LMFDB - Embedded newform 539.2.q.d.410.2

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			410.2
Root			$-0.764115 - 1.19001i$ of defining polynomial
Character	$\chi$	$=$	539.410
Dual form			539.2.q.d.422.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.58102 + 1.14914i) q^{2} +(4.00286 + 4.44563i) q^{4} +(3.47668 + 10.7001i) q^{8} +(-2.74064 - 1.22021i) q^{9} +(0.750967 - 3.23049i) q^{11} +(-2.07198 + 19.7136i) q^{16} +(-5.67144 - 6.29877i) q^{18} +(5.65055 - 7.47498i) q^{22} +(-3.75233 - 6.49922i) q^{23} +(4.89074 + 1.03956i) q^{25} +(1.42225 - 4.37724i) q^{29} +(-16.7508 + 29.0132i) q^{32} +(-5.54579 - 17.0682i) q^{36} +(-8.03151 + 1.70715i) q^{37} +6.59794 q^{43} +(17.3676 - 9.59268i) q^{44} +(-2.21629 - 21.0866i) q^{46} +(11.4285 + 8.30328i) q^{50} +(-0.195430 - 1.85939i) q^{53} +(8.70093 - 9.66336i) q^{58} +(-44.5014 + 32.3322i) q^{64} +(-3.04737 + 5.27819i) q^{67} +(7.95588 + 5.78028i) q^{71} +(3.52808 - 33.5674i) q^{72} +(-22.6912 - 4.82317i) q^{74} +(-14.4367 - 6.42763i) q^{79} +(6.02218 + 6.68830i) q^{81} +(17.0294 + 7.58197i) q^{86} +(37.1775 - 3.19593i) q^{88} +(13.8731 - 42.6969i) q^{92} +(-6.00000 + 7.93725i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/539\mathbb{Z}\right)^\times$.

$n$	$199$	$442$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{4}{5}\right)$

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 $

$2$	2.58102	+	1.14914i	1.82506	+	0.812567i	0.931722	+	0.363173i	$0.118307\pi$
$2$	2.58102	+	1.14914i	1.82506	+	0.812567i	0.893334	+	0.449394i	$0.148360\pi$
$3$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$3$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$4$	4.00286	+	4.44563i	2.00143	+	2.22281i
$5$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$5$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	3.47668	+	10.7001i	1.22919	+	3.78306i
$9$	−2.74064	−	1.22021i	−0.913545	−	0.406737i
$10$		0			0
$11$	0.750967	−	3.23049i	0.226425	−	0.974029i
$11$	0.750967	−	3.23049i	0.226425	−	0.974029i
$12$		0			0
$13$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$13$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$14$		0			0
$15$		0			0
$16$	−2.07198	+	19.7136i	−0.517995	+	4.92839i
$17$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$17$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$18$	−5.67144	−	6.29877i	−1.33677	−	1.48463i
$19$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$19$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$20$		0			0
$21$		0			0
$22$	5.65055	−	7.47498i	1.20470	−	1.59367i
$23$	−3.75233	−	6.49922i	−0.782414	−	1.35518i	−0.930532	−	0.366211i	$-0.880655\pi$
$23$	−3.75233	−	6.49922i	−0.782414	−	1.35518i	0.148117	−	0.988970i	$-0.452679\pi$
$24$		0			0
$25$	4.89074	+	1.03956i	0.978148	+	0.207912i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.42225	−	4.37724i	0.264105	−	0.812833i	−0.727793	−	0.685797i	$-0.759454\pi$
$29$	1.42225	−	4.37724i	0.264105	−	0.812833i	0.991898	−	0.127036i	$-0.0405463\pi$
$30$		0			0
$31$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$31$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$32$	−16.7508	+	29.0132i	−2.96115	+	5.12886i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−5.54579	−	17.0682i	−0.924298	−	2.84470i
$37$	−8.03151	+	1.70715i	−1.32037	+	0.280654i	−0.813599	−	0.581426i	$-0.802495\pi$
$37$	−8.03151	+	1.70715i	−1.32037	+	0.280654i	−0.506772	+	0.862080i	$0.669162\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$41$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$42$		0			0
$43$	6.59794			1.00618			0.503088	−	0.864235i	$-0.332197\pi$
$43$	6.59794			1.00618			0.503088	+	0.864235i	$0.332197\pi$
$44$	17.3676	−	9.59268i	2.61826	−	1.44615i
$45$		0			0
$46$	−2.21629	−	21.0866i	−0.326774	−	3.10904i
$47$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$47$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$48$		0			0
$49$		0			0
$50$	11.4285	+	8.30328i	1.61623	+	1.17426i
$51$		0			0
$52$		0			0
$53$	−0.195430	−	1.85939i	−0.0268444	−	0.255407i	−0.999711	−	0.0240405i	$-0.992347\pi$
$53$	−0.195430	−	1.85939i	−0.0268444	−	0.255407i	0.972867	−	0.231367i	$-0.0743197\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	8.70093	−	9.66336i	1.14249	−	1.26886i
$59$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$59$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$60$		0			0
$61$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$61$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$62$		0			0
$63$		0			0
$64$	−44.5014	+	32.3322i	−5.56268	+	4.04152i
$65$		0			0
$66$		0			0
$67$	−3.04737	+	5.27819i	−0.372295	+	0.644834i	−0.989918	−	0.141640i	$-0.954762\pi$
$67$	−3.04737	+	5.27819i	−0.372295	+	0.644834i	0.617623	+	0.786474i	$0.288096\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	7.95588	+	5.78028i	0.944189	+	0.685993i	0.949425	−	0.313993i	$-0.101667\pi$
