LMFDB - Embedded newform 3332.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3332,2,Mod(1,3332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3332, 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("3332.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3332 = 2^{2} \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3332.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:	$26.6061539535$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 476)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3332.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} -4.00000 q^{5} +6.00000 q^{9} +O(q^{10})$

$q+3.00000 q^{3} -4.00000 q^{5} +6.00000 q^{9} +1.00000 q^{11} -3.00000 q^{13} -12.0000 q^{15} +1.00000 q^{17} +2.00000 q^{19} +4.00000 q^{23} +11.0000 q^{25} +9.00000 q^{27} -8.00000 q^{31} +3.00000 q^{33} +8.00000 q^{37} -9.00000 q^{39} +10.0000 q^{43} -24.0000 q^{45} +10.0000 q^{47} +3.00000 q^{51} +3.00000 q^{53} -4.00000 q^{55} +6.00000 q^{57} +14.0000 q^{59} +8.00000 q^{61} +12.0000 q^{65} -10.0000 q^{67} +12.0000 q^{69} -5.00000 q^{71} +16.0000 q^{73} +33.0000 q^{75} +11.0000 q^{79} +9.00000 q^{81} -12.0000 q^{83} -4.00000 q^{85} -9.00000 q^{89} -24.0000 q^{93} -8.00000 q^{95} -10.0000 q^{97} +6.00000 q^{99} +O(q^{100})$

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
$3$	3.00000	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$3$	3.00000	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$4$	0	0
$5$	−4.00000	−1.78885	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	−4.00000	−1.78885	−0.894427	+	0.447214i	$0.852416\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	6.00000	2.00000
$10$	0	0
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	0	0
$13$	−3.00000	−0.832050	−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000	−0.832050	−0.416025	+	0.909353i	$0.636577\pi$
$14$	0	0
$15$	−12.0000	−3.09839
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	11.0000	2.20000
$26$	0	0
$27$	9.00000	1.73205
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	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
$35$	0	0
$36$	0	0
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	0	0
$39$	−9.00000	−1.44115
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	0	0
$45$	−24.0000	−3.57771
$46$	0	0
$47$	10.0000	1.45865	0.729325	−	0.684167i	$-0.239834\pi$
$47$	10.0000	1.45865	0.729325	+	0.684167i	$0.239834\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	3.00000	0.420084
$52$	0	0
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	0	0
$57$	6.00000	0.794719
$58$	0	0
$59$	14.0000	1.82264	0.911322	−	0.411693i	$-0.135063\pi$
$59$	14.0000	1.82264	0.911322	+	0.411693i	$0.135063\pi$
$60$	0	0
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	12.0000	1.48842
$66$	0	0
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	0	0
$69$	12.0000	1.44463
$70$	0	0
$71$	−5.00000	−0.593391	−0.296695	−	0.954972i	$-0.595885\pi$
$71$	−5.00000	−0.593391	−0.296695	+	0.954972i	$0.595885\pi$
$72$	0	0
$73$	16.0000	1.87266	0.936329	−	0.351123i	$-0.114200\pi$
$73$	16.0000	1.87266	0.936329	+	0.351123i	$0.114200\pi$
$74$	0	0
$75$	33.0000	3.81051
$76$	0	0
$77$	0	0
$78$	0	0
$79$	11.0000	1.23760	0.618798	−	0.785550i	$-0.287620\pi$
$79$	11.0000	1.23760	0.618798	+	0.785550i	$0.287620\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−9.00000	−0.953998	−0.476999	−	0.878904i	$-0.658275\pi$
$89$	−9.00000	−0.953998	−0.476999	+	0.878904i	$0.658275\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−24.0000	−2.48868
$94$	0	0
$95$	−8.00000	−0.820783
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	0	0
$99$	6.00000	0.603023
