LMFDB - Embedded newform 429.1.v.c.38.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [429,1,Mod(38,429)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(429, base_ring=CyclotomicField(10))

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

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

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

chi := DirichletCharacter("429.38");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 429 = 3 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	429.v (of order $10$, 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:	$0.214098890420$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{10})$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{10}$
Projective field:	Galois closure of 10.0.1487719872058563.1

Embedding invariants

Embedding label			38.2
Root			$-0.587785 - 0.809017i$ of defining polynomial
Character	$\chi$	$=$	429.38
Dual form			429.1.v.c.350.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.587785 + 1.80902i) q^{2} +(0.809017 + 0.587785i) q^{3} +(-2.11803 + 1.53884i) q^{4} +(0.363271 - 1.11803i) q^{5} +(-0.587785 + 1.80902i) q^{6} +(-2.48990 - 1.80902i) q^{8} +(0.309017 + 0.951057i) q^{9} +O(q^{10})$	\(q+(0.587785 + 1.80902i) q^{2} +(0.809017 + 0.587785i) q^{3} +(-2.11803 + 1.53884i) q^{4} +(0.363271 - 1.11803i) q^{5} +(-0.587785 + 1.80902i) q^{6} +(-2.48990 - 1.80902i) q^{8} +(0.309017 + 0.951057i) q^{9} +2.23607 q^{10} +(-0.951057 - 0.309017i) q^{11} -2.61803 q^{12} +(-0.309017 - 0.951057i) q^{13} +(0.951057 - 0.690983i) q^{15} +(1.00000 - 3.07768i) q^{16} +(-1.53884 + 1.11803i) q^{18} +(0.951057 + 2.92705i) q^{20} -1.90211i q^{22} +(-0.951057 - 2.92705i) q^{24} +(-0.309017 - 0.224514i) q^{25} +(1.53884 - 1.11803i) q^{26} +(-0.309017 + 0.951057i) q^{27} +(1.80902 + 1.31433i) q^{30} +3.07768 q^{32} +(-0.587785 - 0.809017i) q^{33} +(-2.11803 - 1.53884i) q^{36} +(0.309017 - 0.951057i) q^{39} +(-2.92705 + 2.12663i) q^{40} -1.61803 q^{43} +(2.48990 - 0.809017i) q^{44} +1.17557 q^{45} +(1.53884 + 1.11803i) q^{47} +(2.61803 - 1.90211i) q^{48} +(0.309017 - 0.951057i) q^{49} +(0.224514 - 0.690983i) q^{50} +(2.11803 + 1.53884i) q^{52} -1.90211 q^{54} +(-0.690983 + 0.951057i) q^{55} +(-1.53884 + 1.11803i) q^{59} +(-0.951057 + 2.92705i) q^{60} +(-0.190983 + 0.587785i) q^{61} +(0.809017 + 2.48990i) q^{64} -1.17557 q^{65} +(1.11803 - 1.53884i) q^{66} +(0.951057 - 2.92705i) q^{72} +(-0.118034 - 0.363271i) q^{75} +1.90211 q^{78} +(-0.190983 - 0.587785i) q^{79} +(-3.07768 - 2.23607i) q^{80} +(-0.809017 + 0.587785i) q^{81} +(-0.363271 + 1.11803i) q^{83} +(-0.951057 - 2.92705i) q^{86} +(1.80902 + 2.48990i) q^{88} +(0.690983 + 2.12663i) q^{90} +(-1.11803 + 3.44095i) q^{94} +(2.48990 + 1.80902i) q^{96} +1.90211 q^{98} -1.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{3} - 8 q^{4} - 2 q^{9}+O(q^{10}) $ 8 * q + 2 * q^3 - 8 * q^4 - 2 * q^9	$ 8 q + 2 q^{3} - 8 q^{4} - 2 q^{9} - 12 q^{12} + 2 q^{13} + 8 q^{16} + 2 q^{25} + 2 q^{27} + 10 q^{30} - 8 q^{36} - 2 q^{39} - 10 q^{40} - 4 q^{43} + 12 q^{48} - 2 q^{49} + 8 q^{52} - 10 q^{55} - 6 q^{61} + 2 q^{64} + 8 q^{75} - 6 q^{79} - 2 q^{81} + 10 q^{88} + 10 q^{90}+O(q^{100}) $ 8 * q + 2 * q^3 - 8 * q^4 - 2 * q^9 - 12 * q^12 + 2 * q^13 + 8 * q^16 + 2 * q^25 + 2 * q^27 + 10 * q^30 - 8 * q^36 - 2 * q^39 - 10 * q^40 - 4 * q^43 + 12 * q^48 - 2 * q^49 + 8 * q^52 - 10 * q^55 - 6 * q^61 + 2 * q^64 + 8 * q^75 - 6 * q^79 - 2 * q^81 + 10 * q^88 + 10 * q^90

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/429\mathbb{Z}\right)^\times$.

