LMFDB - Embedded newform 799.1.h.b.281.3

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			281.3
Root			$0.987688 - 0.156434i$ of defining polynomial
Character	$\chi$	$=$	799.281
Dual form			799.1.h.b.563.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.831254 - 0.831254i) q^{2} +(-0.144974 - 0.0600500i) q^{3} -0.381966i q^{4} +(-0.170427 + 0.0705930i) q^{6} +(0.744220 + 1.79671i) q^{7} +(0.513743 + 0.513743i) q^{8} +(-0.689695 - 0.689695i) q^{9} +(-0.0229371 + 0.0553750i) q^{12} +(2.11215 + 0.874883i) q^{14} +1.23607 q^{16} +(-0.309017 - 0.951057i) q^{17} -1.14662 q^{18} -0.305165i q^{21} +(-0.0436289 - 0.105329i) q^{24} +(-0.707107 - 0.707107i) q^{25} +(0.118621 + 0.286377i) q^{27} +(0.686280 - 0.284267i) q^{28} +(0.513743 - 0.513743i) q^{32} +(-1.04744 - 0.533698i) q^{34} +(-0.263440 + 0.263440i) q^{36} +(-1.84206 - 0.763007i) q^{37} +(-0.253670 - 0.253670i) q^{42} -1.00000i q^{47} +(-0.179197 - 0.0742259i) q^{48} +(-1.96718 + 1.96718i) q^{49} -1.17557 q^{50} +(-0.0123117 + 0.156434i) q^{51} +(1.39680 - 1.39680i) q^{53} +(0.336657 + 0.139448i) q^{54} +(-0.540707 + 1.30538i) q^{56} +(-1.14412 - 1.14412i) q^{59} +(0.652583 + 1.57547i) q^{61} +(0.725895 - 1.75246i) q^{63} +0.381966i q^{64} +(-0.363271 + 0.118034i) q^{68} +(1.20002 + 0.497066i) q^{71} -0.708653i q^{72} +(-2.16547 + 0.896969i) q^{74} +(0.0600500 + 0.144974i) q^{75} +(1.20002 - 0.497066i) q^{79} +0.926736i q^{81} +(-1.00000 + 1.00000i) q^{83} -0.116563 q^{84} +1.17557i q^{89} +(-0.831254 - 0.831254i) q^{94} +(-0.105329 + 0.0436289i) q^{96} +(-0.399903 + 0.965451i) q^{97} +3.27045i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	−0.156434	−	0.987688i	$-0.550000\pi$
$2$	0.831254	−	0.831254i	0.831254	−	0.831254i	0.987688	+	0.156434i	$0.0500000\pi$
$3$	−0.144974	−	0.0600500i	−0.144974	−	0.0600500i	0.309017	−	0.951057i	$-0.400000\pi$
$3$	−0.144974	−	0.0600500i	−0.144974	−	0.0600500i	−0.453990	+	0.891007i	$0.650000\pi$
$4$		−	0.381966i		−	0.381966i
$5$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$5$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$6$	−0.170427	+	0.0705930i	−0.170427	+	0.0705930i
$7$	0.744220	+	1.79671i	0.744220	+	1.79671i	0.587785	+	0.809017i	$0.300000\pi$
$7$	0.744220	+	1.79671i	0.744220	+	1.79671i	0.156434	+	0.987688i	$0.450000\pi$
$8$	0.513743	+	0.513743i	0.513743	+	0.513743i
$9$	−0.689695	−	0.689695i	−0.689695	−	0.689695i
$10$		0			0
$11$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$11$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$12$	−0.0229371	+	0.0553750i	−0.0229371	+	0.0553750i
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$	2.11215	+	0.874883i	2.11215	+	0.874883i
$15$		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
$18$	−1.14662			−1.14662
$19$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$19$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$20$		0			0
$21$		−	0.305165i		−	0.305165i
$22$		0			0
$23$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$23$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$24$	−0.0436289	−	0.105329i	−0.0436289	−	0.105329i
$25$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$26$		0			0
$27$	0.118621	+	0.286377i	0.118621	+	0.286377i
$28$	0.686280	−	0.284267i	0.686280	−	0.284267i
$29$		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
$34$	−1.04744	−	0.533698i	−1.04744	−	0.533698i
$35$		0			0
$36$	−0.263440	+	0.263440i	−0.263440	+	0.263440i
$37$	−1.84206	−	0.763007i	−1.84206	−	0.763007i	−0.951057	−	0.309017i	$-0.900000\pi$
$37$	−1.84206	−	0.763007i	−1.84206	−	0.763007i	−0.891007	−	0.453990i	$-0.850000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$41$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$42$	−0.253670	−	0.253670i	−0.253670	−	0.253670i
$43$		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
$48$	−0.179197	−	0.0742259i	−0.179197	−	0.0742259i
$49$	−1.96718	+	1.96718i	−1.96718	+	1.96718i
$50$	−1.17557			−1.17557
$51$	−0.0123117	+	0.156434i	−0.0123117	+	0.156434i
$52$		0			0
$53$	1.39680	−	1.39680i	1.39680	−	1.39680i	0.587785	−	0.809017i	$-0.300000\pi$
$53$	1.39680	−	1.39680i	1.39680	−	1.39680i	0.809017	−	0.587785i	$-0.200000\pi$
$54$	0.336657	+	0.139448i	0.336657	+	0.139448i
$55$		0			0
$56$	−0.540707	+	1.30538i	−0.540707	+	1.30538i
$57$		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
$61$	0.652583	+	1.57547i	0.652583	+	1.57547i	0.809017	+	0.587785i	$0.200000\pi$
$61$	0.652583	+	1.57547i	0.652583	+	1.57547i	−0.156434	+	0.987688i	$0.550000\pi$
$62$		0			0
$63$	0.725895	−	1.75246i	0.725895	−	1.75246i
$64$			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
