LMFDB - Embedded newform 799.1.h.b.281.1

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			281.1
Root			$-0.891007 - 0.453990i$ of defining polynomial
Character	$\chi$	$=$	799.281
Dual form			799.1.h.b.563.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.34500 + 1.34500i) q^{2} +(-1.79671 - 0.744220i) q^{3} -2.61803i q^{4} +(3.41754 - 1.41559i) q^{6} +(-0.497066 - 1.20002i) q^{7} +(2.17625 + 2.17625i) q^{8} +(1.96718 + 1.96718i) q^{9} +(-1.94839 + 4.70384i) q^{12} +(2.28258 + 0.945476i) q^{14} -3.23607 q^{16} +(0.809017 - 0.587785i) q^{17} -5.29170 q^{18} +2.52601i q^{21} +(-2.29047 - 5.52969i) q^{24} +(-0.707107 - 0.707107i) q^{25} +(-1.32621 - 3.20175i) q^{27} +(-3.14170 + 1.30134i) q^{28} +(2.17625 - 2.17625i) q^{32} +(-0.297556 + 1.87869i) q^{34} +(5.15014 - 5.15014i) q^{36} +(-0.431351 - 0.178671i) q^{37} +(-3.39748 - 3.39748i) q^{42} -1.00000i q^{47} +(5.81426 + 2.40835i) q^{48} +(-0.485875 + 0.485875i) q^{49} +1.90211 q^{50} +(-1.89101 + 0.453990i) q^{51} +(-1.26007 + 1.26007i) q^{53} +(6.09009 + 2.52260i) q^{54} +(1.52981 - 3.69329i) q^{56} +(0.437016 + 0.437016i) q^{59} +(-0.763007 - 1.84206i) q^{61} +(1.38284 - 3.33848i) q^{63} +2.61803i q^{64} +(-1.53884 - 2.11803i) q^{68} +(-0.965451 - 0.399903i) q^{71} +8.56216i q^{72} +(0.820478 - 0.339853i) q^{74} +(0.744220 + 1.79671i) q^{75} +(-0.965451 + 0.399903i) q^{79} +3.95758i q^{81} +(-1.00000 + 1.00000i) q^{83} +6.61319 q^{84} -1.90211i q^{89} +(1.34500 + 1.34500i) q^{94} +(-5.52969 + 2.29047i) q^{96} +(-0.0600500 + 0.144974i) q^{97} -1.30700i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$0.650000\pi$
$2$	−1.34500	+	1.34500i	−1.34500	+	1.34500i	−0.891007	+	0.453990i	$0.850000\pi$
$3$	−1.79671	−	0.744220i	−1.79671	−	0.744220i	−0.987688	−	0.156434i	$-0.950000\pi$
$3$	−1.79671	−	0.744220i	−1.79671	−	0.744220i	−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	$0.625000\pi$
$6$	3.41754	−	1.41559i	3.41754	−	1.41559i
$7$	−0.497066	−	1.20002i	−0.497066	−	1.20002i	−0.951057	−	0.309017i	$-0.900000\pi$
$7$	−0.497066	−	1.20002i	−0.497066	−	1.20002i	0.453990	−	0.891007i	$-0.350000\pi$
$8$	2.17625	+	2.17625i	2.17625	+	2.17625i
$9$	1.96718	+	1.96718i	1.96718	+	1.96718i
$10$		0			0
$11$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$11$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$12$	−1.94839	+	4.70384i	−1.94839	+	4.70384i
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$	2.28258	+	0.945476i	2.28258	+	0.945476i
$15$		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
$18$	−5.29170			−5.29170
$19$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$19$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$20$		0			0
$21$			2.52601i			2.52601i
$22$		0			0
$23$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$23$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$24$	−2.29047	−	5.52969i	−2.29047	−	5.52969i
$25$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$26$		0			0
$27$	−1.32621	−	3.20175i	−1.32621	−	3.20175i
$28$	−3.14170	+	1.30134i	−3.14170	+	1.30134i
$29$		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
$34$	−0.297556	+	1.87869i	−0.297556	+	1.87869i
$35$		0			0
$36$	5.15014	−	5.15014i	5.15014	−	5.15014i
$37$	−0.431351	−	0.178671i	−0.431351	−	0.178671i	0.156434	−	0.987688i	$-0.450000\pi$
$37$	−0.431351	−	0.178671i	−0.431351	−	0.178671i	−0.587785	+	0.809017i	$0.700000\pi$
$38$		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	$0.375000\pi$
$42$	−3.39748	−	3.39748i	−3.39748	−	3.39748i
$43$		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
$48$	5.81426	+	2.40835i	5.81426	+	2.40835i
$49$	−0.485875	+	0.485875i	−0.485875	+	0.485875i
$50$	1.90211			1.90211
$51$	−1.89101	+	0.453990i	−1.89101	+	0.453990i
$52$		0			0
$53$	−1.26007	+	1.26007i	−1.26007	+	1.26007i	−0.309017	+	0.951057i	$0.600000\pi$
$53$	−1.26007	+	1.26007i	−1.26007	+	1.26007i	−0.951057	+	0.309017i	$0.900000\pi$
$54$	6.09009	+	2.52260i	6.09009	+	2.52260i
$55$		0			0
$56$	1.52981	−	3.69329i	1.52981	−	3.69329i
$57$		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
$61$	−0.763007	−	1.84206i	−0.763007	−	1.84206i	−0.453990	−	0.891007i	$-0.650000\pi$
$61$	−0.763007	−	1.84206i	−0.763007	−	1.84206i	−0.309017	−	0.951057i	$-0.600000\pi$
$62$		0			0
$63$	1.38284	−	3.33848i	1.38284	−	3.33848i
$64$			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
