LMFDB - Embedded newform 171.2.a.e.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [171,2,Mod(1,171)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("171.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(171, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 171 = 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	171.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,10,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$1.36544187456$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.13068.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} - x + 1 $ x^4 - x^3 - 6*x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.82405$ of defining polynomial
Character	$\chi$	$=$	171.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.27582 q^{2} -0.372281 q^{4} +2.15121 q^{5} +3.37228 q^{7} -3.02661 q^{8} +2.74456 q^{10} -4.70285 q^{11} +2.00000 q^{13} +4.30243 q^{14} -3.11684 q^{16} +2.15121 q^{17} +1.00000 q^{19} -0.800857 q^{20} -6.00000 q^{22} -6.85407 q^{23} -0.372281 q^{25} +2.55164 q^{26} -1.25544 q^{28} -6.85407 q^{29} -6.74456 q^{31} +2.07668 q^{32} +2.74456 q^{34} +7.25450 q^{35} -0.744563 q^{37} +1.27582 q^{38} -6.51087 q^{40} -2.55164 q^{41} +6.11684 q^{43} +1.75079 q^{44} -8.74456 q^{46} +9.00528 q^{47} +4.37228 q^{49} -0.474964 q^{50} -0.744563 q^{52} +11.9574 q^{53} -10.1168 q^{55} -10.2066 q^{56} -8.74456 q^{58} +5.10328 q^{59} +12.1168 q^{61} -8.60485 q^{62} +8.88316 q^{64} +4.30243 q^{65} -4.00000 q^{67} -0.800857 q^{68} +9.25544 q^{70} -13.7081 q^{71} +12.1168 q^{73} -0.949929 q^{74} -0.372281 q^{76} -15.8593 q^{77} -4.00000 q^{79} -6.70500 q^{80} -3.25544 q^{82} +1.75079 q^{83} +4.62772 q^{85} +7.80400 q^{86} +14.2337 q^{88} +11.9574 q^{89} +6.74456 q^{91} +2.55164 q^{92} +11.4891 q^{94} +2.15121 q^{95} -15.4891 q^{97} +5.57825 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 10 q^{4} + 2 q^{7} - 12 q^{10} + 8 q^{13} + 22 q^{16} + 4 q^{19} - 24 q^{22} + 10 q^{25} - 28 q^{28} - 4 q^{31} - 12 q^{34} + 20 q^{37} - 72 q^{40} - 10 q^{43} - 12 q^{46} + 6 q^{49} + 20 q^{52} - 6 q^{55}+ \cdots - 16 q^{97}+O(q^{100}) $ 4 * q + 10 * q^4 + 2 * q^7 - 12 * q^10 + 8 * q^13 + 22 * q^16 + 4 * q^19 - 24 * q^22 + 10 * q^25 - 28 * q^28 - 4 * q^31 - 12 * q^34 + 20 * q^37 - 72 * q^40 - 10 * q^43 - 12 * q^46 + 6 * q^49 + 20 * q^52 - 6 * q^55 - 12 * q^58 + 14 * q^61 + 70 * q^64 - 16 * q^67 + 60 * q^70 + 14 * q^73 + 10 * q^76 - 16 * q^79 - 36 * q^82 + 30 * q^85 - 12 * q^88 + 4 * q^91 - 16 * q^97

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 $

$2$	1.27582	0.902142	0.451071	−	0.892488i	$-0.351042\pi$
$2$	1.27582	0.902142	0.451071	+	0.892488i	$0.351042\pi$
$3$	0	0
$3$	0	0
$4$	−0.372281	−0.186141
$5$	2.15121	0.962052	0.481026	−	0.876706i	$-0.340264\pi$
$5$	2.15121	0.962052	0.481026	+	0.876706i	$0.340264\pi$
$6$	0	0
$7$	3.37228	1.27460	0.637301	−	0.770615i	$-0.280051\pi$
$7$	3.37228	1.27460	0.637301	+	0.770615i	$0.280051\pi$
$8$	−3.02661	−1.07007
$9$	0	0
$10$	2.74456	0.867907
$11$	−4.70285	−1.41796	−0.708982	−	0.705227i	$-0.750845\pi$
$11$	−4.70285	−1.41796	−0.708982	+	0.705227i	$0.750845\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	4.30243	1.14987
$15$	0	0
$16$	−3.11684	−0.779211
$17$	2.15121	0.521746	0.260873	−	0.965373i	$-0.415990\pi$
$17$	2.15121	0.521746	0.260873	+	0.965373i	$0.415990\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−0.800857	−0.179077
$21$	0	0
$22$	−6.00000	−1.27920
$23$	−6.85407	−1.42917	−0.714586	−	0.699548i	$-0.753385\pi$
$23$	−6.85407	−1.42917	−0.714586	+	0.699548i	$0.753385\pi$
$24$	0	0
$25$	−0.372281	−0.0744563
$26$	2.55164	0.500418
$27$	0	0
$28$	−1.25544	−0.237255
$29$	−6.85407	−1.27277	−0.636384	−	0.771372i	$-0.719571\pi$
$29$	−6.85407	−1.27277	−0.636384	+	0.771372i	$0.719571\pi$
$30$	0	0
$31$	−6.74456	−1.21136	−0.605680	−	0.795709i	$-0.707099\pi$
$31$	−6.74456	−1.21136	−0.605680	+	0.795709i	$0.707099\pi$
$32$	2.07668	0.367108
$33$	0	0
$34$	2.74456	0.470689
$35$	7.25450	1.22623
$36$	0	0
$37$	−0.744563	−0.122405	−0.0612027	−	0.998125i	$-0.519494\pi$
$37$	−0.744563	−0.122405	−0.0612027	+	0.998125i	$0.519494\pi$
$38$	1.27582	0.206965
$39$	0	0
$40$	−6.51087	−1.02946
$41$	−2.55164	−0.398499	−0.199250	−	0.979949i	$-0.563850\pi$
$41$	−2.55164	−0.398499	−0.199250	+	0.979949i	$0.563850\pi$
$42$	0	0
$43$	6.11684	0.932810	0.466405	−	0.884571i	$-0.345549\pi$
$43$	6.11684	0.932810	0.466405	+	0.884571i	$0.345549\pi$
$44$	1.75079	0.263941
$45$	0	0
$46$	−8.74456	−1.28932
$47$	9.00528	1.31356	0.656778	−	0.754084i	$-0.271919\pi$
$47$	9.00528	1.31356	0.656778	+	0.754084i	$0.271919\pi$
$48$	0	0
$49$	4.37228	0.624612
$50$	−0.474964	−0.0671701
$51$	0	0
$52$	−0.744563	−0.103252
$53$	11.9574	1.64247	0.821234	−	0.570591i	$-0.193286\pi$
$53$	11.9574	1.64247	0.821234	+	0.570591i	$0.193286\pi$
$54$	0	0
$55$	−10.1168	−1.36415
$56$	−10.2066	−1.36391
$57$	0	0
$58$	−8.74456	−1.14822
$59$	5.10328	0.664391	0.332195	−	0.943211i	$-0.392211\pi$
