LMFDB - Embedded newform 6480.2.a.bl.1.1

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			1.1
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)

$f(q)$	$=$	$q+1.00000 q^{5} -3.44949 q^{7} +2.00000 q^{17} +6.89898 q^{19} -7.44949 q^{23} +1.00000 q^{25} +1.89898 q^{29} -1.10102 q^{31} -3.44949 q^{35} -6.00000 q^{37} +9.89898 q^{41} -11.7980 q^{43} -9.44949 q^{47} +4.89898 q^{49} -7.79796 q^{53} -1.10102 q^{59} -3.00000 q^{61} +13.2474 q^{67} +9.79796 q^{71} +13.7980 q^{73} +6.89898 q^{79} -5.44949 q^{83} +2.00000 q^{85} -2.79796 q^{89} +6.89898 q^{95} +2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−3.44949	−1.30378	−0.651892	−	0.758312i	$-0.726025\pi$
$7$	−3.44949	−1.30378	−0.651892	+	0.758312i	$0.726025\pi$
$8$	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
$19$	6.89898	1.58273	0.791367	−	0.611341i	$-0.209370\pi$
$19$	6.89898	1.58273	0.791367	+	0.611341i	$0.209370\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.44949	−1.55333	−0.776663	−	0.629916i	$-0.783089\pi$
$23$	−7.44949	−1.55333	−0.776663	+	0.629916i	$0.783089\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.89898	0.352632	0.176316	−	0.984334i	$-0.443582\pi$
$29$	1.89898	0.352632	0.176316	+	0.984334i	$0.443582\pi$
$30$	0	0
$31$	−1.10102	−0.197749	−0.0988746	−	0.995100i	$-0.531524\pi$
$31$	−1.10102	−0.197749	−0.0988746	+	0.995100i	$0.531524\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.44949	−0.583070
$36$	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
$41$	9.89898	1.54596	0.772980	−	0.634430i	$-0.218765\pi$
$41$	9.89898	1.54596	0.772980	+	0.634430i	$0.218765\pi$
$42$	0	0
$43$	−11.7980	−1.79917	−0.899586	−	0.436744i	$-0.856132\pi$
$43$	−11.7980	−1.79917	−0.899586	+	0.436744i	$0.856132\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.44949	−1.37835	−0.689175	−	0.724595i	$-0.742027\pi$
$47$	−9.44949	−1.37835	−0.689175	+	0.724595i	$0.742027\pi$
$48$	0	0
$49$	4.89898	0.699854
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−7.79796	−1.07113	−0.535566	−	0.844493i	$-0.679902\pi$
$53$	−7.79796	−1.07113	−0.535566	+	0.844493i	$0.679902\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.10102	−0.143341	−0.0716703	−	0.997428i	$-0.522833\pi$
$59$	−1.10102	−0.143341	−0.0716703	+	0.997428i	$0.522833\pi$
$60$	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
$67$	13.2474	1.61843	0.809217	−	0.587510i	$-0.199892\pi$
$67$	13.2474	1.61843	0.809217	+	0.587510i	$0.199892\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	9.79796	1.16280	0.581402	−	0.813617i	$-0.302504\pi$
$71$	9.79796	1.16280	0.581402	+	0.813617i	$0.302504\pi$
$72$	0	0
$73$	13.7980	1.61493	0.807464	−	0.589916i	$-0.200839\pi$
$73$	13.7980	1.61493	0.807464	+	0.589916i	$0.200839\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	6.89898	0.776196	0.388098	−	0.921618i	$-0.373132\pi$
$79$	6.89898	0.776196	0.388098	+	0.921618i	$0.373132\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−5.44949	−0.598159	−0.299080	−	0.954228i	$-0.596680\pi$
$83$	−5.44949	−0.598159	−0.299080	+	0.954228i	$0.596680\pi$
$84$	0	0
$85$	2.00000	0.216930
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−2.79796	−0.296583	−0.148292	−	0.988944i	$-0.547377\pi$
$89$	−2.79796	−0.296583	−0.148292	+	0.988944i	$0.547377\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	6.89898	0.707820
$96$	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
$107$	−12.3485	−1.19377	−0.596886	−	0.802326i	$-0.703595\pi$
$107$	−12.3485	−1.19377	−0.596886	+	0.802326i	$0.703595\pi$
$108$	0	0
$109$	−10.7980	−1.03426	−0.517128	−	0.855908i	$-0.672999\pi$
$109$	−10.7980	−1.03426	−0.517128	+	0.855908i	$0.672999\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	9.79796	0.921714	0.460857	−	0.887474i	$-0.347542\pi$