$71$	−0.965451	−	0.399903i	−0.965451	−	0.399903i	−0.156434	−	0.987688i	$-0.550000\pi$
$71$	−0.965451	−	0.399903i	−0.965451	−	0.399903i	−0.809017	+	0.587785i	$0.800000\pi$
$72$			8.56216i			8.56216i
$73$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$73$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$74$	0.820478	−	0.339853i	0.820478	−	0.339853i
$75$	0.744220	+	1.79671i	0.744220	+	1.79671i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−0.965451	+	0.399903i	−0.965451	+	0.399903i	−0.809017	−	0.587785i	$-0.800000\pi$
$79$	−0.965451	+	0.399903i	−0.965451	+	0.399903i	−0.156434	+	0.987688i	$0.550000\pi$
$80$		0			0
$81$			3.95758i			3.95758i
$82$		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			$\pi$
$84$	6.61319			6.61319
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	1.90211i		−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$89$		−	1.90211i		−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$	1.34500	+	1.34500i	1.34500	+	1.34500i
$95$		0			0
$96$	−5.52969	+	2.29047i	−5.52969	+	2.29047i
$97$	−0.0600500	+	0.144974i	−0.0600500	+	0.144974i	−0.951057	−	0.309017i	$-0.900000\pi$
$97$	−0.0600500	+	0.144974i	−0.0600500	+	0.144974i	0.891007	+	0.453990i	$0.150000\pi$
$98$		−	1.30700i		−	1.30700i
$99$		0			0
$100$	−1.85123	+	1.85123i	−1.85123	+	1.85123i
$101$	−1.97538			−1.97538			−0.987688	−	0.156434i	$-0.950000\pi$
$101$	−1.97538			−1.97538			−0.987688	+	0.156434i	$0.950000\pi$
$102$	1.93278	−	3.15401i	1.93278	−	3.15401i
$103$	0.312869			0.312869			0.156434	−	0.987688i	$-0.450000\pi$
$103$	0.312869			0.312869			0.156434	+	0.987688i	$0.450000\pi$
$104$		0			0
$105$		0			0
$106$		−	3.38959i		−	3.38959i
$107$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$107$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$108$	−8.38230	+	3.47206i	−8.38230	+	3.47206i
$109$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$109$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$110$		0			0
$111$	0.642040	+	0.642040i	0.642040	+	0.642040i
$112$	1.60854	+	3.88336i	1.60854	+	3.88336i
$113$		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
$119$	−1.10749	−	0.678671i	−1.10749	−	0.678671i
$120$		0			0
$121$	0.707107	−	0.707107i	0.707107	−	0.707107i
$122$	3.50381	+	1.45133i	3.50381	+	1.45133i
$123$		0			0
$124$		0			0
$125$		0			0
$126$	2.63033	+	6.35017i	2.63033	+	6.35017i
$127$		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
$131$	0.0600500	−	0.144974i	0.0600500	−	0.144974i	−0.891007	−	0.453990i	$-0.850000\pi$
$131$	0.0600500	−	0.144974i	0.0600500	−	0.144974i	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		−0.923880	−	0.382683i	$-0.875000\pi$
$139$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$140$		0			0
$141$	−0.744220	+	1.79671i	−0.744220	+	1.79671i
$142$	1.83640	−	0.760661i	1.83640	−	0.760661i
$143$		0			0
$144$	−6.36593	−	6.36593i	−6.36593	−	6.36593i
$145$		0			0
$146$		0			0
$147$	1.23457	−	0.511376i	1.23457	−	0.511376i
$148$	−0.467768	+	1.12929i	−0.467768	+	1.12929i
$149$			0.312869i			0.312869i	0.987688	+	0.156434i	$0.0500000\pi$
$149$			0.312869i			0.312869i	−0.987688	+	0.156434i	$0.950000\pi$
$150$	−3.41754	−	1.41559i	−3.41754	−	1.41559i
$151$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$151$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$152$		0			0
$153$	2.74776	+	0.435203i	2.74776	+	0.435203i
$154$		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	$-0.150000\pi$
$158$	0.760661	−	1.83640i	0.760661	−	1.83640i
$159$	3.20175	−	1.32621i	3.20175	−	1.32621i
$160$		0			0
$161$		0			0
$162$	−5.32294	−	5.32294i	−5.32294	−	5.32294i
$163$		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	$0.125000\pi$
$168$	−5.49724	+	5.49724i	−5.49724	+	5.49724i
$169$	−1.00000			−1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.40505	+	0.581990i	1.40505	+	0.581990i	0.951057	−	0.309017i	$-0.100000\pi$
$173$	1.40505	+	0.581990i	1.40505	+	0.581990i	0.453990	+	0.891007i	$0.350000\pi$
$174$		0			0
$175$	−0.497066	+	1.20002i	−0.497066	+	1.20002i
$176$		0			0
$177$	−0.459953	−	1.11042i	−0.459953	−	1.11042i
$178$	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
$183$			3.87749i			3.87749i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−2.61803			−2.61803
$189$	−3.18296	+	3.18296i	−3.18296	+	3.18296i
$190$		0			0
