Embedded newform 171.2.g.c.121.11

• Label: 171.2.g.c.121.11
• Level: $171$
• Weight: $2$
• Character: 171.121
• Analytic conductor: $1.365$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	171.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.36544187456\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			121.11
Character	\(\chi\)	\(=\)	171.121
Dual form			171.2.g.c.106.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.616796 - 1.06832i) q^{2} +(0.506944 - 1.65620i) q^{3} +(0.239126 + 0.414178i) q^{4} -0.551543 q^{5} +(-1.45668 - 1.56312i) q^{6} +(-1.62156 - 2.80862i) q^{7} +3.05715 q^{8} +(-2.48602 - 1.67920i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + q^{2} - 2 q^{3} - 17 q^{4} - 6 q^{5} + 2 q^{6} + q^{7} - 36 q^{8} - 10 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/171\mathbb{Z}\right)^\times\).

\(n\)	\(20\)	\(154\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.616796	−	1.06832i	0.436140	−	0.755418i	−0.561247	−	0.827648i	\(-0.689678\pi\)
							0.997388	+	0.0722305i	\(0.0230117\pi\)
\(3\)	0.506944	−	1.65620i	0.292684	−	0.956209i
							\(5\)	−0.551543			−0.246658			−0.123329	−	0.992366i	\(-0.539357\pi\)
−0.123329	+	0.992366i	\(0.539357\pi\)
\(7\)	−1.62156	−	2.80862i	−0.612892	−	1.06156i	−0.990750	−	0.135697i	\(-0.956673\pi\)
							0.377858	−	0.925863i	\(-0.376661\pi\)
\(11\)	2.68677	+	4.65363i	0.810093	+	1.40312i	0.912799	+	0.408410i	\(0.133917\pi\)
							−0.102706	+	0.994712i	\(0.532750\pi\)
\(13\)	−1.76195	−	3.05178i	−0.488676	−	0.846411i	0.511239	−	0.859438i	\(-0.329187\pi\)
							−0.999915	+	0.0130271i	\(0.995853\pi\)
\(17\)	2.60168	+	4.50624i	0.630999	+	1.09292i	0.987348	+	0.158570i	\(0.0506882\pi\)
							−0.356349	+	0.934353i	\(0.615978\pi\)
\(19\)	0.164107	+	4.35581i	0.0376488	+	0.999291i
							\(23\)	−1.49233	−	2.58480i	−0.311173	−	0.538967i	0.667444	−	0.744660i	\(-0.267388\pi\)
−0.978617	+	0.205693i	\(0.934055\pi\)
\(29\)	5.49841			1.02103			0.510515	−	0.859869i	\(-0.329455\pi\)
							0.510515	+	0.859869i	\(0.329455\pi\)
\(31\)	−2.54567	+	4.40923i	−0.457216	+	0.791921i	−0.998813	−	0.0487176i	\(-0.984487\pi\)
							0.541597	+	0.840638i	\(0.317820\pi\)
\(37\)	9.20826			1.51383			0.756914	−	0.653514i	\(-0.226706\pi\)
							0.756914	+	0.653514i	\(0.226706\pi\)
\(41\)	1.71003			0.267061			0.133531	−	0.991045i	\(-0.457369\pi\)
							0.133531	+	0.991045i	\(0.457369\pi\)
\(43\)	1.79608	−	3.11091i	0.273900	−	0.474409i	−0.695957	−	0.718083i	\(-0.745020\pi\)
							0.969857	+	0.243675i	\(0.0783529\pi\)
\(47\)	−10.6354			−1.55133			−0.775666	−	0.631143i	\(-0.782586\pi\)
							−0.775666	+	0.631143i	\(0.782586\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.2.g.c.121.11	yes	32
3.2	odd	2		513.2.g.c.64.6		32
9.2	odd	6		513.2.h.c.235.11		32
9.7	even	3		171.2.h.c.7.6	yes	32
19.11	even	3		171.2.h.c.49.6	yes	32
57.11	odd	6		513.2.h.c.334.11		32
171.11	odd	6		513.2.g.c.505.6		32
171.106	even	3	inner	171.2.g.c.106.11	✓	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.2.g.c.106.11	✓	32	171.106	even	3	inner
171.2.g.c.121.11	yes	32	1.1	even	1	trivial
171.2.h.c.7.6	yes	32	9.7	even	3
171.2.h.c.49.6	yes	32	19.11	even	3
513.2.g.c.64.6		32	3.2	odd	2
513.2.g.c.505.6		32	171.11	odd	6
513.2.h.c.235.11		32	9.2	odd	6
513.2.h.c.334.11		32	57.11	odd	6