Properties

Label 3.42.a.a.1.1
Level $3$
Weight $42$
Character 3.1
Self dual yes
Analytic conductor $31.942$
Analytic rank $1$
Dimension $3$
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] = [3,42,Mod(1,3)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 42, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3.1");
 
S:= CuspForms(chi, 42);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 42 \)
Character orbit: \([\chi]\) \(=\) 3.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: \(31.9415011369\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 14982256920x + 433388802120300 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{5}\cdot 7 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(103995.\) of defining polynomial
Character \(\chi\) \(=\) 3.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.34440e6 q^{2} -3.48678e9 q^{3} -3.91623e11 q^{4} +1.06877e14 q^{5} +4.68762e15 q^{6} -3.99469e17 q^{7} +3.48285e18 q^{8} +1.21577e19 q^{9} +O(q^{10})\) \(q-1.34440e6 q^{2} -3.48678e9 q^{3} -3.91623e11 q^{4} +1.06877e14 q^{5} +4.68762e15 q^{6} -3.99469e17 q^{7} +3.48285e18 q^{8} +1.21577e19 q^{9} -1.43685e20 q^{10} +2.59920e21 q^{11} +1.36550e21 q^{12} -3.34490e22 q^{13} +5.37044e23 q^{14} -3.72656e23 q^{15} -3.82115e24 q^{16} +2.78752e25 q^{17} -1.63447e25 q^{18} +2.51816e26 q^{19} -4.18553e25 q^{20} +1.39286e27 q^{21} -3.49435e27 q^{22} -2.03286e27 q^{23} -1.21440e28 q^{24} -3.40521e28 q^{25} +4.49686e28 q^{26} -4.23912e28 q^{27} +1.56441e29 q^{28} -6.39430e29 q^{29} +5.00997e29 q^{30} -3.99382e29 q^{31} -2.52174e30 q^{32} -9.06284e30 q^{33} -3.74753e31 q^{34} -4.26939e31 q^{35} -4.76122e30 q^{36} -1.48588e32 q^{37} -3.38540e32 q^{38} +1.16629e32 q^{39} +3.72236e32 q^{40} +5.94247e31 q^{41} -1.87256e33 q^{42} -1.25575e33 q^{43} -1.01790e33 q^{44} +1.29937e33 q^{45} +2.73297e33 q^{46} +3.54758e33 q^{47} +1.33235e34 q^{48} +1.15008e35 q^{49} +4.57795e34 q^{50} -9.71949e34 q^{51} +1.30994e34 q^{52} +1.54656e34 q^{53} +5.69905e34 q^{54} +2.77793e35 q^{55} -1.39129e36 q^{56} -8.78027e35 q^{57} +8.59647e35 q^{58} -1.46481e35 q^{59} +1.45940e35 q^{60} -4.04266e36 q^{61} +5.36927e35 q^{62} -4.85661e36 q^{63} +1.17930e37 q^{64} -3.57491e36 q^{65} +1.21840e37 q^{66} -8.80253e36 q^{67} -1.09166e37 q^{68} +7.08816e36 q^{69} +5.73975e37 q^{70} +1.53426e38 q^{71} +4.23434e37 q^{72} -2.36946e38 q^{73} +1.99761e38 q^{74} +1.18732e38 q^{75} -9.86167e37 q^{76} -1.03830e39 q^{77} -1.56796e38 q^{78} -4.62673e38 q^{79} -4.08391e38 q^{80} +1.47809e38 q^{81} -7.98903e37 q^{82} +1.09848e39 q^{83} -5.45476e38 q^{84} +2.97921e39 q^{85} +1.68823e39 q^{86} +2.22955e39 q^{87} +9.05262e39 q^{88} -1.60084e40 q^{89} -1.74687e39 q^{90} +1.33618e40 q^{91} +7.96115e38 q^{92} +1.39256e39 q^{93} -4.76936e39 q^{94} +2.69132e40 q^{95} +8.79277e39 q^{96} +8.97977e38 q^{97} -1.54616e41 q^{98} +3.16002e40 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 289380 q^{2} - 10460353203 q^{3} - 2254266178800 q^{4} + 38650546192026 q^{5} + 10\!\cdots\!80 q^{6}+ \cdots + 36\!\cdots\!03 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 289380 q^{2} - 10460353203 q^{3} - 2254266178800 q^{4} + 38650546192026 q^{5} + 10\!\cdots\!80 q^{6}+ \cdots + 26\!\cdots\!04 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.34440e6 −0.906593 −0.453296 0.891360i \(-0.649752\pi\)
−0.453296 + 0.891360i \(0.649752\pi\)
\(3\) −3.48678e9 −0.577350
\(4\) −3.91623e11 −0.178089
\(5\) 1.06877e14 0.501185 0.250592 0.968093i \(-0.419375\pi\)
0.250592 + 0.968093i \(0.419375\pi\)
\(6\) 4.68762e15 0.523422
\(7\) −3.99469e17 −1.89223 −0.946113 0.323835i \(-0.895028\pi\)
−0.946113 + 0.323835i \(0.895028\pi\)
\(8\) 3.48285e18 1.06805
\(9\) 1.21577e19 0.333333
\(10\) −1.43685e20 −0.454370
\(11\) 2.59920e21 1.16490 0.582450 0.812866i \(-0.302094\pi\)
0.582450 + 0.812866i \(0.302094\pi\)
\(12\) 1.36550e21 0.102820
\(13\) −3.34490e22 −0.488139 −0.244070 0.969758i \(-0.578483\pi\)
−0.244070 + 0.969758i \(0.578483\pi\)
\(14\) 5.37044e23 1.71548
\(15\) −3.72656e23 −0.289359
\(16\) −3.82115e24 −0.790195
\(17\) 2.78752e25 1.66347 0.831736 0.555172i \(-0.187348\pi\)
0.831736 + 0.555172i \(0.187348\pi\)
\(18\) −1.63447e25 −0.302198
\(19\) 2.51816e26 1.53686 0.768430 0.639934i \(-0.221038\pi\)
0.768430 + 0.639934i \(0.221038\pi\)
\(20\) −4.18553e25 −0.0892556
\(21\) 1.39286e27 1.09248
\(22\) −3.49435e27 −1.05609
\(23\) −2.03286e27 −0.246995 −0.123497 0.992345i \(-0.539411\pi\)
−0.123497 + 0.992345i \(0.539411\pi\)
\(24\) −1.21440e28 −0.616637
\(25\) −3.40521e28 −0.748814
\(26\) 4.49686e28 0.442544
\(27\) −4.23912e28 −0.192450
\(28\) 1.56441e29 0.336985
\(29\) −6.39430e29 −0.670863 −0.335431 0.942065i \(-0.608882\pi\)
−0.335431 + 0.942065i \(0.608882\pi\)
\(30\) 5.00997e29 0.262331
\(31\) −3.99382e29 −0.106776 −0.0533880 0.998574i \(-0.517002\pi\)
−0.0533880 + 0.998574i \(0.517002\pi\)
\(32\) −2.52174e30 −0.351662
\(33\) −9.06284e30 −0.672556
\(34\) −3.74753e31 −1.50809
\(35\) −4.26939e31 −0.948355
\(36\) −4.76122e30 −0.0593631
\(37\) −1.48588e32 −1.05645 −0.528223 0.849106i \(-0.677141\pi\)
−0.528223 + 0.849106i \(0.677141\pi\)
\(38\) −3.38540e32 −1.39331
\(39\) 1.16629e32 0.281827
\(40\) 3.72236e32 0.535289
\(41\) 5.94247e31 0.0515107 0.0257554 0.999668i \(-0.491801\pi\)
0.0257554 + 0.999668i \(0.491801\pi\)
\(42\) −1.87256e33 −0.990433
\(43\) −1.25575e33 −0.410015 −0.205008 0.978760i \(-0.565722\pi\)
−0.205008 + 0.978760i \(0.565722\pi\)
\(44\) −1.01790e33 −0.207456
\(45\) 1.29937e33 0.167062
\(46\) 2.73297e33 0.223924
\(47\) 3.54758e33 0.187037 0.0935183 0.995618i \(-0.470189\pi\)
0.0935183 + 0.995618i \(0.470189\pi\)
\(48\) 1.33235e34 0.456219
