Properties

Label 177.12.a.b.1.26
Level $177$
Weight $12$
Character 177.1
Self dual yes
Analytic conductor $135.997$
Analytic rank $1$
Dimension $27$
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] = [177,12,Mod(1,177)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(177, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("177.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 177.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: \(135.996742959\)
Analytic rank: \(1\)
Dimension: \(27\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.26
Character \(\chi\) \(=\) 177.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+81.6200 q^{2} +243.000 q^{3} +4613.82 q^{4} +4430.19 q^{5} +19833.6 q^{6} -67642.9 q^{7} +209422. q^{8} +59049.0 q^{9} +O(q^{10})\) \(q+81.6200 q^{2} +243.000 q^{3} +4613.82 q^{4} +4430.19 q^{5} +19833.6 q^{6} -67642.9 q^{7} +209422. q^{8} +59049.0 q^{9} +361592. q^{10} -519775. q^{11} +1.12116e6 q^{12} -2.46231e6 q^{13} -5.52101e6 q^{14} +1.07654e6 q^{15} +7.64391e6 q^{16} -4.35544e6 q^{17} +4.81958e6 q^{18} +3.11830e6 q^{19} +2.04401e7 q^{20} -1.64372e7 q^{21} -4.24240e7 q^{22} -1.04622e7 q^{23} +5.08895e7 q^{24} -2.92016e7 q^{25} -2.00973e8 q^{26} +1.43489e7 q^{27} -3.12092e8 q^{28} -2.13398e8 q^{29} +8.78668e7 q^{30} +1.99220e8 q^{31} +1.94999e8 q^{32} -1.26305e8 q^{33} -3.55491e8 q^{34} -2.99671e8 q^{35} +2.72441e8 q^{36} +7.90601e8 q^{37} +2.54516e8 q^{38} -5.98340e8 q^{39} +9.27778e8 q^{40} -4.32732e8 q^{41} -1.34161e9 q^{42} -4.93130e8 q^{43} -2.39815e9 q^{44} +2.61598e8 q^{45} -8.53921e8 q^{46} -3.47185e8 q^{47} +1.85747e9 q^{48} +2.59824e9 q^{49} -2.38343e9 q^{50} -1.05837e9 q^{51} -1.13606e10 q^{52} +6.14865e8 q^{53} +1.17116e9 q^{54} -2.30270e9 q^{55} -1.41659e10 q^{56} +7.57747e8 q^{57} -1.74175e10 q^{58} -7.14924e8 q^{59} +4.96694e9 q^{60} +9.09541e9 q^{61} +1.62604e10 q^{62} -3.99425e9 q^{63} +2.61119e8 q^{64} -1.09085e10 q^{65} -1.03090e10 q^{66} +2.20186e9 q^{67} -2.00952e10 q^{68} -2.54230e9 q^{69} -2.44591e10 q^{70} +2.45387e10 q^{71} +1.23662e10 q^{72} -8.48573e8 q^{73} +6.45288e10 q^{74} -7.09598e9 q^{75} +1.43873e10 q^{76} +3.51591e10 q^{77} -4.88365e10 q^{78} -3.82381e10 q^{79} +3.38639e10 q^{80} +3.48678e9 q^{81} -3.53196e10 q^{82} +2.31641e10 q^{83} -7.58384e10 q^{84} -1.92954e10 q^{85} -4.02492e10 q^{86} -5.18556e10 q^{87} -1.08852e11 q^{88} -2.16282e10 q^{89} +2.13516e10 q^{90} +1.66558e11 q^{91} -4.82705e10 q^{92} +4.84106e10 q^{93} -2.83372e10 q^{94} +1.38147e10 q^{95} +4.73849e10 q^{96} +8.63799e10 q^{97} +2.12068e11 q^{98} -3.06922e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 27 q - 128 q^{2} + 6561 q^{3} + 26142 q^{4} - 17188 q^{5} - 31104 q^{6} - 126579 q^{7} - 355797 q^{8} + 1594323 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 27 q - 128 q^{2} + 6561 q^{3} + 26142 q^{4} - 17188 q^{5} - 31104 q^{6} - 126579 q^{7} - 355797 q^{8} + 1594323 q^{9} - 383719 q^{10} - 1816556 q^{11} + 6352506 q^{12} - 3951804 q^{13} - 6207867 q^{14} - 4176684 q^{15} + 28295194 q^{16} - 17723275 q^{17} - 7558272 q^{18} - 19573013 q^{19} - 48468099 q^{20} - 30758697 q^{21} - 1729910 q^{22} - 88593797 q^{23} - 86458671 q^{24} + 345714963 q^{25} - 6676346 q^{26} + 387420489 q^{27} + 126954286 q^{28} - 276632427 q^{29} - 93243717 q^{30} - 357680917 q^{31} - 859842334 q^{32} - 441423108 q^{33} + 232730000 q^{34} - 510315139 q^{35} + 1543658958 q^{36} - 660238257 q^{37} - 2067286961 q^{38} - 960288372 q^{39} - 3388951110 q^{40} - 1671147569 q^{41} - 1508511681 q^{42} - 1883107790 q^{43} - 3895687630 q^{44} - 1014934212 q^{45} - 1720344243 q^{46} - 5818572501 q^{47} + 6875732142 q^{48} - 18858180 q^{49} - 21474519647 q^{50} - 4306755825 q^{51} - 42214560062 q^{52} - 11444513368 q^{53} - 1836660096 q^{54} - 24401486484 q^{55} - 50583585764 q^{56} - 4756242159 q^{57} - 45017395090 q^{58} - 19302956073 q^{59} - 11777748057 q^{60} + 408637955 q^{61} - 28543084070 q^{62} - 7474363371 q^{63} + 33067284293 q^{64} - 21656714730 q^{65} - 420368130 q^{66} - 49803132690 q^{67} - 16500749319 q^{68} - 21528292671 q^{69} - 45808890782 q^{70} - 34127492216 q^{71} - 21009457053 q^{72} - 55734362153 q^{73} - 40367816298 q^{74} + 84008736009 q^{75} - 14840406404 q^{76} - 99723443615 q^{77} - 1622352078 q^{78} - 76484916442 q^{79} + 93882788915 q^{80} + 94143178827 q^{81} + 52951239205 q^{82} - 140433865655 q^{83} + 30849891498 q^{84} + 34329063335 q^{85} + 175223869508 q^{86} - 67221679761 q^{87} + 268823645069 q^{88} - 1191878597 q^{89} - 22658223231 q^{90} + 201632581559 q^{91} - 206501888812 q^{92} - 86916462831 q^{93} + 319770144384 q^{94} - 81387074885 q^{95} - 208941687162 q^{96} - 144896178730 q^{97} + 135739195260 q^{98} - 107265815244 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 81.6200 1.80356 0.901782 0.432192i \(-0.142260\pi\)
0.901782 + 0.432192i \(0.142260\pi\)
\(3\) 243.000 0.577350
\(4\) 4613.82 2.25284
\(5\) 4430.19 0.633997 0.316998 0.948426i \(-0.397325\pi\)
0.316998 + 0.948426i \(0.397325\pi\)
\(6\) 19833.6 1.04129
\(7\) −67642.9 −1.52119 −0.760595 0.649227i \(-0.775092\pi\)
−0.760595 + 0.649227i \(0.775092\pi\)
\(8\) 209422. 2.25958
\(9\) 59049.0 0.333333
\(10\) 361592. 1.14345
\(11\) −519775. −0.973096 −0.486548 0.873654i \(-0.661744\pi\)
−0.486548 + 0.873654i \(0.661744\pi\)
\(12\) 1.12116e6 1.30068
\(13\) −2.46231e6 −1.83930 −0.919652 0.392734i \(-0.871529\pi\)
−0.919652 + 0.392734i \(0.871529\pi\)
\(14\) −5.52101e6 −2.74356
\(15\) 1.07654e6 0.366038
\(16\) 7.64391e6 1.82245
\(17\) −4.35544e6 −0.743982 −0.371991 0.928236i \(-0.621325\pi\)
−0.371991 + 0.928236i \(0.621325\pi\)
\(18\) 4.81958e6 0.601188
\(19\) 3.11830e6 0.288917 0.144459 0.989511i \(-0.453856\pi\)
0.144459 + 0.989511i \(0.453856\pi\)
\(20\) 2.04401e7 1.42829
