LMFDB - Embedded newform 9680.2.a.bt.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9680,2,Mod(1,9680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9680, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("9680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9680 = 2^{4} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9680.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:	$77.2951891566$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 110)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.37228$ of defining polynomial
Character	$\chi$	$=$	9680.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.37228 q^{3} +1.00000 q^{5} -2.37228 q^{7} +2.62772 q^{9} -2.00000 q^{13} -2.37228 q^{15} +4.37228 q^{17} +6.37228 q^{19} +5.62772 q^{21} +8.74456 q^{23} +1.00000 q^{25} +0.883156 q^{27} +4.37228 q^{29} +2.37228 q^{31} -2.37228 q^{35} +3.62772 q^{37} +4.74456 q^{39} -11.4891 q^{41} -4.00000 q^{43} +2.62772 q^{45} +8.74456 q^{47} -1.37228 q^{49} -10.3723 q^{51} +13.1168 q^{53} -15.1168 q^{57} -8.74456 q^{59} -0.372281 q^{61} -6.23369 q^{63} -2.00000 q^{65} -8.00000 q^{67} -20.7446 q^{69} +7.11684 q^{71} -7.48913 q^{73} -2.37228 q^{75} -12.7446 q^{79} -9.97825 q^{81} +8.74456 q^{83} +4.37228 q^{85} -10.3723 q^{87} +4.37228 q^{89} +4.74456 q^{91} -5.62772 q^{93} +6.37228 q^{95} -1.25544 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} + 2 q^{5} + q^{7} + 11 q^{9} - 4 q^{13} + q^{15} + 3 q^{17} + 7 q^{19} + 17 q^{21} + 6 q^{23} + 2 q^{25} + 19 q^{27} + 3 q^{29} - q^{31} + q^{35} + 13 q^{37} - 2 q^{39} - 8 q^{43} + 11 q^{45}+ \cdots - 14 q^{97}+O(q^{100}) $ 2 * q + q^3 + 2 * q^5 + q^7 + 11 * q^9 - 4 * q^13 + q^15 + 3 * q^17 + 7 * q^19 + 17 * q^21 + 6 * q^23 + 2 * q^25 + 19 * q^27 + 3 * q^29 - q^31 + q^35 + 13 * q^37 - 2 * q^39 - 8 * q^43 + 11 * q^45 + 6 * q^47 + 3 * q^49 - 15 * q^51 + 9 * q^53 - 13 * q^57 - 6 * q^59 + 5 * q^61 + 22 * q^63 - 4 * q^65 - 16 * q^67 - 30 * q^69 - 3 * q^71 + 8 * q^73 + q^75 - 14 * q^79 + 26 * q^81 + 6 * q^83 + 3 * q^85 - 15 * q^87 + 3 * q^89 - 2 * q^91 - 17 * q^93 + 7 * q^95 - 14 * q^97

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
$3$	−2.37228	−1.36964	−0.684819	−	0.728714i	$-0.740119\pi$
$3$	−2.37228	−1.36964	−0.684819	+	0.728714i	$0.740119\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.37228	−0.896638	−0.448319	−	0.893874i	$-0.647977\pi$
$7$	−2.37228	−0.896638	−0.448319	+	0.893874i	$0.647977\pi$
$8$	0	0
$9$	2.62772	0.875906
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	−2.37228	−0.612520
$16$	0	0
$17$	4.37228	1.06043	0.530217	−	0.847862i	$-0.322110\pi$
$17$	4.37228	1.06043	0.530217	+	0.847862i	$0.322110\pi$
$18$	0	0
$19$	6.37228	1.46190	0.730951	−	0.682430i	$-0.239077\pi$
$19$	6.37228	1.46190	0.730951	+	0.682430i	$0.239077\pi$
$20$	0	0
$21$	5.62772	1.22807
$22$	0	0
$23$	8.74456	1.82337	0.911684	−	0.410893i	$-0.134783\pi$
$23$	8.74456	1.82337	0.911684	+	0.410893i	$0.134783\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0.883156	0.169963
$28$	0	0
$29$	4.37228	0.811912	0.405956	−	0.913893i	$-0.366939\pi$
$29$	4.37228	0.811912	0.405956	+	0.913893i	$0.366939\pi$
$30$	0	0
$31$	2.37228	0.426074	0.213037	−	0.977044i	$-0.431664\pi$
$31$	2.37228	0.426074	0.213037	+	0.977044i	$0.431664\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.37228	−0.400989
$36$	0	0
$37$	3.62772	0.596393	0.298197	−	0.954504i	$-0.403615\pi$
$37$	3.62772	0.596393	0.298197	+	0.954504i	$0.403615\pi$
$38$	0	0
$39$	4.74456	0.759738
$40$	0	0
$41$	−11.4891	−1.79430	−0.897150	−	0.441726i	$-0.854366\pi$
$41$	−11.4891	−1.79430	−0.897150	+	0.441726i	$0.854366\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	2.62772	0.391717
$46$	0	0
$47$	8.74456	1.27553	0.637763	−	0.770233i	$-0.279860\pi$
$47$	8.74456	1.27553	0.637763	+	0.770233i	$0.279860\pi$
$48$	0	0
$49$	−1.37228	−0.196040
$50$	0	0
$51$	−10.3723	−1.45241
$52$	0	0
$53$	13.1168	1.80174	0.900869	−	0.434092i	$-0.142931\pi$
$53$	13.1168	1.80174	0.900869	+	0.434092i	$0.142931\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−15.1168	−2.00227
$58$	0	0
$59$	−8.74456	−1.13845	−0.569223	−	0.822183i	$-0.692756\pi$
$59$	−8.74456	−1.13845	−0.569223	+	0.822183i	$0.692756\pi$
$60$	0	0
$61$	−0.372281	−0.0476657	−0.0238329	−	0.999716i	$-0.507587\pi$
$61$	−0.372281	−0.0476657	−0.0238329	+	0.999716i	$0.507587\pi$
$62$	0	0
$63$	−6.23369	−0.785371
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	−20.7446	−2.49735
