LMFDB - Embedded newform 1620.2.x.b.593.1

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:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{12})$
Coefficient field:	8.0.12960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 $ x^8 - 3x^6 + 8x^4 - 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 60)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			593.1
Root			$-1.40126 - 0.809017i$ of defining polynomial
Character	$\chi$	$=$	1620.593
Dual form			1620.2.x.b.377.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.11803 - 1.93649i) q^{5} +(-0.366025 - 1.36603i) q^{7} +O(q^{10})$	$q+(-1.11803 - 1.93649i) q^{5} +(-0.366025 - 1.36603i) q^{7} +(3.87298 + 2.23607i) q^{11} +(-1.09808 + 4.09808i) q^{13} +(2.23607 + 2.23607i) q^{17} +2.00000i q^{19} +(3.05453 + 0.818458i) q^{23} +(-2.50000 + 4.33013i) q^{25} +(2.23607 - 3.87298i) q^{29} +(-2.00000 - 3.46410i) q^{31} +(-2.23607 + 2.23607i) q^{35} +(-3.00000 + 3.00000i) q^{37} +(7.74597 - 4.47214i) q^{41} +(4.09808 - 1.09808i) q^{43} +(9.16358 - 2.45537i) q^{47} +(4.33013 - 2.50000i) q^{49} +(2.23607 - 2.23607i) q^{53} -10.0000i q^{55} +(-4.47214 - 7.74597i) q^{59} +(3.00000 - 5.19615i) q^{61} +(9.16358 - 2.45537i) q^{65} +(1.36603 + 0.366025i) q^{67} +4.47214i q^{71} +(1.00000 + 1.00000i) q^{73} +(1.63692 - 6.10905i) q^{77} +(-5.19615 - 3.00000i) q^{79} +(2.45537 + 9.16358i) q^{83} +(1.83013 - 6.83013i) q^{85} +4.47214 q^{89} +6.00000 q^{91} +(3.87298 - 2.23607i) q^{95} +(3.29423 + 12.2942i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{7}+O(q^{10}) $ 8 * q + 4 * q^7	$ 8 q + 4 q^{7} + 12 q^{13} - 20 q^{25} - 16 q^{31} - 24 q^{37} + 12 q^{43} + 24 q^{61} + 4 q^{67} + 8 q^{73} - 20 q^{85} + 48 q^{91} - 36 q^{97}+O(q^{100}) $ 8 * q + 4 * q^7 + 12 * q^13 - 20 * q^25 - 16 * q^31 - 24 * q^37 + 12 * q^43 + 24 * q^61 + 4 * q^67 + 8 * q^73 - 20 * q^85 + 48 * q^91 - 36 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$	$e\left(\frac{1}{6}\right)$

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
$5$	−1.11803	−	1.93649i	−0.500000	−	0.866025i
$5$	−1.11803	−	1.93649i	−0.500000	−	0.866025i
$6$		0			0
$7$	−0.366025	−	1.36603i	−0.138345	−	0.516309i	−0.999962	−	0.00875026i	$-0.997215\pi$
$7$	−0.366025	−	1.36603i	−0.138345	−	0.516309i	0.861617	−	0.507559i	$-0.169452\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.87298	+	2.23607i	1.16775	+	0.674200i	0.953149	−	0.302502i	$-0.0978220\pi$
$11$	3.87298	+	2.23607i	1.16775	+	0.674200i	0.214600	+	0.976702i	$0.431155\pi$
$12$		0			0
$13$	−1.09808	+	4.09808i	−0.304552	+	1.13660i	0.628779	+	0.777584i	$0.283555\pi$
$13$	−1.09808	+	4.09808i	−0.304552	+	1.13660i	−0.933331	+	0.359018i	$0.883112\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.23607	+	2.23607i	0.542326	+	0.542326i	0.924210	−	0.381884i	$-0.124725\pi$
$17$	2.23607	+	2.23607i	0.542326	+	0.542326i	−0.381884	+	0.924210i	$0.624725\pi$
$18$		0			0
$19$			2.00000i			0.458831i	0.973329	+	0.229416i	$0.0736815\pi$
$19$			2.00000i			0.458831i	−0.973329	+	0.229416i	$0.926318\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.05453	+	0.818458i	0.636913	+	0.170660i	0.562805	−	0.826590i	$-0.309722\pi$
$23$	3.05453	+	0.818458i	0.636913	+	0.170660i	0.0741081	+	0.997250i	$0.476389\pi$
$24$		0			0
$25$	−2.50000	+	4.33013i	−0.500000	+	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.23607	−	3.87298i	0.415227	−	0.719195i	−0.580225	−	0.814456i	$-0.697035\pi$
$29$	2.23607	−	3.87298i	0.415227	−	0.719195i	0.995452	+	0.0952614i	$0.0303687\pi$
$30$		0			0
$31$	−2.00000	−	3.46410i	−0.359211	−	0.622171i	0.628619	−	0.777714i	$-0.283621\pi$
$31$	−2.00000	−	3.46410i	−0.359211	−	0.622171i	−0.987829	+	0.155543i	$0.950287\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.23607	+	2.23607i	−0.377964	+	0.377964i
$36$		0			0
$37$	−3.00000	+	3.00000i	−0.493197	+	0.493197i	−0.909312	−	0.416115i	$-0.863391\pi$
$37$	−3.00000	+	3.00000i	−0.493197	+	0.493197i	0.416115	+	0.909312i	$0.363391\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	7.74597	−	4.47214i	1.20972	−	0.698430i	0.247019	−	0.969011i	$-0.420549\pi$
$41$	7.74597	−	4.47214i	1.20972	−	0.698430i	0.962697	+	0.270580i	$0.0872155\pi$
$42$		0			0
$43$	4.09808	−	1.09808i	0.624951	−	0.167455i	0.0675734	−	0.997714i	$-0.478474\pi$
$43$	4.09808	−	1.09808i	0.624951	−	0.167455i	0.557377	+	0.830259i	$0.311808\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	9.16358	−	2.45537i	1.33665	−	0.358153i	0.481457	−	0.876470i	$-0.340108\pi$
$47$	9.16358	−	2.45537i	1.33665	−	0.358153i	0.855189	+	0.518317i	$0.173441\pi$
$48$		0			0
$49$	4.33013	−	2.50000i	0.618590	−	0.357143i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	2.23607	−	2.23607i	0.307148	−	0.307148i	−0.536655	−	0.843802i	$-0.680312\pi$
$53$	2.23607	−	2.23607i	0.307148	−	0.307148i	0.843802	+	0.536655i	$0.180312\pi$
$54$		0			0
$55$		−	10.0000i		−	1.34840i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.47214	−	7.74597i	−0.582223	−	1.00844i	−0.995215	−	0.0977047i	$-0.968850\pi$
$59$	−4.47214	−	7.74597i	−0.582223	−	1.00844i	0.412993	−	0.910734i	$-0.364483\pi$
$60$		0			0
$61$	3.00000	−	5.19615i	0.384111	−	0.665299i	−0.607535	−	0.794293i	$-0.707841\pi$
$61$	3.00000	−	5.19615i	0.384111	−	0.665299i	0.991645	+	0.128994i	$0.0411748\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	9.16358	−	2.45537i	1.13660	−	0.304552i
$66$		0			0
$67$	1.36603	+	0.366025i	0.166887	+	0.0447171i	0.341295	−	0.939956i	$-0.389135\pi$
$67$	1.36603	+	0.366025i	0.166887	+	0.0447171i	−0.174408	+	0.984673i	$0.555801\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			4.47214i			0.530745i	0.964146	+	0.265372i	$0.0854949\pi$
$71$			4.47214i			0.530745i	−0.964146	+	0.265372i	$0.914505\pi$
$72$		0			0
$73$	1.00000	+	1.00000i	0.117041	+	0.117041i	0.763202	−	0.646160i	$-0.223626\pi$
$73$	1.00000	+	1.00000i	0.117041	+	0.117041i	−0.646160	+	0.763202i	$0.723626\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.63692	−	6.10905i	0.186544	−	0.696191i
$78$		0			0
$79$	−5.19615	−	3.00000i	−0.584613	−	0.337526i	0.178352	−	0.983967i	$-0.442924\pi$
