LMFDB - Embedded newform 2160.2.q.j.721.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,2,Mod(721,2160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("2160.721");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$17.2476868366$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.954288.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 $ x^6 - x^5 - 2x^4 + 3x^3 - 6x^2 - 9x + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			721.2
Root			$0.403374 - 1.68443i$ of defining polynomial
Character	$\chi$	$=$	2160.721
Dual form			2160.2.q.j.1441.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 + 0.866025i) q^{5} +(-0.596626 + 1.03339i) q^{7} +(1.66044 - 2.87597i) q^{11} +(-0.853695 - 1.47864i) q^{13} +6.34916 q^{17} -1.32088 q^{19} +(-3.43165 - 5.94379i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(1.01414 - 1.75654i) q^{29} +(-1.33956 - 2.32018i) q^{31} -1.19325 q^{35} +3.32088 q^{37} +(1.16044 + 2.00994i) q^{41} +(-3.17458 + 5.49853i) q^{43} +(6.38470 - 11.0586i) q^{47} +(2.78807 + 4.82909i) q^{49} -1.02827 q^{53} +3.32088 q^{55} +(5.83502 + 10.1066i) q^{59} +(4.86783 - 8.43133i) q^{61} +(0.853695 - 1.47864i) q^{65} +(-5.28534 - 9.15448i) q^{67} -1.06562 q^{71} +14.0565 q^{73} +(1.98133 + 3.43176i) q^{77} +(0.707389 - 1.22523i) q^{79} +(5.91751 - 10.2494i) q^{83} +(3.17458 + 5.49853i) q^{85} +11.0000 q^{89} +2.03735 q^{91} +(-0.660442 - 1.14392i) q^{95} +(-8.12763 + 14.0775i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{5} - 5 q^{7} + 2 q^{11} - 4 q^{17} + 8 q^{19} + 7 q^{23} - 3 q^{25} - 7 q^{29} - 16 q^{31} - 10 q^{35} + 4 q^{37} - q^{41} + 2 q^{43} + 13 q^{47} - 10 q^{49} + 20 q^{53} + 4 q^{55} + 6 q^{59}+ \cdots - 30 q^{97}+O(q^{100}) $ 6 * q + 3 * q^5 - 5 * q^7 + 2 * q^11 - 4 * q^17 + 8 * q^19 + 7 * q^23 - 3 * q^25 - 7 * q^29 - 16 * q^31 - 10 * q^35 + 4 * q^37 - q^41 + 2 * q^43 + 13 * q^47 - 10 * q^49 + 20 * q^53 + 4 * q^55 + 6 * q^59 + 11 * q^61 + q^67 - 28 * q^71 + 32 * q^73 - 12 * q^77 - 6 * q^79 + 21 * q^83 - 2 * q^85 + 66 * q^89 + 60 * q^91 + 4 * q^95 - 30 * q^97

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i
$6$		0			0
$7$	−0.596626	+	1.03339i	−0.225504	+	0.390584i	−0.956470	−	0.291829i	$-0.905736\pi$
$7$	−0.596626	+	1.03339i	−0.225504	+	0.390584i	0.730967	+	0.682413i	$0.239069\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.66044	−	2.87597i	0.500642	−	0.867138i	−0.499358	−	0.866396i	$-0.666431\pi$
$11$	1.66044	−	2.87597i	0.500642	−	0.867138i	1.00000		0.000741679i	$-0.000236084\pi$
$12$		0			0
$13$	−0.853695	−	1.47864i	−0.236772	−	0.410102i	0.723014	−	0.690833i	$-0.242756\pi$
$13$	−0.853695	−	1.47864i	−0.236772	−	0.410102i	−0.959786	+	0.280732i	$0.909423\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	6.34916			1.53990			0.769949	−	0.638106i	$-0.220282\pi$
$17$	6.34916			1.53990			0.769949	+	0.638106i	$0.220282\pi$
$18$		0			0
$19$	−1.32088			−0.303032			−0.151516	−	0.988455i	$-0.548415\pi$
$19$	−1.32088			−0.303032			−0.151516	+	0.988455i	$0.548415\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.43165	−	5.94379i	−0.715548	−	1.23937i	−0.962748	−	0.270401i	$-0.912844\pi$
$23$	−3.43165	−	5.94379i	−0.715548	−	1.23937i	0.247200	−	0.968965i	$-0.420490\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.01414	−	1.75654i	0.188321	−	0.326181i	−0.756370	−	0.654144i	$-0.773029\pi$
$29$	1.01414	−	1.75654i	0.188321	−	0.326181i	0.944690	+	0.327963i	$0.106362\pi$
$30$		0			0
$31$	−1.33956	−	2.32018i	−0.240592	−	0.416717i	0.720291	−	0.693672i	$-0.244008\pi$
$31$	−1.33956	−	2.32018i	−0.240592	−	0.416717i	−0.960883	+	0.276955i	$0.910675\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.19325			−0.201696
$36$		0			0
$37$	3.32088			0.545950			0.272975	−	0.962021i	$-0.411992\pi$
$37$	3.32088			0.545950			0.272975	+	0.962021i	$0.411992\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.16044	+	2.00994i	0.181231	+	0.313901i	0.942300	−	0.334770i	$-0.108659\pi$
$41$	1.16044	+	2.00994i	0.181231	+	0.313901i	−0.761069	+	0.648671i	$0.775325\pi$
$42$		0			0
$43$	−3.17458	+	5.49853i	−0.484119	+	0.838518i	−0.999834	−	0.0182420i	$-0.994193\pi$
$43$	−3.17458	+	5.49853i	−0.484119	+	0.838518i	0.515715	+	0.856760i	$0.327526\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	6.38470	−	11.0586i	0.931304	−	1.61307i	0.150209	−	0.988654i	$-0.452005\pi$
$47$	6.38470	−	11.0586i	0.931304	−	1.61307i	0.781095	−	0.624412i	$-0.214661\pi$
$48$		0			0
$49$	2.78807	+	4.82909i	0.398296	+	0.689869i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.02827			−0.141244			−0.0706221	−	0.997503i	$-0.522498\pi$
$53$	−1.02827			−0.141244			−0.0706221	+	0.997503i	$0.522498\pi$
$54$		0			0
$55$	3.32088			0.447788
$56$		0			0
$57$		0			0
$58$		0			0
$59$	5.83502	+	10.1066i	0.759655	+	1.31576i	0.943027	+	0.332718i	$0.107966\pi$
$59$	5.83502	+	10.1066i	0.759655	+	1.31576i	−0.183371	+	0.983044i	$0.558701\pi$
$60$		0			0
$61$	4.86783	−	8.43133i	0.623262	−	1.07952i	−0.365612	−	0.930767i	$-0.619140\pi$
$61$	4.86783	−	8.43133i	0.623262	−	1.07952i	0.988874	−	0.148754i	$-0.0475263\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.853695	−	1.47864i	0.105888	−	0.183403i
$66$		0			0
$67$	−5.28534	−	9.15448i	−0.645707	−	1.11840i	−0.984138	−	0.177406i	$-0.943229\pi$
$67$	−5.28534	−	9.15448i	−0.645707	−	1.11840i	0.338430	−	0.940991i	$-0.390104\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−1.06562			−0.126466			−0.0632329	−	0.997999i	$-0.520141\pi$
$71$	−1.06562			−0.126466			−0.0632329	+	0.997999i	$0.520141\pi$
$72$		0			0
$73$	14.0565			1.64519			0.822597	−	0.568624i	$-0.192524\pi$
$73$	14.0565			1.64519			0.822597	+	0.568624i	$0.192524\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.98133	+	3.43176i	0.225793	+	0.391085i
$78$		0			0
$79$	0.707389	−	1.22523i	0.0795875	−	0.137850i	−0.823485	−	0.567339i	$-0.807973\pi$
