Properties

Label 50.18.a.g.1.2
Level $50$
Weight $18$
Character 50.1
Self dual yes
Analytic conductor $91.611$
Analytic rank $0$
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] = [50,18,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, 18, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("50.1");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 18 \)
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: \(91.6110436723\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{36061}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 9015 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{5}\cdot 5 \)
Twist minimal: no (minimal twist has level 10)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-94.4487\) 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)
 
\(f(q)\) \(=\) \(q+256.000 q^{2} +18345.8 q^{3} +65536.0 q^{4} +4.69652e6 q^{6} -1.60786e7 q^{7} +1.67772e7 q^{8} +2.07428e8 q^{9} +7.88217e8 q^{11} +1.20231e9 q^{12} +2.87409e9 q^{13} -4.11612e9 q^{14} +4.29497e9 q^{16} -2.01570e10 q^{17} +5.31015e10 q^{18} -2.28657e10 q^{19} -2.94975e11 q^{21} +2.01784e11 q^{22} +6.65718e11 q^{23} +3.07791e11 q^{24} +7.35766e11 q^{26} +1.43625e12 q^{27} -1.05373e12 q^{28} -2.42182e12 q^{29} +8.75874e12 q^{31} +1.09951e12 q^{32} +1.44605e13 q^{33} -5.16020e12 q^{34} +1.35940e13 q^{36} -2.75298e13 q^{37} -5.85361e12 q^{38} +5.27274e13 q^{39} +3.91759e13 q^{41} -7.55135e13 q^{42} +1.29981e14 q^{43} +5.16566e13 q^{44} +1.70424e14 q^{46} +3.00620e14 q^{47} +7.87946e13 q^{48} +2.58908e13 q^{49} -3.69796e14 q^{51} +1.88356e14 q^{52} -1.01926e14 q^{53} +3.67679e14 q^{54} -2.69754e14 q^{56} -4.19488e14 q^{57} -6.19987e14 q^{58} -9.67154e14 q^{59} -1.14011e15 q^{61} +2.24224e15 q^{62} -3.33515e15 q^{63} +2.81475e14 q^{64} +3.70188e15 q^{66} +4.04577e15 q^{67} -1.32101e15 q^{68} +1.22131e16 q^{69} -7.35604e15 q^{71} +3.48006e15 q^{72} -6.99946e14 q^{73} -7.04762e15 q^{74} -1.49852e15 q^{76} -1.26734e16 q^{77} +1.34982e16 q^{78} +1.08787e16 q^{79} -4.38165e14 q^{81} +1.00290e16 q^{82} +1.23277e16 q^{83} -1.93315e16 q^{84} +3.32752e16 q^{86} -4.44303e16 q^{87} +1.32241e16 q^{88} -2.21332e16 q^{89} -4.62113e16 q^{91} +4.36285e16 q^{92} +1.60686e17 q^{93} +7.69588e16 q^{94} +2.01714e16 q^{96} -9.41354e16 q^{97} +6.62805e15 q^{98} +1.63498e17 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 512 q^{2} + 6308 q^{3} + 131072 q^{4} + 1614848 q^{6} - 6543844 q^{7} + 33554432 q^{8} + 223195906 q^{9} + 1189408704 q^{11} + 413401088 q^{12} + 2017919228 q^{13} - 1675224064 q^{14} + 8589934592 q^{16}+ \cdots + 16\!\cdots\!12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 256.000 0.707107
\(3\) 18345.8 1.61438 0.807190 0.590292i \(-0.200987\pi\)
0.807190 + 0.590292i \(0.200987\pi\)
\(4\) 65536.0 0.500000
\(5\) 0 0
\(6\) 4.69652e6 1.14154
\(7\) −1.60786e7 −1.05418 −0.527090 0.849809i \(-0.676717\pi\)
−0.527090 + 0.849809i \(0.676717\pi\)
\(8\) 1.67772e7 0.353553
\(9\) 2.07428e8 1.60622
\(10\) 0 0
\(11\) 7.88217e8 1.10868 0.554342 0.832289i \(-0.312970\pi\)
0.554342 + 0.832289i \(0.312970\pi\)
\(12\) 1.20231e9 0.807190
\(13\) 2.87409e9 0.977195 0.488597 0.872509i \(-0.337509\pi\)
0.488597 + 0.872509i \(0.337509\pi\)
\(14\) −4.11612e9 −0.745418
\(15\) 0 0
\(16\) 4.29497e9 0.250000
\(17\) −2.01570e10 −0.700826 −0.350413 0.936595i \(-0.613959\pi\)
−0.350413 + 0.936595i \(0.613959\pi\)
\(18\) 5.31015e10 1.13577
\(19\) −2.28657e10 −0.308872 −0.154436 0.988003i \(-0.549356\pi\)
−0.154436 + 0.988003i \(0.549356\pi\)
\(20\) 0 0
\(21\) −2.94975e11 −1.70185
\(22\) 2.01784e11 0.783959
\(23\) 6.65718e11 1.77257 0.886285 0.463140i \(-0.153277\pi\)
0.886285 + 0.463140i \(0.153277\pi\)
\(24\) 3.07791e11 0.570769
\(25\) 0 0
\(26\) 7.35766e11 0.690981
\(27\) 1.43625e12 0.978672
\(28\) −1.05373e12 −0.527090
\(29\) −2.42182e12 −0.899000 −0.449500 0.893280i \(-0.648398\pi\)
−0.449500 + 0.893280i \(0.648398\pi\)
\(30\) 0 0
\(31\) 8.75874e12 1.84445 0.922226 0.386651i \(-0.126368\pi\)
0.922226 + 0.386651i \(0.126368\pi\)
\(32\) 1.09951e12 0.176777
\(33\) 1.44605e13 1.78984
\(34\) −5.16020e12 −0.495559
\(35\) 0 0
\(36\) 1.35940e13 0.803111
\(37\) −2.75298e13 −1.28851 −0.644255 0.764810i \(-0.722833\pi\)
−0.644255 + 0.764810i \(0.722833\pi\)
\(38\) −5.85361e12 −0.218405
\(39\) 5.27274e13 1.57756
\(40\) 0 0
\(41\) 3.91759e13 0.766225 0.383112 0.923702i \(-0.374852\pi\)
0.383112 + 0.923702i \(0.374852\pi\)
\(42\) −7.55135e13 −1.20339
\(43\) 1.29981e14 1.69589 0.847947 0.530082i \(-0.177839\pi\)
0.847947 + 0.530082i \(0.177839\pi\)
\(44\) 5.16566e13 0.554342
\(45\) 0 0
\(46\) 1.70424e14 1.25340
\(47\) 3.00620e14 1.84156 0.920781 0.390079i \(-0.127552\pi\)
0.920781 + 0.390079i \(0.127552\pi\)
\(48\) 7.87946e13 0.403595
\(49\) 2.58908e13 0.111296
\(50\) 0 0
\(51\) −3.69796e14 −1.13140
\(52\) 1.88356e14 0.488597
\(53\) −1.01926e14 −0.224874 −0.112437 0.993659i \(-0.535866\pi\)
−0.112437 + 0.993659i \(0.535866\pi\)
\(54\) 3.67679e14 0.692026
\(55\) 0 0
\(56\) −2.69754e14 −0.372709
\(57\) −4.19488e14 −0.498636
\(58\) −6.19987e14 −0.635689
\(59\) −9.67154e14 −0.857538 −0.428769 0.903414i \(-0.641053\pi\)
