LMFDB - Embedded newform 539.2.f.c.295.1

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			295.1
Root			$1.10362 + 0.884319i$ of defining polynomial
Character	$\chi$	$=$	539.295
Dual form			539.2.f.c.148.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.730563 + 2.24844i) q^{2} +(-2.90373 - 2.10968i) q^{4} +(3.03958 - 2.20838i) q^{8} +(-0.927051 + 2.85317i) q^{9} +(-1.89823 + 2.71970i) q^{11} +(0.526564 + 1.62060i) q^{16} +(-5.73791 - 4.16884i) q^{18} +(-4.72830 - 6.25495i) q^{22} -9.58240 q^{23} +(4.04508 - 2.93893i) q^{25} +(-8.64279 - 6.27935i) q^{29} +3.48575 q^{32} +(8.71119 - 6.32905i) q^{36} +(1.10547 + 0.803169i) q^{37} -8.74072 q^{43} +(11.2496 - 3.89261i) q^{44} +(7.00055 - 21.5455i) q^{46} +(3.65281 + 11.2422i) q^{50} +(-4.06518 + 12.5113i) q^{53} +(20.4328 - 14.8453i) q^{58} +(-3.59968 + 11.0787i) q^{64} +16.3336 q^{67} +(3.08300 + 9.48849i) q^{71} +(3.48305 + 10.7197i) q^{72} +(-2.61349 + 1.89881i) q^{74} +(-3.90147 + 12.0075i) q^{79} +(-7.28115 - 5.29007i) q^{81} +(6.38565 - 19.6530i) q^{86} +(0.236325 + 12.4587i) q^{88} +(27.8247 + 20.2158i) q^{92} +(-6.00000 - 7.93725i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/539\mathbb{Z}\right)^\times$.

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

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 $

$2$	−0.730563	+	2.24844i	−0.516586	+	1.58989i	0.263792	+	0.964580i	$0.415027\pi$
$2$	−0.730563	+	2.24844i	−0.516586	+	1.58989i	−0.780378	+	0.625308i	$0.784973\pi$
$3$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$3$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$4$	−2.90373	−	2.10968i	−1.45187	−	1.05484i
$5$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$5$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	3.03958	−	2.20838i	1.07465	−	0.780782i
$9$	−0.927051	+	2.85317i	−0.309017	+	0.951057i
$10$		0			0
$11$	−1.89823	+	2.71970i	−0.572336	+	0.820019i
$11$	−1.89823	+	2.71970i	−0.572336	+	0.820019i
$12$		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
$16$	0.526564	+	1.62060i	0.131641	+	0.405150i
$17$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$17$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$18$	−5.73791	−	4.16884i	−1.35244	−	0.982605i
$19$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$19$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$20$		0			0
$21$		0			0
$22$	−4.72830	−	6.25495i	−1.00808	−	1.33356i
$23$	−9.58240			−1.99807			−0.999035	−	0.0439305i	$-0.986012\pi$
$23$	−9.58240			−1.99807			−0.999035	+	0.0439305i	$0.986012\pi$
$24$		0			0
$25$	4.04508	−	2.93893i	0.809017	−	0.587785i
$26$		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		−0.951057	−	0.309017i	$-0.900000\pi$
$31$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$32$	3.48575			0.616199
$33$		0			0
$34$		0			0
$35$		0			0
$36$	8.71119	−	6.32905i	1.45187	−	1.05484i
$37$	1.10547	+	0.803169i	0.181738	+	0.132040i	0.674935	−	0.737878i	$-0.264172\pi$
$37$	1.10547	+	0.803169i	0.181738	+	0.132040i	−0.493197	+	0.869918i	$0.664172\pi$
$38$		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	$0.732197\pi$
$44$	11.2496	−	3.89261i	1.69595	−	0.586833i
$45$		0			0
$46$	7.00055	−	21.5455i	1.03217	−	3.17671i
$47$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$47$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$48$		0			0
$49$		0			0
$50$	3.65281	+	11.2422i	0.516586	+	1.58989i
$51$		0			0
$52$		0			0
$53$	−4.06518	+	12.5113i	−0.558396	+	1.71857i	0.128407	+	0.991722i	$0.459014\pi$
$53$	−4.06518	+	12.5113i	−0.558396	+	1.71857i	−0.686803	+	0.726844i	$0.740986\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	20.4328	−	14.8453i	2.68296	−	1.94929i
$59$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$59$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$60$		0			0
$61$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$61$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$62$		0			0
