LMFDB - Embedded newform 5733.2.a.bv.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5733,2,Mod(1,5733)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5733, 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("5733.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5733 = 3^{2} \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5733.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:	$45.7782354788$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.199374400.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 8x^{4} + 10x^{2} - 1 $ x^6 - 8x^4 + 10x^2 - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 819)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.18733$ of defining polynomial
Character	$\chi$	$=$	5733.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.345110 q^{2} -1.88090 q^{4} -3.42059 q^{5} +1.33934 q^{8} +O(q^{10})$	$q-0.345110 q^{2} -1.88090 q^{4} -3.42059 q^{5} +1.33934 q^{8} +1.18048 q^{10} +1.68445 q^{11} -1.00000 q^{13} +3.29958 q^{16} +4.05911 q^{17} -6.06138 q^{19} +6.43378 q^{20} -0.581319 q^{22} -5.79525 q^{23} +6.70042 q^{25} +0.345110 q^{26} +6.48547 q^{29} -0.299580 q^{31} -3.81739 q^{32} -1.40084 q^{34} +7.76180 q^{37} +2.09184 q^{38} -4.58132 q^{40} -9.90606 q^{41} -6.06138 q^{43} -3.16827 q^{44} +2.00000 q^{46} -7.47970 q^{47} -2.31238 q^{50} +1.88090 q^{52} -9.85437 q^{53} -5.76180 q^{55} -2.23820 q^{58} -3.75511 q^{59} -10.0000 q^{61} +0.103388 q^{62} -5.28174 q^{64} +3.42059 q^{65} +9.76180 q^{67} -7.63478 q^{68} -8.52562 q^{71} +1.70042 q^{73} -2.67867 q^{74} +11.4008 q^{76} +13.4622 q^{79} -11.2865 q^{80} +3.41868 q^{82} +0.638526 q^{83} -13.8846 q^{85} +2.09184 q^{86} +2.25604 q^{88} -12.9193 q^{89} +10.9003 q^{92} +2.58132 q^{94} +20.7335 q^{95} -4.06138 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 12 q^{4}+O(q^{10}) $ 6 * q + 12 * q^4	$ 6 q + 12 q^{4} - 8 q^{10} - 6 q^{13} + 28 q^{16} + 2 q^{19} + 28 q^{22} + 32 q^{25} - 10 q^{31} + 8 q^{34} + 4 q^{40} + 2 q^{43} + 12 q^{46} - 12 q^{52} + 12 q^{55} - 60 q^{58} - 60 q^{61} + 8 q^{64} + 12 q^{67} + 2 q^{73} + 52 q^{76} + 26 q^{79} + 52 q^{82} + 40 q^{85} + 108 q^{88} - 16 q^{94} + 14 q^{97}+O(q^{100}) $ 6 * q + 12 * q^4 - 8 * q^10 - 6 * q^13 + 28 * q^16 + 2 * q^19 + 28 * q^22 + 32 * q^25 - 10 * q^31 + 8 * q^34 + 4 * q^40 + 2 * q^43 + 12 * q^46 - 12 * q^52 + 12 * q^55 - 60 * q^58 - 60 * q^61 + 8 * q^64 + 12 * q^67 + 2 * q^73 + 52 * q^76 + 26 * q^79 + 52 * q^82 + 40 * q^85 + 108 * q^88 - 16 * q^94 + 14 * q^97

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 $

$2$	−0.345110	−0.244030	−0.122015	−	0.992528i	$-0.538936\pi$
$2$	−0.345110	−0.244030	−0.122015	+	0.992528i	$0.538936\pi$
$3$	0	0
$3$	0	0
$4$	−1.88090	−0.940450
$5$	−3.42059	−1.52973	−0.764867	−	0.644189i	$-0.777195\pi$
$5$	−3.42059	−1.52973	−0.764867	+	0.644189i	$0.777195\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.33934	0.473527
$9$	0	0
$10$	1.18048	0.373300
$11$	1.68445	0.507880	0.253940	−	0.967220i	$-0.418273\pi$
$11$	1.68445	0.507880	0.253940	+	0.967220i	$0.418273\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	3.29958	0.824895
$17$	4.05911	0.984480	0.492240	−	0.870460i	$-0.336178\pi$
$17$	4.05911	0.984480	0.492240	+	0.870460i	$0.336178\pi$
$18$	0	0
$19$	−6.06138	−1.39058	−0.695288	−	0.718731i	$-0.744723\pi$
$19$	−6.06138	−1.39058	−0.695288	+	0.718731i	$0.744723\pi$
$20$	6.43378	1.43864
$21$	0	0
$22$	−0.581319	−0.123938
$23$	−5.79525	−1.20839	−0.604197	−	0.796835i	$-0.706506\pi$
$23$	−5.79525	−1.20839	−0.604197	+	0.796835i	$0.706506\pi$
$24$	0	0
$25$	6.70042	1.34008
$26$	0.345110	0.0676816
$27$	0	0
$28$	0	0
$29$	6.48547	1.20432	0.602161	−	0.798375i	$-0.294306\pi$
$29$	6.48547	1.20432	0.602161	+	0.798375i	$0.294306\pi$
$30$	0	0
$31$	−0.299580	−0.0538061	−0.0269031	−	0.999638i	$-0.508565\pi$
$31$	−0.299580	−0.0538061	−0.0269031	+	0.999638i	$0.508565\pi$
$32$	−3.81739	−0.674826
$33$	0	0
$34$	−1.40084	−0.240242
$35$	0	0
$36$	0	0
$37$	7.76180	1.27603	0.638016	−	0.770023i	$-0.279755\pi$
$37$	7.76180	1.27603	0.638016	+	0.770023i	$0.279755\pi$
$38$	2.09184	0.339342
$39$	0	0
$40$	−4.58132	−0.724370
$41$	−9.90606	−1.54707	−0.773533	−	0.633755i	$-0.781513\pi$
$41$	−9.90606	−1.54707	−0.773533	+	0.633755i	$0.781513\pi$
$42$	0	0
$43$	−6.06138	−0.924351	−0.462176	−	0.886788i	$-0.652931\pi$
$43$	−6.06138	−0.924351	−0.462176	+	0.886788i	$0.652931\pi$
$44$	−3.16827	−0.477635
$45$	0	0
$46$	2.00000	0.294884
$47$	−7.47970	−1.09103	−0.545513	−	0.838102i	$-0.683665\pi$
$47$	−7.47970	−1.09103	−0.545513	+	0.838102i	$0.683665\pi$
$48$	0	0
$49$	0	0
$50$	−2.31238	−0.327020
$51$	0	0
$52$	1.88090	0.260834
$53$	−9.85437	−1.35360	−0.676801	−	0.736166i	$-0.736634\pi$
$53$	−9.85437	−1.35360	−0.676801	+	0.736166i	$0.736634\pi$
$54$	0	0
$55$	−5.76180	−0.776921
$56$	0	0
$57$	0	0