−0.428769 + 0.903414i \(0.641053\pi\)
\(60\) 0 0
\(61\) −1.14011e15 −0.761455 −0.380727 0.924687i \(-0.624326\pi\)
−0.380727 + 0.924687i \(0.624326\pi\)
\(62\) 2.24224e15 1.30422
\(63\) −3.33515e15 −1.69325
\(64\) 2.81475e14 0.125000
\(65\) 0 0
\(66\) 3.70188e15 1.26561
\(67\) 4.04577e15 1.21721 0.608605 0.793473i \(-0.291729\pi\)
0.608605 + 0.793473i \(0.291729\pi\)
\(68\) −1.32101e15 −0.350413
\(69\) 1.22131e16 2.86160
\(70\) 0 0
\(71\) −7.35604e15 −1.35191 −0.675955 0.736943i \(-0.736269\pi\)
−0.675955 + 0.736943i \(0.736269\pi\)
\(72\) 3.48006e15 0.567885
\(73\) −6.99946e14 −0.101583 −0.0507914 0.998709i \(-0.516174\pi\)
−0.0507914 + 0.998709i \(0.516174\pi\)
\(74\) −7.04762e15 −0.911115
\(75\) 0 0
\(76\) −1.49852e15 −0.154436
\(77\) −1.26734e16 −1.16875
\(78\) 1.34982e16 1.11551
\(79\) 1.08787e16 0.806761 0.403381 0.915032i \(-0.367835\pi\)
0.403381 + 0.915032i \(0.367835\pi\)
\(80\) 0 0
\(81\) −4.38165e14 −0.0262733
\(82\) 1.00290e16 0.541803
\(83\) 1.23277e16 0.600783 0.300391 0.953816i \(-0.402883\pi\)
0.300391 + 0.953816i \(0.402883\pi\)
\(84\) −1.93315e16 −0.850924
\(85\) 0 0
\(86\) 3.32752e16 1.19918
\(87\) −4.44303e16 −1.45133
\(88\) 1.32241e16 0.391979
\(89\) −2.21332e16 −0.595976 −0.297988 0.954570i \(-0.596316\pi\)
−0.297988 + 0.954570i \(0.596316\pi\)
\(90\) 0 0
\(91\) −4.62113e16 −1.03014
\(92\) 4.36285e16 0.886285
\(93\) 1.60686e17 2.97765
\(94\) 7.69588e16 1.30218
\(95\) 0 0
\(96\) 2.01714e16 0.285385
\(97\) −9.41354e16 −1.21953 −0.609766 0.792581i \(-0.708737\pi\)
−0.609766 + 0.792581i \(0.708737\pi\)
\(98\) 6.62805e15 0.0786981
\(99\) 1.63498e17 1.78079
\(100\) 0 0
\(101\) 3.77746e16 0.347111 0.173556 0.984824i \(-0.444474\pi\)
0.173556 + 0.984824i \(0.444474\pi\)
\(102\) −9.46679e16 −0.800021
\(103\) −1.63051e17 −1.26826 −0.634130 0.773226i \(-0.718642\pi\)
−0.634130 + 0.773226i \(0.718642\pi\)
\(104\) 4.82192e16 0.345491
\(105\) 0 0
\(106\) −2.60930e16 −0.159010
\(107\) 2.33848e17 1.31574 0.657871 0.753131i \(-0.271457\pi\)
0.657871 + 0.753131i \(0.271457\pi\)
\(108\) 9.41259e16 0.489336
\(109\) −4.47257e16 −0.214997 −0.107498 0.994205i \(-0.534284\pi\)
−0.107498 + 0.994205i \(0.534284\pi\)
\(110\) 0 0
\(111\) −5.05055e17 −2.08015
\(112\) −6.90571e16 −0.263545
\(113\) 2.41764e17 0.855510 0.427755 0.903895i \(-0.359305\pi\)
0.427755 + 0.903895i \(0.359305\pi\)
\(114\) −1.07389e17 −0.352589
\(115\) 0 0
\(116\) −1.58717e17 −0.449500
\(117\) 5.96165e17 1.56959
\(118\) −2.47591e17 −0.606371
\(119\) 3.24097e17 0.738797
\(120\) 0 0
\(121\) 1.15840e17 0.229182
\(122\) −2.91869e17 −0.538430
\(123\) 7.18713e17 1.23698
\(124\) 5.74013e17 0.922226
\(125\) 0 0
\(126\) −8.53798e17 −1.19731
\(127\) −1.14453e17 −0.150070 −0.0750352 0.997181i \(-0.523907\pi\)
−0.0750352 + 0.997181i \(0.523907\pi\)
\(128\) 7.20576e16 0.0883883
\(129\) 2.38461e18 2.73782
\(130\) 0 0
\(131\) 4.93656e16 0.0497300 0.0248650 0.999691i \(-0.492084\pi\)
0.0248650 + 0.999691i \(0.492084\pi\)
\(132\) 9.47681e17 0.894919
\(133\) 3.67648e17 0.325606
\(134\) 1.03572e18 0.860698
\(135\) 0 0
\(136\) −3.38179e17 −0.247780
\(137\) 4.16706e17 0.286883 0.143442 0.989659i \(-0.454183\pi\)
0.143442 + 0.989659i \(0.454183\pi\)
\(138\) 3.12656e18 2.02346
\(139\) 5.01647e17 0.305332 0.152666 0.988278i \(-0.451214\pi\)
0.152666 + 0.988278i \(0.451214\pi\)
\(140\) 0 0
\(141\) 5.51512e18 2.97298
\(142\) −1.88315e18 −0.955945
\(143\) 2.26540e18 1.08340
\(144\) 8.90895e17 0.401555
\(145\) 0 0
\(146\) −1.79186e17 −0.0718298
\(147\) 4.74987e17 0.179674
\(148\) −1.80419e18 −0.644255
\(149\) −1.16146e18 −0.391670 −0.195835 0.980637i \(-0.562742\pi\)
−0.195835 + 0.980637i \(0.562742\pi\)
\(150\) 0 0
\(151\) −5.58219e18 −1.68074 −0.840371 0.542012i \(-0.817663\pi\)
−0.840371 + 0.542012i \(0.817663\pi\)
\(152\) −3.83622e17 −0.109203
\(153\) −4.18113e18 −1.12568
\(154\) −3.24440e18 −0.826434
\(155\) 0 0
\(156\) 3.45554e18 0.788782
\(157\) 2.25949e18 0.488499 0.244249 0.969712i \(-0.421458\pi\)
0.244249 + 0.969712i \(0.421458\pi\)
\(158\) 2.78494e18 0.570466
\(159\) −1.86991e18 −0.363033
\(160\) 0 0
\(161\) −1.07038e19 −1.86861
\(162\) −1.12170e17 −0.0185780
\(163\) 2.17080e18 0.341212 0.170606 0.985339i \(-0.445427\pi\)
0.170606 + 0.985339i \(0.445427\pi\)
\(164\) 2.56743e18 0.383112
\(165\) 0 0
\(166\) 3.15589e18 0.424818
\(167\) −2.02306e18 −0.258772 −0.129386 0.991594i \(-0.541301\pi\)
−0.129386 + 0.991594i \(0.541301\pi\)
\(168\) −4.94885e18 −0.601694
\(169\) −3.90047e17 −0.0450900
\(170\) 0 0
\(171\) −4.74297e18 −0.496116
\(172\) 8.51844e18 0.847947
\(173\) −1.22669e19 −1.16237 −0.581183 0.813773i \(-0.697410\pi\)
−0.581183 + 0.813773i \(0.697410\pi\)
\(174\) −1.13741e19 −1.02624
\(175\) 0 0
\(176\) 3.38537e18 0.277171
\(177\) −1.77432e19 −1.38439
\(178\) −5.66610e18 −0.421419
\(179\) −1.77359e19 −1.25778 −0.628888 0.777496i \(-0.716490\pi\)
−0.628888 + 0.777496i \(0.716490\pi\)
\(180\) 0 0
\(181\) −1.40303e19 −0.905315 −0.452657 0.891684i \(-0.649524\pi\)
−0.452657 + 0.891684i \(0.649524\pi\)
\(182\) −1.18301e19 −0.728419
\(183\) −2.09163e19 −1.22928
\(184\) 1.11689e19 0.626698
\(185\) 0 0
\(186\) 4.11356e19 2.10551
