Properties

Label 50.26.a.l.1.2
Level $50$
Weight $26$
Character 50.1
Self dual yes
Analytic conductor $197.998$
Analytic rank $1$
Dimension $6$
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] = [50,26,Mod(1,50)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(50, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 26, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("50.1");
 
S:= CuspForms(chi, 26);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 50.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: \(197.998389976\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2 x^{5} - 27590779188 x^{4} + 26487255863952 x^{3} + \cdots - 30\!\cdots\!08 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{4}\cdot 5^{18} \)
Twist minimal: no (minimal twist has level 10)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(94661.8\) of defining polynomial
Character \(\chi\) \(=\) 50.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4096.00 q^{2} -1.08018e6 q^{3} +1.67772e7 q^{4} -4.42443e9 q^{6} -5.43159e10 q^{7} +6.87195e10 q^{8} +3.19509e11 q^{9} -3.24375e12 q^{11} -1.81225e13 q^{12} -7.83357e13 q^{13} -2.22478e14 q^{14} +2.81475e14 q^{16} +4.33806e15 q^{17} +1.30871e15 q^{18} +8.87783e15 q^{19} +5.86711e16 q^{21} -1.32864e16 q^{22} -7.05451e16 q^{23} -7.42297e16 q^{24} -3.20863e17 q^{26} +5.70099e17 q^{27} -9.11269e17 q^{28} -1.11391e18 q^{29} +6.30227e18 q^{31} +1.15292e18 q^{32} +3.50385e18 q^{33} +1.77687e19 q^{34} +5.36047e18 q^{36} +1.55100e19 q^{37} +3.63636e19 q^{38} +8.46169e19 q^{39} -3.82653e19 q^{41} +2.40317e20 q^{42} -3.32618e20 q^{43} -5.44211e19 q^{44} -2.88953e20 q^{46} +1.01157e21 q^{47} -3.04045e20 q^{48} +1.60914e21 q^{49} -4.68591e21 q^{51} -1.31425e21 q^{52} -3.43480e21 q^{53} +2.33513e21 q^{54} -3.73256e21 q^{56} -9.58969e21 q^{57} -4.56259e21 q^{58} +4.09753e21 q^{59} +3.14378e22 q^{61} +2.58141e22 q^{62} -1.73544e22 q^{63} +4.72237e21 q^{64} +1.43518e22 q^{66} -1.01756e23 q^{67} +7.27806e22 q^{68} +7.62017e22 q^{69} +2.10284e23 q^{71} +2.19565e22 q^{72} -1.56829e23 q^{73} +6.35289e22 q^{74} +1.48945e23 q^{76} +1.76187e23 q^{77} +3.46591e23 q^{78} +6.00925e22 q^{79} -8.86528e23 q^{81} -1.56735e23 q^{82} +7.40331e23 q^{83} +9.84338e23 q^{84} -1.36240e24 q^{86} +1.20323e24 q^{87} -2.22909e23 q^{88} +3.21148e24 q^{89} +4.25487e24 q^{91} -1.18355e24 q^{92} -6.80761e24 q^{93} +4.14341e24 q^{94} -1.24537e24 q^{96} +5.32805e24 q^{97} +6.59105e24 q^{98} -1.03641e24 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 24576 q^{2} - 801416 q^{3} + 100663296 q^{4} - 3282599936 q^{6} - 34007705352 q^{7} + 412316860416 q^{8} + 541468782118 q^{9} + 9861544614312 q^{11} - 13445529337856 q^{12} - 30787386783696 q^{13}+ \cdots - 20\!\cdots\!64 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4096.00 0.707107
\(3\) −1.08018e6 −1.17350 −0.586749 0.809769i \(-0.699592\pi\)
−0.586749 + 0.809769i \(0.699592\pi\)
\(4\) 1.67772e7 0.500000
\(5\) 0 0
\(6\) −4.42443e9 −0.829788
\(7\) −5.43159e10 −1.48320 −0.741602 0.670840i \(-0.765934\pi\)
−0.741602 + 0.670840i \(0.765934\pi\)
\(8\) 6.87195e10 0.353553
\(9\) 3.19509e11 0.377095
\(10\) 0 0
\(11\) −3.24375e12 −0.311630 −0.155815 0.987786i \(-0.549800\pi\)
−0.155815 + 0.987786i \(0.549800\pi\)
\(12\) −1.81225e13 −0.586749
\(13\) −7.83357e13 −0.932541 −0.466270 0.884642i \(-0.654403\pi\)
−0.466270 + 0.884642i \(0.654403\pi\)
\(14\) −2.22478e14 −1.04878
\(15\) 0 0
\(16\) 2.81475e14 0.250000
\(17\) 4.33806e15 1.80586 0.902931 0.429786i \(-0.141411\pi\)
0.902931 + 0.429786i \(0.141411\pi\)
\(18\) 1.30871e15 0.266647
\(19\) 8.87783e15 0.920210 0.460105 0.887865i \(-0.347812\pi\)
0.460105 + 0.887865i \(0.347812\pi\)
\(20\) 0 0
\(21\) 5.86711e16 1.74054
\(22\) −1.32864e16 −0.220355
\(23\) −7.05451e16 −0.671227 −0.335613 0.942000i \(-0.608944\pi\)
−0.335613 + 0.942000i \(0.608944\pi\)
\(24\) −7.42297e16 −0.414894
\(25\) 0 0
\(26\) −3.20863e17 −0.659406
\(27\) 5.70099e17 0.730977
\(28\) −9.11269e17 −0.741602
\(29\) −1.11391e18 −0.584624 −0.292312 0.956323i \(-0.594425\pi\)
−0.292312 + 0.956323i \(0.594425\pi\)
\(30\) 0 0
\(31\) 6.30227e18 1.43706 0.718532 0.695494i \(-0.244815\pi\)
0.718532 + 0.695494i \(0.244815\pi\)
\(32\) 1.15292e18 0.176777
\(33\) 3.50385e18 0.365696
\(34\) 1.77687e19 1.27694
\(35\) 0 0
\(36\) 5.36047e18 0.188548
\(37\) 1.55100e19 0.387338 0.193669 0.981067i \(-0.437961\pi\)
0.193669 + 0.981067i \(0.437961\pi\)
\(38\) 3.63636e19 0.650687
\(39\) 8.46169e19 1.09433
\(40\) 0 0
\(41\) −3.82653e19 −0.264854 −0.132427 0.991193i \(-0.542277\pi\)
−0.132427 + 0.991193i \(0.542277\pi\)
\(42\) 2.40317e20 1.23075
\(43\) −3.32618e20 −1.26937 −0.634687 0.772770i \(-0.718871\pi\)
−0.634687 + 0.772770i \(0.718871\pi\)
\(44\) −5.44211e19 −0.155815
\(45\) 0 0
\(46\) −2.88953e20 −0.474629
\(47\) 1.01157e21 1.26992 0.634958 0.772547i \(-0.281018\pi\)
0.634958 + 0.772547i \(0.281018\pi\)
\(48\) −3.04045e20 −0.293374
\(49\) 1.60914e21 1.19990
\(50\) 0 0
\(51\) −4.68591e21 −2.11917
\(52\) −1.31425e21 −0.466270
\(53\) −3.43480e21 −0.960401 −0.480201 0.877159i \(-0.659436\pi\)
−0.480201 + 0.877159i \(0.659436\pi\)
\(54\) 2.33513e21 0.516879
