LMFDB - Embedded newform 2160.2.by.d.289.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,2,Mod(289,2160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4, 3]))

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

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

chi := DirichletCharacter("2160.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2160.by (of order $6$, degree $2$, not minimal)

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:	no
Analytic conductor:	$17.2476868366$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			289.2
Root			$-0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	2160.289
Dual form			2160.2.by.d.1009.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.917738 + 2.03906i) q^{5} +(-0.389270 - 0.224745i) q^{7} +O(q^{10})$	$q+(0.917738 + 2.03906i) q^{5} +(-0.389270 - 0.224745i) q^{7} +(1.72474 - 2.98735i) q^{11} +(-2.12132 + 1.22474i) q^{13} +5.89898i q^{17} -5.44949 q^{19} +(-5.97469 + 3.44949i) q^{23} +(-3.31552 + 3.74264i) q^{25} +(3.00000 - 5.19615i) q^{29} +(0.775255 + 1.34278i) q^{31} +(0.101021 - 1.00000i) q^{35} +8.00000i q^{37} +(-0.500000 - 0.866025i) q^{41} +(-2.20881 - 1.27526i) q^{43} +(3.85337 + 2.22474i) q^{47} +(-3.39898 - 5.88721i) q^{49} +3.55051i q^{53} +(7.67423 + 0.775255i) q^{55} +(6.62372 + 11.4726i) q^{59} +(-2.22474 + 3.85337i) q^{61} +(-4.44414 - 3.20150i) q^{65} +(-3.94086 + 2.27526i) q^{67} -2.44949 q^{71} -14.7980i q^{73} +(-1.34278 + 0.775255i) q^{77} +(-3.67423 + 6.36396i) q^{79} +(-3.46410 - 2.00000i) q^{83} +(-12.0284 + 5.41372i) q^{85} +3.10102 q^{89} +1.10102 q^{91} +(-5.00120 - 11.1118i) q^{95} +(-11.2583 - 6.50000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{5}+O(q^{10}) $ 8 * q + 4 * q^5	$ 8 q + 4 q^{5} + 4 q^{11} - 24 q^{19} + 24 q^{29} + 16 q^{31} + 40 q^{35} - 4 q^{41} + 12 q^{49} + 32 q^{55} + 4 q^{59} - 8 q^{61} + 12 q^{65} - 20 q^{85} + 64 q^{89} + 48 q^{91} - 24 q^{95}+O(q^{100}) $ 8 * q + 4 * q^5 + 4 * q^11 - 24 * q^19 + 24 * q^29 + 16 * q^31 + 40 * q^35 - 4 * q^41 + 12 * q^49 + 32 * q^55 + 4 * q^59 - 8 * q^61 + 12 * q^65 - 20 * q^85 + 64 * q^89 + 48 * q^91 - 24 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$.

$n$	$271$	$1297$	$1621$	$2081$
$\chi(n)$	$1$	$-1$	$1$	$e\left(\frac{2}{3}\right)$

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
$5$	0.917738	+	2.03906i	0.410425	+	0.911894i
$5$	0.917738	+	2.03906i	0.410425	+	0.911894i
$6$		0			0
$7$	−0.389270	−	0.224745i	−0.147130	−	0.0849456i	0.424628	−	0.905368i	$-0.360405\pi$
$7$	−0.389270	−	0.224745i	−0.147130	−	0.0849456i	−0.571758	+	0.820422i	$0.693738\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.72474	−	2.98735i	0.520030	−	0.900719i	−0.479699	−	0.877433i	$-0.659254\pi$
$11$	1.72474	−	2.98735i	0.520030	−	0.900719i	0.999729	−	0.0232854i	$-0.00741263\pi$
$12$		0			0
$13$	−2.12132	+	1.22474i	−0.588348	+	0.339683i	−0.764444	−	0.644690i	$-0.776986\pi$
$13$	−2.12132	+	1.22474i	−0.588348	+	0.339683i	0.176096	+	0.984373i	$0.443653\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			5.89898i			1.43071i	0.698760	+	0.715356i	$0.253736\pi$
$17$			5.89898i			1.43071i	−0.698760	+	0.715356i	$0.746264\pi$
$18$		0			0
$19$	−5.44949			−1.25020			−0.625099	−	0.780545i	$-0.714942\pi$
$19$	−5.44949			−1.25020			−0.625099	+	0.780545i	$0.714942\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−5.97469	+	3.44949i	−1.24581	+	0.719268i	−0.970271	−	0.242022i	$-0.922189\pi$
$23$	−5.97469	+	3.44949i	−1.24581	+	0.719268i	−0.275538	+	0.961290i	$0.588856\pi$
$24$		0			0
$25$	−3.31552	+	3.74264i	−0.663103	+	0.748528i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.00000	−	5.19615i	0.557086	−	0.964901i	−0.440652	−	0.897678i	$-0.645253\pi$
$29$	3.00000	−	5.19615i	0.557086	−	0.964901i	0.997738	−	0.0672232i	$-0.0214140\pi$
$30$		0			0
$31$	0.775255	+	1.34278i	0.139240	+	0.241171i	0.927209	−	0.374544i	$-0.122201\pi$
$31$	0.775255	+	1.34278i	0.139240	+	0.241171i	−0.787969	+	0.615715i	$0.788867\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.101021	−	1.00000i	0.0170756	−	0.169031i
$36$		0			0
$37$			8.00000i			1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$			8.00000i			1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.500000	−	0.866025i	−0.0780869	−	0.135250i	0.824338	−	0.566099i	$-0.191548\pi$
$41$	−0.500000	−	0.866025i	−0.0780869	−	0.135250i	−0.902424	+	0.430848i	$0.858214\pi$
$42$		0			0
$43$	−2.20881	−	1.27526i	−0.336840	−	0.194475i	0.322034	−	0.946728i	$-0.395634\pi$
$43$	−2.20881	−	1.27526i	−0.336840	−	0.194475i	−0.658874	+	0.752254i	$0.728967\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.85337	+	2.22474i	0.562072	+	0.324512i	0.753977	−	0.656901i	$-0.228133\pi$
$47$	3.85337	+	2.22474i	0.562072	+	0.324512i	−0.191905	+	0.981414i	$0.561466\pi$
$48$		0			0
$49$	−3.39898	−	5.88721i	−0.485568	−	0.841029i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			3.55051i			0.487700i	0.969813	+	0.243850i	$0.0784105\pi$
$53$			3.55051i			0.487700i	−0.969813	+	0.243850i	$0.921590\pi$
$54$		0			0
$55$	7.67423	+	0.775255i	1.03479	+	0.104535i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.62372	+	11.4726i	0.862335	+	1.49361i	0.869669	+	0.493636i	$0.164332\pi$
$59$	6.62372	+	11.4726i	0.862335	+	1.49361i	−0.00733331	+	0.999973i	$0.502334\pi$
$60$		0			0
$61$	−2.22474	+	3.85337i	−0.284849	+	0.493374i	−0.972573	−	0.232599i	$-0.925277\pi$
$61$	−2.22474	+	3.85337i	−0.284849	+	0.493374i	0.687723	+	0.725973i	$0.258610\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−4.44414	−	3.20150i	−0.551228	−	0.397097i
$66$		0			0
$67$	−3.94086	+	2.27526i	−0.481452	+	0.277967i	−0.721022	−	0.692913i	$-0.756327\pi$
$67$	−3.94086	+	2.27526i	−0.481452	+	0.277967i	0.239569	+	0.970879i	$0.422994\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−2.44949			−0.290701			−0.145350	−	0.989380i	$-0.546431\pi$
$71$	−2.44949			−0.290701			−0.145350	+	0.989380i	$0.546431\pi$
$72$		0			0
$73$		−	14.7980i		−	1.73197i	−0.500070	−	0.865985i	$-0.666692\pi$
$73$		−	14.7980i		−	1.73197i	0.500070	−	0.865985i	$-0.333308\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.34278	+	0.775255i	−0.153024	+	0.0883485i
