LMFDB - Embedded newform 6480.2.a.bd.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6480,2,Mod(1,6480)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

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{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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3240)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.37228$ 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)

$f(q)$	$=$	$q-1.00000 q^{5} +2.37228 q^{7} +O(q^{10})$	$q-1.00000 q^{5} +2.37228 q^{7} -3.37228 q^{11} +2.37228 q^{13} -4.74456 q^{17} -1.00000 q^{19} -0.372281 q^{23} +1.00000 q^{25} +3.37228 q^{29} +6.11684 q^{31} -2.37228 q^{35} +6.00000 q^{37} -11.7446 q^{41} -6.74456 q^{43} -3.62772 q^{47} -1.37228 q^{49} +7.11684 q^{53} +3.37228 q^{55} +5.00000 q^{59} +1.25544 q^{61} -2.37228 q^{65} -10.7446 q^{67} +1.37228 q^{71} +3.25544 q^{73} -8.00000 q^{77} -8.74456 q^{79} -10.0000 q^{83} +4.74456 q^{85} -1.37228 q^{89} +5.62772 q^{91} +1.00000 q^{95} +6.74456 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 - q^7	$ 2 q - 2 q^{5} - q^{7} - q^{11} - q^{13} + 2 q^{17} - 2 q^{19} + 5 q^{23} + 2 q^{25} + q^{29} - 5 q^{31} + q^{35} + 12 q^{37} - 12 q^{41} - 2 q^{43} - 13 q^{47} + 3 q^{49} - 3 q^{53} + q^{55} + 10 q^{59} + 14 q^{61} + q^{65} - 10 q^{67} - 3 q^{71} + 18 q^{73} - 16 q^{77} - 6 q^{79} - 20 q^{83} - 2 q^{85} + 3 q^{89} + 17 q^{91} + 2 q^{95} + 2 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - q^7 - q^11 - q^13 + 2 * q^17 - 2 * q^19 + 5 * q^23 + 2 * q^25 + q^29 - 5 * q^31 + q^35 + 12 * q^37 - 12 * q^41 - 2 * q^43 - 13 * q^47 + 3 * q^49 - 3 * q^53 + q^55 + 10 * q^59 + 14 * q^61 + q^65 - 10 * q^67 - 3 * q^71 + 18 * q^73 - 16 * q^77 - 6 * q^79 - 20 * q^83 - 2 * q^85 + 3 * q^89 + 17 * q^91 + 2 * q^95 + 2 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	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
$7$	2.37228	0.896638	0.448319	−	0.893874i	$-0.352023\pi$
$7$	2.37228	0.896638	0.448319	+	0.893874i	$0.352023\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.37228	−1.01678	−0.508391	−	0.861127i	$-0.669759\pi$
$11$	−3.37228	−1.01678	−0.508391	+	0.861127i	$0.669759\pi$
$12$	0	0
$13$	2.37228	0.657952	0.328976	−	0.944338i	$-0.393296\pi$
$13$	2.37228	0.657952	0.328976	+	0.944338i	$0.393296\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.74456	−1.15073	−0.575363	−	0.817898i	$-0.695139\pi$
$17$	−4.74456	−1.15073	−0.575363	+	0.817898i	$0.695139\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.372281	−0.0776260	−0.0388130	−	0.999246i	$-0.512358\pi$
$23$	−0.372281	−0.0776260	−0.0388130	+	0.999246i	$0.512358\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.37228	0.626217	0.313108	−	0.949717i	$-0.398630\pi$
$29$	3.37228	0.626217	0.313108	+	0.949717i	$0.398630\pi$
$30$	0	0
$31$	6.11684	1.09862	0.549309	−	0.835619i	$-0.314891\pi$
$31$	6.11684	1.09862	0.549309	+	0.835619i	$0.314891\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.37228	−0.400989
$36$	0	0
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−11.7446	−1.83419	−0.917096	−	0.398666i	$-0.869473\pi$
$41$	−11.7446	−1.83419	−0.917096	+	0.398666i	$0.869473\pi$
$42$	0	0
$43$	−6.74456	−1.02854	−0.514268	−	0.857629i	$-0.671936\pi$
$43$	−6.74456	−1.02854	−0.514268	+	0.857629i	$0.671936\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.62772	−0.529157	−0.264579	−	0.964364i	$-0.585233\pi$
$47$	−3.62772	−0.529157	−0.264579	+	0.964364i	$0.585233\pi$
$48$	0	0
$49$	−1.37228	−0.196040
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.11684	0.977574	0.488787	−	0.872403i	$-0.337440\pi$
$53$	7.11684	0.977574	0.488787	+	0.872403i	$0.337440\pi$
$54$	0	0
$55$	3.37228	0.454718
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.00000	0.650945	0.325472	−	0.945552i	$-0.394477\pi$
$59$	5.00000	0.650945	0.325472	+	0.945552i	$0.394477\pi$
$60$	0	0
$61$	1.25544	0.160742	0.0803711	−	0.996765i	$-0.474389\pi$
$61$	1.25544	0.160742	0.0803711	+	0.996765i	$0.474389\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.37228	−0.294245
$66$	0	0
$67$	−10.7446	−1.31266	−0.656329	−	0.754475i	$-0.727892\pi$
$67$	−10.7446	−1.31266	−0.656329	+	0.754475i	$0.727892\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.37228	0.162860	0.0814299	−	0.996679i	$-0.474051\pi$
$71$	1.37228	0.162860	0.0814299	+	0.996679i	$0.474051\pi$
$72$	0	0
$73$	3.25544	0.381020	0.190510	−	0.981685i	$-0.438986\pi$
