Embedded newform 115.5.i.a.14.19

• Label: 115.5.i.a.14.19
• 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			14.19
Character	\(\chi\)	\(=\)	115.14
Dual form			115.5.i.a.74.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.617270 - 2.10223i) q^{2} +(-5.53313 + 4.79448i) q^{3} +(9.42171 - 6.05496i) q^{4} +(-12.9412 + 21.3898i) q^{5} +(13.4945 + 8.67241i) q^{6} +(-1.32728 + 2.90635i) q^{7} +(-45.0380 - 39.0256i) q^{8} +(-3.89907 + 27.1186i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 460 q + 306 q^{4} - 11 q^{5} - 182 q^{6} + 944 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.617270	−	2.10223i	−0.154318	−	0.525558i	0.845649	−	0.533740i	\(-0.179214\pi\)
							−0.999967	+	0.00818203i	\(0.997396\pi\)
\(3\)	−5.53313	+	4.79448i	−0.614792	+	0.532720i	−0.905639	−	0.424049i	\(-0.860608\pi\)
							0.290847	+	0.956769i	\(0.406063\pi\)
\(5\)	−12.9412	+	21.3898i	−0.517648	+	0.855594i
							\(7\)	−1.32728	+	2.90635i	−0.0270874	+	0.0593132i	−0.922692	−	0.385538i	\(-0.874016\pi\)
0.895605	+	0.444851i	\(0.146743\pi\)
\(11\)	56.9373	−	193.911i	0.470556	−	1.60257i	−0.292521	−	0.956259i	\(-0.594494\pi\)
							0.763077	−	0.646307i	\(-0.223688\pi\)
\(13\)	−34.0850	+	15.5661i	−0.201687	+	0.0921072i	−0.513698	−	0.857971i	\(-0.671725\pi\)
							0.312011	+	0.950078i	\(0.398997\pi\)
\(17\)	114.391	+	73.5148i	0.395817	+	0.254376i	0.723376	−	0.690455i	\(-0.242589\pi\)
							−0.327558	+	0.944831i	\(0.606226\pi\)
\(19\)	−320.630	−	498.911i	−0.888173	−	1.38202i	−0.923889	−	0.382662i	\(-0.875008\pi\)
							0.0357159	−	0.999362i	\(-0.488629\pi\)
\(23\)	−523.400	+	76.7718i	−0.989413	+	0.145126i
							\(29\)	−769.215	−	494.345i	−0.914643	−	0.587806i	−0.00354466	−	0.999994i	\(-0.501128\pi\)
−0.911099	+	0.412188i	\(0.864765\pi\)
\(31\)	596.389	−	688.270i	0.620592	−	0.716201i	−0.355227	−	0.934780i	\(-0.615597\pi\)
							0.975819	+	0.218579i	\(0.0701420\pi\)
\(37\)	75.9117	−	527.978i	0.0554505	−	0.385667i	−0.943131	−	0.332421i	\(-0.892134\pi\)
							0.998581	−	0.0532453i	\(-0.0169565\pi\)
\(41\)	303.624	+	2111.75i	0.180621	+	1.25625i	0.855299	+	0.518134i	\(0.173373\pi\)
							−0.674678	+	0.738112i	\(0.735718\pi\)
\(43\)	−1600.89	−	1847.53i	−0.865816	−	0.999205i	−0.999966	−	0.00824428i	\(-0.997376\pi\)
							0.134150	−	0.990961i	\(-0.457170\pi\)
\(47\)			1552.06i			0.702609i	0.936261	+	0.351305i	\(0.114262\pi\)
							−0.936261	+	0.351305i	\(0.885738\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.5.i.a.14.19	✓	460
5.4	even	2	inner	115.5.i.a.14.28	yes	460
23.5	odd	22	inner	115.5.i.a.74.28	yes	460
115.74	odd	22	inner	115.5.i.a.74.19	yes	460

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.5.i.a.14.19	✓	460	1.1	even	1	trivial
115.5.i.a.14.28	yes	460	5.4	even	2	inner
115.5.i.a.74.19	yes	460	115.74	odd	22	inner
115.5.i.a.74.28	yes	460	23.5	odd	22	inner