LMFDB - Embedded newform 2160.2.a.ba.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,2,Mod(1,2160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, 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("2160.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2160.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:	$17.2476868366$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 135)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	2160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -4.60555 q^{7} +O(q^{10})$	$q+1.00000 q^{5} -4.60555 q^{7} +2.60555 q^{11} -0.605551 q^{13} -5.60555 q^{17} +3.60555 q^{19} +3.00000 q^{23} +1.00000 q^{25} +8.60555 q^{29} -1.60555 q^{31} -4.60555 q^{35} +2.00000 q^{37} +2.60555 q^{41} +6.60555 q^{43} -5.21110 q^{47} +14.2111 q^{49} -5.60555 q^{53} +2.60555 q^{55} +8.60555 q^{59} +10.2111 q^{61} -0.605551 q^{65} +15.2111 q^{67} -14.6056 q^{71} +5.39445 q^{73} -12.0000 q^{77} +4.39445 q^{79} -3.00000 q^{83} -5.60555 q^{85} +7.81665 q^{89} +2.78890 q^{91} +3.60555 q^{95} +8.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q + 2 * q^5 - 2 * q^7	$ 2 q + 2 q^{5} - 2 q^{7} - 2 q^{11} + 6 q^{13} - 4 q^{17} + 6 q^{23} + 2 q^{25} + 10 q^{29} + 4 q^{31} - 2 q^{35} + 4 q^{37} - 2 q^{41} + 6 q^{43} + 4 q^{47} + 14 q^{49} - 4 q^{53} - 2 q^{55} + 10 q^{59} + 6 q^{61} + 6 q^{65} + 16 q^{67} - 22 q^{71} + 18 q^{73} - 24 q^{77} + 16 q^{79} - 6 q^{83} - 4 q^{85} - 6 q^{89} + 20 q^{91} + 16 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 - 2 * q^7 - 2 * q^11 + 6 * q^13 - 4 * q^17 + 6 * q^23 + 2 * q^25 + 10 * q^29 + 4 * q^31 - 2 * q^35 + 4 * q^37 - 2 * q^41 + 6 * q^43 + 4 * q^47 + 14 * q^49 - 4 * q^53 - 2 * q^55 + 10 * q^59 + 6 * q^61 + 6 * q^65 + 16 * q^67 - 22 * q^71 + 18 * q^73 - 24 * q^77 + 16 * q^79 - 6 * q^83 - 4 * q^85 - 6 * q^89 + 20 * q^91 + 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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.60555	−1.74073	−0.870367	−	0.492403i	$-0.836119\pi$
$7$	−4.60555	−1.74073	−0.870367	+	0.492403i	$0.836119\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.60555	0.785603	0.392802	−	0.919623i	$-0.371506\pi$
$11$	2.60555	0.785603	0.392802	+	0.919623i	$0.371506\pi$
$12$	0	0
$13$	−0.605551	−0.167950	−0.0839749	−	0.996468i	$-0.526762\pi$
$13$	−0.605551	−0.167950	−0.0839749	+	0.996468i	$0.526762\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.60555	−1.35955	−0.679773	−	0.733423i	$-0.737922\pi$
$17$	−5.60555	−1.35955	−0.679773	+	0.733423i	$0.737922\pi$
$18$	0	0
$19$	3.60555	0.827170	0.413585	−	0.910465i	$-0.364276\pi$
$19$	3.60555	0.827170	0.413585	+	0.910465i	$0.364276\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8.60555	1.59801	0.799005	−	0.601324i	$-0.205360\pi$
$29$	8.60555	1.59801	0.799005	+	0.601324i	$0.205360\pi$
$30$	0	0
$31$	−1.60555	−0.288366	−0.144183	−	0.989551i	$-0.546055\pi$
$31$	−1.60555	−0.288366	−0.144183	+	0.989551i	$0.546055\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.60555	−0.778480
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.60555	0.406919	0.203459	−	0.979083i	$-0.434782\pi$
$41$	2.60555	0.406919	0.203459	+	0.979083i	$0.434782\pi$
$42$	0	0
$43$	6.60555	1.00734	0.503669	−	0.863897i	$-0.331983\pi$
$43$	6.60555	1.00734	0.503669	+	0.863897i	$0.331983\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.21110	−0.760117	−0.380059	−	0.924962i	$-0.624096\pi$
$47$	−5.21110	−0.760117	−0.380059	+	0.924962i	$0.624096\pi$
$48$	0	0
$49$	14.2111	2.03016
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−5.60555	−0.769982	−0.384991	−	0.922920i	$-0.625795\pi$
$53$	−5.60555	−0.769982	−0.384991	+	0.922920i	$0.625795\pi$
$54$	0	0
$55$	2.60555	0.351332
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.60555	1.12035	0.560174	−	0.828375i	$-0.310734\pi$
$59$	8.60555	1.12035	0.560174	+	0.828375i	$0.310734\pi$
$60$	0	0
$61$	10.2111	1.30740	0.653699	−	0.756755i	$-0.273216\pi$
$61$	10.2111	1.30740	0.653699	+	0.756755i	$0.273216\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.605551	−0.0751094
$66$	0	0
$67$	15.2111	1.85833	0.929166	−	0.369663i	$-0.120527\pi$
$67$	15.2111	1.85833	0.929166	+	0.369663i	$0.120527\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−14.6056	−1.73336	−0.866680	−	0.498864i	$-0.833751\pi$
$71$	−14.6056	−1.73336	−0.866680	+	0.498864i	$0.833751\pi$
$72$	0	0
$73$	5.39445	0.631372	0.315686	−	0.948864i	$-0.397765\pi$
