LMFDB - Embedded newform 799.1.h.b.93.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [799,1,Mod(93,799)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(799, base_ring=CyclotomicField(8))

chi = DirichletCharacter(H, H._module([5, 4]))

N = Newforms(chi, 1, names="a")

//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);

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: f = N[0] # Warning: the index may be different

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			93.1
Root			$-0.453990 - 0.891007i$ of defining polynomial
Character	$\chi$	$=$	799.93
Dual form			799.1.h.b.610.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.34500 + 1.34500i) q^{2} +(-0.652583 + 1.57547i) q^{3} -2.61803i q^{4} +(-1.24129 - 2.99673i) q^{6} +(1.84206 - 0.763007i) q^{7} +(2.17625 + 2.17625i) q^{8} +(-1.34915 - 1.34915i) q^{9} +O(q^{10})$	\(q+(-1.34500 + 1.34500i) q^{2} +(-0.652583 + 1.57547i) q^{3} -2.61803i q^{4} +(-1.24129 - 2.99673i) q^{6} +(1.84206 - 0.763007i) q^{7} +(2.17625 + 2.17625i) q^{8} +(-1.34915 - 1.34915i) q^{9} +(4.12464 + 1.70848i) q^{12} +(-1.45133 + 3.50381i) q^{14} -3.23607 q^{16} +(0.809017 + 0.587785i) q^{17} +3.62920 q^{18} +3.40005i q^{21} +(-4.84881 + 2.00844i) q^{24} +(0.707107 + 0.707107i) q^{25} +(1.43050 - 0.592533i) q^{27} +(-1.99758 - 4.82258i) q^{28} +(2.17625 - 2.17625i) q^{32} +(-1.87869 + 0.297556i) q^{34} +(-3.53211 + 3.53211i) q^{36} +(-0.399903 + 0.965451i) q^{37} +(-4.57305 - 4.57305i) q^{42} -1.00000i q^{47} +(2.11180 - 5.09834i) q^{48} +(2.10391 - 2.10391i) q^{49} -1.90211 q^{50} +(-1.45399 + 0.891007i) q^{51} +(0.642040 - 0.642040i) q^{53} +(-1.12706 + 2.72097i) q^{54} +(5.66929 + 2.34830i) q^{56} +(-0.437016 - 0.437016i) q^{59} +(-1.20002 + 0.497066i) q^{61} +(-3.51462 - 1.45580i) q^{63} +2.61803i q^{64} +(1.53884 - 2.11803i) q^{68} +(0.178671 - 0.431351i) q^{71} -5.87216i q^{72} +(-0.760661 - 1.83640i) q^{74} +(-1.57547 + 0.652583i) q^{75} +(0.178671 + 0.431351i) q^{79} +0.732410i q^{81} +(-1.00000 + 1.00000i) q^{83} +8.90144 q^{84} +1.90211i q^{89} +(1.34500 + 1.34500i) q^{94} +(2.00844 + 4.84881i) q^{96} +(1.40505 + 0.581990i) q^{97} +5.65950i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{3} - 4 q^{9}+O(q^{10}) $ 16 * q - 4 * q^3 - 4 * q^9	$ 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}+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

Display 100 coefficients 10 coefficients

Character values

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

$n$	$52$	$377$
$\chi(n)$	$-1$	$e\left(\frac{5}{8}\right)$

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	$0.850000\pi$
$3$	−0.652583	+	1.57547i	−0.652583	+	1.57547i	0.156434	+	0.987688i	$0.450000\pi$
$3$	−0.652583	+	1.57547i	−0.652583	+	1.57547i	−0.809017	+	0.587785i	$0.800000\pi$
$4$		−	2.61803i		−	2.61803i
$5$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$5$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$6$	−1.24129	−	2.99673i	−1.24129	−	2.99673i
$7$	1.84206	−	0.763007i	1.84206	−	0.763007i	0.891007	−	0.453990i	$-0.150000\pi$
$7$	1.84206	−	0.763007i	1.84206	−	0.763007i	0.951057	−	0.309017i	$-0.100000\pi$
$8$	2.17625	+	2.17625i	2.17625	+	2.17625i
$9$	−1.34915	−	1.34915i	−1.34915	−	1.34915i
$10$		0			0
$11$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$11$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$12$	4.12464	+	1.70848i	4.12464	+	1.70848i
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$	−1.45133	+	3.50381i	−1.45133	+	3.50381i
$15$		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
$18$	3.62920			3.62920
$19$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$19$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$20$		0			0
$21$			3.40005i			3.40005i
$22$		0			0
$23$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$23$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$24$	−4.84881	+	2.00844i	−4.84881	+	2.00844i
$25$	0.707107	+	0.707107i	0.707107	+	0.707107i
$26$		0			0
$27$	1.43050	−	0.592533i	1.43050	−	0.592533i
$28$	−1.99758	−	4.82258i	−1.99758	−	4.82258i
$29$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$29$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$30$		0			0
$31$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$31$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$32$	2.17625	−	2.17625i	2.17625	−	2.17625i
$33$		0			0
$34$	−1.87869	+	0.297556i	−1.87869	+	0.297556i
$35$		0			0
$36$	−3.53211	+	3.53211i	−3.53211	+	3.53211i
$37$	−0.399903	+	0.965451i	−0.399903	+	0.965451i	0.587785	+	0.809017i	$0.300000\pi$
$37$	−0.399903	+	0.965451i	−0.399903	+	0.965451i	−0.987688	+	0.156434i	$0.950000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$41$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$42$	−4.57305	−	4.57305i	−4.57305	−	4.57305i
$43$		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
$48$	2.11180	−	5.09834i	2.11180	−	5.09834i
