Embedded newform 11.14.c.a.4.4

• Label: 11.14.c.a.4.4
• Level: $11$
• Weight: $14$
• Character: 11.4
• 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			4.4
Character	\(\chi\)	\(=\)	11.4
Dual form			11.14.c.a.3.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-64.6899 - 46.9999i) q^{2} +(-352.519 + 1084.94i) q^{3} +(-555.683 - 1710.22i) q^{4} +(-37053.3 + 26920.8i) q^{5} +(73796.5 - 53616.3i) q^{6} +(24379.1 + 75031.2i) q^{7} +(-246852. + 759732. i) q^{8} +(237007. + 172195. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 59 q^{2} - 864 q^{3} - 44697 q^{4} - 52240 q^{5} - 130115 q^{6} + 390696 q^{7} + 876527 q^{8} - 3520244 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−64.6899	−	46.9999i	−0.714729	−	0.519281i	0.169967	−	0.985450i	\(-0.445634\pi\)
							−0.884696	+	0.466169i	\(0.845634\pi\)
\(3\)	−352.519	+	1084.94i	−0.279186	+	0.859247i	0.708895	+	0.705314i	\(0.249194\pi\)
							−0.988081	+	0.153933i	\(0.950806\pi\)
\(5\)	−37053.3	+	26920.8i	−1.06053	+	0.770519i	−0.974186	−	0.225747i	\(-0.927518\pi\)
							−0.0863418	+	0.996266i	\(0.527518\pi\)
\(7\)	24379.1	+	75031.2i	0.0783215	+	0.241049i	0.982550	−	0.186001i	\(-0.0595528\pi\)
							−0.904228	+	0.427050i	\(0.859553\pi\)
\(11\)	2.03767e6	−	5.51096e6i	0.346801	−	0.937939i
							\(13\)	−2.36261e7	−	1.71653e7i	−1.35756	−	0.986327i	−0.998596	−	0.0529765i	\(-0.983129\pi\)
−0.358967	−	0.933350i	\(-0.616871\pi\)
\(17\)	1.34649e8	−	9.78279e7i	1.35296	−	0.982981i	0.354098	−	0.935208i	\(-0.384788\pi\)
							0.998858	−	0.0477723i	\(-0.0152122\pi\)
\(19\)	−1.15105e7	+	3.54258e7i	−0.0561302	+	0.172751i	−0.975191	−	0.221364i	\(-0.928949\pi\)
							0.919061	+	0.394115i	\(0.128949\pi\)
\(23\)	6.62434e8			0.933065			0.466532	−	0.884504i	\(-0.345503\pi\)
							0.466532	+	0.884504i	\(0.345503\pi\)
\(29\)	−8.31844e8	−	2.56015e9i	−0.259690	−	0.799243i	−0.992869	−	0.119208i	\(-0.961965\pi\)
							0.733180	−	0.680035i	\(-0.238035\pi\)
\(31\)	2.14564e9	+	1.55890e9i	0.434216	+	0.315476i	0.783333	−	0.621603i	\(-0.213518\pi\)
							−0.349116	+	0.937079i	\(0.613518\pi\)
\(37\)	3.98152e9	+	1.22539e10i	0.255116	+	0.785166i	0.993807	+	0.111122i	\(0.0354443\pi\)
							−0.738691	+	0.674044i	\(0.764556\pi\)
\(41\)	1.15972e10	−	3.56927e10i	0.381294	−	1.17350i	−0.557839	−	0.829949i	\(-0.688370\pi\)
							0.939133	−	0.343553i	\(-0.111630\pi\)
\(43\)	−2.38866e9			−0.0576249			−0.0288124	−	0.999585i	\(-0.509173\pi\)
							−0.0288124	+	0.999585i	\(0.509173\pi\)
\(47\)	−1.47373e10	+	4.53567e10i	−0.199426	+	0.613769i	0.800471	+	0.599372i	\(0.204583\pi\)
							−0.999896	+	0.0143973i	\(0.995417\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.14.c.a.4.4	yes	48
11.3	even	5	inner	11.14.c.a.3.4	✓	48
11.5	even	5		121.14.a.h.1.18		24
11.6	odd	10		121.14.a.i.1.7		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.14.c.a.3.4	✓	48	11.3	even	5	inner
11.14.c.a.4.4	yes	48	1.1	even	1	trivial
121.14.a.h.1.18		24	11.5	even	5
121.14.a.i.1.7		24	11.6	odd	10