LMFDB - Embedded newform 820.2.w.a.659.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [820,2,Mod(79,820)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(820, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([4, 4, 7])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("820.79"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 820 = 2^{2} \cdot 5 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	820.w (of order $8$, degree $4$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	no
Analytic conductor:	$6.54773296574$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{8}]$

Embedding invariants

Embedding label			659.1
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	820.659
Dual form			820.2.w.a.219.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.00000 - 1.00000i) q^{2} +2.00000i q^{4} +(0.707107 + 2.12132i) q^{5} +(2.00000 - 2.00000i) q^{8} +(2.12132 + 2.12132i) q^{9} +(1.41421 - 2.82843i) q^{10} +(1.29289 + 0.535534i) q^{13} -4.00000 q^{16} +(-1.12132 + 0.464466i) q^{17} -4.24264i q^{18} +(-4.24264 + 1.41421i) q^{20} +(-4.00000 + 3.00000i) q^{25} +(-0.757359 - 1.82843i) q^{26} +(-0.0502525 + 0.121320i) q^{29} +(4.00000 + 4.00000i) q^{32} +(1.58579 + 0.656854i) q^{34} +(-4.24264 + 4.24264i) q^{36} +7.07107i q^{37} +(5.65685 + 2.82843i) q^{40} +(-5.00000 + 4.00000i) q^{41} +(-3.00000 + 6.00000i) q^{45} +(4.94975 + 4.94975i) q^{49} +(7.00000 + 1.00000i) q^{50} +(-1.07107 + 2.58579i) q^{52} +(-0.636039 + 1.53553i) q^{53} +(0.171573 - 0.0710678i) q^{58} +(1.00000 - 1.00000i) q^{61} -8.00000i q^{64} +(-0.221825 + 3.12132i) q^{65} +(-0.928932 - 2.24264i) q^{68} +8.48528 q^{72} +(4.24264 + 4.24264i) q^{73} +(7.07107 - 7.07107i) q^{74} +(-2.82843 - 8.48528i) q^{80} +9.00000i q^{81} +(9.00000 + 1.00000i) q^{82} +(-1.77817 - 2.05025i) q^{85} +(5.87868 - 14.1924i) q^{89} +(9.00000 - 3.00000i) q^{90} +(5.46447 + 13.1924i) q^{97} -9.89949i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 8 q^{8} + 8 q^{13} - 16 q^{16} + 4 q^{17} - 16 q^{25} - 20 q^{26} - 20 q^{29} + 16 q^{32} + 12 q^{34} - 20 q^{41} - 12 q^{45} + 28 q^{50} + 24 q^{52} - 28 q^{53} + 12 q^{58} + 4 q^{61} - 32 q^{65}+ \cdots + 36 q^{97}+O(q^{100}) $ 4 * q - 4 * q^2 + 8 * q^8 + 8 * q^13 - 16 * q^16 + 4 * q^17 - 16 * q^25 - 20 * q^26 - 20 * q^29 + 16 * q^32 + 12 * q^34 - 20 * q^41 - 12 * q^45 + 28 * q^50 + 24 * q^52 - 28 * q^53 + 12 * q^58 + 4 * q^61 - 32 * q^65 - 32 * q^68 + 36 * q^82 + 24 * q^85 + 32 * q^89 + 36 * q^90 + 36 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/820\mathbb{Z}\right)^\times$.

$n$	$411$	$621$	$657$
$\chi(n)$	$-1$	$e\left(\frac{3}{8}\right)$	$-1$

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 $

$2$	−1.00000	−	1.00000i	−0.707107	−	0.707107i
$2$	−1.00000	−	1.00000i	−0.707107	−	0.707107i
$3$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$3$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$4$			2.00000i			1.00000i
$5$	0.707107	+	2.12132i	0.316228	+	0.948683i
$5$	0.707107	+	2.12132i	0.316228	+	0.948683i
$6$		0			0
$7$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$7$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$8$	2.00000	−	2.00000i	0.707107	−	0.707107i
$9$	2.12132	+	2.12132i	0.707107	+	0.707107i
$10$	1.41421	−	2.82843i	0.447214	−	0.894427i
$11$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$11$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$12$		0			0
$13$	1.29289	+	0.535534i	0.358584	+	0.148530i	0.554700	−	0.832050i	$-0.312833\pi$
$13$	1.29289	+	0.535534i	0.358584	+	0.148530i	−0.196116	+	0.980581i	$0.562833\pi$
$14$		0			0
$15$		0			0
$16$	−4.00000			−1.00000
$17$	−1.12132	+	0.464466i	−0.271960	+	0.112650i	−0.514496	−	0.857493i	$-0.672021\pi$
$17$	−1.12132	+	0.464466i	−0.271960	+	0.112650i	0.242536	+	0.970143i	$0.422021\pi$
$18$		−	4.24264i		−	1.00000i
$19$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$19$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$20$	−4.24264	+	1.41421i	−0.948683	+	0.316228i
$21$		0			0
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	−4.00000	+	3.00000i	−0.800000	+	0.600000i
$26$	−0.757359	−	1.82843i	−0.148530	−	0.358584i
$27$		0			0
$28$		0			0
$29$	−0.0502525	+	0.121320i	−0.00933166	+	0.0225286i	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−0.0502525	+	0.121320i	−0.00933166	+	0.0225286i	0.919145	+	0.393919i	$0.128881\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$	4.00000	+	4.00000i	0.707107	+	0.707107i
$33$		0			0
$34$	1.58579	+	0.656854i	0.271960	+	0.112650i
$35$		0			0
$36$	−4.24264	+	4.24264i	−0.707107	+	0.707107i
$37$			7.07107i			1.16248i	0.813733	+	0.581238i	$0.197432\pi$
$37$			7.07107i			1.16248i	−0.813733	+	0.581238i	$0.802568\pi$
$38$		0			0
$39$		0			0
$40$	5.65685	+	2.82843i	0.894427	+	0.447214i
$41$	−5.00000	+	4.00000i	−0.780869	+	0.624695i
$41$	−5.00000	+	4.00000i	−0.780869	+	0.624695i
$42$		0			0
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$		0			0
$45$	−3.00000	+	6.00000i	−0.447214	+	0.894427i
$46$		0			0
$47$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$47$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$48$		0			0
$49$	4.94975	+	4.94975i	0.707107	+	0.707107i
$50$	7.00000	+	1.00000i	0.989949	+	0.141421i
$51$		0			0
$52$	−1.07107	+	2.58579i	−0.148530	+	0.358584i
$53$	−0.636039	+	1.53553i	−0.0873667	+	0.210922i	−0.961524	−	0.274721i	$-0.911414\pi$
