Properties

Label 1.82.a.a.1.1
Level $1$
Weight $82$
Character 1.1
Self dual yes
Analytic conductor $41.550$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1,82,Mod(1,1)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1, base_ring=CyclotomicField(1))
 
chi = DirichletCharacter(H, H._module([]))
 
N = Newforms(chi, 82, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1.1");
 
S:= CuspForms(chi, 82);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 82 \)
Character orbit: \([\chi]\) \(=\) 1.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: \(41.5501285538\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3 x^{5} + \cdots - 27\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{58}\cdot 3^{26}\cdot 5^{7}\cdot 7^{3}\cdot 13 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.85696e10\) of defining polynomial
Character \(\chi\) \(=\) 1.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.75084e12 q^{2} -2.21125e19 q^{3} +5.14927e24 q^{4} +2.52982e28 q^{5} +6.08279e31 q^{6} -2.62798e34 q^{7} -7.51371e36 q^{8} +4.55346e37 q^{9} +O(q^{10})\) \(q-2.75084e12 q^{2} -2.21125e19 q^{3} +5.14927e24 q^{4} +2.52982e28 q^{5} +6.08279e31 q^{6} -2.62798e34 q^{7} -7.51371e36 q^{8} +4.55346e37 q^{9} -6.95912e40 q^{10} -6.46669e41 q^{11} -1.13863e44 q^{12} +1.64391e45 q^{13} +7.22914e46 q^{14} -5.59405e47 q^{15} +8.21884e48 q^{16} +1.28166e49 q^{17} -1.25259e50 q^{18} -8.42462e51 q^{19} +1.30267e53 q^{20} +5.81110e53 q^{21} +1.77888e54 q^{22} +1.57219e55 q^{23} +1.66147e56 q^{24} +2.26406e56 q^{25} -4.52214e57 q^{26} +8.79837e57 q^{27} -1.35322e59 q^{28} +1.02296e59 q^{29} +1.53883e60 q^{30} +3.33151e59 q^{31} -4.44168e60 q^{32} +1.42994e61 q^{33} -3.52565e61 q^{34} -6.64829e62 q^{35} +2.34470e62 q^{36} +8.95735e62 q^{37} +2.31748e64 q^{38} -3.63510e64 q^{39} -1.90083e65 q^{40} +1.61632e65 q^{41} -1.59854e66 q^{42} +1.76087e66 q^{43} -3.32987e66 q^{44} +1.15194e66 q^{45} -4.32486e67 q^{46} -8.77094e67 q^{47} -1.81739e68 q^{48} +4.06872e68 q^{49} -6.22808e68 q^{50} -2.83407e68 q^{51} +8.46496e69 q^{52} +2.20440e69 q^{53} -2.42029e70 q^{54} -1.63595e70 q^{55} +1.97458e71 q^{56} +1.86289e71 q^{57} -2.81401e71 q^{58} +3.92989e71 q^{59} -2.88053e72 q^{60} -7.29227e71 q^{61} -9.16446e71 q^{62} -1.19664e72 q^{63} -7.65357e72 q^{64} +4.15880e73 q^{65} -3.93355e73 q^{66} +1.78309e73 q^{67} +6.59963e73 q^{68} -3.47651e74 q^{69} +1.82884e75 q^{70} -3.78476e74 q^{71} -3.42134e74 q^{72} -1.18691e75 q^{73} -2.46402e75 q^{74} -5.00640e75 q^{75} -4.33807e76 q^{76} +1.69943e76 q^{77} +9.99958e76 q^{78} -3.54946e76 q^{79} +2.07921e77 q^{80} -2.14745e77 q^{81} -4.44625e77 q^{82} -5.45755e76 q^{83} +2.99230e78 q^{84} +3.24237e77 q^{85} -4.84388e78 q^{86} -2.26202e78 q^{87} +4.85888e78 q^{88} -1.21164e79 q^{89} -3.16881e78 q^{90} -4.32017e79 q^{91} +8.09566e79 q^{92} -7.36680e78 q^{93} +2.41274e80 q^{94} -2.13127e80 q^{95} +9.82165e79 q^{96} +2.69780e80 q^{97} -1.11924e81 q^{98} -2.94458e79 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 460872026640 q^{2} - 15\!\cdots\!60 q^{3}+ \cdots + 11\!\cdots\!98 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 460872026640 q^{2} - 15\!\cdots\!60 q^{3}+ \cdots - 63\!\cdots\!24 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.75084e12 −1.76909 −0.884546 0.466452i \(-0.845532\pi\)
−0.884546 + 0.466452i \(0.845532\pi\)
\(3\) −2.21125e19 −1.05009 −0.525045 0.851075i \(-0.675951\pi\)
−0.525045 + 0.851075i \(0.675951\pi\)
\(4\) 5.14927e24 2.12969
\(5\) 2.52982e28 1.24395 0.621976 0.783036i \(-0.286330\pi\)
0.621976 + 0.783036i \(0.286330\pi\)
\(6\) 6.08279e31 1.85771
\(7\) −2.62798e34 −1.56009 −0.780047 0.625721i \(-0.784805\pi\)
−0.780047 + 0.625721i \(0.784805\pi\)
\(8\) −7.51371e36 −1.99853
\(9\) 4.55346e37 0.102688
\(10\) −6.95912e40 −2.20067
\(11\) −6.46669e41 −0.430802 −0.215401 0.976526i \(-0.569106\pi\)
−0.215401 + 0.976526i \(0.569106\pi\)
\(12\) −1.13863e44 −2.23637
\(13\) 1.64391e45 1.26233 0.631165 0.775648i \(-0.282577\pi\)
0.631165 + 0.775648i \(0.282577\pi\)
\(14\) 7.22914e46 2.75995
\(15\) −5.59405e47 −1.30626
\(16\) 8.21884e48 1.40589
\(17\) 1.28166e49 0.188188 0.0940941 0.995563i \(-0.470005\pi\)
0.0940941 + 0.995563i \(0.470005\pi\)
\(18\) −1.25259e50 −0.181665
\(19\) −8.42462e51 −1.36782 −0.683911 0.729566i \(-0.739722\pi\)
−0.683911 + 0.729566i \(0.739722\pi\)
\(20\) 1.30267e53 2.64923
\(21\) 5.81110e53 1.63824
\(22\) 1.77888e54 0.762129
\(23\) 1.57219e55 1.11309 0.556543 0.830819i \(-0.312127\pi\)
0.556543 + 0.830819i \(0.312127\pi\)
\(24\) 1.66147e56 2.09863
\(25\) 2.26406e56 0.547417
\(26\) −4.52214e57 −2.23318
\(27\) 8.79837e57 0.942258
\(28\) −1.35322e59 −3.32252
\(29\) 1.02296e59 0.606375 0.303187 0.952931i \(-0.401949\pi\)
0.303187 + 0.952931i \(0.401949\pi\)
\(30\) 1.53883e60 2.31090
\(31\) 3.33151e59 0.132585 0.0662923 0.997800i \(-0.478883\pi\)
0.0662923 + 0.997800i \(0.478883\pi\)
\(32\) −4.44168e60 −0.488622
\(33\) 1.42994e61 0.452381
\(34\) −3.52565e61 −0.332923
\(35\) −6.64829e62 −1.94068
\(36\) 2.34470e62 0.218694
\(37\) 8.95735e62 0.275429 0.137715 0.990472i \(-0.456024\pi\)
0.137715 + 0.990472i \(0.456024\pi\)
\(38\) 2.31748e64 2.41980
\(39\) −3.63510e64 −1.32556
\(40\) −1.90083e65 −2.48607
\(41\) 1.61632e65 0.777647 0.388823 0.921312i \(-0.372882\pi\)
0.388823 + 0.921312i \(0.372882\pi\)
\(42\) −1.59854e66 −2.89820
\(43\) 1.76087e66 1.23099 0.615496 0.788140i \(-0.288956\pi\)
0.615496 + 0.788140i \(0.288956\pi\)
\(44\) −3.32987e66 −0.917475
\(45\) 1.15194e66 0.127739
\(46\) −4.32486e67 −1.96915
\(47\) −8.77094e67 −1.67141 −0.835704 0.549179i \(-0.814940\pi\)
−0.835704 + 0.549179i \(0.814940\pi\)
\(48\) −1.81739e68 −1.47631
\(49\) 4.06872e68 1.43389
