LMFDB - Embedded newform 3380.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3380.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$26.9894358832$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 260)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-3.00000 q^{3} +1.00000 q^{5} +3.00000 q^{7} +6.00000 q^{9} +O(q^{10})$

$q-3.00000 q^{3} +1.00000 q^{5} +3.00000 q^{7} +6.00000 q^{9} +3.00000 q^{11} -3.00000 q^{15} -7.00000 q^{17} +1.00000 q^{19} -9.00000 q^{21} -7.00000 q^{23} +1.00000 q^{25} -9.00000 q^{27} -5.00000 q^{29} -4.00000 q^{31} -9.00000 q^{33} +3.00000 q^{35} -3.00000 q^{37} +7.00000 q^{41} -9.00000 q^{43} +6.00000 q^{45} +8.00000 q^{47} +2.00000 q^{49} +21.0000 q^{51} -6.00000 q^{53} +3.00000 q^{55} -3.00000 q^{57} +5.00000 q^{59} -5.00000 q^{61} +18.0000 q^{63} +13.0000 q^{67} +21.0000 q^{69} -3.00000 q^{71} -14.0000 q^{73} -3.00000 q^{75} +9.00000 q^{77} -8.00000 q^{79} +9.00000 q^{81} +12.0000 q^{83} -7.00000 q^{85} +15.0000 q^{87} +7.00000 q^{89} +12.0000 q^{93} +1.00000 q^{95} -11.0000 q^{97} +18.0000 q^{99} +O(q^{100})$

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	$-0.833333\pi$
$3$	−3.00000	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	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
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−3.00000	−0.774597
$16$	0	0
$17$	−7.00000	−1.69775	−0.848875	−	0.528594i	$-0.822719\pi$
$17$	−7.00000	−1.69775	−0.848875	+	0.528594i	$0.822719\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	−9.00000	−1.96396
$22$	0	0
$23$	−7.00000	−1.45960	−0.729800	−	0.683660i	$-0.760387\pi$
$23$	−7.00000	−1.45960	−0.729800	+	0.683660i	$0.760387\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−9.00000	−1.73205
$28$	0	0
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	−9.00000	−1.56670
$34$	0	0
$35$	3.00000	0.507093
$36$	0	0
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.00000	1.09322	0.546608	−	0.837389i	$-0.315919\pi$
$41$	7.00000	1.09322	0.546608	+	0.837389i	$0.315919\pi$
$42$	0	0
$43$	−9.00000	−1.37249	−0.686244	−	0.727372i	$-0.740742\pi$
$43$	−9.00000	−1.37249	−0.686244	+	0.727372i	$0.740742\pi$
$44$	0	0
$45$	6.00000	0.894427
$46$	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
$51$	21.0000	2.94059
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	3.00000	0.404520
$56$	0	0
$57$	−3.00000	−0.397360
$58$	0	0
$59$	5.00000	0.650945	0.325472	−	0.945552i	$-0.394477\pi$
$59$	5.00000	0.650945	0.325472	+	0.945552i	$0.394477\pi$
$60$	0	0
$61$	−5.00000	−0.640184	−0.320092	−	0.947386i	$-0.603714\pi$
$61$	−5.00000	−0.640184	−0.320092	+	0.947386i	$0.603714\pi$
$62$	0	0
$63$	18.0000	2.26779
$64$	0	0
$65$	0	0
$66$	0	0
$67$	13.0000	1.58820	0.794101	−	0.607785i	$-0.207942\pi$
$67$	13.0000	1.58820	0.794101	+	0.607785i	$0.207942\pi$
$68$	0	0
$69$	21.0000	2.52810
$70$	0	0
$71$	−3.00000	−0.356034	−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000	−0.356034	−0.178017	+	0.984027i	$0.556968\pi$
$72$	0	0
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	0	0
$75$	−3.00000	−0.346410
$76$	0	0
$77$	9.00000	1.02565
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	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
$85$	−7.00000	−0.759257
$86$	0	0
$87$	15.0000	1.60817
$88$	0	0
$89$	7.00000	0.741999	0.370999	−	0.928633i	$-0.379015\pi$
$89$	7.00000	0.741999	0.370999	+	0.928633i	$0.379015\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	12.0000	1.24434
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	−11.0000	−1.11688	−0.558440	−	0.829545i	$-0.688600\pi$
$97$	−11.0000	−1.11688	−0.558440	+	0.829545i	$0.688600\pi$
$98$	0	0
$99$	18.0000	1.80907
$100$	0	0
$101$	−9.00000	−0.895533	−0.447767	−	0.894150i	$-0.647781\pi$
$101$	−9.00000	−0.895533	−0.447767	+	0.894150i	$0.647781\pi$
$102$	0	0
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	0	0
$105$	−9.00000	−0.878310
$106$	0	0
$107$	−3.00000	−0.290021	−0.145010	−	0.989430i	$-0.546322\pi$