\(21\) −1.64372e7 −0.878259
\(22\) −4.24240e7 −1.75504
\(23\) −1.04622e7 −0.338936 −0.169468 0.985536i \(-0.554205\pi\)
−0.169468 + 0.985536i \(0.554205\pi\)
\(24\) 5.08895e7 1.30457
\(25\) −2.92016e7 −0.598048
\(26\) −2.00973e8 −3.31730
\(27\) 1.43489e7 0.192450
\(28\) −3.12092e8 −3.42700
\(29\) −2.13398e8 −1.93197 −0.965985 0.258596i \(-0.916740\pi\)
−0.965985 + 0.258596i \(0.916740\pi\)
\(30\) 8.78668e7 0.660173
\(31\) 1.99220e8 1.24981 0.624906 0.780700i \(-0.285137\pi\)
0.624906 + 0.780700i \(0.285137\pi\)
\(32\) 1.94999e8 1.02733
\(33\) −1.26305e8 −0.561817
\(34\) −3.55491e8 −1.34182
\(35\) −2.99671e8 −0.964429
\(36\) 2.72441e8 0.750947
\(37\) 7.90601e8 1.87434 0.937169 0.348876i \(-0.113436\pi\)
0.937169 + 0.348876i \(0.113436\pi\)
\(38\) 2.54516e8 0.521080
\(39\) −5.98340e8 −1.06192
\(40\) 9.27778e8 1.43256
\(41\) −4.32732e8 −0.583321 −0.291661 0.956522i \(-0.594208\pi\)
−0.291661 + 0.956522i \(0.594208\pi\)
\(42\) −1.34161e9 −1.58400
\(43\) −4.93130e8 −0.511546 −0.255773 0.966737i \(-0.582330\pi\)
−0.255773 + 0.966737i \(0.582330\pi\)
\(44\) −2.39815e9 −2.19223
\(45\) 2.61598e8 0.211332
\(46\) −8.53921e8 −0.611293
\(47\) −3.47185e8 −0.220812 −0.110406 0.993887i \(-0.535215\pi\)
−0.110406 + 0.993887i \(0.535215\pi\)
\(48\) 1.85747e9 1.05219
\(49\) 2.59824e9 1.31402
\(50\) −2.38343e9 −1.07862
\(51\) −1.05837e9 −0.429538
\(52\) −1.13606e10 −4.14366
\(53\) 6.14865e8 0.201959 0.100979 0.994889i \(-0.467802\pi\)
0.100979 + 0.994889i \(0.467802\pi\)
\(54\) 1.17116e9 0.347096
\(55\) −2.30270e9 −0.616940
\(56\) −1.41659e10 −3.43724
\(57\) 7.57747e8 0.166806
\(58\) −1.74175e10 −3.48443
\(59\) −7.14924e8 −0.130189
\(60\) 4.96694e9 0.824626
\(61\) 9.09541e9 1.37882 0.689411 0.724370i \(-0.257869\pi\)
0.689411 + 0.724370i \(0.257869\pi\)
\(62\) 1.62604e10 2.25411
\(63\) −3.99425e9 −0.507063
\(64\) 2.61119e8 0.0303983
\(65\) −1.09085e10 −1.16611
\(66\) −1.03090e10 −1.01327
\(67\) 2.20186e9 0.199241 0.0996203 0.995026i \(-0.468237\pi\)
0.0996203 + 0.995026i \(0.468237\pi\)
\(68\) −2.00952e10 −1.67607
\(69\) −2.54230e9 −0.195685
\(70\) −2.44591e10 −1.73941
\(71\) 2.45387e10 1.61410 0.807051 0.590482i \(-0.201062\pi\)
0.807051 + 0.590482i \(0.201062\pi\)
\(72\) 1.23662e10 0.753192
\(73\) −8.48573e8 −0.0479086 −0.0239543 0.999713i \(-0.507626\pi\)
−0.0239543 + 0.999713i \(0.507626\pi\)
\(74\) 6.45288e10 3.38049
\(75\) −7.09598e9 −0.345283
\(76\) 1.43873e10 0.650884
\(77\) 3.51591e10 1.48026
\(78\) −4.88365e10 −1.91525
\(79\) −3.82381e10 −1.39813 −0.699065 0.715058i \(-0.746400\pi\)
−0.699065 + 0.715058i \(0.746400\pi\)
\(80\) 3.38639e10 1.15543
\(81\) 3.48678e9 0.111111
\(82\) −3.53196e10 −1.05206
\(83\) 2.31641e10 0.645485 0.322743 0.946487i \(-0.395395\pi\)
0.322743 + 0.946487i \(0.395395\pi\)
\(84\) −7.58384e10 −1.97858
\(85\) −1.92954e10 −0.471682
\(86\) −4.02492e10 −0.922606
\(87\) −5.18556e10 −1.11542
\(88\) −1.08852e11 −2.19878
\(89\) −2.16282e10 −0.410559 −0.205280 0.978703i \(-0.565810\pi\)
−0.205280 + 0.978703i \(0.565810\pi\)
\(90\) 2.13516e10 0.381151
\(91\) 1.66558e11 2.79793
\(92\) −4.82705e10 −0.763569
\(93\) 4.84106e10 0.721579
\(94\) −2.83372e10 −0.398248
\(95\) 1.38147e10 0.183172
\(96\) 4.73849e10 0.593127
\(97\) 8.63799e10 1.02134 0.510668 0.859778i \(-0.329398\pi\)
0.510668 + 0.859778i \(0.329398\pi\)
\(98\) 2.12068e11 2.36991
\(99\) −3.06922e10 −0.324365
\(100\) −1.34731e11 −1.34731
\(101\) −3.93356e10 −0.372407 −0.186204 0.982511i \(-0.559618\pi\)
−0.186204 + 0.982511i \(0.559618\pi\)
\(102\) −8.63842e10 −0.774700
\(103\) 6.72298e10 0.571422 0.285711 0.958316i \(-0.407770\pi\)
0.285711 + 0.958316i \(0.407770\pi\)
\(104\) −5.15661e11 −4.15605
\(105\) −7.28200e10 −0.556813
\(106\) 5.01852e10 0.364245
\(107\) 1.33422e11 0.919637 0.459818 0.888013i \(-0.347915\pi\)
0.459818 + 0.888013i \(0.347915\pi\)
\(108\) 6.62032e10 0.433559
\(109\) −8.17691e10 −0.509030 −0.254515 0.967069i \(-0.581916\pi\)
−0.254515 + 0.967069i \(0.581916\pi\)
\(110\) −1.87946e11 −1.11269
\(111\) 1.92116e11 1.08215
\(112\) −5.17056e11 −2.77229
\(113\) −1.71104e11 −0.873635 −0.436817 0.899550i \(-0.643894\pi\)
−0.436817 + 0.899550i \(0.643894\pi\)
\(114\) 6.18473e10 0.300846
\(115\) −4.63493e10 −0.214885
\(116\) −9.84578e11 −4.35242
\(117\) −1.45397e11 −0.613102
\(118\) −5.83521e10 −0.234804
\(119\) 2.94615e11 1.13174
\(120\) 2.25450e11 0.827092
\(121\) −1.51456e10 −0.0530845
\(122\) 7.42367e11 2.48679
\(123\) −1.05154e11 −0.336781
\(124\) 9.19167e11 2.81563
\(125\) −3.45686e11 −1.01316
\(126\) −3.26010e11 −0.914520
\(127\) 1.01955e11 0.273835 0.136917 0.990582i \(-0.456281\pi\)
0.136917 + 0.990582i \(0.456281\pi\)
\(128\) −3.78046e11 −0.972501
\(129\) −1.19831e11 −0.295341
\(130\) −8.90350e11 −2.10316
\(131\) −8.49075e10 −0.192289 −0.0961444 0.995367i \(-0.530651\pi\)
−0.0961444 + 0.995367i \(0.530651\pi\)
\(132\) −5.82750e11 −1.26568
\(133\) −2.10931e11 −0.439497
\(134\) 1.79716e11 0.359343
\(135\) 6.35683e10 0.122013
\(136\) −9.12124e11 −1.68109
\(137\) 1.31282e11 0.232404 0.116202 0.993226i \(-0.462928\pi\)
0.116202 + 0.993226i \(0.462928\pi\)
\(138\) −2.07503e11 −0.352930
\(139\) −1.83692e11 −0.300268 −0.150134 0.988666i \(-0.547971\pi\)
−0.150134 + 0.988666i \(0.547971\pi\)
\(140\) −1.38263e12 −2.17270
\(141\) −8.43658e10 −0.127486
\(142\) 2.00285e12 2.91114
\(143\) 1.27985e12 1.78982
\(144\) 4.51365e11 0.607483
\(145\) −9.45391e11 −1.22486
\(146\) −6.92605e10 −0.0864062
\(147\) 6.31372e11 0.758648
\(148\) 3.64769e12 4.22258
\(149\) −1.31670e12 −1.46880 −0.734399 0.678718i \(-0.762536\pi\)
−0.734399 + 0.678718i \(0.762536\pi\)
\(150\) −5.79174e11 −0.622740
\(151\) 1.34356e11 0.139279 0.0696395 0.997572i \(-0.477815\pi\)
0.0696395 + 0.997572i \(0.477815\pi\)
\(152\) 6.53040e11 0.652830
\(153\) −2.57184e11 −0.247994
\(154\) 2.86968e12 2.66975