$49$	2.10391	−	2.10391i	2.10391	−	2.10391i
$50$	−1.90211			−1.90211
$51$	−1.45399	+	0.891007i	−1.45399	+	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	$0.100000\pi$
$54$	−1.12706	+	2.72097i	−1.12706	+	2.72097i
$55$		0			0
$56$	5.66929	+	2.34830i	5.66929	+	2.34830i
$57$		0			0
$58$		0			0
$59$	−0.437016	−	0.437016i	−0.437016	−	0.437016i	0.453990	−	0.891007i	$-0.350000\pi$
$59$	−0.437016	−	0.437016i	−0.437016	−	0.437016i	−0.891007	+	0.453990i	$0.850000\pi$
$60$		0			0
$61$	−1.20002	+	0.497066i	−1.20002	+	0.497066i	−0.891007	−	0.453990i	$-0.850000\pi$
$61$	−1.20002	+	0.497066i	−1.20002	+	0.497066i	−0.309017	+	0.951057i	$0.600000\pi$
$62$		0			0
$63$	−3.51462	−	1.45580i	−3.51462	−	1.45580i
$64$			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
$71$	0.178671	−	0.431351i	0.178671	−	0.431351i	−0.809017	−	0.587785i	$-0.800000\pi$
$71$	0.178671	−	0.431351i	0.178671	−	0.431351i	0.987688	+	0.156434i	$0.0500000\pi$
$72$		−	5.87216i		−	5.87216i
$73$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$73$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$74$	−0.760661	−	1.83640i	−0.760661	−	1.83640i
$75$	−1.57547	+	0.652583i	−1.57547	+	0.652583i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	0.178671	+	0.431351i	0.178671	+	0.431351i	0.987688	−	0.156434i	$-0.0500000\pi$
$79$	0.178671	+	0.431351i	0.178671	+	0.431351i	−0.809017	+	0.587785i	$0.800000\pi$
$80$		0			0
$81$			0.732410i			0.732410i
$82$		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			$\pi$
$84$	8.90144			8.90144
$85$		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
$96$	2.00844	+	4.84881i	2.00844	+	4.84881i
$97$	1.40505	+	0.581990i	1.40505	+	0.581990i	0.951057	−	0.309017i	$-0.100000\pi$
$97$	1.40505	+	0.581990i	1.40505	+	0.581990i	0.453990	+	0.891007i	$0.350000\pi$
$98$			5.65950i			5.65950i
$99$		0			0
$100$	1.85123	−	1.85123i	1.85123	−	1.85123i
$101$	0.312869			0.312869			0.156434	−	0.987688i	$-0.450000\pi$
$101$	0.312869			0.312869			0.156434	+	0.987688i	$0.450000\pi$
$102$	0.757212	−	3.15401i	0.757212	−	3.15401i
$103$	−1.97538			−1.97538			−0.987688	−	0.156434i	$-0.950000\pi$
$103$	−1.97538			−1.97538			−0.987688	+	0.156434i	$0.950000\pi$
$104$		0			0
$105$		0			0
$106$			1.72708i			1.72708i
$107$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$107$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$108$	−1.55127	−	3.74510i	−1.55127	−	3.74510i
$109$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$109$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$110$		0			0
$111$	−1.26007	−	1.26007i	−1.26007	−	1.26007i
$112$	−5.96104	+	2.46914i	−5.96104	+	2.46914i
$113$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$113$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	1.17557			1.17557
$119$	1.93874	+	0.465451i	1.93874	+	0.465451i
$120$		0			0
$121$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$122$	0.945476	−	2.28258i	0.945476	−	2.28258i
$123$		0			0
$124$		0			0
$125$		0			0
$126$	6.68521	−	2.76910i	6.68521	−	2.76910i
$127$		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
$131$	−1.40505	−	0.581990i	−1.40505	−	0.581990i	−0.453990	−	0.891007i	$-0.650000\pi$
$131$	−1.40505	−	0.581990i	−1.40505	−	0.581990i	−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		0.382683	−	0.923880i	$-0.375000\pi$
$139$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$140$		0			0
$141$	1.57547	+	0.652583i	1.57547	+	0.652583i
$142$	0.339853	+	0.820478i	0.339853	+	0.820478i
$143$		0			0
$144$	4.36593	+	4.36593i	4.36593	+	4.36593i
$145$		0			0
$146$		0			0
$147$	1.94168	+	4.68763i	1.94168	+	4.68763i
$148$	2.52758	+	1.04696i	2.52758	+	1.04696i
$149$		−	1.97538i		−	1.97538i	−0.156434	−	0.987688i	$-0.550000\pi$
$149$		−	1.97538i		−	1.97538i	0.156434	−	0.987688i	$-0.450000\pi$
$150$	1.24129	−	2.99673i	1.24129	−	2.99673i
$151$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$151$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$152$		0			0
$153$	−0.298474	−	1.88449i	−0.298474	−	1.88449i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	1.78201i		−	1.78201i	−0.453990	−	0.891007i	$-0.650000\pi$
$157$		−	1.78201i		−	1.78201i	0.453990	−	0.891007i	$-0.350000\pi$
$158$	−0.820478	−	0.339853i	−0.820478	−	0.339853i
$159$	0.592533	+	1.43050i	0.592533	+	1.43050i
$160$		0			0
$161$		0			0
$162$	−0.985090	−	0.985090i	−0.985090	−	0.985090i
$163$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$163$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$164$		0			0
$165$		0			0
$166$		−	2.68999i		−	2.68999i
$167$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$167$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$168$	−7.39936	+	7.39936i	−7.39936	+	7.39936i
$169$	−1.00000			−1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−0.0600500	+	0.144974i	−0.0600500	+	0.144974i	−0.951057	−	0.309017i	$-0.900000\pi$
