LMFDB - Embedded newform 1520.2.q.m.961.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(881,1520)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1520.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1520.q (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$12.1372611072$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + 9x^{6} + 2x^{5} + 65x^{4} - 20x^{3} + 25x^{2} + 6x + 4 $ x^8 - x^7 + 9x^6 + 2x^5 + 65x^4 - 20x^3 + 25x^2 + 6x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			961.3
Root			$0.354609 - 0.614201i$ of defining polynomial
Character	$\chi$	$=$	1520.961
Dual form			1520.2.q.m.881.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.354609 - 0.614201i) q^{3} +(-0.500000 + 0.866025i) q^{5} -3.11079 q^{7} +(1.24850 + 2.16247i) q^{9} +O(q^{10})$	\(q+(0.354609 - 0.614201i) q^{3} +(-0.500000 + 0.866025i) q^{5} -3.11079 q^{7} +(1.24850 + 2.16247i) q^{9} +3.52922 q^{11} +(-0.200785 - 0.347770i) q^{13} +(0.354609 + 0.614201i) q^{15} +(-1.74850 + 3.02850i) q^{17} +(-4.35162 - 0.251824i) q^{19} +(-1.10311 + 1.91065i) q^{21} +(-3.65851 - 6.33672i) q^{23} +(-0.500000 - 0.866025i) q^{25} +3.89858 q^{27} +(3.96540 + 6.86827i) q^{29} -5.73545 q^{31} +(1.25150 - 2.16765i) q^{33} +(1.55539 - 2.69402i) q^{35} -10.5292 q^{37} -0.284801 q^{39} +(0.555394 - 0.961971i) q^{41} +(-4.30390 + 7.45457i) q^{43} -2.49701 q^{45} +(3.76162 + 6.51532i) q^{47} +2.67700 q^{49} +(1.24007 + 2.14787i) q^{51} +(-5.27773 - 9.14130i) q^{53} +(-1.76461 + 3.05640i) q^{55} +(-1.69779 + 2.58347i) q^{57} +(-4.25618 + 7.37192i) q^{59} +(4.61079 + 7.98612i) q^{61} +(-3.88383 - 6.72700i) q^{63} +0.401570 q^{65} +(4.20623 + 7.28540i) q^{67} -5.18936 q^{69} +(-4.31233 + 7.46918i) q^{71} +(-0.870717 + 1.50813i) q^{73} -0.709218 q^{75} -10.9787 q^{77} +(-4.50544 + 7.80366i) q^{79} +(-2.36304 + 4.09291i) q^{81} -4.07857 q^{83} +(-1.74850 - 3.02850i) q^{85} +5.62466 q^{87} +(6.61623 + 11.4596i) q^{89} +(0.624599 + 1.08184i) q^{91} +(-2.03384 + 3.52272i) q^{93} +(2.39390 - 3.64270i) q^{95} +(3.85162 - 6.67120i) q^{97} +(4.40625 + 7.63186i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{3} - 4 q^{5} - 5 q^{9}+O(q^{10}) $ 8 * q + q^3 - 4 * q^5 - 5 * q^9	$ 8 q + q^{3} - 4 q^{5} - 5 q^{9} - 4 q^{11} + 9 q^{13} + q^{15} + q^{17} - 3 q^{19} + 8 q^{21} - 4 q^{25} - 20 q^{27} + 5 q^{29} + 20 q^{31} + 25 q^{33} - 52 q^{37} + 54 q^{39} - 8 q^{41} - 7 q^{43} + 10 q^{45} - 16 q^{47} + 20 q^{49} - 12 q^{51} + 5 q^{53} + 2 q^{55} + 27 q^{57} - 11 q^{59} + 12 q^{61} + 3 q^{63} - 18 q^{65} + 6 q^{69} - 14 q^{71} - 4 q^{73} - 2 q^{75} - 44 q^{77} - 13 q^{79} - 24 q^{81} - 10 q^{83} + q^{85} + 4 q^{87} + 5 q^{89} + 46 q^{91} - 28 q^{93} + 6 q^{95} - q^{97} - 24 q^{99}+O(q^{100}) $ 8 * q + q^3 - 4 * q^5 - 5 * q^9 - 4 * q^11 + 9 * q^13 + q^15 + q^17 - 3 * q^19 + 8 * q^21 - 4 * q^25 - 20 * q^27 + 5 * q^29 + 20 * q^31 + 25 * q^33 - 52 * q^37 + 54 * q^39 - 8 * q^41 - 7 * q^43 + 10 * q^45 - 16 * q^47 + 20 * q^49 - 12 * q^51 + 5 * q^53 + 2 * q^55 + 27 * q^57 - 11 * q^59 + 12 * q^61 + 3 * q^63 - 18 * q^65 + 6 * q^69 - 14 * q^71 - 4 * q^73 - 2 * q^75 - 44 * q^77 - 13 * q^79 - 24 * q^81 - 10 * q^83 + q^85 + 4 * q^87 + 5 * q^89 + 46 * q^91 - 28 * q^93 + 6 * q^95 - q^97 - 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times$.

$n$	$191$	$401$	$1141$	$1217$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$	$1$

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			0
$2$		0			0
$3$	0.354609	−	0.614201i	0.204734	−	0.354609i	−0.745314	−	0.666713i	$-0.767701\pi$
$3$	0.354609	−	0.614201i	0.204734	−	0.354609i	0.950048	+	0.312104i	$0.101034\pi$
$4$		0			0
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$6$		0			0
$7$	−3.11079			−1.17577			−0.587884	−	0.808945i	$-0.700039\pi$
$7$	−3.11079			−1.17577			−0.587884	+	0.808945i	$0.700039\pi$
$8$		0			0
$9$	1.24850	+	2.16247i	0.416168	+	0.720825i
$10$		0			0
$11$	3.52922			1.06410			0.532051	−	0.846713i	$-0.321422\pi$
$11$	3.52922			1.06410			0.532051	+	0.846713i	$0.321422\pi$
$12$		0			0
$13$	−0.200785	−	0.347770i	−0.0556877	−	0.0964540i	0.836838	−	0.547451i	$-0.184402\pi$
$13$	−0.200785	−	0.347770i	−0.0556877	−	0.0964540i	−0.892525	+	0.450997i	$0.851068\pi$
$14$		0			0
$15$	0.354609	+	0.614201i	0.0915597	+	0.158586i
$16$		0			0
$17$	−1.74850	+	3.02850i	−0.424075	+	0.734519i	−0.996334	−	0.0855542i	$-0.972734\pi$
$17$	−1.74850	+	3.02850i	−0.424075	+	0.734519i	0.572259	+	0.820073i	$0.306067\pi$
$18$		0			0
$19$	−4.35162	−	0.251824i	−0.998330	−	0.0577725i
$19$	−4.35162	−	0.251824i	−0.998330	−	0.0577725i
$20$		0			0
$21$	−1.10311	+	1.91065i	−0.240719	+	0.416938i
$22$		0			0
$23$	−3.65851	−	6.33672i	−0.762852	−	1.32130i	−0.941375	−	0.337362i	$-0.890465\pi$
$23$	−3.65851	−	6.33672i	−0.762852	−	1.32130i	0.178523	−	0.983936i	$-0.442868\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$	3.89858			0.750282
$28$		0			0
$29$	3.96540	+	6.86827i	0.736356	+	1.27541i	0.954126	+	0.299406i	$0.0967884\pi$
$29$	3.96540	+	6.86827i	0.736356	+	1.27541i	−0.217770	+	0.976000i	$0.569878\pi$
$30$		0			0
$31$	−5.73545			−1.03012			−0.515059	−	0.857155i	$-0.672230\pi$
$31$	−5.73545			−1.03012			−0.515059	+	0.857155i	$0.672230\pi$
$32$		0			0
$33$	1.25150	−	2.16765i	0.217857	−	0.377340i
$34$		0			0
$35$	1.55539	−	2.69402i	0.262910	−	0.455373i
$36$		0			0
$37$	−10.5292			−1.73099			−0.865497	−	0.500914i	$-0.832997\pi$