$59$	5.10328	0.664391	0.332195	+	0.943211i	$0.392211\pi$
$60$	0	0
$61$	12.1168	1.55140	0.775701	−	0.631100i	$-0.217396\pi$
$61$	12.1168	1.55140	0.775701	+	0.631100i	$0.217396\pi$
$62$	−8.60485	−1.09282
$63$	0	0
$64$	8.88316	1.11039
$65$	4.30243	0.533650
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−0.800857	−0.0971181
$69$	0	0
$70$	9.25544	1.10624
$71$	−13.7081	−1.62686	−0.813428	−	0.581665i	$-0.802401\pi$
$71$	−13.7081	−1.62686	−0.813428	+	0.581665i	$0.802401\pi$
$72$	0	0
$73$	12.1168	1.41817	0.709085	−	0.705123i	$-0.249108\pi$
$73$	12.1168	1.41817	0.709085	+	0.705123i	$0.249108\pi$
$74$	−0.949929	−0.110427
$75$	0	0
$76$	−0.372281	−0.0427036
$77$	−15.8593	−1.80734
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	−6.70500	−0.749641
$81$	0	0
$82$	−3.25544	−0.359503
$83$	1.75079	0.192174	0.0960868	−	0.995373i	$-0.469367\pi$
$83$	1.75079	0.192174	0.0960868	+	0.995373i	$0.469367\pi$
$84$	0	0
$85$	4.62772	0.501947
$86$	7.80400	0.841527
$87$	0	0
$88$	14.2337	1.51732
$89$	11.9574	1.26748	0.633738	−	0.773547i	$-0.281520\pi$
$89$	11.9574	1.26748	0.633738	+	0.773547i	$0.281520\pi$
$90$	0	0
$91$	6.74456	0.707022
$92$	2.55164	0.266027
$93$	0	0
$94$	11.4891	1.18501
$95$	2.15121	0.220710
$96$	0	0
$97$	−15.4891	−1.57268	−0.786341	−	0.617792i	$-0.788027\pi$
$97$	−15.4891	−1.57268	−0.786341	+	0.617792i	$0.788027\pi$
$98$	5.57825	0.563488
$99$	0	0
$100$	0.138593	0.0138593
$101$	−8.60485	−0.856215	−0.428107	−	0.903728i	$-0.640820\pi$
$101$	−8.60485	−0.856215	−0.428107	+	0.903728i	$0.640820\pi$
$102$	0	0
$103$	−18.7446	−1.84696	−0.923478	−	0.383651i	$-0.874667\pi$
$103$	−18.7446	−1.84696	−0.923478	+	0.383651i	$0.874667\pi$
$104$	−6.05321	−0.593566
$105$	0	0
$106$	15.2554	1.48174
$107$	9.40571	0.909284	0.454642	−	0.890674i	$-0.349767\pi$
$107$	9.40571	0.909284	0.454642	+	0.890674i	$0.349767\pi$
$108$	0	0
$109$	16.7446	1.60384	0.801919	−	0.597433i	$-0.203812\pi$
$109$	16.7446	1.60384	0.801919	+	0.597433i	$0.203812\pi$
$110$	−12.9073	−1.23066
$111$	0	0
$112$	−10.5109	−0.993184
$113$	1.75079	0.164700	0.0823500	−	0.996603i	$-0.473757\pi$
$113$	1.75079	0.164700	0.0823500	+	0.996603i	$0.473757\pi$
$114$	0	0
$115$	−14.7446	−1.37494
$116$	2.55164	0.236914
$117$	0	0
$118$	6.51087	0.599375
$119$	7.25450	0.665019
$120$	0	0
$121$	11.1168	1.01062
$122$	15.4589	1.39958
$123$	0	0
$124$	2.51087	0.225483
$125$	−11.5569	−1.03368
$126$	0	0
$127$	−1.25544	−0.111402	−0.0557010	−	0.998447i	$-0.517739\pi$
$127$	−1.25544	−0.111402	−0.0557010	+	0.998447i	$0.517739\pi$
$128$	7.17996	0.634625
$129$	0	0
$130$	5.48913	0.481428
$131$	3.90200	0.340919	0.170460	−	0.985365i	$-0.445475\pi$
$131$	3.90200	0.340919	0.170460	+	0.985365i	$0.445475\pi$
$132$	0	0
$133$	3.37228	0.292414
$134$	−5.10328	−0.440857
$135$	0	0
$136$	−6.51087	−0.558303
$137$	16.6602	1.42338	0.711689	−	0.702495i	$-0.247931\pi$
$137$	16.6602	1.42338	0.711689	+	0.702495i	$0.247931\pi$
$138$	0	0
$139$	−14.1168	−1.19738	−0.598688	−	0.800983i	$-0.704311\pi$
$139$	−14.1168	−1.19738	−0.598688	+	0.800983i	$0.704311\pi$
$140$	−2.70071	−0.228252
$141$	0	0
$142$	−17.4891	−1.46765
$143$	−9.40571	−0.786545
$144$	0	0
$145$	−14.7446	−1.22447
$146$	15.4589	1.27939
$147$	0	0
$148$	0.277187	0.0227846
$149$	−1.35036	−0.110626	−0.0553128	−	0.998469i	$-0.517616\pi$
$149$	−1.35036	−0.110626	−0.0553128	+	0.998469i	$0.517616\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	−3.02661	−0.245490
$153$	0	0
$154$	−20.2337	−1.63048
$155$	−14.5090	−1.16539
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	−5.10328	−0.405995
$159$	0	0
$160$	4.46738	0.353177
$161$	−23.1138	−1.82163
$162$	0	0
$163$	1.48913	0.116637	0.0583186	−	0.998298i	$-0.481426\pi$
$163$	1.48913	0.116637	0.0583186	+	0.998298i	$0.481426\pi$
$164$	0.949929	0.0741770
$165$	0	0
$166$	2.23369	0.173368
$167$	4.30243	0.332932	0.166466	−	0.986047i	$-0.446764\pi$
$167$	4.30243	0.332932	0.166466	+	0.986047i	$0.446764\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	5.90414	0.452827
$171$	0	0
$172$	−2.27719	−0.173634
$173$	6.85407	0.521105	0.260553	−	0.965460i	$-0.416095\pi$
$173$	6.85407	0.521105	0.260553	+	0.965460i	$0.416095\pi$
$174$	0	0
$175$	−1.25544	−0.0949021
$176$	14.6581	1.10489
$177$	0	0
$178$	15.2554	1.14344
$179$	−9.40571	−0.703016	−0.351508	−	0.936185i	$-0.614331\pi$
$179$	−9.40571	−0.703016	−0.351508	+	0.936185i	$0.614331\pi$
$180$	0	0
$181$	−3.48913	−0.259345	−0.129672	−	0.991557i	$-0.541393\pi$
$181$	−3.48913	−0.259345	−0.129672	+	0.991557i	$0.541393\pi$
$182$	8.60485	0.637834
$183$	0	0
$184$	20.7446	1.52931
$185$	−1.60171	−0.117760
$186$	0	0
$187$	−10.1168	−0.739817
$188$	−3.35250	−0.244506
$189$	0	0
$190$	2.74456	0.199112
$191$	8.20442	0.593651	0.296826	−	0.954932i	$-0.404072\pi$
$191$	8.20442	0.593651	0.296826	+	0.954932i	$0.404072\pi$
$192$	0	0
$193$	−18.2337	−1.31249	−0.656245	−	0.754548i	$-0.727856\pi$
$193$	−18.2337	−1.31249	−0.656245	+	0.754548i	$0.727856\pi$
$194$	−19.7613	−1.41878
$195$	0	0
$196$	−1.62772	−0.116266