$n$	$67$	$79$	$287$
$\chi(n)$	$-1$	$e\left(\frac{2}{5}\right)$	$-1$

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 $

$2$	0.587785	+	1.80902i	0.587785	+	1.80902i	0.587785	+	0.809017i	$0.300000\pi$
$2$	0.587785	+	1.80902i	0.587785	+	1.80902i			1.00000i	$0.5\pi$
$3$	0.809017	+	0.587785i	0.809017	+	0.587785i
$3$	0.809017	+	0.587785i	0.809017	+	0.587785i
$4$	−2.11803	+	1.53884i	−2.11803	+	1.53884i
$5$	0.363271	−	1.11803i	0.363271	−	1.11803i	−0.587785	−	0.809017i	$-0.700000\pi$
$5$	0.363271	−	1.11803i	0.363271	−	1.11803i	0.951057	−	0.309017i	$-0.100000\pi$
$6$	−0.587785	+	1.80902i	−0.587785	+	1.80902i
$7$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$7$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$8$	−2.48990	−	1.80902i	−2.48990	−	1.80902i
$9$	0.309017	+	0.951057i	0.309017	+	0.951057i
$10$	2.23607			2.23607
$11$	−0.951057	−	0.309017i	−0.951057	−	0.309017i
$11$	−0.951057	−	0.309017i	−0.951057	−	0.309017i
$12$	−2.61803			−2.61803
$13$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$13$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$14$		0			0
$15$	0.951057	−	0.690983i	0.951057	−	0.690983i
$16$	1.00000	−	3.07768i	1.00000	−	3.07768i
$17$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$17$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$18$	−1.53884	+	1.11803i	−1.53884	+	1.11803i
$19$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$19$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$20$	0.951057	+	2.92705i	0.951057	+	2.92705i
$21$		0			0
$22$		−	1.90211i		−	1.90211i
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$	−0.951057	−	2.92705i	−0.951057	−	2.92705i
$25$	−0.309017	−	0.224514i	−0.309017	−	0.224514i
$26$	1.53884	−	1.11803i	1.53884	−	1.11803i
$27$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$28$		0			0
$29$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$29$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$30$	1.80902	+	1.31433i	1.80902	+	1.31433i
$31$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$31$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$32$	3.07768			3.07768
$33$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$34$		0			0
$35$		0			0
$36$	−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
$39$	0.309017	−	0.951057i	0.309017	−	0.951057i
$40$	−2.92705	+	2.12663i	−2.92705	+	2.12663i
$41$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$41$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$42$		0			0
$43$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$43$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$44$	2.48990	−	0.809017i	2.48990	−	0.809017i
$45$	1.17557			1.17557
$46$		0			0
$47$	1.53884	+	1.11803i	1.53884	+	1.11803i	0.951057	+	0.309017i	$0.100000\pi$
$47$	1.53884	+	1.11803i	1.53884	+	1.11803i	0.587785	+	0.809017i	$0.300000\pi$
$48$	2.61803	−	1.90211i	2.61803	−	1.90211i
$49$	0.309017	−	0.951057i	0.309017	−	0.951057i
$50$	0.224514	−	0.690983i	0.224514	−	0.690983i
$51$		0			0
$52$	2.11803	+	1.53884i	2.11803	+	1.53884i
$53$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$53$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$54$	−1.90211			−1.90211
$55$	−0.690983	+	0.951057i	−0.690983	+	0.951057i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.53884	+	1.11803i	−1.53884	+	1.11803i	−0.587785	+	0.809017i	$0.700000\pi$
$59$	−1.53884	+	1.11803i	−1.53884	+	1.11803i	−0.951057	+	0.309017i	$0.900000\pi$
$60$	−0.951057	+	2.92705i	−0.951057	+	2.92705i
$61$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	0.809017	+	0.587785i	$0.200000\pi$
$61$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	−1.00000			$\pi$
$62$		0			0
$63$		0			0
$64$	0.809017	+	2.48990i	0.809017	+	2.48990i
$65$	−1.17557			−1.17557
$66$	1.11803	−	1.53884i	1.11803	−	1.53884i
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$71$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$72$	0.951057	−	2.92705i	0.951057	−	2.92705i
$73$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$73$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$74$		0			0
$75$	−0.118034	−	0.363271i	−0.118034	−	0.363271i
$76$		0			0
$77$		0			0
$78$	1.90211			1.90211
$79$	−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			$\pi$
$80$	−3.07768	−	2.23607i	−3.07768	−	2.23607i
$81$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$82$		0			0
$83$	−0.363271	+	1.11803i	−0.363271	+	1.11803i	0.587785	+	0.809017i	$0.300000\pi$
$83$	−0.363271	+	1.11803i	−0.363271	+	1.11803i	−0.951057	+	0.309017i	$0.900000\pi$
$84$		0			0
$85$		0			0
$86$	−0.951057	−	2.92705i	−0.951057	−	2.92705i
$87$		0			0
$88$	1.80902	+	2.48990i	1.80902	+	2.48990i
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$	0.690983	+	2.12663i	0.690983	+	2.12663i
$91$		0			0
$92$		0			0
$93$		0			0
$94$	−1.11803	+	3.44095i	−1.11803	+	3.44095i
$95$		0			0
$96$	2.48990	+	1.80902i	2.48990	+	1.80902i
$97$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$97$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$98$	1.90211			1.90211
$99$		−	1.00000i		−	1.00000i
$100$	1.00000			1.00000
$101$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$101$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$102$		0			0
$103$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$103$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$104$	−0.951057	+	2.92705i	−0.951057	+	2.92705i
$105$		0			0
$106$		0			0
$107$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$107$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$108$	−0.809017	−	2.48990i	−0.809017	−	2.48990i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$	−2.12663	−	0.690983i	−2.12663	−	0.690983i
$111$		0			0
$112$		0			0
$113$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$113$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	0.809017	−	0.587785i	0.809017	−	0.587785i
$118$	−2.92705	−	2.12663i	−2.92705	−	2.12663i
$119$		0			0
$120$	−3.61803			−3.61803
$121$	0.809017	+	0.587785i	0.809017	+	0.587785i
