Properties

Label 69.8.a.b.1.1
Level $69$
Weight $8$
Character 69.1
Self dual yes
Analytic conductor $21.555$
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] = [69,8,Mod(1,69)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(69, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("69.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 69.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: \(21.5545667584\)
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: \( x^{6} - 2x^{5} - 466x^{4} + 540x^{3} + 48973x^{2} - 77282x - 1061812 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(18.9294\) of defining polynomial
Character \(\chi\) \(=\) 69.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-19.9294 q^{2} +27.0000 q^{3} +269.181 q^{4} +466.102 q^{5} -538.094 q^{6} -1376.23 q^{7} -2813.66 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-19.9294 q^{2} +27.0000 q^{3} +269.181 q^{4} +466.102 q^{5} -538.094 q^{6} -1376.23 q^{7} -2813.66 q^{8} +729.000 q^{9} -9289.15 q^{10} -6066.56 q^{11} +7267.89 q^{12} +9371.38 q^{13} +27427.5 q^{14} +12584.8 q^{15} +21619.3 q^{16} -31738.9 q^{17} -14528.5 q^{18} -5025.16 q^{19} +125466. q^{20} -37158.2 q^{21} +120903. q^{22} -12167.0 q^{23} -75968.8 q^{24} +139126. q^{25} -186766. q^{26} +19683.0 q^{27} -370455. q^{28} -124306. q^{29} -250807. q^{30} -16099.5 q^{31} -70712.4 q^{32} -163797. q^{33} +632538. q^{34} -641464. q^{35} +196233. q^{36} +101300. q^{37} +100148. q^{38} +253027. q^{39} -1.31145e6 q^{40} -598115. q^{41} +740541. q^{42} -118931. q^{43} -1.63301e6 q^{44} +339789. q^{45} +242481. q^{46} -255226. q^{47} +583722. q^{48} +1.07047e6 q^{49} -2.77271e6 q^{50} -856951. q^{51} +2.52260e6 q^{52} -109847. q^{53} -392271. q^{54} -2.82764e6 q^{55} +3.87224e6 q^{56} -135679. q^{57} +2.47734e6 q^{58} +964886. q^{59} +3.38758e6 q^{60} -1.44972e6 q^{61} +320853. q^{62} -1.00327e6 q^{63} -1.35802e6 q^{64} +4.36803e6 q^{65} +3.26438e6 q^{66} -4.27555e6 q^{67} -8.54353e6 q^{68} -328509. q^{69} +1.27840e7 q^{70} -814451. q^{71} -2.05116e6 q^{72} +6.40210e6 q^{73} -2.01884e6 q^{74} +3.75642e6 q^{75} -1.35268e6 q^{76} +8.34899e6 q^{77} -5.04269e6 q^{78} -6.69183e6 q^{79} +1.00768e7 q^{80} +531441. q^{81} +1.19201e7 q^{82} -7.15533e6 q^{83} -1.00023e7 q^{84} -1.47936e7 q^{85} +2.37022e6 q^{86} -3.35625e6 q^{87} +1.70692e7 q^{88} -8.17909e6 q^{89} -6.77179e6 q^{90} -1.28972e7 q^{91} -3.27513e6 q^{92} -434686. q^{93} +5.08650e6 q^{94} -2.34224e6 q^{95} -1.90923e6 q^{96} +1.53893e7 q^{97} -2.13338e7 q^{98} -4.42252e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 8 q^{2} + 162 q^{3} + 178 q^{4} - 372 q^{5} - 216 q^{6} - 1104 q^{7} - 1956 q^{8} + 4374 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 8 q^{2} + 162 q^{3} + 178 q^{4} - 372 q^{5} - 216 q^{6} - 1104 q^{7} - 1956 q^{8} + 4374 q^{9} - 13042 q^{10} - 14824 q^{11} + 4806 q^{12} - 756 q^{13} - 3926 q^{14} - 10044 q^{15} - 13022 q^{16} - 69484 q^{17} - 5832 q^{18} - 43864 q^{19} + 78886 q^{20} - 29808 q^{21} + 98204 q^{22} - 73002 q^{23} - 52812 q^{24} + 228018 q^{25} - 311956 q^{26} + 118098 q^{27} - 545442 q^{28} - 311100 q^{29} - 352134 q^{30} - 245248 q^{31} - 390156 q^{32} - 400248 q^{33} + 235834 q^{34} - 1331256 q^{35} + 129762 q^{36} - 630044 q^{37} + 80910 q^{38} - 20412 q^{39} - 2153982 q^{40} - 969204 q^{41} - 106002 q^{42} - 1770208 q^{43} - 1749140 q^{44} - 271188 q^{45} + 97336 q^{46} - 1400024 q^{47} - 351594 q^{48} + 1985598 q^{49} - 956660 q^{50} - 1876068 q^{51} + 3217272 q^{52} - 1573516 q^{53} - 157464 q^{54} - 431296 q^{55} + 7740702 q^{56} - 1184328 q^{57} + 5987188 q^{58} - 1410320 q^{59} + 2129922 q^{60} - 942172 q^{61} + 3334412 q^{62} - 804816 q^{63} + 1996866 q^{64} - 420944 q^{65} + 2651508 q^{66} - 452072 q^{67} - 9258254 q^{68} - 1971054 q^{69} + 21981136 q^{70} + 122928 q^{71} - 1425924 q^{72} + 16490716 q^{73} - 600104 q^{74} + 6156486 q^{75} + 7428658 q^{76} + 7239696 q^{77} - 8422812 q^{78} + 2458408 q^{79} + 19440230 q^{80} + 3188646 q^{81} + 20510784 q^{82} - 7566456 q^{83} - 14726934 q^{84} + 5817744 q^{85} - 669666 q^{86} - 8399700 q^{87} + 14775668 q^{88} - 20368036 q^{89} - 9507618 q^{90} + 8815576 q^{91} - 2165726 q^{92} - 6621696 q^{93} + 16952576 q^{94} + 5143832 q^{95} - 10534212 q^{96} + 12586972 q^{97} - 39164812 q^{98} - 10806696 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −19.9294 −1.76153 −0.880764 0.473556i \(-0.842970\pi\)
−0.880764 + 0.473556i \(0.842970\pi\)
\(3\) 27.0000 0.577350
\(4\) 269.181 2.10298
\(5\) 466.102 1.66758 0.833789 0.552083i \(-0.186167\pi\)
0.833789 + 0.552083i \(0.186167\pi\)
\(6\) −538.094 −1.01702
\(7\) −1376.23 −1.51652 −0.758260 0.651953i \(-0.773950\pi\)
−0.758260 + 0.651953i \(0.773950\pi\)
\(8\) −2813.66 −1.94293
\(9\) 729.000 0.333333
\(10\) −9289.15 −2.93749
\(11\) −6066.56 −1.37426 −0.687129 0.726535i \(-0.741129\pi\)
−0.687129 + 0.726535i \(0.741129\pi\)
\(12\) 7267.89 1.21416
\(13\) 9371.38 1.18305 0.591524 0.806288i \(-0.298527\pi\)
0.591524 + 0.806288i \(0.298527\pi\)
\(14\) 27427.5 2.67139
\(15\) 12584.8 0.962777
\(16\) 21619.3 1.31954
\(17\) −31738.9 −1.56683 −0.783414 0.621501i \(-0.786523\pi\)
−0.783414 + 0.621501i \(0.786523\pi\)
\(18\) −14528.5 −0.587176
\(19\) −5025.16 −0.168079 −0.0840393 0.996462i \(-0.526782\pi\)
−0.0840393 + 0.996462i \(0.526782\pi\)
\(20\) 125466. 3.50688
\(21\) −37158.2 −0.875563
\(22\) 120903. 2.42079
\(23\) −12167.0 −0.208514
\(24\) −75968.8 −1.12175
\(25\) 139126. 1.78082
\(26\) −186766. −2.08397
\(27\) 19683.0 0.192450
\(28\) −370455. −3.18921
\(29\) −124306. −0.946449 −0.473225 0.880942i \(-0.656910\pi\)