$73$	5.39445	0.631372	0.315686	+	0.948864i	$0.397765\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−12.0000	−1.36753
$78$	0	0
$79$	4.39445	0.494414	0.247207	−	0.968963i	$-0.420487\pi$
$79$	4.39445	0.494414	0.247207	+	0.968963i	$0.420487\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−3.00000	−0.329293	−0.164646	−	0.986353i	$-0.552648\pi$
$83$	−3.00000	−0.329293	−0.164646	+	0.986353i	$0.552648\pi$
$84$	0	0
$85$	−5.60555	−0.608007
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7.81665	0.828564	0.414282	−	0.910149i	$-0.364033\pi$
$89$	7.81665	0.828564	0.414282	+	0.910149i	$0.364033\pi$
$90$	0	0
$91$	2.78890	0.292356
$92$	0	0
$93$	0	0
$94$	0	0
$95$	3.60555	0.369922
$96$	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
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−7.00000	−0.670478	−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000	−0.670478	−0.335239	+	0.942133i	$0.608817\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−0.788897	−0.0742132	−0.0371066	−	0.999311i	$-0.511814\pi$
$113$	−0.788897	−0.0742132	−0.0371066	+	0.999311i	$0.511814\pi$
$114$	0	0
$115$	3.00000	0.279751
$116$	0	0
$117$	0	0
$118$	0	0
$119$	25.8167	2.36661
$120$	0	0
$121$	−4.21110	−0.382828
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	4.78890	0.424946	0.212473	−	0.977167i	$-0.431848\pi$
$127$	4.78890	0.424946	0.212473	+	0.977167i	$0.431848\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	−16.6056	−1.43988
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.81665	−0.411515	−0.205757	−	0.978603i	$-0.565966\pi$
$137$	−4.81665	−0.411515	−0.205757	+	0.978603i	$0.565966\pi$
$138$	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
$143$	−1.57779	−0.131942
$144$	0	0
$145$	8.60555	0.714652
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−13.0278	−1.06728	−0.533638	−	0.845713i	$-0.679175\pi$
$149$	−13.0278	−1.06728	−0.533638	+	0.845713i	$0.679175\pi$
$150$	0	0
$151$	14.4222	1.17366	0.586831	−	0.809709i	$-0.300375\pi$
$151$	14.4222	1.17366	0.586831	+	0.809709i	$0.300375\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.60555	−0.128961
$156$	0	0
$157$	3.81665	0.304602	0.152301	−	0.988334i	$-0.451332\pi$
$157$	3.81665	0.304602	0.152301	+	0.988334i	$0.451332\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−13.8167	−1.08890
$162$	0	0
$163$	−2.00000	−0.156652	−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000	−0.156652	−0.0783260	+	0.996928i	$0.524958\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.00000	0.232147	0.116073	−	0.993241i	$-0.462969\pi$
$167$	3.00000	0.232147	0.116073	+	0.993241i	$0.462969\pi$
$168$	0	0
$169$	−12.6333	−0.971793
$170$	0	0
$171$	0	0
$172$	0	0
$173$	10.8167	0.822375	0.411187	−	0.911551i	$-0.365114\pi$
$173$	10.8167	0.822375	0.411187	+	0.911551i	$0.365114\pi$
$174$	0	0
$175$	−4.60555	−0.348147
$176$	0	0
$177$	0	0
$178$	0	0
$179$	6.78890	0.507426	0.253713	−	0.967280i	$-0.418348\pi$
$179$	6.78890	0.507426	0.253713	+	0.967280i	$0.418348\pi$
$180$	0	0
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2.00000	0.147043
$186$	0	0
$187$	−14.6056	−1.06806
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−16.4222	−1.18827	−0.594135	−	0.804366i	$-0.702505\pi$
$191$	−16.4222	−1.18827	−0.594135	+	0.804366i	$0.702505\pi$
$192$	0	0
$193$	21.8167	1.57040	0.785199	−	0.619244i	$-0.212561\pi$
$193$	21.8167	1.57040	0.785199	+	0.619244i	$0.212561\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.18335	−0.0843099	−0.0421550	−	0.999111i	$-0.513422\pi$
$197$	−1.18335	−0.0843099	−0.0421550	+	0.999111i	$0.513422\pi$
$198$	0	0
$199$	−13.2111	−0.936510	−0.468255	−	0.883593i	$-0.655117\pi$
$199$	−13.2111	−0.936510	−0.468255	+	0.883593i	$0.655117\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−39.6333	−2.78171
$204$	0	0
$205$	2.60555	0.181980
$206$	0	0
$207$	0	0
$208$	0	0
$209$	9.39445	0.649828
$210$	0	0
$211$	−12.8167	−0.882335	−0.441167	−	0.897425i	$-0.645436\pi$
$211$	−12.8167	−0.882335	−0.441167	+	0.897425i	$0.645436\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	6.60555	0.450495
$216$	0	0
$217$	7.39445	0.501968
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3.39445	0.228335
$222$	0	0
$223$	10.0000	0.669650	0.334825	−	0.942280i	$-0.391323\pi$
$223$	10.0000	0.669650	0.334825	+	0.942280i	$0.391323\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	26.2111	1.73969	0.869846	−	0.493323i	$-0.164218\pi$
$227$	26.2111	1.73969	0.869846	+	0.493323i	$0.164218\pi$
