Embedded newform 171.3.q.a.20.12

• Label: 171.3.q.a.20.12
• Level: $171$
• Weight: $3$
• Character: 171.20
• Analytic conductor: $4.659$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(4.65941252056\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(36\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			20.12
Character	\(\chi\)	\(=\)	171.20
Dual form			171.3.q.a.77.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.77501 - 1.02480i) q^{2} +(2.96735 + 0.441388i) q^{3} +(0.100449 + 0.173983i) q^{4} +(-5.79916 + 3.34815i) q^{5} +(-4.81475 - 3.82443i) q^{6} +(0.488361 - 0.845867i) q^{7} +7.78667i q^{8} +(8.61035 + 2.61951i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 2 q^{3} + 72 q^{4} - 18 q^{5} - 22 q^{6} + 2 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.77501	−	1.02480i	−0.887507	−	0.512402i	−0.0143806	−	0.999897i	\(-0.504578\pi\)
							−0.873126	+	0.487494i	\(0.837911\pi\)
\(3\)	2.96735	+	0.441388i	0.989117	+	0.147129i
							\(5\)	−5.79916	+	3.34815i	−1.15983	+	0.669629i	−0.951264	−	0.308379i	\(-0.900214\pi\)
−0.208568	+	0.978008i	\(0.566880\pi\)
\(7\)	0.488361	−	0.845867i	0.0697659	−	0.120838i	−0.829032	−	0.559201i	\(-0.811108\pi\)
							0.898798	+	0.438363i	\(0.144441\pi\)
\(11\)	11.9163	+	6.87985i	1.08330	+	0.625441i	0.931784	−	0.363014i	\(-0.118252\pi\)
							0.151512	+	0.988455i	\(0.451586\pi\)
\(13\)	3.06264	+	5.30466i	0.235588	+	0.408050i	0.959443	−	0.281901i	\(-0.0909651\pi\)
							−0.723855	+	0.689952i	\(0.757632\pi\)
\(17\)			14.1510i			0.832415i	0.909270	+	0.416207i	\(0.136641\pi\)
							−0.909270	+	0.416207i	\(0.863359\pi\)
\(19\)	4.35890			0.229416
							\(23\)	−10.3484	+	5.97466i	−0.449931	+	0.259768i	−0.707801	−	0.706412i	\(-0.750313\pi\)
0.257870	+	0.966180i	\(0.416979\pi\)
\(29\)	28.2820	+	16.3286i	0.975240	+	0.563055i	0.900830	−	0.434172i	\(-0.142959\pi\)
							0.0744106	+	0.997228i	\(0.476292\pi\)
\(31\)	3.21018	+	5.56020i	0.103554	+	0.179361i	0.913147	−	0.407631i	\(-0.133645\pi\)
							−0.809592	+	0.586993i	\(0.800312\pi\)
\(37\)	3.44857			0.0932046			0.0466023	−	0.998914i	\(-0.485161\pi\)
							0.0466023	+	0.998914i	\(0.485161\pi\)
\(41\)	−12.9330	+	7.46689i	−0.315440	+	0.182119i	−0.649358	−	0.760483i	\(-0.724962\pi\)
							0.333918	+	0.942602i	\(0.391629\pi\)
\(43\)	−1.75334	+	3.03687i	−0.0407753	+	0.0706248i	−0.885693	−	0.464272i	\(-0.846316\pi\)
							0.844918	+	0.534897i	\(0.179649\pi\)
\(47\)	−78.5671	−	45.3607i	−1.67164	−	0.965122i	−0.966721	−	0.255835i	\(-0.917650\pi\)
							−0.704920	−	0.709287i	\(-0.749017\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.3.q.a.20.12	✓	72
3.2	odd	2		513.3.q.a.305.25		72
9.4	even	3		513.3.q.a.476.25		72
9.5	odd	6	inner	171.3.q.a.77.12	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.3.q.a.20.12	✓	72	1.1	even	1	trivial
171.3.q.a.77.12	yes	72	9.5	odd	6	inner
513.3.q.a.305.25		72	3.2	odd	2
513.3.q.a.476.25		72	9.4	even	3