$107$	−3.00000	−0.290021	−0.145010	+	0.989430i	$0.546322\pi$
$108$	0	0
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	9.00000	0.854242
$112$	0	0
$113$	13.0000	1.22294	0.611469	−	0.791269i	$-0.290579\pi$
$113$	13.0000	1.22294	0.611469	+	0.791269i	$0.290579\pi$
$114$	0	0
$115$	−7.00000	−0.652753
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−21.0000	−1.92507
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	−21.0000	−1.89351
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	1.00000	0.0887357	0.0443678	−	0.999015i	$-0.485873\pi$
$127$	1.00000	0.0887357	0.0443678	+	0.999015i	$0.485873\pi$
$128$	0	0
$129$	27.0000	2.37722
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	3.00000	0.260133
$134$	0	0
$135$	−9.00000	−0.774597
$136$	0	0
$137$	−3.00000	−0.256307	−0.128154	−	0.991754i	$-0.540905\pi$
$137$	−3.00000	−0.256307	−0.128154	+	0.991754i	$0.540905\pi$
$138$	0	0
$139$	13.0000	1.10265	0.551323	−	0.834292i	$-0.314123\pi$
$139$	13.0000	1.10265	0.551323	+	0.834292i	$0.314123\pi$
$140$	0	0
$141$	−24.0000	−2.02116
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−5.00000	−0.415227
$146$	0	0
$147$	−6.00000	−0.494872
$148$	0	0
$149$	3.00000	0.245770	0.122885	−	0.992421i	$-0.460785\pi$
$149$	3.00000	0.245770	0.122885	+	0.992421i	$0.460785\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	−42.0000	−3.39550
$154$	0	0
$155$	−4.00000	−0.321288
$156$	0	0
$157$	6.00000	0.478852	0.239426	−	0.970915i	$-0.423041\pi$
$157$	6.00000	0.478852	0.239426	+	0.970915i	$0.423041\pi$
$158$	0	0
$159$	18.0000	1.42749
$160$	0	0
$161$	−21.0000	−1.65503
$162$	0	0
$163$	11.0000	0.861586	0.430793	−	0.902451i	$-0.358234\pi$
$163$	11.0000	0.861586	0.430793	+	0.902451i	$0.358234\pi$
$164$	0	0
$165$	−9.00000	−0.700649
$166$	0	0
$167$	1.00000	0.0773823	0.0386912	−	0.999251i	$-0.487681\pi$
$167$	1.00000	0.0773823	0.0386912	+	0.999251i	$0.487681\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	6.00000	0.458831
$172$	0	0
$173$	−15.0000	−1.14043	−0.570214	−	0.821496i	$-0.693140\pi$
$173$	−15.0000	−1.14043	−0.570214	+	0.821496i	$0.693140\pi$
$174$	0	0
$175$	3.00000	0.226779
$176$	0	0
$177$	−15.0000	−1.12747
$178$	0	0
$179$	19.0000	1.42013	0.710063	−	0.704138i	$-0.248666\pi$
$179$	19.0000	1.42013	0.710063	+	0.704138i	$0.248666\pi$
$180$	0	0
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	0	0
$183$	15.0000	1.10883
$184$	0	0
$185$	−3.00000	−0.220564
$186$	0	0
$187$	−21.0000	−1.53567
$188$	0	0
$189$	−27.0000	−1.96396
$190$	0	0
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	0	0
$193$	−15.0000	−1.07972	−0.539862	−	0.841754i	$-0.681524\pi$
$193$	−15.0000	−1.07972	−0.539862	+	0.841754i	$0.681524\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−23.0000	−1.63868	−0.819341	−	0.573306i	$-0.805660\pi$
$197$	−23.0000	−1.63868	−0.819341	+	0.573306i	$0.805660\pi$
$198$	0	0
$199$	9.00000	0.637993	0.318997	−	0.947756i	$-0.396654\pi$
$199$	9.00000	0.637993	0.318997	+	0.947756i	$0.396654\pi$
$200$	0	0
$201$	−39.0000	−2.75085
$202$	0	0
$203$	−15.0000	−1.05279
$204$	0	0
$205$	7.00000	0.488901
$206$	0	0
$207$	−42.0000	−2.91920
$208$	0	0
$209$	3.00000	0.207514
$210$	0	0
$211$	−5.00000	−0.344214	−0.172107	−	0.985078i	$-0.555058\pi$
$211$	−5.00000	−0.344214	−0.172107	+	0.985078i	$0.555058\pi$
$212$	0	0
$213$	9.00000	0.616670
$214$	0	0
$215$	−9.00000	−0.613795
$216$	0	0
$217$	−12.0000	−0.814613
$218$	0	0
$219$	42.0000	2.83810
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−23.0000	−1.54019	−0.770097	−	0.637927i	$-0.779792\pi$
$223$	−23.0000	−1.54019	−0.770097	+	0.637927i	$0.779792\pi$
$224$	0	0
$225$	6.00000	0.400000
$226$	0	0
$227$	−1.00000	−0.0663723	−0.0331862	−	0.999449i	$-0.510565\pi$
$227$	−1.00000	−0.0663723	−0.0331862	+	0.999449i	$0.510565\pi$
$228$	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
$231$	−27.0000	−1.77647
$232$	0	0
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	8.00000	0.521862
$236$	0	0
$237$	24.0000	1.55897
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	−1.00000	−0.0644157	−0.0322078	−	0.999481i	$-0.510254\pi$