$228$	0	0
$229$	−6.21110	−0.410441	−0.205221	−	0.978716i	$-0.565791\pi$
$229$	−6.21110	−0.410441	−0.205221	+	0.978716i	$0.565791\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	−5.21110	−0.339935
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0.788897	0.0510295	0.0255148	−	0.999674i	$-0.491878\pi$
$239$	0.788897	0.0510295	0.0255148	+	0.999674i	$0.491878\pi$
$240$	0	0
$241$	28.2111	1.81724	0.908618	−	0.417627i	$-0.137138\pi$
$241$	28.2111	1.81724	0.908618	+	0.417627i	$0.137138\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	14.2111	0.907914
$246$	0	0
$247$	−2.18335	−0.138923
$248$	0	0
$249$	0	0
$250$	0	0
$251$	15.6333	0.986766	0.493383	−	0.869812i	$-0.335760\pi$
$251$	15.6333	0.986766	0.493383	+	0.869812i	$0.335760\pi$
$252$	0	0
$253$	7.81665	0.491429
$254$	0	0
$255$	0	0
$256$	0	0
$257$	22.8167	1.42326	0.711632	−	0.702553i	$-0.247956\pi$
$257$	22.8167	1.42326	0.711632	+	0.702553i	$0.247956\pi$
$258$	0	0
$259$	−9.21110	−0.572350
$260$	0	0
$261$	0	0
$262$	0	0
$263$	17.2111	1.06128	0.530641	−	0.847597i	$-0.321951\pi$
$263$	17.2111	1.06128	0.530641	+	0.847597i	$0.321951\pi$
$264$	0	0
$265$	−5.60555	−0.344346
$266$	0	0
$267$	0	0
$268$	0	0
$269$	11.2111	0.683553	0.341776	−	0.939781i	$-0.388971\pi$
$269$	11.2111	0.683553	0.341776	+	0.939781i	$0.388971\pi$
$270$	0	0
$271$	19.2389	1.16868	0.584339	−	0.811510i	$-0.301354\pi$
$271$	19.2389	1.16868	0.584339	+	0.811510i	$0.301354\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.60555	0.157121
$276$	0	0
$277$	−29.0278	−1.74411	−0.872054	−	0.489409i	$-0.837213\pi$
$277$	−29.0278	−1.74411	−0.872054	+	0.489409i	$0.837213\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1.81665	0.108372	0.0541862	−	0.998531i	$-0.482744\pi$
$281$	1.81665	0.108372	0.0541862	+	0.998531i	$0.482744\pi$
$282$	0	0
$283$	−10.6056	−0.630435	−0.315217	−	0.949020i	$-0.602077\pi$
$283$	−10.6056	−0.630435	−0.315217	+	0.949020i	$0.602077\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−12.0000	−0.708338
$288$	0	0
$289$	14.4222	0.848365
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−28.8167	−1.68349	−0.841743	−	0.539878i	$-0.818470\pi$
$293$	−28.8167	−1.68349	−0.841743	+	0.539878i	$0.818470\pi$
$294$	0	0
$295$	8.60555	0.501035
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.81665	−0.105060
$300$	0	0
$301$	−30.4222	−1.75351
$302$	0	0
$303$	0	0
$304$	0	0
$305$	10.2111	0.584686
$306$	0	0
$307$	20.4222	1.16556	0.582778	−	0.812631i	$-0.301966\pi$
$307$	20.4222	1.16556	0.582778	+	0.812631i	$0.301966\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−13.8167	−0.783471	−0.391735	−	0.920078i	$-0.628125\pi$
$311$	−13.8167	−0.783471	−0.391735	+	0.920078i	$0.628125\pi$
$312$	0	0
$313$	23.6333	1.33583	0.667917	−	0.744236i	$-0.267186\pi$
$313$	23.6333	1.33583	0.667917	+	0.744236i	$0.267186\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−0.394449	−0.0221544	−0.0110772	−	0.999939i	$-0.503526\pi$
$317$	−0.394449	−0.0221544	−0.0110772	+	0.999939i	$0.503526\pi$
$318$	0	0
$319$	22.4222	1.25540
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−20.2111	−1.12458
$324$	0	0
$325$	−0.605551	−0.0335899
$326$	0	0
$327$	0	0
$328$	0	0
$329$	24.0000	1.32316
$330$	0	0
$331$	−14.7889	−0.812871	−0.406436	−	0.913679i	$-0.633228\pi$
$331$	−14.7889	−0.812871	−0.406436	+	0.913679i	$0.633228\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	15.2111	0.831071
$336$	0	0
$337$	−0.605551	−0.0329865	−0.0164932	−	0.999864i	$-0.505250\pi$
$337$	−0.605551	−0.0329865	−0.0164932	+	0.999864i	$0.505250\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−4.18335	−0.226541
$342$	0	0
$343$	−33.2111	−1.79323
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.57779	−0.0847005	−0.0423502	−	0.999103i	$-0.513485\pi$
$347$	−1.57779	−0.0847005	−0.0423502	+	0.999103i	$0.513485\pi$
$348$	0	0
$349$	25.8444	1.38342	0.691710	−	0.722176i	$-0.256858\pi$
$349$	25.8444	1.38342	0.691710	+	0.722176i	$0.256858\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−21.6333	−1.15142	−0.575712	−	0.817652i	$-0.695275\pi$
$353$	−21.6333	−1.15142	−0.575712	+	0.817652i	$0.695275\pi$
$354$	0	0
$355$	−14.6056	−0.775182
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−33.6333	−1.77510	−0.887549	−	0.460713i	$-0.847594\pi$
$359$	−33.6333	−1.77510	−0.887549	+	0.460713i	$0.847594\pi$
$360$	0	0
$361$	−6.00000	−0.315789
$362$	0	0
$363$	0	0
$364$	0	0
$365$	5.39445	0.282358
$366$	0	0