$71$	7.95588	+	5.78028i	0.944189	+	0.685993i	−0.00523645	+	0.999986i	$0.501667\pi$
$72$	3.52808	−	33.5674i	0.415788	−	3.95596i
$73$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$73$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$74$	−22.6912	−	4.82317i	−2.63780	−	0.560682i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−14.4367	−	6.42763i	−1.62426	−	0.723165i	−0.625868	−	0.779929i	$-0.715255\pi$
$79$	−14.4367	−	6.42763i	−1.62426	−	0.723165i	−0.998388	+	0.0567635i	$0.981922\pi$
$80$		0			0
$81$	6.02218	+	6.68830i	0.669131	+	0.743145i
$82$		0			0
$83$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$83$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$84$		0			0
$85$		0			0
$86$	17.0294	+	7.58197i	1.83633	+	0.817585i
$87$		0			0
$88$	37.1775	−	3.19593i	3.96313	−	0.340688i
$89$		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
$92$	13.8731	−	42.6969i	1.44637	−	4.45146i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$97$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$98$		0			0
$99$	−6.00000	+	7.93725i	−0.603023	+	0.797724i
$100$	14.9555	+	25.9036i	1.49555	+	2.59036i
$101$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$101$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$102$		0			0
$103$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$103$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$104$		0			0
$105$		0			0
$106$	1.63230	−	5.02371i	0.158543	−	0.487945i
$107$	−0.768043	+	0.852998i	−0.0742495	+	0.0824624i	−0.779125	−	0.626869i	$-0.784336\pi$
$107$	−0.768043	+	0.852998i	−0.0742495	+	0.0824624i	0.704875	+	0.709331i	$0.251003\pi$
$108$		0			0
$109$	2.25137	−	3.89948i	0.215642	−	0.373502i	−0.737829	−	0.674987i	$-0.764149\pi$
$109$	2.25137	−	3.89948i	0.215642	−	0.373502i	0.953471	+	0.301485i	$0.0974824\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−4.34450	−	13.3710i	−0.408696	−	1.25784i	−0.917769	−	0.397114i	$-0.870012\pi$
$113$	−4.34450	−	13.3710i	−0.408696	−	1.25784i	0.509073	−	0.860724i	$-0.329988\pi$
$114$		0			0
$115$		0			0
$116$	25.1527	−	11.1987i	2.33537	−	1.03977i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−9.87210	−	4.85198i	−0.897463	−	0.441089i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	8.21418	+	5.96795i	0.728890	+	0.529570i	0.889212	−	0.457495i	$-0.151253\pi$
$127$	8.21418	+	5.96795i	0.728890	+	0.529570i	−0.160322	+	0.987065i	$0.551253\pi$
$128$	−86.4744	+	18.3807i	−7.64332	+	1.62464i
$129$		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
$134$	−13.9307	+	10.1213i	−1.20343	+	0.874343i
$135$		0			0
$136$		0			0
$137$	−21.2127	+	9.44452i	−1.81233	+	0.806900i	−0.854562	+	0.519350i	$0.826174\pi$
$137$	−21.2127	+	9.44452i	−1.81233	+	0.806900i	−0.957766	+	0.287550i	$0.907159\pi$
$138$		0			0
$139$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$139$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$140$		0			0
$141$		0			0
$142$	13.8919	+	24.0615i	1.16578	+	2.01919i
$143$		0			0
$144$	29.7332	−	51.4995i	2.47777	−	4.29162i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−39.7384	−	28.8716i	−3.26647	−	2.37323i
$149$	−2.29963	+	21.8795i	−0.188393	+	1.79244i	0.336926	+	0.941531i	$0.390613\pi$
$149$	−2.29963	+	21.8795i	−0.188393	+	1.79244i	−0.525318	+	0.850906i	$0.676054\pi$
$150$		0			0
$151$	12.1769	+	2.58828i	0.990941	+	0.210631i	0.674735	−	0.738060i	$-0.264258\pi$
$151$	12.1769	+	2.58828i	0.990941	+	0.210631i	0.316206	+	0.948691i	$0.397591\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$157$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$158$	−29.8751	−	33.1797i	−2.37674	−	2.63963i
$159$		0			0
$160$		0			0
$161$		0			0
$162$	7.85752	+	24.1830i	0.617345	+	1.89999i
$163$	8.14628	+	3.62696i	0.638066	+	0.284085i	0.700161	−	0.713985i	$-0.253112\pi$
$163$	8.14628	+	3.62696i	0.638066	+	0.284085i	−0.0620951	+	0.998070i	$0.519778\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$167$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$168$		0			0
$169$	−4.01722	+	12.3637i	−0.309017	+	0.951057i
$170$		0			0
$171$		0			0
$172$	26.4106	+	29.3320i	2.01379	+	2.23654i
$173$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$173$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$174$		0			0
$175$		0			0
$176$	62.1285	+	21.4977i	4.68311	+	1.62045i
$177$		0			0
$178$		0			0
$179$	23.4037	+	4.97460i	1.74927	+	0.371819i	0.967721	−	0.252022i	$-0.0810956\pi$
$179$	23.4037	+	4.97460i	1.74927	+	0.371819i	0.781551	+	0.623841i	$0.214429\pi$
$180$		0			0
$181$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$181$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$182$		0			0
$183$		0			0
$184$	56.4968	−	62.7461i	4.16500	−	4.62570i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	21.5422	−	4.57894i	1.55874	−	0.331320i	0.653735	−	0.756723i	$-0.273201\pi$