$173$	−0.0600500	+	0.144974i	−0.0600500	+	0.144974i	0.891007	+	0.453990i	$0.150000\pi$
$174$		0			0
$175$	1.84206	+	0.763007i	1.84206	+	0.763007i
$176$		0			0
$177$	0.973696	−	0.403318i	0.973696	−	0.403318i
$178$	−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		−0.382683	−	0.923880i	$-0.625000\pi$
$181$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$182$		0			0
$183$		−	2.21498i		−	2.21498i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−2.61803			−2.61803
$189$	2.18296	−	2.18296i	2.18296	−	2.18296i
$190$		0			0
$191$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$191$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$192$	−4.12464	−	1.70848i	−4.12464	−	1.70848i
$193$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$193$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$194$	−2.67256	+	1.10701i	−2.67256	+	1.10701i
$195$		0			0
$196$	−5.50811	−	5.50811i	−5.50811	−	5.50811i
$197$	1.70711	−	0.707107i	1.70711	−	0.707107i	0.707107	−	0.707107i	$-0.250000\pi$
$197$	1.70711	−	0.707107i	1.70711	−	0.707107i	1.00000			$0$
$198$		0			0
$199$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$199$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$200$			3.07768i			3.07768i
$201$		0			0
$202$	−0.420808	+	0.420808i	−0.420808	+	0.420808i
$203$		0			0
$204$	2.33269	+	3.80660i	2.33269	+	3.80660i
$205$		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		0.923880	−	0.382683i	$-0.125000\pi$
$211$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$212$	−1.68088	−	1.68088i	−1.68088	−	1.68088i
$213$	0.562984	+	0.562984i	0.562984	+	0.562984i
$214$		0			0
$215$		0			0
$216$	4.40263	+	1.82363i	4.40263	+	1.82363i
$217$		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	$0.750000\pi$
$224$	2.34830	−	5.66929i	2.34830	−	5.66929i
$225$		−	1.90798i		−	1.90798i
$226$		0			0
$227$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$227$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$228$		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		−0.923880	−	0.382683i	$-0.875000\pi$
$233$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$234$		0			0
$235$		0			0
$236$	−1.14412	+	1.14412i	−1.14412	+	1.14412i
$237$	−0.796180			−0.796180
$238$	−3.23364	+	1.98158i	−3.23364	+	1.98158i
$239$	−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
$241$	0.399903	−	0.965451i	0.399903	−	0.965451i	−0.587785	−	0.809017i	$-0.700000\pi$
$241$	0.399903	−	0.965451i	0.399903	−	0.965451i	0.987688	−	0.156434i	$-0.0500000\pi$
$242$		−	1.90211i		−	1.90211i
$243$	0.276607	+	0.114574i	0.276607	+	0.114574i
$244$	1.30134	+	3.14170i	1.30134	+	3.14170i
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−0.922891	−	2.22806i	−0.922891	−	2.22806i
$250$		0			0
$251$			0.312869i			0.312869i	0.987688	+	0.156434i	$0.0500000\pi$
$251$			0.312869i			0.312869i	−0.987688	+	0.156434i	$0.950000\pi$
$252$	−3.81134	+	9.20140i	−3.81134	+	9.20140i
$253$		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
$259$			2.08355i			2.08355i
$260$		0			0
$261$		0			0
$262$	2.67256	−	1.10701i	2.67256	−	1.10701i
$263$	0.437016	+	0.437016i	0.437016	+	0.437016i	0.891007	−	0.453990i	$-0.150000\pi$
$263$	0.437016	+	0.437016i	0.437016	+	0.437016i	−0.453990	+	0.891007i	$0.650000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−2.99673	−	1.24129i	−2.99673	−	1.24129i
$268$		0			0
$269$	−0.292893	+	0.707107i	−0.292893	+	0.707107i	0.707107	+	0.707107i	$0.250000\pi$
$269$	−0.292893	+	0.707107i	−0.292893	+	0.707107i	−1.00000			$\pi$
$270$		0			0
$271$	−0.312869			−0.312869			−0.156434	−	0.987688i	$-0.550000\pi$
$271$	−0.312869			−0.312869			−0.156434	+	0.987688i	$0.550000\pi$
$272$	−2.61803	−	1.90211i	−2.61803	−	1.90211i
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	0.431351	+	0.178671i	0.431351	+	0.178671i	0.587785	−	0.809017i	$-0.300000\pi$
$277$	0.431351	+	0.178671i	0.431351	+	0.178671i	−0.156434	+	0.987688i	$0.550000\pi$
$278$		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	$0.250000\pi$
$282$	−2.99673	+	1.24129i	−2.99673	+	1.24129i
$283$	0.763007	+	1.84206i	0.763007	+	1.84206i	0.453990	+	0.891007i	$0.350000\pi$
$283$	0.763007	+	1.84206i	0.763007	+	1.84206i	0.309017	+	0.951057i	$0.400000\pi$
$284$	−1.12929	−	0.467768i	−1.12929	−	0.467768i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−5.87216			−5.87216
$289$	0.309017	+	0.951057i	0.309017	+	0.951057i
$290$		0			0
$291$	−1.83382	+	1.83382i	−1.83382	+	1.83382i
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	−8.91640	−	3.69329i	−8.91640	−	3.69329i
$295$		0			0
$296$	−2.97135	+	1.23078i	−2.97135	+	1.23078i
$297$		0			0
$298$	2.65688	+	2.65688i	2.65688	+	2.65688i
$299$		0			0
$300$	1.70848	+	4.12464i	1.70848	+	4.12464i
$301$		0			0
$302$		0			0
$303$	−0.204173	+	0.492917i	−0.204173	+	0.492917i
$304$		0			0
$305$		0			0
$306$	2.93608	+	2.13319i	2.93608	+	2.13319i
$307$	−0.312869			−0.312869			−0.156434	−	0.987688i	$-0.550000\pi$