$37$	−10.5292			−1.73099			−0.865497	+	0.500914i	$0.832997\pi$
$38$		0			0
$39$	−0.284801			−0.0456046
$40$		0			0
$41$	0.555394	−	0.961971i	0.0867380	−	0.150235i	−0.819392	−	0.573233i	$-0.805689\pi$
$41$	0.555394	−	0.961971i	0.0867380	−	0.150235i	0.906130	+	0.422998i	$0.139022\pi$
$42$		0			0
$43$	−4.30390	+	7.45457i	−0.656338	+	1.13681i	0.325218	+	0.945639i	$0.394562\pi$
$43$	−4.30390	+	7.45457i	−0.656338	+	1.13681i	−0.981556	+	0.191172i	$0.938771\pi$
$44$		0			0
$45$	−2.49701			−0.372232
$46$		0			0
$47$	3.76162	+	6.51532i	0.548689	+	0.950357i	0.998365	+	0.0571653i	$0.0182062\pi$
$47$	3.76162	+	6.51532i	0.548689	+	0.950357i	−0.449676	+	0.893192i	$0.648460\pi$
$48$		0			0
$49$	2.67700			0.382429
$50$		0			0
$51$	1.24007	+	2.14787i	0.173645	+	0.300762i
$52$		0			0
$53$	−5.27773	−	9.14130i	−0.724952	−	1.25565i	−0.958994	−	0.283427i	$-0.908529\pi$
$53$	−5.27773	−	9.14130i	−0.724952	−	1.25565i	0.234042	−	0.972227i	$-0.424805\pi$
$54$		0			0
$55$	−1.76461	+	3.05640i	−0.237940	+	0.412125i
$56$		0			0
$57$	−1.69779	+	2.58347i	−0.224878	+	0.342189i
$58$		0			0
$59$	−4.25618	+	7.37192i	−0.554107	+	0.959742i	0.443865	+	0.896094i	$0.353607\pi$
$59$	−4.25618	+	7.37192i	−0.554107	+	0.959742i	−0.997972	+	0.0636484i	$0.979726\pi$
$60$		0			0
$61$	4.61079	+	7.98612i	0.590351	+	1.02252i	0.994185	+	0.107686i	$0.0343440\pi$
$61$	4.61079	+	7.98612i	0.590351	+	1.02252i	−0.403834	+	0.914832i	$0.632323\pi$
$62$		0			0
$63$	−3.88383	−	6.72700i	−0.489317	−	0.847522i
$64$		0			0
$65$	0.401570			0.0498086
$66$		0			0
$67$	4.20623	+	7.28540i	0.513873	+	0.890053i	0.999870	+	0.0160934i	$0.00512290\pi$
$67$	4.20623	+	7.28540i	0.513873	+	0.890053i	−0.485998	+	0.873960i	$0.661544\pi$
$68$		0			0
$69$	−5.18936			−0.624726
$70$		0			0
$71$	−4.31233	+	7.46918i	−0.511780	+	0.886428i	0.488127	+	0.872773i	$0.337680\pi$
$71$	−4.31233	+	7.46918i	−0.511780	+	0.886428i	−0.999907	+	0.0136558i	$0.995653\pi$
$72$		0			0
$73$	−0.870717	+	1.50813i	−0.101910	+	0.176513i	−0.912471	−	0.409140i	$-0.865829\pi$
$73$	−0.870717	+	1.50813i	−0.101910	+	0.176513i	0.810562	+	0.585653i	$0.199162\pi$
$74$		0			0
$75$	−0.709218			−0.0818935
$76$		0			0
$77$	−10.9787			−1.25114
$78$		0			0
$79$	−4.50544	+	7.80366i	−0.506902	+	0.877980i	0.493066	+	0.869992i	$0.335876\pi$
$79$	−4.50544	+	7.80366i	−0.506902	+	0.877980i	−0.999968	+	0.00798806i	$0.997457\pi$
$80$		0			0
$81$	−2.36304	+	4.09291i	−0.262560	+	0.454768i
$82$		0			0
$83$	−4.07857			−0.447682			−0.223841	−	0.974626i	$-0.571860\pi$
$83$	−4.07857			−0.447682			−0.223841	+	0.974626i	$0.571860\pi$
$84$		0			0
$85$	−1.74850	−	3.02850i	−0.189652	−	0.328487i
$86$		0			0
$87$	5.62466			0.603027
$88$		0			0
$89$	6.61623	+	11.4596i	0.701319	+	1.21472i	0.968004	+	0.250936i	$0.0807385\pi$
$89$	6.61623	+	11.4596i	0.701319	+	1.21472i	−0.266685	+	0.963784i	$0.585928\pi$
$90$		0			0
$91$	0.624599	+	1.08184i	0.0654758	+	0.113407i
$92$		0			0
$93$	−2.03384	+	3.52272i	−0.210900	+	0.365289i
$94$		0			0
$95$	2.39390	−	3.64270i	0.245609	−	0.373733i
$96$		0			0
$97$	3.85162	−	6.67120i	0.391073	−	0.677358i	−0.601519	−	0.798859i	$-0.705437\pi$
$97$	3.85162	−	6.67120i	0.391073	−	0.677358i	0.992591	+	0.121501i	$0.0387708\pi$
$98$		0			0
$99$	4.40625	+	7.63186i	0.442845	+	0.767030i
$100$		0			0
$101$	−8.68773	−	15.0476i	−0.864462	−	1.49729i	−0.867581	−	0.497296i	$-0.834326\pi$
$101$	−8.68773	−	15.0476i	−0.864462	−	1.49729i	0.00311897	−	0.999995i	$-0.499007\pi$
$102$		0			0
$103$	7.12614			0.702159			0.351080	−	0.936346i	$-0.385815\pi$
$103$	7.12614			0.702159			0.351080	+	0.936346i	$0.385815\pi$
$104$		0			0
$105$	−1.10311	−	1.91065i	−0.107653	−	0.186460i
$106$		0			0
$107$	−0.735452			−0.0710989			−0.0355494	−	0.999368i	$-0.511318\pi$
$107$	−0.735452			−0.0710989			−0.0355494	+	0.999368i	$0.511318\pi$
$108$		0			0
$109$	8.00468	−	13.8645i	0.766710	−	1.32798i	−0.172628	−	0.984987i	$-0.555226\pi$
$109$	8.00468	−	13.8645i	0.766710	−	1.32798i	0.939338	−	0.342993i	$-0.111441\pi$
$110$		0			0
$111$	−3.73376	+	6.46706i	−0.354393	+	0.613826i
$112$		0			0
$113$	4.34923			0.409141			0.204571	−	0.978852i	$-0.434420\pi$
$113$	4.34923			0.409141			0.204571	+	0.978852i	$0.434420\pi$
$114$		0			0
$115$	7.31702			0.682315
$116$		0			0
$117$	0.501362	−	0.868384i	0.0463509	−	0.0802822i
$118$		0			0
$119$	5.43923	−	9.42102i	0.498613	−	0.863623i
$120$		0			0
$121$	1.45543			0.132312
$122$		0			0
$123$	−0.393896	−	0.682247i	−0.0355164	−	0.0615162i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	−2.32300	−	4.02355i	−0.206133	−	0.357032i	0.744360	−	0.667778i	$-0.232755\pi$
$127$	−2.32300	−	4.02355i	−0.206133	−	0.357032i	−0.950493	+	0.310746i	$0.899421\pi$
$128$		0			0
$129$	3.05240	+	5.28692i	0.268749	+	0.465487i
$130$		0			0
$131$	5.24775	−	9.08936i	0.458498	−	0.794141i	−0.540384	−	0.841418i	$-0.681721\pi$
$131$	5.24775	−	9.08936i	0.458498	−	0.794141i	0.998882	+	0.0472772i	$0.0150544\pi$
$132$		0			0
$133$	13.5370	+	0.783372i	1.17380	+	0.0679270i
$134$		0			0
$135$	−1.94929	+	3.37627i	−0.167768	+	0.290583i
$136$		0			0
$137$	0.696101	+	1.20568i	0.0594719	+	0.103008i	0.894228	−	0.447611i	$-0.147725\pi$
$137$	0.696101	+	1.20568i	0.0594719	+	0.103008i	−0.834757	+	0.550619i	$0.814392\pi$
$138$		0			0
$139$	−3.03766	−	5.26138i	−0.257651	−	0.446264i	0.707961	−	0.706251i	$-0.249615\pi$
$139$	−3.03766	−	5.26138i	−0.257651	−	0.446264i	−0.965612	+	0.259987i	$0.916282\pi$
