Properties

Label 10.18.a.b.1.1
Level $10$
Weight $18$
Character 10.1
Self dual yes
Analytic conductor $18.322$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [10,18,Mod(1,10)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 18, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.1");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 10.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(18.3222087345\)
Analytic rank: \(1\)
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: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(95.4487\) of defining polynomial
Character \(\chi\) \(=\) 10.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-256.000 q^{2} -18345.8 q^{3} +65536.0 q^{4} -390625. q^{5} +4.69652e6 q^{6} +1.60786e7 q^{7} -1.67772e7 q^{8} +2.07428e8 q^{9} +1.00000e8 q^{10} +7.88217e8 q^{11} -1.20231e9 q^{12} -2.87409e9 q^{13} -4.11612e9 q^{14} +7.16632e9 q^{15} +4.29497e9 q^{16} +2.01570e10 q^{17} -5.31015e10 q^{18} -2.28657e10 q^{19} -2.56000e10 q^{20} -2.94975e11 q^{21} -2.01784e11 q^{22} -6.65718e11 q^{23} +3.07791e11 q^{24} +1.52588e11 q^{25} +7.35766e11 q^{26} -1.43625e12 q^{27} +1.05373e12 q^{28} -2.42182e12 q^{29} -1.83458e12 q^{30} +8.75874e12 q^{31} -1.09951e12 q^{32} -1.44605e13 q^{33} -5.16020e12 q^{34} -6.28070e12 q^{35} +1.35940e13 q^{36} +2.75298e13 q^{37} +5.85361e12 q^{38} +5.27274e13 q^{39} +6.55360e12 q^{40} +3.91759e13 q^{41} +7.55135e13 q^{42} -1.29981e14 q^{43} +5.16566e13 q^{44} -8.10265e13 q^{45} +1.70424e14 q^{46} -3.00620e14 q^{47} -7.87946e13 q^{48} +2.58908e13 q^{49} -3.90625e13 q^{50} -3.69796e14 q^{51} -1.88356e14 q^{52} +1.01926e14 q^{53} +3.67679e14 q^{54} -3.07897e14 q^{55} -2.69754e14 q^{56} +4.19488e14 q^{57} +6.19987e14 q^{58} -9.67154e14 q^{59} +4.69652e14 q^{60} -1.14011e15 q^{61} -2.24224e15 q^{62} +3.33515e15 q^{63} +2.81475e14 q^{64} +1.12269e15 q^{65} +3.70188e15 q^{66} -4.04577e15 q^{67} +1.32101e15 q^{68} +1.22131e16 q^{69} +1.60786e15 q^{70} -7.35604e15 q^{71} -3.48006e15 q^{72} +6.99946e14 q^{73} -7.04762e15 q^{74} -2.79934e15 q^{75} -1.49852e15 q^{76} +1.26734e16 q^{77} -1.34982e16 q^{78} +1.08787e16 q^{79} -1.67772e15 q^{80} -4.38165e14 q^{81} -1.00290e16 q^{82} -1.23277e16 q^{83} -1.93315e16 q^{84} -7.87384e15 q^{85} +3.32752e16 q^{86} +4.44303e16 q^{87} -1.32241e16 q^{88} -2.21332e16 q^{89} +2.07428e16 q^{90} -4.62113e16 q^{91} -4.36285e16 q^{92} -1.60686e17 q^{93} +7.69588e16 q^{94} +8.93190e15 q^{95} +2.01714e16 q^{96} +9.41354e16 q^{97} -6.62805e15 q^{98} +1.63498e17 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 512 q^{2} - 6308 q^{3} + 131072 q^{4} - 781250 q^{5} + 1614848 q^{6} + 6543844 q^{7} - 33554432 q^{8} + 223195906 q^{9} + 200000000 q^{10} + 1189408704 q^{11} - 413401088 q^{12} - 2017919228 q^{13}+ \cdots + 16\!\cdots\!12 q^{99}+O(q^{100}) \) 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 \(-0.799013\pi\)
−0.807190 + 0.590292i \(0.799013\pi\)
\(4\) 65536.0 0.500000
\(5\) −390625. −0.447214
\(6\) 4.69652e6 1.14154
\(7\) 1.60786e7 1.05418 0.527090 0.849809i \(-0.323283\pi\)
0.527090 + 0.849809i \(0.323283\pi\)
\(8\) −1.67772e7 −0.353553
\(9\) 2.07428e8 1.60622
\(10\) 1.00000e8 0.316228
\(11\) 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 \(-0.662491\pi\)
−0.488597 + 0.872509i \(0.662491\pi\)
\(14\) −4.11612e9 −0.745418
\(15\) 7.16632e9 0.721973
\(16\) 4.29497e9 0.250000
\(17\) 2.01570e10 0.700826 0.350413 0.936595i \(-0.386041\pi\)
0.350413 + 0.936595i \(0.386041\pi\)
\(18\) −5.31015e10 −1.13577
\(19\) −2.28657e10 −0.308872 −0.154436 0.988003i \(-0.549356\pi\)
−0.154436 + 0.988003i \(0.549356\pi\)
\(20\) −2.56000e10 −0.223607
\(21\) −2.94975e11 −1.70185
\(22\) −2.01784e11 −0.783959
\(23\) −6.65718e11 −1.77257 −0.886285 0.463140i \(-0.846723\pi\)
−0.886285 + 0.463140i \(0.846723\pi\)
\(24\) 3.07791e11 0.570769
\(25\) 1.52588e11 0.200000
\(26\) 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 \(0.648398\pi\)
\(30\) −1.83458e12 −0.510512
\(31\) 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
\(35\) −6.28070e12 −0.471444
\(36\) 1.35940e13 0.803111
\(37\) 2.75298e13 1.28851 0.644255 0.764810i \(-0.277167\pi\)
0.644255 + 0.764810i \(0.277167\pi\)
\(38\) 5.85361e12 0.218405
\(39\) 5.27274e13 1.57756
\(40\) 6.55360e12 0.158114
\(41\) 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 \(-0.822161\pi\)
−0.847947 + 0.530082i \(0.822161\pi\)
\(44\) 5.16566e13 0.554342
\(45\) −8.10265e13 −0.718324
\(46\) 1.70424e14 1.25340
\(47\) −3.00620e14 −1.84156 −0.920781 0.390079i \(-0.872448\pi\)
−0.920781 + 0.390079i \(0.872448\pi\)
