LMFDB - Embedded newform 3969.2.a.bg.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3969,2,Mod(1,3969)] 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("3969.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3969, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3969 = 3^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3969.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,0,6,0,0,0,0,0,14,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$31.6926245622$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.114612039936.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 11x^{6} + 34x^{4} - 36x^{2} + 9 $ x^8 - 11x^6 + 34x^4 - 36*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 567)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.597336$ of defining polynomial
Character	$\chi$	$=$	3969.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.597336 q^{2} -1.64319 q^{4} +2.09557 q^{5} +2.17621 q^{8} -1.25176 q^{10} +1.65141 q^{11} -0.426044 q^{13} +1.98645 q^{16} +6.07638 q^{17} +5.41249 q^{19} -3.44342 q^{20} -0.986450 q^{22} +7.63457 q^{23} -0.608573 q^{25} +0.254492 q^{26} +3.65377 q^{29} -5.30745 q^{31} -5.53900 q^{32} -3.62964 q^{34} -4.67781 q^{37} -3.23308 q^{38} +4.56041 q^{40} -1.48565 q^{41} +8.48997 q^{43} -2.71359 q^{44} -4.56041 q^{46} -11.3325 q^{47} +0.363522 q^{50} +0.700071 q^{52} -5.48992 q^{53} +3.46066 q^{55} -2.18253 q^{58} +1.55820 q^{59} +5.04817 q^{61} +3.17033 q^{62} -0.664256 q^{64} -0.892807 q^{65} -5.22997 q^{67} -9.98464 q^{68} +12.5604 q^{71} +1.58751 q^{73} +2.79422 q^{74} -8.89375 q^{76} -7.62964 q^{79} +4.16275 q^{80} +0.887434 q^{82} -5.25611 q^{83} +12.7335 q^{85} -5.07137 q^{86} +3.59382 q^{88} +18.5562 q^{89} -12.5450 q^{92} +6.76930 q^{94} +11.3423 q^{95} +13.7546 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 6 q^{4} + 14 q^{10} + 6 q^{13} + 6 q^{16} + 24 q^{19} + 2 q^{22} + 20 q^{31} - 4 q^{37} + 36 q^{40} + 10 q^{43} - 36 q^{46} + 34 q^{52} + 4 q^{55} - 22 q^{58} + 36 q^{61} + 38 q^{64} - 18 q^{67} + 32 q^{73}+ \cdots + 14 q^{97}+O(q^{100}) $ 8 * q + 6 * q^4 + 14 * q^10 + 6 * q^13 + 6 * q^16 + 24 * q^19 + 2 * q^22 + 20 * q^31 - 4 * q^37 + 36 * q^40 + 10 * q^43 - 36 * q^46 + 34 * q^52 + 4 * q^55 - 22 * q^58 + 36 * q^61 + 38 * q^64 - 18 * q^67 + 32 * q^73 + 58 * q^76 - 32 * q^79 - 2 * q^82 + 30 * q^85 - 72 * q^88 + 54 * q^94 + 14 * 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 $

$2$	−0.597336	−0.422381	−0.211190	−	0.977445i	$-0.567734\pi$
$2$	−0.597336	−0.422381	−0.211190	+	0.977445i	$0.567734\pi$
$3$	0	0
$3$	0	0
$4$	−1.64319	−0.821595
$5$	2.09557	0.937169	0.468584	−	0.883419i	$-0.344764\pi$
$5$	2.09557	0.937169	0.468584	+	0.883419i	$0.344764\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	2.17621	0.769406
$9$	0	0
$10$	−1.25176	−0.395842
$11$	1.65141	0.497920	0.248960	−	0.968514i	$-0.419911\pi$
$11$	1.65141	0.497920	0.248960	+	0.968514i	$0.419911\pi$
$12$	0	0
$13$	−0.426044	−0.118163	−0.0590817	−	0.998253i	$-0.518817\pi$
$13$	−0.426044	−0.118163	−0.0590817	+	0.998253i	$0.518817\pi$
$14$	0	0
$15$	0	0
$16$	1.98645	0.496613
$17$	6.07638	1.47374	0.736869	−	0.676036i	$-0.236304\pi$
$17$	6.07638	1.47374	0.736869	+	0.676036i	$0.236304\pi$
$18$	0	0
$19$	5.41249	1.24171	0.620856	−	0.783925i	$-0.286785\pi$
$19$	5.41249	1.24171	0.620856	+	0.783925i	$0.286785\pi$
$20$	−3.44342	−0.769973
$21$	0	0
$22$	−0.986450	−0.210312
$23$	7.63457	1.59192	0.795959	−	0.605351i	$-0.206967\pi$
$23$	7.63457	1.59192	0.795959	+	0.605351i	$0.206967\pi$
$24$	0	0
$25$	−0.608573	−0.121715
$26$	0.254492	0.0499099
$27$	0	0
$28$	0	0
$29$	3.65377	0.678488	0.339244	−	0.940698i	$-0.389829\pi$
$29$	3.65377	0.678488	0.339244	+	0.940698i	$0.389829\pi$
$30$	0	0
$31$	−5.30745	−0.953245	−0.476623	−	0.879108i	$-0.658139\pi$
$31$	−5.30745	−0.953245	−0.476623	+	0.879108i	$0.658139\pi$
$32$	−5.53900	−0.979166
$33$	0	0
$34$	−3.62964	−0.622478
$35$	0	0
$36$	0	0
$37$	−4.67781	−0.769027	−0.384513	−	0.923119i	$-0.625631\pi$
$37$	−4.67781	−0.769027	−0.384513	+	0.923119i	$0.625631\pi$
$38$	−3.23308	−0.524475
$39$	0	0
$40$	4.56041	0.721063
$41$	−1.48565	−0.232020	−0.116010	−	0.993248i	$-0.537010\pi$
$41$	−1.48565	−0.232020	−0.116010	+	0.993248i	$0.537010\pi$
$42$	0	0
$43$	8.48997	1.29471	0.647354	−	0.762189i	$-0.275875\pi$
$43$	8.48997	1.29471	0.647354	+	0.762189i	$0.275875\pi$
$44$	−2.71359	−0.409089
$45$	0	0
$46$	−4.56041	−0.672395
$47$	−11.3325	−1.65301	−0.826506	−	0.562927i	$-0.809675\pi$
$47$	−11.3325	−1.65301	−0.826506	+	0.562927i	$0.809675\pi$
$48$	0	0
$49$	0	0
$50$	0.363522	0.0514098
$51$	0	0
$52$	0.700071	0.0970824
$53$	−5.48992	−0.754099	−0.377050	−	0.926193i	$-0.623061\pi$
$53$	−5.48992	−0.754099	−0.377050	+	0.926193i	$0.623061\pi$
$54$	0	0
$55$	3.46066	0.466635
$56$	0	0
$57$	0	0
$58$	−2.18253	−0.286580
$59$	1.55820	0.202860	0.101430	−	0.994843i	$-0.467658\pi$
$59$	1.55820	0.202860	0.101430	+	0.994843i	$0.467658\pi$
$60$	0	0
