LMFDB - Embedded newform 799.1.h.b.281.2

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			281.2
Root			$0.156434 - 0.987688i$ of defining polynomial
Character	$\chi$	$=$	799.281
Dual form			799.1.h.b.563.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.831254 + 0.831254i) q^{2} +(1.20002 + 0.497066i) q^{3} -0.381966i q^{4} +(-1.41071 + 0.584336i) q^{6} +(0.399903 + 0.965451i) q^{7} +(-0.513743 - 0.513743i) q^{8} +(0.485875 + 0.485875i) q^{9} +(0.189862 - 0.458368i) q^{12} +(-1.13496 - 0.470114i) q^{14} +1.23607 q^{16} +(-0.309017 + 0.951057i) q^{17} -0.807771 q^{18} +1.35734i q^{21} +(-0.361140 - 0.871868i) q^{24} +(-0.707107 - 0.707107i) q^{25} +(-0.155517 - 0.375450i) q^{27} +(0.368770 - 0.152749i) q^{28} +(-0.513743 + 0.513743i) q^{32} +(-0.533698 - 1.04744i) q^{34} +(0.185588 - 0.185588i) q^{36} +(1.40505 + 0.581990i) q^{37} +(-1.12830 - 1.12830i) q^{42} -1.00000i q^{47} +(1.48331 + 0.614407i) q^{48} +(-0.0650673 + 0.0650673i) q^{49} +1.17557 q^{50} +(-0.843566 + 0.987688i) q^{51} +(0.221232 - 0.221232i) q^{53} +(0.441368 + 0.182821i) q^{54} +(0.290547 - 0.701442i) q^{56} +(-1.14412 - 1.14412i) q^{59} +(-0.178671 - 0.431351i) q^{61} +(-0.274786 + 0.663392i) q^{63} +0.381966i q^{64} +(0.363271 + 0.118034i) q^{68} +(-0.144974 - 0.0600500i) q^{71} -0.499230i q^{72} +(-1.65173 + 0.684170i) q^{74} +(-0.497066 - 1.20002i) q^{75} +(-0.144974 + 0.0600500i) q^{79} -1.21498i q^{81} +(-1.00000 + 1.00000i) q^{83} +0.518459 q^{84} -1.17557i q^{89} +(0.831254 + 0.831254i) q^{94} +(-0.871868 + 0.361140i) q^{96} +(-0.744220 + 1.79671i) q^{97} -0.108175i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.831254	+	0.831254i	−0.831254	+	0.831254i	−0.987688	−	0.156434i	$-0.950000\pi$
$2$	−0.831254	+	0.831254i	−0.831254	+	0.831254i	0.156434	+	0.987688i	$0.450000\pi$
$3$	1.20002	+	0.497066i	1.20002	+	0.497066i	0.891007	−	0.453990i	$-0.150000\pi$
$3$	1.20002	+	0.497066i	1.20002	+	0.497066i	0.309017	+	0.951057i	$0.400000\pi$
$4$		−	0.381966i		−	0.381966i
$5$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$5$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$6$	−1.41071	+	0.584336i	−1.41071	+	0.584336i
$7$	0.399903	+	0.965451i	0.399903	+	0.965451i	0.987688	+	0.156434i	$0.0500000\pi$
$7$	0.399903	+	0.965451i	0.399903	+	0.965451i	−0.587785	+	0.809017i	$0.700000\pi$
$8$	−0.513743	−	0.513743i	−0.513743	−	0.513743i
$9$	0.485875	+	0.485875i	0.485875	+	0.485875i
$10$		0			0
$11$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$11$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$12$	0.189862	−	0.458368i	0.189862	−	0.458368i
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$	−1.13496	−	0.470114i	−1.13496	−	0.470114i
$15$		0			0
$16$	1.23607			1.23607
$17$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$17$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$18$	−0.807771			−0.807771
$19$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$19$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$20$		0			0
$21$			1.35734i			1.35734i
$22$		0			0
$23$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$23$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$24$	−0.361140	−	0.871868i	−0.361140	−	0.871868i
$25$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$26$		0			0
$27$	−0.155517	−	0.375450i	−0.155517	−	0.375450i
$28$	0.368770	−	0.152749i	0.368770	−	0.152749i
$29$		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	$0.125000\pi$
$32$	−0.513743	+	0.513743i	−0.513743	+	0.513743i
$33$		0			0
$34$	−0.533698	−	1.04744i	−0.533698	−	1.04744i
$35$		0			0
$36$	0.185588	−	0.185588i	0.185588	−	0.185588i
$37$	1.40505	+	0.581990i	1.40505	+	0.581990i	0.951057	−	0.309017i	$-0.100000\pi$
$37$	1.40505	+	0.581990i	1.40505	+	0.581990i	0.453990	+	0.891007i	$0.350000\pi$
$38$		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	$0.375000\pi$
$42$	−1.12830	−	1.12830i	−1.12830	−	1.12830i
$43$		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
$48$	1.48331	+	0.614407i	1.48331	+	0.614407i
$49$	−0.0650673	+	0.0650673i	−0.0650673	+	0.0650673i
$50$	1.17557			1.17557
$51$	−0.843566	+	0.987688i	−0.843566	+	0.987688i
$52$		0			0
$53$	0.221232	−	0.221232i	0.221232	−	0.221232i	−0.587785	−	0.809017i	$-0.700000\pi$
$53$	0.221232	−	0.221232i	0.221232	−	0.221232i	0.809017	+	0.587785i	$0.200000\pi$
$54$	0.441368	+	0.182821i	0.441368	+	0.182821i
$55$		0			0
$56$	0.290547	−	0.701442i	0.290547	−	0.701442i
$57$		0			0
$58$		0			0
$59$	−1.14412	−	1.14412i	−1.14412	−	1.14412i	−0.987688	−	0.156434i	$-0.950000\pi$
$59$	−1.14412	−	1.14412i	−1.14412	−	1.14412i	−0.156434	−	0.987688i	$-0.550000\pi$
$60$		0			0
$61$	−0.178671	−	0.431351i	−0.178671	−	0.431351i	0.809017	−	0.587785i	$-0.200000\pi$
$61$	−0.178671	−	0.431351i	−0.178671	−	0.431351i	−0.987688	+	0.156434i	$0.950000\pi$
$62$		0			0
$63$	−0.274786	+	0.663392i	−0.274786	+	0.663392i
$64$			0.381966i			0.381966i
$65$		0			0
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$	0.363271	+	0.118034i	0.363271	+	0.118034i
