LMFDB - Embedded newform 1400.1.bs.c.741.1

Newspace parameters

Level:	$ N $	$=$	$ 1400 = 2^{3} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1400.bs (of order $10$, degree $4$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$0.698691017686$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{10})$
Coefficient field:	$\Q(\zeta_{20})$
Defining polynomial:	$x^{8} - x^{6} + x^{4} - x^{2} + 1$
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image	$D_{10}$
Projective field	Galois closure of 10.0.10504375000000000000.2

Embedding invariants

Embedding label			741.1
Root			$0.951057 - 0.309017i$ of defining polynomial
Character	$\chi$	$=$	1400.741
Dual form			1400.1.bs.c.461.1

$q$-expansion

$f(q)$	$=$	$q+(0.809017 + 0.587785i) q^{2} +(-0.587785 - 1.80902i) q^{3} +(0.309017 + 0.951057i) q^{4} +(-0.587785 + 0.809017i) q^{5} +(0.587785 - 1.80902i) q^{6} -1.00000 q^{7} +(-0.309017 + 0.951057i) q^{8} +(-2.11803 + 1.53884i) q^{9} +O(q^{10})$	\(q+(0.809017 + 0.587785i) q^{2} +(-0.587785 - 1.80902i) q^{3} +(0.309017 + 0.951057i) q^{4} +(-0.587785 + 0.809017i) q^{5} +(0.587785 - 1.80902i) q^{6} -1.00000 q^{7} +(-0.309017 + 0.951057i) q^{8} +(-2.11803 + 1.53884i) q^{9} +(-0.951057 + 0.309017i) q^{10} +(1.53884 - 1.11803i) q^{12} +(-1.53884 + 1.11803i) q^{13} +(-0.809017 - 0.587785i) q^{14} +(1.80902 + 0.587785i) q^{15} +(-0.809017 + 0.587785i) q^{16} -2.61803 q^{18} +(-0.951057 - 0.309017i) q^{20} +(0.587785 + 1.80902i) q^{21} +(-0.500000 - 0.363271i) q^{23} +1.90211 q^{24} +(-0.309017 - 0.951057i) q^{25} -1.90211 q^{26} +(2.48990 + 1.80902i) q^{27} +(-0.309017 - 0.951057i) q^{28} +(1.11803 + 1.53884i) q^{30} -1.00000 q^{32} +(0.587785 - 0.809017i) q^{35} +(-2.11803 - 1.53884i) q^{36} +(2.92705 + 2.12663i) q^{39} +(-0.587785 - 0.809017i) q^{40} +(-0.587785 + 1.80902i) q^{42} -2.61803i q^{45} +(-0.190983 - 0.587785i) q^{46} +(1.53884 + 1.11803i) q^{48} +1.00000 q^{49} +(0.309017 - 0.951057i) q^{50} +(-1.53884 - 1.11803i) q^{52} +(0.951057 + 2.92705i) q^{54} +(0.309017 - 0.951057i) q^{56} +(-0.951057 + 0.690983i) q^{59} +1.90211i q^{60} +(0.951057 + 0.690983i) q^{61} +(2.11803 - 1.53884i) q^{63} +(-0.809017 - 0.587785i) q^{64} -1.90211i q^{65} +(-0.363271 + 1.11803i) q^{69} +(0.951057 - 0.309017i) q^{70} +(0.500000 + 1.53884i) q^{71} +(-0.809017 - 2.48990i) q^{72} +(-1.53884 + 1.11803i) q^{75} +(1.11803 + 3.44095i) q^{78} +(-0.190983 - 0.587785i) q^{79} -1.00000i q^{80} +(1.00000 - 3.07768i) q^{81} +(0.363271 - 1.11803i) q^{83} +(-1.53884 + 1.11803i) q^{84} +(1.53884 - 2.11803i) q^{90} +(1.53884 - 1.11803i) q^{91} +(0.190983 - 0.587785i) q^{92} +(0.587785 + 1.80902i) q^{96} +(0.809017 + 0.587785i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8q + 2q^{2} - 2q^{4} - 8q^{7} + 2q^{8} - 8q^{9} + O(q^{10}) $	$ 8q + 2q^{2} - 2q^{4} - 8q^{7} + 2q^{8} - 8q^{9} - 2q^{14} + 10q^{15} - 2q^{16} - 12q^{18} - 4q^{23} + 2q^{25} + 2q^{28} - 8q^{32} - 8q^{36} + 10q^{39} - 6q^{46} + 8q^{49} - 2q^{50} - 2q^{56} + 8q^{63} - 2q^{64} + 4q^{71} - 2q^{72} - 6q^{79} + 8q^{81} + 6q^{92} + 2q^{98} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times$.