$63$		0			0
$64$	−3.59968	+	11.0787i	−0.449961	+	1.38484i
$65$		0			0
$66$		0			0
$67$	16.3336			1.99547			0.997735	−	0.0672706i	$-0.0214291\pi$
$67$	16.3336			1.99547			0.997735	+	0.0672706i	$0.0214291\pi$
$68$		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	$0.698333\pi$
$72$	3.48305	+	10.7197i	0.410481	+	1.26333i
$73$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$73$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$74$	−2.61349	+	1.89881i	−0.303812	+	0.220732i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−3.90147	+	12.0075i	−0.438950	+	1.35095i	0.450035	+	0.893011i	$0.351411\pi$
$79$	−3.90147	+	12.0075i	−0.438950	+	1.35095i	−0.888985	+	0.457937i	$0.848589\pi$
$80$		0			0
$81$	−7.28115	−	5.29007i	−0.809017	−	0.587785i
$82$		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
$86$	6.38565	−	19.6530i	0.688582	−	2.11924i
$87$		0			0
$88$	0.236325	+	12.4587i	0.0251923	+	1.32811i
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		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
$100$	−17.9460			−1.79460
$101$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$101$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$102$		0			0
$103$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$103$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$104$		0			0
$105$		0			0
$106$	−25.1611	−	18.2806i	−2.44387	−	1.77557i
$107$	−10.5739	+	7.68240i	−1.02222	+	0.742685i	−0.966736	−	0.255774i	$-0.917670\pi$
$107$	−10.5739	+	7.68240i	−1.02222	+	0.742685i	−0.0554821	+	0.998460i	$0.517670\pi$
$108$		0			0
$109$	−8.34177			−0.798997			−0.399498	−	0.916734i	$-0.630816\pi$
$109$	−8.34177			−0.798997			−0.399498	+	0.916734i	$0.630816\pi$
$110$		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
$116$	11.8489	+	36.4671i	1.10014	+	3.38588i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−3.79348	−	10.3252i	−0.344862	−	0.938653i
$122$		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	$0.951253\pi$
$128$	−16.6399	−	12.0896i	−1.47078	−	1.06858i
$129$		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
$134$	−11.9327	+	36.7252i	−1.03083	+	3.17257i
$135$		0			0
$136$		0			0
$137$	6.34450	+	19.5264i	0.542047	+	1.66825i	0.727909	+	0.685674i	$0.240493\pi$
$137$	6.34450	+	19.5264i	0.542047	+	1.66825i	−0.185861	+	0.982576i	$0.559507\pi$
$138$		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
$142$	−23.5866			−1.97935
$143$		0			0
$144$	−5.11199			−0.426000
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−1.51555	−	4.66437i	−0.124577	−	0.383409i
$149$	6.79837	+	20.9232i	0.556944	+	1.71410i	0.690752	+	0.723092i	$0.257280\pi$
$149$	6.79837	+	20.9232i	0.556944	+	1.71410i	−0.133808	+	0.991007i	$0.542720\pi$
$150$		0			0
$151$	−18.2245	+	13.2408i	−1.48309	+	1.07752i	−0.506540	+	0.862217i	$0.669076\pi$
$151$	−18.2245	+	13.2408i	−1.48309	+	1.07752i	−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		−0.587785	−	0.809017i	$-0.700000\pi$
$157$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$158$	−24.1479	−	17.5444i	−1.92110	−	1.39576i
$159$		0			0
$160$		0			0
$161$		0			0
$162$	17.2137	−	12.5065i	1.35244	−	0.982605i
$163$	2.11662	−	6.51430i	0.165787	−	0.510239i	−0.833307	−	0.552811i	$-0.813555\pi$
$163$	2.11662	−	6.51430i	0.165787	−	0.510239i	0.999093	+	0.0425718i	$0.0135551\pi$
$164$		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
$172$	25.3807	+	18.4402i	1.93526	+	1.40605i
$173$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$173$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$174$		0			0
$175$		0			0
$176$	−5.40707	−	1.64417i	−0.407573	−	0.123934i
$177$		0			0
$178$		0			0
$179$	−9.96326	+	7.23873i	−0.744689	+	0.541048i	−0.894176	−	0.447715i	$-0.852238\pi$
$179$	−9.96326	+	7.23873i	−0.744689	+	0.541048i	0.149487	+	0.988764i	$0.452238\pi$
$180$		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