\(187\) −1.58881e19 −0.776996
\(188\) 1.97015e19 0.920781
\(189\) −2.30928e19 −1.03170
\(190\) 0 0
\(191\) 1.22487e19 0.500387 0.250194 0.968196i \(-0.419506\pi\)
0.250194 + 0.968196i \(0.419506\pi\)
\(192\) 5.16388e18 0.201797
\(193\) −3.98799e19 −1.49114 −0.745568 0.666430i \(-0.767821\pi\)
−0.745568 + 0.666430i \(0.767821\pi\)
\(194\) −2.40987e19 −0.862340
\(195\) 0 0
\(196\) 1.69678e18 0.0556479
\(197\) 1.32923e19 0.417481 0.208740 0.977971i \(-0.433064\pi\)
0.208740 + 0.977971i \(0.433064\pi\)
\(198\) 4.18555e19 1.25921
\(199\) 6.95301e18 0.200411 0.100206 0.994967i \(-0.468050\pi\)
0.100206 + 0.994967i \(0.468050\pi\)
\(200\) 0 0
\(201\) 7.42229e19 1.96504
\(202\) 9.67031e18 0.245445
\(203\) 3.89395e19 0.947708
\(204\) −2.42350e19 −0.565700
\(205\) 0 0
\(206\) −4.17412e19 −0.896795
\(207\) 1.38088e20 2.84714
\(208\) 1.23441e19 0.244299
\(209\) −1.80231e19 −0.342441
\(210\) 0 0
\(211\) −1.70414e19 −0.298611 −0.149305 0.988791i \(-0.547704\pi\)
−0.149305 + 0.988791i \(0.547704\pi\)
\(212\) −6.67982e18 −0.112437
\(213\) −1.34952e20 −2.18250
\(214\) 5.98650e19 0.930370
\(215\) 0 0
\(216\) 2.40962e19 0.346013
\(217\) −1.40828e20 −1.94439
\(218\) −1.14498e19 −0.152026
\(219\) −1.28411e19 −0.163993
\(220\) 0 0
\(221\) −5.79330e19 −0.684844
\(222\) −1.29294e20 −1.47088
\(223\) −2.89177e19 −0.316645 −0.158322 0.987387i \(-0.550609\pi\)
−0.158322 + 0.987387i \(0.550609\pi\)
\(224\) −1.76786e19 −0.186354
\(225\) 0 0
\(226\) 6.18916e19 0.604937
\(227\) 1.32244e20 1.24496 0.622481 0.782635i \(-0.286125\pi\)
0.622481 + 0.782635i \(0.286125\pi\)
\(228\) −2.74916e19 −0.249318
\(229\) −1.21349e19 −0.106031 −0.0530157 0.998594i \(-0.516883\pi\)
−0.0530157 + 0.998594i \(0.516883\pi\)
\(230\) 0 0
\(231\) −2.32504e20 −1.88681
\(232\) −4.06315e19 −0.317844
\(233\) −1.12741e20 −0.850271 −0.425135 0.905130i \(-0.639773\pi\)
−0.425135 + 0.905130i \(0.639773\pi\)
\(234\) 1.52618e20 1.10987
\(235\) 0 0
\(236\) −6.33834e19 −0.428769
\(237\) 1.99578e20 1.30242
\(238\) 8.29687e19 0.522409
\(239\) −1.64296e20 −0.998261 −0.499131 0.866527i \(-0.666347\pi\)
−0.499131 + 0.866527i \(0.666347\pi\)
\(240\) 0 0
\(241\) 1.25647e20 0.711223 0.355612 0.934634i \(-0.384273\pi\)
0.355612 + 0.934634i \(0.384273\pi\)
\(242\) 2.96549e19 0.162056
\(243\) −1.93516e20 −1.02109
\(244\) −7.47184e19 −0.380727
\(245\) 0 0
\(246\) 1.83990e20 0.874675
\(247\) −6.57178e19 −0.301828
\(248\) 1.46947e20 0.652112
\(249\) 2.26161e20 0.969892
\(250\) 0 0
\(251\) −1.37960e20 −0.552746 −0.276373 0.961050i \(-0.589133\pi\)
−0.276373 + 0.961050i \(0.589133\pi\)
\(252\) −2.18572e20 −0.846624
\(253\) 5.24730e20 1.96522
\(254\) −2.92999e19 −0.106116
\(255\) 0 0
\(256\) 1.84467e19 0.0625000
\(257\) 1.08129e19 0.0354414 0.0177207 0.999843i \(-0.494359\pi\)
0.0177207 + 0.999843i \(0.494359\pi\)
\(258\) 6.10459e20 1.93593
\(259\) 4.42640e20 1.35832
\(260\) 0 0
\(261\) −5.02353e20 −1.44399
\(262\) 1.26376e19 0.0351644
\(263\) 7.17651e20 1.93326 0.966628 0.256185i \(-0.0824658\pi\)
0.966628 + 0.256185i \(0.0824658\pi\)
\(264\) 2.42606e20 0.632803
\(265\) 0 0
\(266\) 9.41178e19 0.230238
\(267\) −4.06051e20 −0.962132
\(268\) 2.65144e20 0.608605
\(269\) −5.80280e20 −1.29046 −0.645228 0.763990i \(-0.723238\pi\)
−0.645228 + 0.763990i \(0.723238\pi\)
\(270\) 0 0
\(271\) 1.64836e20 0.344201 0.172101 0.985079i \(-0.444945\pi\)
0.172101 + 0.985079i \(0.444945\pi\)
\(272\) −8.65737e19 −0.175207
\(273\) −8.47782e20 −1.66304
\(274\) 1.06677e20 0.202857
\(275\) 0 0
\(276\) 8.00399e20 1.43080
\(277\) −1.52391e20 −0.264168 −0.132084 0.991239i \(-0.542167\pi\)
−0.132084 + 0.991239i \(0.542167\pi\)
\(278\) 1.28422e20 0.215902
\(279\) 1.81681e21 2.96260
\(280\) 0 0
\(281\) 1.60032e20 0.245586 0.122793 0.992432i \(-0.460815\pi\)
0.122793 + 0.992432i \(0.460815\pi\)
\(282\) 1.41187e21 2.10222
\(283\) −2.34430e20 −0.338711 −0.169356 0.985555i \(-0.554169\pi\)
−0.169356 + 0.985555i \(0.554169\pi\)
\(284\) −4.82085e20 −0.675955
\(285\) 0 0
\(286\) 5.79943e20 0.766080
\(287\) −6.29894e20 −0.807739
\(288\) 2.28069e20 0.283943
\(289\) −4.20935e20 −0.508842
\(290\) 0 0
\(291\) −1.72699e21 −1.96879
\(292\) −4.58716e19 −0.0507914
\(293\) −1.57885e21 −1.69811 −0.849056 0.528303i \(-0.822828\pi\)
−0.849056 + 0.528303i \(0.822828\pi\)
\(294\) 1.21597e20 0.127049
\(295\) 0 0
\(296\) −4.61873e20 −0.455557
\(297\) 1.13208e21 1.08504
\(298\) −2.97333e20 −0.276952
\(299\) 1.91333e21 1.73215
\(300\) 0 0
\(301\) −2.08991e21 −1.78778
\(302\) −1.42904e21 −1.18846
\(303\) 6.93005e20 0.560369
\(304\) −9.82072e19 −0.0772179
\(305\) 0 0
\(306\) −1.07037e21 −0.795978
\(307\) 2.38391e21 1.72430 0.862152 0.506650i \(-0.169116\pi\)
0.862152 + 0.506650i \(0.169116\pi\)
\(308\) −8.30566e20 −0.584377
\(309\) −2.99131e21 −2.04745
\(310\) 0 0
\(311\) 1.58302e21 1.02570 0.512852 0.858477i \(-0.328589\pi\)
0.512852 + 0.858477i \(0.328589\pi\)
\(312\) 8.84618e20 0.557753
\(313\) 2.98369e20 0.183074 0.0915370 0.995802i \(-0.470822\pi\)
0.0915370 + 0.995802i \(0.470822\pi\)
\(314\) 5.78430e20 0.345421
\(315\) 0 0
\(316\) 7.12944e20 0.403381
\(317\) 1.01873e21 0.561117 0.280559 0.959837i \(-0.409480\pi\)