$71$	1.20002	+	0.497066i	1.20002	+	0.497066i	0.891007	−	0.453990i	$-0.150000\pi$
$71$	1.20002	+	0.497066i	1.20002	+	0.497066i	0.309017	+	0.951057i	$0.400000\pi$
$72$		−	0.708653i		−	0.708653i
$73$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$73$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$74$	−2.16547	+	0.896969i	−2.16547	+	0.896969i
$75$	0.0600500	+	0.144974i	0.0600500	+	0.144974i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	1.20002	−	0.497066i	1.20002	−	0.497066i	0.309017	−	0.951057i	$-0.400000\pi$
$79$	1.20002	−	0.497066i	1.20002	−	0.497066i	0.891007	+	0.453990i	$0.150000\pi$
$80$		0			0
$81$			0.926736i			0.926736i
$82$		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			$\pi$
$84$	−0.116563			−0.116563
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$			1.17557i			1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$89$			1.17557i			1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$	−0.831254	−	0.831254i	−0.831254	−	0.831254i
$95$		0			0
$96$	−0.105329	+	0.0436289i	−0.105329	+	0.0436289i
$97$	−0.399903	+	0.965451i	−0.399903	+	0.965451i	0.587785	+	0.809017i	$0.300000\pi$
$97$	−0.399903	+	0.965451i	−0.399903	+	0.965451i	−0.987688	+	0.156434i	$0.950000\pi$
$98$			3.27045i			3.27045i
$99$		0			0
$100$	−0.270091	+	0.270091i	−0.270091	+	0.270091i
$101$	−0.907981			−0.907981			−0.453990	−	0.891007i	$-0.650000\pi$
$101$	−0.907981			−0.907981			−0.453990	+	0.891007i	$0.650000\pi$
$102$	0.119803	+	0.140271i	0.119803	+	0.140271i
$103$	−1.78201			−1.78201			−0.891007	−	0.453990i	$-0.850000\pi$
$103$	−1.78201			−1.78201			−0.891007	+	0.453990i	$0.850000\pi$
$104$		0			0
$105$		0			0
$106$		−	2.32219i		−	2.32219i
$107$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$107$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$108$	0.109386	−	0.0453093i	0.109386	−	0.0453093i
$109$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$109$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$110$		0			0
$111$	0.221232	+	0.221232i	0.221232	+	0.221232i
$112$	0.919906	+	2.22085i	0.919906	+	2.22085i
$113$		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
$119$	1.47879	−	1.26301i	1.47879	−	1.26301i
$120$		0			0
$121$	0.707107	−	0.707107i	0.707107	−	0.707107i
$122$	1.85208	+	0.767157i	1.85208	+	0.767157i
$123$		0			0
$124$		0			0
$125$		0			0
$126$	−0.853340	−	2.06015i	−0.853340	−	2.06015i
$127$		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
$131$	0.399903	−	0.965451i	0.399903	−	0.965451i	−0.587785	−	0.809017i	$-0.700000\pi$
$131$	0.399903	−	0.965451i	0.399903	−	0.965451i	0.987688	−	0.156434i	$-0.0500000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	0.329843	−	0.647354i	0.329843	−	0.647354i
$137$		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
$141$	−0.0600500	+	0.144974i	−0.0600500	+	0.144974i
$142$	1.41071	−	0.584336i	1.41071	−	0.584336i
$143$		0			0
$144$	−0.852510	−	0.852510i	−0.852510	−	0.852510i
$145$		0			0
$146$		0			0
$147$	0.403318	−	0.167060i	0.403318	−	0.167060i
$148$	−0.291443	+	0.703605i	−0.291443	+	0.703605i
$149$		−	1.78201i		−	1.78201i	−0.453990	−	0.891007i	$-0.650000\pi$
$149$		−	1.78201i		−	1.78201i	0.453990	−	0.891007i	$-0.350000\pi$
$150$	0.170427	+	0.0705930i	0.170427	+	0.0705930i
$151$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$151$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$152$		0			0
$153$	−0.442812	+	0.869067i	−0.442812	+	0.869067i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	0.312869i		−	0.312869i	−0.987688	−	0.156434i	$-0.950000\pi$
$157$		−	0.312869i		−	0.312869i	0.987688	−	0.156434i	$-0.0500000\pi$
$158$	0.584336	−	1.41071i	0.584336	−	1.41071i
$159$	−0.286377	+	0.118621i	−0.286377	+	0.118621i
$160$		0			0
$161$		0			0
$162$	0.770353	+	0.770353i	0.770353	+	0.770353i
$163$		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	$0.125000\pi$
$168$	0.156776	−	0.156776i	0.156776	−	0.156776i
$169$	−1.00000			−1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−0.431351	−	0.178671i	−0.431351	−	0.178671i	0.156434	−	0.987688i	$-0.450000\pi$
$173$	−0.431351	−	0.178671i	−0.431351	−	0.178671i	−0.587785	+	0.809017i	$0.700000\pi$
$174$		0			0
$175$	0.744220	−	1.79671i	0.744220	−	1.79671i
$176$		0			0
$177$	0.0971629	+	0.234572i	0.0971629	+	0.234572i
$178$	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
$183$		−	0.267590i		−	0.267590i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−0.381966			−0.381966
$189$	−0.426255	+	0.426255i	−0.426255	+	0.426255i
$190$		0			0