$53$	−0.636039	+	1.53553i	−0.0873667	+	0.210922i	0.874157	+	0.485643i	$0.161414\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	0.171573	−	0.0710678i	0.0225286	−	0.00933166i
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$	1.00000	−	1.00000i	0.128037	−	0.128037i	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	1.00000	−	1.00000i	0.128037	−	0.128037i	0.768221	+	0.640184i	$0.221142\pi$
$62$		0			0
$63$		0			0
$64$		−	8.00000i		−	1.00000i
$65$	−0.221825	+	3.12132i	−0.0275141	+	0.387152i
$66$		0			0
$67$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$67$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$68$	−0.928932	−	2.24264i	−0.112650	−	0.271960i
$69$		0			0
$70$		0			0
$71$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$71$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$72$	8.48528			1.00000
$73$	4.24264	+	4.24264i	0.496564	+	0.496564i	0.910366	−	0.413803i	$-0.135800\pi$
$73$	4.24264	+	4.24264i	0.496564	+	0.496564i	−0.413803	+	0.910366i	$0.635800\pi$
$74$	7.07107	−	7.07107i	0.821995	−	0.821995i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$79$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$80$	−2.82843	−	8.48528i	−0.316228	−	0.948683i
$81$			9.00000i			1.00000i
$82$	9.00000	+	1.00000i	0.993884	+	0.110432i
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$	−1.77817	−	2.05025i	−0.192870	−	0.222381i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	5.87868	−	14.1924i	0.623139	−	1.50439i	−0.224860	−	0.974391i	$-0.572192\pi$
$89$	5.87868	−	14.1924i	0.623139	−	1.50439i	0.847998	−	0.529999i	$-0.177808\pi$
$90$	9.00000	−	3.00000i	0.948683	−	0.316228i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	5.46447	+	13.1924i	0.554832	+	1.33948i	0.913812	+	0.406138i	$0.133125\pi$
$97$	5.46447	+	13.1924i	0.554832	+	1.33948i	−0.358979	+	0.933346i	$0.616875\pi$
$98$		−	9.89949i		−	1.00000i
$99$		0			0
$100$	−6.00000	−	8.00000i	−0.600000	−	0.800000i
$101$	−2.22183	−	5.36396i	−0.221080	−	0.533734i	0.773957	−	0.633238i	$-0.218274\pi$
$101$	−2.22183	−	5.36396i	−0.221080	−	0.533734i	−0.995037	+	0.0995037i	$0.968274\pi$
$102$		0			0
$103$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$103$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$104$	3.65685	−	1.51472i	0.358584	−	0.148530i
$105$		0			0
$106$	2.17157	−	0.899495i	0.210922	−	0.0873667i
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$	14.9497	−	6.19239i	1.43193	−	0.593123i	0.474100	−	0.880471i	$-0.342774\pi$
$109$	14.9497	−	6.19239i	1.43193	−	0.593123i	0.957826	+	0.287348i	$0.0927736\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.41421			0.133038			0.0665190	−	0.997785i	$-0.478811\pi$
$113$	1.41421			0.133038			0.0665190	+	0.997785i	$0.478811\pi$
$114$		0			0
$115$		0			0
$116$	−0.242641	−	0.100505i	−0.0225286	−	0.00933166i
$117$	1.60660	+	3.87868i	0.148530	+	0.358584i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	7.77817	+	7.77817i	0.707107	+	0.707107i
$122$	−2.00000			−0.181071
$123$		0			0
$124$		0			0
$125$	−9.19239	−	6.36396i	−0.822192	−	0.569210i
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	−8.00000	+	8.00000i	−0.707107	+	0.707107i
$129$		0			0
$130$	3.34315	−	2.89949i	0.293213	−	0.254303i
$131$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$131$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−1.31371	+	3.17157i	−0.112650	+	0.271960i
$137$	−21.6066	+	8.94975i	−1.84598	+	0.764629i	−0.906183	+	0.422885i	$0.861017\pi$
$137$	−21.6066	+	8.94975i	−1.84598	+	0.764629i	−0.939793	+	0.341743i	$0.888983\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−8.48528	−	8.48528i	−0.707107	−	0.707107i
$145$	−0.292893	−	0.0208153i	−0.0243235	−	0.00172861i
$146$		−	8.48528i		−	0.702247i
$147$		0			0
$148$	−14.1421			−1.16248
$149$	−5.02082	−	12.1213i	−0.411321	−	0.993017i	−0.984784	−	0.173785i	$-0.944400\pi$
$149$	−5.02082	−	12.1213i	−0.411321	−	0.993017i	0.573462	−	0.819232i	$-0.305600\pi$
$150$		0			0
$151$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$151$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$152$		0			0
$153$	−3.36396	−	1.39340i	−0.271960	−	0.112650i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−6.02082	+	14.5355i	−0.480513	+	1.16006i	0.478852	+	0.877896i	$0.341053\pi$
$157$	−6.02082	+	14.5355i	−0.480513	+	1.16006i	−0.959366	+	0.282166i	$0.908947\pi$
$158$		0			0
$159$		0			0
$160$	−5.65685	+	11.3137i	−0.447214	+	0.894427i
$161$		0			0
$162$	9.00000	−	9.00000i	0.707107	−	0.707107i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$	−8.00000	−	10.0000i	−0.624695	−	0.780869i
$165$		0			0
$166$		0			0
$167$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$167$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$168$		0			0
$169$	−7.80761	−	7.80761i	−0.600586	−	0.600586i
$170$	−0.272078	+	3.82843i	−0.0208674	+	0.293627i
$171$		0			0
$172$		0			0
$173$	11.0000	−	11.0000i	0.836315	−	0.836315i	−0.152057	−	0.988372i	$-0.548590\pi$
$173$	11.0000	−	11.0000i	0.836315	−	0.836315i	0.988372	+	0.152057i	$0.0485898\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−20.0711	+	8.31371i	−1.50439	+	0.623139i
$179$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$179$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$180$	−12.0000	−	6.00000i	−0.894427	−	0.447214i
$181$	−4.43503	−	10.7071i	−0.329653	−	0.795853i	−0.998618	−	0.0525588i	$-0.983262\pi$
