LMFDB - Embedded newform 6480.2.a.bl.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6480,2,Mod(1,6480)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6480.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6480, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6480 = 2^{4} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6480.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,2,0,-2,0,0,0,0,0,0,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$51.7430605098$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 360)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.44949$ of defining polynomial
Character	$\chi$	$=$	6480.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +1.44949 q^{7} +2.00000 q^{17} -2.89898 q^{19} -2.55051 q^{23} +1.00000 q^{25} -7.89898 q^{29} -10.8990 q^{31} +1.44949 q^{35} -6.00000 q^{37} +0.101021 q^{41} +7.79796 q^{43} -4.55051 q^{47} -4.89898 q^{49} +11.7980 q^{53} -10.8990 q^{59} -3.00000 q^{61} -11.2474 q^{67} -9.79796 q^{71} -5.79796 q^{73} -2.89898 q^{79} -0.550510 q^{83} +2.00000 q^{85} +16.7980 q^{89} -2.89898 q^{95} +2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 2 q^{7} + 4 q^{17} + 4 q^{19} - 10 q^{23} + 2 q^{25} - 6 q^{29} - 12 q^{31} - 2 q^{35} - 12 q^{37} + 10 q^{41} - 4 q^{43} - 14 q^{47} + 4 q^{53} - 12 q^{59} - 6 q^{61} + 2 q^{67} + 8 q^{73}+ \cdots + 4 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 - 2 * q^7 + 4 * q^17 + 4 * q^19 - 10 * q^23 + 2 * q^25 - 6 * q^29 - 12 * q^31 - 2 * q^35 - 12 * q^37 + 10 * q^41 - 4 * q^43 - 14 * q^47 + 4 * q^53 - 12 * q^59 - 6 * q^61 + 2 * q^67 + 8 * q^73 + 4 * q^79 - 6 * q^83 + 4 * q^85 + 14 * q^89 + 4 * q^95 + 4 * 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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.44949	0.547856	0.273928	−	0.961750i	$-0.411677\pi$
$7$	1.44949	0.547856	0.273928	+	0.961750i	$0.411677\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	−2.89898	−0.665072	−0.332536	−	0.943091i	$-0.607904\pi$
$19$	−2.89898	−0.665072	−0.332536	+	0.943091i	$0.607904\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.55051	−0.531818	−0.265909	−	0.963998i	$-0.585672\pi$
$23$	−2.55051	−0.531818	−0.265909	+	0.963998i	$0.585672\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−7.89898	−1.46680	−0.733402	−	0.679795i	$-0.762069\pi$
$29$	−7.89898	−1.46680	−0.733402	+	0.679795i	$0.762069\pi$
$30$	0	0
$31$	−10.8990	−1.95751	−0.978757	−	0.205023i	$-0.934273\pi$
$31$	−10.8990	−1.95751	−0.978757	+	0.205023i	$0.934273\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.44949	0.245008
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.101021	0.0157768	0.00788838	−	0.999969i	$-0.497489\pi$
$41$	0.101021	0.0157768	0.00788838	+	0.999969i	$0.497489\pi$
$42$	0	0
$43$	7.79796	1.18918	0.594589	−	0.804030i	$-0.297315\pi$
$43$	7.79796	1.18918	0.594589	+	0.804030i	$0.297315\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.55051	−0.663760	−0.331880	−	0.943322i	$-0.607683\pi$
$47$	−4.55051	−0.663760	−0.331880	+	0.943322i	$0.607683\pi$
$48$	0	0
$49$	−4.89898	−0.699854
$50$	0	0
$51$	0	0
$52$	0	0
$53$	11.7980	1.62057	0.810287	−	0.586033i	$-0.199311\pi$
$53$	11.7980	1.62057	0.810287	+	0.586033i	$0.199311\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−10.8990	−1.41893	−0.709463	−	0.704743i	$-0.751063\pi$
$59$	−10.8990	−1.41893	−0.709463	+	0.704743i	$0.751063\pi$
$60$	0	0
$61$	−3.00000	−0.384111	−0.192055	−	0.981384i	$-0.561515\pi$
$61$	−3.00000	−0.384111	−0.192055	+	0.981384i	$0.561515\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−11.2474	−1.37409	−0.687047	−	0.726613i	$-0.741093\pi$
$67$	−11.2474	−1.37409	−0.687047	+	0.726613i	$0.741093\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−9.79796	−1.16280	−0.581402	−	0.813617i	$-0.697496\pi$
$71$	−9.79796	−1.16280	−0.581402	+	0.813617i	$0.697496\pi$
$72$	0	0
$73$	−5.79796	−0.678600	−0.339300	−	0.940678i	$-0.610190\pi$
$73$	−5.79796	−0.678600	−0.339300	+	0.940678i	$0.610190\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−2.89898	−0.326161	−0.163080	−	0.986613i	$-0.552143\pi$