$58$	−2.23820	−0.293890
$59$	−3.75511	−0.488873	−0.244437	−	0.969665i	$-0.578603\pi$
$59$	−3.75511	−0.488873	−0.244437	+	0.969665i	$0.578603\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0.103388	0.0131303
$63$	0	0
$64$	−5.28174	−0.660217
$65$	3.42059	0.424272
$66$	0	0
$67$	9.76180	1.19259	0.596297	−	0.802764i	$-0.296638\pi$
$67$	9.76180	1.19259	0.596297	+	0.802764i	$0.296638\pi$
$68$	−7.63478	−0.925853
$69$	0	0
$70$	0	0
$71$	−8.52562	−1.01181	−0.505903	−	0.862591i	$-0.668840\pi$
$71$	−8.52562	−1.01181	−0.505903	+	0.862591i	$0.668840\pi$
$72$	0	0
$73$	1.70042	0.199019	0.0995096	−	0.995037i	$-0.468273\pi$
$73$	1.70042	0.199019	0.0995096	+	0.995037i	$0.468273\pi$
$74$	−2.67867	−0.311390
$75$	0	0
$76$	11.4008	1.30777
$77$	0	0
$78$	0	0
$79$	13.4622	1.51462	0.757309	−	0.653057i	$-0.226514\pi$
$79$	13.4622	1.51462	0.757309	+	0.653057i	$0.226514\pi$
$80$	−11.2865	−1.26187
$81$	0	0
$82$	3.41868	0.377530
$83$	0.638526	0.0700873	0.0350437	−	0.999386i	$-0.488843\pi$
$83$	0.638526	0.0700873	0.0350437	+	0.999386i	$0.488843\pi$
$84$	0	0
$85$	−13.8846	−1.50599
$86$	2.09184	0.225569
$87$	0	0
$88$	2.25604	0.240495
$89$	−12.9193	−1.36944	−0.684719	−	0.728807i	$-0.740075\pi$
$89$	−12.9193	−1.36944	−0.684719	+	0.728807i	$0.740075\pi$
$90$	0	0
$91$	0	0
$92$	10.9003	1.13643
$93$	0	0
$94$	2.58132	0.266243
$95$	20.7335	2.12721
$96$	0	0
$97$	−4.06138	−0.412370	−0.206185	−	0.978513i	$-0.566105\pi$
$97$	−4.06138	−0.412370	−0.206185	+	0.978513i	$0.566105\pi$
$98$	0	0
$99$	0	0
$100$	−12.6028	−1.26028
$101$	9.51985	0.947260	0.473630	−	0.880724i	$-0.342943\pi$
$101$	9.51985	0.947260	0.473630	+	0.880724i	$0.342943\pi$
$102$	0	0
$103$	6.36096	0.626764	0.313382	−	0.949627i	$-0.398538\pi$
$103$	6.36096	0.626764	0.313382	+	0.949627i	$0.398538\pi$
$104$	−1.33934	−0.131333
$105$	0	0
$106$	3.40084	0.330319
$107$	−15.6496	−1.51291	−0.756453	−	0.654048i	$-0.773069\pi$
$107$	−15.6496	−1.51291	−0.756453	+	0.654048i	$0.773069\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	1.98845	0.189592
$111$	0	0
$112$	0	0
$113$	−6.48547	−0.610102	−0.305051	−	0.952336i	$-0.598674\pi$
$113$	−6.48547	−0.610102	−0.305051	+	0.952336i	$0.598674\pi$
$114$	0	0
$115$	19.8232	1.84852
$116$	−12.1985	−1.13260
$117$	0	0
$118$	1.29592	0.119300
$119$	0	0
$120$	0	0
$121$	−8.16264	−0.742058
$122$	3.45110	0.312448
$123$	0	0
$124$	0.563479	0.0506019
$125$	−5.81644	−0.520238
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	9.45756	0.835938
$129$	0	0
$130$	−1.18048	−0.103535
$131$	18.4317	1.61038	0.805192	−	0.593014i	$-0.202062\pi$
$131$	18.4317	1.61038	0.805192	+	0.593014i	$0.202062\pi$
$132$	0	0
$133$	0	0
$134$	−3.36889	−0.291028
$135$	0	0
$136$	5.43652	0.466178
$137$	9.21584	0.787363	0.393681	−	0.919247i	$-0.371201\pi$
$137$	9.21584	0.787363	0.393681	+	0.919247i	$0.371201\pi$
$138$	0	0
$139$	21.2854	1.80540	0.902702	−	0.430267i	$-0.141580\pi$
$139$	21.2854	1.80540	0.902702	+	0.430267i	$0.141580\pi$
$140$	0	0
$141$	0	0
$142$	2.94228	0.246910
$143$	−1.68445	−0.140861
$144$	0	0
$145$	−22.1841	−1.84229
$146$	−0.586832	−0.0485666
$147$	0	0
$148$	−14.5992	−1.20004
$149$	−17.4375	−1.42853	−0.714266	−	0.699874i	$-0.753239\pi$
$149$	−17.4375	−1.42853	−0.714266	+	0.699874i	$0.753239\pi$
$150$	0	0
$151$	6.36096	0.517647	0.258824	−	0.965925i	$-0.416665\pi$
$151$	6.36096	0.517647	0.258824	+	0.965925i	$0.416665\pi$
$152$	−8.11823	−0.658475
$153$	0	0
$154$	0	0
$155$	1.02474	0.0823090
$156$	0	0
$157$	−10.1228	−0.807884	−0.403942	−	0.914785i	$-0.632360\pi$
$157$	−10.1228	−0.807884	−0.403942	+	0.914785i	$0.632360\pi$
$158$	−4.64595	−0.369612
$159$	0	0
$160$	13.0577	1.03230
$161$	0	0
$162$	0	0
$163$	−7.52360	−0.589294	−0.294647	−	0.955606i	$-0.595202\pi$
$163$	−7.52360	−0.589294	−0.294647	+	0.955606i	$0.595202\pi$
$164$	18.6323	1.45494
$165$	0	0
$166$	−0.220362	−0.0171034
$167$	15.7013	1.21500	0.607502	−	0.794318i	$-0.292172\pi$
$167$	15.7013	1.21500	0.607502	+	0.794318i	$0.292172\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	4.79170	0.367506
$171$	0	0
$172$	11.4008	0.869306
$173$	12.9921	0.987773	0.493887	−	0.869526i	$-0.335576\pi$
$173$	12.9921	0.987773	0.493887	+	0.869526i	$0.335576\pi$
$174$	0	0
$175$	0	0
$176$	5.55797	0.418948
$177$	0	0
$178$	4.45856	0.334183
$179$	−10.5446	−0.788140	−0.394070	−	0.919081i	$-0.628933\pi$
$179$	−10.5446	−0.788140	−0.394070	+	0.919081i	$0.628933\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	0	0
$184$	−7.76180	−0.572207
$185$	−26.5499	−1.95199
$186$	0	0
$187$	6.83736	0.499997
$188$	14.0686	1.02606
$189$	0	0
$190$	−7.15533	−0.519102
$191$	−4.16250	−0.301188	−0.150594	−	0.988596i	$-0.548119\pi$
$191$	−4.16250	−0.301188	−0.150594	+	0.988596i	$0.548119\pi$
$192$	0	0
$193$	−16.3610	−1.17769	−0.588844	−	0.808247i	$-0.700417\pi$
$193$	−16.3610	−1.17769	−0.588844	+	0.808247i	$0.700417\pi$
$194$	1.40162	0.100631