$181$	−4.43503	−	10.7071i	−0.329653	−	0.795853i	0.668965	−	0.743294i	$-0.266738\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−15.0000	+	5.00000i	−1.10282	+	0.367607i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$191$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$192$		0			0
$193$	15.5355	+	6.43503i	1.11827	+	0.463204i	0.863779	−	0.503871i	$-0.168091\pi$
$193$	15.5355	+	6.43503i	1.11827	+	0.463204i	0.254493	+	0.967075i	$0.418091\pi$
$194$	7.72792	−	18.6569i	0.554832	−	1.33948i
$195$		0			0
$196$	−9.89949	+	9.89949i	−0.707107	+	0.707107i
$197$	−15.0000	−	15.0000i	−1.06871	−	1.06871i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$	−15.0000	−	15.0000i	−1.06871	−	1.06871i	−0.0712470	−	0.997459i	$-0.522698\pi$
$198$		0			0
$199$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$199$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$200$	−2.00000	+	14.0000i	−0.141421	+	0.989949i
$201$		0			0
$202$	−3.14214	+	7.58579i	−0.221080	+	0.533734i
$203$		0			0
$204$		0			0
$205$	−12.0208	−	7.77817i	−0.839570	−	0.543251i
$206$		0			0
$207$		0			0
$208$	−5.17157	−	2.14214i	−0.358584	−	0.148530i
$209$		0			0
$210$		0			0
$211$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$211$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$212$	−3.07107	−	1.27208i	−0.210922	−	0.0873667i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−21.1421	−	8.75736i	−1.43193	−	0.593123i
$219$		0			0
$220$		0			0
$221$	−1.69848			−0.114252
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$	−14.8492	−	2.12132i	−0.989949	−	0.141421i
$226$	−1.41421	−	1.41421i	−0.0940721	−	0.0940721i
$227$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$227$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$228$		0			0
$229$	14.0208	−	5.80761i	0.926522	−	0.383778i	0.132164	−	0.991228i	$-0.457808\pi$
$229$	14.0208	−	5.80761i	0.926522	−	0.383778i	0.794358	+	0.607450i	$0.207808\pi$
$230$		0			0
$231$		0			0
$232$	0.142136	+	0.343146i	0.00933166	+	0.0225286i
$233$	9.46447	−	22.8492i	0.620038	−	1.49690i	−0.231621	−	0.972806i	$-0.574403\pi$
$233$	9.46447	−	22.8492i	0.620038	−	1.49690i	0.851658	−	0.524097i	$-0.175597\pi$
$234$	2.27208	−	5.48528i	0.148530	−	0.358584i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$239$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$240$		0			0
$241$	−11.0000	+	11.0000i	−0.708572	+	0.708572i	−0.966235	−	0.257663i	$-0.917048\pi$
$241$	−11.0000	+	11.0000i	−0.708572	+	0.708572i	0.257663	+	0.966235i	$0.417048\pi$
$242$		−	15.5563i		−	1.00000i
$243$		0			0
$244$	2.00000	+	2.00000i	0.128037	+	0.128037i
$245$	−7.00000	+	14.0000i	−0.447214	+	0.894427i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	2.82843	+	15.5563i	0.178885	+	0.983870i
$251$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$251$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	16.0000			1.00000
$257$	3.97918	−	9.60660i	0.248215	−	0.599243i	−0.749838	−	0.661622i	$-0.769869\pi$
$257$	3.97918	−	9.60660i	0.248215	−	0.599243i	0.998053	+	0.0623783i	$0.0198685\pi$
$258$		0			0
$259$		0			0
$260$	−6.24264	−	0.443651i	−0.387152	−	0.0275141i
$261$	−0.363961	+	0.150758i	−0.0225286	+	0.00933166i
$262$		0			0
$263$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$263$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$264$		0			0
$265$	−3.70711	−	0.263456i	−0.227726	−	0.0161840i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	26.0000i		−	1.58525i	−0.609711	−	0.792624i	$-0.708714\pi$
$269$		−	26.0000i		−	1.58525i	0.609711	−	0.792624i	$-0.291286\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$	4.48528	−	1.85786i	0.271960	−	0.112650i
$273$		0			0
$274$	30.5563	+	12.6569i	1.84598	+	0.764629i
$275$		0			0
$276$		0			0
$277$	7.07107			0.424859			0.212430	−	0.977176i	$-0.431862\pi$
$277$	7.07107			0.424859			0.212430	+	0.977176i	$0.431862\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	19.8492	−	8.22183i	1.18411	−	0.490473i	0.298275	−	0.954480i	$-0.403589\pi$
$281$	19.8492	−	8.22183i	1.18411	−	0.490473i	0.885832	+	0.464007i	$0.153589\pi$
$282$		0			0
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$			16.9706i			1.00000i
$289$	−10.9792	+	10.9792i	−0.645834	+	0.645834i
$290$	0.272078	+	0.313708i	0.0159770	+	0.0184216i
$291$		0			0
$292$	−8.48528	+	8.48528i	−0.496564	+	0.496564i
$293$	30.4350	−	12.6066i	1.77803	−	0.736486i	0.784883	−	0.619644i	$-0.212723\pi$
$293$	30.4350	−	12.6066i	1.77803	−	0.736486i	0.993151	−	0.116841i	$-0.0372769\pi$
$294$		0			0
$295$		0			0
$296$	14.1421	+	14.1421i	0.821995	+	0.821995i
$297$		0			0
$298$	−7.10051	+	17.1421i	−0.411321	+	0.993017i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2.82843	+	1.41421i	0.161955	+	0.0809776i
$306$	1.97056	+	4.75736i	0.112650	+	0.271960i
$307$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$307$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$311$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$312$		0			0
$313$	29.6777	−	12.2929i	1.67748	−	0.694835i	0.678280	−	0.734803i	$-0.262726\pi$
$313$	29.6777	−	12.2929i	1.67748	−	0.694835i	0.999201	+	0.0399680i	$0.0127256\pi$
$314$	20.5563	−	8.51472i	1.16006	−	0.480513i
$315$		0			0
$316$		0			0
$317$	−31.6777	−	13.1213i	−1.77920	−	0.736967i	−0.992877	−	0.119145i	$-0.961985\pi$