\(50\) −6.22808e68 −0.968432
\(51\) −2.83407e68 −0.197615
\(52\) 8.46496e69 2.68837
\(53\) 2.20440e69 0.323681 0.161841 0.986817i \(-0.448257\pi\)
0.161841 + 0.986817i \(0.448257\pi\)
\(54\) −2.42029e70 −1.66694
\(55\) −1.63595e70 −0.535897
\(56\) 1.97458e71 3.11789
\(57\) 1.86289e71 1.43634
\(58\) −2.81401e71 −1.07273
\(59\) 3.92989e71 0.749676 0.374838 0.927090i \(-0.377698\pi\)
0.374838 + 0.927090i \(0.377698\pi\)
\(60\) −2.88053e72 −2.78193
\(61\) −7.29227e71 −0.360582 −0.180291 0.983613i \(-0.557704\pi\)
−0.180291 + 0.983613i \(0.557704\pi\)
\(62\) −9.16446e71 −0.234554
\(63\) −1.19664e72 −0.160203
\(64\) −7.65357e72 −0.541471
\(65\) 4.15880e73 1.57028
\(66\) −3.93355e73 −0.800304
\(67\) 1.78309e73 0.197308 0.0986538 0.995122i \(-0.468546\pi\)
0.0986538 + 0.995122i \(0.468546\pi\)
\(68\) 6.59963e73 0.400783
\(69\) −3.47651e74 −1.16884
\(70\) 1.82884e75 3.43325
\(71\) −3.78476e74 −0.400014 −0.200007 0.979794i \(-0.564097\pi\)
−0.200007 + 0.979794i \(0.564097\pi\)
\(72\) −3.42134e74 −0.205225
\(73\) −1.18691e75 −0.407234 −0.203617 0.979051i \(-0.565270\pi\)
−0.203617 + 0.979051i \(0.565270\pi\)
\(74\) −2.46402e75 −0.487260
\(75\) −5.00640e75 −0.574837
\(76\) −4.33807e76 −2.91304
\(77\) 1.69943e76 0.672092
\(78\) 9.99958e76 2.34504
\(79\) −3.54946e76 −0.496895 −0.248448 0.968645i \(-0.579920\pi\)
−0.248448 + 0.968645i \(0.579920\pi\)
\(80\) 2.07921e77 1.74886
\(81\) −2.14745e77 −1.09214
\(82\) −4.44625e77 −1.37573
\(83\) −5.45755e76 −0.103356 −0.0516779 0.998664i \(-0.516457\pi\)
−0.0516779 + 0.998664i \(0.516457\pi\)
\(84\) 2.99230e78 3.48894
\(85\) 3.24237e77 0.234097
\(86\) −4.84388e78 −2.17774
\(87\) −2.26202e78 −0.636748
\(88\) 4.85888e78 0.860970
\(89\) −1.21164e79 −1.35856 −0.679278 0.733881i \(-0.737707\pi\)
−0.679278 + 0.733881i \(0.737707\pi\)
\(90\) −3.16881e78 −0.225982
\(91\) −4.32017e79 −1.96935
\(92\) 8.09566e79 2.37053
\(93\) −7.36680e78 −0.139226
\(94\) 2.41274e80 2.95688
\(95\) −2.13127e80 −1.70150
\(96\) 9.82165e79 0.513097
\(97\) 2.69780e80 0.926305 0.463153 0.886279i \(-0.346718\pi\)
0.463153 + 0.886279i \(0.346718\pi\)
\(98\) −1.11924e81 −2.53669
\(99\) −2.94458e79 −0.0442383
\(100\) 1.16583e81 1.16583
\(101\) −1.21458e81 −0.811726 −0.405863 0.913934i \(-0.633029\pi\)
−0.405863 + 0.913934i \(0.633029\pi\)
\(102\) 7.79608e80 0.349598
\(103\) 1.20510e81 0.364011 0.182006 0.983298i \(-0.441741\pi\)
0.182006 + 0.983298i \(0.441741\pi\)
\(104\) −1.23519e82 −2.52280
\(105\) 1.47010e82 2.03789
\(106\) −6.06395e81 −0.572622
\(107\) −2.32176e82 −1.49891 −0.749453 0.662058i \(-0.769683\pi\)
−0.749453 + 0.662058i \(0.769683\pi\)
\(108\) 4.53052e82 2.00672
\(109\) 6.16704e82 1.88063 0.940315 0.340304i \(-0.110530\pi\)
0.940315 + 0.340304i \(0.110530\pi\)
\(110\) 4.50024e82 0.948052
\(111\) −1.98069e82 −0.289225
\(112\) −2.15989e83 −2.19332
\(113\) −3.91748e82 −0.277543 −0.138771 0.990324i \(-0.544315\pi\)
−0.138771 + 0.990324i \(0.544315\pi\)
\(114\) −5.12452e83 −2.54101
\(115\) 3.97736e83 1.38463
\(116\) 5.26752e83 1.29139
\(117\) 7.48550e82 0.129626
\(118\) −1.08105e84 −1.32625
\(119\) −3.36818e83 −0.293591
\(120\) 4.20320e84 2.61060
\(121\) −1.83506e84 −0.814409
\(122\) 2.00599e84 0.637903
\(123\) −3.57409e84 −0.816598
\(124\) 1.71549e84 0.282364
\(125\) −4.73541e84 −0.562992
\(126\) 3.29176e84 0.283414
\(127\) 3.92700e84 0.245477 0.122738 0.992439i \(-0.460832\pi\)
0.122738 + 0.992439i \(0.460832\pi\)
\(128\) 3.17931e85 1.44653
\(129\) −3.89372e85 −1.29265
\(130\) −1.14402e86 −2.77797
\(131\) −3.67830e85 −0.654877 −0.327439 0.944872i \(-0.606185\pi\)
−0.327439 + 0.944872i \(0.606185\pi\)
\(132\) 7.36317e85 0.963431
\(133\) 2.21397e86 2.13393
\(134\) −4.90500e85 −0.349055
\(135\) 2.22583e86 1.17212
\(136\) −9.63004e85 −0.376099
\(137\) 1.69752e86 0.492755 0.246378 0.969174i \(-0.420760\pi\)
0.246378 + 0.969174i \(0.420760\pi\)
\(138\) 9.56332e86 2.06779
\(139\) −1.16624e87 −1.88229 −0.941146 0.337999i \(-0.890250\pi\)
−0.941146 + 0.337999i \(0.890250\pi\)
\(140\) −3.42339e87 −4.13305
\(141\) 1.93947e87 1.75513
\(142\) 1.04113e87 0.707662
\(143\) −1.06307e87 −0.543815
\(144\) 3.74242e86 0.144368
\(145\) 2.58791e87 0.754301
\(146\) 3.26501e87 0.720434
\(147\) −8.99695e87 −1.50572
\(148\) 4.61239e87 0.586579
\(149\) −8.60592e87 −0.833208 −0.416604 0.909088i \(-0.636780\pi\)
−0.416604 + 0.909088i \(0.636780\pi\)
\(150\) 1.37718e88 1.01694
\(151\) −1.22694e88 −0.692243 −0.346121 0.938190i \(-0.612501\pi\)
−0.346121 + 0.938190i \(0.612501\pi\)
\(152\) 6.33002e88 2.73363
\(153\) 5.83601e86 0.0193247
\(154\) −4.67486e88 −1.18899
\(155\) 8.42811e87 0.164929
\(156\) −1.87181e89 −2.82303
\(157\) 6.39779e88 0.744893 0.372447 0.928054i \(-0.378519\pi\)
0.372447 + 0.928054i \(0.378519\pi\)
\(158\) 9.76399e88 0.879054
\(159\) −4.87447e88 −0.339894
\(160\) −1.12366e89 −0.607823
\(161\) −4.13169e89 −1.73652
\(162\) 5.90729e89 1.93210
\(163\) 3.48502e89 0.888398 0.444199 0.895928i \(-0.353488\pi\)
0.444199 + 0.895928i \(0.353488\pi\)
\(164\) 8.32290e89 1.65615
\(165\) 3.61749e89 0.562740
\(166\) 1.50128e89 0.182846
\(167\) −6.12121e89 −0.584548 −0.292274 0.956335i \(-0.594412\pi\)
−0.292274 + 0.956335i \(0.594412\pi\)
\(168\) −4.36629e90 −3.27406
\(169\) 1.00651e90 0.593479
\(170\) −8.91925e89 −0.414140
\(171\) −3.83612e89 −0.140459
\(172\) 9.06722e90 2.62163
\(173\) −7.84695e90 −1.79405 −0.897023 0.441984i \(-0.854275\pi\)
−0.897023 + 0.441984i \(0.854275\pi\)
\(174\) 6.22247e90 1.12647
\(175\) −5.94990e90 −0.854022
\(176\) −5.31486e90 −0.605660
\(177\) −8.68995e90 −0.787227
\(178\) 3.33302e91 2.40341
\(179\) −1.27646e91 −0.733599 −0.366800 0.930300i \(-0.619547\pi\)
−0.366800 + 0.930300i \(0.619547\pi\)
\(180\) 5.93167e90 0.272045