$79$	−5.19615	−	3.00000i	−0.584613	−	0.337526i	−0.762964	+	0.646440i	$0.776257\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	2.45537	+	9.16358i	0.269512	+	1.00583i	0.959430	+	0.281946i	$0.0909798\pi$
$83$	2.45537	+	9.16358i	0.269512	+	1.00583i	−0.689918	+	0.723887i	$0.742353\pi$
$84$		0			0
$85$	1.83013	−	6.83013i	0.198505	−	0.740831i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	4.47214			0.474045			0.237023	−	0.971504i	$-0.423828\pi$
$89$	4.47214			0.474045			0.237023	+	0.971504i	$0.423828\pi$
$90$		0			0
$91$	6.00000			0.628971
$92$		0			0
$93$		0			0
$94$		0			0
$95$	3.87298	−	2.23607i	0.397360	−	0.229416i
$96$		0			0
$97$	3.29423	+	12.2942i	0.334478	+	1.24829i	0.904434	+	0.426614i	$0.140294\pi$
$97$	3.29423	+	12.2942i	0.334478	+	1.24829i	−0.569955	+	0.821676i	$0.693040\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	7.74597	+	4.47214i	0.770752	+	0.444994i	0.833143	−	0.553058i	$-0.186539\pi$
$101$	7.74597	+	4.47214i	0.770752	+	0.444994i	−0.0623905	+	0.998052i	$0.519872\pi$
$102$		0			0
$103$	0.366025	−	1.36603i	0.0360656	−	0.134598i	−0.945546	−	0.325489i	$-0.894471\pi$
$103$	0.366025	−	1.36603i	0.0360656	−	0.134598i	0.981611	+	0.190891i	$0.0611377\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2.23607	+	2.23607i	0.216169	+	0.216169i	0.806882	−	0.590713i	$-0.201153\pi$
$107$	2.23607	+	2.23607i	0.216169	+	0.216169i	−0.590713	+	0.806882i	$0.701153\pi$
$108$		0			0
$109$		−	8.00000i		−	0.766261i	−0.923694	−	0.383131i	$-0.874846\pi$
$109$		−	8.00000i		−	0.766261i	0.923694	−	0.383131i	$-0.125154\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−3.05453	−	0.818458i	−0.287346	−	0.0769940i	0.112267	−	0.993678i	$-0.464189\pi$
$113$	−3.05453	−	0.818458i	−0.287346	−	0.0769940i	−0.399613	+	0.916684i	$0.630855\pi$
$114$		0			0
$115$	−1.83013	−	6.83013i	−0.170660	−	0.636913i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2.23607	−	3.87298i	0.204980	−	0.355036i
$120$		0			0
$121$	4.50000	+	7.79423i	0.409091	+	0.708566i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.1803			1.00000
$126$		0			0
$127$	−13.0000	+	13.0000i	−1.15356	+	1.15356i	−0.167731	+	0.985833i	$0.553644\pi$
$127$	−13.0000	+	13.0000i	−1.15356	+	1.15356i	−0.985833	+	0.167731i	$0.946356\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−3.87298	+	2.23607i	−0.338384	+	0.195366i	−0.659557	−	0.751654i	$-0.729256\pi$
$131$	−3.87298	+	2.23607i	−0.338384	+	0.195366i	0.321173	+	0.947020i	$0.395923\pi$
$132$		0			0
$133$	2.73205	−	0.732051i	0.236899	−	0.0634769i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	9.16358	−	2.45537i	0.782897	−	0.209777i	0.154836	−	0.987940i	$-0.450515\pi$
$137$	9.16358	−	2.45537i	0.782897	−	0.209777i	0.628062	+	0.778164i	$0.283849\pi$
$138$		0			0
$139$	−12.1244	+	7.00000i	−1.02837	+	0.593732i	−0.916519	−	0.399992i	$-0.869013\pi$
$139$	−12.1244	+	7.00000i	−1.02837	+	0.593732i	−0.111856	+	0.993724i	$0.535679\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−13.4164	+	13.4164i	−1.12194	+	1.12194i
$144$		0			0
$145$	−10.0000			−0.830455
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6.70820	+	11.6190i	0.549557	+	0.951861i	0.998305	+	0.0582028i	$0.0185370\pi$
$149$	6.70820	+	11.6190i	0.549557	+	0.951861i	−0.448747	+	0.893659i	$0.648130\pi$
$150$		0			0
$151$	−2.00000	+	3.46410i	−0.162758	+	0.281905i	−0.935857	−	0.352381i	$-0.885372\pi$
$151$	−2.00000	+	3.46410i	−0.162758	+	0.281905i	0.773099	+	0.634285i	$0.218706\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.47214	+	7.74597i	−0.359211	+	0.622171i
$156$		0			0
$157$	−12.2942	−	3.29423i	−0.981186	−	0.262908i	−0.267642	−	0.963518i	$-0.586244\pi$
$157$	−12.2942	−	3.29423i	−0.981186	−	0.262908i	−0.713544	+	0.700610i	$0.752911\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	4.47214i		−	0.352454i
$162$		0			0
$163$	17.0000	+	17.0000i	1.33154	+	1.33154i	0.903991	+	0.427552i	$0.140624\pi$
$163$	17.0000	+	17.0000i	1.33154	+	1.33154i	0.427552	+	0.903991i	$0.359376\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	4.09229	−	15.2726i	0.316671	−	1.18183i	−0.605753	−	0.795653i	$-0.707128\pi$
$167$	4.09229	−	15.2726i	0.316671	−	1.18183i	0.922424	−	0.386179i	$-0.126205\pi$
$168$		0			0
$169$	−4.33013	−	2.50000i	−0.333087	−	0.192308i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−5.72920	−	21.3817i	−0.435583	−	1.62562i	−0.739667	−	0.672974i	$-0.765017\pi$
$173$	−5.72920	−	21.3817i	−0.435583	−	1.62562i	0.304083	−	0.952645i	$-0.401650\pi$
$174$		0			0
$175$	6.83013	+	1.83013i	0.516309	+	0.138345i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−8.94427			−0.668526			−0.334263	−	0.942480i	$-0.608487\pi$
$179$	−8.94427			−0.668526			−0.334263	+	0.942480i	$0.608487\pi$
$180$		0			0
$181$	14.0000			1.04061			0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000			1.04061			0.520306	+	0.853980i	$0.325818\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	9.16358	+	2.45537i	0.673720	+	0.180523i
$186$		0			0
$187$	3.66025	+	13.6603i	0.267664	+	0.998937i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−3.87298	−	2.23607i	−0.280239	−	0.161796i	0.353292	−	0.935513i	$-0.385062\pi$
$191$	−3.87298	−	2.23607i	−0.280239	−	0.161796i	−0.633532	+	0.773717i	$0.718395\pi$
$192$		0			0
$193$	−1.09808	+	4.09808i	−0.0790413	+	0.294986i	−0.994119	−	0.108291i	$-0.965462\pi$
$193$	−1.09808	+	4.09808i	−0.0790413	+	0.294986i	0.915078	+	0.403277i	$0.132129\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−6.70820	−	6.70820i	−0.477940	−	0.477940i	0.426532	−	0.904472i	$-0.359735\pi$
$197$	−6.70820	−	6.70820i	−0.477940	−	0.477940i	−0.904472	+	0.426532i	$0.859735\pi$
$198$		0			0
$199$		−	18.0000i		−	1.27599i	−0.770042	−	0.637993i	$-0.779765\pi$
$199$		−	18.0000i		−	1.27599i	0.770042	−	0.637993i	$-0.220235\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−6.10905	−	1.63692i	−0.428771	−	0.114889i
$204$		0			0
$205$	−17.3205	−	10.0000i	−1.20972	−	0.698430i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−4.47214	+	7.74597i	−0.309344	+	0.535800i
$210$		0			0
$211$	8.00000	+	13.8564i	0.550743	+	0.953914i	0.998221	+	0.0596196i	$0.0189888\pi$
$211$	8.00000	+	13.8564i	0.550743	+	0.953914i	−0.447478	+	0.894295i	$0.647678\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−6.70820	−	6.70820i	−0.457496	−	0.457496i
