Properties

Label 50.26.a.c.1.2
Level $50$
Weight $26$
Character 50.1
Self dual yes
Analytic conductor $197.998$
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] = [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: \(2\)
Coefficient field: \(\Q(\sqrt{106705}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 26676 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7}\cdot 3\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 2)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-162.829\) 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)
 
\(f(q)\) \(=\) \(q-4096.00 q^{2} +1.37803e6 q^{3} +1.67772e7 q^{4} -5.64442e9 q^{6} -3.00388e10 q^{7} -6.87195e10 q^{8} +1.05168e12 q^{9} +5.73599e12 q^{11} +2.31195e13 q^{12} +1.07343e13 q^{13} +1.23039e14 q^{14} +2.81475e14 q^{16} -2.97473e15 q^{17} -4.30769e15 q^{18} -5.42738e15 q^{19} -4.13944e16 q^{21} -2.34946e16 q^{22} +1.04540e17 q^{23} -9.46976e16 q^{24} -4.39677e16 q^{26} +2.81659e17 q^{27} -5.03967e17 q^{28} +3.09182e18 q^{29} -4.26809e18 q^{31} -1.15292e18 q^{32} +7.90437e18 q^{33} +1.21845e19 q^{34} +1.76443e19 q^{36} +4.51028e19 q^{37} +2.22305e19 q^{38} +1.47922e19 q^{39} -7.56724e19 q^{41} +1.69551e20 q^{42} +1.39036e20 q^{43} +9.62339e19 q^{44} -4.28196e20 q^{46} -4.67328e20 q^{47} +3.87881e20 q^{48} -4.38741e20 q^{49} -4.09927e21 q^{51} +1.80092e20 q^{52} +1.24315e21 q^{53} -1.15368e21 q^{54} +2.06425e21 q^{56} -7.47909e21 q^{57} -1.26641e22 q^{58} -1.32468e22 q^{59} -2.36984e20 q^{61} +1.74821e22 q^{62} -3.15912e22 q^{63} +4.72237e21 q^{64} -3.23763e22 q^{66} +5.36922e22 q^{67} -4.99076e22 q^{68} +1.44059e23 q^{69} -2.27032e23 q^{71} -7.22710e22 q^{72} -2.78528e23 q^{73} -1.84741e23 q^{74} -9.10563e22 q^{76} -1.72302e23 q^{77} -6.05889e22 q^{78} +6.36958e23 q^{79} -5.02942e23 q^{81} +3.09954e23 q^{82} -1.19662e24 q^{83} -6.94482e23 q^{84} -5.69491e23 q^{86} +4.26063e24 q^{87} -3.94174e23 q^{88} -3.22134e24 q^{89} -3.22445e23 q^{91} +1.75389e24 q^{92} -5.88156e24 q^{93} +1.91418e24 q^{94} -1.58876e24 q^{96} -3.58465e24 q^{97} +1.79708e24 q^{98} +6.03243e24 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8192 q^{2} - 379848 q^{3} + 33554432 q^{4} + 1555857408 q^{6} + 376536944 q^{7} - 137438953472 q^{8} + 3294531432666 q^{9} + 8323034610264 q^{11} - 6372791943168 q^{12} + 106467053152292 q^{13} - 1542295322624 q^{14}+ \cdots + 11\!\cdots\!12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4096.00 −0.707107
\(3\) 1.37803e6 1.49707 0.748537 0.663093i \(-0.230757\pi\)
0.748537 + 0.663093i \(0.230757\pi\)
\(4\) 1.67772e7 0.500000
\(5\) 0 0
\(6\) −5.64442e9 −1.05859
\(7\) −3.00388e10 −0.820270 −0.410135 0.912025i \(-0.634518\pi\)
−0.410135 + 0.912025i \(0.634518\pi\)
\(8\) −6.87195e10 −0.353553
\(9\) 1.05168e12 1.24123
\(10\) 0 0
\(11\) 5.73599e12 0.551061 0.275531 0.961292i \(-0.411146\pi\)
0.275531 + 0.961292i \(0.411146\pi\)
\(12\) 2.31195e13 0.748537
\(13\) 1.07343e13 0.127786 0.0638928 0.997957i \(-0.479648\pi\)
0.0638928 + 0.997957i \(0.479648\pi\)
\(14\) 1.23039e14 0.580018
\(15\) 0 0
\(16\) 2.81475e14 0.250000
\(17\) −2.97473e15 −1.23833 −0.619164 0.785262i \(-0.712528\pi\)
−0.619164 + 0.785262i \(0.712528\pi\)
\(18\) −4.30769e15 −0.877683
\(19\) −5.42738e15 −0.562561 −0.281281 0.959626i \(-0.590759\pi\)
−0.281281 + 0.959626i \(0.590759\pi\)
\(20\) 0 0
\(21\) −4.13944e16 −1.22800
\(22\) −2.34946e16 −0.389659
\(23\) 1.04540e17 0.994683 0.497341 0.867555i \(-0.334310\pi\)
0.497341 + 0.867555i \(0.334310\pi\)
\(24\) −9.46976e16 −0.529296
\(25\) 0 0
\(26\) −4.39677e16 −0.0903581
\(27\) 2.81659e17 0.361141
\(28\) −5.03967e17 −0.410135
\(29\) 3.09182e18 1.62270 0.811352 0.584558i \(-0.198732\pi\)
0.811352 + 0.584558i \(0.198732\pi\)
\(30\) 0 0
\(31\) −4.26809e18 −0.973222 −0.486611 0.873619i \(-0.661767\pi\)
−0.486611 + 0.873619i \(0.661767\pi\)
\(32\) −1.15292e18 −0.176777
\(33\) 7.90437e18 0.824980
\(34\) 1.21845e19 0.875630
\(35\) 0 0
\(36\) 1.76443e19 0.620616
\(37\) 4.51028e19 1.12637 0.563186 0.826330i \(-0.309576\pi\)
0.563186 + 0.826330i \(0.309576\pi\)
\(38\) 2.22305e19 0.397791
\(39\) 1.47922e19 0.191305
\(40\) 0 0
\(41\) −7.56724e19 −0.523769 −0.261884 0.965099i \(-0.584344\pi\)
−0.261884 + 0.965099i \(0.584344\pi\)
\(42\) 1.69551e20 0.868330
\(43\) 1.39036e20 0.530605 0.265302 0.964165i \(-0.414528\pi\)
0.265302 + 0.964165i \(0.414528\pi\)
\(44\) 9.62339e19 0.275531
\(45\) 0 0
\(46\) −4.28196e20 −0.703347
\(47\) −4.67328e20 −0.586676 −0.293338 0.956009i \(-0.594766\pi\)
−0.293338 + 0.956009i \(0.594766\pi\)
\(48\) 3.87881e20 0.374269
\(49\) −4.38741e20 −0.327158
\(50\) 0 0
\(51\) −4.09927e21 −1.85387
\(52\) 1.80092e20 0.0638928
\(53\) 1.24315e21 0.347595 0.173797 0.984781i \(-0.444396\pi\)
0.173797 + 0.984781i \(0.444396\pi\)
\(54\) −1.15368e21 −0.255365
\(55\) 0 0
\(56\) 2.06425e21 0.290009
\(57\) −7.47909e21 −0.842196
\(58\) −1.26641e22 −1.14743
\(59\) −1.32468e22 −0.969303 −0.484652 0.874707i \(-0.661054\pi\)
−0.484652 + 0.874707i \(0.661054\pi\)
\(60\) 0 0
\(61\) −2.36984e20 −0.0114313 −0.00571565 0.999984i \(-0.501819\pi\)
−0.00571565 + 0.999984i \(0.501819\pi\)
