Embedded newform 115.4.g.a.6.8

• Label: 115.4.g.a.6.8
• 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			6.8
Character	\(\chi\)	\(=\)	115.6
Dual form			115.4.g.a.96.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.356908 - 0.781519i) q^{2} +(5.20863 + 1.52939i) q^{3} +(4.75550 + 5.48814i) q^{4} +(4.20627 - 2.70320i) q^{5} +(3.05425 - 3.52479i) q^{6} +(3.53166 - 24.5632i) q^{7} +(12.5812 - 3.69418i) q^{8} +(2.07692 + 1.33475i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 110 q - 2 q^{2} + 6 q^{3} - 40 q^{4} - 55 q^{5} - 3 q^{6} - 10 q^{7} + 243 q^{8} - 233 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.356908	−	0.781519i	0.126186	−	0.276309i	−0.835986	−	0.548750i	\(-0.815104\pi\)
							0.962172	+	0.272442i	\(0.0878311\pi\)
\(3\)	5.20863	+	1.52939i	1.00240	+	0.294331i	0.741441	−	0.671018i	\(-0.234143\pi\)
							0.260960	+	0.965350i	\(0.415961\pi\)
\(5\)	4.20627	−	2.70320i	0.376220	−	0.241782i
							\(7\)	3.53166	−	24.5632i	0.190692	−	1.32629i	−0.639491	−	0.768798i	\(-0.720855\pi\)
0.830183	−	0.557491i	\(-0.188236\pi\)
\(11\)	0.0443232	+	0.0970542i	0.00121490	+	0.00266027i	0.910238	−	0.414084i	\(-0.135898\pi\)
							−0.909024	+	0.416745i	\(0.863171\pi\)
\(13\)	12.1879	+	84.7689i	0.260025	+	1.80851i	0.532588	+	0.846375i	\(0.321220\pi\)
							−0.272563	+	0.962138i	\(0.587871\pi\)
\(17\)	38.7792	−	44.7536i	0.553255	−	0.638490i	−0.408384	−	0.912810i	\(-0.633907\pi\)
							0.961638	+	0.274320i	\(0.0884528\pi\)
\(19\)	−48.9282	−	56.4661i	−0.590784	−	0.681801i	0.379104	−	0.925354i	\(-0.376232\pi\)
							−0.969888	+	0.243553i	\(0.921687\pi\)
\(23\)	−54.2260	+	96.0549i	−0.491605	+	0.870819i
							\(29\)	−26.6242	+	30.7259i	−0.170482	+	0.196747i	−0.834561	−	0.550916i	\(-0.814279\pi\)
0.664079	+	0.747663i	\(0.268824\pi\)
\(31\)	−33.2448	+	9.76157i	−0.192611	+	0.0565558i	−0.376615	−	0.926370i	\(-0.622912\pi\)
							0.184004	+	0.982925i	\(0.441094\pi\)
\(37\)	21.6659	+	13.9238i	0.0962664	+	0.0618666i	0.587888	−	0.808942i	\(-0.299959\pi\)
							−0.491622	+	0.870809i	\(0.663596\pi\)
\(41\)	−20.3901	+	13.1039i	−0.0776681	+	0.0499143i	−0.578899	−	0.815400i	\(-0.696517\pi\)
							0.501230	+	0.865314i	\(0.332881\pi\)
\(43\)	−310.070	−	91.0447i	−1.09966	−	0.322888i	−0.318940	−	0.947775i	\(-0.603327\pi\)
							−0.780715	+	0.624887i	\(0.785145\pi\)
\(47\)	−159.731			−0.495726			−0.247863	−	0.968795i	\(-0.579728\pi\)
							−0.247863	+	0.968795i	\(0.579728\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.4.g.a.6.8	✓	110
23.4	even	11	inner	115.4.g.a.96.8	yes	110

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.4.g.a.6.8	✓	110	1.1	even	1	trivial
115.4.g.a.96.8	yes	110	23.4	even	11	inner