LMFDB - Embedded newform 7728.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7728,2,Mod(1,7728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7728, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7728.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:	$61.7083906820$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 966)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	7728.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} -3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} -3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +5.00000 q^{13} +3.00000 q^{15} -8.00000 q^{19} +1.00000 q^{21} +1.00000 q^{23} +4.00000 q^{25} -1.00000 q^{27} +3.00000 q^{29} -2.00000 q^{31} +3.00000 q^{35} -7.00000 q^{37} -5.00000 q^{39} +9.00000 q^{41} +1.00000 q^{43} -3.00000 q^{45} +3.00000 q^{47} +1.00000 q^{49} -12.0000 q^{53} +8.00000 q^{57} +6.00000 q^{59} +14.0000 q^{61} -1.00000 q^{63} -15.0000 q^{65} +4.00000 q^{67} -1.00000 q^{69} -6.00000 q^{71} -4.00000 q^{73} -4.00000 q^{75} +16.0000 q^{79} +1.00000 q^{81} +12.0000 q^{83} -3.00000 q^{87} +6.00000 q^{89} -5.00000 q^{91} +2.00000 q^{93} +24.0000 q^{95} -1.00000 q^{97} +O(q^{100})$

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
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	0	0
$15$	3.00000	0.774597
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−8.00000	−1.83533	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−8.00000	−1.83533	−0.917663	+	0.397360i	$0.869927\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.00000	0.507093
$36$	0	0
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	0	0
$39$	−5.00000	−0.800641
$40$	0	0
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	0	0
$45$	−3.00000	−0.447214
$46$	0	0
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−12.0000	−1.64833	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000	−1.64833	−0.824163	+	0.566352i	$0.808354\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	8.00000	1.05963
$58$	0	0
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−15.0000	−1.86052
$66$	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
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	0	0
$75$	−4.00000	−0.461880
$76$	0	0
$77$	0	0
$78$	0	0
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−3.00000	−0.321634
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−5.00000	−0.524142
$92$	0	0
$93$	2.00000	0.207390
$94$	0	0
$95$	24.0000	2.46235
$96$	0	0
$97$	−1.00000	−0.101535	−0.0507673	−	0.998711i	$-0.516167\pi$
$97$	−1.00000	−0.101535	−0.0507673	+	0.998711i	$0.516167\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	1.00000	0.0985329	0.0492665	−	0.998786i	$-0.484312\pi$
$103$	1.00000	0.0985329	0.0492665	+	0.998786i	$0.484312\pi$
$104$	0	0
$105$	−3.00000	−0.292770
$106$	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
$109$	−19.0000	−1.81987	−0.909935	−	0.414751i	$-0.863869\pi$
$109$	−19.0000	−1.81987	−0.909935	+	0.414751i	$0.863869\pi$
$110$	0	0
$111$	7.00000	0.664411
$112$	0	0
$113$	15.0000	1.41108	0.705541	−	0.708669i	$-0.250704\pi$
$113$	15.0000	1.41108	0.705541	+	0.708669i	$0.250704\pi$
$114$	0	0
$115$	−3.00000	−0.279751
$116$	0	0
$117$	5.00000	0.462250
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	−9.00000	−0.811503
$124$	0	0
$125$	3.00000	0.268328
$126$	0	0
$127$	13.0000	1.15356	0.576782	−	0.816898i	$-0.304308\pi$
