LMFDB - Embedded newform 799.1.h.b.281.4

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			281.4
Root			$0.453990 + 0.891007i$ of defining polynomial
Character	$\chi$	$=$	799.281
Dual form			799.1.h.b.563.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.34500 - 1.34500i) q^{2} +(-0.965451 - 0.399903i) q^{3} -2.61803i q^{4} +(-1.83640 + 0.760661i) q^{6} +(0.0600500 + 0.144974i) q^{7} +(-2.17625 - 2.17625i) q^{8} +(0.0650673 + 0.0650673i) q^{9} +(-1.04696 + 2.52758i) q^{12} +(0.275756 + 0.114222i) q^{14} -3.23607 q^{16} +(0.809017 + 0.587785i) q^{17} +0.175031 q^{18} -0.163979i q^{21} +(1.23078 + 2.97135i) q^{24} +(-0.707107 - 0.707107i) q^{25} +(0.363104 + 0.876612i) q^{27} +(0.379546 - 0.157213i) q^{28} +(-2.17625 + 2.17625i) q^{32} +(1.87869 - 0.297556i) q^{34} +(0.170348 - 0.170348i) q^{36} +(1.57547 + 0.652583i) q^{37} +(-0.220551 - 0.220551i) q^{42} -1.00000i q^{47} +(3.12427 + 1.29411i) q^{48} +(0.689695 - 0.689695i) q^{49} -1.90211 q^{50} +(-0.546010 - 0.891007i) q^{51} +(0.642040 - 0.642040i) q^{53} +(1.66741 + 0.690666i) q^{54} +(0.184815 - 0.446183i) q^{56} +(0.437016 + 0.437016i) q^{59} +(0.581990 + 1.40505i) q^{61} +(-0.00552574 + 0.0133403i) q^{63} +2.61803i q^{64} +(1.53884 - 2.11803i) q^{68} +(-1.79671 - 0.744220i) q^{71} -0.283205i q^{72} +(2.99673 - 1.24129i) q^{74} +(0.399903 + 0.965451i) q^{75} +(-1.79671 + 0.744220i) q^{79} -1.08355i q^{81} +(-1.00000 + 1.00000i) q^{83} -0.429303 q^{84} +1.90211i q^{89} +(-1.34500 - 1.34500i) q^{94} +(2.97135 - 1.23078i) q^{96} +(0.497066 - 1.20002i) q^{97} -1.85528i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/799\mathbb{Z}\right)^\times$.

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

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	0.453990	−	0.891007i	$-0.350000\pi$
$2$	1.34500	−	1.34500i	1.34500	−	1.34500i	0.891007	−	0.453990i	$-0.150000\pi$
$3$	−0.965451	−	0.399903i	−0.965451	−	0.399903i	−0.156434	−	0.987688i	$-0.550000\pi$
$3$	−0.965451	−	0.399903i	−0.965451	−	0.399903i	−0.809017	+	0.587785i	$0.800000\pi$
$4$		−	2.61803i		−	2.61803i
$5$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$5$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$6$	−1.83640	+	0.760661i	−1.83640	+	0.760661i
$7$	0.0600500	+	0.144974i	0.0600500	+	0.144974i	0.951057	−	0.309017i	$-0.100000\pi$
$7$	0.0600500	+	0.144974i	0.0600500	+	0.144974i	−0.891007	+	0.453990i	$0.850000\pi$
$8$	−2.17625	−	2.17625i	−2.17625	−	2.17625i
$9$	0.0650673	+	0.0650673i	0.0650673	+	0.0650673i
$10$		0			0
$11$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$11$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$12$	−1.04696	+	2.52758i	−1.04696	+	2.52758i
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$	0.275756	+	0.114222i	0.275756	+	0.114222i
$15$		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
$18$	0.175031			0.175031
$19$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$19$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$20$		0			0
$21$		−	0.163979i		−	0.163979i
$22$		0			0
$23$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$23$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$24$	1.23078	+	2.97135i	1.23078	+	2.97135i
$25$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$26$		0			0
$27$	0.363104	+	0.876612i	0.363104	+	0.876612i
$28$	0.379546	−	0.157213i	0.379546	−	0.157213i
$29$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$29$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$30$		0			0
$31$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$31$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$32$	−2.17625	+	2.17625i	−2.17625	+	2.17625i
$33$		0			0
$34$	1.87869	−	0.297556i	1.87869	−	0.297556i
$35$		0			0
$36$	0.170348	−	0.170348i	0.170348	−	0.170348i
$37$	1.57547	+	0.652583i	1.57547	+	0.652583i	0.987688	−	0.156434i	$-0.0500000\pi$
$37$	1.57547	+	0.652583i	1.57547	+	0.652583i	0.587785	+	0.809017i	$0.300000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$41$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$42$	−0.220551	−	0.220551i	−0.220551	−	0.220551i
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	1.00000i		−	1.00000i
$47$		−	1.00000i		−	1.00000i
$48$	3.12427	+	1.29411i	3.12427	+	1.29411i
$49$	0.689695	−	0.689695i	0.689695	−	0.689695i
$50$	−1.90211			−1.90211
$51$	−0.546010	−	0.891007i	−0.546010	−	0.891007i
$52$		0			0
$53$	0.642040	−	0.642040i	0.642040	−	0.642040i	−0.309017	−	0.951057i	$-0.600000\pi$
$53$	0.642040	−	0.642040i	0.642040	−	0.642040i	0.951057	+	0.309017i	$0.100000\pi$
$54$	1.66741	+	0.690666i	1.66741	+	0.690666i
$55$		0			0
$56$	0.184815	−	0.446183i	0.184815	−	0.446183i
$57$		0			0
$58$		0			0
$59$	0.437016	+	0.437016i	0.437016	+	0.437016i	0.891007	−	0.453990i	$-0.150000\pi$
$59$	0.437016	+	0.437016i	0.437016	+	0.437016i	−0.453990	+	0.891007i	$0.650000\pi$
$60$		0			0
$61$	0.581990	+	1.40505i	0.581990	+	1.40505i	0.891007	+	0.453990i	$0.150000\pi$
$61$	0.581990	+	1.40505i	0.581990	+	1.40505i	−0.309017	+	0.951057i	$0.600000\pi$
$62$		0			0
$63$	−0.00552574	+	0.0133403i	−0.00552574	+	0.0133403i
$64$			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
$71$	−1.79671	−	0.744220i	−1.79671	−	0.744220i	−0.987688	−	0.156434i	$-0.950000\pi$
$71$	−1.79671	−	0.744220i	−1.79671	−	0.744220i	−0.809017	−	0.587785i	$-0.800000\pi$
