Embedded newform 460.2.q.a.11.9

• Label: 460.2.q.a.11.9
• Level: $460$
• Weight: $2$
• Character: 460.11
• Analytic conductor: $3.673$
• Analytic rank: $0$
• Dimension: $480$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 460 = 2^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	460.q (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.67311849298\)
Analytic rank:	\(0\)
Dimension:	\(480\)
Relative dimension:	\(48\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			11.9
Character	\(\chi\)	\(=\)	460.11
Dual form			460.2.q.a.251.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.19089 - 0.762752i) q^{2} +(-2.15085 - 0.309246i) q^{3} +(0.836418 + 1.81670i) q^{4} +(0.281733 + 0.959493i) q^{5} +(2.32554 + 2.00885i) q^{6} +(-2.11824 - 2.44458i) q^{7} +(0.389615 - 2.80146i) q^{8} +(1.65206 + 0.485089i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q - 4 q^{2} + 2 q^{6} + 2 q^{8} + 48 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/460\mathbb{Z}\right)^\times\).

\(n\)	\(231\)	\(277\)	\(281\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{9}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.19089	−	0.762752i	−0.842083	−	0.539347i
							\(3\)	−2.15085	−	0.309246i	−1.24180	−	0.178543i	−0.510064	−	0.860136i	\(-0.670378\pi\)
−0.731732	+	0.681593i	\(0.761288\pi\)
\(5\)	0.281733	+	0.959493i	0.125995	+	0.429098i
							\(7\)	−2.11824	−	2.44458i	−0.800619	−	0.923964i	0.197796	−	0.980243i	\(-0.436622\pi\)
−0.998415	+	0.0562797i	\(0.982076\pi\)
\(11\)	−0.379688	−	0.244011i	−0.114480	−	0.0735720i	0.482148	−	0.876090i	\(-0.339857\pi\)
							−0.596628	+	0.802518i	\(0.703493\pi\)
\(13\)	0.363605	−	0.419623i	0.100846	−	0.116382i	−0.703083	−	0.711108i	\(-0.748194\pi\)
							0.803929	+	0.594725i	\(0.202739\pi\)
\(17\)	−1.41623	+	0.646768i	−0.343485	+	0.156864i	−0.579686	−	0.814840i	\(-0.696825\pi\)
							0.236201	+	0.971704i	\(0.424098\pi\)
\(19\)	−1.36469	+	2.98825i	−0.313081	+	0.685551i	−0.999117	−	0.0420161i	\(-0.986622\pi\)
							0.686036	+	0.727568i	\(0.259349\pi\)
\(23\)	4.26172	+	2.19949i	0.888630	+	0.458624i
							\(29\)	1.94629	+	4.26179i	0.361417	+	0.791394i	0.999766	+	0.0216462i	\(0.00689073\pi\)
−0.638348	+	0.769748i	\(0.720382\pi\)
\(31\)	−0.130890	+	0.0188191i	−0.0235085	+	0.00338001i	−0.154059	−	0.988062i	\(-0.549235\pi\)
							0.130551	+	0.991442i	\(0.458326\pi\)
\(37\)	−1.37905	+	4.69663i	−0.226715	+	0.772121i	0.765041	+	0.643981i	\(0.222719\pi\)
							−0.991756	+	0.128139i	\(0.959100\pi\)
\(41\)	9.50343	−	2.79046i	1.48419	−	0.435796i	0.563505	−	0.826113i	\(-0.309453\pi\)
							0.920681	+	0.390316i	\(0.127634\pi\)
\(43\)	−0.0425217	+	0.295745i	−0.00648451	+	0.0451007i	−0.992807	−	0.119728i	\(-0.961798\pi\)
							0.986322	+	0.164828i	\(0.0527070\pi\)
\(47\)			8.64277i			1.26068i	0.776320	+	0.630339i	\(0.217084\pi\)
							−0.776320	+	0.630339i	\(0.782916\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	460.2.q.a.11.9	✓	480
4.3	odd	2	inner	460.2.q.a.11.27	yes	480
23.21	odd	22	inner	460.2.q.a.251.27	yes	480
92.67	even	22	inner	460.2.q.a.251.9	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.2.q.a.11.9	✓	480	1.1	even	1	trivial
460.2.q.a.11.27	yes	480	4.3	odd	2	inner
460.2.q.a.251.9	yes	480	92.67	even	22	inner
460.2.q.a.251.27	yes	480	23.21	odd	22	inner