LMFDB - Embedded newform 539.2.q.d.361.2

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			361.2
Root			$1.26336 + 0.635539i$ of defining polynomial
Character	$\chi$	$=$	539.361
Dual form			539.2.q.d.324.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.31249 - 0.491535i) q^{2} +(3.27890 - 1.45986i) q^{4} +(3.03958 - 2.20838i) q^{8} +(2.93444 - 0.623735i) q^{9} +(-1.40621 - 3.00376i) q^{11} +(1.14020 - 1.26632i) q^{16} +(6.47928 - 2.88476i) q^{18} +(-4.72830 - 6.25495i) q^{22} +(4.79120 + 8.29860i) q^{23} +(0.522642 + 4.97261i) q^{25} +(-8.64279 - 6.27935i) q^{29} +(-1.74287 + 3.01874i) q^{32} +(8.71119 - 6.32905i) q^{36} +(0.142831 - 1.35895i) q^{37} -8.74072 q^{43} +(-8.99591 - 7.79616i) q^{44} +(15.1586 + 16.8354i) q^{46} +(3.65281 + 11.2422i) q^{50} +(-8.80255 - 9.77622i) q^{53} +(-23.0729 - 10.2727i) q^{58} +(-3.59968 + 11.0787i) q^{64} +(-8.16681 + 14.1453i) q^{67} +(3.08300 + 9.48849i) q^{71} +(7.54203 - 8.37627i) q^{72} +(-0.337674 - 3.21276i) q^{74} +(12.3495 - 2.62497i) q^{79} +(8.22191 - 3.66063i) q^{81} +(-20.2128 + 4.29637i) q^{86} +(-10.9077 - 6.02471i) q^{88} +(27.8247 + 20.2158i) q^{92} +(-6.00000 - 7.93725i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/539\mathbb{Z}\right)^\times$.

$n$	$199$	$442$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{3}{5}\right)$

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 $

$2$	2.31249	−	0.491535i	1.63518	−	0.347567i	0.703454	−	0.710740i	$-0.251640\pi$
$2$	2.31249	−	0.491535i	1.63518	−	0.347567i	0.931722	+	0.363173i	$0.118307\pi$
$3$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$3$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$4$	3.27890	−	1.45986i	1.63945	−	0.729931i
$5$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$5$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	3.03958	−	2.20838i	1.07465	−	0.780782i
$9$	2.93444	−	0.623735i	0.978148	−	0.207912i
$10$		0			0
$11$	−1.40621	−	3.00376i	−0.423989	−	0.905667i
$11$	−1.40621	−	3.00376i	−0.423989	−	0.905667i
$12$		0			0
$13$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$13$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$14$		0			0
$15$		0			0
$16$	1.14020	−	1.26632i	0.285049	−	0.316579i
$17$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$17$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$18$	6.47928	−	2.88476i	1.52718	−	0.679944i
$19$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$19$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$20$		0			0
$21$		0			0
$22$	−4.72830	−	6.25495i	−1.00808	−	1.33356i
$23$	4.79120	+	8.29860i	0.999035	+	1.73038i	0.537562	+	0.843224i	$0.319345\pi$
$23$	4.79120	+	8.29860i	0.999035	+	1.73038i	0.461472	+	0.887155i	$0.347321\pi$
$24$		0			0
$25$	0.522642	+	4.97261i	0.104528	+	0.994522i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−8.64279	−	6.27935i	−1.60492	−	1.16605i	−0.877132	−	0.480249i	$-0.840546\pi$
$29$	−8.64279	−	6.27935i	−1.60492	−	1.16605i	−0.727793	−	0.685797i	$-0.759454\pi$
$30$		0			0
$31$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$31$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$32$	−1.74287	+	3.01874i	−0.308099	+	0.533644i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	8.71119	−	6.32905i	1.45187	−	1.05484i
$37$	0.142831	−	1.35895i	0.0234813	−	0.223410i	−0.976488	−	0.215572i	$-0.930838\pi$
$37$	0.142831	−	1.35895i	0.0234813	−	0.223410i	0.999969	−	0.00783774i	$-0.00249486\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$41$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$42$		0			0
$43$	−8.74072			−1.33295			−0.666474	−	0.745528i	$-0.732197\pi$
$43$	−8.74072			−1.33295			−0.666474	+	0.745528i	$0.732197\pi$
$44$	−8.99591	−	7.79616i	−1.35618	−	1.17532i
$45$		0			0
$46$	15.1586	+	16.8354i	2.23502	+	2.48224i
$47$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$47$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$48$		0			0
$49$		0			0
$50$	3.65281	+	11.2422i	0.516586	+	1.58989i
$51$		0			0
$52$		0			0
$53$	−8.80255	−	9.77622i	−1.20912	−	1.34287i	−0.923060	−	0.384657i	$-0.874320\pi$
$53$	−8.80255	−	9.77622i	−1.20912	−	1.34287i	−0.286064	−	0.958211i	$-0.592347\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−23.0729	−	10.2727i	−3.02961	−	1.34887i
$59$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$59$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$60$		0			0
$61$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$61$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$62$		0			0
$63$		0			0
$64$	−3.59968	+	11.0787i	−0.449961	+	1.38484i
$65$		0			0
$66$		0			0
$67$	−8.16681	+	14.1453i	−0.997735	+	1.72813i	−0.440609	+	0.897699i	$0.645238\pi$
$67$	−8.16681	+	14.1453i	−0.997735	+	1.72813i	−0.557125	+	0.830428i	$0.688096\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	3.08300	+	9.48849i	0.365885	+	1.12608i	0.949425	+	0.313993i	$0.101667\pi$
$71$	3.08300	+	9.48849i	0.365885	+	1.12608i	−0.583541	+	0.812084i	$0.698333\pi$
$72$	7.54203	−	8.37627i	0.888836	−	0.987153i
