LMFDB - Embedded newform 9408.2.a.dk.1.1

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:	$ 2 $
Twist minimal:	no (minimal twist has level 4704)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$	$=$	$q-1.00000 q^{3} -2.82843 q^{5} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.82843 q^{5} +1.00000 q^{9} +2.82843 q^{11} +2.82843 q^{15} -2.82843 q^{17} -4.00000 q^{19} +8.48528 q^{23} +3.00000 q^{25} -1.00000 q^{27} -2.00000 q^{29} -2.82843 q^{33} +6.00000 q^{37} -8.48528 q^{41} -11.3137 q^{43} -2.82843 q^{45} +8.00000 q^{47} +2.82843 q^{51} -6.00000 q^{53} -8.00000 q^{55} +4.00000 q^{57} -12.0000 q^{59} +5.65685 q^{61} +5.65685 q^{67} -8.48528 q^{69} +2.82843 q^{71} +5.65685 q^{73} -3.00000 q^{75} +5.65685 q^{79} +1.00000 q^{81} +4.00000 q^{83} +8.00000 q^{85} +2.00000 q^{87} +2.82843 q^{89} +11.3137 q^{95} +16.9706 q^{97} +2.82843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 8 q^{19} + 6 q^{25} - 2 q^{27} - 4 q^{29} + 12 q^{37} + 16 q^{47} - 12 q^{53} - 16 q^{55} + 8 q^{57} - 24 q^{59} - 6 q^{75} + 2 q^{81} + 8 q^{83} + 16 q^{85} + 4 q^{87}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^9 - 8 * q^19 + 6 * q^25 - 2 * q^27 - 4 * q^29 + 12 * q^37 + 16 * q^47 - 12 * q^53 - 16 * q^55 + 8 * q^57 - 24 * q^59 - 6 * q^75 + 2 * q^81 + 8 * q^83 + 16 * q^85 + 4 * q^87

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
$5$	−2.82843	−1.26491	−0.632456	−	0.774597i	$-0.717953\pi$
$5$	−2.82843	−1.26491	−0.632456	+	0.774597i	$0.717953\pi$
$6$	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	$-0.359781\pi$
$11$	2.82843	0.852803	0.426401	+	0.904534i	$0.359781\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	2.82843	0.730297
$16$	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
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.48528	1.76930	0.884652	−	0.466252i	$-0.154396\pi$
$23$	8.48528	1.76930	0.884652	+	0.466252i	$0.154396\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−2.82843	−0.492366
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.48528	−1.32518	−0.662589	−	0.748983i	$-0.730542\pi$
$41$	−8.48528	−1.32518	−0.662589	+	0.748983i	$0.730542\pi$
$42$	0	0
$43$	−11.3137	−1.72532	−0.862662	−	0.505781i	$-0.831205\pi$
$43$	−11.3137	−1.72532	−0.862662	+	0.505781i	$0.831205\pi$
$44$	0	0
$45$	−2.82843	−0.421637
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	2.82843	0.396059
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	−8.00000	−1.07872
$56$	0	0
$57$	4.00000	0.529813
$58$	0	0
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	5.65685	0.724286	0.362143	−	0.932123i	$-0.382045\pi$
$61$	5.65685	0.724286	0.362143	+	0.932123i	$0.382045\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	5.65685	0.691095	0.345547	−	0.938401i	$-0.387693\pi$
$67$	5.65685	0.691095	0.345547	+	0.938401i	$0.387693\pi$
$68$	0	0
$69$	−8.48528	−1.02151
$70$	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
$73$	5.65685	0.662085	0.331042	−	0.943616i	$-0.392600\pi$
$73$	5.65685	0.662085	0.331042	+	0.943616i	$0.392600\pi$
$74$	0	0
$75$	−3.00000	−0.346410
$76$	0	0
$77$	0	0
$78$	0	0
$79$	5.65685	0.636446	0.318223	−	0.948016i	$-0.396914\pi$
$79$	5.65685	0.636446	0.318223	+	0.948016i	$0.396914\pi$
$80$	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
$85$	8.00000	0.867722
$86$	0	0
$87$	2.00000	0.214423
$88$	0	0
$89$	2.82843	0.299813	0.149906	−	0.988700i	$-0.452103\pi$
$89$	2.82843	0.299813	0.149906	+	0.988700i	$0.452103\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	11.3137	1.16076
