Properties

Label 10.34.a.a.1.2
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 1.2
Root \(-62746.5\) 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)
 
\(f(q)\) \(=\) \(q+65536.0 q^{2} -3.44083e6 q^{3} +4.29497e9 q^{4} +1.52588e11 q^{5} -2.25498e11 q^{6} +9.65954e13 q^{7} +2.81475e14 q^{8} -5.54722e15 q^{9} +1.00000e16 q^{10} -1.25630e17 q^{11} -1.47783e16 q^{12} -3.83776e18 q^{13} +6.33048e18 q^{14} -5.25030e17 q^{15} +1.84467e19 q^{16} -1.12561e19 q^{17} -3.63543e20 q^{18} -1.21644e21 q^{19} +6.55360e20 q^{20} -3.32369e20 q^{21} -8.23330e21 q^{22} +3.30661e22 q^{23} -9.68509e20 q^{24} +2.32831e22 q^{25} -2.51511e23 q^{26} +3.82149e22 q^{27} +4.14874e23 q^{28} +5.37492e23 q^{29} -3.44083e22 q^{30} -4.22167e24 q^{31} +1.20893e24 q^{32} +4.32273e23 q^{33} -7.37680e23 q^{34} +1.47393e25 q^{35} -2.38251e25 q^{36} -1.37889e26 q^{37} -7.97206e25 q^{38} +1.32051e25 q^{39} +4.29497e25 q^{40} +4.39352e26 q^{41} -2.17821e25 q^{42} -1.20456e27 q^{43} -5.39577e26 q^{44} -8.46439e26 q^{45} +2.16702e27 q^{46} -3.56629e27 q^{47} -6.34722e25 q^{48} +1.59968e27 q^{49} +1.52588e27 q^{50} +3.87304e25 q^{51} -1.64830e28 q^{52} +3.20094e28 q^{53} +2.50445e27 q^{54} -1.91696e28 q^{55} +2.71892e28 q^{56} +4.18557e27 q^{57} +3.52251e28 q^{58} +2.08570e29 q^{59} -2.25498e27 q^{60} -1.84408e29 q^{61} -2.76671e29 q^{62} -5.35836e29 q^{63} +7.92282e28 q^{64} -5.85595e29 q^{65} +2.83294e28 q^{66} -3.61772e29 q^{67} -4.83446e28 q^{68} -1.13775e29 q^{69} +9.65954e29 q^{70} -5.64849e30 q^{71} -1.56140e30 q^{72} -6.52836e29 q^{73} -9.03671e30 q^{74} -8.01132e28 q^{75} -5.22457e30 q^{76} -1.21353e31 q^{77} +8.65408e29 q^{78} -2.61534e31 q^{79} +2.81475e30 q^{80} +3.07058e31 q^{81} +2.87934e31 q^{82} -7.88667e31 q^{83} -1.42751e30 q^{84} -1.71755e30 q^{85} -7.89424e31 q^{86} -1.84942e30 q^{87} -3.53617e31 q^{88} +6.51043e31 q^{89} -5.54722e31 q^{90} -3.70710e32 q^{91} +1.42018e32 q^{92} +1.45261e31 q^{93} -2.33721e32 q^{94} -1.85614e32 q^{95} -4.15971e30 q^{96} -7.62730e32 q^{97} +1.04837e32 q^{98} +6.96898e32 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 131072 q^{2} - 74648412 q^{3} + 8589934592 q^{4} + 305175781250 q^{5} - 4892158328832 q^{6} + 20327789415556 q^{7} + 562949953421312 q^{8} - 60\!\cdots\!74 q^{9} + 20\!\cdots\!00 q^{10} - 11\!\cdots\!76 q^{11}+ \cdots + 64\!\cdots\!12 q^{99}+O(q^{100}) \) 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
\(3\) −3.44083e6 −0.0461491 −0.0230745 0.999734i \(-0.507346\pi\)
−0.0230745 + 0.999734i \(0.507346\pi\)
\(4\) 4.29497e9 0.500000
\(5\) 1.52588e11 0.447214
\(6\) −2.25498e11 −0.0326323
\(7\) 9.65954e13 1.09860 0.549299 0.835626i \(-0.314895\pi\)
0.549299 + 0.835626i \(0.314895\pi\)
\(8\) 2.81475e14 0.353553
\(9\) −5.54722e15 −0.997870
\(10\) 1.00000e16 0.316228
\(11\) −1.25630e17 −0.824356 −0.412178 0.911103i \(-0.635232\pi\)
−0.412178 + 0.911103i \(0.635232\pi\)
\(12\) −1.47783e16 −0.0230745
\(13\) −3.83776e18 −1.59960 −0.799801 0.600265i \(-0.795062\pi\)
−0.799801 + 0.600265i \(0.795062\pi\)
\(14\) 6.33048e18 0.776826
\(15\) −5.25030e17 −0.0206385
\(16\) 1.84467e19 0.250000
\(17\) −1.12561e19 −0.0561023 −0.0280512 0.999606i \(-0.508930\pi\)
−0.0280512 + 0.999606i \(0.508930\pi\)
\(18\) −3.63543e20 −0.705601
\(19\) −1.21644e21 −0.967512 −0.483756 0.875203i \(-0.660728\pi\)
−0.483756 + 0.875203i \(0.660728\pi\)
\(20\) 6.55360e20 0.223607
\(21\) −3.32369e20 −0.0506993
\(22\) −8.23330e21 −0.582907
\(23\) 3.30661e22 1.12428 0.562139 0.827043i \(-0.309979\pi\)
0.562139 + 0.827043i \(0.309979\pi\)
\(24\) −9.68509e20 −0.0163162
\(25\) 2.32831e22 0.200000
\(26\) −2.51511e23 −1.13109
\(27\) 3.82149e22 0.0921999
\(28\) 4.14874e23 0.549299
\(29\) 5.37492e23 0.398846 0.199423 0.979914i \(-0.436093\pi\)
0.199423 + 0.979914i \(0.436093\pi\)
\(30\) −3.44083e22 −0.0145936
\(31\) −4.22167e24 −1.04236 −0.521178 0.853448i \(-0.674507\pi\)
−0.521178 + 0.853448i \(0.674507\pi\)
\(32\) 1.20893e24 0.176777
\(33\) 4.32273e23 0.0380432
\(34\) −7.37680e23 −0.0396704
\(35\) 1.47393e25 0.491308
\(36\) −2.38251e25 −0.498935
\(37\) −1.37889e26 −1.83739 −0.918696 0.394966i \(-0.870756\pi\)
−0.918696 + 0.394966i \(0.870756\pi\)
\(38\) −7.97206e25 −0.684134
\(39\) 1.32051e25 0.0738202
\(40\) 4.29497e25 0.158114
\(41\) 4.39352e26 1.07617 0.538083 0.842892i \(-0.319149\pi\)
0.538083 + 0.842892i \(0.319149\pi\)
\(42\) −2.17821e25 −0.0358498
\(43\) −1.20456e27 −1.34462 −0.672312 0.740268i \(-0.734699\pi\)
−0.672312 + 0.740268i \(0.734699\pi\)
\(44\) −5.39577e26 −0.412178
\(45\) −8.46439e26 −0.446261
\(46\) 2.16702e27 0.794984
\(47\) −3.56629e27 −0.917493 −0.458746 0.888567i \(-0.651701\pi\)
−0.458746 + 0.888567i \(0.651701\pi\)
\(48\) −6.34722e25 −0.0115373
\(49\) 1.59968e27 0.206918
\(50\) 1.52588e27 0.141421
\(51\) 3.87304e25 0.00258907
\(52\) −1.64830e28 −0.799801
\(53\) 3.20094e28 1.13429 0.567147 0.823617i \(-0.308047\pi\)
0.567147 + 0.823617i \(0.308047\pi\)
\(54\) 2.50445e27 0.0651952
\(55\) −1.91696e28 −0.368663
\(56\) 2.71892e28 0.388413
\(57\) 4.18557e27 0.0446498
\(58\) 3.52251e28 0.282027
\(59\) 2.08570e29 1.25949 0.629744 0.776803i \(-0.283160\pi\)
0.629744 + 0.776803i \(0.283160\pi\)
\(60\) −2.25498e27 −0.0103192
\(61\) −1.84408e29 −0.642448 −0.321224 0.947003i \(-0.604094\pi\)
−0.321224 + 0.947003i \(0.604094\pi\)
\(62\) −2.76671e29 −0.737057
\(63\) −5.35836e29 −1.09626
\(64\) 7.92282e28 0.125000
\(65\) −5.85595e29 −0.715364
\(66\) 2.83294e28 0.0269006
\(67\) −3.61772e29 −0.268040 −0.134020 0.990979i \(-0.542789\pi\)
−0.134020 + 0.990979i \(0.542789\pi\)