\(181\) −4.31048e91 −1.57959 −0.789794 0.613372i \(-0.789813\pi\)
−0.789794 + 0.613372i \(0.789813\pi\)
\(182\) 1.18841e92 3.48397
\(183\) 1.61250e91 0.378644
\(184\) −1.18130e92 −2.22453
\(185\) 2.26604e91 0.342621
\(186\) 2.02649e91 0.246303
\(187\) −8.28811e90 −0.0810719
\(188\) −4.51640e92 −3.55958
\(189\) −2.31219e92 −1.47001
\(190\) 5.86280e92 3.01012
\(191\) 1.96408e92 0.815280 0.407640 0.913143i \(-0.366352\pi\)
0.407640 + 0.913143i \(0.366352\pi\)
\(192\) 1.69239e92 0.568593
\(193\) 7.05705e92 1.92111 0.960555 0.278090i \(-0.0897013\pi\)
0.960555 + 0.278090i \(0.0897013\pi\)
\(194\) −7.42121e92 −1.63872
\(195\) −9.19613e92 −1.64893
\(196\) 2.09510e93 3.05375
\(197\) −3.03690e92 −0.360204 −0.180102 0.983648i \(-0.557643\pi\)
−0.180102 + 0.983648i \(0.557643\pi\)
\(198\) 8.10008e91 0.0782616
\(199\) −1.81756e93 −1.43199 −0.715995 0.698105i \(-0.754027\pi\)
−0.715995 + 0.698105i \(0.754027\pi\)
\(200\) −1.70115e93 −1.09403
\(201\) −3.94285e92 −0.207191
\(202\) 3.34111e93 1.43602
\(203\) −2.68832e93 −0.946002
\(204\) −1.45934e93 −0.420858
\(205\) 4.08900e93 0.967355
\(206\) −3.31503e93 −0.643969
\(207\) 7.15893e92 0.114301
\(208\) 1.35111e94 1.77470
\(209\) 5.44794e93 0.589261
\(210\) −4.04402e94 −3.60522
\(211\) 4.77860e93 0.351448 0.175724 0.984439i \(-0.443773\pi\)
0.175724 + 0.984439i \(0.443773\pi\)
\(212\) 1.13511e94 0.689340
\(213\) 8.36904e93 0.420051
\(214\) 6.38679e94 2.65170
\(215\) 4.45468e94 1.53130
\(216\) −6.61084e94 −1.88313
\(217\) −8.75513e93 −0.206844
\(218\) −1.69645e95 −3.32701
\(219\) 2.62456e94 0.427632
\(220\) −8.42397e94 −1.14130
\(221\) 2.10694e94 0.237556
\(222\) 5.44857e94 0.511666
\(223\) 1.37268e95 1.07454 0.537268 0.843412i \(-0.319456\pi\)
0.537268 + 0.843412i \(0.319456\pi\)
\(224\) 1.16726e95 0.762296
\(225\) 1.03093e94 0.0562132
\(226\) 1.07764e95 0.490999
\(227\) −4.29567e95 −1.63676 −0.818378 0.574680i \(-0.805127\pi\)
−0.818378 + 0.574680i \(0.805127\pi\)
\(228\) 9.59254e95 3.05895
\(229\) 3.49520e95 0.933545 0.466773 0.884377i \(-0.345417\pi\)
0.466773 + 0.884377i \(0.345417\pi\)
\(230\) −1.09411e96 −2.44953
\(231\) −3.75786e95 −0.705757
\(232\) −7.68625e95 −1.21186
\(233\) 1.12210e96 1.48634 0.743169 0.669104i \(-0.233322\pi\)
0.743169 + 0.669104i \(0.233322\pi\)
\(234\) −2.05914e95 −0.229321
\(235\) −2.21889e96 −2.07915
\(236\) 2.02361e96 1.59658
\(237\) 7.84873e95 0.521784
\(238\) 9.26533e95 0.519390
\(239\) 5.57365e95 0.263648 0.131824 0.991273i \(-0.457917\pi\)
0.131824 + 0.991273i \(0.457917\pi\)
\(240\) −4.59766e96 −1.83646
\(241\) −2.44469e96 −0.825150 −0.412575 0.910924i \(-0.635370\pi\)
−0.412575 + 0.910924i \(0.635370\pi\)
\(242\) 5.04796e96 1.44077
\(243\) 8.47109e95 0.204590
\(244\) −3.75499e96 −0.767928
\(245\) 1.02931e97 1.78369
\(246\) 9.83176e96 1.44464
\(247\) −1.38493e97 −1.72664
\(248\) −2.50320e96 −0.264974
\(249\) 1.20680e96 0.108533
\(250\) 1.30264e97 0.995985
\(251\) −1.07328e97 −0.698114 −0.349057 0.937101i \(-0.613498\pi\)
−0.349057 + 0.937101i \(0.613498\pi\)
\(252\) −6.16182e96 −0.341183
\(253\) −1.01669e97 −0.479520
\(254\) −1.08026e97 −0.434271
\(255\) −7.16968e96 −0.245823
\(256\) −6.89525e97 −2.01758
\(257\) −2.29731e97 −0.574021 −0.287010 0.957928i \(-0.592661\pi\)
−0.287010 + 0.957928i \(0.592661\pi\)
\(258\) 1.07110e98 2.28682
\(259\) −2.35397e97 −0.429695
\(260\) 2.14148e98 3.34421
\(261\) 4.65803e96 0.0622675
\(262\) 1.01184e98 1.15854
\(263\) −1.00706e98 −0.988205 −0.494102 0.869404i \(-0.664503\pi\)
−0.494102 + 0.869404i \(0.664503\pi\)
\(264\) −1.07442e98 −0.904095
\(265\) 5.57672e97 0.402644
\(266\) −6.09028e98 −3.77512
\(267\) 2.67923e98 1.42660
\(268\) 9.18162e97 0.420204
\(269\) 1.53194e98 0.602941 0.301471 0.953475i \(-0.402522\pi\)
0.301471 + 0.953475i \(0.402522\pi\)
\(270\) −6.12289e98 −2.07360
\(271\) 4.06678e98 1.18575 0.592877 0.805293i \(-0.297992\pi\)
0.592877 + 0.805293i \(0.297992\pi\)
\(272\) 1.05338e98 0.264572
\(273\) 9.55295e98 2.06800
\(274\) −4.66960e98 −0.871730
\(275\) −1.46410e98 −0.235829
\(276\) −1.79015e99 −2.48927
\(277\) −1.40686e99 −1.68974 −0.844868 0.534974i \(-0.820321\pi\)
−0.844868 + 0.534974i \(0.820321\pi\)
\(278\) 3.20813e99 3.32995
\(279\) 1.51699e97 0.0136149
\(280\) 4.99533e99 3.87850
\(281\) −2.51972e99 −1.69335 −0.846674 0.532112i \(-0.821398\pi\)
−0.846674 + 0.532112i \(0.821398\pi\)
\(282\) −5.33517e99 −3.10499
\(283\) 2.91238e99 1.46858 0.734289 0.678837i \(-0.237516\pi\)
0.734289 + 0.678837i \(0.237516\pi\)
\(284\) −1.94888e99 −0.851906
\(285\) 4.71277e99 1.78673
\(286\) 2.92433e99 0.962059
\(287\) −4.24766e99 −1.21320
\(288\) −2.02250e98 −0.0501757
\(289\) −4.47407e99 −0.964585
\(290\) −7.11893e99 −1.33443
\(291\) −5.96550e99 −0.972703
\(292\) −6.11175e99 −0.867282
\(293\) 4.29360e99 0.530497 0.265249 0.964180i \(-0.414546\pi\)
0.265249 + 0.964180i \(0.414546\pi\)
\(294\) 2.47492e100 2.66375
\(295\) 9.94189e99 0.932561
\(296\) −6.73029e99 −0.550452
\(297\) −5.68963e99 −0.405927
\(298\) 2.36735e100 1.47402
\(299\) 2.58455e100 1.40508
\(300\) −2.57793e100 −1.22422
\(301\) −4.62753e100 −1.92046
\(302\) 3.37513e100 1.22464
\(303\) 2.68573e100 0.852385
\(304\) −6.92406e100 −1.92301
\(305\) −1.84481e100 −0.448547
\(306\) −1.60539e99 −0.0341872
\(307\) −8.74192e100 −1.63118 −0.815592 0.578628i \(-0.803588\pi\)
−0.815592 + 0.578628i \(0.803588\pi\)
\(308\) 8.75083e100 1.43135
\(309\) −2.66477e100 −0.382244
\(310\) −2.31844e100 −0.291774
\(311\) 7.37638e100 0.814793 0.407397 0.913251i \(-0.366437\pi\)
0.407397 + 0.913251i \(0.366437\pi\)
\(312\) 2.73131e101 2.64917
\(313\) −2.41414e100 −0.205692 −0.102846 0.994697i \(-0.532795\pi\)
−0.102846 + 0.994697i \(0.532795\pi\)
\(314\) −1.75993e101 −1.31779
