LMFDB - Embedded newform 81.10.a.d.1.4

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			1.4
Root			$0.320476$ 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)

$f(q)$	$=$	\(q+1.67952 q^{2} -509.179 q^{4} +1538.61 q^{5} +5656.94 q^{7} -1715.10 q^{8} +2584.13 q^{10} -19816.0 q^{11} +99794.5 q^{13} +9500.96 q^{14} +257819. q^{16} -503911. q^{17} -670691. q^{19} -783426. q^{20} -33281.4 q^{22} +1.93025e6 q^{23} +414183. q^{25} +167607. q^{26} -2.88039e6 q^{28} +4.32636e6 q^{29} +1.02625e6 q^{31} +1.31114e6 q^{32} -846330. q^{34} +8.70380e6 q^{35} +1.81292e7 q^{37} -1.12644e6 q^{38} -2.63886e6 q^{40} -9.30385e6 q^{41} -1.57198e6 q^{43} +1.00899e7 q^{44} +3.24191e6 q^{46} +3.83357e7 q^{47} -8.35267e6 q^{49} +695630. q^{50} -5.08133e7 q^{52} +4.39042e7 q^{53} -3.04890e7 q^{55} -9.70218e6 q^{56} +7.26623e6 q^{58} +1.61803e8 q^{59} +1.45581e8 q^{61} +1.72361e6 q^{62} -1.29801e8 q^{64} +1.53544e8 q^{65} -8.15422e6 q^{67} +2.56581e8 q^{68} +1.46182e7 q^{70} +1.10960e8 q^{71} +1.42423e8 q^{73} +3.04484e7 q^{74} +3.41502e8 q^{76} -1.12098e8 q^{77} -2.00982e8 q^{79} +3.96682e8 q^{80} -1.56260e7 q^{82} +5.11970e8 q^{83} -7.75320e8 q^{85} -2.64018e6 q^{86} +3.39863e7 q^{88} -2.71324e8 q^{89} +5.64531e8 q^{91} -9.82845e8 q^{92} +6.43857e7 q^{94} -1.03193e9 q^{95} -5.98982e8 q^{97} -1.40285e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.67952	0.0742252	0.0371126	−	0.999311i	$-0.488184\pi$
$2$	1.67952	0.0742252	0.0371126	+	0.999311i	$0.488184\pi$
$3$	0	0
$3$	0	0
$4$	−509.179	−0.994491
$5$	1538.61	1.10094	0.550468	−	0.834856i	$-0.314449\pi$
$5$	1538.61	1.10094	0.550468	+	0.834856i	$0.314449\pi$
$6$	0	0
$7$	5656.94	0.890513	0.445256	−	0.895403i	$-0.353113\pi$
$7$	5656.94	0.890513	0.445256	+	0.895403i	$0.353113\pi$
$8$	−1715.10	−0.148041
$9$	0	0
$10$	2584.13	0.0817172
$11$	−19816.0	−0.408083	−0.204042	−	0.978962i	$-0.565408\pi$
$11$	−19816.0	−0.408083	−0.204042	+	0.978962i	$0.565408\pi$
$12$	0	0
$13$	99794.5	0.969084	0.484542	−	0.874768i	$-0.338986\pi$
$13$	99794.5	0.969084	0.484542	+	0.874768i	$0.338986\pi$
$14$	9500.96	0.0660985
$15$	0	0
$16$	257819.	0.983502
$17$	−503911.	−1.46330	−0.731650	−	0.681680i	$-0.761250\pi$
$17$	−503911.	−1.46330	−0.731650	+	0.681680i	$0.761250\pi$
$18$	0	0
$19$	−670691.	−1.18068	−0.590339	−	0.807156i	$-0.701006\pi$
$19$	−670691.	−1.18068	−0.590339	+	0.807156i	$0.701006\pi$
$20$	−783426.	−1.09487
$21$	0	0
$22$	−33281.4	−0.0302901
$23$	1.93025e6	1.43827	0.719133	−	0.694873i	$-0.244539\pi$
$23$	1.93025e6	1.43827	0.719133	+	0.694873i	$0.244539\pi$
$24$	0	0
$25$	414183.	0.212061
$26$	167607.	0.0719305
$27$	0	0
$28$	−2.88039e6	−0.885607
$29$	4.32636e6	1.13588	0.567939	−	0.823070i	$-0.307741\pi$
$29$	4.32636e6	1.13588	0.567939	+	0.823070i	$0.307741\pi$
$30$	0	0
$31$	1.02625e6	0.199584	0.0997920	−	0.995008i	$-0.468182\pi$
$31$	1.02625e6	0.199584	0.0997920	+	0.995008i	$0.468182\pi$
$32$	1.31114e6	0.221042
$33$	0	0
$34$	−846330.	−0.108614
$35$	8.70380e6	0.980398
$36$	0	0
$37$	1.81292e7	1.59027	0.795133	−	0.606435i	$-0.207401\pi$
$37$	1.81292e7	1.59027	0.795133	+	0.606435i	$0.207401\pi$
$38$	−1.12644e6	−0.0876360
$39$	0	0
$40$	−2.63886e6	−0.162984
$41$	−9.30385e6	−0.514204	−0.257102	−	0.966384i	$-0.582768\pi$
$41$	−9.30385e6	−0.514204	−0.257102	+	0.966384i	$0.582768\pi$
$42$	0	0
$43$	−1.57198e6	−0.0701196	−0.0350598	−	0.999385i	$-0.511162\pi$
$43$	−1.57198e6	−0.0701196	−0.0350598	+	0.999385i	$0.511162\pi$
$44$	1.00899e7	0.405835
$45$	0	0
$46$	3.24191e6	0.106756
$47$	3.83357e7	1.14594	0.572972	−	0.819575i	$-0.305790\pi$
$47$	3.83357e7	1.14594	0.572972	+	0.819575i	$0.305790\pi$
$48$	0	0
$49$	−8.35267e6	−0.206987
$50$	695630.	0.0157403
$51$	0	0
$52$	−5.08133e7	−0.963745
$53$	4.39042e7	0.764301	0.382151	−	0.924100i	$-0.375184\pi$
$53$	4.39042e7	0.764301	0.382151	+	0.924100i	$0.375184\pi$
$54$	0	0
$55$	−3.04890e7	−0.449274
$56$	−9.70218e6	−0.131833
$57$	0	0
$58$	7.26623e6	0.0843108
$59$	1.61803e8	1.73841	0.869204	−	0.494453i	$-0.164632\pi$
$59$	1.61803e8	1.73841	0.869204	+	0.494453i	$0.164632\pi$
$60$	0	0
$61$	1.45581e8	1.34623	0.673117	−	0.739536i	$-0.264955\pi$
$61$	1.45581e8	1.34623	0.673117	+	0.739536i	$0.264955\pi$
$62$	1.72361e6	0.0148142
$63$	0	0
$64$	−1.29801e8	−0.967095
$65$	1.53544e8	1.06690
$66$	0	0
$67$	−8.15422e6	−0.0494363	−0.0247181	−	0.999694i	$-0.507869\pi$
$67$	−8.15422e6	−0.0494363	−0.0247181	+	0.999694i	$0.507869\pi$
$68$	2.56581e8	1.45524
$69$	0	0
$70$	1.46182e7	0.0727702
$71$	1.10960e8	0.518208	0.259104	−	0.965849i	$-0.416573\pi$
$71$	1.10960e8	0.518208	0.259104	+	0.965849i	$0.416573\pi$
$72$	0	0
$73$	1.42423e8	0.586986	0.293493	−	0.955961i	$-0.405182\pi$
$73$	1.42423e8	0.586986	0.293493	+	0.955961i	$0.405182\pi$
$74$	3.04484e7	0.118038
$75$	0	0
$76$	3.41502e8	1.17417
$77$	−1.12098e8	−0.363403
$78$	0	0
$79$	−2.00982e8	−0.580545	−0.290273	−	0.956944i	$-0.593746\pi$
$79$	−2.00982e8	−0.580545	−0.290273	+	0.956944i	$0.593746\pi$
$80$	3.96682e8	1.08277
$81$	0	0
$82$	−1.56260e7	−0.0381669
$83$	5.11970e8	1.18411	0.592056	−	0.805897i	$-0.298316\pi$
$83$	5.11970e8	1.18411	0.592056	+	0.805897i	$0.298316\pi$
$84$	0	0
$85$	−7.75320e8	−1.61100
$86$	−2.64018e6	−0.00520464
$87$	0	0