$191$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$191$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$192$	1.94839	−	4.70384i	1.94839	−	4.70384i
$193$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$193$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$194$	−0.114222	−	0.275756i	−0.114222	−	0.275756i
$195$		0			0
$196$	1.27204	+	1.27204i	1.27204	+	1.27204i
$197$	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
$202$	2.65688	−	2.65688i	2.65688	−	2.65688i
$203$		0			0
$204$	1.18856	+	4.95072i	1.18856	+	4.95072i
$205$		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		−0.382683	−	0.923880i	$-0.625000\pi$
$211$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$212$	3.29892	+	3.29892i	3.29892	+	3.29892i
$213$	1.43702	+	1.43702i	1.43702	+	1.43702i
$214$		0			0
$215$		0			0
$216$	4.08165	−	9.85398i	4.08165	−	9.85398i
$217$		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	$-0.250000\pi$
$223$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$224$	−3.69329	−	1.52981i	−3.69329	−	1.52981i
$225$		−	2.78201i		−	2.78201i
$226$		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
$238$	2.40238	−	0.576761i	2.40238	−	0.576761i
$239$	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
$241$	0.431351	+	0.178671i	0.431351	+	0.178671i	0.587785	−	0.809017i	$-0.300000\pi$
$241$	0.431351	+	0.178671i	0.431351	+	0.178671i	−0.156434	+	0.987688i	$0.550000\pi$
$242$			1.90211i			1.90211i
$243$	1.61910	−	3.90886i	1.61910	−	3.90886i
$244$	−4.82258	+	1.99758i	−4.82258	+	1.99758i
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	2.54093	−	1.05249i	2.54093	−	1.05249i
$250$		0			0
$251$		−	1.97538i		−	1.97538i	−0.156434	−	0.987688i	$-0.550000\pi$
$251$		−	1.97538i		−	1.97538i	0.156434	−	0.987688i	$-0.450000\pi$
$252$	−8.74026	−	3.62033i	−8.74026	−	3.62033i
$253$		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
$259$			0.606443i			0.606443i
$260$		0			0
$261$		0			0
$262$	0.114222	+	0.275756i	0.114222	+	0.275756i
$263$	−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
$267$	−1.41559	+	3.41754i	−1.41559	+	3.41754i
$268$		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
$271$	1.97538			1.97538			0.987688	−	0.156434i	$-0.0500000\pi$
$271$	1.97538			1.97538			0.987688	+	0.156434i	$0.0500000\pi$
$272$	−2.61803	+	1.90211i	−2.61803	+	1.90211i
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	0.399903	−	0.965451i	0.399903	−	0.965451i	−0.587785	−	0.809017i	$-0.700000\pi$
$277$	0.399903	−	0.965451i	0.399903	−	0.965451i	0.987688	−	0.156434i	$-0.0500000\pi$
$278$		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	$0.250000\pi$
$282$	−1.41559	−	3.41754i	−1.41559	−	3.41754i
$283$	1.20002	−	0.497066i	1.20002	−	0.497066i	0.309017	−	0.951057i	$-0.400000\pi$
$283$	1.20002	−	0.497066i	1.20002	−	0.497066i	0.891007	+	0.453990i	$0.150000\pi$
$284$	−1.04696	+	2.52758i	−1.04696	+	2.52758i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	8.56216			8.56216
$289$	0.309017	−	0.951057i	0.309017	−	0.951057i
$290$		0			0
$291$	0.215784	−	0.215784i	0.215784	−	0.215784i
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	−0.972696	+	2.34830i	−0.972696	+	2.34830i
$295$		0			0
$296$	−0.549894	−	1.32756i	−0.549894	−	1.32756i
$297$		0			0
$298$	−0.420808	−	0.420808i	−0.420808	−	0.420808i
$299$		0			0
$300$	4.70384	−	1.94839i	4.70384	−	1.94839i
$301$		0			0
$302$		0			0
$303$	3.54917	+	1.47011i	3.54917	+	1.47011i
$304$		0			0
$305$		0			0
$306$	−4.28108	+	3.11039i	−4.28108	+	3.11039i
$307$	1.97538			1.97538			0.987688	−	0.156434i	$-0.0500000\pi$
$307$	1.97538			1.97538			0.987688	+	0.156434i	$0.0500000\pi$
$308$		0			0
$309$	−0.562133	−	0.232843i	−0.562133	−	0.232843i
$310$		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	$0.375000\pi$
$314$	1.22123	+	1.22123i	1.22123	+	1.22123i
$315$		0			0
$316$	1.04696	+	2.52758i	1.04696	+	2.52758i
$317$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$317$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$318$	−2.52260	+	6.09009i	−2.52260	+	6.09009i
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	10.3611			10.3611
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.20002	+	0.497066i	−1.20002	+	0.497066i
$330$		0			0
$331$	−1.34500	−	1.34500i	−1.34500	−	1.34500i	−0.891007	−	0.453990i	$-0.850000\pi$
$331$	−1.34500	−	1.34500i	−1.34500	−	1.34500i	−0.453990	−	0.891007i	$-0.650000\pi$
$332$	2.61803	+	2.61803i	2.61803	+	2.61803i