\(48\) −7.87946e13 −0.403595
\(49\) 2.58908e13 0.111296
\(50\) −3.90625e13 −0.141421
\(51\) −3.69796e14 −1.13140
\(52\) −1.88356e14 −0.488597
\(53\) 1.01926e14 0.224874 0.112437 0.993659i \(-0.464134\pi\)
0.112437 + 0.993659i \(0.464134\pi\)
\(54\) 3.67679e14 0.692026
\(55\) −3.07897e14 −0.495819
\(56\) −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 \(0.641053\pi\)
\(60\) 4.69652e14 0.360986
\(61\) −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
\(65\) 1.12269e15 0.437015
\(66\) 3.70188e15 1.26561
\(67\) −4.04577e15 −1.21721 −0.608605 0.793473i \(-0.708271\pi\)
−0.608605 + 0.793473i \(0.708271\pi\)
\(68\) 1.32101e15 0.350413
\(69\) 1.22131e16 2.86160
\(70\) 1.60786e15 0.333361
\(71\) −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 \(-0.483826\pi\)
0.0507914 + 0.998709i \(0.483826\pi\)
\(74\) −7.04762e15 −0.911115
\(75\) −2.79934e15 −0.322876
\(76\) −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 \(0.367835\pi\)
\(80\) −1.67772e15 −0.111803
\(81\) −4.38165e14 −0.0262733
\(82\) −1.00290e16 −0.541803
\(83\) −1.23277e16 −0.600783 −0.300391 0.953816i \(-0.597117\pi\)
−0.300391 + 0.953816i \(0.597117\pi\)
\(84\) −1.93315e16 −0.850924
\(85\) −7.87384e15 −0.313419
\(86\) 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 \(0.596316\pi\)
\(90\) 2.07428e16 0.507932
\(91\) −4.62113e16 −1.03014
\(92\) −4.36285e16 −0.886285
\(93\) −1.60686e17 −2.97765
\(94\) 7.69588e16 1.30218
\(95\) 8.93190e15 0.138132
\(96\) 2.01714e16 0.285385
\(97\) 9.41354e16 1.21953 0.609766 0.792581i \(-0.291263\pi\)
0.609766 + 0.792581i \(0.291263\pi\)
\(98\) −6.62805e15 −0.0786981
\(99\) 1.63498e17 1.78079
\(100\) 1.00000e16 0.100000
\(101\) 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 \(-0.281358\pi\)
0.634130 + 0.773226i \(0.281358\pi\)
\(104\) 4.82192e16 0.345491
\(105\) 1.15224e17 0.761089
\(106\) −2.60930e16 −0.159010
\(107\) −2.33848e17 −1.31574 −0.657871 0.753131i \(-0.728543\pi\)
−0.657871 + 0.753131i \(0.728543\pi\)
\(108\) −9.41259e16 −0.489336
\(109\) −4.47257e16 −0.214997 −0.107498 0.994205i \(-0.534284\pi\)
−0.107498 + 0.994205i \(0.534284\pi\)
\(110\) 7.88217e16 0.350597
\(111\) −5.05055e17 −2.08015
\(112\) 6.90571e16 0.263545
\(113\) −2.41764e17 −0.855510 −0.427755 0.903895i \(-0.640695\pi\)
−0.427755 + 0.903895i \(0.640695\pi\)
\(114\) −1.07389e17 −0.352589
\(115\) 2.60046e17 0.792717
\(116\) −1.58717e17 −0.449500
\(117\) −5.96165e17 −1.56959
\(118\) 2.47591e17 0.606371
\(119\) 3.24097e17 0.738797
\(120\) −1.20231e17 −0.255256
\(121\) 1.15840e17 0.229182
\(122\) 2.91869e17 0.538430
\(123\) −7.18713e17 −1.23698
\(124\) 5.74013e17 0.922226
\(125\) −5.96046e16 −0.0894427
\(126\) −8.53798e17 −1.19731
\(127\) 1.14453e17 0.150070 0.0750352 0.997181i \(-0.476093\pi\)
0.0750352 + 0.997181i \(0.476093\pi\)
\(128\) −7.20576e16 −0.0883883
\(129\) 2.38461e18 2.73782
\(130\) −2.87409e17 −0.309016
\(131\) 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
\(135\) 5.61034e17 0.437676
\(136\) −3.38179e17 −0.247780
\(137\) −4.16706e17 −0.286883 −0.143442 0.989659i \(-0.545817\pi\)
−0.143442 + 0.989659i \(0.545817\pi\)
\(138\) −3.12656e18 −2.02346
\(139\) 5.01647e17 0.305332 0.152666 0.988278i \(-0.451214\pi\)
0.152666 + 0.988278i \(0.451214\pi\)
\(140\) −4.11612e17 −0.235722
\(141\) 5.51512e18 2.97298
\(142\) 1.88315e18 0.955945
\(143\) −2.26540e18 −1.08340
\(144\) 8.90895e17 0.401555
\(145\) 9.46025e17 0.402045
\(146\) −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 \(0.562742\pi\)
\(150\) 7.16632e17 0.228308
\(151\) −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
\(155\) −3.42138e18 −0.824864
\(156\) 3.45554e18 0.788782
\(157\) −2.25949e18 −0.488499 −0.244249 0.969712i \(-0.578542\pi\)
−0.244249 + 0.969712i \(0.578542\pi\)
\(158\) −2.78494e18 −0.570466
\(159\) −1.86991e18 −0.363033
\(160\) 4.29497e17 0.0790569
\(161\) −1.07038e19 −1.86861
\(162\) 1.12170e17 0.0185780
\(163\) −2.17080e18 −0.341212 −0.170606 0.985339i \(-0.554573\pi\)
−0.170606 + 0.985339i \(0.554573\pi\)
\(164\) 2.56743e18 0.383112
\(165\) 5.64862e18 0.800440
\(166\) 3.15589e18 0.424818
\(167\) 2.02306e18 0.258772 0.129386 0.991594i \(-0.458699\pi\)
0.129386 + 0.991594i \(0.458699\pi\)
\(168\) 4.94885e18 0.601694
\(169\) −3.90047e17 −0.0450900
\(170\) 2.01570e18 0.221621
\(171\) −4.74297e18 −0.496116
\(172\) −8.51844e18 −0.847947
\(173\) 1.22669e19 1.16237 0.581183 0.813773i \(-0.302590\pi\)
0.581183 + 0.813773i \(0.302590\pi\)
\(174\) −1.13741e19 −1.02624
\(175\) 2.45340e18 0.210836