−0.473225 + 0.880942i \(0.656910\pi\)
\(30\) −250807. −1.69596
\(31\) −16099.5 −0.0970614 −0.0485307 0.998822i \(-0.515454\pi\)
−0.0485307 + 0.998822i \(0.515454\pi\)
\(32\) −70712.4 −0.381479
\(33\) −163797. −0.793428
\(34\) 632538. 2.76001
\(35\) −641464. −2.52892
\(36\) 196233. 0.700993
\(37\) 101300. 0.328778 0.164389 0.986396i \(-0.447435\pi\)
0.164389 + 0.986396i \(0.447435\pi\)
\(38\) 100148. 0.296075
\(39\) 253027. 0.683033
\(40\) −1.31145e6 −3.23998
\(41\) −598115. −1.35532 −0.677659 0.735377i \(-0.737005\pi\)
−0.677659 + 0.735377i \(0.737005\pi\)
\(42\) 740541. 1.54233
\(43\) −118931. −0.228115 −0.114058 0.993474i \(-0.536385\pi\)
−0.114058 + 0.993474i \(0.536385\pi\)
\(44\) −1.63301e6 −2.89004
\(45\) 339789. 0.555860
\(46\) 242481. 0.367304
\(47\) −255226. −0.358577 −0.179288 0.983797i \(-0.557380\pi\)
−0.179288 + 0.983797i \(0.557380\pi\)
\(48\) 583722. 0.761837
\(49\) 1.07047e6 1.29983
\(50\) −2.77271e6 −3.13696
\(51\) −856951. −0.904608
\(52\) 2.52260e6 2.48792
\(53\) −109847. −0.101349 −0.0506747 0.998715i \(-0.516137\pi\)
−0.0506747 + 0.998715i \(0.516137\pi\)
\(54\) −392271. −0.339006
\(55\) −2.82764e6 −2.29168
\(56\) 3.87224e6 2.94649
\(57\) −135679. −0.0970402
\(58\) 2.47734e6 1.66720
\(59\) 964886. 0.611637 0.305819 0.952090i \(-0.401070\pi\)
0.305819 + 0.952090i \(0.401070\pi\)
\(60\) 3.38758e6 2.02470
\(61\) −1.44972e6 −0.817767 −0.408884 0.912587i \(-0.634082\pi\)
−0.408884 + 0.912587i \(0.634082\pi\)
\(62\) 320853. 0.170976
\(63\) −1.00327e6 −0.505506
\(64\) −1.35802e6 −0.647554
\(65\) 4.36803e6 1.97282
\(66\) 3.26438e6 1.39765
\(67\) −4.27555e6 −1.73672 −0.868361 0.495932i \(-0.834827\pi\)
−0.868361 + 0.495932i \(0.834827\pi\)
\(68\) −8.54353e6 −3.29500
\(69\) −328509. −0.120386
\(70\) 1.27840e7 4.45475
\(71\) −814451. −0.270060 −0.135030 0.990841i \(-0.543113\pi\)
−0.135030 + 0.990841i \(0.543113\pi\)
\(72\) −2.05116e6 −0.647642
\(73\) 6.40210e6 1.92616 0.963080 0.269214i \(-0.0867639\pi\)
0.963080 + 0.269214i \(0.0867639\pi\)
\(74\) −2.01884e6 −0.579151
\(75\) 3.75642e6 1.02816
\(76\) −1.35268e6 −0.353466
\(77\) 8.34899e6 2.08409
\(78\) −5.04269e6 −1.20318
\(79\) −6.69183e6 −1.52704 −0.763519 0.645785i \(-0.776530\pi\)
−0.763519 + 0.645785i \(0.776530\pi\)
\(80\) 1.00768e7 2.20044
\(81\) 531441. 0.111111
\(82\) 1.19201e7 2.38743
\(83\) −7.15533e6 −1.37359 −0.686793 0.726853i \(-0.740982\pi\)
−0.686793 + 0.726853i \(0.740982\pi\)
\(84\) −1.00023e7 −1.84129
\(85\) −1.47936e7 −2.61281
\(86\) 2.37022e6 0.401831
\(87\) −3.35625e6 −0.546433
\(88\) 1.70692e7 2.67008
\(89\) −8.17909e6 −1.22982 −0.614908 0.788599i \(-0.710807\pi\)
−0.614908 + 0.788599i \(0.710807\pi\)
\(90\) −6.77179e6 −0.979162
\(91\) −1.28972e7 −1.79411
\(92\) −3.27513e6 −0.438501
\(93\) −434686. −0.0560384
\(94\) 5.08650e6 0.631643
\(95\) −2.34224e6 −0.280284
\(96\) −1.90923e6 −0.220247
\(97\) 1.53893e7 1.71205 0.856027 0.516931i \(-0.172926\pi\)
0.856027 + 0.516931i \(0.172926\pi\)
\(98\) −2.13338e7 −2.28969
\(99\) −4.42252e6 −0.458086
\(100\) 3.74502e7 3.74502
\(101\) −7.54010e6 −0.728203 −0.364101 0.931359i \(-0.618624\pi\)
−0.364101 + 0.931359i \(0.618624\pi\)
\(102\) 1.70785e7 1.59349
\(103\) −5.04980e6 −0.455348 −0.227674 0.973737i \(-0.573112\pi\)
−0.227674 + 0.973737i \(0.573112\pi\)
\(104\) −2.63679e7 −2.29857
\(105\) −1.73195e7 −1.46007
\(106\) 2.18918e6 0.178530
\(107\) 9.37807e6 0.740066 0.370033 0.929019i \(-0.379346\pi\)
0.370033 + 0.929019i \(0.379346\pi\)
\(108\) 5.29829e6 0.404718
\(109\) −2.92944e6 −0.216667 −0.108333 0.994115i \(-0.534551\pi\)
−0.108333 + 0.994115i \(0.534551\pi\)
\(110\) 5.63532e7 4.03686
\(111\) 2.73509e6 0.189820
\(112\) −2.97532e7 −2.00111
\(113\) −2.03016e7 −1.32360 −0.661800 0.749681i \(-0.730207\pi\)
−0.661800 + 0.749681i \(0.730207\pi\)
\(114\) 2.70401e6 0.170939
\(115\) −5.67107e6 −0.347714
\(116\) −3.34607e7 −1.99036
\(117\) 6.83174e6 0.394349
\(118\) −1.92296e7 −1.07742
\(119\) 4.36801e7 2.37612
\(120\) −3.54092e7 −1.87061
\(121\) 1.73160e7 0.888586
\(122\) 2.88921e7 1.44052
\(123\) −1.61491e7 −0.782493
\(124\) −4.33368e6 −0.204118
\(125\) 2.84329e7 1.30208
\(126\) 1.99946e7 0.890463
\(127\) −1.35596e7 −0.587400 −0.293700 0.955898i \(-0.594887\pi\)
−0.293700 + 0.955898i \(0.594887\pi\)
\(128\) 3.61157e7 1.52216
\(129\) −3.21113e6 −0.131702
\(130\) −8.70521e7 −3.47518
\(131\) 411901. 0.0160082 0.00800411 0.999968i \(-0.497452\pi\)
0.00800411 + 0.999968i \(0.497452\pi\)
\(132\) −4.40911e7 −1.66856
\(133\) 6.91578e6 0.254894
\(134\) 8.52092e7 3.05928
\(135\) 9.17429e6 0.320926
\(136\) 8.93025e7 3.04423
\(137\) 5.98529e7 1.98867 0.994337 0.106274i \(-0.0338921\pi\)
0.994337 + 0.106274i \(0.0338921\pi\)
\(138\) 6.54699e6 0.212063
\(139\) 5.65384e7 1.78563 0.892815 0.450423i \(-0.148727\pi\)
0.892815 + 0.450423i \(0.148727\pi\)
\(140\) −1.72670e8 −5.31825
\(141\) −6.89110e6 −0.207024
\(142\) 1.62315e7 0.475719
\(143\) −5.68521e7 −1.62581
\(144\) 1.57605e7 0.439847
\(145\) −5.79391e7 −1.57828
\(146\) −1.27590e8 −3.39298
\(147\) 2.89026e7 0.750458
\(148\) 2.72680e7 0.691412
\(149\) −3.55040e7 −0.879275 −0.439638 0.898175i \(-0.644893\pi\)
−0.439638 + 0.898175i \(0.644893\pi\)
\(150\) −7.48631e7 −1.81113
\(151\) 1.69536e7 0.400721 0.200361 0.979722i \(-0.435789\pi\)
0.200361 + 0.979722i \(0.435789\pi\)
\(152\) 1.41391e7 0.326564
\(153\) −2.31377e7 −0.522276
\(154\) −1.66390e8 −3.67118
\(155\) −7.50401e6 −0.161857
\(156\) 6.81102e7 1.43640
\(157\) 5.44237e7 1.12238 0.561189 0.827688i \(-0.310344\pi\)
0.561189 + 0.827688i \(0.310344\pi\)
\(158\) 1.33364e8 2.68992
\(159\) −2.96586e6 −0.0585142
\(160\) −3.29592e7 −0.636146
