LMFDB - Embedded newform 820.2.w.b.79.1

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			79.1
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	820.79
Dual form			820.2.w.b.519.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.00000 + 1.00000i) q^{2} +2.00000i q^{4} +(-2.12132 - 0.707107i) q^{5} +(-2.00000 + 2.00000i) q^{8} +(-2.12132 - 2.12132i) q^{9} +(-1.41421 - 2.82843i) q^{10} +(-2.70711 + 6.53553i) q^{13} -4.00000 q^{16} +(-3.12132 - 7.53553i) q^{17} -4.24264i q^{18} +(1.41421 - 4.24264i) q^{20} +(4.00000 + 3.00000i) q^{25} +(-9.24264 + 3.82843i) q^{26} +(-9.94975 - 4.12132i) q^{29} +(-4.00000 - 4.00000i) q^{32} +(4.41421 - 10.6569i) q^{34} +(4.24264 - 4.24264i) q^{36} +7.07107i q^{37} +(5.65685 - 2.82843i) q^{40} +(-5.00000 + 4.00000i) q^{41} +(3.00000 + 6.00000i) q^{45} +(-4.94975 - 4.94975i) q^{49} +(1.00000 + 7.00000i) q^{50} +(-13.0711 - 5.41421i) q^{52} +(13.3640 + 5.53553i) q^{53} +(-5.82843 - 14.0711i) q^{58} +(1.00000 - 1.00000i) q^{61} -8.00000i q^{64} +(10.3640 - 11.9497i) q^{65} +(15.0711 - 6.24264i) q^{68} +8.48528 q^{72} +(4.24264 + 4.24264i) q^{73} +(-7.07107 + 7.07107i) q^{74} +(8.48528 + 2.82843i) q^{80} +9.00000i q^{81} +(-9.00000 - 1.00000i) q^{82} +(1.29289 + 18.1924i) q^{85} +(10.1213 + 4.19239i) q^{89} +(-3.00000 + 9.00000i) q^{90} +(-12.5355 + 5.19239i) q^{97} -9.89949i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 4 q^{50} - 24 q^{52} + 28 q^{53} - 12 q^{58} + 4 q^{61} + 16 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 + 4 * q^50 - 24 * q^52 + 28 * q^53 - 12 * q^58 + 4 * q^61 + 16 * q^65 + 32 * q^68 - 36 * q^82 + 8 * q^85 + 32 * q^89 - 12 * q^90 - 36 * q^97

Character values

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

$n$	$411$	$621$	$657$
$\chi(n)$	$-1$	$e\left(\frac{7}{8}\right)$	$-1$

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		0.382683	−	0.923880i	$-0.375000\pi$
$3$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$4$			2.00000i			1.00000i
$5$	−2.12132	−	0.707107i	−0.948683	−	0.316228i
$5$	−2.12132	−	0.707107i	−0.948683	−	0.316228i
$6$		0			0
$7$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$7$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$8$	−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		0.382683	−	0.923880i	$-0.375000\pi$
$11$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$12$		0			0
$13$	−2.70711	+	6.53553i	−0.750816	+	1.81263i	−0.196116	+	0.980581i	$0.562833\pi$
$13$	−2.70711	+	6.53553i	−0.750816	+	1.81263i	−0.554700	+	0.832050i	$0.687167\pi$
$14$		0			0
$15$		0			0
$16$	−4.00000			−1.00000
$17$	−3.12132	−	7.53553i	−0.757031	−	1.82764i	−0.514496	−	0.857493i	$-0.672021\pi$
$17$	−3.12132	−	7.53553i	−0.757031	−	1.82764i	−0.242536	−	0.970143i	$-0.577979\pi$
$18$		−	4.24264i		−	1.00000i
$19$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$19$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$20$	1.41421	−	4.24264i	0.316228	−	0.948683i
$21$		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
$26$	−9.24264	+	3.82843i	−1.81263	+	0.750816i
$27$		0			0
$28$		0			0
$29$	−9.94975	−	4.12132i	−1.84762	−	0.765310i	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−9.94975	−	4.12132i	−1.84762	−	0.765310i	−0.919145	−	0.393919i	$-0.871119\pi$
$30$		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
$34$	4.41421	−	10.6569i	0.757031	−	1.82764i
$35$		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		−0.382683	−	0.923880i	$-0.625000\pi$
$47$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$48$		0			0
$49$	−4.94975	−	4.94975i	−0.707107	−	0.707107i
$50$	1.00000	+	7.00000i	0.141421	+	0.989949i
$51$		0			0
$52$	−13.0711	−	5.41421i	−1.81263	−	0.750816i
$53$	13.3640	+	5.53553i	1.83568	+	0.760364i	0.961524	+	0.274721i	$0.0885855\pi$
$53$	13.3640	+	5.53553i	1.83568	+	0.760364i	0.874157	+	0.485643i	$0.161414\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−5.82843	−	14.0711i	−0.765310	−	1.84762i
$59$		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
$65$	10.3640	−	11.9497i	1.28549	−	1.48218i
$66$		0			0
$67$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$67$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$68$	15.0711	−	6.24264i	1.82764	−	0.757031i
$69$		0			0
$70$		0			0