\(68\) −4.83446e28 −0.0280512
\(69\) −1.13775e29 −0.0518844
\(70\) 9.65954e29 0.347407
\(71\) −5.64849e30 −1.60757 −0.803784 0.594921i \(-0.797183\pi\)
−0.803784 + 0.594921i \(0.797183\pi\)
\(72\) −1.56140e30 −0.352800
\(73\) −6.52836e29 −0.117484 −0.0587418 0.998273i \(-0.518709\pi\)
−0.0587418 + 0.998273i \(0.518709\pi\)
\(74\) −9.03671e30 −1.29923
\(75\) −8.01132e28 −0.00922982
\(76\) −5.22457e30 −0.483756
\(77\) −1.21353e31 −0.905636
\(78\) 8.65408e29 0.0521988
\(79\) −2.61534e31 −1.27844 −0.639221 0.769023i \(-0.720743\pi\)
−0.639221 + 0.769023i \(0.720743\pi\)
\(80\) 2.81475e30 0.111803
\(81\) 3.07058e31 0.993615
\(82\) 2.87934e31 0.760965
\(83\) −7.88667e31 −1.70650 −0.853248 0.521505i \(-0.825371\pi\)
−0.853248 + 0.521505i \(0.825371\pi\)
\(84\) −1.42751e30 −0.0253496
\(85\) −1.71755e30 −0.0250897
\(86\) −7.89424e31 −0.950792
\(87\) −1.84942e30 −0.0184064
\(88\) −3.53617e31 −0.291454
\(89\) 6.51043e31 0.445323 0.222661 0.974896i \(-0.428526\pi\)
0.222661 + 0.974896i \(0.428526\pi\)
\(90\) −5.54722e31 −0.315554
\(91\) −3.70710e32 −1.75732
\(92\) 1.42018e32 0.562139
\(93\) 1.45261e31 0.0481038
\(94\) −2.33721e32 −0.648765
\(95\) −1.85614e32 −0.432685
\(96\) −4.15971e30 −0.00815808
\(97\) −7.62730e32 −1.26077 −0.630387 0.776281i \(-0.717104\pi\)
−0.630387 + 0.776281i \(0.717104\pi\)
\(98\) 1.04837e32 0.146313
\(99\) 6.96898e32 0.822600
\(100\) 1.00000e32 0.100000
\(101\) −3.89030e32 −0.330127 −0.165063 0.986283i \(-0.552783\pi\)
−0.165063 + 0.986283i \(0.552783\pi\)
\(102\) 2.53824e30 0.00183075
\(103\) 1.75553e33 1.07794 0.538970 0.842325i \(-0.318814\pi\)
0.538970 + 0.842325i \(0.318814\pi\)
\(104\) −1.08023e33 −0.565545
\(105\) −5.07155e31 −0.0226734
\(106\) 2.09777e33 0.802067
\(107\) 2.84461e33 0.931517 0.465759 0.884912i \(-0.345781\pi\)
0.465759 + 0.884912i \(0.345781\pi\)
\(108\) 1.64132e32 0.0460999
\(109\) 7.03753e33 1.69779 0.848894 0.528564i \(-0.177269\pi\)
0.848894 + 0.528564i \(0.177269\pi\)
\(110\) −1.25630e33 −0.260684
\(111\) 4.74454e32 0.0847939
\(112\) 1.78187e33 0.274650
\(113\) −3.22217e33 −0.428898 −0.214449 0.976735i \(-0.568796\pi\)
−0.214449 + 0.976735i \(0.568796\pi\)
\(114\) 2.74305e32 0.0315722
\(115\) 5.04548e33 0.502792
\(116\) 2.30851e33 0.199423
\(117\) 2.12889e34 1.59620
\(118\) 1.36689e34 0.890593
\(119\) −1.08729e33 −0.0616339
\(120\) −1.47783e32 −0.00729681
\(121\) −7.44222e33 −0.320438
\(122\) −1.20854e34 −0.454279
\(123\) −1.51174e33 −0.0496641
\(124\) −1.81319e34 −0.521178
\(125\) 3.55271e33 0.0894427
\(126\) −3.51166e34 −0.775172
\(127\) −2.14069e33 −0.0414756 −0.0207378 0.999785i \(-0.506602\pi\)
−0.0207378 + 0.999785i \(0.506602\pi\)
\(128\) 5.19230e33 0.0883883
\(129\) 4.14471e33 0.0620531
\(130\) −3.83776e34 −0.505839
\(131\) 7.71149e34 0.895699 0.447850 0.894109i \(-0.352190\pi\)
0.447850 + 0.894109i \(0.352190\pi\)
\(132\) 1.85660e33 0.0190216
\(133\) −1.17503e35 −1.06291
\(134\) −2.37091e34 −0.189533
\(135\) 5.83113e33 0.0412330
\(136\) −3.16831e33 −0.0198352
\(137\) −2.86481e35 −1.58930 −0.794650 0.607068i \(-0.792345\pi\)
−0.794650 + 0.607068i \(0.792345\pi\)
\(138\) −7.45635e33 −0.0366878
\(139\) 4.17053e34 0.182158 0.0910788 0.995844i \(-0.470968\pi\)
0.0910788 + 0.995844i \(0.470968\pi\)
\(140\) 6.33048e34 0.245654
\(141\) 1.22710e34 0.0423414
\(142\) −3.70179e35 −1.13672
\(143\) 4.82138e35 1.31864
\(144\) −1.02328e35 −0.249468
\(145\) 8.20148e34 0.178369
\(146\) −4.27843e34 −0.0830734
\(147\) −5.50424e33 −0.00954907
\(148\) −5.92230e35 −0.918696
\(149\) 2.17706e35 0.302201 0.151101 0.988518i \(-0.451718\pi\)
0.151101 + 0.988518i \(0.451718\pi\)
\(150\) −5.25030e33 −0.00652646
\(151\) −3.80622e35 −0.424008 −0.212004 0.977269i \(-0.567999\pi\)
−0.212004 + 0.977269i \(0.567999\pi\)
\(152\) −3.42398e35 −0.342067
\(153\) 6.24401e34 0.0559829
\(154\) −7.95299e35 −0.640381
\(155\) −6.44175e35 −0.466156
\(156\) 5.67154e34 0.0369101
\(157\) 1.98603e36 1.16316 0.581582 0.813488i \(-0.302434\pi\)
0.581582 + 0.813488i \(0.302434\pi\)
\(158\) −1.71399e36 −0.903995
\(159\) −1.10139e35 −0.0523466
\(160\) 1.84467e35 0.0790569
\(161\) 3.19403e36 1.23513
\(162\) 2.01234e36 0.702592
\(163\) 4.49451e36 1.41771 0.708854 0.705355i \(-0.249212\pi\)
0.708854 + 0.705355i \(0.249212\pi\)
\(164\) 1.88700e36 0.538083
\(165\) 6.59596e34 0.0170135
\(166\) −5.16861e36 −1.20668
\(167\) −1.58924e36 −0.336023 −0.168012 0.985785i \(-0.553735\pi\)
−0.168012 + 0.985785i \(0.553735\pi\)
\(168\) −9.35535e34 −0.0179249
\(169\) 8.97224e36 1.55873
\(170\) −1.12561e35 −0.0177411
\(171\) 6.74786e36 0.965452
\(172\) −5.17357e36 −0.672312
\(173\) 6.81608e36 0.804960 0.402480 0.915429i \(-0.368148\pi\)
0.402480 + 0.915429i \(0.368148\pi\)
\(174\) −1.21204e35 −0.0130153
\(175\) 2.24904e36 0.219720
\(176\) −2.31747e36 −0.206089
\(177\) −7.17655e35 −0.0581242
\(178\) 4.26668e36 0.314891
\(179\) 7.04020e36 0.473707 0.236853 0.971545i \(-0.423884\pi\)
0.236853 + 0.971545i \(0.423884\pi\)
\(180\) −3.63543e36 −0.223131
\(181\) 1.81472e36 0.101651 0.0508257 0.998708i \(-0.483815\pi\)
0.0508257 + 0.998708i \(0.483815\pi\)
\(182\) −2.42948e37 −1.24261
\(183\) 6.34518e35 0.0296484
\(184\) 9.30728e36 0.397492
\(185\) −2.10402e37 −0.821707
\(186\) 9.51980e35 0.0340145
\(187\) 1.41411e36 0.0462483
\(188\) −1.53171e37 −0.458746
\(189\) 3.69138e36 0.101291
\(190\) −1.21644e37 −0.305954
\(191\) −5.47196e37 −1.26209 −0.631047 0.775744i \(-0.717375\pi\)
−0.631047 + 0.775744i \(0.717375\pi\)
\(192\) −2.72611e35 −0.00576863
\(193\) −4.57793e37 −0.889148 −0.444574 0.895742i \(-0.646645\pi\)