$216$		0			0
$217$	−4.00000	+	4.00000i	−0.271538	+	0.271538i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−11.6190	+	6.70820i	−0.781575	+	0.451243i
$222$		0			0
$223$	4.09808	−	1.09808i	0.274427	−	0.0735326i	−0.118981	−	0.992897i	$-0.537963\pi$
$223$	4.09808	−	1.09808i	0.274427	−	0.0735326i	0.393408	+	0.919364i	$0.371296\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−15.2726	+	4.09229i	−1.01368	+	0.271615i	−0.727166	−	0.686461i	$-0.759163\pi$
$227$	−15.2726	+	4.09229i	−1.01368	+	0.271615i	−0.286514	+	0.958076i	$0.592496\pi$
$228$		0			0
$229$	10.3923	−	6.00000i	0.686743	−	0.396491i	−0.115648	−	0.993290i	$-0.536894\pi$
$229$	10.3923	−	6.00000i	0.686743	−	0.396491i	0.802391	+	0.596799i	$0.203561\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	20.1246	−	20.1246i	1.31841	−	1.31841i	0.403370	−	0.915037i	$-0.367839\pi$
$233$	20.1246	−	20.1246i	1.31841	−	1.31841i	0.915037	−	0.403370i	$-0.132161\pi$
$234$		0			0
$235$	−15.0000	−	15.0000i	−0.978492	−	0.978492i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$239$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$240$		0			0
$241$	13.0000	−	22.5167i	0.837404	−	1.45043i	−0.0546547	−	0.998505i	$-0.517406\pi$
$241$	13.0000	−	22.5167i	0.837404	−	1.45043i	0.892058	−	0.451920i	$-0.149261\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−9.68246	−	5.59017i	−0.618590	−	0.357143i
$246$		0			0
$247$	−8.19615	−	2.19615i	−0.521509	−	0.139738i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			13.4164i			0.846836i	0.905934	+	0.423418i	$0.139170\pi$
$251$			13.4164i			0.846836i	−0.905934	+	0.423418i	$0.860830\pi$
$252$		0			0
$253$	10.0000	+	10.0000i	0.628695	+	0.628695i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−5.72920	+	21.3817i	−0.357378	+	1.33375i	0.520088	+	0.854113i	$0.325899\pi$
$257$	−5.72920	+	21.3817i	−0.357378	+	1.33375i	−0.877466	+	0.479640i	$0.840767\pi$
$258$		0			0
$259$	5.19615	+	3.00000i	0.322873	+	0.186411i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	5.72920	+	21.3817i	0.353278	+	1.31845i	0.882638	+	0.470054i	$0.155765\pi$
$263$	5.72920	+	21.3817i	0.353278	+	1.31845i	−0.529360	+	0.848397i	$0.677568\pi$
$264$		0			0
$265$	−6.83013	−	1.83013i	−0.419571	−	0.112424i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	22.3607			1.36335			0.681677	−	0.731653i	$-0.261251\pi$
$269$	22.3607			1.36335			0.681677	+	0.731653i	$0.261251\pi$
$270$		0			0
$271$	12.0000			0.728948			0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000			0.728948			0.364474	+	0.931214i	$0.381249\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−19.3649	+	11.1803i	−1.16775	+	0.674200i
$276$		0			0
$277$	3.29423	+	12.2942i	0.197931	+	0.738689i	0.991489	+	0.130193i	$0.0415598\pi$
$277$	3.29423	+	12.2942i	0.197931	+	0.738689i	−0.793558	+	0.608495i	$0.791773\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−23.2379	−	13.4164i	−1.38626	−	0.800356i	−0.393366	−	0.919382i	$-0.628689\pi$
$281$	−23.2379	−	13.4164i	−1.38626	−	0.800356i	−0.992891	+	0.119026i	$0.962023\pi$
$282$		0			0
$283$	6.22243	−	23.2224i	0.369885	−	1.38043i	−0.490791	−	0.871277i	$-0.663292\pi$
$283$	6.22243	−	23.2224i	0.369885	−	1.38043i	0.860676	−	0.509153i	$-0.170041\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−8.94427	−	8.94427i	−0.527964	−	0.527964i
$288$		0			0
$289$		−	7.00000i		−	0.411765i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	9.16358	+	2.45537i	0.535342	+	0.143444i	0.516354	−	0.856375i	$-0.327289\pi$
$293$	9.16358	+	2.45537i	0.535342	+	0.143444i	0.0189880	+	0.999820i	$0.493956\pi$
$294$		0			0
$295$	−10.0000	+	17.3205i	−0.582223	+	1.00844i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−6.70820	+	11.6190i	−0.387945	+	0.671941i
$300$		0			0
$301$	−3.00000	−	5.19615i	−0.172917	−	0.299501i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−13.4164			−0.768221
$306$		0			0
$307$	−21.0000	+	21.0000i	−1.19853	+	1.19853i	−0.223928	+	0.974606i	$0.571888\pi$
$307$	−21.0000	+	21.0000i	−1.19853	+	1.19853i	−0.974606	+	0.223928i	$0.928112\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3.87298	−	2.23607i	0.219617	−	0.126796i	−0.386156	−	0.922433i	$-0.626197\pi$
$311$	3.87298	−	2.23607i	0.219617	−	0.126796i	0.605773	+	0.795638i	$0.292864\pi$
$312$		0			0
$313$	4.09808	−	1.09808i	0.231637	−	0.0620669i	−0.141133	−	0.989991i	$-0.545075\pi$
$313$	4.09808	−	1.09808i	0.231637	−	0.0620669i	0.372770	+	0.927924i	$0.378408\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−27.4907	+	7.36612i	−1.54403	+	0.413722i	−0.927566	−	0.373658i	$-0.878103\pi$
$317$	−27.4907	+	7.36612i	−1.54403	+	0.413722i	−0.616467	+	0.787381i	$0.711437\pi$
$318$		0			0
$319$	17.3205	−	10.0000i	0.969762	−	0.559893i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−4.47214	+	4.47214i	−0.248836	+	0.248836i
$324$		0			0
$325$	−15.0000	−	15.0000i	−0.832050	−	0.832050i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−6.70820	−	11.6190i	−0.369835	−	0.640573i
$330$		0			0
$331$	4.00000	−	6.92820i	0.219860	−	0.380808i	−0.734905	−	0.678170i	$-0.762773\pi$
$331$	4.00000	−	6.92820i	0.219860	−	0.380808i	0.954765	+	0.297361i	$0.0961066\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−0.818458	−	3.05453i	−0.0447171	−	0.166887i
$336$		0			0
$337$	−12.2942	−	3.29423i	−0.669709	−	0.179448i	−0.0920854	−	0.995751i	$-0.529353\pi$
$337$	−12.2942	−	3.29423i	−0.669709	−	0.179448i	−0.577624	+	0.816303i	$0.696020\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	17.8885i		−	0.968719i
$342$		0			0
$343$	−12.0000	−	12.0000i	−0.647939	−	0.647939i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−9.00303	+	33.5998i	−0.483308	+	1.80373i	0.104253	+	0.994551i	$0.466755\pi$
$347$	−9.00303	+	33.5998i	−0.483308	+	1.80373i	−0.587561	+	0.809180i	$0.699912\pi$
$348$		0			0
$349$	3.46410	+	2.00000i	0.185429	+	0.107058i	0.589841	−	0.807519i	$-0.299190\pi$
$349$	3.46410	+	2.00000i	0.185429	+	0.107058i	−0.404412	+	0.914577i	$0.632524\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−2.45537	−	9.16358i	−0.130686	−	0.487728i	0.869292	−	0.494299i	$-0.164575\pi$