$191$	21.5422	−	4.57894i	1.55874	−	0.331320i	0.905004	+	0.425403i	$0.139868\pi$
$192$		0			0
$193$	−24.6688	+	10.9833i	−1.77570	+	0.790592i	−0.792061	+	0.610442i	$0.790992\pi$
$193$	−24.6688	+	10.9833i	−1.77570	+	0.790592i	−0.983639	+	0.180150i	$0.942342\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−18.0995			−1.28953			−0.644767	−	0.764379i	$-0.723046\pi$
$197$	−18.0995			−1.28953			−0.644767	+	0.764379i	$0.723046\pi$
$198$	−24.6072	+	13.5913i	−1.74875	+	0.965894i
$199$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$199$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$200$	5.88013	+	55.9457i	0.415788	+	3.95596i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	2.35335	+	22.3906i	0.163569	+	1.55626i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−17.3570	+	12.6106i	−1.19490	+	0.868148i	−0.993774	−	0.111417i	$-0.964461\pi$
$211$	−17.3570	+	12.6106i	−1.19490	+	0.868148i	−0.201129	+	0.979565i	$0.564461\pi$
$212$	7.48389	−	8.31171i	0.513996	−	0.570850i
$213$		0			0
$214$	−2.96255	+	1.31901i	−0.202516	+	0.0901658i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	10.2919	−	7.47748i	0.697054	−	0.506439i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$223$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$224$		0			0
$225$	−12.1353	−	8.81678i	−0.809017	−	0.587785i
$226$	4.15196	−	39.5033i	0.276184	−	2.62772i
$227$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$227$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$228$		0			0
$229$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$229$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$230$		0			0
$231$		0			0
$232$	51.7817			3.39963
$233$	20.0980	+	8.94821i	1.31666	+	0.586216i	0.940329	−	0.340267i	$-0.110518\pi$
$233$	20.0980	+	8.94821i	1.31666	+	0.586216i	0.376335	+	0.926484i	$0.377184\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	0.805627	+	2.47947i	0.0521117	+	0.160383i	0.973726	−	0.227725i	$-0.0731287\pi$
$239$	0.805627	+	2.47947i	0.0521117	+	0.160383i	−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	$0.166667\pi$
$242$	−19.9044	−	23.8675i	−1.27951	−	1.53426i
$243$		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	$-0.700000\pi$
$251$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$252$		0			0
$253$	−23.8135	+	7.24115i	−1.49714	+	0.455247i
$254$	14.3429	+	24.8427i	0.899954	+	1.55877i
$255$		0			0
$256$	−136.704	−	29.0574i	−8.54403	−	1.81609i
$257$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$257$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−9.23902	+	10.2610i	−0.571881	+	0.635138i
$262$		0			0
$263$	14.4994	−	25.1137i	0.894072	−	1.54858i	0.0591223	−	0.998251i	$-0.481170\pi$
$263$	14.4994	−	25.1137i	0.894072	−	1.54858i	0.834949	−	0.550327i	$-0.185497\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−35.6631	+	7.58042i	−2.17847	+	0.463048i
$269$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$269$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$270$		0			0
$271$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$271$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$272$		0			0
$273$		0			0
$274$	−65.6036			−3.96326
$275$	7.03106	−	15.0188i	0.423989	−	0.905667i
$276$		0			0
$277$	−2.83324	−	26.9565i	−0.170233	−	1.61966i	−0.662396	−	0.749154i	$-0.730460\pi$
$277$	−2.83324	−	26.9565i	−0.170233	−	1.61966i	0.492164	−	0.870503i	$-0.336206\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−6.95218	−	5.05106i	−0.414733	−	0.301321i	0.360782	−	0.932650i	$-0.382510\pi$
$281$	−6.95218	−	5.05106i	−0.414733	−	0.301321i	−0.775515	+	0.631329i	$0.782510\pi$
$282$		0			0
$283$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$283$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$284$	6.14929	+	58.5066i	0.364893	+	3.47173i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	81.3099	−	59.0751i	4.79123	−	3.48104i
$289$	−11.3752	+	12.6335i	−0.669131	+	0.743145i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$293$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$294$		0			0
$295$		0			0
$296$	−46.1897	−	80.0029i	−2.68472	−	4.65007i
$297$		0			0
$298$	−31.0780	+	53.8287i	−1.80030	+	3.11822i
$299$		0			0
$300$		0			0
$301$		0			0
$302$	28.4545	+	20.6734i	1.63737	+	1.18962i
$303$		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	$-0.433333\pi$
$311$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$312$		0			0
$313$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$313$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$314$		0			0
$315$		0			0
$316$	−29.2133	−	89.9092i	−1.64337	−	5.05779i
$317$	30.8113	+	13.7181i	1.73053	+	0.770483i	0.995742	+	0.0921886i	$0.0293863\pi$
$317$	30.8113	+	13.7181i	1.73053	+	0.770483i	0.734791	+	0.678294i	$0.237280\pi$
$318$		0			0
$319$	−13.0725	−	7.88172i	−0.731922	−	0.441292i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−5.62778	+	53.5447i	−0.312654	+	2.97471i