$88$	3.39863e7	0.0604132
$89$	−2.71324e8	−0.458387	−0.229194	−	0.973381i	$-0.573609\pi$
$89$	−2.71324e8	−0.458387	−0.229194	+	0.973381i	$0.573609\pi$
$90$	0	0
$91$	5.64531e8	0.862982
$92$	−9.82845e8	−1.43034
$93$	0	0
$94$	6.43857e7	0.0850579
$95$	−1.03193e9	−1.29985
$96$	0	0
$97$	−5.98982e8	−0.686975	−0.343488	−	0.939157i	$-0.611608\pi$
$97$	−5.98982e8	−0.686975	−0.343488	+	0.939157i	$0.611608\pi$
$98$	−1.40285e7	−0.0153637
$99$	0	0
$100$	−2.10893e8	−0.210893
$101$	7.35052e8	0.702865	0.351433	−	0.936213i	$-0.385695\pi$
$101$	7.35052e8	0.702865	0.351433	+	0.936213i	$0.385695\pi$
$102$	0	0
$103$	4.11768e8	0.360483	0.180242	−	0.983622i	$-0.442312\pi$
$103$	4.11768e8	0.360483	0.180242	+	0.983622i	$0.442312\pi$
$104$	−1.71157e8	−0.143465
$105$	0	0
$106$	7.37382e7	0.0567304
$107$	2.79607e8	0.206215	0.103108	−	0.994670i	$-0.467121\pi$
$107$	2.79607e8	0.206215	0.103108	+	0.994670i	$0.467121\pi$
$108$	0	0
$109$	−4.87814e8	−0.331006	−0.165503	−	0.986209i	$-0.552925\pi$
$109$	−4.87814e8	−0.331006	−0.165503	+	0.986209i	$0.552925\pi$
$110$	−5.12070e7	−0.0333474
$111$	0	0
$112$	1.45847e9	0.875821
$113$	2.28051e9	1.31577	0.657884	−	0.753119i	$-0.271452\pi$
$113$	2.28051e9	1.31577	0.657884	+	0.753119i	$0.271452\pi$
$114$	0	0
$115$	2.96990e9	1.58344
$116$	−2.20289e9	−1.12962
$117$	0	0
$118$	2.71752e8	0.129034
$119$	−2.85059e9	−1.30309
$120$	0	0
$121$	−1.96527e9	−0.833468
$122$	2.44507e8	0.0999245
$123$	0	0
$124$	−5.22545e8	−0.198484
$125$	−2.36783e9	−0.867470
$126$	0	0
$127$	−3.67474e9	−1.25346	−0.626730	−	0.779237i	$-0.715607\pi$
$127$	−3.67474e9	−1.25346	−0.626730	+	0.779237i	$0.715607\pi$
$128$	−8.89309e8	−0.292825
$129$	0	0
$130$	2.57882e8	0.0791909
$131$	−5.07549e9	−1.50576	−0.752882	−	0.658156i	$-0.771337\pi$
$131$	−5.07549e9	−1.50576	−0.752882	+	0.658156i	$0.771337\pi$
$132$	0	0
$133$	−3.79406e9	−1.05141
$134$	−1.36952e7	−0.00366942
$135$	0	0
$136$	8.64255e8	0.216629
$137$	−2.01688e9	−0.489146	−0.244573	−	0.969631i	$-0.578648\pi$
$137$	−2.01688e9	−0.489146	−0.244573	+	0.969631i	$0.578648\pi$
$138$	0	0
$139$	2.64813e9	0.601690	0.300845	−	0.953673i	$-0.402731\pi$
$139$	2.64813e9	0.601690	0.300845	+	0.953673i	$0.402731\pi$
$140$	−4.43179e9	−0.974997
$141$	0	0
$142$	1.86360e8	0.0384641
$143$	−1.97753e9	−0.395467
$144$	0	0
$145$	6.65657e9	1.25053
$146$	2.39203e8	0.0435691
$147$	0	0
$148$	−9.23099e9	−1.58151
$149$	−1.00926e9	−0.167752	−0.0838758	−	0.996476i	$-0.526730\pi$
$149$	−1.00926e9	−0.167752	−0.0838758	+	0.996476i	$0.526730\pi$
$150$	0	0
$151$	−5.84420e8	−0.0914805	−0.0457402	−	0.998953i	$-0.514565\pi$
$151$	−5.84420e8	−0.0914805	−0.0457402	+	0.998953i	$0.514565\pi$
$152$	1.15030e9	0.174789
$153$	0	0
$154$	−1.88271e8	−0.0269737
$155$	1.57900e9	0.219729
$156$	0	0
$157$	7.48463e9	0.983154	0.491577	−	0.870834i	$-0.336421\pi$
$157$	7.48463e9	0.983154	0.491577	+	0.870834i	$0.336421\pi$
$158$	−3.37555e8	−0.0430911
$159$	0	0
$160$	2.01733e9	0.243353
$161$	1.09193e10	1.28079
$162$	0	0
$163$	4.71277e9	0.522916	0.261458	−	0.965215i	$-0.415797\pi$
$163$	4.71277e9	0.522916	0.261458	+	0.965215i	$0.415797\pi$
$164$	4.73733e9	0.511371
$165$	0	0
$166$	8.59866e8	0.0878910
$167$	1.21767e10	1.21145	0.605725	−	0.795674i	$-0.292883\pi$
$167$	1.21767e10	1.21145	0.605725	+	0.795674i	$0.292883\pi$
$168$	0	0
$169$	−6.45552e8	−0.0608753
$170$	−1.30217e9	−0.119577
$171$	0	0
$172$	8.00421e8	0.0697333
$173$	5.10501e9	0.433301	0.216650	−	0.976249i	$-0.430487\pi$
$173$	5.10501e9	0.433301	0.216650	+	0.976249i	$0.430487\pi$
$174$	0	0
$175$	2.34300e9	0.188843
$176$	−5.10894e9	−0.401351
$177$	0	0
$178$	−4.55694e8	−0.0340239
$179$	1.79471e9	0.130664	0.0653320	−	0.997864i	$-0.479189\pi$
$179$	1.79471e9	0.130664	0.0653320	+	0.997864i	$0.479189\pi$
$180$	0	0
$181$	−2.66921e10	−1.84854	−0.924271	−	0.381738i	$-0.875326\pi$
$181$	−2.66921e10	−1.84854	−0.924271	+	0.381738i	$0.875326\pi$
$182$	9.48144e8	0.0640550
$183$	0	0
$184$	−3.31057e9	−0.212923
$185$	2.78936e10	1.75078
$186$	0	0
$187$	9.98549e9	0.597149
$188$	−1.95197e10	−1.13963
$189$	0	0
$190$	−1.73315e9	−0.0964817
$191$	7.42827e8	0.0403866	0.0201933	−	0.999796i	$-0.493572\pi$
$191$	7.42827e8	0.0403866	0.0201933	+	0.999796i	$0.493572\pi$
$192$	0	0
$193$	−2.03489e9	−0.105568	−0.0527841	−	0.998606i	$-0.516810\pi$
$193$	−2.03489e9	−0.105568	−0.0527841	+	0.998606i	$0.516810\pi$
$194$	−1.00600e9	−0.0509909
$195$	0	0
$196$	4.25301e9	0.205847
$197$	3.40311e10	1.60982	0.804910	−	0.593396i	$-0.202213\pi$
$197$	3.40311e10	1.60982	0.804910	+	0.593396i	$0.202213\pi$
$198$	0	0
$199$	1.03084e9	0.0465963	0.0232982	−	0.999729i	$-0.492583\pi$
$199$	1.03084e9	0.0465963	0.0232982	+	0.999729i	$0.492583\pi$
$200$	−7.10363e8	−0.0313939
$201$	0	0
$202$	1.23454e9	0.0521703
$203$	2.44740e10	1.01151
$204$	0	0
$205$	−1.43150e10	−0.566106
$206$	6.91574e8	0.0267569
$207$	0	0
$208$	2.57289e10	0.953097
$209$	1.32904e10	0.481815
$210$	0	0
$211$	1.48595e9	0.0516100	0.0258050	−	0.999667i	$-0.491785\pi$
$211$	1.48595e9	0.0516100	0.0258050	+	0.999667i	$0.491785\pi$
$212$	−2.23551e10	−0.760091
$213$	0	0
$214$	4.69607e8	0.0153064
$215$	−2.41866e9	−0.0771973
$216$	0	0
$217$	5.80543e9	0.177732
$218$	−8.19296e8	−0.0245689
$219$	0	0
$220$	1.55244e10	0.446799