$127$	13.0000	1.15356	0.576782	+	0.816898i	$0.304308\pi$
$128$	0	0
$129$	−1.00000	−0.0880451
$130$	0	0
$131$	18.0000	1.57267	0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000	1.57267	0.786334	+	0.617802i	$0.211977\pi$
$132$	0	0
$133$	8.00000	0.693688
$134$	0	0
$135$	3.00000	0.258199
$136$	0	0
$137$	−9.00000	−0.768922	−0.384461	−	0.923141i	$-0.625613\pi$
$137$	−9.00000	−0.768922	−0.384461	+	0.923141i	$0.625613\pi$
$138$	0	0
$139$	−5.00000	−0.424094	−0.212047	−	0.977259i	$-0.568013\pi$
$139$	−5.00000	−0.424094	−0.212047	+	0.977259i	$0.568013\pi$
$140$	0	0
$141$	−3.00000	−0.252646
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−9.00000	−0.747409
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−11.0000	−0.895167	−0.447584	−	0.894242i	$-0.647715\pi$
$151$	−11.0000	−0.895167	−0.447584	+	0.894242i	$0.647715\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	6.00000	0.481932
$156$	0	0
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	0	0
$159$	12.0000	0.951662
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	−8.00000	−0.611775
$172$	0	0
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	0	0
$177$	−6.00000	−0.450988
$178$	0	0
$179$	−15.0000	−1.12115	−0.560576	−	0.828103i	$-0.689420\pi$
$179$	−15.0000	−1.12115	−0.560576	+	0.828103i	$0.689420\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	−14.0000	−1.03491
$184$	0	0
$185$	21.0000	1.54395
$186$	0	0
$187$	0	0
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	−1.00000	−0.0719816	−0.0359908	−	0.999352i	$-0.511459\pi$
$193$	−1.00000	−0.0719816	−0.0359908	+	0.999352i	$0.511459\pi$
$194$	0	0
$195$	15.0000	1.07417
$196$	0	0
$197$	9.00000	0.641223	0.320612	−	0.947211i	$-0.396112\pi$
$197$	9.00000	0.641223	0.320612	+	0.947211i	$0.396112\pi$
$198$	0	0
$199$	−17.0000	−1.20510	−0.602549	−	0.798082i	$-0.705848\pi$
$199$	−17.0000	−1.20510	−0.602549	+	0.798082i	$0.705848\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	−3.00000	−0.210559
$204$	0	0
$205$	−27.0000	−1.88576
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	0	0
$213$	6.00000	0.411113
$214$	0	0
$215$	−3.00000	−0.204598
$216$	0	0
$217$	2.00000	0.135769
$218$	0	0
$219$	4.00000	0.270295
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−14.0000	−0.937509	−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000	−0.937509	−0.468755	+	0.883328i	$0.655297\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	0	0
$227$	15.0000	0.995585	0.497792	−	0.867296i	$-0.334144\pi$
$227$	15.0000	0.995585	0.497792	+	0.867296i	$0.334144\pi$
$228$	0	0
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	−9.00000	−0.587095
$236$	0	0
$237$	−16.0000	−1.03931
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	−19.0000	−1.22390	−0.611949	−	0.790897i	$-0.709614\pi$
$241$	−19.0000	−1.22390	−0.611949	+	0.790897i	$0.709614\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−3.00000	−0.191663
$246$	0	0
$247$	−40.0000	−2.54514
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	15.0000	0.946792	0.473396	−	0.880850i	$-0.343028\pi$
$251$	15.0000	0.946792	0.473396	+	0.880850i	$0.343028\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	7.00000	0.434959
$260$	0	0
$261$	3.00000	0.185695
$262$	0	0
$263$	9.00000	0.554964	0.277482	−	0.960731i	$-0.410500\pi$
$263$	9.00000	0.554964	0.277482	+	0.960731i	$0.410500\pi$
$264$	0	0
$265$	36.0000	2.21146
$266$	0	0
$267$	−6.00000	−0.367194
$268$	0	0