$73$	3.25544	0.381020	0.190510	+	0.981685i	$0.438986\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−8.00000	−0.911685
$78$	0	0
$79$	−8.74456	−0.983840	−0.491920	−	0.870640i	$-0.663705\pi$
$79$	−8.74456	−0.983840	−0.491920	+	0.870640i	$0.663705\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−10.0000	−1.09764	−0.548821	−	0.835940i	$-0.684923\pi$
$83$	−10.0000	−1.09764	−0.548821	+	0.835940i	$0.684923\pi$
$84$	0	0
$85$	4.74456	0.514620
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1.37228	−0.145462	−0.0727308	−	0.997352i	$-0.523171\pi$
$89$	−1.37228	−0.145462	−0.0727308	+	0.997352i	$0.523171\pi$
$90$	0	0
$91$	5.62772	0.589945
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	6.74456	0.684807	0.342403	−	0.939553i	$-0.388759\pi$
$97$	6.74456	0.684807	0.342403	+	0.939553i	$0.388759\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−12.8614	−1.27976	−0.639879	−	0.768476i	$-0.721015\pi$
$101$	−12.8614	−1.27976	−0.639879	+	0.768476i	$0.721015\pi$
$102$	0	0
$103$	−17.1168	−1.68657	−0.843286	−	0.537464i	$-0.819382\pi$
$103$	−17.1168	−1.68657	−0.843286	+	0.537464i	$0.819382\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	0	0
$109$	18.1168	1.73528	0.867639	−	0.497194i	$-0.165636\pi$
$109$	18.1168	1.73528	0.867639	+	0.497194i	$0.165636\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−3.25544	−0.306246	−0.153123	−	0.988207i	$-0.548933\pi$
$113$	−3.25544	−0.306246	−0.153123	+	0.988207i	$0.548933\pi$
$114$	0	0
$115$	0.372281	0.0347154
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−11.2554	−1.03178
$120$	0	0
$121$	0.372281	0.0338438
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−15.8614	−1.40747	−0.703736	−	0.710461i	$-0.748486\pi$
$127$	−15.8614	−1.40747	−0.703736	+	0.710461i	$0.748486\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	17.7446	1.55035	0.775175	−	0.631747i	$-0.217662\pi$
$131$	17.7446	1.55035	0.775175	+	0.631747i	$0.217662\pi$
$132$	0	0
$133$	−2.37228	−0.205703
$134$	0	0
$135$	0	0
$136$	0	0
$137$	16.7446	1.43058	0.715292	−	0.698825i	$-0.246294\pi$
$137$	16.7446	1.43058	0.715292	+	0.698825i	$0.246294\pi$
$138$	0	0
$139$	−15.0000	−1.27228	−0.636142	−	0.771572i	$-0.719471\pi$
$139$	−15.0000	−1.27228	−0.636142	+	0.771572i	$0.719471\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−8.00000	−0.668994
$144$	0	0
$145$	−3.37228	−0.280053
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3.48913	0.285840	0.142920	−	0.989734i	$-0.454351\pi$
$149$	3.48913	0.285840	0.142920	+	0.989734i	$0.454351\pi$
$150$	0	0
$151$	−18.6277	−1.51590	−0.757951	−	0.652311i	$-0.773799\pi$
$151$	−18.6277	−1.51590	−0.757951	+	0.652311i	$0.773799\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−6.11684	−0.491317
$156$	0	0
$157$	−5.86141	−0.467791	−0.233896	−	0.972262i	$-0.575147\pi$
$157$	−5.86141	−0.467791	−0.233896	+	0.972262i	$0.575147\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.883156	−0.0696024
$162$	0	0
$163$	13.4891	1.05655	0.528275	−	0.849073i	$-0.322839\pi$
$163$	13.4891	1.05655	0.528275	+	0.849073i	$0.322839\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	−7.37228	−0.567099
$170$	0	0
$171$	0	0
$172$	0	0
$173$	13.8614	1.05386	0.526932	−	0.849908i	$-0.323342\pi$
$173$	13.8614	1.05386	0.526932	+	0.849908i	$0.323342\pi$
$174$	0	0
$175$	2.37228	0.179328
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−15.7446	−1.17680	−0.588402	−	0.808569i	$-0.700243\pi$
$179$	−15.7446	−1.17680	−0.588402	+	0.808569i	$0.700243\pi$
$180$	0	0
$181$	14.8614	1.10464	0.552320	−	0.833632i	$-0.313743\pi$
$181$	14.8614	1.10464	0.552320	+	0.833632i	$0.313743\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−6.00000	−0.441129
$186$	0	0
$187$	16.0000	1.17004
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−11.3723	−0.822869	−0.411435	−	0.911439i	$-0.634972\pi$
$191$	−11.3723	−0.822869	−0.411435	+	0.911439i	$0.634972\pi$
$192$	0	0
$193$	−10.7446	−0.773411	−0.386705	−	0.922203i	$-0.626387\pi$
$193$	−10.7446	−0.773411	−0.386705	+	0.922203i	$0.626387\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.37228	0.454006	0.227003	−	0.973894i	$-0.427107\pi$
$197$	6.37228	0.454006	0.227003	+	0.973894i	$0.427107\pi$
$198$	0	0
$199$	9.48913	0.672666	0.336333	−	0.941743i	$-0.390813\pi$
$199$	9.48913	0.672666	0.336333	+	0.941743i	$0.390813\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	8.00000	0.561490
$204$	0	0
$205$	11.7446	0.820276
$206$	0	0
$207$	0	0
$208$	0	0
$209$	3.37228	0.233266
$210$	0	0
$211$	−17.0000	−1.17033	−0.585164	−	0.810915i	$-0.698970\pi$