\(62\) 1.74821e22 0.688172
\(63\) −3.15912e22 −1.01814
\(64\) 4.72237e21 0.125000
\(65\) 0 0
\(66\) −3.23763e22 −0.583349
\(67\) 5.36922e22 0.801634 0.400817 0.916158i \(-0.368726\pi\)
0.400817 + 0.916158i \(0.368726\pi\)
\(68\) −4.99076e22 −0.619164
\(69\) 1.44059e23 1.48911
\(70\) 0 0
\(71\) −2.27032e23 −1.64194 −0.820971 0.570970i \(-0.806567\pi\)
−0.820971 + 0.570970i \(0.806567\pi\)
\(72\) −7.22710e22 −0.438842
\(73\) −2.78528e23 −1.42342 −0.711710 0.702473i \(-0.752079\pi\)
−0.711710 + 0.702473i \(0.752079\pi\)
\(74\) −1.84741e23 −0.796465
\(75\) 0 0
\(76\) −9.10563e22 −0.281281
\(77\) −1.72302e23 −0.452019
\(78\) −6.05889e22 −0.135273
\(79\) 6.36958e23 1.21275 0.606376 0.795178i \(-0.292622\pi\)
0.606376 + 0.795178i \(0.292622\pi\)
\(80\) 0 0
\(81\) −5.02942e23 −0.700576
\(82\) 3.09954e23 0.370360
\(83\) −1.19662e24 −1.22880 −0.614398 0.788996i \(-0.710601\pi\)
−0.614398 + 0.788996i \(0.710601\pi\)
\(84\) −6.94482e23 −0.614002
\(85\) 0 0
\(86\) −5.69491e23 −0.375194
\(87\) 4.26063e24 2.42931
\(88\) −3.94174e23 −0.194830
\(89\) −3.22134e24 −1.38249 −0.691244 0.722621i \(-0.742937\pi\)
−0.691244 + 0.722621i \(0.742937\pi\)
\(90\) 0 0
\(91\) −3.22445e23 −0.104819
\(92\) 1.75389e24 0.497341
\(93\) −5.88156e24 −1.45699
\(94\) 1.91418e24 0.414843
\(95\) 0 0
\(96\) −1.58876e24 −0.264648
\(97\) −3.58465e24 −0.524566 −0.262283 0.964991i \(-0.584475\pi\)
−0.262283 + 0.964991i \(0.584475\pi\)
\(98\) 1.79708e24 0.231335
\(99\) 6.03243e24 0.683994
\(100\) 0 0
\(101\) −3.40630e24 −0.300792 −0.150396 0.988626i \(-0.548055\pi\)
−0.150396 + 0.988626i \(0.548055\pi\)
\(102\) 1.67906e25 1.31088
\(103\) −8.27075e24 −0.571583 −0.285792 0.958292i \(-0.592257\pi\)
−0.285792 + 0.958292i \(0.592257\pi\)
\(104\) −7.37656e23 −0.0451791
\(105\) 0 0
\(106\) −5.09192e24 −0.245787
\(107\) −1.57986e25 −0.678142 −0.339071 0.940761i \(-0.610113\pi\)
−0.339071 + 0.940761i \(0.610113\pi\)
\(108\) 4.72546e24 0.180571
\(109\) 3.00869e25 1.02458 0.512290 0.858813i \(-0.328797\pi\)
0.512290 + 0.858813i \(0.328797\pi\)
\(110\) 0 0
\(111\) 6.21530e25 1.68626
\(112\) −8.45516e24 −0.205067
\(113\) 8.06745e25 1.75088 0.875439 0.483329i \(-0.160572\pi\)
0.875439 + 0.483329i \(0.160572\pi\)
\(114\) 3.06344e25 0.595523
\(115\) 0 0
\(116\) 5.18722e25 0.811352
\(117\) 1.12891e25 0.158612
\(118\) 5.42587e25 0.685401
\(119\) 8.93571e25 1.01576
\(120\) 0 0
\(121\) −7.54455e25 −0.696332
\(122\) 9.70685e23 0.00808315
\(123\) −1.04279e26 −0.784121
\(124\) −7.16066e25 −0.486611
\(125\) 0 0
\(126\) 1.29398e26 0.719937
\(127\) 1.32973e26 0.670221 0.335111 0.942179i \(-0.391226\pi\)
0.335111 + 0.942179i \(0.391226\pi\)
\(128\) −1.93428e25 −0.0883883
\(129\) 1.91596e26 0.794355
\(130\) 0 0
\(131\) 6.73908e25 0.230520 0.115260 0.993335i \(-0.463230\pi\)
0.115260 + 0.993335i \(0.463230\pi\)
\(132\) 1.32613e26 0.412490
\(133\) 1.63032e26 0.461452
\(134\) −2.19923e26 −0.566841
\(135\) 0 0
\(136\) 2.04422e26 0.437815
\(137\) 3.22963e26 0.631170 0.315585 0.948897i \(-0.397799\pi\)
0.315585 + 0.948897i \(0.397799\pi\)
\(138\) −5.90067e26 −1.05296
\(139\) −9.21581e25 −0.150262 −0.0751309 0.997174i \(-0.523937\pi\)
−0.0751309 + 0.997174i \(0.523937\pi\)
\(140\) 0 0
\(141\) −6.43992e26 −0.878298
\(142\) 9.29923e26 1.16103
\(143\) 6.15719e25 0.0704177
\(144\) 2.96022e26 0.310308
\(145\) 0 0
\(146\) 1.14085e27 1.00651
\(147\) −6.04599e26 −0.489779
\(148\) 7.56699e26 0.563186
\(149\) −7.12667e26 −0.487594 −0.243797 0.969826i \(-0.578393\pi\)
−0.243797 + 0.969826i \(0.578393\pi\)
\(150\) 0 0
\(151\) −3.06161e27 −1.77312 −0.886559 0.462616i \(-0.846911\pi\)
−0.886559 + 0.462616i \(0.846911\pi\)
\(152\) 3.72966e26 0.198895
\(153\) −3.12846e27 −1.53705
\(154\) 7.05749e26 0.319626
\(155\) 0 0
\(156\) 2.48172e26 0.0956523
\(157\) −2.38960e27 −0.850315 −0.425157 0.905119i \(-0.639781\pi\)
−0.425157 + 0.905119i \(0.639781\pi\)
\(158\) −2.60898e27 −0.857546
\(159\) 1.71309e27 0.520375
\(160\) 0 0
\(161\) −3.14025e27 −0.815908
\(162\) 2.06005e27 0.495382
\(163\) −6.63092e27 −1.47648 −0.738242 0.674536i \(-0.764344\pi\)
−0.738242 + 0.674536i \(0.764344\pi\)
\(164\) −1.26957e27 −0.261884
\(165\) 0 0
\(166\) 4.90135e27 0.868890
\(167\) −4.06997e27 −0.669323 −0.334662 0.942338i \(-0.608622\pi\)
−0.334662 + 0.942338i \(0.608622\pi\)
\(168\) 2.84460e27 0.434165
\(169\) −6.94118e27 −0.983671
\(170\) 0 0
\(171\) −5.70787e27 −0.698269
\(172\) 2.33263e27 0.265302
\(173\) 8.68916e27 0.919182 0.459591 0.888131i \(-0.347996\pi\)
0.459591 + 0.888131i \(0.347996\pi\)
\(174\) −1.74515e28 −1.71778
\(175\) 0 0
\(176\) 1.61454e27 0.137765
\(177\) −1.82545e28 −1.45112
\(178\) 1.31946e28 0.977567
\(179\) −4.31406e27 −0.298005 −0.149002 0.988837i \(-0.547606\pi\)
−0.149002 + 0.988837i \(0.547606\pi\)
\(180\) 0 0
\(181\) 2.22969e28 1.34049 0.670243 0.742142i \(-0.266190\pi\)
0.670243 + 0.742142i \(0.266190\pi\)
\(182\) 1.32074e27 0.0741180
\(183\) −3.26571e26 −0.0171135
\(184\) −7.18394e27 −0.351674
\(185\) 0 0
\(186\) 2.40909e28 1.03024
\(187\) −1.70630e28 −0.682394
\(188\) −7.84046e27 −0.293338
\(189\) −8.46070e27 −0.296233
\(190\) 0 0
\(191\) −1.91337e28 −0.587329 −0.293665 0.955908i \(-0.594875\pi\)