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	28.0000	1.70088	0.850439	−	0.526073i	$-0.176336\pi$
$271$	28.0000	1.70088	0.850439	+	0.526073i	$0.176336\pi$
$272$	0	0
$273$	5.00000	0.302614
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−4.00000	−0.240337	−0.120168	−	0.992754i	$-0.538343\pi$
$277$	−4.00000	−0.240337	−0.120168	+	0.992754i	$0.538343\pi$
$278$	0	0
$279$	−2.00000	−0.119737
$280$	0	0
$281$	−15.0000	−0.894825	−0.447412	−	0.894328i	$-0.647654\pi$
$281$	−15.0000	−0.894825	−0.447412	+	0.894328i	$0.647654\pi$
$282$	0	0
$283$	−14.0000	−0.832214	−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000	−0.832214	−0.416107	+	0.909316i	$0.636606\pi$
$284$	0	0
$285$	−24.0000	−1.42164
$286$	0	0
$287$	−9.00000	−0.531253
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	1.00000	0.0586210
$292$	0	0
$293$	−30.0000	−1.75262	−0.876309	−	0.481749i	$-0.840002\pi$
$293$	−30.0000	−1.75262	−0.876309	+	0.481749i	$0.840002\pi$
$294$	0	0
$295$	−18.0000	−1.04800
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.00000	0.289157
$300$	0	0
$301$	−1.00000	−0.0576390
$302$	0	0
$303$	6.00000	0.344691
$304$	0	0
$305$	−42.0000	−2.40491
$306$	0	0
$307$	−11.0000	−0.627803	−0.313902	−	0.949456i	$-0.601636\pi$
$307$	−11.0000	−0.627803	−0.313902	+	0.949456i	$0.601636\pi$
$308$	0	0
$309$	−1.00000	−0.0568880
$310$	0	0
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	0	0
$315$	3.00000	0.169031
$316$	0	0
$317$	−27.0000	−1.51647	−0.758236	−	0.651981i	$-0.773938\pi$
$317$	−27.0000	−1.51647	−0.758236	+	0.651981i	$0.773938\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	0	0
$324$	0	0
$325$	20.0000	1.10940
$326$	0	0
$327$	19.0000	1.05070
$328$	0	0
$329$	−3.00000	−0.165395
$330$	0	0
$331$	16.0000	0.879440	0.439720	−	0.898135i	$-0.355078\pi$
$331$	16.0000	0.879440	0.439720	+	0.898135i	$0.355078\pi$
$332$	0	0
$333$	−7.00000	−0.383598
$334$	0	0
$335$	−12.0000	−0.655630
$336$	0	0
$337$	−4.00000	−0.217894	−0.108947	−	0.994048i	$-0.534748\pi$
$337$	−4.00000	−0.217894	−0.108947	+	0.994048i	$0.534748\pi$
$338$	0	0
$339$	−15.0000	−0.814688
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	3.00000	0.161515
$346$	0	0
$347$	−3.00000	−0.161048	−0.0805242	−	0.996753i	$-0.525659\pi$
$347$	−3.00000	−0.161048	−0.0805242	+	0.996753i	$0.525659\pi$
$348$	0	0
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	−5.00000	−0.266880
$352$	0	0
$353$	27.0000	1.43706	0.718532	−	0.695493i	$-0.244814\pi$
$353$	27.0000	1.43706	0.718532	+	0.695493i	$0.244814\pi$
$354$	0	0
$355$	18.0000	0.955341
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−21.0000	−1.10834	−0.554169	−	0.832404i	$-0.686964\pi$
$359$	−21.0000	−1.10834	−0.554169	+	0.832404i	$0.686964\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	0	0
$363$	11.0000	0.577350
$364$	0	0
$365$	12.0000	0.628109
$366$	0	0
$367$	−29.0000	−1.51379	−0.756894	−	0.653538i	$-0.773284\pi$
$367$	−29.0000	−1.51379	−0.756894	+	0.653538i	$0.773284\pi$
$368$	0	0
$369$	9.00000	0.468521
$370$	0	0
$371$	12.0000	0.623009
$372$	0	0
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	0	0
$375$	−3.00000	−0.154919
$376$	0	0
$377$	15.0000	0.772539
$378$	0	0
$379$	−29.0000	−1.48963	−0.744815	−	0.667271i	$-0.767462\pi$
$379$	−29.0000	−1.48963	−0.744815	+	0.667271i	$0.767462\pi$
$380$	0	0
$381$	−13.0000	−0.666010
$382$	0	0
$383$	−36.0000	−1.83951	−0.919757	−	0.392488i	$-0.871614\pi$
