LMFDB - Embedded newform 3969.2.a.bi.1.10

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 = [12,4,0,12,0,0,0,12,0,0,20] 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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 22 x^{10} + 92 x^{9} + 125 x^{8} - 620 x^{7} - 94 x^{6} + 1280 x^{5} - 234 x^{4} + \cdots - 7 $ x^12 - 4x^11 - 22x^10 + 92x^9 + 125x^8 - 620x^7 - 94x^6 + 1280x^5 - 234x^4 - 736x^3 + 96x^2 + 60*x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 441)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$3.14082$ 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)

$f(q)$	$=$	$q+1.72661 q^{2} +0.981184 q^{4} +3.51231 q^{5} -1.75910 q^{8} +6.06439 q^{10} +6.09064 q^{11} +1.12028 q^{13} -4.99965 q^{16} +1.20396 q^{17} +2.20537 q^{19} +3.44622 q^{20} +10.5162 q^{22} +1.27339 q^{23} +7.33633 q^{25} +1.93429 q^{26} -6.20524 q^{29} +0.188404 q^{31} -5.11425 q^{32} +2.07876 q^{34} +3.57670 q^{37} +3.80782 q^{38} -6.17850 q^{40} -3.36640 q^{41} +3.80551 q^{43} +5.97604 q^{44} +2.19865 q^{46} -5.72070 q^{47} +12.6670 q^{50} +1.09920 q^{52} +8.33827 q^{53} +21.3922 q^{55} -10.7140 q^{58} -11.2685 q^{59} -12.0022 q^{61} +0.325300 q^{62} +1.16898 q^{64} +3.93477 q^{65} -7.91303 q^{67} +1.18130 q^{68} +12.2052 q^{71} +5.31473 q^{73} +6.17557 q^{74} +2.16388 q^{76} +9.21711 q^{79} -17.5603 q^{80} -5.81246 q^{82} -1.24990 q^{83} +4.22867 q^{85} +6.57064 q^{86} -10.7140 q^{88} +5.54131 q^{89} +1.24943 q^{92} -9.87741 q^{94} +7.74596 q^{95} +16.4855 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{2} + 12 q^{4} + 12 q^{8} + 20 q^{11} + 12 q^{16} + 32 q^{23} + 12 q^{25} + 16 q^{29} + 48 q^{32} + 12 q^{37} + 56 q^{44} - 24 q^{46} - 4 q^{50} + 32 q^{53} + 48 q^{64} + 60 q^{65} + 12 q^{67}+ \cdots + 64 q^{95}+O(q^{100}) $ 12 * q + 4 * q^2 + 12 * q^4 + 12 * q^8 + 20 * q^11 + 12 * q^16 + 32 * q^23 + 12 * q^25 + 16 * q^29 + 48 * q^32 + 12 * q^37 + 56 * q^44 - 24 * q^46 - 4 * q^50 + 32 * q^53 + 48 * q^64 + 60 * q^65 + 12 * q^67 + 56 * q^71 + 68 * q^74 - 12 * q^79 - 12 * q^85 + 76 * q^86 + 16 * q^92 + 64 * q^95

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 $

$2$	1.72661	1.22090	0.610449	−	0.792056i	$-0.290989\pi$
$2$	1.72661	1.22090	0.610449	+	0.792056i	$0.290989\pi$
$3$	0	0
$3$	0	0
$4$	0.981184	0.490592
$5$	3.51231	1.57075	0.785377	−	0.619018i	$-0.212469\pi$
$5$	3.51231	1.57075	0.785377	+	0.619018i	$0.212469\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.75910	−0.621935
$9$	0	0
$10$	6.06439	1.91773
$11$	6.09064	1.83640	0.918199	−	0.396120i	$-0.129644\pi$
$11$	6.09064	1.83640	0.918199	+	0.396120i	$0.129644\pi$
$12$	0	0
$13$	1.12028	0.310709	0.155355	−	0.987859i	$-0.450348\pi$
$13$	1.12028	0.310709	0.155355	+	0.987859i	$0.450348\pi$
$14$	0	0
$15$	0	0
$16$	−4.99965	−1.24991
$17$	1.20396	0.292002	0.146001	−	0.989284i	$-0.453360\pi$
$17$	1.20396	0.292002	0.146001	+	0.989284i	$0.453360\pi$
$18$	0	0
$19$	2.20537	0.505947	0.252974	−	0.967473i	$-0.418591\pi$
$19$	2.20537	0.505947	0.252974	+	0.967473i	$0.418591\pi$
$20$	3.44622	0.770599
$21$	0	0
$22$	10.5162	2.24205
$23$	1.27339	0.265520	0.132760	−	0.991148i	$-0.457616\pi$
$23$	1.27339	0.265520	0.132760	+	0.991148i	$0.457616\pi$
$24$	0	0
$25$	7.33633	1.46727
$26$	1.93429	0.379345
$27$	0	0
$28$	0	0
$29$	−6.20524	−1.15228	−0.576142	−	0.817350i	$-0.695443\pi$
$29$	−6.20524	−1.15228	−0.576142	+	0.817350i	$0.695443\pi$
$30$	0	0
$31$	0.188404	0.0338383	0.0169192	−	0.999857i	$-0.494614\pi$
$31$	0.188404	0.0338383	0.0169192	+	0.999857i	$0.494614\pi$
$32$	−5.11425	−0.904079
$33$	0	0
$34$	2.07876	0.356505
$35$	0	0
$36$	0	0
$37$	3.57670	0.588006	0.294003	−	0.955804i	$-0.405012\pi$
$37$	3.57670	0.588006	0.294003	+	0.955804i	$0.405012\pi$
$38$	3.80782	0.617710
$39$	0	0
$40$	−6.17850	−0.976907
$41$	−3.36640	−0.525743	−0.262871	−	0.964831i	$-0.584669\pi$
$41$	−3.36640	−0.525743	−0.262871	+	0.964831i	$0.584669\pi$
$42$	0	0
$43$	3.80551	0.580336	0.290168	−	0.956976i	$-0.406289\pi$
$43$	3.80551	0.580336	0.290168	+	0.956976i	$0.406289\pi$
$44$	5.97604	0.900922
$45$	0	0
$46$	2.19865	0.324173
$47$	−5.72070	−0.834449	−0.417225	−	0.908803i	$-0.636997\pi$
$47$	−5.72070	−0.834449	−0.417225	+	0.908803i	$0.636997\pi$
$48$	0	0
$49$	0	0
$50$	12.6670	1.79138
$51$	0	0
$52$	1.09920	0.152432
$53$	8.33827	1.14535	0.572675	−	0.819783i	$-0.305906\pi$
$53$	8.33827	1.14535	0.572675	+	0.819783i	$0.305906\pi$
$54$	0	0
$55$	21.3922	2.88453
$56$	0	0
$57$	0	0
$58$	−10.7140	−1.40682
$59$	−11.2685	−1.46704	−0.733519	−	0.679669i	$-0.762123\pi$
$59$	−11.2685	−1.46704	−0.733519	+	0.679669i	$0.762123\pi$
$60$	0	0
$61$	−12.0022	−1.53672	−0.768361	−	0.640017i	$-0.778927\pi$
$61$	−12.0022	−1.53672	−0.768361	+	0.640017i	$0.778927\pi$
$62$	0.325300	0.0413131
$63$	0	0
$64$	1.16898	0.146123
$65$	3.93477	0.488048
$66$	0	0
$67$	−7.91303	−0.966731	−0.483366	−	0.875419i	$-0.660586\pi$
$67$	−7.91303	−0.966731	−0.483366	+	0.875419i	$0.660586\pi$
$68$	1.18130	0.143254
$69$	0	0
$70$	0	0