$191$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$191$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$192$	0.0229371	−	0.0553750i	0.0229371	−	0.0553750i
$193$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$193$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$194$	0.470114	+	1.13496i	0.470114	+	1.13496i
$195$		0			0
$196$	0.751396	+	0.751396i	0.751396	+	0.751396i
$197$	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
$202$	−0.754763	+	0.754763i	−0.754763	+	0.754763i
$203$		0			0
$204$	0.0597526	+	0.00470264i	0.0597526	+	0.00470264i
$205$		0			0
$206$	−1.48131	+	1.48131i	−1.48131	+	1.48131i
$207$		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	$0.375000\pi$
$212$	−0.533531	−	0.533531i	−0.533531	−	0.533531i
$213$	−0.144123	−	0.144123i	−0.144123	−	0.144123i
$214$		0			0
$215$		0			0
$216$	−0.0861835	+	0.208065i	−0.0861835	+	0.208065i
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$	0.367799			0.367799
$223$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$223$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$224$	1.30538	+	0.540707i	1.30538	+	0.540707i
$225$			0.975377i			0.975377i
$226$		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
$238$	0.179372	−	2.27913i	0.179372	−	2.27913i
$239$	1.90211			1.90211			0.951057	−	0.309017i	$-0.100000\pi$
$239$	1.90211			1.90211			0.951057	+	0.309017i	$0.100000\pi$
$240$		0			0
$241$	1.84206	+	0.763007i	1.84206	+	0.763007i	0.951057	+	0.309017i	$0.100000\pi$
$241$	1.84206	+	0.763007i	1.84206	+	0.763007i	0.891007	+	0.453990i	$0.150000\pi$
$242$		−	1.17557i		−	1.17557i
$243$	0.174272	−	0.420729i	0.174272	−	0.420729i
$244$	0.601777	−	0.249264i	0.601777	−	0.249264i
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	0.205023	−	0.0849235i	0.205023	−	0.0849235i
$250$		0			0
$251$		−	0.907981i		−	0.907981i	−0.891007	−	0.453990i	$-0.850000\pi$
$251$		−	0.907981i		−	0.907981i	0.891007	−	0.453990i	$-0.150000\pi$
$252$	−0.669382	−	0.277267i	−0.669382	−	0.277267i
$253$		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
$259$		−	3.87749i		−	3.87749i
$260$		0			0
$261$		0			0
$262$	−0.470114	−	1.13496i	−0.470114	−	1.13496i
$263$	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
$267$	0.0705930	−	0.170427i	0.0705930	−	0.170427i
$268$		0			0
$269$	−1.70711	−	0.707107i	−1.70711	−	0.707107i	−0.707107	−	0.707107i	$-0.750000\pi$
$269$	−1.70711	−	0.707107i	−1.70711	−	0.707107i	−1.00000			$\pi$
$270$		0			0
$271$	0.907981			0.907981			0.453990	−	0.891007i	$-0.350000\pi$
$271$	0.907981			0.907981			0.453990	+	0.891007i	$0.350000\pi$
$272$	−0.381966	−	1.17557i	−0.381966	−	1.17557i
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−0.497066	+	1.20002i	−0.497066	+	1.20002i	0.453990	+	0.891007i	$0.350000\pi$
$277$	−0.497066	+	1.20002i	−0.497066	+	1.20002i	−0.951057	+	0.309017i	$0.900000\pi$
$278$		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	$0.250000\pi$
$282$	0.0705930	+	0.170427i	0.0705930	+	0.170427i
$283$	−1.79671	+	0.744220i	−1.79671	+	0.744220i	−0.809017	+	0.587785i	$0.800000\pi$
$283$	−1.79671	+	0.744220i	−1.79671	+	0.744220i	−0.987688	+	0.156434i	$0.950000\pi$
$284$	0.189862	−	0.458368i	0.189862	−	0.458368i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.708653			−0.708653
$289$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$290$		0			0
$291$	0.115951	−	0.115951i	0.115951	−	0.115951i
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	0.196391	−	0.474129i	0.196391	−	0.474129i
$295$		0			0
$296$	−0.554357	−	1.33834i	−0.554357	−	1.33834i
$297$		0			0
$298$	−1.48131	−	1.48131i	−1.48131	−	1.48131i
$299$		0			0
$300$	0.0553750	−	0.0229371i	0.0553750	−	0.0229371i
$301$		0			0
$302$		0			0
$303$	0.131633	+	0.0545243i	0.131633	+	0.0545243i
$304$		0			0
$305$		0			0
$306$	0.354326	+	1.09050i	0.354326	+	1.09050i
$307$	0.907981			0.907981			0.453990	−	0.891007i	$-0.350000\pi$
$307$	0.907981			0.907981			0.453990	+	0.891007i	$0.350000\pi$
$308$		0			0
$309$	0.258345	+	0.107010i	0.258345	+	0.107010i
$310$		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	$0.375000\pi$
$314$	−0.260074	−	0.260074i	−0.260074	−	0.260074i
$315$		0			0
$316$	−0.189862	−	0.458368i	−0.189862	−	0.458368i
$317$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$317$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$318$	−0.139448	+	0.336657i	−0.139448	+	0.336657i
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.353982			0.353982
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1.79671	−	0.744220i	1.79671	−	0.744220i
$330$		0			0
$331$	0.831254	+	0.831254i	0.831254	+	0.831254i	0.987688	−	0.156434i	$-0.0500000\pi$
$331$	0.831254	+	0.831254i	0.831254	+	0.831254i	−0.156434	+	0.987688i	$0.550000\pi$