$195$	0	0
$196$	0	0
$197$	3.75511	0.267540	0.133770	−	0.991012i	$-0.457292\pi$
$197$	3.75511	0.267540	0.133770	+	0.991012i	$0.457292\pi$
$198$	0	0
$199$	27.5236	1.95110	0.975548	−	0.219786i	$-0.0705361\pi$
$199$	27.5236	1.95110	0.975548	+	0.219786i	$0.0705361\pi$
$200$	8.97412	0.634566
$201$	0	0
$202$	−3.28539	−0.231160
$203$	0	0
$204$	0	0
$205$	33.8846	2.36660
$206$	−2.19523	−0.152949
$207$	0	0
$208$	−3.29958	−0.228785
$209$	−10.2101	−0.706245
$210$	0	0
$211$	2.06138	0.141911	0.0709556	−	0.997479i	$-0.477395\pi$
$211$	2.06138	0.141911	0.0709556	+	0.997479i	$0.477395\pi$
$212$	18.5351	1.27299
$213$	0	0
$214$	5.40084	0.369194
$215$	20.7335	1.41401
$216$	0	0
$217$	0	0
$218$	−3.45110	−0.233738
$219$	0	0
$220$	10.8374	0.730655
$221$	−4.05911	−0.273046
$222$	0	0
$223$	2.06138	0.138040	0.0690200	−	0.997615i	$-0.478013\pi$
$223$	2.06138	0.138040	0.0690200	+	0.997615i	$0.478013\pi$
$224$	0	0
$225$	0	0
$226$	2.23820	0.148883
$227$	−17.4375	−1.15736	−0.578682	−	0.815553i	$-0.696433\pi$
$227$	−17.4375	−1.15736	−0.578682	+	0.815553i	$0.696433\pi$
$228$	0	0
$229$	8.36096	0.552508	0.276254	−	0.961085i	$-0.410907\pi$
$229$	8.36096	0.552508	0.276254	+	0.961085i	$0.410907\pi$
$230$	−6.84118	−0.451094
$231$	0	0
$232$	8.68624	0.570279
$233$	25.5252	1.67221	0.836105	−	0.548570i	$-0.184827\pi$
$233$	25.5252	1.67221	0.836105	+	0.548570i	$0.184827\pi$
$234$	0	0
$235$	25.5850	1.66898
$236$	7.06298	0.459761
$237$	0	0
$238$	0	0
$239$	−16.6438	−1.07660	−0.538301	−	0.842753i	$-0.680933\pi$
$239$	−16.6438	−1.07660	−0.538301	+	0.842753i	$0.680933\pi$
$240$	0	0
$241$	−5.10126	−0.328601	−0.164301	−	0.986410i	$-0.552537\pi$
$241$	−5.10126	−0.328601	−0.164301	+	0.986410i	$0.552537\pi$
$242$	2.81701	0.181084
$243$	0	0
$244$	18.8090	1.20412
$245$	0	0
$246$	0	0
$247$	6.06138	0.385676
$248$	−0.401238	−0.0254787
$249$	0	0
$250$	2.00731	0.126953
$251$	2.65749	0.167739	0.0838697	−	0.996477i	$-0.473272\pi$
$251$	2.65749	0.167739	0.0838697	+	0.996477i	$0.473272\pi$
$252$	0	0
$253$	−9.76180	−0.613719
$254$	−1.38044	−0.0866165
$255$	0	0
$256$	7.29958	0.456224
$257$	10.9003	0.679941	0.339971	−	0.940436i	$-0.389583\pi$
$257$	10.9003	0.679941	0.339971	+	0.940436i	$0.389583\pi$
$258$	0	0
$259$	0	0
$260$	−6.43378	−0.399006
$261$	0	0
$262$	−6.36096	−0.392981
$263$	7.88710	0.486339	0.243170	−	0.969984i	$-0.421813\pi$
$263$	7.88710	0.486339	0.243170	+	0.969984i	$0.421813\pi$
$264$	0	0
$265$	33.7077	2.07065
$266$	0	0
$267$	0	0
$268$	−18.3610	−1.12157
$269$	−8.80845	−0.537061	−0.268530	−	0.963271i	$-0.586538\pi$
$269$	−8.80845	−0.537061	−0.268530	+	0.963271i	$0.586538\pi$
$270$	0	0
$271$	1.16264	0.0706253	0.0353126	−	0.999376i	$-0.488757\pi$
$271$	1.16264	0.0706253	0.0353126	+	0.999376i	$0.488757\pi$
$272$	13.3934	0.812092
$273$	0	0
$274$	−3.18048	−0.192140
$275$	11.2865	0.680602
$276$	0	0
$277$	16.0614	0.965035	0.482517	−	0.875886i	$-0.339723\pi$
$277$	16.0614	0.965035	0.482517	+	0.875886i	$0.339723\pi$
$278$	−7.34580	−0.440572
$279$	0	0
$280$	0	0
$281$	9.92724	0.592210	0.296105	−	0.955155i	$-0.404312\pi$
$281$	9.92724	0.592210	0.296105	+	0.955155i	$0.404312\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	16.0358	0.951552
$285$	0	0
$286$	0.581319	0.0343741
$287$	0	0
$288$	0	0
$289$	−0.523596	−0.0307998
$290$	7.65597	0.449574
$291$	0	0
$292$	−3.19832	−0.187167
$293$	6.18147	0.361125	0.180563	−	0.983563i	$-0.442208\pi$
$293$	6.18147	0.361125	0.180563	+	0.983563i	$0.442208\pi$
$294$	0	0
$295$	12.8447	0.747846
$296$	10.3957	0.604236
$297$	0	0
$298$	6.01784	0.348604
$299$	5.79525	0.335148
$300$	0	0
$301$	0	0
$302$	−2.19523	−0.126321
$303$	0	0
$304$	−20.0000	−1.14708
$305$	34.2059	1.95862
$306$	0	0
$307$	26.1841	1.49441	0.747204	−	0.664595i	$-0.231396\pi$
$307$	26.1841	1.49441	0.747204	+	0.664595i	$0.231396\pi$
$308$	0	0
$309$	0	0
$310$	−0.353648	−0.0200858
$311$	−4.08030	−0.231372	−0.115686	−	0.993286i	$-0.536907\pi$
$311$	−4.08030	−0.231372	−0.115686	+	0.993286i	$0.536907\pi$
$312$	0	0
$313$	−2.47640	−0.139975	−0.0699873	−	0.997548i	$-0.522296\pi$
$313$	−2.47640	−0.139975	−0.0699873	+	0.997548i	$0.522296\pi$
$314$	3.49346	0.197148
$315$	0	0
$316$	−25.3211	−1.42442
$317$	−32.3969	−1.81959	−0.909794	−	0.415059i	$-0.863761\pi$
$317$	−32.3969	−1.81959	−0.909794	+	0.415059i	$0.863761\pi$
$318$	0	0
$319$	10.9244	0.611651
$320$	18.0667	1.00996
$321$	0	0
$322$	0	0
$323$	−24.6038	−1.36899
$324$	0	0
$325$	−6.70042	−0.371672
$326$	2.59647	0.143805
$327$	0	0
$328$	−13.2676	−0.732578
$329$	0	0
$330$	0	0
$331$	3.40084	0.186927	0.0934636	−	0.995623i	$-0.470206\pi$
$331$	3.40084	0.186927	0.0934636	+	0.995623i	$0.470206\pi$
$332$	−1.20100	−0.0659136
$333$	0	0
$334$	−5.41868	−0.296497
$335$	−33.3911	−1.82435
$336$	0	0
$337$	29.9459	1.63126	0.815629	−	0.578575i	$-0.196391\pi$
$337$	29.9459	1.63126	0.815629	+	0.578575i	$0.196391\pi$