\(155\) 8.82584e11 0.792377
\(156\) −2.76063e12 −2.39234
\(157\) −9.86879e11 −0.825688 −0.412844 0.910802i \(-0.635465\pi\)
−0.412844 + 0.910802i \(0.635465\pi\)
\(158\) −3.12100e12 −2.52162
\(159\) 1.49412e11 0.116601
\(160\) 8.63884e11 0.651322
\(161\) 7.07691e11 0.515586
\(162\) 2.84591e11 0.200396
\(163\) −2.56794e12 −1.74805 −0.874024 0.485884i \(-0.838498\pi\)
−0.874024 + 0.485884i \(0.838498\pi\)
\(164\) −1.99655e12 −1.31413
\(165\) −5.59556e11 −0.356190
\(166\) 1.89065e12 1.16417
\(167\) −2.97952e11 −0.177503 −0.0887516 0.996054i \(-0.528288\pi\)
−0.0887516 + 0.996054i \(0.528288\pi\)
\(168\) −3.44232e12 −1.98449
\(169\) 4.27079e12 2.38304
\(170\) −1.57489e12 −0.850709
\(171\) 1.84133e11 0.0963057
\(172\) −2.27521e12 −1.15243
\(173\) −3.28574e12 −1.61205 −0.806027 0.591879i \(-0.798386\pi\)
−0.806027 + 0.591879i \(0.798386\pi\)
\(174\) −4.23245e12 −2.01174
\(175\) 1.97528e12 0.909744
\(176\) −3.97311e12 −1.77342
\(177\) −1.73727e11 −0.0751646
\(178\) −1.76529e12 −0.740469
\(179\) −4.57859e12 −1.86226 −0.931130 0.364687i \(-0.881176\pi\)
−0.931130 + 0.364687i \(0.881176\pi\)
\(180\) 1.20697e12 0.476098
\(181\) 4.41485e12 1.68921 0.844606 0.535389i \(-0.179835\pi\)
0.844606 + 0.535389i \(0.179835\pi\)
\(182\) 1.35944e13 5.04624
\(183\) 2.21018e12 0.796063
\(184\) −2.19100e12 −0.765853
\(185\) 3.50251e12 1.18832
\(186\) 3.95127e12 1.30141
\(187\) 2.26385e12 0.723966
\(188\) −1.60185e12 −0.497454
\(189\) −9.70602e11 −0.292753
\(190\) 1.12755e12 0.330363
\(191\) 6.22699e12 1.77253 0.886267 0.463175i \(-0.153290\pi\)
0.886267 + 0.463175i \(0.153290\pi\)
\(192\) 6.34520e10 0.0175504
\(193\) 2.20764e11 0.0593421 0.0296711 0.999560i \(-0.490554\pi\)
0.0296711 + 0.999560i \(0.490554\pi\)
\(194\) 7.05033e12 1.84204
\(195\) −2.65076e12 −0.673256
\(196\) 1.19878e13 2.96027
\(197\) 3.23417e12 0.776603 0.388302 0.921532i \(-0.373062\pi\)
0.388302 + 0.921532i \(0.373062\pi\)
\(198\) −2.50510e12 −0.585013
\(199\) −4.99865e12 −1.13543 −0.567716 0.823225i \(-0.692173\pi\)
−0.567716 + 0.823225i \(0.692173\pi\)
\(200\) −6.11545e12 −1.35134
\(201\) 5.35051e11 0.115032
\(202\) −3.21057e12 −0.671660
\(203\) 1.44348e13 2.93889
\(204\) −4.88313e12 −0.967682
\(205\) −1.91708e12 −0.369824
\(206\) 5.48729e12 1.03060
\(207\) −6.17780e11 −0.112979
\(208\) −1.88216e13 −3.35204
\(209\) −1.62081e12 −0.281144
\(210\) −5.94357e12 −1.00425
\(211\) −7.63999e12 −1.25759 −0.628796 0.777571i \(-0.716452\pi\)
−0.628796 + 0.777571i \(0.716452\pi\)
\(212\) 2.83687e12 0.454981
\(213\) 5.96291e12 0.931902
\(214\) 1.08899e13 1.65862
\(215\) −2.18466e12 −0.324319
\(216\) 3.00498e12 0.434856
\(217\) −1.34759e13 −1.90120
\(218\) −6.67399e12 −0.918068
\(219\) −2.06203e11 −0.0276601
\(220\) −1.06242e13 −1.38987
\(221\) 1.07244e13 1.36841
\(222\) 1.56805e13 1.95173
\(223\) −1.31311e13 −1.59450 −0.797250 0.603649i \(-0.793713\pi\)
−0.797250 + 0.603649i \(0.793713\pi\)
\(224\) −1.31903e13 −1.56276
\(225\) −1.72432e12 −0.199349
\(226\) −1.39655e13 −1.57566
\(227\) 6.16894e11 0.0679311 0.0339656 0.999423i \(-0.489186\pi\)
0.0339656 + 0.999423i \(0.489186\pi\)
\(228\) 3.49611e12 0.375788
\(229\) −1.04390e13 −1.09538 −0.547690 0.836682i \(-0.684492\pi\)
−0.547690 + 0.836682i \(0.684492\pi\)
\(230\) −3.78303e12 −0.387558
\(231\) 8.54366e12 0.854630
\(232\) −4.46901e13 −4.36544
\(233\) −1.43228e13 −1.36637 −0.683187 0.730243i \(-0.739407\pi\)
−0.683187 + 0.730243i \(0.739407\pi\)
\(234\) −1.18673e13 −1.10577
\(235\) −1.53809e12 −0.139994
\(236\) −3.29853e12 −0.293295
\(237\) −9.29187e12 −0.807211
\(238\) 2.40464e13 2.04116
\(239\) −5.85963e11 −0.0486051 −0.0243025 0.999705i \(-0.507737\pi\)
−0.0243025 + 0.999705i \(0.507737\pi\)
\(240\) 8.22894e12 0.667086
\(241\) −5.63661e12 −0.446606 −0.223303 0.974749i \(-0.571684\pi\)
−0.223303 + 0.974749i \(0.571684\pi\)
\(242\) −1.23619e12 −0.0957413
\(243\) 8.47289e11 0.0641500
\(244\) 4.19646e13 3.10627
\(245\) 1.15107e13 0.833082
\(246\) −8.58265e12 −0.607405
\(247\) −7.67821e12 −0.531406
\(248\) 4.17211e13 2.82405
\(249\) 5.62888e12 0.372671
\(250\) −2.82149e13 −1.82729
\(251\) −1.30318e13 −0.825659 −0.412829 0.910808i \(-0.635459\pi\)
−0.412829 + 0.910808i \(0.635459\pi\)
\(252\) −1.84287e13 −1.14233
\(253\) 5.43797e12 0.329818
\(254\) 8.32157e12 0.493878
\(255\) −4.68878e12 −0.272326
\(256\) −3.13909e13 −1.78437
\(257\) 2.06055e12 0.114644 0.0573218 0.998356i \(-0.481744\pi\)
0.0573218 + 0.998356i \(0.481744\pi\)
\(258\) −9.78056e12 −0.532667
\(259\) −5.34786e13 −2.85122
\(260\) −5.03297e13 −2.62707
\(261\) −1.26009e13 −0.643990
\(262\) −6.93015e12 −0.346805
\(263\) −4.23790e12 −0.207680 −0.103840 0.994594i \(-0.533113\pi\)
−0.103840 + 0.994594i \(0.533113\pi\)
\(264\) −2.64511e13 −1.26947
\(265\) 2.72397e12 0.128041
\(266\) −1.72162e13 −0.792661
\(267\) −5.25566e12 −0.237036
\(268\) 1.01590e13 0.448857
\(269\) 1.56676e13 0.678211 0.339105 0.940748i \(-0.389876\pi\)
0.339105 + 0.940748i \(0.389876\pi\)
\(270\) 5.18845e12 0.220058
\(271\) 4.51306e13 1.87560 0.937800 0.347176i \(-0.112859\pi\)
0.937800 + 0.347176i \(0.112859\pi\)
\(272\) −3.32926e13 −1.35587
\(273\) 4.04735e13 1.61539
\(274\) 1.07153e13 0.419155
\(275\) 1.51782e13 0.581958
\(276\) −1.17297e13 −0.440847
\(277\) −2.57908e13 −0.950226 −0.475113 0.879925i \(-0.657593\pi\)
−0.475113 + 0.879925i \(0.657593\pi\)
\(278\) −1.49929e13 −0.541553
\(279\) 1.17638e13 0.416604
\(280\) −6.27576e13 −2.17920
\(281\) −4.46980e13 −1.52196 −0.760980 0.648775i \(-0.775282\pi\)
−0.760980 + 0.648775i \(0.775282\pi\)
\(282\) −6.88594e12 −0.229929
\(283\) 3.44164e13 1.12704 0.563521 0.826102i \(-0.309446\pi\)
0.563521 + 0.826102i \(0.309446\pi\)
\(284\) 1.13217e14 3.63632
\(285\) 3.35696e12 0.105755
\(286\) 1.04461e14 3.22805
\(287\) 2.92712e13 0.887342