\(315\) −3.02728e100 −0.199285
\(316\) −1.82771e101 −1.05823
\(317\) −3.79837e99 −0.0193507 −0.00967536 0.999953i \(-0.503080\pi\)
−0.00967536 + 0.999953i \(0.503080\pi\)
\(318\) 1.34089e101 0.601304
\(319\) −6.61518e100 −0.261228
\(320\) −1.93621e101 −0.673564
\(321\) 5.13398e101 1.57398
\(322\) 1.13656e102 3.07206
\(323\) −1.07975e101 −0.257408
\(324\) −1.10578e102 −2.32593
\(325\) 3.72193e101 0.691021
\(326\) −9.58674e101 −1.57166
\(327\) −1.36368e102 −1.97483
\(328\) −1.21446e102 −1.55415
\(329\) 2.30498e102 2.60755
\(330\) −9.95115e101 −0.995540
\(331\) 1.31632e102 1.16501 0.582503 0.812829i \(-0.302074\pi\)
0.582503 + 0.812829i \(0.302074\pi\)
\(332\) −2.81024e101 −0.220116
\(333\) 4.07870e100 0.0282833
\(334\) 1.68385e102 1.03412
\(335\) 4.51089e101 0.245441
\(336\) 4.77605e102 2.30318
\(337\) −2.45788e102 −1.05087 −0.525435 0.850834i \(-0.676097\pi\)
−0.525435 + 0.850834i \(0.676097\pi\)
\(338\) −2.76874e102 −1.04992
\(339\) 8.66252e101 0.291445
\(340\) 1.66959e102 0.498554
\(341\) −2.15438e101 −0.0571177
\(342\) 1.05526e102 0.248485
\(343\) −3.23553e102 −0.676914
\(344\) −1.32307e103 −2.46017
\(345\) −8.79493e102 −1.45398
\(346\) 2.15857e103 3.17383
\(347\) −4.49961e102 −0.588615 −0.294308 0.955711i \(-0.595089\pi\)
−0.294308 + 0.955711i \(0.595089\pi\)
\(348\) −1.16478e103 −1.35608
\(349\) −2.66970e102 −0.276715 −0.138357 0.990382i \(-0.544182\pi\)
−0.138357 + 0.990382i \(0.544182\pi\)
\(350\) 1.63672e103 1.51084
\(351\) 1.44638e103 1.18944
\(352\) 2.87230e102 0.210500
\(353\) −2.24963e102 −0.146972 −0.0734861 0.997296i \(-0.523412\pi\)
−0.0734861 + 0.997296i \(0.523412\pi\)
\(354\) 2.39047e103 1.39268
\(355\) −9.57475e102 −0.497598
\(356\) −6.23905e103 −2.89330
\(357\) 7.44787e102 0.308297
\(358\) 3.51133e103 1.29781
\(359\) −2.07238e103 −0.684140 −0.342070 0.939674i \(-0.611128\pi\)
−0.342070 + 0.939674i \(0.611128\pi\)
\(360\) −8.65536e102 −0.255290
\(361\) 3.30391e103 0.870937
\(362\) 1.18575e104 2.79444
\(363\) 4.05777e103 0.855203
\(364\) −2.22457e104 −4.19412
\(365\) −3.00268e103 −0.506579
\(366\) −4.43573e103 −0.669856
\(367\) 1.73601e103 0.234734 0.117367 0.993089i \(-0.462555\pi\)
0.117367 + 0.993089i \(0.462555\pi\)
\(368\) 1.29216e104 1.56488
\(369\) 7.35987e102 0.0798551
\(370\) −6.23353e103 −0.606128
\(371\) −5.79311e103 −0.504973
\(372\) −3.79337e103 −0.296507
\(373\) −2.70244e104 −1.89473 −0.947366 0.320152i \(-0.896266\pi\)
−0.947366 + 0.320152i \(0.896266\pi\)
\(374\) 2.27993e103 0.143424
\(375\) 1.04712e104 0.591192
\(376\) 6.59023e104 3.34035
\(377\) 1.68166e104 0.765446
\(378\) 6.36047e104 2.60059
\(379\) 1.10245e104 0.405016 0.202508 0.979281i \(-0.435091\pi\)
0.202508 + 0.979281i \(0.435091\pi\)
\(380\) −1.09745e105 −3.62368
\(381\) −8.68357e103 −0.257773
\(382\) −5.40286e104 −1.44231
\(383\) 4.67521e104 1.12267 0.561333 0.827590i \(-0.310289\pi\)
0.561333 + 0.827590i \(0.310289\pi\)
\(384\) −7.03024e104 −1.51899
\(385\) 4.29924e104 0.836050
\(386\) −1.94128e105 −3.39862
\(387\) 8.01807e103 0.126408
\(388\) 1.38917e105 1.97274
\(389\) 3.56223e103 0.0455788 0.0227894 0.999740i \(-0.492745\pi\)
0.0227894 + 0.999740i \(0.492745\pi\)
\(390\) 2.52971e105 2.91712
\(391\) 2.01502e104 0.209470
\(392\) −3.05712e105 −2.86567
\(393\) 8.13363e104 0.687680
\(394\) 8.35403e104 0.637234
\(395\) −8.97947e104 −0.618114
\(396\) −1.51625e104 −0.0942138
\(397\) 1.81315e104 0.101723 0.0508615 0.998706i \(-0.483803\pi\)
0.0508615 + 0.998706i \(0.483803\pi\)
\(398\) 4.99983e105 2.53332
\(399\) −4.89563e105 −2.24082
\(400\) 1.86080e105 0.769608
\(401\) −3.21134e103 −0.0120044 −0.00600218 0.999982i \(-0.501911\pi\)
−0.00600218 + 0.999982i \(0.501911\pi\)
\(402\) 1.08462e105 0.366539
\(403\) 5.47672e104 0.167366
\(404\) −6.25419e105 −1.72872
\(405\) −5.43265e105 −1.35857
\(406\) 7.39515e105 1.67357
\(407\) −5.79244e104 −0.118655
\(408\) 2.12944e105 0.394938
\(409\) 7.61995e105 1.27985 0.639923 0.768439i \(-0.278966\pi\)
0.639923 + 0.768439i \(0.278966\pi\)
\(410\) −1.12482e106 −1.71134
\(411\) −3.75363e105 −0.517437
\(412\) 6.20538e105 0.775231
\(413\) −1.03276e106 −1.16956
\(414\) −1.96931e105 −0.202209
\(415\) −1.38066e105 −0.128570
\(416\) −7.30174e105 −0.616803
\(417\) 2.57884e106 1.97658
\(418\) −1.49864e106 −1.04246
\(419\) 5.20228e105 0.328493 0.164246 0.986419i \(-0.447481\pi\)
0.164246 + 0.986419i \(0.447481\pi\)
\(420\) 7.56996e106 4.34007
\(421\) −1.35551e106 −0.705792 −0.352896 0.935663i \(-0.614803\pi\)
−0.352896 + 0.935663i \(0.614803\pi\)
\(422\) −1.31452e106 −0.621745
\(423\) −3.99381e105 −0.171634
\(424\) −1.65632e106 −0.646885
\(425\) 2.90177e105 0.103017
\(426\) −2.30219e106 −0.743109
\(427\) 1.91639e106 0.562542
\(428\) −1.19554e107 −3.19220
\(429\) 2.35070e106 0.571055
\(430\) −1.22541e107 −2.70901
\(431\) 9.80300e106 1.97256 0.986278 0.165091i \(-0.0527917\pi\)
0.986278 + 0.165091i \(0.0527917\pi\)
\(432\) 7.23124e106 1.32471
\(433\) −2.54698e106 −0.424879 −0.212439 0.977174i \(-0.568141\pi\)
−0.212439 + 0.977174i \(0.568141\pi\)
\(434\) 2.40840e106 0.365927
\(435\) −5.72251e106 −0.792084
\(436\) 3.17558e107 4.00516
\(437\) −1.32451e107 −1.52250
\(438\) −7.21975e106 −0.756521
\(439\) 2.15473e106 0.205864 0.102932 0.994688i \(-0.467178\pi\)
0.102932 + 0.994688i \(0.467178\pi\)
\(440\) 1.22921e107 1.07101
\(441\) 1.85268e106 0.147244
\(442\) −5.79587e106 −0.420258
\(443\) 1.17940e107 0.780391 0.390195 0.920732i \(-0.372407\pi\)
0.390195 + 0.920732i \(0.372407\pi\)
\(444\) −1.01991e107 −0.615960
\(445\) −3.06522e107 −1.68998
\(446\) −3.77601e107 −1.90095
\(447\) 1.90298e107 0.874943
\(448\) 2.01134e107 0.844746
\(449\) 3.36157e107 1.28993 0.644964 0.764213i \(-0.276872\pi\)
