LMFDB - Embedded newform 1776.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1776,2,Mod(1,1776)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1776, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1776.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1776 = 2^{4} \cdot 3 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1776.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:	$14.1814313990$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 222)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1776.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q-1.00000 q^{3} +4.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} -3.00000 q^{13} -4.00000 q^{15} +3.00000 q^{17} +5.00000 q^{19} -1.00000 q^{21} -5.00000 q^{23} +11.0000 q^{25} -1.00000 q^{27} +4.00000 q^{29} +10.0000 q^{31} -1.00000 q^{33} +4.00000 q^{35} -1.00000 q^{37} +3.00000 q^{39} -6.00000 q^{41} -4.00000 q^{43} +4.00000 q^{45} -2.00000 q^{47} -6.00000 q^{49} -3.00000 q^{51} -11.0000 q^{53} +4.00000 q^{55} -5.00000 q^{57} +12.0000 q^{59} +10.0000 q^{61} +1.00000 q^{63} -12.0000 q^{65} -14.0000 q^{67} +5.00000 q^{69} -11.0000 q^{73} -11.0000 q^{75} +1.00000 q^{77} +10.0000 q^{79} +1.00000 q^{81} +9.00000 q^{83} +12.0000 q^{85} -4.00000 q^{87} +11.0000 q^{89} -3.00000 q^{91} -10.0000 q^{93} +20.0000 q^{95} +10.0000 q^{97} +1.00000 q^{99} +O(q^{100})$

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
$5$	4.00000	1.78885	0.894427	−	0.447214i	$-0.147584\pi$
$5$	4.00000	1.78885	0.894427	+	0.447214i	$0.147584\pi$
$6$	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
$9$	1.00000	0.333333
$10$	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
$15$	−4.00000	−1.03280
$16$	0	0
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−5.00000	−1.04257	−0.521286	−	0.853382i	$-0.674548\pi$
$23$	−5.00000	−1.04257	−0.521286	+	0.853382i	$0.674548\pi$
$24$	0	0
$25$	11.0000	2.20000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	10.0000	1.79605	0.898027	−	0.439941i	$-0.145001\pi$
$31$	10.0000	1.79605	0.898027	+	0.439941i	$0.145001\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	4.00000	0.676123
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	3.00000	0.480384
$40$	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
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	4.00000	0.596285
$46$	0	0
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−3.00000	−0.420084
$52$	0	0
$53$	−11.0000	−1.51097	−0.755483	−	0.655168i	$-0.772598\pi$
$53$	−11.0000	−1.51097	−0.755483	+	0.655168i	$0.772598\pi$
$54$	0	0
$55$	4.00000	0.539360
$56$	0	0
$57$	−5.00000	−0.662266
$58$	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
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−12.0000	−1.48842
$66$	0	0
$67$	−14.0000	−1.71037	−0.855186	−	0.518321i	$-0.826557\pi$
$67$	−14.0000	−1.71037	−0.855186	+	0.518321i	$0.826557\pi$
$68$	0	0
$69$	5.00000	0.601929
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−11.0000	−1.28745	−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000	−1.28745	−0.643726	+	0.765256i	$0.722612\pi$
$74$	0	0
$75$	−11.0000	−1.27017
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	9.00000	0.987878	0.493939	−	0.869496i	$-0.335557\pi$
$83$	9.00000	0.987878	0.493939	+	0.869496i	$0.335557\pi$
$84$	0	0
$85$	12.0000	1.30158
$86$	0	0
$87$	−4.00000	−0.428845
$88$	0	0
$89$	11.0000	1.16600	0.582999	−	0.812473i	$-0.301879\pi$
$89$	11.0000	1.16600	0.582999	+	0.812473i	$0.301879\pi$
$90$	0	0
$91$	−3.00000	−0.314485
$92$	0	0
$93$	−10.0000	−1.03695
$94$	0	0
$95$	20.0000	2.05196
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	2.00000	0.197066	0.0985329	−	0.995134i	$-0.468585\pi$
$103$	2.00000	0.197066	0.0985329	+	0.995134i	$0.468585\pi$
$104$	0	0
$105$	−4.00000	−0.390360