$353$	−2.45537	−	9.16358i	−0.130686	−	0.487728i	−0.999978	+	0.00657077i	$0.997908\pi$
$354$		0			0
$355$	8.66025	−	5.00000i	0.459639	−	0.265372i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−35.7771			−1.88824			−0.944121	−	0.329598i	$-0.893087\pi$
$359$	−35.7771			−1.88824			−0.944121	+	0.329598i	$0.893087\pi$
$360$		0			0
$361$	15.0000			0.789474
$362$		0			0
$363$		0			0
$364$		0			0
$365$	0.818458	−	3.05453i	0.0428400	−	0.159881i
$366$		0			0
$367$	−0.366025	−	1.36603i	−0.0191064	−	0.0713059i	0.955714	−	0.294296i	$-0.0950850\pi$
$367$	−0.366025	−	1.36603i	−0.0191064	−	0.0713059i	−0.974821	+	0.222990i	$0.928418\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−3.87298	−	2.23607i	−0.201075	−	0.116091i
$372$		0			0
$373$	−8.41858	+	31.4186i	−0.435898	+	1.62679i	0.303011	+	0.952987i	$0.402008\pi$
$373$	−8.41858	+	31.4186i	−0.435898	+	1.62679i	−0.738909	+	0.673806i	$0.764659\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	13.4164	+	13.4164i	0.690980	+	0.690980i
$378$		0			0
$379$			2.00000i			0.102733i	0.998680	+	0.0513665i	$0.0163577\pi$
$379$			2.00000i			0.102733i	−0.998680	+	0.0513665i	$0.983642\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−9.16358	−	2.45537i	−0.468237	−	0.125464i	0.0169831	−	0.999856i	$-0.494594\pi$
$383$	−9.16358	−	2.45537i	−0.468237	−	0.125464i	−0.485220	+	0.874392i	$0.661261\pi$
$384$		0			0
$385$	−13.6603	+	3.66025i	−0.696191	+	0.186544i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	6.70820	−	11.6190i	0.340119	−	0.589104i	−0.644335	−	0.764743i	$-0.722866\pi$
$389$	6.70820	−	11.6190i	0.340119	−	0.589104i	0.984455	+	0.175639i	$0.0561992\pi$
$390$		0			0
$391$	5.00000	+	8.66025i	0.252861	+	0.437968i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			13.4164i			0.675053i
$396$		0			0
$397$	−11.0000	+	11.0000i	−0.552074	+	0.552074i	−0.927039	−	0.374965i	$-0.877655\pi$
$397$	−11.0000	+	11.0000i	−0.552074	+	0.552074i	0.374965	+	0.927039i	$0.377655\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$	16.3923	−	4.39230i	0.816559	−	0.218796i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−18.3272	+	4.91075i	−0.908443	+	0.243417i
$408$		0			0
$409$	10.3923	−	6.00000i	0.513866	−	0.296681i	−0.220555	−	0.975375i	$-0.570787\pi$
$409$	10.3923	−	6.00000i	0.513866	−	0.296681i	0.734422	+	0.678694i	$0.237454\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−8.94427	+	8.94427i	−0.440119	+	0.440119i
$414$		0			0
$415$	15.0000	−	15.0000i	0.736321	−	0.736321i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	13.4164	+	23.2379i	0.655434	+	1.13525i	0.981785	+	0.189997i	$0.0608478\pi$
$419$	13.4164	+	23.2379i	0.655434	+	1.13525i	−0.326350	+	0.945249i	$0.605819\pi$
$420$		0			0
$421$	−1.00000	+	1.73205i	−0.0487370	+	0.0844150i	−0.889365	−	0.457198i	$-0.848853\pi$
$421$	−1.00000	+	1.73205i	−0.0487370	+	0.0844150i	0.840628	+	0.541613i	$0.182186\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−15.2726	+	4.09229i	−0.740831	+	0.198505i
$426$		0			0
$427$	−8.19615	−	2.19615i	−0.396640	−	0.106279i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	31.3050i		−	1.50791i	−0.656928	−	0.753953i	$-0.728145\pi$
$431$		−	31.3050i		−	1.50791i	0.656928	−	0.753953i	$-0.271855\pi$
$432$		0			0
$433$	17.0000	+	17.0000i	0.816968	+	0.816968i	0.985668	−	0.168700i	$-0.0539568\pi$
$433$	17.0000	+	17.0000i	0.816968	+	0.816968i	−0.168700	+	0.985668i	$0.553957\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.63692	+	6.10905i	−0.0783043	+	0.292236i
$438$		0			0
$439$	−19.0526	−	11.0000i	−0.909329	−	0.525001i	−0.0291138	−	0.999576i	$-0.509269\pi$
$439$	−19.0526	−	11.0000i	−0.909329	−	0.525001i	−0.880215	+	0.474575i	$0.842602\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−7.36612	−	27.4907i	−0.349975	−	1.30612i	−0.886692	−	0.462361i	$-0.847002\pi$
$443$	−7.36612	−	27.4907i	−0.349975	−	1.30612i	0.536717	−	0.843763i	$-0.319664\pi$
$444$		0			0
$445$	−5.00000	−	8.66025i	−0.237023	−	0.410535i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−4.47214			−0.211053			−0.105527	−	0.994416i	$-0.533653\pi$
$449$	−4.47214			−0.211053			−0.105527	+	0.994416i	$0.533653\pi$
$450$		0			0
$451$	40.0000			1.88353
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−6.70820	−	11.6190i	−0.314485	−	0.544705i
$456$		0			0
$457$	−1.09808	−	4.09808i	−0.0513658	−	0.191700i	0.935475	−	0.353392i	$-0.114972\pi$
$457$	−1.09808	−	4.09808i	−0.0513658	−	0.191700i	−0.986841	+	0.161692i	$0.948305\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−7.74597	−	4.47214i	−0.360766	−	0.208288i	0.308651	−	0.951175i	$-0.400123\pi$
$461$	−7.74597	−	4.47214i	−0.360766	−	0.208288i	−0.669417	+	0.742887i	$0.733456\pi$
$462$		0			0
$463$	−6.95448	+	25.9545i	−0.323202	+	1.20621i	0.592905	+	0.805273i	$0.297981\pi$
$463$	−6.95448	+	25.9545i	−0.323202	+	1.20621i	−0.916107	+	0.400934i	$0.868686\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	2.23607	+	2.23607i	0.103473	+	0.103473i	0.756948	−	0.653475i	$-0.226690\pi$
$467$	2.23607	+	2.23607i	0.103473	+	0.103473i	−0.653475	+	0.756948i	$0.726690\pi$
$468$		0			0
$469$		−	2.00000i		−	0.0923514i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	18.3272	+	4.91075i	0.842683	+	0.225796i
$474$		0			0
$475$	−8.66025	−	5.00000i	−0.397360	−	0.229416i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−17.8885	+	30.9839i	−0.817348	+	1.41569i	0.0902809	+	0.995916i	$0.471224\pi$
$479$	−17.8885	+	30.9839i	−0.817348	+	1.41569i	−0.907629	+	0.419773i	$0.862110\pi$
$480$		0			0
$481$	−9.00000	−	15.5885i	−0.410365	−	0.710772i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	20.1246	−	20.1246i	0.913812	−	0.913812i
$486$		0			0
$487$	−21.0000	+	21.0000i	−0.951601	+	0.951601i	−0.998882	−	0.0472808i	$-0.984944\pi$
$487$	−21.0000	+	21.0000i	−0.951601	+	0.951601i	0.0472808	+	0.998882i	$0.484944\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−19.3649	+	11.1803i	−0.873926	+	0.504562i	−0.868651	−	0.495424i	$-0.835013\pi$
$491$	−19.3649	+	11.1803i	−0.873926	+	0.504562i	−0.00527540	+	0.999986i	$0.501679\pi$
$492$		0			0
$493$	13.6603	−	3.66025i	0.615227	−	0.164850i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	6.10905	−	1.63692i	0.274028	−	0.0734257i
$498$		0			0