$221$	−5.02875e10	−1.41806
$222$	0	0
$223$	−1.85640e9	−0.0502690	−0.0251345	−	0.999684i	$-0.508001\pi$
$223$	−1.85640e9	−0.0502690	−0.0251345	+	0.999684i	$0.508001\pi$
$224$	7.41705e9	0.196841
$225$	0	0
$226$	3.83018e9	0.0976631
$227$	6.96548e10	1.74114	0.870571	−	0.492042i	$-0.163749\pi$
$227$	6.96548e10	1.74114	0.870571	+	0.492042i	$0.163749\pi$
$228$	0	0
$229$	−2.03648e10	−0.489352	−0.244676	−	0.969605i	$-0.578682\pi$
$229$	−2.03648e10	−0.489352	−0.244676	+	0.969605i	$0.578682\pi$
$230$	4.98802e9	0.117531
$231$	0	0
$232$	−7.42012e9	−0.168157
$233$	4.32644e10	0.961676	0.480838	−	0.876809i	$-0.340332\pi$
$233$	4.32644e10	0.961676	0.480838	+	0.876809i	$0.340332\pi$
$234$	0	0
$235$	5.89835e10	1.26161
$236$	−8.23866e10	−1.72883
$237$	0	0
$238$	−4.78764e9	−0.0967219
$239$	3.76752e10	0.746905	0.373452	−	0.927649i	$-0.378174\pi$
$239$	3.76752e10	0.746905	0.373452	+	0.927649i	$0.378174\pi$
$240$	0	0
$241$	−3.34708e10	−0.639130	−0.319565	−	0.947564i	$-0.603537\pi$
$241$	−3.34708e10	−0.639130	−0.319565	+	0.947564i	$0.603537\pi$
$242$	−3.30073e9	−0.0618643
$243$	0	0
$244$	−7.41268e10	−1.33882
$245$	−1.28515e10	−0.227880
$246$	0	0
$247$	−6.69313e10	−1.14418
$248$	−1.76012e9	−0.0295467
$249$	0	0
$250$	−3.97682e9	−0.0643881
$251$	−7.16769e10	−1.13985	−0.569925	−	0.821697i	$-0.693028\pi$
$251$	−7.16769e10	−1.13985	−0.569925	+	0.821697i	$0.693028\pi$
$252$	0	0
$253$	−3.82499e10	−0.586932
$254$	−6.17182e9	−0.0930382
$255$	0	0
$256$	6.49647e10	0.945360
$257$	−4.25870e10	−0.608945	−0.304472	−	0.952521i	$-0.598480\pi$
$257$	−4.25870e10	−0.608945	−0.304472	+	0.952521i	$0.598480\pi$
$258$	0	0
$259$	1.02556e11	1.41615
$260$	−7.81816e10	−1.06102
$261$	0	0
$262$	−8.52440e9	−0.111766
$263$	1.75582e10	0.226297	0.113149	−	0.993578i	$-0.463906\pi$
$263$	1.75582e10	0.226297	0.113149	+	0.993578i	$0.463906\pi$
$264$	0	0
$265$	6.75513e10	0.841447
$266$	−6.37221e9	−0.0780410
$267$	0	0
$268$	4.15196e9	0.0491639
$269$	−6.58769e10	−0.767093	−0.383546	−	0.923522i	$-0.625297\pi$
$269$	−6.58769e10	−0.767093	−0.383546	+	0.923522i	$0.625297\pi$
$270$	0	0
$271$	−2.84327e10	−0.320226	−0.160113	−	0.987099i	$-0.551186\pi$
$271$	−2.84327e10	−0.320226	−0.160113	+	0.987099i	$0.551186\pi$
$272$	−1.29918e11	−1.43916
$273$	0	0
$274$	−3.38741e9	−0.0363069
$275$	−8.20744e9	−0.0865388
$276$	0	0
$277$	−6.72391e10	−0.686220	−0.343110	−	0.939295i	$-0.611480\pi$
$277$	−6.72391e10	−0.686220	−0.343110	+	0.939295i	$0.611480\pi$
$278$	4.44760e9	0.0446605
$279$	0	0
$280$	−1.49278e10	−0.145140
$281$	−1.31148e11	−1.25482	−0.627412	−	0.778687i	$-0.715886\pi$
$281$	−1.31148e11	−1.25482	−0.627412	+	0.778687i	$0.715886\pi$
$282$	0	0
$283$	1.15708e10	0.107232	0.0536159	−	0.998562i	$-0.482925\pi$
$283$	1.15708e10	0.107232	0.0536159	+	0.998562i	$0.482925\pi$
$284$	−5.64985e10	−0.515353
$285$	0	0
$286$	−3.32131e9	−0.0293536
$287$	−5.26313e10	−0.457905
$288$	0	0
$289$	1.35338e11	1.14125
$290$	1.11799e10	0.0928209
$291$	0	0
$292$	−7.25189e10	−0.583752
$293$	−1.58060e10	−0.125291	−0.0626453	−	0.998036i	$-0.519954\pi$
$293$	−1.58060e10	−0.125291	−0.0626453	+	0.998036i	$0.519954\pi$
$294$	0	0
$295$	2.48951e11	1.91388
$296$	−3.10932e10	−0.235425
$297$	0	0
$298$	−1.69508e9	−0.0124514
$299$	1.92629e11	1.39380
$300$	0	0
$301$	−8.89261e9	−0.0624424
$302$	−9.81547e8	−0.00679016
$303$	0	0
$304$	−1.72917e11	−1.16120
$305$	2.23992e11	1.48212
$306$	0	0
$307$	−4.10341e10	−0.263647	−0.131823	−	0.991273i	$-0.542083\pi$
$307$	−4.10341e10	−0.263647	−0.131823	+	0.991273i	$0.542083\pi$
$308$	5.70779e10	0.361401
$309$	0	0
$310$	2.65196e9	0.0163095
$311$	1.98468e10	0.120301	0.0601505	−	0.998189i	$-0.480842\pi$
$311$	1.98468e10	0.120301	0.0601505	+	0.998189i	$0.480842\pi$
$312$	0	0
$313$	−3.03685e11	−1.78844	−0.894219	−	0.447629i	$-0.852269\pi$
$313$	−3.03685e11	−1.78844	−0.894219	+	0.447629i	$0.852269\pi$
$314$	1.25706e10	0.0729748
$315$	0	0
$316$	1.02336e11	0.577347
$317$	8.01362e10	0.445720	0.222860	−	0.974850i	$-0.428461\pi$
$317$	8.01362e10	0.445720	0.222860	+	0.974850i	$0.428461\pi$
$318$	0	0
$319$	−8.57312e10	−0.463533
$320$	−1.99713e11	−1.06471
$321$	0	0
$322$	1.83393e10	0.0950672
$323$	3.37969e11	1.72769
$324$	0	0
$325$	4.13332e10	0.205506
$326$	7.91521e9	0.0388136
$327$	0	0
$328$	1.59570e10	0.0761235
$329$	2.16863e11	1.02048
$330$	0	0
$331$	−1.41037e11	−0.645813	−0.322907	−	0.946431i	$-0.604660\pi$
$331$	−1.41037e11	−0.645813	−0.322907	+	0.946431i	$0.604660\pi$
$332$	−2.60684e11	−1.17759
$333$	0	0
$334$	2.04511e10	0.0899201
$335$	−1.25461e10	−0.0544262
$336$	0	0
$337$	2.71754e10	0.114773	0.0573866	−	0.998352i	$-0.481723\pi$
$337$	2.71754e10	0.114773	0.0573866	+	0.998352i	$0.481723\pi$
$338$	−1.08422e9	−0.00451848
$339$	0	0
$340$	3.94777e11	1.60213
$341$	−2.03362e10	−0.0814469
$342$	0	0
$343$	−2.75528e11	−1.07484
$344$	2.69610e9	0.0103806
$345$	0	0
$346$	8.57400e9	0.0321618
$347$	2.72424e11	1.00870	0.504351	−	0.863498i	$-0.331732\pi$
$347$	2.72424e11	1.00870	0.504351	+	0.863498i	$0.331732\pi$
$348$	0	0
$349$	2.82971e11	1.02100	0.510502	−	0.859877i	$-0.329460\pi$
$349$	2.82971e11	1.02100	0.510502	+	0.859877i	$0.329460\pi$
$350$	3.93513e9	0.0140169
$351$	0	0
$352$	−2.59816e10	−0.0902036
