LMFDB - Embedded newform 799.1.e.b.234.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [799,1,Mod(140,799)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(799, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([3, 2]))

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

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

chi := DirichletCharacter("799.140");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 799 = 17 \cdot 47 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	799.e (of order $4$, degree $2$, 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:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{20})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{20}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{20} - \cdots)$

Embedding invariants

Embedding label			234.3
Root			$0.587785 + 0.809017i$ of defining polynomial
Character	$\chi$	$=$	799.234
Dual form			799.1.e.b.140.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.618034i q^{2} +(-1.39680 - 1.39680i) q^{3} +0.618034 q^{4} +(0.863271 - 0.863271i) q^{6} +(-1.26007 + 1.26007i) q^{7} +1.00000i q^{8} +2.90211i q^{9} +O(q^{10})$	\(q+0.618034i q^{2} +(-1.39680 - 1.39680i) q^{3} +0.618034 q^{4} +(0.863271 - 0.863271i) q^{6} +(-1.26007 + 1.26007i) q^{7} +1.00000i q^{8} +2.90211i q^{9} +(-0.863271 - 0.863271i) q^{12} +(-0.778768 - 0.778768i) q^{14} +(-0.809017 + 0.587785i) q^{17} -1.79360 q^{18} +3.52015 q^{21} +(1.39680 - 1.39680i) q^{24} +1.00000i q^{25} +(2.65688 - 2.65688i) q^{27} +(-0.778768 + 0.778768i) q^{28} +1.00000i q^{32} +(-0.363271 - 0.500000i) q^{34} +1.79360i q^{36} +(1.39680 + 1.39680i) q^{37} +2.17557i q^{42} -1.00000 q^{47} -2.17557i q^{49} -0.618034 q^{50} +(1.95106 + 0.309017i) q^{51} -1.90211i q^{53} +(1.64204 + 1.64204i) q^{54} +(-1.26007 - 1.26007i) q^{56} +0.618034i q^{59} +(-0.642040 + 0.642040i) q^{61} +(-3.65688 - 3.65688i) q^{63} -0.618034 q^{64} +(-0.500000 + 0.363271i) q^{68} +(-0.221232 - 0.221232i) q^{71} -2.90211 q^{72} +(-0.863271 + 0.863271i) q^{74} +(1.39680 - 1.39680i) q^{75} +(-0.221232 + 0.221232i) q^{79} -4.52015 q^{81} +2.17557 q^{84} +0.618034 q^{89} -0.618034i q^{94} +(1.39680 - 1.39680i) q^{96} +(0.642040 + 0.642040i) q^{97} +1.34458 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{3} - 4 q^{4} + 4 q^{6} + 2 q^{7}+O(q^{10}) $ 8 * q - 2 * q^3 - 4 * q^4 + 4 * q^6 + 2 * q^7	$ 8 q - 2 q^{3} - 4 q^{4} + 4 q^{6} + 2 q^{7} - 4 q^{12} - 6 q^{14} - 2 q^{17} + 4 q^{18} + 4 q^{21} + 2 q^{24} - 6 q^{28} + 2 q^{37} - 8 q^{47} + 4 q^{50} + 8 q^{51} + 10 q^{54} + 2 q^{56} - 2 q^{61} - 8 q^{63} + 4 q^{64} - 4 q^{68} - 2 q^{71} - 8 q^{72} - 4 q^{74} + 2 q^{75} - 2 q^{79} - 12 q^{81} + 8 q^{84} - 4 q^{89} + 2 q^{96} + 2 q^{97} - 4 q^{98}+O(q^{100}) $ 8 * q - 2 * q^3 - 4 * q^4 + 4 * q^6 + 2 * q^7 - 4 * q^12 - 6 * q^14 - 2 * q^17 + 4 * q^18 + 4 * q^21 + 2 * q^24 - 6 * q^28 + 2 * q^37 - 8 * q^47 + 4 * q^50 + 8 * q^51 + 10 * q^54 + 2 * q^56 - 2 * q^61 - 8 * q^63 + 4 * q^64 - 4 * q^68 - 2 * q^71 - 8 * q^72 - 4 * q^74 + 2 * q^75 - 2 * q^79 - 12 * q^81 + 8 * q^84 - 4 * q^89 + 2 * q^96 + 2 * q^97 - 4 * q^98

Display 100 coefficients 10 coefficients

Character values

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

$n$	$52$	$377$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$

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 $

$2$			0.618034i			0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$2$			0.618034i			0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$3$	−1.39680	−	1.39680i	−1.39680	−	1.39680i	−0.809017	−	0.587785i	$-0.800000\pi$
$3$	−1.39680	−	1.39680i	−1.39680	−	1.39680i	−0.587785	−	0.809017i	$-0.700000\pi$
$4$	0.618034			0.618034
$5$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$5$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$6$	0.863271	−	0.863271i	0.863271	−	0.863271i
$7$	−1.26007	+	1.26007i	−1.26007	+	1.26007i	−0.309017	+	0.951057i	$0.600000\pi$
$7$	−1.26007	+	1.26007i	−1.26007	+	1.26007i	−0.951057	+	0.309017i	$0.900000\pi$
$8$			1.00000i			1.00000i
$9$			2.90211i			2.90211i
$10$		0			0
$11$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$11$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$12$	−0.863271	−	0.863271i	−0.863271	−	0.863271i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	−0.778768	−	0.778768i	−0.778768	−	0.778768i
$15$		0			0
$16$		0			0
$17$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$17$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$18$	−1.79360			−1.79360
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$	3.52015			3.52015
$22$		0			0
$23$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$23$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$24$	1.39680	−	1.39680i	1.39680	−	1.39680i
$25$			1.00000i			1.00000i
$26$		0			0
$27$	2.65688	−	2.65688i	2.65688	−	2.65688i
$28$	−0.778768	+	0.778768i	−0.778768	+	0.778768i