\(49\) 1.15008e35 2.58052
\(50\) 4.57795e34 0.678869
\(51\) −9.71949e34 −0.960406
\(52\) 1.30994e34 0.0869324
\(53\) 1.54656e34 0.0694564 0.0347282 0.999397i \(-0.488943\pi\)
0.0347282 + 0.999397i \(0.488943\pi\)
\(54\) 5.69905e34 0.174474
\(55\) 2.77793e35 0.583830
\(56\) −1.39129e36 −2.02099
\(57\) −8.78027e35 −0.887306
\(58\) 8.59647e35 0.608200
\(59\) −1.46481e35 −0.0729988 −0.0364994 0.999334i \(-0.511621\pi\)
−0.0364994 + 0.999334i \(0.511621\pi\)
\(60\) 1.45940e35 0.0515318
\(61\) −4.04266e36 −1.01720 −0.508600 0.861003i \(-0.669837\pi\)
−0.508600 + 0.861003i \(0.669837\pi\)
\(62\) 5.36927e35 0.0968023
\(63\) −4.85661e36 −0.630742
\(64\) 1.17930e37 1.10901
\(65\) −3.57491e36 −0.244648
\(66\) 1.21840e37 0.609734
\(67\) −8.80253e36 −0.323649 −0.161824 0.986820i \(-0.551738\pi\)
−0.161824 + 0.986820i \(0.551738\pi\)
\(68\) −1.09166e37 −0.296246
\(69\) 7.08816e36 0.142602
\(70\) 5.73975e37 0.859772
\(71\) 1.53426e38 1.71832 0.859160 0.511707i \(-0.170987\pi\)
0.859160 + 0.511707i \(0.170987\pi\)
\(72\) 4.23434e37 0.356016
\(73\) −2.36946e38 −1.50152 −0.750759 0.660577i \(-0.770312\pi\)
−0.750759 + 0.660577i \(0.770312\pi\)
\(74\) 1.99761e38 0.957766
\(75\) 1.18732e38 0.432328
\(76\) −9.86167e37 −0.273698
\(77\) −1.03830e39 −2.20426
\(78\) −1.56796e38 −0.255503
\(79\) −4.62673e38 −0.580656 −0.290328 0.956927i \(-0.593764\pi\)
−0.290328 + 0.956927i \(0.593764\pi\)
\(80\) −4.08391e38 −0.396033
\(81\) 1.47809e38 0.111111
\(82\) −7.98903e37 −0.0466992
\(83\) 1.09848e39 0.500830 0.250415 0.968139i \(-0.419433\pi\)
0.250415 + 0.968139i \(0.419433\pi\)
\(84\) −5.45476e38 −0.194559
\(85\) 2.97921e39 0.833706
\(86\) 1.68823e39 0.371717
\(87\) 2.22955e39 0.387323
\(88\) 9.05262e39 1.24417
\(89\) −1.60084e40 −1.74523 −0.872617 0.488405i \(-0.837579\pi\)
−0.872617 + 0.488405i \(0.837579\pi\)
\(90\) −1.74687e39 −0.151457
\(91\) 1.33618e40 0.923670
\(92\) 7.96115e38 0.0439871
\(93\) 1.39256e39 0.0616471
\(94\) −4.76936e39 −0.169566
\(95\) 2.69132e40 0.770250
\(96\) 8.79277e39 0.203032
\(97\) 8.97977e38 0.0167666 0.00838329 0.999965i \(-0.497331\pi\)
0.00838329 + 0.999965i \(0.497331\pi\)
\(98\) −1.54616e41 −2.33948
\(99\) 3.16002e40 0.388300
\(100\) 1.33356e40 0.133356
\(101\) −9.88339e40 −0.805968 −0.402984 0.915207i \(-0.632027\pi\)
−0.402984 + 0.915207i \(0.632027\pi\)
\(102\) 1.30668e41 0.870697
\(103\) −2.00844e41 −1.09571 −0.547856 0.836573i \(-0.684556\pi\)
−0.547856 + 0.836573i \(0.684556\pi\)
\(104\) −1.16498e41 −0.521356
\(105\) 1.48864e41 0.547533
\(106\) −2.07919e40 −0.0629687
\(107\) 1.39775e41 0.349189 0.174595 0.984640i \(-0.444138\pi\)
0.174595 + 0.984640i \(0.444138\pi\)
\(108\) 1.66013e40 0.0342733
\(109\) 9.35581e41 1.59896 0.799481 0.600691i \(-0.205108\pi\)
0.799481 + 0.600691i \(0.205108\pi\)
\(110\) −3.73464e41 −0.529296
\(111\) 5.18093e41 0.609939
\(112\) 1.52643e42 1.49523
\(113\) −1.69777e42 −1.38603 −0.693013 0.720925i \(-0.743717\pi\)
−0.693013 + 0.720925i \(0.743717\pi\)
\(114\) 1.18042e42 0.804425
\(115\) −2.17266e41 −0.123790
\(116\) 2.50415e41 0.119474
\(117\) −4.06661e41 −0.162713
\(118\) 1.96929e41 0.0661802
\(119\) −1.11353e43 −3.14767
\(120\) −1.29791e42 −0.309049
\(121\) 1.77730e42 0.356994
\(122\) 5.43494e42 0.922186
\(123\) −2.07201e41 −0.0297397
\(124\) 1.56407e41 0.0190156
\(125\) −8.49956e42 −0.876479
\(126\) 6.52921e42 0.571827
\(127\) 1.24842e43 0.929793 0.464896 0.885365i \(-0.346092\pi\)
0.464896 + 0.885365i \(0.346092\pi\)
\(128\) −1.03091e43 −0.653758
\(129\) 4.37853e42 0.236722
\(130\) 4.80610e42 0.221796
\(131\) 4.74003e43 1.86947 0.934737 0.355339i \(-0.115635\pi\)
0.934737 + 0.355339i \(0.115635\pi\)
\(132\) 3.54921e42 0.119775
\(133\) −1.00593e44 −2.90809
\(134\) 1.18341e43 0.293418
\(135\) −4.53062e42 −0.0964530
\(136\) 9.70853e43 1.77667
\(137\) 4.51190e43 0.710539 0.355269 0.934764i \(-0.384389\pi\)
0.355269 + 0.934764i \(0.384389\pi\)
\(138\) −9.52929e42 −0.129282
\(139\) 1.39477e43 0.163192 0.0815959 0.996665i \(-0.473998\pi\)
0.0815959 + 0.996665i \(0.473998\pi\)
\(140\) 1.67199e43 0.168892
\(141\) −1.23697e43 −0.107986
\(142\) −2.06266e44 −1.55782
\(143\) −8.69404e43 −0.568634
\(144\) −4.64562e43 −0.263398
\(145\) −6.83401e43 −0.336226
\(146\) 3.18549e44 1.36126
\(147\) −4.01007e44 −1.48987
\(148\) 5.81903e43 0.188142
\(149\) −2.52241e44 −0.710391 −0.355196 0.934792i \(-0.615586\pi\)
−0.355196 + 0.934792i \(0.615586\pi\)
\(150\) −1.59623e44 −0.391945
\(151\) −3.68239e44 −0.789046 −0.394523 0.918886i \(-0.629090\pi\)
−0.394523 + 0.918886i \(0.629090\pi\)
\(152\) 8.77037e44 1.64144
\(153\) 3.38898e44 0.554490
\(154\) 1.39588e45 1.99836
\(155\) −4.26846e43 −0.0535144
\(156\) −4.56747e43 −0.0501904
\(157\) −4.32607e44 −0.417014 −0.208507 0.978021i \(-0.566860\pi\)
−0.208507 + 0.978021i \(0.566860\pi\)
\(158\) 6.22016e44 0.526418
\(159\) −5.39253e43 −0.0401007
\(160\) −2.69515e44 −0.176248
\(161\) 8.12066e44 0.467370
\(162\) −1.98714e44 −0.100733
\(163\) −1.63490e45 −0.730541 −0.365270 0.930901i \(-0.619023\pi\)
−0.365270 + 0.930901i \(0.619023\pi\)
\(164\) −2.32720e43 −0.00917351
\(165\) −9.68605e44 −0.337075
\(166\) −1.47679e45 −0.454049
\(167\) −4.35680e45 −1.18435 −0.592175 0.805810i \(-0.701730\pi\)
−0.592175 + 0.805810i \(0.701730\pi\)
\(168\) 4.85114e45 1.16682
\(169\) −3.57662e45 −0.761720
\(170\) −4.00524e45 −0.755832
\(171\) 3.06149e45 0.512286
\(172\) 4.91780e44 0.0730193
\(173\) −2.11282e44 −0.0278559 −0.0139280 0.999903i \(-0.504434\pi\)
−0.0139280 + 0.999903i \(0.504434\pi\)
\(174\) −2.99740e45 −0.351144
\(175\) 1.36028e46 1.41693
\(176\) −9.93191e45 −0.920499
\(177\) 5.10748e44 0.0421459
\(178\) 2.15217e46 1.58222