$241$	−1.00000	−0.0644157	−0.0322078	+	0.999481i	$0.510254\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.00000	0.127775
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−36.0000	−2.28141
$250$	0	0
$251$	5.00000	0.315597	0.157799	−	0.987471i	$-0.449560\pi$
$251$	5.00000	0.315597	0.157799	+	0.987471i	$0.449560\pi$
$252$	0	0
$253$	−21.0000	−1.32026
$254$	0	0
$255$	21.0000	1.31507
$256$	0	0
$257$	−19.0000	−1.18519	−0.592594	−	0.805502i	$-0.701896\pi$
$257$	−19.0000	−1.18519	−0.592594	+	0.805502i	$0.701896\pi$
$258$	0	0
$259$	−9.00000	−0.559233
$260$	0	0
$261$	−30.0000	−1.85695
$262$	0	0
$263$	−7.00000	−0.431638	−0.215819	−	0.976433i	$-0.569242\pi$
$263$	−7.00000	−0.431638	−0.215819	+	0.976433i	$0.569242\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	0	0
$267$	−21.0000	−1.28518
$268$	0	0
$269$	3.00000	0.182913	0.0914566	−	0.995809i	$-0.470848\pi$
$269$	3.00000	0.182913	0.0914566	+	0.995809i	$0.470848\pi$
$270$	0	0
$271$	23.0000	1.39715	0.698575	−	0.715537i	$-0.253818\pi$
$271$	23.0000	1.39715	0.698575	+	0.715537i	$0.253818\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.00000	0.180907
$276$	0	0
$277$	−23.0000	−1.38194	−0.690968	−	0.722885i	$-0.742815\pi$
$277$	−23.0000	−1.38194	−0.690968	+	0.722885i	$0.742815\pi$
$278$	0	0
$279$	−24.0000	−1.43684
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	1.00000	0.0594438	0.0297219	−	0.999558i	$-0.490538\pi$
$283$	1.00000	0.0594438	0.0297219	+	0.999558i	$0.490538\pi$
$284$	0	0
$285$	−3.00000	−0.177705
$286$	0	0
$287$	21.0000	1.23959
$288$	0	0
$289$	32.0000	1.88235
$290$	0	0
$291$	33.0000	1.93449
$292$	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
$295$	5.00000	0.291111
$296$	0	0
$297$	−27.0000	−1.56670
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−27.0000	−1.55625
$302$	0	0
$303$	27.0000	1.55111
$304$	0	0
$305$	−5.00000	−0.286299
$306$	0	0
$307$	28.0000	1.59804	0.799022	−	0.601302i	$-0.205351\pi$
$307$	28.0000	1.59804	0.799022	+	0.601302i	$0.205351\pi$
$308$	0	0
$309$	48.0000	2.73062
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	0	0
$315$	18.0000	1.01419
$316$	0	0
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	0	0
$319$	−15.0000	−0.839839
$320$	0	0
$321$	9.00000	0.502331
$322$	0	0
$323$	−7.00000	−0.389490
$324$	0	0
$325$	0	0
$326$	0	0
$327$	42.0000	2.32261
$328$	0	0
$329$	24.0000	1.32316
$330$	0	0
$331$	13.0000	0.714545	0.357272	−	0.934000i	$-0.383707\pi$
$331$	13.0000	0.714545	0.357272	+	0.934000i	$0.383707\pi$
$332$	0	0
$333$	−18.0000	−0.986394
$334$	0	0
$335$	13.0000	0.710266
$336$	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
$339$	−39.0000	−2.11819
$340$	0	0
$341$	−12.0000	−0.649836
$342$	0	0
$343$	−15.0000	−0.809924
$344$	0	0
$345$	21.0000	1.13060
$346$	0	0
$347$	−13.0000	−0.697877	−0.348938	−	0.937146i	$-0.613458\pi$
$347$	−13.0000	−0.697877	−0.348938	+	0.937146i	$0.613458\pi$
$348$	0	0
$349$	−25.0000	−1.33822	−0.669110	−	0.743164i	$-0.733324\pi$
$349$	−25.0000	−1.33822	−0.669110	+	0.743164i	$0.733324\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	21.0000	1.11772	0.558859	−	0.829263i	$-0.311239\pi$
$353$	21.0000	1.11772	0.558859	+	0.829263i	$0.311239\pi$
$354$	0	0
$355$	−3.00000	−0.159223
$356$	0	0
$357$	63.0000	3.33431
$358$	0	0
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	6.00000	0.314918
$364$	0	0
$365$	−14.0000	−0.732793
$366$	0	0
$367$	9.00000	0.469796	0.234898	−	0.972020i	$-0.424524\pi$
$367$	9.00000	0.469796	0.234898	+	0.972020i	$0.424524\pi$
$368$	0	0
$369$	42.0000	2.18643
$370$	0	0
$371$	−18.0000	−0.934513
$372$	0	0
$373$	−27.0000	−1.39801	−0.699004	−	0.715118i	$-0.746373\pi$
$373$	−27.0000	−1.39801	−0.699004	+	0.715118i	$0.746373\pi$
$374$	0	0
$375$	−3.00000	−0.154919
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−9.00000	−0.462299	−0.231149	−	0.972918i	$-0.574249\pi$
$379$	−9.00000	−0.462299	−0.231149	+	0.972918i	$0.574249\pi$
$380$	0	0
$381$	−3.00000	−0.153695
$382$	0	0