$69$		0			0
$70$		0			0
$71$	−0.144974	−	0.0600500i	−0.144974	−	0.0600500i	0.309017	−	0.951057i	$-0.400000\pi$
$71$	−0.144974	−	0.0600500i	−0.144974	−	0.0600500i	−0.453990	+	0.891007i	$0.650000\pi$
$72$		−	0.499230i		−	0.499230i
$73$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$73$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$74$	−1.65173	+	0.684170i	−1.65173	+	0.684170i
$75$	−0.497066	−	1.20002i	−0.497066	−	1.20002i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−0.144974	+	0.0600500i	−0.144974	+	0.0600500i	−0.453990	−	0.891007i	$-0.650000\pi$
$79$	−0.144974	+	0.0600500i	−0.144974	+	0.0600500i	0.309017	+	0.951057i	$0.400000\pi$
$80$		0			0
$81$		−	1.21498i		−	1.21498i
$82$		0			0
$83$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$83$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$84$	0.518459			0.518459
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	1.17557i		−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$89$		−	1.17557i		−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$	0.831254	+	0.831254i	0.831254	+	0.831254i
$95$		0			0
$96$	−0.871868	+	0.361140i	−0.871868	+	0.361140i
$97$	−0.744220	+	1.79671i	−0.744220	+	1.79671i	−0.156434	+	0.987688i	$0.550000\pi$
$97$	−0.744220	+	1.79671i	−0.744220	+	1.79671i	−0.587785	+	0.809017i	$0.700000\pi$
$98$		−	0.108175i		−	0.108175i
$99$		0			0
$100$	−0.270091	+	0.270091i	−0.270091	+	0.270091i
$101$	1.78201			1.78201			0.891007	−	0.453990i	$-0.150000\pi$
$101$	1.78201			1.78201			0.891007	+	0.453990i	$0.150000\pi$
$102$	−0.119803	−	1.52224i	−0.119803	−	1.52224i
$103$	0.907981			0.907981			0.453990	−	0.891007i	$-0.350000\pi$
$103$	0.907981			0.907981			0.453990	+	0.891007i	$0.350000\pi$
$104$		0			0
$105$		0			0
$106$			0.367799i			0.367799i
$107$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$107$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$108$	−0.143409	+	0.0594020i	−0.143409	+	0.0594020i
$109$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$109$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$110$		0			0
$111$	1.39680	+	1.39680i	1.39680	+	1.39680i
$112$	0.494307	+	1.19336i	0.494307	+	1.19336i
$113$		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
$118$	1.90211			1.90211
$119$	−1.04178	+	0.0819895i	−1.04178	+	0.0819895i
$120$		0			0
$121$	0.707107	−	0.707107i	0.707107	−	0.707107i
$122$	0.507083	+	0.210041i	0.507083	+	0.210041i
$123$		0			0
$124$		0			0
$125$		0			0
$126$	−0.323030	−	0.779864i	−0.323030	−	0.779864i
$127$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$127$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$128$	−0.831254	−	0.831254i	−0.831254	−	0.831254i
$129$		0			0
$130$		0			0
$131$	0.744220	−	1.79671i	0.744220	−	1.79671i	0.156434	−	0.987688i	$-0.450000\pi$
$131$	0.744220	−	1.79671i	0.744220	−	1.79671i	0.587785	−	0.809017i	$-0.300000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	0.647354	−	0.329843i	0.647354	−	0.329843i
$137$		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
$141$	0.497066	−	1.20002i	0.497066	−	1.20002i
$142$	0.170427	−	0.0705930i	0.170427	−	0.0705930i
$143$		0			0
$144$	0.600575	+	0.600575i	0.600575	+	0.600575i
$145$		0			0
$146$		0			0
$147$	−0.110425	+	0.0457395i	−0.110425	+	0.0457395i
$148$	0.222300	−	0.536680i	0.222300	−	0.536680i
$149$			0.907981i			0.907981i	0.891007	+	0.453990i	$0.150000\pi$
$149$			0.907981i			0.907981i	−0.891007	+	0.453990i	$0.850000\pi$
$150$	1.41071	+	0.584336i	1.41071	+	0.584336i
$151$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$151$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$152$		0			0
$153$	−0.612238	+	0.311951i	−0.612238	+	0.311951i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	1.97538i		−	1.97538i	−0.156434	−	0.987688i	$-0.550000\pi$
$157$		−	1.97538i		−	1.97538i	0.156434	−	0.987688i	$-0.450000\pi$
$158$	0.0705930	−	0.170427i	0.0705930	−	0.170427i
$159$	0.375450	−	0.155517i	0.375450	−	0.155517i
$160$		0			0
$161$		0			0
$162$	1.00996	+	1.00996i	1.00996	+	1.00996i
$163$		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
$166$		−	1.66251i		−	1.66251i
$167$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$167$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$168$	0.697326	−	0.697326i	0.697326	−	0.697326i
$169$	−1.00000			−1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.57547	+	0.652583i	1.57547	+	0.652583i	0.987688	−	0.156434i	$-0.0500000\pi$
$173$	1.57547	+	0.652583i	1.57547	+	0.652583i	0.587785	+	0.809017i	$0.300000\pi$
$174$		0			0
$175$	0.399903	−	0.965451i	0.399903	−	0.965451i
$176$		0			0
$177$	−0.804270	−	1.94168i	−0.804270	−	1.94168i
$178$	0.977198	+	0.977198i	0.977198	+	0.977198i
$179$		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
$183$		−	0.606443i		−	0.606443i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−0.381966			−0.381966
$189$	0.300287	−	0.300287i	0.300287	−	0.300287i
$190$		0			0