$71$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$71$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$72$	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		0.382683	−	0.923880i	$-0.375000\pi$
$79$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$80$	8.48528	+	2.82843i	0.948683	+	0.316228i
$81$			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
$85$	1.29289	+	18.1924i	0.140234	+	1.97324i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	10.1213	+	4.19239i	1.07286	+	0.444392i	0.847998	−	0.529999i	$-0.177808\pi$
$89$	10.1213	+	4.19239i	1.07286	+	0.444392i	0.224860	+	0.974391i	$0.427808\pi$
$90$	−3.00000	+	9.00000i	−0.316228	+	0.948683i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−12.5355	+	5.19239i	−1.27279	+	0.527207i	−0.913812	−	0.406138i	$-0.866875\pi$
$97$	−12.5355	+	5.19239i	−1.27279	+	0.527207i	−0.358979	+	0.933346i	$0.616875\pi$
$98$		−	9.89949i		−	1.00000i
$99$		0			0
$100$	−6.00000	+	8.00000i	−0.600000	+	0.800000i
$101$	−17.7782	+	7.36396i	−1.76899	+	0.732742i	−0.773957	+	0.633238i	$0.781726\pi$
$101$	−17.7782	+	7.36396i	−1.76899	+	0.732742i	−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	$0.750000\pi$
$104$	−7.65685	−	18.4853i	−0.750816	−	1.81263i
$105$		0			0
$106$	7.82843	+	18.8995i	0.760364	+	1.83568i
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$	5.05025	+	12.1924i	0.483726	+	1.16782i	0.957826	+	0.287348i	$0.0927736\pi$
$109$	5.05025	+	12.1924i	0.483726	+	1.16782i	−0.474100	+	0.880471i	$0.657226\pi$
$110$		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
$116$	8.24264	−	19.8995i	0.765310	−	1.84762i
$117$	19.6066	−	8.12132i	1.81263	−	0.750816i
$118$		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
$125$	−6.36396	−	9.19239i	−0.569210	−	0.822192i
$126$		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
$130$	22.3137	−	1.58579i	1.95704	−	0.139083i
$131$		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
$136$	21.3137	+	8.82843i	1.82764	+	0.757031i
$137$	0.393398	+	0.949747i	0.0336103	+	0.0811424i	0.939793	−	0.341743i	$-0.111017\pi$
$137$	0.393398	+	0.949747i	0.0336103	+	0.0811424i	−0.906183	+	0.422885i	$0.861017\pi$
$138$		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
$145$	18.1924	+	15.7782i	1.51080	+	1.31031i
$146$			8.48528i			0.702247i
$147$		0			0
$148$	−14.1421			−1.16248
$149$	19.0208	−	7.87868i	1.55825	−	0.645447i	0.573462	−	0.819232i	$-0.305600\pi$
$149$	19.0208	−	7.87868i	1.55825	−	0.645447i	0.984784	+	0.173785i	$0.0555999\pi$
$150$		0			0
$151$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$151$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$152$		0			0
$153$	−9.36396	+	22.6066i	−0.757031	+	1.82764i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−18.0208	−	7.46447i	−1.43822	−	0.595729i	−0.478852	−	0.877896i	$-0.658947\pi$
$157$	−18.0208	−	7.46447i	−1.43822	−	0.595729i	−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		0.382683	−	0.923880i	$-0.375000\pi$
$167$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$168$		0			0
$169$	−26.1924	−	26.1924i	−2.01480	−	2.01480i
$170$	−16.8995	+	19.4853i	−1.29613	+	1.49445i
$171$		0			0
$172$		0			0
$173$	−11.0000	+	11.0000i	−0.836315	+	0.836315i	−0.988372	−	0.152057i	$-0.951410\pi$
$173$	−11.0000	+	11.0000i	−0.836315	+	0.836315i	0.152057	+	0.988372i	$0.451410\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	5.92893	+	14.3137i	0.444392	+	1.07286i
$179$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$179$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$180$	−12.0000	+	6.00000i	−0.894427	+	0.447214i
$181$	22.4350	−	9.29289i	1.66758	−	0.690735i	0.668965	−	0.743294i	$-0.266738\pi$
$181$	22.4350	−	9.29289i	1.66758	−	0.690735i	0.998618	+	0.0525588i	$0.0167377\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	5.00000	−	15.0000i	0.367607	−	1.10282i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$191$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$192$		0			0
$193$	−8.46447	+	20.4350i	−0.609286	+	1.47095i	0.254493	+	0.967075i	$0.418091\pi$
$193$	−8.46447	+	20.4350i	−0.609286	+	1.47095i	−0.863779	+	0.503871i	$0.831909\pi$
$194$	−17.7279	−	7.34315i	−1.27279	−	0.527207i
$195$		0			0
$196$	9.89949	−	9.89949i	0.707107	−	0.707107i
$197$	15.0000	+	15.0000i	1.06871	+	1.06871i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$	15.0000	+	15.0000i	1.06871	+	1.06871i	0.0712470	+	0.997459i	$0.477302\pi$
