LMFDB - Embedded newform 799.1.h.b.610.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [799,1,Mod(93,799)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("799.93"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(799, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([5, 4])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 799 = 17 \cdot 47 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	799.h (of order $8$, degree $4$, minimal)

Newform invariants

comment:select newform

sage:traces = [16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$0.398752945094$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{8})$
Coefficient field:	$\Q(\zeta_{40})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{12} + x^{8} - x^{4} + 1 $ x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{40}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{40} + \cdots)$

Embedding invariants

Embedding label			610.4
Root			$0.891007 - 0.453990i$ of defining polynomial
Character	$\chi$	$=$	799.610
Dual form			799.1.h.b.93.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.34500 + 1.34500i) q^{2} +(0.178671 + 0.431351i) q^{3} +2.61803i q^{4} +(-0.339853 + 0.820478i) q^{6} +(-1.40505 - 0.581990i) q^{7} +(-2.17625 + 2.17625i) q^{8} +(0.552967 - 0.552967i) q^{9} +(-1.12929 + 0.467768i) q^{12} +(-1.10701 - 2.67256i) q^{14} -3.23607 q^{16} +(0.809017 + 0.587785i) q^{17} +1.48748 q^{18} -0.710053i q^{21} +(-1.32756 - 0.549894i) q^{24} +(0.707107 - 0.707107i) q^{25} +(0.768673 + 0.318395i) q^{27} +(1.52367 - 3.67846i) q^{28} +(-2.17625 - 2.17625i) q^{32} +(0.297556 + 1.87869i) q^{34} +(1.44769 + 1.44769i) q^{36} +(-0.744220 - 1.79671i) q^{37} +(0.955019 - 0.955019i) q^{42} +1.00000i q^{47} +(-0.578193 - 1.39588i) q^{48} +(0.928339 + 0.928339i) q^{49} +1.90211 q^{50} +(-0.108993 + 0.453990i) q^{51} +(-1.26007 - 1.26007i) q^{53} +(0.605623 + 1.46210i) q^{54} +(4.32429 - 1.79118i) q^{56} +(-0.437016 + 0.437016i) q^{59} +(0.144974 + 0.0600500i) q^{61} +(-1.09877 + 0.455123i) q^{63} -2.61803i q^{64} +(-1.53884 + 2.11803i) q^{68} +(-0.652583 - 1.57547i) q^{71} +2.40679i q^{72} +(1.41559 - 3.41754i) q^{74} +(0.431351 + 0.178671i) q^{75} +(-0.652583 + 1.57547i) q^{79} -0.393557i q^{81} +(-1.00000 - 1.00000i) q^{83} +1.85894 q^{84} +1.90211i q^{89} +(-1.34500 + 1.34500i) q^{94} +(0.549894 - 1.32756i) q^{96} +(-1.84206 + 0.763007i) q^{97} +2.49723i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{3} - 4 q^{9} - 16 q^{16} + 4 q^{17} - 20 q^{24} - 4 q^{27} - 4 q^{28} + 4 q^{36} - 20 q^{42} + 24 q^{48} + 4 q^{49} - 16 q^{51} + 4 q^{53} + 20 q^{54} + 20 q^{56} + 4 q^{61} - 4 q^{63} - 4 q^{71}+ \cdots + 32 q^{84}+O(q^{100}) $ 16 * q - 4 * q^3 - 4 * q^9 - 16 * q^16 + 4 * q^17 - 20 * q^24 - 4 * q^27 - 4 * q^28 + 4 * q^36 - 20 * q^42 + 24 * q^48 + 4 * q^49 - 16 * q^51 + 4 * q^53 + 20 * q^54 + 20 * q^56 + 4 * q^61 - 4 * q^63 - 4 * q^71 - 4 * q^79 - 16 * q^83 + 32 * q^84

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/799\mathbb{Z}\right)^\times$.

$n$	$52$	$377$
$\chi(n)$	$-1$	$e\left(\frac{3}{8}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	1.34500	+	1.34500i	1.34500	+	1.34500i	0.891007	+	0.453990i	$0.150000\pi$
$2$	1.34500	+	1.34500i	1.34500	+	1.34500i	0.453990	+	0.891007i	$0.350000\pi$
$3$	0.178671	+	0.431351i	0.178671	+	0.431351i	0.987688	−	0.156434i	$-0.0500000\pi$
$3$	0.178671	+	0.431351i	0.178671	+	0.431351i	−0.809017	+	0.587785i	$0.800000\pi$
$4$			2.61803i			2.61803i
$5$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$5$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$6$	−0.339853	+	0.820478i	−0.339853	+	0.820478i
$7$	−1.40505	−	0.581990i	−1.40505	−	0.581990i	−0.453990	−	0.891007i	$-0.650000\pi$
$7$	−1.40505	−	0.581990i	−1.40505	−	0.581990i	−0.951057	+	0.309017i	$0.900000\pi$
$8$	−2.17625	+	2.17625i	−2.17625	+	2.17625i
$9$	0.552967	−	0.552967i	0.552967	−	0.552967i
$10$		0			0
$11$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$11$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$12$	−1.12929	+	0.467768i	−1.12929	+	0.467768i
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$	−1.10701	−	2.67256i	−1.10701	−	2.67256i
$15$		0			0
$16$	−3.23607			−3.23607
$17$	0.809017	+	0.587785i	0.809017	+	0.587785i
$17$	0.809017	+	0.587785i	0.809017	+	0.587785i
$18$	1.48748			1.48748
$19$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$19$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$20$		0			0
$21$		−	0.710053i		−	0.710053i
$22$		0			0
$23$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$23$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$24$	−1.32756	−	0.549894i	−1.32756	−	0.549894i
$25$	0.707107	−	0.707107i	0.707107	−	0.707107i
$26$		0			0
$27$	0.768673	+	0.318395i	0.768673	+	0.318395i
$28$	1.52367	−	3.67846i	1.52367	−	3.67846i
$29$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$29$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$30$		0			0
$31$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$31$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$32$	−2.17625	−	2.17625i	−2.17625	−	2.17625i
$33$		0			0
$34$	0.297556	+	1.87869i	0.297556	+	1.87869i
$35$		0			0
$36$	1.44769	+	1.44769i	1.44769	+	1.44769i
$37$	−0.744220	−	1.79671i	−0.744220	−	1.79671i	−0.587785	−	0.809017i	$-0.700000\pi$
$37$	−0.744220	−	1.79671i	−0.744220	−	1.79671i	−0.156434	−	0.987688i	$-0.550000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$41$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$42$	0.955019	−	0.955019i	0.955019	−	0.955019i
