Properties

Label 50.26.a.k.1.6
Level $50$
Weight $26$
Character 50.1
Self dual yes
Analytic conductor $197.998$
Analytic rank $0$
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: \(0\)
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 1.6
Root \(134113.\) 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)
 
\(f(q)\) \(=\) \(q-4096.00 q^{2} +1.47469e6 q^{3} +1.67772e7 q^{4} -6.04034e9 q^{6} -3.57876e9 q^{7} -6.87195e10 q^{8} +1.32743e12 q^{9} +1.08564e12 q^{11} +2.47412e13 q^{12} -2.13293e13 q^{13} +1.46586e13 q^{14} +2.81475e14 q^{16} +3.79275e15 q^{17} -5.43715e15 q^{18} +6.90611e15 q^{19} -5.27758e15 q^{21} -4.44680e15 q^{22} +1.56245e17 q^{23} -1.01340e17 q^{24} +8.73648e16 q^{26} +7.08060e17 q^{27} -6.00417e16 q^{28} +6.08503e17 q^{29} -3.42961e18 q^{31} -1.15292e18 q^{32} +1.60099e18 q^{33} -1.55351e19 q^{34} +2.22706e19 q^{36} -7.61614e19 q^{37} -2.82874e19 q^{38} -3.14542e19 q^{39} +2.17158e20 q^{41} +2.16169e19 q^{42} -4.70339e20 q^{43} +1.82141e19 q^{44} -6.39979e20 q^{46} +6.53190e20 q^{47} +4.15089e20 q^{48} -1.32826e21 q^{49} +5.59313e21 q^{51} -3.57846e20 q^{52} +4.54358e21 q^{53} -2.90021e21 q^{54} +2.45931e20 q^{56} +1.01844e22 q^{57} -2.49243e21 q^{58} +1.39811e22 q^{59} -4.30120e21 q^{61} +1.40477e22 q^{62} -4.75056e21 q^{63} +4.72237e21 q^{64} -6.55766e21 q^{66} -3.04650e22 q^{67} +6.36317e22 q^{68} +2.30413e23 q^{69} -4.08644e22 q^{71} -9.12202e22 q^{72} +1.14063e23 q^{73} +3.11957e23 q^{74} +1.15865e23 q^{76} -3.88526e21 q^{77} +1.28836e23 q^{78} +6.98160e23 q^{79} -8.05452e22 q^{81} -8.89479e23 q^{82} -1.18110e24 q^{83} -8.85430e22 q^{84} +1.92651e24 q^{86} +8.97354e23 q^{87} -7.46049e22 q^{88} +2.79755e24 q^{89} +7.63325e22 q^{91} +2.62136e24 q^{92} -5.05762e24 q^{93} -2.67546e24 q^{94} -1.70020e24 q^{96} +1.06166e25 q^{97} +5.44056e24 q^{98} +1.44112e24 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 24576 q^{2} + 801416 q^{3} + 100663296 q^{4} - 3282599936 q^{6} + 34007705352 q^{7} - 412316860416 q^{8} + 541468782118 q^{9} + 9861544614312 q^{11} + 13445529337856 q^{12} + 30787386783696 q^{13}+ \cdots - 20\!\cdots\!64 q^{99}+O(q^{100}) \) 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
\(3\) 1.47469e6 1.60209 0.801043 0.598607i \(-0.204279\pi\)
0.801043 + 0.598607i \(0.204279\pi\)
\(4\) 1.67772e7 0.500000
\(5\) 0 0
\(6\) −6.04034e9 −1.13285
\(7\) −3.57876e9 −0.0977254 −0.0488627 0.998806i \(-0.515560\pi\)
−0.0488627 + 0.998806i \(0.515560\pi\)
\(8\) −6.87195e10 −0.353553
\(9\) 1.32743e12 1.56668
\(10\) 0 0
\(11\) 1.08564e12 0.104299 0.0521493 0.998639i \(-0.483393\pi\)
0.0521493 + 0.998639i \(0.483393\pi\)
\(12\) 2.47412e13 0.801043
\(13\) −2.13293e13 −0.253913 −0.126956 0.991908i \(-0.540521\pi\)
−0.126956 + 0.991908i \(0.540521\pi\)
\(14\) 1.46586e13 0.0691023
\(15\) 0 0
\(16\) 2.81475e14 0.250000
\(17\) 3.79275e15 1.57886 0.789428 0.613844i \(-0.210377\pi\)
0.789428 + 0.613844i \(0.210377\pi\)
\(18\) −5.43715e15 −1.10781
\(19\) 6.90611e15 0.715836 0.357918 0.933753i \(-0.383487\pi\)
0.357918 + 0.933753i \(0.383487\pi\)
\(20\) 0 0
\(21\) −5.27758e15 −0.156564
\(22\) −4.44680e15 −0.0737503
\(23\) 1.56245e17 1.48665 0.743324 0.668932i \(-0.233248\pi\)
0.743324 + 0.668932i \(0.233248\pi\)
\(24\) −1.01340e17 −0.566423
\(25\) 0 0
\(26\) 8.73648e16 0.179544
\(27\) 7.08060e17 0.907868
\(28\) −6.00417e16 −0.0488627
\(29\) 6.08503e17 0.319365 0.159683 0.987168i \(-0.448953\pi\)
0.159683 + 0.987168i \(0.448953\pi\)
\(30\) 0 0
\(31\) −3.42961e18 −0.782030 −0.391015 0.920384i \(-0.627876\pi\)
−0.391015 + 0.920384i \(0.627876\pi\)
\(32\) −1.15292e18 −0.176777
\(33\) 1.60099e18 0.167095
\(34\) −1.55351e19 −1.11642
\(35\) 0 0
\(36\) 2.22706e19 0.783339
\(37\) −7.61614e19 −1.90201 −0.951006 0.309173i \(-0.899948\pi\)
−0.951006 + 0.309173i \(0.899948\pi\)
\(38\) −2.82874e19 −0.506172
\(39\) −3.14542e19 −0.406790
\(40\) 0 0
\(41\) 2.17158e20 1.50306 0.751532 0.659696i \(-0.229315\pi\)
0.751532 + 0.659696i \(0.229315\pi\)
\(42\) 2.16169e19 0.110708
\(43\) −4.70339e20 −1.79496 −0.897481 0.441052i \(-0.854605\pi\)
−0.897481 + 0.441052i \(0.854605\pi\)
\(44\) 1.82141e19 0.0521493
\(45\) 0 0
\(46\) −6.39979e20 −1.05122
\(47\) 6.53190e20 0.820004 0.410002 0.912085i \(-0.365528\pi\)
0.410002 + 0.912085i \(0.365528\pi\)
\(48\) 4.15089e20 0.400521
\(49\) −1.32826e21 −0.990450
\(50\) 0 0
\(51\) 5.59313e21 2.52946
\(52\) −3.57846e20 −0.126956
\(53\) 4.54358e21 1.27043 0.635214 0.772336i \(-0.280912\pi\)
0.635214 + 0.772336i \(0.280912\pi\)
\(54\) −2.90021e21 −0.641960
\(55\) 0 0
\(56\) 2.45931e20 0.0345512
\(57\) 1.01844e22 1.14683
\(58\) −2.49243e21 −0.225825
\(59\) 1.39811e22 1.02303 0.511517 0.859273i \(-0.329084\pi\)
0.511517 + 0.859273i \(0.329084\pi\)
\(60\) 0 0
\(61\) −4.30120e21 −0.207475 −0.103738 0.994605i \(-0.533080\pi\)
−0.103738 + 0.994605i \(0.533080\pi\)
\(62\) 1.40477e22 0.552979
\(63\) −4.75056e21 −0.153104
\(64\) 4.72237e21 0.125000
\(65\) 0 0
\(66\) −6.55766e21 −0.118154
\(67\) −3.04650e22 −0.454848 −0.227424 0.973796i \(-0.573030\pi\)
−0.227424 + 0.973796i \(0.573030\pi\)
