LMFDB - Embedded newform 1620.2.x.d.377.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,2,Mod(53,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 10, 9]))

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

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

chi := DirichletCharacter("1620.53");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1620.x (of order $12$, degree $4$, 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:	$12.9357651274$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{12})$
Coefficient field:	16.0.1154223326374133760000.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 12x^{14} + 100x^{12} - 408x^{10} + 1191x^{8} - 2040x^{6} + 2500x^{4} - 1500x^{2} + 625 $ x^16 - 12x^14 + 100x^12 - 408x^10 + 1191x^8 - 2040x^6 + 2500x^4 - 1500*x^2 + 625
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{4}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 540)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			377.1
Root			$0.750156 + 0.433103i$ of defining polynomial
Character	$\chi$	$=$	1620.377
Dual form			1620.2.x.d.593.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.10648 - 0.750156i) q^{5} +(0.0822623 - 0.307007i) q^{7} +O(q^{10})$	$q+(-2.10648 - 0.750156i) q^{5} +(0.0822623 - 0.307007i) q^{7} +(2.41131 - 1.39217i) q^{11} +(-0.448288 - 1.67303i) q^{13} +(-2.78434 + 2.78434i) q^{17} -3.89898i q^{19} +(-8.46177 + 2.26732i) q^{23} +(3.87453 + 3.16038i) q^{25} +(1.39217 + 2.41131i) q^{29} +(-0.224745 + 0.389270i) q^{31} +(-0.403587 + 0.584996i) q^{35} +(3.67423 + 3.67423i) q^{37} +(-8.31779 - 4.80228i) q^{41} +(-0.752011 - 0.201501i) q^{43} +(5.97469 + 3.44949i) q^{49} +(-6.19445 - 6.19445i) q^{53} +(-6.12372 + 1.12372i) q^{55} +(-6.19445 + 10.7291i) q^{59} +(-6.39898 - 11.0834i) q^{61} +(-0.310725 + 3.86050i) q^{65} +(-11.8492 + 3.17499i) q^{67} -9.60455i q^{71} +(-6.22474 + 6.22474i) q^{73} +(-0.229046 - 0.854813i) q^{77} +(-8.00853 + 4.62372i) q^{79} +(-1.24819 + 4.65829i) q^{83} +(7.95385 - 3.77647i) q^{85} -12.3889 q^{89} -0.550510 q^{91} +(-2.92484 + 8.21313i) q^{95} +(-1.74786 + 6.52312i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{7}+O(q^{10}) $ 16 * q + 8 * q^7	$ 16 q + 8 q^{7} + 8 q^{25} + 16 q^{31} - 24 q^{43} - 24 q^{61} - 40 q^{67} - 80 q^{73} - 40 q^{85} - 48 q^{91} + 48 q^{97}+O(q^{100}) $ 16 * q + 8 * q^7 + 8 * q^25 + 16 * q^31 - 24 * q^43 - 24 * q^61 - 40 * q^67 - 80 * q^73 - 40 * q^85 - 48 * q^91 + 48 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$	$e\left(\frac{5}{6}\right)$

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
$5$	−2.10648	−	0.750156i	−0.942047	−	0.335480i
$5$	−2.10648	−	0.750156i	−0.942047	−	0.335480i
$6$		0			0
$7$	0.0822623	−	0.307007i	0.0310922	−	0.116038i	−0.948636	−	0.316371i	$-0.897536\pi$
$7$	0.0822623	−	0.307007i	0.0310922	−	0.116038i	0.979728	+	0.200333i	$0.0642023\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.41131	−	1.39217i	0.727037	−	0.419755i	−0.0903001	−	0.995915i	$-0.528783\pi$
$11$	2.41131	−	1.39217i	0.727037	−	0.419755i	0.817337	+	0.576159i	$0.195449\pi$
$12$		0			0
$13$	−0.448288	−	1.67303i	−0.124333	−	0.464016i	0.875482	−	0.483250i	$-0.160544\pi$
$13$	−0.448288	−	1.67303i	−0.124333	−	0.464016i	−0.999815	+	0.0192343i	$0.993877\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.78434	+	2.78434i	−0.675302	+	0.675302i	−0.958933	−	0.283632i	$-0.908461\pi$
$17$	−2.78434	+	2.78434i	−0.675302	+	0.675302i	0.283632	+	0.958933i	$0.408461\pi$
$18$		0			0
$19$		−	3.89898i		−	0.894487i	−0.894412	−	0.447244i	$-0.852406\pi$
$19$		−	3.89898i		−	0.894487i	0.894412	−	0.447244i	$-0.147594\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−8.46177	+	2.26732i	−1.76440	+	0.472770i	−0.987602	−	0.156978i	$-0.949825\pi$
$23$	−8.46177	+	2.26732i	−1.76440	+	0.472770i	−0.776799	+	0.629748i	$0.783158\pi$
$24$		0			0
$25$	3.87453	+	3.16038i	0.774907	+	0.632076i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.39217	+	2.41131i	0.258520	+	0.447769i	0.965846	−	0.259119i	$-0.0834320\pi$
$29$	1.39217	+	2.41131i	0.258520	+	0.447769i	−0.707326	+	0.706887i	$0.750099\pi$
$30$		0			0
$31$	−0.224745	+	0.389270i	−0.0403654	+	0.0699149i	−0.885502	−	0.464635i	$-0.846186\pi$
$31$	−0.224745	+	0.389270i	−0.0403654	+	0.0699149i	0.845137	+	0.534550i	$0.179519\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.403587	+	0.584996i	−0.0682187	+	0.0988823i
$36$		0			0
$37$	3.67423	+	3.67423i	0.604040	+	0.604040i	0.941382	−	0.337342i	$-0.109528\pi$
$37$	3.67423	+	3.67423i	0.604040	+	0.604040i	−0.337342	+	0.941382i	$0.609528\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−8.31779	−	4.80228i	−1.29902	−	0.749990i	−0.318785	−	0.947827i	$-0.603275\pi$
$41$	−8.31779	−	4.80228i	−1.29902	−	0.749990i	−0.980235	+	0.197837i	$0.936608\pi$
$42$		0			0
$43$	−0.752011	−	0.201501i	−0.114681	−	0.0307286i	0.201022	−	0.979587i	$-0.435574\pi$
$43$	−0.752011	−	0.201501i	−0.114681	−	0.0307286i	−0.315703	+	0.948858i	$0.602240\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$47$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$48$		0			0
$49$	5.97469	+	3.44949i	0.853527	+	0.492784i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−6.19445	−	6.19445i	−0.850873	−	0.850873i	0.139368	−	0.990241i	$-0.455493\pi$
$53$	−6.19445	−	6.19445i	−0.850873	−	0.850873i	−0.990241	+	0.139368i	$0.955493\pi$
$54$		0			0
$55$	−6.12372	+	1.12372i	−0.825723	+	0.151523i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−6.19445	+	10.7291i	−0.806448	+	1.39681i	0.108861	+	0.994057i	$0.465280\pi$
$59$	−6.19445	+	10.7291i	−0.806448	+	1.39681i	−0.915309	+	0.402752i	$0.868054\pi$
$60$		0			0
$61$	−6.39898	−	11.0834i	−0.819305	−	1.41908i	−0.906195	−	0.422861i	$-0.861026\pi$
$61$	−6.39898	−	11.0834i	−0.819305	−	1.41908i	0.0868894	−	0.996218i	$-0.472307\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.310725	+	3.86050i	−0.0385406	+	0.478836i
$66$		0			0
$67$	−11.8492	+	3.17499i	−1.44761	+	0.387887i	−0.895193	−	0.445679i	$-0.852962\pi$
$67$	−11.8492	+	3.17499i	−1.44761	+	0.387887i	−0.552420	+	0.833566i	$0.686296\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	9.60455i		−	1.13985i	−0.821696	−	0.569925i	$-0.806972\pi$