$73$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$73$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$74$	−0.337674	−	3.21276i	−0.0392538	−	0.373475i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	12.3495	−	2.62497i	1.38943	−	0.295332i	0.548352	−	0.836247i	$-0.315255\pi$
$79$	12.3495	−	2.62497i	1.38943	−	0.295332i	0.841077	+	0.540915i	$0.181922\pi$
$80$		0			0
$81$	8.22191	−	3.66063i	0.913545	−	0.406737i
$82$		0			0
$83$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$83$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$84$		0			0
$85$		0			0
$86$	−20.2128	+	4.29637i	−2.17960	+	0.463289i
$87$		0			0
$88$	−10.9077	−	6.02471i	−1.16277	−	0.642236i
$89$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$89$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$90$		0			0
$91$		0			0
$92$	27.8247	+	20.2158i	2.90093	+	2.10765i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$97$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$98$		0			0
$99$	−6.00000	−	7.93725i	−0.603023	−	0.797724i
$100$	8.97302	+	15.5417i	0.897302	+	1.55417i
$101$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$101$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$102$		0			0
$103$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$103$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$104$		0			0
$105$		0			0
$106$	−25.1611	−	18.2806i	−2.44387	−	1.77557i
$107$	11.9401	+	5.31608i	1.15429	+	0.513925i	0.892432	−	0.451181i	$-0.148997\pi$
$107$	11.9401	+	5.31608i	1.15429	+	0.513925i	0.261861	+	0.965106i	$0.415664\pi$
$108$		0			0
$109$	4.17089	−	7.22419i	0.399498	−	0.691952i	−0.594166	−	0.804343i	$-0.702518\pi$
$109$	4.17089	−	7.22419i	0.399498	−	0.691952i	0.993664	+	0.112391i	$0.0358510\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	15.7856	−	11.4689i	1.48498	−	1.07890i	0.509073	−	0.860724i	$-0.329988\pi$
$113$	15.7856	−	11.4689i	1.48498	−	1.07890i	0.975909	−	0.218179i	$-0.0700116\pi$
$114$		0			0
$115$		0			0
$116$	−37.5059	−	7.97212i	−3.48233	−	0.740192i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−7.04513	+	8.44785i	−0.640467	+	0.767986i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−1.11662	−	3.43661i	−0.0990843	−	0.304950i	0.889212	−	0.457495i	$-0.151253\pi$
$127$	−1.11662	−	3.43661i	−0.0990843	−	0.304950i	−0.988297	+	0.152545i	$0.951253\pi$
$128$	−2.14995	+	20.4554i	−0.190031	+	1.80802i
$129$		0			0
$130$		0			0
$131$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$131$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$132$		0			0
$133$		0			0
$134$	−11.9327	+	36.7252i	−1.03083	+	3.17257i
$135$		0			0
$136$		0			0
$137$	−20.0826	−	4.26868i	−1.71577	−	0.364698i	−0.758005	−	0.652249i	$-0.773826\pi$
$137$	−20.0826	−	4.26868i	−1.71577	−	0.364698i	−0.957766	+	0.287550i	$0.907159\pi$
$138$		0			0
$139$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$139$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$140$		0			0
$141$		0			0
$142$	11.7933	+	20.4266i	0.989673	+	1.71416i
$143$		0			0
$144$	2.55600	−	4.42712i	0.213000	−	0.368926i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−1.51555	−	4.66437i	−0.124577	−	0.383409i
$149$	14.7209	−	16.3492i	1.20598	−	1.33938i	0.280840	−	0.959755i	$-0.409387\pi$
$149$	14.7209	−	16.3492i	1.20598	−	1.33938i	0.925141	−	0.379623i	$-0.123946\pi$
$150$		0			0
$151$	−2.35468	−	22.4033i	−0.191621	−	1.82315i	−0.493432	−	0.869785i	$-0.664258\pi$
$151$	−2.35468	−	22.4033i	−0.191621	−	1.82315i	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		0.994522	−	0.104528i	$-0.0333333\pi$
$157$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$158$	27.2679	−	12.1404i	2.16931	−	0.965841i
$159$		0			0
$160$		0			0
$161$		0			0
$162$	17.2137	−	12.5065i	1.35244	−	0.982605i
$163$	−6.69986	+	1.42410i	−0.524773	+	0.111544i	−0.462678	−	0.886526i	$-0.653112\pi$
$163$	−6.69986	+	1.42410i	−0.524773	+	0.111544i	−0.0620951	+	0.998070i	$0.519778\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$167$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$168$		0			0
$169$	10.5172	+	7.64121i	0.809017	+	0.587785i
$170$		0			0
$171$		0			0
$172$	−28.6600	+	12.7603i	−2.18530	+	0.972960i
$173$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$173$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$174$		0			0
$175$		0			0
$176$	−5.40707	−	1.64417i	−0.407573	−	0.123934i
$177$		0			0
$178$		0			0
$179$	−1.28730	−	12.2478i	−0.0962170	−	0.915444i	−0.931038	−	0.364922i	$-0.881096\pi$
$179$	−1.28730	−	12.2478i	−0.0962170	−	0.915444i	0.834821	−	0.550522i	$-0.185571\pi$
$180$		0			0
$181$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$181$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$182$		0			0
$183$		0			0
$184$	32.8897	+	14.6435i	2.42466	+	1.07953i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2.37180	−	22.5662i	0.171617	−	1.63283i	−0.482118	−	0.876106i	$-0.660132\pi$
$191$	2.37180	−	22.5662i	0.171617	−	1.63283i	0.653735	−	0.756723i	$-0.273201\pi$
$192$		0			0
$193$	14.2494	+	3.02881i	1.02570	+	0.218018i	0.689891	−	0.723913i	$-0.257658\pi$
