LMFDB - Embedded newform 9216.2.a.w.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9216,2,Mod(1,9216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9216, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("9216.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9216 = 2^{10} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9216.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$73.5901305028$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{16})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 2 $ x^4 - 4*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 512)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.765367$ of defining polynomial
Character	$\chi$	$=$	9216.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.585786 q^{5} -3.69552 q^{7} +4.14386 q^{11} +3.41421 q^{13} -2.82843 q^{17} +6.30864 q^{19} -6.75699 q^{23} -4.65685 q^{25} -7.41421 q^{29} -3.06147 q^{31} +2.16478 q^{35} +9.07107 q^{37} +4.00000 q^{41} +1.08239 q^{43} +3.06147 q^{47} +6.65685 q^{49} -4.58579 q^{53} -2.42742 q^{55} +1.08239 q^{59} -1.07107 q^{61} -2.00000 q^{65} -1.97908 q^{67} -8.02509 q^{71} -6.48528 q^{73} -15.3137 q^{77} +14.7821 q^{79} +13.6997 q^{83} +1.65685 q^{85} -4.82843 q^{89} -12.6173 q^{91} -3.69552 q^{95} +5.17157 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{5} + 8 q^{13} + 4 q^{25} - 24 q^{29} + 8 q^{37} + 16 q^{41} + 4 q^{49} - 24 q^{53} + 24 q^{61} - 8 q^{65} + 8 q^{73} - 16 q^{77} - 16 q^{85} - 8 q^{89} + 32 q^{97}+O(q^{100}) $ 4 * q - 8 * q^5 + 8 * q^13 + 4 * q^25 - 24 * q^29 + 8 * q^37 + 16 * q^41 + 4 * q^49 - 24 * q^53 + 24 * q^61 - 8 * q^65 + 8 * q^73 - 16 * q^77 - 16 * q^85 - 8 * q^89 + 32 * 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	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−0.585786	−0.261972	−0.130986	−	0.991384i	$-0.541814\pi$
$5$	−0.585786	−0.261972	−0.130986	+	0.991384i	$0.541814\pi$
$6$	0	0
$7$	−3.69552	−1.39677	−0.698387	−	0.715720i	$-0.746099\pi$
$7$	−3.69552	−1.39677	−0.698387	+	0.715720i	$0.746099\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.14386	1.24942	0.624710	−	0.780857i	$-0.285217\pi$
$11$	4.14386	1.24942	0.624710	+	0.780857i	$0.285217\pi$
$12$	0	0
$13$	3.41421	0.946932	0.473466	−	0.880812i	$-0.343003\pi$
$13$	3.41421	0.946932	0.473466	+	0.880812i	$0.343003\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.82843	−0.685994	−0.342997	−	0.939336i	$-0.611442\pi$
$17$	−2.82843	−0.685994	−0.342997	+	0.939336i	$0.611442\pi$
$18$	0	0
$19$	6.30864	1.44730	0.723651	−	0.690166i	$-0.242462\pi$
$19$	6.30864	1.44730	0.723651	+	0.690166i	$0.242462\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.75699	−1.40893	−0.704464	−	0.709739i	$-0.748813\pi$
$23$	−6.75699	−1.40893	−0.704464	+	0.709739i	$0.748813\pi$
$24$	0	0
$25$	−4.65685	−0.931371
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−7.41421	−1.37678	−0.688392	−	0.725338i	$-0.741683\pi$
$29$	−7.41421	−1.37678	−0.688392	+	0.725338i	$0.741683\pi$
$30$	0	0
$31$	−3.06147	−0.549856	−0.274928	−	0.961465i	$-0.588654\pi$
$31$	−3.06147	−0.549856	−0.274928	+	0.961465i	$0.588654\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.16478	0.365915
$36$	0	0
$37$	9.07107	1.49127	0.745637	−	0.666352i	$-0.232145\pi$
$37$	9.07107	1.49127	0.745637	+	0.666352i	$0.232145\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	0	0
$43$	1.08239	0.165063	0.0825316	−	0.996588i	$-0.473699\pi$
$43$	1.08239	0.165063	0.0825316	+	0.996588i	$0.473699\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.06147	0.446561	0.223280	−	0.974754i	$-0.428323\pi$
$47$	3.06147	0.446561	0.223280	+	0.974754i	$0.428323\pi$
$48$	0	0
$49$	6.65685	0.950979
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−4.58579	−0.629906	−0.314953	−	0.949107i	$-0.601989\pi$
$53$	−4.58579	−0.629906	−0.314953	+	0.949107i	$0.601989\pi$
$54$	0	0
$55$	−2.42742	−0.327313
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.08239	0.140915	0.0704577	−	0.997515i	$-0.477554\pi$
$59$	1.08239	0.140915	0.0704577	+	0.997515i	$0.477554\pi$
$60$	0	0
$61$	−1.07107	−0.137136	−0.0685681	−	0.997646i	$-0.521843\pi$
$61$	−1.07107	−0.137136	−0.0685681	+	0.997646i	$0.521843\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	−1.97908	−0.241783	−0.120891	−	0.992666i	$-0.538575\pi$
$67$	−1.97908	−0.241783	−0.120891	+	0.992666i	$0.538575\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8.02509	−0.952403	−0.476201	−	0.879336i	$-0.657987\pi$
$71$	−8.02509	−0.952403	−0.476201	+	0.879336i	$0.657987\pi$
$72$	0	0
$73$	−6.48528	−0.759045	−0.379522	−	0.925183i	$-0.623912\pi$
$73$	−6.48528	−0.759045	−0.379522	+	0.925183i	$0.623912\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−15.3137	−1.74516