−0.293665 + 0.955908i \(0.594875\pi\)
\(192\) 6.50757e27 0.187134
\(193\) 5.96748e28 1.60814 0.804071 0.594533i \(-0.202663\pi\)
0.804071 + 0.594533i \(0.202663\pi\)
\(194\) 1.46827e28 0.370924
\(195\) 0 0
\(196\) −7.36085e27 −0.163579
\(197\) −6.24204e28 −1.30166 −0.650832 0.759222i \(-0.725580\pi\)
−0.650832 + 0.759222i \(0.725580\pi\)
\(198\) −2.47088e28 −0.483657
\(199\) −3.24229e28 −0.595921 −0.297960 0.954578i \(-0.596306\pi\)
−0.297960 + 0.954578i \(0.596306\pi\)
\(200\) 0 0
\(201\) 7.39895e28 1.20011
\(202\) 1.39522e28 0.212692
\(203\) −9.28745e28 −1.33106
\(204\) −6.87743e28 −0.926934
\(205\) 0 0
\(206\) 3.38770e28 0.404171
\(207\) 1.09943e29 1.23463
\(208\) 3.02144e27 0.0319464
\(209\) −3.11314e28 −0.310006
\(210\) 0 0
\(211\) 3.82197e28 0.337875 0.168938 0.985627i \(-0.445966\pi\)
0.168938 + 0.985627i \(0.445966\pi\)
\(212\) 2.08565e28 0.173797
\(213\) −3.12857e29 −2.45811
\(214\) 6.47109e28 0.479519
\(215\) 0 0
\(216\) −1.93555e28 −0.127683
\(217\) 1.28208e29 0.798305
\(218\) −1.23236e29 −0.724487
\(219\) −3.83820e29 −2.13097
\(220\) 0 0
\(221\) −3.19316e28 −0.158241
\(222\) −2.54579e29 −1.19237
\(223\) −3.11100e29 −1.37749 −0.688746 0.725003i \(-0.741839\pi\)
−0.688746 + 0.725003i \(0.741839\pi\)
\(224\) 3.46323e28 0.145005
\(225\) 0 0
\(226\) −3.30443e29 −1.23806
\(227\) 1.05572e29 0.374304 0.187152 0.982331i \(-0.440074\pi\)
0.187152 + 0.982331i \(0.440074\pi\)
\(228\) −1.25478e29 −0.421098
\(229\) −3.70322e29 −1.17662 −0.588310 0.808635i \(-0.700207\pi\)
−0.588310 + 0.808635i \(0.700207\pi\)
\(230\) 0 0
\(231\) −2.37438e29 −0.676706
\(232\) −2.12468e29 −0.573713
\(233\) 6.84781e28 0.175228 0.0876138 0.996155i \(-0.472076\pi\)
0.0876138 + 0.996155i \(0.472076\pi\)
\(234\) −4.62400e28 −0.112155
\(235\) 0 0
\(236\) −2.22244e29 −0.484652
\(237\) 8.77748e29 1.81558
\(238\) −3.66007e29 −0.718253
\(239\) 3.87773e29 0.722111 0.361055 0.932544i \(-0.382417\pi\)
0.361055 + 0.932544i \(0.382417\pi\)
\(240\) 0 0
\(241\) 1.03723e30 1.74046 0.870230 0.492646i \(-0.163970\pi\)
0.870230 + 0.492646i \(0.163970\pi\)
\(242\) 3.09025e29 0.492381
\(243\) −9.31717e29 −1.40996
\(244\) −3.97593e27 −0.00571565
\(245\) 0 0
\(246\) 4.27127e29 0.554457
\(247\) −5.82591e28 −0.0718873
\(248\) 2.93301e29 0.344086
\(249\) −1.64898e30 −1.83960
\(250\) 0 0
\(251\) 1.47990e30 1.49386 0.746930 0.664903i \(-0.231527\pi\)
0.746930 + 0.664903i \(0.231527\pi\)
\(252\) −5.30013e29 −0.509072
\(253\) 5.99641e29 0.548131
\(254\) −5.44659e29 −0.473918
\(255\) 0 0
\(256\) 7.92282e28 0.0625000
\(257\) −6.75384e29 −0.507442 −0.253721 0.967277i \(-0.581655\pi\)
−0.253721 + 0.967277i \(0.581655\pi\)
\(258\) −7.84776e29 −0.561694
\(259\) −1.35483e30 −0.923929
\(260\) 0 0
\(261\) 3.25161e30 2.01415
\(262\) −2.76033e29 −0.163003
\(263\) −1.42102e30 −0.800119 −0.400059 0.916489i \(-0.631011\pi\)
−0.400059 + 0.916489i \(0.631011\pi\)
\(264\) −5.43184e29 −0.291674
\(265\) 0 0
\(266\) −6.67778e29 −0.326296
\(267\) −4.43911e30 −2.06969
\(268\) 9.00805e29 0.400817
\(269\) −1.90129e30 −0.807504 −0.403752 0.914868i \(-0.632294\pi\)
−0.403752 + 0.914868i \(0.632294\pi\)
\(270\) 0 0
\(271\) −1.45800e30 −0.564471 −0.282236 0.959345i \(-0.591076\pi\)
−0.282236 + 0.959345i \(0.591076\pi\)
\(272\) −8.37311e29 −0.309582
\(273\) −4.44340e29 −0.156921
\(274\) −1.32286e30 −0.446304
\(275\) 0 0
\(276\) 2.41692e30 0.744557
\(277\) 5.31088e29 0.156376 0.0781879 0.996939i \(-0.475087\pi\)
0.0781879 + 0.996939i \(0.475087\pi\)
\(278\) 3.77479e29 0.106251
\(279\) −4.48867e30 −1.20799
\(280\) 0 0
\(281\) −3.64974e30 −0.898323 −0.449161 0.893451i \(-0.648277\pi\)
−0.449161 + 0.893451i \(0.648277\pi\)
\(282\) 2.63779e30 0.621050
\(283\) 1.43965e30 0.324285 0.162143 0.986767i \(-0.448160\pi\)
0.162143 + 0.986767i \(0.448160\pi\)
\(284\) −3.80897e30 −0.820971
\(285\) 0 0
\(286\) −2.52199e29 −0.0497929
\(287\) 2.27311e30 0.429632
\(288\) −1.21251e30 −0.219421
\(289\) 3.07837e30 0.533456
\(290\) 0 0
\(291\) −4.93976e30 −0.785315
\(292\) −4.67292e30 −0.711710
\(293\) −1.24504e31 −1.81692 −0.908462 0.417967i \(-0.862743\pi\)
−0.908462 + 0.417967i \(0.862743\pi\)
\(294\) 2.47644e30 0.346326
\(295\) 0 0
\(296\) −3.09944e30 −0.398233
\(297\) 1.61559e30 0.199011
\(298\) 2.91909e30 0.344781
\(299\) 1.12217e30 0.127106
\(300\) 0 0
\(301\) −4.17647e30 −0.435239
\(302\) 1.25403e31 1.25378
\(303\) −4.69399e30 −0.450307
\(304\) −1.52767e30 −0.140640
\(305\) 0 0
\(306\) 1.28142e31 1.08686
\(307\) −2.27193e31 −1.84997 −0.924986 0.380001i \(-0.875924\pi\)
−0.924986 + 0.380001i \(0.875924\pi\)
\(308\) −2.89075e30 −0.226009
\(309\) −1.13974e31 −0.855703
\(310\) 0 0
\(311\) −1.84336e31 −1.27675 −0.638374 0.769726i \(-0.720393\pi\)
−0.638374 + 0.769726i \(0.720393\pi\)
\(312\) −1.01651e30 −0.0676364
\(313\) −4.98693e30 −0.318808 −0.159404 0.987213i \(-0.550957\pi\)
−0.159404 + 0.987213i \(0.550957\pi\)
\(314\) 9.78780e30 0.601263
\(315\) 0 0
\(316\) 1.06864e31 0.606376
\(317\) 8.79478e30 0.479716 0.239858 0.970808i \(-0.422899\pi\)
0.239858 + 0.970808i \(0.422899\pi\)
\(318\) −7.01683e30 −0.367961
\(319\) 1.77347e31 0.894209