$333$	−0.497066	−	1.20002i	−0.497066	−	1.20002i
$334$		0			0
$335$		0			0
$336$		−	8.17436i		−	8.17436i
$337$	−1.40505	−	0.581990i	−1.40505	−	0.581990i	−0.453990	−	0.891007i	$-0.650000\pi$
$337$	−1.40505	−	0.581990i	−1.40505	−	0.581990i	−0.951057	+	0.309017i	$0.900000\pi$
$338$	1.34500	−	1.34500i	1.34500	−	1.34500i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−0.375450	−	0.155517i	−0.375450	−	0.155517i
$344$		0			0
$345$		0			0
$346$	−2.67256	+	1.10701i	−2.67256	+	1.10701i
$347$	−0.744220	−	1.79671i	−0.744220	−	1.79671i	−0.587785	−	0.809017i	$-0.700000\pi$
$347$	−0.744220	−	1.79671i	−0.744220	−	1.79671i	−0.156434	−	0.987688i	$-0.550000\pi$
$348$		0			0
$349$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$349$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$350$	−0.945476	−	2.28258i	−0.945476	−	2.28258i
$351$		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	$0.850000\pi$
$354$	2.11215	+	0.874883i	2.11215	+	0.874883i
$355$		0			0
$356$	−4.97980			−4.97980
$357$	1.48475	+	2.04359i	1.48475	+	2.04359i
$358$		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
$363$	−1.79671	+	0.744220i	−1.79671	+	0.744220i
$364$		0			0
$365$		0			0
$366$	−5.21521	−	5.21521i	−5.21521	−	5.21521i
$367$		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
$371$	2.13846	+	0.885778i	2.13846	+	0.885778i
$372$		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
$378$		−	8.56216i		−	8.56216i
$379$	−0.581990	+	1.40505i	−0.581990	+	1.40505i	0.309017	+	0.951057i	$0.400000\pi$
$379$	−0.581990	+	1.40505i	−0.581990	+	1.40505i	−0.891007	+	0.453990i	$0.850000\pi$
$380$		0			0
$381$		0			0
$382$	−1.90211	−	1.90211i	−1.90211	−	1.90211i
$383$	−0.221232	−	0.221232i	−0.221232	−	0.221232i	0.587785	−	0.809017i	$-0.300000\pi$
$383$	−0.221232	−	0.221232i	−0.221232	−	0.221232i	−0.809017	+	0.587785i	$0.800000\pi$
$384$	1.41559	+	3.41754i	1.41559	+	3.41754i
$385$		0			0
$386$		0			0
$387$		0			0
$388$	0.379546	+	0.157213i	0.379546	+	0.157213i
$389$		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
$392$	−2.11477			−2.11477
$393$	−0.215784	+	0.215784i	−0.215784	+	0.215784i
$394$	−1.34500	−	0.557116i	−1.34500	−	0.557116i
$395$		0			0
$396$		0			0
$397$	−1.57547	+	0.652583i	−1.57547	+	0.652583i	−0.987688	−	0.156434i	$-0.950000\pi$
$397$	−1.57547	+	0.652583i	−1.57547	+	0.652583i	−0.587785	+	0.809017i	$0.700000\pi$
$398$		0			0
$399$		0			0
$400$	2.28825	+	2.28825i	2.28825	+	2.28825i
$401$	0.744220	+	1.79671i	0.744220	+	1.79671i	0.587785	+	0.809017i	$0.300000\pi$
$401$	0.744220	+	1.79671i	0.744220	+	1.79671i	0.156434	+	0.987688i	$0.450000\pi$
$402$		0			0
$403$		0			0
$404$			5.17160i			5.17160i
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−5.10330	−	3.12731i	−5.10330	−	3.12731i
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$		−	0.819101i		−	0.819101i
$413$	0.307204	−	0.741655i	0.307204	−	0.741655i
$414$		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
$423$	1.96718	−	1.96718i	1.96718	−	1.96718i
$424$	−5.48447			−5.48447
$425$	−0.987688	−	0.156434i	−0.987688	−	0.156434i
$426$	−3.86556			−3.86556
$427$	−1.83125	+	1.83125i	−1.83125	+	1.83125i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−0.144974	+	0.0600500i	−0.144974	+	0.0600500i	−0.453990	−	0.891007i	$-0.650000\pi$
$431$	−0.144974	+	0.0600500i	−0.144974	+	0.0600500i	0.309017	+	0.951057i	$0.400000\pi$
$432$	4.29170	+	10.3611i	4.29170	+	10.3611i
$433$		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
$441$	−1.91161			−1.91161
$442$		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
$447$	0.232843	−	0.562133i	0.232843	−	0.562133i
$448$	3.14170	−	1.30134i	3.14170	−	1.30134i
$449$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$449$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$450$	3.74180	+	3.74180i	3.74180	+	3.74180i
$451$		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	0.587785	−	0.809017i	$-0.300000\pi$
$457$	1.39680	−	1.39680i	1.39680	−	1.39680i	0.809017	−	0.587785i	$-0.200000\pi$
$458$		0			0
$459$	−2.95487	−	1.81075i	−2.95487	−	1.81075i
$460$		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
$471$	−0.675737	+	1.63137i	−0.675737	+	1.63137i
$472$			1.90211i			1.90211i
$473$		0			0
$474$	−2.73337	+	2.73337i	−2.73337	+	2.73337i
$475$		0			0
$476$	−1.77678	+	2.89945i	−1.77678	+	2.89945i
$477$	−4.95758			−4.95758