$338$	−0.345110	−0.0187715
$339$	0	0
$340$	26.1154	1.41631
$341$	−0.504626	−0.0273270
$342$	0	0
$343$	0	0
$344$	−8.11823	−0.437705
$345$	0	0
$346$	−4.48371	−0.241046
$347$	−2.78206	−0.149349	−0.0746745	−	0.997208i	$-0.523792\pi$
$347$	−2.78206	−0.149349	−0.0746745	+	0.997208i	$0.523792\pi$
$348$	0	0
$349$	31.1086	1.66520	0.832602	−	0.553872i	$-0.186850\pi$
$349$	31.1086	1.66520	0.832602	+	0.553872i	$0.186850\pi$
$350$	0	0
$351$	0	0
$352$	−6.43019	−0.342730
$353$	32.3147	1.71994	0.859968	−	0.510348i	$-0.170484\pi$
$353$	32.3147	1.71994	0.859968	+	0.510348i	$0.170484\pi$
$354$	0	0
$355$	29.1626	1.54779
$356$	24.2998	1.28789
$357$	0	0
$358$	3.63904	0.192329
$359$	25.5769	1.34990	0.674948	−	0.737866i	$-0.264166\pi$
$359$	25.5769	1.34990	0.674948	+	0.737866i	$0.264166\pi$
$360$	0	0
$361$	17.7403	0.933700
$362$	3.45110	0.181386
$363$	0	0
$364$	0	0
$365$	−5.81644	−0.304446
$366$	0	0
$367$	6.23820	0.325631	0.162816	−	0.986656i	$-0.447942\pi$
$367$	6.23820	0.325631	0.162816	+	0.986656i	$0.447942\pi$
$368$	−19.1219	−0.996798
$369$	0	0
$370$	9.16264	0.476343
$371$	0	0
$372$	0	0
$373$	3.63904	0.188422	0.0942112	−	0.995552i	$-0.469967\pi$
$373$	3.63904	0.188422	0.0942112	+	0.995552i	$0.469967\pi$
$374$	−2.35964	−0.122014
$375$	0	0
$376$	−10.0178	−0.516631
$377$	−6.48547	−0.334019
$378$	0	0
$379$	24.5635	1.26174	0.630871	−	0.775888i	$-0.282698\pi$
$379$	24.5635	1.26174	0.630871	+	0.775888i	$0.282698\pi$
$380$	−38.9976	−2.00053
$381$	0	0
$382$	1.43652	0.0734988
$383$	−25.5557	−1.30583	−0.652917	−	0.757429i	$-0.726455\pi$
$383$	−25.5557	−1.30583	−0.652917	+	0.757429i	$0.726455\pi$
$384$	0	0
$385$	0	0
$386$	5.64633	0.287391
$387$	0	0
$388$	7.63904	0.387814
$389$	2.76088	0.139982	0.0699911	−	0.997548i	$-0.477703\pi$
$389$	2.76088	0.139982	0.0699911	+	0.997548i	$0.477703\pi$
$390$	0	0
$391$	−23.5236	−1.18964
$392$	0	0
$393$	0	0
$394$	−1.29592	−0.0652877
$395$	−46.0487	−2.31696
$396$	0	0
$397$	−25.2240	−1.26596	−0.632979	−	0.774169i	$-0.718168\pi$
$397$	−25.2240	−1.26596	−0.632979	+	0.774169i	$0.718168\pi$
$398$	−9.49867	−0.476125
$399$	0	0
$400$	22.1086	1.10543
$401$	−32.3969	−1.61782	−0.808911	−	0.587931i	$-0.799943\pi$
$401$	−32.3969	−1.61782	−0.808911	+	0.587931i	$0.799943\pi$
$402$	0	0
$403$	0.299580	0.0149231
$404$	−17.9059	−0.890851
$405$	0	0
$406$	0	0
$407$	13.0743	0.648071
$408$	0	0
$409$	7.46222	0.368983	0.184491	−	0.982834i	$-0.440936\pi$
$409$	7.46222	0.368983	0.184491	+	0.982834i	$0.440936\pi$
$410$	−11.6939	−0.577520
$411$	0	0
$412$	−11.9643	−0.589440
$413$	0	0
$414$	0	0
$415$	−2.18413	−0.107215
$416$	3.81739	0.187163
$417$	0	0
$418$	3.52360	0.172345
$419$	36.7600	1.79584	0.897921	−	0.440156i	$-0.145077\pi$
$419$	36.7600	1.79584	0.897921	+	0.440156i	$0.145077\pi$
$420$	0	0
$421$	7.28539	0.355068	0.177534	−	0.984115i	$-0.443188\pi$
$421$	7.28539	0.355068	0.177534	+	0.984115i	$0.443188\pi$
$422$	−0.711402	−0.0346305
$423$	0	0
$424$	−13.1983	−0.640967
$425$	27.1978	1.31929
$426$	0	0
$427$	0	0
$428$	29.4354	1.42281
$429$	0	0
$430$	−7.15533	−0.345061
$431$	−0.510783	−0.0246035	−0.0123018	−	0.999924i	$-0.503916\pi$
$431$	−0.510783	−0.0246035	−0.0123018	+	0.999924i	$0.503916\pi$
$432$	0	0
$433$	−10.1228	−0.486469	−0.243234	−	0.969968i	$-0.578208\pi$
$433$	−10.1228	−0.486469	−0.243234	+	0.969968i	$0.578208\pi$
$434$	0	0
$435$	0	0
$436$	−18.8090	−0.900787
$437$	35.1272	1.68036
$438$	0	0
$439$	−29.2854	−1.39772	−0.698858	−	0.715261i	$-0.746308\pi$
$439$	−29.2854	−1.39772	−0.698858	+	0.715261i	$0.746308\pi$
$440$	−7.71699	−0.367893
$441$	0	0
$442$	1.40084	0.0666312
$443$	38.5173	1.83001	0.915006	−	0.403440i	$-0.132186\pi$
$443$	38.5173	1.83001	0.915006	+	0.403440i	$0.132186\pi$
$444$	0	0
$445$	44.1914	2.09488
$446$	−0.711402	−0.0336859
$447$	0	0
$448$	0	0
$449$	22.0834	1.04218	0.521090	−	0.853502i	$-0.325526\pi$
$449$	22.0834	1.04218	0.521090	+	0.853502i	$0.325526\pi$
$450$	0	0
$451$	−16.6862	−0.785724
$452$	12.1985	0.573770
$453$	0	0
$454$	6.01784	0.282431
$455$	0	0
$456$	0	0
$457$	36.4480	1.70497	0.852484	−	0.522754i	$-0.175095\pi$
$457$	36.4480	1.70497	0.852484	+	0.522754i	$0.175095\pi$
$458$	−2.88545	−0.134828
$459$	0	0
$460$	−37.2854	−1.73844
$461$	−22.7736	−1.06067	−0.530337	−	0.847787i	$-0.677934\pi$
$461$	−22.7736	−1.06067	−0.530337	+	0.847787i	$0.677934\pi$
$462$	0	0
$463$	−13.7618	−0.639565	−0.319783	−	0.947491i	$-0.603610\pi$
$463$	−13.7618	−0.639565	−0.319783	+	0.947491i	$0.603610\pi$
$464$	21.3993	0.993439
$465$	0	0
$466$	−8.80899	−0.408069
$467$	−13.5790	−0.628359	−0.314180	−	0.949364i	$-0.601729\pi$
$467$	−13.5790	−0.628359	−0.314180	+	0.949364i	$0.601729\pi$
$468$	0	0
$469$	0	0
$470$	−8.82963	−0.407280
$471$	0	0
$472$	−5.02935	−0.231495
$473$	−10.2101	−0.469459
$474$	0	0
$475$	−40.6138	−1.86349
$476$	0	0
$477$	0	0
$478$	5.74396	0.262723