−0.444574 + 0.895742i \(0.646645\pi\)
\(194\) −4.99863e37 −0.891502
\(195\) 2.01494e36 0.0330134
\(196\) 6.87058e36 0.103459
\(197\) −7.48669e37 −1.03657 −0.518284 0.855209i \(-0.673429\pi\)
−0.518284 + 0.855209i \(0.673429\pi\)
\(198\) 4.56719e37 0.581666
\(199\) 3.94421e37 0.462258 0.231129 0.972923i \(-0.425758\pi\)
0.231129 + 0.972923i \(0.425758\pi\)
\(200\) 6.55360e36 0.0707107
\(201\) 1.24480e36 0.0123698
\(202\) −2.54955e37 −0.233435
\(203\) 5.19193e37 0.438171
\(204\) 1.66346e35 0.00129454
\(205\) 6.70399e37 0.481276
\(206\) 1.15051e38 0.762219
\(207\) −1.83425e38 −1.12188
\(208\) −7.07941e37 −0.399901
\(209\) 1.52822e38 0.797574
\(210\) −3.32369e36 −0.0160325
\(211\) 2.10514e38 0.938905 0.469452 0.882958i \(-0.344451\pi\)
0.469452 + 0.882958i \(0.344451\pi\)
\(212\) 1.37479e38 0.567147
\(213\) 1.94355e37 0.0741878
\(214\) 1.86425e38 0.658682
\(215\) −1.83802e38 −0.601334
\(216\) 1.07565e37 0.0325976
\(217\) −4.07794e38 −1.14513
\(218\) 4.61212e38 1.20052
\(219\) 2.24630e36 0.00542176
\(220\) −8.23330e37 −0.184332
\(221\) 4.31982e37 0.0897415
\(222\) 3.10938e37 0.0599584
\(223\) 2.49660e38 0.447012 0.223506 0.974703i \(-0.428250\pi\)
0.223506 + 0.974703i \(0.428250\pi\)
\(224\) 1.16777e38 0.194207
\(225\) −1.29156e38 −0.199574
\(226\) −2.11168e38 −0.303276
\(227\) −3.49100e38 −0.466147 −0.233073 0.972459i \(-0.574878\pi\)
−0.233073 + 0.972459i \(0.574878\pi\)
\(228\) 1.79769e37 0.0223249
\(229\) 8.78358e38 1.01481 0.507406 0.861707i \(-0.330604\pi\)
0.507406 + 0.861707i \(0.330604\pi\)
\(230\) 3.30661e38 0.355528
\(231\) 4.17555e37 0.0417942
\(232\) 1.51291e38 0.141013
\(233\) 1.89180e38 0.164249 0.0821245 0.996622i \(-0.473829\pi\)
0.0821245 + 0.996622i \(0.473829\pi\)
\(234\) 1.39519e39 1.12868
\(235\) −5.44173e38 −0.410315
\(236\) 8.95802e38 0.629744
\(237\) 8.99895e37 0.0589989
\(238\) −7.12565e37 −0.0435818
\(239\) −2.46744e39 −1.40825 −0.704126 0.710075i \(-0.748661\pi\)
−0.704126 + 0.710075i \(0.748661\pi\)
\(240\) −9.68509e36 −0.00515962
\(241\) −3.30287e39 −1.64289 −0.821447 0.570284i \(-0.806833\pi\)
−0.821447 + 0.570284i \(0.806833\pi\)
\(242\) −4.87733e38 −0.226584
\(243\) −3.18093e38 −0.138054
\(244\) −7.92027e38 −0.321224
\(245\) 2.44092e38 0.0925365
\(246\) −9.90733e37 −0.0351178
\(247\) 4.66840e39 1.54764
\(248\) −1.18829e39 −0.368529
\(249\) 2.71367e38 0.0787532
\(250\) 2.32831e38 0.0632456
\(251\) 2.80676e39 0.713820 0.356910 0.934139i \(-0.383830\pi\)
0.356910 + 0.934139i \(0.383830\pi\)
\(252\) −2.30140e39 −0.548129
\(253\) −4.15410e39 −0.926804
\(254\) −1.40292e38 −0.0293277
\(255\) 5.90979e36 0.00115787
\(256\) 3.40282e38 0.0625000
\(257\) −7.29261e39 −1.25599 −0.627996 0.778217i \(-0.716124\pi\)
−0.627996 + 0.778217i \(0.716124\pi\)
\(258\) 2.71628e38 0.0438782
\(259\) −1.33195e40 −2.01856
\(260\) −2.51511e39 −0.357682
\(261\) −2.98159e39 −0.397996
\(262\) 5.05380e39 0.633355
\(263\) 8.76662e39 1.03172 0.515861 0.856673i \(-0.327472\pi\)
0.515861 + 0.856673i \(0.327472\pi\)
\(264\) 1.21674e38 0.0134503
\(265\) 4.88425e39 0.507272
\(266\) −7.70065e39 −0.751589
\(267\) −2.24013e38 −0.0205512
\(268\) −1.55380e39 −0.134020
\(269\) 2.09117e40 1.69620 0.848098 0.529839i \(-0.177748\pi\)
0.848098 + 0.529839i \(0.177748\pi\)
\(270\) 3.82149e38 0.0291562
\(271\) 1.15067e39 0.0825958 0.0412979 0.999147i \(-0.486851\pi\)
0.0412979 + 0.999147i \(0.486851\pi\)
\(272\) −2.07639e38 −0.0140256
\(273\) 1.27555e39 0.0810987
\(274\) −1.87748e40 −1.12380
\(275\) −2.92506e39 −0.164871
\(276\) −4.88660e38 −0.0259422
\(277\) 2.78662e40 1.39367 0.696837 0.717229i \(-0.254590\pi\)
0.696837 + 0.717229i \(0.254590\pi\)
\(278\) 2.73320e39 0.128805
\(279\) 2.34185e40 1.04014
\(280\) 4.14874e39 0.173704
\(281\) 4.93528e40 1.94831 0.974153 0.225888i \(-0.0725283\pi\)
0.974153 + 0.225888i \(0.0725283\pi\)
\(282\) 8.04194e38 0.0299399
\(283\) −1.57701e40 −0.553806 −0.276903 0.960898i \(-0.589308\pi\)
−0.276903 + 0.960898i \(0.589308\pi\)
\(284\) −2.42601e40 −0.803784
\(285\) 6.38667e38 0.0199680
\(286\) 3.15974e40 0.932420
\(287\) 4.24394e40 1.18227
\(288\) −6.70618e39 −0.176400
\(289\) −4.01278e40 −0.996853
\(290\) 5.37492e39 0.126126
\(291\) 2.62443e39 0.0581836
\(292\) −2.80391e39 −0.0587418
\(293\) 3.85942e40 0.764199 0.382100 0.924121i \(-0.375201\pi\)
0.382100 + 0.924121i \(0.375201\pi\)
\(294\) −3.60726e38 −0.00675221
\(295\) 3.18253e40 0.563260
\(296\) −3.88124e40 −0.649616
\(297\) −4.80094e39 −0.0760055
\(298\) 1.42676e40 0.213689
\(299\) −1.26900e41 −1.79840
\(300\) −3.44083e38 −0.00461491
\(301\) −1.16355e41 −1.47720
\(302\) −2.49444e40 −0.299819
\(303\) 1.33859e39 0.0152351
\(304\) −2.24394e40 −0.241878
\(305\) −2.81385e40 −0.287311
\(306\) 4.09208e39 0.0395859
\(307\) 8.29043e40 0.759965 0.379983 0.924994i \(-0.375930\pi\)
0.379983 + 0.924994i \(0.375930\pi\)
\(308\) −5.21207e40 −0.452818
\(309\) −6.04050e39 −0.0497460
\(310\) −4.22167e40 −0.329622
\(311\) 1.68520e40 0.124769 0.0623844 0.998052i \(-0.480130\pi\)
0.0623844 + 0.998052i \(0.480130\pi\)
\(312\) 3.71690e39 0.0260994
\(313\) −5.51347e39 −0.0367235 −0.0183617 0.999831i \(-0.505845\pi\)
−0.0183617 + 0.999831i \(0.505845\pi\)
\(314\) 1.30156e41 0.822482
\(315\) −8.17621e40 −0.490262
\(316\) −1.12328e41 −0.639221
\(317\) −2.09450e41 −1.13136 −0.565682 0.824624i \(-0.691387\pi\)
−0.565682 + 0.824624i \(0.691387\pi\)
\(318\) −7.21807e39 −0.0370147
\(319\) −6.75252e40 −0.328791
\(320\) 1.20893e40 0.0559017
\(321\) −9.78784e39 −0.0429887
\(322\) 2.09324e41 0.873368
\(323\) 1.36924e40 0.0542797
\(324\) 1.31881e41 0.496808
\(325\) −8.93547e40 −0.319921
