Embedded newform 115.4.g.a.16.2

• Label: 115.4.g.a.16.2
• Level: $115$
• Weight: $4$
• Character: 115.16
• Analytic conductor: $6.785$
• Analytic rank: $0$
• Dimension: $110$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 115 = 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	115.g (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.78521965066\)
Analytic rank:	\(0\)
Dimension:	\(110\)
Relative dimension:	\(11\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			16.2
Character	\(\chi\)	\(=\)	115.16
Dual form			115.4.g.a.36.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.78120 + 3.20967i) q^{2} +(0.620793 - 0.398960i) q^{3} +(-1.42842 - 9.93487i) q^{4} +(2.07708 + 4.54816i) q^{5} +(-0.446019 + 3.10213i) q^{6} +(-1.32223 + 0.388243i) q^{7} +(7.27793 + 4.67724i) q^{8} +(-10.9900 + 24.0647i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 110 q - 2 q^{2} + 6 q^{3} - 40 q^{4} - 55 q^{5} - 3 q^{6} - 10 q^{7} + 243 q^{8} - 233 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/115\mathbb{Z}\right)^\times\).

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{4}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.78120	+	3.20967i	−0.983301	+	1.13479i	0.00756923	+	0.999971i	\(0.497591\pi\)
							−0.990870	+	0.134819i	\(0.956955\pi\)
\(3\)	0.620793	−	0.398960i	0.119472	−	0.0767798i	−0.479540	−	0.877520i	\(-0.659197\pi\)
							0.599012	+	0.800740i	\(0.295560\pi\)
\(5\)	2.07708	+	4.54816i	0.185779	+	0.406800i
							\(7\)	−1.32223	+	0.388243i	−0.0713940	+	0.0209632i	−0.317234	−	0.948347i	\(-0.602754\pi\)
0.245841	+	0.969310i	\(0.420936\pi\)
\(11\)	−14.8819	−	17.1746i	−0.407915	−	0.470759i	0.514203	−	0.857669i	\(-0.328088\pi\)
							−0.922118	+	0.386910i	\(0.873542\pi\)
\(13\)	−74.9993	−	22.0218i	−1.60008	−	0.469827i	−0.644513	−	0.764593i	\(-0.722940\pi\)
							−0.955569	+	0.294766i	\(0.904758\pi\)
\(17\)	11.2099	−	77.9669i	0.159930	−	1.11234i	−0.738828	−	0.673894i	\(-0.764620\pi\)
							0.898758	−	0.438445i	\(-0.144470\pi\)
\(19\)	15.1094	+	105.088i	0.182439	+	1.26889i	0.850973	+	0.525209i	\(0.176013\pi\)
							−0.668534	+	0.743681i	\(0.733078\pi\)
\(23\)	−34.9765	−	104.612i	−0.317092	−	0.948395i
							\(29\)	37.5529	−	261.186i	0.240462	−	1.67245i	−0.409368	−	0.912369i	\(-0.634251\pi\)
0.649830	−	0.760080i	\(-0.274840\pi\)
\(31\)	−169.450	−	108.899i	−0.981747	−	0.630930i	−0.0518130	−	0.998657i	\(-0.516500\pi\)
							−0.929934	+	0.367726i	\(0.880136\pi\)
\(37\)	55.8043	−	122.194i	0.247951	−	0.542936i	−0.744204	−	0.667952i	\(-0.767171\pi\)
							0.992155	+	0.125016i	\(0.0398983\pi\)
\(41\)	154.173	+	337.593i	0.587265	+	1.28593i	0.937081	+	0.349112i	\(0.113517\pi\)
							−0.349816	+	0.936818i	\(0.613756\pi\)
\(43\)	−376.985	+	242.274i	−1.33697	+	0.859219i	−0.996705	−	0.0811107i	\(-0.974153\pi\)
							−0.340265	+	0.940329i	\(0.610517\pi\)
\(47\)	−376.918			−1.16977			−0.584884	−	0.811117i	\(-0.698860\pi\)
							−0.584884	+	0.811117i	\(0.698860\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.4.g.a.16.2	✓	110
23.13	even	11	inner	115.4.g.a.36.2	yes	110

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.4.g.a.16.2	✓	110	1.1	even	1	trivial
115.4.g.a.36.2	yes	110	23.13	even	11	inner