LMFDB - Embedded newform 539.2.q.d.422.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [539,2,Mod(214,539)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("539.214"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(539, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([20, 12])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 539 = 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	539.q (of order $15$, degree $8$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$4.30393666895$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$2$ over $\Q(\zeta_{15})$
Coefficient field:	16.0.9234096523681640625.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{15} + 2 x^{14} - 5 x^{13} + 5 x^{12} + x^{11} + 6 x^{10} + 5 x^{9} - 21 x^{8} + 10 x^{7} + \cdots + 256 $ x^16 - x^15 + 2x^14 - 5x^13 + 5x^12 + x^11 + 6x^10 + 5x^9 - 21x^8 + 10x^7 + 24x^6 + 8x^5 + 80x^4 - 160x^3 + 128x^2 - 128*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{15}]$

Embedding invariants

Embedding label			422.1
Root			$-0.214032 - 1.39792i$ of defining polynomial
Character	$\chi$	$=$	539.422
Dual form			539.2.q.d.410.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.01642 + 0.897766i) q^{2} +(1.92169 - 2.13425i) q^{4} +(-0.594713 + 1.83034i) q^{8} +(-2.74064 + 1.22021i) q^{9} +(2.90321 - 1.60354i) q^{11} +(0.156367 + 1.48773i) q^{16} +(4.43080 - 4.92090i) q^{18} +(-4.41448 + 5.83981i) q^{22} +(1.28019 - 2.21736i) q^{23} +(4.89074 - 1.03956i) q^{25} +(-2.42225 - 7.45492i) q^{29} +(-3.57547 - 6.19289i) q^{32} +(-2.66241 + 8.19407i) q^{36} +(11.6587 + 2.47813i) q^{37} +12.8185 q^{43} +(2.15671 - 9.27769i) q^{44} +(-0.590730 + 5.62042i) q^{46} +(-8.92848 + 6.48692i) q^{50} +(-1.49588 + 14.2323i) q^{53} +(11.5770 + 12.8576i) q^{58} +(10.3489 + 7.51894i) q^{64} +(6.28343 + 10.8832i) q^{67} +(12.9884 - 9.43662i) q^{71} +(-0.603505 - 5.74197i) q^{72} +(-25.7335 + 5.46983i) q^{74} +(2.61152 - 1.16272i) q^{79} +(6.02218 - 6.68830i) q^{81} +(-25.8474 + 11.5080i) q^{86} +(1.20844 + 6.26752i) q^{88} +(-2.27227 - 6.99331i) q^{92} +(-6.00000 + 7.93725i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 2 q^{2} - 2 q^{4} + 32 q^{8} - 6 q^{9} + 4 q^{11} + 28 q^{16} + 9 q^{18} - 8 q^{22} + 16 q^{23} - 10 q^{25} - 8 q^{29} - 100 q^{32} - 12 q^{36} + 18 q^{37} + 48 q^{43} - 9 q^{44} + 31 q^{46} + 20 q^{50}+ \cdots - 96 q^{99}+O(q^{100}) $ 16 * q + 2 * q^2 - 2 * q^4 + 32 * q^8 - 6 * q^9 + 4 * q^11 + 28 * q^16 + 9 * q^18 - 8 * q^22 + 16 * q^23 - 10 * q^25 - 8 * q^29 - 100 * q^32 - 12 * q^36 + 18 * q^37 + 48 * q^43 - 9 * q^44 + 31 * q^46 + 20 * q^50 + 30 * q^53 - 71 * q^58 - 148 * q^64 + 8 * q^67 + 96 * q^71 - 57 * q^72 - 12 * q^74 - 24 * q^79 + 18 * q^81 - 29 * q^86 + 38 * q^88 + 142 * q^92 - 96 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/539\mathbb{Z}\right)^\times$.

$n$	$199$	$442$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{5}\right)$

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 $

$2$	−2.01642	+	0.897766i	−1.42582	+	0.634817i	−0.967246	−	0.253839i	$-0.918307\pi$
$2$	−2.01642	+	0.897766i	−1.42582	+	0.634817i	−0.458575	+	0.888656i	$0.651640\pi$
$3$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$3$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$4$	1.92169	−	2.13425i	0.960844	−	1.06713i
$5$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$5$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	−0.594713	+	1.83034i	−0.210263	+	0.647123i
$9$	−2.74064	+	1.22021i	−0.913545	+	0.406737i
$10$		0			0
$11$	2.90321	−	1.60354i	0.875352	−	0.483486i
$11$	2.90321	−	1.60354i	0.875352	−	0.483486i
$12$		0			0
$13$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$13$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$14$		0			0
$15$		0			0
$16$	0.156367	+	1.48773i	0.0390917	+	0.371933i
$17$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$17$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$18$	4.43080	−	4.92090i	1.04435	−	1.15987i
$19$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$19$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$20$		0			0
$21$		0			0
$22$	−4.41448	+	5.83981i	−0.941171	+	1.24505i
$23$	1.28019	−	2.21736i	0.266938	−	0.462351i	−0.701131	−	0.713032i	$-0.747321\pi$
$23$	1.28019	−	2.21736i	0.266938	−	0.462351i	0.968070	+	0.250681i	$0.0806547\pi$
$24$		0			0
$25$	4.89074	−	1.03956i	0.978148	−	0.207912i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.42225	−	7.45492i	−0.449801	−	1.38434i	−0.877132	−	0.480249i	$-0.840546\pi$
$29$	−2.42225	−	7.45492i	−0.449801	−	1.38434i	0.427331	−	0.904095i	$-0.359454\pi$
$30$		0			0
$31$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$31$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$32$	−3.57547	−	6.19289i	−0.632059	−	1.09476i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−2.66241	+	8.19407i	−0.443736	+	1.36568i
$37$	11.6587	+	2.47813i	1.91667	+	0.407402i	0.999969	+	0.00783774i	$0.00249486\pi$
$37$	11.6587	+	2.47813i	1.91667	+	0.407402i	0.916705	+	0.399564i	$0.130838\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$41$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$42$		0			0
$43$	12.8185			1.95480			0.977399	−	0.211402i	$-0.0678028\pi$
$43$	12.8185			1.95480			0.977399	+	0.211402i	$0.0678028\pi$
$44$	2.15671	−	9.27769i	0.325137	−	1.39866i