0.280559 + 0.959837i \(0.409480\pi\)
\(318\) −4.78697e20 −0.256703
\(319\) −1.90892e21 −0.996707
\(320\) 0 0
\(321\) 4.29012e21 2.12411
\(322\) −2.74017e21 −1.32131
\(323\) 4.60903e20 0.216465
\(324\) −2.87156e19 −0.0131366
\(325\) 0 0
\(326\) 5.55724e20 0.241273
\(327\) −8.20527e20 −0.347086
\(328\) 6.57263e20 0.270901
\(329\) −4.83355e21 −1.94134
\(330\) 0 0
\(331\) −6.27305e20 −0.239299 −0.119649 0.992816i \(-0.538177\pi\)
−0.119649 + 0.992816i \(0.538177\pi\)
\(332\) 8.07907e20 0.300391
\(333\) −5.71044e21 −2.06963
\(334\) −5.17903e20 −0.182980
\(335\) 0 0
\(336\) −1.26691e21 −0.425462
\(337\) −3.61857e21 −1.18490 −0.592451 0.805606i \(-0.701840\pi\)
−0.592451 + 0.805606i \(0.701840\pi\)
\(338\) −9.98521e19 −0.0318834
\(339\) 4.43535e21 1.38112
\(340\) 0 0
\(341\) 6.90379e21 2.04492
\(342\) −1.21420e21 −0.350807
\(343\) 3.32408e21 0.936854
\(344\) 2.18072e21 0.599589
\(345\) 0 0
\(346\) −3.14033e21 −0.821917
\(347\) −7.74896e21 −1.97899 −0.989493 0.144579i \(-0.953817\pi\)
−0.989493 + 0.144579i \(0.953817\pi\)
\(348\) −2.91178e21 −0.725663
\(349\) −7.13117e21 −1.73438 −0.867191 0.497976i \(-0.834077\pi\)
−0.867191 + 0.497976i \(0.834077\pi\)
\(350\) 0 0
\(351\) 4.12790e21 0.956354
\(352\) 8.66654e20 0.195990
\(353\) 7.70298e21 1.70049 0.850244 0.526388i \(-0.176454\pi\)
0.850244 + 0.526388i \(0.176454\pi\)
\(354\) −4.54226e21 −0.978913
\(355\) 0 0
\(356\) −1.45052e21 −0.297988
\(357\) 5.94581e21 1.19270
\(358\) −4.54040e21 −0.889382
\(359\) −4.55697e21 −0.871712 −0.435856 0.900016i \(-0.643554\pi\)
−0.435856 + 0.900016i \(0.643554\pi\)
\(360\) 0 0
\(361\) −4.95755e21 −0.904598
\(362\) −3.59176e21 −0.640154
\(363\) 2.12517e21 0.369987
\(364\) −3.02850e21 −0.515070
\(365\) 0 0
\(366\) −5.35456e21 −0.869230
\(367\) 6.69643e19 0.0106214 0.00531070 0.999986i \(-0.498310\pi\)
0.00531070 + 0.999986i \(0.498310\pi\)
\(368\) 2.85924e21 0.443142
\(369\) 8.12617e21 1.23073
\(370\) 0 0
\(371\) 1.63883e21 0.237058
\(372\) 1.05307e22 1.48882
\(373\) −5.61068e21 −0.775336 −0.387668 0.921799i \(-0.626719\pi\)
−0.387668 + 0.921799i \(0.626719\pi\)
\(374\) −4.06736e21 −0.549419
\(375\) 0 0
\(376\) 5.04357e21 0.651091
\(377\) −6.96053e21 −0.878498
\(378\) −5.91177e21 −0.729520
\(379\) 6.43379e21 0.776307 0.388153 0.921595i \(-0.373113\pi\)
0.388153 + 0.921595i \(0.373113\pi\)
\(380\) 0 0
\(381\) −2.09973e21 −0.242271
\(382\) 3.13567e21 0.353827
\(383\) −1.11213e22 −1.22735 −0.613674 0.789560i \(-0.710309\pi\)
−0.613674 + 0.789560i \(0.710309\pi\)
\(384\) 1.32195e21 0.142692
\(385\) 0 0
\(386\) −1.02093e22 −1.05439
\(387\) 2.69617e22 2.72398
\(388\) −6.16926e21 −0.609766
\(389\) 6.91791e21 0.668965 0.334483 0.942402i \(-0.391438\pi\)
0.334483 + 0.942402i \(0.391438\pi\)
\(390\) 0 0
\(391\) −1.34189e22 −1.24226
\(392\) 4.34376e20 0.0393490
\(393\) 9.05651e20 0.0802831
\(394\) 3.40282e21 0.295204
\(395\) 0 0
\(396\) 1.07150e22 0.890397
\(397\) −2.17800e22 −1.77149 −0.885746 0.464171i \(-0.846352\pi\)
−0.885746 + 0.464171i \(0.846352\pi\)
\(398\) 1.77997e21 0.141712
\(399\) 6.74479e21 0.525652
\(400\) 0 0
\(401\) 4.05717e21 0.303037 0.151519 0.988454i \(-0.451584\pi\)
0.151519 + 0.988454i \(0.451584\pi\)
\(402\) 1.90011e22 1.38949
\(403\) 2.51734e22 1.80239
\(404\) 2.47560e21 0.173556
\(405\) 0 0
\(406\) 9.96852e21 0.670130
\(407\) −2.16994e22 −1.42855
\(408\) −6.20415e21 −0.400010
\(409\) −6.48833e21 −0.409717 −0.204859 0.978792i \(-0.565673\pi\)
−0.204859 + 0.978792i \(0.565673\pi\)
\(410\) 0 0
\(411\) 7.64480e21 0.463138
\(412\) −1.06857e22 −0.634130
\(413\) 1.55505e22 0.904000
\(414\) 3.53506e22 2.01323
\(415\) 0 0
\(416\) 3.16009e21 0.172745
\(417\) 9.20310e21 0.492922
\(418\) −4.61391e21 −0.242143
\(419\) 2.17453e22 1.11827 0.559135 0.829077i \(-0.311133\pi\)
0.559135 + 0.829077i \(0.311133\pi\)
\(420\) 0 0
\(421\) 1.45920e22 0.720638 0.360319 0.932829i \(-0.382668\pi\)
0.360319 + 0.932829i \(0.382668\pi\)
\(422\) −4.36261e21 −0.211150
\(423\) 6.23570e22 2.95796
\(424\) −1.71003e21 −0.0795051
\(425\) 0 0
\(426\) −3.45478e22 −1.54326
\(427\) 1.83314e22 0.802710
\(428\) 1.53254e22 0.657871
\(429\) 4.15606e22 1.74902
\(430\) 0 0
\(431\) −2.61293e22 −1.05699 −0.528495 0.848937i \(-0.677243\pi\)
−0.528495 + 0.848937i \(0.677243\pi\)
\(432\) 6.16864e21 0.244668
\(433\) 1.27697e22 0.496632 0.248316 0.968679i \(-0.420123\pi\)
0.248316 + 0.968679i \(0.420123\pi\)
\(434\) −3.60521e22 −1.37489
\(435\) 0 0
\(436\) −2.93114e21 −0.107498
\(437\) −1.52221e22 −0.547497
\(438\) −3.28731e21 −0.115961
\(439\) −2.83409e22 −0.980540 −0.490270 0.871571i \(-0.663102\pi\)
−0.490270 + 0.871571i \(0.663102\pi\)
\(440\) 0 0
\(441\) 5.37047e21 0.178766
\(442\) −1.48308e22 −0.484258
\(443\) 1.45222e22 0.465158 0.232579 0.972578i \(-0.425284\pi\)
0.232579 + 0.972578i \(0.425284\pi\)
\(444\) −3.30993e22 −1.04007
\(445\) 0 0
\(446\) −7.40293e21 −0.223902
\(447\) −2.13078e22 −0.632303
\(448\) −4.52572e21 −0.131773
\(449\) 1.38898e22 0.396828 0.198414 0.980118i \(-0.436421\pi\)
0.198414 + 0.980118i \(0.436421\pi\)
\(450\) 0 0
\(451\) 3.08791e22 0.849502
\(452\) 1.58443e22 0.427755
\(453\) −1.02410e23 −2.71335
\(454\) 3.38544e22 0.880321
