LMFDB - Embedded newform 8880.2.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8880,2,Mod(1,8880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8880, 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("8880.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8880 = 2^{4} \cdot 3 \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8880.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:	$70.9071569949$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1110)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8880.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} +1.00000 q^{5} +4.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} +1.00000 q^{5} +4.00000 q^{7} +1.00000 q^{9} -6.00000 q^{11} +2.00000 q^{13} -1.00000 q^{15} -6.00000 q^{17} -2.00000 q^{19} -4.00000 q^{21} +1.00000 q^{25} -1.00000 q^{27} +6.00000 q^{29} -8.00000 q^{31} +6.00000 q^{33} +4.00000 q^{35} +1.00000 q^{37} -2.00000 q^{39} -6.00000 q^{41} -8.00000 q^{43} +1.00000 q^{45} -6.00000 q^{47} +9.00000 q^{49} +6.00000 q^{51} +6.00000 q^{53} -6.00000 q^{55} +2.00000 q^{57} +12.0000 q^{59} +8.00000 q^{61} +4.00000 q^{63} +2.00000 q^{65} +4.00000 q^{67} +14.0000 q^{73} -1.00000 q^{75} -24.0000 q^{77} +16.0000 q^{79} +1.00000 q^{81} +12.0000 q^{83} -6.00000 q^{85} -6.00000 q^{87} +6.00000 q^{89} +8.00000 q^{91} +8.00000 q^{93} -2.00000 q^{95} +8.00000 q^{97} -6.00000 q^{99} +O(q^{100})$

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
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	−4.00000	−0.872872
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	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
$33$	6.00000	1.04447
$34$	0	0
$35$	4.00000	0.676123
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	6.00000	0.840168
$52$	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
$55$	−6.00000	−0.809040
$56$	0	0
$57$	2.00000	0.264906
$58$	0	0
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	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
$63$	4.00000	0.503953
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−24.0000	−2.73505
$78$	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
$85$	−6.00000	−0.650791
$86$	0	0
$87$	−6.00000	−0.643268
$88$	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
$91$	8.00000	0.838628
$92$	0	0
$93$	8.00000	0.829561
$94$	0	0
$95$	−2.00000	−0.205196
$96$	0	0
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	0	0
$99$	−6.00000	−0.603023
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	0	0
$105$	−4.00000	−0.390360
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	20.0000	1.91565	0.957826	−	0.287348i	$-0.0927736\pi$
$109$	20.0000	1.91565	0.957826	+	0.287348i	$0.0927736\pi$
$110$	0	0
$111$	−1.00000	−0.0949158
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	−24.0000	−2.20008
$120$	0	0
$121$	25.0000	2.27273
$122$	0	0
$123$	6.00000	0.541002
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	0	0
$129$	8.00000	0.704361
$130$	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
$133$	−8.00000	−0.693688
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	0	0
$143$	−12.0000	−1.00349
$144$	0	0
$145$	6.00000	0.498273
$146$	0	0
$147$	−9.00000	−0.742307
$148$	0	0
$149$	−12.0000	−0.983078	−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−12.0000	−0.983078	−0.491539	+	0.870855i	$0.663566\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	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
$161$	0	0
$162$	0	0
$163$	16.0000	1.25322	0.626608	−	0.779334i	$-0.284443\pi$
$163$	16.0000	1.25322	0.626608	+	0.779334i	$0.284443\pi$
$164$	0	0
$165$	6.00000	0.467099
$166$	0	0
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−2.00000	−0.152944
$172$	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
$177$	−12.0000	−0.901975
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	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
