LMFDB - Embedded newform 799.1.h.b.93.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("799.93");

S:= CuspForms(chi, 1);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.398752945094$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{8})$
Coefficient field:	$\Q(\zeta_{40})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{12} + x^{8} - x^{4} + 1 $ x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{40}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{40} + \cdots)$

Embedding invariants

Embedding label			93.3
Root			$-0.156434 + 0.987688i$ of defining polynomial
Character	$\chi$	$=$	799.93
Dual form			799.1.h.b.610.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.831254 - 0.831254i) q^{2} +(-0.581990 + 1.40505i) q^{3} -0.381966i q^{4} +(0.684170 + 1.65173i) q^{6} +(-1.57547 + 0.652583i) q^{7} +(0.513743 + 0.513743i) q^{8} +(-0.928339 - 0.928339i) q^{9} +O(q^{10})$	\(q+(0.831254 - 0.831254i) q^{2} +(-0.581990 + 1.40505i) q^{3} -0.381966i q^{4} +(0.684170 + 1.65173i) q^{6} +(-1.57547 + 0.652583i) q^{7} +(0.513743 + 0.513743i) q^{8} +(-0.928339 - 0.928339i) q^{9} +(0.536680 + 0.222300i) q^{12} +(-0.767157 + 1.85208i) q^{14} +1.23607 q^{16} +(-0.309017 + 0.951057i) q^{17} -1.54337 q^{18} -2.59341i q^{21} +(-1.02083 + 0.422840i) q^{24} +(0.707107 + 0.707107i) q^{25} +(0.439596 - 0.182086i) q^{27} +(0.249264 + 0.601777i) q^{28} +(0.513743 - 0.513743i) q^{32} +(0.533698 + 1.04744i) q^{34} +(-0.354594 + 0.354594i) q^{36} +(0.497066 - 1.20002i) q^{37} +(-2.15578 - 2.15578i) q^{42} -1.00000i q^{47} +(-0.719379 + 1.73673i) q^{48} +(1.34915 - 1.34915i) q^{49} +1.17557 q^{50} +(-1.15643 - 0.987688i) q^{51} +(0.221232 - 0.221232i) q^{53} +(0.214055 - 0.516776i) q^{54} +(-1.14465 - 0.474129i) q^{56} +(1.14412 + 1.14412i) q^{59} +(1.79671 - 0.744220i) q^{61} +(2.06839 + 0.856755i) q^{63} +0.381966i q^{64} +(0.363271 + 0.118034i) q^{68} +(0.763007 - 1.84206i) q^{71} -0.953855i q^{72} +(-0.584336 - 1.41071i) q^{74} +(-1.40505 + 0.581990i) q^{75} +(0.763007 + 1.84206i) q^{79} -0.589244i q^{81} +(-1.00000 + 1.00000i) q^{83} -0.990595 q^{84} -1.17557i q^{89} +(-0.831254 - 0.831254i) q^{94} +(0.422840 + 1.02083i) q^{96} +(-0.431351 - 0.178671i) q^{97} -2.24297i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{3} - 4 q^{9}+O(q^{10}) $ 16 * q - 4 * q^3 - 4 * q^9	$ 16 q - 4 q^{3} - 4 q^{9} - 16 q^{16} + 4 q^{17} - 20 q^{24} - 4 q^{27} - 4 q^{28} + 4 q^{36} - 20 q^{42} + 24 q^{48} + 4 q^{49} - 16 q^{51} + 4 q^{53} + 20 q^{54} + 20 q^{56} + 4 q^{61} - 4 q^{63} - 4 q^{71} - 4 q^{79} - 16 q^{83} + 32 q^{84}+O(q^{100}) $ 16 * q - 4 * q^3 - 4 * q^9 - 16 * q^16 + 4 * q^17 - 20 * q^24 - 4 * q^27 - 4 * q^28 + 4 * q^36 - 20 * q^42 + 24 * q^48 + 4 * q^49 - 16 * q^51 + 4 * q^53 + 20 * q^54 + 20 * q^56 + 4 * q^61 - 4 * q^63 - 4 * q^71 - 4 * q^79 - 16 * q^83 + 32 * q^84

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	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	$0.0500000\pi$
$3$	−0.581990	+	1.40505i	−0.581990	+	1.40505i	0.309017	+	0.951057i	$0.400000\pi$
$3$	−0.581990	+	1.40505i	−0.581990	+	1.40505i	−0.891007	+	0.453990i	$0.850000\pi$
$4$		−	0.381966i		−	0.381966i
$5$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$5$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$6$	0.684170	+	1.65173i	0.684170	+	1.65173i
$7$	−1.57547	+	0.652583i	−1.57547	+	0.652583i	−0.987688	−	0.156434i	$-0.950000\pi$
$7$	−1.57547	+	0.652583i	−1.57547	+	0.652583i	−0.587785	+	0.809017i	$0.700000\pi$
$8$	0.513743	+	0.513743i	0.513743	+	0.513743i
$9$	−0.928339	−	0.928339i	−0.928339	−	0.928339i
$10$		0			0
$11$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$11$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$12$	0.536680	+	0.222300i	0.536680	+	0.222300i
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$	−0.767157	+	1.85208i	−0.767157	+	1.85208i
$15$		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
$18$	−1.54337			−1.54337
$19$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$19$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$20$		0			0
$21$		−	2.59341i		−	2.59341i
$22$		0			0
$23$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$23$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$24$	−1.02083	+	0.422840i	−1.02083	+	0.422840i
$25$	0.707107	+	0.707107i	0.707107	+	0.707107i
$26$		0			0
$27$	0.439596	−	0.182086i	0.439596	−	0.182086i
$28$	0.249264	+	0.601777i	0.249264	+	0.601777i
$29$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$29$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$30$		0			0
$31$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$31$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$32$	0.513743	−	0.513743i	0.513743	−	0.513743i
$33$		0			0
$34$	0.533698	+	1.04744i	0.533698	+	1.04744i
$35$		0			0
$36$	−0.354594	+	0.354594i	−0.354594	+	0.354594i
$37$	0.497066	−	1.20002i	0.497066	−	1.20002i	−0.453990	−	0.891007i	$-0.650000\pi$