$211$	−17.0000	−1.17033	−0.585164	+	0.810915i	$0.698970\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	6.74456	0.459975
$216$	0	0
$217$	14.5109	0.985062
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−11.2554	−0.757123
$222$	0	0
$223$	−2.51087	−0.168141	−0.0840703	−	0.996460i	$-0.526792\pi$
$223$	−2.51087	−0.168141	−0.0840703	+	0.996460i	$0.526792\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5.48913	0.364326	0.182163	−	0.983268i	$-0.441690\pi$
$227$	5.48913	0.364326	0.182163	+	0.983268i	$0.441690\pi$
$228$	0	0
$229$	−5.25544	−0.347289	−0.173645	−	0.984808i	$-0.555554\pi$
$229$	−5.25544	−0.347289	−0.173645	+	0.984808i	$0.555554\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−20.2337	−1.32555	−0.662776	−	0.748817i	$-0.730622\pi$
$233$	−20.2337	−1.32555	−0.662776	+	0.748817i	$0.730622\pi$
$234$	0	0
$235$	3.62772	0.236646
$236$	0	0
$237$	0	0
$238$	0	0
$239$	14.2337	0.920701	0.460350	−	0.887737i	$-0.347724\pi$
$239$	14.2337	0.920701	0.460350	+	0.887737i	$0.347724\pi$
$240$	0	0
$241$	29.6060	1.90709	0.953544	−	0.301254i	$-0.0974051\pi$
$241$	29.6060	1.90709	0.953544	+	0.301254i	$0.0974051\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.37228	0.0876718
$246$	0	0
$247$	−2.37228	−0.150945
$248$	0	0
$249$	0	0
$250$	0	0
$251$	15.8614	1.00116	0.500582	−	0.865689i	$-0.333120\pi$
$251$	15.8614	1.00116	0.500582	+	0.865689i	$0.333120\pi$
$252$	0	0
$253$	1.25544	0.0789287
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−23.4891	−1.46521	−0.732606	−	0.680653i	$-0.761696\pi$
$257$	−23.4891	−1.46521	−0.732606	+	0.680653i	$0.761696\pi$
$258$	0	0
$259$	14.2337	0.884438
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−13.1168	−0.808819	−0.404410	−	0.914578i	$-0.632523\pi$
$263$	−13.1168	−0.808819	−0.404410	+	0.914578i	$0.632523\pi$
$264$	0	0
$265$	−7.11684	−0.437184
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−12.1168	−0.738777	−0.369389	−	0.929275i	$-0.620433\pi$
$269$	−12.1168	−0.738777	−0.369389	+	0.929275i	$0.620433\pi$
$270$	0	0
$271$	20.7446	1.26014	0.630071	−	0.776537i	$-0.283026\pi$
$271$	20.7446	1.26014	0.630071	+	0.776537i	$0.283026\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.37228	−0.203356
$276$	0	0
$277$	−17.1168	−1.02845	−0.514226	−	0.857655i	$-0.671921\pi$
$277$	−17.1168	−1.02845	−0.514226	+	0.857655i	$0.671921\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−11.6277	−0.693652	−0.346826	−	0.937930i	$-0.612740\pi$
$281$	−11.6277	−0.693652	−0.346826	+	0.937930i	$0.612740\pi$
$282$	0	0
$283$	−5.25544	−0.312403	−0.156202	−	0.987725i	$-0.549925\pi$
$283$	−5.25544	−0.312403	−0.156202	+	0.987725i	$0.549925\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−27.8614	−1.64461
$288$	0	0
$289$	5.51087	0.324169
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−13.8614	−0.809792	−0.404896	−	0.914363i	$-0.632692\pi$
$293$	−13.8614	−0.809792	−0.404896	+	0.914363i	$0.632692\pi$
$294$	0	0
$295$	−5.00000	−0.291111
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−0.883156	−0.0510742
$300$	0	0
$301$	−16.0000	−0.922225
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1.25544	−0.0718861
$306$	0	0
$307$	10.7446	0.613225	0.306612	−	0.951834i	$-0.400805\pi$
$307$	10.7446	0.613225	0.306612	+	0.951834i	$0.400805\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−10.8614	−0.615894	−0.307947	−	0.951404i	$-0.599642\pi$
$311$	−10.8614	−0.615894	−0.307947	+	0.951404i	$0.599642\pi$
$312$	0	0
$313$	−4.00000	−0.226093	−0.113047	−	0.993590i	$-0.536061\pi$
$313$	−4.00000	−0.226093	−0.113047	+	0.993590i	$0.536061\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.3723	0.919559	0.459779	−	0.888033i	$-0.347928\pi$
$317$	16.3723	0.919559	0.459779	+	0.888033i	$0.347928\pi$
$318$	0	0
$319$	−11.3723	−0.636726
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4.74456	0.263995
$324$	0	0
$325$	2.37228	0.131590
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−8.60597	−0.474462
$330$	0	0
$331$	−19.3723	−1.06480	−0.532398	−	0.846494i	$-0.678709\pi$
$331$	−19.3723	−1.06480	−0.532398	+	0.846494i	$0.678709\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	10.7446	0.587038
$336$	0	0
$337$	0.744563	0.0405589	0.0202795	−	0.999794i	$-0.493544\pi$
$337$	0.744563	0.0405589	0.0202795	+	0.999794i	$0.493544\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−20.6277	−1.11705
$342$	0	0
$343$	−19.8614	−1.07242
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−30.2337	−1.62303	−0.811515	−	0.584332i	$-0.801357\pi$