$353$	−7.23120e10	−0.247870	−0.123935	−	0.992290i	$-0.539551\pi$
$353$	−7.23120e10	−0.247870	−0.123935	+	0.992290i	$0.539551\pi$
$354$	0	0
$355$	1.70724e11	0.570514
$356$	1.38152e11	0.455862
$357$	0	0
$358$	3.01426e9	0.00969856
$359$	−3.88231e11	−1.23358	−0.616788	−	0.787130i	$-0.711566\pi$
$359$	−3.88231e11	−1.23358	−0.616788	+	0.787130i	$0.711566\pi$
$360$	0	0
$361$	1.27139e11	0.394000
$362$	−4.48300e10	−0.137208
$363$	0	0
$364$	−2.87448e11	−0.858228
$365$	2.19133e11	0.646235
$366$	0	0
$367$	−1.57415e11	−0.452949	−0.226475	−	0.974017i	$-0.572720\pi$
$367$	−1.57415e11	−0.452949	−0.226475	+	0.974017i	$0.572720\pi$
$368$	4.97657e11	1.41454
$369$	0	0
$370$	4.68480e10	0.129952
$371$	2.48363e11	0.680620
$372$	0	0
$373$	7.20169e11	1.92639	0.963196	−	0.268801i	$-0.0866273\pi$
$373$	7.20169e11	1.92639	0.963196	+	0.268801i	$0.0866273\pi$
$374$	1.67709e10	0.0443235
$375$	0	0
$376$	−6.57494e10	−0.169647
$377$	4.31747e11	1.10076
$378$	0	0
$379$	−7.20072e11	−1.79267	−0.896333	−	0.443381i	$-0.853779\pi$
$379$	−7.20072e11	−1.79267	−0.896333	+	0.443381i	$0.853779\pi$
$380$	5.25437e11	1.29269
$381$	0	0
$382$	1.24760e9	0.00299770
$383$	1.80684e11	0.429067	0.214533	−	0.976717i	$-0.431177\pi$
$383$	1.80684e11	0.429067	0.214533	+	0.976717i	$0.431177\pi$
$384$	0	0
$385$	−1.72474e11	−0.400084
$386$	−3.41765e9	−0.00783582
$387$	0	0
$388$	3.04989e11	0.683190
$389$	−2.04711e11	−0.453281	−0.226640	−	0.973978i	$-0.572774\pi$
$389$	−2.04711e11	−0.453281	−0.226640	+	0.973978i	$0.572774\pi$
$390$	0	0
$391$	−9.72676e11	−2.10462
$392$	1.43256e10	0.0306427
$393$	0	0
$394$	5.71560e10	0.119489
$395$	−3.09233e11	−0.639143
$396$	0	0
$397$	3.34553e11	0.675939	0.337970	−	0.941157i	$-0.390260\pi$
$397$	3.34553e11	0.675939	0.337970	+	0.941157i	$0.390260\pi$
$398$	1.73132e9	0.00345862
$399$	0	0
$400$	1.06784e11	0.208563
$401$	−4.41279e11	−0.852244	−0.426122	−	0.904666i	$-0.640121\pi$
$401$	−4.41279e11	−0.852244	−0.426122	+	0.904666i	$0.640121\pi$
$402$	0	0
$403$	1.02414e11	0.193414
$404$	−3.74273e11	−0.698993
$405$	0	0
$406$	4.11046e10	0.0750798
$407$	−3.59247e11	−0.648961
$408$	0	0
$409$	−1.92897e11	−0.340856	−0.170428	−	0.985370i	$-0.554515\pi$
$409$	−1.92897e11	−0.340856	−0.170428	+	0.985370i	$0.554515\pi$
$410$	−2.40423e10	−0.0420193
$411$	0	0
$412$	−2.09664e11	−0.358497
$413$	9.15308e11	1.54807
$414$	0	0
$415$	7.87720e11	1.30363
$416$	1.30845e11	0.214208
$417$	0	0
$418$	2.23216e10	0.0357628
$419$	−5.92813e11	−0.939625	−0.469812	−	0.882766i	$-0.655678\pi$
$419$	−5.92813e11	−0.939625	−0.469812	+	0.882766i	$0.655678\pi$
$420$	0	0
$421$	−5.20544e11	−0.807585	−0.403792	−	0.914851i	$-0.632308\pi$
$421$	−5.20544e11	−0.807585	−0.403792	+	0.914851i	$0.632308\pi$
$422$	2.49569e9	0.00383076
$423$	0	0
$424$	−7.52999e10	−0.113148
$425$	−2.08711e11	−0.310310
$426$	0	0
$427$	8.23543e11	1.19884
$428$	−1.42370e11	−0.205079
$429$	0	0
$430$	−4.06220e9	−0.00572998
$431$	−6.97508e11	−0.973647	−0.486824	−	0.873500i	$-0.661845\pi$
$431$	−6.97508e11	−0.973647	−0.486824	+	0.873500i	$0.661845\pi$
$432$	0	0
$433$	−1.30466e11	−0.178361	−0.0891806	−	0.996015i	$-0.528425\pi$
$433$	−1.30466e11	−0.178361	−0.0891806	+	0.996015i	$0.528425\pi$
$434$	9.75037e9	0.0131922
$435$	0	0
$436$	2.48385e11	0.329182
$437$	−1.29460e12	−1.69813
$438$	0	0
$439$	1.03545e12	1.33057	0.665286	−	0.746589i	$-0.268310\pi$
$439$	1.03545e12	1.33057	0.665286	+	0.746589i	$0.268310\pi$
$440$	5.22915e10	0.0665111
$441$	0	0
$442$	−8.44591e10	−0.105256
$443$	4.06754e10	0.0501781	0.0250891	−	0.999685i	$-0.492013\pi$
$443$	4.06754e10	0.0501781	0.0250891	+	0.999685i	$0.492013\pi$
$444$	0	0
$445$	−4.17460e11	−0.504655
$446$	−3.11787e9	−0.00373122
$447$	0	0
$448$	−7.34278e11	−0.861211
$449$	−1.58095e12	−1.83574	−0.917869	−	0.396884i	$-0.870091\pi$
$449$	−1.58095e12	−1.83574	−0.917869	+	0.396884i	$0.870091\pi$
$450$	0	0
$451$	1.84365e11	0.209838
$452$	−1.16119e12	−1.30852
$453$	0	0
$454$	1.16987e11	0.129237
$455$	8.68591e11	0.950089
$456$	0	0
$457$	9.01224e11	0.966518	0.483259	−	0.875478i	$-0.339453\pi$
$457$	9.01224e11	0.966518	0.483259	+	0.875478i	$0.339453\pi$
$458$	−3.42033e10	−0.0363223
$459$	0	0
$460$	−1.51221e12	−1.57472
$461$	−1.11552e12	−1.15033	−0.575164	−	0.818038i	$-0.695062\pi$
$461$	−1.11552e12	−1.15033	−0.575164	+	0.818038i	$0.695062\pi$
$462$	0	0
$463$	−8.68995e11	−0.878826	−0.439413	−	0.898285i	$-0.644813\pi$
$463$	−8.68995e11	−0.878826	−0.439413	+	0.898285i	$0.644813\pi$
$464$	1.11542e12	1.11714
$465$	0	0
$466$	7.26636e10	0.0713806
$467$	−1.02354e12	−0.995820	−0.497910	−	0.867229i	$-0.665899\pi$
$467$	−1.02354e12	−0.995820	−0.497910	+	0.867229i	$0.665899\pi$
$468$	0	0
$469$	−4.61279e10	−0.0440236
$470$	9.90643e10	0.0936433
$471$	0	0
$472$	−2.77507e11	−0.257356
$473$	3.11504e10	0.0286147
$474$	0	0
$475$	−2.77789e11	−0.250376
$476$	1.45146e12	1.29591
$477$	0	0
$478$	6.32765e10	0.0554392
$479$	7.16070e11	0.621507	0.310753	−	0.950491i	$-0.399419\pi$
$479$	7.16070e11	0.621507	0.310753	+	0.950491i	$0.399419\pi$
$480$	0	0
$481$	1.80919e12	1.54110
$482$	−5.62150e10	−0.0474395
$483$	0	0
$484$	1.00068e12	0.828876
$485$	−9.21597e11	−0.756316
$486$	0	0
$487$	−1.04858e12	−0.844739	−0.422369	−	0.906424i	$-0.638802\pi$