$61$	5.04817	0.646352	0.323176	−	0.946339i	$-0.395249\pi$
$61$	5.04817	0.646352	0.323176	+	0.946339i	$0.395249\pi$
$62$	3.17033	0.402632
$63$	0	0
$64$	−0.664256	−0.0830320
$65$	−0.892807	−0.110739
$66$	0	0
$67$	−5.22997	−0.638942	−0.319471	−	0.947596i	$-0.603505\pi$
$67$	−5.22997	−0.638942	−0.319471	+	0.947596i	$0.603505\pi$
$68$	−9.98464	−1.21081
$69$	0	0
$70$	0	0
$71$	12.5604	1.49065	0.745324	−	0.666703i	$-0.232295\pi$
$71$	12.5604	1.49065	0.745324	+	0.666703i	$0.232295\pi$
$72$	0	0
$73$	1.58751	0.185803	0.0929017	−	0.995675i	$-0.470386\pi$
$73$	1.58751	0.185803	0.0929017	+	0.995675i	$0.470386\pi$
$74$	2.79422	0.324822
$75$	0	0
$76$	−8.89375	−1.02018
$77$	0	0
$78$	0	0
$79$	−7.62964	−0.858401	−0.429201	−	0.903209i	$-0.641205\pi$
$79$	−7.62964	−0.858401	−0.429201	+	0.903209i	$0.641205\pi$
$80$	4.16275	0.465410
$81$	0	0
$82$	0.887434	0.0980007
$83$	−5.25611	−0.576933	−0.288467	−	0.957490i	$-0.593145\pi$
$83$	−5.25611	−0.576933	−0.288467	+	0.957490i	$0.593145\pi$
$84$	0	0
$85$	12.7335	1.38114
$86$	−5.07137	−0.546860
$87$	0	0
$88$	3.59382	0.383103
$89$	18.5562	1.96695	0.983474	−	0.181047i	$-0.0579485\pi$
$89$	18.5562	1.96695	0.983474	+	0.181047i	$0.0579485\pi$
$90$	0	0
$91$	0	0
$92$	−12.5450	−1.30791
$93$	0	0
$94$	6.76930	0.698200
$95$	11.3423	1.16369
$96$	0	0
$97$	13.7546	1.39656	0.698282	−	0.715823i	$-0.253948\pi$
$97$	13.7546	1.39656	0.698282	+	0.715823i	$0.253948\pi$
$98$	0	0
$99$	0	0
$100$	1.00000	0.100000
$101$	−15.9885	−1.59091	−0.795456	−	0.606011i	$-0.792769\pi$
$101$	−15.9885	−1.59091	−0.795456	+	0.606011i	$0.792769\pi$
$102$	0	0
$103$	−2.36313	−0.232846	−0.116423	−	0.993200i	$-0.537143\pi$
$103$	−2.36313	−0.232846	−0.116423	+	0.993200i	$0.537143\pi$
$104$	−0.927161	−0.0909156
$105$	0	0
$106$	3.27933	0.318517
$107$	4.54209	0.439100	0.219550	−	0.975601i	$-0.429541\pi$
$107$	4.54209	0.439100	0.219550	+	0.975601i	$0.429541\pi$
$108$	0	0
$109$	17.2445	1.65173	0.825863	−	0.563870i	$-0.190688\pi$
$109$	17.2445	1.65173	0.825863	+	0.563870i	$0.190688\pi$
$110$	−2.06718	−0.197098
$111$	0	0
$112$	0	0
$113$	−10.6752	−1.00424	−0.502119	−	0.864799i	$-0.667446\pi$
$113$	−10.6752	−1.00424	−0.502119	+	0.864799i	$0.667446\pi$
$114$	0	0
$115$	15.9988	1.49190
$116$	−6.00383	−0.557442
$117$	0	0
$118$	−0.930766	−0.0856840
$119$	0	0
$120$	0	0
$121$	−8.27283	−0.752075
$122$	−3.01545	−0.273006
$123$	0	0
$124$	8.72114	0.783181
$125$	−11.7532	−1.05124
$126$	0	0
$127$	14.4060	1.27833	0.639163	−	0.769072i	$-0.279281\pi$
$127$	14.4060	1.27833	0.639163	+	0.769072i	$0.279281\pi$
$128$	11.4748	1.01424
$129$	0	0
$130$	0.533306	0.0467740
$131$	−10.1852	−0.889884	−0.444942	−	0.895559i	$-0.646776\pi$
$131$	−10.1852	−0.889884	−0.444942	+	0.895559i	$0.646776\pi$
$132$	0	0
$133$	0	0
$134$	3.12405	0.269877
$135$	0	0
$136$	13.2235	1.13390
$137$	−2.06231	−0.176195	−0.0880976	−	0.996112i	$-0.528079\pi$
$137$	−2.06231	−0.176195	−0.0880976	+	0.996112i	$0.528079\pi$
$138$	0	0
$139$	6.63789	0.563018	0.281509	−	0.959559i	$-0.409165\pi$
$139$	6.63789	0.563018	0.281509	+	0.959559i	$0.409165\pi$
$140$	0	0
$141$	0	0
$142$	−7.50279	−0.629621
$143$	−0.703576	−0.0588360
$144$	0	0
$145$	7.65674	0.635858
$146$	−0.948275	−0.0784798
$147$	0	0
$148$	7.68652	0.631828
$149$	−9.51701	−0.779664	−0.389832	−	0.920886i	$-0.627467\pi$
$149$	−9.51701	−0.779664	−0.389832	+	0.920886i	$0.627467\pi$
$150$	0	0
$151$	−11.0978	−0.903128	−0.451564	−	0.892239i	$-0.649134\pi$
$151$	−11.0978	−0.903128	−0.451564	+	0.892239i	$0.649134\pi$
$152$	11.7787	0.955380
$153$	0	0
$154$	0	0
$155$	−11.1221	−0.893352
$156$	0	0
$157$	24.2528	1.93558	0.967791	−	0.251754i	$-0.0810075\pi$
$157$	24.2528	1.93558	0.967791	+	0.251754i	$0.0810075\pi$
$158$	4.55746	0.362572
$159$	0	0
$160$	−11.6074	−0.917644
$161$	0	0
$162$	0	0
$163$	−11.0903	−0.868659	−0.434330	−	0.900754i	$-0.643015\pi$
$163$	−11.0903	−0.868659	−0.434330	+	0.900754i	$0.643015\pi$
$164$	2.44121	0.190626
$165$	0	0
$166$	3.13967	0.243685
$167$	−2.45100	−0.189664	−0.0948321	−	0.995493i	$-0.530231\pi$
$167$	−2.45100	−0.189664	−0.0948321	+	0.995493i	$0.530231\pi$
$168$	0	0
$169$	−12.8185	−0.986037
$170$	−7.60618	−0.583367
$171$	0	0
$172$	−13.9506	−1.06373
$173$	−9.28106	−0.705626	−0.352813	−	0.935694i	$-0.614775\pi$
$173$	−9.28106	−0.705626	−0.352813	+	0.935694i	$0.614775\pi$
$174$	0	0
$175$	0	0
$176$	3.28045	0.247273
$177$	0	0
$178$	−11.0843	−0.830801
$179$	−18.5562	−1.38695	−0.693476	−	0.720479i	$-0.743922\pi$
$179$	−18.5562	−1.38695	−0.693476	+	0.720479i	$0.743922\pi$
$180$	0	0
$181$	13.2382	0.983989	0.491994	−	0.870598i	$-0.336268\pi$
$181$	13.2382	0.983989	0.491994	+	0.870598i	$0.336268\pi$
$182$	0	0
$183$	0	0
$184$	16.6144	1.22483
$185$	−9.80269	−0.720708
$186$	0	0
$187$	10.0346	0.733804
$188$	18.6214	1.35811
$189$	0	0
$190$	−6.77515	−0.491521
$191$	10.5973	0.766797	0.383399	−	0.923583i	$-0.374754\pi$
$191$	10.5973	0.766797	0.383399	+	0.923583i	$0.374754\pi$
$192$	0	0
$193$	−8.11961	−0.584463	−0.292231	−	0.956348i	$-0.594398\pi$