$325$		0			0
$326$	16.8578	+	18.7225i	0.933667	+	1.03694i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−3.04605	−	5.27591i	−0.167426	−	0.289990i	0.770088	−	0.637937i	$-0.220212\pi$
$331$	−3.04605	−	5.27591i	−0.167426	−	0.289990i	−0.937514	+	0.347947i	$0.886879\pi$
$332$		0			0
$333$	24.0945	+	5.12145i	1.32037	+	0.280654i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	9.08528	−	27.9616i	0.494907	−	1.52317i	−0.322195	−	0.946673i	$-0.604421\pi$
$337$	9.08528	−	27.9616i	0.494907	−	1.52317i	0.817102	−	0.576493i	$-0.195579\pi$
$338$	−24.5762	+	27.2947i	−1.33677	+	1.48463i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	22.9389	+	70.5987i	1.23678	+	3.80643i
$345$		0			0
$346$		0			0
$347$	−22.8459	+	10.1716i	−1.22643	+	0.546043i	−0.914702	−	0.404128i	$-0.867575\pi$
$347$	−22.8459	+	10.1716i	−1.22643	+	0.546043i	−0.311729	+	0.950171i	$0.600908\pi$
$348$		0			0
$349$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$349$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$350$		0			0
$351$		0			0
$352$	81.1475	+	75.9011i	4.32517	+	4.04554i
$353$		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
$358$	54.6888	+	39.7337i	2.89039	+	2.09999i
$359$	32.0397	−	6.81025i	1.69099	−	0.359431i	0.740951	−	0.671559i	$-0.234375\pi$
$359$	32.0397	−	6.81025i	1.69099	−	0.359431i	0.950040	+	0.312128i	$0.101042\pi$
$360$		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	$-0.233333\pi$
$367$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$368$	135.898	−	60.5055i	7.08415	−	3.15407i
$369$		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
$379$	31.4997	+	22.8859i	1.61803	+	1.17557i	0.809522	+	0.587090i	$0.199726\pi$
$379$	31.4997	+	22.8859i	1.61803	+	1.17557i	0.808511	+	0.588481i	$0.200274\pi$
$380$		0			0
$381$		0			0
$382$	60.8627	+	12.9368i	3.11401	+	0.661902i
$383$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$383$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$384$		0			0
$385$		0			0
$386$	−76.2920			−3.88316
$387$	−18.0825	−	8.05087i	−0.919187	−	0.409249i
$388$		0			0
$389$	−14.5922	−	16.2063i	−0.739853	−	0.821690i	0.249323	−	0.968420i	$-0.419792\pi$
$389$	−14.5922	−	16.2063i	−0.739853	−	0.821690i	−0.989177	+	0.146730i	$0.953125\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	−46.7151	−	20.7989i	−2.35347	−	1.04783i
$395$		0			0
$396$	−59.3033	+	5.09796i	−2.98010	+	0.256182i
$397$		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
$400$	−30.6269	+	94.2600i	−1.53135	+	4.71300i
$401$	−1.00593	+	9.57078i	−0.0502337	+	0.477942i	0.940267	+	0.340437i	$0.110575\pi$
$401$	−1.00593	+	9.57078i	−0.0502337	+	0.477942i	−0.990501	+	0.137505i	$0.956092\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−0.516471	+	27.2277i	−0.0256005	+	1.34963i
$408$		0			0
$409$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$409$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−19.6560	+	60.4950i	−0.966040	+	2.97316i
$415$		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
$421$	−11.8136	−	36.3585i	−0.575759	−	1.77200i	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−11.8136	−	36.3585i	−0.575759	−	1.77200i	0.0578225	−	0.998327i	$-0.481584\pi$
$422$	−59.2900	+	12.6025i	−2.88619	+	0.613479i
$423$		0			0
$424$	19.2163	−	8.55564i	0.933225	−	0.415499i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−6.86648			−0.331904
$429$		0			0
$430$		0			0
$431$	−3.66384	−	34.8591i	−0.176481	−	1.67910i	−0.621369	−	0.783518i	$-0.713423\pi$
$431$	−3.66384	−	34.8591i	−0.176481	−	1.67910i	0.444888	−	0.895586i	$-0.353244\pi$
$432$		0			0
$433$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$433$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$434$		0			0
$435$		0			0
$436$	26.3475	−	5.60034i	1.26182	−	0.268208i
$437$		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
$443$	25.3950	−	28.2040i	1.20655	−	1.34001i	0.281785	−	0.959477i	$-0.409073\pi$
$443$	25.3950	−	28.2040i	1.20655	−	1.34001i	0.924767	−	0.380535i	$-0.124260\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	32.0711	−	23.3010i	1.51353	−	1.09964i	0.548950	−	0.835855i	$-0.315028\pi$
$449$	32.0711	−	23.3010i	1.51353	−	1.09964i	0.964580	−	0.263790i	$-0.0849724\pi$
$450$	−21.1896	−	36.7014i	−0.998886	−	1.73012i
$451$		0			0
$452$	42.0521	−	72.8363i	1.97796	−	3.42593i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−4.01452	+	38.1957i	−0.187792	+	1.78672i	0.343122	+	0.939291i	$0.388516\pi$
$457$	−4.01452	+	38.1957i	−0.187792	+	1.78672i	−0.530913	+	0.847426i	$0.678151\pi$
$458$		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
$463$	41.6915			1.93757			0.968784	−	0.247907i	$-0.0797429\pi$
$463$	41.6915			1.93757			0.968784	+	0.247907i	$0.0797429\pi$
$464$	83.3441	+	37.1072i	3.86915	+	1.72266i
$465$		0			0
$466$	41.5905	+	46.1910i	1.92664	+	2.13976i