$122$	−1.17557			−1.17557
$123$		0			0
$124$		0			0
$125$	0.587785	−	0.427051i	0.587785	−	0.427051i
$126$		0			0
$127$	0.500000	−	1.53884i	0.500000	−	1.53884i	−0.309017	−	0.951057i	$-0.600000\pi$
$127$	0.500000	−	1.53884i	0.500000	−	1.53884i	0.809017	−	0.587785i	$-0.200000\pi$
$128$	−1.53884	+	1.11803i	−1.53884	+	1.11803i
$129$	−1.30902	−	0.951057i	−1.30902	−	0.951057i
$130$	−0.690983	−	2.12663i	−0.690983	−	2.12663i
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$	2.48990	+	0.809017i	2.48990	+	0.809017i
$133$		0			0
$134$		0			0
$135$	0.951057	+	0.690983i	0.951057	+	0.690983i
$136$		0			0
$137$	0.587785	−	1.80902i	0.587785	−	1.80902i		−	1.00000i	$-0.5\pi$
$137$	0.587785	−	1.80902i	0.587785	−	1.80902i	0.587785	−	0.809017i	$-0.300000\pi$
$138$		0			0
$139$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$139$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$140$		0			0
$141$	0.587785	+	1.80902i	0.587785	+	1.80902i
$142$		0			0
$143$			1.00000i			1.00000i
$144$	3.23607			3.23607
$145$		0			0
$146$		0			0
$147$	0.809017	−	0.587785i	0.809017	−	0.587785i
$148$		0			0
$149$	0.363271	−	1.11803i	0.363271	−	1.11803i	−0.587785	−	0.809017i	$-0.700000\pi$
$149$	0.363271	−	1.11803i	0.363271	−	1.11803i	0.951057	−	0.309017i	$-0.100000\pi$
$150$	0.587785	−	0.427051i	0.587785	−	0.427051i
$151$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$151$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	0.809017	+	2.48990i	0.809017	+	2.48990i
$157$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$157$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$158$	0.951057	−	0.690983i	0.951057	−	0.690983i
$159$		0			0
$160$	1.11803	−	3.44095i	1.11803	−	3.44095i
$161$		0			0
$162$	−1.53884	−	1.11803i	−1.53884	−	1.11803i
$163$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$163$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$164$		0			0
$165$	−1.11803	+	0.363271i	−1.11803	+	0.363271i
$166$	−2.23607			−2.23607
$167$	0.363271	+	1.11803i	0.363271	+	1.11803i	0.951057	+	0.309017i	$0.100000\pi$
$167$	0.363271	+	1.11803i	0.363271	+	1.11803i	−0.587785	+	0.809017i	$0.700000\pi$
$168$		0			0
$169$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$170$		0			0
$171$		0			0
$172$	3.42705	−	2.48990i	3.42705	−	2.48990i
$173$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$173$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$174$		0			0
$175$		0			0
$176$	−1.90211	+	2.61803i	−1.90211	+	2.61803i
$177$	−1.90211			−1.90211
$178$		0			0
$179$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$179$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$180$	−2.48990	+	1.80902i	−2.48990	+	1.80902i
$181$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$181$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$182$		0			0
$183$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−4.97980			−4.97980
$189$		0			0
$190$		0			0
$191$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$191$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$192$	−0.809017	+	2.48990i	−0.809017	+	2.48990i
$193$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$193$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$194$		0			0
$195$	−0.951057	−	0.690983i	−0.951057	−	0.690983i
$196$	0.809017	+	2.48990i	0.809017	+	2.48990i
$197$	−1.90211			−1.90211			−0.951057	−	0.309017i	$-0.900000\pi$
$197$	−1.90211			−1.90211			−0.951057	+	0.309017i	$0.900000\pi$
$198$	1.80902	−	0.587785i	1.80902	−	0.587785i
$199$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$199$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$200$	0.363271	+	1.11803i	0.363271	+	1.11803i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$	2.48990	+	1.80902i	2.48990	+	1.80902i
$207$		0			0
$208$	−3.23607			−3.23607
$209$		0			0
$210$		0			0
$211$	0.500000	+	1.53884i	0.500000	+	1.53884i	0.809017	+	0.587785i	$0.200000\pi$
$211$	0.500000	+	1.53884i	0.500000	+	1.53884i	−0.309017	+	0.951057i	$0.600000\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.587785	+	1.80902i	−0.587785	+	1.80902i
$216$	2.48990	−	1.80902i	2.48990	−	1.80902i
$217$		0			0
$218$		0			0
$219$		0			0
$220$		−	3.07768i		−	3.07768i
$221$		0			0
$222$		0			0
$223$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$223$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$224$		0			0
$225$	0.118034	−	0.363271i	0.118034	−	0.363271i
$226$		0			0
$227$	1.53884	−	1.11803i	1.53884	−	1.11803i	0.587785	−	0.809017i	$-0.300000\pi$
$227$	1.53884	−	1.11803i	1.53884	−	1.11803i	0.951057	−	0.309017i	$-0.100000\pi$
$228$		0			0
$229$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$229$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$233$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$234$	1.53884	+	1.11803i	1.53884	+	1.11803i
$235$	1.80902	−	1.31433i	1.80902	−	1.31433i
$236$	1.53884	−	4.73607i	1.53884	−	4.73607i
$237$	0.190983	−	0.587785i	0.190983	−	0.587785i
$238$		0			0
$239$	0.951057	+	0.690983i	0.951057	+	0.690983i	0.951057	−	0.309017i	$-0.100000\pi$
$239$	0.951057	+	0.690983i	0.951057	+	0.690983i			1.00000i	$0.5\pi$
$240$	−1.17557	−	3.61803i	−1.17557	−	3.61803i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	−0.587785	+	1.80902i	−0.587785	+	1.80902i
$243$	−1.00000			−1.00000
$244$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$245$	−0.951057	−	0.690983i	−0.951057	−	0.690983i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−0.951057	+	0.690983i	−0.951057	+	0.690983i
$250$	1.11803	+	0.812299i	1.11803	+	0.812299i
$251$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$251$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$252$		0			0
$253$		0			0
$254$	3.07768			3.07768
$255$		0			0
$256$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$257$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$257$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$258$	0.951057	−	2.92705i	0.951057	−	2.92705i
$259$		0			0
$260$	2.48990	−	1.80902i	2.48990	−	1.80902i
$261$		0			0