$183$	−8.00000	−0.591377
$184$	0	0
$185$	1.00000	0.0735215
$186$	0	0
$187$	36.0000	2.63258
$188$	0	0
$189$	−4.00000	−0.290957
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	0	0
$195$	−2.00000	−0.143223
$196$	0	0
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	24.0000	1.68447
$204$	0	0
$205$	−6.00000	−0.419058
$206$	0	0
$207$	0	0
$208$	0	0
$209$	12.0000	0.830057
$210$	0	0
$211$	16.0000	1.10149	0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000	1.10149	0.550743	+	0.834675i	$0.314345\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−8.00000	−0.545595
$216$	0	0
$217$	−32.0000	−2.17230
$218$	0	0
$219$	−14.0000	−0.946032
$220$	0	0
$221$	−12.0000	−0.807207
$222$	0	0
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	0	0
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	0	0
$231$	24.0000	1.57908
$232$	0	0
$233$	−24.0000	−1.57229	−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000	−1.57229	−0.786146	+	0.618041i	$0.787927\pi$
$234$	0	0
$235$	−6.00000	−0.391397
$236$	0	0
$237$	−16.0000	−1.03931
$238$	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
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	9.00000	0.574989
$246$	0	0
$247$	−4.00000	−0.254514
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	6.00000	0.375735
$256$	0	0
$257$	30.0000	1.87135	0.935674	−	0.352865i	$-0.114792\pi$
$257$	30.0000	1.87135	0.935674	+	0.352865i	$0.114792\pi$
$258$	0	0
$259$	4.00000	0.248548
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	6.00000	0.369976	0.184988	−	0.982741i	$-0.440775\pi$
$263$	6.00000	0.369976	0.184988	+	0.982741i	$0.440775\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	0	0
$267$	−6.00000	−0.367194
$268$	0	0
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	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
$273$	−8.00000	−0.484182
$274$	0	0
$275$	−6.00000	−0.361814
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	0	0
$279$	−8.00000	−0.478947
$280$	0	0
$281$	−30.0000	−1.78965	−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000	−1.78965	−0.894825	+	0.446417i	$0.852700\pi$
$282$	0	0
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	0	0
$285$	2.00000	0.118470
$286$	0	0
$287$	−24.0000	−1.41668
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	−8.00000	−0.468968
$292$	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
$295$	12.0000	0.698667
$296$	0	0
$297$	6.00000	0.348155
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−32.0000	−1.84445
$302$	0	0
$303$	0	0
$304$	0	0
$305$	8.00000	0.458079
$306$	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
$309$	14.0000	0.796432
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−16.0000	−0.904373	−0.452187	−	0.891923i	$-0.649356\pi$
$313$	−16.0000	−0.904373	−0.452187	+	0.891923i	$0.649356\pi$
$314$	0	0
$315$	4.00000	0.225374
$316$	0	0
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	−36.0000	−2.01561
$320$	0	0
$321$	0	0
$322$	0	0
$323$	12.0000	0.667698
$324$	0	0
$325$	2.00000	0.110940
$326$	0	0
$327$	−20.0000	−1.10600
$328$	0	0
$329$	−24.0000	−1.32316
$330$	0	0
$331$	−14.0000	−0.769510	−0.384755	−	0.923019i	$-0.625714\pi$
$331$	−14.0000	−0.769510	−0.384755	+	0.923019i	$0.625714\pi$
$332$	0	0
$333$	1.00000	0.0547997
$334$	0	0
$335$	4.00000	0.218543
$336$	0	0
$337$	−10.0000	−0.544735	−0.272367	−	0.962193i	$-0.587807\pi$
$337$	−10.0000	−0.544735	−0.272367	+	0.962193i	$0.587807\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	48.0000	2.59935
$342$	0	0
$343$	8.00000	0.431959
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	0	0
$353$	30.0000	1.59674	0.798369	−	0.602168i	$-0.205696\pi$
$353$	30.0000	1.59674	0.798369	+	0.602168i	$0.205696\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	24.0000	1.27021
$358$	0	0
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	−25.0000	−1.31216