$70$	0	0
$71$	7.11684	0.844614	0.422307	−	0.906453i	$-0.361220\pi$
$71$	7.11684	0.844614	0.422307	+	0.906453i	$0.361220\pi$
$72$	0	0
$73$	−7.48913	−0.876536	−0.438268	−	0.898844i	$-0.644408\pi$
$73$	−7.48913	−0.876536	−0.438268	+	0.898844i	$0.644408\pi$
$74$	0	0
$75$	−2.37228	−0.273927
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−12.7446	−1.43388	−0.716938	−	0.697137i	$-0.754457\pi$
$79$	−12.7446	−1.43388	−0.716938	+	0.697137i	$0.754457\pi$
$80$	0	0
$81$	−9.97825	−1.10869
$82$	0	0
$83$	8.74456	0.959840	0.479920	−	0.877312i	$-0.340666\pi$
$83$	8.74456	0.959840	0.479920	+	0.877312i	$0.340666\pi$
$84$	0	0
$85$	4.37228	0.474240
$86$	0	0
$87$	−10.3723	−1.11203
$88$	0	0
$89$	4.37228	0.463461	0.231730	−	0.972780i	$-0.425561\pi$
$89$	4.37228	0.463461	0.231730	+	0.972780i	$0.425561\pi$
$90$	0	0
$91$	4.74456	0.497365
$92$	0	0
$93$	−5.62772	−0.583567
$94$	0	0
$95$	6.37228	0.653782
$96$	0	0
$97$	−1.25544	−0.127470	−0.0637352	−	0.997967i	$-0.520301\pi$
$97$	−1.25544	−0.127470	−0.0637352	+	0.997967i	$0.520301\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	−13.4891	−1.32912	−0.664562	−	0.747234i	$-0.731382\pi$
$103$	−13.4891	−1.32912	−0.664562	+	0.747234i	$0.731382\pi$
$104$	0	0
$105$	5.62772	0.549209
$106$	0	0
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	−7.48913	−0.717328	−0.358664	−	0.933467i	$-0.616768\pi$
$109$	−7.48913	−0.717328	−0.358664	+	0.933467i	$0.616768\pi$
$110$	0	0
$111$	−8.60597	−0.816842
$112$	0	0
$113$	14.7446	1.38705	0.693526	−	0.720432i	$-0.256056\pi$
$113$	14.7446	1.38705	0.693526	+	0.720432i	$0.256056\pi$
$114$	0	0
$115$	8.74456	0.815435
$116$	0	0
$117$	−5.25544	−0.485865
$118$	0	0
$119$	−10.3723	−0.950825
$120$	0	0
$121$	0	0
$122$	0	0
$123$	27.2554	2.45754
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	0	0
$129$	9.48913	0.835471
$130$	0	0
$131$	4.88316	0.426643	0.213322	−	0.976982i	$-0.431572\pi$
$131$	4.88316	0.426643	0.213322	+	0.976982i	$0.431572\pi$
$132$	0	0
$133$	−15.1168	−1.31080
$134$	0	0
$135$	0.883156	0.0760100
$136$	0	0
$137$	−2.74456	−0.234484	−0.117242	−	0.993103i	$-0.537405\pi$
$137$	−2.74456	−0.234484	−0.117242	+	0.993103i	$0.537405\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	−20.7446	−1.74701
$142$	0	0
$143$	0	0
$144$	0	0
$145$	4.37228	0.363098
$146$	0	0
$147$	3.25544	0.268504
$148$	0	0
$149$	18.6060	1.52426	0.762130	−	0.647424i	$-0.224154\pi$
$149$	18.6060	1.52426	0.762130	+	0.647424i	$0.224154\pi$
$150$	0	0
$151$	22.2337	1.80935	0.904676	−	0.426100i	$-0.140113\pi$
$151$	22.2337	1.80935	0.904676	+	0.426100i	$0.140113\pi$
$152$	0	0
$153$	11.4891	0.928841
$154$	0	0
$155$	2.37228	0.190546
$156$	0	0
$157$	3.62772	0.289523	0.144762	−	0.989467i	$-0.453758\pi$
$157$	3.62772	0.289523	0.144762	+	0.989467i	$0.453758\pi$
$158$	0	0
$159$	−31.1168	−2.46773
$160$	0	0
$161$	−20.7446	−1.63490
$162$	0	0
$163$	23.1168	1.81065	0.905325	−	0.424718i	$-0.139627\pi$
$163$	23.1168	1.81065	0.905325	+	0.424718i	$0.139627\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−10.3723	−0.802631	−0.401316	−	0.915940i	$-0.631447\pi$
$167$	−10.3723	−0.802631	−0.401316	+	0.915940i	$0.631447\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	16.7446	1.28049
$172$	0	0
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	−2.37228	−0.179328
$176$	0	0
$177$	20.7446	1.55926
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	0.883156	0.0652848
$184$	0	0
$185$	3.62772	0.266715
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−2.09509	−0.152396
$190$	0	0
$191$	−17.4891	−1.26547	−0.632734	−	0.774369i	$-0.718067\pi$
$191$	−17.4891	−1.26547	−0.632734	+	0.774369i	$0.718067\pi$
$192$	0	0
$193$	13.8614	0.997766	0.498883	−	0.866669i	$-0.333744\pi$
$193$	13.8614	0.997766	0.498883	+	0.866669i	$0.333744\pi$
$194$	0	0
$195$	4.74456	0.339765
$196$	0	0
$197$	9.25544	0.659423	0.329711	−	0.944082i	$-0.393049\pi$
$197$	9.25544	0.659423	0.329711	+	0.944082i	$0.393049\pi$
$198$	0	0
$199$	−0.883156	−0.0626053	−0.0313026	−	0.999510i	$-0.509966\pi$
$199$	−0.883156	−0.0626053	−0.0313026	+	0.999510i	$0.509966\pi$
$200$	0	0
$201$	18.9783	1.33862
$202$	0	0
$203$	−10.3723	−0.727991
$204$	0	0
$205$	−11.4891	−0.802435
$206$	0	0
$207$	22.9783	1.59710
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−11.1168	−0.765315	−0.382658	−	0.923890i	$-0.624991\pi$
$211$	−11.1168	−0.765315	−0.382658	+	0.923890i	$0.624991\pi$
$212$	0	0
$213$	−16.8832	−1.15681
$214$	0	0
$215$	−4.00000	−0.272798