$79$	−2.89898	−0.326161	−0.163080	+	0.986613i	$0.552143\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−0.550510	−0.0604264	−0.0302132	−	0.999543i	$-0.509619\pi$
$83$	−0.550510	−0.0604264	−0.0302132	+	0.999543i	$0.509619\pi$
$84$	0	0
$85$	2.00000	0.216930
$86$	0	0
$87$	0	0
$88$	0	0
$89$	16.7980	1.78058	0.890290	−	0.455394i	$-0.150502\pi$
$89$	16.7980	1.78058	0.890290	+	0.455394i	$0.150502\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.89898	−0.297429
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	−10.0000	−0.985329	−0.492665	−	0.870219i	$-0.663977\pi$
$103$	−10.0000	−0.985329	−0.492665	+	0.870219i	$0.663977\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.34847	0.227035	0.113518	−	0.993536i	$-0.463788\pi$
$107$	2.34847	0.227035	0.113518	+	0.993536i	$0.463788\pi$
$108$	0	0
$109$	8.79796	0.842692	0.421346	−	0.906900i	$-0.361558\pi$
$109$	8.79796	0.842692	0.421346	+	0.906900i	$0.361558\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−9.79796	−0.921714	−0.460857	−	0.887474i	$-0.652458\pi$
$113$	−9.79796	−0.921714	−0.460857	+	0.887474i	$0.652458\pi$
$114$	0	0
$115$	−2.55051	−0.237836
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.89898	0.265749
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	14.3485	1.27322	0.636610	−	0.771186i	$-0.280336\pi$
$127$	14.3485	1.27322	0.636610	+	0.771186i	$0.280336\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−6.89898	−0.602767	−0.301383	−	0.953503i	$-0.597448\pi$
$131$	−6.89898	−0.602767	−0.301383	+	0.953503i	$0.597448\pi$
$132$	0	0
$133$	−4.20204	−0.364363
$134$	0	0
$135$	0	0
$136$	0	0
$137$	19.5959	1.67419	0.837096	−	0.547056i	$-0.184251\pi$
$137$	19.5959	1.67419	0.837096	+	0.547056i	$0.184251\pi$
$138$	0	0
$139$	19.5959	1.66210	0.831052	−	0.556195i	$-0.187739\pi$
$139$	19.5959	1.66210	0.831052	+	0.556195i	$0.187739\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−7.89898	−0.655975
$146$	0	0
$147$	0	0
$148$	0	0
$149$	21.0000	1.72039	0.860194	−	0.509968i	$-0.170343\pi$
$149$	21.0000	1.72039	0.860194	+	0.509968i	$0.170343\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−10.8990	−0.875427
$156$	0	0
$157$	−4.20204	−0.335359	−0.167680	−	0.985842i	$-0.553627\pi$
$157$	−4.20204	−0.335359	−0.167680	+	0.985842i	$0.553627\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.69694	−0.291360
$162$	0	0
$163$	−11.7980	−0.924087	−0.462044	−	0.886857i	$-0.652884\pi$
$163$	−11.7980	−0.924087	−0.462044	+	0.886857i	$0.652884\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.34847	−0.491259	−0.245630	−	0.969364i	$-0.578995\pi$
$167$	−6.34847	−0.491259	−0.245630	+	0.969364i	$0.578995\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	12.0000	0.912343	0.456172	−	0.889892i	$-0.349220\pi$
$173$	12.0000	0.912343	0.456172	+	0.889892i	$0.349220\pi$
$174$	0	0
$175$	1.44949	0.109571
$176$	0	0
$177$	0	0
$178$	0	0
$179$	5.79796	0.433360	0.216680	−	0.976243i	$-0.430477\pi$
$179$	5.79796	0.433360	0.216680	+	0.976243i	$0.430477\pi$
$180$	0	0
$181$	19.6969	1.46406	0.732031	−	0.681271i	$-0.238573\pi$
$181$	19.6969	1.46406	0.732031	+	0.681271i	$0.238573\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−6.00000	−0.441129
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−5.10102	−0.369097	−0.184548	−	0.982823i	$-0.559082\pi$
$191$	−5.10102	−0.369097	−0.184548	+	0.982823i	$0.559082\pi$
$192$	0	0
$193$	−21.7980	−1.56905	−0.784526	−	0.620096i	$-0.787094\pi$
$193$	−21.7980	−1.56905	−0.784526	+	0.620096i	$0.787094\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−23.5959	−1.68114	−0.840570	−	0.541703i	$-0.817780\pi$
$197$	−23.5959	−1.68114	−0.840570	+	0.541703i	$0.817780\pi$
$198$	0	0
$199$	−13.1010	−0.928707	−0.464353	−	0.885650i	$-0.653713\pi$
$199$	−13.1010	−0.928707	−0.464353	+	0.885650i	$0.653713\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−11.4495	−0.803597
$204$	0	0
$205$	0.101021	0.00705558
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	7.79796	0.531816
$216$	0	0
$217$	−15.7980	−1.07244
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−0.550510	−0.0368649	−0.0184324	−	0.999830i	$-0.505868\pi$