$106$	0	0
$107$	3.00000	0.290021	0.145010	−	0.989430i	$-0.453678\pi$
$107$	3.00000	0.290021	0.145010	+	0.989430i	$0.453678\pi$
$108$	0	0
$109$	−9.00000	−0.862044	−0.431022	−	0.902342i	$-0.641847\pi$
$109$	−9.00000	−0.862044	−0.431022	+	0.902342i	$0.641847\pi$
$110$	0	0
$111$	1.00000	0.0949158
$112$	0	0
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	0	0
$115$	−20.0000	−1.86501
$116$	0	0
$117$	−3.00000	−0.277350
$118$	0	0
$119$	3.00000	0.275010
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	6.00000	0.541002
$124$	0	0
$125$	24.0000	2.14663
$126$	0	0
$127$	3.00000	0.266207	0.133103	−	0.991102i	$-0.457506\pi$
$127$	3.00000	0.266207	0.133103	+	0.991102i	$0.457506\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	−22.0000	−1.92215	−0.961074	−	0.276289i	$-0.910895\pi$
$131$	−22.0000	−1.92215	−0.961074	+	0.276289i	$0.910895\pi$
$132$	0	0
$133$	5.00000	0.433555
$134$	0	0
$135$	−4.00000	−0.344265
$136$	0	0
$137$	20.0000	1.70872	0.854358	−	0.519685i	$-0.173951\pi$
$137$	20.0000	1.70872	0.854358	+	0.519685i	$0.173951\pi$
$138$	0	0
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	0	0
$141$	2.00000	0.168430
$142$	0	0
$143$	−3.00000	−0.250873
$144$	0	0
$145$	16.0000	1.32873
$146$	0	0
$147$	6.00000	0.494872
$148$	0	0
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	−5.00000	−0.406894	−0.203447	−	0.979086i	$-0.565214\pi$
$151$	−5.00000	−0.406894	−0.203447	+	0.979086i	$0.565214\pi$
$152$	0	0
$153$	3.00000	0.242536
$154$	0	0
$155$	40.0000	3.21288
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	11.0000	0.872357
$160$	0	0
$161$	−5.00000	−0.394055
$162$	0	0
$163$	−9.00000	−0.704934	−0.352467	−	0.935824i	$-0.614657\pi$
$163$	−9.00000	−0.704934	−0.352467	+	0.935824i	$0.614657\pi$
$164$	0	0
$165$	−4.00000	−0.311400
$166$	0	0
$167$	17.0000	1.31550	0.657750	−	0.753237i	$-0.271508\pi$
$167$	17.0000	1.31550	0.657750	+	0.753237i	$0.271508\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	5.00000	0.382360
$172$	0	0
$173$	−13.0000	−0.988372	−0.494186	−	0.869356i	$-0.664534\pi$
$173$	−13.0000	−0.988372	−0.494186	+	0.869356i	$0.664534\pi$
$174$	0	0
$175$	11.0000	0.831522
$176$	0	0
$177$	−12.0000	−0.901975
$178$	0	0
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	−16.0000	−1.18927	−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000	−1.18927	−0.594635	+	0.803996i	$0.702704\pi$
$182$	0	0
$183$	−10.0000	−0.739221
$184$	0	0
$185$	−4.00000	−0.294086
$186$	0	0
$187$	3.00000	0.219382
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−9.00000	−0.651217	−0.325609	−	0.945505i	$-0.605569\pi$
$191$	−9.00000	−0.651217	−0.325609	+	0.945505i	$0.605569\pi$
$192$	0	0
$193$	18.0000	1.29567	0.647834	−	0.761781i	$-0.275675\pi$
$193$	18.0000	1.29567	0.647834	+	0.761781i	$0.275675\pi$
$194$	0	0
$195$	12.0000	0.859338
$196$	0	0
$197$	7.00000	0.498729	0.249365	−	0.968410i	$-0.419778\pi$
$197$	7.00000	0.498729	0.249365	+	0.968410i	$0.419778\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	14.0000	0.987484
$202$	0	0
$203$	4.00000	0.280745
$204$	0	0
$205$	−24.0000	−1.67623
$206$	0	0
$207$	−5.00000	−0.347524
$208$	0	0
$209$	5.00000	0.345857
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−16.0000	−1.09119
$216$	0	0
$217$	10.0000	0.678844
$218$	0	0
$219$	11.0000	0.743311
$220$	0	0
$221$	−9.00000	−0.605406
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	11.0000	0.733333
$226$	0	0
$227$	−16.0000	−1.06196	−0.530979	−	0.847385i	$-0.678176\pi$
$227$	−16.0000	−1.06196	−0.530979	+	0.847385i	$0.678176\pi$
$228$	0	0
$229$	−8.00000	−0.528655	−0.264327	−	0.964433i	$-0.585150\pi$
$229$	−8.00000	−0.528655	−0.264327	+	0.964433i	$0.585150\pi$
$230$	0	0
$231$	−1.00000	−0.0657952
$232$	0	0
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	0	0
$235$	−8.00000	−0.521862
$236$	0	0