\(55\) 0 0
\(56\) −3.73256e21 −0.524392
\(57\) −9.58969e21 −1.07986
\(58\) −4.56259e21 −0.413392
\(59\) 4.09753e21 0.299828 0.149914 0.988699i \(-0.452100\pi\)
0.149914 + 0.988699i \(0.452100\pi\)
\(60\) 0 0
\(61\) 3.14378e22 1.51646 0.758228 0.651990i \(-0.226065\pi\)
0.758228 + 0.651990i \(0.226065\pi\)
\(62\) 2.58141e22 1.01616
\(63\) −1.73544e22 −0.559310
\(64\) 4.72237e21 0.125000
\(65\) 0 0
\(66\) 1.43518e22 0.258586
\(67\) −1.01756e23 −1.51923 −0.759616 0.650372i \(-0.774613\pi\)
−0.759616 + 0.650372i \(0.774613\pi\)
\(68\) 7.27806e22 0.902931
\(69\) 7.62017e22 0.787682
\(70\) 0 0
\(71\) 2.10284e23 1.52081 0.760407 0.649446i \(-0.224999\pi\)
0.760407 + 0.649446i \(0.224999\pi\)
\(72\) 2.19565e22 0.133323
\(73\) −1.56829e23 −0.801478 −0.400739 0.916192i \(-0.631247\pi\)
−0.400739 + 0.916192i \(0.631247\pi\)
\(74\) 6.35289e22 0.273889
\(75\) 0 0
\(76\) 1.48945e23 0.460105
\(77\) 1.76187e23 0.462211
\(78\) 3.46591e23 0.773811
\(79\) 6.00925e22 0.114415 0.0572073 0.998362i \(-0.481780\pi\)
0.0572073 + 0.998362i \(0.481780\pi\)
\(80\) 0 0
\(81\) −8.86528e23 −1.23489
\(82\) −1.56735e23 −0.187280
\(83\) 7.40331e23 0.760238 0.380119 0.924938i \(-0.375883\pi\)
0.380119 + 0.924938i \(0.375883\pi\)
\(84\) 9.84338e23 0.870268
\(85\) 0 0
\(86\) −1.36240e24 −0.897583
\(87\) 1.20323e24 0.686055
\(88\) −2.22909e23 −0.110178
\(89\) 3.21148e24 1.37826 0.689128 0.724640i \(-0.257994\pi\)
0.689128 + 0.724640i \(0.257994\pi\)
\(90\) 0 0
\(91\) 4.25487e24 1.38315
\(92\) −1.18355e24 −0.335613
\(93\) −6.80761e24 −1.68639
\(94\) 4.14341e24 0.897966
\(95\) 0 0
\(96\) −1.24537e24 −0.207447
\(97\) 5.32805e24 0.779689 0.389844 0.920881i \(-0.372529\pi\)
0.389844 + 0.920881i \(0.372529\pi\)
\(98\) 6.59105e24 0.848455
\(99\) −1.03641e24 −0.117514
\(100\) 0 0
\(101\) −9.97515e24 −0.880851 −0.440425 0.897789i \(-0.645172\pi\)
−0.440425 + 0.897789i \(0.645172\pi\)
\(102\) −1.91935e25 −1.49848
\(103\) 6.46906e24 0.447070 0.223535 0.974696i \(-0.428240\pi\)
0.223535 + 0.974696i \(0.428240\pi\)
\(104\) −5.38319e24 −0.329703
\(105\) 0 0
\(106\) −1.40689e25 −0.679106
\(107\) 2.37528e25 1.01957 0.509786 0.860302i \(-0.329725\pi\)
0.509786 + 0.860302i \(0.329725\pi\)
\(108\) 9.56468e24 0.365488
\(109\) −4.56311e25 −1.55393 −0.776963 0.629547i \(-0.783241\pi\)
−0.776963 + 0.629547i \(0.783241\pi\)
\(110\) 0 0
\(111\) −1.67536e25 −0.454540
\(112\) −1.52886e25 −0.370801
\(113\) −4.40956e25 −0.957005 −0.478503 0.878086i \(-0.658820\pi\)
−0.478503 + 0.878086i \(0.658820\pi\)
\(114\) −3.92794e25 −0.763579
\(115\) 0 0
\(116\) −1.86884e25 −0.292312
\(117\) −2.50289e25 −0.351657
\(118\) 1.67835e25 0.212011
\(119\) −2.35626e26 −2.67846
\(120\) 0 0
\(121\) −9.78251e25 −0.902887
\(122\) 1.28769e26 1.07230
\(123\) 4.13335e25 0.310805
\(124\) 1.05735e26 0.718532
\(125\) 0 0
\(126\) −7.10836e25 −0.395492
\(127\) −3.54526e26 −1.78691 −0.893453 0.449157i \(-0.851724\pi\)
−0.893453 + 0.449157i \(0.851724\pi\)
\(128\) 1.93428e25 0.0883883
\(129\) 3.59288e26 1.48961
\(130\) 0 0
\(131\) −3.11451e26 −1.06537 −0.532683 0.846315i \(-0.678816\pi\)
−0.532683 + 0.846315i \(0.678816\pi\)
\(132\) 5.87848e25 0.182848
\(133\) −4.82207e26 −1.36486
\(134\) −4.16792e26 −1.07426
\(135\) 0 0
\(136\) 2.98110e26 0.638468
\(137\) 4.08877e26 0.799072 0.399536 0.916718i \(-0.369171\pi\)
0.399536 + 0.916718i \(0.369171\pi\)
\(138\) 3.12122e26 0.556976
\(139\) 2.66018e26 0.433737 0.216869 0.976201i \(-0.430416\pi\)
0.216869 + 0.976201i \(0.430416\pi\)
\(140\) 0 0
\(141\) −1.09269e27 −1.49024
\(142\) 8.61322e26 1.07538
\(143\) 2.54101e26 0.290607
\(144\) 8.99337e25 0.0942739
\(145\) 0 0
\(146\) −6.42373e26 −0.566730
\(147\) −1.73817e27 −1.40807
\(148\) 2.60214e26 0.193669
\(149\) −1.37183e26 −0.0938579 −0.0469290 0.998898i \(-0.514943\pi\)
−0.0469290 + 0.998898i \(0.514943\pi\)
\(150\) 0 0
\(151\) 2.28415e27 1.32286 0.661428 0.750009i \(-0.269951\pi\)
0.661428 + 0.750009i \(0.269951\pi\)
\(152\) 6.10080e26 0.325343
\(153\) 1.38605e27 0.680982
\(154\) 7.21662e26 0.326832
\(155\) 0 0
\(156\) 1.41964e27 0.547167
\(157\) −5.56736e26 −0.198109 −0.0990544 0.995082i \(-0.531582\pi\)
−0.0990544 + 0.995082i \(0.531582\pi\)
\(158\) 2.46139e26 0.0809034
\(159\) 3.71021e27 1.12703
\(160\) 0 0
\(161\) 3.83172e27 0.995566
\(162\) −3.63122e27 −0.873202
\(163\) −6.45827e27 −1.43804 −0.719020 0.694990i \(-0.755409\pi\)
−0.719020 + 0.694990i \(0.755409\pi\)
\(164\) −6.41985e26 −0.132427
\(165\) 0 0
\(166\) 3.03239e27 0.537569
\(167\) −9.17351e26 −0.150862 −0.0754310 0.997151i \(-0.524033\pi\)
−0.0754310 + 0.997151i \(0.524033\pi\)
\(168\) 4.03185e27 0.615373
\(169\) −9.19932e26 −0.130368
\(170\) 0 0
\(171\) 2.83654e27 0.347007
\(172\) −5.58040e27 −0.634687
\(173\) −1.13072e28 −1.19614 −0.598068 0.801445i \(-0.704065\pi\)
−0.598068 + 0.801445i \(0.704065\pi\)
\(174\) 4.92844e27 0.485114
\(175\) 0 0
\(176\) −9.13034e26 −0.0779074
\(177\) −4.42609e27 −0.351847
\(178\) 1.31542e28 0.974574
\(179\) −3.61567e27 −0.249762 −0.124881 0.992172i \(-0.539855\pi\)
−0.124881 + 0.992172i \(0.539855\pi\)
\(180\) 0 0
\(181\) −1.10646e28 −0.665201 −0.332601 0.943068i \(-0.607926\pi\)
−0.332601 + 0.943068i \(0.607926\pi\)