$78$	0	0
$79$	14.7821	1.66311	0.831557	−	0.555440i	$-0.187450\pi$
$79$	14.7821	1.66311	0.831557	+	0.555440i	$0.187450\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	13.6997	1.50374	0.751868	−	0.659314i	$-0.229153\pi$
$83$	13.6997	1.50374	0.751868	+	0.659314i	$0.229153\pi$
$84$	0	0
$85$	1.65685	0.179711
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−4.82843	−0.511812	−0.255906	−	0.966702i	$-0.582374\pi$
$89$	−4.82843	−0.511812	−0.255906	+	0.966702i	$0.582374\pi$
$90$	0	0
$91$	−12.6173	−1.32265
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.69552	−0.379152
$96$	0	0
$97$	5.17157	0.525094	0.262547	−	0.964919i	$-0.415438\pi$
$97$	5.17157	0.525094	0.262547	+	0.964919i	$0.415438\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−17.0711	−1.69863	−0.849317	−	0.527883i	$-0.822986\pi$
$101$	−17.0711	−1.69863	−0.849317	+	0.527883i	$0.822986\pi$
$102$	0	0
$103$	14.1480	1.39405	0.697023	−	0.717049i	$-0.254508\pi$
$103$	14.1480	1.39405	0.697023	+	0.717049i	$0.254508\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.97908	−0.191324	−0.0956622	−	0.995414i	$-0.530497\pi$
$107$	−1.97908	−0.191324	−0.0956622	+	0.995414i	$0.530497\pi$
$108$	0	0
$109$	−13.0711	−1.25198	−0.625991	−	0.779831i	$-0.715305\pi$
$109$	−13.0711	−1.25198	−0.625991	+	0.779831i	$0.715305\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	7.65685	0.720296	0.360148	−	0.932895i	$-0.382726\pi$
$113$	7.65685	0.720296	0.360148	+	0.932895i	$0.382726\pi$
$114$	0	0
$115$	3.95815	0.369099
$116$	0	0
$117$	0	0
$118$	0	0
$119$	10.4525	0.958179
$120$	0	0
$121$	6.17157	0.561052
$122$	0	0
$123$	0	0
$124$	0	0
$125$	5.65685	0.505964
$126$	0	0
$127$	−11.7206	−1.04004	−0.520018	−	0.854155i	$-0.674075\pi$
$127$	−11.7206	−1.04004	−0.520018	+	0.854155i	$0.674075\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−14.5964	−1.27529	−0.637645	−	0.770330i	$-0.720091\pi$
$131$	−14.5964	−1.27529	−0.637645	+	0.770330i	$0.720091\pi$
$132$	0	0
$133$	−23.3137	−2.02155
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−12.9706	−1.10815	−0.554075	−	0.832467i	$-0.686928\pi$
$137$	−12.9706	−1.10815	−0.554075	+	0.832467i	$0.686928\pi$
$138$	0	0
$139$	20.1940	1.71284	0.856418	−	0.516283i	$-0.172685\pi$
$139$	20.1940	1.71284	0.856418	+	0.516283i	$0.172685\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	14.1480	1.18312
$144$	0	0
$145$	4.34315	0.360679
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.75736	−0.143968	−0.0719842	−	0.997406i	$-0.522933\pi$
$149$	−1.75736	−0.143968	−0.0719842	+	0.997406i	$0.522933\pi$
$150$	0	0
$151$	0.634051	0.0515983	0.0257992	−	0.999667i	$-0.491787\pi$
$151$	0.634051	0.0515983	0.0257992	+	0.999667i	$0.491787\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.79337	0.144047
$156$	0	0
$157$	2.24264	0.178982	0.0894911	−	0.995988i	$-0.471476\pi$
$157$	2.24264	0.178982	0.0894911	+	0.995988i	$0.471476\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	24.9706	1.96796
$162$	0	0
$163$	−5.41196	−0.423898	−0.211949	−	0.977281i	$-0.567981\pi$
$163$	−5.41196	−0.423898	−0.211949	+	0.977281i	$0.567981\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.69552	0.285968	0.142984	−	0.989725i	$-0.454330\pi$
$167$	3.69552	0.285968	0.142984	+	0.989725i	$0.454330\pi$
$168$	0	0
$169$	−1.34315	−0.103319
$170$	0	0
$171$	0	0
$172$	0	0
$173$	19.2132	1.46075	0.730376	−	0.683045i	$-0.239345\pi$
$173$	19.2132	1.46075	0.730376	+	0.683045i	$0.239345\pi$
$174$	0	0
$175$	17.2095	1.30092
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−11.5349	−0.862159	−0.431079	−	0.902314i	$-0.641867\pi$
$179$	−11.5349	−0.862159	−0.431079	+	0.902314i	$0.641867\pi$
$180$	0	0
$181$	−14.2426	−1.05865	−0.529324	−	0.848420i	$-0.677554\pi$
$181$	−14.2426	−1.05865	−0.529324	+	0.848420i	$0.677554\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−5.31371	−0.390672
$186$	0	0
$187$	−11.7206	−0.857096
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−11.7206	−0.848073	−0.424037	−	0.905645i	$-0.639387\pi$
$191$	−11.7206	−0.848073	−0.424037	+	0.905645i	$0.639387\pi$
$192$	0	0
$193$	0.485281	0.0349313	0.0174657	−	0.999847i	$-0.494440\pi$
$193$	0.485281	0.0349313	0.0174657	+	0.999847i	$0.494440\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	10.7279	0.764333	0.382166	−	0.924094i	$-0.375178\pi$
$197$	10.7279	0.764333	0.382166	+	0.924094i	$0.375178\pi$
$198$	0	0
$199$	−12.3547	−0.875798	−0.437899	−	0.899024i	$-0.644277\pi$
$199$	−12.3547	−0.875798	−0.437899	+	0.899024i	$0.644277\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	27.3994	1.92306
$204$	0	0
$205$	−2.34315	−0.163652
$206$	0	0
$207$	0	0
