LMFDB - Embedded newform 9408.2.a.ea.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9408,2,Mod(1,9408)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9408, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9408.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9408 = 2^{6} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9408.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:	$75.1232582216$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4704)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	9408.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.41421 q^{5} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.41421 q^{5} +1.00000 q^{9} -2.82843 q^{11} +1.41421 q^{13} +1.41421 q^{15} +1.41421 q^{17} -2.82843 q^{23} -3.00000 q^{25} +1.00000 q^{27} -4.00000 q^{31} -2.82843 q^{33} -4.00000 q^{37} +1.41421 q^{39} -1.41421 q^{41} -5.65685 q^{43} +1.41421 q^{45} -12.0000 q^{47} +1.41421 q^{51} -10.0000 q^{53} -4.00000 q^{55} +1.41421 q^{61} +2.00000 q^{65} -11.3137 q^{67} -2.82843 q^{69} +2.82843 q^{71} +12.7279 q^{73} -3.00000 q^{75} -11.3137 q^{79} +1.00000 q^{81} +4.00000 q^{83} +2.00000 q^{85} +7.07107 q^{89} -4.00000 q^{93} -9.89949 q^{97} -2.82843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{9} - 6 q^{25} + 2 q^{27} - 8 q^{31} - 8 q^{37} - 24 q^{47} - 20 q^{53} - 8 q^{55} + 4 q^{65} - 6 q^{75} + 2 q^{81} + 8 q^{83} + 4 q^{85} - 8 q^{93}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^9 - 6 * q^25 + 2 * q^27 - 8 * q^31 - 8 * q^37 - 24 * q^47 - 20 * q^53 - 8 * q^55 + 4 * q^65 - 6 * q^75 + 2 * q^81 + 8 * q^83 + 4 * q^85 - 8 * q^93

Display 100 coefficients 10 coefficients

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.41421	0.632456	0.316228	−	0.948683i	$-0.397584\pi$
$5$	1.41421	0.632456	0.316228	+	0.948683i	$0.397584\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.82843	−0.852803	−0.426401	−	0.904534i	$-0.640219\pi$
$11$	−2.82843	−0.852803	−0.426401	+	0.904534i	$0.640219\pi$
$12$	0	0
$13$	1.41421	0.392232	0.196116	−	0.980581i	$-0.437167\pi$
$13$	1.41421	0.392232	0.196116	+	0.980581i	$0.437167\pi$
$14$	0	0
$15$	1.41421	0.365148
$16$	0	0
$17$	1.41421	0.342997	0.171499	−	0.985184i	$-0.445139\pi$
$17$	1.41421	0.342997	0.171499	+	0.985184i	$0.445139\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.82843	−0.589768	−0.294884	−	0.955533i	$-0.595281\pi$
$23$	−2.82843	−0.589768	−0.294884	+	0.955533i	$0.595281\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	−2.82843	−0.492366
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	0	0
$39$	1.41421	0.226455
$40$	0	0
$41$	−1.41421	−0.220863	−0.110432	−	0.993884i	$-0.535223\pi$
$41$	−1.41421	−0.220863	−0.110432	+	0.993884i	$0.535223\pi$
$42$	0	0
$43$	−5.65685	−0.862662	−0.431331	−	0.902194i	$-0.641956\pi$
$43$	−5.65685	−0.862662	−0.431331	+	0.902194i	$0.641956\pi$
$44$	0	0
$45$	1.41421	0.210819
$46$	0	0
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	1.41421	0.198030
$52$	0	0
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	1.41421	0.181071	0.0905357	−	0.995893i	$-0.471142\pi$
$61$	1.41421	0.181071	0.0905357	+	0.995893i	$0.471142\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	−11.3137	−1.38219	−0.691095	−	0.722764i	$-0.742871\pi$
$67$	−11.3137	−1.38219	−0.691095	+	0.722764i	$0.742871\pi$
$68$	0	0
$69$	−2.82843	−0.340503
$70$	0	0
$71$	2.82843	0.335673	0.167836	−	0.985815i	$-0.446322\pi$
$71$	2.82843	0.335673	0.167836	+	0.985815i	$0.446322\pi$
$72$	0	0
$73$	12.7279	1.48969	0.744845	−	0.667237i	$-0.232523\pi$
$73$	12.7279	1.48969	0.744845	+	0.667237i	$0.232523\pi$
$74$	0	0
$75$	−3.00000	−0.346410
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−11.3137	−1.27289	−0.636446	−	0.771321i	$-0.719596\pi$
$79$	−11.3137	−1.27289	−0.636446	+	0.771321i	$0.719596\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	2.00000	0.216930
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7.07107	0.749532	0.374766	−	0.927119i	$-0.377723\pi$