$140$		0			0
$141$	5.33562			0.449340
$142$		0			0
$143$	−0.708615	−	1.22736i	−0.0592574	−	0.102637i
$144$		0			0
$145$	−7.93079			−0.658617
$146$		0			0
$147$	0.949290	−	1.64422i	0.0782961	−	0.135613i
$148$		0			0
$149$	0.0422770	−	0.0732259i	0.00346347	−	0.00599890i	−0.864288	−	0.502996i	$-0.832231\pi$
$149$	0.0422770	−	0.0732259i	0.00346347	−	0.00599890i	0.867752	+	0.496998i	$0.165564\pi$
$150$		0			0
$151$	15.2216			1.23871			0.619357	−	0.785109i	$-0.287393\pi$
$151$	15.2216			1.23871			0.619357	+	0.785109i	$0.287393\pi$
$152$		0			0
$153$	−8.73207			−0.705946
$154$		0			0
$155$	2.86773	−	4.96705i	0.230341	−	0.398963i
$156$		0			0
$157$	−8.12935	+	14.0804i	−0.648793	+	1.12374i	0.334619	+	0.942354i	$0.391392\pi$
$157$	−8.12935	+	14.0804i	−0.648793	+	1.12374i	−0.983412	+	0.181388i	$0.941941\pi$
$158$		0			0
$159$	−7.48612			−0.593688
$160$		0			0
$161$	11.3808	+	19.7122i	0.896936	+	1.55354i
$162$		0			0
$163$	−14.9461			−1.17067			−0.585336	−	0.810791i	$-0.699037\pi$
$163$	−14.9461			−1.17067			−0.585336	+	0.810791i	$0.699037\pi$
$164$		0			0
$165$	1.25150	+	2.16765i	0.0974288	+	0.168752i
$166$		0			0
$167$	6.26461	+	10.8506i	0.484770	+	0.839647i	0.999847	−	0.0174974i	$-0.00556988\pi$
$167$	6.26461	+	10.8506i	0.484770	+	0.839647i	−0.515077	+	0.857144i	$0.672237\pi$
$168$		0			0
$169$	6.41937	−	11.1187i	0.493798	−	0.855283i
$170$		0			0
$171$	−4.88845	−	9.72466i	−0.373829	−	0.743664i
$172$		0			0
$173$	−8.69242	+	15.0557i	−0.660872	+	1.14466i	0.319515	+	0.947581i	$0.396480\pi$
$173$	−8.69242	+	15.0557i	−0.660872	+	1.14466i	−0.980387	+	0.197083i	$0.936853\pi$
$174$		0			0
$175$	1.55539	+	2.69402i	0.117577	+	0.203649i
$176$		0			0
$177$	3.01856	+	5.22830i	0.226889	+	0.392983i
$178$		0			0
$179$	−20.7830			−1.55340			−0.776698	−	0.629873i	$-0.783107\pi$
$179$	−20.7830			−1.55340			−0.776698	+	0.629873i	$0.783107\pi$
$180$		0			0
$181$	−0.814564	−	1.41087i	−0.0605460	−	0.104869i	0.834164	−	0.551517i	$-0.185951\pi$
$181$	−0.814564	−	1.41087i	−0.0605460	−	0.104869i	−0.894710	+	0.446648i	$0.852618\pi$
$182$		0			0
$183$	6.54011			0.483459
$184$		0			0
$185$	5.26461	−	9.11858i	0.387062	−	0.670411i
$186$		0			0
$187$	−6.17087	+	10.6883i	−0.451258	+	0.781603i
$188$		0			0
$189$	−12.1277			−0.882157
$190$		0			0
$191$	13.4016			0.969704			0.484852	−	0.874596i	$-0.338874\pi$
$191$	13.4016			0.969704			0.484852	+	0.874596i	$0.338874\pi$
$192$		0			0
$193$	5.34693	−	9.26116i	0.384881	−	0.666633i	−0.606872	−	0.794800i	$-0.707576\pi$
$193$	5.34693	−	9.26116i	0.384881	−	0.666633i	0.991753	+	0.128167i	$0.0409092\pi$
$194$		0			0
$195$	0.142400	−	0.246645i	0.0101975	−	0.0176626i
$196$		0			0
$197$	−7.00611			−0.499165			−0.249582	−	0.968354i	$-0.580293\pi$
$197$	−7.00611			−0.499165			−0.249582	+	0.968354i	$0.580293\pi$
$198$		0			0
$199$	−10.3909	−	17.9976i	−0.736592	−	1.27581i	−0.954021	−	0.299738i	$-0.903101\pi$
$199$	−10.3909	−	17.9976i	−0.736592	−	1.27581i	0.217430	−	0.976076i	$-0.430233\pi$
$200$		0			0
$201$	5.96627			0.420828
$202$		0			0
$203$	−12.3355	−	21.3657i	−0.865783	−	1.49958i
$204$		0			0
$205$	0.555394	+	0.961971i	0.0387904	+	0.0671870i
$206$		0			0
$207$	9.13533	−	15.8229i	0.634949	−	1.09976i
$208$		0			0
$209$	−15.3578	−	0.888745i	−1.06232	−	0.0614758i
$210$		0			0
$211$	−5.61623	+	9.72760i	−0.386637	+	0.669675i	−0.991995	−	0.126278i	$-0.959697\pi$
$211$	−5.61623	+	9.72760i	−0.386637	+	0.669675i	0.605358	+	0.795954i	$0.293030\pi$
$212$		0			0
$213$	3.05838	+	5.29728i	0.209557	+	0.362963i
$214$		0			0
$215$	−4.30390	−	7.45457i	−0.293523	−	0.508398i
$216$		0			0
$217$	17.8418			1.21118
$218$		0			0
$219$	0.617528	+	1.06959i	0.0417287	+	0.0722762i
$220$		0			0
$221$	1.40429			0.0944630
$222$		0			0
$223$	−5.45005	+	9.43976i	−0.364962	+	0.632133i	−0.988770	−	0.149445i	$-0.952251\pi$
$223$	−5.45005	+	9.43976i	−0.364962	+	0.632133i	0.623808	+	0.781578i	$0.285585\pi$
$224$		0			0
$225$	1.24850	−	2.16247i	0.0832336	−	0.144165i
$226$		0			0
$227$	−3.96627			−0.263250			−0.131625	−	0.991300i	$-0.542020\pi$
$227$	−3.96627			−0.263250			−0.131625	+	0.991300i	$0.542020\pi$
$228$		0			0
$229$	−10.9139			−0.721213			−0.360606	−	0.932718i	$-0.617430\pi$
$229$	−10.9139			−0.721213			−0.360606	+	0.932718i	$0.617430\pi$
$230$		0			0
$231$	−3.89314	+	6.74311i	−0.256150	+	0.443664i
$232$		0			0
$233$	7.79547	−	13.5021i	0.510698	−	0.884555i	−0.489225	−	0.872157i	$-0.662720\pi$
$233$	7.79547	−	13.5021i	0.510698	−	0.884555i	0.999923	−	0.0123973i	$-0.00394628\pi$
$234$		0			0
$235$	−7.52324			−0.490762
$236$		0			0
$237$	3.19534	+	5.53450i	0.207560	+	0.359504i
$238$		0			0
$239$	8.09544			0.523650			0.261825	−	0.965115i	$-0.415676\pi$
$239$	8.09544			0.523650			0.261825	+	0.965115i	$0.415676\pi$
$240$		0			0
$241$	12.6425	+	21.8974i	0.814373	+	1.41054i	0.909777	+	0.415096i	$0.136252\pi$
$241$	12.6425	+	21.8974i	0.814373	+	1.41054i	−0.0954047	+	0.995439i	$0.530415\pi$
$242$		0			0
$243$	7.52378	+	13.0316i	0.482651	+	0.835976i
$244$		0			0
$245$	−1.33850	+	2.31835i	−0.0855137	+	0.148114i
$246$		0			0
$247$	0.786163	+	1.56392i	0.0500223	+	0.0995101i
$248$		0			0
$249$	−1.44630	+	2.50506i	−0.0916555	+	0.158752i
$250$		0			0
$251$	3.94461	+	6.83226i	0.248981	+	0.431248i	0.963243	−	0.268630i	$-0.0865709\pi$
$251$	3.94461	+	6.83226i	0.248981	+	0.431248i	−0.714262	+	0.699878i	$0.753238\pi$
$252$		0			0
$253$	−12.9117	−	22.3637i	−0.811751	−	1.40599i
