Embedded newform 11.13.d.a.6.9

• Label: 11.13.d.a.6.9
• Level: $11$
• Weight: $13$
• Character: 11.6
• Analytic conductor: $10.054$
• Analytic rank: $0$
• Dimension: $44$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 11 \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	11.d (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.0539319900\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(11\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			6.9
Character	\(\chi\)	\(=\)	11.6
Dual form			11.13.d.a.2.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(73.1050 - 23.7533i) q^{2} +(491.809 + 357.320i) q^{3} +(1466.40 - 1065.40i) q^{4} +(-9397.34 + 28922.0i) q^{5} +(44441.2 + 14439.8i) q^{6} +(26666.1 + 36702.7i) q^{7} +(-103169. + 142000. i) q^{8} +(-50026.1 - 153965. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 5 q^{2} - 1083 q^{3} + 16247 q^{4} - 723 q^{5} + 147195 q^{6} - 219605 q^{7} + 921595 q^{8} - 3695582 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{9}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	73.1050	−	23.7533i	1.14227	−	0.371145i	0.324041	−	0.946043i	\(-0.394958\pi\)
							0.818225	+	0.574898i	\(0.194958\pi\)
\(3\)	491.809	+	357.320i	0.674634	+	0.490151i	0.871573	−	0.490265i	\(-0.163100\pi\)
							−0.196939	+	0.980416i	\(0.563100\pi\)
\(5\)	−9397.34	+	28922.0i	−0.601430	+	1.85101i	−0.0817448	+	0.996653i	\(0.526049\pi\)
							−0.519685	+	0.854358i	\(0.673951\pi\)
\(7\)	26666.1	+	36702.7i	0.226658	+	0.311968i	0.907166	−	0.420773i	\(-0.138241\pi\)
							−0.680508	+	0.732740i	\(0.738241\pi\)
\(11\)	216586.	−	1.75827e6i	0.122257	−	0.992498i
							\(13\)	6.06247e6	−	1.96982e6i	1.25600	−	0.408099i	0.395932	−	0.918280i	\(-0.370421\pi\)
0.860067	+	0.510181i	\(0.170421\pi\)
\(17\)	3.22928e7	+	1.04926e7i	1.33787	+	0.434699i	0.888593	−	0.458696i	\(-0.151683\pi\)
							0.449273	+	0.893395i	\(0.351683\pi\)
\(19\)	3.76244e6	−	5.17855e6i	0.0799738	−	0.110074i	−0.767156	−	0.641461i	\(-0.778329\pi\)
							0.847130	+	0.531386i	\(0.178329\pi\)
\(23\)	7.05718e7			0.476721			0.238360	−	0.971177i	\(-0.423390\pi\)
							0.238360	+	0.971177i	\(0.423390\pi\)
\(29\)	−1.53129e7	−	2.10764e7i	−0.0257436	−	0.0354330i	0.795951	−	0.605361i	\(-0.206971\pi\)
							−0.821695	+	0.569928i	\(0.806971\pi\)
\(31\)	2.23990e8	+	6.89369e8i	0.252381	+	0.776750i	0.994334	+	0.106298i	\(0.0338998\pi\)
							−0.741953	+	0.670452i	\(0.766100\pi\)
\(37\)	−2.12527e9	+	1.54410e9i	−0.828330	+	0.601817i	−0.919086	−	0.394056i	\(-0.871072\pi\)
							0.0907564	+	0.995873i	\(0.471072\pi\)
\(41\)	−2.68646e7	+	3.69759e7i	−0.00565557	+	0.00778423i	−0.811835	−	0.583886i	\(-0.801531\pi\)
							0.806180	+	0.591671i	\(0.201531\pi\)
\(43\)			2.66785e9i			0.422037i	0.977482	+	0.211019i	\(0.0676781\pi\)
							−0.977482	+	0.211019i	\(0.932322\pi\)
\(47\)	6.45838e9	+	4.69229e9i	0.599151	+	0.435309i	0.845578	−	0.533853i	\(-0.179256\pi\)
							−0.246426	+	0.969162i	\(0.579256\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.13.d.a.6.9	yes	44
11.2	odd	10	inner	11.13.d.a.2.9	✓	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.13.d.a.2.9	✓	44	11.2	odd	10	inner
11.13.d.a.6.9	yes	44	1.1	even	1	trivial