LMFDB - Embedded newform 81.10.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [81,10,Mod(1,81)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(81, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 10, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("81.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	81.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$41.7179027293$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 2930 x^{6} - 1276 x^{5} + 2487472 x^{4} + 3423248 x^{3} - 586568096 x^{2} + \cdots + 965565184 $ x^8 - x^7 - 2930x^6 - 1276x^5 + 2487472x^4 + 3423248x^3 - 586568096x^2 - 2825363008x + 965565184
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{7}\cdot 3^{18}\cdot 17 $
Twist minimal:	no (minimal twist has level 9)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$27.6540$ of defining polynomial
Character	$\chi$	$=$	81.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-25.6540 q^{2} +146.127 q^{4} -706.451 q^{5} -4568.42 q^{7} +9386.09 q^{8} +18123.3 q^{10} -71433.4 q^{11} -26402.1 q^{13} +117198. q^{14} -315608. q^{16} +314759. q^{17} -904019. q^{19} -103232. q^{20} +1.83255e6 q^{22} +111842. q^{23} -1.45405e6 q^{25} +677318. q^{26} -667571. q^{28} -4.38803e6 q^{29} -9.88785e6 q^{31} +3.29093e6 q^{32} -8.07483e6 q^{34} +3.22736e6 q^{35} -6.32122e6 q^{37} +2.31917e7 q^{38} -6.63082e6 q^{40} -5.36038e6 q^{41} +2.24550e7 q^{43} -1.04384e7 q^{44} -2.86920e6 q^{46} +4.47288e7 q^{47} -1.94832e7 q^{49} +3.73022e7 q^{50} -3.85806e6 q^{52} +4.00972e7 q^{53} +5.04642e7 q^{55} -4.28796e7 q^{56} +1.12571e8 q^{58} +9.58236e7 q^{59} +2.02225e7 q^{61} +2.53663e8 q^{62} +7.71659e7 q^{64} +1.86518e7 q^{65} +1.82498e8 q^{67} +4.59950e7 q^{68} -8.27948e7 q^{70} -1.38543e8 q^{71} -2.17970e8 q^{73} +1.62165e8 q^{74} -1.32102e8 q^{76} +3.26337e8 q^{77} +3.94587e8 q^{79} +2.22962e8 q^{80} +1.37515e8 q^{82} +4.73277e8 q^{83} -2.22362e8 q^{85} -5.76060e8 q^{86} -6.70481e8 q^{88} -2.60507e8 q^{89} +1.20616e8 q^{91} +1.63432e7 q^{92} -1.14747e9 q^{94} +6.38646e8 q^{95} +5.74842e8 q^{97} +4.99822e8 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 15 q^{2} + 1793 q^{4} + 453 q^{5} + 343 q^{7} + 7239 q^{8} + 510 q^{10} + 99150 q^{11} - 32435 q^{13} + 394824 q^{14} + 328193 q^{16} + 415539 q^{17} - 85277 q^{19} + 1855164 q^{20} - 529359 q^{22}+ \cdots - 2413650159 q^{98}+O(q^{100}) $ 8 * q + 15 * q^2 + 1793 * q^4 + 453 * q^5 + 343 * q^7 + 7239 * q^8 + 510 * q^10 + 99150 * q^11 - 32435 * q^13 + 394824 * q^14 + 328193 * q^16 + 415539 * q^17 - 85277 * q^19 + 1855164 * q^20 - 529359 * q^22 + 1064559 * q^23 + 2293229 * q^25 - 1218156 * q^26 + 612862 * q^28 - 1309053 * q^29 + 2359819 * q^31 + 5760063 * q^32 - 981801 * q^34 + 15533277 * q^35 + 8195758 * q^37 + 39490203 * q^38 + 16760496 * q^40 + 54747318 * q^41 - 15249608 * q^43 + 166254963 * q^44 + 1195260 * q^46 + 156295545 * q^47 - 15239583 * q^49 + 315590163 * q^50 + 19773358 * q^52 + 262758114 * q^53 - 3789885 * q^55 + 470339790 * q^56 - 55408560 * q^58 + 307774074 * q^59 - 69192125 * q^61 + 457218462 * q^62 - 201794239 * q^64 + 482470359 * q^65 - 14328044 * q^67 + 915409575 * q^68 + 229271934 * q^70 + 619800696 * q^71 + 299306599 * q^73 + 1022736000 * q^74 - 119954093 * q^76 + 717995541 * q^77 - 30257531 * q^79 + 1463913264 * q^80 - 101188011 * q^82 + 1176168291 * q^83 - 4818366 * q^85 + 1426944009 * q^86 - 911312427 * q^88 + 1658520648 * q^89 - 369615061 * q^91 - 76813998 * q^92 + 1954316784 * q^94 - 391400652 * q^95 + 267311278 * q^97 - 2413650159 * q^98

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 $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−25.6540	−1.13376	−0.566879	−	0.823801i	$-0.691849\pi$
$2$	−25.6540	−1.13376	−0.566879	+	0.823801i	$0.691849\pi$
$3$	0	0
$3$	0	0
$4$	146.127	0.285405
$5$	−706.451	−0.505495	−0.252748	−	0.967532i	$-0.581334\pi$
$5$	−706.451	−0.505495	−0.252748	+	0.967532i	$0.581334\pi$
$6$	0	0
$7$	−4568.42	−0.719158	−0.359579	−	0.933115i	$-0.617080\pi$
$7$	−4568.42	−0.719158	−0.359579	+	0.933115i	$0.617080\pi$
$8$	9386.09	0.810177
$9$	0	0
$10$	18123.3	0.573109
$11$	−71433.4	−1.47107	−0.735537	−	0.677485i	$-0.763070\pi$
$11$	−71433.4	−1.47107	−0.735537	+	0.677485i	$0.763070\pi$
$12$	0	0
$13$	−26402.1	−0.256385	−0.128193	−	0.991749i	$-0.540918\pi$
$13$	−26402.1	−0.256385	−0.128193	+	0.991749i	$0.540918\pi$
$14$	117198.	0.815351
$15$	0	0
$16$	−315608.	−1.20395
$17$	314759.	0.914026	0.457013	−	0.889460i	$-0.348919\pi$
$17$	314759.	0.914026	0.457013	+	0.889460i	$0.348919\pi$
$18$	0	0
$19$	−904019.	−1.59143	−0.795713	−	0.605674i	$-0.792904\pi$
$19$	−904019.	−1.59143	−0.795713	+	0.605674i	$0.792904\pi$
$20$	−103232.	−0.144271
$21$	0	0
$22$	1.83255e6	1.66784
$23$	111842.	0.0833357	0.0416678	−	0.999132i	$-0.486733\pi$
$23$	111842.	0.0833357	0.0416678	+	0.999132i	$0.486733\pi$
$24$	0	0
$25$	−1.45405e6	−0.744474
$26$	677318.	0.290678
$27$	0	0
$28$	−667571.	−0.205251
$29$	−4.38803e6	−1.15207	−0.576035	−	0.817425i	$-0.695401\pi$
$29$	−4.38803e6	−1.15207	−0.576035	+	0.817425i	$0.695401\pi$
$30$	0	0
$31$	−9.88785e6	−1.92298	−0.961488	−	0.274845i	$-0.911373\pi$
$31$	−9.88785e6	−1.92298	−0.961488	+	0.274845i	$0.911373\pi$
$32$	3.29093e6	0.554809
$33$	0	0
$34$	−8.07483e6	−1.03628
$35$	3.22736e6	0.363531
$36$	0	0
$37$	−6.32122e6	−0.554489	−0.277245	−	0.960799i	$-0.589421\pi$
$37$	−6.32122e6	−0.554489	−0.277245	+	0.960799i	$0.589421\pi$
$38$	2.31917e7	1.80429
$39$	0	0
$40$	−6.63082e6	−0.409541
$41$	−5.36038e6	−0.296257	−0.148128	−	0.988968i	$-0.547325\pi$
$41$	−5.36038e6	−0.296257	−0.148128	+	0.988968i	$0.547325\pi$
$42$	0	0