$198$		0			0
$199$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$199$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$200$	−14.0000	+	2.00000i	−0.989949	+	0.141421i
$201$		0			0
$202$	−25.1421	−	10.4142i	−1.76899	−	0.732742i
$203$		0			0
$204$		0			0
$205$	13.4350	−	4.94975i	0.938343	−	0.345705i
$206$		0			0
$207$		0			0
$208$	10.8284	−	26.1421i	0.750816	−	1.81263i
$209$		0			0
$210$		0			0
$211$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$211$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$212$	−11.0711	+	26.7279i	−0.760364	+	1.83568i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−7.14214	+	17.2426i	−0.483726	+	1.16782i
$219$		0			0
$220$		0			0
$221$	57.6985			3.88122
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$	−2.12132	−	14.8492i	−0.141421	−	0.989949i
$226$	1.41421	+	1.41421i	0.0940721	+	0.0940721i
$227$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$227$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$228$		0			0
$229$	−10.0208	−	24.1924i	−0.662194	−	1.59868i	−0.794358	−	0.607450i	$-0.792192\pi$
$229$	−10.0208	−	24.1924i	−0.662194	−	1.59868i	0.132164	−	0.991228i	$-0.457808\pi$
$230$		0			0
$231$		0			0
$232$	28.1421	−	11.6569i	1.84762	−	0.765310i
$233$	−16.5355	−	6.84924i	−1.08328	−	0.448709i	−0.231621	−	0.972806i	$-0.574403\pi$
$233$	−16.5355	−	6.84924i	−1.08328	−	0.448709i	−0.851658	+	0.524097i	$0.824403\pi$
$234$	27.7279	+	11.4853i	1.81263	+	0.750816i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$239$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$240$		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
$257$	−28.0208	−	11.6066i	−1.74789	−	0.724000i	−0.998053	−	0.0623783i	$-0.980131\pi$
$257$	−28.0208	−	11.6066i	−1.74789	−	0.724000i	−0.749838	−	0.661622i	$-0.769869\pi$
$258$		0			0
$259$		0			0
$260$	23.8995	+	20.7279i	1.48218	+	1.28549i
$261$	12.3640	+	29.8492i	0.765310	+	1.84762i
$262$		0			0
$263$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$263$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$264$		0			0
$265$	−24.4350	−	21.1924i	−1.50103	−	1.30184i
$266$		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	$0.5\pi$
$272$	12.4853	+	30.1421i	0.757031	+	1.82764i
$273$		0			0
$274$	−0.556349	+	1.34315i	−0.0336103	+	0.0811424i
$275$		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
$281$	−9.84924	−	23.7782i	−0.587557	−	1.41849i	−0.885832	−	0.464007i	$-0.846411\pi$
$281$	−9.84924	−	23.7782i	−0.587557	−	1.41849i	0.298275	−	0.954480i	$-0.403589\pi$
$282$		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
$289$	−35.0208	+	35.0208i	−2.06005	+	2.06005i
$290$	2.41421	+	33.9706i	0.141768	+	1.99482i
$291$		0			0
$292$	−8.48528	+	8.48528i	−0.496564	+	0.496564i
$293$	−3.56497	−	8.60660i	−0.208268	−	0.502803i	0.784883	−	0.619644i	$-0.212723\pi$
$293$	−3.56497	−	8.60660i	−0.208268	−	0.502803i	−0.993151	+	0.116841i	$0.962723\pi$
$294$		0			0
$295$		0			0
$296$	−14.1421	−	14.1421i	−0.821995	−	0.821995i
$297$		0			0
$298$	26.8995	+	11.1421i	1.55825	+	0.645447i
$299$		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
$306$	−31.9706	+	13.2426i	−1.82764	+	0.757031i
$307$		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		−0.382683	−	0.923880i	$-0.625000\pi$
$311$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$312$		0			0
$313$	5.67767	+	13.7071i	0.320921	+	0.774771i	0.999201	+	0.0399680i	$0.0127256\pi$
$313$	5.67767	+	13.7071i	0.320921	+	0.774771i	−0.678280	+	0.734803i	$0.737274\pi$
$314$	−10.5563	−	25.4853i	−0.595729	−	1.43822i
$315$		0			0
$316$		0			0
$317$	−3.67767	+	8.87868i	−0.206558	+	0.498676i	−0.992877	−	0.119145i	$-0.961985\pi$
$317$	−3.67767	+	8.87868i	−0.206558	+	0.498676i	0.786318	+	0.617822i	$0.211985\pi$
$318$		0			0
$319$		0			0
$320$	−5.65685	+	16.9706i	−0.316228	+	0.948683i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−18.0000			−1.00000
$325$	−30.4350	+	18.0208i	−1.68823	+	0.999615i
$326$		0			0
$327$		0			0
$328$	2.00000	−	18.0000i	0.110432	−	0.993884i
$329$		0			0
$330$		0			0
$331$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$331$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$332$		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	$-0.0869014\pi$