$479$	14.4243	0.659061	0.329531	−	0.944145i	$-0.393109\pi$
$479$	14.4243	0.659061	0.329531	+	0.944145i	$0.393109\pi$
$480$	0	0
$481$	−7.76180	−0.353908
$482$	1.76050	0.0801884
$483$	0	0
$484$	15.3531	0.697868
$485$	13.8923	0.630817
$486$	0	0
$487$	−25.2854	−1.14579	−0.572895	−	0.819629i	$-0.694180\pi$
$487$	−25.2854	−1.14579	−0.572895	+	0.819629i	$0.694180\pi$
$488$	−13.3934	−0.606289
$489$	0	0
$490$	0	0
$491$	12.8887	0.581661	0.290830	−	0.956775i	$-0.406068\pi$
$491$	12.8887	0.581661	0.290830	+	0.956775i	$0.406068\pi$
$492$	0	0
$493$	26.3253	1.18563
$494$	−2.09184	−0.0941164
$495$	0	0
$496$	−0.988487	−0.0443844
$497$	0	0
$498$	0	0
$499$	20.2455	0.906314	0.453157	−	0.891431i	$-0.350298\pi$
$499$	20.2455	0.906314	0.453157	+	0.891431i	$0.350298\pi$
$500$	10.9401	0.489258
$501$	0	0
$502$	−0.917127	−0.0409334
$503$	1.48383	0.0661606	0.0330803	−	0.999453i	$-0.489468\pi$
$503$	1.48383	0.0661606	0.0330803	+	0.999453i	$0.489468\pi$
$504$	0	0
$505$	−32.5635	−1.44906
$506$	3.36889	0.149766
$507$	0	0
$508$	−7.52360	−0.333806
$509$	−25.8902	−1.14756	−0.573782	−	0.819008i	$-0.694524\pi$
$509$	−25.8902	−1.14756	−0.573782	+	0.819008i	$0.694524\pi$
$510$	0	0
$511$	0	0
$512$	−21.4343	−0.947271
$513$	0	0
$514$	−3.76180	−0.165926
$515$	−21.7582	−0.958781
$516$	0	0
$517$	−12.5992	−0.554110
$518$	0	0
$519$	0	0
$520$	4.58132	0.200904
$521$	38.9340	1.70573	0.852865	−	0.522131i	$-0.174863\pi$
$521$	38.9340	1.70573	0.852865	+	0.522131i	$0.174863\pi$
$522$	0	0
$523$	25.7618	1.12648	0.563242	−	0.826292i	$-0.309554\pi$
$523$	25.7618	1.12648	0.563242	+	0.826292i	$0.309554\pi$
$524$	−34.6681	−1.51448
$525$	0	0
$526$	−2.72192	−0.118681
$527$	−1.21603	−0.0529710
$528$	0	0
$529$	10.5850	0.460216
$530$	−11.6329	−0.505300
$531$	0	0
$532$	0	0
$533$	9.90606	0.429079
$534$	0	0
$535$	53.5309	2.31434
$536$	13.0743	0.564725
$537$	0	0
$538$	3.03988	0.131059
$539$	0	0
$540$	0	0
$541$	24.2382	1.04208	0.521041	−	0.853532i	$-0.325544\pi$
$541$	24.2382	1.04208	0.521041	+	0.853532i	$0.325544\pi$
$542$	−0.401238	−0.0172347
$543$	0	0
$544$	−15.4952	−0.664352
$545$	−34.2059	−1.46522
$546$	0	0
$547$	−21.1086	−0.902537	−0.451269	−	0.892388i	$-0.649028\pi$
$547$	−21.1086	−0.902537	−0.451269	+	0.892388i	$0.649028\pi$
$548$	−17.3341	−0.740475
$549$	0	0
$550$	−3.89508	−0.166087
$551$	−39.3109	−1.67470
$552$	0	0
$553$	0	0
$554$	−5.54294	−0.235497
$555$	0	0
$556$	−40.0357	−1.69789
$557$	−30.2016	−1.27968	−0.639842	−	0.768506i	$-0.721000\pi$
$557$	−30.2016	−1.27968	−0.639842	+	0.768506i	$0.721000\pi$
$558$	0	0
$559$	6.06138	0.256369
$560$	0	0
$561$	0	0
$562$	−3.42599	−0.144517
$563$	−17.8237	−0.751178	−0.375589	−	0.926786i	$-0.622560\pi$
$563$	−17.8237	−0.751178	−0.375589	+	0.926786i	$0.622560\pi$
$564$	0	0
$565$	22.1841	0.933294
$566$	−1.38044	−0.0580242
$567$	0	0
$568$	−11.4187	−0.479117
$569$	0.921351	0.0386250	0.0193125	−	0.999813i	$-0.493852\pi$
$569$	0.921351	0.0386250	0.0193125	+	0.999813i	$0.493852\pi$
$570$	0	0
$571$	17.4622	0.730771	0.365386	−	0.930856i	$-0.380937\pi$
$571$	17.4622	0.730771	0.365386	+	0.930856i	$0.380937\pi$
$572$	3.16827	0.132472
$573$	0	0
$574$	0	0
$575$	−38.8306	−1.61935
$576$	0	0
$577$	11.6390	0.484540	0.242270	−	0.970209i	$-0.422108\pi$
$577$	11.6390	0.484540	0.242270	+	0.970209i	$0.422108\pi$
$578$	0.180698	0.00751606
$579$	0	0
$580$	41.7261	1.73258
$581$	0	0
$582$	0	0
$583$	−16.5992	−0.687467
$584$	2.27744	0.0942410
$585$	0	0
$586$	−2.13329	−0.0881252
$587$	25.9114	1.06948	0.534739	−	0.845017i	$-0.320410\pi$
$587$	25.9114	1.06948	0.534739	+	0.845017i	$0.320410\pi$
$588$	0	0
$589$	1.81587	0.0748215
$590$	−4.43282	−0.182496
$591$	0	0
$592$	25.6107	1.05259
$593$	38.0887	1.56412	0.782058	−	0.623205i	$-0.214170\pi$
$593$	38.0887	1.56412	0.782058	+	0.623205i	$0.214170\pi$
$594$	0	0
$595$	0	0
$596$	32.7981	1.34346
$597$	0	0
$598$	−2.00000	−0.0817861
$599$	−35.0450	−1.43190	−0.715950	−	0.698151i	$-0.754006\pi$
$599$	−35.0450	−1.43190	−0.715950	+	0.698151i	$0.754006\pi$
$600$	0	0
$601$	−40.5708	−1.65492	−0.827459	−	0.561527i	$-0.810214\pi$
$601$	−40.5708	−1.65492	−0.827459	+	0.561527i	$0.810214\pi$
$602$	0	0
$603$	0	0
$604$	−11.9643	−0.486821
$605$	27.9210	1.13515
$606$	0	0
$607$	30.4837	1.23730	0.618648	−	0.785668i	$-0.287681\pi$
$607$	30.4837	1.23730	0.618648	+	0.785668i	$0.287681\pi$
$608$	23.1387	0.938396
$609$	0	0
$610$	−11.8048	−0.477962
$611$	7.47970	0.302596
$612$	0	0
$613$	−12.9244	−0.522013	−0.261007	−	0.965337i	$-0.584054\pi$
$613$	−12.9244	−0.522013	−0.261007	+	0.965337i	$0.584054\pi$
$614$	−9.03641	−0.364680
$615$	0	0
$616$	0	0
$617$	38.6300	1.55519	0.777593	−	0.628768i	$-0.216440\pi$
$617$	38.6300	1.55519	0.777593	+	0.628768i	$0.216440\pi$
$618$	0	0
$619$	17.8846	0.718841	0.359420	−	0.933176i	$-0.382974\pi$
$619$	17.8846	0.718841	0.359420	+	0.933176i	$0.382974\pi$
$620$	−1.92743	−0.0774075