$184$	−29.1265	+	21.1616i	−2.14723	+	1.56006i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	18.3570	+	13.3371i	1.32826	+	0.965040i	0.999789	+	0.0205267i	$0.00653431\pi$
$191$	18.3570	+	13.3371i	1.32826	+	0.965040i	0.328474	+	0.944513i	$0.393466\pi$
$192$		0			0
$193$	−4.50169	−	13.8548i	−0.324039	−	0.997288i	−0.971873	−	0.235507i	$-0.924325\pi$
$193$	−4.50169	−	13.8548i	−0.324039	−	0.997288i	0.647834	−	0.761781i	$-0.275675\pi$
$194$		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	$0.0769543\pi$
$198$	22.2298	−	7.69198i	1.57980	−	0.546645i
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	5.80508	−	17.8662i	0.410481	−	1.26333i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	8.88338	−	27.3402i	0.617437	−	1.90028i
$208$		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	$0.164461\pi$
$212$	38.1992	−	27.7533i	2.62353	−	1.90611i
$213$		0			0
$214$	−9.54851	−	29.3873i	−0.652723	−	2.00887i
$215$		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
$226$	14.2548	+	43.8717i	0.948213	+	2.91830i
$227$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$227$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$228$		0			0
$229$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$229$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$230$		0			0
$231$		0			0
$232$	−40.1377			−2.63517
$233$	6.79837	−	20.9232i	0.445376	−	1.37073i	−0.436694	−	0.899610i	$-0.643851\pi$
$233$	6.79837	−	20.9232i	0.445376	−	1.37073i	0.882071	−	0.471117i	$-0.156149\pi$
$234$		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			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$	25.9869	−	0.986227i	1.67050	−	0.0633970i
$243$		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
$254$	8.54279			0.536022
$255$		0			0
$256$	20.4911	−	14.8877i	1.28069	−	0.930479i
$257$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$257$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	25.9284	−	18.8381i	1.60492	−	1.16605i
$262$		0			0
$263$	4.85603			0.299435			0.149718	−	0.988729i	$-0.452164\pi$
$263$	4.85603			0.299435			0.149718	+	0.988729i	$0.452164\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−47.4284	−	34.4588i	−2.89715	−	2.10490i
$269$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$269$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$270$		0			0
$271$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$271$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$272$		0			0
$273$		0			0
$274$	−48.5389			−2.93234
$275$	0.314502	+	16.5801i	0.0189652	+	0.999820i
$276$		0			0
$277$	−3.26675	+	10.0540i	−0.196280	+	0.604088i	0.803679	+	0.595063i	$0.202873\pi$
$277$	−3.26675	+	10.0540i	−0.196280	+	0.604088i	−0.999959	+	0.00902525i	$0.997127\pi$
$278$		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		−0.587785	−	0.809017i	$-0.700000\pi$
$283$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$284$	11.0655	−	34.0562i	0.656618	−	2.02086i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−3.23146	+	9.94542i	−0.190416	+	0.586040i
$289$	13.7533	−	9.99235i	0.809017	−	0.587785i
$290$		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
$296$	5.13386			0.298400
$297$		0			0
$298$	−52.0113			−3.01293
$299$		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		−0.587785	−	0.809017i	$-0.700000\pi$
$311$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$312$		0			0
$313$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$313$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$314$		0			0
$315$		0			0
$316$	36.6608	−	26.6356i	2.06233	−	1.49837i
$317$	−0.136444	+	0.419931i	−0.00766345	+	0.0235857i	−0.954815	−	0.297200i	$-0.903947\pi$
$317$	−0.136444	+	0.419931i	−0.00766345	+	0.0235857i	0.947152	+	0.320786i	$0.103947\pi$
$318$		0			0
$319$	33.4839	−	11.5861i	1.87474	−	0.648698i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	9.98214	+	30.7219i	0.554563	+	1.70677i