$223$	−0.550510	−0.0368649	−0.0184324	+	0.999830i	$0.505868\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−27.7980	−1.84502	−0.922508	−	0.385979i	$-0.873864\pi$
$227$	−27.7980	−1.84502	−0.922508	+	0.385979i	$0.873864\pi$
$228$	0	0
$229$	−13.8990	−0.918470	−0.459235	−	0.888315i	$-0.651876\pi$
$229$	−13.8990	−0.918470	−0.459235	+	0.888315i	$0.651876\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0.202041	0.0132361	0.00661807	−	0.999978i	$-0.497893\pi$
$233$	0.202041	0.0132361	0.00661807	+	0.999978i	$0.497893\pi$
$234$	0	0
$235$	−4.55051	−0.296843
$236$	0	0
$237$	0	0
$238$	0	0
$239$	21.7980	1.40999	0.704996	−	0.709211i	$-0.250949\pi$
$239$	21.7980	1.40999	0.704996	+	0.709211i	$0.250949\pi$
$240$	0	0
$241$	25.6969	1.65529	0.827643	−	0.561255i	$-0.189681\pi$
$241$	25.6969	1.65529	0.827643	+	0.561255i	$0.189681\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−4.89898	−0.312984
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−6.89898	−0.435460	−0.217730	−	0.976009i	$-0.569865\pi$
$251$	−6.89898	−0.435460	−0.217730	+	0.976009i	$0.569865\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−8.20204	−0.511629	−0.255815	−	0.966726i	$-0.582344\pi$
$257$	−8.20204	−0.511629	−0.255815	+	0.966726i	$0.582344\pi$
$258$	0	0
$259$	−8.69694	−0.540401
$260$	0	0
$261$	0	0
$262$	0	0
$263$	18.0000	1.10993	0.554964	−	0.831875i	$-0.312732\pi$
$263$	18.0000	1.10993	0.554964	+	0.831875i	$0.312732\pi$
$264$	0	0
$265$	11.7980	0.724743
$266$	0	0
$267$	0	0
$268$	0	0
$269$	16.5959	1.01187	0.505935	−	0.862571i	$-0.331147\pi$
$269$	16.5959	1.01187	0.505935	+	0.862571i	$0.331147\pi$
$270$	0	0
$271$	−21.1010	−1.28180	−0.640898	−	0.767626i	$-0.721438\pi$
$271$	−21.1010	−1.28180	−0.640898	+	0.767626i	$0.721438\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−14.0000	−0.841178	−0.420589	−	0.907251i	$-0.638177\pi$
$277$	−14.0000	−0.841178	−0.420589	+	0.907251i	$0.638177\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	17.8990	1.06776	0.533882	−	0.845559i	$-0.320733\pi$
$281$	17.8990	1.06776	0.533882	+	0.845559i	$0.320733\pi$
$282$	0	0
$283$	18.3485	1.09070	0.545352	−	0.838207i	$-0.316396\pi$
$283$	18.3485	1.09070	0.545352	+	0.838207i	$0.316396\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.146428	0.00864338
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	21.5959	1.26165	0.630823	−	0.775926i	$-0.282717\pi$
$293$	21.5959	1.26165	0.630823	+	0.775926i	$0.282717\pi$
$294$	0	0
$295$	−10.8990	−0.634563
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	11.3031	0.651498
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−3.00000	−0.171780
$306$	0	0
$307$	29.2474	1.66924	0.834620	−	0.550826i	$-0.185687\pi$
$307$	29.2474	1.66924	0.834620	+	0.550826i	$0.185687\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2.89898	0.164386	0.0821930	−	0.996616i	$-0.473808\pi$
$311$	2.89898	0.164386	0.0821930	+	0.996616i	$0.473808\pi$
$312$	0	0
$313$	23.5959	1.33372	0.666860	−	0.745183i	$-0.267638\pi$
$313$	23.5959	1.33372	0.666860	+	0.745183i	$0.267638\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.20204	0.348341	0.174171	−	0.984715i	$-0.444276\pi$
$317$	6.20204	0.348341	0.174171	+	0.984715i	$0.444276\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−5.79796	−0.322607
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−6.59592	−0.363645
$330$	0	0
$331$	−2.89898	−0.159342	−0.0796712	−	0.996821i	$-0.525387\pi$
$331$	−2.89898	−0.159342	−0.0796712	+	0.996821i	$0.525387\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−11.2474	−0.614514
$336$	0	0
$337$	−10.2020	−0.555741	−0.277870	−	0.960619i	$-0.589629\pi$
$337$	−10.2020	−0.555741	−0.277870	+	0.960619i	$0.589629\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−17.2474	−0.931275
$344$	0	0
$345$	0	0
$346$	0	0
$347$	13.5959	0.729867	0.364934	−	0.931034i	$-0.381092\pi$
$347$	13.5959	0.729867	0.364934	+	0.931034i	$0.381092\pi$
$348$	0	0
$349$	−31.6969	−1.69670	−0.848349	−	0.529437i	$-0.822403\pi$
$349$	−31.6969	−1.69670	−0.848349	+	0.529437i	$0.822403\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	−9.79796	−0.520022
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−0.696938	−0.0367830	−0.0183915	−	0.999831i	$-0.505855\pi$