$487$	−1.04858e12	−0.844739	−0.422369	+	0.906424i	$0.638802\pi$
$488$	−2.49685e11	−0.199298
$489$	0	0
$490$	−2.15844e10	−0.0169144
$491$	−1.82538e12	−1.41738	−0.708691	−	0.705519i	$-0.750714\pi$
$491$	−1.82538e12	−1.41738	−0.708691	+	0.705519i	$0.750714\pi$
$492$	0	0
$493$	−2.18010e12	−1.66213
$494$	−1.12413e11	−0.0849267
$495$	0	0
$496$	2.64587e11	0.196291
$497$	6.27694e11	0.461471
$498$	0	0
$499$	−2.15436e12	−1.55548	−0.777742	−	0.628584i	$-0.783635\pi$
$499$	−2.15436e12	−1.55548	−0.777742	+	0.628584i	$0.783635\pi$
$500$	1.20565e12	0.862691
$501$	0	0
$502$	−1.20383e11	−0.0846055
$503$	9.53366e11	0.664055	0.332027	−	0.943270i	$-0.392267\pi$
$503$	9.53366e11	0.664055	0.332027	+	0.943270i	$0.392267\pi$
$504$	0	0
$505$	1.13096e12	0.773810
$506$	−6.42416e10	−0.0435652
$507$	0	0
$508$	1.87110e12	1.24655
$509$	1.69887e10	0.0112184	0.00560918	−	0.999984i	$-0.498215\pi$
$509$	1.69887e10	0.0112184	0.00560918	+	0.999984i	$0.498215\pi$
$510$	0	0
$511$	8.05679e11	0.522719
$512$	5.64436e11	0.362994
$513$	0	0
$514$	−7.15259e10	−0.0451990
$515$	6.33549e11	0.396869
$516$	0	0
$517$	−7.59660e11	−0.467640
$518$	1.72244e11	0.105114
$519$	0	0
$520$	−2.63343e11	−0.157945
$521$	1.80540e12	1.07350	0.536751	−	0.843741i	$-0.319651\pi$
$521$	1.80540e12	1.07350	0.536751	+	0.843741i	$0.319651\pi$
$522$	0	0
$523$	2.69922e12	1.57754	0.788770	−	0.614688i	$-0.210718\pi$
$523$	2.69922e12	1.57754	0.788770	+	0.614688i	$0.210718\pi$
$524$	2.58433e12	1.49747
$525$	0	0
$526$	2.94895e10	0.0167970
$527$	−5.17139e11	−0.292051
$528$	0	0
$529$	1.92473e12	1.06861
$530$	1.13454e11	0.0624566
$531$	0	0
$532$	1.93186e12	1.04562
$533$	−9.28473e11	−0.498307
$534$	0	0
$535$	4.30205e11	0.227030
$536$	1.39853e10	0.00731862
$537$	0	0
$538$	−1.10642e11	−0.0569376
$539$	1.65517e11	0.0844680
$540$	0	0
$541$	−2.63643e12	−1.32321	−0.661604	−	0.749854i	$-0.730124\pi$
$541$	−2.63643e12	−1.32321	−0.661604	+	0.749854i	$0.730124\pi$
$542$	−4.77534e10	−0.0237688
$543$	0	0
$544$	−6.60699e11	−0.323451
$545$	−7.50554e11	−0.364416
$546$	0	0
$547$	−3.53652e12	−1.68901	−0.844507	−	0.535544i	$-0.820107\pi$
$547$	−3.53652e12	−1.68901	−0.844507	+	0.535544i	$0.820107\pi$
$548$	1.02696e12	0.486451
$549$	0	0
$550$	−1.37846e10	−0.00642335
$551$	−2.90165e12	−1.34111
$552$	0	0
$553$	−1.13694e12	−0.516983
$554$	−1.12930e11	−0.0509348
$555$	0	0
$556$	−1.34837e12	−0.598375
$557$	−1.41497e12	−0.622870	−0.311435	−	0.950267i	$-0.600810\pi$
$557$	−1.41497e12	−0.622870	−0.311435	+	0.950267i	$0.600810\pi$
$558$	0	0
$559$	−1.56875e11	−0.0679519
$560$	2.24401e12	0.964224
$561$	0	0
$562$	−2.20266e11	−0.0931396
$563$	1.79761e12	0.754063	0.377032	−	0.926200i	$-0.376945\pi$
$563$	1.79761e12	0.754063	0.377032	+	0.926200i	$0.376945\pi$
$564$	0	0
$565$	3.50881e12	1.44858
$566$	1.94334e10	0.00795930
$567$	0	0
$568$	−1.90307e11	−0.0767162
$569$	2.52812e12	1.01110	0.505549	−	0.862798i	$-0.331290\pi$
$569$	2.52812e12	1.01110	0.505549	+	0.862798i	$0.331290\pi$
$570$	0	0
$571$	4.70093e12	1.85064	0.925318	−	0.379191i	$-0.123798\pi$
$571$	4.70093e12	1.85064	0.925318	+	0.379191i	$0.123798\pi$
$572$	1.00692e12	0.393288
$573$	0	0
$574$	−8.83955e10	−0.0339881
$575$	7.99478e11	0.305001
$576$	0	0
$577$	−2.61526e12	−0.982253	−0.491126	−	0.871088i	$-0.663415\pi$
$577$	−2.61526e12	−0.982253	−0.491126	+	0.871088i	$0.663415\pi$
$578$	2.27304e11	0.0847094
$579$	0	0
$580$	−3.38939e12	−1.24364
$581$	2.89618e12	1.05447
$582$	0	0
$583$	−8.70005e11	−0.311899
$584$	−2.44269e11	−0.0868983
$585$	0	0
$586$	−2.65466e10	−0.00929972
$587$	1.18649e12	0.412471	0.206236	−	0.978502i	$-0.433879\pi$
$587$	1.18649e12	0.412471	0.206236	+	0.978502i	$0.433879\pi$
$588$	0	0
$589$	−6.88297e11	−0.235644
$590$	4.18119e11	0.142058
$591$	0	0
$592$	4.67405e12	1.56403
$593$	6.62653e11	0.220060	0.110030	−	0.993928i	$-0.464905\pi$
$593$	6.62653e11	0.220060	0.110030	+	0.993928i	$0.464905\pi$
$594$	0	0
$595$	−4.38594e12	−1.43462
$596$	5.13897e11	0.166827
$597$	0	0
$598$	3.23525e11	0.103455
$599$	5.71521e12	1.81389	0.906946	−	0.421248i	$-0.138408\pi$
$599$	5.71521e12	1.81389	0.906946	+	0.421248i	$0.138408\pi$
$600$	0	0
$601$	−3.95677e12	−1.23710	−0.618551	−	0.785745i	$-0.712280\pi$
$601$	−3.95677e12	−1.23710	−0.618551	+	0.785745i	$0.712280\pi$
$602$	−1.49353e10	−0.00463480
$603$	0	0
$604$	2.97574e11	0.0909765
$605$	−3.02378e12	−0.917595
$606$	0	0
$607$	3.13313e12	0.936762	0.468381	−	0.883527i	$-0.344837\pi$
$607$	3.13313e12	0.936762	0.468381	+	0.883527i	$0.344837\pi$
$608$	−8.79371e11	−0.260979
$609$	0	0
$610$	3.76200e11	0.110010
$611$	3.82569e12	1.11052
$612$	0	0
$613$	5.12506e12	1.46598	0.732988	−	0.680242i	$-0.238125\pi$
$613$	5.12506e12	1.46598	0.732988	+	0.680242i	$0.238125\pi$
$614$	−6.89177e10	−0.0195692
$615$	0	0
$616$	1.92258e11	0.0537988
$617$	−2.32957e12	−0.647131	−0.323565	−	0.946206i	$-0.604882\pi$
$617$	−2.32957e12	−0.647131	−0.323565	+	0.946206i	$0.604882\pi$
$618$	0	0
$619$	5.78623e12	1.58412	0.792059	−	0.610444i	$-0.209009\pi$
$619$	5.78623e12	1.58412	0.792059	+	0.610444i	$0.209009\pi$
$620$	−8.03991e11	−0.218519
$621$	0	0
$622$	3.33332e10	0.00892936
$623$	−1.53486e12	−0.408200
$624$	0	0
$625$	−4.45210e12	−1.16709
$626$	−5.10047e11	−0.132747
$627$	0	0