$113$	9.79796	0.921714	0.460857	+	0.887474i	$0.347542\pi$
$114$	0	0
$115$	−7.44949	−0.694669
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.89898	−0.632428
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−0.348469	−0.0309216	−0.0154608	−	0.999880i	$-0.504922\pi$
$127$	−0.348469	−0.0309216	−0.0154608	+	0.999880i	$0.504922\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2.89898	0.253285	0.126643	−	0.991948i	$-0.459580\pi$
$131$	2.89898	0.253285	0.126643	+	0.991948i	$0.459580\pi$
$132$	0	0
$133$	−23.7980	−2.06354
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−19.5959	−1.67419	−0.837096	−	0.547056i	$-0.815749\pi$
$137$	−19.5959	−1.67419	−0.837096	+	0.547056i	$0.815749\pi$
$138$	0	0
$139$	−19.5959	−1.66210	−0.831052	−	0.556195i	$-0.812261\pi$
$139$	−19.5959	−1.66210	−0.831052	+	0.556195i	$0.812261\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	1.89898	0.157702
$146$	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
$155$	−1.10102	−0.0884361
$156$	0	0
$157$	−23.7980	−1.89928	−0.949642	−	0.313337i	$-0.898553\pi$
$157$	−23.7980	−1.89928	−0.949642	+	0.313337i	$0.898553\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	25.6969	2.02520
$162$	0	0
$163$	7.79796	0.610783	0.305392	−	0.952227i	$-0.401213\pi$
$163$	7.79796	0.610783	0.305392	+	0.952227i	$0.401213\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.34847	0.646024	0.323012	−	0.946395i	$-0.395305\pi$
$167$	8.34847	0.646024	0.323012	+	0.946395i	$0.395305\pi$
$168$	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
$175$	−3.44949	−0.260757
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−13.7980	−1.03131	−0.515654	−	0.856797i	$-0.672451\pi$
$179$	−13.7980	−1.03131	−0.515654	+	0.856797i	$0.672451\pi$
$180$	0	0
$181$	−9.69694	−0.720768	−0.360384	−	0.932804i	$-0.617354\pi$
$181$	−9.69694	−0.720768	−0.360384	+	0.932804i	$0.617354\pi$
$182$	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
$191$	−14.8990	−1.07805	−0.539026	−	0.842289i	$-0.681208\pi$
$191$	−14.8990	−1.07805	−0.539026	+	0.842289i	$0.681208\pi$
$192$	0	0
$193$	−2.20204	−0.158506	−0.0792532	−	0.996855i	$-0.525254\pi$
$193$	−2.20204	−0.158506	−0.0792532	+	0.996855i	$0.525254\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	15.5959	1.11116	0.555582	−	0.831462i	$-0.312496\pi$
$197$	15.5959	1.11116	0.555582	+	0.831462i	$0.312496\pi$
$198$	0	0
$199$	−22.8990	−1.62327	−0.811633	−	0.584168i	$-0.801421\pi$
$199$	−22.8990	−1.62327	−0.811633	+	0.584168i	$0.801421\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−6.55051	−0.459756
$204$	0	0
$205$	9.89898	0.691375
$206$	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
$215$	−11.7980	−0.804614
$216$	0	0
$217$	3.79796	0.257822
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−5.44949	−0.364925	−0.182462	−	0.983213i	$-0.558407\pi$
$223$	−5.44949	−0.364925	−0.182462	+	0.983213i	$0.558407\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−8.20204	−0.544389	−0.272194	−	0.962242i	$-0.587749\pi$
$227$	−8.20204	−0.544389	−0.272194	+	0.962242i	$0.587749\pi$
$228$	0	0
$229$	−4.10102	−0.271003	−0.135502	−	0.990777i	$-0.543265\pi$
$229$	−4.10102	−0.271003	−0.135502	+	0.990777i	$0.543265\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	19.7980	1.29701	0.648504	−	0.761211i	$-0.275395\pi$
$233$	19.7980	1.29701	0.648504	+	0.761211i	$0.275395\pi$
$234$	0	0
$235$	−9.44949	−0.616417
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2.20204	0.142438	0.0712191	−	0.997461i	$-0.477311\pi$
$239$	2.20204	0.142438	0.0712191	+	0.997461i	$0.477311\pi$
$240$	0	0
$241$	−3.69694	−0.238141	−0.119070	−	0.992886i	$-0.537991\pi$