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	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
$107$	−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
$109$	8.00000	0.766261	0.383131	−	0.923694i	$-0.374846\pi$
$109$	8.00000	0.766261	0.383131	+	0.923694i	$0.374846\pi$
$110$	0	0
$111$	24.0000	2.27798
$112$	0	0
$113$	−8.00000	−0.752577	−0.376288	−	0.926503i	$-0.622800\pi$
$113$	−8.00000	−0.752577	−0.376288	+	0.926503i	$0.622800\pi$
$114$	0	0
$115$	−16.0000	−1.49201
$116$	0	0
$117$	−18.0000	−1.66410
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−24.0000	−2.14663
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	30.0000	2.64135
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−36.0000	−3.09839
$136$	0	0
$137$	17.0000	1.45241	0.726204	−	0.687479i	$-0.241283\pi$
$137$	17.0000	1.45241	0.726204	+	0.687479i	$0.241283\pi$
$138$	0	0
$139$	21.0000	1.78120	0.890598	−	0.454791i	$-0.150286\pi$
$139$	21.0000	1.78120	0.890598	+	0.454791i	$0.150286\pi$
$140$	0	0
$141$	30.0000	2.52646
$142$	0	0
$143$	−3.00000	−0.250873
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−11.0000	−0.901155	−0.450578	−	0.892737i	$-0.648782\pi$
$149$	−11.0000	−0.901155	−0.450578	+	0.892737i	$0.648782\pi$
$150$	0	0
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	0	0
$155$	32.0000	2.57030
$156$	0	0
$157$	−17.0000	−1.35675	−0.678374	−	0.734717i	$-0.737315\pi$
$157$	−17.0000	−1.35675	−0.678374	+	0.734717i	$0.737315\pi$
$158$	0	0
$159$	9.00000	0.713746
$160$	0	0
$161$	0	0
$162$	0	0
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	0	0
$165$	−12.0000	−0.934199
$166$	0	0
$167$	−17.0000	−1.31550	−0.657750	−	0.753237i	$-0.728492\pi$
$167$	−17.0000	−1.31550	−0.657750	+	0.753237i	$0.728492\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	12.0000	0.917663
$172$	0	0
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	42.0000	3.15691
$178$	0	0
$179$	16.0000	1.19590	0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000	1.19590	0.597948	+	0.801535i	$0.295983\pi$
$180$	0	0
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	24.0000	1.77413
$184$	0	0
$185$	−32.0000	−2.35269
$186$	0	0
$187$	1.00000	0.0731272
$188$	0	0
$189$	0	0
$190$	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
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	0	0
$195$	36.0000	2.57801
$196$	0	0
$197$	−14.0000	−0.997459	−0.498729	−	0.866758i	$-0.666200\pi$
$197$	−14.0000	−0.997459	−0.498729	+	0.866758i	$0.666200\pi$
$198$	0	0
$199$	3.00000	0.212664	0.106332	−	0.994331i	$-0.466089\pi$
$199$	3.00000	0.212664	0.106332	+	0.994331i	$0.466089\pi$
$200$	0	0
$201$	−30.0000	−2.11604
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	24.0000	1.66812
$208$	0	0
$209$	2.00000	0.138343
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	0	0
$213$	−15.0000	−1.02778
$214$	0	0
$215$	−40.0000	−2.72798
$216$	0	0
$217$	0	0
$218$	0	0
$219$	48.0000	3.24354
$220$	0	0
$221$	−3.00000	−0.201802
$222$	0	0
$223$	6.00000	0.401790	0.200895	−	0.979613i	$-0.435615\pi$
$223$	6.00000	0.401790	0.200895	+	0.979613i	$0.435615\pi$
$224$	0	0
$225$	66.0000	4.40000
$226$	0	0
$227$	3.00000	0.199117	0.0995585	−	0.995032i	$-0.468257\pi$
$227$	3.00000	0.199117	0.0995585	+	0.995032i	$0.468257\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−20.0000	−1.31024	−0.655122	−	0.755523i	$-0.727383\pi$
$233$	−20.0000	−1.31024	−0.655122	+	0.755523i	$0.727383\pi$
$234$	0	0
$235$	−40.0000	−2.60931
$236$	0	0
$237$	33.0000	2.14358
$238$	0	0
$239$	2.00000	0.129369	0.0646846	−	0.997906i	$-0.479396\pi$
$239$	2.00000	0.129369	0.0646846	+	0.997906i	$0.479396\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−6.00000	−0.381771