$325$		0			0
$326$	13.1007	+	9.51821i	0.725580	+	0.527165i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−26.0143			−1.42988			−0.714939	−	0.699187i	$-0.753545\pi$
$331$	−26.0143			−1.42988			−0.714939	+	0.699187i	$0.753545\pi$
$332$		0			0
$333$	−3.31640	+	2.40951i	−0.181738	+	0.132040i
$334$		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	$0.204421\pi$
$338$	−24.8643	+	18.0650i	−1.35244	+	0.982605i
$339$		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
$347$	−10.5040	−	32.3279i	−0.563883	−	1.73545i	−0.671248	−	0.741233i	$-0.734242\pi$
$347$	−10.5040	−	32.3279i	−0.563883	−	1.73545i	0.107366	−	0.994220i	$-0.465758\pi$
$348$		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
$352$	−6.61673	+	9.48017i	−0.352673	+	0.505295i
$353$		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
$358$	−8.99708	−	27.6902i	−0.475510	−	1.46347i
$359$	−12.3777	−	8.99296i	−0.653272	−	0.474630i	0.211112	−	0.977462i	$-0.432292\pi$
$359$	−12.3777	−	8.99296i	−0.653272	−	0.474630i	−0.864384	+	0.502832i	$0.832292\pi$
$360$		0			0
$361$	−5.87132	+	18.0701i	−0.309017	+	0.951057i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$367$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$368$	−5.04575	−	15.5292i	−0.263028	−	0.809517i
$369$		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
$379$	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
$382$	−43.3986	+	31.5309i	−2.22047	+	1.61326i
$383$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$383$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$384$		0			0
$385$		0			0
$386$	34.4404			1.75297
$387$	8.10309	−	24.9388i	0.411903	−	1.26771i
$388$		0			0
$389$	−29.9038	−	21.7264i	−1.51618	−	1.10157i	−0.963338	−	0.268290i	$-0.913542\pi$
$389$	−29.9038	−	21.7264i	−1.51618	−	1.10157i	−0.552847	−	0.833283i	$-0.686458\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	−19.9115	+	61.2812i	−1.00313	+	3.08730i
$395$		0			0
$396$	0.677288	+	35.7057i	0.0340350	+	1.79428i
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$		0			0
$399$		0			0
$400$	6.89282	+	5.00793i	0.344641	+	0.250396i
$401$	4.65550	+	14.3282i	0.232484	+	0.715514i	0.997445	+	0.0714367i	$0.0227584\pi$
$401$	4.65550	+	14.3282i	0.232484	+	0.715514i	−0.764961	+	0.644077i	$0.777242\pi$
$402$		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		−0.951057	−	0.309017i	$-0.900000\pi$
$409$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$410$		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	$0.881584\pi$
$422$	−11.1758	−	8.11969i	−0.544029	−	0.395260i
$423$		0			0
$424$	15.2734	+	47.0067i	0.741742	+	2.28285i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	46.9112			2.26754
$429$		0			0
$430$		0			0
$431$	−12.8056	+	39.4117i	−0.616825	+	1.89839i	−0.248963	+	0.968513i	$0.580090\pi$
$431$	−12.8056	+	39.4117i	−0.616825	+	1.89839i	−0.367862	+	0.929880i	$0.619910\pi$
$432$		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
$436$	24.2223	+	17.5985i	1.16004	+	0.842815i
$437$		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
$443$	−23.4997	+	17.0736i	−1.11651	+	0.811190i	−0.983676	−	0.179949i	$-0.942407\pi$
$443$	−23.4997	+	17.0736i	−1.11651	+	0.811190i	−0.132831	+	0.991139i	$0.542407\pi$
$444$		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	$-0.0849724\pi$
$450$	−35.4623			−1.67171
$451$		0			0
$452$	−70.0328			−3.29406
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−6.18900	−	19.0478i	−0.289509	−	0.891018i	−0.985011	−	0.172493i	$-0.944818\pi$
$457$	−6.18900	−	19.0478i	−0.289509	−	0.891018i	0.695501	−	0.718525i	$-0.255182\pi$
$458$		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	$0.479743\pi$
$464$	5.62532	−	17.3130i	0.261149	−	0.803734i
$465$		0			0
$466$	42.0780	+	30.5715i	1.94923	+	1.41620i