$79$	0.707389	−	1.22523i	0.0795875	−	0.137850i	0.903072	+	0.429489i	$0.141306\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	5.91751	−	10.2494i	0.649531	−	1.12502i	−0.333704	−	0.942678i	$-0.608299\pi$
$83$	5.91751	−	10.2494i	0.649531	−	1.12502i	0.983235	−	0.182343i	$-0.0583681\pi$
$84$		0			0
$85$	3.17458	+	5.49853i	0.344331	+	0.596400i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	11.0000			1.16600			0.582999	−	0.812473i	$-0.301879\pi$
$89$	11.0000			1.16600			0.582999	+	0.812473i	$0.301879\pi$
$90$		0			0
$91$	2.03735			0.213572
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.660442	−	1.14392i	−0.0677599	−	0.117364i
$96$		0			0
$97$	−8.12763	+	14.0775i	−0.825236	+	1.42935i	0.0765028	+	0.997069i	$0.475625\pi$
$97$	−8.12763	+	14.0775i	−0.825236	+	1.42935i	−0.901739	+	0.432281i	$0.857709\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.19325	+	2.06677i	−0.118733	+	0.205652i	−0.919266	−	0.393637i	$-0.871217\pi$
$101$	−1.19325	+	2.06677i	−0.118733	+	0.205652i	0.800533	+	0.599289i	$0.204550\pi$
$102$		0			0
$103$	4.88197	+	8.45582i	0.481035	+	0.833176i	0.999763	−	0.0217626i	$-0.00692779\pi$
$103$	4.88197	+	8.45582i	0.481035	+	0.833176i	−0.518729	+	0.854939i	$0.673594\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.57976			−0.732763			−0.366381	−	0.930465i	$-0.619403\pi$
$107$	−7.57976			−0.732763			−0.366381	+	0.930465i	$0.619403\pi$
$108$		0			0
$109$	16.5761			1.58771			0.793854	−	0.608109i	$-0.208072\pi$
$109$	16.5761			1.58771			0.793854	+	0.608109i	$0.208072\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−10.5424	−	18.2600i	−0.991747	−	1.71776i	−0.606909	−	0.794771i	$-0.707591\pi$
$113$	−10.5424	−	18.2600i	−0.991747	−	1.71776i	−0.384837	−	0.922985i	$-0.625742\pi$
$114$		0			0
$115$	3.43165	−	5.94379i	0.320003	−	0.554261i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−3.78807	+	6.56114i	−0.347252	+	0.601458i
$120$		0			0
$121$	−0.0141369	−	0.0244859i	−0.00128518	−	0.00222599i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	15.4112			1.36752			0.683760	−	0.729707i	$-0.260343\pi$
$127$	15.4112			1.36752			0.683760	+	0.729707i	$0.260343\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1.67912	+	2.90831i	0.146705	+	0.254101i	0.930008	−	0.367540i	$-0.119800\pi$
$131$	1.67912	+	2.90831i	0.146705	+	0.254101i	−0.783303	+	0.621640i	$0.786467\pi$
$132$		0			0
$133$	0.788074	−	1.36498i	0.0683347	−	0.118359i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−3.80675	+	6.59348i	−0.325232	+	0.563319i	−0.981559	−	0.191158i	$-0.938776\pi$
$137$	−3.80675	+	6.59348i	−0.325232	+	0.563319i	0.656327	+	0.754477i	$0.272109\pi$
$138$		0			0
$139$	6.64177	+	11.5039i	0.563347	+	0.975746i	0.997201	+	0.0747632i	$0.0238201\pi$
$139$	6.64177	+	11.5039i	0.563347	+	0.975746i	−0.433854	+	0.900983i	$0.642847\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−5.67004			−0.474153
$144$		0			0
$145$	2.02827			0.168439
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−8.99546	−	15.5806i	−0.736937	−	1.27641i	−0.953868	−	0.300226i	$-0.902938\pi$
$149$	−8.99546	−	15.5806i	−0.736937	−	1.27641i	0.216931	−	0.976187i	$-0.430395\pi$
$150$		0			0
$151$	−4.33956	+	7.51633i	−0.353148	+	0.611671i	−0.986799	−	0.161948i	$-0.948222\pi$
$151$	−4.33956	+	7.51633i	−0.353148	+	0.611671i	0.633651	+	0.773619i	$0.281556\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.33956	−	2.32018i	0.107596	−	0.186362i
$156$		0			0
$157$	−9.44852	−	16.3653i	−0.754074	−	1.30609i	−0.945834	−	0.324652i	$-0.894753\pi$
$157$	−9.44852	−	16.3653i	−0.754074	−	1.30609i	0.191760	−	0.981442i	$-0.438581\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	8.18964			0.645434
$162$		0			0
$163$	0.990927			0.0776154			0.0388077	−	0.999247i	$-0.487644\pi$
$163$	0.990927			0.0776154			0.0388077	+	0.999247i	$0.487644\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−6.79948	−	11.7770i	−0.526160	−	0.911335i	−0.999536	−	0.0304745i	$-0.990298\pi$
$167$	−6.79948	−	11.7770i	−0.526160	−	0.911335i	0.473376	−	0.880860i	$-0.343035\pi$
$168$		0			0
$169$	5.04241	−	8.73371i	0.387878	−	0.671824i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−0.0938942	+	0.162630i	−0.00713865	+	0.0123645i	−0.869573	−	0.493805i	$-0.835606\pi$
$173$	−0.0938942	+	0.162630i	−0.00713865	+	0.0123645i	0.862434	+	0.506170i	$0.168939\pi$
$174$		0			0
$175$	−0.596626	−	1.03339i	−0.0451007	−	0.0781167i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	9.76394			0.729791			0.364895	−	0.931048i	$-0.381105\pi$
$179$	9.76394			0.729791			0.364895	+	0.931048i	$0.381105\pi$
$180$		0			0
$181$	−16.0848			−1.19558			−0.597788	−	0.801654i	$-0.703953\pi$
$181$	−16.0848			−1.19558			−0.597788	+	0.801654i	$0.703953\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.66044	+	2.87597i	0.122078	+	0.211446i
$186$		0			0
$187$	10.5424	−	18.2600i	0.770937	−	1.33530i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1.53281	−	2.65491i	0.110910	−	0.192102i	−0.805227	−	0.592966i	$-0.797957\pi$
$191$	1.53281	−	2.65491i	0.110910	−	0.192102i	0.916138	+	0.400864i	$0.131290\pi$
$192$		0			0
$193$	−0.118031	−	0.204436i	−0.00849609	−	0.0147157i	0.861746	−	0.507340i	$-0.169371\pi$
$193$	−0.118031	−	0.204436i	−0.00849609	−	0.0147157i	−0.870242	+	0.492624i	$0.836038\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1.74474			0.124307			0.0621536	−	0.998067i	$-0.480203\pi$
$197$	1.74474			0.124307			0.0621536	+	0.998067i	$0.480203\pi$
$198$		0			0
$199$	22.7922			1.61570			0.807848	−	0.589390i	$-0.200632\pi$
$199$	22.7922			1.61570			0.807848	+	0.589390i	$0.200632\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1.21012	+	2.09599i	0.0849339	+	0.147110i
$204$		0			0
$205$	−1.16044	+	2.00994i	−0.0810488	+	0.140381i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2.19325	+	3.79882i	−0.151710	+	0.262770i