$72$		−	0.283205i		−	0.283205i
$73$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$73$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$74$	2.99673	−	1.24129i	2.99673	−	1.24129i
$75$	0.399903	+	0.965451i	0.399903	+	0.965451i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−1.79671	+	0.744220i	−1.79671	+	0.744220i	−0.809017	+	0.587785i	$0.800000\pi$
$79$	−1.79671	+	0.744220i	−1.79671	+	0.744220i	−0.987688	+	0.156434i	$0.950000\pi$
$80$		0			0
$81$		−	1.08355i		−	1.08355i
$82$		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			$\pi$
$84$	−0.429303			−0.429303
$85$		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
$96$	2.97135	−	1.23078i	2.97135	−	1.23078i
$97$	0.497066	−	1.20002i	0.497066	−	1.20002i	−0.453990	−	0.891007i	$-0.650000\pi$
$97$	0.497066	−	1.20002i	0.497066	−	1.20002i	0.951057	−	0.309017i	$-0.100000\pi$
$98$		−	1.85528i		−	1.85528i
$99$		0			0
$100$	−1.85123	+	1.85123i	−1.85123	+	1.85123i
$101$	−0.312869			−0.312869			−0.156434	−	0.987688i	$-0.550000\pi$
$101$	−0.312869			−0.312869			−0.156434	+	0.987688i	$0.550000\pi$
$102$	−1.93278	−	0.464020i	−1.93278	−	0.464020i
$103$	1.97538			1.97538			0.987688	−	0.156434i	$-0.0500000\pi$
$103$	1.97538			1.97538			0.987688	+	0.156434i	$0.0500000\pi$
$104$		0			0
$105$		0			0
$106$		−	1.72708i		−	1.72708i
$107$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$107$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$108$	2.29500	−	0.950620i	2.29500	−	0.950620i
$109$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$109$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$110$		0			0
$111$	−1.26007	−	1.26007i	−1.26007	−	1.26007i
$112$	−0.194326	−	0.469144i	−0.194326	−	0.469144i
$113$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$113$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	1.17557			1.17557
$119$	−0.0366318	+	0.152583i	−0.0366318	+	0.152583i
$120$		0			0
$121$	0.707107	−	0.707107i	0.707107	−	0.707107i
$122$	2.67256	+	1.10701i	2.67256	+	1.10701i
$123$		0			0
$124$		0			0
$125$		0			0
$126$	0.0105106	+	0.0253748i	0.0105106	+	0.0253748i
$127$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$127$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$128$	1.34500	+	1.34500i	1.34500	+	1.34500i
$129$		0			0
$130$		0			0
$131$	−0.497066	+	1.20002i	−0.497066	+	1.20002i	0.453990	+	0.891007i	$0.350000\pi$
$131$	−0.497066	+	1.20002i	−0.497066	+	1.20002i	−0.951057	+	0.309017i	$0.900000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−0.481456	−	3.03979i	−0.481456	−	3.03979i
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$139$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$140$		0			0
$141$	−0.399903	+	0.965451i	−0.399903	+	0.965451i
$142$	−3.41754	+	1.41559i	−3.41754	+	1.41559i
$143$		0			0
$144$	−0.210562	−	0.210562i	−0.210562	−	0.210562i
$145$		0			0
$146$		0			0
$147$	−0.941679	+	0.390056i	−0.941679	+	0.390056i
$148$	1.70848	−	4.12464i	1.70848	−	4.12464i
$149$			1.97538i			1.97538i	0.156434	+	0.987688i	$0.450000\pi$
$149$			1.97538i			1.97538i	−0.156434	+	0.987688i	$0.550000\pi$
$150$	1.83640	+	0.760661i	1.83640	+	0.760661i
$151$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$151$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$152$		0			0
$153$	0.0143949	+	0.0908861i	0.0143949	+	0.0908861i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			1.78201i			1.78201i	0.453990	+	0.891007i	$0.350000\pi$
$157$			1.78201i			1.78201i	−0.453990	+	0.891007i	$0.650000\pi$
$158$	−1.41559	+	3.41754i	−1.41559	+	3.41754i
$159$	−0.876612	+	0.363104i	−0.876612	+	0.363104i
$160$		0			0
$161$		0			0
$162$	−1.45737	−	1.45737i	−1.45737	−	1.45737i
$163$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$163$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$164$		0			0
$165$		0			0
$166$			2.68999i			2.68999i
$167$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$167$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$168$	−0.356860	+	0.356860i	−0.356860	+	0.356860i
$169$	−1.00000			−1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.84206	−	0.763007i	−1.84206	−	0.763007i	−0.951057	−	0.309017i	$-0.900000\pi$
$173$	−1.84206	−	0.763007i	−1.84206	−	0.763007i	−0.891007	−	0.453990i	$-0.850000\pi$
$174$		0			0
$175$	0.0600500	−	0.144974i	0.0600500	−	0.144974i
$176$		0			0
$177$	−0.247154	−	0.596682i	−0.247154	−	0.596682i
$178$	2.55834	+	2.55834i	2.55834	+	2.55834i
$179$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$179$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$180$		0			0
$181$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$181$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$182$		0			0
$183$		−	1.58924i		−	1.58924i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−2.61803			−2.61803
$189$	−0.105281	+	0.105281i	−0.105281	+	0.105281i
$190$		0			0
$191$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$191$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$192$	1.04696	−	2.52758i	1.04696	−	2.52758i
$193$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$193$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$194$	−0.945476	−	2.28258i	−0.945476	−	2.28258i
$195$		0			0
$196$	−1.80565	−	1.80565i	−1.80565	−	1.80565i