\(288\) 1.15145e13 0.342442
\(289\) −1.53021e13 −0.446490
\(290\) −7.71628e13 −2.20912
\(291\) 2.09903e13 0.589668
\(292\) −3.91516e12 −0.107930
\(293\) 2.91846e13 0.789554 0.394777 0.918777i \(-0.370822\pi\)
0.394777 + 0.918777i \(0.370822\pi\)
\(294\) 5.15326e13 1.36827
\(295\) −3.16725e12 −0.0825394
\(296\) 1.65569e14 4.23521
\(297\) −7.45820e12 −0.187272
\(298\) −1.07469e14 −2.64907
\(299\) 2.57610e13 0.623407
\(300\) −3.27396e13 −0.777868
\(301\) 3.33567e13 0.778158
\(302\) 1.09662e13 0.251198
\(303\) −9.55854e12 −0.215009
\(304\) 2.38360e13 0.526537
\(305\) 4.02944e13 0.874169
\(306\) −2.09914e13 −0.447273
\(307\) −1.28328e13 −0.268572 −0.134286 0.990943i \(-0.542874\pi\)
−0.134286 + 0.990943i \(0.542874\pi\)
\(308\) 1.62218e14 3.33480
\(309\) 1.63368e13 0.329911
\(310\) 7.20365e13 1.42910
\(311\) −7.63079e13 −1.48726 −0.743631 0.668590i \(-0.766898\pi\)
−0.743631 + 0.668590i \(0.766898\pi\)
\(312\) −1.25306e14 −2.39950
\(313\) −4.98603e13 −0.938126 −0.469063 0.883165i \(-0.655408\pi\)
−0.469063 + 0.883165i \(0.655408\pi\)
\(314\) −8.05490e13 −1.48918
\(315\) −1.76953e13 −0.321476
\(316\) −1.76424e14 −3.14977
\(317\) 9.01966e13 1.58257 0.791287 0.611444i \(-0.209411\pi\)
0.791287 + 0.611444i \(0.209411\pi\)
\(318\) 1.21950e13 0.210297
\(319\) 1.10919e14 1.87999
\(320\) 1.15681e12 0.0192724
\(321\) 3.24215e13 0.530952
\(322\) 5.77617e13 0.929892
\(323\) −1.35816e13 −0.214949
\(324\) 1.60874e13 0.250316
\(325\) 7.19032e13 1.09999
\(326\) −2.09595e14 −3.15271
\(327\) −1.98699e13 −0.293889
\(328\) −9.06235e13 −1.31806
\(329\) 2.34846e13 0.335896
\(330\) −4.56710e13 −0.642412
\(331\) 4.61834e13 0.638899 0.319449 0.947603i \(-0.396502\pi\)
0.319449 + 0.947603i \(0.396502\pi\)
\(332\) 1.06875e14 1.45417
\(333\) 4.66842e13 0.624779
\(334\) −2.43189e13 −0.320138
\(335\) 9.75464e12 0.126318
\(336\) −1.25645e14 −1.60058
\(337\) 1.14155e14 1.43065 0.715323 0.698794i \(-0.246279\pi\)
0.715323 + 0.698794i \(0.246279\pi\)
\(338\) 3.48582e14 4.29797
\(339\) −4.15784e13 −0.504393
\(340\) −8.90255e13 −1.06263
\(341\) −1.03550e14 −1.21619
\(342\) 1.50289e13 0.173693
\(343\) −4.20003e13 −0.477678
\(344\) −1.03272e14 −1.15588
\(345\) −1.12629e13 −0.124064
\(346\) −2.68182e14 −2.90744
\(347\) 1.57017e14 1.67546 0.837728 0.546087i \(-0.183883\pi\)
0.837728 + 0.546087i \(0.183883\pi\)
\(348\) −2.39252e14 −2.51287
\(349\) −4.41868e13 −0.456828 −0.228414 0.973564i \(-0.573354\pi\)
−0.228414 + 0.973564i \(0.573354\pi\)
\(350\) 1.61222e14 1.64078
\(351\) −3.53314e13 −0.353974
\(352\) −1.01356e14 −0.999687
\(353\) 2.97334e12 0.0288724 0.0144362 0.999896i \(-0.495405\pi\)
0.0144362 + 0.999896i \(0.495405\pi\)
\(354\) −1.41796e13 −0.135564
\(355\) 1.08711e14 1.02334
\(356\) −9.97886e13 −0.924924
\(357\) 7.15913e13 0.653409
\(358\) −3.73705e14 −3.35870
\(359\) 5.01767e13 0.444101 0.222051 0.975035i \(-0.428725\pi\)
0.222051 + 0.975035i \(0.428725\pi\)
\(360\) 5.47844e13 0.477522
\(361\) −1.06766e14 −0.916527
\(362\) 3.60340e14 3.04660
\(363\) −3.68039e12 −0.0306484
\(364\) 7.68466e14 6.30329
\(365\) −3.75934e12 −0.0303739
\(366\) 1.80395e14 1.43575
\(367\) −2.52774e14 −1.98184 −0.990922 0.134440i \(-0.957076\pi\)
−0.990922 + 0.134440i \(0.957076\pi\)
\(368\) −7.99718e13 −0.617694
\(369\) −2.55524e13 −0.194440
\(370\) 2.85875e14 2.14322
\(371\) −4.15913e13 −0.307217
\(372\) 2.23358e14 1.62560
\(373\) 2.51671e14 1.80483 0.902413 0.430873i \(-0.141794\pi\)
0.902413 + 0.430873i \(0.141794\pi\)
\(374\) 1.84775e14 1.30572
\(375\) −8.40017e13 −0.584947
\(376\) −7.27080e13 −0.498941
\(377\) 5.25450e14 3.55348
\(378\) −7.92205e13 −0.527999
\(379\) −9.29507e12 −0.0610572 −0.0305286 0.999534i \(-0.509719\pi\)
−0.0305286 + 0.999534i \(0.509719\pi\)
\(380\) 6.37383e13 0.412658
\(381\) 2.47751e13 0.158098
\(382\) 5.08246e14 3.19688
\(383\) −1.92786e13 −0.119532 −0.0597658 0.998212i \(-0.519035\pi\)
−0.0597658 + 0.998212i \(0.519035\pi\)
\(384\) −9.18652e13 −0.561474
\(385\) 1.55761e14 0.938482
\(386\) 1.80187e13 0.107027
\(387\) −2.91188e13 −0.170515
\(388\) 3.98541e14 2.30091
\(389\) −7.46505e13 −0.424923 −0.212461 0.977169i \(-0.568148\pi\)
−0.212461 + 0.977169i \(0.568148\pi\)
\(390\) −2.16355e14 −1.21426
\(391\) 4.55673e13 0.252163
\(392\) 5.44128e14 2.96912
\(393\) −2.06325e13 −0.111018
\(394\) 2.63973e14 1.40065
\(395\) −1.69402e14 −0.886410
\(396\) −1.41608e14 −0.730743
\(397\) 2.36210e14 1.20213 0.601064 0.799201i \(-0.294744\pi\)
0.601064 + 0.799201i \(0.294744\pi\)
\(398\) −4.07990e14 −2.04782
\(399\) −5.12562e13 −0.253744
\(400\) −2.23214e14 −1.08991
\(401\) 2.20343e14 1.06122 0.530609 0.847616i \(-0.321963\pi\)
0.530609 + 0.847616i \(0.321963\pi\)
\(402\) 4.36709e13 0.207467
\(403\) −4.90542e14 −2.29878
\(404\) −1.81487e14 −0.838974
\(405\) 1.54471e13 0.0704441
\(406\) 1.17817e15 5.30048
\(407\) −4.10935e14 −1.82391
\(408\) −2.21646e14 −0.970575
\(409\) 3.88815e13 0.167983 0.0839915 0.996466i \(-0.473233\pi\)
0.0839915 + 0.996466i \(0.473233\pi\)
\(410\) −1.56472e14 −0.667000
\(411\) 3.19016e13 0.134178
\(412\) 3.10186e14 1.28732
\(413\) 4.83596e13 0.198042
\(414\) −5.04232e13 −0.203764
\(415\) 1.02621e14 0.409235
\(416\) −4.80148e14 −1.88957
\(417\) −4.46372e13 −0.173360
\(418\) −1.32291e14 −0.507061
\(419\) −3.55079e14 −1.34322 −0.671611 0.740904i \(-0.734398\pi\)
−0.671611 + 0.740904i \(0.734398\pi\)
\(420\) −3.35978e14 −1.25441
\(421\) −1.71850e14 −0.633282 −0.316641 0.948545i \(-0.602555\pi\)
−0.316641 + 0.948545i \(0.602555\pi\)
\(422\) −6.23576e14 −2.26815
\(423\) −2.05009e13 −0.0736039
\(424\) 1.28766e14 0.456341
\(425\) 1.27186e14 0.444937
\(426\) 4.86692e14 1.68074
\(427\) −6.15240e14 −2.09745
\(428\) 6.15584e14 2.07179
\(429\) 3.11002e14 1.03335
\(430\) −1.78312e14 −0.584929
\(431\) −6.14914e13 −0.199154 −0.0995771 0.995030i \(-0.531749\pi\)