$71$	12.2052	1.44850	0.724248	−	0.689540i	$-0.242187\pi$
$71$	12.2052	1.44850	0.724248	+	0.689540i	$0.242187\pi$
$72$	0	0
$73$	5.31473	0.622042	0.311021	−	0.950403i	$-0.399329\pi$
$73$	5.31473	0.622042	0.311021	+	0.950403i	$0.399329\pi$
$74$	6.17557	0.717896
$75$	0	0
$76$	2.16388	0.248214
$77$	0	0
$78$	0	0
$79$	9.21711	1.03701	0.518503	−	0.855076i	$-0.326490\pi$
$79$	9.21711	1.03701	0.518503	+	0.855076i	$0.326490\pi$
$80$	−17.5603	−1.96330
$81$	0	0
$82$	−5.81246	−0.641878
$83$	−1.24990	−0.137194	−0.0685972	−	0.997644i	$-0.521852\pi$
$83$	−1.24990	−0.137194	−0.0685972	+	0.997644i	$0.521852\pi$
$84$	0	0
$85$	4.22867	0.458663
$86$	6.57064	0.708531
$87$	0	0
$88$	−10.7140	−1.14212
$89$	5.54131	0.587378	0.293689	−	0.955901i	$-0.405117\pi$
$89$	5.54131	0.587378	0.293689	+	0.955901i	$0.405117\pi$
$90$	0	0
$91$	0	0
$92$	1.24943	0.130262
$93$	0	0
$94$	−9.87741	−1.01878
$95$	7.74596	0.794718
$96$	0	0
$97$	16.4855	1.67385	0.836926	−	0.547316i	$-0.184350\pi$
$97$	16.4855	1.67385	0.836926	+	0.547316i	$0.184350\pi$
$98$	0	0
$99$	0	0
$100$	7.19829	0.719829
$101$	−12.9638	−1.28995	−0.644975	−	0.764203i	$-0.723132\pi$
$101$	−12.9638	−1.28995	−0.644975	+	0.764203i	$0.723132\pi$
$102$	0	0
$103$	−2.70182	−0.266218	−0.133109	−	0.991101i	$-0.542496\pi$
$103$	−2.70182	−0.266218	−0.133109	+	0.991101i	$0.542496\pi$
$104$	−1.97068	−0.193241
$105$	0	0
$106$	14.3969	1.39835
$107$	0.178480	0.0172544	0.00862718	−	0.999963i	$-0.497254\pi$
$107$	0.178480	0.0172544	0.00862718	+	0.999963i	$0.497254\pi$
$108$	0	0
$109$	9.35853	0.896385	0.448192	−	0.893937i	$-0.352068\pi$
$109$	9.35853	0.896385	0.448192	+	0.893937i	$0.352068\pi$
$110$	36.9360	3.52171
$111$	0	0
$112$	0	0
$113$	8.42038	0.792123	0.396061	−	0.918224i	$-0.370377\pi$
$113$	8.42038	0.792123	0.396061	+	0.918224i	$0.370377\pi$
$114$	0	0
$115$	4.47254	0.417067
$116$	−6.08848	−0.565302
$117$	0	0
$118$	−19.4564	−1.79110
$119$	0	0
$120$	0	0
$121$	26.0959	2.37235
$122$	−20.7231	−1.87618
$123$	0	0
$124$	0.184859	0.0166008
$125$	8.20593	0.733960
$126$	0	0
$127$	−9.92438	−0.880647	−0.440323	−	0.897839i	$-0.645136\pi$
$127$	−9.92438	−0.880647	−0.440323	+	0.897839i	$0.645136\pi$
$128$	12.2469	1.08248
$129$	0	0
$130$	6.79381	0.595857
$131$	−15.2467	−1.33211	−0.666055	−	0.745902i	$-0.732019\pi$
$131$	−15.2467	−1.33211	−0.666055	+	0.745902i	$0.732019\pi$
$132$	0	0
$133$	0	0
$134$	−13.6627	−1.18028
$135$	0	0
$136$	−2.11788	−0.181606
$137$	−6.14700	−0.525174	−0.262587	−	0.964908i	$-0.584576\pi$
$137$	−6.14700	−0.525174	−0.262587	+	0.964908i	$0.584576\pi$
$138$	0	0
$139$	0.877375	0.0744179	0.0372090	−	0.999308i	$-0.488153\pi$
$139$	0.877375	0.0744179	0.0372090	+	0.999308i	$0.488153\pi$
$140$	0	0
$141$	0	0
$142$	21.0737	1.76847
$143$	6.82321	0.570586
$144$	0	0
$145$	−21.7947	−1.80995
$146$	9.17647	0.759450
$147$	0	0
$148$	3.50940	0.288471
$149$	−5.77553	−0.473150	−0.236575	−	0.971613i	$-0.576025\pi$
$149$	−5.77553	−0.473150	−0.236575	+	0.971613i	$0.576025\pi$
$150$	0	0
$151$	−2.02643	−0.164908	−0.0824541	−	0.996595i	$-0.526276\pi$
$151$	−2.02643	−0.164908	−0.0824541	+	0.996595i	$0.526276\pi$
$152$	−3.87947	−0.314666
$153$	0	0
$154$	0	0
$155$	0.661733	0.0531516
$156$	0	0
$157$	3.04756	0.243222	0.121611	−	0.992578i	$-0.461194\pi$
$157$	3.04756	0.243222	0.121611	+	0.992578i	$0.461194\pi$
$158$	15.9144	1.26608
$159$	0	0
$160$	−17.9628	−1.42009
$161$	0	0
$162$	0	0
$163$	−5.38891	−0.422092	−0.211046	−	0.977476i	$-0.567687\pi$
$163$	−5.38891	−0.422092	−0.211046	+	0.977476i	$0.567687\pi$
$164$	−3.30306	−0.257925
$165$	0	0
$166$	−2.15809	−0.167500
$167$	16.6096	1.28529	0.642645	−	0.766164i	$-0.277837\pi$
$167$	16.6096	1.28529	0.642645	+	0.766164i	$0.277837\pi$
$168$	0	0
$169$	−11.7450	−0.903460
$170$	7.30126	0.559981
$171$	0	0
$172$	3.73391	0.284708
$173$	−17.6503	−1.34193	−0.670965	−	0.741489i	$-0.734120\pi$
$173$	−17.6503	−1.34193	−0.670965	+	0.741489i	$0.734120\pi$
$174$	0	0
$175$	0	0
$176$	−30.4510	−2.29533
$177$	0	0
$178$	9.56769	0.717128
$179$	−2.62844	−0.196459	−0.0982294	−	0.995164i	$-0.531318\pi$
$179$	−2.62844	−0.196459	−0.0982294	+	0.995164i	$0.531318\pi$
$180$	0	0
$181$	−3.97391	−0.295378	−0.147689	−	0.989034i	$-0.547184\pi$
$181$	−3.97391	−0.295378	−0.147689	+	0.989034i	$0.547184\pi$
$182$	0	0
$183$	0	0
$184$	−2.24002	−0.165136
$185$	12.5625	0.923613
$186$	0	0
$187$	7.33286	0.536232
$188$	−5.61306	−0.409374
$189$	0	0
$190$	13.3743	0.970270
$191$	18.2059	1.31733	0.658666	−	0.752435i	$-0.271121\pi$
$191$	18.2059	1.31733	0.658666	+	0.752435i	$0.271121\pi$
$192$	0	0
$193$	−0.202385	−0.0145680	−0.00728401	−	0.999973i	$-0.502319\pi$
$193$	−0.202385	−0.0145680	−0.00728401	+	0.999973i	$0.502319\pi$
$194$	28.4641	2.04360
$195$	0	0
$196$	0	0
$197$	1.63136	0.116229	0.0581147	−	0.998310i	$-0.481491\pi$
$197$	1.63136	0.116229	0.0581147	+	0.998310i	$0.481491\pi$
$198$	0	0
$199$	−6.29211	−0.446036	−0.223018	−	0.974814i	$-0.571591\pi$
$199$	−6.29211	−0.446036	−0.223018	+	0.974814i	$0.571591\pi$
$200$	−12.9053	−0.912544
$201$	0	0
$202$	−22.3835	−1.57490