$96$	0	0
$97$	16.9706	1.72310	0.861550	−	0.507673i	$-0.169494\pi$
$97$	16.9706	1.72310	0.861550	+	0.507673i	$0.169494\pi$
$98$	0	0
$99$	2.82843	0.284268
$100$	0	0
$101$	14.1421	1.40720	0.703598	−	0.710599i	$-0.251576\pi$
$101$	14.1421	1.40720	0.703598	+	0.710599i	$0.251576\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.82843	0.273434	0.136717	−	0.990610i	$-0.456345\pi$
$107$	2.82843	0.273434	0.136717	+	0.990610i	$0.456345\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	0	0
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	0	0
$115$	−24.0000	−2.23801
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−3.00000	−0.272727
$122$	0	0
$123$	8.48528	0.765092
$124$	0	0
$125$	5.65685	0.505964
$126$	0	0
$127$	−16.9706	−1.50589	−0.752947	−	0.658081i	$-0.771368\pi$
$127$	−16.9706	−1.50589	−0.752947	+	0.658081i	$0.771368\pi$
$128$	0	0
$129$	11.3137	0.996116
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	2.82843	0.243432
$136$	0	0
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	0	0
$143$	0	0
$144$	0	0
$145$	5.65685	0.469776
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	−2.82843	−0.228665
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−16.9706	−1.35440	−0.677199	−	0.735800i	$-0.736806\pi$
$157$	−16.9706	−1.35440	−0.677199	+	0.735800i	$0.736806\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	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
$165$	8.00000	0.622799
$166$	0	0
$167$	−24.0000	−1.85718	−0.928588	−	0.371113i	$-0.878976\pi$
$167$	−24.0000	−1.85718	−0.928588	+	0.371113i	$0.878976\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	−4.00000	−0.305888
$172$	0	0
$173$	−8.48528	−0.645124	−0.322562	−	0.946548i	$-0.604544\pi$
$173$	−8.48528	−0.645124	−0.322562	+	0.946548i	$0.604544\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	12.0000	0.901975
$178$	0	0
$179$	−2.82843	−0.211407	−0.105703	−	0.994398i	$-0.533709\pi$
$179$	−2.82843	−0.211407	−0.105703	+	0.994398i	$0.533709\pi$
$180$	0	0
$181$	22.6274	1.68188	0.840941	−	0.541126i	$-0.182002\pi$
$181$	22.6274	1.68188	0.840941	+	0.541126i	$0.182002\pi$
$182$	0	0
$183$	−5.65685	−0.418167
$184$	0	0
$185$	−16.9706	−1.24770
$186$	0	0
$187$	−8.00000	−0.585018
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−19.7990	−1.43260	−0.716302	−	0.697790i	$-0.754167\pi$
$191$	−19.7990	−1.43260	−0.716302	+	0.697790i	$0.754167\pi$
$192$	0	0
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	−5.65685	−0.399004
$202$	0	0
$203$	0	0
$204$	0	0
$205$	24.0000	1.67623
$206$	0	0
$207$	8.48528	0.589768
$208$	0	0
$209$	−11.3137	−0.782586
$210$	0	0
$211$	22.6274	1.55774	0.778868	−	0.627188i	$-0.215794\pi$
$211$	22.6274	1.55774	0.778868	+	0.627188i	$0.215794\pi$
$212$	0	0
$213$	−2.82843	−0.193801
$214$	0	0
$215$	32.0000	2.18238
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−5.65685	−0.382255
$220$	0	0
$221$	0	0
$222$	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
$227$	−28.0000	−1.85843	−0.929213	−	0.369546i	$-0.879513\pi$
$227$	−28.0000	−1.85843	−0.929213	+	0.369546i	$0.879513\pi$
$228$	0	0
$229$	11.3137	0.747631	0.373815	−	0.927503i	$-0.378049\pi$
$229$	11.3137	0.747631	0.373815	+	0.927503i	$0.378049\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	−22.6274	−1.47605
$236$	0	0
$237$	−5.65685	−0.367452
$238$	0	0
$239$	−14.1421	−0.914779	−0.457389	−	0.889267i	$-0.651215\pi$
$239$	−14.1421	−0.914779	−0.457389	+	0.889267i	$0.651215\pi$
$240$	0	0
$241$	−16.9706	−1.09317	−0.546585	−	0.837404i	$-0.684072\pi$