\(176\) 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 \(0.716490\pi\)
\(180\) −5.31015e18 −0.359162
\(181\) −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
\(185\) −1.07538e19 −0.576239
\(186\) 4.11356e19 2.10551
\(187\) 1.58881e19 0.776996
\(188\) −1.97015e19 −0.920781
\(189\) −2.30928e19 −1.03170
\(190\) −2.28657e18 −0.0976738
\(191\) 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 \(-0.232179\pi\)
0.745568 + 0.666430i \(0.232179\pi\)
\(194\) −2.40987e19 −0.862340
\(195\) −2.05966e19 −0.705508
\(196\) 1.69678e18 0.0556479
\(197\) −1.32923e19 −0.417481 −0.208740 0.977971i \(-0.566936\pi\)
−0.208740 + 0.977971i \(0.566936\pi\)
\(198\) −4.18555e19 −1.25921
\(199\) 6.95301e18 0.200411 0.100206 0.994967i \(-0.468050\pi\)
0.100206 + 0.994967i \(0.468050\pi\)
\(200\) −2.56000e18 −0.0707107
\(201\) 7.42229e19 1.96504
\(202\) −9.67031e18 −0.245445
\(203\) −3.89395e19 −0.947708
\(204\) −2.42350e19 −0.565700
\(205\) −1.53031e19 −0.342666
\(206\) −4.17412e19 −0.896795
\(207\) −1.38088e20 −2.84714
\(208\) −1.23441e19 −0.244299
\(209\) −1.80231e19 −0.342441
\(210\) −2.94975e19 −0.538171
\(211\) −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
\(215\) 5.07739e19 0.758426
\(216\) 2.40962e19 0.346013
\(217\) 1.40828e20 1.94439
\(218\) 1.14498e19 0.152026
\(219\) −1.28411e19 −0.163993
\(220\) −2.01784e19 −0.247909
\(221\) −5.79330e19 −0.684844
\(222\) 1.29294e20 1.47088
\(223\) 2.89177e19 0.316645 0.158322 0.987387i \(-0.449391\pi\)
0.158322 + 0.987387i \(0.449391\pi\)
\(224\) −1.76786e19 −0.186354
\(225\) 3.16510e19 0.321244
\(226\) 6.18916e19 0.604937
\(227\) −1.32244e20 −1.24496 −0.622481 0.782635i \(-0.713875\pi\)
−0.622481 + 0.782635i \(0.713875\pi\)
\(228\) 2.74916e19 0.249318
\(229\) −1.21349e19 −0.106031 −0.0530157 0.998594i \(-0.516883\pi\)
−0.0530157 + 0.998594i \(0.516883\pi\)
\(230\) −6.65718e19 −0.560536
\(231\) −2.32504e20 −1.88681
\(232\) 4.06315e19 0.317844
\(233\) 1.12741e20 0.850271 0.425135 0.905130i \(-0.360227\pi\)
0.425135 + 0.905130i \(0.360227\pi\)
\(234\) 1.52618e20 1.10987
\(235\) 1.17430e20 0.823572
\(236\) −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 \(0.666347\pi\)
\(240\) 3.07791e19 0.180493
\(241\) 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
\(245\) −1.01136e19 −0.0497730
\(246\) 1.83990e20 0.874675
\(247\) 6.57178e19 0.301828
\(248\) −1.46947e20 −0.652112
\(249\) 2.26161e20 0.969892
\(250\) 1.52588e19 0.0632456
\(251\) −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
\(255\) 1.44452e20 0.505977
\(256\) 1.84467e19 0.0625000
\(257\) −1.08129e19 −0.0354414 −0.0177207 0.999843i \(-0.505641\pi\)
−0.0177207 + 0.999843i \(0.505641\pi\)
\(258\) −6.10459e20 −1.93593
\(259\) 4.42640e20 1.35832
\(260\) 7.35766e19 0.218507
\(261\) −5.02353e20 −1.44399
\(262\) −1.26376e19 −0.0351644
\(263\) −7.17651e20 −1.93326 −0.966628 0.256185i \(-0.917534\pi\)
−0.966628 + 0.256185i \(0.917534\pi\)
\(264\) 2.42606e20 0.632803
\(265\) −3.98148e19 −0.100567
\(266\) 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 \(0.723238\pi\)
\(270\) −1.43625e20 −0.309483
\(271\) 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
\(275\) 1.20272e20 0.221737
\(276\) 8.00399e20 1.43080
\(277\) 1.52391e20 0.264168 0.132084 0.991239i \(-0.457833\pi\)
0.132084 + 0.991239i \(0.457833\pi\)
\(278\) −1.28422e20 −0.215902
\(279\) 1.81681e21 2.96260
\(280\) 1.05373e20 0.166681
\(281\) 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 \(-0.445831\pi\)
0.169356 + 0.985555i \(0.445831\pi\)
\(284\) −4.82085e20 −0.675955
\(285\) −1.63863e20 −0.222997
\(286\) 5.79943e20 0.766080
\(287\) 6.29894e20 0.807739
\(288\) −2.28069e20 −0.283943
\(289\) −4.20935e20 −0.508842
\(290\) −2.42182e20 −0.284289
\(291\) −1.72699e21 −1.96879
\(292\) 4.58716e19 0.0507914
\(293\) 1.57885e21 1.69811 0.849056 0.528303i \(-0.177172\pi\)
0.849056 + 0.528303i \(0.177172\pi\)
\(294\) 1.21597e20 0.127049
\(295\) 3.77795e20 0.383503
\(296\) −4.61873e20 −0.455557
\(297\) −1.13208e21 −1.08504
\(298\) 2.97333e20 0.276952
\(299\) 1.91333e21 1.73215
\(300\) −1.83458e20 −0.161438
\(301\) −2.08991e21 −1.78778
\(302\) 1.42904e21 1.18846
\(303\) −6.93005e20 −0.560369
\(304\) −9.82072e19 −0.0772179
\(305\) 4.45356e20 0.340533
\(306\) −1.07037e21 −0.795978
\(307\) −2.38391e21 −1.72430 −0.862152 0.506650i \(-0.830884\pi\)
−0.862152 + 0.506650i \(0.830884\pi\)
\(308\) 8.30566e20 0.584377
\(309\) −2.99131e21 −2.04745
\(310\) 8.75874e20 0.583267
\(311\) 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 \(-0.529178\pi\)
−0.0915370 + 0.995802i \(0.529178\pi\)