$n$	$351$	$701$	$801$	$1177$
$\chi(n)$	$1$	$-1$	$-1$	$e\left(\frac{1}{5}\right)$

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 $

$2$	0.809017	+	0.587785i	0.809017	+	0.587785i
$2$	0.809017	+	0.587785i	0.809017	+	0.587785i
$3$	−0.587785	−	1.80902i	−0.587785	−	1.80902i	−0.587785	−	0.809017i	$-0.700000\pi$
$3$	−0.587785	−	1.80902i	−0.587785	−	1.80902i		−	1.00000i	$-0.5\pi$
$4$	0.309017	+	0.951057i	0.309017	+	0.951057i
$5$	−0.587785	+	0.809017i	−0.587785	+	0.809017i
$5$	−0.587785	+	0.809017i	−0.587785	+	0.809017i
$6$	0.587785	−	1.80902i	0.587785	−	1.80902i
$7$	−1.00000			−1.00000
$7$	−1.00000			−1.00000
$8$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$9$	−2.11803	+	1.53884i	−2.11803	+	1.53884i
$10$	−0.951057	+	0.309017i	−0.951057	+	0.309017i
$11$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$11$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$12$	1.53884	−	1.11803i	1.53884	−	1.11803i
$13$	−1.53884	+	1.11803i	−1.53884	+	1.11803i	−0.587785	+	0.809017i	$0.700000\pi$
$13$	−1.53884	+	1.11803i	−1.53884	+	1.11803i	−0.951057	+	0.309017i	$0.900000\pi$
$14$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$15$	1.80902	+	0.587785i	1.80902	+	0.587785i
$16$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$17$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$17$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$18$	−2.61803			−2.61803
$19$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$19$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$20$	−0.951057	−	0.309017i	−0.951057	−	0.309017i
$21$	0.587785	+	1.80902i	0.587785	+	1.80902i
$22$		0			0
$23$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$23$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$24$	1.90211			1.90211
$25$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$26$	−1.90211			−1.90211
$27$	2.48990	+	1.80902i	2.48990	+	1.80902i
$28$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$29$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$29$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$30$	1.11803	+	1.53884i	1.11803	+	1.53884i
$31$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$31$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$32$	−1.00000			−1.00000
$33$		0			0
$34$		0			0
$35$	0.587785	−	0.809017i	0.587785	−	0.809017i
$36$	−2.11803	−	1.53884i	−2.11803	−	1.53884i
$37$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$37$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$38$		0			0
$39$	2.92705	+	2.12663i	2.92705	+	2.12663i
$40$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$41$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$41$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$42$	−0.587785	+	1.80902i	−0.587785	+	1.80902i
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$		−	2.61803i		−	2.61803i
$46$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$47$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$47$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$48$	1.53884	+	1.11803i	1.53884	+	1.11803i
$49$	1.00000			1.00000
$50$	0.309017	−	0.951057i	0.309017	−	0.951057i
$51$		0			0
$52$	−1.53884	−	1.11803i	−1.53884	−	1.11803i
$53$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$53$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$54$	0.951057	+	2.92705i	0.951057	+	2.92705i
$55$		0			0
$56$	0.309017	−	0.951057i	0.309017	−	0.951057i
$57$		0			0
$58$		0			0
$59$	−0.951057	+	0.690983i	−0.951057	+	0.690983i	−0.951057	−	0.309017i	$-0.900000\pi$
$59$	−0.951057	+	0.690983i	−0.951057	+	0.690983i			1.00000i	$0.5\pi$
$60$			1.90211i			1.90211i
$61$	0.951057	+	0.690983i	0.951057	+	0.690983i	0.951057	−	0.309017i	$-0.100000\pi$
$61$	0.951057	+	0.690983i	0.951057	+	0.690983i			1.00000i	$0.5\pi$
$62$		0			0
$63$	2.11803	−	1.53884i	2.11803	−	1.53884i
$64$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$65$		−	1.90211i		−	1.90211i
$66$		0			0
$67$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$67$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$68$		0			0
$69$	−0.363271	+	1.11803i	−0.363271	+	1.11803i
$70$	0.951057	−	0.309017i	0.951057	−	0.309017i
$71$	0.500000	+	1.53884i	0.500000	+	1.53884i	0.809017	+	0.587785i	$0.200000\pi$
$71$	0.500000	+	1.53884i	0.500000	+	1.53884i	−0.309017	+	0.951057i	$0.600000\pi$
$72$	−0.809017	−	2.48990i	−0.809017	−	2.48990i
$73$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$73$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$74$		0			0
$75$	−1.53884	+	1.11803i	−1.53884	+	1.11803i
$76$		0			0
$77$		0			0
$78$	1.11803	+	3.44095i	1.11803	+	3.44095i
$79$	−0.190983	−	0.587785i	−0.190983	−	0.587785i	0.809017	−	0.587785i	$-0.200000\pi$
$79$	−0.190983	−	0.587785i	−0.190983	−	0.587785i	−1.00000			$\pi$
$80$		−	1.00000i		−	1.00000i
$81$	1.00000	−	3.07768i	1.00000	−	3.07768i
$82$		0			0
$83$	0.363271	−	1.11803i	0.363271	−	1.11803i	−0.587785	−	0.809017i	$-0.700000\pi$
$83$	0.363271	−	1.11803i	0.363271	−	1.11803i	0.951057	−	0.309017i	$-0.100000\pi$
