LMFDB - Embedded newform 9216.2.a.f.1.1

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:	$2$
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 1536)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q-1.41421 q^{5} +O(q^{10})$	$q-1.41421 q^{5} -4.00000 q^{11} -1.41421 q^{13} +4.00000 q^{19} -5.65685 q^{23} -3.00000 q^{25} +7.07107 q^{29} +5.65685 q^{31} +4.24264 q^{37} +12.0000 q^{43} +11.3137 q^{47} -7.00000 q^{49} -1.41421 q^{53} +5.65685 q^{55} +4.00000 q^{59} +12.7279 q^{61} +2.00000 q^{65} +4.00000 q^{67} -5.65685 q^{71} -10.0000 q^{73} -16.9706 q^{79} -12.0000 q^{83} -6.00000 q^{89} -5.65685 q^{95} +8.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q - 8 q^{11} + 8 q^{19} - 6 q^{25} + 24 q^{43} - 14 q^{49} + 8 q^{59} + 4 q^{65} + 8 q^{67} - 20 q^{73} - 24 q^{83} - 12 q^{89} + 16 q^{97}+O(q^{100}) $ 2 * q - 8 * q^11 + 8 * q^19 - 6 * q^25 + 24 * q^43 - 14 * q^49 + 8 * q^59 + 4 * q^65 + 8 * q^67 - 20 * q^73 - 24 * q^83 - 12 * q^89 + 16 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.41421	−0.632456	−0.316228	−	0.948683i	$-0.602416\pi$
$5$	−1.41421	−0.632456	−0.316228	+	0.948683i	$0.602416\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	−1.41421	−0.392232	−0.196116	−	0.980581i	$-0.562833\pi$
$13$	−1.41421	−0.392232	−0.196116	+	0.980581i	$0.562833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.65685	−1.17954	−0.589768	−	0.807573i	$-0.700781\pi$
$23$	−5.65685	−1.17954	−0.589768	+	0.807573i	$0.700781\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.07107	1.31306	0.656532	−	0.754298i	$-0.272023\pi$
$29$	7.07107	1.31306	0.656532	+	0.754298i	$0.272023\pi$
$30$	0	0
$31$	5.65685	1.01600	0.508001	−	0.861357i	$-0.330385\pi$
$31$	5.65685	1.01600	0.508001	+	0.861357i	$0.330385\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.24264	0.697486	0.348743	−	0.937218i	$-0.386609\pi$
$37$	4.24264	0.697486	0.348743	+	0.937218i	$0.386609\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	12.0000	1.82998	0.914991	−	0.403473i	$-0.132197\pi$
$43$	12.0000	1.82998	0.914991	+	0.403473i	$0.132197\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.3137	1.65027	0.825137	−	0.564933i	$-0.191098\pi$
$47$	11.3137	1.65027	0.825137	+	0.564933i	$0.191098\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1.41421	−0.194257	−0.0971286	−	0.995272i	$-0.530966\pi$
$53$	−1.41421	−0.194257	−0.0971286	+	0.995272i	$0.530966\pi$
$54$	0	0
$55$	5.65685	0.762770
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	12.7279	1.62964	0.814822	−	0.579712i	$-0.196835\pi$
$61$	12.7279	1.62964	0.814822	+	0.579712i	$0.196835\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−5.65685	−0.671345	−0.335673	−	0.941979i	$-0.608964\pi$
$71$	−5.65685	−0.671345	−0.335673	+	0.941979i	$0.608964\pi$
$72$	0	0
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−16.9706	−1.90934	−0.954669	−	0.297670i	$-0.903790\pi$
$79$	−16.9706	−1.90934	−0.954669	+	0.297670i	$0.903790\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−5.65685	−0.580381
$96$	0	0
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.41421	0.140720	0.0703598	−	0.997522i	$-0.477585\pi$
$101$	1.41421	0.140720	0.0703598	+	0.997522i	$0.477585\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	−9.89949	−0.948200	−0.474100	−	0.880471i	$-0.657226\pi$
$109$	−9.89949	−0.948200	−0.474100	+	0.880471i	$0.657226\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	8.00000	0.746004
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.3137	1.01193