$78$		0			0
$79$	−3.67423	+	6.36396i	−0.413384	+	0.716002i	−0.995257	−	0.0972777i	$-0.968987\pi$
$79$	−3.67423	+	6.36396i	−0.413384	+	0.716002i	0.581874	+	0.813279i	$0.302320\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−3.46410	−	2.00000i	−0.380235	−	0.219529i	0.297686	−	0.954664i	$-0.403785\pi$
$83$	−3.46410	−	2.00000i	−0.380235	−	0.219529i	−0.677920	+	0.735135i	$0.737119\pi$
$84$		0			0
$85$	−12.0284	+	5.41372i	−1.30466	+	0.587200i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	3.10102			0.328708			0.164354	−	0.986401i	$-0.447446\pi$
$89$	3.10102			0.328708			0.164354	+	0.986401i	$0.447446\pi$
$90$		0			0
$91$	1.10102			0.115418
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−5.00120	−	11.1118i	−0.513112	−	1.14005i
$96$		0			0
$97$	−11.2583	−	6.50000i	−1.14311	−	0.659975i	−0.195911	−	0.980622i	$-0.562766\pi$
$97$	−11.2583	−	6.50000i	−1.14311	−	0.659975i	−0.947199	+	0.320647i	$0.896100\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−4.00000	+	6.92820i	−0.398015	+	0.689382i	−0.993481	−	0.113998i	$-0.963634\pi$
$101$	−4.00000	+	6.92820i	−0.398015	+	0.689382i	0.595466	+	0.803380i	$0.296967\pi$
$102$		0			0
$103$	−12.3387	+	7.12372i	−1.21576	+	0.701921i	−0.964009	−	0.265870i	$-0.914341\pi$
$103$	−12.3387	+	7.12372i	−1.21576	+	0.701921i	−0.251755	+	0.967791i	$0.581008\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			16.3485i			1.58047i	0.612806	+	0.790233i	$0.290041\pi$
$107$			16.3485i			1.58047i	−0.612806	+	0.790233i	$0.709959\pi$
$108$		0			0
$109$	8.00000			0.766261			0.383131	−	0.923694i	$-0.374846\pi$
$109$	8.00000			0.766261			0.383131	+	0.923694i	$0.374846\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−4.24264	+	2.44949i	−0.399114	+	0.230429i	−0.686102	−	0.727506i	$-0.740679\pi$
$113$	−4.24264	+	2.44949i	−0.399114	+	0.230429i	0.286988	+	0.957934i	$0.407346\pi$
$114$		0			0
$115$	−12.5169	−	9.01702i	−1.16721	−	0.840841i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1.32577	−	2.29629i	0.121533	−	0.210501i
$120$		0			0
$121$	−0.449490	−	0.778539i	−0.0408627	−	0.0707763i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−10.6742	−	3.32577i	−0.954733	−	0.297465i
$126$		0			0
$127$		−	6.89898i		−	0.612185i	−0.952002	−	0.306093i	$-0.900978\pi$
$127$		−	6.89898i		−	0.612185i	0.952002	−	0.306093i	$-0.0990218\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−2.44949	−	4.24264i	−0.214013	−	0.370681i	0.738954	−	0.673756i	$-0.235320\pi$
$131$	−2.44949	−	4.24264i	−0.214013	−	0.370681i	−0.952967	+	0.303075i	$0.901987\pi$
$132$		0			0
$133$	2.12132	+	1.22474i	0.183942	+	0.106199i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2.59808	+	1.50000i	0.221969	+	0.128154i	0.606861	−	0.794808i	$-0.292428\pi$
$137$	2.59808	+	1.50000i	0.221969	+	0.128154i	−0.384893	+	0.922961i	$0.625762\pi$
$138$		0			0
$139$	−6.62372	−	11.4726i	−0.561817	−	0.973096i	−0.997338	−	0.0729170i	$-0.976769\pi$
$139$	−6.62372	−	11.4726i	−0.561817	−	0.973096i	0.435521	−	0.900179i	$-0.356564\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			8.44949i			0.706582i
$144$		0			0
$145$	13.3485	+	1.34847i	1.10853	+	0.111984i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	4.12372	+	7.14250i	0.337829	+	0.585136i	0.984024	−	0.178036i	$-0.0569742\pi$
$149$	4.12372	+	7.14250i	0.337829	+	0.585136i	−0.646195	+	0.763172i	$0.723641\pi$
$150$		0			0
$151$	1.44949	−	2.51059i	0.117958	−	0.204309i	−0.801000	−	0.598664i	$-0.795699\pi$
$151$	1.44949	−	2.51059i	0.117958	−	0.204309i	0.918958	+	0.394355i	$0.129032\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.02653	+	2.81311i	−0.162775	+	0.225955i
$156$		0			0
$157$	13.8564	−	8.00000i	1.10586	−	0.638470i	0.168107	−	0.985769i	$-0.446235\pi$
$157$	13.8564	−	8.00000i	1.10586	−	0.638470i	0.937754	+	0.347299i	$0.112901\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	3.10102			0.244395
$162$		0			0
$163$			8.89898i			0.697022i	0.937305	+	0.348511i	$0.113313\pi$
$163$			8.89898i			0.697022i	−0.937305	+	0.348511i	$0.886687\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	0.214297	−	0.123724i	0.0165828	−	0.00957408i	−0.491686	−	0.870773i	$-0.663619\pi$
$167$	0.214297	−	0.123724i	0.0165828	−	0.00957408i	0.508268	+	0.861199i	$0.330286\pi$
$168$		0			0
$169$	−3.50000	+	6.06218i	−0.269231	+	0.466321i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	10.2173	+	5.89898i	0.776809	+	0.448491i	0.835298	−	0.549797i	$-0.185295\pi$
$173$	10.2173	+	5.89898i	0.776809	+	0.448491i	−0.0584890	+	0.998288i	$0.518628\pi$
$174$		0			0
$175$	2.13177	−	0.711751i	0.161147	−	0.0538033i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	0.898979			0.0671929			0.0335964	−	0.999435i	$-0.489304\pi$
$179$	0.898979			0.0671929			0.0335964	+	0.999435i	$0.489304\pi$
$180$		0			0
$181$	5.55051			0.412566			0.206283	−	0.978492i	$-0.433863\pi$
$181$	5.55051			0.412566			0.206283	+	0.978492i	$0.433863\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−16.3125	+	7.34190i	−1.19932	+	0.539787i
$186$		0			0
$187$	17.6223	+	10.1742i	1.28867	+	0.744014i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−9.12372	+	15.8028i	−0.660170	+	1.14345i	0.320401	+	0.947282i	$0.396182\pi$
$191$	−9.12372	+	15.8028i	−0.660170	+	1.14345i	−0.980571	+	0.196165i	$0.937151\pi$
$192$		0			0
$193$	−11.8619	+	6.84847i	−0.853838	+	0.492964i	−0.861944	−	0.507004i	$-0.830753\pi$
$193$	−11.8619	+	6.84847i	−0.853838	+	0.492964i	0.00810596	+	0.999967i	$0.497420\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			8.00000i			0.569976i	0.958531	+	0.284988i	$0.0919897\pi$
$197$			8.00000i			0.569976i	−0.958531	+	0.284988i	$0.908010\pi$
$198$		0			0
$199$	15.5505			1.10235			0.551173	−	0.834391i	$-0.314180\pi$
$199$	15.5505			1.10235			0.551173	+	0.834391i	$0.314180\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−2.33562	+	1.34847i	−0.163928	+	0.0946440i
$204$		0			0
$205$	1.30701	−	1.81431i	0.0912853	−	0.126717i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−9.39898	+	16.2795i	−0.650141	+	1.12608i
$210$		0			0
$211$	1.89898	+	3.28913i	0.130731	+	0.226433i	0.923959	−	0.382492i	$-0.124934\pi$