$43$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$43$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			1.00000i			1.00000i
$47$			1.00000i			1.00000i
$48$	−0.578193	−	1.39588i	−0.578193	−	1.39588i
$49$	0.928339	+	0.928339i	0.928339	+	0.928339i
$50$	1.90211			1.90211
$51$	−0.108993	+	0.453990i	−0.108993	+	0.453990i
$52$		0			0
$53$	−1.26007	−	1.26007i	−1.26007	−	1.26007i	−0.951057	−	0.309017i	$-0.900000\pi$
$53$	−1.26007	−	1.26007i	−1.26007	−	1.26007i	−0.309017	−	0.951057i	$-0.600000\pi$
$54$	0.605623	+	1.46210i	0.605623	+	1.46210i
$55$		0			0
$56$	4.32429	−	1.79118i	4.32429	−	1.79118i
$57$		0			0
$58$		0			0
$59$	−0.437016	+	0.437016i	−0.437016	+	0.437016i	−0.891007	−	0.453990i	$-0.850000\pi$
$59$	−0.437016	+	0.437016i	−0.437016	+	0.437016i	0.453990	+	0.891007i	$0.350000\pi$
$60$		0			0
$61$	0.144974	+	0.0600500i	0.144974	+	0.0600500i	0.453990	−	0.891007i	$-0.350000\pi$
$61$	0.144974	+	0.0600500i	0.144974	+	0.0600500i	−0.309017	+	0.951057i	$0.600000\pi$
$62$		0			0
$63$	−1.09877	+	0.455123i	−1.09877	+	0.455123i
$64$		−	2.61803i		−	2.61803i
$65$		0			0
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$	−1.53884	+	2.11803i	−1.53884	+	2.11803i
$69$		0			0
$70$		0			0
$71$	−0.652583	−	1.57547i	−0.652583	−	1.57547i	−0.809017	−	0.587785i	$-0.800000\pi$
$71$	−0.652583	−	1.57547i	−0.652583	−	1.57547i	0.156434	−	0.987688i	$-0.450000\pi$
$72$			2.40679i			2.40679i
$73$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$73$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$74$	1.41559	−	3.41754i	1.41559	−	3.41754i
$75$	0.431351	+	0.178671i	0.431351	+	0.178671i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−0.652583	+	1.57547i	−0.652583	+	1.57547i	0.156434	+	0.987688i	$0.450000\pi$
$79$	−0.652583	+	1.57547i	−0.652583	+	1.57547i	−0.809017	+	0.587785i	$0.800000\pi$
$80$		0			0
$81$		−	0.393557i		−	0.393557i
$82$		0			0
$83$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$83$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$84$	1.85894			1.85894
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$			1.90211i			1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$89$			1.90211i			1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$	−1.34500	+	1.34500i	−1.34500	+	1.34500i
$95$		0			0
$96$	0.549894	−	1.32756i	0.549894	−	1.32756i
$97$	−1.84206	+	0.763007i	−1.84206	+	0.763007i	−0.891007	+	0.453990i	$0.850000\pi$
$97$	−1.84206	+	0.763007i	−1.84206	+	0.763007i	−0.951057	+	0.309017i	$0.900000\pi$
$98$			2.49723i			2.49723i
$99$		0			0
$100$	1.85123	+	1.85123i	1.85123	+	1.85123i
$101$	1.97538			1.97538			0.987688	−	0.156434i	$-0.0500000\pi$
$101$	1.97538			1.97538			0.987688	+	0.156434i	$0.0500000\pi$
$102$	−0.757212	+	0.464020i	−0.757212	+	0.464020i
$103$	−0.312869			−0.312869			−0.156434	−	0.987688i	$-0.550000\pi$
$103$	−0.312869			−0.312869			−0.156434	+	0.987688i	$0.550000\pi$
$104$		0			0
$105$		0			0
$106$		−	3.38959i		−	3.38959i
$107$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$107$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$108$	−0.833568	+	2.01241i	−0.833568	+	2.01241i
$109$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$109$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$110$		0			0
$111$	0.642040	−	0.642040i	0.642040	−	0.642040i
$112$	4.54683	+	1.88336i	4.54683	+	1.88336i
$113$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$113$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−1.17557			−1.17557
$119$	−0.794622	−	1.29671i	−0.794622	−	1.29671i
$120$		0			0
$121$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$122$	0.114222	+	0.275756i	0.114222	+	0.275756i
$123$		0			0
$124$		0			0
$125$		0			0
$126$	−2.08998	−	0.865696i	−2.08998	−	0.865696i
$127$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$127$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$128$	1.34500	−	1.34500i	1.34500	−	1.34500i
$129$		0			0
$130$		0			0
$131$	1.84206	−	0.763007i	1.84206	−	0.763007i	0.891007	−	0.453990i	$-0.150000\pi$
$131$	1.84206	−	0.763007i	1.84206	−	0.763007i	0.951057	−	0.309017i	$-0.100000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−3.03979	+	0.481456i	−3.03979	+	0.481456i
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$139$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$140$		0			0
$141$	−0.431351	+	0.178671i	−0.431351	+	0.178671i
$142$	1.24129	−	2.99673i	1.24129	−	2.99673i
$143$		0			0
$144$	−1.78944	+	1.78944i	−1.78944	+	1.78944i
$145$		0			0
$146$		0			0
$147$	−0.234572	+	0.566307i	−0.234572	+	0.566307i
$148$	4.70384	−	1.94839i	4.70384	−	1.94839i
$149$			0.312869i			0.312869i	0.987688	+	0.156434i	$0.0500000\pi$
$149$			0.312869i			0.312869i	−0.987688	+	0.156434i	$0.950000\pi$
$150$	0.339853	+	0.820478i	0.339853	+	0.820478i
$151$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$151$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$152$		0			0
$153$	0.772385	−	0.122334i	0.772385	−	0.122334i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	0.907981i		−	0.907981i	−0.891007	−	0.453990i	$-0.850000\pi$
$157$		−	0.907981i		−	0.907981i	0.891007	−	0.453990i	$-0.150000\pi$
$158$	−2.99673	+	1.24129i	−2.99673	+	1.24129i
$159$	0.318395	−	0.768673i	0.318395	−	0.768673i
$160$		0			0
$161$		0			0
$162$	0.529334	−	0.529334i	0.529334	−	0.529334i
$163$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$163$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$164$		0			0
$165$		0			0
$166$		−	2.68999i		−	2.68999i
$167$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$167$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$168$	1.54525	+	1.54525i	1.54525	+	1.54525i