$499$	−32.9090	+	19.0000i	−1.47321	+	0.850557i	−0.999545	−	0.0301498i	$-0.990402\pi$
$499$	−32.9090	+	19.0000i	−1.47321	+	0.850557i	−0.473662	+	0.880707i	$0.657068\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	6.70820	−	6.70820i	0.299104	−	0.299104i	−0.541559	−	0.840663i	$-0.682166\pi$
$503$	6.70820	−	6.70820i	0.299104	−	0.299104i	0.840663	+	0.541559i	$0.182166\pi$
$504$		0			0
$505$		−	20.0000i		−	0.889988i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	2.23607	+	3.87298i	0.0991120	+	0.171667i	0.911317	−	0.411705i	$-0.135066\pi$
$509$	2.23607	+	3.87298i	0.0991120	+	0.171667i	−0.812205	+	0.583372i	$0.801733\pi$
$510$		0			0
$511$	1.00000	−	1.73205i	0.0442374	−	0.0766214i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−3.05453	+	0.818458i	−0.134598	+	0.0360656i
$516$		0			0
$517$	40.9808	+	10.9808i	1.80233	+	0.482933i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	−3.00000	−	3.00000i	−0.131181	−	0.131181i	0.638468	−	0.769649i	$-0.279569\pi$
$523$	−3.00000	−	3.00000i	−0.131181	−	0.131181i	−0.769649	+	0.638468i	$0.779569\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	3.27383	−	12.2181i	0.142610	−	0.532229i
$528$		0			0
$529$	−11.2583	−	6.50000i	−0.489493	−	0.282609i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	9.82149	+	36.6543i	0.425416	+	1.58767i
$534$		0			0
$535$	1.83013	−	6.83013i	0.0791233	−	0.295292i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	22.3607			0.963143
$540$		0			0
$541$	−26.0000			−1.11783			−0.558914	−	0.829226i	$-0.688782\pi$
$541$	−26.0000			−1.11783			−0.558914	+	0.829226i	$0.688782\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−15.4919	+	8.94427i	−0.663602	+	0.383131i
$546$		0			0
$547$	9.88269	+	36.8827i	0.422553	+	1.57699i	0.769209	+	0.638997i	$0.220651\pi$
$547$	9.88269	+	36.8827i	0.422553	+	1.57699i	−0.346656	+	0.937992i	$0.612683\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	7.74597	+	4.47214i	0.329989	+	0.190519i
$552$		0			0
$553$	−2.19615	+	8.19615i	−0.0933899	+	0.348536i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−6.70820	−	6.70820i	−0.284236	−	0.284236i	0.550560	−	0.834796i	$-0.314414\pi$
$557$	−6.70820	−	6.70820i	−0.284236	−	0.284236i	−0.834796	+	0.550560i	$0.814414\pi$
$558$		0			0
$559$			18.0000i			0.761319i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−9.16358	−	2.45537i	−0.386199	−	0.103482i	0.0604952	−	0.998168i	$-0.480732\pi$
$563$	−9.16358	−	2.45537i	−0.386199	−	0.103482i	−0.446694	+	0.894687i	$0.647399\pi$
$564$		0			0
$565$	1.83013	+	6.83013i	0.0769940	+	0.287346i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	11.1803	−	19.3649i	0.468704	−	0.811820i	−0.530656	−	0.847587i	$-0.678054\pi$
$569$	11.1803	−	19.3649i	0.468704	−	0.811820i	0.999360	+	0.0357678i	$0.0113877\pi$
$570$		0			0
$571$	8.00000	+	13.8564i	0.334790	+	0.579873i	0.983444	−	0.181210i	$-0.0580014\pi$
$571$	8.00000	+	13.8564i	0.334790	+	0.579873i	−0.648655	+	0.761083i	$0.724668\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−11.1803	+	11.1803i	−0.466252	+	0.466252i
$576$		0			0
$577$	−11.0000	+	11.0000i	−0.457936	+	0.457936i	−0.897977	−	0.440041i	$-0.854964\pi$
$577$	−11.0000	+	11.0000i	−0.457936	+	0.457936i	0.440041	+	0.897977i	$0.354964\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	11.6190	−	6.70820i	0.482035	−	0.278303i
$582$		0			0
$583$	13.6603	−	3.66025i	0.565750	−	0.151592i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	9.16358	−	2.45537i	0.378221	−	0.101344i	−0.0646996	−	0.997905i	$-0.520609\pi$
$587$	9.16358	−	2.45537i	0.378221	−	0.101344i	0.442921	+	0.896561i	$0.353942\pi$
$588$		0			0
$589$	6.92820	−	4.00000i	0.285472	−	0.164817i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	11.1803	−	11.1803i	0.459122	−	0.459122i	−0.439246	−	0.898367i	$-0.644754\pi$
$593$	11.1803	−	11.1803i	0.459122	−	0.459122i	0.898367	+	0.439246i	$0.144754\pi$
$594$		0			0
$595$	−10.0000			−0.409960
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−17.8885	−	30.9839i	−0.730906	−	1.26597i	−0.956496	−	0.291744i	$-0.905764\pi$
$599$	−17.8885	−	30.9839i	−0.730906	−	1.26597i	0.225590	−	0.974222i	$-0.427569\pi$
$600$		0			0
$601$	−7.00000	+	12.1244i	−0.285536	+	0.494563i	−0.972739	−	0.231903i	$-0.925505\pi$
$601$	−7.00000	+	12.1244i	−0.285536	+	0.494563i	0.687203	+	0.726465i	$0.258838\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	10.0623	−	17.4284i	0.409091	−	0.708566i
$606$		0			0
$607$	−36.8827	−	9.88269i	−1.49702	−	0.401126i	−0.584921	−	0.811091i	$-0.698874\pi$
$607$	−36.8827	−	9.88269i	−1.49702	−	0.401126i	−0.912101	+	0.409965i	$0.865541\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			40.2492i			1.62831i
$612$		0			0
$613$	21.0000	+	21.0000i	0.848182	+	0.848182i	0.989906	−	0.141724i	$-0.0452646\pi$
$613$	21.0000	+	21.0000i	0.848182	+	0.848182i	−0.141724	+	0.989906i	$0.545265\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	7.36612	−	27.4907i	0.296549	−	1.10674i	−0.643431	−	0.765504i	$-0.722490\pi$
$617$	7.36612	−	27.4907i	0.296549	−	1.10674i	0.939980	−	0.341231i	$-0.110844\pi$
$618$		0			0
$619$	−36.3731	−	21.0000i	−1.46196	−	0.844061i	−0.462855	−	0.886434i	$-0.653175\pi$
$619$	−36.3731	−	21.0000i	−1.46196	−	0.844061i	−0.999102	+	0.0423727i	$0.986508\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1.63692	−	6.10905i	−0.0655816	−	0.244754i
$624$		0			0
$625$	−12.5000	−	21.6506i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−13.4164			−0.534947
$630$		0			0
$631$	4.00000			0.159237			0.0796187	−	0.996825i	$-0.474630\pi$
$631$	4.00000			0.159237			0.0796187	+	0.996825i	$0.474630\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	39.7088	+	10.6399i	1.57580	+	0.422234i
$636$		0			0
$637$	5.49038	+	20.4904i	0.217537	+	0.811858i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	7.74597	+	4.47214i	0.305947	+	0.176639i	0.645112	−	0.764088i	$-0.276811\pi$
$641$	7.74597	+	4.47214i	0.305947	+	0.176639i	−0.339164	+	0.940727i	$0.610144\pi$
$642$		0			0
$643$	7.68653	−	28.6865i	0.303127	−	1.13129i	−0.631418	−	0.775443i	$-0.717527\pi$
$643$	7.68653	−	28.6865i	0.303127	−	1.13129i	0.934545	−	0.355844i	$-0.115807\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−15.6525	−	15.6525i	−0.615362	−	0.615362i	0.328976	−	0.944338i	$-0.393296\pi$