$126$	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
$129$	0	0
$130$	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
$135$	0	0
$136$	0	0
$137$	−16.0000	−1.36697	−0.683486	−	0.729964i	$-0.739537\pi$
$137$	−16.0000	−1.36697	−0.683486	+	0.729964i	$0.739537\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	5.65685	0.473050
$144$	0	0
$145$	−10.0000	−0.830455
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−21.2132	−1.73785	−0.868927	−	0.494941i	$-0.835190\pi$
$149$	−21.2132	−1.73785	−0.868927	+	0.494941i	$0.835190\pi$
$150$	0	0
$151$	11.3137	0.920697	0.460348	−	0.887738i	$-0.347725\pi$
$151$	11.3137	0.920697	0.460348	+	0.887738i	$0.347725\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−8.00000	−0.642575
$156$	0	0
$157$	12.7279	1.01580	0.507899	−	0.861416i	$-0.330422\pi$
$157$	12.7279	1.01580	0.507899	+	0.861416i	$0.330422\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.65685	0.437741	0.218870	−	0.975754i	$-0.429763\pi$
$167$	5.65685	0.437741	0.218870	+	0.975754i	$0.429763\pi$
$168$	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
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	−18.3848	−1.36653	−0.683265	−	0.730171i	$-0.739441\pi$
$181$	−18.3848	−1.36653	−0.683265	+	0.730171i	$0.739441\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−6.00000	−0.441129
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	22.6274	1.63726	0.818631	−	0.574320i	$-0.194733\pi$
$191$	22.6274	1.63726	0.818631	+	0.574320i	$0.194733\pi$
$192$	0	0
$193$	24.0000	1.72756	0.863779	−	0.503871i	$-0.168091\pi$
$193$	24.0000	1.72756	0.863779	+	0.503871i	$0.168091\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−12.7279	−0.906827	−0.453413	−	0.891300i	$-0.649794\pi$
$197$	−12.7279	−0.906827	−0.453413	+	0.891300i	$0.649794\pi$
$198$	0	0
$199$	22.6274	1.60402	0.802008	−	0.597314i	$-0.203765\pi$
$199$	22.6274	1.60402	0.802008	+	0.597314i	$0.203765\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−16.0000	−1.10674
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−16.9706	−1.15738
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	5.65685	0.378811	0.189405	−	0.981899i	$-0.439344\pi$
$223$	5.65685	0.378811	0.189405	+	0.981899i	$0.439344\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−20.0000	−1.32745	−0.663723	−	0.747978i	$-0.731025\pi$
$227$	−20.0000	−1.32745	−0.663723	+	0.747978i	$0.731025\pi$
$228$	0	0
$229$	−4.24264	−0.280362	−0.140181	−	0.990126i	$-0.544768\pi$
$229$	−4.24264	−0.280362	−0.140181	+	0.990126i	$0.544768\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−22.0000	−1.44127	−0.720634	−	0.693316i	$-0.756149\pi$
$233$	−22.0000	−1.44127	−0.720634	+	0.693316i	$0.756149\pi$
$234$	0	0
$235$	−16.0000	−1.04372
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−11.3137	−0.731823	−0.365911	−	0.930650i	$-0.619243\pi$
$239$	−11.3137	−0.731823	−0.365911	+	0.930650i	$0.619243\pi$
$240$	0	0
$241$	−24.0000	−1.54598	−0.772988	−	0.634421i	$-0.781239\pi$
$241$	−24.0000	−1.54598	−0.772988	+	0.634421i	$0.781239\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	9.89949	0.632456
$246$	0	0
$247$	−5.65685	−0.359937
$248$	0	0
$249$	0	0
$250$	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
$253$	22.6274	1.42257
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−16.9706	−1.04645	−0.523225	−	0.852195i	$-0.675271\pi$
$263$	−16.9706	−1.04645	−0.523225	+	0.852195i	$0.675271\pi$
$264$	0	0
$265$	2.00000	0.122859
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−7.07107	−0.431131	−0.215565	−	0.976489i	$-0.569159\pi$