\(455\) 0 0
\(456\) −7.03785e21 −0.176294
\(457\) 3.72555e22 0.916016 0.458008 0.888948i \(-0.348563\pi\)
0.458008 + 0.888948i \(0.348563\pi\)
\(458\) −3.10653e21 −0.0749755
\(459\) −2.89505e22 −0.685879
\(460\) 0 0
\(461\) 1.60242e22 0.365864 0.182932 0.983126i \(-0.441441\pi\)
0.182932 + 0.983126i \(0.441441\pi\)
\(462\) −5.95210e22 −1.33418
\(463\) −6.01797e22 −1.32438 −0.662189 0.749337i \(-0.730372\pi\)
−0.662189 + 0.749337i \(0.730372\pi\)
\(464\) −1.04017e22 −0.224750
\(465\) 0 0
\(466\) −2.88617e22 −0.601232
\(467\) 4.21105e22 0.861384 0.430692 0.902499i \(-0.358270\pi\)
0.430692 + 0.902499i \(0.358270\pi\)
\(468\) 3.90703e22 0.784796
\(469\) −6.50504e22 −1.28316
\(470\) 0 0
\(471\) 4.14521e22 0.788622
\(472\) −1.62262e22 −0.303186
\(473\) 1.02453e23 1.88021
\(474\) 5.10919e22 0.920949
\(475\) 0 0
\(476\) 2.12400e22 0.369399
\(477\) −2.11423e22 −0.361198
\(478\) −4.20597e22 −0.705877
\(479\) −4.13412e22 −0.681603 −0.340801 0.940135i \(-0.610698\pi\)
−0.340801 + 0.940135i \(0.610698\pi\)
\(480\) 0 0
\(481\) −7.91229e22 −1.25913
\(482\) 3.21655e22 0.502911
\(483\) −1.96370e23 −3.01664
\(484\) 7.59166e21 0.114591
\(485\) 0 0
\(486\) −4.95400e22 −0.722018
\(487\) −9.82373e22 −1.40696 −0.703478 0.710717i \(-0.748370\pi\)
−0.703478 + 0.710717i \(0.748370\pi\)
\(488\) −1.91279e22 −0.269215
\(489\) 3.98250e22 0.550846
\(490\) 0 0
\(491\) −2.86911e21 −0.0383314 −0.0191657 0.999816i \(-0.506101\pi\)
−0.0191657 + 0.999816i \(0.506101\pi\)
\(492\) 4.71016e22 0.618489
\(493\) 4.88167e22 0.630043
\(494\) −1.68238e22 −0.213424
\(495\) 0 0
\(496\) 3.76185e22 0.461113
\(497\) 1.18275e23 1.42516
\(498\) 5.78972e22 0.685817
\(499\) −8.68275e22 −1.01112 −0.505560 0.862791i \(-0.668714\pi\)
−0.505560 + 0.862791i \(0.668714\pi\)
\(500\) 0 0
\(501\) −3.71146e22 −0.417757
\(502\) −3.53177e22 −0.390851
\(503\) −2.63166e22 −0.286353 −0.143177 0.989697i \(-0.545732\pi\)
−0.143177 + 0.989697i \(0.545732\pi\)
\(504\) −5.59545e22 −0.598653
\(505\) 0 0
\(506\) 1.34331e23 1.38962
\(507\) −7.15572e21 −0.0727924
\(508\) −7.50078e21 −0.0750352
\(509\) 2.52122e22 0.248033 0.124017 0.992280i \(-0.460422\pi\)
0.124017 + 0.992280i \(0.460422\pi\)
\(510\) 0 0
\(511\) 1.12541e22 0.107087
\(512\) 4.72237e21 0.0441942
\(513\) −3.28407e22 −0.302284
\(514\) 2.76810e21 0.0250608
\(515\) 0 0
\(516\) 1.56278e23 1.36891
\(517\) 2.36954e23 2.04171
\(518\) 1.13316e23 0.960479
\(519\) −2.25046e23 −1.87650
\(520\) 0 0
\(521\) 8.93522e22 0.721082 0.360541 0.932743i \(-0.382592\pi\)
0.360541 + 0.932743i \(0.382592\pi\)
\(522\) −1.28602e23 −1.02106
\(523\) −9.62048e22 −0.751506 −0.375753 0.926720i \(-0.622616\pi\)
−0.375753 + 0.926720i \(0.622616\pi\)
\(524\) 3.23523e21 0.0248650
\(525\) 0 0
\(526\) 1.83719e23 1.36702
\(527\) −1.76550e23 −1.29264
\(528\) 6.21072e22 0.447460
\(529\) 3.02130e23 2.14200
\(530\) 0 0
\(531\) −2.00615e23 −1.37740
\(532\) 2.40942e22 0.162803
\(533\) 1.12595e23 0.748751
\(534\) −1.03949e23 −0.680330
\(535\) 0 0
\(536\) 6.78768e22 0.430349
\(537\) −3.25380e23 −2.03053
\(538\) −1.48552e23 −0.912490
\(539\) 2.04076e22 0.123392
\(540\) 0 0
\(541\) −1.09224e23 −0.639944 −0.319972 0.947427i \(-0.603673\pi\)
−0.319972 + 0.947427i \(0.603673\pi\)
\(542\) 4.21979e22 0.243387
\(543\) −2.57397e23 −1.46152
\(544\) −2.21629e22 −0.123890
\(545\) 0 0
\(546\) −2.17032e23 −1.17594
\(547\) 2.52518e23 1.34710 0.673551 0.739140i \(-0.264768\pi\)
0.673551 + 0.739140i \(0.264768\pi\)
\(548\) 2.73092e22 0.143442
\(549\) −2.36491e23 −1.22307
\(550\) 0 0
\(551\) 5.53766e22 0.277675
\(552\) 2.04902e23 1.01173
\(553\) −1.74914e23 −0.850472
\(554\) −3.90120e22 −0.186795
\(555\) 0 0
\(556\) 3.28759e22 0.152666
\(557\) −1.44437e23 −0.660554 −0.330277 0.943884i \(-0.607142\pi\)
−0.330277 + 0.943884i \(0.607142\pi\)
\(558\) 4.65102e23 2.09487
\(559\) 3.73577e23 1.65722
\(560\) 0 0
\(561\) −2.91480e23 −1.25437
\(562\) 4.09682e22 0.173655
\(563\) −7.42080e22 −0.309834 −0.154917 0.987927i \(-0.549511\pi\)
−0.154917 + 0.987927i \(0.549511\pi\)
\(564\) 3.61439e23 1.48649
\(565\) 0 0
\(566\) −6.00142e22 −0.239505
\(567\) 7.04507e21 0.0276968
\(568\) −1.23414e23 −0.477972
\(569\) 1.99386e23 0.760747 0.380373 0.924833i \(-0.375795\pi\)
0.380373 + 0.924833i \(0.375795\pi\)
\(570\) 0 0
\(571\) 5.76243e22 0.213402 0.106701 0.994291i \(-0.465971\pi\)
0.106701 + 0.994291i \(0.465971\pi\)
\(572\) 1.48466e23 0.541701
\(573\) 2.24712e23 0.807815
\(574\) −1.61253e23 −0.571158
\(575\) 0 0
\(576\) 5.83857e22 0.200778
\(577\) 2.51358e23 0.851722 0.425861 0.904789i \(-0.359971\pi\)
0.425861 + 0.904789i \(0.359971\pi\)
\(578\) −1.07759e23 −0.359806
\(579\) −7.31628e23 −2.40726
\(580\) 0 0
\(581\) −1.98212e23 −0.633333
\(582\) −4.42109e23 −1.39214
\(583\) −8.03398e22 −0.249315
\(584\) −1.17431e22 −0.0359149
\(585\) 0 0
\(586\) −4.04186e23 −1.20075
\(587\) −2.04262e23 −0.598087 −0.299043 0.954240i \(-0.596668\pi\)
−0.299043 + 0.954240i \(0.596668\pi\)
\(588\) 3.11288e22 0.0898369
\(589\) −2.00274e23 −0.569699
\(590\) 0 0
\(591\) 2.43857e23 0.673973
\(592\) −1.18239e23 −0.322128
\(593\) 2.40118e23 0.644851 0.322425 0.946595i \(-0.395502\pi\)