$208$	0	0
$209$	26.1421	1.80829
$210$	0	0
$211$	13.3283	0.917555	0.458778	−	0.888551i	$-0.348287\pi$
$211$	13.3283	0.917555	0.458778	+	0.888551i	$0.348287\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−0.634051	−0.0432419
$216$	0	0
$217$	11.3137	0.768025
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−9.65685	−0.649590
$222$	0	0
$223$	−3.06147	−0.205011	−0.102506	−	0.994732i	$-0.532686\pi$
$223$	−3.06147	−0.205011	−0.102506	+	0.994732i	$0.532686\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.08239	0.0718409	0.0359204	−	0.999355i	$-0.488564\pi$
$227$	1.08239	0.0718409	0.0359204	+	0.999355i	$0.488564\pi$
$228$	0	0
$229$	3.89949	0.257686	0.128843	−	0.991665i	$-0.458874\pi$
$229$	3.89949	0.257686	0.128843	+	0.991665i	$0.458874\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−20.8284	−1.36452	−0.682258	−	0.731112i	$-0.739002\pi$
$233$	−20.8284	−1.36452	−0.682258	+	0.731112i	$0.739002\pi$
$234$	0	0
$235$	−1.79337	−0.116986
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−12.2459	−0.792119	−0.396060	−	0.918225i	$-0.629623\pi$
$239$	−12.2459	−0.792119	−0.396060	+	0.918225i	$0.629623\pi$
$240$	0	0
$241$	−13.1716	−0.848456	−0.424228	−	0.905556i	$-0.639454\pi$
$241$	−13.1716	−0.848456	−0.424228	+	0.905556i	$0.639454\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3.89949	−0.249130
$246$	0	0
$247$	21.5391	1.37050
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−18.0292	−1.13800	−0.568998	−	0.822339i	$-0.692669\pi$
$251$	−18.0292	−1.13800	−0.568998	+	0.822339i	$0.692669\pi$
$252$	0	0
$253$	−28.0000	−1.76034
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.9706	0.933838	0.466919	−	0.884300i	$-0.345364\pi$
$257$	14.9706	0.933838	0.466919	+	0.884300i	$0.345364\pi$
$258$	0	0
$259$	−33.5223	−2.08297
$260$	0	0
$261$	0	0
$262$	0	0
$263$	12.8799	0.794210	0.397105	−	0.917773i	$-0.370015\pi$
$263$	12.8799	0.794210	0.397105	+	0.917773i	$0.370015\pi$
$264$	0	0
$265$	2.68629	0.165018
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−25.5563	−1.55820	−0.779099	−	0.626901i	$-0.784323\pi$
$269$	−25.5563	−1.55820	−0.779099	+	0.626901i	$0.784323\pi$
$270$	0	0
$271$	5.59767	0.340034	0.170017	−	0.985441i	$-0.445618\pi$
$271$	5.59767	0.340034	0.170017	+	0.985441i	$0.445618\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−19.2974	−1.16367
$276$	0	0
$277$	−3.41421	−0.205140	−0.102570	−	0.994726i	$-0.532707\pi$
$277$	−3.41421	−0.205140	−0.102570	+	0.994726i	$0.532707\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−17.7990	−1.06180	−0.530899	−	0.847435i	$-0.678146\pi$
$281$	−17.7990	−1.06180	−0.530899	+	0.847435i	$0.678146\pi$
$282$	0	0
$283$	14.5964	0.867664	0.433832	−	0.900994i	$-0.357161\pi$
$283$	14.5964	0.867664	0.433832	+	0.900994i	$0.357161\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−14.7821	−0.872558
$288$	0	0
$289$	−9.00000	−0.529412
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−0.585786	−0.0342220	−0.0171110	−	0.999854i	$-0.505447\pi$
$293$	−0.585786	−0.0342220	−0.0171110	+	0.999854i	$0.505447\pi$
$294$	0	0
$295$	−0.634051	−0.0369159
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−23.0698	−1.33416
$300$	0	0
$301$	−4.00000	−0.230556
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0.627417	0.0359258
$306$	0	0
$307$	4.51528	0.257701	0.128850	−	0.991664i	$-0.458871\pi$
$307$	4.51528	0.257701	0.128850	+	0.991664i	$0.458871\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−24.6005	−1.39497	−0.697484	−	0.716600i	$-0.745697\pi$
$311$	−24.6005	−1.39497	−0.697484	+	0.716600i	$0.745697\pi$
$312$	0	0
$313$	−31.3137	−1.76996	−0.884978	−	0.465633i	$-0.845827\pi$
$313$	−31.3137	−1.76996	−0.884978	+	0.465633i	$0.845827\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−15.4142	−0.865748	−0.432874	−	0.901454i	$-0.642501\pi$
$317$	−15.4142	−0.865748	−0.432874	+	0.901454i	$0.642501\pi$
$318$	0	0
$319$	−30.7235	−1.72018
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−17.8435	−0.992841
$324$	0	0
$325$	−15.8995	−0.881945
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−11.3137	−0.623745
$330$	0	0
$331$	−8.47343	−0.465742	−0.232871	−	0.972508i	$-0.574812\pi$
$331$	−8.47343	−0.465742	−0.232871	+	0.972508i	$0.574812\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.15932	0.0633402
$336$	0	0
$337$	−6.00000	−0.326841	−0.163420	−	0.986557i	$-0.552253\pi$
$337$	−6.00000	−0.326841	−0.163420	+	0.986557i	$0.552253\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−12.6863	−0.687001
$342$	0	0
$343$	1.26810	0.0684710
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−15.4930	−0.831710	−0.415855	−	0.909431i	$-0.636518\pi$
