LMFDB - Embedded newform 539.2.f.c.246.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [539,2,Mod(148,539)] 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("539.148"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(539, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 539 = 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	539.f (of order $5$, degree $4$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,-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:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{5})$
Coefficient field:	8.0.37515625.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - x^{6} + 3x^{5} - x^{4} + 6x^{3} - 4x^{2} - 8x + 16 $ x^8 - x^7 - x^6 + 3x^5 - x^4 + 6x^3 - 4x^2 - 8x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{5}]$

Embedding invariants

Embedding label			246.1
Root			$1.18208 - 0.776336i$ of defining polynomial
Character	$\chi$	$=$	539.246
Dual form			539.2.f.c.344.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.28570 - 1.66066i) q^{2} +(1.84860 + 5.68940i) q^{4} +(3.47668 - 10.7001i) q^{8} +(2.42705 + 1.76336i) q^{9} +(-3.17317 - 0.964887i) q^{11} +(-16.0365 + 11.6512i) q^{16} +(-2.61917 - 8.06099i) q^{18} +(5.65055 + 7.47498i) q^{22} +7.50465 q^{23} +(-1.54508 + 4.75528i) q^{25} +(1.42225 + 4.37724i) q^{29} +33.5015 q^{32} +(-5.54579 + 17.0682i) q^{36} +(2.53732 + 7.80906i) q^{37} +6.59794 q^{43} +(-0.376282 - 19.8371i) q^{44} +(-17.1534 - 12.4626i) q^{46} +(11.4285 - 8.30328i) q^{50} +(-1.51257 - 1.09894i) q^{53} +(4.01825 - 12.3669i) q^{58} +(-44.5014 - 32.3322i) q^{64} +6.09473 q^{67} +(7.95588 - 5.78028i) q^{71} +(27.3062 - 19.8391i) q^{72} +(7.16862 - 22.0628i) q^{74} +(12.7848 + 9.28873i) q^{79} +(2.78115 + 8.55951i) q^{81} +(-15.0809 - 10.9569i) q^{86} +(-21.3565 + 30.5987i) q^{88} +(13.8731 + 42.6969i) q^{92} +(-6.00000 - 7.93725i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} + 2 q^{4} + 16 q^{8} + 6 q^{9} - 4 q^{11} - 28 q^{16} - 9 q^{18} - 4 q^{22} - 16 q^{23} + 10 q^{25} - 4 q^{29} + 100 q^{32} - 6 q^{36} - 18 q^{37} + 24 q^{43} + 9 q^{44} - 31 q^{46} + 10 q^{50}+ \cdots - 48 q^{99}+O(q^{100}) $ 8 * q - 2 * q^2 + 2 * q^4 + 16 * q^8 + 6 * q^9 - 4 * q^11 - 28 * q^16 - 9 * q^18 - 4 * q^22 - 16 * q^23 + 10 * q^25 - 4 * q^29 + 100 * q^32 - 6 * q^36 - 18 * q^37 + 24 * q^43 + 9 * q^44 - 31 * q^46 + 10 * q^50 - 30 * q^53 + 71 * q^58 - 74 * q^64 - 8 * q^67 + 48 * q^71 + 57 * q^72 + 12 * q^74 + 24 * q^79 - 18 * q^81 + 29 * q^86 - 38 * q^88 + 71 * q^92 - 48 * q^99

Character values

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

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

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 $

$2$	−2.28570	−	1.66066i	−1.61623	−	1.17426i	−0.835853	−	0.548953i	$-0.815027\pi$
$2$	−2.28570	−	1.66066i	−1.61623	−	1.17426i	−0.780378	−	0.625308i	$-0.784973\pi$
$3$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$3$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$4$	1.84860	+	5.68940i	0.924298	+	2.84470i
$5$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$5$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	3.47668	−	10.7001i	1.22919	−	3.78306i
$9$	2.42705	+	1.76336i	0.809017	+	0.587785i
$10$		0			0
$11$	−3.17317	−	0.964887i	−0.956746	−	0.290924i
$11$	−3.17317	−	0.964887i	−0.956746	−	0.290924i
$12$		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
$16$	−16.0365	+	11.6512i	−4.00912	+	2.91279i
$17$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$17$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$18$	−2.61917	−	8.06099i	−0.617345	−	1.89999i
$19$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$19$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$20$		0			0
$21$		0			0
$22$	5.65055	+	7.47498i	1.20470	+	1.59367i
$23$	7.50465			1.56483			0.782414	−	0.622758i	$-0.213988\pi$
$23$	7.50465			1.56483			0.782414	+	0.622758i	$0.213988\pi$
$24$		0			0
$25$	−1.54508	+	4.75528i	−0.309017	+	0.951057i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.42225	+	4.37724i	0.264105	+	0.812833i	0.991898	+	0.127036i	$0.0405463\pi$
$29$	1.42225	+	4.37724i	0.264105	+	0.812833i	−0.727793	+	0.685797i	$0.759454\pi$
$30$		0			0
$31$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$31$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$32$	33.5015			5.92229
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−5.54579	+	17.0682i	−0.924298	+	2.84470i
$37$	2.53732	+	7.80906i	0.417133	+	1.28380i	0.910330	+	0.413884i	$0.135828\pi$
$37$	2.53732	+	7.80906i	0.417133	+	1.28380i	−0.493197	+	0.869918i	$0.664172\pi$