$71$		−	9.60455i		−	1.13985i	0.821696	−	0.569925i	$-0.193028\pi$
$72$		0			0
$73$	−6.22474	+	6.22474i	−0.728551	+	0.728551i	−0.970331	−	0.241780i	$-0.922269\pi$
$73$	−6.22474	+	6.22474i	−0.728551	+	0.728551i	0.241780	+	0.970331i	$0.422269\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.229046	−	0.854813i	−0.0261023	−	0.0974149i
$78$		0			0
$79$	−8.00853	+	4.62372i	−0.901030	+	0.520210i	−0.877534	−	0.479514i	$-0.840813\pi$
$79$	−8.00853	+	4.62372i	−0.901030	+	0.520210i	−0.0234955	+	0.999724i	$0.507480\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1.24819	+	4.65829i	−0.137006	+	0.511314i	0.862975	+	0.505246i	$0.168598\pi$
$83$	−1.24819	+	4.65829i	−0.137006	+	0.511314i	−0.999982	+	0.00606809i	$0.998068\pi$
$84$		0			0
$85$	7.95385	−	3.77647i	0.862716	−	0.409616i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−12.3889			−1.31322			−0.656610	−	0.754230i	$-0.728010\pi$
$89$	−12.3889			−1.31322			−0.656610	+	0.754230i	$0.728010\pi$
$90$		0			0
$91$	−0.550510			−0.0577092
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.92484	+	8.21313i	−0.300082	+	0.842649i
$96$		0			0
$97$	−1.74786	+	6.52312i	−0.177469	+	0.662322i	0.818649	+	0.574294i	$0.194723\pi$
$97$	−1.74786	+	6.52312i	−0.177469	+	0.662322i	−0.996118	+	0.0880285i	$0.971943\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−15.5517	+	8.97879i	−1.54745	+	0.893423i	−0.549118	+	0.835745i	$0.685036\pi$
$101$	−15.5517	+	8.97879i	−1.54745	+	0.893423i	−0.998335	+	0.0576781i	$0.981630\pi$
$102$		0			0
$103$	−2.27841	−	8.50316i	−0.224499	−	0.837841i	−0.982605	−	0.185710i	$-0.940542\pi$
$103$	−2.27841	−	8.50316i	−0.224499	−	0.837841i	0.758106	−	0.652132i	$-0.226125\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	8.97879	−	8.97879i	0.868012	−	0.868012i	−0.124240	−	0.992252i	$-0.539649\pi$
$107$	8.97879	−	8.97879i	0.868012	−	0.868012i	0.992252	+	0.124240i	$0.0396493\pi$
$108$		0			0
$109$		−	11.3485i		−	1.08699i	−0.839414	−	0.543493i	$-0.817101\pi$
$109$		−	11.3485i		−	1.08699i	0.839414	−	0.543493i	$-0.182899\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.854813	+	0.229046i	−0.0804140	+	0.0215469i	−0.298802	−	0.954315i	$-0.596587\pi$
$113$	−0.854813	+	0.229046i	−0.0804140	+	0.0215469i	0.218388	+	0.975862i	$0.429920\pi$
$114$		0			0
$115$	19.5254	+	1.57157i	1.82075	+	0.146549i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	0.625766	+	1.08386i	0.0573639	+	0.0993572i
$120$		0			0
$121$	−1.62372	+	2.81237i	−0.147611	+	0.255670i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−5.79086	−	9.56378i	−0.517950	−	0.855411i
$126$		0			0
$127$	9.89898	+	9.89898i	0.878392	+	0.878392i	0.993368	−	0.114976i	$-0.0366791\pi$
$127$	9.89898	+	9.89898i	0.878392	+	0.878392i	−0.114976	+	0.993368i	$0.536679\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	4.82262	+	2.78434i	0.421354	+	0.243269i	0.695657	−	0.718375i	$-0.255114\pi$
$131$	4.82262	+	2.78434i	0.421354	+	0.243269i	−0.274302	+	0.961644i	$0.588447\pi$
$132$		0			0
$133$	−1.19701	−	0.320739i	−0.103794	−	0.0278116i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.65829	−	1.24819i	−0.397985	−	0.106640i	0.0542749	−	0.998526i	$-0.482715\pi$
$137$	−4.65829	−	1.24819i	−0.397985	−	0.106640i	−0.452260	+	0.891886i	$0.649382\pi$
$138$		0			0
$139$	16.6688	+	9.62372i	1.41383	+	0.816274i	0.995746	−	0.0921361i	$-0.0293695\pi$
$139$	16.6688	+	9.62372i	1.41383	+	0.816274i	0.418081	+	0.908410i	$0.362703\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3.41011	−	3.41011i	−0.285167	−	0.285167i
$144$		0			0
$145$	−1.12372	−	6.12372i	−0.0933202	−	0.508548i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.41011	−	5.90648i	0.279367	−	0.483878i	−0.691861	−	0.722031i	$-0.743209\pi$
$149$	3.41011	−	5.90648i	0.279367	−	0.483878i	0.971228	+	0.238153i	$0.0765421\pi$
$150$		0			0
$151$	−5.27526	−	9.13701i	−0.429294	−	0.743559i	0.567517	−	0.823362i	$-0.307904\pi$
$151$	−5.27526	−	9.13701i	−0.429294	−	0.743559i	−0.996811	+	0.0798027i	$0.974571\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.765434	−	0.651396i	0.0614811	−	0.0523214i
$156$		0			0
$157$	−18.2343	+	4.88588i	−1.45526	+	0.389936i	−0.897850	−	0.440302i	$-0.854871\pi$
$157$	−18.2343	+	4.88588i	−1.45526	+	0.389936i	−0.557410	+	0.830238i	$0.688205\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			2.78434i			0.219437i
$162$		0			0
$163$	3.77526	−	3.77526i	0.295701	−	0.295701i	−0.543626	−	0.839327i	$-0.682949\pi$
$163$	3.77526	−	3.77526i	0.295701	−	0.295701i	0.839327	+	0.543626i	$0.182949\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	3.51551	+	13.1201i	0.272038	+	1.01526i	0.957800	+	0.287435i	$0.0928025\pi$
$167$	3.51551	+	13.1201i	0.272038	+	1.01526i	−0.685762	+	0.727826i	$0.740531\pi$
$168$		0			0
$169$	8.66025	−	5.00000i	0.666173	−	0.384615i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−2.03828	+	7.60696i	−0.154967	+	0.578346i	0.844141	+	0.536122i	$0.180111\pi$
$173$	−2.03828	+	7.60696i	−0.154967	+	0.578346i	−0.999108	+	0.0422247i	$0.986555\pi$
$174$		0			0
$175$	1.28899	−	0.929530i	0.0974383	−	0.0702658i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	9.60455			0.717878			0.358939	−	0.933361i	$-0.383139\pi$
$179$	9.60455			0.717878			0.358939	+	0.933361i	$0.383139\pi$
$180$		0			0
$181$	11.2474			0.836016			0.418008	−	0.908443i	$-0.362728\pi$
$181$	11.2474			0.836016			0.418008	+	0.908443i	$0.362728\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−4.98346	−	10.4960i	−0.366391	−	0.771678i
$186$		0			0
$187$	−2.83763	+	10.5902i	−0.207508	+	0.774431i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	10.7291	−	6.19445i	0.776330	−	0.448214i	−0.0587979	−	0.998270i	$-0.518727\pi$
$191$	10.7291	−	6.19445i	0.776330	−	0.448214i	0.835128	+	0.550055i	$0.185393\pi$
$192$		0			0
$193$	−4.23609	−	15.8093i	−0.304920	−	1.13798i	−0.933014	−	0.359841i	$-0.882831\pi$
$193$	−4.23609	−	15.8093i	−0.304920	−	1.13798i	0.628093	−	0.778138i	$-0.283836\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−6.82021	+	6.82021i	−0.485920	+	0.485920i	−0.907016	−	0.421096i	$-0.861646\pi$
$197$	−6.82021	+	6.82021i	−0.485920	+	0.485920i	0.421096	+	0.907016i	$0.361646\pi$
$198$		0			0
$199$		−	9.24745i		−	0.655534i	−0.944759	−	0.327767i	$-0.893704\pi$