$216$	0	0
$217$	−5.62772	−0.382034
$218$	0	0
$219$	17.7663	1.20054
$220$	0	0
$221$	−8.74456	−0.588223
$222$	0	0
$223$	7.25544	0.485860	0.242930	−	0.970044i	$-0.421891\pi$
$223$	7.25544	0.485860	0.242930	+	0.970044i	$0.421891\pi$
$224$	0	0
$225$	2.62772	0.175181
$226$	0	0
$227$	8.74456	0.580397	0.290199	−	0.956966i	$-0.406279\pi$
$227$	8.74456	0.580397	0.290199	+	0.956966i	$0.406279\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4.37228	0.286438	0.143219	−	0.989691i	$-0.454255\pi$
$233$	4.37228	0.286438	0.143219	+	0.989691i	$0.454255\pi$
$234$	0	0
$235$	8.74456	0.570432
$236$	0	0
$237$	30.2337	1.96389
$238$	0	0
$239$	−3.25544	−0.210577	−0.105288	−	0.994442i	$-0.533577\pi$
$239$	−3.25544	−0.210577	−0.105288	+	0.994442i	$0.533577\pi$
$240$	0	0
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	0	0
$243$	21.0217	1.34855
$244$	0	0
$245$	−1.37228	−0.0876718
$246$	0	0
$247$	−12.7446	−0.810917
$248$	0	0
$249$	−20.7446	−1.31463
$250$	0	0
$251$	8.74456	0.551952	0.275976	−	0.961165i	$-0.410999\pi$
$251$	8.74456	0.551952	0.275976	+	0.961165i	$0.410999\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−10.3723	−0.649537
$256$	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
$259$	−8.60597	−0.534749
$260$	0	0
$261$	11.4891	0.711159
$262$	0	0
$263$	3.86141	0.238105	0.119052	−	0.992888i	$-0.462014\pi$
$263$	3.86141	0.238105	0.119052	+	0.992888i	$0.462014\pi$
$264$	0	0
$265$	13.1168	0.805761
$266$	0	0
$267$	−10.3723	−0.634773
$268$	0	0
$269$	2.74456	0.167339	0.0836695	−	0.996494i	$-0.473336\pi$
$269$	2.74456	0.167339	0.0836695	+	0.996494i	$0.473336\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	0	0
$273$	−11.2554	−0.681210
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.25544	0.0754319	0.0377160	−	0.999289i	$-0.487992\pi$
$277$	1.25544	0.0754319	0.0377160	+	0.999289i	$0.487992\pi$
$278$	0	0
$279$	6.23369	0.373201
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	16.7446	0.995361	0.497680	−	0.867360i	$-0.334185\pi$
$283$	16.7446	0.995361	0.497680	+	0.867360i	$0.334185\pi$
$284$	0	0
$285$	−15.1168	−0.895445
$286$	0	0
$287$	27.2554	1.60884
$288$	0	0
$289$	2.11684	0.124520
$290$	0	0
$291$	2.97825	0.174588
$292$	0	0
$293$	−0.510875	−0.0298456	−0.0149228	−	0.999889i	$-0.504750\pi$
$293$	−0.510875	−0.0298456	−0.0149228	+	0.999889i	$0.504750\pi$
$294$	0	0
$295$	−8.74456	−0.509128
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−17.4891	−1.01142
$300$	0	0
$301$	9.48913	0.546944
$302$	0	0
$303$	14.2337	0.817704
$304$	0	0
$305$	−0.372281	−0.0213168
$306$	0	0
$307$	16.7446	0.955663	0.477831	−	0.878452i	$-0.341423\pi$
$307$	16.7446	0.955663	0.477831	+	0.878452i	$0.341423\pi$
$308$	0	0
$309$	32.0000	1.82042
$310$	0	0
$311$	−13.6277	−0.772757	−0.386379	−	0.922340i	$-0.626274\pi$
$311$	−13.6277	−0.772757	−0.386379	+	0.922340i	$0.626274\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	0	0
$315$	−6.23369	−0.351229
$316$	0	0
$317$	27.3505	1.53616	0.768079	−	0.640355i	$-0.221213\pi$
$317$	27.3505	1.53616	0.768079	+	0.640355i	$0.221213\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	28.4674	1.58889
$322$	0	0
$323$	27.8614	1.55025
$324$	0	0
$325$	−2.00000	−0.110940
$326$	0	0
$327$	17.7663	0.982479
$328$	0	0
$329$	−20.7446	−1.14368
$330$	0	0
$331$	14.9783	0.823279	0.411640	−	0.911347i	$-0.364956\pi$
$331$	14.9783	0.823279	0.411640	+	0.911347i	$0.364956\pi$
$332$	0	0
$333$	9.53262	0.522385
$334$	0	0
$335$	−8.00000	−0.437087
$336$	0	0
$337$	−6.88316	−0.374949	−0.187475	−	0.982269i	$-0.560030\pi$
$337$	−6.88316	−0.374949	−0.187475	+	0.982269i	$0.560030\pi$
$338$	0	0
$339$	−34.9783	−1.89976
$340$	0	0
$341$	0	0
$342$	0	0
$343$	19.8614	1.07242
$344$	0	0
$345$	−20.7446	−1.11685
$346$	0	0
$347$	2.23369	0.119911	0.0599553	−	0.998201i	$-0.480904\pi$
$347$	2.23369	0.119911	0.0599553	+	0.998201i	$0.480904\pi$
$348$	0	0
$349$	3.48913	0.186769	0.0933843	−	0.995630i	$-0.470231\pi$
$349$	3.48913	0.186769	0.0933843	+	0.995630i	$0.470231\pi$
$350$	0	0
$351$	−1.76631	−0.0942788
$352$	0	0
$353$	−23.4891	−1.25020	−0.625100	−	0.780545i	$-0.714942\pi$
$353$	−23.4891	−1.25020	−0.625100	+	0.780545i	$0.714942\pi$
$354$	0	0
$355$	7.11684	0.377723
$356$	0	0
$357$	24.6060	1.30229
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	21.6060	1.13716
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−7.48913	−0.391999
$366$	0	0
$367$	4.00000	0.208798	0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000	0.208798	0.104399	+	0.994535i	$0.466708\pi$