$169$	−1.00000			−1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$	0.497066	+	1.20002i	0.497066	+	1.20002i	0.951057	+	0.309017i	$0.100000\pi$
$173$	0.497066	+	1.20002i	0.497066	+	1.20002i	−0.453990	+	0.891007i	$0.650000\pi$
$174$		0			0
$175$	−1.40505	+	0.581990i	−1.40505	+	0.581990i
$176$		0			0
$177$	−0.266589	−	0.110425i	−0.266589	−	0.110425i
$178$	−2.55834	+	2.55834i	−2.55834	+	2.55834i
$179$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$179$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$180$		0			0
$181$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$181$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$182$		0			0
$183$			0.0732636i			0.0732636i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−2.61803			−2.61803
$189$	−0.894719	−	0.894719i	−0.894719	−	0.894719i
$190$		0			0
$191$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$191$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$192$	1.12929	−	0.467768i	1.12929	−	0.467768i
$193$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$193$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$194$	−3.50381	−	1.45133i	−3.50381	−	1.45133i
$195$		0			0
$196$	−2.43042	+	2.43042i	−2.43042	+	2.43042i
$197$	1.70711	+	0.707107i	1.70711	+	0.707107i	1.00000			$0$
$197$	1.70711	+	0.707107i	1.70711	+	0.707107i	0.707107	+	0.707107i	$0.250000\pi$
$198$		0			0
$199$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$199$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$200$			3.07768i			3.07768i
$201$		0			0
$202$	2.65688	+	2.65688i	2.65688	+	2.65688i
$203$		0			0
$204$	−1.18856	−	0.285349i	−1.18856	−	0.285349i
$205$		0			0
$206$	−0.420808	−	0.420808i	−0.420808	−	0.420808i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$211$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$212$	3.29892	−	3.29892i	3.29892	−	3.29892i
$213$	0.562984	−	0.562984i	0.562984	−	0.562984i
$214$		0			0
$215$		0			0
$216$	−2.36573	+	0.979918i	−2.36573	+	0.979918i
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$	1.72708			1.72708
$223$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$223$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$224$	1.79118	+	4.32429i	1.79118	+	4.32429i
$225$		−	0.782013i		−	0.782013i
$226$		0			0
$227$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$227$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$228$		0			0
$229$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$229$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$233$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$234$		0			0
$235$		0			0
$236$	−1.14412	−	1.14412i	−1.14412	−	1.14412i
$237$	−0.796180			−0.796180
$238$	0.675301	−	2.81283i	0.675301	−	2.81283i
$239$	1.17557			1.17557			0.587785	−	0.809017i	$-0.300000\pi$
$239$	1.17557			1.17557			0.587785	+	0.809017i	$0.300000\pi$
$240$		0			0
$241$	0.744220	+	1.79671i	0.744220	+	1.79671i	0.587785	+	0.809017i	$0.300000\pi$
$241$	0.744220	+	1.79671i	0.744220	+	1.79671i	0.156434	+	0.987688i	$0.450000\pi$
$242$		−	1.90211i		−	1.90211i
$243$	0.938434	−	0.388712i	0.938434	−	0.388712i
$244$	−0.157213	+	0.379546i	−0.157213	+	0.379546i
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	0.252679	−	0.610022i	0.252679	−	0.610022i
$250$		0			0
$251$		−	1.97538i		−	1.97538i	−0.156434	−	0.987688i	$-0.550000\pi$
$251$		−	1.97538i		−	1.97538i	0.156434	−	0.987688i	$-0.450000\pi$
$252$	−1.19153	−	2.87660i	−1.19153	−	2.87660i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	1.00000			1.00000
$257$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$257$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$258$		0			0
$259$			2.95758i			2.95758i
$260$		0			0
$261$		0			0
$262$	3.50381	+	1.45133i	3.50381	+	1.45133i
$263$	0.437016	−	0.437016i	0.437016	−	0.437016i	−0.453990	−	0.891007i	$-0.650000\pi$
$263$	0.437016	−	0.437016i	0.437016	−	0.437016i	0.891007	+	0.453990i	$0.150000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−0.820478	+	0.339853i	−0.820478	+	0.339853i
$268$		0			0
$269$	−0.292893	−	0.707107i	−0.292893	−	0.707107i	0.707107	−	0.707107i	$-0.250000\pi$
$269$	−0.292893	−	0.707107i	−0.292893	−	0.707107i	−1.00000			$\pi$
$270$		0			0
$271$	−1.97538			−1.97538			−0.987688	−	0.156434i	$-0.950000\pi$
$271$	−1.97538			−1.97538			−0.987688	+	0.156434i	$0.950000\pi$
$272$	−2.61803	−	1.90211i	−2.61803	−	1.90211i
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1.57547	+	0.652583i	−1.57547	+	0.652583i	−0.987688	−	0.156434i	$-0.950000\pi$
$277$	−1.57547	+	0.652583i	−1.57547	+	0.652583i	−0.587785	+	0.809017i	$0.700000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$281$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$282$	−0.820478	−	0.339853i	−0.820478	−	0.339853i
$283$	−0.581990	+	1.40505i	−0.581990	+	1.40505i	0.309017	+	0.951057i	$0.400000\pi$
$283$	−0.581990	+	1.40505i	−0.581990	+	1.40505i	−0.891007	+	0.453990i	$0.850000\pi$
$284$	4.12464	−	1.70848i	4.12464	−	1.70848i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−2.40679			−2.40679
$289$	0.309017	+	0.951057i	0.309017	+	0.951057i
$290$		0			0
$291$	−0.658248	−	0.658248i	−0.658248	−	0.658248i
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	−1.07718	+	0.446183i	−1.07718	+	0.446183i
$295$		0			0
$296$	5.52969	+	2.29047i	5.52969	+	2.29047i
$297$		0			0
$298$	−0.420808	+	0.420808i	−0.420808	+	0.420808i
$299$		0			0
$300$	−0.467768	+	1.12929i	−0.467768	+	1.12929i
$301$		0			0
$302$		0			0
$303$	0.352943	+	0.852080i	0.352943	+	0.852080i
$304$		0			0
$305$		0			0
$306$	1.20339	+	0.874317i	1.20339	+	0.874317i