$317$	−31.6777	−	13.1213i	−1.77920	−	0.736967i	−0.786318	−	0.617822i	$-0.788015\pi$
$318$		0			0
$319$		0			0
$320$	16.9706	−	5.65685i	0.948683	−	0.316228i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−18.0000			−1.00000
$325$	−6.77817	+	1.73654i	−0.375985	+	0.0963261i
$326$		0			0
$327$		0			0
$328$	−2.00000	+	18.0000i	−0.110432	+	0.993884i
$329$		0			0
$330$		0			0
$331$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$331$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$332$		0			0
$333$	−15.0000	+	15.0000i	−0.821995	+	0.821995i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	22.6274	−	22.6274i	1.23259	−	1.23259i	0.269630	−	0.962964i	$-0.413099\pi$
$337$	22.6274	−	22.6274i	1.23259	−	1.23259i	0.962964	−	0.269630i	$-0.0869014\pi$
$338$			15.6152i			0.849356i
$339$		0			0
$340$	4.10051	−	3.55635i	0.222381	−	0.192870i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−22.0000			−1.18273
$347$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$347$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$348$		0			0
$349$	−7.07107	−	7.07107i	−0.378506	−	0.378506i	0.492057	−	0.870563i	$-0.336245\pi$
$349$	−7.07107	−	7.07107i	−0.378506	−	0.378506i	−0.870563	+	0.492057i	$0.836245\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	12.7279			0.677439			0.338719	−	0.940887i	$-0.390006\pi$
$353$	12.7279			0.677439			0.338719	+	0.940887i	$0.390006\pi$
$354$		0			0
$355$		0			0
$356$	28.3848	+	11.7574i	1.50439	+	0.623139i
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$	6.00000	+	18.0000i	0.316228	+	0.948683i
$361$	−13.4350	+	13.4350i	−0.707107	+	0.707107i
$362$	−6.27208	+	15.1421i	−0.329653	+	0.795853i
$363$		0			0
$364$		0			0
$365$	−6.00000	+	12.0000i	−0.314054	+	0.628109i
$366$		0			0
$367$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$367$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$368$		0			0
$369$	−19.0919	−	2.12132i	−0.993884	−	0.110432i
$370$	20.0000	+	10.0000i	1.03975	+	0.519875i
$371$		0			0
$372$		0			0
$373$	36.0000			1.86401			0.932005	−	0.362446i	$-0.118058\pi$
$373$	36.0000			1.86401			0.932005	+	0.362446i	$0.118058\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−0.129942	+	0.129942i	−0.00669237	+	0.00669237i
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$383$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$384$		0			0
$385$		0			0
$386$	−9.10051	−	21.9706i	−0.463204	−	1.11827i
$387$		0			0
$388$	−26.3848	+	10.9289i	−1.33948	+	0.554832i
$389$	7.00000	−	7.00000i	0.354914	−	0.354914i	−0.507020	−	0.861934i	$-0.669253\pi$
$389$	7.00000	−	7.00000i	0.354914	−	0.354914i	0.861934	+	0.507020i	$0.169253\pi$
$390$		0			0
$391$		0			0
$392$	19.7990			1.00000
$393$		0			0
$394$			30.0000i			1.51138i
$395$		0			0
$396$		0			0
$397$	15.1924	−	36.6777i	0.762484	−	1.84080i	0.301131	−	0.953583i	$-0.402636\pi$
$397$	15.1924	−	36.6777i	0.762484	−	1.84080i	0.461353	−	0.887217i	$-0.347364\pi$
$398$		0			0
$399$		0			0
$400$	16.0000	−	12.0000i	0.800000	−	0.600000i
$401$	−1.41421	+	1.41421i	−0.0706225	+	0.0706225i	−0.741536	−	0.670913i	$-0.765902\pi$
$401$	−1.41421	+	1.41421i	−0.0706225	+	0.0706225i	0.670913	+	0.741536i	$0.265902\pi$
$402$		0			0
$403$		0			0
$404$	10.7279	−	4.44365i	0.533734	−	0.221080i
$405$	−19.0919	+	6.36396i	−0.948683	+	0.316228i
$406$		0			0
$407$		0			0
$408$		0			0
$409$			24.0416i			1.18878i	0.804176	+	0.594391i	$0.202607\pi$
$409$			24.0416i			1.18878i	−0.804176	+	0.594391i	$0.797393\pi$
$410$	4.24264	+	19.7990i	0.209529	+	0.977802i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	3.02944	+	7.31371i	0.148530	+	0.358584i
$417$		0			0
$418$		0			0
$419$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$419$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$420$		0			0
$421$	−35.5061	−	14.7071i	−1.73046	−	0.716781i	−0.999406	−	0.0344623i	$-0.989028\pi$
$421$	−35.5061	−	14.7071i	−1.73046	−	0.716781i	−0.731055	−	0.682318i	$-0.760972\pi$
$422$		0			0
$423$		0			0
$424$	1.79899	+	4.34315i	0.0873667	+	0.210922i
$425$	3.09188	−	5.22183i	0.149978	−	0.253296i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$431$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$432$		0			0
$433$	24.0000			1.15337			0.576683	−	0.816968i	$-0.304347\pi$
$433$	24.0000			1.15337			0.576683	+	0.816968i	$0.304347\pi$
$434$		0			0
$435$		0			0
$436$	12.3848	+	29.8995i	0.593123	+	1.43193i
$437$		0			0
$438$		0			0
$439$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$439$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$440$		0			0
$441$			21.0000i			1.00000i
$442$	1.69848	+	1.69848i	0.0807887	+	0.0807887i
$443$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$443$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$444$		0			0
$445$	34.2635	+	2.43503i	1.62424	+	0.115431i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	27.0000	+	27.0000i	1.27421	+	1.27421i	0.943858	+	0.330350i	$0.107167\pi$
$449$	27.0000	+	27.0000i	1.27421	+	1.27421i	0.330350	+	0.943858i	$0.392833\pi$
$450$	12.7279	+	16.9706i	0.600000	+	0.800000i
$451$		0			0
$452$			2.82843i			0.133038i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	33.0208	−	13.6777i	1.54465	−	0.639814i	0.562310	−	0.826927i	$-0.309913\pi$
$457$	33.0208	−	13.6777i	1.54465	−	0.639814i	0.982339	+	0.187112i	$0.0599128\pi$
$458$	−19.8284	−	8.21320i	−0.926522	−	0.383778i
$459$		0			0
$460$		0			0