$38$		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
$43$	6.59794			1.00618			0.503088	−	0.864235i	$-0.332197\pi$
$43$	6.59794			1.00618			0.503088	+	0.864235i	$0.332197\pi$
$44$	−0.376282	−	19.8371i	−0.0567266	−	2.99055i
$45$		0			0
$46$	−17.1534	−	12.4626i	−2.52912	−	1.83752i
$47$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$47$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$48$		0			0
$49$		0			0
$50$	11.4285	−	8.30328i	1.61623	−	1.17426i
$51$		0			0
$52$		0			0
$53$	−1.51257	−	1.09894i	−0.207767	−	0.150952i	0.479036	−	0.877795i	$-0.340986\pi$
$53$	−1.51257	−	1.09894i	−0.207767	−	0.150952i	−0.686803	+	0.726844i	$0.740986\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	4.01825	−	12.3669i	0.527622	−	1.62385i
$59$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$59$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$60$		0			0
$61$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$61$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$62$		0			0
$63$		0			0
$64$	−44.5014	−	32.3322i	−5.56268	−	4.04152i
$65$		0			0
$66$		0			0
$67$	6.09473			0.744590			0.372295	−	0.928114i	$-0.378571\pi$
$67$	6.09473			0.744590			0.372295	+	0.928114i	$0.378571\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	7.95588	−	5.78028i	0.944189	−	0.685993i	−0.00523645	−	0.999986i	$-0.501667\pi$
$71$	7.95588	−	5.78028i	0.944189	−	0.685993i	0.949425	+	0.313993i	$0.101667\pi$
$72$	27.3062	−	19.8391i	3.21807	−	2.33806i
$73$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$73$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$74$	7.16862	−	22.0628i	0.833336	−	2.56474i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	12.7848	+	9.28873i	1.43841	+	1.04506i	0.988372	+	0.152053i	$0.0485886\pi$
$79$	12.7848	+	9.28873i	1.43841	+	1.04506i	0.450035	+	0.893011i	$0.351411\pi$
$80$		0			0
$81$	2.78115	+	8.55951i	0.309017	+	0.951057i
$82$		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
$86$	−15.0809	−	10.9569i	−1.62621	−	1.18151i
$87$		0			0
$88$	−21.3565	+	30.5987i	−2.27661	+	3.26183i
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		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	$-0.300000\pi$
$97$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$98$		0			0
$99$	−6.00000	−	7.93725i	−0.603023	−	0.797724i
$100$	−29.9109			−2.99109
$101$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$101$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$102$		0			0
$103$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$103$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$104$		0			0
$105$		0			0
$106$	1.63230	+	5.02371i	0.158543	+	0.487945i
$107$	−0.354696	+	1.09164i	−0.0342898	+	0.105533i	−0.966736	−	0.255774i	$-0.917670\pi$
$107$	−0.354696	+	1.09164i	−0.0342898	+	0.105533i	0.932447	+	0.361308i	$0.117670\pi$
$108$		0			0
$109$	−4.50273			−0.431283			−0.215642	−	0.976473i	$-0.569184\pi$
$109$	−4.50273			−0.431283			−0.215642	+	0.976473i	$0.569184\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−4.34450	+	13.3710i	−0.408696	+	1.25784i	0.509073	+	0.860724i	$0.329988\pi$
$113$	−4.34450	+	13.3710i	−0.408696	+	1.25784i	−0.917769	+	0.397114i	$0.870012\pi$
$114$		0			0
$115$		0			0
$116$	−22.2747	+	16.1835i	−2.06815	+	1.50260i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	9.13799	+	6.12350i	0.830726	+	0.556682i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	8.21418	−	5.96795i	0.728890	−	0.529570i	−0.160322	−	0.987065i	$-0.551253\pi$
$127$	8.21418	−	5.96795i	0.728890	−	0.529570i	0.889212	+	0.457495i	$0.151253\pi$
$128$	27.3190	+	84.0793i	2.41468	+	7.43163i
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$	−13.9307	−	10.1213i	−1.20343	−	0.874343i
$135$		0			0
$136$		0			0
$137$	18.7856	−	13.6485i	1.60496	−	1.16607i	0.727909	−	0.685674i	$-0.240493\pi$
$137$	18.7856	−	13.6485i	1.60496	−	1.16607i	0.877051	−	0.480397i	$-0.159507\pi$
$138$		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
$142$	−27.7838			−2.33156
$143$		0			0
$144$	−59.4665			−4.95554
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−39.7384	+	28.8716i	−3.26647	+	2.37323i
$149$	−17.7984	+	12.9313i	−1.45810	+	1.05937i	−0.474247	+	0.880392i	$0.657280\pi$
$149$	−17.7984	+	12.9313i	−1.45810	+	1.05937i	−0.983853	+	0.178979i	$0.942720\pi$
$150$		0			0
$151$	−3.84693	+	11.8396i	−0.313059	+	0.963496i	0.663487	+	0.748187i	$0.269076\pi$
$151$	−3.84693	+	11.8396i	−0.313059	+	0.963496i	−0.976546	+	0.215308i	$0.930924\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$157$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$158$	−13.7969	−	42.4624i	−1.09762	−	3.37813i