$307$	−1.97538			−1.97538			−0.987688	−	0.156434i	$-0.950000\pi$
$307$	−1.97538			−1.97538			−0.987688	+	0.156434i	$0.950000\pi$
$308$		0			0
$309$	−0.0559007	−	0.134956i	−0.0559007	−	0.134956i
$310$		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
$313$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$313$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$314$	1.22123	−	1.22123i	1.22123	−	1.22123i
$315$		0			0
$316$	−4.12464	−	1.70848i	−4.12464	−	1.70848i
$317$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$317$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$318$	1.46210	−	0.605623i	1.46210	−	0.605623i
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	1.03035			1.03035
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	0.581990	−	1.40505i	0.581990	−	1.40505i
$330$		0			0
$331$	1.34500	−	1.34500i	1.34500	−	1.34500i	0.453990	−	0.891007i	$-0.350000\pi$
$331$	1.34500	−	1.34500i	1.34500	−	1.34500i	0.891007	−	0.453990i	$-0.150000\pi$
$332$	2.61803	−	2.61803i	2.61803	−	2.61803i
$333$	−1.40505	−	0.581990i	−1.40505	−	0.581990i
$334$		0			0
$335$		0			0
$336$			2.29778i			2.29778i
$337$	−0.497066	−	1.20002i	−0.497066	−	1.20002i	−0.951057	−	0.309017i	$-0.900000\pi$
$337$	−0.497066	−	1.20002i	−0.497066	−	1.20002i	0.453990	−	0.891007i	$-0.350000\pi$
$338$	−1.34500	−	1.34500i	−1.34500	−	1.34500i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−0.182086	−	0.439596i	−0.182086	−	0.439596i
$344$		0			0
$345$		0			0
$346$	−0.945476	+	2.28258i	−0.945476	+	2.28258i
$347$	−0.431351	−	0.178671i	−0.431351	−	0.178671i	0.156434	−	0.987688i	$-0.450000\pi$
$347$	−0.431351	−	0.178671i	−0.431351	−	0.178671i	−0.587785	+	0.809017i	$0.700000\pi$
$348$		0			0
$349$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$349$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$350$	−2.67256	−	1.10701i	−2.67256	−	1.10701i
$351$		0			0
$352$		0			0
$353$			0.907981i			0.907981i	0.891007	+	0.453990i	$0.150000\pi$
$353$			0.907981i			0.907981i	−0.891007	+	0.453990i	$0.850000\pi$
$354$	−0.210041	−	0.507083i	−0.210041	−	0.507083i
$355$		0			0
$356$	−4.97980			−4.97980
$357$	0.417359	−	0.574445i	0.417359	−	0.574445i
$358$		0			0
$359$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$359$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$360$		0			0
$361$			1.00000i			1.00000i
$362$		0			0
$363$	0.178671	−	0.431351i	0.178671	−	0.431351i
$364$		0			0
$365$		0			0
$366$	−0.0985394	+	0.0985394i	−0.0985394	+	0.0985394i
$367$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$367$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.03711	+	2.50381i	1.03711	+	2.50381i
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$	−2.17625	−	2.17625i	−2.17625	−	2.17625i
$377$		0			0
$378$		−	2.40679i		−	2.40679i
$379$	1.20002	−	0.497066i	1.20002	−	0.497066i	0.309017	−	0.951057i	$-0.400000\pi$
$379$	1.20002	−	0.497066i	1.20002	−	0.497066i	0.891007	+	0.453990i	$0.150000\pi$
$380$		0			0
$381$		0			0
$382$	−1.90211	+	1.90211i	−1.90211	+	1.90211i
$383$	−0.221232	+	0.221232i	−0.221232	+	0.221232i	−0.809017	−	0.587785i	$-0.800000\pi$
$383$	−0.221232	+	0.221232i	−0.221232	+	0.221232i	0.587785	+	0.809017i	$0.300000\pi$
$384$	0.820478	+	0.339853i	0.820478	+	0.339853i
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−1.99758	−	4.82258i	−1.99758	−	4.82258i
$389$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$389$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$390$		0			0
$391$		0			0
$392$	−4.04060			−4.04060
$393$	0.658248	+	0.658248i	0.658248	+	0.658248i
$394$	1.34500	+	3.24711i	1.34500	+	3.24711i
$395$		0			0
$396$		0			0
$397$	0.399903	−	0.965451i	0.399903	−	0.965451i	−0.587785	−	0.809017i	$-0.700000\pi$
$397$	0.399903	−	0.965451i	0.399903	−	0.965451i	0.987688	−	0.156434i	$-0.0500000\pi$
$398$		0			0
$399$		0			0
$400$	−2.28825	+	2.28825i	−2.28825	+	2.28825i
$401$	0.431351	+	0.178671i	0.431351	+	0.178671i	0.587785	−	0.809017i	$-0.300000\pi$
$401$	0.431351	+	0.178671i	0.431351	+	0.178671i	−0.156434	+	0.987688i	$0.550000\pi$
$402$		0			0
$403$		0			0
$404$			5.17160i			5.17160i
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−0.750800	−	1.22519i	−0.750800	−	1.22519i
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$		−	0.819101i		−	0.819101i
$413$	0.868367	−	0.359689i	0.868367	−	0.359689i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$419$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$420$		0			0
$421$		0			0			−	1.00000i	$-0.5\pi$
$421$		0			0				1.00000i	$0.5\pi$
$422$		0			0
$423$	0.552967	+	0.552967i	0.552967	+	0.552967i
$424$	5.48447			5.48447
$425$	0.987688	−	0.156434i	0.987688	−	0.156434i
$426$	1.51442			1.51442
$427$	−0.168746	−	0.168746i	−0.168746	−	0.168746i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	0.763007	−	1.84206i	0.763007	−	1.84206i	0.309017	−	0.951057i	$-0.400000\pi$
$431$	0.763007	−	1.84206i	0.763007	−	1.84206i	0.453990	−	0.891007i	$-0.350000\pi$
$432$	−2.48748	−	1.03035i	−2.48748	−	1.03035i
$433$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$433$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	0.707107	+	1.70711i	0.707107	+	1.70711i	0.707107	+	0.707107i	$0.250000\pi$
$439$	0.707107	+	1.70711i	0.707107	+	1.70711i			1.00000i	$0.5\pi$
$440$		0			0
$441$	1.02668			1.02668
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$	1.68088	+	1.68088i	1.68088	+	1.68088i
$445$		0			0
$446$		0			0
$447$	−0.134956	+	0.0559007i	−0.134956	+	0.0559007i
$448$	−1.52367	+	3.67846i	−1.52367	+	3.67846i