\(68\) 6.36317e22 0.789428
\(69\) 2.30413e23 2.38174
\(70\) 0 0
\(71\) −4.08644e22 −0.295540 −0.147770 0.989022i \(-0.547209\pi\)
−0.147770 + 0.989022i \(0.547209\pi\)
\(72\) −9.12202e22 −0.553905
\(73\) 1.14063e23 0.582918 0.291459 0.956583i \(-0.405859\pi\)
0.291459 + 0.956583i \(0.405859\pi\)
\(74\) 3.11957e23 1.34493
\(75\) 0 0
\(76\) 1.15865e23 0.357918
\(77\) −3.88526e21 −0.0101926
\(78\) 1.28836e23 0.287644
\(79\) 6.98160e23 1.32928 0.664640 0.747164i \(-0.268585\pi\)
0.664640 + 0.747164i \(0.268585\pi\)
\(80\) 0 0
\(81\) −8.05452e22 −0.112196
\(82\) −8.89479e23 −1.06283
\(83\) −1.18110e24 −1.21286 −0.606428 0.795139i \(-0.707398\pi\)
−0.606428 + 0.795139i \(0.707398\pi\)
\(84\) −8.85430e22 −0.0782822
\(85\) 0 0
\(86\) 1.92651e24 1.26923
\(87\) 8.97354e23 0.511650
\(88\) −7.46049e22 −0.0368751
\(89\) 2.79755e24 1.20061 0.600307 0.799770i \(-0.295045\pi\)
0.600307 + 0.799770i \(0.295045\pi\)
\(90\) 0 0
\(91\) 7.63325e22 0.0248137
\(92\) 2.62136e24 0.743324
\(93\) −5.05762e24 −1.25288
\(94\) −2.67546e24 −0.579831
\(95\) 0 0
\(96\) −1.70020e24 −0.283211
\(97\) 1.06166e25 1.55360 0.776798 0.629750i \(-0.216843\pi\)
0.776798 + 0.629750i \(0.216843\pi\)
\(98\) 5.44056e24 0.700354
\(99\) 1.44112e24 0.163403
\(100\) 0 0
\(101\) 7.61024e24 0.672019 0.336009 0.941859i \(-0.390923\pi\)
0.336009 + 0.941859i \(0.390923\pi\)
\(102\) −2.29095e25 −1.78860
\(103\) 1.50264e25 1.03846 0.519229 0.854635i \(-0.326219\pi\)
0.519229 + 0.854635i \(0.326219\pi\)
\(104\) 1.46574e24 0.0897718
\(105\) 0 0
\(106\) −1.86105e25 −0.898328
\(107\) −3.03661e24 −0.130344 −0.0651721 0.997874i \(-0.520760\pi\)
−0.0651721 + 0.997874i \(0.520760\pi\)
\(108\) 1.18793e25 0.453934
\(109\) −3.16922e25 −1.07925 −0.539625 0.841906i \(-0.681434\pi\)
−0.539625 + 0.841906i \(0.681434\pi\)
\(110\) 0 0
\(111\) −1.12315e26 −3.04719
\(112\) −1.00733e24 −0.0244314
\(113\) −6.24206e25 −1.35471 −0.677357 0.735655i \(-0.736875\pi\)
−0.677357 + 0.735655i \(0.736875\pi\)
\(114\) −4.17152e25 −0.810931
\(115\) 0 0
\(116\) 1.02090e25 0.159683
\(117\) −2.83131e25 −0.397800
\(118\) −5.72664e25 −0.723394
\(119\) −1.35733e25 −0.154294
\(120\) 0 0
\(121\) −1.07168e26 −0.989122
\(122\) 1.76177e25 0.146707
\(123\) 3.20241e26 2.40804
\(124\) −5.75393e25 −0.391015
\(125\) 0 0
\(126\) 1.94583e25 0.108261
\(127\) 1.67435e26 0.843917 0.421959 0.906615i \(-0.361343\pi\)
0.421959 + 0.906615i \(0.361343\pi\)
\(128\) −1.93428e25 −0.0883883
\(129\) −6.93606e26 −2.87568
\(130\) 0 0
\(131\) −1.90057e26 −0.650118 −0.325059 0.945694i \(-0.605384\pi\)
−0.325059 + 0.945694i \(0.605384\pi\)
\(132\) 2.68602e25 0.0835477
\(133\) −2.47153e25 −0.0699553
\(134\) 1.24785e26 0.321626
\(135\) 0 0
\(136\) −2.60636e26 −0.558210
\(137\) 3.51717e26 0.687364 0.343682 0.939086i \(-0.388326\pi\)
0.343682 + 0.939086i \(0.388326\pi\)
\(138\) −9.43773e26 −1.68414
\(139\) 3.91803e26 0.638827 0.319414 0.947615i \(-0.396514\pi\)
0.319414 + 0.947615i \(0.396514\pi\)
\(140\) 0 0
\(141\) 9.63254e26 1.31372
\(142\) 1.67381e26 0.208978
\(143\) −2.31560e25 −0.0264828
\(144\) 3.73638e26 0.391670
\(145\) 0 0
\(146\) −4.67201e26 −0.412186
\(147\) −1.95878e27 −1.58679
\(148\) −1.27778e27 −0.951006
\(149\) 1.80648e27 1.23596 0.617979 0.786194i \(-0.287952\pi\)
0.617979 + 0.786194i \(0.287952\pi\)
\(150\) 0 0
\(151\) −1.68416e26 −0.0975373 −0.0487687 0.998810i \(-0.515530\pi\)
−0.0487687 + 0.998810i \(0.515530\pi\)
\(152\) −4.74584e26 −0.253086
\(153\) 5.03460e27 2.47356
\(154\) 1.59140e25 0.00720728
\(155\) 0 0
\(156\) −5.27713e26 −0.203395
\(157\) 1.56946e27 0.558475 0.279237 0.960222i \(-0.409918\pi\)
0.279237 + 0.960222i \(0.409918\pi\)
\(158\) −2.85967e27 −0.939943
\(159\) 6.70039e27 2.03533
\(160\) 0 0
\(161\) −5.59164e26 −0.145283
\(162\) 3.29913e26 0.0793345
\(163\) 5.35307e27 1.19195 0.595974 0.803004i \(-0.296766\pi\)
0.595974 + 0.803004i \(0.296766\pi\)
\(164\) 3.64331e27 0.751532
\(165\) 0 0
\(166\) 4.83777e27 0.857618
\(167\) 1.01057e28 1.66192 0.830959 0.556334i \(-0.187793\pi\)
0.830959 + 0.556334i \(0.187793\pi\)
\(168\) 3.62672e26 0.0553539
\(169\) −6.60147e27 −0.935528
\(170\) 0 0
\(171\) 9.16737e27 1.12148
\(172\) −7.89098e27 −0.897481
\(173\) 8.66506e26 0.0916633 0.0458316 0.998949i \(-0.485406\pi\)
0.0458316 + 0.998949i \(0.485406\pi\)
\(174\) −3.67556e27 −0.361791
\(175\) 0 0
\(176\) 3.05581e26 0.0260747
\(177\) 2.06178e28 1.63899
\(178\) −1.14588e28 −0.848962
\(179\) −2.03208e28 −1.40371 −0.701856 0.712319i \(-0.747645\pi\)
−0.701856 + 0.712319i \(0.747645\pi\)
\(180\) 0 0
\(181\) −6.11555e27 −0.367666 −0.183833 0.982957i \(-0.558851\pi\)
−0.183833 + 0.982957i \(0.558851\pi\)
\(182\) −3.12658e26 −0.0175460
\(183\) −6.34295e27 −0.332393
\(184\) −1.07371e28 −0.525609
\(185\) 0 0
\(186\) 2.07160e28 0.885919
\(187\) 4.11757e27 0.164672
\(188\) 1.09587e28 0.410002
\(189\) −2.53398e27 −0.0887218
\(190\) 0 0
\(191\) 1.25038e28 0.383819 0.191909 0.981413i \(-0.438532\pi\)
0.191909 + 0.981413i \(0.438532\pi\)
\(192\) 6.96404e27 0.200261
\(193\) 4.00087e28 1.07817 0.539086 0.842251i \(-0.318770\pi\)
0.539086 + 0.842251i \(0.318770\pi\)