$461$	41.0122			1.91013			0.955064	−	0.296399i	$-0.0957859\pi$
$461$	41.0122			1.91013			0.955064	+	0.296399i	$0.0957859\pi$
$462$		0			0
$463$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$463$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$464$	0.201010	−	0.485281i	0.00933166	−	0.0225286i
$465$		0			0
$466$	−32.3137	+	13.3848i	−1.49690	+	0.620038i
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$	−7.75736	+	3.21320i	−0.358584	+	0.148530i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−4.60660	+	1.90812i	−0.210922	+	0.0873667i
$478$		0			0
$479$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$479$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$480$		0			0
$481$	−3.78680	+	9.14214i	−0.172663	+	0.416846i
$482$	22.0000			1.00207
$483$		0			0
$484$	−15.5563	+	15.5563i	−0.707107	+	0.707107i
$485$	−24.1213	+	20.9203i	−1.09529	+	0.949942i
$486$		0			0
$487$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$487$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$488$		−	4.00000i		−	0.181071i
$489$		0			0
$490$	21.0000	−	7.00000i	0.948683	−	0.316228i
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		−	0.159380i		−	0.00717809i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$499$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$500$	12.7279	−	18.3848i	0.569210	−	0.822192i
$501$		0			0
$502$		0			0
$503$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$503$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$504$		0			0
$505$	9.80761	−	8.50610i	0.436433	−	0.378516i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−2.90812	−	7.02082i	−0.128900	−	0.311192i	0.846233	−	0.532813i	$-0.178865\pi$
$509$	−2.90812	−	7.02082i	−0.128900	−	0.311192i	−0.975133	+	0.221621i	$0.928865\pi$
$510$		0			0
$511$		0			0
$512$	−16.0000	−	16.0000i	−0.707107	−	0.707107i
$513$		0			0
$514$	−13.5858	+	5.62742i	−0.599243	+	0.248215i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	5.79899	+	6.68629i	0.254303	+	0.293213i
$521$	1.92031	−	4.63604i	0.0841303	−	0.203109i	−0.876216	−	0.481919i	$-0.839940\pi$
$521$	1.92031	−	4.63604i	0.0841303	−	0.203109i	0.960346	+	0.278810i	$0.0899400\pi$
$522$	0.514719	+	0.213203i	0.0225286	+	0.00933166i
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	23.0000			1.00000
$530$	3.44365	+	3.97056i	0.149583	+	0.172470i
$531$		0			0
$532$		0			0
$533$	−8.60660	+	2.49390i	−0.372793	+	0.108023i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−26.0000	+	26.0000i	−1.12094	+	1.12094i
$539$		0			0
$540$		0			0
$541$	−29.6985	−	29.6985i	−1.27684	−	1.27684i	−0.942428	−	0.334410i	$-0.891463\pi$
$541$	−29.6985	−	29.6985i	−1.27684	−	1.27684i	−0.334410	−	0.942428i	$-0.608537\pi$
$542$		0			0
$543$		0			0
$544$	−6.34315	−	2.62742i	−0.271960	−	0.112650i
$545$	23.7071	+	27.3345i	1.01550	+	1.17088i
$546$		0			0
$547$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$547$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$548$	−17.8995	−	43.2132i	−0.764629	−	1.84598i
$549$	4.24264			0.181071
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−7.07107	−	7.07107i	−0.300421	−	0.300421i
$555$		0			0
$556$		0			0
$557$	−4.33452	−	10.4645i	−0.183660	−	0.443394i	0.805056	−	0.593199i	$-0.202135\pi$
$557$	−4.33452	−	10.4645i	−0.183660	−	0.443394i	−0.988716	+	0.149805i	$0.952135\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−28.0711	−	11.6274i	−1.18411	−	0.490473i
$563$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$563$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$564$		0			0
$565$	1.00000	+	3.00000i	0.0420703	+	0.126211i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−28.2843	+	28.2843i	−1.18574	+	1.18574i	−0.207504	+	0.978234i	$0.566534\pi$
$569$	−28.2843	+	28.2843i	−1.18574	+	1.18574i	−0.978234	+	0.207504i	$0.933466\pi$
$570$		0			0
$571$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$571$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	16.9706	−	16.9706i	0.707107	−	0.707107i
$577$	−15.2635	−	6.32233i	−0.635426	−	0.263202i	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	−15.2635	−	6.32233i	−0.635426	−	0.263202i	−0.677057	+	0.735931i	$0.736745\pi$
$578$	21.9584			0.913348
$579$		0			0
$580$	0.0416306	−	0.585786i	0.00172861	−	0.0243235i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	16.9706			0.702247
$585$	−7.09188	+	6.15076i	−0.293213	+	0.254303i
$586$	−43.0416	−	17.8284i	−1.77803	−	0.736486i
$587$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$587$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		−	28.2843i		−	1.16248i
$593$	−2.60660	+	1.07969i	−0.107040	+	0.0443375i	−0.435561	−	0.900159i	$-0.643450\pi$
$593$	−2.60660	+	1.07969i	−0.107040	+	0.0443375i	0.328521	+	0.944497i	$0.393450\pi$
$594$		0			0
$595$		0			0
$596$	24.2426	−	10.0416i	0.993017	−	0.411321i
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	8.43503	−	3.49390i	0.344072	−	0.142519i	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	8.43503	−	3.49390i	0.344072	−	0.142519i	0.548026	+	0.836461i	$0.315379\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−11.0000	+	22.0000i	−0.447214	+	0.894427i
$606$		0			0
$607$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$607$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$608$		0			0
$609$		0			0
$610$	−1.41421	−	4.24264i	−0.0572598	−	0.171780i
$611$		0			0
$612$	2.78680	−	6.72792i	0.112650	−	0.271960i