$203$	0	0
$204$	0	0
$205$	−11.8238	−0.825812
$206$	−4.66499	−0.325025
$207$	0	0
$208$	−5.60100	−0.388359
$209$	13.4321	0.929120
$210$	0	0
$211$	−16.2874	−1.12127	−0.560634	−	0.828064i	$-0.689442\pi$
$211$	−16.2874	−1.12127	−0.560634	+	0.828064i	$0.689442\pi$
$212$	8.18138	0.561899
$213$	0	0
$214$	0.308166	0.0210658
$215$	13.3662	0.911564
$216$	0	0
$217$	0	0
$218$	16.1585	1.09439
$219$	0	0
$220$	20.9897	1.41513
$221$	1.34877	0.0907278
$222$	0	0
$223$	−19.9694	−1.33725	−0.668626	−	0.743599i	$-0.733117\pi$
$223$	−19.9694	−1.33725	−0.668626	+	0.743599i	$0.733117\pi$
$224$	0	0
$225$	0	0
$226$	14.5387	0.967101
$227$	3.61283	0.239792	0.119896	−	0.992786i	$-0.461744\pi$
$227$	3.61283	0.239792	0.119896	+	0.992786i	$0.461744\pi$
$228$	0	0
$229$	−13.7147	−0.906290	−0.453145	−	0.891437i	$-0.649698\pi$
$229$	−13.7147	−0.906290	−0.453145	+	0.891437i	$0.649698\pi$
$230$	7.72234	0.509196
$231$	0	0
$232$	10.9156	0.716646
$233$	25.2542	1.65445	0.827227	−	0.561867i	$-0.189917\pi$
$233$	25.2542	1.65445	0.827227	+	0.561867i	$0.189917\pi$
$234$	0	0
$235$	−20.0929	−1.31071
$236$	−11.0565	−0.719717
$237$	0	0
$238$	0	0
$239$	−8.98991	−0.581509	−0.290754	−	0.956798i	$-0.593906\pi$
$239$	−8.98991	−0.581509	−0.290754	+	0.956798i	$0.593906\pi$
$240$	0	0
$241$	−9.25724	−0.596311	−0.298156	−	0.954517i	$-0.596371\pi$
$241$	−9.25724	−0.596311	−0.298156	+	0.954517i	$0.596371\pi$
$242$	45.0575	2.89640
$243$	0	0
$244$	−11.7763	−0.753903
$245$	0	0
$246$	0	0
$247$	2.47063	0.157203
$248$	−0.331421	−0.0210452
$249$	0	0
$250$	14.1684	0.896091
$251$	20.6517	1.30353	0.651763	−	0.758422i	$-0.274030\pi$
$251$	20.6517	1.30353	0.651763	+	0.758422i	$0.274030\pi$
$252$	0	0
$253$	7.75576	0.487600
$254$	−17.1355	−1.07518
$255$	0	0
$256$	18.8076	1.17548
$257$	−2.44579	−0.152564	−0.0762819	−	0.997086i	$-0.524305\pi$
$257$	−2.44579	−0.152564	−0.0762819	+	0.997086i	$0.524305\pi$
$258$	0	0
$259$	0	0
$260$	3.86073	0.239432
$261$	0	0
$262$	−26.3251	−1.62637
$263$	24.5628	1.51460	0.757302	−	0.653065i	$-0.226517\pi$
$263$	24.5628	1.51460	0.757302	+	0.653065i	$0.226517\pi$
$264$	0	0
$265$	29.2866	1.79906
$266$	0	0
$267$	0	0
$268$	−7.76415	−0.474271
$269$	−29.5703	−1.80293	−0.901466	−	0.432849i	$-0.857508\pi$
$269$	−29.5703	−1.80293	−0.901466	+	0.432849i	$0.857508\pi$
$270$	0	0
$271$	24.7915	1.50598	0.752989	−	0.658034i	$-0.228612\pi$
$271$	24.7915	1.50598	0.752989	+	0.658034i	$0.228612\pi$
$272$	−6.01935	−0.364977
$273$	0	0
$274$	−10.6135	−0.641184
$275$	44.6830	2.69448
$276$	0	0
$277$	1.87850	0.112868	0.0564340	−	0.998406i	$-0.482027\pi$
$277$	1.87850	0.112868	0.0564340	+	0.998406i	$0.482027\pi$
$278$	1.51488	0.0908567
$279$	0	0
$280$	0	0
$281$	12.0793	0.720590	0.360295	−	0.932838i	$-0.382676\pi$
$281$	12.0793	0.720590	0.360295	+	0.932838i	$0.382676\pi$
$282$	0	0
$283$	27.9719	1.66276	0.831378	−	0.555708i	$-0.187553\pi$
$283$	27.9719	1.66276	0.831378	+	0.555708i	$0.187553\pi$
$284$	11.9756	0.710620
$285$	0	0
$286$	11.7810	0.696627
$287$	0	0
$288$	0	0
$289$	−15.5505	−0.914735
$290$	−37.6310	−2.20977
$291$	0	0
$292$	5.21473	0.305169
$293$	8.82326	0.515460	0.257730	−	0.966217i	$-0.417025\pi$
$293$	8.82326	0.515460	0.257730	+	0.966217i	$0.417025\pi$
$294$	0	0
$295$	−39.5786	−2.30435
$296$	−6.29177	−0.365702
$297$	0	0
$298$	−9.97209	−0.577668
$299$	1.42655	0.0824996
$300$	0	0
$301$	0	0
$302$	−3.49885	−0.201336
$303$	0	0
$304$	−11.0261	−0.632389
$305$	−42.1554	−2.41381
$306$	0	0
$307$	−1.05532	−0.0602304	−0.0301152	−	0.999546i	$-0.509587\pi$
$307$	−1.05532	−0.0602304	−0.0301152	+	0.999546i	$0.509587\pi$
$308$	0	0
$309$	0	0
$310$	1.14255	0.0648927
$311$	3.07215	0.174206	0.0871029	−	0.996199i	$-0.472239\pi$
$311$	3.07215	0.174206	0.0871029	+	0.996199i	$0.472239\pi$
$312$	0	0
$313$	−28.1621	−1.59181	−0.795907	−	0.605419i	$-0.793006\pi$
$313$	−28.1621	−1.59181	−0.795907	+	0.605419i	$0.793006\pi$
$314$	5.26196	0.296949
$315$	0	0
$316$	9.04368	0.508747
$317$	−12.8465	−0.721530	−0.360765	−	0.932657i	$-0.617484\pi$
$317$	−12.8465	−0.721530	−0.360765	+	0.932657i	$0.617484\pi$
$318$	0	0
$319$	−37.7939	−2.11605
$320$	4.10582	0.229523
$321$	0	0
$322$	0	0
$323$	2.65517	0.147738
$324$	0	0
$325$	8.21874	0.455893
$326$	−9.30454	−0.515331
$327$	0	0
$328$	5.92182	0.326978
$329$	0	0
$330$	0	0
$331$	−21.5560	−1.18483	−0.592413	−	0.805634i	$-0.701825\pi$
$331$	−21.5560	−1.18483	−0.592413	+	0.805634i	$0.701825\pi$
$332$	−1.22638	−0.0673065
$333$	0	0
$334$	28.6783	1.56921
$335$	−27.7930	−1.51850
$336$	0	0
$337$	−12.6068	−0.686736	−0.343368	−	0.939201i	$-0.611568\pi$
$337$	−12.6068	−0.686736	−0.343368	+	0.939201i	$0.611568\pi$
$338$	−20.2790	−1.10303
$339$	0	0
$340$	4.14910	0.225017
$341$	1.14750	0.0621406
$342$	0	0
$343$	0	0
$344$	−6.69427	−0.360931
$345$	0	0
$346$	−30.4752	−1.63836
$347$	−23.1366	−1.24204	−0.621020	−	0.783795i	$-0.713281\pi$
$347$	−23.1366	−1.24204	−0.621020	+	0.783795i	$0.713281\pi$
$348$	0	0
$349$	16.4869	0.882524	0.441262	−	0.897378i	$-0.354531\pi$