$383$	−13.0000	−0.664269	−0.332134	−	0.943232i	$-0.607769\pi$
$383$	−13.0000	−0.664269	−0.332134	+	0.943232i	$0.607769\pi$
$384$	0	0
$385$	9.00000	0.458682
$386$	0	0
$387$	−54.0000	−2.74497
$388$	0	0
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	49.0000	2.47804
$392$	0	0
$393$	−12.0000	−0.605320
$394$	0	0
$395$	−8.00000	−0.402524
$396$	0	0
$397$	33.0000	1.65622	0.828111	−	0.560564i	$-0.189416\pi$
$397$	33.0000	1.65622	0.828111	+	0.560564i	$0.189416\pi$
$398$	0	0
$399$	−9.00000	−0.450564
$400$	0	0
$401$	15.0000	0.749064	0.374532	−	0.927214i	$-0.377803\pi$
$401$	15.0000	0.749064	0.374532	+	0.927214i	$0.377803\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	9.00000	0.447214
$406$	0	0
$407$	−9.00000	−0.446113
$408$	0	0
$409$	−1.00000	−0.0494468	−0.0247234	−	0.999694i	$-0.507871\pi$
$409$	−1.00000	−0.0494468	−0.0247234	+	0.999694i	$0.507871\pi$
$410$	0	0
$411$	9.00000	0.443937
$412$	0	0
$413$	15.0000	0.738102
$414$	0	0
$415$	12.0000	0.589057
$416$	0	0
$417$	−39.0000	−1.90984
$418$	0	0
$419$	−33.0000	−1.61216	−0.806078	−	0.591810i	$-0.798414\pi$
$419$	−33.0000	−1.61216	−0.806078	+	0.591810i	$0.798414\pi$
$420$	0	0
$421$	34.0000	1.65706	0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000	1.65706	0.828529	+	0.559946i	$0.189178\pi$
$422$	0	0
$423$	48.0000	2.33384
$424$	0	0
$425$	−7.00000	−0.339550
$426$	0	0
$427$	−15.0000	−0.725901
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−9.00000	−0.433515	−0.216757	−	0.976226i	$-0.569548\pi$
$431$	−9.00000	−0.433515	−0.216757	+	0.976226i	$0.569548\pi$
$432$	0	0
$433$	1.00000	0.0480569	0.0240285	−	0.999711i	$-0.492351\pi$
$433$	1.00000	0.0480569	0.0240285	+	0.999711i	$0.492351\pi$
$434$	0	0
$435$	15.0000	0.719195
$436$	0	0
$437$	−7.00000	−0.334855
$438$	0	0
$439$	3.00000	0.143182	0.0715911	−	0.997434i	$-0.477192\pi$
$439$	3.00000	0.143182	0.0715911	+	0.997434i	$0.477192\pi$
$440$	0	0
$441$	12.0000	0.571429
$442$	0	0
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	0	0
$445$	7.00000	0.331832
$446$	0	0
$447$	−9.00000	−0.425685
$448$	0	0
$449$	−21.0000	−0.991051	−0.495526	−	0.868593i	$-0.665025\pi$
$449$	−21.0000	−0.991051	−0.495526	+	0.868593i	$0.665025\pi$
$450$	0	0
$451$	21.0000	0.988851
$452$	0	0
$453$	24.0000	1.12762
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−23.0000	−1.07589	−0.537947	−	0.842978i	$-0.680800\pi$
$457$	−23.0000	−1.07589	−0.537947	+	0.842978i	$0.680800\pi$
$458$	0	0
$459$	63.0000	2.94059
$460$	0	0
$461$	11.0000	0.512321	0.256161	−	0.966634i	$-0.417542\pi$
$461$	11.0000	0.512321	0.256161	+	0.966634i	$0.417542\pi$
$462$	0	0
$463$	−28.0000	−1.30127	−0.650635	−	0.759390i	$-0.725497\pi$
$463$	−28.0000	−1.30127	−0.650635	+	0.759390i	$0.725497\pi$
$464$	0	0
$465$	12.0000	0.556487
$466$	0	0
$467$	20.0000	0.925490	0.462745	−	0.886492i	$-0.346865\pi$
$467$	20.0000	0.925490	0.462745	+	0.886492i	$0.346865\pi$
$468$	0	0
$469$	39.0000	1.80085
$470$	0	0
$471$	−18.0000	−0.829396
$472$	0	0
$473$	−27.0000	−1.24146
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	−36.0000	−1.64833
$478$	0	0
$479$	−1.00000	−0.0456912	−0.0228456	−	0.999739i	$-0.507273\pi$
$479$	−1.00000	−0.0456912	−0.0228456	+	0.999739i	$0.507273\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	63.0000	2.86660
$484$	0	0
$485$	−11.0000	−0.499484
$486$	0	0
$487$	−17.0000	−0.770344	−0.385172	−	0.922845i	$-0.625858\pi$
$487$	−17.0000	−0.770344	−0.385172	+	0.922845i	$0.625858\pi$
$488$	0	0
$489$	−33.0000	−1.49231
$490$	0	0
$491$	23.0000	1.03798	0.518988	−	0.854782i	$-0.326309\pi$
$491$	23.0000	1.03798	0.518988	+	0.854782i	$0.326309\pi$
$492$	0	0
$493$	35.0000	1.57632
$494$	0	0
$495$	18.0000	0.809040
$496$	0	0
$497$	−9.00000	−0.403705
$498$	0	0
$499$	−16.0000	−0.716258	−0.358129	−	0.933672i	$-0.616585\pi$
$499$	−16.0000	−0.716258	−0.358129	+	0.933672i	$0.616585\pi$
$500$	0	0
$501$	−3.00000	−0.134030
$502$	0	0
$503$	11.0000	0.490466	0.245233	−	0.969464i	$-0.421136\pi$
$503$	11.0000	0.490466	0.245233	+	0.969464i	$0.421136\pi$
$504$	0	0