$84$	−1.53884	+	1.11803i	−1.53884	+	1.11803i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$89$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$90$	1.53884	−	2.11803i	1.53884	−	2.11803i
$91$	1.53884	−	1.11803i	1.53884	−	1.11803i
$92$	0.190983	−	0.587785i	0.190983	−	0.587785i
$93$		0			0
$94$		0			0
$95$		0			0
$96$	0.587785	+	1.80902i	0.587785	+	1.80902i
$97$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$97$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$98$	0.809017	+	0.587785i	0.809017	+	0.587785i
$99$		0			0
$100$	0.809017	−	0.587785i	0.809017	−	0.587785i
$101$	−1.90211			−1.90211			−0.951057	−	0.309017i	$-0.900000\pi$
$101$	−1.90211			−1.90211			−0.951057	+	0.309017i	$0.900000\pi$
$102$		0			0
$103$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$103$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$104$	−0.587785	−	1.80902i	−0.587785	−	1.80902i
$105$	−1.80902	−	0.587785i	−1.80902	−	0.587785i
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$	−0.951057	+	2.92705i	−0.951057	+	2.92705i
$109$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$109$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$110$		0			0
$111$		0			0
$112$	0.809017	−	0.587785i	0.809017	−	0.587785i
$113$	−1.30902	+	0.951057i	−1.30902	+	0.951057i	−0.309017	+	0.951057i	$0.600000\pi$
$113$	−1.30902	+	0.951057i	−1.30902	+	0.951057i	−1.00000			$\pi$
$114$		0			0
$115$	0.587785	−	0.190983i	0.587785	−	0.190983i
$116$		0			0
$117$	1.53884	−	4.73607i	1.53884	−	4.73607i
$118$	−1.17557			−1.17557
$119$		0			0
$120$	−1.11803	+	1.53884i	−1.11803	+	1.53884i
$121$	0.309017	+	0.951057i	0.309017	+	0.951057i
$122$	0.363271	+	1.11803i	0.363271	+	1.11803i
$123$		0			0
$124$		0			0
$125$	0.951057	+	0.309017i	0.951057	+	0.309017i
$126$	2.61803			2.61803
$127$	−1.61803	−	1.17557i	−1.61803	−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$127$	−1.61803	−	1.17557i	−1.61803	−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$128$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$129$		0			0
$130$	1.11803	−	1.53884i	1.11803	−	1.53884i
$131$	−0.363271	+	1.11803i	−0.363271	+	1.11803i	0.587785	+	0.809017i	$0.300000\pi$
$131$	−0.363271	+	1.11803i	−0.363271	+	1.11803i	−0.951057	+	0.309017i	$0.900000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−2.92705	+	0.951057i	−2.92705	+	0.951057i
$136$		0			0
$137$	−1.61803	+	1.17557i	−1.61803	+	1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$137$	−1.61803	+	1.17557i	−1.61803	+	1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$138$	−0.951057	+	0.690983i	−0.951057	+	0.690983i
$139$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$139$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$140$	0.951057	+	0.309017i	0.951057	+	0.309017i
$141$		0			0
$142$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$143$		0			0
$144$	0.809017	−	2.48990i	0.809017	−	2.48990i
$145$		0			0
$146$		0			0
$147$	−0.587785	−	1.80902i	−0.587785	−	1.80902i
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$	−1.90211			−1.90211
$151$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$151$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−1.11803	+	3.44095i	−1.11803	+	3.44095i
$157$	1.17557			1.17557			0.587785	−	0.809017i	$-0.300000\pi$
$157$	1.17557			1.17557			0.587785	+	0.809017i	$0.300000\pi$
$158$	0.190983	−	0.587785i	0.190983	−	0.587785i
$159$		0			0
$160$	0.587785	−	0.809017i	0.587785	−	0.809017i
$161$	0.500000	+	0.363271i	0.500000	+	0.363271i
$162$	2.61803	−	1.90211i	2.61803	−	1.90211i
$163$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$163$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$164$		0			0
$165$		0			0
$166$	0.951057	−	0.690983i	0.951057	−	0.690983i
$167$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$167$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$168$	−1.90211			−1.90211
$169$	0.809017	−	2.48990i	0.809017	−	2.48990i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	0.951057	+	0.690983i	0.951057	+	0.690983i	0.951057	−	0.309017i	$-0.100000\pi$
$173$	0.951057	+	0.690983i	0.951057	+	0.690983i			1.00000i	$0.5\pi$
$174$		0			0
$175$	0.309017	+	0.951057i	0.309017	+	0.951057i
$176$		0			0
$177$	1.80902	+	1.31433i	1.80902	+	1.31433i
$178$		0			0
$179$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$179$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$180$	2.48990	−	0.809017i	2.48990	−	0.809017i
$181$	0.363271	−	1.11803i	0.363271	−	1.11803i	−0.587785	−	0.809017i	$-0.700000\pi$
$181$	0.363271	−	1.11803i	0.363271	−	1.11803i	0.951057	−	0.309017i	$-0.100000\pi$
$182$	1.90211			1.90211
$183$	0.690983	−	2.12663i	0.690983	−	2.12663i
$184$	0.500000	−	0.363271i	0.500000	−	0.363271i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−2.48990	−	1.80902i	−2.48990	−	1.80902i
$190$		0			0
$191$	0.500000	−	0.363271i	0.500000	−	0.363271i	−0.309017	−	0.951057i	$-0.600000\pi$
$191$	0.500000	−	0.363271i	0.500000	−	0.363271i	0.809017	+	0.587785i	$0.200000\pi$
$192$	−0.587785	+	1.80902i	−0.587785	+	1.80902i
$193$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$193$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$194$		0			0
$195$	−3.44095	+	1.11803i	−3.44095	+	1.11803i