\(320\) 0 0
\(321\) −2.17709e31 −1.01523
\(322\) 1.28625e31 0.576934
\(323\) 1.61450e31 0.696635
\(324\) −8.43797e30 −0.350288
\(325\) 0 0
\(326\) 2.71602e31 1.04403
\(327\) 4.14606e31 1.53387
\(328\) 5.20017e30 0.185180
\(329\) 1.40380e31 0.481233
\(330\) 0 0
\(331\) 5.97122e31 1.89764 0.948818 0.315822i \(-0.102280\pi\)
0.948818 + 0.315822i \(0.102280\pi\)
\(332\) −2.00759e31 −0.614398
\(333\) 4.74337e31 1.39809
\(334\) 1.66706e31 0.473283
\(335\) 0 0
\(336\) −1.16515e31 −0.307001
\(337\) −1.08428e31 −0.275275 −0.137638 0.990483i \(-0.543951\pi\)
−0.137638 + 0.990483i \(0.543951\pi\)
\(338\) 2.84311e31 0.695560
\(339\) 1.11172e32 2.62119
\(340\) 0 0
\(341\) −2.44817e31 −0.536305
\(342\) 2.33794e31 0.493751
\(343\) 5.34633e31 1.08863
\(344\) −9.55447e30 −0.187597
\(345\) 0 0
\(346\) −3.55908e31 −0.649960
\(347\) −7.20193e30 −0.126862 −0.0634308 0.997986i \(-0.520204\pi\)
−0.0634308 + 0.997986i \(0.520204\pi\)
\(348\) 7.14814e31 1.21465
\(349\) 5.27779e31 0.865235 0.432618 0.901577i \(-0.357590\pi\)
0.432618 + 0.901577i \(0.357590\pi\)
\(350\) 0 0
\(351\) 3.02342e30 0.0461487
\(352\) −6.61315e30 −0.0974148
\(353\) 5.67715e31 0.807135 0.403567 0.914950i \(-0.367770\pi\)
0.403567 + 0.914950i \(0.367770\pi\)
\(354\) 7.47702e31 1.02610
\(355\) 0 0
\(356\) −5.40451e31 −0.691244
\(357\) 1.23137e32 1.52067
\(358\) 1.76704e31 0.210721
\(359\) 2.18925e31 0.252125 0.126062 0.992022i \(-0.459766\pi\)
0.126062 + 0.992022i \(0.459766\pi\)
\(360\) 0 0
\(361\) −6.36201e31 −0.683525
\(362\) −9.13280e31 −0.947866
\(363\) −1.03966e32 −1.04246
\(364\) −5.40974e30 −0.0524094
\(365\) 0 0
\(366\) 1.33763e30 0.0121011
\(367\) −1.00500e32 −0.878699 −0.439349 0.898316i \(-0.644791\pi\)
−0.439349 + 0.898316i \(0.644791\pi\)
\(368\) 2.94254e31 0.248671
\(369\) −7.95833e31 −0.650118
\(370\) 0 0
\(371\) −3.73425e31 −0.285122
\(372\) −9.86761e31 −0.728493
\(373\) −6.31052e31 −0.450510 −0.225255 0.974300i \(-0.572322\pi\)
−0.225255 + 0.974300i \(0.572322\pi\)
\(374\) 6.98901e31 0.482526
\(375\) 0 0
\(376\) 3.21145e31 0.207421
\(377\) 3.31886e31 0.207358
\(378\) 3.46550e31 0.209468
\(379\) −8.85755e31 −0.517993 −0.258996 0.965878i \(-0.583392\pi\)
−0.258996 + 0.965878i \(0.583392\pi\)
\(380\) 0 0
\(381\) 1.83241e32 1.00337
\(382\) 7.83716e31 0.415305
\(383\) −9.08315e31 −0.465856 −0.232928 0.972494i \(-0.574831\pi\)
−0.232928 + 0.972494i \(0.574831\pi\)
\(384\) −2.66550e31 −0.132324
\(385\) 0 0
\(386\) −2.44428e32 −1.13713
\(387\) 1.46221e32 0.658603
\(388\) −6.01405e31 −0.262283
\(389\) 7.13043e31 0.301124 0.150562 0.988601i \(-0.451892\pi\)
0.150562 + 0.988601i \(0.451892\pi\)
\(390\) 0 0
\(391\) −3.10978e32 −1.23174
\(392\) 3.01500e31 0.115668
\(393\) 9.28666e31 0.345106
\(394\) 2.55674e32 0.920416
\(395\) 0 0
\(396\) 1.01207e32 0.341997
\(397\) −2.99427e31 −0.0980413 −0.0490206 0.998798i \(-0.515610\pi\)
−0.0490206 + 0.998798i \(0.515610\pi\)
\(398\) 1.32804e32 0.421380
\(399\) 2.24663e32 0.690828
\(400\) 0 0
\(401\) −5.00588e32 −1.44603 −0.723013 0.690835i \(-0.757243\pi\)
−0.723013 + 0.690835i \(0.757243\pi\)
\(402\) −3.03061e32 −0.848603
\(403\) −4.58150e31 −0.124364
\(404\) −5.71482e31 −0.150396
\(405\) 0 0
\(406\) 3.80414e32 0.941198
\(407\) 2.58709e32 0.620700
\(408\) 2.81699e32 0.655441
\(409\) −2.78379e32 −0.628195 −0.314097 0.949391i \(-0.601702\pi\)
−0.314097 + 0.949391i \(0.601702\pi\)
\(410\) 0 0
\(411\) 4.45054e32 0.944908
\(412\) −1.38760e32 −0.285792
\(413\) 3.97916e32 0.795090
\(414\) −4.50326e32 −0.873016
\(415\) 0 0
\(416\) −1.23758e31 −0.0225895
\(417\) −1.26997e32 −0.224953
\(418\) 1.27514e32 0.219207
\(419\) −6.36828e32 −1.06254 −0.531272 0.847201i \(-0.678286\pi\)
−0.531272 + 0.847201i \(0.678286\pi\)
\(420\) 0 0
\(421\) −3.92275e32 −0.616686 −0.308343 0.951275i \(-0.599775\pi\)
−0.308343 + 0.951275i \(0.599775\pi\)
\(422\) −1.56548e32 −0.238914
\(423\) −4.91480e32 −0.728201
\(424\) −8.54283e31 −0.122893
\(425\) 0 0
\(426\) 1.28146e33 1.73815
\(427\) 7.11870e30 0.00937675
\(428\) −2.65056e32 −0.339071
\(429\) 8.48480e31 0.105421
\(430\) 0 0
\(431\) 2.44178e32 0.286247 0.143124 0.989705i \(-0.454285\pi\)
0.143124 + 0.989705i \(0.454285\pi\)
\(432\) 7.92800e31 0.0902853
\(433\) −9.12872e32 −1.00998 −0.504988 0.863126i \(-0.668503\pi\)
−0.504988 + 0.863126i \(0.668503\pi\)
\(434\) −5.25140e32 −0.564487
\(435\) 0 0
\(436\) 5.04774e32 0.512290
\(437\) −5.67378e32 −0.559570
\(438\) 1.57213e33 1.50682
\(439\) 6.07601e32 0.565995 0.282997 0.959121i \(-0.408671\pi\)
0.282997 + 0.959121i \(0.408671\pi\)
\(440\) 0 0
\(441\) −4.61416e32 −0.406078
\(442\) 1.30792e32 0.111893
\(443\) −9.06289e32 −0.753737 −0.376869 0.926267i \(-0.622999\pi\)
−0.376869 + 0.926267i \(0.622999\pi\)
\(444\) 1.04275e33 0.843131
\(445\) 0 0
\(446\) 1.27427e33 0.974034
\(447\) −9.82078e32 −0.729965
\(448\) −1.41854e32 −0.102534
\(449\) −1.21579e33 −0.854630 −0.427315 0.904103i \(-0.640540\pi\)
−0.427315 + 0.904103i \(0.640540\pi\)
\(450\) 0 0
\(451\) −4.34056e32 −0.288629
\(452\) 1.35349e33 0.875439
\(453\) −4.21899e33 −2.65449
\(454\) −4.32422e32 −0.264673
\(455\) 0 0
\(456\) 5.13959e32 0.297761