$347$	−15.4930	−0.831710	−0.415855	+	0.909431i	$0.636518\pi$
$348$	0	0
$349$	2.72792	0.146022	0.0730112	−	0.997331i	$-0.476739\pi$
$349$	2.72792	0.146022	0.0730112	+	0.997331i	$0.476739\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−22.9706	−1.22260	−0.611300	−	0.791399i	$-0.709353\pi$
$353$	−22.9706	−1.22260	−0.611300	+	0.791399i	$0.709353\pi$
$354$	0	0
$355$	4.70099	0.249502
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0.634051	0.0334639	0.0167320	−	0.999860i	$-0.494674\pi$
$359$	0.634051	0.0334639	0.0167320	+	0.999860i	$0.494674\pi$
$360$	0	0
$361$	20.7990	1.09468
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3.79899	0.198848
$366$	0	0
$367$	9.18440	0.479422	0.239711	−	0.970844i	$-0.422947\pi$
$367$	9.18440	0.479422	0.239711	+	0.970844i	$0.422947\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	16.9469	0.879837
$372$	0	0
$373$	−6.72792	−0.348359	−0.174179	−	0.984714i	$-0.555727\pi$
$373$	−6.72792	−0.348359	−0.174179	+	0.984714i	$0.555727\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−25.3137	−1.30372
$378$	0	0
$379$	−26.3170	−1.35181	−0.675906	−	0.736988i	$-0.736247\pi$
$379$	−26.3170	−1.35181	−0.675906	+	0.736988i	$0.736247\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−20.9050	−1.06820	−0.534098	−	0.845423i	$-0.679349\pi$
$383$	−20.9050	−1.06820	−0.534098	+	0.845423i	$0.679349\pi$
$384$	0	0
$385$	8.97056	0.457182
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−20.3848	−1.03355	−0.516775	−	0.856121i	$-0.672867\pi$
$389$	−20.3848	−1.03355	−0.516775	+	0.856121i	$0.672867\pi$
$390$	0	0
$391$	19.1116	0.966517
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−8.65914	−0.435688
$396$	0	0
$397$	−5.07107	−0.254510	−0.127255	−	0.991870i	$-0.540617\pi$
$397$	−5.07107	−0.254510	−0.127255	+	0.991870i	$0.540617\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−7.51472	−0.375267	−0.187634	−	0.982239i	$-0.560082\pi$
$401$	−7.51472	−0.375267	−0.187634	+	0.982239i	$0.560082\pi$
$402$	0	0
$403$	−10.4525	−0.520676
$404$	0	0
$405$	0	0
$406$	0	0
$407$	37.5892	1.86323
$408$	0	0
$409$	−31.3137	−1.54836	−0.774182	−	0.632964i	$-0.781838\pi$
$409$	−31.3137	−1.54836	−0.774182	+	0.632964i	$0.781838\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−4.00000	−0.196827
$414$	0	0
$415$	−8.02509	−0.393936
$416$	0	0
$417$	0	0
$418$	0	0
$419$	12.4316	0.607322	0.303661	−	0.952780i	$-0.401791\pi$
$419$	12.4316	0.607322	0.303661	+	0.952780i	$0.401791\pi$
$420$	0	0
$421$	8.58579	0.418446	0.209223	−	0.977868i	$-0.432907\pi$
$421$	8.58579	0.418446	0.209223	+	0.977868i	$0.432907\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	13.1716	0.638915
$426$	0	0
$427$	3.95815	0.191548
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	25.4558	1.22333	0.611665	−	0.791117i	$-0.290500\pi$
$433$	25.4558	1.22333	0.611665	+	0.791117i	$0.290500\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−42.6274	−2.03915
$438$	0	0
$439$	−38.1145	−1.81911	−0.909553	−	0.415588i	$-0.863576\pi$
$439$	−38.1145	−1.81911	−0.909553	+	0.415588i	$0.863576\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.97908	0.0940287	0.0470144	−	0.998894i	$-0.485029\pi$
$443$	1.97908	0.0940287	0.0470144	+	0.998894i	$0.485029\pi$
$444$	0	0
$445$	2.82843	0.134080
$446$	0	0
$447$	0	0
$448$	0	0
$449$	6.14214	0.289865	0.144933	−	0.989442i	$-0.453703\pi$
$449$	6.14214	0.289865	0.144933	+	0.989442i	$0.453703\pi$
$450$	0	0
$451$	16.5754	0.780507
$452$	0	0
$453$	0	0
$454$	0	0
$455$	7.39104	0.346497
$456$	0	0
$457$	−26.6274	−1.24558	−0.622789	−	0.782390i	$-0.714001\pi$
$457$	−26.6274	−1.24558	−0.622789	+	0.782390i	$0.714001\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−38.2426	−1.78114	−0.890569	−	0.454849i	$-0.849693\pi$
$461$	−38.2426	−1.78114	−0.890569	+	0.454849i	$0.849693\pi$
$462$	0	0
$463$	−35.6871	−1.65852	−0.829260	−	0.558864i	$-0.811238\pi$
$463$	−35.6871	−1.65852	−0.829260	+	0.558864i	$0.811238\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	38.0376	1.76017	0.880084	−	0.474817i	$-0.157486\pi$
$467$	38.0376	1.76017	0.880084	+	0.474817i	$0.157486\pi$
$468$	0	0
$469$	7.31371	0.337716
$470$	0	0
$471$	0	0
$472$	0	0
$473$	4.48528	0.206233
$474$	0	0
$475$	−29.3784	−1.34798
$476$	0	0
$477$	0	0
$478$	0	0
$479$	35.1618	1.60658	0.803292	−	0.595585i	$-0.203080\pi$
$479$	35.1618	1.60658	0.803292	+	0.595585i	$0.203080\pi$
$480$	0	0
$481$	30.9706	1.41214
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3.02944	−0.137560
$486$	0	0
$487$	−6.23172	−0.282386	−0.141193	−	0.989982i	$-0.545094\pi$
$487$	−6.23172	−0.282386	−0.141193	+	0.989982i	$0.545094\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	5.04054	0.227477	0.113738	−	0.993511i	$-0.463717\pi$