$193$	14.2494	+	3.02881i	1.02570	+	0.218018i	0.335805	+	0.941932i	$0.390992\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	27.2550			1.94184			0.970918	−	0.239411i	$-0.0769543\pi$
$197$	27.2550			1.94184			0.970918	+	0.239411i	$0.0769543\pi$
$198$	−17.7764	−	15.4056i	−1.26331	−	1.09483i
$199$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$199$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$200$	12.5700	+	13.9604i	0.888836	+	0.987153i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	19.2356	+	21.3633i	1.33697	+	1.48485i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−1.80563	+	5.55715i	−0.124304	+	0.382570i	−0.993774	−	0.111417i	$-0.964461\pi$
$211$	−1.80563	+	5.55715i	−0.124304	+	0.382570i	0.869469	+	0.493987i	$0.164461\pi$
$212$	−43.1347	−	19.2048i	−2.96250	−	1.31899i
$213$		0			0
$214$	30.2244	+	6.42439i	2.06610	+	0.439162i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	6.09419	−	18.7560i	0.412750	−	1.27032i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$223$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$224$		0			0
$225$	4.63525	+	14.2658i	0.309017	+	0.951057i
$226$	30.8666	−	34.2808i	2.05322	−	2.28033i
$227$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$227$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$228$		0			0
$229$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$229$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$230$		0			0
$231$		0			0
$232$	−40.1377			−2.63517
$233$	−21.5192	+	4.57406i	−1.40977	+	0.299656i	−0.849035	−	0.528337i	$-0.822816\pi$
$233$	−21.5192	+	4.57406i	−1.40977	+	0.299656i	−0.560738	+	0.827993i	$0.689482\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−24.3570	+	17.6964i	−1.57552	+	1.14468i	−0.653907	+	0.756575i	$0.726871\pi$
$239$	−24.3570	+	17.6964i	−1.57552	+	1.14468i	−0.921614	+	0.388108i	$0.873129\pi$
$240$		0			0
$241$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$241$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$242$	−12.1394	+	22.9985i	−0.780349	+	1.47840i
$243$		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		−0.951057	−	0.309017i	$-0.900000\pi$
$251$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$252$		0			0
$253$	18.1896	−	26.0612i	1.14357	−	1.63845i
$254$	−4.27139	−	7.39827i	−0.268011	−	0.464209i
$255$		0			0
$256$	2.64754	+	25.1896i	0.165471	+	1.57435i
$257$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$257$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−29.2784	−	13.0356i	−1.81229	−	0.806883i
$262$		0			0
$263$	−2.42801	+	4.20544i	−0.149718	+	0.259319i	−0.931123	−	0.364705i	$-0.881170\pi$
$263$	−2.42801	+	4.20544i	−0.149718	+	0.259319i	0.781405	+	0.624024i	$0.214503\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−6.12796	+	58.3036i	−0.374325	+	3.56146i
$269$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$269$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$270$		0			0
$271$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$271$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$272$		0			0
$273$		0			0
$274$	−48.5389			−2.93234
$275$	14.2016	−	8.56244i	0.856387	−	0.516334i
$276$		0			0
$277$	−7.07367	−	7.85611i	−0.425016	−	0.472028i	0.492164	−	0.870503i	$-0.336206\pi$
$277$	−7.07367	−	7.85611i	−0.425016	−	0.472028i	−0.917179	+	0.398475i	$0.869540\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−8.70331	−	26.7860i	−0.519196	−	1.59792i	−0.775515	−	0.631329i	$-0.782510\pi$
$281$	−8.70331	−	26.7860i	−0.519196	−	1.59792i	0.256319	−	0.966592i	$-0.417490\pi$
$282$		0			0
$283$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$283$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$284$	23.9607	+	26.6111i	1.42181	+	1.57908i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−3.23146	+	9.94542i	−0.190416	+	0.586040i
$289$	−15.5303	−	6.91452i	−0.913545	−	0.406737i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$293$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$294$		0			0
$295$		0			0
$296$	−2.56693	−	4.44605i	−0.149200	−	0.258422i
$297$		0			0
$298$	26.0057	−	45.0431i	1.50647	−	2.60928i
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−16.4571	−	50.6499i	−0.947002	−	2.91457i
$303$		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		0.994522	−	0.104528i	$-0.0333333\pi$
$311$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$312$		0			0
$313$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$313$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$314$		0			0
$315$		0			0
$316$	36.6608	−	26.6356i	2.06233	−	1.49837i
$317$	0.431893	−	0.0918017i	0.0242575	−	0.00515610i	−0.195767	−	0.980650i	$-0.562720\pi$
$317$	0.431893	−	0.0918017i	0.0242575	−	0.00515610i	0.220024	+	0.975494i	$0.429386\pi$
$318$		0			0
$319$	−6.70806	+	34.7909i	−0.375579	+	1.94792i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	21.6148	−	24.0057i	1.20082	−	1.33365i
$325$		0			0
$326$	−14.7934	+	6.58642i	−0.819328	+	0.364788i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	13.0072	+	22.5291i	0.714939	+	1.23831i	0.962983	+	0.269562i	$0.0868788\pi$
