LMFDB - Embedded newform 3380.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,2,Mod(1,3380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, 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("3380.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3380 = 2^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3380.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:	$26.9894358832$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 260)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3380.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.00000 q^{9} +O(q^{10})$

$q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.00000 q^{9} -3.00000 q^{11} +1.00000 q^{15} -3.00000 q^{17} -5.00000 q^{19} +1.00000 q^{21} +9.00000 q^{23} +1.00000 q^{25} -5.00000 q^{27} -9.00000 q^{29} -8.00000 q^{31} -3.00000 q^{33} +1.00000 q^{35} +7.00000 q^{37} -3.00000 q^{41} -1.00000 q^{43} -2.00000 q^{45} -6.00000 q^{49} -3.00000 q^{51} +6.00000 q^{53} -3.00000 q^{55} -5.00000 q^{57} -9.00000 q^{59} -1.00000 q^{61} -2.00000 q^{63} -5.00000 q^{67} +9.00000 q^{69} -9.00000 q^{71} -2.00000 q^{73} +1.00000 q^{75} -3.00000 q^{77} +8.00000 q^{79} +1.00000 q^{81} -3.00000 q^{85} -9.00000 q^{87} -3.00000 q^{89} -8.00000 q^{93} -5.00000 q^{95} -17.0000 q^{97} +6.00000 q^{99} +O(q^{100})$

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	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	9.00000	1.87663	0.938315	−	0.345782i	$-0.112386\pi$
$23$	9.00000	1.87663	0.938315	+	0.345782i	$0.112386\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	−9.00000	−1.67126	−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000	−1.67126	−0.835629	+	0.549294i	$0.814897\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	0	0
$33$	−3.00000	−0.522233
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	7.00000	1.15079	0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000	1.15079	0.575396	+	0.817875i	$0.304848\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	0	0
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	0	0
$45$	−2.00000	−0.298142
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−3.00000	−0.420084
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	−3.00000	−0.404520
$56$	0	0
$57$	−5.00000	−0.662266
$58$	0	0
$59$	−9.00000	−1.17170	−0.585850	−	0.810419i	$-0.699239\pi$
$59$	−9.00000	−1.17170	−0.585850	+	0.810419i	$0.699239\pi$
$60$	0	0
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−5.00000	−0.610847	−0.305424	−	0.952217i	$-0.598798\pi$
$67$	−5.00000	−0.610847	−0.305424	+	0.952217i	$0.598798\pi$
$68$	0	0
$69$	9.00000	1.08347
$70$	0	0
$71$	−9.00000	−1.06810	−0.534052	−	0.845452i	$-0.679331\pi$
$71$	−9.00000	−1.06810	−0.534052	+	0.845452i	$0.679331\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−3.00000	−0.341882
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	−3.00000	−0.325396
$86$	0	0
$87$	−9.00000	−0.964901
$88$	0	0
$89$	−3.00000	−0.317999	−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000	−0.317999	−0.159000	+	0.987279i	$0.550827\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−8.00000	−0.829561
$94$	0	0
$95$	−5.00000	−0.512989
$96$	0	0
$97$	−17.0000	−1.72609	−0.863044	−	0.505128i	$-0.831445\pi$
$97$	−17.0000	−1.72609	−0.863044	+	0.505128i	$0.831445\pi$
$98$	0	0
$99$	6.00000	0.603023
$100$	0	0
$101$	15.0000	1.49256	0.746278	−	0.665635i	$-0.231839\pi$
$101$	15.0000	1.49256	0.746278	+	0.665635i	$0.231839\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	0	0
$107$	−3.00000	−0.290021	−0.145010	−	0.989430i	$-0.546322\pi$
$107$	−3.00000	−0.290021	−0.145010	+	0.989430i	$0.546322\pi$
$108$	0	0
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	7.00000	0.664411
$112$	0	0
$113$	−15.0000	−1.41108	−0.705541	−	0.708669i	$-0.749296\pi$