0.644964 + 0.764213i \(0.276872\pi\)
\(450\) −2.83593e106 −0.0994464
\(451\) −1.04523e107 −0.335012
\(452\) −2.01722e107 −0.591081
\(453\) 2.71308e107 0.726917
\(454\) 1.18167e108 2.89558
\(455\) −1.09292e108 −2.44978
\(456\) −1.39972e108 −2.87055
\(457\) −7.65904e106 −0.143737 −0.0718685 0.997414i \(-0.522896\pi\)
−0.0718685 + 0.997414i \(0.522896\pi\)
\(458\) −9.61475e107 −1.65153
\(459\) 1.12765e107 0.177322
\(460\) 2.04805e108 2.94882
\(461\) 7.66641e107 1.01089 0.505446 0.862858i \(-0.331328\pi\)
0.505446 + 0.862858i \(0.331328\pi\)
\(462\) 1.03373e108 1.24855
\(463\) 4.28640e107 0.474310 0.237155 0.971472i \(-0.423785\pi\)
0.237155 + 0.971472i \(0.423785\pi\)
\(464\) 8.40757e107 0.852496
\(465\) −1.86366e107 −0.173190
\(466\) −3.08672e108 −2.62947
\(467\) −9.14459e107 −0.714218 −0.357109 0.934063i \(-0.616237\pi\)
−0.357109 + 0.934063i \(0.616237\pi\)
\(468\) 3.85449e107 0.276064
\(469\) −4.68592e107 −0.307818
\(470\) 6.10380e108 3.67821
\(471\) −1.41471e108 −0.782205
\(472\) −2.95280e108 −1.49825
\(473\) −1.13870e108 −0.530315
\(474\) −2.15906e108 −0.923085
\(475\) −1.90739e108 −0.748769
\(476\) −1.73437e108 −0.625259
\(477\) 1.00376e107 0.0332382
\(478\) −1.53322e108 −0.466418
\(479\) 1.30765e108 0.365514 0.182757 0.983158i \(-0.441498\pi\)
0.182757 + 0.983158i \(0.441498\pi\)
\(480\) 2.48470e108 0.638268
\(481\) 1.47251e108 0.347683
\(482\) 6.72494e108 1.45977
\(483\) 9.13618e108 1.82350
\(484\) −9.44923e108 −1.73444
\(485\) 6.82493e108 1.15228
\(486\) −2.33026e108 −0.361939
\(487\) 1.34851e109 1.92722 0.963612 0.267303i \(-0.0861326\pi\)
0.963612 + 0.267303i \(0.0861326\pi\)
\(488\) 5.47920e108 0.720633
\(489\) −7.70624e108 −0.932898
\(490\) −2.83147e109 −3.15552
\(491\) −1.05930e109 −1.08698 −0.543488 0.839417i \(-0.682897\pi\)
−0.543488 + 0.839417i \(0.682897\pi\)
\(492\) −1.84040e109 −1.73910
\(493\) 1.31109e108 0.114113
\(494\) 3.80974e109 3.05459
\(495\) −7.44925e107 −0.0550303
\(496\) 2.73812e108 0.186399
\(497\) 9.94626e108 0.624060
\(498\) −3.31971e108 −0.192005
\(499\) −1.24975e109 −0.666427 −0.333213 0.942851i \(-0.608133\pi\)
−0.333213 + 0.942851i \(0.608133\pi\)
\(500\) −2.43839e109 −1.19900
\(501\) 1.35355e109 0.613828
\(502\) 2.95242e109 1.23503
\(503\) −2.57395e109 −0.993337 −0.496669 0.867940i \(-0.665444\pi\)
−0.496669 + 0.867940i \(0.665444\pi\)
\(504\) 8.99120e108 0.320170
\(505\) −3.07266e109 −1.00975
\(506\) 2.79675e109 0.848316
\(507\) −2.22564e109 −0.623207
\(508\) 2.02212e109 0.522790
\(509\) −2.29496e109 −0.547905 −0.273953 0.961743i \(-0.588331\pi\)
−0.273953 + 0.961743i \(0.588331\pi\)
\(510\) 1.97227e109 0.434884
\(511\) 3.11918e109 0.635323
\(512\) 1.12806e110 2.12276
\(513\) −7.41229e109 −1.28884
\(514\) 6.31952e109 1.01550
\(515\) 3.04867e109 0.452812
\(516\) −2.00499e110 −2.75295
\(517\) 5.67189e109 0.720047
\(518\) 6.47540e109 0.760171
\(519\) 1.73515e110 1.88391
\(520\) −3.12480e110 −3.13824
\(521\) −1.75441e110 −1.63006 −0.815030 0.579418i \(-0.803280\pi\)
−0.815030 + 0.579418i \(0.803280\pi\)
\(522\) −1.28135e109 −0.110157
\(523\) −5.76234e109 −0.458438 −0.229219 0.973375i \(-0.573617\pi\)
−0.229219 + 0.973375i \(0.573617\pi\)
\(524\) −1.89406e110 −1.39469
\(525\) 1.31567e110 0.896800
\(526\) 2.77026e110 1.74823
\(527\) 4.26988e108 0.0249509
\(528\) 1.17525e110 0.635998
\(529\) 4.76740e109 0.238961
\(530\) −1.53407e110 −0.712314
\(531\) 1.78946e109 0.0769828
\(532\) 1.14003e111 4.54461
\(533\) 2.65710e110 0.981647
\(534\) −7.37013e110 −2.52380
\(535\) −5.87362e110 −1.86457
\(536\) −1.33976e110 −0.394324
\(537\) 2.82256e110 0.770345
\(538\) −4.21413e110 −1.06666
\(539\) −2.63111e110 −0.617724
\(540\) 1.14614e111 2.49626
\(541\) −7.18664e110 −1.45223 −0.726117 0.687571i \(-0.758677\pi\)
−0.726117 + 0.687571i \(0.758677\pi\)
\(542\) −1.11871e111 −2.09771
\(543\) 9.53154e110 1.65871
\(544\) −5.69274e109 −0.0919530
\(545\) 1.56015e111 2.33941
\(546\) −2.62786e111 −3.65848
\(547\) −6.52155e110 −0.843070 −0.421535 0.906812i \(-0.638509\pi\)
−0.421535 + 0.906812i \(0.638509\pi\)
\(548\) 8.74098e110 1.04942
\(549\) −3.32051e109 −0.0370275
\(550\) 4.02750e110 0.417203
\(551\) −8.61808e110 −0.829413
\(552\) 2.61215e111 2.33596
\(553\) 9.32789e110 0.775203
\(554\) 3.87004e111 2.98930
\(555\) −5.01078e110 −0.359782
\(556\) −6.00527e111 −4.00870
\(557\) −1.59621e111 −0.990731 −0.495366 0.868685i \(-0.664966\pi\)
−0.495366 + 0.868685i \(0.664966\pi\)
\(558\) −4.17300e109 −0.0240859
\(559\) 2.89472e111 1.55392
\(560\) −5.46412e111 −2.72838
\(561\) 1.83271e110 0.0851328
\(562\) 6.93135e111 2.99569
\(563\) −2.62548e111 −1.05589 −0.527945 0.849278i \(-0.677037\pi\)
−0.527945 + 0.849278i \(0.677037\pi\)
\(564\) 9.98686e111 3.73788
\(565\) −9.91051e110 −0.345250
\(566\) −8.01148e111 −2.59805
\(567\) 5.64344e111 1.70385
\(568\) 2.84376e111 0.799439
\(569\) −4.82520e111 −1.26319 −0.631594 0.775299i \(-0.717599\pi\)
−0.631594 + 0.775299i \(0.717599\pi\)
\(570\) −1.29641e112 −3.16090
\(571\) −2.19654e111 −0.498858 −0.249429 0.968393i \(-0.580243\pi\)
−0.249429 + 0.968393i \(0.580243\pi\)
\(572\) −5.47403e111 −1.15816
\(573\) −4.34306e111 −0.856117
\(574\) 1.16846e112 2.14627
\(575\) 3.55955e111 0.609322
\(576\) −3.48503e110 −0.0556026
\(577\) 1.35541e111 0.201581 0.100790 0.994908i \(-0.467863\pi\)
0.100790 + 0.994908i \(0.467863\pi\)
\(578\) 1.23075e112 1.70644
\(579\) −1.56049e112 −2.01734
\(580\) 1.33259e112 1.60643
\(581\) 1.43423e111 0.161245
\(582\) 1.64101e112 1.72080
\(583\) −1.42552e111 −0.139443
\(584\) 8.91813e111 0.813867
\(585\) 1.89369e111 0.161249
\(586\) −1.18110e112 −0.938499
\(587\) 1.67568e112 1.24265 0.621325 0.783553i \(-0.286595\pi\)
0.621325 + 0.783553i \(0.286595\pi\)
\(588\) −4.63277e112 −3.20671