$383$	−36.0000	−1.83951	−0.919757	+	0.392488i	$0.871614\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.00000	0.0508329
$388$	0	0
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−18.0000	−0.907980
$394$	0	0
$395$	−48.0000	−2.41514
$396$	0	0
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	0	0
$399$	−8.00000	−0.400501
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	−10.0000	−0.498135
$404$	0	0
$405$	−3.00000	−0.149071
$406$	0	0
$407$	0	0
$408$	0	0
$409$	32.0000	1.58230	0.791149	−	0.611623i	$-0.209483\pi$
$409$	32.0000	1.58230	0.791149	+	0.611623i	$0.209483\pi$
$410$	0	0
$411$	9.00000	0.443937
$412$	0	0
$413$	−6.00000	−0.295241
$414$	0	0
$415$	−36.0000	−1.76717
$416$	0	0
$417$	5.00000	0.244851
$418$	0	0
$419$	−36.0000	−1.75872	−0.879358	−	0.476162i	$-0.842028\pi$
$419$	−36.0000	−1.75872	−0.879358	+	0.476162i	$0.842028\pi$
$420$	0	0
$421$	−19.0000	−0.926003	−0.463002	−	0.886357i	$-0.653228\pi$
$421$	−19.0000	−0.926003	−0.463002	+	0.886357i	$0.653228\pi$
$422$	0	0
$423$	3.00000	0.145865
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−14.0000	−0.677507
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−15.0000	−0.722525	−0.361262	−	0.932464i	$-0.617654\pi$
$431$	−15.0000	−0.722525	−0.361262	+	0.932464i	$0.617654\pi$
$432$	0	0
$433$	17.0000	0.816968	0.408484	−	0.912766i	$-0.366058\pi$
$433$	17.0000	0.816968	0.408484	+	0.912766i	$0.366058\pi$
$434$	0	0
$435$	9.00000	0.431517
$436$	0	0
$437$	−8.00000	−0.382692
$438$	0	0
$439$	−26.0000	−1.24091	−0.620456	−	0.784241i	$-0.713053\pi$
$439$	−26.0000	−1.24091	−0.620456	+	0.784241i	$0.713053\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	27.0000	1.28281	0.641404	−	0.767203i	$-0.278352\pi$
$443$	27.0000	1.28281	0.641404	+	0.767203i	$0.278352\pi$
$444$	0	0
$445$	−18.0000	−0.853282
$446$	0	0
$447$	0	0
$448$	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
$451$	0	0
$452$	0	0
$453$	11.0000	0.516825
$454$	0	0
$455$	15.0000	0.703211
$456$	0	0
$457$	−28.0000	−1.30978	−0.654892	−	0.755722i	$-0.727286\pi$
$457$	−28.0000	−1.30978	−0.654892	+	0.755722i	$0.727286\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	31.0000	1.44069	0.720346	−	0.693615i	$-0.243983\pi$
$463$	31.0000	1.44069	0.720346	+	0.693615i	$0.243983\pi$
$464$	0	0
$465$	−6.00000	−0.278243
$466$	0	0
$467$	33.0000	1.52706	0.763529	−	0.645774i	$-0.223465\pi$
$467$	33.0000	1.52706	0.763529	+	0.645774i	$0.223465\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	4.00000	0.184310
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−32.0000	−1.46826
$476$	0	0
$477$	−12.0000	−0.549442
$478$	0	0
$479$	6.00000	0.274147	0.137073	−	0.990561i	$-0.456230\pi$
$479$	6.00000	0.274147	0.137073	+	0.990561i	$0.456230\pi$
$480$	0	0
$481$	−35.0000	−1.59586
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	3.00000	0.136223
$486$	0	0
$487$	31.0000	1.40474	0.702372	−	0.711810i	$-0.252124\pi$
$487$	31.0000	1.40474	0.702372	+	0.711810i	$0.252124\pi$
$488$	0	0
$489$	20.0000	0.904431
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.00000	0.269137
$498$	0	0
$499$	−14.0000	−0.626726	−0.313363	−	0.949633i	$-0.601456\pi$
$499$	−14.0000	−0.626726	−0.313363	+	0.949633i	$0.601456\pi$
$500$	0	0
$501$	−12.0000	−0.536120
$502$	0	0
$503$	−18.0000	−0.802580	−0.401290	−	0.915951i	$-0.631438\pi$
$503$	−18.0000	−0.802580	−0.401290	+	0.915951i	$0.631438\pi$
$504$	0	0
$505$	18.0000	0.800989
$506$	0	0
$507$	−12.0000	−0.532939
$508$	0	0