$197$	0.292893	+	0.707107i	0.292893	+	0.707107i	1.00000			$0$
$197$	0.292893	+	0.707107i	0.292893	+	0.707107i	−0.707107	+	0.707107i	$0.750000\pi$
$198$		0			0
$199$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$199$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$200$			3.07768i			3.07768i
$201$		0			0
$202$	−0.420808	+	0.420808i	−0.420808	+	0.420808i
$203$		0			0
$204$	−2.33269	+	1.42947i	−2.33269	+	1.42947i
$205$		0			0
$206$	2.65688	−	2.65688i	2.65688	−	2.65688i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$211$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$212$	−1.68088	−	1.68088i	−1.68088	−	1.68088i
$213$	1.43702	+	1.43702i	1.43702	+	1.43702i
$214$		0			0
$215$		0			0
$216$	1.11752	−	2.69793i	1.11752	−	2.69793i
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$	−3.38959			−3.38959
$223$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$223$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$224$	−0.446183	−	0.184815i	−0.446183	−	0.184815i
$225$		−	0.0920190i		−	0.0920190i
$226$		0			0
$227$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$227$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$228$		0			0
$229$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$229$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$233$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$234$		0			0
$235$		0			0
$236$	1.14412	−	1.14412i	1.14412	−	1.14412i
$237$	2.03225			2.03225
$238$	0.155953	+	0.254493i	0.155953	+	0.254493i
$239$	−1.17557			−1.17557			−0.587785	−	0.809017i	$-0.700000\pi$
$239$	−1.17557			−1.17557			−0.587785	+	0.809017i	$0.700000\pi$
$240$		0			0
$241$	−1.57547	−	0.652583i	−1.57547	−	0.652583i	−0.587785	−	0.809017i	$-0.700000\pi$
$241$	−1.57547	−	0.652583i	−1.57547	−	0.652583i	−0.987688	+	0.156434i	$0.950000\pi$
$242$		−	1.90211i		−	1.90211i
$243$	−0.0702112	+	0.169505i	−0.0702112	+	0.169505i
$244$	3.67846	−	1.52367i	3.67846	−	1.52367i
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	1.36535	−	0.565548i	1.36535	−	0.565548i
$250$		0			0
$251$		−	0.312869i		−	0.312869i	−0.987688	−	0.156434i	$-0.950000\pi$
$251$		−	0.312869i		−	0.312869i	0.987688	−	0.156434i	$-0.0500000\pi$
$252$	0.0349254	+	0.0144666i	0.0349254	+	0.0144666i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	1.00000			1.00000
$257$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$257$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$258$		0			0
$259$			0.267590i			0.267590i
$260$		0			0
$261$		0			0
$262$	0.945476	+	2.28258i	0.945476	+	2.28258i
$263$	−0.437016	−	0.437016i	−0.437016	−	0.437016i	0.453990	−	0.891007i	$-0.350000\pi$
$263$	−0.437016	−	0.437016i	−0.437016	−	0.437016i	−0.891007	+	0.453990i	$0.850000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	0.760661	−	1.83640i	0.760661	−	1.83640i
$268$		0			0
$269$	−1.70711	−	0.707107i	−1.70711	−	0.707107i	−0.707107	−	0.707107i	$-0.750000\pi$
$269$	−1.70711	−	0.707107i	−1.70711	−	0.707107i	−1.00000			$\pi$
$270$		0			0
$271$	0.312869			0.312869			0.156434	−	0.987688i	$-0.450000\pi$
$271$	0.312869			0.312869			0.156434	+	0.987688i	$0.450000\pi$
$272$	−2.61803	−	1.90211i	−2.61803	−	1.90211i
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	0.744220	−	1.79671i	0.744220	−	1.79671i	0.156434	−	0.987688i	$-0.450000\pi$
$277$	0.744220	−	1.79671i	0.744220	−	1.79671i	0.587785	−	0.809017i	$-0.300000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$281$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$282$	0.760661	+	1.83640i	0.760661	+	1.83640i
$283$	−0.144974	+	0.0600500i	−0.144974	+	0.0600500i	−0.453990	−	0.891007i	$-0.650000\pi$
$283$	−0.144974	+	0.0600500i	−0.144974	+	0.0600500i	0.309017	+	0.951057i	$0.400000\pi$
$284$	−1.94839	+	4.70384i	−1.94839	+	4.70384i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.283205			−0.283205
$289$	0.309017	+	0.951057i	0.309017	+	0.951057i
$290$		0			0
$291$	−0.959786	+	0.959786i	−0.959786	+	0.959786i
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	−0.741931	+	1.79118i	−0.741931	+	1.79118i
$295$		0			0
$296$	−2.00844	−	4.84881i	−2.00844	−	4.84881i
$297$		0			0
$298$	2.65688	+	2.65688i	2.65688	+	2.65688i
$299$		0			0
$300$	2.52758	−	1.04696i	2.52758	−	1.04696i
$301$		0			0
$302$		0			0
$303$	0.302060	+	0.125117i	0.302060	+	0.125117i
$304$		0			0
$305$		0			0
$306$	0.141603	+	0.102880i	0.141603	+	0.102880i
$307$	0.312869			0.312869			0.156434	−	0.987688i	$-0.450000\pi$
$307$	0.312869			0.312869			0.156434	+	0.987688i	$0.450000\pi$
$308$		0			0
$309$	−1.90713	−	0.789959i	−1.90713	−	0.789959i
$310$		0			0
$311$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$311$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$312$		0			0
$313$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$313$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$314$	2.39680	+	2.39680i	2.39680	+	2.39680i
$315$		0			0
$316$	1.94839	+	4.70384i	1.94839	+	4.70384i
$317$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$317$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$318$	−0.690666	+	1.66741i	−0.690666	+	1.66741i
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−2.83677			−2.83677
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	0.144974	−	0.0600500i	0.144974	−	0.0600500i
$330$		0			0