$193$	−8.11961	−0.584463	−0.292231	+	0.956348i	$0.594398\pi$
$194$	−8.21610	−0.589881
$195$	0	0
$196$	0	0
$197$	9.05534	0.645166	0.322583	−	0.946541i	$-0.395449\pi$
$197$	9.05534	0.645166	0.322583	+	0.946541i	$0.395449\pi$
$198$	0	0
$199$	7.42604	0.526418	0.263209	−	0.964739i	$-0.415219\pi$
$199$	7.42604	0.526418	0.263209	+	0.964739i	$0.415219\pi$
$200$	−1.32438	−0.0936479
$201$	0	0
$202$	9.55049	0.671970
$203$	0	0
$204$	0	0
$205$	−3.11329	−0.217442
$206$	1.41158	0.0983496
$207$	0	0
$208$	−0.846316	−0.0586814
$209$	8.93827	0.618273
$210$	0	0
$211$	2.67781	0.184348	0.0921739	−	0.995743i	$-0.470618\pi$
$211$	2.67781	0.184348	0.0921739	+	0.995743i	$0.470618\pi$
$212$	9.02099	0.619564
$213$	0	0
$214$	−2.71315	−0.185467
$215$	17.7914	1.21336
$216$	0	0
$217$	0	0
$218$	−10.3008	−0.697657
$219$	0	0
$220$	−5.68652	−0.383385
$221$	−2.58880	−0.174142
$222$	0	0
$223$	21.5369	1.44222	0.721110	−	0.692820i	$-0.243632\pi$
$223$	21.5369	1.44222	0.721110	+	0.692820i	$0.243632\pi$
$224$	0	0
$225$	0	0
$226$	6.37668	0.424170
$227$	13.1010	0.869546	0.434773	−	0.900540i	$-0.356829\pi$
$227$	13.1010	0.869546	0.434773	+	0.900540i	$0.356829\pi$
$228$	0	0
$229$	16.8245	1.11180	0.555898	−	0.831251i	$-0.312375\pi$
$229$	16.8245	1.11180	0.555898	+	0.831251i	$0.312375\pi$
$230$	−9.55666	−0.630148
$231$	0	0
$232$	7.95136	0.522033
$233$	−22.7684	−1.49161	−0.745804	−	0.666165i	$-0.767935\pi$
$233$	−22.7684	−1.49161	−0.745804	+	0.666165i	$0.767935\pi$
$234$	0	0
$235$	−23.7481	−1.54915
$236$	−2.56041	−0.166668
$237$	0	0
$238$	0	0
$239$	−9.51810	−0.615675	−0.307837	−	0.951439i	$-0.599605\pi$
$239$	−9.51810	−0.615675	−0.307837	+	0.951439i	$0.599605\pi$
$240$	0	0
$241$	1.17308	0.0755650	0.0377825	−	0.999286i	$-0.487971\pi$
$241$	1.17308	0.0755650	0.0377825	+	0.999286i	$0.487971\pi$
$242$	4.94166	0.317662
$243$	0	0
$244$	−8.29509	−0.531039
$245$	0	0
$246$	0	0
$247$	−2.30596	−0.146725
$248$	−11.5501	−0.733433
$249$	0	0
$250$	7.02060	0.444022
$251$	7.59042	0.479103	0.239552	−	0.970884i	$-0.423000\pi$
$251$	7.59042	0.479103	0.239552	+	0.970884i	$0.423000\pi$
$252$	0	0
$253$	12.6078	0.792648
$254$	−8.60522	−0.539940
$255$	0	0
$256$	−5.52579	−0.345362
$257$	23.2862	1.45255	0.726277	−	0.687402i	$-0.241249\pi$
$257$	23.2862	1.45255	0.726277	+	0.687402i	$0.241249\pi$
$258$	0	0
$259$	0	0
$260$	1.46705	0.0909826
$261$	0	0
$262$	6.08398	0.375870
$263$	10.4446	0.644042	0.322021	−	0.946733i	$-0.395638\pi$
$263$	10.4446	0.644042	0.322021	+	0.946733i	$0.395638\pi$
$264$	0	0
$265$	−11.5045	−0.706718
$266$	0	0
$267$	0	0
$268$	8.59382	0.524951
$269$	28.4698	1.73583	0.867917	−	0.496710i	$-0.165459\pi$
$269$	28.4698	1.73583	0.867917	+	0.496710i	$0.165459\pi$
$270$	0	0
$271$	27.3924	1.66397	0.831986	−	0.554796i	$-0.187204\pi$
$271$	27.3924	1.66397	0.831986	+	0.554796i	$0.187204\pi$
$272$	12.0704	0.731876
$273$	0	0
$274$	1.23189	0.0744214
$275$	−1.00501	−0.0606041
$276$	0	0
$277$	4.11257	0.247100	0.123550	−	0.992338i	$-0.460572\pi$
$277$	4.11257	0.247100	0.123550	+	0.992338i	$0.460572\pi$
$278$	−3.96505	−0.237808
$279$	0	0
$280$	0	0
$281$	−7.26126	−0.433170	−0.216585	−	0.976264i	$-0.569492\pi$
$281$	−7.26126	−0.433170	−0.216585	+	0.976264i	$0.569492\pi$
$282$	0	0
$283$	−1.28035	−0.0761086	−0.0380543	−	0.999276i	$-0.512116\pi$
$283$	−1.28035	−0.0761086	−0.0380543	+	0.999276i	$0.512116\pi$
$284$	−20.6391	−1.22471
$285$	0	0
$286$	0.420271	0.0248512
$287$	0	0
$288$	0	0
$289$	19.9223	1.17190
$290$	−4.57365	−0.268574
$291$	0	0
$292$	−2.60857	−0.152655
$293$	20.0134	1.16920	0.584598	−	0.811323i	$-0.301252\pi$
$293$	20.0134	1.16920	0.584598	+	0.811323i	$0.301252\pi$
$294$	0	0
$295$	3.26531	0.190114
$296$	−10.1799	−0.591694
$297$	0	0
$298$	5.68485	0.329315
$299$	−3.25266	−0.188106
$300$	0	0
$301$	0	0
$302$	6.62913	0.381464
$303$	0	0
$304$	10.7516	0.616649
$305$	10.5788	0.605741
$306$	0	0
$307$	−2.19415	−0.125227	−0.0626134	−	0.998038i	$-0.519944\pi$
$307$	−2.19415	−0.125227	−0.0626134	+	0.998038i	$0.519944\pi$
$308$	0	0
$309$	0	0
$310$	6.64366	0.377334
$311$	−8.50877	−0.482488	−0.241244	−	0.970464i	$-0.577555\pi$
$311$	−8.50877	−0.482488	−0.241244	+	0.970464i	$0.577555\pi$
$312$	0	0
$313$	3.03462	0.171527	0.0857633	−	0.996316i	$-0.472667\pi$
$313$	3.03462	0.171527	0.0857633	+	0.996316i	$0.472667\pi$
$314$	−14.4871	−0.817552
$315$	0	0
$316$	12.5369	0.705258
$317$	5.28451	0.296807	0.148404	−	0.988927i	$-0.452587\pi$
$317$	5.28451	0.296807	0.148404	+	0.988927i	$0.452587\pi$
$318$	0	0
$319$	6.03389	0.337833
$320$	−1.39200	−0.0778150
$321$	0	0
$322$	0	0
$323$	32.8883	1.82996
$324$	0	0
$325$	0.259279	0.0143822
$326$	6.62464	0.366905
$327$	0	0
$328$	−3.23309	−0.178518
$329$	0	0
$330$	0	0
$331$	21.2017	1.16535	0.582675	−	0.812706i	$-0.302006\pi$
$331$	21.2017	1.16535	0.582675	+	0.812706i	$0.302006\pi$
$332$	8.63678	0.474005
$333$	0	0
$334$	1.46407	0.0801105
$335$	−10.9598	−0.598797
$336$	0	0
$337$	−14.0497	−0.765333	−0.382667	−	0.923886i	$-0.624994\pi$