\(457\) 5.59530e32 0.315407 0.157703 0.987487i \(-0.449591\pi\)
0.157703 + 0.987487i \(0.449591\pi\)
\(458\) 1.51684e33 0.831997
\(459\) −8.37859e32 −0.447211
\(460\) 0 0
\(461\) −3.78592e32 −0.191386 −0.0956930 0.995411i \(-0.530507\pi\)
−0.0956930 + 0.995411i \(0.530507\pi\)
\(462\) 9.72545e32 0.478503
\(463\) −2.77039e33 −1.32672 −0.663359 0.748301i \(-0.730870\pi\)
−0.663359 + 0.748301i \(0.730870\pi\)
\(464\) 8.70270e32 0.405676
\(465\) 0 0
\(466\) −2.80486e32 −0.123905
\(467\) 2.31922e33 0.997428 0.498714 0.866767i \(-0.333806\pi\)
0.498714 + 0.866767i \(0.333806\pi\)
\(468\) 1.89399e32 0.0793058
\(469\) −1.61285e33 −0.657556
\(470\) 0 0
\(471\) −3.29294e33 −1.27298
\(472\) 9.10311e32 0.342700
\(473\) 7.97508e32 0.292396
\(474\) −3.59526e33 −1.28381
\(475\) 0 0
\(476\) 1.49916e33 0.507881
\(477\) 1.30739e33 0.431446
\(478\) −1.58832e33 −0.510609
\(479\) 2.24376e33 0.702720 0.351360 0.936240i \(-0.385719\pi\)
0.351360 + 0.936240i \(0.385719\pi\)
\(480\) 0 0
\(481\) 4.84147e32 0.143934
\(482\) −4.24851e33 −1.23069
\(483\) −4.32737e33 −1.22148
\(484\) −1.26577e33 −0.348166
\(485\) 0 0
\(486\) 3.81631e33 0.996989
\(487\) −3.71393e32 −0.0945632 −0.0472816 0.998882i \(-0.515056\pi\)
−0.0472816 + 0.998882i \(0.515056\pi\)
\(488\) 1.62854e31 0.00404157
\(489\) −9.13761e33 −2.21041
\(490\) 0 0
\(491\) −2.65508e33 −0.610323 −0.305161 0.952301i \(-0.598710\pi\)
−0.305161 + 0.952301i \(0.598710\pi\)
\(492\) −1.74951e33 −0.392060
\(493\) −9.19732e33 −2.00944
\(494\) 2.38629e32 0.0508320
\(495\) 0 0
\(496\) −1.20136e33 −0.243306
\(497\) 6.81976e33 1.34684
\(498\) 6.75421e33 1.30079
\(499\) −2.39117e33 −0.449111 −0.224556 0.974461i \(-0.572093\pi\)
−0.224556 + 0.974461i \(0.572093\pi\)
\(500\) 0 0
\(501\) −5.60855e33 −1.00203
\(502\) −6.06166e33 −1.05632
\(503\) 6.34146e33 1.07793 0.538964 0.842329i \(-0.318816\pi\)
0.538964 + 0.842329i \(0.318816\pi\)
\(504\) 2.17093e33 0.359968
\(505\) 0 0
\(506\) −2.45613e33 −0.387587
\(507\) −9.56517e33 −1.47263
\(508\) 2.23092e33 0.335111
\(509\) −3.62528e32 −0.0531337 −0.0265668 0.999647i \(-0.508457\pi\)
−0.0265668 + 0.999647i \(0.508457\pi\)
\(510\) 0 0
\(511\) 8.36664e33 1.16759
\(512\) −3.24519e32 −0.0441942
\(513\) −1.52867e33 −0.203164
\(514\) 2.76637e33 0.358816
\(515\) 0 0
\(516\) 3.21444e33 0.397177
\(517\) −2.68059e33 −0.323295
\(518\) 5.54939e33 0.653316
\(519\) 1.19739e34 1.37608
\(520\) 0 0
\(521\) 8.23573e33 0.902050 0.451025 0.892511i \(-0.351058\pi\)
0.451025 + 0.892511i \(0.351058\pi\)
\(522\) −1.33186e34 −1.42422
\(523\) 1.20443e34 1.25750 0.628752 0.777606i \(-0.283566\pi\)
0.628752 + 0.777606i \(0.283566\pi\)
\(524\) 1.13063e33 0.115260
\(525\) 0 0
\(526\) 5.82051e33 0.565770
\(527\) 1.26964e34 1.20517
\(528\) 2.22488e33 0.206245
\(529\) −1.17149e32 −0.0106057
\(530\) 0 0
\(531\) −1.39314e34 −1.20313
\(532\) 2.73522e33 0.230726
\(533\) −8.12291e32 −0.0669301
\(534\) 1.81826e34 1.46349
\(535\) 0 0
\(536\) −3.68970e33 −0.283420
\(537\) −5.94490e33 −0.446135
\(538\) 7.78769e33 0.570992
\(539\) −2.51661e33 −0.180284
\(540\) 0 0
\(541\) 1.30366e34 0.891657 0.445829 0.895118i \(-0.352909\pi\)
0.445829 + 0.895118i \(0.352909\pi\)
\(542\) 5.97197e33 0.399141
\(543\) 3.07258e34 2.00681
\(544\) 3.42963e33 0.218907
\(545\) 0 0
\(546\) 1.82002e33 0.110960
\(547\) 1.78334e34 1.06266 0.531328 0.847166i \(-0.321693\pi\)
0.531328 + 0.847166i \(0.321693\pi\)
\(548\) 5.41843e33 0.315585
\(549\) −2.49231e32 −0.0141889
\(550\) 0 0
\(551\) −1.67805e34 −0.912871
\(552\) −9.89969e33 −0.526481
\(553\) −1.91334e34 −0.994784
\(554\) −2.17534e33 −0.110574
\(555\) 0 0
\(556\) −1.54616e33 −0.0751309
\(557\) −8.73339e33 −0.414947 −0.207474 0.978241i \(-0.566524\pi\)
−0.207474 + 0.978241i \(0.566524\pi\)
\(558\) 1.83856e34 0.854181
\(559\) 1.49245e33 0.0678037
\(560\) 0 0
\(561\) −2.35134e34 −1.02160
\(562\) 1.49493e34 0.635210
\(563\) −3.82416e34 −1.58921 −0.794606 0.607126i \(-0.792322\pi\)
−0.794606 + 0.607126i \(0.792322\pi\)
\(564\) −1.08044e34 −0.439149
\(565\) 0 0
\(566\) −5.89682e33 −0.229304
\(567\) 1.51078e34 0.574661
\(568\) 1.56015e34 0.580514
\(569\) 3.49591e34 1.27250 0.636248 0.771484i \(-0.280485\pi\)
0.636248 + 0.771484i \(0.280485\pi\)
\(570\) 0 0
\(571\) −3.60524e34 −1.25598 −0.627989 0.778222i \(-0.716122\pi\)
−0.627989 + 0.778222i \(0.716122\pi\)
\(572\) 1.03301e33 0.0352089
\(573\) −2.63668e34 −0.879276
\(574\) −9.31065e33 −0.303795
\(575\) 0 0
\(576\) 4.96642e33 0.155154
\(577\) 4.45926e34 1.36322 0.681609 0.731717i \(-0.261281\pi\)
0.681609 + 0.731717i \(0.261281\pi\)
\(578\) −1.26090e34 −0.377210
\(579\) 8.22337e34 2.40751
\(580\) 0 0
\(581\) 3.59450e34 1.00794
\(582\) 2.02333e34 0.555301
\(583\) 7.13067e33 0.191546
\(584\) 1.91403e34 0.503255
\(585\) 0 0
\(586\) 5.09967e34 1.28476
\(587\) 2.50958e34 0.618905 0.309452 0.950915i \(-0.399854\pi\)
0.309452 + 0.950915i \(0.399854\pi\)
\(588\) −1.01435e34 −0.244890
\(589\) 2.31645e34 0.547497
\(590\) 0 0
\(591\) −8.60173e34 −1.94869
\(592\) 1.26953e34 0.281593
\(593\) 4.88078e33 0.106000 0.0530000 0.998595i \(-0.483122\pi\)
0.0530000 + 0.998595i \(0.483122\pi\)
\(594\) −6.61748e33 −0.140722
\(595\) 0 0