$367$	−4.60555	−0.240408	−0.120204	−	0.992749i	$-0.538355\pi$
$367$	−4.60555	−0.240408	−0.120204	+	0.992749i	$0.538355\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	25.8167	1.34033
$372$	0	0
$373$	−10.7889	−0.558628	−0.279314	−	0.960200i	$-0.590107\pi$
$373$	−10.7889	−0.558628	−0.279314	+	0.960200i	$0.590107\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5.21110	−0.268385
$378$	0	0
$379$	−14.3944	−0.739393	−0.369697	−	0.929153i	$-0.620538\pi$
$379$	−14.3944	−0.739393	−0.369697	+	0.929153i	$0.620538\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	18.6333	0.952118	0.476059	−	0.879413i	$-0.342065\pi$
$383$	18.6333	0.952118	0.476059	+	0.879413i	$0.342065\pi$
$384$	0	0
$385$	−12.0000	−0.611577
$386$	0	0
$387$	0	0
$388$	0	0
$389$	4.18335	0.212104	0.106052	−	0.994361i	$-0.466179\pi$
$389$	4.18335	0.212104	0.106052	+	0.994361i	$0.466179\pi$
$390$	0	0
$391$	−16.8167	−0.850455
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4.39445	0.221109
$396$	0	0
$397$	−12.6056	−0.632654	−0.316327	−	0.948650i	$-0.602450\pi$
$397$	−12.6056	−0.632654	−0.316327	+	0.948650i	$0.602450\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	0	0
$403$	0.972244	0.0484309
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5.21110	0.258305
$408$	0	0
$409$	5.00000	0.247234	0.123617	−	0.992330i	$-0.460551\pi$
$409$	5.00000	0.247234	0.123617	+	0.992330i	$0.460551\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−39.6333	−1.95023
$414$	0	0
$415$	−3.00000	−0.147264
$416$	0	0
$417$	0	0
$418$	0	0
$419$	13.0278	0.636448	0.318224	−	0.948016i	$-0.396914\pi$
$419$	13.0278	0.636448	0.318224	+	0.948016i	$0.396914\pi$
$420$	0	0
$421$	−23.4222	−1.14153	−0.570764	−	0.821114i	$-0.693353\pi$
$421$	−23.4222	−1.14153	−0.570764	+	0.821114i	$0.693353\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−5.60555	−0.271909
$426$	0	0
$427$	−47.0278	−2.27583
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−25.8167	−1.24354	−0.621772	−	0.783198i	$-0.713587\pi$
$431$	−25.8167	−1.24354	−0.621772	+	0.783198i	$0.713587\pi$
$432$	0	0
$433$	−28.2389	−1.35707	−0.678536	−	0.734567i	$-0.737385\pi$
$433$	−28.2389	−1.35707	−0.678536	+	0.734567i	$0.737385\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	10.8167	0.517431
$438$	0	0
$439$	−20.3944	−0.973374	−0.486687	−	0.873576i	$-0.661795\pi$
$439$	−20.3944	−0.973374	−0.486687	+	0.873576i	$0.661795\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−18.6333	−0.885295	−0.442648	−	0.896696i	$-0.645961\pi$
$443$	−18.6333	−0.885295	−0.442648	+	0.896696i	$0.645961\pi$
$444$	0	0
$445$	7.81665	0.370545
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−12.2389	−0.577587	−0.288794	−	0.957391i	$-0.593254\pi$
$449$	−12.2389	−0.577587	−0.288794	+	0.957391i	$0.593254\pi$
$450$	0	0
$451$	6.78890	0.319677
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.78890	0.130746
$456$	0	0
$457$	1.21110	0.0566530	0.0283265	−	0.999599i	$-0.490982\pi$
$457$	1.21110	0.0566530	0.0283265	+	0.999599i	$0.490982\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−21.6333	−1.00756	−0.503782	−	0.863831i	$-0.668058\pi$
$461$	−21.6333	−1.00756	−0.503782	+	0.863831i	$0.668058\pi$
$462$	0	0
$463$	15.2111	0.706920	0.353460	−	0.935450i	$-0.385005\pi$
$463$	15.2111	0.706920	0.353460	+	0.935450i	$0.385005\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2.21110	0.102318	0.0511588	−	0.998691i	$-0.483709\pi$
$467$	2.21110	0.102318	0.0511588	+	0.998691i	$0.483709\pi$
$468$	0	0
$469$	−70.0555	−3.23486
$470$	0	0
$471$	0	0
$472$	0	0
$473$	17.2111	0.791367
$474$	0	0
$475$	3.60555	0.165434
$476$	0	0
$477$	0	0
$478$	0	0
$479$	16.1833	0.739436	0.369718	−	0.929144i	$-0.379454\pi$
$479$	16.1833	0.739436	0.369718	+	0.929144i	$0.379454\pi$
$480$	0	0
$481$	−1.21110	−0.0552215
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8.00000	0.363261
$486$	0	0
$487$	8.18335	0.370823	0.185411	−	0.982661i	$-0.440638\pi$
$487$	8.18335	0.370823	0.185411	+	0.982661i	$0.440638\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−12.7889	−0.577155	−0.288577	−	0.957457i	$-0.593182\pi$
$491$	−12.7889	−0.577155	−0.288577	+	0.957457i	$0.593182\pi$
$492$	0	0
$493$	−48.2389	−2.17257
$494$	0	0
$495$	0	0
$496$	0	0
$497$	67.2666	3.01732
$498$	0	0
$499$	27.6056	1.23579	0.617897	−	0.786259i	$-0.287985\pi$
$499$	27.6056	1.23579	0.617897	+	0.786259i	$0.287985\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−31.4222	−1.40105	−0.700523	−	0.713629i	$-0.747050\pi$