$647$	−15.6525	−	15.6525i	−0.615362	−	0.615362i	−0.944338	+	0.328976i	$0.893296\pi$
$648$		0			0
$649$		−	40.0000i		−	1.57014i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−27.4907	−	7.36612i	−1.07580	−	0.288259i	−0.322923	−	0.946425i	$-0.604666\pi$
$653$	−27.4907	−	7.36612i	−1.07580	−	0.288259i	−0.752872	+	0.658167i	$0.771332\pi$
$654$		0			0
$655$	8.66025	+	5.00000i	0.338384	+	0.195366i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	4.47214	−	7.74597i	0.174210	−	0.301740i	−0.765678	−	0.643224i	$-0.777596\pi$
$659$	4.47214	−	7.74597i	0.174210	−	0.301740i	0.939887	+	0.341484i	$0.110930\pi$
$660$		0			0
$661$	−1.00000	−	1.73205i	−0.0388955	−	0.0673690i	0.845922	−	0.533306i	$-0.179051\pi$
$661$	−1.00000	−	1.73205i	−0.0388955	−	0.0673690i	−0.884818	+	0.465937i	$0.845717\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−4.47214	−	4.47214i	−0.173422	−	0.173422i
$666$		0			0
$667$	10.0000	−	10.0000i	0.387202	−	0.387202i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	23.2379	−	13.4164i	0.897089	−	0.517935i
$672$		0			0
$673$	−1.36603	+	0.366025i	−0.0526564	+	0.0141092i	−0.285051	−	0.958512i	$-0.592011\pi$
$673$	−1.36603	+	0.366025i	−0.0526564	+	0.0141092i	0.232395	+	0.972622i	$0.425344\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	33.5998	−	9.00303i	1.29134	−	0.346015i	0.453174	−	0.891422i	$-0.350292\pi$
$677$	33.5998	−	9.00303i	1.29134	−	0.346015i	0.838171	+	0.545408i	$0.183625\pi$
$678$		0			0
$679$	15.5885	−	9.00000i	0.598230	−	0.345388i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−20.1246	+	20.1246i	−0.770047	+	0.770047i	−0.978114	−	0.208068i	$-0.933283\pi$
$683$	−20.1246	+	20.1246i	−0.770047	+	0.770047i	0.208068	+	0.978114i	$0.433283\pi$
$684$		0			0
$685$	−15.0000	−	15.0000i	−0.573121	−	0.573121i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	6.70820	+	11.6190i	0.255562	+	0.442647i
$690$		0			0
$691$	24.0000	−	41.5692i	0.913003	−	1.58137i	0.103204	−	0.994660i	$-0.467091\pi$
$691$	24.0000	−	41.5692i	0.913003	−	1.58137i	0.809799	−	0.586707i	$-0.199576\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	27.1109	+	15.6525i	1.02837	+	0.593732i
$696$		0			0
$697$	27.3205	+	7.32051i	1.03484	+	0.277284i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			35.7771i			1.35128i	0.737231	+	0.675641i	$0.236133\pi$
$701$			35.7771i			1.35128i	−0.737231	+	0.675641i	$0.763867\pi$
$702$		0			0
$703$	−6.00000	−	6.00000i	−0.226294	−	0.226294i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	3.27383	−	12.2181i	0.123125	−	0.459509i
$708$		0			0
$709$	24.2487	+	14.0000i	0.910679	+	0.525781i	0.880650	−	0.473768i	$-0.157106\pi$
$709$	24.2487	+	14.0000i	0.910679	+	0.525781i	0.0300298	+	0.999549i	$0.490440\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−3.27383	−	12.2181i	−0.122606	−	0.457572i
$714$		0			0
$715$	40.9808	+	10.9808i	1.53259	+	0.410657i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	35.7771			1.33426			0.667130	−	0.744941i	$-0.267522\pi$
$719$	35.7771			1.33426			0.667130	+	0.744941i	$0.267522\pi$
$720$		0			0
$721$	−2.00000			−0.0744839
$722$		0			0
$723$		0			0
$724$		0			0
$725$	11.1803	+	19.3649i	0.415227	+	0.719195i
$726$		0			0
$727$	−7.68653	−	28.6865i	−0.285078	−	1.06392i	−0.948783	−	0.315930i	$-0.897684\pi$
$727$	−7.68653	−	28.6865i	−0.285078	−	1.06392i	0.663705	−	0.747995i	$-0.268983\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	11.6190	+	6.70820i	0.429742	+	0.248112i
$732$		0			0
$733$	0.366025	−	1.36603i	0.0135195	−	0.0504553i	−0.958837	−	0.283958i	$-0.908352\pi$
$733$	0.366025	−	1.36603i	0.0135195	−	0.0504553i	0.972356	+	0.233503i	$0.0750189\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	4.47214	+	4.47214i	0.164733	+	0.164733i
$738$		0			0
$739$		−	14.0000i		−	0.514998i	−0.966279	−	0.257499i	$-0.917102\pi$
$739$		−	14.0000i		−	0.514998i	0.966279	−	0.257499i	$-0.0828985\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	15.2726	+	4.09229i	0.560298	+	0.150132i	0.527843	−	0.849342i	$-0.323001\pi$
$743$	15.2726	+	4.09229i	0.560298	+	0.150132i	0.0324550	+	0.999473i	$0.489667\pi$
$744$		0			0
$745$	15.0000	−	25.9808i	0.549557	−	0.951861i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	2.23607	−	3.87298i	0.0817041	−	0.141516i
$750$		0			0
$751$	−6.00000	−	10.3923i	−0.218943	−	0.379221i	0.735542	−	0.677479i	$-0.236928\pi$
$751$	−6.00000	−	10.3923i	−0.218943	−	0.379221i	−0.954485	+	0.298259i	$0.903594\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	8.94427			0.325515
$756$		0			0
$757$	9.00000	−	9.00000i	0.327111	−	0.327111i	−0.524376	−	0.851487i	$-0.675701\pi$
$757$	9.00000	−	9.00000i	0.327111	−	0.327111i	0.851487	+	0.524376i	$0.175701\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−15.4919	+	8.94427i	−0.561582	+	0.324230i	−0.753780	−	0.657127i	$-0.771772\pi$
$761$	−15.4919	+	8.94427i	−0.561582	+	0.324230i	0.192198	+	0.981356i	$0.438438\pi$
$762$		0			0
$763$	−10.9282	+	2.92820i	−0.395628	+	0.106008i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	36.6543	−	9.82149i	1.32351	−	0.354634i
$768$		0			0
$769$	−6.92820	+	4.00000i	−0.249837	+	0.144244i	−0.619690	−	0.784847i	$-0.712742\pi$
$769$	−6.92820	+	4.00000i	−0.249837	+	0.144244i	0.369852	+	0.929091i	$0.379408\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−6.70820	+	6.70820i	−0.241277	+	0.241277i	−0.817378	−	0.576101i	$-0.804573\pi$
$773$	−6.70820	+	6.70820i	−0.241277	+	0.241277i	0.576101	+	0.817378i	$0.304573\pi$
$774$		0			0
$775$	20.0000			0.718421
$776$		0			0
$777$		0			0
$778$		0			0
$779$	8.94427	+	15.4919i	0.320462	+	0.555056i
$780$		0			0
$781$	−10.0000	+	17.3205i	−0.357828	+	0.619777i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	7.36612	+	27.4907i	0.262908	+	0.981186i
$786$		0			0
$787$	−25.9545	−	6.95448i	−0.925177	−	0.247901i	−0.235380	−	0.971903i	$-0.575634\pi$
$787$	−25.9545	−	6.95448i	−0.925177	−	0.247901i	−0.689797	+	0.724003i	$0.742300\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			4.47214i			0.159011i
$792$		0			0