$45$		0			0
$46$	−0.590730	+	5.62042i	−0.0870984	+	0.828686i
$47$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$47$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$48$		0			0
$49$		0			0
$50$	−8.92848	+	6.48692i	−1.26268	+	0.917389i
$51$		0			0
$52$		0			0
$53$	−1.49588	+	14.2323i	−0.205474	+	1.95496i	0.0805894	+	0.996747i	$0.474320\pi$
$53$	−1.49588	+	14.2323i	−0.205474	+	1.95496i	−0.286064	+	0.958211i	$0.592347\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	11.5770	+	12.8576i	1.52014	+	1.68829i
$59$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$59$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$60$		0			0
$61$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$61$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$62$		0			0
$63$		0			0
$64$	10.3489	+	7.51894i	1.29362	+	0.939868i
$65$		0			0
$66$		0			0
$67$	6.28343	+	10.8832i	0.767644	+	1.32960i	0.938837	+	0.344361i	$0.111904\pi$
$67$	6.28343	+	10.8832i	0.767644	+	1.32960i	−0.171194	+	0.985237i	$0.554762\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	12.9884	−	9.43662i	1.54144	−	1.11992i	0.592014	−	0.805928i	$-0.298333\pi$
$71$	12.9884	−	9.43662i	1.54144	−	1.11992i	0.949425	−	0.313993i	$-0.101667\pi$
$72$	−0.603505	−	5.74197i	−0.0711238	−	0.676698i
$73$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$73$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$74$	−25.7335	+	5.46983i	−2.99146	+	0.635855i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	2.61152	−	1.16272i	0.293819	−	0.130817i	−0.254533	−	0.967064i	$-0.581922\pi$
$79$	2.61152	−	1.16272i	0.293819	−	0.130817i	0.548352	+	0.836247i	$0.315255\pi$
$80$		0			0
$81$	6.02218	−	6.68830i	0.669131	−	0.743145i
$82$		0			0
$83$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$83$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$84$		0			0
$85$		0			0
$86$	−25.8474	+	11.5080i	−2.78719	+	1.24094i
$87$		0			0
$88$	1.20844	+	6.26752i	0.128820	+	0.668119i
$89$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$89$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$90$		0			0
$91$		0			0
$92$	−2.27227	−	6.99331i	−0.236900	−	0.729103i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$97$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$98$		0			0
$99$	−6.00000	+	7.93725i	−0.603023	+	0.797724i
$100$	7.17979	−	12.4358i	0.717979	−	1.24358i
$101$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$101$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$102$		0			0
$103$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$103$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$104$		0			0
$105$		0			0
$106$	−9.76098	−	30.0412i	−0.948070	−	2.91786i
$107$	−7.50287	−	8.33278i	−0.725329	−	0.805560i	0.261861	−	0.965106i	$-0.415664\pi$
$107$	−7.50287	−	8.33278i	−0.725329	−	0.805560i	−0.987191	+	0.159546i	$0.948997\pi$
$108$		0			0
$109$	−7.81367	−	13.5337i	−0.748414	−	1.29629i	−0.948583	−	0.316530i	$-0.897482\pi$
$109$	−7.81367	−	13.5337i	−0.748414	−	1.29629i	0.200169	−	0.979761i	$-0.435851\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.34450	−	10.2933i	0.314624	−	0.968314i	−0.661285	−	0.750135i	$-0.729988\pi$
$113$	3.34450	−	10.2933i	0.314624	−	0.968314i	0.975909	−	0.218179i	$-0.0700116\pi$
$114$		0			0
$115$		0			0
$116$	−20.5655	−	9.15634i	−1.90946	−	0.850145i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	5.85731	−	9.31085i	0.532483	−	0.846441i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−16.2142	+	11.7803i	−1.43878	+	1.04533i	−0.450479	+	0.892787i	$0.648747\pi$
$127$	−16.2142	+	11.7803i	−1.43878	+	1.04533i	−0.988297	+	0.152545i	$0.951253\pi$
$128$	−13.6287	−	2.89687i	−1.20462	−	0.256049i
$129$		0			0
$130$		0			0
$131$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$131$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$132$		0			0
$133$		0			0
$134$	−22.4406	−	16.3041i	−1.93857	−	1.40846i
$135$		0			0
$136$		0			0
$137$	15.5667	+	6.93075i	1.32996	+	0.592134i	0.943866	−	0.330327i	$-0.107159\pi$
$137$	15.5667	+	6.93075i	1.32996	+	0.592134i	0.386089	+	0.922462i	$0.373826\pi$
$138$		0			0
$139$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$139$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$140$		0			0
$141$		0			0
$142$	−17.7181	+	30.6887i	−1.48687	+	2.57534i
$143$		0			0
$144$	−2.24389	−	3.88653i	−0.186991	−	0.323877i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	27.6933	−	20.1203i	2.27637	−	1.65388i
$149$	−2.29963	−	21.8795i	−0.188393	−	1.79244i	−0.525318	−	0.850906i	$-0.676054\pi$
$149$	−2.29963	−	21.8795i	−0.188393	−	1.79244i	0.336926	−	0.941531i	$-0.390613\pi$
$150$		0			0
$151$	2.33180	−	0.495639i	0.189759	−	0.0403345i	−0.112052	−	0.993702i	$-0.535742\pi$
$151$	2.33180	−	0.495639i	0.189759	−	0.0403345i	0.301811	+	0.953368i	$0.402409\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$157$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$158$	−4.22206	+	4.68907i	−0.335889	+	0.373042i
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−6.13868	+	18.8929i	−0.482300	+	1.48437i
$163$	−19.4383	+	8.65450i	−1.52253	+	0.677873i	−0.986112	−	0.166082i	$-0.946888\pi$
$163$	−19.4383	+	8.65450i	−1.52253	+	0.677873i	−0.536415	+	0.843954i	$0.680222\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$167$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$168$		0			0