$467$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$467$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	16.5919	−	23.7721i	0.762894	−	1.09304i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−31.9284	−	23.1973i	−1.46190	−	1.06213i
$478$	−21.9950	−	67.6935i	−1.00603	−	3.09623i
$479$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$479$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$480$		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
$487$	34.4997	−	25.0655i	1.56333	−	1.13583i	0.630123	−	0.776495i	$-0.283004\pi$
$487$	34.4997	−	25.0655i	1.56333	−	1.13583i	0.933210	−	0.359333i	$-0.116996\pi$
$488$		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
$499$	−29.3570	−	21.3291i	−1.31420	−	0.954821i	−0.999985	−	0.00546838i	$-0.998259\pi$
$499$	−29.3570	−	21.3291i	−1.31420	−	0.954821i	−0.314213	−	0.949352i	$-0.601741\pi$
$500$		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
$506$	45.3085	+	59.9375i	2.01421	+	2.66455i
$507$		0			0
$508$	−4.00779	+	12.3347i	−0.177817	+	0.547265i
$509$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$509$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$510$		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		0.587785	−	0.809017i	$-0.300000\pi$
$521$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$522$	23.4140	+	72.0608i	1.02480	+	3.15401i
$523$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$523$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$524$		0			0
$525$		0			0
$526$	−3.54763	+	10.9185i	−0.154684	+	0.476069i
$527$		0			0
$528$		0			0
$529$	68.8225			2.99228
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	49.6473	−	36.0709i	2.14444	−	1.55803i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	14.2668	−	43.9085i	0.613376	−	1.88778i	0.190138	−	0.981757i	$-0.439107\pi$
$541$	14.2668	−	43.9085i	0.613376	−	1.88778i	0.423238	−	0.906019i	$-0.360893\pi$
$542$		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	$0.910215\pi$
$548$	22.7717	−	70.0842i	0.972760	−	2.99385i
$549$		0			0
$550$	−37.5092	−	11.4057i	−1.59940	−	0.486340i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−20.2193	−	14.6902i	−0.859037	−	0.624127i
$555$		0			0
$556$		0			0
$557$	−19.6428	−	14.2713i	−0.832292	−	0.604695i	0.0879152	−	0.996128i	$-0.471980\pi$
$557$	−19.6428	−	14.2713i	−0.832292	−	0.604695i	−0.920207	+	0.391433i	$0.871980\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	66.5852			2.80873
$563$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$563$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	30.3252	+	22.0326i	1.27242	+	0.924467i
$569$	−17.7984	+	12.9313i	−0.746147	+	0.542107i	−0.894630	−	0.446808i	$-0.852561\pi$
$569$	−17.7984	+	12.9313i	−0.746147	+	0.542107i	0.148483	+	0.988915i	$0.452561\pi$
$570$		0			0
$571$	24.7563			1.03602			0.518010	−	0.855374i	$-0.326673\pi$
$571$	24.7563			1.03602			0.518010	+	0.855374i	$0.326673\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−38.7616	+	28.1620i	−1.61647	+	1.17444i
$576$	−28.2723	−	20.5410i	−1.17801	−	0.855876i
$577$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$577$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$578$	12.4196	+	38.2235i	0.516586	+	1.58989i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−26.3104	−	34.8054i	−1.08967	−	1.44149i
$584$		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
$592$	−0.719515	+	2.21444i	−0.0295719	+	0.0910129i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$	24.4008	−	75.0979i	0.999495	−	3.07613i
$597$		0			0
$598$		0			0
$599$	−1.27212	−	3.91519i	−0.0519775	−	0.159970i	0.921698	−	0.387907i	$-0.126802\pi$
$599$	−1.27212	−	3.91519i	−0.0519775	−	0.159970i	−0.973676	+	0.227937i	$0.926802\pi$
$600$		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
$604$	80.8529			3.28986
$605$		0			0
$606$		0			0
$607$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$607$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−14.9284	+	10.8461i	−0.602951	+	0.438069i	−0.846925	−	0.531712i	$-0.821549\pi$