$505$	−9.00000	−0.400495
$506$	0	0
$507$	0	0
$508$	0	0
$509$	15.0000	0.664863	0.332432	−	0.943127i	$-0.392131\pi$
$509$	15.0000	0.664863	0.332432	+	0.943127i	$0.392131\pi$
$510$	0	0
$511$	−42.0000	−1.85797
$512$	0	0
$513$	−9.00000	−0.397360
$514$	0	0
$515$	−16.0000	−0.705044
$516$	0	0
$517$	24.0000	1.05552
$518$	0	0
$519$	45.0000	1.97528
$520$	0	0
$521$	−34.0000	−1.48957	−0.744784	−	0.667306i	$-0.767447\pi$
$521$	−34.0000	−1.48957	−0.744784	+	0.667306i	$0.767447\pi$
$522$	0	0
$523$	−23.0000	−1.00572	−0.502860	−	0.864368i	$-0.667719\pi$
$523$	−23.0000	−1.00572	−0.502860	+	0.864368i	$0.667719\pi$
$524$	0	0
$525$	−9.00000	−0.392792
$526$	0	0
$527$	28.0000	1.21970
$528$	0	0
$529$	26.0000	1.13043
$530$	0	0
$531$	30.0000	1.30189
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−3.00000	−0.129701
$536$	0	0
$537$	−57.0000	−2.45973
$538$	0	0
$539$	6.00000	0.258438
$540$	0	0
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	0	0
$543$	42.0000	1.80239
$544$	0	0
$545$	−14.0000	−0.599694
$546$	0	0
$547$	−16.0000	−0.684111	−0.342055	−	0.939680i	$-0.611123\pi$
$547$	−16.0000	−0.684111	−0.342055	+	0.939680i	$0.611123\pi$
$548$	0	0
$549$	−30.0000	−1.28037
$550$	0	0
$551$	−5.00000	−0.213007
$552$	0	0
$553$	−24.0000	−1.02058
$554$	0	0
$555$	9.00000	0.382029
$556$	0	0
$557$	−19.0000	−0.805056	−0.402528	−	0.915408i	$-0.631868\pi$
$557$	−19.0000	−0.805056	−0.402528	+	0.915408i	$0.631868\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	63.0000	2.65986
$562$	0	0
$563$	−45.0000	−1.89652	−0.948262	−	0.317489i	$-0.897160\pi$
$563$	−45.0000	−1.89652	−0.948262	+	0.317489i	$0.897160\pi$
$564$	0	0
$565$	13.0000	0.546914
$566$	0	0
$567$	27.0000	1.13389
$568$	0	0
$569$	19.0000	0.796521	0.398261	−	0.917272i	$-0.369614\pi$
$569$	19.0000	0.796521	0.398261	+	0.917272i	$0.369614\pi$
$570$	0	0
$571$	−40.0000	−1.67395	−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000	−1.67395	−0.836974	+	0.547243i	$0.815677\pi$
$572$	0	0
$573$	9.00000	0.375980
$574$	0	0
$575$	−7.00000	−0.291920
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	0	0
$579$	45.0000	1.87014
$580$	0	0
$581$	36.0000	1.49353
$582$	0	0
$583$	−18.0000	−0.745484
$584$	0	0
$585$	0	0
$586$	0	0
$587$	37.0000	1.52715	0.763577	−	0.645717i	$-0.223441\pi$
$587$	37.0000	1.52715	0.763577	+	0.645717i	$0.223441\pi$
$588$	0	0
$589$	−4.00000	−0.164817
$590$	0	0
$591$	69.0000	2.83828
$592$	0	0
$593$	26.0000	1.06769	0.533846	−	0.845582i	$-0.320746\pi$
$593$	26.0000	1.06769	0.533846	+	0.845582i	$0.320746\pi$
$594$	0	0
$595$	−21.0000	−0.860916
$596$	0	0
$597$	−27.0000	−1.10504
$598$	0	0
$599$	8.00000	0.326871	0.163436	−	0.986554i	$-0.447742\pi$
$599$	8.00000	0.326871	0.163436	+	0.986554i	$0.447742\pi$
$600$	0	0
$601$	−13.0000	−0.530281	−0.265141	−	0.964210i	$-0.585418\pi$
$601$	−13.0000	−0.530281	−0.265141	+	0.964210i	$0.585418\pi$
$602$	0	0
$603$	78.0000	3.17641
$604$	0	0
$605$	−2.00000	−0.0813116
$606$	0	0
$607$	−9.00000	−0.365299	−0.182649	−	0.983178i	$-0.558467\pi$
$607$	−9.00000	−0.365299	−0.182649	+	0.983178i	$0.558467\pi$
$608$	0	0
$609$	45.0000	1.82349
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−31.0000	−1.25208	−0.626039	−	0.779792i	$-0.715325\pi$
$613$	−31.0000	−1.25208	−0.626039	+	0.779792i	$0.715325\pi$
$614$	0	0
$615$	−21.0000	−0.846802
$616$	0	0
$617$	29.0000	1.16750	0.583748	−	0.811935i	$-0.301586\pi$
$617$	29.0000	1.16750	0.583748	+	0.811935i	$0.301586\pi$
$618$	0	0
$619$	12.0000	0.482321	0.241160	−	0.970485i	$-0.422472\pi$
$619$	12.0000	0.482321	0.241160	+	0.970485i	$0.422472\pi$
$620$	0	0
$621$	63.0000	2.52810
$622$	0	0
$623$	21.0000	0.841347
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−9.00000	−0.359425
$628$	0	0
$629$	21.0000	0.837325
$630$	0	0
$631$	−15.0000	−0.597141	−0.298570	−	0.954388i	$-0.596510\pi$
$631$	−15.0000	−0.597141	−0.298570	+	0.954388i	$0.596510\pi$
$632$	0	0
$633$	15.0000	0.596196
$634$	0	0
$635$	1.00000	0.0396838
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−18.0000	−0.712069