\(194\) −4.34855e28 −1.09856
\(195\) 0 0
\(196\) −2.22845e28 −0.495225
\(197\) −1.19953e28 −0.250141 −0.125070 0.992148i \(-0.539916\pi\)
−0.125070 + 0.992148i \(0.539916\pi\)
\(198\) −5.90281e27 −0.115543
\(199\) 3.34370e28 0.614559 0.307280 0.951619i \(-0.400581\pi\)
0.307280 + 0.951619i \(0.400581\pi\)
\(200\) 0 0
\(201\) −4.49265e28 −0.728705
\(202\) −3.11715e28 −0.475189
\(203\) −2.17769e27 −0.0312101
\(204\) 9.38372e28 1.26473
\(205\) 0 0
\(206\) −6.15480e28 −0.734301
\(207\) 2.07404e29 2.32910
\(208\) −6.00366e27 −0.0634782
\(209\) 7.49757e27 0.0746607
\(210\) 0 0
\(211\) 6.92258e28 0.611980 0.305990 0.952035i \(-0.401013\pi\)
0.305990 + 0.952035i \(0.401013\pi\)
\(212\) 7.62287e28 0.635214
\(213\) −6.02624e28 −0.473480
\(214\) 1.24379e28 0.0921672
\(215\) 0 0
\(216\) −4.86575e28 −0.320980
\(217\) 1.22738e28 0.0764242
\(218\) 1.29811e29 0.763144
\(219\) 1.68207e29 0.933885
\(220\) 0 0
\(221\) −8.08966e28 −0.400892
\(222\) 4.60040e29 2.15469
\(223\) 4.37006e29 1.93498 0.967490 0.252908i \(-0.0813869\pi\)
0.967490 + 0.252908i \(0.0813869\pi\)
\(224\) 4.12603e27 0.0172756
\(225\) 0 0
\(226\) 2.55675e29 0.957927
\(227\) −4.51616e28 −0.160120 −0.0800600 0.996790i \(-0.525511\pi\)
−0.0800600 + 0.996790i \(0.525511\pi\)
\(228\) 1.70866e29 0.573415
\(229\) −3.96675e29 −1.26035 −0.630175 0.776454i \(-0.717017\pi\)
−0.630175 + 0.776454i \(0.717017\pi\)
\(230\) 0 0
\(231\) −5.72957e27 −0.0163295
\(232\) −4.18160e28 −0.112913
\(233\) −2.52711e29 −0.646658 −0.323329 0.946287i \(-0.604802\pi\)
−0.323329 + 0.946287i \(0.604802\pi\)
\(234\) 1.15971e29 0.281287
\(235\) 0 0
\(236\) 2.34563e29 0.511517
\(237\) 1.02957e30 2.12962
\(238\) 5.55964e28 0.109103
\(239\) 7.75729e29 1.44456 0.722281 0.691600i \(-0.243094\pi\)
0.722281 + 0.691600i \(0.243094\pi\)
\(240\) 0 0
\(241\) −1.98502e28 −0.0333082 −0.0166541 0.999861i \(-0.505301\pi\)
−0.0166541 + 0.999861i \(0.505301\pi\)
\(242\) 4.38962e29 0.699415
\(243\) −7.18710e29 −1.08762
\(244\) −7.21621e28 −0.103738
\(245\) 0 0
\(246\) −1.31171e30 −1.70274
\(247\) −1.47302e29 −0.181760
\(248\) 2.35681e29 0.276489
\(249\) −1.74175e30 −1.94310
\(250\) 0 0
\(251\) −9.10004e29 −0.918590 −0.459295 0.888284i \(-0.651898\pi\)
−0.459295 + 0.888284i \(0.651898\pi\)
\(252\) −7.97011e28 −0.0765522
\(253\) 1.69626e29 0.155055
\(254\) −6.85814e29 −0.596740
\(255\) 0 0
\(256\) 7.92282e28 0.0625000
\(257\) 1.13259e30 0.850961 0.425480 0.904968i \(-0.360105\pi\)
0.425480 + 0.904968i \(0.360105\pi\)
\(258\) 2.84101e30 2.03342
\(259\) 2.72563e29 0.185875
\(260\) 0 0
\(261\) 8.07744e29 0.500343
\(262\) 7.78473e29 0.459703
\(263\) 7.83926e29 0.441396 0.220698 0.975342i \(-0.429167\pi\)
0.220698 + 0.975342i \(0.429167\pi\)
\(264\) −1.10019e29 −0.0590771
\(265\) 0 0
\(266\) 1.01234e29 0.0494659
\(267\) 4.12553e30 1.92349
\(268\) −5.11118e29 −0.227424
\(269\) 2.21562e30 0.941005 0.470502 0.882399i \(-0.344073\pi\)
0.470502 + 0.882399i \(0.344073\pi\)
\(270\) 0 0
\(271\) −3.34900e29 −0.129658 −0.0648289 0.997896i \(-0.520650\pi\)
−0.0648289 + 0.997896i \(0.520650\pi\)
\(272\) 1.06756e30 0.394714
\(273\) 1.12567e29 0.0397537
\(274\) −1.44063e30 −0.486040
\(275\) 0 0
\(276\) 3.86569e30 1.19087
\(277\) 3.78301e30 1.11388 0.556942 0.830552i \(-0.311975\pi\)
0.556942 + 0.830552i \(0.311975\pi\)
\(278\) −1.60483e30 −0.451719
\(279\) −4.55256e30 −1.22519
\(280\) 0 0
\(281\) −2.06047e30 −0.507151 −0.253576 0.967316i \(-0.581607\pi\)
−0.253576 + 0.967316i \(0.581607\pi\)
\(282\) −3.94549e30 −0.928938
\(283\) 1.40735e30 0.317010 0.158505 0.987358i \(-0.449333\pi\)
0.158505 + 0.987358i \(0.449333\pi\)
\(284\) −6.85591e29 −0.147770
\(285\) 0 0
\(286\) 9.48471e28 0.0187261
\(287\) −7.77157e29 −0.146888
\(288\) −1.53042e30 −0.276952
\(289\) 8.61430e30 1.49278
\(290\) 0 0
\(291\) 1.56562e31 2.48899
\(292\) 1.91365e30 0.291459
\(293\) 3.14320e30 0.458698 0.229349 0.973344i \(-0.426340\pi\)
0.229349 + 0.973344i \(0.426340\pi\)
\(294\) 8.02315e30 1.12203
\(295\) 0 0
\(296\) 5.23377e30 0.672463
\(297\) 7.68701e29 0.0946894
\(298\) −7.39933e30 −0.873955
\(299\) −3.33260e30 −0.377479
\(300\) 0 0
\(301\) 1.68323e30 0.175413
\(302\) 6.89831e29 0.0689693
\(303\) 1.12228e31 1.07663
\(304\) 1.94390e30 0.178959
\(305\) 0 0
\(306\) −2.06217e31 −1.74907
\(307\) 6.65869e29 0.0542200 0.0271100 0.999632i \(-0.491370\pi\)
0.0271100 + 0.999632i \(0.491370\pi\)
\(308\) −6.51839e28 −0.00509631
\(309\) 2.21593e31 1.66370
\(310\) 0 0
\(311\) −1.30158e31 −0.901497 −0.450749 0.892651i \(-0.648843\pi\)
−0.450749 + 0.892651i \(0.648843\pi\)
\(312\) 2.16151e30 0.143822
\(313\) 1.48840e31 0.951514 0.475757 0.879577i \(-0.342174\pi\)
0.475757 + 0.879577i \(0.342174\pi\)
\(314\) −6.42849e30 −0.394901
\(315\) 0 0
\(316\) 1.17132e31 0.664640
\(317\) −4.63362e30 −0.252743 −0.126372 0.991983i \(-0.540333\pi\)
−0.126372 + 0.991983i \(0.540333\pi\)
\(318\) −2.74448e31 −1.43920
\(319\) 6.60617e29 0.0333093
\(320\) 0 0
\(321\) −4.47806e30 −0.208822
\(322\) 2.29034e30 0.102731
\(323\) 2.61931e31 1.13020
\(324\) −1.35132e30 −0.0560980
\(325\) 0 0
\(326\) −2.19262e31 −0.842835