$613$	−24.0416	+	24.0416i	−0.971032	+	0.971032i	−0.999592	−	0.0285598i	$-0.990908\pi$
$613$	−24.0416	+	24.0416i	−0.971032	+	0.971032i	0.0285598	+	0.999592i	$0.490908\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−35.0000	−	35.0000i	−1.40905	−	1.40905i	−0.764911	−	0.644136i	$-0.777217\pi$
$617$	−35.0000	−	35.0000i	−1.40905	−	1.40905i	−0.644136	−	0.764911i	$-0.722783\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	7.00000	−	24.0000i	0.280000	−	0.960000i
$626$	−41.9706	−	17.3848i	−1.67748	−	0.694835i
$627$		0			0
$628$	−29.0711	−	12.0416i	−1.16006	−	0.480513i
$629$	−3.28427	−	7.92893i	−0.130952	−	0.316147i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$	18.5563	+	44.7990i	0.736967	+	1.77920i
$635$		0			0
$636$		0			0
$637$	3.74874	+	9.05025i	0.148530	+	0.358584i
$638$		0			0
$639$		0			0
$640$	−22.6274	−	11.3137i	−0.894427	−	0.447214i
$641$	16.5061	+	39.8492i	0.651952	+	1.57395i	0.809942	+	0.586510i	$0.199498\pi$
$641$	16.5061	+	39.8492i	0.651952	+	1.57395i	−0.157991	+	0.987441i	$0.550502\pi$
$642$		0			0
$643$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$643$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$647$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$648$	18.0000	+	18.0000i	0.707107	+	0.707107i
$649$		0			0
$650$	8.51472	+	5.04163i	0.333975	+	0.197749i
$651$		0			0
$652$		0			0
$653$	2.74874	−	6.63604i	0.107566	−	0.259688i	−0.860927	−	0.508729i	$-0.830115\pi$
$653$	2.74874	−	6.63604i	0.107566	−	0.259688i	0.968493	+	0.249041i	$0.0801154\pi$
$654$		0			0
$655$		0			0
$656$	20.0000	−	16.0000i	0.780869	−	0.624695i
$657$			18.0000i			0.702247i
$658$		0			0
$659$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$659$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$660$		0			0
$661$	−35.3553	−	35.3553i	−1.37516	−	1.37516i	−0.852601	−	0.522562i	$-0.824976\pi$
$661$	−35.3553	−	35.3553i	−1.37516	−	1.37516i	−0.522562	−	0.852601i	$-0.675024\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	30.0000			1.16248
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	36.7487	−	15.2218i	1.41656	−	0.586758i	0.462566	−	0.886585i	$-0.346929\pi$
$673$	36.7487	−	15.2218i	1.41656	−	0.586758i	0.953994	+	0.299827i	$0.0969288\pi$
$674$	−45.2548			−1.74315
$675$		0			0
$676$	15.6152	−	15.6152i	0.600586	−	0.600586i
$677$	−36.7696	+	36.7696i	−1.41317	+	1.41317i	−0.679408	+	0.733761i	$0.737763\pi$
$677$	−36.7696	+	36.7696i	−1.41317	+	1.41317i	−0.733761	+	0.679408i	$0.762237\pi$
$678$		0			0
$679$		0			0
$680$	−7.65685	−	0.544156i	−0.293627	−	0.0208674i
$681$		0			0
$682$		0			0
$683$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$683$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$684$		0			0
$685$	−34.2635	−	39.5061i	−1.30914	−	1.50945i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1.64466	+	1.64466i	−0.0626566	+	0.0626566i
$690$		0			0
$691$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$691$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$692$	22.0000	+	22.0000i	0.836315	+	0.836315i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	3.74874	−	6.80761i	0.141994	−	0.257857i
$698$			14.1421i			0.535288i
$699$		0			0
$700$		0			0
$701$		−	43.8406i		−	1.65584i	−0.560848	−	0.827919i	$-0.689525\pi$
$701$		−	43.8406i		−	1.65584i	0.560848	−	0.827919i	$-0.310475\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−12.7279	−	12.7279i	−0.479022	−	0.479022i
$707$		0			0
$708$		0			0
$709$	−26.9497	−	11.1630i	−1.01212	−	0.419233i	−0.185892	−	0.982570i	$-0.559517\pi$
$709$	−26.9497	−	11.1630i	−1.01212	−	0.419233i	−0.826227	+	0.563337i	$0.809517\pi$
$710$		0			0
$711$		0			0
$712$	−16.6274	−	40.1421i	−0.623139	−	1.50439i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$719$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$720$	12.0000	−	24.0000i	0.447214	−	0.894427i
$721$		0			0
$722$	26.8701			1.00000
$723$		0			0
$724$	21.4142	−	8.87006i	0.795853	−	0.329653i
$725$	−0.162951	−	0.636039i	−0.00605184	−	0.0236219i
$726$		0			0
$727$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$727$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$728$		0			0
$729$	−19.0919	+	19.0919i	−0.707107	+	0.707107i
$730$	18.0000	−	6.00000i	0.666210	−	0.222070i
$731$		0			0
$732$		0			0
$733$	−29.0000	−	29.0000i	−1.07114	−	1.07114i	−0.997268	−	0.0738717i	$-0.976464\pi$
$733$	−29.0000	−	29.0000i	−1.07114	−	1.07114i	−0.0738717	−	0.997268i	$-0.523536\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	16.9706	+	21.2132i	0.624695	+	0.780869i
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$	−10.0000	−	30.0000i	−0.367607	−	1.10282i
$741$		0			0
$742$		0			0
$743$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$743$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$744$		0			0
$745$	22.1630	−	19.2218i	0.811988	−	0.704233i
$746$	−36.0000	−	36.0000i	−1.31805	−	1.31805i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$751$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$752$		0			0
$753$		0			0
$754$	0.259885			0.00946444
$755$		0			0
$756$		0			0
$757$	−38.0208	+	15.7487i	−1.38189	+	0.572398i	−0.944986	−	0.327111i	$-0.893925\pi$
$757$	−38.0208	+	15.7487i	−1.38189	+	0.572398i	−0.436904	+	0.899508i	$0.643925\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			1.41421i			0.0512652i	0.999671	+	0.0256326i	$0.00816000\pi$
