Embedded newform 11.10.c.a.9.7

• Label: 11.10.c.a.9.7
• 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			9.7
Character	\(\chi\)	\(=\)	11.9
Dual form			11.10.c.a.5.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(6.35502 - 19.5587i) q^{2} +(-114.115 + 82.9098i) q^{3} +(72.0591 + 52.3540i) q^{4} +(411.773 + 1267.31i) q^{5} +(896.404 + 2758.85i) q^{6} +(2149.58 + 1561.76i) q^{7} +(10000.4 - 7265.71i) q^{8} +(65.9352 - 202.928i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 21 q^{2} + 69 q^{3} - 1625 q^{4} + 225 q^{5} + 8173 q^{6} - 10675 q^{7} + 22863 q^{8} - 93587 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	6.35502	−	19.5587i	0.280855	−	0.864382i	−0.706756	−	0.707457i	\(-0.749842\pi\)
							0.987611	−	0.156924i	\(-0.0501579\pi\)
\(3\)	−114.115	+	82.9098i	−0.813390	+	0.590963i	−0.914812	−	0.403881i	\(-0.867661\pi\)
							0.101421	+	0.994844i	\(0.467661\pi\)
\(5\)	411.773	+	1267.31i	0.294641	+	0.906811i	0.983342	+	0.181766i	\(0.0581812\pi\)
							−0.688701	+	0.725045i	\(0.741819\pi\)
\(7\)	2149.58	+	1561.76i	0.338386	+	0.245852i	0.743981	−	0.668201i	\(-0.232936\pi\)
							−0.405595	+	0.914053i	\(0.632936\pi\)
\(11\)	2318.46	+	48503.3i	0.0477454	+	0.998860i
							\(13\)	−32407.6	+	99740.3i	−0.314704	+	0.968558i	0.661173	+	0.750234i	\(0.270059\pi\)
−0.975876	+	0.218324i	\(0.929941\pi\)
\(17\)	−108507.	−	333951.i	−0.315093	−	0.969758i	−0.975716	−	0.219039i	\(-0.929708\pi\)
							0.660623	−	0.750718i	\(-0.270292\pi\)
\(19\)	6359.00	−	4620.09i	0.0111943	−	0.00813315i	−0.582174	−	0.813064i	\(-0.697798\pi\)
							0.593368	+	0.804931i	\(0.297798\pi\)
\(23\)	−62525.7			−0.0465890			−0.0232945	−	0.999729i	\(-0.507416\pi\)
							−0.0232945	+	0.999729i	\(0.507416\pi\)
\(29\)	5.25555e6	+	3.81838e6i	1.37983	+	1.00251i	0.996896	+	0.0787341i	\(0.0250878\pi\)
							0.382938	+	0.923774i	\(0.374912\pi\)
\(31\)	1.93693e6	−	5.96126e6i	0.376692	−	1.15934i	−0.565637	−	0.824654i	\(-0.691370\pi\)
							0.942330	−	0.334686i	\(-0.108630\pi\)
\(37\)	−6.80695e6	−	4.94554e6i	−0.597097	−	0.433816i	0.247750	−	0.968824i	\(-0.420309\pi\)
							−0.844847	+	0.535008i	\(0.820309\pi\)
\(41\)	−1.32056e7	+	9.59444e6i	−0.729846	+	0.530264i	−0.889515	−	0.456906i	\(-0.848957\pi\)
							0.159669	+	0.987171i	\(0.448957\pi\)
\(43\)	1.60731e7			0.716954			0.358477	−	0.933539i	\(-0.383296\pi\)
							0.358477	+	0.933539i	\(0.383296\pi\)
\(47\)	3.36207e7	−	2.44269e7i	1.00500	−	0.730175i	0.0418456	−	0.999124i	\(-0.486676\pi\)
							0.963154	+	0.268949i	\(0.0866763\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.10.c.a.9.7	yes	32
3.2	odd	2		99.10.f.a.64.2		32
11.4	even	5		121.10.a.h.1.13		16
11.5	even	5	inner	11.10.c.a.5.7	✓	32
11.7	odd	10		121.10.a.i.1.4		16
33.5	odd	10		99.10.f.a.82.2		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.10.c.a.5.7	✓	32	11.5	even	5	inner
11.10.c.a.9.7	yes	32	1.1	even	1	trivial
99.10.f.a.64.2		32	3.2	odd	2
99.10.f.a.82.2		32	33.5	odd	10
121.10.a.h.1.13		16	11.4	even	5
121.10.a.i.1.4		16	11.7	odd	10