$331$	1.34500	+	1.34500i	1.34500	+	1.34500i	0.891007	+	0.453990i	$0.150000\pi$
$331$	1.34500	+	1.34500i	1.34500	+	1.34500i	0.453990	+	0.891007i	$0.350000\pi$
$332$	2.61803	+	2.61803i	2.61803	+	2.61803i
$333$	0.0600500	+	0.144974i	0.0600500	+	0.144974i
$334$		0			0
$335$		0			0
$336$			0.530647i			0.530647i
$337$	1.84206	+	0.763007i	1.84206	+	0.763007i	0.951057	+	0.309017i	$0.100000\pi$
$337$	1.84206	+	0.763007i	1.84206	+	0.763007i	0.891007	+	0.453990i	$0.150000\pi$
$338$	−1.34500	+	1.34500i	−1.34500	+	1.34500i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.286377	+	0.118621i	0.286377	+	0.118621i
$344$		0			0
$345$		0			0
$346$	−3.50381	+	1.45133i	−3.50381	+	1.45133i
$347$	−0.399903	−	0.965451i	−0.399903	−	0.965451i	−0.987688	−	0.156434i	$-0.950000\pi$
$347$	−0.399903	−	0.965451i	−0.399903	−	0.965451i	0.587785	−	0.809017i	$-0.300000\pi$
$348$		0			0
$349$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$349$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$350$	−0.114222	−	0.275756i	−0.114222	−	0.275756i
$351$		0			0
$352$		0			0
$353$		−	1.78201i		−	1.78201i	−0.453990	−	0.891007i	$-0.650000\pi$
$353$		−	1.78201i		−	1.78201i	0.453990	−	0.891007i	$-0.350000\pi$
$354$	−1.13496	−	0.470114i	−1.13496	−	0.470114i
$355$		0			0
$356$	4.97980			4.97980
$357$	0.0963845	−	0.132662i	0.0963845	−	0.132662i
$358$		0			0
$359$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$359$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$360$		0			0
$361$		−	1.00000i		−	1.00000i
$362$		0			0
$363$	−0.965451	+	0.399903i	−0.965451	+	0.399903i
$364$		0			0
$365$		0			0
$366$	−2.13753	−	2.13753i	−2.13753	−	2.13753i
$367$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$367$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	0.131633	+	0.0545243i	0.131633	+	0.0545243i
$372$		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
$378$			0.283205i			0.283205i
$379$	0.763007	−	1.84206i	0.763007	−	1.84206i	0.309017	−	0.951057i	$-0.400000\pi$
$379$	0.763007	−	1.84206i	0.763007	−	1.84206i	0.453990	−	0.891007i	$-0.350000\pi$
$380$		0			0
$381$		0			0
$382$	1.90211	+	1.90211i	1.90211	+	1.90211i
$383$	−1.39680	−	1.39680i	−1.39680	−	1.39680i	−0.809017	−	0.587785i	$-0.800000\pi$
$383$	−1.39680	−	1.39680i	−1.39680	−	1.39680i	−0.587785	−	0.809017i	$-0.700000\pi$
$384$	−0.760661	−	1.83640i	−0.760661	−	1.83640i
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−3.14170	−	1.30134i	−3.14170	−	1.30134i
$389$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$389$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$390$		0			0
$391$		0			0
$392$	−3.00190			−3.00190
$393$	0.959786	−	0.959786i	0.959786	−	0.959786i
$394$	1.34500	+	0.557116i	1.34500	+	0.557116i
$395$		0			0
$396$		0			0
$397$	0.431351	−	0.178671i	0.431351	−	0.178671i	−0.156434	−	0.987688i	$-0.550000\pi$
$397$	0.431351	−	0.178671i	0.431351	−	0.178671i	0.587785	+	0.809017i	$0.300000\pi$
$398$		0			0
$399$		0			0
$400$	2.28825	+	2.28825i	2.28825	+	2.28825i
$401$	0.399903	+	0.965451i	0.399903	+	0.965451i	0.987688	+	0.156434i	$0.0500000\pi$
$401$	0.399903	+	0.965451i	0.399903	+	0.965451i	−0.587785	+	0.809017i	$0.700000\pi$
$402$		0			0
$403$		0			0
$404$			0.819101i			0.819101i
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−0.750800	+	3.12731i	−0.750800	+	3.12731i
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$		−	5.17160i		−	5.17160i
$413$	−0.0371129	+	0.0895986i	−0.0371129	+	0.0895986i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$419$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$420$		0			0
$421$		0			0			−	1.00000i	$-0.5\pi$
$421$		0			0				1.00000i	$0.5\pi$
$422$		0			0
$423$	0.0650673	−	0.0650673i	0.0650673	−	0.0650673i
$424$	−2.79448			−2.79448
$425$	−0.156434	−	0.987688i	−0.156434	−	0.987688i
$426$	3.86556			3.86556
$427$	−0.168746	+	0.168746i	−0.168746	+	0.168746i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.20002	−	0.497066i	1.20002	−	0.497066i	0.309017	−	0.951057i	$-0.400000\pi$
$431$	1.20002	−	0.497066i	1.20002	−	0.497066i	0.891007	+	0.453990i	$0.150000\pi$
$432$	−1.17503	−	2.83677i	−1.17503	−	2.83677i
$433$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$433$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−0.707107	−	0.292893i	−0.707107	−	0.292893i		−	1.00000i	$-0.5\pi$
$439$	−0.707107	−	0.292893i	−0.707107	−	0.292893i	−0.707107	+	0.707107i	$0.750000\pi$
$440$		0			0
$441$	0.0897532			0.0897532
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$	−3.29892	+	3.29892i	−3.29892	+	3.29892i
$445$		0			0
$446$		0			0
$447$	0.789959	−	1.90713i	0.789959	−	1.90713i
$448$	−0.379546	+	0.157213i	−0.379546	+	0.157213i
$449$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$449$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$450$	−0.123765	−	0.123765i	−0.123765	−	0.123765i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.221232	−	0.221232i	0.221232	−	0.221232i	−0.587785	−	0.809017i	$-0.700000\pi$
$457$	0.221232	−	0.221232i	0.221232	−	0.221232i	0.809017	+	0.587785i	$0.200000\pi$
$458$		0			0
$459$	−0.221502	+	0.922621i	−0.221502	+	0.922621i
$460$		0			0
$461$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$461$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$462$		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	$-0.750000\pi$