$368$	0	0
$369$	−30.1902	−1.57164
$370$	0	0
$371$	−31.1168	−1.61551
$372$	0	0
$373$	−8.51087	−0.440676	−0.220338	−	0.975424i	$-0.570716\pi$
$373$	−8.51087	−0.440676	−0.220338	+	0.975424i	$0.570716\pi$
$374$	0	0
$375$	−2.37228	−0.122504
$376$	0	0
$377$	−8.74456	−0.450368
$378$	0	0
$379$	−34.2337	−1.75847	−0.879233	−	0.476392i	$-0.841944\pi$
$379$	−34.2337	−1.75847	−0.879233	+	0.476392i	$0.841944\pi$
$380$	0	0
$381$	−18.9783	−0.972285
$382$	0	0
$383$	2.23369	0.114136	0.0570681	−	0.998370i	$-0.481825\pi$
$383$	2.23369	0.114136	0.0570681	+	0.998370i	$0.481825\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−10.5109	−0.534298
$388$	0	0
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	38.2337	1.93356
$392$	0	0
$393$	−11.5842	−0.584347
$394$	0	0
$395$	−12.7446	−0.641249
$396$	0	0
$397$	−20.9783	−1.05287	−0.526434	−	0.850216i	$-0.676471\pi$
$397$	−20.9783	−1.05287	−0.526434	+	0.850216i	$0.676471\pi$
$398$	0	0
$399$	35.8614	1.79532
$400$	0	0
$401$	7.62772	0.380910	0.190455	−	0.981696i	$-0.439004\pi$
$401$	7.62772	0.380910	0.190455	+	0.981696i	$0.439004\pi$
$402$	0	0
$403$	−4.74456	−0.236343
$404$	0	0
$405$	−9.97825	−0.495823
$406$	0	0
$407$	0	0
$408$	0	0
$409$	36.2337	1.79164	0.895820	−	0.444417i	$-0.146589\pi$
$409$	36.2337	1.79164	0.895820	+	0.444417i	$0.146589\pi$
$410$	0	0
$411$	6.51087	0.321158
$412$	0	0
$413$	20.7446	1.02077
$414$	0	0
$415$	8.74456	0.429254
$416$	0	0
$417$	9.48913	0.464684
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	−24.2337	−1.18108	−0.590539	−	0.807009i	$-0.701085\pi$
$421$	−24.2337	−1.18108	−0.590539	+	0.807009i	$0.701085\pi$
$422$	0	0
$423$	22.9783	1.11724
$424$	0	0
$425$	4.37228	0.212087
$426$	0	0
$427$	0.883156	0.0427389
$428$	0	0
$429$	0	0
$430$	0	0
$431$	10.9783	0.528804	0.264402	−	0.964413i	$-0.414825\pi$
$431$	10.9783	0.528804	0.264402	+	0.964413i	$0.414825\pi$
$432$	0	0
$433$	−29.7228	−1.42839	−0.714194	−	0.699948i	$-0.753206\pi$
$433$	−29.7228	−1.42839	−0.714194	+	0.699948i	$0.753206\pi$
$434$	0	0
$435$	−10.3723	−0.497313
$436$	0	0
$437$	55.7228	2.66558
$438$	0	0
$439$	−26.9783	−1.28760	−0.643801	−	0.765193i	$-0.722643\pi$
$439$	−26.9783	−1.28760	−0.643801	+	0.765193i	$0.722643\pi$
$440$	0	0
$441$	−3.60597	−0.171713
$442$	0	0
$443$	−6.51087	−0.309341	−0.154670	−	0.987966i	$-0.549432\pi$
$443$	−6.51087	−0.309341	−0.154670	+	0.987966i	$0.549432\pi$
$444$	0	0
$445$	4.37228	0.207266
$446$	0	0
$447$	−44.1386	−2.08768
$448$	0	0
$449$	−16.9783	−0.801253	−0.400627	−	0.916241i	$-0.631208\pi$
$449$	−16.9783	−0.801253	−0.400627	+	0.916241i	$0.631208\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−52.7446	−2.47816
$454$	0	0
$455$	4.74456	0.222429
$456$	0	0
$457$	−35.3505	−1.65363	−0.826814	−	0.562475i	$-0.809849\pi$
$457$	−35.3505	−1.65363	−0.826814	+	0.562475i	$0.809849\pi$
$458$	0	0
$459$	3.86141	0.180235
$460$	0	0
$461$	1.11684	0.0520166	0.0260083	−	0.999662i	$-0.491720\pi$
$461$	1.11684	0.0520166	0.0260083	+	0.999662i	$0.491720\pi$
$462$	0	0
$463$	−34.2337	−1.59097	−0.795487	−	0.605970i	$-0.792785\pi$
$463$	−34.2337	−1.59097	−0.795487	+	0.605970i	$0.792785\pi$
$464$	0	0
$465$	−5.62772	−0.260979
$466$	0	0
$467$	13.6277	0.630616	0.315308	−	0.948989i	$-0.397892\pi$
$467$	13.6277	0.630616	0.315308	+	0.948989i	$0.397892\pi$
$468$	0	0
$469$	18.9783	0.876334
$470$	0	0
$471$	−8.60597	−0.396542
$472$	0	0
$473$	0	0
$474$	0	0
$475$	6.37228	0.292380
$476$	0	0
$477$	34.4674	1.57815
$478$	0	0
$479$	−17.4891	−0.799099	−0.399549	−	0.916712i	$-0.630833\pi$
$479$	−17.4891	−0.799099	−0.399549	+	0.916712i	$0.630833\pi$
$480$	0	0
$481$	−7.25544	−0.330819
$482$	0	0
$483$	49.2119	2.23922
$484$	0	0
$485$	−1.25544	−0.0570065
$486$	0	0
$487$	−20.0000	−0.906287	−0.453143	−	0.891438i	$-0.649697\pi$
$487$	−20.0000	−0.906287	−0.453143	+	0.891438i	$0.649697\pi$
$488$	0	0
$489$	−54.8397	−2.47994
$490$	0	0
$491$	−1.62772	−0.0734579	−0.0367290	−	0.999325i	$-0.511694\pi$
$491$	−1.62772	−0.0734579	−0.0367290	+	0.999325i	$0.511694\pi$
$492$	0	0
$493$	19.1168	0.860979
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−16.8832	−0.757313
$498$	0	0
$499$	10.5109	0.470531	0.235266	−	0.971931i	$-0.424404\pi$
$499$	10.5109	0.470531	0.235266	+	0.971931i	$0.424404\pi$
$500$	0	0
$501$	24.6060	1.09931
$502$	0	0
$503$	10.9783	0.489496	0.244748	−	0.969587i	$-0.421295\pi$
$503$	10.9783	0.489496	0.244748	+	0.969587i	$0.421295\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	0	0