$337$	−14.0497	−0.765333	−0.382667	+	0.923886i	$0.624994\pi$
$338$	7.65695	0.416483
$339$	0	0
$340$	−20.9235	−1.13474
$341$	−8.76479	−0.474640
$342$	0	0
$343$	0	0
$344$	18.4760	0.996157
$345$	0	0
$346$	5.54391	0.298043
$347$	20.0580	1.07677	0.538385	−	0.842699i	$-0.319035\pi$
$347$	20.0580	1.07677	0.538385	+	0.842699i	$0.319035\pi$
$348$	0	0
$349$	−27.9294	−1.49503	−0.747513	−	0.664247i	$-0.768752\pi$
$349$	−27.9294	−1.49503	−0.747513	+	0.664247i	$0.768752\pi$
$350$	0	0
$351$	0	0
$352$	−9.14718	−0.487546
$353$	−5.99251	−0.318949	−0.159475	−	0.987202i	$-0.550980\pi$
$353$	−5.99251	−0.318949	−0.159475	+	0.987202i	$0.550980\pi$
$354$	0	0
$355$	26.3213	1.39699
$356$	−30.4913	−1.61603
$357$	0	0
$358$	11.0843	0.585822
$359$	16.6142	0.876864	0.438432	−	0.898764i	$-0.355534\pi$
$359$	16.6142	0.876864	0.438432	+	0.898764i	$0.355534\pi$
$360$	0	0
$361$	10.2951	0.541847
$362$	−7.90766	−0.415618
$363$	0	0
$364$	0	0
$365$	3.32673	0.174129
$366$	0	0
$367$	6.49528	0.339051	0.169525	−	0.985526i	$-0.445777\pi$
$367$	6.49528	0.339051	0.169525	+	0.985526i	$0.445777\pi$
$368$	15.1657	0.790566
$369$	0	0
$370$	5.85550	0.304413
$371$	0	0
$372$	0	0
$373$	−20.1610	−1.04390	−0.521949	−	0.852977i	$-0.674795\pi$
$373$	−20.1610	−1.04390	−0.521949	+	0.852977i	$0.674795\pi$
$374$	−5.99404	−0.309944
$375$	0	0
$376$	−24.6619	−1.27184
$377$	−1.55667	−0.0801724
$378$	0	0
$379$	−15.3014	−0.785981	−0.392990	−	0.919543i	$-0.628559\pi$
$379$	−15.3014	−0.785981	−0.392990	+	0.919543i	$0.628559\pi$
$380$	−18.6375	−0.956084
$381$	0	0
$382$	−6.33018	−0.323880
$383$	6.78591	0.346744	0.173372	−	0.984856i	$-0.444534\pi$
$383$	6.78591	0.346744	0.173372	+	0.984856i	$0.444534\pi$
$384$	0	0
$385$	0	0
$386$	4.85014	0.246866
$387$	0	0
$388$	−22.6013	−1.14741
$389$	−5.75913	−0.292000	−0.146000	−	0.989285i	$-0.546640\pi$
$389$	−5.75913	−0.292000	−0.146000	+	0.989285i	$0.546640\pi$
$390$	0	0
$391$	46.3905	2.34607
$392$	0	0
$393$	0	0
$394$	−5.40908	−0.272506
$395$	−15.9885	−0.804467
$396$	0	0
$397$	−0.495747	−0.0248808	−0.0124404	−	0.999923i	$-0.503960\pi$
$397$	−0.495747	−0.0248808	−0.0124404	+	0.999923i	$0.503960\pi$
$398$	−4.43585	−0.222349
$399$	0	0
$400$	−1.20890	−0.0604450
$401$	10.3829	0.518500	0.259250	−	0.965810i	$-0.416525\pi$
$401$	10.3829	0.518500	0.259250	+	0.965810i	$0.416525\pi$
$402$	0	0
$403$	2.26121	0.112639
$404$	26.2721	1.30708
$405$	0	0
$406$	0	0
$407$	−7.72500	−0.382914
$408$	0	0
$409$	−30.0194	−1.48436	−0.742182	−	0.670199i	$-0.766209\pi$
$409$	−30.0194	−1.48436	−0.742182	+	0.670199i	$0.766209\pi$
$410$	1.85968	0.0918432
$411$	0	0
$412$	3.88307	0.191305
$413$	0	0
$414$	0	0
$415$	−11.0146	−0.540684
$416$	2.35986	0.115702
$417$	0	0
$418$	−5.33916	−0.261147
$419$	27.6495	1.35077	0.675384	−	0.737467i	$-0.263978\pi$
$419$	27.6495	1.35077	0.675384	+	0.737467i	$0.263978\pi$
$420$	0	0
$421$	12.4621	0.607368	0.303684	−	0.952773i	$-0.401783\pi$
$421$	12.4621	0.607368	0.303684	+	0.952773i	$0.401783\pi$
$422$	−1.59955	−0.0778649
$423$	0	0
$424$	−11.9472	−0.580208
$425$	−3.69792	−0.179375
$426$	0	0
$427$	0	0
$428$	−7.46351	−0.360762
$429$	0	0
$430$	−10.6274	−0.512500
$431$	−14.1032	−0.679329	−0.339665	−	0.940547i	$-0.610314\pi$
$431$	−14.1032	−0.679329	−0.339665	+	0.940547i	$0.610314\pi$
$432$	0	0
$433$	−15.3684	−0.738560	−0.369280	−	0.929318i	$-0.620396\pi$
$433$	−15.3684	−0.738560	−0.369280	+	0.929318i	$0.620396\pi$
$434$	0	0
$435$	0	0
$436$	−28.3360	−1.35705
$437$	41.3221	1.97670
$438$	0	0
$439$	19.2457	0.918549	0.459274	−	0.888295i	$-0.348110\pi$
$439$	19.2457	0.918549	0.459274	+	0.888295i	$0.348110\pi$
$440$	7.53112	0.359032
$441$	0	0
$442$	1.54639	0.0735541
$443$	24.5698	1.16735	0.583673	−	0.811989i	$-0.301615\pi$
$443$	24.5698	1.16735	0.583673	+	0.811989i	$0.301615\pi$
$444$	0	0
$445$	38.8858	1.84336
$446$	−12.8648	−0.609166
$447$	0	0
$448$	0	0
$449$	−40.0045	−1.88793	−0.943965	−	0.330046i	$-0.892936\pi$
$449$	−40.0045	−1.88793	−0.943965	+	0.330046i	$0.892936\pi$
$450$	0	0
$451$	−2.45343	−0.115527
$452$	17.5414	0.825076
$453$	0	0
$454$	−7.82572	−0.367279
$455$	0	0
$456$	0	0
$457$	−10.6361	−0.497538	−0.248769	−	0.968563i	$-0.580026\pi$
$457$	−10.6361	−0.497538	−0.248769	+	0.968563i	$0.580026\pi$
$458$	−10.0499	−0.469601
$459$	0	0
$460$	−26.2891	−1.22573
$461$	−16.4092	−0.764251	−0.382125	−	0.924111i	$-0.624808\pi$
$461$	−16.4092	−0.764251	−0.382125	+	0.924111i	$0.624808\pi$
$462$	0	0
$463$	6.06393	0.281815	0.140907	−	0.990023i	$-0.454998\pi$
$463$	6.06393	0.281815	0.140907	+	0.990023i	$0.454998\pi$
$464$	7.25803	0.336946
$465$	0	0
$466$	13.6004	0.630027
$467$	−25.2866	−1.17012	−0.585062	−	0.810988i	$-0.698930\pi$
$467$	−25.2866	−1.17012	−0.585062	+	0.810988i	$0.698930\pi$
$468$	0	0
$469$	0	0
$470$	14.1856	0.654332
$471$	0	0
$472$	3.39096	0.156082
$473$	14.0205	0.644662
$474$	0	0
$475$	−3.29390	−0.151134
$476$	0	0
$477$	0	0
$478$	5.68551	0.260049
$479$	3.34249	0.152722	0.0763611	−	0.997080i	$-0.475670\pi$
