Embedded newform 115.4.g.a.6.4

• Label: 115.4.g.a.6.4
• 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			6.4
Character	\(\chi\)	\(=\)	115.6
Dual form			115.4.g.a.96.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.13426 + 2.48368i) q^{2} +(-4.84173 - 1.42166i) q^{3} +(0.356777 + 0.411743i) q^{4} +(4.20627 - 2.70320i) q^{5} +(9.02270 - 10.4128i) q^{6} +(-3.26809 + 22.7301i) q^{7} +(-22.3859 + 6.57308i) q^{8} +(-1.29264 - 0.830727i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 110 q - 2 q^{2} + 6 q^{3} - 40 q^{4} - 55 q^{5} - 3 q^{6} - 10 q^{7} + 243 q^{8} - 233 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.13426	+	2.48368i	−0.401020	+	0.878112i	0.596145	+	0.802876i	\(0.296698\pi\)
							−0.997166	+	0.0752357i	\(0.976029\pi\)
\(3\)	−4.84173	−	1.42166i	−0.931791	−	0.273598i	−0.219605	−	0.975589i	\(-0.570477\pi\)
							−0.712186	+	0.701990i	\(0.752295\pi\)
\(5\)	4.20627	−	2.70320i	0.376220	−	0.241782i
							\(7\)	−3.26809	+	22.7301i	−0.176460	+	1.22731i	0.688413	+	0.725319i	\(0.258308\pi\)
−0.864873	+	0.501990i	\(0.832601\pi\)
\(11\)	−19.7103	−	43.1595i	−0.540261	−	1.18301i	−0.961183	−	0.275910i	\(-0.911021\pi\)
							0.420922	−	0.907097i	\(-0.361707\pi\)
\(13\)	3.63873	+	25.3079i	0.0776309	+	0.539935i	0.991110	+	0.133048i	\(0.0424765\pi\)
							−0.913479	+	0.406887i	\(0.866614\pi\)
\(17\)	56.7638	−	65.5090i	0.809838	−	0.934603i	−0.189039	−	0.981970i	\(-0.560537\pi\)
							0.998878	+	0.0473663i	\(0.0150828\pi\)
\(19\)	−76.3911	−	88.1600i	−0.922385	−	1.06449i	−0.997731	−	0.0673316i	\(-0.978551\pi\)
							0.0753457	−	0.997157i	\(-0.475994\pi\)
\(23\)	−87.1373	−	67.6320i	−0.789973	−	0.613141i
							\(29\)	−32.8084	+	37.8629i	−0.210082	+	0.242447i	−0.851005	−	0.525158i	\(-0.824006\pi\)
0.640923	+	0.767605i	\(0.278552\pi\)
\(31\)	−149.245	+	43.8223i	−0.864684	+	0.253894i	−0.683853	−	0.729620i	\(-0.739697\pi\)
							−0.180831	+	0.983514i	\(0.557879\pi\)
\(37\)	−229.801	−	147.684i	−1.02106	−	0.656193i	−0.0808246	−	0.996728i	\(-0.525755\pi\)
							−0.940232	+	0.340535i	\(0.889392\pi\)
\(41\)	20.7001	−	13.3032i	0.0788492	−	0.0506733i	−0.500622	−	0.865666i	\(-0.666895\pi\)
							0.579471	+	0.814993i	\(0.303259\pi\)
\(43\)	−287.156	−	84.3167i	−1.01839	−	0.299027i	−0.270410	−	0.962745i	\(-0.587159\pi\)
							−0.747983	+	0.663718i	\(0.768978\pi\)
\(47\)	249.467			0.774222			0.387111	−	0.922033i	\(-0.373473\pi\)
							0.387111	+	0.922033i	\(0.373473\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.4.g.a.6.4	✓	110
23.4	even	11	inner	115.4.g.a.96.4	yes	110

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.4.g.a.6.4	✓	110	1.1	even	1	trivial
115.4.g.a.96.4	yes	110	23.4	even	11	inner