$210$		0			0
$211$	−6.68872	−	11.5852i	−0.460470	−	0.797558i	0.538514	−	0.842616i	$-0.318986\pi$
$211$	−6.68872	−	11.5852i	−0.460470	−	0.797558i	−0.998984	+	0.0450587i	$0.985653\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−6.34916			−0.433009
$216$		0			0
$217$	3.19686			0.217017
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−5.42024	−	9.38814i	−0.364605	−	0.631514i
$222$		0			0
$223$	−4.74293	+	8.21500i	−0.317610	+	0.550117i	−0.979989	−	0.199052i	$-0.936214\pi$
$223$	−4.74293	+	8.21500i	−0.317610	+	0.550117i	0.662379	+	0.749169i	$0.269547\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1.75434	+	3.03860i	−0.116439	+	0.201679i	−0.918354	−	0.395759i	$-0.870481\pi$
$227$	−1.75434	+	3.03860i	−0.116439	+	0.201679i	0.801915	+	0.597438i	$0.203815\pi$
$228$		0			0
$229$	−9.36330	−	16.2177i	−0.618744	−	1.07170i	−0.989715	−	0.143052i	$-0.954308\pi$
$229$	−9.36330	−	16.2177i	−0.618744	−	1.07170i	0.370971	−	0.928644i	$-0.379025\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−2.15044			−0.140880			−0.0704401	−	0.997516i	$-0.522440\pi$
$233$	−2.15044			−0.140880			−0.0704401	+	0.997516i	$0.522440\pi$
$234$		0			0
$235$	12.7694			0.832984
$236$		0			0
$237$		0			0
$238$		0			0
$239$	12.7977	+	22.1662i	0.827813	+	1.43381i	0.899751	+	0.436405i	$0.143748\pi$
$239$	12.7977	+	22.1662i	0.827813	+	1.43381i	−0.0719377	+	0.997409i	$0.522918\pi$
$240$		0			0
$241$	−2.38197	+	4.12569i	−0.153436	+	0.265759i	−0.932488	−	0.361200i	$-0.882367\pi$
$241$	−2.38197	+	4.12569i	−0.153436	+	0.265759i	0.779052	+	0.626959i	$0.215701\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.78807	+	4.82909i	−0.178124	+	0.308519i
$246$		0			0
$247$	1.12763	+	1.95312i	0.0717495	+	0.124274i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	7.76394			0.490055			0.245028	−	0.969516i	$-0.421203\pi$
$251$	7.76394			0.490055			0.245028	+	0.969516i	$0.421203\pi$
$252$		0			0
$253$	−22.7922			−1.43293
$254$		0			0
$255$		0			0
$256$		0			0
$257$	9.64177	+	16.7000i	0.601437	+	1.04172i	0.992604	+	0.121400i	$0.0387383\pi$
$257$	9.64177	+	16.7000i	0.601437	+	1.04172i	−0.391167	+	0.920320i	$0.627928\pi$
$258$		0			0
$259$	−1.98133	+	3.43176i	−0.123114	+	0.213239i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−3.85369	+	6.67479i	−0.237629	+	0.411585i	−0.960033	−	0.279885i	$-0.909704\pi$
$263$	−3.85369	+	6.67479i	−0.237629	+	0.411585i	0.722404	+	0.691471i	$0.243037\pi$
$264$		0			0
$265$	−0.514137	−	0.890511i	−0.0315832	−	0.0547037i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−14.1222			−0.861044			−0.430522	−	0.902580i	$-0.641670\pi$
$269$	−14.1222			−0.861044			−0.430522	+	0.902580i	$0.641670\pi$
$270$		0			0
$271$	3.26434			0.198294			0.0991472	−	0.995073i	$-0.468389\pi$
$271$	3.26434			0.198294			0.0991472	+	0.995073i	$0.468389\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.66044	+	2.87597i	0.100128	+	0.173428i
$276$		0			0
$277$	−13.2311	+	22.9170i	−0.794981	+	1.37695i	0.127870	+	0.991791i	$0.459186\pi$
$277$	−13.2311	+	22.9170i	−0.794981	+	1.37695i	−0.922851	+	0.385157i	$0.874147\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	9.70285	−	16.8058i	0.578824	−	1.00255i	−0.416791	−	0.909002i	$-0.636845\pi$
$281$	9.70285	−	16.8058i	0.578824	−	1.00255i	0.995615	−	0.0935497i	$-0.0298214\pi$
$282$		0			0
$283$	−2.69598	−	4.66958i	−0.160260	−	0.277578i	0.774702	−	0.632326i	$-0.217900\pi$
$283$	−2.69598	−	4.66958i	−0.160260	−	0.277578i	−0.934962	+	0.354749i	$0.884566\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−2.76940			−0.163473
$288$		0			0
$289$	23.3118			1.37128
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−16.4581	−	28.5063i	−0.961493	−	1.66536i	−0.718755	−	0.695264i	$-0.755288\pi$
$293$	−16.4581	−	28.5063i	−0.961493	−	1.66536i	−0.242739	−	0.970092i	$-0.578046\pi$
$294$		0			0
$295$	−5.83502	+	10.1066i	−0.339728	+	0.588426i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−5.85916	+	10.1484i	−0.338844	+	0.586895i
$300$		0			0
$301$	−3.78807	−	6.56114i	−0.218341	−	0.378178i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	9.73566			0.557462
$306$		0			0
$307$	−6.27807			−0.358309			−0.179154	−	0.983821i	$-0.557336\pi$
$307$	−6.27807			−0.358309			−0.179154	+	0.983821i	$0.557336\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	10.2311	+	17.7208i	0.580154	+	1.00486i	0.995461	+	0.0951747i	$0.0303410\pi$
$311$	10.2311	+	17.7208i	0.580154	+	1.00486i	−0.415307	+	0.909681i	$0.636326\pi$
$312$		0			0
$313$	−11.6887	+	20.2455i	−0.660685	+	1.14434i	0.319751	+	0.947502i	$0.396401\pi$
$313$	−11.6887	+	20.2455i	−0.660685	+	1.14434i	−0.980436	+	0.196839i	$0.936932\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.04695	+	1.81337i	−0.0588024	+	0.101849i	−0.893928	−	0.448210i	$-0.852062\pi$
$317$	−1.04695	+	1.81337i	−0.0588024	+	0.101849i	0.835126	+	0.550059i	$0.185395\pi$
$318$		0			0
$319$	−3.36783	−	5.83326i	−0.188562	−	0.326600i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−8.38650			−0.466638
$324$		0			0
$325$	1.70739			0.0947089
$326$		0			0
$327$		0			0
$328$		0			0
$329$	7.61856	+	13.1957i	0.420025	+	0.727504i
$330$		0			0
$331$	−13.1559	+	22.7867i	−0.723114	+	1.25247i	0.236632	+	0.971599i	$0.423957\pi$
$331$	−13.1559	+	22.7867i	−0.723114	+	1.25247i	−0.959746	+	0.280871i	$0.909377\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	5.28534	−	9.15448i	0.288769	−	0.500163i
$336$		0			0
$337$	11.5333	+	19.9763i	0.628261	+	1.08818i	0.987901	+	0.155089i	$0.0495663\pi$
$337$	11.5333	+	19.9763i	0.628261	+	1.08818i	−0.359640	+	0.933091i	$0.617100\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−8.89703			−0.481801
$342$		0			0
$343$	−15.0065			−0.810276
$344$		0			0
$345$		0			0
$346$		0			0
$347$	7.26847	+	12.5894i	0.390192	+	0.675833i	0.992475	−	0.122450i	$-0.0390753\pi$
