LMFDB - Embedded newform 5776.2.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5776,2,Mod(1,5776)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5776, 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("5776.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5776 = 2^{4} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5776.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:	$46.1215922075$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 722)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5776.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} +2.00000 q^{5} +3.00000 q^{7} +6.00000 q^{9} +O(q^{10})$

$q+3.00000 q^{3} +2.00000 q^{5} +3.00000 q^{7} +6.00000 q^{9} +2.00000 q^{11} +3.00000 q^{13} +6.00000 q^{15} -1.00000 q^{17} +9.00000 q^{21} -5.00000 q^{23} -1.00000 q^{25} +9.00000 q^{27} +3.00000 q^{29} -6.00000 q^{31} +6.00000 q^{33} +6.00000 q^{35} -6.00000 q^{37} +9.00000 q^{39} -12.0000 q^{41} +10.0000 q^{43} +12.0000 q^{45} +8.00000 q^{47} +2.00000 q^{49} -3.00000 q^{51} +3.00000 q^{53} +4.00000 q^{55} +3.00000 q^{59} +18.0000 q^{63} +6.00000 q^{65} +15.0000 q^{67} -15.0000 q^{69} -11.0000 q^{73} -3.00000 q^{75} +6.00000 q^{77} -12.0000 q^{79} +9.00000 q^{81} -2.00000 q^{83} -2.00000 q^{85} +9.00000 q^{87} -6.00000 q^{89} +9.00000 q^{91} -18.0000 q^{93} -12.0000 q^{97} +12.0000 q^{99} +O(q^{100})$

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
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	0	0
$9$	6.00000	2.00000
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	0	0
$15$	6.00000	1.54919
$16$	0	0
$17$	−1.00000	−0.242536	−0.121268	−	0.992620i	$-0.538696\pi$
$17$	−1.00000	−0.242536	−0.121268	+	0.992620i	$0.538696\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	9.00000	1.96396
$22$	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
$25$	−1.00000	−0.200000
$26$	0	0
$27$	9.00000	1.73205
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	0	0
$33$	6.00000	1.04447
$34$	0	0
$35$	6.00000	1.01419
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	0	0
$39$	9.00000	1.44115
$40$	0	0
$41$	−12.0000	−1.87409	−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000	−1.87409	−0.937043	+	0.349215i	$0.886448\pi$
$42$	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
$45$	12.0000	1.78885
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	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
$57$	0	0
$58$	0	0
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	18.0000	2.26779
$64$	0	0
$65$	6.00000	0.744208
$66$	0	0
$67$	15.0000	1.83254	0.916271	−	0.400559i	$-0.131184\pi$
$67$	15.0000	1.83254	0.916271	+	0.400559i	$0.131184\pi$
$68$	0	0
$69$	−15.0000	−1.80579
$70$	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
$75$	−3.00000	−0.346410
$76$	0	0
$77$	6.00000	0.683763
$78$	0	0
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	−2.00000	−0.219529	−0.109764	−	0.993958i	$-0.535010\pi$
$83$	−2.00000	−0.219529	−0.109764	+	0.993958i	$0.535010\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	0	0
$87$	9.00000	0.964901
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	9.00000	0.943456
$92$	0	0
$93$	−18.0000	−1.86651
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−12.0000	−1.21842	−0.609208	−	0.793011i	$-0.708512\pi$
$97$	−12.0000	−1.21842	−0.609208	+	0.793011i	$0.708512\pi$
$98$	0	0
$99$	12.0000	1.20605
$100$	0	0
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	0	0
$103$	6.00000	0.591198	0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000	0.591198	0.295599	+	0.955312i	$0.404481\pi$
$104$	0	0
$105$	18.0000	1.75662
$106$	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
$109$	−3.00000	−0.287348	−0.143674	−	0.989625i	$-0.545892\pi$
$109$	−3.00000	−0.287348	−0.143674	+	0.989625i	$0.545892\pi$
$110$	0	0