\(179\) 4.71116e45 0.308774 0.154387 0.988010i \(-0.450660\pi\)
0.154387 + 0.988010i \(0.450660\pi\)
\(180\) −5.08863e44 −0.0297519
\(181\) −6.10024e45 −0.318373 −0.159186 0.987249i \(-0.550887\pi\)
−0.159186 + 0.987249i \(0.550887\pi\)
\(182\) −1.79636e46 −0.837393
\(183\) 1.40959e46 0.587280
\(184\) −7.08017e45 −0.263802
\(185\) −1.58805e46 −0.529474
\(186\) −1.87215e45 −0.0558888
\(187\) 7.24531e46 1.93778
\(188\) −1.38931e45 −0.0333092
\(189\) 1.69340e46 0.364159
\(190\) −3.61820e46 −0.698303
\(191\) 1.46963e46 0.254697 0.127348 0.991858i \(-0.459353\pi\)
0.127348 + 0.991858i \(0.459353\pi\)
\(192\) −4.11197e46 −0.640287
\(193\) −8.24885e46 −1.15470 −0.577349 0.816497i \(-0.695913\pi\)
−0.577349 + 0.816497i \(0.695913\pi\)
\(194\) −1.20724e45 −0.0152005
\(195\) 1.24649e46 0.141248
\(196\) −4.50397e46 −0.459564
\(197\) 6.11379e46 0.562022 0.281011 0.959705i \(-0.409330\pi\)
0.281011 + 0.959705i \(0.409330\pi\)
\(198\) −4.24831e46 −0.352030
\(199\) −2.06724e47 −1.54491 −0.772453 0.635072i \(-0.780971\pi\)
−0.772453 + 0.635072i \(0.780971\pi\)
\(200\) −1.18599e47 −0.799769
\(201\) 3.06925e46 0.186859
\(202\) 1.32872e47 0.730685
\(203\) 2.55432e47 1.26942
\(204\) 3.80637e46 0.171038
\(205\) 6.35111e45 0.0258164
\(206\) 2.70014e47 0.993365
\(207\) −2.47149e46 −0.0823316
\(208\) 1.27813e47 0.385725
\(209\) 6.54518e47 1.79029
\(210\) −2.00133e47 −0.496390
\(211\) 3.08234e45 0.00693570 0.00346785 0.999994i \(-0.498896\pi\)
0.00346785 + 0.999994i \(0.498896\pi\)
\(212\) −6.05669e45 −0.0123694
\(213\) −5.34965e47 −0.992072
\(214\) −1.87913e47 −0.316573
\(215\) −1.34210e47 −0.205493
\(216\) −1.47642e47 −0.205546
\(217\) 1.59541e47 0.202044
\(218\) −1.25779e48 −1.44961
\(219\) 8.26179e47 0.866901
\(220\) −1.08790e47 −0.103974
\(221\) −9.32397e47 −0.812006
\(222\) −6.96522e47 −0.552967
\(223\) −5.03674e47 −0.364670 −0.182335 0.983236i \(-0.558365\pi\)
−0.182335 + 0.983236i \(0.558365\pi\)
\(224\) 1.00736e48 0.665425
\(225\) −4.13994e47 −0.249605
\(226\) 2.28248e48 1.25656
\(227\) −1.43782e47 −0.0723058 −0.0361529 0.999346i \(-0.511510\pi\)
−0.0361529 + 0.999346i \(0.511510\pi\)
\(228\) 3.43855e47 0.158020
\(229\) −2.29667e48 −0.964874 −0.482437 0.875931i \(-0.660248\pi\)
−0.482437 + 0.875931i \(0.660248\pi\)
\(230\) 2.92091e47 0.112227
\(231\) 3.62032e48 1.27263
\(232\) −2.22704e48 −0.716513
\(233\) 2.20055e47 0.0648239 0.0324119 0.999475i \(-0.489681\pi\)
0.0324119 + 0.999475i \(0.489681\pi\)
\(234\) 5.46714e47 0.147515
\(235\) 3.79154e47 0.0937398
\(236\) 5.73653e46 0.0130003
\(237\) 1.61324e48 0.335242
\(238\) 1.49702e49 2.85365
\(239\) −1.04096e49 −1.82087 −0.910434 0.413655i \(-0.864252\pi\)
−0.910434 + 0.413655i \(0.864252\pi\)
\(240\) 1.42397e48 0.228650
\(241\) −8.83597e48 −1.30288 −0.651440 0.758700i \(-0.725835\pi\)
−0.651440 + 0.758700i \(0.725835\pi\)
\(242\) −2.38940e48 −0.323648
\(243\) −5.15378e47 −0.0641500
\(244\) 1.58320e48 0.181152
\(245\) 1.22916e49 1.29332
\(246\) 2.78560e47 0.0269618
\(247\) −8.42297e48 −0.750201
\(248\) −1.39099e48 −0.114042
\(249\) −3.83016e48 −0.289155
\(250\) 1.14268e49 0.794609
\(251\) −2.56380e49 −1.64275 −0.821377 0.570386i \(-0.806794\pi\)
−0.821377 + 0.570386i \(0.806794\pi\)
\(252\) 1.90196e48 0.112328
\(253\) −5.28381e48 −0.287724
\(254\) −1.67838e49 −0.842943
\(255\) −1.03879e49 −0.481340
\(256\) −1.20736e49 −0.516317
\(257\) 3.19009e49 1.25943 0.629715 0.776827i \(-0.283172\pi\)
0.629715 + 0.776827i \(0.283172\pi\)
\(258\) −5.88648e48 −0.214611
\(259\) 5.93561e49 1.99904
\(260\) 1.40002e48 0.0435692
\(261\) −7.77397e48 −0.223621
\(262\) −6.37248e49 −1.69485
\(263\) 2.05862e49 0.506389 0.253194 0.967415i \(-0.418519\pi\)
0.253194 + 0.967415i \(0.418519\pi\)
\(264\) −3.15645e49 −0.718321
\(265\) 1.65292e48 0.0348105
\(266\) 1.35236e50 2.63645
\(267\) 5.58180e49 1.00761
\(268\) 3.44727e48 0.0576384
\(269\) 9.01933e49 1.39718 0.698590 0.715522i \(-0.253811\pi\)
0.698590 + 0.715522i \(0.253811\pi\)
\(270\) 6.09095e48 0.0874436
\(271\) −2.40475e49 −0.320037 −0.160018 0.987114i \(-0.551155\pi\)
−0.160018 + 0.987114i \(0.551155\pi\)
\(272\) −1.06515e50 −1.31447
\(273\) −4.65898e49 −0.533281
\(274\) −6.06578e49 −0.644169
\(275\) −8.85081e49 −0.872294
\(276\) −2.77588e48 −0.0253960
\(277\) −7.23233e49 −0.614388 −0.307194 0.951647i \(-0.599390\pi\)
−0.307194 + 0.951647i \(0.599390\pi\)
\(278\) −1.87512e49 −0.147949
\(279\) −4.85555e48 −0.0355920
\(280\) −1.48697e50 −1.01289
\(281\) 1.01780e50 0.644440 0.322220 0.946665i \(-0.395571\pi\)
0.322220 + 0.946665i \(0.395571\pi\)
\(282\) 1.66297e49 0.0978990
\(283\) 6.28092e49 0.343875 0.171938 0.985108i \(-0.444997\pi\)
0.171938 + 0.985108i \(0.444997\pi\)
\(284\) −6.00852e49 −0.306014
\(285\) −9.38406e49 −0.444704
\(286\) 1.16882e50 0.515519
\(287\) −2.37383e49 −0.0974700
\(288\) −3.06585e49 −0.117221
\(289\) 4.96222e50 1.76714
\(290\) 9.18762e49 0.304820
\(291\) −3.13105e48 −0.00968019
\(292\) 9.27933e49 0.267404
\(293\) −5.08876e50 −1.36718 −0.683590 0.729866i \(-0.739582\pi\)
−0.683590 + 0.729866i \(0.739582\pi\)
\(294\) 5.39113e50 1.35070
\(295\) −1.56554e49 −0.0365859
\(296\) −5.17509e50 −1.12833
\(297\) −1.10183e50 −0.224185
\(298\) 3.39112e50 0.644036
\(299\) 6.79972e49 0.120568
\(300\) −4.64983e49 −0.0769930
\(301\) 5.01633e50 0.775842
\(302\) 4.95059e50 0.715344
\(303\) 3.44612e50 0.465326
\(304\) −9.62225e50 −1.21442
\(305\) −4.32066e50 −0.509805
\(306\) −4.55612e50 −0.502697
\(307\) −1.50028e50 −0.154823 −0.0774113 0.996999i \(-0.524665\pi\)
−0.0774113 + 0.996999i \(0.524665\pi\)
\(308\) 4.06621e50 0.392555
\(309\) 7.00301e50 0.632610
\(310\) 5.73849e49 0.0485158