\(327\) −4.67363e31 −1.72905
\(328\) −1.49230e31 −0.531414
\(329\) −2.33761e30 −0.0801352
\(330\) 0 0
\(331\) −1.64006e31 −0.521207 −0.260604 0.965446i \(-0.583922\pi\)
−0.260604 + 0.965446i \(0.583922\pi\)
\(332\) −1.98155e31 −0.606428
\(333\) −1.01099e32 −2.97984
\(334\) −4.13928e31 −1.17515
\(335\) 0 0
\(336\) −1.48551e30 −0.0391411
\(337\) 8.69817e30 0.220828 0.110414 0.993886i \(-0.464782\pi\)
0.110414 + 0.993886i \(0.464782\pi\)
\(338\) 2.70396e31 0.661518
\(339\) −9.20512e31 −2.17037
\(340\) 0 0
\(341\) −3.72333e30 −0.0815647
\(342\) −3.75495e31 −0.793009
\(343\) 9.55290e30 0.194518
\(344\) 3.23215e31 0.634615
\(345\) 0 0
\(346\) −3.54921e30 −0.0648157
\(347\) −8.06238e31 −1.42019 −0.710093 0.704108i \(-0.751347\pi\)
−0.710093 + 0.704108i \(0.751347\pi\)
\(348\) 1.50551e31 0.255825
\(349\) −9.75945e31 −1.59995 −0.799977 0.600031i \(-0.795155\pi\)
−0.799977 + 0.600031i \(0.795155\pi\)
\(350\) 0 0
\(351\) −1.51024e31 −0.230519
\(352\) −1.25166e30 −0.0184376
\(353\) 1.35792e31 0.193060 0.0965298 0.995330i \(-0.469226\pi\)
0.0965298 + 0.995330i \(0.469226\pi\)
\(354\) −8.44503e31 −1.15894
\(355\) 0 0
\(356\) 4.69351e31 0.600307
\(357\) −2.00165e31 −0.247193
\(358\) 8.32340e31 0.992574
\(359\) 1.23431e32 1.42148 0.710742 0.703453i \(-0.248359\pi\)
0.710742 + 0.703453i \(0.248359\pi\)
\(360\) 0 0
\(361\) −4.53822e31 −0.487579
\(362\) 2.50493e31 0.259979
\(363\) −1.58040e32 −1.58466
\(364\) 1.28065e30 0.0124069
\(365\) 0 0
\(366\) 2.59807e31 0.235038
\(367\) −1.94163e31 −0.169762 −0.0848811 0.996391i \(-0.527051\pi\)
−0.0848811 + 0.996391i \(0.527051\pi\)
\(368\) 4.39791e31 0.371662
\(369\) 2.88262e32 2.35482
\(370\) 0 0
\(371\) −1.62604e31 −0.124153
\(372\) −8.48527e31 −0.626439
\(373\) −1.75681e32 −1.25420 −0.627098 0.778941i \(-0.715757\pi\)
−0.627098 + 0.778941i \(0.715757\pi\)
\(374\) −1.68656e31 −0.116441
\(375\) 0 0
\(376\) −4.48869e31 −0.289915
\(377\) −1.29789e31 −0.0810909
\(378\) 1.03792e31 0.0627358
\(379\) 2.09100e32 1.22283 0.611413 0.791312i \(-0.290601\pi\)
0.611413 + 0.791312i \(0.290601\pi\)
\(380\) 0 0
\(381\) 2.46915e32 1.35203
\(382\) −5.12157e31 −0.271401
\(383\) 1.47981e32 0.758966 0.379483 0.925199i \(-0.376102\pi\)
0.379483 + 0.925199i \(0.376102\pi\)
\(384\) −2.85247e31 −0.141606
\(385\) 0 0
\(386\) −1.63876e32 −0.762383
\(387\) −6.24342e32 −2.81213
\(388\) 1.78117e32 0.776798
\(389\) 2.19739e32 0.927978 0.463989 0.885841i \(-0.346418\pi\)
0.463989 + 0.885841i \(0.346418\pi\)
\(390\) 0 0
\(391\) 5.92598e32 2.34720
\(392\) 9.12774e31 0.350177
\(393\) −2.80275e32 −1.04155
\(394\) 4.91329e31 0.176876
\(395\) 0 0
\(396\) 2.41779e31 0.0817013
\(397\) 1.95462e32 0.640003 0.320001 0.947417i \(-0.396317\pi\)
0.320001 + 0.947417i \(0.396317\pi\)
\(398\) −1.36958e32 −0.434559
\(399\) −3.64475e31 −0.112074
\(400\) 0 0
\(401\) −4.71177e32 −1.36107 −0.680534 0.732717i \(-0.738252\pi\)
−0.680534 + 0.732717i \(0.738252\pi\)
\(402\) 1.84019e32 0.515273
\(403\) 7.31511e31 0.198567
\(404\) 1.27679e32 0.336009
\(405\) 0 0
\(406\) 8.91980e30 0.0220689
\(407\) −8.26841e31 −0.198377
\(408\) −3.84357e32 −0.894300
\(409\) −4.32845e32 −0.976767 −0.488384 0.872629i \(-0.662413\pi\)
−0.488384 + 0.872629i \(0.662413\pi\)
\(410\) 0 0
\(411\) 5.18675e32 1.10122
\(412\) 2.52101e32 0.519229
\(413\) −5.00349e31 −0.0999763
\(414\) −8.49528e32 −1.64692
\(415\) 0 0
\(416\) 2.45910e31 0.0448859
\(417\) 5.77790e32 1.02346
\(418\) −3.07101e31 −0.0527931
\(419\) 9.61858e31 0.160485 0.0802427 0.996775i \(-0.474430\pi\)
0.0802427 + 0.996775i \(0.474430\pi\)
\(420\) 0 0
\(421\) 1.59673e31 0.0251018 0.0125509 0.999921i \(-0.496005\pi\)
0.0125509 + 0.999921i \(0.496005\pi\)
\(422\) −2.83549e32 −0.432735
\(423\) 8.67063e32 1.28468
\(424\) −3.12233e32 −0.449164
\(425\) 0 0
\(426\) 2.46835e32 0.334801
\(427\) 1.53930e31 0.0202756
\(428\) −5.09458e31 −0.0651721
\(429\) −3.41480e31 −0.0424277
\(430\) 0 0
\(431\) −7.11839e32 −0.834479 −0.417240 0.908796i \(-0.637002\pi\)
−0.417240 + 0.908796i \(0.637002\pi\)
\(432\) 1.99301e32 0.226967
\(433\) 9.69753e32 1.07291 0.536454 0.843930i \(-0.319764\pi\)
0.536454 + 0.843930i \(0.319764\pi\)
\(434\) −5.02733e31 −0.0540401
\(435\) 0 0
\(436\) −5.31708e32 −0.539625
\(437\) 1.07904e33 1.06420
\(438\) −6.88977e32 −0.660357
\(439\) −6.89367e32 −0.642161 −0.321081 0.947052i \(-0.604046\pi\)
−0.321081 + 0.947052i \(0.604046\pi\)
\(440\) 0 0
\(441\) −1.76317e33 −1.55172
\(442\) 3.31353e32 0.283473
\(443\) 1.47442e33 1.22624 0.613120 0.789990i \(-0.289914\pi\)
0.613120 + 0.789990i \(0.289914\pi\)
\(444\) −1.88433e33 −1.52359
\(445\) 0 0
\(446\) −1.78998e33 −1.36824
\(447\) 2.66400e33 1.98011
\(448\) −1.69002e31 −0.0122157
\(449\) 2.43553e33 1.71204 0.856020 0.516942i \(-0.172930\pi\)
0.856020 + 0.516942i \(0.172930\pi\)
\(450\) 0 0
\(451\) 2.35756e32 0.156768
\(452\) −1.04724e33 −0.677357
\(453\) −2.48361e32 −0.156263
\(454\) 1.84982e32 0.113222
\(455\) 0 0
\(456\) −6.99866e32 −0.405466
\(457\) 1.75801e33 0.990990 0.495495 0.868611i \(-0.334987\pi\)
0.495495 + 0.868611i \(0.334987\pi\)
\(458\) 1.62478e33 0.891201