$111$	−18.0000	−1.70848
$112$	0	0
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	0	0
$115$	−10.0000	−0.932505
$116$	0	0
$117$	18.0000	1.66410
$118$	0	0
$119$	−3.00000	−0.275010
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−36.0000	−3.24601
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	0	0
$129$	30.0000	2.64135
$130$	0	0
$131$	−14.0000	−1.22319	−0.611593	−	0.791173i	$-0.709471\pi$
$131$	−14.0000	−1.22319	−0.611593	+	0.791173i	$0.709471\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	18.0000	1.54919
$136$	0	0
$137$	19.0000	1.62328	0.811640	−	0.584158i	$-0.198575\pi$
$137$	19.0000	1.62328	0.811640	+	0.584158i	$0.198575\pi$
$138$	0	0
$139$	−6.00000	−0.508913	−0.254457	−	0.967084i	$-0.581897\pi$
$139$	−6.00000	−0.508913	−0.254457	+	0.967084i	$0.581897\pi$
$140$	0	0
$141$	24.0000	2.02116
$142$	0	0
$143$	6.00000	0.501745
$144$	0	0
$145$	6.00000	0.498273
$146$	0	0
$147$	6.00000	0.494872
$148$	0	0
$149$	−8.00000	−0.655386	−0.327693	−	0.944784i	$-0.606271\pi$
$149$	−8.00000	−0.655386	−0.327693	+	0.944784i	$0.606271\pi$
$150$	0	0
$151$	−18.0000	−1.46482	−0.732410	−	0.680864i	$-0.761604\pi$
$151$	−18.0000	−1.46482	−0.732410	+	0.680864i	$0.761604\pi$
$152$	0	0
$153$	−6.00000	−0.485071
$154$	0	0
$155$	−12.0000	−0.963863
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	9.00000	0.713746
$160$	0	0
$161$	−15.0000	−1.18217
$162$	0	0
$163$	6.00000	0.469956	0.234978	−	0.972001i	$-0.424498\pi$
$163$	6.00000	0.469956	0.234978	+	0.972001i	$0.424498\pi$
$164$	0	0
$165$	12.0000	0.934199
$166$	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
$169$	−4.00000	−0.307692
$170$	0	0
$171$	0	0
$172$	0	0
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	0	0
$175$	−3.00000	−0.226779
$176$	0	0
$177$	9.00000	0.676481
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−12.0000	−0.882258
$186$	0	0
$187$	−2.00000	−0.146254
$188$	0	0
$189$	27.0000	1.96396
$190$	0	0
$191$	11.0000	0.795932	0.397966	−	0.917400i	$-0.369716\pi$
$191$	11.0000	0.795932	0.397966	+	0.917400i	$0.369716\pi$
$192$	0	0
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	0	0
$195$	18.0000	1.28901
$196$	0	0
$197$	4.00000	0.284988	0.142494	−	0.989796i	$-0.454488\pi$
$197$	4.00000	0.284988	0.142494	+	0.989796i	$0.454488\pi$
$198$	0	0
$199$	7.00000	0.496217	0.248108	−	0.968732i	$-0.420191\pi$
$199$	7.00000	0.496217	0.248108	+	0.968732i	$0.420191\pi$
$200$	0	0
$201$	45.0000	3.17406
$202$	0	0
$203$	9.00000	0.631676
$204$	0	0
$205$	−24.0000	−1.67623
$206$	0	0
$207$	−30.0000	−2.08514
$208$	0	0
$209$	0	0
$210$	0	0
$211$	3.00000	0.206529	0.103264	−	0.994654i	$-0.467071\pi$
$211$	3.00000	0.206529	0.103264	+	0.994654i	$0.467071\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	20.0000	1.36399
$216$	0	0
$217$	−18.0000	−1.22192
$218$	0	0
$219$	−33.0000	−2.22993
$220$	0	0
$221$	−3.00000	−0.201802
$222$	0	0
$223$	18.0000	1.20537	0.602685	−	0.797980i	$-0.294098\pi$
$223$	18.0000	1.20537	0.602685	+	0.797980i	$0.294098\pi$
$224$	0	0
$225$	−6.00000	−0.400000
$226$	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
$229$	−12.0000	−0.792982	−0.396491	−	0.918039i	$-0.629772\pi$
$229$	−12.0000	−0.792982	−0.396491	+	0.918039i	$0.629772\pi$
$230$	0	0
$231$	18.0000	1.18431
$232$	0	0
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	0	0
$235$	16.0000	1.04372
$236$	0	0
$237$	−36.0000	−2.33845
$238$	0	0
$239$	−1.00000	−0.0646846	−0.0323423	−	0.999477i	$-0.510297\pi$
$239$	−1.00000	−0.0646846	−0.0323423	+	0.999477i	$0.510297\pi$
$240$	0	0
$241$	−24.0000	−1.54598	−0.772988	−	0.634421i	$-0.781239\pi$
$241$	−24.0000	−1.54598	−0.772988	+	0.634421i	$0.781239\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4.00000	0.255551
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−6.00000	−0.380235
$250$	0	0
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	0	0
