Embedded newform 171.3.bf.a.23.13

• Label: 171.3.bf.a.23.13
• Level: $171$
• Weight: $3$
• Character: 171.23
• 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.bf (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			23.13
Character	\(\chi\)	\(=\)	171.23
Dual form			171.3.bf.a.119.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.52687 + 0.269229i) q^{2} +(-0.833307 - 2.88194i) q^{3} +(-1.49991 + 0.545922i) q^{4} +(-7.11297 + 1.25421i) q^{5} +(2.04826 + 4.17602i) q^{6} +2.04957 q^{7} +(7.51404 - 4.33823i) q^{8} +(-7.61120 + 4.80309i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 228 q - 9 q^{2} - 12 q^{3} - 3 q^{4} - 9 q^{5} + 6 q^{6} - 6 q^{7} - 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(20\)	\(154\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.52687	+	0.269229i	−0.763437	+	0.134615i	−0.541792	−	0.840513i	\(-0.682254\pi\)
							−0.221645	+	0.975127i	\(0.571143\pi\)
\(3\)	−0.833307	−	2.88194i	−0.277769	−	0.960648i
							\(5\)	−7.11297	+	1.25421i	−1.42259	+	0.250842i	−0.831394	−	0.555683i	\(-0.812457\pi\)
−0.591200	+	0.806525i	\(0.701346\pi\)
\(7\)	2.04957			0.292795			0.146398	−	0.989226i	\(-0.453232\pi\)
							0.146398	+	0.989226i	\(0.453232\pi\)
\(11\)	−1.84961	−	1.06787i	−0.168146	−	0.0970792i	0.413565	−	0.910474i	\(-0.364283\pi\)
							−0.581711	+	0.813395i	\(0.697617\pi\)
\(13\)	14.6571	+	12.2987i	1.12747	+	0.946058i	0.998957	−	0.0456505i	\(-0.0145361\pi\)
							0.128510	+	0.991708i	\(0.458981\pi\)
\(17\)	3.96271	−	10.8874i	0.233100	−	0.640438i	−0.766899	−	0.641768i	\(-0.778201\pi\)
							0.999999	+	0.00133020i	\(0.000423414\pi\)
\(19\)	14.7519	−	11.9742i	0.776414	−	0.630223i
							\(23\)	5.96920	+	16.4002i	0.259530	+	0.713054i	0.999196	+	0.0400796i	\(0.0127612\pi\)
−0.739666	+	0.672974i	\(0.765017\pi\)
\(29\)	−15.5356	+	18.5146i	−0.535709	+	0.638433i	−0.964220	−	0.265103i	\(-0.914594\pi\)
							0.428511	+	0.903537i	\(0.359038\pi\)
\(31\)	24.8236	+	42.9957i	0.800761	+	1.38696i	0.919116	+	0.393987i	\(0.128905\pi\)
							−0.118355	+	0.992971i	\(0.537762\pi\)
\(37\)	−8.46501			−0.228784			−0.114392	−	0.993436i	\(-0.536492\pi\)
							−0.114392	+	0.993436i	\(0.536492\pi\)
\(41\)	7.91237	−	21.7390i	0.192985	−	0.530221i	−0.805028	−	0.593237i	\(-0.797850\pi\)
							0.998013	+	0.0630162i	\(0.0200720\pi\)
\(43\)	48.2217	+	17.5512i	1.12143	+	0.408169i	0.835175	−	0.549984i	\(-0.185366\pi\)
							0.286259	+	0.958152i	\(0.407588\pi\)
\(47\)	−27.3206	+	32.5594i	−0.581290	+	0.692754i	−0.973907	−	0.226948i	\(-0.927125\pi\)
							0.392617	+	0.919702i	\(0.371570\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.3.bf.a.23.13	yes	228
9.2	odd	6		171.3.z.a.137.13	yes	228
19.5	even	9		171.3.z.a.5.13	✓	228
171.119	odd	18	inner	171.3.bf.a.119.13	yes	228

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.3.z.a.5.13	✓	228	19.5	even	9
171.3.z.a.137.13	yes	228	9.2	odd	6
171.3.bf.a.23.13	yes	228	1.1	even	1	trivial
171.3.bf.a.119.13	yes	228	171.119	odd	18	inner