$241$	−3.69694	−0.238141	−0.119070	+	0.992886i	$0.537991\pi$
$242$	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
$251$	2.89898	0.182982	0.0914910	−	0.995806i	$-0.470837\pi$
$251$	2.89898	0.182982	0.0914910	+	0.995806i	$0.470837\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−27.7980	−1.73399	−0.866995	−	0.498318i	$-0.833951\pi$
$257$	−27.7980	−1.73399	−0.866995	+	0.498318i	$0.833951\pi$
$258$	0	0
$259$	20.6969	1.28605
$260$	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
$265$	−7.79796	−0.479025
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−22.5959	−1.37770	−0.688849	−	0.724905i	$-0.741884\pi$
$269$	−22.5959	−1.37770	−0.688849	+	0.724905i	$0.741884\pi$
$270$	0	0
$271$	−30.8990	−1.87698	−0.938490	−	0.345307i	$-0.887775\pi$
$271$	−30.8990	−1.87698	−0.938490	+	0.345307i	$0.887775\pi$
$272$	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
$281$	8.10102	0.483266	0.241633	−	0.970368i	$-0.422317\pi$
$281$	8.10102	0.483266	0.241633	+	0.970368i	$0.422317\pi$
$282$	0	0
$283$	3.65153	0.217061	0.108530	−	0.994093i	$-0.465385\pi$
$283$	3.65153	0.217061	0.108530	+	0.994093i	$0.465385\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−34.1464	−2.01560
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−17.5959	−1.02796	−0.513982	−	0.857801i	$-0.671830\pi$
$293$	−17.5959	−1.02796	−0.513982	+	0.857801i	$0.671830\pi$
$294$	0	0
$295$	−1.10102	−0.0641039
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	40.6969	2.34573
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−3.00000	−0.171780
$306$	0	0
$307$	4.75255	0.271242	0.135621	−	0.990761i	$-0.456697\pi$
$307$	4.75255	0.271242	0.135621	+	0.990761i	$0.456697\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−6.89898	−0.391205	−0.195603	−	0.980683i	$-0.562666\pi$
$311$	−6.89898	−0.391205	−0.195603	+	0.980683i	$0.562666\pi$
$312$	0	0
$313$	−15.5959	−0.881533	−0.440767	−	0.897622i	$-0.645293\pi$
$313$	−15.5959	−0.881533	−0.440767	+	0.897622i	$0.645293\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	25.7980	1.44896	0.724479	−	0.689297i	$-0.242080\pi$
$317$	25.7980	1.44896	0.724479	+	0.689297i	$0.242080\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	13.7980	0.767739
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	32.5959	1.79707
$330$	0	0
$331$	6.89898	0.379202	0.189601	−	0.981861i	$-0.439281\pi$
$331$	6.89898	0.379202	0.189601	+	0.981861i	$0.439281\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	13.2474	0.723785
$336$	0	0
$337$	−29.7980	−1.62320	−0.811599	−	0.584215i	$-0.801403\pi$
$337$	−29.7980	−1.62320	−0.811599	+	0.584215i	$0.801403\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	7.24745	0.391325
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−25.5959	−1.37406	−0.687030	−	0.726629i	$-0.741086\pi$
$347$	−25.5959	−1.37406	−0.687030	+	0.726629i	$0.741086\pi$
$348$	0	0
$349$	−2.30306	−0.123280	−0.0616400	−	0.998098i	$-0.519633\pi$
$349$	−2.30306	−0.123280	−0.0616400	+	0.998098i	$0.519633\pi$
$350$	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
$359$	28.6969	1.51457	0.757283	−	0.653087i	$-0.226526\pi$
$359$	28.6969	1.51457	0.757283	+	0.653087i	$0.226526\pi$
$360$	0	0
$361$	28.5959	1.50505
$362$	0	0
$363$	0	0
$364$	0	0
$365$	13.7980	0.722218
$366$	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
$371$	26.8990	1.39653
$372$	0	0
$373$	21.5959	1.11819	0.559097	−	0.829102i	$-0.311148\pi$
$373$	21.5959	1.11819	0.559097	+	0.829102i	$0.311148\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−26.8990	−1.38171	−0.690854	−	0.722994i	$-0.742765\pi$
$379$	−26.8990	−1.38171	−0.690854	+	0.722994i	$0.742765\pi$
$380$	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