$478$	−1.58114	+	1.58114i	−1.58114	+	1.58114i
$479$	−1.20002	−	0.497066i	−1.20002	−	0.497066i	−0.309017	−	0.951057i	$-0.600000\pi$
$479$	−1.20002	−	0.497066i	−1.20002	−	0.497066i	−0.891007	+	0.453990i	$0.850000\pi$
$480$		0			0
$481$		0			0
$482$	−0.820478	+	0.339853i	−0.820478	+	0.339853i
$483$		0			0
$484$	−1.85123	−	1.85123i	−1.85123	−	1.85123i
$485$		0			0
$486$	3.07972	+	7.43509i	3.07972	+	7.43509i
$487$	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	$0.250000\pi$
$488$	2.34830	−	5.66929i	2.34830	−	5.66929i
$489$		0			0
$490$		0			0
$491$	−0.642040	+	0.642040i	−0.642040	+	0.642040i	−0.951057	−	0.309017i	$-0.900000\pi$
$491$	−0.642040	+	0.642040i	−0.642040	+	0.642040i	0.309017	+	0.951057i	$0.400000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$			1.35734i			1.35734i
$498$	−2.00195	+	4.83313i	−2.00195	+	4.83313i
$499$		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
$502$	2.65688	+	2.65688i	2.65688	+	2.65688i
$503$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$503$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$504$	10.2748	−	4.25596i	10.2748	−	4.25596i
$505$		0			0
$506$		0			0
$507$	1.79671	+	0.744220i	1.79671	+	0.744220i
$508$		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
$518$	−0.815663	−	0.815663i	−0.815663	−	0.815663i
$519$	−2.09133	−	2.09133i	−2.09133	−	2.09133i
$520$		0			0
$521$	1.40505	−	0.581990i	1.40505	−	0.581990i	0.453990	−	0.891007i	$-0.350000\pi$
$521$	1.40505	−	0.581990i	1.40505	−	0.581990i	0.951057	+	0.309017i	$0.100000\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$	−0.379546	−	0.157213i	−0.379546	−	0.157213i
$525$	1.78616	−	1.78616i	1.78616	−	1.78616i
$526$	1.17557			1.17557
$527$		0			0
$528$		0			0
$529$	0.707107	−	0.707107i	0.707107	−	0.707107i
$530$		0			0
$531$			1.71938i			1.71938i
$532$		0			0
$533$		0			0
$534$	−2.69261	−	6.50054i	−2.69261	−	6.50054i
$535$		0			0
$536$		0			0
$537$		0			0
$538$	3.24711	−	1.34500i	3.24711	−	1.34500i
$539$		0			0
$540$		0			0
$541$	1.40505	+	0.581990i	1.40505	+	0.581990i	0.951057	−	0.309017i	$-0.100000\pi$
$541$	1.40505	+	0.581990i	1.40505	+	0.581990i	0.453990	+	0.891007i	$0.350000\pi$
$542$	−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		−0.923880	−	0.382683i	$-0.875000\pi$
$547$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$548$		0			0
$549$	2.12270	−	5.12464i	2.12270	−	5.12464i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	0.959786	+	0.959786i	0.959786	+	0.959786i
$554$	0.760661	+	1.83640i	0.760661	+	1.83640i
$555$		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	$0.750000\pi$
$564$	4.70384	+	1.94839i	4.70384	+	1.94839i
$565$		0			0
$566$	−0.945476	+	2.28258i	−0.945476	+	2.28258i
$567$	4.74919	−	1.96718i	4.74919	−	1.96718i
$568$	−1.23078	−	2.97135i	−1.23078	−	2.97135i
$569$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$569$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$570$		0			0
$571$	0.652583	+	1.57547i	0.652583	+	1.57547i	0.809017	+	0.587785i	$0.200000\pi$
$571$	0.652583	+	1.57547i	0.652583	+	1.57547i	−0.156434	+	0.987688i	$0.550000\pi$
$572$		0			0
$573$	1.05249	−	2.54093i	1.05249	−	2.54093i
$574$		0			0
$575$		0			0
$576$	−5.15014	+	5.15014i	−5.15014	+	5.15014i
$577$		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
$581$	1.69709	+	0.702958i	1.69709	+	0.702958i
$582$			0.580458i			0.580458i
$583$		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	$0.250000\pi$
$588$	−1.33880	−	3.23215i	−1.33880	−	3.23215i
$589$		0			0
$590$		0			0
$591$		−	1.48844i		−	1.48844i
$592$	1.39588	+	0.578193i	1.39588	+	0.578193i
$593$		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
$596$	0.819101			0.819101
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$	−2.29047	+	5.52969i	−2.29047	+	5.52969i
$601$	1.84206	−	0.763007i	1.84206	−	0.763007i	0.891007	−	0.453990i	$-0.150000\pi$
$601$	1.84206	−	0.763007i	1.84206	−	0.763007i	0.951057	−	0.309017i	$-0.100000\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$	−6.75092	+	2.79632i	−6.75092	+	2.79632i
$607$		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
$612$	1.13938	−	7.19373i	1.13938	−	7.19373i
$613$	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
$617$	−0.652583	+	1.57547i	−0.652583	+	1.57547i	0.156434	+	0.987688i	$0.450000\pi$
$617$	−0.652583	+	1.57547i	−0.652583	+	1.57547i	−0.809017	+	0.587785i	$0.800000\pi$