\(161\) 1.67446e7 0.316216
\(162\) −1.05913e7 −0.195725
\(163\) −1.51739e7 −0.274436 −0.137218 0.990541i \(-0.543816\pi\)
−0.137218 + 0.990541i \(0.543816\pi\)
\(164\) −1.61001e8 −2.85020
\(165\) −7.63463e7 −1.32310
\(166\) 1.42601e8 2.41961
\(167\) −2.89893e7 −0.481649 −0.240824 0.970569i \(-0.577418\pi\)
−0.240824 + 0.970569i \(0.577418\pi\)
\(168\) 1.04551e8 1.70115
\(169\) 2.50743e7 0.399600
\(170\) 2.94828e8 4.60253
\(171\) −3.66334e6 −0.0560262
\(172\) −3.20139e7 −0.479721
\(173\) 1.52831e7 0.224413 0.112207 0.993685i \(-0.464208\pi\)
0.112207 + 0.993685i \(0.464208\pi\)
\(174\) 6.68880e7 0.962556
\(175\) −1.91470e8 −2.70065
\(176\) −1.31155e8 −1.81339
\(177\) 2.60519e7 0.353129
\(178\) 1.63004e8 2.16635
\(179\) −1.01079e8 −1.31727 −0.658635 0.752463i \(-0.728866\pi\)
−0.658635 + 0.752463i \(0.728866\pi\)
\(180\) 9.14647e7 1.16896
\(181\) 4.91643e6 0.0616276 0.0308138 0.999525i \(-0.490190\pi\)
0.0308138 + 0.999525i \(0.490190\pi\)
\(182\) 2.57033e8 3.16038
\(183\) −3.91424e7 −0.472138
\(184\) 3.42338e7 0.405128
\(185\) 4.72160e7 0.548263
\(186\) 8.66304e6 0.0987132
\(187\) 1.92546e8 2.15323
\(188\) −6.87020e7 −0.754080
\(189\) −2.70883e7 −0.291854
\(190\) 4.66794e7 0.493728
\(191\) −6.54949e7 −0.680128 −0.340064 0.940402i \(-0.610449\pi\)
−0.340064 + 0.940402i \(0.610449\pi\)
\(192\) −3.66665e7 −0.373866
\(193\) 2.61850e7 0.262182 0.131091 0.991370i \(-0.458152\pi\)
0.131091 + 0.991370i \(0.458152\pi\)
\(194\) −3.06699e8 −3.01583
\(195\) 1.17937e8 1.13901
\(196\) 2.88150e8 2.73352
\(197\) 4.80412e7 0.447695 0.223848 0.974624i \(-0.428138\pi\)
0.223848 + 0.974624i \(0.428138\pi\)
\(198\) 8.81383e7 0.806931
\(199\) 1.05462e7 0.0948664 0.0474332 0.998874i \(-0.484896\pi\)
0.0474332 + 0.998874i \(0.484896\pi\)
\(200\) −3.91454e8 −3.46000
\(201\) −1.15440e8 −1.00270
\(202\) 1.50270e8 1.28275
\(203\) 1.71073e8 1.43531
\(204\) −2.30675e8 −1.90237
\(205\) −2.78783e8 −2.26010
\(206\) 1.00639e8 0.802108
\(207\) −8.86974e6 −0.0695048
\(208\) 2.02603e8 1.56108
\(209\) 3.04854e7 0.230983
\(210\) 3.45168e8 2.57195
\(211\) 9.30256e7 0.681733 0.340866 0.940112i \(-0.389280\pi\)
0.340866 + 0.940112i \(0.389280\pi\)
\(212\) −2.95687e7 −0.213136
\(213\) −2.19902e7 −0.155919
\(214\) −1.86899e8 −1.30365
\(215\) −5.54339e7 −0.380400
\(216\) −5.53812e7 −0.373916
\(217\) 2.21566e7 0.147195
\(218\) 5.83821e7 0.381665
\(219\) 1.72857e8 1.11207
\(220\) −7.61148e8 −4.81936
\(221\) −2.97438e8 −1.85363
\(222\) −5.45088e7 −0.334373
\(223\) −1.33339e8 −0.805176 −0.402588 0.915381i \(-0.631889\pi\)
−0.402588 + 0.915381i \(0.631889\pi\)
\(224\) 9.73165e7 0.578520
\(225\) 1.01423e8 0.593606
\(226\) 4.04600e8 2.33156
\(227\) 2.24477e8 1.27374 0.636872 0.770970i \(-0.280228\pi\)
0.636872 + 0.770970i \(0.280228\pi\)
\(228\) −3.65223e7 −0.204073
\(229\) −2.59953e8 −1.43045 −0.715223 0.698897i \(-0.753675\pi\)
−0.715223 + 0.698897i \(0.753675\pi\)
\(230\) 1.13021e8 0.612508
\(231\) 2.25423e8 1.20325
\(232\) 3.49753e8 1.83888
\(233\) −5.23601e7 −0.271178 −0.135589 0.990765i \(-0.543293\pi\)
−0.135589 + 0.990765i \(0.543293\pi\)
\(234\) −1.36153e8 −0.694657
\(235\) −1.18961e8 −0.597955
\(236\) 2.59729e8 1.28626
\(237\) −1.80679e8 −0.881636
\(238\) −8.70518e8 −4.18561
\(239\) 3.85956e8 1.82871 0.914357 0.404910i \(-0.132697\pi\)
0.914357 + 0.404910i \(0.132697\pi\)
\(240\) 2.72074e8 1.27042
\(241\) 4.72252e7 0.217327 0.108664 0.994079i \(-0.465343\pi\)
0.108664 + 0.994079i \(0.465343\pi\)
\(242\) −3.45098e8 −1.56527
\(243\) 1.43489e7 0.0641500
\(244\) −3.90237e8 −1.71975
\(245\) 4.98947e8 2.16757
\(246\) 3.21842e8 1.37838
\(247\) −4.70927e7 −0.198845
\(248\) 4.52985e7 0.188583
\(249\) −1.93194e8 −0.793041
\(250\) −5.66652e8 −2.29364
\(251\) 4.32121e7 0.172484 0.0862418 0.996274i \(-0.472514\pi\)
0.0862418 + 0.996274i \(0.472514\pi\)
\(252\) −2.70062e8 −1.06307
\(253\) 7.38119e7 0.286553
\(254\) 2.70235e8 1.03472
\(255\) −3.99427e8 −1.50851
\(256\) −5.45938e8 −2.03378
\(257\) −3.37491e8 −1.24021 −0.620106 0.784518i \(-0.712911\pi\)
−0.620106 + 0.784518i \(0.712911\pi\)
\(258\) 6.39959e7 0.231997
\(259\) −1.39412e8 −0.498598
\(260\) 1.17579e9 4.14881
\(261\) −9.06187e7 −0.315483
\(262\) −8.20894e6 −0.0281989
\(263\) −1.49351e8 −0.506249 −0.253125 0.967434i \(-0.581458\pi\)
−0.253125 + 0.967434i \(0.581458\pi\)
\(264\) 4.60869e8 1.54157
\(265\) −5.11998e7 −0.169008
\(266\) −1.37827e8 −0.449003
\(267\) −2.20836e8 −0.710035
\(268\) −1.15090e9 −3.65229
\(269\) 3.20872e8 1.00508 0.502538 0.864555i \(-0.332400\pi\)
0.502538 + 0.864555i \(0.332400\pi\)
\(270\) −1.82838e8 −0.565319
\(271\) 2.45542e7 0.0749433 0.0374716 0.999298i \(-0.488070\pi\)
0.0374716 + 0.999298i \(0.488070\pi\)
\(272\) −6.86175e8 −2.06749
\(273\) −3.48224e8 −1.03583
\(274\) −1.19283e9 −3.50310
\(275\) −8.44020e8 −2.44731
\(276\) −8.84285e7 −0.253169
\(277\) 2.99640e8 0.847072 0.423536 0.905879i \(-0.360789\pi\)
0.423536 + 0.905879i \(0.360789\pi\)
\(278\) −1.12678e9 −3.14544
\(279\) −1.17365e7 −0.0323538
\(280\) 1.80486e9 4.91350
\(281\) 2.26970e8 0.610235 0.305117 0.952315i \(-0.401304\pi\)
0.305117 + 0.952315i \(0.401304\pi\)
\(282\) 1.37336e8 0.364679
\(283\) 7.55768e7 0.198215 0.0991073 0.995077i \(-0.468401\pi\)
0.0991073 + 0.995077i \(0.468401\pi\)
\(284\) −2.19235e8 −0.567931
\(285\) −6.32405e7 −0.161822
\(286\) 1.13303e9 2.86391
\(287\) 8.23144e8 2.05536
\(288\) −5.15493e7 −0.127160
\(289\) 5.97021e8 1.45495
\(290\) 1.15469e9 2.78018
\(291\) 4.15511e8 0.988455
\(292\) 1.72333e9 4.05067
\(293\) 7.83295e8 1.81923 0.909617 0.415448i \(-0.136375\pi\)
0.909617 + 0.415448i \(0.136375\pi\)
\(294\) −5.76012e8 −1.32195