$269$	−7.07107	−0.431131	−0.215565	+	0.976489i	$0.569159\pi$
$270$	0	0
$271$	5.65685	0.343629	0.171815	−	0.985129i	$-0.445037\pi$
$271$	5.65685	0.343629	0.171815	+	0.985129i	$0.445037\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	12.0000	0.723627
$276$	0	0
$277$	−15.5563	−0.934690	−0.467345	−	0.884075i	$-0.654789\pi$
$277$	−15.5563	−0.934690	−0.467345	+	0.884075i	$0.654789\pi$
$278$	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	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	9.89949	0.578335	0.289167	−	0.957279i	$-0.406622\pi$
$293$	9.89949	0.578335	0.289167	+	0.957279i	$0.406622\pi$
$294$	0	0
$295$	−5.65685	−0.329355
$296$	0	0
$297$	0	0
$298$	0	0
$299$	8.00000	0.462652
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−18.0000	−1.03068
$306$	0	0
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	28.2843	1.60385	0.801927	−	0.597422i	$-0.203808\pi$
$311$	28.2843	1.60385	0.801927	+	0.597422i	$0.203808\pi$
$312$	0	0
$313$	8.00000	0.452187	0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000	0.452187	0.226093	+	0.974106i	$0.427405\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	29.6985	1.66803	0.834017	−	0.551739i	$-0.186036\pi$
$317$	29.6985	1.66803	0.834017	+	0.551739i	$0.186036\pi$
$318$	0	0
$319$	−28.2843	−1.58362
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	4.24264	0.235339
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−5.65685	−0.309067
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−22.6274	−1.22534
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−4.00000	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	0	0
$349$	−1.41421	−0.0757011	−0.0378506	−	0.999283i	$-0.512051\pi$
$349$	−1.41421	−0.0757011	−0.0378506	+	0.999283i	$0.512051\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−30.0000	−1.59674	−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000	−1.59674	−0.798369	+	0.602168i	$0.794304\pi$
$354$	0	0
$355$	8.00000	0.424596
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5.65685	−0.298557	−0.149279	−	0.988795i	$-0.547695\pi$
$359$	−5.65685	−0.298557	−0.149279	+	0.988795i	$0.547695\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	0	0
$364$	0	0
$365$	14.1421	0.740233
$366$	0	0
$367$	28.2843	1.47643	0.738213	−	0.674567i	$-0.235670\pi$
$367$	28.2843	1.47643	0.738213	+	0.674567i	$0.235670\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−4.24264	−0.219676	−0.109838	−	0.993950i	$-0.535033\pi$
$373$	−4.24264	−0.219676	−0.109838	+	0.993950i	$0.535033\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−10.0000	−0.515026
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	11.3137	0.578103	0.289052	−	0.957313i	$-0.406660\pi$
$383$	11.3137	0.578103	0.289052	+	0.957313i	$0.406660\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−32.5269	−1.64918	−0.824590	−	0.565731i	$-0.808594\pi$
$389$	−32.5269	−1.64918	−0.824590	+	0.565731i	$0.808594\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	24.0000	1.20757
$396$	0	0
$397$	−21.2132	−1.06466	−0.532330	−	0.846537i	$-0.678683\pi$
$397$	−21.2132	−1.06466	−0.532330	+	0.846537i	$0.678683\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−16.0000	−0.799002	−0.399501	−	0.916733i	$-0.630817\pi$
$401$	−16.0000	−0.799002	−0.399501	+	0.916733i	$0.630817\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−16.9706	−0.841200
$408$	0	0
$409$	−8.00000	−0.395575	−0.197787	−	0.980245i	$-0.563376\pi$