$197$	3.50157	0.249477	0.124738	−	0.992190i	$-0.460191\pi$
$197$	3.50157	0.249477	0.124738	+	0.992190i	$0.460191\pi$
$198$	0	0
$199$	18.1168	1.28427	0.642135	−	0.766592i	$-0.278049\pi$
$199$	18.1168	1.28427	0.642135	+	0.766592i	$0.278049\pi$
$200$	1.12675	0.0796732
$201$	0	0
$202$	−10.9783	−0.772427
$203$	−23.1138	−1.62227
$204$	0	0
$205$	−5.48913	−0.383377
$206$	−23.9147	−1.66622
$207$	0	0
$208$	−6.23369	−0.432228
$209$	−4.70285	−0.325303
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	−4.45150	−0.305730
$213$	0	0
$214$	12.0000	0.820303
$215$	13.1586	0.897412
$216$	0	0
$217$	−22.7446	−1.54400
$218$	21.3631	1.44689
$219$	0	0
$220$	3.76631	0.253925
$221$	4.30243	0.289413
$222$	0	0
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	7.00314	0.467917
$225$	0	0
$226$	2.23369	0.148583
$227$	−12.9073	−0.856686	−0.428343	−	0.903616i	$-0.640903\pi$
$227$	−12.9073	−0.856686	−0.428343	+	0.903616i	$0.640903\pi$
$228$	0	0
$229$	21.3723	1.41232	0.706160	−	0.708052i	$-0.250426\pi$
$229$	21.3723	1.41232	0.706160	+	0.708052i	$0.250426\pi$
$230$	−18.8114	−1.24039
$231$	0	0
$232$	20.7446	1.36195
$233$	−7.25450	−0.475258	−0.237629	−	0.971356i	$-0.576370\pi$
$233$	−7.25450	−0.475258	−0.237629	+	0.971356i	$0.576370\pi$
$234$	0	0
$235$	19.3723	1.26371
$236$	−1.89986	−0.123670
$237$	0	0
$238$	9.25544	0.599941
$239$	27.0158	1.74751	0.873755	−	0.486367i	$-0.161678\pi$
$239$	27.0158	1.74751	0.873755	+	0.486367i	$0.161678\pi$
$240$	0	0
$241$	10.2337	0.659210	0.329605	−	0.944119i	$-0.393084\pi$
$241$	10.2337	0.659210	0.329605	+	0.944119i	$0.393084\pi$
$242$	14.1831	0.911724
$243$	0	0
$244$	−4.51087	−0.288779
$245$	9.40571	0.600909
$246$	0	0
$247$	2.00000	0.127257
$248$	20.4131	1.29624
$249$	0	0
$250$	−14.7446	−0.932528
$251$	14.9094	0.941074	0.470537	−	0.882380i	$-0.344060\pi$
$251$	14.9094	0.941074	0.470537	+	0.882380i	$0.344060\pi$
$252$	0	0
$253$	32.2337	2.02651
$254$	−1.60171	−0.100500
$255$	0	0
$256$	−8.60597	−0.537873
$257$	2.55164	0.159167	0.0795835	−	0.996828i	$-0.474641\pi$
$257$	2.55164	0.159167	0.0795835	+	0.996828i	$0.474641\pi$
$258$	0	0
$259$	−2.51087	−0.156018
$260$	−1.60171	−0.0993340
$261$	0	0
$262$	4.97825	0.307557
$263$	9.80614	0.604672	0.302336	−	0.953201i	$-0.402233\pi$
$263$	9.80614	0.604672	0.302336	+	0.953201i	$0.402233\pi$
$264$	0	0
$265$	25.7228	1.58014
$266$	4.30243	0.263799
$267$	0	0
$268$	1.48913	0.0909628
$269$	25.6655	1.56485	0.782426	−	0.622743i	$-0.213982\pi$
$269$	25.6655	1.56485	0.782426	+	0.622743i	$0.213982\pi$
$270$	0	0
$271$	2.51087	0.152525	0.0762624	−	0.997088i	$-0.475701\pi$
$271$	2.51087	0.152525	0.0762624	+	0.997088i	$0.475701\pi$
$272$	−6.70500	−0.406550
$273$	0	0
$274$	21.2554	1.28409
$275$	1.75079	0.105576
$276$	0	0
$277$	−8.11684	−0.487694	−0.243847	−	0.969814i	$-0.578409\pi$
$277$	−8.11684	−0.487694	−0.243847	+	0.969814i	$0.578409\pi$
$278$	−18.0106	−1.08020
$279$	0	0
$280$	−21.9565	−1.31215
$281$	−10.3556	−0.617766	−0.308883	−	0.951100i	$-0.599955\pi$
$281$	−10.3556	−0.617766	−0.308883	+	0.951100i	$0.599955\pi$
$282$	0	0
$283$	0.627719	0.0373140	0.0186570	−	0.999826i	$-0.494061\pi$
$283$	0.627719	0.0373140	0.0186570	+	0.999826i	$0.494061\pi$
$284$	5.10328	0.302824
$285$	0	0
$286$	−12.0000	−0.709575
$287$	−8.60485	−0.507928
$288$	0	0
$289$	−12.3723	−0.727781
$290$	−18.8114	−1.10464
$291$	0	0
$292$	−4.51087	−0.263979
$293$	−26.4663	−1.54618	−0.773090	−	0.634296i	$-0.781290\pi$
$293$	−26.4663	−1.54618	−0.773090	+	0.634296i	$0.781290\pi$
$294$	0	0
$295$	10.9783	0.639178
$296$	2.25350	0.130982
$297$	0	0
$298$	−1.72281	−0.0997999
$299$	−13.7081	−0.792762
$300$	0	0
$301$	20.6277	1.18896
$302$	−5.10328	−0.293661
$303$	0	0
$304$	−3.11684	−0.178763
$305$	26.0659	1.49253
$306$	0	0
$307$	2.51087	0.143303	0.0716516	−	0.997430i	$-0.477173\pi$
$307$	2.51087	0.143303	0.0716516	+	0.997430i	$0.477173\pi$
$308$	5.90414	0.336420
$309$	0	0
$310$	−18.5109	−1.05135
$311$	−27.0158	−1.53193	−0.765964	−	0.642883i	$-0.777738\pi$
$311$	−27.0158	−1.53193	−0.765964	+	0.642883i	$0.777738\pi$
$312$	0	0
$313$	−15.4891	−0.875497	−0.437749	−	0.899097i	$-0.644224\pi$
$313$	−15.4891	−0.875497	−0.437749	+	0.899097i	$0.644224\pi$
$314$	2.55164	0.143997
$315$	0	0
$316$	1.48913	0.0837698
$317$	−35.0712	−1.96979	−0.984897	−	0.173139i	$-0.944609\pi$
$317$	−35.0712	−1.96979	−0.984897	+	0.173139i	$0.944609\pi$
$318$	0	0
$319$	32.2337	1.80474
$320$	19.1096	1.06826
$321$	0	0
$322$	−29.4891	−1.64336
$323$	2.15121	0.119697
$324$	0	0
$325$	−0.744563	−0.0413009
$326$	1.89986	0.105223
$327$	0	0
$328$	7.72281	0.426421
$329$	30.3683	1.67426
$330$	0	0
$331$	−26.9783	−1.48286	−0.741429	−	0.671031i	$-0.765852\pi$
$331$	−26.9783	−1.48286	−0.741429	+	0.671031i	$0.765852\pi$
$332$	−0.651785	−0.0357713
$333$	0	0
$334$	5.48913	0.300352
$335$	−8.60485	−0.470133
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	−11.4824	−0.624560
$339$	0	0
$340$	−1.72281	−0.0934327
$341$	31.7187	1.71766
$342$	0	0
$343$	−8.86141	−0.478471
$344$	−18.5133	−0.998169
$345$	0	0