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$			3.07768i			3.07768i
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$269$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$270$	−0.690983	+	2.12663i	−0.690983	+	2.12663i
$271$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$271$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$272$		0			0
$273$		0			0
$274$	3.61803			3.61803
$275$	0.224514	+	0.309017i	0.224514	+	0.309017i
$276$		0			0
$277$	0.618034	+	1.90211i	0.618034	+	1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$277$	0.618034	+	1.90211i	0.618034	+	1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$278$	−0.951057	−	0.690983i	−0.951057	−	0.690983i
$279$		0			0
$280$		0			0
$281$	−0.587785	+	1.80902i	−0.587785	+	1.80902i			1.00000i	$0.5\pi$
$281$	−0.587785	+	1.80902i	−0.587785	+	1.80902i	−0.587785	+	0.809017i	$0.700000\pi$
$282$	−2.92705	+	2.12663i	−2.92705	+	2.12663i
$283$	0.500000	+	0.363271i	0.500000	+	0.363271i	0.809017	−	0.587785i	$-0.200000\pi$
$283$	0.500000	+	0.363271i	0.500000	+	0.363271i	−0.309017	+	0.951057i	$0.600000\pi$
$284$		0			0
$285$		0			0
$286$	−1.80902	+	0.587785i	−1.80902	+	0.587785i
$287$		0			0
$288$	0.951057	+	2.92705i	0.951057	+	2.92705i
$289$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	0.951057	−	0.690983i	0.951057	−	0.690983i		−	1.00000i	$-0.5\pi$
$293$	0.951057	−	0.690983i	0.951057	−	0.690983i	0.951057	+	0.309017i	$0.100000\pi$
$294$	1.53884	+	1.11803i	1.53884	+	1.11803i
$295$	0.690983	+	2.12663i	0.690983	+	2.12663i
$296$		0			0
$297$	0.587785	−	0.809017i	0.587785	−	0.809017i
$298$	2.23607			2.23607
$299$		0			0
$300$	0.809017	+	0.587785i	0.809017	+	0.587785i
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	0.587785	+	0.427051i	0.587785	+	0.427051i
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$	1.61803			1.61803
$310$		0			0
$311$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$311$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$312$	−2.48990	+	1.80902i	−2.48990	+	1.80902i
$313$	0.500000	−	1.53884i	0.500000	−	1.53884i	−0.309017	−	0.951057i	$-0.600000\pi$
$313$	0.500000	−	1.53884i	0.500000	−	1.53884i	0.809017	−	0.587785i	$-0.200000\pi$
$314$	0.363271	−	1.11803i	0.363271	−	1.11803i
$315$		0			0
$316$	1.30902	+	0.951057i	1.30902	+	0.951057i
$317$	−0.363271	−	1.11803i	−0.363271	−	1.11803i	−0.951057	−	0.309017i	$-0.900000\pi$
$317$	−0.363271	−	1.11803i	−0.363271	−	1.11803i	0.587785	−	0.809017i	$-0.300000\pi$
$318$		0			0
$319$		0			0
$320$	3.07768			3.07768
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.809017	−	2.48990i	0.809017	−	2.48990i
$325$	−0.118034	+	0.363271i	−0.118034	+	0.363271i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$	−1.31433	−	1.80902i	−1.31433	−	1.80902i
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$	−0.951057	−	2.92705i	−0.951057	−	2.92705i
$333$		0			0
$334$	−1.80902	+	1.31433i	−1.80902	+	1.31433i
$335$		0			0
$336$		0			0
$337$	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	$0.200000\pi$
$338$	−1.53884	−	1.11803i	−1.53884	−	1.11803i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	4.02874	+	2.92705i	4.02874	+	2.92705i
$345$		0			0
$346$		0			0
$347$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$347$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$348$		0			0
$349$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$349$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$350$		0			0
$351$	1.00000			1.00000
$352$	−2.92705	−	0.951057i	−2.92705	−	0.951057i
$353$	−1.17557			−1.17557			−0.587785	−	0.809017i	$-0.700000\pi$
$353$	−1.17557			−1.17557			−0.587785	+	0.809017i	$0.700000\pi$
$354$	−1.11803	−	3.44095i	−1.11803	−	3.44095i
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$359$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$360$	−2.92705	−	2.12663i	−2.92705	−	2.12663i
$361$	0.309017	+	0.951057i	0.309017	+	0.951057i
$362$	−3.07768			−3.07768
$363$	0.309017	+	0.951057i	0.309017	+	0.951057i
$364$		0			0
$365$		0			0
$366$	−0.951057	−	0.690983i	−0.951057	−	0.690983i
$367$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$367$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$373$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$374$		0			0
$375$	0.726543			0.726543
$376$	−1.80902	−	5.56758i	−1.80902	−	5.56758i
$377$		0			0
$378$		0			0
$379$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$379$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$380$		0			0
$381$	1.30902	−	0.951057i	1.30902	−	0.951057i
$382$		0			0
$383$	0.587785	+	1.80902i	0.587785	+	1.80902i	0.587785	+	0.809017i	$0.300000\pi$
$383$	0.587785	+	1.80902i	0.587785	+	1.80902i			1.00000i	$0.5\pi$
$384$	−1.90211			−1.90211
$385$		0			0
$386$		0			0
$387$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$388$		0			0
$389$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$389$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$390$	0.690983	−	2.12663i	0.690983	−	2.12663i
$391$		0			0
$392$	−2.48990	+	1.80902i	−2.48990	+	1.80902i
$393$		0			0
$394$	−1.11803	−	3.44095i	−1.11803	−	3.44095i
$395$	−0.726543			−0.726543
$396$	1.53884	+	2.11803i	1.53884	+	2.11803i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$	−0.363271	−	1.11803i	−0.363271	−	1.11803i
$399$		0			0
$400$	−1.00000	+	0.726543i	−1.00000	+	0.726543i
$401$	−0.363271	+	1.11803i	−0.363271	+	1.11803i	0.587785	+	0.809017i	$0.300000\pi$
$401$	−0.363271	+	1.11803i	−0.363271	+	1.11803i	−0.951057	+	0.309017i	$0.900000\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	0.363271	+	1.11803i	0.363271	+	1.11803i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$409$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$410$		0			0
$411$	1.53884	−	1.11803i	1.53884	−	1.11803i
$412$	−1.30902	+	4.02874i	−1.30902	+	4.02874i
$413$		0			0
$414$		0			0
$415$	1.11803	+	0.812299i	1.11803	+	0.812299i
$416$	−0.951057	−	2.92705i	−0.951057	−	2.92705i
$417$	−0.618034			−0.618034