$211$	1.89898	+	3.28913i	0.130731	+	0.226433i	−0.793227	+	0.608925i	$0.791601\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	0.573214	−	5.67423i	0.0390929	−	0.386980i
$216$		0			0
$217$		−	0.696938i		−	0.0473113i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−7.22474	−	12.5136i	−0.485989	−	0.841758i
$222$		0			0
$223$	−7.88171	−	4.55051i	−0.527799	−	0.304725i	0.212321	−	0.977200i	$-0.431898\pi$
$223$	−7.88171	−	4.55051i	−0.527799	−	0.304725i	−0.740120	+	0.672475i	$0.765231\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2.98735	+	1.72474i	0.198277	+	0.114475i	0.595852	−	0.803095i	$-0.296815\pi$
$227$	2.98735	+	1.72474i	0.198277	+	0.114475i	−0.397574	+	0.917570i	$0.630148\pi$
$228$		0			0
$229$	−9.22474	−	15.9777i	−0.609588	−	1.05584i	−0.991308	−	0.131560i	$-0.958001\pi$
$229$	−9.22474	−	15.9777i	−0.609588	−	1.05584i	0.381720	−	0.924278i	$-0.375332\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	13.6969i		−	0.897316i	−0.893703	−	0.448658i	$-0.851902\pi$
$233$		−	13.6969i		−	0.897316i	0.893703	−	0.448658i	$-0.148098\pi$
$234$		0			0
$235$	−1.00000	+	9.89898i	−0.0652328	+	0.645738i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−0.348469	−	0.603566i	−0.0225406	−	0.0390415i	0.854535	−	0.519394i	$-0.173842\pi$
$239$	−0.348469	−	0.603566i	−0.0225406	−	0.0390415i	−0.877076	+	0.480352i	$0.840509\pi$
$240$		0			0
$241$	−0.500000	+	0.866025i	−0.0322078	+	0.0557856i	−0.881680	−	0.471848i	$-0.843587\pi$
$241$	−0.500000	+	0.866025i	−0.0322078	+	0.0557856i	0.849472	+	0.527633i	$0.176921\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	8.88498	−	12.3336i	0.567641	−	0.787966i
$246$		0			0
$247$	11.5601	−	6.67423i	0.735552	−	0.424671i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	6.55051			0.413465			0.206732	−	0.978398i	$-0.433717\pi$
$251$	6.55051			0.413465			0.206732	+	0.978398i	$0.433717\pi$
$252$		0			0
$253$			23.7980i			1.49616i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−8.74774	+	5.05051i	−0.545669	+	0.315042i	−0.747373	−	0.664404i	$-0.768685\pi$
$257$	−8.74774	+	5.05051i	−0.545669	+	0.315042i	0.201704	+	0.979446i	$0.435352\pi$
$258$		0			0
$259$	1.79796	−	3.11416i	0.111720	−	0.193504i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−10.7816	−	6.22474i	−0.664820	−	0.383834i	0.129291	−	0.991607i	$-0.458730\pi$
$263$	−10.7816	−	6.22474i	−0.664820	−	0.383834i	−0.794111	+	0.607773i	$0.792063\pi$
$264$		0			0
$265$	−7.23970	+	3.25844i	−0.444731	+	0.200164i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	16.0454			0.978306			0.489153	−	0.872198i	$-0.337306\pi$
$269$	16.0454			0.978306			0.489153	+	0.872198i	$0.337306\pi$
$270$		0			0
$271$	15.5959			0.947385			0.473692	−	0.880690i	$-0.342921\pi$
$271$	15.5959			0.947385			0.473692	+	0.880690i	$0.342921\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	5.46214	+	16.3597i	0.329380	+	0.986526i
$276$		0			0
$277$	25.6308	+	14.7980i	1.54001	+	0.889123i	0.998837	+	0.0482095i	$0.0153515\pi$
$277$	25.6308	+	14.7980i	1.54001	+	0.889123i	0.541169	+	0.840914i	$0.317982\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	6.00000	−	10.3923i	0.357930	−	0.619953i	−0.629685	−	0.776851i	$-0.716816\pi$
$281$	6.00000	−	10.3923i	0.357930	−	0.619953i	0.987615	+	0.156898i	$0.0501493\pi$
$282$		0			0
$283$	3.46410	−	2.00000i	0.205919	−	0.118888i	−0.393494	−	0.919327i	$-0.628734\pi$
$283$	3.46410	−	2.00000i	0.205919	−	0.118888i	0.599414	+	0.800439i	$0.295400\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			0.449490i			0.0265325i
$288$		0			0
$289$	−17.7980			−1.04694
$290$		0			0
$291$		0			0
$292$		0			0
$293$	15.5885	−	9.00000i	0.910687	−	0.525786i	0.0300351	−	0.999549i	$-0.490438\pi$
$293$	15.5885	−	9.00000i	0.910687	−	0.525786i	0.880652	+	0.473763i	$0.157105\pi$
$294$		0			0
$295$	−17.3145	+	24.0350i	−1.00809	+	1.39937i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	8.44949	−	14.6349i	0.488647	−	0.846361i
$300$		0			0
$301$	0.573214	+	0.992836i	0.0330395	+	0.0572261i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−9.89898	−	1.00000i	−0.566814	−	0.0572598i
$306$		0			0
$307$		−	29.9444i		−	1.70902i	−0.519438	−	0.854508i	$-0.673859\pi$
$307$		−	29.9444i		−	1.70902i	0.519438	−	0.854508i	$-0.326141\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−6.55051	−	11.3458i	−0.371445	−	0.643362i	0.618343	−	0.785909i	$-0.287804\pi$
$311$	−6.55051	−	11.3458i	−0.371445	−	0.643362i	−0.989788	+	0.142546i	$0.954471\pi$
$312$		0			0
$313$	−18.7901	−	10.8485i	−1.06208	−	0.613192i	−0.136073	−	0.990699i	$-0.543448\pi$
$313$	−18.7901	−	10.8485i	−1.06208	−	0.613192i	−0.926007	+	0.377507i	$0.876781\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	19.8704	+	11.4722i	1.11603	+	0.644343i	0.940386	−	0.340110i	$-0.110464\pi$
$317$	19.8704	+	11.4722i	1.11603	+	0.644343i	0.175649	+	0.984453i	$0.443798\pi$
$318$		0			0
$319$	−10.3485	−	17.9241i	−0.579403	−	1.00356i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	32.1464i		−	1.78868i
$324$		0			0
$325$	2.44949	−	12.0000i	0.135873	−	0.665640i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.00000	−	1.73205i	−0.0551318	−	0.0954911i
$330$		0			0
$331$	−12.6969	+	21.9917i	−0.697887	+	1.20878i	0.271311	+	0.962492i	$0.412543\pi$
$331$	−12.6969	+	21.9917i	−0.697887	+	1.20878i	−0.969198	+	0.246284i	$0.920790\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−8.25605	−	5.94755i	−0.451076	−	0.324949i
$336$		0			0
$337$	16.1045	−	9.29796i	0.877270	−	0.506492i	0.00751272	−	0.999972i	$-0.497609\pi$
$337$	16.1045	−	9.29796i	0.877270	−	0.506492i	0.869757	+	0.493480i	$0.164275\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	5.34847			0.289636
$342$		0			0
$343$			6.20204i			0.334879i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	8.00853	−	4.62372i	0.429920	−	0.248215i	−0.269392	−	0.963030i	$-0.586823\pi$
$347$	8.00853	−	4.62372i	0.429920	−	0.248215i	0.699313	+	0.714816i	$0.253490\pi$
$348$		0			0
$349$	−13.7980	+	23.8988i	−0.738588	+	1.27927i	0.214543	+	0.976714i	$0.431174\pi$