$467$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$467$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	4.95483	−	21.3146i	0.227823	−	0.980044i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−1.73325	+	5.33439i	−0.0793599	+	0.244245i
$478$	−0.769923	+	7.32533i	−0.0352154	+	0.335053i
$479$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$479$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−17.9466	−	63.3095i	−0.815752	−	2.87770i
$485$		0			0
$486$		0			0
$487$	2.30399	+	0.489728i	0.104404	+	0.0221917i	0.259817	−	0.965658i	$-0.416338\pi$
$487$	2.30399	+	0.489728i	0.104404	+	0.0221917i	−0.155414	+	0.987850i	$0.549671\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	13.5967	−	41.8465i	0.613613	−	1.88851i	0.193249	−	0.981150i	$-0.438097\pi$
$491$	13.5967	−	41.8465i	0.613613	−	1.88851i	0.420363	−	0.907356i	$-0.361903\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	13.2767	−	2.82204i	0.594345	−	0.126332i	0.0990885	−	0.995079i	$-0.468407\pi$
$499$	13.2767	−	2.82204i	0.594345	−	0.126332i	0.495257	+	0.868747i	$0.335074\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$503$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$504$		0			0
$505$		0			0
$506$	−69.7843	−	8.67563i	−3.10229	−	0.385679i
$507$		0			0
$508$	6.34893	+	60.4061i	0.281688	+	2.68009i
$509$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$509$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$510$		0			0
$511$		0			0
$512$	−176.401	−	128.163i	−7.79590	−	5.66405i
$513$		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	$-0.233333\pi$
$521$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$522$	−35.6374	+	15.8668i	−1.55981	+	0.694471i
$523$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$523$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$524$		0			0
$525$		0			0
$526$	66.2825	−	48.1570i	2.89005	−	2.09975i
$527$		0			0
$528$		0			0
$529$	−16.6599	+	28.8558i	−0.724344	+	1.25460i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−67.0720	−	14.2566i	−2.89707	−	0.615791i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−37.1828	−	16.5549i	−1.59862	−	0.711749i	−0.602361	−	0.798224i	$-0.705773\pi$
$541$	−37.1828	−	16.5549i	−1.59862	−	0.711749i	−0.996254	+	0.0864744i	$0.972440\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	13.5967	+	41.8465i	0.581355	+	1.78923i	0.613440	+	0.789741i	$0.289785\pi$
$547$	13.5967	+	41.8465i	0.581355	+	1.78923i	−0.0320849	+	0.999485i	$0.510215\pi$
$548$	−126.899	−	56.4989i	−5.42084	−	2.41351i
$549$		0			0
$550$	35.4060	−	30.6841i	1.50972	−	1.30837i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	23.6642	−	72.8309i	1.00540	−	3.09429i
$555$		0			0
$556$		0			0
$557$	−20.7392	−	23.0332i	−0.878748	−	0.975949i	0.121113	−	0.992639i	$-0.461354\pi$
$557$	−20.7392	−	23.0332i	−0.878748	−	0.975949i	−0.999861	+	0.0166902i	$0.994687\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−12.1393	−	21.0259i	−0.512067	−	0.886925i
$563$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$563$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	−34.1897	+	105.225i	−1.43457	+	4.41514i
$569$	14.7209	−	16.3492i	0.617131	−	0.685394i	−0.350846	−	0.936433i	$-0.614106\pi$
$569$	14.7209	−	16.3492i	0.617131	−	0.685394i	0.967977	+	0.251040i	$0.0807725\pi$
$570$		0			0
$571$	22.0283	−	38.1541i	0.921856	−	1.59670i	0.125314	−	0.992117i	$-0.460006\pi$
$571$	22.0283	−	38.1541i	0.921856	−	1.59670i	0.796542	−	0.604583i	$-0.206660\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−11.5953	−	35.6868i	−0.483559	−	1.48824i
$576$	161.414	−	34.3097i	6.72559	−	1.42957i
$577$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$577$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$578$	−43.8773	+	19.5354i	−1.82506	+	0.812567i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−6.15351	−	0.765008i	−0.254852	−	0.0316834i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$587$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−17.0129	−	161.867i	−0.699226	−	6.65269i
$593$		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
$596$	−106.473	+	77.3573i	−4.36131	+	3.16868i
$597$		0			0
$598$		0			0
$599$	−23.1484	+	10.3063i	−0.945818	+	0.421105i	−0.820907	−	0.571063i	$-0.806531\pi$
$599$	−23.1484	+	10.3063i	−0.945818	+	0.421105i	−0.124912	+	0.992168i	$0.539865\pi$
$600$		0			0
$601$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$601$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$602$		0			0
$603$	14.7922	−	10.7472i	0.602386	−	0.437659i
$604$	37.2359	+	64.4944i	1.51511	+	2.62424i
$605$		0			0
$606$		0			0
$607$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$607$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−48.3246	−	10.2717i	−1.95181	−	0.414871i	−0.989971	−	0.141268i	$-0.954882\pi$