\(596\) −1.19566e34 −0.243797
\(597\) −4.46798e34 −0.892138
\(598\) −4.59639e33 −0.0898777
\(599\) 4.61838e34 0.884412 0.442206 0.896914i \(-0.354196\pi\)
0.442206 + 0.896914i \(0.354196\pi\)
\(600\) 0 0
\(601\) −3.07405e34 −0.564651 −0.282325 0.959319i \(-0.591106\pi\)
−0.282325 + 0.959319i \(0.591106\pi\)
\(602\) 1.71068e34 0.307760
\(603\) 5.64671e34 0.995013
\(604\) −5.13652e34 −0.886559
\(605\) 0 0
\(606\) 1.92266e34 0.318415
\(607\) −1.03737e35 −1.68297 −0.841484 0.540282i \(-0.818318\pi\)
−0.841484 + 0.540282i \(0.818318\pi\)
\(608\) 6.25734e33 0.0994477
\(609\) −1.27984e35 −1.99269
\(610\) 0 0
\(611\) −5.01644e33 −0.0749688
\(612\) −5.24869e34 −0.768526
\(613\) −2.85744e34 −0.409941 −0.204970 0.978768i \(-0.565710\pi\)
−0.204970 + 0.978768i \(0.565710\pi\)
\(614\) 9.30582e34 1.30813
\(615\) 0 0
\(616\) 1.18405e34 0.159813
\(617\) −5.36336e34 −0.709370 −0.354685 0.934986i \(-0.615412\pi\)
−0.354685 + 0.934986i \(0.615412\pi\)
\(618\) 4.66836e34 0.605073
\(619\) 6.60909e34 0.839477 0.419738 0.907645i \(-0.362122\pi\)
0.419738 + 0.907645i \(0.362122\pi\)
\(620\) 0 0
\(621\) 2.94447e34 0.359221
\(622\) 7.55041e34 0.902798
\(623\) 9.67651e34 1.13401
\(624\) 4.16364e33 0.0478262
\(625\) 0 0
\(626\) 2.04264e34 0.225431
\(627\) −4.29000e34 −0.464102
\(628\) −4.00908e34 −0.425157
\(629\) −1.34168e35 −1.39482
\(630\) 0 0
\(631\) 8.62816e34 0.862087 0.431043 0.902331i \(-0.358146\pi\)
0.431043 + 0.902331i \(0.358146\pi\)
\(632\) −4.37714e34 −0.428773
\(633\) 5.26679e34 0.505824
\(634\) −3.60234e34 −0.339210
\(635\) 0 0
\(636\) 2.87409e34 0.260188
\(637\) −4.70958e33 −0.0418061
\(638\) −7.26412e34 −0.632302
\(639\) −2.38765e35 −2.03803
\(640\) 0 0
\(641\) 1.77093e35 1.45370 0.726850 0.686796i \(-0.240983\pi\)
0.726850 + 0.686796i \(0.240983\pi\)
\(642\) 8.91737e34 0.717875
\(643\) 7.98617e34 0.630524 0.315262 0.949005i \(-0.397908\pi\)
0.315262 + 0.949005i \(0.397908\pi\)
\(644\) −5.26847e34 −0.407954
\(645\) 0 0
\(646\) −6.61298e34 −0.492596
\(647\) 2.16671e35 1.58306 0.791530 0.611130i \(-0.209285\pi\)
0.791530 + 0.611130i \(0.209285\pi\)
\(648\) 3.45619e34 0.247691
\(649\) −7.59833e34 −0.534145
\(650\) 0 0
\(651\) 1.76675e35 1.19512
\(652\) −1.11248e35 −0.738242
\(653\) −2.41184e35 −1.57013 −0.785063 0.619416i \(-0.787369\pi\)
−0.785063 + 0.619416i \(0.787369\pi\)
\(654\) −1.69823e35 −1.08461
\(655\) 0 0
\(656\) −2.12999e34 −0.130942
\(657\) −2.92923e35 −1.76679
\(658\) −5.74995e34 −0.340283
\(659\) 2.18788e35 1.27045 0.635223 0.772329i \(-0.280908\pi\)
0.635223 + 0.772329i \(0.280908\pi\)
\(660\) 0 0
\(661\) −8.90411e34 −0.497820 −0.248910 0.968527i \(-0.580072\pi\)
−0.248910 + 0.968527i \(0.580072\pi\)
\(662\) −2.44581e35 −1.34183
\(663\) −4.40028e34 −0.236898
\(664\) 8.22310e34 0.434445
\(665\) 0 0
\(666\) −1.94289e35 −0.988597
\(667\) 3.23219e35 1.61408
\(668\) −6.82828e34 −0.334662
\(669\) −4.28705e35 −2.06221
\(670\) 0 0
\(671\) −1.35934e33 −0.00629934
\(672\) 4.77244e34 0.217083
\(673\) −2.39665e35 −1.07008 −0.535040 0.844827i \(-0.679703\pi\)
−0.535040 + 0.844827i \(0.679703\pi\)
\(674\) 4.44120e34 0.194649
\(675\) 0 0
\(676\) −1.16454e35 −0.491835
\(677\) 3.79216e34 0.157227 0.0786137 0.996905i \(-0.474951\pi\)
0.0786137 + 0.996905i \(0.474951\pi\)
\(678\) −4.55360e35 −1.85346
\(679\) 1.07679e35 0.430286
\(680\) 0 0
\(681\) 1.45481e35 0.560361
\(682\) 1.00277e35 0.379225
\(683\) 3.21208e35 1.19269 0.596345 0.802729i \(-0.296619\pi\)
0.596345 + 0.802729i \(0.296619\pi\)
\(684\) −9.57621e34 −0.349134
\(685\) 0 0
\(686\) −2.18986e35 −0.769776
\(687\) −5.10316e35 −1.76149
\(688\) 3.91351e34 0.132651
\(689\) 1.33443e34 0.0444177
\(690\) 0 0
\(691\) 8.34540e34 0.267899 0.133950 0.990988i \(-0.457234\pi\)
0.133950 + 0.990988i \(0.457234\pi\)
\(692\) 1.45780e35 0.459591
\(693\) −1.81207e35 −0.561060
\(694\) 2.94991e34 0.0897047
\(695\) 0 0
\(696\) −2.92788e35 −0.858890
\(697\) 2.25105e35 0.648597
\(698\) −2.16178e35 −0.611814
\(699\) 9.43649e34 0.262329
\(700\) 0 0
\(701\) 1.50008e35 0.402383 0.201191 0.979552i \(-0.435519\pi\)
0.201191 + 0.979552i \(0.435519\pi\)
\(702\) −1.23839e34 −0.0326320
\(703\) −2.44790e35 −0.633653
\(704\) 2.70874e34 0.0688827
\(705\) 0 0
\(706\) −2.32536e35 −0.570731
\(707\) 1.02321e35 0.246730
\(708\) −3.06259e35 −0.725559
\(709\) −2.12042e35 −0.493564 −0.246782 0.969071i \(-0.579373\pi\)
−0.246782 + 0.969071i \(0.579373\pi\)
\(710\) 0 0
\(711\) 6.69877e35 1.50531
\(712\) 2.21369e35 0.488783
\(713\) −4.46186e35 −0.968048
\(714\) −5.04369e35 −1.07528
\(715\) 0 0
\(716\) −7.23778e34 −0.149002
\(717\) 5.34363e35 1.08105
\(718\) −8.96718e34 −0.178279
\(719\) 3.27911e35 0.640686 0.320343 0.947302i \(-0.396202\pi\)
0.320343 + 0.947302i \(0.396202\pi\)
\(720\) 0 0
\(721\) 2.48443e35 0.468853
\(722\) 2.60588e35 0.483325
\(723\) 1.42934e36 2.60560
\(724\) 3.74080e35 0.670243
\(725\) 0 0
\(726\) 4.25846e35 0.737131
\(727\) −2.40795e35 −0.409701 −0.204850 0.978793i \(-0.565671\pi\)
−0.204850 + 0.978793i \(0.565671\pi\)
\(728\) 2.21583e34 0.0370590
\(729\) −8.57797e35 −1.41023
\(730\) 0 0
\(731\) −4.13594e35 −0.657063
\(732\) −5.47895e33 −0.00855675