$332$	0.381966	+	0.381966i	0.381966	+	0.381966i
$333$	0.744220	+	1.79671i	0.744220	+	1.79671i
$334$		0			0
$335$		0			0
$336$		−	0.377205i		−	0.377205i
$337$	0.431351	+	0.178671i	0.431351	+	0.178671i	0.587785	−	0.809017i	$-0.300000\pi$
$337$	0.431351	+	0.178671i	0.431351	+	0.178671i	−0.156434	+	0.987688i	$0.550000\pi$
$338$	−0.831254	+	0.831254i	−0.831254	+	0.831254i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−3.20175	−	1.32621i	−3.20175	−	1.32621i
$344$		0			0
$345$		0			0
$346$	−0.507083	+	0.210041i	−0.507083	+	0.210041i
$347$	−0.0600500	−	0.144974i	−0.0600500	−	0.144974i	0.891007	−	0.453990i	$-0.150000\pi$
$347$	−0.0600500	−	0.144974i	−0.0600500	−	0.144974i	−0.951057	+	0.309017i	$0.900000\pi$
$348$		0			0
$349$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$349$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$350$	−0.874883	−	2.11215i	−0.874883	−	2.11215i
$351$		0			0
$352$		0			0
$353$			0.312869i			0.312869i	0.987688	+	0.156434i	$0.0500000\pi$
$353$			0.312869i			0.312869i	−0.987688	+	0.156434i	$0.950000\pi$
$354$	0.275756	+	0.114222i	0.275756	+	0.114222i
$355$		0			0
$356$	0.449028			0.449028
$357$	−0.290229	+	0.0943012i	−0.290229	+	0.0943012i
$358$		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
$363$	−0.144974	+	0.0600500i	−0.144974	+	0.0600500i
$364$		0			0
$365$		0			0
$366$	−0.222435	−	0.222435i	−0.222435	−	0.222435i
$367$		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
$371$	3.54917	+	1.47011i	3.54917	+	1.47011i
$372$		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
$378$			0.708653i			0.708653i
$379$	0.178671	−	0.431351i	0.178671	−	0.431351i	−0.809017	−	0.587785i	$-0.800000\pi$
$379$	0.178671	−	0.431351i	0.178671	−	0.431351i	0.987688	+	0.156434i	$0.0500000\pi$
$380$		0			0
$381$		0			0
$382$	1.17557	+	1.17557i	1.17557	+	1.17557i
$383$	1.26007	+	1.26007i	1.26007	+	1.26007i	0.951057	+	0.309017i	$0.100000\pi$
$383$	1.26007	+	1.26007i	1.26007	+	1.26007i	0.309017	+	0.951057i	$0.400000\pi$
$384$	−0.0705930	−	0.170427i	−0.0705930	−	0.170427i
$385$		0			0
$386$		0			0
$387$		0			0
$388$	0.368770	+	0.152749i	0.368770	+	0.152749i
$389$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$389$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$390$		0			0
$391$		0			0
$392$	−2.02125			−2.02125
$393$	−0.115951	+	0.115951i	−0.115951	+	0.115951i
$394$	0.831254	+	0.344317i	0.831254	+	0.344317i
$395$		0			0
$396$		0			0
$397$	−1.40505	+	0.581990i	−1.40505	+	0.581990i	−0.951057	−	0.309017i	$-0.900000\pi$
$397$	−1.40505	+	0.581990i	−1.40505	+	0.581990i	−0.453990	+	0.891007i	$0.650000\pi$
$398$		0			0
$399$		0			0
$400$	−0.874032	−	0.874032i	−0.874032	−	0.874032i
$401$	0.0600500	+	0.144974i	0.0600500	+	0.144974i	0.951057	−	0.309017i	$-0.100000\pi$
$401$	0.0600500	+	0.144974i	0.0600500	+	0.144974i	−0.891007	+	0.453990i	$0.850000\pi$
$402$		0			0
$403$		0			0
$404$			0.346818i			0.346818i
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−0.0866922	+	0.0740421i	−0.0866922	+	0.0740421i
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$			0.680668i			0.680668i
$413$	1.20417	−	2.90713i	1.20417	−	2.90713i
$414$		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
$423$	−0.689695	+	0.689695i	−0.689695	+	0.689695i
$424$	1.43520			1.43520
$425$	−0.453990	+	0.891007i	−0.453990	+	0.891007i
$426$	−0.239605			−0.239605
$427$	−2.34500	+	2.34500i	−2.34500	+	2.34500i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−0.965451	+	0.399903i	−0.965451	+	0.399903i	−0.809017	−	0.587785i	$-0.800000\pi$
$431$	−0.965451	+	0.399903i	−0.965451	+	0.399903i	−0.156434	+	0.987688i	$0.550000\pi$
$432$	0.146624	+	0.353982i	0.146624	+	0.353982i
$433$		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
$441$	2.71351			2.71351
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$	0.0845030	−	0.0845030i	0.0845030	−	0.0845030i
$445$		0			0
$446$		0			0
$447$	−0.107010	+	0.258345i	−0.107010	+	0.258345i
$448$	−0.686280	+	0.284267i	−0.686280	+	0.284267i
$449$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$449$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$450$	0.810786	+	0.810786i	0.810786	+	0.810786i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.642040	−	0.642040i	0.642040	−	0.642040i	−0.309017	−	0.951057i	$-0.600000\pi$
$457$	0.642040	−	0.642040i	0.642040	−	0.642040i	0.951057	+	0.309017i	$0.100000\pi$
$458$		0			0
$459$	0.235705	−	0.201311i	0.235705	−	0.201311i
$460$		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
$471$	−0.0187878	+	0.0453577i	−0.0187878	+	0.0453577i
$472$		−	1.17557i		−	1.17557i