$467$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	0.712633	−	1.72045i	0.712633	−	1.72045i
$472$		−	1.90211i		−	1.90211i
$473$		0			0
$474$	2.73337	−	2.73337i	2.73337	−	2.73337i
$475$		0			0
$476$	0.399466	+	0.0959034i	0.399466	+	0.0959034i
$477$	0.0835515			0.0835515
$478$	−1.58114	+	1.58114i	−1.58114	+	1.58114i
$479$	0.144974	+	0.0600500i	0.144974	+	0.0600500i	0.453990	−	0.891007i	$-0.350000\pi$
$479$	0.144974	+	0.0600500i	0.144974	+	0.0600500i	−0.309017	+	0.951057i	$0.600000\pi$
$480$		0			0
$481$		0			0
$482$	−2.99673	+	1.24129i	−2.99673	+	1.24129i
$483$		0			0
$484$	−1.85123	−	1.85123i	−1.85123	−	1.85123i
$485$		0			0
$486$	0.133550	+	0.322417i	0.133550	+	0.322417i
$487$	0.707107	−	0.292893i	0.707107	−	0.292893i		−	1.00000i	$-0.5\pi$
$487$	0.707107	−	0.292893i	0.707107	−	0.292893i	0.707107	+	0.707107i	$0.250000\pi$
$488$	1.79118	−	4.32429i	1.79118	−	4.32429i
$489$		0			0
$490$		0			0
$491$	1.26007	−	1.26007i	1.26007	−	1.26007i	0.309017	−	0.951057i	$-0.400000\pi$
$491$	1.26007	−	1.26007i	1.26007	−	1.26007i	0.951057	−	0.309017i	$-0.100000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	0.305165i		−	0.305165i
$498$	1.07574	−	2.59706i	1.07574	−	2.59706i
$499$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$499$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$500$		0			0
$501$		0			0
$502$	−0.420808	−	0.420808i	−0.420808	−	0.420808i
$503$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$503$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$504$	0.0410573	−	0.0170065i	0.0410573	−	0.0170065i
$505$		0			0
$506$		0			0
$507$	0.965451	+	0.399903i	0.965451	+	0.399903i
$508$		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
$518$	0.359907	+	0.359907i	0.359907	+	0.359907i
$519$	1.47329	+	1.47329i	1.47329	+	1.47329i
$520$		0			0
$521$	−1.84206	+	0.763007i	−1.84206	+	0.763007i	−0.891007	+	0.453990i	$0.850000\pi$
$521$	−1.84206	+	0.763007i	−1.84206	+	0.763007i	−0.951057	+	0.309017i	$0.900000\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$	3.14170	+	1.30134i	3.14170	+	1.30134i
$525$	−0.115951	+	0.115951i	−0.115951	+	0.115951i
$526$	−1.17557			−1.17557
$527$		0			0
$528$		0			0
$529$	0.707107	−	0.707107i	0.707107	−	0.707107i
$530$		0			0
$531$			0.0568709i			0.0568709i
$532$		0			0
$533$		0			0
$534$	−1.44686	−	3.49304i	−1.44686	−	3.49304i
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−3.24711	+	1.34500i	−3.24711	+	1.34500i
$539$		0			0
$540$		0			0
$541$	−1.84206	−	0.763007i	−1.84206	−	0.763007i	−0.951057	−	0.309017i	$-0.900000\pi$
$541$	−1.84206	−	0.763007i	−1.84206	−	0.763007i	−0.891007	−	0.453990i	$-0.850000\pi$
$542$	0.420808	−	0.420808i	0.420808	−	0.420808i
$543$		0			0
$544$	−3.03979	+	0.481456i	−3.03979	+	0.481456i
$545$		0			0
$546$		0			0
$547$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$547$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$548$		0			0
$549$	−0.0535541	+	0.129291i	−0.0535541	+	0.129291i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−0.215784	−	0.215784i	−0.215784	−	0.215784i
$554$	−1.41559	−	3.41754i	−1.41559	−	3.41754i
$555$		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	$-0.250000\pi$
$563$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$564$	2.52758	+	1.04696i	2.52758	+	1.04696i
$565$		0			0
$566$	−0.114222	+	0.275756i	−0.114222	+	0.275756i
$567$	0.157086	−	0.0650673i	0.157086	−	0.0650673i
$568$	2.29047	+	5.52969i	2.29047	+	5.52969i
$569$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$569$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$570$		0			0
$571$	−0.178671	−	0.431351i	−0.178671	−	0.431351i	0.809017	−	0.587785i	$-0.200000\pi$
$571$	−0.178671	−	0.431351i	−0.178671	−	0.431351i	−0.987688	+	0.156434i	$0.950000\pi$
$572$		0			0
$573$	0.565548	−	1.36535i	0.565548	−	1.36535i
$574$		0			0
$575$		0			0
$576$	−0.170348	+	0.170348i	−0.170348	+	0.170348i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	1.69480	+	0.863541i	1.69480	+	0.863541i
$579$		0			0
$580$		0			0
$581$	−0.205023	−	0.0849235i	−0.205023	−	0.0849235i
$582$			2.58182i			2.58182i
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$587$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$588$	1.02118	+	2.46535i	1.02118	+	2.46535i
$589$		0			0
$590$		0			0
$591$		−	0.799806i		−	0.799806i
$592$	−5.09834	−	2.11180i	−5.09834	−	2.11180i
$593$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$593$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$594$		0			0
$595$		0			0
$596$	5.17160			5.17160
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$	1.23078	−	2.97135i	1.23078	−	2.97135i
$601$	−1.40505	+	0.581990i	−1.40505	+	0.581990i	−0.951057	−	0.309017i	$-0.900000\pi$
$601$	−1.40505	+	0.581990i	−1.40505	+	0.581990i	−0.453990	+	0.891007i	$0.650000\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$	0.574552	−	0.237987i	0.574552	−	0.237987i
$607$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$607$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	0.237943	−	0.0376865i	0.237943	−	0.0376865i
$613$	−1.90211			−1.90211			−0.951057	−	0.309017i	$-0.900000\pi$
$613$	−1.90211			−1.90211			−0.951057	+	0.309017i	$0.900000\pi$
$614$	0.420808	−	0.420808i	0.420808	−	0.420808i
$615$		0			0
$616$		0			0