\(589\) −2.80667e111 −0.181352
\(590\) −2.73485e112 −1.64979
\(591\) 6.71534e111 0.378247
\(592\) 7.36190e111 0.387223
\(593\) 7.97650e111 0.391830 0.195915 0.980621i \(-0.437232\pi\)
0.195915 + 0.980621i \(0.437232\pi\)
\(594\) 1.56513e112 0.718122
\(595\) −8.52087e111 −0.365214
\(596\) −4.43142e112 −1.77447
\(597\) 4.01908e112 1.50372
\(598\) −7.10969e112 −2.48572
\(599\) 2.32037e112 0.758178 0.379089 0.925360i \(-0.376237\pi\)
0.379089 + 0.925360i \(0.376237\pi\)
\(600\) 3.76167e112 1.14883
\(601\) 3.08514e112 0.880762 0.440381 0.897811i \(-0.354843\pi\)
0.440381 + 0.897811i \(0.354843\pi\)
\(602\) 1.27296e113 3.39748
\(603\) 8.11923e110 0.0202611
\(604\) −6.31787e112 −1.47426
\(605\) −4.64236e112 −1.01309
\(606\) −7.38801e112 −1.50795
\(607\) −1.27038e112 −0.242544 −0.121272 0.992619i \(-0.538697\pi\)
−0.121272 + 0.992619i \(0.538697\pi\)
\(608\) 3.74195e112 0.668348
\(609\) 5.94455e112 0.993387
\(610\) 5.07478e112 0.793521
\(611\) −1.44187e113 −2.10987
\(612\) 3.00512e111 0.0411556
\(613\) 5.88625e112 0.754553 0.377276 0.926101i \(-0.376861\pi\)
0.377276 + 0.926101i \(0.376861\pi\)
\(614\) 2.40476e113 2.88571
\(615\) −9.04179e112 −1.01581
\(616\) −1.27690e113 −1.34319
\(617\) −1.09249e113 −1.07614 −0.538071 0.842900i \(-0.680847\pi\)
−0.538071 + 0.842900i \(0.680847\pi\)
\(618\) 7.33035e112 0.676225
\(619\) −1.28134e113 −1.10711 −0.553556 0.832812i \(-0.686730\pi\)
−0.553556 + 0.832812i \(0.686730\pi\)
\(620\) 4.33987e112 0.351247
\(621\) 1.38327e113 1.04881
\(622\) −2.02912e113 −1.44144
\(623\) 3.18415e113 2.11947
\(624\) −2.98763e113 −1.86359
\(625\) −2.13437e113 −1.24775
\(626\) 6.64092e112 0.363888
\(627\) −1.20467e113 −0.618777
\(628\) 3.29440e113 1.58639
\(629\) 1.14803e112 0.0518325
\(630\) 8.32756e112 0.352554
\(631\) −3.06535e112 −0.121700 −0.0608500 0.998147i \(-0.519381\pi\)
−0.0608500 + 0.998147i \(0.519381\pi\)
\(632\) 2.66696e113 0.993058
\(633\) −1.05667e113 −0.369052
\(634\) 1.04487e112 0.0342332
\(635\) 9.93459e112 0.305361
\(636\) −2.51000e113 −0.723869
\(637\) 6.68863e113 1.81005
\(638\) 1.81973e113 0.462136
\(639\) −1.72338e112 −0.0410767
\(640\) 8.04306e113 1.79942
\(641\) 1.95149e113 0.409841 0.204920 0.978779i \(-0.434306\pi\)
0.204920 + 0.978779i \(0.434306\pi\)
\(642\) −1.41228e114 −2.78453
\(643\) 2.08964e112 0.0386837 0.0193418 0.999813i \(-0.493843\pi\)
0.0193418 + 0.999813i \(0.493843\pi\)
\(644\) −2.12752e114 −3.69825
\(645\) −9.85040e113 −1.60800
\(646\) 2.97023e113 0.455379
\(647\) −3.65213e113 −0.525924 −0.262962 0.964806i \(-0.584699\pi\)
−0.262962 + 0.964806i \(0.584699\pi\)
\(648\) 1.61353e114 2.18268
\(649\) −2.54133e113 −0.322962
\(650\) −1.02384e114 −1.22248
\(651\) 1.93598e113 0.217205
\(652\) 1.79453e114 1.89201
\(653\) 1.17868e114 1.16792 0.583958 0.811784i \(-0.301503\pi\)
0.583958 + 0.811784i \(0.301503\pi\)
\(654\) 3.75128e114 3.49366
\(655\) −9.30542e113 −0.814636
\(656\) 1.32843e114 1.09328
\(657\) −5.40457e112 −0.0418181
\(658\) −6.34064e114 −4.61301
\(659\) −1.22498e114 −0.838047 −0.419023 0.907975i \(-0.637627\pi\)
−0.419023 + 0.907975i \(0.637627\pi\)
\(660\) 1.86275e114 1.19846
\(661\) 2.44914e114 1.48202 0.741011 0.671492i \(-0.234346\pi\)
0.741011 + 0.671492i \(0.234346\pi\)
\(662\) −3.62099e114 −2.06100
\(663\) −4.65897e113 −0.249455
\(664\) 4.10064e113 0.206559
\(665\) 5.60094e114 2.65451
\(666\) −1.12198e113 −0.0500358
\(667\) 1.60830e114 0.674948
\(668\) −3.15198e114 −1.24491
\(669\) −3.03532e114 −1.12836
\(670\) −1.24087e114 −0.434208
\(671\) 4.71568e113 0.155340
\(672\) −2.58111e114 −0.800480
\(673\) −3.94004e114 −1.15051 −0.575256 0.817974i \(-0.695098\pi\)
−0.575256 + 0.817974i \(0.695098\pi\)
\(674\) 6.76122e114 1.85909
\(675\) 1.99201e114 0.515808
\(676\) 5.18279e114 1.26393
\(677\) −3.10016e114 −0.712104 −0.356052 0.934466i \(-0.615877\pi\)
−0.356052 + 0.934466i \(0.615877\pi\)
\(678\) −2.38292e114 −0.515593
\(679\) −7.08975e114 −1.44512
\(680\) −2.43622e114 −0.467849
\(681\) 9.49880e114 1.71874
\(682\) 5.92637e113 0.101047
\(683\) −8.16946e114 −1.31267 −0.656334 0.754470i \(-0.727894\pi\)
−0.656334 + 0.754470i \(0.727894\pi\)
\(684\) −1.97532e114 −0.299134
\(685\) 4.29441e114 0.612964
\(686\) 8.90043e114 1.19752
\(687\) −7.72875e114 −0.980306
\(688\) 1.44723e115 1.73064
\(689\) 3.62384e114 0.408593
\(690\) 2.41934e115 2.57223
\(691\) 5.12579e114 0.513925 0.256963 0.966421i \(-0.417278\pi\)
0.256963 + 0.966421i \(0.417278\pi\)
\(692\) −4.04061e115 −3.82076
\(693\) 7.73829e113 0.0690159
\(694\) 1.23777e115 1.04131
\(695\) −2.95036e115 −2.34148
\(696\) 1.69962e115 1.27256
\(697\) 2.07158e114 0.146344
\(698\) 7.34391e114 0.489534
\(699\) −2.48124e115 −1.56079
\(700\) −3.06377e115 −1.81880
\(701\) 2.06857e115 1.15902 0.579509 0.814966i \(-0.303244\pi\)
0.579509 + 0.814966i \(0.303244\pi\)
\(702\) −3.97875e115 −2.10423
\(703\) −7.54623e114 −0.376738
\(704\) 4.94933e114 0.233267
\(705\) 4.90650e115 2.18330
\(706\) 6.18837e114 0.260007
\(707\) 3.19188e115 1.26637
\(708\) −4.47469e115 −1.67655
\(709\) 1.30065e115 0.460243 0.230121 0.973162i \(-0.426088\pi\)
0.230121 + 0.973162i \(0.426088\pi\)
\(710\) 2.63386e115 0.880298
\(711\) −1.61623e114 −0.0510252
\(712\) 9.10389e115 2.71511
\(713\) 5.23778e114 0.147578
\(714\) −2.04879e115 −0.545406
\(715\) −2.68936e115 −0.676480
\(716\) −6.57284e115 −1.56234
\(717\) −1.23247e115 −0.276854
\(718\) 5.70080e115 1.21031
\(719\) 2.80859e115 0.563596 0.281798 0.959474i \(-0.409069\pi\)
0.281798 + 0.959474i \(0.409069\pi\)
\(720\) 9.46763e114 0.179587
\(721\) −3.16697e115 −0.567891
\(722\) −9.08853e115 −1.54077
\(723\) 5.40580e115 0.866481
\(724\) −2.21959e116 −3.36403
\(725\) 2.31605e115 0.331940
\(726\) −1.11623e116 −1.51293