$248$	0	0
$249$	−36.0000	−2.28141
$250$	0	0
$251$	−8.00000	−0.504956	−0.252478	−	0.967603i	$-0.581245\pi$
$251$	−8.00000	−0.504956	−0.252478	+	0.967603i	$0.581245\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	0	0
$255$	−12.0000	−0.751469
$256$	0	0
$257$	−11.0000	−0.686161	−0.343081	−	0.939306i	$-0.611470\pi$
$257$	−11.0000	−0.686161	−0.343081	+	0.939306i	$0.611470\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	0	0
$265$	−12.0000	−0.737154
$266$	0	0
$267$	−27.0000	−1.65237
$268$	0	0
$269$	2.00000	0.121942	0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000	0.121942	0.0609711	+	0.998140i	$0.480580\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	11.0000	0.663325
$276$	0	0
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	0	0
$279$	−48.0000	−2.87368
$280$	0	0
$281$	−3.00000	−0.178965	−0.0894825	−	0.995988i	$-0.528521\pi$
$281$	−3.00000	−0.178965	−0.0894825	+	0.995988i	$0.528521\pi$
$282$	0	0
$283$	−19.0000	−1.12943	−0.564716	−	0.825285i	$-0.691014\pi$
$283$	−19.0000	−1.12943	−0.564716	+	0.825285i	$0.691014\pi$
$284$	0	0
$285$	−24.0000	−1.42164
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−30.0000	−1.75863
$292$	0	0
$293$	−1.00000	−0.0584206	−0.0292103	−	0.999573i	$-0.509299\pi$
$293$	−1.00000	−0.0584206	−0.0292103	+	0.999573i	$0.509299\pi$
$294$	0	0
$295$	−56.0000	−3.26045
$296$	0	0
$297$	9.00000	0.522233
$298$	0	0
$299$	−12.0000	−0.693978
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−30.0000	−1.72345
$304$	0	0
$305$	−32.0000	−1.83231
$306$	0	0
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	0	0
$309$	12.0000	0.682656
$310$	0	0
$311$	9.00000	0.510343	0.255172	−	0.966896i	$-0.417868\pi$
$311$	9.00000	0.510343	0.255172	+	0.966896i	$0.417868\pi$
$312$	0	0
$313$	18.0000	1.01742	0.508710	−	0.860938i	$-0.330123\pi$
$313$	18.0000	1.01742	0.508710	+	0.860938i	$0.330123\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−9.00000	−0.502331
$322$	0	0
$323$	2.00000	0.111283
$324$	0	0
$325$	−33.0000	−1.83051
$326$	0	0
$327$	24.0000	1.32720
$328$	0	0
$329$	0	0
$330$	0	0
$331$	10.0000	0.549650	0.274825	−	0.961494i	$-0.411380\pi$
$331$	10.0000	0.549650	0.274825	+	0.961494i	$0.411380\pi$
$332$	0	0
$333$	48.0000	2.63038
$334$	0	0
$335$	40.0000	2.18543
$336$	0	0
$337$	−8.00000	−0.435788	−0.217894	−	0.975972i	$-0.569919\pi$
$337$	−8.00000	−0.435788	−0.217894	+	0.975972i	$0.569919\pi$
$338$	0	0
$339$	−24.0000	−1.30350
$340$	0	0
$341$	−8.00000	−0.433224
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−48.0000	−2.58423
$346$	0	0
$347$	−4.00000	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	0	0
$349$	−30.0000	−1.60586	−0.802932	−	0.596071i	$-0.796728\pi$
$349$	−30.0000	−1.60586	−0.802932	+	0.596071i	$0.796728\pi$
$350$	0	0
$351$	−27.0000	−1.44115
$352$	0	0
$353$	−11.0000	−0.585471	−0.292735	−	0.956193i	$-0.594566\pi$
$353$	−11.0000	−0.585471	−0.292735	+	0.956193i	$0.594566\pi$
$354$	0	0
$355$	20.0000	1.06149
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−10.0000	−0.527780	−0.263890	−	0.964553i	$-0.585006\pi$
$359$	−10.0000	−0.527780	−0.263890	+	0.964553i	$0.585006\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	−30.0000	−1.57459
$364$	0	0
$365$	−64.0000	−3.34991
$366$	0	0
$367$	−3.00000	−0.156599	−0.0782994	−	0.996930i	$-0.524949\pi$
$367$	−3.00000	−0.156599	−0.0782994	+	0.996930i	$0.524949\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	7.00000	0.362446	0.181223	−	0.983442i	$-0.441994\pi$
$373$	7.00000	0.362446	0.181223	+	0.983442i	$0.441994\pi$
$374$	0	0
$375$	−72.0000	−3.71806
$376$	0	0
$377$	0	0
$378$	0	0
$379$	13.0000	0.667765	0.333883	−	0.942615i	$-0.391641\pi$