\(295\) 4.49736e8 1.01995
\(296\) −2.85023e8 −0.638791
\(297\) −1.19408e8 −0.264476
\(298\) 7.07573e8 1.54887
\(299\) −1.14022e8 −0.246682
\(300\) 1.01116e9 2.16219
\(301\) 1.63676e8 0.345941
\(302\) −3.37875e8 −0.705881
\(303\) −2.03583e8 −0.420428
\(304\) −1.08641e8 −0.221786
\(305\) −6.75718e8 −1.36369
\(306\) 4.61120e8 0.920003
\(307\) 4.64667e7 0.0916554 0.0458277 0.998949i \(-0.485407\pi\)
0.0458277 + 0.998949i \(0.485407\pi\)
\(308\) 2.24739e9 4.38279
\(309\) −1.36344e8 −0.262895
\(310\) 1.49551e8 0.285116
\(311\) 9.79489e7 0.184645 0.0923227 0.995729i \(-0.470571\pi\)
0.0923227 + 0.995729i \(0.470571\pi\)
\(312\) −7.11933e8 −1.32708
\(313\) −7.03146e8 −1.29610 −0.648052 0.761596i \(-0.724416\pi\)
−0.648052 + 0.761596i \(0.724416\pi\)
\(314\) −1.08463e9 −1.97710
\(315\) −4.67627e8 −0.842972
\(316\) −1.80131e9 −3.21133
\(317\) 1.51468e8 0.267064 0.133532 0.991045i \(-0.457368\pi\)
0.133532 + 0.991045i \(0.457368\pi\)
\(318\) 5.91079e7 0.103074
\(319\) 7.54107e8 1.30067
\(320\) −6.32976e8 −1.07985
\(321\) 2.53208e8 0.427277
\(322\) −3.33710e8 −0.557023
\(323\) 1.59493e8 0.263350
\(324\) 1.43054e8 0.233664
\(325\) 1.30381e9 2.10679
\(326\) 3.02408e8 0.483427
\(327\) −7.90950e7 −0.125093
\(328\) 1.68289e9 2.63328
\(329\) 3.51250e8 0.543789
\(330\) 1.52154e9 2.33068
\(331\) −3.67566e8 −0.557105 −0.278552 0.960421i \(-0.589855\pi\)
−0.278552 + 0.960421i \(0.589855\pi\)
\(332\) −1.92608e9 −2.88862
\(333\) 7.38475e7 0.109593
\(334\) 5.77740e8 0.848438
\(335\) −1.99285e9 −2.89612
\(336\) −8.03336e8 −1.15534
\(337\) −3.50623e8 −0.499041 −0.249521 0.968370i \(-0.580273\pi\)
−0.249521 + 0.968370i \(0.580273\pi\)
\(338\) −4.99716e8 −0.703907
\(339\) −5.48144e8 −0.764180
\(340\) −3.98216e9 −5.49468
\(341\) 9.76686e7 0.133387
\(342\) 7.30082e7 0.0986916
\(343\) −3.39824e8 −0.454699
\(344\) 3.34630e8 0.443211
\(345\) −1.53119e8 −0.200753
\(346\) −3.04582e8 −0.395310
\(347\) 8.21515e7 0.105551 0.0527755 0.998606i \(-0.483193\pi\)
0.0527755 + 0.998606i \(0.483193\pi\)
\(348\) −9.03439e8 −1.14914
\(349\) 5.47670e8 0.689652 0.344826 0.938667i \(-0.387938\pi\)
0.344826 + 0.938667i \(0.387938\pi\)
\(350\) 3.81589e9 4.75726
\(351\) 1.84457e8 0.227678
\(352\) 4.28981e8 0.524251
\(353\) 7.05195e8 0.853292 0.426646 0.904419i \(-0.359695\pi\)
0.426646 + 0.904419i \(0.359695\pi\)
\(354\) −5.19199e8 −0.622046
\(355\) −3.79618e8 −0.450347
\(356\) −2.20166e9 −2.58628
\(357\) 1.17936e9 1.37186
\(358\) 2.01444e9 2.32041
\(359\) −6.05279e8 −0.690438 −0.345219 0.938522i \(-0.612195\pi\)
−0.345219 + 0.938522i \(0.612195\pi\)
\(360\) −9.56049e8 −1.07999
\(361\) −8.68620e8 −0.971750
\(362\) −9.79816e7 −0.108559
\(363\) 4.67533e8 0.513025
\(364\) −3.47168e9 −3.77298
\(365\) 2.98404e9 3.21202
\(366\) 7.80086e8 0.831684
\(367\) −4.00347e8 −0.422771 −0.211386 0.977403i \(-0.567798\pi\)
−0.211386 + 0.977403i \(0.567798\pi\)
\(368\) −2.63043e8 −0.275143
\(369\) −4.36026e8 −0.451772
\(370\) −9.40988e8 −0.965779
\(371\) 1.51174e8 0.153698
\(372\) −1.17009e8 −0.117848
\(373\) 7.43740e8 0.742063 0.371031 0.928620i \(-0.379004\pi\)
0.371031 + 0.928620i \(0.379004\pi\)
\(374\) −3.83733e9 −3.79297
\(375\) 7.67689e8 0.751755
\(376\) 7.18119e8 0.696689
\(377\) −1.16491e9 −1.11969
\(378\) 5.39855e8 0.514109
\(379\) 1.08034e9 1.01935 0.509676 0.860366i \(-0.329765\pi\)
0.509676 + 0.860366i \(0.329765\pi\)
\(380\) −6.30487e8 −0.589432
\(381\) −3.66109e8 −0.339135
\(382\) 1.30528e9 1.19806
\(383\) 1.83043e9 1.66478 0.832389 0.554192i \(-0.186973\pi\)
0.832389 + 0.554192i \(0.186973\pi\)
\(384\) 9.75124e8 0.878822
\(385\) 3.89148e9 3.47538
\(386\) −5.21852e8 −0.461840
\(387\) −8.67005e7 −0.0760384
\(388\) 4.14251e9 3.60041
\(389\) −9.20484e8 −0.792853 −0.396426 0.918066i \(-0.629750\pi\)
−0.396426 + 0.918066i \(0.629750\pi\)
\(390\) −2.35041e9 −2.00640
\(391\) 3.86168e8 0.326706
\(392\) −3.01193e9 −2.52548
\(393\) 1.11213e7 0.00924235
\(394\) −9.57433e8 −0.788627
\(395\) −3.11908e9 −2.54646
\(396\) −1.19046e9 −0.963345
\(397\) 1.68258e9 1.34961 0.674807 0.737995i \(-0.264227\pi\)
0.674807 + 0.737995i \(0.264227\pi\)
\(398\) −2.10180e8 −0.167110
\(399\) 1.86726e8 0.147163
\(400\) 3.00782e9 2.34986
\(401\) 1.78289e9 1.38076 0.690381 0.723446i \(-0.257443\pi\)
0.690381 + 0.723446i \(0.257443\pi\)
\(402\) 2.30065e9 1.76628
\(403\) −1.50875e8 −0.114828
\(404\) −2.02965e9 −1.53139
\(405\) 2.47706e8 0.185287
\(406\) −3.40938e9 −2.52834
\(407\) −6.14541e8 −0.451825
\(408\) 2.41117e9 1.75759
\(409\) −1.62476e9 −1.17425 −0.587123 0.809498i \(-0.699739\pi\)
−0.587123 + 0.809498i \(0.699739\pi\)
\(410\) 5.55597e9 3.98123
\(411\) 1.61603e9 1.14816
\(412\) −1.35931e9 −0.957587
\(413\) −1.32790e9 −0.927559
\(414\) 1.76769e8 0.122435
\(415\) −3.33512e9 −2.29056
\(416\) −6.62673e8 −0.451308
\(417\) 1.52654e9 1.03093
\(418\) −6.07557e8 −0.406883
\(419\) 2.24799e9 1.49295 0.746476 0.665412i \(-0.231744\pi\)
0.746476 + 0.665412i \(0.231744\pi\)
\(420\) −4.66209e9 −3.07050
\(421\) 3.33475e8 0.217809 0.108904 0.994052i \(-0.465266\pi\)
0.108904 + 0.994052i \(0.465266\pi\)
\(422\) −1.85395e9 −1.20089
\(423\) −1.86060e8 −0.119526
\(424\) 3.09071e8 0.196915
\(425\) −4.41573e9 −2.79024
\(426\) 4.38251e8 0.274656
\(427\) 1.99515e9 1.24016
\(428\) 2.52440e9 1.55634
\(429\) −1.53501e9 −0.938663
\(430\) 1.10476e9 0.670085
\(431\) −2.33004e9 −1.40182 −0.700910 0.713249i \(-0.747223\pi\)
−0.700910 + 0.713249i \(0.747223\pi\)
\(432\) 4.25534e8 0.253946
\(433\) 1.82844e8 0.108237 0.0541183 0.998535i \(-0.482765\pi\)
0.0541183 + 0.998535i \(0.482765\pi\)
\(434\) −4.41568e8 −0.259289
\(435\) −1.56436e9 −0.911220
\(436\) −7.88551e8 −0.455646