$237$	−10.0000	−0.649570
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	28.0000	1.80364	0.901819	−	0.432113i	$-0.142232\pi$
$241$	28.0000	1.80364	0.901819	+	0.432113i	$0.142232\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−24.0000	−1.53330
$246$	0	0
$247$	−15.0000	−0.954427
$248$	0	0
$249$	−9.00000	−0.570352
$250$	0	0
$251$	6.00000	0.378717	0.189358	−	0.981908i	$-0.439359\pi$
$251$	6.00000	0.378717	0.189358	+	0.981908i	$0.439359\pi$
$252$	0	0
$253$	−5.00000	−0.314347
$254$	0	0
$255$	−12.0000	−0.751469
$256$	0	0
$257$	27.0000	1.68421	0.842107	−	0.539311i	$-0.181315\pi$
$257$	27.0000	1.68421	0.842107	+	0.539311i	$0.181315\pi$
$258$	0	0
$259$	−1.00000	−0.0621370
$260$	0	0
$261$	4.00000	0.247594
$262$	0	0
$263$	−18.0000	−1.10993	−0.554964	−	0.831875i	$-0.687268\pi$
$263$	−18.0000	−1.10993	−0.554964	+	0.831875i	$0.687268\pi$
$264$	0	0
$265$	−44.0000	−2.70290
$266$	0	0
$267$	−11.0000	−0.673189
$268$	0	0
$269$	27.0000	1.64622	0.823110	−	0.567883i	$-0.192237\pi$
$269$	27.0000	1.64622	0.823110	+	0.567883i	$0.192237\pi$
$270$	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
$273$	3.00000	0.181568
$274$	0	0
$275$	11.0000	0.663325
$276$	0	0
$277$	−5.00000	−0.300421	−0.150210	−	0.988654i	$-0.547995\pi$
$277$	−5.00000	−0.300421	−0.150210	+	0.988654i	$0.547995\pi$
$278$	0	0
$279$	10.0000	0.598684
$280$	0	0
$281$	−9.00000	−0.536895	−0.268447	−	0.963294i	$-0.586511\pi$
$281$	−9.00000	−0.536895	−0.268447	+	0.963294i	$0.586511\pi$
$282$	0	0
$283$	−21.0000	−1.24832	−0.624160	−	0.781296i	$-0.714559\pi$
$283$	−21.0000	−1.24832	−0.624160	+	0.781296i	$0.714559\pi$
$284$	0	0
$285$	−20.0000	−1.18470
$286$	0	0
$287$	−6.00000	−0.354169
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−10.0000	−0.586210
$292$	0	0
$293$	21.0000	1.22683	0.613417	−	0.789760i	$-0.289795\pi$
$293$	21.0000	1.22683	0.613417	+	0.789760i	$0.289795\pi$
$294$	0	0
$295$	48.0000	2.79467
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	15.0000	0.867472
$300$	0	0
$301$	−4.00000	−0.230556
$302$	0	0
$303$	−6.00000	−0.344691
$304$	0	0
$305$	40.0000	2.29039
$306$	0	0
$307$	−32.0000	−1.82634	−0.913168	−	0.407583i	$-0.866372\pi$
$307$	−32.0000	−1.82634	−0.913168	+	0.407583i	$0.866372\pi$
$308$	0	0
$309$	−2.00000	−0.113776
$310$	0	0
$311$	16.0000	0.907277	0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000	0.907277	0.453638	+	0.891186i	$0.350126\pi$
$312$	0	0
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	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
$319$	4.00000	0.223957
$320$	0	0
$321$	−3.00000	−0.167444
$322$	0	0
$323$	15.0000	0.834622
$324$	0	0
$325$	−33.0000	−1.83051
$326$	0	0
$327$	9.00000	0.497701
$328$	0	0
$329$	−2.00000	−0.110264
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	0	0
$333$	−1.00000	−0.0547997
$334$	0	0
$335$	−56.0000	−3.05961
$336$	0	0
$337$	5.00000	0.272367	0.136184	−	0.990684i	$-0.456516\pi$
$337$	5.00000	0.272367	0.136184	+	0.990684i	$0.456516\pi$
$338$	0	0
$339$	−14.0000	−0.760376
$340$	0	0
$341$	10.0000	0.541530
$342$	0	0
$343$	−13.0000	−0.701934
$344$	0	0
$345$	20.0000	1.07676
$346$	0	0
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	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
$351$	3.00000	0.160128
$352$	0	0
$353$	−26.0000	−1.38384	−0.691920	−	0.721974i	$-0.743235\pi$
$353$	−26.0000	−1.38384	−0.691920	+	0.721974i	$0.743235\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−3.00000	−0.158777
$358$	0	0
$359$	−6.00000	−0.316668	−0.158334	−	0.987386i	$-0.550612\pi$
$359$	−6.00000	−0.316668	−0.158334	+	0.987386i	$0.550612\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	0	0
$363$	10.0000	0.524864
$364$	0	0
$365$	−44.0000	−2.30307
$366$	0	0
$367$	−15.0000	−0.782994	−0.391497	−	0.920179i	$-0.628043\pi$