$113$	−15.0000	−1.41108	−0.705541	+	0.708669i	$0.749296\pi$
$114$	0	0
$115$	9.00000	0.839254
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3.00000	−0.275010
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	−3.00000	−0.270501
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	17.0000	1.50851	0.754253	−	0.656584i	$-0.227999\pi$
$127$	17.0000	1.50851	0.754253	+	0.656584i	$0.227999\pi$
$128$	0	0
$129$	−1.00000	−0.0880451
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−5.00000	−0.433555
$134$	0	0
$135$	−5.00000	−0.430331
$136$	0	0
$137$	3.00000	0.256307	0.128154	−	0.991754i	$-0.459095\pi$
$137$	3.00000	0.256307	0.128154	+	0.991754i	$0.459095\pi$
$138$	0	0
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−9.00000	−0.747409
$146$	0	0
$147$	−6.00000	−0.494872
$148$	0	0
$149$	9.00000	0.737309	0.368654	−	0.929567i	$-0.379819\pi$
$149$	9.00000	0.737309	0.368654	+	0.929567i	$0.379819\pi$
$150$	0	0
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	0	0
$155$	−8.00000	−0.642575
$156$	0	0
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	9.00000	0.709299
$162$	0	0
$163$	1.00000	0.0783260	0.0391630	−	0.999233i	$-0.487531\pi$
$163$	1.00000	0.0783260	0.0391630	+	0.999233i	$0.487531\pi$
$164$	0	0
$165$	−3.00000	−0.233550
$166$	0	0
$167$	15.0000	1.16073	0.580367	−	0.814355i	$-0.302909\pi$
$167$	15.0000	1.16073	0.580367	+	0.814355i	$0.302909\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	10.0000	0.764719
$172$	0	0
$173$	21.0000	1.59660	0.798300	−	0.602260i	$-0.205733\pi$
$173$	21.0000	1.59660	0.798300	+	0.602260i	$0.205733\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−9.00000	−0.676481
$178$	0	0
$179$	15.0000	1.12115	0.560576	−	0.828103i	$-0.310580\pi$
$179$	15.0000	1.12115	0.560576	+	0.828103i	$0.310580\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	−1.00000	−0.0739221
$184$	0	0
$185$	7.00000	0.514650
$186$	0	0
$187$	9.00000	0.658145
$188$	0	0
$189$	−5.00000	−0.363696
$190$	0	0
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	0	0
$193$	−5.00000	−0.359908	−0.179954	−	0.983675i	$-0.557595\pi$
$193$	−5.00000	−0.359908	−0.179954	+	0.983675i	$0.557595\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.00000	0.213741	0.106871	−	0.994273i	$-0.465917\pi$
$197$	3.00000	0.213741	0.106871	+	0.994273i	$0.465917\pi$
$198$	0	0
$199$	−7.00000	−0.496217	−0.248108	−	0.968732i	$-0.579809\pi$
$199$	−7.00000	−0.496217	−0.248108	+	0.968732i	$0.579809\pi$
$200$	0	0
$201$	−5.00000	−0.352673
$202$	0	0
$203$	−9.00000	−0.631676
$204$	0	0
$205$	−3.00000	−0.209529
$206$	0	0
$207$	−18.0000	−1.25109
$208$	0	0
$209$	15.0000	1.03757
$210$	0	0
$211$	−25.0000	−1.72107	−0.860535	−	0.509390i	$-0.829871\pi$
$211$	−25.0000	−1.72107	−0.860535	+	0.509390i	$0.829871\pi$
$212$	0	0
$213$	−9.00000	−0.616670
$214$	0	0
$215$	−1.00000	−0.0681994
$216$	0	0
$217$	−8.00000	−0.543075
$218$	0	0
$219$	−2.00000	−0.135147
$220$	0	0
$221$	0	0
$222$	0	0
$223$	19.0000	1.27233	0.636167	−	0.771551i	$-0.280519\pi$
$223$	19.0000	1.27233	0.636167	+	0.771551i	$0.280519\pi$
$224$	0	0
$225$	−2.00000	−0.133333
$226$	0	0
$227$	−15.0000	−0.995585	−0.497792	−	0.867296i	$-0.665856\pi$
$227$	−15.0000	−0.995585	−0.497792	+	0.867296i	$0.665856\pi$
$228$	0	0
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	0	0
$231$	−3.00000	−0.197386
$232$	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
$235$	0	0
$236$	0	0
$237$	8.00000	0.519656
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	−23.0000	−1.48156	−0.740780	−	0.671748i	$-0.765544\pi$
$241$	−23.0000	−1.48156	−0.740780	+	0.671748i	$0.765544\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	0	0
$245$	−6.00000	−0.383326
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	9.00000	0.568075	0.284037	−	0.958813i	$-0.408326\pi$
$251$	9.00000	0.568075	0.284037	+	0.958813i	$0.408326\pi$