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$421$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$422$	−2.48990	+	1.80902i	−2.48990	+	1.80902i
$423$	−0.587785	+	1.80902i	−0.587785	+	1.80902i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−0.587785	+	0.809017i	−0.587785	+	0.809017i
$430$	−3.61803			−3.61803
$431$	−0.587785	−	1.80902i	−0.587785	−	1.80902i	−0.587785	−	0.809017i	$-0.700000\pi$
$431$	−0.587785	−	1.80902i	−0.587785	−	1.80902i		−	1.00000i	$-0.5\pi$
$432$	2.61803	+	1.90211i	2.61803	+	1.90211i
$433$	−1.30902	+	0.951057i	−1.30902	+	0.951057i	−0.309017	+	0.951057i	$0.600000\pi$
$433$	−1.30902	+	0.951057i	−1.30902	+	0.951057i	−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$439$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$440$	3.44095	−	1.11803i	3.44095	−	1.11803i
$441$	1.00000			1.00000
$442$		0			0
$443$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$443$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	0.951057	−	0.690983i	0.951057	−	0.690983i
$448$		0			0
$449$	−0.587785	−	1.80902i	−0.587785	−	1.80902i	−0.587785	−	0.809017i	$-0.700000\pi$
$449$	−0.587785	−	1.80902i	−0.587785	−	1.80902i		−	1.00000i	$-0.5\pi$
$450$	0.726543			0.726543
$451$		0			0
$452$		0			0
$453$		0			0
$454$	2.92705	+	2.12663i	2.92705	+	2.12663i
$455$		0			0
$456$		0			0
$457$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$457$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$458$		0			0
$459$		0			0
$460$		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		0.309017	−	0.951057i	$-0.400000\pi$
$467$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$468$	−0.809017	+	2.48990i	−0.809017	+	2.48990i
$469$		0			0
$470$	3.44095	+	2.50000i	3.44095	+	2.50000i
$471$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$472$	5.85410			5.85410
$473$	1.53884	+	0.500000i	1.53884	+	0.500000i
$474$	1.17557			1.17557
$475$		0			0
$476$		0			0
$477$		0			0
$478$	−0.690983	+	2.12663i	−0.690983	+	2.12663i
$479$	0.363271	−	1.11803i	0.363271	−	1.11803i	−0.587785	−	0.809017i	$-0.700000\pi$
$479$	0.363271	−	1.11803i	0.363271	−	1.11803i	0.951057	−	0.309017i	$-0.100000\pi$
$480$	2.92705	−	2.12663i	2.92705	−	2.12663i
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−2.61803			−2.61803
$485$		0			0
$486$	−0.587785	−	1.80902i	−0.587785	−	1.80902i
$487$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$487$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$488$	1.53884	−	1.11803i	1.53884	−	1.11803i
$489$		0			0
$490$	0.690983	−	2.12663i	0.690983	−	2.12663i
$491$		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
$495$	−1.11803	−	0.363271i	−1.11803	−	0.363271i
$496$		0			0
$497$		0			0
$498$	−1.80902	−	1.31433i	−1.80902	−	1.31433i
$499$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$499$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$500$	−0.587785	+	1.80902i	−0.587785	+	1.80902i
$501$	−0.363271	+	1.11803i	−0.363271	+	1.11803i
$502$		0			0
$503$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$503$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−1.00000			−1.00000
$508$	1.30902	+	4.02874i	1.30902	+	4.02874i
$509$	0.951057	+	0.690983i	0.951057	+	0.690983i	0.951057	−	0.309017i	$-0.100000\pi$
$509$	0.951057	+	0.690983i	0.951057	+	0.690983i			1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−0.587785	−	1.80902i	−0.587785	−	1.80902i
$516$	4.23607			4.23607
$517$	−1.11803	−	1.53884i	−1.11803	−	1.53884i
$518$		0			0
$519$		0			0
$520$	2.92705	+	2.12663i	2.92705	+	2.12663i
$521$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$521$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$522$		0			0
$523$	0.618034	−	1.90211i	0.618034	−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$523$	0.618034	−	1.90211i	0.618034	−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−3.07768	+	1.00000i	−3.07768	+	1.00000i
$529$	1.00000			1.00000
$530$		0			0
$531$	−1.53884	−	1.11803i	−1.53884	−	1.11803i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−0.587785	+	0.809017i	−0.587785	+	0.809017i
$540$	−3.07768			−3.07768
$541$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$541$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$542$		0			0
$543$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$547$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$548$	1.53884	+	4.73607i	1.53884	+	4.73607i
$549$	−0.618034			−0.618034
$550$	−0.427051	+	0.587785i	−0.427051	+	0.587785i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−3.07768	+	2.23607i	−3.07768	+	2.23607i
$555$		0			0
$556$	0.500000	−	1.53884i	0.500000	−	1.53884i
$557$	1.53884	−	1.11803i	1.53884	−	1.11803i	0.587785	−	0.809017i	$-0.300000\pi$
$557$	1.53884	−	1.11803i	1.53884	−	1.11803i	0.951057	−	0.309017i	$-0.100000\pi$
$558$		0			0
$559$	0.500000	+	1.53884i	0.500000	+	1.53884i
$560$		0			0
$561$		0			0
$562$	−3.61803			−3.61803
$563$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$563$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$564$	−4.02874	−	2.92705i	−4.02874	−	2.92705i
$565$		0			0
$566$	−0.363271	+	1.11803i	−0.363271	+	1.11803i
$567$		0			0
$568$		0			0
$569$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$569$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$570$		0			0
$571$	−2.00000			−2.00000			−1.00000			$\pi$
$571$	−2.00000			−2.00000			−1.00000			$\pi$
$572$	−1.53884	−	2.11803i	−1.53884	−	2.11803i
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−2.11803	+	1.53884i	−2.11803	+	1.53884i
$577$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$577$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$578$	0.587785	−	1.80902i	0.587785	−	1.80902i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−0.363271	−	1.11803i	−0.363271	−	1.11803i
$586$	1.80902	+	1.31433i	1.80902	+	1.31433i
$587$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$587$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$588$	−0.809017	+	2.48990i	−0.809017	+	2.48990i
$589$		0			0
$590$	−3.44095	+	2.50000i	−3.44095	+	2.50000i