$349$	−13.7980	+	23.8988i	−0.738588	+	1.27927i	−0.953131	+	0.302557i	$0.902160\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−28.2289	−	16.2980i	−1.50247	−	0.867453i	−0.999996	−	0.00286194i	$-0.999089\pi$
$353$	−28.2289	−	16.2980i	−1.50247	−	0.867453i	−0.502476	−	0.864591i	$-0.667578\pi$
$354$		0			0
$355$	−2.24799	−	4.99465i	−0.119311	−	0.265089i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−3.55051			−0.187389			−0.0936944	−	0.995601i	$-0.529868\pi$
$359$	−3.55051			−0.187389			−0.0936944	+	0.995601i	$0.529868\pi$
$360$		0			0
$361$	10.6969			0.562997
$362$		0			0
$363$		0			0
$364$		0			0
$365$	30.1739	−	13.5806i	1.57937	−	0.710843i
$366$		0			0
$367$	23.8988	+	13.7980i	1.24751	+	0.720248i	0.970611	−	0.240653i	$-0.0773615\pi$
$367$	23.8988	+	13.7980i	1.24751	+	0.720248i	0.276894	+	0.960900i	$0.410695\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	0.797959	−	1.38211i	0.0414280	−	0.0717553i
$372$		0			0
$373$	11.7744	−	6.79796i	0.609656	−	0.351985i	−0.163175	−	0.986597i	$-0.552173\pi$
$373$	11.7744	−	6.79796i	0.609656	−	0.351985i	0.772831	+	0.634612i	$0.218840\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			14.6969i			0.756931i
$378$		0			0
$379$	4.14643			0.212988			0.106494	−	0.994313i	$-0.466038\pi$
$379$	4.14643			0.212988			0.106494	+	0.994313i	$0.466038\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	15.4135	−	8.89898i	0.787592	−	0.454717i	−0.0515220	−	0.998672i	$-0.516407\pi$
$383$	15.4135	−	8.89898i	0.787592	−	0.454717i	0.839114	+	0.543955i	$0.183074\pi$
$384$		0			0
$385$	−2.81311	−	2.02653i	−0.143369	−	0.103281i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−8.77526	+	15.1992i	−0.444923	+	0.770629i	−0.998047	−	0.0624697i	$-0.980102\pi$
$389$	−8.77526	+	15.1992i	−0.444923	+	0.770629i	0.553124	+	0.833099i	$0.313436\pi$
$390$		0			0
$391$	−20.3485	−	35.2446i	−1.02907	−	1.78240i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−16.3485	−	1.65153i	−0.822581	−	0.0830975i
$396$		0			0
$397$		−	1.79796i		−	0.0902370i	−0.998982	−	0.0451185i	$-0.985633\pi$
$397$		−	1.79796i		−	0.0902370i	0.998982	−	0.0451185i	$-0.0143665\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	4.60102	+	7.96920i	0.229764	+	0.397963i	0.957738	−	0.287642i	$-0.0928712\pi$
$401$	4.60102	+	7.96920i	0.229764	+	0.397963i	−0.727974	+	0.685605i	$0.759538\pi$
$402$		0			0
$403$	−3.28913	−	1.89898i	−0.163843	−	0.0945949i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	23.8988	+	13.7980i	1.18462	+	0.683939i
$408$		0			0
$409$	7.94949	+	13.7689i	0.393077	+	0.680829i	0.992854	−	0.119338i	$-0.0380773\pi$
$409$	7.94949	+	13.7689i	0.393077	+	0.680829i	−0.599777	+	0.800167i	$0.704744\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	5.95459i		−	0.293006i
$414$		0			0
$415$	0.898979	−	8.89898i	0.0441292	−	0.436834i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−4.44949	−	7.70674i	−0.217372	−	0.376499i	0.736632	−	0.676294i	$-0.236415\pi$
$419$	−4.44949	−	7.70674i	−0.217372	−	0.376499i	−0.954004	+	0.299795i	$0.903082\pi$
$420$		0			0
$421$	5.77526	−	10.0030i	0.281469	−	0.487518i	−0.690278	−	0.723544i	$-0.742512\pi$
$421$	5.77526	−	10.0030i	0.281469	−	0.487518i	0.971747	+	0.236026i	$0.0758451\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−22.0778	−	19.5582i	−1.07093	−	0.948710i
$426$		0			0
$427$	1.73205	−	1.00000i	0.0838198	−	0.0483934i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	38.2474			1.84231			0.921157	−	0.389190i	$-0.127245\pi$
$431$	38.2474			1.84231			0.921157	+	0.389190i	$0.127245\pi$
$432$		0			0
$433$		−	23.0000i		−	1.10531i	−0.833410	−	0.552655i	$-0.813615\pi$
$433$		−	23.0000i		−	1.10531i	0.833410	−	0.552655i	$-0.186385\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	32.5590	−	18.7980i	1.55751	−	0.899228i
$438$		0			0
$439$	−11.0227	+	19.0919i	−0.526085	+	0.911206i	0.473453	+	0.880819i	$0.343007\pi$
$439$	−11.0227	+	19.0919i	−0.526085	+	0.911206i	−0.999538	+	0.0303869i	$0.990326\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	18.4008	+	10.6237i	0.874250	+	0.504748i	0.868758	−	0.495237i	$-0.164919\pi$
$443$	18.4008	+	10.6237i	0.874250	+	0.504748i	0.00549166	+	0.999985i	$0.498252\pi$
$444$		0			0
$445$	2.84592	+	6.32316i	0.134910	+	0.299747i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	18.7980			0.887131			0.443565	−	0.896242i	$-0.353713\pi$
$449$	18.7980			0.887131			0.443565	+	0.896242i	$0.353713\pi$
$450$		0			0
$451$	−3.44949			−0.162430
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1.01045	+	2.24504i	0.0473705	+	0.105249i
$456$		0			0
$457$	15.5010	+	8.94949i	0.725105	+	0.418639i	0.816629	−	0.577163i	$-0.195840\pi$
$457$	15.5010	+	8.94949i	0.725105	+	0.418639i	−0.0915238	+	0.995803i	$0.529174\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1.22474	+	2.12132i	−0.0570421	+	0.0987997i	−0.893136	−	0.449786i	$-0.851500\pi$
$461$	−1.22474	+	2.12132i	−0.0570421	+	0.0987997i	0.836094	+	0.548586i	$0.184834\pi$
$462$		0			0
$463$	20.7846	−	12.0000i	0.965943	−	0.557687i	0.0679458	−	0.997689i	$-0.478356\pi$
$463$	20.7846	−	12.0000i	0.965943	−	0.557687i	0.897997	+	0.440002i	$0.145022\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	10.3485i		−	0.478870i	−0.970912	−	0.239435i	$-0.923038\pi$
$467$		−	10.3485i		−	0.478870i	0.970912	−	0.239435i	$-0.0769622\pi$
$468$		0			0
$469$	2.04541			0.0944482
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−7.61926	+	4.39898i	−0.350334	+	0.202265i
$474$		0			0
$475$	18.0679	−	20.3955i	0.829011	−	0.935809i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	8.34847	−	14.4600i	0.381451	−	0.660693i	−0.609819	−	0.792541i	$-0.708758\pi$
$479$	8.34847	−	14.4600i	0.381451	−	0.660693i	0.991270	+	0.131848i	$0.0420911\pi$
$480$		0			0
$481$	−9.79796	−	16.9706i	−0.446748	−	0.773791i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.92168	−	28.9217i	0.132667	−	1.31327i
$486$		0			0
$487$			25.1010i			1.13744i	0.822533	+	0.568718i	$0.192560\pi$
$487$			25.1010i			1.13744i	−0.822533	+	0.568718i	$0.807440\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−9.27526	−	16.0652i	−0.418586	−	0.725013i	0.577211	−	0.816595i	$-0.304141\pi$
$491$	−9.27526	−	16.0652i	−0.418586	−	0.725013i	−0.995798	+	0.0915820i	$0.970808\pi$