$379$	13.0000	0.667765	0.333883	+	0.942615i	$0.391641\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−34.0000	−1.73732	−0.868659	−	0.495410i	$-0.835018\pi$
$383$	−34.0000	−1.73732	−0.868659	+	0.495410i	$0.835018\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	60.0000	3.04997
$388$	0	0
$389$	−3.00000	−0.152106	−0.0760530	−	0.997104i	$-0.524232\pi$
$389$	−3.00000	−0.152106	−0.0760530	+	0.997104i	$0.524232\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	0	0
$393$	36.0000	1.81596
$394$	0	0
$395$	−44.0000	−2.21388
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−26.0000	−1.29838	−0.649189	−	0.760627i	$-0.724892\pi$
$401$	−26.0000	−1.29838	−0.649189	+	0.760627i	$0.724892\pi$
$402$	0	0
$403$	24.0000	1.19553
$404$	0	0
$405$	−36.0000	−1.78885
$406$	0	0
$407$	8.00000	0.396545
$408$	0	0
$409$	−11.0000	−0.543915	−0.271957	−	0.962309i	$-0.587671\pi$
$409$	−11.0000	−0.543915	−0.271957	+	0.962309i	$0.587671\pi$
$410$	0	0
$411$	51.0000	2.51564
$412$	0	0
$413$	0	0
$414$	0	0
$415$	48.0000	2.35623
$416$	0	0
$417$	63.0000	3.08512
$418$	0	0
$419$	−9.00000	−0.439679	−0.219839	−	0.975536i	$-0.570553\pi$
$419$	−9.00000	−0.439679	−0.219839	+	0.975536i	$0.570553\pi$
$420$	0	0
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	0	0
$423$	60.0000	2.91730
$424$	0	0
$425$	11.0000	0.533578
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−9.00000	−0.434524
$430$	0	0
$431$	−19.0000	−0.915198	−0.457599	−	0.889159i	$-0.651290\pi$
$431$	−19.0000	−0.915198	−0.457599	+	0.889159i	$0.651290\pi$
$432$	0	0
$433$	−6.00000	−0.288342	−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000	−0.288342	−0.144171	+	0.989553i	$0.546051\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8.00000	0.382692
$438$	0	0
$439$	29.0000	1.38409	0.692047	−	0.721852i	$-0.256709\pi$
$439$	29.0000	1.38409	0.692047	+	0.721852i	$0.256709\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−6.00000	−0.285069	−0.142534	−	0.989790i	$-0.545525\pi$
$443$	−6.00000	−0.285069	−0.142534	+	0.989790i	$0.545525\pi$
$444$	0	0
$445$	36.0000	1.70656
$446$	0	0
$447$	−33.0000	−1.56085
$448$	0	0
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	6.00000	0.281905
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	0	0
$459$	9.00000	0.420084
$460$	0	0
$461$	17.0000	0.791769	0.395884	−	0.918300i	$-0.370438\pi$
$461$	17.0000	0.791769	0.395884	+	0.918300i	$0.370438\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	0	0
$465$	96.0000	4.45189
$466$	0	0
$467$	36.0000	1.66588	0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000	1.66588	0.832941	+	0.553362i	$0.186655\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−51.0000	−2.34996
$472$	0	0
$473$	10.0000	0.459800
$474$	0	0
$475$	22.0000	1.00943
$476$	0	0
$477$	18.0000	0.824163
$478$	0	0
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	−24.0000	−1.09431
$482$	0	0
$483$	0	0
$484$	0	0
$485$	40.0000	1.81631
$486$	0	0
$487$	23.0000	1.04223	0.521115	−	0.853487i	$-0.325516\pi$
$487$	23.0000	1.04223	0.521115	+	0.853487i	$0.325516\pi$
$488$	0	0
$489$	36.0000	1.62798
$490$	0	0
$491$	−26.0000	−1.17336	−0.586682	−	0.809818i	$-0.699566\pi$
$491$	−26.0000	−1.17336	−0.586682	+	0.809818i	$0.699566\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−24.0000	−1.07872
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−5.00000	−0.223831	−0.111915	−	0.993718i	$-0.535699\pi$
$499$	−5.00000	−0.223831	−0.111915	+	0.993718i	$0.535699\pi$
$500$	0	0
$501$	−51.0000	−2.27851
$502$	0	0
$503$	−33.0000	−1.47140	−0.735699	−	0.677309i	$-0.763146\pi$
$503$	−33.0000	−1.47140	−0.735699	+	0.677309i	$0.763146\pi$
$504$	0	0
$505$	40.0000	1.77998
$506$	0	0
