Embedded newform 11.10.c.a.5.4

• Label: 11.10.c.a.5.4
• Level: $11$
• Weight: $10$
• Character: 11.5
• Analytic conductor: $5.665$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• 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			5.4
Character	\(\chi\)	\(=\)	11.5
Dual form			11.10.c.a.9.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.28817 - 13.1976i) q^{2} +(55.2813 + 40.1642i) q^{3} +(258.428 - 187.759i) q^{4} +(-237.559 + 731.131i) q^{5} +(293.017 - 901.813i) q^{6} +(9125.19 - 6629.84i) q^{7} +(-9334.16 - 6781.66i) q^{8} +(-4639.52 - 14279.0i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 21 q^{2} + 69 q^{3} - 1625 q^{4} + 225 q^{5} + 8173 q^{6} - 10675 q^{7} + 22863 q^{8} - 93587 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.28817	−	13.1976i	−0.189512	−	0.583258i	0.810485	−	0.585760i	\(-0.199204\pi\)
							−0.999997	+	0.00250145i	\(0.999204\pi\)
\(3\)	55.2813	+	40.1642i	0.394033	+	0.286282i	0.767106	−	0.641520i	\(-0.221696\pi\)
							−0.373073	+	0.927802i	\(0.621696\pi\)
\(5\)	−237.559	+	731.131i	−0.169983	+	0.523155i	−0.999369	−	0.0355218i	\(-0.988691\pi\)
							0.829386	+	0.558677i	\(0.188691\pi\)
\(7\)	9125.19	−	6629.84i	1.43648	−	1.04367i	0.447720	−	0.894174i	\(-0.352236\pi\)
							0.988763	−	0.149493i	\(-0.0477640\pi\)
\(11\)	19694.5	−	44385.5i	0.405582	−	0.914059i
							\(13\)	59114.6	+	181936.i	0.574050	+	1.76674i	0.639393	+	0.768880i	\(0.279186\pi\)
−0.0653428	+	0.997863i	\(0.520814\pi\)
\(17\)	−81333.0	+	250317.i	−0.236182	+	0.726893i	0.760781	+	0.649009i	\(0.224816\pi\)
							−0.996962	+	0.0778838i	\(0.975184\pi\)
\(19\)	−207599.	−	150830.i	−0.365456	−	0.265519i	0.389868	−	0.920871i	\(-0.372520\pi\)
							−0.755324	+	0.655352i	\(0.772520\pi\)
\(23\)	−61217.9			−0.0456145			−0.0228072	−	0.999740i	\(-0.507260\pi\)
							−0.0228072	+	0.999740i	\(0.507260\pi\)
\(29\)	−4.93644e6	+	3.58653e6i	−1.29605	+	0.941637i	−0.999909	−	0.0135061i	\(-0.995701\pi\)
							−0.296144	+	0.955143i	\(0.595701\pi\)
\(31\)	571018.	+	1.75741e6i	0.111051	+	0.341780i	0.991103	−	0.133098i	\(-0.0424924\pi\)
							−0.880052	+	0.474877i	\(0.842492\pi\)
\(37\)	−6.69504e6	+	4.86423e6i	−0.587280	+	0.426684i	−0.841341	−	0.540504i	\(-0.818234\pi\)
							0.254061	+	0.967188i	\(0.418234\pi\)
\(41\)	9.32639e6	+	6.77602e6i	0.515450	+	0.374496i	0.814887	−	0.579620i	\(-0.196799\pi\)
							−0.299437	+	0.954116i	\(0.596799\pi\)
\(43\)	−7.83354e6			−0.349422			−0.174711	−	0.984620i	\(-0.555899\pi\)
							−0.174711	+	0.984620i	\(0.555899\pi\)
\(47\)	−1.34758e7	−	9.79077e6i	−0.402824	−	0.292669i	0.367866	−	0.929879i	\(-0.380088\pi\)
							−0.770690	+	0.637210i	\(0.780088\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.10.c.a.5.4	✓	32
3.2	odd	2		99.10.f.a.82.5		32
11.3	even	5		121.10.a.h.1.7		16
11.8	odd	10		121.10.a.i.1.10		16
11.9	even	5	inner	11.10.c.a.9.4	yes	32
33.20	odd	10		99.10.f.a.64.5		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.10.c.a.5.4	✓	32	1.1	even	1	trivial
11.10.c.a.9.4	yes	32	11.9	even	5	inner
99.10.f.a.64.5		32	33.20	odd	10
99.10.f.a.82.5		32	3.2	odd	2
121.10.a.h.1.7		16	11.3	even	5
121.10.a.i.1.10		16	11.8	odd	10