$307$	−0.312869			−0.312869			−0.156434	+	0.987688i	$0.550000\pi$
$308$		0			0
$309$	1.28910	−	3.11215i	1.28910	−	3.11215i
$310$		0			0
$311$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$311$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$312$		0			0
$313$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$313$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$314$	2.39680	+	2.39680i	2.39680	+	2.39680i
$315$		0			0
$316$	1.12929	−	0.467768i	1.12929	−	0.467768i
$317$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$317$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$318$	−2.72097	−	1.12706i	−2.72097	−	1.12706i
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	1.91748			1.91748
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−0.763007	−	1.84206i	−0.763007	−	1.84206i
$330$		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
$333$	1.84206	−	0.763007i	1.84206	−	0.763007i
$334$		0			0
$335$		0			0
$336$		−	11.0028i		−	11.0028i
$337$	0.0600500	−	0.144974i	0.0600500	−	0.144974i	−0.891007	−	0.453990i	$-0.850000\pi$
$337$	0.0600500	−	0.144974i	0.0600500	−	0.144974i	0.951057	+	0.309017i	$0.100000\pi$
$338$	1.34500	−	1.34500i	1.34500	−	1.34500i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	1.50723	−	3.63877i	1.50723	−	3.63877i
$344$		0			0
$345$		0			0
$346$	−0.114222	−	0.275756i	−0.114222	−	0.275756i
$347$	1.57547	−	0.652583i	1.57547	−	0.652583i	0.587785	−	0.809017i	$-0.300000\pi$
$347$	1.57547	−	0.652583i	1.57547	−	0.652583i	0.987688	+	0.156434i	$0.0500000\pi$
$348$		0			0
$349$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$349$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$350$	−3.50381	+	1.45133i	−3.50381	+	1.45133i
$351$		0			0
$352$		0			0
$353$			1.78201i			1.78201i	0.453990	+	0.891007i	$0.350000\pi$
$353$			1.78201i			1.78201i	−0.453990	+	0.891007i	$0.650000\pi$
$354$	−0.767157	+	1.85208i	−0.767157	+	1.85208i
$355$		0			0
$356$	4.97980			4.97980
$357$	−1.99850	+	2.75070i	−1.99850	+	2.75070i
$358$		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
$363$	−0.652583	−	1.57547i	−0.652583	−	1.57547i
$364$		0			0
$365$		0			0
$366$	2.97914	+	2.97914i	2.97914	+	2.97914i
$367$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$367$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	0.692796	−	1.67256i	0.692796	−	1.67256i
$372$		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
$378$			5.87216i			5.87216i
$379$	−0.144974	−	0.0600500i	−0.144974	−	0.0600500i	0.309017	−	0.951057i	$-0.400000\pi$
$379$	−0.144974	−	0.0600500i	−0.144974	−	0.0600500i	−0.453990	+	0.891007i	$0.650000\pi$
$380$		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	$-0.700000\pi$
$384$	2.99673	−	1.24129i	2.99673	−	1.24129i
$385$		0			0
$386$		0			0
$387$		0			0
$388$	1.52367	−	3.67846i	1.52367	−	3.67846i
$389$		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
$392$	9.15727			9.15727
$393$	1.83382	−	1.83382i	1.83382	−	1.83382i
$394$	−1.34500	+	3.24711i	−1.34500	+	3.24711i
$395$		0			0
$396$		0			0
$397$	0.744220	+	1.79671i	0.744220	+	1.79671i	0.587785	+	0.809017i	$0.300000\pi$
$397$	0.744220	+	1.79671i	0.744220	+	1.79671i	0.156434	+	0.987688i	$0.450000\pi$
$398$		0			0
$399$		0			0
$400$	−2.28825	−	2.28825i	−2.28825	−	2.28825i
$401$	−1.57547	+	0.652583i	−1.57547	+	0.652583i	−0.987688	−	0.156434i	$-0.950000\pi$
$401$	−1.57547	+	0.652583i	−1.57547	+	0.652583i	−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
$408$	−5.10330	−	1.22519i	−5.10330	−	1.22519i
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$			5.17160i			5.17160i
$413$	−1.13846	−	0.471565i	−1.13846	−	0.471565i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$419$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$420$		0			0
$421$		0			0			−	1.00000i	$-0.5\pi$
$421$		0			0				1.00000i	$0.5\pi$
$422$		0			0
$423$	−1.34915	+	1.34915i	−1.34915	+	1.34915i
$424$	2.79448			2.79448
$425$	0.156434	+	0.987688i	0.156434	+	0.987688i
$426$	−1.51442			−1.51442
$427$	−1.83125	+	1.83125i	−1.83125	+	1.83125i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−0.581990	−	1.40505i	−0.581990	−	1.40505i	−0.891007	−	0.453990i	$-0.850000\pi$
$431$	−0.581990	−	1.40505i	−0.581990	−	1.40505i	0.309017	−	0.951057i	$-0.400000\pi$
$432$	−4.62920	+	1.91748i	−4.62920	+	1.91748i
$433$		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	−	1.70711i	0.707107	−	1.70711i		−	1.00000i	$-0.5\pi$
$439$	0.707107	−	1.70711i	0.707107	−	1.70711i	0.707107	−	0.707107i	$-0.250000\pi$
$440$		0			0
$441$	−5.67696			−5.67696
$442$		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
$447$	3.11215	+	1.28910i	3.11215	+	1.28910i
$448$	1.99758	+	4.82258i	1.99758	+	4.82258i
$449$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$449$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$450$	2.56623	+	2.56623i	2.56623	+	2.56623i
$451$		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