\(314\) 5.78430e20 0.345421
\(315\) −1.30279e21 −0.757243
\(316\) 7.12944e20 0.403381
\(317\) −1.01873e21 −0.561117 −0.280559 0.959837i \(-0.590520\pi\)
−0.280559 + 0.959837i \(0.590520\pi\)
\(318\) 4.78697e20 0.256703
\(319\) −1.90892e21 −0.996707
\(320\) −1.09951e20 −0.0559017
\(321\) 4.29012e21 2.12411
\(322\) 2.74017e21 1.32131
\(323\) −4.60903e20 −0.216465
\(324\) −2.87156e19 −0.0131366
\(325\) −4.38551e20 −0.195439
\(326\) 5.55724e20 0.241273
\(327\) 8.20527e20 0.347086
\(328\) −6.57263e20 −0.270901
\(329\) −4.83355e21 −1.94134
\(330\) −1.44605e21 −0.565997
\(331\) −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
\(335\) 1.58038e21 0.544353
\(336\) −1.26691e21 −0.425462
\(337\) 3.61857e21 1.18490 0.592451 0.805606i \(-0.298160\pi\)
0.592451 + 0.805606i \(0.298160\pi\)
\(338\) 9.98521e19 0.0318834
\(339\) 4.43535e21 1.38112
\(340\) −5.16020e20 −0.156710
\(341\) 6.90379e21 2.04492
\(342\) 1.21420e21 0.350807
\(343\) −3.32408e21 −0.936854
\(344\) 2.18072e21 0.599589
\(345\) −4.77075e21 −1.27975
\(346\) −3.14033e21 −0.821917
\(347\) 7.74896e21 1.97899 0.989493 0.144579i \(-0.0461827\pi\)
0.989493 + 0.144579i \(0.0461827\pi\)
\(348\) 2.91178e21 0.725663
\(349\) −7.13117e21 −1.73438 −0.867191 0.497976i \(-0.834077\pi\)
−0.867191 + 0.497976i \(0.834077\pi\)
\(350\) −6.28070e20 −0.149084
\(351\) 4.12790e21 0.956354
\(352\) −8.66654e20 −0.195990
\(353\) −7.70298e21 −1.70049 −0.850244 0.526388i \(-0.823546\pi\)
−0.850244 + 0.526388i \(0.823546\pi\)
\(354\) −4.54226e21 −0.978913
\(355\) 2.87345e21 0.604593
\(356\) −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 \(0.643554\pi\)
\(360\) 1.35940e21 0.253966
\(361\) −4.95755e21 −0.904598
\(362\) 3.59176e21 0.640154
\(363\) −2.12517e21 −0.369987
\(364\) −3.02850e21 −0.515070
\(365\) −2.73416e20 −0.0454292
\(366\) −5.35456e21 −0.869230
\(367\) −6.69643e19 −0.0106214 −0.00531070 0.999986i \(-0.501690\pi\)
−0.00531070 + 0.999986i \(0.501690\pi\)
\(368\) −2.85924e21 −0.443142
\(369\) 8.12617e21 1.23073
\(370\) 2.75298e21 0.407463
\(371\) 1.63883e21 0.237058
\(372\) −1.05307e22 −1.48882
\(373\) 5.61068e21 0.775336 0.387668 0.921799i \(-0.373281\pi\)
0.387668 + 0.921799i \(0.373281\pi\)
\(374\) −4.06736e21 −0.549419
\(375\) 1.09349e21 0.144395
\(376\) 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 \(0.373113\pi\)
\(380\) 5.85361e20 0.0690658
\(381\) −2.09973e21 −0.242271
\(382\) −3.13567e21 −0.353827
\(383\) 1.11213e22 1.22735 0.613674 0.789560i \(-0.289691\pi\)
0.613674 + 0.789560i \(0.289691\pi\)
\(384\) 1.32195e21 0.142692
\(385\) −4.95056e21 −0.522683
\(386\) −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 \(0.391438\pi\)
\(390\) 5.27274e21 0.498869
\(391\) −1.34189e22 −1.24226
\(392\) −4.34376e20 −0.0393490
\(393\) −9.05651e20 −0.0802831
\(394\) 3.40282e21 0.295204
\(395\) −4.24948e21 −0.360795
\(396\) 1.07150e22 0.890397
\(397\) 2.17800e22 1.77149 0.885746 0.464171i \(-0.153648\pi\)
0.885746 + 0.464171i \(0.153648\pi\)
\(398\) −1.77997e21 −0.141712
\(399\) 6.74479e21 0.525652
\(400\) 6.55360e20 0.0500000
\(401\) 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
\(405\) 1.71158e20 0.0117498
\(406\) 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 \(0.565673\pi\)
\(410\) 3.91759e21 0.242302
\(411\) 7.64480e21 0.463138
\(412\) 1.06857e22 0.634130
\(413\) −1.55505e22 −0.904000
\(414\) 3.53506e22 2.01323
\(415\) 4.81550e21 0.268678
\(416\) 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 \(0.311133\pi\)
\(420\) 7.55135e21 0.380545
\(421\) 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
\(425\) 3.07572e21 0.140165
\(426\) −3.45478e22 −1.54326
\(427\) −1.83314e22 −0.802710
\(428\) −1.53254e22 −0.657871
\(429\) 4.15606e22 1.74902
\(430\) −1.29981e22 −0.536288
\(431\) −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 \(-0.579877\pi\)
−0.248316 + 0.968679i \(0.579877\pi\)
\(434\) −3.60521e22 −1.37489
\(435\) −1.73556e22 −0.649053
\(436\) −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 \(0.663102\pi\)
\(440\) 5.16566e21 0.175298
\(441\) 5.37047e21 0.178766
\(442\) 1.48308e22 0.484258
\(443\) −1.45222e22 −0.465158 −0.232579 0.972578i \(-0.574716\pi\)
−0.232579 + 0.972578i \(0.574716\pi\)
\(444\) −3.30993e22 −1.04007
\(445\) 8.64578e21 0.266529
\(446\) −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 \(0.436421\pi\)
\(450\) −8.10265e21 −0.227154
\(451\) 3.08791e22 0.849502
\(452\) −1.58443e22 −0.427755
\(453\) 1.02410e23 2.71335
\(454\) 3.38544e22 0.880321
\(455\) 1.80513e22 0.460692
\(456\) −7.03785e21 −0.176294
