Embedded newform 171.2.g.c.106.1

• Label: 171.2.g.c.106.1
• Level: $171$
• Weight: $2$
• Character: 171.106
• 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			106.1
Character	\(\chi\)	\(=\)	171.106
Dual form			171.2.g.c.121.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.23404 - 2.13742i) q^{2} +(1.62292 + 0.605076i) q^{3} +(-2.04570 + 3.54325i) q^{4} +1.91524 q^{5} +(-0.709450 - 4.21555i) q^{6} +(-0.324708 + 0.562412i) q^{7} +5.16173 q^{8} +(2.26777 + 1.96399i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + q^{2} - 2 q^{3} - 17 q^{4} - 6 q^{5} + 2 q^{6} + q^{7} - 36 q^{8} - 10 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(20\)	\(154\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.23404	−	2.13742i	−0.872597	−	1.51138i	−0.859301	−	0.511470i	\(-0.829101\pi\)
							−0.0132953	−	0.999912i	\(-0.504232\pi\)
\(3\)	1.62292	+	0.605076i	0.936996	+	0.349341i
							\(5\)	1.91524			0.856521			0.428260	−	0.903655i	\(-0.359127\pi\)
0.428260	+	0.903655i	\(0.359127\pi\)
\(7\)	−0.324708	+	0.562412i	−0.122728	+	0.212572i	−0.920843	−	0.389934i	\(-0.872498\pi\)
							0.798114	+	0.602506i	\(0.205831\pi\)
\(11\)	2.93240	−	5.07906i	0.884151	−	1.53140i	0.0374684	−	0.999298i	\(-0.488071\pi\)
							0.846683	−	0.532097i	\(-0.178596\pi\)
\(13\)	0.327691	−	0.567577i	0.0908851	−	0.157418i	−0.816999	−	0.576639i	\(-0.804364\pi\)
							0.907884	+	0.419222i	\(0.137697\pi\)
\(17\)	−1.93700	+	3.35499i	−0.469792	+	0.813704i	−0.999403	−	0.0345362i	\(-0.989005\pi\)
							0.529611	+	0.848241i	\(0.322338\pi\)
\(19\)	−4.28438	−	0.802530i	−0.982905	−	0.184113i
							\(23\)	0.961768	−	1.66583i	0.200542	−	0.347350i	−0.748161	−	0.663517i	\(-0.769063\pi\)
0.948703	+	0.316168i	\(0.102396\pi\)
\(29\)	−6.53639			−1.21378			−0.606888	−	0.794787i	\(-0.707582\pi\)
							−0.606888	+	0.794787i	\(0.707582\pi\)
\(31\)	1.54544	+	2.67678i	0.277569	+	0.480764i	0.970780	−	0.239971i	\(-0.0771380\pi\)
							−0.693211	+	0.720735i	\(0.743805\pi\)
\(37\)	2.23125			0.366816			0.183408	−	0.983037i	\(-0.441287\pi\)
							0.183408	+	0.983037i	\(0.441287\pi\)
\(41\)	−6.96810			−1.08823			−0.544117	−	0.839009i	\(-0.683135\pi\)
							−0.544117	+	0.839009i	\(0.683135\pi\)
\(43\)	4.46940	+	7.74122i	0.681576	+	1.18052i	0.974500	+	0.224389i	\(0.0720386\pi\)
							−0.292923	+	0.956136i	\(0.594628\pi\)
\(47\)	11.5497			1.68470			0.842352	−	0.538928i	\(-0.181171\pi\)
							0.842352	+	0.538928i	\(0.181171\pi\)