$621$	0	0
$622$	1.40815	0.0564617
$623$	0	0
$624$	0	0
$625$	−13.6065	−0.544259
$626$	0.854632	0.0341579
$627$	0	0
$628$	19.0399	0.759774
$629$	31.5060	1.25623
$630$	0	0
$631$	−43.1699	−1.71857	−0.859284	−	0.511498i	$-0.829091\pi$
$631$	−43.1699	−1.71857	−0.859284	+	0.511498i	$0.829091\pi$
$632$	18.0304	0.717213
$633$	0	0
$634$	11.1805	0.444034
$635$	−13.6824	−0.542968
$636$	0	0
$637$	0	0
$638$	−3.77013	−0.149261
$639$	0	0
$640$	−32.3504	−1.27876
$641$	−43.2455	−1.70809	−0.854046	−	0.520197i	$-0.825859\pi$
$641$	−43.2455	−1.70809	−0.854046	+	0.520197i	$0.825859\pi$
$642$	0	0
$643$	47.5236	1.87415	0.937074	−	0.349131i	$-0.113523\pi$
$643$	47.5236	1.87415	0.937074	+	0.349131i	$0.113523\pi$
$644$	0	0
$645$	0	0
$646$	8.49102	0.334075
$647$	−11.5905	−0.455670	−0.227835	−	0.973700i	$-0.573165\pi$
$647$	−11.5905	−0.455670	−0.227835	+	0.973700i	$0.573165\pi$
$648$	0	0
$649$	−6.32528	−0.248289
$650$	2.31238	0.0906991
$651$	0	0
$652$	14.1511	0.554201
$653$	−39.0586	−1.52848	−0.764241	−	0.644931i	$-0.776886\pi$
$653$	−39.0586	−1.52848	−0.764241	+	0.644931i	$0.776886\pi$
$654$	0	0
$655$	−63.0472	−2.46346
$656$	−32.6858	−1.27617
$657$	0	0
$658$	0	0
$659$	10.5446	0.410759	0.205379	−	0.978682i	$-0.434157\pi$
$659$	10.5446	0.410759	0.205379	+	0.978682i	$0.434157\pi$
$660$	0	0
$661$	1.25970	0.0489965	0.0244983	−	0.999700i	$-0.492201\pi$
$661$	1.25970	0.0489965	0.0244983	+	0.999700i	$0.492201\pi$
$662$	−1.17366	−0.0456157
$663$	0	0
$664$	0.855201	0.0331882
$665$	0	0
$666$	0	0
$667$	−37.5850	−1.45530
$668$	−29.5326	−1.14265
$669$	0	0
$670$	11.5236	0.445195
$671$	−16.8445	−0.650274
$672$	0	0
$673$	7.46222	0.287647	0.143824	−	0.989603i	$-0.454060\pi$
$673$	7.46222	0.287647	0.143824	+	0.989603i	$0.454060\pi$
$674$	−10.3346	−0.398075
$675$	0	0
$676$	−1.88090	−0.0723423
$677$	11.7151	0.450247	0.225124	−	0.974330i	$-0.427721\pi$
$677$	11.7151	0.450247	0.225124	+	0.974330i	$0.427721\pi$
$678$	0	0
$679$	0	0
$680$	−18.5961	−0.713128
$681$	0	0
$682$	0.174152	0.00666861
$683$	−4.38430	−0.167761	−0.0838803	−	0.996476i	$-0.526731\pi$
$683$	−4.38430	−0.167761	−0.0838803	+	0.996476i	$0.526731\pi$
$684$	0	0
$685$	−31.5236	−1.20445
$686$	0	0
$687$	0	0
$688$	−20.0000	−0.762493
$689$	9.85437	0.375422
$690$	0	0
$691$	0.176824	0.00672669	0.00336335	−	0.999994i	$-0.498929\pi$
$691$	0.176824	0.00672669	0.00336335	+	0.999994i	$0.498929\pi$
$692$	−24.4369	−0.928951
$693$	0	0
$694$	0.960117	0.0364456
$695$	−72.8086	−2.76179
$696$	0	0
$697$	−40.2098	−1.52306
$698$	−10.7359	−0.406359
$699$	0	0
$700$	0	0
$701$	−41.1536	−1.55435	−0.777175	−	0.629284i	$-0.783348\pi$
$701$	−41.1536	−1.55435	−0.777175	+	0.629284i	$0.783348\pi$
$702$	0	0
$703$	−47.0472	−1.77442
$704$	−8.89681	−0.335311
$705$	0	0
$706$	−11.1521	−0.419715
$707$	0	0
$708$	0	0
$709$	34.2455	1.28612	0.643059	−	0.765817i	$-0.277665\pi$
$709$	34.2455	1.28612	0.643059	+	0.765817i	$0.277665\pi$
$710$	−10.0643	−0.377707
$711$	0	0
$712$	−17.3032	−0.648466
$713$	1.73614	0.0650190
$714$	0	0
$715$	5.76180	0.215479
$716$	19.8333	0.741205
$717$	0	0
$718$	−8.82683	−0.329414
$719$	17.7203	0.660855	0.330428	−	0.943831i	$-0.392807\pi$
$719$	17.7203	0.660855	0.330428	+	0.943831i	$0.392807\pi$
$720$	0	0
$721$	0	0
$722$	−6.12236	−0.227850
$723$	0	0
$724$	18.8090	0.699031
$725$	43.4554	1.61389
$726$	0	0
$727$	−0.122756	−0.00455277	−0.00227638	−	0.999997i	$-0.500725\pi$
$727$	−0.122756	−0.00455277	−0.00227638	+	0.999997i	$0.500725\pi$
$728$	0	0
$729$	0	0
$730$	2.00731	0.0742939
$731$	−24.6038	−0.910005
$732$	0	0
$733$	17.2240	0.636184	0.318092	−	0.948060i	$-0.396958\pi$
$733$	17.2240	0.636184	0.318092	+	0.948060i	$0.396958\pi$
$734$	−2.15287	−0.0794637
$735$	0	0
$736$	22.1228	0.815456
$737$	16.4432	0.605694
$738$	0	0
$739$	−13.0399	−0.479680	−0.239840	−	0.970812i	$-0.577095\pi$
$739$	−13.0399	−0.479680	−0.239840	+	0.970812i	$0.577095\pi$
$740$	49.9377	1.83575
$741$	0	0
$742$	0	0
$743$	28.3377	1.03961	0.519806	−	0.854285i	$-0.326004\pi$
$743$	28.3377	1.03961	0.519806	+	0.854285i	$0.326004\pi$
$744$	0	0
$745$	59.6464	2.18527
$746$	−1.25587	−0.0459807
$747$	0	0
$748$	−12.8604	−0.470222
$749$	0	0
$750$	0	0
$751$	−1.81587	−0.0662619	−0.0331309	−	0.999451i	$-0.510548\pi$
$751$	−1.81587	−0.0662619	−0.0331309	+	0.999451i	$0.510548\pi$
$752$	−24.6799	−0.899982
$753$	0	0
$754$	2.23820	0.0815105
$755$	−21.7582	−0.791862
$756$	0	0
$757$	−34.0687	−1.23825	−0.619124	−	0.785293i	$-0.712512\pi$
$757$	−34.0687	−1.23825	−0.619124	+	0.785293i	$0.712512\pi$
$758$	−8.47710	−0.307902
$759$	0	0
$760$	27.7691	1.00729
$761$	−28.7968	−1.04388	−0.521942	−	0.852981i	$-0.674792\pi$
$761$	−28.7968	−1.04388	−0.521942	+	0.852981i	$0.674792\pi$
$762$	0	0
$763$	0	0
$764$	7.82925	0.283252
$765$	0	0
$766$	8.81952	0.318662
$767$	3.75511	0.135589
$768$	0	0
$769$	−19.3395	−0.697399	−0.348699	−	0.937235i	$-0.613377\pi$