$338$		−	52.3848i		−	2.84936i
$339$		0			0
$340$	−36.3848	+	2.58579i	−1.97324	+	0.140234i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−22.0000			−1.18273
$347$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$347$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$348$		0			0
$349$	7.07107	+	7.07107i	0.378506	+	0.378506i	0.870563	−	0.492057i	$-0.163755\pi$
$349$	7.07107	+	7.07107i	0.378506	+	0.378506i	−0.492057	+	0.870563i	$0.663755\pi$
$350$		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
$356$	−8.38478	+	20.2426i	−0.444392	+	1.07286i
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$	−18.0000	−	6.00000i	−0.948683	−	0.316228i
$361$	13.4350	−	13.4350i	0.707107	−	0.707107i
$362$	31.7279	+	13.1421i	1.66758	+	0.690735i
$363$		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	$-0.881942\pi$
$373$	−36.0000			−1.86401			−0.932005	+	0.362446i	$0.881942\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	53.8701	−	53.8701i	2.77445	−	2.77445i
$378$		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		−0.382683	−	0.923880i	$-0.625000\pi$
$383$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$384$		0			0
$385$		0			0
$386$	−28.8995	+	11.9706i	−1.47095	+	0.609286i
$387$		0			0
$388$	−10.3848	−	25.0711i	−0.527207	−	1.27279i
$389$	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
$397$	3.19239	+	1.32233i	0.160221	+	0.0663659i	0.461353	−	0.887217i	$-0.347364\pi$
$397$	3.19239	+	1.32233i	0.160221	+	0.0663659i	−0.301131	+	0.953583i	$0.597364\pi$
$398$		0			0
$399$		0			0
$400$	−16.0000	−	12.0000i	−0.800000	−	0.600000i
$401$	1.41421	−	1.41421i	0.0706225	−	0.0706225i	−0.670913	−	0.741536i	$-0.734098\pi$
$401$	1.41421	−	1.41421i	0.0706225	−	0.0706225i	0.741536	+	0.670913i	$0.234098\pi$
$402$		0			0
$403$		0			0
$404$	−14.7279	−	35.5563i	−0.732742	−	1.76899i
$405$	6.36396	−	19.0919i	0.316228	−	0.948683i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	24.0416i		−	1.18878i	−0.804176	−	0.594391i	$-0.797393\pi$
$409$		−	24.0416i		−	1.18878i	0.804176	−	0.594391i	$-0.202607\pi$
$410$	18.3848	+	8.48528i	0.907959	+	0.419058i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	36.9706	−	15.3137i	1.81263	−	0.750816i
$417$		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
$421$	5.50610	−	13.2929i	0.268351	−	0.647856i	−0.731055	−	0.682318i	$-0.760972\pi$
$421$	5.50610	−	13.2929i	0.268351	−	0.647856i	0.999406	+	0.0344623i	$0.0109719\pi$
$422$		0			0
$423$		0			0
$424$	−37.7990	+	15.6569i	−1.83568	+	0.760364i
$425$	10.1213	−	39.5061i	0.490956	−	1.91633i
$426$		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	$-0.695653\pi$
$433$	−24.0000			−1.15337			−0.576683	+	0.816968i	$0.695653\pi$
$434$		0			0
$435$		0			0
$436$	−24.3848	+	10.1005i	−1.16782	+	0.483726i
$437$		0			0
$438$		0			0
$439$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$439$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$440$		0			0
$441$			21.0000i			1.00000i
$442$	57.6985	+	57.6985i	2.74444	+	2.74444i
$443$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$443$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$444$		0			0
$445$	−18.5061	−	16.0503i	−0.877273	−	0.760855i
$446$		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
$457$	−8.97918	−	21.6777i	−0.420029	−	1.01404i	−0.982339	−	0.187112i	$-0.940087\pi$
$457$	−8.97918	−	21.6777i	−0.420029	−	1.01404i	0.562310	−	0.826927i	$-0.309913\pi$
$458$	14.1716	−	34.2132i	0.662194	−	1.59868i
$459$		0			0
$460$		0			0
$461$	−41.0122			−1.91013			−0.955064	−	0.296399i	$-0.904214\pi$
$461$	−41.0122			−1.91013			−0.955064	+	0.296399i	$0.904214\pi$
$462$		0			0
$463$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$463$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$464$	39.7990	+	16.4853i	1.84762	+	0.765310i
$465$		0			0
$466$	−9.68629	−	23.3848i	−0.448709	−	1.08328i
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$	16.2426	+	39.2132i	0.750816	+	1.81263i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−16.6066	−	40.0919i	−0.760364	−	1.83568i
$478$		0			0
$479$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$479$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$480$		0			0
$481$	−46.2132	−	19.1421i	−2.10714	−	0.872806i
$482$	−22.0000			−1.00207
$483$		0			0