\(182\) 1.74279e28 0.978034
\(183\) −3.39587e28 −1.77956
\(184\) −4.84782e27 −0.237314
\(185\) 0 0
\(186\) −2.78840e28 −1.19246
\(187\) −1.40716e28 −0.562760
\(188\) 1.69714e28 0.634958
\(189\) −3.09654e28 −1.08419
\(190\) 0 0
\(191\) 5.84877e27 0.179534 0.0897671 0.995963i \(-0.471388\pi\)
0.0897671 + 0.995963i \(0.471388\pi\)
\(192\) −5.10102e27 −0.146687
\(193\) 1.61649e28 0.435620 0.217810 0.975991i \(-0.430109\pi\)
0.217810 + 0.975991i \(0.430109\pi\)
\(194\) 2.18237e28 0.551323
\(195\) 0 0
\(196\) 2.69969e28 0.599948
\(197\) −5.33679e28 −1.11289 −0.556445 0.830885i \(-0.687835\pi\)
−0.556445 + 0.830885i \(0.687835\pi\)
\(198\) −4.24512e27 −0.0830950
\(199\) 7.95288e27 0.146171 0.0730855 0.997326i \(-0.476715\pi\)
0.0730855 + 0.997326i \(0.476715\pi\)
\(200\) 0 0
\(201\) 1.09915e29 1.78281
\(202\) −4.08582e28 −0.622855
\(203\) 6.05032e28 0.867118
\(204\) −7.86165e28 −1.05959
\(205\) 0 0
\(206\) 2.64973e28 0.316127
\(207\) −2.25398e28 −0.253116
\(208\) −2.20495e28 −0.233135
\(209\) −2.87975e28 −0.286765
\(210\) 0 0
\(211\) −6.17913e28 −0.546257 −0.273128 0.961978i \(-0.588058\pi\)
−0.273128 + 0.961978i \(0.588058\pi\)
\(212\) −5.76263e28 −0.480201
\(213\) −2.27145e29 −1.78467
\(214\) 9.72914e28 0.720946
\(215\) 0 0
\(216\) 3.91769e28 0.258439
\(217\) −3.42313e29 −2.13146
\(218\) −1.86905e29 −1.09879
\(219\) 1.69404e29 0.940532
\(220\) 0 0
\(221\) −3.39825e29 −1.68404
\(222\) −6.86229e28 −0.321408
\(223\) 3.94158e29 1.74526 0.872630 0.488382i \(-0.162413\pi\)
0.872630 + 0.488382i \(0.162413\pi\)
\(224\) −6.26219e28 −0.262196
\(225\) 0 0
\(226\) −1.80615e29 −0.676705
\(227\) −1.43566e29 −0.509012 −0.254506 0.967071i \(-0.581913\pi\)
−0.254506 + 0.967071i \(0.581913\pi\)
\(228\) −1.60888e29 −0.539932
\(229\) 3.75130e29 1.19189 0.595947 0.803023i \(-0.296777\pi\)
0.595947 + 0.803023i \(0.296777\pi\)
\(230\) 0 0
\(231\) −1.90314e29 −0.542403
\(232\) −7.65476e28 −0.206696
\(233\) −4.51422e29 −1.15514 −0.577569 0.816342i \(-0.695999\pi\)
−0.577569 + 0.816342i \(0.695999\pi\)
\(234\) −1.02518e29 −0.248659
\(235\) 0 0
\(236\) 6.87452e28 0.149914
\(237\) −6.49109e28 −0.134265
\(238\) −9.65123e29 −1.89396
\(239\) −1.77838e29 −0.331169 −0.165585 0.986196i \(-0.552951\pi\)
−0.165585 + 0.986196i \(0.552951\pi\)
\(240\) 0 0
\(241\) −5.18328e28 −0.0869745 −0.0434872 0.999054i \(-0.513847\pi\)
−0.0434872 + 0.999054i \(0.513847\pi\)
\(242\) −4.00692e29 −0.638438
\(243\) 4.74575e29 0.718168
\(244\) 5.27439e29 0.758228
\(245\) 0 0
\(246\) 1.69302e29 0.219773
\(247\) −6.95451e29 −0.858133
\(248\) 4.33089e29 0.508079
\(249\) −7.99693e29 −0.892137
\(250\) 0 0
\(251\) 9.22037e29 0.930737 0.465368 0.885117i \(-0.345922\pi\)
0.465368 + 0.885117i \(0.345922\pi\)
\(252\) −2.91158e29 −0.279655
\(253\) 2.28831e29 0.209174
\(254\) −1.45214e30 −1.26353
\(255\) 0 0
\(256\) 7.92282e28 0.0625000
\(257\) −2.62664e30 −1.97350 −0.986749 0.162254i \(-0.948123\pi\)
−0.986749 + 0.162254i \(0.948123\pi\)
\(258\) 1.47164e30 1.05331
\(259\) −8.42438e29 −0.574501
\(260\) 0 0
\(261\) −3.55905e29 −0.220459
\(262\) −1.27570e30 −0.753328
\(263\) −1.28488e30 −0.723465 −0.361732 0.932282i \(-0.617815\pi\)
−0.361732 + 0.932282i \(0.617815\pi\)
\(264\) 2.40782e29 0.129293
\(265\) 0 0
\(266\) −1.97512e30 −0.965102
\(267\) −3.46899e30 −1.61738
\(268\) −1.70718e30 −0.759616
\(269\) −2.87870e30 −1.22262 −0.611312 0.791390i \(-0.709358\pi\)
−0.611312 + 0.791390i \(0.709358\pi\)
\(270\) 0 0
\(271\) 6.47478e29 0.250674 0.125337 0.992114i \(-0.459999\pi\)
0.125337 + 0.992114i \(0.459999\pi\)
\(272\) 1.22106e30 0.451465
\(273\) −4.59604e30 −1.62312
\(274\) 1.67476e30 0.565029
\(275\) 0 0
\(276\) 1.27845e30 0.393841
\(277\) 3.36035e30 0.989436 0.494718 0.869054i \(-0.335271\pi\)
0.494718 + 0.869054i \(0.335271\pi\)
\(278\) 1.08961e30 0.306699
\(279\) 2.01363e30 0.541910
\(280\) 0 0
\(281\) 3.79608e30 0.934343 0.467172 0.884167i \(-0.345273\pi\)
0.467172 + 0.884167i \(0.345273\pi\)
\(282\) −4.47564e30 −1.05376
\(283\) 2.26117e30 0.509334 0.254667 0.967029i \(-0.418034\pi\)
0.254667 + 0.967029i \(0.418034\pi\)
\(284\) 3.52798e30 0.760407
\(285\) 0 0
\(286\) 1.04080e30 0.205490
\(287\) 2.07841e30 0.392833
\(288\) 3.68368e29 0.0666617
\(289\) 1.30482e31 2.26114
\(290\) 0 0
\(291\) −5.75527e30 −0.914962
\(292\) −2.63116e30 −0.400739
\(293\) −8.20237e30 −1.19700 −0.598500 0.801123i \(-0.704236\pi\)
−0.598500 + 0.801123i \(0.704236\pi\)
\(294\) −7.11955e30 −0.995659
\(295\) 0 0
\(296\) 1.06584e30 0.136945
\(297\) −1.84926e30 −0.227794
\(298\) −5.61900e29 −0.0663676
\(299\) 5.52620e30 0.625946
\(300\) 0 0
\(301\) 1.80664e31 1.88274
\(302\) 9.35588e30 0.935401
\(303\) 1.07750e31 1.03368
\(304\) 2.49889e30 0.230052
\(305\) 0 0
\(306\) 5.67726e30 0.481527
\(307\) −1.16923e31 −0.952072 −0.476036 0.879426i \(-0.657927\pi\)
−0.476036 + 0.879426i \(0.657927\pi\)
\(308\) 2.95593e30 0.231105
\(309\) −6.98778e30 −0.524636
\(310\) 0 0
\(311\) −9.75410e30 −0.675588 −0.337794 0.941220i \(-0.609681\pi\)
−0.337794 + 0.941220i \(0.609681\pi\)
\(312\) 5.81483e30 0.386905
\(313\) −2.13999e31 −1.36807 −0.684035 0.729449i \(-0.739776\pi\)
−0.684035 + 0.729449i \(0.739776\pi\)