$347$	−30.2337	−1.62303	−0.811515	+	0.584332i	$0.801357\pi$
$348$	0	0
$349$	−25.3723	−1.35815	−0.679074	−	0.734070i	$-0.737618\pi$
$349$	−25.3723	−1.35815	−0.679074	+	0.734070i	$0.737618\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−24.9783	−1.32946	−0.664729	−	0.747085i	$-0.731453\pi$
$353$	−24.9783	−1.32946	−0.664729	+	0.747085i	$0.731453\pi$
$354$	0	0
$355$	−1.37228	−0.0728331
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−10.6277	−0.560910	−0.280455	−	0.959867i	$-0.590485\pi$
$359$	−10.6277	−0.560910	−0.280455	+	0.959867i	$0.590485\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−3.25544	−0.170397
$366$	0	0
$367$	−4.23369	−0.220997	−0.110498	−	0.993876i	$-0.535245\pi$
$367$	−4.23369	−0.220997	−0.110498	+	0.993876i	$0.535245\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	16.8832	0.876530
$372$	0	0
$373$	0.744563	0.0385520	0.0192760	−	0.999814i	$-0.493864\pi$
$373$	0.744563	0.0385520	0.0192760	+	0.999814i	$0.493864\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.00000	0.412021
$378$	0	0
$379$	−25.3505	−1.30217	−0.651085	−	0.759005i	$-0.725686\pi$
$379$	−25.3505	−1.30217	−0.651085	+	0.759005i	$0.725686\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−13.6277	−0.696344	−0.348172	−	0.937431i	$-0.613197\pi$
$383$	−13.6277	−0.696344	−0.348172	+	0.937431i	$0.613197\pi$
$384$	0	0
$385$	8.00000	0.407718
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−9.25544	−0.469269	−0.234635	−	0.972084i	$-0.575389\pi$
$389$	−9.25544	−0.469269	−0.234635	+	0.972084i	$0.575389\pi$
$390$	0	0
$391$	1.76631	0.0893262
$392$	0	0
$393$	0	0
$394$	0	0
$395$	8.74456	0.439987
$396$	0	0
$397$	34.0000	1.70641	0.853206	−	0.521575i	$-0.174655\pi$
$397$	34.0000	1.70641	0.853206	+	0.521575i	$0.174655\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	15.3505	0.766569	0.383284	−	0.923630i	$-0.374793\pi$
$401$	15.3505	0.766569	0.383284	+	0.923630i	$0.374793\pi$
$402$	0	0
$403$	14.5109	0.722838
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−20.2337	−1.00295
$408$	0	0
$409$	−15.6277	−0.772741	−0.386370	−	0.922344i	$-0.626271\pi$
$409$	−15.6277	−0.772741	−0.386370	+	0.922344i	$0.626271\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	11.8614	0.583662
$414$	0	0
$415$	10.0000	0.490881
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−10.9783	−0.536323	−0.268161	−	0.963374i	$-0.586416\pi$
$419$	−10.9783	−0.536323	−0.268161	+	0.963374i	$0.586416\pi$
$420$	0	0
$421$	22.6277	1.10281	0.551404	−	0.834239i	$-0.314092\pi$
$421$	22.6277	1.10281	0.551404	+	0.834239i	$0.314092\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−4.74456	−0.230145
$426$	0	0
$427$	2.97825	0.144128
$428$	0	0
$429$	0	0
$430$	0	0
$431$	15.3723	0.740457	0.370228	−	0.928941i	$-0.379279\pi$
$431$	15.3723	0.740457	0.370228	+	0.928941i	$0.379279\pi$
$432$	0	0
$433$	39.2119	1.88441	0.942203	−	0.335043i	$-0.108751\pi$
$433$	39.2119	1.88441	0.942203	+	0.335043i	$0.108751\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.372281	0.0178086
$438$	0	0
$439$	−31.0951	−1.48409	−0.742044	−	0.670351i	$-0.766143\pi$
$439$	−31.0951	−1.48409	−0.742044	+	0.670351i	$0.766143\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	16.9783	0.806661	0.403331	−	0.915054i	$-0.367852\pi$
$443$	16.9783	0.806661	0.403331	+	0.915054i	$0.367852\pi$
$444$	0	0
$445$	1.37228	0.0650524
$446$	0	0
$447$	0	0
$448$	0	0
$449$	22.7228	1.07236	0.536178	−	0.844105i	$-0.319868\pi$
$449$	22.7228	1.07236	0.536178	+	0.844105i	$0.319868\pi$
$450$	0	0
$451$	39.6060	1.86497
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−5.62772	−0.263832
$456$	0	0
$457$	−40.2337	−1.88205	−0.941026	−	0.338334i	$-0.890137\pi$
$457$	−40.2337	−1.88205	−0.941026	+	0.338334i	$0.890137\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−13.8832	−0.646603	−0.323302	−	0.946296i	$-0.604793\pi$
$461$	−13.8832	−0.646603	−0.323302	+	0.946296i	$0.604793\pi$
$462$	0	0
$463$	21.8614	1.01599	0.507993	−	0.861361i	$-0.330388\pi$
$463$	21.8614	1.01599	0.507993	+	0.861361i	$0.330388\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−18.2337	−0.843754	−0.421877	−	0.906653i	$-0.638629\pi$
$467$	−18.2337	−0.843754	−0.421877	+	0.906653i	$0.638629\pi$
$468$	0	0
$469$	−25.4891	−1.17698
$470$	0	0
$471$	0	0
$472$	0	0
$473$	22.7446	1.04580
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−24.1168	−1.10193	−0.550963	−	0.834529i	$-0.685740\pi$
$479$	−24.1168	−1.10193	−0.550963	+	0.834529i	$0.685740\pi$
$480$	0	0