$364$	0	0
$365$	14.0000	0.732793
$366$	0	0
$367$	28.0000	1.46159	0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000	1.46159	0.730794	+	0.682598i	$0.239150\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	24.0000	1.24602
$372$	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
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	12.0000	0.618031
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	0	0
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	0	0
$385$	−24.0000	−1.22315
$386$	0	0
$387$	−8.00000	−0.406663
$388$	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
$393$	−12.0000	−0.605320
$394$	0	0
$395$	16.0000	0.805047
$396$	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
$403$	−16.0000	−0.797017
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−6.00000	−0.297409
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	0	0
$413$	48.0000	2.36193
$414$	0	0
$415$	12.0000	0.589057
$416$	0	0
$417$	−4.00000	−0.195881
$418$	0	0
$419$	−6.00000	−0.293119	−0.146560	−	0.989202i	$-0.546820\pi$
$419$	−6.00000	−0.293119	−0.146560	+	0.989202i	$0.546820\pi$
$420$	0	0
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	0	0
$423$	−6.00000	−0.291730
$424$	0	0
$425$	−6.00000	−0.291043
$426$	0	0
$427$	32.0000	1.54859
$428$	0	0
$429$	12.0000	0.579365
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	38.0000	1.82616	0.913082	−	0.407777i	$-0.133696\pi$
$433$	38.0000	1.82616	0.913082	+	0.407777i	$0.133696\pi$
$434$	0	0
$435$	−6.00000	−0.287678
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	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
$445$	6.00000	0.284427
$446$	0	0
$447$	12.0000	0.567581
$448$	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
$451$	36.0000	1.69517
$452$	0	0
$453$	−16.0000	−0.751746
$454$	0	0
$455$	8.00000	0.375046
$456$	0	0
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	0	0
$459$	6.00000	0.280056
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	34.0000	1.58011	0.790057	−	0.613033i	$-0.210051\pi$
$463$	34.0000	1.58011	0.790057	+	0.613033i	$0.210051\pi$
$464$	0	0
$465$	8.00000	0.370991
$466$	0	0
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	0	0
$471$	−14.0000	−0.645086
$472$	0	0
$473$	48.0000	2.20704
$474$	0	0
$475$	−2.00000	−0.0917663
$476$	0	0
$477$	6.00000	0.274721
$478$	0	0
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8.00000	0.363261
$486$	0	0
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	0	0
$489$	−16.0000	−0.723545
$490$	0	0
$491$	6.00000	0.270776	0.135388	−	0.990793i	$-0.456772\pi$
$491$	6.00000	0.270776	0.135388	+	0.990793i	$0.456772\pi$
$492$	0	0
$493$	−36.0000	−1.62136
$494$	0	0
$495$	−6.00000	−0.269680
$496$	0	0
$497$	0	0
$498$	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
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	9.00000	0.399704
$508$	0	0
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	56.0000	2.47729
$512$	0	0
$513$	2.00000	0.0883022
$514$	0	0
$515$	−14.0000	−0.616914
$516$	0	0
$517$	36.0000	1.58328
$518$	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
$523$	−32.0000	−1.39926	−0.699631	−	0.714504i	$-0.746652\pi$
$523$	−32.0000	−1.39926	−0.699631	+	0.714504i	$0.746652\pi$
$524$	0	0
$525$	−4.00000	−0.174574
$526$	0	0
$527$	48.0000	2.09091
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	12.0000	0.520756
$532$	0	0
$533$	−12.0000	−0.519778
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−12.0000	−0.517838
$538$	0	0
$539$	−54.0000	−2.32594
$540$	0	0
$541$	8.00000	0.343947	0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000	0.343947	0.171973	+	0.985102i	$0.444986\pi$
$542$	0	0
$543$	10.0000	0.429141
$544$	0	0
$545$	20.0000	0.856706
$546$	0	0
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	0	0
$549$	8.00000	0.341432
$550$	0	0
$551$	−12.0000	−0.511217
$552$	0	0
$553$	64.0000	2.72156
$554$	0	0