\(314\) −2.28039e30 −0.140084
\(315\) 0 0
\(316\) 1.00818e30 0.0572073
\(317\) −2.87837e31 −1.57002 −0.785011 0.619482i \(-0.787343\pi\)
−0.785011 + 0.619482i \(0.787343\pi\)
\(318\) 1.51970e31 0.796929
\(319\) 3.61326e30 0.182186
\(320\) 0 0
\(321\) −2.56574e31 −1.19646
\(322\) 1.56947e31 0.703972
\(323\) 3.85126e31 1.66177
\(324\) −1.48735e31 −0.617447
\(325\) 0 0
\(326\) −2.64531e31 −1.01685
\(327\) 4.92900e31 1.82353
\(328\) −2.62957e30 −0.0936400
\(329\) −5.49445e31 −1.88354
\(330\) 0 0
\(331\) 7.95904e30 0.252936 0.126468 0.991971i \(-0.459636\pi\)
0.126468 + 0.991971i \(0.459636\pi\)
\(332\) 1.24207e31 0.380119
\(333\) 4.95557e30 0.146063
\(334\) −3.75747e30 −0.106675
\(335\) 0 0
\(336\) 1.65144e31 0.435134
\(337\) 5.91626e31 1.50201 0.751006 0.660295i \(-0.229569\pi\)
0.751006 + 0.660295i \(0.229569\pi\)
\(338\) −3.76804e30 −0.0921842
\(339\) 4.76313e31 1.12304
\(340\) 0 0
\(341\) −2.04430e31 −0.447832
\(342\) 1.16185e31 0.245371
\(343\) −1.45607e31 −0.296487
\(344\) −2.28573e31 −0.448791
\(345\) 0 0
\(346\) −4.63145e31 −0.845796
\(347\) −2.62292e31 −0.462026 −0.231013 0.972951i \(-0.574204\pi\)
−0.231013 + 0.972951i \(0.574204\pi\)
\(348\) 2.01869e31 0.343028
\(349\) 6.42126e31 1.05269 0.526347 0.850270i \(-0.323561\pi\)
0.526347 + 0.850270i \(0.323561\pi\)
\(350\) 0 0
\(351\) −4.46591e31 −0.681665
\(352\) −3.73979e30 −0.0550889
\(353\) −3.67958e31 −0.523136 −0.261568 0.965185i \(-0.584240\pi\)
−0.261568 + 0.965185i \(0.584240\pi\)
\(354\) −1.81293e31 −0.248794
\(355\) 0 0
\(356\) 5.38797e31 0.689128
\(357\) 2.54519e32 3.14317
\(358\) −1.48098e31 −0.176608
\(359\) −1.54521e32 −1.77953 −0.889766 0.456418i \(-0.849132\pi\)
−0.889766 + 0.456418i \(0.849132\pi\)
\(360\) 0 0
\(361\) −1.42606e31 −0.153214
\(362\) −4.53205e31 −0.470368
\(363\) 1.05669e32 1.05954
\(364\) 7.13849e31 0.691574
\(365\) 0 0
\(366\) −1.39095e32 −1.25834
\(367\) 3.45113e31 0.301742 0.150871 0.988553i \(-0.451792\pi\)
0.150871 + 0.988553i \(0.451792\pi\)
\(368\) −1.98567e31 −0.167807
\(369\) −1.22261e31 −0.0998752
\(370\) 0 0
\(371\) 1.86564e32 1.42447
\(372\) −1.14213e32 −0.843195
\(373\) 5.25010e31 0.374807 0.187403 0.982283i \(-0.439993\pi\)
0.187403 + 0.982283i \(0.439993\pi\)
\(374\) −5.76372e31 −0.397931
\(375\) 0 0
\(376\) 6.95149e31 0.448983
\(377\) 8.72593e31 0.545186
\(378\) −1.26834e32 −0.766637
\(379\) 3.14351e32 1.83834 0.919169 0.393864i \(-0.128862\pi\)
0.919169 + 0.393864i \(0.128862\pi\)
\(380\) 0 0
\(381\) 3.82953e32 2.09693
\(382\) 2.39566e31 0.126950
\(383\) 2.45971e32 1.26154 0.630768 0.775972i \(-0.282740\pi\)
0.630768 + 0.775972i \(0.282740\pi\)
\(384\) −2.08938e31 −0.103723
\(385\) 0 0
\(386\) 6.62116e31 0.308030
\(387\) −1.06274e32 −0.478675
\(388\) 8.93898e31 0.389844
\(389\) 2.32919e32 0.983638 0.491819 0.870698i \(-0.336332\pi\)
0.491819 + 0.870698i \(0.336332\pi\)
\(390\) 0 0
\(391\) −3.06029e32 −1.21214
\(392\) 1.10579e32 0.424227
\(393\) 3.36425e32 1.25020
\(394\) −2.18595e32 −0.786932
\(395\) 0 0
\(396\) −1.73880e31 −0.0587571
\(397\) 2.38017e32 0.779338 0.389669 0.920955i \(-0.372589\pi\)
0.389669 + 0.920955i \(0.372589\pi\)
\(398\) 3.25750e31 0.103358
\(399\) 5.20872e32 1.60166
\(400\) 0 0
\(401\) 5.61815e32 1.62289 0.811444 0.584430i \(-0.198682\pi\)
0.811444 + 0.584430i \(0.198682\pi\)
\(402\) 4.50212e32 1.26064
\(403\) −4.93693e32 −1.34012
\(404\) −1.67355e32 −0.440425
\(405\) 0 0
\(406\) 2.47821e32 0.613145
\(407\) −5.03105e31 −0.120706
\(408\) −3.22013e32 −0.749241
\(409\) −3.90144e32 −0.880408 −0.440204 0.897898i \(-0.645094\pi\)
−0.440204 + 0.897898i \(0.645094\pi\)
\(410\) 0 0
\(411\) −4.41662e32 −0.937708
\(412\) 1.08533e32 0.223535
\(413\) −2.22561e32 −0.444707
\(414\) −9.23229e31 −0.178980
\(415\) 0 0
\(416\) −9.03149e31 −0.164851
\(417\) −2.87349e32 −0.508989
\(418\) −1.17954e32 −0.202773
\(419\) 5.77394e32 0.963378 0.481689 0.876342i \(-0.340023\pi\)
0.481689 + 0.876342i \(0.340023\pi\)
\(420\) 0 0
\(421\) −9.43709e32 −1.48358 −0.741792 0.670630i \(-0.766024\pi\)
−0.741792 + 0.670630i \(0.766024\pi\)
\(422\) −2.53097e32 −0.386262
\(423\) 3.23207e32 0.478879
\(424\) −2.36037e32 −0.339553
\(425\) 0 0
\(426\) −9.30386e32 −1.26195
\(427\) −1.70757e33 −2.24921
\(428\) 3.98506e32 0.509786
\(429\) −2.74476e32 −0.341027
\(430\) 0 0
\(431\) 3.90741e32 0.458061 0.229030 0.973419i \(-0.426444\pi\)
0.229030 + 0.973419i \(0.426444\pi\)
\(432\) 1.60469e32 0.182744
\(433\) −7.17692e32 −0.794034 −0.397017 0.917811i \(-0.629955\pi\)
−0.397017 + 0.917811i \(0.629955\pi\)
\(434\) −1.40211e33 −1.50717
\(435\) 0 0
\(436\) −7.65564e32 −0.776963
\(437\) −6.26288e32 −0.617669
\(438\) 6.93881e32 0.665056
\(439\) −4.73065e32 −0.440671 −0.220336 0.975424i \(-0.570715\pi\)
−0.220336 + 0.975424i \(0.570715\pi\)
\(440\) 0 0
\(441\) 5.14135e32 0.452475
\(442\) −1.39192e33 −1.19080
\(443\) −1.16496e33 −0.968871 −0.484436 0.874827i \(-0.660975\pi\)
−0.484436 + 0.874827i \(0.660975\pi\)
\(444\) −2.81079e32 −0.227270
\(445\) 0 0
\(446\) 1.61447e33 1.23408
\(447\) 1.48183e32 0.110142
\(448\) −2.56499e32 −0.185401
\(449\) −1.31459e33 −0.924082 −0.462041 0.886859i \(-0.652883\pi\)