$617$	0.178671	−	0.431351i	0.178671	−	0.431351i	−0.809017	−	0.587785i	$-0.800000\pi$
$617$	0.178671	−	0.431351i	0.178671	−	0.431351i	0.987688	+	0.156434i	$0.0500000\pi$
$618$	−3.62758	+	1.50259i	−3.62758	+	1.50259i
$619$	0.707107	+	1.70711i	0.707107	+	1.70711i	0.707107	+	0.707107i	$0.250000\pi$
$619$	0.707107	+	1.70711i	0.707107	+	1.70711i			1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−0.275756	+	0.114222i	−0.275756	+	0.114222i
$624$		0			0
$625$			1.00000i			1.00000i
$626$		0			0
$627$		0			0
$628$	4.66537			4.66537
$629$	0.891007	+	1.45399i	0.891007	+	1.45399i
$630$		0			0
$631$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$631$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$632$	5.52969	+	2.29047i	5.52969	+	2.29047i
$633$		0			0
$634$		0			0
$635$		0			0
$636$	0.950620	+	2.29500i	0.950620	+	2.29500i
$637$		0			0
$638$		0			0
$639$	−0.0684824	−	0.165331i	−0.0684824	−	0.165331i
$640$		0			0
$641$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$641$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$642$		0			0
$643$	1.40505	+	0.581990i	1.40505	+	0.581990i	0.951057	−	0.309017i	$-0.100000\pi$
$643$	1.40505	+	0.581990i	1.40505	+	0.581990i	0.453990	+	0.891007i	$0.350000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1.78201			1.78201			0.891007	−	0.453990i	$-0.150000\pi$
$647$	1.78201			1.78201			0.891007	+	0.453990i	$0.150000\pi$
$648$	−2.35808	+	2.35808i	−2.35808	+	2.35808i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	0.652583	+	1.57547i	0.652583	+	1.57547i	0.809017	+	0.587785i	$0.200000\pi$
$653$	0.652583	+	1.57547i	0.652583	+	1.57547i	−0.156434	+	0.987688i	$0.550000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	0.114222	−	0.275756i	0.114222	−	0.275756i
$659$		−	1.61803i		−	1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$659$		−	1.61803i		−	1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$660$		0			0
$661$	−0.831254	+	0.831254i	−0.831254	+	0.831254i	−0.987688	−	0.156434i	$-0.950000\pi$
$661$	−0.831254	+	0.831254i	−0.831254	+	0.831254i	0.156434	+	0.987688i	$0.450000\pi$
$662$	3.61803			3.61803
$663$		0			0
$664$	4.35250			4.35250
$665$		0			0
$666$	0.275756	+	0.114222i	0.275756	+	0.114222i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$	0.356860	+	0.356860i	0.356860	+	0.356860i
$673$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$673$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$674$	3.50381	−	1.45133i	3.50381	−	1.45133i
$675$	0.363104	−	0.876612i	0.363104	−	0.876612i
$676$			2.61803i			2.61803i
$677$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$677$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$678$		0			0
$679$	0.203820			0.203820
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.40505	−	0.581990i	−1.40505	−	0.581990i	−0.453990	−	0.891007i	$-0.650000\pi$
$683$	−1.40505	−	0.581990i	−1.40505	−	0.581990i	−0.951057	+	0.309017i	$0.900000\pi$
$684$		0			0
$685$		0			0
$686$	0.544722	−	0.225631i	0.544722	−	0.225631i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$691$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$692$	−1.99758	+	4.82258i	−1.99758	+	4.82258i
$693$		0			0
$694$	−1.83640	−	0.760661i	−1.83640	−	0.760661i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.379546	−	0.157213i	−0.379546	−	0.157213i
$701$		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
$706$	−2.39680	−	2.39680i	−2.39680	−	2.39680i
$707$	−0.0187878	−	0.0453577i	−0.0187878	−	0.0453577i
$708$	−1.56213	+	0.647057i	−1.56213	+	0.647057i
$709$	0.0600500	−	0.144974i	0.0600500	−	0.144974i	−0.891007	−	0.453990i	$-0.850000\pi$
$709$	0.0600500	−	0.144974i	0.0600500	−	0.144974i	0.951057	+	0.309017i	$0.100000\pi$
$710$		0			0
$711$	−0.165331	−	0.0684824i	−0.165331	−	0.0684824i
$712$	4.13948	−	4.13948i	4.13948	−	4.13948i
$713$		0			0
$714$	−0.0487930	−	0.308067i	−0.0487930	−	0.308067i
$715$		0			0
$716$		0			0
$717$	1.13496	+	0.470114i	1.13496	+	0.470114i
$718$		0			0
$719$	−0.581990	+	1.40505i	−0.581990	+	1.40505i	0.309017	+	0.951057i	$0.400000\pi$
$719$	−0.581990	+	1.40505i	−0.581990	+	1.40505i	−0.891007	+	0.453990i	$0.850000\pi$
$720$		0			0
$721$	0.118621	+	0.286377i	0.118621	+	0.286377i
$722$	−1.34500	−	1.34500i	−1.34500	−	1.34500i
$723$	1.26007	+	1.26007i	1.26007	+	1.26007i
$724$		0			0
$725$		0			0
$726$	−0.760661	+	1.83640i	−0.760661	+	1.83640i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	−0.630616	+	0.630616i	−0.630616	+	0.630616i
$730$		0			0
$731$		0			0
$732$	−4.16070			−4.16070
$733$	−0.831254	+	0.831254i	−0.831254	+	0.831254i	−0.987688	−	0.156434i	$-0.950000\pi$
$733$	−0.831254	+	0.831254i	−0.831254	+	0.831254i	0.156434	+	0.987688i	$0.450000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−1.26007	−	1.26007i	−1.26007	−	1.26007i	−0.951057	−	0.309017i	$-0.900000\pi$
$739$	−1.26007	−	1.26007i	−1.26007	−	1.26007i	−0.309017	−	0.951057i	$-0.600000\pi$
$740$		0			0
$741$		0			0
$742$	0.250381	−	0.103711i	0.250381	−	0.103711i
$743$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$743$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−0.130135			−0.130135
$748$		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	$0.125000\pi$
$752$			3.23607i			3.23607i
$753$	−0.125117	+	0.302060i	−0.125117	+	0.302060i
$754$		0			0
$755$		0			0