\(457\) −3.72555e22 −0.916016 −0.458008 0.888948i \(-0.651437\pi\)
−0.458008 + 0.888948i \(0.651437\pi\)
\(458\) 3.10653e21 0.0749755
\(459\) −2.89505e22 −0.685879
\(460\) 1.70424e22 0.396359
\(461\) 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 \(-0.269628\pi\)
0.662189 + 0.749337i \(0.269628\pi\)
\(464\) −1.04017e22 −0.224750
\(465\) 6.27680e22 1.33164
\(466\) −2.88617e22 −0.601232
\(467\) −4.21105e22 −0.861384 −0.430692 0.902499i \(-0.641730\pi\)
−0.430692 + 0.902499i \(0.641730\pi\)
\(468\) −3.90703e22 −0.784796
\(469\) −6.50504e22 −1.28316
\(470\) −3.00620e22 −0.582353
\(471\) 4.14521e22 0.788622
\(472\) 1.62262e22 0.303186
\(473\) −1.02453e23 −1.88021
\(474\) 5.10919e22 0.920949
\(475\) −3.48902e21 −0.0617743
\(476\) 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 \(0.610698\pi\)
\(480\) −7.87946e21 −0.127628
\(481\) −7.91229e22 −1.25913
\(482\) −3.21655e22 −0.502911
\(483\) 1.96370e23 3.01664
\(484\) 7.59166e21 0.114591
\(485\) −3.67716e22 −0.545391
\(486\) −4.95400e22 −0.722018
\(487\) 9.82373e22 1.40696 0.703478 0.710717i \(-0.251630\pi\)
0.703478 + 0.710717i \(0.251630\pi\)
\(488\) 1.91279e22 0.269215
\(489\) 3.98250e22 0.550846
\(490\) 2.58908e21 0.0351949
\(491\) −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
\(495\) −6.38665e22 −0.796395
\(496\) 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 \(0.668714\pi\)
\(500\) −3.90625e21 −0.0447214
\(501\) −3.71146e22 −0.417757
\(502\) 3.53177e22 0.390851
\(503\) 2.63166e22 0.286353 0.143177 0.989697i \(-0.454268\pi\)
0.143177 + 0.989697i \(0.454268\pi\)
\(504\) −5.59545e22 −0.598653
\(505\) −1.47557e22 −0.155233
\(506\) 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 \(0.460422\pi\)
\(510\) −3.69796e22 −0.357780
\(511\) 1.12541e22 0.107087
\(512\) −4.72237e21 −0.0441942
\(513\) 3.28407e22 0.302284
\(514\) 2.76810e21 0.0250608
\(515\) −6.36920e22 −0.567183
\(516\) 1.56278e23 1.36891
\(517\) −2.36954e23 −2.04171
\(518\) −1.13316e23 −0.960479
\(519\) −2.25046e23 −1.87650
\(520\) −1.88356e22 −0.154508
\(521\) 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 \(-0.377384\pi\)
0.375753 + 0.926720i \(0.377384\pi\)
\(524\) 3.23523e21 0.0248650
\(525\) −4.50095e22 −0.340369
\(526\) 1.83719e23 1.36702
\(527\) 1.76550e23 1.29264
\(528\) −6.21072e22 −0.447460
\(529\) 3.02130e23 2.14200
\(530\) 1.01926e22 0.0711115
\(531\) −2.00615e23 −1.37740
\(532\) −2.40942e22 −0.162803
\(533\) −1.12595e23 −0.748751
\(534\) −1.03949e23 −0.680330
\(535\) 9.13467e22 0.588418
\(536\) 6.78768e22 0.430349
\(537\) 3.25380e23 2.03053
\(538\) 1.48552e23 0.912490
\(539\) 2.04076e22 0.123392
\(540\) 3.67679e22 0.218838
\(541\) −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
\(545\) 1.74710e22 0.0961494
\(546\) −2.17032e23 −1.17594
\(547\) −2.52518e23 −1.34710 −0.673551 0.739140i \(-0.735232\pi\)
−0.673551 + 0.739140i \(0.735232\pi\)
\(548\) −2.73092e22 −0.143442
\(549\) −2.36491e23 −1.22307
\(550\) −3.07897e22 −0.156792
\(551\) 5.53766e22 0.277675
\(552\) −2.04902e23 −1.01173
\(553\) 1.74914e23 0.850472
\(554\) −3.90120e22 −0.186795
\(555\) 1.97287e23 0.930269
\(556\) 3.28759e22 0.152666
\(557\) 1.44437e23 0.660554 0.330277 0.943884i \(-0.392858\pi\)
0.330277 + 0.943884i \(0.392858\pi\)
\(558\) −4.65102e23 −2.09487
\(559\) 3.73577e23 1.65722
\(560\) −2.69754e22 −0.117861
\(561\) −2.91480e23 −1.25437
\(562\) −4.09682e22 −0.173655
\(563\) 7.42080e22 0.309834 0.154917 0.987927i \(-0.450489\pi\)
0.154917 + 0.987927i \(0.450489\pi\)
\(564\) 3.61439e23 1.48649
\(565\) 9.44391e22 0.382596
\(566\) −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 \(0.375795\pi\)
\(570\) 4.19488e22 0.157683
\(571\) 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
\(575\) −1.01580e23 −0.354514
\(576\) 5.83857e22 0.200778
\(577\) −2.51358e23 −0.851722 −0.425861 0.904789i \(-0.640029\pi\)
−0.425861 + 0.904789i \(0.640029\pi\)
\(578\) 1.07759e23 0.359806
\(579\) −7.31628e23 −2.40726
\(580\) 6.19987e22 0.201022
\(581\) −1.98212e23 −0.633333
\(582\) 4.42109e23 1.39214
\(583\) 8.03398e22 0.249315
\(584\) −1.17431e22 −0.0359149
\(585\) 2.32877e23 0.701943
\(586\) −4.04186e23 −1.20075
\(587\) 2.04262e23 0.598087 0.299043 0.954240i \(-0.403332\pi\)
0.299043 + 0.954240i \(0.403332\pi\)
\(588\) −3.11288e22 −0.0898369
\(589\) −2.00274e23 −0.569699
\(590\) −9.67154e22 −0.271177
\(591\) 2.43857e23 0.673973
\(592\) 1.18239e23 0.322128
\(593\) −2.40118e23 −0.644851 −0.322425 0.946595i \(-0.604498\pi\)
−0.322425 + 0.946595i \(0.604498\pi\)
\(594\) 2.89811e23 0.767239
\(595\) −1.26600e23 −0.330400