$37$	0.497066	−	1.20002i	0.497066	−	1.20002i	0.951057	−	0.309017i	$-0.100000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$41$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$42$	−2.15578	−	2.15578i	−2.15578	−	2.15578i
$43$		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
$48$	−0.719379	+	1.73673i	−0.719379	+	1.73673i
$49$	1.34915	−	1.34915i	1.34915	−	1.34915i
$50$	1.17557			1.17557
$51$	−1.15643	−	0.987688i	−1.15643	−	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	$0.200000\pi$
$54$	0.214055	−	0.516776i	0.214055	−	0.516776i
$55$		0			0
$56$	−1.14465	−	0.474129i	−1.14465	−	0.474129i
$57$		0			0
$58$		0			0
$59$	1.14412	+	1.14412i	1.14412	+	1.14412i	0.987688	+	0.156434i	$0.0500000\pi$
$59$	1.14412	+	1.14412i	1.14412	+	1.14412i	0.156434	+	0.987688i	$0.450000\pi$
$60$		0			0
$61$	1.79671	−	0.744220i	1.79671	−	0.744220i	0.809017	−	0.587785i	$-0.200000\pi$
$61$	1.79671	−	0.744220i	1.79671	−	0.744220i	0.987688	−	0.156434i	$-0.0500000\pi$
$62$		0			0
$63$	2.06839	+	0.856755i	2.06839	+	0.856755i
$64$			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
$71$	0.763007	−	1.84206i	0.763007	−	1.84206i	0.309017	−	0.951057i	$-0.400000\pi$
$71$	0.763007	−	1.84206i	0.763007	−	1.84206i	0.453990	−	0.891007i	$-0.350000\pi$
$72$		−	0.953855i		−	0.953855i
$73$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$73$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$74$	−0.584336	−	1.41071i	−0.584336	−	1.41071i
$75$	−1.40505	+	0.581990i	−1.40505	+	0.581990i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	0.763007	+	1.84206i	0.763007	+	1.84206i	0.453990	+	0.891007i	$0.350000\pi$
$79$	0.763007	+	1.84206i	0.763007	+	1.84206i	0.309017	+	0.951057i	$0.400000\pi$
$80$		0			0
$81$		−	0.589244i		−	0.589244i
$82$		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			$\pi$
$84$	−0.990595			−0.990595
$85$		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
$96$	0.422840	+	1.02083i	0.422840	+	1.02083i
$97$	−0.431351	−	0.178671i	−0.431351	−	0.178671i	0.156434	−	0.987688i	$-0.450000\pi$
$97$	−0.431351	−	0.178671i	−0.431351	−	0.178671i	−0.587785	+	0.809017i	$0.700000\pi$
$98$		−	2.24297i		−	2.24297i
$99$		0			0
$100$	0.270091	−	0.270091i	0.270091	−	0.270091i
$101$	−1.78201			−1.78201			−0.891007	−	0.453990i	$-0.850000\pi$
$101$	−1.78201			−1.78201			−0.891007	+	0.453990i	$0.850000\pi$
$102$	−1.78231	+	0.140271i	−1.78231	+	0.140271i
$103$	−0.907981			−0.907981			−0.453990	−	0.891007i	$-0.650000\pi$
$103$	−0.907981			−0.907981			−0.453990	+	0.891007i	$0.650000\pi$
$104$		0			0
$105$		0			0
$106$		−	0.367799i		−	0.367799i
$107$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$107$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$108$	−0.0695508	−	0.167911i	−0.0695508	−	0.167911i
$109$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$109$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$110$		0			0
$111$	1.39680	+	1.39680i	1.39680	+	1.39680i
$112$	−1.94739	+	0.806636i	−1.94739	+	0.806636i
$113$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$113$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	1.90211			1.90211
$119$	−0.133795	−	1.70002i	−0.133795	−	1.70002i
$120$		0			0
$121$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$122$	0.874883	−	2.11215i	0.874883	−	2.11215i
$123$		0			0
$124$		0			0
$125$		0			0
$126$	2.43154	−	1.00718i	2.43154	−	1.00718i
$127$		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
$131$	0.431351	+	0.178671i	0.431351	+	0.178671i	0.587785	−	0.809017i	$-0.300000\pi$
$131$	0.431351	+	0.178671i	0.431351	+	0.178671i	−0.156434	+	0.987688i	$0.550000\pi$
$132$		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		0.382683	−	0.923880i	$-0.375000\pi$
$139$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$140$		0			0
$141$	1.40505	+	0.581990i	1.40505	+	0.581990i
$142$	−0.896969	−	2.16547i	−0.896969	−	2.16547i
$143$		0			0
$144$	−1.14749	−	1.14749i	−1.14749	−	1.14749i
$145$		0			0
$146$		0			0
$147$	1.11042	+	2.68080i	1.11042	+	2.68080i
$148$	−0.458368	−	0.189862i	−0.458368	−	0.189862i
$149$		−	0.907981i		−	0.907981i	−0.891007	−	0.453990i	$-0.850000\pi$
$149$		−	0.907981i		−	0.907981i	0.891007	−	0.453990i	$-0.150000\pi$
$150$	−0.684170	+	1.65173i	−0.684170	+	1.65173i
$151$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$151$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$152$		0			0
$153$	1.16977	−	0.596030i	1.16977	−	0.596030i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			1.97538i			1.97538i	0.156434	+	0.987688i	$0.450000\pi$
$157$			1.97538i			1.97538i	−0.156434	+	0.987688i	$0.550000\pi$
$158$	2.16547	+	0.896969i	2.16547	+	0.896969i
$159$	0.182086	+	0.439596i	0.182086	+	0.439596i
$160$		0			0
$161$		0			0
$162$	−0.489811	−	0.489811i	−0.489811	−	0.489811i
$163$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$163$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$164$		0			0
$165$		0			0
$166$			1.66251i			1.66251i
$167$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$167$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$168$	1.33235	−	1.33235i	1.33235	−	1.33235i