$484$	15.5563	−	15.5563i	0.707107	−	0.707107i
$485$	30.2635	−	2.15076i	1.37419	−	0.0976609i
$486$		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
$490$	−7.00000	+	21.0000i	−0.316228	+	0.948683i
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$			87.8406i			3.95614i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$499$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$500$	18.3848	−	12.7279i	0.822192	−	0.569210i
$501$		0			0
$502$		0			0
$503$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$503$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$504$		0			0
$505$	42.9203	−	3.05025i	1.90993	−	0.135734i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−41.0919	+	17.0208i	−1.82137	+	0.754434i	−0.846233	+	0.532813i	$0.821135\pi$
$509$	−41.0919	+	17.0208i	−1.82137	+	0.754434i	−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
$514$	−16.4142	−	39.6274i	−0.724000	−	1.74789i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	3.17157	+	44.6274i	0.139083	+	1.95704i
$521$	−41.9203	−	17.3640i	−1.83656	−	0.760729i	−0.960346	−	0.278810i	$-0.910060\pi$
$521$	−41.9203	−	17.3640i	−1.83656	−	0.760729i	−0.876216	−	0.481919i	$-0.839940\pi$
$522$	−17.4853	+	42.2132i	−0.765310	+	1.84762i
$523$		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
$530$	−3.24264	−	45.6274i	−0.140851	−	1.98193i
$531$		0			0
$532$		0			0
$533$	−12.6066	−	43.5061i	−0.546053	−	1.88446i
$534$		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	$0.108537\pi$
$541$	29.6985	+	29.6985i	1.27684	+	1.27684i	0.334410	+	0.942428i	$0.391463\pi$
$542$		0			0
$543$		0			0
$544$	−17.6569	+	42.6274i	−0.757031	+	1.82764i
$545$	−2.09188	−	29.4350i	−0.0896064	−	1.26086i
$546$		0			0
$547$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$547$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$548$	−1.89949	+	0.786797i	−0.0811424	+	0.0336103i
$549$	−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
$557$	−42.3345	+	17.5355i	−1.79377	+	0.743004i	−0.805056	+	0.593199i	$0.797865\pi$
$557$	−42.3345	+	17.5355i	−1.79377	+	0.743004i	−0.988716	+	0.149805i	$0.952135\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	13.9289	−	33.6274i	0.587557	−	1.41849i
$563$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$563$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$564$		0			0
$565$	−3.00000	−	1.00000i	−0.126211	−	0.0420703i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	28.2843	−	28.2843i	1.18574	−	1.18574i	0.207504	−	0.978234i	$-0.433466\pi$
$569$	28.2843	−	28.2843i	1.18574	−	1.18574i	0.978234	−	0.207504i	$-0.0665341\pi$
$570$		0			0
$571$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$571$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−16.9706	+	16.9706i	−0.707107	+	0.707107i
$577$	−17.2635	+	41.6777i	−0.718687	+	1.73506i	−0.0416305	+	0.999133i	$0.513255\pi$
$577$	−17.2635	+	41.6777i	−0.718687	+	1.73506i	−0.677057	+	0.735931i	$0.736745\pi$
$578$	−70.0416			−2.91335
$579$		0			0
$580$	−31.5563	+	36.3848i	−1.31031	+	1.51080i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−16.9706			−0.702247
$585$	−47.3345	+	3.36396i	−1.95704	+	0.139083i
$586$	5.04163	−	12.1716i	0.208268	−	0.502803i
$587$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$587$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		−	28.2843i		−	1.16248i
$593$	−18.6066	−	44.9203i	−0.764082	−	1.84466i	−0.435561	−	0.900159i	$-0.643450\pi$
$593$	−18.6066	−	44.9203i	−0.764082	−	1.84466i	−0.328521	−	0.944497i	$-0.606550\pi$
$594$		0			0
$595$		0			0
$596$	15.7574	+	38.0416i	0.645447	+	1.55825i
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−18.4350	−	44.5061i	−0.751981	−	1.81544i	−0.548026	−	0.836461i	$-0.684621\pi$
$601$	−18.4350	−	44.5061i	−0.751981	−	1.81544i	−0.203954	−	0.978980i	$-0.565379\pi$
$602$		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
$610$	−4.24264	−	1.41421i	−0.171780	−	0.0572598i
$611$		0			0
$612$	−45.2132	−	18.7279i	−1.82764	−	0.757031i
$613$	−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	$0.222783\pi$
$617$	35.0000	+	35.0000i	1.40905	+	1.40905i	0.644136	+	0.764911i	$0.277217\pi$
$618$		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
$626$	−8.02944	+	19.3848i	−0.320921	+	0.774771i
$627$		0			0
$628$	14.9289	−	36.0416i	0.595729	−	1.43822i