$793$	18.0000	+	18.0000i	0.639199	+	0.639199i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	7.36612	−	27.4907i	0.260921	−	0.973772i	−0.703778	−	0.710420i	$-0.748505\pi$
$797$	7.36612	−	27.4907i	0.260921	−	0.973772i	0.964700	−	0.263352i	$-0.0848281\pi$
$798$		0			0
$799$	25.9808	+	15.0000i	0.919133	+	0.530662i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1.63692	+	6.10905i	0.0577655	+	0.215584i
$804$		0			0
$805$	−8.66025	+	5.00000i	−0.305234	+	0.176227i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	4.47214			0.157232			0.0786160	−	0.996905i	$-0.474950\pi$
$809$	4.47214			0.157232			0.0786160	+	0.996905i	$0.474950\pi$
$810$		0			0
$811$	−16.0000			−0.561836			−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000			−0.561836			−0.280918	+	0.959732i	$0.590639\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	13.9138	−	51.9269i	0.487378	−	1.81892i
$816$		0			0
$817$	2.19615	+	8.19615i	0.0768336	+	0.286747i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	15.4919	+	8.94427i	0.540672	+	0.312157i	0.745351	−	0.666672i	$-0.232282\pi$
$821$	15.4919	+	8.94427i	0.540672	+	0.312157i	−0.204679	+	0.978829i	$0.565615\pi$
$822$		0			0
$823$	6.22243	−	23.2224i	0.216900	−	0.809483i	−0.768589	−	0.639743i	$-0.779041\pi$
$823$	6.22243	−	23.2224i	0.216900	−	0.809483i	0.985489	−	0.169740i	$-0.0542927\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	20.1246	+	20.1246i	0.699801	+	0.699801i	0.964368	−	0.264566i	$-0.0852288\pi$
$827$	20.1246	+	20.1246i	0.699801	+	0.699801i	−0.264566	+	0.964368i	$0.585229\pi$
$828$		0			0
$829$			16.0000i			0.555703i	0.960624	+	0.277851i	$0.0896223\pi$
$829$			16.0000i			0.555703i	−0.960624	+	0.277851i	$0.910378\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	15.2726	+	4.09229i	0.529165	+	0.141789i
$834$		0			0
$835$	−34.1506	+	9.15064i	−1.18183	+	0.316671i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	17.8885	−	30.9839i	0.617581	−	1.06968i	−0.372345	−	0.928095i	$-0.621446\pi$
$839$	17.8885	−	30.9839i	0.617581	−	1.06968i	0.989926	−	0.141587i	$-0.0452206\pi$
$840$		0			0
$841$	4.50000	+	7.79423i	0.155172	+	0.268767i
$842$		0			0
$843$		0			0
$844$		0			0
$845$			11.1803i			0.384615i
$846$		0			0
$847$	9.00000	−	9.00000i	0.309244	−	0.309244i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−11.6190	+	6.70820i	−0.398292	+	0.229954i
$852$		0			0
$853$	−50.5429	+	13.5429i	−1.73056	+	0.463701i	−0.980311	−	0.197458i	$-0.936732\pi$
$853$	−50.5429	+	13.5429i	−1.73056	+	0.463701i	−0.750246	+	0.661159i	$0.770065\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	9.16358	−	2.45537i	0.313022	−	0.0838739i	−0.0988880	−	0.995099i	$-0.531529\pi$
$857$	9.16358	−	2.45537i	0.313022	−	0.0838739i	0.411910	+	0.911225i	$0.364862\pi$
$858$		0			0
$859$	−12.1244	+	7.00000i	−0.413678	+	0.238837i	−0.692369	−	0.721544i	$-0.743433\pi$
$859$	−12.1244	+	7.00000i	−0.413678	+	0.238837i	0.278691	+	0.960381i	$0.410099\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	24.5967	−	24.5967i	0.837283	−	0.837283i	−0.151217	−	0.988501i	$-0.548319\pi$
$863$	24.5967	−	24.5967i	0.837283	−	0.837283i	0.988501	+	0.151217i	$0.0483194\pi$
$864$		0			0
$865$	−35.0000	+	35.0000i	−1.19004	+	1.19004i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−13.4164	−	23.2379i	−0.455120	−	0.788292i
$870$		0			0
$871$	−3.00000	+	5.19615i	−0.101651	+	0.176065i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−4.09229	−	15.2726i	−0.138345	−	0.516309i
$876$		0			0
$877$	−12.2942	−	3.29423i	−0.415147	−	0.111238i	0.0451990	−	0.998978i	$-0.485608\pi$
$877$	−12.2942	−	3.29423i	−0.415147	−	0.111238i	−0.460346	+	0.887740i	$0.652274\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	44.7214i		−	1.50670i	−0.657619	−	0.753350i	$-0.728436\pi$
$881$		−	44.7214i		−	1.50670i	0.657619	−	0.753350i	$-0.271564\pi$
$882$		0			0
$883$	−23.0000	−	23.0000i	−0.774012	−	0.774012i	0.204794	−	0.978805i	$-0.434348\pi$
$883$	−23.0000	−	23.0000i	−0.774012	−	0.774012i	−0.978805	+	0.204794i	$0.934348\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	7.36612	−	27.4907i	0.247330	−	0.923048i	−0.724868	−	0.688888i	$-0.758099\pi$
$887$	7.36612	−	27.4907i	0.247330	−	0.923048i	0.972198	−	0.234160i	$-0.0752341\pi$
$888$		0			0
$889$	22.5167	+	13.0000i	0.755185	+	0.436006i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	4.91075	+	18.3272i	0.164332	+	0.613295i
$894$		0			0
$895$	10.0000	+	17.3205i	0.334263	+	0.578961i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−17.8885			−0.596616
$900$		0			0
$901$	10.0000			0.333148
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−15.6525	−	27.1109i	−0.520306	−	0.901196i
$906$		0			0
$907$	2.56218	+	9.56218i	0.0850757	+	0.317507i	0.995329	−	0.0965460i	$-0.0307795\pi$
$907$	2.56218	+	9.56218i	0.0850757	+	0.317507i	−0.910253	+	0.414053i	$0.864113\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	42.6028	+	24.5967i	1.41149	+	0.814927i	0.995529	−	0.0944540i	$-0.0301105\pi$
$911$	42.6028	+	24.5967i	1.41149	+	0.814927i	0.415965	+	0.909381i	$0.363444\pi$
$912$		0			0
$913$	−10.9808	+	40.9808i	−0.363410	+	1.35627i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	4.47214	+	4.47214i	0.147683	+	0.147683i
$918$		0			0
$919$			22.0000i			0.725713i	0.931845	+	0.362857i	$0.118198\pi$
$919$			22.0000i			0.725713i	−0.931845	+	0.362857i	$0.881802\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−18.3272	−	4.91075i	−0.603246	−	0.161639i
$924$		0			0
$925$	−5.49038	−	20.4904i	−0.180523	−	0.673720i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	2.23607	−	3.87298i	0.0733630	−	0.127068i	−0.827010	−	0.562187i	$-0.809960\pi$
$929$	2.23607	−	3.87298i	0.0733630	−	0.127068i	0.900373	+	0.435118i	$0.143293\pi$
$930$		0			0
$931$	5.00000	+	8.66025i	0.163868	+	0.283828i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	22.3607	−	22.3607i	0.731272	−	0.731272i
$936$		0			0
$937$	−11.0000	+	11.0000i	−0.359354	+	0.359354i	−0.863575	−	0.504221i	$-0.831780\pi$
$937$	−11.0000	+	11.0000i	−0.359354	+	0.359354i	0.504221	+	0.863575i	$0.331780\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	23.2379	−	13.4164i	0.757534	−	0.437362i	−0.0708757	−	0.997485i	$-0.522579\pi$