$89$	7.07107	0.749532	0.374766	+	0.927119i	$0.377723\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−4.00000	−0.414781
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−9.89949	−1.00514	−0.502571	−	0.864536i	$-0.667612\pi$
$97$	−9.89949	−1.00514	−0.502571	+	0.864536i	$0.667612\pi$
$98$	0	0
$99$	−2.82843	−0.284268
$100$	0	0
$101$	−7.07107	−0.703598	−0.351799	−	0.936076i	$-0.614430\pi$
$101$	−7.07107	−0.703598	−0.351799	+	0.936076i	$0.614430\pi$
$102$	0	0
$103$	−12.0000	−1.18240	−0.591198	−	0.806527i	$-0.701345\pi$
$103$	−12.0000	−1.18240	−0.591198	+	0.806527i	$0.701345\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	19.7990	1.91404	0.957020	−	0.290021i	$-0.0936623\pi$
$107$	19.7990	1.91404	0.957020	+	0.290021i	$0.0936623\pi$
$108$	0	0
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	0	0
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	−4.00000	−0.373002
$116$	0	0
$117$	1.41421	0.130744
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−3.00000	−0.272727
$122$	0	0
$123$	−1.41421	−0.127515
$124$	0	0
$125$	−11.3137	−1.01193
$126$	0	0
$127$	11.3137	1.00393	0.501965	−	0.864888i	$-0.332611\pi$
$127$	11.3137	1.00393	0.501965	+	0.864888i	$0.332611\pi$
$128$	0	0
$129$	−5.65685	−0.498058
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	1.41421	0.121716
$136$	0	0
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	−12.0000	−1.01058
$142$	0	0
$143$	−4.00000	−0.334497
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−11.3137	−0.920697	−0.460348	−	0.887738i	$-0.652275\pi$
$151$	−11.3137	−0.920697	−0.460348	+	0.887738i	$0.652275\pi$
$152$	0	0
$153$	1.41421	0.114332
$154$	0	0
$155$	−5.65685	−0.454369
$156$	0	0
$157$	4.24264	0.338600	0.169300	−	0.985565i	$-0.445849\pi$
$157$	4.24264	0.338600	0.169300	+	0.985565i	$0.445849\pi$
$158$	0	0
$159$	−10.0000	−0.793052
$160$	0	0
$161$	0	0
$162$	0	0
$163$	16.9706	1.32924	0.664619	−	0.747183i	$-0.268594\pi$
$163$	16.9706	1.32924	0.664619	+	0.747183i	$0.268594\pi$
$164$	0	0
$165$	−4.00000	−0.311400
$166$	0	0
$167$	−20.0000	−1.54765	−0.773823	−	0.633402i	$-0.781658\pi$
$167$	−20.0000	−1.54765	−0.773823	+	0.633402i	$0.781658\pi$
$168$	0	0
$169$	−11.0000	−0.846154
$170$	0	0
$171$	0	0
$172$	0	0
$173$	7.07107	0.537603	0.268802	−	0.963196i	$-0.413372\pi$
$173$	7.07107	0.537603	0.268802	+	0.963196i	$0.413372\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−14.1421	−1.05703	−0.528516	−	0.848923i	$-0.677252\pi$
$179$	−14.1421	−1.05703	−0.528516	+	0.848923i	$0.677252\pi$
$180$	0	0
$181$	15.5563	1.15629	0.578147	−	0.815933i	$-0.303776\pi$
$181$	15.5563	1.15629	0.578147	+	0.815933i	$0.303776\pi$
$182$	0	0
$183$	1.41421	0.104542
$184$	0	0
$185$	−5.65685	−0.415900
$186$	0	0
$187$	−4.00000	−0.292509
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−8.48528	−0.613973	−0.306987	−	0.951714i	$-0.599321\pi$
$191$	−8.48528	−0.613973	−0.306987	+	0.951714i	$0.599321\pi$
$192$	0	0
$193$	−26.0000	−1.87152	−0.935760	−	0.352636i	$-0.885285\pi$
$193$	−26.0000	−1.87152	−0.935760	+	0.352636i	$0.885285\pi$
$194$	0	0
$195$	2.00000	0.143223
$196$	0	0
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	−11.3137	−0.798007
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−2.00000	−0.139686
$206$	0	0
$207$	−2.82843	−0.196589
$208$	0	0
$209$	0	0
$210$	0	0
$211$	5.65685	0.389434	0.194717	−	0.980859i	$-0.437621\pi$
$211$	5.65685	0.389434	0.194717	+	0.980859i	$0.437621\pi$
$212$	0	0
$213$	2.82843	0.193801
$214$	0	0
$215$	−8.00000	−0.545595
$216$	0	0
$217$	0	0
$218$	0	0
$219$	12.7279	0.860073
$220$	0	0
$221$	2.00000	0.134535
$222$	0	0
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	0	0
$225$	−3.00000	−0.200000
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	−7.07107	−0.467269	−0.233635	−	0.972324i	$-0.575062\pi$
$229$	−7.07107	−0.467269	−0.233635	+	0.972324i	$0.575062\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	12.0000	0.786146	0.393073	−	0.919507i	$-0.371412\pi$