$349$	16.4869	0.882524	0.441262	+	0.897378i	$0.354531\pi$
$350$	0	0
$351$	0	0
$352$	−31.1490	−1.66025
$353$	−24.4875	−1.30334	−0.651669	−	0.758503i	$-0.725931\pi$
$353$	−24.4875	−1.30334	−0.651669	+	0.758503i	$0.725931\pi$
$354$	0	0
$355$	42.8686	2.27523
$356$	5.43705	0.288163
$357$	0	0
$358$	−4.53829	−0.239856
$359$	−20.4777	−1.08077	−0.540386	−	0.841417i	$-0.681722\pi$
$359$	−20.4777	−1.08077	−0.540386	+	0.841417i	$0.681722\pi$
$360$	0	0
$361$	−14.1363	−0.744017
$362$	−6.86139	−0.360627
$363$	0	0
$364$	0	0
$365$	18.6670	0.977075
$366$	0	0
$367$	−22.2539	−1.16164	−0.580821	−	0.814031i	$-0.697268\pi$
$367$	−22.2539	−1.16164	−0.580821	+	0.814031i	$0.697268\pi$
$368$	−6.36650	−0.331877
$369$	0	0
$370$	21.6905	1.12764
$371$	0	0
$372$	0	0
$373$	−32.5369	−1.68469	−0.842347	−	0.538935i	$-0.818827\pi$
$373$	−32.5369	−1.68469	−0.842347	+	0.538935i	$0.818827\pi$
$374$	12.6610	0.654685
$375$	0	0
$376$	10.0633	0.518973
$377$	−6.95160	−0.358026
$378$	0	0
$379$	1.54440	0.0793306	0.0396653	−	0.999213i	$-0.487371\pi$
$379$	1.54440	0.0793306	0.0396653	+	0.999213i	$0.487371\pi$
$380$	7.60021	0.389883
$381$	0	0
$382$	31.4345	1.60833
$383$	31.6294	1.61619	0.808093	−	0.589055i	$-0.200500\pi$
$383$	31.6294	1.61619	0.808093	+	0.589055i	$0.200500\pi$
$384$	0	0
$385$	0	0
$386$	−0.349441	−0.0177861
$387$	0	0
$388$	16.1753	0.821179
$389$	5.24626	0.265996	0.132998	−	0.991116i	$-0.457540\pi$
$389$	5.24626	0.265996	0.132998	+	0.991116i	$0.457540\pi$
$390$	0	0
$391$	1.53310	0.0775324
$392$	0	0
$393$	0	0
$394$	2.81672	0.141904
$395$	32.3734	1.62888
$396$	0	0
$397$	−0.0276349	−0.00138696	−0.000693478	−	1.00000i	$-0.500221\pi$
$397$	−0.0276349	−0.00138696	−0.000693478		1.00000i	$0.500221\pi$
$398$	−10.8640	−0.544564
$399$	0	0
$400$	−36.6791	−1.83395
$401$	−12.1377	−0.606127	−0.303064	−	0.952970i	$-0.598009\pi$
$401$	−12.1377	−0.606127	−0.303064	+	0.952970i	$0.598009\pi$
$402$	0	0
$403$	0.211065	0.0105139
$404$	−12.7199	−0.632840
$405$	0	0
$406$	0	0
$407$	21.7844	1.07981
$408$	0	0
$409$	−31.3453	−1.54993	−0.774963	−	0.632007i	$-0.782231\pi$
$409$	−31.3453	−1.54993	−0.774963	+	0.632007i	$0.782231\pi$
$410$	−20.4152	−1.00823
$411$	0	0
$412$	−2.65098	−0.130604
$413$	0	0
$414$	0	0
$415$	−4.39004	−0.215499
$416$	−5.72938	−0.280906
$417$	0	0
$418$	23.1921	1.13436
$419$	−14.8864	−0.727247	−0.363623	−	0.931546i	$-0.618461\pi$
$419$	−14.8864	−0.727247	−0.363623	+	0.931546i	$0.618461\pi$
$420$	0	0
$421$	9.08427	0.442740	0.221370	−	0.975190i	$-0.428947\pi$
$421$	9.08427	0.442740	0.221370	+	0.975190i	$0.428947\pi$
$422$	−28.1219	−1.36895
$423$	0	0
$424$	−14.6678	−0.712333
$425$	8.83262	0.428445
$426$	0	0
$427$	0	0
$428$	0.175122	0.00846485
$429$	0	0
$430$	23.0781	1.11293
$431$	16.6355	0.801305	0.400652	−	0.916230i	$-0.368784\pi$
$431$	16.6355	0.801305	0.400652	+	0.916230i	$0.368784\pi$
$432$	0	0
$433$	−19.7423	−0.948756	−0.474378	−	0.880321i	$-0.657327\pi$
$433$	−19.7423	−0.948756	−0.474378	+	0.880321i	$0.657327\pi$
$434$	0	0
$435$	0	0
$436$	9.18244	0.439759
$437$	2.80830	0.134339
$438$	0	0
$439$	6.73514	0.321451	0.160725	−	0.986999i	$-0.448617\pi$
$439$	6.73514	0.321451	0.160725	+	0.986999i	$0.448617\pi$
$440$	−37.6310	−1.79399
$441$	0	0
$442$	2.32879	0.110769
$443$	−28.6403	−1.36074	−0.680372	−	0.732867i	$-0.738182\pi$
$443$	−28.6403	−1.36074	−0.680372	+	0.732867i	$0.738182\pi$
$444$	0	0
$445$	19.4628	0.922626
$446$	−34.4794	−1.63265
$447$	0	0
$448$	0	0
$449$	6.66872	0.314716	0.157358	−	0.987542i	$-0.449702\pi$
$449$	6.66872	0.314716	0.157358	+	0.987542i	$0.449702\pi$
$450$	0	0
$451$	−20.5035	−0.965473
$452$	8.26194	0.388609
$453$	0	0
$454$	6.23795	0.292762
$455$	0	0
$456$	0	0
$457$	−28.6573	−1.34053	−0.670266	−	0.742121i	$-0.733820\pi$
$457$	−28.6573	−1.34053	−0.670266	+	0.742121i	$0.733820\pi$
$458$	−23.6799	−1.10649
$459$	0	0
$460$	4.38839	0.204610
$461$	20.0173	0.932299	0.466150	−	0.884706i	$-0.345641\pi$
$461$	20.0173	0.932299	0.466150	+	0.884706i	$0.345641\pi$
$462$	0	0
$463$	9.91578	0.460825	0.230413	−	0.973093i	$-0.425992\pi$
$463$	9.91578	0.460825	0.230413	+	0.973093i	$0.425992\pi$
$464$	31.0240	1.44025
$465$	0	0
$466$	43.6041	2.01992
$467$	−16.0807	−0.744126	−0.372063	−	0.928207i	$-0.621349\pi$
$467$	−16.0807	−0.744126	−0.372063	+	0.928207i	$0.621349\pi$
$468$	0	0
$469$	0	0
$470$	−34.6926	−1.60025
$471$	0	0
$472$	19.8225	0.912402
$473$	23.1780	1.06573
$474$	0	0
$475$	16.1793	0.742359
$476$	0	0
$477$	0	0
$478$	−15.5221	−0.709963
$479$	8.20255	0.374784	0.187392	−	0.982285i	$-0.439997\pi$
$479$	8.20255	0.374784	0.187392	+	0.982285i	$0.439997\pi$
$480$	0	0
$481$	4.00690	0.182699
$482$	−15.9837	−0.728036
$483$	0	0
$484$	25.6049	1.16386
$485$	57.9023	2.62921
$486$	0	0
$487$	2.73680	0.124016	0.0620081	−	0.998076i	$-0.480250\pi$
$487$	2.73680	0.124016	0.0620081	+	0.998076i	$0.480250\pi$
$488$	21.1130	0.955741
$489$	0	0
$490$	0	0
$491$	19.7096	0.889483	0.444742	−	0.895659i	$-0.353295\pi$
$491$	19.7096	0.889483	0.444742	+	0.895659i	$0.353295\pi$
$492$	0	0
$493$	−7.47084	−0.336470