\(437\) 6.11411e7 0.0350468
\(438\) −3.44493e9 −1.95894
\(439\) −1.41207e9 −0.796580 −0.398290 0.917260i \(-0.630396\pi\)
−0.398290 + 0.917260i \(0.630396\pi\)
\(440\) 7.95601e9 4.45257
\(441\) 7.80370e8 0.433277
\(442\) 5.92776e9 3.26522
\(443\) 6.88910e8 0.376487 0.188243 0.982122i \(-0.439721\pi\)
0.188243 + 0.982122i \(0.439721\pi\)
\(444\) 7.36236e8 0.399187
\(445\) −3.81230e9 −2.05082
\(446\) 2.65737e9 1.41834
\(447\) −9.58607e8 −0.507650
\(448\) 1.86895e9 0.982029
\(449\) −6.50678e8 −0.339238 −0.169619 0.985510i \(-0.554254\pi\)
−0.169619 + 0.985510i \(0.554254\pi\)
\(450\) −2.02130e9 −1.04565
\(451\) 3.62850e9 1.86256
\(452\) −5.46482e9 −2.78350
\(453\) 4.57747e8 0.231356
\(454\) −4.47370e9 −2.24373
\(455\) −6.01141e9 −2.99183
\(456\) 3.81755e8 0.188542
\(457\) −1.61945e9 −0.793708 −0.396854 0.917882i \(-0.629898\pi\)
−0.396854 + 0.917882i \(0.629898\pi\)
\(458\) 5.18072e9 2.51977
\(459\) −6.24717e8 −0.301536
\(460\) −1.52655e9 −0.731236
\(461\) −6.21807e8 −0.295599 −0.147799 0.989017i \(-0.547219\pi\)
−0.147799 + 0.989017i \(0.547219\pi\)
\(462\) −4.49254e9 −2.11956
\(463\) 2.85900e9 1.33869 0.669347 0.742950i \(-0.266574\pi\)
0.669347 + 0.742950i \(0.266574\pi\)
\(464\) −2.68740e9 −1.24888
\(465\) −2.02608e8 −0.0934484
\(466\) 1.04351e9 0.477688
\(467\) 1.28603e9 0.584309 0.292154 0.956371i \(-0.405628\pi\)
0.292154 + 0.956371i \(0.405628\pi\)
\(468\) 1.83898e9 0.829307
\(469\) 5.88415e9 2.63377
\(470\) 2.37083e9 1.05331
\(471\) 1.46944e9 0.648005
\(472\) −2.71486e9 −1.18837
\(473\) 7.21500e8 0.313489
\(474\) 3.60083e9 1.55303
\(475\) −6.99133e8 −0.299317
\(476\) 1.17579e10 4.99694
\(477\) −8.00783e7 −0.0337832
\(478\) −7.69188e9 −3.22133
\(479\) −3.90154e9 −1.62204 −0.811021 0.585017i \(-0.801088\pi\)
−0.811021 + 0.585017i \(0.801088\pi\)
\(480\) −8.89899e8 −0.367279
\(481\) 9.49319e8 0.388959
\(482\) −9.41170e8 −0.382828
\(483\) 4.52104e8 0.182567
\(484\) 4.66115e9 1.86868
\(485\) 7.17299e9 2.85499
\(486\) −2.85965e8 −0.113002
\(487\) −2.21345e9 −0.868397 −0.434199 0.900817i \(-0.642968\pi\)
−0.434199 + 0.900817i \(0.642968\pi\)
\(488\) 4.07902e9 1.58886
\(489\) −4.09696e8 −0.158446
\(490\) −9.94372e9 −3.81823
\(491\) −4.33060e9 −1.65106 −0.825530 0.564358i \(-0.809124\pi\)
−0.825530 + 0.564358i \(0.809124\pi\)
\(492\) −4.34703e9 −1.64557
\(493\) 3.94532e9 1.48292
\(494\) 9.38530e8 0.350271
\(495\) −2.06135e9 −0.763895
\(496\) −3.48061e8 −0.128076
\(497\) 1.12087e9 0.409552
\(498\) 3.85024e9 1.39696
\(499\) −1.11601e9 −0.402085 −0.201042 0.979583i \(-0.564433\pi\)
−0.201042 + 0.979583i \(0.564433\pi\)
\(500\) 7.65362e9 2.73824
\(501\) −7.82712e8 −0.278080
\(502\) −8.61192e8 −0.303834
\(503\) −5.61394e9 −1.96689 −0.983444 0.181212i \(-0.941998\pi\)
−0.983444 + 0.181212i \(0.941998\pi\)
\(504\) 2.82286e9 0.982162
\(505\) −3.51446e9 −1.21434
\(506\) −1.47103e9 −0.504770
\(507\) 6.77007e8 0.230709
\(508\) −3.64999e9 −1.23529
\(509\) 9.54298e8 0.320754 0.160377 0.987056i \(-0.448729\pi\)
0.160377 + 0.987056i \(0.448729\pi\)
\(510\) 7.96034e9 2.65727
\(511\) −8.81077e9 −2.92106
\(512\) 6.25741e9 2.06039
\(513\) −9.89102e7 −0.0323467
\(514\) 6.72600e9 2.18467
\(515\) −2.35372e9 −0.759329
\(516\) −8.64375e8 −0.276967
\(517\) 1.54834e9 0.492777
\(518\) 2.77839e9 0.878293
\(519\) 4.12642e8 0.129565
\(520\) −1.22901e10 −3.83305
\(521\) −3.27349e9 −1.01409 −0.507047 0.861918i \(-0.669263\pi\)
−0.507047 + 0.861918i \(0.669263\pi\)
\(522\) 1.80598e9 0.555732
\(523\) 3.35615e9 1.02585 0.512927 0.858432i \(-0.328561\pi\)
0.512927 + 0.858432i \(0.328561\pi\)
\(524\) 1.10876e8 0.0336649
\(525\) −5.16969e9 −1.55922
\(526\) 2.97649e9 0.891772
\(527\) 5.10981e8 0.152078
\(528\) −3.54119e9 −1.04696
\(529\) 1.48036e8 0.0434783
\(530\) 1.02038e9 0.297713
\(531\) 7.03402e8 0.203879
\(532\) 1.86160e9 0.536037
\(533\) −5.60516e9 −1.60340
\(534\) 4.40112e9 1.25075
\(535\) 4.37114e9 1.23412
\(536\) 1.20299e10 3.37433
\(537\) −2.72913e9 −0.760526
\(538\) −6.39479e9 −1.77047
\(539\) −6.49405e9 −1.78630
\(540\) 2.46955e9 0.674900
\(541\) −2.62460e9 −0.712643 −0.356321 0.934363i \(-0.615969\pi\)
−0.356321 + 0.934363i \(0.615969\pi\)
\(542\) −4.89350e8 −0.132015
\(543\) 1.32744e8 0.0355807
\(544\) 2.24434e9 0.597712
\(545\) −1.36542e9 −0.361309
\(546\) 6.93990e9 1.82465
\(547\) 1.01462e9 0.265063 0.132532 0.991179i \(-0.457689\pi\)
0.132532 + 0.991179i \(0.457689\pi\)
\(548\) 1.61113e10 4.18214
\(549\) −1.05685e9 −0.272589
\(550\) 1.68208e10 4.31100
\(551\) 6.24655e8 0.159078
\(552\) 9.24312e8 0.233901
\(553\) 9.20949e9 2.31578
\(554\) −5.97164e9 −1.49214
\(555\) 1.27483e9 0.316540
\(556\) 1.52191e10 3.75514
\(557\) 2.67923e9 0.656926 0.328463 0.944517i \(-0.393469\pi\)
0.328463 + 0.944517i \(0.393469\pi\)
\(558\) 2.33902e8 0.0569921
\(559\) −1.11454e9 −0.269871
\(560\) −1.38680e10 −3.33700
\(561\) 5.19875e9 1.24317
\(562\) −4.52338e9 −1.07494
\(563\) 3.37265e9 0.796512 0.398256 0.917274i \(-0.369616\pi\)
0.398256 + 0.917274i \(0.369616\pi\)
\(564\) −1.85496e9 −0.435368
\(565\) −9.46264e9 −2.20721
\(566\) −1.50620e9 −0.349161
\(567\) −7.31385e8 −0.168502
\(568\) 2.29159e9 0.524707
\(569\) 5.59018e9 1.27213 0.636067 0.771634i \(-0.280560\pi\)
0.636067 + 0.771634i \(0.280560\pi\)
\(570\) 1.26034e9 0.285054
\(571\) −1.53052e9 −0.344043 −0.172022 0.985093i \(-0.555030\pi\)
−0.172022 + 0.985093i \(0.555030\pi\)
\(572\) −1.53035e10 −3.41905
\(573\) −1.76836e9 −0.392672
\(574\) −1.64048e10 −3.62058
\(575\) −1.69275e9 −0.371326
\(576\) −9.89996e8 −0.215851
\(577\) −2.33744e9 −0.506554 −0.253277 0.967394i \(-0.581508\pi\)
−0.253277 + 0.967394i \(0.581508\pi\)
\(578\) −1.18983e10 −2.56293
