40282e38 0.0625000
\(257\) 1.10808e40 1.90843 0.954213 0.299129i \(-0.0966960\pi\)
0.954213 + 0.299129i \(0.0966960\pi\)
\(258\) −2.78077e39 −0.449200
\(259\) 3.52738e39 0.534572
\(260\) 2.56804e39 0.365209
\(261\) −6.46925e37 −0.00863547
\(262\) −9.69684e39 −1.21523
\(263\) −1.40170e40 −1.64963 −0.824815 0.565403i \(-0.808721\pi\)
−0.824815 + 0.565403i \(0.808721\pi\)
\(264\) −2.29189e39 −0.253355
\(265\) −8.47798e39 −0.880512
\(266\) 6.09153e39 0.594538
\(267\) 1.54174e40 1.41441
\(268\) −9.29562e39 −0.801778
\(269\) −6.31508e39 −0.512232 −0.256116 0.966646i \(-0.582443\pi\)
−0.256116 + 0.966646i \(0.582443\pi\)
\(270\) 4.30635e39 0.328554
\(271\) 1.08648e40 0.779879 0.389939 0.920841i \(-0.372496\pi\)
0.389939 + 0.920841i \(0.372496\pi\)
\(272\) −3.59171e39 −0.242613
\(273\) 2.12808e40 1.35302
\(274\) −1.27213e40 −0.761460
\(275\) 2.66237e39 0.150065
\(276\) 1.23473e39 0.0655497
\(277\) 2.33494e40 1.16778 0.583889 0.811834i \(-0.301530\pi\)
0.583889 + 0.811834i \(0.301530\pi\)
\(278\) −1.61536e40 −0.761257
\(279\) −2.22723e39 −0.0989225
\(280\) −3.27567e39 −0.137149
\(281\) 3.50713e40 1.38451 0.692257 0.721652i \(-0.256617\pi\)
0.692257 + 0.721652i \(0.256617\pi\)
\(282\) 1.43758e40 0.535208
\(283\) 2.34001e40 0.821753 0.410876 0.911691i \(-0.365223\pi\)
0.410876 + 0.911691i \(0.365223\pi\)
\(284\) −1.33535e40 −0.442428
\(285\) 1.32420e40 0.414011
\(286\) 2.93649e40 0.866542
\(287\) 3.24178e40 0.903093
\(288\) −5.90610e38 −0.0155355
\(289\) −2.34364e39 −0.0582205
\(290\) 1.32420e39 0.0310732
\(291\) −1.04482e40 −0.231637
\(292\) −3.16955e40 −0.664020
\(293\) −3.11041e40 −0.615889 −0.307944 0.951404i \(-0.599641\pi\)
−0.307944 + 0.951404i \(0.599641\pi\)
\(294\) 8.93312e39 0.167214
\(295\) −2.83762e40 −0.502216
\(296\) −1.30182e40 −0.217891
\(297\) 4.92421e40 0.779571
\(298\) 2.04071e40 0.305642
\(299\) −1.58200e40 −0.224198
\(300\) −7.12076e39 −0.0955049
\(301\) −4.54463e40 −0.576968
\(302\) 5.94646e39 0.0714733
\(303\) −1.23513e40 −0.140575
\(304\) −2.24815e40 −0.242333
\(305\) −2.36962e40 −0.241953
\(306\) 6.23395e39 0.0603059
\(307\) −1.56830e41 −1.43762 −0.718811 0.695206i \(-0.755313\pi\)
−0.718811 + 0.695206i \(0.755313\pi\)
\(308\) −3.74565e40 −0.325417
\(309\) −1.45891e41 −1.20147
\(310\) 4.55893e40 0.355955
\(311\) −1.90389e41 −1.40960 −0.704798 0.709408i \(-0.748963\pi\)
−0.704798 + 0.709408i \(0.748963\pi\)
\(312\) −7.85394e40 −0.551489
\(313\) 3.43316e40 0.228671 0.114336 0.993442i \(-0.463526\pi\)
0.114336 + 0.993442i \(0.463526\pi\)
\(314\) −1.09919e41 −0.694599
\(315\) 5.68541e39 0.0340908
\(316\) 2.96088e40 0.168494
\(317\) 2.69572e41 1.45612 0.728060 0.685514i \(-0.240422\pi\)
0.728060 + 0.685514i \(0.240422\pi\)
\(318\) 2.59286e41 1.32963
\(319\) 1.51419e40 0.0737283
\(320\) 1.20893e40 0.0559017
\(321\) −2.13006e41 −0.935531
\(322\) 2.01792e40 0.0841944
\(323\) 2.37295e41 0.940689
\(324\) −1.20039e41 −0.452197
\(325\) 9.12351e40 0.326653
\(326\) 1.05626e41 0.359483
\(327\) −7.20602e40 −0.233162
\(328\) −1.19642e41 −0.368099
\(329\) 2.34946e41 0.687440
\(330\) −8.14242e40 −0.226607
\(331\) 1.12474e41 0.297777 0.148888 0.988854i \(-0.452430\pi\)
0.148888 + 0.988854i \(0.452430\pi\)
\(332\) 2.18671e41 0.550823
\(333\) 2.25950e40 0.0541606
\(334\) 3.13394e41 0.714947
\(335\) −3.30247e41 −0.717132
\(336\) 1.00181e41 0.207104
\(337\) −8.06548e41 −1.58759 −0.793793 0.608188i \(-0.791897\pi\)
−0.793793 + 0.608188i \(0.791897\pi\)
\(338\) 6.29057e41 1.17914
\(339\) −7.96139e41 −1.42133
\(340\) −1.27603e41 −0.217000
\(341\) 5.21304e41 0.844585
\(342\) 3.90200e40 0.0602361
\(343\) 7.35619e41 1.08218
\(344\) 1.67725e41 0.235171
\(345\) 4.38663e40 0.0586295
\(346\) 2.10906e41 0.268741
\(347\) −6.20227e41 −0.753553 −0.376777 0.926304i \(-0.622968\pi\)
−0.376777 + 0.926304i \(0.622968\pi\)