$507$	21.3505	0.948210
$508$	0	0
$509$	44.2337	1.96062	0.980312	−	0.197455i	$-0.0632678\pi$
$509$	44.2337	1.96062	0.980312	+	0.197455i	$0.0632678\pi$
$510$	0	0
$511$	17.7663	0.785935
$512$	0	0
$513$	5.62772	0.248470
$514$	0	0
$515$	−13.4891	−0.594402
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−14.2337	−0.624790
$520$	0	0
$521$	35.4891	1.55481	0.777403	−	0.629002i	$-0.216536\pi$
$521$	35.4891	1.55481	0.777403	+	0.629002i	$0.216536\pi$
$522$	0	0
$523$	−14.9783	−0.654953	−0.327477	−	0.944859i	$-0.606198\pi$
$523$	−14.9783	−0.654953	−0.327477	+	0.944859i	$0.606198\pi$
$524$	0	0
$525$	5.62772	0.245614
$526$	0	0
$527$	10.3723	0.451824
$528$	0	0
$529$	53.4674	2.32467
$530$	0	0
$531$	−22.9783	−0.997171
$532$	0	0
$533$	22.9783	0.995299
$534$	0	0
$535$	−12.0000	−0.518805
$536$	0	0
$537$	−28.4674	−1.22846
$538$	0	0
$539$	0	0
$540$	0	0
$541$	2.88316	0.123957	0.0619783	−	0.998077i	$-0.480259\pi$
$541$	2.88316	0.123957	0.0619783	+	0.998077i	$0.480259\pi$
$542$	0	0
$543$	23.7228	1.01804
$544$	0	0
$545$	−7.48913	−0.320799
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	0	0
$549$	−0.978251	−0.0417507
$550$	0	0
$551$	27.8614	1.18694
$552$	0	0
$553$	30.2337	1.28567
$554$	0	0
$555$	−8.60597	−0.365303
$556$	0	0
$557$	40.9783	1.73630	0.868152	−	0.496298i	$-0.165308\pi$
$557$	40.9783	1.73630	0.868152	+	0.496298i	$0.165308\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	0	0
$563$	26.2337	1.10562	0.552809	−	0.833308i	$-0.313556\pi$
$563$	26.2337	1.10562	0.552809	+	0.833308i	$0.313556\pi$
$564$	0	0
$565$	14.7446	0.620308
$566$	0	0
$567$	23.6712	0.994098
$568$	0	0
$569$	26.7446	1.12119	0.560595	−	0.828090i	$-0.310572\pi$
$569$	26.7446	1.12119	0.560595	+	0.828090i	$0.310572\pi$
$570$	0	0
$571$	9.62772	0.402907	0.201454	−	0.979498i	$-0.435433\pi$
$571$	9.62772	0.402907	0.201454	+	0.979498i	$0.435433\pi$
$572$	0	0
$573$	41.4891	1.73323
$574$	0	0
$575$	8.74456	0.364673
$576$	0	0
$577$	−8.97825	−0.373769	−0.186885	−	0.982382i	$-0.559839\pi$
$577$	−8.97825	−0.373769	−0.186885	+	0.982382i	$0.559839\pi$
$578$	0	0
$579$	−32.8832	−1.36658
$580$	0	0
$581$	−20.7446	−0.860629
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−5.25544	−0.217286
$586$	0	0
$587$	3.86141	0.159377	0.0796887	−	0.996820i	$-0.474607\pi$
$587$	3.86141	0.159377	0.0796887	+	0.996820i	$0.474607\pi$
$588$	0	0
$589$	15.1168	0.622879
$590$	0	0
$591$	−21.9565	−0.903170
$592$	0	0
$593$	35.4891	1.45736	0.728682	−	0.684852i	$-0.240133\pi$
$593$	35.4891	1.45736	0.728682	+	0.684852i	$0.240133\pi$
$594$	0	0
$595$	−10.3723	−0.425222
$596$	0	0
$597$	2.09509	0.0857465
$598$	0	0
$599$	−0.605969	−0.0247592	−0.0123796	−	0.999923i	$-0.503941\pi$
$599$	−0.605969	−0.0247592	−0.0123796	+	0.999923i	$0.503941\pi$
$600$	0	0
$601$	39.4891	1.61080	0.805398	−	0.592735i	$-0.201952\pi$
$601$	39.4891	1.61080	0.805398	+	0.592735i	$0.201952\pi$
$602$	0	0
$603$	−21.0217	−0.856072
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−23.1168	−0.938284	−0.469142	−	0.883123i	$-0.655437\pi$
$607$	−23.1168	−0.938284	−0.469142	+	0.883123i	$0.655437\pi$
$608$	0	0
$609$	24.6060	0.997084
$610$	0	0
$611$	−17.4891	−0.707534
$612$	0	0
$613$	−43.4891	−1.75651	−0.878255	−	0.478193i	$-0.841292\pi$
$613$	−43.4891	−1.75651	−0.878255	+	0.478193i	$0.841292\pi$
$614$	0	0
$615$	27.2554	1.09905
$616$	0	0
$617$	32.2337	1.29768	0.648840	−	0.760925i	$-0.275255\pi$
$617$	32.2337	1.29768	0.648840	+	0.760925i	$0.275255\pi$
$618$	0	0
$619$	−24.4674	−0.983427	−0.491713	−	0.870757i	$-0.663629\pi$
$619$	−24.4674	−0.983427	−0.491713	+	0.870757i	$0.663629\pi$
$620$	0	0
$621$	7.72281	0.309906
$622$	0	0
$623$	−10.3723	−0.415557
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	15.8614	0.632436
$630$	0	0
$631$	−24.8832	−0.990583	−0.495291	−	0.868727i	$-0.664939\pi$
$631$	−24.8832	−0.990583	−0.495291	+	0.868727i	$0.664939\pi$
$632$	0	0
$633$	26.3723	1.04820
$634$	0	0
$635$	8.00000	0.317470
$636$	0	0
$637$	2.74456	0.108744
$638$	0	0
$639$	18.7011	0.739803
$640$	0	0
$641$	36.0951	1.42567	0.712835	−	0.701332i	$-0.247411\pi$
$641$	36.0951	1.42567	0.712835	+	0.701332i	$0.247411\pi$
$642$	0	0
$643$	23.1168	0.911639	0.455820	−	0.890072i	$-0.349346\pi$
$643$	23.1168	0.911639	0.455820	+	0.890072i	$0.349346\pi$
$644$	0	0
$645$	9.48913	0.373634
$646$	0	0
$647$	−19.7228	−0.775384	−0.387692	−	0.921789i	$-0.626728\pi$
$647$	−19.7228	−0.775384	−0.387692	+	0.921789i	$0.626728\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	13.3505	0.523249