$331$	13.0072	+	22.5291i	0.714939	+	1.23831i	−0.248044	+	0.968749i	$0.579788\pi$
$332$		0			0
$333$	−0.428493	−	4.07684i	−0.0234813	−	0.223410i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	29.7003	+	21.5785i	1.61788	+	1.17546i	0.817102	+	0.576493i	$0.195579\pi$
$337$	29.7003	+	21.5785i	1.61788	+	1.17546i	0.800776	+	0.598964i	$0.204421\pi$
$338$	28.0769	+	12.5006i	1.52718	+	0.679944i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	−26.5681	+	19.3029i	−1.43246	+	1.04074i
$345$		0			0
$346$		0			0
$347$	33.2488	+	7.06724i	1.78489	+	0.379389i	0.977550	−	0.210702i	$-0.0675749\pi$
$347$	33.2488	+	7.06724i	1.78489	+	0.379389i	0.807337	+	0.590091i	$0.200908\pi$
$348$		0			0
$349$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$349$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$350$		0			0
$351$		0			0
$352$	11.5184	+	0.990173i	0.613934	+	0.0527764i
$353$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$353$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−8.99708	−	27.6902i	−0.475510	−	1.46347i
$359$	−1.59926	+	15.2159i	−0.0844056	+	0.803066i	0.867657	+	0.497163i	$0.165625\pi$
$359$	−1.59926	+	15.2159i	−0.0844056	+	0.803066i	−0.952063	+	0.305903i	$0.901042\pi$
$360$		0			0
$361$	−12.7135	−	14.1198i	−0.669131	−	0.743145i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$367$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$368$	15.9716	+	3.39486i	0.832576	+	0.176970i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−11.0000	−	19.0526i	−0.569558	−	0.986504i	−0.996610	−	0.0822766i	$-0.973781\pi$
$373$	−11.0000	−	19.0526i	−0.569558	−	0.986504i	0.427051	−	0.904227i	$-0.359552\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	9.72788	+	29.9393i	0.499688	+	1.53788i	0.809522	+	0.587090i	$0.199726\pi$
$379$	9.72788	+	29.9393i	0.499688	+	1.53788i	−0.309834	+	0.950791i	$0.600274\pi$
$380$		0			0
$381$		0			0
$382$	−5.60729	−	53.3498i	−0.286894	−	2.72961i
$383$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$383$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$384$		0			0
$385$		0			0
$386$	34.4404			1.75297
$387$	−25.6491	+	5.45189i	−1.30382	+	0.277135i
$388$		0			0
$389$	33.7675	−	15.0343i	1.71208	−	0.762268i	0.714015	−	0.700130i	$-0.246875\pi$
$389$	33.7675	−	15.0343i	1.71208	−	0.762268i	0.998068	−	0.0621378i	$-0.0197918\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	63.0268	−	13.3968i	3.17524	−	0.674919i
$395$		0			0
$396$	−31.2607	−	17.2663i	−1.57091	−	0.867665i
$397$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$397$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$398$		0			0
$399$		0			0
$400$	6.89282	+	5.00793i	0.344641	+	0.250396i
$401$	10.0808	−	11.1959i	0.503411	−	0.559094i	−0.436857	−	0.899531i	$-0.643908\pi$
$401$	10.0808	−	11.1959i	0.503411	−	0.559094i	0.940267	+	0.340437i	$0.110575\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−4.28280	+	1.48194i	−0.212290	+	0.0734570i
$408$		0			0
$409$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$409$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	54.9830	+	39.9475i	2.70227	+	1.96331i
$415$		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
$421$	−32.1148	+	23.3327i	−1.56518	+	1.13717i	−0.633581	+	0.773676i	$0.718416\pi$
$421$	−32.1148	+	23.3327i	−1.56518	+	1.13717i	−0.931597	+	0.363492i	$0.881584\pi$
$422$	−1.44396	+	13.7384i	−0.0702909	+	0.668773i
$423$		0			0
$424$	−48.3457	−	10.2762i	−2.34788	−	0.499056i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	46.9112			2.26754
$429$		0			0
$430$		0			0
$431$	−27.7287	−	30.7958i	−1.33564	−	1.48338i	−0.714275	−	0.699865i	$-0.753244\pi$
$431$	−27.7287	−	30.7958i	−1.33564	−	1.48338i	−0.621369	−	0.783518i	$-0.713423\pi$
$432$		0			0
$433$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$433$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$434$		0			0
$435$		0			0
$436$	3.12962	−	29.7763i	0.149882	−	1.42603i
$437$		0			0
$438$		0			0
$439$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$439$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	26.5360	+	11.8146i	1.26076	+	0.561328i	0.924767	−	0.380535i	$-0.124260\pi$
$443$	26.5360	+	11.8146i	1.26076	+	0.561328i	0.335997	+	0.941863i	$0.390927\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	7.18900	−	22.1255i	0.339270	−	1.04417i	−0.625310	−	0.780376i	$-0.715028\pi$
$449$	7.18900	−	22.1255i	0.339270	−	1.04417i	0.964580	−	0.263790i	$-0.0849724\pi$
$450$	17.7311	+	30.7112i	0.835853	+	1.44774i
$451$		0			0
$452$	35.0164	−	60.6502i	1.64703	−	2.85274i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−13.4014	+	14.8837i	−0.626890	+	0.696232i	−0.970011	−	0.243059i	$-0.921849\pi$
$457$	−13.4014	+	14.8837i	−0.626890	+	0.696232i	0.343122	+	0.939291i	$0.388516\pi$
$458$		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
$463$	2.73687			0.127193			0.0635967	−	0.997976i	$-0.479743\pi$
$463$	2.73687			0.127193			0.0635967	+	0.997976i	$0.479743\pi$
$464$	−17.8061	+	3.78481i	−0.826629	+	0.175705i
$465$		0			0
$466$	−47.5147	+	21.1549i	−2.20108	+	0.979982i