$555$	−1.00000	−0.0424476
$556$	0	0
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	0	0
$559$	−16.0000	−0.676728
$560$	0	0
$561$	−36.0000	−1.51992
$562$	0	0
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	0	0
$565$	6.00000	0.252422
$566$	0	0
$567$	4.00000	0.167984
$568$	0	0
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	16.0000	0.669579	0.334790	−	0.942293i	$-0.391335\pi$
$571$	16.0000	0.669579	0.334790	+	0.942293i	$0.391335\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	20.0000	0.832611	0.416305	−	0.909225i	$-0.363325\pi$
$577$	20.0000	0.832611	0.416305	+	0.909225i	$0.363325\pi$
$578$	0	0
$579$	16.0000	0.664937
$580$	0	0
$581$	48.0000	1.99138
$582$	0	0
$583$	−36.0000	−1.49097
$584$	0	0
$585$	2.00000	0.0826898
$586$	0	0
$587$	−36.0000	−1.48588	−0.742940	−	0.669359i	$-0.766569\pi$
$587$	−36.0000	−1.48588	−0.742940	+	0.669359i	$0.766569\pi$
$588$	0	0
$589$	16.0000	0.659269
$590$	0	0
$591$	−18.0000	−0.740421
$592$	0	0
$593$	12.0000	0.492781	0.246390	−	0.969171i	$-0.420755\pi$
$593$	12.0000	0.492781	0.246390	+	0.969171i	$0.420755\pi$
$594$	0	0
$595$	−24.0000	−0.983904
$596$	0	0
$597$	−4.00000	−0.163709
$598$	0	0
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	0	0
$605$	25.0000	1.01639
$606$	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
$609$	−24.0000	−0.972529
$610$	0	0
$611$	−12.0000	−0.485468
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	0	0
$615$	6.00000	0.241943
$616$	0	0
$617$	−12.0000	−0.483102	−0.241551	−	0.970388i	$-0.577656\pi$
$617$	−12.0000	−0.483102	−0.241551	+	0.970388i	$0.577656\pi$
$618$	0	0
$619$	−32.0000	−1.28619	−0.643094	−	0.765787i	$-0.722350\pi$
$619$	−32.0000	−1.28619	−0.643094	+	0.765787i	$0.722350\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	24.0000	0.961540
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−12.0000	−0.479234
$628$	0	0
$629$	−6.00000	−0.239236
$630$	0	0
$631$	28.0000	1.11466	0.557331	−	0.830290i	$-0.311825\pi$
$631$	28.0000	1.11466	0.557331	+	0.830290i	$0.311825\pi$
$632$	0	0
$633$	−16.0000	−0.635943
$634$	0	0
$635$	4.00000	0.158735
$636$	0	0
$637$	18.0000	0.713186
$638$	0	0
$639$	0	0
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	0	0
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	0	0
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	0	0
$649$	−72.0000	−2.82625
$650$	0	0
$651$	32.0000	1.25418
$652$	0	0
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	0	0
$655$	12.0000	0.468879
$656$	0	0
$657$	14.0000	0.546192
$658$	0	0
$659$	−6.00000	−0.233727	−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000	−0.233727	−0.116863	+	0.993148i	$0.537284\pi$
$660$	0	0
$661$	20.0000	0.777910	0.388955	−	0.921257i	$-0.372836\pi$
$661$	20.0000	0.777910	0.388955	+	0.921257i	$0.372836\pi$
$662$	0	0
$663$	12.0000	0.466041
$664$	0	0
$665$	−8.00000	−0.310227
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−4.00000	−0.154649
$670$	0	0
$671$	−48.0000	−1.85302
$672$	0	0
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	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
$679$	32.0000	1.22805
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	0	0
$685$	−12.0000	−0.458496
$686$	0	0
$687$	−26.0000	−0.991962
$688$	0	0
$689$	12.0000	0.457164
$690$	0	0
$691$	40.0000	1.52167	0.760836	−	0.648944i	$-0.224789\pi$
$691$	40.0000	1.52167	0.760836	+	0.648944i	$0.224789\pi$
$692$	0	0
$693$	−24.0000	−0.911685
$694$	0	0
$695$	4.00000	0.151729
$696$	0	0
$697$	36.0000	1.36360
$698$	0	0
$699$	24.0000	0.907763
$700$	0	0
$701$	−6.00000	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$702$	0	0
$703$	−2.00000	−0.0754314
$704$	0	0
$705$	6.00000	0.225973
$706$	0	0
$707$	0	0
$708$	0	0
$709$	44.0000	1.65245	0.826227	−	0.563337i	$-0.190483\pi$
$709$	44.0000	1.65245	0.826227	+	0.563337i	$0.190483\pi$
$710$	0	0
$711$	16.0000	0.600047
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−12.0000	−0.448775