$169$	−4.01722	−	12.3637i	−0.309017	−	0.951057i
$170$		0			0
$171$		0			0
$172$	24.6331	−	27.3578i	1.87826	−	2.08601i
$173$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$173$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$174$		0			0
$175$		0			0
$176$	2.83960	+	4.06846i	0.214043	+	0.306672i
$177$		0			0
$178$		0			0
$179$	−25.8218	+	5.48859i	−1.93001	+	0.410236i	−0.931038	+	0.364922i	$0.881096\pi$
$179$	−25.8218	+	5.48859i	−1.93001	+	0.410236i	−0.998973	+	0.0453142i	$0.985571\pi$
$180$		0			0
$181$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$181$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$182$		0			0
$183$		0			0
$184$	3.29717	+	3.66188i	0.243070	+	0.269957i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−8.88080	−	1.88767i	−0.642592	−	0.136587i	−0.124921	−	0.992167i	$-0.539868\pi$
$191$	−8.88080	−	1.88767i	−0.642592	−	0.136587i	−0.517671	+	0.855580i	$0.673201\pi$
$192$		0			0
$193$	−1.93785	−	0.862785i	−0.139489	−	0.0621046i	0.335805	−	0.941932i	$-0.390992\pi$
$193$	−1.93785	−	0.862785i	−0.139489	−	0.0621046i	−0.475294	+	0.879827i	$0.657658\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2.03059			0.144674			0.0723369	−	0.997380i	$-0.476954\pi$
$197$	2.03059			0.144674			0.0723369	+	0.997380i	$0.476954\pi$
$198$	4.97270	−	21.3914i	0.353394	−	1.52022i
$199$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$199$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$200$	−1.00584	+	9.56995i	−0.0711238	+	0.676698i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−0.802898	+	7.63907i	−0.0558053	+	0.530952i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	23.3570	+	16.9698i	1.60796	+	1.16825i	0.869469	+	0.493987i	$0.164461\pi$
$211$	23.3570	+	16.9698i	1.60796	+	1.16825i	0.738490	+	0.674264i	$0.235539\pi$
$212$	27.5007	+	30.5426i	1.88876	+	2.09768i
$213$		0			0
$214$	22.6098	+	10.0665i	1.54557	+	0.688133i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	27.9057	+	20.2747i	1.89001	+	1.37317i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$223$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$224$		0			0
$225$	−12.1353	+	8.81678i	−0.809017	+	0.587785i
$226$	2.49709	+	23.7582i	0.166104	+	1.58037i
$227$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$227$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$228$		0			0
$229$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$229$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$230$		0			0
$231$		0			0
$232$	15.0856			0.990417
$233$	20.0980	−	8.94821i	1.31666	−	0.586216i	0.376335	−	0.926484i	$-0.377184\pi$
$233$	20.0980	−	8.94821i	1.31666	−	0.586216i	0.940329	+	0.340267i	$0.110518\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−8.80563	+	27.1009i	−0.569589	+	1.75301i	0.0843185	+	0.996439i	$0.473129\pi$
$239$	−8.80563	+	27.1009i	−0.569589	+	1.75301i	−0.653907	+	0.756575i	$0.726871\pi$
$240$		0			0
$241$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$241$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$242$	−3.45182	+	24.0330i	−0.221891	+	1.54490i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$251$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$252$		0			0
$253$	0.161049	−	8.49030i	0.0101251	−	0.533781i
$254$	22.1186	−	38.3105i	1.38784	−	2.40381i
$255$		0			0
$256$	5.05689	−	1.07487i	0.316055	−	0.0671796i
$257$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$257$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	15.7351	+	17.4756i	0.973977	+	1.08171i
$262$		0			0
$263$	11.3891	+	19.7266i	0.702284	+	1.21639i	0.967663	+	0.252248i	$0.0811698\pi$
$263$	11.3891	+	19.7266i	0.702284	+	1.21639i	−0.265378	+	0.964144i	$0.585497\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	35.3023	+	7.50374i	2.15643	+	0.458364i
$269$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$269$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$270$		0			0
$271$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$271$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$272$		0			0
$273$		0			0
$274$	−37.6112			−2.27217
$275$	12.5319	−	10.8606i	0.755701	−	0.654916i
$276$		0			0
$277$	3.47926	−	33.1029i	0.209048	−	1.98896i	0.0955484	−	0.995425i	$-0.469540\pi$
$277$	3.47926	−	33.1029i	0.209048	−	1.98896i	0.113500	−	0.993538i	$-0.463794\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−27.0823	+	19.6764i	−1.61559	+	1.17380i	−0.775515	+	0.631329i	$0.782510\pi$
$281$	−27.0823	+	19.6764i	−1.61559	+	1.17380i	−0.840077	+	0.542467i	$0.817490\pi$
$282$		0			0
$283$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$283$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$284$	4.81953	−	45.8547i	0.285986	−	2.72098i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	17.3557	+	12.6096i	1.02269	+	0.743030i
$289$	−11.3752	−	12.6335i	−0.669131	−	0.743145i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$293$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$294$		0			0
$295$		0			0
$296$	−11.4694	+	19.8656i	−0.666644	+	1.15466i
$297$		0			0
$298$	24.2797	+	42.0536i	1.40648	+	2.43610i
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−4.25690	+	3.09282i	−0.244957	+	0.177972i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$311$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$312$		0			0