$252$	0	0
$253$	−27.0000	−1.69748
$254$	0	0
$255$	−3.00000	−0.187867
$256$	0	0
$257$	−27.0000	−1.68421	−0.842107	−	0.539311i	$-0.818685\pi$
$257$	−27.0000	−1.68421	−0.842107	+	0.539311i	$0.818685\pi$
$258$	0	0
$259$	7.00000	0.434959
$260$	0	0
$261$	18.0000	1.11417
$262$	0	0
$263$	−3.00000	−0.184988	−0.0924940	−	0.995713i	$-0.529484\pi$
$263$	−3.00000	−0.184988	−0.0924940	+	0.995713i	$0.529484\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	0	0
$267$	−3.00000	−0.183597
$268$	0	0
$269$	−21.0000	−1.28039	−0.640196	−	0.768211i	$-0.721147\pi$
$269$	−21.0000	−1.28039	−0.640196	+	0.768211i	$0.721147\pi$
$270$	0	0
$271$	13.0000	0.789694	0.394847	−	0.918747i	$-0.370798\pi$
$271$	13.0000	0.789694	0.394847	+	0.918747i	$0.370798\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.00000	−0.180907
$276$	0	0
$277$	−19.0000	−1.14160	−0.570800	−	0.821089i	$-0.693367\pi$
$277$	−19.0000	−1.14160	−0.570800	+	0.821089i	$0.693367\pi$
$278$	0	0
$279$	16.0000	0.957895
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	17.0000	1.01055	0.505273	−	0.862960i	$-0.331392\pi$
$283$	17.0000	1.01055	0.505273	+	0.862960i	$0.331392\pi$
$284$	0	0
$285$	−5.00000	−0.296174
$286$	0	0
$287$	−3.00000	−0.177084
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−17.0000	−0.996558
$292$	0	0
$293$	3.00000	0.175262	0.0876309	−	0.996153i	$-0.472070\pi$
$293$	3.00000	0.175262	0.0876309	+	0.996153i	$0.472070\pi$
$294$	0	0
$295$	−9.00000	−0.524000
$296$	0	0
$297$	15.0000	0.870388
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−1.00000	−0.0576390
$302$	0	0
$303$	15.0000	0.861727
$304$	0	0
$305$	−1.00000	−0.0572598
$306$	0	0
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	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
$315$	−2.00000	−0.112687
$316$	0	0
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	0	0
$319$	27.0000	1.51171
$320$	0	0
$321$	−3.00000	−0.167444
$322$	0	0
$323$	15.0000	0.834622
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−14.0000	−0.774202
$328$	0	0
$329$	0	0
$330$	0	0
$331$	19.0000	1.04433	0.522167	−	0.852843i	$-0.325124\pi$
$331$	19.0000	1.04433	0.522167	+	0.852843i	$0.325124\pi$
$332$	0	0
$333$	−14.0000	−0.767195
$334$	0	0
$335$	−5.00000	−0.273179
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	0	0
$339$	−15.0000	−0.814688
$340$	0	0
$341$	24.0000	1.29967
$342$	0	0
$343$	−13.0000	−0.701934
$344$	0	0
$345$	9.00000	0.484544
$346$	0	0
$347$	−9.00000	−0.483145	−0.241573	−	0.970383i	$-0.577663\pi$
$347$	−9.00000	−0.483145	−0.241573	+	0.970383i	$0.577663\pi$
$348$	0	0
$349$	−35.0000	−1.87351	−0.936754	−	0.349990i	$-0.886185\pi$
$349$	−35.0000	−1.87351	−0.936754	+	0.349990i	$0.886185\pi$
$350$	0	0
$351$	0	0
$352$	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
$355$	−9.00000	−0.477670
$356$	0	0
$357$	−3.00000	−0.158777
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	0	0
$363$	−2.00000	−0.104973
$364$	0	0
$365$	−2.00000	−0.104685
$366$	0	0
$367$	17.0000	0.887393	0.443696	−	0.896177i	$-0.353667\pi$
$367$	17.0000	0.887393	0.443696	+	0.896177i	$0.353667\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	6.00000	0.311504
$372$	0	0
$373$	5.00000	0.258890	0.129445	−	0.991587i	$-0.458680\pi$
$373$	5.00000	0.258890	0.129445	+	0.991587i	$0.458680\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−11.0000	−0.565032	−0.282516	−	0.959263i	$-0.591169\pi$
$379$	−11.0000	−0.565032	−0.282516	+	0.959263i	$0.591169\pi$
$380$	0	0
$381$	17.0000	0.870936
$382$	0	0
$383$	21.0000	1.07305	0.536525	−	0.843884i	$-0.319737\pi$
$383$	21.0000	1.07305	0.536525	+	0.843884i	$0.319737\pi$
$384$	0	0
$385$	−3.00000	−0.152894
$386$	0	0
$387$	2.00000	0.101666
$388$	0	0
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	−27.0000	−1.36545
$392$	0	0
$393$	0	0
$394$	0	0