$492$		0			0
$493$	30.6520	+	17.6969i	1.38050	+	0.797030i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	0.953512	+	0.550510i	0.0427708	+	0.0246938i
$498$		0			0
$499$	10.6237	+	18.4008i	0.475583	+	0.823734i	0.999609	−	0.0279682i	$-0.00890372\pi$
$499$	10.6237	+	18.4008i	0.475583	+	0.823734i	−0.524026	+	0.851703i	$0.675570\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	14.4495i		−	0.644271i	−0.946694	−	0.322135i	$-0.895599\pi$
$503$		−	14.4495i		−	0.644271i	0.946694	−	0.322135i	$-0.104401\pi$
$504$		0			0
$505$	−17.7980	−	1.79796i	−0.791999	−	0.0800081i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−15.7980	−	27.3629i	−0.700232	−	1.21284i	−0.968385	−	0.249461i	$-0.919746\pi$
$509$	−15.7980	−	27.3629i	−0.700232	−	1.21284i	0.268153	−	0.963376i	$-0.413587\pi$
$510$		0			0
$511$	−3.32577	+	5.76039i	−0.147123	+	0.254825i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−25.8493	−	18.6215i	−1.13906	−	0.820562i
$516$		0			0
$517$	13.2922	−	7.67423i	0.584589	−	0.337512i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−21.6969			−0.950560			−0.475280	−	0.879835i	$-0.657653\pi$
$521$	−21.6969			−0.950560			−0.475280	+	0.879835i	$0.657653\pi$
$522$		0			0
$523$		−	10.2020i		−	0.446104i	−0.974807	−	0.223052i	$-0.928398\pi$
$523$		−	10.2020i		−	0.446104i	0.974807	−	0.223052i	$-0.0716020\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−7.92104	+	4.57321i	−0.345046	+	0.199212i
$528$		0			0
$529$	12.2980	−	21.3007i	0.534694	−	0.926117i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	2.12132	+	1.22474i	0.0918846	+	0.0530496i
$534$		0			0
$535$	−33.3355	+	15.0036i	−1.44122	+	0.648662i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−23.4495			−1.01004
$540$		0			0
$541$	−0.404082			−0.0173728			−0.00868642	−	0.999962i	$-0.502765\pi$
$541$	−0.404082			−0.0173728			−0.00868642	+	0.999962i	$0.502765\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	7.34190	+	16.3125i	0.314493	+	0.698749i
$546$		0			0
$547$	14.9367	+	8.62372i	0.638648	+	0.368724i	0.784094	−	0.620642i	$-0.213128\pi$
$547$	14.9367	+	8.62372i	0.638648	+	0.368724i	−0.145445	+	0.989366i	$0.546461\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−16.3485	+	28.3164i	−0.696468	+	1.20632i
$552$		0			0
$553$	2.86054	−	1.65153i	0.121642	−	0.0702302i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	14.9444i		−	0.633214i	−0.948557	−	0.316607i	$-0.897456\pi$
$557$		−	14.9444i		−	0.633214i	0.948557	−	0.316607i	$-0.102544\pi$
$558$		0			0
$559$	6.24745			0.264239
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1.60524	+	0.926786i	−0.0676528	+	0.0390594i	−0.533445	−	0.845835i	$-0.679103\pi$
$563$	−1.60524	+	0.926786i	−0.0676528	+	0.0390594i	0.465792	+	0.884894i	$0.345769\pi$
$564$		0			0
$565$	−8.88828	−	6.40300i	−0.373933	−	0.269376i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	15.7474	−	27.2754i	0.660167	−	1.14344i	−0.320404	−	0.947281i	$-0.603819\pi$
$569$	15.7474	−	27.2754i	0.660167	−	1.14344i	0.980571	−	0.196162i	$-0.0628480\pi$
$570$		0			0
$571$	18.6237	+	32.2572i	0.779379	+	1.34992i	0.932300	+	0.361685i	$0.117798\pi$
$571$	18.6237	+	32.2572i	0.779379	+	1.34992i	−0.152922	+	0.988238i	$0.548868\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	6.89898	−	33.7980i	0.287707	−	1.40947i
$576$		0			0
$577$		−	15.6969i		−	0.653472i	−0.945116	−	0.326736i	$-0.894051\pi$
$577$		−	15.6969i		−	0.653472i	0.945116	−	0.326736i	$-0.105949\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	0.898979	+	1.55708i	0.0372960	+	0.0645985i
$582$		0			0
$583$	10.6066	+	6.12372i	0.439281	+	0.253619i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	20.7364	+	11.9722i	0.855885	+	0.494145i	0.862632	−	0.505832i	$-0.168814\pi$
$587$	20.7364	+	11.9722i	0.855885	+	0.494145i	−0.00674727	+	0.999977i	$0.502148\pi$
$588$		0			0
$589$	−4.22474	−	7.31747i	−0.174078	−	0.301511i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	17.3939i		−	0.714281i	−0.934051	−	0.357140i	$-0.883752\pi$
$593$		−	17.3939i		−	0.714281i	0.934051	−	0.357140i	$-0.116248\pi$
$594$		0			0
$595$	5.89898	+	0.595918i	0.241835	+	0.0244303i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	16.8990	+	29.2699i	0.690474	+	1.19594i	0.971683	+	0.236289i	$0.0759312\pi$
$599$	16.8990	+	29.2699i	0.690474	+	1.19594i	−0.281209	+	0.959646i	$0.590736\pi$
$600$		0			0
$601$	−19.3990	+	33.6000i	−0.791301	+	1.37057i	0.133861	+	0.991000i	$0.457262\pi$
$601$	−19.3990	+	33.6000i	−0.791301	+	1.37057i	−0.925162	+	0.379573i	$0.876071\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.17497	−	1.63103i	0.0477694	−	0.0663108i
$606$		0			0
$607$	23.8988	−	13.7980i	0.970021	−	0.560042i	0.0707783	−	0.997492i	$-0.477452\pi$
$607$	23.8988	−	13.7980i	0.970021	−	0.560042i	0.899243	+	0.437450i	$0.144118\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−10.8990			−0.440926
$612$		0			0
$613$			36.9444i			1.49217i	0.665851	+	0.746085i	$0.268069\pi$
$613$			36.9444i			1.49217i	−0.665851	+	0.746085i	$0.731931\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−7.19066	+	4.15153i	−0.289485	+	0.167134i	−0.637710	−	0.770277i	$-0.720118\pi$
$617$	−7.19066	+	4.15153i	−0.289485	+	0.167134i	0.348224	+	0.937411i	$0.386785\pi$
$618$		0			0
$619$	14.2753	−	24.7255i	0.573771	−	0.993800i	−0.422403	−	0.906408i	$-0.638813\pi$
$619$	14.2753	−	24.7255i	0.573771	−	0.993800i	0.996174	−	0.0873923i	$-0.0278534\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1.20713	−	0.696938i	−0.0483628	−	0.0279222i
$624$		0			0
$625$	−3.01472	−	24.8176i	−0.120589	−	0.992703i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−47.1918			−1.88166
$630$		0			0
$631$	11.3485			0.451775			0.225888	−	0.974153i	$-0.427472\pi$
$631$	11.3485			0.451775			0.225888	+	0.974153i	$0.427472\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	14.0674	−	6.33145i	0.558249	−	0.251256i
$636$		0			0
$637$	14.4206	+	8.32577i	0.571367	+	0.329879i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−18.5000	+	32.0429i	−0.730706	+	1.26562i	0.225876	+	0.974156i	$0.427476\pi$
$641$	−18.5000	+	32.0429i	−0.730706	+	1.26562i	−0.956582	+	0.291464i	$0.905858\pi$
$642$		0			0