$169$	−1.00000			−1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−0.399903	+	0.965451i	−0.399903	+	0.965451i	0.587785	+	0.809017i	$0.300000\pi$
$173$	−0.399903	+	0.965451i	−0.399903	+	0.965451i	−0.987688	+	0.156434i	$0.950000\pi$
$174$		0			0
$175$	−1.57547	−	0.652583i	−1.57547	−	0.652583i
$176$		0			0
$177$	−2.27341	+	0.941679i	−2.27341	+	0.941679i
$178$	−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		−0.382683	−	0.923880i	$-0.625000\pi$
$181$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$182$		0			0
$183$			2.95758i			2.95758i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−0.381966			−0.381966
$189$	−0.573745	+	0.573745i	−0.573745	+	0.573745i
$190$		0			0
$191$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$191$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$192$	−0.536680	−	0.222300i	−0.536680	−	0.222300i
$193$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$193$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$194$	−0.507083	+	0.210041i	−0.507083	+	0.210041i
$195$		0			0
$196$	−0.515328	−	0.515328i	−0.515328	−	0.515328i
$197$	1.70711	−	0.707107i	1.70711	−	0.707107i	0.707107	−	0.707107i	$-0.250000\pi$
$197$	1.70711	−	0.707107i	1.70711	−	0.707107i	1.00000			$0$
$198$		0			0
$199$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$199$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$200$			0.726543i			0.726543i
$201$		0			0
$202$	−1.48131	+	1.48131i	−1.48131	+	1.48131i
$203$		0			0
$204$	−0.377263	+	0.441719i	−0.377263	+	0.441719i
$205$		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		0.923880	−	0.382683i	$-0.125000\pi$
$211$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$212$	−0.0845030	−	0.0845030i	−0.0845030	−	0.0845030i
$213$	2.14412	+	2.14412i	2.14412	+	2.14412i
$214$		0			0
$215$		0			0
$216$	0.319385	+	0.132294i	0.319385	+	0.132294i
$217$		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	$0.750000\pi$
$224$	−0.474129	+	1.14465i	−0.474129	+	1.14465i
$225$		−	1.31287i		−	1.31287i
$226$		0			0
$227$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$227$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$228$		0			0
$229$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$229$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$233$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$234$		0			0
$235$		0			0
$236$	0.437016	−	0.437016i	0.437016	−	0.437016i
$237$	−3.03225			−3.03225
$238$	−1.52437	−	1.30193i	−1.52437	−	1.30193i
$239$	−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
$241$	−0.497066	+	1.20002i	−0.497066	+	1.20002i	0.453990	+	0.891007i	$0.350000\pi$
$241$	−0.497066	+	1.20002i	−0.497066	+	1.20002i	−0.951057	+	0.309017i	$0.900000\pi$
$242$			1.17557i			1.17557i
$243$	1.26751	+	0.525020i	1.26751	+	0.525020i
$244$	−0.284267	−	0.686280i	−0.284267	−	0.686280i
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−0.823057	−	1.98704i	−0.823057	−	1.98704i
$250$		0			0
$251$		−	1.78201i		−	1.78201i	−0.453990	−	0.891007i	$-0.650000\pi$
$251$		−	1.78201i		−	1.78201i	0.453990	−	0.891007i	$-0.350000\pi$
$252$	0.327251	−	0.790055i	0.327251	−	0.790055i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	1.00000			1.00000
$257$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$257$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$258$		0			0
$259$			2.21498i			2.21498i
$260$		0			0
$261$		0			0
$262$	0.507083	−	0.210041i	0.507083	−	0.210041i
$263$	−1.14412	−	1.14412i	−1.14412	−	1.14412i	−0.987688	−	0.156434i	$-0.950000\pi$
$263$	−1.14412	−	1.14412i	−1.14412	−	1.14412i	−0.156434	−	0.987688i	$-0.550000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	1.65173	+	0.684170i	1.65173	+	0.684170i
$268$		0			0
$269$	−0.292893	+	0.707107i	−0.292893	+	0.707107i	0.707107	+	0.707107i	$0.250000\pi$
$269$	−0.292893	+	0.707107i	−0.292893	+	0.707107i	−1.00000			$\pi$
$270$		0			0
$271$	1.78201			1.78201			0.891007	−	0.453990i	$-0.150000\pi$
$271$	1.78201			1.78201			0.891007	+	0.453990i	$0.150000\pi$
$272$	−0.381966	+	1.17557i	−0.381966	+	1.17557i
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	1.84206	+	0.763007i	1.84206	+	0.763007i	0.951057	+	0.309017i	$0.100000\pi$
$277$	1.84206	+	0.763007i	1.84206	+	0.763007i	0.891007	+	0.453990i	$0.150000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$281$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$282$	1.65173	−	0.684170i	1.65173	−	0.684170i
$283$	−0.652583	−	1.57547i	−0.652583	−	1.57547i	−0.809017	−	0.587785i	$-0.800000\pi$
$283$	−0.652583	−	1.57547i	−0.652583	−	1.57547i	0.156434	−	0.987688i	$-0.450000\pi$
$284$	−0.703605	−	0.291443i	−0.703605	−	0.291443i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.953855			−0.953855
$289$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$290$		0			0
$291$	0.502083	−	0.502083i	0.502083	−	0.502083i
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	3.15147	+	1.30538i	3.15147	+	1.30538i
$295$		0			0
$296$	0.871868	−	0.361140i	0.871868	−	0.361140i
$297$		0			0
$298$	−0.754763	−	0.754763i	−0.754763	−	0.754763i
$299$		0			0
$300$	0.222300	+	0.536680i	0.222300	+	0.536680i