\(727\) −3.89805e115 −0.499694 −0.249847 0.968285i \(-0.580380\pi\)
−0.249847 + 0.968285i \(0.580380\pi\)
\(728\) 3.24605e116 3.93581
\(729\) 7.64919e115 0.877305
\(730\) 8.25988e115 0.896186
\(731\) 2.25685e115 0.231658
\(732\) 8.30320e115 0.806393
\(733\) −6.67683e115 −0.613563 −0.306782 0.951780i \(-0.599252\pi\)
−0.306782 + 0.951780i \(0.599252\pi\)
\(734\) −4.77547e115 −0.415266
\(735\) −2.27606e116 −1.87304
\(736\) −6.98318e115 −0.543879
\(737\) −1.15307e115 −0.0850005
\(738\) −2.02458e115 −0.141271
\(739\) −1.41381e116 −0.933883 −0.466941 0.884288i \(-0.654644\pi\)
−0.466941 + 0.884288i \(0.654644\pi\)
\(740\) 1.16685e116 0.729676
\(741\) 3.06243e116 1.81313
\(742\) 1.59359e116 0.893344
\(743\) −3.10050e116 −1.64583 −0.822915 0.568164i \(-0.807654\pi\)
−0.822915 + 0.568164i \(0.807654\pi\)
\(744\) 5.53520e115 0.278246
\(745\) −2.17714e116 −1.03647
\(746\) 7.43397e116 3.35196
\(747\) −2.48507e114 −0.0106134
\(748\) −4.26778e115 −0.172658
\(749\) 6.10153e116 2.33843
\(750\) −2.88045e116 −1.04587
\(751\) −2.35328e116 −0.809573 −0.404786 0.914411i \(-0.632654\pi\)
−0.404786 + 0.914411i \(0.632654\pi\)
\(752\) −7.20869e116 −2.34982
\(753\) 2.37328e116 0.733083
\(754\) −4.62599e116 −1.35414
\(755\) −3.10394e116 −0.861117
\(756\) −1.19061e117 −3.13067
\(757\) 7.10783e116 1.77155 0.885776 0.464113i \(-0.153627\pi\)
0.885776 + 0.464113i \(0.153627\pi\)
\(758\) −3.03267e116 −0.716511
\(759\) 2.24815e116 0.503539
\(760\) 1.60138e117 3.40050
\(761\) 1.38307e116 0.278462 0.139231 0.990260i \(-0.455537\pi\)
0.139231 + 0.990260i \(0.455537\pi\)
\(762\) 2.38871e116 0.456024
\(763\) −1.62068e117 −2.93396
\(764\) 1.01136e117 1.73629
\(765\) 1.47640e115 0.0240390
\(766\) −1.28608e117 −1.98610
\(767\) 6.46039e116 0.946339
\(768\) 1.52471e117 2.11864
\(769\) 9.90906e115 0.130622 0.0653109 0.997865i \(-0.479196\pi\)
0.0653109 + 0.997865i \(0.479196\pi\)
\(770\) −1.18265e117 −1.47905
\(771\) 5.07991e116 0.602773
\(772\) 3.63387e117 4.09137
\(773\) −1.27598e117 −1.36325 −0.681624 0.731703i \(-0.738726\pi\)
−0.681624 + 0.731703i \(0.738726\pi\)
\(774\) −2.20564e116 −0.223628
\(775\) 7.54276e115 0.0725790
\(776\) −2.02705e117 −1.85125
\(777\) 5.20521e116 0.451219
\(778\) −9.79912e115 −0.0806331
\(779\) −1.36169e117 −1.06368
\(780\) −4.73534e117 −3.51172
\(781\) 2.44749e116 0.172327
\(782\) −5.54301e116 −0.370571
\(783\) 9.00041e116 0.571362
\(784\) 3.34402e117 2.01589
\(785\) 1.61852e117 0.926612
\(786\) −2.23743e117 −1.21657
\(787\) 9.41474e116 0.486219 0.243109 0.969999i \(-0.421833\pi\)
0.243109 + 0.969999i \(0.421833\pi\)
\(788\) −1.56378e117 −0.767123
\(789\) 2.22686e117 1.03770
\(790\) 2.47011e117 1.09350
\(791\) 1.02951e117 0.432993
\(792\) 2.21247e116 0.0884114
\(793\) −1.19879e117 −0.455174
\(794\) −4.98770e116 −0.179957
\(795\) −1.23315e117 −0.422812
\(796\) −9.35913e117 −3.04969
\(797\) 2.29481e117 0.710696 0.355348 0.934734i \(-0.384362\pi\)
0.355348 + 0.934734i \(0.384362\pi\)
\(798\) 1.34671e118 3.96422
\(799\) −1.12414e117 −0.314540
\(800\) −1.00562e117 −0.267480
\(801\) −5.51714e116 −0.139507
\(802\) 8.83388e115 0.0212368
\(803\) 7.67541e116 0.175437
\(804\) −2.03028e117 −0.441252
\(805\) −1.04524e118 −2.16015
\(806\) −1.50656e117 −0.296085
\(807\) −3.38750e117 −0.633142
\(808\) 9.12598e117 1.62226
\(809\) 1.11320e118 1.88217 0.941084 0.338174i \(-0.109809\pi\)
0.941084 + 0.338174i \(0.109809\pi\)
\(810\) 1.49444e118 2.40344
\(811\) −6.69075e117 −1.02360 −0.511800 0.859105i \(-0.671021\pi\)
−0.511800 + 0.859105i \(0.671021\pi\)
\(812\) −1.38429e118 −2.01469
\(813\) −8.99266e117 −1.24515
\(814\) 1.59341e117 0.209913
\(815\) 8.81646e117 1.10512
\(816\) −2.32928e117 −0.277824
\(817\) −1.48347e118 −1.68378
\(818\) −2.09613e118 −2.26417
\(819\) −1.96717e117 −0.202229
\(820\) 2.10554e118 2.06017
\(821\) 1.07493e117 0.100111 0.0500555 0.998746i \(-0.484060\pi\)
0.0500555 + 0.998746i \(0.484060\pi\)
\(822\) 1.03256e118 0.915394
\(823\) −8.97693e117 −0.757591 −0.378795 0.925480i \(-0.623662\pi\)
−0.378795 + 0.925480i \(0.623662\pi\)
\(824\) −9.05475e117 −0.727486
\(825\) 3.23748e117 0.247641
\(826\) 2.84097e118 2.06907
\(827\) 4.59941e117 0.318955 0.159477 0.987202i \(-0.449019\pi\)
0.159477 + 0.987202i \(0.449019\pi\)
\(828\) 3.68633e117 0.243425
\(829\) 2.87941e118 1.81070 0.905348 0.424671i \(-0.139610\pi\)
0.905348 + 0.424671i \(0.139610\pi\)
\(830\) 3.79797e117 0.227452
\(831\) 3.11091e118 1.77437
\(832\) −1.25818e118 −0.683516
\(833\) 5.21473e117 0.269842
\(834\) −7.09397e118 −3.49675
\(835\) −1.54855e118 −0.727150
\(836\) 2.80529e118 1.25494
\(837\) 2.93119e117 0.124929
\(838\) −1.43107e118 −0.581135
\(839\) −2.90748e118 −1.12501 −0.562506 0.826793i \(-0.690163\pi\)
−0.562506 + 0.826793i \(0.690163\pi\)
\(840\) −1.10459e119 −4.07278
\(841\) −1.79956e118 −0.632309
\(842\) 3.72878e118 1.24861
\(843\) 5.57172e118 1.77817
\(844\) 2.46063e118 0.748476
\(845\) 2.54628e118 0.738260
\(846\) 1.09863e118 0.303636
\(847\) 4.82249e118 1.27056
\(848\) 1.81176e118 0.455060
\(849\) −6.43998e118 −1.54214
\(850\) −7.98230e117 −0.182247
\(851\) 1.40827e118 0.306576
\(852\) 4.30945e118 0.894578
\(853\) −6.52434e118 −1.29152 −0.645759 0.763541i \(-0.723459\pi\)
−0.645759 + 0.763541i \(0.723459\pi\)
\(854\) −5.27168e118 −0.995189
\(855\) −9.70468e117 −0.174724
\(856\) 1.74450e119 2.99560
\(857\) −1.76150e118 −0.288508 −0.144254 0.989541i \(-0.546078\pi\)
−0.144254 + 0.989541i \(0.546078\pi\)
\(858\) −6.46641e118 −1.01025
\(859\) −1.96639e118 −0.293053 −0.146526 0.989207i \(-0.546809\pi\)
−0.146526 + 0.989207i \(0.546809\pi\)
\(860\) 2.29384e119 3.26119
\(861\) 9.39262e118 1.27397
\(862\) −2.69665e119 −3.48964
\(863\) −1.80815e118 −0.223253 −0.111626 0.993750i \(-0.535606\pi\)