$43$	2.24550e7	1.00162	0.500812	−	0.865556i	$-0.333035\pi$
$43$	2.24550e7	1.00162	0.500812	+	0.865556i	$0.333035\pi$
$44$	−1.04384e7	−0.419852
$45$	0	0
$46$	−2.86920e6	−0.0944824
$47$	4.47288e7	1.33705	0.668524	−	0.743691i	$-0.266926\pi$
$47$	4.47288e7	1.33705	0.668524	+	0.743691i	$0.266926\pi$
$48$	0	0
$49$	−1.94832e7	−0.482812
$50$	3.73022e7	0.844053
$51$	0	0
$52$	−3.85806e6	−0.0731736
$53$	4.00972e7	0.698027	0.349013	−	0.937118i	$-0.386517\pi$
$53$	4.00972e7	0.698027	0.349013	+	0.937118i	$0.386517\pi$
$54$	0	0
$55$	5.04642e7	0.743621
$56$	−4.28796e7	−0.582645
$57$	0	0
$58$	1.12571e8	1.30617
$59$	9.58236e7	1.02953	0.514764	−	0.857332i	$-0.327879\pi$
$59$	9.58236e7	1.02953	0.514764	+	0.857332i	$0.327879\pi$
$60$	0	0
$61$	2.02225e7	0.187004	0.0935020	−	0.995619i	$-0.470194\pi$
$61$	2.02225e7	0.187004	0.0935020	+	0.995619i	$0.470194\pi$
$62$	2.53663e8	2.18019
$63$	0	0
$64$	7.71659e7	0.574931
$65$	1.86518e7	0.129601
$66$	0	0
$67$	1.82498e8	1.10642	0.553211	−	0.833041i	$-0.313402\pi$
$67$	1.82498e8	1.10642	0.553211	+	0.833041i	$0.313402\pi$
$68$	4.59950e7	0.260868
$69$	0	0
$70$	−8.27948e7	−0.412156
$71$	−1.38543e8	−0.647025	−0.323512	−	0.946224i	$-0.604864\pi$
$71$	−1.38543e8	−0.647025	−0.323512	+	0.946224i	$0.604864\pi$
$72$	0	0
$73$	−2.17970e8	−0.898347	−0.449173	−	0.893445i	$-0.648281\pi$
$73$	−2.17970e8	−0.898347	−0.449173	+	0.893445i	$0.648281\pi$
$74$	1.62165e8	0.628656
$75$	0	0
$76$	−1.32102e8	−0.454201
$77$	3.26337e8	1.05793
$78$	0	0
$79$	3.94587e8	1.13978	0.569890	−	0.821721i	$-0.306986\pi$
$79$	3.94587e8	1.13978	0.569890	+	0.821721i	$0.306986\pi$
$80$	2.22962e8	0.608591
$81$	0	0
$82$	1.37515e8	0.335883
$83$	4.73277e8	1.09462	0.547310	−	0.836930i	$-0.315652\pi$
$83$	4.73277e8	1.09462	0.547310	+	0.836930i	$0.315652\pi$
$84$	0	0
$85$	−2.22362e8	−0.462036
$86$	−5.76060e8	−1.13560
$87$	0	0
$88$	−6.70481e8	−1.19183
$89$	−2.60507e8	−0.440113	−0.220057	−	0.975487i	$-0.570624\pi$
$89$	−2.60507e8	−0.440113	−0.220057	+	0.975487i	$0.570624\pi$
$90$	0	0
$91$	1.20616e8	0.184381
$92$	1.63432e7	0.0237844
$93$	0	0
$94$	−1.14747e9	−1.51589
$95$	6.38646e8	0.804459
$96$	0	0
$97$	5.74842e8	0.659289	0.329644	−	0.944105i	$-0.393071\pi$
$97$	5.74842e8	0.659289	0.329644	+	0.944105i	$0.393071\pi$
$98$	4.99822e8	0.547391
$99$	0	0
$100$	−2.12477e8	−0.212477
$101$	−1.40772e8	−0.134608	−0.0673039	−	0.997733i	$-0.521440\pi$
$101$	−1.40772e8	−0.134608	−0.0673039	+	0.997733i	$0.521440\pi$
$102$	0	0
$103$	−3.41637e7	−0.0299087	−0.0149544	−	0.999888i	$-0.504760\pi$
$103$	−3.41637e7	−0.0299087	−0.0149544	+	0.999888i	$0.504760\pi$
$104$	−2.47812e8	−0.207717
$105$	0	0
$106$	−1.02865e9	−0.791393
$107$	−1.61837e9	−1.19358	−0.596790	−	0.802397i	$-0.703557\pi$
$107$	−1.61837e9	−1.19358	−0.596790	+	0.802397i	$0.703557\pi$
$108$	0	0
$109$	−8.56262e8	−0.581015	−0.290508	−	0.956873i	$-0.593824\pi$
$109$	−8.56262e8	−0.581015	−0.290508	+	0.956873i	$0.593824\pi$
$110$	−1.29461e9	−0.843085
$111$	0	0
$112$	1.44183e9	0.865830
$113$	−1.93262e9	−1.11505	−0.557524	−	0.830160i	$-0.688249\pi$
$113$	−1.93262e9	−1.11505	−0.557524	+	0.830160i	$0.688249\pi$
$114$	0	0
$115$	−7.90112e7	−0.0421258
$116$	−6.41212e8	−0.328807
$117$	0	0
$118$	−2.45826e9	−1.16724
$119$	−1.43795e9	−0.657329
$120$	0	0
$121$	2.74478e9	1.16406
$122$	−5.18789e8	−0.212017
$123$	0	0
$124$	−1.44489e9	−0.548827
$125$	2.40700e9	0.881824
$126$	0	0
$127$	4.98705e9	1.70109	0.850545	−	0.525903i	$-0.176272\pi$
$127$	4.98705e9	1.70109	0.850545	+	0.525903i	$0.176272\pi$
$128$	−3.66457e9	−1.20664
$129$	0	0
$130$	−4.78492e8	−0.146937
$131$	1.91672e8	0.0568641	0.0284320	−	0.999596i	$-0.490949\pi$
$131$	1.91672e8	0.0568641	0.0284320	+	0.999596i	$0.490949\pi$
$132$	0	0
$133$	4.12994e9	1.14449
$134$	−4.68180e9	−1.25441
$135$	0	0
$136$	2.95436e9	0.740523
$137$	−1.00077e9	−0.242712	−0.121356	−	0.992609i	$-0.538724\pi$
$137$	−1.00077e9	−0.242712	−0.121356	+	0.992609i	$0.538724\pi$
$138$	0	0
$139$	−1.35024e9	−0.306792	−0.153396	−	0.988165i	$-0.549021\pi$
$139$	−1.35024e9	−0.306792	−0.153396	+	0.988165i	$0.549021\pi$
$140$	4.71606e8	0.103754
$141$	0	0
$142$	3.55417e9	0.733569
$143$	1.88599e9	0.377161
$144$	0	0
$145$	3.09993e9	0.582366
$146$	5.59180e9	1.01851
$147$	0	0
$148$	−9.23704e8	−0.158254
$149$	−3.52992e9	−0.586714	−0.293357	−	0.956003i	$-0.594773\pi$
$149$	−3.52992e9	−0.586714	−0.293357	+	0.956003i	$0.594773\pi$
$150$	0	0
$151$	5.72646e8	0.0896376	0.0448188	−	0.998995i	$-0.485729\pi$
$151$	5.72646e8	0.0896376	0.0448188	+	0.998995i	$0.485729\pi$
$152$	−8.48521e9	−1.28934
$153$	0	0
$154$	−8.37186e9	−1.19944
$155$	6.98528e9	0.972056
$156$	0	0
$157$	8.40171e8	0.110362	0.0551810	−	0.998476i	$-0.482426\pi$
$157$	8.40171e8	0.110362	0.0551810	+	0.998476i	$0.482426\pi$
$158$	−1.01227e10	−1.29223
$159$	0	0
$160$	−2.32488e9	−0.280453
$161$	−5.10942e8	−0.0599315
$162$	0	0
$163$	−7.00425e9	−0.777173	−0.388586	−	0.921412i	$-0.627037\pi$
$163$	−7.00425e9	−0.777173	−0.388586	+	0.921412i	$0.627037\pi$
$164$	−7.83299e8	−0.0845532
$165$	0	0
$166$	−1.21414e10	−1.24103
$167$	9.27017e9	0.922281	0.461141	−	0.887327i	$-0.347440\pi$
$167$	9.27017e9	0.922281	0.461141	+	0.887327i	$0.347440\pi$
$168$	0	0
$169$	−9.90743e9	−0.934267
$170$	5.70448e9	0.523836
$171$	0	0
$172$	3.28129e9	0.285869
$173$	−8.27334e9	−0.702220	−0.351110	−	0.936334i	$-0.614196\pi$
$173$	−8.27334e9	−0.702220	−0.351110	+	0.936334i	$0.614196\pi$
$174$	0	0
$175$	6.64271e9	0.535395
$176$	2.25450e10	1.77110
$177$	0	0
$178$	6.68304e9	0.498981
$179$	2.36310e10	1.72046	0.860229	−	0.509907i	$-0.170320\pi$
$179$	2.36310e10	1.72046	0.860229	+	0.509907i	$0.170320\pi$
$180$	0	0