$301$		0			0
$302$		0			0
$303$	1.03711	−	2.50381i	1.03711	−	2.50381i
$304$		0			0
$305$		0			0
$306$	0.476928	−	1.46783i	0.476928	−	1.46783i
$307$	1.78201			1.78201			0.891007	−	0.453990i	$-0.150000\pi$
$307$	1.78201			1.78201			0.891007	+	0.453990i	$0.150000\pi$
$308$		0			0
$309$	0.528435	−	1.27576i	0.528435	−	1.27576i
$310$		0			0
$311$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$311$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$312$		0			0
$313$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$313$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$314$	1.64204	+	1.64204i	1.64204	+	1.64204i
$315$		0			0
$316$	0.703605	−	0.291443i	0.703605	−	0.291443i
$317$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$317$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$318$	0.516776	+	0.214055i	0.516776	+	0.214055i
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.225071			−0.225071
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	0.652583	+	1.57547i	0.652583	+	1.57547i
$330$		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
$333$	−1.57547	+	0.652583i	−1.57547	+	0.652583i
$334$		0			0
$335$		0			0
$336$		−	3.20563i		−	3.20563i
$337$	0.399903	−	0.965451i	0.399903	−	0.965451i	−0.587785	−	0.809017i	$-0.700000\pi$
$337$	0.399903	−	0.965451i	0.399903	−	0.965451i	0.987688	−	0.156434i	$-0.0500000\pi$
$338$	−0.831254	+	0.831254i	−0.831254	+	0.831254i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−0.592533	+	1.43050i	−0.592533	+	1.43050i
$344$		0			0
$345$		0			0
$346$	0.470114	+	1.13496i	0.470114	+	1.13496i
$347$	1.40505	−	0.581990i	1.40505	−	0.581990i	0.453990	−	0.891007i	$-0.350000\pi$
$347$	1.40505	−	0.581990i	1.40505	−	0.581990i	0.951057	+	0.309017i	$0.100000\pi$
$348$		0			0
$349$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$349$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$350$	−1.85208	+	0.767157i	−1.85208	+	0.767157i
$351$		0			0
$352$		0			0
$353$		−	1.97538i		−	1.97538i	−0.156434	−	0.987688i	$-0.550000\pi$
$353$		−	1.97538i		−	1.97538i	0.156434	−	0.987688i	$-0.450000\pi$
$354$	−1.10701	+	2.67256i	−1.10701	+	2.67256i
$355$		0			0
$356$	−0.449028			−0.449028
$357$	2.46648	+	0.801408i	2.46648	+	0.801408i
$358$		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
$363$	−0.581990	−	1.40505i	−0.581990	−	1.40505i
$364$		0			0
$365$		0			0
$366$	2.45850	+	2.45850i	2.45850	+	2.45850i
$367$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$367$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−0.204173	+	0.492917i	−0.204173	+	0.492917i
$372$		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
$378$			0.953855i			0.953855i
$379$	−0.965451	−	0.399903i	−0.965451	−	0.399903i	−0.156434	−	0.987688i	$-0.550000\pi$
$379$	−0.965451	−	0.399903i	−0.965451	−	0.399903i	−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	$0.900000\pi$
$384$	−1.65173	+	0.684170i	−1.65173	+	0.684170i
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−0.0682464	+	0.164761i	−0.0682464	+	0.164761i
$389$		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
$392$	1.38623			1.38623
$393$	−0.502083	+	0.502083i	−0.502083	+	0.502083i
$394$	0.831254	−	2.00682i	0.831254	−	2.00682i
$395$		0			0
$396$		0			0
$397$	0.0600500	+	0.144974i	0.0600500	+	0.144974i	0.951057	−	0.309017i	$-0.100000\pi$
$397$	0.0600500	+	0.144974i	0.0600500	+	0.144974i	−0.891007	+	0.453990i	$0.850000\pi$
$398$		0			0
$399$		0			0
$400$	0.874032	+	0.874032i	0.874032	+	0.874032i
$401$	−1.40505	+	0.581990i	−1.40505	+	0.581990i	−0.951057	−	0.309017i	$-0.900000\pi$
$401$	−1.40505	+	0.581990i	−1.40505	+	0.581990i	−0.453990	+	0.891007i	$0.650000\pi$
$402$		0			0
$403$		0			0
$404$			0.680668i			0.680668i
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−0.0866922	−	1.10153i	−0.0866922	−	1.10153i
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$			0.346818i			0.346818i
$413$	−2.54917	−	1.05590i	−2.54917	−	1.05590i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$419$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$420$		0			0
$421$		0			0			−	1.00000i	$-0.5\pi$
$421$		0			0				1.00000i	$0.5\pi$
$422$		0			0
$423$	−0.928339	+	0.928339i	−0.928339	+	0.928339i
$424$	0.227313			0.227313
$425$	−0.891007	+	0.453990i	−0.891007	+	0.453990i
$426$	3.56462			3.56462
$427$	−2.34500	+	2.34500i	−2.34500	+	2.34500i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	0.178671	+	0.431351i	0.178671	+	0.431351i	0.987688	−	0.156434i	$-0.0500000\pi$
$431$	0.178671	+	0.431351i	0.178671	+	0.431351i	−0.809017	+	0.587785i	$0.800000\pi$
$432$	0.543370	−	0.225071i	0.543370	−	0.225071i
$433$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$433$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	0.707107	−	1.70711i	0.707107	−	1.70711i		−	1.00000i	$-0.5\pi$
$439$	0.707107	−	1.70711i	0.707107	−	1.70711i	0.707107	−	0.707107i	$-0.250000\pi$
$440$		0			0
$441$	−2.50493			−2.50493
$442$		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
$447$	1.27576	+	0.528435i	1.27576	+	0.528435i