$640$	0	0
$641$	−21.0000	−0.829450	−0.414725	−	0.909947i	$-0.636122\pi$
$641$	−21.0000	−0.829450	−0.414725	+	0.909947i	$0.636122\pi$
$642$	0	0
$643$	7.00000	0.276053	0.138027	−	0.990429i	$-0.455924\pi$
$643$	7.00000	0.276053	0.138027	+	0.990429i	$0.455924\pi$
$644$	0	0
$645$	27.0000	1.06312
$646$	0	0
$647$	17.0000	0.668339	0.334169	−	0.942513i	$-0.391544\pi$
$647$	17.0000	0.668339	0.334169	+	0.942513i	$0.391544\pi$
$648$	0	0
$649$	15.0000	0.588802
$650$	0	0
$651$	36.0000	1.41095
$652$	0	0
$653$	−11.0000	−0.430463	−0.215232	−	0.976563i	$-0.569051\pi$
$653$	−11.0000	−0.430463	−0.215232	+	0.976563i	$0.569051\pi$
$654$	0	0
$655$	4.00000	0.156293
$656$	0	0
$657$	−84.0000	−3.27715
$658$	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
$661$	−17.0000	−0.661223	−0.330612	−	0.943767i	$-0.607255\pi$
$661$	−17.0000	−0.661223	−0.330612	+	0.943767i	$0.607255\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.00000	0.116335
$666$	0	0
$667$	35.0000	1.35521
$668$	0	0
$669$	69.0000	2.66769
$670$	0	0
$671$	−15.0000	−0.579069
$672$	0	0
$673$	13.0000	0.501113	0.250557	−	0.968102i	$-0.419386\pi$
$673$	13.0000	0.501113	0.250557	+	0.968102i	$0.419386\pi$
$674$	0	0
$675$	−9.00000	−0.346410
$676$	0	0
$677$	22.0000	0.845529	0.422764	−	0.906240i	$-0.361060\pi$
$677$	22.0000	0.845529	0.422764	+	0.906240i	$0.361060\pi$
$678$	0	0
$679$	−33.0000	−1.26642
$680$	0	0
$681$	3.00000	0.114960
$682$	0	0
$683$	31.0000	1.18618	0.593091	−	0.805135i	$-0.297907\pi$
$683$	31.0000	1.18618	0.593091	+	0.805135i	$0.297907\pi$
$684$	0	0
$685$	−3.00000	−0.114624
$686$	0	0
$687$	−78.0000	−2.97589
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−25.0000	−0.951045	−0.475522	−	0.879704i	$-0.657741\pi$
$691$	−25.0000	−0.951045	−0.475522	+	0.879704i	$0.657741\pi$
$692$	0	0
$693$	54.0000	2.05129
$694$	0	0
$695$	13.0000	0.493118
$696$	0	0
$697$	−49.0000	−1.85601
$698$	0	0
$699$	−54.0000	−2.04247
$700$	0	0
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	0	0
$703$	−3.00000	−0.113147
$704$	0	0
$705$	−24.0000	−0.903892
$706$	0	0
$707$	−27.0000	−1.01544
$708$	0	0
$709$	35.0000	1.31445	0.657226	−	0.753693i	$-0.271730\pi$
$709$	35.0000	1.31445	0.657226	+	0.753693i	$0.271730\pi$
$710$	0	0
$711$	−48.0000	−1.80014
$712$	0	0
$713$	28.0000	1.04861
$714$	0	0
$715$	0	0
$716$	0	0
$717$	48.0000	1.79259
$718$	0	0
$719$	1.00000	0.0372937	0.0186469	−	0.999826i	$-0.494064\pi$
$719$	1.00000	0.0372937	0.0186469	+	0.999826i	$0.494064\pi$
$720$	0	0
$721$	−48.0000	−1.78761
$722$	0	0
$723$	3.00000	0.111571
$724$	0	0
$725$	−5.00000	−0.185695
$726$	0	0
$727$	−28.0000	−1.03846	−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000	−1.03846	−0.519231	+	0.854634i	$0.673782\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	63.0000	2.33014
$732$	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
$735$	−6.00000	−0.221313
$736$	0	0
$737$	39.0000	1.43658
$738$	0	0
$739$	39.0000	1.43464	0.717319	−	0.696745i	$-0.245369\pi$
$739$	39.0000	1.43464	0.717319	+	0.696745i	$0.245369\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.00000	0.0366864	0.0183432	−	0.999832i	$-0.494161\pi$
$743$	1.00000	0.0366864	0.0183432	+	0.999832i	$0.494161\pi$
$744$	0	0
$745$	3.00000	0.109911
$746$	0	0
$747$	72.0000	2.63434
$748$	0	0
$749$	−9.00000	−0.328853
$750$	0	0
$751$	−13.0000	−0.474377	−0.237188	−	0.971464i	$-0.576226\pi$
$751$	−13.0000	−0.474377	−0.237188	+	0.971464i	$0.576226\pi$
$752$	0	0
$753$	−15.0000	−0.546630
$754$	0	0
$755$	−8.00000	−0.291150
$756$	0	0
$757$	1.00000	0.0363456	0.0181728	−	0.999835i	$-0.494215\pi$
$757$	1.00000	0.0363456	0.0181728	+	0.999835i	$0.494215\pi$
$758$	0	0
$759$	63.0000	2.28676
$760$	0	0
$761$	51.0000	1.84875	0.924374	−	0.381487i	$-0.124588\pi$
$761$	51.0000	1.84875	0.924374	+	0.381487i	$0.124588\pi$
$762$	0	0
$763$	−42.0000	−1.52050
$764$	0	0
$765$	−42.0000	−1.51851
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−5.00000	−0.180305	−0.0901523	−	0.995928i	$-0.528735\pi$
$769$	−5.00000	−0.180305	−0.0901523	+	0.995928i	$0.528735\pi$