$503$	−31.4222	−1.40105	−0.700523	+	0.713629i	$0.747050\pi$
$504$	0	0
$505$	12.0000	0.533993
$506$	0	0
$507$	0	0
$508$	0	0
$509$	32.8444	1.45580	0.727901	−	0.685682i	$-0.240496\pi$
$509$	32.8444	1.45580	0.727901	+	0.685682i	$0.240496\pi$
$510$	0	0
$511$	−24.8444	−1.09905
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4.00000	0.176261
$516$	0	0
$517$	−13.5778	−0.597151
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−21.3944	−0.937308	−0.468654	−	0.883382i	$-0.655261\pi$
$521$	−21.3944	−0.937308	−0.468654	+	0.883382i	$0.655261\pi$
$522$	0	0
$523$	−23.3944	−1.02297	−0.511484	−	0.859293i	$-0.670904\pi$
$523$	−23.3944	−1.02297	−0.511484	+	0.859293i	$0.670904\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.00000	0.392046
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1.57779	−0.0683419
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	37.0278	1.59490
$540$	0	0
$541$	−11.5778	−0.497768	−0.248884	−	0.968533i	$-0.580064\pi$
$541$	−11.5778	−0.497768	−0.248884	+	0.968533i	$0.580064\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−7.00000	−0.299847
$546$	0	0
$547$	6.60555	0.282433	0.141216	−	0.989979i	$-0.454899\pi$
$547$	6.60555	0.282433	0.141216	+	0.989979i	$0.454899\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	31.0278	1.32183
$552$	0	0
$553$	−20.2389	−0.860644
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−33.6333	−1.42509	−0.712544	−	0.701627i	$-0.752457\pi$
$557$	−33.6333	−1.42509	−0.712544	+	0.701627i	$0.752457\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	0	0
$565$	−0.788897	−0.0331892
$566$	0	0
$567$	0	0
$568$	0	0
$569$	37.8167	1.58536	0.792678	−	0.609640i	$-0.208686\pi$
$569$	37.8167	1.58536	0.792678	+	0.609640i	$0.208686\pi$
$570$	0	0
$571$	36.4500	1.52538	0.762692	−	0.646762i	$-0.223877\pi$
$571$	36.4500	1.52538	0.762692	+	0.646762i	$0.223877\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.00000	0.125109
$576$	0	0
$577$	27.8167	1.15802	0.579011	−	0.815320i	$-0.303439\pi$
$577$	27.8167	1.15802	0.579011	+	0.815320i	$0.303439\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	13.8167	0.573211
$582$	0	0
$583$	−14.6056	−0.604900
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−21.0000	−0.866763	−0.433381	−	0.901211i	$-0.642680\pi$
$587$	−21.0000	−0.866763	−0.433381	+	0.901211i	$0.642680\pi$
$588$	0	0
$589$	−5.78890	−0.238527
$590$	0	0
$591$	0	0
$592$	0	0
$593$	21.2389	0.872175	0.436088	−	0.899904i	$-0.356364\pi$
$593$	21.2389	0.872175	0.436088	+	0.899904i	$0.356364\pi$
$594$	0	0
$595$	25.8167	1.05838
$596$	0	0
$597$	0	0
$598$	0	0
$599$	15.3944	0.629000	0.314500	−	0.949257i	$-0.398163\pi$
$599$	15.3944	0.629000	0.314500	+	0.949257i	$0.398163\pi$
$600$	0	0
$601$	32.6333	1.33114	0.665570	−	0.746335i	$-0.268188\pi$
$601$	32.6333	1.33114	0.665570	+	0.746335i	$0.268188\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−4.21110	−0.171206
$606$	0	0
$607$	−17.3944	−0.706019	−0.353009	−	0.935620i	$-0.614842\pi$
$607$	−17.3944	−0.706019	−0.353009	+	0.935620i	$0.614842\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3.15559	0.127661
$612$	0	0
$613$	28.8444	1.16501	0.582507	−	0.812825i	$-0.302072\pi$
$613$	28.8444	1.16501	0.582507	+	0.812825i	$0.302072\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−26.4500	−1.06484	−0.532418	−	0.846482i	$-0.678716\pi$
$617$	−26.4500	−1.06484	−0.532418	+	0.846482i	$0.678716\pi$
$618$	0	0
$619$	7.63331	0.306809	0.153404	−	0.988164i	$-0.450976\pi$
$619$	7.63331	0.306809	0.153404	+	0.988164i	$0.450976\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−36.0000	−1.44231
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−11.2111	−0.447016
$630$	0	0
$631$	−30.0278	−1.19539	−0.597693	−	0.801725i	$-0.703916\pi$
$631$	−30.0278	−1.19539	−0.597693	+	0.801725i	$0.703916\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4.78890	0.190042
$636$	0	0
$637$	−8.60555	−0.340964
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−24.0000	−0.947943	−0.473972	−	0.880540i	$-0.657180\pi$
$641$	−24.0000	−0.947943	−0.473972	+	0.880540i	$0.657180\pi$
$642$	0	0
$643$	23.0278	0.908126	0.454063	−	0.890970i	$-0.349974\pi$
$643$	23.0278	0.908126	0.454063	+	0.890970i	$0.349974\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−38.2111	−1.50223	−0.751117	−	0.660169i	$-0.770484\pi$
$647$	−38.2111	−1.50223	−0.751117	+	0.660169i	$0.770484\pi$
$648$	0	0
$649$	22.4222	0.880149
$650$	0	0
$651$	0	0
$652$	0	0