\(348\) −4.04985e40 −0.0469225
\(349\) −8.80550e41 −0.973048 −0.486524 0.873667i \(-0.661735\pi\)
−0.486524 + 0.873667i \(0.661735\pi\)
\(350\) −1.16375e41 −0.122670
\(351\) 1.68745e42 1.69693
\(352\) 1.38238e41 0.132640
\(353\) 1.63312e42 1.49533 0.747663 0.664078i \(-0.231176\pi\)
0.747663 + 0.664078i \(0.231176\pi\)
\(354\) 8.67840e41 0.758378
\(355\) −4.74412e41 −0.395720
\(356\) −9.29915e41 −0.740489
\(357\) −1.05742e42 −0.803938
\(358\) 4.87357e40 0.0353816
\(359\) −4.69226e41 −0.325330 −0.162665 0.986681i \(-0.552009\pi\)
−0.162665 + 0.986681i \(0.552009\pi\)
\(360\) −2.09827e40 −0.0138954
\(361\) −9.54774e40 −0.0603993
\(362\) 9.74709e41 0.589092
\(363\) 7.22739e41 0.417371
\(364\) −1.28358e42 −0.708352
\(365\) −1.12605e42 −0.593918
\(366\) 7.24709e41 0.365364
\(367\) 2.12758e42 1.02541 0.512704 0.858565i \(-0.328644\pi\)
0.512704 + 0.858565i \(0.328644\pi\)
\(368\) −7.44740e40 −0.0343175
\(369\) 2.07656e41 0.0914977
\(370\) −4.62500e41 −0.194887
\(371\) 4.23753e42 1.70782
\(372\) −1.39428e42 −0.537515
\(373\) −3.78223e42 −1.39493 −0.697465 0.716619i \(-0.745689\pi\)
−0.697465 + 0.716619i \(0.745689\pi\)
\(374\) −1.45911e42 −0.514882
\(375\) −2.52980e41 −0.0854222
\(376\) −8.67095e41 −0.280199
\(377\) 5.18889e41 0.160488
\(378\) −2.15243e42 −0.637257
\(379\) −6.08338e42 −1.72424 −0.862121 0.506702i \(-0.830864\pi\)
−0.862121 + 0.506702i \(0.830864\pi\)
\(380\) −7.98704e41 −0.216749
\(381\) 2.80487e41 0.0728870
\(382\) −3.19671e41 −0.0795532
\(383\) 8.15229e41 0.194312 0.0971560 0.995269i \(-0.469025\pi\)
0.0971560 + 0.995269i \(0.469025\pi\)
\(384\) −3.69731e41 −0.0844152
\(385\) −1.33072e42 −0.291062
\(386\) −4.66168e42 −0.976905
\(387\) −2.91112e41 −0.0584560
\(388\) 6.30195e41 0.121270
\(389\) −2.71183e42 −0.500143 −0.250071 0.968227i \(-0.580454\pi\)
−0.250071 + 0.968227i \(0.580454\pi\)
\(390\) −2.79028e42 −0.493267
\(391\) 7.86080e41 0.133214
\(392\) −5.38811e41 −0.0875420
\(393\) 1.05360e43 1.64134
\(394\) −7.96404e42 −1.18972
\(395\) 1.05192e42 0.150705
\(396\) −2.39932e41 −0.0329699
\(397\) 3.40205e42 0.448432 0.224216 0.974539i \(-0.428018\pi\)
0.224216 + 0.974539i \(0.428018\pi\)
\(398\) 1.04430e43 1.32055
\(399\) −6.61870e42 −0.803008
\(400\) 4.29497e41 0.0500000
\(401\) 5.96117e42 0.665961 0.332981 0.942934i \(-0.391946\pi\)
0.332981 + 0.942934i \(0.391946\pi\)
\(402\) 1.01001e43 1.08292
\(403\) 1.78643e43 1.83845
\(404\) 7.44981e41 0.0735957
\(405\) −4.26463e42 −0.404458
\(406\) −6.61870e41 −0.0602689
\(407\) −5.28858e42 −0.462415
\(408\) 3.90254e42 0.327684
\(409\) −2.07282e43 −1.67158 −0.835788 0.549052i \(-0.814989\pi\)
−0.835788 + 0.549052i \(0.814989\pi\)
\(410\) −4.25054e42 −0.329238
\(411\) 1.38222e43 1.02846
\(412\) 8.79958e42 0.629010
\(413\) 1.41832e43 0.974087
\(414\) 1.29261e41 0.00853022
\(415\) 7.76874e42 0.492671
\(416\) 4.73720e42 0.288723
\(417\) 1.75516e43 1.02819
\(418\) −9.13300e42 −0.514286
\(419\) −7.96843e42 −0.431362 −0.215681 0.976464i \(-0.569197\pi\)
−0.215681 + 0.976464i \(0.569197\pi\)
\(420\) 3.55915e42 0.185239
\(421\) −1.40035e43 −0.700779 −0.350390 0.936604i \(-0.613951\pi\)
−0.350390 + 0.936604i \(0.613951\pi\)
\(422\) −1.21936e43 −0.586780
\(423\) 1.50497e42 0.0696486
\(424\) −1.56391e43 −0.696106
\(425\) −4.53338e42 −0.194091
\(426\) 1.45091e43 0.597562
\(427\) 1.18440e43 0.469287
\(428\) 1.28477e43 0.489782
\(429\) −3.19062e43 −1.17039
\(430\) 5.95880e42 0.210344
\(431\) −2.19278e42 −0.0744937 −0.0372468 0.999306i \(-0.511859\pi\)
−0.0372468 + 0.999306i \(0.511859\pi\)
\(432\) 7.94382e42 0.259745
\(433\) 4.94849e43 1.55748 0.778739 0.627348i \(-0.215859\pi\)
0.778739 + 0.627348i \(0.215859\pi\)
\(434\) −2.27868e43 −0.690403
\(435\) −1.43880e42 −0.0419688
\(436\) 4.34639e42 0.122068
\(437\) 4.92029e42 0.133060
\(438\) 3.44385e43 0.896854