$196$	0.309017	+	0.951057i	0.309017	+	0.951057i
$197$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$197$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	1.00000			1.00000
$201$		0			0
$202$	−1.53884	−	1.11803i	−1.53884	−	1.11803i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	1.61803			1.61803
$208$	0.587785	−	1.80902i	0.587785	−	1.80902i
$209$		0			0
$210$	−1.11803	−	1.53884i	−1.11803	−	1.53884i
$211$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$211$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$212$		0			0
$213$	2.48990	−	1.80902i	2.48990	−	1.80902i
$214$		0			0
$215$		0			0
$216$	−2.48990	+	1.80902i	−2.48990	+	1.80902i
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$223$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$224$	1.00000			1.00000
$225$	2.11803	+	1.53884i	2.11803	+	1.53884i
$226$	−1.61803			−1.61803
$227$	1.53884	+	1.11803i	1.53884	+	1.11803i	0.951057	+	0.309017i	$0.100000\pi$
$227$	1.53884	+	1.11803i	1.53884	+	1.11803i	0.587785	+	0.809017i	$0.300000\pi$
$228$		0			0
$229$	0.363271	+	1.11803i	0.363271	+	1.11803i	0.951057	+	0.309017i	$0.100000\pi$
$229$	0.363271	+	1.11803i	0.363271	+	1.11803i	−0.587785	+	0.809017i	$0.700000\pi$
$230$	0.587785	+	0.190983i	0.587785	+	0.190983i
$231$		0			0
$232$		0			0
$233$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$233$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$234$	4.02874	−	2.92705i	4.02874	−	2.92705i
$235$		0			0
$236$	−0.951057	−	0.690983i	−0.951057	−	0.690983i
$237$	−0.951057	+	0.690983i	−0.951057	+	0.690983i
$238$		0			0
$239$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$239$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$240$	−1.80902	+	0.587785i	−1.80902	+	0.587785i
$241$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$241$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$242$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$243$	−3.07768			−3.07768
$244$	−0.363271	+	1.11803i	−0.363271	+	1.11803i
$245$	−0.587785	+	0.809017i	−0.587785	+	0.809017i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−2.23607			−2.23607
$250$	0.587785	+	0.809017i	0.587785	+	0.809017i
$251$	−1.17557			−1.17557			−0.587785	−	0.809017i	$-0.700000\pi$
$251$	−1.17557			−1.17557			−0.587785	+	0.809017i	$0.700000\pi$
$252$	2.11803	+	1.53884i	2.11803	+	1.53884i
$253$		0			0
$254$	−0.618034	−	1.90211i	−0.618034	−	1.90211i
$255$		0			0
$256$	0.309017	−	0.951057i	0.309017	−	0.951057i
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$		0			0
$260$	1.80902	−	0.587785i	1.80902	−	0.587785i
$261$		0			0
$262$	−0.951057	+	0.690983i	−0.951057	+	0.690983i
$263$	0.500000	−	0.363271i	0.500000	−	0.363271i	−0.309017	−	0.951057i	$-0.600000\pi$
$263$	0.500000	−	0.363271i	0.500000	−	0.363271i	0.809017	+	0.587785i	$0.200000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	0.363271	−	1.11803i	0.363271	−	1.11803i	−0.587785	−	0.809017i	$-0.700000\pi$
$269$	0.363271	−	1.11803i	0.363271	−	1.11803i	0.951057	−	0.309017i	$-0.100000\pi$
$270$	−2.92705	−	0.951057i	−2.92705	−	0.951057i
$271$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$271$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$272$		0			0
$273$	−2.92705	−	2.12663i	−2.92705	−	2.12663i
$274$	−2.00000			−2.00000
$275$		0			0
$276$	−1.17557			−1.17557
$277$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$277$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$278$		0			0
$279$		0			0
$280$	0.587785	+	0.809017i	0.587785	+	0.809017i
$281$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$281$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$282$		0			0
$283$	−0.587785	+	1.80902i	−0.587785	+	1.80902i			1.00000i	$0.5\pi$
$283$	−0.587785	+	1.80902i	−0.587785	+	1.80902i	−0.587785	+	0.809017i	$0.700000\pi$
$284$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	2.11803	−	1.53884i	2.11803	−	1.53884i
$289$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	0.587785	−	1.80902i	0.587785	−	1.80902i
$295$		−	1.17557i		−	1.17557i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.17557			1.17557
$300$	−1.53884	−	1.11803i	−1.53884	−	1.11803i
$301$		0			0
$302$	0.500000	+	0.363271i	0.500000	+	0.363271i
$303$	1.11803	+	3.44095i	1.11803	+	3.44095i
$304$		0			0
$305$	−1.11803	+	0.363271i	−1.11803	+	0.363271i
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$311$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$312$	−2.92705	+	2.12663i	−2.92705	+	2.12663i
$313$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$313$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$314$	0.951057	+	0.690983i	0.951057	+	0.690983i
$315$			2.61803i			2.61803i
$316$	0.500000	−	0.363271i	0.500000	−	0.363271i
$317$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$317$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$318$		0			0
$319$		0			0
$320$	0.951057	−	0.309017i	0.951057	−	0.309017i
$321$		0			0
$322$	0.190983	+	0.587785i	0.190983	+	0.587785i
$323$		0			0
$324$	3.23607			3.23607