$347$	7.26847	+	12.5894i	0.390192	+	0.675833i	−0.602283	+	0.798283i	$0.705742\pi$
$348$		0			0
$349$	11.4909	−	19.9029i	0.615095	−	1.06538i	−0.375273	−	0.926915i	$-0.622451\pi$
$349$	11.4909	−	19.9029i	0.615095	−	1.06538i	0.990368	−	0.138462i	$-0.0442158\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	10.6983	−	18.5300i	0.569414	−	0.986254i	−0.427210	−	0.904152i	$-0.640504\pi$
$353$	10.6983	−	18.5300i	0.569414	−	0.986254i	0.996624	−	0.0821015i	$-0.0261632\pi$
$354$		0			0
$355$	−0.532810	−	0.922854i	−0.0282786	−	0.0489800i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	22.5935			1.19244			0.596220	−	0.802821i	$-0.296669\pi$
$359$	22.5935			1.19244			0.596220	+	0.802821i	$0.296669\pi$
$360$		0			0
$361$	−17.2553			−0.908172
$362$		0			0
$363$		0			0
$364$		0			0
$365$	7.02827	+	12.1733i	0.367877	+	0.637181i
$366$		0			0
$367$	4.14631	−	7.18161i	0.216435	−	0.374877i	−0.737280	−	0.675587i	$-0.763890\pi$
$367$	4.14631	−	7.18161i	0.216435	−	0.374877i	0.953716	+	0.300710i	$0.0972236\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	0.613495	−	1.06260i	0.0318511	−	0.0551677i
$372$		0			0
$373$	−9.74113	−	16.8721i	−0.504376	−	0.873606i	−0.999987	−	0.00506088i	$-0.998389\pi$
$373$	−9.74113	−	16.8721i	−0.504376	−	0.873606i	0.495611	−	0.868545i	$-0.334944\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−3.46305			−0.178356
$378$		0			0
$379$	−22.0000			−1.13006			−0.565032	−	0.825069i	$-0.691136\pi$
$379$	−22.0000			−1.13006			−0.565032	+	0.825069i	$0.691136\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−5.75980	−	9.97627i	−0.294312	−	0.509763i	0.680513	−	0.732736i	$-0.261757\pi$
$383$	−5.75980	−	9.97627i	−0.294312	−	0.509763i	−0.974825	+	0.222973i	$0.928424\pi$
$384$		0			0
$385$	−1.98133	+	3.43176i	−0.100978	+	0.174899i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−11.1887	+	19.3794i	−0.567290	+	0.982576i	0.429542	+	0.903047i	$0.358675\pi$
$389$	−11.1887	+	19.3794i	−0.567290	+	0.982576i	−0.996833	+	0.0795290i	$0.974658\pi$
$390$		0			0
$391$	−21.7881	−	37.7381i	−1.10187	−	1.90850i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.41478			0.0711852
$396$		0			0
$397$	−9.34009			−0.468765			−0.234383	−	0.972144i	$-0.575307\pi$
$397$	−9.34009			−0.468765			−0.234383	+	0.972144i	$0.575307\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−4.86330	−	8.42347i	−0.242861	−	0.420648i	0.718667	−	0.695355i	$-0.244753\pi$
$401$	−4.86330	−	8.42347i	−0.242861	−	0.420648i	−0.961528	+	0.274706i	$0.911419\pi$
$402$		0			0
$403$	−2.28715	+	3.96145i	−0.113931	+	0.197334i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	5.51414	−	9.55077i	0.273326	−	0.473414i
$408$		0			0
$409$	−5.09389	−	8.82288i	−0.251877	−	0.436264i	0.712166	−	0.702011i	$-0.247714\pi$
$409$	−5.09389	−	8.82288i	−0.251877	−	0.436264i	−0.964043	+	0.265748i	$0.914381\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−13.9253			−0.685220
$414$		0			0
$415$	11.8350			0.580958
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−6.00960	−	10.4089i	−0.293588	−	0.508510i	0.681067	−	0.732221i	$-0.261516\pi$
$419$	−6.00960	−	10.4089i	−0.293588	−	0.508510i	−0.974655	+	0.223711i	$0.928183\pi$
$420$		0			0
$421$	−9.73566	+	16.8627i	−0.474487	+	0.821836i	−0.999573	−	0.0292132i	$-0.990700\pi$
$421$	−9.73566	+	16.8627i	−0.474487	+	0.821836i	0.525086	+	0.851049i	$0.324033\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3.17458	+	5.49853i	−0.153990	+	0.266718i
$426$		0			0
$427$	5.80855	+	10.0607i	0.281096	+	0.486872i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−16.8296			−0.810651			−0.405326	−	0.914172i	$-0.632842\pi$
$431$	−16.8296			−0.810651			−0.405326	+	0.914172i	$0.632842\pi$
$432$		0			0
$433$	−8.40571			−0.403952			−0.201976	−	0.979390i	$-0.564736\pi$
$433$	−8.40571			−0.403952			−0.201976	+	0.979390i	$0.564736\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	4.53281	+	7.85106i	0.216834	+	0.375567i
$438$		0			0
$439$	−14.5424	+	25.1882i	−0.694071	+	1.20217i	0.276421	+	0.961037i	$0.410851\pi$
$439$	−14.5424	+	25.1882i	−0.694071	+	1.20217i	−0.970493	+	0.241130i	$0.922482\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−7.09209	+	12.2839i	−0.336955	+	0.583624i	−0.983858	−	0.178948i	$-0.942730\pi$
$443$	−7.09209	+	12.2839i	−0.336955	+	0.583624i	0.646903	+	0.762572i	$0.276064\pi$
$444$		0			0
$445$	5.50000	+	9.52628i	0.260725	+	0.451589i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	26.6874			1.25946			0.629728	−	0.776816i	$-0.283166\pi$
$449$	26.6874			1.25946			0.629728	+	0.776816i	$0.283166\pi$
$450$		0			0
$451$	7.70739			0.362927
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1.01867	+	1.76439i	0.0477561	+	0.0827161i
$456$		0			0
$457$	−7.95305	+	13.7751i	−0.372028	+	0.644372i	−0.989877	−	0.141925i	$-0.954671\pi$
$457$	−7.95305	+	13.7751i	−0.372028	+	0.644372i	0.617849	+	0.786297i	$0.288004\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−19.0333	+	32.9667i	−0.886471	+	1.53541i	−0.0424525	+	0.999098i	$0.513517\pi$
$461$	−19.0333	+	32.9667i	−0.886471	+	1.53541i	−0.844018	+	0.536314i	$0.819816\pi$
$462$		0			0
$463$	−17.9723	−	31.1289i	−0.835241	−	1.44668i	−0.893834	−	0.448399i	$-0.851994\pi$
$463$	−17.9723	−	31.1289i	−0.835241	−	1.44668i	0.0585922	−	0.998282i	$-0.481339\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−26.4623			−1.22453			−0.612264	−	0.790654i	$-0.709741\pi$
$467$	−26.4623			−1.22453			−0.612264	+	0.790654i	$0.709741\pi$
$468$		0			0
$469$	12.6135			0.582437
$470$		0			0
$471$		0			0
$472$		0			0
$473$	10.5424	+	18.2600i	0.484741	+	0.839595i
$474$		0			0
$475$	0.660442	−	1.14392i	0.0303032	−	0.0524866i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−19.0565	+	33.0069i	−0.870716	+	1.50812i	−0.00945845	+	0.999955i	$0.503011\pi$
$479$	−19.0565	+	33.0069i	−0.870716	+	1.50812i	−0.861257	+	0.508169i	$0.830323\pi$
$480$		0			0