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	0	0
$513$	8.00000	0.353209
$514$	0	0
$515$	−3.00000	−0.132196
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−6.00000	−0.263371
$520$	0	0
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	0	0
$523$	−26.0000	−1.13690	−0.568450	−	0.822718i	$-0.692457\pi$
$523$	−26.0000	−1.13690	−0.568450	+	0.822718i	$0.692457\pi$
$524$	0	0
$525$	4.00000	0.174574
$526$	0	0
$527$	0	0
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	6.00000	0.260378
$532$	0	0
$533$	45.0000	1.94917
$534$	0	0
$535$	36.0000	1.55642
$536$	0	0
$537$	15.0000	0.647298
$538$	0	0
$539$	0	0
$540$	0	0
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	0	0
$543$	−2.00000	−0.0858282
$544$	0	0
$545$	57.0000	2.44161
$546$	0	0
$547$	10.0000	0.427569	0.213785	−	0.976881i	$-0.431421\pi$
$547$	10.0000	0.427569	0.213785	+	0.976881i	$0.431421\pi$
$548$	0	0
$549$	14.0000	0.597505
$550$	0	0
$551$	−24.0000	−1.02243
$552$	0	0
$553$	−16.0000	−0.680389
$554$	0	0
$555$	−21.0000	−0.891400
$556$	0	0
$557$	24.0000	1.01691	0.508456	−	0.861088i	$-0.330216\pi$
$557$	24.0000	1.01691	0.508456	+	0.861088i	$0.330216\pi$
$558$	0	0
$559$	5.00000	0.211477
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3.00000	0.126435	0.0632175	−	0.998000i	$-0.479864\pi$
$563$	3.00000	0.126435	0.0632175	+	0.998000i	$0.479864\pi$
$564$	0	0
$565$	−45.0000	−1.89316
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	45.0000	1.88650	0.943249	−	0.332086i	$-0.107752\pi$
$569$	45.0000	1.88650	0.943249	+	0.332086i	$0.107752\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	0	0
$573$	−12.0000	−0.501307
$574$	0	0
$575$	4.00000	0.166812
$576$	0	0
$577$	26.0000	1.08239	0.541197	−	0.840896i	$-0.317971\pi$
$577$	26.0000	1.08239	0.541197	+	0.840896i	$0.317971\pi$
$578$	0	0
$579$	1.00000	0.0415586
$580$	0	0
$581$	−12.0000	−0.497844
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−15.0000	−0.620174
$586$	0	0
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	0	0
$589$	16.0000	0.659269
$590$	0	0
$591$	−9.00000	−0.370211
$592$	0	0
$593$	−27.0000	−1.10876	−0.554379	−	0.832265i	$-0.687044\pi$
$593$	−27.0000	−1.10876	−0.554379	+	0.832265i	$0.687044\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	17.0000	0.695764
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	−16.0000	−0.652654	−0.326327	−	0.945257i	$-0.605811\pi$
$601$	−16.0000	−0.652654	−0.326327	+	0.945257i	$0.605811\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	0	0
$605$	33.0000	1.34164
$606$	0	0
$607$	−14.0000	−0.568242	−0.284121	−	0.958788i	$-0.591702\pi$
$607$	−14.0000	−0.568242	−0.284121	+	0.958788i	$0.591702\pi$
$608$	0	0
$609$	3.00000	0.121566
$610$	0	0
$611$	15.0000	0.606835
$612$	0	0
$613$	−1.00000	−0.0403896	−0.0201948	−	0.999796i	$-0.506429\pi$
$613$	−1.00000	−0.0403896	−0.0201948	+	0.999796i	$0.506429\pi$
$614$	0	0
$615$	27.0000	1.08875
$616$	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
$619$	10.0000	0.401934	0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000	0.401934	0.200967	+	0.979598i	$0.435592\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−6.00000	−0.240385
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−20.0000	−0.796187	−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000	−0.796187	−0.398094	+	0.917345i	$0.630328\pi$
$632$	0	0
$633$	8.00000	0.317971
$634$	0	0
$635$	−39.0000	−1.54767
$636$	0	0
$637$	5.00000	0.198107
$638$	0	0