$756$	0.275629	+	0.275629i	0.275629	+	0.275629i
$757$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$757$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$758$	−1.45133	−	3.50381i	−1.45133	−	3.50381i
$759$		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
$766$	−3.75739			−3.75739
$767$		0			0
$768$	−0.965451	−	0.399903i	−0.965451	−	0.399903i
$769$		−	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	0.987688	−	0.156434i	$-0.0500000\pi$
$773$	0.831254	+	0.831254i	0.831254	+	0.831254i	−0.156434	+	0.987688i	$0.550000\pi$
$774$		0			0
$775$		0			0
$776$	−3.69329	+	1.52981i	−3.69329	+	1.52981i
$777$	0.107010	−	0.258345i	0.107010	−	0.258345i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−2.23190	+	2.23190i	−2.23190	+	2.23190i
$785$		0			0
$786$		−	2.58182i		−	2.58182i
$787$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$787$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$788$	1.85123	−	0.766804i	1.85123	−	0.766804i
$789$	0.247154	+	0.596682i	0.247154	+	0.596682i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	0.339853	−	0.820478i	0.339853	−	0.820478i
$795$		0			0
$796$		0			0
$797$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$797$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$798$		0			0
$799$	0.587785	−	0.809017i	0.587785	−	0.809017i
$800$	3.07768			3.07768
$801$	−0.123765	+	0.123765i	−0.123765	+	0.123765i
$802$	1.83640	+	0.760661i	1.83640	+	0.760661i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$	1.36535	+	1.36535i	1.36535	+	1.36535i
$808$	0.680881	+	0.680881i	0.680881	+	0.680881i
$809$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$809$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$810$		0			0
$811$	−0.744220	+	1.79671i	−0.744220	+	1.79671i	−0.156434	+	0.987688i	$0.550000\pi$
$811$	−0.744220	+	1.79671i	−0.744220	+	1.79671i	−0.587785	+	0.809017i	$0.700000\pi$
$812$		0			0
$813$	−0.302060	−	0.125117i	−0.302060	−	0.125117i
$814$		0			0
$815$		0			0
$816$	1.76692	+	2.88336i	1.76692	+	2.88336i
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$821$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$822$		0			0
$823$	0.707107	+	1.70711i	0.707107	+	1.70711i	0.707107	+	0.707107i	$0.250000\pi$
$823$	0.707107	+	1.70711i	0.707107	+	1.70711i			1.00000i	$0.5\pi$
$824$	−4.29892	−	4.29892i	−4.29892	−	4.29892i
$825$		0			0
$826$	0.0705930	+	0.170427i	0.0705930	+	0.170427i
$827$	−1.40505	+	0.581990i	−1.40505	+	0.581990i	−0.951057	−	0.309017i	$-0.900000\pi$
$827$	−1.40505	+	0.581990i	−1.40505	+	0.581990i	−0.453990	+	0.891007i	$0.650000\pi$
$828$		0			0
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		0			0
$831$	−1.43702	+	1.43702i	−1.43702	+	1.43702i
$832$		0			0
$833$	0.963368	−	0.152583i	0.963368	−	0.152583i
$834$		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
$841$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		−	0.175031i		−	0.175031i
$847$	0.144974	+	0.0600500i	0.144974	+	0.0600500i
$848$	−2.07768	+	2.07768i	−2.07768	+	2.07768i
$849$	0.163979			0.163979
$850$	−1.53884	−	1.11803i	−1.53884	−	1.11803i
$851$		0			0
$852$	3.76216	−	3.76216i	3.76216	−	3.76216i
$853$	−0.431351	−	0.178671i	−0.431351	−	0.178671i	0.156434	−	0.987688i	$-0.450000\pi$
$853$	−0.431351	−	0.178671i	−0.431351	−	0.178671i	−0.587785	+	0.809017i	$0.700000\pi$
$854$			0.453926i			0.453926i
$855$		0			0
$856$		0			0
$857$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$857$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$858$		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
$862$	0.945476	−	2.28258i	0.945476	−	2.28258i
$863$			1.61803i			1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$863$			1.61803i			1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$864$	−2.69793	−	1.11752i	−2.69793	−	1.11752i
$865$		0			0
$866$		0			0
$867$	0.0819895	−	1.04178i	0.0819895	−	1.04178i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	0.110425	−	0.0457395i	0.110425	−	0.0457395i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$877$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$878$	−1.34500	+	0.557116i	−1.34500	+	0.557116i
$879$		0			0
$880$		0			0
$881$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$881$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$882$	0.120718	−	0.120718i	0.120718	−	0.120718i
$883$	1.78201			1.78201			0.891007	−	0.453990i	$-0.150000\pi$
$883$	1.78201			1.78201			0.891007	+	0.453990i	$0.150000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$887$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$888$			5.48447i			5.48447i
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$	−1.50259	−	3.62758i	−1.50259	−	3.62758i
$895$		0			0
$896$	−0.114222	+	0.275756i	−0.114222	+	0.275756i
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.240909			−0.240909
$901$	0.896802	−	0.142040i	0.896802	−	0.142040i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1.70711	−	0.707107i	1.70711	−	0.707107i	0.707107	−	0.707107i	$-0.250000\pi$
$907$	1.70711	−	0.707107i	1.70711	−	0.707107i	1.00000			$0$
$908$		0			0
$909$	−0.0203575	−	0.0203575i	−0.0203575	−	0.0203575i
$910$		0			0
$911$	−0.581990	−	1.40505i	−0.581990	−	1.40505i	−0.891007	−	0.453990i	$-0.850000\pi$
$911$	−0.581990	−	1.40505i	−0.581990	−	1.40505i	0.309017	−	0.951057i	$-0.400000\pi$