$359$	−0.696938	−0.0367830	−0.0183915	+	0.999831i	$0.505855\pi$
$360$	0	0
$361$	−10.5959	−0.557680
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−5.79796	−0.303479
$366$	0	0
$367$	−22.0000	−1.14839	−0.574195	−	0.818718i	$-0.694685\pi$
$367$	−22.0000	−1.14839	−0.574195	+	0.818718i	$0.694685\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	17.1010	0.887841
$372$	0	0
$373$	−17.5959	−0.911082	−0.455541	−	0.890215i	$-0.650554\pi$
$373$	−17.5959	−0.911082	−0.455541	+	0.890215i	$0.650554\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−17.1010	−0.878420	−0.439210	−	0.898384i	$-0.644742\pi$
$379$	−17.1010	−0.878420	−0.439210	+	0.898384i	$0.644742\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	2.00000	0.102195	0.0510976	−	0.998694i	$-0.483728\pi$
$383$	2.00000	0.102195	0.0510976	+	0.998694i	$0.483728\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−22.5959	−1.14566	−0.572829	−	0.819675i	$-0.694154\pi$
$389$	−22.5959	−1.14566	−0.572829	+	0.819675i	$0.694154\pi$
$390$	0	0
$391$	−5.10102	−0.257970
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2.89898	−0.145863
$396$	0	0
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	13.5959	0.678948	0.339474	−	0.940615i	$-0.389751\pi$
$401$	13.5959	0.678948	0.339474	+	0.940615i	$0.389751\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	29.5959	1.46342	0.731712	−	0.681614i	$-0.238722\pi$
$409$	29.5959	1.46342	0.731712	+	0.681614i	$0.238722\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−15.7980	−0.777367
$414$	0	0
$415$	−0.550510	−0.0270235
$416$	0	0
$417$	0	0
$418$	0	0
$419$	20.0000	0.977064	0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000	0.977064	0.488532	+	0.872546i	$0.337533\pi$
$420$	0	0
$421$	−13.5959	−0.662624	−0.331312	−	0.943521i	$-0.607491\pi$
$421$	−13.5959	−0.662624	−0.331312	+	0.943521i	$0.607491\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2.00000	0.0970143
$426$	0	0
$427$	−4.34847	−0.210437
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5.10102	0.245708	0.122854	−	0.992425i	$-0.460795\pi$
$431$	5.10102	0.245708	0.122854	+	0.992425i	$0.460795\pi$
$432$	0	0
$433$	7.79796	0.374746	0.187373	−	0.982289i	$-0.440003\pi$
$433$	7.79796	0.374746	0.187373	+	0.982289i	$0.440003\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.39388	0.353697
$438$	0	0
$439$	−1.79796	−0.0858119	−0.0429059	−	0.999079i	$-0.513662\pi$
$439$	−1.79796	−0.0858119	−0.0429059	+	0.999079i	$0.513662\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	26.1464	1.24225	0.621127	−	0.783710i	$-0.286675\pi$
$443$	26.1464	1.24225	0.621127	+	0.783710i	$0.286675\pi$
$444$	0	0
$445$	16.7980	0.796300
$446$	0	0
$447$	0	0
$448$	0	0
$449$	17.5959	0.830403	0.415201	−	0.909730i	$-0.363711\pi$
$449$	17.5959	0.830403	0.415201	+	0.909730i	$0.363711\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−21.5959	−1.01021	−0.505107	−	0.863057i	$-0.668547\pi$
$457$	−21.5959	−1.01021	−0.505107	+	0.863057i	$0.668547\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−21.8990	−1.01994	−0.509969	−	0.860193i	$-0.670343\pi$
$461$	−21.8990	−1.01994	−0.509969	+	0.860193i	$0.670343\pi$
$462$	0	0
$463$	−31.7980	−1.47778	−0.738888	−	0.673828i	$-0.764649\pi$
$463$	−31.7980	−1.47778	−0.738888	+	0.673828i	$0.764649\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−11.7980	−0.545944	−0.272972	−	0.962022i	$-0.588007\pi$
$467$	−11.7980	−0.545944	−0.272972	+	0.962022i	$0.588007\pi$
$468$	0	0
$469$	−16.3031	−0.752805
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−2.89898	−0.133014
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−18.4949	−0.845053	−0.422527	−	0.906350i	$-0.638857\pi$
$479$	−18.4949	−0.845053	−0.422527	+	0.906350i	$0.638857\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.00000	0.0908153
$486$	0	0
$487$	−25.5959	−1.15986	−0.579931	−	0.814666i	$-0.696920\pi$
$487$	−25.5959	−1.15986	−0.579931	+	0.814666i	$0.696920\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	5.79796	0.261658	0.130829	−	0.991405i	$-0.458236\pi$
$491$	5.79796	0.261658	0.130829	+	0.991405i	$0.458236\pi$
$492$	0	0
$493$	−15.7980	−0.711504