$643$	13.2047	−	7.62372i	0.520742	−	0.300650i	−0.216496	−	0.976283i	$-0.569463\pi$
$643$	13.2047	−	7.62372i	0.520742	−	0.300650i	0.737238	+	0.675633i	$0.236130\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			34.8990i			1.37202i	0.727592	+	0.686010i	$0.240639\pi$
$647$			34.8990i			1.37202i	−0.727592	+	0.686010i	$0.759361\pi$
$648$		0			0
$649$	45.6969			1.79376
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−29.4449	+	17.0000i	−1.15227	+	0.665261i	−0.949439	−	0.313953i	$-0.898347\pi$
$653$	−29.4449	+	17.0000i	−1.15227	+	0.665261i	−0.202828	+	0.979214i	$0.565013\pi$
$654$		0			0
$655$	6.40300	−	8.88828i	0.250186	−	0.347294i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−17.8990	+	31.0019i	−0.697245	+	1.20766i	0.272173	+	0.962248i	$0.412258\pi$
$659$	−17.8990	+	31.0019i	−0.697245	+	1.20766i	−0.969418	+	0.245416i	$0.921076\pi$
$660$		0			0
$661$	−2.89898	−	5.02118i	−0.112757	−	0.195301i	0.804124	−	0.594462i	$-0.202635\pi$
$661$	−2.89898	−	5.02118i	−0.112757	−	0.195301i	−0.916881	+	0.399161i	$0.869302\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−0.550510	+	5.44949i	−0.0213479	+	0.211322i
$666$		0			0
$667$			41.3939i			1.60278i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	7.67423	+	13.2922i	0.296261	+	0.513138i
$672$		0			0
$673$	−25.0273	−	14.4495i	−0.964730	−	0.556987i	−0.0671042	−	0.997746i	$-0.521376\pi$
$673$	−25.0273	−	14.4495i	−0.964730	−	0.556987i	−0.897625	+	0.440759i	$0.854709\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	34.2911	+	19.7980i	1.31791	+	0.760897i	0.983393	−	0.181491i	$-0.0580924\pi$
$677$	34.2911	+	19.7980i	1.31791	+	0.760897i	0.334520	+	0.942389i	$0.391426\pi$
$678$		0			0
$679$	2.92168	+	5.06050i	0.112124	+	0.194204i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			45.4495i			1.73908i	0.493866	+	0.869538i	$0.335583\pi$
$683$			45.4495i			1.73908i	−0.493866	+	0.869538i	$0.664417\pi$
$684$		0			0
$685$	−0.674235	+	6.67423i	−0.0257612	+	0.255009i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−4.34847	−	7.53177i	−0.165663	−	0.286938i
$690$		0			0
$691$	8.79796	−	15.2385i	0.334690	−	0.579700i	−0.648735	−	0.761014i	$-0.724702\pi$
$691$	8.79796	−	15.2385i	0.334690	−	0.579700i	0.983425	+	0.181314i	$0.0580350\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	17.3145	−	24.0350i	0.656777	−	0.911700i
$696$		0			0
$697$	5.10867	−	2.94949i	0.193505	−	0.111720i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	39.3939			1.48789			0.743943	−	0.668243i	$-0.232953\pi$
$701$	39.3939			1.48789			0.743943	+	0.668243i	$0.232953\pi$
$702$		0			0
$703$		−	43.5959i		−	1.64425i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	3.11416	−	1.79796i	0.117120	−	0.0676192i
$708$		0			0
$709$	18.6742	−	32.3447i	0.701326	−	1.21473i	−0.266676	−	0.963786i	$-0.585925\pi$
$709$	18.6742	−	32.3447i	0.701326	−	1.21473i	0.968001	−	0.250945i	$-0.0807414\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−9.26382	−	5.34847i	−0.346933	−	0.200302i
$714$		0			0
$715$	−17.2290	+	7.75442i	−0.644328	+	0.289999i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−41.7980			−1.55880			−0.779400	−	0.626526i	$-0.784476\pi$
$719$	−41.7980			−1.55880			−0.779400	+	0.626526i	$0.784476\pi$
$720$		0			0
$721$	6.40408			0.238500
$722$		0			0
$723$		0			0
$724$		0			0
$725$	9.50079	+	28.4558i	0.352850	+	1.05682i
$726$		0			0
$727$	−21.9524	−	12.6742i	−0.814170	−	0.470061i	0.0342318	−	0.999414i	$-0.489102\pi$
$727$	−21.9524	−	12.6742i	−0.814170	−	0.470061i	−0.848402	+	0.529353i	$0.822435\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	7.52270	−	13.0297i	0.278237	−	0.481921i
$732$		0			0
$733$	−18.3133	+	10.5732i	−0.676419	+	0.390531i	−0.798504	−	0.601989i	$-0.794375\pi$
$733$	−18.3133	+	10.5732i	−0.676419	+	0.390531i	0.122086	+	0.992520i	$0.461042\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			15.6969i			0.578204i
$738$		0			0
$739$	−17.2474			−0.634458			−0.317229	−	0.948349i	$-0.602752\pi$
$739$	−17.2474			−0.634458			−0.317229	+	0.948349i	$0.602752\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	19.8311	−	11.4495i	0.727532	−	0.420041i	−0.0899863	−	0.995943i	$-0.528682\pi$
$743$	19.8311	−	11.4495i	0.727532	−	0.420041i	0.817519	+	0.575902i	$0.195349\pi$
$744$		0			0
$745$	−10.7795	+	14.9635i	−0.394929	+	0.548219i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	3.67423	−	6.36396i	0.134254	−	0.232534i
$750$		0			0
$751$	−26.4949	−	45.8905i	−0.966813	−	1.67457i	−0.704664	−	0.709541i	$-0.748902\pi$
$751$	−26.4949	−	45.8905i	−0.966813	−	1.67457i	−0.262148	−	0.965028i	$-0.584431\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	6.44949	+	0.651531i	0.234721	+	0.0237116i
$756$		0			0
$757$		−	10.0000i		−	0.363456i	−0.983349	−	0.181728i	$-0.941831\pi$
$757$		−	10.0000i		−	0.363456i	0.983349	−	0.181728i	$-0.0581691\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.247449	+	0.428594i	0.00897001	+	0.0155365i	0.870476	−	0.492212i	$-0.163811\pi$
$761$	0.247449	+	0.428594i	0.00897001	+	0.0155365i	−0.861506	+	0.507748i	$0.830478\pi$
$762$		0			0
$763$	−3.11416	−	1.79796i	−0.112740	−	0.0650905i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−28.1021	−	16.2247i	−1.01471	−	0.585842i
$768$		0			0
$769$	12.2474	+	21.2132i	0.441654	+	0.764968i	0.997812	−	0.0661088i	$-0.0210584\pi$
$769$	12.2474	+	21.2132i	0.441654	+	0.764968i	−0.556158	+	0.831076i	$0.687725\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			35.3939i			1.27303i	0.771265	+	0.636515i	$0.219625\pi$
$773$			35.3939i			1.27303i	−0.771265	+	0.636515i	$0.780375\pi$
$774$		0			0
$775$	−7.59592	−	1.55051i	−0.272853	−	0.0556960i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	2.72474	+	4.71940i	0.0976241	+	0.169090i
$780$		0			0
$781$	−4.22474	+	7.31747i	−0.151173	+	0.261840i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	29.0290	+	20.9121i	1.03609	+	0.746385i
$786$		0			0
$787$	44.5084	−	25.6969i	1.58655	−	0.915997i	0.592684	−	0.805435i	$-0.298068\pi$
$787$	44.5084	−	25.6969i	1.58655	−	0.915997i	0.993869	−	0.110562i	$-0.0352650\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	2.20204			0.0782956