$591$	−1.53884	−	1.11803i	−1.53884	−	1.11803i
$592$		0			0
$593$	1.90211			1.90211			0.951057	−	0.309017i	$-0.100000\pi$
$593$	1.90211			1.90211			0.951057	+	0.309017i	$0.100000\pi$
$594$	1.80902	+	0.587785i	1.80902	+	0.587785i
$595$		0			0
$596$	0.951057	+	2.92705i	0.951057	+	2.92705i
$597$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$598$		0			0
$599$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$599$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$600$	−0.363271	+	1.11803i	−0.363271	+	1.11803i
$601$	−1.30902	+	0.951057i	−1.30902	+	0.951057i	−0.309017	+	0.951057i	$0.600000\pi$
$601$	−1.30902	+	0.951057i	−1.30902	+	0.951057i	−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.951057	−	0.690983i	0.951057	−	0.690983i
$606$		0			0
$607$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$607$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$608$		0			0
$609$		0			0
$610$	−0.427051	+	1.31433i	−0.427051	+	1.31433i
$611$	0.587785	−	1.80902i	0.587785	−	1.80902i
$612$		0			0
$613$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$613$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.90211			1.90211			0.951057	−	0.309017i	$-0.100000\pi$
$617$	1.90211			1.90211			0.951057	+	0.309017i	$0.100000\pi$
$618$	0.951057	+	2.92705i	0.951057	+	2.92705i
$619$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$619$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−2.61803	−	1.90211i	−2.61803	−	1.90211i
$625$	−0.381966	−	1.17557i	−0.381966	−	1.17557i
$626$	3.07768			3.07768
$627$		0			0
$628$	1.61803			1.61803
$629$		0			0
$630$		0			0
$631$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$631$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$632$	−0.587785	+	1.80902i	−0.587785	+	1.80902i
$633$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$634$	1.80902	−	1.31433i	1.80902	−	1.31433i
$635$	−1.53884	−	1.11803i	−1.53884	−	1.11803i
$636$		0			0
$637$	−1.00000			−1.00000
$638$		0			0
$639$		0			0
$640$	0.690983	+	2.12663i	0.690983	+	2.12663i
$641$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$641$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$642$		0			0
$643$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$643$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$644$		0			0
$645$	−1.53884	+	1.11803i	−1.53884	+	1.11803i
$646$		0			0
$647$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$647$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$648$	3.07768			3.07768
$649$	1.80902	−	0.587785i	1.80902	−	0.587785i
$650$	−0.726543			−0.726543
$651$		0			0
$652$		0			0
$653$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$653$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$	1.80902	−	2.48990i	1.80902	−	2.48990i
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$	2.92705	−	2.12663i	2.92705	−	2.12663i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−2.48990	−	1.80902i	−2.48990	−	1.80902i
$669$		0			0
$670$		0			0
$671$	0.363271	−	0.500000i	0.363271	−	0.500000i
$672$		0			0
$673$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$673$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$674$	0.951057	+	0.690983i	0.951057	+	0.690983i
$675$	0.309017	−	0.224514i	0.309017	−	0.224514i
$676$	0.809017	−	2.48990i	0.809017	−	2.48990i
$677$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$677$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	1.90211			1.90211
$682$		0			0
$683$	−1.90211			−1.90211			−0.951057	−	0.309017i	$-0.900000\pi$
$683$	−1.90211			−1.90211			−0.951057	+	0.309017i	$0.900000\pi$
$684$		0			0
$685$	−1.80902	−	1.31433i	−1.80902	−	1.31433i
$686$		0			0
$687$		0			0
$688$	−1.61803	+	4.97980i	−1.61803	+	4.97980i
$689$		0			0
$690$		0			0
$691$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$691$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	0.224514	+	0.690983i	0.224514	+	0.690983i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$701$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$702$	0.587785	+	1.80902i	0.587785	+	1.80902i
$703$		0			0
$704$		−	2.61803i		−	2.61803i
$705$	2.23607			2.23607
$706$	−0.690983	−	2.12663i	−0.690983	−	2.12663i
$707$		0			0
$708$	4.02874	−	2.92705i	4.02874	−	2.92705i
$709$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$709$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$710$		0			0
$711$	0.500000	−	0.363271i	0.500000	−	0.363271i
$712$		0			0
$713$		0			0
$714$		0			0
$715$	1.11803	+	0.363271i	1.11803	+	0.363271i
$716$		0			0
$717$	0.363271	+	1.11803i	0.363271	+	1.11803i
$718$		0			0
$719$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$719$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$720$	1.17557	−	3.61803i	1.17557	−	3.61803i
$721$		0			0
$722$	−1.53884	+	1.11803i	−1.53884	+	1.11803i
$723$		0			0
$724$	−1.30902	−	4.02874i	−1.30902	−	4.02874i
$725$		0			0
$726$	−1.53884	+	1.11803i	−1.53884	+	1.11803i
$727$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$727$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$728$		0			0
$729$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$730$		0			0
$731$		0			0
$732$	0.500000	−	1.53884i	0.500000	−	1.53884i
$733$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$733$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$734$	−0.951057	−	0.690983i	−0.951057	−	0.690983i
$735$	−0.363271	−	1.11803i	−0.363271	−	1.11803i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$739$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	0.587785	−	1.80902i	0.587785	−	1.80902i		−	1.00000i	$-0.5\pi$
$743$	0.587785	−	1.80902i	0.587785	−	1.80902i	0.587785	−	0.809017i	$-0.300000\pi$
$744$		0			0
$745$	−1.11803	−	0.812299i	−1.11803	−	0.812299i
$746$	0.951057	+	2.92705i	0.951057	+	2.92705i
$747$	−1.17557			−1.17557
$748$		0			0
$749$		0			0
$750$	0.427051	+	1.31433i	0.427051	+	1.31433i