$481$	14.2337	0.649000
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6.74456	−0.306255
$486$	0	0
$487$	20.6060	0.933746	0.466873	−	0.884324i	$-0.345381\pi$
$487$	20.6060	0.933746	0.466873	+	0.884324i	$0.345381\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	19.9783	0.901606	0.450803	−	0.892624i	$-0.351138\pi$
$491$	19.9783	0.901606	0.450803	+	0.892624i	$0.351138\pi$
$492$	0	0
$493$	−16.0000	−0.720604
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.25544	0.146026
$498$	0	0
$499$	−8.25544	−0.369564	−0.184782	−	0.982780i	$-0.559158\pi$
$499$	−8.25544	−0.369564	−0.184782	+	0.982780i	$0.559158\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−34.9783	−1.55960	−0.779802	−	0.626027i	$-0.784680\pi$
$503$	−34.9783	−1.55960	−0.779802	+	0.626027i	$0.784680\pi$
$504$	0	0
$505$	12.8614	0.572325
$506$	0	0
$507$	0	0
$508$	0	0
$509$	38.0000	1.68432	0.842160	−	0.539227i	$-0.181284\pi$
$509$	38.0000	1.68432	0.842160	+	0.539227i	$0.181284\pi$
$510$	0	0
$511$	7.72281	0.341637
$512$	0	0
$513$	0	0
$514$	0	0
$515$	17.1168	0.754258
$516$	0	0
$517$	12.2337	0.538037
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−17.1168	−0.749903	−0.374951	−	0.927045i	$-0.622341\pi$
$521$	−17.1168	−0.749903	−0.374951	+	0.927045i	$0.622341\pi$
$522$	0	0
$523$	5.76631	0.252143	0.126072	−	0.992021i	$-0.459763\pi$
$523$	5.76631	0.252143	0.126072	+	0.992021i	$0.459763\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−29.0217	−1.26421
$528$	0	0
$529$	−22.8614	−0.993974
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−27.8614	−1.20681
$534$	0	0
$535$	6.00000	0.259403
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4.62772	0.199330
$540$	0	0
$541$	−38.3505	−1.64882	−0.824409	−	0.565994i	$-0.808492\pi$
$541$	−38.3505	−1.64882	−0.824409	+	0.565994i	$0.808492\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−18.1168	−0.776040
$546$	0	0
$547$	40.0000	1.71028	0.855138	−	0.518400i	$-0.173472\pi$
$547$	40.0000	1.71028	0.855138	+	0.518400i	$0.173472\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3.37228	−0.143664
$552$	0	0
$553$	−20.7446	−0.882149
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−8.60597	−0.364647	−0.182323	−	0.983239i	$-0.558362\pi$
$557$	−8.60597	−0.364647	−0.182323	+	0.983239i	$0.558362\pi$
$558$	0	0
$559$	−16.0000	−0.676728
$560$	0	0
$561$	0	0
$562$	0	0
$563$	24.2337	1.02133	0.510664	−	0.859780i	$-0.329400\pi$
$563$	24.2337	1.02133	0.510664	+	0.859780i	$0.329400\pi$
$564$	0	0
$565$	3.25544	0.136957
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−43.2337	−1.81245	−0.906225	−	0.422795i	$-0.861049\pi$
$569$	−43.2337	−1.81245	−0.906225	+	0.422795i	$0.861049\pi$
$570$	0	0
$571$	−14.1168	−0.590772	−0.295386	−	0.955378i	$-0.595448\pi$
$571$	−14.1168	−0.590772	−0.295386	+	0.955378i	$0.595448\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−0.372281	−0.0155252
$576$	0	0
$577$	13.4891	0.561560	0.280780	−	0.959772i	$-0.409407\pi$
$577$	13.4891	0.561560	0.280780	+	0.959772i	$0.409407\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−23.7228	−0.984188
$582$	0	0
$583$	−24.0000	−0.993978
$584$	0	0
$585$	0	0
$586$	0	0
$587$	30.0000	1.23823	0.619116	−	0.785299i	$-0.287491\pi$
$587$	30.0000	1.23823	0.619116	+	0.785299i	$0.287491\pi$
$588$	0	0
$589$	−6.11684	−0.252040
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−16.9783	−0.697213	−0.348607	−	0.937269i	$-0.613345\pi$
$593$	−16.9783	−0.697213	−0.348607	+	0.937269i	$0.613345\pi$
$594$	0	0
$595$	11.2554	0.461428
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−36.3505	−1.48524	−0.742621	−	0.669712i	$-0.766418\pi$
$599$	−36.3505	−1.48524	−0.742621	+	0.669712i	$0.766418\pi$
$600$	0	0
$601$	13.0000	0.530281	0.265141	−	0.964210i	$-0.414582\pi$
$601$	13.0000	0.530281	0.265141	+	0.964210i	$0.414582\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−0.372281	−0.0151354
$606$	0	0
$607$	33.2554	1.34980	0.674898	−	0.737911i	$-0.264187\pi$
$607$	33.2554	1.34980	0.674898	+	0.737911i	$0.264187\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−8.60597	−0.348160
$612$	0	0
$613$	−33.3505	−1.34702	−0.673508	−	0.739180i	$-0.735213\pi$
$613$	−33.3505	−1.34702	−0.673508	+	0.739180i	$0.735213\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	43.7228	1.76021	0.880107	−	0.474775i	$-0.157471\pi$
$617$	43.7228	1.76021	0.880107	+	0.474775i	$0.157471\pi$
$618$	0	0
$619$	38.3723	1.54231	0.771156	−	0.636646i	$-0.219679\pi$