\(311\) −1.32377e51 −1.04767 −0.523836 0.851819i \(-0.675500\pi\)
−0.523836 + 0.851819i \(0.675500\pi\)
\(312\) 4.06203e50 0.301005
\(313\) −1.00439e51 −0.697018 −0.348509 0.937305i \(-0.613312\pi\)
−0.348509 + 0.937305i \(0.613312\pi\)
\(314\) 5.81595e50 0.378062
\(315\) −5.19058e50 −0.316118
\(316\) 1.81193e50 0.103409
\(317\) −2.46384e51 −1.31794 −0.658972 0.752167i \(-0.729009\pi\)
−0.658972 + 0.752167i \(0.729009\pi\)
\(318\) 7.24970e49 0.0363550
\(319\) −1.66200e51 −0.781489
\(320\) 1.26040e51 0.555818
\(321\) −4.87364e50 −0.201605
\(322\) −1.09174e51 −0.423714
\(323\) 7.01942e51 2.55652
\(324\) −5.78853e49 −0.0197877
\(325\) 1.13901e51 0.365526
\(326\) 2.19795e51 0.662303
\(327\) −3.26217e51 −0.923161
\(328\) 2.06967e50 0.0550159
\(329\) −1.41715e51 −0.353916
\(330\) 1.30219e51 0.305589
\(331\) 7.34740e51 1.62054 0.810270 0.586057i \(-0.199320\pi\)
0.810270 + 0.586057i \(0.199320\pi\)
\(332\) −4.30189e50 −0.0891925
\(333\) −1.80648e51 −0.352149
\(334\) 5.85726e51 1.07372
\(335\) −9.40785e50 −0.162208
\(336\) −5.32233e51 −0.863270
\(337\) −6.00807e51 −0.916900 −0.458450 0.888720i \(-0.651595\pi\)
−0.458450 + 0.888720i \(0.651595\pi\)
\(338\) 4.80839e51 0.690570
\(339\) 5.91977e51 0.800223
\(340\) −1.16673e51 −0.148474
\(341\) −1.03807e51 −0.124383
\(342\) −4.11586e51 −0.464435
\(343\) −2.81387e52 −2.99071
\(344\) −4.37359e51 −0.437916
\(345\) 7.57559e50 0.0714702
\(346\) 2.84047e50 0.0252540
\(347\) 8.44413e51 0.707621 0.353811 0.935317i \(-0.384886\pi\)
0.353811 + 0.935317i \(0.384886\pi\)
\(348\) −8.73143e50 −0.0689781
\(349\) 1.62504e52 1.21044 0.605219 0.796059i \(-0.293086\pi\)
0.605219 + 0.796059i \(0.293086\pi\)
\(350\) −1.82875e52 −1.28458
\(351\) 1.41794e51 0.0939425
\(352\) −6.55450e51 −0.409652
\(353\) −3.57047e50 −0.0210544 −0.0105272 0.999945i \(-0.503351\pi\)
−0.0105272 + 0.999945i \(0.503351\pi\)
\(354\) −6.86648e50 −0.0382091
\(355\) 1.63977e52 0.861195
\(356\) 6.26926e51 0.310808
\(357\) 3.88263e52 1.81731
\(358\) −6.33367e51 −0.279932
\(359\) −8.95225e51 −0.373677 −0.186838 0.982391i \(-0.559824\pi\)
−0.186838 + 0.982391i \(0.559824\pi\)
\(360\) 4.52552e51 0.178430
\(361\) 3.65641e52 1.36194
\(362\) 8.20113e51 0.288634
\(363\) −6.19706e51 −0.206110
\(364\) −5.23279e51 −0.164496
\(365\) −2.53240e52 −0.752537
\(366\) −1.89505e52 −0.532424
\(367\) −2.14287e52 −0.569301 −0.284651 0.958631i \(-0.591878\pi\)
−0.284651 + 0.958631i \(0.591878\pi\)
\(368\) 7.76787e51 0.195174
\(369\) 7.22465e50 0.0171702
\(370\) 2.13497e52 0.480018
\(371\) −6.17804e51 −0.131427
\(372\) −5.45357e50 −0.0109787
\(373\) 1.75320e52 0.334041 0.167020 0.985953i \(-0.446585\pi\)
0.167020 + 0.985953i \(0.446585\pi\)
\(374\) −9.74057e52 −1.75678
\(375\) 2.96361e52 0.506035
\(376\) 1.23557e52 0.199764
\(377\) 2.13883e52 0.327475
\(378\) −2.27659e52 −0.330144
\(379\) 5.01381e52 0.688754 0.344377 0.938831i \(-0.388090\pi\)
0.344377 + 0.938831i \(0.388090\pi\)
\(380\) −1.05398e52 −0.137173
\(381\) −4.35299e52 −0.536816
\(382\) −1.97576e52 −0.230906
\(383\) −2.69866e52 −0.298933 −0.149466 0.988767i \(-0.547756\pi\)
−0.149466 + 0.988767i \(0.547756\pi\)
\(384\) 3.59456e52 0.377447
\(385\) −1.10970e53 −1.10474
\(386\) 1.10897e53 1.04684
\(387\) −1.52670e52 −0.136672
\(388\) −3.51668e50 −0.00298595
\(389\) −1.55739e53 −1.25438 −0.627190 0.778866i \(-0.715795\pi\)
−0.627190 + 0.778866i \(0.715795\pi\)
\(390\) −1.67578e52 −0.128054
\(391\) −5.66665e52 −0.410869
\(392\) 4.00555e53 2.75612
\(393\) −1.65275e53 −1.07934
\(394\) −8.21936e52 −0.509525
\(395\) −4.94489e52 −0.291016
\(396\) −1.23753e52 −0.0691521
\(397\) −3.01135e52 −0.159792 −0.0798961 0.996803i \(-0.525459\pi\)
−0.0798961 + 0.996803i \(0.525459\pi\)
\(398\) 2.77919e53 1.40060
\(399\) 3.50745e53 1.67898
\(400\) 1.30118e53 0.591709
\(401\) 2.30382e53 0.995378 0.497689 0.867355i \(-0.334182\pi\)
0.497689 + 0.867355i \(0.334182\pi\)
\(402\) −4.12629e52 −0.169405
\(403\) 1.33589e52 0.0521215
\(404\) 3.87056e52 0.143534
\(405\) 1.57973e52 0.0556872
\(406\) −3.43402e53 −1.15085
\(407\) −3.86208e53 −1.23065
\(408\) −3.38515e53 −1.02576
\(409\) 1.23126e53 0.354831 0.177415 0.984136i \(-0.443226\pi\)
0.177415 + 0.984136i \(0.443226\pi\)
\(410\) −8.53840e51 −0.0234049
\(411\) −1.57320e53 −0.410230
\(412\) 7.86552e52 0.195135
\(413\) 5.85147e52 0.138130
\(414\) 3.32266e52 0.0746412
\(415\) 1.17402e53 0.251008
\(416\) 8.43496e52 0.171660
\(417\) −4.86325e52 −0.0942188
\(418\) −8.79932e53 −1.62306
\(419\) −1.08562e54 −1.90673 −0.953367 0.301814i \(-0.902408\pi\)
−0.953367 + 0.301814i \(0.902408\pi\)
\(420\) −5.82987e52 −0.0975098
\(421\) 9.36709e53 1.49218 0.746091 0.665844i \(-0.231928\pi\)
0.746091 + 0.665844i \(0.231928\pi\)
\(422\) −4.14388e51 −0.00628785
\(423\) 4.31303e52 0.0623455
\(424\) 5.38645e52 0.0741828
\(425\) −9.49210e53 −1.24563
\(426\) 7.19205e53 0.899406
\(427\) 1.61492e54 1.92477
\(428\) −5.47389e52 −0.0621869
\(429\) 3.03142e53 0.328301
\(430\) 1.80432e53 0.186299
\(431\) 1.72107e54 1.69440 0.847198 0.531278i \(-0.178288\pi\)
0.847198 + 0.531278i \(0.178288\pi\)
\(432\) 1.61983e53 0.152073
\(433\) −1.20576e54 −1.07959 −0.539795 0.841796i \(-0.681498\pi\)
−0.539795 + 0.841796i \(0.681498\pi\)
\(434\) −2.14486e53 −0.183172
\(435\) 2.38287e53 0.194120
\(436\) −3.66395e53 −0.284758
\(437\) −5.11907e53 −0.379596
\(438\) −1.11071e54 −0.785927
\(439\) 1.69765e54 1.14637 0.573187 0.819424i \(-0.305707\pi\)
0.573187 + 0.819424i \(0.305707\pi\)
\(440\) 9.67514e53 0.623558
\(441\) 1.39823e54 0.860174
\(442\) 1.25351e54 0.736159
\(443\) −3.00986e54 −1.68760 −0.843800 0.536657i \(-0.819687\pi\)
−0.843800 + 0.536657i \(0.819687\pi\)
\(444\) −2.02897e53 −0.108624