$313$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$313$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$314$		0			0
$315$		0			0
$316$	2.53698	−	7.80803i	0.142716	−	0.439236i
$317$	19.4458	−	8.65781i	1.09218	−	0.486271i	0.220024	−	0.975494i	$-0.429386\pi$
$317$	19.4458	−	8.65781i	1.09218	−	0.486271i	0.872159	+	0.489223i	$0.162720\pi$
$318$		0			0
$319$	−18.9866	−	17.7591i	−1.06304	−	0.994317i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−2.70177	−	25.7057i	−0.150099	−	1.42809i
$325$		0			0
$326$	31.4260	−	34.9021i	1.74053	−	1.93305i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−8.07856	+	13.9925i	−0.444038	+	0.769097i	−0.997985	−	0.0634557i	$-0.979788\pi$
$331$	−8.07856	+	13.9925i	−0.444038	+	0.769097i	0.553947	+	0.832552i	$0.313121\pi$
$332$		0			0
$333$	−34.9760	+	7.43438i	−1.91667	+	0.407402i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−3.35579	−	10.3281i	−0.182801	−	0.562605i	0.817102	−	0.576493i	$-0.195579\pi$
$337$	−3.35579	−	10.3281i	−0.182801	−	0.562605i	−0.999904	+	0.0138879i	$0.995579\pi$
$338$	19.2001	+	21.3239i	1.04435	+	1.15987i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	−7.62332	+	23.4622i	−0.411022	+	1.26499i
$345$		0			0
$346$		0			0
$347$	16.9333	+	7.53919i	0.909027	+	0.404725i	0.807337	−	0.590091i	$-0.200908\pi$
$347$	16.9333	+	7.53919i	0.909027	+	0.404725i	0.101690	+	0.994816i	$0.467575\pi$
$348$		0			0
$349$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$349$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$350$		0			0
$351$		0			0
$352$	−20.3109	−	12.2459i	−1.08257	−	0.652708i
$353$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$353$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	47.1400	−	34.2492i	2.49143	−	1.81013i
$359$	−36.8759	−	7.83822i	−1.94624	−	0.413685i	−0.994174	−	0.107783i	$-0.965625\pi$
$359$	−36.8759	−	7.83822i	−1.94624	−	0.413685i	−0.952063	−	0.305903i	$-0.901042\pi$
$360$		0			0
$361$	1.98604	−	18.8959i	0.104528	−	0.994522i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$367$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$368$	3.49901	+	1.55786i	0.182398	+	0.0812090i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−11.0000	+	19.0526i	−0.569558	+	0.986504i	0.427051	+	0.904227i	$0.359552\pi$
$373$	−11.0000	+	19.0526i	−0.569558	+	0.986504i	−0.996610	+	0.0822766i	$0.973781\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−25.4997	+	18.5267i	−1.30983	+	0.951650i	−0.309834	+	0.950791i	$0.600274\pi$
$379$	−25.4997	+	18.5267i	−1.30983	+	0.951650i	−1.00000		0.000859657i	$0.999726\pi$
$380$		0			0
$381$		0			0
$382$	19.6021	−	4.16655i	1.00293	−	0.213179i
$383$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$383$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$384$		0			0
$385$		0			0
$386$	4.68208			0.238312
$387$	−35.1308	+	15.6412i	−1.78580	+	0.795088i
$388$		0			0
$389$	−1.12254	+	1.24671i	−0.0569150	+	0.0632106i	−0.770930	−	0.636920i	$-0.780208\pi$
$389$	−1.12254	+	1.24671i	−0.0569150	+	0.0632106i	0.714015	+	0.700130i	$0.246875\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	−4.09452	+	1.82300i	−0.206279	+	0.0918414i
$395$		0			0
$396$	5.40996	+	28.0584i	0.271861	+	1.40999i
$397$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$397$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$398$		0			0
$399$		0			0
$400$	2.31133	+	7.11355i	0.115567	+	0.355677i
$401$	3.20240	+	30.4688i	0.159920	+	1.52154i	0.720509	+	0.693446i	$0.243908\pi$
$401$	3.20240	+	30.4688i	0.159920	+	1.52154i	−0.560589	+	0.828094i	$0.689425\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	37.8214	−	11.5006i	1.87474	−	0.570065i
$408$		0			0
$409$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$409$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−5.23912	−	16.1244i	−0.257489	−	0.792469i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	6.84802	−	21.0760i	0.333752	−	1.02718i	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	6.84802	−	21.0760i	0.333752	−	1.02718i	0.967333	−	0.253507i	$-0.0815842\pi$
$422$	−62.3323	−	13.2491i	−3.03429	−	0.644958i
$423$		0			0
$424$	−25.1603	−	11.2021i	−1.22189	−	0.544022i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−32.2024			−1.55656
$429$		0			0
$430$		0			0
$431$	1.59657	−	15.1904i	0.0769042	−	0.731695i	−0.886334	−	0.463046i	$-0.846756\pi$
$431$	1.59657	−	15.1904i	0.0769042	−	0.731695i	0.963238	−	0.268648i	$-0.0865769\pi$
$432$		0			0
$433$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$433$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$434$		0			0
$435$		0			0
$436$	−43.8997	−	9.33117i	−2.10241	−	0.446882i
$437$		0			0
$438$		0			0
$439$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$439$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−3.74146	−	4.15532i	−0.177762	−	0.197425i	0.647678	−	0.761914i	$-0.275740\pi$
$443$	−3.74146	−	4.15532i	−0.177762	−	0.197425i	−0.825441	+	0.564489i	$0.809073\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−33.0711	−	24.0276i	−1.56072	−	1.13393i	−0.935413	−	0.353556i	$-0.884972\pi$
$449$	−33.0711	−	24.0276i	−1.56072	−	1.13393i	−0.625310	−	0.780376i	$-0.715028\pi$
$450$	16.5543	−	28.6729i	0.780378	−	1.35165i
$451$		0			0