$254$		0			0
$255$	−2.48014			−0.155313
$256$		0			0
$257$	4.59305	+	7.95540i	0.286507	+	0.496244i	0.972973	−	0.230917i	$-0.0741726\pi$
$257$	4.59305	+	7.95540i	0.286507	+	0.496244i	−0.686467	+	0.727161i	$0.740839\pi$
$258$		0			0
$259$	32.7542			2.03525
$260$		0			0
$261$	−9.90163	+	17.1501i	−0.612896	+	1.06157i
$262$		0			0
$263$	12.1217	−	20.9954i	0.747454	−	1.29463i	−0.201585	−	0.979471i	$-0.564609\pi$
$263$	12.1217	−	20.9954i	0.747454	−	1.29463i	0.949039	−	0.315158i	$-0.102058\pi$
$264$		0			0
$265$	10.5555			0.648417
$266$		0			0
$267$	9.38470			0.574335
$268$		0			0
$269$	−0.712209	+	1.23358i	−0.0434241	+	0.0752128i	−0.886921	−	0.461922i	$-0.847160\pi$
$269$	−0.712209	+	1.23358i	−0.0434241	+	0.0752128i	0.843496	+	0.537135i	$0.180493\pi$
$270$		0			0
$271$	−5.19617	+	9.00002i	−0.315645	+	0.546712i	−0.979574	−	0.201083i	$-0.935554\pi$
$271$	−5.19617	+	9.00002i	−0.315645	+	0.546712i	0.663930	+	0.747795i	$0.268887\pi$
$272$		0			0
$273$	0.885955			0.0536204
$274$		0			0
$275$	−1.76461	−	3.05640i	−0.106410	−	0.184308i
$276$		0			0
$277$	20.3078			1.22018			0.610088	−	0.792334i	$-0.291134\pi$
$277$	20.3078			1.22018			0.610088	+	0.792334i	$0.291134\pi$
$278$		0			0
$279$	−7.16074	−	12.4028i	−0.428702	−	0.742534i
$280$		0			0
$281$	−7.79547	−	13.5021i	−0.465038	−	0.805470i	0.534165	−	0.845380i	$-0.320626\pi$
$281$	−7.79547	−	13.5021i	−0.465038	−	0.805470i	−0.999203	+	0.0399101i	$0.987293\pi$
$282$		0			0
$283$	13.6323	−	23.6119i	0.810358	−	1.40358i	−0.102255	−	0.994758i	$-0.532606\pi$
$283$	13.6323	−	23.6119i	0.810358	−	1.40358i	0.912613	−	0.408824i	$-0.134061\pi$
$284$		0			0
$285$	−1.38845	−	2.76207i	−0.0822448	−	0.163611i
$286$		0			0
$287$	−1.72771	+	2.99249i	−0.101984	+	0.176641i
$288$		0			0
$289$	2.38546	+	4.13174i	0.140321	+	0.243044i
$290$		0			0
$291$	−2.73164	−	4.73134i	−0.160131	−	0.277356i
$292$		0			0
$293$	−12.6710			−0.740249			−0.370125	−	0.928982i	$-0.620685\pi$
$293$	−12.6710			−0.740249			−0.370125	+	0.928982i	$0.620685\pi$
$294$		0			0
$295$	−4.25618	−	7.37192i	−0.247804	−	0.429210i
$296$		0			0
$297$	13.7590			0.798376
$298$		0			0
$299$	−1.46915	+	2.54464i	−0.0849630	+	0.147160i
$300$		0			0
$301$	13.3885	−	23.1896i	0.771701	−	1.33663i
$302$		0			0
$303$	−12.3230			−0.707938
$304$		0			0
$305$	−9.22158			−0.528026
$306$		0			0
$307$	1.28540	−	2.22638i	0.0733619	−	0.127066i	−0.827011	−	0.562186i	$-0.809961\pi$
$307$	1.28540	−	2.22638i	0.0733619	−	0.127066i	0.900373	+	0.435119i	$0.143294\pi$
$308$		0			0
$309$	2.52699	−	4.37688i	0.143756	−	0.248992i
$310$		0			0
$311$	−19.7568			−1.12030			−0.560152	−	0.828390i	$-0.689257\pi$
$311$	−19.7568			−1.12030			−0.560152	+	0.828390i	$0.689257\pi$
$312$		0			0
$313$	−11.1877	−	19.3777i	−0.632368	−	1.09529i	−0.987066	−	0.160313i	$-0.948750\pi$
$313$	−11.1877	−	19.3777i	−0.632368	−	1.09529i	0.354698	−	0.934981i	$-0.384584\pi$
$314$		0			0
$315$	7.76767			0.437658
$316$		0			0
$317$	−6.65313	−	11.5236i	−0.373677	−	0.647228i	0.616451	−	0.787393i	$-0.288570\pi$
$317$	−6.65313	−	11.5236i	−0.373677	−	0.647228i	−0.990128	+	0.140166i	$0.955236\pi$
$318$		0			0
$319$	13.9948	+	24.2397i	0.783557	+	1.35716i
$320$		0			0
$321$	−0.260798	+	0.451716i	−0.0145563	+	0.0252123i
$322$		0			0
$323$	8.37148	−	12.7386i	0.465801	−	0.708792i
$324$		0			0
$325$	−0.200785	+	0.347770i	−0.0111375	+	0.0192908i
$326$		0			0
$327$	−5.67707	−	9.83297i	−0.313943	−	0.543764i
$328$		0			0
$329$	−11.7016	−	20.2678i	−0.645131	−	1.11740i
$330$		0			0
$331$	19.6173			1.07826			0.539132	−	0.842221i	$-0.318752\pi$
$331$	19.6173			1.07826			0.539132	+	0.842221i	$0.318752\pi$
$332$		0			0
$333$	−13.1458	−	22.7692i	−0.720385	−	1.24774i
$334$		0			0
$335$	−8.41246			−0.459622
$336$		0			0
$337$	14.5909	−	25.2722i	0.794819	−	1.37667i	−0.128136	−	0.991757i	$-0.540899\pi$
$337$	14.5909	−	25.2722i	0.794819	−	1.37667i	0.922954	−	0.384910i	$-0.125767\pi$
$338$		0			0
$339$	1.54228	−	2.67130i	0.0837650	−	0.145085i
$340$		0			0
$341$	−20.2417			−1.09615
$342$		0			0
$343$	13.4479			0.726120
$344$		0			0
$345$	2.59468	−	4.49412i	0.139693	−	0.241955i
$346$		0			0
$347$	−6.03935	+	10.4605i	−0.324209	+	0.561547i	−0.981352	−	0.192219i	$-0.938432\pi$
$347$	−6.03935	+	10.4605i	−0.324209	+	0.561547i	0.657143	+	0.753766i	$0.271765\pi$
$348$		0			0
$349$	20.2894			1.08607			0.543033	−	0.839711i	$-0.317276\pi$
$349$	20.2894			1.08607			0.543033	+	0.839711i	$0.317276\pi$
$350$		0			0
$351$	−0.782776	−	1.35581i	−0.0417815	−	0.0723677i
$352$		0			0
$353$	6.69387			0.356279			0.178139	−	0.984005i	$-0.442992\pi$
$353$	6.69387			0.356279			0.178139	+	0.984005i	$0.442992\pi$
$354$		0			0
$355$	−4.31233	−	7.46918i	−0.228875	−	0.396423i
$356$		0			0
$357$	−3.85760	−	6.68156i	−0.204166	−	0.353626i
$358$		0			0
$359$	4.94536	−	8.56562i	0.261006	−	0.452076i	−0.705503	−	0.708707i	$-0.749279\pi$
$359$	4.94536	−	8.56562i	0.261006	−	0.452076i	0.966510	+	0.256630i	$0.0826123\pi$
$360$		0			0
$361$	18.8732	+	2.19169i	0.993325	+	0.115352i
$362$		0			0
$363$	0.516108	−	0.893925i	0.0270886	−	0.0469189i
$364$		0			0
$365$	−0.870717	−	1.50813i	−0.0455754	−	0.0789389i
$366$		0			0
$367$	8.09936	+	14.0285i	0.422783	+	0.732282i	0.996211	−	0.0869742i	$-0.0277198\pi$
$367$	8.09936	+	14.0285i	0.422783	+	0.732282i	−0.573427	+	0.819257i	$0.694386\pi$
$368$		0			0
$369$	2.77365			0.144390
$370$		0			0