$159$		0			0
$160$		0			0
$161$		0			0
$162$	7.85752	−	24.1830i	0.617345	−	1.89999i
$163$	−7.21418	−	5.24141i	−0.565058	−	0.410539i	0.268249	−	0.963350i	$-0.413555\pi$
$163$	−7.21418	−	5.24141i	−0.565058	−	0.410539i	−0.833307	+	0.552811i	$0.813555\pi$
$164$		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
$172$	12.1969	+	37.5383i	0.930007	+	2.86227i
$173$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$173$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$174$		0			0
$175$		0			0
$176$	62.1285	−	21.4977i	4.68311	−	1.62045i
$177$		0			0
$178$		0			0
$179$	−7.39370	+	22.7555i	−0.552631	+	1.70082i	0.149487	+	0.988764i	$0.452238\pi$
$179$	−7.39370	+	22.7555i	−0.552631	+	1.70082i	−0.702118	+	0.712060i	$0.747762\pi$
$180$		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
$184$	26.0913	−	80.3007i	1.92347	−	5.91985i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−6.80563	−	20.9456i	−0.492438	−	1.51557i	−0.820912	−	0.571055i	$-0.806534\pi$
$191$	−6.80563	−	20.9456i	−0.492438	−	1.51557i	0.328474	−	0.944513i	$-0.393466\pi$
$192$		0			0
$193$	21.8462	−	15.8722i	1.57252	−	1.14250i	0.647834	−	0.761781i	$-0.275675\pi$
$193$	21.8462	−	15.8722i	1.57252	−	1.14250i	0.924689	−	0.380724i	$-0.124325\pi$
$194$		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	$0.723046\pi$
$198$	0.533133	+	28.1061i	0.0378881	+	1.99741i
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	45.5103	+	33.0652i	3.21807	+	2.33806i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	18.2142	+	13.2334i	1.26597	+	0.919783i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−17.3570	−	12.6106i	−1.19490	−	0.868148i	−0.201129	−	0.979565i	$-0.564461\pi$
$211$	−17.3570	−	12.6106i	−1.19490	−	0.868148i	−0.993774	+	0.111417i	$0.964461\pi$
$212$	3.45620	−	10.6371i	0.237373	−	0.730559i
$213$		0			0
$214$	2.62357	−	1.90614i	0.179344	−	0.130301i
$215$		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	$-0.900000\pi$
$223$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$224$		0			0
$225$	−12.1353	+	8.81678i	−0.809017	+	0.587785i
$226$	32.1348	−	23.3473i	2.13758	−	1.55304i
$227$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$227$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$228$		0			0
$229$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$229$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$230$		0			0
$231$		0			0
$232$	51.7817			3.39963
$233$	−17.7984	−	12.9313i	−1.16601	−	0.847156i	−0.175484	−	0.984482i	$-0.556149\pi$
$233$	−17.7984	−	12.9313i	−1.16601	−	0.847156i	−0.990526	+	0.137326i	$0.956149\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	0.805627	−	2.47947i	0.0521117	−	0.160383i	−0.921614	−	0.388108i	$-0.873129\pi$
$239$	0.805627	−	2.47947i	0.0521117	−	0.160383i	0.973726	+	0.227725i	$0.0731287\pi$
$240$		0			0
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$	−10.7176	−	29.1715i	−0.688956	−	1.87522i
$243$		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
$253$	−23.8135	−	7.24115i	−1.49714	−	0.455247i
$254$	−28.6858			−1.79991
$255$		0			0
$256$	43.1877	−	132.918i	2.69923	−	8.30739i
$257$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$257$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−4.26675	+	13.1317i	−0.264105	+	0.812833i
$262$		0			0
$263$	−28.9988			−1.78814			−0.894072	−	0.447924i	$-0.852164\pi$
$263$	−28.9988			−1.78814			−0.894072	+	0.447924i	$0.852164\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	11.2667	+	34.6754i	0.688224	+	2.11813i
$269$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$269$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$270$		0			0
$271$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$271$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$272$		0			0
$273$		0			0
$274$	−65.6036			−3.96326
$275$	9.49113	−	13.5985i	0.572336	−	0.820019i
$276$		0			0
$277$	−21.9284	−	15.9319i	−1.31755	−	0.957254i	−0.999959	−	0.00902525i	$-0.997127\pi$
$277$	−21.9284	−	15.9319i	−1.31755	−	0.957254i	−0.317588	−	0.948229i	$-0.602873\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−6.95218	+	5.05106i	−0.414733	+	0.301321i	−0.775515	−	0.631329i	$-0.782510\pi$
$281$	−6.95218	+	5.05106i	−0.414733	+	0.301321i	0.360782	+	0.932650i	$0.382510\pi$
$282$		0			0
$283$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$283$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$284$	47.5935	+	34.5787i	2.82416	+	2.05187i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	81.3099	+	59.0751i	4.79123	+	3.48104i
$289$	−5.25329	+	16.1680i	−0.309017	+	0.951057i