$346$	8.74456	0.470111
$347$	−3.10114	−0.166478	−0.0832390	−	0.996530i	$-0.526526\pi$
$347$	−3.10114	−0.166478	−0.0832390	+	0.996530i	$0.526526\pi$
$348$	0	0
$349$	−10.8614	−0.581398	−0.290699	−	0.956815i	$-0.593888\pi$
$349$	−10.8614	−0.581398	−0.290699	+	0.956815i	$0.593888\pi$
$350$	−1.60171	−0.0856152
$351$	0	0
$352$	−9.76631	−0.520546
$353$	−22.3130	−1.18760	−0.593800	−	0.804612i	$-0.702373\pi$
$353$	−22.3130	−1.18760	−0.593800	+	0.804612i	$0.702373\pi$
$354$	0	0
$355$	−29.4891	−1.56512
$356$	−4.45150	−0.235929
$357$	0	0
$358$	−12.0000	−0.634220
$359$	−27.8167	−1.46811	−0.734055	−	0.679090i	$-0.762374\pi$
$359$	−27.8167	−1.46811	−0.734055	+	0.679090i	$0.762374\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−4.45150	−0.233966
$363$	0	0
$364$	−2.51087	−0.131606
$365$	26.0659	1.36435
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	21.3631	1.11363
$369$	0	0
$370$	−2.04350	−0.106236
$371$	40.3236	2.09349
$372$	0	0
$373$	10.2337	0.529880	0.264940	−	0.964265i	$-0.414648\pi$
$373$	10.2337	0.529880	0.264940	+	0.964265i	$0.414648\pi$
$374$	−12.9073	−0.667420
$375$	0	0
$376$	−27.2554	−1.40559
$377$	−13.7081	−0.706005
$378$	0	0
$379$	17.2554	0.886352	0.443176	−	0.896435i	$-0.353852\pi$
$379$	17.2554	0.886352	0.443176	+	0.896435i	$0.353852\pi$
$380$	−0.800857	−0.0410831
$381$	0	0
$382$	10.4674	0.535558
$383$	−0.800857	−0.0409219	−0.0204609	−	0.999791i	$-0.506513\pi$
$383$	−0.800857	−0.0409219	−0.0204609	+	0.999791i	$0.506513\pi$
$384$	0	0
$385$	−34.1168	−1.73876
$386$	−23.2629	−1.18405
$387$	0	0
$388$	5.76631	0.292740
$389$	16.6602	0.844706	0.422353	−	0.906431i	$-0.361204\pi$
$389$	16.6602	0.844706	0.422353	+	0.906431i	$0.361204\pi$
$390$	0	0
$391$	−14.7446	−0.745665
$392$	−13.2332	−0.668376
$393$	0	0
$394$	4.46738	0.225063
$395$	−8.60485	−0.432957
$396$	0	0
$397$	0.116844	0.00586423	0.00293212	−	0.999996i	$-0.499067\pi$
$397$	0.116844	0.00586423	0.00293212	+	0.999996i	$0.499067\pi$
$398$	23.1138	1.15859
$399$	0	0
$400$	1.16034	0.0580171
$401$	16.2598	0.811975	0.405987	−	0.913879i	$-0.366928\pi$
$401$	16.2598	0.811975	0.405987	+	0.913879i	$0.366928\pi$
$402$	0	0
$403$	−13.4891	−0.671941
$404$	3.20343	0.159376
$405$	0	0
$406$	−29.4891	−1.46352
$407$	3.50157	0.173566
$408$	0	0
$409$	−15.4891	−0.765888	−0.382944	−	0.923772i	$-0.625090\pi$
$409$	−15.4891	−0.765888	−0.382944	+	0.923772i	$0.625090\pi$
$410$	−7.00314	−0.345860
$411$	0	0
$412$	6.97825	0.343794
$413$	17.2097	0.846834
$414$	0	0
$415$	3.76631	0.184881
$416$	4.15335	0.203635
$417$	0	0
$418$	−6.00000	−0.293470
$419$	−1.75079	−0.0855314	−0.0427657	−	0.999085i	$-0.513617\pi$
$419$	−1.75079	−0.0855314	−0.0427657	+	0.999085i	$0.513617\pi$
$420$	0	0
$421$	36.9783	1.80221	0.901105	−	0.433601i	$-0.142757\pi$
$421$	36.9783	1.80221	0.901105	+	0.433601i	$0.142757\pi$
$422$	−5.10328	−0.248424
$423$	0	0
$424$	−36.1902	−1.75755
$425$	−0.800857	−0.0388472
$426$	0	0
$427$	40.8614	1.97742
$428$	−3.50157	−0.169255
$429$	0	0
$430$	16.7881	0.809592
$431$	7.80400	0.375905	0.187953	−	0.982178i	$-0.439815\pi$
$431$	7.80400	0.375905	0.187953	+	0.982178i	$0.439815\pi$
$432$	0	0
$433$	−22.0000	−1.05725	−0.528626	−	0.848855i	$-0.677293\pi$
$433$	−22.0000	−1.05725	−0.528626	+	0.848855i	$0.677293\pi$
$434$	−29.0180	−1.39291
$435$	0	0
$436$	−6.23369	−0.298540
$437$	−6.85407	−0.327875
$438$	0	0
$439$	16.2337	0.774792	0.387396	−	0.921913i	$-0.373375\pi$
$439$	16.2337	0.774792	0.387396	+	0.921913i	$0.373375\pi$
$440$	30.6197	1.45974
$441$	0	0
$442$	5.48913	0.261091
$443$	12.5069	0.594218	0.297109	−	0.954843i	$-0.403977\pi$
$443$	12.5069	0.594218	0.297109	+	0.954843i	$0.403977\pi$
$444$	0	0
$445$	25.7228	1.21938
$446$	−5.10328	−0.241647
$447$	0	0
$448$	29.9565	1.41531
$449$	−33.4695	−1.57952	−0.789761	−	0.613414i	$-0.789796\pi$
$449$	−33.4695	−1.57952	−0.789761	+	0.613414i	$0.789796\pi$
$450$	0	0
$451$	12.0000	0.565058
$452$	−0.651785	−0.0306574
$453$	0	0
$454$	−16.4674	−0.772852
$455$	14.5090	0.680192
$456$	0	0
$457$	−2.62772	−0.122919	−0.0614597	−	0.998110i	$-0.519576\pi$
$457$	−2.62772	−0.122919	−0.0614597	+	0.998110i	$0.519576\pi$
$458$	27.2672	1.27411
$459$	0	0
$460$	5.48913	0.255932
$461$	5.65278	0.263276	0.131638	−	0.991298i	$-0.457976\pi$
$461$	5.65278	0.263276	0.131638	+	0.991298i	$0.457976\pi$
$462$	0	0
$463$	3.37228	0.156723	0.0783616	−	0.996925i	$-0.475031\pi$
$463$	3.37228	0.156723	0.0783616	+	0.996925i	$0.475031\pi$
$464$	21.3631	0.991755
$465$	0	0
$466$	−9.25544	−0.428750
$467$	30.5174	1.41218	0.706089	−	0.708123i	$-0.250458\pi$
$467$	30.5174	1.41218	0.706089	+	0.708123i	$0.250458\pi$
$468$	0	0
$469$	−13.4891	−0.622870
$470$	24.7156	1.14004
$471$	0	0
$472$	−15.4456	−0.710943
$473$	−28.7666	−1.32269
$474$	0	0
$475$	−0.372281	−0.0170814
$476$	−2.70071	−0.123787
$477$	0	0
$478$	34.4674	1.57650
$479$	8.45578	0.386355	0.193177	−	0.981164i	$-0.438121\pi$
$479$	8.45578	0.386355	0.193177	+	0.981164i	$0.438121\pi$
$480$	0	0
$481$	−1.48913	−0.0678983
$482$	13.0564	0.594701