$652$	0	0
$653$	16.3723	0.640697	0.320348	−	0.947300i	$-0.396200\pi$
$653$	16.3723	0.640697	0.320348	+	0.947300i	$0.396200\pi$
$654$	0	0
$655$	4.88316	0.190801
$656$	0	0
$657$	−19.6793	−0.767763
$658$	0	0
$659$	−15.8614	−0.617873	−0.308936	−	0.951083i	$-0.599973\pi$
$659$	−15.8614	−0.617873	−0.308936	+	0.951083i	$0.599973\pi$
$660$	0	0
$661$	31.4891	1.22479	0.612393	−	0.790554i	$-0.290207\pi$
$661$	31.4891	1.22479	0.612393	+	0.790554i	$0.290207\pi$
$662$	0	0
$663$	20.7446	0.805652
$664$	0	0
$665$	−15.1168	−0.586206
$666$	0	0
$667$	38.2337	1.48041
$668$	0	0
$669$	−17.2119	−0.665452
$670$	0	0
$671$	0	0
$672$	0	0
$673$	13.8614	0.534318	0.267159	−	0.963652i	$-0.413915\pi$
$673$	13.8614	0.534318	0.267159	+	0.963652i	$0.413915\pi$
$674$	0	0
$675$	0.883156	0.0339927
$676$	0	0
$677$	−14.7446	−0.566680	−0.283340	−	0.959020i	$-0.591442\pi$
$677$	−14.7446	−0.566680	−0.283340	+	0.959020i	$0.591442\pi$
$678$	0	0
$679$	2.97825	0.114295
$680$	0	0
$681$	−20.7446	−0.794933
$682$	0	0
$683$	34.3723	1.31522	0.657609	−	0.753359i	$-0.271568\pi$
$683$	34.3723	1.31522	0.657609	+	0.753359i	$0.271568\pi$
$684$	0	0
$685$	−2.74456	−0.104864
$686$	0	0
$687$	23.7228	0.905082
$688$	0	0
$689$	−26.2337	−0.999424
$690$	0	0
$691$	−5.76631	−0.219361	−0.109680	−	0.993967i	$-0.534983\pi$
$691$	−5.76631	−0.219361	−0.109680	+	0.993967i	$0.534983\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4.00000	−0.151729
$696$	0	0
$697$	−50.2337	−1.90274
$698$	0	0
$699$	−10.3723	−0.392316
$700$	0	0
$701$	31.6277	1.19456	0.597281	−	0.802032i	$-0.296248\pi$
$701$	31.6277	1.19456	0.597281	+	0.802032i	$0.296248\pi$
$702$	0	0
$703$	23.1168	0.871868
$704$	0	0
$705$	−20.7446	−0.781285
$706$	0	0
$707$	14.2337	0.535313
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	−33.4891	−1.25594
$712$	0	0
$713$	20.7446	0.776890
$714$	0	0
$715$	0	0
$716$	0	0
$717$	7.72281	0.288414
$718$	0	0
$719$	−31.1168	−1.16046	−0.580231	−	0.814452i	$-0.697038\pi$
$719$	−31.1168	−1.16046	−0.580231	+	0.814452i	$0.697038\pi$
$720$	0	0
$721$	32.0000	1.19174
$722$	0	0
$723$	−52.1902	−1.94097
$724$	0	0
$725$	4.37228	0.162382
$726$	0	0
$727$	−10.2337	−0.379546	−0.189773	−	0.981828i	$-0.560775\pi$
$727$	−10.2337	−0.379546	−0.189773	+	0.981828i	$0.560775\pi$
$728$	0	0
$729$	−19.9348	−0.738324
$730$	0	0
$731$	−17.4891	−0.646859
$732$	0	0
$733$	−11.7663	−0.434599	−0.217299	−	0.976105i	$-0.569725\pi$
$733$	−11.7663	−0.434599	−0.217299	+	0.976105i	$0.569725\pi$
$734$	0	0
$735$	3.25544	0.120079
$736$	0	0
$737$	0	0
$738$	0	0
$739$	48.4674	1.78290	0.891451	−	0.453118i	$-0.149688\pi$
$739$	48.4674	1.78290	0.891451	+	0.453118i	$0.149688\pi$
$740$	0	0
$741$	30.2337	1.11066
$742$	0	0
$743$	−10.3723	−0.380522	−0.190261	−	0.981734i	$-0.560933\pi$
$743$	−10.3723	−0.380522	−0.190261	+	0.981734i	$0.560933\pi$
$744$	0	0
$745$	18.6060	0.681670
$746$	0	0
$747$	22.9783	0.840730
$748$	0	0
$749$	28.4674	1.04018
$750$	0	0
$751$	19.8614	0.724753	0.362377	−	0.932032i	$-0.381965\pi$
$751$	19.8614	0.724753	0.362377	+	0.932032i	$0.381965\pi$
$752$	0	0
$753$	−20.7446	−0.755974
$754$	0	0
$755$	22.2337	0.809167
$756$	0	0
$757$	24.9783	0.907850	0.453925	−	0.891040i	$-0.350023\pi$
$757$	24.9783	0.907850	0.453925	+	0.891040i	$0.350023\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	40.9783	1.48546	0.742730	−	0.669591i	$-0.233531\pi$
$761$	40.9783	1.48546	0.742730	+	0.669591i	$0.233531\pi$
$762$	0	0
$763$	17.7663	0.643184
$764$	0	0
$765$	11.4891	0.415390
$766$	0	0
$767$	17.4891	0.631496
$768$	0	0
$769$	22.0000	0.793340	0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000	0.793340	0.396670	+	0.917961i	$0.370166\pi$
$770$	0	0
$771$	−42.7011	−1.53784
$772$	0	0
$773$	6.60597	0.237600	0.118800	−	0.992918i	$-0.462095\pi$
$773$	6.60597	0.237600	0.118800	+	0.992918i	$0.462095\pi$
$774$	0	0
$775$	2.37228	0.0852149
$776$	0	0
$777$	20.4158	0.732412
$778$	0	0
$779$	−73.2119	−2.62309
$780$	0	0
$781$	0	0
$782$	0	0
$783$	3.86141	0.137995
$784$	0	0
$785$	3.62772	0.129479
$786$	0	0
$787$	24.4674	0.872168	0.436084	−	0.899906i	$-0.356365\pi$
$787$	24.4674	0.872168	0.436084	+	0.899906i	$0.356365\pi$
$788$	0	0
$789$	−9.16034	−0.326117
$790$	0	0
$791$	−34.9783	−1.24368
$792$	0	0
$793$	0.744563	0.0264402
$794$	0	0
$795$	−31.1168	−1.10360
$796$	0	0
$797$	−11.4891	−0.406966	−0.203483	−	0.979079i	$-0.565226\pi$
$797$	−11.4891	−0.406966	−0.203483	+	0.979079i	$0.565226\pi$
$798$	0	0