$199$		−	9.24745i		−	0.655534i	0.944759	−	0.327767i	$-0.106296\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	0.854813	−	0.229046i	0.0599961	−	0.0160759i
$204$		0			0
$205$	13.9188	+	16.3555i	0.972132	+	1.14232i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−5.42804	−	9.40165i	−0.375466	−	0.650325i
$210$		0			0
$211$	6.62372	−	11.4726i	0.455996	−	0.789808i	−0.542749	−	0.839895i	$-0.682616\pi$
$211$	6.62372	−	11.4726i	0.455996	−	0.789808i	0.998745	+	0.0500868i	$0.0159498\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.43294	+	0.988583i	0.0977257	+	0.0674208i
$216$		0			0
$217$	0.101021	+	0.101021i	0.00685772	+	0.00685772i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	5.90648	+	3.41011i	0.397313	+	0.229389i
$222$		0			0
$223$	−17.4823	−	4.68438i	−1.17070	−	0.313689i	−0.379471	−	0.925204i	$-0.623894\pi$
$223$	−17.4823	−	4.68438i	−1.17070	−	0.313689i	−0.791233	+	0.611515i	$0.790561\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	21.5818	+	5.78284i	1.43244	+	0.383820i	0.889878	−	0.456198i	$-0.150789\pi$
$227$	21.5818	+	5.78284i	1.43244	+	0.383820i	0.542558	+	0.840018i	$0.317456\pi$
$228$		0			0
$229$	1.16781	+	0.674235i	0.0771710	+	0.0445547i	0.538089	−	0.842888i	$-0.319146\pi$
$229$	1.16781	+	0.674235i	0.0771710	+	0.0445547i	−0.460918	+	0.887443i	$0.652480\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	15.1732	+	15.1732i	0.994032	+	0.994032i	0.999982	−	0.00595067i	$-0.00189417\pi$
$233$	15.1732	+	15.1732i	0.994032	+	0.994032i	−0.00595067	+	0.999982i	$0.501894\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	4.17651	−	7.23393i	0.270156	−	0.467924i	−0.698746	−	0.715370i	$-0.746258\pi$
$239$	4.17651	−	7.23393i	0.270156	−	0.467924i	0.968902	+	0.247446i	$0.0795914\pi$
$240$		0			0
$241$	−9.50000	−	16.4545i	−0.611949	−	1.05993i	−0.990912	−	0.134515i	$-0.957053\pi$
$241$	−9.50000	−	16.4545i	−0.611949	−	1.05993i	0.378963	−	0.925412i	$-0.376281\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−9.99793	−	11.7482i	−0.638744	−	0.750567i
$246$		0			0
$247$	−6.52312	+	1.74786i	−0.415056	+	0.111214i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			21.9934i			1.38821i	0.719872	+	0.694107i	$0.244201\pi$
$251$			21.9934i			1.38821i	−0.719872	+	0.694107i	$0.755799\pi$
$252$		0			0
$253$	−17.2474	+	17.2474i	−1.08434	+	1.08434i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−7.59207	−	28.3340i	−0.473580	−	1.76743i	−0.626745	−	0.779225i	$-0.715613\pi$
$257$	−7.59207	−	28.3340i	−0.473580	−	1.76743i	0.153165	−	0.988201i	$-0.451054\pi$
$258$		0			0
$259$	1.43027	−	0.825765i	0.0888725	−	0.0513106i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	4.53465	−	16.9235i	0.279618	−	1.04355i	−0.673064	−	0.739584i	$-0.735022\pi$
$263$	4.53465	−	16.9235i	0.279618	−	1.04355i	0.952683	−	0.303966i	$-0.0983111\pi$
$264$		0			0
$265$	8.40169	+	17.6953i	0.516112	+	1.08701i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	12.3889			0.755364			0.377682	−	0.925935i	$-0.376721\pi$
$269$	12.3889			0.755364			0.377682	+	0.925935i	$0.376721\pi$
$270$		0			0
$271$	−21.0000			−1.27566			−0.637830	−	0.770178i	$-0.720168\pi$
$271$	−21.0000			−1.27566			−0.637830	+	0.770178i	$0.720168\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	13.7425	+	2.22664i	0.828703	+	0.134271i
$276$		0			0
$277$	−5.08738	+	18.9864i	−0.305671	+	1.14078i	0.626695	+	0.779264i	$0.284407\pi$
$277$	−5.08738	+	18.9864i	−0.305671	+	1.14078i	−0.932366	+	0.361515i	$0.882260\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	20.3743	−	11.7631i	1.21543	−	0.701729i	0.251494	−	0.967859i	$-0.419078\pi$
$281$	20.3743	−	11.7631i	1.21543	−	0.701729i	0.963937	+	0.266130i	$0.0857449\pi$
$282$		0			0
$283$	1.02412	+	3.82208i	0.0608779	+	0.227199i	0.989661	−	0.143423i	$-0.0458110\pi$
$283$	1.02412	+	3.82208i	0.0608779	+	0.227199i	−0.928784	+	0.370623i	$0.879144\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−2.15857	+	2.15857i	−0.127417	+	0.127417i
$288$		0			0
$289$			1.49490i			0.0879351i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	13.9749	−	3.74456i	0.816421	−	0.218759i	0.173640	−	0.984809i	$-0.444447\pi$
$293$	13.9749	−	3.74456i	0.816421	−	0.218759i	0.642781	+	0.766050i	$0.277780\pi$
$294$		0			0
$295$	21.0970	−	17.9539i	1.22831	−	1.04531i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	7.58662	+	13.1404i	0.438745	+	0.759929i
$300$		0			0
$301$	−0.123724	+	0.214297i	−0.00713135	+	0.0123519i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	5.16509	+	28.1471i	0.295752	+	1.61170i
$306$		0			0
$307$	13.3485	+	13.3485i	0.761837	+	0.761837i	0.976654	−	0.214817i	$-0.0689155\pi$
$307$	13.3485	+	13.3485i	0.761837	+	0.761837i	−0.214817	+	0.976654i	$0.568915\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1.08386	+	0.625766i	0.0614600	+	0.0354839i	0.530415	−	0.847738i	$-0.322036\pi$
$311$	1.08386	+	0.625766i	0.0614600	+	0.0354839i	−0.468955	+	0.883222i	$0.655369\pi$
$312$		0			0
$313$	−18.8173	−	5.04209i	−1.06362	−	0.284996i	−0.315750	−	0.948842i	$-0.602256\pi$
$313$	−18.8173	−	5.04209i	−1.06362	−	0.284996i	−0.747869	+	0.663846i	$0.768923\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−30.0436	−	8.05016i	−1.68742	−	0.452142i	−0.717696	−	0.696357i	$-0.754803\pi$
$317$	−30.0436	−	8.05016i	−1.68742	−	0.452142i	−0.969721	+	0.244215i	$0.921470\pi$
$318$		0			0
$319$	6.71391	+	3.87628i	0.375907	+	0.217030i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	10.8561	+	10.8561i	0.604049	+	0.604049i
$324$		0			0
$325$	3.55051	−	7.89898i	0.196947	−	0.438157i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−3.37628	−	5.84788i	−0.185577	−	0.321429i	0.758194	−	0.652029i	$-0.226082\pi$
$331$	−3.37628	−	5.84788i	−0.185577	−	0.321429i	−0.943771	+	0.330601i	$0.892749\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	27.3419	+	2.20070i	1.49385	+	0.120237i
$336$		0			0
$337$	−8.77915	+	2.35237i	−0.478231	+	0.128142i	−0.489879	−	0.871791i	$-0.662959\pi$
$337$	−8.77915	+	2.35237i	−0.478231	+	0.128142i	0.0116478	+	0.999932i	$0.496292\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			1.25153i			0.0677743i
$342$		0			0