$483$	0	0
$484$	−4.13859	−0.188118
$485$	−33.3204	−1.51300
$486$	0	0
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	−36.6729	−1.66010
$489$	0	0
$490$	12.0000	0.542105
$491$	6.85407	0.309320	0.154660	−	0.987968i	$-0.450572\pi$
$491$	6.85407	0.309320	0.154660	+	0.987968i	$0.450572\pi$
$492$	0	0
$493$	−14.7446	−0.664062
$494$	2.55164	0.114804
$495$	0	0
$496$	21.0217	0.943904
$497$	−46.2277	−2.07360
$498$	0	0
$499$	6.11684	0.273828	0.136914	−	0.990583i	$-0.456282\pi$
$499$	6.11684	0.273828	0.136914	+	0.990583i	$0.456282\pi$
$500$	4.30243	0.192410
$501$	0	0
$502$	19.0217	0.848982
$503$	−22.1639	−0.988240	−0.494120	−	0.869394i	$-0.664510\pi$
$503$	−22.1639	−0.988240	−0.494120	+	0.869394i	$0.664510\pi$
$504$	0	0
$505$	−18.5109	−0.823723
$506$	41.1244	1.82820
$507$	0	0
$508$	0.467376	0.0207365
$509$	−10.3556	−0.459006	−0.229503	−	0.973308i	$-0.573710\pi$
$509$	−10.3556	−0.459006	−0.229503	+	0.973308i	$0.573710\pi$
$510$	0	0
$511$	40.8614	1.80760
$512$	−25.3396	−1.11986
$513$	0	0
$514$	3.25544	0.143591
$515$	−40.3236	−1.77687
$516$	0	0
$517$	−42.3505	−1.86257
$518$	−3.20343	−0.140750
$519$	0	0
$520$	−13.0217	−0.571041
$521$	7.65492	0.335368	0.167684	−	0.985841i	$-0.446371\pi$
$521$	7.65492	0.335368	0.167684	+	0.985841i	$0.446371\pi$
$522$	0	0
$523$	16.2337	0.709850	0.354925	−	0.934895i	$-0.384506\pi$
$523$	16.2337	0.709850	0.354925	+	0.934895i	$0.384506\pi$
$524$	−1.45264	−0.0634589
$525$	0	0
$526$	12.5109	0.545500
$527$	−14.5090	−0.632022
$528$	0	0
$529$	23.9783	1.04253
$530$	32.8177	1.42551
$531$	0	0
$532$	−1.25544	−0.0544301
$533$	−5.10328	−0.221048
$534$	0	0
$535$	20.2337	0.874779
$536$	12.1064	0.522918
$537$	0	0
$538$	32.7446	1.41172
$539$	−20.5622	−0.885677
$540$	0	0
$541$	3.88316	0.166950	0.0834750	−	0.996510i	$-0.473398\pi$
$541$	3.88316	0.166950	0.0834750	+	0.996510i	$0.473398\pi$
$542$	3.20343	0.137599
$543$	0	0
$544$	4.46738	0.191537
$545$	36.0211	1.54298
$546$	0	0
$547$	−12.2337	−0.523075	−0.261537	−	0.965193i	$-0.584229\pi$
$547$	−12.2337	−0.523075	−0.261537	+	0.965193i	$0.584229\pi$
$548$	−6.20228	−0.264948
$549$	0	0
$550$	2.23369	0.0952448
$551$	−6.85407	−0.291993
$552$	0	0
$553$	−13.4891	−0.573616
$554$	−10.3556	−0.439969
$555$	0	0
$556$	5.25544	0.222880
$557$	1.35036	0.0572165	0.0286082	−	0.999591i	$-0.490892\pi$
$557$	1.35036	0.0572165	0.0286082	+	0.999591i	$0.490892\pi$
$558$	0	0
$559$	12.2337	0.517430
$560$	−22.6111	−0.955495
$561$	0	0
$562$	−13.2119	−0.557312
$563$	25.8146	1.08795	0.543977	−	0.839100i	$-0.316918\pi$
$563$	25.8146	1.08795	0.543977	+	0.839100i	$0.316918\pi$
$564$	0	0
$565$	3.76631	0.158450
$566$	0.800857	0.0336625
$567$	0	0
$568$	41.4891	1.74085
$569$	−20.5622	−0.862012	−0.431006	−	0.902349i	$-0.641841\pi$
$569$	−20.5622	−0.862012	−0.431006	+	0.902349i	$0.641841\pi$
$570$	0	0
$571$	25.4891	1.06669	0.533343	−	0.845899i	$-0.320935\pi$
$571$	25.4891	1.06669	0.533343	+	0.845899i	$0.320935\pi$
$572$	3.50157	0.146408
$573$	0	0
$574$	−10.9783	−0.458223
$575$	2.55164	0.106411
$576$	0	0
$577$	−23.8832	−0.994269	−0.497134	−	0.867674i	$-0.665614\pi$
$577$	−23.8832	−0.994269	−0.497134	+	0.867674i	$0.665614\pi$
$578$	−15.7848	−0.656562
$579$	0	0
$580$	5.48913	0.227924
$581$	5.90414	0.244945
$582$	0	0
$583$	−56.2337	−2.32896
$584$	−36.6729	−1.51754
$585$	0	0
$586$	−33.7663	−1.39487
$587$	32.1191	1.32570	0.662849	−	0.748753i	$-0.269347\pi$
$587$	32.1191	1.32570	0.662849	+	0.748753i	$0.269347\pi$
$588$	0	0
$589$	−6.74456	−0.277905
$590$	14.0063	0.576629
$591$	0	0
$592$	2.32069	0.0953796
$593$	−27.4163	−1.12585	−0.562926	−	0.826508i	$-0.690324\pi$
$593$	−27.4163	−1.12585	−0.562926	+	0.826508i	$0.690324\pi$
$594$	0	0
$595$	15.6060	0.639782
$596$	0.502713	0.0205919
$597$	0	0
$598$	−17.4891	−0.715184
$599$	−8.60485	−0.351585	−0.175792	−	0.984427i	$-0.556249\pi$
$599$	−8.60485	−0.351585	−0.175792	+	0.984427i	$0.556249\pi$
$600$	0	0
$601$	−36.7446	−1.49884	−0.749421	−	0.662094i	$-0.769668\pi$
$601$	−36.7446	−1.49884	−0.749421	+	0.662094i	$0.769668\pi$
$602$	26.3173	1.07261
$603$	0	0
$604$	1.48913	0.0605916
$605$	23.9147	0.972271
$606$	0	0
$607$	17.2554	0.700377	0.350188	−	0.936679i	$-0.386118\pi$
$607$	17.2554	0.700377	0.350188	+	0.936679i	$0.386118\pi$
$608$	2.07668	0.0842204
$609$	0	0
$610$	33.2554	1.34647
$611$	18.0106	0.728629
$612$	0	0
$613$	−37.6060	−1.51889	−0.759445	−	0.650571i	$-0.774530\pi$
$613$	−37.6060	−1.51889	−0.759445	+	0.650571i	$0.774530\pi$
$614$	3.20343	0.129280
$615$	0	0
$616$	48.0000	1.93398
$617$	−2.15121	−0.0866046	−0.0433023	−	0.999062i	$-0.513788\pi$
$617$	−2.15121	−0.0866046	−0.0433023	+	0.999062i	$0.513788\pi$
$618$	0	0
$619$	−38.9783	−1.56667	−0.783334	−	0.621601i	$-0.786483\pi$
$619$	−38.9783	−1.56667	−0.783334	+	0.621601i	$0.786483\pi$
$620$	5.40143	0.216927
$621$	0	0
$622$	−34.4674	−1.38202
$623$	40.3236	1.61553
$624$	0	0
$625$	−23.0000	−0.920000
$626$	−19.7613	−0.789822
$627$	0	0
$628$	−0.744563	−0.0297113
$629$	−1.60171	−0.0638645
$630$	0	0