$389$	16.5959	0.841446	0.420723	−	0.907189i	$-0.361776\pi$
$389$	16.5959	0.841446	0.420723	+	0.907189i	$0.361776\pi$
$390$	0	0
$391$	−14.8990	−0.753474
$392$	0	0
$393$	0	0
$394$	0	0
$395$	6.89898	0.347125
$396$	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
$401$	−25.5959	−1.27820	−0.639100	−	0.769124i	$-0.720693\pi$
$401$	−25.5959	−1.27820	−0.639100	+	0.769124i	$0.720693\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−9.59592	−0.474488	−0.237244	−	0.971450i	$-0.576244\pi$
$409$	−9.59592	−0.474488	−0.237244	+	0.971450i	$0.576244\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3.79796	0.186885
$414$	0	0
$415$	−5.44949	−0.267505
$416$	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
$421$	25.5959	1.24747	0.623734	−	0.781636i	$-0.285615\pi$
$421$	25.5959	1.24747	0.623734	+	0.781636i	$0.285615\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2.00000	0.0970143
$426$	0	0
$427$	10.3485	0.500798
$428$	0	0
$429$	0	0
$430$	0	0
$431$	14.8990	0.717659	0.358829	−	0.933403i	$-0.383176\pi$
$431$	14.8990	0.717659	0.358829	+	0.933403i	$0.383176\pi$
$432$	0	0
$433$	−11.7980	−0.566974	−0.283487	−	0.958976i	$-0.591491\pi$
$433$	−11.7980	−0.566974	−0.283487	+	0.958976i	$0.591491\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−51.3939	−2.45850
$438$	0	0
$439$	17.7980	0.849450	0.424725	−	0.905322i	$-0.360371\pi$
$439$	17.7980	0.849450	0.424725	+	0.905322i	$0.360371\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−8.14643	−0.387048	−0.193524	−	0.981095i	$-0.561992\pi$
$443$	−8.14643	−0.387048	−0.193524	+	0.981095i	$0.561992\pi$
$444$	0	0
$445$	−2.79796	−0.132636
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−21.5959	−1.01917	−0.509587	−	0.860419i	$-0.670202\pi$
$449$	−21.5959	−1.01917	−0.509587	+	0.860419i	$0.670202\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	17.5959	0.823102	0.411551	−	0.911387i	$-0.364987\pi$
$457$	17.5959	0.823102	0.411551	+	0.911387i	$0.364987\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−12.1010	−0.563601	−0.281800	−	0.959473i	$-0.590932\pi$
$461$	−12.1010	−0.563601	−0.281800	+	0.959473i	$0.590932\pi$
$462$	0	0
$463$	−12.2020	−0.567077	−0.283538	−	0.958961i	$-0.591508\pi$
$463$	−12.2020	−0.567077	−0.283538	+	0.958961i	$0.591508\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7.79796	0.360847	0.180423	−	0.983589i	$-0.442253\pi$
$467$	7.79796	0.360847	0.180423	+	0.983589i	$0.442253\pi$
$468$	0	0
$469$	−45.6969	−2.11009
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	6.89898	0.316547
$476$	0	0
$477$	0	0
$478$	0	0
$479$	30.4949	1.39335	0.696674	−	0.717388i	$-0.254663\pi$
$479$	30.4949	1.39335	0.696674	+	0.717388i	$0.254663\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.00000	0.0908153
$486$	0	0
$487$	13.5959	0.616090	0.308045	−	0.951372i	$-0.400325\pi$
$487$	13.5959	0.616090	0.308045	+	0.951372i	$0.400325\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−13.7980	−0.622693	−0.311347	−	0.950296i	$-0.600780\pi$
$491$	−13.7980	−0.622693	−0.311347	+	0.950296i	$0.600780\pi$
$492$	0	0
$493$	3.79796	0.171051
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−33.7980	−1.51605
$498$	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
$503$	25.0454	1.11672	0.558360	−	0.829599i	$-0.311431\pi$
$503$	25.0454	1.11672	0.558360	+	0.829599i	$0.311431\pi$
$504$	0	0
$505$	2.00000	0.0889988
$506$	0	0
$507$	0	0
$508$	0	0
$509$	4.10102	0.181775	0.0908873	−	0.995861i	$-0.471030\pi$
$509$	4.10102	0.181775	0.0908873	+	0.995861i	$0.471030\pi$
$510$	0	0
$511$	−47.5959	−2.10552
$512$	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
$521$	−9.20204	−0.403149	−0.201574	−	0.979473i	$-0.564606\pi$