$449$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$449$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$450$	1.05181	−	1.05181i	1.05181	−	1.05181i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.39680	+	1.39680i	1.39680	+	1.39680i	0.809017	+	0.587785i	$0.200000\pi$
$457$	1.39680	+	1.39680i	1.39680	+	1.39680i	0.587785	+	0.809017i	$0.300000\pi$
$458$		0			0
$459$	0.434722	+	0.709401i	0.434722	+	0.709401i
$460$		0			0
$461$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$461$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$467$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	0.391658	−	0.162230i	0.391658	−	0.162230i
$472$		−	1.90211i		−	1.90211i
$473$		0			0
$474$	−1.07086	−	1.07086i	−1.07086	−	1.07086i
$475$		0			0
$476$	3.39482	−	2.08035i	3.39482	−	2.08035i
$477$	−1.39356			−1.39356
$478$	1.58114	+	1.58114i	1.58114	+	1.58114i
$479$	0.581990	+	1.40505i	0.581990	+	1.40505i	0.891007	+	0.453990i	$0.150000\pi$
$479$	0.581990	+	1.40505i	0.581990	+	1.40505i	−0.309017	+	0.951057i	$0.600000\pi$
$480$		0			0
$481$		0			0
$482$	−1.41559	+	3.41754i	−1.41559	+	3.41754i
$483$		0			0
$484$	1.85123	−	1.85123i	1.85123	−	1.85123i
$485$		0			0
$486$	1.78501	+	0.739374i	1.78501	+	0.739374i
$487$	−0.707107	+	1.70711i	−0.707107	+	1.70711i			1.00000i	$0.5\pi$
$487$	−0.707107	+	1.70711i	−0.707107	+	1.70711i	−0.707107	+	0.707107i	$0.750000\pi$
$488$	−0.446183	+	0.184815i	−0.446183	+	0.184815i
$489$		0			0
$490$		0			0
$491$	−0.642040	−	0.642040i	−0.642040	−	0.642040i	0.309017	−	0.951057i	$-0.400000\pi$
$491$	−0.642040	−	0.642040i	−0.642040	−	0.642040i	−0.951057	+	0.309017i	$0.900000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$			2.59341i			2.59341i
$498$	1.16033	−	0.480625i	1.16033	−	0.480625i
$499$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$499$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$500$		0			0
$501$		0			0
$502$	2.65688	−	2.65688i	2.65688	−	2.65688i
$503$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$503$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$504$	1.40073	−	3.38165i	1.40073	−	3.38165i
$505$		0			0
$506$		0			0
$507$	−0.178671	−	0.431351i	−0.178671	−	0.431351i
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	−3.97794	+	3.97794i	−3.97794	+	3.97794i
$519$	−0.428820	+	0.428820i	−0.428820	+	0.428820i
$520$		0			0
$521$	0.497066	−	1.20002i	0.497066	−	1.20002i	−0.453990	−	0.891007i	$-0.650000\pi$
$521$	0.497066	−	1.20002i	0.497066	−	1.20002i	0.951057	−	0.309017i	$-0.100000\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$	1.99758	+	4.82258i	1.99758	+	4.82258i
$525$	−0.502083	−	0.502083i	−0.502083	−	0.502083i
$526$	1.17557			1.17557
$527$		0			0
$528$		0			0
$529$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$530$		0			0
$531$			0.483311i			0.483311i
$532$		0			0
$533$		0			0
$534$	−1.56064	−	0.646439i	−1.56064	−	0.646439i
$535$		0			0
$536$		0			0
$537$		0			0
$538$	0.557116	−	1.34500i	0.557116	−	1.34500i
$539$		0			0
$540$		0			0
$541$	0.497066	+	1.20002i	0.497066	+	1.20002i	0.951057	+	0.309017i	$0.100000\pi$
$541$	0.497066	+	1.20002i	0.497066	+	1.20002i	−0.453990	+	0.891007i	$0.650000\pi$
$542$	−2.65688	−	2.65688i	−2.65688	−	2.65688i
$543$		0			0
$544$	−0.481456	−	3.03979i	−0.481456	−	3.03979i
$545$		0			0
$546$		0			0
$547$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$547$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$548$		0			0
$549$	0.113371	−	0.0469599i	0.113371	−	0.0469599i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	1.83382	−	1.83382i	1.83382	−	1.83382i
$554$	−2.99673	−	1.24129i	−2.99673	−	1.24129i
$555$		0			0
$556$		0			0
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$563$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$564$	−0.467768	−	1.12929i	−0.467768	−	1.12929i
$565$		0			0
$566$	−2.67256	+	1.10701i	−2.67256	+	1.10701i
$567$	−0.229046	+	0.552967i	−0.229046	+	0.552967i
$568$	4.84881	+	2.00844i	4.84881	+	2.00844i
$569$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$569$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$570$		0			0
$571$	0.965451	+	0.399903i	0.965451	+	0.399903i	0.809017	−	0.587785i	$-0.200000\pi$
$571$	0.965451	+	0.399903i	0.965451	+	0.399903i	0.156434	+	0.987688i	$0.450000\pi$
$572$		0			0
$573$	−0.610022	+	0.252679i	−0.610022	+	0.252679i
$574$		0			0
$575$		0			0
$576$	−1.44769	−	1.44769i	−1.44769	−	1.44769i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−0.863541	+	1.69480i	−0.863541	+	1.69480i
$579$		0			0
$580$		0			0
$581$	0.823057	+	1.98704i	0.823057	+	1.98704i
$582$		−	1.77068i		−	1.77068i
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$587$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$588$	−1.48261	−	0.614118i	−1.48261	−	0.614118i
$589$		0			0
$590$		0			0
$591$			0.862702i			0.862702i
$592$	2.40835	+	5.81426i	2.40835	+	5.81426i
$593$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$593$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$594$		0			0
$595$		0			0
$596$	−0.819101			−0.819101
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$	−1.32756	+	0.549894i	−1.32756	+	0.549894i
$601$	0.0600500	−	0.144974i	0.0600500	−	0.144974i	−0.891007	−	0.453990i	$-0.850000\pi$
$601$	0.0600500	−	0.144974i	0.0600500	−	0.144974i	0.951057	+	0.309017i	$0.100000\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$	−0.671338	+	1.62075i	−0.671338	+	1.62075i
$607$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$607$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	0.320274	+	2.02213i	0.320274	+	2.02213i
$613$	1.90211			1.90211			0.951057	−	0.309017i	$-0.100000\pi$