$631$	−2.11684	−0.0842702	−0.0421351	−	0.999112i	$-0.513416\pi$
$631$	−2.11684	−0.0842702	−0.0421351	+	0.999112i	$0.513416\pi$
$632$	12.1064	0.481568
$633$	0	0
$634$	−44.7446	−1.77703
$635$	−2.70071	−0.107175
$636$	0	0
$637$	8.74456	0.346472
$638$	41.1244	1.62813
$639$	0	0
$640$	15.4456	0.610542
$641$	10.3556	0.409023	0.204512	−	0.978864i	$-0.434439\pi$
$641$	10.3556	0.409023	0.204512	+	0.978864i	$0.434439\pi$
$642$	0	0
$643$	−17.8832	−0.705243	−0.352621	−	0.935766i	$-0.614710\pi$
$643$	−17.8832	−0.705243	−0.352621	+	0.935766i	$0.614710\pi$
$644$	8.60485	0.339079
$645$	0	0
$646$	2.74456	0.107983
$647$	17.6101	0.692326	0.346163	−	0.938174i	$-0.387484\pi$
$647$	17.6101	0.692326	0.346163	+	0.938174i	$0.387484\pi$
$648$	0	0
$649$	−24.0000	−0.942082
$650$	−0.949929	−0.0372593
$651$	0	0
$652$	−0.554374	−0.0217109
$653$	2.95207	0.115523	0.0577617	−	0.998330i	$-0.481604\pi$
$653$	2.95207	0.115523	0.0577617	+	0.998330i	$0.481604\pi$
$654$	0	0
$655$	8.39403	0.327982
$656$	7.95307	0.310515
$657$	0	0
$658$	38.7446	1.51042
$659$	41.9253	1.63318	0.816588	−	0.577221i	$-0.195863\pi$
$659$	41.9253	1.63318	0.816588	+	0.577221i	$0.195863\pi$
$660$	0	0
$661$	4.74456	0.184542	0.0922710	−	0.995734i	$-0.470587\pi$
$661$	4.74456	0.184542	0.0922710	+	0.995734i	$0.470587\pi$
$662$	−34.4194	−1.33775
$663$	0	0
$664$	−5.29894	−0.205639
$665$	7.25450	0.281317
$666$	0	0
$667$	46.9783	1.81901
$668$	−1.60171	−0.0619721
$669$	0	0
$670$	−10.9783	−0.424127
$671$	−56.9838	−2.19983
$672$	0	0
$673$	−36.7446	−1.41640	−0.708199	−	0.706012i	$-0.750492\pi$
$673$	−36.7446	−1.41640	−0.708199	+	0.706012i	$0.750492\pi$
$674$	17.8615	0.687999
$675$	0	0
$676$	3.35053	0.128867
$677$	13.5591	0.521117	0.260559	−	0.965458i	$-0.416093\pi$
$677$	13.5591	0.521117	0.260559	+	0.965458i	$0.416093\pi$
$678$	0	0
$679$	−52.2337	−2.00454
$680$	−14.0063	−0.537116
$681$	0	0
$682$	40.4674	1.54958
$683$	−35.2203	−1.34767	−0.673833	−	0.738884i	$-0.735353\pi$
$683$	−35.2203	−1.34767	−0.673833	+	0.738884i	$0.735353\pi$
$684$	0	0
$685$	35.8397	1.36936
$686$	−11.3056	−0.431649
$687$	0	0
$688$	−19.0652	−0.726856
$689$	23.9147	0.911078
$690$	0	0
$691$	9.88316	0.375973	0.187986	−	0.982172i	$-0.439804\pi$
$691$	9.88316	0.375973	0.187986	+	0.982172i	$0.439804\pi$
$692$	−2.55164	−0.0969989
$693$	0	0
$694$	−3.95650	−0.150187
$695$	−30.3683	−1.15194
$696$	0	0
$697$	−5.48913	−0.207915
$698$	−13.8572	−0.524503
$699$	0	0
$700$	0.467376	0.0176651
$701$	−29.0180	−1.09599	−0.547997	−	0.836480i	$-0.684610\pi$
$701$	−29.0180	−1.09599	−0.547997	+	0.836480i	$0.684610\pi$
$702$	0	0
$703$	−0.744563	−0.0280817
$704$	−41.7762	−1.57450
$705$	0	0
$706$	−28.4674	−1.07138
$707$	−29.0180	−1.09133
$708$	0	0
$709$	38.0000	1.42712	0.713560	−	0.700594i	$-0.247082\pi$
$709$	38.0000	1.42712	0.713560	+	0.700594i	$0.247082\pi$
$710$	−37.6228	−1.41196
$711$	0	0
$712$	−36.1902	−1.35628
$713$	46.2277	1.73124
$714$	0	0
$715$	−20.2337	−0.756697
$716$	3.50157	0.130860
$717$	0	0
$718$	−35.4891	−1.32444
$719$	7.40357	0.276107	0.138053	−	0.990425i	$-0.455915\pi$
$719$	7.40357	0.276107	0.138053	+	0.990425i	$0.455915\pi$
$720$	0	0
$721$	−63.2119	−2.35414
$722$	1.27582	0.0474811
$723$	0	0
$724$	1.29894	0.0482746
$725$	2.55164	0.0947656
$726$	0	0
$727$	14.3505	0.532232	0.266116	−	0.963941i	$-0.414260\pi$
$727$	14.3505	0.532232	0.266116	+	0.963941i	$0.414260\pi$
$728$	−20.4131	−0.756561
$729$	0	0
$730$	33.2554	1.23084
$731$	13.1586	0.486690
$732$	0	0
$733$	−50.4674	−1.86406	−0.932028	−	0.362387i	$-0.881962\pi$
$733$	−50.4674	−1.86406	−0.932028	+	0.362387i	$0.881962\pi$
$734$	10.2066	0.376731
$735$	0	0
$736$	−14.2337	−0.524661
$737$	18.8114	0.692928
$738$	0	0
$739$	9.88316	0.363558	0.181779	−	0.983339i	$-0.441814\pi$
$739$	9.88316	0.363558	0.181779	+	0.983339i	$0.441814\pi$
$740$	0.596288	0.0219200
$741$	0	0
$742$	51.4456	1.88863
$743$	42.7261	1.56747	0.783735	−	0.621096i	$-0.213312\pi$
$743$	42.7261	1.56747	0.783735	+	0.621096i	$0.213312\pi$
$744$	0	0
$745$	−2.90491	−0.106428
$746$	13.0564	0.478027
$747$	0	0
$748$	3.76631	0.137710
$749$	31.7187	1.15898
$750$	0	0
$751$	−44.4674	−1.62264	−0.811319	−	0.584604i	$-0.801250\pi$
$751$	−44.4674	−1.62264	−0.811319	+	0.584604i	$0.801250\pi$
$752$	−28.0681	−1.02354
$753$	0	0
$754$	−17.4891	−0.636916
$755$	−8.60485	−0.313163
$756$	0	0
$757$	12.1168	0.440394	0.220197	−	0.975455i	$-0.429330\pi$
$757$	12.1168	0.440394	0.220197	+	0.975455i	$0.429330\pi$
$758$	22.0148	0.799615
$759$	0	0
$760$	−6.51087	−0.236174
$761$	−25.2651	−0.915858	−0.457929	−	0.888989i	$-0.651409\pi$
$761$	−25.2651	−0.915858	−0.457929	+	0.888989i	$0.651409\pi$
$762$	0	0
$763$	56.4674	2.04426
$764$	−3.05435	−0.110503
$765$	0	0
$766$	−1.02175	−0.0369173
$767$	10.2066	0.368538
$768$	0	0
$769$	24.1168	0.869676	0.434838	−	0.900509i	$-0.356806\pi$
$769$	24.1168	0.869676	0.434838	+	0.900509i	$0.356806\pi$
$770$	−43.5270	−1.56860
$771$	0	0
$772$	6.78806	0.244308
$773$	2.55164	0.0917762	0.0458881	−	0.998947i	$-0.485388\pi$