$619$	38.3723	1.54231	0.771156	+	0.636646i	$0.219679\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−3.25544	−0.130426
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−28.4674	−1.13507
$630$	0	0
$631$	36.6277	1.45813	0.729063	−	0.684446i	$-0.239956\pi$
$631$	36.6277	1.45813	0.729063	+	0.684446i	$0.239956\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	15.8614	0.629441
$636$	0	0
$637$	−3.25544	−0.128985
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−2.39403	−0.0945585	−0.0472793	−	0.998882i	$-0.515055\pi$
$641$	−2.39403	−0.0945585	−0.0472793	+	0.998882i	$0.515055\pi$
$642$	0	0
$643$	−21.2554	−0.838233	−0.419116	−	0.907933i	$-0.637660\pi$
$643$	−21.2554	−0.838233	−0.419116	+	0.907933i	$0.637660\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	25.7228	1.01127	0.505634	−	0.862748i	$-0.331259\pi$
$647$	25.7228	1.01127	0.505634	+	0.862748i	$0.331259\pi$
$648$	0	0
$649$	−16.8614	−0.661868
$650$	0	0
$651$	0	0
$652$	0	0
$653$	24.7446	0.968330	0.484165	−	0.874977i	$-0.339124\pi$
$653$	24.7446	0.968330	0.484165	+	0.874977i	$0.339124\pi$
$654$	0	0
$655$	−17.7446	−0.693337
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−47.5842	−1.85362	−0.926809	−	0.375533i	$-0.877460\pi$
$659$	−47.5842	−1.85362	−0.926809	+	0.375533i	$0.877460\pi$
$660$	0	0
$661$	−45.0951	−1.75400	−0.876998	−	0.480494i	$-0.840457\pi$
$661$	−45.0951	−1.75400	−0.876998	+	0.480494i	$0.840457\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.37228	0.0919931
$666$	0	0
$667$	−1.25544	−0.0486107
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.23369	−0.163440
$672$	0	0
$673$	−0.510875	−0.0196928	−0.00984639	−	0.999952i	$-0.503134\pi$
$673$	−0.510875	−0.0196928	−0.00984639	+	0.999952i	$0.503134\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	21.1168	0.811586	0.405793	−	0.913965i	$-0.366995\pi$
$677$	21.1168	0.811586	0.405793	+	0.913965i	$0.366995\pi$
$678$	0	0
$679$	16.0000	0.614024
$680$	0	0
$681$	0	0
$682$	0	0
$683$	38.2337	1.46297	0.731486	−	0.681857i	$-0.238827\pi$
$683$	38.2337	1.46297	0.731486	+	0.681857i	$0.238827\pi$
$684$	0	0
$685$	−16.7446	−0.639777
$686$	0	0
$687$	0	0
$688$	0	0
$689$	16.8832	0.643197
$690$	0	0
$691$	−41.3505	−1.57305	−0.786524	−	0.617559i	$-0.788121\pi$
$691$	−41.3505	−1.57305	−0.786524	+	0.617559i	$0.788121\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	15.0000	0.568982
$696$	0	0
$697$	55.7228	2.11065
$698$	0	0
$699$	0	0
$700$	0	0
$701$	8.11684	0.306569	0.153284	−	0.988182i	$-0.451015\pi$
$701$	8.11684	0.306569	0.153284	+	0.988182i	$0.451015\pi$
$702$	0	0
$703$	−6.00000	−0.226294
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−30.5109	−1.14748
$708$	0	0
$709$	18.4674	0.693557	0.346778	−	0.937947i	$-0.387276\pi$
$709$	18.4674	0.693557	0.346778	+	0.937947i	$0.387276\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−2.27719	−0.0852813
$714$	0	0
$715$	8.00000	0.299183
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−10.3940	−0.387632	−0.193816	−	0.981038i	$-0.562086\pi$
$719$	−10.3940	−0.387632	−0.193816	+	0.981038i	$0.562086\pi$
$720$	0	0
$721$	−40.6060	−1.51225
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3.37228	0.125243
$726$	0	0
$727$	47.1168	1.74747	0.873734	−	0.486405i	$-0.161692\pi$
$727$	47.1168	1.74747	0.873734	+	0.486405i	$0.161692\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	32.0000	1.18356
$732$	0	0
$733$	−28.9783	−1.07034	−0.535168	−	0.844746i	$-0.679752\pi$
$733$	−28.9783	−1.07034	−0.535168	+	0.844746i	$0.679752\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	36.2337	1.33469
$738$	0	0
$739$	12.8614	0.473114	0.236557	−	0.971618i	$-0.423981\pi$
$739$	12.8614	0.473114	0.236557	+	0.971618i	$0.423981\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−29.4891	−1.08185	−0.540926	−	0.841070i	$-0.681926\pi$
$743$	−29.4891	−1.08185	−0.540926	+	0.841070i	$0.681926\pi$
$744$	0	0
$745$	−3.48913	−0.127832
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−14.2337	−0.520088
$750$	0	0
$751$	38.2337	1.39517	0.697584	−	0.716503i	$-0.254259\pi$
$751$	38.2337	1.39517	0.697584	+	0.716503i	$0.254259\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	18.6277	0.677932
$756$	0	0
$757$	27.1168	0.985578	0.492789	−	0.870149i	$-0.335977\pi$
$757$	27.1168	0.985578	0.492789	+	0.870149i	$0.335977\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	17.2337	0.624721	0.312360	−	0.949964i	$-0.398880\pi$
$761$	17.2337	0.624721	0.312360	+	0.949964i	$0.398880\pi$
$762$	0	0
$763$	42.9783	1.55592
$764$	0	0
$765$	0	0
$766$	0	0
$767$	11.8614	0.428291