$618$	1.06924	−	0.442894i	1.06924	−	0.442894i
$619$	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
$623$	−2.28258	+	0.945476i	−2.28258	+	0.945476i
$624$		0			0
$625$			1.00000i			1.00000i
$626$		0			0
$627$		0			0
$628$	−2.37713			−2.37713
$629$	−0.453990	+	0.108993i	−0.453990	+	0.108993i
$630$		0			0
$631$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$631$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$632$	−2.97135	−	1.23078i	−2.97135	−	1.23078i
$633$		0			0
$634$		0			0
$635$		0			0
$636$	−3.47206	−	8.38230i	−3.47206	−	8.38230i
$637$		0			0
$638$		0			0
$639$	−1.11254	−	2.68590i	−1.11254	−	2.68590i
$640$		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
$643$	−1.84206	−	0.763007i	−1.84206	−	0.763007i	−0.951057	−	0.309017i	$-0.900000\pi$
$643$	−1.84206	−	0.763007i	−1.84206	−	0.763007i	−0.891007	−	0.453990i	$-0.850000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−0.907981			−0.907981			−0.453990	−	0.891007i	$-0.650000\pi$
$647$	−0.907981			−0.907981			−0.453990	+	0.891007i	$0.650000\pi$
$648$	−8.61269	+	8.61269i	−8.61269	+	8.61269i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−0.178671	−	0.431351i	−0.178671	−	0.431351i	0.809017	−	0.587785i	$-0.200000\pi$
$653$	−0.178671	−	0.431351i	−0.178671	−	0.431351i	−0.987688	+	0.156434i	$0.950000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	0.945476	−	2.28258i	0.945476	−	2.28258i
$659$		−	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	−0.156434	−	0.987688i	$-0.550000\pi$
$661$	0.831254	−	0.831254i	0.831254	−	0.831254i	0.987688	+	0.156434i	$0.0500000\pi$
$662$	3.61803			3.61803
$663$		0			0
$664$	−4.35250			−4.35250
$665$		0			0
$666$	2.28258	+	0.945476i	2.28258	+	0.945476i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$	5.49724	+	5.49724i	5.49724	+	5.49724i
$673$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$673$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$674$	2.67256	−	1.10701i	2.67256	−	1.10701i
$675$	−1.32621	+	3.20175i	−1.32621	+	3.20175i
$676$			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
$683$	1.84206	+	0.763007i	1.84206	+	0.763007i	0.951057	+	0.309017i	$0.100000\pi$
$683$	1.84206	+	0.763007i	1.84206	+	0.763007i	0.891007	+	0.453990i	$0.150000\pi$
$684$		0			0
$685$		0			0
$686$	0.714148	−	0.295810i	0.714148	−	0.295810i
$687$		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	$0.875000\pi$
$692$	1.52367	−	3.67846i	1.52367	−	3.67846i
$693$		0			0
$694$	3.41754	+	1.41559i	3.41754	+	1.41559i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	3.14170	+	1.30134i	3.14170	+	1.30134i
$701$		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
$707$	0.981893	+	2.37050i	0.981893	+	2.37050i
$708$	−2.90713	+	1.20417i	−2.90713	+	1.20417i
$709$	−0.497066	+	1.20002i	−0.497066	+	1.20002i	0.453990	+	0.891007i	$0.350000\pi$
$709$	−0.497066	+	1.20002i	−0.497066	+	1.20002i	−0.951057	+	0.309017i	$0.900000\pi$
$710$		0			0
$711$	−2.68590	−	1.11254i	−2.68590	−	1.11254i
$712$	4.13948	−	4.13948i	4.13948	−	4.13948i
$713$		0			0
$714$	−4.74561	−	0.751631i	−4.74561	−	0.751631i
$715$		0			0
$716$		0			0
$717$	−2.11215	−	0.874883i	−2.11215	−	0.874883i
$718$		0			0
$719$	0.763007	−	1.84206i	0.763007	−	1.84206i	0.309017	−	0.951057i	$-0.400000\pi$
$719$	0.763007	−	1.84206i	0.763007	−	1.84206i	0.453990	−	0.891007i	$-0.350000\pi$
$720$		0			0
$721$	−0.155517	−	0.375450i	−0.155517	−	0.375450i
$722$	1.34500	+	1.34500i	1.34500	+	1.34500i
$723$	−0.642040	−	0.642040i	−0.642040	−	0.642040i
$724$		0			0
$725$		0			0
$726$	1.41559	−	3.41754i	1.41559	−	3.41754i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	−3.01967	+	3.01967i	−3.01967	+	3.01967i
$730$		0			0
$731$		0			0
$732$	10.1514			10.1514
$733$	0.831254	−	0.831254i	0.831254	−	0.831254i	−0.156434	−	0.987688i	$-0.550000\pi$
$733$	0.831254	−	0.831254i	0.831254	−	0.831254i	0.987688	+	0.156434i	$0.0500000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	0.642040	+	0.642040i	0.642040	+	0.642040i	0.951057	−	0.309017i	$-0.100000\pi$
$739$	0.642040	+	0.642040i	0.642040	+	0.642040i	−0.309017	+	0.951057i	$0.600000\pi$
$740$		0			0
$741$		0			0
$742$	−4.06759	+	1.68485i	−4.06759	+	1.68485i
$743$		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
$747$	−3.93436			−3.93436
$748$		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
$753$	−1.47011	+	3.54917i	−1.47011	+	3.54917i