$409$	−8.00000	−0.395575	−0.197787	+	0.980245i	$0.563376\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	16.9706	0.833052
$416$	0	0
$417$	0	0
$418$	0	0
$419$	36.0000	1.75872	0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000	1.75872	0.879358	+	0.476162i	$0.157972\pi$
$420$	0	0
$421$	−4.24264	−0.206774	−0.103387	−	0.994641i	$-0.532968\pi$
$421$	−4.24264	−0.206774	−0.103387	+	0.994641i	$0.532968\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−22.6274	−1.08992	−0.544962	−	0.838461i	$-0.683456\pi$
$431$	−22.6274	−1.08992	−0.544962	+	0.838461i	$0.683456\pi$
$432$	0	0
$433$	−8.00000	−0.384455	−0.192228	−	0.981350i	$-0.561571\pi$
$433$	−8.00000	−0.384455	−0.192228	+	0.981350i	$0.561571\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−22.6274	−1.08242
$438$	0	0
$439$	−33.9411	−1.61992	−0.809961	−	0.586484i	$-0.800512\pi$
$439$	−33.9411	−1.61992	−0.809961	+	0.586484i	$0.800512\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	0	0
$445$	8.48528	0.402241
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−8.00000	−0.374224	−0.187112	−	0.982339i	$-0.559913\pi$
$457$	−8.00000	−0.374224	−0.187112	+	0.982339i	$0.559913\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	4.24264	0.197599	0.0987997	−	0.995107i	$-0.468500\pi$
$461$	4.24264	0.197599	0.0987997	+	0.995107i	$0.468500\pi$
$462$	0	0
$463$	16.9706	0.788689	0.394344	−	0.918963i	$-0.370972\pi$
$463$	16.9706	0.788689	0.394344	+	0.918963i	$0.370972\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−48.0000	−2.20704
$474$	0	0
$475$	−12.0000	−0.550598
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−22.6274	−1.03387	−0.516937	−	0.856024i	$-0.672928\pi$
$479$	−22.6274	−1.03387	−0.516937	+	0.856024i	$0.672928\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−11.3137	−0.513729
$486$	0	0
$487$	−33.9411	−1.53802	−0.769010	−	0.639237i	$-0.779250\pi$
$487$	−33.9411	−1.53802	−0.769010	+	0.639237i	$0.779250\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	28.0000	1.26362	0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000	1.26362	0.631811	+	0.775122i	$0.282312\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	28.2843	1.26113	0.630567	−	0.776135i	$-0.282823\pi$
$503$	28.2843	1.26113	0.630567	+	0.776135i	$0.282823\pi$
$504$	0	0
$505$	−2.00000	−0.0889988
$506$	0	0
$507$	0	0
$508$	0	0
$509$	26.8701	1.19099	0.595497	−	0.803357i	$-0.296955\pi$
$509$	26.8701	1.19099	0.595497	+	0.803357i	$0.296955\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−45.2548	−1.99031
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	9.00000	0.391304
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	16.9706	0.733701
$536$	0	0
$537$	0	0
$538$	0	0
$539$	28.0000	1.20605
$540$	0	0
$541$	−21.2132	−0.912027	−0.456013	−	0.889973i	$-0.650723\pi$
$541$	−21.2132	−0.912027	−0.456013	+	0.889973i	$0.650723\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	14.0000	0.599694
$546$	0	0
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	28.2843	1.20495
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7.07107	0.299611	0.149805	−	0.988716i	$-0.452135\pi$
$557$	7.07107	0.299611	0.149805	+	0.988716i	$0.452135\pi$
$558$	0	0
$559$	−16.9706	−0.717778
$560$	0	0
$561$	0	0
$562$	0	0
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	0	0
$565$	19.7990	0.832950
$566$	0	0
$567$	0	0
$568$	0	0
$569$	32.0000	1.34151	0.670755	−	0.741679i	$-0.265970\pi$
$569$	32.0000	1.34151	0.670755	+	0.741679i	$0.265970\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	16.9706	0.707721
$576$	0	0