$29$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$29$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$30$		0			0
$31$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$31$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$32$			1.00000i			1.00000i
$33$		0			0
$34$	−0.363271	−	0.500000i	−0.363271	−	0.500000i
$35$		0			0
$36$			1.79360i			1.79360i
$37$	1.39680	+	1.39680i	1.39680	+	1.39680i	0.809017	+	0.587785i	$0.200000\pi$
$37$	1.39680	+	1.39680i	1.39680	+	1.39680i	0.587785	+	0.809017i	$0.300000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$41$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$42$			2.17557i			2.17557i
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.00000			−1.00000
$47$	−1.00000			−1.00000
$48$		0			0
$49$		−	2.17557i		−	2.17557i
$50$	−0.618034			−0.618034
$51$	1.95106	+	0.309017i	1.95106	+	0.309017i
$52$		0			0
$53$		−	1.90211i		−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$53$		−	1.90211i		−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$54$	1.64204	+	1.64204i	1.64204	+	1.64204i
$55$		0			0
$56$	−1.26007	−	1.26007i	−1.26007	−	1.26007i
$57$		0			0
$58$		0			0
$59$			0.618034i			0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$59$			0.618034i			0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$60$		0			0
$61$	−0.642040	+	0.642040i	−0.642040	+	0.642040i	−0.951057	−	0.309017i	$-0.900000\pi$
$61$	−0.642040	+	0.642040i	−0.642040	+	0.642040i	0.309017	+	0.951057i	$0.400000\pi$
$62$		0			0
$63$	−3.65688	−	3.65688i	−3.65688	−	3.65688i
$64$	−0.618034			−0.618034
$65$		0			0
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$69$		0			0
$70$		0			0
$71$	−0.221232	−	0.221232i	−0.221232	−	0.221232i	0.587785	−	0.809017i	$-0.300000\pi$
$71$	−0.221232	−	0.221232i	−0.221232	−	0.221232i	−0.809017	+	0.587785i	$0.800000\pi$
$72$	−2.90211			−2.90211
$73$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$73$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$74$	−0.863271	+	0.863271i	−0.863271	+	0.863271i
$75$	1.39680	−	1.39680i	1.39680	−	1.39680i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−0.221232	+	0.221232i	−0.221232	+	0.221232i	−0.809017	−	0.587785i	$-0.800000\pi$
$79$	−0.221232	+	0.221232i	−0.221232	+	0.221232i	0.587785	+	0.809017i	$0.300000\pi$
$80$		0			0
$81$	−4.52015			−4.52015
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$	2.17557			2.17557
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$89$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		−	0.618034i		−	0.618034i
$95$		0			0
$96$	1.39680	−	1.39680i	1.39680	−	1.39680i
$97$	0.642040	+	0.642040i	0.642040	+	0.642040i	0.951057	−	0.309017i	$-0.100000\pi$
$97$	0.642040	+	0.642040i	0.642040	+	0.642040i	−0.309017	+	0.951057i	$0.600000\pi$
$98$	1.34458			1.34458
$99$		0			0
$100$			0.618034i			0.618034i
$101$	−1.17557			−1.17557			−0.587785	−	0.809017i	$-0.700000\pi$
$101$	−1.17557			−1.17557			−0.587785	+	0.809017i	$0.700000\pi$
$102$	−0.190983	+	1.20582i	−0.190983	+	1.20582i
$103$	1.17557			1.17557			0.587785	−	0.809017i	$-0.300000\pi$
$103$	1.17557			1.17557			0.587785	+	0.809017i	$0.300000\pi$
$104$		0			0
$105$		0			0
$106$	1.17557			1.17557
$107$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$107$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$108$	1.64204	−	1.64204i	1.64204	−	1.64204i
$109$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$109$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$110$		0			0
$111$		−	3.90211i		−	3.90211i
$112$		0			0
$113$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$113$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−0.381966			−0.381966
$119$	0.278768	−	1.76007i	0.278768	−	1.76007i
$120$		0			0
$121$		−	1.00000i		−	1.00000i
$122$	−0.396802	−	0.396802i	−0.396802	−	0.396802i
$123$		0			0
$124$		0			0
$125$		0			0
$126$	2.26007	−	2.26007i	2.26007	−	2.26007i
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$			0.618034i			0.618034i
$129$		0			0
$130$		0			0
$131$	0.642040	+	0.642040i	0.642040	+	0.642040i	0.951057	−	0.309017i	$-0.100000\pi$
$131$	0.642040	+	0.642040i	0.642040	+	0.642040i	−0.309017	+	0.951057i	$0.600000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$139$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$140$		0			0
$141$	1.39680	+	1.39680i	1.39680	+	1.39680i
$142$	0.136729	−	0.136729i	0.136729	−	0.136729i
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−3.03884	+	3.03884i	−3.03884	+	3.03884i
$148$	0.863271	+	0.863271i	0.863271	+	0.863271i
$149$	−1.17557			−1.17557			−0.587785	−	0.809017i	$-0.700000\pi$
$149$	−1.17557			−1.17557			−0.587785	+	0.809017i	$0.700000\pi$
$150$	0.863271	+	0.863271i	0.863271	+	0.863271i
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$	−1.70582	−	2.34786i	−1.70582	−	2.34786i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	1.90211			1.90211			0.951057	−	0.309017i	$-0.100000\pi$