$233$	12.0000	0.786146	0.393073	+	0.919507i	$0.371412\pi$
$234$	0	0
$235$	−16.9706	−1.10704
$236$	0	0
$237$	−11.3137	−0.734904
$238$	0	0
$239$	19.7990	1.28069	0.640345	−	0.768087i	$-0.278791\pi$
$239$	19.7990	1.28069	0.640345	+	0.768087i	$0.278791\pi$
$240$	0	0
$241$	−1.41421	−0.0910975	−0.0455488	−	0.998962i	$-0.514504\pi$
$241$	−1.41421	−0.0910975	−0.0455488	+	0.998962i	$0.514504\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	4.00000	0.253490
$250$	0	0
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	0	0
$253$	8.00000	0.502956
$254$	0	0
$255$	2.00000	0.125245
$256$	0	0
$257$	9.89949	0.617514	0.308757	−	0.951141i	$-0.400087\pi$
$257$	9.89949	0.617514	0.308757	+	0.951141i	$0.400087\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−14.1421	−0.872041	−0.436021	−	0.899937i	$-0.643613\pi$
$263$	−14.1421	−0.872041	−0.436021	+	0.899937i	$0.643613\pi$
$264$	0	0
$265$	−14.1421	−0.868744
$266$	0	0
$267$	7.07107	0.432742
$268$	0	0
$269$	9.89949	0.603583	0.301791	−	0.953374i	$-0.402415\pi$
$269$	9.89949	0.603583	0.301791	+	0.953374i	$0.402415\pi$
$270$	0	0
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	8.48528	0.511682
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	12.0000	0.715860	0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000	0.715860	0.357930	+	0.933748i	$0.383483\pi$
$282$	0	0
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−15.0000	−0.882353
$290$	0	0
$291$	−9.89949	−0.580319
$292$	0	0
$293$	−29.6985	−1.73500	−0.867502	−	0.497434i	$-0.834276\pi$
$293$	−29.6985	−1.73500	−0.867502	+	0.497434i	$0.834276\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−2.82843	−0.164122
$298$	0	0
$299$	−4.00000	−0.231326
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−7.07107	−0.406222
$304$	0	0
$305$	2.00000	0.114520
$306$	0	0
$307$	−8.00000	−0.456584	−0.228292	−	0.973593i	$-0.573314\pi$
$307$	−8.00000	−0.456584	−0.228292	+	0.973593i	$0.573314\pi$
$308$	0	0
$309$	−12.0000	−0.682656
$310$	0	0
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	12.7279	0.719425	0.359712	−	0.933063i	$-0.382875\pi$
$313$	12.7279	0.719425	0.359712	+	0.933063i	$0.382875\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	19.7990	1.10507
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−4.24264	−0.235339
$326$	0	0
$327$	4.00000	0.221201
$328$	0	0
$329$	0	0
$330$	0	0
$331$	33.9411	1.86557	0.932786	−	0.360429i	$-0.117370\pi$
$331$	33.9411	1.86557	0.932786	+	0.360429i	$0.117370\pi$
$332$	0	0
$333$	−4.00000	−0.219199
$334$	0	0
$335$	−16.0000	−0.874173
$336$	0	0
$337$	−8.00000	−0.435788	−0.217894	−	0.975972i	$-0.569919\pi$
$337$	−8.00000	−0.435788	−0.217894	+	0.975972i	$0.569919\pi$
$338$	0	0
$339$	−2.00000	−0.108625
$340$	0	0
$341$	11.3137	0.612672
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−4.00000	−0.215353
$346$	0	0
$347$	−25.4558	−1.36654	−0.683271	−	0.730165i	$-0.739443\pi$
$347$	−25.4558	−1.36654	−0.683271	+	0.730165i	$0.739443\pi$
$348$	0	0
$349$	26.8701	1.43832	0.719161	−	0.694844i	$-0.244527\pi$
$349$	26.8701	1.43832	0.719161	+	0.694844i	$0.244527\pi$
$350$	0	0
$351$	1.41421	0.0754851
$352$	0	0
$353$	−18.3848	−0.978523	−0.489261	−	0.872137i	$-0.662734\pi$
$353$	−18.3848	−0.978523	−0.489261	+	0.872137i	$0.662734\pi$
$354$	0	0
$355$	4.00000	0.212298
$356$	0	0
$357$	0	0
$358$	0	0
$359$	14.1421	0.746393	0.373197	−	0.927752i	$-0.378262\pi$
$359$	14.1421	0.746393	0.373197	+	0.927752i	$0.378262\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	−3.00000	−0.157459
$364$	0	0
$365$	18.0000	0.942163
$366$	0	0
$367$	−24.0000	−1.25279	−0.626395	−	0.779506i	$-0.715470\pi$
$367$	−24.0000	−1.25279	−0.626395	+	0.779506i	$0.715470\pi$
$368$	0	0
$369$	−1.41421	−0.0736210
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−2.00000	−0.103556	−0.0517780	−	0.998659i	$-0.516489\pi$
$373$	−2.00000	−0.103556	−0.0517780	+	0.998659i	$0.516489\pi$
$374$	0	0
$375$	−11.3137	−0.584237