$577$	−34.0000	−1.41544	−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000	−1.41544	−0.707719	+	0.706494i	$0.750276\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	5.65685	0.234283
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−28.0000	−1.15568	−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000	−1.15568	−0.577842	+	0.816149i	$0.696105\pi$
$588$	0	0
$589$	22.6274	0.932346
$590$	0	0
$591$	0	0
$592$	0	0
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	5.65685	0.231133	0.115566	−	0.993300i	$-0.463132\pi$
$599$	5.65685	0.231133	0.115566	+	0.993300i	$0.463132\pi$
$600$	0	0
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−7.07107	−0.287480
$606$	0	0
$607$	39.5980	1.60723	0.803616	−	0.595148i	$-0.202907\pi$
$607$	39.5980	1.60723	0.803616	+	0.595148i	$0.202907\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−16.0000	−0.647291
$612$	0	0
$613$	29.6985	1.19951	0.599755	−	0.800184i	$-0.295265\pi$
$613$	29.6985	1.19951	0.599755	+	0.800184i	$0.295265\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	22.0000	0.885687	0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000	0.885687	0.442843	+	0.896599i	$0.353970\pi$
$618$	0	0
$619$	−36.0000	−1.44696	−0.723481	−	0.690344i	$-0.757459\pi$
$619$	−36.0000	−1.44696	−0.723481	+	0.690344i	$0.757459\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−33.9411	−1.35117	−0.675587	−	0.737280i	$-0.736110\pi$
$631$	−33.9411	−1.35117	−0.675587	+	0.737280i	$0.736110\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	24.0000	0.952411
$636$	0	0
$637$	9.89949	0.392232
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−32.0000	−1.26392	−0.631962	−	0.774999i	$-0.717750\pi$
$641$	−32.0000	−1.26392	−0.631962	+	0.774999i	$0.717750\pi$
$642$	0	0
$643$	−20.0000	−0.788723	−0.394362	−	0.918955i	$-0.629034\pi$
$643$	−20.0000	−0.788723	−0.394362	+	0.918955i	$0.629034\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	28.2843	1.11197	0.555985	−	0.831193i	$-0.312341\pi$
$647$	28.2843	1.11197	0.555985	+	0.831193i	$0.312341\pi$
$648$	0	0
$649$	−16.0000	−0.628055
$650$	0	0
$651$	0	0
$652$	0	0
$653$	26.8701	1.05151	0.525753	−	0.850637i	$-0.323784\pi$
$653$	26.8701	1.05151	0.525753	+	0.850637i	$0.323784\pi$
$654$	0	0
$655$	5.65685	0.221032
$656$	0	0
$657$	0	0
$658$	0	0
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	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
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−40.0000	−1.54881
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−50.9117	−1.96542
$672$	0	0
$673$	−8.00000	−0.308377	−0.154189	−	0.988041i	$-0.549276\pi$
$673$	−8.00000	−0.308377	−0.154189	+	0.988041i	$0.549276\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−9.89949	−0.380468	−0.190234	−	0.981739i	$-0.560925\pi$
$677$	−9.89949	−0.380468	−0.190234	+	0.981739i	$0.560925\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	0	0
$685$	22.6274	0.864549
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.00000	0.0761939
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−5.65685	−0.214577
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	4.24264	0.160242	0.0801212	−	0.996785i	$-0.474469\pi$
$701$	4.24264	0.160242	0.0801212	+	0.996785i	$0.474469\pi$
$702$	0	0
$703$	16.9706	0.640057
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−7.07107	−0.265560	−0.132780	−	0.991146i	$-0.542390\pi$
$709$	−7.07107	−0.265560	−0.132780	+	0.991146i	$0.542390\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−32.0000	−1.19841
$714$	0	0