$395$	8.00000	0.402524
$396$	0	0
$397$	−29.0000	−1.45547	−0.727734	−	0.685859i	$-0.759427\pi$
$397$	−29.0000	−1.45547	−0.727734	+	0.685859i	$0.759427\pi$
$398$	0	0
$399$	−5.00000	−0.250313
$400$	0	0
$401$	−15.0000	−0.749064	−0.374532	−	0.927214i	$-0.622197\pi$
$401$	−15.0000	−0.749064	−0.374532	+	0.927214i	$0.622197\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−21.0000	−1.04093
$408$	0	0
$409$	25.0000	1.23617	0.618085	−	0.786111i	$-0.287909\pi$
$409$	25.0000	1.23617	0.618085	+	0.786111i	$0.287909\pi$
$410$	0	0
$411$	3.00000	0.147979
$412$	0	0
$413$	−9.00000	−0.442861
$414$	0	0
$415$	0	0
$416$	0	0
$417$	5.00000	0.244851
$418$	0	0
$419$	−21.0000	−1.02592	−0.512959	−	0.858413i	$-0.671451\pi$
$419$	−21.0000	−1.02592	−0.512959	+	0.858413i	$0.671451\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−3.00000	−0.145521
$426$	0	0
$427$	−1.00000	−0.0483934
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−3.00000	−0.144505	−0.0722525	−	0.997386i	$-0.523019\pi$
$431$	−3.00000	−0.144505	−0.0722525	+	0.997386i	$0.523019\pi$
$432$	0	0
$433$	5.00000	0.240285	0.120142	−	0.992757i	$-0.461665\pi$
$433$	5.00000	0.240285	0.120142	+	0.992757i	$0.461665\pi$
$434$	0	0
$435$	−9.00000	−0.431517
$436$	0	0
$437$	−45.0000	−2.15264
$438$	0	0
$439$	−1.00000	−0.0477274	−0.0238637	−	0.999715i	$-0.507597\pi$
$439$	−1.00000	−0.0477274	−0.0238637	+	0.999715i	$0.507597\pi$
$440$	0	0
$441$	12.0000	0.571429
$442$	0	0
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	0	0
$445$	−3.00000	−0.142214
$446$	0	0
$447$	9.00000	0.425685
$448$	0	0
$449$	−15.0000	−0.707894	−0.353947	−	0.935266i	$-0.615161\pi$
$449$	−15.0000	−0.707894	−0.353947	+	0.935266i	$0.615161\pi$
$450$	0	0
$451$	9.00000	0.423793
$452$	0	0
$453$	−20.0000	−0.939682
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−41.0000	−1.91790	−0.958950	−	0.283577i	$-0.908479\pi$
$457$	−41.0000	−1.91790	−0.958950	+	0.283577i	$0.908479\pi$
$458$	0	0
$459$	15.0000	0.700140
$460$	0	0
$461$	−15.0000	−0.698620	−0.349310	−	0.937007i	$-0.613584\pi$
$461$	−15.0000	−0.698620	−0.349310	+	0.937007i	$0.613584\pi$
$462$	0	0
$463$	40.0000	1.85896	0.929479	−	0.368875i	$-0.120257\pi$
$463$	40.0000	1.85896	0.929479	+	0.368875i	$0.120257\pi$
$464$	0	0
$465$	−8.00000	−0.370991
$466$	0	0
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	0	0
$469$	−5.00000	−0.230879
$470$	0	0
$471$	14.0000	0.645086
$472$	0	0
$473$	3.00000	0.137940
$474$	0	0
$475$	−5.00000	−0.229416
$476$	0	0
$477$	−12.0000	−0.549442
$478$	0	0
$479$	−15.0000	−0.685367	−0.342684	−	0.939451i	$-0.611336\pi$
$479$	−15.0000	−0.685367	−0.342684	+	0.939451i	$0.611336\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	9.00000	0.409514
$484$	0	0
$485$	−17.0000	−0.771930
$486$	0	0
$487$	−35.0000	−1.58600	−0.793001	−	0.609221i	$-0.791482\pi$
$487$	−35.0000	−1.58600	−0.793001	+	0.609221i	$0.791482\pi$
$488$	0	0
$489$	1.00000	0.0452216
$490$	0	0
$491$	15.0000	0.676941	0.338470	−	0.940977i	$-0.390091\pi$
$491$	15.0000	0.676941	0.338470	+	0.940977i	$0.390091\pi$
$492$	0	0
$493$	27.0000	1.21602
$494$	0	0
$495$	6.00000	0.269680
$496$	0	0
$497$	−9.00000	−0.403705
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	15.0000	0.670151
$502$	0	0
$503$	39.0000	1.73892	0.869462	−	0.494000i	$-0.164466\pi$
$503$	39.0000	1.73892	0.869462	+	0.494000i	$0.164466\pi$
$504$	0	0
$505$	15.0000	0.667491
$506$	0	0
$507$	0	0
$508$	0	0
$509$	33.0000	1.46270	0.731350	−	0.682003i	$-0.238891\pi$
$509$	33.0000	1.46270	0.731350	+	0.682003i	$0.238891\pi$
$510$	0	0
$511$	−2.00000	−0.0884748
$512$	0	0
$513$	25.0000	1.10378
$514$	0	0
$515$	8.00000	0.352522
$516$	0	0
$517$	0	0
$518$	0	0
$519$	21.0000	0.921798
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	5.00000	0.218635	0.109317	−	0.994007i	$-0.465134\pi$
$523$	5.00000	0.218635	0.109317	+	0.994007i	$0.465134\pi$