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	11.3137	0.579619
$382$	0	0
$383$	−16.0000	−0.817562	−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000	−0.817562	−0.408781	+	0.912633i	$0.634046\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−5.65685	−0.287554
$388$	0	0
$389$	−8.00000	−0.405616	−0.202808	−	0.979219i	$-0.565007\pi$
$389$	−8.00000	−0.405616	−0.202808	+	0.979219i	$0.565007\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	0	0
$393$	−4.00000	−0.201773
$394$	0	0
$395$	−16.0000	−0.805047
$396$	0	0
$397$	−38.1838	−1.91639	−0.958194	−	0.286119i	$-0.907635\pi$
$397$	−38.1838	−1.91639	−0.958194	+	0.286119i	$0.907635\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−12.0000	−0.599251	−0.299626	−	0.954057i	$-0.596862\pi$
$401$	−12.0000	−0.599251	−0.299626	+	0.954057i	$0.596862\pi$
$402$	0	0
$403$	−5.65685	−0.281788
$404$	0	0
$405$	1.41421	0.0702728
$406$	0	0
$407$	11.3137	0.560800
$408$	0	0
$409$	29.6985	1.46850	0.734248	−	0.678882i	$-0.237535\pi$
$409$	29.6985	1.46850	0.734248	+	0.678882i	$0.237535\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	0	0
$413$	0	0
$414$	0	0
$415$	5.65685	0.277684
$416$	0	0
$417$	20.0000	0.979404
$418$	0	0
$419$	−24.0000	−1.17248	−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000	−1.17248	−0.586238	+	0.810139i	$0.699392\pi$
$420$	0	0
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	0	0
$423$	−12.0000	−0.583460
$424$	0	0
$425$	−4.24264	−0.205798
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−4.00000	−0.193122
$430$	0	0
$431$	−2.82843	−0.136241	−0.0681203	−	0.997677i	$-0.521700\pi$
$431$	−2.82843	−0.136241	−0.0681203	+	0.997677i	$0.521700\pi$
$432$	0	0
$433$	4.24264	0.203888	0.101944	−	0.994790i	$-0.467494\pi$
$433$	4.24264	0.203888	0.101944	+	0.994790i	$0.467494\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	19.7990	0.940678	0.470339	−	0.882486i	$-0.344132\pi$
$443$	19.7990	0.940678	0.470339	+	0.882486i	$0.344132\pi$
$444$	0	0
$445$	10.0000	0.474045
$446$	0	0
$447$	6.00000	0.283790
$448$	0	0
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	4.00000	0.188353
$452$	0	0
$453$	−11.3137	−0.531564
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	0	0
$459$	1.41421	0.0660098
$460$	0	0
$461$	−4.24264	−0.197599	−0.0987997	−	0.995107i	$-0.531500\pi$
$461$	−4.24264	−0.197599	−0.0987997	+	0.995107i	$0.531500\pi$
$462$	0	0
$463$	39.5980	1.84027	0.920137	−	0.391596i	$-0.128077\pi$
$463$	39.5980	1.84027	0.920137	+	0.391596i	$0.128077\pi$
$464$	0	0
$465$	−5.65685	−0.262330
$466$	0	0
$467$	−16.0000	−0.740392	−0.370196	−	0.928954i	$-0.620709\pi$
$467$	−16.0000	−0.740392	−0.370196	+	0.928954i	$0.620709\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	4.24264	0.195491
$472$	0	0
$473$	16.0000	0.735681
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−10.0000	−0.457869
$478$	0	0
$479$	−20.0000	−0.913823	−0.456912	−	0.889512i	$-0.651044\pi$
$479$	−20.0000	−0.913823	−0.456912	+	0.889512i	$0.651044\pi$
$480$	0	0
$481$	−5.65685	−0.257930
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−14.0000	−0.635707
$486$	0	0
$487$	−11.3137	−0.512673	−0.256337	−	0.966588i	$-0.582516\pi$
$487$	−11.3137	−0.512673	−0.256337	+	0.966588i	$0.582516\pi$
$488$	0	0
$489$	16.9706	0.767435
$490$	0	0
$491$	14.1421	0.638226	0.319113	−	0.947717i	$-0.396615\pi$
$491$	14.1421	0.638226	0.319113	+	0.947717i	$0.396615\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−4.00000	−0.179787
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−11.3137	−0.506471	−0.253236	−	0.967405i	$-0.581495\pi$
$499$	−11.3137	−0.506471	−0.253236	+	0.967405i	$0.581495\pi$
$500$	0	0
$501$	−20.0000	−0.893534
$502$	0	0
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	0	0
$505$	−10.0000	−0.444994
$506$	0	0
$507$	−11.0000	−0.488527
$508$	0	0
$509$	32.5269	1.44173	0.720865	−	0.693075i	$-0.243745\pi$
$509$	32.5269	1.44173	0.720865	+	0.693075i	$0.243745\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−16.9706	−0.747812