$473$		0			0
$474$	−0.169427	+	0.169427i	−0.169427	+	0.169427i
$475$		0			0
$476$	−0.482426	−	0.564848i	−0.482426	−	0.564848i
$477$	−1.92674			−1.92674
$478$	1.58114	−	1.58114i	1.58114	−	1.58114i
$479$	1.79671	+	0.744220i	1.79671	+	0.744220i	0.987688	+	0.156434i	$0.0500000\pi$
$479$	1.79671	+	0.744220i	1.79671	+	0.744220i	0.809017	+	0.587785i	$0.200000\pi$
$480$		0			0
$481$		0			0
$482$	2.16547	−	0.896969i	2.16547	−	0.896969i
$483$		0			0
$484$	−0.270091	−	0.270091i	−0.270091	−	0.270091i
$485$		0			0
$486$	−0.204869	−	0.494597i	−0.204869	−	0.494597i
$487$	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	$0.250000\pi$
$488$	−0.474129	+	1.14465i	−0.474129	+	1.14465i
$489$		0			0
$490$		0			0
$491$	−0.221232	+	0.221232i	−0.221232	+	0.221232i	−0.809017	−	0.587785i	$-0.800000\pi$
$491$	−0.221232	+	0.221232i	−0.221232	+	0.221232i	0.587785	+	0.809017i	$0.300000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$			2.52601i			2.52601i
$498$	0.0998336	−	0.241020i	0.0998336	−	0.241020i
$499$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$499$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$500$		0			0
$501$		0			0
$502$	−0.754763	−	0.754763i	−0.754763	−	0.754763i
$503$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$503$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$504$	1.27324	−	0.527393i	1.27324	−	0.527393i
$505$		0			0
$506$		0			0
$507$	0.144974	+	0.0600500i	0.144974	+	0.0600500i
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	−3.22318	−	3.22318i	−3.22318	−	3.22318i
$519$	0.0518052	+	0.0518052i	0.0518052	+	0.0518052i
$520$		0			0
$521$	−0.431351	+	0.178671i	−0.431351	+	0.178671i	−0.587785	−	0.809017i	$-0.700000\pi$
$521$	−0.431351	+	0.178671i	−0.431351	+	0.178671i	0.156434	+	0.987688i	$0.450000\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$	−0.368770	−	0.152749i	−0.368770	−	0.152749i
$525$	−0.215784	+	0.215784i	−0.215784	+	0.215784i
$526$	1.90211			1.90211
$527$		0			0
$528$		0			0
$529$	0.707107	−	0.707107i	0.707107	−	0.707107i
$530$		0			0
$531$			1.57819i			1.57819i
$532$		0			0
$533$		0			0
$534$	−0.0829870	−	0.200348i	−0.0829870	−	0.200348i
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−2.00682	+	0.831254i	−2.00682	+	0.831254i
$539$		0			0
$540$		0			0
$541$	−0.431351	−	0.178671i	−0.431351	−	0.178671i	0.156434	−	0.987688i	$-0.450000\pi$
$541$	−0.431351	−	0.178671i	−0.431351	−	0.178671i	−0.587785	+	0.809017i	$0.700000\pi$
$542$	0.754763	−	0.754763i	0.754763	−	0.754763i
$543$		0			0
$544$	−0.647354	−	0.329843i	−0.647354	−	0.329843i
$545$		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
$549$	0.636514	−	1.53668i	0.636514	−	1.53668i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	1.78616	+	1.78616i	1.78616	+	1.78616i
$554$	0.584336	+	1.41071i	0.584336	+	1.41071i
$555$		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	$0.750000\pi$
$564$	0.0553750	+	0.0229371i	0.0553750	+	0.0229371i
$565$		0			0
$566$	−0.874883	+	2.11215i	−0.874883	+	2.11215i
$567$	−1.66507	+	0.689695i	−1.66507	+	0.689695i
$568$	0.361140	+	0.871868i	0.361140	+	0.871868i
$569$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$569$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$570$		0			0
$571$	0.581990	+	1.40505i	0.581990	+	1.40505i	0.891007	+	0.453990i	$0.150000\pi$
$571$	0.581990	+	1.40505i	0.581990	+	1.40505i	−0.309017	+	0.951057i	$0.600000\pi$
$572$		0			0
$573$	0.0849235	−	0.205023i	0.0849235	−	0.205023i
$574$		0			0
$575$		0			0
$576$	0.263440	−	0.263440i	0.263440	−	0.263440i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−0.183900	+	1.16110i	−0.183900	+	1.16110i
$579$		0			0
$580$		0			0
$581$	−2.54093	−	1.05249i	−2.54093	−	1.05249i
$582$		−	0.192769i		−	0.192769i
$583$		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	$0.250000\pi$
$588$	−0.0638112	−	0.154054i	−0.0638112	−	0.154054i
$589$		0			0
$590$		0			0
$591$		−	0.120100i		−	0.120100i
$592$	−2.27692	−	0.943129i	−2.27692	−	0.943129i
$593$		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
$596$	−0.680668			−0.680668
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$	−0.0436289	+	0.105329i	−0.0436289	+	0.105329i
$601$	−1.57547	+	0.652583i	−1.57547	+	0.652583i	−0.987688	−	0.156434i	$-0.950000\pi$
$601$	−1.57547	+	0.652583i	−1.57547	+	0.652583i	−0.587785	+	0.809017i	$0.700000\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$	0.154744	−	0.0640971i	0.154744	−	0.0640971i
$607$		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
$612$	0.331954	+	0.169139i	0.331954	+	0.169139i
$613$	−1.17557			−1.17557			−0.587785	−	0.809017i	$-0.700000\pi$