\(439\) −5.29628e43 −1.32833 −0.664165 0.747586i \(-0.731213\pi\)
−0.664165 + 0.747586i \(0.731213\pi\)
\(440\) 4.91120e42 0.118636
\(441\) 9.35187e41 0.0217601
\(442\) −5.00016e43 −1.12077
\(443\) −6.57483e43 −1.41979 −0.709893 0.704309i \(-0.751257\pi\)
−0.709893 + 0.704309i \(0.751257\pi\)
\(444\) 1.41448e43 0.294292
\(445\) −3.30372e43 −0.662313
\(446\) 1.54625e43 0.298713
\(447\) −2.21731e43 −0.412813
\(448\) −6.04254e42 −0.108426
\(449\) −7.00108e43 −1.21088 −0.605439 0.795892i \(-0.707002\pi\)
−0.605439 + 0.795892i \(0.707002\pi\)
\(450\) −7.45455e41 −0.0124284
\(451\) −4.86039e43 −0.781193
\(452\) 4.80200e43 0.744112
\(453\) −6.46108e42 −0.0965349
\(454\) 2.55214e43 0.367692
\(455\) −4.56018e43 −0.633569
\(456\) 2.44271e43 0.327305
\(457\) −4.08696e42 −0.0528181 −0.0264091 0.999651i \(-0.508407\pi\)
−0.0264091 + 0.999651i \(0.508407\pi\)
\(458\) −4.21099e43 −0.524933
\(459\) −8.38477e43 −1.00828
\(460\) −2.64585e42 −0.0306945
\(461\) 5.94001e43 0.664848 0.332424 0.943130i \(-0.392134\pi\)
0.332424 + 0.943130i \(0.392134\pi\)
\(462\) 4.06981e43 0.439523
\(463\) 1.30594e44 1.36093 0.680466 0.732780i \(-0.261777\pi\)
0.680466 + 0.732780i \(0.261777\pi\)
\(464\) 2.44271e42 0.0245655
\(465\) −4.95347e43 −0.480768
\(466\) 6.53120e43 0.611823
\(467\) −3.66775e43 −0.331643 −0.165822 0.986156i \(-0.553028\pi\)
−0.165822 + 0.986156i \(0.553028\pi\)
\(468\) −8.22210e42 −0.0717673
\(469\) 1.65066e44 1.39093
\(470\) −3.08054e43 −0.250618
\(471\) 1.19432e44 0.938155
\(472\) −5.23448e43 −0.397036
\(473\) 6.81375e43 0.499088
\(474\) −3.21712e43 −0.227575
\(475\) −2.83757e43 −0.193866
\(476\) 6.37796e43 0.420889
\(477\) 2.71440e43 0.173030
\(478\) 1.35486e44 0.834324
\(479\) 1.79143e44 1.06577 0.532885 0.846188i \(-0.321108\pi\)
0.532885 + 0.846188i \(0.321108\pi\)
\(480\) −1.31355e43 −0.0755032
\(481\) −1.81231e44 −1.00656
\(482\) −1.93329e44 −1.03758
\(483\) −2.19256e43 −0.113717
\(484\) −4.35928e43 −0.218508
\(485\) 2.23890e43 0.108467
\(486\) −2.64615e43 −0.123913
\(487\) 2.85315e44 1.29151 0.645753 0.763546i \(-0.276543\pi\)
0.645753 + 0.763546i \(0.276543\pi\)
\(488\) −4.37117e43 −0.191280
\(489\) −1.14767e44 −0.485533
\(490\) −1.91424e43 −0.0783000
\(491\) 1.91812e44 0.758631 0.379316 0.925267i \(-0.376159\pi\)
0.379316 + 0.925267i \(0.376159\pi\)
\(492\) 1.29996e44 0.497171
\(493\) −2.57831e43 −0.0953587
\(494\) −3.12974e44 −1.11947
\(495\) −8.52411e42 −0.0294892
\(496\) 8.40975e43 0.281407
\(497\) 2.37124e44 0.767530
\(498\) −2.37595e44 −0.743965
\(499\) 2.99230e44 0.906456 0.453228 0.891395i \(-0.350272\pi\)
0.453228 + 0.891395i \(0.350272\pi\)
\(500\) 1.52588e43 0.0447214
\(501\) −3.40515e44 −0.965638
\(502\) −3.57997e44 −0.982355
\(503\) 2.22024e44 0.589559 0.294780 0.955565i \(-0.404754\pi\)
0.294780 + 0.955565i \(0.404754\pi\)
\(504\) 1.04877e43 0.0269512
\(505\) 2.64670e43 0.0658260
\(506\) −3.02547e43 −0.0728297
\(507\) −6.83496e44 −1.59259
\(508\) −1.69179e43 −0.0381588
\(509\) 7.25178e44 1.58344 0.791720 0.610885i \(-0.209186\pi\)
0.791720 + 0.610885i \(0.209186\pi\)
\(510\) 1.38646e44 0.293089
\(511\) 5.62832e44 1.15195
\(512\) 2.23007e43 0.0441942
\(513\) −5.24826e44 −1.00711
\(514\) 7.26192e44 1.34946
\(515\) 3.12624e44 0.562604
\(516\) −1.82240e44 −0.317632
\(517\) −3.52253e44 −0.594649
\(518\) 2.31170e44 0.377999
\(519\) −2.29158e44 −0.362973
\(520\) 1.68299e44 0.258242
\(521\) −1.11989e45 −1.66476 −0.832382 0.554202i \(-0.813023\pi\)
−0.832382 + 0.554202i \(0.813023\pi\)
\(522\) −4.23969e42 −0.00610620
\(523\) 1.32139e45 1.84396 0.921981 0.387234i \(-0.126569\pi\)
0.921981 + 0.387234i \(0.126569\pi\)
\(524\) −6.35492e44 −0.859298
\(525\) 1.26446e44 0.165683
\(526\) −9.18621e44 −1.16646
\(527\) −8.87657e44 −1.09237
\(528\) −1.50201e44 −0.179149
\(529\) −8.48706e44 −0.981157
\(530\) −5.55613e44 −0.622616