$524$	0	0
$525$	1.00000	0.0436436
$526$	0	0
$527$	24.0000	1.04546
$528$	0	0
$529$	58.0000	2.52174
$530$	0	0
$531$	18.0000	0.781133
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−3.00000	−0.129701
$536$	0	0
$537$	15.0000	0.647298
$538$	0	0
$539$	18.0000	0.775315
$540$	0	0
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	0	0
$543$	−10.0000	−0.429141
$544$	0	0
$545$	−14.0000	−0.599694
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	0	0
$549$	2.00000	0.0853579
$550$	0	0
$551$	45.0000	1.91706
$552$	0	0
$553$	8.00000	0.340195
$554$	0	0
$555$	7.00000	0.297133
$556$	0	0
$557$	39.0000	1.65248	0.826242	−	0.563316i	$-0.190475\pi$
$557$	39.0000	1.65248	0.826242	+	0.563316i	$0.190475\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	9.00000	0.379980
$562$	0	0
$563$	−33.0000	−1.39078	−0.695392	−	0.718631i	$-0.744769\pi$
$563$	−33.0000	−1.39078	−0.695392	+	0.718631i	$0.744769\pi$
$564$	0	0
$565$	−15.0000	−0.631055
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−33.0000	−1.38343	−0.691716	−	0.722170i	$-0.743145\pi$
$569$	−33.0000	−1.38343	−0.691716	+	0.722170i	$0.743145\pi$
$570$	0	0
$571$	44.0000	1.84134	0.920671	−	0.390339i	$-0.127642\pi$
$571$	44.0000	1.84134	0.920671	+	0.390339i	$0.127642\pi$
$572$	0	0
$573$	−3.00000	−0.125327
$574$	0	0
$575$	9.00000	0.375326
$576$	0	0
$577$	22.0000	0.915872	0.457936	−	0.888985i	$-0.348589\pi$
$577$	22.0000	0.915872	0.457936	+	0.888985i	$0.348589\pi$
$578$	0	0
$579$	−5.00000	−0.207793
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−18.0000	−0.745484
$584$	0	0
$585$	0	0
$586$	0	0
$587$	27.0000	1.11441	0.557205	−	0.830375i	$-0.311874\pi$
$587$	27.0000	1.11441	0.557205	+	0.830375i	$0.311874\pi$
$588$	0	0
$589$	40.0000	1.64817
$590$	0	0
$591$	3.00000	0.123404
$592$	0	0
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	−3.00000	−0.122988
$596$	0	0
$597$	−7.00000	−0.286491
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−1.00000	−0.0407909	−0.0203954	−	0.999792i	$-0.506493\pi$
$601$	−1.00000	−0.0407909	−0.0203954	+	0.999792i	$0.506493\pi$
$602$	0	0
$603$	10.0000	0.407231
$604$	0	0
$605$	−2.00000	−0.0813116
$606$	0	0
$607$	23.0000	0.933541	0.466771	−	0.884378i	$-0.345417\pi$
$607$	23.0000	0.933541	0.466771	+	0.884378i	$0.345417\pi$
$608$	0	0
$609$	−9.00000	−0.364698
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−29.0000	−1.17130	−0.585649	−	0.810564i	$-0.699160\pi$
$613$	−29.0000	−1.17130	−0.585649	+	0.810564i	$0.699160\pi$
$614$	0	0
$615$	−3.00000	−0.120972
$616$	0	0
$617$	27.0000	1.08698	0.543490	−	0.839416i	$-0.317103\pi$
$617$	27.0000	1.08698	0.543490	+	0.839416i	$0.317103\pi$
$618$	0	0
$619$	28.0000	1.12542	0.562708	−	0.826656i	$-0.309760\pi$
$619$	28.0000	1.12542	0.562708	+	0.826656i	$0.309760\pi$
$620$	0	0
$621$	−45.0000	−1.80579
$622$	0	0
$623$	−3.00000	−0.120192
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	15.0000	0.599042
$628$	0	0
$629$	−21.0000	−0.837325
$630$	0	0
$631$	7.00000	0.278666	0.139333	−	0.990246i	$-0.455504\pi$
$631$	7.00000	0.278666	0.139333	+	0.990246i	$0.455504\pi$
$632$	0	0
$633$	−25.0000	−0.993661
$634$	0	0
$635$	17.0000	0.674624
$636$	0	0
$637$	0	0
$638$	0	0
$639$	18.0000	0.712069
$640$	0	0
$641$	3.00000	0.118493	0.0592464	−	0.998243i	$-0.481130\pi$
$641$	3.00000	0.118493	0.0592464	+	0.998243i	$0.481130\pi$
$642$	0	0
$643$	25.0000	0.985904	0.492952	−	0.870057i	$-0.335918\pi$
$643$	25.0000	0.985904	0.492952	+	0.870057i	$0.335918\pi$
$644$	0	0
$645$	−1.00000	−0.0393750
$646$	0	0
$647$	−27.0000	−1.06148	−0.530740	−	0.847535i	$-0.678086\pi$
$647$	−27.0000	−1.06148	−0.530740	+	0.847535i	$0.678086\pi$
$648$	0	0
$649$	27.0000	1.05984
$650$	0	0
$651$	−8.00000	−0.313545
$652$	0	0
$653$	−27.0000	−1.05659	−0.528296	−	0.849060i	$-0.677169\pi$
