Embedded newform 171.3.i.a.103.38

• Label: 171.3.i.a.103.38
• 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			103.38
Character	\(\chi\)	\(=\)	171.103
Dual form			171.3.i.a.88.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+3.78076i q^{2} +(-2.81408 - 1.03968i) q^{3} -10.2942 q^{4} +(-0.565984 - 0.980312i) q^{5} +(3.93080 - 10.6394i) q^{6} +(0.290777 + 0.503641i) q^{7} -23.7968i q^{8} +(6.83811 + 5.85151i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - 3 q^{3} - 146 q^{4} + q^{5} + 7 q^{6} - 3 q^{7} - 13 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)			3.78076i			1.89038i	0.326517	+	0.945191i	\(0.394125\pi\)
							−0.326517	+	0.945191i	\(0.605875\pi\)
\(3\)	−2.81408	−	1.03968i	−0.938027	−	0.346561i
							\(5\)	−0.565984	−	0.980312i	−0.113197	−	0.196062i	0.803861	−	0.594818i	\(-0.202776\pi\)
−0.917057	+	0.398755i	\(0.869442\pi\)
\(7\)	0.290777	+	0.503641i	0.0415396	+	0.0719487i	0.886048	−	0.463594i	\(-0.153440\pi\)
							−0.844508	+	0.535543i	\(0.820107\pi\)
\(11\)	−5.05824	−	8.76112i	−0.459840	−	0.796466i	0.539112	−	0.842234i	\(-0.318760\pi\)
							−0.998952	+	0.0457681i	\(0.985426\pi\)
\(13\)		−	2.42354i		−	0.186426i	−0.995646	−	0.0932130i	\(-0.970286\pi\)
							0.995646	−	0.0932130i	\(-0.0297138\pi\)
\(17\)	8.04843	−	13.9403i	0.473437	−	0.820017i	−0.526101	−	0.850422i	\(-0.676346\pi\)
							0.999538	+	0.0304053i	\(0.00967981\pi\)
\(19\)	14.0351	+	12.8069i	0.738687	+	0.674049i
							\(23\)	−22.3347			−0.971075			−0.485537	−	0.874216i	\(-0.661376\pi\)
−0.485537	+	0.874216i	\(0.661376\pi\)
\(29\)	−45.7680	−	26.4242i	−1.57821	−	0.911178i	−0.995110	−	0.0987752i	\(-0.968508\pi\)
							−0.583097	−	0.812403i	\(-0.698159\pi\)
\(31\)	−34.3947	−	19.8578i	−1.10951	−	0.640573i	−0.170804	−	0.985305i	\(-0.554636\pi\)
							−0.938701	+	0.344732i	\(0.887970\pi\)
\(37\)		−	11.5673i		−	0.312629i	−0.987707	−	0.156315i	\(-0.950039\pi\)
							0.987707	−	0.156315i	\(-0.0499613\pi\)
\(41\)	53.9877	−	31.1698i	1.31677	−	0.760239i	0.333565	−	0.942727i	\(-0.391748\pi\)
							0.983208	+	0.182488i	\(0.0584151\pi\)
\(43\)	9.67316			0.224957			0.112479	−	0.993654i	\(-0.464121\pi\)
							0.112479	+	0.993654i	\(0.464121\pi\)
\(47\)	6.28979	−	10.8942i	0.133825	−	0.231792i	−0.791323	−	0.611399i	\(-0.790607\pi\)
							0.925148	+	0.379606i	\(0.123941\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.3.i.a.103.38	yes	76
3.2	odd	2		513.3.i.a.388.1		76
9.2	odd	6		513.3.s.a.46.1		76
9.7	even	3		171.3.s.a.160.38	yes	76
19.12	odd	6		171.3.s.a.31.38	yes	76
57.50	even	6		513.3.s.a.145.1		76
171.88	odd	6	inner	171.3.i.a.88.1	✓	76
171.164	even	6		513.3.i.a.316.38		76

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.3.i.a.88.1	✓	76	171.88	odd	6	inner
171.3.i.a.103.38	yes	76	1.1	even	1	trivial
171.3.s.a.31.38	yes	76	19.12	odd	6
171.3.s.a.160.38	yes	76	9.7	even	3
513.3.i.a.316.38		76	171.164	even	6
513.3.i.a.388.1		76	3.2	odd	2
513.3.s.a.46.1		76	9.2	odd	6
513.3.s.a.145.1		76	57.50	even	6