Embedded newform 11.10.c.a.9.8

• Label: 11.10.c.a.9.8
• 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			9.8
Character	\(\chi\)	\(=\)	11.9
Dual form			11.10.c.a.5.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(11.8250 - 36.3935i) q^{2} +(17.5248 - 12.7325i) q^{3} +(-770.441 - 559.758i) q^{4} +(-276.050 - 849.593i) q^{5} +(-256.151 - 788.352i) q^{6} +(42.1653 + 30.6349i) q^{7} +(-13631.4 + 9903.80i) q^{8} +(-5937.38 + 18273.4i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 21 q^{2} + 69 q^{3} - 1625 q^{4} + 225 q^{5} + 8173 q^{6} - 10675 q^{7} + 22863 q^{8} - 93587 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	11.8250	−	36.3935i	0.522595	−	1.60838i	−0.246429	−	0.969161i	\(-0.579257\pi\)
							0.769024	−	0.639220i	\(-0.220743\pi\)
\(3\)	17.5248	−	12.7325i	0.124913	−	0.0907548i	−0.523574	−	0.851980i	\(-0.675402\pi\)
							0.648488	+	0.761225i	\(0.275402\pi\)
\(5\)	−276.050	−	849.593i	−0.197525	−	0.607920i	−0.999938	−	0.0111502i	\(-0.996451\pi\)
							0.802413	−	0.596769i	\(-0.203549\pi\)
\(7\)	42.1653	+	30.6349i	0.00663765	+	0.00482253i	0.591099	−	0.806599i	\(-0.298694\pi\)
							−0.584461	+	0.811421i	\(0.698694\pi\)
\(11\)	26773.8	−	40510.6i	0.551371	−	0.834260i
							\(13\)	36495.6	−	112322.i	0.354401	−	1.09074i	−0.601955	−	0.798530i	\(-0.705611\pi\)
0.956356	−	0.292205i	\(-0.0943888\pi\)
\(17\)	40243.1	+	123856.i	0.116862	+	0.359663i	0.992331	−	0.123611i	\(-0.0394475\pi\)
							−0.875469	+	0.483274i	\(0.839448\pi\)
\(19\)	663650.	−	482170.i	1.16828	−	0.848807i	0.177480	−	0.984124i	\(-0.443206\pi\)
							0.990802	+	0.135318i	\(0.0432055\pi\)
\(23\)	−951650.			−0.709091			−0.354546	−	0.935039i	\(-0.615364\pi\)
							−0.354546	+	0.935039i	\(0.615364\pi\)
\(29\)	3.81219e6	+	2.76972e6i	1.00088	+	0.727184i	0.962277	−	0.272070i	\(-0.0877082\pi\)
							0.0386061	+	0.999255i	\(0.487708\pi\)
\(31\)	−2.42493e6	+	7.46316e6i	−0.471597	+	1.45143i	0.378895	+	0.925440i	\(0.376304\pi\)
							−0.850492	+	0.525987i	\(0.823696\pi\)
\(37\)	9.82897e6	+	7.14117e6i	0.862185	+	0.626414i	0.928478	−	0.371386i	\(-0.121117\pi\)
							−0.0662937	+	0.997800i	\(0.521117\pi\)
\(41\)	−2.77890e7	+	2.01899e7i	−1.53584	+	1.11585i	−0.582963	+	0.812499i	\(0.698107\pi\)
							−0.952878	+	0.303355i	\(0.901893\pi\)
\(43\)	2.51272e7			1.12082			0.560411	−	0.828215i	\(-0.310643\pi\)
							0.560411	+	0.828215i	\(0.310643\pi\)
\(47\)	1.72727e7	−	1.25494e7i	0.516321	−	0.375129i	−0.298895	−	0.954286i	\(-0.596618\pi\)
							0.815216	+	0.579157i	\(0.196618\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.10.c.a.9.8	yes	32
3.2	odd	2		99.10.f.a.64.1		32
11.4	even	5		121.10.a.h.1.15		16
11.5	even	5	inner	11.10.c.a.5.8	✓	32
11.7	odd	10		121.10.a.i.1.2		16
33.5	odd	10		99.10.f.a.82.1		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.10.c.a.5.8	✓	32	11.5	even	5	inner
11.10.c.a.9.8	yes	32	1.1	even	1	trivial
99.10.f.a.64.1		32	3.2	odd	2
99.10.f.a.82.1		32	33.5	odd	10
121.10.a.h.1.15		16	11.4	even	5
121.10.a.i.1.2		16	11.7	odd	10