$494$	4.26582	0.191928
$495$	0	0
$496$	−0.941952	−0.0422949
$497$	0	0
$498$	0	0
$499$	−33.0960	−1.48158	−0.740789	−	0.671737i	$-0.765548\pi$
$499$	−33.0960	−1.48158	−0.740789	+	0.671737i	$0.765548\pi$
$500$	8.05153	0.360075
$501$	0	0
$502$	35.6575	1.59147
$503$	−12.1860	−0.543346	−0.271673	−	0.962390i	$-0.587577\pi$
$503$	−12.1860	−0.543346	−0.271673	+	0.962390i	$0.587577\pi$
$504$	0	0
$505$	−45.5331	−2.02619
$506$	13.3912	0.595310
$507$	0	0
$508$	−9.73765	−0.432038
$509$	−13.6393	−0.604551	−0.302276	−	0.953221i	$-0.597746\pi$
$509$	−13.6393	−0.604551	−0.302276	+	0.953221i	$0.597746\pi$
$510$	0	0
$511$	0	0
$512$	7.97968	0.352656
$513$	0	0
$514$	−4.22292	−0.186265
$515$	−9.48962	−0.418163
$516$	0	0
$517$	−34.8427	−1.53238
$518$	0	0
$519$	0	0
$520$	−6.92164	−0.303534
$521$	35.5490	1.55743	0.778714	−	0.627379i	$-0.215872\pi$
$521$	35.5490	1.55743	0.778714	+	0.627379i	$0.215872\pi$
$522$	0	0
$523$	26.7187	1.16833	0.584163	−	0.811636i	$-0.301423\pi$
$523$	26.7187	1.16833	0.584163	+	0.811636i	$0.301423\pi$
$524$	−14.9598	−0.653523
$525$	0	0
$526$	42.4103	1.84918
$527$	0.226830	0.00988086
$528$	0	0
$529$	−21.3785	−0.929499
$530$	50.5665	2.19647
$531$	0	0
$532$	0	0
$533$	−3.77130	−0.163353
$534$	0	0
$535$	0.626879	0.0271023
$536$	13.9198	0.601244
$537$	0	0
$538$	−51.0564	−2.20120
$539$	0	0
$540$	0	0
$541$	37.5855	1.61593	0.807963	−	0.589233i	$-0.200570\pi$
$541$	37.5855	1.61593	0.807963	+	0.589233i	$0.200570\pi$
$542$	42.8053	1.83864
$543$	0	0
$544$	−6.15733	−0.263993
$545$	32.8701	1.40800
$546$	0	0
$547$	18.2676	0.781067	0.390533	−	0.920589i	$-0.372291\pi$
$547$	18.2676	0.781067	0.390533	+	0.920589i	$0.372291\pi$
$548$	−6.03134	−0.257646
$549$	0	0
$550$	77.1501	3.28969
$551$	−13.6849	−0.582995
$552$	0	0
$553$	0	0
$554$	3.24343	0.137800
$555$	0	0
$556$	0.860866	0.0365089
$557$	3.89272	0.164940	0.0824698	−	0.996594i	$-0.473719\pi$
$557$	3.89272	0.164940	0.0824698	+	0.996594i	$0.473719\pi$
$558$	0	0
$559$	4.26324	0.180316
$560$	0	0
$561$	0	0
$562$	20.8562	0.879767
$563$	3.32855	0.140282	0.0701409	−	0.997537i	$-0.477655\pi$
$563$	3.32855	0.140282	0.0701409	+	0.997537i	$0.477655\pi$
$564$	0	0
$565$	29.5750	1.24423
$566$	48.2965	2.03006
$567$	0	0
$568$	−21.4702	−0.900870
$569$	36.6244	1.53538	0.767688	−	0.640824i	$-0.221407\pi$
$569$	36.6244	1.53538	0.767688	+	0.640824i	$0.221407\pi$
$570$	0	0
$571$	−22.5824	−0.945044	−0.472522	−	0.881319i	$-0.656656\pi$
$571$	−22.5824	−0.945044	−0.472522	+	0.881319i	$0.656656\pi$
$572$	6.69483	0.279925
$573$	0	0
$574$	0	0
$575$	9.34201	0.389589
$576$	0	0
$577$	−22.5449	−0.938557	−0.469279	−	0.883050i	$-0.655486\pi$
$577$	−22.5449	−0.938557	−0.469279	+	0.883050i	$0.655486\pi$
$578$	−26.8496	−1.11680
$579$	0	0
$580$	−21.3847	−0.887949
$581$	0	0
$582$	0	0
$583$	50.7854	2.10332
$584$	−9.34913	−0.386870
$585$	0	0
$586$	15.2343	0.629325
$587$	24.2396	1.00047	0.500237	−	0.865888i	$-0.333246\pi$
$587$	24.2396	1.00047	0.500237	+	0.865888i	$0.333246\pi$
$588$	0	0
$589$	0.415501	0.0171204
$590$	−68.3368	−2.81338
$591$	0	0
$592$	−17.8822	−0.734956
$593$	−45.7326	−1.87801	−0.939007	−	0.343898i	$-0.888253\pi$
$593$	−45.7326	−1.87801	−0.939007	+	0.343898i	$0.888253\pi$
$594$	0	0
$595$	0	0
$596$	−5.66686	−0.232123
$597$	0	0
$598$	2.46310	0.100724
$599$	30.1668	1.23258	0.616290	−	0.787519i	$-0.288635\pi$
$599$	30.1668	1.23258	0.616290	+	0.787519i	$0.288635\pi$
$600$	0	0
$601$	14.7387	0.601202	0.300601	−	0.953750i	$-0.402813\pi$
$601$	14.7387	0.601202	0.300601	+	0.953750i	$0.402813\pi$
$602$	0	0
$603$	0	0
$604$	−1.98830	−0.0809027
$605$	91.6569	3.72638
$606$	0	0
$607$	6.07836	0.246713	0.123356	−	0.992362i	$-0.460634\pi$
$607$	6.07836	0.246713	0.123356	+	0.992362i	$0.460634\pi$
$608$	−11.2788	−0.457417
$609$	0	0
$610$	−72.7859	−2.94702
$611$	−6.40877	−0.259271
$612$	0	0
$613$	11.7734	0.475522	0.237761	−	0.971324i	$-0.423587\pi$
$613$	11.7734	0.475522	0.237761	+	0.971324i	$0.423587\pi$
$614$	−1.82213	−0.0735352
$615$	0	0
$616$	0	0
$617$	−32.0637	−1.29084	−0.645418	−	0.763829i	$-0.723317\pi$
$617$	−32.0637	−1.29084	−0.645418	+	0.763829i	$0.723317\pi$
$618$	0	0
$619$	12.5518	0.504498	0.252249	−	0.967662i	$-0.418830\pi$
$619$	12.5518	0.504498	0.252249	+	0.967662i	$0.418830\pi$
$620$	0.649282	0.0260758
$621$	0	0
$622$	5.30441	0.212688
$623$	0	0
$624$	0	0
$625$	−7.85989	−0.314396
$626$	−48.6249	−1.94344
$627$	0	0
$628$	2.99022	0.119323
$629$	4.30619	0.171699
$630$	0	0
$631$	33.4642	1.33219	0.666095	−	0.745867i	$-0.267964\pi$
$631$	33.4642	1.33219	0.666095	+	0.745867i	$0.267964\pi$
$632$	−16.2138	−0.644950
$633$	0	0
$634$	−22.1809	−0.880915
$635$	−34.8575	−1.38328
$636$	0	0
$637$	0	0
$638$	−65.2553	−2.58348
$639$	0	0
$640$	43.0148	1.70031
$641$	18.9837	0.749809	0.374905	−	0.927063i	$-0.377675\pi$
$641$	18.9837	0.749809	0.374905	+	0.927063i	$0.377675\pi$
$642$	0	0
$643$	9.62695	0.379650	0.189825	−	0.981818i	$-0.439208\pi$
$643$	9.62695	0.379650	0.189825	+	0.981818i	$0.439208\pi$
$644$	0	0
$645$	0	0
$646$	4.58445	0.180373