$157$	1.90211			1.90211			0.951057	+	0.309017i	$0.100000\pi$
$158$	−0.136729	−	0.136729i	−0.136729	−	0.136729i
$159$	−2.65688	+	2.65688i	−2.65688	+	2.65688i
$160$		0			0
$161$		0			0
$162$		−	2.79360i		−	2.79360i
$163$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$163$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$167$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$168$			3.52015i			3.52015i
$169$	1.00000			1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.26007	−	1.26007i	−1.26007	−	1.26007i	−0.951057	−	0.309017i	$-0.900000\pi$
$173$	−1.26007	−	1.26007i	−1.26007	−	1.26007i	−0.309017	−	0.951057i	$-0.600000\pi$
$174$		0			0
$175$	−1.26007	−	1.26007i	−1.26007	−	1.26007i
$176$		0			0
$177$	0.863271	−	0.863271i	0.863271	−	0.863271i
$178$			0.381966i			0.381966i
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$181$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$182$		0			0
$183$	1.79360			1.79360
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−0.618034			−0.618034
$189$			6.69572i			6.69572i
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	0.863271	+	0.863271i	0.863271	+	0.863271i
$193$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$193$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$194$	−0.396802	+	0.396802i	−0.396802	+	0.396802i
$195$		0			0
$196$		−	1.34458i		−	1.34458i
$197$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$197$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$198$		0			0
$199$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$199$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$200$	−1.00000			−1.00000
$201$		0			0
$202$		−	0.726543i		−	0.726543i
$203$		0			0
$204$	1.20582	+	0.190983i	1.20582	+	0.190983i
$205$		0			0
$206$			0.726543i			0.726543i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$211$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$212$		−	1.17557i		−	1.17557i
$213$			0.618034i			0.618034i
$214$		0			0
$215$		0			0
$216$	2.65688	+	2.65688i	2.65688	+	2.65688i
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$	2.41164			2.41164
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$	−1.26007	−	1.26007i	−1.26007	−	1.26007i
$225$	−2.90211			−2.90211
$226$		0			0
$227$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$227$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$228$		0			0
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$233$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$234$		0			0
$235$		0			0
$236$			0.381966i			0.381966i
$237$	0.618034			0.618034
$238$	1.08779	+	0.172288i	1.08779	+	0.172288i
$239$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$239$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$240$		0			0
$241$	1.39680	+	1.39680i	1.39680	+	1.39680i	0.809017	+	0.587785i	$0.200000\pi$
$241$	1.39680	+	1.39680i	1.39680	+	1.39680i	0.587785	+	0.809017i	$0.300000\pi$
$242$	0.618034			0.618034
$243$	3.65688	+	3.65688i	3.65688	+	3.65688i
$244$	−0.396802	+	0.396802i	−0.396802	+	0.396802i
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1.17557			1.17557			0.587785	−	0.809017i	$-0.300000\pi$
$251$	1.17557			1.17557			0.587785	+	0.809017i	$0.300000\pi$
$252$	−2.26007	−	2.26007i	−2.26007	−	2.26007i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−1.00000			−1.00000
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$	−3.52015			−3.52015
$260$		0			0
$261$		0			0
$262$	−0.396802	+	0.396802i	−0.396802	+	0.396802i
$263$			0.618034i			0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$263$			0.618034i			0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−0.863271	−	0.863271i	−0.863271	−	0.863271i
$268$		0			0
$269$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$269$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$270$		0			0
$271$	−1.17557			−1.17557			−0.587785	−	0.809017i	$-0.700000\pi$
$271$	−1.17557			−1.17557			−0.587785	+	0.809017i	$0.700000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	0.221232	+	0.221232i	0.221232	+	0.221232i	0.809017	−	0.587785i	$-0.200000\pi$
$277$	0.221232	+	0.221232i	0.221232	+	0.221232i	−0.587785	+	0.809017i	$0.700000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$	−0.863271	+	0.863271i	−0.863271	+	0.863271i
$283$	1.26007	−	1.26007i	1.26007	−	1.26007i	0.309017	−	0.951057i	$-0.400000\pi$
$283$	1.26007	−	1.26007i	1.26007	−	1.26007i	0.951057	−	0.309017i	$-0.100000\pi$
$284$	−0.136729	−	0.136729i	−0.136729	−	0.136729i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−2.90211			−2.90211
$289$	0.309017	−	0.951057i	0.309017	−	0.951057i
$290$		0			0
$291$		−	1.79360i		−	1.79360i
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$	−1.87811	−	1.87811i	−1.87811	−	1.87811i
$295$		0			0
$296$	−1.39680	+	1.39680i	−1.39680	+	1.39680i
$297$		0			0
$298$		−	0.726543i		−	0.726543i
$299$		0			0