\(459\) 2.68549e33 1.43339
\(460\) 0 0
\(461\) 2.02960e33 1.02600 0.513002 0.858387i \(-0.328533\pi\)
0.513002 + 0.858387i \(0.328533\pi\)
\(462\) 2.34683e31 0.0115467
\(463\) −5.63047e32 −0.269639 −0.134819 0.990870i \(-0.543045\pi\)
−0.134819 + 0.990870i \(0.543045\pi\)
\(464\) 1.71278e32 0.0798413
\(465\) 0 0
\(466\) 1.03510e33 0.457256
\(467\) −3.35205e33 −1.44162 −0.720808 0.693134i \(-0.756229\pi\)
−0.720808 + 0.693134i \(0.756229\pi\)
\(468\) −4.75016e32 −0.198900
\(469\) 1.09027e32 0.0444502
\(470\) 0 0
\(471\) 2.31447e33 0.894725
\(472\) −9.60771e32 −0.361697
\(473\) −5.10621e32 −0.187212
\(474\) −4.21713e33 −1.50587
\(475\) 0 0
\(476\) −2.27723e32 −0.0771471
\(477\) 6.03128e33 1.99035
\(478\) −3.17739e33 −1.02146
\(479\) 3.28090e33 1.02754 0.513770 0.857928i \(-0.328248\pi\)
0.513770 + 0.857928i \(0.328248\pi\)
\(480\) 0 0
\(481\) 1.62447e33 0.482945
\(482\) 8.13063e31 0.0235525
\(483\) −8.24595e32 −0.232756
\(484\) −1.79799e33 −0.494561
\(485\) 0 0
\(486\) 2.94384e33 0.769061
\(487\) 2.83575e33 0.722032 0.361016 0.932560i \(-0.382430\pi\)
0.361016 + 0.932560i \(0.382430\pi\)
\(488\) 2.95576e32 0.0733536
\(489\) 7.89413e33 1.90960
\(490\) 0 0
\(491\) −6.18264e33 −1.42120 −0.710600 0.703596i \(-0.751576\pi\)
−0.710600 + 0.703596i \(0.751576\pi\)
\(492\) 5.37276e33 1.20402
\(493\) 2.30790e33 0.504231
\(494\) 6.03351e32 0.128524
\(495\) 0 0
\(496\) −9.65349e32 −0.195507
\(497\) 1.46244e32 0.0288817
\(498\) 7.13422e33 1.37398
\(499\) −2.73531e33 −0.513748 −0.256874 0.966445i \(-0.582693\pi\)
−0.256874 + 0.966445i \(0.582693\pi\)
\(500\) 0 0
\(501\) 1.49028e34 2.66253
\(502\) 3.72738e33 0.649542
\(503\) −1.17377e33 −0.199519 −0.0997597 0.995012i \(-0.531807\pi\)
−0.0997597 + 0.995012i \(0.531807\pi\)
\(504\) 3.26456e32 0.0541306
\(505\) 0 0
\(506\) −6.94790e32 −0.109641
\(507\) −9.73514e33 −1.49880
\(508\) 2.80909e33 0.421959
\(509\) −4.20998e33 −0.617031 −0.308516 0.951219i \(-0.599832\pi\)
−0.308516 + 0.951219i \(0.599832\pi\)
\(510\) 0 0
\(511\) −4.08203e32 −0.0569659
\(512\) −3.24519e32 −0.0441942
\(513\) 4.88994e33 0.649885
\(514\) −4.63910e33 −0.601720
\(515\) 0 0
\(516\) −1.16368e34 −1.43784
\(517\) 7.09131e32 0.0855253
\(518\) −1.11642e33 −0.131433
\(519\) 1.27783e33 0.146852
\(520\) 0 0
\(521\) −1.42619e34 −1.56208 −0.781042 0.624478i \(-0.785312\pi\)
−0.781042 + 0.624478i \(0.785312\pi\)
\(522\) −3.30852e33 −0.353796
\(523\) −1.07993e34 −1.12752 −0.563760 0.825939i \(-0.690646\pi\)
−0.563760 + 0.825939i \(0.690646\pi\)
\(524\) −3.18863e33 −0.325059
\(525\) 0 0
\(526\) −3.21096e33 −0.312114
\(527\) −1.30076e34 −1.23471
\(528\) 4.50639e32 0.0417738
\(529\) 1.33667e34 1.21012
\(530\) 0 0
\(531\) 1.85589e34 1.60276
\(532\) −4.14654e32 −0.0349777
\(533\) −4.63183e33 −0.381647
\(534\) −1.68982e34 −1.36011
\(535\) 0 0
\(536\) 2.09354e33 0.160813
\(537\) −2.99669e34 −2.24887
\(538\) −9.07518e33 −0.665391
\(539\) −1.44202e33 −0.103303
\(540\) 0 0
\(541\) −9.36097e32 −0.0640258 −0.0320129 0.999487i \(-0.510192\pi\)
−0.0320129 + 0.999487i \(0.510192\pi\)
\(542\) 1.37175e33 0.0916819
\(543\) −9.01856e33 −0.589033
\(544\) −4.37274e33 −0.279105
\(545\) 0 0
\(546\) −4.61074e32 −0.0281101
\(547\) 1.05246e34 0.627137 0.313568 0.949566i \(-0.398475\pi\)
0.313568 + 0.949566i \(0.398475\pi\)
\(548\) 5.90084e33 0.343682
\(549\) −5.70954e33 −0.325047
\(550\) 0 0
\(551\) 4.20238e33 0.228613
\(552\) −1.58339e34 −0.842071
\(553\) −2.49855e33 −0.129904
\(554\) −1.54952e34 −0.787635
\(555\) 0 0
\(556\) 6.57337e33 0.319414
\(557\) 9.65307e33 0.458644 0.229322 0.973351i \(-0.426349\pi\)
0.229322 + 0.973351i \(0.426349\pi\)
\(558\) 1.86473e34 0.866340
\(559\) 1.00320e34 0.455764
\(560\) 0 0
\(561\) 6.07215e33 0.263819
\(562\) 8.43969e33 0.358610
\(563\) −4.08658e34 −1.69826 −0.849132 0.528180i \(-0.822875\pi\)
−0.849132 + 0.528180i \(0.822875\pi\)
\(564\) 1.61607e34 0.656859
\(565\) 0 0
\(566\) −5.76452e33 −0.224160
\(567\) 2.88252e32 0.0109644
\(568\) 2.80818e33 0.104489
\(569\) 8.49519e33 0.309221 0.154611 0.987975i \(-0.450588\pi\)
0.154611 + 0.987975i \(0.450588\pi\)
\(570\) 0 0
\(571\) −2.91693e34 −1.01619 −0.508095 0.861301i \(-0.669650\pi\)
−0.508095 + 0.861301i \(0.669650\pi\)
\(572\) −3.88494e32 −0.0132414
\(573\) 1.84393e34 0.614910
\(574\) 3.18324e33 0.103865
\(575\) 0 0
\(576\) 6.26861e33 0.195835
\(577\) −5.12659e33 −0.156722 −0.0783611 0.996925i \(-0.524969\pi\)
−0.0783611 + 0.996925i \(0.524969\pi\)
\(578\) −3.52842e34 −1.05556
\(579\) 5.90005e34 1.72732
\(580\) 0 0
\(581\) 4.22686e33 0.118527
\(582\) −6.41278e34 −1.75998
\(583\) 4.93271e33 0.132504
\(584\) −7.83833e33 −0.206093
\(585\) 0 0
\(586\) −1.28746e34 −0.324349
\(587\) −6.37641e34 −1.57253 −0.786266 0.617888i \(-0.787989\pi\)
−0.786266 + 0.617888i \(0.787989\pi\)
\(588\) −3.28628e34 −0.793393
\(589\) −2.36852e34 −0.559805
\(590\) 0 0
\(591\) −1.76894e34 −0.400747
\(592\) −2.14375e34 −0.475503
\(593\) 7.56532e34 1.64302 0.821511 0.570192i \(-0.193131\pi\)
0.821511 + 0.570192i \(0.193131\pi\)
\(594\) −3.14860e33 −0.0669556
\(595\) 0 0
\(596\) 3.03076e34 0.617979
\(597\) 4.93093e34 0.984577
