Embedded newform 11.12.c.a.3.6

• Label: 11.12.c.a.3.6
• 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			3.6
Character	\(\chi\)	\(=\)	11.3
Dual form			11.12.c.a.4.6

$q$-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 \(\left(\mathbb{Z}/11\mathbb{Z}\right)^\times\).

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.12.c.a.3.6	✓	40	1.1	even	1	trivial
11.12.c.a.4.6	yes	40	11.4	even	5	inner
99.12.f.a.37.5		40	33.26	odd	10
99.12.f.a.91.5		40	3.2	odd	2
121.12.a.h.1.11		20	11.2	odd	10
121.12.a.j.1.10		20	11.9	even	5