Embedded newform 171.3.i.a.103.35

• Label: 171.3.i.a.103.35
• Level: $171$
• Weight: $3$
• Character: 171.103
• Analytic conductor: $4.659$
• Analytic rank: $0$
• Dimension: $76$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(4.65941252056\)
Analytic rank:	\(0\)
Dimension:	\(76\)
Relative dimension:	\(38\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			103.35
Character	\(\chi\)	\(=\)	171.103
Dual form			171.3.i.a.88.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+3.38557i q^{2} +(2.99959 + 0.0494451i) q^{3} -7.46206 q^{4} +(0.678756 + 1.17564i) q^{5} +(-0.167400 + 10.1553i) q^{6} +(5.25485 + 9.10167i) q^{7} -11.7210i q^{8} +(8.99511 + 0.296630i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - 3 q^{3} - 146 q^{4} + q^{5} + 7 q^{6} - 3 q^{7} - 13 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(20\)	\(154\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)			3.38557i			1.69278i	0.532561	+	0.846392i	\(0.321230\pi\)
							−0.532561	+	0.846392i	\(0.678770\pi\)
\(3\)	2.99959	+	0.0494451i	0.999864	+	0.0164817i
							\(5\)	0.678756	+	1.17564i	0.135751	+	0.235128i	0.925884	−	0.377807i	\(-0.123322\pi\)
−0.790133	+	0.612936i	\(0.789989\pi\)
\(7\)	5.25485	+	9.10167i	0.750693	+	1.30024i	0.947487	+	0.319794i	\(0.103614\pi\)
							−0.196794	+	0.980445i	\(0.563053\pi\)
\(11\)	−5.42743	−	9.40059i	−0.493403	−	0.854599i	0.506568	−	0.862200i	\(-0.330914\pi\)
							−0.999971	+	0.00760074i	\(0.997581\pi\)
\(13\)		−	1.91583i		−	0.147372i	−0.997282	−	0.0736858i	\(-0.976524\pi\)
							0.997282	−	0.0736858i	\(-0.0234762\pi\)
\(17\)	−9.35624	+	16.2055i	−0.550367	+	0.953264i	0.447881	+	0.894093i	\(0.352179\pi\)
							−0.998248	+	0.0591705i	\(0.981154\pi\)
\(19\)	2.39994	−	18.8478i	0.126313	−	0.991990i
							\(23\)	−19.6518			−0.854426			−0.427213	−	0.904151i	\(-0.640505\pi\)
−0.427213	+	0.904151i	\(0.640505\pi\)
\(29\)	−3.95244	−	2.28194i	−0.136291	−	0.0786876i	0.430304	−	0.902684i	\(-0.358406\pi\)
							−0.566595	+	0.823996i	\(0.691740\pi\)
\(31\)	47.3688	+	27.3484i	1.52803	+	0.882206i	0.999445	+	0.0333224i	\(0.0106088\pi\)
							0.528580	+	0.848883i	\(0.322725\pi\)
\(37\)		−	4.15148i		−	0.112202i	−0.998425	−	0.0561011i	\(-0.982133\pi\)
							0.998425	−	0.0561011i	\(-0.0178669\pi\)
\(41\)	30.1980	−	17.4348i	0.736537	−	0.425240i	−0.0842720	−	0.996443i	\(-0.526856\pi\)
							0.820809	+	0.571203i	\(0.193523\pi\)
\(43\)	25.0073			0.581564			0.290782	−	0.956789i	\(-0.406085\pi\)
							0.290782	+	0.956789i	\(0.406085\pi\)
\(47\)	−7.36259	+	12.7524i	−0.156651	+	0.271327i	−0.933659	−	0.358163i	\(-0.883403\pi\)
							0.777008	+	0.629491i	\(0.216736\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.3.i.a.103.35	yes	76
3.2	odd	2		513.3.i.a.388.4		76
9.2	odd	6		513.3.s.a.46.4		76
9.7	even	3		171.3.s.a.160.35	yes	76
19.12	odd	6		171.3.s.a.31.35	yes	76
57.50	even	6		513.3.s.a.145.4		76
171.88	odd	6	inner	171.3.i.a.88.4	✓	76
171.164	even	6		513.3.i.a.316.35		76

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.3.i.a.88.4	✓	76	171.88	odd	6	inner
171.3.i.a.103.35	yes	76	1.1	even	1	trivial
171.3.s.a.31.35	yes	76	19.12	odd	6
171.3.s.a.160.35	yes	76	9.7	even	3
513.3.i.a.316.35		76	171.164	even	6
513.3.i.a.388.4		76	3.2	odd	2
513.3.s.a.46.4		76	9.2	odd	6
513.3.s.a.145.4		76	57.50	even	6