$448$	−0.249264	−	0.601777i	−0.249264	−	0.601777i
$449$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$449$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$450$	−1.09133	−	1.09133i	−1.09133	−	1.09133i
$451$		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
$459$	0.0373320	+	0.474348i	0.0373320	+	0.474348i
$460$		0			0
$461$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$461$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$467$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−2.77550	−	1.14965i	−2.77550	−	1.14965i
$472$			1.17557i			1.17557i
$473$		0			0
$474$	−2.52057	+	2.52057i	−2.52057	+	2.52057i
$475$		0			0
$476$	−0.649351	+	0.0511050i	−0.649351	+	0.0511050i
$477$	−0.410756			−0.410756
$478$	−1.58114	+	1.58114i	−1.58114	+	1.58114i
$479$	0.652583	−	1.57547i	0.652583	−	1.57547i	−0.156434	−	0.987688i	$-0.550000\pi$
$479$	0.652583	−	1.57547i	0.652583	−	1.57547i	0.809017	−	0.587785i	$-0.200000\pi$
$480$		0			0
$481$		0			0
$482$	0.584336	+	1.41071i	0.584336	+	1.41071i
$483$		0			0
$484$	0.270091	+	0.270091i	0.270091	+	0.270091i
$485$		0			0
$486$	1.49005	−	0.617198i	1.49005	−	0.617198i
$487$	−0.707107	−	1.70711i	−0.707107	−	1.70711i	−0.707107	−	0.707107i	$-0.750000\pi$
$487$	−0.707107	−	1.70711i	−0.707107	−	1.70711i		−	1.00000i	$-0.5\pi$
$488$	1.30538	+	0.540707i	1.30538	+	0.540707i
$489$		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
$497$			3.40005i			3.40005i
$498$	−2.33590	−	0.967562i	−2.33590	−	0.967562i
$499$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$499$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$500$		0			0
$501$		0			0
$502$	−1.48131	−	1.48131i	−1.48131	−	1.48131i
$503$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$503$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$504$	0.622469	+	1.50277i	0.622469	+	1.50277i
$505$		0			0
$506$		0			0
$507$	0.581990	−	1.40505i	0.581990	−	1.40505i
$508$		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
$518$	1.84121	+	1.84121i	1.84121	+	1.84121i
$519$	−1.12377	−	1.12377i	−1.12377	−	1.12377i
$520$		0			0
$521$	−0.399903	−	0.965451i	−0.399903	−	0.965451i	−0.987688	−	0.156434i	$-0.950000\pi$
$521$	−0.399903	−	0.965451i	−0.399903	−	0.965451i	0.587785	−	0.809017i	$-0.300000\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$	0.0682464	−	0.164761i	0.0682464	−	0.164761i
$525$	1.83382	−	1.83382i	1.83382	−	1.83382i
$526$	−1.90211			−1.90211
$527$		0			0
$528$		0			0
$529$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$530$		0			0
$531$		−	2.12427i		−	2.12427i
$532$		0			0
$533$		0			0
$534$	1.94173	−	0.804290i	1.94173	−	0.804290i
$535$		0			0
$536$		0			0
$537$		0			0
$538$	0.344317	+	0.831254i	0.344317	+	0.831254i
$539$		0			0
$540$		0			0
$541$	−0.399903	+	0.965451i	−0.399903	+	0.965451i	0.587785	+	0.809017i	$0.300000\pi$
$541$	−0.399903	+	0.965451i	−0.399903	+	0.965451i	−0.987688	+	0.156434i	$0.950000\pi$
$542$	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		0.382683	−	0.923880i	$-0.375000\pi$
$547$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$548$		0			0
$549$	−2.35884	−	0.977063i	−2.35884	−	0.977063i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−2.40420	−	2.40420i	−2.40420	−	2.40420i
$554$	2.16547	−	0.896969i	2.16547	−	0.896969i
$555$		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	$0.750000\pi$
$564$	0.222300	−	0.536680i	0.222300	−	0.536680i
$565$		0			0
$566$	−1.85208	−	0.767157i	−1.85208	−	0.767157i
$567$	0.384530	+	0.928339i	0.384530	+	0.928339i
$568$	1.33834	−	0.554357i	1.33834	−	0.554357i
$569$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$569$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$570$		0			0
$571$	0.144974	−	0.0600500i	0.144974	−	0.0600500i	−0.309017	−	0.951057i	$-0.600000\pi$
$571$	0.144974	−	0.0600500i	0.144974	−	0.0600500i	0.453990	+	0.891007i	$0.350000\pi$
$572$		0			0
$573$	1.98704	+	0.823057i	1.98704	+	0.823057i
$574$		0			0
$575$		0			0
$576$	0.354594	−	0.354594i	0.354594	−	0.354594i
$577$		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
$581$	0.922891	−	2.22806i	0.922891	−	2.22806i
$582$		−	0.834717i		−	0.834717i
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$587$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$588$	1.02398	−	0.424145i	1.02398	−	0.424145i
$589$		0			0
$590$		0			0
$591$			2.81009i			2.81009i
$592$	0.614407	−	1.48331i	0.614407	−	1.48331i
$593$		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	$0.5\pi$
$600$	−1.02083	−	0.422840i	−1.02083	−	0.422840i
$601$	0.744220	+	1.79671i	0.744220	+	1.79671i	0.587785	+	0.809017i	$0.300000\pi$
$601$	0.744220	+	1.79671i	0.744220	+	1.79671i	0.156434	+	0.987688i	$0.450000\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$	−1.21920	−	2.94341i	−1.21920	−	2.94341i
$607$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$607$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−0.227663	−	0.446814i	−0.227663	−	0.446814i
$613$	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
$617$	−0.144974	−	0.0600500i	−0.144974	−	0.0600500i	0.309017	−	0.951057i	$-0.400000\pi$
