Embedded newform 115.4.g.a.6.9

• Label: 115.4.g.a.6.9
• Level: $115$
• Weight: $4$
• Character: 115.6
• 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			6.9
Character	\(\chi\)	\(=\)	115.6
Dual form			115.4.g.a.96.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.857442 - 1.87754i) q^{2} +(-4.53505 - 1.33161i) q^{3} +(2.44895 + 2.82624i) q^{4} +(4.20627 - 2.70320i) q^{5} +(-6.38869 + 7.37294i) q^{6} +(1.83914 - 12.7915i) q^{7} +(23.2498 - 6.82676i) q^{8} +(-3.92038 - 2.51948i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 110 q - 2 q^{2} + 6 q^{3} - 40 q^{4} - 55 q^{5} - 3 q^{6} - 10 q^{7} + 243 q^{8} - 233 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{9}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.857442	−	1.87754i	0.303152	−	0.663809i	−0.695342	−	0.718679i	\(-0.744747\pi\)
							0.998494	+	0.0548697i	\(0.0174743\pi\)
\(3\)	−4.53505	−	1.33161i	−0.872770	−	0.256268i	−0.185477	−	0.982649i	\(-0.559383\pi\)
							−0.687293	+	0.726380i	\(0.741201\pi\)
\(5\)	4.20627	−	2.70320i	0.376220	−	0.241782i
							\(7\)	1.83914	−	12.7915i	0.0993041	−	0.690675i	−0.877973	−	0.478710i	\(-0.841105\pi\)
0.977277	−	0.211965i	\(-0.0679864\pi\)
\(11\)	−0.872387	−	1.91026i	−0.0239122	−	0.0523605i	0.897299	−	0.441423i	\(-0.145526\pi\)
							−0.921211	+	0.389063i	\(0.872799\pi\)
\(13\)	−8.93876	−	62.1704i	−0.190705	−	1.32638i	−0.830148	−	0.557544i	\(-0.811744\pi\)
							0.639443	−	0.768839i	\(-0.279165\pi\)
\(17\)	11.3782	−	13.1312i	0.162331	−	0.187340i	−0.668757	−	0.743481i	\(-0.733173\pi\)
							0.831088	+	0.556141i	\(0.187719\pi\)
\(19\)	−38.9485	−	44.9490i	−0.470285	−	0.542737i	0.470206	−	0.882557i	\(-0.344180\pi\)
							−0.940491	+	0.339819i	\(0.889634\pi\)
\(23\)	−65.8584	−	88.4854i	−0.597062	−	0.802195i
							\(29\)	5.07537	−	5.85729i	0.0324991	−	0.0375059i	−0.739267	−	0.673412i	\(-0.764828\pi\)
0.771766	+	0.635906i	\(0.219373\pi\)
\(31\)	12.9633	−	3.80635i	0.0751055	−	0.0220530i	−0.243964	−	0.969784i	\(-0.578448\pi\)
							0.319069	+	0.947731i	\(0.396630\pi\)
\(37\)	311.189	+	199.989i	1.38268	+	0.888595i	0.999386	−	0.0350317i	\(-0.0111532\pi\)
							0.383294	+	0.923626i	\(0.374790\pi\)
\(41\)	33.4303	−	21.4843i	0.127340	−	0.0818364i	−0.475420	−	0.879759i	\(-0.657704\pi\)
							0.602760	+	0.797922i	\(0.294068\pi\)
\(43\)	213.228	+	62.6094i	0.756209	+	0.222043i	0.637040	−	0.770831i	\(-0.280159\pi\)
							0.119169	+	0.992874i	\(0.461977\pi\)
\(47\)	−85.8592			−0.266465			−0.133232	−	0.991085i	\(-0.542536\pi\)
							−0.133232	+	0.991085i	\(0.542536\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.4.g.a.6.9	✓	110
23.4	even	11	inner	115.4.g.a.96.9	yes	110

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.4.g.a.6.9	✓	110	1.1	even	1	trivial
115.4.g.a.96.9	yes	110	23.4	even	11	inner