$459$	1.50558	+	0.361458i	1.50558	+	0.361458i
$460$		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
$471$	2.80751	+	1.16291i	2.80751	+	1.16291i
$472$		−	1.90211i		−	1.90211i
$473$		0			0
$474$	1.07086	−	1.07086i	1.07086	−	1.07086i
$475$		0			0
$476$	1.21857	−	5.07570i	1.21857	−	5.07570i
$477$	−1.73241			−1.73241
$478$	1.58114	−	1.58114i	1.58114	−	1.58114i
$479$	−0.763007	+	1.84206i	−0.763007	+	1.84206i	−0.309017	+	0.951057i	$0.600000\pi$
$479$	−0.763007	+	1.84206i	−0.763007	+	1.84206i	−0.453990	+	0.891007i	$0.650000\pi$
$480$		0			0
$481$		0			0
$482$	0.760661	+	1.83640i	0.760661	+	1.83640i
$483$		0			0
$484$	1.85123	+	1.85123i	1.85123	+	1.85123i
$485$		0			0
$486$	−0.526137	+	0.217933i	−0.526137	+	0.217933i
$487$	−0.707107	−	1.70711i	−0.707107	−	1.70711i	−0.707107	−	0.707107i	$-0.750000\pi$
$487$	−0.707107	−	1.70711i	−0.707107	−	1.70711i		−	1.00000i	$-0.5\pi$
$488$	−3.69329	−	1.52981i	−3.69329	−	1.52981i
$489$		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
$497$		−	0.930903i		−	0.930903i
$498$	4.23801	+	1.75544i	4.23801	+	1.75544i
$499$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$499$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$500$		0			0
$501$		0			0
$502$	−0.420808	−	0.420808i	−0.420808	−	0.420808i
$503$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$503$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$504$	−4.48050	−	10.8169i	−4.48050	−	10.8169i
$505$		0			0
$506$		0			0
$507$	0.652583	−	1.57547i	0.652583	−	1.57547i
$508$		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
$518$	−2.80237	−	2.80237i	−2.80237	−	2.80237i
$519$	−0.189214	−	0.189214i	−0.189214	−	0.189214i
$520$		0			0
$521$	−0.0600500	−	0.144974i	−0.0600500	−	0.144974i	0.891007	−	0.453990i	$-0.150000\pi$
$521$	−0.0600500	−	0.144974i	−0.0600500	−	0.144974i	−0.951057	+	0.309017i	$0.900000\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$	−1.52367	+	3.67846i	−1.52367	+	3.67846i
$525$	−2.40420	+	2.40420i	−2.40420	+	2.40420i
$526$	−1.17557			−1.17557
$527$		0			0
$528$		0			0
$529$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$530$		0			0
$531$			1.17920i			1.17920i
$532$		0			0
$533$		0			0
$534$	5.70012	−	2.36107i	5.70012	−	2.36107i
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−0.557116	−	1.34500i	−0.557116	−	1.34500i
$539$		0			0
$540$		0			0
$541$	−0.0600500	+	0.144974i	−0.0600500	+	0.144974i	−0.951057	−	0.309017i	$-0.900000\pi$
$541$	−0.0600500	+	0.144974i	−0.0600500	+	0.144974i	0.891007	+	0.453990i	$0.150000\pi$
$542$	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		0.382683	−	0.923880i	$-0.375000\pi$
$547$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$548$		0			0
$549$	2.28962	+	0.948393i	2.28962	+	0.948393i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	0.658248	+	0.658248i	0.658248	+	0.658248i
$554$	−0.820478	+	0.339853i	−0.820478	+	0.339853i
$555$		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	$0.750000\pi$
$564$	1.70848	−	4.12464i	1.70848	−	4.12464i
$565$		0			0
$566$	−3.50381	−	1.45133i	−3.50381	−	1.45133i
$567$	0.558835	+	1.34915i	0.558835	+	1.34915i
$568$	1.32756	−	0.549894i	1.32756	−	0.549894i
$569$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$569$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$570$		0			0
$571$	1.79671	−	0.744220i	1.79671	−	0.744220i	0.809017	−	0.587785i	$-0.200000\pi$
$571$	1.79671	−	0.744220i	1.79671	−	0.744220i	0.987688	−	0.156434i	$-0.0500000\pi$
$572$		0			0
$573$	2.22806	+	0.922891i	2.22806	+	0.922891i
$574$		0			0
$575$		0			0
$576$	3.53211	−	3.53211i	3.53211	−	3.53211i
$577$		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
$581$	−1.07906	+	2.60507i	−1.07906	+	2.60507i
$582$		−	4.93296i		−	4.93296i
$583$		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	$0.250000\pi$
$588$	12.2724	−	5.08338i	12.2724	−	5.08338i
$589$		0			0
$590$		0			0
$591$			3.15095i			3.15095i
$592$	1.29411	−	3.12427i	1.29411	−	3.12427i
$593$		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	$0.5\pi$
$600$	−4.84881	−	2.00844i	−4.84881	−	2.00844i
$601$	−0.497066	−	1.20002i	−0.497066	−	1.20002i	−0.951057	−	0.309017i	$-0.900000\pi$
$601$	−0.497066	−	1.20002i	−0.497066	−	1.20002i	0.453990	−	0.891007i	$-0.350000\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$	−0.388360	−	0.937583i	−0.388360	−	0.937583i
$607$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$607$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−4.93366	+	0.781415i	−4.93366	+	0.781415i
$613$	−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
$617$	−1.79671	−	0.744220i	−1.79671	−	0.744220i	−0.987688	−	0.156434i	$-0.950000\pi$
$617$	−1.79671	−	0.744220i	−1.79671	−	0.744220i	−0.809017	−	0.587785i	$-0.800000\pi$
$618$	2.45201	+	5.91967i	2.45201	+	5.91967i
$619$	−0.707107	+	0.292893i	−0.707107	+	0.292893i	−0.707107	−	0.707107i	$-0.750000\pi$