$647$	−7.81214	−0.307127	−0.153564	−	0.988139i	$-0.549075\pi$
$647$	−7.81214	−0.307127	−0.153564	+	0.988139i	$0.549075\pi$
$648$	0	0
$649$	−68.6326	−2.69406
$650$	14.1906	0.556599
$651$	0	0
$652$	−5.28751	−0.207075
$653$	−31.7429	−1.24219	−0.621097	−	0.783734i	$-0.713313\pi$
$653$	−31.7429	−1.24219	−0.621097	+	0.783734i	$0.713313\pi$
$654$	0	0
$655$	−53.5512	−2.09242
$656$	16.8308	0.657132
$657$	0	0
$658$	0	0
$659$	6.21370	0.242051	0.121026	−	0.992649i	$-0.461382\pi$
$659$	6.21370	0.242051	0.121026	+	0.992649i	$0.461382\pi$
$660$	0	0
$661$	−27.5263	−1.07065	−0.535324	−	0.844647i	$-0.679810\pi$
$661$	−27.5263	−1.07065	−0.535324	+	0.844647i	$0.679810\pi$
$662$	−37.2189	−1.44655
$663$	0	0
$664$	2.19870	0.0853260
$665$	0	0
$666$	0	0
$667$	−7.90169	−0.305955
$668$	16.2971	0.630553
$669$	0	0
$670$	−47.9878	−1.85393
$671$	−73.1010	−2.82203
$672$	0	0
$673$	16.2179	0.625154	0.312577	−	0.949892i	$-0.398808\pi$
$673$	16.2179	0.625154	0.312577	+	0.949892i	$0.398808\pi$
$674$	−21.7670	−0.838434
$675$	0	0
$676$	−11.5240	−0.443230
$677$	−20.5090	−0.788225	−0.394112	−	0.919062i	$-0.628948\pi$
$677$	−20.5090	−0.788225	−0.394112	+	0.919062i	$0.628948\pi$
$678$	0	0
$679$	0	0
$680$	−7.43864	−0.285259
$681$	0	0
$682$	1.98128	0.0758673
$683$	0.112308	0.00429736	0.00214868	−	0.999998i	$-0.499316\pi$
$683$	0.112308	0.00429736	0.00214868	+	0.999998i	$0.499316\pi$
$684$	0	0
$685$	−21.5902	−0.824918
$686$	0	0
$687$	0	0
$688$	−19.0262	−0.725368
$689$	9.34118	0.355871
$690$	0	0
$691$	−18.8670	−0.717735	−0.358868	−	0.933388i	$-0.616837\pi$
$691$	−18.8670	−0.717735	−0.358868	+	0.933388i	$0.616837\pi$
$692$	−17.3182	−0.658340
$693$	0	0
$694$	−39.9480	−1.51640
$695$	3.08161	0.116892
$696$	0	0
$697$	−4.05299	−0.153518
$698$	28.4665	1.07747
$699$	0	0
$700$	0	0
$701$	3.16006	0.119354	0.0596770	−	0.998218i	$-0.480993\pi$
$701$	3.16006	0.119354	0.0596770	+	0.998218i	$0.480993\pi$
$702$	0	0
$703$	7.88796	0.297500
$704$	7.11984	0.268339
$705$	0	0
$706$	−42.2804	−1.59124
$707$	0	0
$708$	0	0
$709$	−21.5211	−0.808243	−0.404121	−	0.914705i	$-0.632423\pi$
$709$	−21.5211	−0.808243	−0.404121	+	0.914705i	$0.632423\pi$
$710$	74.0174	2.77782
$711$	0	0
$712$	−9.74771	−0.365311
$713$	0.239911	0.00898475
$714$	0	0
$715$	23.9653	0.896250
$716$	−2.57898	−0.0963812
$717$	0	0
$718$	−35.3570	−1.31951
$719$	−18.8302	−0.702246	−0.351123	−	0.936329i	$-0.614200\pi$
$719$	−18.8302	−0.702246	−0.351123	+	0.936329i	$0.614200\pi$
$720$	0	0
$721$	0	0
$722$	−24.4079	−0.908369
$723$	0	0
$724$	−3.89914	−0.144910
$725$	−45.5237	−1.69071
$726$	0	0
$727$	−39.0853	−1.44959	−0.724797	−	0.688963i	$-0.758066\pi$
$727$	−39.0853	−1.44959	−0.724797	+	0.688963i	$0.758066\pi$
$728$	0	0
$729$	0	0
$730$	32.2306	1.19291
$731$	4.58167	0.169459
$732$	0	0
$733$	−18.5985	−0.686951	−0.343475	−	0.939162i	$-0.611604\pi$
$733$	−18.5985	−0.686951	−0.343475	+	0.939162i	$0.611604\pi$
$734$	−38.4237	−1.41825
$735$	0	0
$736$	−6.51243	−0.240051
$737$	−48.1954	−1.77530
$738$	0	0
$739$	5.50136	0.202371	0.101185	−	0.994868i	$-0.467736\pi$
$739$	5.50136	0.202371	0.101185	+	0.994868i	$0.467736\pi$
$740$	12.3261	0.453117
$741$	0	0
$742$	0	0
$743$	20.4653	0.750797	0.375399	−	0.926863i	$-0.377506\pi$
$743$	20.4653	0.750797	0.375399	+	0.926863i	$0.377506\pi$
$744$	0	0
$745$	−20.2855	−0.743201
$746$	−56.1785	−2.05684
$747$	0	0
$748$	7.19489	0.263071
$749$	0	0
$750$	0	0
$751$	38.0460	1.38832	0.694159	−	0.719822i	$-0.255777\pi$
$751$	38.0460	1.38832	0.694159	+	0.719822i	$0.255777\pi$
$752$	28.6015	1.04299
$753$	0	0
$754$	−12.0027	−0.437113
$755$	−7.11744	−0.259030
$756$	0	0
$757$	−51.0780	−1.85646	−0.928230	−	0.372006i	$-0.878670\pi$
$757$	−51.0780	−1.85646	−0.928230	+	0.372006i	$0.878670\pi$
$758$	2.66658	0.0968546
$759$	0	0
$760$	−13.6259	−0.494263
$761$	−40.0749	−1.45271	−0.726357	−	0.687317i	$-0.758788\pi$
$761$	−40.0749	−1.45271	−0.726357	+	0.687317i	$0.758788\pi$
$762$	0	0
$763$	0	0
$764$	17.8633	0.646273
$765$	0	0
$766$	54.6116	1.97320
$767$	−12.6239	−0.455822
$768$	0	0
$769$	44.9656	1.62150	0.810751	−	0.585391i	$-0.199059\pi$
$769$	44.9656	1.62150	0.810751	+	0.585391i	$0.199059\pi$
$770$	0	0
$771$	0	0
$772$	−0.198577	−0.00714695
$773$	−24.3561	−0.876029	−0.438014	−	0.898968i	$-0.644318\pi$
$773$	−24.3561	−0.876029	−0.438014	+	0.898968i	$0.644318\pi$
$774$	0	0
$775$	1.38219	0.0496498
$776$	−28.9997	−1.04103
$777$	0	0
$778$	9.05826	0.324754
$779$	−7.42416	−0.265998
$780$	0	0
$781$	74.3377	2.66001
$782$	2.64707	0.0946592
$783$	0	0
$784$	0	0
$785$	10.7040	0.382042
$786$	0	0
$787$	−41.5233	−1.48015	−0.740073	−	0.672526i	$-0.765209\pi$
$787$	−41.5233	−1.48015	−0.740073	+	0.672526i	$0.765209\pi$
$788$	1.60066	0.0570212
$789$	0	0
$790$	55.8962	1.98870
$791$	0	0
$792$	0	0
$793$	−13.4458	−0.477474
$794$	−0.0477147	−0.00169333
$795$	0	0
$796$	−6.17372	−0.218822
$797$	34.6036	1.22572	0.612861	−	0.790191i	$-0.290019\pi$
$797$	34.6036	1.22572	0.612861	+	0.790191i	$0.290019\pi$
$798$	0	0
$799$	−6.88747	−0.243661
$800$	−37.5198	−1.32653