$770$	0	0
$771$	57.0000	2.05280
$772$	0	0
$773$	25.0000	0.899188	0.449594	−	0.893233i	$-0.351569\pi$
$773$	25.0000	0.899188	0.449594	+	0.893233i	$0.351569\pi$
$774$	0	0
$775$	−4.00000	−0.143684
$776$	0	0
$777$	27.0000	0.968620
$778$	0	0
$779$	7.00000	0.250801
$780$	0	0
$781$	−9.00000	−0.322045
$782$	0	0
$783$	45.0000	1.60817
$784$	0	0
$785$	6.00000	0.214149
$786$	0	0
$787$	−1.00000	−0.0356462	−0.0178231	−	0.999841i	$-0.505674\pi$
$787$	−1.00000	−0.0356462	−0.0178231	+	0.999841i	$0.505674\pi$
$788$	0	0
$789$	21.0000	0.747620
$790$	0	0
$791$	39.0000	1.38668
$792$	0	0
$793$	0	0
$794$	0	0
$795$	18.0000	0.638394
$796$	0	0
$797$	−7.00000	−0.247953	−0.123976	−	0.992285i	$-0.539565\pi$
$797$	−7.00000	−0.247953	−0.123976	+	0.992285i	$0.539565\pi$
$798$	0	0
$799$	−56.0000	−1.98114
$800$	0	0
$801$	42.0000	1.48400
$802$	0	0
$803$	−42.0000	−1.48215
$804$	0	0
$805$	−21.0000	−0.740153
$806$	0	0
$807$	−9.00000	−0.316815
$808$	0	0
$809$	−25.0000	−0.878953	−0.439477	−	0.898254i	$-0.644836\pi$
$809$	−25.0000	−0.878953	−0.439477	+	0.898254i	$0.644836\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	−69.0000	−2.41994
$814$	0	0
$815$	11.0000	0.385313
$816$	0	0
$817$	−9.00000	−0.314870
$818$	0	0
$819$	0	0
$820$	0	0
$821$	15.0000	0.523504	0.261752	−	0.965135i	$-0.415700\pi$
$821$	15.0000	0.523504	0.261752	+	0.965135i	$0.415700\pi$
$822$	0	0
$823$	47.0000	1.63832	0.819159	−	0.573567i	$-0.194441\pi$
$823$	47.0000	1.63832	0.819159	+	0.573567i	$0.194441\pi$
$824$	0	0
$825$	−9.00000	−0.313340
$826$	0	0
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	−13.0000	−0.451509	−0.225754	−	0.974184i	$-0.572485\pi$
$829$	−13.0000	−0.451509	−0.225754	+	0.974184i	$0.572485\pi$
$830$	0	0
$831$	69.0000	2.39358
$832$	0	0
$833$	−14.0000	−0.485071
$834$	0	0
$835$	1.00000	0.0346064
$836$	0	0
$837$	36.0000	1.24434
$838$	0	0
$839$	13.0000	0.448810	0.224405	−	0.974496i	$-0.427956\pi$
$839$	13.0000	0.448810	0.224405	+	0.974496i	$0.427956\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	0	0
$843$	30.0000	1.03325
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−6.00000	−0.206162
$848$	0	0
$849$	−3.00000	−0.102960
$850$	0	0
$851$	21.0000	0.719871
$852$	0	0
$853$	10.0000	0.342393	0.171197	−	0.985237i	$-0.445237\pi$
$853$	10.0000	0.342393	0.171197	+	0.985237i	$0.445237\pi$
$854$	0	0
$855$	6.00000	0.205196
$856$	0	0
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	0	0
$859$	44.0000	1.50126	0.750630	−	0.660722i	$-0.229750\pi$
$859$	44.0000	1.50126	0.750630	+	0.660722i	$0.229750\pi$
$860$	0	0
$861$	−63.0000	−2.14703
$862$	0	0
$863$	16.0000	0.544646	0.272323	−	0.962206i	$-0.412208\pi$
$863$	16.0000	0.544646	0.272323	+	0.962206i	$0.412208\pi$
$864$	0	0
$865$	−15.0000	−0.510015
$866$	0	0
$867$	−96.0000	−3.26033
$868$	0	0
$869$	−24.0000	−0.814144
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−66.0000	−2.23376
$874$	0	0
$875$	3.00000	0.101419
$876$	0	0
$877$	5.00000	0.168838	0.0844190	−	0.996430i	$-0.473097\pi$
$877$	5.00000	0.168838	0.0844190	+	0.996430i	$0.473097\pi$
$878$	0	0
$879$	−27.0000	−0.910687
$880$	0	0
$881$	11.0000	0.370599	0.185300	−	0.982682i	$-0.440674\pi$
$881$	11.0000	0.370599	0.185300	+	0.982682i	$0.440674\pi$
$882$	0	0
$883$	−44.0000	−1.48072	−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000	−1.48072	−0.740359	+	0.672212i	$0.765344\pi$
$884$	0	0
$885$	−15.0000	−0.504219
$886$	0	0
$887$	−19.0000	−0.637958	−0.318979	−	0.947762i	$-0.603340\pi$
$887$	−19.0000	−0.637958	−0.318979	+	0.947762i	$0.603340\pi$
$888$	0	0
$889$	3.00000	0.100617
$890$	0	0
$891$	27.0000	0.904534
$892$	0	0
$893$	8.00000	0.267710
$894$	0	0
$895$	19.0000	0.635100
$896$	0	0
$897$	0	0
$898$	0	0
$899$	20.0000	0.667037
$900$	0	0
$901$	42.0000	1.39922
$902$	0	0
$903$	81.0000	2.69551
$904$	0	0
$905$	−14.0000	−0.465376
$906$	0	0
$907$	17.0000	0.564476	0.282238	−	0.959344i	$-0.408923\pi$
$907$	17.0000	0.564476	0.282238	+	0.959344i	$0.408923\pi$
$908$	0	0
$909$	−54.0000	−1.79107