$507$	−12.0000	−0.532939
$508$	0	0
$509$	−38.0000	−1.68432	−0.842160	−	0.539227i	$-0.818716\pi$
$509$	−38.0000	−1.68432	−0.842160	+	0.539227i	$0.818716\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	18.0000	0.794719
$514$	0	0
$515$	−16.0000	−0.705044
$516$	0	0
$517$	10.0000	0.439799
$518$	0	0
$519$	−18.0000	−0.790112
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	−22.0000	−0.961993	−0.480996	−	0.876723i	$-0.659725\pi$
$523$	−22.0000	−0.961993	−0.480996	+	0.876723i	$0.659725\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8.00000	−0.348485
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	84.0000	3.64529
$532$	0	0
$533$	0	0
$534$	0	0
$535$	12.0000	0.518805
$536$	0	0
$537$	48.0000	2.07135
$538$	0	0
$539$	0	0
$540$	0	0
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	0	0
$543$	18.0000	0.772454
$544$	0	0
$545$	−32.0000	−1.37073
$546$	0	0
$547$	−1.00000	−0.0427569	−0.0213785	−	0.999771i	$-0.506805\pi$
$547$	−1.00000	−0.0427569	−0.0213785	+	0.999771i	$0.506805\pi$
$548$	0	0
$549$	48.0000	2.04859
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−96.0000	−4.07497
$556$	0	0
$557$	15.0000	0.635570	0.317785	−	0.948163i	$-0.397061\pi$
$557$	15.0000	0.635570	0.317785	+	0.948163i	$0.397061\pi$
$558$	0	0
$559$	−30.0000	−1.26886
$560$	0	0
$561$	3.00000	0.126660
$562$	0	0
$563$	2.00000	0.0842900	0.0421450	−	0.999112i	$-0.486581\pi$
$563$	2.00000	0.0842900	0.0421450	+	0.999112i	$0.486581\pi$
$564$	0	0
$565$	32.0000	1.34625
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−15.0000	−0.628833	−0.314416	−	0.949285i	$-0.601809\pi$
$569$	−15.0000	−0.628833	−0.314416	+	0.949285i	$0.601809\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	0	0
$573$	36.0000	1.50392
$574$	0	0
$575$	44.0000	1.83493
$576$	0	0
$577$	43.0000	1.79011	0.895057	−	0.445952i	$-0.147135\pi$
$577$	43.0000	1.79011	0.895057	+	0.445952i	$0.147135\pi$
$578$	0	0
$579$	42.0000	1.74546
$580$	0	0
$581$	0	0
$582$	0	0
$583$	3.00000	0.124247
$584$	0	0
$585$	72.0000	2.97683
$586$	0	0
$587$	10.0000	0.412744	0.206372	−	0.978474i	$-0.433834\pi$
$587$	10.0000	0.412744	0.206372	+	0.978474i	$0.433834\pi$
$588$	0	0
$589$	−16.0000	−0.659269
$590$	0	0
$591$	−42.0000	−1.72765
$592$	0	0
$593$	−21.0000	−0.862367	−0.431183	−	0.902264i	$-0.641904\pi$
$593$	−21.0000	−0.862367	−0.431183	+	0.902264i	$0.641904\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	9.00000	0.368345
$598$	0	0
$599$	30.0000	1.22577	0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000	1.22577	0.612883	+	0.790173i	$0.290010\pi$
$600$	0	0
$601$	46.0000	1.87638	0.938190	−	0.346122i	$-0.112502\pi$
$601$	46.0000	1.87638	0.938190	+	0.346122i	$0.112502\pi$
$602$	0	0
$603$	−60.0000	−2.44339
$604$	0	0
$605$	40.0000	1.62623
$606$	0	0
$607$	−29.0000	−1.17707	−0.588537	−	0.808470i	$-0.700296\pi$
$607$	−29.0000	−1.17707	−0.588537	+	0.808470i	$0.700296\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−30.0000	−1.21367
$612$	0	0
$613$	17.0000	0.686624	0.343312	−	0.939222i	$-0.388451\pi$
$613$	17.0000	0.686624	0.343312	+	0.939222i	$0.388451\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−44.0000	−1.77137	−0.885687	−	0.464283i	$-0.846312\pi$
$617$	−44.0000	−1.77137	−0.885687	+	0.464283i	$0.846312\pi$
$618$	0	0
$619$	29.0000	1.16561	0.582804	−	0.812613i	$-0.301955\pi$
$619$	29.0000	1.16561	0.582804	+	0.812613i	$0.301955\pi$
$620$	0	0
$621$	36.0000	1.44463
$622$	0	0
$623$	0	0
$624$	0	0
$625$	41.0000	1.64000
$626$	0	0
$627$	6.00000	0.239617
$628$	0	0
$629$	8.00000	0.318981
$630$	0	0
$631$	−46.0000	−1.83123	−0.915616	−	0.402055i	$-0.868296\pi$
$631$	−46.0000	−1.83123	−0.915616	+	0.402055i	$0.868296\pi$
$632$	0	0