$801$	0	0
$802$	−20.9571	−0.740020
$803$	32.3701	1.14232
$804$	0	0
$805$	0	0
$806$	0.364427	0.0128364
$807$	0	0
$808$	22.8047	0.802266
$809$	11.2519	0.395597	0.197799	−	0.980243i	$-0.436621\pi$
$809$	11.2519	0.395597	0.197799	+	0.980243i	$0.436621\pi$
$810$	0	0
$811$	29.6803	1.04222	0.521108	−	0.853491i	$-0.325519\pi$
$811$	29.6803	1.04222	0.521108	+	0.853491i	$0.325519\pi$
$812$	0	0
$813$	0	0
$814$	37.6132	1.31834
$815$	−18.9275	−0.663002
$816$	0	0
$817$	8.39258	0.293619
$818$	−54.1211	−1.89230
$819$	0	0
$820$	−11.6014	−0.405137
$821$	−34.6431	−1.20905	−0.604526	−	0.796585i	$-0.706638\pi$
$821$	−34.6431	−1.20905	−0.604526	+	0.796585i	$0.706638\pi$
$822$	0	0
$823$	36.3870	1.26837	0.634186	−	0.773180i	$-0.281335\pi$
$823$	36.3870	1.26837	0.634186	+	0.773180i	$0.281335\pi$
$824$	4.75276	0.165570
$825$	0	0
$826$	0	0
$827$	24.3576	0.846997	0.423498	−	0.905897i	$-0.360802\pi$
$827$	24.3576	0.846997	0.423498	+	0.905897i	$0.360802\pi$
$828$	0	0
$829$	−39.1702	−1.36044	−0.680219	−	0.733009i	$-0.738115\pi$
$829$	−39.1702	−1.36044	−0.680219	+	0.733009i	$0.738115\pi$
$830$	−7.57989	−0.263102
$831$	0	0
$832$	1.30958	0.0454017
$833$	0	0
$834$	0	0
$835$	58.3381	2.01887
$836$	13.1794	0.455819
$837$	0	0
$838$	−25.7030	−0.887894
$839$	34.3477	1.18581	0.592907	−	0.805271i	$-0.297980\pi$
$839$	34.3477	1.18581	0.592907	+	0.805271i	$0.297980\pi$
$840$	0	0
$841$	9.50500	0.327759
$842$	15.6850	0.540541
$843$	0	0
$844$	−15.9809	−0.550085
$845$	−41.2520	−1.41911
$846$	0	0
$847$	0	0
$848$	−41.6884	−1.43159
$849$	0	0
$850$	15.2505	0.523088
$851$	4.55454	0.156127
$852$	0	0
$853$	−32.6743	−1.11875	−0.559373	−	0.828916i	$-0.688958\pi$
$853$	−32.6743	−1.11875	−0.559373	+	0.828916i	$0.688958\pi$
$854$	0	0
$855$	0	0
$856$	−0.313965	−0.0107311
$857$	−57.6679	−1.96990	−0.984950	−	0.172841i	$-0.944705\pi$
$857$	−57.6679	−1.96990	−0.984950	+	0.172841i	$0.944705\pi$
$858$	0	0
$859$	−29.9768	−1.02279	−0.511397	−	0.859344i	$-0.670872\pi$
$859$	−29.9768	−1.02279	−0.511397	+	0.859344i	$0.670872\pi$
$860$	13.1147	0.447206
$861$	0	0
$862$	28.7231	0.978311
$863$	23.1776	0.788974	0.394487	−	0.918901i	$-0.370922\pi$
$863$	23.1776	0.788974	0.394487	+	0.918901i	$0.370922\pi$
$864$	0	0
$865$	−61.9934	−2.10784
$866$	−34.0873	−1.15833
$867$	0	0
$868$	0	0
$869$	56.1381	1.90435
$870$	0	0
$871$	−8.86480	−0.300372
$872$	−16.4626	−0.557493
$873$	0	0
$874$	4.84884	0.164014
$875$	0	0
$876$	0	0
$877$	−0.739956	−0.0249865	−0.0124933	−	0.999922i	$-0.503977\pi$
$877$	−0.739956	−0.0249865	−0.0124933	+	0.999922i	$0.503977\pi$
$878$	11.6290	0.392458
$879$	0	0
$880$	−106.954	−3.60540
$881$	−18.0285	−0.607395	−0.303697	−	0.952769i	$-0.598221\pi$
$881$	−18.0285	−0.607395	−0.303697	+	0.952769i	$0.598221\pi$
$882$	0	0
$883$	43.0928	1.45019	0.725095	−	0.688649i	$-0.241796\pi$
$883$	43.0928	1.45019	0.725095	+	0.688649i	$0.241796\pi$
$884$	1.32339	0.0445104
$885$	0	0
$886$	−49.4507	−1.66133
$887$	−30.9527	−1.03929	−0.519645	−	0.854382i	$-0.673936\pi$
$887$	−30.9527	−1.03929	−0.519645	+	0.854382i	$0.673936\pi$
$888$	0	0
$889$	0	0
$890$	33.6047	1.12643
$891$	0	0
$892$	−19.5937	−0.656046
$893$	−12.6163	−0.422187
$894$	0	0
$895$	−9.23190	−0.308588
$896$	0	0
$897$	0	0
$898$	11.5143	0.384236
$899$	−1.16909	−0.0389913
$900$	0	0
$901$	10.0389	0.334444
$902$	−35.4016	−1.17874
$903$	0	0
$904$	−14.8123	−0.492649
$905$	−13.9576	−0.463966
$906$	0	0
$907$	−42.8083	−1.42143	−0.710714	−	0.703481i	$-0.751628\pi$
$907$	−42.8083	−1.42143	−0.710714	+	0.703481i	$0.751628\pi$
$908$	3.54485	0.117640
$909$	0	0
$910$	0	0
$911$	6.09738	0.202015	0.101008	−	0.994886i	$-0.467793\pi$
$911$	6.09738	0.202015	0.101008	+	0.994886i	$0.467793\pi$
$912$	0	0
$913$	−7.61269	−0.251943
$914$	−49.4800	−1.63665
$915$	0	0
$916$	−13.4566	−0.444619
$917$	0	0
$918$	0	0
$919$	24.8613	0.820100	0.410050	−	0.912063i	$-0.365511\pi$
$919$	24.8613	0.820100	0.410050	+	0.912063i	$0.365511\pi$
$920$	−7.86764	−0.259388
$921$	0	0
$922$	34.5621	1.13824
$923$	13.6733	0.450061
$924$	0	0
$925$	26.2399	0.862762
$926$	17.1207	0.562621
$927$	0	0
$928$	31.7351	1.04176
$929$	−41.8402	−1.37273	−0.686366	−	0.727256i	$-0.740795\pi$
$929$	−41.8402	−1.37273	−0.686366	+	0.727256i	$0.740795\pi$
$930$	0	0
$931$	0	0
$932$	24.7790	0.811663
$933$	0	0
$934$	−27.7651	−0.908502
$935$	25.7553	0.842288
$936$	0	0
$937$	29.2537	0.955676	0.477838	−	0.878448i	$-0.341421\pi$
$937$	29.2537	0.955676	0.477838	+	0.878448i	$0.341421\pi$
$938$	0	0
$939$	0	0
$940$	−19.7148	−0.643026
$941$	−3.35737	−0.109447	−0.0547236	−	0.998502i	$-0.517428\pi$
$941$	−3.35737	−0.109447	−0.0547236	+	0.998502i	$0.517428\pi$
$942$	0	0
$943$	−4.28673	−0.139595
$944$	56.3387	1.83367
$945$	0	0
$946$	40.0194	1.30114
$947$	−5.26601	−0.171122	−0.0855612	−	0.996333i	$-0.527268\pi$
$947$	−5.26601	−0.171122	−0.0855612	+	0.996333i	$0.527268\pi$
$948$	0	0
$949$	5.95398	0.193274
$950$	27.9354	0.906345
$951$	0	0
$952$	0	0
$953$	−56.2520	−1.82218	−0.911090	−	0.412208i	$-0.864758\pi$