$491$	5.04054	0.227477	0.113738	+	0.993511i	$0.463717\pi$
$492$	0	0
$493$	20.9706	0.944467
$494$	0	0
$495$	0	0
$496$	0	0
$497$	29.6569	1.33029
$498$	0	0
$499$	17.6578	0.790473	0.395237	−	0.918579i	$-0.370663\pi$
$499$	17.6578	0.790473	0.395237	+	0.918579i	$0.370663\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	22.8072	1.01692	0.508460	−	0.861085i	$-0.330215\pi$
$503$	22.8072	1.01692	0.508460	+	0.861085i	$0.330215\pi$
$504$	0	0
$505$	10.0000	0.444994
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−26.2426	−1.16318	−0.581592	−	0.813480i	$-0.697570\pi$
$509$	−26.2426	−1.16318	−0.581592	+	0.813480i	$0.697570\pi$
$510$	0	0
$511$	23.9665	1.06021
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−8.28772	−0.365201
$516$	0	0
$517$	12.6863	0.557942
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−6.34315	−0.277898	−0.138949	−	0.990300i	$-0.544372\pi$
$521$	−6.34315	−0.277898	−0.138949	+	0.990300i	$0.544372\pi$
$522$	0	0
$523$	14.2249	0.622013	0.311007	−	0.950408i	$-0.399334\pi$
$523$	14.2249	0.622013	0.311007	+	0.950408i	$0.399334\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.65914	0.377198
$528$	0	0
$529$	22.6569	0.985081
$530$	0	0
$531$	0	0
$532$	0	0
$533$	13.6569	0.591544
$534$	0	0
$535$	1.15932	0.0501216
$536$	0	0
$537$	0	0
$538$	0	0
$539$	27.5851	1.18817
$540$	0	0
$541$	10.2426	0.440366	0.220183	−	0.975459i	$-0.429335\pi$
$541$	10.2426	0.440366	0.220183	+	0.975459i	$0.429335\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	7.65685	0.327984
$546$	0	0
$547$	18.9259	0.809214	0.404607	−	0.914491i	$-0.367408\pi$
$547$	18.9259	0.809214	0.404607	+	0.914491i	$0.367408\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−46.7736	−1.99262
$552$	0	0
$553$	−54.6274	−2.32299
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−11.4142	−0.483636	−0.241818	−	0.970322i	$-0.577744\pi$
$557$	−11.4142	−0.483636	−0.241818	+	0.970322i	$0.577744\pi$
$558$	0	0
$559$	3.69552	0.156304
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−14.9678	−0.630817	−0.315408	−	0.948956i	$-0.602141\pi$
$563$	−14.9678	−0.630817	−0.315408	+	0.948956i	$0.602141\pi$
$564$	0	0
$565$	−4.48528	−0.188697
$566$	0	0
$567$	0	0
$568$	0	0
$569$	25.6569	1.07559	0.537796	−	0.843075i	$-0.319257\pi$
$569$	25.6569	1.07559	0.537796	+	0.843075i	$0.319257\pi$
$570$	0	0
$571$	27.5851	1.15440	0.577200	−	0.816603i	$-0.304145\pi$
$571$	27.5851	1.15440	0.577200	+	0.816603i	$0.304145\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	31.4663	1.31224
$576$	0	0
$577$	17.3137	0.720779	0.360390	−	0.932802i	$-0.382644\pi$
$577$	17.3137	0.720779	0.360390	+	0.932802i	$0.382644\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−50.6274	−2.10038
$582$	0	0
$583$	−19.0029	−0.787018
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−18.5545	−0.765827	−0.382913	−	0.923784i	$-0.625079\pi$
$587$	−18.5545	−0.765827	−0.382913	+	0.923784i	$0.625079\pi$
$588$	0	0
$589$	−19.3137	−0.795807
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−9.31371	−0.382468	−0.191234	−	0.981544i	$-0.561249\pi$
$593$	−9.31371	−0.382468	−0.191234	+	0.981544i	$0.561249\pi$
$594$	0	0
$595$	−6.12293	−0.251016
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−45.5055	−1.85931	−0.929653	−	0.368437i	$-0.879893\pi$
$599$	−45.5055	−1.85931	−0.929653	+	0.368437i	$0.879893\pi$
$600$	0	0
$601$	20.8284	0.849609	0.424805	−	0.905285i	$-0.360343\pi$
$601$	20.8284	0.849609	0.424805	+	0.905285i	$0.360343\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3.61522	−0.146980
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	10.4525	0.422863
$612$	0	0
$613$	−39.4142	−1.59193	−0.795963	−	0.605346i	$-0.793035\pi$
$613$	−39.4142	−1.59193	−0.795963	+	0.605346i	$0.793035\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−38.4853	−1.54936	−0.774680	−	0.632354i	$-0.782089\pi$
$617$	−38.4853	−1.54936	−0.774680	+	0.632354i	$0.782089\pi$
$618$	0	0
$619$	−4.14386	−0.166556	−0.0832779	−	0.996526i	$-0.526539\pi$
$619$	−4.14386	−0.166556	−0.0832779	+	0.996526i	$0.526539\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	17.8435	0.714886
$624$	0	0
$625$	19.9706	0.798823
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−25.6569	−1.02301
$630$	0	0
$631$	−14.1480	−0.563224	−0.281612	−	0.959528i	$-0.590869\pi$
$631$	−14.1480	−0.563224	−0.281612	+	0.959528i	$0.590869\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	6.86577	0.272460
$636$	0	0
$637$	22.7279	0.900513
$638$	0	0
$639$	0	0
$640$	0	0
$641$	38.1421	1.50652	0.753262	−	0.657721i	$-0.228479\pi$
$641$	38.1421	1.50652	0.753262	+	0.657721i	$0.228479\pi$
$642$	0	0
$643$	18.9259	0.746366	0.373183	−	0.927758i	$-0.378266\pi$
