Embedded newform 115.5.i.a.14.8

• Label: 115.5.i.a.14.8
• Level: $115$
• Weight: $5$
• Character: 115.14
• Analytic conductor: $11.888$
• Analytic rank: $0$
• Dimension: $460$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 115 = 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	115.i (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8875457546\)
Analytic rank:	\(0\)
Dimension:	\(460\)
Relative dimension:	\(46\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			14.8
Character	\(\chi\)	\(=\)	115.14
Dual form			115.5.i.a.74.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.72741 - 5.88302i) q^{2} +(-0.556901 + 0.482558i) q^{3} +(-18.1659 + 11.6745i) q^{4} +(22.6244 - 10.6365i) q^{5} +(3.80090 + 2.44269i) q^{6} +(-11.9218 + 26.1051i) q^{7} +(25.9210 + 22.4607i) q^{8} +(-11.4502 + 79.6381i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 460 q + 306 q^{4} - 11 q^{5} - 182 q^{6} + 944 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{21}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.72741	−	5.88302i	−0.431853	−	1.47076i	−0.832243	−	0.554411i	\(-0.812944\pi\)
							0.400390	−	0.916345i	\(-0.368875\pi\)
\(3\)	−0.556901	+	0.482558i	−0.0618779	+	0.0536175i	−0.685251	−	0.728307i	\(-0.740307\pi\)
							0.623373	+	0.781925i	\(0.285762\pi\)
\(5\)	22.6244	−	10.6365i	0.904976	−	0.425462i
							\(7\)	−11.9218	+	26.1051i	−0.243302	+	0.532756i	−0.991405	−	0.130825i	\(-0.958237\pi\)
0.748104	+	0.663582i	\(0.230965\pi\)
\(11\)	−53.9345	+	183.684i	−0.445740	+	1.51805i	0.364077	+	0.931369i	\(0.381384\pi\)
							−0.809817	+	0.586682i	\(0.800434\pi\)
\(13\)	−72.7506	+	33.2241i	−0.430477	+	0.196592i	−0.618856	−	0.785504i	\(-0.712404\pi\)
							0.188380	+	0.982096i	\(0.439676\pi\)
\(17\)	274.824	+	176.619i	0.950949	+	0.611138i	0.921479	−	0.388428i	\(-0.126982\pi\)
							0.0294701	+	0.999566i	\(0.490618\pi\)
\(19\)	−1.42215	−	2.21292i	−0.00393949	−	0.00612996i	0.839278	−	0.543702i	\(-0.182978\pi\)
							−0.843218	+	0.537572i	\(0.819342\pi\)
\(23\)	−521.358	+	89.5900i	−0.985555	+	0.169357i
							\(29\)	−290.830	−	186.905i	−0.345815	−	0.222242i	0.356190	−	0.934413i	\(-0.384075\pi\)
−0.702005	+	0.712172i	\(0.747712\pi\)
\(31\)	128.098	−	147.833i	0.133296	−	0.153832i	−0.685177	−	0.728376i	\(-0.740275\pi\)
							0.818474	+	0.574544i	\(0.194821\pi\)
\(37\)	82.4666	−	573.568i	0.0602386	−	0.418969i	−0.937281	−	0.348575i	\(-0.886666\pi\)
							0.997519	−	0.0703932i	\(-0.0224254\pi\)
\(41\)	−7.95870	−	55.3540i	−0.00473450	−	0.0329292i	0.987317	−	0.158760i	\(-0.0507495\pi\)
							−0.992052	+	0.125831i	\(0.959840\pi\)
\(43\)	144.029	+	166.219i	0.0778957	+	0.0898964i	0.793360	−	0.608753i	\(-0.208330\pi\)
							−0.715464	+	0.698650i	\(0.753785\pi\)
\(47\)			1331.26i			0.602653i	0.953521	+	0.301326i	\(0.0974294\pi\)
							−0.953521	+	0.301326i	\(0.902571\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.5.i.a.14.8	✓	460
5.4	even	2	inner	115.5.i.a.14.39	yes	460
23.5	odd	22	inner	115.5.i.a.74.39	yes	460
115.74	odd	22	inner	115.5.i.a.74.8	yes	460

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.5.i.a.14.8	✓	460	1.1	even	1	trivial
115.5.i.a.14.39	yes	460	5.4	even	2	inner
115.5.i.a.74.8	yes	460	115.74	odd	22	inner
115.5.i.a.74.39	yes	460	23.5	odd	22	inner