\(596\) −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 \(0.194704\pi\)
\(600\) 4.69652e22 0.114154
\(601\) −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
\(605\) −4.52498e22 −0.102493
\(606\) 1.77409e23 0.396241
\(607\) −2.77451e23 −0.611057 −0.305529 0.952183i \(-0.598833\pi\)
−0.305529 + 0.952183i \(0.598833\pi\)
\(608\) 2.51411e22 0.0546013
\(609\) 7.14376e23 1.52996
\(610\) −1.14011e23 −0.240793
\(611\) 8.64009e23 1.79957
\(612\) 2.74014e23 0.562841
\(613\) −3.66032e23 −0.741491 −0.370745 0.928735i \(-0.620898\pi\)
−0.370745 + 0.928735i \(0.620898\pi\)
\(614\) 6.10281e23 1.21927
\(615\) 2.80747e23 0.553193
\(616\) −2.12625e23 −0.413217
\(617\) −5.54311e23 −1.06250 −0.531251 0.847215i \(-0.678278\pi\)
−0.531251 + 0.847215i \(0.678278\pi\)
\(618\) 7.65775e23 1.44777
\(619\) 7.96802e23 1.48587 0.742934 0.669365i \(-0.233434\pi\)
0.742934 + 0.669365i \(0.233434\pi\)
\(620\) −2.24224e23 −0.412432
\(621\) 9.56135e23 1.73476
\(622\) −4.05252e23 −0.725282
\(623\) −3.55871e23 −0.628266
\(624\) 2.26462e23 0.394391
\(625\) 2.32831e22 0.0400000
\(626\) 7.63825e22 0.129453
\(627\) 3.30648e23 0.552830
\(628\) −1.48078e23 −0.244249
\(629\) 5.54918e23 0.903022
\(630\) 3.33515e23 0.535452
\(631\) 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
\(635\) −4.47081e22 −0.0671135
\(636\) −1.22547e23 −0.181516
\(637\) −7.44124e22 −0.108758
\(638\) 4.88684e23 0.704779
\(639\) −1.52585e24 −2.17147
\(640\) 2.81475e22 0.0395285
\(641\) 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 \(-0.887697\pi\)
−0.938405 + 0.345536i \(0.887697\pi\)
\(644\) −7.01485e23 −0.934304
\(645\) −9.31487e23 −1.22439
\(646\) 1.17991e23 0.153064
\(647\) −1.93162e23 −0.247306 −0.123653 0.992325i \(-0.539461\pi\)
−0.123653 + 0.992325i \(0.539461\pi\)
\(648\) 7.35118e21 0.00928901
\(649\) −7.62328e23 −0.950740
\(650\) 1.12269e23 0.138196
\(651\) −2.58361e24 −3.13898
\(652\) −1.42265e23 −0.170606
\(653\) 1.41423e24 1.67401 0.837007 0.547192i \(-0.184303\pi\)
0.837007 + 0.547192i \(0.184303\pi\)
\(654\) −2.10055e23 −0.245427
\(655\) −1.92835e22 −0.0222399
\(656\) 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 \(0.566313\pi\)
\(660\) 3.70188e23 0.400220
\(661\) 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
\(665\) 1.43612e23 0.145616
\(666\) −1.46187e24 −1.46345
\(667\) 1.61225e24 1.59354
\(668\) 1.32583e23 0.129386
\(669\) −5.30518e23 −0.511185
\(670\) −4.04577e23 −0.384916
\(671\) −8.98656e23 −0.844213
\(672\) 3.24328e23 0.300847
\(673\) 1.46706e24 1.34375 0.671876 0.740663i \(-0.265489\pi\)
0.671876 + 0.740663i \(0.265489\pi\)
\(674\) −9.26353e23 −0.837852
\(675\) −2.19154e23 −0.195734
\(676\) −2.55621e22 −0.0225450
\(677\) 8.89634e22 0.0774832 0.0387416 0.999249i \(-0.487665\pi\)
0.0387416 + 0.999249i \(0.487665\pi\)
\(678\) −1.13545e24 −0.976598
\(679\) 1.51357e24 1.28561
\(680\) 1.32101e23 0.110810
\(681\) 2.42612e24 2.00984
\(682\) −1.76737e24 −1.44597
\(683\) −8.93594e23 −0.722045 −0.361022 0.932557i \(-0.617572\pi\)
−0.361022 + 0.932557i \(0.617572\pi\)
\(684\) −3.10835e23 −0.248058
\(685\) 1.62776e23 0.128298
\(686\) 8.50966e23 0.662456
\(687\) 2.22624e23 0.171175
\(688\) −5.58265e23 −0.423973
\(689\) −2.92944e23 −0.219746
\(690\) 1.22131e24 0.904918
\(691\) −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
\(695\) −1.95956e23 −0.136549
\(696\) −7.45416e23 −0.513121
\(697\) 7.89669e23 0.536991
\(698\) 1.82558e24 1.22639
\(699\) −2.06833e24 −1.37266
\(700\) 1.60786e23 0.105418
\(701\) 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
\(705\) −2.15434e24 −1.32956
\(706\) 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 \(0.458101\pi\)
\(710\) −7.35604e23 −0.427512
\(711\) 2.25654e24 1.29584
\(712\) 3.71333e23 0.210709
\(713\) −5.83085e24 −3.26942
\(714\) 1.52213e24 0.843366
\(715\) 8.84924e23 0.484512
\(716\) −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 \(0.367150\pi\)
\(720\) −3.48006e23 −0.179581
\(721\) 2.62164e24 1.33697
\(722\) 1.26913e24 0.639648
\(723\) −2.30509e24 −1.14818
\(724\) −9.19491e23 −0.452657
\(725\) −3.69541e23 −0.179800
\(726\) 5.44043e23 0.261621
\(727\) 1.38114e23 0.0656439 0.0328219 0.999461i \(-0.489551\pi\)
0.0328219 + 0.999461i \(0.489551\pi\)
\(728\) 7.75296e23 0.364209
\(729\) −3.49361e24 −1.62215
\(730\) 6.99946e22 0.0321233
\(731\) −2.62003e24 −1.18853
\(732\) 1.37077e24 0.614639
\(733\) 1.51067e24 0.669555 0.334778 0.942297i \(-0.391339\pi\)
0.334778 + 0.942297i \(0.391339\pi\)
\(734\) 1.71428e22 0.00751046
\(735\) 1.85542e23 0.0803526