$653$	−27.0000	−1.05659	−0.528296	+	0.849060i	$0.677169\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	4.00000	0.156055
$658$	0	0
$659$	45.0000	1.75295	0.876476	−	0.481446i	$-0.159888\pi$
$659$	45.0000	1.75295	0.876476	+	0.481446i	$0.159888\pi$
$660$	0	0
$661$	−35.0000	−1.36134	−0.680671	−	0.732589i	$-0.738312\pi$
$661$	−35.0000	−1.36134	−0.680671	+	0.732589i	$0.738312\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−5.00000	−0.193892
$666$	0	0
$667$	−81.0000	−3.13633
$668$	0	0
$669$	19.0000	0.734582
$670$	0	0
$671$	3.00000	0.115814
$672$	0	0
$673$	−19.0000	−0.732396	−0.366198	−	0.930537i	$-0.619341\pi$
$673$	−19.0000	−0.732396	−0.366198	+	0.930537i	$0.619341\pi$
$674$	0	0
$675$	−5.00000	−0.192450
$676$	0	0
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	0	0
$679$	−17.0000	−0.652400
$680$	0	0
$681$	−15.0000	−0.574801
$682$	0	0
$683$	21.0000	0.803543	0.401771	−	0.915740i	$-0.368395\pi$
$683$	21.0000	0.803543	0.401771	+	0.915740i	$0.368395\pi$
$684$	0	0
$685$	3.00000	0.114624
$686$	0	0
$687$	22.0000	0.839352
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−23.0000	−0.874961	−0.437481	−	0.899228i	$-0.644129\pi$
$691$	−23.0000	−0.874961	−0.437481	+	0.899228i	$0.644129\pi$
$692$	0	0
$693$	6.00000	0.227921
$694$	0	0
$695$	5.00000	0.189661
$696$	0	0
$697$	9.00000	0.340899
$698$	0	0
$699$	−6.00000	−0.226941
$700$	0	0
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	0	0
$703$	−35.0000	−1.32005
$704$	0	0
$705$	0	0
$706$	0	0
$707$	15.0000	0.564133
$708$	0	0
$709$	−35.0000	−1.31445	−0.657226	−	0.753693i	$-0.728270\pi$
$709$	−35.0000	−1.31445	−0.657226	+	0.753693i	$0.728270\pi$
$710$	0	0
$711$	−16.0000	−0.600047
$712$	0	0
$713$	−72.0000	−2.69642
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−24.0000	−0.896296
$718$	0	0
$719$	45.0000	1.67822	0.839108	−	0.543964i	$-0.183077\pi$
$719$	45.0000	1.67822	0.839108	+	0.543964i	$0.183077\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	0	0
$723$	−23.0000	−0.855379
$724$	0	0
$725$	−9.00000	−0.334252
$726$	0	0
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	3.00000	0.110959
$732$	0	0
$733$	34.0000	1.25582	0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000	1.25582	0.627909	+	0.778287i	$0.283911\pi$
$734$	0	0
$735$	−6.00000	−0.221313
$736$	0	0
$737$	15.0000	0.552532
$738$	0	0
$739$	−11.0000	−0.404642	−0.202321	−	0.979319i	$-0.564848\pi$
$739$	−11.0000	−0.404642	−0.202321	+	0.979319i	$0.564848\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−33.0000	−1.21065	−0.605326	−	0.795977i	$-0.706957\pi$
$743$	−33.0000	−1.21065	−0.605326	+	0.795977i	$0.706957\pi$
$744$	0	0
$745$	9.00000	0.329734
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−3.00000	−0.109618
$750$	0	0
$751$	−13.0000	−0.474377	−0.237188	−	0.971464i	$-0.576226\pi$
$751$	−13.0000	−0.474377	−0.237188	+	0.971464i	$0.576226\pi$
$752$	0	0
$753$	9.00000	0.327978
$754$	0	0
$755$	−20.0000	−0.727875
$756$	0	0
$757$	5.00000	0.181728	0.0908640	−	0.995863i	$-0.471037\pi$
$757$	5.00000	0.181728	0.0908640	+	0.995863i	$0.471037\pi$
$758$	0	0
$759$	−27.0000	−0.980038
$760$	0	0
$761$	9.00000	0.326250	0.163125	−	0.986605i	$-0.447843\pi$
$761$	9.00000	0.326250	0.163125	+	0.986605i	$0.447843\pi$
$762$	0	0
$763$	−14.0000	−0.506834
$764$	0	0
$765$	6.00000	0.216930
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−11.0000	−0.396670	−0.198335	−	0.980134i	$-0.563553\pi$
$769$	−11.0000	−0.396670	−0.198335	+	0.980134i	$0.563553\pi$
$770$	0	0
$771$	−27.0000	−0.972381
$772$	0	0
$773$	15.0000	0.539513	0.269756	−	0.962929i	$-0.413057\pi$
$773$	15.0000	0.539513	0.269756	+	0.962929i	$0.413057\pi$
$774$	0	0
$775$	−8.00000	−0.287368
$776$	0	0
$777$	7.00000	0.251124
$778$	0	0
$779$	15.0000	0.537431
$780$	0	0
$781$	27.0000	0.966136
$782$	0	0
$783$	45.0000	1.60817
$784$	0	0
$785$	14.0000	0.499681
$786$	0	0