$367$	−15.0000	−0.782994	−0.391497	+	0.920179i	$0.628043\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	−11.0000	−0.571092
$372$	0	0
$373$	−28.0000	−1.44979	−0.724893	−	0.688862i	$-0.758111\pi$
$373$	−28.0000	−1.44979	−0.724893	+	0.688862i	$0.758111\pi$
$374$	0	0
$375$	−24.0000	−1.23935
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	−3.00000	−0.153695
$382$	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
$385$	4.00000	0.203859
$386$	0	0
$387$	−4.00000	−0.203331
$388$	0	0
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	−15.0000	−0.758583
$392$	0	0
$393$	22.0000	1.10975
$394$	0	0
$395$	40.0000	2.01262
$396$	0	0
$397$	−32.0000	−1.60603	−0.803017	−	0.595956i	$-0.796773\pi$
$397$	−32.0000	−1.60603	−0.803017	+	0.595956i	$0.796773\pi$
$398$	0	0
$399$	−5.00000	−0.250313
$400$	0	0
$401$	19.0000	0.948815	0.474407	−	0.880305i	$-0.342662\pi$
$401$	19.0000	0.948815	0.474407	+	0.880305i	$0.342662\pi$
$402$	0	0
$403$	−30.0000	−1.49441
$404$	0	0
$405$	4.00000	0.198762
$406$	0	0
$407$	−1.00000	−0.0495682
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	−20.0000	−0.986527
$412$	0	0
$413$	12.0000	0.590481
$414$	0	0
$415$	36.0000	1.76717
$416$	0	0
$417$	−14.0000	−0.685583
$418$	0	0
$419$	−5.00000	−0.244266	−0.122133	−	0.992514i	$-0.538973\pi$
$419$	−5.00000	−0.244266	−0.122133	+	0.992514i	$0.538973\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	0	0
$423$	−2.00000	−0.0972433
$424$	0	0
$425$	33.0000	1.60074
$426$	0	0
$427$	10.0000	0.483934
$428$	0	0
$429$	3.00000	0.144841
$430$	0	0
$431$	−15.0000	−0.722525	−0.361262	−	0.932464i	$-0.617654\pi$
$431$	−15.0000	−0.722525	−0.361262	+	0.932464i	$0.617654\pi$
$432$	0	0
$433$	−25.0000	−1.20142	−0.600712	−	0.799466i	$-0.705116\pi$
$433$	−25.0000	−1.20142	−0.600712	+	0.799466i	$0.705116\pi$
$434$	0	0
$435$	−16.0000	−0.767141
$436$	0	0
$437$	−25.0000	−1.19591
$438$	0	0
$439$	14.0000	0.668184	0.334092	−	0.942541i	$-0.391570\pi$
$439$	14.0000	0.668184	0.334092	+	0.942541i	$0.391570\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	0	0
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	0	0
$445$	44.0000	2.08580
$446$	0	0
$447$	10.0000	0.472984
$448$	0	0
$449$	2.00000	0.0943858	0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000	0.0943858	0.0471929	+	0.998886i	$0.484972\pi$
$450$	0	0
$451$	−6.00000	−0.282529
$452$	0	0
$453$	5.00000	0.234920
$454$	0	0
$455$	−12.0000	−0.562569
$456$	0	0
$457$	28.0000	1.30978	0.654892	−	0.755722i	$-0.272714\pi$
$457$	28.0000	1.30978	0.654892	+	0.755722i	$0.272714\pi$
$458$	0	0
$459$	−3.00000	−0.140028
$460$	0	0
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	0	0
$463$	22.0000	1.02243	0.511213	−	0.859454i	$-0.329196\pi$
$463$	22.0000	1.02243	0.511213	+	0.859454i	$0.329196\pi$
$464$	0	0
$465$	−40.0000	−1.85496
$466$	0	0
$467$	34.0000	1.57333	0.786666	−	0.617379i	$-0.211805\pi$
$467$	34.0000	1.57333	0.786666	+	0.617379i	$0.211805\pi$
$468$	0	0
$469$	−14.0000	−0.646460
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−4.00000	−0.183920
$474$	0	0
$475$	55.0000	2.52357
$476$	0	0
$477$	−11.0000	−0.503655
$478$	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
$481$	3.00000	0.136788
$482$	0	0
$483$	5.00000	0.227508
$484$	0	0
$485$	40.0000	1.81631
$486$	0	0
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	0	0
$489$	9.00000	0.406994
$490$	0	0
$491$	29.0000	1.30875	0.654376	−	0.756169i	$-0.272931\pi$
$491$	29.0000	1.30875	0.654376	+	0.756169i	$0.272931\pi$
$492$	0	0
$493$	12.0000	0.540453
$494$	0	0
$495$	4.00000	0.179787
$496$	0	0
$497$	0	0
$498$	0	0
$499$	15.0000	0.671492	0.335746	−	0.941953i	$-0.391012\pi$
$499$	15.0000	0.671492	0.335746	+	0.941953i	$0.391012\pi$
$500$	0	0
$501$	−17.0000	−0.759504
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	24.0000	1.06799
$506$	0	0
$507$	4.00000	0.177646
$508$	0	0