\(326\) 2.94552e41 1.00247
\(327\) −2.42150e40 −0.0783513
\(328\) 1.23667e41 0.380482
\(329\) −3.44488e41 −1.00796
\(330\) 4.32273e39 0.0120303
\(331\) 1.39181e41 0.368485 0.184242 0.982881i \(-0.441017\pi\)
0.184242 + 0.982881i \(0.441017\pi\)
\(332\) −3.38730e41 −0.853248
\(333\) 7.64902e41 1.83348
\(334\) −1.04153e41 −0.237604
\(335\) −5.52020e40 −0.119871
\(336\) −6.13112e39 −0.0126748
\(337\) −4.64639e41 −0.914582 −0.457291 0.889317i \(-0.651180\pi\)
−0.457291 + 0.889317i \(0.651180\pi\)
\(338\) 5.88005e41 1.10219
\(339\) 1.10869e40 0.0197932
\(340\) −7.37680e39 −0.0125449
\(341\) 5.30369e41 0.859272
\(342\) 4.42228e41 0.682677
\(343\) −5.92257e41 −0.871279
\(344\) −3.39055e41 −0.475396
\(345\) −1.73607e40 −0.0232034
\(346\) 4.46699e41 0.569192
\(347\) 3.74401e41 0.454884 0.227442 0.973792i \(-0.426964\pi\)
0.227442 + 0.973792i \(0.426964\pi\)
\(348\) −7.94320e39 −0.00920318
\(349\) −3.17132e41 −0.350446 −0.175223 0.984529i \(-0.556065\pi\)
−0.175223 + 0.984529i \(0.556065\pi\)
\(350\) 1.47393e41 0.155365
\(351\) −1.46659e41 −0.147483
\(352\) −1.51878e41 −0.145727
\(353\) 1.29547e42 1.18616 0.593080 0.805144i \(-0.297912\pi\)
0.593080 + 0.805144i \(0.297912\pi\)
\(354\) −4.70323e40 −0.0411000
\(355\) −8.61891e41 −0.718926
\(356\) 2.79621e41 0.222661
\(357\) 3.74118e39 0.00284435
\(358\) 4.61386e41 0.334961
\(359\) 1.19616e42 0.829338 0.414669 0.909972i \(-0.363897\pi\)
0.414669 + 0.909972i \(0.363897\pi\)
\(360\) −2.38251e41 −0.157777
\(361\) −1.01043e41 −0.0639202
\(362\) 1.18930e41 0.0718784
\(363\) 2.56074e40 0.0147879
\(364\) −1.59219e42 −0.878660
\(365\) −9.96149e40 −0.0525402
\(366\) 4.15838e40 0.0209646
\(367\) −2.24871e42 −1.08378 −0.541892 0.840448i \(-0.682292\pi\)
−0.541892 + 0.840448i \(0.682292\pi\)
\(368\) 6.09962e41 0.281069
\(369\) −2.43719e42 −1.07387
\(370\) −1.37889e42 −0.581034
\(371\) 3.09196e42 1.24613
\(372\) 6.23890e40 0.0240519
\(373\) 4.82851e42 1.78081 0.890404 0.455171i \(-0.150422\pi\)
0.890404 + 0.455171i \(0.150422\pi\)
\(374\) 9.26749e40 0.0327025
\(375\) −1.22243e40 −0.00412770
\(376\) −1.00382e42 −0.324383
\(377\) −2.06276e42 −0.637995
\(378\) 2.41918e41 0.0716233
\(379\) −4.26076e42 −1.20765 −0.603823 0.797118i \(-0.706357\pi\)
−0.603823 + 0.797118i \(0.706357\pi\)
\(380\) −7.97206e41 −0.216342
\(381\) 7.36576e39 0.00191406
\(382\) −3.58610e42 −0.892436
\(383\) 8.60164e41 0.205023 0.102511 0.994732i \(-0.467312\pi\)
0.102511 + 0.994732i \(0.467312\pi\)
\(384\) −1.78658e40 −0.00407904
\(385\) −1.85170e42 −0.405013
\(386\) −3.00020e42 −0.628722
\(387\) 6.68199e42 1.34176
\(388\) −3.27590e42 −0.630387
\(389\) −1.00931e43 −1.86148 −0.930739 0.365685i \(-0.880835\pi\)
−0.930739 + 0.365685i \(0.880835\pi\)
\(390\) 1.32051e41 0.0233440
\(391\) −3.72195e41 −0.0630746
\(392\) 4.50270e41 0.0731565
\(393\) −2.65340e41 −0.0413357
\(394\) −4.90648e42 −0.732964
\(395\) −3.99069e42 −0.571737
\(396\) 2.99316e42 0.411300
\(397\) 8.51038e42 1.12177 0.560887 0.827892i \(-0.310460\pi\)
0.560887 + 0.827892i \(0.310460\pi\)
\(398\) 2.58488e42 0.326866
\(399\) 4.04307e41 0.0490522
\(400\) 4.29497e41 0.0500000
\(401\) 5.69759e42 0.636516 0.318258 0.948004i \(-0.396902\pi\)
0.318258 + 0.948004i \(0.396902\pi\)
\(402\) 8.15789e40 0.00874678
\(403\) 1.62017e43 1.66736
\(404\) −1.67087e42 −0.165063
\(405\) 4.68534e42 0.444358
\(406\) 3.40258e42 0.309834
\(407\) 1.73230e43 1.51466
\(408\) 1.09016e40 0.000915375 0
\(409\) −1.52688e43 −1.23132 −0.615660 0.788012i \(-0.711111\pi\)
−0.615660 + 0.788012i \(0.711111\pi\)
\(410\) 4.39352e42 0.340314
\(411\) 9.85732e41 0.0733447
\(412\) 7.53996e42 0.538970
\(413\) 2.01469e43 1.38367
\(414\) −1.20209e43 −0.793291
\(415\) −1.20341e43 −0.763168
\(416\) −4.63956e42 −0.282772
\(417\) −1.43501e41 −0.00840640
\(418\) 1.00153e43 0.563970
\(419\) −2.95628e43 −1.60035 −0.800173 0.599769i \(-0.795259\pi\)
−0.800173 + 0.599769i \(0.795259\pi\)
\(420\) −2.17821e41 −0.0113367
\(421\) 2.45835e43 1.23024 0.615119 0.788434i \(-0.289108\pi\)
0.615119 + 0.788434i \(0.289108\pi\)
\(422\) 1.37963e43 0.663906
\(423\) 1.97830e43 0.915539
\(424\) 9.00985e42 0.401034
\(425\) −2.62077e41 −0.0112205
\(426\) 1.27373e42 0.0524587
\(427\) −1.78130e43 −0.705792
\(428\) 1.22175e43 0.465759
\(429\) −1.65896e42 −0.0608541
\(430\) −1.20456e43 −0.425207
\(431\) 1.54742e43 0.525693 0.262846 0.964838i \(-0.415339\pi\)
0.262846 + 0.964838i \(0.415339\pi\)
\(432\) 7.04940e41 0.0230500
\(433\) −4.36385e43 −1.37347 −0.686735 0.726907i \(-0.740957\pi\)
−0.686735 + 0.726907i \(0.740957\pi\)
\(434\) −2.67252e43 −0.809730
\(435\) −2.82199e41 −0.00823158
\(436\) 3.02260e43 0.848894
\(437\) −4.02229e43 −1.08775
\(438\) 1.47214e41 0.00383376
\(439\) −6.44930e43 −1.61751 −0.808757 0.588144i \(-0.799859\pi\)
−0.808757 + 0.588144i \(0.799859\pi\)
\(440\) −5.39577e42 −0.130342
\(441\) −8.87379e42 −0.206477
\(442\) 2.83104e42 0.0634568
\(443\) 3.04125e43 0.656736 0.328368 0.944550i \(-0.393501\pi\)
0.328368 + 0.944550i \(0.393501\pi\)
\(444\) 2.03776e42 0.0423970
\(445\) 9.93413e42 0.199154
\(446\) 1.63617e43 0.316085
\(447\) −7.49089e41 −0.0139463
\(448\) 7.65308e42 0.137325
\(449\) −6.70336e43 −1.15939 −0.579693 0.814835i \(-0.696827\pi\)
−0.579693 + 0.814835i \(0.696827\pi\)
\(450\) −8.46439e42 −0.141120
\(451\) −5.51959e43 −0.887144
\(452\) −1.38391e43 −0.214449
\(453\) 1.30966e42 0.0195676
\(454\) −2.28786e43 −0.329616
\(455\) −5.65658e43 −0.785898
\(456\) 1.17813e42 0.0157861
\(457\) 5.09970e43 0.659064 0.329532 0.944144i \(-0.393109\pi\)
0.329532 + 0.944144i \(0.393109\pi\)