$628$	−3.81102e12	−0.977738
$629$	−9.13548e12	−2.32704
$630$	0	0
$631$	3.68878e12	0.926297	0.463148	−	0.886281i	$-0.346720\pi$
$631$	3.68878e12	0.926297	0.463148	+	0.886281i	$0.346720\pi$
$632$	3.44704e11	0.0859447
$633$	0	0
$634$	1.34591e11	0.0330837
$635$	−5.65398e12	−1.37998
$636$	0	0
$637$	−8.33551e11	−0.200588
$638$	−1.43988e11	−0.0344058
$639$	0	0
$640$	−1.36830e12	−0.322382
$641$	8.08197e11	0.189085	0.0945423	−	0.995521i	$-0.469861\pi$
$641$	8.08197e11	0.189085	0.0945423	+	0.995521i	$0.469861\pi$
$642$	0	0
$643$	6.87954e12	1.58712	0.793561	−	0.608491i	$-0.208225\pi$
$643$	6.87954e12	1.58712	0.793561	+	0.608491i	$0.208225\pi$
$644$	−5.55989e12	−1.27374
$645$	0	0
$646$	5.67626e11	0.128238
$647$	6.68555e12	1.49992	0.749959	−	0.661484i	$-0.230073\pi$
$647$	6.68555e12	1.49992	0.749959	+	0.661484i	$0.230073\pi$
$648$	0	0
$649$	−3.20628e12	−0.709415
$650$	6.94200e10	0.0152537
$651$	0	0
$652$	−2.39965e12	−0.520035
$653$	−5.10086e12	−1.09783	−0.548913	−	0.835879i	$-0.684958\pi$
$653$	−5.10086e12	−1.09783	−0.548913	+	0.835879i	$0.684958\pi$
$654$	0	0
$655$	−7.80917e12	−1.65775
$656$	−2.39871e12	−0.505721
$657$	0	0
$658$	3.64226e11	0.0757451
$659$	−1.25632e12	−0.259488	−0.129744	−	0.991548i	$-0.541416\pi$
$659$	−1.25632e12	−0.259488	−0.129744	+	0.991548i	$0.541416\pi$
$660$	0	0
$661$	−5.23779e12	−1.06719	−0.533595	−	0.845740i	$-0.679159\pi$
$661$	−5.23779e12	−1.06719	−0.533595	+	0.845740i	$0.679159\pi$
$662$	−2.36875e11	−0.0479356
$663$	0	0
$664$	−8.78077e11	−0.175298
$665$	−5.83756e12	−1.15753
$666$	0	0
$667$	8.35098e12	1.63370
$668$	−6.20012e12	−1.20478
$669$	0	0
$670$	−2.10715e10	−0.00403980
$671$	−2.88483e12	−0.549376
$672$	0	0
$673$	2.41160e12	0.453146	0.226573	−	0.973994i	$-0.427248\pi$
$673$	2.41160e12	0.453146	0.226573	+	0.973994i	$0.427248\pi$
$674$	4.56417e10	0.00851907
$675$	0	0
$676$	3.28702e11	0.0605400
$677$	−4.27462e12	−0.782076	−0.391038	−	0.920375i	$-0.627884\pi$
$677$	−4.27462e12	−0.782076	−0.391038	+	0.920375i	$0.627884\pi$
$678$	0	0
$679$	−3.38840e12	−0.611760
$680$	1.32975e12	0.238495
$681$	0	0
$682$	−3.41551e10	−0.00604541
$683$	6.17544e12	1.08586	0.542931	−	0.839777i	$-0.317315\pi$
$683$	6.17544e12	1.08586	0.542931	+	0.839777i	$0.317315\pi$
$684$	0	0
$685$	−3.10319e12	−0.538519
$686$	−4.62757e11	−0.0797800
$687$	0	0
$688$	−4.05287e11	−0.0689628
$689$	4.38140e12	0.740673
$690$	0	0
$691$	1.41466e12	0.236049	0.118024	−	0.993011i	$-0.462344\pi$
$691$	1.41466e12	0.236049	0.118024	+	0.993011i	$0.462344\pi$
$692$	−2.59937e12	−0.430914
$693$	0	0
$694$	4.57543e11	0.0748712
$695$	4.07443e12	0.662423
$696$	0	0
$697$	4.68831e12	0.752435
$698$	4.75256e11	0.0757842
$699$	0	0
$700$	−1.19301e12	−0.187803
$701$	−7.79387e12	−1.21905	−0.609526	−	0.792766i	$-0.708640\pi$
$701$	−7.79387e12	−1.21905	−0.609526	+	0.792766i	$0.708640\pi$
$702$	0	0
$703$	−1.21591e13	−1.87759
$704$	2.57214e12	0.394655
$705$	0	0
$706$	−1.21450e11	−0.0183982
$707$	4.15814e12	0.625910
$708$	0	0
$709$	1.07942e13	1.60428	0.802142	−	0.597134i	$-0.203694\pi$
$709$	1.07942e13	1.60428	0.802142	+	0.597134i	$0.203694\pi$
$710$	2.86735e11	0.0423465
$711$	0	0
$712$	4.65346e11	0.0678603
$713$	1.98092e12	0.287055
$714$	0	0
$715$	−3.04264e12	−0.435384
$716$	−9.13830e11	−0.129944
$717$	0	0
$718$	−6.52044e11	−0.0915623
$719$	1.54014e12	0.214921	0.107461	−	0.994209i	$-0.465728\pi$
$719$	1.54014e12	0.214921	0.107461	+	0.994209i	$0.465728\pi$
$720$	0	0
$721$	2.32935e12	0.321015
$722$	2.13533e11	0.0292447
$723$	0	0
$724$	1.35911e13	1.83836
$725$	1.79190e12	0.240876
$726$	0	0
$727$	−1.83997e12	−0.244291	−0.122145	−	0.992512i	$-0.538977\pi$
$727$	−1.83997e12	−0.244291	−0.122145	+	0.992512i	$0.538977\pi$
$728$	−9.68225e11	−0.127757
$729$	0	0
$730$	3.68039e11	0.0479669
$731$	7.92139e11	0.102606
$732$	0	0
$733$	4.00426e12	0.512335	0.256168	−	0.966632i	$-0.417540\pi$
$733$	4.00426e12	0.512335	0.256168	+	0.966632i	$0.417540\pi$
$734$	−2.64383e11	−0.0336202
$735$	0	0
$736$	2.53084e12	0.317917
$737$	1.61584e11	0.0201741
$738$	0	0
$739$	7.68031e12	0.947281	0.473641	−	0.880718i	$-0.342940\pi$
$739$	7.68031e12	0.947281	0.473641	+	0.880718i	$0.342940\pi$
$740$	−1.42029e13	−1.74114
$741$	0	0
$742$	4.17132e11	0.0505192
$743$	−4.02322e11	−0.0484310	−0.0242155	−	0.999707i	$-0.507709\pi$
$743$	−4.02322e11	−0.0484310	−0.0242155	+	0.999707i	$0.507709\pi$
$744$	0	0
$745$	−1.55286e12	−0.184684
$746$	1.20954e12	0.142987
$747$	0	0
$748$	−5.08441e12	−0.593859
$749$	1.58172e12	0.183637
$750$	0	0
$751$	−8.58822e12	−0.985198	−0.492599	−	0.870256i	$-0.663953\pi$
$751$	−8.58822e12	−0.985198	−0.492599	+	0.870256i	$0.663953\pi$
$752$	9.88368e12	1.12704
$753$	0	0
$754$	7.25130e11	0.0817043
$755$	−8.99191e11	−0.100714
$756$	0	0
$757$	−5.70537e12	−0.631470	−0.315735	−	0.948847i	$-0.602251\pi$
$757$	−5.70537e12	−0.631470	−0.315735	+	0.948847i	$0.602251\pi$
$758$	−1.20938e12	−0.133061
$759$	0	0
$760$	1.76986e12	0.192432
$761$	−7.46605e11	−0.0806974	−0.0403487	−	0.999186i	$-0.512847\pi$
$761$	−7.46605e11	−0.0806974	−0.0403487	+	0.999186i	$0.512847\pi$
$762$	0	0
$763$	−2.75953e12	−0.294765
$764$	−3.78232e11	−0.0401641
$765$	0	0
$766$	3.03463e11	0.0318476
$767$	1.61470e13	1.68466
$768$	0	0