$371$	16.4179	+	28.4366i	0.852375	+	1.47636i
$372$		0			0
$373$	10.3003			0.533328			0.266664	−	0.963790i	$-0.414079\pi$
$373$	10.3003			0.533328			0.266664	+	0.963790i	$0.414079\pi$
$374$		0			0
$375$	0.354609	−	0.614201i	0.0183119	−	0.0317172i
$376$		0			0
$377$	1.59238	−	2.75809i	0.0820120	−	0.142049i
$378$		0			0
$379$	21.2587			1.09199			0.545993	−	0.837790i	$-0.316153\pi$
$379$	21.2587			1.09199			0.545993	+	0.837790i	$0.316153\pi$
$380$		0			0
$381$	−3.29502			−0.168809
$382$		0			0
$383$	−6.79002	+	11.7607i	−0.346954	+	0.600942i	−0.985707	−	0.168470i	$-0.946117\pi$
$383$	−6.79002	+	11.7607i	−0.346954	+	0.600942i	0.638753	+	0.769412i	$0.279451\pi$
$384$		0			0
$385$	5.48934	−	9.50781i	0.279762	−	0.484563i
$386$		0			0
$387$	−21.4938			−1.09259
$388$		0			0
$389$	−8.98771	−	15.5672i	−0.455695	−	0.789287i	0.543033	−	0.839711i	$-0.317276\pi$
$389$	−8.98771	−	15.5672i	−0.455695	−	0.789287i	−0.998728	+	0.0504248i	$0.983942\pi$
$390$		0			0
$391$	25.5877			1.29402
$392$		0			0
$393$	−3.72180	−	6.44634i	−0.187740	−	0.325175i
$394$		0			0
$395$	−4.50544	−	7.80366i	−0.226693	−	0.392645i
$396$		0			0
$397$	−12.0169	+	20.8139i	−0.603112	+	1.04462i	0.389234	+	0.921139i	$0.372740\pi$
$397$	−12.0169	+	20.8139i	−0.603112	+	1.04462i	−0.992347	+	0.123483i	$0.960594\pi$
$398$		0			0
$399$	5.28148	−	8.03663i	0.264405	−	0.402335i
$400$		0			0
$401$	5.91076	−	10.2377i	0.295169	−	0.511248i	−0.679855	−	0.733347i	$-0.737957\pi$
$401$	5.91076	−	10.2377i	0.295169	−	0.511248i	0.975024	+	0.222098i	$0.0712906\pi$
$402$		0			0
$403$	1.15159	+	1.99462i	0.0573649	+	0.0993589i
$404$		0			0
$405$	−2.36304	−	4.09291i	−0.117421	−	0.203378i
$406$		0			0
$407$	−37.1600			−1.84195
$408$		0			0
$409$	15.6300	+	27.0719i	0.772851	+	1.33862i	0.935994	+	0.352015i	$0.114504\pi$
$409$	15.6300	+	27.0719i	0.772851	+	1.33862i	−0.163143	+	0.986602i	$0.552163\pi$
$410$		0			0
$411$	0.987375			0.0487036
$412$		0			0
$413$	13.2401	−	22.9325i	0.651501	−	1.12843i
$414$		0			0
$415$	2.03929	−	3.53215i	0.100105	−	0.173386i
$416$		0			0
$417$	−4.30872			−0.210999
$418$		0			0
$419$	22.4217			1.09537			0.547686	−	0.836684i	$-0.315509\pi$
$419$	22.4217			1.09537			0.547686	+	0.836684i	$0.315509\pi$
$420$		0			0
$421$	−13.6431	+	23.6305i	−0.664922	+	1.15168i	0.314384	+	0.949296i	$0.398202\pi$
$421$	−13.6431	+	23.6305i	−0.664922	+	1.15168i	−0.979306	+	0.202384i	$0.935131\pi$
$422$		0			0
$423$	−9.39281	+	16.2688i	−0.456694	+	0.791017i
$424$		0			0
$425$	3.49701			0.169630
$426$		0			0
$427$	−14.3432	−	24.8431i	−0.694115	−	1.20224i
$428$		0			0
$429$	−1.00513			−0.0485279
$430$		0			0
$431$	13.6686	+	23.6748i	0.658395	+	1.14037i	0.981031	+	0.193850i	$0.0620977\pi$
$431$	13.6686	+	23.6748i	0.658395	+	1.14037i	−0.322636	+	0.946523i	$0.604569\pi$
$432$		0			0
$433$	3.67761	+	6.36980i	0.176734	+	0.306113i	0.940760	−	0.339073i	$-0.110113\pi$
$433$	3.67761	+	6.36980i	0.176734	+	0.306113i	−0.764026	+	0.645186i	$0.776780\pi$
$434$		0			0
$435$	−2.81233	+	4.87110i	−0.134841	+	0.233551i
$436$		0			0
$437$	14.3247	+	28.4963i	0.685243	+	1.36316i
$438$		0			0
$439$	2.11862	−	3.66955i	0.101116	−	0.175138i	−0.811029	−	0.585006i	$-0.801092\pi$
$439$	2.11862	−	3.66955i	0.101116	−	0.175138i	0.912145	+	0.409868i	$0.134425\pi$
$440$		0			0
$441$	3.34225	+	5.78895i	0.159155	+	0.275664i
$442$		0			0
$443$	−11.1310	−	19.2794i	−0.528849	−	0.915993i	−0.999434	−	0.0336383i	$-0.989291\pi$
$443$	−11.1310	−	19.2794i	−0.528849	−	0.915993i	0.470585	−	0.882354i	$-0.344043\pi$
$444$		0			0
$445$	−13.2325			−0.627279
$446$		0			0
$447$	−0.0299836	−	0.0519332i	−0.00141818	−	0.00245635i
$448$		0			0
$449$	4.46328			0.210635			0.105318	−	0.994439i	$-0.466414\pi$
$449$	4.46328			0.210635			0.105318	+	0.994439i	$0.466414\pi$
$450$		0			0
$451$	1.96011	−	3.39501i	0.0922980	−	0.159865i
$452$		0			0
$453$	5.39771	−	9.34911i	0.253607	−	0.439259i
$454$		0			0
$455$	−1.24920			−0.0585634
$456$		0			0
$457$	10.7463			0.502692			0.251346	−	0.967897i	$-0.419127\pi$
$457$	10.7463			0.502692			0.251346	+	0.967897i	$0.419127\pi$
$458$		0			0
$459$	−6.81668	+	11.8068i	−0.318176	+	0.551096i
$460$		0			0
$461$	−11.9163	+	20.6397i	−0.554998	+	0.961285i	0.442906	+	0.896568i	$0.353948\pi$
$461$	−11.9163	+	20.6397i	−0.554998	+	0.961285i	−0.997904	+	0.0647167i	$0.979386\pi$
$462$		0			0
$463$	25.3695			1.17902			0.589510	−	0.807761i	$-0.299321\pi$
$463$	25.3695			1.17902			0.589510	+	0.807761i	$0.299321\pi$
$464$		0			0
$465$	−2.03384	−	3.52272i	−0.0943172	−	0.163362i
$466$		0			0
$467$	−5.62020			−0.260072			−0.130036	−	0.991509i	$-0.541509\pi$
$467$	−5.62020			−0.260072			−0.130036	+	0.991509i	$0.541509\pi$
$468$		0			0
$469$	−13.0847	−	22.6633i	−0.604195	−	1.04650i
$470$		0			0
$471$	5.76548	+	9.98611i	0.265659	+	0.460136i
$472$		0			0
$473$	−15.1894	+	26.3089i	−0.698411	+	1.20968i
$474$		0			0
$475$	1.95772	+	3.89452i	0.0898265	+	0.178693i
$476$		0			0
$477$	13.1785	−	22.8259i	0.603404	−	1.04513i
$478$		0			0
$479$	6.83611	+	11.8405i	0.312350	+	0.541006i	0.978871	−	0.204480i	$-0.0655505\pi$
$479$	6.83611	+	11.8405i	0.312350	+	0.541006i	−0.666521	+	0.745487i	$0.732217\pi$
$480$		0			0
$481$	2.11411	+	3.66175i	0.0963951	+	0.166961i
$482$		0			0
$483$	16.1430			0.734532
$484$		0			0
$485$	3.85162	+	6.67120i	0.174893	+	0.302924i
$486$		0			0
$487$	7.39233			0.334979			0.167489	−	0.985874i	$-0.446434\pi$
$487$	7.39233			0.334979			0.167489	+	0.985874i	$0.446434\pi$