$516$	0	0
$517$	33.9411	1.49273
$518$	0	0
$519$	7.07107	0.310385
$520$	0	0
$521$	−29.6985	−1.30111	−0.650557	−	0.759457i	$-0.725465\pi$
$521$	−29.6985	−1.30111	−0.650557	+	0.759457i	$0.725465\pi$
$522$	0	0
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.65685	−0.246416
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−2.00000	−0.0866296
$534$	0	0
$535$	28.0000	1.21055
$536$	0	0
$537$	−14.1421	−0.610278
$538$	0	0
$539$	0	0
$540$	0	0
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	0	0
$543$	15.5563	0.667587
$544$	0	0
$545$	5.65685	0.242313
$546$	0	0
$547$	22.6274	0.967478	0.483739	−	0.875212i	$-0.339278\pi$
$547$	22.6274	0.967478	0.483739	+	0.875212i	$0.339278\pi$
$548$	0	0
$549$	1.41421	0.0603572
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−5.65685	−0.240120
$556$	0	0
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	0	0
$561$	−4.00000	−0.168880
$562$	0	0
$563$	−16.0000	−0.674320	−0.337160	−	0.941447i	$-0.609466\pi$
$563$	−16.0000	−0.674320	−0.337160	+	0.941447i	$0.609466\pi$
$564$	0	0
$565$	−2.82843	−0.118993
$566$	0	0
$567$	0	0
$568$	0	0
$569$	28.0000	1.17382	0.586911	−	0.809652i	$-0.300344\pi$
$569$	28.0000	1.17382	0.586911	+	0.809652i	$0.300344\pi$
$570$	0	0
$571$	−22.6274	−0.946928	−0.473464	−	0.880813i	$-0.656997\pi$
$571$	−22.6274	−0.946928	−0.473464	+	0.880813i	$0.656997\pi$
$572$	0	0
$573$	−8.48528	−0.354478
$574$	0	0
$575$	8.48528	0.353861
$576$	0	0
$577$	21.2132	0.883117	0.441559	−	0.897232i	$-0.354426\pi$
$577$	21.2132	0.883117	0.441559	+	0.897232i	$0.354426\pi$
$578$	0	0
$579$	−26.0000	−1.08052
$580$	0	0
$581$	0	0
$582$	0	0
$583$	28.2843	1.17141
$584$	0	0
$585$	2.00000	0.0826898
$586$	0	0
$587$	−8.00000	−0.330195	−0.165098	−	0.986277i	$-0.552794\pi$
$587$	−8.00000	−0.330195	−0.165098	+	0.986277i	$0.552794\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−6.00000	−0.246807
$592$	0	0
$593$	29.6985	1.21957	0.609785	−	0.792567i	$-0.291256\pi$
$593$	29.6985	1.21957	0.609785	+	0.792567i	$0.291256\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−8.00000	−0.327418
$598$	0	0
$599$	−31.1127	−1.27123	−0.635615	−	0.772006i	$-0.719253\pi$
$599$	−31.1127	−1.27123	−0.635615	+	0.772006i	$0.719253\pi$
$600$	0	0
$601$	−15.5563	−0.634557	−0.317278	−	0.948332i	$-0.602769\pi$
$601$	−15.5563	−0.634557	−0.317278	+	0.948332i	$0.602769\pi$
$602$	0	0
$603$	−11.3137	−0.460730
$604$	0	0
$605$	−4.24264	−0.172488
$606$	0	0
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−16.9706	−0.686555
$612$	0	0
$613$	44.0000	1.77714	0.888572	−	0.458738i	$-0.151698\pi$
$613$	44.0000	1.77714	0.888572	+	0.458738i	$0.151698\pi$
$614$	0	0
$615$	−2.00000	−0.0806478
$616$	0	0
$617$	36.0000	1.44931	0.724653	−	0.689114i	$-0.242000\pi$
$617$	36.0000	1.44931	0.724653	+	0.689114i	$0.242000\pi$
$618$	0	0
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	0	0
$621$	−2.82843	−0.113501
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−5.65685	−0.225554
$630$	0	0
$631$	−16.9706	−0.675587	−0.337794	−	0.941220i	$-0.609681\pi$
$631$	−16.9706	−0.675587	−0.337794	+	0.941220i	$0.609681\pi$
$632$	0	0
$633$	5.65685	0.224840
$634$	0	0
$635$	16.0000	0.634941
$636$	0	0
$637$	0	0
$638$	0	0
$639$	2.82843	0.111891
$640$	0	0
$641$	4.00000	0.157991	0.0789953	−	0.996875i	$-0.474829\pi$
$641$	4.00000	0.157991	0.0789953	+	0.996875i	$0.474829\pi$
$642$	0	0
$643$	−40.0000	−1.57745	−0.788723	−	0.614749i	$-0.789257\pi$
$643$	−40.0000	−1.57745	−0.788723	+	0.614749i	$0.789257\pi$
$644$	0	0
$645$	−8.00000	−0.315000
$646$	0	0
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−8.00000	−0.313064	−0.156532	−	0.987673i	$-0.550031\pi$
$653$	−8.00000	−0.313064	−0.156532	+	0.987673i	$0.550031\pi$
$654$	0	0
$655$	−5.65685	−0.221032
$656$	0	0
$657$	12.7279	0.496564
$658$	0	0
$659$	19.7990	0.771259	0.385630	−	0.922654i	$-0.373984\pi$
$659$	19.7990	0.771259	0.385630	+	0.922654i	$0.373984\pi$
$660$	0	0
$661$	−29.6985	−1.15514	−0.577569	−	0.816342i	$-0.695998\pi$