−0.111626 + 0.993750i \(0.535606\pi\)
\(864\) −3.90795e118 −0.460408
\(865\) −1.98513e119 −2.23171
\(866\) 7.00633e118 0.751650
\(867\) 9.89328e118 1.01290
\(868\) −4.50826e118 −0.440514
\(869\) 2.29532e118 0.214064
\(870\) 1.57417e119 1.40127
\(871\) 2.93125e118 0.249067
\(872\) −4.63373e119 −3.75849
\(873\) 1.22843e118 0.0951205
\(874\) 3.64353e119 2.69345
\(875\) 1.24445e119 0.878320
\(876\) 1.35146e119 0.910723
\(877\) −2.45811e119 −1.58168 −0.790838 0.612026i \(-0.790355\pi\)
−0.790838 + 0.612026i \(0.790355\pi\)
\(878\) −5.92733e118 −0.364193
\(879\) −9.49421e118 −0.557070
\(880\) −1.34456e119 −0.753412
\(881\) −1.20622e119 −0.645509 −0.322754 0.946483i \(-0.604609\pi\)
−0.322754 + 0.946483i \(0.604609\pi\)
\(882\) −5.09642e118 −0.260488
\(883\) 3.61723e119 1.76590 0.882952 0.469463i \(-0.155552\pi\)
0.882952 + 0.469463i \(0.155552\pi\)
\(884\) 1.08492e119 0.505920
\(885\) −2.19840e119 −0.979272
\(886\) −3.24436e119 −1.38058
\(887\) 9.24436e118 0.375812 0.187906 0.982187i \(-0.439830\pi\)
0.187906 + 0.982187i \(0.439830\pi\)
\(888\) 1.48823e119 0.578024
\(889\) −1.03201e119 −0.382967
\(890\) 8.43193e119 2.98973
\(891\) 1.38869e119 0.470498
\(892\) 7.06828e119 2.28843
\(893\) 7.38918e119 2.28619
\(894\) −5.23480e119 −1.54785
\(895\) −3.22921e119 −0.912563
\(896\) −8.35515e119 −2.25673
\(897\) −5.71508e119 −1.47546
\(898\) −9.24715e119 −2.28200
\(899\) 3.40802e118 0.0803959
\(900\) 5.30856e118 0.119717
\(901\) 2.82530e118 0.0609130
\(902\) 2.87525e119 0.592667
\(903\) 1.02326e120 2.01666
\(904\) 2.94348e119 0.554677
\(905\) −1.09047e120 −1.96493
\(906\) −7.46324e119 −1.28598
\(907\) 3.32537e119 0.547954 0.273977 0.961736i \(-0.411661\pi\)
0.273977 + 0.961736i \(0.411661\pi\)
\(908\) −2.21196e120 −3.48578
\(909\) −5.53053e118 −0.0833546
\(910\) 3.00645e120 4.33389
\(911\) −1.17467e120 −1.61966 −0.809829 0.586666i \(-0.800440\pi\)
−0.809829 + 0.586666i \(0.800440\pi\)
\(912\) 1.53108e120 2.01933
\(913\) 3.52922e118 0.0445259
\(914\) 2.10688e119 0.254284
\(915\) 4.07933e119 0.471014
\(916\) 1.79978e120 1.98816
\(917\) 9.66649e119 1.02167
\(918\) −3.10200e119 −0.313699
\(919\) 7.96795e119 0.771025 0.385512 0.922703i \(-0.374025\pi\)
0.385512 + 0.922703i \(0.374025\pi\)
\(920\) −2.98847e120 −2.76721
\(921\) 1.93305e120 1.71289
\(922\) −2.10891e120 −1.78836
\(923\) −6.22182e119 −0.504950
\(924\) −1.93502e120 −1.50304
\(925\) 2.02800e119 0.150775
\(926\) −1.17912e120 −0.839098
\(927\) 5.48737e118 0.0373796
\(928\) −4.54368e119 −0.296288
\(929\) 1.46480e120 0.914413 0.457207 0.889360i \(-0.348850\pi\)
0.457207 + 0.889360i \(0.348850\pi\)
\(930\) 5.12664e119 0.306389
\(931\) −3.42774e120 −1.96131
\(932\) 5.77801e120 3.16544
\(933\) −1.63110e120 −0.855606
\(934\) 2.51553e120 1.26352
\(935\) −2.09674e119 −0.100850
\(936\) −5.62439e119 −0.259062
\(937\) 1.21230e118 0.00534758 0.00267379 0.999996i \(-0.499149\pi\)
0.00267379 + 0.999996i \(0.499149\pi\)
\(938\) 1.28902e120 0.544559
\(939\) 5.33827e119 0.215995
\(940\) −1.14256e121 −4.42795
\(941\) 1.76392e118 0.00654785 0.00327393 0.999995i \(-0.498958\pi\)
0.00327393 + 0.999995i \(0.498958\pi\)
\(942\) 3.89164e120 1.38379
\(943\) 2.54118e120 0.865588
\(944\) 3.22991e120 1.05396
\(945\) −5.84942e120 −1.82862
\(946\) 3.13239e120 0.938176
\(947\) −1.62111e120 −0.465200 −0.232600 0.972573i \(-0.574723\pi\)
−0.232600 + 0.972573i \(0.574723\pi\)
\(948\) 4.04152e120 1.11124
\(949\) −1.95119e120 −0.514064
\(950\) 5.24692e120 1.32464
\(951\) 8.39913e118 0.0203200
\(952\) 2.53075e120 0.586750
\(953\) −4.61855e120 −1.02623 −0.513113 0.858321i \(-0.671508\pi\)
−0.513113 + 0.858321i \(0.671508\pi\)
\(954\) −2.76120e119 −0.0588015
\(955\) 4.96875e120 1.01417
\(956\) 2.87002e120 0.561489
\(957\) 1.46278e120 0.274313
\(958\) −3.59714e120 −0.646628
\(959\) −4.46103e120 −0.768744
\(960\) 4.28144e120 0.707303
\(961\) −6.20290e120 −0.982421
\(962\) −4.05064e120 −0.615083
\(963\) −1.05720e120 −0.153920
\(964\) −1.25884e121 −1.75731
\(965\) 1.78530e121 2.38977
\(966\) −2.51322e121 −3.22594
\(967\) 1.32196e121 1.62722 0.813611 0.581410i \(-0.197499\pi\)
0.813611 + 0.581410i \(0.197499\pi\)
\(968\) 1.37881e121 1.62762
\(969\) 2.38760e120 0.270301
\(970\) −1.87743e121 −2.03849
\(971\) −1.00799e121 −1.04973 −0.524867 0.851185i \(-0.675885\pi\)
−0.524867 + 0.851185i \(0.675885\pi\)
\(972\) 4.36200e120 0.435714
\(973\) 3.06484e121 2.93655
\(974\) −3.70955e121 −3.40944
\(975\) −8.23009e120 −0.725634
\(976\) −5.99339e120 −0.506938
\(977\) 1.80749e120 0.146671 0.0733357 0.997307i \(-0.476636\pi\)
0.0733357 + 0.997307i \(0.476636\pi\)
\(978\) 2.11987e121 1.65038
\(979\) 7.83528e120 0.585269
\(980\) 5.30021e121 3.79872
\(981\) 2.80814e120 0.193118
\(982\) 2.91398e121 1.92296
\(983\) −2.03021e121 −1.28566 −0.642828 0.766011i \(-0.722239\pi\)
−0.642828 + 0.766011i \(0.722239\pi\)
\(984\) 2.68547e121 1.63199
\(985\) −7.68280e120 −0.448077
\(986\) −3.60661e120 −0.201876
\(987\) −5.09688e121 −2.73817
\(988\) −7.13141e121 −3.67722
\(989\) 2.76843e121 1.37020
\(990\) 2.04917e120 0.0973537
\(991\) 1.62263e121 0.740008 0.370004 0.929030i \(-0.379356\pi\)
0.370004 + 0.929030i \(0.379356\pi\)
\(992\) −1.47975e120 −0.0647837
\(993\) −2.91071e121 −1.22336
\(994\) −2.73606e121 −1.10402
\(995\) −4.59810e121 −1.78133
\(996\) 6.21413e120 0.231141
\(997\) 5.12490e121 1.83034 0.915170 0.403068i \(-0.132056\pi\)
0.915170 + 0.403068i \(0.132056\pi\)
\(998\) 3.43788e121 1.17897
\(999\) 7.88101e120 0.259525
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 1.82.a.a.1.1 6
3.2 odd 2 9.82.a.b.1.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.82.a.a.1.1 6 1.1 even 1 trivial
9.82.a.b.1.6 6 3.2 odd 2