\(579\) 7.06995e8 0.151371
\(580\) −1.55961e10 −3.31909
\(581\) 9.84738e9 2.08307
\(582\) −8.28088e9 −1.74119
\(583\) 6.66392e8 0.139280
\(584\) −1.80133e10 −3.74239
\(585\) 3.18429e9 0.657608
\(586\) −1.56106e10 −3.20463
\(587\) −3.31061e9 −0.675578 −0.337789 0.941222i \(-0.609679\pi\)
−0.337789 + 0.941222i \(0.609679\pi\)
\(588\) 7.78004e9 1.57820
\(589\) 8.09025e7 0.0163139
\(590\) −8.96296e9 −1.79668
\(591\) 1.29711e9 0.258477
\(592\) 2.19003e9 0.433835
\(593\) 8.44516e8 0.166309 0.0831546 0.996537i \(-0.473500\pi\)
0.0831546 + 0.996537i \(0.473500\pi\)
\(594\) 2.37973e9 0.465882
\(595\) 2.03594e10 3.96237
\(596\) −9.55700e9 −1.84910
\(597\) 2.84749e8 0.0547711
\(598\) 2.27238e9 0.434538
\(599\) 5.48397e9 1.04256 0.521280 0.853386i \(-0.325455\pi\)
0.521280 + 0.853386i \(0.325455\pi\)
\(600\) −1.05693e10 −1.99763
\(601\) −1.01235e10 −1.90225 −0.951127 0.308801i \(-0.900072\pi\)
−0.951127 + 0.308801i \(0.900072\pi\)
\(602\) −3.26196e9 −0.609385
\(603\) −3.11688e9 −0.578908
\(604\) 4.56359e9 0.842708
\(605\) 8.07104e9 1.48179
\(606\) 4.05728e9 0.740596
\(607\) −6.01963e9 −1.09247 −0.546235 0.837632i \(-0.683939\pi\)
−0.546235 + 0.837632i \(0.683939\pi\)
\(608\) 3.55341e8 0.0641184
\(609\) 4.61897e9 0.828676
\(610\) 1.34667e10 2.40218
\(611\) −2.39182e9 −0.424213
\(612\) −6.22823e9 −1.09833
\(613\) 1.95203e9 0.342274 0.171137 0.985247i \(-0.445256\pi\)
0.171137 + 0.985247i \(0.445256\pi\)
\(614\) −9.26055e8 −0.161453
\(615\) −7.52713e9 −1.30487
\(616\) −2.34912e10 −4.04923
\(617\) 9.76165e8 0.167311 0.0836556 0.996495i \(-0.473340\pi\)
0.0836556 + 0.996495i \(0.473340\pi\)
\(618\) 2.71726e9 0.463097
\(619\) −2.48694e9 −0.421452 −0.210726 0.977545i \(-0.567583\pi\)
−0.210726 + 0.977545i \(0.567583\pi\)
\(620\) −2.01994e9 −0.340383
\(621\) −2.39483e8 −0.0401286
\(622\) −1.95206e9 −0.325258
\(623\) 1.12563e10 1.86504
\(624\) 5.47029e9 0.901289
\(625\) 2.38341e9 0.390498
\(626\) 1.40133e10 2.28312
\(627\) 8.23107e8 0.133358
\(628\) 1.46498e10 2.36034
\(629\) −3.21515e9 −0.515138
\(630\) 9.31954e9 1.48492
\(631\) 3.03958e9 0.481627 0.240813 0.970571i \(-0.422586\pi\)
0.240813 + 0.970571i \(0.422586\pi\)
\(632\) 1.88285e10 2.96692
\(633\) 2.51169e9 0.393598
\(634\) −3.01868e9 −0.470440
\(635\) −6.32016e9 −0.979536
\(636\) −7.98354e8 −0.123054
\(637\) 1.00318e10 1.53776
\(638\) −1.50289e10 −2.29116
\(639\) −5.93735e8 −0.0900201
\(640\) 1.68336e10 2.53833
\(641\) −1.19765e10 −1.79609 −0.898046 0.439902i \(-0.855013\pi\)
−0.898046 + 0.439902i \(0.855013\pi\)
\(642\) −5.04628e9 −0.752660
\(643\) 4.75355e9 0.705146 0.352573 0.935784i \(-0.385307\pi\)
0.352573 + 0.935784i \(0.385307\pi\)
\(644\) 4.50733e9 0.664996
\(645\) −1.49671e9 −0.219624
\(646\) −3.17860e9 −0.463898
\(647\) 2.68173e9 0.389269 0.194635 0.980876i \(-0.437648\pi\)
0.194635 + 0.980876i \(0.437648\pi\)
\(648\) −1.49529e9 −0.215881
\(649\) −5.85354e9 −0.840547
\(650\) −2.59841e10 −3.71117
\(651\) 5.98229e8 0.0849833
\(652\) −4.08454e9 −0.577134
\(653\) 2.10661e9 0.296065 0.148033 0.988982i \(-0.452706\pi\)
0.148033 + 0.988982i \(0.452706\pi\)
\(654\) 1.57632e9 0.220354
\(655\) 1.91988e8 0.0266950
\(656\) −1.29308e10 −1.78840
\(657\) 4.66713e9 0.642054
\(658\) −7.00020e9 −0.957899
\(659\) 1.10631e10 1.50583 0.752916 0.658117i \(-0.228647\pi\)
0.752916 + 0.658117i \(0.228647\pi\)
\(660\) −2.05510e10 −2.78246
\(661\) −1.11027e10 −1.49528 −0.747641 0.664103i \(-0.768814\pi\)
−0.747641 + 0.664103i \(0.768814\pi\)
\(662\) 7.32537e9 0.981356
\(663\) −8.03082e9 −1.07019
\(664\) 2.01326e10 2.66878
\(665\) 3.22346e9 0.425056
\(666\) −1.47174e9 −0.193050
\(667\) 1.51243e9 0.197348
\(668\) −7.80338e9 −1.01290
\(669\) −3.60016e9 −0.464868
\(670\) 3.97162e10 5.10160
\(671\) 8.79482e9 1.12382
\(672\) 2.62755e9 0.334009
\(673\) 2.54772e9 0.322180 0.161090 0.986940i \(-0.448499\pi\)
0.161090 + 0.986940i \(0.448499\pi\)
\(674\) 6.98772e9 0.879074
\(675\) 2.73843e9 0.342719
\(676\) 6.74954e9 0.840351
\(677\) −1.21766e10 −1.50822 −0.754112 0.656745i \(-0.771933\pi\)
−0.754112 + 0.656745i \(0.771933\pi\)
\(678\) 1.09242e10 1.34612
\(679\) −2.11792e10 −2.59636
\(680\) 4.16241e10 5.07649
\(681\) 6.06089e9 0.735396
\(682\) −1.94648e9 −0.234965
\(683\) −9.17376e9 −1.10173 −0.550864 0.834595i \(-0.685702\pi\)
−0.550864 + 0.834595i \(0.685702\pi\)
\(684\) −9.86103e8 −0.117822
\(685\) 2.78976e10 3.31627
\(686\) 6.77248e9 0.800965
\(687\) −7.01874e9 −0.825868
\(688\) −2.57120e9 −0.301007
\(689\) −1.02942e9 −0.119901
\(690\) 3.05157e9 0.353632
\(691\) 2.31775e8 0.0267235 0.0133617 0.999911i \(-0.495747\pi\)
0.0133617 + 0.999911i \(0.495747\pi\)
\(692\) 4.11391e9 0.471936
\(693\) 6.08641e9 0.694696
\(694\) −1.63723e9 −0.185931
\(695\) 2.63527e10 2.97768
\(696\) 9.44334e9 1.06168
\(697\) 1.89835e10 2.12355
\(698\) −1.09147e10 −1.21484
\(699\) −1.41372e9 −0.156565
\(700\) −5.15402e10 −5.67940
\(701\) −5.45607e9 −0.598228 −0.299114 0.954217i \(-0.596691\pi\)
−0.299114 + 0.954217i \(0.596691\pi\)
\(702\) −3.67612e9 −0.401060
\(703\) −5.09047e8 −0.0552604
\(704\) 8.23851e9 0.889907
\(705\) −3.21196e9 −0.345230
\(706\) −1.40541e10 −1.50310
\(707\) 1.03769e10 1.10433
\(708\) 7.01269e9 0.742622
\(709\) −1.56400e9 −0.164807 −0.0824034 0.996599i \(-0.526260\pi\)
−0.0824034 + 0.996599i \(0.526260\pi\)
\(710\) 7.56556e9 0.793298
\(711\) −4.87834e9 −0.509013
\(712\) 2.30132e10 2.38944
\(713\) 1.95883e8 0.0202387
\(714\) −2.35040e10 −2.41656
\(715\) −2.64989e10 −2.71117
\(716\) −2.72085e10 −2.77019
\(717\) 1.04208e10 1.05581
\(718\) 1.20628e10 1.21623
\(719\) −1.01004e9 −0.101341 −0.0506706 0.998715i \(-0.516136\pi\)
