Embedded newform 171.3.z.a.101.6

• Label: 171.3.z.a.101.6
• Level: $171$
• Weight: $3$
• Character: 171.101
• Analytic conductor: $4.659$
• Analytic rank: $0$
• Dimension: $228$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	171.z (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.65941252056\)
Analytic rank:	\(0\)
Dimension:	\(228\)
Relative dimension:	\(38\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			101.6
Character	\(\chi\)	\(=\)	171.101
Dual form			171.3.z.a.149.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.85834 + 0.504002i) q^{2} +(-0.999824 + 2.82849i) q^{3} +(4.15730 - 1.51313i) q^{4} +(-2.82853 - 3.37092i) q^{5} +(1.43227 - 8.58869i) q^{6} +(2.39095 + 4.14125i) q^{7} +(-1.06602 + 0.615465i) q^{8} +(-7.00071 - 5.65598i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 228 q - 9 q^{2} + 6 q^{3} - 3 q^{4} - 9 q^{5} - 30 q^{6} + 3 q^{7} + 30 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(20\)	\(154\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{7}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.85834	+	0.504002i	−1.42917	+	0.252001i	−0.834073	−	0.551654i	\(-0.813997\pi\)
							−0.595095	+	0.803655i	\(0.702886\pi\)
\(3\)	−0.999824	+	2.82849i	−0.333275	+	0.942830i
							\(5\)	−2.82853	−	3.37092i	−0.565707	−	0.674183i	0.405037	−	0.914300i	\(-0.367259\pi\)
−0.970744	+	0.240117i	\(0.922814\pi\)
\(7\)	2.39095	+	4.14125i	0.341565	+	0.591608i	0.984724	−	0.174125i	\(-0.0557098\pi\)
							−0.643159	+	0.765733i	\(0.722376\pi\)
\(11\)		−	18.5310i		−	1.68464i	−0.538980	−	0.842319i	\(-0.681190\pi\)
							0.538980	−	0.842319i	\(-0.318810\pi\)
\(13\)	0.468954	+	0.393499i	0.0360734	+	0.0302691i	0.660646	−	0.750698i	\(-0.270282\pi\)
							−0.624573	+	0.780967i	\(0.714727\pi\)
\(17\)	11.1441	+	13.2810i	0.655533	+	0.781233i	0.986737	−	0.162325i	\(-0.0518995\pi\)
							−0.331205	+	0.943559i	\(0.607455\pi\)
\(19\)	17.2818	+	7.89554i	0.909568	+	0.415555i
							\(23\)	3.67111	+	10.0863i	0.159614	+	0.438535i	0.993560	−	0.113311i	\(-0.0361458\pi\)
−0.833946	+	0.551846i	\(0.813924\pi\)
\(29\)	9.48025	+	26.0468i	0.326905	+	0.898165i	0.988890	+	0.148648i	\(0.0474920\pi\)
							−0.661985	+	0.749517i	\(0.730286\pi\)
\(31\)	30.9075			0.997015			0.498507	−	0.866886i	\(-0.333882\pi\)
							0.498507	+	0.866886i	\(0.333882\pi\)
\(37\)	68.1360			1.84151			0.920757	−	0.390138i	\(-0.127573\pi\)
							0.920757	+	0.390138i	\(0.127573\pi\)
\(41\)	−46.4110	+	8.18351i	−1.13198	+	0.199598i	−0.708093	−	0.706120i	\(-0.750444\pi\)
							−0.423882	+	0.905717i	\(0.639333\pi\)
\(43\)	−44.2413	−	16.1025i	−1.02887	−	0.374477i	−0.228218	−	0.973610i	\(-0.573290\pi\)
							−0.800649	+	0.599133i	\(0.795512\pi\)
\(47\)	−4.92725	−	13.5375i	−0.104835	−	0.288032i	0.876174	−	0.481996i	\(-0.160088\pi\)
							−0.981009	+	0.193964i	\(0.937866\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.3.z.a.101.6	✓	228
9.5	odd	6		171.3.bf.a.158.6	yes	228
19.16	even	9		171.3.bf.a.92.6	yes	228
171.149	odd	18	inner	171.3.z.a.149.6	yes	228

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.3.z.a.101.6	✓	228	1.1	even	1	trivial
171.3.z.a.149.6	yes	228	171.149	odd	18	inner
171.3.bf.a.92.6	yes	228	19.16	even	9
171.3.bf.a.158.6	yes	228	9.5	odd	6