$300$	0.863271	−	0.863271i	0.863271	−	0.863271i
$301$		0			0
$302$		0			0
$303$	1.64204	+	1.64204i	1.64204	+	1.64204i
$304$		0			0
$305$		0			0
$306$	1.45106	−	1.05425i	1.45106	−	1.05425i
$307$	−1.17557			−1.17557			−0.587785	−	0.809017i	$-0.700000\pi$
$307$	−1.17557			−1.17557			−0.587785	+	0.809017i	$0.700000\pi$
$308$		0			0
$309$	−1.64204	−	1.64204i	−1.64204	−	1.64204i
$310$		0			0
$311$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$311$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$312$		0			0
$313$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$313$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$314$			1.17557i			1.17557i
$315$		0			0
$316$	−0.136729	+	0.136729i	−0.136729	+	0.136729i
$317$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$317$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$318$	−1.64204	−	1.64204i	−1.64204	−	1.64204i
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−2.79360			−2.79360
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1.26007	−	1.26007i	1.26007	−	1.26007i
$330$		0			0
$331$		−	0.618034i		−	0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$331$		−	0.618034i		−	0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$332$		0			0
$333$	−4.05368	+	4.05368i	−4.05368	+	4.05368i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−1.26007	−	1.26007i	−1.26007	−	1.26007i	−0.951057	−	0.309017i	$-0.900000\pi$
$337$	−1.26007	−	1.26007i	−1.26007	−	1.26007i	−0.309017	−	0.951057i	$-0.600000\pi$
$338$			0.618034i			0.618034i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	1.48131	+	1.48131i	1.48131	+	1.48131i
$344$		0			0
$345$		0			0
$346$	0.778768	−	0.778768i	0.778768	−	0.778768i
$347$	1.39680	−	1.39680i	1.39680	−	1.39680i	0.587785	−	0.809017i	$-0.300000\pi$
$347$	1.39680	−	1.39680i	1.39680	−	1.39680i	0.809017	−	0.587785i	$-0.200000\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$	0.778768	−	0.778768i	0.778768	−	0.778768i
$351$		0			0
$352$		0			0
$353$	1.90211			1.90211			0.951057	−	0.309017i	$-0.100000\pi$
$353$	1.90211			1.90211			0.951057	+	0.309017i	$0.100000\pi$
$354$	0.533531	+	0.533531i	0.533531	+	0.533531i
$355$		0			0
$356$	0.381966			0.381966
$357$	−2.84786	+	2.06909i	−2.84786	+	2.06909i
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−1.00000			−1.00000
$362$		0			0
$363$	−1.39680	+	1.39680i	−1.39680	+	1.39680i
$364$		0			0
$365$		0			0
$366$			1.10851i			1.10851i
$367$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$367$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	2.39680	+	2.39680i	2.39680	+	2.39680i
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$		−	1.00000i		−	1.00000i
$377$		0			0
$378$	−4.13818			−4.13818
$379$	1.26007	+	1.26007i	1.26007	+	1.26007i	0.951057	+	0.309017i	$0.100000\pi$
$379$	1.26007	+	1.26007i	1.26007	+	1.26007i	0.309017	+	0.951057i	$0.400000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			1.17557i			1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$383$			1.17557i			1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$384$	0.863271	−	0.863271i	0.863271	−	0.863271i
$385$		0			0
$386$		0			0
$387$		0			0
$388$	0.396802	+	0.396802i	0.396802	+	0.396802i
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$	2.17557			2.17557
$393$		−	1.79360i		−	1.79360i
$394$	0.618034	+	0.618034i	0.618034	+	0.618034i
$395$		0			0
$396$		0			0
$397$	0.221232	−	0.221232i	0.221232	−	0.221232i	−0.587785	−	0.809017i	$-0.700000\pi$
$397$	0.221232	−	0.221232i	0.221232	−	0.221232i	0.809017	+	0.587785i	$0.200000\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.39680	−	1.39680i	1.39680	−	1.39680i	0.587785	−	0.809017i	$-0.300000\pi$
$401$	1.39680	−	1.39680i	1.39680	−	1.39680i	0.809017	−	0.587785i	$-0.200000\pi$
$402$		0			0
$403$		0			0
$404$	−0.726543			−0.726543
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−0.309017	+	1.95106i	−0.309017	+	1.95106i
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$	0.726543			0.726543
$413$	−0.778768	−	0.778768i	−0.778768	−	0.778768i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$419$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		−	2.90211i		−	2.90211i
$424$	1.90211			1.90211
$425$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$426$	−0.381966			−0.381966
$427$		−	1.61803i		−	1.61803i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−0.642040	+	0.642040i	−0.642040	+	0.642040i	−0.951057	−	0.309017i	$-0.900000\pi$
$431$	−0.642040	+	0.642040i	−0.642040	+	0.642040i	0.309017	+	0.951057i	$0.400000\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$439$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$440$		0			0
$441$	6.31375			6.31375
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		−	2.41164i		−	2.41164i
$445$		0			0
$446$		0			0
$447$	1.64204	+	1.64204i	1.64204	+	1.64204i
$448$	0.778768	−	0.778768i	0.778768	−	0.778768i