\(531\) 9.08521e43 0.0986905
\(532\) 3.99214e44 0.420401
\(533\) −1.66558e45 −1.70046
\(534\) 1.01039e45 1.00014
\(535\) 4.56442e44 0.438074
\(536\) −6.09198e44 −0.566943
\(537\) −5.29534e43 −0.0477879
\(538\) −4.13865e44 −0.362203
\(539\) −2.18889e44 −0.185785
\(540\) 2.82221e44 0.232323
\(541\) 7.54315e42 0.00602279 0.00301140 0.999995i \(-0.499041\pi\)
0.00301140 + 0.999995i \(0.499041\pi\)
\(542\) 7.12033e44 0.551458
\(543\) −1.05906e45 −0.795654
\(544\) −2.35386e44 −0.171554
\(545\) 1.54415e44 0.109181
\(546\) 1.39466e45 0.956730
\(547\) 1.65387e45 1.10080 0.550402 0.834900i \(-0.314475\pi\)
0.550402 + 0.834900i \(0.314475\pi\)
\(548\) −8.33703e44 −0.538433
\(549\) 7.58681e43 0.0475462
\(550\) 1.74481e44 0.106112
\(551\) −1.61383e44 −0.0952484
\(552\) 8.09191e43 0.0463507
\(553\) −5.25776e44 −0.292305
\(554\) 1.53023e45 0.825744
\(555\) 5.02526e44 0.263223
\(556\) −1.05864e45 −0.538290
\(557\) 8.23792e44 0.406637 0.203318 0.979113i \(-0.434827\pi\)
0.203318 + 0.979113i \(0.434827\pi\)
\(558\) −1.45964e44 −0.0699488
\(559\) 2.33497e45 1.08639
\(560\) −2.14674e44 −0.0969790
\(561\) 1.58539e45 0.695422
\(562\) 2.29843e45 0.978999
\(563\) −2.90289e45 −1.20072 −0.600360 0.799730i \(-0.704976\pi\)
−0.600360 + 0.799730i \(0.704976\pi\)
\(564\) 9.42135e44 0.378449
\(565\) 1.70601e45 0.665554
\(566\) 1.53355e45 0.581067
\(567\) 2.13158e45 0.784478
\(568\) −8.75135e44 −0.312844
\(569\) 3.12951e45 1.08674 0.543369 0.839494i \(-0.317149\pi\)
0.543369 + 0.839494i \(0.317149\pi\)
\(570\) 8.67825e44 0.292750
\(571\) −3.84526e45 −1.26017 −0.630086 0.776525i \(-0.716980\pi\)
−0.630086 + 0.776525i \(0.716980\pi\)
\(572\) 1.92446e45 0.612738
\(573\) 3.47336e44 0.107448
\(574\) 2.12454e45 0.638583
\(575\) −9.39994e43 −0.0274540
\(576\) −3.87062e43 −0.0109853
\(577\) −2.53135e45 −0.698154 −0.349077 0.937094i \(-0.613505\pi\)
−0.349077 + 0.937094i \(0.613505\pi\)
\(578\) −1.53593e44 −0.0411681
\(579\) 5.06511e45 1.31945
\(580\) 8.67826e43 0.0219721
\(581\) −3.88303e45 −0.955575
\(582\) −6.84733e44 −0.163792
\(583\) −6.35331e45 −1.47730
\(584\) −2.07720e45 −0.469533
\(585\) −2.92108e44 −0.0641906
\(586\) −2.03844e45 −0.435499
\(587\) −1.89473e45 −0.393568 −0.196784 0.980447i \(-0.563050\pi\)
−0.196784 + 0.980447i \(0.563050\pi\)
\(588\) 5.85441e44 0.118238
\(589\) −5.55609e45 −1.09111
\(590\) −1.85966e45 −0.355120
\(591\) 8.65326e45 1.60689
\(592\) −8.53162e44 −0.154072
\(593\) −9.96728e45 −1.75055 −0.875275 0.483626i \(-0.839320\pi\)
−0.875275 + 0.483626i \(0.839320\pi\)
\(594\) 3.22713e45 0.551240
\(595\) 2.26591e45 0.376454
\(596\) 1.33740e45 0.216121
\(597\) −1.13467e46 −1.78359
\(598\) −1.03678e45 −0.158532
\(599\) 5.94982e45 0.885037 0.442518 0.896759i \(-0.354085\pi\)
0.442518 + 0.896759i \(0.354085\pi\)
\(600\) −4.66666e44 −0.0675321
\(601\) 1.42873e45 0.201150 0.100575 0.994929i \(-0.467932\pi\)
0.100575 + 0.994929i \(0.467932\pi\)
\(602\) −2.97837e45 −0.407978
\(603\) 1.05735e45 0.140924
\(604\) 3.89707e44 0.0505393
\(605\) −1.54873e45 −0.195439
\(606\) −8.09452e44 −0.0994015
\(607\) 5.41904e45 0.647603 0.323801 0.946125i \(-0.395039\pi\)
0.323801 + 0.946125i \(0.395039\pi\)
\(608\) −1.47335e45 −0.171355
\(609\) 7.19150e44 0.0814018
\(610\) −1.55295e45 −0.171086
\(611\) −1.20712e46 −1.29440
\(612\) 4.08548e44 0.0426427
\(613\) 1.66868e46 1.69541 0.847705 0.530467i \(-0.177984\pi\)
0.847705 + 0.530467i \(0.177984\pi\)
\(614\) −1.02780e46 −1.01655
\(615\) 4.61838e45 0.444683
\(616\) −2.45475e45 −0.230105
\(617\) −5.91960e45 −0.540240 −0.270120 0.962827i \(-0.587063\pi\)
−0.270120 + 0.962827i \(0.587063\pi\)
\(618\) −9.56111e45 −0.849568
\(619\) 8.04200e45 0.695774 0.347887 0.937536i \(-0.386899\pi\)
0.347887 + 0.937536i \(0.386899\pi\)
\(620\) 2.98774e45 0.251698
\(621\) −1.73858e45 −0.142621
\(622\) −1.24773e46 −0.996736