$787$	49.0000	1.74666	0.873331	−	0.487128i	$-0.161955\pi$
$787$	49.0000	1.74666	0.873331	+	0.487128i	$0.161955\pi$
$788$	0	0
$789$	−3.00000	−0.106803
$790$	0	0
$791$	−15.0000	−0.533339
$792$	0	0
$793$	0	0
$794$	0	0
$795$	6.00000	0.212798
$796$	0	0
$797$	−27.0000	−0.956389	−0.478195	−	0.878254i	$-0.658709\pi$
$797$	−27.0000	−0.956389	−0.478195	+	0.878254i	$0.658709\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	6.00000	0.212000
$802$	0	0
$803$	6.00000	0.211735
$804$	0	0
$805$	9.00000	0.317208
$806$	0	0
$807$	−21.0000	−0.739235
$808$	0	0
$809$	39.0000	1.37117	0.685583	−	0.727994i	$-0.259547\pi$
$809$	39.0000	1.37117	0.685583	+	0.727994i	$0.259547\pi$
$810$	0	0
$811$	16.0000	0.561836	0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000	0.561836	0.280918	+	0.959732i	$0.409361\pi$
$812$	0	0
$813$	13.0000	0.455930
$814$	0	0
$815$	1.00000	0.0350285
$816$	0	0
$817$	5.00000	0.174928
$818$	0	0
$819$	0	0
$820$	0	0
$821$	21.0000	0.732905	0.366453	−	0.930437i	$-0.380572\pi$
$821$	21.0000	0.732905	0.366453	+	0.930437i	$0.380572\pi$
$822$	0	0
$823$	47.0000	1.63832	0.819159	−	0.573567i	$-0.194441\pi$
$823$	47.0000	1.63832	0.819159	+	0.573567i	$0.194441\pi$
$824$	0	0
$825$	−3.00000	−0.104447
$826$	0	0
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	11.0000	0.382046	0.191023	−	0.981586i	$-0.438820\pi$
$829$	11.0000	0.382046	0.191023	+	0.981586i	$0.438820\pi$
$830$	0	0
$831$	−19.0000	−0.659103
$832$	0	0
$833$	18.0000	0.623663
$834$	0	0
$835$	15.0000	0.519096
$836$	0	0
$837$	40.0000	1.38260
$838$	0	0
$839$	15.0000	0.517858	0.258929	−	0.965896i	$-0.416631\pi$
$839$	15.0000	0.517858	0.258929	+	0.965896i	$0.416631\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	0	0
$843$	18.0000	0.619953
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−2.00000	−0.0687208
$848$	0	0
$849$	17.0000	0.583438
$850$	0	0
$851$	63.0000	2.15961
$852$	0	0
$853$	−14.0000	−0.479351	−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000	−0.479351	−0.239675	+	0.970853i	$0.577041\pi$
$854$	0	0
$855$	10.0000	0.341993
$856$	0	0
$857$	−42.0000	−1.43469	−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000	−1.43469	−0.717346	+	0.696717i	$0.754643\pi$
$858$	0	0
$859$	8.00000	0.272956	0.136478	−	0.990643i	$-0.456422\pi$
$859$	8.00000	0.272956	0.136478	+	0.990643i	$0.456422\pi$
$860$	0	0
$861$	−3.00000	−0.102240
$862$	0	0
$863$	−48.0000	−1.63394	−0.816970	−	0.576681i	$-0.804348\pi$
$863$	−48.0000	−1.63394	−0.816970	+	0.576681i	$0.804348\pi$
$864$	0	0
$865$	21.0000	0.714021
$866$	0	0
$867$	−8.00000	−0.271694
$868$	0	0
$869$	−24.0000	−0.814144
$870$	0	0
$871$	0	0
$872$	0	0
$873$	34.0000	1.15073
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−17.0000	−0.574049	−0.287025	−	0.957923i	$-0.592666\pi$
$877$	−17.0000	−0.574049	−0.287025	+	0.957923i	$0.592666\pi$
$878$	0	0
$879$	3.00000	0.101187
$880$	0	0
$881$	3.00000	0.101073	0.0505363	−	0.998722i	$-0.483907\pi$
$881$	3.00000	0.101073	0.0505363	+	0.998722i	$0.483907\pi$
$882$	0	0
$883$	56.0000	1.88455	0.942275	−	0.334840i	$-0.108682\pi$
$883$	56.0000	1.88455	0.942275	+	0.334840i	$0.108682\pi$
$884$	0	0
$885$	−9.00000	−0.302532
$886$	0	0
$887$	33.0000	1.10803	0.554016	−	0.832506i	$-0.313095\pi$
$887$	33.0000	1.10803	0.554016	+	0.832506i	$0.313095\pi$
$888$	0	0
$889$	17.0000	0.570162
$890$	0	0
$891$	−3.00000	−0.100504
$892$	0	0
$893$	0	0
$894$	0	0
$895$	15.0000	0.501395
$896$	0	0
$897$	0	0
$898$	0	0
$899$	72.0000	2.40133
$900$	0	0
$901$	−18.0000	−0.599667
$902$	0	0
$903$	−1.00000	−0.0332779
$904$	0	0
$905$	−10.0000	−0.332411
$906$	0	0
$907$	−55.0000	−1.82625	−0.913123	−	0.407685i	$-0.866336\pi$
$907$	−55.0000	−1.82625	−0.913123	+	0.407685i	$0.866336\pi$
$908$	0	0
$909$	−30.0000	−0.995037
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−1.00000	−0.0330590