$769$	−1.74218e13	−1.79649	−0.898243	−	0.439499i	$-0.855156\pi$
$769$	−1.74218e13	−1.79649	−0.898243	+	0.439499i	$0.855156\pi$
$770$	−2.89675e11	−0.0296963
$771$	0	0
$772$	1.03612e12	0.104987
$773$	8.20681e12	0.826736	0.413368	−	0.910564i	$-0.364352\pi$
$773$	8.20681e12	0.826736	0.413368	+	0.910564i	$0.364352\pi$
$774$	0	0
$775$	4.25055e11	0.0423241
$776$	1.02731e12	0.101701
$777$	0	0
$778$	−3.43817e11	−0.0336449
$779$	6.24001e12	0.607109
$780$	0	0
$781$	−2.19878e12	−0.211472
$782$	−1.63363e12	−0.156215
$783$	0	0
$784$	−2.15348e12	−0.203572
$785$	1.15159e13	1.08239
$786$	0	0
$787$	6.41360e11	0.0595958	0.0297979	−	0.999556i	$-0.490514\pi$
$787$	6.41360e11	0.0595958	0.0297979	+	0.999556i	$0.490514\pi$
$788$	−1.73279e13	−1.60095
$789$	0	0
$790$	−5.19363e11	−0.0474405
$791$	1.29007e13	1.17171
$792$	0	0
$793$	1.45282e13	1.30461
$794$	5.61890e11	0.0501717
$795$	0	0
$796$	−5.24881e11	−0.0463396
$797$	−3.41418e12	−0.299725	−0.149863	−	0.988707i	$-0.547883\pi$
$797$	−3.41418e12	−0.299725	−0.149863	+	0.988707i	$0.547883\pi$
$798$	0	0
$799$	−1.93178e13	−1.67686
$800$	5.43052e11	0.0468745
$801$	0	0
$802$	−7.41139e11	−0.0632579
$803$	−2.82226e12	−0.239539
$804$	0	0
$805$	1.68005e13	1.41007
$806$	1.72007e11	0.0143562
$807$	0	0
$808$	−1.26068e12	−0.104053
$809$	1.64198e13	1.34772	0.673859	−	0.738860i	$-0.264636\pi$
$809$	1.64198e13	1.34772	0.673859	+	0.738860i	$0.264636\pi$
$810$	0	0
$811$	1.11691e13	0.906616	0.453308	−	0.891354i	$-0.350244\pi$
$811$	1.11691e13	0.906616	0.453308	+	0.891354i	$0.350244\pi$
$812$	−1.24616e13	−1.00594
$813$	0	0
$814$	−6.03365e11	−0.0481693
$815$	7.25110e12	0.575698
$816$	0	0
$817$	1.05431e12	0.0827887
$818$	−3.23975e11	−0.0253001
$819$	0	0
$820$	7.28888e12	0.562987
$821$	2.25396e13	1.73142	0.865710	−	0.500545i	$-0.166867\pi$
$821$	2.25396e13	1.73142	0.865710	+	0.500545i	$0.166867\pi$
$822$	0	0
$823$	−1.13893e12	−0.0865364	−0.0432682	−	0.999063i	$-0.513777\pi$
$823$	−1.13893e12	−0.0865364	−0.0432682	+	0.999063i	$0.513777\pi$
$824$	−7.06221e11	−0.0533665
$825$	0	0
$826$	1.53728e12	0.114906
$827$	−3.06302e11	−0.0227706	−0.0113853	−	0.999935i	$-0.503624\pi$
$827$	−3.06302e11	−0.0227706	−0.0113853	+	0.999935i	$0.503624\pi$
$828$	0	0
$829$	−1.50391e13	−1.10593	−0.552964	−	0.833205i	$-0.686503\pi$
$829$	−1.50391e13	−1.10593	−0.552964	+	0.833205i	$0.686503\pi$
$830$	1.32299e12	0.0967624
$831$	0	0
$832$	−1.29535e13	−0.937197
$833$	4.20900e12	0.302884
$834$	0	0
$835$	1.87351e13	1.33373
$836$	−6.76720e12	−0.479160
$837$	0	0
$838$	−9.95643e11	−0.0697438
$839$	4.04209e12	0.281629	0.140814	−	0.990036i	$-0.455028\pi$
$839$	4.04209e12	0.281629	0.140814	+	0.990036i	$0.455028\pi$
$840$	0	0
$841$	4.21027e12	0.290221
$842$	−8.74266e11	−0.0599431
$843$	0	0
$844$	−7.56616e11	−0.0513257
$845$	−9.93251e11	−0.0670199
$846$	0	0
$847$	−1.11174e13	−0.742214
$848$	1.13193e13	0.751692
$849$	0	0
$850$	−3.50535e11	−0.0230328
$851$	3.49939e13	2.28723
$852$	0	0
$853$	1.99594e13	1.29085	0.645426	−	0.763823i	$-0.276680\pi$
$853$	1.99594e13	1.29085	0.645426	+	0.763823i	$0.276680\pi$
$854$	1.38316e12	0.0889840
$855$	0	0
$856$	−4.79553e11	−0.0305284
$857$	−2.85272e13	−1.80653	−0.903266	−	0.429082i	$-0.858837\pi$
$857$	−2.85272e13	−1.80653	−0.903266	+	0.429082i	$0.858837\pi$
$858$	0	0
$859$	1.04738e13	0.656351	0.328176	−	0.944617i	$-0.393566\pi$
$859$	1.04738e13	0.656351	0.328176	+	0.944617i	$0.393566\pi$
$860$	1.23153e12	0.0767720
$861$	0	0
$862$	−1.17148e12	−0.0722692
$863$	−1.55631e13	−0.955096	−0.477548	−	0.878606i	$-0.658474\pi$
$863$	−1.55631e13	−0.955096	−0.477548	+	0.878606i	$0.658474\pi$
$864$	0	0
$865$	7.85461e12	0.477037
$866$	−2.19120e11	−0.0132389
$867$	0	0
$868$	−2.95601e12	−0.176753
$869$	3.98266e12	0.236911
$870$	0	0
$871$	−8.13746e11	−0.0479079
$872$	8.36648e11	0.0490025
$873$	0	0
$874$	−2.17432e12	−0.126044
$875$	−1.33946e13	−0.772493
$876$	0	0
$877$	8.28334e12	0.472832	0.236416	−	0.971652i	$-0.424027\pi$
$877$	8.28334e12	0.472832	0.236416	+	0.971652i	$0.424027\pi$
$878$	1.73906e12	0.0987619
$879$	0	0
$880$	−7.86065e12	−0.441862
$881$	3.05766e12	0.171001	0.0855004	−	0.996338i	$-0.472751\pi$
$881$	3.05766e12	0.171001	0.0855004	+	0.996338i	$0.472751\pi$
$882$	0	0
$883$	−1.81997e13	−1.00749	−0.503745	−	0.863852i	$-0.668045\pi$
$883$	−1.81997e13	−1.00749	−0.503745	+	0.863852i	$0.668045\pi$
$884$	2.56054e13	1.41025
$885$	0	0
$886$	6.83152e10	0.00372448
$887$	−1.79103e13	−0.971506	−0.485753	−	0.874096i	$-0.661455\pi$
$887$	−1.79103e13	−0.971506	−0.485753	+	0.874096i	$0.661455\pi$
$888$	0	0
$889$	−2.07878e13	−1.11622
$890$	−7.01134e11	−0.0374581
$891$	0	0
$892$	9.45241e11	0.0499920
$893$	−2.57114e13	−1.35299
$894$	0	0
$895$	2.76135e12	0.143853
$896$	−5.03077e12	−0.260764
$897$	0	0
$898$	−2.65525e12	−0.136258
$899$	4.43993e12	0.226703
$900$	0	0
$901$	−2.21238e13	−1.11840
$902$	3.09645e11	0.0155753
$903$	0	0
$904$	−3.91130e12	−0.194788
$905$	−4.10686e13	−2.03513
$906$	0	0
$907$	9.17815e12	0.450321	0.225161	−	0.974322i	$-0.427709\pi$
$907$	9.17815e12	0.450321	0.225161	+	0.974322i	$0.427709\pi$
$908$	−3.54668e13	−1.73155
$909$	0	0
$910$	1.45882e12	0.0705205
$911$	−2.14634e12	−0.103244	−0.0516222	−	0.998667i	$-0.516439\pi$