$633$	60.0000	2.38479
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−30.0000	−1.18678
$640$	0	0
$641$	−26.0000	−1.02694	−0.513469	−	0.858108i	$-0.671640\pi$
$641$	−26.0000	−1.02694	−0.513469	+	0.858108i	$0.671640\pi$
$642$	0	0
$643$	−27.0000	−1.06478	−0.532388	−	0.846500i	$-0.678705\pi$
$643$	−27.0000	−1.06478	−0.532388	+	0.846500i	$0.678705\pi$
$644$	0	0
$645$	−120.000	−4.72500
$646$	0	0
$647$	−28.0000	−1.10079	−0.550397	−	0.834903i	$-0.685524\pi$
$647$	−28.0000	−1.10079	−0.550397	+	0.834903i	$0.685524\pi$
$648$	0	0
$649$	14.0000	0.549548
$650$	0	0
$651$	0	0
$652$	0	0
$653$	46.0000	1.80012	0.900060	−	0.435767i	$-0.143523\pi$
$653$	46.0000	1.80012	0.900060	+	0.435767i	$0.143523\pi$
$654$	0	0
$655$	−48.0000	−1.87552
$656$	0	0
$657$	96.0000	3.74532
$658$	0	0
$659$	2.00000	0.0779089	0.0389545	−	0.999241i	$-0.487597\pi$
$659$	2.00000	0.0779089	0.0389545	+	0.999241i	$0.487597\pi$
$660$	0	0
$661$	30.0000	1.16686	0.583432	−	0.812162i	$-0.301709\pi$
$661$	30.0000	1.16686	0.583432	+	0.812162i	$0.301709\pi$
$662$	0	0
$663$	−9.00000	−0.349531
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	18.0000	0.695920
$670$	0	0
$671$	8.00000	0.308837
$672$	0	0
$673$	−44.0000	−1.69608	−0.848038	−	0.529936i	$-0.822216\pi$
$673$	−44.0000	−1.69608	−0.848038	+	0.529936i	$0.822216\pi$
$674$	0	0
$675$	99.0000	3.81051
$676$	0	0
$677$	48.0000	1.84479	0.922395	−	0.386248i	$-0.126229\pi$
$677$	48.0000	1.84479	0.922395	+	0.386248i	$0.126229\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	9.00000	0.344881
$682$	0	0
$683$	−9.00000	−0.344375	−0.172188	−	0.985064i	$-0.555084\pi$
$683$	−9.00000	−0.344375	−0.172188	+	0.985064i	$0.555084\pi$
$684$	0	0
$685$	−68.0000	−2.59815
$686$	0	0
$687$	30.0000	1.14457
$688$	0	0
$689$	−9.00000	−0.342873
$690$	0	0
$691$	−16.0000	−0.608669	−0.304334	−	0.952565i	$-0.598434\pi$
$691$	−16.0000	−0.608669	−0.304334	+	0.952565i	$0.598434\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−84.0000	−3.18630
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−60.0000	−2.26941
$700$	0	0
$701$	39.0000	1.47301	0.736505	−	0.676432i	$-0.236475\pi$
$701$	39.0000	1.47301	0.736505	+	0.676432i	$0.236475\pi$
$702$	0	0
$703$	16.0000	0.603451
$704$	0	0
$705$	−120.000	−4.51946
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−2.00000	−0.0751116	−0.0375558	−	0.999295i	$-0.511957\pi$
$709$	−2.00000	−0.0751116	−0.0375558	+	0.999295i	$0.511957\pi$
$710$	0	0
$711$	66.0000	2.47519
$712$	0	0
$713$	−32.0000	−1.19841
$714$	0	0
$715$	12.0000	0.448775
$716$	0	0
$717$	6.00000	0.224074
$718$	0	0
$719$	15.0000	0.559406	0.279703	−	0.960087i	$-0.409764\pi$
$719$	15.0000	0.559406	0.279703	+	0.960087i	$0.409764\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	14.0000	0.519231	0.259616	−	0.965712i	$-0.416404\pi$
$727$	14.0000	0.519231	0.259616	+	0.965712i	$0.416404\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	10.0000	0.369863
$732$	0	0
$733$	13.0000	0.480166	0.240083	−	0.970752i	$-0.422825\pi$
$733$	13.0000	0.480166	0.240083	+	0.970752i	$0.422825\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−10.0000	−0.368355
$738$	0	0
$739$	−34.0000	−1.25071	−0.625355	−	0.780340i	$-0.715046\pi$
$739$	−34.0000	−1.25071	−0.625355	+	0.780340i	$0.715046\pi$
$740$	0	0
$741$	−18.0000	−0.661247
$742$	0	0
$743$	−27.0000	−0.990534	−0.495267	−	0.868741i	$-0.664930\pi$
$743$	−27.0000	−0.990534	−0.495267	+	0.868741i	$0.664930\pi$
$744$	0	0
$745$	44.0000	1.61204
$746$	0	0
$747$	−72.0000	−2.63434
$748$	0	0
$749$	0	0
$750$	0	0
$751$	23.0000	0.839282	0.419641	−	0.907690i	$-0.362156\pi$
$751$	23.0000	0.839282	0.419641	+	0.907690i	$0.362156\pi$
$752$	0	0
