Embedded newform 231.2.y.a.16.5

• Label: 231.2.y.a.16.5
• Level: $231$
• Weight: $2$
• Character: 231.16
• Analytic conductor: $1.845$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 231 = 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	231.y (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.84454428669\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(8\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			16.5
Character	\(\chi\)	\(=\)	231.16
Dual form			231.2.y.a.130.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.17765 + 0.250317i) q^{2} +(0.104528 - 0.994522i) q^{3} +(-0.502893 - 0.223903i) q^{4} +(-2.18435 - 2.42597i) q^{5} +(0.372043 - 1.14503i) q^{6} +(1.90032 - 1.84087i) q^{7} +(-2.48423 - 1.80490i) q^{8} +(-0.978148 - 0.207912i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 8 q^{3} + 10 q^{4} - q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/231\mathbb{Z}\right)^\times\).

\(n\)	\(155\)	\(199\)	\(211\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.17765	+	0.250317i	0.832723	+	0.177001i	0.604497	−	0.796607i	\(-0.293374\pi\)
							0.228226	+	0.973608i	\(0.426707\pi\)
\(3\)	0.104528	−	0.994522i	0.0603495	−	0.574187i
							\(5\)	−2.18435	−	2.42597i	−0.976873	−	1.08493i	−0.996371	−	0.0851176i	\(-0.972873\pi\)
0.0194978	−	0.999810i	\(-0.493793\pi\)
\(7\)	1.90032	−	1.84087i	0.718252	−	0.695783i
							\(11\)	−1.79664	+	2.78785i	−0.541706	+	0.840568i
\(13\)	0.920706	+	2.83364i	0.255358	+	0.785911i								0.993759	+	0.111549i	\(0.0355812\pi\)
							−0.738401	+	0.674362i	\(0.764419\pi\)
\(17\)	6.84067	−	1.45403i	1.65911	−	0.352654i	0.719390	−	0.694606i	\(-0.244421\pi\)
							0.939717	+	0.341952i	\(0.111088\pi\)
\(19\)	6.81727	−	3.03524i	1.56399	−	0.696333i	0.571721	−	0.820448i	\(-0.306276\pi\)
							0.992268	+	0.124116i	\(0.0396094\pi\)
\(23\)	1.57455	−	2.72720i	0.328316	−	0.568661i	−0.653862	−	0.756614i	\(-0.726852\pi\)
							0.982178	+	0.187954i	\(0.0601854\pi\)
\(29\)	−1.62105	+	1.17776i	−0.301021	+	0.218705i	−0.728034	−	0.685541i	\(-0.759566\pi\)
							0.427013	+	0.904245i	\(0.359566\pi\)
\(31\)	−0.746561	+	0.829140i	−0.134086	+	0.148918i	−0.806448	−	0.591304i	\(-0.798613\pi\)
							0.672362	+	0.740222i	\(0.265280\pi\)
\(37\)	0.300218	+	2.85639i	0.0493556	+	0.469587i	0.991087	+	0.133220i	\(0.0425317\pi\)
							−0.941731	+	0.336367i	\(0.890802\pi\)
\(41\)	−8.66274	−	6.29385i	−1.35289	−	0.982934i	−0.998862	−	0.0476997i	\(-0.984811\pi\)
							−0.354030	−	0.935234i	\(-0.615189\pi\)
\(43\)	0.903777			0.137825			0.0689124	−	0.997623i	\(-0.478047\pi\)
							0.0689124	+	0.997623i	\(0.478047\pi\)
\(47\)	−4.05724	+	1.80640i	−0.591809	+	0.263490i	−0.680713	−	0.732550i	\(-0.738330\pi\)
							0.0889046	+	0.996040i	\(0.471663\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	231.2.y.a.16.5	✓	64
3.2	odd	2		693.2.by.d.478.4		64
7.4	even	3	inner	231.2.y.a.214.4	yes	64
11.9	even	5	inner	231.2.y.a.163.4	yes	64
21.11	odd	6		693.2.by.d.676.5		64
33.20	odd	10		693.2.by.d.163.5		64
77.53	even	15	inner	231.2.y.a.130.5	yes	64
231.53	odd	30		693.2.by.d.361.4		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.y.a.16.5	✓	64	1.1	even	1	trivial
231.2.y.a.130.5	yes	64	77.53	even	15	inner
231.2.y.a.163.4	yes	64	11.9	even	5	inner
231.2.y.a.214.4	yes	64	7.4	even	3	inner
693.2.by.d.163.5		64	33.20	odd	10
693.2.by.d.361.4		64	231.53	odd	30
693.2.by.d.478.4		64	3.2	odd	2
693.2.by.d.676.5		64	21.11	odd	6