$911$	−2.14634e12	−0.103244	−0.0516222	+	0.998667i	$0.516439\pi$
$912$	0	0
$913$	−1.01452e13	−0.483217
$914$	1.51363e12	0.0717400
$915$	0	0
$916$	1.03694e13	0.486656
$917$	−2.87117e13	−1.34090
$918$	0	0
$919$	1.70600e13	0.788967	0.394483	−	0.918903i	$-0.370924\pi$
$919$	1.70600e13	0.788967	0.394483	+	0.918903i	$0.370924\pi$
$920$	−5.09366e12	−0.234415
$921$	0	0
$922$	−1.87354e12	−0.0853833
$923$	1.10732e13	0.502187
$924$	0	0
$925$	7.50878e12	0.337234
$926$	−1.45950e12	−0.0652310
$927$	0	0
$928$	5.67248e12	0.251077
$929$	−4.17325e13	−1.83825	−0.919123	−	0.393971i	$-0.871101\pi$
$929$	−4.17325e13	−1.83825	−0.919123	+	0.393971i	$0.871101\pi$
$930$	0	0
$931$	5.60206e12	0.244385
$932$	−2.20293e13	−0.956378
$933$	0	0
$934$	−1.71907e12	−0.0739149
$935$	1.53637e13	0.657423
$936$	0	0
$937$	3.48481e13	1.47690	0.738451	−	0.674308i	$-0.235558\pi$
$937$	3.48481e13	1.47690	0.738451	+	0.674308i	$0.235558\pi$
$938$	−7.74729e10	−0.00326766
$939$	0	0
$940$	−3.00332e13	−1.25466
$941$	−7.74164e12	−0.321869	−0.160935	−	0.986965i	$-0.551451\pi$
$941$	−7.74164e12	−0.321869	−0.160935	+	0.986965i	$0.551451\pi$
$942$	0	0
$943$	−1.79588e13	−0.739562
$944$	4.17158e13	1.70973
$945$	0	0
$946$	5.23179e10	0.00212393
$947$	4.06195e13	1.64119	0.820596	−	0.571508i	$-0.193642\pi$
$947$	4.06195e13	1.64119	0.820596	+	0.571508i	$0.193642\pi$
$948$	0	0
$949$	1.42131e13	0.568839
$950$	−4.66553e11	−0.0185842
$951$	0	0
$952$	4.88904e12	0.192911
$953$	−3.21151e13	−1.26122	−0.630611	−	0.776099i	$-0.717196\pi$
$953$	−3.21151e13	−1.26122	−0.630611	+	0.776099i	$0.717196\pi$
$954$	0	0
$955$	1.14292e12	0.0444631
$956$	−1.91834e13	−0.742790
$957$	0	0
$958$	1.20266e12	0.0461314
$959$	−1.14094e13	−0.435591
$960$	0	0
$961$	−2.53864e13	−0.960166
$962$	3.03858e12	0.114389
$963$	0	0
$964$	1.70426e13	0.635608
$965$	−3.13090e12	−0.116224
$966$	0	0
$967$	−2.18017e13	−0.801811	−0.400905	−	0.916120i	$-0.631304\pi$
$967$	−2.18017e13	−0.801811	−0.400905	+	0.916120i	$0.631304\pi$
$968$	3.37063e12	0.123388
$969$	0	0
$970$	−1.54784e12	−0.0561377
$971$	2.45155e13	0.885024	0.442512	−	0.896763i	$-0.354087\pi$
$971$	2.45155e13	0.885024	0.442512	+	0.896763i	$0.354087\pi$
$972$	0	0
$973$	1.49803e13	0.535813
$974$	−1.76112e12	−0.0627009
$975$	0	0
$976$	3.75336e13	1.32402
$977$	−2.38100e13	−0.836054	−0.418027	−	0.908435i	$-0.637278\pi$
$977$	−2.38100e13	−0.836054	−0.418027	+	0.908435i	$0.637278\pi$
$978$	0	0
$979$	5.37655e12	0.187060
$980$	6.54370e12	0.226624
$981$	0	0
$982$	−3.06577e12	−0.105205
$983$	4.26737e13	1.45770	0.728852	−	0.684671i	$-0.240054\pi$
$983$	4.26737e13	1.45770	0.728852	+	0.684671i	$0.240054\pi$
$984$	0	0
$985$	5.23604e13	1.77231
$986$	−3.66153e12	−0.123372
$987$	0	0
$988$	3.40800e13	1.13787
$989$	−3.03433e12	−0.100851
$990$	0	0
$991$	−5.04883e13	−1.66287	−0.831437	−	0.555620i	$-0.812481\pi$
$991$	−5.04883e13	−1.66287	−0.831437	+	0.555620i	$0.812481\pi$
$992$	1.34556e12	0.0441165
$993$	0	0
$994$	1.05423e12	0.0342527
$995$	1.58605e12	0.0512996
$996$	0	0
$997$	−3.53339e13	−1.13257	−0.566283	−	0.824211i	$-0.691619\pi$
$997$	−3.53339e13	−1.13257	−0.566283	+	0.824211i	$0.691619\pi$
$998$	−3.61830e12	−0.115456
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.10.a.d.1.4		8
3.2	odd	2		81.10.a.c.1.5		8
9.2	odd	6		9.10.c.a.4.4	✓	16
9.4	even	3		27.10.c.a.19.5		16
9.5	odd	6		9.10.c.a.7.4	yes	16
9.7	even	3		27.10.c.a.10.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.10.c.a.4.4	✓	16	9.2	odd	6
9.10.c.a.7.4	yes	16	9.5	odd	6
27.10.c.a.10.5		16	9.7	even	3
27.10.c.a.19.5		16	9.4	even	3
81.10.a.c.1.5		8	3.2	odd	2
81.10.a.d.1.4		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)

\(f(q)\)	\(=\)	\(q+1.67952 q^{2} -509.179 q^{4} +1538.61 q^{5} +5656.94 q^{7} -1715.10 q^{8} +2584.13 q^{10} -19816.0 q^{11} +99794.5 q^{13} +9500.96 q^{14} +257819. q^{16} -503911. q^{17} -670691. q^{19} -783426. q^{20} -33281.4 q^{22} +1.93025e6 q^{23} +414183. q^{25} +167607. q^{26} -2.88039e6 q^{28} +4.32636e6 q^{29} +1.02625e6 q^{31} +1.31114e6 q^{32} -846330. q^{34} +8.70380e6 q^{35} +1.81292e7 q^{37} -1.12644e6 q^{38} -2.63886e6 q^{40} -9.30385e6 q^{41} -1.57198e6 q^{43} +1.00899e7 q^{44} +3.24191e6 q^{46} +3.83357e7 q^{47} -8.35267e6 q^{49} +695630. q^{50} -5.08133e7 q^{52} +4.39042e7 q^{53} -3.04890e7 q^{55} -9.70218e6 q^{56} +7.26623e6 q^{58} +1.61803e8 q^{59} +1.45581e8 q^{61} +1.72361e6 q^{62} -1.29801e8 q^{64} +1.53544e8 q^{65} -8.15422e6 q^{67} +2.56581e8 q^{68} +1.46182e7 q^{70} +1.10960e8 q^{71} +1.42423e8 q^{73} +3.04484e7 q^{74} +3.41502e8 q^{76} -1.12098e8 q^{77} -2.00982e8 q^{79} +3.96682e8 q^{80} -1.56260e7 q^{82} +5.11970e8 q^{83} -7.75320e8 q^{85} -2.64018e6 q^{86} +3.39863e7 q^{88} -2.71324e8 q^{89} +5.64531e8 q^{91} -9.82845e8 q^{92} +6.43857e7 q^{94} -1.03193e9 q^{95} -5.98982e8 q^{97} -1.40285e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 15 q^{2} + 1793 q^{4} + 453 q^{5} + 343 q^{7} + 7239 q^{8} + 510 q^{10} + 99150 q^{11} - 32435 q^{13} + 394824 q^{14} + 328193 q^{16} + 415539 q^{17} - 85277 q^{19} + 1855164 q^{20} - 529359 q^{22}+ \cdots - 2413650159 q^{98}+O(q^{100}) \) 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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