$521$	−9.20204	−0.403149	−0.201574	+	0.979473i	$0.564606\pi$
$522$	0	0
$523$	34.3485	1.50195	0.750977	−	0.660329i	$-0.229583\pi$
$523$	34.3485	1.50195	0.750977	+	0.660329i	$0.229583\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.20204	−0.0959224
$528$	0	0
$529$	32.4949	1.41282
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−12.3485	−0.533871
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	12.1010	0.520264	0.260132	−	0.965573i	$-0.416234\pi$
$541$	12.1010	0.520264	0.260132	+	0.965573i	$0.416234\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−10.7980	−0.462534
$546$	0	0
$547$	16.7526	0.716287	0.358144	−	0.933666i	$-0.383410\pi$
$547$	16.7526	0.716287	0.358144	+	0.933666i	$0.383410\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	13.1010	0.558122
$552$	0	0
$553$	−23.7980	−1.01199
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−43.7980	−1.85578	−0.927890	−	0.372855i	$-0.878379\pi$
$557$	−43.7980	−1.85578	−0.927890	+	0.372855i	$0.878379\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−10.3485	−0.436136	−0.218068	−	0.975934i	$-0.569975\pi$
$563$	−10.3485	−0.436136	−0.218068	+	0.975934i	$0.569975\pi$
$564$	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
$571$	−25.7980	−1.07961	−0.539805	−	0.841790i	$-0.681502\pi$
$571$	−25.7980	−1.07961	−0.539805	+	0.841790i	$0.681502\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−7.44949	−0.310665
$576$	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
$581$	18.7980	0.779871
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7.65153	0.315812	0.157906	−	0.987454i	$-0.449526\pi$
$587$	7.65153	0.315812	0.157906	+	0.987454i	$0.449526\pi$
$588$	0	0
$589$	−7.59592	−0.312984
$590$	0	0
$591$	0	0
$592$	0	0
$593$	31.3939	1.28919	0.644596	−	0.764523i	$-0.277026\pi$
$593$	31.3939	1.28919	0.644596	+	0.764523i	$0.277026\pi$
$594$	0	0
$595$	−6.89898	−0.282831
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−24.6969	−1.00909	−0.504545	−	0.863386i	$-0.668340\pi$
$599$	−24.6969	−1.00909	−0.504545	+	0.863386i	$0.668340\pi$
$600$	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
$607$	−2.14643	−0.0871208	−0.0435604	−	0.999051i	$-0.513870\pi$
$607$	−2.14643	−0.0871208	−0.0435604	+	0.999051i	$0.513870\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−29.5959	−1.19537	−0.597684	−	0.801732i	$-0.703912\pi$
$613$	−29.5959	−1.19537	−0.597684	+	0.801732i	$0.703912\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	35.1918	1.41677	0.708385	−	0.705826i	$-0.249424\pi$
$617$	35.1918	1.41677	0.708385	+	0.705826i	$0.249424\pi$
$618$	0	0
$619$	24.6969	0.992654	0.496327	−	0.868136i	$-0.334682\pi$
$619$	24.6969	0.992654	0.496327	+	0.868136i	$0.334682\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	9.65153	0.386680
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−12.0000	−0.478471
$630$	0	0
$631$	15.5959	0.620864	0.310432	−	0.950596i	$-0.399526\pi$
$631$	15.5959	0.620864	0.310432	+	0.950596i	$0.399526\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−0.348469	−0.0138286
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	13.8990	0.548977	0.274488	−	0.961590i	$-0.411492\pi$
$641$	13.8990	0.548977	0.274488	+	0.961590i	$0.411492\pi$
$642$	0	0
$643$	20.1464	0.794498	0.397249	−	0.917711i	$-0.369965\pi$
$643$	20.1464	0.794498	0.397249	+	0.917711i	$0.369965\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6.55051	0.257527	0.128764	−	0.991675i	$-0.458899\pi$
$647$	6.55051	0.257527	0.128764	+	0.991675i	$0.458899\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−27.7980	−1.08782	−0.543909	−	0.839144i	$-0.683056\pi$
$653$	−27.7980	−1.08782	−0.543909	+	0.839144i	$0.683056\pi$
$654$	0	0
$655$	2.89898	0.113273
