Embedded newform 11.10.c.a.3.4

• Label: 11.10.c.a.3.4
• Level: $11$
• Weight: $10$
• Character: 11.3
• 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			3.4
Character	\(\chi\)	\(=\)	11.3
Dual form			11.10.c.a.4.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.91186 + 5.74830i) q^{2} +(-77.1605 - 237.476i) q^{3} +(-128.662 + 395.981i) q^{4} +(690.317 + 501.544i) q^{5} +(1975.57 + 1435.33i) q^{6} +(-2608.39 + 8027.81i) q^{7} +(-2805.56 - 8634.62i) q^{8} +(-34517.1 + 25078.1i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 21 q^{2} + 69 q^{3} - 1625 q^{4} + 225 q^{5} + 8173 q^{6} - 10675 q^{7} + 22863 q^{8} - 93587 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.91186	+	5.74830i	−0.349658	+	0.254042i	−0.748726	−	0.662880i	\(-0.769334\pi\)
							0.399067	+	0.916922i	\(0.369334\pi\)
\(3\)	−77.1605	−	237.476i	−0.549983	−	1.69267i	−0.708837	−	0.705373i	\(-0.750780\pi\)
							0.158853	−	0.987302i	\(-0.449220\pi\)
\(5\)	690.317	+	501.544i	0.493950	+	0.358876i	0.806701	−	0.590959i	\(-0.201251\pi\)
							−0.312751	+	0.949835i	\(0.601251\pi\)
\(7\)	−2608.39	+	8027.81i	−0.410612	+	1.26373i	0.505505	+	0.862824i	\(0.331306\pi\)
							−0.916117	+	0.400911i	\(0.868694\pi\)
\(11\)	40689.0	+	26501.9i	0.837935	+	0.545771i
							\(13\)	−76554.5	+	55620.1i	−0.743405	+	0.540115i	−0.893776	−	0.448514i	\(-0.851953\pi\)
0.150371	+	0.988630i	\(0.451953\pi\)
\(17\)	−133577.	−	97049.7i	−0.387894	−	0.281821i	0.376698	−	0.926336i	\(-0.377059\pi\)
							−0.764592	+	0.644515i	\(0.777059\pi\)
\(19\)	115077.	+	354171.i	0.202581	+	0.623480i	0.999804	+	0.0197951i	\(0.00630137\pi\)
							−0.797223	+	0.603685i	\(0.793699\pi\)
\(23\)	−1.52505e6			−1.13634			−0.568170	−	0.822911i	\(-0.692348\pi\)
							−0.568170	+	0.822911i	\(0.692348\pi\)
\(29\)	−1.22948e6	+	3.78394e6i	−0.322797	+	0.993466i	0.649629	+	0.760252i	\(0.274924\pi\)
							−0.972425	+	0.233214i	\(0.925076\pi\)
\(31\)	−4.54693e6	+	3.30354e6i	−0.884281	+	0.642468i	−0.934381	−	0.356276i	\(-0.884046\pi\)
							0.0500996	+	0.998744i	\(0.484046\pi\)
\(37\)	2.72428e6	−	8.38448e6i	0.238970	−	0.735476i	−0.757599	−	0.652720i	\(-0.773628\pi\)
							0.996570	−	0.0827557i	\(-0.0263721\pi\)
\(41\)	769029.	+	2.36683e6i	0.0425026	+	0.130810i	0.970056	−	0.242881i	\(-0.0780923\pi\)
							−0.927554	+	0.373690i	\(0.878092\pi\)
\(43\)	3.18822e7			1.42213			0.711067	−	0.703125i	\(-0.248212\pi\)
							0.711067	+	0.703125i	\(0.248212\pi\)
\(47\)	6.40469e6	+	1.97116e7i	0.191451	+	0.589226i	1.00000		0.000796328i	\(0.000253479\pi\)
							−0.808549	+	0.588429i	\(0.799747\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.10.c.a.3.4	✓	32
3.2	odd	2		99.10.f.a.91.5		32
11.2	odd	10		121.10.a.i.1.7		16
11.4	even	5	inner	11.10.c.a.4.4	yes	32
11.9	even	5		121.10.a.h.1.10		16
33.26	odd	10		99.10.f.a.37.5		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.10.c.a.3.4	✓	32	1.1	even	1	trivial
11.10.c.a.4.4	yes	32	11.4	even	5	inner
99.10.f.a.37.5		32	33.26	odd	10
99.10.f.a.91.5		32	3.2	odd	2
121.10.a.h.1.10		16	11.9	even	5
121.10.a.i.1.7		16	11.2	odd	10