$241$	−16.9706	−1.09317	−0.546585	+	0.837404i	$0.684072\pi$
$242$	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
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	0	0
$253$	24.0000	1.50887
$254$	0	0
$255$	−8.00000	−0.500979
$256$	0	0
$257$	−8.48528	−0.529297	−0.264649	−	0.964345i	$-0.585256\pi$
$257$	−8.48528	−0.529297	−0.264649	+	0.964345i	$0.585256\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−2.00000	−0.123797
$262$	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
$265$	16.9706	1.04249
$266$	0	0
$267$	−2.82843	−0.173097
$268$	0	0
$269$	8.48528	0.517357	0.258678	−	0.965964i	$-0.416713\pi$
$269$	8.48528	0.517357	0.258678	+	0.965964i	$0.416713\pi$
$270$	0	0
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	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
$279$	0	0
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	0	0
$285$	−11.3137	−0.670166
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−9.00000	−0.529412
$290$	0	0
$291$	−16.9706	−0.994832
$292$	0	0
$293$	−8.48528	−0.495715	−0.247858	−	0.968796i	$-0.579727\pi$
$293$	−8.48528	−0.495715	−0.247858	+	0.968796i	$0.579727\pi$
$294$	0	0
$295$	33.9411	1.97613
$296$	0	0
$297$	−2.82843	−0.164122
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−14.1421	−0.812444
$304$	0	0
$305$	−16.0000	−0.916157
$306$	0	0
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	0	0
$309$	−16.0000	−0.910208
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	22.6274	1.27898	0.639489	−	0.768801i	$-0.279146\pi$
$313$	22.6274	1.27898	0.639489	+	0.768801i	$0.279146\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−30.0000	−1.68497	−0.842484	−	0.538721i	$-0.818908\pi$
$317$	−30.0000	−1.68497	−0.842484	+	0.538721i	$0.818908\pi$
$318$	0	0
$319$	−5.65685	−0.316723
$320$	0	0
$321$	−2.82843	−0.157867
$322$	0	0
$323$	11.3137	0.629512
$324$	0	0
$325$	0	0
$326$	0	0
$327$	10.0000	0.553001
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−22.6274	−1.24372	−0.621858	−	0.783130i	$-0.713622\pi$
$331$	−22.6274	−1.24372	−0.621858	+	0.783130i	$0.713622\pi$
$332$	0	0
$333$	6.00000	0.328798
$334$	0	0
$335$	−16.0000	−0.874173
$336$	0	0
$337$	−34.0000	−1.85210	−0.926049	−	0.377403i	$-0.876817\pi$
$337$	−34.0000	−1.85210	−0.926049	+	0.377403i	$0.876817\pi$
$338$	0	0
$339$	−18.0000	−0.977626
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	24.0000	1.29212
$346$	0	0
$347$	19.7990	1.06287	0.531433	−	0.847100i	$-0.321654\pi$
$347$	19.7990	1.06287	0.531433	+	0.847100i	$0.321654\pi$
$348$	0	0
$349$	−16.9706	−0.908413	−0.454207	−	0.890896i	$-0.650077\pi$
$349$	−16.9706	−0.908413	−0.454207	+	0.890896i	$0.650077\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	31.1127	1.65596	0.827981	−	0.560756i	$-0.189490\pi$
$353$	31.1127	1.65596	0.827981	+	0.560756i	$0.189490\pi$
$354$	0	0
$355$	−8.00000	−0.424596
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.82843	0.149279	0.0746393	−	0.997211i	$-0.476219\pi$
$359$	2.82843	0.149279	0.0746393	+	0.997211i	$0.476219\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	3.00000	0.157459
$364$	0	0
$365$	−16.0000	−0.837478
$366$	0	0
$367$	−32.0000	−1.67039	−0.835193	−	0.549957i	$-0.814644\pi$
$367$	−32.0000	−1.67039	−0.835193	+	0.549957i	$0.814644\pi$
$368$	0	0
$369$	−8.48528	−0.441726
$370$	0	0
$371$	0	0
$372$	0	0
$373$	18.0000	0.932005	0.466002	−	0.884783i	$-0.345694\pi$
$373$	18.0000	0.932005	0.466002	+	0.884783i	$0.345694\pi$
$374$	0	0
$375$	−5.65685	−0.292119
$376$	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
$381$	16.9706	0.869428
