Embedded newform 115.5.i.a.14.13

• Label: 115.5.i.a.14.13
• 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			14.13
Character	\(\chi\)	\(=\)	115.14
Dual form			115.5.i.a.74.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.13341 - 3.86005i) q^{2} +(2.81764 - 2.44150i) q^{3} +(-0.155323 + 0.0998199i) q^{4} +(1.67039 + 24.9441i) q^{5} +(-12.6179 - 8.10901i) q^{6} +(-7.78355 + 17.0436i) q^{7} +(-48.0849 - 41.6658i) q^{8} +(-9.54933 + 66.4170i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 460 q + 306 q^{4} - 11 q^{5} - 182 q^{6} + 944 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.13341	−	3.86005i	−0.283353	−	0.965013i	−0.971022	−	0.238989i	\(-0.923184\pi\)
							0.687669	−	0.726024i	\(-0.258634\pi\)
\(3\)	2.81764	−	2.44150i	0.313071	−	0.271278i	−0.484124	−	0.874999i	\(-0.660862\pi\)
							0.797195	+	0.603722i	\(0.206316\pi\)
\(5\)	1.67039	+	24.9441i	0.0668158	+	0.997765i
							\(7\)	−7.78355	+	17.0436i	−0.158848	+	0.347829i	−0.972276	−	0.233837i	\(-0.924872\pi\)
0.813428	+	0.581666i	\(0.197599\pi\)
\(11\)	−33.2401	+	113.206i	−0.274712	+	0.935583i	0.700378	+	0.713772i	\(0.253015\pi\)
							−0.975090	+	0.221810i	\(0.928803\pi\)
\(13\)	−297.336	+	135.789i	−1.75938	+	0.803483i	−0.773987	+	0.633201i	\(0.781740\pi\)
							−0.985395	+	0.170282i	\(0.945532\pi\)
\(17\)	−451.445	−	290.126i	−1.56209	−	1.00390i	−0.981896	−	0.189422i	\(-0.939339\pi\)
							−0.580199	−	0.814475i	\(-0.697025\pi\)
\(19\)	55.9256	+	87.0219i	0.154919	+	0.241058i	0.910034	−	0.414534i	\(-0.136055\pi\)
							−0.755115	+	0.655592i	\(0.772419\pi\)
\(23\)	528.055	−	31.6084i	0.998213	−	0.0597512i
							\(29\)	499.274	+	320.864i	0.593667	+	0.381526i	0.802701	−	0.596382i	\(-0.203396\pi\)
−0.209034	+	0.977908i	\(0.567032\pi\)
\(31\)	796.186	−	918.847i	0.828497	−	0.956137i	−0.171079	−	0.985257i	\(-0.554725\pi\)
							0.999576	+	0.0291207i	\(0.00927072\pi\)
\(37\)	53.3388	−	370.980i	0.0389619	−	0.270986i	−0.961023	−	0.276468i	\(-0.910836\pi\)
							0.999985	+	0.00548226i	\(0.00174507\pi\)
\(41\)	−26.7916	−	186.339i	−0.0159379	−	0.110850i	0.980300	−	0.197517i	\(-0.0632877\pi\)
							−0.996237	+	0.0866664i	\(0.972379\pi\)
\(43\)	−83.5247	−	96.3926i	−0.0451729	−	0.0521323i	0.732713	−	0.680537i	\(-0.238254\pi\)
							−0.777886	+	0.628405i	\(0.783708\pi\)
\(47\)			2179.69i			0.986733i	0.869822	+	0.493367i	\(0.164234\pi\)
							−0.869822	+	0.493367i	\(0.835766\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.5.i.a.14.13	✓	460
5.4	even	2	inner	115.5.i.a.14.34	yes	460
23.5	odd	22	inner	115.5.i.a.74.34	yes	460
115.74	odd	22	inner	115.5.i.a.74.13	yes	460

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.5.i.a.14.13	✓	460	1.1	even	1	trivial
115.5.i.a.14.34	yes	460	5.4	even	2	inner
115.5.i.a.74.13	yes	460	115.74	odd	22	inner
115.5.i.a.74.34	yes	460	23.5	odd	22	inner