$754$		0			0
$755$		0			0
$756$	8.33311	+	8.33311i	8.33311	+	8.33311i
$757$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$757$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$758$	−1.10701	−	2.67256i	−1.10701	−	2.67256i
$759$		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
$768$	−1.79671	−	0.744220i	−1.79671	−	0.744220i
$769$			1.17557i			1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$769$			1.17557i			1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−0.831254	−	0.831254i	−0.831254	−	0.831254i	0.156434	−	0.987688i	$-0.450000\pi$
$773$	−0.831254	−	0.831254i	−0.831254	−	0.831254i	−0.987688	+	0.156434i	$0.950000\pi$
$774$		0			0
$775$		0			0
$776$	−0.446183	+	0.184815i	−0.446183	+	0.184815i
$777$	0.451326	−	1.08960i	0.451326	−	1.08960i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	1.57232	−	1.57232i	1.57232	−	1.57232i
$785$		0			0
$786$		−	0.580458i		−	0.580458i
$787$		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
$789$	0.459953	+	1.11042i	0.459953	+	1.11042i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	1.24129	−	2.99673i	1.24129	−	2.99673i
$795$		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
$801$	3.74180	−	3.74180i	3.74180	−	3.74180i
$802$	−3.41754	−	1.41559i	−3.41754	−	1.41559i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$	2.54093	+	2.54093i	2.54093	+	2.54093i
$808$	−4.29892	−	4.29892i	−4.29892	−	4.29892i
$809$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$809$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$810$		0			0
$811$	−0.399903	+	0.965451i	−0.399903	+	0.965451i	0.587785	+	0.809017i	$0.300000\pi$
$811$	−0.399903	+	0.965451i	−0.399903	+	0.965451i	−0.987688	+	0.156434i	$0.950000\pi$
$812$		0			0
$813$	−3.54917	−	1.47011i	−3.54917	−	1.47011i
$814$		0			0
$815$		0			0
$816$	6.11943	−	1.46914i	6.11943	−	1.46914i
$817$		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	$0.5\pi$
$824$	0.680881	+	0.680881i	0.680881	+	0.680881i
$825$		0			0
$826$	0.584336	+	1.41071i	0.584336	+	1.41071i
$827$	1.84206	−	0.763007i	1.84206	−	0.763007i	0.891007	−	0.453990i	$-0.150000\pi$
$827$	1.84206	−	0.763007i	1.84206	−	0.763007i	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
$831$	−1.43702	+	1.43702i	−1.43702	+	1.43702i
$832$		0			0
$833$	−0.107491	+	0.678671i	−0.107491	+	0.678671i
$834$		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
$846$			5.29170i			5.29170i
$847$	−1.20002	−	0.497066i	−1.20002	−	0.497066i
$848$	4.07768	−	4.07768i	4.07768	−	4.07768i
$849$	−2.52601			−2.52601
$850$	1.53884	−	1.11803i	1.53884	−	1.11803i
$851$		0			0
$852$	3.76216	−	3.76216i	3.76216	−	3.76216i
$853$	1.57547	+	0.652583i	1.57547	+	0.652583i	0.987688	−	0.156434i	$-0.0500000\pi$
$853$	1.57547	+	0.652583i	1.57547	+	0.652583i	0.587785	+	0.809017i	$0.300000\pi$
$854$		−	4.92606i		−	4.92606i
$855$		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
$862$	0.114222	−	0.275756i	0.114222	−	0.275756i
$863$			1.61803i			1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$863$			1.61803i			1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$864$	−9.85398	−	4.08165i	−9.85398	−	4.08165i
$865$		0			0
$866$		0			0
$867$	−1.26301	+	1.47879i	−1.26301	+	1.47879i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−0.403318	+	0.167060i	−0.403318	+	0.167060i
$874$		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	$0.125000\pi$
$882$	2.57111	−	2.57111i	2.57111	−	2.57111i
$883$	−0.907981			−0.907981			−0.453990	−	0.891007i	$-0.650000\pi$
$883$	−0.907981			−0.907981			−0.453990	+	0.891007i	$0.650000\pi$
$884$		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	$0.125000\pi$
$888$			2.79448i			2.79448i
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$	0.442894	+	1.06924i	0.442894	+	1.06924i
$895$		0			0
$896$	−0.945476	+	2.28258i	−0.945476	+	2.28258i
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−7.28340			−7.28340
$901$	−0.278768	+	1.76007i	−0.278768	+	1.76007i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	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
$909$	−3.88592	−	3.88592i	−3.88592	−	3.88592i
$910$		0			0
$911$	0.763007	+	1.84206i	0.763007	+	1.84206i	0.453990	+	0.891007i	$0.350000\pi$
$911$	0.763007	+	1.84206i	0.763007	+	1.84206i	0.309017	+	0.951057i	$0.400000\pi$
$912$		0			0
$913$		0			0
$914$			3.75739i			3.75739i