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−14.2020	−0.637049
$498$	0	0
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−19.0454	−0.849193	−0.424596	−	0.905383i	$-0.639584\pi$
$503$	−19.0454	−0.849193	−0.424596	+	0.905383i	$0.639584\pi$
$504$	0	0
$505$	2.00000	0.0889988
$506$	0	0
$507$	0	0
$508$	0	0
$509$	13.8990	0.616061	0.308031	−	0.951376i	$-0.400330\pi$
$509$	13.8990	0.616061	0.308031	+	0.951376i	$0.400330\pi$
$510$	0	0
$511$	−8.40408	−0.371775
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−10.0000	−0.440653
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−28.7980	−1.26166	−0.630831	−	0.775920i	$-0.717286\pi$
$521$	−28.7980	−1.26166	−0.630831	+	0.775920i	$0.717286\pi$
$522$	0	0
$523$	19.6515	0.859301	0.429651	−	0.902995i	$-0.358637\pi$
$523$	19.6515	0.859301	0.429651	+	0.902995i	$0.358637\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−21.7980	−0.949534
$528$	0	0
$529$	−16.4949	−0.717169
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	2.34847	0.101533
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	21.8990	0.941511	0.470755	−	0.882264i	$-0.343981\pi$
$541$	21.8990	0.941511	0.470755	+	0.882264i	$0.343981\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	8.79796	0.376863
$546$	0	0
$547$	41.2474	1.76361	0.881807	−	0.471611i	$-0.156327\pi$
$547$	41.2474	1.76361	0.881807	+	0.471611i	$0.156327\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	22.8990	0.975529
$552$	0	0
$553$	−4.20204	−0.178689
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−24.2020	−1.02547	−0.512737	−	0.858546i	$-0.671368\pi$
$557$	−24.2020	−1.02547	−0.512737	+	0.858546i	$0.671368\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	4.34847	0.183266	0.0916331	−	0.995793i	$-0.470791\pi$
$563$	4.34847	0.183266	0.0916331	+	0.995793i	$0.470791\pi$
$564$	0	0
$565$	−9.79796	−0.412203
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	−6.20204	−0.259547	−0.129774	−	0.991544i	$-0.541425\pi$
$571$	−6.20204	−0.259547	−0.129774	+	0.991544i	$0.541425\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2.55051	−0.106364
$576$	0	0
$577$	−26.0000	−1.08239	−0.541197	−	0.840896i	$-0.682029\pi$
$577$	−26.0000	−1.08239	−0.541197	+	0.840896i	$0.682029\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−0.797959	−0.0331049
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	22.3485	0.922420	0.461210	−	0.887291i	$-0.347415\pi$
$587$	22.3485	0.922420	0.461210	+	0.887291i	$0.347415\pi$
$588$	0	0
$589$	31.5959	1.30189
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−27.3939	−1.12493	−0.562466	−	0.826821i	$-0.690147\pi$
$593$	−27.3939	−1.12493	−0.562466	+	0.826821i	$0.690147\pi$
$594$	0	0
$595$	2.89898	0.118847
$596$	0	0
$597$	0	0
$598$	0	0
$599$	4.69694	0.191912	0.0959559	−	0.995386i	$-0.469409\pi$
$599$	4.69694	0.191912	0.0959559	+	0.995386i	$0.469409\pi$
$600$	0	0
$601$	14.0000	0.571072	0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000	0.571072	0.285536	+	0.958368i	$0.407828\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−11.0000	−0.447214
$606$	0	0
$607$	32.1464	1.30478	0.652392	−	0.757882i	$-0.273766\pi$
$607$	32.1464	1.30478	0.652392	+	0.757882i	$0.273766\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	9.59592	0.387575	0.193788	−	0.981043i	$-0.437923\pi$
$613$	9.59592	0.387575	0.193788	+	0.981043i	$0.437923\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−43.1918	−1.73884	−0.869419	−	0.494076i	$-0.835507\pi$
$617$	−43.1918	−1.73884	−0.869419	+	0.494076i	$0.835507\pi$
$618$	0	0
$619$	−4.69694	−0.188786	−0.0943929	−	0.995535i	$-0.530091\pi$
$619$	−4.69694	−0.188786	−0.0943929	+	0.995535i	$0.530091\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	24.3485	0.975501
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−12.0000	−0.478471
$630$	0	0
$631$	−23.5959	−0.939339	−0.469669	−	0.882842i	$-0.655627\pi$
$631$	−23.5959	−0.939339	−0.469669	+	0.882842i	$0.655627\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	14.3485	0.569402
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	4.10102	0.161981	0.0809903	−	0.996715i	$-0.474192\pi$
$641$	4.10102	0.161981	0.0809903	+	0.996715i	$0.474192\pi$