$643$	18.9259	0.746366	0.373183	+	0.927758i	$0.378266\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	21.5391	0.846788	0.423394	−	0.905946i	$-0.360839\pi$
$647$	21.5391	0.846788	0.423394	+	0.905946i	$0.360839\pi$
$648$	0	0
$649$	4.48528	0.176063
$650$	0	0
$651$	0	0
$652$	0	0
$653$	6.44365	0.252160	0.126080	−	0.992020i	$-0.459760\pi$
$653$	6.44365	0.252160	0.126080	+	0.992020i	$0.459760\pi$
$654$	0	0
$655$	8.55035	0.334090
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1.45381	0.0566324	0.0283162	−	0.999599i	$-0.490985\pi$
$659$	1.45381	0.0566324	0.0283162	+	0.999599i	$0.490985\pi$
$660$	0	0
$661$	29.0711	1.13073	0.565367	−	0.824840i	$-0.308735\pi$
$661$	29.0711	1.13073	0.565367	+	0.824840i	$0.308735\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	13.6569	0.529590
$666$	0	0
$667$	50.0977	1.93979
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.43835	−0.171341
$672$	0	0
$673$	−26.8284	−1.03416	−0.517080	−	0.855937i	$-0.672981\pi$
$673$	−26.8284	−1.03416	−0.517080	+	0.855937i	$0.672981\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	13.5563	0.521013	0.260506	−	0.965472i	$-0.416110\pi$
$677$	13.5563	0.521013	0.260506	+	0.965472i	$0.416110\pi$
$678$	0	0
$679$	−19.1116	−0.733437
$680$	0	0
$681$	0	0
$682$	0	0
$683$	32.8113	1.25549	0.627745	−	0.778419i	$-0.283978\pi$
$683$	32.8113	1.25549	0.627745	+	0.778419i	$0.283978\pi$
$684$	0	0
$685$	7.59798	0.290304
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−15.6569	−0.596479
$690$	0	0
$691$	−40.7276	−1.54935	−0.774676	−	0.632358i	$-0.782087\pi$
$691$	−40.7276	−1.54935	−0.774676	+	0.632358i	$0.782087\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−11.8294	−0.448714
$696$	0	0
$697$	−11.3137	−0.428537
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−24.8701	−0.939329	−0.469665	−	0.882845i	$-0.655625\pi$
$701$	−24.8701	−0.939329	−0.469665	+	0.882845i	$0.655625\pi$
$702$	0	0
$703$	57.2261	2.15832
$704$	0	0
$705$	0	0
$706$	0	0
$707$	63.0864	2.37261
$708$	0	0
$709$	23.6985	0.890015	0.445008	−	0.895527i	$-0.353201\pi$
$709$	23.6985	0.890015	0.445008	+	0.895527i	$0.353201\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	20.6863	0.774708
$714$	0	0
$715$	−8.28772	−0.309943
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−29.5641	−1.10256	−0.551278	−	0.834321i	$-0.685860\pi$
$719$	−29.5641	−1.10256	−0.551278	+	0.834321i	$0.685860\pi$
$720$	0	0
$721$	−52.2843	−1.94717
$722$	0	0
$723$	0	0
$724$	0	0
$725$	34.5269	1.28230
$726$	0	0
$727$	22.0643	0.818320	0.409160	−	0.912463i	$-0.365822\pi$
$727$	22.0643	0.818320	0.409160	+	0.912463i	$0.365822\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−3.06147	−0.113232
$732$	0	0
$733$	−18.5269	−0.684307	−0.342154	−	0.939644i	$-0.611156\pi$
$733$	−18.5269	−0.684307	−0.342154	+	0.939644i	$0.611156\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8.20101	−0.302088
$738$	0	0
$739$	−49.7582	−1.83038	−0.915192	−	0.403018i	$-0.867961\pi$
$739$	−49.7582	−1.83038	−0.915192	+	0.403018i	$0.867961\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	27.1367	0.995550	0.497775	−	0.867306i	$-0.334151\pi$
$743$	27.1367	0.995550	0.497775	+	0.867306i	$0.334151\pi$
$744$	0	0
$745$	1.02944	0.0377157
$746$	0	0
$747$	0	0
$748$	0	0
$749$	7.31371	0.267237
$750$	0	0
$751$	−14.7821	−0.539405	−0.269703	−	0.962944i	$-0.586925\pi$
$751$	−14.7821	−0.539405	−0.269703	+	0.962944i	$0.586925\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−0.371418	−0.0135173
$756$	0	0
$757$	−18.9289	−0.687984	−0.343992	−	0.938973i	$-0.611779\pi$
$757$	−18.9289	−0.687984	−0.343992	+	0.938973i	$0.611779\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−34.6274	−1.25524	−0.627621	−	0.778519i	$-0.715971\pi$
$761$	−34.6274	−1.25524	−0.627621	+	0.778519i	$0.715971\pi$
$762$	0	0
$763$	48.3044	1.74874
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.69552	0.133437
$768$	0	0
$769$	5.17157	0.186492	0.0932458	−	0.995643i	$-0.470276\pi$
$769$	5.17157	0.186492	0.0932458	+	0.995643i	$0.470276\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−39.6985	−1.42786	−0.713928	−	0.700219i	$-0.753085\pi$
$773$	−39.6985	−1.42786	−0.713928	+	0.700219i	$0.753085\pi$
$774$	0	0
$775$	14.2568	0.512120
$776$	0	0
$777$	0	0
$778$	0	0
$779$	25.2346	0.904123
$780$	0	0
$781$	−33.2548	−1.18995
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−1.31371	−0.0468883
$786$	0	0
$787$	7.94816	0.283321	0.141661	−	0.989915i	$-0.454756\pi$
$787$	7.94816	0.283321	0.141661	+	0.989915i	$0.454756\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−28.2960	−1.00609
$792$	0	0
$793$	−3.65685	−0.129859
$794$	0	0
$795$	0	0
$796$	0	0