$613$	−48.3246	−	10.2717i	−1.95181	−	0.414871i	−0.961843	−	0.273603i	$-0.911785\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	45.9166			1.84853			0.924266	−	0.381749i	$-0.124678\pi$
$617$	45.9166			1.84853			0.924266	+	0.381749i	$0.124678\pi$
$618$		0			0
$619$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$619$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$620$		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
$631$	−12.4683	+	38.3736i	−0.496357	+	1.52763i	0.318475	+	0.947931i	$0.396829\pi$
$631$	−12.4683	+	38.3736i	−0.496357	+	1.52763i	−0.814832	+	0.579698i	$0.803171\pi$
$632$	18.5847	−	176.821i	0.739258	−	7.03357i
$633$		0			0
$634$	63.7604	+	70.8131i	2.53225	+	2.81235i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−24.6833	−	35.3651i	−0.977219	−	1.40012i
$639$	−14.7510	−	25.5495i	−0.583541	−	1.01072i
$640$		0			0
$641$	48.5708	+	10.3240i	1.91843	+	0.407775i	0.999924	+	0.0123672i	$0.00393672\pi$
$641$	48.5708	+	10.3240i	1.91843	+	0.407775i	0.918506	+	0.395407i	$0.129397\pi$
$642$		0			0
$643$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$643$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$647$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$648$	−50.6285	+	87.6911i	−1.98887	+	3.44483i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	16.4843	+	50.7336i	0.645576	+	1.98688i
$653$	−5.26812	+	1.11977i	−0.206157	+	0.0438201i	−0.309833	−	0.950791i	$-0.600273\pi$
$653$	−5.26812	+	1.11977i	−0.206157	+	0.0438201i	0.103675	+	0.994611i	$0.466940\pi$
$654$		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	$0.166667\pi$
$662$	−1.79913	−	17.1176i	−0.0699251	−	0.665293i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	56.3031	+	40.9066i	2.18170	+	1.58510i
$667$	−33.7854	+	7.18131i	−1.30818	+	0.278061i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.494817	+	0.359506i	−0.0190738	+	0.0138579i	−0.597281	−	0.802032i	$-0.703752\pi$
$673$	−0.494817	+	0.359506i	−0.0190738	+	0.0138579i	0.578208	+	0.815890i	$0.303752\pi$
$674$	55.5812	−	61.7291i	2.14091	−	2.37772i
$675$		0			0
$676$	−71.0450	+	31.6313i	−2.73250	+	1.21659i
$677$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$677$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	10.5507	−	18.2743i	0.403711	−	0.699249i	−0.590459	−	0.807067i	$-0.701053\pi$
$683$	10.5507	−	18.2743i	0.403711	−	0.699249i	0.994171	+	0.107819i	$0.0343867\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−13.6708	+	130.069i	−0.521194	+	4.95883i
$689$		0			0
$690$		0			0
$691$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$691$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$692$		0			0
$693$		0			0
$694$	−70.6543			−2.68200
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−15.3604	−	47.2743i	−0.580153	−	1.78553i	−0.617922	−	0.786239i	$-0.712025\pi$
$701$	−15.3604	−	47.2743i	−0.580153	−	1.78553i	0.0377695	−	0.999286i	$-0.487975\pi$
$702$		0			0
$703$		0			0
$704$	71.0296	+	168.042i	2.67703	+	6.33331i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	2.74372	−	26.1048i	0.103043	−	0.980386i	−0.813802	−	0.581142i	$-0.802606\pi$
$709$	2.74372	−	26.1048i	0.103043	−	0.980386i	0.916845	−	0.399244i	$-0.130727\pi$
$710$		0			0
$711$	31.7227	+	35.2316i	1.18969	+	1.32129i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	71.5664	+	123.957i	2.67456	+	4.63248i
$717$		0			0
$718$	90.5210	+	19.2408i	3.37821	+	0.718062i
$719$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$719$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$720$		0			0
$721$		0			0
$722$	−16.5881	+	51.0529i	−0.617345	+	1.89999i
$723$		0			0
$724$		0			0
$725$	11.5063	−	19.9294i	0.427331	−	0.740160i
$726$		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	$-0.566667\pi$
$733$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$734$		0			0
$735$		0			0
$736$	251.417			9.26737
$737$	14.7627	+	13.8082i	0.543790	+	0.508633i
$738$		0			0
$739$	−5.37273	−	51.1181i	−0.197639	−	1.88041i	−0.422993	−	0.906133i	$-0.639020\pi$
$739$	−5.37273	−	51.1181i	−0.197639	−	1.88041i	0.225354	−	0.974277i	$-0.427646\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−8.56653	−	6.22395i	−0.314275	−	0.228334i	0.419453	−	0.907777i	$-0.362222\pi$
$743$	−8.56653	−	6.22395i	−0.314275	−	0.228334i	−0.733729	+	0.679442i	$0.762222\pi$
$744$		0			0
$745$		0			0
$746$	−6.49708	−	61.8156i	−0.237875	−	2.26323i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	6.91197	−	7.67652i	0.252221	−	0.280120i	−0.603716	−	0.797199i	$-0.706314\pi$
$751$	6.91197	−	7.67652i	0.252221	−	0.280120i	0.855938	+	0.517079i	$0.172981\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−30.3109	+	22.0222i	−1.10167	+	0.800410i	−0.981332	−	0.192323i	$-0.938398\pi$
$757$	−30.3109	+	22.0222i	−1.10167	+	0.800410i	−0.120338	+	0.992733i	$0.538398\pi$
$758$	55.0022	+	95.2667i	1.99777	+	3.46024i
$759$		0			0
$760$		0			0