$661$	−29.6985	−1.15514	−0.577569	+	0.816342i	$0.695998\pi$
$662$	0	0
$663$	2.00000	0.0776736
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−16.0000	−0.618596
$670$	0	0
$671$	−4.00000	−0.154418
$672$	0	0
$673$	−32.0000	−1.23351	−0.616755	−	0.787155i	$-0.711553\pi$
$673$	−32.0000	−1.23351	−0.616755	+	0.787155i	$0.711553\pi$
$674$	0	0
$675$	−3.00000	−0.115470
$676$	0	0
$677$	46.6690	1.79364	0.896819	−	0.442398i	$-0.145872\pi$
$677$	46.6690	1.79364	0.896819	+	0.442398i	$0.145872\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	19.7990	0.757587	0.378794	−	0.925481i	$-0.376339\pi$
$683$	19.7990	0.757587	0.378794	+	0.925481i	$0.376339\pi$
$684$	0	0
$685$	16.9706	0.648412
$686$	0	0
$687$	−7.07107	−0.269778
$688$	0	0
$689$	−14.1421	−0.538772
$690$	0	0
$691$	−36.0000	−1.36950	−0.684752	−	0.728776i	$-0.740090\pi$
$691$	−36.0000	−1.36950	−0.684752	+	0.728776i	$0.740090\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	28.2843	1.07288
$696$	0	0
$697$	−2.00000	−0.0757554
$698$	0	0
$699$	12.0000	0.453882
$700$	0	0
$701$	16.0000	0.604312	0.302156	−	0.953259i	$-0.402294\pi$
$701$	16.0000	0.604312	0.302156	+	0.953259i	$0.402294\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−16.9706	−0.639148
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−36.0000	−1.35201	−0.676004	−	0.736898i	$-0.736290\pi$
$709$	−36.0000	−1.35201	−0.676004	+	0.736898i	$0.736290\pi$
$710$	0	0
$711$	−11.3137	−0.424297
$712$	0	0
$713$	11.3137	0.423702
$714$	0	0
$715$	−5.65685	−0.211554
$716$	0	0
$717$	19.7990	0.739407
$718$	0	0
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−1.41421	−0.0525952
$724$	0	0
$725$	0	0
$726$	0	0
$727$	28.0000	1.03846	0.519231	−	0.854634i	$-0.326218\pi$
$727$	28.0000	1.03846	0.519231	+	0.854634i	$0.326218\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−8.00000	−0.295891
$732$	0	0
$733$	−29.6985	−1.09694	−0.548469	−	0.836171i	$-0.684789\pi$
$733$	−29.6985	−1.09694	−0.548469	+	0.836171i	$0.684789\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	32.0000	1.17874
$738$	0	0
$739$	16.9706	0.624272	0.312136	−	0.950037i	$-0.398955\pi$
$739$	16.9706	0.624272	0.312136	+	0.950037i	$0.398955\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−36.7696	−1.34894	−0.674472	−	0.738300i	$-0.735629\pi$
$743$	−36.7696	−1.34894	−0.674472	+	0.738300i	$0.735629\pi$
$744$	0	0
$745$	8.48528	0.310877
$746$	0	0
$747$	4.00000	0.146352
$748$	0	0
$749$	0	0
$750$	0	0
$751$	28.2843	1.03211	0.516054	−	0.856556i	$-0.327400\pi$
$751$	28.2843	1.03211	0.516054	+	0.856556i	$0.327400\pi$
$752$	0	0
$753$	24.0000	0.874609
$754$	0	0
$755$	−16.0000	−0.582300
$756$	0	0
$757$	−28.0000	−1.01768	−0.508839	−	0.860862i	$-0.669925\pi$
$757$	−28.0000	−1.01768	−0.508839	+	0.860862i	$0.669925\pi$
$758$	0	0
$759$	8.00000	0.290382
$760$	0	0
$761$	−43.8406	−1.58922	−0.794611	−	0.607119i	$-0.792325\pi$
$761$	−43.8406	−1.58922	−0.794611	+	0.607119i	$0.792325\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	2.00000	0.0723102
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−18.3848	−0.662972	−0.331486	−	0.943460i	$-0.607550\pi$
$769$	−18.3848	−0.662972	−0.331486	+	0.943460i	$0.607550\pi$
$770$	0	0
$771$	9.89949	0.356522
$772$	0	0
$773$	−15.5563	−0.559523	−0.279761	−	0.960070i	$-0.590255\pi$
$773$	−15.5563	−0.559523	−0.279761	+	0.960070i	$0.590255\pi$
$774$	0	0
$775$	12.0000	0.431053
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−8.00000	−0.286263
$782$	0	0
$783$	0	0
$784$	0	0
$785$	6.00000	0.214149
$786$	0	0
$787$	−20.0000	−0.712923	−0.356462	−	0.934310i	$-0.616017\pi$
$787$	−20.0000	−0.712923	−0.356462	+	0.934310i	$0.616017\pi$
$788$	0	0
$789$	−14.1421	−0.503473
$790$	0	0
$791$	0	0
$792$	0	0
$793$	2.00000	0.0710221
$794$	0	0
$795$	−14.1421	−0.501570
$796$	0	0
$797$	−18.3848	−0.651222	−0.325611	−	0.945504i	$-0.605570\pi$
$797$	−18.3848	−0.651222	−0.325611	+	0.945504i	$0.605570\pi$
$798$	0	0
$799$	−16.9706	−0.600375
$800$	0	0