$191$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$191$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$192$	−0.189862	+	0.458368i	−0.189862	+	0.458368i
$193$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$193$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$194$	−0.874883	−	2.11215i	−0.874883	−	2.11215i
$195$		0			0
$196$	0.0248535	+	0.0248535i	0.0248535	+	0.0248535i
$197$	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	$0.625000\pi$
$200$			0.726543i			0.726543i
$201$		0			0
$202$	−1.48131	+	1.48131i	−1.48131	+	1.48131i
$203$		0			0
$204$	0.377263	+	0.322213i	0.377263	+	0.322213i
$205$		0			0
$206$	−0.754763	+	0.754763i	−0.754763	+	0.754763i
$207$		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	$0.375000\pi$
$212$	−0.0845030	−	0.0845030i	−0.0845030	−	0.0845030i
$213$	−0.144123	−	0.144123i	−0.144123	−	0.144123i
$214$		0			0
$215$		0			0
$216$	−0.112989	+	0.272780i	−0.112989	+	0.272780i
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$	−2.32219			−2.32219
$223$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$223$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$224$	−0.701442	−	0.290547i	−0.701442	−	0.290547i
$225$		−	0.687131i		−	0.687131i
$226$		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
$236$	−0.437016	+	0.437016i	−0.437016	+	0.437016i
$237$	−0.203820			−0.203820
$238$	0.797826	−	0.934134i	0.797826	−	0.934134i
$239$	−1.90211			−1.90211			−0.951057	−	0.309017i	$-0.900000\pi$
$239$	−1.90211			−1.90211			−0.951057	+	0.309017i	$0.900000\pi$
$240$		0			0
$241$	−1.40505	−	0.581990i	−1.40505	−	0.581990i	−0.453990	−	0.891007i	$-0.650000\pi$
$241$	−1.40505	−	0.581990i	−1.40505	−	0.581990i	−0.951057	+	0.309017i	$0.900000\pi$
$242$			1.17557i			1.17557i
$243$	0.448410	−	1.08256i	0.448410	−	1.08256i
$244$	−0.164761	+	0.0682464i	−0.164761	+	0.0682464i
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−1.69709	+	0.702958i	−1.69709	+	0.702958i
$250$		0			0
$251$			1.78201i			1.78201i	0.453990	+	0.891007i	$0.350000\pi$
$251$			1.78201i			1.78201i	−0.453990	+	0.891007i	$0.650000\pi$
$252$	0.253393	+	0.104959i	0.253393	+	0.104959i
$253$		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
$259$			1.58924i			1.58924i
$260$		0			0
$261$		0			0
$262$	0.874883	+	2.11215i	0.874883	+	2.11215i
$263$	1.14412	+	1.14412i	1.14412	+	1.14412i	0.987688	+	0.156434i	$0.0500000\pi$
$263$	1.14412	+	1.14412i	1.14412	+	1.14412i	0.156434	+	0.987688i	$0.450000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	0.584336	−	1.41071i	0.584336	−	1.41071i
$268$		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
$271$	−1.78201			−1.78201			−0.891007	−	0.453990i	$-0.850000\pi$
$271$	−1.78201			−1.78201			−0.891007	+	0.453990i	$0.850000\pi$
$272$	−0.381966	+	1.17557i	−0.381966	+	1.17557i
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	0.0600500	−	0.144974i	0.0600500	−	0.144974i	−0.891007	−	0.453990i	$-0.850000\pi$
$277$	0.0600500	−	0.144974i	0.0600500	−	0.144974i	0.951057	+	0.309017i	$0.100000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$281$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$282$	0.584336	+	1.41071i	0.584336	+	1.41071i
$283$	−0.965451	+	0.399903i	−0.965451	+	0.399903i	−0.809017	−	0.587785i	$-0.800000\pi$
$283$	−0.965451	+	0.399903i	−0.965451	+	0.399903i	−0.156434	+	0.987688i	$0.550000\pi$
$284$	−0.0229371	+	0.0553750i	−0.0229371	+	0.0553750i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.499230			−0.499230
$289$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$290$		0			0
$291$	−1.78616	+	1.78616i	−1.78616	+	1.78616i
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	0.0537700	−	0.129812i	0.0537700	−	0.129812i
$295$		0			0
$296$	−0.422840	−	1.02083i	−0.422840	−	1.02083i
$297$		0			0
$298$	−0.754763	−	0.754763i	−0.754763	−	0.754763i
$299$		0			0
$300$	−0.458368	+	0.189862i	−0.458368	+	0.189862i
$301$		0			0
$302$		0			0
$303$	2.13846	+	0.885778i	2.13846	+	0.885778i
$304$		0			0
$305$		0			0
$306$	0.249615	−	0.768236i	0.249615	−	0.768236i
$307$	−1.78201			−1.78201			−0.891007	−	0.453990i	$-0.850000\pi$
$307$	−1.78201			−1.78201			−0.891007	+	0.453990i	$0.850000\pi$
$308$		0			0
$309$	1.08960	+	0.451326i	1.08960	+	0.451326i
$310$		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	$0.375000\pi$
$314$	1.64204	+	1.64204i	1.64204	+	1.64204i
$315$		0			0
$316$	0.0229371	+	0.0553750i	0.0229371	+	0.0553750i
$317$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$317$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$318$	−0.182821	+	0.441368i	−0.182821	+	0.441368i
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.464082			−0.464082
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	0.965451	−	0.399903i	0.965451	−	0.399903i
$330$		0			0
$331$	−0.831254	−	0.831254i	−0.831254	−	0.831254i	0.156434	−	0.987688i	$-0.450000\pi$
$331$	−0.831254	−	0.831254i	−0.831254	−	0.831254i	−0.987688	+	0.156434i	$0.950000\pi$
$332$	0.381966	+	0.381966i	0.381966	+	0.381966i