$799$	38.2337	1.35261
$800$	0	0
$801$	11.4891	0.405948
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−20.7446	−0.731150
$806$	0	0
$807$	−6.51087	−0.229194
$808$	0	0
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	0	0
$811$	16.1386	0.566703	0.283351	−	0.959016i	$-0.408554\pi$
$811$	16.1386	0.566703	0.283351	+	0.959016i	$0.408554\pi$
$812$	0	0
$813$	37.9565	1.33119
$814$	0	0
$815$	23.1168	0.809748
$816$	0	0
$817$	−25.4891	−0.891752
$818$	0	0
$819$	12.4674	0.435645
$820$	0	0
$821$	11.4891	0.400973	0.200487	−	0.979696i	$-0.435748\pi$
$821$	11.4891	0.400973	0.200487	+	0.979696i	$0.435748\pi$
$822$	0	0
$823$	28.0000	0.976019	0.488009	−	0.872838i	$-0.337723\pi$
$823$	28.0000	0.976019	0.488009	+	0.872838i	$0.337723\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.02175	0.0355297	0.0177649	−	0.999842i	$-0.494345\pi$
$827$	1.02175	0.0355297	0.0177649	+	0.999842i	$0.494345\pi$
$828$	0	0
$829$	−13.2554	−0.460380	−0.230190	−	0.973146i	$-0.573935\pi$
$829$	−13.2554	−0.460380	−0.230190	+	0.973146i	$0.573935\pi$
$830$	0	0
$831$	−2.97825	−0.103314
$832$	0	0
$833$	−6.00000	−0.207888
$834$	0	0
$835$	−10.3723	−0.358948
$836$	0	0
$837$	2.09509	0.0724171
$838$	0	0
$839$	−34.9783	−1.20758	−0.603792	−	0.797142i	$-0.706344\pi$
$839$	−34.9783	−1.20758	−0.603792	+	0.797142i	$0.706344\pi$
$840$	0	0
$841$	−9.88316	−0.340798
$842$	0	0
$843$	42.7011	1.47070
$844$	0	0
$845$	−9.00000	−0.309609
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−39.7228	−1.36328
$850$	0	0
$851$	31.7228	1.08744
$852$	0	0
$853$	−30.4674	−1.04318	−0.521592	−	0.853195i	$-0.674661\pi$
$853$	−30.4674	−1.04318	−0.521592	+	0.853195i	$0.674661\pi$
$854$	0	0
$855$	16.7446	0.572652
$856$	0	0
$857$	15.3505	0.524364	0.262182	−	0.965018i	$-0.415558\pi$
$857$	15.3505	0.524364	0.262182	+	0.965018i	$0.415558\pi$
$858$	0	0
$859$	31.2554	1.06642	0.533211	−	0.845982i	$-0.320985\pi$
$859$	31.2554	1.06642	0.533211	+	0.845982i	$0.320985\pi$
$860$	0	0
$861$	−64.6576	−2.20352
$862$	0	0
$863$	−32.7446	−1.11464	−0.557319	−	0.830299i	$-0.688170\pi$
$863$	−32.7446	−1.11464	−0.557319	+	0.830299i	$0.688170\pi$
$864$	0	0
$865$	6.00000	0.204006
$866$	0	0
$867$	−5.02175	−0.170548
$868$	0	0
$869$	0	0
$870$	0	0
$871$	16.0000	0.542139
$872$	0	0
$873$	−3.29894	−0.111652
$874$	0	0
$875$	−2.37228	−0.0801977
$876$	0	0
$877$	8.97825	0.303174	0.151587	−	0.988444i	$-0.451562\pi$
$877$	8.97825	0.303174	0.151587	+	0.988444i	$0.451562\pi$
$878$	0	0
$879$	1.21194	0.0408777
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	2.37228	0.0798336	0.0399168	−	0.999203i	$-0.487291\pi$
$883$	2.37228	0.0798336	0.0399168	+	0.999203i	$0.487291\pi$
$884$	0	0
$885$	20.7446	0.697321
$886$	0	0
$887$	34.9783	1.17445	0.587227	−	0.809422i	$-0.300219\pi$
$887$	34.9783	1.17445	0.587227	+	0.809422i	$0.300219\pi$
$888$	0	0
$889$	−18.9783	−0.636510
$890$	0	0
$891$	0	0
$892$	0	0
$893$	55.7228	1.86469
$894$	0	0
$895$	12.0000	0.401116
$896$	0	0
$897$	41.4891	1.38528
$898$	0	0
$899$	10.3723	0.345935
$900$	0	0
$901$	57.3505	1.91062
$902$	0	0
$903$	−22.5109	−0.749115
$904$	0	0
$905$	−10.0000	−0.332411
$906$	0	0
$907$	40.6060	1.34830	0.674150	−	0.738595i	$-0.264510\pi$
$907$	40.6060	1.34830	0.674150	+	0.738595i	$0.264510\pi$
$908$	0	0
$909$	−15.7663	−0.522936
$910$	0	0
$911$	48.6060	1.61039	0.805194	−	0.593012i	$-0.202061\pi$
$911$	48.6060	1.61039	0.805194	+	0.593012i	$0.202061\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0.883156	0.0291962
$916$	0	0
$917$	−11.5842	−0.382545
$918$	0	0
$919$	−26.9783	−0.889930	−0.444965	−	0.895548i	$-0.646784\pi$
$919$	−26.9783	−0.889930	−0.444965	+	0.895548i	$0.646784\pi$
$920$	0	0
$921$	−39.7228	−1.30891
$922$	0	0
$923$	−14.2337	−0.468508
$924$	0	0
$925$	3.62772	0.119279
$926$	0	0
$927$	−35.4456	−1.16419
$928$	0	0
$929$	−27.3505	−0.897342	−0.448671	−	0.893697i	$-0.648102\pi$
$929$	−27.3505	−0.897342	−0.448671	+	0.893697i	$0.648102\pi$
$930$	0	0
$931$	−8.74456	−0.286591
$932$	0	0
$933$	32.3288	1.05840
$934$	0	0
$935$	0	0
$936$	0	0
$937$	51.4891	1.68208	0.841038	−	0.540976i	$-0.181945\pi$
$937$	51.4891	1.68208	0.841038	+	0.540976i	$0.181945\pi$
$938$	0	0
$939$	52.1902	1.70316
$940$	0	0
$941$	−48.0951	−1.56786	−0.783928	−	0.620852i	$-0.786787\pi$
$941$	−48.0951	−1.56786	−0.783928	+	0.620852i	$0.786787\pi$
$942$	0	0
$943$	−100.467	−3.27167
$944$	0	0
$945$	−2.09509	−0.0681534
$946$	0	0
$947$	−20.1386	−0.654416	−0.327208	−	0.944952i	$-0.606108\pi$