$449$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$449$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$450$		−	1.79360i		−	1.79360i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$			1.17557i			1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$457$			1.17557i			1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$458$		0			0
$459$	−0.587785	+	3.71113i	−0.587785	+	3.71113i
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−2.65688	−	2.65688i	−2.65688	−	2.65688i
$472$	−0.618034			−0.618034
$473$		0			0
$474$			0.381966i			0.381966i
$475$		0			0
$476$	0.172288	−	1.08779i	0.172288	−	1.08779i
$477$	5.52015			5.52015
$478$			1.00000i			1.00000i
$479$	1.26007	+	1.26007i	1.26007	+	1.26007i	0.951057	+	0.309017i	$0.100000\pi$
$479$	1.26007	+	1.26007i	1.26007	+	1.26007i	0.309017	+	0.951057i	$0.400000\pi$
$480$		0			0
$481$		0			0
$482$	−0.863271	+	0.863271i	−0.863271	+	0.863271i
$483$		0			0
$484$		−	0.618034i		−	0.618034i
$485$		0			0
$486$	−2.26007	+	2.26007i	−2.26007	+	2.26007i
$487$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$487$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$488$	−0.642040	−	0.642040i	−0.642040	−	0.642040i
$489$		0			0
$490$		0			0
$491$			1.90211i			1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$491$			1.90211i			1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	0.557537			0.557537
$498$		0			0
$499$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$499$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$500$		0			0
$501$		0			0
$502$			0.726543i			0.726543i
$503$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$503$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$504$	3.65688	−	3.65688i	3.65688	−	3.65688i
$505$		0			0
$506$		0			0
$507$	−1.39680	−	1.39680i	−1.39680	−	1.39680i
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		−	2.17557i		−	2.17557i
$519$			3.52015i			3.52015i
$520$		0			0
$521$	−1.26007	+	1.26007i	−1.26007	+	1.26007i	−0.309017	+	0.951057i	$0.600000\pi$
$521$	−1.26007	+	1.26007i	−1.26007	+	1.26007i	−0.951057	+	0.309017i	$0.900000\pi$
$522$		0			0
$523$	2.00000			2.00000			1.00000			$0$
$523$	2.00000			2.00000			1.00000			$0$
$524$	0.396802	+	0.396802i	0.396802	+	0.396802i
$525$			3.52015i			3.52015i
$526$	−0.381966			−0.381966
$527$		0			0
$528$		0			0
$529$		−	1.00000i		−	1.00000i
$530$		0			0
$531$	−1.79360			−1.79360
$532$		0			0
$533$		0			0
$534$	0.533531	−	0.533531i	0.533531	−	0.533531i
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−0.618034	+	0.618034i	−0.618034	+	0.618034i
$539$		0			0
$540$		0			0
$541$	−1.26007	−	1.26007i	−1.26007	−	1.26007i	−0.951057	−	0.309017i	$-0.900000\pi$
$541$	−1.26007	−	1.26007i	−1.26007	−	1.26007i	−0.309017	−	0.951057i	$-0.600000\pi$
$542$		−	0.726543i		−	0.726543i
$543$		0			0
$544$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$545$		0			0
$546$		0			0
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$		0			0
$549$	−1.86327	−	1.86327i	−1.86327	−	1.86327i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		−	0.557537i		−	0.557537i
$554$	−0.136729	+	0.136729i	−0.136729	+	0.136729i
$555$		0			0
$556$		0			0
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$	0.863271	+	0.863271i	0.863271	+	0.863271i
$565$		0			0
$566$	0.778768	+	0.778768i	0.778768	+	0.778768i
$567$	5.69572	−	5.69572i	5.69572	−	5.69572i
$568$	0.221232	−	0.221232i	0.221232	−	0.221232i
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$	−0.221232	+	0.221232i	−0.221232	+	0.221232i	−0.809017	−	0.587785i	$-0.800000\pi$
$571$	−0.221232	+	0.221232i	−0.221232	+	0.221232i	0.587785	+	0.809017i	$0.300000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		−	1.79360i		−	1.79360i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	0.587785	+	0.190983i	0.587785	+	0.190983i
$579$		0			0
$580$		0			0
$581$		0			0
$582$	1.10851			1.10851
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	−1.87811	+	1.87811i	−1.87811	+	1.87811i
$589$		0			0
$590$		0			0
$591$	−2.79360			−2.79360
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$	−0.726543			−0.726543
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$	1.39680	+	1.39680i	1.39680	+	1.39680i
$601$	0.642040	−	0.642040i	0.642040	−	0.642040i	−0.309017	−	0.951057i	$-0.600000\pi$
$601$	0.642040	−	0.642040i	0.642040	−	0.642040i	0.951057	+	0.309017i	$0.100000\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$	−1.01484	+	1.01484i	−1.01484	+	1.01484i
$607$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$607$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−1.05425	−	1.45106i	−1.05425	−	1.45106i
$613$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$613$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$614$		−	0.726543i		−	0.726543i
$615$		0			0
$616$		0			0
$617$	−0.221232	−	0.221232i	−0.221232	−	0.221232i	0.587785	−	0.809017i	$-0.300000\pi$