$753$	−24.0000	−0.874609
$754$	0	0
$755$	−8.00000	−0.291150
$756$	0	0
$757$	17.0000	0.617876	0.308938	−	0.951082i	$-0.400027\pi$
$757$	17.0000	0.617876	0.308938	+	0.951082i	$0.400027\pi$
$758$	0	0
$759$	12.0000	0.435572
$760$	0	0
$761$	27.0000	0.978749	0.489375	−	0.872074i	$-0.337225\pi$
$761$	27.0000	0.978749	0.489375	+	0.872074i	$0.337225\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−24.0000	−0.867722
$766$	0	0
$767$	−42.0000	−1.51653
$768$	0	0
$769$	−25.0000	−0.901523	−0.450762	−	0.892644i	$-0.648848\pi$
$769$	−25.0000	−0.901523	−0.450762	+	0.892644i	$0.648848\pi$
$770$	0	0
$771$	−33.0000	−1.18847
$772$	0	0
$773$	−33.0000	−1.18693	−0.593464	−	0.804861i	$-0.702240\pi$
$773$	−33.0000	−1.18693	−0.593464	+	0.804861i	$0.702240\pi$
$774$	0	0
$775$	−88.0000	−3.16105
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−5.00000	−0.178914
$782$	0	0
$783$	0	0
$784$	0	0
$785$	68.0000	2.42702
$786$	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
$789$	36.0000	1.28163
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−24.0000	−0.852265
$794$	0	0
$795$	−36.0000	−1.27679
$796$	0	0
$797$	37.0000	1.31061	0.655304	−	0.755366i	$-0.272541\pi$
$797$	37.0000	1.31061	0.655304	+	0.755366i	$0.272541\pi$
$798$	0	0
$799$	10.0000	0.353775
$800$	0	0
$801$	−54.0000	−1.90800
$802$	0	0
$803$	16.0000	0.564628
$804$	0	0
$805$	0	0
$806$	0	0
$807$	6.00000	0.211210
$808$	0	0
$809$	−42.0000	−1.47664	−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000	−1.47664	−0.738321	+	0.674450i	$0.764381\pi$
$810$	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
$813$	−24.0000	−0.841717
$814$	0	0
$815$	−48.0000	−1.68137
$816$	0	0
$817$	20.0000	0.699711
$818$	0	0
$819$	0	0
$820$	0	0
$821$	24.0000	0.837606	0.418803	−	0.908077i	$-0.362450\pi$
$821$	24.0000	0.837606	0.418803	+	0.908077i	$0.362450\pi$
$822$	0	0
$823$	−7.00000	−0.244005	−0.122002	−	0.992530i	$-0.538932\pi$
$823$	−7.00000	−0.244005	−0.122002	+	0.992530i	$0.538932\pi$
$824$	0	0
$825$	33.0000	1.14891
$826$	0	0
$827$	−27.0000	−0.938882	−0.469441	−	0.882964i	$-0.655545\pi$
$827$	−27.0000	−0.938882	−0.469441	+	0.882964i	$0.655545\pi$
$828$	0	0
$829$	−25.0000	−0.868286	−0.434143	−	0.900844i	$-0.642949\pi$
$829$	−25.0000	−0.868286	−0.434143	+	0.900844i	$0.642949\pi$
$830$	0	0
$831$	24.0000	0.832551
$832$	0	0
$833$	0	0
$834$	0	0
$835$	68.0000	2.35324
$836$	0	0
$837$	−72.0000	−2.48868
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	−9.00000	−0.309976
$844$	0	0
$845$	16.0000	0.550417
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−57.0000	−1.95623
$850$	0	0
$851$	32.0000	1.09695
$852$	0	0
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	0	0
$855$	−48.0000	−1.64157
$856$	0	0
$857$	2.00000	0.0683187	0.0341593	−	0.999416i	$-0.489125\pi$
$857$	2.00000	0.0683187	0.0341593	+	0.999416i	$0.489125\pi$
$858$	0	0
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	20.0000	0.680808	0.340404	−	0.940279i	$-0.389436\pi$
$863$	20.0000	0.680808	0.340404	+	0.940279i	$0.389436\pi$
$864$	0	0
$865$	24.0000	0.816024
$866$	0	0
$867$	3.00000	0.101885
$868$	0	0
$869$	11.0000	0.373149
$870$	0	0
$871$	30.0000	1.01651
$872$	0	0
$873$	−60.0000	−2.03069
$874$	0	0
$875$	0	0
$876$	0	0
$877$	6.00000	0.202606	0.101303	−	0.994856i	$-0.467699\pi$
$877$	6.00000	0.202606	0.101303	+	0.994856i	$0.467699\pi$
$878$	0	0
$879$	−3.00000	−0.101187
$880$	0	0
$881$	54.0000	1.81931	0.909653	−	0.415369i	$-0.136347\pi$
$881$	54.0000	1.81931	0.909653	+	0.415369i	$0.136347\pi$
$882$	0	0
$883$	34.0000	1.14419	0.572096	−	0.820187i	$-0.306131\pi$
$883$	34.0000	1.14419	0.572096	+	0.820187i	$0.306131\pi$
$884$	0	0
$885$	−168.000	−5.64726
$886$	0	0