\(623\) 1.65129e46 1.28461
\(624\) −5.14716e45 −0.389962
\(625\) 5.42101e44 0.0400000
\(626\) 2.24995e45 0.161695
\(627\) 9.92339e45 0.694617
\(628\) −7.20366e45 −0.491156
\(629\) 9.00521e45 0.598078
\(630\) 3.72599e44 0.0241059
\(631\) −1.17738e46 −0.742050 −0.371025 0.928623i \(-0.620994\pi\)
−0.371025 + 0.928623i \(0.620994\pi\)
\(632\) 1.94044e45 0.119143
\(633\) 1.32488e46 0.792530
\(634\) 1.76667e46 1.02963
\(635\) −6.01044e44 −0.0341303
\(636\) 1.69925e46 0.940191
\(637\) −7.50099e45 −0.404406
\(638\) 9.92339e44 0.0521338
\(639\) 1.51893e45 0.0777629
\(640\) 7.92282e44 0.0395285
\(641\) −1.99312e46 −0.969116 −0.484558 0.874759i \(-0.661019\pi\)
−0.484558 + 0.874759i \(0.661019\pi\)
\(642\) −1.39595e46 −0.661520
\(643\) 3.26909e46 1.50989 0.754946 0.655787i \(-0.227663\pi\)
0.754946 + 0.655787i \(0.227663\pi\)
\(644\) 1.32247e45 0.0595344
\(645\) −6.47448e45 −0.284099
\(646\) 1.55513e46 0.665168
\(647\) 1.84958e46 0.771174 0.385587 0.922672i \(-0.373999\pi\)
0.385587 + 0.922672i \(0.373999\pi\)
\(648\) −7.86685e45 −0.319752
\(649\) −2.12648e46 −0.842604
\(650\) 5.97918e45 0.230978
\(651\) 2.47588e46 0.932488
\(652\) 6.92228e45 0.254193
\(653\) −2.33266e46 −0.835189 −0.417594 0.908634i \(-0.637127\pi\)
−0.417594 + 0.908634i \(0.637127\pi\)
\(654\) −4.72254e45 −0.164870
\(655\) −2.25772e46 −0.768580
\(656\) −7.84086e45 −0.260286
\(657\) 3.60529e45 0.116711
\(658\) 1.53974e46 0.486094
\(659\) 1.31702e46 0.405493 0.202747 0.979231i \(-0.435013\pi\)
0.202747 + 0.979231i \(0.435013\pi\)
\(660\) −5.33622e45 −0.160236
\(661\) −8.17736e45 −0.239491 −0.119746 0.992805i \(-0.538208\pi\)
−0.119746 + 0.992805i \(0.538208\pi\)
\(662\) 7.37111e45 0.210560
\(663\) 5.43288e46 1.51376
\(664\) 1.43308e46 0.389491
\(665\) 1.41829e46 0.376019
\(666\) 1.48079e45 0.0382973
\(667\) −5.34611e44 −0.0134884
\(668\) 2.05386e46 0.505544
\(669\) −1.68006e46 −0.403455
\(670\) −2.16431e46 −0.507089
\(671\) −1.77576e46 −0.405942
\(672\) 6.56547e45 0.146445
\(673\) −4.48159e46 −0.975401 −0.487701 0.873011i \(-0.662164\pi\)
−0.487701 + 0.873011i \(0.662164\pi\)
\(674\) −5.28580e46 −1.12259
\(675\) 1.00265e46 0.207796
\(676\) 4.12259e46 0.833776
\(677\) −5.83822e46 −1.15231 −0.576153 0.817342i \(-0.695447\pi\)
−0.576153 + 0.817342i \(0.695447\pi\)
\(678\) −5.21757e46 −1.00503
\(679\) −1.11907e46 −0.210380
\(680\) −8.36261e45 −0.153442
\(681\) −2.77301e46 −0.496620
\(682\) 3.41642e46 0.597212
\(683\) 2.10568e46 0.359295 0.179647 0.983731i \(-0.442504\pi\)
0.179647 + 0.983731i \(0.442504\pi\)
\(684\) 2.55722e45 0.0425933
\(685\) −2.96191e46 −0.481589
\(686\) 4.82095e46 0.765218
\(687\) 4.57542e46 0.708997
\(688\) 1.09920e46 0.166291
\(689\) −2.17718e47 −3.21571
\(690\) 2.87482e45 0.0414573
\(691\) 8.94914e46 1.26007 0.630033 0.776568i \(-0.283041\pi\)
0.630033 + 0.776568i \(0.283041\pi\)
\(692\) 1.38220e46 0.190029
\(693\) 4.26058e45 0.0571967
\(694\) −4.06472e46 −0.532843
\(695\) −3.76106e46 −0.481461
\(696\) −2.65411e45 −0.0331792
\(697\) 8.27610e46 1.01038
\(698\) −5.77077e46 −0.688049
\(699\) −7.09642e46 −0.826354
\(700\) −7.62676e45 −0.0867406
\(701\) −6.16333e45 −0.0684650 −0.0342325 0.999414i \(-0.510899\pi\)
−0.0342325 + 0.999414i \(0.510899\pi\)
\(702\) 1.10589e47 1.19991
\(703\) 5.63661e46 0.597386
\(704\) 9.05956e45 0.0937904
\(705\) 3.34714e46 0.338495
\(706\) 1.07028e47 1.05736
\(707\) −1.32290e46 −0.127675
\(708\) 5.68748e46 0.536254
\(709\) −4.95391e46 −0.456336 −0.228168 0.973622i \(-0.573274\pi\)
−0.228168 + 0.973622i \(0.573274\pi\)
\(710\) −3.10911e46 −0.279816
\(711\) −3.36792e45 −0.0296151
\(712\) −6.09429e46 −0.523605
\(713\) −1.84055e46 −0.154515
\(714\) −6.92992e46 −0.568470
\(715\) 6.83706e46 0.548049
\(716\) 3.19395e45 0.0250186
\(717\) −1.47211e47 −1.12687
\(718\) −3.07512e46 −0.230043
\(719\) −2.13187e47 −1.55860 −0.779300 0.626651i \(-0.784425\pi\)