$467$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$467$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	12.2913	+	26.2550i	0.565155	+	1.20721i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−31.9284	−	23.1973i	−1.46190	−	1.06213i
$478$	−47.6268	+	52.8949i	−2.17840	+	2.41936i
$479$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$479$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−10.7676	+	37.9846i	−0.489437	+	1.72657i
$485$		0			0
$486$		0			0
$487$	4.45752	+	42.4104i	0.201989	+	1.92180i	0.357403	+	0.933950i	$0.383662\pi$
$487$	4.45752	+	42.4104i	0.201989	+	1.92180i	−0.155414	+	0.987850i	$0.549671\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−35.5967	−	25.8626i	−1.60646	−	1.16716i	−0.873412	−	0.486983i	$-0.838097\pi$
$491$	−35.5967	−	25.8626i	−1.60646	−	1.16716i	−0.733047	−	0.680178i	$-0.761903\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−3.79305	+	36.0884i	−0.169800	+	1.61554i	0.495257	+	0.868747i	$0.335074\pi$
$499$	−3.79305	+	36.0884i	−0.169800	+	1.61554i	−0.665057	+	0.746793i	$0.731593\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$503$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$504$		0			0
$505$		0			0
$506$	29.2532	−	69.2070i	1.30046	−	3.07663i
$507$		0			0
$508$	−8.67829	−	9.63821i	−0.385037	−	0.427627i
$509$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$509$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$510$		0			0
$511$		0			0
$512$	5.79221	+	17.8266i	0.255982	+	0.787831i
$513$		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		−0.406737	−	0.913545i	$-0.633333\pi$
$521$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$522$	−74.1134	−	15.7533i	−3.24386	−	0.689503i
$523$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$523$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$524$		0			0
$525$		0			0
$526$	−3.54763	+	10.9185i	−0.154684	+	0.476069i
$527$		0			0
$528$		0			0
$529$	−34.4112	+	59.6020i	−1.49614	+	2.59139i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	6.41465	+	61.0313i	0.277071	+	2.63615i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−45.1593	+	9.59890i	−1.94155	+	0.412689i	−0.945296	+	0.326215i	$0.894227\pi$
$541$	−45.1593	+	9.59890i	−1.94155	+	0.412689i	−0.996254	+	0.0864744i	$0.972440\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−35.5967	+	25.8626i	−1.52201	+	1.10580i	−0.561525	+	0.827460i	$0.689785\pi$
$547$	−35.5967	+	25.8626i	−1.52201	+	1.10580i	−0.960482	+	0.278343i	$0.910215\pi$
$548$	−72.0806	+	15.3212i	−3.07913	+	0.654489i
$549$		0			0
$550$	28.6322	−	26.7811i	1.22088	−	1.14195i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−20.2193	−	14.6902i	−0.859037	−	0.624127i
$555$		0			0
$556$		0			0
$557$	22.1807	−	9.87549i	0.939827	−	0.418438i	0.121113	−	0.992639i	$-0.461354\pi$
$557$	22.1807	−	9.87549i	0.939827	−	0.418438i	0.818715	+	0.574201i	$0.194687\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−33.2926	−	57.6644i	−1.40436	−	2.43243i
$563$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$563$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	30.3252	+	22.0326i	1.27242	+	0.924467i
$569$	20.0980	+	8.94821i	0.842552	+	0.375128i	0.782184	−	0.623048i	$-0.214106\pi$
$569$	20.0980	+	8.94821i	0.842552	+	0.375128i	0.0603683	+	0.998176i	$0.480772\pi$
$570$		0			0
$571$	−12.3782	+	21.4396i	−0.518010	+	0.897220i	0.481771	+	0.876297i	$0.339994\pi$
$571$	−12.3782	+	21.4396i	−0.518010	+	0.897220i	−0.999781	+	0.0209228i	$0.993340\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−38.7616	+	28.1620i	−1.61647	+	1.17444i
$576$	−3.65290	+	34.7550i	−0.152204	+	1.44813i
$577$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$577$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$578$	−39.3123	−	8.35609i	−1.63518	−	0.347567i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−16.9872	+	40.1882i	−0.703536	+	1.66442i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$587$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−1.55800	−	1.73034i	−0.0640335	−	0.0711164i
$593$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$593$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$594$		0			0
$595$		0			0
$596$	24.4008	−	75.0979i	0.999495	−	3.07613i
$597$		0			0
$598$		0			0
$599$	4.02671	+	0.855905i	0.164527	+	0.0349713i	0.289439	−	0.957197i	$-0.406531\pi$
$599$	4.02671	+	0.855905i	0.164527	+	0.0349713i	−0.124912	+	0.992168i	$0.539865\pi$
$600$		0			0
$601$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$601$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$602$		0			0
$603$	−15.1421	+	46.6026i	−0.616634	+	1.89780i
$604$	−40.4265	−	70.0207i	−1.64493	−	2.84910i
$605$		0			0
$606$		0			0
$607$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$607$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−1.92881	−	18.3514i	−0.0779038	−	0.741205i	−0.961843	−	0.273603i	$-0.911785\pi$
$613$	−1.92881	−	18.3514i	−0.0779038	−	0.741205i	0.883939	−	0.467602i	$-0.154882\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	32.2257			1.29736			0.648679	−	0.761062i	$-0.275322\pi$