$479$	3.34249	0.152722	0.0763611	+	0.997080i	$0.475670\pi$
$480$	0	0
$481$	1.99295	0.0908708
$482$	−0.700726	−0.0319172
$483$	0	0
$484$	13.5938	0.617901
$485$	28.8237	1.30882
$486$	0	0
$487$	7.27886	0.329837	0.164918	−	0.986307i	$-0.447264\pi$
$487$	7.27886	0.329837	0.164918	+	0.986307i	$0.447264\pi$
$488$	10.9859	0.497307
$489$	0	0
$490$	0	0
$491$	2.97131	0.134093	0.0670466	−	0.997750i	$-0.478642\pi$
$491$	2.97131	0.134093	0.0670466	+	0.997750i	$0.478642\pi$
$492$	0	0
$493$	22.2017	0.999913
$494$	1.37743	0.0619737
$495$	0	0
$496$	−10.5430	−0.473394
$497$	0	0
$498$	0	0
$499$	12.4859	0.558944	0.279472	−	0.960154i	$-0.409841\pi$
$499$	12.4859	0.558944	0.279472	+	0.960154i	$0.409841\pi$
$500$	19.3127	0.863690
$501$	0	0
$502$	−4.53404	−0.202364
$503$	17.4657	0.778755	0.389377	−	0.921078i	$-0.372690\pi$
$503$	17.4657	0.778755	0.389377	+	0.921078i	$0.372690\pi$
$504$	0	0
$505$	−33.5050	−1.49095
$506$	−7.53112	−0.334799
$507$	0	0
$508$	−23.6718	−1.05027
$509$	−27.5341	−1.22043	−0.610215	−	0.792236i	$-0.708917\pi$
$509$	−27.5341	−1.22043	−0.610215	+	0.792236i	$0.708917\pi$
$510$	0	0
$511$	0	0
$512$	−19.6488	−0.868363
$513$	0	0
$514$	−13.9097	−0.613531
$515$	−4.95211	−0.218216
$516$	0	0
$517$	−18.7146	−0.823069
$518$	0	0
$519$	0	0
$520$	−1.94293	−0.0852033
$521$	6.92504	0.303391	0.151696	−	0.988427i	$-0.451527\pi$
$521$	6.92504	0.303391	0.151696	+	0.988427i	$0.451527\pi$
$522$	0	0
$523$	−41.5936	−1.81876	−0.909381	−	0.415963i	$-0.863445\pi$
$523$	−41.5936	−1.81876	−0.909381	+	0.415963i	$0.863445\pi$
$524$	16.7362	0.731124
$525$	0	0
$526$	−6.23894	−0.272031
$527$	−32.2500	−1.40483
$528$	0	0
$529$	35.2867	1.53420
$530$	6.87208	0.298504
$531$	0	0
$532$	0	0
$533$	0.632954	0.0274163
$534$	0	0
$535$	9.51827	0.411511
$536$	−11.3815	−0.491606
$537$	0	0
$538$	−17.0060	−0.733182
$539$	0	0
$540$	0	0
$541$	−33.7887	−1.45269	−0.726345	−	0.687330i	$-0.758782\pi$
$541$	−33.7887	−1.45269	−0.726345	+	0.687330i	$0.758782\pi$
$542$	−16.3625	−0.702830
$543$	0	0
$544$	−33.6570	−1.44303
$545$	36.1372	1.54795
$546$	0	0
$547$	−14.1310	−0.604196	−0.302098	−	0.953277i	$-0.597687\pi$
$547$	−14.1310	−0.604196	−0.302098	+	0.953277i	$0.597687\pi$
$548$	3.38877	0.144761
$549$	0	0
$550$	0.600326	0.0255980
$551$	19.7760	0.842486
$552$	0	0
$553$	0	0
$554$	−2.45658	−0.104370
$555$	0	0
$556$	−10.9073	−0.462573
$557$	−7.49568	−0.317602	−0.158801	−	0.987311i	$-0.550763\pi$
$557$	−7.49568	−0.317602	−0.158801	+	0.987311i	$0.550763\pi$
$558$	0	0
$559$	−3.61710	−0.152987
$560$	0	0
$561$	0	0
$562$	4.33741	0.182963
$563$	10.4224	0.439252	0.219626	−	0.975584i	$-0.429516\pi$
$563$	10.4224	0.439252	0.219626	+	0.975584i	$0.429516\pi$
$564$	0	0
$565$	−22.3706	−0.941140
$566$	0.764797	0.0321468
$567$	0	0
$568$	27.3341	1.14691
$569$	−29.9665	−1.25626	−0.628130	−	0.778108i	$-0.716180\pi$
$569$	−29.9665	−1.25626	−0.628130	+	0.778108i	$0.716180\pi$
$570$	0	0
$571$	31.4336	1.31545	0.657727	−	0.753257i	$-0.271518\pi$
$571$	31.4336	1.31545	0.657727	+	0.753257i	$0.271518\pi$
$572$	1.15611	0.0483393
$573$	0	0
$574$	0	0
$575$	−4.64619	−0.193760
$576$	0	0
$577$	−2.71141	−0.112877	−0.0564387	−	0.998406i	$-0.517975\pi$
$577$	−2.71141	−0.112877	−0.0564387	+	0.998406i	$0.517975\pi$
$578$	−11.9003	−0.494989
$579$	0	0
$580$	−12.5815	−0.522417
$581$	0	0
$582$	0	0
$583$	−9.06614	−0.375481
$584$	3.45474	0.142958
$585$	0	0
$586$	−11.9547	−0.493846
$587$	10.7040	0.441800	0.220900	−	0.975296i	$-0.429101\pi$
$587$	10.7040	0.441800	0.220900	+	0.975296i	$0.429101\pi$
$588$	0	0
$589$	−28.7265	−1.18366
$590$	−1.95049	−0.0803004
$591$	0	0
$592$	−9.29223	−0.381908
$593$	−29.6979	−1.21955	−0.609773	−	0.792576i	$-0.708739\pi$
$593$	−29.6979	−1.21955	−0.609773	+	0.792576i	$0.708739\pi$
$594$	0	0
$595$	0	0
$596$	15.6382	0.640568
$597$	0	0
$598$	1.94293	0.0794525
$599$	14.3983	0.588298	0.294149	−	0.955760i	$-0.404964\pi$
$599$	14.3983	0.588298	0.294149	+	0.955760i	$0.404964\pi$
$600$	0	0
$601$	17.7120	0.722486	0.361243	−	0.932472i	$-0.382353\pi$
$601$	17.7120	0.722486	0.361243	+	0.932472i	$0.382353\pi$
$602$	0	0
$603$	0	0
$604$	18.2358	0.742005
$605$	−17.3363	−0.704822
$606$	0	0
$607$	−5.15543	−0.209252	−0.104626	−	0.994512i	$-0.533365\pi$
$607$	−5.15543	−0.209252	−0.104626	+	0.994512i	$0.533365\pi$
$608$	−29.9798	−1.21584
$609$	0	0
$610$	−6.31910	−0.255853
$611$	4.82814	0.195326
$612$	0	0
$613$	11.2588	0.454739	0.227370	−	0.973809i	$-0.426987\pi$
$613$	11.2588	0.454739	0.227370	+	0.973809i	$0.426987\pi$
$614$	1.31065	0.0528934
$615$	0	0
$616$	0	0
$617$	−0.520304	−0.0209467	−0.0104733	−	0.999945i	$-0.503334\pi$
$617$	−0.520304	−0.0209467	−0.0104733	+	0.999945i	$0.503334\pi$
$618$	0	0
$619$	−37.4740	−1.50621	−0.753104	−	0.657901i	$-0.771444\pi$
$619$	−37.4740	−1.50621	−0.753104	+	0.657901i	$0.771444\pi$
$620$	18.2758	0.733973
$621$	0	0
$622$	5.08260	0.203794
$623$	0	0
$624$	0	0
$625$	−21.5868	−0.863471
$626$	−1.81269	−0.0724495
$627$	0	0
$628$	−39.8519	−1.59026
$629$	−28.4241	−1.13334
$630$	0	0