$253$	−10.0000	−0.628695
$254$	0	0
$255$	−6.00000	−0.375735
$256$	0	0
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	−18.0000	−1.11847
$260$	0	0
$261$	18.0000	1.11417
$262$	0	0
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	0	0
$267$	−18.0000	−1.10158
$268$	0	0
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	11.0000	0.668202	0.334101	−	0.942537i	$-0.391567\pi$
$271$	11.0000	0.668202	0.334101	+	0.942537i	$0.391567\pi$
$272$	0	0
$273$	27.0000	1.63411
$274$	0	0
$275$	−2.00000	−0.120605
$276$	0	0
$277$	30.0000	1.80253	0.901263	−	0.433273i	$-0.142641\pi$
$277$	30.0000	1.80253	0.901263	+	0.433273i	$0.142641\pi$
$278$	0	0
$279$	−36.0000	−2.15526
$280$	0	0
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	0	0
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−36.0000	−2.12501
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	−36.0000	−2.11036
$292$	0	0
$293$	9.00000	0.525786	0.262893	−	0.964825i	$-0.415323\pi$
$293$	9.00000	0.525786	0.262893	+	0.964825i	$0.415323\pi$
$294$	0	0
$295$	6.00000	0.349334
$296$	0	0
$297$	18.0000	1.04447
$298$	0	0
$299$	−15.0000	−0.867472
$300$	0	0
$301$	30.0000	1.72917
$302$	0	0
$303$	30.0000	1.72345
$304$	0	0
$305$	0	0
$306$	0	0
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	0	0
$309$	18.0000	1.02398
$310$	0	0
$311$	11.0000	0.623753	0.311876	−	0.950123i	$-0.399043\pi$
$311$	11.0000	0.623753	0.311876	+	0.950123i	$0.399043\pi$
$312$	0	0
$313$	21.0000	1.18699	0.593495	−	0.804838i	$-0.297748\pi$
$313$	21.0000	1.18699	0.593495	+	0.804838i	$0.297748\pi$
$314$	0	0
$315$	36.0000	2.02837
$316$	0	0
$317$	33.0000	1.85346	0.926732	−	0.375722i	$-0.122605\pi$
$317$	33.0000	1.85346	0.926732	+	0.375722i	$0.122605\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	0	0
$321$	−9.00000	−0.502331
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−3.00000	−0.166410
$326$	0	0
$327$	−9.00000	−0.497701
$328$	0	0
$329$	24.0000	1.32316
$330$	0	0
$331$	9.00000	0.494685	0.247342	−	0.968928i	$-0.420443\pi$
$331$	9.00000	0.494685	0.247342	+	0.968928i	$0.420443\pi$
$332$	0	0
$333$	−36.0000	−1.97279
$334$	0	0
$335$	30.0000	1.63908
$336$	0	0
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	0	0
$339$	−36.0000	−1.95525
$340$	0	0
$341$	−12.0000	−0.649836
$342$	0	0
$343$	−15.0000	−0.809924
$344$	0	0
$345$	−30.0000	−1.61515
$346$	0	0
$347$	−16.0000	−0.858925	−0.429463	−	0.903085i	$-0.641297\pi$
$347$	−16.0000	−0.858925	−0.429463	+	0.903085i	$0.641297\pi$
$348$	0	0
$349$	28.0000	1.49881	0.749403	−	0.662114i	$-0.230341\pi$
$349$	28.0000	1.49881	0.749403	+	0.662114i	$0.230341\pi$
$350$	0	0
$351$	27.0000	1.44115
$352$	0	0
$353$	−31.0000	−1.64996	−0.824982	−	0.565159i	$-0.808815\pi$
$353$	−31.0000	−1.64996	−0.824982	+	0.565159i	$0.808815\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−9.00000	−0.476331
$358$	0	0
$359$	−19.0000	−1.00278	−0.501391	−	0.865221i	$-0.667178\pi$
$359$	−19.0000	−1.00278	−0.501391	+	0.865221i	$0.667178\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−21.0000	−1.10221
$364$	0	0
$365$	−22.0000	−1.15153
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	0	0
$369$	−72.0000	−3.74817
$370$	0	0
$371$	9.00000	0.467257
$372$	0	0
$373$	21.0000	1.08734	0.543669	−	0.839299i	$-0.317035\pi$
$373$	21.0000	1.08734	0.543669	+	0.839299i	$0.317035\pi$
$374$	0	0
$375$	−36.0000	−1.85903
$376$	0	0
$377$	9.00000	0.463524
$378$	0	0
$379$	3.00000	0.154100	0.0770498	−	0.997027i	$-0.475450\pi$
$379$	3.00000	0.154100	0.0770498	+	0.997027i	$0.475450\pi$
$380$	0	0
$381$	36.0000	1.84434
$382$	0	0
$383$	18.0000	0.919757	0.459879	−	0.887982i	$-0.347893\pi$
$383$	18.0000	0.919757	0.459879	+	0.887982i	$0.347893\pi$
$384$	0	0
$385$	12.0000	0.611577
$386$	0	0
$387$	60.0000	3.04997
$388$	0	0
$389$	26.0000	1.31825	0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000	1.31825	0.659126	+	0.752032i	$0.270926\pi$