$797$	39.6985	1.40619	0.703096	−	0.711095i	$-0.251800\pi$
$797$	39.6985	1.40619	0.703096	+	0.711095i	$0.251800\pi$
$798$	0	0
$799$	−8.65914	−0.306338
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−26.8741	−0.948366
$804$	0	0
$805$	−14.6274	−0.515549
$806$	0	0
$807$	0	0
$808$	0	0
$809$	24.6863	0.867924	0.433962	−	0.900931i	$-0.357115\pi$
$809$	24.6863	0.867924	0.433962	+	0.900931i	$0.357115\pi$
$810$	0	0
$811$	−2.35049	−0.0825370	−0.0412685	−	0.999148i	$-0.513140\pi$
$811$	−2.35049	−0.0825370	−0.0412685	+	0.999148i	$0.513140\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	3.17025	0.111049
$816$	0	0
$817$	6.82843	0.238896
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−38.5269	−1.34460	−0.672299	−	0.740279i	$-0.734693\pi$
$821$	−38.5269	−1.34460	−0.672299	+	0.740279i	$0.734693\pi$
$822$	0	0
$823$	22.0643	0.769114	0.384557	−	0.923101i	$-0.374354\pi$
$823$	22.0643	0.769114	0.384557	+	0.923101i	$0.374354\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	18.5545	0.645204	0.322602	−	0.946535i	$-0.395443\pi$
$827$	18.5545	0.645204	0.322602	+	0.946535i	$0.395443\pi$
$828$	0	0
$829$	−12.3848	−0.430141	−0.215071	−	0.976599i	$-0.568998\pi$
$829$	−12.3848	−0.430141	−0.215071	+	0.976599i	$0.568998\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−18.8284	−0.652366
$834$	0	0
$835$	−2.16478	−0.0749155
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−29.4554	−1.01691	−0.508456	−	0.861088i	$-0.669784\pi$
$839$	−29.4554	−1.01691	−0.508456	+	0.861088i	$0.669784\pi$
$840$	0	0
$841$	25.9706	0.895537
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0.786797	0.0270666
$846$	0	0
$847$	−22.8072	−0.783663
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−61.2931	−2.10110
$852$	0	0
$853$	−32.1005	−1.09910	−0.549550	−	0.835461i	$-0.685201\pi$
$853$	−32.1005	−1.09910	−0.549550	+	0.835461i	$0.685201\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−24.2843	−0.829535	−0.414767	−	0.909927i	$-0.636137\pi$
$857$	−24.2843	−0.829535	−0.414767	+	0.909927i	$0.636137\pi$
$858$	0	0
$859$	11.9063	0.406238	0.203119	−	0.979154i	$-0.434892\pi$
$859$	11.9063	0.406238	0.203119	+	0.979154i	$0.434892\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0.525265	0.0178802	0.00894011	−	0.999960i	$-0.497154\pi$
$863$	0.525265	0.0178802	0.00894011	+	0.999960i	$0.497154\pi$
$864$	0	0
$865$	−11.2548	−0.382676
$866$	0	0
$867$	0	0
$868$	0	0
$869$	61.2548	2.07793
$870$	0	0
$871$	−6.75699	−0.228952
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−20.9050	−0.706718
$876$	0	0
$877$	49.5563	1.67340	0.836700	−	0.547662i	$-0.184482\pi$
$877$	49.5563	1.67340	0.836700	+	0.547662i	$0.184482\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	39.9411	1.34565	0.672825	−	0.739801i	$-0.265081\pi$
$881$	39.9411	1.34565	0.672825	+	0.739801i	$0.265081\pi$
$882$	0	0
$883$	−29.3784	−0.988663	−0.494332	−	0.869273i	$-0.664587\pi$
$883$	−29.3784	−0.988663	−0.494332	+	0.869273i	$0.664587\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	17.2095	0.577838	0.288919	−	0.957354i	$-0.406704\pi$
$887$	17.2095	0.577838	0.288919	+	0.957354i	$0.406704\pi$
$888$	0	0
$889$	43.3137	1.45270
$890$	0	0
$891$	0	0
$892$	0	0
$893$	19.3137	0.646309
$894$	0	0
$895$	6.75699	0.225861
$896$	0	0
$897$	0	0
$898$	0	0
$899$	22.6984	0.757033
$900$	0	0
$901$	12.9706	0.432112
$902$	0	0
$903$	0	0
$904$	0	0
$905$	8.34315	0.277336
$906$	0	0
$907$	−45.0572	−1.49610	−0.748050	−	0.663643i	$-0.769010\pi$
$907$	−45.0572	−1.49610	−0.748050	+	0.663643i	$0.769010\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	23.9665	0.794045	0.397022	−	0.917809i	$-0.370044\pi$
$911$	23.9665	0.794045	0.397022	+	0.917809i	$0.370044\pi$
$912$	0	0
$913$	56.7696	1.87880
$914$	0	0
$915$	0	0
$916$	0	0
$917$	53.9411	1.78129
$918$	0	0
$919$	19.0029	0.626846	0.313423	−	0.949614i	$-0.398524\pi$
$919$	19.0029	0.626846	0.313423	+	0.949614i	$0.398524\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−27.3994	−0.901861
$924$	0	0
$925$	−42.2426	−1.38893
$926$	0	0
$927$	0	0
$928$	0	0
$929$	42.8284	1.40516	0.702578	−	0.711607i	$-0.252032\pi$
$929$	42.8284	1.40516	0.702578	+	0.711607i	$0.252032\pi$
$930$	0	0
$931$	41.9957	1.37635
$932$	0	0
$933$	0	0
$934$	0	0
$935$	6.86577	0.224535
$936$	0	0
$937$	43.4558	1.41964	0.709820	−	0.704383i	$-0.248776\pi$
$937$	43.4558	1.41964	0.709820	+	0.704383i	$0.248776\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−2.92893	−0.0954805	−0.0477402	−	0.998860i	$-0.515202\pi$
$941$	−2.92893	−0.0954805	−0.0477402	+	0.998860i	$0.515202\pi$
$942$	0	0
$943$	−27.0279	−0.880151
$944$	0	0
$945$	0	0
$946$	0	0
$947$	21.4621	0.697426	0.348713	−	0.937230i	$-0.386619\pi$