$642$	0	0
$643$	−14.1464	−0.557881	−0.278940	−	0.960308i	$-0.589983\pi$
$643$	−14.1464	−0.557881	−0.278940	+	0.960308i	$0.589983\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	11.4495	0.450126	0.225063	−	0.974344i	$-0.427741\pi$
$647$	11.4495	0.450126	0.225063	+	0.974344i	$0.427741\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−8.20204	−0.320971	−0.160485	−	0.987038i	$-0.551306\pi$
$653$	−8.20204	−0.320971	−0.160485	+	0.987038i	$0.551306\pi$
$654$	0	0
$655$	−6.89898	−0.269565
$656$	0	0
$657$	0	0
$658$	0	0
$659$	29.7980	1.16076	0.580382	−	0.814344i	$-0.302903\pi$
$659$	29.7980	1.16076	0.580382	+	0.814344i	$0.302903\pi$
$660$	0	0
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−4.20204	−0.162948
$666$	0	0
$667$	20.1464	0.780073
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−16.0000	−0.616755	−0.308377	−	0.951264i	$-0.599786\pi$
$673$	−16.0000	−0.616755	−0.308377	+	0.951264i	$0.599786\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−33.7980	−1.29896	−0.649481	−	0.760378i	$-0.725014\pi$
$677$	−33.7980	−1.29896	−0.649481	+	0.760378i	$0.725014\pi$
$678$	0	0
$679$	2.89898	0.111253
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−35.3939	−1.35431	−0.677155	−	0.735841i	$-0.736787\pi$
$683$	−35.3939	−1.35431	−0.677155	+	0.735841i	$0.736787\pi$
$684$	0	0
$685$	19.5959	0.748722
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−29.1010	−1.10705	−0.553527	−	0.832831i	$-0.686719\pi$
$691$	−29.1010	−1.10705	−0.553527	+	0.832831i	$0.686719\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	19.5959	0.743316
$696$	0	0
$697$	0.202041	0.00765285
$698$	0	0
$699$	0	0
$700$	0	0
$701$	28.3939	1.07242	0.536211	−	0.844084i	$-0.319855\pi$
$701$	28.3939	1.07242	0.536211	+	0.844084i	$0.319855\pi$
$702$	0	0
$703$	17.3939	0.656022
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.89898	0.109027
$708$	0	0
$709$	−19.6969	−0.739734	−0.369867	−	0.929085i	$-0.620597\pi$
$709$	−19.6969	−0.739734	−0.369867	+	0.929085i	$0.620597\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	27.7980	1.04104
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	40.2929	1.50267	0.751335	−	0.659921i	$-0.229410\pi$
$719$	40.2929	1.50267	0.751335	+	0.659921i	$0.229410\pi$
$720$	0	0
$721$	−14.4949	−0.539818
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−7.89898	−0.293361
$726$	0	0
$727$	6.75255	0.250438	0.125219	−	0.992129i	$-0.460037\pi$
$727$	6.75255	0.250438	0.125219	+	0.992129i	$0.460037\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	15.5959	0.576836
$732$	0	0
$733$	9.59592	0.354433	0.177217	−	0.984172i	$-0.443291\pi$
$733$	9.59592	0.354433	0.177217	+	0.984172i	$0.443291\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	42.8990	1.57806	0.789032	−	0.614352i	$-0.210582\pi$
$739$	42.8990	1.57806	0.789032	+	0.614352i	$0.210582\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	37.0454	1.35906	0.679532	−	0.733646i	$-0.262183\pi$
$743$	37.0454	1.35906	0.679532	+	0.733646i	$0.262183\pi$
$744$	0	0
$745$	21.0000	0.769380
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3.40408	0.124382
$750$	0	0
$751$	−45.7980	−1.67119	−0.835596	−	0.549345i	$-0.814877\pi$
$751$	−45.7980	−1.67119	−0.835596	+	0.549345i	$0.814877\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−12.0000	−0.436725
$756$	0	0
$757$	−7.59592	−0.276078	−0.138039	−	0.990427i	$-0.544080\pi$
$757$	−7.59592	−0.276078	−0.138039	+	0.990427i	$0.544080\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−35.0000	−1.26875	−0.634375	−	0.773026i	$-0.718742\pi$
$761$	−35.0000	−1.26875	−0.634375	+	0.773026i	$0.718742\pi$
$762$	0	0
$763$	12.7526	0.461673
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−24.7980	−0.894237	−0.447119	−	0.894475i	$-0.647550\pi$
$769$	−24.7980	−0.894237	−0.447119	+	0.894475i	$0.647550\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−25.7980	−0.927888	−0.463944	−	0.885865i	$-0.653566\pi$
$773$	−25.7980	−0.927888	−0.463944	+	0.885865i	$0.653566\pi$
$774$	0	0
$775$	−10.8990	−0.391503
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−0.292856	−0.0104927
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−4.20204	−0.149977