$509$	−15.0000	−0.664863	−0.332432	−	0.943127i	$-0.607869\pi$
$509$	−15.0000	−0.664863	−0.332432	+	0.943127i	$0.607869\pi$
$510$	0	0
$511$	−11.0000	−0.486611
$512$	0	0
$513$	−5.00000	−0.220755
$514$	0	0
$515$	8.00000	0.352522
$516$	0	0
$517$	−2.00000	−0.0879599
$518$	0	0
$519$	13.0000	0.570637
$520$	0	0
$521$	2.00000	0.0876216	0.0438108	−	0.999040i	$-0.486050\pi$
$521$	2.00000	0.0876216	0.0438108	+	0.999040i	$0.486050\pi$
$522$	0	0
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	0	0
$525$	−11.0000	−0.480079
$526$	0	0
$527$	30.0000	1.30682
$528$	0	0
$529$	2.00000	0.0869565
$530$	0	0
$531$	12.0000	0.520756
$532$	0	0
$533$	18.0000	0.779667
$534$	0	0
$535$	12.0000	0.518805
$536$	0	0
$537$	4.00000	0.172613
$538$	0	0
$539$	−6.00000	−0.258438
$540$	0	0
$541$	1.00000	0.0429934	0.0214967	−	0.999769i	$-0.493157\pi$
$541$	1.00000	0.0429934	0.0214967	+	0.999769i	$0.493157\pi$
$542$	0	0
$543$	16.0000	0.686626
$544$	0	0
$545$	−36.0000	−1.54207
$546$	0	0
$547$	−17.0000	−0.726868	−0.363434	−	0.931620i	$-0.618396\pi$
$547$	−17.0000	−0.726868	−0.363434	+	0.931620i	$0.618396\pi$
$548$	0	0
$549$	10.0000	0.426790
$550$	0	0
$551$	20.0000	0.852029
$552$	0	0
$553$	10.0000	0.425243
$554$	0	0
$555$	4.00000	0.169791
$556$	0	0
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	0	0
$559$	12.0000	0.507546
$560$	0	0
$561$	−3.00000	−0.126660
$562$	0	0
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	0	0
$565$	56.0000	2.35594
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−3.00000	−0.125767	−0.0628833	−	0.998021i	$-0.520030\pi$
$569$	−3.00000	−0.125767	−0.0628833	+	0.998021i	$0.520030\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	0	0
$573$	9.00000	0.375980
$574$	0	0
$575$	−55.0000	−2.29366
$576$	0	0
$577$	−34.0000	−1.41544	−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000	−1.41544	−0.707719	+	0.706494i	$0.750276\pi$
$578$	0	0
$579$	−18.0000	−0.748054
$580$	0	0
$581$	9.00000	0.373383
$582$	0	0
$583$	−11.0000	−0.455573
$584$	0	0
$585$	−12.0000	−0.496139
$586$	0	0
$587$	−28.0000	−1.15568	−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000	−1.15568	−0.577842	+	0.816149i	$0.696105\pi$
$588$	0	0
$589$	50.0000	2.06021
$590$	0	0
$591$	−7.00000	−0.287942
$592$	0	0
$593$	28.0000	1.14982	0.574911	−	0.818216i	$-0.305037\pi$
$593$	28.0000	1.14982	0.574911	+	0.818216i	$0.305037\pi$
$594$	0	0
$595$	12.0000	0.491952
$596$	0	0
$597$	16.0000	0.654836
$598$	0	0
$599$	−42.0000	−1.71607	−0.858037	−	0.513588i	$-0.828316\pi$
$599$	−42.0000	−1.71607	−0.858037	+	0.513588i	$0.828316\pi$
$600$	0	0
$601$	−9.00000	−0.367118	−0.183559	−	0.983009i	$-0.558762\pi$
$601$	−9.00000	−0.367118	−0.183559	+	0.983009i	$0.558762\pi$
$602$	0	0
$603$	−14.0000	−0.570124
$604$	0	0
$605$	−40.0000	−1.62623
$606$	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
$609$	−4.00000	−0.162088
$610$	0	0
$611$	6.00000	0.242734
$612$	0	0
$613$	−10.0000	−0.403896	−0.201948	−	0.979396i	$-0.564727\pi$
$613$	−10.0000	−0.403896	−0.201948	+	0.979396i	$0.564727\pi$
$614$	0	0
$615$	24.0000	0.967773
$616$	0	0
$617$	−28.0000	−1.12724	−0.563619	−	0.826035i	$-0.690591\pi$
$617$	−28.0000	−1.12724	−0.563619	+	0.826035i	$0.690591\pi$
$618$	0	0
$619$	−26.0000	−1.04503	−0.522514	−	0.852631i	$-0.675006\pi$
$619$	−26.0000	−1.04503	−0.522514	+	0.852631i	$0.675006\pi$
$620$	0	0
$621$	5.00000	0.200643
$622$	0	0
$623$	11.0000	0.440706
$624$	0	0
$625$	41.0000	1.64000
$626$	0	0
$627$	−5.00000	−0.199681
$628$	0	0
$629$	−3.00000	−0.119618
$630$	0	0
$631$	−44.0000	−1.75161	−0.875806	−	0.482663i	$-0.839670\pi$
$631$	−44.0000	−1.75161	−0.875806	+	0.482663i	$0.839670\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	12.0000	0.476205
$636$	0	0
$637$	18.0000	0.713186
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	0	0
