Embedded newform 11.13.d.a.2.3

• Label: 11.13.d.a.2.3
• Level: $11$
• Weight: $13$
• Character: 11.2
• 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			2.3
Character	\(\chi\)	\(=\)	11.2
Dual form			11.13.d.a.6.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-79.4376 - 25.8108i) q^{2} +(-641.631 + 466.172i) q^{3} +(2330.39 + 1693.13i) q^{4} +(6734.78 + 20727.5i) q^{5} +(63001.9 - 20470.5i) q^{6} +(-24839.2 + 34188.3i) q^{7} +(59673.9 + 82134.1i) q^{8} +(30149.4 - 92790.4i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 5 q^{2} - 1083 q^{3} + 16247 q^{4} - 723 q^{5} + 147195 q^{6} - 219605 q^{7} + 921595 q^{8} - 3695582 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−79.4376	−	25.8108i	−1.24121	−	0.403294i	−0.386448	−	0.922311i	\(-0.626298\pi\)
							−0.854764	+	0.519017i	\(0.826298\pi\)
\(3\)	−641.631	+	466.172i	−0.880152	+	0.639468i	−0.933292	−	0.359119i	\(-0.883077\pi\)
							0.0531397	+	0.998587i	\(0.483077\pi\)
\(5\)	6734.78	+	20727.5i	0.431026	+	1.32656i	0.897105	+	0.441818i	\(0.145666\pi\)
							−0.466079	+	0.884743i	\(0.654334\pi\)
\(7\)	−24839.2	+	34188.3i	−0.211130	+	0.290596i	−0.901427	−	0.432930i	\(-0.857480\pi\)
							0.690297	+	0.723526i	\(0.257480\pi\)
\(11\)	−1.53680e6	+	881292.i	−0.867483	+	0.497466i
							\(13\)	−4.87983e6	−	1.58555e6i	−1.01098	−	0.328489i	−0.243738	−	0.969841i	\(-0.578374\pi\)
−0.767247	+	0.641352i	\(0.778374\pi\)
\(17\)	1.85868e7	−	6.03921e6i	0.770035	−	0.250199i	0.102455	−	0.994738i	\(-0.467330\pi\)
							0.667580	+	0.744538i	\(0.267330\pi\)
\(19\)	3.34155e7	+	4.59925e7i	0.710275	+	0.977610i	0.999791	+	0.0204369i	\(0.00650571\pi\)
							−0.289516	+	0.957173i	\(0.593494\pi\)
\(23\)	−9.37518e7			−0.633304			−0.316652	−	0.948542i	\(-0.602559\pi\)
							−0.316652	+	0.948542i	\(0.602559\pi\)
\(29\)	3.56503e8	−	4.90684e8i	0.599342	−	0.824924i	−0.396306	−	0.918119i	\(-0.629708\pi\)
							0.995648	+	0.0931947i	\(0.0297079\pi\)
\(31\)	1.38899e7	−	4.27486e7i	0.0156505	−	0.0481672i	−0.942926	−	0.333002i	\(-0.891939\pi\)
							0.958577	+	0.284834i	\(0.0919386\pi\)
\(37\)	−3.47917e9	−	2.52777e9i	−1.35602	−	0.985204i	−0.998687	−	0.0512264i	\(-0.983687\pi\)
							−0.357330	−	0.933978i	\(-0.616313\pi\)
\(41\)	−1.73598e9	−	2.38937e9i	−0.365461	−	0.503014i	0.586199	−	0.810167i	\(-0.300624\pi\)
							−0.951660	+	0.307153i	\(0.900624\pi\)
\(43\)			9.17938e9i			1.45212i	0.687631	+	0.726060i	\(0.258651\pi\)
							−0.687631	+	0.726060i	\(0.741349\pi\)
\(47\)	6.98094e9	−	5.07195e9i	0.647630	−	0.470530i	−0.214833	−	0.976651i	\(-0.568921\pi\)
							0.862463	+	0.506120i	\(0.168921\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.13.d.a.2.3	✓	44
11.6	odd	10	inner	11.13.d.a.6.3	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.13.d.a.2.3	✓	44	1.1	even	1	trivial
11.13.d.a.6.3	yes	44	11.6	odd	10	inner