$617$	32.2257			1.29736			0.648679	+	0.761062i	$0.275322\pi$
$618$		0			0
$619$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$619$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−24.4537	+	5.19779i	−0.978148	+	0.207912i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−12.1742	−	8.84507i	−0.484647	−	0.352117i	0.318475	−	0.947931i	$-0.396829\pi$
$631$	−12.1742	−	8.84507i	−0.484647	−	0.352117i	−0.803122	+	0.595815i	$0.796829\pi$
$632$	31.7404	−	35.2513i	1.26257	−	1.40222i
$633$		0			0
$634$	0.953624	−	0.424581i	0.0378732	−	0.0168623i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	1.58864	+	83.7509i	0.0628947	+	3.31573i
$639$	14.9652	+	25.9205i	0.592014	+	1.02540i
$640$		0			0
$641$	0.618316	+	5.88289i	0.0244220	+	0.232360i	0.999924	+	0.0123672i	$0.00393672\pi$
$641$	0.618316	+	5.88289i	0.0244220	+	0.232360i	−0.975502	+	0.219993i	$0.929397\pi$
$642$		0			0
$643$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$643$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$647$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$648$	16.9071	−	29.2839i	0.664173	−	1.15038i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−19.8892	+	14.4504i	−0.778922	+	0.565920i
$653$	3.57804	−	34.0428i	0.140020	−	1.33220i	−0.668493	−	0.743719i	$-0.733060\pi$
$653$	3.57804	−	34.0428i	0.140020	−	1.33220i	0.808512	−	0.588479i	$-0.200273\pi$
$654$		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	$-0.833333\pi$
$661$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$662$	41.1528	+	45.7048i	1.59945	+	1.77637i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−2.99479	−	9.21703i	−0.116046	−	0.357153i
$667$	10.7005	−	101.809i	0.414326	−	3.94205i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	15.3058	−	47.1065i	0.589996	−	1.81582i	0.0117883	−	0.999931i	$-0.496248\pi$
$673$	15.3058	−	47.1065i	0.589996	−	1.81582i	0.578208	−	0.815890i	$-0.303752\pi$
$674$	79.2882	+	35.3014i	3.05407	+	1.35976i
$675$		0			0
$676$	45.6401	+	9.70110i	1.75539	+	0.373119i
$677$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$677$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−22.5896	+	39.1263i	−0.864366	+	1.49713i	0.00330916	+	0.999995i	$0.498947\pi$
$683$	−22.5896	+	39.1263i	−0.864366	+	1.49713i	−0.867675	+	0.497131i	$0.834387\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−9.96615	+	11.0685i	−0.379956	+	0.421984i
$689$		0			0
$690$		0			0
$691$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$691$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$692$		0			0
$693$		0			0
$694$	80.3612			3.05047
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−23.8536	+	17.3306i	−0.900937	+	0.654569i	−0.938707	−	0.344717i	$-0.887975\pi$
$701$	−23.8536	+	17.3306i	−0.900937	+	0.654569i	0.0377695	+	0.999286i	$0.487975\pi$
$702$		0			0
$703$		0			0
$704$	38.3396	−	4.76641i	1.44498	−	0.179641i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−34.9148	+	38.7768i	−1.31125	+	1.45629i	−0.507074	+	0.861903i	$0.669273\pi$
$709$	−34.9148	+	38.7768i	−1.31125	+	1.45629i	−0.804178	+	0.594389i	$0.797394\pi$
$710$		0			0
$711$	34.6017	−	15.4057i	1.29766	−	0.577757i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−22.1010	−	38.2801i	−0.825954	−	1.43059i
$717$		0			0
$718$	3.78089	+	35.9727i	0.141101	+	1.34249i
$719$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$719$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$720$		0			0
$721$		0			0
$722$	−36.3401	−	26.4026i	−1.35244	−	0.982605i
$723$		0			0
$724$		0			0
$725$	26.7077	−	46.2591i	0.991898	−	1.71802i
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	21.8435	−	15.8702i	0.809017	−	0.587785i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$733$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$734$		0			0
$735$		0			0
$736$	−33.4018			−1.23121
$737$	53.9734	+	4.63979i	1.98814	+	0.170909i
$738$		0			0
$739$	−0.649946	−	0.721839i	−0.0239087	−	0.0265533i	0.731072	−	0.682300i	$-0.239020\pi$
$739$	−0.649946	−	0.721839i	−0.0239087	−	0.0265533i	−0.754981	+	0.655747i	$0.772354\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−14.7056	−	45.2592i	−0.539496	−	1.66040i	−0.733729	−	0.679442i	$-0.762222\pi$
$743$	−14.7056	−	45.2592i	−0.539496	−	1.66040i	0.194233	−	0.980955i	$-0.437778\pi$
$744$		0			0
$745$		0			0
$746$	−34.8024	−	38.6519i	−1.27420	−	1.41515i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	21.2687	+	9.46943i	0.776106	+	0.345544i	0.756271	−	0.654259i	$-0.227019\pi$
$751$	21.2687	+	9.46943i	0.776106	+	0.345544i	0.0198348	+	0.999803i	$0.493686\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−8.26681	+	25.4426i	−0.300462	+	0.924728i	0.680869	+	0.732405i	$0.261602\pi$
$757$	−8.26681	+	25.4426i	−0.300462	+	0.924728i	−0.981332	+	0.192323i	$0.938398\pi$
$758$	37.2118	+	64.4528i	1.35159	+	2.34103i
$759$		0			0
$760$		0			0
$761$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$761$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$762$		0			0
$763$		0			0
$764$	−25.1666	−	77.4548i	−0.910495	−	2.80222i