$613$	−14.9284	+	10.8461i	−0.602951	+	0.438069i	0.243974	+	0.969782i	$0.421549\pi$
$614$		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		−0.587785	−	0.809017i	$-0.700000\pi$
$619$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	7.72542	−	23.7764i	0.309017	−	0.951057i
$626$		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	$0.796829\pi$
$632$	14.6583	+	45.1137i	0.583076	+	1.79452i
$633$		0			0
$634$	−0.844509	−	0.613572i	−0.0335398	−	0.0243681i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	1.58864	+	83.7509i	0.0628947	+	3.31573i
$639$	−29.9304			−1.18403
$640$		0			0
$641$	4.78557	−	3.47692i	0.189019	−	0.137330i	−0.489251	−	0.872143i	$-0.662730\pi$
$641$	4.78557	−	3.47692i	0.189019	−	0.137330i	0.678270	+	0.734813i	$0.262730\pi$
$642$		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		−0.951057	−	0.309017i	$-0.900000\pi$
$647$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$648$	−33.8141			−1.32835
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−19.8892	+	14.4504i	−0.778922	+	0.565920i
$653$	27.6929	+	20.1201i	1.08371	+	0.787359i	0.978326	−	0.207072i	$-0.0663936\pi$
$653$	27.6929	+	20.1201i	1.08371	+	0.787359i	0.105382	+	0.994432i	$0.466394\pi$
$654$		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	$0.5\pi$
$662$	19.0051	−	58.4917i	0.738655	−	2.27335i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−2.99479	−	9.21703i	−0.116046	−	0.357153i
$667$	82.8187	+	60.1713i	3.20675	+	2.32984i
$668$		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	$-0.303752\pi$
$674$	−70.2160	+	51.0149i	−2.70462	+	1.96502i
$675$		0			0
$676$	−14.4186	−	44.3760i	−0.554563	−	1.70677i
$677$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$677$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	45.1792			1.72873			0.864366	−	0.502863i	$-0.167720\pi$
$683$	45.1792			1.72873			0.864366	+	0.502863i	$0.167720\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−4.60255	−	14.1652i	−0.175471	−	0.540043i
$689$		0			0
$690$		0			0
$691$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$691$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$692$		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
$704$	−23.2976	−	30.8199i	−0.878063	−	1.16157i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−16.1243	−	49.6254i	−0.605560	−	1.86372i	−0.492893	−	0.870090i	$-0.664061\pi$
$709$	−16.1243	−	49.6254i	−0.605560	−	1.86372i	−0.112667	−	0.993633i	$-0.535939\pi$
$710$		0			0
$711$	−30.6425	−	22.2631i	−1.14918	−	0.834932i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	44.2021			1.65191
$717$		0			0
$718$	29.2629	−	21.2607i	1.09208	−	0.793443i
$719$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$719$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$720$		0			0
$721$		0			0
$722$	−36.3401	−	26.4026i	−1.35244	−	0.982605i
$723$		0			0
$724$		0			0
$725$	−53.4154			−1.98380
$726$		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		0.587785	−	0.809017i	$-0.300000\pi$
$733$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$734$		0			0
$735$		0			0
$736$	−33.4018			−1.23121
$737$	−31.0049	+	44.4225i	−1.14208	+	1.63632i
$738$		0			0
$739$	−0.300157	+	0.923789i	−0.0110415	+	0.0339821i	−0.956425	−	0.291977i	$-0.905687\pi$
$739$	−0.300157	+	0.923789i	−0.0110415	+	0.0339821i	0.945384	+	0.325959i	$0.105687\pi$
$740$		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
$746$	−16.0724	+	49.4657i	−0.588452	+	1.81107i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−18.8351	+	13.6845i	−0.687303	+	0.499355i	−0.875772	−	0.482724i	$-0.839647\pi$
$751$	−18.8351	+	13.6845i	−0.687303	+	0.499355i	0.188469	+	0.982079i	$0.439647\pi$
$752$		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	$0.938398\pi$
$758$	−74.4236			−2.70319
$759$		0			0
$760$		0			0