$181$	1.40618e10	0.973841	0.486921	−	0.873446i	$-0.338120\pi$
$181$	1.40618e10	0.973841	0.486921	+	0.873446i	$0.338120\pi$
$182$	−3.09427e9	−0.209044
$183$	0	0
$184$	1.04976e9	0.0675166
$185$	4.46563e9	0.280292
$186$	0	0
$187$	−2.24843e10	−1.34460
$188$	6.53611e9	0.381600
$189$	0	0
$190$	−1.63838e10	−0.912061
$191$	9.75192e9	0.530201	0.265100	−	0.964221i	$-0.414595\pi$
$191$	9.75192e9	0.530201	0.265100	+	0.964221i	$0.414595\pi$
$192$	0	0
$193$	9.95176e9	0.516288	0.258144	−	0.966106i	$-0.416889\pi$
$193$	9.95176e9	0.516288	0.258144	+	0.966106i	$0.416889\pi$
$194$	−1.47470e10	−0.747473
$195$	0	0
$196$	−2.84703e9	−0.137797
$197$	3.23117e9	0.152849	0.0764244	−	0.997075i	$-0.475650\pi$
$197$	3.23117e9	0.152849	0.0764244	+	0.997075i	$0.475650\pi$
$198$	0	0
$199$	−2.90155e10	−1.31157	−0.655785	−	0.754947i	$-0.727662\pi$
$199$	−2.90155e10	−1.31157	−0.655785	+	0.754947i	$0.727662\pi$
$200$	−1.36479e10	−0.603156
$201$	0	0
$202$	3.61137e9	0.152613
$203$	2.00464e10	0.828521
$204$	0	0
$205$	3.78685e9	0.149757
$206$	8.76437e8	0.0339092
$207$	0	0
$208$	8.33270e9	0.308675
$209$	6.45772e10	2.34110
$210$	0	0
$211$	−4.25536e9	−0.147797	−0.0738985	−	0.997266i	$-0.523544\pi$
$211$	−4.25536e9	−0.147797	−0.0738985	+	0.997266i	$0.523544\pi$
$212$	5.85929e9	0.199220
$213$	0	0
$214$	4.15177e10	1.35323
$215$	−1.58634e10	−0.506317
$216$	0	0
$217$	4.51718e10	1.38292
$218$	2.19665e10	0.658730
$219$	0	0
$220$	7.37421e9	0.212233
$221$	−8.31029e9	−0.234343
$222$	0	0
$223$	4.23518e10	1.14683	0.573416	−	0.819265i	$-0.305618\pi$
$223$	4.23518e10	1.14683	0.573416	+	0.819265i	$0.305618\pi$
$224$	−1.50343e10	−0.398995
$225$	0	0
$226$	4.95795e10	1.26419
$227$	−5.92602e10	−1.48131	−0.740656	−	0.671884i	$-0.765485\pi$
$227$	−5.92602e10	−1.48131	−0.740656	+	0.671884i	$0.765485\pi$
$228$	0	0
$229$	−6.15045e10	−1.47791	−0.738954	−	0.673756i	$-0.764680\pi$
$229$	−6.15045e10	−1.47791	−0.738954	+	0.673756i	$0.764680\pi$
$230$	2.02695e9	0.0477604
$231$	0	0
$232$	−4.11865e10	−0.933381
$233$	4.69389e10	1.04335	0.521677	−	0.853143i	$-0.325307\pi$
$233$	4.69389e10	1.04335	0.521677	+	0.853143i	$0.325307\pi$
$234$	0	0
$235$	−3.15987e10	−0.675872
$236$	1.40025e10	0.293833
$237$	0	0
$238$	3.68892e10	0.745251
$239$	−1.04488e10	−0.207146	−0.103573	−	0.994622i	$-0.533027\pi$
$239$	−1.04488e10	−0.207146	−0.103573	+	0.994622i	$0.533027\pi$
$240$	0	0
$241$	2.99400e10	0.571708	0.285854	−	0.958273i	$-0.407723\pi$
$241$	2.99400e10	0.571708	0.285854	+	0.958273i	$0.407723\pi$
$242$	−7.04147e10	−1.31976
$243$	0	0
$244$	2.95507e9	0.0533719
$245$	1.37639e10	0.244059
$246$	0	0
$247$	2.38680e10	0.408018
$248$	−9.28082e10	−1.55795
$249$	0	0
$250$	−6.17493e10	−0.999774
$251$	1.50652e9	0.0239576	0.0119788	−	0.999928i	$-0.496187\pi$
$251$	1.50652e9	0.0239576	0.0119788	+	0.999928i	$0.496187\pi$
$252$	0	0
$253$	−7.98928e9	−0.122593
$254$	−1.27938e11	−1.92862
$255$	0	0
$256$	5.45019e10	0.793106
$257$	8.71580e10	1.24626	0.623130	−	0.782119i	$-0.285861\pi$
$257$	8.71580e10	1.24626	0.623130	+	0.782119i	$0.285861\pi$
$258$	0	0
$259$	2.88780e10	0.398765
$260$	2.72553e9	0.0369889
$261$	0	0
$262$	−4.91715e9	−0.0644700
$263$	4.21286e10	0.542970	0.271485	−	0.962443i	$-0.412485\pi$
$263$	4.21286e10	0.542970	0.271485	+	0.962443i	$0.412485\pi$
$264$	0	0
$265$	−2.83267e10	−0.352849
$266$	−1.05949e11	−1.29757
$267$	0	0
$268$	2.66679e10	0.315779
$269$	−6.32241e10	−0.736202	−0.368101	−	0.929786i	$-0.619992\pi$
$269$	−6.32241e10	−0.736202	−0.368101	+	0.929786i	$0.619992\pi$
$270$	0	0
$271$	−1.63811e11	−1.84493	−0.922467	−	0.386075i	$-0.873831\pi$
$271$	−1.63811e11	−1.84493	−0.922467	+	0.386075i	$0.873831\pi$
$272$	−9.93406e10	−1.10044
$273$	0	0
$274$	2.56738e10	0.275177
$275$	1.03868e11	1.09518
$276$	0	0
$277$	−1.72604e11	−1.76154	−0.880771	−	0.473542i	$-0.842975\pi$
$277$	−1.72604e11	−1.76154	−0.880771	+	0.473542i	$0.842975\pi$
$278$	3.46390e10	0.347827
$279$	0	0
$280$	3.02923e10	0.294525
$281$	2.55429e10	0.244394	0.122197	−	0.992506i	$-0.461006\pi$
$281$	2.55429e10	0.244394	0.122197	+	0.992506i	$0.461006\pi$
$282$	0	0
$283$	−7.38934e10	−0.684805	−0.342402	−	0.939553i	$-0.611241\pi$
$283$	−7.38934e10	−0.684805	−0.342402	+	0.939553i	$0.611241\pi$
$284$	−2.02449e10	−0.184664
$285$	0	0
$286$	−4.83831e10	−0.427609
$287$	2.44885e10	0.213056
$288$	0	0
$289$	−1.95145e10	−0.164557
$290$	−7.95256e10	−0.660262
$291$	0	0
$292$	−3.18514e10	−0.256393
$293$	−2.03535e11	−1.61337	−0.806687	−	0.590979i	$-0.798742\pi$
$293$	−2.03535e11	−1.61337	−0.806687	+	0.590979i	$0.798742\pi$
$294$	0	0
$295$	−6.76947e10	−0.520422
$296$	−5.93316e10	−0.449234
$297$	0	0
$298$	9.05566e10	0.665192
$299$	−2.95287e9	−0.0213660
$300$	0	0
$301$	−1.02584e11	−0.720326
$302$	−1.46907e10	−0.101627
$303$	0	0
$304$	2.85316e11	1.91600
$305$	−1.42862e10	−0.0945297
$306$	0	0
$307$	8.22523e10	0.528476	0.264238	−	0.964457i	$-0.414880\pi$
$307$	8.22523e10	0.528476	0.264238	+	0.964457i	$0.414880\pi$
$308$	4.76869e10	0.301940
$309$	0	0
$310$	−1.79200e11	−1.10208
$311$	2.32094e11	1.40683	0.703415	−	0.710780i	$-0.251658\pi$
$311$	2.32094e11	1.40683	0.703415	+	0.710780i	$0.251658\pi$
$312$	0	0
$313$	6.30080e10	0.371062	0.185531	−	0.982638i	$-0.440600\pi$
$313$	6.30080e10	0.371062	0.185531	+	0.982638i	$0.440600\pi$
$314$	−2.15537e10	−0.125124
$315$	0	0
$316$	5.76600e10	0.325299
$317$	2.31470e11	1.28744	0.643722	−	0.765260i	$-0.277389\pi$
$317$	2.31470e11	1.28744	0.643722	+	0.765260i	$0.277389\pi$
$318$	0	0
$319$	3.13452e11	1.69478
$320$	−5.45140e10	−0.290625
$321$	0	0
$322$	1.31077e10	0.0679478
$323$	−2.84548e11	−1.45460
$324$	0	0
$325$	3.83899e10	0.190872
$326$	1.79687e11	0.881125
