Embedded newform 11.10.c.a.9.2

• Label: 11.10.c.a.9.2
• Level: $11$
• Weight: $10$
• Character: 11.9
• Analytic conductor: $5.665$
• Analytic rank: $0$
• Dimension: $32$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 11 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	11.c (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.66539419780\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			9.2
Character	\(\chi\)	\(=\)	11.9
Dual form			11.10.c.a.5.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-10.8459 + 33.3802i) q^{2} +(-181.934 + 132.183i) q^{3} +(-582.388 - 423.130i) q^{4} +(311.221 + 957.840i) q^{5} +(-2439.05 - 7506.62i) q^{6} +(7268.43 + 5280.82i) q^{7} +(5902.49 - 4288.41i) q^{8} +(9545.25 - 29377.3i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 21 q^{2} + 69 q^{3} - 1625 q^{4} + 225 q^{5} + 8173 q^{6} - 10675 q^{7} + 22863 q^{8} - 93587 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/11\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{3}{5}\right)\)

Coefficient data

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

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−10.8459	+	33.3802i	−0.479325	+	1.47521i	0.360710	+	0.932678i	\(0.382535\pi\)
							−0.840035	+	0.542533i	\(0.817465\pi\)
\(3\)	−181.934	+	132.183i	−1.29678	+	0.942168i	−0.999919	−	0.0127512i	\(-0.995941\pi\)
							−0.296865	+	0.954920i	\(0.595941\pi\)
\(5\)	311.221	+	957.840i	0.222692	+	0.685375i	0.998518	+	0.0544277i	\(0.0173334\pi\)
							−0.775826	+	0.630947i	\(0.782667\pi\)
\(7\)	7268.43	+	5280.82i	1.14419	+	0.831305i	0.987698	−	0.156374i	\(-0.0499804\pi\)
							0.156495	+	0.987679i	\(0.449980\pi\)
\(11\)	−38536.7	−	29544.3i	−0.793611	−	0.608425i
							\(13\)	−6776.90	+	20857.2i	−0.0658091	+	0.202540i	−0.978554	−	0.205990i	\(-0.933958\pi\)
0.912745	+	0.408530i	\(0.133958\pi\)
\(17\)	176454.	+	543071.i	0.512404	+	1.57702i	0.787956	+	0.615731i	\(0.211139\pi\)
							−0.275552	+	0.961286i	\(0.588861\pi\)
\(19\)	397648.	−	288908.i	0.700016	−	0.508591i	−0.179922	−	0.983681i	\(-0.557584\pi\)
							0.879937	+	0.475090i	\(0.157584\pi\)
\(23\)	−1.91937e6			−1.43016			−0.715079	−	0.699044i	\(-0.753609\pi\)
							−0.715079	+	0.699044i	\(0.753609\pi\)
\(29\)	159610.	+	115963.i	0.0419053	+	0.0304460i	0.608541	−	0.793523i	\(-0.291755\pi\)
							−0.566635	+	0.823969i	\(0.691755\pi\)
\(31\)	−698160.	+	2.14872e6i	−0.135777	+	0.417880i	−0.995710	−	0.0925276i	\(-0.970505\pi\)
							0.859933	+	0.510407i	\(0.170505\pi\)
\(37\)	3.94612e6	+	2.86702e6i	0.346148	+	0.251491i	0.747251	−	0.664541i	\(-0.231373\pi\)
							−0.401103	+	0.916033i	\(0.631373\pi\)
\(41\)	−5.49295e6	+	3.99087e6i	−0.303584	+	0.220567i	−0.729139	−	0.684366i	\(-0.760079\pi\)
							0.425555	+	0.904933i	\(0.360079\pi\)
\(43\)	−3.04965e7			−1.36032			−0.680162	−	0.733062i	\(-0.738091\pi\)
							−0.680162	+	0.733062i	\(0.738091\pi\)
\(47\)	−2.13828e7	+	1.55355e7i	−0.639183	+	0.464394i	−0.859569	−	0.511019i	\(-0.829268\pi\)
							0.220386	+	0.975413i	\(0.429268\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.10.c.a.9.2	yes	32
3.2	odd	2		99.10.f.a.64.7		32
11.4	even	5		121.10.a.h.1.3		16
11.5	even	5	inner	11.10.c.a.5.2	✓	32
11.7	odd	10		121.10.a.i.1.14		16
33.5	odd	10		99.10.f.a.82.7		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.10.c.a.5.2	✓	32	11.5	even	5	inner
11.10.c.a.9.2	yes	32	1.1	even	1	trivial
99.10.f.a.64.7		32	3.2	odd	2
99.10.f.a.82.7		32	33.5	odd	10
121.10.a.h.1.3		16	11.4	even	5
121.10.a.i.1.14		16	11.7	odd	10