−0.0506706 + 0.998715i \(0.516136\pi\)
\(720\) 7.34601e9 0.733479
\(721\) 6.94968e9 0.690544
\(722\) 1.73111e10 1.71176
\(723\) 1.27508e9 0.125474
\(724\) 1.32341e9 0.129601
\(725\) −1.72942e10 −1.68546
\(726\) −9.31765e9 −0.903708
\(727\) 2.06395e8 0.0199218 0.00996091 0.999950i \(-0.496829\pi\)
0.00996091 + 0.999950i \(0.496829\pi\)
\(728\) 3.62883e10 3.48583
\(729\) 3.87420e8 0.0370370
\(730\) −5.94700e10 −5.65807
\(731\) 3.77473e9 0.357417
\(732\) −1.05364e10 −0.992896
\(733\) −1.37499e10 −1.28954 −0.644772 0.764375i \(-0.723048\pi\)
−0.644772 + 0.764375i \(0.723048\pi\)
\(734\) 7.97868e9 0.744723
\(735\) 1.34716e10 1.25145
\(736\) 8.60358e8 0.0795439
\(737\) 2.59379e10 2.38671
\(738\) 8.68973e9 0.795809
\(739\) −1.08060e10 −0.984937 −0.492469 0.870330i \(-0.663905\pi\)
−0.492469 + 0.870330i \(0.663905\pi\)
\(740\) 1.27097e10 1.15298
\(741\) −1.27150e9 −0.114803
\(742\) −3.01282e9 −0.270744
\(743\) −6.51922e9 −0.583088 −0.291544 0.956557i \(-0.594169\pi\)
−0.291544 + 0.956557i \(0.594169\pi\)
\(744\) 1.22306e9 0.108879
\(745\) −1.65485e10 −1.46626
\(746\) −1.48223e10 −1.30716
\(747\) −5.21623e9 −0.457862
\(748\) 5.18298e10 4.52819
\(749\) −1.29064e10 −1.12232
\(750\) −1.52996e10 −1.32424
\(751\) −4.66797e9 −0.402150 −0.201075 0.979576i \(-0.564443\pi\)
−0.201075 + 0.979576i \(0.564443\pi\)
\(752\) −5.51782e9 −0.473157
\(753\) 1.16673e9 0.0995834
\(754\) 2.32161e10 1.97237
\(755\) 7.90211e9 0.668234
\(756\) −7.29167e9 −0.613763
\(757\) 1.84301e10 1.54416 0.772079 0.635526i \(-0.219217\pi\)
0.772079 + 0.635526i \(0.219217\pi\)
\(758\) −2.15306e10 −1.79562
\(759\) 1.99292e9 0.165441
\(760\) 6.59026e9 0.544572
\(761\) 6.60838e9 0.543561 0.271781 0.962359i \(-0.412387\pi\)
0.271781 + 0.962359i \(0.412387\pi\)
\(762\) 7.29634e9 0.597396
\(763\) 4.03159e9 0.328579
\(764\) −1.76300e10 −1.43030
\(765\) −1.07845e10 −0.870936
\(766\) −3.64793e10 −2.93255
\(767\) 9.04231e9 0.723595
\(768\) −1.47403e10 −1.17420
\(769\) 1.51293e10 1.19971 0.599857 0.800107i \(-0.295224\pi\)
0.599857 + 0.800107i \(0.295224\pi\)
\(770\) −7.75550e10 −6.12198
\(771\) −9.11226e9 −0.716037
\(772\) 7.04851e9 0.551362
\(773\) −1.46251e10 −1.13886 −0.569432 0.822039i \(-0.692837\pi\)
−0.569432 + 0.822039i \(0.692837\pi\)
\(774\) 1.72789e9 0.133944
\(775\) −2.23987e9 −0.172849
\(776\) −4.33002e10 −3.32640
\(777\) −3.76412e9 −0.287865
\(778\) 1.83447e10 1.39663
\(779\) 3.00562e9 0.227800
\(780\) 3.17463e10 2.39531
\(781\) 4.94092e9 0.371133
\(782\) −7.69609e9 −0.575502
\(783\) −2.44671e9 −0.182144
\(784\) 2.31428e10 1.71518
\(785\) 2.53670e10 1.87165
\(786\) −2.21641e8 −0.0162807
\(787\) −7.19809e9 −0.526387 −0.263194 0.964743i \(-0.584776\pi\)
−0.263194 + 0.964743i \(0.584776\pi\)
\(788\) 1.29318e10 0.941493
\(789\) −4.03249e9 −0.292283
\(790\) 6.21613e10 4.48565
\(791\) 2.79397e10 2.00726
\(792\) 1.24435e10 0.890028
\(793\) −1.35859e10 −0.967457
\(794\) −3.35328e10 −2.37738
\(795\) −1.38240e9 −0.0975770
\(796\) 2.83885e9 0.199502
\(797\) −8.60373e9 −0.601981 −0.300990 0.953627i \(-0.597317\pi\)
−0.300990 + 0.953627i \(0.597317\pi\)
\(798\) −3.72134e9 −0.259232
\(799\) 8.10060e9 0.561828
\(800\) −9.83797e9 −0.679345
\(801\) −5.96256e9 −0.409939
\(802\) −3.55319e10 −2.43225
\(803\) −3.88388e10 −2.64704
\(804\) −3.10743e10 −2.10865
\(805\) 7.80470e9 0.527315
\(806\) 3.00684e9 0.202273
\(807\) 8.66354e9 0.580281
\(808\) 2.12153e10 1.41484
\(809\) 2.23111e10 1.48150 0.740748 0.671783i \(-0.234471\pi\)
0.740748 + 0.671783i \(0.234471\pi\)
\(810\) −4.93663e9 −0.326387
\(811\) 6.50028e9 0.427917 0.213958 0.976843i \(-0.431364\pi\)
0.213958 + 0.976843i \(0.431364\pi\)
\(812\) 4.60496e10 3.01842
\(813\) 6.62963e8 0.0432685
\(814\) 1.22474e10 0.795903
\(815\) −7.07261e9 −0.457644
\(816\) −1.85267e10 −1.19367
\(817\) 5.97646e8 0.0383413
\(818\) 3.23806e10 2.06846
\(819\) −9.40205e9 −0.598038
\(820\) −7.50431e10 −4.75294
\(821\) 7.01294e9 0.442282 0.221141 0.975242i \(-0.429022\pi\)
0.221141 + 0.975242i \(0.429022\pi\)
\(822\) −3.22065e10 −2.02252
\(823\) −1.67931e10 −1.05010 −0.525051 0.851071i \(-0.675954\pi\)
−0.525051 + 0.851071i \(0.675954\pi\)
\(824\) 1.42084e10 0.884708
\(825\) −2.27885e10 −1.41295
\(826\) 2.64644e10 1.63392
\(827\) 5.65627e9 0.347745 0.173872 0.984768i \(-0.444372\pi\)
0.173872 + 0.984768i \(0.444372\pi\)
\(828\) −2.38757e9 −0.146167
\(829\) 8.59914e9 0.524220 0.262110 0.965038i \(-0.415582\pi\)
0.262110 + 0.965038i \(0.415582\pi\)
\(830\) 6.64669e10 4.03489
\(831\) 8.09027e9 0.489057
\(832\) −1.27265e10 −0.766087
\(833\) −3.39755e10 −2.03661
\(834\) −3.04230e10 −1.81602
\(835\) −1.35120e10 −0.803187
\(836\) 8.20611e9 0.485753
\(837\) −3.16886e8 −0.0186795
\(838\) −4.48012e10 −2.62988
\(839\) −1.50648e10 −0.880636 −0.440318 0.897842i \(-0.645134\pi\)
−0.440318 + 0.897842i \(0.645134\pi\)
\(840\) 4.87313e10 2.83681
\(841\) −1.79802e9 −0.104234
\(842\) −6.64595e9 −0.383676
\(843\) 6.12820e9 0.352319
\(844\) 2.50408e10 1.43367
\(845\) 1.16872e10 0.666365
\(846\) 3.70806e9 0.210548
\(847\) −2.38308e10 −1.34756
\(848\) −2.37481e9 −0.133735
\(849\) 2.04057e9 0.114439
\(850\) 8.80028e10 4.91508
\(851\) −1.23251e9 −0.0685549
\(852\) −5.91934e9 −0.327895
\(853\) −1.71226e10 −0.944598 −0.472299 0.881438i \(-0.656576\pi\)
−0.472299 + 0.881438i \(0.656576\pi\)
\(854\) −3.97621e10 −2.18458
\(855\) −1.70749e9 −0.0934281
\(856\) −2.63867e10 −1.43789
\(857\) −2.12967e10 −1.15579 −0.577897 0.816110i \(-0.696126\pi\)
−0.577897 + 0.816110i \(0.696126\pi\)
\(858\) 3.05918e10 1.65348
\(859\) −1.57530e10 −0.847981 −0.423990 0.905667i \(-0.639371\pi\)
−0.423990 + 0.905667i \(0.639371\pi\)
\(860\) −1.49218e10 −0.799973