$481$	−2.83502	−	4.91040i	−0.129266	−	0.223895i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−16.2553			−0.738114
$486$		0			0
$487$	−38.6610			−1.75190			−0.875948	−	0.482406i	$-0.839763\pi$
$487$	−38.6610			−1.75190			−0.875948	+	0.482406i	$0.839763\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−15.8633	−	27.4760i	−0.715900	−	1.23998i	−0.962611	−	0.270887i	$-0.912683\pi$
$491$	−15.8633	−	27.4760i	−0.715900	−	1.23998i	0.246711	−	0.969089i	$-0.420650\pi$
$492$		0			0
$493$	6.43892	−	11.1525i	0.289994	−	0.502285i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	0.635777	−	1.10120i	0.0285185	−	0.0493955i
$498$		0			0
$499$	19.2407	+	33.3259i	0.861333	+	1.49187i	0.870642	+	0.491916i	$0.163703\pi$
$499$	19.2407	+	33.3259i	0.861333	+	1.49187i	−0.00930924	+	0.999957i	$0.502963\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	11.0994			0.494896			0.247448	−	0.968901i	$-0.420408\pi$
$503$	11.0994			0.494896			0.247448	+	0.968901i	$0.420408\pi$
$504$		0			0
$505$	−2.38650			−0.106198
$506$		0			0
$507$		0			0
$508$		0			0
$509$	13.2639	+	22.9738i	0.587914	+	1.01830i	0.994505	+	0.104686i	$0.0333839\pi$
$509$	13.2639	+	22.9738i	0.587914	+	1.01830i	−0.406592	+	0.913610i	$0.633283\pi$
$510$		0			0
$511$	−8.38650	+	14.5259i	−0.370997	+	0.642586i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−4.88197	+	8.45582i	−0.215125	+	0.372608i
$516$		0			0
$517$	−21.2029	−	36.7244i	−0.932500	−	1.61514i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	9.89703			0.433597			0.216798	−	0.976216i	$-0.430439\pi$
$521$	9.89703			0.433597			0.216798	+	0.976216i	$0.430439\pi$
$522$		0			0
$523$	−28.4394			−1.24357			−0.621785	−	0.783188i	$-0.713592\pi$
$523$	−28.4394			−1.24357			−0.621785	+	0.783188i	$0.713592\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−8.50506	−	14.7312i	−0.370486	−	0.641701i
$528$		0			0
$529$	−12.0524	+	20.8754i	−0.524018	+	0.907626i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.98133	−	3.43176i	0.0858208	−	0.148646i
$534$		0			0
$535$	−3.78988	−	6.56426i	−0.163851	−	0.283798i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	18.5177			0.797616
$540$		0			0
$541$	13.1131			0.563776			0.281888	−	0.959447i	$-0.409039\pi$
$541$	13.1131			0.563776			0.281888	+	0.959447i	$0.409039\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	8.28807	+	14.3554i	0.355022	+	0.614916i
$546$		0			0
$547$	12.3660	−	21.4186i	0.528733	−	0.915793i	−0.470705	−	0.882290i	$-0.656001\pi$
$547$	12.3660	−	21.4186i	0.528733	−	0.915793i	0.999439	−	0.0335023i	$-0.0106661\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1.33956	+	2.32018i	−0.0570671	+	0.0988431i
$552$		0			0
$553$	0.844094	+	1.46201i	0.0358945	+	0.0621712i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−22.7730			−0.964923			−0.482462	−	0.875917i	$-0.660257\pi$
$557$	−22.7730			−0.964923			−0.482462	+	0.875917i	$0.660257\pi$
$558$		0			0
$559$	10.8405			0.458504
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−14.7056	−	25.4708i	−0.619767	−	1.07347i	−0.989528	−	0.144340i	$-0.953894\pi$
$563$	−14.7056	−	25.4708i	−0.619767	−	1.07347i	0.369762	−	0.929127i	$-0.379439\pi$
$564$		0			0
$565$	10.5424	−	18.2600i	0.443523	−	0.768204i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−18.2835	+	31.6680i	−0.766486	+	1.32759i	0.172972	+	0.984927i	$0.444663\pi$
$569$	−18.2835	+	31.6680i	−0.766486	+	1.32759i	−0.939457	+	0.342666i	$0.888670\pi$
$570$		0			0
$571$	2.18779	+	3.78936i	0.0915561	+	0.158580i	0.908166	−	0.418610i	$-0.137483\pi$
$571$	2.18779	+	3.78936i	0.0915561	+	0.158580i	−0.816610	+	0.577190i	$0.804149\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	6.86330			0.286219
$576$		0			0
$577$	9.02827			0.375852			0.187926	−	0.982183i	$-0.439823\pi$
$577$	9.02827			0.375852			0.187926	+	0.982183i	$0.439823\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	7.06108	+	12.2302i	0.292943	+	0.507392i
$582$		0			0
$583$	−1.70739	+	2.95729i	−0.0707128	+	0.122478i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	1.63397	−	2.83012i	0.0674413	−	0.116812i	−0.830333	−	0.557267i	$-0.811850\pi$
$587$	1.63397	−	2.83012i	0.0674413	−	0.116812i	0.897774	+	0.440456i	$0.145183\pi$
$588$		0			0
$589$	1.76940	+	3.06469i	0.0729069	+	0.126278i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−3.39558			−0.139440			−0.0697198	−	0.997567i	$-0.522211\pi$
$593$	−3.39558			−0.139440			−0.0697198	+	0.997567i	$0.522211\pi$
$594$		0			0
$595$	−7.57615			−0.310592
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−0.301683	−	0.522531i	−0.0123264	−	0.0213500i	0.859796	−	0.510637i	$-0.170590\pi$
$599$	−0.301683	−	0.522531i	−0.0123264	−	0.0213500i	−0.872123	+	0.489287i	$0.837257\pi$
$600$		0			0
$601$	20.9627	−	36.3084i	0.855084	−	1.48105i	−0.0214822	−	0.999769i	$-0.506839\pi$
$601$	20.9627	−	36.3084i	0.855084	−	1.48105i	0.876567	−	0.481281i	$-0.159828\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.0141369	−	0.0244859i	0.000574748	−	0.000995493i
$606$		0			0
$607$	−2.71466	−	4.70193i	−0.110185	−	0.190845i	0.805660	−	0.592378i	$-0.201811\pi$
$607$	−2.71466	−	4.70193i	−0.110185	−	0.190845i	−0.915845	+	0.401533i	$0.868478\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−21.8023			−0.882028
$612$		0			0
$613$	2.42385			0.0978984			0.0489492	−	0.998801i	$-0.484413\pi$
$613$	2.42385			0.0978984			0.0489492	+	0.998801i	$0.484413\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	14.5424	+	25.1882i	0.585455	+	1.01404i	0.994818	+	0.101667i	$0.0324176\pi$
$617$	14.5424	+	25.1882i	0.585455	+	1.01404i	−0.409363	+	0.912372i	$0.634249\pi$
$618$		0			0
$619$	3.72606	−	6.45373i	0.149763	−	0.259397i	−0.781377	−	0.624060i	$-0.785482\pi$
