Embedded newform 115.5.i.a.14.17

• Label: 115.5.i.a.14.17
• 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			14.17
Character	\(\chi\)	\(=\)	115.14
Dual form			115.5.i.a.74.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.674293 - 2.29643i) q^{2} +(0.167586 - 0.145214i) q^{3} +(8.64114 - 5.55332i) q^{4} +(-14.8202 - 20.1336i) q^{5} +(-0.446476 - 0.286933i) q^{6} +(-39.0504 + 85.5085i) q^{7} +(-47.5202 - 41.1765i) q^{8} +(-11.5205 + 80.1269i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 460 q + 306 q^{4} - 11 q^{5} - 182 q^{6} + 944 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.674293	−	2.29643i	−0.168573	−	0.574108i	−0.999834	−	0.0182248i	\(-0.994199\pi\)
							0.831261	−	0.555883i	\(-0.187620\pi\)
\(3\)	0.167586	−	0.145214i	0.0186207	−	0.0161349i	−0.645501	−	0.763760i	\(-0.723351\pi\)
							0.664121	+	0.747625i	\(0.268806\pi\)
\(5\)	−14.8202	−	20.1336i	−0.592807	−	0.805344i
							\(7\)	−39.0504	+	85.5085i	−0.796948	+	1.74507i	−0.141352	+	0.989959i	\(0.545145\pi\)
−0.655595	+	0.755112i	\(0.727582\pi\)
\(11\)	20.3215	−	69.2086i	0.167946	−	0.571972i	−0.831908	−	0.554913i	\(-0.812752\pi\)
							0.999854	−	0.0170589i	\(-0.00543027\pi\)
\(13\)	−107.160	+	48.9385i	−0.634085	+	0.289577i	−0.706409	−	0.707804i	\(-0.749686\pi\)
							0.0723233	+	0.997381i	\(0.476959\pi\)
\(17\)	−50.0938	−	32.1933i	−0.173335	−	0.111396i	0.451097	−	0.892475i	\(-0.351033\pi\)
							−0.624432	+	0.781079i	\(0.714669\pi\)
\(19\)	283.988	+	441.893i	0.786669	+	1.22408i	0.970495	+	0.241122i	\(0.0775155\pi\)
							−0.183825	+	0.982959i	\(0.558848\pi\)
\(23\)	−524.361	−	69.9050i	−0.991230	−	0.132145i
							\(29\)	252.729	+	162.419i	0.300510	+	0.193126i	0.682202	−	0.731163i	\(-0.261022\pi\)
−0.381692	+	0.924289i	\(0.624659\pi\)
\(31\)	−455.547	+	525.729i	−0.474034	+	0.547064i	−0.941529	−	0.336931i	\(-0.890611\pi\)
							0.467495	+	0.883996i	\(0.345156\pi\)
\(37\)	−27.0061	+	187.831i	−0.0197268	+	0.137203i	−0.997305	−	0.0733710i	\(-0.976624\pi\)
							0.977578	+	0.210574i	\(0.0675334\pi\)
\(41\)	176.054	+	1224.48i	0.104732	+	0.728426i	0.972744	+	0.231882i	\(0.0744885\pi\)
							−0.868012	+	0.496543i	\(0.834602\pi\)
\(43\)	79.8015	+	92.0958i	0.0431593	+	0.0498084i	0.776919	−	0.629601i	\(-0.216782\pi\)
							−0.733759	+	0.679409i	\(0.762236\pi\)
\(47\)		−	1741.67i		−	0.788444i	−0.919015	−	0.394222i	\(-0.871014\pi\)
							0.919015	−	0.394222i	\(-0.128986\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.5.i.a.14.17	✓	460
5.4	even	2	inner	115.5.i.a.14.30	yes	460
23.5	odd	22	inner	115.5.i.a.74.30	yes	460
115.74	odd	22	inner	115.5.i.a.74.17	yes	460

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.5.i.a.14.17	✓	460	1.1	even	1	trivial
115.5.i.a.14.30	yes	460	5.4	even	2	inner
115.5.i.a.74.17	yes	460	115.74	odd	22	inner
115.5.i.a.74.30	yes	460	23.5	odd	22	inner