$325$	1.53884	+	1.11803i	1.53884	+	1.11803i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$331$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$332$	1.17557			1.17557
$333$		0			0
$334$		0			0
$335$		0			0
$336$	−1.53884	−	1.11803i	−1.53884	−	1.11803i
$337$	0.500000	−	0.363271i	0.500000	−	0.363271i	−0.309017	−	0.951057i	$-0.600000\pi$
$337$	0.500000	−	0.363271i	0.500000	−	0.363271i	0.809017	+	0.587785i	$0.200000\pi$
$338$	2.11803	−	1.53884i	2.11803	−	1.53884i
$339$	2.48990	+	1.80902i	2.48990	+	1.80902i
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−1.00000			−1.00000
$344$		0			0
$345$	−0.690983	−	0.951057i	−0.690983	−	0.951057i
$346$	0.363271	+	1.11803i	0.363271	+	1.11803i
$347$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$347$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$348$		0			0
$349$	−1.90211			−1.90211			−0.951057	−	0.309017i	$-0.900000\pi$
$349$	−1.90211			−1.90211			−0.951057	+	0.309017i	$0.900000\pi$
$350$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$351$	−5.85410			−5.85410
$352$		0			0
$353$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$353$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$354$	0.690983	+	2.12663i	0.690983	+	2.12663i
$355$	−1.53884	−	0.500000i	−1.53884	−	0.500000i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$359$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$360$	2.48990	+	0.809017i	2.48990	+	0.809017i
$361$	0.809017	+	0.587785i	0.809017	+	0.587785i
$362$	0.951057	−	0.690983i	0.951057	−	0.690983i
$363$	1.53884	−	1.11803i	1.53884	−	1.11803i
$364$	1.53884	+	1.11803i	1.53884	+	1.11803i
$365$		0			0
$366$	1.80902	−	1.31433i	1.80902	−	1.31433i
$367$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$367$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$368$	0.618034			0.618034
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$373$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$374$		0			0
$375$		−	1.90211i		−	1.90211i
$376$		0			0
$377$		0			0
$378$	−0.951057	−	2.92705i	−0.951057	−	2.92705i
$379$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$379$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$380$		0			0
$381$	−1.17557	+	3.61803i	−1.17557	+	3.61803i
$382$	0.618034			0.618034
$383$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$383$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$384$	−1.53884	+	1.11803i	−1.53884	+	1.11803i
$385$		0			0
$386$	1.30902	+	0.951057i	1.30902	+	0.951057i
$387$		0			0
$388$		0			0
$389$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$389$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$390$	−3.44095	−	1.11803i	−3.44095	−	1.11803i
$391$		0			0
$392$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$393$	2.23607			2.23607
$394$		0			0
$395$	0.587785	+	0.190983i	0.587785	+	0.190983i
$396$		0			0
$397$	0.587785	+	1.80902i	0.587785	+	1.80902i	0.587785	+	0.809017i	$0.300000\pi$
$397$	0.587785	+	1.80902i	0.587785	+	1.80902i			1.00000i	$0.5\pi$
$398$		0			0
$399$		0			0
$400$	0.809017	+	0.587785i	0.809017	+	0.587785i
$401$	−2.00000			−2.00000			−1.00000			$\pi$
$401$	−2.00000			−2.00000			−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$	−0.587785	−	1.80902i	−0.587785	−	1.80902i
$405$	1.90211	+	2.61803i	1.90211	+	2.61803i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$409$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$410$		0			0
$411$	3.07768	+	2.23607i	3.07768	+	2.23607i
$412$		0			0
$413$	0.951057	−	0.690983i	0.951057	−	0.690983i
$414$	1.30902	+	0.951057i	1.30902	+	0.951057i
$415$	0.690983	+	0.951057i	0.690983	+	0.951057i
$416$	1.53884	−	1.11803i	1.53884	−	1.11803i
$417$		0			0
$418$		0			0
$419$	0.587785	−	1.80902i	0.587785	−	1.80902i		−	1.00000i	$-0.5\pi$
$419$	0.587785	−	1.80902i	0.587785	−	1.80902i	0.587785	−	0.809017i	$-0.300000\pi$
$420$		−	1.90211i		−	1.90211i
$421$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$421$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$	3.07768			3.07768
$427$	−0.951057	−	0.690983i	−0.951057	−	0.690983i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$431$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$432$	−3.07768			−3.07768
$433$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$433$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$439$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$440$		0			0
$441$	−2.11803	+	1.53884i	−2.11803	+	1.53884i
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	0.809017	+	0.587785i	0.809017	+	0.587785i
$449$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$449$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$450$	0.809017	+	2.48990i	0.809017	+	2.48990i
$451$		0			0
$452$	−1.30902	−	0.951057i	−1.30902	−	0.951057i
$453$	−0.363271	−	1.11803i	−0.363271	−	1.11803i
$454$	0.587785	+	1.80902i	0.587785	+	1.80902i
$455$			1.90211i			1.90211i
$456$		0			0
$457$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$457$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$458$	−0.363271	+	1.11803i	−0.363271	+	1.11803i
$459$		0			0
$460$	0.363271	+	0.500000i	0.363271	+	0.500000i