$887$	47.0000	1.57811	0.789053	−	0.614325i	$-0.210572\pi$
$887$	47.0000	1.57811	0.789053	+	0.614325i	$0.210572\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	9.00000	0.301511
$892$	0	0
$893$	20.0000	0.669274
$894$	0	0
$895$	−64.0000	−2.13928
$896$	0	0
$897$	−36.0000	−1.20201
$898$	0	0
$899$	0	0
$900$	0	0
$901$	3.00000	0.0999445
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−24.0000	−0.797787
$906$	0	0
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	0	0
$909$	−60.0000	−1.99007
$910$	0	0
$911$	28.0000	0.927681	0.463841	−	0.885919i	$-0.346471\pi$
$911$	28.0000	0.927681	0.463841	+	0.885919i	$0.346471\pi$
$912$	0	0
$913$	−12.0000	−0.397142
$914$	0	0
$915$	−96.0000	−3.17366
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−4.00000	−0.131948	−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000	−0.131948	−0.0659739	+	0.997821i	$0.521015\pi$
$920$	0	0
$921$	−12.0000	−0.395413
$922$	0	0
$923$	15.0000	0.493731
$924$	0	0
$925$	88.0000	2.89342
$926$	0	0
$927$	24.0000	0.788263
$928$	0	0
$929$	44.0000	1.44359	0.721797	−	0.692105i	$-0.243317\pi$
$929$	44.0000	1.44359	0.721797	+	0.692105i	$0.243317\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	27.0000	0.883940
$934$	0	0
$935$	−4.00000	−0.130814
$936$	0	0
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	0	0
$939$	54.0000	1.76222
$940$	0	0
$941$	48.0000	1.56476	0.782378	−	0.622804i	$-0.214007\pi$
$941$	48.0000	1.56476	0.782378	+	0.622804i	$0.214007\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	9.00000	0.292461	0.146230	−	0.989251i	$-0.453286\pi$
$947$	9.00000	0.292461	0.146230	+	0.989251i	$0.453286\pi$
$948$	0	0
$949$	−48.0000	−1.55815
$950$	0	0
$951$	18.0000	0.583690
$952$	0	0
$953$	−51.0000	−1.65205	−0.826026	−	0.563632i	$-0.809404\pi$
$953$	−51.0000	−1.65205	−0.826026	+	0.563632i	$0.809404\pi$
$954$	0	0
$955$	−48.0000	−1.55324
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	−18.0000	−0.580042
$964$	0	0
$965$	−56.0000	−1.80270
$966$	0	0
$967$	−22.0000	−0.707472	−0.353736	−	0.935345i	$-0.615089\pi$
$967$	−22.0000	−0.707472	−0.353736	+	0.935345i	$0.615089\pi$
$968$	0	0
$969$	6.00000	0.192748
$970$	0	0
$971$	−6.00000	−0.192549	−0.0962746	−	0.995355i	$-0.530693\pi$
$971$	−6.00000	−0.192549	−0.0962746	+	0.995355i	$0.530693\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−99.0000	−3.17054
$976$	0	0
$977$	34.0000	1.08776	0.543878	−	0.839164i	$-0.316955\pi$
$977$	34.0000	1.08776	0.543878	+	0.839164i	$0.316955\pi$
$978$	0	0
$979$	−9.00000	−0.287641
$980$	0	0
$981$	48.0000	1.53252
$982$	0	0
$983$	37.0000	1.18012	0.590058	−	0.807361i	$-0.299105\pi$
$983$	37.0000	1.18012	0.590058	+	0.807361i	$0.299105\pi$
$984$	0	0
$985$	56.0000	1.78431
$986$	0	0
$987$	0	0
$988$	0	0
$989$	40.0000	1.27193
$990$	0	0
$991$	13.0000	0.412959	0.206479	−	0.978451i	$-0.433799\pi$
$991$	13.0000	0.412959	0.206479	+	0.978451i	$0.433799\pi$
$992$	0	0
$993$	30.0000	0.952021
$994$	0	0
$995$	−12.0000	−0.380426
$996$	0	0
$997$	30.0000	0.950110	0.475055	−	0.879956i	$-0.342428\pi$
$997$	30.0000	0.950110	0.475055	+	0.879956i	$0.342428\pi$
$998$	0	0
$999$	72.0000	2.27798

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3332.2.a.f.1.1		1
7.3	odd	6		476.2.i.c.205.1	yes	2
7.5	odd	6		476.2.i.c.137.1	✓	2
7.6	odd	2		3332.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
476.2.i.c.137.1	✓	2	7.5	odd	6
476.2.i.c.205.1	yes	2	7.3	odd	6
3332.2.a.a.1.1		1	7.6	odd	2
3332.2.a.f.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:	\( N \)	\(=\)	\( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3332.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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