$290$		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
$296$	92.3793			5.36944
$297$		0			0
$298$	62.1561			3.60060
$299$		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		−0.951057	−	0.309017i	$-0.900000\pi$
$311$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$312$		0			0
$313$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$313$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$314$		0			0
$315$		0			0
$316$	−29.2133	+	89.9092i	−1.64337	+	5.05779i
$317$	−27.2858	−	19.8243i	−1.53252	−	1.11344i	−0.954815	−	0.297200i	$-0.903947\pi$
$317$	−27.2858	−	19.8243i	−1.53252	−	1.11344i	−0.577708	−	0.816243i	$-0.696053\pi$
$318$		0			0
$319$	−0.289499	−	15.2620i	−0.0162088	−	0.854509i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−43.5572	+	31.6462i	−2.41984	+	1.75812i
$325$		0			0
$326$	7.78525	+	23.9605i	0.431185	+	1.32705i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	6.09209			0.334852			0.167426	−	0.985885i	$-0.446455\pi$
$331$	6.09209			0.334852			0.167426	+	0.985885i	$0.446455\pi$
$332$		0			0
$333$	−7.61195	+	23.4272i	−0.417133	+	1.28380i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	9.08528	+	27.9616i	0.494907	+	1.52317i	0.817102	+	0.576493i	$0.195579\pi$
$337$	9.08528	+	27.9616i	0.494907	+	1.52317i	−0.322195	+	0.946673i	$0.604421\pi$
$338$	−11.3498	+	34.9310i	−0.617345	+	1.89999i
$339$		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
$347$	20.2318	−	14.6993i	1.08610	−	0.789100i	0.107366	−	0.994220i	$-0.465758\pi$
$347$	20.2318	−	14.6993i	1.08610	−	0.789100i	0.978737	+	0.205120i	$0.0657585\pi$
$348$		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
$352$	−106.306	−	32.3252i	−5.66613	−	1.72294i
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	54.6888	−	39.7337i	2.89039	−	2.09999i
$359$	−10.1220	−	31.1523i	−0.534219	−	1.64416i	−0.745331	−	0.666695i	$-0.767708\pi$
$359$	−10.1220	−	31.1523i	−0.534219	−	1.64416i	0.211112	−	0.977462i	$-0.432292\pi$
$360$		0			0
$361$	15.3713	+	11.1679i	0.809017	+	0.587785i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$367$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$368$	−120.348	+	87.4380i	−6.27358	+	4.55802i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	22.0000			1.13912			0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000			1.13912			0.569558	+	0.821951i	$0.307114\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	31.4997	−	22.8859i	1.61803	−	1.17557i	0.808511	−	0.588481i	$-0.200274\pi$
$379$	31.4997	−	22.8859i	1.61803	−	1.17557i	0.809522	−	0.587090i	$-0.199726\pi$
$380$		0			0
$381$		0			0
$382$	−19.2278	+	59.1770i	−0.983779	+	3.02776i
$383$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$383$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$384$		0			0
$385$		0			0
$386$	−76.2920			−3.88316
$387$	16.0135	+	11.6345i	0.814013	+	0.591415i
$388$		0			0
$389$	−6.73894	−	20.7403i	−0.341678	−	1.05158i	−0.963338	−	0.268290i	$-0.913542\pi$
$389$	−6.73894	−	20.7403i	−0.341678	−	1.05158i	0.621660	−	0.783287i	$-0.286458\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	41.3699	+	30.0570i	2.08419	+	1.51425i
$395$		0			0
$396$	34.0666	−	48.8092i	1.71191	−	2.45275i
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$		0			0
$399$		0			0
$400$	−30.6269	−	94.2600i	−1.53135	−	4.71300i
$401$	−7.78557	+	5.65655i	−0.388793	+	0.282475i	−0.764961	−	0.644077i	$-0.777242\pi$
$401$	−7.78557	+	5.65655i	−0.388793	+	0.282475i	0.376168	+	0.926552i	$0.377242\pi$
$402$		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		0.587785	−	0.809017i	$-0.300000\pi$
$409$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$410$		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	0.0578225	+	0.998327i	$0.481584\pi$
$421$	−11.8136	+	36.3585i	−0.575759	+	1.77200i	−0.633581	+	0.773676i	$0.718416\pi$
$422$	18.7309	+	57.6479i	0.911807	+	2.80625i
$423$		0			0
$424$	−17.0175	+	12.3640i	−0.826445	+	0.600447i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−6.86648			−0.331904
$429$		0			0
$430$		0			0
$431$	−28.3570	−	20.6025i	−1.36591	−	0.992389i	−0.998044	−	0.0625092i	$-0.980090\pi$
$431$	−28.3570	−	20.6025i	−1.36591	−	0.992389i	−0.367862	−	0.929880i	$-0.619910\pi$
$432$		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
$436$	−8.32373	−	25.6178i	−0.398634	−	1.22687i
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	11.7279	−	36.0947i	0.557208	−	1.71491i	−0.132831	−	0.991139i	$-0.542407\pi$
$443$	11.7279	−	36.0947i	0.557208	−	1.71491i	0.690039	−	0.723772i	$-0.257593\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	32.0711	+	23.3010i	1.51353	+	1.09964i	0.964580	+	0.263790i	$0.0849724\pi$