$333$	0.399903	+	0.965451i	0.399903	+	0.965451i
$334$		0			0
$335$		0			0
$336$			1.67777i			1.67777i
$337$	−1.57547	−	0.652583i	−1.57547	−	0.652583i	−0.587785	−	0.809017i	$-0.700000\pi$
$337$	−1.57547	−	0.652583i	−1.57547	−	0.652583i	−0.987688	+	0.156434i	$0.950000\pi$
$338$	0.831254	−	0.831254i	0.831254	−	0.831254i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.876612	+	0.363104i	0.876612	+	0.363104i
$344$		0			0
$345$		0			0
$346$	−1.85208	+	0.767157i	−1.85208	+	0.767157i
$347$	0.497066	+	1.20002i	0.497066	+	1.20002i	0.951057	+	0.309017i	$0.100000\pi$
$347$	0.497066	+	1.20002i	0.497066	+	1.20002i	−0.453990	+	0.891007i	$0.650000\pi$
$348$		0			0
$349$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$349$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$350$	0.470114	+	1.13496i	0.470114	+	1.13496i
$351$		0			0
$352$		0			0
$353$			1.97538i			1.97538i	0.156434	+	0.987688i	$0.450000\pi$
$353$			1.97538i			1.97538i	−0.156434	+	0.987688i	$0.550000\pi$
$354$	2.28258	+	0.945476i	2.28258	+	0.945476i
$355$		0			0
$356$	−0.449028			−0.449028
$357$	−1.29091	−	0.419442i	−1.29091	−	0.419442i
$358$		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
$363$	1.20002	−	0.497066i	1.20002	−	0.497066i
$364$		0			0
$365$		0			0
$366$	0.504108	+	0.504108i	0.504108	+	0.504108i
$367$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$367$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	0.302060	+	0.125117i	0.302060	+	0.125117i
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$	−0.513743	+	0.513743i	−0.513743	+	0.513743i
$377$		0			0
$378$			0.499230i			0.499230i
$379$	−0.652583	+	1.57547i	−0.652583	+	1.57547i	0.156434	+	0.987688i	$0.450000\pi$
$379$	−0.652583	+	1.57547i	−0.652583	+	1.57547i	−0.809017	+	0.587785i	$0.800000\pi$
$380$		0			0
$381$		0			0
$382$	−1.17557	−	1.17557i	−1.17557	−	1.17557i
$383$	−0.642040	−	0.642040i	−0.642040	−	0.642040i	0.309017	−	0.951057i	$-0.400000\pi$
$383$	−0.642040	−	0.642040i	−0.642040	−	0.642040i	−0.951057	+	0.309017i	$0.900000\pi$
$384$	−0.584336	−	1.41071i	−0.584336	−	1.41071i
$385$		0			0
$386$		0			0
$387$		0			0
$388$	0.686280	+	0.284267i	0.686280	+	0.284267i
$389$		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
$392$	0.0668557			0.0668557
$393$	1.78616	−	1.78616i	1.78616	−	1.78616i
$394$	−0.831254	−	0.344317i	−0.831254	−	0.344317i
$395$		0			0
$396$		0			0
$397$	1.84206	−	0.763007i	1.84206	−	0.763007i	0.891007	−	0.453990i	$-0.150000\pi$
$397$	1.84206	−	0.763007i	1.84206	−	0.763007i	0.951057	−	0.309017i	$-0.100000\pi$
$398$		0			0
$399$		0			0
$400$	−0.874032	−	0.874032i	−0.874032	−	0.874032i
$401$	−0.497066	−	1.20002i	−0.497066	−	1.20002i	−0.951057	−	0.309017i	$-0.900000\pi$
$401$	−0.497066	−	1.20002i	−0.497066	−	1.20002i	0.453990	−	0.891007i	$-0.350000\pi$
$402$		0			0
$403$		0			0
$404$		−	0.680668i		−	0.680668i
$405$		0			0
$406$		0			0
$407$		0			0
$408$	0.940794	−	0.0740421i	0.940794	−	0.0740421i
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$		−	0.346818i		−	0.346818i
$413$	0.647057	−	1.56213i	0.647057	−	1.56213i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$419$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$420$		0			0
$421$		0			0			−	1.00000i	$-0.5\pi$
$421$		0			0				1.00000i	$0.5\pi$
$422$		0			0
$423$	0.485875	−	0.485875i	0.485875	−	0.485875i
$424$	−0.227313			−0.227313
$425$	0.891007	−	0.453990i	0.891007	−	0.453990i
$426$	0.239605			0.239605
$427$	0.344997	−	0.344997i	0.344997	−	0.344997i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−1.79671	+	0.744220i	−1.79671	+	0.744220i	−0.809017	+	0.587785i	$0.800000\pi$
$431$	−1.79671	+	0.744220i	−1.79671	+	0.744220i	−0.987688	+	0.156434i	$0.950000\pi$
$432$	−0.192229	−	0.464082i	−0.192229	−	0.464082i
$433$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$433$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−0.707107	−	0.292893i	−0.707107	−	0.292893i		−	1.00000i	$-0.5\pi$
$439$	−0.707107	−	0.292893i	−0.707107	−	0.292893i	−0.707107	+	0.707107i	$0.750000\pi$
$440$		0			0
$441$	−0.0632291			−0.0632291
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$	0.533531	−	0.533531i	0.533531	−	0.533531i
$445$		0			0
$446$		0			0
$447$	−0.451326	+	1.08960i	−0.451326	+	1.08960i
$448$	−0.368770	+	0.152749i	−0.368770	+	0.152749i
$449$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$449$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$450$	0.571180	+	0.571180i	0.571180	+	0.571180i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.26007	+	1.26007i	−1.26007	+	1.26007i	−0.309017	+	0.951057i	$0.600000\pi$
$457$	−1.26007	+	1.26007i	−1.26007	+	1.26007i	−0.951057	+	0.309017i	$0.900000\pi$
$458$		0			0
$459$	0.405131	−	0.0318845i	0.405131	−	0.0318845i
$460$		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
$471$	0.981893	−	2.37050i	0.981893	−	2.37050i
$472$			1.17557i			1.17557i
$473$		0			0
$474$	0.169427	−	0.169427i	0.169427	−	0.169427i
$475$		0			0
$476$	0.0313172	+	0.397923i	0.0313172	+	0.397923i