$656$	0	0
$657$	0	0
$658$	0	0
$659$	10.2020	0.397415	0.198708	−	0.980059i	$-0.436326\pi$
$659$	10.2020	0.397415	0.198708	+	0.980059i	$0.436326\pi$
$660$	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
$665$	−23.7980	−0.922845
$666$	0	0
$667$	−14.1464	−0.547752
$668$	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
$677$	−14.2020	−0.545829	−0.272914	−	0.962038i	$-0.587988\pi$
$677$	−14.2020	−0.545829	−0.272914	+	0.962038i	$0.587988\pi$
$678$	0	0
$679$	−6.89898	−0.264759
$680$	0	0
$681$	0	0
$682$	0	0
$683$	23.3939	0.895142	0.447571	−	0.894248i	$-0.352289\pi$
$683$	23.3939	0.895142	0.447571	+	0.894248i	$0.352289\pi$
$684$	0	0
$685$	−19.5959	−0.748722
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−38.8990	−1.47979	−0.739893	−	0.672724i	$-0.765124\pi$
$691$	−38.8990	−1.47979	−0.739893	+	0.672724i	$0.765124\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−19.5959	−0.743316
$696$	0	0
$697$	19.7980	0.749901
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−30.3939	−1.14796	−0.573980	−	0.818869i	$-0.694601\pi$
$701$	−30.3939	−1.14796	−0.573980	+	0.818869i	$0.694601\pi$
$702$	0	0
$703$	−41.3939	−1.56120
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−6.89898	−0.259463
$708$	0	0
$709$	9.69694	0.364176	0.182088	−	0.983282i	$-0.441714\pi$
$709$	9.69694	0.364176	0.182088	+	0.983282i	$0.441714\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8.20204	0.307169
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−28.2929	−1.05515	−0.527573	−	0.849510i	$-0.676898\pi$
$719$	−28.2929	−1.05515	−0.527573	+	0.849510i	$0.676898\pi$
$720$	0	0
$721$	34.4949	1.28466
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.89898	0.0705263
$726$	0	0
$727$	31.2474	1.15890	0.579452	−	0.815006i	$-0.303267\pi$
$727$	31.2474	1.15890	0.579452	+	0.815006i	$0.303267\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−23.5959	−0.872727
$732$	0	0
$733$	−29.5959	−1.09315	−0.546575	−	0.837410i	$-0.684069\pi$
$733$	−29.5959	−1.09315	−0.546575	+	0.837410i	$0.684069\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	33.1010	1.21764	0.608820	−	0.793308i	$-0.291643\pi$
$739$	33.1010	1.21764	0.608820	+	0.793308i	$0.291643\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−7.04541	−0.258471	−0.129235	−	0.991614i	$-0.541252\pi$
$743$	−7.04541	−0.258471	−0.129235	+	0.991614i	$0.541252\pi$
$744$	0	0
$745$	21.0000	0.769380
$746$	0	0
$747$	0	0
$748$	0	0
$749$	42.5959	1.55642
$750$	0	0
$751$	−26.2020	−0.956126	−0.478063	−	0.878326i	$-0.658661\pi$
$751$	−26.2020	−0.956126	−0.478063	+	0.878326i	$0.658661\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−12.0000	−0.436725
$756$	0	0
$757$	31.5959	1.14837	0.574187	−	0.818724i	$-0.305318\pi$
$757$	31.5959	1.14837	0.574187	+	0.818724i	$0.305318\pi$
$758$	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
$763$	37.2474	1.34845
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−5.20204	−0.187590	−0.0937952	−	0.995592i	$-0.529900\pi$
$769$	−5.20204	−0.187590	−0.0937952	+	0.995592i	$0.529900\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−6.20204	−0.223072	−0.111536	−	0.993760i	$-0.535577\pi$
$773$	−6.20204	−0.223072	−0.111536	+	0.993760i	$0.535577\pi$
$774$	0	0
$775$	−1.10102	−0.0395498
$776$	0	0
$777$	0	0
$778$	0	0
$779$	68.2929	2.44685
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−23.7980	−0.849386
$786$	0	0
$787$	49.5959	1.76790	0.883952	−	0.467578i	$-0.154873\pi$
$787$	49.5959	1.76790	0.883952	+	0.467578i	$0.154873\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−33.7980	−1.20172
$792$	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
$799$	−18.8990	−0.668598
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	25.6969	0.905698