$449$	32.0711	+	23.3010i	1.51353	+	1.09964i	0.548950	+	0.835855i	$0.315028\pi$
$450$	42.3791			1.99777
$451$		0			0
$452$	−84.1041			−3.95593
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−31.0711	+	22.5745i	−1.45345	+	1.05599i	−0.468436	+	0.883497i	$0.655182\pi$
$457$	−31.0711	+	22.5745i	−1.45345	+	1.05599i	−0.985011	+	0.172493i	$0.944818\pi$
$458$		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	$0.0797429\pi$
$464$	−73.8078	−	53.6245i	−3.42644	−	2.48946i
$465$		0			0
$466$	19.2073	+	59.1139i	0.889760	+	2.73840i
$467$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$467$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−20.9364	−	6.36627i	−0.962655	−	0.292721i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−1.73325	−	5.33439i	−0.0793599	−	0.244245i
$478$	−5.95896	+	4.32944i	−0.272556	+	0.198024i
$479$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$479$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$480$		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
$487$	−0.727878	+	2.24018i	−0.0329833	+	0.101512i	−0.966193	−	0.257821i	$-0.916996\pi$
$487$	−0.727878	+	2.24018i	−0.0329833	+	0.101512i	0.933210	+	0.359333i	$0.116996\pi$
$488$		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
$499$	−4.19437	−	12.9090i	−0.187766	−	0.577884i	0.812219	−	0.583352i	$-0.198259\pi$
$499$	−4.19437	−	12.9090i	−0.187766	−	0.577884i	−0.999985	+	0.00546838i	$0.998259\pi$
$500$		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
$506$	42.4054	+	56.0971i	1.88515	+	2.49382i
$507$		0			0
$508$	49.1387	+	35.7014i	2.18018	+	1.58399i
$509$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$509$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$510$		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		0.951057	−	0.309017i	$-0.100000\pi$
$521$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$522$	31.5598	−	22.9295i	1.38133	−	1.00360i
$523$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$523$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$524$		0			0
$525$		0			0
$526$	66.2825	+	48.1570i	2.89005	+	2.09975i
$527$		0			0
$528$		0			0
$529$	33.3198			1.44869
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	21.1894	−	65.2144i	0.915244	−	2.81683i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	32.9284	+	23.9239i	1.41570	+	1.02857i	0.992463	+	0.122548i	$0.0391066\pi$
$541$	32.9284	+	23.9239i	1.41570	+	1.02857i	0.423238	+	0.906019i	$0.360893\pi$
$542$		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	$-0.289785\pi$
$548$	112.379	+	81.6479i	4.80058	+	3.48783i
$549$		0			0
$550$	−44.2762	+	15.3205i	−1.88794	+	0.653268i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	23.6642	+	72.8309i	1.00540	+	3.09429i
$555$		0			0
$556$		0			0
$557$	−9.57775	−	29.4773i	−0.405822	−	1.24899i	−0.920207	−	0.391433i	$-0.871980\pi$
$557$	−9.57775	−	29.4773i	−0.405822	−	1.24899i	0.514384	−	0.857560i	$-0.328020\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	24.2787			1.02413
$563$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$563$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	−34.1897	−	105.225i	−1.43457	−	4.41514i
$569$	6.79837	−	20.9232i	0.285003	−	0.877148i	−0.701395	−	0.712773i	$-0.747439\pi$
$569$	6.79837	−	20.9232i	0.285003	−	0.877148i	0.986398	−	0.164375i	$-0.0525608\pi$
$570$		0			0
$571$	−44.0566			−1.84371			−0.921856	−	0.387534i	$-0.873327\pi$
$571$	−44.0566			−1.84371			−0.921856	+	0.387534i	$0.873327\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−11.5953	+	35.6868i	−0.483559	+	1.48824i
$576$	−50.9941	−	156.944i	−2.12475	−	6.53932i
$577$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$577$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$578$	38.8568	−	28.2311i	1.61623	−	1.17426i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	3.73927	+	4.94659i	0.154865	+	0.204867i
$584$		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
$592$	−131.674	−	95.6670i	−5.41178	−	3.93189i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$	−106.473	−	77.3573i	−4.36131	−	3.16868i
$597$		0			0
$598$		0			0
$599$	20.4997	−	14.8939i	0.837597	−	0.608550i	−0.0841014	−	0.996457i	$-0.526802\pi$
$599$	20.4997	−	14.8939i	0.837597	−	0.608550i	0.921698	+	0.387907i	$0.126802\pi$
$600$		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
$603$	14.7922	+	10.7472i	0.602386	+	0.437659i
$604$	−74.4718			−3.03021
$605$		0			0
$606$		0			0
$607$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$607$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	15.2668	−	46.9862i	0.616618	−	1.89776i	0.243974	−	0.969782i	$-0.421549\pi$