$617$	−0.221232	−	0.221232i	−0.221232	−	0.221232i	−0.809017	+	0.587785i	$0.800000\pi$
$618$	1.01484	−	1.01484i	1.01484	−	1.01484i
$619$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$619$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−0.778768	+	0.778768i	−0.778768	+	0.778768i
$624$		0			0
$625$	−1.00000			−1.00000
$626$		0			0
$627$		0			0
$628$	1.17557			1.17557
$629$	−1.95106	−	0.309017i	−1.95106	−	0.309017i
$630$		0			0
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$	−0.221232	−	0.221232i	−0.221232	−	0.221232i
$633$		0			0
$634$		0			0
$635$		0			0
$636$	−1.64204	+	1.64204i	−1.64204	+	1.64204i
$637$		0			0
$638$		0			0
$639$	0.642040	−	0.642040i	0.642040	−	0.642040i
$640$		0			0
$641$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$641$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$642$		0			0
$643$	0.642040	+	0.642040i	0.642040	+	0.642040i	0.951057	−	0.309017i	$-0.100000\pi$
$643$	0.642040	+	0.642040i	0.642040	+	0.642040i	−0.309017	+	0.951057i	$0.600000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.90211			−1.90211			−0.951057	−	0.309017i	$-0.900000\pi$
$647$	−1.90211			−1.90211			−0.951057	+	0.309017i	$0.900000\pi$
$648$		−	4.52015i		−	4.52015i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.39680	+	1.39680i	−1.39680	+	1.39680i	−0.587785	+	0.809017i	$0.700000\pi$
$653$	−1.39680	+	1.39680i	−1.39680	+	1.39680i	−0.809017	+	0.587785i	$0.800000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	0.778768	+	0.778768i	0.778768	+	0.778768i
$659$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$659$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$660$		0			0
$661$		−	1.61803i		−	1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$661$		−	1.61803i		−	1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$662$	0.381966			0.381966
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−2.50531	−	2.50531i	−2.50531	−	2.50531i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$			3.52015i			3.52015i
$673$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$673$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$674$	0.778768	−	0.778768i	0.778768	−	0.778768i
$675$	2.65688	+	2.65688i	2.65688	+	2.65688i
$676$	0.618034			0.618034
$677$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$677$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$678$		0			0
$679$	−1.61803			−1.61803
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.642040	+	0.642040i	0.642040	+	0.642040i	0.951057	−	0.309017i	$-0.100000\pi$
$683$	0.642040	+	0.642040i	0.642040	+	0.642040i	−0.309017	+	0.951057i	$0.600000\pi$
$684$		0			0
$685$		0			0
$686$	−0.915497	+	0.915497i	−0.915497	+	0.915497i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$691$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$692$	−0.778768	−	0.778768i	−0.778768	−	0.778768i
$693$		0			0
$694$	0.863271	+	0.863271i	0.863271	+	0.863271i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.778768	−	0.778768i	−0.778768	−	0.778768i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$			1.17557i			1.17557i
$707$	1.48131	−	1.48131i	1.48131	−	1.48131i
$708$	0.533531	−	0.533531i	0.533531	−	0.533531i
$709$	−1.26007	−	1.26007i	−1.26007	−	1.26007i	−0.951057	−	0.309017i	$-0.900000\pi$
$709$	−1.26007	−	1.26007i	−1.26007	−	1.26007i	−0.309017	−	0.951057i	$-0.600000\pi$
$710$		0			0
$711$	−0.642040	−	0.642040i	−0.642040	−	0.642040i
$712$			0.618034i			0.618034i
$713$		0			0
$714$	−1.27877	−	1.76007i	−1.27877	−	1.76007i
$715$		0			0
$716$		0			0
$717$	−2.26007	−	2.26007i	−2.26007	−	2.26007i
$718$		0			0
$719$	−0.642040	−	0.642040i	−0.642040	−	0.642040i	0.309017	−	0.951057i	$-0.400000\pi$
$719$	−0.642040	−	0.642040i	−0.642040	−	0.642040i	−0.951057	+	0.309017i	$0.900000\pi$
$720$		0			0
$721$	−1.48131	+	1.48131i	−1.48131	+	1.48131i
$722$		−	0.618034i		−	0.618034i
$723$		−	3.90211i		−	3.90211i
$724$		0			0
$725$		0			0
$726$	−0.863271	−	0.863271i	−0.863271	−	0.863271i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$		−	5.69572i		−	5.69572i
$730$		0			0
$731$		0			0
$732$	1.10851			1.10851
$733$		−	1.61803i		−	1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$733$		−	1.61803i		−	1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		−	1.90211i		−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$739$		−	1.90211i		−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$740$		0			0
$741$		0			0
$742$	−1.48131	+	1.48131i	−1.48131	+	1.48131i
$743$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$743$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$751$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$752$		0			0
$753$	−1.64204	−	1.64204i	−1.64204	−	1.64204i
$754$		0			0
$755$		0			0
$756$			4.13818i			4.13818i
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$	−0.778768	+	0.778768i	−0.778768	+	0.778768i
$759$		0			0