\(458\) 5.75640e43 0.717581
\(459\) −4.30151e41 −0.00517263
\(460\) 2.16702e43 0.251396
\(461\) −7.70670e43 −0.862588 −0.431294 0.902211i \(-0.641943\pi\)
−0.431294 + 0.902211i \(0.641943\pi\)
\(462\) 2.73649e42 0.0295530
\(463\) 4.19125e43 0.436774 0.218387 0.975862i \(-0.429921\pi\)
0.218387 + 0.975862i \(0.429921\pi\)
\(464\) 9.91498e42 0.0997115
\(465\) 2.21650e42 0.0215127
\(466\) 1.23981e43 0.116142
\(467\) 1.16618e44 1.05448 0.527238 0.849717i \(-0.323227\pi\)
0.527238 + 0.849717i \(0.323227\pi\)
\(468\) 9.14351e43 0.798098
\(469\) −3.49455e43 −0.294469
\(470\) −3.56629e43 −0.290137
\(471\) −6.83359e42 −0.0536790
\(472\) 5.87073e43 0.445296
\(473\) 1.51330e44 1.10845
\(474\) 5.89755e42 0.0417185
\(475\) −2.83225e43 −0.193502
\(476\) −4.66987e42 −0.0308170
\(477\) −1.77563e44 −1.13188
\(478\) −1.61706e44 −0.995785
\(479\) 1.47459e44 0.877276 0.438638 0.898664i \(-0.355461\pi\)
0.438638 + 0.898664i \(0.355461\pi\)
\(480\) −6.34722e41 −0.00364840
\(481\) 5.29185e44 2.93910
\(482\) −2.16457e44 −1.16170
\(483\) −1.09901e43 −0.0570001
\(484\) −3.19641e43 −0.160219
\(485\) −1.16383e44 −0.563835
\(486\) −2.08465e43 −0.0976191
\(487\) 1.46242e44 0.661979 0.330990 0.943634i \(-0.392618\pi\)
0.330990 + 0.943634i \(0.392618\pi\)
\(488\) −5.19063e43 −0.227140
\(489\) −1.54649e43 −0.0654260
\(490\) 1.59968e43 0.0654332
\(491\) −1.96551e44 −0.777374 −0.388687 0.921370i \(-0.627071\pi\)
−0.388687 + 0.921370i \(0.627071\pi\)
\(492\) −6.49287e42 −0.0248320
\(493\) −6.05007e42 −0.0223762
\(494\) 3.05948e44 1.09434
\(495\) 1.06338e44 0.367878
\(496\) −7.78760e43 −0.260589
\(497\) −5.45618e44 −1.76607
\(498\) 1.77843e43 0.0556869
\(499\) 5.06608e44 1.53466 0.767332 0.641250i \(-0.221584\pi\)
0.767332 + 0.641250i \(0.221584\pi\)
\(500\) 1.52588e43 0.0447214
\(501\) 5.46833e42 0.0155072
\(502\) 1.83944e44 0.504747
\(503\) 6.45463e44 1.71395 0.856977 0.515354i \(-0.172340\pi\)
0.856977 + 0.515354i \(0.172340\pi\)
\(504\) −1.50824e44 −0.387586
\(505\) −5.93613e43 −0.147637
\(506\) −2.72243e44 −0.655350
\(507\) −3.08720e43 −0.0719339
\(508\) −9.19419e42 −0.0207378
\(509\) −1.22512e44 −0.267507 −0.133753 0.991015i \(-0.542703\pi\)
−0.133753 + 0.991015i \(0.542703\pi\)
\(510\) 3.87304e41 0.000818736 0
\(511\) −6.30610e43 −0.129067
\(512\) 2.23007e43 0.0441942
\(513\) −4.64861e43 −0.0892045
\(514\) −4.77929e44 −0.888120
\(515\) 2.67873e44 0.482070
\(516\) 1.78014e43 0.0310266
\(517\) 4.48034e44 0.756340
\(518\) −8.72904e44 −1.42733
\(519\) −2.34530e43 −0.0371481
\(520\) −1.64830e44 −0.252919
\(521\) 4.86872e43 0.0723757 0.0361878 0.999345i \(-0.488479\pi\)
0.0361878 + 0.999345i \(0.488479\pi\)
\(522\) −1.95401e44 −0.281426
\(523\) −1.13407e45 −1.58257 −0.791283 0.611450i \(-0.790587\pi\)
−0.791283 + 0.611450i \(0.790587\pi\)
\(524\) 3.31206e44 0.447850
\(525\) −7.73856e42 −0.0101399
\(526\) 5.74529e44 0.729537
\(527\) 4.75196e43 0.0584786
\(528\) 7.97402e42 0.00951081
\(529\) 2.28361e44 0.264000
\(530\) 3.20094e44 0.358695
\(531\) −1.15699e45 −1.25681
\(532\) −5.04670e44 −0.531454
\(533\) −1.68613e45 −1.72144
\(534\) −1.46809e43 −0.0145319
\(535\) 4.34054e44 0.416587
\(536\) −1.01830e44 −0.0947666
\(537\) −2.42241e43 −0.0218611
\(538\) 1.37047e45 1.19939
\(539\) −2.00968e44 −0.170574
\(540\) 2.50445e43 0.0206165
\(541\) −6.78227e44 −0.541527 −0.270763 0.962646i \(-0.587276\pi\)
−0.270763 + 0.962646i \(0.587276\pi\)
\(542\) 7.54104e43 0.0584041
\(543\) −6.24415e42 −0.00469112
\(544\) −1.36078e43 −0.00991759
\(545\) 1.07384e45 0.759273
\(546\) 8.35945e43 0.0573455
\(547\) 6.38198e44 0.424780 0.212390 0.977185i \(-0.431875\pi\)
0.212390 + 0.977185i \(0.431875\pi\)
\(548\) −1.23042e45 −0.794650
\(549\) 1.02295e45 0.641080
\(550\) −1.91696e44 −0.116581
\(551\) −6.53827e44 −0.385888
\(552\) −3.20248e43 −0.0183439
\(553\) −2.52630e45 −1.40449
\(554\) 1.82624e45 0.985477
\(555\) 7.23959e43 0.0379210
\(556\) 1.79123e44 0.0910788
\(557\) −1.02802e45 −0.507447 −0.253724 0.967277i \(-0.581655\pi\)
−0.253724 + 0.967277i \(0.581655\pi\)
\(558\) 1.53476e45 0.735487
\(559\) 4.62283e45 2.15086
\(560\) 2.71892e44 0.122827
\(561\) −4.86571e42 −0.00213432
\(562\) 3.23438e45 1.37766
\(563\) 1.16743e45 0.482885 0.241442 0.970415i \(-0.422380\pi\)
0.241442 + 0.970415i \(0.422380\pi\)
\(564\) 5.27037e43 0.0211707
\(565\) −4.91664e44 −0.191809
\(566\) −1.03351e45 −0.391600
\(567\) 2.96604e45 1.09158
\(568\) −1.58991e45 −0.568361
\(569\) −1.70096e45 −0.590666 −0.295333 0.955394i \(-0.595431\pi\)
−0.295333 + 0.955394i \(0.595431\pi\)
\(570\) 4.18557e43 0.0141195
\(571\) 2.65488e45 0.870060 0.435030 0.900416i \(-0.356738\pi\)
0.435030 + 0.900416i \(0.356738\pi\)
\(572\) 2.07077e45 0.659321
\(573\) 1.88281e44 0.0582445
\(574\) 2.78131e45 0.835994
\(575\) 7.69880e44 0.224855
\(576\) −4.39496e44 −0.124734
\(577\) −2.36511e45 −0.652304 −0.326152 0.945317i \(-0.605752\pi\)
−0.326152 + 0.945317i \(0.605752\pi\)
\(578\) −2.62982e45 −0.704881
\(579\) 1.57519e44 0.0410334
\(580\) 3.52251e44 0.0891846
\(581\) −7.61817e45 −1.87475
\(582\) 1.71994e44 0.0411420
\(583\) −4.02135e45 −0.935062
\(584\) −1.83757e44 −0.0415367
\(585\) 3.24843e45 0.713840
\(586\) 2.52931e45 0.540370
\(587\) −1.20601e45 −0.250508 −0.125254 0.992125i \(-0.539975\pi\)
−0.125254 + 0.992125i \(0.539975\pi\)
\(588\) −2.36405e43 −0.00477454
\(589\) 5.13541e45 1.00849
\(590\) 2.08570e45 0.398285
\(591\) 2.57605e44 0.0478366
\(592\) −2.54361e45 −0.459348
\(593\) −2.17738e45 −0.382413 −0.191206 0.981550i \(-0.561240\pi\)
−0.191206 + 0.981550i \(0.561240\pi\)
\(594\) −3.14634e44 −0.0537440
