Embedded newform 11.12.c.a.3.5

• Label: 11.12.c.a.3.5
• 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			3.5
Character	\(\chi\)	\(=\)	11.3
Dual form			11.12.c.a.4.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.97513 + 5.06773i) q^{2} +(187.776 + 577.916i) q^{3} +(-609.896 + 1877.07i) q^{4} +(-462.160 - 335.779i) q^{5} +(-4238.49 - 3079.44i) q^{6} +(5696.63 - 17532.4i) q^{7} +(-10714.8 - 32976.7i) q^{8} +(-155412. + 112914. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 11 q^{2} - 276 q^{3} - 6137 q^{4} + 6038 q^{5} - 30371 q^{6} + 55440 q^{7} - 347809 q^{8} - 240812 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.97513	+	5.06773i	−0.154130	+	0.111982i	−0.662177	−	0.749347i	\(-0.730367\pi\)
							0.508047	+	0.861329i	\(0.330367\pi\)
\(3\)	187.776	+	577.916i	0.446143	+	1.37309i	0.881225	+	0.472696i	\(0.156719\pi\)
							−0.435082	+	0.900391i	\(0.643281\pi\)
\(5\)	−462.160	−	335.779i	−0.0661390	−	0.0480528i	0.554225	−	0.832367i	\(-0.313015\pi\)
							−0.620363	+	0.784314i	\(0.713015\pi\)
\(7\)	5696.63	−	17532.4i	0.128109	−	0.394278i	−0.866346	−	0.499444i	\(-0.833538\pi\)
							0.994455	+	0.105166i	\(0.0335375\pi\)
\(11\)	−470610.	+	252662.i	−0.881051	+	0.473021i
							\(13\)	−30852.6	+	22415.7i	−0.0230464	+	0.0167442i	−0.599249	−	0.800563i	\(-0.704534\pi\)
0.576202	+	0.817307i	\(0.304534\pi\)
\(17\)	1.70096e6	+	1.23582e6i	0.290553	+	0.211099i	0.723507	−	0.690317i	\(-0.242529\pi\)
							−0.432954	+	0.901416i	\(0.642529\pi\)
\(19\)	2.97433e6	+	9.15406e6i	0.275578	+	0.848143i	0.989066	+	0.147475i	\(0.0471146\pi\)
							−0.713487	+	0.700668i	\(0.752885\pi\)
\(23\)	2.80641e7			0.909175			0.454587	−	0.890702i	\(-0.349787\pi\)
							0.454587	+	0.890702i	\(0.349787\pi\)
\(29\)	−4.22761e7	+	1.30113e8i	−0.382742	+	1.17796i	0.555363	+	0.831608i	\(0.312579\pi\)
							−0.938105	+	0.346351i	\(0.887421\pi\)
\(31\)	−1.58676e8	+	1.15285e8i	−0.995456	+	0.723241i	−0.961109	−	0.276169i	\(-0.910935\pi\)
							−0.0343466	+	0.999410i	\(0.510935\pi\)
\(37\)	−4.14690e7	+	1.27629e8i	−0.0983137	+	0.302579i	−0.988103	−	0.153793i	\(-0.950851\pi\)
							0.889789	+	0.456371i	\(0.150851\pi\)
\(41\)	4.48210e8	+	1.37945e9i	0.604186	+	1.85949i	0.502292	+	0.864698i	\(0.332490\pi\)
							0.101894	+	0.994795i	\(0.467510\pi\)
\(43\)	1.67079e9			1.73318			0.866591	−	0.499019i	\(-0.166306\pi\)
							0.866591	+	0.499019i	\(0.166306\pi\)
\(47\)	−4.24217e8	−	1.30560e9i	−0.269805	−	0.830374i	−0.990547	−	0.137171i	\(-0.956199\pi\)
							0.720743	−	0.693203i	\(-0.243801\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.12.c.a.3.5	✓	40
3.2	odd	2		99.12.f.a.91.6		40
11.2	odd	10		121.12.a.h.1.10		20
11.4	even	5	inner	11.12.c.a.4.5	yes	40
11.9	even	5		121.12.a.j.1.11		20
33.26	odd	10		99.12.f.a.37.6		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.12.c.a.3.5	✓	40	1.1	even	1	trivial
11.12.c.a.4.5	yes	40	11.4	even	5	inner
99.12.f.a.37.6		40	33.26	odd	10
99.12.f.a.91.6		40	3.2	odd	2
121.12.a.h.1.10		20	11.2	odd	10
121.12.a.j.1.11		20	11.9	even	5