Embedded newform 170.3.p.b.61.3

• Label: 170.3.p.b.61.3
• 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			61.3
Character	\(\chi\)	\(=\)	170.61
Dual form			170.3.p.b.131.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.541196 - 1.30656i) q^{2} +(-0.183174 - 0.122393i) q^{3} +(-1.41421 - 1.41421i) q^{4} +(2.19310 - 0.436235i) q^{5} +(-0.259048 + 0.173090i) q^{6} +(10.6204 + 2.11252i) q^{7} +(-2.61313 + 1.08239i) q^{8} +(-3.42558 - 8.27008i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 16 q^{3} - 16 q^{6} + 16 q^{7} - 32 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.183174	−	0.122393i	−0.0610581	−	0.0407977i	0.524667	−	0.851308i	\(-0.324190\pi\)
−0.585725	+	0.810510i	\(0.699190\pi\)
\(5\)	2.19310	−	0.436235i	0.438621	−	0.0872470i
							\(7\)	10.6204	+	2.11252i	1.51720	+	0.301789i	0.882255	−	0.470771i	\(-0.156024\pi\)
0.634942	+	0.772560i	\(0.281024\pi\)
\(11\)	0.724213	+	1.08386i	0.0658375	+	0.0985328i	0.862928	−	0.505328i	\(-0.168628\pi\)
							−0.797090	+	0.603861i	\(0.793628\pi\)
\(13\)	6.85381	−	6.85381i	0.527216	−	0.527216i	−0.392525	−	0.919741i	\(-0.628398\pi\)
							0.919741	+	0.392525i	\(0.128398\pi\)
\(17\)	−16.8964	−	1.87390i	−0.993906	−	0.110230i
							\(19\)	10.9226	−	26.3694i	0.574873	−	1.38787i	−0.322490	−	0.946573i	\(-0.604520\pi\)
0.897363	−	0.441293i	\(-0.145480\pi\)
\(23\)	7.65764	−	5.11667i	0.332941	−	0.222464i	−0.377851	−	0.925866i	\(-0.623337\pi\)
							0.710792	+	0.703403i	\(0.248337\pi\)
\(29\)	10.2231	+	51.3951i	0.352521	+	1.77224i	0.596650	+	0.802501i	\(0.296498\pi\)
							−0.244129	+	0.969743i	\(0.578502\pi\)
\(31\)	−23.8828	+	35.7431i	−0.770413	+	1.15300i	0.213949	+	0.976845i	\(0.431367\pi\)
							−0.984362	+	0.176159i	\(0.943633\pi\)
\(37\)	8.32264	+	5.56101i	0.224936	+	0.150298i	0.662932	−	0.748679i	\(-0.269312\pi\)
							−0.437996	+	0.898977i	\(0.644312\pi\)
\(41\)	38.8405	+	7.72585i	0.947328	+	0.188435i	0.644495	−	0.764608i	\(-0.277068\pi\)
							0.302833	+	0.953044i	\(0.402068\pi\)
\(43\)	−18.0552	−	43.5892i	−0.419889	−	1.01370i	−0.982379	−	0.186898i	\(-0.940157\pi\)
							0.562490	−	0.826804i	\(-0.309843\pi\)
\(47\)	−30.9664	+	30.9664i	−0.658860	+	0.658860i	−0.955110	−	0.296250i	\(-0.904264\pi\)
							0.296250	+	0.955110i	\(0.404264\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	170.3.p.b.61.3	✓	48
17.12	odd	16	inner	170.3.p.b.131.3	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
170.3.p.b.61.3	✓	48	1.1	even	1	trivial
170.3.p.b.131.3	yes	48	17.12	odd	16	inner