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−0.726543			−0.726543
$767$		0			0
$768$	1.39680	+	1.39680i	1.39680	+	1.39680i
$769$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$769$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			1.61803i			1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$773$			1.61803i			1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$774$		0			0
$775$		0			0
$776$	−0.642040	+	0.642040i	−0.642040	+	0.642040i
$777$	4.91695	+	4.91695i	4.91695	+	4.91695i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$	1.10851			1.10851
$787$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$787$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$788$	0.618034	−	0.618034i	0.618034	−	0.618034i
$789$	0.863271	−	0.863271i	0.863271	−	0.863271i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	0.136729	+	0.136729i	0.136729	+	0.136729i
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$	0.809017	−	0.587785i	0.809017	−	0.587785i
$800$	−1.00000			−1.00000
$801$			1.79360i			1.79360i
$802$	0.863271	+	0.863271i	0.863271	+	0.863271i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	2.79360i		−	2.79360i
$808$		−	1.17557i		−	1.17557i
$809$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$809$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$810$		0			0
$811$	0.221232	+	0.221232i	0.221232	+	0.221232i	0.809017	−	0.587785i	$-0.200000\pi$
$811$	0.221232	+	0.221232i	0.221232	+	0.221232i	−0.587785	+	0.809017i	$0.700000\pi$
$812$		0			0
$813$	1.64204	+	1.64204i	1.64204	+	1.64204i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$821$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$822$		0			0
$823$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$823$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$824$			1.17557i			1.17557i
$825$		0			0
$826$	0.481305	−	0.481305i	0.481305	−	0.481305i
$827$	0.642040	−	0.642040i	0.642040	−	0.642040i	−0.309017	−	0.951057i	$-0.600000\pi$
$827$	0.642040	−	0.642040i	0.642040	−	0.642040i	0.951057	+	0.309017i	$0.100000\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		−	0.618034i		−	0.618034i
$832$		0			0
$833$	1.27877	+	1.76007i	1.27877	+	1.76007i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$839$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$840$		0			0
$841$			1.00000i			1.00000i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$	1.79360			1.79360
$847$	1.26007	+	1.26007i	1.26007	+	1.26007i
$848$		0			0
$849$	−3.52015			−3.52015
$850$	0.500000	−	0.363271i	0.500000	−	0.363271i
$851$		0			0
$852$			0.381966i			0.381966i
$853$	0.221232	+	0.221232i	0.221232	+	0.221232i	0.809017	−	0.587785i	$-0.200000\pi$
$853$	0.221232	+	0.221232i	0.221232	+	0.221232i	−0.587785	+	0.809017i	$0.700000\pi$
$854$	1.00000			1.00000
$855$		0			0
$856$		0			0
$857$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$857$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$		0			0
$862$	−0.396802	−	0.396802i	−0.396802	−	0.396802i
$863$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$863$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$864$	2.65688	+	2.65688i	2.65688	+	2.65688i
$865$		0			0
$866$		0			0
$867$	−1.76007	+	0.896802i	−1.76007	+	0.896802i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−1.86327	+	1.86327i	−1.86327	+	1.86327i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$877$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$878$	0.618034	−	0.618034i	0.618034	−	0.618034i
$879$		0			0
$880$		0			0
$881$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$881$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$882$			3.90211i			3.90211i
$883$	−1.90211			−1.90211			−0.951057	−	0.309017i	$-0.900000\pi$
$883$	−1.90211			−1.90211			−0.951057	+	0.309017i	$0.900000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$887$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$888$	3.90211			3.90211
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$	−1.01484	+	1.01484i	−1.01484	+	1.01484i
$895$		0			0
$896$	−0.778768	−	0.778768i	−0.778768	−	0.778768i
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−1.79360			−1.79360
$901$	1.11803	+	1.53884i	1.11803	+	1.53884i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$907$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$908$		0			0
$909$		−	3.41164i		−	3.41164i
$910$		0			0
$911$	−0.642040	+	0.642040i	−0.642040	+	0.642040i	−0.951057	−	0.309017i	$-0.900000\pi$
$911$	−0.642040	+	0.642040i	−0.642040	+	0.642040i	0.309017	+	0.951057i	$0.400000\pi$
$912$		0			0
$913$		0			0
$914$	−0.726543			−0.726543
$915$		0			0
$916$		0			0
$917$	−1.61803			−1.61803
$918$	−2.29360	−	0.363271i	−2.29360	−	0.363271i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	1.64204	+	1.64204i	1.64204	+	1.64204i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1.39680	+	1.39680i	−1.39680	+	1.39680i
$926$		0			0