$619$	3.72606	−	6.45373i	0.149763	−	0.259397i	0.931140	+	0.364662i	$0.118816\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−6.56289	+	11.3673i	−0.262937	+	0.455419i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.0200000	−	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	21.0848			0.840707
$630$		0			0
$631$	27.8013			1.10675			0.553376	−	0.832932i	$-0.313339\pi$
$631$	27.8013			1.10675			0.553376	+	0.832932i	$0.313339\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	7.70559	+	13.3465i	0.305787	+	0.529638i
$636$		0			0
$637$	4.76033	−	8.24513i	0.188611	−	0.326684i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	17.0237	−	29.4860i	0.672397	−	1.16463i	−0.304825	−	0.952408i	$-0.598598\pi$
$641$	17.0237	−	29.4860i	0.672397	−	1.16463i	0.977222	−	0.212218i	$-0.0680686\pi$
$642$		0			0
$643$	−6.92711	−	11.9981i	−0.273179	−	0.473159i	0.696495	−	0.717561i	$-0.254742\pi$
$643$	−6.92711	−	11.9981i	−0.273179	−	0.473159i	−0.969674	+	0.244402i	$0.921408\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−5.11750			−0.201190			−0.100595	−	0.994927i	$-0.532075\pi$
$647$	−5.11750			−0.201190			−0.100595	+	0.994927i	$0.532075\pi$
$648$		0			0
$649$	38.7549			1.52126
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−3.69779	−	6.40476i	−0.144706	−	0.250638i	0.784557	−	0.620056i	$-0.212890\pi$
$653$	−3.69779	−	6.40476i	−0.144706	−	0.250638i	−0.929263	+	0.369419i	$0.879557\pi$
$654$		0			0
$655$	−1.67912	+	2.90831i	−0.0656085	+	0.113637i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	8.54241	−	14.7959i	0.332765	−	0.576366i	−0.650288	−	0.759688i	$-0.725352\pi$
$659$	8.54241	−	14.7959i	0.332765	−	0.576366i	0.983053	+	0.183322i	$0.0586851\pi$
$660$		0			0
$661$	19.3118	+	33.4490i	0.751142	+	1.30102i	0.947269	+	0.320438i	$0.103830\pi$
$661$	19.3118	+	33.4490i	0.751142	+	1.30102i	−0.196127	+	0.980579i	$0.562836\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1.57615			0.0611204
$666$		0			0
$667$	−13.9206			−0.539009
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−16.1655	−	27.9995i	−0.624062	−	1.08091i
$672$		0			0
$673$	−13.2649	+	22.9754i	−0.511323	+	0.885637i	0.488591	+	0.872513i	$0.337511\pi$
$673$	−13.2649	+	22.9754i	−0.511323	+	0.885637i	−0.999914	+	0.0131244i	$0.995822\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	19.0096	−	32.9256i	0.730598	−	1.26543i	−0.226030	−	0.974120i	$-0.572575\pi$
$677$	19.0096	−	32.9256i	0.730598	−	1.26543i	0.956628	−	0.291313i	$-0.0940921\pi$
$678$		0			0
$679$	−9.69832	−	16.7980i	−0.372187	−	0.644647i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	6.23606			0.238616			0.119308	−	0.992857i	$-0.461932\pi$
$683$	6.23606			0.238616			0.119308	+	0.992857i	$0.461932\pi$
$684$		0			0
$685$	−7.61350			−0.290897
$686$		0			0
$687$		0			0
$688$		0			0
$689$	0.877832	+	1.52045i	0.0334427	+	0.0579245i
$690$		0			0
$691$	1.58522	−	2.74568i	0.0603047	−	0.104451i	−0.834297	−	0.551315i	$-0.814126\pi$
$691$	1.58522	−	2.74568i	0.0603047	−	0.104451i	0.894602	+	0.446865i	$0.147459\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−6.64177	+	11.5039i	−0.251937	+	0.436367i
$696$		0			0
$697$	7.36783	+	12.7615i	0.279077	+	0.483375i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−51.5863			−1.94839			−0.974193	−	0.225715i	$-0.927528\pi$
$701$	−51.5863			−1.94839			−0.974193	+	0.225715i	$0.927528\pi$
$702$		0			0
$703$	−4.38650			−0.165440
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1.42385	−	2.46618i	−0.0535494	−	0.0927504i
$708$		0			0
$709$	−6.42892	+	11.1352i	−0.241443	+	0.418192i	−0.961126	−	0.276112i	$-0.910954\pi$
$709$	−6.42892	+	11.1352i	−0.241443	+	0.418192i	0.719683	+	0.694303i	$0.244287\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−9.19378	+	15.9241i	−0.344310	+	0.596362i
$714$		0			0
$715$	−2.83502	−	4.91040i	−0.106024	−	0.183639i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	4.84049			0.180520			0.0902598	−	0.995918i	$-0.471230\pi$
$719$	4.84049			0.180520			0.0902598	+	0.995918i	$0.471230\pi$
$720$		0			0
$721$	−11.6508			−0.433900
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.01414	+	1.75654i	0.0376641	+	0.0652361i
$726$		0			0
$727$	−20.5351	+	35.5679i	−0.761606	+	1.31914i	0.180416	+	0.983590i	$0.442256\pi$
$727$	−20.5351	+	35.5679i	−0.761606	+	1.31914i	−0.942022	+	0.335550i	$0.891078\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−20.1559	+	34.9111i	−0.745493	+	1.29123i
$732$		0			0
$733$	−17.6983	−	30.6544i	−0.653702	−	1.13225i	−0.982217	−	0.187747i	$-0.939882\pi$
$733$	−17.6983	−	30.6544i	−0.653702	−	1.13225i	0.328515	−	0.944499i	$-0.393452\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−35.1040			−1.29307
$738$		0			0
$739$	19.4823			0.716666			0.358333	−	0.933594i	$-0.383345\pi$
$739$	19.4823			0.716666			0.358333	+	0.933594i	$0.383345\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	8.10169	+	14.0325i	0.297222	+	0.514804i	0.975499	−	0.220002i	$-0.0706065\pi$
$743$	8.10169	+	14.0325i	0.297222	+	0.514804i	−0.678277	+	0.734806i	$0.737273\pi$
$744$		0			0
$745$	8.99546	−	15.5806i	0.329568	−	0.570829i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	4.52228	−	7.83282i	0.165241	−	0.286205i
$750$		0			0
$751$	7.17458	+	12.4267i	0.261804	+	0.453458i	0.966721	−	0.255832i	$-0.0823492\pi$
$751$	7.17458	+	12.4267i	0.261804	+	0.453458i	−0.704917	+	0.709290i	$0.749016\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−8.67912			−0.315865
$756$		0			0
$757$	16.7466			0.608665			0.304333	−	0.952566i	$-0.401567\pi$
$757$	16.7466			0.608665			0.304333	+	0.952566i	$0.401567\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−9.81542	−	17.0008i	−0.355809	−	0.616279i	0.631447	−	0.775419i	$-0.282461\pi$
$761$	−9.81542	−	17.0008i	−0.355809	−	0.616279i	−0.987256	+	0.159140i	$0.949128\pi$
$762$		0			0
$763$	−9.88976	+	17.1296i	−0.358034	+	0.620132i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	9.96265	−	17.2558i	0.359731	−	0.623072i
$768$		0			0