−0.0995771 + 0.995030i \(0.531749\pi\)
\(432\) 1.09682e14 0.350731
\(433\) −7.82609e13 −0.247094 −0.123547 0.992339i \(-0.539427\pi\)
−0.123547 + 0.992339i \(0.539427\pi\)
\(434\) −1.09990e15 −3.42893
\(435\) −2.29730e14 −0.707175
\(436\) −3.77268e14 −1.14676
\(437\) −3.26241e13 −0.0979245
\(438\) −1.68303e13 −0.0498867
\(439\) 3.28708e14 0.962179 0.481089 0.876672i \(-0.340241\pi\)
0.481089 + 0.876672i \(0.340241\pi\)
\(440\) −4.82236e14 −1.39402
\(441\) 1.53423e14 0.438005
\(442\) 8.75327e14 2.46801
\(443\) 4.77298e14 1.32914 0.664568 0.747228i \(-0.268616\pi\)
0.664568 + 0.747228i \(0.268616\pi\)
\(444\) 8.86388e14 2.43791
\(445\) −9.58170e13 −0.260293
\(446\) −1.07176e15 −2.87578
\(447\) −3.19958e14 −0.848011
\(448\) −1.76629e13 −0.0462415
\(449\) 5.08724e14 1.31561 0.657806 0.753187i \(-0.271485\pi\)
0.657806 + 0.753187i \(0.271485\pi\)
\(450\) −1.40739e14 −0.359539
\(451\) 2.24923e14 0.567627
\(452\) −7.89445e14 −1.96816
\(453\) 3.26486e13 0.0804127
\(454\) 5.03509e13 0.122518
\(455\) 7.37881e14 1.77388
\(456\) 1.58689e14 0.376912
\(457\) 3.88523e14 0.911754 0.455877 0.890043i \(-0.349326\pi\)
0.455877 + 0.890043i \(0.349326\pi\)
\(458\) −8.52032e14 −1.97559
\(459\) −6.24958e13 −0.143179
\(460\) −2.13847e14 −0.484101
\(461\) 6.82062e12 0.0152570 0.00762849 0.999971i \(-0.497572\pi\)
0.00762849 + 0.999971i \(0.497572\pi\)
\(462\) 6.97333e14 1.54138
\(463\) 1.72115e14 0.375943 0.187972 0.982174i \(-0.439809\pi\)
0.187972 + 0.982174i \(0.439809\pi\)
\(464\) −1.63119e15 −3.52092
\(465\) 2.14468e14 0.457479
\(466\) −1.16902e15 −2.46434
\(467\) 7.08424e14 1.47588 0.737939 0.674868i \(-0.235799\pi\)
0.737939 + 0.674868i \(0.235799\pi\)
\(468\) −6.70834e14 −1.38122
\(469\) −1.48940e14 −0.303083
\(470\) −1.25539e14 −0.252488
\(471\) −2.39812e14 −0.476711
\(472\) −1.49721e14 −0.294172
\(473\) 2.56317e14 0.497783
\(474\) −7.58402e14 −1.45586
\(475\) −9.10593e13 −0.172786
\(476\) 1.35930e15 2.54963
\(477\) 3.63072e13 0.0673195
\(478\) −4.78263e13 −0.0876623
\(479\) −8.72334e13 −0.158066 −0.0790328 0.996872i \(-0.525183\pi\)
−0.0790328 + 0.996872i \(0.525183\pi\)
\(480\) 2.09924e14 0.376041
\(481\) −1.94670e15 −3.44748
\(482\) −4.60060e14 −0.805482
\(483\) 1.71969e14 0.297674
\(484\) −6.98792e13 −0.119591
\(485\) 3.82679e14 0.647523
\(486\) 6.91557e13 0.115699
\(487\) 4.43086e14 0.732958 0.366479 0.930426i \(-0.380563\pi\)
0.366479 + 0.930426i \(0.380563\pi\)
\(488\) 1.90478e15 3.11555
\(489\) −6.24010e14 −1.00924
\(490\) 9.39502e14 1.50252
\(491\) −2.84084e14 −0.449260 −0.224630 0.974444i \(-0.572117\pi\)
−0.224630 + 0.974444i \(0.572117\pi\)
\(492\) −4.85161e14 −0.758713
\(493\) 9.29440e14 1.43735
\(494\) −6.26695e14 −0.958425
\(495\) −1.35972e14 −0.205647
\(496\) 1.52282e15 2.27772
\(497\) −1.65987e15 −2.45535
\(498\) 4.59429e14 0.672136
\(499\) 9.50905e14 1.37589 0.687946 0.725762i \(-0.258513\pi\)
0.687946 + 0.725762i \(0.258513\pi\)
\(500\) −1.59493e15 −2.28248
\(501\) −7.24024e13 −0.102482
\(502\) −1.06366e15 −1.48913
\(503\) −1.16004e15 −1.60638 −0.803188 0.595725i \(-0.796865\pi\)
−0.803188 + 0.595725i \(0.796865\pi\)
\(504\) −8.36483e14 −1.14575
\(505\) −1.74264e14 −0.236105
\(506\) 4.43847e14 0.594847
\(507\) 1.03780e15 1.37585
\(508\) 4.70402e14 0.616906
\(509\) 7.97015e14 1.03400 0.516998 0.855987i \(-0.327050\pi\)
0.516998 + 0.855987i \(0.327050\pi\)
\(510\) −3.82698e14 −0.491157
\(511\) 5.74000e13 0.0728781
\(512\) −1.78788e15 −2.24572
\(513\) 4.47442e13 0.0556021
\(514\) 1.68182e14 0.206767
\(515\) 2.97841e14 0.362280
\(516\) −5.52876e14 −0.665357
\(517\) 1.80458e14 0.214871
\(518\) −4.36492e15 −5.14236
\(519\) −7.98434e14 −0.930719
\(520\) −2.28447e15 −2.63492
\(521\) 1.25262e15 1.42960 0.714798 0.699331i \(-0.246519\pi\)
0.714798 + 0.699331i \(0.246519\pi\)
\(522\) −1.02849e15 −1.16148
\(523\) −8.79438e14 −0.982756 −0.491378 0.870946i \(-0.663507\pi\)
−0.491378 + 0.870946i \(0.663507\pi\)
\(524\) −3.91748e14 −0.433196
\(525\) 4.79993e14 0.525241
\(526\) −3.45897e14 −0.374564
\(527\) −8.67692e14 −0.929838
\(528\) −9.65466e14 −1.02388
\(529\) −8.43353e14 −0.885122
\(530\) 2.22330e14 0.230930
\(531\) −4.22156e13 −0.0433963
\(532\) −9.73197e14 −0.990118
\(533\) 1.06552e15 1.07291
\(534\) −4.28966e14 −0.427510
\(535\) 5.91084e14 0.583047
\(536\) 4.61117e14 0.450200
\(537\) −1.11260e15 −1.07518
\(538\) 1.27879e15 1.22320
\(539\) −1.35050e15 −1.27866
\(540\) 2.93293e14 0.274875
\(541\) −4.33896e14 −0.402533 −0.201266 0.979537i \(-0.564506\pi\)
−0.201266 + 0.979537i \(0.564506\pi\)
\(542\) 3.68356e15 3.38276
\(543\) 1.07281e15 0.975267
\(544\) −8.49308e14 −0.764313
\(545\) −3.62252e14 −0.322723
\(546\) 3.30345e15 2.91345
\(547\) −1.45755e15 −1.27260 −0.636302 0.771440i \(-0.719537\pi\)
−0.636302 + 0.771440i \(0.719537\pi\)
\(548\) 6.05713e14 0.523569
\(549\) 5.37075e14 0.459607
\(550\) 1.23885e15 1.04960
\(551\) −6.65438e14 −0.558179
\(552\) −5.32414e14 −0.442165
\(553\) 2.58654e15 2.12682
\(554\) −2.10505e15 −1.71379
\(555\) 8.51110e14 0.686079
\(556\) −8.47522e14 −0.676456
\(557\) −1.26701e15 −1.00133 −0.500666 0.865641i \(-0.666912\pi\)
−0.500666 + 0.865641i \(0.666912\pi\)
\(558\) 9.60158e14 0.751371
\(559\) 1.21424e15 0.940889
\(560\) −2.29066e15 −1.75762
\(561\) 5.50115e14 0.417982
\(562\) −3.64825e15 −2.74495
\(563\) 2.07754e15 1.54794 0.773969 0.633224i \(-0.218269\pi\)
0.773969 + 0.633224i \(0.218269\pi\)
\(564\) −3.89249e14 −0.287205
\(565\) −7.58025e14 −0.553882
\(566\) 2.80906e15 2.03269
\(567\) −2.35856e14 −0.169021
\(568\) 5.13895e15 3.64719
\(569\) 1.41887e15 0.997301 0.498650 0.866803i \(-0.333829\pi\)
0.498650 + 0.866803i \(0.333829\pi\)
\(570\) 2.73995e14 0.190735
\(571\) 1.21211e15 0.835686 0.417843 0.908519i \(-0.362786\pi\)
0.417843 + 0.908519i \(0.362786\pi\)