\(445\) −1.71093e54 −0.874685
\(446\) 6.77137e53 0.330607
\(447\) 8.79511e53 0.410145
\(448\) −4.71094e54 −2.09850
\(449\) −1.71494e54 −0.729793 −0.364896 0.931048i \(-0.618896\pi\)
−0.364896 + 0.931048i \(0.618896\pi\)
\(450\) 5.56572e53 0.226290
\(451\) 1.54456e53 0.0600049
\(452\) 6.64886e53 0.246837
\(453\) 1.28397e54 0.455556
\(454\) 1.93299e53 0.0655520
\(455\) 1.42807e54 0.462929
\(456\) −3.05804e54 −0.947685
\(457\) 1.21867e54 0.361081 0.180541 0.983568i \(-0.442215\pi\)
0.180541 + 0.983568i \(0.442215\pi\)
\(458\) 3.08763e54 0.874748
\(459\) −1.18166e54 −0.320135
\(460\) 8.50861e52 0.0220457
\(461\) −1.07257e54 −0.265801 −0.132901 0.991129i \(-0.542429\pi\)
−0.132901 + 0.991129i \(0.542429\pi\)
\(462\) −4.86715e54 −1.15376
\(463\) 7.33040e53 0.166233 0.0831165 0.996540i \(-0.473513\pi\)
0.0831165 + 0.996540i \(0.473513\pi\)
\(464\) 2.44336e54 0.530112
\(465\) 1.48832e53 0.0308966
\(466\) −2.95841e53 −0.0587688
\(467\) 4.62154e54 0.878598 0.439299 0.898341i \(-0.355227\pi\)
0.439299 + 0.898341i \(0.355227\pi\)
\(468\) 1.59258e53 0.0289775
\(469\) 3.51634e54 0.612417
\(470\) −5.09733e53 −0.0849839
\(471\) 1.50841e54 0.240763
\(472\) −5.10173e53 −0.0779662
\(473\) −3.26394e54 −0.477627
\(474\) −2.16883e54 −0.303928
\(475\) −8.57486e54 −1.15082
\(476\) 4.36083e54 0.560566
\(477\) 1.88026e53 0.0231521
\(478\) 1.39947e55 1.65079
\(479\) 1.06254e54 0.120079 0.0600394 0.998196i \(-0.480877\pi\)
0.0600394 + 0.998196i \(0.480877\pi\)
\(480\) 9.39741e53 0.101757
\(481\) 4.97010e54 0.515693
\(482\) 1.18790e55 1.18118
\(483\) −2.83150e54 −0.269836
\(484\) −6.96031e53 −0.0635768
\(485\) 9.59727e52 0.00840315
\(486\) 6.92871e53 0.0581580
\(487\) −1.81852e55 −1.46344 −0.731720 0.681605i \(-0.761282\pi\)
−0.731720 + 0.681605i \(0.761282\pi\)
\(488\) −1.40800e55 −1.08642
\(489\) 5.70053e54 0.421778
\(490\) −1.65248e55 −1.17251
\(491\) 3.62159e54 0.246450 0.123225 0.992379i \(-0.460676\pi\)
0.123225 + 0.992379i \(0.460676\pi\)
\(492\) 8.11446e52 0.00529633
\(493\) −1.78242e55 −1.11596
\(494\) 1.13238e55 0.680127
\(495\) 3.37732e54 0.194610
\(496\) 1.52610e54 0.0843738
\(497\) −6.12891e55 −3.25145
\(498\) 5.14925e54 0.262145
\(499\) 2.06983e55 1.01128 0.505642 0.862744i \(-0.331256\pi\)
0.505642 + 0.862744i \(0.331256\pi\)
\(500\) 3.32862e54 0.156091
\(501\) 1.51912e55 0.683784
\(502\) 3.44676e55 1.48931
\(503\) −1.21024e55 −0.502026 −0.251013 0.967984i \(-0.580764\pi\)
−0.251013 + 0.967984i \(0.580764\pi\)
\(504\) −1.69149e55 −0.673663
\(505\) −1.05630e55 −0.403939
\(506\) 7.10354e54 0.260849
\(507\) 1.24709e55 0.439779
\(508\) −4.88911e54 −0.165586
\(509\) 2.83445e55 0.922051 0.461025 0.887387i \(-0.347482\pi\)
0.461025 + 0.887387i \(0.347482\pi\)
\(510\) 1.39654e55 0.436380
\(511\) 9.46525e55 2.84121
\(512\) 3.89016e55 1.12185
\(513\) −1.06748e55 −0.295769
\(514\) −4.28874e55 −1.14179
\(515\) −2.14656e55 −0.549154
\(516\) −1.71473e54 −0.0421577
\(517\) 9.22086e54 0.217879
\(518\) −7.97982e55 −1.81231
\(519\) 7.36695e53 0.0160826
\(520\) −1.24509e55 −0.261296
\(521\) 2.07621e55 0.418889 0.209444 0.977821i \(-0.432835\pi\)
0.209444 + 0.977821i \(0.432835\pi\)
\(522\) 1.04513e55 0.202733
\(523\) 3.43281e55 0.640273 0.320136 0.947371i \(-0.396271\pi\)
0.320136 + 0.947371i \(0.396271\pi\)
\(524\) −1.85630e55 −0.332934
\(525\) −4.74299e55 −0.818063
\(526\) −2.76761e55 −0.459088
\(527\) −1.11328e55 −0.177619
\(528\) 3.46304e55 0.531450
\(529\) −6.36069e55 −0.938994
\(530\) −2.22217e54 −0.0315589
\(531\) −1.78087e54 −0.0243329
\(532\) 3.93943e55 0.517899
\(533\) −1.98769e54 −0.0251444
\(534\) −7.50414e55 −0.913493
\(535\) 1.49386e55 0.175008
\(536\) −3.06579e55 −0.345672
\(537\) −1.64268e55 −0.178271
\(538\) −1.21255e56 −1.26667
\(539\) 2.98928e56 3.00605
\(540\) 1.77429e54 0.0171773
\(541\) −1.41875e56 −1.32240 −0.661200 0.750209i \(-0.729953\pi\)
−0.661200 + 0.750209i \(0.729953\pi\)
\(542\) 3.23294e55 0.290143
\(543\) 2.12702e55 0.183813
\(544\) −7.02941e55 −0.584980
\(545\) 9.99917e55 0.801375
\(546\) 6.26351e55 0.483469
\(547\) −6.85690e54 −0.0509785 −0.0254893 0.999675i \(-0.508114\pi\)
−0.0254893 + 0.999675i \(0.508114\pi\)
\(548\) −1.76696e55 −0.126539
\(549\) −4.91494e55 −0.339066
\(550\) 1.18990e56 0.790816
\(551\) −1.61018e56 −1.03102
\(552\) 2.46870e55 0.152306
\(553\) 1.84823e56 1.09873
\(554\) 9.72312e55 0.557000
\(555\) 5.53720e55 0.305692
\(556\) −5.46222e54 −0.0290627
\(557\) −4.86641e55 −0.249562 −0.124781 0.992184i \(-0.539823\pi\)
−0.124781 + 0.992184i \(0.539823\pi\)
\(558\) 6.52778e54 0.0322674
\(559\) 4.20035e55 0.200145
\(560\) 1.63140e56 0.749385
\(561\) −2.52628e56 −1.11878
\(562\) −1.36832e56 −0.584245
\(563\) 1.31993e56 0.543412 0.271706 0.962380i \(-0.412412\pi\)
0.271706 + 0.962380i \(0.412412\pi\)
\(564\) 4.84424e54 0.0192311
\(565\) −1.81452e56 −0.694655
\(566\) −8.44404e55 −0.311755
\(567\) −5.90450e55 −0.210247
\(568\) 5.34362e56 1.83525
\(569\) −5.17929e56 −1.71581 −0.857903 0.513811i \(-0.828233\pi\)
−0.857903 + 0.513811i \(0.828233\pi\)
\(570\) 1.26159e56 0.403166
\(571\) 1.79414e56 0.553115 0.276558 0.960997i \(-0.410806\pi\)
0.276558 + 0.960997i \(0.410806\pi\)
\(572\) 3.40478e55 0.101268
\(573\) −5.12428e55 −0.147049
\(574\) 3.19137e55 0.0883656
\(575\) 6.92233e55 0.184953
\(576\) 1.43375e56 0.369670
\(577\) −6.52560e56 −1.62374 −0.811870 0.583838i \(-0.801550\pi\)
−0.811870 + 0.583838i \(0.801550\pi\)
\(578\) −6.67119e56 −1.60207
\(579\) 2.87620e56 0.666666
\(580\) 2.67635e55 0.0598783
\(581\) −4.38808e56 −0.947685
\(582\) 4.20937e54 0.00877599
\(583\) 4.01982e55 0.0809098
\(584\) −8.25247e56 −1.60369
\(585\) −4.34626e55 −0.0815493
\(586\) 6.84131e56 1.23948