\(595\) −1.65907e44 −0.0275635
\(596\) 9.35039e44 0.151101
\(597\) −1.35714e44 −0.0213328
\(598\) −8.31649e45 −1.27166
\(599\) −4.67713e45 −0.695724 −0.347862 0.937546i \(-0.613092\pi\)
−0.347862 + 0.937546i \(0.613092\pi\)
\(600\) −2.25498e43 −0.00326323
\(601\) −1.01814e46 −1.43344 −0.716722 0.697359i \(-0.754358\pi\)
−0.716722 + 0.697359i \(0.754358\pi\)
\(602\) −7.62547e45 −1.04454
\(603\) 2.00683e45 0.267470
\(604\) −1.63476e45 −0.212004
\(605\) −1.13559e45 −0.143304
\(606\) 8.77258e43 0.0107728
\(607\) −8.73331e45 −1.04368 −0.521838 0.853045i \(-0.674753\pi\)
−0.521838 + 0.853045i \(0.674753\pi\)
\(608\) −1.47059e45 −0.171034
\(609\) −1.78646e44 −0.0202212
\(610\) −1.84408e45 −0.203160
\(611\) 1.36866e46 1.46762
\(612\) 2.68178e44 0.0279914
\(613\) −8.45222e44 −0.0858763 −0.0429381 0.999078i \(-0.513672\pi\)
−0.0429381 + 0.999078i \(0.513672\pi\)
\(614\) 5.43321e45 0.537377
\(615\) −2.30673e44 −0.0222105
\(616\) −3.41578e45 −0.320191
\(617\) −6.99843e44 −0.0638698 −0.0319349 0.999490i \(-0.510167\pi\)
−0.0319349 + 0.999490i \(0.510167\pi\)
\(618\) −3.95870e44 −0.0351757
\(619\) 1.82399e46 1.57807 0.789037 0.614345i \(-0.210580\pi\)
0.789037 + 0.614345i \(0.210580\pi\)
\(620\) −2.76671e45 −0.233078
\(621\) 1.26362e45 0.103658
\(622\) 1.10442e45 0.0882249
\(623\) 6.28878e45 0.489231
\(624\) 2.43591e44 0.0184550
\(625\) 5.42101e44 0.0400000
\(626\) −3.61331e44 −0.0259674
\(627\) −5.25834e44 −0.0368073
\(628\) 8.52993e45 0.581582
\(629\) 1.55210e45 0.103082
\(630\) −5.35836e45 −0.346667
\(631\) 1.71214e46 1.07908 0.539540 0.841960i \(-0.318598\pi\)
0.539540 + 0.841960i \(0.318598\pi\)
\(632\) −7.36153e45 −0.451998
\(633\) −7.24345e44 −0.0433296
\(634\) −1.37265e46 −0.799995
\(635\) −3.26643e44 −0.0185484
\(636\) −4.73044e44 −0.0261733
\(637\) −6.13919e45 −0.330986
\(638\) −4.42533e45 −0.232490
\(639\) 3.13334e46 1.60414
\(640\) 7.92282e44 0.0395285
\(641\) −3.25948e46 −1.58486 −0.792429 0.609964i \(-0.791184\pi\)
−0.792429 + 0.609964i \(0.791184\pi\)
\(642\) −6.41456e44 −0.0303976
\(643\) 3.81037e46 1.75989 0.879947 0.475073i \(-0.157578\pi\)
0.879947 + 0.475073i \(0.157578\pi\)
\(644\) 1.37183e46 0.617565
\(645\) 6.32432e44 0.0277510
\(646\) 8.97344e44 0.0383815
\(647\) 1.23656e46 0.515580 0.257790 0.966201i \(-0.417006\pi\)
0.257790 + 0.966201i \(0.417006\pi\)
\(648\) 8.64293e45 0.351296
\(649\) −2.62027e46 −1.03827
\(650\) −5.85595e45 −0.226218
\(651\) 1.40315e45 0.0528467
\(652\) 1.93038e46 0.708854
\(653\) 4.34456e45 0.155553 0.0777764 0.996971i \(-0.475218\pi\)
0.0777764 + 0.996971i \(0.475218\pi\)
\(654\) −1.58695e45 −0.0554027
\(655\) 1.17668e46 0.400569
\(656\) 8.10462e45 0.269042
\(657\) 3.62143e45 0.117233
\(658\) −2.25763e46 −0.712732
\(659\) −3.61863e46 −1.11413 −0.557063 0.830470i \(-0.688072\pi\)
−0.557063 + 0.830470i \(0.688072\pi\)
\(660\) 2.83294e44 0.00850673
\(661\) −1.31841e46 −0.386124 −0.193062 0.981187i \(-0.561842\pi\)
−0.193062 + 0.981187i \(0.561842\pi\)
\(662\) 9.12140e45 0.260558
\(663\) −1.48638e44 −0.00414149
\(664\) −2.21990e46 −0.603338
\(665\) −1.79295e46 −0.475347
\(666\) 5.01286e46 1.29647
\(667\) 1.77728e46 0.448413
\(668\) −6.82575e45 −0.168012
\(669\) −8.59040e44 −0.0206292
\(670\) −3.61772e45 −0.0847618
\(671\) 2.31672e46 0.529606
\(672\) −4.01809e44 −0.00896245
\(673\) 4.84617e46 1.05475 0.527375 0.849633i \(-0.323176\pi\)
0.527375 + 0.849633i \(0.323176\pi\)
\(674\) −3.04506e46 −0.646707
\(675\) 8.89759e44 0.0184400
\(676\) 3.85355e46 0.779364
\(677\) −4.32560e46 −0.853755 −0.426877 0.904310i \(-0.640386\pi\)
−0.426877 + 0.904310i \(0.640386\pi\)
\(678\) 7.26594e44 0.0139959
\(679\) −7.36762e46 −1.38508
\(680\) −4.83446e44 −0.00887056
\(681\) 1.20119e45 0.0215122
\(682\) 3.47583e46 0.607597
\(683\) −6.34451e45 −0.108257 −0.0541286 0.998534i \(-0.517238\pi\)
−0.0541286 + 0.998534i \(0.517238\pi\)
\(684\) 2.89819e46 0.482726
\(685\) −4.37135e46 −0.710756
\(686\) −3.88141e46 −0.616087
\(687\) −3.02228e45 −0.0468327
\(688\) −2.22203e46 −0.336156
\(689\) −1.22844e47 −1.81442
\(690\) −1.13775e45 −0.0164073
\(691\) −5.27291e46 −0.742442 −0.371221 0.928545i \(-0.621061\pi\)
−0.371221 + 0.928545i \(0.621061\pi\)
\(692\) 2.92749e46 0.402480
\(693\) 6.73172e46 0.903707
\(694\) 2.45367e46 0.321652
\(695\) 6.36373e45 0.0814633
\(696\) −5.20566e44 −0.00650763
\(697\) −4.94540e45 −0.0603755
\(698\) −2.07836e46 −0.247803
\(699\) −6.50937e44 −0.00757994
\(700\) 9.65954e45 0.109860
\(701\) −1.32738e47 −1.47451 −0.737255 0.675614i \(-0.763879\pi\)
−0.737255 + 0.675614i \(0.763879\pi\)
\(702\) −9.61147e45 −0.104286
\(703\) 1.67734e47 1.77770
\(704\) −9.95345e45 −0.103044
\(705\) 1.87241e45 0.0189357
\(706\) 8.48996e46 0.838741
\(707\) −3.75786e46 −0.362677
\(708\) −3.08231e45 −0.0290621
\(709\) −1.89943e47 −1.74968 −0.874841 0.484409i \(-0.839035\pi\)
−0.874841 + 0.484409i \(0.839035\pi\)
\(710\) −5.64849e46 −0.508358
\(711\) 1.45079e47 1.27572
\(712\) 1.83252e46 0.157445
\(713\) −1.39594e47 −1.17190
\(714\) 2.45182e44 0.00201126
\(715\) 7.35684e46 0.589714
\(716\) 3.02374e46 0.236853
\(717\) 8.49004e45 0.0649895
\(718\) 7.83917e46 0.586430
\(719\) −1.12722e47 −0.824104 −0.412052 0.911160i \(-0.635188\pi\)
−0.412052 + 0.911160i \(0.635188\pi\)
\(720\) −1.56140e46 −0.111565
\(721\) 1.69576e47 1.18422
\(722\) −6.62196e45 −0.0451984
\(723\) 1.13646e46 0.0758181
\(724\) 7.79417e45 0.0508257
\(725\) 1.25145e46 0.0797692
\(726\) 1.67821e45 0.0104566
\(727\) 1.60341e47 0.976626 0.488313 0.872669i \(-0.337613\pi\)
0.488313 + 0.872669i \(0.337613\pi\)