$617$	−0.144974	−	0.0600500i	−0.144974	−	0.0600500i	−0.453990	+	0.891007i	$0.650000\pi$
$618$	−0.621213	−	1.49974i	−0.621213	−	1.49974i
$619$	−0.707107	+	0.292893i	−0.707107	+	0.292893i	−0.707107	−	0.707107i	$-0.750000\pi$
$619$	−0.707107	+	0.292893i	−0.707107	+	0.292893i			1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	0.767157	+	1.85208i	0.767157	+	1.85208i
$624$		0			0
$625$			1.00000i			1.00000i
$626$		0			0
$627$		0			0
$628$	0.754527			0.754527
$629$	0.987688	+	0.843566i	0.987688	+	0.843566i
$630$		0			0
$631$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$631$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$632$	−0.554357	+	1.33834i	−0.554357	+	1.33834i
$633$		0			0
$634$		0			0
$635$		0			0
$636$	0.167911	−	0.0695508i	0.167911	−	0.0695508i
$637$		0			0
$638$		0			0
$639$	−2.41839	+	1.00173i	−2.41839	+	1.00173i
$640$		0			0
$641$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$641$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$642$		0			0
$643$	−0.744220	+	1.79671i	−0.744220	+	1.79671i	−0.156434	+	0.987688i	$0.550000\pi$
$643$	−0.744220	+	1.79671i	−0.744220	+	1.79671i	−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	$-0.0500000\pi$
$647$	1.97538			1.97538			0.987688	+	0.156434i	$0.0500000\pi$
$648$	0.302720	−	0.302720i	0.302720	−	0.302720i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.20002	+	0.497066i	−1.20002	+	0.497066i	−0.891007	−	0.453990i	$-0.850000\pi$
$653$	−1.20002	+	0.497066i	−1.20002	+	0.497066i	−0.309017	+	0.951057i	$0.600000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	1.85208	+	0.767157i	1.85208	+	0.767157i
$659$			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
$666$	−0.767157	+	1.85208i	−0.767157	+	1.85208i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$	−1.33235	−	1.33235i	−1.33235	−	1.33235i
$673$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$673$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$674$	−0.470114	−	1.13496i	−0.470114	−	1.13496i
$675$	0.439596	+	0.182086i	0.439596	+	0.182086i
$676$			0.381966i			0.381966i
$677$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$677$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$678$		0			0
$679$	0.796180			0.796180
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.744220	−	1.79671i	0.744220	−	1.79671i	0.156434	−	0.987688i	$-0.450000\pi$
$683$	0.744220	−	1.79671i	0.744220	−	1.79671i	0.587785	−	0.809017i	$-0.300000\pi$
$684$		0			0
$685$		0			0
$686$	0.696564	+	1.68165i	0.696564	+	1.68165i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$691$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$692$	0.368770	+	0.152749i	0.368770	+	0.152749i
$693$		0			0
$694$	0.684170	−	1.65173i	0.684170	−	1.65173i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.249264	+	0.601777i	−0.249264	+	0.601777i
$701$		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
$707$	2.80751	−	1.16291i	2.80751	−	1.16291i
$708$	0.359689	+	0.868367i	0.359689	+	0.868367i
$709$	−1.57547	−	0.652583i	−1.57547	−	0.652583i	−0.587785	−	0.809017i	$-0.700000\pi$
$709$	−1.57547	−	0.652583i	−1.57547	−	0.652583i	−0.987688	+	0.156434i	$0.950000\pi$
$710$		0			0
$711$	1.00173	−	2.41839i	1.00173	−	2.41839i
$712$	0.603941	−	0.603941i	0.603941	−	0.603941i
$713$		0			0
$714$	2.71644	−	1.38410i	2.71644	−	1.38410i
$715$		0			0
$716$		0			0
$717$	1.10701	−	2.67256i	1.10701	−	2.67256i
$718$		0			0
$719$	−1.79671	−	0.744220i	−1.79671	−	0.744220i	−0.987688	−	0.156434i	$-0.950000\pi$
$719$	−1.79671	−	0.744220i	−1.79671	−	0.744220i	−0.809017	−	0.587785i	$-0.800000\pi$
$720$		0			0
$721$	1.43050	−	0.592533i	1.43050	−	0.592533i
$722$	−0.831254	−	0.831254i	−0.831254	−	0.831254i
$723$	−1.39680	−	1.39680i	−1.39680	−	1.39680i
$724$		0			0
$725$		0			0
$726$	−1.65173	−	0.684170i	−1.65173	−	0.684170i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	−1.05870	+	1.05870i	−1.05870	+	1.05870i
$730$		0			0
$731$		0			0
$732$	1.12970			1.12970
$733$	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
$739$	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
$742$	0.240020	+	0.579458i	0.240020	+	0.579458i
$743$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$743$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	1.85668			1.85668
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$751$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$752$		−	1.23607i		−	1.23607i
$753$	2.50381	+	1.03711i	2.50381	+	1.03711i
$754$		0			0
$755$		0			0
$756$	0.219151	+	0.219151i	0.219151	+	0.219151i
$757$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$757$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$758$	−1.13496	+	0.470114i	−1.13496	+	0.470114i
$759$		0			0
$760$		0			0
$761$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$761$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$762$		0			0
$763$		0			0
$764$	−0.540182			−0.540182
$765$		0			0
$766$	−1.06740			−1.06740
$767$		0			0
$768$	−0.581990	+	1.40505i	−0.581990	+	1.40505i