−0.779300 + 0.626651i \(0.784425\pi\)
\(720\) −1.37512e45 −0.00982551
\(721\) −1.56258e47 −1.09122
\(722\) −6.25721e45 −0.0427088
\(723\) 2.10060e47 1.40139
\(724\) 6.38785e46 0.416551
\(725\) 3.08314e45 0.0196524
\(726\) 4.73654e46 0.295126
\(727\) 1.31060e47 0.798273 0.399137 0.916891i \(-0.369310\pi\)
0.399137 + 0.916891i \(0.369310\pi\)
\(728\) −8.41205e46 −0.500880
\(729\) 1.84120e47 1.07176
\(730\) −7.37969e46 −0.419963
\(731\) −1.16022e47 −0.645511
\(732\) 4.74946e46 0.258352
\(733\) 3.54336e47 1.88451 0.942257 0.334892i \(-0.108700\pi\)
0.942257 + 0.334892i \(0.108700\pi\)
\(734\) 1.39433e47 0.725073
\(735\) 2.07990e46 0.105755
\(736\) −4.88073e45 −0.0242661
\(737\) −2.47483e47 −1.20319
\(738\) 1.36090e46 0.0646986
\(739\) 2.64104e47 1.22784 0.613919 0.789369i \(-0.289592\pi\)
0.613919 + 0.789369i \(0.289592\pi\)
\(740\) −3.03104e46 −0.137806
\(741\) 3.40059e47 1.51201
\(742\) 2.77711e47 1.20761
\(743\) 1.68076e47 0.714809 0.357404 0.933950i \(-0.383662\pi\)
0.357404 + 0.933950i \(0.383662\pi\)
\(744\) −9.13754e46 −0.380081
\(745\) 4.75139e46 0.193305
\(746\) −2.47872e47 −0.986364
\(747\) −2.48732e46 −0.0968149
\(748\) −9.56245e46 −0.364077
\(749\) −2.28142e47 −0.849680
\(750\) −1.65793e46 −0.0604026
\(751\) 4.23831e47 1.51055 0.755273 0.655410i \(-0.227504\pi\)
0.755273 + 0.655410i \(0.227504\pi\)
\(752\) −5.68259e46 −0.198131
\(753\) 3.88979e47 1.32681
\(754\) 3.40059e46 0.113482
\(755\) 1.38452e46 0.0452037
\(756\) −1.41062e47 −0.450609
\(757\) 4.49973e47 1.40639 0.703193 0.710999i \(-0.251757\pi\)
0.703193 + 0.710999i \(0.251757\pi\)
\(758\) −3.98681e47 −1.21922
\(759\) 3.28729e46 0.0983670
\(760\) −5.23439e46 −0.153265
\(761\) −3.43548e47 −0.984329 −0.492164 0.870502i \(-0.663794\pi\)
−0.492164 + 0.870502i \(0.663794\pi\)
\(762\) 1.83820e46 0.0515389
\(763\) −7.71808e46 −0.211765
\(764\) −2.09499e46 −0.0562526
\(765\) 1.45145e46 0.0381408
\(766\) 5.34268e46 0.137399
\(767\) −7.28711e47 −1.83414
\(768\) −2.42307e46 −0.0596905
\(769\) −7.98461e47 −1.92517 −0.962585 0.270981i \(-0.912652\pi\)
−0.962585 + 0.270981i \(0.912652\pi\)
\(770\) −8.72103e46 −0.205812
\(771\) −7.89038e47 −1.82264
\(772\) −3.05508e47 −0.690776
\(773\) 4.85460e47 1.07446 0.537232 0.843435i \(-0.319470\pi\)
0.537232 + 0.843435i \(0.319470\pi\)
\(774\) −1.90783e46 −0.0413346
\(775\) 1.06146e47 0.225126
\(776\) 4.13005e46 0.0857505
\(777\) −2.51176e47 −0.510542
\(778\) −1.77723e47 −0.353654
\(779\) 5.18024e47 1.00921
\(780\) −1.82864e47 −0.348792
\(781\) −3.55519e47 −0.663928
\(782\) 5.15165e46 0.0941966
\(783\) 5.70246e46 0.102092
\(784\) −3.53115e46 −0.0619016
\(785\) −2.55925e47 −0.439303
\(786\) 6.90489e47 1.16061
\(787\) −4.81834e47 −0.793075 −0.396538 0.918018i \(-0.629788\pi\)
−0.396538 + 0.918018i \(0.629788\pi\)
\(788\) −5.21931e47 −0.841261
\(789\) 9.98120e47 1.57548
\(790\) 6.89383e46 0.106565
\(791\) −8.52713e47 −1.29089
\(792\) −1.57242e46 −0.0233133
\(793\) −6.08527e47 −0.883633
\(794\) 2.22956e47 0.317089
\(795\) 6.03696e47 0.840932
\(796\) 6.84392e47 0.933768
\(797\) −2.06006e47 −0.275307 −0.137654 0.990480i \(-0.543956\pi\)
−0.137654 + 0.990480i \(0.543956\pi\)
\(798\) −4.33763e47 −0.567812
\(799\) 5.99803e47 0.769107
\(800\) 2.81475e46 0.0353553
\(801\) 1.05775e47 0.130151
\(802\) 3.90671e47 0.470906
\(803\) −8.43851e47 −0.996459
\(804\) 6.61919e47 0.765737
\(805\) 4.69835e46 0.0532492
\(806\) 1.17075e48 1.29998
\(807\) 4.49682e47 0.489206
\(808\) 4.88231e46 0.0520400
\(809\) 9.08440e47 0.948737 0.474368 0.880326i \(-0.342676\pi\)
0.474368 + 0.880326i \(0.342676\pi\)
\(810\) −2.79487e47 −0.285995
\(811\) 1.62541e48 1.62974 0.814869 0.579646i \(-0.196809\pi\)
0.814869 + 0.579646i \(0.196809\pi\)
\(812\) −4.33763e46 −0.0426166
\(813\) −7.73654e47 −0.744822
\(814\) −3.46593e47 −0.326977
