Embedded newform 11.13.d.a.6.8

• Label: 11.13.d.a.6.8
• 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			6.8
Character	\(\chi\)	\(=\)	11.6
Dual form			11.13.d.a.2.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(59.3643 - 19.2886i) q^{2} +(-999.365 - 726.081i) q^{3} +(-161.661 + 117.454i) q^{4} +(875.230 - 2693.68i) q^{5} +(-73331.7 - 23826.9i) q^{6} +(110980. + 152751. i) q^{7} +(-157610. + 216932. i) q^{8} +(307312. + 945809. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 5 q^{2} - 1083 q^{3} + 16247 q^{4} - 723 q^{5} + 147195 q^{6} - 219605 q^{7} + 921595 q^{8} - 3695582 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	59.3643	−	19.2886i	0.927568	−	0.301385i	0.194000	−	0.981002i	\(-0.437854\pi\)
							0.733568	+	0.679616i	\(0.237854\pi\)
\(3\)	−999.365	−	726.081i	−1.37087	−	0.995996i	−0.997669	−	0.0682458i	\(-0.978260\pi\)
							−0.373202	−	0.927750i	\(-0.621740\pi\)
\(5\)	875.230	−	2693.68i	0.0560147	−	0.172396i	−0.919135	−	0.393943i	\(-0.871111\pi\)
							0.975150	+	0.221547i	\(0.0711108\pi\)
\(7\)	110980.	+	152751.i	0.943318	+	1.29837i	0.954432	+	0.298427i	\(0.0964620\pi\)
							−0.0111149	+	0.999938i	\(0.503538\pi\)
\(11\)	−726651.	−	1.61568e6i	−0.410176	−	0.912007i
							\(13\)	−4.23740e6	+	1.37681e6i	−0.877888	+	0.285243i	−0.713080	−	0.701082i	\(-0.752700\pi\)
−0.164808	+	0.986326i	\(0.552700\pi\)
\(17\)	−1.01467e7	−	3.29686e6i	−0.420369	−	0.136586i	0.0911916	−	0.995833i	\(-0.470932\pi\)
							−0.511561	+	0.859247i	\(0.670932\pi\)
\(19\)	−4.40950e7	+	6.06916e7i	−0.937277	+	1.29005i	0.0196756	+	0.999806i	\(0.493737\pi\)
							−0.956953	+	0.290245i	\(0.906263\pi\)
\(23\)	−6.86801e7			−0.463942			−0.231971	−	0.972723i	\(-0.574517\pi\)
							−0.231971	+	0.972723i	\(0.574517\pi\)
\(29\)	−2.80538e8	−	3.86128e8i	−0.471633	−	0.649147i	0.505237	−	0.862980i	\(-0.331405\pi\)
							−0.976870	+	0.213834i	\(0.931405\pi\)
\(31\)	3.10656e7	+	9.56100e7i	0.0350033	+	0.107729i	0.967032	−	0.254656i	\(-0.0819624\pi\)
							−0.932028	+	0.362386i	\(0.881962\pi\)
\(37\)	1.04904e9	−	7.62169e8i	0.408865	−	0.297058i	−0.364277	−	0.931291i	\(-0.618684\pi\)
							0.773142	+	0.634233i	\(0.218684\pi\)
\(41\)	−2.00504e9	+	2.75970e9i	−0.422105	+	0.580977i	−0.966118	−	0.258099i	\(-0.916904\pi\)
							0.544014	+	0.839076i	\(0.316904\pi\)
\(43\)			1.67046e9i			0.264256i	0.991233	+	0.132128i	\(0.0421811\pi\)
							−0.991233	+	0.132128i	\(0.957819\pi\)
\(47\)	−1.34379e10	−	9.76323e9i	−1.24665	−	0.905746i	−0.248630	−	0.968599i	\(-0.579980\pi\)
							−0.998023	+	0.0628525i	\(0.979980\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.13.d.a.6.8	yes	44
11.2	odd	10	inner	11.13.d.a.2.8	✓	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.13.d.a.2.8	✓	44	11.2	odd	10	inner
11.13.d.a.6.8	yes	44	1.1	even	1	trivial