$452$	−15.5414	−	26.9185i	−0.731007	−	1.26614i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	4.40214	+	41.8835i	0.205923	+	1.95923i	0.274510	+	0.961584i	$0.411484\pi$
$457$	4.40214	+	41.8835i	0.205923	+	1.95923i	−0.0685868	+	0.997645i	$0.521849\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	23.0299			1.07029			0.535145	−	0.844760i	$-0.320257\pi$
$463$	23.0299			1.07029			0.535145	+	0.844760i	$0.320257\pi$
$464$	10.7122	−	4.76936i	0.497299	−	0.221412i
$465$		0			0
$466$	−32.4925	+	36.0866i	−1.50519	+	1.67168i
$467$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$467$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	37.2148	−	20.5549i	1.71114	−	0.945117i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−13.2668	−	40.8309i	−0.607443	−	1.86952i
$478$	−6.57450	−	62.5521i	−0.300710	−	2.86107i
$479$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$479$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−8.61575	−	30.3935i	−0.391625	−	1.38152i
$485$		0			0
$486$		0			0
$487$	−40.2882	+	8.56352i	−1.82563	+	0.388050i	−0.987526	−	0.157455i	$-0.949671\pi$
$487$	−40.2882	+	8.56352i	−1.82563	+	0.388050i	−0.838107	+	0.545506i	$0.816338\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	13.5967	+	41.8465i	0.613613	+	1.88851i	0.420363	+	0.907356i	$0.361903\pi$
$491$	13.5967	+	41.8465i	0.613613	+	1.88851i	0.193249	+	0.981150i	$0.438097\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	43.6997	+	9.28865i	1.95627	+	0.415817i	0.979270	+	0.202560i	$0.0649260\pi$
$499$	43.6997	+	9.28865i	1.95627	+	0.415817i	0.976996	+	0.213257i	$0.0684073\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$503$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$504$		0			0
$505$		0			0
$506$	7.29756	+	17.2646i	0.324416	+	0.767503i
$507$		0			0
$508$	−6.01650	+	57.2432i	−0.266939	+	2.53975i
$509$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$509$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$510$		0			0
$511$		0			0
$512$	13.3125	−	9.67211i	0.588336	−	0.427451i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$521$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$522$	−47.4174	−	21.1116i	−2.07541	−	0.924030i
$523$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$523$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$524$		0			0
$525$		0			0
$526$	−40.6751	−	29.5522i	−1.77352	−	1.28854i
$527$		0			0
$528$		0			0
$529$	8.22222	+	14.2413i	0.357488	+	0.619187i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−23.6568	+	5.02842i	−1.02182	+	0.217195i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	17.9864	−	8.00805i	0.773294	−	0.344293i	0.0181362	−	0.999836i	$-0.494227\pi$
$541$	17.9864	−	8.00805i	0.773294	−	0.344293i	0.755158	+	0.655543i	$0.227560\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	13.5967	−	41.8465i	0.581355	−	1.78923i	−0.0320849	−	0.999485i	$-0.510215\pi$
$547$	13.5967	−	41.8465i	0.581355	−	1.78923i	0.613440	−	0.789741i	$-0.289785\pi$
$548$	44.7063	−	19.9045i	1.90976	−	0.850280i
$549$		0			0
$550$	−15.5193	+	33.1501i	−0.661743	+	1.41353i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	22.7031	+	69.8729i	0.964561	+	2.96861i
$555$		0			0
$556$		0			0
$557$	−29.0639	+	32.2787i	−1.23148	+	1.36769i	−0.324847	+	0.945767i	$0.605313\pi$
$557$	−29.0639	+	32.2787i	−1.23148	+	1.36769i	−0.906630	+	0.421927i	$0.861354\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	36.9443	−	63.9894i	1.55840	−	2.69923i
$563$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$563$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	9.54785	+	29.3853i	0.400619	+	1.23298i
$569$	14.7209	+	16.3492i	0.617131	+	0.685394i	0.967977	−	0.251040i	$-0.0807725\pi$
$569$	14.7209	+	16.3492i	0.617131	+	0.685394i	−0.350846	+	0.936433i	$0.614106\pi$
$570$		0			0
$571$	−23.2644	−	40.2951i	−0.973583	−	1.68630i	−0.684533	−	0.728982i	$-0.739994\pi$
$571$	−23.2644	−	40.2951i	−0.973583	−	1.68630i	−0.289051	−	0.957314i	$-0.593340\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	3.95601	−	12.1753i	0.164977	−	0.507747i
$576$	−37.5374	−	7.97881i	−1.56406	−	0.332451i
$577$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$577$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$578$	34.2791	+	15.2620i	1.42582	+	0.634817i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	18.4792	+	43.7181i	0.765332	+	1.81062i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$587$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−1.86376	+	17.7325i	−0.0765999	+	0.728800i
$593$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$593$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$594$		0			0
$595$		0			0
$596$	−51.1155	−	37.1376i	−2.09377	−	1.52121i
$597$		0			0
$598$		0			0
$599$	41.2157	+	18.3504i	1.68403	+	0.749777i	0.999789	+	0.0205175i	$0.00653138\pi$
$599$	41.2157	+	18.3504i	1.68403	+	0.749777i	0.684237	+	0.729260i	$0.260135\pi$
$600$		0			0
$601$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$601$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$602$		0			0
$603$	−30.5004	−	22.1599i	−1.24207	−	0.902419i
$604$	3.42317	−	5.92910i	0.139287	−	0.241252i
$605$		0			0
$606$		0			0
$607$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$607$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−11.8170	+	2.51179i	−0.477286	+	0.101450i	−0.440272	−	0.897865i	$-0.645118\pi$