$806$	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
$811$	−13.1010	−0.460039	−0.230020	−	0.973186i	$-0.573879\pi$
$811$	−13.1010	−0.460039	−0.230020	+	0.973186i	$0.573879\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	7.79796	0.273151
$816$	0	0
$817$	−81.3939	−2.84761
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−2.79796	−0.0976494	−0.0488247	−	0.998807i	$-0.515548\pi$
$821$	−2.79796	−0.0976494	−0.0488247	+	0.998807i	$0.515548\pi$
$822$	0	0
$823$	8.34847	0.291009	0.145505	−	0.989358i	$-0.453519\pi$
$823$	8.34847	0.291009	0.145505	+	0.989358i	$0.453519\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	53.5403	1.86178	0.930889	−	0.365301i	$-0.119034\pi$
$827$	53.5403	1.86178	0.930889	+	0.365301i	$0.119034\pi$
$828$	0	0
$829$	50.1918	1.74323	0.871617	−	0.490187i	$-0.163072\pi$
$829$	50.1918	1.74323	0.871617	+	0.490187i	$0.163072\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	9.79796	0.339479
$834$	0	0
$835$	8.34847	0.288911
$836$	0	0
$837$	0	0
$838$	0	0
$839$	4.69694	0.162156	0.0810782	−	0.996708i	$-0.474164\pi$
$839$	4.69694	0.162156	0.0810782	+	0.996708i	$0.474164\pi$
$840$	0	0
$841$	−25.3939	−0.875651
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−13.0000	−0.447214
$846$	0	0
$847$	37.9444	1.30378
$848$	0	0
$849$	0	0
$850$	0	0
$851$	44.6969	1.53219
$852$	0	0
$853$	18.2020	0.623226	0.311613	−	0.950209i	$-0.399131\pi$
$853$	18.2020	0.623226	0.311613	+	0.950209i	$0.399131\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−0.404082	−0.0138032	−0.00690159	−	0.999976i	$-0.502197\pi$
$857$	−0.404082	−0.0138032	−0.00690159	+	0.999976i	$0.502197\pi$
$858$	0	0
$859$	−39.1918	−1.33721	−0.668604	−	0.743619i	$-0.733108\pi$
$859$	−39.1918	−1.33721	−0.668604	+	0.743619i	$0.733108\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	57.4495	1.95560	0.977802	−	0.209532i	$-0.0671942\pi$
$863$	57.4495	1.95560	0.977802	+	0.209532i	$0.0671942\pi$
$864$	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
$875$	−3.44949	−0.116614
$876$	0	0
$877$	−0.404082	−0.0136449	−0.00682244	−	0.999977i	$-0.502172\pi$
$877$	−0.404082	−0.0136449	−0.00682244	+	0.999977i	$0.502172\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−6.10102	−0.205549	−0.102774	−	0.994705i	$-0.532772\pi$
$881$	−6.10102	−0.205549	−0.102774	+	0.994705i	$0.532772\pi$
$882$	0	0
$883$	0.954592	0.0321246	0.0160623	−	0.999871i	$-0.494887\pi$
$883$	0.954592	0.0321246	0.0160623	+	0.999871i	$0.494887\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.59592	0.0535857	0.0267928	−	0.999641i	$-0.491471\pi$
$887$	1.59592	0.0535857	0.0267928	+	0.999641i	$0.491471\pi$
$888$	0	0
$889$	1.20204	0.0403152
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−65.1918	−2.18156
$894$	0	0
$895$	−13.7980	−0.461215
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−2.09082	−0.0697326
$900$	0	0
$901$	−15.5959	−0.519575
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−9.69694	−0.322337
$906$	0	0
$907$	21.6515	0.718927	0.359464	−	0.933159i	$-0.382960\pi$
$907$	21.6515	0.718927	0.359464	+	0.933159i	$0.382960\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	10.4949	0.347711	0.173856	−	0.984771i	$-0.444377\pi$
$911$	10.4949	0.347711	0.173856	+	0.984771i	$0.444377\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−10.0000	−0.330229
$918$	0	0
$919$	−24.6969	−0.814677	−0.407338	−	0.913277i	$-0.633543\pi$
$919$	−24.6969	−0.814677	−0.407338	+	0.913277i	$0.633543\pi$
$920$	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
$929$	−37.5959	−1.23348	−0.616741	−	0.787166i	$-0.711547\pi$
$929$	−37.5959	−1.23348	−0.616741	+	0.787166i	$0.711547\pi$
$930$	0	0
$931$	33.7980	1.10768