$613$	15.2668	−	46.9862i	0.616618	−	1.89776i	0.372644	−	0.927974i	$-0.378451\pi$
$614$		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		−0.951057	−	0.309017i	$-0.900000\pi$
$619$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−20.2254	−	14.6946i	−0.809017	−	0.587785i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−12.4683	−	38.3736i	−0.496357	−	1.52763i	−0.814832	−	0.579698i	$-0.803171\pi$
$631$	−12.4683	−	38.3736i	−0.496357	−	1.52763i	0.318475	−	0.947931i	$-0.396829\pi$
$632$	143.839	−	104.505i	5.72162	−	4.15700i
$633$		0			0
$634$	29.4457	+	90.6247i	1.16944	+	3.59916i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−24.6833	+	35.3651i	−0.977219	+	1.40012i
$639$	29.5020			1.16708
$640$		0			0
$641$	−15.3445	+	47.2255i	−0.606071	+	1.86530i	−0.116820	+	0.993153i	$0.537270\pi$
$641$	−15.3445	+	47.2255i	−0.606071	+	1.86530i	−0.489251	+	0.872143i	$0.662730\pi$
$642$		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		0.587785	−	0.809017i	$-0.300000\pi$
$647$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$648$	101.257			3.97775
$649$		0			0
$650$		0			0
$651$		0			0
$652$	16.4843	−	50.7336i	0.645576	−	1.98688i
$653$	1.66431	+	5.12221i	0.0651294	+	0.200448i	0.978326	−	0.207072i	$-0.0663936\pi$
$653$	1.66431	+	5.12221i	0.0651294	+	0.200448i	−0.913196	+	0.407520i	$0.866394\pi$
$654$		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			−	1.00000i	$-0.5\pi$
$661$		0			0				1.00000i	$0.5\pi$
$662$	−13.9247	−	10.1169i	−0.541198	−	0.393203i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	56.3031	−	40.9066i	2.18170	−	1.58510i
$667$	10.6735	+	32.8497i	0.413280	+	1.27194i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.494817	−	0.359506i	−0.0190738	−	0.0138579i	0.578208	−	0.815890i	$-0.303752\pi$
$673$	−0.494817	−	0.359506i	−0.0190738	−	0.0138579i	−0.597281	+	0.802032i	$0.703752\pi$
$674$	25.6684	−	78.9993i	0.988711	−	3.04294i
$675$		0			0
$676$	62.9160	−	45.7111i	2.41984	−	1.75812i
$677$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$677$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−21.1014			−0.807423			−0.403711	−	0.914886i	$-0.632280\pi$
$683$	−21.1014			−0.807423			−0.403711	+	0.914886i	$0.632280\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−105.808	+	76.8737i	−4.03388	+	2.93078i
$689$		0			0
$690$		0			0
$691$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$691$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$692$		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	0.0377695	+	0.999286i	$0.487975\pi$
$701$	−15.3604	+	47.2743i	−0.580153	+	1.78553i	−0.617922	+	0.786239i	$0.712025\pi$
$702$		0			0
$703$		0			0
$704$	110.014	+	145.534i	4.14629	+	5.48503i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	21.2355	−	15.4285i	0.797517	−	0.579430i	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	21.2355	−	15.4285i	0.797517	−	0.579430i	0.910185	+	0.414202i	$0.135939\pi$
$710$		0			0
$711$	14.6501	+	45.0885i	0.549423	+	1.69095i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−143.133			−5.34913
$717$		0			0
$718$	−28.5975	+	88.0139i	−1.06725	+	3.28465i
$719$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$719$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$720$		0			0
$721$		0			0
$722$	−16.5881	−	51.0529i	−0.617345	−	1.89999i
$723$		0			0
$724$		0			0
$725$	−23.0125			−0.854663
$726$		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		0.951057	−	0.309017i	$-0.100000\pi$
$733$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$734$		0			0
$735$		0			0
$736$	251.417			9.26737
$737$	−19.3396	−	5.88073i	−0.712384	−	0.216620i
$738$		0			0
$739$	−41.5832	−	30.2120i	−1.52966	−	1.11137i	−0.956425	−	0.291977i	$-0.905687\pi$
$739$	−41.5832	−	30.2120i	−1.52966	−	1.11137i	−0.573238	−	0.819389i	$-0.694313\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−8.56653	+	6.22395i	−0.314275	+	0.228334i	−0.733729	−	0.679442i	$-0.762222\pi$
$743$	−8.56653	+	6.22395i	−0.314275	+	0.228334i	0.419453	+	0.907777i	$0.362222\pi$
$744$		0			0
$745$		0			0
$746$	−50.2853	−	36.5344i	−1.84108	−	1.33762i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	3.19208	−	9.82420i	0.116481	−	0.358490i	−0.875772	−	0.482724i	$-0.839647\pi$
$751$	3.19208	−	9.82420i	0.116481	−	0.358490i	0.992253	+	0.124234i	$0.0396474\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−30.3109	−	22.0222i	−1.10167	−	0.800410i	−0.120338	−	0.992733i	$-0.538398\pi$
$757$	−30.3109	−	22.0222i	−1.10167	−	0.800410i	−0.981332	+	0.192323i	$0.938398\pi$