\(587\) 7.86168e56 1.37542 0.687708 0.725987i \(-0.258617\pi\)
0.687708 + 0.725987i \(0.258617\pi\)
\(588\) 1.57044e56 0.265329
\(589\) −1.00571e56 −0.164100
\(590\) 2.10471e55 0.0331685
\(591\) −2.13175e56 −0.324483
\(592\) 5.67775e56 0.834798
\(593\) 1.07653e57 1.52898 0.764492 0.644633i \(-0.222990\pi\)
0.764492 + 0.644633i \(0.222990\pi\)
\(594\) 1.48129e56 0.203245
\(595\) −1.19010e57 −1.57756
\(596\) 9.87834e55 0.126513
\(597\) 7.20801e56 0.891952
\(598\) −9.14151e55 −0.109306
\(599\) 8.88910e56 1.02709 0.513544 0.858063i \(-0.328332\pi\)
0.513544 + 0.858063i \(0.328332\pi\)
\(600\) 4.13528e56 0.461747
\(601\) 5.98671e56 0.646043 0.323021 0.946392i \(-0.395301\pi\)
0.323021 + 0.946392i \(0.395301\pi\)
\(602\) −6.74394e56 −0.703373
\(603\) −1.07018e56 −0.107883
\(604\) 1.44211e56 0.140521
\(605\) 1.89952e56 0.178920
\(606\) −4.63296e56 −0.421861
\(607\) 6.99287e56 0.615584 0.307792 0.951454i \(-0.400410\pi\)
0.307792 + 0.951454i \(0.400410\pi\)
\(608\) −6.35014e56 −0.540455
\(609\) −8.90637e56 −0.732903
\(610\) 5.80868e56 0.462185
\(611\) −1.18663e56 −0.0912999
\(612\) −1.32720e56 −0.0987488
\(613\) −1.99522e57 −1.43566 −0.717829 0.696219i \(-0.754864\pi\)
−0.717829 + 0.696219i \(0.754864\pi\)
\(614\) 2.01697e56 0.140361
\(615\) −2.21449e55 −0.0149051
\(616\) −3.61624e57 −2.35425
\(617\) −2.01107e57 −1.26643 −0.633216 0.773975i \(-0.718265\pi\)
−0.633216 + 0.773975i \(0.718265\pi\)
\(618\) −9.41482e56 −0.573519
\(619\) 1.27917e56 0.0753823 0.0376911 0.999289i \(-0.488000\pi\)
0.0376911 + 0.999289i \(0.488000\pi\)
\(620\) 1.67162e55 0.00953035
\(621\) 8.61755e55 0.0475342
\(622\) 1.77968e57 0.949813
\(623\) 6.39487e57 3.30238
\(624\) −4.45658e56 −0.222699
\(625\) 6.40106e56 0.309536
\(626\) 1.35030e57 0.631911
\(627\) −2.28216e57 −1.03362
\(628\) 1.69419e56 0.0742658
\(629\) −4.14191e57 −1.75737
\(630\) 6.97820e56 0.286591
\(631\) 2.77600e57 1.10361 0.551807 0.833972i \(-0.313939\pi\)
0.551807 + 0.833972i \(0.313939\pi\)
\(632\) −1.61142e57 −0.620168
\(633\) −1.07475e55 −0.00400433
\(634\) 3.31237e57 1.19484
\(635\) 1.33427e57 0.465998
\(636\) 2.11184e55 0.00714150
\(637\) −3.84689e57 −1.25965
\(638\) 2.23439e57 0.708492
\(639\) 1.86531e57 0.572773
\(640\) −1.10180e57 −0.327653
\(641\) 1.50182e57 0.432541 0.216271 0.976333i \(-0.430611\pi\)
0.216271 + 0.976333i \(0.430611\pi\)
\(642\) 6.55210e56 0.182773
\(643\) 3.44585e57 0.931046 0.465523 0.885036i \(-0.345866\pi\)
0.465523 + 0.885036i \(0.345866\pi\)
\(644\) −3.18023e56 −0.0832336
\(645\) 4.67963e56 0.118642
\(646\) −9.43688e57 −2.31772
\(647\) 6.27028e56 0.149193 0.0745966 0.997214i \(-0.476233\pi\)
0.0745966 + 0.997214i \(0.476233\pi\)
\(648\) 5.14797e56 0.118672
\(649\) −3.80733e56 −0.0850363
\(650\) −1.53128e57 −0.331383
\(651\) −5.56283e56 −0.116650
\(652\) 6.40263e56 0.130102
\(653\) 4.27681e57 0.842170 0.421085 0.907021i \(-0.361649\pi\)
0.421085 + 0.907021i \(0.361649\pi\)
\(654\) 4.38565e57 0.836931
\(655\) 5.06599e57 0.936952
\(656\) −2.27070e56 −0.0407035
\(657\) −2.88071e57 −0.500506
\(658\) 1.90521e57 0.320857
\(659\) −7.18638e57 −1.17317 −0.586583 0.809889i \(-0.699527\pi\)
−0.586583 + 0.809889i \(0.699527\pi\)
\(660\) 3.79328e56 0.0600294
\(661\) −4.82359e57 −0.740014 −0.370007 0.929029i \(-0.620645\pi\)
−0.370007 + 0.929029i \(0.620645\pi\)
\(662\) −9.87782e57 −1.46917
\(663\) 3.25107e57 0.468812
\(664\) 3.82584e57 0.534911
\(665\) −1.07510e58 −1.45749
\(666\) 2.42862e57 0.319255
\(667\) 1.29987e57 0.165700
\(668\) 1.70622e57 0.210920
\(669\) 1.75620e57 0.210542
\(670\) 1.26479e57 0.147056
\(671\) −1.05077e58 −1.18494
\(672\) −3.51244e57 −0.384183
\(673\) 2.10611e57 0.223446 0.111723 0.993739i \(-0.464363\pi\)
0.111723 + 0.993739i \(0.464363\pi\)
\(674\) 8.07722e57 0.831255
\(675\) 1.44351e57 0.144109
\(676\) 1.40069e57 0.135654
\(677\) 5.40673e57 0.508004 0.254002 0.967204i \(-0.418253\pi\)
0.254002 + 0.967204i \(0.418253\pi\)
\(678\) −7.95852e57 −0.725477
\(679\) −3.58714e56 −0.0317262
\(680\) 1.03761e58 0.890438
\(681\) 5.01336e56 0.0417458
\(682\) 1.39558e57 0.112765
\(683\) −2.46559e58 −1.93329 −0.966644 0.256125i \(-0.917554\pi\)
−0.966644 + 0.256125i \(0.917554\pi\)
\(684\) −1.19895e57 −0.0912327
\(685\) 4.82217e57 0.356111
\(686\) 3.78295e58 2.71135
\(687\) 8.00798e57 0.557070
\(688\) 4.79841e57 0.323992
\(689\) −5.17309e56 −0.0339044
\(690\) −1.01846e57 −0.0647943
\(691\) 2.01787e58 1.24621 0.623107 0.782136i \(-0.285870\pi\)
0.623107 + 0.782136i \(0.285870\pi\)
\(692\) 8.27428e55 0.00496084
\(693\) −1.26233e58 −0.734752
\(694\) −1.13523e58 −0.641524
\(695\) 1.49068e57 0.0817892
\(696\) 7.76521e57 0.413679
\(697\) 1.65647e57 0.0856866
\(698\) −2.18470e58 −1.09737
\(699\) −7.67284e56 −0.0374261
\(700\) −5.32715e57 −0.252339
\(701\) 1.63846e57 0.0753729 0.0376864 0.999290i \(-0.488001\pi\)
0.0376864 + 0.999290i \(0.488001\pi\)
\(702\) −1.90627e57 −0.0851676
\(703\) −3.74167e58 −1.62361
\(704\) 3.06523e58 1.29189
\(705\) −1.32203e57 −0.0541207
\(706\) 4.80012e56 0.0190878
\(707\) 3.94811e58 1.52507
\(708\) −2.00021e56 −0.00750573
\(709\) −3.78678e58 −1.38046 −0.690228 0.723592i \(-0.742490\pi\)
−0.690228 + 0.723592i \(0.742490\pi\)
\(710\) −2.20450e58 −0.780754
\(711\) −5.62502e57 −0.193552
\(712\) −5.57550e58 −1.86399
\(713\) 8.11888e56 0.0263731
\(714\) −5.21980e58 −1.64756
\(715\) −9.29189e57 −0.284991
\(716\) −1.84500e57 −0.0549894
\(717\) 3.62961e58 1.05128
\(718\) 1.20354e58 0.338773
\(719\) 2.43656e58 0.666553 0.333277 0.942829i \(-0.391846\pi\)
0.333277 + 0.942829i \(0.391846\pi\)
\(720\) −4.96509e57 −0.132011
\(721\) 8.02311e58 2.07334
\(722\) −4.91566e58 −1.23472
\(723\) 3.08091e58 0.752219