$382$	0	0
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−11.3137	−0.575108
$388$	0	0
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	−24.0000	−1.21373
$392$	0	0
$393$	12.0000	0.605320
$394$	0	0
$395$	−16.0000	−0.805047
$396$	0	0
$397$	28.2843	1.41955	0.709773	−	0.704430i	$-0.248797\pi$
$397$	28.2843	1.41955	0.709773	+	0.704430i	$0.248797\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−2.82843	−0.140546
$406$	0	0
$407$	16.9706	0.841200
$408$	0	0
$409$	5.65685	0.279713	0.139857	−	0.990172i	$-0.455336\pi$
$409$	5.65685	0.279713	0.139857	+	0.990172i	$0.455336\pi$
$410$	0	0
$411$	10.0000	0.493264
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−11.3137	−0.555368
$416$	0	0
$417$	4.00000	0.195881
$418$	0	0
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	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
$423$	8.00000	0.388973
$424$	0	0
$425$	−8.48528	−0.411597
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−19.7990	−0.953684	−0.476842	−	0.878989i	$-0.658219\pi$
$431$	−19.7990	−0.953684	−0.476842	+	0.878989i	$0.658219\pi$
$432$	0	0
$433$	−33.9411	−1.63111	−0.815553	−	0.578682i	$-0.803567\pi$
$433$	−33.9411	−1.63111	−0.815553	+	0.578682i	$0.803567\pi$
$434$	0	0
$435$	−5.65685	−0.271225
$436$	0	0
$437$	−33.9411	−1.62362
$438$	0	0
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−36.7696	−1.74697	−0.873487	−	0.486847i	$-0.838147\pi$
$443$	−36.7696	−1.74697	−0.873487	+	0.486847i	$0.838147\pi$
$444$	0	0
$445$	−8.00000	−0.379236
$446$	0	0
$447$	6.00000	0.283790
$448$	0	0
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	−24.0000	−1.13012
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−6.00000	−0.280668	−0.140334	−	0.990104i	$-0.544818\pi$
$457$	−6.00000	−0.280668	−0.140334	+	0.990104i	$0.544818\pi$
$458$	0	0
$459$	2.82843	0.132020
$460$	0	0
$461$	−19.7990	−0.922131	−0.461065	−	0.887366i	$-0.652533\pi$
$461$	−19.7990	−0.922131	−0.461065	+	0.887366i	$0.652533\pi$
$462$	0	0
$463$	−5.65685	−0.262896	−0.131448	−	0.991323i	$-0.541963\pi$
$463$	−5.65685	−0.262896	−0.131448	+	0.991323i	$0.541963\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	16.9706	0.781962
$472$	0	0
$473$	−32.0000	−1.47136
$474$	0	0
$475$	−12.0000	−0.550598
$476$	0	0
$477$	−6.00000	−0.274721
$478$	0	0
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−48.0000	−2.17957
$486$	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
$491$	−36.7696	−1.65939	−0.829693	−	0.558219i	$-0.811485\pi$
$491$	−36.7696	−1.65939	−0.829693	+	0.558219i	$0.811485\pi$
$492$	0	0
$493$	5.65685	0.254772
$494$	0	0
$495$	−8.00000	−0.359573
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−22.6274	−1.01294	−0.506471	−	0.862257i	$-0.669050\pi$
$499$	−22.6274	−1.01294	−0.506471	+	0.862257i	$0.669050\pi$
$500$	0	0
$501$	24.0000	1.07224
$502$	0	0
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	0	0
$505$	−40.0000	−1.77998
$506$	0	0
$507$	13.0000	0.577350
$508$	0	0
$509$	−14.1421	−0.626839	−0.313420	−	0.949615i	$-0.601475\pi$
$509$	−14.1421	−0.626839	−0.313420	+	0.949615i	$0.601475\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	4.00000	0.176604
$514$	0	0
$515$	−45.2548	−1.99417
$516$	0	0
$517$	22.6274	0.995153
$518$	0	0
$519$	8.48528	0.372463
$520$	0	0
$521$	19.7990	0.867409	0.433705	−	0.901055i	$-0.357206\pi$
$521$	19.7990	0.867409	0.433705	+	0.901055i	$0.357206\pi$
$522$	0	0
$523$	36.0000	1.57417	0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000	1.57417	0.787085	+	0.616844i	$0.211589\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	49.0000	2.13043