$488$		0			0
$489$	−5.30004	+	9.17994i	−0.239676	+	0.415131i
$490$		0			0
$491$	0.979053	−	1.69577i	0.0441840	−	0.0765290i	−0.843088	−	0.537776i	$-0.819265\pi$
$491$	0.979053	−	1.69577i	0.0441840	−	0.0765290i	0.887272	+	0.461247i	$0.152598\pi$
$492$		0			0
$493$	−27.7341			−1.24908
$494$		0			0
$495$	−8.81251			−0.396093
$496$		0			0
$497$	13.4148	−	23.2350i	0.601734	−	1.04223i
$498$		0			0
$499$	2.72234	−	4.71522i	0.121868	−	0.211082i	−0.798636	−	0.601814i	$-0.794445\pi$
$499$	2.72234	−	4.71522i	0.121868	−	0.211082i	0.920505	+	0.390732i	$0.127778\pi$
$500$		0			0
$501$	8.88595			0.396995
$502$		0			0
$503$	−8.29629	−	14.3696i	−0.369913	−	0.640709i	0.619638	−	0.784887i	$-0.287279\pi$
$503$	−8.29629	−	14.3696i	−0.369913	−	0.640709i	−0.989552	+	0.144179i	$0.953946\pi$
$504$		0			0
$505$	17.3755			0.773198
$506$		0			0
$507$	−4.55273	−	7.88557i	−0.202194	−	0.350210i
$508$		0			0
$509$	−6.06552	−	10.5058i	−0.268849	−	0.465661i	0.699716	−	0.714422i	$-0.253310\pi$
$509$	−6.06552	−	10.5058i	−0.268849	−	0.465661i	−0.968565	+	0.248761i	$0.919977\pi$
$510$		0			0
$511$	2.70862	−	4.69146i	0.119822	−	0.207538i
$512$		0			0
$513$	−16.9651	−	0.981757i	−0.749029	−	0.0433456i
$514$		0			0
$515$	−3.56307	+	6.17142i	−0.157008	+	0.271945i
$516$		0			0
$517$	13.2756	+	22.9940i	0.583861	+	1.01128i
$518$		0			0
$519$	6.16482	+	10.6778i	0.270606	+	0.468703i
$520$		0			0
$521$	14.1249			0.618824			0.309412	−	0.950928i	$-0.399868\pi$
$521$	14.1249			0.618824			0.309412	+	0.950928i	$0.399868\pi$
$522$		0			0
$523$	−13.1840	−	22.8353i	−0.576495	−	0.998519i	−0.995877	−	0.0907094i	$-0.971087\pi$
$523$	−13.1840	−	22.8353i	−0.576495	−	0.998519i	0.419382	−	0.907810i	$-0.362247\pi$
$524$		0			0
$525$	2.20623			0.0962877
$526$		0			0
$527$	10.0285	−	17.3698i	0.436847	−	0.756641i
$528$		0			0
$529$	−15.2694	+	26.4473i	−0.663885	+	1.14988i
$530$		0			0
$531$	−21.2554			−0.922407
$532$		0			0
$533$	−0.446059			−0.0193210
$534$		0			0
$535$	0.367726	−	0.636920i	0.0158982	−	0.0275365i
$536$		0			0
$537$	−7.36985	+	12.7649i	−0.318032	+	0.550848i
$538$		0			0
$539$	9.44775			0.406943
$540$		0			0
$541$	14.6986	+	25.4586i	0.631940	+	1.09455i	0.987155	+	0.159768i	$0.0510745\pi$
$541$	14.6986	+	25.4586i	0.631940	+	1.09455i	−0.355214	+	0.934785i	$0.615592\pi$
$542$		0			0
$543$	−1.15541			−0.0495833
$544$		0			0
$545$	8.00468	+	13.8645i	0.342883	+	0.593891i
$546$		0			0
$547$	10.8762	+	18.8381i	0.465031	+	0.805457i	0.999203	−	0.0399185i	$-0.0127098\pi$
$547$	10.8762	+	18.8381i	0.465031	+	0.805457i	−0.534172	+	0.845376i	$0.679377\pi$
$548$		0			0
$549$	−11.5132	+	19.9414i	−0.491371	+	0.851079i
$550$		0			0
$551$	−15.5263	−	30.8867i	−0.661443	−	1.31582i
$552$		0			0
$553$	14.0155	−	24.2755i	0.595999	−	1.03230i
$554$		0			0
$555$	−3.73376	−	6.46706i	−0.158489	−	0.274511i
$556$		0			0
$557$	12.0716	+	20.9086i	0.511489	+	0.885924i	0.999911	+	0.0133172i	$0.00423912\pi$
$557$	12.0716	+	20.9086i	0.511489	+	0.885924i	−0.488423	+	0.872607i	$0.662428\pi$
$558$		0			0
$559$	3.45663			0.146200
$560$		0			0
$561$	4.37649	+	7.58030i	0.184776	+	0.320041i
$562$		0			0
$563$	15.0693			0.635097			0.317548	−	0.948242i	$-0.397140\pi$
$563$	15.0693			0.635097			0.317548	+	0.948242i	$0.397140\pi$
$564$		0			0
$565$	−2.17462	+	3.76654i	−0.0914868	+	0.158460i
$566$		0			0
$567$	7.35092	−	12.7322i	0.308710	−	0.534701i
$568$		0			0
$569$	0.302873			0.0126971			0.00634855	−	0.999980i	$-0.497979\pi$
$569$	0.302873			0.0126971			0.00634855	+	0.999980i	$0.497979\pi$
$570$		0			0
$571$	−30.3107			−1.26846			−0.634232	−	0.773143i	$-0.718684\pi$
$571$	−30.3107			−1.26846			−0.634232	+	0.773143i	$0.718684\pi$
$572$		0			0
$573$	4.75232	−	8.23126i	0.198531	−	0.343866i
$574$		0			0
$575$	−3.65851	+	6.33672i	−0.152570	+	0.264260i
$576$		0			0
$577$	18.6601			0.776832			0.388416	−	0.921484i	$-0.373022\pi$
$577$	18.6601			0.776832			0.388416	+	0.921484i	$0.373022\pi$
$578$		0			0
$579$	−3.79214	−	6.56819i	−0.157596	−	0.272964i
$580$		0			0
$581$	12.6876			0.526369
$582$		0			0
$583$	−18.6263	−	32.2617i	−0.771422	−	1.33614i
$584$		0			0
$585$	0.501362	+	0.868384i	0.0207288	+	0.0359033i
$586$		0			0
$587$	−13.5986	+	23.5535i	−0.561275	+	0.972156i	0.436111	+	0.899893i	$0.356355\pi$
$587$	−13.5986	+	23.5535i	−0.561275	+	0.972156i	−0.997386	+	0.0722631i	$0.976978\pi$
$588$		0			0
$589$	24.9585	+	1.44433i	1.02840	+	0.0595124i
$590$		0			0
$591$	−2.48443	+	4.30316i	−0.102196	+	0.177008i
$592$		0			0
$593$	14.7987	+	25.6321i	0.607709	+	1.05258i	0.991617	+	0.129211i	$0.0412446\pi$
$593$	14.7987	+	25.6321i	0.607709	+	1.05258i	−0.383908	+	0.923371i	$0.625422\pi$
$594$		0			0
$595$	5.43923	+	9.42102i	0.222987	+	0.386224i
$596$		0			0
$597$	−14.7388			−0.603221
$598$		0			0
$599$	7.54404	+	13.0667i	0.308241	+	0.533889i	0.977978	−	0.208710i	$-0.0669264\pi$
$599$	7.54404	+	13.0667i	0.308241	+	0.533889i	−0.669737	+	0.742599i	$0.733593\pi$
$600$		0			0
$601$	−44.9249			−1.83253			−0.916263	−	0.400576i	$-0.868810\pi$
$601$	−44.9249			−1.83253			−0.916263	+	0.400576i	$0.868810\pi$
$602$		0			0
$603$	−10.5030	+	18.1917i	−0.427715	+	0.740824i
$604$		0			0
$605$	−0.727713	+	1.26044i	−0.0295858	+	0.0512440i
$606$		0			0
$607$	24.9570			1.01297			0.506487	−	0.862247i	$-0.330944\pi$
$607$	24.9570			1.01297			0.506487	+	0.862247i	$0.330944\pi$
$608$		0			0
$609$	−17.4971			−0.709020