$953$	−56.2520	−1.82218	−0.911090	+	0.412208i	$0.864758\pi$
$954$	0	0
$955$	63.9448	2.06921
$956$	−8.82075	−0.285284
$957$	0	0
$958$	14.1626	0.457573
$959$	0	0
$960$	0	0
$961$	−30.9645	−0.998855
$962$	6.91836	0.223057
$963$	0	0
$964$	−9.08306	−0.292546
$965$	−0.710840	−0.0228828
$966$	0	0
$967$	13.7728	0.442904	0.221452	−	0.975171i	$-0.428920\pi$
$967$	13.7728	0.442904	0.221452	+	0.975171i	$0.428920\pi$
$968$	−45.9052	−1.47545
$969$	0	0
$970$	99.9748	3.21000
$971$	51.3254	1.64711	0.823555	−	0.567236i	$-0.191987\pi$
$971$	51.3254	1.64711	0.823555	+	0.567236i	$0.191987\pi$
$972$	0	0
$973$	0	0
$974$	4.72538	0.151411
$975$	0	0
$976$	60.0066	1.92077
$977$	−17.6850	−0.565794	−0.282897	−	0.959150i	$-0.591296\pi$
$977$	−17.6850	−0.565794	−0.282897	+	0.959150i	$0.591296\pi$
$978$	0	0
$979$	33.7501	1.07866
$980$	0	0
$981$	0	0
$982$	34.0309	1.08597
$983$	16.0041	0.510453	0.255226	−	0.966881i	$-0.417850\pi$
$983$	16.0041	0.510453	0.255226	+	0.966881i	$0.417850\pi$
$984$	0	0
$985$	5.72984	0.182568
$986$	−12.8992	−0.410795
$987$	0	0
$988$	2.42415	0.0771224
$989$	4.84590	0.154091
$990$	0	0
$991$	−10.8664	−0.345182	−0.172591	−	0.984994i	$-0.555214\pi$
$991$	−10.8664	−0.345182	−0.172591	+	0.984994i	$0.555214\pi$
$992$	−0.963543	−0.0305925
$993$	0	0
$994$	0	0
$995$	−22.0998	−0.700612
$996$	0	0
$997$	41.0375	1.29967	0.649835	−	0.760075i	$-0.274838\pi$
$997$	41.0375	1.29967	0.649835	+	0.760075i	$0.274838\pi$
$998$	−57.1438	−1.80886
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3969.2.a.bi.1.10		12
3.2	odd	2		3969.2.a.bh.1.3		12
7.6	odd	2	inner	3969.2.a.bi.1.9		12
9.2	odd	6		441.2.f.h.148.9	✓	24
9.4	even	3		1323.2.f.h.883.3		24
9.5	odd	6		441.2.f.h.295.9	yes	24
9.7	even	3		1323.2.f.h.442.3		24
21.20	even	2		3969.2.a.bh.1.4		12
63.2	odd	6		441.2.g.h.67.9		24
63.4	even	3		1323.2.g.h.667.4		24
63.5	even	6		441.2.h.h.214.3		24
63.11	odd	6		441.2.h.h.373.4		24
63.13	odd	6		1323.2.f.h.883.4		24
63.16	even	3		1323.2.g.h.361.4		24
63.20	even	6		441.2.f.h.148.10	yes	24
63.23	odd	6		441.2.h.h.214.4		24
63.25	even	3		1323.2.h.h.226.10		24
63.31	odd	6		1323.2.g.h.667.3		24
63.32	odd	6		441.2.g.h.79.9		24
63.34	odd	6		1323.2.f.h.442.4		24
63.38	even	6		441.2.h.h.373.3		24
63.40	odd	6		1323.2.h.h.802.9		24
63.41	even	6		441.2.f.h.295.10	yes	24
63.47	even	6		441.2.g.h.67.10		24
63.52	odd	6		1323.2.h.h.226.9		24
63.58	even	3		1323.2.h.h.802.10		24
63.59	even	6		441.2.g.h.79.10		24
63.61	odd	6		1323.2.g.h.361.3		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.f.h.148.9	✓	24	9.2	odd	6
441.2.f.h.148.10	yes	24	63.20	even	6
441.2.f.h.295.9	yes	24	9.5	odd	6
441.2.f.h.295.10	yes	24	63.41	even	6
441.2.g.h.67.9		24	63.2	odd	6
441.2.g.h.67.10		24	63.47	even	6
441.2.g.h.79.9		24	63.32	odd	6
441.2.g.h.79.10		24	63.59	even	6
441.2.h.h.214.3		24	63.5	even	6
441.2.h.h.214.4		24	63.23	odd	6
441.2.h.h.373.3		24	63.38	even	6
441.2.h.h.373.4		24	63.11	odd	6
1323.2.f.h.442.3		24	9.7	even	3
1323.2.f.h.442.4		24	63.34	odd	6
1323.2.f.h.883.3		24	9.4	even	3
1323.2.f.h.883.4		24	63.13	odd	6
1323.2.g.h.361.3		24	63.61	odd	6
1323.2.g.h.361.4		24	63.16	even	3
1323.2.g.h.667.3		24	63.31	odd	6
1323.2.g.h.667.4		24	63.4	even	3
1323.2.h.h.226.9		24	63.52	odd	6
1323.2.h.h.226.10		24	63.25	even	3
1323.2.h.h.802.9		24	63.40	odd	6
1323.2.h.h.802.10		24	63.58	even	3
3969.2.a.bh.1.3		12	3.2	odd	2
3969.2.a.bh.1.4		12	21.20	even	2
3969.2.a.bi.1.9		12	7.6	odd	2	inner
3969.2.a.bi.1.10		12	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+1.72661 q^{2} +0.981184 q^{4} +3.51231 q^{5} -1.75910 q^{8} +6.06439 q^{10} +6.09064 q^{11} +1.12028 q^{13} -4.99965 q^{16} +1.20396 q^{17} +2.20537 q^{19} +3.44622 q^{20} +10.5162 q^{22} +1.27339 q^{23} +7.33633 q^{25} +1.93429 q^{26} -6.20524 q^{29} +0.188404 q^{31} -5.11425 q^{32} +2.07876 q^{34} +3.57670 q^{37} +3.80782 q^{38} -6.17850 q^{40} -3.36640 q^{41} +3.80551 q^{43} +5.97604 q^{44} +2.19865 q^{46} -5.72070 q^{47} +12.6670 q^{50} +1.09920 q^{52} +8.33827 q^{53} +21.3922 q^{55} -10.7140 q^{58} -11.2685 q^{59} -12.0022 q^{61} +0.325300 q^{62} +1.16898 q^{64} +3.93477 q^{65} -7.91303 q^{67} +1.18130 q^{68} +12.2052 q^{71} +5.31473 q^{73} +6.17557 q^{74} +2.16388 q^{76} +9.21711 q^{79} -17.5603 q^{80} -5.81246 q^{82} -1.24990 q^{83} +4.22867 q^{85} +6.57064 q^{86} -10.7140 q^{88} +5.54131 q^{89} +1.24943 q^{92} -9.87741 q^{94} +7.74596 q^{95} +16.4855 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{2} + 12 q^{4} + 12 q^{8} + 20 q^{11} + 12 q^{16} + 32 q^{23} + 12 q^{25} + 16 q^{29} + 48 q^{32} + 12 q^{37} + 56 q^{44} - 24 q^{46} - 4 q^{50} + 32 q^{53} + 48 q^{64} + 60 q^{65} + 12 q^{67}+ \cdots + 64 q^{95}+O(q^{100}) \) 12 * q + 4 * q^2 + 12 * q^4 + 12 * q^8 + 20 * q^11 + 12 * q^16 + 32 * q^23 + 12 * q^25 + 16 * q^29 + 48 * q^32 + 12 * q^37 + 56 * q^44 - 24 * q^46 - 4 * q^50 + 32 * q^53 + 48 * q^64 + 60 * q^65 + 12 * q^67 + 56 * q^71 + 68 * q^74 - 12 * q^79 - 12 * q^85 + 76 * q^86 + 16 * q^92 + 64 * q^95

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