\(572\) 5.90497e15 4.03218
\(573\) 1.51316e15 1.02337
\(574\) 2.38912e15 1.60038
\(575\) 3.05511e14 0.202700
\(576\) 1.54188e13 0.0101328
\(577\) −1.12144e15 −0.729974 −0.364987 0.931013i \(-0.618927\pi\)
−0.364987 + 0.931013i \(0.618927\pi\)
\(578\) −1.24895e15 −0.805273
\(579\) 5.36457e13 0.0342612
\(580\) −4.36186e15 −2.75942
\(581\) −1.56689e15 −0.981905
\(582\) 1.71323e15 1.06350
\(583\) −3.19591e14 −0.196525
\(584\) −1.77710e14 −0.108253
\(585\) −6.44135e14 −0.388704
\(586\) 2.38205e15 1.42401
\(587\) −4.87270e14 −0.288576 −0.144288 0.989536i \(-0.546089\pi\)
−0.144288 + 0.989536i \(0.546089\pi\)
\(588\) 2.91304e15 1.70911
\(589\) 6.21229e14 0.361092
\(590\) −2.58511e14 −0.148865
\(591\) 7.85904e14 0.448372
\(592\) 6.04328e15 3.41589
\(593\) 1.61540e14 0.0904648 0.0452324 0.998976i \(-0.485597\pi\)
0.0452324 + 0.998976i \(0.485597\pi\)
\(594\) −6.08738e14 −0.337758
\(595\) 1.30520e15 0.717518
\(596\) −6.07501e15 −3.30897
\(597\) −1.21467e15 −0.655542
\(598\) 2.10261e15 1.12435
\(599\) −3.16140e15 −1.67507 −0.837533 0.546386i \(-0.816003\pi\)
−0.837533 + 0.546386i \(0.816003\pi\)
\(600\) −1.48605e15 −0.780194
\(601\) 3.23614e15 1.68352 0.841759 0.539853i \(-0.181520\pi\)
0.841759 + 0.539853i \(0.181520\pi\)
\(602\) 2.72258e15 1.40346
\(603\) 1.30018e14 0.0664135
\(604\) 6.19896e14 0.313773
\(605\) −6.70980e13 −0.0336554
\(606\) −7.80168e14 −0.387783
\(607\) −2.17513e15 −1.07139 −0.535695 0.844412i \(-0.679950\pi\)
−0.535695 + 0.844412i \(0.679950\pi\)
\(608\) 6.08067e14 0.296812
\(609\) 3.50767e15 1.69677
\(610\) 3.28883e15 1.57662
\(611\) 8.54875e14 0.406140
\(612\) −1.18660e15 −0.558691
\(613\) 2.25791e15 1.05359 0.526797 0.849991i \(-0.323393\pi\)
0.526797 + 0.849991i \(0.323393\pi\)
\(614\) −1.04741e15 −0.484387
\(615\) −4.65851e14 −0.213518
\(616\) 7.36309e15 3.34477
\(617\) −3.28604e14 −0.147946 −0.0739732 0.997260i \(-0.523568\pi\)
−0.0739732 + 0.997260i \(0.523568\pi\)
\(618\) 1.33341e15 0.595015
\(619\) 4.52186e14 0.199995 0.0999974 0.994988i \(-0.468117\pi\)
0.0999974 + 0.994988i \(0.468117\pi\)
\(620\) 4.07208e15 1.78510
\(621\) −1.50121e14 −0.0652283
\(622\) −6.22825e15 −2.68237
\(623\) 1.46300e15 0.624538
\(624\) −4.57366e15 −1.93530
\(625\) −1.05597e14 −0.0442904
\(626\) −4.06960e15 −1.69197
\(627\) −3.93858e14 −0.162319
\(628\) −4.55328e15 −1.86014
\(629\) −3.44341e15 −1.39447
\(630\) −1.44429e15 −0.579803
\(631\) −2.72204e15 −1.08326 −0.541630 0.840617i \(-0.682192\pi\)
−0.541630 + 0.840617i \(0.682192\pi\)
\(632\) −8.00790e15 −3.15918
\(633\) −1.85652e15 −0.726071
\(634\) 7.36184e15 2.85427
\(635\) 4.51680e14 0.173610
\(636\) 6.89360e14 0.262683
\(637\) −6.39766e15 −2.41688
\(638\) 9.05318e15 3.39069
\(639\) 1.44899e15 0.538034
\(640\) −1.67482e15 −0.616563
\(641\) −2.25361e15 −0.822544 −0.411272 0.911513i \(-0.634915\pi\)
−0.411272 + 0.911513i \(0.634915\pi\)
\(642\) 2.64624e15 0.957606
\(643\) 2.05762e15 0.738252 0.369126 0.929379i \(-0.379657\pi\)
0.369126 + 0.929379i \(0.379657\pi\)
\(644\) 3.26516e15 1.16153
\(645\) −5.30872e14 −0.187245
\(646\) −1.10853e15 −0.387674
\(647\) −4.94916e15 −1.71616 −0.858081 0.513515i \(-0.828343\pi\)
−0.858081 + 0.513515i \(0.828343\pi\)
\(648\) 7.30209e14 0.251064
\(649\) 3.71600e14 0.126686
\(650\) 5.86874e15 1.98391
\(651\) −3.27463e15 −1.09766
\(652\) −1.18480e16 −3.93807
\(653\) 3.07508e14 0.101352 0.0506762 0.998715i \(-0.483862\pi\)
0.0506762 + 0.998715i \(0.483862\pi\)
\(654\) −1.62178e15 −0.530047
\(655\) −3.76156e14 −0.121910
\(656\) −3.30776e15 −1.06307
\(657\) −5.01074e13 −0.0159695
\(658\) 1.91681e15 0.605810
\(659\) −5.82789e15 −1.82659 −0.913297 0.407295i \(-0.866472\pi\)
−0.913297 + 0.407295i \(0.866472\pi\)
\(660\) −2.58169e15 −0.802440
\(661\) 1.33207e15 0.410601 0.205300 0.978699i \(-0.434183\pi\)
0.205300 + 0.978699i \(0.434183\pi\)
\(662\) 3.76949e15 1.15229
\(663\) 2.60603e15 0.790052
\(664\) 4.85107e15 1.45852
\(665\) −9.34464e14 −0.278640
\(666\) 3.81036e15 1.12683
\(667\) 2.23260e15 0.654815
\(668\) −1.37470e15 −0.399887
\(669\) −3.19086e15 −0.920585
\(670\) 7.96174e14 0.227822
\(671\) −4.72757e15 −1.34173
\(672\) −3.20525e15 −0.902258
\(673\) −1.52829e15 −0.426700 −0.213350 0.976976i \(-0.568437\pi\)
−0.213350 + 0.976976i \(0.568437\pi\)
\(674\) 9.31737e15 2.58026
\(675\) −4.19011e14 −0.115094
\(676\) 1.97047e16 5.36861
\(677\) −5.12640e15 −1.38540 −0.692700 0.721226i \(-0.743579\pi\)
−0.692700 + 0.721226i \(0.743579\pi\)
\(678\) −3.39363e15 −0.909705
\(679\) −5.84299e15 −1.55364
\(680\) −4.04088e15 −1.06580
\(681\) 1.49905e14 0.0392200
\(682\) −8.45173e15 −2.19347
\(683\) 1.46991e14 0.0378423 0.0189211 0.999821i \(-0.493977\pi\)
0.0189211 + 0.999821i \(0.493977\pi\)
\(684\) 8.49554e14 0.216961
\(685\) 5.81605e14 0.147343
\(686\) −3.42807e15 −0.861522
\(687\) −2.53668e15 −0.632417
\(688\) −3.76944e15 −0.932267
\(689\) −1.51399e15 −0.371463
\(690\) −9.19276e14 −0.223757
\(691\) −1.07587e15 −0.259796 −0.129898 0.991527i \(-0.541465\pi\)
−0.129898 + 0.991527i \(0.541465\pi\)
\(692\) −1.51598e16 −3.63170
\(693\) 2.07611e15 0.493421
\(694\) 1.28157e16 3.02179
\(695\) −8.13791e14 −0.190369
\(696\) −1.08597e16 −2.52039
\(697\) 1.88474e15 0.433981
\(698\) −3.60653e15 −0.823919
\(699\) −3.48044e15 −0.788877
\(700\) 9.11358e15 2.04951
\(701\) 2.38968e15 0.533201 0.266600 0.963807i \(-0.414100\pi\)
0.266600 + 0.963807i \(0.414100\pi\)
\(702\) −2.88375e15 −0.638415
\(703\) 2.46533e15 0.541528
\(704\) −1.35723e14 −0.0295804
\(705\) −3.73757e14 −0.0808255
\(706\) 2.42684e14 0.0520732
\(707\) 2.66077e15 0.566501
\(708\) −8.01543e14 −0.169334
\(709\) −7.42029e15 −1.55549 −0.777743 0.628582i \(-0.783636\pi\)
−0.777743 + 0.628582i \(0.783636\pi\)
\(710\) 8.87300e15 1.84565
\(711\) −2.25792e15 −0.466044
\(712\) −4.52942e15 −0.927690
\(713\) −2.08428e15 −0.423607
