Embedded newform 11.12.c.a.4.6

• Label: 11.12.c.a.4.6
• Level: $11$
• Weight: $12$
• Character: 11.4
• 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			4.6
Character	\(\chi\)	\(=\)	11.4
Dual form			11.12.c.a.3.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.91043 + 3.56764i) q^{2} +(-108.526 + 334.010i) q^{3} +(-621.482 - 1912.73i) q^{4} +(6091.05 - 4425.40i) q^{5} +(-1724.54 + 1252.95i) q^{6} +(20002.0 + 61559.7i) q^{7} +(7613.44 - 23431.7i) q^{8} +(43530.5 + 31626.8i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 11 q^{2} - 276 q^{3} - 6137 q^{4} + 6038 q^{5} - 30371 q^{6} + 55440 q^{7} - 347809 q^{8} - 240812 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.91043	+	3.56764i	0.108506	+	0.0788344i	0.640715	−	0.767779i	\(-0.278638\pi\)
							−0.532209	+	0.846613i	\(0.678638\pi\)
\(3\)	−108.526	+	334.010i	−0.257851	+	0.793582i	0.735404	+	0.677629i	\(0.236992\pi\)
							−0.993255	+	0.115954i	\(0.963008\pi\)
\(5\)	6091.05	−	4425.40i	0.871680	−	0.633312i	−0.0593574	−	0.998237i	\(-0.518905\pi\)
							0.931037	+	0.364924i	\(0.118905\pi\)
\(7\)	20002.0	+	61559.7i	0.449814	+	1.38439i	0.877117	+	0.480277i	\(0.159464\pi\)
							−0.427302	+	0.904109i	\(0.640536\pi\)
\(11\)	524411.	+	101514.i	0.981775	+	0.190048i
							\(13\)	395476.	+	287330.i	0.295414	+	0.214631i	0.725613	−	0.688103i	\(-0.241556\pi\)
−0.430198	+	0.902734i	\(0.641556\pi\)
\(17\)	7.90193e6	−	5.74109e6i	1.34978	−	0.980675i	0.350761	−	0.936465i	\(-0.385923\pi\)
							0.999022	−	0.0442099i	\(-0.0140770\pi\)
\(19\)	892797.	−	2.74775e6i	0.0827195	−	0.254585i	−0.901140	−	0.433529i	\(-0.857268\pi\)
							0.983859	+	0.178944i	\(0.0572682\pi\)
\(23\)	−1.93020e7			−0.625316			−0.312658	−	0.949866i	\(-0.601219\pi\)
							−0.312658	+	0.949866i	\(0.601219\pi\)
\(29\)	−3.52139e7	−	1.08377e8i	−0.318805	−	0.981182i	−0.974160	−	0.225860i	\(-0.927481\pi\)
							0.655355	−	0.755321i	\(-0.272519\pi\)
\(31\)	2.40329e8	+	1.74609e8i	1.50770	+	1.09541i	0.967183	+	0.254080i	\(0.0817728\pi\)
							0.540521	+	0.841331i	\(0.318227\pi\)
\(37\)	−1.59954e8	−	4.92288e8i	−0.379215	−	1.16711i	−0.940590	−	0.339544i	\(-0.889727\pi\)
							0.561375	−	0.827562i	\(-0.310273\pi\)
\(41\)	−1.81967e8	+	5.60036e8i	−0.245291	+	0.754927i	0.750298	+	0.661100i	\(0.229910\pi\)
							−0.995589	+	0.0938271i	\(0.970090\pi\)
\(43\)	−8.27570e8			−0.858476			−0.429238	−	0.903191i	\(-0.641218\pi\)
							−0.429238	+	0.903191i	\(0.641218\pi\)
\(47\)	9.32356e7	−	2.86950e8i	0.0592985	−	0.182502i	−0.917020	−	0.398842i	\(-0.869412\pi\)
							0.976318	+	0.216340i	\(0.0694121\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.12.c.a.4.6	yes	40
3.2	odd	2		99.12.f.a.37.5		40
11.3	even	5	inner	11.12.c.a.3.6	✓	40
11.5	even	5		121.12.a.j.1.10		20
11.6	odd	10		121.12.a.h.1.11		20
33.14	odd	10		99.12.f.a.91.5		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.12.c.a.3.6	✓	40	11.3	even	5	inner
11.12.c.a.4.6	yes	40	1.1	even	1	trivial
99.12.f.a.37.5		40	3.2	odd	2
99.12.f.a.91.5		40	33.14	odd	10
121.12.a.h.1.11		20	11.6	odd	10
121.12.a.j.1.10		20	11.5	even	5