$765$		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
$772$	51.1442	−	10.8710i	1.84072	−	0.391257i
$773$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$773$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$774$	−56.6336	+	25.2149i	−2.03565	+	0.906330i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	70.6972	−	51.3645i	2.53462	−	1.84151i
$779$		0			0
$780$		0			0
$781$	24.1658	−	22.6034i	0.864720	−	0.808814i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$787$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$788$	89.3665	−	39.7885i	3.18355	−	1.41741i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−35.7660	−	10.8756i	−1.27089	−	0.386448i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$797$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$798$		0			0
$799$		0			0
$800$	−15.9219	−	7.08890i	−0.562925	−	0.250631i
$801$		0			0
$802$	17.8086	−	30.8453i	0.628842	−	1.08919i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−54.4092	−	11.5650i	−1.91293	−	0.406605i	−0.912928	−	0.408120i	$-0.866185\pi$
$809$	−54.4092	−	11.5650i	−1.91293	−	0.406605i	−0.999999	+	0.00151439i	$0.999518\pi$
$810$		0			0
$811$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$811$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$812$		0			0
$813$		0			0
$814$	−9.17550	+	5.53211i	−0.321601	+	0.193900i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−2.29963	+	21.8795i	−0.0802575	+	0.763599i	0.878186	+	0.478318i	$0.158754\pi$
$821$	−2.29963	+	21.8795i	−0.0802575	+	0.763599i	−0.958444	+	0.285281i	$0.907913\pi$
$822$		0			0
$823$	−23.6900	−	26.3104i	−0.825781	−	0.917122i	0.171905	−	0.985114i	$-0.445008\pi$
$823$	−23.6900	−	26.3104i	−0.825781	−	0.917122i	−0.997686	+	0.0679910i	$0.978341\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	13.5967	−	41.8465i	0.472805	−	1.45514i	−0.376090	−	0.926583i	$-0.622732\pi$
$827$	13.5967	−	41.8465i	0.472805	−	1.45514i	0.848895	−	0.528562i	$-0.177268\pi$
$828$	94.2594	+	41.9670i	3.27574	+	1.45845i
$829$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$829$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$830$		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		−0.587785	−	0.809017i	$-0.700000\pi$
$839$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$840$		0			0
$841$	26.3060	+	80.9615i	0.907103	+	2.79178i
$842$	−62.7962	+	69.7422i	−2.16410	+	2.40348i
$843$		0			0
$844$	2.19219	+	20.8573i	0.0754584	+	0.717939i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	−22.4164			−0.769784
$849$		0			0
$850$		0			0
$851$	11.9617	−	5.32569i	0.410042	−	0.182562i
$852$		0			0
$853$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$853$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$854$		0			0
$855$		0			0
$856$	48.0328	−	10.2097i	1.64173	−	0.348960i
$857$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$857$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$858$		0			0
$859$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$859$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$860$		0			0
$861$		0			0
$862$	−79.2595	−	57.5854i	−2.69959	−	1.96137i
$863$	5.35304	−	5.94516i	0.182220	−	0.202376i	−0.645114	−	0.764087i	$-0.723190\pi$
$863$	5.35304	−	5.94516i	0.182220	−	0.202376i	0.827334	+	0.561711i	$0.189857\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−25.2508	−	33.4037i	−0.856576	−	1.13314i
$870$		0			0
$871$		0			0
$872$	−3.27604	−	31.1694i	−0.110941	−	1.05553i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	41.6997	+	18.5659i	1.40810	+	0.626925i	0.963232	−	0.268672i	$-0.0865848\pi$
$877$	41.6997	+	18.5659i	1.40810	+	0.626925i	0.444866	+	0.895597i	$0.353251\pi$
$878$		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
$883$	9.70820	−	7.05342i	0.326707	−	0.237367i	−0.412325	−	0.911037i	$-0.635283\pi$
$883$	9.70820	−	7.05342i	0.326707	−	0.237367i	0.739032	+	0.673670i	$0.235283\pi$
$884$		0			0
$885$		0			0
$886$	67.1715	+	14.2777i	2.25667	+	0.479670i
$887$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$887$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−22.5574	−	19.5490i	−0.755701	−	0.654916i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	5.74905	−	54.6985i	0.191848	−	1.82531i
$899$		0			0
$900$	36.0247	+	40.0095i	1.20082	+	1.33365i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	22.6538	−	69.7212i	0.753454	−	2.31889i
$905$		0			0
$906$		0			0
$907$	−23.0584	−	4.90121i	−0.765642	−	0.162742i	−0.191494	−	0.981494i	$-0.561333\pi$
$907$	−23.0584	−	4.90121i	−0.765642	−	0.162742i	−0.574148	+	0.818752i	$0.694667\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	4.94427	−	15.2169i	0.163811	−	0.504159i	−0.835136	−	0.550044i	$-0.814611\pi$
$911$	4.94427	−	15.2169i	0.163811	−	0.504159i	0.998947	+	0.0458855i	$0.0146109\pi$
$912$		0			0
$913$		0			0
$914$	−23.6746	+	41.0057i	−0.783088	+	1.35635i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	13.6468	−	15.1563i	0.450166	−	0.499960i	−0.474756	−	0.880118i	$-0.657464\pi$
$919$	13.6468	−	15.1563i	0.450166	−	0.499960i	0.924922	+	0.380158i	$0.124130\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	6.83216			0.224640