$619$	−0.707107	+	0.292893i	−0.707107	+	0.292893i			1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1.45133	+	3.50381i	1.45133	+	3.50381i
$624$		0			0
$625$			1.00000i			1.00000i
$626$		0			0
$627$		0			0
$628$	−4.66537			−4.66537
$629$	−0.891007	+	0.546010i	−0.891007	+	0.546010i
$630$		0			0
$631$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$631$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$632$	−0.549894	+	1.32756i	−0.549894	+	1.32756i
$633$		0			0
$634$		0			0
$635$		0			0
$636$	3.74510	−	1.55127i	3.74510	−	1.55127i
$637$		0			0
$638$		0			0
$639$	−0.823009	+	0.340902i	−0.823009	+	0.340902i
$640$		0			0
$641$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$641$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$642$		0			0
$643$	0.497066	−	1.20002i	0.497066	−	1.20002i	−0.453990	−	0.891007i	$-0.650000\pi$
$643$	0.497066	−	1.20002i	0.497066	−	1.20002i	0.951057	−	0.309017i	$-0.100000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.78201			−1.78201			−0.891007	−	0.453990i	$-0.850000\pi$
$647$	−1.78201			−1.78201			−0.891007	+	0.453990i	$0.850000\pi$
$648$	−1.59391	+	1.59391i	−1.59391	+	1.59391i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	0.965451	−	0.399903i	0.965451	−	0.399903i	0.156434	−	0.987688i	$-0.450000\pi$
$653$	0.965451	−	0.399903i	0.965451	−	0.399903i	0.809017	+	0.587785i	$0.200000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	3.50381	+	1.45133i	3.50381	+	1.45133i
$659$		−	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
$666$	−1.45133	+	3.50381i	−1.45133	+	3.50381i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$	7.39936	+	7.39936i	7.39936	+	7.39936i
$673$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$673$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$674$	0.114222	+	0.275756i	0.114222	+	0.275756i
$675$	1.43050	+	0.592533i	1.43050	+	0.592533i
$676$			2.61803i			2.61803i
$677$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$677$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$678$		0			0
$679$	3.03225			3.03225
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−0.497066	+	1.20002i	−0.497066	+	1.20002i	0.453990	+	0.891007i	$0.350000\pi$
$683$	−0.497066	+	1.20002i	−0.497066	+	1.20002i	−0.951057	+	0.309017i	$0.900000\pi$
$684$		0			0
$685$		0			0
$686$	2.86692	+	6.92135i	2.86692	+	6.92135i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$691$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$692$	0.379546	+	0.157213i	0.379546	+	0.157213i
$693$		0			0
$694$	−1.24129	+	2.99673i	−1.24129	+	2.99673i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	1.99758	−	4.82258i	1.99758	−	4.82258i
$701$		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
$707$	0.576324	−	0.238721i	0.576324	−	0.238721i
$708$	−1.05590	−	2.54917i	−1.05590	−	2.54917i
$709$	1.84206	+	0.763007i	1.84206	+	0.763007i	0.951057	+	0.309017i	$0.100000\pi$
$709$	1.84206	+	0.763007i	1.84206	+	0.763007i	0.891007	+	0.453990i	$0.150000\pi$
$710$		0			0
$711$	0.340902	−	0.823009i	0.340902	−	0.823009i
$712$	−4.13948	+	4.13948i	−4.13948	+	4.13948i
$713$		0			0
$714$	−1.01170	−	6.38765i	−1.01170	−	6.38765i
$715$		0			0
$716$		0			0
$717$	0.767157	−	1.85208i	0.767157	−	1.85208i
$718$		0			0
$719$	1.20002	+	0.497066i	1.20002	+	0.497066i	0.891007	−	0.453990i	$-0.150000\pi$
$719$	1.20002	+	0.497066i	1.20002	+	0.497066i	0.309017	+	0.951057i	$0.400000\pi$
$720$		0			0
$721$	−3.63877	+	1.50723i	−3.63877	+	1.50723i
$722$	1.34500	+	1.34500i	1.34500	+	1.34500i
$723$	1.26007	+	1.26007i	1.26007	+	1.26007i
$724$		0			0
$725$		0			0
$726$	2.99673	+	1.24129i	2.99673	+	1.24129i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	−0.878910	+	0.878910i	−0.878910	+	0.878910i
$730$		0			0
$731$		0			0
$732$	−5.79890			−5.79890
$733$	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
$739$	−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
$742$	1.31778	+	3.18139i	1.31778	+	3.18139i
$743$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$743$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	2.69829			2.69829
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$751$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$752$			3.23607i			3.23607i
$753$	−0.492917	−	0.204173i	−0.492917	−	0.204173i
$754$		0			0
$755$		0			0
$756$	−5.71508	−	5.71508i	−5.71508	−	5.71508i
$757$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$757$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$758$	0.275756	−	0.114222i	0.275756	−	0.114222i
$759$		0			0
$760$		0			0
$761$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$761$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$762$		0			0
$763$		0			0
$764$	−3.70246			−3.70246
$765$		0			0
$766$	3.75739			3.75739
$767$		0			0
$768$	−0.652583	+	1.57547i	−0.652583	+	1.57547i