$653$	27.2389	1.06594	0.532969	−	0.846134i	$-0.321076\pi$
$653$	27.2389	1.06594	0.532969	+	0.846134i	$0.321076\pi$
$654$	0	0
$655$	−6.00000	−0.234439
$656$	0	0
$657$	0	0
$658$	0	0
$659$	20.6056	0.802678	0.401339	−	0.915930i	$-0.368545\pi$
$659$	20.6056	0.802678	0.401339	+	0.915930i	$0.368545\pi$
$660$	0	0
$661$	22.8444	0.888545	0.444272	−	0.895892i	$-0.353462\pi$
$661$	22.8444	0.888545	0.444272	+	0.895892i	$0.353462\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−16.6056	−0.643936
$666$	0	0
$667$	25.8167	0.999625
$668$	0	0
$669$	0	0
$670$	0	0
$671$	26.6056	1.02710
$672$	0	0
$673$	−11.0278	−0.425089	−0.212544	−	0.977151i	$-0.568175\pi$
$673$	−11.0278	−0.425089	−0.212544	+	0.977151i	$0.568175\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−21.6333	−0.831436	−0.415718	−	0.909494i	$-0.636470\pi$
$677$	−21.6333	−0.831436	−0.415718	+	0.909494i	$0.636470\pi$
$678$	0	0
$679$	−36.8444	−1.41396
$680$	0	0
$681$	0	0
$682$	0	0
$683$	9.00000	0.344375	0.172188	−	0.985064i	$-0.444916\pi$
$683$	9.00000	0.344375	0.172188	+	0.985064i	$0.444916\pi$
$684$	0	0
$685$	−4.81665	−0.184035
$686$	0	0
$687$	0	0
$688$	0	0
$689$	3.39445	0.129318
$690$	0	0
$691$	−2.39445	−0.0910891	−0.0455446	−	0.998962i	$-0.514502\pi$
$691$	−2.39445	−0.0910891	−0.0455446	+	0.998962i	$0.514502\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.00000	0.151729
$696$	0	0
$697$	−14.6056	−0.553225
$698$	0	0
$699$	0	0
$700$	0	0
$701$	40.4222	1.52673	0.763363	−	0.645970i	$-0.223547\pi$
$701$	40.4222	1.52673	0.763363	+	0.645970i	$0.223547\pi$
$702$	0	0
$703$	7.21110	0.271972
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−55.2666	−2.07851
$708$	0	0
$709$	34.8444	1.30861	0.654305	−	0.756231i	$-0.272961\pi$
$709$	34.8444	1.30861	0.654305	+	0.756231i	$0.272961\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−4.81665	−0.180385
$714$	0	0
$715$	−1.57779	−0.0590062
$716$	0	0
$717$	0	0
$718$	0	0
$719$	49.2666	1.83733	0.918667	−	0.395032i	$-0.129267\pi$
$719$	49.2666	1.83733	0.918667	+	0.395032i	$0.129267\pi$
$720$	0	0
$721$	−18.4222	−0.686079
$722$	0	0
$723$	0	0
$724$	0	0
$725$	8.60555	0.319602
$726$	0	0
$727$	7.63331	0.283104	0.141552	−	0.989931i	$-0.454791\pi$
$727$	7.63331	0.283104	0.141552	+	0.989931i	$0.454791\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−37.0278	−1.36952
$732$	0	0
$733$	32.0000	1.18195	0.590973	−	0.806691i	$-0.298744\pi$
$733$	32.0000	1.18195	0.590973	+	0.806691i	$0.298744\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	39.6333	1.45991
$738$	0	0
$739$	−30.0278	−1.10459	−0.552294	−	0.833649i	$-0.686248\pi$
$739$	−30.0278	−1.10459	−0.552294	+	0.833649i	$0.686248\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	34.4222	1.26283	0.631414	−	0.775446i	$-0.282475\pi$
$743$	34.4222	1.26283	0.631414	+	0.775446i	$0.282475\pi$
$744$	0	0
$745$	−13.0278	−0.477300
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−6.02776	−0.219956	−0.109978	−	0.993934i	$-0.535078\pi$
$751$	−6.02776	−0.219956	−0.109978	+	0.993934i	$0.535078\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	14.4222	0.524878
$756$	0	0
$757$	31.2111	1.13439	0.567193	−	0.823585i	$-0.308029\pi$
$757$	31.2111	1.13439	0.567193	+	0.823585i	$0.308029\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−53.4500	−1.93756	−0.968780	−	0.247923i	$-0.920252\pi$
$761$	−53.4500	−1.93756	−0.968780	+	0.247923i	$0.920252\pi$
$762$	0	0
$763$	32.2389	1.16713
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5.21110	−0.188162
$768$	0	0
$769$	41.0000	1.47850	0.739249	−	0.673432i	$-0.235181\pi$
$769$	41.0000	1.47850	0.739249	+	0.673432i	$0.235181\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	16.8167	0.604853	0.302426	−	0.953173i	$-0.402203\pi$
$773$	16.8167	0.604853	0.302426	+	0.953173i	$0.402203\pi$
$774$	0	0
$775$	−1.60555	−0.0576731
$776$	0	0
$777$	0	0
$778$	0	0
$779$	9.39445	0.336591
$780$	0	0
$781$	−38.0555	−1.36173
$782$	0	0
$783$	0	0
$784$	0	0
$785$	3.81665	0.136222
$786$	0	0
$787$	2.97224	0.105949	0.0529745	−	0.998596i	$-0.483130\pi$
$787$	2.97224	0.105949	0.0529745	+	0.998596i	$0.483130\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	3.63331	0.129186
$792$	0	0
$793$	−6.18335	−0.219577
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−37.6611	−1.33402	−0.667012	−	0.745047i	$-0.732427\pi$
$797$	−37.6611	−1.33402	−0.667012	+	0.745047i	$0.732427\pi$
$798$	0	0
$799$	29.2111	1.03341
$800$	0	0
$801$	0	0
$802$	0	0
$803$	14.0555	0.496008
$804$	0	0