−0.462041 + 0.886859i \(0.652883\pi\)
\(450\) 0 0
\(451\) 1.24123e32 0.0825363
\(452\) −7.39801e32 −0.478503
\(453\) −2.46730e33 −1.55237
\(454\) −5.88046e32 −0.359926
\(455\) 0 0
\(456\) −6.58999e32 −0.381789
\(457\) 4.66527e32 0.262981 0.131490 0.991317i \(-0.458024\pi\)
0.131490 + 0.991317i \(0.458024\pi\)
\(458\) 1.53653e33 0.842797
\(459\) 2.47313e33 1.32004
\(460\) 0 0
\(461\) 1.54013e33 0.778566 0.389283 0.921118i \(-0.372723\pi\)
0.389283 + 0.921118i \(0.372723\pi\)
\(462\) −7.79528e32 −0.383537
\(463\) 2.04979e33 0.981628 0.490814 0.871264i \(-0.336699\pi\)
0.490814 + 0.871264i \(0.336699\pi\)
\(464\) −3.13539e32 −0.146156
\(465\) 0 0
\(466\) −1.84903e33 −0.816806
\(467\) −1.89201e33 −0.813699 −0.406849 0.913495i \(-0.633373\pi\)
−0.406849 + 0.913495i \(0.633373\pi\)
\(468\) −4.19916e32 −0.175828
\(469\) 5.52695e33 2.25333
\(470\) 0 0
\(471\) 6.01378e32 0.232480
\(472\) 2.81580e32 0.106005
\(473\) 1.07893e33 0.395574
\(474\) −2.65875e32 −0.0949399
\(475\) 0 0
\(476\) −3.95314e33 −1.33923
\(477\) −1.09745e33 −0.362163
\(478\) −7.28423e32 −0.234172
\(479\) 9.31244e32 0.291655 0.145828 0.989310i \(-0.453416\pi\)
0.145828 + 0.989310i \(0.453416\pi\)
\(480\) 0 0
\(481\) −1.21499e33 −0.361208
\(482\) −2.12307e32 −0.0615003
\(483\) −4.13896e33 −1.16829
\(484\) −1.64123e33 −0.451443
\(485\) 0 0
\(486\) 1.94386e33 0.507822
\(487\) 1.90079e33 0.483973 0.241987 0.970280i \(-0.422201\pi\)
0.241987 + 0.970280i \(0.422201\pi\)
\(488\) 2.16039e33 0.536148
\(489\) 6.97611e33 1.68754
\(490\) 0 0
\(491\) −1.53089e32 −0.0351905 −0.0175953 0.999845i \(-0.505601\pi\)
−0.0175953 + 0.999845i \(0.505601\pi\)
\(492\) 6.93461e32 0.155403
\(493\) −4.83223e33 −1.05575
\(494\) −2.84857e33 −0.606792
\(495\) 0 0
\(496\) 1.77393e33 0.359266
\(497\) −1.14217e34 −2.25568
\(498\) −3.27554e33 −0.630836
\(499\) −3.99567e33 −0.750469 −0.375234 0.926930i \(-0.622438\pi\)
−0.375234 + 0.926930i \(0.622438\pi\)
\(500\) 0 0
\(501\) 9.90908e32 0.177036
\(502\) 3.77666e33 0.658130
\(503\) 7.03745e33 1.19623 0.598116 0.801410i \(-0.295916\pi\)
0.598116 + 0.801410i \(0.295916\pi\)
\(504\) −1.19258e33 −0.197746
\(505\) 0 0
\(506\) 9.37291e32 0.147908
\(507\) 9.93695e32 0.152987
\(508\) −5.94796e33 −0.893453
\(509\) 5.62123e32 0.0823870 0.0411935 0.999151i \(-0.486884\pi\)
0.0411935 + 0.999151i \(0.486884\pi\)
\(510\) 0 0
\(511\) 8.51832e33 1.18876
\(512\) 3.24519e32 0.0441942
\(513\) 5.06125e33 0.672652
\(514\) −1.07587e34 −1.39547
\(515\) 0 0
\(516\) 6.02785e33 0.744803
\(517\) −3.28129e33 −0.395743
\(518\) −3.45063e33 −0.406234
\(519\) 1.22139e34 1.40366
\(520\) 0 0
\(521\) 1.15250e34 1.26232 0.631162 0.775651i \(-0.282578\pi\)
0.631162 + 0.775651i \(0.282578\pi\)
\(522\) −1.45779e33 −0.155888
\(523\) 1.37858e34 1.43933 0.719663 0.694323i \(-0.244296\pi\)
0.719663 + 0.694323i \(0.244296\pi\)
\(524\) −5.22529e33 −0.532683
\(525\) 0 0
\(526\) −5.26289e33 −0.511567
\(527\) 2.73397e34 2.59514
\(528\) 9.86245e32 0.0914241
\(529\) −6.06915e33 −0.549455
\(530\) 0 0
\(531\) 1.30920e33 0.113064
\(532\) −8.09009e33 −0.682430
\(533\) 2.99754e33 0.246987
\(534\) −1.42090e34 −1.14366
\(535\) 0 0
\(536\) −6.99260e33 −0.537129
\(537\) 3.90559e33 0.293095
\(538\) −1.17911e34 −0.864525
\(539\) −5.21966e33 −0.373923
\(540\) 0 0
\(541\) −2.20638e34 −1.50909 −0.754543 0.656251i \(-0.772141\pi\)
−0.754543 + 0.656251i \(0.772141\pi\)
\(542\) 2.65207e33 0.177253
\(543\) 1.19518e34 0.780612
\(544\) 5.00145e33 0.319234
\(545\) 0 0
\(546\) −1.88254e34 −1.14772
\(547\) 1.59179e34 0.948514 0.474257 0.880387i \(-0.342717\pi\)
0.474257 + 0.880387i \(0.342717\pi\)
\(548\) 6.85982e33 0.399536
\(549\) 1.00447e34 0.571849
\(550\) 0 0
\(551\) −9.88915e33 −0.537977
\(552\) 5.23654e33 0.278488
\(553\) −3.26398e33 −0.169700
\(554\) 1.37640e34 0.699637
\(555\) 0 0
\(556\) 4.46305e33 0.216869
\(557\) 1.37117e34 0.651479 0.325739 0.945460i \(-0.394387\pi\)
0.325739 + 0.945460i \(0.394387\pi\)
\(558\) 8.24783e33 0.383188
\(559\) 2.60558e34 1.18374
\(560\) 0 0
\(561\) 1.51999e34 0.660397
\(562\) 1.55488e34 0.660681
\(563\) −3.01185e34 −1.25164 −0.625818 0.779969i \(-0.715235\pi\)
−0.625818 + 0.779969i \(0.715235\pi\)
\(564\) −1.83322e34 −0.745121
\(565\) 0 0
\(566\) 9.26175e33 0.360153
\(567\) 4.81525e34 1.83160
\(568\) 1.44506e34 0.537689
\(569\) 1.37959e34 0.502166 0.251083 0.967966i \(-0.419213\pi\)
0.251083 + 0.967966i \(0.419213\pi\)
\(570\) 0 0
\(571\) 1.46730e34 0.511173 0.255587 0.966786i \(-0.417731\pi\)
0.255587 + 0.966786i \(0.417731\pi\)
\(572\) 4.26311e33 0.145304
\(573\) −6.31775e33 −0.210683
\(574\) 8.51317e33 0.277775
\(575\) 0 0
\(576\) 1.50884e33 0.0471369
\(577\) −5.61780e34 −1.71739 −0.858694 0.512488i \(-0.828724\pi\)
−0.858694 + 0.512488i \(0.828724\pi\)
\(578\) 5.34453e34 1.59886
\(579\) −1.74611e34 −0.511199
\(580\) 0 0
\(581\) −4.02117e34 −1.12759
\(582\) −2.35736e34 −0.646976
\(583\) 1.11416e34 0.299290
\(584\) −1.07772e34 −0.283365
\(585\) 0 0
\(586\) −3.35969e34 −0.846407
\(587\) 3.79085e34 0.934889 0.467444 0.884022i \(-0.345175\pi\)
0.467444 + 0.884022i \(0.345175\pi\)
\(588\) −2.91617e34 −0.704037
\(589\) 5.59505e34 1.32240
\(590\) 0 0
\(591\) 5.76471e34 1.30597