\(714\) 5.84328e15 1.17846
\(715\) 5.66995e15 1.13474
\(716\) −2.11248e16 −4.19538
\(717\) −1.42389e14 −0.0280622
\(718\) 4.09542e15 0.800965
\(719\) −7.33878e15 −1.42434 −0.712172 0.702005i \(-0.752288\pi\)
−0.712172 + 0.702005i \(0.752288\pi\)
\(720\) 1.99963e15 0.385142
\(721\) −4.54762e15 −0.869241
\(722\) −8.71427e15 −1.65301
\(723\) −1.36970e15 −0.257848
\(724\) 2.03693e16 3.80552
\(725\) 6.23154e15 1.15541
\(726\) −3.00393e14 −0.0552763
\(727\) −5.82793e15 −1.06433 −0.532163 0.846642i \(-0.678621\pi\)
−0.532163 + 0.846642i \(0.678621\pi\)
\(728\) 3.48808e16 6.32214
\(729\) 2.05891e14 0.0370370
\(730\) −3.06837e14 −0.0547813
\(731\) 2.14780e15 0.380581
\(732\) 1.01974e16 1.79340
\(733\) −6.11773e15 −1.06787 −0.533935 0.845525i \(-0.679287\pi\)
−0.533935 + 0.845525i \(0.679287\pi\)
\(734\) −2.06314e16 −3.57438
\(735\) 2.79710e15 0.480980
\(736\) −2.04011e15 −0.348198
\(737\) −1.14447e15 −0.193880
\(738\) −2.08558e15 −0.350685
\(739\) 1.98400e13 0.00331128 0.00165564 0.999999i \(-0.499473\pi\)
0.00165564 + 0.999999i \(0.499473\pi\)
\(740\) 1.61599e16 2.67710
\(741\) −1.86581e15 −0.306808
\(742\) −3.39468e15 −0.554086
\(743\) −1.16649e15 −0.188992 −0.0944959 0.995525i \(-0.530124\pi\)
−0.0944959 + 0.995525i \(0.530124\pi\)
\(744\) 1.01382e16 1.63046
\(745\) −5.83322e15 −0.931213
\(746\) 2.05414e16 3.25512
\(747\) 1.36782e15 0.215162
\(748\) 1.04450e16 1.63098
\(749\) −9.02505e15 −1.39894
\(750\) −6.85622e15 −1.05499
\(751\) 7.85668e15 1.20010 0.600052 0.799961i \(-0.295146\pi\)
0.600052 + 0.799961i \(0.295146\pi\)
\(752\) −2.65385e15 −0.402418
\(753\) −3.16674e15 −0.476694
\(754\) 4.28872e16 6.40893
\(755\) 5.95224e14 0.0883024
\(756\) −4.47818e15 −0.659526
\(757\) −9.69063e15 −1.41685 −0.708427 0.705784i \(-0.750595\pi\)
−0.708427 + 0.705784i \(0.750595\pi\)
\(758\) −7.58663e14 −0.110121
\(759\) 1.32143e15 0.190420
\(760\) 2.89309e15 0.413892
\(761\) −3.74357e15 −0.531704 −0.265852 0.964014i \(-0.585653\pi\)
−0.265852 + 0.964014i \(0.585653\pi\)
\(762\) 2.02214e15 0.285141
\(763\) 5.53110e15 0.774331
\(764\) 2.87302e16 3.99324
\(765\) −1.13937e15 −0.157227
\(766\) −1.57352e15 −0.215583
\(767\) 1.76036e15 0.239457
\(768\) −7.62799e15 −1.03020
\(769\) 9.86595e15 1.32295 0.661476 0.749967i \(-0.269930\pi\)
0.661476 + 0.749967i \(0.269930\pi\)
\(770\) 1.27132e16 1.69261
\(771\) 5.00713e14 0.0661896
\(772\) 1.01856e15 0.133688
\(773\) −1.10280e16 −1.43717 −0.718585 0.695439i \(-0.755210\pi\)
−0.718585 + 0.695439i \(0.755210\pi\)
\(774\) −2.37668e15 −0.307535
\(775\) −5.81755e15 −0.747447
\(776\) 1.80899e16 2.30779
\(777\) −1.29953e16 −1.64615
\(778\) −6.09297e15 −0.766375
\(779\) −1.34939e15 −0.168531
\(780\) −1.22301e16 −1.51674
\(781\) −1.27546e16 −1.57068
\(782\) 3.71920e15 0.454791
\(783\) −3.06202e15 −0.371808
\(784\) 1.98607e16 2.39473
\(785\) −4.37206e15 −0.523483
\(786\) −1.68403e15 −0.200228
\(787\) −8.72132e15 −1.02972 −0.514862 0.857273i \(-0.672157\pi\)
−0.514862 + 0.857273i \(0.672157\pi\)
\(788\) 1.49219e16 1.74956
\(789\) −1.02981e15 −0.119904
\(790\) −1.38266e16 −1.59870
\(791\) 1.15740e16 1.32896
\(792\) −6.42762e15 −0.732928
\(793\) −2.23957e16 −2.53607
\(794\) 1.92795e16 2.16811
\(795\) 6.61924e14 0.0739246
\(796\) −2.30629e16 −2.55795
\(797\) 4.87124e15 0.536561 0.268280 0.963341i \(-0.413545\pi\)
0.268280 + 0.963341i \(0.413545\pi\)
\(798\) −4.18353e15 −0.457643
\(799\) 1.51214e15 0.164280
\(800\) −5.69429e15 −0.614391
\(801\) −1.27712e15 −0.136853
\(802\) 1.79844e16 1.91398
\(803\) 4.41067e14 0.0466197
\(804\) 2.46863e15 0.259148
\(805\) 3.13520e15 0.326880
\(806\) −4.00380e16 −4.14600
\(807\) 3.80723e15 0.391565
\(808\) −8.23773e15 −0.841482
\(809\) 1.27539e16 1.29398 0.646989 0.762499i \(-0.276028\pi\)
0.646989 + 0.762499i \(0.276028\pi\)
\(810\) 1.26079e15 0.127050
\(811\) 6.39555e15 0.640123 0.320061 0.947397i \(-0.396296\pi\)
0.320061 + 0.947397i \(0.396296\pi\)
\(812\) 6.65997e16 6.62086
\(813\) 1.09667e16 1.08288
\(814\) −3.35405e16 −3.28954
\(815\) −1.13765e16 −1.10826
\(816\) −8.09009e15 −0.782812
\(817\) −1.53773e15 −0.147794
\(818\) 3.17351e15 0.302968
\(819\) 9.83506e15 0.932643
\(820\) −8.84507e15 −0.833154
\(821\) 3.80524e15 0.356036 0.178018 0.984027i \(-0.443031\pi\)
0.178018 + 0.984027i \(0.443031\pi\)
\(822\) 2.60381e15 0.241999
\(823\) 8.80374e15 0.812771 0.406386 0.913702i \(-0.366789\pi\)
0.406386 + 0.913702i \(0.366789\pi\)
\(824\) 1.40794e16 1.29117
\(825\) 3.68831e15 0.335994
\(826\) 3.94711e15 0.357181
\(827\) −1.45550e16 −1.30838 −0.654188 0.756332i \(-0.726990\pi\)
−0.654188 + 0.756332i \(0.726990\pi\)
\(828\) −2.85032e15 −0.254523
\(829\) 1.55479e16 1.37918 0.689590 0.724200i \(-0.257791\pi\)
0.689590 + 0.724200i \(0.257791\pi\)
\(830\) 8.37595e15 0.738082
\(831\) −6.26717e15 −0.548613
\(832\) −6.42955e14 −0.0559117
\(833\) −1.13165e16 −0.977605
\(834\) −3.64329e15 −0.312666
\(835\) −1.31998e15 −0.112536
\(836\) −7.47814e15 −0.633372
\(837\) 2.85860e15 0.240526
\(838\) −2.89815e16 −2.42259
\(839\) 1.35558e15 0.112573 0.0562863 0.998415i \(-0.482074\pi\)
0.0562863 + 0.998415i \(0.482074\pi\)
\(840\) −1.52501e16 −1.25816
\(841\) 3.33380e16 2.73251
\(842\) −1.40264e16 −1.14216
\(843\) −1.08616e16 −0.878704
\(844\) −3.52495e16 −2.83315
\(845\) 1.89204e16 1.51084
\(846\) −1.67328e15 −0.132749
\(847\) 1.02450e15 0.0807516
\(848\) 4.69997e15 0.368059
\(849\) 8.36318e15 0.650698
\(850\) 1.03809e16 0.802473
\(851\) −8.27139e15 −0.635281
\(852\) 2.75118e16 2.09943
\(853\) 1.78055e15 0.135000 0.0675002 0.997719i \(-0.478498\pi\)
0.0675002 + 0.997719i \(0.478498\pi\)
\(854\) −5.02159e16 −3.78288
\(855\) 8.15741e14 0.0610575
\(856\) 2.79415e16 2.07799
\(857\) −1.27734e16 −0.943867 −0.471934 0.881634i \(-0.656444\pi\)
−0.471934 + 0.881634i \(0.656444\pi\)