\(598\) 1.36503e34 0.266918
\(599\) 6.58590e34 1.26119 0.630594 0.776113i \(-0.282811\pi\)
0.630594 + 0.776113i \(0.282811\pi\)
\(600\) 0 0
\(601\) −3.63692e34 −0.668041 −0.334020 0.942566i \(-0.608405\pi\)
−0.334020 + 0.942566i \(0.608405\pi\)
\(602\) −6.89452e33 −0.124036
\(603\) −4.04401e34 −0.712601
\(604\) −2.82555e33 −0.0487687
\(605\) 0 0
\(606\) −4.59684e34 −0.761293
\(607\) −1.13535e35 −1.84193 −0.920963 0.389650i \(-0.872596\pi\)
−0.920963 + 0.389650i \(0.872596\pi\)
\(608\) −7.96220e33 −0.126543
\(609\) −3.21142e33 −0.0500012
\(610\) 0 0
\(611\) −1.39321e34 −0.208210
\(612\) 8.44666e34 1.23678
\(613\) 8.49768e34 1.21911 0.609557 0.792742i \(-0.291347\pi\)
0.609557 + 0.792742i \(0.291347\pi\)
\(614\) −2.72740e33 −0.0383393
\(615\) 0 0
\(616\) 2.66993e32 0.00360364
\(617\) 1.49456e35 1.97674 0.988368 0.152079i \(-0.0485967\pi\)
0.988368 + 0.152079i \(0.0485967\pi\)
\(618\) −9.07644e34 −1.17641
\(619\) −5.75885e34 −0.731481 −0.365740 0.930717i \(-0.619184\pi\)
−0.365740 + 0.930717i \(0.619184\pi\)
\(620\) 0 0
\(621\) 1.10631e35 1.34968
\(622\) 5.33125e34 0.637455
\(623\) −1.00118e34 −0.117330
\(624\) −8.85356e33 −0.101698
\(625\) 0 0
\(626\) −6.09648e34 −0.672822
\(627\) 1.10566e34 0.119613
\(628\) 2.63311e34 0.279237
\(629\) −2.88861e35 −3.00300
\(630\) 0 0
\(631\) 3.54832e34 0.354532 0.177266 0.984163i \(-0.443275\pi\)
0.177266 + 0.984163i \(0.443275\pi\)
\(632\) −4.79772e34 −0.469972
\(633\) 1.02087e35 0.980445
\(634\) 1.89793e34 0.178716
\(635\) 0 0
\(636\) 1.12414e35 1.01767
\(637\) 2.83309e34 0.251488
\(638\) −2.70589e33 −0.0235533
\(639\) −5.42446e34 −0.463016
\(640\) 0 0
\(641\) 1.20958e35 0.992909 0.496454 0.868063i \(-0.334635\pi\)
0.496454 + 0.868063i \(0.334635\pi\)
\(642\) 1.83421e34 0.147660
\(643\) 4.32278e34 0.341292 0.170646 0.985332i \(-0.445415\pi\)
0.170646 + 0.985332i \(0.445415\pi\)
\(644\) −9.38121e33 −0.0726416
\(645\) 0 0
\(646\) −1.07287e35 −0.799173
\(647\) 4.93254e34 0.360385 0.180193 0.983631i \(-0.442328\pi\)
0.180193 + 0.983631i \(0.442328\pi\)
\(648\) 5.53503e33 0.0396673
\(649\) 1.51784e34 0.106701
\(650\) 0 0
\(651\) 1.81000e34 0.122438
\(652\) 8.98095e34 0.595974
\(653\) −2.81796e35 −1.83451 −0.917255 0.398301i \(-0.869600\pi\)
−0.917255 + 0.398301i \(0.869600\pi\)
\(654\) 1.91432e35 1.22262
\(655\) 0 0
\(656\) 6.11245e34 0.375766
\(657\) 1.51410e35 0.913246
\(658\) 9.57486e33 0.0566642
\(659\) −1.27642e35 −0.741185 −0.370592 0.928796i \(-0.620845\pi\)
−0.370592 + 0.928796i \(0.620845\pi\)
\(660\) 0 0
\(661\) −3.04753e35 −1.70384 −0.851921 0.523671i \(-0.824562\pi\)
−0.851921 + 0.523671i \(0.824562\pi\)
\(662\) 6.71770e34 0.368549
\(663\) −1.19298e35 −0.642263
\(664\) 8.11643e34 0.428809
\(665\) 0 0
\(666\) 4.14101e35 2.10707
\(667\) 9.50755e34 0.474783
\(668\) 1.69545e35 0.830959
\(669\) 6.44450e35 3.10001
\(670\) 0 0
\(671\) −4.66957e33 −0.0216394
\(672\) 6.08463e33 0.0276770
\(673\) 2.27652e35 1.01644 0.508221 0.861227i \(-0.330303\pi\)
0.508221 + 0.861227i \(0.330303\pi\)
\(674\) −3.56277e34 −0.156149
\(675\) 0 0
\(676\) −1.10754e35 −0.467764
\(677\) 1.12736e35 0.467416 0.233708 0.972307i \(-0.424914\pi\)
0.233708 + 0.972307i \(0.424914\pi\)
\(678\) 3.77042e35 1.53468
\(679\) −3.79942e34 −0.151826
\(680\) 0 0
\(681\) −6.65994e34 −0.256526
\(682\) 1.52508e34 0.0576749
\(683\) 1.62407e35 0.603039 0.301520 0.953460i \(-0.402506\pi\)
0.301520 + 0.953460i \(0.402506\pi\)
\(684\) 1.53803e35 0.560742
\(685\) 0 0
\(686\) −3.91287e34 −0.137545
\(687\) −5.84973e35 −2.01919
\(688\) −1.32389e35 −0.448741
\(689\) −9.69114e34 −0.322578
\(690\) 0 0
\(691\) −2.17749e35 −0.699006 −0.349503 0.936935i \(-0.613649\pi\)
−0.349503 + 0.936935i \(0.613649\pi\)
\(692\) 1.45376e34 0.0458316
\(693\) −5.15741e33 −0.0159686
\(694\) 3.30235e35 1.00422
\(695\) 0 0
\(696\) −6.16657e34 −0.180896
\(697\) 8.23625e35 2.37312
\(698\) 3.99747e35 1.13134
\(699\) −3.72671e35 −1.03600
\(700\) 0 0
\(701\) −1.31197e35 −0.351923 −0.175961 0.984397i \(-0.556303\pi\)
−0.175961 + 0.984397i \(0.556303\pi\)
\(702\) 6.18595e34 0.163002
\(703\) −5.25979e35 −1.36153
\(704\) 5.12681e33 0.0130373
\(705\) 0 0
\(706\) −5.56205e34 −0.136514
\(707\) −2.72353e34 −0.0656733
\(708\) 3.45908e35 0.819493
\(709\) 1.07011e35 0.249088 0.124544 0.992214i \(-0.460253\pi\)
0.124544 + 0.992214i \(0.460253\pi\)
\(710\) 0 0
\(711\) 9.26759e35 2.08256
\(712\) −1.92246e35 −0.424481
\(713\) −5.35859e35 −1.16260
\(714\) 8.19876e34 0.174792
\(715\) 0 0
\(716\) −3.40927e35 −0.701856
\(717\) 1.14396e36 2.31431
\(718\) −5.05572e35 −1.00514
\(719\) −7.33489e35 −1.43312 −0.716560 0.697526i \(-0.754284\pi\)
−0.716560 + 0.697526i \(0.754284\pi\)
\(720\) 0 0
\(721\) −5.37759e34 −0.101484
\(722\) 1.85885e35 0.344771
\(723\) −2.92729e34 −0.0533626
\(724\) −1.02602e35 −0.183833
\(725\) 0 0
\(726\) 6.47334e35 1.12052
\(727\) −1.00868e36 −1.71622 −0.858111 0.513465i \(-0.828362\pi\)
−0.858111 + 0.513465i \(0.828362\pi\)
\(728\) −5.24553e33 −0.00877298
\(729\) −9.91632e35 −1.63026
\(730\) 0 0
\(731\) −1.78388e36 −2.83399
\(732\) −1.06417e35 −0.166197