$613$	−1.17557			−1.17557			−0.587785	+	0.809017i	$0.700000\pi$
$614$	0.754763	−	0.754763i	0.754763	−	0.754763i
$615$		0			0
$616$		0			0
$617$	−0.581990	+	1.40505i	−0.581990	+	1.40505i	0.309017	+	0.951057i	$0.400000\pi$
$617$	−0.581990	+	1.40505i	−0.581990	+	1.40505i	−0.891007	+	0.453990i	$0.850000\pi$
$618$	0.303702	−	0.125798i	0.303702	−	0.125798i
$619$	0.707107	+	1.70711i	0.707107	+	1.70711i	0.707107	+	0.707107i	$0.250000\pi$
$619$	0.707107	+	1.70711i	0.707107	+	1.70711i			1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−2.11215	+	0.874883i	−2.11215	+	0.874883i
$624$		0			0
$625$			1.00000i			1.00000i
$626$		0			0
$627$		0			0
$628$	−0.119505			−0.119505
$629$	−0.156434	+	1.98769i	−0.156434	+	1.98769i
$630$		0			0
$631$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$631$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$632$	0.871868	+	0.361140i	0.871868	+	0.361140i
$633$		0			0
$634$		0			0
$635$		0			0
$636$	0.0453093	+	0.109386i	0.0453093	+	0.109386i
$637$		0			0
$638$		0			0
$639$	−0.484827	−	1.17047i	−0.484827	−	1.17047i
$640$		0			0
$641$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$641$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$642$		0			0
$643$	1.57547	+	0.652583i	1.57547	+	0.652583i	0.987688	−	0.156434i	$-0.0500000\pi$
$643$	1.57547	+	0.652583i	1.57547	+	0.652583i	0.587785	+	0.809017i	$0.300000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−0.312869			−0.312869			−0.156434	−	0.987688i	$-0.550000\pi$
$647$	−0.312869			−0.312869			−0.156434	+	0.987688i	$0.550000\pi$
$648$	−0.476104	+	0.476104i	−0.476104	+	0.476104i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−0.763007	−	1.84206i	−0.763007	−	1.84206i	−0.453990	−	0.891007i	$-0.650000\pi$
$653$	−0.763007	−	1.84206i	−0.763007	−	1.84206i	−0.309017	−	0.951057i	$-0.600000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	0.874883	−	2.11215i	0.874883	−	2.11215i
$659$			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	$-0.350000\pi$
$661$	1.34500	−	1.34500i	1.34500	−	1.34500i	0.891007	−	0.453990i	$-0.150000\pi$
$662$	1.38197			1.38197
$663$		0			0
$664$	−1.02749			−1.02749
$665$		0			0
$666$	2.11215	+	0.874883i	2.11215	+	0.874883i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$	−0.156776	−	0.156776i	−0.156776	−	0.156776i
$673$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$673$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$674$	0.507083	−	0.210041i	0.507083	−	0.210041i
$675$	0.118621	−	0.286377i	0.118621	−	0.286377i
$676$			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
$683$	−1.57547	−	0.652583i	−1.57547	−	0.652583i	−0.587785	−	0.809017i	$-0.700000\pi$
$683$	−1.57547	−	0.652583i	−1.57547	−	0.652583i	−0.987688	+	0.156434i	$0.950000\pi$
$684$		0			0
$685$		0			0
$686$	−3.76389	+	1.55905i	−3.76389	+	1.55905i
$687$		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	$0.875000\pi$
$692$	−0.0682464	+	0.164761i	−0.0682464	+	0.164761i
$693$		0			0
$694$	−0.170427	−	0.0705930i	−0.170427	−	0.0705930i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.686280	−	0.284267i	−0.686280	−	0.284267i
$701$		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
$706$	0.260074	+	0.260074i	0.260074	+	0.260074i
$707$	−0.675737	−	1.63137i	−0.675737	−	1.63137i
$708$	0.0895986	−	0.0371129i	0.0895986	−	0.0371129i
$709$	0.744220	−	1.79671i	0.744220	−	1.79671i	0.156434	−	0.987688i	$-0.450000\pi$
$709$	0.744220	−	1.79671i	0.744220	−	1.79671i	0.587785	−	0.809017i	$-0.300000\pi$
$710$		0			0
$711$	−1.17047	−	0.484827i	−1.17047	−	0.484827i
$712$	−0.603941	+	0.603941i	−0.603941	+	0.603941i
$713$		0			0
$714$	−0.162866	+	0.319642i	−0.162866	+	0.319642i
$715$		0			0
$716$		0			0
$717$	−0.275756	−	0.114222i	−0.275756	−	0.114222i
$718$		0			0
$719$	−0.652583	+	1.57547i	−0.652583	+	1.57547i	0.156434	+	0.987688i	$0.450000\pi$
$719$	−0.652583	+	1.57547i	−0.652583	+	1.57547i	−0.809017	+	0.587785i	$0.800000\pi$
$720$		0			0
$721$	−1.32621	−	3.20175i	−1.32621	−	3.20175i
$722$	−0.831254	−	0.831254i	−0.831254	−	0.831254i
$723$	−0.221232	−	0.221232i	−0.221232	−	0.221232i
$724$		0			0
$725$		0			0
$726$	−0.0705930	+	0.170427i	−0.0705930	+	0.170427i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	0.604772	−	0.604772i	0.604772	−	0.604772i
$730$		0			0
$731$		0			0
$732$	−0.102210			−0.102210
$733$	1.34500	−	1.34500i	1.34500	−	1.34500i	0.453990	−	0.891007i	$-0.350000\pi$
$733$	1.34500	−	1.34500i	1.34500	−	1.34500i	0.891007	−	0.453990i	$-0.150000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	0.221232	+	0.221232i	0.221232	+	0.221232i	0.809017	−	0.587785i	$-0.200000\pi$