$941$	23.2379	−	13.4164i	0.757534	−	0.437362i	0.828410	+	0.560123i	$0.189246\pi$
$942$		0			0
$943$	27.3205	−	7.32051i	0.889678	−	0.238389i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−15.2726	+	4.09229i	−0.496294	+	0.132982i	−0.498279	−	0.867017i	$-0.666035\pi$
$947$	−15.2726	+	4.09229i	−0.496294	+	0.132982i	0.00198565	+	0.999998i	$0.499368\pi$
$948$		0			0
$949$	−5.19615	+	3.00000i	−0.168674	+	0.0973841i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−33.5410	+	33.5410i	−1.08650	+	1.08650i	−0.0906141	+	0.995886i	$0.528883\pi$
$953$	−33.5410	+	33.5410i	−1.08650	+	1.08650i	−0.995886	+	0.0906141i	$0.971117\pi$
$954$		0			0
$955$			10.0000i			0.323592i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−6.70820	−	11.6190i	−0.216619	−	0.375195i
$960$		0			0
$961$	7.50000	−	12.9904i	0.241935	−	0.419045i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	9.16358	−	2.45537i	0.294986	−	0.0790413i
$966$		0			0
$967$	−9.56218	−	2.56218i	−0.307499	−	0.0823941i	0.101770	−	0.994808i	$-0.467549\pi$
$967$	−9.56218	−	2.56218i	−0.307499	−	0.0823941i	−0.409269	+	0.912414i	$0.634216\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	22.3607i		−	0.717588i	−0.933417	−	0.358794i	$-0.883188\pi$
$971$		−	22.3607i		−	0.717588i	0.933417	−	0.358794i	$-0.116812\pi$
$972$		0			0
$973$	14.0000	+	14.0000i	0.448819	+	0.448819i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−2.45537	+	9.16358i	−0.0785543	+	0.293169i	−0.994016	−	0.109237i	$-0.965159\pi$
$977$	−2.45537	+	9.16358i	−0.0785543	+	0.293169i	0.915461	+	0.402406i	$0.131826\pi$
$978$		0			0
$979$	17.3205	+	10.0000i	0.553566	+	0.319601i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−4.09229	−	15.2726i	−0.130524	−	0.487121i	0.869452	−	0.494017i	$-0.164472\pi$
$983$	−4.09229	−	15.2726i	−0.130524	−	0.487121i	−0.999976	+	0.00689536i	$0.997805\pi$
$984$		0			0
$985$	−5.49038	+	20.4904i	−0.174938	+	0.652878i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	13.4164			0.426617
$990$		0			0
$991$	4.00000			0.127064			0.0635321	−	0.997980i	$-0.479763\pi$
$991$	4.00000			0.127064			0.0635321	+	0.997980i	$0.479763\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−34.8569	+	20.1246i	−1.10504	+	0.637993i
$996$		0			0
$997$	−8.41858	−	31.4186i	−0.266619	−	0.995037i	−0.961252	−	0.275672i	$-0.911100\pi$
$997$	−8.41858	−	31.4186i	−0.266619	−	0.995037i	0.694633	−	0.719365i	$-0.255567\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.2.x.b.593.1		8
3.2	odd	2	inner	1620.2.x.b.593.2		8
5.2	odd	4	inner	1620.2.x.b.917.2		8
9.2	odd	6		60.2.i.a.53.2	yes	4
9.4	even	3	inner	1620.2.x.b.53.1		8
9.5	odd	6	inner	1620.2.x.b.53.2		8
9.7	even	3		60.2.i.a.53.1	yes	4
15.2	even	4	inner	1620.2.x.b.917.1		8
36.7	odd	6		240.2.v.b.113.2		4
36.11	even	6		240.2.v.b.113.1		4
45.2	even	12		60.2.i.a.17.1	✓	4
45.7	odd	12		60.2.i.a.17.2	yes	4
45.22	odd	12	inner	1620.2.x.b.377.2		8
45.29	odd	6		300.2.i.a.293.1		4
45.32	even	12	inner	1620.2.x.b.377.1		8
45.34	even	6		300.2.i.a.293.2		4
45.38	even	12		300.2.i.a.257.2		4
45.43	odd	12		300.2.i.a.257.1		4
72.11	even	6		960.2.v.h.833.2		4
72.29	odd	6		960.2.v.e.833.1		4
72.43	odd	6		960.2.v.h.833.1		4
72.61	even	6		960.2.v.e.833.2		4
180.7	even	12		240.2.v.b.17.1		4
180.43	even	12		1200.2.v.i.257.2		4
180.47	odd	12		240.2.v.b.17.2		4
180.79	odd	6		1200.2.v.i.593.1		4
180.83	odd	12		1200.2.v.i.257.1		4
180.119	even	6		1200.2.v.i.593.2		4
360.187	even	12		960.2.v.h.257.2		4
360.227	odd	12		960.2.v.h.257.1		4
360.277	odd	12		960.2.v.e.257.1		4
360.317	even	12		960.2.v.e.257.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.2.i.a.17.1	✓	4	45.2	even	12
60.2.i.a.17.2	yes	4	45.7	odd	12
60.2.i.a.53.1	yes	4	9.7	even	3
60.2.i.a.53.2	yes	4	9.2	odd	6
240.2.v.b.17.1		4	180.7	even	12
240.2.v.b.17.2		4	180.47	odd	12
240.2.v.b.113.1		4	36.11	even	6
240.2.v.b.113.2		4	36.7	odd	6
300.2.i.a.257.1		4	45.43	odd	12
300.2.i.a.257.2		4	45.38	even	12
300.2.i.a.293.1		4	45.29	odd	6
300.2.i.a.293.2		4	45.34	even	6
960.2.v.e.257.1		4	360.277	odd	12
960.2.v.e.257.2		4	360.317	even	12
960.2.v.e.833.1		4	72.29	odd	6
960.2.v.e.833.2		4	72.61	even	6
960.2.v.h.257.1		4	360.227	odd	12
960.2.v.h.257.2		4	360.187	even	12
960.2.v.h.833.1		4	72.43	odd	6
960.2.v.h.833.2		4	72.11	even	6
1200.2.v.i.257.1		4	180.83	odd	12
1200.2.v.i.257.2		4	180.43	even	12
1200.2.v.i.593.1		4	180.79	odd	6
1200.2.v.i.593.2		4	180.119	even	6
1620.2.x.b.53.1		8	9.4	even	3	inner
1620.2.x.b.53.2		8	9.5	odd	6	inner
1620.2.x.b.377.1		8	45.32	even	12	inner
1620.2.x.b.377.2		8	45.22	odd	12	inner
1620.2.x.b.593.1		8	1.1	even	1	trivial
1620.2.x.b.593.2		8	3.2	odd	2	inner
1620.2.x.b.917.1		8	15.2	even	4	inner
1620.2.x.b.917.2		8	5.2	odd	4	inner

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)

\(f(q)\)	\(=\)	\(q+(-1.11803 - 1.93649i) q^{5} +(-0.366025 - 1.36603i) q^{7} +O(q^{10})\)	\(q+(-1.11803 - 1.93649i) q^{5} +(-0.366025 - 1.36603i) q^{7} +(3.87298 + 2.23607i) q^{11} +(-1.09808 + 4.09808i) q^{13} +(2.23607 + 2.23607i) q^{17} +2.00000i q^{19} +(3.05453 + 0.818458i) q^{23} +(-2.50000 + 4.33013i) q^{25} +(2.23607 - 3.87298i) q^{29} +(-2.00000 - 3.46410i) q^{31} +(-2.23607 + 2.23607i) q^{35} +(-3.00000 + 3.00000i) q^{37} +(7.74597 - 4.47214i) q^{41} +(4.09808 - 1.09808i) q^{43} +(9.16358 - 2.45537i) q^{47} +(4.33013 - 2.50000i) q^{49} +(2.23607 - 2.23607i) q^{53} -10.0000i q^{55} +(-4.47214 - 7.74597i) q^{59} +(3.00000 - 5.19615i) q^{61} +(9.16358 - 2.45537i) q^{65} +(1.36603 + 0.366025i) q^{67} +4.47214i q^{71} +(1.00000 + 1.00000i) q^{73} +(1.63692 - 6.10905i) q^{77} +(-5.19615 - 3.00000i) q^{79} +(2.45537 + 9.16358i) q^{83} +(1.83013 - 6.83013i) q^{85} +4.47214 q^{89} +6.00000 q^{91} +(3.87298 - 2.23607i) q^{95} +(3.29423 + 12.2942i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{7}+O(q^{10}) \) 8 * q + 4 * q^7	\( 8 q + 4 q^{7} + 12 q^{13} - 20 q^{25} - 16 q^{31} - 24 q^{37} + 12 q^{43} + 24 q^{61} + 4 q^{67} + 8 q^{73} - 20 q^{85} + 48 q^{91} - 36 q^{97}+O(q^{100}) \) 8 * q + 4 * q^7 + 12 * q^13 - 20 * q^25 - 16 * q^31 - 24 * q^37 + 12 * q^43 + 24 * q^61 + 4 * q^67 + 8 * q^73 - 20 * q^85 + 48 * q^91 - 36 * q^97

Introduction

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