\(733\) 1.19811e36 1.83948 0.919741 0.392525i \(-0.128398\pi\)
0.919741 + 0.392525i \(0.128398\pi\)
\(734\) 7.95292e34 0.120040
\(735\) 0 0
\(736\) −1.80138e35 −0.262805
\(737\) −3.30741e34 −0.0474400
\(738\) −1.18072e36 −1.66511
\(739\) −7.47298e35 −1.03619 −0.518094 0.855324i \(-0.673358\pi\)
−0.518094 + 0.855324i \(0.673358\pi\)
\(740\) 0 0
\(741\) −2.17226e35 −0.291195
\(742\) 6.66026e34 0.0877895
\(743\) 1.09617e36 1.42075 0.710376 0.703822i \(-0.248525\pi\)
0.710376 + 0.703822i \(0.248525\pi\)
\(744\) 3.47557e35 0.442960
\(745\) 0 0
\(746\) 7.19590e35 0.886850
\(747\) −1.56782e36 −1.90015
\(748\) 6.90814e34 0.0823362
\(749\) 1.08673e34 0.0127379
\(750\) 0 0
\(751\) −4.75595e34 −0.0539186 −0.0269593 0.999637i \(-0.508582\pi\)
−0.0269593 + 0.999637i \(0.508582\pi\)
\(752\) 1.83857e35 0.205001
\(753\) −1.34198e36 −1.47166
\(754\) 5.31617e34 0.0573399
\(755\) 0 0
\(756\) −4.25131e34 −0.0443609
\(757\) −1.35534e36 −1.39107 −0.695537 0.718490i \(-0.744834\pi\)
−0.695537 + 0.718490i \(0.744834\pi\)
\(758\) −8.56475e35 −0.864669
\(759\) 2.50147e35 0.248412
\(760\) 0 0
\(761\) −4.04146e35 −0.388356 −0.194178 0.980966i \(-0.562204\pi\)
−0.194178 + 0.980966i \(0.562204\pi\)
\(762\) −1.01136e36 −0.956028
\(763\) 1.13419e35 0.105470
\(764\) 2.09779e35 0.191909
\(765\) 0 0
\(766\) −6.06132e35 −0.536670
\(767\) −2.98206e35 −0.259761
\(768\) 1.16837e35 0.100130
\(769\) −1.59517e36 −1.34502 −0.672509 0.740089i \(-0.734783\pi\)
−0.672509 + 0.740089i \(0.734783\pi\)
\(770\) 0 0
\(771\) 1.67023e36 1.36331
\(772\) 6.71234e35 0.539086
\(773\) 5.09562e35 0.402674 0.201337 0.979522i \(-0.435471\pi\)
0.201337 + 0.979522i \(0.435471\pi\)
\(774\) 2.55730e36 1.98848
\(775\) 0 0
\(776\) −7.29566e35 −0.549279
\(777\) 4.01947e35 0.297787
\(778\) −9.00052e35 −0.656180
\(779\) 1.49972e36 1.07595
\(780\) 0 0
\(781\) −4.43642e34 −0.0308244
\(782\) −2.42728e36 −1.65972
\(783\) 4.30856e35 0.289941
\(784\) −3.73872e35 −0.247612
\(785\) 0 0
\(786\) 1.14801e36 0.736484
\(787\) −1.58597e36 −1.00141 −0.500703 0.865619i \(-0.666925\pi\)
−0.500703 + 0.865619i \(0.666925\pi\)
\(788\) −2.01248e35 −0.125070
\(789\) 1.15605e36 0.707154
\(790\) 0 0
\(791\) 2.23389e35 0.132390
\(792\) −9.90327e34 −0.0577715
\(793\) 9.17416e34 0.0526807
\(794\) −8.00614e35 −0.452550
\(795\) 0 0
\(796\) 5.60980e35 0.307280
\(797\) 1.54676e36 0.834053 0.417027 0.908894i \(-0.363072\pi\)
0.417027 + 0.908894i \(0.363072\pi\)
\(798\) 1.49289e35 0.0792486
\(799\) 2.47738e36 1.29467
\(800\) 0 0
\(801\) 3.71355e36 1.88098
\(802\) 1.92994e36 0.962420
\(803\) 1.23831e35 0.0607976
\(804\) −7.53742e35 −0.364353
\(805\) 0 0
\(806\) −2.99627e35 −0.140408
\(807\) 3.26736e36 1.50757
\(808\) −5.22972e35 −0.237594
\(809\) 6.21020e35 0.277811 0.138905 0.990306i \(-0.455642\pi\)
0.138905 + 0.990306i \(0.455642\pi\)
\(810\) 0 0
\(811\) 4.69897e35 0.203818 0.101909 0.994794i \(-0.467505\pi\)
0.101909 + 0.994794i \(0.467505\pi\)
\(812\) −3.65355e34 −0.0156050
\(813\) −4.93874e35 −0.207723
\(814\) 3.38674e35 0.140274
\(815\) 0 0
\(816\) 1.57433e36 0.632365
\(817\) −3.24821e36 −1.28490
\(818\) 1.77293e36 0.690679
\(819\) 1.01326e35 0.0388752
\(820\) 0 0
\(821\) −3.77384e36 −1.40441 −0.702205 0.711975i \(-0.747801\pi\)
−0.702205 + 0.711975i \(0.747801\pi\)
\(822\) −2.12449e36 −0.778677
\(823\) −1.31964e36 −0.476385 −0.238192 0.971218i \(-0.576555\pi\)
−0.238192 + 0.971218i \(0.576555\pi\)
\(824\) −1.03260e36 −0.367150
\(825\) 0 0
\(826\) 2.04943e35 0.0706939
\(827\) −2.35273e36 −0.799380 −0.399690 0.916650i \(-0.630882\pi\)
−0.399690 + 0.916650i \(0.630882\pi\)
\(828\) 3.47966e36 1.16455
\(829\) −1.44960e36 −0.477878 −0.238939 0.971035i \(-0.576800\pi\)
−0.238939 + 0.971035i \(0.576800\pi\)
\(830\) 0 0
\(831\) 5.57877e36 1.78454
\(832\) −1.00725e35 −0.0317391
\(833\) −5.03776e36 −1.56378
\(834\) −2.36663e36 −0.723692
\(835\) 0 0
\(836\) 1.25788e35 0.0373304
\(837\) −2.42837e36 −0.709980
\(838\) −3.93977e35 −0.113480
\(839\) −6.60849e35 −0.187533 −0.0937665 0.995594i \(-0.529891\pi\)
−0.0937665 + 0.995594i \(0.529891\pi\)
\(840\) 0 0
\(841\) −3.26009e36 −0.898006
\(842\) −6.54019e34 −0.0177496
\(843\) −3.03856e36 −0.812500
\(844\) 1.16142e36 0.305990
\(845\) 0 0
\(846\) −3.55149e36 −0.908408
\(847\) 3.83530e35 0.0966623
\(848\) 1.27890e36 0.317607
\(849\) 2.07541e36 0.507877
\(850\) 0 0
\(851\) −1.18998e37 −2.82762
\(852\) −1.01104e36 −0.236740
\(853\) −5.07169e35 −0.117028 −0.0585140 0.998287i \(-0.518636\pi\)
−0.0585140 + 0.998287i \(0.518636\pi\)
\(854\) −6.30496e34 −0.0143370
\(855\) 0 0
\(856\) 2.08674e35 0.0460836
\(857\) 2.45792e36 0.534943 0.267471 0.963566i \(-0.413812\pi\)
0.267471 + 0.963566i \(0.413812\pi\)
\(858\) 1.39870e35 0.0300009
\(859\) −7.15238e36 −1.51194 −0.755972 0.654603i \(-0.772836\pi\)
−0.755972 + 0.654603i \(0.772836\pi\)
\(860\) 0 0
\(861\) −1.14607e36 −0.235327
\(862\) 2.91569e36 0.590066
\(863\) −2.16205e36 −0.431252 −0.215626 0.976476i \(-0.569179\pi\)
−0.215626 + 0.976476i \(0.569179\pi\)
\(864\) −8.16337e35 −0.160490
\(865\) 0 0
\(866\) −3.97211e36 −0.758660
\(867\) 1.27034e37 2.39157