$631$	0.502795	0.0200159	0.0100080	−	0.999950i	$-0.496814\pi$
$631$	0.502795	0.0200159	0.0100080	+	0.999950i	$0.496814\pi$
$632$	−16.6037	−0.660459
$633$	0	0
$634$	−3.15663	−0.125366
$635$	30.1888	1.19801
$636$	0	0
$637$	0	0
$638$	−3.60426	−0.142694
$639$	0	0
$640$	24.0462	0.950511
$641$	−9.96117	−0.393442	−0.196721	−	0.980459i	$-0.563029\pi$
$641$	−9.96117	−0.393442	−0.196721	+	0.980459i	$0.563029\pi$
$642$	0	0
$643$	21.6236	0.852752	0.426376	−	0.904546i	$-0.359790\pi$
$643$	21.6236	0.852752	0.426376	+	0.904546i	$0.359790\pi$
$644$	0	0
$645$	0	0
$646$	−19.6454	−0.772938
$647$	−38.7358	−1.52286	−0.761430	−	0.648247i	$-0.775502\pi$
$647$	−38.7358	−1.52286	−0.761430	+	0.648247i	$0.775502\pi$
$648$	0	0
$649$	2.57323	0.101008
$650$	−0.154877	−0.00607476
$651$	0	0
$652$	18.2235	0.713686
$653$	−21.7978	−0.853013	−0.426506	−	0.904485i	$-0.640256\pi$
$653$	−21.7978	−0.853013	−0.426506	+	0.904485i	$0.640256\pi$
$654$	0	0
$655$	−21.3438	−0.833972
$656$	−2.95118	−0.115224
$657$	0	0
$658$	0	0
$659$	5.13619	0.200078	0.100039	−	0.994984i	$-0.468103\pi$
$659$	5.13619	0.200078	0.100039	+	0.994984i	$0.468103\pi$
$660$	0	0
$661$	13.1617	0.511933	0.255966	−	0.966686i	$-0.417606\pi$
$661$	13.1617	0.511933	0.255966	+	0.966686i	$0.417606\pi$
$662$	−12.6645	−0.492221
$663$	0	0
$664$	−11.4384	−0.443896
$665$	0	0
$666$	0	0
$667$	27.8950	1.08010
$668$	4.02746	0.155827
$669$	0	0
$670$	6.54667	0.252920
$671$	8.33662	0.321832
$672$	0	0
$673$	−7.01209	−0.270296	−0.135148	−	0.990825i	$-0.543151\pi$
$673$	−7.01209	−0.270296	−0.135148	+	0.990825i	$0.543151\pi$
$674$	8.39237	0.323262
$675$	0	0
$676$	21.0632	0.810123
$677$	36.8962	1.41804	0.709018	−	0.705190i	$-0.249138\pi$
$677$	36.8962	1.41804	0.709018	+	0.705190i	$0.249138\pi$
$678$	0	0
$679$	0	0
$680$	27.7107	1.06266
$681$	0	0
$682$	5.23553	0.200479
$683$	−5.59939	−0.214255	−0.107127	−	0.994245i	$-0.534165\pi$
$683$	−5.59939	−0.214255	−0.107127	+	0.994245i	$0.534165\pi$
$684$	0	0
$685$	−4.32172	−0.165125
$686$	0	0
$687$	0	0
$688$	16.8649	0.642969
$689$	2.33895	0.0891069
$690$	0	0
$691$	−6.38438	−0.242873	−0.121437	−	0.992599i	$-0.538750\pi$
$691$	−6.38438	−0.242873	−0.121437	+	0.992599i	$0.538750\pi$
$692$	15.2505	0.579738
$693$	0	0
$694$	−11.9814	−0.454807
$695$	13.9102	0.527643
$696$	0	0
$697$	−9.02739	−0.341937
$698$	16.6832	0.631470
$699$	0	0
$700$	0	0
$701$	−10.9860	−0.414937	−0.207468	−	0.978242i	$-0.566522\pi$
$701$	−10.9860	−0.414937	−0.207468	+	0.978242i	$0.566522\pi$
$702$	0	0
$703$	−25.3186	−0.954909
$704$	−1.09696	−0.0413433
$705$	0	0
$706$	3.57955	0.134718
$707$	0	0
$708$	0	0
$709$	26.5316	0.996417	0.498208	−	0.867057i	$-0.333991\pi$
$709$	26.5316	0.996417	0.498208	+	0.867057i	$0.333991\pi$
$710$	−15.7227	−0.590061
$711$	0	0
$712$	40.3821	1.51338
$713$	−40.5201	−1.51749
$714$	0	0
$715$	−1.47439	−0.0551392
$716$	30.4913	1.13951
$717$	0	0
$718$	−9.92426	−0.370370
$719$	−20.9960	−0.783020	−0.391510	−	0.920174i	$-0.628047\pi$
$719$	−20.9960	−0.783020	−0.391510	+	0.920174i	$0.628047\pi$
$720$	0	0
$721$	0	0
$722$	−6.14963	−0.228866
$723$	0	0
$724$	−21.7529	−0.808440
$725$	−2.22358	−0.0825818
$726$	0	0
$727$	17.0632	0.632839	0.316420	−	0.948619i	$-0.397519\pi$
$727$	17.0632	0.632839	0.316420	+	0.948619i	$0.397519\pi$
$728$	0	0
$729$	0	0
$730$	−1.98718	−0.0735488
$731$	51.5883	1.90806
$732$	0	0
$733$	−34.4805	−1.27357	−0.636784	−	0.771042i	$-0.719736\pi$
$733$	−34.4805	−1.27357	−0.636784	+	0.771042i	$0.719736\pi$
$734$	−3.87987	−0.143208
$735$	0	0
$736$	−42.2879	−1.55875
$737$	−8.63684	−0.318142
$738$	0	0
$739$	−1.37184	−0.0504641	−0.0252321	−	0.999682i	$-0.508032\pi$
$739$	−1.37184	−0.0504641	−0.0252321	+	0.999682i	$0.508032\pi$
$740$	16.1077	0.592130
$741$	0	0
$742$	0	0
$743$	−8.94248	−0.328068	−0.164034	−	0.986455i	$-0.552451\pi$
$743$	−8.94248	−0.328068	−0.164034	+	0.986455i	$0.552451\pi$
$744$	0	0
$745$	−19.9436	−0.730676
$746$	12.0429	0.440922
$747$	0	0
$748$	−16.4888	−0.602889
$749$	0	0
$750$	0	0
$751$	50.5711	1.84537	0.922683	−	0.385559i	$-0.125991\pi$
$751$	50.5711	1.84537	0.922683	+	0.385559i	$0.125991\pi$
$752$	−22.5114	−0.820907
$753$	0	0
$754$	0.929854	0.0338633
$755$	−23.2563	−0.846383
$756$	0	0
$757$	9.56041	0.347479	0.173739	−	0.984792i	$-0.444415\pi$
$757$	9.56041	0.347479	0.173739	+	0.984792i	$0.444415\pi$
$758$	9.14009	0.331983
$759$	0	0
$760$	24.6832	0.895353
$761$	37.8812	1.37319	0.686595	−	0.727040i	$-0.259104\pi$
$761$	37.8812	1.37319	0.686595	+	0.727040i	$0.259104\pi$
$762$	0	0
$763$	0	0
$764$	−17.4134	−0.629996
$765$	0	0
$766$	−4.05347	−0.146458
$767$	−0.663860	−0.0239706
$768$	0	0
$769$	−28.5588	−1.02986	−0.514928	−	0.857234i	$-0.672181\pi$
$769$	−28.5588	−1.02986	−0.514928	+	0.857234i	$0.672181\pi$
$770$	0	0
$771$	0	0
$772$	13.3421	0.480191
$773$	−10.7030	−0.384959	−0.192480	−	0.981301i	$-0.561653\pi$
$773$	−10.7030	−0.384959	−0.192480	+	0.981301i	$0.561653\pi$
$774$	0	0
$775$	3.22997	0.116024
$776$	29.9328	1.07452
$777$	0	0
$778$	3.44014	0.123335
