Embedded newform 11.12.c.a.3.1

• Label: 11.12.c.a.3.1
• Level: $11$
• Weight: $12$
• Character: 11.3
• Analytic conductor: $8.452$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 11 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	11.c (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.45177498616\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(10\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			3.1
Character	\(\chi\)	\(=\)	11.3
Dual form			11.12.c.a.4.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-66.2723 + 48.1496i) q^{2} +(155.409 + 478.301i) q^{3} +(1440.76 - 4434.21i) q^{4} +(-3908.19 - 2839.47i) q^{5} +(-33329.3 - 24215.2i) q^{6} +(15024.9 - 46241.9i) q^{7} +(66180.4 + 203682. i) q^{8} +(-61304.7 + 44540.5i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 11 q^{2} - 276 q^{3} - 6137 q^{4} + 6038 q^{5} - 30371 q^{6} + 55440 q^{7} - 347809 q^{8} - 240812 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−66.2723	+	48.1496i	−1.46442	+	1.06397i	−0.482242	+	0.876038i	\(0.660177\pi\)
							−0.982183	+	0.187929i	\(0.939823\pi\)
\(3\)	155.409	+	478.301i	0.369241	+	1.13641i	0.947282	+	0.320400i	\(0.103817\pi\)
							−0.578041	+	0.816008i	\(0.696183\pi\)
\(5\)	−3908.19	−	2839.47i	−0.559295	−	0.406351i	0.271906	−	0.962324i	\(-0.412346\pi\)
							−0.831201	+	0.555972i	\(0.812346\pi\)
\(7\)	15024.9	−	46241.9i	0.337888	−	1.03991i	−0.627394	−	0.778702i	\(-0.715878\pi\)
							0.965282	−	0.261210i	\(-0.0841215\pi\)
\(11\)	346976.	−	406103.i	0.649590	−	0.760284i
							\(13\)	−1.50664e6	+	1.09464e6i	−1.12544	+	0.817679i	−0.985025	−	0.172414i	\(-0.944843\pi\)
−0.140413	+	0.990093i	\(0.544843\pi\)
\(17\)	3.49865e6	+	2.54192e6i	0.597628	+	0.434202i	0.845036	−	0.534709i	\(-0.179579\pi\)
							−0.247408	+	0.968911i	\(0.579579\pi\)
\(19\)	−6.47653e6	−	1.99327e7i	−0.600064	−	1.84681i	−0.527701	−	0.849430i	\(-0.676946\pi\)
							−0.0723632	−	0.997378i	\(-0.523054\pi\)
\(23\)	2.70056e7			0.874884			0.437442	−	0.899247i	\(-0.355884\pi\)
							0.437442	+	0.899247i	\(0.355884\pi\)
\(29\)	−8.03568e6	+	2.47313e7i	−0.0727501	+	0.223902i	−0.980820	−	0.194918i	\(-0.937556\pi\)
							0.908070	+	0.418819i	\(0.137556\pi\)
\(31\)	1.57211e8	−	1.14220e8i	0.986265	−	0.716563i	0.0271649	−	0.999631i	\(-0.491352\pi\)
							0.959100	+	0.283068i	\(0.0913521\pi\)
\(37\)	1.24593e8	−	3.83459e8i	0.295383	−	0.909095i	−0.687710	−	0.725986i	\(-0.741384\pi\)
							0.983093	−	0.183109i	\(-0.0586162\pi\)
\(41\)	−5.08461e7	−	1.56488e8i	−0.0685404	−	0.210946i	0.910920	−	0.412584i	\(-0.135374\pi\)
							−0.979460	+	0.201638i	\(0.935374\pi\)
\(43\)	−3.85055e8			−0.399436			−0.199718	−	0.979853i	\(-0.564003\pi\)
							−0.199718	+	0.979853i	\(0.564003\pi\)
\(47\)	−5.45509e7	−	1.67890e8i	−0.0346947	−	0.106779i	0.932209	−	0.361919i	\(-0.117878\pi\)
							−0.966904	+	0.255140i	\(0.917878\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.12.c.a.3.1	✓	40
3.2	odd	2		99.12.f.a.91.10		40
11.2	odd	10		121.12.a.h.1.2		20
11.4	even	5	inner	11.12.c.a.4.1	yes	40
11.9	even	5		121.12.a.j.1.19		20
33.26	odd	10		99.12.f.a.37.10		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.12.c.a.3.1	✓	40	1.1	even	1	trivial
11.12.c.a.4.1	yes	40	11.4	even	5	inner
99.12.f.a.37.10		40	33.26	odd	10
99.12.f.a.91.10		40	3.2	odd	2
121.12.a.h.1.2		20	11.2	odd	10
121.12.a.j.1.19		20	11.9	even	5