$639$	−6.00000	−0.237356
$640$	0	0
$641$	−15.0000	−0.592464	−0.296232	−	0.955116i	$-0.595730\pi$
$641$	−15.0000	−0.592464	−0.296232	+	0.955116i	$0.595730\pi$
$642$	0	0
$643$	−14.0000	−0.552106	−0.276053	−	0.961142i	$-0.589027\pi$
$643$	−14.0000	−0.552106	−0.276053	+	0.961142i	$0.589027\pi$
$644$	0	0
$645$	3.00000	0.118125
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−2.00000	−0.0783862
$652$	0	0
$653$	−9.00000	−0.352197	−0.176099	−	0.984373i	$-0.556348\pi$
$653$	−9.00000	−0.352197	−0.176099	+	0.984373i	$0.556348\pi$
$654$	0	0
$655$	−54.0000	−2.10995
$656$	0	0
$657$	−4.00000	−0.156055
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−24.0000	−0.930680
$666$	0	0
$667$	3.00000	0.116160
$668$	0	0
$669$	14.0000	0.541271
$670$	0	0
$671$	0	0
$672$	0	0
$673$	35.0000	1.34915	0.674575	−	0.738206i	$-0.264327\pi$
$673$	35.0000	1.34915	0.674575	+	0.738206i	$0.264327\pi$
$674$	0	0
$675$	−4.00000	−0.153960
$676$	0	0
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	0	0
$679$	1.00000	0.0383765
$680$	0	0
$681$	−15.0000	−0.574801
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	27.0000	1.03162
$686$	0	0
$687$	4.00000	0.152610
$688$	0	0
$689$	−60.0000	−2.28582
$690$	0	0
$691$	49.0000	1.86405	0.932024	−	0.362397i	$-0.118041\pi$
$691$	49.0000	1.86405	0.932024	+	0.362397i	$0.118041\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	15.0000	0.568982
$696$	0	0
$697$	0	0
$698$	0	0
$699$	6.00000	0.226941
$700$	0	0
$701$	36.0000	1.35970	0.679851	−	0.733351i	$-0.262045\pi$
$701$	36.0000	1.35970	0.679851	+	0.733351i	$0.262045\pi$
$702$	0	0
$703$	56.0000	2.11208
$704$	0	0
$705$	9.00000	0.338960
$706$	0	0
$707$	6.00000	0.225653
$708$	0	0
$709$	−22.0000	−0.826227	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000	−0.826227	−0.413114	+	0.910679i	$0.635559\pi$
$710$	0	0
$711$	16.0000	0.600047
$712$	0	0
$713$	−2.00000	−0.0749006
$714$	0	0
$715$	0	0
$716$	0	0
$717$	12.0000	0.448148
$718$	0	0
$719$	−45.0000	−1.67822	−0.839108	−	0.543964i	$-0.816923\pi$
$719$	−45.0000	−1.67822	−0.839108	+	0.543964i	$0.816923\pi$
$720$	0	0
$721$	−1.00000	−0.0372419
$722$	0	0
$723$	19.0000	0.706618
$724$	0	0
$725$	12.0000	0.445669
$726$	0	0
$727$	28.0000	1.03846	0.519231	−	0.854634i	$-0.326218\pi$
$727$	28.0000	1.03846	0.519231	+	0.854634i	$0.326218\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	8.00000	0.295487	0.147743	−	0.989026i	$-0.452799\pi$
$733$	8.00000	0.295487	0.147743	+	0.989026i	$0.452799\pi$
$734$	0	0
$735$	3.00000	0.110657
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−50.0000	−1.83928	−0.919640	−	0.392763i	$-0.871519\pi$
$739$	−50.0000	−1.83928	−0.919640	+	0.392763i	$0.871519\pi$
$740$	0	0
$741$	40.0000	1.46944
$742$	0	0
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	12.0000	0.439057
$748$	0	0
$749$	12.0000	0.438470
$750$	0	0
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	0	0
$753$	−15.0000	−0.546630
$754$	0	0
$755$	33.0000	1.20099
$756$	0	0
$757$	−34.0000	−1.23575	−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000	−1.23575	−0.617876	+	0.786276i	$0.712006\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	0	0
$763$	19.0000	0.687846
$764$	0	0
$765$	0	0
$766$	0	0
$767$	30.0000	1.08324
$768$	0	0
$769$	−37.0000	−1.33425	−0.667127	−	0.744944i	$-0.732476\pi$
$769$	−37.0000	−1.33425	−0.667127	+	0.744944i	$0.732476\pi$
$770$	0	0
$771$	−6.00000	−0.216085
$772$	0	0
$773$	−51.0000	−1.83434	−0.917171	−	0.398493i	$-0.869533\pi$