$805$	−13.8167	−0.486973
$806$	0	0
$807$	0	0
$808$	0	0
$809$	50.6056	1.77920	0.889598	−	0.456744i	$-0.150984\pi$
$809$	50.6056	1.77920	0.889598	+	0.456744i	$0.150984\pi$
$810$	0	0
$811$	−42.4222	−1.48965	−0.744823	−	0.667263i	$-0.767466\pi$
$811$	−42.4222	−1.48965	−0.744823	+	0.667263i	$0.767466\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−2.00000	−0.0700569
$816$	0	0
$817$	23.8167	0.833239
$818$	0	0
$819$	0	0
$820$	0	0
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	0	0
$823$	−3.81665	−0.133040	−0.0665201	−	0.997785i	$-0.521190\pi$
$823$	−3.81665	−0.133040	−0.0665201	+	0.997785i	$0.521190\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	33.7889	1.17496	0.587478	−	0.809240i	$-0.300121\pi$
$827$	33.7889	1.17496	0.587478	+	0.809240i	$0.300121\pi$
$828$	0	0
$829$	−27.2111	−0.945081	−0.472540	−	0.881309i	$-0.656663\pi$
$829$	−27.2111	−0.945081	−0.472540	+	0.881309i	$0.656663\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−79.6611	−2.76009
$834$	0	0
$835$	3.00000	0.103819
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−18.7889	−0.648665	−0.324332	−	0.945943i	$-0.605140\pi$
$839$	−18.7889	−0.648665	−0.324332	+	0.945943i	$0.605140\pi$
$840$	0	0
$841$	45.0555	1.55364
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−12.6333	−0.434599
$846$	0	0
$847$	19.3944	0.666401
$848$	0	0
$849$	0	0
$850$	0	0
$851$	6.00000	0.205677
$852$	0	0
$853$	14.7889	0.506362	0.253181	−	0.967419i	$-0.418523\pi$
$853$	14.7889	0.506362	0.253181	+	0.967419i	$0.418523\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−45.2389	−1.54533	−0.772665	−	0.634814i	$-0.781077\pi$
$857$	−45.2389	−1.54533	−0.772665	+	0.634814i	$0.781077\pi$
$858$	0	0
$859$	−6.02776	−0.205664	−0.102832	−	0.994699i	$-0.532790\pi$
$859$	−6.02776	−0.205664	−0.102832	+	0.994699i	$0.532790\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−15.7889	−0.537460	−0.268730	−	0.963216i	$-0.586604\pi$
$863$	−15.7889	−0.537460	−0.268730	+	0.963216i	$0.586604\pi$
$864$	0	0
$865$	10.8167	0.367777
$866$	0	0
$867$	0	0
$868$	0	0
$869$	11.4500	0.388413
$870$	0	0
$871$	−9.21110	−0.312106
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−4.60555	−0.155696
$876$	0	0
$877$	−56.6611	−1.91331	−0.956654	−	0.291227i	$-0.905937\pi$
$877$	−56.6611	−1.91331	−0.956654	+	0.291227i	$0.905937\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−32.6056	−1.09851	−0.549254	−	0.835655i	$-0.685088\pi$
$881$	−32.6056	−1.09851	−0.549254	+	0.835655i	$0.685088\pi$
$882$	0	0
$883$	5.81665	0.195746	0.0978730	−	0.995199i	$-0.468796\pi$
$883$	5.81665	0.195746	0.0978730	+	0.995199i	$0.468796\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−12.6333	−0.424185	−0.212092	−	0.977250i	$-0.568028\pi$
$887$	−12.6333	−0.424185	−0.212092	+	0.977250i	$0.568028\pi$
$888$	0	0
$889$	−22.0555	−0.739718
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−18.7889	−0.628746
$894$	0	0
$895$	6.78890	0.226928
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−13.8167	−0.460811
$900$	0	0
$901$	31.4222	1.04683
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−7.00000	−0.232688
$906$	0	0
$907$	−30.4222	−1.01015	−0.505076	−	0.863075i	$-0.668536\pi$
$907$	−30.4222	−1.01015	−0.505076	+	0.863075i	$0.668536\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−31.0278	−1.02800	−0.513998	−	0.857792i	$-0.671836\pi$
$911$	−31.0278	−1.02800	−0.513998	+	0.857792i	$0.671836\pi$
$912$	0	0
$913$	−7.81665	−0.258693
$914$	0	0
$915$	0	0
$916$	0	0
$917$	27.6333	0.912532
$918$	0	0
$919$	2.42221	0.0799012	0.0399506	−	0.999202i	$-0.487280\pi$
$919$	2.42221	0.0799012	0.0399506	+	0.999202i	$0.487280\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	8.84441	0.291117
$924$	0	0
$925$	2.00000	0.0657596
$926$	0	0
$927$	0	0
$928$	0	0
$929$	26.6056	0.872900	0.436450	−	0.899729i	$-0.356236\pi$
$929$	26.6056	0.872900	0.436450	+	0.899729i	$0.356236\pi$
$930$	0	0
$931$	51.2389	1.67929
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−14.6056	−0.477653
$936$	0	0
$937$	26.7889	0.875155	0.437578	−	0.899181i	$-0.355837\pi$
$937$	26.7889	0.875155	0.437578	+	0.899181i	$0.355837\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	28.4222	0.926537	0.463269	−	0.886218i	$-0.346676\pi$
$941$	28.4222	0.926537	0.463269	+	0.886218i	$0.346676\pi$
$942$	0	0
$943$	7.81665	0.254545
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−39.0000	−1.26733	−0.633665	−	0.773608i	$-0.718450\pi$
$947$	−39.0000	−1.26733	−0.633665	+	0.773608i	$0.718450\pi$