$947$	−20.1386	−0.654416	−0.327208	+	0.944952i	$0.606108\pi$
$948$	0	0
$949$	14.9783	0.486215
$950$	0	0
$951$	−64.8832	−2.10398
$952$	0	0
$953$	−22.8832	−0.741258	−0.370629	−	0.928781i	$-0.620858\pi$
$953$	−22.8832	−0.741258	−0.370629	+	0.928781i	$0.620858\pi$
$954$	0	0
$955$	−17.4891	−0.565935
$956$	0	0
$957$	0	0
$958$	0	0
$959$	6.51087	0.210247
$960$	0	0
$961$	−25.3723	−0.818461
$962$	0	0
$963$	−31.5326	−1.01612
$964$	0	0
$965$	13.8614	0.446214
$966$	0	0
$967$	7.39403	0.237776	0.118888	−	0.992908i	$-0.462067\pi$
$967$	7.39403	0.237776	0.118888	+	0.992908i	$0.462067\pi$
$968$	0	0
$969$	−66.0951	−2.12328
$970$	0	0
$971$	46.9783	1.50760	0.753802	−	0.657102i	$-0.228218\pi$
$971$	46.9783	1.50760	0.753802	+	0.657102i	$0.228218\pi$
$972$	0	0
$973$	9.48913	0.304207
$974$	0	0
$975$	4.74456	0.151948
$976$	0	0
$977$	−20.2337	−0.647333	−0.323667	−	0.946171i	$-0.604916\pi$
$977$	−20.2337	−0.647333	−0.323667	+	0.946171i	$0.604916\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−19.6793	−0.628312
$982$	0	0
$983$	−43.7228	−1.39454	−0.697271	−	0.716808i	$-0.745602\pi$
$983$	−43.7228	−1.39454	−0.697271	+	0.716808i	$0.745602\pi$
$984$	0	0
$985$	9.25544	0.294903
$986$	0	0
$987$	49.2119	1.56643
$988$	0	0
$989$	−34.9783	−1.11224
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	0	0
$993$	−35.5326	−1.12759
$994$	0	0
$995$	−0.883156	−0.0279979
$996$	0	0
$997$	12.2337	0.387445	0.193722	−	0.981056i	$-0.437944\pi$
$997$	12.2337	0.387445	0.193722	+	0.981056i	$0.437944\pi$
$998$	0	0
$999$	3.20384	0.101365

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9680.2.a.bt.1.1		2
4.3	odd	2		1210.2.a.r.1.2		2
11.10	odd	2		880.2.a.n.1.1		2
20.19	odd	2		6050.2.a.cb.1.1		2
33.32	even	2		7920.2.a.bq.1.2		2
44.43	even	2		110.2.a.d.1.2	✓	2
55.32	even	4		4400.2.b.p.4049.3		4
55.43	even	4		4400.2.b.p.4049.2		4
55.54	odd	2		4400.2.a.bl.1.2		2
88.21	odd	2		3520.2.a.bj.1.2		2
88.43	even	2		3520.2.a.bq.1.1		2
132.131	odd	2		990.2.a.m.1.1		2
220.43	odd	4		550.2.b.f.199.4		4
220.87	odd	4		550.2.b.f.199.1		4
220.219	even	2		550.2.a.n.1.1		2
308.307	odd	2		5390.2.a.bp.1.1		2
660.263	even	4		4950.2.c.bc.199.2		4
660.527	even	4		4950.2.c.bc.199.3		4
660.659	odd	2		4950.2.a.bw.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
110.2.a.d.1.2	✓	2	44.43	even	2
550.2.a.n.1.1		2	220.219	even	2
550.2.b.f.199.1		4	220.87	odd	4
550.2.b.f.199.4		4	220.43	odd	4
880.2.a.n.1.1		2	11.10	odd	2
990.2.a.m.1.1		2	132.131	odd	2
1210.2.a.r.1.2		2	4.3	odd	2
3520.2.a.bj.1.2		2	88.21	odd	2
3520.2.a.bq.1.1		2	88.43	even	2
4400.2.a.bl.1.2		2	55.54	odd	2
4400.2.b.p.4049.2		4	55.43	even	4
4400.2.b.p.4049.3		4	55.32	even	4
4950.2.a.bw.1.2		2	660.659	odd	2
4950.2.c.bc.199.2		4	660.263	even	4
4950.2.c.bc.199.3		4	660.527	even	4
5390.2.a.bp.1.1		2	308.307	odd	2
6050.2.a.cb.1.1		2	20.19	odd	2
7920.2.a.bq.1.2		2	33.32	even	2
9680.2.a.bt.1.1		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9680.a (trivial)

\(f(q)\)	\(=\)	\(q-2.37228 q^{3} +1.00000 q^{5} -2.37228 q^{7} +2.62772 q^{9} -2.00000 q^{13} -2.37228 q^{15} +4.37228 q^{17} +6.37228 q^{19} +5.62772 q^{21} +8.74456 q^{23} +1.00000 q^{25} +0.883156 q^{27} +4.37228 q^{29} +2.37228 q^{31} -2.37228 q^{35} +3.62772 q^{37} +4.74456 q^{39} -11.4891 q^{41} -4.00000 q^{43} +2.62772 q^{45} +8.74456 q^{47} -1.37228 q^{49} -10.3723 q^{51} +13.1168 q^{53} -15.1168 q^{57} -8.74456 q^{59} -0.372281 q^{61} -6.23369 q^{63} -2.00000 q^{65} -8.00000 q^{67} -20.7446 q^{69} +7.11684 q^{71} -7.48913 q^{73} -2.37228 q^{75} -12.7446 q^{79} -9.97825 q^{81} +8.74456 q^{83} +4.37228 q^{85} -10.3723 q^{87} +4.37228 q^{89} +4.74456 q^{91} -5.62772 q^{93} +6.37228 q^{95} -1.25544 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + 2 q^{5} + q^{7} + 11 q^{9} - 4 q^{13} + q^{15} + 3 q^{17} + 7 q^{19} + 17 q^{21} + 6 q^{23} + 2 q^{25} + 19 q^{27} + 3 q^{29} - q^{31} + q^{35} + 13 q^{37} - 2 q^{39} - 8 q^{43} + 11 q^{45}+ \cdots - 14 q^{97}+O(q^{100}) \) 2 * q + q^3 + 2 * q^5 + q^7 + 11 * q^9 - 4 * q^13 + q^15 + 3 * q^17 + 7 * q^19 + 17 * q^21 + 6 * q^23 + 2 * q^25 + 19 * q^27 + 3 * q^29 - q^31 + q^35 + 13 * q^37 - 2 * q^39 - 8 * q^43 + 11 * q^45 + 6 * q^47 + 3 * q^49 - 15 * q^51 + 9 * q^53 - 13 * q^57 - 6 * q^59 + 5 * q^61 + 22 * q^63 - 4 * q^65 - 16 * q^67 - 30 * q^69 - 3 * q^71 + 8 * q^73 + q^75 - 14 * q^79 + 26 * q^81 + 6 * q^83 + 3 * q^85 - 15 * q^87 + 3 * q^89 - 2 * q^91 - 17 * q^93 + 7 * q^95 - 14 * q^97

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