$769$		−	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	$-0.850000\pi$
$773$	−1.34500	−	1.34500i	−1.34500	−	1.34500i	−0.453990	−	0.891007i	$-0.650000\pi$
$774$		0			0
$775$		0			0
$776$	−0.129812	−	0.313395i	−0.129812	−	0.313395i
$777$	−3.11215	−	1.28910i	−3.11215	−	1.28910i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	1.66764	−	1.66764i	1.66764	−	1.66764i
$785$		0			0
$786$			0.834717i			0.834717i
$787$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$787$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$788$	−0.270091	−	0.652057i	−0.270091	−	0.652057i
$789$	2.27341	−	0.941679i	2.27341	−	0.941679i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	0.170427	+	0.0705930i	0.170427	+	0.0705930i
$795$		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
$801$	−1.09133	+	1.09133i	−1.09133	+	1.09133i
$802$	−0.684170	+	1.65173i	−0.684170	+	1.65173i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−0.823057	−	0.823057i	−0.823057	−	0.823057i
$808$	−0.915497	−	0.915497i	−0.915497	−	0.915497i
$809$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$809$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$810$		0			0
$811$	−1.84206	−	0.763007i	−1.84206	−	0.763007i	−0.951057	−	0.309017i	$-0.900000\pi$
$811$	−1.84206	−	0.763007i	−1.84206	−	0.763007i	−0.891007	−	0.453990i	$-0.850000\pi$
$812$		0			0
$813$	−1.03711	+	2.50381i	−1.03711	+	2.50381i
$814$		0			0
$815$		0			0
$816$	−1.42943	−	1.22085i	−1.42943	−	1.22085i
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$821$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$822$		0			0
$823$	−0.707107	+	0.292893i	−0.707107	+	0.292893i	−0.707107	−	0.707107i	$-0.750000\pi$
$823$	−0.707107	+	0.292893i	−0.707107	+	0.292893i			1.00000i	$0.5\pi$
$824$	−0.466469	−	0.466469i	−0.466469	−	0.466469i
$825$		0			0
$826$	−2.99673	+	1.24129i	−2.99673	+	1.24129i
$827$	0.744220	+	1.79671i	0.744220	+	1.79671i	0.587785	+	0.809017i	$0.300000\pi$
$827$	0.744220	+	1.79671i	0.744220	+	1.79671i	0.156434	+	0.987688i	$0.450000\pi$
$828$		0			0
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		0			0
$831$	−2.14412	+	2.14412i	−2.14412	+	2.14412i
$832$		0			0
$833$	0.866205	+	1.70002i	0.866205	+	1.70002i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$839$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$840$		0			0
$841$	0.707107	+	0.707107i	0.707107	+	0.707107i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$			1.54337i			1.54337i
$847$	0.652583	−	1.57547i	0.652583	−	1.57547i
$848$	0.273457	−	0.273457i	0.273457	−	0.273457i
$849$	2.59341			2.59341
$850$	−0.363271	+	1.11803i	−0.363271	+	1.11803i
$851$		0			0
$852$	0.818982	−	0.818982i	0.818982	−	0.818982i
$853$	−0.0600500	+	0.144974i	−0.0600500	+	0.144974i	−0.951057	−	0.309017i	$-0.900000\pi$
$853$	−0.0600500	+	0.144974i	−0.0600500	+	0.144974i	0.891007	+	0.453990i	$0.150000\pi$
$854$			3.89858i			3.89858i
$855$		0			0
$856$		0			0
$857$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$857$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$858$		0			0
$859$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$859$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$860$		0			0
$861$		0			0
$862$	0.507083	+	0.210041i	0.507083	+	0.210041i
$863$		−	0.618034i		−	0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$863$		−	0.618034i		−	0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$864$	0.132294	−	0.319385i	0.132294	−	0.319385i
$865$		0			0
$866$		0			0
$867$	1.29671	−	0.794622i	1.29671	−	0.794622i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	0.234572	+	0.566307i	0.234572	+	0.566307i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$877$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$878$	−0.831254	−	2.00682i	−0.831254	−	2.00682i
$879$		0			0
$880$		0			0
$881$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$881$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$882$	−2.08223	+	2.08223i	−2.08223	+	2.08223i
$883$	1.97538			1.97538			0.987688	−	0.156434i	$-0.0500000\pi$
$883$	1.97538			1.97538			0.987688	+	0.156434i	$0.0500000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$887$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$888$			1.43520i			1.43520i
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$	1.49974	−	0.621213i	1.49974	−	0.621213i
$895$		0			0
$896$	−1.85208	−	0.767157i	−1.85208	−	0.767157i
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.501471			−0.501471
$901$	0.142040	+	0.278768i	0.142040	+	0.278768i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	0.292893	+	0.707107i	0.292893	+	0.707107i	1.00000			$0$
$907$	0.292893	+	0.707107i	0.292893	+	0.707107i	−0.707107	+	0.707107i	$0.750000\pi$
$908$		0			0
$909$	1.65431	+	1.65431i	1.65431	+	1.65431i
$910$		0			0
$911$	−1.79671	+	0.744220i	−1.79671	+	0.744220i	−0.809017	+	0.587785i	$0.800000\pi$
$911$	−1.79671	+	0.744220i	−1.79671	+	0.744220i	−0.987688	+	0.156434i	$0.950000\pi$
$912$		0			0
$913$		0			0
$914$			2.09488i			2.09488i
$915$		0			0
$916$		0			0
$917$	−0.796180			−0.796180