\(592\) 4.36567e33 0.0968344
\(593\) 1.87221e34 0.406604 0.203302 0.979116i \(-0.434833\pi\)
0.203302 + 0.979116i \(0.434833\pi\)
\(594\) −7.57457e33 −0.161075
\(595\) 0 0
\(596\) −2.30154e33 −0.0469290
\(597\) −8.59057e33 −0.171531
\(598\) 2.26353e34 0.442611
\(599\) −5.76089e34 −1.10320 −0.551601 0.834108i \(-0.685983\pi\)
−0.551601 + 0.834108i \(0.685983\pi\)
\(600\) 0 0
\(601\) 4.67680e34 0.859049 0.429525 0.903055i \(-0.358681\pi\)
0.429525 + 0.903055i \(0.358681\pi\)
\(602\) 7.40000e34 1.33130
\(603\) −3.25118e34 −0.572895
\(604\) 3.83217e34 0.661428
\(605\) 0 0
\(606\) 4.41344e34 0.730919
\(607\) 3.27875e33 0.0531924 0.0265962 0.999646i \(-0.491533\pi\)
0.0265962 + 0.999646i \(0.491533\pi\)
\(608\) 1.02354e34 0.162672
\(609\) −6.53546e34 −1.01756
\(610\) 0 0
\(611\) −7.92424e34 −1.18425
\(612\) 2.32540e34 0.340491
\(613\) 3.82074e34 0.548140 0.274070 0.961710i \(-0.411630\pi\)
0.274070 + 0.961710i \(0.411630\pi\)
\(614\) −4.78916e34 −0.673216
\(615\) 0 0
\(616\) 1.21075e34 0.163416
\(617\) −8.69280e34 −1.14973 −0.574864 0.818249i \(-0.694945\pi\)
−0.574864 + 0.818249i \(0.694945\pi\)
\(618\) −2.86219e34 −0.370974
\(619\) −3.61889e34 −0.459666 −0.229833 0.973230i \(-0.573818\pi\)
−0.229833 + 0.973230i \(0.573818\pi\)
\(620\) 0 0
\(621\) −4.02177e34 −0.490651
\(622\) −3.99528e34 −0.477713
\(623\) −1.74434e35 −2.04424
\(624\) 2.38176e34 0.273583
\(625\) 0 0
\(626\) −8.76542e34 −0.967372
\(627\) 3.11066e34 0.336517
\(628\) −9.34049e33 −0.0990544
\(629\) 6.72833e34 0.699478
\(630\) 0 0
\(631\) −6.79781e34 −0.679207 −0.339604 0.940569i \(-0.610293\pi\)
−0.339604 + 0.940569i \(0.610293\pi\)
\(632\) 4.12953e33 0.0404517
\(633\) 6.67460e34 0.641030
\(634\) −1.17898e35 −1.11017
\(635\) 0 0
\(636\) 6.22470e34 0.563514
\(637\) −1.26053e35 −1.11895
\(638\) 1.47999e34 0.128825
\(639\) 6.71875e34 0.573492
\(640\) 0 0
\(641\) −8.90435e34 −0.730932 −0.365466 0.930825i \(-0.619090\pi\)
−0.365466 + 0.930825i \(0.619090\pi\)
\(642\) −1.05093e35 −0.846028
\(643\) 1.28997e35 1.01846 0.509230 0.860630i \(-0.329930\pi\)
0.509230 + 0.860630i \(0.329930\pi\)
\(644\) 6.42856e34 0.497783
\(645\) 0 0
\(646\) 1.57748e35 1.17505
\(647\) 1.39383e34 0.101837 0.0509186 0.998703i \(-0.483785\pi\)
0.0509186 + 0.998703i \(0.483785\pi\)
\(648\) −6.09218e34 −0.436601
\(649\) −1.32914e34 −0.0934353
\(650\) 0 0
\(651\) 3.69761e35 2.50126
\(652\) −1.08352e35 −0.719020
\(653\) −1.27030e35 −0.826970 −0.413485 0.910511i \(-0.635689\pi\)
−0.413485 + 0.910511i \(0.635689\pi\)
\(654\) 2.01892e35 1.28943
\(655\) 0 0
\(656\) −1.07707e34 −0.0662135
\(657\) −5.01083e34 −0.302234
\(658\) −2.25053e35 −1.33187
\(659\) 4.18479e34 0.243000 0.121500 0.992591i \(-0.461230\pi\)
0.121500 + 0.992591i \(0.461230\pi\)
\(660\) 0 0
\(661\) −2.05843e34 −0.115085 −0.0575424 0.998343i \(-0.518326\pi\)
−0.0575424 + 0.998343i \(0.518326\pi\)
\(662\) 3.26002e34 0.178853
\(663\) 3.67074e35 1.97621
\(664\) 5.08751e34 0.268785
\(665\) 0 0
\(666\) 2.02980e34 0.103282
\(667\) 7.85813e34 0.392415
\(668\) −1.53906e34 −0.0754310
\(669\) −4.25764e35 −2.04806
\(670\) 0 0
\(671\) −1.01976e35 −0.472573
\(672\) 6.76432e34 0.307686
\(673\) −1.99051e35 −0.888741 −0.444370 0.895843i \(-0.646573\pi\)
−0.444370 + 0.895843i \(0.646573\pi\)
\(674\) 2.42330e35 1.06208
\(675\) 0 0
\(676\) −1.54339e34 −0.0651841
\(677\) −3.44905e35 −1.43002 −0.715008 0.699116i \(-0.753577\pi\)
−0.715008 + 0.699116i \(0.753577\pi\)
\(678\) 1.95098e35 0.794111
\(679\) −2.89397e35 −1.15644
\(680\) 0 0
\(681\) 1.55078e35 0.597324
\(682\) −8.37345e34 −0.316665
\(683\) −3.76388e35 −1.39758 −0.698791 0.715326i \(-0.746278\pi\)
−0.698791 + 0.715326i \(0.746278\pi\)
\(684\) 4.75893e34 0.173503
\(685\) 0 0
\(686\) −5.96407e34 −0.209648
\(687\) −4.05209e35 −1.39869
\(688\) −9.36235e34 −0.317343
\(689\) 2.69067e35 0.895613
\(690\) 0 0
\(691\) −5.21059e35 −1.67268 −0.836338 0.548215i \(-0.815308\pi\)
−0.836338 + 0.548215i \(0.815308\pi\)
\(692\) −1.89704e35 −0.598068
\(693\) 5.62933e34 0.174297
\(694\) −1.07435e35 −0.326702
\(695\) 0 0
\(696\) 8.26855e34 0.242557
\(697\) −1.65997e35 −0.478290
\(698\) 2.63015e35 0.744368
\(699\) 4.87619e35 1.35555
\(700\) 0 0
\(701\) 2.95057e35 0.791462 0.395731 0.918366i \(-0.370491\pi\)
0.395731 + 0.918366i \(0.370491\pi\)
\(702\) −1.82924e35 −0.482010
\(703\) 1.37695e35 0.356432
\(704\) −1.53182e34 −0.0389537
\(705\) 0 0
\(706\) −1.50716e35 −0.369913
\(707\) 5.41809e35 1.30648
\(708\) −7.42575e34 −0.175924
\(709\) −6.60823e35 −1.53818 −0.769090 0.639141i \(-0.779290\pi\)
−0.769090 + 0.639141i \(0.779290\pi\)
\(710\) 0 0
\(711\) 1.92001e34 0.0431452
\(712\) 2.20691e35 0.487287
\(713\) −4.44595e35 −0.964595
\(714\) 1.04251e36 2.22256
\(715\) 0 0
\(716\) −6.06609e34 −0.124881
\(717\) 1.92097e35 0.388626
\(718\) −6.32916e35 −1.25832
\(719\) −5.66102e35 −1.10607 −0.553036 0.833157i \(-0.686531\pi\)
−0.553036 + 0.833157i \(0.686531\pi\)
\(720\) 0 0
\(721\) −3.51373e35 −0.663097
\(722\) −5.84114e34 −0.108339
\(723\) 5.59890e34 0.102064
\(724\) −1.85633e35 −0.332601
\(725\) 0 0
\(726\) 4.32821e35 0.749205
\(727\) 1.18938e35 0.202368 0.101184 0.994868i \(-0.467737\pi\)