$947$	21.4621	0.697426	0.348713	+	0.937230i	$0.386619\pi$
$948$	0	0
$949$	−22.1421	−0.718764
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−11.0294	−0.357279	−0.178639	−	0.983915i	$-0.557170\pi$
$953$	−11.0294	−0.357279	−0.178639	+	0.983915i	$0.557170\pi$
$954$	0	0
$955$	6.86577	0.222171
$956$	0	0
$957$	0	0
$958$	0	0
$959$	47.9329	1.54784
$960$	0	0
$961$	−21.6274	−0.697659
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−0.284271	−0.00915102
$966$	0	0
$967$	11.0866	0.356520	0.178260	−	0.983983i	$-0.442953\pi$
$967$	11.0866	0.356520	0.178260	+	0.983983i	$0.442953\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−57.6745	−1.85086	−0.925431	−	0.378916i	$-0.876297\pi$
$971$	−57.6745	−1.85086	−0.925431	+	0.378916i	$0.876297\pi$
$972$	0	0
$973$	−74.6274	−2.39245
$974$	0	0
$975$	0	0
$976$	0	0
$977$	47.1127	1.50727	0.753634	−	0.657294i	$-0.228299\pi$
$977$	47.1127	1.50727	0.753634	+	0.657294i	$0.228299\pi$
$978$	0	0
$979$	−20.0083	−0.639469
$980$	0	0
$981$	0	0
$982$	0	0
$983$	46.7736	1.49185	0.745924	−	0.666031i	$-0.232008\pi$
$983$	46.7736	1.49185	0.745924	+	0.666031i	$0.232008\pi$
$984$	0	0
$985$	−6.28427	−0.200234
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−7.31371	−0.232562
$990$	0	0
$991$	27.0279	0.858571	0.429285	−	0.903169i	$-0.358765\pi$
$991$	27.0279	0.858571	0.429285	+	0.903169i	$0.358765\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	7.23719	0.229434
$996$	0	0
$997$	28.3848	0.898955	0.449477	−	0.893292i	$-0.351610\pi$
$997$	28.3848	0.898955	0.449477	+	0.893292i	$0.351610\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9216.2.a.w.1.3		4
3.2	odd	2		1024.2.a.i.1.2		4
4.3	odd	2	inner	9216.2.a.w.1.4		4
8.3	odd	2		9216.2.a.bp.1.2		4
8.5	even	2		9216.2.a.bp.1.1		4
12.11	even	2		1024.2.a.i.1.3		4
24.5	odd	2		1024.2.a.h.1.3		4
24.11	even	2		1024.2.a.h.1.2		4
32.3	odd	8		4608.2.k.bd.1153.3		8
32.5	even	8		4608.2.k.bi.3457.1		8
32.11	odd	8		4608.2.k.bd.3457.4		8
32.13	even	8		4608.2.k.bi.1153.2		8
32.19	odd	8		4608.2.k.bi.1153.1		8
32.21	even	8		4608.2.k.bd.3457.3		8
32.27	odd	8		4608.2.k.bi.3457.2		8
32.29	even	8		4608.2.k.bd.1153.4		8
48.5	odd	4		1024.2.b.g.513.6		8
48.11	even	4		1024.2.b.g.513.4		8
48.29	odd	4		1024.2.b.g.513.3		8
48.35	even	4		1024.2.b.g.513.5		8
96.5	odd	8		512.2.e.i.385.3	yes	8
96.11	even	8		512.2.e.j.385.3	yes	8
96.29	odd	8		512.2.e.j.129.2	yes	8
96.35	even	8		512.2.e.j.129.3	yes	8
96.53	odd	8		512.2.e.j.385.2	yes	8
96.59	even	8		512.2.e.i.385.2	yes	8
96.77	odd	8		512.2.e.i.129.3	yes	8
96.83	even	8		512.2.e.i.129.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
512.2.e.i.129.2	✓	8	96.83	even	8
512.2.e.i.129.3	yes	8	96.77	odd	8
512.2.e.i.385.2	yes	8	96.59	even	8
512.2.e.i.385.3	yes	8	96.5	odd	8
512.2.e.j.129.2	yes	8	96.29	odd	8
512.2.e.j.129.3	yes	8	96.35	even	8
512.2.e.j.385.2	yes	8	96.53	odd	8
512.2.e.j.385.3	yes	8	96.11	even	8
1024.2.a.h.1.2		4	24.11	even	2
1024.2.a.h.1.3		4	24.5	odd	2
1024.2.a.i.1.2		4	3.2	odd	2
1024.2.a.i.1.3		4	12.11	even	2
1024.2.b.g.513.3		8	48.29	odd	4
1024.2.b.g.513.4		8	48.11	even	4
1024.2.b.g.513.5		8	48.35	even	4
1024.2.b.g.513.6		8	48.5	odd	4
4608.2.k.bd.1153.3		8	32.3	odd	8
4608.2.k.bd.1153.4		8	32.29	even	8
4608.2.k.bd.3457.3		8	32.21	even	8
4608.2.k.bd.3457.4		8	32.11	odd	8
4608.2.k.bi.1153.1		8	32.19	odd	8
4608.2.k.bi.1153.2		8	32.13	even	8
4608.2.k.bi.3457.1		8	32.5	even	8
4608.2.k.bi.3457.2		8	32.27	odd	8
9216.2.a.w.1.3		4	1.1	even	1	trivial
9216.2.a.w.1.4		4	4.3	odd	2	inner
9216.2.a.bp.1.1		4	8.5	even	2
9216.2.a.bp.1.2		4	8.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 9216 = 2^{10} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9216.a (trivial)

\(f(q)\)	\(=\)	\(q-0.585786 q^{5} -3.69552 q^{7} +4.14386 q^{11} +3.41421 q^{13} -2.82843 q^{17} +6.30864 q^{19} -6.75699 q^{23} -4.65685 q^{25} -7.41421 q^{29} -3.06147 q^{31} +2.16478 q^{35} +9.07107 q^{37} +4.00000 q^{41} +1.08239 q^{43} +3.06147 q^{47} +6.65685 q^{49} -4.58579 q^{53} -2.42742 q^{55} +1.08239 q^{59} -1.07107 q^{61} -2.00000 q^{65} -1.97908 q^{67} -8.02509 q^{71} -6.48528 q^{73} -15.3137 q^{77} +14.7821 q^{79} +13.6997 q^{83} +1.65685 q^{85} -4.82843 q^{89} -12.6173 q^{91} -3.69552 q^{95} +5.17157 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{5} + 8 q^{13} + 4 q^{25} - 24 q^{29} + 8 q^{37} + 16 q^{41} + 4 q^{49} - 24 q^{53} + 24 q^{61} - 8 q^{65} + 8 q^{73} - 16 q^{77} - 16 q^{85} - 8 q^{89} + 32 q^{97}+O(q^{100}) \) 4 * q - 8 * q^5 + 8 * q^13 + 4 * q^25 - 24 * q^29 + 8 * q^37 + 16 * q^41 + 4 * q^49 - 24 * q^53 + 24 * q^61 - 8 * q^65 + 8 * q^73 - 16 * q^77 - 16 * q^85 - 8 * q^89 + 32 * q^97

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