$613$	1.90211			1.90211			0.951057	+	0.309017i	$0.100000\pi$
$614$	−2.65688	−	2.65688i	−2.65688	−	2.65688i
$615$		0			0
$616$		0			0
$617$	−0.965451	+	0.399903i	−0.965451	+	0.399903i	−0.809017	−	0.587785i	$-0.800000\pi$
$617$	−0.965451	+	0.399903i	−0.965451	+	0.399903i	−0.156434	+	0.987688i	$0.550000\pi$
$618$	0.106329	−	0.256702i	0.106329	−	0.256702i
$619$	−0.707107	−	0.292893i	−0.707107	−	0.292893i		−	1.00000i	$-0.5\pi$
$619$	−0.707107	−	0.292893i	−0.707107	−	0.292893i	−0.707107	+	0.707107i	$0.750000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1.10701	−	2.67256i	1.10701	−	2.67256i
$624$		0			0
$625$		−	1.00000i		−	1.00000i
$626$		0			0
$627$		0			0
$628$	2.37713			2.37713
$629$	0.453990	−	1.89101i	0.453990	−	1.89101i
$630$		0			0
$631$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$631$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$632$	−2.00844	−	4.84881i	−2.00844	−	4.84881i
$633$		0			0
$634$		0			0
$635$		0			0
$636$	2.01241	+	0.833568i	2.01241	+	0.833568i
$637$		0			0
$638$		0			0
$639$	−1.23204	−	0.510328i	−1.23204	−	0.510328i
$640$		0			0
$641$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$641$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$642$		0			0
$643$	−0.0600500	−	0.144974i	−0.0600500	−	0.144974i	0.891007	−	0.453990i	$-0.150000\pi$
$643$	−0.0600500	−	0.144974i	−0.0600500	−	0.144974i	−0.951057	+	0.309017i	$0.900000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	0.907981			0.907981			0.453990	−	0.891007i	$-0.350000\pi$
$647$	0.907981			0.907981			0.453990	+	0.891007i	$0.350000\pi$
$648$	0.856480	+	0.856480i	0.856480	+	0.856480i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1.79671	+	0.744220i	1.79671	+	0.744220i	0.987688	+	0.156434i	$0.0500000\pi$
$653$	1.79671	+	0.744220i	1.79671	+	0.744220i	0.809017	+	0.587785i	$0.200000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	2.67256	−	1.10701i	2.67256	−	1.10701i
$659$			1.61803i			1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$659$			1.61803i			1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$660$		0			0
$661$	−0.831254	−	0.831254i	−0.831254	−	0.831254i	0.156434	−	0.987688i	$-0.450000\pi$
$661$	−0.831254	−	0.831254i	−0.831254	−	0.831254i	−0.987688	+	0.156434i	$0.950000\pi$
$662$	3.61803			3.61803
$663$		0			0
$664$	4.35250			4.35250
$665$		0			0
$666$	−1.10701	−	2.67256i	−1.10701	−	2.67256i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$	−1.54525	+	1.54525i	−1.54525	+	1.54525i
$673$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$673$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$674$	0.945476	−	2.28258i	0.945476	−	2.28258i
$675$	0.768673	−	0.318395i	0.768673	−	0.318395i
$676$		−	2.61803i		−	2.61803i
$677$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$677$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$678$		0			0
$679$	3.03225			3.03225
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.0600500	+	0.144974i	0.0600500	+	0.144974i	0.951057	−	0.309017i	$-0.100000\pi$
$683$	0.0600500	+	0.144974i	0.0600500	+	0.144974i	−0.891007	+	0.453990i	$0.850000\pi$
$684$		0			0
$685$		0			0
$686$	0.346349	−	0.836160i	0.346349	−	0.836160i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$691$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$692$	−3.14170	+	1.30134i	−3.14170	+	1.30134i
$693$		0			0
$694$	−0.339853	−	0.820478i	−0.339853	−	0.820478i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−1.52367	−	3.67846i	−1.52367	−	3.67846i
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−1.22123	+	1.22123i	−1.22123	+	1.22123i
$707$	−2.77550	−	1.14965i	−2.77550	−	1.14965i
$708$	0.289096	−	0.697940i	0.289096	−	0.697940i
$709$	−1.40505	+	0.581990i	−1.40505	+	0.581990i	−0.951057	−	0.309017i	$-0.900000\pi$
$709$	−1.40505	+	0.581990i	−1.40505	+	0.581990i	−0.453990	+	0.891007i	$0.650000\pi$
$710$		0			0
$711$	0.510328	+	1.23204i	0.510328	+	1.23204i
$712$	−4.13948	−	4.13948i	−4.13948	−	4.13948i
$713$		0			0
$714$	1.33397	−	0.211281i	1.33397	−	0.211281i
$715$		0			0
$716$		0			0
$717$	0.210041	+	0.507083i	0.210041	+	0.507083i
$718$		0			0
$719$	−0.144974	+	0.0600500i	−0.144974	+	0.0600500i	−0.453990	−	0.891007i	$-0.650000\pi$
$719$	−0.144974	+	0.0600500i	−0.144974	+	0.0600500i	0.309017	+	0.951057i	$0.400000\pi$
$720$		0			0
$721$	0.439596	+	0.182086i	0.439596	+	0.182086i
$722$	−1.34500	+	1.34500i	−1.34500	+	1.34500i
$723$	−0.642040	+	0.642040i	−0.642040	+	0.642040i
$724$		0			0
$725$		0			0
$726$	0.820478	−	0.339853i	0.820478	−	0.339853i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	0.0570554	+	0.0570554i	0.0570554	+	0.0570554i
$730$		0			0
$731$		0			0
$732$	−0.191807			−0.191807
$733$	−0.831254	−	0.831254i	−0.831254	−	0.831254i	0.156434	−	0.987688i	$-0.450000\pi$
$733$	−0.831254	−	0.831254i	−0.831254	−	0.831254i	−0.987688	+	0.156434i	$0.950000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	0.642040	−	0.642040i	0.642040	−	0.642040i	−0.309017	−	0.951057i	$-0.600000\pi$
$739$	0.642040	−	0.642040i	0.642040	−	0.642040i	0.951057	+	0.309017i	$0.100000\pi$
$740$		0			0
$741$		0			0
$742$	−1.97271	+	4.76253i	−1.97271	+	4.76253i
$743$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$743$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−1.10593			−1.10593
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$751$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$752$		−	3.23607i		−	3.23607i
$753$	0.852080	−	0.352943i	0.852080	−	0.352943i
$754$		0			0
$755$		0			0
$756$	2.34240	−	2.34240i	2.34240	−	2.34240i