$613$	−11.8170	+	2.51179i	−0.477286	+	0.101450i	−0.0370140	+	0.999315i	$0.511785\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−3.84770			−0.154902			−0.0774512	−	0.996996i	$-0.524678\pi$
$617$	−3.84770			−0.154902			−0.0774512	+	0.996996i	$0.524678\pi$
$618$		0			0
$619$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$619$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	22.8386	−	10.1684i	0.913545	−	0.406737i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	15.5241	+	47.7782i	0.618003	+	1.90202i	0.318475	+	0.947931i	$0.396829\pi$
$631$	15.5241	+	47.7782i	0.618003	+	1.90202i	0.299528	+	0.954087i	$0.403171\pi$
$632$	0.575073	+	5.47146i	0.0228752	+	0.217643i
$633$		0			0
$634$	−31.4381	+	34.9155i	−1.24857	+	1.38667i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	54.2283	+	18.7641i	2.14692	+	0.742879i
$639$	−24.0818	+	41.7109i	−0.952662	+	1.65006i
$640$		0			0
$641$	24.2323	−	5.15074i	0.957120	−	0.203442i	0.297231	−	0.954805i	$-0.403937\pi$
$641$	24.2323	−	5.15074i	0.957120	−	0.203442i	0.659889	+	0.751363i	$0.270603\pi$
$642$		0			0
$643$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$643$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$647$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$648$	8.66040	+	15.0002i	0.340213	+	0.589265i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−18.8835	+	58.1175i	−0.739536	+	2.27606i
$653$	−24.9583	−	5.30505i	−0.976694	−	0.207603i	−0.308201	−	0.951321i	$-0.599727\pi$
$653$	−24.9583	−	5.30505i	−0.976694	−	0.207603i	−0.668493	+	0.743719i	$0.733060\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	44.0000			1.71400			0.856998	−	0.515319i	$-0.172327\pi$
$659$	44.0000			1.71400			0.856998	+	0.515319i	$0.172327\pi$
$660$		0			0
$661$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$661$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$662$	3.72777	−	35.4673i	0.144884	−	1.37848i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	63.8519	−	46.3911i	2.47421	−	1.79762i
$667$	−19.6312	−	4.17273i	−0.760121	−	0.161569i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	39.7653	+	28.8912i	1.53284	+	1.11367i	0.954633	+	0.297784i	$0.0962476\pi$
$673$	39.7653	+	28.8912i	1.53284	+	1.11367i	0.578208	+	0.815890i	$0.303752\pi$
$674$	16.0388	+	17.8129i	0.617793	+	0.686129i
$675$		0			0
$676$	−34.1072	−	15.1855i	−1.31181	−	0.584057i
$677$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$677$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	5.51818	+	9.55777i	0.211147	+	0.365718i	0.952074	−	0.305868i	$-0.0989467\pi$
$683$	5.51818	+	9.55777i	0.211147	+	0.365718i	−0.740926	+	0.671586i	$0.765613\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	2.00438	+	19.0704i	0.0764164	+	0.727053i
$689$		0			0
$690$		0			0
$691$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$691$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$692$		0			0
$693$		0			0
$694$	−40.9130			−1.55304
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	15.7423	−	48.4499i	0.594579	−	1.82993i	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	15.7423	−	48.4499i	0.594579	−	1.82993i	0.556810	−	0.830640i	$-0.312025\pi$
$702$		0			0
$703$		0			0
$704$	42.1021	+	5.23417i	1.58678	+	0.197270i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−3.75851	−	35.7598i	−0.141154	−	1.34299i	−0.804178	−	0.594389i	$-0.797394\pi$
$709$	−3.75851	−	35.7598i	−0.141154	−	1.34299i	0.663024	−	0.748598i	$-0.269273\pi$
$710$		0			0
$711$	−5.73846	+	6.37321i	−0.215209	+	0.239014i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−37.9074	+	65.6575i	−1.41667	+	2.45374i
$717$		0			0
$718$	81.3941	−	17.3009i	3.03760	−	0.645662i
$719$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$719$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$720$		0			0
$721$		0			0
$722$	12.9594	+	39.8850i	0.482300	+	1.48437i
$723$		0			0
$724$		0			0
$725$	−19.5964	−	33.9420i	−0.727793	−	1.26057i
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	−8.34346	+	25.6785i	−0.309017	+	0.951057i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$733$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$734$		0			0
$735$		0			0
$736$	−18.3091			−0.674883
$737$	35.6939	+	21.5206i	1.31480	+	0.792722i
$738$		0			0
$739$	−3.42206	+	32.5587i	−0.125883	+	1.19769i	0.731072	+	0.682300i	$0.239020\pi$
$739$	−3.42206	+	32.5587i	−0.125883	+	1.19769i	−0.856954	+	0.515392i	$0.827646\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−43.7942	+	31.8183i	−1.60665	+	1.16730i	−0.733729	+	0.679442i	$0.762222\pi$
$743$	−43.7942	+	31.8183i	−1.60665	+	1.16730i	−0.872923	+	0.487858i	$0.837778\pi$
$744$		0			0
$745$		0			0
$746$	5.07583	−	48.2933i	0.185839	−	1.76814i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−26.7622	−	29.7224i	−0.976565	−	1.08459i	−0.996400	−	0.0847817i	$-0.972981\pi$
$751$	−26.7622	−	29.7224i	−0.976565	−	1.08459i	0.0198348	−	0.999803i	$-0.493686\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−40.3760	−	29.3349i	−1.46749	−	1.06619i	−0.981332	−	0.192323i	$-0.938398\pi$
$757$	−40.3760	−	29.3349i	−1.46749	−	1.06619i	−0.486158	−	0.873871i	$-0.661602\pi$
$758$	34.7855	−	60.2502i	1.26347	−	2.18839i
$759$		0			0
$760$		0			0
$761$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$761$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$762$		0			0