\(815\) 2.45929e47 0.227357
\(816\) 2.55757e47 0.231708
\(817\) −7.26214e47 −0.644764
\(818\) −1.35844e48 −1.18198
\(819\) 1.46004e47 0.124503
\(820\) −2.78563e47 −0.232807
\(821\) −1.89976e48 −1.55610 −0.778048 0.628205i \(-0.783790\pi\)
−0.778048 + 0.628205i \(0.783790\pi\)
\(822\) 9.05852e47 0.727231
\(823\) 1.15896e48 0.911950 0.455975 0.889992i \(-0.349291\pi\)
0.455975 + 0.889992i \(0.349291\pi\)
\(824\) 5.76689e47 0.444778
\(825\) −1.89581e47 −0.143319
\(826\) 9.29509e47 0.688784
\(827\) −7.78559e47 −0.565524 −0.282762 0.959190i \(-0.591251\pi\)
−0.282762 + 0.959190i \(0.591251\pi\)
\(828\) 8.47122e45 0.00603178
\(829\) 2.22673e48 1.55424 0.777120 0.629353i \(-0.216680\pi\)
0.777120 + 0.629353i \(0.216680\pi\)
\(830\) 5.09132e47 0.348371
\(831\) −1.66266e48 −1.11528
\(832\) 3.10457e47 0.204158
\(833\) 3.72717e47 0.240290
\(834\) 1.15026e48 0.727037
\(835\) 7.29677e47 0.452172
\(836\) −5.98540e47 −0.363655
\(837\) 1.96324e48 1.16951
\(838\) −5.22219e47 −0.305019
\(839\) −1.12708e48 −0.645478 −0.322739 0.946488i \(-0.604604\pi\)
−0.322739 + 0.946488i \(0.604604\pi\)
\(840\) 2.33253e47 0.130984
\(841\) −1.79854e48 −0.990345
\(842\) −9.17732e47 −0.495526
\(843\) −2.49734e48 −1.32228
\(844\) −7.99118e47 −0.414916
\(845\) 1.46464e48 0.745752
\(846\) 9.86298e46 0.0492490
\(847\) 7.74097e47 0.379070
\(848\) −1.02492e48 −0.492221
\(849\) −1.66626e48 −0.784814
\(850\) −2.97099e47 −0.137243
\(851\) 1.86722e47 0.0845978
\(852\) 9.50871e47 0.422540
\(853\) 3.65470e48 1.59292 0.796459 0.604693i \(-0.206704\pi\)
0.796459 + 0.604693i \(0.206704\pi\)
\(854\) 7.76208e47 0.331836
\(855\) 9.08505e46 0.0380966
\(856\) 8.41986e47 0.346328
\(857\) 1.20507e48 0.486214 0.243107 0.969999i \(-0.421833\pi\)
0.243107 + 0.969999i \(0.421833\pi\)
\(858\) −2.09101e48 −0.827590
\(859\) −1.13686e48 −0.441389 −0.220694 0.975343i \(-0.570832\pi\)
−0.220694 + 0.975343i \(0.570832\pi\)
\(860\) 3.90516e47 0.148735
\(861\) −2.30840e48 −0.862498
\(862\) −1.43706e47 −0.0526750
\(863\) 3.08439e46 0.0110915 0.00554576 0.999985i \(-0.498235\pi\)
0.00554576 + 0.999985i \(0.498235\pi\)
\(864\) 5.20606e47 0.183668
\(865\) 4.91055e47 0.169967
\(866\) 3.24304e48 1.10130
\(867\) 1.66885e47 0.0556034
\(868\) −1.49336e48 −0.488189
\(869\) 7.88294e47 0.252849
\(870\) −9.42929e46 −0.0296764
\(871\) −8.48087e48 −2.61903
\(872\) 2.84845e47 0.0863151
\(873\) −7.16831e46 −0.0213148
\(874\) 3.22456e47 0.0940875
\(875\) −2.70957e47 −0.0775832
\(876\) 2.25696e48 0.634171
\(877\) −3.96964e48 −1.09461 −0.547303 0.836935i \(-0.684345\pi\)
−0.547303 + 0.836935i \(0.684345\pi\)
\(878\) −3.47097e48 −0.939272
\(879\) 2.21485e48 0.588204
\(880\) 3.21860e47 0.0838887
\(881\) −1.10851e48 −0.283556 −0.141778 0.989898i \(-0.545282\pi\)
−0.141778 + 0.989898i \(0.545282\pi\)
\(882\) 6.12884e46 0.0153867
\(883\) 3.17298e48 0.781836 0.390918 0.920426i \(-0.372158\pi\)
0.390918 + 0.920426i \(0.372158\pi\)
\(884\) −3.27690e48 −0.792503
\(885\) 2.02060e48 0.479641
\(886\) −4.30888e48 −1.00394
\(887\) 5.58665e47 0.127765 0.0638824 0.997957i \(-0.479652\pi\)
0.0638824 + 0.997957i \(0.479652\pi\)
\(888\) 9.26996e47 0.208096
\(889\) 3.00418e47 0.0661984
\(890\) −2.16513e48 −0.468326
\(891\) −3.19586e48 −0.678588
\(892\) 1.01335e48 0.211222
\(893\) 3.75434e48 0.768217
\(894\) −1.45314e48 −0.291903
\(895\) 1.13472e47 0.0223773
\(896\) −3.96004e47 −0.0766686
\(897\) 1.12650e48 0.214120
\(898\) −4.58823e48 −0.856220
\(899\) 6.03693e47 0.110607
\(900\) −4.88541e46 −0.00878820
\(901\) 1.08182e49 1.91071
\(902\) −3.18531e48 −0.552387
\(903\) 3.23612e48 0.551033
\(904\) 3.14704e48 0.526166
\(905\) 2.26942e48 0.372575
\(906\) −4.23433e47 −0.0682605
\(907\) −1.20858e49 −1.91318 −0.956589 0.291441i \(-0.905865\pi\)
−0.956589 + 0.291441i \(0.905865\pi\)
\(908\) 1.67257e48 0.259997