0.322425 + 0.946595i \(0.395502\pi\)
\(594\) 2.89811e23 0.767239
\(595\) 0 0
\(596\) −7.61172e22 −0.195835
\(597\) 1.27558e23 0.323539
\(598\) 4.89812e23 1.22481
\(599\) 6.64163e23 1.63737 0.818684 0.574244i \(-0.194704\pi\)
0.818684 + 0.574244i \(0.194704\pi\)
\(600\) 0 0
\(601\) −5.64984e23 −1.35395 −0.676975 0.736006i \(-0.736710\pi\)
−0.676975 + 0.736006i \(0.736710\pi\)
\(602\) −5.35018e23 −1.26415
\(603\) 8.39206e23 1.95511
\(604\) −3.65835e23 −0.840371
\(605\) 0 0
\(606\) 1.77409e23 0.396241
\(607\) 2.77451e23 0.611057 0.305529 0.952183i \(-0.401167\pi\)
0.305529 + 0.952183i \(0.401167\pi\)
\(608\) −2.51411e22 −0.0546013
\(609\) 7.14376e23 1.52996
\(610\) 0 0
\(611\) 8.64009e23 1.79957
\(612\) −2.74014e23 −0.562841
\(613\) 3.66032e23 0.741491 0.370745 0.928735i \(-0.379102\pi\)
0.370745 + 0.928735i \(0.379102\pi\)
\(614\) 6.10281e23 1.21927
\(615\) 0 0
\(616\) −2.12625e23 −0.413217
\(617\) 5.54311e23 1.06250 0.531251 0.847215i \(-0.321722\pi\)
0.531251 + 0.847215i \(0.321722\pi\)
\(618\) −7.65775e23 −1.44777
\(619\) 7.96802e23 1.48587 0.742934 0.669365i \(-0.233434\pi\)
0.742934 + 0.669365i \(0.233434\pi\)
\(620\) 0 0
\(621\) 9.56135e23 1.73476
\(622\) 4.05252e23 0.725282
\(623\) 3.55871e23 0.628266
\(624\) 2.26462e23 0.394391
\(625\) 0 0
\(626\) 7.63825e22 0.129453
\(627\) −3.30648e23 −0.552830
\(628\) 1.48078e23 0.244249
\(629\) 5.54918e23 0.903022
\(630\) 0 0
\(631\) 4.62634e23 0.732804 0.366402 0.930457i \(-0.380590\pi\)
0.366402 + 0.930457i \(0.380590\pi\)
\(632\) 1.82514e23 0.285233
\(633\) −3.12639e23 −0.482071
\(634\) 2.60794e23 0.396770
\(635\) 0 0
\(636\) −1.22547e23 −0.181516
\(637\) 7.44124e22 0.108758
\(638\) −4.88684e23 −0.704779
\(639\) −1.52585e24 −2.17147
\(640\) 0 0
\(641\) 2.38553e23 0.330592 0.165296 0.986244i \(-0.447142\pi\)
0.165296 + 0.986244i \(0.447142\pi\)
\(642\) 1.09827e24 1.50197
\(643\) 1.39064e24 1.87681 0.938405 0.345536i \(-0.112303\pi\)
0.938405 + 0.345536i \(0.112303\pi\)
\(644\) −7.01485e23 −0.934304
\(645\) 0 0
\(646\) 1.17991e23 0.153064
\(647\) 1.93162e23 0.247306 0.123653 0.992325i \(-0.460539\pi\)
0.123653 + 0.992325i \(0.460539\pi\)
\(648\) −7.35118e21 −0.00928901
\(649\) −7.62328e23 −0.950740
\(650\) 0 0
\(651\) −2.58361e24 −3.13898
\(652\) 1.42265e23 0.170606
\(653\) −1.41423e24 −1.67401 −0.837007 0.547192i \(-0.815697\pi\)
−0.837007 + 0.547192i \(0.815697\pi\)
\(654\) −2.10055e23 −0.245427
\(655\) 0 0
\(656\) 1.68259e23 0.191556
\(657\) −1.45188e23 −0.163164
\(658\) −1.23739e24 −1.37273
\(659\) −3.77712e23 −0.413651 −0.206826 0.978378i \(-0.566313\pi\)
−0.206826 + 0.978378i \(0.566313\pi\)
\(660\) 0 0
\(661\) 6.30952e23 0.673416 0.336708 0.941609i \(-0.390686\pi\)
0.336708 + 0.941609i \(0.390686\pi\)
\(662\) −1.60590e23 −0.169210
\(663\) −1.06283e24 −1.10560
\(664\) 2.06824e23 0.212409
\(665\) 0 0
\(666\) −1.46187e24 −1.46345
\(667\) −1.61225e24 −1.59354
\(668\) −1.32583e23 −0.129386
\(669\) −5.30518e23 −0.511185
\(670\) 0 0
\(671\) −8.98656e23 −0.844213
\(672\) −3.24328e23 −0.300847
\(673\) −1.46706e24 −1.34375 −0.671876 0.740663i \(-0.734511\pi\)
−0.671876 + 0.740663i \(0.734511\pi\)
\(674\) −9.26353e23 −0.837852
\(675\) 0 0
\(676\) −2.55621e22 −0.0225450
\(677\) −8.89634e22 −0.0774832 −0.0387416 0.999249i \(-0.512335\pi\)
−0.0387416 + 0.999249i \(0.512335\pi\)
\(678\) 1.13545e24 0.976598
\(679\) 1.51357e24 1.28561
\(680\) 0 0
\(681\) 2.42612e24 2.00984
\(682\) 1.76737e24 1.44597
\(683\) 8.93594e23 0.722045 0.361022 0.932557i \(-0.382428\pi\)
0.361022 + 0.932557i \(0.382428\pi\)
\(684\) −3.10835e23 −0.248058
\(685\) 0 0
\(686\) 8.50966e23 0.662456
\(687\) −2.22624e23 −0.171175
\(688\) 5.58265e23 0.423973
\(689\) −2.92944e23 −0.219746
\(690\) 0 0
\(691\) −7.89975e23 −0.578162 −0.289081 0.957305i \(-0.593350\pi\)
−0.289081 + 0.957305i \(0.593350\pi\)
\(692\) −8.03924e23 −0.581183
\(693\) −2.62882e24 −1.87728
\(694\) −1.98373e24 −1.39935
\(695\) 0 0
\(696\) −7.45416e23 −0.513121
\(697\) −7.89669e23 −0.536991
\(698\) −1.82558e24 −1.22639
\(699\) −2.06833e24 −1.37266
\(700\) 0 0
\(701\) 3.61705e23 0.234288 0.117144 0.993115i \(-0.462626\pi\)
0.117144 + 0.993115i \(0.462626\pi\)
\(702\) 1.05674e24 0.676244
\(703\) 6.29486e23 0.397984
\(704\) 2.21863e23 0.138586
\(705\) 0 0
\(706\) 1.97196e24 1.20243
\(707\) −6.07363e23 −0.365918
\(708\) −1.16282e24 −0.692196
\(709\) 4.46298e23 0.262501 0.131251 0.991349i \(-0.458101\pi\)
0.131251 + 0.991349i \(0.458101\pi\)
\(710\) 0 0
\(711\) 2.25654e24 1.29584
\(712\) −3.71333e23 −0.210709
\(713\) 5.83085e24 3.26942
\(714\) 1.52213e24 0.843366
\(715\) 0 0
\(716\) −1.16234e24 −0.628888
\(717\) −3.01414e24 −1.61157
\(718\) −1.16658e24 −0.616393
\(719\) 1.55258e24 0.810698 0.405349 0.914162i \(-0.367150\pi\)
0.405349 + 0.914162i \(0.367150\pi\)
\(720\) 0 0
\(721\) 2.62164e24 1.33697
\(722\) −1.26913e24 −0.639648
\(723\) 2.30509e24 1.14818
\(724\) −9.19491e23 −0.452657
\(725\) 0 0
\(726\) 5.44043e23 0.261621
\(727\) −1.38114e23 −0.0656439 −0.0328219 0.999461i \(-0.510449\pi\)
−0.0328219 + 0.999461i \(0.510449\pi\)
\(728\) −7.75296e23 −0.364209
\(729\) −3.49361e24 −1.62215
\(730\) 0 0
\(731\) −2.62003e24 −1.18853