$610$		0			0
$611$	1.51055	−	2.61636i	0.0611105	−	0.105846i
$612$		0			0
$613$	9.64316	−	16.7024i	0.389484	−	0.674605i	−0.602897	−	0.797819i	$-0.705987\pi$
$613$	9.64316	−	16.7024i	0.389484	−	0.674605i	0.992380	+	0.123214i	$0.0393202\pi$
$614$		0			0
$615$	0.787791			0.0317668
$616$		0			0
$617$	−9.72294	−	16.8406i	−0.391431	−	0.677978i	0.601208	−	0.799093i	$-0.294686\pi$
$617$	−9.72294	−	16.8406i	−0.391431	−	0.677978i	−0.992639	+	0.121115i	$0.961353\pi$
$618$		0			0
$619$	1.70324			0.0684589			0.0342294	−	0.999414i	$-0.489102\pi$
$619$	1.70324			0.0684589			0.0342294	+	0.999414i	$0.489102\pi$
$620$		0			0
$621$	−14.2630	−	24.7042i	−0.572354	−	0.991346i
$622$		0			0
$623$	−20.5817	−	35.6485i	−0.824588	−	1.42823i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$	−5.99190	+	9.11764i	−0.239293	+	0.364124i
$628$		0			0
$629$	18.4104	−	31.8877i	0.734071	−	1.27145i
$630$		0			0
$631$	−4.39314	−	7.60914i	−0.174888	−	0.302915i	0.765235	−	0.643752i	$-0.222623\pi$
$631$	−4.39314	−	7.60914i	−0.174888	−	0.302915i	−0.940123	+	0.340837i	$0.889290\pi$
$632$		0			0
$633$	3.98313	+	6.89899i	0.158315	+	0.274210i
$634$		0			0
$635$	4.64599			0.184371
$636$		0			0
$637$	−0.537502	−	0.930981i	−0.0212966	−	0.0368868i
$638$		0			0
$639$	−21.5359			−0.851946
$640$		0			0
$641$	17.9033	−	31.0095i	0.707139	−	1.22480i	−0.258775	−	0.965938i	$-0.583319\pi$
$641$	17.9033	−	31.0095i	0.707139	−	1.22480i	0.965914	−	0.258863i	$-0.0833478\pi$
$642$		0			0
$643$	1.30758	−	2.26480i	0.0515661	−	0.0893150i	−0.839090	−	0.543992i	$-0.816912\pi$
$643$	1.30758	−	2.26480i	0.0515661	−	0.0893150i	0.890656	+	0.454677i	$0.150245\pi$
$644$		0			0
$645$	−6.10481			−0.240377
$646$		0			0
$647$	−30.8849			−1.21421			−0.607105	−	0.794622i	$-0.707669\pi$
$647$	−30.8849			−1.21421			−0.607105	+	0.794622i	$0.707669\pi$
$648$		0			0
$649$	−15.0210	+	26.0172i	−0.589626	+	1.02126i
$650$		0			0
$651$	6.32686	−	10.9584i	0.247969	−	0.429495i
$652$		0			0
$653$	−17.2198			−0.673864			−0.336932	−	0.941529i	$-0.609389\pi$
$653$	−17.2198			−0.673864			−0.336932	+	0.941529i	$0.609389\pi$
$654$		0			0
$655$	5.24775	+	9.08936i	0.205046	+	0.355151i
$656$		0			0
$657$	−4.34838			−0.169646
$658$		0			0
$659$	−0.0769444	−	0.133272i	−0.00299733	−	0.00519153i	0.864523	−	0.502594i	$-0.167621\pi$
$659$	−0.0769444	−	0.133272i	−0.00299733	−	0.00519153i	−0.867520	+	0.497402i	$0.834287\pi$
$660$		0			0
$661$	13.8595	+	24.0054i	0.539073	+	0.933701i	0.998954	+	0.0457209i	$0.0145585\pi$
$661$	13.8595	+	24.0054i	0.539073	+	0.933701i	−0.459882	+	0.887980i	$0.652108\pi$
$662$		0			0
$663$	0.497975	−	0.862519i	0.0193398	−	0.0334975i
$664$		0			0
$665$	−7.44690	+	11.3317i	−0.288778	+	0.439423i
$666$		0			0
$667$	29.0149	−	50.2552i	1.12346	−	1.94589i
$668$		0			0
$669$	3.86527	+	6.69485i	0.149440	+	0.258838i
$670$		0			0
$671$	16.2725	+	28.1848i	0.628193	+	1.08806i
$672$		0			0
$673$	−48.0820			−1.85342			−0.926712	−	0.375773i	$-0.877377\pi$
$673$	−48.0820			−1.85342			−0.926712	+	0.375773i	$0.877377\pi$
$674$		0			0
$675$	−1.94929	−	3.37627i	−0.0750282	−	0.129953i
$676$		0			0
$677$	−42.8392			−1.64644			−0.823222	−	0.567720i	$-0.807826\pi$
$677$	−42.8392			−1.64644			−0.823222	+	0.567720i	$0.807826\pi$
$678$		0			0
$679$	−11.9816	+	20.7527i	−0.459810	+	0.796415i
$680$		0			0
$681$	−1.40647	+	2.43609i	−0.0538962	+	0.0933510i
$682$		0			0
$683$	−41.4022			−1.58421			−0.792106	−	0.610383i	$-0.791015\pi$
$683$	−41.4022			−1.58421			−0.792106	+	0.610383i	$0.791015\pi$
$684$		0			0
$685$	−1.39220			−0.0531933
$686$		0			0
$687$	−3.87018	+	6.70335i	−0.147657	+	0.255749i
$688$		0			0
$689$	−2.11938	+	3.67087i	−0.0807418	+	0.139849i
$690$		0			0
$691$	−16.9248			−0.643850			−0.321925	−	0.946765i	$-0.604330\pi$
$691$	−16.9248			−0.643850			−0.321925	+	0.946765i	$0.604330\pi$
$692$		0			0
$693$	−13.7069	−	23.7411i	−0.520683	−	0.901849i
$694$		0			0
$695$	6.07532			0.230450
$696$		0			0
$697$	1.94222	+	3.36402i	0.0735668	+	0.127421i
$698$		0			0
$699$	−5.52869	−	9.57597i	−0.209114	−	0.362196i
$700$		0			0
$701$	18.7780	−	32.5244i	0.709233	−	1.22843i	−0.255908	−	0.966701i	$-0.582375\pi$
$701$	18.7780	−	32.5244i	0.709233	−	1.22843i	0.965142	−	0.261727i	$-0.0842921\pi$
$702$		0			0
$703$	45.8192	+	2.65152i	1.72810	+	0.100004i
$704$		0			0
$705$	−2.66781	+	4.62078i	−0.100476	+	0.174029i
$706$		0			0
$707$	27.0257	+	46.8099i	1.01641	+	1.76047i
$708$		0			0
$709$	15.3872	+	26.6514i	0.577879	+	1.00092i	0.995722	+	0.0923969i	$0.0294529\pi$
$709$	15.3872	+	26.6514i	0.577879	+	1.00092i	−0.417843	+	0.908519i	$0.637214\pi$
$710$		0			0
$711$	−22.5003			−0.843826
$712$		0			0
$713$	20.9832	+	36.3440i	0.785827	+	1.36109i
$714$		0			0
$715$	1.41723			0.0530014
$716$		0			0
$717$	2.87072	−	4.97223i	0.107209	−	0.185691i
$718$		0			0
$719$	−23.1509	+	40.0985i	−0.863383	+	1.49542i	0.00526119	+	0.999986i	$0.498325\pi$
$719$	−23.1509	+	40.0985i	−0.863383	+	1.49542i	−0.868644	+	0.495437i	$0.835008\pi$
$720$		0			0
$721$	−22.1679			−0.825576
$722$		0			0
$723$	17.9325			0.666918
$724$		0			0
$725$	3.96540	−	6.86827i	0.147271	−	0.255081i
$726$		0			0
$727$	−2.93454	+	5.08278i	−0.108836	+	0.188510i	−0.915299	−	0.402775i	$-0.868046\pi$
$727$	−2.93454	+	5.08278i	−0.108836	+	0.188510i	0.806463	+	0.591285i	$0.201379\pi$
$728$		0			0
$729$	−3.50625			−0.129861
$730$		0			0
$731$	−15.0508	−	26.0687i	−0.556673	−	0.964186i
$732$		0			0