$769$	21.4572	+	37.1649i	0.773766	+	1.34020i	0.935485	+	0.353365i	$0.114963\pi$
$769$	21.4572	+	37.1649i	0.773766	+	1.34020i	−0.161719	+	0.986837i	$0.551704\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	7.22699			0.259937			0.129968	−	0.991518i	$-0.458512\pi$
$773$	7.22699			0.259937			0.129968	+	0.991518i	$0.458512\pi$
$774$		0			0
$775$	2.67912			0.0962367
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1.53281	−	2.65491i	−0.0549186	−	0.0951218i
$780$		0			0
$781$	−1.76940	+	3.06469i	−0.0633141	+	0.109663i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	9.44852	−	16.3653i	0.337232	−	0.584103i
$786$		0			0
$787$	15.1180	+	26.1852i	0.538900	+	0.933402i	0.998964	+	0.0455158i	$0.0144931\pi$
$787$	15.1180	+	26.1852i	0.538900	+	0.933402i	−0.460064	+	0.887886i	$0.652174\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	25.1595			0.894569
$792$		0			0
$793$	−16.6226			−0.590285
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−13.7079	−	23.7428i	−0.485559	−	0.841013i	0.514303	−	0.857609i	$-0.328051\pi$
$797$	−13.7079	−	23.7428i	−0.485559	−	0.841013i	−0.999862	+	0.0165951i	$0.994717\pi$
$798$		0			0
$799$	40.5375	−	70.2130i	1.43411	−	2.48396i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	23.3401	−	40.4262i	0.823654	−	1.42661i
$804$		0			0
$805$	4.09482	+	7.09244i	0.144324	+	0.249976i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−28.1806			−0.990776			−0.495388	−	0.868672i	$-0.664974\pi$
$809$	−28.1806			−0.990776			−0.495388	+	0.868672i	$0.664974\pi$
$810$		0			0
$811$	−20.1312			−0.706903			−0.353452	−	0.935453i	$-0.614992\pi$
$811$	−20.1312			−0.706903			−0.353452	+	0.935453i	$0.614992\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	0.495464	+	0.858168i	0.0173553	+	0.0300603i
$816$		0			0
$817$	4.19325	−	7.26293i	0.146703	−	0.254098i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	6.12310	−	10.6055i	0.213698	−	0.370135i	−0.739171	−	0.673517i	$-0.764783\pi$
$821$	6.12310	−	10.6055i	0.213698	−	0.370135i	0.952869	+	0.303382i	$0.0981160\pi$
$822$		0			0
$823$	−13.9645	−	24.1872i	−0.486770	−	0.843111i	0.513114	−	0.858321i	$-0.328492\pi$
$823$	−13.9645	−	24.1872i	−0.486770	−	0.843111i	−0.999884	+	0.0152094i	$0.995158\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	8.57068			0.298032			0.149016	−	0.988835i	$-0.452389\pi$
$827$	8.57068			0.298032			0.149016	+	0.988835i	$0.452389\pi$
$828$		0			0
$829$	−19.7357			−0.685448			−0.342724	−	0.939436i	$-0.611350\pi$
$829$	−19.7357			−0.685448			−0.342724	+	0.939436i	$0.611350\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	17.7019	+	30.6606i	0.613335	+	1.06233i
$834$		0			0
$835$	6.79948	−	11.7770i	0.235306	−	0.407561i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−23.3588	+	40.4586i	−0.806434	+	1.39678i	0.108885	+	0.994054i	$0.465272\pi$
$839$	−23.3588	+	40.4586i	−0.806434	+	1.39678i	−0.915319	+	0.402730i	$0.868061\pi$
$840$		0			0
$841$	12.4431	+	21.5520i	0.429071	+	0.743172i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	10.0848			0.346928
$846$		0			0
$847$	0.0337379			0.00115925
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−11.3961	−	19.7386i	−0.390653	−	0.676632i
$852$		0			0
$853$	−2.69779	+	4.67271i	−0.0923705	+	0.159990i	−0.908508	−	0.417867i	$-0.862778\pi$
$853$	−2.69779	+	4.67271i	−0.0923705	+	0.159990i	0.816138	+	0.577858i	$0.196111\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	14.3359	−	24.8306i	0.489707	−	0.848197i	−0.510223	−	0.860042i	$-0.670437\pi$
$857$	14.3359	−	24.8306i	0.489707	−	0.848197i	0.999930	+	0.0118452i	$0.00377053\pi$
$858$		0			0
$859$	11.3733	+	19.6991i	0.388052	+	0.672125i	0.992187	−	0.124756i	$-0.0398148\pi$
$859$	11.3733	+	19.6991i	0.388052	+	0.672125i	−0.604136	+	0.796882i	$0.706481\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−0.202325			−0.00688723			−0.00344361	−	0.999994i	$-0.501096\pi$
$863$	−0.202325			−0.00688723			−0.00344361	+	0.999994i	$0.501096\pi$
$864$		0			0
$865$	−0.187788			−0.00638500
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−2.34916	−	4.06886i	−0.0796897	−	0.138027i
$870$		0			0
$871$	−9.02414	+	15.6303i	−0.305771	+	0.529611i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	0.596626	−	1.03339i	0.0201696	−	0.0349349i
$876$		0			0
$877$	−6.48639	−	11.2348i	−0.219030	−	0.379371i	0.735482	−	0.677545i	$-0.236956\pi$
$877$	−6.48639	−	11.2348i	−0.219030	−	0.379371i	−0.954512	+	0.298174i	$0.903623\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−16.4905			−0.555580			−0.277790	−	0.960642i	$-0.589602\pi$
$881$	−16.4905			−0.555580			−0.277790	+	0.960642i	$0.589602\pi$
$882$		0			0
$883$	8.38290			0.282107			0.141053	−	0.990002i	$-0.454951\pi$
$883$	8.38290			0.282107			0.141053	+	0.990002i	$0.454951\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	29.2311	+	50.6298i	0.981485	+	1.69998i	0.656619	+	0.754222i	$0.271986\pi$
$887$	29.2311	+	50.6298i	0.981485	+	1.69998i	0.324866	+	0.945760i	$0.394681\pi$
$888$		0			0
$889$	−9.19471	+	15.9257i	−0.308381	+	0.534131i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−8.43345	+	14.6072i	−0.282215	+	0.488810i
$894$		0			0
$895$	4.88197	+	8.45582i	0.163186	+	0.282647i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−5.43398			−0.181233
$900$		0			0
$901$	−6.52867			−0.217502
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−8.04241	−	13.9299i	−0.267339	−	0.463044i
$906$		0			0
$907$	15.3510	−	26.5886i	0.509720	−	0.882862i	−0.490216	−	0.871601i	$-0.663082\pi$
$907$	15.3510	−	26.5886i	0.509720	−	0.882862i	0.999937	−	0.0112607i	$-0.00358447\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	17.1559	−	29.7149i	0.568401	−	0.984499i	−0.428324	−	0.903625i	$-0.640896\pi$
$911$	17.1559	−	29.7149i	0.568401	−	0.984499i	0.996724	−	0.0808733i	$-0.0257709\pi$
$912$		0			0