$751$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$751$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$752$	4.97980	−	3.61803i	4.97980	−	3.61803i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−0.190983	−	0.587785i	−0.190983	−	0.587785i	0.809017	−	0.587785i	$-0.200000\pi$
$757$	−0.190983	−	0.587785i	−0.190983	−	0.587785i	−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.587785	+	1.80902i	0.587785	+	1.80902i	0.587785	+	0.809017i	$0.300000\pi$
$761$	0.587785	+	1.80902i	0.587785	+	1.80902i			1.00000i	$0.5\pi$
$762$	2.48990	+	1.80902i	2.48990	+	1.80902i
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−2.92705	+	2.12663i	−2.92705	+	2.12663i
$767$	1.53884	+	1.11803i	1.53884	+	1.11803i
$768$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1.53884	−	1.11803i	−1.53884	−	1.11803i	−0.951057	−	0.309017i	$-0.900000\pi$
$773$	−1.53884	−	1.11803i	−1.53884	−	1.11803i	−0.587785	−	0.809017i	$-0.700000\pi$
$774$	2.48990	−	1.80902i	2.48990	−	1.80902i
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$	3.07768			3.07768
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−2.61803	−	1.90211i	−2.61803	−	1.90211i
$785$	−0.587785	+	0.427051i	−0.587785	+	0.427051i
$786$		0			0
$787$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$787$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$788$	4.02874	−	2.92705i	4.02874	−	2.92705i
$789$		0			0
$790$	−0.427051	−	1.31433i	−0.427051	−	1.31433i
$791$		0			0
$792$	−1.80902	+	2.48990i	−1.80902	+	2.48990i
$793$	0.618034			0.618034
$794$		0			0
$795$		0			0
$796$	1.30902	−	0.951057i	1.30902	−	0.951057i
$797$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$797$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$798$		0			0
$799$		0			0
$800$	−0.951057	−	0.690983i	−0.951057	−	0.690983i
$801$		0			0
$802$	−2.23607			−2.23607
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$809$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$810$	−1.80902	+	1.31433i	−1.80902	+	1.31433i
$811$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$811$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.53884	+	1.11803i	−1.53884	+	1.11803i	−0.587785	+	0.809017i	$0.700000\pi$
$821$	−1.53884	+	1.11803i	−1.53884	+	1.11803i	−0.951057	+	0.309017i	$0.900000\pi$
$822$	2.92705	+	2.12663i	2.92705	+	2.12663i
$823$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$823$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$824$	−4.97980			−4.97980
$825$			0.381966i			0.381966i
$826$		0			0
$827$	−0.587785	−	1.80902i	−0.587785	−	1.80902i	−0.587785	−	0.809017i	$-0.700000\pi$
$827$	−0.587785	−	1.80902i	−0.587785	−	1.80902i		−	1.00000i	$-0.5\pi$
$828$		0			0
$829$	0.500000	−	0.363271i	0.500000	−	0.363271i	−0.309017	−	0.951057i	$-0.600000\pi$
$829$	0.500000	−	0.363271i	0.500000	−	0.363271i	0.809017	+	0.587785i	$0.200000\pi$
$830$	−0.812299	+	2.50000i	−0.812299	+	2.50000i
$831$	−0.618034	+	1.90211i	−0.618034	+	1.90211i
$832$	2.11803	−	1.53884i	2.11803	−	1.53884i
$833$		0			0
$834$	−0.363271	−	1.11803i	−0.363271	−	1.11803i
$835$	1.38197			1.38197
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−1.53884	−	1.11803i	−1.53884	−	1.11803i	−0.951057	−	0.309017i	$-0.900000\pi$
$839$	−1.53884	−	1.11803i	−1.53884	−	1.11803i	−0.587785	−	0.809017i	$-0.700000\pi$
$840$		0			0
$841$	0.309017	−	0.951057i	0.309017	−	0.951057i
$842$		0			0
$843$	−1.53884	+	1.11803i	−1.53884	+	1.11803i
$844$	−3.42705	−	2.48990i	−3.42705	−	2.48990i
$845$	0.363271	+	1.11803i	0.363271	+	1.11803i
$846$	−3.61803			−3.61803
$847$		0			0
$848$		0			0
$849$	0.190983	+	0.587785i	0.190983	+	0.587785i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$853$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$	−1.80902	−	0.587785i	−1.80902	−	0.587785i
$859$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$859$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$860$	−1.53884	−	4.73607i	−1.53884	−	4.73607i
$861$		0			0
$862$	2.92705	−	2.12663i	2.92705	−	2.12663i
$863$	0.587785	−	1.80902i	0.587785	−	1.80902i		−	1.00000i	$-0.5\pi$
$863$	0.587785	−	1.80902i	0.587785	−	1.80902i	0.587785	−	0.809017i	$-0.300000\pi$
$864$	−0.951057	+	2.92705i	−0.951057	+	2.92705i
$865$		0			0
$866$	−2.48990	−	1.80902i	−2.48990	−	1.80902i
$867$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$868$		0			0
$869$			0.618034i			0.618034i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$877$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$878$	−0.951057	−	2.92705i	−0.951057	−	2.92705i
$879$	1.17557			1.17557
$880$	2.23607	+	3.07768i	2.23607	+	3.07768i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	0.587785	+	1.80902i	0.587785	+	1.80902i
$883$	−1.30902	−	0.951057i	−1.30902	−	0.951057i	−0.309017	−	0.951057i	$-0.600000\pi$
$883$	−1.30902	−	0.951057i	−1.30902	−	0.951057i	−1.00000			$\pi$
$884$		0			0
$885$	−0.690983	+	2.12663i	−0.690983	+	2.12663i
$886$		0			0
$887$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$887$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	0.951057	−	0.309017i	0.951057	−	0.309017i
$892$		0			0
$893$		0			0
$894$	1.80902	+	1.31433i	1.80902	+	1.31433i
$895$		0			0
$896$		0			0
$897$		0			0
$898$	2.92705	−	2.12663i	2.92705	−	2.12663i
$899$		0			0
$900$	0.309017	+	0.951057i	0.309017	+	0.951057i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1.53884	+	1.11803i	1.53884	+	1.11803i
$906$		0			0
$907$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	0.809017	+	0.587785i	$0.200000\pi$
$907$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	−1.00000			$\pi$
$908$	−1.53884	+	4.73607i	−1.53884	+	4.73607i
$909$		0			0
$910$		0			0
$911$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$911$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$912$		0			0
$913$	0.690983	−	0.951057i	0.690983	−	0.951057i
$914$		0			0
$915$	0.224514	+	0.690983i	0.224514	+	0.690983i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	0.500000	−	1.53884i	0.500000	−	1.53884i	−0.309017	−	0.951057i	$-0.600000\pi$
$919$	0.500000	−	1.53884i	0.500000	−	1.53884i	0.809017	−	0.587785i	$-0.200000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	1.30902	+	0.951057i	1.30902	+	0.951057i