$926$	6.32899	−	1.34527i	0.207984	−	0.0442083i
$927$		0			0
$928$	34.0190	−	15.1462i	1.11673	−	0.497200i
$929$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$929$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$930$		0			0
$931$		0			0
$932$	−63.8821	+	46.4130i	−2.09253	+	1.52031i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$937$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$941$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	41.3288	+	54.6728i	1.34371	+	1.77757i
$947$	10.0000	+	17.3205i	0.324956	+	0.562841i	0.981504	−	0.191444i	$-0.0613171\pi$
$947$	10.0000	+	17.3205i	0.324956	+	0.562841i	−0.656547	+	0.754285i	$0.727984\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	27.8965	+	20.2680i	0.903655	+	0.656544i	0.939402	−	0.342817i	$-0.111381\pi$
$953$	27.8965	+	20.2680i	0.903655	+	0.656544i	−0.0357473	+	0.999361i	$0.511381\pi$
$954$	−85.2362	−	37.9496i	−2.75962	−	1.22866i
$955$		0			0
$956$	−54.0299	+	93.5825i	−1.74745	+	3.02668i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	3.24038	−	30.8302i	0.104528	−	0.994522i
$962$		0			0
$963$	38.3534	+	8.15226i	1.23592	+	0.262703i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	60.3531			1.94082			0.970412	−	0.241454i	$-0.0776244\pi$
$967$	60.3531			1.94082			0.970412	+	0.241454i	$0.0776244\pi$
$968$	−2.75815	+	41.2363i	−0.0886504	+	1.32538i
$969$		0			0
$970$		0			0
$971$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$971$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$972$		0			0
$973$		0			0
$974$	31.1541	+	95.8826i	0.998243	+	3.07228i
$975$		0			0
$976$		0			0
$977$	−8.25215	−	9.16494i	−0.264010	−	0.293212i	0.596535	−	0.802587i	$-0.296544\pi$
$977$	−8.25215	−	9.16494i	−0.264010	−	0.293212i	−0.860545	+	0.509374i	$0.829877\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	7.73325	−	23.8005i	0.246904	−	0.759891i
$982$	−95.0294	−	42.3098i	−3.03251	−	1.35016i
$983$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$983$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−41.8786	−	72.5358i	−1.33166	−	2.30650i
$990$		0			0
$991$	12.0000	−	20.7846i	0.381193	−	0.660245i	−0.610040	−	0.792370i	$-0.708847\pi$
$991$	12.0000	−	20.7846i	0.381193	−	0.660245i	0.991233	+	0.132125i	$0.0421802\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$997$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$998$	8.96733	+	85.3185i	0.283856	+	2.70071i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.q.d.361.2		16
7.2	even	3	inner	539.2.q.d.471.1		16
7.3	odd	6		539.2.f.c.295.1	yes	8
7.4	even	3		539.2.f.c.295.1	yes	8
7.5	odd	6	inner	539.2.q.d.471.1		16
7.6	odd	2	CM	539.2.q.d.361.2		16
11.5	even	5	inner	539.2.q.d.214.1		16
77.4	even	15		5929.2.a.bc.1.1		4
77.5	odd	30	inner	539.2.q.d.324.2		16
77.16	even	15	inner	539.2.q.d.324.2		16
77.18	odd	30		5929.2.a.bg.1.4		4
77.27	odd	10	inner	539.2.q.d.214.1		16
77.38	odd	30		539.2.f.c.148.1	✓	8
77.59	odd	30		5929.2.a.bc.1.1		4
77.60	even	15		539.2.f.c.148.1	✓	8
77.73	even	30		5929.2.a.bg.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.f.c.148.1	✓	8	77.38	odd	30
539.2.f.c.148.1	✓	8	77.60	even	15
539.2.f.c.295.1	yes	8	7.3	odd	6
539.2.f.c.295.1	yes	8	7.4	even	3
539.2.q.d.214.1		16	11.5	even	5	inner
539.2.q.d.214.1		16	77.27	odd	10	inner
539.2.q.d.324.2		16	77.5	odd	30	inner
539.2.q.d.324.2		16	77.16	even	15	inner
539.2.q.d.361.2		16	1.1	even	1	trivial
539.2.q.d.361.2		16	7.6	odd	2	CM
539.2.q.d.471.1		16	7.2	even	3	inner
539.2.q.d.471.1		16	7.5	odd	6	inner
5929.2.a.bc.1.1		4	77.4	even	15
5929.2.a.bc.1.1		4	77.59	odd	30
5929.2.a.bg.1.4		4	77.18	odd	30
5929.2.a.bg.1.4		4	77.73	even	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)

\(f(q)\)	\(=\)	\(q+(2.31249 - 0.491535i) q^{2} +(3.27890 - 1.45986i) q^{4} +(3.03958 - 2.20838i) q^{8} +(2.93444 - 0.623735i) q^{9} +(-1.40621 - 3.00376i) q^{11} +(1.14020 - 1.26632i) q^{16} +(6.47928 - 2.88476i) q^{18} +(-4.72830 - 6.25495i) q^{22} +(4.79120 + 8.29860i) q^{23} +(0.522642 + 4.97261i) q^{25} +(-8.64279 - 6.27935i) q^{29} +(-1.74287 + 3.01874i) q^{32} +(8.71119 - 6.32905i) q^{36} +(0.142831 - 1.35895i) q^{37} -8.74072 q^{43} +(-8.99591 - 7.79616i) q^{44} +(15.1586 + 16.8354i) q^{46} +(3.65281 + 11.2422i) q^{50} +(-8.80255 - 9.77622i) q^{53} +(-23.0729 - 10.2727i) q^{58} +(-3.59968 + 11.0787i) q^{64} +(-8.16681 + 14.1453i) q^{67} +(3.08300 + 9.48849i) q^{71} +(7.54203 - 8.37627i) q^{72} +(-0.337674 - 3.21276i) q^{74} +(12.3495 - 2.62497i) q^{79} +(8.22191 - 3.66063i) q^{81} +(-20.2128 + 4.29637i) q^{86} +(-10.9077 - 6.02471i) q^{88} +(27.8247 + 20.2158i) q^{92} +(-6.00000 - 7.93725i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{2} - 2 q^{4} + 32 q^{8} - 6 q^{9} + 4 q^{11} + 28 q^{16} + 9 q^{18} - 8 q^{22} + 16 q^{23} - 10 q^{25} - 8 q^{29} - 100 q^{32} - 12 q^{36} + 18 q^{37} + 48 q^{43} - 9 q^{44} + 31 q^{46} + 20 q^{50}+ \cdots - 96 q^{99}+O(q^{100}) \) 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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