$343$	3.12372	−	3.12372i	0.168665	−	0.168665i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	0.229046	+	0.854813i	0.0122959	+	0.0458887i	0.971801	−	0.235802i	$-0.0757716\pi$
$347$	0.229046	+	0.854813i	0.0122959	+	0.0458887i	−0.959505	+	0.281690i	$0.909105\pi$
$348$		0			0
$349$	11.2583	−	6.50000i	0.602645	−	0.347937i	−0.167437	−	0.985883i	$-0.553549\pi$
$349$	11.2583	−	6.50000i	0.602645	−	0.347937i	0.770081	+	0.637946i	$0.220216\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	5.55379	−	20.7270i	0.295598	−	1.10319i	−0.645143	−	0.764062i	$-0.723202\pi$
$353$	5.55379	−	20.7270i	0.295598	−	1.10319i	0.940741	−	0.339126i	$-0.110131\pi$
$354$		0			0
$355$	−7.20491	+	20.2318i	−0.382397	+	1.07379i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−4.03587			−0.213005			−0.106503	−	0.994312i	$-0.533965\pi$
$359$	−4.03587			−0.213005			−0.106503	+	0.994312i	$0.533965\pi$
$360$		0			0
$361$	3.79796			0.199893
$362$		0			0
$363$		0			0
$364$		0			0
$365$	17.7818	−	8.44279i	0.930744	−	0.441916i
$366$		0			0
$367$	−6.79827	+	25.3715i	−0.354867	+	1.32438i	0.525785	+	0.850617i	$0.323771\pi$
$367$	−6.79827	+	25.3715i	−0.354867	+	1.32438i	−0.880652	+	0.473763i	$0.842895\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−2.41131	+	1.39217i	−0.125189	+	0.0722779i
$372$		0			0
$373$	−1.58334	−	5.90911i	−0.0819822	−	0.305962i	0.912743	−	0.408533i	$-0.133960\pi$
$373$	−1.58334	−	5.90911i	−0.0819822	−	0.305962i	−0.994726	+	0.102571i	$0.967293\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3.41011	−	3.41011i	0.175629	−	0.175629i
$378$		0			0
$379$			13.0000i			0.667765i	0.942615	+	0.333883i	$0.108359\pi$
$379$			13.0000i			0.667765i	−0.942615	+	0.333883i	$0.891641\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	32.1375	−	8.61121i	1.64215	−	0.440012i	0.684748	−	0.728780i	$-0.259912\pi$
$383$	32.1375	−	8.61121i	1.64215	−	0.440012i	0.957399	+	0.288767i	$0.0932454\pi$
$384$		0			0
$385$	−0.158760	+	1.97247i	−0.00809118	+	0.100526i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−8.35302	−	14.4679i	−0.423515	−	0.733549i	0.572765	−	0.819719i	$-0.305871\pi$
$389$	−8.35302	−	14.4679i	−0.423515	−	0.733549i	−0.996280	+	0.0861698i	$0.972537\pi$
$390$		0			0
$391$	17.2474	−	29.8735i	0.872241	−	1.51077i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	20.3383	−	3.73215i	1.02333	−	0.187785i
$396$		0			0
$397$	−16.1464	−	16.1464i	−0.810366	−	0.810366i	0.174323	−	0.984689i	$-0.444226\pi$
$397$	−16.1464	−	16.1464i	−0.810366	−	0.810366i	−0.984689	+	0.174323i	$0.944226\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1.32745	−	0.766404i	−0.0662897	−	0.0382724i	0.466489	−	0.884527i	$-0.345519\pi$
$401$	−1.32745	−	0.766404i	−0.0662897	−	0.0382724i	−0.532779	+	0.846255i	$0.678852\pi$
$402$		0			0
$403$	0.752011	+	0.201501i	0.0374603	+	0.0100375i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	13.9749	+	3.74456i	0.692709	+	0.185611i
$408$		0			0
$409$	−23.8113	−	13.7474i	−1.17739	−	0.679768i	−0.221982	−	0.975051i	$-0.571253\pi$
$409$	−23.8113	−	13.7474i	−1.17739	−	0.679768i	−0.955410	+	0.295283i	$0.904586\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2.78434	+	2.78434i	0.137008	+	0.137008i
$414$		0			0
$415$	6.12372	−	8.87628i	0.300602	−	0.435719i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	15.1732	−	26.2808i	0.741261	−	1.28390i	−0.210660	−	0.977559i	$-0.567561\pi$
$419$	15.1732	−	26.2808i	0.741261	−	1.28390i	0.951921	−	0.306342i	$-0.0991052\pi$
$420$		0			0
$421$	−5.62372	−	9.74058i	−0.274084	−	0.474727i	0.695820	−	0.718216i	$-0.255041\pi$
$421$	−5.62372	−	9.74058i	−0.274084	−	0.474727i	−0.969904	+	0.243490i	$0.921708\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−19.5876	+	1.98845i	−0.950138	+	0.0964540i
$426$		0			0
$427$	−3.92907	+	1.05279i	−0.190141	+	0.0509481i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	20.7419i		−	0.999103i	−0.866284	−	0.499551i	$-0.833498\pi$
$431$		−	20.7419i		−	0.999103i	0.866284	−	0.499551i	$-0.166502\pi$
$432$		0			0
$433$	2.79796	−	2.79796i	0.134461	−	0.134461i	−0.636673	−	0.771134i	$-0.719690\pi$
$433$	2.79796	−	2.79796i	0.134461	−	0.134461i	0.771134	+	0.636673i	$0.219690\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	8.84025	+	32.9923i	0.422887	+	1.57823i
$438$		0			0
$439$	11.3851	−	6.57321i	0.543383	−	0.313722i	−0.203066	−	0.979165i	$-0.565091\pi$
$439$	11.3851	−	6.57321i	0.543383	−	0.313722i	0.746449	+	0.665443i	$0.231757\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2.49637	+	9.31658i	−0.118606	+	0.442644i	−0.999531	−	0.0306113i	$-0.990255\pi$
$443$	−2.49637	+	9.31658i	−0.118606	+	0.442644i	0.880925	+	0.473256i	$0.156921\pi$
$444$		0			0
$445$	26.0970	+	9.29360i	1.23712	+	0.440559i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	12.3889			0.584668			0.292334	−	0.956316i	$-0.405568\pi$
$449$	12.3889			0.584668			0.292334	+	0.956316i	$0.405568\pi$
$450$		0			0
$451$	−26.7423			−1.25925
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1.15964	+	0.412968i	0.0543648	+	0.0193603i
$456$		0			0
$457$	−8.18011	+	30.5286i	−0.382649	+	1.42807i	0.459190	+	0.888338i	$0.348140\pi$
$457$	−8.18011	+	30.5286i	−0.382649	+	1.42807i	−0.841839	+	0.539728i	$0.818527\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−25.1970	+	14.5475i	−1.17354	+	0.677543i	−0.954511	−	0.298175i	$-0.903622\pi$
$461$	−25.1970	+	14.5475i	−1.17354	+	0.677543i	−0.219028	+	0.975719i	$0.570289\pi$
$462$		0			0
$463$	0.374336	+	1.39704i	0.0173969	+	0.0649259i	0.974079	−	0.226210i	$-0.0726335\pi$
$463$	0.374336	+	1.39704i	0.0173969	+	0.0649259i	−0.956682	+	0.291136i	$0.905967\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−16.4248	+	16.4248i	−0.760048	+	0.760048i	−0.976331	−	0.216283i	$-0.930607\pi$
$467$	−16.4248	+	16.4248i	−0.760048	+	0.760048i	0.216283	+	0.976331i	$0.430607\pi$
$468$		0			0
$469$			3.89898i			0.180038i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−2.09385	+	0.561047i	−0.0962755	+	0.0257970i
$474$		0			0
$475$	12.3223	−	15.1067i	0.565384	−	0.693144i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−2.01794	−	3.49517i	−0.0922019	−	0.159698i	0.816235	−	0.577719i	$-0.196057\pi$
$479$	−2.01794	−	3.49517i	−0.0922019	−	0.159698i	−0.908437	+	0.418021i	$0.862724\pi$
$480$		0			0