$792$		0			0
$793$		−	10.8990i		−	0.387034i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−3.11416	+	1.79796i	−0.110309	+	0.0636870i	−0.554139	−	0.832424i	$-0.686953\pi$
$797$	−3.11416	+	1.79796i	−0.110309	+	0.0636870i	0.443830	+	0.896111i	$0.353619\pi$
$798$		0			0
$799$	−13.1237	+	22.7310i	−0.464284	+	0.804163i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−44.2066	−	25.5227i	−1.56002	−	0.900677i
$804$		0			0
$805$	2.84592	+	6.32316i	0.100306	+	0.222862i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−41.0908			−1.44468			−0.722338	−	0.691540i	$-0.756933\pi$
$809$	−41.0908			−1.44468			−0.722338	+	0.691540i	$0.756933\pi$
$810$		0			0
$811$	−7.24745			−0.254492			−0.127246	−	0.991871i	$-0.540614\pi$
$811$	−7.24745			−0.254492			−0.127246	+	0.991871i	$0.540614\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−18.1455	+	8.16693i	−0.635610	+	0.286075i
$816$		0			0
$817$	12.0369	+	6.94949i	0.421117	+	0.243132i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.02270	−	1.77138i	0.0356926	−	0.0618214i	−0.847627	−	0.530592i	$-0.821970\pi$
$821$	1.02270	−	1.77138i	0.0356926	−	0.0618214i	0.883320	+	0.468771i	$0.155303\pi$
$822$		0			0
$823$	−27.3629	+	15.7980i	−0.953810	+	0.550682i	−0.894262	−	0.447543i	$-0.852299\pi$
$823$	−27.3629	+	15.7980i	−0.953810	+	0.550682i	−0.0595473	+	0.998225i	$0.518966\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			5.79796i			0.201615i	0.994906	+	0.100807i	$0.0321426\pi$
$827$			5.79796i			0.201615i	−0.994906	+	0.100807i	$0.967857\pi$
$828$		0			0
$829$	26.4495			0.918629			0.459314	−	0.888274i	$-0.348095\pi$
$829$	26.4495			0.918629			0.459314	+	0.888274i	$0.348095\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	34.7285	−	20.0505i	1.20327	−	0.694709i
$834$		0			0
$835$	0.448949	+	0.323417i	0.0155365	+	0.0111923i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	20.1237	−	34.8553i	0.694748	−	1.20334i	−0.275517	−	0.961296i	$-0.588849\pi$
$839$	20.1237	−	34.8553i	0.694748	−	1.20334i	0.970266	−	0.242043i	$-0.0778175\pi$
$840$		0			0
$841$	−3.50000	−	6.06218i	−0.120690	−	0.209041i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−15.5732	−	1.57321i	−0.535735	−	0.0541202i
$846$		0			0
$847$			0.404082i			0.0138844i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−27.5959	−	47.7975i	−0.945976	−	1.63848i
$852$		0			0
$853$	7.92104	+	4.57321i	0.271211	+	0.156584i	0.629438	−	0.777051i	$-0.283285\pi$
$853$	7.92104	+	4.57321i	0.271211	+	0.156584i	−0.358227	+	0.933635i	$0.616619\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	4.67123	+	2.69694i	0.159566	+	0.0921257i	0.577657	−	0.816280i	$-0.303967\pi$
$857$	4.67123	+	2.69694i	0.159566	+	0.0921257i	−0.418091	+	0.908405i	$0.637301\pi$
$858$		0			0
$859$	18.8712	+	32.6858i	0.643876	+	1.11523i	0.984560	+	0.175048i	$0.0560081\pi$
$859$	18.8712	+	32.6858i	0.643876	+	1.11523i	−0.340684	+	0.940178i	$0.610659\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	26.4495i		−	0.900351i	−0.892940	−	0.450176i	$-0.851361\pi$
$863$		−	26.4495i		−	0.900351i	0.892940	−	0.450176i	$-0.148639\pi$
$864$		0			0
$865$	−2.65153	+	26.2474i	−0.0901548	+	0.892440i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	12.6742	+	21.9524i	0.429944	+	0.744685i
$870$		0			0
$871$	5.57321	−	9.65309i	0.188841	−	0.327082i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	3.40771	+	3.69360i	0.115201	+	0.124866i
$876$		0			0
$877$	−18.0597	+	10.4268i	−0.609834	+	0.352088i	−0.772900	−	0.634527i	$-0.781195\pi$
$877$	−18.0597	+	10.4268i	−0.609834	+	0.352088i	0.163067	+	0.986615i	$0.447861\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	30.0000			1.01073			0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000			1.01073			0.505363	+	0.862907i	$0.331359\pi$
$882$		0			0
$883$		−	6.55051i		−	0.220442i	−0.993907	−	0.110221i	$-0.964844\pi$
$883$		−	6.55051i		−	0.220442i	0.993907	−	0.110221i	$-0.0351559\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	16.7563	−	9.67423i	0.562620	−	0.324829i	−0.191576	−	0.981478i	$-0.561360\pi$
$887$	16.7563	−	9.67423i	0.562620	−	0.324829i	0.754197	+	0.656649i	$0.228027\pi$
$888$		0			0
$889$	−1.55051	+	2.68556i	−0.0520024	+	0.0900709i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−20.9989	−	12.1237i	−0.702702	−	0.405705i
$894$		0			0
$895$	0.825027	+	1.83307i	0.0275776	+	0.0612728i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	9.30306			0.310274
$900$		0			0
$901$	−20.9444			−0.697759
$902$		0			0
$903$		0			0
$904$		0			0
$905$	5.09391	+	11.3178i	0.169327	+	0.376217i
$906$		0			0
$907$	−34.4179	−	19.8712i	−1.14283	−	0.659811i	−0.195698	−	0.980664i	$-0.562697\pi$
$907$	−34.4179	−	19.8712i	−1.14283	−	0.659811i	−0.947129	+	0.320853i	$0.896031\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−12.1237	+	20.9989i	−0.401677	+	0.695725i	−0.993928	−	0.110028i	$-0.964906\pi$
$911$	−12.1237	+	20.9989i	−0.401677	+	0.695725i	0.592252	+	0.805753i	$0.298239\pi$
$912$		0			0
$913$	−11.9494	+	6.89898i	−0.395467	+	0.228323i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			2.20204i			0.0727178i
$918$		0			0
$919$	1.10102			0.0363193			0.0181597	−	0.999835i	$-0.494219\pi$
$919$	1.10102			0.0363193			0.0181597	+	0.999835i	$0.494219\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	5.19615	−	3.00000i	0.171033	−	0.0987462i
$924$		0			0
$925$	−29.9411	−	26.5241i	−0.984458	−	0.872108i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	8.20204	−	14.2064i	0.269100	−	0.466095i	−0.699529	−	0.714604i	$-0.746607\pi$
$929$	8.20204	−	14.2064i	0.269100	−	0.466095i	0.968630	+	0.248508i	$0.0799404\pi$
$930$		0			0
$931$	18.5227	+	32.0823i	0.607057	+	1.05145i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−4.57321	+	45.2702i	−0.149560	+	1.48049i
$936$		0			0
$937$		−	0.404082i		−	0.0132008i	−0.999978	−	0.00660039i	$-0.997899\pi$
$937$		−	0.404082i		−	0.0132008i	0.999978	−	0.00660039i	$-0.00210099\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−15.1010	−	26.1557i	−0.492279	−	0.852653i	0.507681	−	0.861545i	$-0.330503\pi$
$941$	−15.1010	−	26.1557i	−0.492279	−	0.852653i	−0.999960	+	0.00889239i	$0.997169\pi$