\(728\) −1.04346e47 −0.621307
\(729\) −1.69601e47 −0.987244
\(730\) −6.52836e45 −0.0371516
\(731\) 1.35587e46 0.0754365
\(732\) 2.72523e45 0.0148242
\(733\) 3.44847e47 1.83405 0.917024 0.398832i \(-0.130584\pi\)
0.917024 + 0.398832i \(0.130584\pi\)
\(734\) −1.47371e47 −0.766351
\(735\) −8.39880e44 −0.00427047
\(736\) 3.99744e46 0.198746
\(737\) 4.54494e46 0.220961
\(738\) −1.59723e47 −0.759344
\(739\) 2.87632e47 1.33722 0.668611 0.743612i \(-0.266889\pi\)
0.668611 + 0.743612i \(0.266889\pi\)
\(740\) −9.03671e46 −0.410853
\(741\) −1.60632e46 −0.0714219
\(742\) 2.02635e47 0.881149
\(743\) 3.46436e47 1.47335 0.736677 0.676244i \(-0.236394\pi\)
0.736677 + 0.676244i \(0.236394\pi\)
\(744\) 4.08872e45 0.0170073
\(745\) 3.32193e46 0.135149
\(746\) 3.16441e47 1.25922
\(747\) 4.37491e47 1.70286
\(748\) 6.07354e45 0.0231241
\(749\) 2.74777e47 1.02336
\(750\) −8.01132e44 −0.00291872
\(751\) −4.94603e47 −1.76278 −0.881389 0.472391i \(-0.843391\pi\)
−0.881389 + 0.472391i \(0.843391\pi\)
\(752\) −6.57865e46 −0.229373
\(753\) −9.65759e45 −0.0329421
\(754\) −1.35185e47 −0.451131
\(755\) −5.80783e46 −0.189622
\(756\) 1.58544e46 0.0506453
\(757\) −5.29375e47 −1.65456 −0.827278 0.561793i \(-0.810112\pi\)
−0.827278 + 0.561793i \(0.810112\pi\)
\(758\) −2.79233e47 −0.853935
\(759\) 1.42936e46 0.0427712
\(760\) −5.22457e46 −0.152977
\(761\) −5.45469e47 −1.56287 −0.781436 0.623985i \(-0.785513\pi\)
−0.781436 + 0.623985i \(0.785513\pi\)
\(762\) 4.82722e44 0.00135344
\(763\) 6.79793e47 1.86519
\(764\) −2.35019e47 −0.631047
\(765\) 9.52761e45 0.0250363
\(766\) 5.63717e46 0.144973
\(767\) −8.00442e47 −2.01468
\(768\) −1.17086e45 −0.00288432
\(769\) 6.99140e47 1.68570 0.842848 0.538151i \(-0.180877\pi\)
0.842848 + 0.538151i \(0.180877\pi\)
\(770\) −1.21353e47 −0.286387
\(771\) 2.50927e46 0.0579628
\(772\) −1.96621e47 −0.444574
\(773\) −7.98533e47 −1.76738 −0.883692 0.468069i \(-0.844950\pi\)
−0.883692 + 0.468069i \(0.844950\pi\)
\(774\) 4.37911e47 0.948767
\(775\) −9.82934e46 −0.208471
\(776\) −2.14689e47 −0.445751
\(777\) 4.58301e46 0.0931545
\(778\) −6.61464e47 −1.31626
\(779\) −5.34446e47 −1.04120
\(780\) 8.65408e45 0.0165067
\(781\) 7.09620e47 1.32521
\(782\) −2.43922e46 −0.0446005
\(783\) 2.05402e46 0.0367735
\(784\) 2.95089e46 0.0517295
\(785\) 3.03044e47 0.520183
\(786\) −1.73893e46 −0.0292287
\(787\) −3.34859e47 −0.551161 −0.275580 0.961278i \(-0.588870\pi\)
−0.275580 + 0.961278i \(0.588870\pi\)
\(788\) −3.21551e47 −0.518284
\(789\) −3.01645e46 −0.0476130
\(790\) −2.61534e47 −0.404279
\(791\) −3.11246e47 −0.471186
\(792\) 1.96159e47 0.290833
\(793\) 7.07714e47 1.02766
\(794\) 5.57736e47 0.793214
\(795\) −1.68059e46 −0.0234101
\(796\) 1.69403e47 0.231129
\(797\) −9.43634e47 −1.26108 −0.630538 0.776159i \(-0.717166\pi\)
−0.630538 + 0.776159i \(0.717166\pi\)
\(798\) 2.64967e46 0.0346851
\(799\) 4.01426e46 0.0514735
\(800\) 2.81475e46 0.0353553
\(801\) −3.61148e47 −0.444374
\(802\) 3.73397e47 0.450084
\(803\) 8.20159e46 0.0968482
\(804\) 5.34636e45 0.00618491
\(805\) 4.87371e47 0.552367
\(806\) 1.06180e48 1.17900
\(807\) −7.19536e46 −0.0782779
\(808\) −1.09502e47 −0.116717
\(809\) −2.25513e47 −0.235516 −0.117758 0.993042i \(-0.537571\pi\)
−0.117758 + 0.993042i \(0.537571\pi\)
\(810\) 3.07058e47 0.314209
\(811\) −1.37037e48 −1.37402 −0.687009 0.726649i \(-0.741077\pi\)
−0.687009 + 0.726649i \(0.741077\pi\)
\(812\) 2.22992e47 0.219086
\(813\) −3.95927e45 −0.00381172
\(814\) 1.13528e48 1.07103
\(815\) 6.85808e47 0.634019
\(816\) 7.14450e44 0.000647268 0
\(817\) 1.46528e48 1.30094
\(818\) −1.00066e48 −0.870675
\(819\) 2.05641e48 1.75358
\(820\) 2.87934e47 0.240638
\(821\) −4.56387e47 −0.373827 −0.186914 0.982376i \(-0.559848\pi\)
−0.186914 + 0.982376i \(0.559848\pi\)
\(822\) 6.46009e46 0.0518625
\(823\) −8.98179e47 −0.706750 −0.353375 0.935482i \(-0.614966\pi\)
−0.353375 + 0.935482i \(0.614966\pi\)
\(824\) 4.94139e47 0.381109
\(825\) 1.00646e46 0.00760865
\(826\) 1.32035e48 0.978403
\(827\) 4.00288e47 0.290758 0.145379 0.989376i \(-0.453560\pi\)
0.145379 + 0.989376i \(0.453560\pi\)
\(828\) −7.87804e47 −0.560942
\(829\) −7.44189e47 −0.519437 −0.259719 0.965684i \(-0.583630\pi\)
−0.259719 + 0.965684i \(0.583630\pi\)
\(830\) −7.88667e47 −0.539642
\(831\) −9.58829e46 −0.0643168
\(832\) −3.04058e47 −0.199950
\(833\) −1.80062e46 −0.0116086
\(834\) −9.40449e45 −0.00594422
\(835\) −2.42500e47 −0.150274
\(836\) 6.56364e47 0.398787
\(837\) −1.61331e47 −0.0961051
\(838\) −1.93743e48 −1.13162
\(839\) 2.38585e48 1.36638 0.683188 0.730243i \(-0.260593\pi\)
0.683188 + 0.730243i \(0.260593\pi\)
\(840\) −1.42751e46 −0.00801626
\(841\) −1.52718e48 −0.840922
\(842\) 1.61111e48 0.869910
\(843\) −1.69815e47 −0.0899126
\(844\) 9.04153e47 0.469452
\(845\) 1.36906e48 0.697085
\(846\) 1.29650e48 0.647384
\(847\) −7.18884e47 −0.352032
\(848\) 5.90469e47 0.283574
\(849\) 5.42622e46 0.0255576
\(850\) −1.71755e46 −0.00793407
\(851\) −4.55946e48 −2.06574
\(852\) 8.34749e46 0.0370939
\(853\) 1.91914e48 0.836466 0.418233 0.908340i \(-0.362650\pi\)
0.418233 + 0.908340i \(0.362650\pi\)
\(854\) −1.16739e48 −0.499070
\(855\) 1.02964e48 0.431763
\(856\) 8.00687e47 0.329341
\(857\) −4.27431e48 −1.72458 −0.862289 0.506417i \(-0.830970\pi\)
−0.862289 + 0.506417i \(0.830970\pi\)
\(858\) −1.08721e47 −0.0430303
\(859\) −2.08295e48 −0.808707 −0.404353 0.914603i \(-0.632503\pi\)
−0.404353 + 0.914603i \(0.632503\pi\)
\(860\) −7.89424e47 −0.300667
\(861\) −1.46027e47 −0.0545609
\(862\) 1.01411e48 0.371721
\(863\) −2.59078e48 −0.931650 −0.465825 0.884877i \(-0.654242\pi\)