$769$		−	1.17557i		−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$769$		−	1.17557i		−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−0.831254	−	0.831254i	−0.831254	−	0.831254i	0.156434	−	0.987688i	$-0.450000\pi$
$773$	−0.831254	−	0.831254i	−0.831254	−	0.831254i	−0.987688	+	0.156434i	$0.950000\pi$
$774$		0			0
$775$		0			0
$776$	1.79118	+	4.32429i	1.79118	+	4.32429i
$777$	−3.28258	−	1.35969i	−3.28258	−	1.35969i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−6.80839	+	6.80839i	−6.80839	+	6.80839i
$785$		0			0
$786$			4.93296i			4.93296i
$787$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$787$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$788$	−1.85123	−	4.46926i	−1.85123	−	4.46926i
$789$	−0.973696	+	0.403318i	−0.973696	+	0.403318i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	−3.41754	−	1.41559i	−3.41754	−	1.41559i
$795$		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
$801$	2.56623	−	2.56623i	2.56623	−	2.56623i
$802$	1.24129	−	2.99673i	1.24129	−	2.99673i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−0.922891	−	0.922891i	−0.922891	−	0.922891i
$808$	0.680881	+	0.680881i	0.680881	+	0.680881i
$809$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$809$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$810$		0			0
$811$	−0.431351	−	0.178671i	−0.431351	−	0.178671i	0.156434	−	0.987688i	$-0.450000\pi$
$811$	−0.431351	−	0.178671i	−0.431351	−	0.178671i	−0.587785	+	0.809017i	$0.700000\pi$
$812$		0			0
$813$	0.204173	−	0.492917i	0.204173	−	0.492917i
$814$		0			0
$815$		0			0
$816$	4.70521	−	2.88336i	4.70521	−	2.88336i
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$821$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$822$		0			0
$823$	−0.707107	+	0.292893i	−0.707107	+	0.292893i	−0.707107	−	0.707107i	$-0.750000\pi$
$823$	−0.707107	+	0.292893i	−0.707107	+	0.292893i			1.00000i	$0.5\pi$
$824$	−4.29892	−	4.29892i	−4.29892	−	4.29892i
$825$		0			0
$826$	2.16547	−	0.896969i	2.16547	−	0.896969i
$827$	−0.497066	−	1.20002i	−0.497066	−	1.20002i	−0.951057	−	0.309017i	$-0.900000\pi$
$827$	−0.497066	−	1.20002i	−0.497066	−	1.20002i	0.453990	−	0.891007i	$-0.350000\pi$
$828$		0			0
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		0			0
$831$	−0.562984	+	0.562984i	−0.562984	+	0.562984i
$832$		0			0
$833$	2.93874	−	0.465451i	2.93874	−	0.465451i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$839$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$840$		0			0
$841$	0.707107	+	0.707107i	0.707107	+	0.707107i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		−	3.62920i		−	3.62920i
$847$	−0.763007	+	1.84206i	−0.763007	+	1.84206i
$848$	−2.07768	+	2.07768i	−2.07768	+	2.07768i
$849$	−3.40005			−3.40005
$850$	−1.53884	−	1.11803i	−1.53884	−	1.11803i
$851$		0			0
$852$	1.47391	−	1.47391i	1.47391	−	1.47391i
$853$	−0.744220	+	1.79671i	−0.744220	+	1.79671i	−0.156434	+	0.987688i	$0.550000\pi$
$853$	−0.744220	+	1.79671i	−0.744220	+	1.79671i	−0.587785	+	0.809017i	$0.700000\pi$
$854$		−	4.92606i		−	4.92606i
$855$		0			0
$856$		0			0
$857$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$857$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$858$		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
$862$	2.67256	+	1.10701i	2.67256	+	1.10701i
$863$			1.61803i			1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$863$			1.61803i			1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$864$	1.82363	−	4.40263i	1.82363	−	4.40263i
$865$		0			0
$866$		0			0
$867$	−1.70002	−	0.133795i	−1.70002	−	0.133795i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−1.11042	−	2.68080i	−1.11042	−	2.68080i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$877$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$878$	1.34500	+	3.24711i	1.34500	+	3.24711i
$879$		0			0
$880$		0			0
$881$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$881$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$882$	7.63550	−	7.63550i	7.63550	−	7.63550i
$883$	−1.78201			−1.78201			−0.891007	−	0.453990i	$-0.850000\pi$
$883$	−1.78201			−1.78201			−0.891007	+	0.453990i	$0.850000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$887$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$888$		−	5.48447i		−	5.48447i
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$	−5.91967	+	2.45201i	−5.91967	+	2.45201i
$895$		0			0
$896$	−3.50381	−	1.45133i	−3.50381	−	1.45133i
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−4.99516			−4.99516
$901$	0.896802	−	0.142040i	0.896802	−	0.142040i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	0.292893	+	0.707107i	0.292893	+	0.707107i	1.00000			$0$
$907$	0.292893	+	0.707107i	0.292893	+	0.707107i	−0.707107	+	0.707107i	$0.750000\pi$
$908$		0			0
$909$	−0.422106	−	0.422106i	−0.422106	−	0.422106i
$910$		0			0
$911$	1.20002	−	0.497066i	1.20002	−	0.497066i	0.309017	−	0.951057i	$-0.400000\pi$
$911$	1.20002	−	0.497066i	1.20002	−	0.497066i	0.891007	+	0.453990i	$0.150000\pi$
$912$		0			0
$913$		0			0