$481$	4.50000	−	7.79423i	0.205182	−	0.355386i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	8.57520	−	12.4297i	0.389380	−	0.564402i
$486$		0			0
$487$	18.6742	+	18.6742i	0.846210	+	0.846210i	0.989658	−	0.143448i	$-0.0458188\pi$
$487$	18.6742	+	18.6742i	0.846210	+	0.846210i	−0.143448	+	0.989658i	$0.545819\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	34.5986	+	19.9755i	1.56141	+	0.901482i	0.997115	+	0.0759113i	$0.0241866\pi$
$491$	34.5986	+	19.9755i	1.56141	+	0.901482i	0.564298	+	0.825571i	$0.309147\pi$
$492$		0			0
$493$	−10.5902	−	2.83763i	−0.476958	−	0.127800i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−2.94867	−	0.790093i	−0.132266	−	0.0354405i
$498$		0			0
$499$	−0.778539	−	0.449490i	−0.0348522	−	0.0201219i	0.482473	−	0.875911i	$-0.339739\pi$
$499$	−0.778539	−	0.449490i	−0.0348522	−	0.0201219i	−0.517325	+	0.855789i	$0.673072\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−18.5833	−	18.5833i	−0.828590	−	0.828590i	0.158732	−	0.987322i	$-0.449260\pi$
$503$	−18.5833	−	18.5833i	−0.828590	−	0.828590i	−0.987322	+	0.158732i	$0.949260\pi$
$504$		0			0
$505$	39.4949	−	7.24745i	1.75750	−	0.322507i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	8.21238	−	14.2243i	0.364007	−	0.630479i	−0.624609	−	0.780938i	$-0.714742\pi$
$509$	8.21238	−	14.2243i	0.364007	−	0.630479i	0.988616	+	0.150458i	$0.0480749\pi$
$510$		0			0
$511$	1.39898	+	2.42310i	0.0618872	+	0.107192i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.57925	+	19.6209i	−0.0695902	+	0.864601i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$			15.1732i			0.664751i	0.943147	+	0.332376i	$0.107850\pi$
$521$			15.1732i			0.664751i	−0.943147	+	0.332376i	$0.892150\pi$
$522$		0			0
$523$	27.9217	−	27.9217i	1.22093	−	1.22093i	0.253628	−	0.967302i	$-0.418376\pi$
$523$	27.9217	−	27.9217i	1.22093	−	1.22093i	0.967302	−	0.253628i	$-0.0816240\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−0.458093	−	1.70963i	−0.0199548	−	0.0744724i
$528$		0			0
$529$	46.5422	−	26.8712i	2.02358	−	1.16831i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−4.30560	+	16.0687i	−0.186496	+	0.696014i
$534$		0			0
$535$	−25.6491	+	12.1782i	−1.10891	+	0.526508i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	19.2091			0.827395
$540$		0			0
$541$	10.5505			0.453602			0.226801	−	0.973941i	$-0.427173\pi$
$541$	10.5505			0.453602			0.226801	+	0.973941i	$0.427173\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−8.51312	+	23.9053i	−0.364662	+	1.02399i
$546$		0			0
$547$	−8.62840	+	32.2016i	−0.368924	+	1.37684i	0.493100	+	0.869973i	$0.335864\pi$
$547$	−8.62840	+	32.2016i	−0.368924	+	1.37684i	−0.862023	+	0.506869i	$0.830803\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	9.40165	−	5.42804i	0.400524	−	0.231242i
$552$		0			0
$553$	0.760717	+	2.83903i	0.0323490	+	0.120728i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	23.5263	−	23.5263i	0.996839	−	0.996839i	−0.00315559	−	0.999995i	$-0.501004\pi$
$557$	23.5263	−	23.5263i	0.996839	−	0.996839i	0.999995	+	0.00315559i	$0.00100446\pi$
$558$		0			0
$559$			1.34847i			0.0570342i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	11.4104	−	3.05742i	0.480893	−	0.128855i	−0.0102256	−	0.999948i	$-0.503255\pi$
$563$	11.4104	−	3.05742i	0.480893	−	0.128855i	0.491118	+	0.871093i	$0.336588\pi$
$564$		0			0
$565$	1.97247	+	0.158760i	0.0829823	+	0.00667910i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−8.97879	−	15.5517i	−0.376410	−	0.651962i	0.614127	−	0.789207i	$-0.289508\pi$
$569$	−8.97879	−	15.5517i	−0.376410	−	0.651962i	−0.990537	+	0.137246i	$0.956175\pi$
$570$		0			0
$571$	4.72474	−	8.18350i	0.197724	−	0.342469i	−0.750066	−	0.661363i	$-0.769978\pi$
$571$	4.72474	−	8.18350i	0.197724	−	0.342469i	0.947790	+	0.318894i	$0.103312\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−39.9510	−	17.9576i	−1.66607	−	0.748883i
$576$		0			0
$577$	−22.0227	−	22.0227i	−0.916817	−	0.916817i	0.0799794	−	0.996797i	$-0.474515\pi$
$577$	−22.0227	−	22.0227i	−0.916817	−	0.916817i	−0.996797	+	0.0799794i	$0.974515\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1.32745	+	0.766404i	0.0550719	+	0.0317958i
$582$		0			0
$583$	−23.5605	−	6.31300i	−0.975774	−	0.261458i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−13.9749	−	3.74456i	−0.576805	−	0.154554i	−0.0413892	−	0.999143i	$-0.513178\pi$
$587$	−13.9749	−	3.74456i	−0.576805	−	0.154554i	−0.535416	+	0.844589i	$0.679845\pi$
$588$		0			0
$589$	1.51775	+	0.876276i	0.0625380	+	0.0361063i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−8.97879	−	8.97879i	−0.368715	−	0.368715i	0.498294	−	0.867008i	$-0.333960\pi$
$593$	−8.97879	−	8.97879i	−0.368715	−	0.368715i	−0.867008	+	0.498294i	$0.833960\pi$
$594$		0			0
$595$	−0.505103	−	2.75255i	−0.0207072	−	0.112844i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−17.1912	+	29.7760i	−0.702412	+	1.21661i	0.265205	+	0.964192i	$0.414560\pi$
$599$	−17.1912	+	29.7760i	−0.702412	+	1.21661i	−0.967617	+	0.252422i	$0.918773\pi$
$600$		0			0
$601$	7.02270	+	12.1637i	0.286462	+	0.496167i	0.972963	−	0.230962i	$-0.0741874\pi$
$601$	7.02270	+	12.1637i	0.286462	+	0.496167i	−0.686501	+	0.727129i	$0.740854\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	5.53006	−	4.70617i	0.224829	−	0.191333i
$606$		0			0
$607$	−5.01910	+	1.34486i	−0.203719	+	0.0545863i	−0.359235	−	0.933247i	$-0.616962\pi$
$607$	−5.01910	+	1.34486i	−0.203719	+	0.0545863i	0.155517	+	0.987833i	$0.450296\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−31.2247	+	31.2247i	−1.26116	+	1.26116i	−0.310622	+	0.950533i	$0.600537\pi$
$613$	−31.2247	+	31.2247i	−1.26116	+	1.26116i	−0.950533	+	0.310622i	$0.899463\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−4.30560	−	16.0687i	−0.173337	−	0.646903i	−0.996829	−	0.0795746i	$-0.974644\pi$
$617$	−4.30560	−	16.0687i	−0.173337	−	0.646903i	0.823492	−	0.567328i	$-0.192023\pi$
$618$		0			0
$619$	−1.43027	+	0.825765i	−0.0574873	+	0.0331903i	−0.528468	−	0.848953i	$-0.677233\pi$
$619$	−1.43027	+	0.825765i	−0.0574873	+	0.0331903i	0.470981	+	0.882143i	$0.343900\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1.01914	+	3.80348i	−0.0408310	+	0.152383i
$624$		0			0
