Embedded newform 11.10.c.a.5.1

• Label: 11.10.c.a.5.1
• Level: $11$
• Weight: $10$
• Character: 11.5
• 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			5.1
Character	\(\chi\)	\(=\)	11.5
Dual form			11.10.c.a.9.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-13.1726 - 40.5409i) q^{2} +(121.812 + 88.5013i) q^{3} +(-1055.84 + 767.109i) q^{4} +(-602.641 + 1854.74i) q^{5} +(1983.36 - 6104.15i) q^{6} +(-3576.09 + 2598.18i) q^{7} +(27350.5 + 19871.3i) q^{8} +(923.206 + 2841.34i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 21 q^{2} + 69 q^{3} - 1625 q^{4} + 225 q^{5} + 8173 q^{6} - 10675 q^{7} + 22863 q^{8} - 93587 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−13.1726	−	40.5409i	−0.582150	−	1.79167i	−0.610426	−	0.792073i	\(-0.709002\pi\)
							0.0282759	−	0.999600i	\(-0.490998\pi\)
\(3\)	121.812	+	88.5013i	0.868247	+	0.630818i	0.930116	−	0.367266i	\(-0.119706\pi\)
							−0.0618693	+	0.998084i	\(0.519706\pi\)
\(5\)	−602.641	+	1854.74i	−0.431215	+	1.32714i	0.465702	+	0.884942i	\(0.345802\pi\)
							−0.896916	+	0.442200i	\(0.854198\pi\)
\(7\)	−3576.09	+	2598.18i	−0.562946	+	0.409004i	−0.832536	−	0.553971i	\(-0.813112\pi\)
							0.269590	+	0.962975i	\(0.413112\pi\)
\(11\)	−48558.4	−	157.079i	−0.999995	−	0.00323482i
							\(13\)	−8755.21	−	26945.8i	−0.0850201	−	0.261665i	0.899504	−	0.436912i	\(-0.143928\pi\)
−0.984525	+	0.175247i	\(0.943928\pi\)
\(17\)	−95160.6	+	292874.i	−0.276336	+	0.850474i	0.712527	+	0.701645i	\(0.247551\pi\)
							−0.988863	+	0.148830i	\(0.952449\pi\)
\(19\)	619230.	+	449897.i	1.09009	+	0.791993i	0.979414	−	0.201864i	\(-0.0646997\pi\)
							0.110672	+	0.993857i	\(0.464700\pi\)
\(23\)	−697927.			−0.520038			−0.260019	−	0.965604i	\(-0.583729\pi\)
							−0.260019	+	0.965604i	\(0.583729\pi\)
\(29\)	1.46083e6	−	1.06136e6i	0.383539	−	0.278657i	−0.379264	−	0.925288i	\(-0.623823\pi\)
							0.762803	+	0.646631i	\(0.223823\pi\)
\(31\)	1.26462e6	+	3.89210e6i	0.245942	+	0.756932i	0.995480	+	0.0949705i	\(0.0302757\pi\)
							−0.749538	+	0.661961i	\(0.769724\pi\)
\(37\)	−9.42482e6	+	6.84753e6i	−0.826733	+	0.600657i	−0.918633	−	0.395112i	\(-0.870706\pi\)
							0.0919001	+	0.995768i	\(0.470706\pi\)
\(41\)	−3.45018e6	−	2.50670e6i	−0.190684	−	0.138540i	0.488347	−	0.872649i	\(-0.337600\pi\)
							−0.679031	+	0.734109i	\(0.737600\pi\)
\(43\)	2.59611e7			1.15802			0.579008	−	0.815322i	\(-0.303440\pi\)
							0.579008	+	0.815322i	\(0.303440\pi\)
\(47\)	−2.88438e7	−	2.09562e7i	−0.862207	−	0.626430i	0.0662773	−	0.997801i	\(-0.478888\pi\)
							−0.928485	+	0.371371i	\(0.878888\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.10.c.a.5.1	✓	32
3.2	odd	2		99.10.f.a.82.8		32
11.3	even	5		121.10.a.h.1.2		16
11.8	odd	10		121.10.a.i.1.15		16
11.9	even	5	inner	11.10.c.a.9.1	yes	32
33.20	odd	10		99.10.f.a.64.8		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.10.c.a.5.1	✓	32	1.1	even	1	trivial
11.10.c.a.9.1	yes	32	11.9	even	5	inner
99.10.f.a.64.8		32	33.20	odd	10
99.10.f.a.82.8		32	3.2	odd	2
121.10.a.h.1.2		16	11.3	even	5
121.10.a.i.1.15		16	11.8	odd	10