$779$	−8.04109	−0.288102
$780$	0	0
$781$	20.7425	0.742224
$782$	−27.7107	−0.990934
$783$	0	0
$784$	0	0
$785$	50.8235	1.81397
$786$	0	0
$787$	−36.6367	−1.30596	−0.652978	−	0.757377i	$-0.726481\pi$
$787$	−36.6367	−1.30596	−0.652978	+	0.757377i	$0.726481\pi$
$788$	−14.8796	−0.530065
$789$	0	0
$790$	9.55049	0.339791
$791$	0	0
$792$	0	0
$793$	−2.15074	−0.0763751
$794$	0.296128	0.0105092
$795$	0	0
$796$	−12.2024	−0.432502
$797$	19.0481	0.674717	0.337359	−	0.941376i	$-0.390467\pi$
$797$	19.0481	0.674717	0.337359	+	0.941376i	$0.390467\pi$
$798$	0	0
$799$	−68.8604	−2.43611
$800$	3.37088	0.119179
$801$	0	0
$802$	−6.20211	−0.219004
$803$	2.62163	0.0925153
$804$	0	0
$805$	0	0
$806$	−1.35070	−0.0475764
$807$	0	0
$808$	−34.7943	−1.22406
$809$	18.4913	0.650121	0.325060	−	0.945693i	$-0.394615\pi$
$809$	18.4913	0.650121	0.325060	+	0.945693i	$0.394615\pi$
$810$	0	0
$811$	41.5033	1.45738	0.728690	−	0.684844i	$-0.240130\pi$
$811$	41.5033	1.45738	0.728690	+	0.684844i	$0.240130\pi$
$812$	0	0
$813$	0	0
$814$	4.61442	0.161735
$815$	−23.2405	−0.814081
$816$	0	0
$817$	45.9519	1.60765
$818$	17.9317	0.626966
$819$	0	0
$820$	5.11573	0.178649
$821$	−32.4887	−1.13386	−0.566932	−	0.823765i	$-0.691870\pi$
$821$	−32.4887	−1.13386	−0.566932	+	0.823765i	$0.691870\pi$
$822$	0	0
$823$	−15.7393	−0.548636	−0.274318	−	0.961639i	$-0.588452\pi$
$823$	−15.7393	−0.548636	−0.274318	+	0.961639i	$0.588452\pi$
$824$	−5.14266	−0.179153
$825$	0	0
$826$	0	0
$827$	−40.6095	−1.41213	−0.706066	−	0.708146i	$-0.749532\pi$
$827$	−40.6095	−1.41213	−0.706066	+	0.708146i	$0.749532\pi$
$828$	0	0
$829$	25.4580	0.884193	0.442097	−	0.896967i	$-0.354235\pi$
$829$	25.4580	0.884193	0.442097	+	0.896967i	$0.354235\pi$
$830$	6.57940	0.228374
$831$	0	0
$832$	0.283003	0.00981135
$833$	0	0
$834$	0	0
$835$	−5.13625	−0.177747
$836$	−14.6873	−0.507970
$837$	0	0
$838$	−16.5161	−0.570538
$839$	48.3492	1.66920	0.834600	−	0.550856i	$-0.185699\pi$
$839$	48.3492	1.66920	0.834600	+	0.550856i	$0.185699\pi$
$840$	0	0
$841$	−15.6500	−0.539654
$842$	−7.44409	−0.256540
$843$	0	0
$844$	−4.40014	−0.151459
$845$	−26.8621	−0.924084
$846$	0	0
$847$	0	0
$848$	−10.9055	−0.374495
$849$	0	0
$850$	2.20890	0.0757646
$851$	−35.7130	−1.22423
$852$	0	0
$853$	−2.65867	−0.0910310	−0.0455155	−	0.998964i	$-0.514493\pi$
$853$	−2.65867	−0.0910310	−0.0455155	+	0.998964i	$0.514493\pi$
$854$	0	0
$855$	0	0
$856$	9.88453	0.337846
$857$	14.0664	0.480498	0.240249	−	0.970711i	$-0.422771\pi$
$857$	14.0664	0.480498	0.240249	+	0.970711i	$0.422771\pi$
$858$	0	0
$859$	−19.2927	−0.658258	−0.329129	−	0.944285i	$-0.606755\pi$
$859$	−19.2927	−0.658258	−0.329129	+	0.944285i	$0.606755\pi$
$860$	−29.2346	−0.996891
$861$	0	0
$862$	8.42438	0.286935
$863$	−42.3384	−1.44122	−0.720608	−	0.693343i	$-0.756137\pi$
$863$	−42.3384	−1.44122	−0.720608	+	0.693343i	$0.756137\pi$
$864$	0	0
$865$	−19.4491	−0.661291
$866$	9.18012	0.311953
$867$	0	0
$868$	0	0
$869$	−12.5997	−0.427416
$870$	0	0
$871$	2.22820	0.0754996
$872$	37.5277	1.27085
$873$	0	0
$874$	−24.6832	−0.834921
$875$	0	0
$876$	0	0
$877$	−10.0508	−0.339393	−0.169697	−	0.985496i	$-0.554279\pi$
$877$	−10.0508	−0.339393	−0.169697	+	0.985496i	$0.554279\pi$
$878$	−11.4962	−0.387977
$879$	0	0
$880$	6.87443	0.231737
$881$	−25.7602	−0.867885	−0.433942	−	0.900941i	$-0.642878\pi$
$881$	−25.7602	−0.867885	−0.433942	+	0.900941i	$0.642878\pi$
$882$	0	0
$883$	−53.1876	−1.78991	−0.894953	−	0.446160i	$-0.852791\pi$
$883$	−53.1876	−1.78991	−0.894953	+	0.446160i	$0.852791\pi$
$884$	4.25390	0.143074
$885$	0	0
$886$	−14.6764	−0.493064
$887$	20.9517	0.703489	0.351745	−	0.936096i	$-0.385589\pi$
$887$	20.9517	0.703489	0.351745	+	0.936096i	$0.385589\pi$
$888$	0	0
$889$	0	0
$890$	−23.2279	−0.778601
$891$	0	0
$892$	−35.3893	−1.18492
$893$	−61.3370	−2.05256
$894$	0	0
$895$	−38.8858	−1.29981
$896$	0	0
$897$	0	0
$898$	23.8961	0.797425
$899$	−19.3922	−0.646765
$900$	0	0
$901$	−33.3588	−1.11134
$902$	1.46552	0.0487966
$903$	0	0
$904$	−23.2314	−0.772667
$905$	27.7416	0.922163
$906$	0	0
$907$	47.5854	1.58005	0.790024	−	0.613077i	$-0.210068\pi$
$907$	47.5854	1.58005	0.790024	+	0.613077i	$0.210068\pi$
$908$	−21.5275	−0.714414
$909$	0	0
$910$	0	0
$911$	−29.7720	−0.986389	−0.493194	−	0.869919i	$-0.664171\pi$
$911$	−29.7720	−0.986389	−0.493194	+	0.869919i	$0.664171\pi$
$912$	0	0
$913$	−8.68002	−0.287267
$914$	6.35335	0.210150
$915$	0	0
$916$	−27.6459	−0.913445
$917$	0	0
$918$	0	0
$919$	30.4585	1.00473	0.502367	−	0.864654i	$-0.332462\pi$
$919$	30.4585	1.00473	0.502367	+	0.864654i	$0.332462\pi$
$920$	34.8167	1.14787
$921$	0	0
$922$	9.80179	0.322805
$923$	−5.35129	−0.176140
$924$	0	0
$925$	2.84678	0.0936017
$926$	−3.62221	−0.119033
$927$	0	0
$928$	−20.2382	−0.664352
$929$	−21.1661	−0.694436	−0.347218	−	0.937784i	$-0.612874\pi$
$929$	−21.1661	−0.694436	−0.347218	+	0.937784i	$0.612874\pi$
$930$	0	0
$931$	0	0
$932$	37.4128	1.22550
$933$	0	0
$934$	15.1046	0.494238
$935$	21.0283	0.687698
$936$	0	0