$625$	5.02402	+	24.4900i	0.200961	+	0.979599i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−20.4606			−0.815819
$630$		0			0
$631$	−35.4949			−1.41303			−0.706515	−	0.707698i	$-0.749734\pi$
$631$	−35.4949			−1.41303			−0.706515	+	0.707698i	$0.749734\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−13.4262	−	28.2778i	−0.532804	−	1.12217i
$636$		0			0
$637$	3.09273	−	11.5422i	0.122538	−	0.457319i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−6.99034	+	4.03587i	−0.276102	+	0.159407i	−0.631657	−	0.775248i	$-0.717625\pi$
$641$	−6.99034	+	4.03587i	−0.276102	+	0.159407i	0.355556	+	0.934655i	$0.384292\pi$
$642$		0			0
$643$	−4.59381	−	17.1443i	−0.181162	−	0.676106i	−0.995420	−	0.0956021i	$-0.969522\pi$
$643$	−4.59381	−	17.1443i	−0.181162	−	0.676106i	0.814258	−	0.580504i	$-0.197144\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	13.0147	−	13.0147i	0.511659	−	0.511659i	−0.403375	−	0.915035i	$-0.632163\pi$
$647$	13.0147	−	13.0147i	0.511659	−	0.511659i	0.915035	+	0.403375i	$0.132163\pi$
$648$		0			0
$649$			34.4949i			1.35404i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−30.0436	+	8.05016i	−1.17570	+	0.315027i	−0.793218	−	0.608937i	$-0.791596\pi$
$653$	−30.0436	+	8.05016i	−1.17570	+	0.315027i	−0.382479	+	0.923964i	$0.624929\pi$
$654$		0			0
$655$	−8.07007	−	9.48288i	−0.315324	−	0.370527i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	24.7778	+	42.9164i	0.965206	+	1.67179i	0.709062	+	0.705146i	$0.249119\pi$
$659$	24.7778	+	42.9164i	0.965206	+	1.67179i	0.256144	+	0.966639i	$0.417548\pi$
$660$		0			0
$661$	0.500000	−	0.866025i	0.0194477	−	0.0336845i	−0.856138	−	0.516748i	$-0.827143\pi$
$661$	0.500000	−	0.866025i	0.0194477	−	0.0336845i	0.875585	+	0.483063i	$0.160476\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	2.28089	+	1.57358i	0.0884490	+	0.0610208i
$666$		0			0
$667$	−17.2474	−	17.2474i	−0.667824	−	0.667824i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−30.8598	−	17.8169i	−1.19133	−	0.687815i
$672$		0			0
$673$	−24.9575	−	6.68734i	−0.962041	−	0.257778i	−0.256577	−	0.966524i	$-0.582595\pi$
$673$	−24.9575	−	6.68734i	−0.962041	−	0.257778i	−0.705464	+	0.708746i	$0.749261\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	15.2139	+	4.07656i	0.584718	+	0.156675i	0.539038	−	0.842281i	$-0.318788\pi$
$677$	15.2139	+	4.07656i	0.584718	+	0.156675i	0.0456802	+	0.998956i	$0.485454\pi$
$678$		0			0
$679$	1.85886	+	1.07321i	0.0713366	+	0.0411862i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	20.1161	+	20.1161i	0.769723	+	0.769723i	0.978058	−	0.208335i	$-0.0668043\pi$
$683$	20.1161	+	20.1161i	0.769723	+	0.769723i	−0.208335	+	0.978058i	$0.566804\pi$
$684$		0			0
$685$	8.87628	+	6.12372i	0.339145	+	0.233975i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−7.58662	+	13.1404i	−0.289027	+	0.500610i
$690$		0			0
$691$	0.123724	+	0.214297i	0.00470670	+	0.00815224i	0.868369	−	0.495918i	$-0.165168\pi$
$691$	0.123724	+	0.214297i	0.00470670	+	0.00815224i	−0.863662	+	0.504071i	$0.831835\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−27.8932	−	32.7764i	−1.05805	−	1.24328i
$696$		0			0
$697$	36.5307	−	9.78838i	1.38370	−	0.370761i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	17.6763i		−	0.667625i	−0.942640	−	0.333812i	$-0.891665\pi$
$701$		−	17.6763i		−	0.667625i	0.942640	−	0.333812i	$-0.108335\pi$
$702$		0			0
$703$	14.3258	−	14.3258i	0.540306	−	0.540306i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.47723	+	5.51310i	0.0555570	+	0.207342i
$708$		0			0
$709$	7.22999	−	4.17423i	0.271528	−	0.156767i	−0.358054	−	0.933701i	$-0.616560\pi$
$709$	7.22999	−	4.17423i	0.271528	−	0.156767i	0.629582	+	0.776934i	$0.283226\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1.01914	−	3.80348i	0.0381671	−	0.142441i
$714$		0			0
$715$	4.62522	+	9.74144i	0.172973	+	0.364309i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−49.8369			−1.85860			−0.929300	−	0.369325i	$-0.879589\pi$
$719$	−49.8369			−1.85860			−0.929300	+	0.369325i	$0.879589\pi$
$720$		0			0
$721$	−2.79796			−0.104201
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2.22664	+	13.7425i	−0.0826954	+	0.510383i
$726$		0			0
$727$	−5.08738	+	18.9864i	−0.188680	+	0.704165i	0.805132	+	0.593095i	$0.202094\pi$
$727$	−5.08738	+	18.9864i	−0.188680	+	0.704165i	−0.993813	+	0.111070i	$0.964572\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	2.65490	−	1.53281i	0.0981951	−	0.0566929i
$732$		0			0
$733$	−0.933552	−	3.48406i	−0.0344815	−	0.128687i	0.946540	−	0.322588i	$-0.104553\pi$
$733$	−0.933552	−	3.48406i	−0.0344815	−	0.128687i	−0.981021	+	0.193901i	$0.937886\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−24.1520	+	24.1520i	−0.889651	+	0.889651i
$738$		0			0
$739$			26.2474i			0.965528i	0.875750	+	0.482764i	$0.160367\pi$
$739$			26.2474i			0.965528i	−0.875750	+	0.482764i	$0.839633\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−46.9672	+	12.5848i	−1.72306	+	0.461692i	−0.978565	−	0.205939i	$-0.933975\pi$
$743$	−46.9672	+	12.5848i	−1.72306	+	0.461692i	−0.744493	+	0.667631i	$0.767309\pi$
$744$		0			0
$745$	−11.6141	+	9.88378i	−0.425508	+	0.362114i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−2.01794	−	3.49517i	−0.0737338	−	0.127711i
$750$		0			0
$751$	−4.50000	+	7.79423i	−0.164207	+	0.284415i	−0.936374	−	0.351005i	$-0.885840\pi$
$751$	−4.50000	+	7.79423i	−0.164207	+	0.284415i	0.772166	+	0.635421i	$0.219173\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	4.25805	+	23.2042i	0.154966	+	0.844488i
$756$		0			0
$757$	19.7196	+	19.7196i	0.716723	+	0.716723i	0.967933	−	0.251210i	$-0.0808285\pi$
$757$	19.7196	+	19.7196i	0.716723	+	0.716723i	−0.251210	+	0.967933i	$0.580828\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−20.1307	−	11.6225i	−0.729739	−	0.421315i	0.0885878	−	0.996068i	$-0.471765\pi$
$761$	−20.1307	−	11.6225i	−0.729739	−	0.421315i	−0.818327	+	0.574753i	$0.805098\pi$
$762$		0			0
$763$	−3.48406	−	0.933552i	−0.126132	−	0.0337968i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	20.7270	+	5.55379i	0.748409	+	0.200536i
$768$		0			0
$769$	−25.7576	−	14.8712i	−0.928844	−	0.536268i	−0.0423981	−	0.999101i	$-0.513500\pi$