$942$		0			0
$943$	5.97469	+	3.44949i	0.194563	+	0.112331i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−2.81237	−	1.62372i	−0.0913898	−	0.0527640i	0.453609	−	0.891201i	$-0.350136\pi$
$947$	−2.81237	−	1.62372i	−0.0913898	−	0.0527640i	−0.544998	+	0.838437i	$0.683470\pi$
$948$		0			0
$949$	18.1237	+	31.3912i	0.588321	+	1.01900i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	31.2020i		−	1.01073i	−0.862905	−	0.505367i	$-0.831357\pi$
$953$		−	31.2020i		−	1.01073i	0.862905	−	0.505367i	$-0.168643\pi$
$954$		0			0
$955$	−40.5959	−	4.10102i	−1.31365	−	0.132706i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−0.674235	−	1.16781i	−0.0217722	−	0.0377105i
$960$		0			0
$961$	14.2980	−	24.7648i	0.461224	−	0.798864i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−24.8505	−	17.9020i	−0.799967	−	0.576286i
$966$		0			0
$967$	−0.603566	+	0.348469i	−0.0194094	+	0.0112060i	−0.509673	−	0.860368i	$-0.670234\pi$
$967$	−0.603566	+	0.348469i	−0.0194094	+	0.0112060i	0.490264	+	0.871574i	$0.336900\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	35.3939			1.13584			0.567922	−	0.823083i	$-0.307748\pi$
$971$	35.3939			1.13584			0.567922	+	0.823083i	$0.307748\pi$
$972$		0			0
$973$			5.95459i			0.190895i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−24.3362	+	14.0505i	−0.778584	+	0.449516i	−0.835928	−	0.548839i	$-0.815070\pi$
$977$	−24.3362	+	14.0505i	−0.778584	+	0.449516i	0.0573443	+	0.998354i	$0.481737\pi$
$978$		0			0
$979$	5.34847	−	9.26382i	0.170938	−	0.296073i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	18.5276	+	10.6969i	0.590940	+	0.341179i	0.765469	−	0.643473i	$-0.222507\pi$
$983$	18.5276	+	10.6969i	0.590940	+	0.341179i	−0.174529	+	0.984652i	$0.555840\pi$
$984$		0			0
$985$	−16.3125	+	7.34190i	−0.519758	+	0.233932i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	17.5959			0.559518
$990$		0			0
$991$	−4.00000			−0.127064			−0.0635321	−	0.997980i	$-0.520237\pi$
$991$	−4.00000			−0.127064			−0.0635321	+	0.997980i	$0.520237\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	14.2713	+	31.7084i	0.452430	+	1.00522i
$996$		0			0
$997$	−18.1384	−	10.4722i	−0.574448	−	0.331658i	0.184476	−	0.982837i	$-0.440941\pi$
$997$	−18.1384	−	10.4722i	−0.574448	−	0.331658i	−0.758924	+	0.651179i	$0.774275\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.2.by.d.289.2		8
3.2	odd	2		720.2.by.c.529.2		8
4.3	odd	2		270.2.i.b.19.1		8
5.4	even	2	inner	2160.2.by.d.289.3		8
9.4	even	3	inner	2160.2.by.d.1009.3		8
9.5	odd	6		720.2.by.c.49.3		8
12.11	even	2		90.2.i.b.79.3	yes	8
15.14	odd	2		720.2.by.c.529.3		8
20.3	even	4		1350.2.e.j.451.2		4
20.7	even	4		1350.2.e.m.451.1		4
20.19	odd	2		270.2.i.b.19.4		8
36.7	odd	6		810.2.c.e.649.3		4
36.11	even	6		810.2.c.f.649.2		4
36.23	even	6		90.2.i.b.49.2	✓	8
36.31	odd	6		270.2.i.b.199.4		8
45.4	even	6	inner	2160.2.by.d.1009.2		8
45.14	odd	6		720.2.by.c.49.2		8
60.23	odd	4		450.2.e.n.151.1		4
60.47	odd	4		450.2.e.k.151.2		4
60.59	even	2		90.2.i.b.79.2	yes	8
180.7	even	12		4050.2.a.bm.1.2		2
180.23	odd	12		450.2.e.n.301.1		4
180.43	even	12		4050.2.a.bz.1.1		2
180.47	odd	12		4050.2.a.bs.1.2		2
180.59	even	6		90.2.i.b.49.3	yes	8
180.67	even	12		1350.2.e.m.901.1		4
180.79	odd	6		810.2.c.e.649.1		4
180.83	odd	12		4050.2.a.bq.1.1		2
180.103	even	12		1350.2.e.j.901.2		4
180.119	even	6		810.2.c.f.649.4		4
180.139	odd	6		270.2.i.b.199.1		8
180.167	odd	12		450.2.e.k.301.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.2.i.b.49.2	✓	8	36.23	even	6
90.2.i.b.49.3	yes	8	180.59	even	6
90.2.i.b.79.2	yes	8	60.59	even	2
90.2.i.b.79.3	yes	8	12.11	even	2
270.2.i.b.19.1		8	4.3	odd	2
270.2.i.b.19.4		8	20.19	odd	2
270.2.i.b.199.1		8	180.139	odd	6
270.2.i.b.199.4		8	36.31	odd	6
450.2.e.k.151.2		4	60.47	odd	4
450.2.e.k.301.2		4	180.167	odd	12
450.2.e.n.151.1		4	60.23	odd	4
450.2.e.n.301.1		4	180.23	odd	12
720.2.by.c.49.2		8	45.14	odd	6
720.2.by.c.49.3		8	9.5	odd	6
720.2.by.c.529.2		8	3.2	odd	2
720.2.by.c.529.3		8	15.14	odd	2
810.2.c.e.649.1		4	180.79	odd	6
810.2.c.e.649.3		4	36.7	odd	6
810.2.c.f.649.2		4	36.11	even	6
810.2.c.f.649.4		4	180.119	even	6
1350.2.e.j.451.2		4	20.3	even	4
1350.2.e.j.901.2		4	180.103	even	12
1350.2.e.m.451.1		4	20.7	even	4
1350.2.e.m.901.1		4	180.67	even	12
2160.2.by.d.289.2		8	1.1	even	1	trivial
2160.2.by.d.289.3		8	5.4	even	2	inner
2160.2.by.d.1009.2		8	45.4	even	6	inner
2160.2.by.d.1009.3		8	9.4	even	3	inner
4050.2.a.bm.1.2		2	180.7	even	12
4050.2.a.bq.1.1		2	180.83	odd	12
4050.2.a.bs.1.2		2	180.47	odd	12
4050.2.a.bz.1.1		2	180.43	even	12

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2160.by (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.917738 + 2.03906i) q^{5} +(-0.389270 - 0.224745i) q^{7} +O(q^{10})\)	\(q+(0.917738 + 2.03906i) q^{5} +(-0.389270 - 0.224745i) q^{7} +(1.72474 - 2.98735i) q^{11} +(-2.12132 + 1.22474i) q^{13} +5.89898i q^{17} -5.44949 q^{19} +(-5.97469 + 3.44949i) q^{23} +(-3.31552 + 3.74264i) q^{25} +(3.00000 - 5.19615i) q^{29} +(0.775255 + 1.34278i) q^{31} +(0.101021 - 1.00000i) q^{35} +8.00000i q^{37} +(-0.500000 - 0.866025i) q^{41} +(-2.20881 - 1.27526i) q^{43} +(3.85337 + 2.22474i) q^{47} +(-3.39898 - 5.88721i) q^{49} +3.55051i q^{53} +(7.67423 + 0.775255i) q^{55} +(6.62372 + 11.4726i) q^{59} +(-2.22474 + 3.85337i) q^{61} +(-4.44414 - 3.20150i) q^{65} +(-3.94086 + 2.27526i) q^{67} -2.44949 q^{71} -14.7980i q^{73} +(-1.34278 + 0.775255i) q^{77} +(-3.67423 + 6.36396i) q^{79} +(-3.46410 - 2.00000i) q^{83} +(-12.0284 + 5.41372i) q^{85} +3.10102 q^{89} +1.10102 q^{91} +(-5.00120 - 11.1118i) q^{95} +(-11.2583 - 6.50000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{5}+O(q^{10}) \) 8 * q + 4 * q^5	\( 8 q + 4 q^{5} + 4 q^{11} - 24 q^{19} + 24 q^{29} + 16 q^{31} + 40 q^{35} - 4 q^{41} + 12 q^{49} + 32 q^{55} + 4 q^{59} - 8 q^{61} + 12 q^{65} - 20 q^{85} + 64 q^{89} + 48 q^{91} - 24 q^{95}+O(q^{100}) \) 8 * q + 4 * q^5 + 4 * q^11 - 24 * q^19 + 24 * q^29 + 16 * q^31 + 40 * q^35 - 4 * q^41 + 12 * q^49 + 32 * q^55 + 4 * q^59 - 8 * q^61 + 12 * q^65 - 20 * q^85 + 64 * q^89 + 48 * q^91 - 24 * q^95

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