$461$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$461$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$462$		0			0
$463$	−1.30902	+	0.951057i	−1.30902	+	0.951057i	−0.309017	+	0.951057i	$0.600000\pi$
$463$	−1.30902	+	0.951057i	−1.30902	+	0.951057i	−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$467$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$467$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$468$	4.97980			4.97980
$469$		0			0
$470$		0			0
$471$	−0.690983	−	2.12663i	−0.690983	−	2.12663i
$472$	−0.363271	−	1.11803i	−0.363271	−	1.11803i
$473$		0			0
$474$	−1.17557			−1.17557
$475$		0			0
$476$		0			0
$477$		0			0
$478$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$479$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$479$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$480$	−1.80902	−	0.587785i	−1.80902	−	0.587785i
$481$		0			0
$482$		0			0
$483$	0.363271	−	1.11803i	0.363271	−	1.11803i
$484$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$485$		0			0
$486$	−2.48990	−	1.80902i	−2.48990	−	1.80902i
$487$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$487$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$488$	−0.951057	+	0.690983i	−0.951057	+	0.690983i
$489$		0			0
$490$	−0.951057	+	0.309017i	−0.951057	+	0.309017i
$491$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$491$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$498$	−1.80902	−	1.31433i	−1.80902	−	1.31433i
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$			1.00000i			1.00000i
$501$		0			0
$502$	−0.951057	−	0.690983i	−0.951057	−	0.690983i
$503$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$503$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$504$	0.809017	+	2.48990i	0.809017	+	2.48990i
$505$	1.11803	−	1.53884i	1.11803	−	1.53884i
$506$		0			0
$507$	−4.97980			−4.97980
$508$	0.618034	−	1.90211i	0.618034	−	1.90211i
$509$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$509$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$510$		0			0
$511$		0			0
$512$	0.809017	−	0.587785i	0.809017	−	0.587785i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	0.690983	−	2.12663i	0.690983	−	2.12663i
$520$	1.80902	+	0.587785i	1.80902	+	0.587785i
$521$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$521$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$522$		0			0
$523$	−1.53884	−	1.11803i	−1.53884	−	1.11803i	−0.951057	−	0.309017i	$-0.900000\pi$
$523$	−1.53884	−	1.11803i	−1.53884	−	1.11803i	−0.587785	−	0.809017i	$-0.700000\pi$
$524$	−1.17557			−1.17557
$525$	1.53884	−	1.11803i	1.53884	−	1.11803i
$526$	0.618034			0.618034
$527$		0			0
$528$		0			0
$529$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$530$		0			0
$531$	0.951057	−	2.92705i	0.951057	−	2.92705i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	0.951057	−	0.690983i	0.951057	−	0.690983i
$539$		0			0
$540$	−1.80902	−	2.48990i	−1.80902	−	2.48990i
$541$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$541$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$542$		0			0
$543$	−2.23607			−2.23607
$544$		0			0
$545$		0			0
$546$	−1.11803	−	3.44095i	−1.11803	−	3.44095i
$547$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$547$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$548$	−1.61803	−	1.17557i	−1.61803	−	1.17557i
$549$	−3.07768			−3.07768
$550$		0			0
$551$		0			0
$552$	−0.951057	−	0.690983i	−0.951057	−	0.690983i
$553$	0.190983	+	0.587785i	0.190983	+	0.587785i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$			1.00000i			1.00000i
$561$		0			0
$562$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$563$	−1.53884	+	1.11803i	−1.53884	+	1.11803i	−0.587785	+	0.809017i	$0.700000\pi$
$563$	−1.53884	+	1.11803i	−1.53884	+	1.11803i	−0.951057	+	0.309017i	$0.900000\pi$
$564$		0			0
$565$		−	1.61803i		−	1.61803i
$566$	−1.53884	+	1.11803i	−1.53884	+	1.11803i
$567$	−1.00000	+	3.07768i	−1.00000	+	3.07768i
$568$	−1.61803			−1.61803
$569$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	0.809017	+	0.587785i	$0.200000\pi$
$569$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	−1.00000			$\pi$
$570$		0			0
$571$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$571$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$572$		0			0
$573$	−0.951057	−	0.690983i	−0.951057	−	0.690983i
$574$		0			0
$575$	−0.190983	+	0.587785i	−0.190983	+	0.587785i
$576$	2.61803			2.61803
$577$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$577$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$578$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$579$	−0.951057	−	2.92705i	−0.951057	−	2.92705i
$580$		0			0
$581$	−0.363271	+	1.11803i	−0.363271	+	1.11803i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	2.92705	+	4.02874i	2.92705	+	4.02874i
$586$		0			0
$587$	0.951057	−	0.690983i	0.951057	−	0.690983i		−	1.00000i	$-0.5\pi$
$587$	0.951057	−	0.690983i	0.951057	−	0.690983i	0.951057	+	0.309017i	$0.100000\pi$
$588$	1.53884	−	1.11803i	1.53884	−	1.11803i
$589$		0			0
$590$	0.690983	−	0.951057i	0.690983	−	0.951057i
$591$		0			0
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	0.951057	+	0.690983i	0.951057	+	0.690983i