$530$	0	0
$531$	−12.0000	−0.520756
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−8.00000	−0.345870
$536$	0	0
$537$	2.82843	0.122056
$538$	0	0
$539$	0	0
$540$	0	0
$541$	34.0000	1.46177	0.730887	−	0.682498i	$-0.239107\pi$
$541$	34.0000	1.46177	0.730887	+	0.682498i	$0.239107\pi$
$542$	0	0
$543$	−22.6274	−0.971035
$544$	0	0
$545$	28.2843	1.21157
$546$	0	0
$547$	−16.9706	−0.725609	−0.362804	−	0.931865i	$-0.618181\pi$
$547$	−16.9706	−0.725609	−0.362804	+	0.931865i	$0.618181\pi$
$548$	0	0
$549$	5.65685	0.241429
$550$	0	0
$551$	8.00000	0.340811
$552$	0	0
$553$	0	0
$554$	0	0
$555$	16.9706	0.720360
$556$	0	0
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	8.00000	0.337760
$562$	0	0
$563$	−20.0000	−0.842900	−0.421450	−	0.906852i	$-0.638479\pi$
$563$	−20.0000	−0.842900	−0.421450	+	0.906852i	$0.638479\pi$
$564$	0	0
$565$	−50.9117	−2.14187
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−26.0000	−1.08998	−0.544988	−	0.838444i	$-0.683466\pi$
$569$	−26.0000	−1.08998	−0.544988	+	0.838444i	$0.683466\pi$
$570$	0	0
$571$	39.5980	1.65712	0.828562	−	0.559897i	$-0.189159\pi$
$571$	39.5980	1.65712	0.828562	+	0.559897i	$0.189159\pi$
$572$	0	0
$573$	19.7990	0.827115
$574$	0	0
$575$	25.4558	1.06158
$576$	0	0
$577$	−33.9411	−1.41299	−0.706494	−	0.707719i	$-0.749724\pi$
$577$	−33.9411	−1.41299	−0.706494	+	0.707719i	$0.749724\pi$
$578$	0	0
$579$	−10.0000	−0.415586
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−16.9706	−0.702849
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−20.0000	−0.825488	−0.412744	−	0.910847i	$-0.635430\pi$
$587$	−20.0000	−0.825488	−0.412744	+	0.910847i	$0.635430\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−10.0000	−0.411345
$592$	0	0
$593$	−2.82843	−0.116150	−0.0580748	−	0.998312i	$-0.518496\pi$
$593$	−2.82843	−0.116150	−0.0580748	+	0.998312i	$0.518496\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−8.00000	−0.327418
$598$	0	0
$599$	14.1421	0.577832	0.288916	−	0.957354i	$-0.406705\pi$
$599$	14.1421	0.577832	0.288916	+	0.957354i	$0.406705\pi$
$600$	0	0
$601$	33.9411	1.38449	0.692244	−	0.721664i	$-0.256622\pi$
$601$	33.9411	1.38449	0.692244	+	0.721664i	$0.256622\pi$
$602$	0	0
$603$	5.65685	0.230365
$604$	0	0
$605$	8.48528	0.344976
$606$	0	0
$607$	−32.0000	−1.29884	−0.649420	−	0.760430i	$-0.724988\pi$
$607$	−32.0000	−1.29884	−0.649420	+	0.760430i	$0.724988\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	0	0
$615$	−24.0000	−0.967773
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	0	0
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	−8.48528	−0.340503
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	11.3137	0.451826
$628$	0	0
$629$	−16.9706	−0.676661
$630$	0	0
$631$	5.65685	0.225196	0.112598	−	0.993641i	$-0.464083\pi$
$631$	5.65685	0.225196	0.112598	+	0.993641i	$0.464083\pi$
$632$	0	0
$633$	−22.6274	−0.899359
$634$	0	0
$635$	48.0000	1.90482
$636$	0	0
$637$	0	0
$638$	0	0
$639$	2.82843	0.111891
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	−12.0000	−0.473234	−0.236617	−	0.971603i	$-0.576039\pi$
$643$	−12.0000	−0.473234	−0.236617	+	0.971603i	$0.576039\pi$
$644$	0	0
$645$	−32.0000	−1.26000
$646$	0	0
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	0	0
$649$	−33.9411	−1.33231
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	0	0
$655$	33.9411	1.32619
$656$	0	0
$657$	5.65685	0.220695
$658$	0	0
$659$	25.4558	0.991619	0.495809	−	0.868431i	$-0.334871\pi$
$659$	25.4558	0.991619	0.495809	+	0.868431i	$0.334871\pi$