$327$	0	0
$328$	−5.03131e10	−0.240021
$329$	−2.04340e11	−0.961549
$330$	0	0
$331$	1.54455e11	0.707254	0.353627	−	0.935386i	$-0.384948\pi$
$331$	1.54455e11	0.707254	0.353627	+	0.935386i	$0.384948\pi$
$332$	6.91587e10	0.312410
$333$	0	0
$334$	−2.37817e11	−1.04564
$335$	−1.28926e11	−0.559291
$336$	0	0
$337$	1.35250e11	0.571218	0.285609	−	0.958346i	$-0.407804\pi$
$337$	1.35250e11	0.571218	0.285609	+	0.958346i	$0.407804\pi$
$338$	2.54165e11	1.05923
$339$	0	0
$340$	−3.24932e10	−0.131867
$341$	7.06323e11	2.82884
$342$	0	0
$343$	2.73359e11	1.06638
$344$	2.10765e11	0.811493
$345$	0	0
$346$	2.12244e11	0.796147
$347$	2.83506e11	1.04973	0.524867	−	0.851184i	$-0.324115\pi$
$347$	2.83506e11	1.04973	0.524867	+	0.851184i	$0.324115\pi$
$348$	0	0
$349$	9.73597e10	0.351289	0.175645	−	0.984454i	$-0.443799\pi$
$349$	9.73597e10	0.351289	0.175645	+	0.984454i	$0.443799\pi$
$350$	−1.70412e11	−0.607008
$351$	0	0
$352$	−2.35082e11	−0.816164
$353$	−3.83723e10	−0.131532	−0.0657660	−	0.997835i	$-0.520949\pi$
$353$	−3.83723e10	−0.131532	−0.0657660	+	0.997835i	$0.520949\pi$
$354$	0	0
$355$	9.78737e10	0.327068
$356$	−3.80672e10	−0.125611
$357$	0	0
$358$	−6.06231e11	−1.95058
$359$	4.48036e11	1.42360	0.711800	−	0.702383i	$-0.247880\pi$
$359$	4.48036e11	1.42360	0.711800	+	0.702383i	$0.247880\pi$
$360$	0	0
$361$	4.94563e11	1.53264
$362$	−3.60742e11	−1.10410
$363$	0	0
$364$	1.76252e10	0.0526234
$365$	1.53985e11	0.454110
$366$	0	0
$367$	−7.48073e10	−0.215252	−0.107626	−	0.994191i	$-0.534325\pi$
$367$	−7.48073e10	−0.215252	−0.107626	+	0.994191i	$0.534325\pi$
$368$	−3.52983e10	−0.100332
$369$	0	0
$370$	−1.14561e11	−0.317783
$371$	−1.83180e11	−0.501992
$372$	0	0
$373$	−2.79392e11	−0.747351	−0.373675	−	0.927560i	$-0.621903\pi$
$373$	−2.79392e11	−0.747351	−0.373675	+	0.927560i	$0.621903\pi$
$374$	5.76813e11	1.52445
$375$	0	0
$376$	4.19829e11	1.08325
$377$	1.15853e11	0.295374
$378$	0	0
$379$	−6.40616e11	−1.59485	−0.797427	−	0.603415i	$-0.793806\pi$
$379$	−6.40616e11	−1.59485	−0.797427	+	0.603415i	$0.793806\pi$
$380$	9.33236e10	0.229597
$381$	0	0
$382$	−2.50176e11	−0.601119
$383$	−3.33056e11	−0.790903	−0.395451	−	0.918487i	$-0.629412\pi$
$383$	−3.33056e11	−0.790903	−0.395451	+	0.918487i	$0.629412\pi$
$384$	0	0
$385$	−2.30542e11	−0.534781
$386$	−2.55302e11	−0.585345
$387$	0	0
$388$	8.40002e10	0.188164
$389$	1.63159e11	0.361275	0.180638	−	0.983550i	$-0.442184\pi$
$389$	1.63159e11	0.361275	0.180638	+	0.983550i	$0.442184\pi$
$390$	0	0
$391$	3.52034e10	0.0761709
$392$	−1.82871e11	−0.391163
$393$	0	0
$394$	−8.28925e10	−0.173293
$395$	−2.78757e11	−0.576153
$396$	0	0
$397$	2.79119e11	0.563939	0.281969	−	0.959423i	$-0.409012\pi$
$397$	2.79119e11	0.563939	0.281969	+	0.959423i	$0.409012\pi$
$398$	7.44364e11	1.48700
$399$	0	0
$400$	4.58910e11	0.896309
$401$	−7.59932e11	−1.46766	−0.733830	−	0.679333i	$-0.762269\pi$
$401$	−7.59932e11	−1.46766	−0.733830	+	0.679333i	$0.762269\pi$
$402$	0	0
$403$	2.61059e11	0.493023
$404$	−2.05707e10	−0.0384178
$405$	0	0
$406$	−5.14269e11	−0.939341
$407$	4.51546e11	0.815694
$408$	0	0
$409$	−3.79941e10	−0.0671369	−0.0335685	−	0.999436i	$-0.510687\pi$
$409$	−3.79941e10	−0.0671369	−0.0335685	+	0.999436i	$0.510687\pi$
$410$	−9.71478e10	−0.169788
$411$	0	0
$412$	−4.99226e9	−0.00853610
$413$	−4.37762e11	−0.740394
$414$	0	0
$415$	−3.34347e11	−0.553326
$416$	−8.68872e10	−0.142245
$417$	0	0
$418$	−1.65666e12	−2.65424
$419$	7.33645e11	1.16285	0.581424	−	0.813601i	$-0.302496\pi$
$419$	7.33645e11	1.16285	0.581424	+	0.813601i	$0.302496\pi$
$420$	0	0
$421$	−3.21432e11	−0.498677	−0.249339	−	0.968416i	$-0.580213\pi$
$421$	−3.21432e11	−0.498677	−0.249339	+	0.968416i	$0.580213\pi$
$422$	1.09167e11	0.167566
$423$	0	0
$424$	3.76356e11	0.565525
$425$	−4.57676e11	−0.680469
$426$	0	0
$427$	−9.23849e10	−0.134486
$428$	−2.36489e11	−0.340654
$429$	0	0
$430$	4.06959e11	0.574040
$431$	−3.86098e11	−0.538952	−0.269476	−	0.963007i	$-0.586850\pi$
$431$	−3.86098e11	−0.538952	−0.269476	+	0.963007i	$0.586850\pi$
$432$	0	0
$433$	5.48808e11	0.750282	0.375141	−	0.926968i	$-0.377594\pi$
$433$	5.48808e11	0.750282	0.375141	+	0.926968i	$0.377594\pi$
$434$	−1.15884e12	−1.56790
$435$	0	0
$436$	−1.25123e11	−0.165825
$437$	−1.01108e11	−0.132623
$438$	0	0
$439$	−3.00100e11	−0.385635	−0.192817	−	0.981235i	$-0.561762\pi$
$439$	−3.00100e11	−0.385635	−0.192817	+	0.981235i	$0.561762\pi$
$440$	4.73662e11	0.602465
$441$	0	0
$442$	2.13192e11	0.265687
$443$	−4.22660e11	−0.521404	−0.260702	−	0.965419i	$-0.583954\pi$
$443$	−4.22660e11	−0.521404	−0.260702	+	0.965419i	$0.583954\pi$
$444$	0	0
$445$	1.84036e11	0.222475
$446$	−1.08649e12	−1.30023
$447$	0	0
$448$	−3.52526e11	−0.413466
$449$	−1.63406e12	−1.89741	−0.948704	−	0.316165i	$-0.897605\pi$
$449$	−1.63406e12	−1.89741	−0.948704	+	0.316165i	$0.897605\pi$
$450$	0	0
$451$	3.82911e11	0.435816
$452$	−2.82409e11	−0.318241
$453$	0	0
$454$	1.52026e12	1.67945
$455$	−8.52090e10	−0.0932040
$456$	0	0
$457$	1.24637e12	1.33667	0.668334	−	0.743862i	$-0.267008\pi$
$457$	1.24637e12	1.33667	0.668334	+	0.743862i	$0.267008\pi$
$458$	1.57784e12	1.67559
$459$	0	0
$460$	−1.15457e10	−0.0120229
$461$	−2.88051e11	−0.297040	−0.148520	−	0.988909i	$-0.547451\pi$
$461$	−2.88051e11	−0.297040	−0.148520	+	0.988909i	$0.547451\pi$
$462$	0	0
$463$	1.57266e12	1.59045	0.795225	−	0.606315i	$-0.207353\pi$
$463$	1.57266e12	1.59045	0.795225	+	0.606315i	$0.207353\pi$
$464$	1.38490e12	1.38703
$465$	0	0
$466$	−1.20417e12	−1.18291
$467$	−2.12882e11	−0.207116	−0.103558	−	0.994623i	$-0.533023\pi$
$467$	−2.12882e11	−0.207116	−0.103558	+	0.994623i	$0.533023\pi$
$468$	0	0
$469$	−8.33726e11	−0.795693