$643$	41.0000	1.61688	0.808441	−	0.588577i	$-0.200312\pi$
$643$	41.0000	1.61688	0.808441	+	0.588577i	$0.200312\pi$
$644$	0	0
$645$	16.0000	0.629999
$646$	0	0
$647$	11.0000	0.432455	0.216227	−	0.976343i	$-0.430625\pi$
$647$	11.0000	0.432455	0.216227	+	0.976343i	$0.430625\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	−10.0000	−0.391931
$652$	0	0
$653$	−20.0000	−0.782660	−0.391330	−	0.920250i	$-0.627985\pi$
$653$	−20.0000	−0.782660	−0.391330	+	0.920250i	$0.627985\pi$
$654$	0	0
$655$	−88.0000	−3.43844
$656$	0	0
$657$	−11.0000	−0.429151
$658$	0	0
$659$	−16.0000	−0.623272	−0.311636	−	0.950202i	$-0.600877\pi$
$659$	−16.0000	−0.623272	−0.311636	+	0.950202i	$0.600877\pi$
$660$	0	0
$661$	5.00000	0.194477	0.0972387	−	0.995261i	$-0.468999\pi$
$661$	5.00000	0.194477	0.0972387	+	0.995261i	$0.468999\pi$
$662$	0	0
$663$	9.00000	0.349531
$664$	0	0
$665$	20.0000	0.775567
$666$	0	0
$667$	−20.0000	−0.774403
$668$	0	0
$669$	0	0
$670$	0	0
$671$	10.0000	0.386046
$672$	0	0
$673$	37.0000	1.42625	0.713123	−	0.701039i	$-0.247280\pi$
$673$	37.0000	1.42625	0.713123	+	0.701039i	$0.247280\pi$
$674$	0	0
$675$	−11.0000	−0.423390
$676$	0	0
$677$	15.0000	0.576497	0.288248	−	0.957556i	$-0.406927\pi$
$677$	15.0000	0.576497	0.288248	+	0.957556i	$0.406927\pi$
$678$	0	0
$679$	10.0000	0.383765
$680$	0	0
$681$	16.0000	0.613121
$682$	0	0
$683$	22.0000	0.841807	0.420903	−	0.907106i	$-0.361713\pi$
$683$	22.0000	0.841807	0.420903	+	0.907106i	$0.361713\pi$
$684$	0	0
$685$	80.0000	3.05664
$686$	0	0
$687$	8.00000	0.305219
$688$	0	0
$689$	33.0000	1.25720
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	0	0
$693$	1.00000	0.0379869
$694$	0	0
$695$	56.0000	2.12420
$696$	0	0
$697$	−18.0000	−0.681799
$698$	0	0
$699$	−10.0000	−0.378235
$700$	0	0
$701$	−40.0000	−1.51078	−0.755390	−	0.655276i	$-0.772552\pi$
$701$	−40.0000	−1.51078	−0.755390	+	0.655276i	$0.772552\pi$
$702$	0	0
$703$	−5.00000	−0.188579
$704$	0	0
$705$	8.00000	0.301297
$706$	0	0
$707$	6.00000	0.225653
$708$	0	0
$709$	−19.0000	−0.713560	−0.356780	−	0.934188i	$-0.616125\pi$
$709$	−19.0000	−0.713560	−0.356780	+	0.934188i	$0.616125\pi$
$710$	0	0
$711$	10.0000	0.375029
$712$	0	0
$713$	−50.0000	−1.87251
$714$	0	0
$715$	−12.0000	−0.448775
$716$	0	0
$717$	0	0
$718$	0	0
$719$	8.00000	0.298350	0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000	0.298350	0.149175	+	0.988811i	$0.452338\pi$
$720$	0	0
$721$	2.00000	0.0744839
$722$	0	0
$723$	−28.0000	−1.04133
$724$	0	0
$725$	44.0000	1.63412
$726$	0	0
$727$	−20.0000	−0.741759	−0.370879	−	0.928681i	$-0.620944\pi$
$727$	−20.0000	−0.741759	−0.370879	+	0.928681i	$0.620944\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−12.0000	−0.443836
$732$	0	0
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	0	0
$735$	24.0000	0.885253
$736$	0	0
$737$	−14.0000	−0.515697
$738$	0	0
$739$	38.0000	1.39785	0.698926	−	0.715194i	$-0.253662\pi$
$739$	38.0000	1.39785	0.698926	+	0.715194i	$0.253662\pi$
$740$	0	0
$741$	15.0000	0.551039
$742$	0	0
$743$	−8.00000	−0.293492	−0.146746	−	0.989174i	$-0.546880\pi$
$743$	−8.00000	−0.293492	−0.146746	+	0.989174i	$0.546880\pi$
$744$	0	0
$745$	−40.0000	−1.46549
$746$	0	0
$747$	9.00000	0.329293
$748$	0	0
$749$	3.00000	0.109618
$750$	0	0
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	0	0
$753$	−6.00000	−0.218652
$754$	0	0
$755$	−20.0000	−0.727875
$756$	0	0
$757$	15.0000	0.545184	0.272592	−	0.962130i	$-0.412119\pi$
$757$	15.0000	0.545184	0.272592	+	0.962130i	$0.412119\pi$
$758$	0	0
$759$	5.00000	0.181489
$760$	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
$763$	−9.00000	−0.325822
$764$	0	0
$765$	12.0000	0.433861
$766$	0	0
$767$	−36.0000	−1.29988
$768$	0	0
$769$	−4.00000	−0.144244	−0.0721218	−	0.997396i	$-0.522977\pi$
$769$	−4.00000	−0.144244	−0.0721218	+	0.997396i	$0.522977\pi$
$770$	0	0
$771$	−27.0000	−0.972381