$763$		0			0
$764$	−21.0949	+	15.3263i	−0.763186	+	0.554487i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$		0			0
$772$	−5.56534	+	2.47785i	−0.200301	+	0.0891797i
$773$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$773$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$774$	56.7961	−	63.0784i	2.04149	−	2.26731i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	1.14426	−	3.52166i	0.0410236	−	0.126258i
$779$		0			0
$780$		0			0
$781$	22.5761	−	48.2240i	0.807836	−	1.72559i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$787$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$788$	3.90217	−	4.33380i	0.139009	−	0.154385i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−10.9596	−	15.7024i	−0.389432	−	0.557961i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$797$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$798$		0			0
$799$		0			0
$800$	−23.9245	−	26.5709i	−0.845860	−	0.939423i
$801$		0			0
$802$	−33.8113	−	58.5628i	−1.19392	−	2.06792i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	26.0520	+	11.5991i	0.915941	+	0.407803i	0.809906	−	0.586559i	$-0.199518\pi$
$809$	26.0520	+	11.5991i	0.915941	+	0.407803i	0.106034	+	0.994362i	$0.466185\pi$
$810$		0			0
$811$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$811$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$812$		0			0
$813$		0			0
$814$	−65.9388	+	57.1448i	−2.31116	+	2.00293i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−21.5192	−	4.57406i	−0.751027	−	0.159636i	−0.183533	−	0.983013i	$-0.558754\pi$
$821$	−21.5192	−	4.57406i	−0.751027	−	0.159636i	−0.567494	+	0.823378i	$0.692087\pi$
$822$		0			0
$823$	−0.219913	+	2.09234i	−0.00766570	+	0.0729342i	−0.997686	−	0.0679910i	$-0.978341\pi$
$823$	−0.219913	+	2.09234i	−0.00766570	+	0.0729342i	0.990020	+	0.140925i	$0.0450077\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−35.5967	−	25.8626i	−1.23782	−	0.899329i	−0.240369	−	0.970682i	$-0.577268\pi$
$827$	−35.5967	−	25.8626i	−1.23782	−	0.899329i	−0.997451	+	0.0713526i	$0.977268\pi$
$828$	14.7608	+	16.3935i	0.512972	+	0.569713i
$829$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$829$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$839$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$840$		0			0
$841$	−26.2471	+	19.0696i	−0.905071	+	0.657573i
$842$	5.11290	+	48.6460i	0.176202	+	1.67645i
$843$		0			0
$844$	81.1027	−	17.2389i	2.79167	−	0.593387i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	−21.4077			−0.735145
$849$		0			0
$850$		0			0
$851$	20.4202	−	22.6790i	0.699996	−	0.777425i
$852$		0			0
$853$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$853$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$854$		0			0
$855$		0			0
$856$	19.7139	−	8.77718i	0.673806	−	0.299998i
$857$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$857$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$858$		0			0
$859$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$859$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$860$		0			0
$861$		0			0
$862$	10.4180	+	32.0635i	0.354840	+	1.09209i
$863$	−0.836228	−	7.95618i	−0.0284655	−	0.270831i	−0.999492	−	0.0318607i	$-0.989857\pi$
$863$	−0.836228	−	7.95618i	−0.0284655	−	0.270831i	0.971027	−	0.238971i	$-0.0768100\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	5.71733	−	7.56332i	0.193947	−	0.256568i
$870$		0			0
$871$		0			0
$872$	29.4181	−	6.25301i	0.996223	−	0.211754i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−39.5540	−	43.9291i	−1.33564	−	1.48338i	−0.714293	−	0.699847i	$-0.753251\pi$
$877$	−39.5540	−	43.9291i	−1.33564	−	1.48338i	−0.621349	−	0.783534i	$-0.713415\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	−3.70820	+	11.4127i	−0.124791	+	0.384067i	−0.993863	−	0.110619i	$-0.964717\pi$
$883$	−3.70820	+	11.4127i	−0.124791	+	0.384067i	0.869072	+	0.494686i	$0.164717\pi$
$884$		0			0
$885$		0			0
$886$	11.2749	+	5.01989i	0.378786	+	0.168646i
$887$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$887$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	6.75870	−	29.0744i	0.226425	−	0.974029i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	88.2563	+	18.7595i	2.94515	+	0.626011i
$899$		0			0
$900$	−4.50296	+	42.8428i	−0.150099	+	1.42809i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	16.8512	+	12.2431i	0.560464	+	0.407201i
$905$		0			0
$906$		0			0
$907$	−47.1858	−	21.0085i	−1.56678	−	0.697575i	−0.574148	−	0.818752i	$-0.694667\pi$
$907$	−47.1858	−	21.0085i	−1.56678	−	0.697575i	−0.992631	+	0.121177i	$0.961333\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−12.9443	−	9.40456i	−0.428863	−	0.311587i	0.352331	−	0.935875i	$-0.385389\pi$
$911$	−12.9443	−	9.40456i	−0.428863	−	0.311587i	−0.781194	+	0.624288i	$0.785389\pi$
$912$		0			0
$913$		0			0
$914$	−46.4782	−	80.5025i	−1.53736	−	2.66279i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−1.78335	−	16.9675i	−0.0588274	−	0.559706i	−0.983749	−	0.179548i	$-0.942536\pi$
$919$	−1.78335	−	16.9675i	−0.0588274	−	0.559706i	0.924922	−	0.380158i	$-0.124130\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	59.5957			1.95949
$926$	−46.4378	+	20.6754i	−1.52604	+	0.679437i
$927$		0			0
$928$	−37.5068	+	41.6556i	−1.23122	+	1.36741i
$929$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$929$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$930$		0			0