$477$	0.214982			0.214982
$478$	1.58114	−	1.58114i	1.58114	−	1.58114i
$479$	0.965451	+	0.399903i	0.965451	+	0.399903i	0.809017	−	0.587785i	$-0.200000\pi$
$479$	0.965451	+	0.399903i	0.965451	+	0.399903i	0.156434	+	0.987688i	$0.450000\pi$
$480$		0			0
$481$		0			0
$482$	1.65173	−	0.684170i	1.65173	−	0.684170i
$483$		0			0
$484$	−0.270091	−	0.270091i	−0.270091	−	0.270091i
$485$		0			0
$486$	0.527137	+	1.27262i	0.527137	+	1.27262i
$487$	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	$0.250000\pi$
$488$	−0.129812	+	0.313395i	−0.129812	+	0.313395i
$489$		0			0
$490$		0			0
$491$	−1.39680	+	1.39680i	−1.39680	+	1.39680i	−0.587785	+	0.809017i	$0.700000\pi$
$491$	−1.39680	+	1.39680i	−1.39680	+	1.39680i	−0.809017	+	0.587785i	$0.800000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	0.163979i		−	0.163979i
$498$	0.826376	−	1.99505i	0.826376	−	1.99505i
$499$		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
$502$	−1.48131	−	1.48131i	−1.48131	−	1.48131i
$503$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$503$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$504$	0.481982	−	0.199644i	0.481982	−	0.199644i
$505$		0			0
$506$		0			0
$507$	−1.20002	−	0.497066i	−1.20002	−	0.497066i
$508$		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
$518$	−1.32107	−	1.32107i	−1.32107	−	1.32107i
$519$	1.56623	+	1.56623i	1.56623	+	1.56623i
$520$		0			0
$521$	1.57547	−	0.652583i	1.57547	−	0.652583i	0.587785	−	0.809017i	$-0.300000\pi$
$521$	1.57547	−	0.652583i	1.57547	−	0.652583i	0.987688	+	0.156434i	$0.0500000\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$	−0.686280	−	0.284267i	−0.686280	−	0.284267i
$525$	0.959786	−	0.959786i	0.959786	−	0.959786i
$526$	−1.90211			−1.90211
$527$		0			0
$528$		0			0
$529$	0.707107	−	0.707107i	0.707107	−	0.707107i
$530$		0			0
$531$		−	1.11180i		−	1.11180i
$532$		0			0
$533$		0			0
$534$	0.686928	+	1.65839i	0.686928	+	1.65839i
$535$		0			0
$536$		0			0
$537$		0			0
$538$	2.00682	−	0.831254i	2.00682	−	0.831254i
$539$		0			0
$540$		0			0
$541$	1.57547	+	0.652583i	1.57547	+	0.652583i	0.987688	−	0.156434i	$-0.0500000\pi$
$541$	1.57547	+	0.652583i	1.57547	+	0.652583i	0.587785	+	0.809017i	$0.300000\pi$
$542$	1.48131	−	1.48131i	1.48131	−	1.48131i
$543$		0			0
$544$	−0.329843	−	0.647354i	−0.329843	−	0.647354i
$545$		0			0
$546$		0			0
$547$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$547$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$548$		0			0
$549$	0.122771	−	0.296395i	0.122771	−	0.296395i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−0.115951	−	0.115951i	−0.115951	−	0.115951i
$554$	0.0705930	+	0.170427i	0.0705930	+	0.170427i
$555$		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	$0.750000\pi$
$564$	−0.458368	−	0.189862i	−0.458368	−	0.189862i
$565$		0			0
$566$	0.470114	−	1.13496i	0.470114	−	1.13496i
$567$	1.17301	−	0.485875i	1.17301	−	0.485875i
$568$	0.0436289	+	0.105329i	0.0436289	+	0.105329i
$569$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$569$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$570$		0			0
$571$	−0.763007	−	1.84206i	−0.763007	−	1.84206i	−0.453990	−	0.891007i	$-0.650000\pi$
$571$	−0.763007	−	1.84206i	−0.763007	−	1.84206i	−0.309017	−	0.951057i	$-0.600000\pi$
$572$		0			0
$573$	−0.702958	+	1.69709i	−0.702958	+	1.69709i
$574$		0			0
$575$		0			0
$576$	−0.185588	+	0.185588i	−0.185588	+	0.185588i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	1.16110	−	0.183900i	1.16110	−	0.183900i
$579$		0			0
$580$		0			0
$581$	−1.36535	−	0.565548i	−1.36535	−	0.565548i
$582$		−	2.96951i		−	2.96951i
$583$		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	$0.250000\pi$
$588$	0.0174709	+	0.0421786i	0.0174709	+	0.0421786i
$589$		0			0
$590$		0			0
$591$			0.994132i			0.994132i
$592$	1.73673	+	0.719379i	1.73673	+	0.719379i
$593$		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
$596$	0.346818			0.346818
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$	−0.361140	+	0.871868i	−0.361140	+	0.871868i
$601$	0.431351	−	0.178671i	0.431351	−	0.178671i	−0.156434	−	0.987688i	$-0.550000\pi$
$601$	0.431351	−	0.178671i	0.431351	−	0.178671i	0.587785	+	0.809017i	$0.300000\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$	−2.51391	+	1.04129i	−2.51391	+	1.04129i
$607$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$607$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	0.119155	+	0.233854i	0.119155	+	0.233854i
$613$	1.17557			1.17557			0.587785	−	0.809017i	$-0.300000\pi$
$613$	1.17557			1.17557			0.587785	+	0.809017i	$0.300000\pi$
$614$	1.48131	−	1.48131i	1.48131	−	1.48131i
$615$		0			0
$616$		0			0
$617$	0.763007	−	1.84206i	0.763007	−	1.84206i	0.309017	−	0.951057i	$-0.400000\pi$
$617$	0.763007	−	1.84206i	0.763007	−	1.84206i	0.453990	−	0.891007i	$-0.350000\pi$
$618$	−1.28090	+	0.530566i	−1.28090	+	0.530566i
$619$	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
$623$	1.13496	−	0.470114i	1.13496	−	0.470114i