$390$	0	0
$391$	5.00000	0.252861
$392$	0	0
$393$	−42.0000	−2.11862
$394$	0	0
$395$	−24.0000	−1.20757
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−36.0000	−1.79775	−0.898877	−	0.438201i	$-0.855616\pi$
$401$	−36.0000	−1.79775	−0.898877	+	0.438201i	$0.855616\pi$
$402$	0	0
$403$	−18.0000	−0.896644
$404$	0	0
$405$	18.0000	0.894427
$406$	0	0
$407$	−12.0000	−0.594818
$408$	0	0
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	0	0
$411$	57.0000	2.81160
$412$	0	0
$413$	9.00000	0.442861
$414$	0	0
$415$	−4.00000	−0.196352
$416$	0	0
$417$	−18.0000	−0.881464
$418$	0	0
$419$	14.0000	0.683945	0.341972	−	0.939710i	$-0.388905\pi$
$419$	14.0000	0.683945	0.341972	+	0.939710i	$0.388905\pi$
$420$	0	0
$421$	27.0000	1.31590	0.657950	−	0.753062i	$-0.271424\pi$
$421$	27.0000	1.31590	0.657950	+	0.753062i	$0.271424\pi$
$422$	0	0
$423$	48.0000	2.33384
$424$	0	0
$425$	1.00000	0.0485071
$426$	0	0
$427$	0	0
$428$	0	0
$429$	18.0000	0.869048
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	0	0
$433$	−30.0000	−1.44171	−0.720854	−	0.693087i	$-0.756250\pi$
$433$	−30.0000	−1.44171	−0.720854	+	0.693087i	$0.756250\pi$
$434$	0	0
$435$	18.0000	0.863034
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	12.0000	0.571429
$442$	0	0
$443$	22.0000	1.04525	0.522626	−	0.852562i	$-0.324953\pi$
$443$	22.0000	1.04525	0.522626	+	0.852562i	$0.324953\pi$
$444$	0	0
$445$	−12.0000	−0.568855
$446$	0	0
$447$	−24.0000	−1.13516
$448$	0	0
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	−24.0000	−1.13012
$452$	0	0
$453$	−54.0000	−2.53714
$454$	0	0
$455$	18.0000	0.843853
$456$	0	0
$457$	1.00000	0.0467780	0.0233890	−	0.999726i	$-0.492554\pi$
$457$	1.00000	0.0467780	0.0233890	+	0.999726i	$0.492554\pi$
$458$	0	0
$459$	−9.00000	−0.420084
$460$	0	0
$461$	4.00000	0.186299	0.0931493	−	0.995652i	$-0.470307\pi$
$461$	4.00000	0.186299	0.0931493	+	0.995652i	$0.470307\pi$
$462$	0	0
$463$	−32.0000	−1.48717	−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000	−1.48717	−0.743583	+	0.668644i	$0.766875\pi$
$464$	0	0
$465$	−36.0000	−1.66946
$466$	0	0
$467$	−8.00000	−0.370196	−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000	−0.370196	−0.185098	+	0.982720i	$0.559260\pi$
$468$	0	0
$469$	45.0000	2.07791
$470$	0	0
$471$	0	0
$472$	0	0
$473$	20.0000	0.919601
$474$	0	0
$475$	0	0
$476$	0	0
$477$	18.0000	0.824163
$478$	0	0
$479$	40.0000	1.82765	0.913823	−	0.406112i	$-0.133116\pi$
$479$	40.0000	1.82765	0.913823	+	0.406112i	$0.133116\pi$
$480$	0	0
$481$	−18.0000	−0.820729
$482$	0	0
$483$	−45.0000	−2.04757
$484$	0	0
$485$	−24.0000	−1.08978
$486$	0	0
$487$	−18.0000	−0.815658	−0.407829	−	0.913058i	$-0.633714\pi$
$487$	−18.0000	−0.815658	−0.407829	+	0.913058i	$0.633714\pi$
$488$	0	0
$489$	18.0000	0.813988
$490$	0	0
$491$	−8.00000	−0.361035	−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000	−0.361035	−0.180517	+	0.983572i	$0.557777\pi$
$492$	0	0
$493$	−3.00000	−0.135113
$494$	0	0
$495$	24.0000	1.07872
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−18.0000	−0.805791	−0.402895	−	0.915246i	$-0.631996\pi$
$499$	−18.0000	−0.805791	−0.402895	+	0.915246i	$0.631996\pi$
$500$	0	0
$501$	−36.0000	−1.60836
$502$	0	0
$503$	−1.00000	−0.0445878	−0.0222939	−	0.999751i	$-0.507097\pi$
$503$	−1.00000	−0.0445878	−0.0222939	+	0.999751i	$0.507097\pi$
$504$	0	0
$505$	20.0000	0.889988
$506$	0	0
$507$	−12.0000	−0.532939
$508$	0	0
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0	0
$511$	−33.0000	−1.45983
$512$	0	0
$513$	0	0
$514$	0	0
$515$	12.0000	0.528783
$516$	0	0
$517$	16.0000	0.703679
$518$	0	0
$519$	54.0000	2.37034
$520$	0	0
$521$	12.0000	0.525730	0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000	0.525730	0.262865	+	0.964833i	$0.415333\pi$
$522$	0	0
$523$	9.00000	0.393543	0.196771	−	0.980449i	$-0.436954\pi$