$931$		0			0
$932$	19.5244	−	60.0898i	0.639542	−	1.96831i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$937$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$941$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	−56.5869	+	74.8575i	−1.83980	+	2.43383i
$947$	10.0000	−	17.3205i	0.324956	−	0.562841i	−0.656547	−	0.754285i	$-0.727984\pi$
$947$	10.0000	−	17.3205i	0.324956	−	0.562841i	0.981504	+	0.191444i	$0.0613171\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	11.7590	+	36.1906i	0.380913	+	1.17233i	0.939402	+	0.342817i	$0.111381\pi$
$953$	11.7590	+	36.1906i	0.380913	+	1.17233i	−0.558489	+	0.829512i	$0.688619\pi$
$954$	63.4079	+	70.4216i	2.05291	+	2.27998i
$955$		0			0
$956$	40.9185	+	70.8729i	1.32340	+	2.29219i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	30.3226	+	6.44526i	0.978148	+	0.207912i
$962$		0			0
$963$	30.7304	+	13.6820i	0.990272	+	0.440898i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	32.9320			1.05902			0.529511	−	0.848303i	$-0.322376\pi$
$967$	32.9320			1.05902			0.529511	+	0.848303i	$0.322376\pi$
$968$	13.5586	+	16.2582i	0.435790	+	0.522557i
$969$		0			0
$970$		0			0
$971$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$971$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$972$		0			0
$973$		0			0
$974$	73.5498	−	53.4370i	2.35669	−	1.71223i
$975$		0			0
$976$		0			0
$977$	−5.69418	+	54.1765i	−0.182173	+	1.73326i	0.396793	+	0.917908i	$0.370123\pi$
$977$	−5.69418	+	54.1765i	−0.182173	+	1.73326i	−0.578966	+	0.815352i	$0.696544\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	37.9284	+	27.5566i	1.21096	+	0.879813i
$982$	−64.9851	−	72.1732i	−2.07376	−	2.30314i
$983$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$983$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	16.4101	−	28.4231i	0.521811	−	0.903802i
$990$		0			0
$991$	12.0000	+	20.7846i	0.381193	+	0.660245i	0.991233	−	0.132125i	$-0.0421802\pi$
$991$	12.0000	+	20.7846i	0.381193	+	0.660245i	−0.610040	+	0.792370i	$0.708847\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$997$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$998$	−96.4558	+	20.5023i	−3.05325	+	0.648989i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.q.d.422.1		16
7.2	even	3		539.2.f.c.246.2	✓	8
7.3	odd	6	inner	539.2.q.d.312.2		16
7.4	even	3	inner	539.2.q.d.312.2		16
7.5	odd	6		539.2.f.c.246.2	✓	8
7.6	odd	2	CM	539.2.q.d.422.1		16
11.3	even	5	inner	539.2.q.d.520.2		16
77.3	odd	30	inner	539.2.q.d.410.1		16
77.5	odd	30		5929.2.a.bc.1.2		4
77.16	even	15		5929.2.a.bc.1.2		4
77.25	even	15	inner	539.2.q.d.410.1		16
77.47	odd	30		539.2.f.c.344.2	yes	8
77.58	even	15		539.2.f.c.344.2	yes	8
77.61	even	30		5929.2.a.bg.1.3		4
77.69	odd	10	inner	539.2.q.d.520.2		16
77.72	odd	30		5929.2.a.bg.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.f.c.246.2	✓	8	7.2	even	3
539.2.f.c.246.2	✓	8	7.5	odd	6
539.2.f.c.344.2	yes	8	77.47	odd	30
539.2.f.c.344.2	yes	8	77.58	even	15
539.2.q.d.312.2		16	7.3	odd	6	inner
539.2.q.d.312.2		16	7.4	even	3	inner
539.2.q.d.410.1		16	77.3	odd	30	inner
539.2.q.d.410.1		16	77.25	even	15	inner
539.2.q.d.422.1		16	1.1	even	1	trivial
539.2.q.d.422.1		16	7.6	odd	2	CM
539.2.q.d.520.2		16	11.3	even	5	inner
539.2.q.d.520.2		16	77.69	odd	10	inner
5929.2.a.bc.1.2		4	77.5	odd	30
5929.2.a.bc.1.2		4	77.16	even	15
5929.2.a.bg.1.3		4	77.61	even	30
5929.2.a.bg.1.3		4	77.72	odd	30

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 539 = 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	539.q (of order \(15\), degree \(8\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.01642 + 0.897766i) q^{2} +(1.92169 - 2.13425i) q^{4} +(-0.594713 + 1.83034i) q^{8} +(-2.74064 + 1.22021i) q^{9} +(2.90321 - 1.60354i) q^{11} +(0.156367 + 1.48773i) q^{16} +(4.43080 - 4.92090i) q^{18} +(-4.41448 + 5.83981i) q^{22} +(1.28019 - 2.21736i) q^{23} +(4.89074 - 1.03956i) q^{25} +(-2.42225 - 7.45492i) q^{29} +(-3.57547 - 6.19289i) q^{32} +(-2.66241 + 8.19407i) q^{36} +(11.6587 + 2.47813i) q^{37} +12.8185 q^{43} +(2.15671 - 9.27769i) q^{44} +(-0.590730 + 5.62042i) q^{46} +(-8.92848 + 6.48692i) q^{50} +(-1.49588 + 14.2323i) q^{53} +(11.5770 + 12.8576i) q^{58} +(10.3489 + 7.51894i) q^{64} +(6.28343 + 10.8832i) q^{67} +(12.9884 - 9.43662i) q^{71} +(-0.603505 - 5.74197i) q^{72} +(-25.7335 + 5.46983i) q^{74} +(2.61152 - 1.16272i) q^{79} +(6.02218 - 6.68830i) q^{81} +(-25.8474 + 11.5080i) q^{86} +(1.20844 + 6.26752i) q^{88} +(-2.27227 - 6.99331i) q^{92} +(-6.00000 + 7.93725i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{2} - 2 q^{4} + 32 q^{8} - 6 q^{9} + 4 q^{11} + 28 q^{16} + 9 q^{18} - 8 q^{22} + 16 q^{23} - 10 q^{25} - 8 q^{29} - 100 q^{32} - 12 q^{36} + 18 q^{37} + 48 q^{43} - 9 q^{44} + 31 q^{46} + 20 q^{50}+ \cdots - 96 q^{99}+O(q^{100}) \) 16 * q + 2 * q^2 - 2 * q^4 + 32 * q^8 - 6 * q^9 + 4 * q^11 + 28 * q^16 + 9 * q^18 - 8 * q^22 + 16 * q^23 - 10 * q^25 - 8 * q^29 - 100 * q^32 - 12 * q^36 + 18 * q^37 + 48 * q^43 - 9 * q^44 + 31 * q^46 + 20 * q^50 + 30 * q^53 - 71 * q^58 - 148 * q^64 + 8 * q^67 + 96 * q^71 - 57 * q^72 - 12 * q^74 - 24 * q^79 + 18 * q^81 - 29 * q^86 + 38 * q^88 + 142 * q^92 - 96 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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