\(861\) 2.22249e10 1.18667
\(862\) 4.64363e10 2.46935
\(863\) 2.32739e10 1.23263 0.616313 0.787501i \(-0.288626\pi\)
0.616313 + 0.787501i \(0.288626\pi\)
\(864\) −1.39183e9 −0.0734157
\(865\) 7.12347e9 0.374227
\(866\) −3.64398e9 −0.190662
\(867\) 1.61196e10 0.840015
\(868\) 5.96414e9 0.309549
\(869\) 4.05964e10 2.09854
\(870\) 3.11767e10 1.60514
\(871\) −4.00679e10 −2.05462
\(872\) 8.24245e9 0.420968
\(873\) 1.12188e10 0.570685
\(874\) −1.21851e9 −0.0617359
\(875\) −3.91303e10 −1.97463
\(876\) 4.65298e10 2.33866
\(877\) −9.84011e9 −0.492608 −0.246304 0.969193i \(-0.579216\pi\)
−0.246304 + 0.969193i \(0.579216\pi\)
\(878\) 2.81416e10 1.40320
\(879\) 2.11490e10 1.05034
\(880\) −6.11317e10 −3.02397
\(881\) −2.40556e10 −1.18522 −0.592611 0.805489i \(-0.701903\pi\)
−0.592611 + 0.805489i \(0.701903\pi\)
\(882\) −1.55523e10 −0.763229
\(883\) 4.99537e9 0.244177 0.122089 0.992519i \(-0.461041\pi\)
0.122089 + 0.992519i \(0.461041\pi\)
\(884\) −8.00647e10 −3.89814
\(885\) 1.21429e10 0.588870
\(886\) −1.37296e10 −0.663192
\(887\) −4.68695e9 −0.225506 −0.112753 0.993623i \(-0.535967\pi\)
−0.112753 + 0.993623i \(0.535967\pi\)
\(888\) −7.69562e9 −0.368806
\(889\) 1.86611e10 0.890803
\(890\) 7.59768e10 3.61257
\(891\) −3.22402e9 −0.152695
\(892\) −3.58924e10 −1.69327
\(893\) 1.28255e9 0.0602691
\(894\) 1.91045e10 0.894239
\(895\) −4.71131e10 −2.19665
\(896\) −4.97035e10 −2.30839
\(897\) −3.07858e9 −0.142422
\(898\) 1.29676e10 0.597577
\(899\) 2.00126e9 0.0918637
\(900\) 2.73012e10 1.24834
\(901\) 3.48642e9 0.158797
\(902\) −7.23139e10 −3.28094
\(903\) 4.41925e9 0.199729
\(904\) 5.71219e10 2.57166
\(905\) 2.29156e9 0.102769
\(906\) −9.12262e9 −0.407541
\(907\) 2.14893e10 0.956304 0.478152 0.878277i \(-0.341307\pi\)
0.478152 + 0.878277i \(0.341307\pi\)
\(908\) 6.04251e10 2.67865
\(909\) −5.49673e9 −0.242734
\(910\) 1.19804e11 5.27018
\(911\) 3.15925e10 1.38443 0.692214 0.721693i \(-0.256636\pi\)
0.692214 + 0.721693i \(0.256636\pi\)
\(912\) −2.93330e9 −0.128048
\(913\) 4.34082e10 1.88766
\(914\) 3.22747e10 1.39814
\(915\) −1.82444e10 −0.787327
\(916\) −6.99746e10 −3.00820
\(917\) −5.66870e8 −0.0242768
\(918\) 1.24502e10 0.531164
\(919\) −1.58869e10 −0.675205 −0.337603 0.941289i \(-0.609616\pi\)
−0.337603 + 0.941289i \(0.609616\pi\)
\(920\) 1.59564e10 0.675583
\(921\) 1.25460e9 0.0529172
\(922\) 1.23922e10 0.520705
\(923\) −7.63254e9 −0.319494
\(924\) 6.06796e10 2.53041
\(925\) 1.40935e10 0.585493
\(926\) −5.69782e10 −2.35815
\(927\) −3.68130e9 −0.151783
\(928\) 8.78994e9 0.361051
\(929\) −4.25155e10 −1.73977 −0.869886 0.493253i \(-0.835808\pi\)
−0.869886 + 0.493253i \(0.835808\pi\)
\(930\) 4.03786e9 0.164612
\(931\) −5.37927e9 −0.218474
\(932\) −1.40944e10 −0.570282
\(933\) 2.64462e9 0.106605
\(934\) −2.56298e10 −1.02928
\(935\) 8.97463e10 3.59067
\(936\) −1.92222e10 −0.766191
\(937\) −1.52382e9 −0.0605126 −0.0302563 0.999542i \(-0.509632\pi\)
−0.0302563 + 0.999542i \(0.509632\pi\)
\(938\) −1.17268e11 −4.63946
\(939\) −1.89849e10 −0.748306
\(940\) −3.20222e10 −1.25749
\(941\) 3.98090e10 1.55746 0.778732 0.627357i \(-0.215863\pi\)
0.778732 + 0.627357i \(0.215863\pi\)
\(942\) −2.92850e10 −1.14148
\(943\) 7.27726e9 0.282603
\(944\) 2.08602e10 0.807080
\(945\) −1.26259e10 −0.486690
\(946\) −1.43791e10 −0.552220
\(947\) 1.71695e9 0.0656949 0.0328475 0.999460i \(-0.489542\pi\)
0.0328475 + 0.999460i \(0.489542\pi\)
\(948\) −4.86355e10 −1.85406
\(949\) 5.99966e10 2.27874
\(950\) 1.39333e10 0.527256
\(951\) 4.08965e9 0.154189
\(952\) −1.22901e11 −4.61663
\(953\) −2.93461e10 −1.09831 −0.549155 0.835721i \(-0.685050\pi\)
−0.549155 + 0.835721i \(0.685050\pi\)
\(954\) 1.59591e9 0.0595100
\(955\) −3.05274e10 −1.13417
\(956\) 1.03892e11 3.84574
\(957\) 2.03609e10 0.750940
\(958\) 7.77554e10 2.85727
\(959\) −8.23714e10 −3.01586
\(960\) −1.70904e10 −0.623450
\(961\) −2.72534e10 −0.990579
\(962\) −1.89194e10 −0.685163
\(963\) 6.83661e9 0.246689
\(964\) 1.27121e10 0.457034
\(965\) 1.22049e10 0.437209
\(966\) −9.01016e9 −0.321598
\(967\) 1.69145e10 0.601543 0.300771 0.953696i \(-0.402756\pi\)
0.300771 + 0.953696i \(0.402756\pi\)
\(968\) −4.87214e10 −1.72646
\(969\) 4.30632e9 0.152045
\(970\) −1.42953e11 −5.02913
\(971\) −3.82953e9 −0.134239 −0.0671195 0.997745i \(-0.521381\pi\)
−0.0671195 + 0.997745i \(0.521381\pi\)
\(972\) 3.86246e9 0.134906
\(973\) −7.78098e10 −2.70794
\(974\) 4.41127e10 1.52971
\(975\) 3.52028e10 1.21636
\(976\) −3.13420e10 −1.07908
\(977\) −3.33271e10 −1.14332 −0.571658 0.820492i \(-0.693700\pi\)
−0.571658 + 0.820492i \(0.693700\pi\)
\(978\) 8.16500e9 0.279107
\(979\) 4.96190e10 1.69008
\(980\) 1.34307e11 4.55835
\(981\) −2.13556e9 −0.0722223
\(982\) 8.63063e10 2.90839
\(983\) 1.71429e10 0.575634 0.287817 0.957685i \(-0.407070\pi\)
0.287817 + 0.957685i \(0.407070\pi\)
\(984\) 4.54380e10 1.52033
\(985\) 2.23921e10 0.746567
\(986\) −7.86280e10 −2.61221
\(987\) 9.48374e9 0.313957
\(988\) −1.26765e10 −0.418166
\(989\) 1.44703e9 0.0475653
\(990\) 4.10815e10 1.34562
\(991\) 4.73510e10 1.54551 0.772755 0.634705i \(-0.218878\pi\)
0.772755 + 0.634705i \(0.218878\pi\)
\(992\) 1.13843e9 0.0370269
\(993\) −9.92427e9 −0.321645
\(994\) −2.23383e10 −0.721437
\(995\) 4.91563e9 0.158197
\(996\) −5.20042e10 −1.66775
\(997\) −2.46977e10 −0.789267 −0.394634 0.918839i \(-0.629128\pi\)
−0.394634 + 0.918839i \(0.629128\pi\)
\(998\) 2.22415e10 0.708283
\(999\) 1.99388e9 0.0632733
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 69.8.a.b.1.1 6
3.2 odd 2 207.8.a.c.1.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
69.8.a.b.1.1 6 1.1 even 1 trivial
207.8.a.c.1.6 6 3.2 odd 2