\(732\) −1.37077e24 −0.614639
\(733\) −1.51067e24 −0.669555 −0.334778 0.942297i \(-0.608661\pi\)
−0.334778 + 0.942297i \(0.608661\pi\)
\(734\) 1.71428e22 0.00751046
\(735\) 0 0
\(736\) 7.31964e23 0.313349
\(737\) 3.18895e24 1.34950
\(738\) 2.08030e24 0.870255
\(739\) −3.13836e24 −1.29785 −0.648926 0.760851i \(-0.724782\pi\)
−0.648926 + 0.760851i \(0.724782\pi\)
\(740\) 0 0
\(741\) −1.20565e24 −0.487265
\(742\) 4.19540e23 0.167625
\(743\) 4.99845e23 0.197438 0.0987190 0.995115i \(-0.468526\pi\)
0.0987190 + 0.995115i \(0.468526\pi\)
\(744\) 2.69586e24 1.05276
\(745\) 0 0
\(746\) −1.43633e24 −0.548246
\(747\) 2.55710e24 0.964991
\(748\) −1.04124e24 −0.388498
\(749\) −3.75994e24 −1.38703
\(750\) 0 0
\(751\) 4.01494e24 1.44790 0.723952 0.689851i \(-0.242324\pi\)
0.723952 + 0.689851i \(0.242324\pi\)
\(752\) 1.29115e24 0.460391
\(753\) −2.53098e24 −0.892342
\(754\) −1.78190e24 −0.621192
\(755\) 0 0
\(756\) −1.51341e24 −0.515848
\(757\) 3.79032e24 1.27750 0.638750 0.769414i \(-0.279452\pi\)
0.638750 + 0.769414i \(0.279452\pi\)
\(758\) 1.64705e24 0.548932
\(759\) 9.62659e24 3.17261
\(760\) 0 0
\(761\) −1.23826e24 −0.399065 −0.199532 0.979891i \(-0.563942\pi\)
−0.199532 + 0.979891i \(0.563942\pi\)
\(762\) −5.37530e23 −0.171311
\(763\) 7.19126e23 0.226645
\(764\) 8.02731e23 0.250194
\(765\) 0 0
\(766\) −2.84706e24 −0.867866
\(767\) −2.77968e24 −0.837982
\(768\) 3.38420e23 0.100899
\(769\) 3.58282e24 1.05646 0.528228 0.849102i \(-0.322857\pi\)
0.528228 + 0.849102i \(0.322857\pi\)
\(770\) 0 0
\(771\) 1.98371e23 0.0572158
\(772\) −2.61357e24 −0.745568
\(773\) −1.21994e24 −0.344203 −0.172101 0.985079i \(-0.555056\pi\)
−0.172101 + 0.985079i \(0.555056\pi\)
\(774\) 6.90219e24 1.92615
\(775\) 0 0
\(776\) −1.57933e24 −0.431170
\(777\) 8.12058e24 2.19285
\(778\) 1.77099e24 0.473030
\(779\) −8.95783e23 −0.236665
\(780\) 0 0
\(781\) −5.79815e24 −1.49884
\(782\) −3.43523e24 −0.878413
\(783\) −3.47834e24 −0.879826
\(784\) 1.11200e23 0.0278240
\(785\) 0 0
\(786\) 2.31847e23 0.0567688
\(787\) 1.07097e24 0.259414 0.129707 0.991552i \(-0.458596\pi\)
0.129707 + 0.991552i \(0.458596\pi\)
\(788\) 8.71123e23 0.208740
\(789\) 1.31659e25 3.12101
\(790\) 0 0
\(791\) −3.88723e24 −0.901862
\(792\) 2.74304e24 0.629606
\(793\) −3.27678e24 −0.744090
\(794\) −5.57569e24 −1.25263
\(795\) 0 0
\(796\) 4.55672e23 0.100206
\(797\) −5.86608e22 −0.0127630 −0.00638149 0.999980i \(-0.502031\pi\)
−0.00638149 + 0.999980i \(0.502031\pi\)
\(798\) 1.72667e24 0.371692
\(799\) −6.05961e24 −1.29062
\(800\) 0 0
\(801\) −4.59104e24 −0.957270
\(802\) 1.03864e24 0.214280
\(803\) −5.51709e23 −0.112623
\(804\) 4.86427e24 0.982520
\(805\) 0 0
\(806\) 6.44439e24 1.27448
\(807\) −1.06457e25 −2.08329
\(808\) 6.33753e23 0.122722
\(809\) 2.47056e24 0.473406 0.236703 0.971582i \(-0.423933\pi\)
0.236703 + 0.971582i \(0.423933\pi\)
\(810\) 0 0
\(811\) 9.80457e24 1.83972 0.919859 0.392249i \(-0.128303\pi\)
0.919859 + 0.392249i \(0.128303\pi\)
\(812\) 2.55194e24 0.473854
\(813\) 3.02404e24 0.555671
\(814\) −5.55506e24 −1.01014
\(815\) 0 0
\(816\) −1.58826e24 −0.282850
\(817\) −2.97210e24 −0.523813
\(818\) −1.66101e24 −0.289714
\(819\) −9.58550e24 −1.65463
\(820\) 0 0
\(821\) 3.65428e23 0.0617853 0.0308927 0.999523i \(-0.490165\pi\)
0.0308927 + 0.999523i \(0.490165\pi\)
\(822\) 1.95707e24 0.327488
\(823\) 9.20886e24 1.52513 0.762565 0.646911i \(-0.223940\pi\)
0.762565 + 0.646911i \(0.223940\pi\)
\(824\) −2.73555e24 −0.448398
\(825\) 0 0
\(826\) 3.98092e24 0.639224
\(827\) −2.05375e24 −0.326400 −0.163200 0.986593i \(-0.552182\pi\)
−0.163200 + 0.986593i \(0.552182\pi\)
\(828\) 9.04975e24 1.42357
\(829\) 2.35707e24 0.366994 0.183497 0.983020i \(-0.441258\pi\)
0.183497 + 0.983020i \(0.441258\pi\)
\(830\) 0 0
\(831\) −2.79573e24 −0.426468
\(832\) 8.08983e23 0.122149
\(833\) −5.21882e23 −0.0779991
\(834\) 2.35599e24 0.348548
\(835\) 0 0
\(836\) −1.18116e24 −0.171221
\(837\) 1.25797e25 1.80511
\(838\) 5.56680e24 0.790736
\(839\) 4.69193e24 0.659743 0.329872 0.944026i \(-0.392994\pi\)
0.329872 + 0.944026i \(0.392994\pi\)
\(840\) 0 0
\(841\) −1.39192e24 −0.191800
\(842\) 3.73555e24 0.509568
\(843\) 2.93591e24 0.396469
\(844\) −1.11683e24 −0.149305
\(845\) 0 0
\(846\) 1.59634e25 2.09159
\(847\) −1.86254e24 −0.241600
\(848\) −4.37769e23 −0.0562186
\(849\) −4.30081e24 −0.546808
\(850\) 0 0
\(851\) −1.83271e25 −2.28398
\(852\) −8.84423e24 −1.09125
\(853\) 8.89565e24 1.08670 0.543352 0.839505i \(-0.317155\pi\)
0.543352 + 0.839505i \(0.317155\pi\)
\(854\) 4.69284e24 0.567602
\(855\) 0 0
\(856\) 3.92331e24 0.465185
\(857\) −1.23863e25 −1.45413 −0.727066 0.686567i \(-0.759117\pi\)
−0.727066 + 0.686567i \(0.759117\pi\)
\(858\) 1.06395e25 1.23674
\(859\) 3.17743e24 0.365708 0.182854 0.983140i \(-0.441466\pi\)
0.182854 + 0.983140i \(0.441466\pi\)
\(860\) 0 0
\(861\) −1.15559e25 −1.30400
\(862\) −6.68909e24 −0.747404
\(863\) −1.42021e25 −1.57130 −0.785652 0.618669i \(-0.787672\pi\)
−0.785652 + 0.618669i \(0.787672\pi\)
\(864\) 1.57917e24 0.173006
\(865\) 0 0
\(866\) 3.26905e24 0.351172
\(867\) −7.72238e24 −0.821465
\(868\) −9.22933e24 −0.972193
\(869\) 8.57475e24 0.894444