$801$	7.07107	0.249844
$802$	0	0
$803$	−36.0000	−1.27041
$804$	0	0
$805$	0	0
$806$	0	0
$807$	9.89949	0.348479
$808$	0	0
$809$	26.0000	0.914111	0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000	0.914111	0.457056	+	0.889438i	$0.348904\pi$
$810$	0	0
$811$	12.0000	0.421377	0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000	0.421377	0.210688	+	0.977553i	$0.432429\pi$
$812$	0	0
$813$	−20.0000	−0.701431
$814$	0	0
$815$	24.0000	0.840683
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	26.0000	0.907406	0.453703	−	0.891153i	$-0.350103\pi$
$821$	26.0000	0.907406	0.453703	+	0.891153i	$0.350103\pi$
$822$	0	0
$823$	−33.9411	−1.18311	−0.591557	−	0.806263i	$-0.701486\pi$
$823$	−33.9411	−1.18311	−0.591557	+	0.806263i	$0.701486\pi$
$824$	0	0
$825$	8.48528	0.295420
$826$	0	0
$827$	19.7990	0.688478	0.344239	−	0.938882i	$-0.388137\pi$
$827$	19.7990	0.688478	0.344239	+	0.938882i	$0.388137\pi$
$828$	0	0
$829$	−38.1838	−1.32618	−0.663089	−	0.748541i	$-0.730755\pi$
$829$	−38.1838	−1.32618	−0.663089	+	0.748541i	$0.730755\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−28.2843	−0.978818
$836$	0	0
$837$	−4.00000	−0.138260
$838$	0	0
$839$	36.0000	1.24286	0.621429	−	0.783470i	$-0.286552\pi$
$839$	36.0000	1.24286	0.621429	+	0.783470i	$0.286552\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	12.0000	0.413302
$844$	0	0
$845$	−15.5563	−0.535155
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−16.0000	−0.549119
$850$	0	0
$851$	11.3137	0.387829
$852$	0	0
$853$	55.1543	1.88845	0.944224	−	0.329304i	$-0.106814\pi$
$853$	55.1543	1.88845	0.944224	+	0.329304i	$0.106814\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−9.89949	−0.338160	−0.169080	−	0.985602i	$-0.554080\pi$
$857$	−9.89949	−0.338160	−0.169080	+	0.985602i	$0.554080\pi$
$858$	0	0
$859$	−8.00000	−0.272956	−0.136478	−	0.990643i	$-0.543578\pi$
$859$	−8.00000	−0.272956	−0.136478	+	0.990643i	$0.543578\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−8.48528	−0.288842	−0.144421	−	0.989516i	$-0.546132\pi$
$863$	−8.48528	−0.288842	−0.144421	+	0.989516i	$0.546132\pi$
$864$	0	0
$865$	10.0000	0.340010
$866$	0	0
$867$	−15.0000	−0.509427
$868$	0	0
$869$	32.0000	1.08553
$870$	0	0
$871$	−16.0000	−0.542139
$872$	0	0
$873$	−9.89949	−0.335047
$874$	0	0
$875$	0	0
$876$	0	0
$877$	4.00000	0.135070	0.0675352	−	0.997717i	$-0.478487\pi$
$877$	4.00000	0.135070	0.0675352	+	0.997717i	$0.478487\pi$
$878$	0	0
$879$	−29.6985	−1.00171
$880$	0	0
$881$	−15.5563	−0.524107	−0.262053	−	0.965053i	$-0.584400\pi$
$881$	−15.5563	−0.524107	−0.262053	+	0.965053i	$0.584400\pi$
$882$	0	0
$883$	11.3137	0.380737	0.190368	−	0.981713i	$-0.439032\pi$
$883$	11.3137	0.380737	0.190368	+	0.981713i	$0.439032\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	36.0000	1.20876	0.604381	−	0.796696i	$-0.293421\pi$
$887$	36.0000	1.20876	0.604381	+	0.796696i	$0.293421\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−2.82843	−0.0947559
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−20.0000	−0.668526
$896$	0	0
$897$	−4.00000	−0.133556
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−14.1421	−0.471143
$902$	0	0
$903$	0	0
$904$	0	0
$905$	22.0000	0.731305
$906$	0	0
$907$	11.3137	0.375666	0.187833	−	0.982201i	$-0.439854\pi$
$907$	11.3137	0.375666	0.187833	+	0.982201i	$0.439854\pi$
$908$	0	0
$909$	−7.07107	−0.234533
$910$	0	0
$911$	−25.4558	−0.843390	−0.421695	−	0.906738i	$-0.638565\pi$
$911$	−25.4558	−0.843390	−0.421695	+	0.906738i	$0.638565\pi$
$912$	0	0
$913$	−11.3137	−0.374429
$914$	0	0
$915$	2.00000	0.0661180
$916$	0	0
$917$	0	0
$918$	0	0
$919$	16.9706	0.559807	0.279904	−	0.960028i	$-0.409697\pi$
$919$	16.9706	0.559807	0.279904	+	0.960028i	$0.409697\pi$
$920$	0	0
$921$	−8.00000	−0.263609
$922$	0	0
$923$	4.00000	0.131662
$924$	0	0
$925$	12.0000	0.394558
$926$	0	0
$927$	−12.0000	−0.394132
$928$	0	0
$929$	41.0122	1.34557	0.672783	−	0.739840i	$-0.265099\pi$