$927$			3.41164i			3.41164i
$928$		0			0
$929$	1.39680	−	1.39680i	1.39680	−	1.39680i	0.587785	−	0.809017i	$-0.300000\pi$
$929$	1.39680	−	1.39680i	1.39680	−	1.39680i	0.809017	−	0.587785i	$-0.200000\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−0.642040	+	0.642040i	−0.642040	+	0.642040i	−0.951057	−	0.309017i	$-0.900000\pi$
$941$	−0.642040	+	0.642040i	−0.642040	+	0.642040i	0.309017	+	0.951057i	$0.400000\pi$
$942$	1.64204	−	1.64204i	1.64204	−	1.64204i
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$947$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$948$	0.381966			0.381966
$949$		0			0
$950$		0			0
$951$		0			0
$952$	1.76007	+	0.278768i	1.76007	+	0.278768i
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$			3.41164i			3.41164i
$955$		0			0
$956$	1.00000			1.00000
$957$		0			0
$958$	−0.778768	+	0.778768i	−0.778768	+	0.778768i
$959$		0			0
$960$		0			0
$961$			1.00000i			1.00000i
$962$		0			0
$963$		0			0
$964$	0.863271	+	0.863271i	0.863271	+	0.863271i
$965$		0			0
$966$		0			0
$967$			1.90211i			1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$967$			1.90211i			1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$968$	1.00000			1.00000
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	2.26007	+	2.26007i	2.26007	+	2.26007i
$973$		0			0
$974$	−0.618034	−	0.618034i	−0.618034	−	0.618034i
$975$		0			0
$976$		0			0
$977$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$977$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	−1.17557			−1.17557
$983$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$983$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−3.52015			−3.52015
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−0.642040	−	0.642040i	−0.642040	−	0.642040i	0.309017	−	0.951057i	$-0.400000\pi$
$991$	−0.642040	−	0.642040i	−0.642040	−	0.642040i	−0.951057	+	0.309017i	$0.900000\pi$
$992$		0			0
$993$	−0.863271	+	0.863271i	−0.863271	+	0.863271i
$994$			0.344577i			0.344577i
$995$		0			0
$996$		0			0
$997$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$997$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$998$		0			0
$999$	7.42226			7.42226

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	799.1.e.b.234.3	yes	8
17.4	even	4	inner	799.1.e.b.140.1	✓	8
47.46	odd	2	CM	799.1.e.b.234.3	yes	8
799.140	odd	4	inner	799.1.e.b.140.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
799.1.e.b.140.1	✓	8	17.4	even	4	inner
799.1.e.b.140.1	✓	8	799.140	odd	4	inner
799.1.e.b.234.3	yes	8	1.1	even	1	trivial
799.1.e.b.234.3	yes	8	47.46	odd	2	CM

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.e (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+0.618034i q^{2} +(-1.39680 - 1.39680i) q^{3} +0.618034 q^{4} +(0.863271 - 0.863271i) q^{6} +(-1.26007 + 1.26007i) q^{7} +1.00000i q^{8} +2.90211i q^{9} +O(q^{10})\)	\(q+0.618034i q^{2} +(-1.39680 - 1.39680i) q^{3} +0.618034 q^{4} +(0.863271 - 0.863271i) q^{6} +(-1.26007 + 1.26007i) q^{7} +1.00000i q^{8} +2.90211i q^{9} +(-0.863271 - 0.863271i) q^{12} +(-0.778768 - 0.778768i) q^{14} +(-0.809017 + 0.587785i) q^{17} -1.79360 q^{18} +3.52015 q^{21} +(1.39680 - 1.39680i) q^{24} +1.00000i q^{25} +(2.65688 - 2.65688i) q^{27} +(-0.778768 + 0.778768i) q^{28} +1.00000i q^{32} +(-0.363271 - 0.500000i) q^{34} +1.79360i q^{36} +(1.39680 + 1.39680i) q^{37} +2.17557i q^{42} -1.00000 q^{47} -2.17557i q^{49} -0.618034 q^{50} +(1.95106 + 0.309017i) q^{51} -1.90211i q^{53} +(1.64204 + 1.64204i) q^{54} +(-1.26007 - 1.26007i) q^{56} +0.618034i q^{59} +(-0.642040 + 0.642040i) q^{61} +(-3.65688 - 3.65688i) q^{63} -0.618034 q^{64} +(-0.500000 + 0.363271i) q^{68} +(-0.221232 - 0.221232i) q^{71} -2.90211 q^{72} +(-0.863271 + 0.863271i) q^{74} +(1.39680 - 1.39680i) q^{75} +(-0.221232 + 0.221232i) q^{79} -4.52015 q^{81} +2.17557 q^{84} +0.618034 q^{89} -0.618034i q^{94} +(1.39680 - 1.39680i) q^{96} +(0.642040 + 0.642040i) q^{97} +1.34458 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{3} - 4 q^{4} + 4 q^{6} + 2 q^{7}+O(q^{10}) \) 8 * q - 2 * q^3 - 4 * q^4 + 4 * q^6 + 2 * q^7	\( 8 q - 2 q^{3} - 4 q^{4} + 4 q^{6} + 2 q^{7} - 4 q^{12} - 6 q^{14} - 2 q^{17} + 4 q^{18} + 4 q^{21} + 2 q^{24} - 6 q^{28} + 2 q^{37} - 8 q^{47} + 4 q^{50} + 8 q^{51} + 10 q^{54} + 2 q^{56} - 2 q^{61} - 8 q^{63} + 4 q^{64} - 4 q^{68} - 2 q^{71} - 8 q^{72} - 4 q^{74} + 2 q^{75} - 2 q^{79} - 12 q^{81} + 8 q^{84} - 4 q^{89} + 2 q^{96} + 2 q^{97} - 4 q^{98}+O(q^{100}) \) 8 * q - 2 * q^3 - 4 * q^4 + 4 * q^6 + 2 * q^7 - 4 * q^12 - 6 * q^14 - 2 * q^17 + 4 * q^18 + 4 * q^21 + 2 * q^24 - 6 * q^28 + 2 * q^37 - 8 * q^47 + 4 * q^50 + 8 * q^51 + 10 * q^54 + 2 * q^56 - 2 * q^61 - 8 * q^63 + 4 * q^64 - 4 * q^68 - 2 * q^71 - 8 * q^72 - 4 * q^74 + 2 * q^75 - 2 * q^79 - 12 * q^81 + 8 * q^84 - 4 * q^89 + 2 * q^96 + 2 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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