$786$	0	0
$787$	10.4041	0.370865	0.185433	−	0.982657i	$-0.440631\pi$
$787$	10.4041	0.370865	0.185433	+	0.982657i	$0.440631\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−14.2020	−0.504966
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	8.00000	0.283375	0.141687	−	0.989911i	$-0.454747\pi$
$797$	8.00000	0.283375	0.141687	+	0.989911i	$0.454747\pi$
$798$	0	0
$799$	−9.10102	−0.321971
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−3.69694	−0.130300
$806$	0	0
$807$	0	0
$808$	0	0
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	−22.8990	−0.804092	−0.402046	−	0.915619i	$-0.631701\pi$
$811$	−22.8990	−0.804092	−0.402046	+	0.915619i	$0.631701\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−11.7980	−0.413264
$816$	0	0
$817$	−22.6061	−0.790888
$818$	0	0
$819$	0	0
$820$	0	0
$821$	16.7980	0.586253	0.293126	−	0.956074i	$-0.405304\pi$
$821$	16.7980	0.586253	0.293126	+	0.956074i	$0.405304\pi$
$822$	0	0
$823$	−6.34847	−0.221294	−0.110647	−	0.993860i	$-0.535292\pi$
$823$	−6.34847	−0.221294	−0.110647	+	0.993860i	$0.535292\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−39.5403	−1.37495	−0.687476	−	0.726208i	$-0.741281\pi$
$827$	−39.5403	−1.37495	−0.687476	+	0.726208i	$0.741281\pi$
$828$	0	0
$829$	−28.1918	−0.979143	−0.489571	−	0.871963i	$-0.662847\pi$
$829$	−28.1918	−0.979143	−0.489571	+	0.871963i	$0.662847\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−9.79796	−0.339479
$834$	0	0
$835$	−6.34847	−0.219698
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−24.6969	−0.852633	−0.426317	−	0.904574i	$-0.640189\pi$
$839$	−24.6969	−0.852633	−0.426317	+	0.904574i	$0.640189\pi$
$840$	0	0
$841$	33.3939	1.15151
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−13.0000	−0.447214
$846$	0	0
$847$	−15.9444	−0.547856
$848$	0	0
$849$	0	0
$850$	0	0
$851$	15.3031	0.524582
$852$	0	0
$853$	37.7980	1.29418	0.647089	−	0.762415i	$-0.275986\pi$
$853$	37.7980	1.29418	0.647089	+	0.762415i	$0.275986\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−39.5959	−1.35257	−0.676285	−	0.736640i	$-0.736411\pi$
$857$	−39.5959	−1.35257	−0.676285	+	0.736640i	$0.736411\pi$
$858$	0	0
$859$	39.1918	1.33721	0.668604	−	0.743619i	$-0.266892\pi$
$859$	39.1918	1.33721	0.668604	+	0.743619i	$0.266892\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	52.5505	1.78884	0.894420	−	0.447228i	$-0.147589\pi$
$863$	52.5505	1.78884	0.894420	+	0.447228i	$0.147589\pi$
$864$	0	0
$865$	12.0000	0.408012
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.44949	0.0490017
$876$	0	0
$877$	−39.5959	−1.33706	−0.668530	−	0.743686i	$-0.733076\pi$
$877$	−39.5959	−1.33706	−0.668530	+	0.743686i	$0.733076\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−15.8990	−0.535650	−0.267825	−	0.963468i	$-0.586305\pi$
$881$	−15.8990	−0.535650	−0.267825	+	0.963468i	$0.586305\pi$
$882$	0	0
$883$	45.0454	1.51590	0.757949	−	0.652313i	$-0.226201\pi$
$883$	45.0454	1.51590	0.757949	+	0.652313i	$0.226201\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−37.5959	−1.26235	−0.631174	−	0.775642i	$-0.717426\pi$
$887$	−37.5959	−1.26235	−0.631174	+	0.775642i	$0.717426\pi$
$888$	0	0
$889$	20.7980	0.697541
$890$	0	0
$891$	0	0
$892$	0	0
$893$	13.1918	0.441448
$894$	0	0
$895$	5.79796	0.193804
$896$	0	0
$897$	0	0
$898$	0	0
$899$	86.0908	2.87129
$900$	0	0
$901$	23.5959	0.786094
$902$	0	0
$903$	0	0
$904$	0	0
$905$	19.6969	0.654748
$906$	0	0
$907$	36.3485	1.20693	0.603466	−	0.797389i	$-0.293786\pi$
$907$	36.3485	1.20693	0.603466	+	0.797389i	$0.293786\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−38.4949	−1.27539	−0.637696	−	0.770288i	$-0.720113\pi$
$911$	−38.4949	−1.27539	−0.637696	+	0.770288i	$0.720113\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−10.0000	−0.330229
$918$	0	0
$919$	4.69694	0.154938	0.0774689	−	0.996995i	$-0.475316\pi$
$919$	4.69694	0.154938	0.0774689	+	0.996995i	$0.475316\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−6.00000	−0.197279
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.59592	0.0523604	0.0261802	−	0.999657i	$-0.491666\pi$
$929$	1.59592	0.0523604	0.0261802	+	0.999657i	$0.491666\pi$