0.101184 + 0.994868i \(0.467737\pi\)
\(728\) 2.92392e35 0.489017
\(729\) 2.38517e35 0.392126
\(730\) 0 0
\(731\) −1.44292e36 −2.29231
\(732\) −5.69732e35 −0.889778
\(733\) 2.75860e35 0.423534 0.211767 0.977320i \(-0.432078\pi\)
0.211767 + 0.977320i \(0.432078\pi\)
\(734\) 1.41358e35 0.213364
\(735\) 0 0
\(736\) −8.13330e34 −0.118657
\(737\) 3.30070e35 0.473438
\(738\) −5.00780e34 −0.0706224
\(739\) −7.44302e35 −1.03203 −0.516017 0.856579i \(-0.672586\pi\)
−0.516017 + 0.856579i \(0.672586\pi\)
\(740\) 0 0
\(741\) 7.51215e35 1.00702
\(742\) 7.64166e35 1.00725
\(743\) 9.71605e35 1.25930 0.629651 0.776878i \(-0.283198\pi\)
0.629651 + 0.776878i \(0.283198\pi\)
\(744\) −4.67816e35 −0.596229
\(745\) 0 0
\(746\) 2.15044e35 0.265028
\(747\) 2.36542e35 0.286682
\(748\) −2.36082e35 −0.281380
\(749\) −1.29015e36 −1.51223
\(750\) 0 0
\(751\) −3.81212e35 −0.432183 −0.216091 0.976373i \(-0.569331\pi\)
−0.216091 + 0.976373i \(0.569331\pi\)
\(752\) 2.84733e35 0.317479
\(753\) −9.95969e35 −1.09222
\(754\) 3.57414e35 0.385505
\(755\) 0 0
\(756\) −5.19514e35 −0.542094
\(757\) 1.20614e35 0.123794 0.0618968 0.998083i \(-0.480285\pi\)
0.0618968 + 0.998083i \(0.480285\pi\)
\(758\) 1.28758e36 1.29990
\(759\) −2.47179e35 −0.245465
\(760\) 0 0
\(761\) −1.39110e36 −1.33675 −0.668374 0.743825i \(-0.733010\pi\)
−0.668374 + 0.743825i \(0.733010\pi\)
\(762\) 1.56858e36 1.48275
\(763\) 2.47849e36 2.30479
\(764\) 9.81260e34 0.0897671
\(765\) 0 0
\(766\) 1.00750e36 0.892040
\(767\) −3.20983e35 −0.279602
\(768\) −8.55810e34 −0.0733436
\(769\) 5.62814e35 0.474554 0.237277 0.971442i \(-0.423745\pi\)
0.237277 + 0.971442i \(0.423745\pi\)
\(770\) 0 0
\(771\) 2.83726e36 2.31589
\(772\) 2.71203e35 0.217810
\(773\) −1.08729e36 −0.859211 −0.429605 0.903017i \(-0.641347\pi\)
−0.429605 + 0.903017i \(0.641347\pi\)
\(774\) −4.35299e35 −0.338474
\(775\) 0 0
\(776\) 3.66141e35 0.275662
\(777\) 9.09988e35 0.674175
\(778\) 9.54037e35 0.695537
\(779\) −3.39713e35 −0.243721
\(780\) 0 0
\(781\) −6.82108e35 −0.473931
\(782\) −1.25350e36 −0.857114
\(783\) −6.35042e35 −0.427347
\(784\) 4.52934e35 0.299974
\(785\) 0 0
\(786\) 1.37800e36 0.884028
\(787\) 5.64847e35 0.356654 0.178327 0.983971i \(-0.442932\pi\)
0.178327 + 0.983971i \(0.442932\pi\)
\(788\) −8.95364e35 −0.556445
\(789\) 1.38791e36 0.848984
\(790\) 0 0
\(791\) 2.39509e36 1.41943
\(792\) −7.12213e34 −0.0415475
\(793\) −2.46270e36 −1.41416
\(794\) 9.74916e35 0.551075
\(795\) 0 0
\(796\) 1.33427e35 0.0730855
\(797\) 2.16083e34 0.0116517 0.00582587 0.999983i \(-0.498146\pi\)
0.00582587 + 0.999983i \(0.498146\pi\)
\(798\) 2.13349e36 1.13254
\(799\) 4.38828e36 2.29329
\(800\) 0 0
\(801\) 1.02610e36 0.519734
\(802\) 2.30119e36 1.14756
\(803\) 5.08715e35 0.249764
\(804\) 1.84407e36 0.891407
\(805\) 0 0
\(806\) −2.02217e36 −0.947608
\(807\) 3.10952e36 1.43474
\(808\) −6.85487e35 −0.311428
\(809\) −3.34498e36 −1.49637 −0.748183 0.663493i \(-0.769073\pi\)
−0.748183 + 0.663493i \(0.769073\pi\)
\(810\) 0 0
\(811\) −3.06093e36 −1.32768 −0.663838 0.747876i \(-0.731074\pi\)
−0.663838 + 0.747876i \(0.731074\pi\)
\(812\) 1.01508e36 0.433559
\(813\) −6.99395e35 −0.294165
\(814\) −2.06072e35 −0.0853520
\(815\) 0 0
\(816\) −1.31897e36 −0.529793
\(817\) −2.95292e36 −1.16809
\(818\) −1.59803e36 −0.622542
\(819\) 1.35947e36 0.521579
\(820\) 0 0
\(821\) 4.35079e36 1.61912 0.809559 0.587039i \(-0.199706\pi\)
0.809559 + 0.587039i \(0.199706\pi\)
\(822\) −1.80905e36 −0.663060
\(823\) 2.15419e35 0.0777655 0.0388827 0.999244i \(-0.487620\pi\)
0.0388827 + 0.999244i \(0.487620\pi\)
\(824\) 4.44550e35 0.158063
\(825\) 0 0
\(826\) −9.11610e35 −0.314455
\(827\) 1.29774e36 0.440929 0.220465 0.975395i \(-0.429243\pi\)
0.220465 + 0.975395i \(0.429243\pi\)
\(828\) −3.78155e35 −0.126558
\(829\) 2.45539e36 0.809447 0.404723 0.914439i \(-0.367368\pi\)
0.404723 + 0.914439i \(0.367368\pi\)
\(830\) 0 0
\(831\) −3.62980e36 −1.16110
\(832\) −3.69930e35 −0.116568
\(833\) 6.98057e36 2.16685
\(834\) −1.17698e36 −0.359910
\(835\) 0 0
\(836\) −4.83141e35 −0.143382
\(837\) 3.59292e36 1.05046
\(838\) 2.36500e36 0.681211
\(839\) −4.28477e36 −1.21591 −0.607957 0.793970i \(-0.708011\pi\)
−0.607957 + 0.793970i \(0.708011\pi\)
\(840\) 0 0
\(841\) −2.38956e36 −0.658214
\(842\) −3.86543e36 −1.04905
\(843\) −4.10047e36 −1.09645
\(844\) −1.03669e36 −0.273128
\(845\) 0 0
\(846\) 1.32386e36 0.338619
\(847\) 5.31346e36 1.33917
\(848\) −9.66810e35 −0.240100
\(849\) −2.44248e36 −0.597702
\(850\) 0 0
\(851\) −1.09415e36 −0.259991
\(852\) −3.81086e36 −0.892336
\(853\) 5.04634e36 1.16443 0.582215 0.813035i \(-0.302186\pi\)
0.582215 + 0.813035i \(0.302186\pi\)
\(854\) −6.99422e36 −1.59043
\(855\) 0 0
\(856\) 1.63228e36 0.360473
\(857\) 1.55987e35 0.0339491 0.0169745 0.999856i \(-0.494597\pi\)
0.0169745 + 0.999856i \(0.494597\pi\)
\(858\) −1.12425e36 −0.241142
\(859\) −6.31632e36 −1.33521 −0.667605 0.744516i \(-0.732680\pi\)
−0.667605 + 0.744516i \(0.732680\pi\)
\(860\) 0 0
\(861\) −2.24507e36 −0.460988
\(862\) 1.60048e36 0.323898
\(863\) −3.19641e36 −0.637569 −0.318785 0.947827i \(-0.603275\pi\)
−0.318785 + 0.947827i \(0.603275\pi\)
\(864\) 6.57280e35 0.129220
\(865\) 0 0