$629$	53.2843	−	22.0711i	2.12458	−	0.880031i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$	−12.5563	+	5.20101i	−0.498676	+	0.206558i
$635$		0			0
$636$		0			0
$637$	45.7487	−	18.9497i	1.81263	−	0.750816i
$638$		0			0
$639$		0			0
$640$	−22.6274	+	11.3137i	−0.894427	+	0.447214i
$641$	−24.5061	+	10.1508i	−0.967933	+	0.400931i	−0.809942	−	0.586510i	$-0.800502\pi$
$641$	−24.5061	+	10.1508i	−0.967933	+	0.400931i	−0.157991	+	0.987441i	$0.550502\pi$
$642$		0			0
$643$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$643$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$644$		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
$650$	−48.4558	−	12.4142i	−1.90059	−	0.486926i
$651$		0			0
$652$		0			0
$653$	46.7487	+	19.3640i	1.82942	+	0.757770i	0.968493	+	0.249041i	$0.0801154\pi$
$653$	46.7487	+	19.3640i	1.82942	+	0.757770i	0.860927	+	0.508729i	$0.169885\pi$
$654$		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		−0.923880	−	0.382683i	$-0.875000\pi$
$659$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$660$		0			0
$661$	35.3553	+	35.3553i	1.37516	+	1.37516i	0.852601	+	0.522562i	$0.175024\pi$
$661$	35.3553	+	35.3553i	1.37516	+	1.37516i	0.522562	+	0.852601i	$0.324976\pi$
$662$		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
$673$	12.7487	+	30.7782i	0.491428	+	1.18641i	0.953994	+	0.299827i	$0.0969288\pi$
$673$	12.7487	+	30.7782i	0.491428	+	1.18641i	−0.462566	+	0.886585i	$0.653071\pi$
$674$	45.2548			1.74315
$675$		0			0
$676$	52.3848	−	52.3848i	2.01480	−	2.01480i
$677$	−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
$680$	−38.9706	−	33.7990i	−1.49445	−	1.29613i
$681$		0			0
$682$		0			0
$683$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$683$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$684$		0			0
$685$	−0.162951	−	2.29289i	−0.00622603	−	0.0876069i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−72.3553	+	72.3553i	−2.75652	+	2.75652i
$690$		0			0
$691$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$691$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$692$	−22.0000	−	22.0000i	−0.836315	−	0.836315i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	45.7487	+	25.1924i	1.73286	+	0.954230i
$698$			14.1421i			0.535288i
$699$		0			0
$700$		0			0
$701$			43.8406i			1.65584i	0.560848	+	0.827919i	$0.310475\pi$
$701$			43.8406i			1.65584i	−0.560848	+	0.827919i	$0.689525\pi$
$702$		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
$709$	−17.0503	+	41.1630i	−0.640336	+	1.54591i	0.185892	+	0.982570i	$0.440483\pi$
$709$	−17.0503	+	41.1630i	−0.640336	+	1.54591i	−0.826227	+	0.563337i	$0.809517\pi$
$710$		0			0
$711$		0			0
$712$	−28.6274	+	11.8579i	−1.07286	+	0.444392i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$719$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$720$	−12.0000	−	24.0000i	−0.447214	−	0.894427i
$721$		0			0
$722$	26.8701			1.00000
$723$		0			0
$724$	18.5858	+	44.8701i	0.690735	+	1.66758i
$725$	−27.4350	−	46.3345i	−1.01891	−	1.72082i
$726$		0			0
$727$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$727$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$728$		0			0
$729$	19.0919	−	19.0919i	0.707107	−	0.707107i
$730$	6.00000	−	18.0000i	0.222070	−	0.666210i
$731$		0			0
$732$		0			0
$733$	29.0000	+	29.0000i	1.07114	+	1.07114i	0.997268	+	0.0738717i	$0.0235355\pi$
$733$	29.0000	+	29.0000i	1.07114	+	1.07114i	0.0738717	+	0.997268i	$0.476464\pi$
$734$		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			$\pi$
$740$	30.0000	+	10.0000i	1.10282	+	0.367607i
$741$		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
$745$	−45.9203	+	3.26346i	−1.68239	+	0.119564i
$746$	−36.0000	−	36.0000i	−1.31805	−	1.31805i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$751$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$752$		0			0
$753$		0			0
$754$	107.740			3.92366
$755$		0			0
$756$		0			0
$757$	13.9792	+	33.7487i	0.508082	+	1.22662i	0.944986	+	0.327111i	$0.106075\pi$
$757$	13.9792	+	33.7487i	0.508082	+	1.22662i	−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	$-0.991840\pi$
$761$		−	1.41421i		−	0.0512652i	0.999671	−	0.0256326i	$-0.00816000\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	35.8492	−	41.3345i	1.29613	−	1.49445i