$739$	0.221232	+	0.221232i	0.221232	+	0.221232i	−0.587785	+	0.809017i	$0.700000\pi$
$740$		0			0
$741$		0			0
$742$	4.17230	−	1.72822i	4.17230	−	1.72822i
$743$		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
$747$	1.37939			1.37939
$748$		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
$753$	−0.0545243	+	0.131633i	−0.0545243	+	0.131633i
$754$		0			0
$755$		0			0
$756$	0.162815	+	0.162815i	0.162815	+	0.162815i
$757$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$757$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$758$	−0.210041	−	0.507083i	−0.210041	−	0.507083i
$759$		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
$766$	2.09488			2.09488
$767$		0			0
$768$	−0.144974	−	0.0600500i	−0.144974	−	0.0600500i
$769$			1.90211i			1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$769$			1.90211i			1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1.34500	−	1.34500i	−1.34500	−	1.34500i	−0.891007	−	0.453990i	$-0.850000\pi$
$773$	−1.34500	−	1.34500i	−1.34500	−	1.34500i	−0.453990	−	0.891007i	$-0.650000\pi$
$774$		0			0
$775$		0			0
$776$	−0.701442	+	0.290547i	−0.701442	+	0.290547i
$777$	−0.232843	+	0.562133i	−0.232843	+	0.562133i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−2.43157	+	2.43157i	−2.43157	+	2.43157i
$785$		0			0
$786$			0.192769i			0.192769i
$787$		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
$789$	−0.0971629	−	0.234572i	−0.0971629	−	0.234572i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	−0.684170	+	1.65173i	−0.684170	+	1.65173i
$795$		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
$801$	0.810786	−	0.810786i	0.810786	−	0.810786i
$802$	0.170427	+	0.0705930i	0.170427	+	0.0705930i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$	0.205023	+	0.205023i	0.205023	+	0.205023i
$808$	−0.466469	−	0.466469i	−0.466469	−	0.466469i
$809$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$809$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$810$		0			0
$811$	0.497066	−	1.20002i	0.497066	−	1.20002i	−0.453990	−	0.891007i	$-0.650000\pi$
$811$	0.497066	−	1.20002i	0.497066	−	1.20002i	0.951057	−	0.309017i	$-0.100000\pi$
$812$		0			0
$813$	−0.131633	−	0.0545243i	−0.131633	−	0.0545243i
$814$		0			0
$815$		0			0
$816$	−0.0152180	+	0.193364i	−0.0152180	+	0.193364i
$817$		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	$0.5\pi$
$824$	−0.915497	−	0.915497i	−0.915497	−	0.915497i
$825$		0			0
$826$	−1.41559	−	3.41754i	−1.41559	−	3.41754i
$827$	−1.57547	+	0.652583i	−1.57547	+	0.652583i	−0.987688	−	0.156434i	$-0.950000\pi$
$827$	−1.57547	+	0.652583i	−1.57547	+	0.652583i	−0.587785	+	0.809017i	$0.700000\pi$
$828$		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
$833$	2.47879	+	1.26301i	2.47879	+	1.26301i
$834$		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
$846$			1.14662i			1.14662i
$847$	1.79671	+	0.744220i	1.79671	+	0.744220i
$848$	1.72654	−	1.72654i	1.72654	−	1.72654i
$849$	0.305165			0.305165
$850$	0.363271	+	1.11803i	0.363271	+	1.11803i
$851$		0			0
$852$	−0.0550500	+	0.0550500i	−0.0550500	+	0.0550500i
$853$	1.40505	+	0.581990i	1.40505	+	0.581990i	0.951057	−	0.309017i	$-0.100000\pi$
$853$	1.40505	+	0.581990i	1.40505	+	0.581990i	0.453990	+	0.891007i	$0.350000\pi$
$854$			3.89858i			3.89858i
$855$		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
$862$	−0.470114	+	1.13496i	−0.470114	+	1.13496i
$863$		−	0.618034i		−	0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$863$		−	0.618034i		−	0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$864$	0.208065	+	0.0861835i	0.208065	+	0.0861835i
$865$		0			0
$866$		0			0
$867$	0.152583	−	0.0366318i	0.152583	−	0.0366318i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	0.941679	−	0.390056i	0.941679	−	0.390056i
$874$		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	$0.125000\pi$
$882$	2.25562	−	2.25562i	2.25562	−	2.25562i
$883$	−0.312869			−0.312869			−0.156434	−	0.987688i	$-0.550000\pi$
$883$	−0.312869			−0.312869			−0.156434	+	0.987688i	$0.550000\pi$
$884$		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	$0.125000\pi$
$888$			0.227313i			0.227313i
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$	0.125798	+	0.303702i	0.125798	+	0.303702i
$895$		0			0
$896$	−0.874883	+	2.11215i	−0.874883	+	2.11215i
$897$		0			0
$898$		0			0
$899$		0			0
$900$	0.372561			0.372561
$901$	−1.76007	−	0.896802i	−1.76007	−	0.896802i
$902$		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
$909$	0.626230	+	0.626230i	0.626230	+	0.626230i
$910$		0			0
$911$	−0.652583	−	1.57547i	−0.652583	−	1.57547i	−0.809017	−	0.587785i	$-0.800000\pi$
