Embedded newform 11.10.c.a.9.3

• Label: 11.10.c.a.9.3
• 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			9.3
Character	\(\chi\)	\(=\)	11.9
Dual form			11.10.c.a.5.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.86703 + 21.1346i) q^{2} +(42.7536 - 31.0623i) q^{3} +(14.7033 + 10.6826i) q^{4} +(597.948 + 1840.30i) q^{5} +(362.897 + 1116.88i) q^{6} +(-7706.91 - 5599.40i) q^{7} +(-9531.54 + 6925.07i) q^{8} +(-5219.38 + 16063.6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 21 q^{2} + 69 q^{3} - 1625 q^{4} + 225 q^{5} + 8173 q^{6} - 10675 q^{7} + 22863 q^{8} - 93587 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.86703	+	21.1346i	−0.303483	+	0.934024i	0.676756	+	0.736207i	\(0.263385\pi\)
							−0.980239	+	0.197817i	\(0.936615\pi\)
\(3\)	42.7536	−	31.0623i	0.304738	−	0.221405i	−0.424897	−	0.905242i	\(-0.639690\pi\)
							0.729635	+	0.683836i	\(0.239690\pi\)
\(5\)	597.948	+	1840.30i	0.427857	+	1.31681i	0.900232	+	0.435411i	\(0.143397\pi\)
							−0.472375	+	0.881398i	\(0.656603\pi\)
\(7\)	−7706.91	−	5599.40i	−1.21322	−	0.881455i	−0.217699	−	0.976016i	\(-0.569855\pi\)
							−0.995519	+	0.0945610i	\(0.969855\pi\)
\(11\)	47283.6	−	11055.0i	0.973740	−	0.227662i
							\(13\)	−37984.7	+	116905.i	−0.368862	+	1.13524i	0.578665	+	0.815565i	\(0.303574\pi\)
−0.947527	+	0.319675i	\(0.896426\pi\)
\(17\)	25414.6	+	78218.1i	0.0738012	+	0.227137i	0.981152	−	0.193238i	\(-0.0618988\pi\)
							−0.907351	+	0.420374i	\(0.861899\pi\)
\(19\)	677469.	−	492210.i	1.19261	−	0.866482i	0.199072	−	0.979985i	\(-0.436207\pi\)
							0.993538	+	0.113503i	\(0.0362071\pi\)
\(23\)	1.33428e6			0.994199			0.497099	−	0.867694i	\(-0.334398\pi\)
							0.497099	+	0.867694i	\(0.334398\pi\)
\(29\)	−234529.	−	170395.i	−0.0615751	−	0.0447369i	0.556572	−	0.830799i	\(-0.312116\pi\)
							−0.618147	+	0.786062i	\(0.712116\pi\)
\(31\)	354836.	−	1.09207e6i	0.0690081	−	0.212385i	−0.910605	−	0.413277i	\(-0.864384\pi\)
							0.979613	+	0.200892i	\(0.0643840\pi\)
\(37\)	1.44541e7	+	1.05015e7i	1.26790	+	0.921182i	0.999117	−	0.0420193i	\(-0.0133791\pi\)
							0.268781	+	0.963201i	\(0.413379\pi\)
\(41\)	1.95177e7	−	1.41804e7i	1.07870	−	0.783721i	0.101243	−	0.994862i	\(-0.467718\pi\)
							0.977456	+	0.211141i	\(0.0677179\pi\)
\(43\)	−1.26584e7			−0.564640			−0.282320	−	0.959320i	\(-0.591104\pi\)
							−0.282320	+	0.959320i	\(0.591104\pi\)
\(47\)	−1.76663e7	+	1.28353e7i	−0.528087	+	0.383678i	−0.819642	−	0.572876i	\(-0.805827\pi\)
							0.291555	+	0.956554i	\(0.405827\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.10.c.a.9.3	yes	32
3.2	odd	2		99.10.f.a.64.6		32
11.4	even	5		121.10.a.h.1.5		16
11.5	even	5	inner	11.10.c.a.5.3	✓	32
11.7	odd	10		121.10.a.i.1.12		16
33.5	odd	10		99.10.f.a.82.6		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.10.c.a.5.3	✓	32	11.5	even	5	inner
11.10.c.a.9.3	yes	32	1.1	even	1	trivial
99.10.f.a.64.6		32	3.2	odd	2
99.10.f.a.82.6		32	33.5	odd	10
121.10.a.h.1.5		16	11.4	even	5
121.10.a.i.1.12		16	11.7	odd	10