$913$	−19.6514	−	34.0372i	−0.650365	−	1.12647i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−4.00722			−0.132330
$918$		0			0
$919$	18.8861			0.622995			0.311498	−	0.950247i	$-0.399169\pi$
$919$	18.8861			0.622995			0.311498	+	0.950247i	$0.399169\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	0.909714	+	1.57567i	0.0299436	+	0.0518639i
$924$		0			0
$925$	−1.66044	+	2.87597i	−0.0545950	+	0.0945613i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	4.04748	−	7.01043i	0.132793	−	0.230005i	−0.791959	−	0.610574i	$-0.790939\pi$
$929$	4.04748	−	7.01043i	0.132793	−	0.230005i	0.924752	+	0.380569i	$0.124272\pi$
$930$		0			0
$931$	−3.68272	−	6.37867i	−0.120696	−	0.209052i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	21.0848			0.689547
$936$		0			0
$937$	29.7084			0.970533			0.485266	−	0.874366i	$-0.338723\pi$
$937$	29.7084			0.970533			0.485266	+	0.874366i	$0.338723\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	5.24113	+	9.07790i	0.170856	+	0.295931i	0.938719	−	0.344682i	$-0.112013\pi$
$941$	5.24113	+	9.07790i	0.170856	+	0.295931i	−0.767863	+	0.640614i	$0.778680\pi$
$942$		0			0
$943$	7.96446	−	13.7948i	0.259358	−	0.449222i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−2.32815	+	4.03248i	−0.0756548	+	0.131038i	−0.901371	−	0.433048i	$-0.857438\pi$
$947$	−2.32815	+	4.03248i	−0.0756548	+	0.131038i	0.825716	+	0.564086i	$0.190771\pi$
$948$		0			0
$949$	−12.0000	−	20.7846i	−0.389536	−	0.674697i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	21.9445			0.710852			0.355426	−	0.934704i	$-0.384336\pi$
$953$	21.9445			0.710852			0.355426	+	0.934704i	$0.384336\pi$
$954$		0			0
$955$	3.06562			0.0992011
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−4.54241	−	7.86769i	−0.146682	−	0.254061i
$960$		0			0
$961$	11.9112	−	20.6308i	0.384231	−	0.665508i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	0.118031	−	0.204436i	0.00379957	−	0.00658104i
$966$		0			0
$967$	30.3515	+	52.5703i	0.976038	+	1.69055i	0.676466	+	0.736474i	$0.263511\pi$
$967$	30.3515	+	52.5703i	0.976038	+	1.69055i	0.299572	+	0.954074i	$0.403156\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	38.9053			1.24853			0.624265	−	0.781212i	$-0.285398\pi$
$971$	38.9053			1.24853			0.624265	+	0.781212i	$0.285398\pi$
$972$		0			0
$973$	−15.8506			−0.508147
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−7.20832	−	12.4852i	−0.230614	−	0.399436i	0.727375	−	0.686241i	$-0.240740\pi$
$977$	−7.20832	−	12.4852i	−0.230614	−	0.399436i	−0.957989	+	0.286805i	$0.907407\pi$
$978$		0			0
$979$	18.2649	−	31.6357i	0.583748	−	1.01108i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−12.9512	+	22.4322i	−0.413081	+	0.715477i	−0.995225	−	0.0976089i	$-0.968881\pi$
$983$	−12.9512	+	22.4322i	−0.413081	+	0.715477i	0.582144	+	0.813086i	$0.302214\pi$
$984$		0			0
$985$	0.872368	+	1.51099i	0.0277960	+	0.0481440i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	43.5761			1.38564
$990$		0			0
$991$	26.1987			0.832230			0.416115	−	0.909312i	$-0.363391\pi$
$991$	26.1987			0.832230			0.416115	+	0.909312i	$0.363391\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	11.3961	+	19.7386i	0.361281	+	0.625757i
$996$		0			0
$997$	12.3829	−	21.4478i	0.392170	−	0.679259i	−0.600565	−	0.799576i	$-0.705058\pi$
$997$	12.3829	−	21.4478i	0.392170	−	0.679259i	0.992736	+	0.120317i	$0.0383911\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.2.q.j.721.2		6
3.2	odd	2		720.2.q.j.241.2		6
4.3	odd	2		1080.2.q.d.721.2		6
9.2	odd	6		6480.2.a.bx.1.2		3
9.4	even	3	inner	2160.2.q.j.1441.2		6
9.5	odd	6		720.2.q.j.481.2		6
9.7	even	3		6480.2.a.bu.1.2		3
12.11	even	2		360.2.q.d.241.2	yes	6
36.7	odd	6		3240.2.a.q.1.2		3
36.11	even	6		3240.2.a.r.1.2		3
36.23	even	6		360.2.q.d.121.2	✓	6
36.31	odd	6		1080.2.q.d.361.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.q.d.121.2	✓	6	36.23	even	6
360.2.q.d.241.2	yes	6	12.11	even	2
720.2.q.j.241.2		6	3.2	odd	2
720.2.q.j.481.2		6	9.5	odd	6
1080.2.q.d.361.2		6	36.31	odd	6
1080.2.q.d.721.2		6	4.3	odd	2
2160.2.q.j.721.2		6	1.1	even	1	trivial
2160.2.q.j.1441.2		6	9.4	even	3	inner
3240.2.a.q.1.2		3	36.7	odd	6
3240.2.a.r.1.2		3	36.11	even	6
6480.2.a.bu.1.2		3	9.7	even	3
6480.2.a.bx.1.2		3	9.2	odd	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:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2160.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{5} +(-0.596626 + 1.03339i) q^{7} +(1.66044 - 2.87597i) q^{11} +(-0.853695 - 1.47864i) q^{13} +6.34916 q^{17} -1.32088 q^{19} +(-3.43165 - 5.94379i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(1.01414 - 1.75654i) q^{29} +(-1.33956 - 2.32018i) q^{31} -1.19325 q^{35} +3.32088 q^{37} +(1.16044 + 2.00994i) q^{41} +(-3.17458 + 5.49853i) q^{43} +(6.38470 - 11.0586i) q^{47} +(2.78807 + 4.82909i) q^{49} -1.02827 q^{53} +3.32088 q^{55} +(5.83502 + 10.1066i) q^{59} +(4.86783 - 8.43133i) q^{61} +(0.853695 - 1.47864i) q^{65} +(-5.28534 - 9.15448i) q^{67} -1.06562 q^{71} +14.0565 q^{73} +(1.98133 + 3.43176i) q^{77} +(0.707389 - 1.22523i) q^{79} +(5.91751 - 10.2494i) q^{83} +(3.17458 + 5.49853i) q^{85} +11.0000 q^{89} +2.03735 q^{91} +(-0.660442 - 1.14392i) q^{95} +(-8.12763 + 14.0775i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{5} - 5 q^{7} + 2 q^{11} - 4 q^{17} + 8 q^{19} + 7 q^{23} - 3 q^{25} - 7 q^{29} - 16 q^{31} - 10 q^{35} + 4 q^{37} - q^{41} + 2 q^{43} + 13 q^{47} - 10 q^{49} + 20 q^{53} + 4 q^{55} + 6 q^{59}+ \cdots - 30 q^{97}+O(q^{100}) \) 6 * q + 3 * q^5 - 5 * q^7 + 2 * q^11 - 4 * q^17 + 8 * q^19 + 7 * q^23 - 3 * q^25 - 7 * q^29 - 16 * q^31 - 10 * q^35 + 4 * q^37 - q^41 + 2 * q^43 + 13 * q^47 - 10 * q^49 + 20 * q^53 + 4 * q^55 + 6 * q^59 + 11 * q^61 + q^67 - 28 * q^71 + 32 * q^73 - 12 * q^77 - 6 * q^79 + 21 * q^83 - 2 * q^85 + 66 * q^89 + 60 * q^91 + 4 * q^95 - 30 * q^97

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