$915$		0			0
$916$		0			0
$917$	−0.203820			−0.203820
$918$	6.40974	−	1.53884i	6.40974	−	1.53884i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	−3.54917	−	1.47011i	−3.54917	−	1.47011i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	0.178671	+	0.431351i	0.178671	+	0.431351i
$926$		0			0
$927$	0.615470	+	0.615470i	0.615470	+	0.615470i
$928$		0			0
$929$	−0.431351	+	0.178671i	−0.431351	+	0.178671i	−0.587785	−	0.809017i	$-0.700000\pi$
$929$	−0.431351	+	0.178671i	−0.431351	+	0.178671i	0.156434	+	0.987688i	$0.450000\pi$
$930$		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
$941$	0.144974	−	0.0600500i	0.144974	−	0.0600500i	−0.309017	−	0.951057i	$-0.600000\pi$
$941$	0.144974	−	0.0600500i	0.144974	−	0.0600500i	0.453990	+	0.891007i	$0.350000\pi$
$942$	−1.28533	−	3.10306i	−1.28533	−	3.10306i
$943$		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
$952$	−0.933219	−	3.88714i	−0.933219	−	3.88714i
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$	6.66794	−	6.66794i	6.66794	−	6.66794i
$955$		0			0
$956$		−	3.07768i		−	3.07768i
$957$		0			0
$958$	2.28258	−	0.945476i	2.28258	−	0.945476i
$959$		0			0
$960$		0			0
$961$	0.707107	+	0.707107i	0.707107	+	0.707107i
$962$		0			0
$963$		0			0
$964$	0.467768	−	1.12929i	0.467768	−	1.12929i
$965$		0			0
$966$		0			0
$967$	0.642040	−	0.642040i	0.642040	−	0.642040i	−0.309017	−	0.951057i	$-0.600000\pi$
$967$	0.642040	−	0.642040i	0.642040	−	0.642040i	0.951057	+	0.309017i	$0.100000\pi$
$968$	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	$0.750000\pi$
$972$	−10.2335	−	4.23887i	−10.2335	−	4.23887i
$973$		0			0
$974$	−0.557116	+	1.34500i	−0.557116	+	1.34500i
$975$		0			0
$976$	2.46914	+	5.96104i	2.46914	+	5.96104i
$977$		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
$982$		−	1.72708i		−	1.72708i
$983$		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
$987$	2.52601			2.52601
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−0.763007	+	1.84206i	−0.763007	+	1.84206i	−0.309017	+	0.951057i	$0.600000\pi$
$991$	−0.763007	+	1.84206i	−0.763007	+	1.84206i	−0.453990	+	0.891007i	$0.650000\pi$
$992$		0			0
$993$	1.41559	+	3.41754i	1.41559	+	3.41754i
$994$	−1.82562	−	1.82562i	−1.82562	−	1.82562i
$995$		0			0
$996$	−2.75544	−	6.65223i	−2.75544	−	6.65223i
$997$		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.281.1	✓	16
17.2	even	8	inner	799.1.h.b.563.1	yes	16
47.46	odd	2	CM	799.1.h.b.281.1	✓	16
799.563	odd	8	inner	799.1.h.b.563.1	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
799.1.h.b.281.1	✓	16	1.1	even	1	trivial
799.1.h.b.281.1	✓	16	47.46	odd	2	CM
799.1.h.b.563.1	yes	16	17.2	even	8	inner
799.1.h.b.563.1	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)

\(f(q)\)	\(=\)	\(q+(-1.34500 + 1.34500i) q^{2} +(-1.79671 - 0.744220i) q^{3} -2.61803i q^{4} +(3.41754 - 1.41559i) q^{6} +(-0.497066 - 1.20002i) q^{7} +(2.17625 + 2.17625i) q^{8} +(1.96718 + 1.96718i) q^{9} +(-1.94839 + 4.70384i) q^{12} +(2.28258 + 0.945476i) q^{14} -3.23607 q^{16} +(0.809017 - 0.587785i) q^{17} -5.29170 q^{18} +2.52601i q^{21} +(-2.29047 - 5.52969i) q^{24} +(-0.707107 - 0.707107i) q^{25} +(-1.32621 - 3.20175i) q^{27} +(-3.14170 + 1.30134i) q^{28} +(2.17625 - 2.17625i) q^{32} +(-0.297556 + 1.87869i) q^{34} +(5.15014 - 5.15014i) q^{36} +(-0.431351 - 0.178671i) q^{37} +(-3.39748 - 3.39748i) q^{42} -1.00000i q^{47} +(5.81426 + 2.40835i) q^{48} +(-0.485875 + 0.485875i) q^{49} +1.90211 q^{50} +(-1.89101 + 0.453990i) q^{51} +(-1.26007 + 1.26007i) q^{53} +(6.09009 + 2.52260i) q^{54} +(1.52981 - 3.69329i) q^{56} +(0.437016 + 0.437016i) q^{59} +(-0.763007 - 1.84206i) q^{61} +(1.38284 - 3.33848i) q^{63} +2.61803i q^{64} +(-1.53884 - 2.11803i) q^{68} +(-0.965451 - 0.399903i) q^{71} +8.56216i q^{72} +(0.820478 - 0.339853i) q^{74} +(0.744220 + 1.79671i) q^{75} +(-0.965451 + 0.399903i) q^{79} +3.95758i q^{81} +(-1.00000 + 1.00000i) q^{83} +6.61319 q^{84} -1.90211i q^{89} +(1.34500 + 1.34500i) q^{94} +(-5.52969 + 2.29047i) q^{96} +(-0.0600500 + 0.144974i) q^{97} -1.30700i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{3} - 4 q^{9} - 16 q^{16} + 4 q^{17} - 20 q^{24} - 4 q^{27} - 4 q^{28} + 4 q^{36} - 20 q^{42} + 24 q^{48} + 4 q^{49} - 16 q^{51} + 4 q^{53} + 20 q^{54} + 20 q^{56} + 4 q^{61} - 4 q^{63} - 4 q^{71}+ \cdots + 32 q^{84}+O(q^{100}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more