$766$		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
$772$	−40.8701	−	16.9289i	−1.47095	−	0.609286i
$773$	−13.4645	+	5.57716i	−0.484283	+	0.200597i	−0.611448	−	0.791285i	$-0.709412\pi$
$773$	−13.4645	+	5.57716i	−0.484283	+	0.200597i	0.127164	+	0.991882i	$0.459412\pi$
$774$		0			0
$775$		0			0
$776$	14.6863	−	35.4558i	0.527207	−	1.27279i
$777$		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
$785$	32.9497	+	28.5772i	1.17603	+	1.01996i
$786$		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
$793$	3.82843	+	9.24264i	0.135951	+	0.328216i
$794$	1.87006	+	4.51472i	0.0663659	+	0.160221i
$795$		0			0
$796$		0			0
$797$		−	52.0000i		−	1.84193i	−0.389640	−	0.920967i	$-0.627401\pi$
$797$		−	52.0000i		−	1.84193i	0.389640	−	0.920967i	$-0.372599\pi$
$798$		0			0
$799$		0			0
$800$	−4.00000	−	28.0000i	−0.141421	−	0.989949i
$801$	−12.5772	−	30.3640i	−0.444392	−	1.07286i
$802$	2.82843			0.0998752
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	20.8284	−	50.2843i	0.732742	−	1.76899i
$809$	11.2635	+	4.66548i	0.396002	+	0.164029i	0.571793	−	0.820398i	$-0.306248\pi$
$809$	11.2635	+	4.66548i	0.396002	+	0.164029i	−0.175791	+	0.984428i	$0.556248\pi$
$810$	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
$820$	9.89949	+	26.8701i	0.345705	+	0.938343i
$821$	15.5563			0.542920			0.271460	−	0.962450i	$-0.412493\pi$
$821$	15.5563			0.542920			0.271460	+	0.962450i	$0.412493\pi$
$822$		0			0
$823$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$823$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$827$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$828$		0			0
$829$	14.1421	+	14.1421i	0.491177	+	0.491177i	0.908677	−	0.417500i	$-0.137094\pi$
$829$	14.1421	+	14.1421i	0.491177	+	0.491177i	−0.417500	+	0.908677i	$0.637094\pi$
$830$		0			0
$831$		0			0
$832$	52.2843	+	21.6569i	1.81263	+	0.750816i
$833$	−21.8492	+	52.7487i	−0.757031	+	1.82764i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$839$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$840$		0			0
$841$	61.5061	+	61.5061i	2.12090	+	2.12090i
$842$	18.7990	−	7.78680i	0.647856	−	0.268351i
$843$		0			0
$844$		0			0
$845$	37.0416	+	74.0833i	1.27427	+	2.54854i
$846$		0			0
$847$		0			0
$848$	−53.4558	−	22.1421i	−1.83568	−	0.760364i
$849$		0			0
$850$	49.6274	−	29.3848i	1.70221	−	1.00789i
$851$		0			0
$852$		0			0
$853$	−5.00000	−	5.00000i	−0.171197	−	0.171197i	0.616308	−	0.787505i	$-0.288628\pi$
$853$	−5.00000	−	5.00000i	−0.171197	−	0.171197i	−0.787505	+	0.616308i	$0.788628\pi$
$854$		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
$872$	−34.4853	−	14.2843i	−1.16782	−	0.483726i
$873$	37.6066	+	15.5772i	1.27279	+	0.527207i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	12.0000i		−	0.405211i	−0.979260	−	0.202606i	$-0.935059\pi$
$877$		−	12.0000i		−	0.405211i	0.979260	−	0.202606i	$-0.0649409\pi$
$878$		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		0.923880	−	0.382683i	$-0.125000\pi$
$883$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$884$			115.397i			3.88122i
$885$		0			0
$886$		0			0
$887$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$887$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$888$		0			0
$889$		0			0
$890$	−2.45584	−	34.5563i	−0.0823201	−	1.15833i
$891$		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
$900$	29.6985	−	4.24264i	0.989949	−	0.141421i
$901$		−	117.983i		−	3.93058i
$902$		0			0
$903$		0			0
$904$	−2.82843	+	2.82843i	−0.0940721	+	0.0940721i
$905$	−54.1630	+	3.84924i	−1.80044	+	0.127953i
$906$		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
$909$	53.3345	+	22.0919i	1.76899	+	0.732742i
$910$		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
$914$	12.6985	−	30.6569i	0.420029	−	1.01404i
$915$		0			0
$916$	48.3848	−	20.0416i	1.59868	−	0.662194i
$917$		0			0
$918$		0			0
$919$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$919$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$920$		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
$928$	23.3137	+	56.2843i	0.765310	+	1.84762i
$929$	−10.4056	−	25.1213i	−0.341396	−	0.824204i	−0.997575	−	0.0695983i	$-0.977828\pi$
$929$	−10.4056	−	25.1213i	−0.341396	−	0.824204i	0.656179	−	0.754606i	$-0.272172\pi$
$930$		0			0
$931$		0			0