$599$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$599$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$600$	−0.587785	−	1.80902i	−0.587785	−	1.80902i
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$	0.190983	+	0.587785i	0.190983	+	0.587785i
$605$	−0.951057	−	0.309017i	−0.951057	−	0.309017i
$606$	−1.11803	+	3.44095i	−1.11803	+	3.44095i
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$	−1.11803	−	0.363271i	−1.11803	−	0.363271i
$611$		0			0
$612$		0			0
$613$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$613$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	0.809017	+	0.587785i	$0.200000\pi$
$617$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	−1.00000			$\pi$
$618$		0			0
$619$	0.363271	−	1.11803i	0.363271	−	1.11803i	−0.587785	−	0.809017i	$-0.700000\pi$
$619$	0.363271	−	1.11803i	0.363271	−	1.11803i	0.951057	−	0.309017i	$-0.100000\pi$
$620$		0			0
$621$	−0.587785	−	1.80902i	−0.587785	−	1.80902i
$622$		0			0
$623$		0			0
$624$	−3.61803			−3.61803
$625$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$626$		0			0
$627$		0			0
$628$	0.363271	+	1.11803i	0.363271	+	1.11803i
$629$		0			0
$630$	−1.53884	+	2.11803i	−1.53884	+	2.11803i
$631$	0.500000	−	1.53884i	0.500000	−	1.53884i	−0.309017	−	0.951057i	$-0.600000\pi$
$631$	0.500000	−	1.53884i	0.500000	−	1.53884i	0.809017	−	0.587785i	$-0.200000\pi$
$632$	0.618034			0.618034
$633$		0			0
$634$		0			0
$635$	1.90211	−	0.618034i	1.90211	−	0.618034i
$636$		0			0
$637$	−1.53884	+	1.11803i	−1.53884	+	1.11803i
$638$		0			0
$639$	−3.42705	−	2.48990i	−3.42705	−	2.48990i
$640$	0.951057	+	0.309017i	0.951057	+	0.309017i
$641$	0.500000	−	0.363271i	0.500000	−	0.363271i	−0.309017	−	0.951057i	$-0.600000\pi$
$641$	0.500000	−	0.363271i	0.500000	−	0.363271i	0.809017	+	0.587785i	$0.200000\pi$
$642$		0			0
$643$	1.17557			1.17557			0.587785	−	0.809017i	$-0.300000\pi$
$643$	1.17557			1.17557			0.587785	+	0.809017i	$0.300000\pi$
$644$	−0.190983	+	0.587785i	−0.190983	+	0.587785i
$645$		0			0
$646$		0			0
$647$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$647$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$648$	2.61803	+	1.90211i	2.61803	+	1.90211i
$649$		0			0
$650$	0.587785	+	1.80902i	0.587785	+	1.80902i
$651$		0			0
$652$		0			0
$653$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$653$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$654$		0			0
$655$	−0.690983	−	0.951057i	−0.690983	−	0.951057i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$659$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$660$		0			0
$661$	0.951057	+	0.690983i	0.951057	+	0.690983i	0.951057	−	0.309017i	$-0.100000\pi$
$661$	0.951057	+	0.690983i	0.951057	+	0.690983i			1.00000i	$0.5\pi$
$662$		0			0
$663$		0			0
$664$	0.951057	+	0.690983i	0.951057	+	0.690983i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$	−0.587785	−	1.80902i	−0.587785	−	1.80902i
$673$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$673$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$674$	0.618034			0.618034
$675$	0.951057	−	2.92705i	0.951057	−	2.92705i
$676$	2.61803			2.61803
$677$	−0.951057	−	0.690983i	−0.951057	−	0.690983i		−	1.00000i	$-0.5\pi$
$677$	−0.951057	−	0.690983i	−0.951057	−	0.690983i	−0.951057	+	0.309017i	$0.900000\pi$
$678$	0.951057	+	2.92705i	0.951057	+	2.92705i
$679$		0			0
$680$		0			0
$681$	1.11803	−	3.44095i	1.11803	−	3.44095i
$682$		0			0
$683$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$683$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$684$		0			0
$685$		−	2.00000i		−	2.00000i
$686$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$687$	1.80902	−	1.31433i	1.80902	−	1.31433i
$688$		0			0
$689$		0			0
$690$		−	1.17557i		−	1.17557i
$691$	1.53884	−	1.11803i	1.53884	−	1.11803i	0.587785	−	0.809017i	$-0.300000\pi$
$691$	1.53884	−	1.11803i	1.53884	−	1.11803i	0.951057	−	0.309017i	$-0.100000\pi$
$692$	−0.363271	+	1.11803i	−0.363271	+	1.11803i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$	−1.53884	−	1.11803i	−1.53884	−	1.11803i
$699$	3.07768			3.07768
$700$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$	−4.73607	−	3.44095i	−4.73607	−	3.44095i
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.90211			1.90211
$708$	−0.690983	+	2.12663i	−0.690983	+	2.12663i
$709$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$709$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$710$	−0.951057	−	1.30902i	−0.951057	−	1.30902i
$711$	1.30902	+	0.951057i	1.30902	+	0.951057i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−0.363271	+	1.11803i	−0.363271	+	1.11803i
$718$	−0.618034			−0.618034
$719$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$719$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$720$	1.53884	+	2.11803i	1.53884	+	2.11803i
$721$		0			0
$722$	0.309017	+	0.951057i	0.309017	+	0.951057i
$723$		0			0
$724$	1.17557			1.17557
$725$		0			0
$726$	1.90211			1.90211
$727$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$727$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$728$	0.587785	+	1.80902i	0.587785	+	1.80902i