$757$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$757$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$758$	2.28258	+	0.945476i	2.28258	+	0.945476i
$759$		0			0
$760$		0			0
$761$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$761$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$762$		0			0
$763$		0			0
$764$	−3.70246			−3.70246
$765$		0			0
$766$	−0.595112			−0.595112
$767$		0			0
$768$	0.178671	+	0.431351i	0.178671	+	0.431351i
$769$		−	1.17557i		−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$769$		−	1.17557i		−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	0.831254	−	0.831254i	0.831254	−	0.831254i	−0.156434	−	0.987688i	$-0.550000\pi$
$773$	0.831254	−	0.831254i	0.831254	−	0.831254i	0.987688	+	0.156434i	$0.0500000\pi$
$774$		0			0
$775$		0			0
$776$	2.34830	−	5.66929i	2.34830	−	5.66929i
$777$	−1.27576	+	0.528435i	−1.27576	+	0.528435i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−3.00417	−	3.00417i	−3.00417	−	3.00417i
$785$		0			0
$786$			1.77068i			1.77068i
$787$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$787$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$788$	−1.85123	+	4.46926i	−1.85123	+	4.46926i
$789$	0.266589	+	0.110425i	0.266589	+	0.110425i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	1.83640	−	0.760661i	1.83640	−	0.760661i
$795$		0			0
$796$		0			0
$797$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$797$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$798$		0			0
$799$	−0.587785	+	0.809017i	−0.587785	+	0.809017i
$800$	−3.07768			−3.07768
$801$	1.05181	+	1.05181i	1.05181	+	1.05181i
$802$	0.339853	+	0.820478i	0.339853	+	0.820478i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$	0.252679	−	0.252679i	0.252679	−	0.252679i
$808$	−4.29892	+	4.29892i	−4.29892	+	4.29892i
$809$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$809$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$810$		0			0
$811$	1.57547	−	0.652583i	1.57547	−	0.652583i	0.587785	−	0.809017i	$-0.300000\pi$
$811$	1.57547	−	0.652583i	1.57547	−	0.652583i	0.987688	+	0.156434i	$0.0500000\pi$
$812$		0			0
$813$	−0.352943	−	0.852080i	−0.352943	−	0.852080i
$814$		0			0
$815$		0			0
$816$	0.352710	−	1.46914i	0.352710	−	1.46914i
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$821$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$822$		0			0
$823$	−0.707107	−	0.292893i	−0.707107	−	0.292893i		−	1.00000i	$-0.5\pi$
$823$	−0.707107	−	0.292893i	−0.707107	−	0.292893i	−0.707107	+	0.707107i	$0.750000\pi$
$824$	0.680881	−	0.680881i	0.680881	−	0.680881i
$825$		0			0
$826$	1.65173	+	0.684170i	1.65173	+	0.684170i
$827$	0.0600500	−	0.144974i	0.0600500	−	0.144974i	−0.891007	−	0.453990i	$-0.850000\pi$
$827$	0.0600500	−	0.144974i	0.0600500	−	0.144974i	0.951057	+	0.309017i	$0.100000\pi$
$828$		0			0
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		0			0
$831$	−0.562984	−	0.562984i	−0.562984	−	0.562984i
$832$		0			0
$833$	0.205378	+	1.29671i	0.205378	+	1.29671i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$839$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$840$		0			0
$841$	0.707107	−	0.707107i	0.707107	−	0.707107i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$			1.48748i			1.48748i
$847$	0.581990	+	1.40505i	0.581990	+	1.40505i
$848$	4.07768	+	4.07768i	4.07768	+	4.07768i
$849$	−0.710053			−0.710053
$850$	1.53884	+	1.11803i	1.53884	+	1.11803i
$851$		0			0
$852$	1.47391	+	1.47391i	1.47391	+	1.47391i
$853$	−0.399903	−	0.965451i	−0.399903	−	0.965451i	−0.987688	−	0.156434i	$-0.950000\pi$
$853$	−0.399903	−	0.965451i	−0.399903	−	0.965451i	0.587785	−	0.809017i	$-0.300000\pi$
$854$		−	0.453926i		−	0.453926i
$855$		0			0
$856$		0			0
$857$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$857$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$858$		0			0
$859$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$859$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$860$		0			0
$861$		0			0
$862$	3.50381	−	1.45133i	3.50381	−	1.45133i
$863$		−	1.61803i		−	1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$863$		−	1.61803i		−	1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$864$	−0.979918	−	2.36573i	−0.979918	−	2.36573i
$865$		0			0
$866$		0			0
$867$	−0.355026	+	0.303221i	−0.355026	+	0.303221i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−0.596682	+	1.44052i	−0.596682	+	1.44052i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$877$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$878$	−1.34500	+	3.24711i	−1.34500	+	3.24711i
$879$		0			0
$880$		0			0
$881$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$881$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$882$	1.38088	+	1.38088i	1.38088	+	1.38088i
$883$	0.907981			0.907981			0.453990	−	0.891007i	$-0.350000\pi$
$883$	0.907981			0.907981			0.453990	+	0.891007i	$0.350000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$887$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$888$			2.79448i			2.79448i
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$	−0.256702	−	0.106329i	−0.256702	−	0.106329i
$895$		0			0
$896$	−2.67256	+	1.10701i	−2.67256	+	1.10701i
$897$		0			0
$898$		0			0
$899$		0			0
$900$	2.04734			2.04734
$901$	−0.278768	−	1.76007i	−0.278768	−	1.76007i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	0.292893	−	0.707107i	0.292893	−	0.707107i	−0.707107	−	0.707107i	$-0.750000\pi$
$907$	0.292893	−	0.707107i	0.292893	−	0.707107i	1.00000			$0$
$908$		0			0
$909$	1.09232	−	1.09232i	1.09232	−	1.09232i
$910$		0			0
$911$	−0.144974	−	0.0600500i	−0.144974	−	0.0600500i	0.309017	−	0.951057i	$-0.400000\pi$