$761$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$761$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$762$		0			0
$763$		0			0
$764$	106.587	+	77.4398i	3.85617	+	2.80167i
$765$		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
$772$	−147.573	−	65.7039i	−5.31128	−	2.36474i
$773$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$773$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$774$	−37.4198	−	41.5589i	−1.34503	−	1.49380i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−19.0394	−	58.5972i	−0.682595	−	2.10081i
$779$		0			0
$780$		0			0
$781$	24.6477	−	21.3606i	0.881965	−	0.764341i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$787$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$788$	−72.4497	−	80.4636i	−2.58091	−	2.86640i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−105.790	−	36.6054i	−3.75907	−	1.30072i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$797$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$798$		0			0
$799$		0			0
$800$	−112.085	+	124.482i	−3.96279	+	4.40112i
$801$		0			0
$802$	−13.5945	+	23.5464i	−0.480039	+	0.831452i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−47.5069	+	21.1514i	−1.67025	+	0.743645i	−0.670255	+	0.742131i	$0.733815\pi$
$809$	−47.5069	+	21.1514i	−1.67025	+	0.743645i	−0.999999	+	0.00151439i	$0.999518\pi$
$810$		0			0
$811$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$811$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$812$		0			0
$813$		0			0
$814$	−32.6215	+	69.6817i	−1.14338	+	2.44234i
$815$		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	−0.567494	−	0.823378i	$-0.692087\pi$
$821$	−21.5192	+	4.57406i	−0.751027	+	0.159636i	−0.183533	+	0.983013i	$0.558754\pi$
$822$		0			0
$823$	5.63209	+	53.5858i	0.196322	+	1.86788i	0.439961	+	0.898017i	$0.354992\pi$
$823$	5.63209	+	53.5858i	0.196322	+	1.86788i	−0.243639	+	0.969866i	$0.578341\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−35.5967	+	25.8626i	−1.23782	+	0.899329i	−0.997451	−	0.0713526i	$-0.977268\pi$
$827$	−35.5967	+	25.8626i	−1.23782	+	0.899329i	−0.240369	+	0.970682i	$0.577268\pi$
$828$	−90.1203	+	100.089i	−3.13190	+	3.47832i
$829$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$829$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$830$		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	$-0.100000\pi$
$839$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$840$		0			0
$841$	6.32408	+	4.59471i	0.218072	+	0.158438i
$842$	11.2900	−	107.417i	0.389080	−	3.70185i
$843$		0			0
$844$	−125.539	−	26.6842i	−4.32125	−	0.918509i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	37.0602			1.27265
$849$		0			0
$850$		0			0
$851$	41.2320	+	45.7927i	1.41341	+	1.56976i
$852$		0			0
$853$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$853$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$854$		0			0
$855$		0			0
$856$	−11.7974	−	5.25255i	−0.403227	−	0.179528i
$857$		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
$862$	30.6017	−	94.1823i	1.04230	−	3.20786i
$863$	−0.836228	+	7.95618i	−0.0284655	+	0.270831i	0.971027	+	0.238971i	$0.0768100\pi$
$863$	−0.836228	+	7.95618i	−0.0284655	+	0.270831i	−0.999492	+	0.0318607i	$0.989857\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−31.6059	+	41.8106i	−1.07216	+	1.41833i
$870$		0			0
$871$		0			0
$872$	49.5522	+	10.5326i	1.67805	+	0.356680i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−14.5799	+	16.1926i	−0.492327	+	0.546784i	−0.937192	−	0.348813i	$-0.886585\pi$
$877$	−14.5799	+	16.1926i	−0.492327	+	0.546784i	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
$883$	−3.70820	−	11.4127i	−0.124791	−	0.384067i	0.869072	−	0.494686i	$-0.164717\pi$
$883$	−3.70820	−	11.4127i	−0.124791	−	0.384067i	−0.993863	+	0.110619i	$0.964717\pi$
$884$		0			0
$885$		0			0
$886$	97.9553	−	43.6125i	3.29087	−	1.46519i
$887$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$887$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	26.1289	−	14.4319i	0.875352	−	0.483486i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	109.552	−	23.2861i	3.65581	−	0.777067i
$899$		0			0
$900$	−9.37963	−	89.2412i	−0.312654	−	2.97471i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	127.967	−	92.9734i	4.25611	−	3.09225i
$905$		0			0
$906$		0			0
$907$	−41.5031	+	18.4784i	−1.37809	+	0.613564i	−0.956101	−	0.293039i	$-0.905333\pi$
$907$	−41.5031	+	18.4784i	−1.37809	+	0.613564i	−0.421986	+	0.906602i	$0.638667\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−12.9443	+	9.40456i	−0.428863	+	0.311587i	−0.781194	−	0.624288i	$-0.785389\pi$
$911$	−12.9443	+	9.40456i	−0.428863	+	0.311587i	0.352331	+	0.935875i	$0.385389\pi$
$912$		0			0
$913$		0			0
$914$	−54.2538	+	93.9704i	−1.79456	+	3.10826i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−6.33491	+	60.2727i	−0.208970	+	1.98821i	−0.0757347	+	0.997128i	$0.524130\pi$