$624$		0			0
$625$			1.00000i			1.00000i
$626$		0			0
$627$		0			0
$628$	−0.754527			−0.754527
$629$	−0.987688	+	1.15643i	−0.987688	+	1.15643i
$630$		0			0
$631$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$631$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$632$	0.105329	+	0.0436289i	0.105329	+	0.0436289i
$633$		0			0
$634$		0			0
$635$		0			0
$636$	−0.0594020	−	0.143409i	−0.0594020	−	0.143409i
$637$		0			0
$638$		0			0
$639$	−0.0412622	−	0.0996158i	−0.0412622	−	0.0996158i
$640$		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
$643$	−0.431351	−	0.178671i	−0.431351	−	0.178671i	0.156434	−	0.987688i	$-0.450000\pi$
$643$	−0.431351	−	0.178671i	−0.431351	−	0.178671i	−0.587785	+	0.809017i	$0.700000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.97538			−1.97538			−0.987688	−	0.156434i	$-0.950000\pi$
$647$	−1.97538			−1.97538			−0.987688	+	0.156434i	$0.950000\pi$
$648$	−0.624189	+	0.624189i	−0.624189	+	0.624189i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	0.581990	+	1.40505i	0.581990	+	1.40505i	0.891007	+	0.453990i	$0.150000\pi$
$653$	0.581990	+	1.40505i	0.581990	+	1.40505i	−0.309017	+	0.951057i	$0.600000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	−0.470114	+	1.13496i	−0.470114	+	1.13496i
$659$			0.618034i			0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$659$			0.618034i			0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$660$		0			0
$661$	−1.34500	+	1.34500i	−1.34500	+	1.34500i	−0.453990	+	0.891007i	$0.650000\pi$
$661$	−1.34500	+	1.34500i	−1.34500	+	1.34500i	−0.891007	+	0.453990i	$0.850000\pi$
$662$	1.38197			1.38197
$663$		0			0
$664$	1.02749			1.02749
$665$		0			0
$666$	−1.13496	−	0.470114i	−1.13496	−	0.470114i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$	−0.697326	−	0.697326i	−0.697326	−	0.697326i
$673$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$673$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$674$	1.85208	−	0.767157i	1.85208	−	0.767157i
$675$	−0.155517	+	0.375450i	−0.155517	+	0.375450i
$676$			0.381966i			0.381966i
$677$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$677$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$678$		0			0
$679$	−2.03225			−2.03225
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.431351	+	0.178671i	0.431351	+	0.178671i	0.587785	−	0.809017i	$-0.300000\pi$
$683$	0.431351	+	0.178671i	0.431351	+	0.178671i	−0.156434	+	0.987688i	$0.550000\pi$
$684$		0			0
$685$		0			0
$686$	−1.03052	+	0.426855i	−1.03052	+	0.426855i
$687$		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	$0.875000\pi$
$692$	0.249264	−	0.601777i	0.249264	−	0.601777i
$693$		0			0
$694$	−1.41071	−	0.584336i	−1.41071	−	0.584336i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.368770	−	0.152749i	−0.368770	−	0.152749i
$701$		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
$706$	−1.64204	−	1.64204i	−1.64204	−	1.64204i
$707$	0.712633	+	1.72045i	0.712633	+	1.72045i
$708$	−0.741655	+	0.307204i	−0.741655	+	0.307204i
$709$	0.399903	−	0.965451i	0.399903	−	0.965451i	−0.587785	−	0.809017i	$-0.700000\pi$
$709$	0.399903	−	0.965451i	0.399903	−	0.965451i	0.987688	−	0.156434i	$-0.0500000\pi$
$710$		0			0
$711$	−0.0996158	−	0.0412622i	−0.0996158	−	0.0412622i
$712$	−0.603941	+	0.603941i	−0.603941	+	0.603941i
$713$		0			0
$714$	1.42174	−	0.724411i	1.42174	−	0.724411i
$715$		0			0
$716$		0			0
$717$	−2.28258	−	0.945476i	−2.28258	−	0.945476i
$718$		0			0
$719$	0.178671	−	0.431351i	0.178671	−	0.431351i	−0.809017	−	0.587785i	$-0.800000\pi$
$719$	0.178671	−	0.431351i	0.178671	−	0.431351i	0.987688	+	0.156434i	$0.0500000\pi$
$720$		0			0
$721$	0.363104	+	0.876612i	0.363104	+	0.876612i
$722$	0.831254	+	0.831254i	0.831254	+	0.831254i
$723$	−1.39680	−	1.39680i	−1.39680	−	1.39680i
$724$		0			0
$725$		0			0
$726$	−0.584336	+	1.41071i	−0.584336	+	1.41071i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	0.217082	−	0.217082i	0.217082	−	0.217082i
$730$		0			0
$731$		0			0
$732$	−0.231640			−0.231640
$733$	−1.34500	+	1.34500i	−1.34500	+	1.34500i	−0.453990	+	0.891007i	$0.650000\pi$
$733$	−1.34500	+	1.34500i	−1.34500	+	1.34500i	−0.891007	+	0.453990i	$0.850000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	1.39680	+	1.39680i	1.39680	+	1.39680i	0.809017	+	0.587785i	$0.200000\pi$
$739$	1.39680	+	1.39680i	1.39680	+	1.39680i	0.587785	+	0.809017i	$0.300000\pi$
$740$		0			0
$741$		0			0
$742$	−0.355093	+	0.147084i	−0.355093	+	0.147084i
$743$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$743$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−0.971750			−0.971750
$748$		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	$0.125000\pi$
$752$		−	1.23607i		−	1.23607i
$753$	−0.885778	+	2.13846i	−0.885778	+	2.13846i
$754$		0			0
$755$		0			0
$756$	−0.114700	−	0.114700i	−0.114700	−	0.114700i
$757$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$757$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$758$	−0.767157	−	1.85208i	−0.767157	−	1.85208i