−0.465825 + 0.884877i \(0.654242\pi\)
\(864\) 4.61989e46 0.0162988
\(865\) 1.04005e48 0.359989
\(866\) −2.85989e48 −0.971191
\(867\) 1.38073e47 0.0460038
\(868\) −1.75146e48 −0.572565
\(869\) 3.28566e48 1.05389
\(870\) −1.84942e46 −0.00582060
\(871\) 1.38839e48 0.428758
\(872\) 1.98089e48 0.600258
\(873\) 4.23103e48 1.25809
\(874\) −2.63605e48 −0.769157
\(875\) 3.43176e47 0.0982616
\(876\) 9.64779e45 0.00271088
\(877\) 5.50587e48 1.51821 0.759107 0.650966i \(-0.225636\pi\)
0.759107 + 0.650966i \(0.225636\pi\)
\(878\) −4.22661e48 −1.14375
\(879\) −1.32796e47 −0.0352671
\(880\) −3.53617e47 −0.0921658
\(881\) −1.00108e48 −0.256076 −0.128038 0.991769i \(-0.540868\pi\)
−0.128038 + 0.991769i \(0.540868\pi\)
\(882\) −5.81552e47 −0.146001
\(883\) −7.50570e48 −1.84944 −0.924719 0.380652i \(-0.875700\pi\)
−0.924719 + 0.380652i \(0.875700\pi\)
\(884\) 1.85535e47 0.0448707
\(885\) −1.09506e47 −0.0259939
\(886\) 1.99311e48 0.464382
\(887\) −1.00895e48 −0.230744 −0.115372 0.993322i \(-0.536806\pi\)
−0.115372 + 0.993322i \(0.536806\pi\)
\(888\) 1.33547e47 0.0299792
\(889\) −2.06781e47 −0.0455650
\(890\) 6.51043e47 0.140823
\(891\) −3.85758e48 −0.819092
\(892\) 1.07228e48 0.223506
\(893\) 4.33819e48 0.887685
\(894\) −4.90923e46 −0.00986154
\(895\) 1.07425e48 0.211848
\(896\) 5.01552e47 0.0971033
\(897\) 4.36640e47 0.0829944
\(898\) −4.39311e48 −0.819809
\(899\) −2.26911e48 −0.415739
\(900\) −5.54722e47 −0.0997870
\(901\) −3.60301e47 −0.0636366
\(902\) −3.61732e48 −0.627305
\(903\) 4.00360e47 0.0681715
\(904\) −9.06959e47 −0.151638
\(905\) 2.76904e47 0.0454599
\(906\) 8.58296e46 0.0138364
\(907\) −4.59452e47 −0.0727310 −0.0363655 0.999339i \(-0.511578\pi\)
−0.0363655 + 0.999339i \(0.511578\pi\)
\(908\) −1.49937e48 −0.233073
\(909\) 2.15804e48 0.329424
\(910\) −3.70710e48 −0.555714
\(911\) 6.61697e48 0.974105 0.487053 0.873373i \(-0.338072\pi\)
0.487053 + 0.873373i \(0.338072\pi\)
\(912\) 7.72101e46 0.0111624
\(913\) 9.90804e48 1.40676
\(914\) 3.34214e48 0.466028
\(915\) 9.68198e46 0.0132592
\(916\) 3.77252e48 0.507406
\(917\) 7.44895e48 0.984013
\(918\) −2.81904e46 −0.00365760
\(919\) −1.20821e49 −1.53971 −0.769854 0.638220i \(-0.779671\pi\)
−0.769854 + 0.638220i \(0.779671\pi\)
\(920\) 1.42018e48 0.177764
\(921\) −2.85260e47 −0.0350717
\(922\) −5.05066e48 −0.609942
\(923\) 2.16775e49 2.57147
\(924\) 1.79339e47 0.0208971
\(925\) −3.21048e48 −0.367478
\(926\) 2.74677e48 0.308846
\(927\) −9.73833e48 −1.07564
\(928\) 6.49788e47 0.0705067
\(929\) 5.93684e48 0.632843 0.316422 0.948619i \(-0.397519\pi\)
0.316422 + 0.948619i \(0.397519\pi\)
\(930\) 1.45261e47 0.0152118
\(931\) −1.94592e48 −0.200196
\(932\) 8.12522e47 0.0821245
\(933\) −5.79851e46 −0.00575797
\(934\) 7.64268e48 0.745628
\(935\) 2.15776e47 0.0206829
\(936\) 5.99229e48 0.564340
\(937\) −1.17439e49 −1.08670 −0.543349 0.839507i \(-0.682844\pi\)
−0.543349 + 0.839507i \(0.682844\pi\)
\(938\) −2.29019e48 −0.208221
\(939\) 1.89709e46 0.00169475
\(940\) −2.33721e48 −0.205158
\(941\) 1.07666e49 0.928646 0.464323 0.885666i \(-0.346298\pi\)
0.464323 + 0.885666i \(0.346298\pi\)
\(942\) −4.47846e47 −0.0379568
\(943\) 1.45277e49 1.20991
\(944\) 3.84744e48 0.314872
\(945\) 5.63260e47 0.0452985
\(946\) 9.91754e48 0.783791
\(947\) −2.67196e48 −0.207518 −0.103759 0.994602i \(-0.533087\pi\)
−0.103759 + 0.994602i \(0.533087\pi\)
\(948\) 3.86502e47 0.0294995
\(949\) 2.50543e48 0.187927
\(950\) −1.85614e48 −0.136827
\(951\) 7.20682e47 0.0522114
\(952\) −3.06044e47 −0.0217909
\(953\) −5.75060e48 −0.402421 −0.201210 0.979548i \(-0.564487\pi\)
−0.201210 + 0.979548i \(0.564487\pi\)
\(954\) −1.16368e49 −0.800359
\(955\) −8.34954e48 −0.564426
\(956\) −1.05976e49 −0.704126
\(957\) 2.32343e47 0.0151734
\(958\) 9.66390e48 0.620328
\(959\) −2.76727e49 −1.74600
\(960\) −4.15971e46 −0.00257981
\(961\) 1.41901e48 0.0865066
\(962\) 3.46807e49 2.07826
\(963\) −1.57797e49 −0.929533
\(964\) −1.41857e49 −0.821447
\(965\) −6.98537e48 −0.397639
\(966\) −7.20250e47 −0.0403051
\(967\) −1.91342e49 −1.05262 −0.526311 0.850292i \(-0.676425\pi\)
−0.526311 + 0.850292i \(0.676425\pi\)
\(968\) −2.09480e48 −0.113292
\(969\) −4.71132e46 −0.00250496
\(970\) −7.62730e48 −0.398692
\(971\) −1.56571e49 −0.804625 −0.402313 0.915502i \(-0.631794\pi\)
−0.402313 + 0.915502i \(0.631794\pi\)
\(972\) −1.36620e48 −0.0690271
\(973\) 4.02854e48 0.200118
\(974\) 9.58412e48 0.468090
\(975\) 3.07455e47 0.0147640
\(976\) −3.40173e48 −0.160612
\(977\) 1.35623e49 0.629614 0.314807 0.949156i \(-0.398060\pi\)
0.314807 + 0.949156i \(0.398060\pi\)
\(978\) −1.01351e48 −0.0462631
\(979\) −8.17907e48 −0.367104
\(980\) 1.04837e48 0.0462683
\(981\) −3.90388e49 −1.69417
\(982\) −1.28811e49 −0.549686
\(983\) 1.58986e49 0.667155 0.333578 0.942723i \(-0.391744\pi\)
0.333578 + 0.942723i \(0.391744\pi\)
\(984\) −4.25517e47 −0.0175589
\(985\) −1.14238e49 −0.463567
\(986\) −3.96497e47 −0.0158224
\(987\) 1.18533e48 0.0465162
\(988\) 2.00506e49 0.773818
\(989\) −3.98302e49 −1.51173
\(990\) 6.96898e48 0.260129
\(991\) 1.36513e49 0.501141 0.250570 0.968098i \(-0.419382\pi\)
0.250570 + 0.968098i \(0.419382\pi\)
\(992\) −5.10368e48 −0.184264
\(993\) −4.78900e47 −0.0170052
\(994\) −3.57576e49 −1.24880
\(995\) 6.01839e48 0.206728
\(996\) 1.16551e48 0.0393766
\(997\) 3.08204e49 1.02416 0.512081 0.858937i \(-0.328875\pi\)
0.512081 + 0.858937i \(0.328875\pi\)
\(998\) 3.32011e49 1.08517
\(999\) −5.26942e48 −0.169407
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.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.34.a.a.1.2 2 1.1 even 1 trivial