\(870\) 0 0
\(871\) 1.16279e25 1.18945
\(872\) −7.50372e23 −0.0760128
\(873\) −1.95263e25 −1.95884
\(874\) −3.89685e24 −0.387139
\(875\) 0 0
\(876\) −8.41551e23 −0.0819966
\(877\) 8.28800e24 0.799748 0.399874 0.916570i \(-0.369054\pi\)
0.399874 + 0.916570i \(0.369054\pi\)
\(878\) −7.25527e24 −0.693347
\(879\) −2.89653e25 −2.74140
\(880\) 0 0
\(881\) 7.02937e24 0.652561 0.326281 0.945273i \(-0.394205\pi\)
0.326281 + 0.945273i \(0.394205\pi\)
\(882\) 1.37484e24 0.126407
\(883\) 3.64222e23 0.0331665 0.0165833 0.999862i \(-0.494721\pi\)
0.0165833 + 0.999862i \(0.494721\pi\)
\(884\) −3.79670e24 −0.342422
\(885\) 0 0
\(886\) 3.71768e24 0.328916
\(887\) −1.85309e25 −1.62385 −0.811923 0.583765i \(-0.801579\pi\)
−0.811923 + 0.583765i \(0.801579\pi\)
\(888\) −8.47342e24 −0.735442
\(889\) 1.84024e24 0.158201
\(890\) 0 0
\(891\) −3.45369e23 −0.0291288
\(892\) −1.89515e24 −0.158322
\(893\) −6.87388e24 −0.568807
\(894\) −5.45481e24 −0.447106
\(895\) 0 0
\(896\) −1.15859e24 −0.0931772
\(897\) 3.51015e25 2.79634
\(898\) 3.55579e24 0.280600
\(899\) −2.12121e25 −1.65816
\(900\) 0 0
\(901\) 2.05452e24 0.157598
\(902\) 7.90506e24 0.600689
\(903\) −3.83411e25 −2.88615
\(904\) 4.05613e24 0.302469
\(905\) 0 0
\(906\) −2.62169e25 −1.91863
\(907\) 1.87024e25 1.35592 0.677962 0.735097i \(-0.262863\pi\)
0.677962 + 0.735097i \(0.262863\pi\)
\(908\) 8.66674e24 0.622481
\(909\) 7.83551e24 0.557538
\(910\) 0 0
\(911\) −1.41129e25 −0.985620 −0.492810 0.870137i \(-0.664030\pi\)
−0.492810 + 0.870137i \(0.664030\pi\)
\(912\) −1.80169e24 −0.124659
\(913\) 9.71689e24 0.666079
\(914\) 9.53741e24 0.647721
\(915\) 0 0
\(916\) −7.95273e23 −0.0530157
\(917\) −7.93730e23 −0.0524244
\(918\) −7.41132e24 −0.484990
\(919\) −2.33638e25 −1.51482 −0.757411 0.652938i \(-0.773536\pi\)
−0.757411 + 0.652938i \(0.773536\pi\)
\(920\) 0 0
\(921\) 4.37347e25 2.78368
\(922\) 4.10221e24 0.258705
\(923\) −2.11419e25 −1.32108
\(924\) −1.52374e25 −0.943406
\(925\) 0 0
\(926\) −1.54060e25 −0.936476
\(927\) −3.38214e25 −2.03711
\(928\) −2.66282e24 −0.158922
\(929\) 2.98004e25 1.76234 0.881169 0.472802i \(-0.156757\pi\)
0.881169 + 0.472802i \(0.156757\pi\)
\(930\) 0 0
\(931\) −5.92010e23 −0.0343761
\(932\) −7.38860e24 −0.425135
\(933\) 2.90417e25 1.65588
\(934\) 1.07803e25 0.609090
\(935\) 0 0
\(936\) 1.00020e25 0.554935
\(937\) 2.48730e25 1.36755 0.683774 0.729694i \(-0.260337\pi\)
0.683774 + 0.729694i \(0.260337\pi\)
\(938\) −1.66529e25 −0.907331
\(939\) 5.47381e24 0.295551
\(940\) 0 0
\(941\) −3.05795e25 −1.62150 −0.810752 0.585390i \(-0.800941\pi\)
−0.810752 + 0.585390i \(0.800941\pi\)
\(942\) 1.06117e25 0.557640
\(943\) 2.60801e25 1.35819
\(944\) −4.15389e24 −0.214385
\(945\) 0 0
\(946\) 2.62281e25 1.32951
\(947\) 3.09044e25 1.55255 0.776276 0.630393i \(-0.217106\pi\)
0.776276 + 0.630393i \(0.217106\pi\)
\(948\) 1.30795e25 0.651210
\(949\) −2.01170e24 −0.0992661
\(950\) 0 0
\(951\) 1.86893e25 0.905856
\(952\) 5.43744e24 0.261204
\(953\) −2.50647e24 −0.119337 −0.0596683 0.998218i \(-0.519004\pi\)
−0.0596683 + 0.998218i \(0.519004\pi\)
\(954\) −5.41242e24 −0.255406
\(955\) 0 0
\(956\) −1.07673e25 −0.499131
\(957\) −3.50207e25 −1.60906
\(958\) −1.05834e25 −0.481966
\(959\) −6.70005e24 −0.302426
\(960\) 0 0
\(961\) 5.41655e25 2.40200
\(962\) −2.02555e25 −0.890337
\(963\) 4.85065e25 2.11337
\(964\) 8.23438e24 0.355612
\(965\) 0 0
\(966\) −5.02707e25 −2.13309
\(967\) 1.13278e25 0.476455 0.238228 0.971209i \(-0.423434\pi\)
0.238228 + 0.971209i \(0.423434\pi\)
\(968\) 1.94347e24 0.0810282
\(969\) 8.45564e24 0.349457
\(970\) 0 0
\(971\) 4.00712e25 1.62731 0.813653 0.581351i \(-0.197476\pi\)
0.813653 + 0.581351i \(0.197476\pi\)
\(972\) −1.26822e25 −0.510544
\(973\) −8.06578e24 −0.321875
\(974\) −2.51488e25 −0.994868
\(975\) 0 0
\(976\) −4.89675e24 −0.190364
\(977\) −8.86356e24 −0.341589 −0.170795 0.985307i \(-0.554633\pi\)
−0.170795 + 0.985307i \(0.554633\pi\)
\(978\) 1.01952e25 0.389507
\(979\) −1.74458e25 −0.660750
\(980\) 0 0
\(981\) −9.27734e24 −0.345332
\(982\) −7.34492e23 −0.0271044
\(983\) 1.56885e25 0.573954 0.286977 0.957938i \(-0.407350\pi\)
0.286977 + 0.957938i \(0.407350\pi\)
\(984\) 1.20580e25 0.437338
\(985\) 0 0
\(986\) 1.24971e25 0.445507
\(987\) −8.86754e25 −3.13406
\(988\) −4.30688e24 −0.150914
\(989\) 8.65307e25 3.00609
\(990\) 0 0
\(991\) −2.85830e25 −0.976073 −0.488037 0.872823i \(-0.662287\pi\)
−0.488037 + 0.872823i \(0.662287\pi\)
\(992\) 9.63034e24 0.326056
\(993\) −1.15084e25 −0.386319
\(994\) 3.02783e25 1.00774
\(995\) 0 0
\(996\) 1.48217e25 0.484946
\(997\) 6.58977e24 0.213777 0.106889 0.994271i \(-0.465911\pi\)
0.106889 + 0.994271i \(0.465911\pi\)
\(998\) −2.22278e25 −0.714970
\(999\) −3.95396e25 −1.26103
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 50.18.a.g.1.2 2
5.2 odd 4 50.18.b.e.49.3 4
5.3 odd 4 50.18.b.e.49.2 4
5.4 even 2 10.18.a.b.1.1 2
15.14 odd 2 90.18.a.n.1.2 2
20.19 odd 2 80.18.a.e.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.18.a.b.1.1 2 5.4 even 2
50.18.a.g.1.2 2 1.1 even 1 trivial
50.18.b.e.49.2 4 5.3 odd 4
50.18.b.e.49.3 4 5.2 odd 4
80.18.a.e.1.2 2 20.19 odd 2
90.18.a.n.1.2 2 15.14 odd 2