Embedded newform 11.14.c.a.3.9

• Label: 11.14.c.a.3.9
• 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			3.9
Character	\(\chi\)	\(=\)	11.3
Dual form			11.14.c.a.4.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(80.7641 - 58.6786i) q^{2} +(-728.875 - 2243.25i) q^{3} +(548.202 - 1687.19i) q^{4} +(-12915.0 - 9383.32i) q^{5} +(-190497. - 138404. i) q^{6} +(94448.4 - 290682. i) q^{7} +(197989. + 609348. i) q^{8} +(-3.21106e6 + 2.33297e6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 59 q^{2} - 864 q^{3} - 44697 q^{4} - 52240 q^{5} - 130115 q^{6} + 390696 q^{7} + 876527 q^{8} - 3520244 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	80.7641	−	58.6786i	0.892326	−	0.648313i	−0.0441575	−	0.999025i	\(-0.514060\pi\)
							0.936483	+	0.350712i	\(0.114060\pi\)
\(3\)	−728.875	−	2243.25i	−0.577251	−	1.77660i	−0.628384	−	0.777904i	\(-0.716283\pi\)
							0.0511326	−	0.998692i	\(-0.483717\pi\)
\(5\)	−12915.0	−	9383.32i	−0.369650	−	0.268566i	0.387416	−	0.921905i	\(-0.373368\pi\)
							−0.757066	+	0.653339i	\(0.773368\pi\)
\(7\)	94448.4	−	290682.i	0.303429	−	0.933859i	−0.676830	−	0.736140i	\(-0.736647\pi\)
							0.980259	−	0.197719i	\(-0.0633533\pi\)
\(11\)	5.82861e6	−	741614.i	0.992002	−	0.126219i
							\(13\)	−8.46129e6	+	6.14749e6i	−0.486189	+	0.353237i	−0.803717	−	0.595012i	\(-0.797147\pi\)
0.317528	+	0.948249i	\(0.397147\pi\)
\(17\)	−1.20241e8	−	8.73602e7i	−1.20819	−	0.877800i	−0.213123	−	0.977025i	\(-0.568364\pi\)
							−0.995065	+	0.0992250i	\(0.968364\pi\)
\(19\)	−2.64147e7	−	8.12962e7i	−0.128810	−	0.396435i	0.865766	−	0.500448i	\(-0.166832\pi\)
							−0.994576	+	0.104013i	\(0.966832\pi\)
\(23\)	−3.67179e8			−0.517186			−0.258593	−	0.965986i	\(-0.583259\pi\)
							−0.258593	+	0.965986i	\(0.583259\pi\)
\(29\)	1.35741e9	−	4.17767e9i	0.423764	−	1.30421i	−0.480410	−	0.877044i	\(-0.659512\pi\)
							0.904173	−	0.427166i	\(-0.140488\pi\)
\(31\)	2.90434e9	−	2.11012e9i	0.587754	−	0.427028i	−0.253757	−	0.967268i	\(-0.581666\pi\)
							0.841511	+	0.540240i	\(0.181666\pi\)
\(37\)	−1.51257e9	+	4.65522e9i	−0.0969181	+	0.298283i	−0.987749	−	0.156052i	\(-0.950123\pi\)
							0.890831	+	0.454335i	\(0.150123\pi\)
\(41\)	9.54030e9	+	2.93620e10i	0.313666	+	0.965364i	0.976300	+	0.216421i	\(0.0694383\pi\)
							−0.662635	+	0.748943i	\(0.730562\pi\)
\(43\)	−2.41572e9			−0.0582776			−0.0291388	−	0.999575i	\(-0.509276\pi\)
							−0.0291388	+	0.999575i	\(0.509276\pi\)
\(47\)	−3.45079e10	−	1.06204e11i	−0.466963	−	1.43717i	−0.856496	−	0.516153i	\(-0.827364\pi\)
							0.389533	−	0.921013i	\(-0.372636\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.14.c.a.3.9	✓	48
11.2	odd	10		121.14.a.i.1.18		24
11.4	even	5	inner	11.14.c.a.4.9	yes	48
11.9	even	5		121.14.a.h.1.7		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.14.c.a.3.9	✓	48	1.1	even	1	trivial
11.14.c.a.4.9	yes	48	11.4	even	5	inner
121.14.a.h.1.7		24	11.9	even	5
121.14.a.i.1.18		24	11.2	odd	10