$930$	0	0
$931$	14.2020	0.465453
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	56.9898	1.86178	0.930888	−	0.365305i	$-0.119035\pi$
$937$	56.9898	1.86178	0.930888	+	0.365305i	$0.119035\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−27.6969	−0.902894	−0.451447	−	0.892298i	$-0.649092\pi$
$941$	−27.6969	−0.902894	−0.451447	+	0.892298i	$0.649092\pi$
$942$	0	0
$943$	−0.257654	−0.00839036
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−46.1464	−1.49956	−0.749779	−	0.661689i	$-0.769840\pi$
$947$	−46.1464	−1.49956	−0.749779	+	0.661689i	$0.769840\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	33.7980	1.09482	0.547412	−	0.836863i	$-0.315613\pi$
$953$	33.7980	1.09482	0.547412	+	0.836863i	$0.315613\pi$
$954$	0	0
$955$	−5.10102	−0.165065
$956$	0	0
$957$	0	0
$958$	0	0
$959$	28.4041	0.917216
$960$	0	0
$961$	87.7878	2.83186
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−21.7980	−0.701701
$966$	0	0
$967$	14.8434	0.477330	0.238665	−	0.971102i	$-0.423290\pi$
$967$	14.8434	0.477330	0.238665	+	0.971102i	$0.423290\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	60.6969	1.94786	0.973929	−	0.226854i	$-0.0728441\pi$
$971$	60.6969	1.94786	0.973929	+	0.226854i	$0.0728441\pi$
$972$	0	0
$973$	28.4041	0.910593
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−45.3939	−1.45228	−0.726139	−	0.687548i	$-0.758687\pi$
$977$	−45.3939	−1.45228	−0.726139	+	0.687548i	$0.758687\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−6.55051	−0.208929	−0.104464	−	0.994529i	$-0.533313\pi$
$983$	−6.55051	−0.208929	−0.104464	+	0.994529i	$0.533313\pi$
$984$	0	0
$985$	−23.5959	−0.751828
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−19.8888	−0.632426
$990$	0	0
$991$	0.696938	0.0221390	0.0110695	−	0.999939i	$-0.496476\pi$
$991$	0.696938	0.0221390	0.0110695	+	0.999939i	$0.496476\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−13.1010	−0.415330
$996$	0	0
$997$	35.3939	1.12094	0.560468	−	0.828176i	$-0.310621\pi$
$997$	35.3939	1.12094	0.560468	+	0.828176i	$0.310621\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6480.2.a.bl.1.2		2
3.2	odd	2		6480.2.a.bc.1.2		2
4.3	odd	2		3240.2.a.o.1.1		2
9.2	odd	6		720.2.q.g.481.1		4
9.4	even	3		2160.2.q.g.721.1		4
9.5	odd	6		720.2.q.g.241.1		4
9.7	even	3		2160.2.q.g.1441.1		4
12.11	even	2		3240.2.a.j.1.1		2
36.7	odd	6		1080.2.q.c.361.2		4
36.11	even	6		360.2.q.c.121.2	✓	4
36.23	even	6		360.2.q.c.241.2	yes	4
36.31	odd	6		1080.2.q.c.721.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.q.c.121.2	✓	4	36.11	even	6
360.2.q.c.241.2	yes	4	36.23	even	6
720.2.q.g.241.1		4	9.5	odd	6
720.2.q.g.481.1		4	9.2	odd	6
1080.2.q.c.361.2		4	36.7	odd	6
1080.2.q.c.721.2		4	36.31	odd	6
2160.2.q.g.721.1		4	9.4	even	3
2160.2.q.g.1441.1		4	9.7	even	3
3240.2.a.j.1.1		2	12.11	even	2
3240.2.a.o.1.1		2	4.3	odd	2
6480.2.a.bc.1.2		2	3.2	odd	2
6480.2.a.bl.1.2		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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6480 = 2^{4} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6480.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +1.44949 q^{7} +2.00000 q^{17} -2.89898 q^{19} -2.55051 q^{23} +1.00000 q^{25} -7.89898 q^{29} -10.8990 q^{31} +1.44949 q^{35} -6.00000 q^{37} +0.101021 q^{41} +7.79796 q^{43} -4.55051 q^{47} -4.89898 q^{49} +11.7980 q^{53} -10.8990 q^{59} -3.00000 q^{61} -11.2474 q^{67} -9.79796 q^{71} -5.79796 q^{73} -2.89898 q^{79} -0.550510 q^{83} +2.00000 q^{85} +16.7980 q^{89} -2.89898 q^{95} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 2 q^{7} + 4 q^{17} + 4 q^{19} - 10 q^{23} + 2 q^{25} - 6 q^{29} - 12 q^{31} - 2 q^{35} - 12 q^{37} + 10 q^{41} - 4 q^{43} - 14 q^{47} + 4 q^{53} - 12 q^{59} - 6 q^{61} + 2 q^{67} + 8 q^{73}+ \cdots + 4 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 - 2 * q^7 + 4 * q^17 + 4 * q^19 - 10 * q^23 + 2 * q^25 - 6 * q^29 - 12 * q^31 - 2 * q^35 - 12 * q^37 + 10 * q^41 - 4 * q^43 - 14 * q^47 + 4 * q^53 - 12 * q^59 - 6 * q^61 + 2 * q^67 + 8 * q^73 + 4 * q^79 - 6 * q^83 + 4 * q^85 + 14 * q^89 + 4 * q^95 + 4 * q^97

Introduction

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