$773$	2.55164	0.0917762	0.0458881	+	0.998947i	$0.485388\pi$
$774$	0	0
$775$	2.51087	0.0901933
$776$	46.8795	1.68288
$777$	0	0
$778$	21.2554	0.762044
$779$	−2.55164	−0.0914220
$780$	0	0
$781$	64.4674	2.30682
$782$	−18.8114	−0.672695
$783$	0	0
$784$	−13.6277	−0.486704
$785$	4.30243	0.153560
$786$	0	0
$787$	41.9565	1.49559	0.747794	−	0.663931i	$-0.231113\pi$
$787$	41.9565	1.49559	0.747794	+	0.663931i	$0.231113\pi$
$788$	−1.30357	−0.0464377
$789$	0	0
$790$	−10.9783	−0.390589
$791$	5.90414	0.209927
$792$	0	0
$793$	24.2337	0.860563
$794$	0.149072	0.00529037
$795$	0	0
$796$	−6.74456	−0.239055
$797$	11.1565	0.395183	0.197592	−	0.980284i	$-0.436688\pi$
$797$	11.1565	0.395183	0.197592	+	0.980284i	$0.436688\pi$
$798$	0	0
$799$	19.3723	0.685342
$800$	−0.773108	−0.0273335
$801$	0	0
$802$	20.7446	0.732516
$803$	−56.9838	−2.01091
$804$	0	0
$805$	−49.7228	−1.75250
$806$	−17.2097	−0.606186
$807$	0	0
$808$	26.0435	0.916207
$809$	−34.6708	−1.21896	−0.609480	−	0.792802i	$-0.708622\pi$
$809$	−34.6708	−1.21896	−0.609480	+	0.792802i	$0.708622\pi$
$810$	0	0
$811$	−25.2554	−0.886838	−0.443419	−	0.896314i	$-0.646235\pi$
$811$	−25.2554	−0.886838	−0.443419	+	0.896314i	$0.646235\pi$
$812$	8.60485	0.301971
$813$	0	0
$814$	4.46738	0.156581
$815$	3.20343	0.112211
$816$	0	0
$817$	6.11684	0.214001
$818$	−19.7613	−0.690939
$819$	0	0
$820$	2.04350	0.0713621
$821$	−17.4611	−0.609395	−0.304698	−	0.952449i	$-0.598555\pi$
$821$	−17.4611	−0.609395	−0.304698	+	0.952449i	$0.598555\pi$
$822$	0	0
$823$	−31.6060	−1.10171	−0.550857	−	0.834599i	$-0.685699\pi$
$823$	−31.6060	−1.10171	−0.550857	+	0.834599i	$0.685699\pi$
$824$	56.7324	1.97637
$825$	0	0
$826$	21.9565	0.763964
$827$	−42.7261	−1.48573	−0.742866	−	0.669440i	$-0.766534\pi$
$827$	−42.7261	−1.48573	−0.742866	+	0.669440i	$0.766534\pi$
$828$	0	0
$829$	−12.7446	−0.442637	−0.221318	−	0.975202i	$-0.571036\pi$
$829$	−12.7446	−0.442637	−0.221318	+	0.975202i	$0.571036\pi$
$830$	4.80514	0.166789
$831$	0	0
$832$	17.7663	0.615936
$833$	9.40571	0.325889
$834$	0	0
$835$	9.25544	0.320298
$836$	1.75079	0.0605522
$837$	0	0
$838$	−2.23369	−0.0771615
$839$	−7.80400	−0.269424	−0.134712	−	0.990885i	$-0.543011\pi$
$839$	−7.80400	−0.269424	−0.134712	+	0.990885i	$0.543011\pi$
$840$	0	0
$841$	17.9783	0.619940
$842$	47.1776	1.62585
$843$	0	0
$844$	1.48913	0.0512578
$845$	−19.3609	−0.666036
$846$	0	0
$847$	37.4891	1.28814
$848$	−37.2692	−1.27983
$849$	0	0
$850$	−1.02175	−0.0350457
$851$	5.10328	0.174938
$852$	0	0
$853$	30.4674	1.04318	0.521592	−	0.853195i	$-0.325339\pi$
$853$	30.4674	1.04318	0.521592	+	0.853195i	$0.325339\pi$
$854$	52.1318	1.78391
$855$	0	0
$856$	−28.4674	−0.972995
$857$	42.0743	1.43723	0.718616	−	0.695407i	$-0.244776\pi$
$857$	42.0743	1.43723	0.718616	+	0.695407i	$0.244776\pi$
$858$	0	0
$859$	−14.1168	−0.481661	−0.240830	−	0.970567i	$-0.577420\pi$
$859$	−14.1168	−0.481661	−0.240830	+	0.970567i	$0.577420\pi$
$860$	−4.89871	−0.167045
$861$	0	0
$862$	9.95650	0.339120
$863$	22.3130	0.759543	0.379771	−	0.925080i	$-0.376003\pi$
$863$	22.3130	0.759543	0.379771	+	0.925080i	$0.376003\pi$
$864$	0	0
$865$	14.7446	0.501330
$866$	−28.0681	−0.953791
$867$	0	0
$868$	8.46738	0.287401
$869$	18.8114	0.638134
$870$	0	0
$871$	−8.00000	−0.271070
$872$	−50.6792	−1.71621
$873$	0	0
$874$	−8.74456	−0.295789
$875$	−38.9732	−1.31753
$876$	0	0
$877$	14.0000	0.472746	0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000	0.472746	0.236373	+	0.971662i	$0.424041\pi$
$878$	20.7113	0.698972
$879$	0	0
$880$	31.5326	1.06296
$881$	25.2651	0.851201	0.425601	−	0.904911i	$-0.360063\pi$
$881$	25.2651	0.851201	0.425601	+	0.904911i	$0.360063\pi$
$882$	0	0
$883$	−14.1168	−0.475070	−0.237535	−	0.971379i	$-0.576339\pi$
$883$	−14.1168	−0.475070	−0.237535	+	0.971379i	$0.576339\pi$
$884$	−1.60171	−0.0538714
$885$	0	0
$886$	15.9565	0.536069
$887$	10.2066	0.342703	0.171351	−	0.985210i	$-0.445187\pi$
$887$	10.2066	0.342703	0.171351	+	0.985210i	$0.445187\pi$
$888$	0	0
$889$	−4.23369	−0.141993
$890$	32.8177	1.10005
$891$	0	0
$892$	1.48913	0.0498596
$893$	9.00528	0.301350
$894$	0	0
$895$	−20.2337	−0.676338
$896$	24.2128	0.808894
$897$	0	0
$898$	−42.7011	−1.42495
$899$	46.2277	1.54178
$900$	0	0
$901$	25.7228	0.856951
$902$	15.3098	0.509762
$903$	0	0
$904$	−5.29894	−0.176240
$905$	−7.50585	−0.249503
$906$	0	0
$907$	34.7446	1.15367	0.576837	−	0.816859i	$-0.304287\pi$
$907$	34.7446	1.15367	0.576837	+	0.816859i	$0.304287\pi$
$908$	4.80514	0.159464
$909$	0	0
$910$	18.5109	0.613630
$911$	−24.7156	−0.818863	−0.409432	−	0.912341i	$-0.634273\pi$
$911$	−24.7156	−0.818863	−0.409432	+	0.912341i	$0.634273\pi$
$912$	0	0
$913$	−8.23369	−0.272495
$914$	−3.35250	−0.110891
$915$	0	0
$916$	−7.95650	−0.262890
$917$	13.1586	0.434536
$918$	0	0
$919$	−26.9783	−0.889930	−0.444965	−	0.895548i	$-0.646784\pi$
$919$	−26.9783	−0.889930	−0.444965	+	0.895548i	$0.646784\pi$
$920$	44.6260	1.47127
$921$	0	0
$922$	7.21194	0.237513
$923$	−27.4163	−0.902418
$924$	0	0
$925$	0.277187	0.00911384