$769$	−25.7576	−	14.8712i	−0.928844	−	0.536268i	−0.886446	+	0.462833i	$0.846833\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−3.41011	−	3.41011i	−0.122653	−	0.122653i	0.643116	−	0.765769i	$-0.277641\pi$
$773$	−3.41011	−	3.41011i	−0.122653	−	0.122653i	−0.765769	+	0.643116i	$0.777641\pi$
$774$		0			0
$775$	−2.10102	+	0.797959i	−0.0754709	+	0.0286635i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−18.7240	+	32.4309i	−0.670856	+	1.16196i
$780$		0			0
$781$	−13.3712	−	23.1596i	−0.478458	−	0.828714i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	42.0755	+	3.38658i	1.50174	+	0.120872i
$786$		0			0
$787$	−23.8675	+	6.39527i	−0.850783	+	0.227967i	−0.657760	−	0.753228i	$-0.728496\pi$
$787$	−23.8675	+	6.39527i	−0.850783	+	0.227967i	−0.193023	+	0.981194i	$0.561829\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			0.281275i			0.0100010i
$792$		0			0
$793$	−15.6742	+	15.6742i	−0.556608	+	0.556608i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−8.05016	−	30.0436i	−0.285151	−	1.06420i	−0.948729	−	0.316092i	$-0.897629\pi$
$797$	−8.05016	−	30.0436i	−0.285151	−	1.06420i	0.663577	−	0.748108i	$-0.269037\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−6.34388	+	23.6757i	−0.223871	+	0.835497i
$804$		0			0
$805$	2.08869	−	5.86516i	0.0736166	−	0.206720i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.25153			0.0440015			0.0220008	−	0.999758i	$-0.492996\pi$
$809$	1.25153			0.0440015			0.0220008	+	0.999758i	$0.492996\pi$
$810$		0			0
$811$	−19.5505			−0.686511			−0.343256	−	0.939242i	$-0.611530\pi$
$811$	−19.5505			−0.686511			−0.343256	+	0.939242i	$0.611530\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−10.7845	+	5.12048i	−0.377766	+	0.179363i
$816$		0			0
$817$	−0.785647	+	2.93208i	−0.0274863	+	0.102580i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	15.5517	−	8.97879i	0.542759	−	0.313362i	−0.203438	−	0.979088i	$-0.565211\pi$
$821$	15.5517	−	8.97879i	0.542759	−	0.313362i	0.746196	+	0.665726i	$0.231878\pi$
$822$		0			0
$823$	11.3551	+	42.3778i	0.395814	+	1.47720i	0.820390	+	0.571805i	$0.193757\pi$
$823$	11.3551	+	42.3778i	0.395814	+	1.47720i	−0.424576	+	0.905392i	$0.639577\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	6.82021	−	6.82021i	0.237162	−	0.237162i	−0.578512	−	0.815674i	$-0.696366\pi$
$827$	6.82021	−	6.82021i	0.237162	−	0.237162i	0.815674	+	0.578512i	$0.196366\pi$
$828$		0			0
$829$		−	49.2474i		−	1.71043i	−0.518270	−	0.855217i	$-0.673424\pi$
$829$		−	49.2474i		−	1.71043i	0.518270	−	0.855217i	$-0.326576\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−26.2401	+	7.03102i	−0.909167	+	0.243610i
$834$		0			0
$835$	2.43673	−	30.2744i	0.0843265	−	1.04769i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−12.3889	−	21.4582i	−0.427712	−	0.740819i	0.568957	−	0.822367i	$-0.307347\pi$
$839$	−12.3889	−	21.4582i	−0.427712	−	0.740819i	−0.996669	+	0.0815479i	$0.974014\pi$
$840$		0			0
$841$	10.6237	−	18.4008i	0.366335	−	0.634511i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−21.9934	+	4.03587i	−0.756598	+	0.138838i
$846$		0			0
$847$	0.729847	+	0.729847i	0.0250779	+	0.0250779i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−39.4212	−	22.7599i	−1.35134	−	0.780198i
$852$		0			0
$853$	4.74310	+	1.27091i	0.162401	+	0.0435152i	0.339103	−	0.940749i	$-0.389876\pi$
$853$	4.74310	+	1.27091i	0.162401	+	0.0435152i	−0.176702	+	0.984264i	$0.556543\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−4.65829	−	1.24819i	−0.159124	−	0.0426372i	0.178377	−	0.983962i	$-0.442915\pi$
$857$	−4.65829	−	1.24819i	−0.159124	−	0.0426372i	−0.337502	+	0.941325i	$0.609582\pi$
$858$		0			0
$859$	−4.54442	−	2.62372i	−0.155054	−	0.0895203i	0.420466	−	0.907308i	$-0.361867\pi$
$859$	−4.54442	−	2.62372i	−0.155054	−	0.0895203i	−0.575519	+	0.817788i	$0.695200\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−10.8561	−	10.8561i	−0.369545	−	0.369545i	0.497766	−	0.867311i	$-0.334154\pi$
$863$	−10.8561	−	10.8561i	−0.369545	−	0.369545i	−0.867311	+	0.497766i	$0.834154\pi$
$864$		0			0
$865$	10.0000	−	14.4949i	0.340010	−	0.492841i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−12.8740	+	22.2985i	−0.436721	+	0.756423i
$870$		0			0
$871$	10.6237	+	18.4008i	0.359971	+	0.623488i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−3.41252	+	0.991097i	−0.115364	+	0.0335052i
$876$		0			0
$877$	−13.9673	+	3.74252i	−0.471641	+	0.126376i	−0.486808	−	0.873509i	$-0.661839\pi$
$877$	−13.9673	+	3.74252i	−0.471641	+	0.126376i	0.0151668	+	0.999885i	$0.495172\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			2.78434i			0.0938068i	0.998899	+	0.0469034i	$0.0149353\pi$
$881$			2.78434i			0.0938068i	−0.998899	+	0.0469034i	$0.985065\pi$
$882$		0			0
$883$	22.9217	−	22.9217i	0.771376	−	0.771376i	−0.206971	−	0.978347i	$-0.566361\pi$
$883$	22.9217	−	22.9217i	0.771376	−	0.771376i	0.978347	+	0.206971i	$0.0663606\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−4.99274	−	18.6332i	−0.167640	−	0.625641i	−0.997689	−	0.0679499i	$-0.978354\pi$
$887$	−4.99274	−	18.6332i	−0.167640	−	0.625641i	0.830049	−	0.557691i	$-0.188312\pi$
$888$		0			0
$889$	3.85337	−	2.22474i	0.129238	−	0.0746155i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−20.2318	−	7.20491i	−0.676275	−	0.240834i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1.25153			−0.0417409
$900$		0			0
$901$	34.4949			1.14919
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−23.6925	−	8.43734i	−0.787567	−	0.280467i
$906$		0			0
$907$	5.00512	−	18.6794i	0.166192	−	0.620238i	−0.831693	−	0.555236i	$-0.812628\pi$
$907$	5.00512	−	18.6794i	0.166192	−	0.620238i	0.997885	−	0.0650019i	$-0.0207053\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−3.73876	+	2.15857i	−0.123871	+	0.0715168i	−0.560655	−	0.828049i	$-0.689451\pi$
$911$	−3.73876	+	2.15857i	−0.123871	+	0.0715168i	0.436784	+	0.899566i	$0.356117\pi$
$912$		0			0
$913$	3.47537	+	12.9703i	0.115018	+	0.429253i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1.25153	−	1.25153i	0.0413292	−	0.0413292i
$918$		0			0
$919$			18.6515i			0.615257i	0.951507	+	0.307629i	$0.0995354\pi$
$919$			18.6515i			0.615257i	−0.951507	+	0.307629i	$0.900465\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−16.0687	+	4.30560i	−0.528909	+	0.141721i
$924$		0			0