$715$	−8.00000	−0.299183
$716$	0	0
$717$	0	0
$718$	0	0
$719$	33.9411	1.26579	0.632895	−	0.774237i	$-0.281866\pi$
$719$	33.9411	1.26579	0.632895	+	0.774237i	$0.281866\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−21.2132	−0.787839
$726$	0	0
$727$	33.9411	1.25881	0.629403	−	0.777079i	$-0.283299\pi$
$727$	33.9411	1.25881	0.629403	+	0.777079i	$0.283299\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−12.7279	−0.470117	−0.235058	−	0.971981i	$-0.575528\pi$
$733$	−12.7279	−0.470117	−0.235058	+	0.971981i	$0.575528\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−16.0000	−0.589368
$738$	0	0
$739$	−36.0000	−1.32428	−0.662141	−	0.749380i	$-0.730352\pi$
$739$	−36.0000	−1.32428	−0.662141	+	0.749380i	$0.730352\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	28.2843	1.03765	0.518825	−	0.854881i	$-0.326370\pi$
$743$	28.2843	1.03765	0.518825	+	0.854881i	$0.326370\pi$
$744$	0	0
$745$	30.0000	1.09911
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−28.2843	−1.03211	−0.516054	−	0.856556i	$-0.672600\pi$
$751$	−28.2843	−1.03211	−0.516054	+	0.856556i	$0.672600\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−16.0000	−0.582300
$756$	0	0
$757$	26.8701	0.976609	0.488304	−	0.872673i	$-0.337616\pi$
$757$	26.8701	0.976609	0.488304	+	0.872673i	$0.337616\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−48.0000	−1.74000	−0.869999	−	0.493053i	$-0.835881\pi$
$761$	−48.0000	−1.74000	−0.869999	+	0.493053i	$0.835881\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5.65685	−0.204257
$768$	0	0
$769$	−8.00000	−0.288487	−0.144244	−	0.989542i	$-0.546075\pi$
$769$	−8.00000	−0.288487	−0.144244	+	0.989542i	$0.546075\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−43.8406	−1.57684	−0.788419	−	0.615139i	$-0.789100\pi$
$773$	−43.8406	−1.57684	−0.788419	+	0.615139i	$0.789100\pi$
$774$	0	0
$775$	−16.9706	−0.609601
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	22.6274	0.809673
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−18.0000	−0.642448
$786$	0	0
$787$	−52.0000	−1.85360	−0.926800	−	0.375555i	$-0.877452\pi$
$787$	−52.0000	−1.85360	−0.926800	+	0.375555i	$0.877452\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−18.0000	−0.639199
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−52.3259	−1.85348	−0.926739	−	0.375705i	$-0.877401\pi$
$797$	−52.3259	−1.85348	−0.926739	+	0.375705i	$0.877401\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	40.0000	1.41157
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−48.0000	−1.68759	−0.843795	−	0.536666i	$-0.819684\pi$
$809$	−48.0000	−1.68759	−0.843795	+	0.536666i	$0.819684\pi$
$810$	0	0
$811$	52.0000	1.82597	0.912983	−	0.407997i	$-0.133772\pi$
$811$	52.0000	1.82597	0.912983	+	0.407997i	$0.133772\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−5.65685	−0.198151
$816$	0	0
$817$	48.0000	1.67931
$818$	0	0
$819$	0	0
$820$	0	0
$821$	43.8406	1.53005	0.765024	−	0.644002i	$-0.222727\pi$
$821$	43.8406	1.53005	0.765024	+	0.644002i	$0.222727\pi$
$822$	0	0
$823$	−22.6274	−0.788742	−0.394371	−	0.918951i	$-0.629038\pi$
$823$	−22.6274	−0.788742	−0.394371	+	0.918951i	$0.629038\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	28.0000	0.973655	0.486828	−	0.873498i	$-0.338154\pi$
$827$	28.0000	0.973655	0.486828	+	0.873498i	$0.338154\pi$
$828$	0	0
$829$	−43.8406	−1.52265	−0.761324	−	0.648372i	$-0.775450\pi$
$829$	−43.8406	−1.52265	−0.761324	+	0.648372i	$0.775450\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−8.00000	−0.276851