\(866\) −2.93967e36 −0.561467
\(867\) −1.40944e37 −2.65344
\(868\) −5.74306e36 −1.06573
\(869\) −1.94925e35 −0.0356550
\(870\) 0 0
\(871\) 7.97111e36 1.41674
\(872\) −3.13575e36 −0.549396
\(873\) 1.70236e36 0.294017
\(874\) −2.56527e36 −0.436758
\(875\) 0 0
\(876\) 2.84214e36 0.470266
\(877\) −1.75514e36 −0.286297 −0.143148 0.989701i \(-0.545723\pi\)
−0.143148 + 0.989701i \(0.545723\pi\)
\(878\) −1.93768e36 −0.311602
\(879\) 8.86007e36 1.40468
\(880\) 0 0
\(881\) −1.15533e36 −0.178035 −0.0890177 0.996030i \(-0.528373\pi\)
−0.0890177 + 0.996030i \(0.528373\pi\)
\(882\) 2.10590e36 0.319948
\(883\) 2.62306e35 0.0392916 0.0196458 0.999807i \(-0.493746\pi\)
0.0196458 + 0.999807i \(0.493746\pi\)
\(884\) −5.70132e36 −0.842020
\(885\) 0 0
\(886\) −4.77170e36 −0.685096
\(887\) −1.26656e37 −1.79299 −0.896497 0.443049i \(-0.853897\pi\)
−0.896497 + 0.443049i \(0.853897\pi\)
\(888\) −1.15130e36 −0.160704
\(889\) 1.92564e37 2.65035
\(890\) 0 0
\(891\) 2.87568e36 0.384830
\(892\) 6.61288e36 0.872630
\(893\) 8.98059e36 1.16859
\(894\) 6.06956e35 0.0778822
\(895\) 0 0
\(896\) −1.05062e36 −0.131098
\(897\) −5.96931e36 −0.734546
\(898\) −5.38455e36 −0.653424
\(899\) −7.02019e36 −0.840142
\(900\) 0 0
\(901\) −1.49004e37 −1.73435
\(902\) 5.08408e35 0.0583620
\(903\) −1.95150e37 −2.20939
\(904\) −3.03022e36 −0.338352
\(905\) 0 0
\(906\) −1.01061e37 −1.09769
\(907\) −1.10945e37 −1.18855 −0.594275 0.804262i \(-0.702561\pi\)
−0.594275 + 0.804262i \(0.702561\pi\)
\(908\) −2.40864e36 −0.254506
\(909\) −3.18715e36 −0.332165
\(910\) 0 0
\(911\) −6.00569e36 −0.608952 −0.304476 0.952520i \(-0.598481\pi\)
−0.304476 + 0.952520i \(0.598481\pi\)
\(912\) −2.69926e36 −0.269966
\(913\) −2.40145e36 −0.236913
\(914\) 1.91089e36 0.185956
\(915\) 0 0
\(916\) 6.29363e36 0.595947
\(917\) 1.69167e37 1.58016
\(918\) 1.01299e37 0.933411
\(919\) 8.07538e36 0.734039 0.367020 0.930213i \(-0.380378\pi\)
0.367020 + 0.930213i \(0.380378\pi\)
\(920\) 0 0
\(921\) 1.26298e37 1.11725
\(922\) 6.30837e36 0.550529
\(923\) −1.64727e37 −1.41822
\(924\) −3.19295e36 −0.271201
\(925\) 0 0
\(926\) 8.39594e36 0.694116
\(927\) 2.06692e36 0.168588
\(928\) −1.28426e36 −0.103348
\(929\) 1.77412e37 1.40860 0.704299 0.709903i \(-0.251261\pi\)
0.704299 + 0.709903i \(0.251261\pi\)
\(930\) 0 0
\(931\) 1.42857e37 1.10416
\(932\) −7.57361e36 −0.577569
\(933\) 1.05362e37 0.792801
\(934\) −7.74969e36 −0.575372
\(935\) 0 0
\(936\) −1.71997e36 −0.124329
\(937\) −1.54851e37 −1.10451 −0.552254 0.833676i \(-0.686232\pi\)
−0.552254 + 0.833676i \(0.686232\pi\)
\(938\) 2.26384e37 1.59335
\(939\) 2.31159e37 1.60543
\(940\) 0 0
\(941\) 6.87286e36 0.464801 0.232400 0.972620i \(-0.425342\pi\)
0.232400 + 0.972620i \(0.425342\pi\)
\(942\) 2.46324e36 0.164388
\(943\) 2.69943e36 0.177777
\(944\) 1.15335e36 0.0749570
\(945\) 0 0
\(946\) 4.41929e36 0.279713
\(947\) 6.52055e36 0.407295 0.203648 0.979044i \(-0.434720\pi\)
0.203648 + 0.979044i \(0.434720\pi\)
\(948\) −1.08903e36 −0.0671326
\(949\) 1.22853e37 0.747410
\(950\) 0 0
\(951\) 3.10917e37 1.84242
\(952\) −1.61921e37 −0.946979
\(953\) −8.60992e36 −0.496979 −0.248489 0.968635i \(-0.579934\pi\)
−0.248489 + 0.968635i \(0.579934\pi\)
\(954\) −4.49515e36 −0.256088
\(955\) 0 0
\(956\) −2.98362e36 −0.165585
\(957\) −3.90299e36 −0.213795
\(958\) 3.81438e36 0.206231
\(959\) −2.22085e37 −1.18519
\(960\) 0 0
\(961\) 2.04858e37 1.06515
\(962\) −4.97658e36 −0.255413
\(963\) 7.58922e36 0.384476
\(964\) −8.69611e35 −0.0434872
\(965\) 0 0
\(966\) −1.69532e37 −0.826109
\(967\) −1.52261e37 −0.732415 −0.366208 0.930533i \(-0.619344\pi\)
−0.366208 + 0.930533i \(0.619344\pi\)
\(968\) −6.72249e36 −0.319219
\(969\) −4.16007e37 −1.95008
\(970\) 0 0
\(971\) −1.36186e37 −0.622145 −0.311072 0.950386i \(-0.600688\pi\)
−0.311072 + 0.950386i \(0.600688\pi\)
\(972\) 7.96204e36 0.359084
\(973\) −1.44490e37 −0.643321
\(974\) 7.78562e36 0.342221
\(975\) 0 0
\(976\) 8.84897e36 0.379114
\(977\) 1.77242e37 0.749696 0.374848 0.927086i \(-0.377695\pi\)
0.374848 + 0.927086i \(0.377695\pi\)
\(978\) 2.85742e37 1.19327
\(979\) −1.04172e37 −0.429505
\(980\) 0 0
\(981\) −1.45795e37 −0.585978
\(982\) −6.27053e35 −0.0248835
\(983\) 1.62613e37 0.637140 0.318570 0.947899i \(-0.396797\pi\)
0.318570 + 0.947899i \(0.396797\pi\)
\(984\) 2.84042e36 0.109886
\(985\) 0 0
\(986\) −1.97928e37 −0.746528
\(987\) 5.93502e37 2.21033
\(988\) −1.16677e37 −0.429066
\(989\) 2.34645e37 0.852037
\(990\) 0 0
\(991\) −2.62346e37 −0.928869 −0.464434 0.885608i \(-0.653742\pi\)
−0.464434 + 0.885608i \(0.653742\pi\)
\(992\) 7.26602e36 0.254039
\(993\) −8.59723e36 −0.296820
\(994\) −4.67835e37 −1.59501
\(995\) 0 0
\(996\) −1.34166e37 −0.446069
\(997\) 4.75249e37 1.56038 0.780192 0.625540i \(-0.215121\pi\)
0.780192 + 0.625540i \(0.215121\pi\)
\(998\) −1.63663e37 −0.530662
\(999\) 8.84223e36 0.283135
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 50.26.a.l.1.2 6
5.2 odd 4 10.26.b.a.9.11 yes 12
5.3 odd 4 10.26.b.a.9.2 12
5.4 even 2 50.26.a.k.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.26.b.a.9.2 12 5.3 odd 4
10.26.b.a.9.11 yes 12 5.2 odd 4
50.26.a.k.1.5 6 5.4 even 2
50.26.a.l.1.2 6 1.1 even 1 trivial