$773$	−51.0000	−1.83434	−0.917171	+	0.398493i	$0.869533\pi$
$774$	0	0
$775$	−8.00000	−0.287368
$776$	0	0
$777$	−7.00000	−0.251124
$778$	0	0
$779$	−72.0000	−2.57967
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−3.00000	−0.107211
$784$	0	0
$785$	12.0000	0.428298
$786$	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
$789$	−9.00000	−0.320408
$790$	0	0
$791$	−15.0000	−0.533339
$792$	0	0
$793$	70.0000	2.48577
$794$	0	0
$795$	−36.0000	−1.27679
$796$	0	0
$797$	33.0000	1.16892	0.584460	−	0.811423i	$-0.301306\pi$
$797$	33.0000	1.16892	0.584460	+	0.811423i	$0.301306\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	6.00000	0.212000
$802$	0	0
$803$	0	0
$804$	0	0
$805$	3.00000	0.105736
$806$	0	0
$807$	18.0000	0.633630
$808$	0	0
$809$	−12.0000	−0.421898	−0.210949	−	0.977497i	$-0.567655\pi$
$809$	−12.0000	−0.421898	−0.210949	+	0.977497i	$0.567655\pi$
$810$	0	0
$811$	−5.00000	−0.175574	−0.0877869	−	0.996139i	$-0.527979\pi$
$811$	−5.00000	−0.175574	−0.0877869	+	0.996139i	$0.527979\pi$
$812$	0	0
$813$	−28.0000	−0.982003
$814$	0	0
$815$	60.0000	2.10171
$816$	0	0
$817$	−8.00000	−0.279885
$818$	0	0
$819$	−5.00000	−0.174714
$820$	0	0
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	0	0
$823$	13.0000	0.453152	0.226576	−	0.973994i	$-0.427247\pi$
$823$	13.0000	0.453152	0.226576	+	0.973994i	$0.427247\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−6.00000	−0.208640	−0.104320	−	0.994544i	$-0.533267\pi$
$827$	−6.00000	−0.208640	−0.104320	+	0.994544i	$0.533267\pi$
$828$	0	0
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	0	0
$831$	4.00000	0.138758
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−36.0000	−1.24583
$836$	0	0
$837$	2.00000	0.0691301
$838$	0	0
$839$	−6.00000	−0.207143	−0.103572	−	0.994622i	$-0.533027\pi$
$839$	−6.00000	−0.207143	−0.103572	+	0.994622i	$0.533027\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	15.0000	0.516627
$844$	0	0
$845$	−36.0000	−1.23844
$846$	0	0
$847$	11.0000	0.377964
$848$	0	0
$849$	14.0000	0.480479
$850$	0	0
$851$	−7.00000	−0.239957
$852$	0	0
$853$	−49.0000	−1.67773	−0.838864	−	0.544341i	$-0.816780\pi$
$853$	−49.0000	−1.67773	−0.838864	+	0.544341i	$0.816780\pi$
$854$	0	0
$855$	24.0000	0.820783
$856$	0	0
$857$	−15.0000	−0.512390	−0.256195	−	0.966625i	$-0.582469\pi$
$857$	−15.0000	−0.512390	−0.256195	+	0.966625i	$0.582469\pi$
$858$	0	0
$859$	31.0000	1.05771	0.528853	−	0.848713i	$-0.322622\pi$
$859$	31.0000	1.05771	0.528853	+	0.848713i	$0.322622\pi$
$860$	0	0
$861$	9.00000	0.306719
$862$	0	0
$863$	−12.0000	−0.408485	−0.204242	−	0.978920i	$-0.565473\pi$
$863$	−12.0000	−0.408485	−0.204242	+	0.978920i	$0.565473\pi$
$864$	0	0
$865$	−18.0000	−0.612018
$866$	0	0
$867$	17.0000	0.577350
$868$	0	0
$869$	0	0
$870$	0	0
$871$	20.0000	0.677674
$872$	0	0
$873$	−1.00000	−0.0338449
$874$	0	0
$875$	−3.00000	−0.101419
$876$	0	0
$877$	−40.0000	−1.35070	−0.675352	−	0.737496i	$-0.736008\pi$
$877$	−40.0000	−1.35070	−0.675352	+	0.737496i	$0.736008\pi$
$878$	0	0
$879$	30.0000	1.01187
$880$	0	0
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	0	0
$883$	−26.0000	−0.874970	−0.437485	−	0.899226i	$-0.644131\pi$
$883$	−26.0000	−0.874970	−0.437485	+	0.899226i	$0.644131\pi$
$884$	0	0
$885$	18.0000	0.605063
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	−13.0000	−0.436006
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−24.0000	−0.803129
$894$	0	0