$911$	−0.652583	−	1.57547i	−0.652583	−	1.57547i	0.156434	−	0.987688i	$-0.450000\pi$
$912$		0			0
$913$		0			0
$914$		−	1.06740i		−	1.06740i
$915$		0			0
$916$		0			0
$917$	2.03225			2.03225
$918$	0.0285901	−	0.363271i	0.0285901	−	0.363271i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	−0.131633	−	0.0545243i	−0.131633	−	0.0545243i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	0.763007	+	1.84206i	0.763007	+	1.84206i
$926$		0			0
$927$	1.22905	+	1.22905i	1.22905	+	1.22905i
$928$		0			0
$929$	−1.84206	+	0.763007i	−1.84206	+	0.763007i	−0.891007	+	0.453990i	$0.850000\pi$
$929$	−1.84206	+	0.763007i	−1.84206	+	0.763007i	−0.951057	+	0.309017i	$0.900000\pi$
$930$		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
$941$	0.965451	−	0.399903i	0.965451	−	0.399903i	0.156434	−	0.987688i	$-0.450000\pi$
$941$	0.965451	−	0.399903i	0.965451	−	0.399903i	0.809017	+	0.587785i	$0.200000\pi$
$942$	0.0220864	+	0.0533212i	0.0220864	+	0.0533212i
$943$		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
$952$	1.40858	+	0.110858i	1.40858	+	0.110858i
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$	−1.60161	+	1.60161i	−1.60161	+	1.60161i
$955$		0			0
$956$		−	0.726543i		−	0.726543i
$957$		0			0
$958$	2.11215	−	0.874883i	2.11215	−	0.874883i
$959$		0			0
$960$		0			0
$961$	0.707107	+	0.707107i	0.707107	+	0.707107i
$962$		0			0
$963$		0			0
$964$	0.291443	−	0.703605i	0.291443	−	0.703605i
$965$		0			0
$966$		0			0
$967$	0.221232	−	0.221232i	0.221232	−	0.221232i	−0.587785	−	0.809017i	$-0.700000\pi$
$967$	0.221232	−	0.221232i	0.221232	−	0.221232i	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	$0.750000\pi$
$972$	−0.160704	−	0.0665659i	−0.160704	−	0.0665659i
$973$		0			0
$974$	0.344317	−	0.831254i	0.344317	−	0.831254i
$975$		0			0
$976$	0.806636	+	1.94739i	0.806636	+	1.94739i
$977$		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
$982$			0.367799i			0.367799i
$983$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$983$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−0.305165			−0.305165
$988$		0			0
$989$		0			0
$990$		0			0
$991$	0.652583	−	1.57547i	0.652583	−	1.57547i	−0.156434	−	0.987688i	$-0.550000\pi$
$991$	0.652583	−	1.57547i	0.652583	−	1.57547i	0.809017	−	0.587785i	$-0.200000\pi$
$992$		0			0
$993$	−0.0705930	−	0.170427i	−0.0705930	−	0.170427i
$994$	2.09976	+	2.09976i	2.09976	+	2.09976i
$995$		0			0
$996$	−0.0324379	−	0.0783120i	−0.0324379	−	0.0783120i
$997$		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.281.3	✓	16
17.2	even	8	inner	799.1.h.b.563.3	yes	16
47.46	odd	2	CM	799.1.h.b.281.3	✓	16
799.563	odd	8	inner	799.1.h.b.563.3	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
799.1.h.b.281.3	✓	16	1.1	even	1	trivial
799.1.h.b.281.3	✓	16	47.46	odd	2	CM
799.1.h.b.563.3	yes	16	17.2	even	8	inner
799.1.h.b.563.3	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)

\(f(q)\)	\(=\)	\(q+(0.831254 - 0.831254i) q^{2} +(-0.144974 - 0.0600500i) q^{3} -0.381966i q^{4} +(-0.170427 + 0.0705930i) q^{6} +(0.744220 + 1.79671i) q^{7} +(0.513743 + 0.513743i) q^{8} +(-0.689695 - 0.689695i) q^{9} +(-0.0229371 + 0.0553750i) q^{12} +(2.11215 + 0.874883i) q^{14} +1.23607 q^{16} +(-0.309017 - 0.951057i) q^{17} -1.14662 q^{18} -0.305165i q^{21} +(-0.0436289 - 0.105329i) q^{24} +(-0.707107 - 0.707107i) q^{25} +(0.118621 + 0.286377i) q^{27} +(0.686280 - 0.284267i) q^{28} +(0.513743 - 0.513743i) q^{32} +(-1.04744 - 0.533698i) q^{34} +(-0.263440 + 0.263440i) q^{36} +(-1.84206 - 0.763007i) q^{37} +(-0.253670 - 0.253670i) q^{42} -1.00000i q^{47} +(-0.179197 - 0.0742259i) q^{48} +(-1.96718 + 1.96718i) q^{49} -1.17557 q^{50} +(-0.0123117 + 0.156434i) q^{51} +(1.39680 - 1.39680i) q^{53} +(0.336657 + 0.139448i) q^{54} +(-0.540707 + 1.30538i) q^{56} +(-1.14412 - 1.14412i) q^{59} +(0.652583 + 1.57547i) q^{61} +(0.725895 - 1.75246i) q^{63} +0.381966i q^{64} +(-0.363271 + 0.118034i) q^{68} +(1.20002 + 0.497066i) q^{71} -0.708653i q^{72} +(-2.16547 + 0.896969i) q^{74} +(0.0600500 + 0.144974i) q^{75} +(1.20002 - 0.497066i) q^{79} +0.926736i q^{81} +(-1.00000 + 1.00000i) q^{83} -0.116563 q^{84} +1.17557i q^{89} +(-0.831254 - 0.831254i) q^{94} +(-0.105329 + 0.0436289i) q^{96} +(-0.399903 + 0.965451i) q^{97} +3.27045i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{3} - 4 q^{9} - 16 q^{16} + 4 q^{17} - 20 q^{24} - 4 q^{27} - 4 q^{28} + 4 q^{36} - 20 q^{42} + 24 q^{48} + 4 q^{49} - 16 q^{51} + 4 q^{53} + 20 q^{54} + 20 q^{56} + 4 q^{61} - 4 q^{63} - 4 q^{71}+ \cdots + 32 q^{84}+O(q^{100}) \) 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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