$761$			1.41421i			0.0512652i	−0.999671	+	0.0256326i	$0.991840\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	0.577164	−	8.12132i	0.0208674	−	0.293627i
$766$		0			0
$767$		0			0
$768$		0			0
$769$			24.0000i			0.865462i	0.901523	+	0.432731i	$0.142450\pi$
$769$			24.0000i			0.865462i	−0.901523	+	0.432731i	$0.857550\pi$
$770$		0			0
$771$		0			0
$772$	−12.8701	+	31.0711i	−0.463204	+	1.11827i
$773$	20.5355	+	49.5772i	0.738612	+	1.78317i	0.611448	+	0.791285i	$0.290588\pi$
$773$	20.5355	+	49.5772i	0.738612	+	1.78317i	0.127164	+	0.991882i	$0.459412\pi$
$774$		0			0
$775$		0			0
$776$	37.3137	+	15.4558i	1.33948	+	0.554832i
$777$		0			0
$778$	−14.0000			−0.501924
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−19.7990	−	19.7990i	−0.707107	−	0.707107i
$785$	−35.0919	−	2.49390i	−1.25248	−	0.0890112i
$786$		0			0
$787$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$787$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$788$	30.0000	−	30.0000i	1.06871	−	1.06871i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	1.82843	−	0.757359i	0.0649294	−	0.0268946i
$794$	−51.8701	+	21.4853i	−1.84080	+	0.762484i
$795$		0			0
$796$		0			0
$797$			52.0000i			1.84193i	0.389640	+	0.920967i	$0.372599\pi$
$797$			52.0000i			1.84193i	−0.389640	+	0.920967i	$0.627401\pi$
$798$		0			0
$799$		0			0
$800$	−28.0000	−	4.00000i	−0.989949	−	0.141421i
$801$	42.5772	−	17.6360i	1.50439	−	0.623139i
$802$	2.82843			0.0998752
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−15.1716	−	6.28427i	−0.533734	−	0.221080i
$809$	−21.2635	+	51.3345i	−0.747583	+	1.80483i	−0.175791	+	0.984428i	$0.556248\pi$
$809$	−21.2635	+	51.3345i	−0.747583	+	1.80483i	−0.571793	+	0.820398i	$0.693752\pi$
$810$	25.4558	+	12.7279i	0.894427	+	0.447214i
$811$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$811$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	24.0416	−	24.0416i	0.840596	−	0.840596i
$819$		0			0
$820$	15.5563	−	24.0416i	0.543251	−	0.839570i
$821$	−15.5563			−0.542920			−0.271460	−	0.962450i	$-0.587507\pi$
$821$	−15.5563			−0.542920			−0.271460	+	0.962450i	$0.587507\pi$
$822$		0			0
$823$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$823$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$827$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$828$		0			0
$829$	−14.1421	−	14.1421i	−0.491177	−	0.491177i	0.417500	−	0.908677i	$-0.362906\pi$
$829$	−14.1421	−	14.1421i	−0.491177	−	0.491177i	−0.908677	+	0.417500i	$0.862906\pi$
$830$		0			0
$831$		0			0
$832$	4.28427	−	10.3431i	0.148530	−	0.358584i
$833$	−7.84924	−	3.25126i	−0.271960	−	0.112650i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$839$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$840$		0			0
$841$	20.4939	+	20.4939i	0.706686	+	0.706686i
$842$	20.7990	+	50.2132i	0.716781	+	1.73046i
$843$		0			0
$844$		0			0
$845$	11.0416	−	22.0833i	0.379844	−	0.759687i
$846$		0			0
$847$		0			0
$848$	2.54416	−	6.14214i	0.0873667	−	0.210922i
$849$		0			0
$850$	−8.31371	+	2.12994i	−0.285158	+	0.0730564i
$851$		0			0
$852$		0			0
$853$	5.00000	+	5.00000i	0.171197	+	0.171197i	0.787505	−	0.616308i	$-0.211372\pi$
$853$	5.00000	+	5.00000i	0.171197	+	0.171197i	−0.616308	+	0.787505i	$0.711372\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			35.3553i			1.20772i	0.797092	+	0.603858i	$0.206370\pi$
$857$			35.3553i			1.20772i	−0.797092	+	0.603858i	$0.793630\pi$
$858$		0			0
$859$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$859$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$863$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$864$		0			0
$865$	31.1127	+	15.5563i	1.05786	+	0.528932i
$866$	−24.0000	−	24.0000i	−0.815553	−	0.815553i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	17.5147	−	42.2843i	0.593123	−	1.43193i
$873$	−16.3934	+	39.5772i	−0.554832	+	1.33948i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			12.0000i			0.405211i	0.979260	+	0.202606i	$0.0649409\pi$
$877$			12.0000i			0.405211i	−0.979260	+	0.202606i	$0.935059\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	41.0000	−	41.0000i	1.38133	−	1.38133i	0.539054	−	0.842271i	$-0.318782\pi$
$881$	41.0000	−	41.0000i	1.38133	−	1.38133i	0.842271	−	0.539054i	$-0.181218\pi$
$882$	21.0000	−	21.0000i	0.707107	−	0.707107i
$883$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$883$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$884$		−	3.39697i		−	0.114252i
$885$		0			0
$886$		0			0
$887$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$887$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$888$		0			0
$889$		0			0
$890$	−31.8284	−	36.6985i	−1.06689	−	1.23014i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		−	54.0000i		−	1.80200i
$899$		0			0
$900$	4.24264	−	29.6985i	0.141421	−	0.989949i
$901$		−	2.01724i		−	0.0672041i
$902$		0			0
$903$		0			0
$904$	2.82843	−	2.82843i	0.0940721	−	0.0940721i
$905$	19.5772	−	16.9792i	0.650767	−	0.564407i
$906$		0			0
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$		0			0
$909$	6.66548	−	16.0919i	0.221080	−	0.533734i
$910$		0			0
$911$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$911$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$912$		0			0
$913$		0			0
$914$	−46.6985	−	19.3431i	−1.54465	−	0.639814i
$915$		0			0
$916$	11.6152	+	28.0416i	0.383778	+	0.926522i
$917$		0			0
$918$		0			0