$914$			0.595112i			0.595112i
$915$		0			0
$916$		0			0
$917$	−3.03225			−3.03225
$918$	−2.51116	+	1.53884i	−2.51116	+	1.53884i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	0.204173	−	0.492917i	0.204173	−	0.492917i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.965451	+	0.399903i	−0.965451	+	0.399903i
$926$		0			0
$927$	2.66507	+	2.66507i	2.66507	+	2.66507i
$928$		0			0
$929$	−0.399903	−	0.965451i	−0.399903	−	0.965451i	−0.987688	−	0.156434i	$-0.950000\pi$
$929$	−0.399903	−	0.965451i	−0.399903	−	0.965451i	0.587785	−	0.809017i	$-0.300000\pi$
$930$		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
$941$	0.581990	+	1.40505i	0.581990	+	1.40505i	0.891007	+	0.453990i	$0.150000\pi$
$941$	0.581990	+	1.40505i	0.581990	+	1.40505i	−0.309017	+	0.951057i	$0.600000\pi$
$942$	−5.34021	+	2.21199i	−5.34021	+	2.21199i
$943$		0			0
$944$	1.41421	+	1.41421i	1.41421	+	1.41421i
$945$		0			0
$946$		0			0
$947$	−0.707107	−	0.292893i	−0.707107	−	0.292893i		−	1.00000i	$-0.5\pi$
$947$	−0.707107	−	0.292893i	−0.707107	−	0.292893i	−0.707107	+	0.707107i	$0.750000\pi$
$948$			2.08443i			2.08443i
$949$		0			0
$950$		0			0
$951$		0			0
$952$	3.20626	+	5.23213i	3.20626	+	5.23213i
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$	2.33009	−	2.33009i	2.33009	−	2.33009i
$955$		0			0
$956$			3.07768i			3.07768i
$957$		0			0
$958$	−1.45133	−	3.50381i	−1.45133	−	3.50381i
$959$		0			0
$960$		0			0
$961$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$962$		0			0
$963$		0			0
$964$	−2.52758	−	1.04696i	−2.52758	−	1.04696i
$965$		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	$0.750000\pi$
$972$	0.299959	−	0.724166i	0.299959	−	0.724166i
$973$		0			0
$974$	3.24711	+	1.34500i	3.24711	+	1.34500i
$975$		0			0
$976$	3.88336	−	1.60854i	3.88336	−	1.60854i
$977$		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		0.382683	−	0.923880i	$-0.375000\pi$
$983$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	3.40005			3.40005
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−1.20002	−	0.497066i	−1.20002	−	0.497066i	−0.309017	−	0.951057i	$-0.600000\pi$
$991$	−1.20002	−	0.497066i	−1.20002	−	0.497066i	−0.891007	+	0.453990i	$0.850000\pi$
$992$		0			0
$993$	2.99673	−	1.24129i	2.99673	−	1.24129i
$994$	1.25206	+	1.25206i	1.25206	+	1.25206i
$995$		0			0
$996$	−5.83313	+	2.41616i	−5.83313	+	2.41616i
$997$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$997$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$998$		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.93.1	✓	16
17.15	even	8	inner	799.1.h.b.610.1	yes	16
47.46	odd	2	CM	799.1.h.b.93.1	✓	16
799.610	odd	8	inner	799.1.h.b.610.1	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
799.1.h.b.93.1	✓	16	1.1	even	1	trivial
799.1.h.b.93.1	✓	16	47.46	odd	2	CM
799.1.h.b.610.1	yes	16	17.15	even	8	inner
799.1.h.b.610.1	yes	16	799.610	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(-1.34500 + 1.34500i) q^{2} +(-0.652583 + 1.57547i) q^{3} -2.61803i q^{4} +(-1.24129 - 2.99673i) q^{6} +(1.84206 - 0.763007i) q^{7} +(2.17625 + 2.17625i) q^{8} +(-1.34915 - 1.34915i) q^{9} +O(q^{10})\)	\(q+(-1.34500 + 1.34500i) q^{2} +(-0.652583 + 1.57547i) q^{3} -2.61803i q^{4} +(-1.24129 - 2.99673i) q^{6} +(1.84206 - 0.763007i) q^{7} +(2.17625 + 2.17625i) q^{8} +(-1.34915 - 1.34915i) q^{9} +(4.12464 + 1.70848i) q^{12} +(-1.45133 + 3.50381i) q^{14} -3.23607 q^{16} +(0.809017 + 0.587785i) q^{17} +3.62920 q^{18} +3.40005i q^{21} +(-4.84881 + 2.00844i) q^{24} +(0.707107 + 0.707107i) q^{25} +(1.43050 - 0.592533i) q^{27} +(-1.99758 - 4.82258i) q^{28} +(2.17625 - 2.17625i) q^{32} +(-1.87869 + 0.297556i) q^{34} +(-3.53211 + 3.53211i) q^{36} +(-0.399903 + 0.965451i) q^{37} +(-4.57305 - 4.57305i) q^{42} -1.00000i q^{47} +(2.11180 - 5.09834i) q^{48} +(2.10391 - 2.10391i) q^{49} -1.90211 q^{50} +(-1.45399 + 0.891007i) q^{51} +(0.642040 - 0.642040i) q^{53} +(-1.12706 + 2.72097i) q^{54} +(5.66929 + 2.34830i) q^{56} +(-0.437016 - 0.437016i) q^{59} +(-1.20002 + 0.497066i) q^{61} +(-3.51462 - 1.45580i) q^{63} +2.61803i q^{64} +(1.53884 - 2.11803i) q^{68} +(0.178671 - 0.431351i) q^{71} -5.87216i q^{72} +(-0.760661 - 1.83640i) q^{74} +(-1.57547 + 0.652583i) q^{75} +(0.178671 + 0.431351i) q^{79} +0.732410i q^{81} +(-1.00000 + 1.00000i) q^{83} +8.90144 q^{84} +1.90211i q^{89} +(1.34500 + 1.34500i) q^{94} +(2.00844 + 4.84881i) q^{96} +(1.40505 + 0.581990i) q^{97} +5.65950i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{3} - 4 q^{9}+O(q^{10}) \) 16 * q - 4 * q^3 - 4 * q^9	\( 16 q - 4 q^{3} - 4 q^{9} - 16 q^{16} + 4 q^{17} - 20 q^{24} - 4 q^{27} - 4 q^{28} + 4 q^{36} - 20 q^{42} + 24 q^{48} + 4 q^{49} - 16 q^{51} + 4 q^{53} + 20 q^{54} + 20 q^{56} + 4 q^{61} - 4 q^{63} - 4 q^{71} - 4 q^{79} - 16 q^{83} + 32 q^{84}+O(q^{100}) \) 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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