$729$	0.809017	+	2.48990i	0.809017	+	2.48990i
$730$		0			0
$731$		0			0
$732$	2.23607			2.23607
$733$	−0.363271	+	1.11803i	−0.363271	+	1.11803i	0.587785	+	0.809017i	$0.300000\pi$
$733$	−0.363271	+	1.11803i	−0.363271	+	1.11803i	−0.951057	+	0.309017i	$0.900000\pi$
$734$		0			0
$735$	1.80902	+	0.587785i	1.80902	+	0.587785i
$736$	0.500000	+	0.363271i	0.500000	+	0.363271i
$737$		0			0
$738$		0			0
$739$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$739$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$743$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	0.951057	+	2.92705i	0.951057	+	2.92705i
$748$		0			0
$749$		0			0
$750$	1.11803	−	1.53884i	1.11803	−	1.53884i
$751$	2.00000			2.00000			1.00000			$0$
$751$	2.00000			2.00000			1.00000			$0$
$752$		0			0
$753$	0.690983	+	2.12663i	0.690983	+	2.12663i
$754$		0			0
$755$	−0.363271	+	0.500000i	−0.363271	+	0.500000i
$756$	0.951057	−	2.92705i	0.951057	−	2.92705i
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$761$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$762$	−3.07768	+	2.23607i	−3.07768	+	2.23607i
$763$		0			0
$764$	0.500000	+	0.363271i	0.500000	+	0.363271i
$765$		0			0
$766$		0			0
$767$	0.690983	−	2.12663i	0.690983	−	2.12663i
$768$	−1.90211			−1.90211
$769$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$769$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$770$		0			0
$771$		0			0
$772$	0.500000	+	1.53884i	0.500000	+	1.53884i
$773$	1.53884	+	1.11803i	1.53884	+	1.11803i	0.951057	+	0.309017i	$0.100000\pi$
$773$	1.53884	+	1.11803i	1.53884	+	1.11803i	0.587785	+	0.809017i	$0.300000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$	−2.12663	−	2.92705i	−2.12663	−	2.92705i
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$785$	−0.690983	+	0.951057i	−0.690983	+	0.951057i
$786$	1.80902	+	1.31433i	1.80902	+	1.31433i
$787$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$787$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$788$		0			0
$789$	−0.951057	−	0.690983i	−0.951057	−	0.690983i
$790$	0.363271	+	0.500000i	0.363271	+	0.500000i
$791$	1.30902	−	0.951057i	1.30902	−	0.951057i
$792$		0			0
$793$	−2.23607			−2.23607
$794$	−0.587785	+	1.80902i	−0.587785	+	1.80902i
$795$		0			0
$796$		0			0
$797$	−0.363271	−	1.11803i	−0.363271	−	1.11803i	−0.951057	−	0.309017i	$-0.900000\pi$
$797$	−0.363271	−	1.11803i	−0.363271	−	1.11803i	0.587785	−	0.809017i	$-0.300000\pi$
$798$

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:	\( N \)	\(=\)	\( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1400.bs (of order \(10\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(0.809017 + 0.587785i) q^{2} +(-0.587785 - 1.80902i) q^{3} +(0.309017 + 0.951057i) q^{4} +(-0.587785 + 0.809017i) q^{5} +(0.587785 - 1.80902i) q^{6} -1.00000 q^{7} +(-0.309017 + 0.951057i) q^{8} +(-2.11803 + 1.53884i) q^{9} +O(q^{10})\)	\(q+(0.809017 + 0.587785i) q^{2} +(-0.587785 - 1.80902i) q^{3} +(0.309017 + 0.951057i) q^{4} +(-0.587785 + 0.809017i) q^{5} +(0.587785 - 1.80902i) q^{6} -1.00000 q^{7} +(-0.309017 + 0.951057i) q^{8} +(-2.11803 + 1.53884i) q^{9} +(-0.951057 + 0.309017i) q^{10} +(1.53884 - 1.11803i) q^{12} +(-1.53884 + 1.11803i) q^{13} +(-0.809017 - 0.587785i) q^{14} +(1.80902 + 0.587785i) q^{15} +(-0.809017 + 0.587785i) q^{16} -2.61803 q^{18} +(-0.951057 - 0.309017i) q^{20} +(0.587785 + 1.80902i) q^{21} +(-0.500000 - 0.363271i) q^{23} +1.90211 q^{24} +(-0.309017 - 0.951057i) q^{25} -1.90211 q^{26} +(2.48990 + 1.80902i) q^{27} +(-0.309017 - 0.951057i) q^{28} +(1.11803 + 1.53884i) q^{30} -1.00000 q^{32} +(0.587785 - 0.809017i) q^{35} +(-2.11803 - 1.53884i) q^{36} +(2.92705 + 2.12663i) q^{39} +(-0.587785 - 0.809017i) q^{40} +(-0.587785 + 1.80902i) q^{42} -2.61803i q^{45} +(-0.190983 - 0.587785i) q^{46} +(1.53884 + 1.11803i) q^{48} +1.00000 q^{49} +(0.309017 - 0.951057i) q^{50} +(-1.53884 - 1.11803i) q^{52} +(0.951057 + 2.92705i) q^{54} +(0.309017 - 0.951057i) q^{56} +(-0.951057 + 0.690983i) q^{59} +1.90211i q^{60} +(0.951057 + 0.690983i) q^{61} +(2.11803 - 1.53884i) q^{63} +(-0.809017 - 0.587785i) q^{64} -1.90211i q^{65} +(-0.363271 + 1.11803i) q^{69} +(0.951057 - 0.309017i) q^{70} +(0.500000 + 1.53884i) q^{71} +(-0.809017 - 2.48990i) q^{72} +(-1.53884 + 1.11803i) q^{75} +(1.11803 + 3.44095i) q^{78} +(-0.190983 - 0.587785i) q^{79} -1.00000i q^{80} +(1.00000 - 3.07768i) q^{81} +(0.363271 - 1.11803i) q^{83} +(-1.53884 + 1.11803i) q^{84} +(1.53884 - 2.11803i) q^{90} +(1.53884 - 1.11803i) q^{91} +(0.190983 - 0.587785i) q^{92} +(0.587785 + 1.80902i) q^{96} +(0.809017 + 0.587785i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8q + 2q^{2} - 2q^{4} - 8q^{7} + 2q^{8} - 8q^{9} + O(q^{10}) \)	\( 8q + 2q^{2} - 2q^{4} - 8q^{7} + 2q^{8} - 8q^{9} - 2q^{14} + 10q^{15} - 2q^{16} - 12q^{18} - 4q^{23} + 2q^{25} + 2q^{28} - 8q^{32} - 8q^{36} + 10q^{39} - 6q^{46} + 8q^{49} - 2q^{50} - 2q^{56} + 8q^{63} - 2q^{64} + 4q^{71} - 2q^{72} - 6q^{79} + 8q^{81} + 6q^{92} + 2q^{98} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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