$919$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$919$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$920$		0			0
$921$		0			0
$922$	−41.0122	−	41.0122i	−1.35066	−	1.35066i
$923$		0			0
$924$		0			0
$925$	−21.2132	−	28.2843i	−0.697486	−	0.929981i
$926$		0			0
$927$		0			0
$928$	−0.686292	+	0.284271i	−0.0225286	+	0.00933166i
$929$	50.4056	−	20.8787i	1.65375	−	0.685007i	0.656179	−	0.754606i	$-0.272172\pi$
$929$	50.4056	−	20.8787i	1.65375	−	0.685007i	0.997575	+	0.0695983i	$0.0221717\pi$
$930$		0			0
$931$		0			0
$932$	45.6985	+	18.9289i	1.49690	+	0.620038i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	10.9706	+	4.54416i	0.358584	+	0.148530i
$937$	20.4645	+	49.4056i	0.668545	+	1.61401i	0.784046	+	0.620703i	$0.213153\pi$
$937$	20.4645	+	49.4056i	0.668545	+	1.61401i	−0.115501	+	0.993307i	$0.536847\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	39.0000	+	39.0000i	1.27136	+	1.27136i	0.945373	+	0.325991i	$0.105698\pi$
$941$	39.0000	+	39.0000i	1.27136	+	1.27136i	0.325991	+	0.945373i	$0.394302\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$		0			0
$949$	3.21320	+	7.75736i	0.104305	+	0.251815i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	21.2132i		−	0.687163i	−0.939123	−	0.343582i	$-0.888360\pi$
$953$		−	21.2132i		−	0.687163i	0.939123	−	0.343582i	$-0.111640\pi$
$954$	6.51472	+	2.69848i	0.210922	+	0.0873667i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−31.0000			−1.00000
$962$	12.9289	−	5.35534i	0.416846	−	0.172663i
$963$		0			0
$964$	−22.0000	−	22.0000i	−0.708572	−	0.708572i
$965$	−2.66548	+	37.5061i	−0.0858047	+	1.20736i
$966$		0			0
$967$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$967$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$968$	31.1127			1.00000
$969$		0			0
$970$	45.0416	+	3.20101i	1.44620	+	0.102778i
$971$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$971$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−4.00000	+	4.00000i	−0.128037	+	0.128037i
$977$	−20.7487	+	50.0919i	−0.663811	+	1.60258i	0.127971	+	0.991778i	$0.459153\pi$
$977$	−20.7487	+	50.0919i	−0.663811	+	1.60258i	−0.791782	+	0.610803i	$0.790847\pi$
$978$		0			0
$979$		0			0
$980$	−28.0000	−	14.0000i	−0.894427	−	0.447214i
$981$	44.8492	+	18.5772i	1.43193	+	0.593123i
$982$		0			0
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		0			0
$985$	21.2132	−	42.4264i	0.675909	−	1.35182i
$986$	−0.159380	+	0.159380i	−0.00507568	+	0.00507568i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$991$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	48.6777	+	20.1630i	1.54164	+	0.638567i	0.981780	−	0.190022i	$-0.0608559\pi$
$997$	48.6777	+	20.1630i	1.54164	+	0.638567i	0.559857	+	0.828589i	$0.310856\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	820.2.w.a.659.1	yes	4
4.3	odd	2	CM	820.2.w.a.659.1	yes	4
5.4	even	2		820.2.w.b.659.1	yes	4
20.19	odd	2		820.2.w.b.659.1	yes	4
41.14	odd	8		820.2.w.b.219.1	yes	4
164.55	even	8		820.2.w.b.219.1	yes	4
205.14	odd	8	inner	820.2.w.a.219.1	✓	4
820.219	even	8	inner	820.2.w.a.219.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
820.2.w.a.219.1	✓	4	205.14	odd	8	inner
820.2.w.a.219.1	✓	4	820.219	even	8	inner
820.2.w.a.659.1	yes	4	1.1	even	1	trivial
820.2.w.a.659.1	yes	4	4.3	odd	2	CM
820.2.w.b.219.1	yes	4	41.14	odd	8
820.2.w.b.219.1	yes	4	164.55	even	8
820.2.w.b.659.1	yes	4	5.4	even	2
820.2.w.b.659.1	yes	4	20.19	odd	2

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:	\( N \)	\(=\)	\( 820 = 2^{2} \cdot 5 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	820.w (of order \(8\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.00000i) q^{2} +2.00000i q^{4} +(0.707107 + 2.12132i) q^{5} +(2.00000 - 2.00000i) q^{8} +(2.12132 + 2.12132i) q^{9} +(1.41421 - 2.82843i) q^{10} +(1.29289 + 0.535534i) q^{13} -4.00000 q^{16} +(-1.12132 + 0.464466i) q^{17} -4.24264i q^{18} +(-4.24264 + 1.41421i) q^{20} +(-4.00000 + 3.00000i) q^{25} +(-0.757359 - 1.82843i) q^{26} +(-0.0502525 + 0.121320i) q^{29} +(4.00000 + 4.00000i) q^{32} +(1.58579 + 0.656854i) q^{34} +(-4.24264 + 4.24264i) q^{36} +7.07107i q^{37} +(5.65685 + 2.82843i) q^{40} +(-5.00000 + 4.00000i) q^{41} +(-3.00000 + 6.00000i) q^{45} +(4.94975 + 4.94975i) q^{49} +(7.00000 + 1.00000i) q^{50} +(-1.07107 + 2.58579i) q^{52} +(-0.636039 + 1.53553i) q^{53} +(0.171573 - 0.0710678i) q^{58} +(1.00000 - 1.00000i) q^{61} -8.00000i q^{64} +(-0.221825 + 3.12132i) q^{65} +(-0.928932 - 2.24264i) q^{68} +8.48528 q^{72} +(4.24264 + 4.24264i) q^{73} +(7.07107 - 7.07107i) q^{74} +(-2.82843 - 8.48528i) q^{80} +9.00000i q^{81} +(9.00000 + 1.00000i) q^{82} +(-1.77817 - 2.05025i) q^{85} +(5.87868 - 14.1924i) q^{89} +(9.00000 - 3.00000i) q^{90} +(5.46447 + 13.1924i) q^{97} -9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 8 q^{8} + 8 q^{13} - 16 q^{16} + 4 q^{17} - 16 q^{25} - 20 q^{26} - 20 q^{29} + 16 q^{32} + 12 q^{34} - 20 q^{41} - 12 q^{45} + 28 q^{50} + 24 q^{52} - 28 q^{53} + 12 q^{58} + 4 q^{61} - 32 q^{65}+ \cdots + 36 q^{97}+O(q^{100}) \) 4 * q - 4 * q^2 + 8 * q^8 + 8 * q^13 - 16 * q^16 + 4 * q^17 - 16 * q^25 - 20 * q^26 - 20 * q^29 + 16 * q^32 + 12 * q^34 - 20 * q^41 - 12 * q^45 + 28 * q^50 + 24 * q^52 - 28 * q^53 + 12 * q^58 + 4 * q^61 - 32 * q^65 - 32 * q^68 + 36 * q^82 + 24 * q^85 + 32 * q^89 + 36 * q^90 + 36 * q^97

Introduction

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