$948$	0	0
$949$	−3.26662	−0.106039
$950$	0	0
$951$	0	0
$952$	0	0
$953$	26.8444	0.869576	0.434788	−	0.900533i	$-0.356823\pi$
$953$	26.8444	0.869576	0.434788	+	0.900533i	$0.356823\pi$
$954$	0	0
$955$	−16.4222	−0.531410
$956$	0	0
$957$	0	0
$958$	0	0
$959$	22.1833	0.716338
$960$	0	0
$961$	−28.4222	−0.916845
$962$	0	0
$963$	0	0
$964$	0	0
$965$	21.8167	0.702303
$966$	0	0
$967$	−50.0000	−1.60789	−0.803946	−	0.594703i	$-0.797270\pi$
$967$	−50.0000	−1.60789	−0.803946	+	0.594703i	$0.797270\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	18.0000	0.577647	0.288824	−	0.957382i	$-0.406736\pi$
$971$	18.0000	0.577647	0.288824	+	0.957382i	$0.406736\pi$
$972$	0	0
$973$	−18.4222	−0.590589
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−0.788897	−0.0252391	−0.0126195	−	0.999920i	$-0.504017\pi$
$977$	−0.788897	−0.0252391	−0.0126195	+	0.999920i	$0.504017\pi$
$978$	0	0
$979$	20.3667	0.650922
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−0.633308	−0.0201994	−0.0100997	−	0.999949i	$-0.503215\pi$
$983$	−0.633308	−0.0201994	−0.0100997	+	0.999949i	$0.503215\pi$
$984$	0	0
$985$	−1.18335	−0.0377045
$986$	0	0
$987$	0	0
$988$	0	0
$989$	19.8167	0.630133
$990$	0	0
$991$	50.8167	1.61424	0.807122	−	0.590385i	$-0.201024\pi$
$991$	50.8167	1.61424	0.807122	+	0.590385i	$0.201024\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−13.2111	−0.418820
$996$	0	0
$997$	53.8722	1.70615	0.853074	−	0.521789i	$-0.174735\pi$
$997$	53.8722	1.70615	0.853074	+	0.521789i	$0.174735\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.2.a.ba.1.1		2
3.2	odd	2		2160.2.a.y.1.1		2
4.3	odd	2		135.2.a.c.1.2	✓	2
8.3	odd	2		8640.2.a.cr.1.2		2
8.5	even	2		8640.2.a.ck.1.1		2
12.11	even	2		135.2.a.d.1.1	yes	2
20.3	even	4		675.2.b.i.649.2		4
20.7	even	4		675.2.b.i.649.3		4
20.19	odd	2		675.2.a.p.1.1		2
24.5	odd	2		8640.2.a.cy.1.1		2
24.11	even	2		8640.2.a.df.1.2		2
28.27	even	2		6615.2.a.p.1.2		2
36.7	odd	6		405.2.e.k.271.1		4
36.11	even	6		405.2.e.j.271.2		4
36.23	even	6		405.2.e.j.136.2		4
36.31	odd	6		405.2.e.k.136.1		4
60.23	odd	4		675.2.b.h.649.3		4
60.47	odd	4		675.2.b.h.649.2		4
60.59	even	2		675.2.a.k.1.2		2
84.83	odd	2		6615.2.a.v.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.2.a.c.1.2	✓	2	4.3	odd	2
135.2.a.d.1.1	yes	2	12.11	even	2
405.2.e.j.136.2		4	36.23	even	6
405.2.e.j.271.2		4	36.11	even	6
405.2.e.k.136.1		4	36.31	odd	6
405.2.e.k.271.1		4	36.7	odd	6
675.2.a.k.1.2		2	60.59	even	2
675.2.a.p.1.1		2	20.19	odd	2
675.2.b.h.649.2		4	60.47	odd	4
675.2.b.h.649.3		4	60.23	odd	4
675.2.b.i.649.2		4	20.3	even	4
675.2.b.i.649.3		4	20.7	even	4
2160.2.a.y.1.1		2	3.2	odd	2
2160.2.a.ba.1.1		2	1.1	even	1	trivial
6615.2.a.p.1.2		2	28.27	even	2
6615.2.a.v.1.1		2	84.83	odd	2
8640.2.a.ck.1.1		2	8.5	even	2
8640.2.a.cr.1.2		2	8.3	odd	2
8640.2.a.cy.1.1		2	24.5	odd	2
8640.2.a.df.1.2		2	24.11	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:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2160.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -4.60555 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -4.60555 q^{7} +2.60555 q^{11} -0.605551 q^{13} -5.60555 q^{17} +3.60555 q^{19} +3.00000 q^{23} +1.00000 q^{25} +8.60555 q^{29} -1.60555 q^{31} -4.60555 q^{35} +2.00000 q^{37} +2.60555 q^{41} +6.60555 q^{43} -5.21110 q^{47} +14.2111 q^{49} -5.60555 q^{53} +2.60555 q^{55} +8.60555 q^{59} +10.2111 q^{61} -0.605551 q^{65} +15.2111 q^{67} -14.6056 q^{71} +5.39445 q^{73} -12.0000 q^{77} +4.39445 q^{79} -3.00000 q^{83} -5.60555 q^{85} +7.81665 q^{89} +2.78890 q^{91} +3.60555 q^{95} +8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q + 2 * q^5 - 2 * q^7	\( 2 q + 2 q^{5} - 2 q^{7} - 2 q^{11} + 6 q^{13} - 4 q^{17} + 6 q^{23} + 2 q^{25} + 10 q^{29} + 4 q^{31} - 2 q^{35} + 4 q^{37} - 2 q^{41} + 6 q^{43} + 4 q^{47} + 14 q^{49} - 4 q^{53} - 2 q^{55} + 10 q^{59} + 6 q^{61} + 6 q^{65} + 16 q^{67} - 22 q^{71} + 18 q^{73} - 24 q^{77} + 16 q^{79} - 6 q^{83} - 4 q^{85} - 6 q^{89} + 20 q^{91} + 16 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 - 2 * q^7 - 2 * q^11 + 6 * q^13 - 4 * q^17 + 6 * q^23 + 2 * q^25 + 10 * q^29 + 4 * q^31 - 2 * q^35 + 4 * q^37 - 2 * q^41 + 6 * q^43 + 4 * q^47 + 14 * q^49 - 4 * q^53 - 2 * q^55 + 10 * q^59 + 6 * q^61 + 6 * q^65 + 16 * q^67 - 22 * q^71 + 18 * q^73 - 24 * q^77 + 16 * q^79 - 6 * q^83 - 4 * q^85 - 6 * q^89 + 20 * q^91 + 16 * q^97

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