$523$	9.00000	0.393543	0.196771	+	0.980449i	$0.436954\pi$
$524$	0	0
$525$	−9.00000	−0.392792
$526$	0	0
$527$	6.00000	0.261364
$528$	0	0
$529$	2.00000	0.0869565
$530$	0	0
$531$	18.0000	0.781133
$532$	0	0
$533$	−36.0000	−1.55933
$534$	0	0
$535$	−6.00000	−0.259403
$536$	0	0
$537$	−36.0000	−1.55351
$538$	0	0
$539$	4.00000	0.172292
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	0	0
$543$	−54.0000	−2.31736
$544$	0	0
$545$	−6.00000	−0.257012
$546$	0	0
$547$	−36.0000	−1.53925	−0.769624	−	0.638497i	$-0.779557\pi$
$547$	−36.0000	−1.53925	−0.769624	+	0.638497i	$0.779557\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−36.0000	−1.53088
$554$	0	0
$555$	−36.0000	−1.52811
$556$	0	0
$557$	22.0000	0.932170	0.466085	−	0.884740i	$-0.345664\pi$
$557$	22.0000	0.932170	0.466085	+	0.884740i	$0.345664\pi$
$558$	0	0
$559$	30.0000	1.26886
$560$	0	0
$561$	−6.00000	−0.253320
$562$	0	0
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	0	0
$565$	−24.0000	−1.00969
$566$	0	0
$567$	27.0000	1.13389
$568$	0	0
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	0	0
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	0	0
$573$	33.0000	1.37859
$574$	0	0
$575$	5.00000	0.208514
$576$	0	0
$577$	15.0000	0.624458	0.312229	−	0.950007i	$-0.398924\pi$
$577$	15.0000	0.624458	0.312229	+	0.950007i	$0.398924\pi$
$578$	0	0
$579$	−18.0000	−0.748054
$580$	0	0
$581$	−6.00000	−0.248922
$582$	0	0
$583$	6.00000	0.248495
$584$	0	0
$585$	36.0000	1.48842
$586$	0	0
$587$	28.0000	1.15568	0.577842	−	0.816149i	$-0.303895\pi$
$587$	28.0000	1.15568	0.577842	+	0.816149i	$0.303895\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	12.0000	0.493614
$592$	0	0
$593$	−2.00000	−0.0821302	−0.0410651	−	0.999156i	$-0.513075\pi$
$593$	−2.00000	−0.0821302	−0.0410651	+	0.999156i	$0.513075\pi$
$594$	0	0
$595$	−6.00000	−0.245976
$596$	0	0
$597$	21.0000	0.859473
$598$	0	0
$599$	−36.0000	−1.47092	−0.735460	−	0.677568i	$-0.763034\pi$
$599$	−36.0000	−1.47092	−0.735460	+	0.677568i	$0.763034\pi$
$600$	0	0
$601$	6.00000	0.244745	0.122373	−	0.992484i	$-0.460950\pi$
$601$	6.00000	0.244745	0.122373	+	0.992484i	$0.460950\pi$
$602$	0	0
$603$	90.0000	3.66508
$604$	0	0
$605$	−14.0000	−0.569181
$606$	0	0
$607$	−12.0000	−0.487065	−0.243532	−	0.969893i	$-0.578306\pi$
$607$	−12.0000	−0.487065	−0.243532	+	0.969893i	$0.578306\pi$
$608$	0	0
$609$	27.0000	1.09410
$610$	0	0
$611$	24.0000	0.970936
$612$	0	0
$613$	18.0000	0.727013	0.363507	−	0.931592i	$-0.381579\pi$
$613$	18.0000	0.727013	0.363507	+	0.931592i	$0.381579\pi$
$614$	0	0
$615$	−72.0000	−2.90332
$616$	0	0
$617$	10.0000	0.402585	0.201292	−	0.979531i	$-0.435486\pi$
$617$	10.0000	0.402585	0.201292	+	0.979531i	$0.435486\pi$
$618$	0	0
$619$	24.0000	0.964641	0.482321	−	0.875995i	$-0.339794\pi$
$619$	24.0000	0.964641	0.482321	+	0.875995i	$0.339794\pi$
$620$	0	0
$621$	−45.0000	−1.80579
$622$	0	0
$623$	−18.0000	−0.721155
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6.00000	0.239236
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	9.00000	0.357718
$634$	0	0
$635$	24.0000	0.952411
$636$	0	0
$637$	6.00000	0.237729
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	−32.0000	−1.26196	−0.630978	−	0.775800i	$-0.717346\pi$
$643$	−32.0000	−1.26196	−0.630978	+	0.775800i	$0.717346\pi$
$644$	0	0
$645$	60.0000	2.36250
$646$	0	0
$647$	23.0000	0.904223	0.452112	−	0.891961i	$-0.350671\pi$
$647$	23.0000	0.904223	0.452112	+	0.891961i	$0.350671\pi$
$648$	0	0
$649$	6.00000	0.235521
$650$	0	0
$651$	−54.0000	−2.11643
$652$	0	0
$653$	−10.0000	−0.391330	−0.195665	−	0.980671i	$-0.562687\pi$
$653$	−10.0000	−0.391330	−0.195665	+	0.980671i	$0.562687\pi$
$654$	0	0
$655$	−28.0000	−1.09405
$656$	0	0
$657$	−66.0000	−2.57491
$658$	0	0
$659$	−15.0000	−0.584317	−0.292159	−	0.956370i	$-0.594373\pi$
