Embedded newform 11.10.c.a.9.6

• Label: 11.10.c.a.9.6
• 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			9.6
Character	\(\chi\)	\(=\)	11.9
Dual form			11.10.c.a.5.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.40341 - 13.5523i) q^{2} +(169.166 - 122.906i) q^{3} +(249.942 + 181.594i) q^{4} +(63.1636 + 194.398i) q^{5} +(-920.753 - 2833.79i) q^{6} +(-1584.15 - 1150.95i) q^{7} +(9464.08 - 6876.06i) q^{8} +(7428.73 - 22863.3i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 21 q^{2} + 69 q^{3} - 1625 q^{4} + 225 q^{5} + 8173 q^{6} - 10675 q^{7} + 22863 q^{8} - 93587 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.40341	−	13.5523i	0.194605	−	0.598932i	−0.805376	−	0.592764i	\(-0.798037\pi\)
							0.999981	−	0.00616815i	\(-0.00196340\pi\)
\(3\)	169.166	−	122.906i	1.20578	−	0.876047i	0.210935	−	0.977500i	\(-0.432349\pi\)
							0.994840	+	0.101453i	\(0.0323492\pi\)
\(5\)	63.1636	+	194.398i	0.0451962	+	0.139100i	0.971108	−	0.238640i	\(-0.0767015\pi\)
							−0.925912	+	0.377739i	\(0.876702\pi\)
\(7\)	−1584.15	−	1150.95i	−0.249377	−	0.181183i	0.456074	−	0.889942i	\(-0.349255\pi\)
							−0.705450	+	0.708759i	\(0.749255\pi\)
\(11\)	−46215.4	−	14902.5i	−0.951743	−	0.306897i
							\(13\)	−16780.6	+	51645.5i	−0.162953	+	0.501519i	−0.998880	−	0.0473229i	\(-0.984931\pi\)
0.835926	+	0.548842i	\(0.184931\pi\)
\(17\)	124751.	+	383945.i	0.362264	+	1.11493i	0.951677	+	0.307101i	\(0.0993590\pi\)
							−0.589413	+	0.807832i	\(0.700641\pi\)
\(19\)	−433072.	+	314645.i	−0.762375	+	0.553898i	−0.899638	−	0.436637i	\(-0.856170\pi\)
							0.137263	+	0.990535i	\(0.456170\pi\)
\(23\)	1.65471e6			1.23296			0.616478	−	0.787372i	\(-0.288559\pi\)
							0.616478	+	0.787372i	\(0.288559\pi\)
\(29\)	−4.60377e6	−	3.34484e6i	−1.20871	−	0.878181i	−0.213599	−	0.976921i	\(-0.568518\pi\)
							−0.995113	+	0.0987409i	\(0.968518\pi\)
\(31\)	−2.77307e6	+	8.53464e6i	−0.539304	+	1.65981i	0.194858	+	0.980831i	\(0.437575\pi\)
							−0.734162	+	0.678975i	\(0.762425\pi\)
\(37\)	−3.75865e6	−	2.73082e6i	−0.329704	−	0.239544i	0.410601	−	0.911815i	\(-0.365319\pi\)
							−0.740305	+	0.672271i	\(0.765319\pi\)
\(41\)	−6.01478e6	+	4.36999e6i	−0.332424	+	0.241520i	−0.741459	−	0.670999i	\(-0.765866\pi\)
							0.409034	+	0.912519i	\(0.365866\pi\)
\(43\)	−2.24348e7			−1.00072			−0.500362	−	0.865816i	\(-0.666800\pi\)
							−0.500362	+	0.865816i	\(0.666800\pi\)
\(47\)	3.63844e7	−	2.64348e7i	1.08762	−	0.790199i	0.108621	−	0.994083i	\(-0.465357\pi\)
							0.978995	+	0.203884i	\(0.0653567\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.10.c.a.9.6	yes	32
3.2	odd	2		99.10.f.a.64.3		32
11.4	even	5		121.10.a.h.1.11		16
11.5	even	5	inner	11.10.c.a.5.6	✓	32
11.7	odd	10		121.10.a.i.1.6		16
33.5	odd	10		99.10.f.a.82.3		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.10.c.a.5.6	✓	32	11.5	even	5	inner
11.10.c.a.9.6	yes	32	1.1	even	1	trivial
99.10.f.a.64.3		32	3.2	odd	2
99.10.f.a.82.3		32	33.5	odd	10
121.10.a.h.1.11		16	11.4	even	5
121.10.a.i.1.6		16	11.7	odd	10