$769$	−19.3395	−0.697399	−0.348699	+	0.937235i	$0.613377\pi$
$770$	0	0
$771$	0	0
$772$	30.7733	1.10756
$773$	−22.2080	−0.798765	−0.399383	−	0.916784i	$-0.630775\pi$
$773$	−22.2080	−0.798765	−0.399383	+	0.916784i	$0.630775\pi$
$774$	0	0
$775$	−2.00731	−0.0721047
$776$	−5.43955	−0.195269
$777$	0	0
$778$	−0.952807	−0.0341598
$779$	60.0444	2.15131
$780$	0	0
$781$	−14.3610	−0.513875
$782$	8.11823	0.290307
$783$	0	0
$784$	0	0
$785$	34.6258	1.23585
$786$	0	0
$787$	−35.9459	−1.28133	−0.640667	−	0.767819i	$-0.721342\pi$
$787$	−35.9459	−1.28133	−0.640667	+	0.767819i	$0.721342\pi$
$788$	−7.06298	−0.251608
$789$	0	0
$790$	15.8919	0.565407
$791$	0	0
$792$	0	0
$793$	10.0000	0.355110
$794$	8.70506	0.308931
$795$	0	0
$796$	−51.7691	−1.83491
$797$	−11.4659	−0.406144	−0.203072	−	0.979164i	$-0.565093\pi$
$797$	−11.4659	−0.406144	−0.203072	+	0.979164i	$0.565093\pi$
$798$	0	0
$799$	−30.3610	−1.07409
$800$	−25.5781	−0.904323
$801$	0	0
$802$	11.1805	0.394796
$803$	2.86427	0.101078
$804$	0	0
$805$	0	0
$806$	−0.103388	−0.00364169
$807$	0	0
$808$	12.7503	0.448553
$809$	45.9029	1.61386	0.806931	−	0.590646i	$-0.201127\pi$
$809$	45.9029	1.61386	0.806931	+	0.590646i	$0.201127\pi$
$810$	0	0
$811$	17.8846	0.628012	0.314006	−	0.949421i	$-0.398329\pi$
$811$	17.8846	0.628012	0.314006	+	0.949421i	$0.398329\pi$
$812$	0	0
$813$	0	0
$814$	−4.51208	−0.158148
$815$	25.7351	0.901462
$816$	0	0
$817$	36.7403	1.28538
$818$	−2.57529	−0.0900428
$819$	0	0
$820$	−63.7334	−2.22567
$821$	−6.51599	−0.227409	−0.113705	−	0.993515i	$-0.536272\pi$
$821$	−6.51599	−0.227409	−0.113705	+	0.993515i	$0.536272\pi$
$822$	0	0
$823$	−28.7219	−1.00118	−0.500592	−	0.865684i	$-0.666884\pi$
$823$	−28.7219	−1.00118	−0.500592	+	0.865684i	$0.666884\pi$
$824$	8.51947	0.296790
$825$	0	0
$826$	0	0
$827$	40.3905	1.40452	0.702258	−	0.711923i	$-0.252175\pi$
$827$	40.3905	1.40452	0.702258	+	0.711923i	$0.252175\pi$
$828$	0	0
$829$	18.0000	0.625166	0.312583	−	0.949890i	$-0.398806\pi$
$829$	18.0000	0.625166	0.312583	+	0.949890i	$0.398806\pi$
$830$	0.753766	0.0261636
$831$	0	0
$832$	5.28174	0.183111
$833$	0	0
$834$	0	0
$835$	−53.7077	−1.85863
$836$	19.2041	0.664188
$837$	0	0
$838$	−12.6862	−0.438239
$839$	−20.1983	−0.697324	−0.348662	−	0.937249i	$-0.613364\pi$
$839$	−20.1983	−0.697324	−0.348662	+	0.937249i	$0.613364\pi$
$840$	0	0
$841$	13.0614	0.450392
$842$	−2.51426	−0.0866472
$843$	0	0
$844$	−3.87724	−0.133460
$845$	−3.42059	−0.117672
$846$	0	0
$847$	0	0
$848$	−32.5153	−1.11658
$849$	0	0
$850$	−9.38622	−0.321945
$851$	−44.9816	−1.54195
$852$	0	0
$853$	−22.2996	−0.763523	−0.381762	−	0.924261i	$-0.624682\pi$
$853$	−22.2996	−0.763523	−0.381762	+	0.924261i	$0.624682\pi$
$854$	0	0
$855$	0	0
$856$	−20.9601	−0.716402
$857$	28.7663	0.982639	0.491319	−	0.870979i	$-0.336515\pi$
$857$	28.7663	0.982639	0.491319	+	0.870979i	$0.336515\pi$
$858$	0	0
$859$	10.2025	0.348106	0.174053	−	0.984736i	$-0.444314\pi$
$859$	10.2025	0.348106	0.174053	+	0.984736i	$0.444314\pi$
$860$	−38.9976	−1.32981
$861$	0	0
$862$	0.176276	0.00600399
$863$	−21.3932	−0.728232	−0.364116	−	0.931354i	$-0.618629\pi$
$863$	−21.3932	−0.728232	−0.364116	+	0.931354i	$0.618629\pi$
$864$	0	0
$865$	−44.4407	−1.51103
$866$	3.49346	0.118713
$867$	0	0
$868$	0	0
$869$	22.6764	0.769244
$870$	0	0
$871$	−9.76180	−0.330766
$872$	13.3934	0.453557
$873$	0	0
$874$	−12.1228	−0.410058
$875$	0	0
$876$	0	0
$877$	9.87724	0.333531	0.166765	−	0.985997i	$-0.446668\pi$
$877$	9.87724	0.333531	0.166765	+	0.985997i	$0.446668\pi$
$878$	10.1067	0.341084
$879$	0	0
$880$	−19.0115	−0.640878
$881$	48.9373	1.64874	0.824370	−	0.566051i	$-0.191530\pi$
$881$	48.9373	1.64874	0.824370	+	0.566051i	$0.191530\pi$
$882$	0	0
$883$	40.2455	1.35437	0.677185	−	0.735813i	$-0.263200\pi$
$883$	40.2455	1.35437	0.677185	+	0.735813i	$0.263200\pi$
$884$	7.63478	0.256786
$885$	0	0
$886$	−13.2927	−0.446577
$887$	−42.4275	−1.42458	−0.712288	−	0.701887i	$-0.752341\pi$
$887$	−42.4275	−1.42458	−0.712288	+	0.701887i	$0.752341\pi$
$888$	0	0
$889$	0	0
$890$	−15.2509	−0.511212
$891$	0	0
$892$	−3.87724	−0.129820
$893$	45.3373	1.51715
$894$	0	0
$895$	36.0687	1.20564
$896$	0	0
$897$	0	0
$898$	−7.62120	−0.254323
$899$	−1.94292	−0.0647999
$900$	0	0
$901$	−40.0000	−1.33259
$902$	5.75859	0.191740
$903$	0	0
$904$	−8.68624	−0.288900
$905$	34.2059	1.13704
$906$	0	0
$907$	−42.9060	−1.42467	−0.712336	−	0.701839i	$-0.752363\pi$
$907$	−42.9060	−1.42467	−0.712336	+	0.701839i	$0.752363\pi$
$908$	32.7981	1.08844
$909$	0	0
$910$	0	0
$911$	37.1979	1.23242	0.616210	−	0.787582i	$-0.288667\pi$
$911$	37.1979	1.23242	0.616210	+	0.787582i	$0.288667\pi$
$912$	0	0
$913$	1.07556	0.0355959
$914$	−12.5786	−0.416062
$915$	0	0
$916$	−15.7261	−0.519606
$917$	0	0
$918$	0	0
$919$	52.2455	1.72342	0.861710	−	0.507401i	$-0.169394\pi$
$919$	52.2455	1.72342	0.861710	+	0.507401i	$0.169394\pi$