$768$	0	0
$769$	−32.3505	−1.16659	−0.583295	−	0.812260i	$-0.698237\pi$
$769$	−32.3505	−1.16659	−0.583295	+	0.812260i	$0.698237\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	0	0
$775$	6.11684	0.219724
$776$	0	0
$777$	0	0
$778$	0	0
$779$	11.7446	0.420793
$780$	0	0
$781$	−4.62772	−0.165593
$782$	0	0
$783$	0	0
$784$	0	0
$785$	5.86141	0.209203
$786$	0	0
$787$	−15.7228	−0.560458	−0.280229	−	0.959933i	$-0.590410\pi$
$787$	−15.7228	−0.560458	−0.280229	+	0.959933i	$0.590410\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−7.72281	−0.274592
$792$	0	0
$793$	2.97825	0.105761
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−14.2337	−0.504183	−0.252092	−	0.967703i	$-0.581118\pi$
$797$	−14.2337	−0.504183	−0.252092	+	0.967703i	$0.581118\pi$
$798$	0	0
$799$	17.2119	0.608915
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−10.9783	−0.387414
$804$	0	0
$805$	0.883156	0.0311272
$806$	0	0
$807$	0	0
$808$	0	0
$809$	34.2119	1.20283	0.601414	−	0.798938i	$-0.294604\pi$
$809$	34.2119	1.20283	0.601414	+	0.798938i	$0.294604\pi$
$810$	0	0
$811$	−14.3505	−0.503915	−0.251958	−	0.967738i	$-0.581074\pi$
$811$	−14.3505	−0.503915	−0.251958	+	0.967738i	$0.581074\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−13.4891	−0.472503
$816$	0	0
$817$	6.74456	0.235962
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−41.3723	−1.44390	−0.721951	−	0.691944i	$-0.756755\pi$
$821$	−41.3723	−1.44390	−0.721951	+	0.691944i	$0.756755\pi$
$822$	0	0
$823$	34.7446	1.21112	0.605560	−	0.795800i	$-0.292949\pi$
$823$	34.7446	1.21112	0.605560	+	0.795800i	$0.292949\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	45.9565	1.59806	0.799032	−	0.601288i	$-0.205346\pi$
$827$	45.9565	1.59806	0.799032	+	0.601288i	$0.205346\pi$
$828$	0	0
$829$	42.5842	1.47901	0.739506	−	0.673150i	$-0.235059\pi$
$829$	42.5842	1.47901	0.739506	+	0.673150i	$0.235059\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	6.51087	0.225588
$834$	0	0
$835$	12.0000	0.415277
$836$	0	0
$837$	0	0
$838$	0	0
$839$	11.6060	0.400683	0.200341	−	0.979726i	$-0.435795\pi$
$839$	11.6060	0.400683	0.200341	+	0.979726i	$0.435795\pi$
$840$	0	0
$841$	−17.6277	−0.607852
$842$	0	0
$843$	0	0
$844$	0	0
$845$	7.37228	0.253614
$846$	0	0
$847$	0.883156	0.0303456
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−2.23369	−0.0765698
$852$	0	0
$853$	−0.744563	−0.0254933	−0.0127467	−	0.999919i	$-0.504058\pi$
$853$	−0.744563	−0.0254933	−0.0127467	+	0.999919i	$0.504058\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	44.7446	1.52845	0.764223	−	0.644953i	$-0.223123\pi$
$857$	44.7446	1.52845	0.764223	+	0.644953i	$0.223123\pi$
$858$	0	0
$859$	−50.3505	−1.71794	−0.858969	−	0.512028i	$-0.828895\pi$
$859$	−50.3505	−1.71794	−0.858969	+	0.512028i	$0.828895\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−11.3940	−0.387857	−0.193929	−	0.981016i	$-0.562123\pi$
$863$	−11.3940	−0.387857	−0.193929	+	0.981016i	$0.562123\pi$
$864$	0	0
$865$	−13.8614	−0.471302
$866$	0	0
$867$	0	0
$868$	0	0
$869$	29.4891	1.00035
$870$	0	0
$871$	−25.4891	−0.863666
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2.37228	−0.0801977
$876$	0	0
$877$	8.88316	0.299963	0.149981	−	0.988689i	$-0.452079\pi$
$877$	8.88316	0.299963	0.149981	+	0.988689i	$0.452079\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−25.3723	−0.854814	−0.427407	−	0.904059i	$-0.640573\pi$
$881$	−25.3723	−0.854814	−0.427407	+	0.904059i	$0.640573\pi$
$882$	0	0
$883$	−5.76631	−0.194052	−0.0970259	−	0.995282i	$-0.530933\pi$
$883$	−5.76631	−0.194052	−0.0970259	+	0.995282i	$0.530933\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−41.8614	−1.40557	−0.702784	−	0.711403i	$-0.748060\pi$
$887$	−41.8614	−1.40557	−0.702784	+	0.711403i	$0.748060\pi$
$888$	0	0
$889$	−37.6277	−1.26199
$890$	0	0
$891$	0	0
$892$	0	0
$893$	3.62772	0.121397
$894$	0	0
$895$	15.7446	0.526283
$896$	0	0
$897$	0	0
$898$	0	0
$899$	20.6277	0.687973
$900$	0	0
$901$	−33.7663	−1.12492
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−14.8614	−0.494010
$906$	0	0
$907$	35.4891	1.17840	0.589199	−	0.807988i	$-0.299444\pi$
$907$	35.4891	1.17840	0.589199	+	0.807988i	$0.299444\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	54.0733	1.79153	0.895765	−	0.444528i	$-0.146629\pi$
$911$	54.0733	1.79153	0.895765	+	0.444528i	$0.146629\pi$
$912$	0	0
$913$	33.7228	1.11606
$914$	0	0
$915$	0	0
$916$	0	0
$917$	42.0951	1.39010
$918$	0	0
$919$	−42.1168	−1.38931	−0.694653	−	0.719345i	$-0.744442\pi$