\(868\) 2.05919e35 0.0382121
\(869\) 7.57953e35 0.138642
\(870\) 0 0
\(871\) 6.49797e35 0.115492
\(872\) 2.17787e36 0.381572
\(873\) 1.40928e37 2.43399
\(874\) −4.41977e36 −0.752500
\(875\) 0 0
\(876\) 2.82205e36 0.466943
\(877\) −6.84625e36 −1.11676 −0.558378 0.829587i \(-0.688576\pi\)
−0.558378 + 0.829587i \(0.688576\pi\)
\(878\) 2.82365e36 0.454077
\(879\) 4.63526e36 0.734874
\(880\) 0 0
\(881\) 2.59663e36 0.400139 0.200069 0.979782i \(-0.435883\pi\)
0.200069 + 0.979782i \(0.435883\pi\)
\(882\) 7.22196e36 1.09723
\(883\) −1.13962e36 −0.170707 −0.0853533 0.996351i \(-0.527202\pi\)
−0.0853533 + 0.996351i \(0.527202\pi\)
\(884\) −1.35722e36 −0.200446
\(885\) 0 0
\(886\) −6.03924e36 −0.867083
\(887\) 2.38557e36 0.337713 0.168856 0.985641i \(-0.445993\pi\)
0.168856 + 0.985641i \(0.445993\pi\)
\(888\) 7.71820e36 1.07734
\(889\) −5.99211e35 −0.0824722
\(890\) 0 0
\(891\) −8.74434e34 −0.0117019
\(892\) 7.33175e36 0.967490
\(893\) 4.51100e36 0.586988
\(894\) −1.09117e37 −1.40015
\(895\) 0 0
\(896\) 6.92234e34 0.00863779
\(897\) −4.91456e36 −0.604754
\(898\) −9.97593e36 −1.21060
\(899\) −2.08693e36 −0.249753
\(900\) 0 0
\(901\) 1.72327e37 2.00582
\(902\) −9.65657e35 −0.110851
\(903\) 2.48225e36 0.281027
\(904\) 4.28951e36 0.478964
\(905\) 0 0
\(906\) 1.01729e36 0.110495
\(907\) 1.67063e37 1.78974 0.894870 0.446326i \(-0.147268\pi\)
0.894870 + 0.446326i \(0.147268\pi\)
\(908\) −7.57685e35 −0.0800600
\(909\) 1.01021e37 1.05284
\(910\) 0 0
\(911\) 1.13215e37 1.14795 0.573976 0.818872i \(-0.305400\pi\)
0.573976 + 0.818872i \(0.305400\pi\)
\(912\) 2.86665e36 0.286708
\(913\) −1.28225e36 −0.126499
\(914\) −7.20082e36 −0.700736
\(915\) 0 0
\(916\) −6.65510e36 −0.630175
\(917\) 6.80169e35 0.0635331
\(918\) −1.09998e37 −1.01356
\(919\) 1.76508e37 1.60443 0.802215 0.597035i \(-0.203655\pi\)
0.802215 + 0.597035i \(0.203655\pi\)
\(920\) 0 0
\(921\) 9.81953e35 0.0868651
\(922\) −8.31325e36 −0.725495
\(923\) 8.71609e35 0.0750413
\(924\) −9.61262e34 −0.00816473
\(925\) 0 0
\(926\) 2.30624e36 0.190663
\(927\) 1.99465e37 1.62693
\(928\) −7.01556e35 −0.0564563
\(929\) 2.88984e36 0.229445 0.114722 0.993398i \(-0.463402\pi\)
0.114722 + 0.993398i \(0.463402\pi\)
\(930\) 0 0
\(931\) −9.17311e36 −0.708999
\(932\) −4.23978e36 −0.323329
\(933\) −1.91942e37 −1.44428
\(934\) 1.37300e37 1.01938
\(935\) 0 0
\(936\) 1.94566e36 0.140644
\(937\) −1.29200e37 −0.921545 −0.460772 0.887518i \(-0.652428\pi\)
−0.460772 + 0.887518i \(0.652428\pi\)
\(938\) −4.46575e35 −0.0314310
\(939\) 2.19493e37 1.52441
\(940\) 0 0
\(941\) 2.06414e37 1.39595 0.697973 0.716124i \(-0.254086\pi\)
0.697973 + 0.716124i \(0.254086\pi\)
\(942\) −9.48005e36 −0.632666
\(943\) 3.39299e37 2.23453
\(944\) 3.93532e36 0.255758
\(945\) 0 0
\(946\) 2.09150e36 0.132379
\(947\) −2.09426e36 −0.130815 −0.0654073 0.997859i \(-0.520835\pi\)
−0.0654073 + 0.997859i \(0.520835\pi\)
\(948\) 1.72734e37 1.06481
\(949\) −2.43288e36 −0.148010
\(950\) 0 0
\(951\) −6.83316e36 −0.404916
\(952\) 9.32753e35 0.0545513
\(953\) 1.70726e37 0.985461 0.492731 0.870182i \(-0.335999\pi\)
0.492731 + 0.870182i \(0.335999\pi\)
\(954\) −2.47041e37 −1.40739
\(955\) 0 0
\(956\) 1.30146e37 0.722281
\(957\) 9.74207e35 0.0533644
\(958\) −1.34386e37 −0.726581
\(959\) −1.25871e36 −0.0671729
\(960\) 0 0
\(961\) −7.47058e36 −0.388429
\(962\) −6.65382e36 −0.341494
\(963\) −4.03088e36 −0.204207
\(964\) −3.33031e35 −0.0166541
\(965\) 0 0
\(966\) 3.37754e36 0.164584
\(967\) 1.41585e37 0.681063 0.340532 0.940233i \(-0.389393\pi\)
0.340532 + 0.940233i \(0.389393\pi\)
\(968\) 7.36456e36 0.349707
\(969\) 3.86268e37 1.81068
\(970\) 0 0
\(971\) −1.45645e37 −0.665357 −0.332678 0.943040i \(-0.607952\pi\)
−0.332678 + 0.943040i \(0.607952\pi\)
\(972\) −1.20580e37 −0.543808
\(973\) −1.40217e36 −0.0624296
\(974\) −1.16153e37 −0.510554
\(975\) 0 0
\(976\) −1.21068e36 −0.0518688
\(977\) −2.61043e36 −0.110415 −0.0552077 0.998475i \(-0.517582\pi\)
−0.0552077 + 0.998475i \(0.517582\pi\)
\(978\) −3.23343e37 −1.35029
\(979\) 3.03714e36 0.125222
\(980\) 0 0
\(981\) −4.20692e37 −1.69084
\(982\) 2.53241e37 1.00494
\(983\) −4.36771e37 −1.71133 −0.855667 0.517527i \(-0.826853\pi\)
−0.855667 + 0.517527i \(0.826853\pi\)
\(984\) −2.20068e37 −0.851370
\(985\) 0 0
\(986\) −9.45314e36 −0.356545
\(987\) −3.44726e36 −0.128384
\(988\) −2.47133e36 −0.0908800
\(989\) −7.34881e37 −2.66848
\(990\) 0 0
\(991\) 5.30135e37 1.87701 0.938503 0.345271i \(-0.112213\pi\)
0.938503 + 0.345271i \(0.112213\pi\)
\(992\) 3.95407e36 0.138245
\(993\) −2.41859e37 −0.835019
\(994\) −5.99015e35 −0.0204225
\(995\) 0 0
\(996\) −2.92218e37 −0.971549
\(997\) −4.76523e37 −1.56457 −0.782284 0.622921i \(-0.785946\pi\)
−0.782284 + 0.622921i \(0.785946\pi\)
\(998\) 1.12038e37 0.363275
\(999\) −5.39268e37 −1.72678
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.k.1.6 6
5.2 odd 4 10.26.b.a.9.1 12
5.3 odd 4 10.26.b.a.9.12 yes 12
5.4 even 2 50.26.a.l.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.26.b.a.9.1 12 5.2 odd 4
10.26.b.a.9.12 yes 12 5.3 odd 4
50.26.a.k.1.6 6 1.1 even 1 trivial
50.26.a.l.1.1 6 5.4 even 2