$659$	−15.0000	−0.584317	−0.292159	+	0.956370i	$0.594373\pi$
$660$	0	0
$661$	−15.0000	−0.583432	−0.291716	−	0.956505i	$-0.594226\pi$
$661$	−15.0000	−0.583432	−0.291716	+	0.956505i	$0.594226\pi$
$662$	0	0
$663$	−9.00000	−0.349531
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−15.0000	−0.580802
$668$	0	0
$669$	54.0000	2.08776
$670$	0	0
$671$	0	0
$672$	0	0
$673$	48.0000	1.85026	0.925132	−	0.379646i	$-0.123954\pi$
$673$	48.0000	1.85026	0.925132	+	0.379646i	$0.123954\pi$
$674$	0	0
$675$	−9.00000	−0.346410
$676$	0	0
$677$	−3.00000	−0.115299	−0.0576497	−	0.998337i	$-0.518361\pi$
$677$	−3.00000	−0.115299	−0.0576497	+	0.998337i	$0.518361\pi$
$678$	0	0
$679$	−36.0000	−1.38155
$680$	0	0
$681$	9.00000	0.344881
$682$	0	0
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	0	0
$685$	38.0000	1.45191
$686$	0	0
$687$	−36.0000	−1.37349
$688$	0	0
$689$	9.00000	0.342873
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	0	0
$693$	36.0000	1.36753
$694$	0	0
$695$	−12.0000	−0.455186
$696$	0	0
$697$	12.0000	0.454532
$698$	0	0
$699$	42.0000	1.58859
$700$	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
$703$	0	0
$704$	0	0
$705$	48.0000	1.80778
$706$	0	0
$707$	30.0000	1.12827
$708$	0	0
$709$	−8.00000	−0.300446	−0.150223	−	0.988652i	$-0.547999\pi$
$709$	−8.00000	−0.300446	−0.150223	+	0.988652i	$0.547999\pi$
$710$	0	0
$711$	−72.0000	−2.70021
$712$	0	0
$713$	30.0000	1.12351
$714$	0	0
$715$	12.0000	0.448775
$716$	0	0
$717$	−3.00000	−0.112037
$718$	0	0
$719$	43.0000	1.60363	0.801815	−	0.597573i	$-0.203868\pi$
$719$	43.0000	1.60363	0.801815	+	0.597573i	$0.203868\pi$
$720$	0	0
$721$	18.0000	0.670355
$722$	0	0
$723$	−72.0000	−2.67771
$724$	0	0
$725$	−3.00000	−0.111417
$726$	0	0
$727$	35.0000	1.29808	0.649039	−	0.760755i	$-0.275171\pi$
$727$	35.0000	1.29808	0.649039	+	0.760755i	$0.275171\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	−10.0000	−0.369863
$732$	0	0
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	0	0
$735$	12.0000	0.442627
$736$	0	0
$737$	30.0000	1.10506
$738$	0	0
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	0	0
$741$	0	0
$742$	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
$745$	−16.0000	−0.586195
$746$	0	0
$747$	−12.0000	−0.439057
$748$	0	0
$749$	−9.00000	−0.328853
$750$	0	0
$751$	12.0000	0.437886	0.218943	−	0.975738i	$-0.429739\pi$
$751$	12.0000	0.437886	0.218943	+	0.975738i	$0.429739\pi$
$752$	0	0
$753$	60.0000	2.18652
$754$	0	0
$755$	−36.0000	−1.31017
$756$	0	0
$757$	12.0000	0.436147	0.218074	−	0.975932i	$-0.430023\pi$
$757$	12.0000	0.436147	0.218074	+	0.975932i	$0.430023\pi$
$758$	0	0
$759$	−30.0000	−1.08893
$760$	0	0
$761$	−13.0000	−0.471250	−0.235625	−	0.971844i	$-0.575714\pi$
$761$	−13.0000	−0.471250	−0.235625	+	0.971844i	$0.575714\pi$
$762$	0	0
$763$	−9.00000	−0.325822
$764$	0	0
$765$	−12.0000	−0.433861
$766$	0	0
$767$	9.00000	0.324971
$768$	0	0
$769$	−15.0000	−0.540914	−0.270457	−	0.962732i	$-0.587175\pi$
$769$	−15.0000	−0.540914	−0.270457	+	0.962732i	$0.587175\pi$
$770$	0	0
$771$	54.0000	1.94476
$772$	0	0
$773$	−15.0000	−0.539513	−0.269756	−	0.962929i	$-0.586943\pi$
$773$	−15.0000	−0.539513	−0.269756	+	0.962929i	$0.586943\pi$
$774$	0	0
$775$	6.00000	0.215526
$776$	0	0
$777$	−54.0000	−1.93724
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	27.0000	0.964901
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−9.00000	−0.320815	−0.160408	−	0.987051i	$-0.551281\pi$
$787$	−9.00000	−0.320815	−0.160408	+	0.987051i	$0.551281\pi$
$788$	0	0
$789$	24.0000	0.854423
$790$	0	0
$791$	−36.0000	−1.28001
$792$	0	0
$793$	0	0
$794$	0	0
$795$	18.0000	0.638394
$796$	0	0
$797$	−33.0000	−1.16892	−0.584460	−	0.811423i	$-0.698694\pi$
$797$	−33.0000	−1.16892	−0.584460	+	0.811423i	$0.698694\pi$
$798$	0	0
$799$	−8.00000	−0.283020
$800$	0	0
$801$	−36.0000	−1.27200
$802$	0	0
$803$	−22.0000	−0.776363
$804$	0	0