$926$	4.30243	0.141387
$927$	0	0
$928$	−14.2337	−0.467244
$929$	46.2277	1.51668	0.758341	−	0.651858i	$-0.226010\pi$
$929$	46.2277	1.51668	0.758341	+	0.651858i	$0.226010\pi$
$930$	0	0
$931$	4.37228	0.143296
$932$	2.70071	0.0884648
$933$	0	0
$934$	38.9348	1.27398
$935$	−21.7635	−0.711742
$936$	0	0
$937$	12.1168	0.395840	0.197920	−	0.980218i	$-0.436581\pi$
$937$	12.1168	0.395840	0.197920	+	0.980218i	$0.436581\pi$
$938$	−17.2097	−0.561917
$939$	0	0
$940$	−7.21194	−0.235227
$941$	−10.3556	−0.337584	−0.168792	−	0.985652i	$-0.553987\pi$
$941$	−10.3556	−0.337584	−0.168792	+	0.985652i	$0.553987\pi$
$942$	0	0
$943$	17.4891	0.569524
$944$	−15.9061	−0.517701
$945$	0	0
$946$	−36.7011	−1.19325
$947$	−11.9574	−0.388562	−0.194281	−	0.980946i	$-0.562237\pi$
$947$	−11.9574	−0.388562	−0.194281	+	0.980946i	$0.562237\pi$
$948$	0	0
$949$	24.2337	0.786659
$950$	−0.474964	−0.0154099
$951$	0	0
$952$	−21.9565	−0.711614
$953$	−24.8646	−0.805444	−0.402722	−	0.915322i	$-0.631936\pi$
$953$	−24.8646	−0.805444	−0.402722	+	0.915322i	$0.631936\pi$
$954$	0	0
$955$	17.6495	0.571123
$956$	−10.0575	−0.325283
$957$	0	0
$958$	10.7881	0.348546
$959$	56.1829	1.81424
$960$	0	0
$961$	14.4891	0.467391
$962$	−1.89986	−0.0612538
$963$	0	0
$964$	−3.80981	−0.122706
$965$	−39.2246	−1.26268
$966$	0	0
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	−33.6463	−1.08143
$969$	0	0
$970$	−42.5109	−1.36494
$971$	34.4194	1.10457	0.552286	−	0.833655i	$-0.313756\pi$
$971$	34.4194	1.10457	0.552286	+	0.833655i	$0.313756\pi$
$972$	0	0
$973$	−47.6060	−1.52618
$974$	10.2066	0.327039
$975$	0	0
$976$	−37.7663	−1.20887
$977$	−17.0606	−0.545818	−0.272909	−	0.962040i	$-0.587986\pi$
$977$	−17.0606	−0.545818	−0.272909	+	0.962040i	$0.587986\pi$
$978$	0	0
$979$	−56.2337	−1.79724
$980$	−3.50157	−0.111854
$981$	0	0
$982$	8.74456	0.279050
$983$	39.2246	1.25107	0.625534	−	0.780197i	$-0.284881\pi$
$983$	39.2246	1.25107	0.625534	+	0.780197i	$0.284881\pi$
$984$	0	0
$985$	7.53262	0.240009
$986$	−18.8114	−0.599078
$987$	0	0
$988$	−0.744563	−0.0236877
$989$	−41.9253	−1.33315
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	−14.0063	−0.444700
$993$	0	0
$994$	−58.9783	−1.87068
$995$	38.9732	1.23553
$996$	0	0
$997$	32.3505	1.02455	0.512276	−	0.858821i	$-0.328803\pi$
$997$	32.3505	1.02455	0.512276	+	0.858821i	$0.328803\pi$
$998$	7.80400	0.247031
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.2.a.e.1.3	yes	4
3.2	odd	2	inner	171.2.a.e.1.2	✓	4
4.3	odd	2		2736.2.a.bf.1.3		4
5.4	even	2		4275.2.a.bp.1.2		4
7.6	odd	2		8379.2.a.bw.1.3		4
12.11	even	2		2736.2.a.bf.1.2		4
15.14	odd	2		4275.2.a.bp.1.3		4
19.18	odd	2		3249.2.a.bf.1.2		4
21.20	even	2		8379.2.a.bw.1.2		4
57.56	even	2		3249.2.a.bf.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.2.a.e.1.2	✓	4	3.2	odd	2	inner
171.2.a.e.1.3	yes	4	1.1	even	1	trivial
2736.2.a.bf.1.2		4	12.11	even	2
2736.2.a.bf.1.3		4	4.3	odd	2
3249.2.a.bf.1.2		4	19.18	odd	2
3249.2.a.bf.1.3		4	57.56	even	2
4275.2.a.bp.1.2		4	5.4	even	2
4275.2.a.bp.1.3		4	15.14	odd	2
8379.2.a.bw.1.2		4	21.20	even	2
8379.2.a.bw.1.3		4	7.6	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	171.a (trivial)

\(f(q)\)	\(=\)	\(q+1.27582 q^{2} -0.372281 q^{4} +2.15121 q^{5} +3.37228 q^{7} -3.02661 q^{8} +2.74456 q^{10} -4.70285 q^{11} +2.00000 q^{13} +4.30243 q^{14} -3.11684 q^{16} +2.15121 q^{17} +1.00000 q^{19} -0.800857 q^{20} -6.00000 q^{22} -6.85407 q^{23} -0.372281 q^{25} +2.55164 q^{26} -1.25544 q^{28} -6.85407 q^{29} -6.74456 q^{31} +2.07668 q^{32} +2.74456 q^{34} +7.25450 q^{35} -0.744563 q^{37} +1.27582 q^{38} -6.51087 q^{40} -2.55164 q^{41} +6.11684 q^{43} +1.75079 q^{44} -8.74456 q^{46} +9.00528 q^{47} +4.37228 q^{49} -0.474964 q^{50} -0.744563 q^{52} +11.9574 q^{53} -10.1168 q^{55} -10.2066 q^{56} -8.74456 q^{58} +5.10328 q^{59} +12.1168 q^{61} -8.60485 q^{62} +8.88316 q^{64} +4.30243 q^{65} -4.00000 q^{67} -0.800857 q^{68} +9.25544 q^{70} -13.7081 q^{71} +12.1168 q^{73} -0.949929 q^{74} -0.372281 q^{76} -15.8593 q^{77} -4.00000 q^{79} -6.70500 q^{80} -3.25544 q^{82} +1.75079 q^{83} +4.62772 q^{85} +7.80400 q^{86} +14.2337 q^{88} +11.9574 q^{89} +6.74456 q^{91} +2.55164 q^{92} +11.4891 q^{94} +2.15121 q^{95} -15.4891 q^{97} +5.57825 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 10 q^{4} + 2 q^{7} - 12 q^{10} + 8 q^{13} + 22 q^{16} + 4 q^{19} - 24 q^{22} + 10 q^{25} - 28 q^{28} - 4 q^{31} - 12 q^{34} + 20 q^{37} - 72 q^{40} - 10 q^{43} - 12 q^{46} + 6 q^{49} + 20 q^{52} - 6 q^{55}+ \cdots - 16 q^{97}+O(q^{100}) \) 4 * q + 10 * q^4 + 2 * q^7 - 12 * q^10 + 8 * q^13 + 22 * q^16 + 4 * q^19 - 24 * q^22 + 10 * q^25 - 28 * q^28 - 4 * q^31 - 12 * q^34 + 20 * q^37 - 72 * q^40 - 10 * q^43 - 12 * q^46 + 6 * q^49 + 20 * q^52 - 6 * q^55 - 12 * q^58 + 14 * q^61 + 70 * q^64 - 16 * q^67 + 60 * q^70 + 14 * q^73 + 10 * q^76 - 16 * q^79 - 36 * q^82 + 30 * q^85 - 12 * q^88 + 4 * q^91 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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