$716$	0	0
$717$	24.0000	0.896296
$718$	0	0
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	0	0
$721$	−56.0000	−2.08555
$722$	0	0
$723$	−2.00000	−0.0743808
$724$	0	0
$725$	6.00000	0.222834
$726$	0	0
$727$	−38.0000	−1.40934	−0.704671	−	0.709534i	$-0.748905\pi$
$727$	−38.0000	−1.40934	−0.704671	+	0.709534i	$0.748905\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	48.0000	1.77534
$732$	0	0
$733$	2.00000	0.0738717	0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000	0.0738717	0.0369358	+	0.999318i	$0.488240\pi$
$734$	0	0
$735$	−9.00000	−0.331970
$736$	0	0
$737$	−24.0000	−0.884051
$738$	0	0
$739$	16.0000	0.588570	0.294285	−	0.955718i	$-0.404919\pi$
$739$	16.0000	0.588570	0.294285	+	0.955718i	$0.404919\pi$
$740$	0	0
$741$	4.00000	0.146944
$742$	0	0
$743$	6.00000	0.220119	0.110059	−	0.993925i	$-0.464896\pi$
$743$	6.00000	0.220119	0.110059	+	0.993925i	$0.464896\pi$
$744$	0	0
$745$	−12.0000	−0.439646
$746$	0	0
$747$	12.0000	0.439057
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	16.0000	0.582300
$756$	0	0
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	0	0
$763$	80.0000	2.89619
$764$	0	0
$765$	−6.00000	−0.216930
$766$	0	0
$767$	24.0000	0.866590
$768$	0	0
$769$	26.0000	0.937584	0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000	0.937584	0.468792	+	0.883309i	$0.344689\pi$
$770$	0	0
$771$	−30.0000	−1.08042
$772$	0	0
$773$	30.0000	1.07903	0.539513	−	0.841978i	$-0.318609\pi$
$773$	30.0000	1.07903	0.539513	+	0.841978i	$0.318609\pi$
$774$	0	0
$775$	−8.00000	−0.287368
$776$	0	0
$777$	−4.00000	−0.143499
$778$	0	0
$779$	12.0000	0.429945
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−6.00000	−0.214423
$784$	0	0
$785$	14.0000	0.499681
$786$	0	0
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	0	0
$789$	−6.00000	−0.213606
$790$	0	0
$791$	24.0000	0.853342
$792$	0	0
$793$	16.0000	0.568177
$794$	0	0
$795$	−6.00000	−0.212798
$796$	0	0
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	0	0
$799$	36.0000	1.27359
$800$	0	0
$801$	6.00000	0.212000
$802$	0	0
$803$	−84.0000	−2.96430
$804$	0	0
$805$	0	0
$806$	0	0
$807$	24.0000	0.844840
$808$	0	0
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	−20.0000	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$812$	0	0
$813$	8.00000	0.280572
$814$	0	0
$815$	16.0000	0.560456
$816$	0	0
$817$	16.0000	0.559769
$818$	0	0
$819$	8.00000	0.279543
$820$	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
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	0	0
$825$	6.00000	0.208893
$826$	0	0
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	0	0
$829$	−28.0000	−0.972480	−0.486240	−	0.873825i	$-0.661632\pi$
$829$	−28.0000	−0.972480	−0.486240	+	0.873825i	$0.661632\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	0	0
$833$	−54.0000	−1.87099
$834$	0	0
$835$	−12.0000	−0.415277
$836$	0	0
$837$	8.00000	0.276520
$838$	0	0
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	30.0000	1.03325
$844$	0	0
$845$	−9.00000	−0.309609
$846$	0	0
$847$	100.000	3.43604
$848$	0	0
$849$	−16.0000	−0.549119
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−10.0000	−0.342393	−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000	−0.342393	−0.171197	+	0.985237i	$0.554763\pi$
$854$	0	0
$855$	−2.00000	−0.0683986
$856$	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
$859$	−14.0000	−0.477674	−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000	−0.477674	−0.238837	+	0.971060i	$0.576766\pi$
$860$	0	0
$861$	24.0000	0.817918
$862$	0	0
$863$	−30.0000	−1.02121	−0.510606	−	0.859815i	$-0.670579\pi$
$863$	−30.0000	−1.02121	−0.510606	+	0.859815i	$0.670579\pi$
$864$	0	0
$865$	6.00000	0.204006
$866$	0	0
$867$	−19.0000	−0.645274
$868$	0	0
$869$	−96.0000	−3.25658
$870$	0	0
$871$	8.00000	0.271070
$872$	0	0
$873$	8.00000	0.270759
$874$	0	0
$875$	4.00000	0.135225
$876$	0	0