$758$	−110.004			−3.99554
$759$		0			0
$760$		0			0
$761$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$761$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$762$		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
$772$	130.688	+	94.9503i	4.70356	+	3.41734i
$773$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$773$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$774$	−17.2812	−	53.1859i	−0.621158	−	1.91173i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−19.0394	+	58.5972i	−0.682595	+	2.10081i
$779$		0			0
$780$		0			0
$781$	−30.8227	+	10.6653i	−1.10292	+	0.381634i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$787$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$788$	−33.4586	−	102.975i	−1.19191	−	3.66834i
$789$		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	$-0.700000\pi$
$797$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$798$		0			0
$799$		0			0
$800$	−51.7627	+	159.309i	−1.83009	+	5.63243i
$801$		0			0
$802$	27.1890			0.960078
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	42.0711	−	30.5665i	1.47914	−	1.07466i	0.501311	−	0.865267i	$-0.332851\pi$
$809$	42.0711	−	30.5665i	1.47914	−	1.07466i	0.977832	−	0.209393i	$-0.0671487\pi$
$810$		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
$814$	−44.0353	+	63.0919i	−1.54344	+	2.21137i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	6.79837	+	20.9232i	0.237265	+	0.730226i	0.996813	+	0.0797750i	$0.0254202\pi$
$821$	6.79837	+	20.9232i	0.237265	+	0.730226i	−0.759548	+	0.650451i	$0.774580\pi$
$822$		0			0
$823$	43.5906	+	31.6704i	1.51947	+	1.10396i	0.961748	+	0.273936i	$0.0883256\pi$
$823$	43.5906	+	31.6704i	1.51947	+	1.10396i	0.557725	+	0.830026i	$0.311674\pi$
$824$		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	$0.977268\pi$
$828$	−41.6192	+	128.091i	−1.44637	+	4.45146i
$829$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$829$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$830$		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
$841$	6.32408	−	4.59471i	0.218072	−	0.158438i
$842$	87.3811	−	63.4861i	3.01135	−	2.18788i
$843$		0			0
$844$	39.6605	−	122.062i	1.36517	−	4.20156i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	37.0602			1.27265
$849$		0			0
$850$		0			0
$851$	19.0417	+	58.6043i	0.652741	+	2.00893i
$852$		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
$856$	10.4475	+	7.59059i	0.357090	+	0.259441i
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$		0			0
$862$	30.6017	+	94.1823i	1.04230	+	3.20786i
$863$	−6.47214	+	4.70228i	−0.220314	+	0.160068i	−0.692468	−	0.721449i	$-0.743477\pi$
$863$	−6.47214	+	4.70228i	−0.220314	+	0.160068i	0.472154	+	0.881516i	$0.343477\pi$
$864$		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
$872$	−15.6545	+	48.1797i	−0.530130	+	1.63157i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−6.73325	+	20.7228i	−0.227366	+	0.699759i	0.770677	+	0.637226i	$0.219918\pi$
$877$	−6.73325	+	20.7228i	−0.227366	+	0.699759i	−0.998043	+	0.0625337i	$0.980082\pi$
$878$		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
$886$	−86.7472	+	63.0255i	−2.91433	+	2.11738i
$887$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$887$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−0.566103	−	29.8443i	−0.0189652	−	0.999820i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−34.6099	−	106.518i	−1.15495	−	3.55456i
$899$		0			0
$900$	−72.5953	−	52.7436i	−2.41984	−	1.75812i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	127.967	+	92.9734i	4.25611	+	3.09225i
$905$		0			0
$906$		0			0
$907$	36.7543	−	26.7035i	1.22040	−	0.886676i	0.224271	−	0.974527i	$-0.428000\pi$
$907$	36.7543	−	26.7035i	1.22040	−	0.886676i	0.996134	+	0.0878507i	$0.0279999\pi$
$908$		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
$914$	108.508			3.58911
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−49.0302	+	35.6225i	−1.61736	+	1.17508i	−0.791687	+	0.610927i	$0.790797\pi$
$919$	−49.0302	+	35.6225i	−1.61736	+	1.17508i	−0.825671	+	0.564152i	$0.809203\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−41.0547			−1.34987
$926$	−95.2941	−	69.2352i	−3.13156	−	2.27521i
$927$		0			0
$928$	47.6476	+	146.644i	1.56411	+	4.81383i
$929$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$929$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$930$		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	$-0.300000\pi$
$937$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$941$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	37.2820	+	49.3194i	1.21214	+	1.60351i
$947$	−20.0000			−0.649913			−0.324956	−	0.945729i	$-0.605350\pi$