$470$	8.10634e11	0.766274
$471$	0	0
$472$	8.99409e11	0.834100
$473$	−1.60404e12	−1.47346
$474$	0	0
$475$	1.31449e12	1.18478
$476$	−2.10124e11	−0.187605
$477$	0	0
$478$	2.68053e11	0.234853
$479$	−1.52484e12	−1.32347	−0.661733	−	0.749739i	$-0.730179\pi$
$479$	−1.52484e12	−1.32347	−0.661733	+	0.749739i	$0.730179\pi$
$480$	0	0
$481$	1.66893e11	0.142163
$482$	−7.68079e11	−0.648178
$483$	0	0
$484$	4.01088e11	0.332228
$485$	−4.06098e11	−0.333267
$486$	0	0
$487$	−1.81008e12	−1.45820	−0.729101	−	0.684406i	$-0.760062\pi$
$487$	−1.81008e12	−1.45820	−0.729101	+	0.684406i	$0.760062\pi$
$488$	1.89810e11	0.151506
$489$	0	0
$490$	−3.53100e11	−0.276704
$491$	1.15383e12	0.895931	0.447966	−	0.894051i	$-0.352149\pi$
$491$	1.15383e12	0.895931	0.447966	+	0.894051i	$0.352149\pi$
$492$	0	0
$493$	−1.38117e12	−1.05302
$494$	−6.12309e11	−0.462593
$495$	0	0
$496$	3.12068e12	2.31517
$497$	6.32921e11	0.465313
$498$	0	0
$499$	−2.01943e12	−1.45806	−0.729032	−	0.684479i	$-0.760030\pi$
$499$	−2.01943e12	−1.45806	−0.729032	+	0.684479i	$0.760030\pi$
$500$	3.51729e11	0.251677
$501$	0	0
$502$	−3.86482e10	−0.0271621
$503$	7.16624e11	0.499155	0.249578	−	0.968355i	$-0.419708\pi$
$503$	7.16624e11	0.499155	0.249578	+	0.968355i	$0.419708\pi$
$504$	0	0
$505$	9.94486e10	0.0680437
$506$	2.04957e11	0.138991
$507$	0	0
$508$	7.28745e11	0.485500
$509$	1.58134e12	1.04423	0.522114	−	0.852876i	$-0.325144\pi$
$509$	1.58134e12	1.04423	0.522114	+	0.852876i	$0.325144\pi$
$510$	0	0
$511$	9.95778e11	0.646053
$512$	4.78068e11	0.307450
$513$	0	0
$514$	−2.23595e12	−1.41296
$515$	2.41350e10	0.0151187
$516$	0	0
$517$	−3.19513e12	−1.96690
$518$	−7.40835e11	−0.452103
$519$	0	0
$520$	1.75067e11	0.105000
$521$	−1.03065e12	−0.612834	−0.306417	−	0.951897i	$-0.599130\pi$
$521$	−1.03065e12	−0.612834	−0.306417	+	0.951897i	$0.599130\pi$
$522$	0	0
$523$	2.03811e12	1.19116	0.595579	−	0.803297i	$-0.296923\pi$
$523$	2.03811e12	1.19116	0.595579	+	0.803297i	$0.296923\pi$
$524$	2.80085e10	0.0162293
$525$	0	0
$526$	−1.08077e12	−0.615596
$527$	−3.11229e12	−1.75765
$528$	0	0
$529$	−1.78864e12	−0.993055
$530$	7.26693e11	0.400046
$531$	0	0
$532$	6.03497e11	0.326642
$533$	1.41525e11	0.0759559
$534$	0	0
$535$	1.14330e12	0.603349
$536$	1.71294e12	0.896398
$537$	0	0
$538$	1.62195e12	0.834675
$539$	1.39175e12	0.710251
$540$	0	0
$541$	−1.53998e12	−0.772905	−0.386453	−	0.922309i	$-0.626300\pi$
$541$	−1.53998e12	−0.772905	−0.386453	+	0.922309i	$0.626300\pi$
$542$	4.20240e12	2.09171
$543$	0	0
$544$	1.03585e12	0.507109
$545$	6.04907e11	0.293700
$546$	0	0
$547$	−1.91921e12	−0.916598	−0.458299	−	0.888798i	$-0.651541\pi$
$547$	−1.91921e12	−0.916598	−0.458299	+	0.888798i	$0.651541\pi$
$548$	−1.46240e11	−0.0692713
$549$	0	0
$550$	−2.66463e12	−1.24166
$551$	3.96687e12	1.83343
$552$	0	0
$553$	−1.80264e12	−0.819682
$554$	4.42799e12	1.99716
$555$	0	0
$556$	−1.97307e11	−0.0875599
$557$	2.71058e11	0.119320	0.0596601	−	0.998219i	$-0.480998\pi$
$557$	2.71058e11	0.119320	0.0596601	+	0.998219i	$0.480998\pi$
$558$	0	0
$559$	−5.92858e11	−0.256802
$560$	−1.01858e12	−0.437673
$561$	0	0
$562$	−6.55277e11	−0.277084
$563$	−3.08234e11	−0.129298	−0.0646492	−	0.997908i	$-0.520593\pi$
$563$	−3.08234e11	−0.129298	−0.0646492	+	0.997908i	$0.520593\pi$
$564$	0	0
$565$	1.36530e12	0.563652
$566$	1.89566e12	0.776402
$567$	0	0
$568$	−1.30037e12	−0.524205
$569$	−3.58165e12	−1.43244	−0.716222	−	0.697872i	$-0.754130\pi$
$569$	−3.58165e12	−1.43244	−0.716222	+	0.697872i	$0.754130\pi$
$570$	0	0
$571$	−8.90822e10	−0.0350694	−0.0175347	−	0.999846i	$-0.505582\pi$
$571$	−8.90822e10	−0.0350694	−0.0175347	+	0.999846i	$0.505582\pi$
$572$	2.75595e11	0.107644
$573$	0	0
$574$	−6.28227e11	−0.241553
$575$	−1.62624e11	−0.0620413
$576$	0	0
$577$	−4.29975e11	−0.161492	−0.0807462	−	0.996735i	$-0.525730\pi$
$577$	−4.29975e11	−0.161492	−0.0807462	+	0.996735i	$0.525730\pi$
$578$	5.00624e11	0.186568
$579$	0	0
$580$	4.52985e11	0.166210
$581$	−2.16212e12	−0.787205
$582$	0	0
$583$	−2.86428e12	−1.02685
$584$	−2.04589e12	−0.727820
$585$	0	0
$586$	5.22149e12	1.82917
$587$	−2.27007e12	−0.789164	−0.394582	−	0.918861i	$-0.629111\pi$
$587$	−2.27007e12	−0.789164	−0.394582	+	0.918861i	$0.629111\pi$
$588$	0	0
$589$	8.93880e12	3.06028
$590$	1.73664e12	0.590032
$591$	0	0
$592$	1.99503e12	0.667577
$593$	4.09944e12	1.36138	0.680688	−	0.732573i	$-0.261681\pi$
$593$	4.09944e12	1.36138	0.680688	+	0.732573i	$0.261681\pi$
$594$	0	0
$595$	1.01584e12	0.332277
$596$	−5.15818e11	−0.167451
$597$	0	0
$598$	7.57528e10	0.0242239
$599$	1.61831e12	0.513618	0.256809	−	0.966462i	$-0.417329\pi$
$599$	1.61831e12	0.513618	0.256809	+	0.966462i	$0.417329\pi$
$600$	0	0
$601$	1.72161e12	0.538269	0.269134	−	0.963103i	$-0.413262\pi$
$601$	1.72161e12	0.538269	0.269134	+	0.963103i	$0.413262\pi$
$602$	2.63168e12	0.816675
$603$	0	0
$604$	8.36793e10	0.0255830
$605$	−1.93906e12	−0.588425
$606$	0	0
$607$	−4.14056e12	−1.23797	−0.618985	−	0.785402i	$-0.712456\pi$
$607$	−4.14056e12	−1.23797	−0.618985	+	0.785402i	$0.712456\pi$
$608$	−2.97506e12	−0.882937
$609$	0	0
$610$	3.66499e11	0.107174
$611$	−1.18093e12	−0.342799
$612$	0	0
$613$	1.88290e12	0.538585	0.269292	−	0.963058i	$-0.413210\pi$
$613$	1.88290e12	0.538585	0.269292	+	0.963058i	$0.413210\pi$
$614$	−2.11010e12	−0.599163
$615$	0	0
$616$	3.06303e12	0.857114
$617$	2.91305e12	0.809217	0.404609	−	0.914490i	$-0.367408\pi$
$617$	2.91305e12	0.809217	0.404609	+	0.914490i	$0.367408\pi$
$618$	0	0
$619$	−2.47495e12	−0.677577	−0.338789	−	0.940863i	$-0.610017\pi$
$619$	−2.47495e12	−0.677577	−0.338789	+	0.940863i	$0.610017\pi$
$620$	1.02074e12	0.277430
$621$	0	0