\(736\) 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 \(0.724782\pi\)
\(740\) −7.04762e23 −0.288120
\(741\) −1.20565e24 −0.487265
\(742\) −4.19540e23 −0.167625
\(743\) −4.99845e23 −0.197438 −0.0987190 0.995115i \(-0.531474\pi\)
−0.0987190 + 0.995115i \(0.531474\pi\)
\(744\) 2.69586e24 1.05276
\(745\) 4.53694e23 0.175160
\(746\) −1.43633e24 −0.548246
\(747\) −2.55710e24 −0.964991
\(748\) 1.04124e24 0.388498
\(749\) −3.75994e24 −1.38703
\(750\) −2.79934e23 −0.102102
\(751\) 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
\(755\) 2.18054e24 0.751650
\(756\) −1.51341e24 −0.515848
\(757\) −3.79032e24 −1.27750 −0.638750 0.769414i \(-0.720548\pi\)
−0.638750 + 0.769414i \(0.720548\pi\)
\(758\) −1.64705e24 −0.548932
\(759\) 9.62659e24 3.17261
\(760\) −1.49852e23 −0.0488369
\(761\) −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
\(765\) −1.63325e24 −0.503421
\(766\) −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 \(0.322857\pi\)
\(770\) 1.26734e24 0.369592
\(771\) 1.98371e23 0.0572158
\(772\) 2.61357e24 0.745568
\(773\) 1.21994e24 0.344203 0.172101 0.985079i \(-0.444944\pi\)
0.172101 + 0.985079i \(0.444944\pi\)
\(774\) 6.90219e24 1.92615
\(775\) 1.33648e24 0.368890
\(776\) −1.57933e24 −0.431170
\(777\) −8.12058e24 −2.19285
\(778\) −1.77099e24 −0.473030
\(779\) −8.95783e23 −0.236665
\(780\) −1.34982e24 −0.352754
\(781\) −5.79815e24 −1.49884
\(782\) 3.43523e24 0.878413
\(783\) 3.47834e24 0.879826
\(784\) 1.11200e23 0.0278240
\(785\) 8.82613e23 0.218463
\(786\) 2.31847e23 0.0567688
\(787\) −1.07097e24 −0.259414 −0.129707 0.991552i \(-0.541404\pi\)
−0.129707 + 0.991552i \(0.541404\pi\)
\(788\) −8.71123e23 −0.208740
\(789\) 1.31659e25 3.12101
\(790\) 1.08787e24 0.255120
\(791\) −3.88723e24 −0.901862
\(792\) −2.74304e24 −0.629606
\(793\) 3.27678e24 0.744090
\(794\) −5.57569e24 −1.25263
\(795\) 7.30434e23 0.162353
\(796\) 4.55672e23 0.100206
\(797\) 5.86608e22 0.0127630 0.00638149 0.999980i \(-0.497969\pi\)
0.00638149 + 0.999980i \(0.497969\pi\)
\(798\) −1.72667e24 −0.371692
\(799\) −6.05961e24 −1.29062
\(800\) −1.67772e23 −0.0353553
\(801\) −4.59104e24 −0.957270
\(802\) −1.03864e24 −0.214280
\(803\) 5.51709e23 0.112623
\(804\) 4.86427e24 0.982520
\(805\) 4.18117e24 0.835667
\(806\) 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 \(0.423933\pi\)
\(810\) −4.38165e22 −0.00830835
\(811\) 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
\(815\) 8.47967e23 0.152595
\(816\) −1.58826e24 −0.282850
\(817\) 2.97210e24 0.523813
\(818\) 1.66101e24 0.289714
\(819\) −9.58550e24 −1.65463
\(820\) −1.00290e24 −0.171333
\(821\) 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 \(-0.776060\pi\)
−0.762565 + 0.646911i \(0.776060\pi\)
\(824\) −2.73555e24 −0.448398
\(825\) −2.20649e24 −0.357968
\(826\) 3.98092e24 0.639224
\(827\) 2.05375e24 0.326400 0.163200 0.986593i \(-0.447818\pi\)
0.163200 + 0.986593i \(0.447818\pi\)
\(828\) −9.04975e24 −1.42357
\(829\) 2.35707e24 0.366994 0.183497 0.983020i \(-0.441258\pi\)
0.183497 + 0.983020i \(0.441258\pi\)
\(830\) −1.23277e24 −0.189984
\(831\) −2.79573e24 −0.426468
\(832\) −8.08983e23 −0.122149
\(833\) 5.21882e23 0.0779991
\(834\) 2.35599e24 0.348548
\(835\) −7.90257e23 −0.115727
\(836\) −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 \(0.392994\pi\)
\(840\) −1.93315e24 −0.269086
\(841\) −1.39192e24 −0.191800
\(842\) −3.73555e24 −0.509568
\(843\) −2.93591e24 −0.396469
\(844\) −1.11683e24 −0.149305
\(845\) 1.52362e23 0.0201649
\(846\) 1.59634e25 2.09159
\(847\) 1.86254e24 0.241600
\(848\) 4.37769e23 0.0562186
\(849\) −4.30081e24 −0.546808
\(850\) −7.87384e23 −0.0991118
\(851\) −1.83271e25 −2.28398
\(852\) 8.84423e24 1.09125
\(853\) −8.89565e24 −1.08670 −0.543352 0.839505i \(-0.682845\pi\)
−0.543352 + 0.839505i \(0.682845\pi\)
\(854\) 4.69284e24 0.567602
\(855\) 1.85272e24 0.221870
\(856\) 3.92331e24 0.465185
\(857\) 1.23863e25 1.45413 0.727066 0.686567i \(-0.240883\pi\)
0.727066 + 0.686567i \(0.240883\pi\)
\(858\) −1.06395e25 −1.23674
\(859\) 3.17743e24 0.365708 0.182854 0.983140i \(-0.441466\pi\)
0.182854 + 0.983140i \(0.441466\pi\)
\(860\) 3.32752e24 0.379213
\(861\) −1.15559e25 −1.30400
\(862\) 6.68909e24 0.747404
\(863\) 1.42021e25 1.57130 0.785652 0.618669i \(-0.212328\pi\)
0.785652 + 0.618669i \(0.212328\pi\)
\(864\) 1.57917e24 0.173006
\(865\) −4.79176e24 −0.519826
\(866\) 3.26905e24 0.351172
\(867\) 7.72238e24 0.821465
\(868\) 9.22933e24 0.972193
\(869\) 8.57475e24 0.894444
\(870\) 4.44303e24 0.458950
\(871\) 1.16279e25 1.18945
\(872\) 7.50372e23 0.0760128
\(873\) 1.95263e25 1.95884