\(909\) −8.47396e46 −0.0129355
\(910\) −2.98856e48 −0.448001
\(911\) 2.56704e48 0.377902 0.188951 0.981986i \(-0.439491\pi\)
0.188951 + 0.981986i \(0.439491\pi\)
\(912\) 1.60086e48 0.231439
\(913\) 5.82181e48 0.826590
\(914\) −2.67843e47 −0.0373480
\(915\) 1.68735e48 0.231077
\(916\) −2.75972e48 −0.371184
\(917\) 1.12847e49 1.49072
\(918\) −5.49504e48 −0.712963
\(919\) −5.63209e47 −0.0717734 −0.0358867 0.999356i \(-0.511426\pi\)
−0.0358867 + 0.999356i \(0.511426\pi\)
\(920\) −1.73398e47 −0.0217043
\(921\) 1.11675e49 1.37300
\(922\) 3.89285e48 0.470118
\(923\) −1.21831e49 −1.44520
\(924\) 2.66719e48 0.310790
\(925\) −1.07684e48 −0.123258
\(926\) 8.55860e48 0.962324
\(927\) −1.00093e48 −0.110557
\(928\) 1.60086e47 0.0173704
\(929\) 3.29801e47 0.0351555 0.0175777 0.999845i \(-0.494405\pi\)
0.0175777 + 0.999845i \(0.494405\pi\)
\(930\) −3.24631e48 −0.339954
\(931\) 2.33294e48 0.240012
\(932\) 4.28029e48 0.432624
\(933\) 1.35571e49 1.34623
\(934\) −2.40370e48 −0.234507
\(935\) −3.39727e48 −0.325640
\(936\) −5.38844e47 −0.0507471
\(937\) −3.91522e48 −0.362287 −0.181144 0.983457i \(-0.557980\pi\)
−0.181144 + 0.983457i \(0.557980\pi\)
\(938\) 1.08178e49 0.983540
\(939\) −2.44467e48 −0.218392
\(940\) −2.01886e48 −0.177214
\(941\) 1.08951e49 0.939731 0.469866 0.882738i \(-0.344302\pi\)
0.469866 + 0.882738i \(0.344302\pi\)
\(942\) 7.82708e48 0.663376
\(943\) 1.71604e48 0.142918
\(944\) −3.43047e48 −0.280747
\(945\) −5.01152e48 −0.403037
\(946\) 4.46546e48 0.352909
\(947\) −1.05116e49 −0.816382 −0.408191 0.912897i \(-0.633840\pi\)
−0.408191 + 0.912897i \(0.633840\pi\)
\(948\) −2.10837e48 −0.160920
\(949\) −2.89175e49 −2.16904
\(950\) −1.85963e48 −0.137084
\(951\) −1.91956e49 −1.39067
\(952\) 4.17986e48 0.297613
\(953\) 1.01486e49 0.710190 0.355095 0.934830i \(-0.384449\pi\)
0.355095 + 0.934830i \(0.384449\pi\)
\(954\) 1.77891e48 0.122350
\(955\) −7.44292e47 −0.0503138
\(956\) 8.87922e48 0.589956
\(957\) −1.07822e48 −0.0704141
\(958\) 1.17403e49 0.753613
\(959\) 1.48044e49 0.934081
\(960\) −8.60847e47 −0.0533888
\(961\) 4.38039e48 0.267040
\(962\) −1.18772e49 −0.711746
\(963\) −1.46139e48 −0.0860860
\(964\) −1.26700e49 −0.733677
\(965\) −1.08538e49 −0.617849
\(966\) −1.43692e48 −0.0804097
\(967\) 7.23400e47 0.0397962 0.0198981 0.999802i \(-0.493666\pi\)
0.0198981 + 0.999802i \(0.493666\pi\)
\(968\) −2.85690e48 −0.154508
\(969\) −1.68972e49 −0.898404
\(970\) 1.46729e48 0.0766976
\(971\) 7.86578e47 0.0404226 0.0202113 0.999796i \(-0.493566\pi\)
0.0202113 + 0.999796i \(0.493566\pi\)
\(972\) −1.73418e48 −0.0876196
\(973\) 1.87988e49 0.933832
\(974\) 1.86984e49 0.913233
\(975\) −6.49663e48 −0.311969
\(976\) −2.86469e48 −0.135256
\(977\) −2.61058e49 −1.21193 −0.605964 0.795492i \(-0.707213\pi\)
−0.605964 + 0.795492i \(0.707213\pi\)
\(978\) −7.52134e48 −0.343324
\(979\) −2.47577e49 −1.11121
\(980\) −1.25452e48 −0.0553664
\(981\) −4.94391e47 −0.0214552
\(982\) 1.25706e49 0.536433
\(983\) −7.33769e48 −0.307912 −0.153956 0.988078i \(-0.549201\pi\)
−0.153956 + 0.988078i \(0.549201\pi\)
\(984\) 8.51941e48 0.351553
\(985\) −1.85427e49 −0.752446
\(986\) −1.68972e48 −0.0674288
\(987\) −1.67299e49 −0.656539
\(988\) −2.05110e49 −0.791586
\(989\) −2.40571e48 −0.0913071
\(990\) −5.58636e47 −0.0208520
\(991\) −2.29393e49 −0.842102 −0.421051 0.907037i \(-0.638339\pi\)
−0.421051 + 0.907037i \(0.638339\pi\)
\(992\) 5.51141e48 0.198985
\(993\) −8.00902e48 −0.284391
\(994\) 1.55402e49 0.542726
\(995\) 2.43145e49 0.835188
\(996\) −1.55710e49 −0.526063
\(997\) 1.96710e49 0.653665 0.326833 0.945082i \(-0.394019\pi\)
0.326833 + 0.945082i \(0.394019\pi\)
\(998\) 1.96104e49 0.640961
\(999\) −1.99169e49 −0.640311
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 10.34.a.a.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.34.a.a.1.1 2 1.1 even 1 trivial