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−7.00000	−0.230909	−0.115454	−	0.993313i	$-0.536832\pi$
$919$	−7.00000	−0.230909	−0.115454	+	0.993313i	$0.536832\pi$
$920$	0	0
$921$	4.00000	0.131804
$922$	0	0
$923$	0	0
$924$	0	0
$925$	7.00000	0.230159
$926$	0	0
$927$	−16.0000	−0.525509
$928$	0	0
$929$	−3.00000	−0.0984268	−0.0492134	−	0.998788i	$-0.515671\pi$
$929$	−3.00000	−0.0984268	−0.0492134	+	0.998788i	$0.515671\pi$
$930$	0	0
$931$	30.0000	0.983210
$932$	0	0
$933$	12.0000	0.392862
$934$	0	0
$935$	9.00000	0.294331
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	−10.0000	−0.326338
$940$	0	0
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	0	0
$943$	−27.0000	−0.879241
$944$	0	0
$945$	−5.00000	−0.162650
$946$	0	0
$947$	51.0000	1.65728	0.828639	−	0.559784i	$-0.189116\pi$
$947$	51.0000	1.65728	0.828639	+	0.559784i	$0.189116\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	18.0000	0.583690
$952$	0	0
$953$	21.0000	0.680257	0.340128	−	0.940379i	$-0.389529\pi$
$953$	21.0000	0.680257	0.340128	+	0.940379i	$0.389529\pi$
$954$	0	0
$955$	−3.00000	−0.0970777
$956$	0	0
$957$	27.0000	0.872786
$958$	0	0
$959$	3.00000	0.0968751
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	6.00000	0.193347
$964$	0	0
$965$	−5.00000	−0.160956
$966$	0	0
$967$	4.00000	0.128631	0.0643157	−	0.997930i	$-0.479514\pi$
$967$	4.00000	0.128631	0.0643157	+	0.997930i	$0.479514\pi$
$968$	0	0
$969$	15.0000	0.481869
$970$	0	0
$971$	−15.0000	−0.481373	−0.240686	−	0.970603i	$-0.577373\pi$
$971$	−15.0000	−0.481373	−0.240686	+	0.970603i	$0.577373\pi$
$972$	0	0
$973$	5.00000	0.160293
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.00000	0.0959785	0.0479893	−	0.998848i	$-0.484719\pi$
$977$	3.00000	0.0959785	0.0479893	+	0.998848i	$0.484719\pi$
$978$	0	0
$979$	9.00000	0.287641
$980$	0	0
$981$	28.0000	0.893971
$982$	0	0
$983$	48.0000	1.53096	0.765481	−	0.643458i	$-0.222501\pi$
$983$	48.0000	1.53096	0.765481	+	0.643458i	$0.222501\pi$
$984$	0	0
$985$	3.00000	0.0955879
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−9.00000	−0.286183
$990$	0	0
$991$	−49.0000	−1.55654	−0.778268	−	0.627932i	$-0.783902\pi$
$991$	−49.0000	−1.55654	−0.778268	+	0.627932i	$0.783902\pi$
$992$	0	0
$993$	19.0000	0.602947
$994$	0	0
$995$	−7.00000	−0.221915
$996$	0	0
$997$	5.00000	0.158352	0.0791758	−	0.996861i	$-0.474771\pi$
$997$	5.00000	0.158352	0.0791758	+	0.996861i	$0.474771\pi$
$998$	0	0
$999$	−35.0000	−1.10735

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.2.a.i.1.1		1
13.4	even	6		260.2.i.a.81.1	yes	2
13.5	odd	4		3380.2.f.d.3041.1		2
13.8	odd	4		3380.2.f.d.3041.2		2
13.10	even	6		260.2.i.a.61.1	✓	2
13.12	even	2		3380.2.a.f.1.1		1
39.17	odd	6		2340.2.q.f.2161.1		2
39.23	odd	6		2340.2.q.f.1621.1		2
52.23	odd	6		1040.2.q.i.321.1		2
52.43	odd	6		1040.2.q.i.81.1		2
65.4	even	6		1300.2.i.d.601.1		2
65.17	odd	12		1300.2.bb.b.549.1		4
65.23	odd	12		1300.2.bb.b.1049.1		4
65.43	odd	12		1300.2.bb.b.549.2		4
65.49	even	6		1300.2.i.d.1101.1		2
65.62	odd	12		1300.2.bb.b.1049.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.i.a.61.1	✓	2	13.10	even	6
260.2.i.a.81.1	yes	2	13.4	even	6
1040.2.q.i.81.1		2	52.43	odd	6
1040.2.q.i.321.1		2	52.23	odd	6
1300.2.i.d.601.1		2	65.4	even	6
1300.2.i.d.1101.1		2	65.49	even	6
1300.2.bb.b.549.1		4	65.17	odd	12
1300.2.bb.b.549.2		4	65.43	odd	12
1300.2.bb.b.1049.1		4	65.23	odd	12
1300.2.bb.b.1049.2		4	65.62	odd	12
2340.2.q.f.1621.1		2	39.23	odd	6
2340.2.q.f.2161.1		2	39.17	odd	6
3380.2.a.f.1.1		1	13.12	even	2
3380.2.a.i.1.1		1	1.1	even	1	trivial
3380.2.f.d.3041.1		2	13.5	odd	4
3380.2.f.d.3041.2		2	13.8	odd	4

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:	\( N \)	\(=\)	\( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3380.a (trivial)

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