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	0	0
$913$	36.0000	1.19143
$914$	0	0
$915$	15.0000	0.495885
$916$	0	0
$917$	12.0000	0.396275
$918$	0	0
$919$	−43.0000	−1.41844	−0.709220	−	0.704988i	$-0.750953\pi$
$919$	−43.0000	−1.41844	−0.709220	+	0.704988i	$0.750953\pi$
$920$	0	0
$921$	−84.0000	−2.76789
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−3.00000	−0.0986394
$926$	0	0
$927$	−96.0000	−3.15305
$928$	0	0
$929$	−53.0000	−1.73887	−0.869437	−	0.494044i	$-0.835518\pi$
$929$	−53.0000	−1.73887	−0.869437	+	0.494044i	$0.835518\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	0	0
$933$	72.0000	2.35717
$934$	0	0
$935$	−21.0000	−0.686773
$936$	0	0
$937$	50.0000	1.63343	0.816714	−	0.577042i	$-0.195793\pi$
$937$	50.0000	1.63343	0.816714	+	0.577042i	$0.195793\pi$
$938$	0	0
$939$	18.0000	0.587408
$940$	0	0
$941$	42.0000	1.36916	0.684580	−	0.728937i	$-0.259985\pi$
$941$	42.0000	1.36916	0.684580	+	0.728937i	$0.259985\pi$
$942$	0	0
$943$	−49.0000	−1.59566
$944$	0	0
$945$	−27.0000	−0.878310
$946$	0	0
$947$	−55.0000	−1.78726	−0.893630	−	0.448805i	$-0.851850\pi$
$947$	−55.0000	−1.78726	−0.893630	+	0.448805i	$0.851850\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	6.00000	0.194563
$952$	0	0
$953$	−31.0000	−1.00419	−0.502094	−	0.864813i	$-0.667437\pi$
$953$	−31.0000	−1.00419	−0.502094	+	0.864813i	$0.667437\pi$
$954$	0	0
$955$	−3.00000	−0.0970777
$956$	0	0
$957$	45.0000	1.45464
$958$	0	0
$959$	−9.00000	−0.290625
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−18.0000	−0.580042
$964$	0	0
$965$	−15.0000	−0.482867
$966$	0	0
$967$	−48.0000	−1.54358	−0.771788	−	0.635880i	$-0.780637\pi$
$967$	−48.0000	−1.54358	−0.771788	+	0.635880i	$0.780637\pi$
$968$	0	0
$969$	21.0000	0.674617
$970$	0	0
$971$	−43.0000	−1.37994	−0.689968	−	0.723840i	$-0.742375\pi$
$971$	−43.0000	−1.37994	−0.689968	+	0.723840i	$0.742375\pi$
$972$	0	0
$973$	39.0000	1.25028
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−3.00000	−0.0959785	−0.0479893	−	0.998848i	$-0.515281\pi$
$977$	−3.00000	−0.0959785	−0.0479893	+	0.998848i	$0.515281\pi$
$978$	0	0
$979$	21.0000	0.671163
$980$	0	0
$981$	−84.0000	−2.68191
$982$	0	0
$983$	36.0000	1.14822	0.574111	−	0.818778i	$-0.305348\pi$
$983$	36.0000	1.14822	0.574111	+	0.818778i	$0.305348\pi$
$984$	0	0
$985$	−23.0000	−0.732841
$986$	0	0
$987$	−72.0000	−2.29179
$988$	0	0
$989$	63.0000	2.00328
$990$	0	0
$991$	−13.0000	−0.412959	−0.206479	−	0.978451i	$-0.566201\pi$
$991$	−13.0000	−0.412959	−0.206479	+	0.978451i	$0.566201\pi$
$992$	0	0
$993$	−39.0000	−1.23763
$994$	0	0
$995$	9.00000	0.285319
$996$	0	0
$997$	13.0000	0.411714	0.205857	−	0.978582i	$-0.434002\pi$
$997$	13.0000	0.411714	0.205857	+	0.978582i	$0.434002\pi$
$998$	0	0
$999$	27.0000	0.854242

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.2.a.b.1.1		1
13.3	even	3		260.2.i.d.61.1	✓	2
13.5	odd	4		3380.2.f.a.3041.1		2
13.8	odd	4		3380.2.f.a.3041.2		2
13.9	even	3		260.2.i.d.81.1	yes	2
13.12	even	2		3380.2.a.a.1.1		1
39.29	odd	6		2340.2.q.a.1621.1		2
39.35	odd	6		2340.2.q.a.2161.1		2
52.3	odd	6		1040.2.q.b.321.1		2
52.35	odd	6		1040.2.q.b.81.1		2
65.3	odd	12		1300.2.bb.e.1049.2		4
65.9	even	6		1300.2.i.a.601.1		2
65.22	odd	12		1300.2.bb.e.549.2		4
65.29	even	6		1300.2.i.a.1101.1		2
65.42	odd	12		1300.2.bb.e.1049.1		4
65.48	odd	12		1300.2.bb.e.549.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.i.d.61.1	✓	2	13.3	even	3
260.2.i.d.81.1	yes	2	13.9	even	3
1040.2.q.b.81.1		2	52.35	odd	6
1040.2.q.b.321.1		2	52.3	odd	6
1300.2.i.a.601.1		2	65.9	even	6
1300.2.i.a.1101.1		2	65.29	even	6
1300.2.bb.e.549.1		4	65.48	odd	12
1300.2.bb.e.549.2		4	65.22	odd	12
1300.2.bb.e.1049.1		4	65.42	odd	12
1300.2.bb.e.1049.2		4	65.3	odd	12
2340.2.q.a.1621.1		2	39.29	odd	6
2340.2.q.a.2161.1		2	39.35	odd	6
3380.2.a.a.1.1		1	13.12	even	2
3380.2.a.b.1.1		1	1.1	even	1	trivial
3380.2.f.a.3041.1		2	13.5	odd	4
3380.2.f.a.3041.2		2	13.8	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

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