$805$	−30.0000	−1.05736
$806$	0	0
$807$	18.0000	0.633630
$808$	0	0
$809$	−11.0000	−0.386739	−0.193370	−	0.981126i	$-0.561942\pi$
$809$	−11.0000	−0.386739	−0.193370	+	0.981126i	$0.561942\pi$
$810$	0	0
$811$	33.0000	1.15879	0.579393	−	0.815048i	$-0.303290\pi$
$811$	33.0000	1.15879	0.579393	+	0.815048i	$0.303290\pi$
$812$	0	0
$813$	33.0000	1.15736
$814$	0	0
$815$	12.0000	0.420342
$816$	0	0
$817$	0	0
$818$	0	0
$819$	54.0000	1.88691
$820$	0	0
$821$	2.00000	0.0698005	0.0349002	−	0.999391i	$-0.488889\pi$
$821$	2.00000	0.0698005	0.0349002	+	0.999391i	$0.488889\pi$
$822$	0	0
$823$	−3.00000	−0.104573	−0.0522867	−	0.998632i	$-0.516651\pi$
$823$	−3.00000	−0.104573	−0.0522867	+	0.998632i	$0.516651\pi$
$824$	0	0
$825$	−6.00000	−0.208893
$826$	0	0
$827$	15.0000	0.521601	0.260801	−	0.965393i	$-0.416014\pi$
$827$	15.0000	0.521601	0.260801	+	0.965393i	$0.416014\pi$
$828$	0	0
$829$	−51.0000	−1.77130	−0.885652	−	0.464350i	$-0.846288\pi$
$829$	−51.0000	−1.77130	−0.885652	+	0.464350i	$0.846288\pi$
$830$	0	0
$831$	90.0000	3.12207
$832$	0	0
$833$	−2.00000	−0.0692959
$834$	0	0
$835$	−24.0000	−0.830554
$836$	0	0
$837$	−54.0000	−1.86651
$838$	0	0
$839$	−18.0000	−0.621429	−0.310715	−	0.950503i	$-0.600568\pi$
$839$	−18.0000	−0.621429	−0.310715	+	0.950503i	$0.600568\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	−36.0000	−1.23991
$844$	0	0
$845$	−8.00000	−0.275208
$846$	0	0
$847$	−21.0000	−0.721569
$848$	0	0
$849$	42.0000	1.44144
$850$	0	0
$851$	30.0000	1.02839
$852$	0	0
$853$	46.0000	1.57501	0.787505	−	0.616308i	$-0.211372\pi$
$853$	46.0000	1.57501	0.787505	+	0.616308i	$0.211372\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−24.0000	−0.819824	−0.409912	−	0.912125i	$-0.634441\pi$
$857$	−24.0000	−0.819824	−0.409912	+	0.912125i	$0.634441\pi$
$858$	0	0
$859$	−54.0000	−1.84246	−0.921228	−	0.389023i	$-0.872813\pi$
$859$	−54.0000	−1.84246	−0.921228	+	0.389023i	$0.872813\pi$
$860$	0	0
$861$	−108.000	−3.68063
$862$	0	0
$863$	48.0000	1.63394	0.816970	−	0.576681i	$-0.195652\pi$
$863$	48.0000	1.63394	0.816970	+	0.576681i	$0.195652\pi$
$864$	0	0
$865$	36.0000	1.22404
$866$	0	0
$867$	−48.0000	−1.63017
$868$	0	0
$869$	−24.0000	−0.814144
$870$	0	0
$871$	45.0000	1.52477
$872$	0	0
$873$	−72.0000	−2.43683
$874$	0	0
$875$	−36.0000	−1.21702
$876$	0	0
$877$	−27.0000	−0.911725	−0.455863	−	0.890050i	$-0.650669\pi$
$877$	−27.0000	−0.911725	−0.455863	+	0.890050i	$0.650669\pi$
$878$	0	0
$879$	27.0000	0.910687
$880$	0	0
$881$	34.0000	1.14549	0.572745	−	0.819734i	$-0.305879\pi$
$881$	34.0000	1.14549	0.572745	+	0.819734i	$0.305879\pi$
$882$	0	0
$883$	24.0000	0.807664	0.403832	−	0.914833i	$-0.367678\pi$
$883$	24.0000	0.807664	0.403832	+	0.914833i	$0.367678\pi$
$884$	0	0
$885$	18.0000	0.605063
$886$	0	0
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	0	0
$889$	36.0000	1.20740
$890$	0	0
$891$	18.0000	0.603023
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−24.0000	−0.802232
$896$	0	0
$897$	−45.0000	−1.50251
$898$	0	0
$899$	−18.0000	−0.600334
$900$	0	0
$901$	−3.00000	−0.0999445
$902$	0	0
$903$	90.0000	2.99501
$904$	0	0
$905$	−36.0000	−1.19668
$906$	0	0
$907$	−15.0000	−0.498067	−0.249033	−	0.968495i	$-0.580113\pi$
$907$	−15.0000	−0.498067	−0.249033	+	0.968495i	$0.580113\pi$
$908$	0	0
$909$	60.0000	1.99007
$910$	0	0
$911$	−30.0000	−0.993944	−0.496972	−	0.867766i	$-0.665555\pi$
$911$	−30.0000	−0.993944	−0.496972	+	0.867766i	$0.665555\pi$
$912$	0	0
$913$	−4.00000	−0.132381
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−42.0000	−1.38696
$918$	0	0
$919$	−15.0000	−0.494804	−0.247402	−	0.968913i	$-0.579577\pi$
$919$	−15.0000	−0.494804	−0.247402	+	0.968913i	$0.579577\pi$
$920$	0	0
$921$	36.0000	1.18624
$922$	0	0
$923$	0	0
$924$	0	0
$925$	6.00000	0.197279
$926$	0	0
$927$	36.0000	1.18240
$928$	0	0
$929$	41.0000	1.34517	0.672583	−	0.740022i	$-0.265185\pi$