$660$	0	0
$661$	28.2843	1.10013	0.550065	−	0.835122i	$-0.314603\pi$
$661$	28.2843	1.10013	0.550065	+	0.835122i	$0.314603\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−16.9706	−0.657103
$668$	0	0
$669$	16.0000	0.618596
$670$	0	0
$671$	16.0000	0.617673
$672$	0	0
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	0	0
$675$	−3.00000	−0.115470
$676$	0	0
$677$	36.7696	1.41317	0.706584	−	0.707629i	$-0.250235\pi$
$677$	36.7696	1.41317	0.706584	+	0.707629i	$0.250235\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	28.0000	1.07296
$682$	0	0
$683$	−25.4558	−0.974041	−0.487020	−	0.873391i	$-0.661916\pi$
$683$	−25.4558	−0.974041	−0.487020	+	0.873391i	$0.661916\pi$
$684$	0	0
$685$	28.2843	1.08069
$686$	0	0
$687$	−11.3137	−0.431645
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−12.0000	−0.456502	−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000	−0.456502	−0.228251	+	0.973602i	$0.573301\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	11.3137	0.429153
$696$	0	0
$697$	24.0000	0.909065
$698$	0	0
$699$	−6.00000	−0.226941
$700$	0	0
$701$	−34.0000	−1.28416	−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000	−1.28416	−0.642081	+	0.766637i	$0.721929\pi$
$702$	0	0
$703$	−24.0000	−0.905177
$704$	0	0
$705$	22.6274	0.852198
$706$	0	0
$707$	0	0
$708$	0	0
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	0	0
$711$	5.65685	0.212149
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	14.1421	0.528148
$718$	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
$723$	16.9706	0.631142
$724$	0	0
$725$	−6.00000	−0.222834
$726$	0	0
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	32.0000	1.18356
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	16.0000	0.589368
$738$	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
$743$	−31.1127	−1.14141	−0.570707	−	0.821154i	$-0.693331\pi$
$743$	−31.1127	−1.14141	−0.570707	+	0.821154i	$0.693331\pi$
$744$	0	0
$745$	16.9706	0.621753
$746$	0	0
$747$	4.00000	0.146352
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	20.0000	0.728841
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	0	0
$759$	−24.0000	−0.871145
$760$	0	0
$761$	−19.7990	−0.717713	−0.358856	−	0.933393i	$-0.616833\pi$
$761$	−19.7990	−0.717713	−0.358856	+	0.933393i	$0.616833\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	8.00000	0.289241
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−33.9411	−1.22395	−0.611974	−	0.790878i	$-0.709624\pi$
$769$	−33.9411	−1.22395	−0.611974	+	0.790878i	$0.709624\pi$
$770$	0	0
$771$	8.48528	0.305590
$772$	0	0
$773$	−31.1127	−1.11905	−0.559523	−	0.828815i	$-0.689016\pi$
$773$	−31.1127	−1.11905	−0.559523	+	0.828815i	$0.689016\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	33.9411	1.21607
$780$	0	0
$781$	8.00000	0.286263
$782$	0	0
$783$	2.00000	0.0714742
$784$	0	0
$785$	48.0000	1.71319
$786$	0	0
$787$	4.00000	0.142585	0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000	0.142585	0.0712923	+	0.997455i	$0.477288\pi$
$788$	0	0
$789$	14.1421	0.503473
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−16.9706	−0.601884
$796$	0	0
$797$	25.4558	0.901692	0.450846	−	0.892602i	$-0.351122\pi$
$797$	25.4558	0.901692	0.450846	+	0.892602i	$0.351122\pi$
$798$	0	0
$799$	−22.6274	−0.800500
$800$	0	0
$801$	2.82843	0.0999376
$802$	0	0
$803$	16.0000	0.564628
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−8.48528	−0.298696
$808$	0	0
$809$	10.0000	0.351581	0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000	0.351581	0.175791	+	0.984428i	$0.443752\pi$