$733$	11.3955			0.420901			0.210450	−	0.977605i	$-0.432507\pi$
$733$	11.3955			0.420901			0.210450	+	0.977605i	$0.432507\pi$
$734$		0			0
$735$	0.949290	+	1.64422i	0.0350151	+	0.0606479i
$736$		0			0
$737$	14.8447	+	25.7118i	0.546812	+	0.947107i
$738$		0			0
$739$	6.86621	−	11.8926i	0.252578	−	0.437477i	−0.711657	−	0.702527i	$-0.752055\pi$
$739$	6.86621	−	11.8926i	0.252578	−	0.437477i	0.964235	+	0.265050i	$0.0853884\pi$
$740$		0			0
$741$	1.23934	+	0.0717198i	0.0455284	+	0.00263469i
$742$		0			0
$743$	−4.33763	+	7.51300i	−0.159132	+	0.275625i	−0.934556	−	0.355816i	$-0.884203\pi$
$743$	−4.33763	+	7.51300i	−0.159132	+	0.275625i	0.775424	+	0.631441i	$0.217536\pi$
$744$		0			0
$745$	0.0422770	+	0.0732259i	0.00154891	+	0.00268279i
$746$		0			0
$747$	−5.09212	−	8.81981i	−0.186311	−	0.322700i
$748$		0			0
$749$	2.28784			0.0835957
$750$		0			0
$751$	3.35314	+	5.80780i	0.122358	+	0.211930i	0.920697	−	0.390278i	$-0.127621\pi$
$751$	3.35314	+	5.80780i	0.122358	+	0.211930i	−0.798339	+	0.602208i	$0.794288\pi$
$752$		0			0
$753$	5.59517			0.203899
$754$		0			0
$755$	−7.61079	+	13.1823i	−0.276985	+	0.479752i
$756$		0			0
$757$	−7.19480	+	12.4618i	−0.261500	+	0.452931i	−0.966641	−	0.256136i	$-0.917550\pi$
$757$	−7.19480	+	12.4618i	−0.261500	+	0.452931i	0.705141	+	0.709067i	$0.250884\pi$
$758$		0			0
$759$	−18.3144			−0.664771
$760$		0			0
$761$	−50.9650			−1.84748			−0.923740	−	0.383020i	$-0.874884\pi$
$761$	−50.9650			−1.84748			−0.923740	+	0.383020i	$0.874884\pi$
$762$		0			0
$763$	−24.9009	+	43.1296i	−0.901472	+	1.56140i
$764$		0			0
$765$	4.36603	−	7.56219i	0.157854	−	0.273412i
$766$		0			0
$767$	3.41831			0.123428
$768$		0			0
$769$	−11.9988	−	20.7825i	−0.432687	−	0.749435i	0.564417	−	0.825490i	$-0.309101\pi$
$769$	−11.9988	−	20.7825i	−0.432687	−	0.749435i	−0.997104	+	0.0760546i	$0.975768\pi$
$770$		0			0
$771$	6.51495			0.234630
$772$		0			0
$773$	21.9925	+	38.0920i	0.791014	+	1.37008i	0.925340	+	0.379139i	$0.123780\pi$
$773$	21.9925	+	38.0920i	0.791014	+	1.37008i	−0.134326	+	0.990937i	$0.542887\pi$
$774$		0			0
$775$	2.86773	+	4.96705i	0.103012	+	0.178422i
$776$		0			0
$777$	11.6149	−	20.1177i	0.416683	−	0.721717i
$778$		0			0
$779$	−2.65911	+	4.04627i	−0.0952725	+	0.144973i
$780$		0			0
$781$	−15.2192	+	26.3604i	−0.544585	+	0.943250i
$782$		0			0
$783$	15.4594	+	26.7765i	0.552474	+	0.956914i
$784$		0			0
$785$	−8.12935	−	14.0804i	−0.290149	−	0.502553i
$786$		0			0
$787$	−16.4815			−0.587501			−0.293751	−	0.955882i	$-0.594904\pi$
$787$	−16.4815			−0.587501			−0.293751	+	0.955882i	$0.594904\pi$
$788$		0			0
$789$	−8.59691	−	14.8903i	−0.306058	−	0.530108i
$790$		0			0
$791$	−13.5295			−0.481055

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.354609 - 0.614201i) q^{3} +(-0.500000 + 0.866025i) q^{5} -3.11079 q^{7} +(1.24850 + 2.16247i) q^{9} +O(q^{10})\)	\(q+(0.354609 - 0.614201i) q^{3} +(-0.500000 + 0.866025i) q^{5} -3.11079 q^{7} +(1.24850 + 2.16247i) q^{9} +3.52922 q^{11} +(-0.200785 - 0.347770i) q^{13} +(0.354609 + 0.614201i) q^{15} +(-1.74850 + 3.02850i) q^{17} +(-4.35162 - 0.251824i) q^{19} +(-1.10311 + 1.91065i) q^{21} +(-3.65851 - 6.33672i) q^{23} +(-0.500000 - 0.866025i) q^{25} +3.89858 q^{27} +(3.96540 + 6.86827i) q^{29} -5.73545 q^{31} +(1.25150 - 2.16765i) q^{33} +(1.55539 - 2.69402i) q^{35} -10.5292 q^{37} -0.284801 q^{39} +(0.555394 - 0.961971i) q^{41} +(-4.30390 + 7.45457i) q^{43} -2.49701 q^{45} +(3.76162 + 6.51532i) q^{47} +2.67700 q^{49} +(1.24007 + 2.14787i) q^{51} +(-5.27773 - 9.14130i) q^{53} +(-1.76461 + 3.05640i) q^{55} +(-1.69779 + 2.58347i) q^{57} +(-4.25618 + 7.37192i) q^{59} +(4.61079 + 7.98612i) q^{61} +(-3.88383 - 6.72700i) q^{63} +0.401570 q^{65} +(4.20623 + 7.28540i) q^{67} -5.18936 q^{69} +(-4.31233 + 7.46918i) q^{71} +(-0.870717 + 1.50813i) q^{73} -0.709218 q^{75} -10.9787 q^{77} +(-4.50544 + 7.80366i) q^{79} +(-2.36304 + 4.09291i) q^{81} -4.07857 q^{83} +(-1.74850 - 3.02850i) q^{85} +5.62466 q^{87} +(6.61623 + 11.4596i) q^{89} +(0.624599 + 1.08184i) q^{91} +(-2.03384 + 3.52272i) q^{93} +(2.39390 - 3.64270i) q^{95} +(3.85162 - 6.67120i) q^{97} +(4.40625 + 7.63186i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{3} - 4 q^{5} - 5 q^{9}+O(q^{10}) \) 8 * q + q^3 - 4 * q^5 - 5 * q^9	\( 8 q + q^{3} - 4 q^{5} - 5 q^{9} - 4 q^{11} + 9 q^{13} + q^{15} + q^{17} - 3 q^{19} + 8 q^{21} - 4 q^{25} - 20 q^{27} + 5 q^{29} + 20 q^{31} + 25 q^{33} - 52 q^{37} + 54 q^{39} - 8 q^{41} - 7 q^{43} + 10 q^{45} - 16 q^{47} + 20 q^{49} - 12 q^{51} + 5 q^{53} + 2 q^{55} + 27 q^{57} - 11 q^{59} + 12 q^{61} + 3 q^{63} - 18 q^{65} + 6 q^{69} - 14 q^{71} - 4 q^{73} - 2 q^{75} - 44 q^{77} - 13 q^{79} - 24 q^{81} - 10 q^{83} + q^{85} + 4 q^{87} + 5 q^{89} + 46 q^{91} - 28 q^{93} + 6 q^{95} - q^{97} - 24 q^{99}+O(q^{100}) \) 8 * q + q^3 - 4 * q^5 - 5 * q^9 - 4 * q^11 + 9 * q^13 + q^15 + q^17 - 3 * q^19 + 8 * q^21 - 4 * q^25 - 20 * q^27 + 5 * q^29 + 20 * q^31 + 25 * q^33 - 52 * q^37 + 54 * q^39 - 8 * q^41 - 7 * q^43 + 10 * q^45 - 16 * q^47 + 20 * q^49 - 12 * q^51 + 5 * q^53 + 2 * q^55 + 27 * q^57 - 11 * q^59 + 12 * q^61 + 3 * q^63 - 18 * q^65 + 6 * q^69 - 14 * q^71 - 4 * q^73 - 2 * q^75 - 44 * q^77 - 13 * q^79 - 24 * q^81 - 10 * q^83 + q^85 + 4 * q^87 + 5 * q^89 + 46 * q^91 - 28 * q^93 + 6 * q^95 - q^97 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

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