$622$	−5.95413e12	−1.59500
$623$	1.19010e12	0.316511
$624$	0	0
$625$	1.13951e12	0.298716
$626$	−1.61641e12	−0.420694
$627$	0	0
$628$	1.22772e11	0.0314979
$629$	−1.98966e12	−0.506817
$630$	0	0
$631$	5.10645e12	1.28229	0.641146	−	0.767419i	$-0.278459\pi$
$631$	5.10645e12	1.28229	0.641146	+	0.767419i	$0.278459\pi$
$632$	3.70363e12	0.923423
$633$	0	0
$634$	−5.93813e12	−1.45965
$635$	−3.52311e12	−0.859893
$636$	0	0
$637$	5.14396e11	0.123786
$638$	−8.04130e12	−1.92147
$639$	0	0
$640$	2.58884e12	0.609951
$641$	6.06494e11	0.141894	0.0709472	−	0.997480i	$-0.477398\pi$
$641$	6.06494e11	0.141894	0.0709472	+	0.997480i	$0.477398\pi$
$642$	0	0
$643$	−2.50984e12	−0.579024	−0.289512	−	0.957174i	$-0.593493\pi$
$643$	−2.50984e12	−0.579024	−0.289512	+	0.957174i	$0.593493\pi$
$644$	−7.46627e10	−0.0171048
$645$	0	0
$646$	7.29980e12	1.64917
$647$	−2.81592e12	−0.631758	−0.315879	−	0.948799i	$-0.602299\pi$
$647$	−2.81592e12	−0.631758	−0.315879	+	0.948799i	$0.602299\pi$
$648$	0	0
$649$	−6.84501e12	−1.51451
$650$	−9.84856e11	−0.216403
$651$	0	0
$652$	−1.02351e12	−0.221809
$653$	−1.08372e11	−0.0233242	−0.0116621	−	0.999932i	$-0.503712\pi$
$653$	−1.08372e11	−0.0233242	−0.0116621	+	0.999932i	$0.503712\pi$
$654$	0	0
$655$	−1.35407e11	−0.0287445
$656$	1.69178e12	0.356678
$657$	0	0
$658$	5.24213e12	1.09016
$659$	6.52712e12	1.34815	0.674073	−	0.738665i	$-0.264543\pi$
$659$	6.52712e12	1.34815	0.674073	+	0.738665i	$0.264543\pi$
$660$	0	0
$661$	4.28485e12	0.873029	0.436515	−	0.899697i	$-0.356213\pi$
$661$	4.28485e12	0.873029	0.436515	+	0.899697i	$0.356213\pi$
$662$	−3.96238e12	−0.801855
$663$	0	0
$664$	4.44222e12	0.886836
$665$	−2.91760e12	−0.578533
$666$	0	0
$667$	−4.90768e11	−0.0960085
$668$	1.35463e12	0.263224
$669$	0	0
$670$	3.30746e12	0.634101
$671$	−1.44456e12	−0.275097
$672$	0	0
$673$	4.97112e12	0.934086	0.467043	−	0.884235i	$-0.345319\pi$
$673$	4.97112e12	0.934086	0.467043	+	0.884235i	$0.345319\pi$
$674$	−3.46970e12	−0.647623
$675$	0	0
$676$	−1.44775e12	−0.266644
$677$	−7.64288e12	−1.39833	−0.699163	−	0.714963i	$-0.746444\pi$
$677$	−7.64288e12	−1.39833	−0.699163	+	0.714963i	$0.746444\pi$
$678$	0	0
$679$	−2.62612e12	−0.474133
$680$	−2.08711e12	−0.374331
$681$	0	0
$682$	−1.81200e13	−3.20722
$683$	4.20646e12	0.739646	0.369823	−	0.929102i	$-0.379418\pi$
$683$	4.20646e12	0.739646	0.369823	+	0.929102i	$0.379418\pi$
$684$	0	0
$685$	7.06995e11	0.122690
$686$	−7.01276e12	−1.20901
$687$	0	0
$688$	−7.08698e12	−1.20590
$689$	−1.05865e12	−0.178964
$690$	0	0
$691$	−2.33689e12	−0.389931	−0.194966	−	0.980810i	$-0.562460\pi$
$691$	−2.33689e12	−0.389931	−0.194966	+	0.980810i	$0.562460\pi$
$692$	−1.20896e12	−0.200417
$693$	0	0
$694$	−7.27305e12	−1.19014
$695$	9.53877e11	0.155082
$696$	0	0
$697$	−1.68723e12	−0.270786
$698$	−2.49767e12	−0.398277
$699$	0	0
$700$	9.70682e11	0.152804
$701$	−1.10126e12	−0.172250	−0.0861251	−	0.996284i	$-0.527448\pi$
$701$	−1.10126e12	−0.172250	−0.0861251	+	0.996284i	$0.527448\pi$
$702$	0	0
$703$	5.71450e12	0.882429
$704$	−5.51222e12	−0.845765
$705$	0	0
$706$	9.84403e11	0.149125
$707$	6.43105e11	0.0968043
$708$	0	0
$709$	1.31550e12	0.195516	0.0977578	−	0.995210i	$-0.468833\pi$
$709$	1.31550e12	0.195516	0.0977578	+	0.995210i	$0.468833\pi$
$710$	−2.51085e12	−0.370816
$711$	0	0
$712$	−2.44514e12	−0.356570
$713$	−1.10588e12	−0.160253
$714$	0	0
$715$	−1.33236e12	−0.190653
$716$	3.45314e12	0.491028
$717$	0	0
$718$	−1.14939e13	−1.61402
$719$	1.32450e13	1.84830	0.924151	−	0.382027i	$-0.124774\pi$
$719$	1.32450e13	1.84830	0.924151	+	0.382027i	$0.124774\pi$
$720$	0	0
$721$	1.56074e11	0.0215091
$722$	−1.26875e13	−1.73764
$723$	0	0
$724$	2.05482e12	0.277939
$725$	6.38043e12	0.857687
$726$	0	0
$727$	4.71696e12	0.626264	0.313132	−	0.949710i	$-0.398622\pi$
$727$	4.71696e12	0.626264	0.313132	+	0.949710i	$0.398622\pi$
$728$	1.13211e12	0.149382
$729$	0	0
$730$	−3.95034e12	−0.514851
$731$	7.06792e12	0.915511
$732$	0	0
$733$	1.65912e12	0.212280	0.106140	−	0.994351i	$-0.466151\pi$
$733$	1.65912e12	0.212280	0.106140	+	0.994351i	$0.466151\pi$
$734$	1.91911e12	0.244043
$735$	0	0
$736$	3.68065e11	0.0462353
$737$	−1.30364e13	−1.62763
$738$	0	0
$739$	−5.72028e12	−0.705533	−0.352767	−	0.935711i	$-0.614759\pi$
$739$	−5.72028e12	−0.705533	−0.352767	+	0.935711i	$0.614759\pi$
$740$	6.52552e11	0.0799967
$741$	0	0
$742$	4.69931e12	0.569137
$743$	6.71743e12	0.808637	0.404318	−	0.914618i	$-0.367509\pi$
$743$	6.71743e12	0.808637	0.404318	+	0.914618i	$0.367509\pi$
$744$	0	0
$745$	2.49372e12	0.296581
$746$	7.16752e12	0.847314
$747$	0	0
$748$	−3.28558e12	−0.383755
$749$	7.39340e12	0.858373
$750$	0	0
$751$	9.11015e12	1.04507	0.522536	−	0.852618i	$-0.324986\pi$
$751$	9.11015e12	1.04507	0.522536	+	0.852618i	$0.324986\pi$
$752$	−1.41168e13	−1.60974
$753$	0	0
$754$	−2.97209e12	−0.334882
$755$	−4.04547e11	−0.0453114
$756$	0	0
$757$	−4.86606e12	−0.538575	−0.269288	−	0.963060i	$-0.586788\pi$
$757$	−4.86606e12	−0.538575	−0.269288	+	0.963060i	$0.586788\pi$
$758$	1.64343e13	1.80818
$759$	0	0
$760$	5.99439e12	0.651754
$761$	−3.71178e12	−0.401191	−0.200596	−	0.979674i	$-0.564288\pi$
$761$	−3.71178e12	−0.401191	−0.200596	+	0.979674i	$0.564288\pi$
$762$	0	0
$763$	3.91176e12	0.417842
$764$	1.42502e12	0.151322
$765$	0	0
$766$	8.54423e12	0.896692
$767$	−2.52994e12	−0.263956
$768$	0	0
$769$	−6.63302e12	−0.683979	−0.341989	−	0.939704i	$-0.611101\pi$
$769$	−6.63302e12	−0.683979	−0.341989	+	0.939704i	$0.611101\pi$
$770$	5.91431e12	0.606312
$771$	0	0
$772$	1.45422e12	0.147351
$773$	7.34772e12	0.740193	0.370096	−	0.928993i	$-0.379325\pi$
$773$	7.34772e12	0.740193	0.370096	+	0.928993i	$0.379325\pi$
$774$	0	0