$918$	0.425336	+	0.363271i	0.425336	+	0.363271i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	−1.03711	+	2.50381i	−1.03711	+	2.50381i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	1.20002	−	0.497066i	1.20002	−	0.497066i
$926$		0			0
$927$	0.842914	+	0.842914i	0.842914	+	0.842914i
$928$		0			0
$929$	0.497066	+	1.20002i	0.497066	+	1.20002i	0.951057	+	0.309017i	$0.100000\pi$
$929$	0.497066	+	1.20002i	0.497066	+	1.20002i	−0.453990	+	0.891007i	$0.650000\pi$
$930$		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
$941$	−0.178671	−	0.431351i	−0.178671	−	0.431351i	0.809017	−	0.587785i	$-0.200000\pi$
$941$	−0.178671	−	0.431351i	−0.178671	−	0.431351i	−0.987688	+	0.156434i	$0.950000\pi$
$942$	−3.26279	+	1.35149i	−3.26279	+	1.35149i
$943$		0			0
$944$	1.41421	+	1.41421i	1.41421	+	1.41421i
$945$		0			0
$946$		0			0
$947$	−0.707107	−	0.292893i	−0.707107	−	0.292893i		−	1.00000i	$-0.5\pi$
$947$	−0.707107	−	0.292893i	−0.707107	−	0.292893i	−0.707107	+	0.707107i	$0.750000\pi$
$948$			1.15822i			1.15822i
$949$		0			0
$950$		0			0
$951$		0			0
$952$	0.804639	−	0.942112i	0.804639	−	0.942112i
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$	−0.341442	+	0.341442i	−0.341442	+	0.341442i
$955$		0			0
$956$			0.726543i			0.726543i
$957$		0			0
$958$	−0.767157	−	1.85208i	−0.767157	−	1.85208i
$959$		0			0
$960$		0			0
$961$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$962$		0			0
$963$		0			0
$964$	0.458368	+	0.189862i	0.458368	+	0.189862i
$965$		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	$0.750000\pi$
$972$	0.200540	−	0.484146i	0.200540	−	0.484146i
$973$		0			0
$974$	−2.00682	−	0.831254i	−2.00682	−	0.831254i
$975$		0			0
$976$	2.22085	−	0.919906i	2.22085	−	0.919906i
$977$		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		0.382683	−	0.923880i	$-0.375000\pi$
$983$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−2.59341			−2.59341
$988$		0			0
$989$		0			0
$990$		0			0
$991$	1.79671	+	0.744220i	1.79671	+	0.744220i	0.987688	+	0.156434i	$0.0500000\pi$
$991$	1.79671	+	0.744220i	1.79671	+	0.744220i	0.809017	+	0.587785i	$0.200000\pi$
$992$		0			0
$993$	−1.65173	+	0.684170i	−1.65173	+	0.684170i
$994$	2.82630	+	2.82630i	2.82630	+	2.82630i
$995$		0			0
$996$	−0.758980	+	0.314380i	−0.758980	+	0.314380i
$997$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$997$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$998$		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.93.3	✓	16
17.15	even	8	inner	799.1.h.b.610.3	yes	16
47.46	odd	2	CM	799.1.h.b.93.3	✓	16
799.610	odd	8	inner	799.1.h.b.610.3	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
799.1.h.b.93.3	✓	16	1.1	even	1	trivial
799.1.h.b.93.3	✓	16	47.46	odd	2	CM
799.1.h.b.610.3	yes	16	17.15	even	8	inner
799.1.h.b.610.3	yes	16	799.610	odd	8	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(0.831254 - 0.831254i) q^{2} +(-0.581990 + 1.40505i) q^{3} -0.381966i q^{4} +(0.684170 + 1.65173i) q^{6} +(-1.57547 + 0.652583i) q^{7} +(0.513743 + 0.513743i) q^{8} +(-0.928339 - 0.928339i) q^{9} +O(q^{10})\)	\(q+(0.831254 - 0.831254i) q^{2} +(-0.581990 + 1.40505i) q^{3} -0.381966i q^{4} +(0.684170 + 1.65173i) q^{6} +(-1.57547 + 0.652583i) q^{7} +(0.513743 + 0.513743i) q^{8} +(-0.928339 - 0.928339i) q^{9} +(0.536680 + 0.222300i) q^{12} +(-0.767157 + 1.85208i) q^{14} +1.23607 q^{16} +(-0.309017 + 0.951057i) q^{17} -1.54337 q^{18} -2.59341i q^{21} +(-1.02083 + 0.422840i) q^{24} +(0.707107 + 0.707107i) q^{25} +(0.439596 - 0.182086i) q^{27} +(0.249264 + 0.601777i) q^{28} +(0.513743 - 0.513743i) q^{32} +(0.533698 + 1.04744i) q^{34} +(-0.354594 + 0.354594i) q^{36} +(0.497066 - 1.20002i) q^{37} +(-2.15578 - 2.15578i) q^{42} -1.00000i q^{47} +(-0.719379 + 1.73673i) q^{48} +(1.34915 - 1.34915i) q^{49} +1.17557 q^{50} +(-1.15643 - 0.987688i) q^{51} +(0.221232 - 0.221232i) q^{53} +(0.214055 - 0.516776i) q^{54} +(-1.14465 - 0.474129i) q^{56} +(1.14412 + 1.14412i) q^{59} +(1.79671 - 0.744220i) q^{61} +(2.06839 + 0.856755i) q^{63} +0.381966i q^{64} +(0.363271 + 0.118034i) q^{68} +(0.763007 - 1.84206i) q^{71} -0.953855i q^{72} +(-0.584336 - 1.41071i) q^{74} +(-1.40505 + 0.581990i) q^{75} +(0.763007 + 1.84206i) q^{79} -0.589244i q^{81} +(-1.00000 + 1.00000i) q^{83} -0.990595 q^{84} -1.17557i q^{89} +(-0.831254 - 0.831254i) q^{94} +(0.422840 + 1.02083i) q^{96} +(-0.431351 - 0.178671i) q^{97} -2.24297i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{3} - 4 q^{9}+O(q^{10}) \) 16 * q - 4 * q^3 - 4 * q^9	\( 16 q - 4 q^{3} - 4 q^{9} - 16 q^{16} + 4 q^{17} - 20 q^{24} - 4 q^{27} - 4 q^{28} + 4 q^{36} - 20 q^{42} + 24 q^{48} + 4 q^{49} - 16 q^{51} + 4 q^{53} + 20 q^{54} + 20 q^{56} + 4 q^{61} - 4 q^{63} - 4 q^{71} - 4 q^{79} - 16 q^{83} + 32 q^{84}+O(q^{100}) \) 16 * q - 4 * q^3 - 4 * q^9 - 16 * q^16 + 4 * q^17 - 20 * q^24 - 4 * q^27 - 4 * q^28 + 4 * q^36 - 20 * q^42 + 24 * q^48 + 4 * q^49 - 16 * q^51 + 4 * q^53 + 20 * q^54 + 20 * q^56 + 4 * q^61 - 4 * q^63 - 4 * q^71 - 4 * q^79 - 16 * q^83 + 32 * q^84

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