$895$	45.0000	1.50418
$896$	0	0
$897$	−5.00000	−0.166945
$898$	0	0
$899$	−6.00000	−0.200111
$900$	0	0
$901$	0	0
$902$	0	0
$903$	1.00000	0.0332779
$904$	0	0
$905$	−6.00000	−0.199447
$906$	0	0
$907$	−41.0000	−1.36138	−0.680691	−	0.732570i	$-0.738320\pi$
$907$	−41.0000	−1.36138	−0.680691	+	0.732570i	$0.738320\pi$
$908$	0	0
$909$	−6.00000	−0.199007
$910$	0	0
$911$	−21.0000	−0.695761	−0.347881	−	0.937539i	$-0.613099\pi$
$911$	−21.0000	−0.695761	−0.347881	+	0.937539i	$0.613099\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	42.0000	1.38848
$916$	0	0
$917$	−18.0000	−0.594412
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	11.0000	0.362462
$922$	0	0
$923$	−30.0000	−0.987462
$924$	0	0
$925$	−28.0000	−0.920634
$926$	0	0
$927$	1.00000	0.0328443
$928$	0	0
$929$	33.0000	1.08269	0.541347	−	0.840799i	$-0.317914\pi$
$929$	33.0000	1.08269	0.541347	+	0.840799i	$0.317914\pi$
$930$	0	0
$931$	−8.00000	−0.262189
$932$	0	0
$933$	12.0000	0.392862
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−25.0000	−0.816714	−0.408357	−	0.912822i	$-0.633898\pi$
$937$	−25.0000	−0.816714	−0.408357	+	0.912822i	$0.633898\pi$
$938$	0	0
$939$	10.0000	0.326338
$940$	0	0
$941$	−51.0000	−1.66255	−0.831276	−	0.555860i	$-0.812389\pi$
$941$	−51.0000	−1.66255	−0.831276	+	0.555860i	$0.812389\pi$
$942$	0	0
$943$	9.00000	0.293080
$944$	0	0
$945$	−3.00000	−0.0975900
$946$	0	0
$947$	3.00000	0.0974869	0.0487435	−	0.998811i	$-0.484478\pi$
$947$	3.00000	0.0974869	0.0487435	+	0.998811i	$0.484478\pi$
$948$	0	0
$949$	−20.0000	−0.649227
$950$	0	0
$951$	27.0000	0.875535
$952$	0	0
$953$	−18.0000	−0.583077	−0.291539	−	0.956559i	$-0.594167\pi$
$953$	−18.0000	−0.583077	−0.291539	+	0.956559i	$0.594167\pi$
$954$	0	0
$955$	−36.0000	−1.16493
$956$	0	0
$957$	0	0
$958$	0	0
$959$	9.00000	0.290625
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	−12.0000	−0.386695
$964$	0	0
$965$	3.00000	0.0965734
$966$	0	0
$967$	−44.0000	−1.41494	−0.707472	−	0.706741i	$-0.750165\pi$
$967$	−44.0000	−1.41494	−0.707472	+	0.706741i	$0.750165\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	5.00000	0.160293
$974$	0	0
$975$	−20.0000	−0.640513
$976$	0	0
$977$	−21.0000	−0.671850	−0.335925	−	0.941889i	$-0.609049\pi$
$977$	−21.0000	−0.671850	−0.335925	+	0.941889i	$0.609049\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−19.0000	−0.606623
$982$	0	0
$983$	−30.0000	−0.956851	−0.478426	−	0.878128i	$-0.658792\pi$
$983$	−30.0000	−0.956851	−0.478426	+	0.878128i	$0.658792\pi$
$984$	0	0
$985$	−27.0000	−0.860292
$986$	0	0
$987$	3.00000	0.0954911
$988$	0	0
$989$	1.00000	0.0317982
$990$	0	0
$991$	28.0000	0.889449	0.444725	−	0.895667i	$-0.353302\pi$
$991$	28.0000	0.889449	0.444725	+	0.895667i	$0.353302\pi$
$992$	0	0
$993$	−16.0000	−0.507745
$994$	0	0
$995$	51.0000	1.61681
$996$	0	0
$997$	26.0000	0.823428	0.411714	−	0.911313i	$-0.364930\pi$
$997$	26.0000	0.823428	0.411714	+	0.911313i	$0.364930\pi$
$998$	0	0
$999$	7.00000	0.221470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.a.1.1		1
4.3	odd	2		966.2.a.i.1.1	✓	1
12.11	even	2		2898.2.a.j.1.1		1
28.27	even	2		6762.2.a.bd.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
966.2.a.i.1.1	✓	1	4.3	odd	2
2898.2.a.j.1.1		1	12.11	even	2
6762.2.a.bd.1.1		1	28.27	even	2
7728.2.a.a.1.1		1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7728.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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