$775$	1.43774e13	1.43161
$776$	5.39552e12	0.534140
$777$	0	0
$778$	−4.18568e12	−0.409598
$779$	4.84589e12	0.471471
$780$	0	0
$781$	9.89658e12	0.951821
$782$	−9.03108e11	−0.0863593
$783$	0	0
$784$	6.14905e12	0.581281
$785$	−5.93540e11	−0.0557875
$786$	0	0
$787$	−3.81365e12	−0.354368	−0.177184	−	0.984178i	$-0.556699\pi$
$787$	−3.81365e12	−0.354368	−0.177184	+	0.984178i	$0.556699\pi$
$788$	4.72163e11	0.0436238
$789$	0	0
$790$	7.15122e12	0.653218
$791$	8.82902e12	0.801897
$792$	0	0
$793$	−5.33916e11	−0.0479451
$794$	−7.16051e12	−0.639369
$795$	0	0
$796$	−4.23996e12	−0.374329
$797$	1.78732e13	1.56906	0.784529	−	0.620092i	$-0.212905\pi$
$797$	1.78732e13	1.56906	0.784529	+	0.620092i	$0.212905\pi$
$798$	0	0
$799$	1.40788e13	1.22210
$800$	−4.78518e12	−0.413041
$801$	0	0
$802$	1.94953e13	1.66397
$803$	1.55703e13	1.32153
$804$	0	0
$805$	3.60956e11	0.0302951
$806$	−6.69722e12	−0.558968
$807$	0	0
$808$	−1.32130e12	−0.109056
$809$	−1.16949e13	−0.959908	−0.479954	−	0.877294i	$-0.659347\pi$
$809$	−1.16949e13	−0.959908	−0.479954	+	0.877294i	$0.659347\pi$
$810$	0	0
$811$	−3.78089e12	−0.306902	−0.153451	−	0.988156i	$-0.549039\pi$
$811$	−3.78089e12	−0.306902	−0.153451	+	0.988156i	$0.549039\pi$
$812$	2.92932e12	0.236464
$813$	0	0
$814$	−1.15840e13	−0.924799
$815$	4.94817e12	0.392857
$816$	0	0
$817$	−2.02997e13	−1.59401
$818$	9.74701e11	0.0761170
$819$	0	0
$820$	5.53363e11	0.0427413
$821$	−7.32598e12	−0.562757	−0.281379	−	0.959597i	$-0.590792\pi$
$821$	−7.32598e12	−0.562757	−0.281379	+	0.959597i	$0.590792\pi$
$822$	0	0
$823$	6.46115e12	0.490920	0.245460	−	0.969407i	$-0.421061\pi$
$823$	6.46115e12	0.490920	0.245460	+	0.969407i	$0.421061\pi$
$824$	−3.20664e11	−0.0242314
$825$	0	0
$826$	1.12303e13	0.839427
$827$	1.17187e13	0.871176	0.435588	−	0.900146i	$-0.356540\pi$
$827$	1.17187e13	0.871176	0.435588	+	0.900146i	$0.356540\pi$
$828$	0	0
$829$	−2.81041e12	−0.206669	−0.103334	−	0.994647i	$-0.532951\pi$
$829$	−2.81041e12	−0.206669	−0.103334	+	0.994647i	$0.532951\pi$
$830$	8.57733e12	0.627337
$831$	0	0
$832$	−2.03734e12	−0.147404
$833$	−6.13251e12	−0.441302
$834$	0	0
$835$	−6.54892e12	−0.466209
$836$	9.43650e12	0.668163
$837$	0	0
$838$	−1.88209e13	−1.31839
$839$	1.60278e13	1.11672	0.558361	−	0.829598i	$-0.311430\pi$
$839$	1.60278e13	1.11672	0.558361	+	0.829598i	$0.311430\pi$
$840$	0	0
$841$	4.74769e12	0.327266
$842$	8.24601e12	0.565379
$843$	0	0
$844$	−6.21825e11	−0.0421820
$845$	6.99912e12	0.472268
$846$	0	0
$847$	−1.25393e13	−0.837141
$848$	−1.26550e13	−0.840389
$849$	0	0
$850$	1.17412e13	0.771486
$851$	−7.06980e11	−0.0462087
$852$	0	0
$853$	−1.79323e13	−1.15975	−0.579875	−	0.814705i	$-0.696899\pi$
$853$	−1.79323e13	−1.15975	−0.579875	+	0.814705i	$0.696899\pi$
$854$	2.37004e12	0.152474
$855$	0	0
$856$	−1.51902e13	−0.967011
$857$	4.36164e12	0.276208	0.138104	−	0.990418i	$-0.455899\pi$
$857$	4.36164e12	0.276208	0.138104	+	0.990418i	$0.455899\pi$
$858$	0	0
$859$	7.35160e12	0.460694	0.230347	−	0.973109i	$-0.426014\pi$
$859$	7.35160e12	0.460694	0.230347	+	0.973109i	$0.426014\pi$
$860$	−2.31807e12	−0.144505
$861$	0	0
$862$	9.90496e12	0.611041
$863$	−1.41639e13	−0.869231	−0.434615	−	0.900616i	$-0.643116\pi$
$863$	−1.41639e13	−0.869231	−0.434615	+	0.900616i	$0.643116\pi$
$864$	0	0
$865$	5.84471e12	0.354969
$866$	−1.40791e13	−0.850638
$867$	0	0
$868$	6.60084e12	0.394694
$869$	−2.81867e13	−1.67670
$870$	0	0
$871$	−4.81832e12	−0.283670
$872$	−8.03695e12	−0.470725
$873$	0	0
$874$	2.59381e12	0.150362
$875$	−1.09962e13	−0.634171
$876$	0	0
$877$	−4.68660e12	−0.267522	−0.133761	−	0.991014i	$-0.542706\pi$
$877$	−4.68660e12	−0.267522	−0.133761	+	0.991014i	$0.542706\pi$
$878$	7.69877e12	0.437216
$879$	0	0
$880$	−1.59269e13	−0.895282
$881$	−1.22598e13	−0.685635	−0.342817	−	0.939402i	$-0.611381\pi$
$881$	−1.22598e13	−0.685635	−0.342817	+	0.939402i	$0.611381\pi$
$882$	0	0
$883$	−6.77194e12	−0.374878	−0.187439	−	0.982276i	$-0.560019\pi$
$883$	−6.77194e12	−0.374878	−0.187439	+	0.982276i	$0.560019\pi$
$884$	−1.21436e12	−0.0668826
$885$	0	0
$886$	1.08429e13	0.591145
$887$	2.17402e13	1.17925	0.589627	−	0.807675i	$-0.299275\pi$
$887$	2.17402e13	1.17925	0.589627	+	0.807675i	$0.299275\pi$
$888$	0	0
$889$	−2.27829e13	−1.22335
$890$	−4.72125e12	−0.252233
$891$	0	0
$892$	6.18875e12	0.327311
$893$	−4.04357e13	−2.12781
$894$	0	0
$895$	−1.66942e13	−0.869684
$896$	1.67413e13	0.867765
$897$	0	0
$898$	4.19203e13	2.15120
$899$	4.33882e13	2.21540
$900$	0	0
$901$	1.26210e13	0.638015
$902$	−9.82318e12	−0.494109
$903$	0	0
$904$	−1.81398e13	−0.903387
$905$	−9.93399e12	−0.492272
$906$	0	0
$907$	−2.27896e13	−1.11816	−0.559079	−	0.829114i	$-0.688845\pi$
$907$	−2.27896e13	−1.11816	−0.559079	+	0.829114i	$0.688845\pi$
$908$	−8.65954e12	−0.422774
$909$	0	0
$910$	2.18595e12	0.105671
$911$	1.12776e13	0.542482	0.271241	−	0.962511i	$-0.412566\pi$
$911$	1.12776e13	0.542482	0.271241	+	0.962511i	$0.412566\pi$
$912$	0	0
$913$	−3.38078e13	−1.61027
$914$	−3.19743e13	−1.51546
$915$	0	0
$916$	−8.98750e12	−0.421803
$917$	−8.75637e11	−0.0408942
$918$	0	0
$919$	−2.27954e12	−0.105421	−0.0527105	−	0.998610i	$-0.516786\pi$
$919$	−2.27954e12	−0.105421	−0.0527105	+	0.998610i	$0.516786\pi$
$920$	−7.41606e11	−0.0341294
$921$	0	0
$922$	7.38966e12	0.336771
$923$	3.65781e12	0.165888
$924$	0	0
$925$	9.19138e12	0.412803
$926$	−4.03450e13	−1.80318
$927$	0	0
$928$	−1.44407e13	−0.639179
$929$	1.88735e13	0.831344	0.415672	−	0.909514i	$-0.363546\pi$
$929$	1.88735e13	0.831344	0.415672	+	0.909514i	$0.363546\pi$
$930$	0	0
$931$	1.76132e13	0.768359
$932$	6.85906e12	0.297778
$933$	0	0
$934$	5.46128e12	0.234819
$935$	1.58841e13	0.679689