$810$	0	0
$811$	−44.0000	−1.54505	−0.772524	−	0.634985i	$-0.781006\pi$
$811$	−44.0000	−1.54505	−0.772524	+	0.634985i	$0.781006\pi$
$812$	0	0
$813$	−24.0000	−0.841717
$814$	0	0
$815$	−48.0000	−1.68137
$816$	0	0
$817$	45.2548	1.58327
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	0	0
$823$	−5.65685	−0.197186	−0.0985928	−	0.995128i	$-0.531434\pi$
$823$	−5.65685	−0.197186	−0.0985928	+	0.995128i	$0.531434\pi$
$824$	0	0
$825$	−8.48528	−0.295420
$826$	0	0
$827$	36.7696	1.27860	0.639301	−	0.768956i	$-0.279224\pi$
$827$	36.7696	1.27860	0.639301	+	0.768956i	$0.279224\pi$
$828$	0	0
$829$	−22.6274	−0.785883	−0.392941	−	0.919564i	$-0.628542\pi$
$829$	−22.6274	−0.785883	−0.392941	+	0.919564i	$0.628542\pi$
$830$	0	0
$831$	−10.0000	−0.346896
$832$	0	0
$833$	0	0
$834$	0	0
$835$	67.8823	2.34916
$836$	0	0
$837$	0	0
$838$	0	0
$839$	8.00000	0.276191	0.138095	−	0.990419i	$-0.455902\pi$
$839$	8.00000	0.276191	0.138095	+	0.990419i	$0.455902\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−6.00000	−0.206651
$844$	0	0
$845$	36.7696	1.26491
$846$	0	0
$847$	0	0
$848$	0	0
$849$	20.0000	0.686398
$850$	0	0
$851$	50.9117	1.74523
$852$	0	0
$853$	−39.5980	−1.35581	−0.677905	−	0.735150i	$-0.737112\pi$
$853$	−39.5980	−1.35581	−0.677905	+	0.735150i	$0.737112\pi$
$854$	0	0
$855$	11.3137	0.386921
$856$	0	0
$857$	2.82843	0.0966172	0.0483086	−	0.998832i	$-0.484617\pi$
$857$	2.82843	0.0966172	0.0483086	+	0.998832i	$0.484617\pi$
$858$	0	0
$859$	52.0000	1.77422	0.887109	−	0.461561i	$-0.152710\pi$
$859$	52.0000	1.77422	0.887109	+	0.461561i	$0.152710\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	31.1127	1.05909	0.529544	−	0.848282i	$-0.322363\pi$
$863$	31.1127	1.05909	0.529544	+	0.848282i	$0.322363\pi$
$864$	0	0
$865$	24.0000	0.816024
$866$	0	0
$867$	9.00000	0.305656
$868$	0	0
$869$	16.0000	0.542763
$870$	0	0
$871$	0	0
$872$	0	0
$873$	16.9706	0.574367
$874$	0	0
$875$	0	0
$876$	0	0
$877$	38.0000	1.28317	0.641584	−	0.767052i	$-0.278277\pi$
$877$	38.0000	1.28317	0.641584	+	0.767052i	$0.278277\pi$
$878$	0	0
$879$	8.48528	0.286201
$880$	0	0
$881$	31.1127	1.04821	0.524107	−	0.851653i	$-0.324399\pi$
$881$	31.1127	1.04821	0.524107	+	0.851653i	$0.324399\pi$
$882$	0	0
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	0	0
$885$	−33.9411	−1.14092
$886$	0	0
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	2.82843	0.0947559
$892$	0	0
$893$	−32.0000	−1.07084
$894$	0	0
$895$	8.00000	0.267411
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	16.9706	0.565371
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−64.0000	−2.12743
$906$	0	0
$907$	−33.9411	−1.12700	−0.563498	−	0.826117i	$-0.690545\pi$
$907$	−33.9411	−1.12700	−0.563498	+	0.826117i	$0.690545\pi$
$908$	0	0
$909$	14.1421	0.469065
$910$	0	0
$911$	−42.4264	−1.40565	−0.702825	−	0.711363i	$-0.748078\pi$
$911$	−42.4264	−1.40565	−0.702825	+	0.711363i	$0.748078\pi$
$912$	0	0
$913$	11.3137	0.374429
$914$	0	0
$915$	16.0000	0.528944
$916$	0	0
$917$	0	0
$918$	0	0
$919$	11.3137	0.373205	0.186602	−	0.982436i	$-0.440252\pi$
$919$	11.3137	0.373205	0.186602	+	0.982436i	$0.440252\pi$
$920$	0	0
$921$	4.00000	0.131804
$922$	0	0
$923$	0	0
$924$	0	0
$925$	18.0000	0.591836
$926$	0	0
$927$	16.0000	0.525509
$928$	0	0
$929$	36.7696	1.20637	0.603185	−	0.797601i	$-0.293898\pi$
$929$	36.7696	1.20637	0.603185	+	0.797601i	$0.293898\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	22.6274	0.739996
$936$	0	0
$937$	−22.6274	−0.739205	−0.369603	−	0.929190i	$-0.620506\pi$