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−39.5980	−1.36707	−0.683537	−	0.729916i	$-0.739559\pi$
$839$	−39.5980	−1.36707	−0.683537	+	0.729916i	$0.739559\pi$
$840$	0	0
$841$	21.0000	0.724138
$842$	0	0
$843$	0	0
$844$	0	0
$845$	15.5563	0.535155
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−24.0000	−0.822709
$852$	0	0
$853$	−4.24264	−0.145265	−0.0726326	−	0.997359i	$-0.523140\pi$
$853$	−4.24264	−0.145265	−0.0726326	+	0.997359i	$0.523140\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	32.0000	1.09310	0.546550	−	0.837427i	$-0.315941\pi$
$857$	32.0000	1.09310	0.546550	+	0.837427i	$0.315941\pi$
$858$	0	0
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−45.2548	−1.54049	−0.770246	−	0.637747i	$-0.779867\pi$
$863$	−45.2548	−1.54049	−0.770246	+	0.637747i	$0.779867\pi$
$864$	0	0
$865$	−10.0000	−0.340010
$866$	0	0
$867$	0	0
$868$	0	0
$869$	67.8823	2.30275
$870$	0	0
$871$	−5.65685	−0.191675
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	46.6690	1.57590	0.787951	−	0.615738i	$-0.211142\pi$
$877$	46.6690	1.57590	0.787951	+	0.615738i	$0.211142\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	14.0000	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$882$	0	0
$883$	36.0000	1.21150	0.605748	−	0.795656i	$-0.292874\pi$
$883$	36.0000	1.21150	0.605748	+	0.795656i	$0.292874\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−16.9706	−0.569816	−0.284908	−	0.958555i	$-0.591963\pi$
$887$	−16.9706	−0.569816	−0.284908	+	0.958555i	$0.591963\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	45.2548	1.51440
$894$	0	0
$895$	5.65685	0.189088
$896$	0	0
$897$	0	0
$898$	0	0
$899$	40.0000	1.33407
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	26.0000	0.864269
$906$	0	0
$907$	−4.00000	−0.132818	−0.0664089	−	0.997792i	$-0.521154\pi$
$907$	−4.00000	−0.132818	−0.0664089	+	0.997792i	$0.521154\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−22.6274	−0.749680	−0.374840	−	0.927090i	$-0.622302\pi$
$911$	−22.6274	−0.749680	−0.374840	+	0.927090i	$0.622302\pi$
$912$	0	0
$913$	48.0000	1.58857
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	33.9411	1.11961	0.559807	−	0.828623i	$-0.310875\pi$
$919$	33.9411	1.11961	0.559807	+	0.828623i	$0.310875\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	8.00000	0.263323
$924$	0	0
$925$	−12.7279	−0.418491
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−16.0000	−0.524943	−0.262471	−	0.964940i	$-0.584538\pi$
$929$	−16.0000	−0.524943	−0.262471	+	0.964940i	$0.584538\pi$
$930$	0	0
$931$	−28.0000	−0.917663
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−42.0000	−1.37208	−0.686040	−	0.727564i	$-0.740653\pi$
$937$	−42.0000	−1.37208	−0.686040	+	0.727564i	$0.740653\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	26.8701	0.875939	0.437969	−	0.898990i	$-0.355698\pi$
$941$	26.8701	0.875939	0.437969	+	0.898990i	$0.355698\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	20.0000	0.649913	0.324956	−	0.945729i	$-0.394650\pi$
$947$	20.0000	0.649913	0.324956	+	0.945729i	$0.394650\pi$
$948$	0	0
$949$	14.1421	0.459073
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	−32.0000	−1.03550
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−33.9411	−1.09260
$966$	0	0
$967$	−22.6274	−0.727649	−0.363824	−	0.931468i	$-0.618529\pi$
$967$	−22.6274	−0.727649	−0.363824	+	0.931468i	$0.618529\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	16.0000	0.511885	0.255943	−	0.966692i	$-0.417614\pi$
$977$	16.0000	0.511885	0.255943	+	0.966692i	$0.417614\pi$
