Embedded newform 11.14.c.a.3.8

• Label: 11.14.c.a.3.8
• Level: $11$
• Weight: $14$
• Character: 11.3
• Analytic conductor: $11.795$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 11 \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	11.c (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.7954021847\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(12\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			3.8
Character	\(\chi\)	\(=\)	11.3
Dual form			11.14.c.a.4.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(55.3587 - 40.2205i) q^{2} +(-140.614 - 432.765i) q^{3} +(-1084.57 + 3337.95i) q^{4} +(7579.93 + 5507.14i) q^{5} +(-25190.2 - 18301.8i) q^{6} +(-99964.7 + 307660. i) q^{7} +(247435. + 761526. i) q^{8} +(1.12232e6 - 815414. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 59 q^{2} - 864 q^{3} - 44697 q^{4} - 52240 q^{5} - 130115 q^{6} + 390696 q^{7} + 876527 q^{8} - 3520244 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	55.3587	−	40.2205i	0.611633	−	0.444377i	−0.238356	−	0.971178i	\(-0.576608\pi\)
							0.849989	+	0.526800i	\(0.176608\pi\)
\(3\)	−140.614	−	432.765i	−0.111363	−	0.342740i	0.879808	−	0.475329i	\(-0.157671\pi\)
							−0.991171	+	0.132589i	\(0.957671\pi\)
\(5\)	7579.93	+	5507.14i	0.216950	+	0.157624i	0.690953	−	0.722900i	\(-0.257191\pi\)
							−0.474003	+	0.880523i	\(0.657191\pi\)
\(7\)	−99964.7	+	307660.i	−0.321151	+	0.988401i	0.651997	+	0.758221i	\(0.273931\pi\)
							−0.973148	+	0.230180i	\(0.926069\pi\)
\(11\)	−823144.	+	5.81766e6i	−0.140095	+	0.990138i
							\(13\)	9.41148e6	−	6.83784e6i	0.540787	−	0.392905i	−0.283590	−	0.958946i	\(-0.591526\pi\)
0.824377	+	0.566041i	\(0.191526\pi\)
\(17\)	7.79620e7	+	5.66427e7i	0.783367	+	0.569149i	0.905988	−	0.423304i	\(-0.139130\pi\)
							−0.122621	+	0.992454i	\(0.539130\pi\)
\(19\)	1.07482e8	+	3.30795e8i	0.524128	+	1.61310i	0.766034	+	0.642800i	\(0.222227\pi\)
							−0.241907	+	0.970300i	\(0.577773\pi\)
\(23\)	−9.03790e8			−1.27302			−0.636512	−	0.771267i	\(-0.719624\pi\)
							−0.636512	+	0.771267i	\(0.719624\pi\)
\(29\)	7.78356e8	−	2.39553e9i	0.242992	−	0.747851i	−0.752969	−	0.658056i	\(-0.771379\pi\)
							0.995960	−	0.0897948i	\(-0.0286211\pi\)
\(31\)	−3.70048e9	+	2.68856e9i	−0.748871	+	0.544087i	−0.895477	−	0.445108i	\(-0.853165\pi\)
							0.146606	+	0.989195i	\(0.453165\pi\)
\(37\)	−3.77533e8	+	1.16193e9i	−0.0241905	+	0.0744506i	−0.962423	−	0.271555i	\(-0.912462\pi\)
							0.938232	+	0.346005i	\(0.112462\pi\)
\(41\)	−2.18184e9	−	6.71501e9i	−0.0717344	−	0.220776i	0.908761	−	0.417316i	\(-0.137029\pi\)
							−0.980496	+	0.196541i	\(0.937029\pi\)
\(43\)	5.11284e10			1.23344			0.616719	−	0.787183i	\(-0.288462\pi\)
							0.616719	+	0.787183i	\(0.288462\pi\)
\(47\)	8.05989e9	+	2.48058e10i	0.109067	+	0.335673i	0.990663	−	0.136330i	\(-0.0435308\pi\)
							−0.881597	+	0.472004i	\(0.843531\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.14.c.a.3.8	✓	48
11.2	odd	10		121.14.a.i.1.15		24
11.4	even	5	inner	11.14.c.a.4.8	yes	48
11.9	even	5		121.14.a.h.1.10		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.14.c.a.3.8	✓	48	1.1	even	1	trivial
11.14.c.a.4.8	yes	48	11.4	even	5	inner
121.14.a.h.1.10		24	11.9	even	5
121.14.a.i.1.15		24	11.2	odd	10