$929$	41.0000	1.34517	0.672583	+	0.740022i	$0.265185\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	33.0000	1.08037
$934$	0	0
$935$	−4.00000	−0.130814
$936$	0	0
$937$	−47.0000	−1.53542	−0.767712	−	0.640796i	$-0.778605\pi$
$937$	−47.0000	−1.53542	−0.767712	+	0.640796i	$0.778605\pi$
$938$	0	0
$939$	63.0000	2.05593
$940$	0	0
$941$	27.0000	0.880175	0.440087	−	0.897955i	$-0.354947\pi$
$941$	27.0000	0.880175	0.440087	+	0.897955i	$0.354947\pi$
$942$	0	0
$943$	60.0000	1.95387
$944$	0	0
$945$	54.0000	1.75662
$946$	0	0
$947$	46.0000	1.49480	0.747400	−	0.664375i	$-0.231302\pi$
$947$	46.0000	1.49480	0.747400	+	0.664375i	$0.231302\pi$
$948$	0	0
$949$	−33.0000	−1.07123
$950$	0	0
$951$	99.0000	3.21029
$952$	0	0
$953$	30.0000	0.971795	0.485898	−	0.874016i	$-0.338493\pi$
$953$	30.0000	0.971795	0.485898	+	0.874016i	$0.338493\pi$
$954$	0	0
$955$	22.0000	0.711903
$956$	0	0
$957$	18.0000	0.581857
$958$	0	0
$959$	57.0000	1.84063
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	−18.0000	−0.580042
$964$	0	0
$965$	−12.0000	−0.386294
$966$	0	0
$967$	24.0000	0.771788	0.385894	−	0.922543i	$-0.373893\pi$
$967$	24.0000	0.771788	0.385894	+	0.922543i	$0.373893\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−48.0000	−1.54039	−0.770197	−	0.637806i	$-0.779842\pi$
$971$	−48.0000	−1.54039	−0.770197	+	0.637806i	$0.779842\pi$
$972$	0	0
$973$	−18.0000	−0.577054
$974$	0	0
$975$	−9.00000	−0.288231
$976$	0	0
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	0	0
$979$	−12.0000	−0.383522
$980$	0	0
$981$	−18.0000	−0.574696
$982$	0	0
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	0	0
$985$	8.00000	0.254901
$986$	0	0
$987$	72.0000	2.29179
$988$	0	0
$989$	−50.0000	−1.58991
$990$	0	0
$991$	6.00000	0.190596	0.0952981	−	0.995449i	$-0.469620\pi$
$991$	6.00000	0.190596	0.0952981	+	0.995449i	$0.469620\pi$
$992$	0	0
$993$	27.0000	0.856819
$994$	0	0
$995$	14.0000	0.443830
$996$	0	0
$997$	6.00000	0.190022	0.0950110	−	0.995476i	$-0.469711\pi$
$997$	6.00000	0.190022	0.0950110	+	0.995476i	$0.469711\pi$
$998$	0	0
$999$	−54.0000	−1.70848

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5776.2.a.q.1.1		1
4.3	odd	2		722.2.a.a.1.1	✓	1
12.11	even	2		6498.2.a.m.1.1		1
19.18	odd	2		5776.2.a.a.1.1		1
76.3	even	18		722.2.e.g.389.1		6
76.7	odd	6		722.2.c.g.429.1		2
76.11	odd	6		722.2.c.g.653.1		2
76.15	even	18		722.2.e.g.415.1		6
76.23	odd	18		722.2.e.h.415.1		6
76.27	even	6		722.2.c.a.653.1		2
76.31	even	6		722.2.c.a.429.1		2
76.35	odd	18		722.2.e.h.389.1		6
76.43	odd	18		722.2.e.h.595.1		6
76.47	odd	18		722.2.e.h.423.1		6
76.51	even	18		722.2.e.g.245.1		6
76.55	odd	18		722.2.e.h.99.1		6
76.59	even	18		722.2.e.g.99.1		6
76.63	odd	18		722.2.e.h.245.1		6
76.67	even	18		722.2.e.g.423.1		6
76.71	even	18		722.2.e.g.595.1		6
76.75	even	2		722.2.a.f.1.1	yes	1
228.227	odd	2		6498.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
722.2.a.a.1.1	✓	1	4.3	odd	2
722.2.a.f.1.1	yes	1	76.75	even	2
722.2.c.a.429.1		2	76.31	even	6
722.2.c.a.653.1		2	76.27	even	6
722.2.c.g.429.1		2	76.7	odd	6
722.2.c.g.653.1		2	76.11	odd	6
722.2.e.g.99.1		6	76.59	even	18
722.2.e.g.245.1		6	76.51	even	18
722.2.e.g.389.1		6	76.3	even	18
722.2.e.g.415.1		6	76.15	even	18
722.2.e.g.423.1		6	76.67	even	18
722.2.e.g.595.1		6	76.71	even	18
722.2.e.h.99.1		6	76.55	odd	18
722.2.e.h.245.1		6	76.63	odd	18
722.2.e.h.389.1		6	76.35	odd	18
722.2.e.h.415.1		6	76.23	odd	18
722.2.e.h.423.1		6	76.47	odd	18
722.2.e.h.595.1		6	76.43	odd	18
5776.2.a.a.1.1		1	19.18	odd	2
5776.2.a.q.1.1		1	1.1	even	1	trivial
6498.2.a.a.1.1		1	228.227	odd	2
6498.2.a.m.1.1		1	12.11	even	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:	\( N \)	\(=\)	\( 5776 = 2^{4} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5776.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more