$772$	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
$775$	110.000	3.95132
$776$	0	0
$777$	1.00000	0.0358748
$778$	0	0
$779$	−30.0000	−1.07486
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−4.00000	−0.142948
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−38.0000	−1.35455	−0.677277	−	0.735728i	$-0.736840\pi$
$787$	−38.0000	−1.35455	−0.677277	+	0.735728i	$0.736840\pi$
$788$	0	0
$789$	18.0000	0.640817
$790$	0	0
$791$	14.0000	0.497783
$792$	0	0
$793$	−30.0000	−1.06533
$794$	0	0
$795$	44.0000	1.56052
$796$	0	0
$797$	4.00000	0.141687	0.0708436	−	0.997487i	$-0.477431\pi$
$797$	4.00000	0.141687	0.0708436	+	0.997487i	$0.477431\pi$
$798$	0	0
$799$	−6.00000	−0.212265
$800$	0	0
$801$	11.0000	0.388666
$802$	0	0
$803$	−11.0000	−0.388182
$804$	0	0
$805$	−20.0000	−0.704907
$806$	0	0
$807$	−27.0000	−0.950445
$808$	0	0
$809$	19.0000	0.668004	0.334002	−	0.942572i	$-0.391601\pi$
$809$	19.0000	0.668004	0.334002	+	0.942572i	$0.391601\pi$
$810$	0	0
$811$	32.0000	1.12367	0.561836	−	0.827249i	$-0.310095\pi$
$811$	32.0000	1.12367	0.561836	+	0.827249i	$0.310095\pi$
$812$	0	0
$813$	8.00000	0.280572
$814$	0	0
$815$	−36.0000	−1.26102
$816$	0	0
$817$	−20.0000	−0.699711
$818$	0	0
$819$	−3.00000	−0.104828
$820$	0	0
$821$	33.0000	1.15171	0.575854	−	0.817553i	$-0.304670\pi$
$821$	33.0000	1.15171	0.575854	+	0.817553i	$0.304670\pi$
$822$	0	0
$823$	−9.00000	−0.313720	−0.156860	−	0.987621i	$-0.550137\pi$
$823$	−9.00000	−0.313720	−0.156860	+	0.987621i	$0.550137\pi$
$824$	0	0
$825$	−11.0000	−0.382971
$826$	0	0
$827$	14.0000	0.486828	0.243414	−	0.969923i	$-0.421733\pi$
$827$	14.0000	0.486828	0.243414	+	0.969923i	$0.421733\pi$
$828$	0	0
$829$	19.0000	0.659897	0.329949	−	0.943999i	$-0.392969\pi$
$829$	19.0000	0.659897	0.329949	+	0.943999i	$0.392969\pi$
$830$	0	0
$831$	5.00000	0.173448
$832$	0	0
$833$	−18.0000	−0.623663
$834$	0	0
$835$	68.0000	2.35324
$836$	0	0
$837$	−10.0000	−0.345651
$838$	0	0
$839$	4.00000	0.138095	0.0690477	−	0.997613i	$-0.478004\pi$
$839$	4.00000	0.138095	0.0690477	+	0.997613i	$0.478004\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	0	0
$843$	9.00000	0.309976
$844$	0	0
$845$	−16.0000	−0.550417
$846$	0	0
$847$	−10.0000	−0.343604
$848$	0	0
$849$	21.0000	0.720718
$850$	0	0
$851$	5.00000	0.171398
$852$	0	0
$853$	−55.0000	−1.88316	−0.941582	−	0.336784i	$-0.890661\pi$
$853$	−55.0000	−1.88316	−0.941582	+	0.336784i	$0.890661\pi$
$854$	0	0
$855$	20.0000	0.683986
$856$	0	0
$857$	−3.00000	−0.102478	−0.0512390	−	0.998686i	$-0.516317\pi$
$857$	−3.00000	−0.102478	−0.0512390	+	0.998686i	$0.516317\pi$
$858$	0	0
$859$	23.0000	0.784750	0.392375	−	0.919805i	$-0.371654\pi$
$859$	23.0000	0.784750	0.392375	+	0.919805i	$0.371654\pi$
$860$	0	0
$861$	6.00000	0.204479
$862$	0	0
$863$	22.0000	0.748889	0.374444	−	0.927249i	$-0.377833\pi$
$863$	22.0000	0.748889	0.374444	+	0.927249i	$0.377833\pi$
$864$	0	0
$865$	−52.0000	−1.76805
$866$	0	0
$867$	8.00000	0.271694
$868$	0	0
$869$	10.0000	0.339227
$870$	0	0
$871$	42.0000	1.42312
$872$	0	0
$873$	10.0000	0.338449
$874$	0	0
$875$	24.0000	0.811348
$876$	0	0
$877$	−14.0000	−0.472746	−0.236373	−	0.971662i	$-0.575959\pi$
$877$	−14.0000	−0.472746	−0.236373	+	0.971662i	$0.575959\pi$
$878$	0	0
$879$	−21.0000	−0.708312
$880$	0	0
$881$	−36.0000	−1.21287	−0.606435	−	0.795133i	$-0.707401\pi$
$881$	−36.0000	−1.21287	−0.606435	+	0.795133i	$0.707401\pi$
$882$	0	0
$883$	11.0000	0.370179	0.185090	−	0.982722i	$-0.440742\pi$
$883$	11.0000	0.370179	0.185090	+	0.982722i	$0.440742\pi$
$884$	0	0
$885$	−48.0000	−1.61350
$886$	0	0
$887$	−6.00000	−0.201460	−0.100730	−	0.994914i	$-0.532118\pi$
$887$	−6.00000	−0.201460	−0.100730	+	0.994914i	$0.532118\pi$
$888$	0	0
$889$	3.00000	0.100617
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	−10.0000	−0.334637
$894$	0	0