$759$		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
$764$	0.540182			0.540182
$765$		0			0
$766$	1.06740			1.06740
$767$		0			0
$768$	1.20002	+	0.497066i	1.20002	+	0.497066i
$769$		−	1.90211i		−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$769$		−	1.90211i		−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1.34500	+	1.34500i	1.34500	+	1.34500i	0.891007	+	0.453990i	$0.150000\pi$
$773$	1.34500	+	1.34500i	1.34500	+	1.34500i	0.453990	+	0.891007i	$0.350000\pi$
$774$		0			0
$775$		0			0
$776$	1.30538	−	0.540707i	1.30538	−	0.540707i
$777$	−0.789959	+	1.90713i	−0.789959	+	1.90713i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.0804276	+	0.0804276i	−0.0804276	+	0.0804276i
$785$		0			0
$786$			2.96951i			2.96951i
$787$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$787$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$788$	0.270091	−	0.111875i	0.270091	−	0.111875i
$789$	0.804270	+	1.94168i	0.804270	+	1.94168i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	−0.896969	+	2.16547i	−0.896969	+	2.16547i
$795$		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
$799$	0.951057	+	0.309017i	0.951057	+	0.309017i
$800$	0.726543			0.726543
$801$	0.571180	−	0.571180i	0.571180	−	0.571180i
$802$	1.41071	+	0.584336i	1.41071	+	0.584336i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−1.69709	−	1.69709i	−1.69709	−	1.69709i
$808$	−0.915497	−	0.915497i	−0.915497	−	0.915497i
$809$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$809$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$810$		0			0
$811$	−0.0600500	+	0.144974i	−0.0600500	+	0.144974i	−0.951057	−	0.309017i	$-0.900000\pi$
$811$	−0.0600500	+	0.144974i	−0.0600500	+	0.144974i	0.891007	+	0.453990i	$0.150000\pi$
$812$		0			0
$813$	−2.13846	−	0.885778i	−2.13846	−	0.885778i
$814$		0			0
$815$		0			0
$816$	−1.04270	+	1.22085i	−1.04270	+	1.22085i
$817$		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	$0.5\pi$
$824$	−0.466469	−	0.466469i	−0.466469	−	0.466469i
$825$		0			0
$826$	0.760661	+	1.83640i	0.760661	+	1.83640i
$827$	0.431351	−	0.178671i	0.431351	−	0.178671i	−0.156434	−	0.987688i	$-0.550000\pi$
$827$	0.431351	−	0.178671i	0.431351	−	0.178671i	0.587785	+	0.809017i	$0.300000\pi$
$828$		0			0
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		0			0
$831$	0.144123	−	0.144123i	0.144123	−	0.144123i
$832$		0			0
$833$	−0.0417758	−	0.0819895i	−0.0417758	−	0.0819895i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$839$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$840$		0			0
$841$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$			0.807771i			0.807771i
$847$	0.965451	+	0.399903i	0.965451	+	0.399903i
$848$	0.273457	−	0.273457i	0.273457	−	0.273457i
$849$	−1.35734			−1.35734
$850$	−0.363271	+	1.11803i	−0.363271	+	1.11803i
$851$		0			0
$852$	−0.0550500	+	0.0550500i	−0.0550500	+	0.0550500i
$853$	−1.84206	−	0.763007i	−1.84206	−	0.763007i	−0.951057	−	0.309017i	$-0.900000\pi$
$853$	−1.84206	−	0.763007i	−1.84206	−	0.763007i	−0.891007	−	0.453990i	$-0.850000\pi$
$854$			0.573560i			0.573560i
$855$		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
$862$	0.874883	−	2.11215i	0.874883	−	2.11215i
$863$		−	0.618034i		−	0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$863$		−	0.618034i		−	0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$864$	0.272780	+	0.112989i	0.272780	+	0.112989i
$865$		0			0
$866$		0			0
$867$	−0.678671	−	1.10749i	−0.678671	−	1.10749i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−1.23457	+	0.511376i	−1.23457	+	0.511376i
$874$		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	$0.375000\pi$
$878$	0.831254	−	0.344317i	0.831254	−	0.344317i
$879$		0			0
$880$		0			0
$881$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$881$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$882$	0.0525594	−	0.0525594i	0.0525594	−	0.0525594i
$883$	−1.97538			−1.97538			−0.987688	−	0.156434i	$-0.950000\pi$
$883$	−1.97538			−1.97538			−0.987688	+	0.156434i	$0.950000\pi$
$884$		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	$0.125000\pi$
$888$		−	1.43520i		−	1.43520i
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$	−0.530566	−	1.28090i	−0.530566	−	1.28090i
$895$		0			0
$896$	0.470114	−	1.13496i	0.470114	−	1.13496i
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.262461			−0.262461
$901$	0.142040	+	0.278768i	0.142040	+	0.278768i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1.70711	−	0.707107i	1.70711	−	0.707107i	0.707107	−	0.707107i	$-0.250000\pi$
$907$	1.70711	−	0.707107i	1.70711	−	0.707107i	1.00000			$0$
$908$		0			0
$909$	0.865836	+	0.865836i	0.865836	+	0.865836i
$910$		0			0
$911$	0.178671	+	0.431351i	0.178671	+	0.431351i	0.987688	−	0.156434i	$-0.0500000\pi$
$911$	0.178671	+	0.431351i	0.178671	+	0.431351i	−0.809017	+	0.587785i	$0.800000\pi$
$912$		0			0
$913$		0			0
$914$		−	2.09488i		−	2.09488i
$915$		0			0
$916$		0			0
$917$	2.03225			2.03225
