Embedded newform 473.2.k.b.128.6

• Label: 473.2.k.b.128.6
• Level: $473$
• Weight: $2$
• Character: 473.128
• Analytic conductor: $3.777$
• Analytic rank: $0$
• Dimension: $160$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 473 = 11 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	473.k (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.77692401561\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(40\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			128.6
Character	\(\chi\)	\(=\)	473.128
Dual form			473.2.k.b.85.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.89979 - 1.38028i) q^{2} +(-1.10873 - 0.360248i) q^{3} +(1.08599 + 3.34234i) q^{4} +(0.367347 + 0.505610i) q^{5} +(1.60911 + 2.21475i) q^{6} +(-0.481036 - 1.48048i) q^{7} +(1.09889 - 3.38204i) q^{8} +(-1.32755 - 0.964520i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 46 q^{4} - 20 q^{6} + 50 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/473\mathbb{Z}\right)^\times\).

\(n\)	\(89\)	\(431\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{7}{10}\right)\)

Coefficient data

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

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.89979	−	1.38028i	−1.34335	−	0.976002i	−0.999314	−	0.0370465i	\(-0.988205\pi\)
							−0.344038	−	0.938956i	\(-0.611795\pi\)
\(3\)	−1.10873	−	0.360248i	−0.640125	−	0.207989i	−0.0290702	−	0.999577i	\(-0.509255\pi\)
							−0.611055	+	0.791588i	\(0.709255\pi\)
\(5\)	0.367347	+	0.505610i	0.164283	+	0.226116i	0.883220	−	0.468960i	\(-0.155371\pi\)
							−0.718937	+	0.695075i	\(0.755371\pi\)
\(7\)	−0.481036	−	1.48048i	−0.181815	−	0.559568i	0.818064	−	0.575127i	\(-0.195047\pi\)
							−0.999879	+	0.0155590i	\(0.995047\pi\)
\(11\)	−1.26327	+	3.06662i	−0.380889	+	0.924621i
							\(13\)	−0.137295	+	0.188970i	−0.0380788	+	0.0524109i	−0.827632	−	0.561271i	\(-0.810312\pi\)
0.789553	+	0.613682i	\(0.210312\pi\)
\(17\)	1.33309	+	1.83484i	0.323322	+	0.445015i	0.939478	−	0.342610i	\(-0.111311\pi\)
							−0.616156	+	0.787624i	\(0.711311\pi\)
\(19\)	−1.23266	+	3.79372i	−0.282791	+	0.870340i	0.704262	+	0.709941i	\(0.251278\pi\)
							−0.987052	+	0.160399i	\(0.948722\pi\)
\(23\)	7.38615			1.54012			0.770059	−	0.637972i	\(-0.220227\pi\)
							0.770059	+	0.637972i	\(0.220227\pi\)
\(29\)	1.53137	+	4.71307i	0.284368	+	0.875195i	0.986587	+	0.163234i	\(0.0521926\pi\)
							−0.702219	+	0.711961i	\(0.747807\pi\)
\(31\)	−0.871701	−	0.633328i	−0.156562	−	0.113749i	0.506746	−	0.862095i	\(-0.330848\pi\)
							−0.663308	+	0.748346i	\(0.730848\pi\)
\(37\)	6.42485	−	2.08756i	1.05624	−	0.343193i	0.271125	−	0.962544i	\(-0.412604\pi\)
							0.785114	+	0.619351i	\(0.212604\pi\)
\(41\)	−8.19089	−	2.66138i	−1.27920	−	0.415638i	−0.410903	−	0.911679i	\(-0.634787\pi\)
							−0.868299	+	0.496041i	\(0.834787\pi\)
\(43\)	6.13533	−	2.31467i	0.935629	−	0.352984i
							\(47\)	0.521184	−	1.60404i	0.0760224	−	0.233973i	−0.905823	−	0.423657i	\(-0.860746\pi\)
0.981845	+	0.189684i	\(0.0607463\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	473.2.k.b.128.6	yes	160
11.8	odd	10	inner	473.2.k.b.85.35	yes	160
43.42	odd	2	inner	473.2.k.b.128.35	yes	160
473.85	even	10	inner	473.2.k.b.85.6	✓	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
473.2.k.b.85.6	✓	160	473.85	even	10	inner
473.2.k.b.85.35	yes	160	11.8	odd	10	inner
473.2.k.b.128.6	yes	160	1.1	even	1	trivial
473.2.k.b.128.35	yes	160	43.42	odd	2	inner