$895$	−16.0000	−0.534821
$896$	0	0
$897$	−15.0000	−0.500835
$898$	0	0
$899$	40.0000	1.33407
$900$	0	0
$901$	−33.0000	−1.09939
$902$	0	0
$903$	4.00000	0.133112
$904$	0	0
$905$	−64.0000	−2.12743
$906$	0	0
$907$	11.0000	0.365249	0.182625	−	0.983183i	$-0.441541\pi$
$907$	11.0000	0.365249	0.182625	+	0.983183i	$0.441541\pi$
$908$	0	0
$909$	6.00000	0.199007
$910$	0	0
$911$	−40.0000	−1.32526	−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000	−1.32526	−0.662630	+	0.748947i	$0.730560\pi$
$912$	0	0
$913$	9.00000	0.297857
$914$	0	0
$915$	−40.0000	−1.32236
$916$	0	0
$917$	−22.0000	−0.726504
$918$	0	0
$919$	−46.0000	−1.51740	−0.758700	−	0.651440i	$-0.774165\pi$
$919$	−46.0000	−1.51740	−0.758700	+	0.651440i	$0.774165\pi$
$920$	0	0
$921$	32.0000	1.05444
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−11.0000	−0.361678
$926$	0	0
$927$	2.00000	0.0656886
$928$	0	0
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	−30.0000	−0.983210
$932$	0	0
$933$	−16.0000	−0.523816
$934$	0	0
$935$	12.0000	0.392442
$936$	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
$939$	14.0000	0.456873
$940$	0	0
$941$	50.0000	1.62995	0.814977	−	0.579494i	$-0.196750\pi$
$941$	50.0000	1.62995	0.814977	+	0.579494i	$0.196750\pi$
$942$	0	0
$943$	30.0000	0.976934
$944$	0	0
$945$	−4.00000	−0.130120
$946$	0	0
$947$	26.0000	0.844886	0.422443	−	0.906389i	$-0.361173\pi$
$947$	26.0000	0.844886	0.422443	+	0.906389i	$0.361173\pi$
$948$	0	0
$949$	33.0000	1.07123
$950$	0	0
$951$	18.0000	0.583690
$952$	0	0
$953$	8.00000	0.259145	0.129573	−	0.991570i	$-0.458639\pi$
$953$	8.00000	0.259145	0.129573	+	0.991570i	$0.458639\pi$
$954$	0	0
$955$	−36.0000	−1.16493
$956$	0	0
$957$	−4.00000	−0.129302
$958$	0	0
$959$	20.0000	0.645834
$960$	0	0
$961$	69.0000	2.22581
$962$	0	0
$963$	3.00000	0.0966736
$964$	0	0
$965$	72.0000	2.31776
$966$	0	0
$967$	12.0000	0.385894	0.192947	−	0.981209i	$-0.438195\pi$
$967$	12.0000	0.385894	0.192947	+	0.981209i	$0.438195\pi$
$968$	0	0
$969$	−15.0000	−0.481869
$970$	0	0
$971$	60.0000	1.92549	0.962746	−	0.270408i	$-0.0871586\pi$
$971$	60.0000	1.92549	0.962746	+	0.270408i	$0.0871586\pi$
$972$	0	0
$973$	14.0000	0.448819
$974$	0	0
$975$	33.0000	1.05685
$976$	0	0
$977$	27.0000	0.863807	0.431903	−	0.901920i	$-0.357842\pi$
$977$	27.0000	0.863807	0.431903	+	0.901920i	$0.357842\pi$
$978$	0	0
$979$	11.0000	0.351562
$980$	0	0
$981$	−9.00000	−0.287348
$982$	0	0
$983$	14.0000	0.446531	0.223265	−	0.974758i	$-0.428328\pi$
$983$	14.0000	0.446531	0.223265	+	0.974758i	$0.428328\pi$
$984$	0	0
$985$	28.0000	0.892154
$986$	0	0
$987$	2.00000	0.0636607
$988$	0	0
$989$	20.0000	0.635963
$990$	0	0
$991$	−20.0000	−0.635321	−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000	−0.635321	−0.317660	+	0.948205i	$0.602897\pi$
$992$	0	0
$993$	28.0000	0.888553
$994$	0	0
$995$	−64.0000	−2.02894
$996$	0	0
$997$	−9.00000	−0.285033	−0.142516	−	0.989792i	$-0.545519\pi$
$997$	−9.00000	−0.285033	−0.142516	+	0.989792i	$0.545519\pi$
$998$	0	0
$999$	1.00000	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1776.2.a.e.1.1		1
3.2	odd	2		5328.2.a.b.1.1		1
4.3	odd	2		222.2.a.c.1.1	✓	1
8.3	odd	2		7104.2.a.a.1.1		1
8.5	even	2		7104.2.a.p.1.1		1
12.11	even	2		666.2.a.d.1.1		1
20.19	odd	2		5550.2.a.z.1.1		1
148.147	odd	2		8214.2.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
222.2.a.c.1.1	✓	1	4.3	odd	2
666.2.a.d.1.1		1	12.11	even	2
1776.2.a.e.1.1		1	1.1	even	1	trivial
5328.2.a.b.1.1		1	3.2	odd	2
5550.2.a.z.1.1		1	20.19	odd	2
7104.2.a.a.1.1		1	8.3	odd	2
7104.2.a.p.1.1		1	8.5	even	2
8214.2.a.j.1.1		1	148.147	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1776 = 2^{4} \cdot 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1776.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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