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	0	0
$879$	30.0000	1.01187
$880$	0	0
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	0	0
$883$	−20.0000	−0.673054	−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	0	0
$885$	−12.0000	−0.403376
$886$	0	0
$887$	6.00000	0.201460	0.100730	−	0.994914i	$-0.467882\pi$
$887$	6.00000	0.201460	0.100730	+	0.994914i	$0.467882\pi$
$888$	0	0
$889$	16.0000	0.536623
$890$	0	0
$891$	−6.00000	−0.201008
$892$	0	0
$893$	12.0000	0.401565
$894$	0	0
$895$	12.0000	0.401116
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−48.0000	−1.60089
$900$	0	0
$901$	−36.0000	−1.19933
$902$	0	0
$903$	32.0000	1.06489
$904$	0	0
$905$	−10.0000	−0.332411
$906$	0	0
$907$	40.0000	1.32818	0.664089	−	0.747653i	$-0.268820\pi$
$907$	40.0000	1.32818	0.664089	+	0.747653i	$0.268820\pi$
$908$	0	0
$909$	0	0
$910$	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
$913$	−72.0000	−2.38285
$914$	0	0
$915$	−8.00000	−0.264472
$916$	0	0
$917$	48.0000	1.58510
$918$	0	0
$919$	4.00000	0.131948	0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000	0.131948	0.0659739	+	0.997821i	$0.478985\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	0	0
$923$	0	0
$924$	0	0
$925$	1.00000	0.0328798
$926$	0	0
$927$	−14.0000	−0.459820
$928$	0	0
$929$	−42.0000	−1.37798	−0.688988	−	0.724773i	$-0.741945\pi$
$929$	−42.0000	−1.37798	−0.688988	+	0.724773i	$0.741945\pi$
$930$	0	0
$931$	−18.0000	−0.589926
$932$	0	0
$933$	0	0
$934$	0	0
$935$	36.0000	1.17733
$936$	0	0
$937$	14.0000	0.457360	0.228680	−	0.973502i	$-0.426559\pi$
$937$	14.0000	0.457360	0.228680	+	0.973502i	$0.426559\pi$
$938$	0	0
$939$	16.0000	0.522140
$940$	0	0
$941$	−48.0000	−1.56476	−0.782378	−	0.622804i	$-0.785993\pi$
$941$	−48.0000	−1.56476	−0.782378	+	0.622804i	$0.785993\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−4.00000	−0.130120
$946$	0	0
$947$	36.0000	1.16984	0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000	1.16984	0.584921	+	0.811090i	$0.301125\pi$
$948$	0	0
$949$	28.0000	0.908918
$950$	0	0
$951$	18.0000	0.583690
$952$	0	0
$953$	36.0000	1.16615	0.583077	−	0.812417i	$-0.301849\pi$
$953$	36.0000	1.16615	0.583077	+	0.812417i	$0.301849\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	36.0000	1.16371
$958$	0	0
$959$	−48.0000	−1.55000
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−16.0000	−0.515058
$966$	0	0
$967$	46.0000	1.47926	0.739630	−	0.673014i	$-0.235000\pi$
$967$	46.0000	1.47926	0.739630	+	0.673014i	$0.235000\pi$
$968$	0	0
$969$	−12.0000	−0.385496
$970$	0	0
$971$	54.0000	1.73294	0.866471	−	0.499227i	$-0.166383\pi$
$971$	54.0000	1.73294	0.866471	+	0.499227i	$0.166383\pi$
$972$	0	0
$973$	16.0000	0.512936
$974$	0	0
$975$	−2.00000	−0.0640513
$976$	0	0
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	0	0
$979$	−36.0000	−1.15056
$980$	0	0
$981$	20.0000	0.638551
$982$	0	0
$983$	30.0000	0.956851	0.478426	−	0.878128i	$-0.341208\pi$
$983$	30.0000	0.956851	0.478426	+	0.878128i	$0.341208\pi$
$984$	0	0
$985$	18.0000	0.573528
$986$	0	0
$987$	24.0000	0.763928
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−56.0000	−1.77890	−0.889449	−	0.457034i	$-0.848912\pi$
$991$	−56.0000	−1.77890	−0.889449	+	0.457034i	$0.848912\pi$
$992$	0	0
$993$	14.0000	0.444277
$994$	0	0
$995$	4.00000	0.126809
$996$	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
$999$	−1.00000	−0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8880.2.a.n.1.1		1
4.3	odd	2		1110.2.a.h.1.1	✓	1
12.11	even	2		3330.2.a.m.1.1		1
20.19	odd	2		5550.2.a.bd.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.a.h.1.1	✓	1	4.3	odd	2
3330.2.a.m.1.1		1	12.11	even	2
5550.2.a.bd.1.1		1	20.19	odd	2
8880.2.a.n.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:	\( N \)	\(=\)	\( 8880 = 2^{4} \cdot 3 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8880.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more