\(874\) −3.89685e24 −0.387139
\(875\) −9.58359e23 −0.0942887
\(876\) −8.41551e23 −0.0819966
\(877\) −8.28800e24 −0.799748 −0.399874 0.916570i \(-0.630946\pi\)
−0.399874 + 0.916570i \(0.630946\pi\)
\(878\) 7.25527e24 0.693347
\(879\) −2.89653e25 −2.74140
\(880\) −1.32241e24 −0.123955
\(881\) 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 \(-0.505279\pi\)
−0.0165833 + 0.999862i \(0.505279\pi\)
\(884\) −3.79670e24 −0.342422
\(885\) −6.93094e24 −0.619119
\(886\) 3.71768e24 0.328916
\(887\) 1.85309e25 1.62385 0.811923 0.583765i \(-0.198421\pi\)
0.811923 + 0.583765i \(0.198421\pi\)
\(888\) 8.47342e24 0.735442
\(889\) 1.84024e24 0.158201
\(890\) −2.21332e24 −0.188464
\(891\) −3.45369e23 −0.0291288
\(892\) 1.89515e24 0.158322
\(893\) 6.87388e24 0.568807
\(894\) −5.45481e24 −0.447106
\(895\) 6.92810e24 0.562495
\(896\) −1.15859e24 −0.0931772
\(897\) −3.51015e25 −2.79634
\(898\) −3.55579e24 −0.280600
\(899\) −2.12121e25 −1.65816
\(900\) 2.07428e24 0.160622
\(901\) 2.05452e24 0.157598
\(902\) −7.90506e24 −0.600689
\(903\) 3.83411e25 2.88615
\(904\) 4.05613e24 0.302469
\(905\) 5.48059e24 0.404869
\(906\) −2.62169e25 −1.91863
\(907\) −1.87024e25 −1.35592 −0.677962 0.735097i \(-0.737137\pi\)
−0.677962 + 0.735097i \(0.737137\pi\)
\(908\) −8.66674e24 −0.622481
\(909\) 7.83551e24 0.557538
\(910\) −4.62113e24 −0.325759
\(911\) −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
\(915\) −8.17041e24 −0.549749
\(916\) −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 \(0.773536\pi\)
\(920\) −4.36285e24 −0.280268
\(921\) 4.37347e25 2.78368
\(922\) −4.10221e24 −0.258705
\(923\) 2.11419e25 1.32108
\(924\) −1.52374e25 −0.943406
\(925\) 4.20071e24 0.257702
\(926\) −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 \(0.156757\pi\)
\(930\) −1.60686e25 −0.941614
\(931\) −5.92010e23 −0.0343761
\(932\) 7.38860e24 0.425135
\(933\) −2.90417e25 −1.65588
\(934\) 1.07803e25 0.609090
\(935\) −6.20629e24 −0.347483
\(936\) 1.00020e25 0.554935
\(937\) −2.48730e25 −1.36755 −0.683774 0.729694i \(-0.739663\pi\)
−0.683774 + 0.729694i \(0.739663\pi\)
\(938\) 1.66529e25 0.907331
\(939\) 5.47381e24 0.295551
\(940\) 7.69588e24 0.411786
\(941\) −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
\(945\) 9.02064e24 0.461389
\(946\) 2.62281e25 1.32951
\(947\) −3.09044e25 −1.55255 −0.776276 0.630393i \(-0.782894\pi\)
−0.776276 + 0.630393i \(0.782894\pi\)
\(948\) −1.30795e25 −0.651210
\(949\) −2.01170e24 −0.0992661
\(950\) 8.93190e23 0.0436810
\(951\) 1.86893e25 0.905856
\(952\) −5.43744e24 −0.261204
\(953\) 2.50647e24 0.119337 0.0596683 0.998218i \(-0.480996\pi\)
0.0596683 + 0.998218i \(0.480996\pi\)
\(954\) −5.41242e24 −0.255406
\(955\) −4.78465e24 −0.223780
\(956\) −1.07673e25 −0.499131
\(957\) 3.50207e25 1.60906
\(958\) 1.05834e25 0.481966
\(959\) −6.70005e24 −0.302426
\(960\) 2.01714e24 0.0902466
\(961\) 5.41655e25 2.40200
\(962\) 2.02555e25 0.890337
\(963\) −4.85065e25 −2.11337
\(964\) 8.23438e24 0.355612
\(965\) −1.55781e25 −0.666856
\(966\) −5.02707e25 −2.13309
\(967\) −1.13278e25 −0.476455 −0.238228 0.971209i \(-0.576566\pi\)
−0.238228 + 0.971209i \(0.576566\pi\)
\(968\) −1.94347e24 −0.0810282
\(969\) 8.45564e24 0.349457
\(970\) 9.41354e24 0.385650
\(971\) 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
\(975\) 8.04556e24 0.315513
\(976\) −4.89675e24 −0.190364
\(977\) 8.86356e24 0.341589 0.170795 0.985307i \(-0.445367\pi\)
0.170795 + 0.985307i \(0.445367\pi\)
\(978\) −1.01952e25 −0.389507
\(979\) −1.74458e25 −0.660750
\(980\) −6.62805e23 −0.0248865
\(981\) −9.27734e24 −0.345332
\(982\) 7.34492e23 0.0271044
\(983\) −1.56885e25 −0.573954 −0.286977 0.957938i \(-0.592650\pi\)
−0.286977 + 0.957938i \(0.592650\pi\)
\(984\) 1.20580e25 0.437338
\(985\) 5.19230e24 0.186703
\(986\) 1.24971e25 0.445507
\(987\) 8.86754e25 3.13406
\(988\) 4.30688e24 0.150914
\(989\) 8.65307e25 3.00609
\(990\) 1.63498e25 0.563137
\(991\) −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
\(995\) −2.71602e24 −0.0896265
\(996\) 1.48217e25 0.484946
\(997\) −6.58977e24 −0.213777 −0.106889 0.994271i \(-0.534089\pi\)
−0.106889 + 0.994271i \(0.534089\pi\)
\(998\) 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 10.18.a.b.1.1 2
3.2 odd 2 90.18.a.n.1.2 2
4.3 odd 2 80.18.a.e.1.2 2
5.2 odd 4 50.18.b.e.49.2 4
5.3 odd 4 50.18.b.e.49.3 4
5.4 even 2 50.18.a.g.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.18.a.b.1.1 2 1.1 even 1 trivial
50.18.a.g.1.2 2 5.4 even 2
50.18.b.e.49.2 4 5.2 odd 4
50.18.b.e.49.3 4 5.3 odd 4
80.18.a.e.1.2 2 4.3 odd 2
90.18.a.n.1.2 2 3.2 odd 2