$936$	0	0
$937$	−1.08798e13	−0.461096	−0.230548	−	0.973061i	$-0.574052\pi$
$937$	−1.08798e13	−0.461096	−0.230548	+	0.973061i	$0.574052\pi$
$938$	2.13884e13	0.902122
$939$	0	0
$940$	−4.61744e12	−0.192897
$941$	−2.83800e13	−1.17994	−0.589970	−	0.807426i	$-0.700860\pi$
$941$	−2.83800e13	−1.17994	−0.589970	+	0.807426i	$0.700860\pi$
$942$	0	0
$943$	−5.99518e11	−0.0246888
$944$	−3.02427e13	−1.23950
$945$	0	0
$946$	4.11500e13	1.67055
$947$	−3.58374e12	−0.144798	−0.0723988	−	0.997376i	$-0.523065\pi$
$947$	−3.58374e12	−0.144798	−0.0723988	+	0.997376i	$0.523065\pi$
$948$	0	0
$949$	5.75486e12	0.230323
$950$	−3.37219e13	−1.34325
$951$	0	0
$952$	−1.34967e13	−0.532553
$953$	−2.91182e12	−0.114353	−0.0571763	−	0.998364i	$-0.518210\pi$
$953$	−2.91182e12	−0.114353	−0.0571763	+	0.998364i	$0.518210\pi$
$954$	0	0
$955$	−6.88926e12	−0.268014
$956$	−1.52686e12	−0.0591204
$957$	0	0
$958$	3.91181e13	1.50049
$959$	4.57193e12	0.174549
$960$	0	0
$961$	7.13299e13	2.69784
$962$	−4.28148e12	−0.161178
$963$	0	0
$964$	4.37505e12	0.163168
$965$	−7.03043e12	−0.260981
$966$	0	0
$967$	−2.68326e13	−0.986833	−0.493417	−	0.869793i	$-0.664252\pi$
$967$	−2.68326e13	−0.986833	−0.493417	+	0.869793i	$0.664252\pi$
$968$	2.57628e13	0.943092
$969$	0	0
$970$	1.04180e13	0.377844
$971$	−5.18797e13	−1.87288	−0.936442	−	0.350824i	$-0.885902\pi$
$971$	−5.18797e13	−1.87288	−0.936442	+	0.350824i	$0.885902\pi$
$972$	0	0
$973$	6.16844e12	0.220632
$974$	4.64358e13	1.65325
$975$	0	0
$976$	−6.38239e12	−0.225143
$977$	−5.55949e13	−1.95213	−0.976067	−	0.217469i	$-0.930220\pi$
$977$	−5.55949e13	−1.95213	−0.976067	+	0.217469i	$0.930220\pi$
$978$	0	0
$979$	1.86089e13	0.647439
$980$	2.01129e12	0.0696557
$981$	0	0
$982$	−2.96003e13	−1.01577
$983$	−4.02241e13	−1.37403	−0.687015	−	0.726644i	$-0.741079\pi$
$983$	−4.02241e13	−1.37403	−0.687015	+	0.726644i	$0.741079\pi$
$984$	0	0
$985$	−2.28267e12	−0.0772644
$986$	3.54326e13	1.19387
$987$	0	0
$988$	3.48776e12	0.116450
$989$	2.51142e12	0.0834710
$990$	0	0
$991$	1.34682e13	0.443588	0.221794	−	0.975094i	$-0.428809\pi$
$991$	1.34682e13	0.443588	0.221794	+	0.975094i	$0.428809\pi$
$992$	−3.25402e13	−1.06688
$993$	0	0
$994$	−1.62369e13	−0.527552
$995$	2.04981e13	0.662993
$996$	0	0
$997$	−3.74584e13	−1.20066	−0.600331	−	0.799752i	$-0.704964\pi$
$997$	−3.74584e13	−1.20066	−0.600331	+	0.799752i	$0.704964\pi$
$998$	5.18065e13	1.65309
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.10.a.d.1.2		8
3.2	odd	2		81.10.a.c.1.7		8
9.2	odd	6		9.10.c.a.4.2	✓	16
9.4	even	3		27.10.c.a.19.7		16
9.5	odd	6		9.10.c.a.7.2	yes	16
9.7	even	3		27.10.c.a.10.7		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.10.c.a.4.2	✓	16	9.2	odd	6
9.10.c.a.7.2	yes	16	9.5	odd	6
27.10.c.a.10.7		16	9.7	even	3
27.10.c.a.19.7		16	9.4	even	3
81.10.a.c.1.7		8	3.2	odd	2
81.10.a.d.1.2		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	81.a (trivial)

\(f(q)\)	\(=\)	\(q-25.6540 q^{2} +146.127 q^{4} -706.451 q^{5} -4568.42 q^{7} +9386.09 q^{8} +18123.3 q^{10} -71433.4 q^{11} -26402.1 q^{13} +117198. q^{14} -315608. q^{16} +314759. q^{17} -904019. q^{19} -103232. q^{20} +1.83255e6 q^{22} +111842. q^{23} -1.45405e6 q^{25} +677318. q^{26} -667571. q^{28} -4.38803e6 q^{29} -9.88785e6 q^{31} +3.29093e6 q^{32} -8.07483e6 q^{34} +3.22736e6 q^{35} -6.32122e6 q^{37} +2.31917e7 q^{38} -6.63082e6 q^{40} -5.36038e6 q^{41} +2.24550e7 q^{43} -1.04384e7 q^{44} -2.86920e6 q^{46} +4.47288e7 q^{47} -1.94832e7 q^{49} +3.73022e7 q^{50} -3.85806e6 q^{52} +4.00972e7 q^{53} +5.04642e7 q^{55} -4.28796e7 q^{56} +1.12571e8 q^{58} +9.58236e7 q^{59} +2.02225e7 q^{61} +2.53663e8 q^{62} +7.71659e7 q^{64} +1.86518e7 q^{65} +1.82498e8 q^{67} +4.59950e7 q^{68} -8.27948e7 q^{70} -1.38543e8 q^{71} -2.17970e8 q^{73} +1.62165e8 q^{74} -1.32102e8 q^{76} +3.26337e8 q^{77} +3.94587e8 q^{79} +2.22962e8 q^{80} +1.37515e8 q^{82} +4.73277e8 q^{83} -2.22362e8 q^{85} -5.76060e8 q^{86} -6.70481e8 q^{88} -2.60507e8 q^{89} +1.20616e8 q^{91} +1.63432e7 q^{92} -1.14747e9 q^{94} +6.38646e8 q^{95} +5.74842e8 q^{97} +4.99822e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 15 q^{2} + 1793 q^{4} + 453 q^{5} + 343 q^{7} + 7239 q^{8} + 510 q^{10} + 99150 q^{11} - 32435 q^{13} + 394824 q^{14} + 328193 q^{16} + 415539 q^{17} - 85277 q^{19} + 1855164 q^{20} - 529359 q^{22}+ \cdots - 2413650159 q^{98}+O(q^{100}) \) 8 * q + 15 * q^2 + 1793 * q^4 + 453 * q^5 + 343 * q^7 + 7239 * q^8 + 510 * q^10 + 99150 * q^11 - 32435 * q^13 + 394824 * q^14 + 328193 * q^16 + 415539 * q^17 - 85277 * q^19 + 1855164 * q^20 - 529359 * q^22 + 1064559 * q^23 + 2293229 * q^25 - 1218156 * q^26 + 612862 * q^28 - 1309053 * q^29 + 2359819 * q^31 + 5760063 * q^32 - 981801 * q^34 + 15533277 * q^35 + 8195758 * q^37 + 39490203 * q^38 + 16760496 * q^40 + 54747318 * q^41 - 15249608 * q^43 + 166254963 * q^44 + 1195260 * q^46 + 156295545 * q^47 - 15239583 * q^49 + 315590163 * q^50 + 19773358 * q^52 + 262758114 * q^53 - 3789885 * q^55 + 470339790 * q^56 - 55408560 * q^58 + 307774074 * q^59 - 69192125 * q^61 + 457218462 * q^62 - 201794239 * q^64 + 482470359 * q^65 - 14328044 * q^67 + 915409575 * q^68 + 229271934 * q^70 + 619800696 * q^71 + 299306599 * q^73 + 1022736000 * q^74 - 119954093 * q^76 + 717995541 * q^77 - 30257531 * q^79 + 1463913264 * q^80 - 101188011 * q^82 + 1176168291 * q^83 - 4818366 * q^85 + 1426944009 * q^86 - 911312427 * q^88 + 1658520648 * q^89 - 369615061 * q^91 - 76813998 * q^92 + 1954316784 * q^94 - 391400652 * q^95 + 267311278 * q^97 - 2413650159 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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