$761$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$761$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$762$		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
$772$	−16.1575	+	49.7276i	−0.581521	+	1.78974i
$773$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$773$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$774$	50.1535	+	36.4387i	1.80273	+	1.30976i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	70.6972	−	51.3645i	2.53462	−	1.84151i
$779$		0			0
$780$		0			0
$781$	−31.6580	−	9.62648i	−1.13281	−	0.344462i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$787$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$788$	−79.1411	−	57.4994i	−2.81928	−	2.04833i
$789$		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
$800$	14.1001	−	10.2443i	0.498515	−	0.362192i
$801$		0			0
$802$	−35.6171			−1.25768
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	17.1890	+	52.9023i	0.604333	+	1.85995i	0.501311	+	0.865267i	$0.332851\pi$
$809$	17.1890	+	52.9023i	0.604333	+	1.85995i	0.103022	+	0.994679i	$0.467149\pi$
$810$		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
$814$	−0.203197	−	10.7123i	−0.00712204	−	0.375465i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−17.7984	−	12.9313i	−0.621168	−	0.451305i	0.232162	−	0.972677i	$-0.425420\pi$
$821$	−17.7984	−	12.9313i	−0.621168	−	0.451305i	−0.853329	+	0.521373i	$0.825420\pi$
$822$		0			0
$823$	−10.9405	+	33.6713i	−0.381361	+	1.17371i	0.557725	+	0.830026i	$0.311674\pi$
$823$	−10.9405	+	33.6713i	−0.381361	+	1.17371i	−0.939086	+	0.343683i	$0.888326\pi$
$824$		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	$-0.177268\pi$
$828$	−83.4741	+	60.6475i	−2.90093	+	2.10765i
$829$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$829$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$830$		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
$842$	−29.0005	−	89.2542i	−0.999422	−	3.07590i
$843$		0			0
$844$	16.9669	−	12.3272i	0.584024	−	0.424318i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	−22.4164			−0.769784
$849$		0			0
$850$		0			0
$851$	−10.5930	−	7.69629i	−0.363124	−	0.263825i
$852$		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
$856$	−15.1746	+	46.7025i	−0.518656	+	1.59626i
$857$		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
$862$	−79.2595	−	57.5854i	−2.69959	−	1.96137i
$863$	2.47214	+	7.60845i	0.0841525	+	0.258995i	0.984275	−	0.176642i	$-0.0565234\pi$
$863$	2.47214	+	7.60845i	0.0841525	+	0.258995i	−0.900123	+	0.435636i	$0.856523\pi$
$864$		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
$872$	−25.3555	+	18.4218i	−0.858645	+	0.623842i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−36.9284	+	26.8300i	−1.24698	+	0.905985i	−0.998043	−	0.0625337i	$-0.980082\pi$
$877$	−36.9284	+	26.8300i	−1.24698	+	0.905985i	−0.248939	+	0.968519i	$0.580082\pi$
$878$		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
$886$	−21.2209	−	65.3111i	−0.712929	−	2.19417i
$887$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$887$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	28.2086	−	9.76078i	0.945025	−	0.326998i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	44.4958	+	32.3281i	1.48484	+	1.07880i
$899$		0			0
$900$	16.6369	−	51.2031i	0.554563	−	1.70677i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	22.6538	−	69.7212i	0.753454	−	2.31889i
$905$		0			0
$906$		0			0
$907$	7.28462	+	22.4198i	0.241882	+	0.744436i	0.996134	+	0.0878507i	$0.0279999\pi$
$907$	7.28462	+	22.4198i	0.241882	+	0.744436i	−0.754252	+	0.656585i	$0.772000\pi$
$908$		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
$914$	47.3493			1.56618
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	6.30234	+	19.3966i	0.207895	+	0.639835i	0.999582	+	0.0289084i	$0.00920311\pi$
$919$	6.30234	+	19.3966i	0.207895	+	0.639835i	−0.791687	+	0.610927i	$0.790797\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	6.83216			0.224640