$947$	−20.0000			−0.649913			−0.324956	+	0.945729i	$0.605350\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.682027	−	2.09906i	−0.0220930	−	0.0679954i	0.939402	−	0.342817i	$-0.111381\pi$
$953$	−0.682027	−	2.09906i	−0.0220930	−	0.0679954i	−0.961495	+	0.274822i	$0.911381\pi$
$954$	−4.89690	+	15.0711i	−0.158543	+	0.487945i
$955$		0			0
$956$	15.5959			0.504409
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−9.57953	−	29.4828i	−0.309017	−	0.951057i
$962$		0			0
$963$	−2.78582	+	2.02402i	−0.0897719	+	0.0652231i
$964$		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	$0.877624\pi$
$968$	97.2920	−	76.4881i	3.12708	−	2.45842i
$969$		0			0
$970$		0			0
$971$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$971$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$972$		0			0
$973$		0			0
$974$	5.38387	−	3.91161i	0.172510	−	0.125336i
$975$		0			0
$976$		0			0
$977$	21.0711	+	15.3091i	0.674126	+	0.489781i	0.871404	−	0.490567i	$-0.163210\pi$
$977$	21.0711	+	15.3091i	0.674126	+	0.489781i	−0.197278	+	0.980348i	$0.563210\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−10.9284	−	7.93992i	−0.348916	−	0.253502i
$982$	38.4146	−	118.228i	1.22586	−	3.77280i
$983$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$983$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	49.5152			1.57449
$990$		0			0
$991$	−24.0000			−0.762385			−0.381193	−	0.924496i	$-0.624487\pi$
$991$	−24.0000			−0.762385			−0.381193	+	0.924496i	$0.624487\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$997$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$998$	−11.8503	+	36.4714i	−0.375113	+	1.15448i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.f.c.246.1	✓	8
7.2	even	3		539.2.q.d.312.1		16
7.3	odd	6		539.2.q.d.422.2		16
7.4	even	3		539.2.q.d.422.2		16
7.5	odd	6		539.2.q.d.312.1		16
7.6	odd	2	CM	539.2.f.c.246.1	✓	8
11.3	even	5	inner	539.2.f.c.344.1	yes	8
11.5	even	5		5929.2.a.bc.1.4		4
11.6	odd	10		5929.2.a.bg.1.1		4
77.3	odd	30		539.2.q.d.520.1		16
77.6	even	10		5929.2.a.bg.1.1		4
77.25	even	15		539.2.q.d.520.1		16
77.27	odd	10		5929.2.a.bc.1.4		4
77.47	odd	30		539.2.q.d.410.2		16
77.58	even	15		539.2.q.d.410.2		16
77.69	odd	10	inner	539.2.f.c.344.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.f.c.246.1	✓	8	1.1	even	1	trivial
539.2.f.c.246.1	✓	8	7.6	odd	2	CM
539.2.f.c.344.1	yes	8	11.3	even	5	inner
539.2.f.c.344.1	yes	8	77.69	odd	10	inner
539.2.q.d.312.1		16	7.2	even	3
539.2.q.d.312.1		16	7.5	odd	6
539.2.q.d.410.2		16	77.47	odd	30
539.2.q.d.410.2		16	77.58	even	15
539.2.q.d.422.2		16	7.3	odd	6
539.2.q.d.422.2		16	7.4	even	3
539.2.q.d.520.1		16	77.3	odd	30
539.2.q.d.520.1		16	77.25	even	15
5929.2.a.bc.1.4		4	11.5	even	5
5929.2.a.bc.1.4		4	77.27	odd	10
5929.2.a.bg.1.1		4	11.6	odd	10
5929.2.a.bg.1.1		4	77.6	even	10

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.f (of order \(5\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.28570 - 1.66066i) q^{2} +(1.84860 + 5.68940i) q^{4} +(3.47668 - 10.7001i) q^{8} +(2.42705 + 1.76336i) q^{9} +(-3.17317 - 0.964887i) q^{11} +(-16.0365 + 11.6512i) q^{16} +(-2.61917 - 8.06099i) q^{18} +(5.65055 + 7.47498i) q^{22} +7.50465 q^{23} +(-1.54508 + 4.75528i) q^{25} +(1.42225 + 4.37724i) q^{29} +33.5015 q^{32} +(-5.54579 + 17.0682i) q^{36} +(2.53732 + 7.80906i) q^{37} +6.59794 q^{43} +(-0.376282 - 19.8371i) q^{44} +(-17.1534 - 12.4626i) q^{46} +(11.4285 - 8.30328i) q^{50} +(-1.51257 - 1.09894i) q^{53} +(4.01825 - 12.3669i) q^{58} +(-44.5014 - 32.3322i) q^{64} +6.09473 q^{67} +(7.95588 - 5.78028i) q^{71} +(27.3062 - 19.8391i) q^{72} +(7.16862 - 22.0628i) q^{74} +(12.7848 + 9.28873i) q^{79} +(2.78115 + 8.55951i) q^{81} +(-15.0809 - 10.9569i) q^{86} +(-21.3565 + 30.5987i) q^{88} +(13.8731 + 42.6969i) q^{92} +(-6.00000 - 7.93725i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 2 q^{4} + 16 q^{8} + 6 q^{9} - 4 q^{11} - 28 q^{16} - 9 q^{18} - 4 q^{22} - 16 q^{23} + 10 q^{25} - 4 q^{29} + 100 q^{32} - 6 q^{36} - 18 q^{37} + 24 q^{43} + 9 q^{44} - 31 q^{46} + 10 q^{50}+ \cdots - 48 q^{99}+O(q^{100}) \) 8 * q - 2 * q^2 + 2 * q^4 + 16 * q^8 + 6 * q^9 - 4 * q^11 - 28 * q^16 - 9 * q^18 - 4 * q^22 - 16 * q^23 + 10 * q^25 - 4 * q^29 + 100 * q^32 - 6 * q^36 - 18 * q^37 + 24 * q^43 + 9 * q^44 - 31 * q^46 + 10 * q^50 - 30 * q^53 + 71 * q^58 - 74 * q^64 - 8 * q^67 + 48 * q^71 + 57 * q^72 + 12 * q^74 + 24 * q^79 - 18 * q^81 + 29 * q^86 - 38 * q^88 + 71 * q^92 - 48 * q^99

Introduction

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Database

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