$932$	13.6985	−	33.0711i	0.448709	−	1.08328i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−22.9706	+	55.4558i	−0.750816	+	1.81263i
$937$	−27.5355	+	11.4056i	−0.899547	+	0.372604i	−0.784046	−	0.620703i	$-0.786847\pi$
$937$	−27.5355	+	11.4056i	−0.899547	+	0.372604i	−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
$949$	−39.2132	+	16.2426i	−1.27291	+	0.527258i
$950$		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	$-0.111640\pi$
$954$	23.4853	−	56.6985i	0.760364	−	1.83568i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−31.0000			−1.00000
$962$	−27.0711	−	65.3553i	−0.872806	−	2.10714i
$963$		0			0
$964$	−22.0000	−	22.0000i	−0.708572	−	0.708572i
$965$	32.4056	−	37.3640i	1.04317	−	1.20279i
$966$		0			0
$967$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$967$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$968$	31.1127			1.00000
$969$		0			0
$970$	32.4142	+	28.1127i	1.04076	+	0.902644i
$971$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$971$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−4.00000	+	4.00000i	−0.128037	+	0.128037i
$977$	−28.7487	−	11.9081i	−0.919754	−	0.380974i	−0.127971	−	0.991778i	$-0.540847\pi$
$977$	−28.7487	−	11.9081i	−0.919754	−	0.380974i	−0.791782	+	0.610803i	$0.790847\pi$
$978$		0			0
$979$		0			0
$980$	−28.0000	+	14.0000i	−0.894427	+	0.447214i
$981$	15.1508	−	36.5772i	0.483726	−	1.16782i
$982$		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
$986$	−87.8406	+	87.8406i	−2.79742	+	2.79742i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$991$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−13.3223	+	32.1630i	−0.421922	+	1.01861i	0.559857	+	0.828589i	$0.310856\pi$
$997$	−13.3223	+	32.1630i	−0.421922	+	1.01861i	−0.981780	+	0.190022i	$0.939144\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	820.2.w.b.79.1	yes	4
4.3	odd	2	CM	820.2.w.b.79.1	yes	4
5.4	even	2		820.2.w.a.79.1	✓	4
20.19	odd	2		820.2.w.a.79.1	✓	4
41.27	odd	8		820.2.w.a.519.1	yes	4
164.27	even	8		820.2.w.a.519.1	yes	4
205.109	odd	8	inner	820.2.w.b.519.1	yes	4
820.519	even	8	inner	820.2.w.b.519.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
820.2.w.a.79.1	✓	4	5.4	even	2
820.2.w.a.79.1	✓	4	20.19	odd	2
820.2.w.a.519.1	yes	4	41.27	odd	8
820.2.w.a.519.1	yes	4	164.27	even	8
820.2.w.b.79.1	yes	4	1.1	even	1	trivial
820.2.w.b.79.1	yes	4	4.3	odd	2	CM
820.2.w.b.519.1	yes	4	205.109	odd	8	inner
820.2.w.b.519.1	yes	4	820.519	even	8	inner

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)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.00000i) q^{2} +2.00000i q^{4} +(-2.12132 - 0.707107i) q^{5} +(-2.00000 + 2.00000i) q^{8} +(-2.12132 - 2.12132i) q^{9} +(-1.41421 - 2.82843i) q^{10} +(-2.70711 + 6.53553i) q^{13} -4.00000 q^{16} +(-3.12132 - 7.53553i) q^{17} -4.24264i q^{18} +(1.41421 - 4.24264i) q^{20} +(4.00000 + 3.00000i) q^{25} +(-9.24264 + 3.82843i) q^{26} +(-9.94975 - 4.12132i) q^{29} +(-4.00000 - 4.00000i) q^{32} +(4.41421 - 10.6569i) q^{34} +(4.24264 - 4.24264i) q^{36} +7.07107i q^{37} +(5.65685 - 2.82843i) q^{40} +(-5.00000 + 4.00000i) q^{41} +(3.00000 + 6.00000i) q^{45} +(-4.94975 - 4.94975i) q^{49} +(1.00000 + 7.00000i) q^{50} +(-13.0711 - 5.41421i) q^{52} +(13.3640 + 5.53553i) q^{53} +(-5.82843 - 14.0711i) q^{58} +(1.00000 - 1.00000i) q^{61} -8.00000i q^{64} +(10.3640 - 11.9497i) q^{65} +(15.0711 - 6.24264i) q^{68} +8.48528 q^{72} +(4.24264 + 4.24264i) q^{73} +(-7.07107 + 7.07107i) q^{74} +(8.48528 + 2.82843i) q^{80} +9.00000i q^{81} +(-9.00000 - 1.00000i) q^{82} +(1.29289 + 18.1924i) q^{85} +(10.1213 + 4.19239i) q^{89} +(-3.00000 + 9.00000i) q^{90} +(-12.5355 + 5.19239i) q^{97} -9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 8 q^{8} - 8 q^{13} - 16 q^{16} - 4 q^{17} + 16 q^{25} - 20 q^{26} - 20 q^{29} - 16 q^{32} + 12 q^{34} - 20 q^{41} + 12 q^{45} + 4 q^{50} - 24 q^{52} + 28 q^{53} - 12 q^{58} + 4 q^{61} + 16 q^{65}+ \cdots - 36 q^{97}+O(q^{100}) \) 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 + 4 * q^50 - 24 * q^52 + 28 * q^53 - 12 * q^58 + 4 * q^61 + 16 * q^65 + 32 * q^68 - 36 * q^82 + 8 * q^85 + 32 * q^89 - 12 * q^90 - 36 * q^97

Introduction

L-functions

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Varieties

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Groups

Database

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