$920$	26.5499	0.875325
$921$	0	0
$922$	7.85940	0.258836
$923$	8.52562	0.280624
$924$	0	0
$925$	52.0073	1.70999
$926$	4.74933	0.156073
$927$	0	0
$928$	−24.7576	−0.812708
$929$	9.48934	0.311335	0.155668	−	0.987810i	$-0.450247\pi$
$929$	9.48934	0.311335	0.155668	+	0.987810i	$0.450247\pi$
$930$	0	0
$931$	0	0
$932$	−48.0103	−1.57263
$933$	0	0
$934$	4.68624	0.153338
$935$	−23.3878	−0.764863
$936$	0	0
$937$	−57.1699	−1.86766	−0.933830	−	0.357716i	$-0.883556\pi$
$937$	−57.1699	−1.86766	−0.933830	+	0.357716i	$0.883556\pi$
$938$	0	0
$939$	0	0
$940$	−48.1228	−1.56959
$941$	16.4339	0.535730	0.267865	−	0.963456i	$-0.413682\pi$
$941$	16.4339	0.535730	0.267865	+	0.963456i	$0.413682\pi$
$942$	0	0
$943$	57.4082	1.86947
$944$	−12.3903	−0.403269
$945$	0	0
$946$	3.52360	0.114562
$947$	6.43378	0.209070	0.104535	−	0.994521i	$-0.466665\pi$
$947$	6.43378	0.209070	0.104535	+	0.994521i	$0.466665\pi$
$948$	0	0
$949$	−1.70042	−0.0551980
$950$	14.0162	0.454746
$951$	0	0
$952$	0	0
$953$	20.1678	0.653300	0.326650	−	0.945145i	$-0.394080\pi$
$953$	20.1678	0.653300	0.326650	+	0.945145i	$0.394080\pi$
$954$	0	0
$955$	14.2382	0.460737
$956$	31.3054	1.01249
$957$	0	0
$958$	−4.97796	−0.160831
$959$	0	0
$960$	0	0
$961$	−30.9103	−0.997105
$962$	2.67867	0.0863639
$963$	0	0
$964$	9.59496	0.309033
$965$	55.9641	1.80155
$966$	0	0
$967$	8.59916	0.276530	0.138265	−	0.990395i	$-0.455847\pi$
$967$	8.59916	0.276530	0.138265	+	0.990395i	$0.455847\pi$
$968$	−10.9325	−0.351385
$969$	0	0
$970$	−4.79437	−0.153938
$971$	54.0180	1.73352	0.866760	−	0.498725i	$-0.166198\pi$
$971$	54.0180	1.73352	0.866760	+	0.498725i	$0.166198\pi$
$972$	0	0
$973$	0	0
$974$	8.72624	0.279607
$975$	0	0
$976$	−32.9958	−1.05617
$977$	37.8576	1.21117	0.605586	−	0.795780i	$-0.292939\pi$
$977$	37.8576	1.21117	0.605586	+	0.795780i	$0.292939\pi$
$978$	0	0
$979$	−21.7618	−0.695510
$980$	0	0
$981$	0	0
$982$	−4.44803	−0.141942
$983$	−11.0554	−0.352612	−0.176306	−	0.984335i	$-0.556415\pi$
$983$	−11.0554	−0.352612	−0.176306	+	0.984335i	$0.556415\pi$
$984$	0	0
$985$	−12.8447	−0.409265
$986$	−9.08512	−0.289329
$987$	0	0
$988$	−11.4008	−0.362709
$989$	35.1272	1.11698
$990$	0	0
$991$	−4.47640	−0.142198	−0.0710988	−	0.997469i	$-0.522651\pi$
$991$	−4.47640	−0.142198	−0.0710988	+	0.997469i	$0.522651\pi$
$992$	1.14361	0.0363098
$993$	0	0
$994$	0	0
$995$	−94.1469	−2.98466
$996$	0	0
$997$	16.4480	0.520914	0.260457	−	0.965485i	$-0.416127\pi$
$997$	16.4480	0.520914	0.260457	+	0.965485i	$0.416127\pi$
$998$	−6.98693	−0.221167
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5733.2.a.bv.1.3		6
3.2	odd	2	inner	5733.2.a.bv.1.4		6
7.6	odd	2		819.2.a.m.1.3	✓	6
21.20	even	2		819.2.a.m.1.4	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.a.m.1.3	✓	6	7.6	odd	2
819.2.a.m.1.4	yes	6	21.20	even	2
5733.2.a.bv.1.3		6	1.1	even	1	trivial
5733.2.a.bv.1.4		6	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 5733 = 3^{2} \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5733.a (trivial)

\(f(q)\)	\(=\)	\(q-0.345110 q^{2} -1.88090 q^{4} -3.42059 q^{5} +1.33934 q^{8} +O(q^{10})\)	\(q-0.345110 q^{2} -1.88090 q^{4} -3.42059 q^{5} +1.33934 q^{8} +1.18048 q^{10} +1.68445 q^{11} -1.00000 q^{13} +3.29958 q^{16} +4.05911 q^{17} -6.06138 q^{19} +6.43378 q^{20} -0.581319 q^{22} -5.79525 q^{23} +6.70042 q^{25} +0.345110 q^{26} +6.48547 q^{29} -0.299580 q^{31} -3.81739 q^{32} -1.40084 q^{34} +7.76180 q^{37} +2.09184 q^{38} -4.58132 q^{40} -9.90606 q^{41} -6.06138 q^{43} -3.16827 q^{44} +2.00000 q^{46} -7.47970 q^{47} -2.31238 q^{50} +1.88090 q^{52} -9.85437 q^{53} -5.76180 q^{55} -2.23820 q^{58} -3.75511 q^{59} -10.0000 q^{61} +0.103388 q^{62} -5.28174 q^{64} +3.42059 q^{65} +9.76180 q^{67} -7.63478 q^{68} -8.52562 q^{71} +1.70042 q^{73} -2.67867 q^{74} +11.4008 q^{76} +13.4622 q^{79} -11.2865 q^{80} +3.41868 q^{82} +0.638526 q^{83} -13.8846 q^{85} +2.09184 q^{86} +2.25604 q^{88} -12.9193 q^{89} +10.9003 q^{92} +2.58132 q^{94} +20.7335 q^{95} -4.06138 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 12 q^{4}+O(q^{10}) \) 6 * q + 12 * q^4	\( 6 q + 12 q^{4} - 8 q^{10} - 6 q^{13} + 28 q^{16} + 2 q^{19} + 28 q^{22} + 32 q^{25} - 10 q^{31} + 8 q^{34} + 4 q^{40} + 2 q^{43} + 12 q^{46} - 12 q^{52} + 12 q^{55} - 60 q^{58} - 60 q^{61} + 8 q^{64} + 12 q^{67} + 2 q^{73} + 52 q^{76} + 26 q^{79} + 52 q^{82} + 40 q^{85} + 108 q^{88} - 16 q^{94} + 14 q^{97}+O(q^{100}) \) 6 * q + 12 * q^4 - 8 * q^10 - 6 * q^13 + 28 * q^16 + 2 * q^19 + 28 * q^22 + 32 * q^25 - 10 * q^31 + 8 * q^34 + 4 * q^40 + 2 * q^43 + 12 * q^46 - 12 * q^52 + 12 * q^55 - 60 * q^58 - 60 * q^61 + 8 * q^64 + 12 * q^67 + 2 * q^73 + 52 * q^76 + 26 * q^79 + 52 * q^82 + 40 * q^85 + 108 * q^88 - 16 * q^94 + 14 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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