$937$	40.2779	1.31582	0.657912	−	0.753095i	$-0.271440\pi$
$937$	40.2779	1.31582	0.657912	+	0.753095i	$0.271440\pi$
$938$	0	0
$939$	0	0
$940$	39.0226	1.27278
$941$	1.82095	0.0593613	0.0296807	−	0.999559i	$-0.490551\pi$
$941$	1.82095	0.0593613	0.0296807	+	0.999559i	$0.490551\pi$
$942$	0	0
$943$	−11.3423	−0.369357
$944$	3.09528	0.100743
$945$	0	0
$946$	−8.37494	−0.272293
$947$	13.4918	0.438424	0.219212	−	0.975677i	$-0.429651\pi$
$947$	13.4918	0.438424	0.219212	+	0.975677i	$0.429651\pi$
$948$	0	0
$949$	−0.676348	−0.0219552
$950$	1.96756	0.0638362
$951$	0	0
$952$	0	0
$953$	−4.95309	−0.160446	−0.0802232	−	0.996777i	$-0.525563\pi$
$953$	−4.95309	−0.160446	−0.0802232	+	0.996777i	$0.525563\pi$
$954$	0	0
$955$	22.2075	0.718618
$956$	15.6400	0.505835
$957$	0	0
$958$	−1.99659	−0.0645069
$959$	0	0
$960$	0	0
$961$	−2.83102	−0.0913233
$962$	−1.19046	−0.0383821
$963$	0	0
$964$	−1.92760	−0.0620838
$965$	−17.0152	−0.547740
$966$	0	0
$967$	17.2566	0.554936	0.277468	−	0.960735i	$-0.410505\pi$
$967$	17.2566	0.554936	0.277468	+	0.960735i	$0.410505\pi$
$968$	−18.0034	−0.578651
$969$	0	0
$970$	−17.2174	−0.552818
$971$	−46.9064	−1.50530	−0.752649	−	0.658422i	$-0.771224\pi$
$971$	−46.9064	−1.50530	−0.752649	+	0.658422i	$0.771224\pi$
$972$	0	0
$973$	0	0
$974$	−4.34793	−0.139317
$975$	0	0
$976$	10.0279	0.320986
$977$	−35.1634	−1.12498	−0.562488	−	0.826806i	$-0.690156\pi$
$977$	−35.1634	−1.12498	−0.562488	+	0.826806i	$0.690156\pi$
$978$	0	0
$979$	30.6439	0.979384
$980$	0	0
$981$	0	0
$982$	−1.77487	−0.0566383
$983$	12.0860	0.385482	0.192741	−	0.981250i	$-0.438262\pi$
$983$	12.0860	0.385482	0.192741	+	0.981250i	$0.438262\pi$
$984$	0	0
$985$	18.9761	0.604630
$986$	−13.2619	−0.422344
$987$	0	0
$988$	3.78913	0.120548
$989$	64.8173	2.06107
$990$	0	0
$991$	32.3748	1.02842	0.514209	−	0.857665i	$-0.328086\pi$
$991$	32.3748	1.02842	0.514209	+	0.857665i	$0.328086\pi$
$992$	29.3979	0.933385
$993$	0	0
$994$	0	0
$995$	15.5618	0.493343
$996$	0	0
$997$	−14.7332	−0.466605	−0.233303	−	0.972404i	$-0.574953\pi$
$997$	−14.7332	−0.466605	−0.233303	+	0.972404i	$0.574953\pi$
$998$	−7.45826	−0.236087
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3969.2.a.bg.1.4		8
3.2	odd	2	inner	3969.2.a.bg.1.5		8
7.2	even	3		567.2.e.g.487.5	yes	16
7.4	even	3		567.2.e.g.163.5	yes	16
7.6	odd	2		3969.2.a.bf.1.4		8
21.2	odd	6		567.2.e.g.487.4	yes	16
21.11	odd	6		567.2.e.g.163.4	✓	16
21.20	even	2		3969.2.a.bf.1.5		8
63.2	odd	6		567.2.g.l.109.4		16
63.4	even	3		567.2.g.l.541.5		16
63.11	odd	6		567.2.h.l.352.5		16
63.16	even	3		567.2.g.l.109.5		16
63.23	odd	6		567.2.h.l.298.5		16
63.25	even	3		567.2.h.l.352.4		16
63.32	odd	6		567.2.g.l.541.4		16
63.58	even	3		567.2.h.l.298.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
567.2.e.g.163.4	✓	16	21.11	odd	6
567.2.e.g.163.5	yes	16	7.4	even	3
567.2.e.g.487.4	yes	16	21.2	odd	6
567.2.e.g.487.5	yes	16	7.2	even	3
567.2.g.l.109.4		16	63.2	odd	6
567.2.g.l.109.5		16	63.16	even	3
567.2.g.l.541.4		16	63.32	odd	6
567.2.g.l.541.5		16	63.4	even	3
567.2.h.l.298.4		16	63.58	even	3
567.2.h.l.298.5		16	63.23	odd	6
567.2.h.l.352.4		16	63.25	even	3
567.2.h.l.352.5		16	63.11	odd	6
3969.2.a.bf.1.4		8	7.6	odd	2
3969.2.a.bf.1.5		8	21.20	even	2
3969.2.a.bg.1.4		8	1.1	even	1	trivial
3969.2.a.bg.1.5		8	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 3969 = 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3969.a (trivial)

\(f(q)\)	\(=\)	\(q-0.597336 q^{2} -1.64319 q^{4} +2.09557 q^{5} +2.17621 q^{8} -1.25176 q^{10} +1.65141 q^{11} -0.426044 q^{13} +1.98645 q^{16} +6.07638 q^{17} +5.41249 q^{19} -3.44342 q^{20} -0.986450 q^{22} +7.63457 q^{23} -0.608573 q^{25} +0.254492 q^{26} +3.65377 q^{29} -5.30745 q^{31} -5.53900 q^{32} -3.62964 q^{34} -4.67781 q^{37} -3.23308 q^{38} +4.56041 q^{40} -1.48565 q^{41} +8.48997 q^{43} -2.71359 q^{44} -4.56041 q^{46} -11.3325 q^{47} +0.363522 q^{50} +0.700071 q^{52} -5.48992 q^{53} +3.46066 q^{55} -2.18253 q^{58} +1.55820 q^{59} +5.04817 q^{61} +3.17033 q^{62} -0.664256 q^{64} -0.892807 q^{65} -5.22997 q^{67} -9.98464 q^{68} +12.5604 q^{71} +1.58751 q^{73} +2.79422 q^{74} -8.89375 q^{76} -7.62964 q^{79} +4.16275 q^{80} +0.887434 q^{82} -5.25611 q^{83} +12.7335 q^{85} -5.07137 q^{86} +3.59382 q^{88} +18.5562 q^{89} -12.5450 q^{92} +6.76930 q^{94} +11.3423 q^{95} +13.7546 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{4} + 14 q^{10} + 6 q^{13} + 6 q^{16} + 24 q^{19} + 2 q^{22} + 20 q^{31} - 4 q^{37} + 36 q^{40} + 10 q^{43} - 36 q^{46} + 34 q^{52} + 4 q^{55} - 22 q^{58} + 36 q^{61} + 38 q^{64} - 18 q^{67} + 32 q^{73}+ \cdots + 14 q^{97}+O(q^{100}) \) 8 * q + 6 * q^4 + 14 * q^10 + 6 * q^13 + 6 * q^16 + 24 * q^19 + 2 * q^22 + 20 * q^31 - 4 * q^37 + 36 * q^40 + 10 * q^43 - 36 * q^46 + 34 * q^52 + 4 * q^55 - 22 * q^58 + 36 * q^61 + 38 * q^64 - 18 * q^67 + 32 * q^73 + 58 * q^76 - 32 * q^79 - 2 * q^82 + 30 * q^85 - 72 * q^88 + 54 * q^94 + 14 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more