\(724\) 2.38899e57 0.0566988
\(725\) 2.17739e58 0.502352
\(726\) 8.33131e57 0.186858
\(727\) 8.38626e57 0.182858 0.0914288 0.995812i \(-0.470857\pi\)
0.0914288 + 0.995812i \(0.470857\pi\)
\(728\) 4.65373e58 0.986524
\(729\) 1.79701e57 0.0370370
\(730\) 3.40454e58 0.682245
\(731\) −3.50043e58 −0.682048
\(732\) −5.52027e57 −0.104588
\(733\) 2.90666e58 0.535505 0.267752 0.963488i \(-0.413719\pi\)
0.267752 + 0.963488i \(0.413719\pi\)
\(734\) 2.88087e58 0.516125
\(735\) −4.28583e58 −0.746698
\(736\) 5.12636e57 0.0868587
\(737\) −2.28795e58 −0.377019
\(738\) −9.71279e56 −0.0155664
\(739\) 3.92079e58 0.611170 0.305585 0.952165i \(-0.401148\pi\)
0.305585 + 0.952165i \(0.401148\pi\)
\(740\) 6.21918e57 0.0942937
\(741\) 2.93691e58 0.433129
\(742\) 8.30573e57 0.119151
\(743\) −1.22168e59 −1.70485 −0.852426 0.522848i \(-0.824870\pi\)
−0.852426 + 0.522848i \(0.824870\pi\)
\(744\) 4.85007e57 0.0658420
\(745\) −2.69587e58 −0.356037
\(746\) −2.35699e58 −0.302839
\(747\) 1.33549e58 0.166943
\(748\) −2.83743e58 −0.345098
\(749\) −5.58356e58 −0.660746
\(750\) −3.98427e58 −0.458768
\(751\) 3.73440e58 0.418410 0.209205 0.977872i \(-0.432912\pi\)
0.209205 + 0.977872i \(0.432912\pi\)
\(752\) −1.35558e58 −0.147795
\(753\) 8.93941e58 0.948444
\(754\) −2.87543e58 −0.296886
\(755\) −3.93562e58 −0.395458
\(756\) −6.63172e57 −0.0648529
\(757\) 1.05552e59 1.00461 0.502307 0.864690i \(-0.332485\pi\)
0.502307 + 0.864690i \(0.332485\pi\)
\(758\) −6.74055e58 −0.624419
\(759\) 1.84235e58 0.166118
\(760\) 9.37348e58 0.822664
\(761\) 4.90222e58 0.418800 0.209400 0.977830i \(-0.432849\pi\)
0.209400 + 0.977830i \(0.432849\pi\)
\(762\) 5.85214e58 0.486674
\(763\) −3.73736e59 −3.02560
\(764\) −5.75540e57 −0.0453588
\(765\) 3.62202e58 0.277902
\(766\) 3.62807e58 0.271010
\(767\) 4.89964e57 0.0356336
\(768\) 4.20980e58 0.298096
\(769\) −1.48791e59 −1.02586 −0.512930 0.858431i \(-0.671440\pi\)
−0.512930 + 0.858431i \(0.671440\pi\)
\(770\) 1.49187e59 1.00155
\(771\) −1.11232e59 −0.727132
\(772\) 3.23044e58 0.205639
\(773\) 4.35850e58 0.270183 0.135091 0.990833i \(-0.456867\pi\)
0.135091 + 0.990833i \(0.456867\pi\)
\(774\) 2.05249e58 0.123906
\(775\) 1.35998e58 0.0799553
\(776\) 3.12752e57 0.0179075
\(777\) −2.06962e59 −1.15414
\(778\) 2.09374e59 1.13721
\(779\) 1.49641e58 0.0791647
\(780\) −4.88155e57 −0.0251547
\(781\) 3.98785e59 2.00167
\(782\) 7.61822e58 0.372491
\(783\) 2.71062e58 0.129108
\(784\) −4.39462e59 −2.03912
\(785\) −4.62356e58 −0.209001
\(786\) 2.22195e59 0.978524
\(787\) −1.24674e59 −0.534924 −0.267462 0.963568i \(-0.586185\pi\)
−0.267462 + 0.963568i \(0.586185\pi\)
\(788\) −2.39430e58 −0.100090
\(789\) −7.17798e58 −0.292364
\(790\) 6.64789e58 0.263833
\(791\) 6.78208e59 2.62268
\(792\) 1.10059e59 0.414723
\(793\) 1.35223e59 0.496535
\(794\) 4.04845e58 0.144867
\(795\) −5.76336e57 −0.0200978
\(796\) 8.09577e58 0.275131
\(797\) −2.54215e59 −0.841988 −0.420994 0.907063i \(-0.638319\pi\)
−0.420994 + 0.907063i \(0.638319\pi\)
\(798\) −4.71540e59 −1.52216
\(799\) 9.88896e58 0.311130
\(800\) 8.58706e58 0.263330
\(801\) −1.94625e59 −0.581745
\(802\) −3.09724e59 −0.902403
\(803\) −6.15868e59 −1.74912
\(804\) −1.20199e58 −0.0332775
\(805\) 8.67909e58 0.234239
\(806\) −1.79596e58 −0.0472530
\(807\) −3.14484e59 −0.806662
\(808\) −3.44224e59 −0.860812
\(809\) 5.56333e59 1.35641 0.678203 0.734874i \(-0.262759\pi\)
0.678203 + 0.734874i \(0.262759\pi\)
\(810\) −2.12378e58 −0.0504856
\(811\) −4.60895e59 −1.06825 −0.534127 0.845404i \(-0.679360\pi\)
−0.534127 + 0.845404i \(0.679360\pi\)
\(812\) −1.00033e59 −0.226071
\(813\) 8.38486e58 0.184773
\(814\) 5.19217e59 1.11570
\(815\) −1.74732e59 −0.366136
\(816\) 3.71396e59 0.758908
\(817\) −3.16218e59 −0.630136
\(818\) −1.65530e59 −0.321687
\(819\) 1.62449e59 0.307890
\(820\) −2.48724e57 −0.00459762
\(821\) 4.31375e59 0.777715 0.388857 0.921298i \(-0.372870\pi\)
0.388857 + 0.921298i \(0.372870\pi\)
\(822\) 2.11501e59 0.371911
\(823\) 6.85688e59 1.17606 0.588030 0.808839i \(-0.299904\pi\)
0.588030 + 0.808839i \(0.299904\pi\)
\(824\) −6.99511e59 −1.17027
\(825\) 3.08609e59 0.503619
\(826\) −7.86669e58 −0.125228
\(827\) −6.48288e59 −1.00671 −0.503356 0.864079i \(-0.667902\pi\)
−0.503356 + 0.864079i \(0.667902\pi\)
\(828\) 9.67890e57 0.0146624
\(829\) 3.30221e59 0.488018 0.244009 0.969773i \(-0.421537\pi\)
0.244009 + 0.969773i \(0.421537\pi\)
\(830\) −1.57834e59 −0.227562
\(831\) 2.52176e59 0.354717
\(832\) −3.94464e59 −0.541351
\(833\) 3.20587e60 4.29263
\(834\) 6.53814e58 0.0854181
\(835\) −4.65640e59 −0.593578
\(836\) −2.56324e59 −0.318831
\(837\) 1.69302e58 0.0205490
\(838\) 1.45950e60 1.72863
\(839\) −1.34134e60 −1.55031 −0.775153 0.631773i \(-0.782327\pi\)
−0.775153 + 0.631773i \(0.782327\pi\)
\(840\) 5.18473e59 0.584791
\(841\) −4.99615e59 −0.549943
\(842\) −1.25931e60 −1.35280
\(843\) −3.54885e59 −0.372068
\(844\) −1.20711e57 −0.00123517
\(845\) −3.82257e59 −0.381762
\(846\) −5.79842e58 −0.0565220
\(847\) −7.09976e59 −0.675513
\(848\) −5.90965e58 −0.0548841
\(849\) −2.19002e59 −0.198537
\(850\) 1.27611e60 1.12928
\(851\) 3.02058e59 0.260937
\(852\) 2.09504e59 0.176678
\(853\) −1.01106e60 −0.832382 −0.416191 0.909277i \(-0.636635\pi\)
−0.416191 + 0.909277i \(0.636635\pi\)
\(854\) −2.17109e60 −1.74498
\(855\) 3.27202e59 0.256750
\(856\) 4.86815e59 0.372951
\(857\) −1.59400e60 −1.19229 −0.596143 0.802878i \(-0.703301\pi\)
−0.596143 + 0.802878i \(0.703301\pi\)
\(858\) −4.07543e59 −0.297635
\(859\) 2.28498e60 1.62938 0.814692 0.579894i \(-0.196906\pi\)
0.814692 + 0.579894i \(0.196906\pi\)
\(860\) 5.25598e58 0.0365962
\(861\) 8.27704e58 0.0562743
\(862\) −2.31380e60 −1.53613