$929$	41.0122	1.34557	0.672783	+	0.739840i	$0.265099\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−12.0000	−0.392862
$934$	0	0
$935$	−5.65685	−0.184999
$936$	0	0
$937$	−26.8701	−0.877807	−0.438903	−	0.898534i	$-0.644633\pi$
$937$	−26.8701	−0.877807	−0.438903	+	0.898534i	$0.644633\pi$
$938$	0	0
$939$	12.7279	0.415360
$940$	0	0
$941$	−29.6985	−0.968143	−0.484071	−	0.875028i	$-0.660843\pi$
$941$	−29.6985	−0.968143	−0.484071	+	0.875028i	$0.660843\pi$
$942$	0	0
$943$	4.00000	0.130258
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−53.7401	−1.74632	−0.873160	−	0.487435i	$-0.837933\pi$
$947$	−53.7401	−1.74632	−0.873160	+	0.487435i	$0.837933\pi$
$948$	0	0
$949$	18.0000	0.584305
$950$	0	0
$951$	18.0000	0.583690
$952$	0	0
$953$	58.0000	1.87880	0.939402	−	0.342817i	$-0.111381\pi$
$953$	58.0000	1.87880	0.939402	+	0.342817i	$0.111381\pi$
$954$	0	0
$955$	−12.0000	−0.388311
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	19.7990	0.638014
$964$	0	0
$965$	−36.7696	−1.18365
$966$	0	0
$967$	−45.2548	−1.45530	−0.727649	−	0.685950i	$-0.759387\pi$
$967$	−45.2548	−1.45530	−0.727649	+	0.685950i	$0.759387\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−20.0000	−0.641831	−0.320915	−	0.947108i	$-0.603990\pi$
$971$	−20.0000	−0.641831	−0.320915	+	0.947108i	$0.603990\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−4.24264	−0.135873
$976$	0	0
$977$	12.0000	0.383914	0.191957	−	0.981403i	$-0.438517\pi$
$977$	12.0000	0.383914	0.191957	+	0.981403i	$0.438517\pi$
$978$	0	0
$979$	−20.0000	−0.639203
$980$	0	0
$981$	4.00000	0.127710
$982$	0	0
$983$	48.0000	1.53096	0.765481	−	0.643458i	$-0.222501\pi$
$983$	48.0000	1.53096	0.765481	+	0.643458i	$0.222501\pi$
$984$	0	0
$985$	−8.48528	−0.270364
$986$	0	0
$987$	0	0
$988$	0	0
$989$	16.0000	0.508770
$990$	0	0
$991$	−5.65685	−0.179696	−0.0898479	−	0.995955i	$-0.528638\pi$
$991$	−5.65685	−0.179696	−0.0898479	+	0.995955i	$0.528638\pi$
$992$	0	0
$993$	33.9411	1.07709
$994$	0	0
$995$	−11.3137	−0.358669
$996$	0	0
$997$	4.24264	0.134366	0.0671829	−	0.997741i	$-0.478599\pi$
$997$	4.24264	0.134366	0.0671829	+	0.997741i	$0.478599\pi$
$998$	0	0
$999$	−4.00000	−0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9408.2.a.ea.1.2		2
4.3	odd	2		9408.2.a.dl.1.2		2
7.6	odd	2		9408.2.a.dl.1.1		2
8.3	odd	2		4704.2.a.bq.1.1	yes	2
8.5	even	2		4704.2.a.bj.1.1	✓	2
28.27	even	2	inner	9408.2.a.ea.1.1		2
56.13	odd	2		4704.2.a.bq.1.2	yes	2
56.27	even	2		4704.2.a.bj.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4704.2.a.bj.1.1	✓	2	8.5	even	2
4704.2.a.bj.1.2	yes	2	56.27	even	2
4704.2.a.bq.1.1	yes	2	8.3	odd	2
4704.2.a.bq.1.2	yes	2	56.13	odd	2
9408.2.a.dl.1.1		2	7.6	odd	2
9408.2.a.dl.1.2		2	4.3	odd	2
9408.2.a.ea.1.1		2	28.27	even	2	inner
9408.2.a.ea.1.2		2	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 9408 = 2^{6} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9408.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.41421 q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.41421 q^{5} +1.00000 q^{9} -2.82843 q^{11} +1.41421 q^{13} +1.41421 q^{15} +1.41421 q^{17} -2.82843 q^{23} -3.00000 q^{25} +1.00000 q^{27} -4.00000 q^{31} -2.82843 q^{33} -4.00000 q^{37} +1.41421 q^{39} -1.41421 q^{41} -5.65685 q^{43} +1.41421 q^{45} -12.0000 q^{47} +1.41421 q^{51} -10.0000 q^{53} -4.00000 q^{55} +1.41421 q^{61} +2.00000 q^{65} -11.3137 q^{67} -2.82843 q^{69} +2.82843 q^{71} +12.7279 q^{73} -3.00000 q^{75} -11.3137 q^{79} +1.00000 q^{81} +4.00000 q^{83} +2.00000 q^{85} +7.07107 q^{89} -4.00000 q^{93} -9.89949 q^{97} -2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{9} - 6 q^{25} + 2 q^{27} - 8 q^{31} - 8 q^{37} - 24 q^{47} - 20 q^{53} - 8 q^{55} + 4 q^{65} - 6 q^{75} + 2 q^{81} + 8 q^{83} + 4 q^{85} - 8 q^{93}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^9 - 6 * q^25 + 2 * q^27 - 8 * q^31 - 8 * q^37 - 24 * q^47 - 20 * q^53 - 8 * q^55 + 4 * q^65 - 6 * q^75 + 2 * q^81 + 8 * q^83 + 4 * q^85 - 8 * q^93

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