$919$	−6.33491	+	60.2727i	−0.208970	+	1.98821i	−0.133235	+	0.991084i	$0.542536\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−41.0547			−1.34987
$926$	107.606	+	47.9095i	3.53617	+	1.57440i
$927$		0			0
$928$	103.174	+	114.586i	3.38685	+	3.76147i
$929$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$929$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$930$		0			0
$931$		0			0
$932$	40.6691	+	125.167i	1.33216	+	4.09997i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$937$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$941$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	37.2820	−	49.3194i	1.21214	−	1.60351i
$947$	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
$953$	−0.682027	+	2.09906i	−0.0220930	+	0.0679954i	−0.961495	−	0.274822i	$-0.911381\pi$
$953$	−0.682027	+	2.09906i	−0.0220930	+	0.0679954i	0.939402	+	0.342817i	$0.111381\pi$
$954$	−10.6035	+	11.7764i	−0.343302	+	0.381275i
$955$		0			0
$956$	−7.79797	+	13.5065i	−0.252204	+	0.436831i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	30.3226	−	6.44526i	0.978148	−	0.207912i
$962$		0			0
$963$	3.14576	−	1.40058i	0.101371	−	0.0451332i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−57.6533			−1.85401			−0.927003	−	0.375053i	$-0.877624\pi$
$967$	−57.6533			−1.85401			−0.927003	+	0.375053i	$0.877624\pi$
$968$	17.5946	−	122.501i	0.565513	−	3.93734i
$969$		0			0
$970$		0			0
$971$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$971$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$972$		0			0
$973$		0			0
$974$	5.38387	+	3.91161i	0.172510	+	0.125336i
$975$		0			0
$976$		0			0
$977$	2.72248	+	25.9027i	0.0870999	+	0.828700i	0.947645	+	0.319326i	$0.103456\pi$
$977$	2.72248	+	25.9027i	0.0870999	+	0.828700i	−0.860545	+	0.509374i	$0.829877\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−10.9284	+	7.93992i	−0.348916	+	0.253502i
$982$	83.1811	−	92.3819i	2.65441	−	2.94803i
$983$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$983$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−24.7576	−	42.8814i	−0.787246	−	1.36355i
$990$		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		0.743145	−	0.669131i	$-0.233333\pi$
$997$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$998$	37.5102	+	7.97305i	1.18737	+	0.252382i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.q.d.410.2		16
7.2	even	3	inner	539.2.q.d.520.1		16
7.3	odd	6		539.2.f.c.344.1	yes	8
7.4	even	3		539.2.f.c.344.1	yes	8
7.5	odd	6	inner	539.2.q.d.520.1		16
7.6	odd	2	CM	539.2.q.d.410.2		16
11.4	even	5	inner	539.2.q.d.312.1		16
77.4	even	15		539.2.f.c.246.1	✓	8
77.24	even	30		5929.2.a.bg.1.1		4
77.26	odd	30	inner	539.2.q.d.422.2		16
77.31	odd	30		5929.2.a.bc.1.4		4
77.37	even	15	inner	539.2.q.d.422.2		16
77.46	odd	30		5929.2.a.bg.1.1		4
77.48	odd	10	inner	539.2.q.d.312.1		16
77.53	even	15		5929.2.a.bc.1.4		4
77.59	odd	30		539.2.f.c.246.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.f.c.246.1	✓	8	77.4	even	15
539.2.f.c.246.1	✓	8	77.59	odd	30
539.2.f.c.344.1	yes	8	7.3	odd	6
539.2.f.c.344.1	yes	8	7.4	even	3
539.2.q.d.312.1		16	11.4	even	5	inner
539.2.q.d.312.1		16	77.48	odd	10	inner
539.2.q.d.410.2		16	1.1	even	1	trivial
539.2.q.d.410.2		16	7.6	odd	2	CM
539.2.q.d.422.2		16	77.26	odd	30	inner
539.2.q.d.422.2		16	77.37	even	15	inner
539.2.q.d.520.1		16	7.2	even	3	inner
539.2.q.d.520.1		16	7.5	odd	6	inner
5929.2.a.bc.1.4		4	77.31	odd	30
5929.2.a.bc.1.4		4	77.53	even	15
5929.2.a.bg.1.1		4	77.24	even	30
5929.2.a.bg.1.1		4	77.46	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)

\(f(q)\)	\(=\)	\(q+(2.58102 + 1.14914i) q^{2} +(4.00286 + 4.44563i) q^{4} +(3.47668 + 10.7001i) q^{8} +(-2.74064 - 1.22021i) q^{9} +(0.750967 - 3.23049i) q^{11} +(-2.07198 + 19.7136i) q^{16} +(-5.67144 - 6.29877i) q^{18} +(5.65055 - 7.47498i) q^{22} +(-3.75233 - 6.49922i) q^{23} +(4.89074 + 1.03956i) q^{25} +(1.42225 - 4.37724i) q^{29} +(-16.7508 + 29.0132i) q^{32} +(-5.54579 - 17.0682i) q^{36} +(-8.03151 + 1.70715i) q^{37} +6.59794 q^{43} +(17.3676 - 9.59268i) q^{44} +(-2.21629 - 21.0866i) q^{46} +(11.4285 + 8.30328i) q^{50} +(-0.195430 - 1.85939i) q^{53} +(8.70093 - 9.66336i) q^{58} +(-44.5014 + 32.3322i) q^{64} +(-3.04737 + 5.27819i) q^{67} +(7.95588 + 5.78028i) q^{71} +(3.52808 - 33.5674i) q^{72} +(-22.6912 - 4.82317i) q^{74} +(-14.4367 - 6.42763i) q^{79} +(6.02218 + 6.68830i) q^{81} +(17.0294 + 7.58197i) q^{86} +(37.1775 - 3.19593i) q^{88} +(13.8731 - 42.6969i) q^{92} +(-6.00000 + 7.93725i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{2} - 2 q^{4} + 32 q^{8} - 6 q^{9} + 4 q^{11} + 28 q^{16} + 9 q^{18} - 8 q^{22} + 16 q^{23} - 10 q^{25} - 8 q^{29} - 100 q^{32} - 12 q^{36} + 18 q^{37} + 48 q^{43} - 9 q^{44} + 31 q^{46} + 20 q^{50}+ \cdots - 96 q^{99}+O(q^{100}) \) 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

Related objects

Downloads

Learn more