\(863\) 1.07116e60 0.694439 0.347220 0.937784i \(-0.387126\pi\)
0.347220 + 0.937784i \(0.387126\pi\)
\(864\) 1.06900e59 0.0676774
\(865\) −2.25811e58 −0.0139609
\(866\) 1.62102e60 0.978749
\(867\) −1.73022e60 −1.02026
\(868\) −6.24797e58 −0.0359819
\(869\) −1.20258e60 −0.676406
\(870\) −3.20352e59 −0.175988
\(871\) 2.94435e59 0.157986
\(872\) 3.25849e60 1.70777
\(873\) 1.09173e58 0.00558886
\(874\) 6.88206e59 0.344139
\(875\) 3.39531e60 1.65850
\(876\) −3.23550e59 −0.154386
\(877\) 1.15644e60 0.539054 0.269527 0.962993i \(-0.413133\pi\)
0.269527 + 0.962993i \(0.413133\pi\)
\(878\) −2.28232e60 −1.03930
\(879\) 1.77434e60 0.789342
\(880\) −1.06149e60 −0.461340
\(881\) −3.12498e59 −0.132691 −0.0663454 0.997797i \(-0.521134\pi\)
−0.0663454 + 0.997797i \(0.521134\pi\)
\(882\) −1.87977e60 −0.779828
\(883\) −3.90070e60 −1.58106 −0.790530 0.612424i \(-0.790195\pi\)
−0.790530 + 0.612424i \(0.790195\pi\)
\(884\) 3.65148e59 0.144610
\(885\) 5.45871e58 0.0211229
\(886\) 4.04644e60 1.52997
\(887\) 7.42130e58 0.0274186 0.0137093 0.999906i \(-0.495636\pi\)
0.0137093 + 0.999906i \(0.495636\pi\)
\(888\) 1.80444e60 0.651444
\(889\) −4.98707e60 −1.75938
\(890\) 2.30016e60 0.792983
\(891\) 3.84184e59 0.129433
\(892\) 1.97250e59 0.0649438
\(893\) 8.93337e59 0.287449
\(894\) −1.18241e60 −0.371834
\(895\) 5.03513e59 0.154753
\(896\) 4.11817e60 1.23706
\(897\) −2.37091e59 −0.0696099
\(898\) 2.30556e60 0.661625
\(899\) 2.55376e59 0.0716320
\(900\) 1.62129e59 0.0444519
\(901\) 4.31108e59 0.115539
\(902\) −2.07650e59 −0.0544000
\(903\) −1.74909e60 −0.447932
\(904\) −5.91310e60 −1.48034
\(905\) −6.51973e59 −0.159564
\(906\) −1.72616e60 −0.413004
\(907\) −1.27830e60 −0.299009 −0.149505 0.988761i \(-0.547768\pi\)
−0.149505 + 0.988761i \(0.547768\pi\)
\(908\) 5.63081e58 0.0128769
\(909\) −1.20159e60 −0.268656
\(910\) −1.91989e60 −0.419688
\(911\) 3.74088e60 0.799552 0.399776 0.916613i \(-0.369088\pi\)
0.399776 + 0.916613i \(0.369088\pi\)
\(912\) 3.35507e60 0.701145
\(913\) 2.85516e60 0.583418
\(914\) −1.63838e60 −0.327354
\(915\) 1.50652e60 0.294336
\(916\) 8.99426e59 0.171834
\(917\) −1.89350e61 −3.53747
\(918\) 1.58862e60 0.290232
\(919\) −4.88524e60 −0.872806 −0.436403 0.899751i \(-0.643748\pi\)
−0.436403 + 0.899751i \(0.643748\pi\)
\(920\) −7.56705e59 −0.132214
\(921\) 5.23114e59 0.0893869
\(922\) 1.44196e60 0.240973
\(923\) −5.13195e60 −0.838780
\(924\) −1.41780e60 −0.226641
\(925\) 5.05972e60 0.791081
\(926\) −9.85496e59 −0.150706
\(927\) −2.44180e60 −0.365237
\(928\) 1.61248e60 0.235917
\(929\) 7.45596e60 1.06704 0.533521 0.845787i \(-0.320868\pi\)
0.533521 + 0.845787i \(0.320868\pi\)
\(930\) −2.00089e59 −0.0280106
\(931\) 2.89608e61 3.96590
\(932\) −8.61785e58 −0.0115444
\(933\) 4.61571e60 0.604874
\(934\) −6.21318e60 −0.796531
\(935\) 7.74355e60 0.971185
\(936\) −1.41634e60 −0.173785
\(937\) −1.56022e60 −0.187294 −0.0936469 0.995605i \(-0.529852\pi\)
−0.0936469 + 0.995605i \(0.529852\pi\)
\(938\) −4.72735e60 −0.555213
\(939\) 3.50210e60 0.402423
\(940\) −1.48485e59 −0.0166941
\(941\) −5.75717e60 −0.633318 −0.316659 0.948539i \(-0.602561\pi\)
−0.316659 + 0.948539i \(0.602561\pi\)
\(942\) −2.02790e60 −0.218274
\(943\) −1.20802e59 −0.0127229
\(944\) 5.59726e59 0.0576833
\(945\) 1.80984e60 0.182511
\(946\) 4.38803e60 0.433013
\(947\) 4.40342e60 0.425222 0.212611 0.977137i \(-0.431803\pi\)
0.212611 + 0.977137i \(0.431803\pi\)
\(948\) −6.31781e59 −0.0597030
\(949\) 7.92559e60 0.732950
\(950\) 1.15280e61 1.04333
\(951\) 8.59087e60 0.760916
\(952\) −3.87826e61 −3.36186
\(953\) −1.39562e61 −1.18403 −0.592013 0.805928i \(-0.701667\pi\)
−0.592013 + 0.805928i \(0.701667\pi\)
\(954\) −2.52781e59 −0.0209896
\(955\) 1.57069e60 0.127650
\(956\) 4.07665e60 0.324277
\(957\) 5.79505e60 0.451193
\(958\) −1.42847e60 −0.108863
\(959\) −1.80236e61 −1.34450
\(960\) −4.39473e60 −0.320902
\(961\) −1.38309e61 −0.988599
\(962\) −6.68178e60 −0.467523
\(963\) 1.69933e60 0.116396
\(964\) 3.46036e60 0.232029
\(965\) −8.81610e60 −0.578717
\(966\) 3.80666e60 0.244632
\(967\) 1.12963e61 0.710713 0.355356 0.934731i \(-0.384360\pi\)
0.355356 + 0.934731i \(0.384360\pi\)
\(968\) 6.19008e60 0.381286
\(969\) −2.44752e61 −1.47601
\(970\) −1.29025e59 −0.00761824
\(971\) 1.69719e61 0.981152 0.490576 0.871398i \(-0.336786\pi\)
0.490576 + 0.871398i \(0.336786\pi\)
\(972\) 2.01833e59 0.0114244
\(973\) −5.57166e60 −0.308796
\(974\) 2.44481e61 1.32674
\(975\) −3.97147e60 −0.211036
\(976\) 1.54476e61 0.803786
\(977\) 3.22632e61 1.64388 0.821938 0.569577i \(-0.192893\pi\)
0.821938 + 0.569577i \(0.192893\pi\)
\(978\) −7.66378e60 −0.382381
\(979\) −4.16091e61 −2.03302
\(980\) −4.81369e60 −0.230326
\(981\) 1.13745e61 0.532987
\(982\) −4.86885e60 −0.223430
\(983\) −1.33938e61 −0.601947 −0.300973 0.953633i \(-0.597312\pi\)
−0.300973 + 0.953633i \(0.597312\pi\)
\(984\) −7.21651e59 −0.0317634
\(985\) 6.53422e60 0.281677
\(986\) 2.39628e61 1.01172
\(987\) 4.94129e60 0.204333
\(988\) 3.29863e60 0.133603
\(989\) 2.55277e60 0.101272
\(990\) −4.54045e60 −0.176432
\(991\) 2.73292e61 1.04020 0.520101 0.854105i \(-0.325894\pi\)
0.520101 + 0.854105i \(0.325894\pi\)
\(992\) 1.00714e60 0.0375491
\(993\) −2.56188e61 −0.935619
\(994\) 8.23968e61 2.94774
\(995\) −2.20939e61 −0.774283
\(996\) 1.49998e60 0.0514953
\(997\) −2.91866e61 −0.981595 −0.490798 0.871274i \(-0.663295\pi\)
−0.490798 + 0.871274i \(0.663295\pi\)
\(998\) −2.78267e61 −0.916822
\(999\) 6.29880e60 0.203313
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 3.42.a.a.1.1 3
3.2 odd 2 9.42.a.a.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.42.a.a.1.1 3 1.1 even 1 trivial
9.42.a.a.1.3 3 3.2 odd 2