$937$	−22.6274	−0.739205	−0.369603	+	0.929190i	$0.620506\pi$
$938$	0	0
$939$	−22.6274	−0.738418
$940$	0	0
$941$	−14.1421	−0.461020	−0.230510	−	0.973070i	$-0.574040\pi$
$941$	−14.1421	−0.461020	−0.230510	+	0.973070i	$0.574040\pi$
$942$	0	0
$943$	−72.0000	−2.34464
$944$	0	0
$945$	0	0
$946$	0	0
$947$	14.1421	0.459558	0.229779	−	0.973243i	$-0.426200\pi$
$947$	14.1421	0.459558	0.229779	+	0.973243i	$0.426200\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	30.0000	0.972817
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	0	0
$955$	56.0000	1.81212
$956$	0	0
$957$	5.65685	0.182860
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	2.82843	0.0911448
$964$	0	0
$965$	−28.2843	−0.910503
$966$	0	0
$967$	−39.5980	−1.27339	−0.636693	−	0.771118i	$-0.719698\pi$
$967$	−39.5980	−1.27339	−0.636693	+	0.771118i	$0.719698\pi$
$968$	0	0
$969$	−11.3137	−0.363449
$970$	0	0
$971$	52.0000	1.66876	0.834380	−	0.551190i	$-0.185826\pi$
$971$	52.0000	1.66876	0.834380	+	0.551190i	$0.185826\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−34.0000	−1.08776	−0.543878	−	0.839164i	$-0.683045\pi$
$977$	−34.0000	−1.08776	−0.543878	+	0.839164i	$0.683045\pi$
$978$	0	0
$979$	8.00000	0.255681
$980$	0	0
$981$	−10.0000	−0.319275
$982$	0	0
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	0	0
$985$	−28.2843	−0.901212
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−96.0000	−3.05262
$990$	0	0
$991$	33.9411	1.07818	0.539088	−	0.842250i	$-0.318769\pi$
$991$	33.9411	1.07818	0.539088	+	0.842250i	$0.318769\pi$
$992$	0	0
$993$	22.6274	0.718059
$994$	0	0
$995$	−22.6274	−0.717337
$996$	0	0
$997$	50.9117	1.61239	0.806195	−	0.591650i	$-0.201523\pi$
$997$	50.9117	1.61239	0.806195	+	0.591650i	$0.201523\pi$
$998$	0	0
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9408.2.a.dk.1.1		2
4.3	odd	2		9408.2.a.eb.1.1		2
7.6	odd	2		9408.2.a.eb.1.2		2
8.3	odd	2		4704.2.a.bi.1.2	yes	2
8.5	even	2		4704.2.a.br.1.2	yes	2
28.27	even	2	inner	9408.2.a.dk.1.2		2
56.13	odd	2		4704.2.a.bi.1.1	✓	2
56.27	even	2		4704.2.a.br.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4704.2.a.bi.1.1	✓	2	56.13	odd	2
4704.2.a.bi.1.2	yes	2	8.3	odd	2
4704.2.a.br.1.1	yes	2	56.27	even	2
4704.2.a.br.1.2	yes	2	8.5	even	2
9408.2.a.dk.1.1		2	1.1	even	1	trivial
9408.2.a.dk.1.2		2	28.27	even	2	inner
9408.2.a.eb.1.1		2	4.3	odd	2
9408.2.a.eb.1.2		2	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.82843 q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.82843 q^{5} +1.00000 q^{9} +2.82843 q^{11} +2.82843 q^{15} -2.82843 q^{17} -4.00000 q^{19} +8.48528 q^{23} +3.00000 q^{25} -1.00000 q^{27} -2.00000 q^{29} -2.82843 q^{33} +6.00000 q^{37} -8.48528 q^{41} -11.3137 q^{43} -2.82843 q^{45} +8.00000 q^{47} +2.82843 q^{51} -6.00000 q^{53} -8.00000 q^{55} +4.00000 q^{57} -12.0000 q^{59} +5.65685 q^{61} +5.65685 q^{67} -8.48528 q^{69} +2.82843 q^{71} +5.65685 q^{73} -3.00000 q^{75} +5.65685 q^{79} +1.00000 q^{81} +4.00000 q^{83} +8.00000 q^{85} +2.00000 q^{87} +2.82843 q^{89} +11.3137 q^{95} +16.9706 q^{97} +2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{9} - 8 q^{19} + 6 q^{25} - 2 q^{27} - 4 q^{29} + 12 q^{37} + 16 q^{47} - 12 q^{53} - 16 q^{55} + 8 q^{57} - 24 q^{59} - 6 q^{75} + 2 q^{81} + 8 q^{83} + 16 q^{85} + 4 q^{87}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^9 - 8 * q^19 + 6 * q^25 - 2 * q^27 - 4 * q^29 + 12 * q^37 + 16 * q^47 - 12 * q^53 - 16 * q^55 + 8 * q^57 - 24 * q^59 - 6 * q^75 + 2 * q^81 + 8 * q^83 + 16 * q^85 + 4 * q^87

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