$911$	−0.144974	−	0.0600500i	−0.144974	−	0.0600500i	−0.453990	+	0.891007i	$0.650000\pi$
$912$		0			0
$913$		0			0
$914$			3.75739i			3.75739i
$915$		0			0
$916$		0			0
$917$	−3.03225			−3.03225
$918$	−0.369443	+	1.53884i	−0.369443	+	1.53884i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	−0.352943	−	0.852080i	−0.352943	−	0.852080i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1.79671	−	0.744220i	−1.79671	−	0.744220i
$926$		0			0
$927$	−0.173006	+	0.173006i	−0.173006	+	0.173006i
$928$		0			0
$929$	−0.744220	+	1.79671i	−0.744220	+	1.79671i	−0.156434	+	0.987688i	$0.550000\pi$
$929$	−0.744220	+	1.79671i	−0.744220	+	1.79671i	−0.587785	+	0.809017i	$0.700000\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$937$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−0.763007	+	1.84206i	−0.763007	+	1.84206i	−0.309017	+	0.951057i	$0.600000\pi$
$941$	−0.763007	+	1.84206i	−0.763007	+	1.84206i	−0.453990	+	0.891007i	$0.650000\pi$
$942$	0.744978	+	0.308580i	0.744978	+	0.308580i
$943$		0			0
$944$	1.41421	−	1.41421i	1.41421	−	1.41421i
$945$		0			0
$946$		0			0
$947$	−0.707107	+	0.292893i	−0.707107	+	0.292893i	−0.707107	−	0.707107i	$-0.750000\pi$
$947$	−0.707107	+	0.292893i	−0.707107	+	0.292893i			1.00000i	$0.5\pi$
$948$		−	2.08443i		−	2.08443i
$949$		0			0
$950$		0			0
$951$		0			0
$952$	4.55125	+	1.09266i	4.55125	+	1.09266i
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$	−1.87433	−	1.87433i	−1.87433	−	1.87433i
$955$		0			0
$956$			3.07768i			3.07768i
$957$		0			0
$958$	−1.10701	+	2.67256i	−1.10701	+	2.67256i
$959$		0			0
$960$		0			0
$961$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$962$		0			0
$963$		0			0
$964$	−4.70384	+	1.94839i	−4.70384	+	1.94839i
$965$		0			0
$966$		0			0
$967$	0.642040	+	0.642040i	0.642040	+	0.642040i	0.951057	−	0.309017i	$-0.100000\pi$
$967$	0.642040	+	0.642040i	0.642040	+	0.642040i	−0.309017	+	0.951057i	$0.600000\pi$
$968$	3.07768			3.07768
$969$		0			0
$970$		0			0
$971$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$971$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$972$	1.01766	+	2.45685i	1.01766	+	2.45685i
$973$		0			0
$974$	−3.24711	+	1.34500i	−3.24711	+	1.34500i
$975$		0			0
$976$	−0.469144	−	0.194326i	−0.469144	−	0.194326i
$977$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$977$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		−	1.72708i		−	1.72708i
$983$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$983$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	0.710053			0.710053
$988$		0			0
$989$		0			0
$990$		0			0
$991$	0.144974	−	0.0600500i	0.144974	−	0.0600500i	−0.309017	−	0.951057i	$-0.600000\pi$
$991$	0.144974	−	0.0600500i	0.144974	−	0.0600500i	0.453990	+	0.891007i	$0.350000\pi$
$992$		0			0
$993$	0.820478	+	0.339853i	0.820478	+	0.339853i
$994$	−3.48813	+	3.48813i	−3.48813	+	3.48813i
$995$		0			0
$996$	1.59706	+	0.661523i	1.59706	+	0.661523i
$997$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$997$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$998$		0			0
$999$		−	1.61803i		−	1.61803i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	799.1.h.b.610.4	yes	16
17.8	even	8	inner	799.1.h.b.93.4	✓	16
47.46	odd	2	CM	799.1.h.b.610.4	yes	16
799.93	odd	8	inner	799.1.h.b.93.4	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
799.1.h.b.93.4	✓	16	17.8	even	8	inner
799.1.h.b.93.4	✓	16	799.93	odd	8	inner
799.1.h.b.610.4	yes	16	1.1	even	1	trivial
799.1.h.b.610.4	yes	16	47.46	odd	2	CM

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:	\( N \)	\(=\)	\( 799 = 17 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	799.h (of order \(8\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(1.34500 + 1.34500i) q^{2} +(0.178671 + 0.431351i) q^{3} +2.61803i q^{4} +(-0.339853 + 0.820478i) q^{6} +(-1.40505 - 0.581990i) q^{7} +(-2.17625 + 2.17625i) q^{8} +(0.552967 - 0.552967i) q^{9} +(-1.12929 + 0.467768i) q^{12} +(-1.10701 - 2.67256i) q^{14} -3.23607 q^{16} +(0.809017 + 0.587785i) q^{17} +1.48748 q^{18} -0.710053i q^{21} +(-1.32756 - 0.549894i) q^{24} +(0.707107 - 0.707107i) q^{25} +(0.768673 + 0.318395i) q^{27} +(1.52367 - 3.67846i) q^{28} +(-2.17625 - 2.17625i) q^{32} +(0.297556 + 1.87869i) q^{34} +(1.44769 + 1.44769i) q^{36} +(-0.744220 - 1.79671i) q^{37} +(0.955019 - 0.955019i) q^{42} +1.00000i q^{47} +(-0.578193 - 1.39588i) q^{48} +(0.928339 + 0.928339i) q^{49} +1.90211 q^{50} +(-0.108993 + 0.453990i) q^{51} +(-1.26007 - 1.26007i) q^{53} +(0.605623 + 1.46210i) q^{54} +(4.32429 - 1.79118i) q^{56} +(-0.437016 + 0.437016i) q^{59} +(0.144974 + 0.0600500i) q^{61} +(-1.09877 + 0.455123i) q^{63} -2.61803i q^{64} +(-1.53884 + 2.11803i) q^{68} +(-0.652583 - 1.57547i) q^{71} +2.40679i q^{72} +(1.41559 - 3.41754i) q^{74} +(0.431351 + 0.178671i) q^{75} +(-0.652583 + 1.57547i) q^{79} -0.393557i q^{81} +(-1.00000 - 1.00000i) q^{83} +1.85894 q^{84} +1.90211i q^{89} +(-1.34500 + 1.34500i) q^{94} +(0.549894 - 1.32756i) q^{96} +(-1.84206 + 0.763007i) q^{97} +2.49723i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{3} - 4 q^{9} - 16 q^{16} + 4 q^{17} - 20 q^{24} - 4 q^{27} - 4 q^{28} + 4 q^{36} - 20 q^{42} + 24 q^{48} + 4 q^{49} - 16 q^{51} + 4 q^{53} + 20 q^{54} + 20 q^{56} + 4 q^{61} - 4 q^{63} - 4 q^{71}+ \cdots + 32 q^{84}+O(q^{100}) \) 16 * q - 4 * q^3 - 4 * q^9 - 16 * q^16 + 4 * q^17 - 20 * q^24 - 4 * q^27 - 4 * q^28 + 4 * q^36 - 20 * q^42 + 24 * q^48 + 4 * q^49 - 16 * q^51 + 4 * q^53 + 20 * q^54 + 20 * q^56 + 4 * q^61 - 4 * q^63 - 4 * q^71 - 4 * q^79 - 16 * q^83 + 32 * q^84

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