$925$	2.62397	+	25.8479i	0.0862757	+	0.849874i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	28.3285	+	49.0665i	0.929429	+	1.60982i	0.784279	+	0.620409i	$0.213033\pi$
$929$	28.3285	+	49.0665i	0.929429	+	1.60982i	0.145150	+	0.989410i	$0.453633\pi$
$930$		0			0
$931$	13.4495	−	23.2952i	0.440789	−	0.763469i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	13.9217	−	20.1794i	0.455288	−	0.659936i
$936$		0			0
$937$	−9.77526	−	9.77526i	−0.319344	−	0.319344i	0.529171	−	0.848515i	$-0.322503\pi$
$937$	−9.77526	−	9.77526i	−0.319344	−	0.319344i	−0.848515	+	0.529171i	$0.822503\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−13.1404	−	7.58662i	−0.428365	−	0.247317i	0.270285	−	0.962780i	$-0.412882\pi$
$941$	−13.1404	−	7.58662i	−0.428365	−	0.247317i	−0.698650	+	0.715464i	$0.746215\pi$
$942$		0			0
$943$	81.2715	+	21.7766i	2.64657	+	0.709145i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−22.4366	−	6.01188i	−0.729093	−	0.195360i	−0.124868	−	0.992173i	$-0.539851\pi$
$947$	−22.4366	−	6.01188i	−0.729093	−	0.195360i	−0.604225	+	0.796813i	$0.706517\pi$
$948$		0			0
$949$	13.2047	+	7.62372i	0.428642	+	0.247477i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−23.5263	−	23.5263i	−0.762090	−	0.762090i	0.214610	−	0.976700i	$-0.431152\pi$
$953$	−23.5263	−	23.5263i	−0.762090	−	0.762090i	−0.976700	+	0.214610i	$0.931152\pi$
$954$		0			0
$955$	−27.2474	+	5.00000i	−0.881707	+	0.161796i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−0.766404	+	1.32745i	−0.0247485	+	0.0428656i
$960$		0			0
$961$	15.3990	+	26.6718i	0.496741	+	0.860381i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−2.93619	+	36.4797i	−0.0945193	+	1.17432i
$966$		0			0
$967$	42.7918	−	11.4660i	1.37609	−	0.368723i	0.506391	−	0.862304i	$-0.330979\pi$
$967$	42.7918	−	11.4660i	1.37609	−	0.368723i	0.869700	+	0.493581i	$0.164312\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			2.50306i			0.0803272i	0.999193	+	0.0401636i	$0.0127879\pi$
$971$			2.50306i			0.0803272i	−0.999193	+	0.0401636i	$0.987212\pi$
$972$		0			0
$973$	4.32577	−	4.32577i	0.138678	−	0.138678i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	12.3558	+	46.1123i	0.395296	+	1.47526i	0.821276	+	0.570532i	$0.193263\pi$
$977$	12.3558	+	46.1123i	0.395296	+	1.47526i	−0.425980	+	0.904733i	$0.640070\pi$
$978$		0			0
$979$	−29.8735	+	17.2474i	−0.954760	+	0.551231i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	9.40130	−	35.0861i	0.299855	−	1.11907i	−0.637430	−	0.770509i	$-0.720002\pi$
$983$	9.40130	−	35.0861i	0.299855	−	1.11907i	0.937285	−	0.348565i	$-0.113331\pi$
$984$		0			0
$985$	19.4829	−	9.25044i	0.620776	−	0.294743i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	6.82021			0.216870
$990$		0			0
$991$	35.0454			1.11325			0.556627	−	0.830763i	$-0.312095\pi$
$991$	35.0454			1.11325			0.556627	+	0.830763i	$0.312095\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−6.93702	+	19.4796i	−0.219918	+	0.617544i
$996$		0			0
$997$	−0.0369761	+	0.137997i	−0.00117104	+	0.00437040i	−0.966509	−	0.256633i	$-0.917387\pi$
$997$	−0.0369761	+	0.137997i	−0.00117104	+	0.00437040i	0.965338	+	0.261004i	$0.0840534\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.2.x.d.377.1		16
3.2	odd	2	inner	1620.2.x.d.377.4		16
5.3	odd	4	inner	1620.2.x.d.53.1		16
9.2	odd	6	inner	1620.2.x.d.917.1		16
9.4	even	3		540.2.j.a.377.3	yes	8
9.5	odd	6		540.2.j.a.377.2	yes	8
9.7	even	3	inner	1620.2.x.d.917.4		16
15.8	even	4	inner	1620.2.x.d.53.4		16
36.23	even	6		2160.2.w.e.1457.2		8
36.31	odd	6		2160.2.w.e.1457.3		8
45.4	even	6		2700.2.j.j.1457.2		8
45.13	odd	12		540.2.j.a.53.2	✓	8
45.14	odd	6		2700.2.j.j.1457.1		8
45.22	odd	12		2700.2.j.j.593.2		8
45.23	even	12		540.2.j.a.53.3	yes	8
45.32	even	12		2700.2.j.j.593.1		8
45.38	even	12	inner	1620.2.x.d.593.1		16
45.43	odd	12	inner	1620.2.x.d.593.4		16
180.23	odd	12		2160.2.w.e.593.3		8
180.103	even	12		2160.2.w.e.593.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
540.2.j.a.53.2	✓	8	45.13	odd	12
540.2.j.a.53.3	yes	8	45.23	even	12
540.2.j.a.377.2	yes	8	9.5	odd	6
540.2.j.a.377.3	yes	8	9.4	even	3
1620.2.x.d.53.1		16	5.3	odd	4	inner
1620.2.x.d.53.4		16	15.8	even	4	inner
1620.2.x.d.377.1		16	1.1	even	1	trivial
1620.2.x.d.377.4		16	3.2	odd	2	inner
1620.2.x.d.593.1		16	45.38	even	12	inner
1620.2.x.d.593.4		16	45.43	odd	12	inner
1620.2.x.d.917.1		16	9.2	odd	6	inner
1620.2.x.d.917.4		16	9.7	even	3	inner
2160.2.w.e.593.2		8	180.103	even	12
2160.2.w.e.593.3		8	180.23	odd	12
2160.2.w.e.1457.2		8	36.23	even	6
2160.2.w.e.1457.3		8	36.31	odd	6
2700.2.j.j.593.1		8	45.32	even	12
2700.2.j.j.593.2		8	45.22	odd	12
2700.2.j.j.1457.1		8	45.14	odd	6
2700.2.j.j.1457.2		8	45.4	even	6

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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1620.x (of order \(12\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.10648 - 0.750156i) q^{5} +(0.0822623 - 0.307007i) q^{7} +O(q^{10})\)	\(q+(-2.10648 - 0.750156i) q^{5} +(0.0822623 - 0.307007i) q^{7} +(2.41131 - 1.39217i) q^{11} +(-0.448288 - 1.67303i) q^{13} +(-2.78434 + 2.78434i) q^{17} -3.89898i q^{19} +(-8.46177 + 2.26732i) q^{23} +(3.87453 + 3.16038i) q^{25} +(1.39217 + 2.41131i) q^{29} +(-0.224745 + 0.389270i) q^{31} +(-0.403587 + 0.584996i) q^{35} +(3.67423 + 3.67423i) q^{37} +(-8.31779 - 4.80228i) q^{41} +(-0.752011 - 0.201501i) q^{43} +(5.97469 + 3.44949i) q^{49} +(-6.19445 - 6.19445i) q^{53} +(-6.12372 + 1.12372i) q^{55} +(-6.19445 + 10.7291i) q^{59} +(-6.39898 - 11.0834i) q^{61} +(-0.310725 + 3.86050i) q^{65} +(-11.8492 + 3.17499i) q^{67} -9.60455i q^{71} +(-6.22474 + 6.22474i) q^{73} +(-0.229046 - 0.854813i) q^{77} +(-8.00853 + 4.62372i) q^{79} +(-1.24819 + 4.65829i) q^{83} +(7.95385 - 3.77647i) q^{85} -12.3889 q^{89} -0.550510 q^{91} +(-2.92484 + 8.21313i) q^{95} +(-1.74786 + 6.52312i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{7}+O(q^{10}) \) 16 * q + 8 * q^7	\( 16 q + 8 q^{7} + 8 q^{25} + 16 q^{31} - 24 q^{43} - 24 q^{61} - 40 q^{67} - 80 q^{73} - 40 q^{85} - 48 q^{91} + 48 q^{97}+O(q^{100}) \) 16 * q + 8 * q^7 + 8 * q^25 + 16 * q^31 - 24 * q^43 - 24 * q^61 - 40 * q^67 - 80 * q^73 - 40 * q^85 - 48 * q^91 + 48 * q^97

Introduction

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