\(733\) 3.39196e35 0.520776 0.260388 0.965504i \(-0.416150\pi\)
0.260388 + 0.965504i \(0.416150\pi\)
\(734\) 4.11648e35 0.621334
\(735\) 0 0
\(736\) −1.20526e35 −0.175837
\(737\) 3.07978e35 0.441749
\(738\) 3.25973e35 0.459703
\(739\) 1.23366e36 1.71057 0.855284 0.518159i \(-0.173383\pi\)
0.855284 + 0.518159i \(0.173383\pi\)
\(740\) 0 0
\(741\) −8.02829e34 −0.107621
\(742\) 1.52955e35 0.201611
\(743\) 6.24478e35 0.809389 0.404694 0.914452i \(-0.367378\pi\)
0.404694 + 0.914452i \(0.367378\pi\)
\(744\) 4.04177e35 0.515122
\(745\) 0 0
\(746\) 2.58479e35 0.318559
\(747\) −1.25846e36 −1.52522
\(748\) −2.86270e35 −0.341197
\(749\) 4.74570e35 0.556259
\(750\) 0 0
\(751\) −4.41291e35 −0.500294 −0.250147 0.968208i \(-0.580479\pi\)
−0.250147 + 0.968208i \(0.580479\pi\)
\(752\) −1.31541e35 −0.146669
\(753\) 2.03934e36 2.23642
\(754\) −1.35940e35 −0.146625
\(755\) 0 0
\(756\) −1.41947e35 −0.148117
\(757\) −1.83266e35 −0.188098 −0.0940489 0.995568i \(-0.529981\pi\)
−0.0940489 + 0.995568i \(0.529981\pi\)
\(758\) 3.62805e35 0.366276
\(759\) 8.26323e35 0.820593
\(760\) 0 0
\(761\) 1.88027e36 1.80681 0.903406 0.428787i \(-0.141059\pi\)
0.903406 + 0.428787i \(0.141059\pi\)
\(762\) −7.50557e35 −0.709490
\(763\) −9.03772e35 −0.840431
\(764\) −3.21010e35 −0.293665
\(765\) 0 0
\(766\) 3.72046e35 0.329410
\(767\) −1.42195e35 −0.123863
\(768\) 1.09179e35 0.0935671
\(769\) 1.28453e36 1.08309 0.541546 0.840671i \(-0.317839\pi\)
0.541546 + 0.840671i \(0.317839\pi\)
\(770\) 0 0
\(771\) −9.30700e35 −0.759679
\(772\) 1.00118e36 0.804071
\(773\) 1.04808e36 0.828226 0.414113 0.910225i \(-0.364092\pi\)
0.414113 + 0.910225i \(0.364092\pi\)
\(774\) −5.98923e35 −0.465703
\(775\) 0 0
\(776\) 2.46335e35 0.185462
\(777\) −1.86700e36 −1.38319
\(778\) −2.92063e35 −0.212927
\(779\) 4.10703e35 0.294652
\(780\) 0 0
\(781\) −1.30225e36 −0.904811
\(782\) 1.27377e36 0.870974
\(783\) 8.70840e35 0.586025
\(784\) −1.23495e35 −0.0817894
\(785\) 0 0
\(786\) −3.80382e35 −0.244027
\(787\) 8.93146e35 0.563947 0.281974 0.959422i \(-0.409011\pi\)
0.281974 + 0.959422i \(0.409011\pi\)
\(788\) −1.04724e36 −0.650832
\(789\) −1.95822e36 −1.19784
\(790\) 0 0
\(791\) −2.42336e36 −1.43619
\(792\) −4.14546e35 −0.241829
\(793\) −2.54386e33 −0.00146076
\(794\) 1.22645e35 0.0693257
\(795\) 0 0
\(796\) −5.43966e35 −0.297960
\(797\) 1.02840e36 0.554540 0.277270 0.960792i \(-0.410570\pi\)
0.277270 + 0.960792i \(0.410570\pi\)
\(798\) −9.20219e35 −0.488489
\(799\) 1.39017e36 0.726498
\(800\) 0 0
\(801\) −3.38782e36 −1.71599
\(802\) 2.05041e36 1.02249
\(803\) −1.59763e36 −0.784392
\(804\) 1.24134e36 0.600053
\(805\) 0 0
\(806\) 1.87658e35 0.0879385
\(807\) −2.62004e36 −1.20889
\(808\) 2.34079e35 0.106346
\(809\) 1.33897e36 0.598981 0.299490 0.954099i \(-0.403183\pi\)
0.299490 + 0.954099i \(0.403183\pi\)
\(810\) 0 0
\(811\) 1.29485e36 0.561643 0.280821 0.959760i \(-0.409393\pi\)
0.280821 + 0.959760i \(0.409393\pi\)
\(812\) −1.55818e36 −0.665528
\(813\) −2.00917e36 −0.845055
\(814\) −1.05967e36 −0.438901
\(815\) 0 0
\(816\) −1.15384e36 −0.463467
\(817\) −7.54600e35 −0.298498
\(818\) 1.14024e36 0.444201
\(819\) −3.39110e35 −0.130104
\(820\) 0 0
\(821\) 3.91664e34 0.0145755 0.00728777 0.999973i \(-0.497680\pi\)
0.00728777 + 0.999973i \(0.497680\pi\)
\(822\) −1.82294e36 −0.668151
\(823\) 3.40400e36 1.22883 0.614414 0.788984i \(-0.289392\pi\)
0.614414 + 0.788984i \(0.289392\pi\)
\(824\) 5.68362e35 0.202085
\(825\) 0 0
\(826\) −1.62987e36 −0.562213
\(827\) 5.02186e36 1.70626 0.853132 0.521696i \(-0.174700\pi\)
0.853132 + 0.521696i \(0.174700\pi\)
\(828\) 1.84453e36 0.617316
\(829\) 8.20485e35 0.270482 0.135241 0.990813i \(-0.456819\pi\)
0.135241 + 0.990813i \(0.456819\pi\)
\(830\) 0 0
\(831\) 7.31856e35 0.234106
\(832\) 5.06913e34 0.0159732
\(833\) 1.30513e36 0.405128
\(834\) 5.20178e35 0.159066
\(835\) 0 0
\(836\) −5.22298e35 −0.155003
\(837\) −1.20215e36 −0.351471
\(838\) 2.60845e36 0.751332
\(839\) −5.38664e36 −1.52860 −0.764300 0.644861i \(-0.776915\pi\)
−0.764300 + 0.644861i \(0.776915\pi\)
\(840\) 0 0
\(841\) 5.92900e36 1.63317
\(842\) 1.60676e36 0.436063
\(843\) −5.02945e36 −1.34486
\(844\) 6.41220e35 0.168938
\(845\) 0 0
\(846\) 2.01310e36 0.514916
\(847\) 2.26629e36 0.571180
\(848\) 3.49914e35 0.0868987
\(849\) 1.98389e36 0.485479
\(850\) 0 0
\(851\) 4.71505e36 1.12038
\(852\) −5.24887e36 −1.22905
\(853\) −9.55369e35 −0.220449 −0.110225 0.993907i \(-0.535157\pi\)
−0.110225 + 0.993907i \(0.535157\pi\)
\(854\) −2.91582e34 −0.00663036
\(855\) 0 0
\(856\) 1.08567e36 0.239759
\(857\) 5.39118e36 1.17334 0.586669 0.809827i \(-0.300439\pi\)
0.586669 + 0.809827i \(0.300439\pi\)
\(858\) −3.47537e35 −0.0745436
\(859\) 5.48992e36 1.16052 0.580258 0.814432i \(-0.302952\pi\)
0.580258 + 0.814432i \(0.302952\pi\)
\(860\) 0 0
\(861\) 3.13241e36 0.643190
\(862\) −1.00016e36 −0.202407
\(863\) 2.89510e36 0.577469 0.288735 0.957409i \(-0.406766\pi\)
0.288735 + 0.957409i \(0.406766\pi\)
\(864\) −3.24731e35 −0.0638413
\(865\) 0 0
\(866\) 3.73912e36 0.714161
\(867\) 4.24209e36 0.798623
\(868\) 2.15097e36 0.399152
\(869\) 3.65359e36 0.668301
\(870\) 0 0
\(871\) 5.76349e35 0.102437
