Embedded newform 11.4.c.a.4.2

• Label: 11.4.c.a.4.2
• Level: $11$
• Weight: $4$
• Character: 11.4
• Analytic conductor: $0.649$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(0.649021010063\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{5})\)
Coefficient field:	8.0.29283765625.1
Defining polynomial:	\( x^{8} - x^{7} + 10x^{6} - 19x^{5} + 109x^{4} + 171x^{3} + 810x^{2} + 729x + 6561 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			4.2
Root			\(2.86504 + 2.08157i\) of defining polynomial
Character	\(\chi\)	\(=\)	11.4
Dual form			11.4.c.a.3.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.747004 + 0.542730i) q^{2} +(-0.476313 + 1.46594i) q^{3} +(-2.20868 - 6.79761i) q^{4} +(-7.05908 + 5.12872i) q^{5} +(-1.15142 + 0.836554i) q^{6} +(0.239524 + 0.737179i) q^{7} +(4.32202 - 13.3018i) q^{8} +(19.9214 + 14.4737i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 7 q^{2} - 3 q^{3} + 3 q^{4} - 7 q^{5} - 29 q^{6} - 35 q^{7} + 47 q^{8} + 31 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.747004	+	0.542730i	0.264106	+	0.191884i	0.711955	−	0.702225i	\(-0.247810\pi\)
							−0.447849	+	0.894109i	\(0.647810\pi\)
\(3\)	−0.476313	+	1.46594i	−0.0916665	+	0.282120i	−0.986371	−	0.164539i	\(-0.947386\pi\)
							0.894704	+	0.446659i	\(0.147386\pi\)
\(5\)	−7.05908	+	5.12872i	−0.631383	+	0.458727i	−0.856879	−	0.515518i	\(-0.827600\pi\)
							0.225496	+	0.974244i	\(0.427600\pi\)
\(7\)	0.239524	+	0.737179i	0.0129331	+	0.0398039i	0.957315	−	0.289047i	\(-0.0933385\pi\)
							−0.944382	+	0.328851i	\(0.893339\pi\)
\(11\)	32.3688	+	16.8303i	0.887233	+	0.461321i
							\(13\)	−52.2644	−	37.9723i	−1.11504	−	0.810124i	−0.131590	−	0.991304i	\(-0.542008\pi\)
−0.983450	+	0.181180i	\(0.942008\pi\)
\(17\)	−56.7325	+	41.2186i	−0.809391	+	0.588057i	−0.913654	−	0.406493i	\(-0.866752\pi\)
							0.104263	+	0.994550i	\(0.466752\pi\)
\(19\)	40.0535	−	123.272i	0.483627	−	1.48845i	−0.350333	−	0.936625i	\(-0.613932\pi\)
							0.833960	−	0.551825i	\(-0.186068\pi\)
\(23\)	103.308			0.936572			0.468286	−	0.883577i	\(-0.344872\pi\)
							0.468286	+	0.883577i	\(0.344872\pi\)
\(29\)	−21.2472	−	65.3921i	−0.136052	−	0.418724i	0.859700	−	0.510799i	\(-0.170650\pi\)
							−0.995752	+	0.0920744i	\(0.970650\pi\)
\(31\)	−4.63339	−	3.36636i	−0.0268446	−	0.0195037i	0.574282	−	0.818658i	\(-0.305281\pi\)
							−0.601127	+	0.799154i	\(0.705281\pi\)
\(37\)	53.2416	+	163.861i	0.236564	+	0.728069i	0.996910	+	0.0785508i	\(0.0250293\pi\)
							−0.760346	+	0.649518i	\(0.774971\pi\)
\(41\)	−65.3332	+	201.075i	−0.248862	+	0.765917i	0.746116	+	0.665816i	\(0.231917\pi\)
							−0.994977	+	0.100101i	\(0.968083\pi\)
\(43\)	300.793			1.06676			0.533378	−	0.845877i	\(-0.320922\pi\)
							0.533378	+	0.845877i	\(0.320922\pi\)
\(47\)	−59.0277	+	181.669i	−0.183193	+	0.563811i	−0.999913	−	0.0132245i	\(-0.995790\pi\)
							0.816719	+	0.577035i	\(0.195790\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.4.c.a.4.2	yes	8
3.2	odd	2		99.4.f.c.37.1		8
4.3	odd	2		176.4.m.c.81.2		8
11.2	odd	10		121.4.c.b.27.2		8
11.3	even	5	inner	11.4.c.a.3.2	✓	8
11.4	even	5		121.4.c.h.9.1		8
11.5	even	5		121.4.a.g.1.2		4
11.6	odd	10		121.4.a.f.1.3		4
11.7	odd	10		121.4.c.b.9.2		8
11.8	odd	10		121.4.c.i.3.1		8
11.9	even	5		121.4.c.h.27.1		8
11.10	odd	2		121.4.c.i.81.1		8
33.5	odd	10		1089.4.a.y.1.3		4
33.14	odd	10		99.4.f.c.91.1		8
33.17	even	10		1089.4.a.bh.1.2		4
44.3	odd	10		176.4.m.c.113.2		8
44.27	odd	10		1936.4.a.bk.1.4		4
44.39	even	10		1936.4.a.bl.1.4		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.4.c.a.3.2	✓	8	11.3	even	5	inner
11.4.c.a.4.2	yes	8	1.1	even	1	trivial
99.4.f.c.37.1		8	3.2	odd	2
99.4.f.c.91.1		8	33.14	odd	10
121.4.a.f.1.3		4	11.6	odd	10
121.4.a.g.1.2		4	11.5	even	5
121.4.c.b.9.2		8	11.7	odd	10
121.4.c.b.27.2		8	11.2	odd	10
121.4.c.h.9.1		8	11.4	even	5
121.4.c.h.27.1		8	11.9	even	5
121.4.c.i.3.1		8	11.8	odd	10
121.4.c.i.81.1		8	11.10	odd	2
176.4.m.c.81.2		8	4.3	odd	2
176.4.m.c.113.2		8	44.3	odd	10
1089.4.a.y.1.3		4	33.5	odd	10
1089.4.a.bh.1.2		4	33.17	even	10
1936.4.a.bk.1.4		4	44.27	odd	10
1936.4.a.bl.1.4		4	44.39	even	10