Embedded newform 170.3.p.b.61.2

• Label: 170.3.p.b.61.2
• Level: $170$
• Weight: $3$
• Character: 170.61
• Analytic conductor: $4.632$
• Analytic rank: $0$
• Dimension: $48$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 170 = 2 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	170.p (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.63216449413\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(6\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			61.2
Character	\(\chi\)	\(=\)	170.61
Dual form			170.3.p.b.131.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.541196 - 1.30656i) q^{2} +(-1.29925 - 0.868131i) q^{3} +(-1.41421 - 1.41421i) q^{4} +(2.19310 - 0.436235i) q^{5} +(-1.83742 + 1.22772i) q^{6} +(-6.29247 - 1.25165i) q^{7} +(-2.61313 + 1.08239i) q^{8} +(-2.50975 - 6.05908i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 16 q^{3} - 16 q^{6} + 16 q^{7} - 32 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/170\mathbb{Z}\right)^\times\).

\(n\)	\(71\)	\(137\)
\(\chi(n)\)	\(e\left(\frac{3}{16}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.541196	−	1.30656i	0.270598	−	0.653281i
							\(3\)	−1.29925	−	0.868131i	−0.433083	−	0.289377i	0.319857	−	0.947466i	\(-0.396365\pi\)
−0.752940	+	0.658089i	\(0.771365\pi\)
\(5\)	2.19310	−	0.436235i	0.438621	−	0.0872470i
							\(7\)	−6.29247	−	1.25165i	−0.898924	−	0.178807i	−0.276063	−	0.961140i	\(-0.589030\pi\)
−0.622861	+	0.782333i	\(0.714030\pi\)
\(11\)	−4.85587	−	7.26732i	−0.441442	−	0.660665i	0.542314	−	0.840176i	\(-0.317548\pi\)
							−0.983756	+	0.179511i	\(0.942548\pi\)
\(13\)	−4.42275	+	4.42275i	−0.340212	+	0.340212i	−0.856447	−	0.516235i	\(-0.827333\pi\)
							0.516235	+	0.856447i	\(0.327333\pi\)
\(17\)	3.63335	−	16.6072i	0.213726	−	0.976894i
							\(19\)	−6.09318	+	14.7102i	−0.320693	+	0.774223i	0.678521	+	0.734581i	\(0.262621\pi\)
−0.999214	+	0.0396411i	\(0.987379\pi\)
\(23\)	−22.4622	+	15.0087i	−0.976616	+	0.652554i	−0.937976	−	0.346700i	\(-0.887302\pi\)
							−0.0386394	+	0.999253i	\(0.512302\pi\)
\(29\)	−9.72228	−	48.8772i	−0.335251	−	1.68542i	−0.669401	−	0.742901i	\(-0.733449\pi\)
							0.334150	−	0.942520i	\(-0.391551\pi\)
\(31\)	33.4001	−	49.9868i	1.07742	−	1.61248i	0.335436	−	0.942063i	\(-0.391116\pi\)
							0.741987	−	0.670414i	\(-0.233884\pi\)
\(37\)	41.8490	+	27.9626i	1.13105	+	0.755746i	0.972799	−	0.231652i	\(-0.0744131\pi\)
							0.158255	+	0.987398i	\(0.449413\pi\)
\(41\)	79.0803	+	15.7300i	1.92879	+	0.383660i	0.999912	+	0.0132659i	\(0.00422279\pi\)
							0.928875	+	0.370394i	\(0.120777\pi\)
\(43\)	−10.2237	−	24.6822i	−0.237760	−	0.574004i	0.759290	−	0.650752i	\(-0.225546\pi\)
							−0.997051	+	0.0767481i	\(0.975546\pi\)
\(47\)	9.66286	−	9.66286i	0.205593	−	0.205593i	−0.596798	−	0.802391i	\(-0.703561\pi\)
							0.802391	+	0.596798i	\(0.203561\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	170.3.p.b.61.2	✓	48
17.12	odd	16	inner	170.3.p.b.131.2	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
170.3.p.b.61.2	✓	48	1.1	even	1	trivial
170.3.p.b.131.2	yes	48	17.12	odd	16	inner