$918$	−0.310263	+	0.363271i	−0.310263	+	0.363271i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	−2.13846	−	0.885778i	−2.13846	−	0.885778i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.581990	−	1.40505i	−0.581990	−	1.40505i
$926$		0			0
$927$	0.441165	+	0.441165i	0.441165	+	0.441165i
$928$		0			0
$929$	1.40505	−	0.581990i	1.40505	−	0.581990i	0.453990	−	0.891007i	$-0.350000\pi$
$929$	1.40505	−	0.581990i	1.40505	−	0.581990i	0.951057	+	0.309017i	$0.100000\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$937$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.79671	−	0.744220i	1.79671	−	0.744220i	0.809017	−	0.587785i	$-0.200000\pi$
$941$	1.79671	−	0.744220i	1.79671	−	0.744220i	0.987688	−	0.156434i	$-0.0500000\pi$
$942$	1.15428	+	2.78669i	1.15428	+	2.78669i
$943$		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	$-0.250000\pi$
$948$			0.0778525i			0.0778525i
$949$		0			0
$950$		0			0
$951$		0			0
$952$	0.577327	+	0.493084i	0.577327	+	0.493084i
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$	−0.178705	+	0.178705i	−0.178705	+	0.178705i
$955$		0			0
$956$			0.726543i			0.726543i
$957$		0			0
$958$	−1.13496	+	0.470114i	−1.13496	+	0.470114i
$959$		0			0
$960$		0			0
$961$	0.707107	+	0.707107i	0.707107	+	0.707107i
$962$		0			0
$963$		0			0
$964$	−0.222300	+	0.536680i	−0.222300	+	0.536680i
$965$		0			0
$966$		0			0
$967$	1.39680	−	1.39680i	1.39680	−	1.39680i	0.587785	−	0.809017i	$-0.300000\pi$
$967$	1.39680	−	1.39680i	1.39680	−	1.39680i	0.809017	−	0.587785i	$-0.200000\pi$
$968$	−0.726543			−0.726543
$969$		0			0
$970$		0			0
$971$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$971$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$972$	−0.413500	−	0.171277i	−0.413500	−	0.171277i
$973$		0			0
$974$	−0.344317	+	0.831254i	−0.344317	+	0.831254i
$975$		0			0
$976$	−0.220850	−	0.533179i	−0.220850	−	0.533179i
$977$		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
$982$		−	2.32219i		−	2.32219i
$983$		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
$987$	1.35734			1.35734
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−0.178671	+	0.431351i	−0.178671	+	0.431351i	−0.987688	−	0.156434i	$-0.950000\pi$
$991$	−0.178671	+	0.431351i	−0.178671	+	0.431351i	0.809017	+	0.587785i	$0.200000\pi$
$992$		0			0
$993$	−0.584336	−	1.41071i	−0.584336	−	1.41071i
$994$	0.136308	+	0.136308i	0.136308	+	0.136308i
$995$		0			0
$996$	0.268506	+	0.648231i	0.268506	+	0.648231i
$997$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$997$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$998$		0			0
$999$		−	0.618034i		−	0.618034i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	799.1.h.b.281.2	✓	16
17.2	even	8	inner	799.1.h.b.563.2	yes	16
47.46	odd	2	CM	799.1.h.b.281.2	✓	16
799.563	odd	8	inner	799.1.h.b.563.2	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
799.1.h.b.281.2	✓	16	1.1	even	1	trivial
799.1.h.b.281.2	✓	16	47.46	odd	2	CM
799.1.h.b.563.2	yes	16	17.2	even	8	inner
799.1.h.b.563.2	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)

\(f(q)\)	\(=\)	\(q+(-0.831254 + 0.831254i) q^{2} +(1.20002 + 0.497066i) q^{3} -0.381966i q^{4} +(-1.41071 + 0.584336i) q^{6} +(0.399903 + 0.965451i) q^{7} +(-0.513743 - 0.513743i) q^{8} +(0.485875 + 0.485875i) q^{9} +(0.189862 - 0.458368i) q^{12} +(-1.13496 - 0.470114i) q^{14} +1.23607 q^{16} +(-0.309017 + 0.951057i) q^{17} -0.807771 q^{18} +1.35734i q^{21} +(-0.361140 - 0.871868i) q^{24} +(-0.707107 - 0.707107i) q^{25} +(-0.155517 - 0.375450i) q^{27} +(0.368770 - 0.152749i) q^{28} +(-0.513743 + 0.513743i) q^{32} +(-0.533698 - 1.04744i) q^{34} +(0.185588 - 0.185588i) q^{36} +(1.40505 + 0.581990i) q^{37} +(-1.12830 - 1.12830i) q^{42} -1.00000i q^{47} +(1.48331 + 0.614407i) q^{48} +(-0.0650673 + 0.0650673i) q^{49} +1.17557 q^{50} +(-0.843566 + 0.987688i) q^{51} +(0.221232 - 0.221232i) q^{53} +(0.441368 + 0.182821i) q^{54} +(0.290547 - 0.701442i) q^{56} +(-1.14412 - 1.14412i) q^{59} +(-0.178671 - 0.431351i) q^{61} +(-0.274786 + 0.663392i) q^{63} +0.381966i q^{64} +(0.363271 + 0.118034i) q^{68} +(-0.144974 - 0.0600500i) q^{71} -0.499230i q^{72} +(-1.65173 + 0.684170i) q^{74} +(-0.497066 - 1.20002i) q^{75} +(-0.144974 + 0.0600500i) q^{79} -1.21498i q^{81} +(-1.00000 + 1.00000i) q^{83} +0.518459 q^{84} -1.17557i q^{89} +(0.831254 + 0.831254i) q^{94} +(-0.871868 + 0.361140i) q^{96} +(-0.744220 + 1.79671i) q^{97} -0.108175i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{3} - 4 q^{9} - 16 q^{16} + 4 q^{17} - 20 q^{24} - 4 q^{27} - 4 q^{28} + 4 q^{36} - 20 q^{42} + 24 q^{48} + 4 q^{49} - 16 q^{51} + 4 q^{53} + 20 q^{54} + 20 q^{56} + 4 q^{61} - 4 q^{63} - 4 q^{71}+ \cdots + 32 q^{84}+O(q^{100}) \) 16 * q - 4 * q^3 - 4 * q^9 - 16 * q^16 + 4 * q^17 - 20 * q^24 - 4 * q^27 - 4 * q^28 + 4 * q^36 - 20 * q^42 + 24 * q^48 + 4 * q^49 - 16 * q^51 + 4 * q^53 + 20 * q^54 + 20 * q^56 + 4 * q^61 - 4 * q^63 - 4 * q^71 - 4 * q^79 - 16 * q^83 + 32 * q^84

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