\(858\) 2.53840e16 1.86372
\(859\) −8.71174e15 −0.635539 −0.317770 0.948168i \(-0.602934\pi\)
−0.317770 + 0.948168i \(0.602934\pi\)
\(860\) −1.00796e16 −0.730638
\(861\) 7.11291e15 0.512307
\(862\) −5.01893e15 −0.359187
\(863\) −7.36490e15 −0.523730 −0.261865 0.965104i \(-0.584338\pi\)
−0.261865 + 0.965104i \(0.584338\pi\)
\(864\) 2.79803e15 0.197709
\(865\) −1.45564e16 −1.02204
\(866\) −6.38765e15 −0.445649
\(867\) −3.71840e15 −0.257781
\(868\) −6.21751e16 −4.28310
\(869\) 1.98752e16 1.36052
\(870\) −1.87506e16 −1.27544
\(871\) −5.42165e15 −0.366464
\(872\) −1.71242e16 −1.15019
\(873\) 5.10065e15 0.340445
\(874\) −2.66278e15 −0.176613
\(875\) 2.33832e16 1.54120
\(876\) −9.51384e14 −0.0623137
\(877\) −5.63043e15 −0.366475 −0.183237 0.983069i \(-0.558658\pi\)
−0.183237 + 0.983069i \(0.558658\pi\)
\(878\) 2.68291e16 1.73535
\(879\) 7.09186e15 0.455849
\(880\) −1.76016e16 −1.12434
\(881\) −9.45939e15 −0.600476 −0.300238 0.953864i \(-0.597066\pi\)
−0.300238 + 0.953864i \(0.597066\pi\)
\(882\) 1.25224e16 0.789970
\(883\) 5.51462e15 0.345725 0.172863 0.984946i \(-0.444698\pi\)
0.172863 + 0.984946i \(0.444698\pi\)
\(884\) 4.94805e16 3.08281
\(885\) −7.69641e14 −0.0476541
\(886\) 3.89571e16 2.39718
\(887\) 4.65928e14 0.0284931 0.0142465 0.999899i \(-0.495465\pi\)
0.0142465 + 0.999899i \(0.495465\pi\)
\(888\) 4.02333e16 2.44520
\(889\) −6.89654e15 −0.416554
\(890\) −7.82058e15 −0.469455
\(891\) −1.81234e15 −0.108122
\(892\) −6.05845e16 −3.59215
\(893\) −1.08263e15 −0.0637963
\(894\) −2.61149e16 −1.52944
\(895\) −2.02840e16 −1.18067
\(896\) 2.55722e16 1.47936
\(897\) 6.25993e15 0.359924
\(898\) 4.15221e16 2.37279
\(899\) −4.25132e16 −2.41460
\(900\) −7.95571e15 −0.449102
\(901\) −2.67801e15 −0.150254
\(902\) 1.83582e16 1.02375
\(903\) 8.10569e15 0.449270
\(904\) −3.58330e16 −1.97404
\(905\) 1.95586e16 1.07095
\(906\) 2.66478e15 0.145029
\(907\) −1.01609e16 −0.549657 −0.274829 0.961493i \(-0.588621\pi\)
−0.274829 + 0.961493i \(0.588621\pi\)
\(908\) 2.84624e15 0.153038
\(909\) −2.32273e15 −0.124136
\(910\) 6.02259e16 3.19930
\(911\) 2.38551e16 1.25959 0.629796 0.776761i \(-0.283139\pi\)
0.629796 + 0.776761i \(0.283139\pi\)
\(912\) 5.79215e15 0.303996
\(913\) −1.20401e16 −0.628119
\(914\) 3.17112e16 1.64441
\(915\) 9.79153e15 0.504702
\(916\) −4.81637e16 −2.46771
\(917\) 5.74339e15 0.292508
\(918\) −5.10090e15 −0.258233
\(919\) −1.02830e16 −0.517469 −0.258734 0.965949i \(-0.583305\pi\)
−0.258734 + 0.965949i \(0.583305\pi\)
\(920\) −9.70656e15 −0.485548
\(921\) −3.11838e15 −0.155060
\(922\) 5.56698e14 0.0275169
\(923\) −6.04219e16 −2.96883
\(924\) 3.94189e16 1.92535
\(925\) −2.30868e16 −1.12094
\(926\) 1.40480e16 0.678037
\(927\) 3.96985e15 0.190474
\(928\) −4.16124e16 −1.98476
\(929\) 1.53630e16 0.728433 0.364217 0.931314i \(-0.381337\pi\)
0.364217 + 0.931314i \(0.381337\pi\)
\(930\) 1.75049e16 0.825092
\(931\) 8.10209e15 0.379642
\(932\) −6.60827e16 −3.07822
\(933\) −1.85428e16 −0.858671
\(934\) 5.78215e16 2.66184
\(935\) 1.00293e16 0.458992
\(936\) −3.04493e16 −1.38535
\(937\) −4.34110e16 −1.96351 −0.981753 0.190163i \(-0.939099\pi\)
−0.981753 + 0.190163i \(0.939099\pi\)
\(938\) −1.21565e16 −0.546629
\(939\) −1.21161e16 −0.541627
\(940\) −7.09648e15 −0.315384
\(941\) −1.63325e16 −0.721624 −0.360812 0.932639i \(-0.617500\pi\)
−0.360812 + 0.932639i \(0.617500\pi\)
\(942\) −1.95734e16 −0.859779
\(943\) 4.52731e15 0.197709
\(944\) −5.46482e15 −0.237263
\(945\) −4.29995e15 −0.185604
\(946\) 2.09205e16 0.897784
\(947\) 2.61485e16 1.11563 0.557817 0.829964i \(-0.311639\pi\)
0.557817 + 0.829964i \(0.311639\pi\)
\(948\) −4.28710e16 −1.81852
\(949\) 2.08945e15 0.0881185
\(950\) −7.43225e15 −0.311631
\(951\) 2.19178e16 0.913700
\(952\) 6.16987e16 2.55725
\(953\) −1.41902e16 −0.584761 −0.292380 0.956302i \(-0.594447\pi\)
−0.292380 + 0.956302i \(0.594447\pi\)
\(954\) 2.96339e15 0.121415
\(955\) 2.75867e16 1.12378
\(956\) −2.70353e15 −0.109499
\(957\) 2.69533e16 1.08541
\(958\) −7.11999e15 −0.285081
\(959\) −8.88032e15 −0.353530
\(960\) 2.81104e14 0.0111269
\(961\) 1.42803e16 0.562029
\(962\) −1.58890e17 −6.21775
\(963\) 7.87843e15 0.306546
\(964\) −2.60063e16 −1.00613
\(965\) 9.78026e14 0.0376227
\(966\) 1.40361e16 0.536874
\(967\) 1.42672e16 0.542618 0.271309 0.962492i \(-0.412544\pi\)
0.271309 + 0.962492i \(0.412544\pi\)
\(968\) −3.17183e15 −0.119949
\(969\) −3.30032e15 −0.124101
\(970\) 3.12343e16 1.16785
\(971\) 4.02883e16 1.49787 0.748933 0.662645i \(-0.230566\pi\)
0.748933 + 0.662645i \(0.230566\pi\)
\(972\) 3.90923e15 0.144520
\(973\) 1.24255e16 0.456765
\(974\) 3.61647e16 1.32194
\(975\) 1.74725e16 0.635081
\(976\) 6.95245e16 2.51283
\(977\) −3.48577e16 −1.25279 −0.626396 0.779505i \(-0.715471\pi\)
−0.626396 + 0.779505i \(0.715471\pi\)
\(978\) −5.09316e16 −1.82022
\(979\) 1.12418e16 0.399513
\(980\) 5.31082e16 1.87680
\(981\) −4.82838e15 −0.169677
\(982\) −2.31869e16 −0.810270
\(983\) −1.36013e15 −0.0472645 −0.0236323 0.999721i \(-0.507523\pi\)
−0.0236323 + 0.999721i \(0.507523\pi\)
\(984\) −2.20215e16 −0.760982
\(985\) 1.43280e16 0.492364
\(986\) 7.58608e16 2.59236
\(987\) 5.70675e15 0.193930
\(988\) −3.54259e16 −1.19717
\(989\) 5.15920e15 0.173382
\(990\) −1.10980e16 −0.370897
\(991\) −4.70157e16 −1.56256 −0.781282 0.624178i \(-0.785434\pi\)
−0.781282 + 0.624178i \(0.785434\pi\)
\(992\) 3.88479e16 1.28396
\(993\) 1.12226e16 0.368868
\(994\) −1.35479e17 −4.42839
\(995\) −2.21450e16 −0.719860
\(996\) 2.59706e16 0.839568
\(997\) −1.00204e16 −0.322154 −0.161077 0.986942i \(-0.551497\pi\)
−0.161077 + 0.986942i \(0.551497\pi\)
\(998\) 7.76129e16 2.48151
\(999\) 1.13443e16 0.360717
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 177.12.a.b.1.26 27
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.b.1.26 27 1.1 even 1 trivial