$928$		0			0
$929$	−0.363271	+	1.11803i	−0.363271	+	1.11803i	0.587785	+	0.809017i	$0.300000\pi$
$929$	−0.363271	+	1.11803i	−0.363271	+	1.11803i	−0.951057	+	0.309017i	$0.900000\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−3.07768			−3.07768
$937$	0.500000	+	1.53884i	0.500000	+	1.53884i	0.809017	+	0.587785i	$0.200000\pi$
$937$	0.500000	+	1.53884i	0.500000	+	1.53884i	−0.309017	+	0.951057i	$0.600000\pi$
$938$		0			0
$939$	1.30902	−	0.951057i	1.30902	−	0.951057i
$940$	−1.80902	+	5.56758i	−1.80902	+	5.56758i
$941$	0.587785	−	1.80902i	0.587785	−	1.80902i		−	1.00000i	$-0.5\pi$
$941$	0.587785	−	1.80902i	0.587785	−	1.80902i	0.587785	−	0.809017i	$-0.300000\pi$
$942$	0.951057	−	0.690983i	0.951057	−	0.690983i
$943$		0			0
$944$	1.90211	+	5.85410i	1.90211	+	5.85410i
$945$		0			0
$946$			3.07768i			3.07768i
$947$	1.17557			1.17557			0.587785	−	0.809017i	$-0.300000\pi$
$947$	1.17557			1.17557			0.587785	+	0.809017i	$0.300000\pi$
$948$	0.500000	+	1.53884i	0.500000	+	1.53884i
$949$		0			0
$950$		0			0
$951$	0.363271	−	1.11803i	0.363271	−	1.11803i
$952$		0			0
$953$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$953$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$954$		0			0
$955$		0			0
$956$	−3.07768			−3.07768
$957$		0			0
$958$	2.23607			2.23607
$959$		0			0
$960$	2.48990	+	1.80902i	2.48990	+	1.80902i
$961$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$	−0.951057	−	2.92705i	−0.951057	−	2.92705i
$969$		0			0
$970$		0			0
$971$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$971$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$972$	2.11803	−	1.53884i	2.11803	−	1.53884i
$973$		0			0
$974$		0			0
$975$	−0.309017	+	0.224514i	−0.309017	+	0.224514i
$976$	1.61803	+	1.17557i	1.61803	+	1.17557i
$977$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$977$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$978$		0			0
$979$		0			0
$980$	3.07768			3.07768
$981$		0			0
$982$		0			0
$983$	−0.951057	+	0.690983i	−0.951057	+	0.690983i	−0.951057	−	0.309017i	$-0.900000\pi$
$983$	−0.951057	+	0.690983i	−0.951057	+	0.690983i			1.00000i	$0.5\pi$
$984$		0			0
$985$	−0.690983	+	2.12663i	−0.690983	+	2.12663i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		−	2.23607i		−	2.23607i
$991$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$991$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−0.224514	+	0.690983i	−0.224514	+	0.690983i
$996$	0.951057	−	2.92705i	0.951057	−	2.92705i
$997$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$997$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.1.v.c.38.2	yes	8
3.2	odd	2	inner	429.1.v.c.38.1	✓	8
11.9	even	5	inner	429.1.v.c.350.2	yes	8
13.12	even	2	inner	429.1.v.c.38.1	✓	8
33.20	odd	10	inner	429.1.v.c.350.1	yes	8
39.38	odd	2	CM	429.1.v.c.38.2	yes	8
143.64	even	10	inner	429.1.v.c.350.1	yes	8
429.350	odd	10	inner	429.1.v.c.350.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.1.v.c.38.1	✓	8	3.2	odd	2	inner
429.1.v.c.38.1	✓	8	13.12	even	2	inner
429.1.v.c.38.2	yes	8	1.1	even	1	trivial
429.1.v.c.38.2	yes	8	39.38	odd	2	CM
429.1.v.c.350.1	yes	8	33.20	odd	10	inner
429.1.v.c.350.1	yes	8	143.64	even	10	inner
429.1.v.c.350.2	yes	8	11.9	even	5	inner
429.1.v.c.350.2	yes	8	429.350	odd	10	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:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	429.v (of order \(10\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(0.587785 + 1.80902i) q^{2} +(0.809017 + 0.587785i) q^{3} +(-2.11803 + 1.53884i) q^{4} +(0.363271 - 1.11803i) q^{5} +(-0.587785 + 1.80902i) q^{6} +(-2.48990 - 1.80902i) q^{8} +(0.309017 + 0.951057i) q^{9} +O(q^{10})\)	\(q+(0.587785 + 1.80902i) q^{2} +(0.809017 + 0.587785i) q^{3} +(-2.11803 + 1.53884i) q^{4} +(0.363271 - 1.11803i) q^{5} +(-0.587785 + 1.80902i) q^{6} +(-2.48990 - 1.80902i) q^{8} +(0.309017 + 0.951057i) q^{9} +2.23607 q^{10} +(-0.951057 - 0.309017i) q^{11} -2.61803 q^{12} +(-0.309017 - 0.951057i) q^{13} +(0.951057 - 0.690983i) q^{15} +(1.00000 - 3.07768i) q^{16} +(-1.53884 + 1.11803i) q^{18} +(0.951057 + 2.92705i) q^{20} -1.90211i q^{22} +(-0.951057 - 2.92705i) q^{24} +(-0.309017 - 0.224514i) q^{25} +(1.53884 - 1.11803i) q^{26} +(-0.309017 + 0.951057i) q^{27} +(1.80902 + 1.31433i) q^{30} +3.07768 q^{32} +(-0.587785 - 0.809017i) q^{33} +(-2.11803 - 1.53884i) q^{36} +(0.309017 - 0.951057i) q^{39} +(-2.92705 + 2.12663i) q^{40} -1.61803 q^{43} +(2.48990 - 0.809017i) q^{44} +1.17557 q^{45} +(1.53884 + 1.11803i) q^{47} +(2.61803 - 1.90211i) q^{48} +(0.309017 - 0.951057i) q^{49} +(0.224514 - 0.690983i) q^{50} +(2.11803 + 1.53884i) q^{52} -1.90211 q^{54} +(-0.690983 + 0.951057i) q^{55} +(-1.53884 + 1.11803i) q^{59} +(-0.951057 + 2.92705i) q^{60} +(-0.190983 + 0.587785i) q^{61} +(0.809017 + 2.48990i) q^{64} -1.17557 q^{65} +(1.11803 - 1.53884i) q^{66} +(0.951057 - 2.92705i) q^{72} +(-0.118034 - 0.363271i) q^{75} +1.90211 q^{78} +(-0.190983 - 0.587785i) q^{79} +(-3.07768 - 2.23607i) q^{80} +(-0.809017 + 0.587785i) q^{81} +(-0.363271 + 1.11803i) q^{83} +(-0.951057 - 2.92705i) q^{86} +(1.80902 + 2.48990i) q^{88} +(0.690983 + 2.12663i) q^{90} +(-1.11803 + 3.44095i) q^{94} +(2.48990 + 1.80902i) q^{96} +1.90211 q^{98} -1.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{3} - 8 q^{4} - 2 q^{9}+O(q^{10}) \) 8 * q + 2 * q^3 - 8 * q^4 - 2 * q^9	\( 8 q + 2 q^{3} - 8 q^{4} - 2 q^{9} - 12 q^{12} + 2 q^{13} + 8 q^{16} + 2 q^{25} + 2 q^{27} + 10 q^{30} - 8 q^{36} - 2 q^{39} - 10 q^{40} - 4 q^{43} + 12 q^{48} - 2 q^{49} + 8 q^{52} - 10 q^{55} - 6 q^{61} + 2 q^{64} + 8 q^{75} - 6 q^{79} - 2 q^{81} + 10 q^{88} + 10 q^{90}+O(q^{100}) \) 8 * q + 2 * q^3 - 8 * q^4 - 2 * q^9 - 12 * q^12 + 2 * q^13 + 8 * q^16 + 2 * q^25 + 2 * q^27 + 10 * q^30 - 8 * q^36 - 2 * q^39 - 10 * q^40 - 4 * q^43 + 12 * q^48 - 2 * q^49 + 8 * q^52 - 10 * q^55 - 6 * q^61 + 2 * q^64 + 8 * q^75 - 6 * q^79 - 2 * q^81 + 10 * q^88 + 10 * q^90

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more