$919$	−42.1168	−1.38931	−0.694653	+	0.719345i	$0.744442\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	3.25544	0.107154
$924$	0	0
$925$	6.00000	0.197279
$926$	0	0
$927$	0	0
$928$	0	0
$929$	2.86141	0.0938797	0.0469399	−	0.998898i	$-0.485053\pi$
$929$	2.86141	0.0938797	0.0469399	+	0.998898i	$0.485053\pi$
$930$	0	0
$931$	1.37228	0.0449747
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−16.0000	−0.523256
$936$	0	0
$937$	42.7446	1.39640	0.698202	−	0.715901i	$-0.253984\pi$
$937$	42.7446	1.39640	0.698202	+	0.715901i	$0.253984\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−26.4674	−0.862812	−0.431406	−	0.902158i	$-0.641982\pi$
$941$	−26.4674	−0.862812	−0.431406	+	0.902158i	$0.641982\pi$
$942$	0	0
$943$	4.37228	0.142381
$944$	0	0
$945$	0	0
$946$	0	0
$947$	37.2554	1.21064	0.605320	−	0.795983i	$-0.293045\pi$
$947$	37.2554	1.21064	0.605320	+	0.795983i	$0.293045\pi$
$948$	0	0
$949$	7.72281	0.250693
$950$	0	0
$951$	0	0
$952$	0	0
$953$	12.0000	0.388718	0.194359	−	0.980930i	$-0.437737\pi$
$953$	12.0000	0.388718	0.194359	+	0.980930i	$0.437737\pi$
$954$	0	0
$955$	11.3723	0.367998
$956$	0	0
$957$	0	0
$958$	0	0
$959$	39.7228	1.28272
$960$	0	0
$961$	6.41578	0.206961
$962$	0	0
$963$	0	0
$964$	0	0
$965$	10.7446	0.345880
$966$	0	0
$967$	18.5109	0.595270	0.297635	−	0.954680i	$-0.403802\pi$
$967$	18.5109	0.595270	0.297635	+	0.954680i	$0.403802\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−8.72281	−0.279928	−0.139964	−	0.990157i	$-0.544699\pi$
$971$	−8.72281	−0.279928	−0.139964	+	0.990157i	$0.544699\pi$
$972$	0	0
$973$	−35.5842	−1.14078
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−17.7663	−0.568395	−0.284197	−	0.958766i	$-0.591727\pi$
$977$	−17.7663	−0.568395	−0.284197	+	0.958766i	$0.591727\pi$
$978$	0	0
$979$	4.62772	0.147903
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−5.48913	−0.175076	−0.0875380	−	0.996161i	$-0.527900\pi$
$983$	−5.48913	−0.175076	−0.0875380	+	0.996161i	$0.527900\pi$
$984$	0	0
$985$	−6.37228	−0.203038
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.51087	0.0798412
$990$	0	0
$991$	−3.37228	−0.107124	−0.0535620	−	0.998565i	$-0.517057\pi$
$991$	−3.37228	−0.107124	−0.0535620	+	0.998565i	$0.517057\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−9.48913	−0.300825
$996$	0	0
$997$	9.35053	0.296134	0.148067	−	0.988977i	$-0.452695\pi$
$997$	9.35053	0.296134	0.148067	+	0.988977i	$0.452695\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6480.2.a.bd.1.2		2
3.2	odd	2		6480.2.a.bo.1.2		2
4.3	odd	2		3240.2.a.i.1.1	✓	2
12.11	even	2		3240.2.a.m.1.1	yes	2
36.7	odd	6		3240.2.q.bd.1081.2		4
36.11	even	6		3240.2.q.ba.1081.2		4
36.23	even	6		3240.2.q.ba.2161.2		4
36.31	odd	6		3240.2.q.bd.2161.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3240.2.a.i.1.1	✓	2	4.3	odd	2
3240.2.a.m.1.1	yes	2	12.11	even	2
3240.2.q.ba.1081.2		4	36.11	even	6
3240.2.q.ba.2161.2		4	36.23	even	6
3240.2.q.bd.1081.2		4	36.7	odd	6
3240.2.q.bd.2161.2		4	36.31	odd	6
6480.2.a.bd.1.2		2	1.1	even	1	trivial
6480.2.a.bo.1.2		2	3.2	odd	2

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +2.37228 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} +2.37228 q^{7} -3.37228 q^{11} +2.37228 q^{13} -4.74456 q^{17} -1.00000 q^{19} -0.372281 q^{23} +1.00000 q^{25} +3.37228 q^{29} +6.11684 q^{31} -2.37228 q^{35} +6.00000 q^{37} -11.7446 q^{41} -6.74456 q^{43} -3.62772 q^{47} -1.37228 q^{49} +7.11684 q^{53} +3.37228 q^{55} +5.00000 q^{59} +1.25544 q^{61} -2.37228 q^{65} -10.7446 q^{67} +1.37228 q^{71} +3.25544 q^{73} -8.00000 q^{77} -8.74456 q^{79} -10.0000 q^{83} +4.74456 q^{85} -1.37228 q^{89} +5.62772 q^{91} +1.00000 q^{95} +6.74456 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 - q^7	\( 2 q - 2 q^{5} - q^{7} - q^{11} - q^{13} + 2 q^{17} - 2 q^{19} + 5 q^{23} + 2 q^{25} + q^{29} - 5 q^{31} + q^{35} + 12 q^{37} - 12 q^{41} - 2 q^{43} - 13 q^{47} + 3 q^{49} - 3 q^{53} + q^{55} + 10 q^{59} + 14 q^{61} + q^{65} - 10 q^{67} - 3 q^{71} + 18 q^{73} - 16 q^{77} - 6 q^{79} - 20 q^{83} - 2 q^{85} + 3 q^{89} + 17 q^{91} + 2 q^{95} + 2 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - q^7 - q^11 - q^13 + 2 * q^17 - 2 * q^19 + 5 * q^23 + 2 * q^25 + q^29 - 5 * q^31 + q^35 + 12 * q^37 - 12 * q^41 - 2 * q^43 - 13 * q^47 + 3 * q^49 - 3 * q^53 + q^55 + 10 * q^59 + 14 * q^61 + q^65 - 10 * q^67 - 3 * q^71 + 18 * q^73 - 16 * q^77 - 6 * q^79 - 20 * q^83 - 2 * q^85 + 3 * q^89 + 17 * q^91 + 2 * q^95 + 2 * q^97

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