$926$	−1.99946	+	6.15370i	−0.0657063	+	0.202223i
$927$		0			0
$928$	−30.1266	−	21.8882i	−0.988953	−	0.718516i
$929$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$929$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$930$		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		0.951057	−	0.309017i	$-0.100000\pi$
$941$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	41.3288	+	54.6728i	1.34371	+	1.77757i
$947$	−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
$953$	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	$0.511381\pi$
$954$	75.4834	−	54.8419i	2.44387	−	1.77557i
$955$		0			0
$956$	108.060			3.49490
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	25.0795	+	18.2213i	0.809017	+	0.587785i
$962$		0			0
$963$	−12.1166	−	37.2911i	−0.390453	−	1.20169i
$964$		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	$0.0776244\pi$
$968$	−34.3326	−	23.0068i	−1.10349	−	0.739465i
$969$		0			0
$970$		0			0
$971$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$971$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$972$		0			0
$973$		0			0
$974$	31.1541	+	95.8826i	0.998243	+	3.07228i
$975$		0			0
$976$		0			0
$977$	−3.81100	+	11.7290i	−0.121925	+	0.375245i	−0.993328	−	0.115321i	$-0.963210\pi$
$977$	−3.81100	+	11.7290i	−0.121925	+	0.375245i	0.871404	+	0.490567i	$0.163210\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	7.73325	−	23.8005i	0.246904	−	0.759891i
$982$	84.1561	−	61.1430i	2.68553	−	1.95115i
$983$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$983$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	83.7571			2.66332
$990$		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		0.587785	−	0.809017i	$-0.300000\pi$
$997$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$998$	69.4043	−	50.4252i	2.19695	−	1.59618i
$999$		0			0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.f.c.148.1	✓	8	11.5	even	5	inner
539.2.f.c.148.1	✓	8	77.27	odd	10	inner
539.2.f.c.295.1	yes	8	1.1	even	1	trivial
539.2.f.c.295.1	yes	8	7.6	odd	2	CM
539.2.q.d.214.1		16	77.5	odd	30
539.2.q.d.214.1		16	77.16	even	15
539.2.q.d.324.2		16	77.38	odd	30
539.2.q.d.324.2		16	77.60	even	15
539.2.q.d.361.2		16	7.2	even	3
539.2.q.d.361.2		16	7.5	odd	6
539.2.q.d.471.1		16	7.3	odd	6
539.2.q.d.471.1		16	7.4	even	3
5929.2.a.bc.1.1		4	11.4	even	5
5929.2.a.bc.1.1		4	77.48	odd	10
5929.2.a.bg.1.4		4	11.7	odd	10
5929.2.a.bg.1.4		4	77.62	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)

\(f(q)\)	\(=\)	\(q+(-0.730563 + 2.24844i) q^{2} +(-2.90373 - 2.10968i) q^{4} +(3.03958 - 2.20838i) q^{8} +(-0.927051 + 2.85317i) q^{9} +(-1.89823 + 2.71970i) q^{11} +(0.526564 + 1.62060i) q^{16} +(-5.73791 - 4.16884i) q^{18} +(-4.72830 - 6.25495i) q^{22} -9.58240 q^{23} +(4.04508 - 2.93893i) q^{25} +(-8.64279 - 6.27935i) q^{29} +3.48575 q^{32} +(8.71119 - 6.32905i) q^{36} +(1.10547 + 0.803169i) q^{37} -8.74072 q^{43} +(11.2496 - 3.89261i) q^{44} +(7.00055 - 21.5455i) q^{46} +(3.65281 + 11.2422i) q^{50} +(-4.06518 + 12.5113i) q^{53} +(20.4328 - 14.8453i) q^{58} +(-3.59968 + 11.0787i) q^{64} +16.3336 q^{67} +(3.08300 + 9.48849i) q^{71} +(3.48305 + 10.7197i) q^{72} +(-2.61349 + 1.89881i) q^{74} +(-3.90147 + 12.0075i) q^{79} +(-7.28115 - 5.29007i) q^{81} +(6.38565 - 19.6530i) q^{86} +(0.236325 + 12.4587i) q^{88} +(27.8247 + 20.2158i) q^{92} +(-6.00000 - 7.93725i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 2 q^{4} + 16 q^{8} + 6 q^{9} - 4 q^{11} - 28 q^{16} - 9 q^{18} - 4 q^{22} - 16 q^{23} + 10 q^{25} - 4 q^{29} + 100 q^{32} - 6 q^{36} - 18 q^{37} + 24 q^{43} + 9 q^{44} - 31 q^{46} + 10 q^{50}+ \cdots - 48 q^{99}+O(q^{100}) \) 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

L-functions

Modular forms

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Database

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