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−40.9898	−1.33908	−0.669539	−	0.742777i	$-0.733508\pi$
$937$	−40.9898	−1.33908	−0.669539	+	0.742777i	$0.733508\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.69694	0.0553186	0.0276593	−	0.999617i	$-0.491195\pi$
$941$	1.69694	0.0553186	0.0276593	+	0.999617i	$0.491195\pi$
$942$	0	0
$943$	−73.7423	−2.40138
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−11.8536	−0.385189	−0.192595	−	0.981278i	$-0.561690\pi$
$947$	−11.8536	−0.385189	−0.192595	+	0.981278i	$0.561690\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	14.2020	0.460049	0.230025	−	0.973185i	$-0.426119\pi$
$953$	14.2020	0.460049	0.230025	+	0.973185i	$0.426119\pi$
$954$	0	0
$955$	−14.8990	−0.482120
$956$	0	0
$957$	0	0
$958$	0	0
$959$	67.5959	2.18279
$960$	0	0
$961$	−29.7878	−0.960895
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−2.20204	−0.0708862
$966$	0	0
$967$	−48.8434	−1.57070	−0.785348	−	0.619054i	$-0.787516\pi$
$967$	−48.8434	−1.57070	−0.785348	+	0.619054i	$0.787516\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	31.3031	1.00456	0.502282	−	0.864704i	$-0.332494\pi$
$971$	31.3031	1.00456	0.502282	+	0.864704i	$0.332494\pi$
$972$	0	0
$973$	67.5959	2.16703
$974$	0	0
$975$	0	0
$976$	0	0
$977$	13.3939	0.428508	0.214254	−	0.976778i	$-0.431268\pi$
$977$	13.3939	0.428508	0.214254	+	0.976778i	$0.431268\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−11.4495	−0.365182	−0.182591	−	0.983189i	$-0.558448\pi$
$983$	−11.4495	−0.365182	−0.182591	+	0.983189i	$0.558448\pi$
$984$	0	0
$985$	15.5959	0.496927
$986$	0	0
$987$	0	0
$988$	0	0
$989$	87.8888	2.79470
$990$	0	0
$991$	−28.6969	−0.911588	−0.455794	−	0.890085i	$-0.650645\pi$
$991$	−28.6969	−0.911588	−0.455794	+	0.890085i	$0.650645\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−22.8990	−0.725946
$996$	0	0
$997$	−23.3939	−0.740892	−0.370446	−	0.928854i	$-0.620795\pi$
$997$	−23.3939	−0.740892	−0.370446	+	0.928854i	$0.620795\pi$
$998$	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.1.1		2
3.2	odd	2		6480.2.a.bc.1.1		2
4.3	odd	2		3240.2.a.o.1.2		2
9.2	odd	6		720.2.q.g.481.2		4
9.4	even	3		2160.2.q.g.721.2		4
9.5	odd	6		720.2.q.g.241.2		4
9.7	even	3		2160.2.q.g.1441.2		4
12.11	even	2		3240.2.a.j.1.2		2
36.7	odd	6		1080.2.q.c.361.1		4
36.11	even	6		360.2.q.c.121.1	✓	4
36.23	even	6		360.2.q.c.241.1	yes	4
36.31	odd	6		1080.2.q.c.721.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.q.c.121.1	✓	4	36.11	even	6
360.2.q.c.241.1	yes	4	36.23	even	6
720.2.q.g.241.2		4	9.5	odd	6
720.2.q.g.481.2		4	9.2	odd	6
1080.2.q.c.361.1		4	36.7	odd	6
1080.2.q.c.721.1		4	36.31	odd	6
2160.2.q.g.721.2		4	9.4	even	3
2160.2.q.g.1441.2		4	9.7	even	3
3240.2.a.j.1.2		2	12.11	even	2
3240.2.a.o.1.2		2	4.3	odd	2
6480.2.a.bc.1.1		2	3.2	odd	2
6480.2.a.bl.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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -3.44949 q^{7} +2.00000 q^{17} +6.89898 q^{19} -7.44949 q^{23} +1.00000 q^{25} +1.89898 q^{29} -1.10102 q^{31} -3.44949 q^{35} -6.00000 q^{37} +9.89898 q^{41} -11.7980 q^{43} -9.44949 q^{47} +4.89898 q^{49} -7.79796 q^{53} -1.10102 q^{59} -3.00000 q^{61} +13.2474 q^{67} +9.79796 q^{71} +13.7980 q^{73} +6.89898 q^{79} -5.44949 q^{83} +2.00000 q^{85} -2.79796 q^{89} +6.89898 q^{95} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 2 q^{7} + 4 q^{17} + 4 q^{19} - 10 q^{23} + 2 q^{25} - 6 q^{29} - 12 q^{31} - 2 q^{35} - 12 q^{37} + 10 q^{41} - 4 q^{43} - 14 q^{47} + 4 q^{53} - 12 q^{59} - 6 q^{61} + 2 q^{67} + 8 q^{73}+ \cdots + 4 q^{97}+O(q^{100}) \) 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

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