\(872\) −2.06755e36 −0.362244
\(873\) −3.76991e36 −0.651108
\(874\) 2.32398e36 0.395676
\(875\) 0 0
\(876\) −6.43943e36 −1.06548
\(877\) −2.04429e36 −0.333463 −0.166731 0.986002i \(-0.553321\pi\)
−0.166731 + 0.986002i \(0.553321\pi\)
\(878\) −2.48874e36 −0.400219
\(879\) −1.71570e37 −2.72007
\(880\) 0 0
\(881\) −1.76163e36 −0.271466 −0.135733 0.990745i \(-0.543339\pi\)
−0.135733 + 0.990745i \(0.543339\pi\)
\(882\) 1.88996e36 0.287141
\(883\) −1.25672e37 −1.88248 −0.941240 0.337738i \(-0.890338\pi\)
−0.941240 + 0.337738i \(0.890338\pi\)
\(884\) −5.35724e35 −0.0791203
\(885\) 0 0
\(886\) 3.71216e36 0.532973
\(887\) −8.81003e36 −1.24719 −0.623595 0.781748i \(-0.714328\pi\)
−0.623595 + 0.781748i \(0.714328\pi\)
\(888\) −4.27112e36 −0.596184
\(889\) −3.99436e36 −0.549762
\(890\) 0 0
\(891\) −2.88487e36 −0.386060
\(892\) −5.21939e36 −0.688746
\(893\) 2.53636e36 0.330041
\(894\) 4.02259e36 0.516163
\(895\) 0 0
\(896\) 5.81034e35 0.0725023
\(897\) 1.54638e36 0.190288
\(898\) 4.97986e36 0.604315
\(899\) −1.31962e37 −1.57925
\(900\) 0 0
\(901\) −3.69802e36 −0.430436
\(902\) 1.77789e36 0.204091
\(903\) −5.75530e36 −0.651585
\(904\) −5.54391e36 −0.619029
\(905\) 0 0
\(906\) 1.72810e37 1.87701
\(907\) 1.59523e37 1.70896 0.854482 0.519481i \(-0.173875\pi\)
0.854482 + 0.519481i \(0.173875\pi\)
\(908\) 1.77120e36 0.187152
\(909\) −3.58234e36 −0.373352
\(910\) 0 0
\(911\) −1.71775e36 −0.174172 −0.0870862 0.996201i \(-0.527756\pi\)
−0.0870862 + 0.996201i \(0.527756\pi\)
\(912\) −2.10518e36 −0.210549
\(913\) −6.86379e36 −0.677142
\(914\) −2.29183e36 −0.223026
\(915\) 0 0
\(916\) −6.21298e36 −0.588310
\(917\) −2.02434e36 −0.189089
\(918\) 3.43187e36 0.316226
\(919\) −7.70373e35 −0.0700256 −0.0350128 0.999387i \(-0.511147\pi\)
−0.0350128 + 0.999387i \(0.511147\pi\)
\(920\) 0 0
\(921\) −3.13079e37 −2.76955
\(922\) 1.55071e36 0.135330
\(923\) −2.43703e36 −0.209817
\(924\) −3.98354e36 −0.338353
\(925\) 0 0
\(926\) 1.13475e37 0.938132
\(927\) −8.69819e36 −0.709467
\(928\) −3.56463e36 −0.286856
\(929\) 1.66420e37 1.32132 0.660662 0.750684i \(-0.270276\pi\)
0.660662 + 0.750684i \(0.270276\pi\)
\(930\) 0 0
\(931\) 2.38121e36 0.184046
\(932\) 1.14887e36 0.0876138
\(933\) −2.54021e37 −1.91139
\(934\) −9.49954e36 −0.705288
\(935\) 0 0
\(936\) −7.75779e35 −0.0560777
\(937\) −9.66427e36 −0.689325 −0.344663 0.938727i \(-0.612007\pi\)
−0.344663 + 0.938727i \(0.612007\pi\)
\(938\) 6.60622e36 0.464962
\(939\) −6.87214e36 −0.477279
\(940\) 0 0
\(941\) −1.33808e37 −0.904923 −0.452461 0.891784i \(-0.649454\pi\)
−0.452461 + 0.891784i \(0.649454\pi\)
\(942\) 1.34879e37 0.900136
\(943\) −7.91080e36 −0.520984
\(944\) −3.72863e36 −0.242326
\(945\) 0 0
\(946\) −3.26659e36 −0.206755
\(947\) −2.63078e37 −1.64328 −0.821638 0.570010i \(-0.806939\pi\)
−0.821638 + 0.570010i \(0.806939\pi\)
\(948\) 1.47262e37 0.907791
\(949\) −2.98981e36 −0.181893
\(950\) 0 0
\(951\) 1.21195e37 0.718170
\(952\) −6.14058e36 −0.359126
\(953\) 9.10378e36 0.525485 0.262742 0.964866i \(-0.415373\pi\)
0.262742 + 0.964866i \(0.415373\pi\)
\(954\) −5.35508e36 −0.305078
\(955\) 0 0
\(956\) 6.50575e36 0.361055
\(957\) 2.44389e37 1.33870
\(958\) −9.19044e36 −0.496898
\(959\) −9.70142e36 −0.517729
\(960\) 0 0
\(961\) −1.01623e36 −0.0528386
\(962\) −1.98307e36 −0.101777
\(963\) −1.66151e37 −0.841731
\(964\) 1.74019e37 0.870230
\(965\) 0 0
\(966\) 1.77249e37 0.863713
\(967\) 1.34794e36 0.0648394 0.0324197 0.999474i \(-0.489679\pi\)
0.0324197 + 0.999474i \(0.489679\pi\)
\(968\) 5.18457e36 0.246190
\(969\) 2.22483e37 1.04291
\(970\) 0 0
\(971\) 3.82504e36 0.174741 0.0873707 0.996176i \(-0.472154\pi\)
0.0873707 + 0.996176i \(0.472154\pi\)
\(972\) −1.56316e37 −0.704978
\(973\) 2.76831e36 0.123255
\(974\) 1.52123e36 0.0668663
\(975\) 0 0
\(976\) −6.67050e34 −0.00285782
\(977\) −1.16914e37 −0.494522 −0.247261 0.968949i \(-0.579530\pi\)
−0.247261 + 0.968949i \(0.579530\pi\)
\(978\) 3.74277e37 1.56299
\(979\) −1.84776e37 −0.761835
\(980\) 0 0
\(981\) 3.16418e37 1.27174
\(982\) 1.08752e37 0.431563
\(983\) −1.38138e37 −0.541244 −0.270622 0.962686i \(-0.587229\pi\)
−0.270622 + 0.962686i \(0.587229\pi\)
\(984\) 7.16600e36 0.277228
\(985\) 0 0
\(986\) 3.76722e37 1.42089
\(987\) 1.93447e37 0.720441
\(988\) −9.77426e35 −0.0359436
\(989\) 1.45348e37 0.527784
\(990\) 0 0
\(991\) −3.79186e37 −1.34255 −0.671276 0.741207i \(-0.734254\pi\)
−0.671276 + 0.741207i \(0.734254\pi\)
\(992\) 4.92077e36 0.172043
\(993\) 8.22853e37 2.84090
\(994\) −2.79338e37 −0.952356
\(995\) 0 0
\(996\) −2.76653e37 −0.919799
\(997\) −5.34619e37 −1.75531 −0.877657 0.479289i \(-0.840894\pi\)
−0.877657 + 0.479289i \(0.840894\pi\)
\(998\) 9.79424e36 0.317570
\(999\) 1.27036e37 0.406779
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.c.1.2 2
5.2 odd 4 50.26.b.e.49.1 4
5.3 odd 4 50.26.b.e.49.4 4
5.4 even 2 2.26.a.b.1.1 2
15.14 odd 2 18.26.a.e.1.1 2
20.19 odd 2 16.26.a.c.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.26.a.b.1.1 2 5.4 even 2
16.26.a.c.1.2 2 20.19 odd 2
18.26.a.e.1.1 2 15.14 odd 2
50.26.a.c.1.2 2 1.1 even 1 trivial
50.26.b.e.49.1 4 5.2 odd 4
50.26.b.e.49.4 4 5.3 odd 4