$912$		0			0
$913$		0			0
$914$		−	0.595112i		−	0.595112i
$915$		0			0
$916$		0			0
$917$	−0.203820			−0.203820
$918$	0.943003	+	1.53884i	0.943003	+	1.53884i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	−0.302060	−	0.125117i	−0.302060	−	0.125117i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.652583	−	1.57547i	−0.652583	−	1.57547i
$926$		0			0
$927$	0.128532	+	0.128532i	0.128532	+	0.128532i
$928$		0			0
$929$	1.57547	−	0.652583i	1.57547	−	0.652583i	0.587785	−	0.809017i	$-0.300000\pi$
$929$	1.57547	−	0.652583i	1.57547	−	0.652583i	0.987688	+	0.156434i	$0.0500000\pi$
$930$		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	$-0.250000\pi$
$937$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1.20002	+	0.497066i	−1.20002	+	0.497066i	−0.891007	−	0.453990i	$-0.850000\pi$
$941$	−1.20002	+	0.497066i	−1.20002	+	0.497066i	−0.309017	+	0.951057i	$0.600000\pi$
$942$	−1.35551	−	3.27248i	−1.35551	−	3.27248i
$943$		0			0
$944$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$945$		0			0
$946$		0			0
$947$	0.707107	−	1.70711i	0.707107	−	1.70711i		−	1.00000i	$-0.5\pi$
$947$	0.707107	−	1.70711i	0.707107	−	1.70711i	0.707107	−	0.707107i	$-0.250000\pi$
$948$		−	5.32049i		−	5.32049i
$949$		0			0
$950$		0			0
$951$		0			0
$952$	0.411778	−	0.252338i	0.411778	−	0.252338i
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$	0.112377	−	0.112377i	0.112377	−	0.112377i
$955$		0			0
$956$			3.07768i			3.07768i
$957$		0			0
$958$	0.275756	−	0.114222i	0.275756	−	0.114222i
$959$		0			0
$960$		0			0
$961$	0.707107	+	0.707107i	0.707107	+	0.707107i
$962$		0			0
$963$		0			0
$964$	−1.70848	+	4.12464i	−1.70848	+	4.12464i
$965$		0			0
$966$		0			0
$967$	−1.26007	+	1.26007i	−1.26007	+	1.26007i	−0.309017	+	0.951057i	$0.600000\pi$
$967$	−1.26007	+	1.26007i	−1.26007	+	1.26007i	−0.951057	+	0.309017i	$0.900000\pi$
$968$	−3.07768			−3.07768
$969$		0			0
$970$		0			0
$971$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$971$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$972$	0.443769	+	0.183815i	0.443769	+	0.183815i
$973$		0			0
$974$	0.557116	−	1.34500i	0.557116	−	1.34500i
$975$		0			0
$976$	−1.88336	−	4.54683i	−1.88336	−	4.54683i
$977$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$977$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		−	3.38959i		−	3.38959i
$983$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$983$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−0.163979			−0.163979
$988$		0			0
$989$		0			0
$990$		0			0
$991$	0.581990	−	1.40505i	0.581990	−	1.40505i	−0.309017	−	0.951057i	$-0.600000\pi$
$991$	0.581990	−	1.40505i	0.581990	−	1.40505i	0.891007	−	0.453990i	$-0.150000\pi$
$992$		0			0
$993$	−0.760661	−	1.83640i	−0.760661	−	1.83640i
$994$	−0.410446	−	0.410446i	−0.410446	−	0.410446i
$995$		0			0
$996$	−1.48062	−	3.57454i	−1.48062	−	3.57454i
$997$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$997$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$998$		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.281.4	✓	16
17.2	even	8	inner	799.1.h.b.563.4	yes	16
47.46	odd	2	CM	799.1.h.b.281.4	✓	16
799.563	odd	8	inner	799.1.h.b.563.4	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
799.1.h.b.281.4	✓	16	1.1	even	1	trivial
799.1.h.b.281.4	✓	16	47.46	odd	2	CM
799.1.h.b.563.4	yes	16	17.2	even	8	inner
799.1.h.b.563.4	yes	16	799.563	odd	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:	\( N \)	\(=\)	\( 799 = 17 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	799.h (of order \(8\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(1.34500 - 1.34500i) q^{2} +(-0.965451 - 0.399903i) q^{3} -2.61803i q^{4} +(-1.83640 + 0.760661i) q^{6} +(0.0600500 + 0.144974i) q^{7} +(-2.17625 - 2.17625i) q^{8} +(0.0650673 + 0.0650673i) q^{9} +(-1.04696 + 2.52758i) q^{12} +(0.275756 + 0.114222i) q^{14} -3.23607 q^{16} +(0.809017 + 0.587785i) q^{17} +0.175031 q^{18} -0.163979i q^{21} +(1.23078 + 2.97135i) q^{24} +(-0.707107 - 0.707107i) q^{25} +(0.363104 + 0.876612i) q^{27} +(0.379546 - 0.157213i) q^{28} +(-2.17625 + 2.17625i) q^{32} +(1.87869 - 0.297556i) q^{34} +(0.170348 - 0.170348i) q^{36} +(1.57547 + 0.652583i) q^{37} +(-0.220551 - 0.220551i) q^{42} -1.00000i q^{47} +(3.12427 + 1.29411i) q^{48} +(0.689695 - 0.689695i) q^{49} -1.90211 q^{50} +(-0.546010 - 0.891007i) q^{51} +(0.642040 - 0.642040i) q^{53} +(1.66741 + 0.690666i) q^{54} +(0.184815 - 0.446183i) q^{56} +(0.437016 + 0.437016i) q^{59} +(0.581990 + 1.40505i) q^{61} +(-0.00552574 + 0.0133403i) q^{63} +2.61803i q^{64} +(1.53884 - 2.11803i) q^{68} +(-1.79671 - 0.744220i) q^{71} -0.283205i q^{72} +(2.99673 - 1.24129i) q^{74} +(0.399903 + 0.965451i) q^{75} +(-1.79671 + 0.744220i) q^{79} -1.08355i q^{81} +(-1.00000 + 1.00000i) q^{83} -0.429303 q^{84} +1.90211i q^{89} +(-1.34500 - 1.34500i) q^{94} +(2.97135 - 1.23078i) q^{96} +(0.497066 - 1.20002i) q^{97} -1.85528i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{3} - 4 q^{9} - 16 q^{16} + 4 q^{17} - 20 q^{24} - 4 q^{27} - 4 q^{28} + 4 q^{36} - 20 q^{42} + 24 q^{48} + 4 q^{49} - 16 q^{51} + 4 q^{53} + 20 q^{54} + 20 q^{56} + 4 q^{61} - 4 q^{63} - 4 q^{71}+ \cdots + 32 q^{84}+O(q^{100}) \) 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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