$978$	0	0
$979$	24.0000	0.767043
$980$	0	0
$981$	0	0
$982$	0	0
$983$	28.2843	0.902128	0.451064	−	0.892492i	$-0.351045\pi$
$983$	28.2843	0.902128	0.451064	+	0.892492i	$0.351045\pi$
$984$	0	0
$985$	18.0000	0.573528
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−67.8823	−2.15853
$990$	0	0
$991$	16.9706	0.539088	0.269544	−	0.962988i	$-0.413127\pi$
$991$	16.9706	0.539088	0.269544	+	0.962988i	$0.413127\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−32.0000	−1.01447
$996$	0	0
$997$	−18.3848	−0.582252	−0.291126	−	0.956685i	$-0.594030\pi$
$997$	−18.3848	−0.582252	−0.291126	+	0.956685i	$0.594030\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9216.2.a.f.1.1		2
3.2	odd	2		3072.2.a.c.1.2		2
4.3	odd	2		9216.2.a.r.1.1		2
8.3	odd	2	inner	9216.2.a.f.1.2		2
8.5	even	2		9216.2.a.r.1.2		2
12.11	even	2		3072.2.a.e.1.2		2
24.5	odd	2		3072.2.a.e.1.1		2
24.11	even	2		3072.2.a.c.1.1		2
32.3	odd	8		4608.2.k.ba.1153.2		4
32.5	even	8		4608.2.k.z.3457.2		4
32.11	odd	8		4608.2.k.ba.3457.2		4
32.13	even	8		4608.2.k.z.1153.2		4
32.19	odd	8		4608.2.k.z.1153.1		4
32.21	even	8		4608.2.k.ba.3457.1		4
32.27	odd	8		4608.2.k.z.3457.1		4
32.29	even	8		4608.2.k.ba.1153.1		4
48.5	odd	4		3072.2.d.b.1537.4		4
48.11	even	4		3072.2.d.b.1537.2		4
48.29	odd	4		3072.2.d.b.1537.1		4
48.35	even	4		3072.2.d.b.1537.3		4
96.5	odd	8		1536.2.j.d.385.2	yes	4
96.11	even	8		1536.2.j.a.385.2	yes	4
96.29	odd	8		1536.2.j.a.1153.1	yes	4
96.35	even	8		1536.2.j.a.1153.2	yes	4
96.53	odd	8		1536.2.j.a.385.1	✓	4
96.59	even	8		1536.2.j.d.385.1	yes	4
96.77	odd	8		1536.2.j.d.1153.2	yes	4
96.83	even	8		1536.2.j.d.1153.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.j.a.385.1	✓	4	96.53	odd	8
1536.2.j.a.385.2	yes	4	96.11	even	8
1536.2.j.a.1153.1	yes	4	96.29	odd	8
1536.2.j.a.1153.2	yes	4	96.35	even	8
1536.2.j.d.385.1	yes	4	96.59	even	8
1536.2.j.d.385.2	yes	4	96.5	odd	8
1536.2.j.d.1153.1	yes	4	96.83	even	8
1536.2.j.d.1153.2	yes	4	96.77	odd	8
3072.2.a.c.1.1		2	24.11	even	2
3072.2.a.c.1.2		2	3.2	odd	2
3072.2.a.e.1.1		2	24.5	odd	2
3072.2.a.e.1.2		2	12.11	even	2
3072.2.d.b.1537.1		4	48.29	odd	4
3072.2.d.b.1537.2		4	48.11	even	4
3072.2.d.b.1537.3		4	48.35	even	4
3072.2.d.b.1537.4		4	48.5	odd	4
4608.2.k.z.1153.1		4	32.19	odd	8
4608.2.k.z.1153.2		4	32.13	even	8
4608.2.k.z.3457.1		4	32.27	odd	8
4608.2.k.z.3457.2		4	32.5	even	8
4608.2.k.ba.1153.1		4	32.29	even	8
4608.2.k.ba.1153.2		4	32.3	odd	8
4608.2.k.ba.3457.1		4	32.21	even	8
4608.2.k.ba.3457.2		4	32.11	odd	8
9216.2.a.f.1.1		2	1.1	even	1	trivial
9216.2.a.f.1.2		2	8.3	odd	2	inner
9216.2.a.r.1.1		2	4.3	odd	2
9216.2.a.r.1.2		2	8.5	even	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)

\(f(q)\)	\(=\)	\(q-1.41421 q^{5} +O(q^{10})\)	\(q-1.41421 q^{5} -4.00000 q^{11} -1.41421 q^{13} +4.00000 q^{19} -5.65685 q^{23} -3.00000 q^{25} +7.07107 q^{29} +5.65685 q^{31} +4.24264 q^{37} +12.0000 q^{43} +11.3137 q^{47} -7.00000 q^{49} -1.41421 q^{53} +5.65685 q^{55} +4.00000 q^{59} +12.7279 q^{61} +2.00000 q^{65} +4.00000 q^{67} -5.65685 q^{71} -10.0000 q^{73} -16.9706 q^{79} -12.0000 q^{83} -6.00000 q^{89} -5.65685 q^{95} +8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q - 8 q^{11} + 8 q^{19} - 6 q^{25} + 24 q^{43} - 14 q^{49} + 8 q^{59} + 4 q^{65} + 8 q^{67} - 20 q^{73} - 24 q^{83} - 12 q^{89} + 16 q^{97}+O(q^{100}) \) 2 * q - 8 * q^11 + 8 * q^19 - 6 * q^25 + 24 * q^43 - 14 * q^49 + 8 * q^59 + 4 * q^65 + 8 * q^67 - 20 * q^73 - 24 * q^83 - 12 * q^89 + 16 * q^97

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